id	sid	tid	token	lemma	pos
cana-1260	1	1	communications	communication	NOUN
cana-1260	1	2	on	on	ADP
cana-1260	1	3	applied	apply	VERB
cana-1260	1	4	nonlinear	nonlinear	ADJ
cana-1260	1	5	analysis	analysis	NOUN
cana-1260	1	6	issn	issn	NOUN
cana-1260	1	7	:	:	PUNCT
cana-1260	1	8	1074	1074	NUM
cana-1260	1	9	-	-	PUNCT
cana-1260	1	10	133x	133x	NUM
cana-1260	1	11	vol	vol	NOUN
cana-1260	1	12	31	31	NUM
cana-1260	1	13	no	no	NOUN
cana-1260	1	14	.	.	PUNCT
cana-1260	2	1	6s	6s	NUM
cana-1260	2	2	(	(	PUNCT
cana-1260	2	3	2024	2024	NUM
cana-1260	2	4	)	)	PUNCT
cana-1260	2	5	670	670	NUM
cana-1260	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	2	7	relation	relation	NOUN
cana-1260	2	8	between	between	ADP
cana-1260	2	9	the	the	DET
cana-1260	2	10	fractional	fractional	ADJ
cana-1260	2	11	domination	domination	NOUN
cana-1260	2	12	number	number	NOUN
cana-1260	2	13	of	of	ADP
cana-1260	2	14	some	some	DET
cana-1260	2	15	graphs	graph	NOUN
cana-1260	2	16	and	and	CCONJ
cana-1260	2	17	their	their	PRON
cana-1260	2	18	line	line	NOUN
cana-1260	2	19	graphs	graph	VERB
cana-1260	2	20	mahesh	mahesh	PROPN
cana-1260	2	21	sarada1,2	sarada1,2	PROPN
cana-1260	2	22	,	,	PUNCT
cana-1260	2	23	rekha	rekha	ADJ
cana-1260	2	24	jain1	jain1	NOUN
cana-1260	2	25	,	,	PUNCT
cana-1260	2	26	ganesh	ganesh	NOUN
cana-1260	2	27	mundhe3	mundhe3	NOUN
cana-1260	3	1	1department	1department	NUM
cana-1260	3	2	of	of	ADP
cana-1260	3	3	mathematics	mathematic	NOUN
cana-1260	3	4	,	,	PUNCT
cana-1260	3	5	medi	medi	NOUN
cana-1260	3	6	-	-	PUNCT
cana-1260	3	7	caps	cap	NOUN
cana-1260	3	8	university	university	NOUN
cana-1260	3	9	,	,	PUNCT
cana-1260	3	10	pigdamber	pigdamber	NOUN
cana-1260	3	11	,	,	PUNCT
cana-1260	3	12	rau-453331	rau-453331	NOUN
cana-1260	3	13	,	,	PUNCT
cana-1260	3	14	indore	indore	PROPN
cana-1260	3	15	,	,	PUNCT
cana-1260	3	16	madhya	madhya	PROPN
cana-1260	3	17	pradesh	pradesh	PROPN
cana-1260	3	18	,	,	PUNCT
cana-1260	3	19	india	india	PROPN
cana-1260	3	20	.	.	PUNCT
cana-1260	4	1	2department	2department	NUM
cana-1260	4	2	of	of	ADP
cana-1260	4	3	engineering	engineering	NOUN
cana-1260	4	4	mathematics	mathematic	NOUN
cana-1260	4	5	,	,	PUNCT
cana-1260	4	6	pimpri	pimpri	NOUN
cana-1260	4	7	chinchwad	chinchwad	PROPN
cana-1260	4	8	college	college	PROPN
cana-1260	4	9	of	of	ADP
cana-1260	4	10	engineering	engineering	PROPN
cana-1260	4	11	&	&	CCONJ
cana-1260	4	12	research	research	PROPN
cana-1260	4	13	,	,	PUNCT
cana-1260	4	14	ravet-412101	ravet-412101	NOUN
cana-1260	4	15	,	,	PUNCT
cana-1260	4	16	pune	pune	NOUN
cana-1260	4	17	,	,	PUNCT
cana-1260	4	18	maharashtra	maharashtra	PROPN
cana-1260	4	19	,	,	PUNCT
cana-1260	4	20	india	india	PROPN
cana-1260	4	21	.	.	PUNCT
cana-1260	5	1	3department	3department	NUM
cana-1260	5	2	of	of	ADP
cana-1260	5	3	engineering	engineering	NOUN
cana-1260	5	4	mathematics	mathematic	NOUN
cana-1260	5	5	,	,	PUNCT
cana-1260	5	6	army	army	PROPN
cana-1260	5	7	institute	institute	PROPN
cana-1260	5	8	of	of	ADP
cana-1260	5	9	technology	technology	PROPN
cana-1260	5	10	,	,	PUNCT
cana-1260	5	11	dighi-411015	dighi-411015	ADJ
cana-1260	5	12	,	,	PUNCT
cana-1260	5	13	pune	pune	NOUN
cana-1260	5	14	,	,	PUNCT
cana-1260	5	15	maharashtra	maharashtra	PROPN
cana-1260	5	16	,	,	PUNCT
cana-1260	5	17	india	india	PROPN
cana-1260	5	18	.	.	PUNCT
cana-1260	6	1	article	article	PROPN
cana-1260	6	2	history	history	NOUN
cana-1260	6	3	:	:	PUNCT
cana-1260	6	4	received	receive	VERB
cana-1260	6	5	:	:	PUNCT
cana-1260	6	6	27	27	NUM
cana-1260	6	7	-	-	SYM
cana-1260	6	8	05	05	NUM
cana-1260	6	9	-	-	PUNCT
cana-1260	6	10	2024	2024	NUM
cana-1260	6	11	revised	revise	VERB
cana-1260	6	12	:	:	PUNCT
cana-1260	6	13	20	20	NUM
cana-1260	6	14	-	-	SYM
cana-1260	6	15	07	07	NUM
cana-1260	6	16	-	-	PUNCT
cana-1260	6	17	2024	2024	NUM
cana-1260	6	18	accepted	accept	VERB
cana-1260	6	19	:	:	PUNCT
cana-1260	6	20	30	30	NUM
cana-1260	6	21	-	-	SYM
cana-1260	6	22	07	07	NUM
cana-1260	6	23	-	-	PUNCT
cana-1260	6	24	2024	2024	NUM
cana-1260	6	25	abstract	abstract	NOUN
cana-1260	6	26	:	:	PUNCT
cana-1260	6	27	let	let	VERB
cana-1260	6	28	g	g	PROPN
cana-1260	6	29	is	be	AUX
cana-1260	6	30	the	the	DET
cana-1260	6	31	simple	simple	ADJ
cana-1260	6	32	connected	connected	ADJ
cana-1260	6	33	graph	graph	NOUN
cana-1260	6	34	with	with	ADP
cana-1260	6	35	order	order	NOUN
cana-1260	6	36	n	n	NOUN
cana-1260	6	37	and	and	CCONJ
cana-1260	6	38	its	its	PRON
cana-1260	6	39	line	line	NOUN
cana-1260	6	40	graph	graph	NOUN
cana-1260	6	41	noted	note	VERB
cana-1260	6	42	as	as	ADP
cana-1260	6	43	l(g	l(g	NOUN
cana-1260	6	44	)	)	PUNCT
cana-1260	6	45	.	.	PUNCT
cana-1260	7	1	let	let	VERB
cana-1260	7	2	the	the	DET
cana-1260	7	3	fractional	fractional	ADJ
cana-1260	7	4	dominating	dominating	NOUN
cana-1260	7	5	number	number	NOUN
cana-1260	7	6	denoted	denote	VERB
cana-1260	7	7	by	by	ADP
cana-1260	7	8	γ_(f	γ_(f	PROPN
cana-1260	7	9	)	)	PUNCT
cana-1260	7	10	(	(	PUNCT
cana-1260	7	11	g	g	NOUN
cana-1260	7	12	)	)	PUNCT
cana-1260	7	13	of	of	ADP
cana-1260	7	14	graph	graph	NOUN
cana-1260	7	15	g	g	PROPN
cana-1260	7	16	and	and	CCONJ
cana-1260	7	17	the	the	DET
cana-1260	7	18	upper	upper	ADJ
cana-1260	7	19	fractional	fractional	ADJ
cana-1260	7	20	dominating	dominating	NOUN
cana-1260	7	21	number	number	NOUN
cana-1260	7	22	г_(f	г_(f	PUNCT
cana-1260	7	23	)	)	PUNCT
cana-1260	8	1	(	(	PUNCT
cana-1260	8	2	g	g	NOUN
cana-1260	8	3	)	)	PUNCT
cana-1260	8	4	.	.	PUNCT
cana-1260	9	1	in	in	ADP
cana-1260	9	2	this	this	DET
cana-1260	9	3	study	study	NOUN
cana-1260	9	4	we	we	PRON
cana-1260	9	5	have	have	AUX
cana-1260	9	6	obtained	obtain	VERB
cana-1260	9	7	union	union	NOUN
cana-1260	9	8	and	and	CCONJ
cana-1260	9	9	join	join	VERB
cana-1260	9	10	on〖	on〖	PROPN
cana-1260	9	11	γ〗_(f	γ〗_(f	PROPN
cana-1260	9	12	)	)	PUNCT
cana-1260	9	13	(	(	PUNCT
cana-1260	9	14	g	g	NOUN
cana-1260	9	15	)	)	PUNCT
cana-1260	9	16	and	and	CCONJ
cana-1260	9	17	γ_(f	γ_(f	NUM
cana-1260	9	18	)	)	PUNCT
cana-1260	9	19	(	(	PUNCT
cana-1260	9	20	l(g	l(g	X
cana-1260	9	21	)	)	PUNCT
cana-1260	9	22	)	)	PUNCT
cana-1260	9	23	exploring	explore	VERB
cana-1260	9	24	the	the	DET
cana-1260	9	25	graphs	graph	NOUN
cana-1260	9	26	for	for	ADP
cana-1260	9	27	upper	upper	ADJ
cana-1260	9	28	fractional	fractional	ADJ
cana-1260	9	29	domination	domination	NOUN
cana-1260	9	30	number	number	NOUN
cana-1260	9	31	including	include	VERB
cana-1260	9	32	cycle	cycle	NOUN
cana-1260	9	33	,	,	PUNCT
cana-1260	9	34	complete	complete	ADJ
cana-1260	9	35	,	,	PUNCT
cana-1260	9	36	star	star	NOUN
cana-1260	9	37	,	,	PUNCT
cana-1260	9	38	bi	bi	ADJ
cana-1260	9	39	-	-	ADJ
cana-1260	9	40	star	star	ADJ
cana-1260	9	41	graph	graph	NOUN
cana-1260	9	42	,	,	PUNCT
cana-1260	9	43	wheel	wheel	NOUN
cana-1260	9	44	graph	graph	NOUN
cana-1260	9	45	,	,	PUNCT
cana-1260	9	46	cubic	cubic	ADJ
cana-1260	9	47	graph	graph	NOUN
cana-1260	9	48	,	,	PUNCT
cana-1260	9	49	graphs	graph	NOUN
cana-1260	9	50	of	of	ADP
cana-1260	9	51	cartesian	cartesian	ADJ
cana-1260	9	52	product	product	NOUN
cana-1260	9	53	like	like	ADP
cana-1260	9	54	(	(	PUNCT
cana-1260	9	55	k_2	k_2	PROPN
cana-1260	9	56	×〖	×〖	PROPN
cana-1260	9	57	p〗_(n	p〗_(n	PROPN
cana-1260	9	58	)	)	PUNCT
cana-1260	9	59	)	)	PUNCT
cana-1260	9	60	,	,	PUNCT
cana-1260	9	61	(	(	PUNCT
cana-1260	9	62	k_3	k_3	PROPN
cana-1260	9	63	×〖	×〖	PROPN
cana-1260	9	64	p〗_(n	p〗_(n	PROPN
cana-1260	9	65	)	)	PUNCT
cana-1260	9	66	)	)	PUNCT
cana-1260	10	1	and	and	CCONJ
cana-1260	10	2	(	(	PUNCT
cana-1260	10	3	cm×cn	cm×cn	PROPN
cana-1260	10	4	)	)	PUNCT
cana-1260	10	5	with	with	ADP
cana-1260	10	6	consideration	consideration	NOUN
cana-1260	10	7	of	of	ADP
cana-1260	10	8	the	the	DET
cana-1260	10	9	computational	computational	ADJ
cana-1260	10	10	complexity	complexity	NOUN
cana-1260	10	11	.	.	PUNCT
cana-1260	11	1	we	we	PRON
cana-1260	11	2	have	have	AUX
cana-1260	11	3	taken	take	VERB
cana-1260	11	4	parameters	parameter	NOUN
cana-1260	11	5	related	relate	VERB
cana-1260	11	6	to	to	ADP
cana-1260	11	7	fractional	fractional	ADJ
cana-1260	11	8	domination	domination	NOUN
cana-1260	11	9	in	in	ADP
cana-1260	11	10	line	line	NOUN
cana-1260	11	11	graphs	graph	NOUN
cana-1260	11	12	towards	towards	ADP
cana-1260	11	13	generalization	generalization	NOUN
cana-1260	11	14	.	.	PUNCT
cana-1260	12	1	the	the	DET
cana-1260	12	2	goal	goal	NOUN
cana-1260	12	3	of	of	ADP
cana-1260	12	4	this	this	DET
cana-1260	12	5	paper	paper	NOUN
cana-1260	12	6	is	be	AUX
cana-1260	12	7	to	to	PART
cana-1260	12	8	provide	provide	VERB
cana-1260	12	9	a	a	DET
cana-1260	12	10	generalized	generalized	ADJ
cana-1260	12	11	results	result	NOUN
cana-1260	12	12	of	of	ADP
cana-1260	12	13	sum	sum	NOUN
cana-1260	12	14	〖	〖	PROPN
cana-1260	12	15	γ〗_(f	γ〗_(f	PROPN
cana-1260	12	16	)	)	PUNCT
cana-1260	13	1	(	(	PUNCT
cana-1260	13	2	g)+γ_(f	g)+γ_(f	PROPN
cana-1260	13	3	)	)	PUNCT
cana-1260	13	4	(	(	PUNCT
cana-1260	13	5	l(g	l(g	PROPN
cana-1260	13	6	)	)	PUNCT
cana-1260	13	7	)	)	PUNCT
cana-1260	13	8	,	,	PUNCT
cana-1260	13	9	г_(f	г_(f	PROPN
cana-1260	13	10	)	)	PUNCT
cana-1260	14	1	(	(	PUNCT
cana-1260	14	2	g)+г_(f	g)+г_(f	NOUN
cana-1260	14	3	)	)	PUNCT
cana-1260	14	4	l(g	l(g	PROPN
cana-1260	14	5	)	)	PUNCT
cana-1260	14	6	and	and	CCONJ
cana-1260	14	7	product	product	NOUN
cana-1260	14	8	〖	〖	PROPN
cana-1260	14	9	γ〗_(f	γ〗_(f	PROPN
cana-1260	14	10	)	)	PUNCT
cana-1260	14	11	(	(	PUNCT
cana-1260	14	12	g)*γ_(f	g)*γ_(f	PROPN
cana-1260	14	13	)	)	PUNCT
cana-1260	14	14	(	(	PUNCT
cana-1260	14	15	l(g	l(g	PROPN
cana-1260	14	16	)	)	PUNCT
cana-1260	14	17	)	)	PUNCT
cana-1260	14	18	,	,	PUNCT
cana-1260	14	19	г_(f	г_(f	PROPN
cana-1260	14	20	)	)	PUNCT
cana-1260	14	21	(	(	PUNCT
cana-1260	14	22	g)*г_(f	g)*г_(f	X
cana-1260	14	23	)	)	PUNCT
cana-1260	14	24	l(g	l(g	PROPN
cana-1260	14	25	)	)	PUNCT
cana-1260	14	26	for	for	ADP
cana-1260	14	27	some	some	DET
cana-1260	14	28	specific	specific	ADJ
cana-1260	14	29	graph	graph	NOUN
cana-1260	14	30	classes	class	NOUN
cana-1260	14	31	.	.	PUNCT
cana-1260	15	1	keywords	keyword	NOUN
cana-1260	15	2	:	:	PUNCT
cana-1260	15	3	domination	domination	NOUN
cana-1260	15	4	set	set	NOUN
cana-1260	15	5	,	,	PUNCT
cana-1260	15	6	fractional	fractional	ADJ
cana-1260	15	7	dominating	dominating	NOUN
cana-1260	15	8	number	number	NOUN
cana-1260	15	9	,	,	PUNCT
cana-1260	15	10	line	line	NOUN
cana-1260	15	11	graph	graph	NOUN
cana-1260	15	12	,	,	PUNCT
cana-1260	15	13	union	union	NOUN
cana-1260	15	14	and	and	CCONJ
cana-1260	15	15	join	join	VERB
cana-1260	15	16	.	.	PUNCT
cana-1260	16	1	2020	2020	NUM
cana-1260	16	2	mathematical	mathematical	ADJ
cana-1260	16	3	sciences	science	NOUN
cana-1260	16	4	classification	classification	NOUN
cana-1260	16	5	:	:	PUNCT
cana-1260	16	6	05c72	05c72	NOUN
cana-1260	16	7	,	,	PUNCT
cana-1260	16	8	05c76	05c76	PRON
cana-1260	16	9	,	,	PUNCT
cana-1260	16	10	05c90	05c90	NOUN
cana-1260	16	11	.	.	PUNCT
cana-1260	17	1	1	1	X
cana-1260	17	2	.	.	X
cana-1260	17	3	introduction	introduction	NOUN
cana-1260	17	4	initially	initially	ADV
cana-1260	17	5	,	,	PUNCT
cana-1260	17	6	the	the	DET
cana-1260	17	7	idea	idea	NOUN
cana-1260	17	8	of	of	ADP
cana-1260	17	9	graph	graph	NOUN
cana-1260	17	10	domination	domination	NOUN
cana-1260	17	11	was	be	AUX
cana-1260	17	12	presented	present	VERB
cana-1260	17	13	by	by	ADP
cana-1260	17	14	author	author	NOUN
cana-1260	17	15	ore	ore	NOUN
cana-1260	17	16	in	in	ADP
cana-1260	17	17	the	the	DET
cana-1260	17	18	year	year	NOUN
cana-1260	17	19	1960	1960	NUM
cana-1260	17	20	,	,	PUNCT
cana-1260	17	21	and	and	CCONJ
cana-1260	17	22	after	after	ADP
cana-1260	17	23	this	this	PRON
cana-1260	17	24	it	it	PRON
cana-1260	17	25	is	be	AUX
cana-1260	17	26	very	very	ADV
cana-1260	17	27	important	important	ADJ
cana-1260	17	28	and	and	CCONJ
cana-1260	17	29	vital	vital	ADJ
cana-1260	17	30	area	area	NOUN
cana-1260	17	31	of	of	ADP
cana-1260	17	32	research	research	NOUN
cana-1260	17	33	in	in	ADP
cana-1260	17	34	theory	theory	NOUN
cana-1260	17	35	of	of	ADP
cana-1260	17	36	graph	graph	NOUN
cana-1260	17	37	.	.	PUNCT
cana-1260	18	1	the	the	DET
cana-1260	18	2	research	research	NOUN
cana-1260	18	3	regarding	regard	VERB
cana-1260	18	4	fractional	fractional	ADJ
cana-1260	18	5	domination	domination	NOUN
cana-1260	18	6	number	number	NOUN
cana-1260	18	7	presented	present	VERB
cana-1260	18	8	by	by	ADP
cana-1260	18	9	some	some	DET
cana-1260	18	10	researchers	researcher	NOUN
cana-1260	18	11	in	in	ADP
cana-1260	18	12	the	the	DET
cana-1260	18	13	late	late	ADJ
cana-1260	18	14	1980s	1980	NOUN
cana-1260	18	15	as	as	ADP
cana-1260	18	16	a	a	DET
cana-1260	18	17	generalized	generalized	ADJ
cana-1260	18	18	concept	concept	NOUN
cana-1260	18	19	of	of	ADP
cana-1260	18	20	domination	domination	NOUN
cana-1260	18	21	.	.	PUNCT
cana-1260	19	1	a	a	DET
cana-1260	19	2	graph	graph	NOUN
cana-1260	19	3	's	's	PART
cana-1260	19	4	fractional	fractional	ADJ
cana-1260	19	5	dominating	dominating	NOUN
cana-1260	19	6	number	number	NOUN
cana-1260	19	7	is	be	AUX
cana-1260	19	8	described	describe	VERB
cana-1260	19	9	as	as	ADP
cana-1260	19	10	minimum	minimum	ADJ
cana-1260	19	11	total	total	ADJ
cana-1260	19	12	weights	weight	NOUN
cana-1260	19	13	assigned	assign	VERB
cana-1260	19	14	to	to	ADP
cana-1260	19	15	all	all	DET
cana-1260	19	16	the	the	DET
cana-1260	19	17	vertices	vertex	NOUN
cana-1260	19	18	of	of	ADP
cana-1260	19	19	graph	graph	NOUN
cana-1260	19	20	from	from	ADP
cana-1260	19	21	fractional	fractional	ADJ
cana-1260	19	22	dominating	dominating	NOUN
cana-1260	19	23	set	set	NOUN
cana-1260	19	24	,	,	PUNCT
cana-1260	19	25	where	where	SCONJ
cana-1260	19	26	a	a	DET
cana-1260	19	27	fractional	fractional	ADJ
cana-1260	19	28	dominating	dominating	NOUN
cana-1260	19	29	set	set	NOUN
cana-1260	19	30	is	be	AUX
cana-1260	19	31	a	a	DET
cana-1260	19	32	method	method	NOUN
cana-1260	19	33	that	that	PRON
cana-1260	19	34	gives	give	VERB
cana-1260	19	35	each	each	DET
cana-1260	19	36	vertex	vertex	NOUN
cana-1260	19	37	a	a	DET
cana-1260	19	38	weight	weight	NOUN
cana-1260	19	39	such	such	ADJ
cana-1260	19	40	that	that	SCONJ
cana-1260	19	41	the	the	DET
cana-1260	19	42	addition	addition	NOUN
cana-1260	19	43	of	of	ADP
cana-1260	19	44	the	the	DET
cana-1260	19	45	weights	weight	NOUN
cana-1260	19	46	of	of	ADP
cana-1260	19	47	all	all	DET
cana-1260	19	48	vertices	vertex	NOUN
cana-1260	19	49	is	be	AUX
cana-1260	19	50	at	at	ADP
cana-1260	19	51	least	least	ADJ
cana-1260	19	52	one	one	NUM
cana-1260	19	53	.	.	PUNCT
cana-1260	20	1	at	at	ADP
cana-1260	20	2	the	the	DET
cana-1260	20	3	18th	18th	ADJ
cana-1260	20	4	south	south	ADJ
cana-1260	20	5	-	-	PUNCT
cana-1260	20	6	eastern	eastern	ADJ
cana-1260	20	7	international	international	ADJ
cana-1260	20	8	conference	conference	NOUN
cana-1260	20	9	based	base	VERB
cana-1260	20	10	on	on	ADP
cana-1260	20	11	combinatorics	combinatoric	NOUN
cana-1260	20	12	,	,	PUNCT
cana-1260	20	13	graph	graph	NOUN
cana-1260	20	14	theory	theory	NOUN
cana-1260	20	15	and	and	CCONJ
cana-1260	20	16	computing	compute	VERB
cana-1260	20	17	s.t	s.t	PROPN
cana-1260	20	18	.	.	PROPN
cana-1260	20	19	hedetniemi	hedetniemi	ADP
cana-1260	20	20	initially	initially	ADV
cana-1260	20	21	defined	define	VERB
cana-1260	20	22	fractional	fractional	ADJ
cana-1260	20	23	dominating	dominating	NOUN
cana-1260	20	24	as	as	ADP
cana-1260	20	25	a	a	DET
cana-1260	20	26	function	function	NOUN
cana-1260	20	27	𝑓:𝑉	𝑓:𝑉	PROPN
cana-1260	20	28	→	→	PUNCT
cana-1260	21	1	[	[	X
cana-1260	21	2	0	0	NUM
cana-1260	21	3	,	,	PUNCT
cana-1260	21	4	1	1	NUM
cana-1260	21	5	]	]	PUNCT
cana-1260	21	6	is	be	AUX
cana-1260	21	7	said	say	VERB
cana-1260	21	8	to	to	PART
cana-1260	21	9	be	be	AUX
cana-1260	21	10	dominating	dominate	VERB
cana-1260	21	11	function	function	NOUN
cana-1260	21	12	of	of	ADP
cana-1260	21	13	the	the	DET
cana-1260	21	14	graph	graph	NOUN
cana-1260	21	15	𝐺	𝐺	NOUN
cana-1260	21	16	=	=	SYM
cana-1260	21	17	(	(	PUNCT
cana-1260	21	18	𝑉	𝑉	PROPN
cana-1260	21	19	,	,	PUNCT
cana-1260	21	20	𝐸	𝐸	PROPN
cana-1260	21	21	)	)	PUNCT
cana-1260	21	22	if	if	SCONJ
cana-1260	21	23	𝑓(𝑁[𝑣	𝑓(𝑁[𝑣	PRON
cana-1260	21	24	]	]	X
cana-1260	21	25	)	)	PUNCT
cana-1260	21	26	≥	≥	NOUN
cana-1260	21	27	1	1	NUM
cana-1260	21	28	for	for	ADP
cana-1260	21	29	all	all	PRON
cana-1260	21	30	𝑣	𝑣	DET
cana-1260	21	31	∈	∈	PROPN
cana-1260	21	32	𝑉.	𝑉.	NOUN
cana-1260	21	33	the	the	DET
cana-1260	21	34	fractional	fractional	ADJ
cana-1260	21	35	dominating	dominating	NOUN
cana-1260	21	36	number	number	NOUN
cana-1260	21	37	of	of	ADP
cana-1260	21	38	𝐺	𝐺	PROPN
cana-1260	21	39	is	be	AUX
cana-1260	21	40	given	give	VERB
cana-1260	21	41	by	by	ADP
cana-1260	21	42	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	21	43	(	(	PUNCT
cana-1260	21	44	𝐺	𝐺	PROPN
cana-1260	21	45	)	)	PUNCT
cana-1260	21	46	.	.	PUNCT
cana-1260	22	1	m.	m.	PROPN
cana-1260	22	2	farber	farber	PROPN
cana-1260	22	3	examined	examine	VERB
cana-1260	22	4	the	the	DET
cana-1260	22	5	issue	issue	NOUN
cana-1260	22	6	of	of	ADP
cana-1260	22	7	figuring	figure	VERB
cana-1260	22	8	out	out	ADP
cana-1260	22	9	whether	whether	SCONJ
cana-1260	22	10	to	to	PART
cana-1260	22	11	use	use	VERB
cana-1260	22	12	the	the	DET
cana-1260	22	13	linear	linear	ADJ
cana-1260	22	14	programming	programming	NOUN
cana-1260	22	15	formulation	formulation	NOUN
cana-1260	22	16	of	of	ADP
cana-1260	22	17	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	22	18	(	(	PUNCT
cana-1260	22	19	𝐺	𝐺	NOUN
cana-1260	22	20	)	)	PUNCT
cana-1260	22	21	could	could	AUX
cana-1260	22	22	provide	provide	VERB
cana-1260	22	23	an	an	DET
cana-1260	22	24	integer	integer	NOUN
cana-1260	22	25	solution	solution	NOUN
cana-1260	22	26	to	to	ADP
cana-1260	22	27	lpp	lpp	PROPN
cana-1260	22	28	.	.	PUNCT
cana-1260	23	1	communications	communication	NOUN
cana-1260	23	2	on	on	ADP
cana-1260	23	3	applied	apply	VERB
cana-1260	23	4	nonlinear	nonlinear	ADJ
cana-1260	23	5	analysis	analysis	NOUN
cana-1260	23	6	issn	issn	NOUN
cana-1260	23	7	:	:	PUNCT
cana-1260	23	8	1074	1074	NUM
cana-1260	23	9	-	-	PUNCT
cana-1260	23	10	133x	133x	NUM
cana-1260	23	11	vol	vol	NOUN
cana-1260	23	12	31	31	NUM
cana-1260	23	13	no	no	NOUN
cana-1260	23	14	.	.	PUNCT
cana-1260	24	1	6s	6s	NUM
cana-1260	24	2	(	(	PUNCT
cana-1260	24	3	2024	2024	NUM
cana-1260	24	4	)	)	PUNCT
cana-1260	24	5	671	671	NUM
cana-1260	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	24	7	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	24	8	(	(	PUNCT
cana-1260	24	9	𝐺	𝐺	NOUN
cana-1260	24	10	)	)	PUNCT
cana-1260	24	11	=	=	SYM
cana-1260	24	12	𝑀𝐼𝑁	𝑀𝐼𝑁	PROPN
cana-1260	24	13	∑	∑	PUNCT
cana-1260	24	14	𝑥𝑖	𝑥𝑖	PROPN
cana-1260	24	15	,	,	PUNCT
cana-1260	24	16	𝑛	𝑛	DET
cana-1260	24	17	𝑖=1	𝑖=1	PROPN
cana-1260	24	18	subject	subject	NOUN
cana-1260	24	19	to	to	ADP
cana-1260	24	20	𝑁.𝑋	𝑁.𝑋	PROPN
cana-1260	24	21	≥	≥	NUM
cana-1260	24	22	1𝑛	1𝑛	NOUN
cana-1260	24	23	,	,	PUNCT
cana-1260	24	24	where	where	SCONJ
cana-1260	24	25	𝑥𝑖	𝑥𝑖	ADV
cana-1260	24	26	>	>	X
cana-1260	24	27	0	0	X
cana-1260	24	28	.	.	PUNCT
cana-1260	25	1	the	the	DET
cana-1260	25	2	constraint	constraint	NOUN
cana-1260	25	3	system	system	NOUN
cana-1260	25	4	for	for	ADP
cana-1260	25	5	domination	domination	NOUN
cana-1260	25	6	is	be	AUX
cana-1260	25	7	𝑁.𝑋	𝑁.𝑋	PROPN
cana-1260	25	8	≥	≥	NUM
cana-1260	25	9	1𝑛	1𝑛	NOUN
cana-1260	25	10	and	and	CCONJ
cana-1260	25	11	using	use	VERB
cana-1260	25	12	a	a	DET
cana-1260	25	13	closed	closed	ADJ
cana-1260	25	14	neighborhood	neighborhood	NOUN
cana-1260	25	15	matrix	matrix	NOUN
cana-1260	25	16	n	n	CCONJ
cana-1260	25	17	we	we	PRON
cana-1260	25	18	aim	aim	VERB
cana-1260	25	19	to	to	PART
cana-1260	25	20	reduce	reduce	VERB
cana-1260	25	21	the	the	DET
cana-1260	25	22	weight	weight	NOUN
cana-1260	25	23	of	of	ADP
cana-1260	25	24	the	the	DET
cana-1260	25	25	function	function	NOUN
cana-1260	25	26	that	that	PRON
cana-1260	25	27	accomplishes	accomplish	VERB
cana-1260	25	28	this	this	PRON
cana-1260	25	29	with	with	ADP
cana-1260	25	30	𝑛-by-𝑛	𝑛-by-𝑛	NOUN
cana-1260	25	31	binary	binary	ADJ
cana-1260	25	32	matrix	matrix	NOUN
cana-1260	25	33	𝑁	𝑁	PROPN
cana-1260	25	34	=	=	SYM
cana-1260	25	35	𝐴	𝐴	PROPN
cana-1260	25	36	+	+	CCONJ
cana-1260	26	1	𝐼𝑛	𝐼𝑛	PROPN
cana-1260	26	2	,	,	PUNCT
cana-1260	26	3	where	where	SCONJ
cana-1260	26	4	𝐼𝑛	𝐼𝑛	PROPN
cana-1260	26	5	be	be	AUX
cana-1260	26	6	the	the	DET
cana-1260	26	7	𝑛-by-𝑛	𝑛-by-𝑛	NOUN
cana-1260	26	8	identity	identity	NOUN
cana-1260	26	9	matrix	matrix	NOUN
cana-1260	26	10	and	and	CCONJ
cana-1260	26	11	𝐴	𝐴	PROPN
cana-1260	26	12	is	be	AUX
cana-1260	26	13	adjacency	adjacency	NOUN
cana-1260	26	14	matrix	matrix	NOUN
cana-1260	26	15	.	.	PUNCT
cana-1260	27	1	the	the	DET
cana-1260	27	2	authors	author	NOUN
cana-1260	27	3	[	[	X
cana-1260	27	4	12	12	NUM
cana-1260	27	5	]	]	PUNCT
cana-1260	27	6	have	have	AUX
cana-1260	27	7	discuss	discuss	VERB
cana-1260	27	8	topological	topological	ADJ
cana-1260	27	9	spaces	space	NOUN
cana-1260	27	10	generated	generate	VERB
cana-1260	27	11	by	by	ADP
cana-1260	27	12	simple	simple	ADJ
cana-1260	27	13	graphs	graph	NOUN
cana-1260	27	14	using	use	VERB
cana-1260	27	15	adjacency	adjacency	PROPN
cana-1260	27	16	relation	relation	NOUN
cana-1260	27	17	and	and	CCONJ
cana-1260	27	18	non	non	ADJ
cana-1260	27	19	adjaceny	adjaceny	PROPN
cana-1260	27	20	relation	relation	NOUN
cana-1260	27	21	on	on	ADP
cana-1260	27	22	vertices	vertex	NOUN
cana-1260	27	23	.	.	PUNCT
cana-1260	28	1	they	they	PRON
cana-1260	28	2	establish	establish	VERB
cana-1260	28	3	important	important	ADJ
cana-1260	28	4	results	result	NOUN
cana-1260	28	5	showing	show	VERB
cana-1260	28	6	relations	relation	NOUN
cana-1260	28	7	between	between	ADP
cana-1260	28	8	complete	complete	ADJ
cana-1260	28	9	graph	graph	NOUN
cana-1260	28	10	and	and	CCONJ
cana-1260	28	11	discrete	discrete	ADJ
cana-1260	28	12	topological	topological	ADJ
cana-1260	28	13	space	space	NOUN
cana-1260	28	14	.	.	PUNCT
cana-1260	29	1	also	also	ADV
cana-1260	29	2	discuss	discuss	VERB
cana-1260	29	3	the	the	DET
cana-1260	29	4	topological	topological	ADJ
cana-1260	29	5	spaces	space	NOUN
cana-1260	29	6	related	relate	VERB
cana-1260	29	7	to	to	ADP
cana-1260	29	8	complete	complete	ADJ
cana-1260	29	9	graphs	graph	NOUN
cana-1260	29	10	,	,	PUNCT
cana-1260	29	11	isomorphic	isomorphic	ADJ
cana-1260	29	12	graphs	graph	NOUN
cana-1260	29	13	and	and	CCONJ
cana-1260	29	14	study	study	VERB
cana-1260	29	15	their	their	PRON
cana-1260	29	16	properties	property	NOUN
cana-1260	29	17	.	.	PUNCT
cana-1260	30	1	the	the	DET
cana-1260	30	2	author	author	NOUN
cana-1260	30	3	[	[	X
cana-1260	30	4	8	8	NUM
cana-1260	30	5	]	]	PUNCT
cana-1260	30	6	have	have	AUX
cana-1260	30	7	taken	take	VERB
cana-1260	30	8	the	the	DET
cana-1260	30	9	simple	simple	ADJ
cana-1260	30	10	connected	connected	ADJ
cana-1260	30	11	graph	graph	NOUN
cana-1260	30	12	𝐺	𝐺	PROPN
cana-1260	30	13	of	of	ADP
cana-1260	30	14	𝑛	𝑛	PROPN
cana-1260	30	15	vertices	vertex	NOUN
cana-1260	30	16	and	and	CCONJ
cana-1260	30	17	𝑚	𝑚	ADP
cana-1260	30	18	edges	edge	NOUN
cana-1260	30	19	.	.	PUNCT
cana-1260	31	1	let	let	VERB
cana-1260	31	2	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	31	3	)	)	PUNCT
cana-1260	31	4	is	be	AUX
cana-1260	31	5	its	its	PRON
cana-1260	31	6	line	line	NOUN
cana-1260	31	7	graph	graph	NOUN
cana-1260	31	8	,	,	PUNCT
cana-1260	31	9	in	in	ADP
cana-1260	31	10	this	this	DET
cana-1260	31	11	paper	paper	NOUN
cana-1260	31	12	they	they	PRON
cana-1260	31	13	have	have	AUX
cana-1260	31	14	obtained	obtain	VERB
cana-1260	31	15	lower	lower	ADV
cana-1260	31	16	bound	bind	VERB
cana-1260	31	17	and	and	CCONJ
cana-1260	31	18	upper	upper	ADJ
cana-1260	31	19	bound	bind	VERB
cana-1260	31	20	for	for	ADP
cana-1260	31	21	addition	addition	NOUN
cana-1260	31	22	of	of	ADP
cana-1260	31	23	graph	graph	NOUN
cana-1260	31	24	and	and	CCONJ
cana-1260	31	25	its	its	PRON
cana-1260	31	26	line	line	NOUN
cana-1260	31	27	graph	graph	NOUN
cana-1260	31	28	’s	’s	PART
cana-1260	31	29	domination	domination	NOUN
cana-1260	31	30	number	number	NOUN
cana-1260	31	31	.	.	PUNCT
cana-1260	32	1	they	they	PRON
cana-1260	32	2	continue	continue	VERB
cana-1260	32	3	to	to	PART
cana-1260	32	4	work	work	VERB
cana-1260	32	5	on	on	ADP
cana-1260	32	6	the	the	DET
cana-1260	32	7	block	block	NOUN
cana-1260	32	8	graph	graph	NOUN
cana-1260	32	9	and	and	CCONJ
cana-1260	32	10	further	further	ADJ
cana-1260	32	11	domination	domination	NOUN
cana-1260	32	12	number	number	NOUN
cana-1260	32	13	of	of	ADP
cana-1260	32	14	graph	graph	NOUN
cana-1260	32	15	.	.	PUNCT
cana-1260	33	1	in	in	ADP
cana-1260	33	2	the	the	DET
cana-1260	33	3	paper	paper	NOUN
cana-1260	33	4	[	[	X
cana-1260	33	5	9	9	X
cana-1260	33	6	]	]	PUNCT
cana-1260	33	7	they	they	PRON
cana-1260	33	8	characterize	characterize	VERB
cana-1260	33	9	regular	regular	ADJ
cana-1260	33	10	graphs	graph	NOUN
cana-1260	33	11	and	and	CCONJ
cana-1260	33	12	unicyclic	unicyclic	ADJ
cana-1260	33	13	graphs	graph	NOUN
cana-1260	33	14	of	of	ADP
cana-1260	33	15	odd	odd	ADJ
cana-1260	33	16	order	order	NOUN
cana-1260	33	17	for	for	ADP
cana-1260	33	18	which	which	PRON
cana-1260	33	19	𝛾	𝛾	ADP
cana-1260	33	20	(	(	PUNCT
cana-1260	33	21	𝐺	𝐺	NOUN
cana-1260	33	22	)	)	PUNCT
cana-1260	33	23	+	+	NUM
cana-1260	33	24	𝛾(𝐿	𝛾(𝐿	NOUN
cana-1260	33	25	(	(	PUNCT
cana-1260	33	26	𝐺	𝐺	NOUN
cana-1260	33	27	)	)	PUNCT
cana-1260	33	28	)	)	PUNCT
cana-1260	34	1	=	=	SYM
cana-1260	35	1	𝑛	𝑛	DET
cana-1260	35	2	−	−	NUM
cana-1260	35	3	2	2	NUM
cana-1260	35	4	.	.	PUNCT
cana-1260	36	1	the	the	DET
cana-1260	36	2	authors	author	NOUN
cana-1260	36	3	[	[	X
cana-1260	36	4	1	1	X
cana-1260	36	5	]	]	PUNCT
cana-1260	36	6	have	have	AUX
cana-1260	36	7	studied	study	VERB
cana-1260	36	8	another	another	DET
cana-1260	36	9	fractional	fractional	ADJ
cana-1260	36	10	version	version	NOUN
cana-1260	36	11	of	of	ADP
cana-1260	36	12	kdomination	kdomination	NOUN
cana-1260	36	13	in	in	ADP
cana-1260	36	14	the	the	DET
cana-1260	36	15	graphs	graph	NOUN
cana-1260	36	16	and	and	CCONJ
cana-1260	36	17	related	related	ADJ
cana-1260	36	18	parameters	parameter	NOUN
cana-1260	36	19	.	.	PUNCT
cana-1260	37	1	let	let	VERB
cana-1260	37	2	𝐺	𝐺	PROPN
cana-1260	37	3	is	be	AUX
cana-1260	37	4	the	the	DET
cana-1260	37	5	connected	connected	ADJ
cana-1260	37	6	graph	graph	NOUN
cana-1260	37	7	and	and	CCONJ
cana-1260	37	8	k	k	PROPN
cana-1260	37	9	is	be	AUX
cana-1260	37	10	the	the	DET
cana-1260	37	11	positive	positive	ADJ
cana-1260	37	12	number	number	NOUN
cana-1260	37	13	with	with	ADP
cana-1260	37	14	𝑘	𝑘	DET
cana-1260	37	15	≤	≤	NUM
cana-1260	37	16	𝑟𝑎𝑑(𝐺	𝑟𝑎𝑑(𝐺	NUM
cana-1260	37	17	)	)	PUNCT
cana-1260	37	18	.	.	PUNCT
cana-1260	38	1	a	a	DET
cana-1260	38	2	subset	subset	NOUN
cana-1260	38	3	𝐷	𝐷	PROPN
cana-1260	38	4	⊆	⊆	PROPN
cana-1260	38	5	𝑉	𝑉	PROPN
cana-1260	38	6	is	be	AUX
cana-1260	38	7	known	know	VERB
cana-1260	38	8	as	as	ADP
cana-1260	38	9	a	a	DET
cana-1260	38	10	distance	distance	NOUN
cana-1260	38	11	k	k	NOUN
cana-1260	38	12	-	-	PUNCT
cana-1260	38	13	domination	domination	NOUN
cana-1260	38	14	set	set	NOUN
cana-1260	38	15	of	of	ADP
cana-1260	38	16	graph	graph	NOUN
cana-1260	38	17	𝐺	𝐺	PROPN
cana-1260	38	18	if	if	SCONJ
cana-1260	38	19	f	f	PROPN
cana-1260	38	20	every	every	DET
cana-1260	38	21	𝑣	𝑣	PROPN
cana-1260	38	22	∈	∈	PROPN
cana-1260	38	23	𝑉	𝑉	PROPN
cana-1260	38	24	−	−	PROPN
cana-1260	38	25	𝐷	𝐷	PROPN
cana-1260	38	26	,	,	PUNCT
cana-1260	38	27	there	there	PRON
cana-1260	38	28	exists	exist	VERB
cana-1260	38	29	a	a	DET
cana-1260	38	30	vertex	vertex	NOUN
cana-1260	38	31	𝑢	𝑢	PROPN
cana-1260	38	32	∈	∈	PROPN
cana-1260	38	33	𝐷	𝐷	NOUN
cana-1260	38	34	such	such	ADJ
cana-1260	38	35	that	that	SCONJ
cana-1260	38	36	𝑑	𝑑	PROPN
cana-1260	38	37	(	(	PUNCT
cana-1260	38	38	𝑢	𝑢	X
cana-1260	38	39	,	,	PUNCT
cana-1260	38	40	𝑣	𝑣	NOUN
cana-1260	38	41	)	)	PUNCT
cana-1260	38	42	≤	≤	NOUN
cana-1260	38	43	𝑘.	𝑘.	VERB
cana-1260	38	44	the	the	DET
cana-1260	38	45	line	line	NOUN
cana-1260	38	46	graph	graph	NOUN
cana-1260	38	47	of	of	ADP
cana-1260	38	48	a	a	DET
cana-1260	38	49	graph	graph	NOUN
cana-1260	38	50	represents	represent	VERB
cana-1260	38	51	the	the	DET
cana-1260	38	52	adjacencies	adjacency	NOUN
cana-1260	38	53	between	between	ADP
cana-1260	38	54	the	the	DET
cana-1260	38	55	line	line	NOUN
cana-1260	38	56	connections	connection	NOUN
cana-1260	38	57	of	of	ADP
cana-1260	38	58	the	the	DET
cana-1260	38	59	original	original	ADJ
cana-1260	38	60	graph	graph	NOUN
cana-1260	38	61	.	.	PUNCT
cana-1260	39	1	this	this	DET
cana-1260	39	2	study	study	NOUN
cana-1260	39	3	investigates	investigate	VERB
cana-1260	39	4	relationship	relationship	NOUN
cana-1260	39	5	between	between	ADP
cana-1260	39	6	the	the	DET
cana-1260	39	7	fractional	fractional	ADJ
cana-1260	39	8	domination	domination	NOUN
cana-1260	39	9	number	number	NOUN
cana-1260	39	10	of	of	ADP
cana-1260	39	11	a	a	DET
cana-1260	39	12	graph	graph	NOUN
cana-1260	39	13	and	and	CCONJ
cana-1260	39	14	their	their	PRON
cana-1260	39	15	line	line	NOUN
cana-1260	39	16	graph	graph	NOUN
cana-1260	39	17	for	for	ADP
cana-1260	39	18	various	various	ADJ
cana-1260	39	19	graph	graph	NOUN
cana-1260	39	20	classes	class	NOUN
cana-1260	39	21	.	.	PUNCT
cana-1260	40	1	our	our	PRON
cana-1260	40	2	results	result	NOUN
cana-1260	40	3	show	show	VERB
cana-1260	40	4	that	that	SCONJ
cana-1260	40	5	there	there	PRON
cana-1260	40	6	is	be	VERB
cana-1260	40	7	a	a	DET
cana-1260	40	8	correlation	correlation	NOUN
cana-1260	40	9	between	between	ADP
cana-1260	40	10	the	the	DET
cana-1260	40	11	sum	sum	NOUN
cana-1260	40	12	of	of	ADP
cana-1260	40	13	fractional	fractional	ADJ
cana-1260	40	14	domination	domination	NOUN
cana-1260	40	15	number	number	NOUN
cana-1260	40	16	of	of	ADP
cana-1260	40	17	a	a	DET
cana-1260	40	18	graph	graph	NOUN
cana-1260	40	19	and	and	CCONJ
cana-1260	40	20	its	its	PRON
cana-1260	40	21	line	line	NOUN
cana-1260	40	22	graph	graph	NOUN
cana-1260	40	23	,	,	PUNCT
cana-1260	40	24	and	and	CCONJ
cana-1260	40	25	this	this	DET
cana-1260	40	26	sum	sum	NOUN
cana-1260	40	27	is	be	AUX
cana-1260	40	28	np	np	NOUN
cana-1260	40	29	-	-	PUNCT
cana-1260	40	30	hard	hard	ADJ
cana-1260	40	31	in	in	ADP
cana-1260	40	32	general	general	ADJ
cana-1260	40	33	,	,	PUNCT
cana-1260	40	34	however	however	ADV
cana-1260	40	35	the	the	DET
cana-1260	40	36	correlation	correlation	NOUN
cana-1260	40	37	between	between	ADP
cana-1260	40	38	the	the	DET
cana-1260	40	39	sum	sum	NOUN
cana-1260	40	40	of	of	ADP
cana-1260	40	41	the	the	DET
cana-1260	40	42	fractional	fractional	ADJ
cana-1260	40	43	domination	domination	NOUN
cana-1260	40	44	number	number	NOUN
cana-1260	40	45	of	of	ADP
cana-1260	40	46	a	a	DET
cana-1260	40	47	graph	graph	NOUN
cana-1260	40	48	𝐺	𝐺	NOUN
cana-1260	40	49	and	and	CCONJ
cana-1260	40	50	its	its	PRON
cana-1260	40	51	line	line	NOUN
cana-1260	40	52	graph	graph	NOUN
cana-1260	40	53	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	40	54	)	)	PUNCT
cana-1260	40	55	is	be	AUX
cana-1260	40	56	not	not	PART
cana-1260	40	57	straightforward	straightforward	ADJ
cana-1260	40	58	.	.	PUNCT
cana-1260	41	1	generally	generally	ADV
cana-1260	41	2	,	,	PUNCT
cana-1260	41	3	depends	depend	VERB
cana-1260	41	4	on	on	ADP
cana-1260	41	5	the	the	DET
cana-1260	41	6	specific	specific	ADJ
cana-1260	41	7	graphs	graph	NOUN
cana-1260	41	8	involved	involve	VERB
cana-1260	41	9	or	or	CCONJ
cana-1260	41	10	some	some	DET
cana-1260	41	11	graph	graph	NOUN
cana-1260	41	12	classes	class	NOUN
cana-1260	41	13	,	,	PUNCT
cana-1260	41	14	it	it	PRON
cana-1260	41	15	can	can	AUX
cana-1260	41	16	be	be	AUX
cana-1260	41	17	solved	solve	VERB
cana-1260	41	18	in	in	ADP
cana-1260	41	19	polynomial	polynomial	ADJ
cana-1260	41	20	time	time	NOUN
cana-1260	41	21	.	.	PUNCT
cana-1260	42	1	these	these	DET
cana-1260	42	2	findings	finding	NOUN
cana-1260	42	3	contribute	contribute	VERB
cana-1260	42	4	to	to	ADP
cana-1260	42	5	our	our	PRON
cana-1260	42	6	understanding	understanding	NOUN
cana-1260	42	7	of	of	ADP
cana-1260	42	8	the	the	DET
cana-1260	42	9	properties	property	NOUN
cana-1260	42	10	of	of	ADP
cana-1260	42	11	graphs	graph	NOUN
cana-1260	42	12	.	.	PUNCT
cana-1260	43	1	in	in	ADP
cana-1260	43	2	some	some	DET
cana-1260	43	3	graph	graph	NOUN
cana-1260	43	4	classes	class	NOUN
cana-1260	43	5	,	,	PUNCT
cana-1260	43	6	the	the	DET
cana-1260	43	7	fractional	fractional	ADJ
cana-1260	43	8	domination	domination	NOUN
cana-1260	43	9	number	number	NOUN
cana-1260	43	10	of	of	ADP
cana-1260	43	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	43	12	)	)	PUNCT
cana-1260	43	13	can	can	AUX
cana-1260	43	14	be	be	AUX
cana-1260	43	15	related	relate	VERB
cana-1260	43	16	to	to	ADP
cana-1260	43	17	the	the	DET
cana-1260	43	18	fractional	fractional	ADJ
cana-1260	43	19	domination	domination	NOUN
cana-1260	43	20	number	number	NOUN
cana-1260	43	21	of	of	ADP
cana-1260	43	22	𝐺.	𝐺.	NOUN
cana-1260	43	23	if	if	SCONJ
cana-1260	43	24	𝐺	𝐺	PROPN
cana-1260	43	25	has	have	VERB
cana-1260	43	26	a	a	DET
cana-1260	43	27	high	high	ADJ
cana-1260	43	28	fractional	fractional	ADJ
cana-1260	43	29	domination	domination	NOUN
cana-1260	43	30	number	number	NOUN
cana-1260	43	31	,	,	PUNCT
cana-1260	43	32	this	this	PRON
cana-1260	43	33	might	might	AUX
cana-1260	43	34	imply	imply	VERB
cana-1260	43	35	that	that	DET
cana-1260	43	36	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	43	37	)	)	PUNCT
cana-1260	43	38	also	also	ADV
cana-1260	43	39	has	have	VERB
cana-1260	43	40	a	a	DET
cana-1260	43	41	relatively	relatively	ADV
cana-1260	43	42	high	high	ADJ
cana-1260	43	43	fractional	fractional	ADJ
cana-1260	43	44	domination	domination	NOUN
cana-1260	43	45	number	number	NOUN
cana-1260	43	46	due	due	ADP
cana-1260	43	47	to	to	ADP
cana-1260	43	48	the	the	DET
cana-1260	43	49	structure	structure	NOUN
cana-1260	43	50	and	and	CCONJ
cana-1260	43	51	relationship	relationship	NOUN
cana-1260	43	52	between	between	ADP
cana-1260	43	53	the	the	DET
cana-1260	43	54	two	two	NUM
cana-1260	43	55	graphs	graph	NOUN
cana-1260	43	56	.	.	PUNCT
cana-1260	44	1	positive	positive	ADJ
cana-1260	44	2	correlation	correlation	NOUN
cana-1260	44	3	:	:	PUNCT
cana-1260	44	4	in	in	ADP
cana-1260	44	5	some	some	DET
cana-1260	44	6	cases	case	NOUN
cana-1260	44	7	,	,	PUNCT
cana-1260	44	8	the	the	DET
cana-1260	44	9	fractional	fractional	ADJ
cana-1260	44	10	domination	domination	NOUN
cana-1260	44	11	number	number	NOUN
cana-1260	44	12	of	of	ADP
cana-1260	44	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	44	14	)	)	PUNCT
cana-1260	44	15	can	can	AUX
cana-1260	44	16	be	be	AUX
cana-1260	44	17	related	relate	VERB
cana-1260	44	18	to	to	ADP
cana-1260	44	19	the	the	DET
cana-1260	44	20	fractional	fractional	ADJ
cana-1260	44	21	domination	domination	NOUN
cana-1260	44	22	number	number	NOUN
cana-1260	44	23	of	of	ADP
cana-1260	44	24	𝐺.	𝐺.	NOUN
cana-1260	44	25	if	if	SCONJ
cana-1260	44	26	𝐺	𝐺	PROPN
cana-1260	44	27	has	have	VERB
cana-1260	44	28	a	a	DET
cana-1260	44	29	high	high	ADJ
cana-1260	44	30	fractional	fractional	ADJ
cana-1260	44	31	domination	domination	NOUN
cana-1260	44	32	number	number	NOUN
cana-1260	44	33	,	,	PUNCT
cana-1260	44	34	this	this	PRON
cana-1260	44	35	might	might	AUX
cana-1260	44	36	imply	imply	VERB
cana-1260	44	37	that	that	DET
cana-1260	44	38	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	44	39	)	)	PUNCT
cana-1260	44	40	also	also	ADV
cana-1260	44	41	has	have	VERB
cana-1260	44	42	a	a	DET
cana-1260	44	43	relatively	relatively	ADV
cana-1260	44	44	high	high	ADJ
cana-1260	44	45	fractional	fractional	ADJ
cana-1260	44	46	domination	domination	NOUN
cana-1260	44	47	number	number	NOUN
cana-1260	44	48	due	due	ADP
cana-1260	44	49	to	to	ADP
cana-1260	44	50	the	the	DET
cana-1260	44	51	structure	structure	NOUN
cana-1260	44	52	and	and	CCONJ
cana-1260	44	53	relationship	relationship	NOUN
cana-1260	44	54	between	between	ADP
cana-1260	44	55	the	the	DET
cana-1260	44	56	two	two	NUM
cana-1260	44	57	graphs	graph	NOUN
cana-1260	44	58	.	.	PUNCT
cana-1260	45	1	no	no	DET
cana-1260	45	2	simple	simple	ADJ
cana-1260	45	3	correlation	correlation	NOUN
cana-1260	45	4	:	:	PUNCT
cana-1260	45	5	however	however	ADV
cana-1260	45	6	,	,	PUNCT
cana-1260	45	7	there	there	PRON
cana-1260	45	8	is	be	VERB
cana-1260	45	9	no	no	DET
cana-1260	45	10	universal	universal	ADJ
cana-1260	45	11	rule	rule	NOUN
cana-1260	45	12	stating	state	VERB
cana-1260	45	13	that	that	SCONJ
cana-1260	45	14	the	the	DET
cana-1260	45	15	sum	sum	NOUN
cana-1260	45	16	of	of	ADP
cana-1260	45	17	the	the	DET
cana-1260	45	18	fractional	fractional	ADJ
cana-1260	45	19	domination	domination	NOUN
cana-1260	45	20	numbers	number	NOUN
cana-1260	45	21	of	of	ADP
cana-1260	45	22	𝐺	𝐺	PROPN
cana-1260	45	23	and	and	CCONJ
cana-1260	45	24	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	45	25	)	)	PUNCT
cana-1260	45	26	must	must	AUX
cana-1260	45	27	be	be	AUX
cana-1260	45	28	correlated	correlate	VERB
cana-1260	45	29	in	in	ADP
cana-1260	45	30	a	a	DET
cana-1260	45	31	simple	simple	ADJ
cana-1260	45	32	manner	manner	NOUN
cana-1260	45	33	.	.	PUNCT
cana-1260	46	1	the	the	DET
cana-1260	46	2	relationship	relationship	NOUN
cana-1260	46	3	is	be	AUX
cana-1260	46	4	influenced	influence	VERB
cana-1260	46	5	by	by	ADP
cana-1260	46	6	various	various	ADJ
cana-1260	46	7	factors	factor	NOUN
cana-1260	46	8	such	such	ADJ
cana-1260	46	9	as	as	ADP
cana-1260	46	10	the	the	DET
cana-1260	46	11	size	size	NOUN
cana-1260	46	12	of	of	ADP
cana-1260	46	13	𝐺	𝐺	PROPN
cana-1260	46	14	,	,	PUNCT
cana-1260	46	15	its	its	PRON
cana-1260	46	16	connectivity	connectivity	NOUN
cana-1260	46	17	,	,	PUNCT
cana-1260	46	18	and	and	CCONJ
cana-1260	46	19	how	how	SCONJ
cana-1260	46	20	vertices	vertex	NOUN
cana-1260	46	21	in	in	ADP
cana-1260	46	22	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	46	23	)	)	PUNCT
cana-1260	46	24	are	be	AUX
cana-1260	46	25	dominated	dominate	VERB
cana-1260	46	26	by	by	ADP
cana-1260	46	27	sets	set	NOUN
cana-1260	46	28	of	of	ADP
cana-1260	46	29	vertices	vertex	NOUN
cana-1260	46	30	in	in	ADP
cana-1260	46	31	𝐺.	𝐺.	NOUN
cana-1260	46	32	the	the	DET
cana-1260	46	33	goal	goal	NOUN
cana-1260	46	34	of	of	ADP
cana-1260	46	35	this	this	DET
cana-1260	46	36	paper	paper	NOUN
cana-1260	46	37	is	be	AUX
cana-1260	46	38	to	to	PART
cana-1260	46	39	investigate	investigate	VERB
cana-1260	46	40	a	a	DET
cana-1260	46	41	generalization	generalization	NOUN
cana-1260	46	42	of	of	ADP
cana-1260	46	43	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	46	44	(	(	PUNCT
cana-1260	46	45	𝐺	𝐺	PROPN
cana-1260	46	46	)	)	PUNCT
cana-1260	46	47	,	,	PUNCT
cana-1260	46	48	communications	communication	NOUN
cana-1260	46	49	on	on	ADP
cana-1260	46	50	applied	apply	VERB
cana-1260	46	51	nonlinear	nonlinear	ADJ
cana-1260	46	52	analysis	analysis	NOUN
cana-1260	46	53	issn	issn	NOUN
cana-1260	46	54	:	:	PUNCT
cana-1260	46	55	1074	1074	NUM
cana-1260	46	56	-	-	PUNCT
cana-1260	46	57	133x	133x	NUM
cana-1260	46	58	vol	vol	NOUN
cana-1260	46	59	31	31	NUM
cana-1260	46	60	no	no	NOUN
cana-1260	46	61	.	.	PUNCT
cana-1260	47	1	6s	6s	NUM
cana-1260	47	2	(	(	PUNCT
cana-1260	47	3	2024	2024	NUM
cana-1260	47	4	)	)	PUNCT
cana-1260	47	5	672	672	NUM
cana-1260	47	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	47	7	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	47	8	(	(	PUNCT
cana-1260	47	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	47	10	)	)	PUNCT
cana-1260	47	11	)	)	PUNCT
cana-1260	47	12	and	and	CCONJ
cana-1260	47	13	г𝑓	г𝑓	INTJ
cana-1260	47	14	(	(	PUNCT
cana-1260	47	15	𝐺	𝐺	PROPN
cana-1260	47	16	)	)	PUNCT
cana-1260	47	17	,	,	PUNCT
cana-1260	47	18	г𝑓	г𝑓	DET
cana-1260	47	19	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	47	20	)	)	PUNCT
cana-1260	47	21	for	for	ADP
cana-1260	47	22	some	some	DET
cana-1260	47	23	specific	specific	ADJ
cana-1260	47	24	graphs	graph	NOUN
cana-1260	47	25	.	.	PUNCT
cana-1260	48	1	in	in	ADP
cana-1260	48	2	this	this	DET
cana-1260	48	3	section	section	NOUN
cana-1260	48	4	we	we	PRON
cana-1260	48	5	are	be	AUX
cana-1260	48	6	going	go	VERB
cana-1260	48	7	to	to	PART
cana-1260	48	8	give	give	VERB
cana-1260	48	9	definitions	definition	NOUN
cana-1260	48	10	which	which	PRON
cana-1260	48	11	are	be	AUX
cana-1260	48	12	useful	useful	ADJ
cana-1260	48	13	in	in	ADP
cana-1260	48	14	further	further	ADJ
cana-1260	48	15	sections	section	NOUN
cana-1260	48	16	.	.	PUNCT
cana-1260	49	1	the	the	DET
cana-1260	49	2	definitions	definition	NOUN
cana-1260	49	3	are	be	AUX
cana-1260	49	4	given	give	VERB
cana-1260	49	5	[	[	X
cana-1260	49	6	5	5	NUM
cana-1260	49	7	,	,	PUNCT
cana-1260	49	8	7	7	NUM
cana-1260	49	9	]	]	PUNCT
cana-1260	49	10	like	like	ADP
cana-1260	49	11	the	the	DET
cana-1260	49	12	following	following	NOUN
cana-1260	49	13	:	:	PUNCT
cana-1260	49	14	definition	definition	NOUN
cana-1260	49	15	1.1	1.1	NUM
cana-1260	49	16	.	.	PUNCT
cana-1260	50	1	domination	domination	NOUN
cana-1260	50	2	set	set	NOUN
cana-1260	50	3	and	and	CCONJ
cana-1260	50	4	domination	domination	NOUN
cana-1260	50	5	number	number	NOUN
cana-1260	50	6	:	:	PUNCT
cana-1260	50	7	let	let	VERB
cana-1260	50	8	𝐺	𝐺	PROPN
cana-1260	50	9	=	=	SYM
cana-1260	50	10	(	(	PUNCT
cana-1260	50	11	𝑉	𝑉	PROPN
cana-1260	50	12	,	,	PUNCT
cana-1260	50	13	𝐸	𝐸	PROPN
cana-1260	50	14	)	)	PUNCT
cana-1260	50	15	is	be	AUX
cana-1260	50	16	a	a	DET
cana-1260	50	17	graph	graph	NOUN
cana-1260	50	18	and	and	CCONJ
cana-1260	50	19	subset	subset	VERB
cana-1260	50	20	𝐷	𝐷	PROPN
cana-1260	50	21	of	of	ADP
cana-1260	50	22	vertex	vertex	NOUN
cana-1260	50	23	set	set	NOUN
cana-1260	50	24	𝑉	𝑉	PROPN
cana-1260	50	25	is	be	AUX
cana-1260	50	26	known	know	VERB
cana-1260	50	27	as	as	ADP
cana-1260	50	28	a	a	DET
cana-1260	50	29	dominating	dominating	NOUN
cana-1260	50	30	set	set	NOUN
cana-1260	50	31	of	of	ADP
cana-1260	50	32	graph	graph	NOUN
cana-1260	50	33	if	if	SCONJ
cana-1260	50	34	every	every	DET
cana-1260	50	35	vertex	vertex	NOUN
cana-1260	50	36	in	in	ADP
cana-1260	50	37	𝑉	𝑉	PROPN
cana-1260	50	38	−	−	PROPN
cana-1260	50	39	𝐷	𝐷	PROPN
cana-1260	50	40	is	be	AUX
cana-1260	50	41	adjacent	adjacent	ADJ
cana-1260	50	42	to	to	ADP
cana-1260	50	43	at	at	ADV
cana-1260	50	44	least	least	ADV
cana-1260	50	45	one	one	NUM
cana-1260	50	46	vertex	vertex	NOUN
cana-1260	50	47	in	in	ADP
cana-1260	50	48	subset	subset	ADJ
cana-1260	50	49	𝐷.	𝐷.	PROPN
cana-1260	50	50	a	a	DET
cana-1260	50	51	dominating	dominating	NOUN
cana-1260	50	52	set	set	NOUN
cana-1260	50	53	𝐷	𝐷	NOUN
cana-1260	50	54	is	be	AUX
cana-1260	50	55	known	know	VERB
cana-1260	50	56	as	as	ADP
cana-1260	50	57	a	a	DET
cana-1260	50	58	minimal	minimal	ADJ
cana-1260	50	59	dominating	dominating	NOUN
cana-1260	50	60	set	set	NOUN
cana-1260	50	61	if	if	SCONJ
cana-1260	50	62	there	there	PRON
cana-1260	50	63	is	be	VERB
cana-1260	50	64	n't	not	PART
cana-1260	50	65	a	a	DET
cana-1260	50	66	dominating	dominating	NOUN
cana-1260	50	67	set	set	NOUN
cana-1260	50	68	of	of	ADP
cana-1260	50	69	𝐺	𝐺	PROPN
cana-1260	50	70	that	that	PRON
cana-1260	50	71	is	be	AUX
cana-1260	50	72	a	a	DET
cana-1260	50	73	proper	proper	ADJ
cana-1260	50	74	subset	subset	NOUN
cana-1260	50	75	of	of	ADP
cana-1260	50	76	d.	d.	PROPN
cana-1260	50	77	the	the	DET
cana-1260	50	78	size	size	NOUN
cana-1260	50	79	of	of	ADP
cana-1260	50	80	smallest	small	ADJ
cana-1260	50	81	dominating	dominating	NOUN
cana-1260	50	82	set	set	NOUN
cana-1260	50	83	of	of	ADP
cana-1260	50	84	graph	graph	NOUN
cana-1260	50	85	𝐺	𝐺	PROPN
cana-1260	50	86	is	be	AUX
cana-1260	50	87	called	call	VERB
cana-1260	50	88	the	the	DET
cana-1260	50	89	domination	domination	NOUN
cana-1260	50	90	number	number	NOUN
cana-1260	50	91	of	of	ADP
cana-1260	50	92	graph	graph	NOUN
cana-1260	50	93	𝐺	𝐺	PROPN
cana-1260	50	94	and	and	CCONJ
cana-1260	50	95	it	it	PRON
cana-1260	50	96	is	be	AUX
cana-1260	50	97	denoted	denote	VERB
cana-1260	50	98	by	by	ADP
cana-1260	50	99	𝛾	𝛾	PROPN
cana-1260	50	100	(	(	PUNCT
cana-1260	50	101	𝐺	𝐺	NOUN
cana-1260	50	102	)	)	PUNCT
cana-1260	50	103	.	.	PUNCT
cana-1260	51	1	the	the	DET
cana-1260	51	2	maximum	maximum	ADJ
cana-1260	51	3	number	number	NOUN
cana-1260	51	4	of	of	ADP
cana-1260	51	5	elements	element	NOUN
cana-1260	51	6	of	of	ADP
cana-1260	51	7	minimal	minimal	ADJ
cana-1260	51	8	dominating	dominating	NOUN
cana-1260	51	9	set	set	NOUN
cana-1260	51	10	of	of	ADP
cana-1260	51	11	graph	graph	NOUN
cana-1260	51	12	𝐺	𝐺	PROPN
cana-1260	51	13	is	be	AUX
cana-1260	51	14	known	know	VERB
cana-1260	51	15	as	as	ADP
cana-1260	51	16	upper	upper	ADJ
cana-1260	51	17	domination	domination	NOUN
cana-1260	51	18	number	number	NOUN
cana-1260	51	19	of	of	ADP
cana-1260	51	20	graph	graph	NOUN
cana-1260	51	21	and	and	CCONJ
cana-1260	51	22	it	it	PRON
cana-1260	51	23	is	be	AUX
cana-1260	51	24	denoted	denote	VERB
cana-1260	51	25	by	by	ADP
cana-1260	51	26	г(g	г(g	NOUN
cana-1260	51	27	)	)	PUNCT
cana-1260	51	28	the	the	DET
cana-1260	51	29	domination	domination	NOUN
cana-1260	51	30	number	number	NOUN
cana-1260	51	31	noted	note	VERB
cana-1260	51	32	as	as	ADP
cana-1260	51	33	𝛾	𝛾	PROPN
cana-1260	51	34	(	(	PUNCT
cana-1260	51	35	𝐺	𝐺	NOUN
cana-1260	51	36	)	)	PUNCT
cana-1260	51	37	and	and	CCONJ
cana-1260	51	38	upper	upper	ADJ
cana-1260	51	39	domination	domination	NOUN
cana-1260	51	40	number	number	NOUN
cana-1260	51	41	noted	note	VERB
cana-1260	51	42	as	as	ADP
cana-1260	51	43	г(g	г(g	NOUN
cana-1260	51	44	)	)	PUNCT
cana-1260	51	45	are	be	AUX
cana-1260	51	46	described	describe	VERB
cana-1260	51	47	as	as	ADP
cana-1260	51	48	:	:	PUNCT
cana-1260	51	49	γ(g	γ(g	PROPN
cana-1260	51	50	)	)	PUNCT
cana-1260	52	1	=	=	SYM
cana-1260	52	2	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-1260	52	3	{	{	PUNCT
cana-1260	52	4	|𝐷|	|𝐷|	NOUN
cana-1260	52	5	:	:	PUNCT
cana-1260	52	6	𝐷	𝐷	NOUN
cana-1260	52	7	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	52	8	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	NOUN
cana-1260	52	9	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	NOUN
cana-1260	52	10	𝑠𝑒𝑡	𝑠𝑒𝑡	PROPN
cana-1260	52	11	𝑜ƒ	𝑜ƒ	ADP
cana-1260	52	12	𝐺	𝐺	PROPN
cana-1260	52	13	}	}	PUNCT
cana-1260	52	14	,	,	PUNCT
cana-1260	52	15	and	and	CCONJ
cana-1260	52	16	г(g	г(g	PROPN
cana-1260	52	17	)	)	PUNCT
cana-1260	52	18	=	=	SYM
cana-1260	52	19	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1260	52	20	{	{	PUNCT
cana-1260	52	21	|𝐷|	|𝐷|	NOUN
cana-1260	52	22	:	:	PUNCT
cana-1260	52	23	𝐷	𝐷	NOUN
cana-1260	52	24	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	52	25	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	NOUN
cana-1260	52	26	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	NOUN
cana-1260	52	27	𝑠𝑒𝑡	𝑠𝑒𝑡	PROPN
cana-1260	52	28	𝑜ƒ	𝑜ƒ	ADP
cana-1260	52	29	𝐺	𝐺	PROPN
cana-1260	52	30	}	}	PUNCT
cana-1260	52	31	.	.	PUNCT
cana-1260	53	1	definition	definition	NOUN
cana-1260	53	2	1.2	1.2	NUM
cana-1260	53	3	.	.	PUNCT
cana-1260	54	1	fractional	fractional	ADJ
cana-1260	54	2	dominating	dominating	NOUN
cana-1260	54	3	function	function	NOUN
cana-1260	54	4	:	:	PUNCT
cana-1260	54	5	a	a	DET
cana-1260	54	6	dominating	dominating	NOUN
cana-1260	54	7	function	function	NOUN
cana-1260	54	8	𝑓	𝑓	PRON
cana-1260	54	9	to	to	PART
cana-1260	54	10	be	be	AUX
cana-1260	54	11	any	any	DET
cana-1260	54	12	𝑓	𝑓	PRON
cana-1260	54	13	:	:	PUNCT
cana-1260	54	14	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	54	15	)	)	PUNCT
cana-1260	54	16	→	→	PUNCT
cana-1260	55	1	[	[	X
cana-1260	55	2	0	0	NUM
cana-1260	55	3	,	,	PUNCT
cana-1260	55	4	1	1	NUM
cana-1260	55	5	]	]	PUNCT
cana-1260	55	6	is	be	AUX
cana-1260	55	7	a	a	DET
cana-1260	55	8	function	function	NOUN
cana-1260	55	9	of	of	ADP
cana-1260	55	10	graph	graph	NOUN
cana-1260	55	11	𝐺	𝐺	PROPN
cana-1260	55	12	which	which	PRON
cana-1260	55	13	allocate	allocate	VERB
cana-1260	55	14	values	value	NOUN
cana-1260	55	15	for	for	ADP
cana-1260	55	16	every	every	DET
cana-1260	55	17	vertex	vertex	NOUN
cana-1260	55	18	𝑣	𝑣	ADP
cana-1260	55	19	∈	∈	PROPN
cana-1260	55	20	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	55	21	)	)	PUNCT
cana-1260	55	22	in	in	ADP
cana-1260	55	23	unit	unit	NOUN
cana-1260	55	24	interval	interval	NOUN
cana-1260	55	25	[	[	X
cana-1260	55	26	0,1	0,1	NUM
cana-1260	55	27	]	]	PUNCT
cana-1260	55	28	.	.	PUNCT
cana-1260	56	1	the	the	DET
cana-1260	56	2	function	function	NOUN
cana-1260	56	3	𝑓	𝑓	PROPN
cana-1260	56	4	is	be	AUX
cana-1260	56	5	known	know	VERB
cana-1260	56	6	as	as	ADP
cana-1260	56	7	fractional	fractional	ADJ
cana-1260	56	8	dominating	dominating	NOUN
cana-1260	56	9	function	function	NOUN
cana-1260	56	10	if	if	SCONJ
cana-1260	56	11	for	for	ADP
cana-1260	56	12	every	every	DET
cana-1260	56	13	vertex	vertex	NOUN
cana-1260	56	14	v	v	ADP
cana-1260	56	15	∈	∈	PROPN
cana-1260	56	16	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	56	17	)	)	PUNCT
cana-1260	56	18	,	,	PUNCT
cana-1260	56	19	𝑓(𝑁[𝑣	𝑓(𝑁[𝑣	PRON
cana-1260	56	20	]	]	X
cana-1260	56	21	)	)	PUNCT
cana-1260	56	22	=	=	SYM
cana-1260	56	23	∑	∑	PUNCT
cana-1260	56	24	𝑓(𝑣	𝑓(𝑣	NUM
cana-1260	56	25	)	)	PUNCT
cana-1260	56	26	⬚	⬚	PROPN
cana-1260	56	27	𝑣∈𝑁[𝑣	𝑣∈𝑁[𝑣	PROPN
cana-1260	56	28	]	]	PUNCT
cana-1260	56	29	≥	≥	NUM
cana-1260	56	30	1	1	NUM
cana-1260	56	31	.	.	PUNCT
cana-1260	57	1	it	it	PRON
cana-1260	57	2	means	mean	VERB
cana-1260	57	3	the	the	DET
cana-1260	57	4	addition	addition	NOUN
cana-1260	57	5	of	of	ADP
cana-1260	57	6	the	the	DET
cana-1260	57	7	values	value	NOUN
cana-1260	57	8	allocated	allocate	VERB
cana-1260	57	9	to	to	ADP
cana-1260	57	10	the	the	DET
cana-1260	57	11	vertices	vertex	NOUN
cana-1260	57	12	in	in	ADP
cana-1260	57	13	closed	closed	ADJ
cana-1260	57	14	neighborhood	neighborhood	NOUN
cana-1260	57	15	of	of	ADP
cana-1260	57	16	𝑣	𝑣	DET
cana-1260	57	17	∈	∈	PROPN
cana-1260	57	18	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	57	19	)	)	PUNCT
cana-1260	57	20	such	such	ADJ
cana-1260	57	21	that	that	SCONJ
cana-1260	57	22	𝑁[𝑣	𝑁[𝑣	PROPN
cana-1260	57	23	]	]	X
cana-1260	57	24	is	be	AUX
cana-1260	57	25	at	at	ADP
cana-1260	57	26	least	least	ADJ
cana-1260	57	27	one	one	NUM
cana-1260	57	28	,	,	PUNCT
cana-1260	57	29	i.e.	i.e.	X
cana-1260	57	30	(	(	PUNCT
cana-1260	57	31	𝑁[𝑣	𝑁[𝑣	PROPN
cana-1260	57	32	]	]	PUNCT
cana-1260	57	33	)	)	PUNCT
cana-1260	57	34	≥	≥	NOUN
cana-1260	57	35	1	1	NUM
cana-1260	57	36	.	.	PUNCT
cana-1260	58	1	(	(	PUNCT
cana-1260	58	2	since	since	SCONJ
cana-1260	58	3	then	then	ADV
cana-1260	58	4	any	any	DET
cana-1260	58	5	vertex	vertex	NOUN
cana-1260	58	6	𝑣	𝑣	ADP
cana-1260	58	7	∈	∈	PROPN
cana-1260	58	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	58	9	)	)	PUNCT
cana-1260	58	10	is	be	AUX
cana-1260	58	11	in	in	ADP
cana-1260	58	12	the	the	DET
cana-1260	58	13	closed	closed	ADJ
cana-1260	58	14	neighborhood	neighborhood	NOUN
cana-1260	58	15	of	of	ADP
cana-1260	58	16	at	at	ADV
cana-1260	58	17	least	least	ADV
cana-1260	58	18	one	one	NUM
cana-1260	58	19	vertex	vertex	NOUN
cana-1260	58	20	in	in	ADP
cana-1260	58	21	𝐷	𝐷	PROPN
cana-1260	58	22	,	,	PUNCT
cana-1260	58	23	where	where	SCONJ
cana-1260	58	24	𝐷	𝐷	PROPN
cana-1260	58	25	is	be	AUX
cana-1260	58	26	subset	subset	VERB
cana-1260	58	27	of	of	ADP
cana-1260	58	28	vertex	vertex	NOUN
cana-1260	58	29	set	set	VERB
cana-1260	58	30	𝑉	𝑉	PROPN
cana-1260	58	31	)	)	PUNCT
cana-1260	58	32	.	.	PUNCT
cana-1260	59	1	definition	definition	NOUN
cana-1260	59	2	1.3	1.3	NUM
cana-1260	59	3	.	.	PUNCT
cana-1260	60	1	fractional	fractional	ADJ
cana-1260	60	2	domination	domination	NOUN
cana-1260	60	3	number	number	NOUN
cana-1260	60	4	:	:	PUNCT
cana-1260	60	5	the	the	DET
cana-1260	60	6	dominating	dominating	NOUN
cana-1260	60	7	function	function	NOUN
cana-1260	60	8	𝑓	𝑓	PROPN
cana-1260	60	9	is	be	AUX
cana-1260	60	10	known	know	VERB
cana-1260	60	11	as	as	ADP
cana-1260	60	12	a	a	DET
cana-1260	60	13	minimal	minimal	ADJ
cana-1260	60	14	fractional	fractional	ADJ
cana-1260	60	15	domination	domination	NOUN
cana-1260	60	16	function	function	NOUN
cana-1260	60	17	if	if	SCONJ
cana-1260	60	18	there	there	PRON
cana-1260	60	19	does	do	AUX
cana-1260	60	20	n’t	not	PART
cana-1260	60	21	exist	exist	VERB
cana-1260	60	22	another	another	DET
cana-1260	60	23	domination	domination	NOUN
cana-1260	60	24	function	function	NOUN
cana-1260	60	25	𝑔	𝑔	PROPN
cana-1260	60	26	≠	≠	PROPN
cana-1260	60	27	𝑓	𝑓	PROPN
cana-1260	60	28	for	for	ADP
cana-1260	60	29	which	which	PRON
cana-1260	60	30	𝑔(𝑣	𝑔(𝑣	PROPN
cana-1260	60	31	)	)	PUNCT
cana-1260	60	32	≤	≤	NOUN
cana-1260	60	33	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1260	60	34	)	)	PUNCT
cana-1260	60	35	for	for	ADP
cana-1260	60	36	all	all	PRON
cana-1260	60	37	𝑣	𝑣	PRON
cana-1260	60	38	∈	∈	PROPN
cana-1260	60	39	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	60	40	)	)	PUNCT
cana-1260	60	41	equivalently	equivalently	ADV
cana-1260	60	42	𝑓	𝑓	PRON
cana-1260	60	43	as	as	ADP
cana-1260	60	44	an	an	DET
cana-1260	60	45	minimal	minimal	ADJ
cana-1260	60	46	fractional	fractional	ADJ
cana-1260	60	47	dominating	dominating	NOUN
cana-1260	60	48	function	function	NOUN
cana-1260	60	49	if	if	SCONJ
cana-1260	60	50	for	for	ADP
cana-1260	60	51	each	each	DET
cana-1260	60	52	vertex	vertex	NOUN
cana-1260	60	53	𝑣	𝑣	X
cana-1260	60	54	with	with	ADP
cana-1260	60	55	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1260	60	56	)	)	PUNCT
cana-1260	60	57	>	>	X
cana-1260	60	58	0	0	NUM
cana-1260	60	59	,	,	PUNCT
cana-1260	60	60	there	there	PRON
cana-1260	60	61	exist	exist	VERB
cana-1260	60	62	a	a	DET
cana-1260	60	63	vertex	vertex	NOUN
cana-1260	60	64	𝑤	𝑤	ADP
cana-1260	60	65	∈	∈	PROPN
cana-1260	60	66	𝑁[𝑣	𝑁[𝑣	PROPN
cana-1260	60	67	]	]	X
cana-1260	60	68	such	such	ADJ
cana-1260	60	69	that	that	SCONJ
cana-1260	60	70	∑	∑	PUNCT
cana-1260	60	71	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	60	72	)	)	PUNCT
cana-1260	60	73	⬚	⬚	PROPN
cana-1260	60	74	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	60	75	]	]	X
cana-1260	60	76	=	=	SYM
cana-1260	60	77	1	1	NUM
cana-1260	60	78	by	by	ADP
cana-1260	60	79	lemma	lemma	PROPN
cana-1260	60	80	[	[	X
cana-1260	60	81	8	8	NUM
cana-1260	60	82	]	]	PUNCT
cana-1260	60	83	let	let	VERB
cana-1260	60	84	𝑓	𝑓	PRON
cana-1260	60	85	be	be	AUX
cana-1260	60	86	a	a	DET
cana-1260	60	87	dominating	dominating	NOUN
cana-1260	60	88	function	function	NOUN
cana-1260	60	89	for	for	ADP
cana-1260	60	90	a	a	DET
cana-1260	60	91	graph	graph	NOUN
cana-1260	60	92	𝐺	𝐺	NOUN
cana-1260	60	93	=	=	SYM
cana-1260	60	94	(	(	PUNCT
cana-1260	60	95	𝑉	𝑉	PROPN
cana-1260	60	96	,	,	PUNCT
cana-1260	60	97	𝐸	𝐸	PROPN
cana-1260	60	98	)	)	PUNCT
cana-1260	60	99	then	then	ADV
cana-1260	60	100	𝑓	𝑓	PRON
cana-1260	60	101	is	be	AUX
cana-1260	60	102	minimal	minimal	ADJ
cana-1260	60	103	dominating	dominating	NOUN
cana-1260	60	104	if	if	SCONJ
cana-1260	60	105	and	and	CCONJ
cana-1260	60	106	only	only	ADV
cana-1260	60	107	if	if	SCONJ
cana-1260	60	108	for	for	ADP
cana-1260	60	109	any	any	DET
cana-1260	60	110	𝑣	𝑣	PRON
cana-1260	60	111	∈	∈	PROPN
cana-1260	60	112	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	60	113	)	)	PUNCT
cana-1260	60	114	whenever	whenever	SCONJ
cana-1260	60	115	𝑓	𝑓	DET
cana-1260	60	116	(	(	PUNCT
cana-1260	60	117	𝑣	𝑣	NOUN
cana-1260	60	118	)	)	PUNCT
cana-1260	60	119	>	>	X
cana-1260	60	120	0	0	PUNCT
cana-1260	61	1	there	there	PRON
cana-1260	61	2	exists	exist	VERB
cana-1260	61	3	some	some	DET
cana-1260	61	4	𝑢	𝑢	PROPN
cana-1260	61	5	∈	∈	PROPN
cana-1260	61	6	𝑁[𝑣	𝑁[𝑣	PROPN
cana-1260	61	7	]	]	X
cana-1260	61	8	such	such	ADJ
cana-1260	61	9	that	that	SCONJ
cana-1260	61	10	𝑓(𝑁[𝑣	𝑓(𝑁[𝑣	NOUN
cana-1260	61	11	]	]	X
cana-1260	61	12	)	)	PUNCT
cana-1260	62	1	=	=	SYM
cana-1260	62	2	1	1	X
cana-1260	62	3	.	.	PUNCT
cana-1260	63	1	(	(	PUNCT
cana-1260	63	2	if	if	SCONJ
cana-1260	63	3	there	there	PRON
cana-1260	63	4	is	be	VERB
cana-1260	63	5	a	a	DET
cana-1260	63	6	vertex	vertex	NOUN
cana-1260	63	7	𝑣	𝑣	X
cana-1260	63	8	for	for	ADP
cana-1260	63	9	which	which	PRON
cana-1260	63	10	given	give	VERB
cana-1260	63	11	condition	condition	NOUN
cana-1260	63	12	is	be	AUX
cana-1260	63	13	not	not	PART
cana-1260	63	14	true	true	ADJ
cana-1260	63	15	,	,	PUNCT
cana-1260	63	16	means	mean	VERB
cana-1260	63	17	if	if	SCONJ
cana-1260	63	18	every	every	DET
cana-1260	63	19	vertex	vertex	NOUN
cana-1260	63	20	in	in	ADP
cana-1260	63	21	the	the	DET
cana-1260	63	22	closed	closed	ADJ
cana-1260	63	23	neighborhood	neighborhood	NOUN
cana-1260	63	24	of	of	ADP
cana-1260	63	25	𝑣	𝑣	PART
cana-1260	63	26	carry	carry	VERB
cana-1260	63	27	out	out	ADP
cana-1260	63	28	(	(	PUNCT
cana-1260	63	29	𝑁[𝑣	𝑁[𝑣	PROPN
cana-1260	63	30	]	]	X
cana-1260	63	31	)	)	PUNCT
cana-1260	63	32	>	>	X
cana-1260	64	1	1	1	NUM
cana-1260	64	2	,	,	PUNCT
cana-1260	64	3	then	then	ADV
cana-1260	64	4	we	we	PRON
cana-1260	64	5	are	be	AUX
cana-1260	64	6	in	in	ADP
cana-1260	64	7	a	a	DET
cana-1260	64	8	position	position	NOUN
cana-1260	64	9	to	to	PART
cana-1260	64	10	decrease	decrease	VERB
cana-1260	64	11	𝑓	𝑓	DET
cana-1260	64	12	(	(	PUNCT
cana-1260	64	13	𝑣	𝑣	NOUN
cana-1260	64	14	)	)	PUNCT
cana-1260	64	15	to	to	PART
cana-1260	64	16	obtain	obtain	VERB
cana-1260	64	17	a	a	DET
cana-1260	64	18	smaller	small	ADJ
cana-1260	64	19	fractional	fractional	ADJ
cana-1260	64	20	dominating	dominating	NOUN
cana-1260	64	21	function	function	NOUN
cana-1260	64	22	and	and	CCONJ
cana-1260	64	23	so	so	ADV
cana-1260	64	24	𝑓	𝑓	PRON
cana-1260	64	25	is	be	AUX
cana-1260	64	26	not	not	PART
cana-1260	64	27	a	a	DET
cana-1260	64	28	minimal	minimal	ADJ
cana-1260	64	29	fractional	fractional	ADJ
cana-1260	64	30	dominating	dominating	NOUN
cana-1260	64	31	function	function	NOUN
cana-1260	64	32	.	.	PUNCT
cana-1260	65	1	the	the	DET
cana-1260	65	2	fractional	fractional	ADJ
cana-1260	65	3	domination	domination	NOUN
cana-1260	65	4	number	number	NOUN
cana-1260	65	5	of	of	ADP
cana-1260	65	6	𝐺	𝐺	PROPN
cana-1260	65	7	,	,	PUNCT
cana-1260	65	8	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	65	9	(	(	PUNCT
cana-1260	65	10	𝐺	𝐺	NOUN
cana-1260	65	11	)	)	PUNCT
cana-1260	65	12	and	and	CCONJ
cana-1260	65	13	upper	upper	ADJ
cana-1260	65	14	fractional	fractional	ADJ
cana-1260	65	15	domination	domination	NOUN
cana-1260	65	16	number	number	NOUN
cana-1260	65	17	of	of	ADP
cana-1260	65	18	𝐺	𝐺	PROPN
cana-1260	65	19	,	,	PUNCT
cana-1260	65	20	г𝑓	г𝑓	X
cana-1260	65	21	(	(	PUNCT
cana-1260	65	22	𝐺	𝐺	NOUN
cana-1260	65	23	)	)	PUNCT
cana-1260	65	24	are	be	AUX
cana-1260	65	25	described	describe	VERB
cana-1260	65	26	as	as	ADP
cana-1260	65	27	,	,	PUNCT
cana-1260	65	28	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	65	29	(	(	PUNCT
cana-1260	65	30	𝐺	𝐺	NOUN
cana-1260	65	31	)	)	PUNCT
cana-1260	65	32	=	=	SYM
cana-1260	65	33	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-1260	65	34	{	{	PUNCT
cana-1260	65	35	|𝑓|	|𝑓|	NOUN
cana-1260	65	36	:	:	PUNCT
cana-1260	65	37	𝑓	𝑓	DET
cana-1260	65	38	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	65	39	𝑎𝑛	𝑎𝑛	NUM
cana-1260	65	40	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	NOUN
cana-1260	65	41	𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙	𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙	NOUN
cana-1260	65	42	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	NOUN
cana-1260	65	43	𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛	𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛	ADV
cana-1260	65	44	𝑜𝑓	𝑜𝑓	ADP
cana-1260	65	45	𝐺	𝐺	PROPN
cana-1260	65	46	}	}	PUNCT
cana-1260	65	47	,	,	PUNCT
cana-1260	65	48	г𝑓	г𝑓	PRON
cana-1260	65	49	(	(	PUNCT
cana-1260	65	50	𝐺	𝐺	NOUN
cana-1260	65	51	)	)	PUNCT
cana-1260	65	52	=	=	PUNCT
cana-1260	65	53	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1260	65	54	{	{	PUNCT
cana-1260	65	55	|𝑓|	|𝑓|	NOUN
cana-1260	65	56	:	:	PUNCT
cana-1260	65	57	𝑓	𝑓	PRON
cana-1260	65	58	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	65	59	𝑎𝑛	𝑎𝑛	NUM
cana-1260	65	60	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	NOUN
cana-1260	65	61	𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙	𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙	NOUN
cana-1260	65	62	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	NOUN
cana-1260	65	63	𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛	𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛	ADV
cana-1260	65	64	𝑜𝑓	𝑜𝑓	ADP
cana-1260	65	65	𝐺	𝐺	PROPN
cana-1260	65	66	}	}	PUNCT
cana-1260	65	67	.	.	PUNCT
cana-1260	66	1	where	where	SCONJ
cana-1260	66	2	|𝑓|	|𝑓|	NOUN
cana-1260	66	3	=	=	SYM
cana-1260	66	4	∑	∑	NOUN
cana-1260	66	5	𝑓(𝑣	𝑓(𝑣	NUM
cana-1260	66	6	)	)	PUNCT
cana-1260	66	7	⬚	⬚	PROPN
cana-1260	66	8	𝑣∈𝑉	𝑣∈𝑉	PROPN
cana-1260	66	9	.	.	PUNCT
cana-1260	67	1	communications	communication	NOUN
cana-1260	67	2	on	on	ADP
cana-1260	67	3	applied	apply	VERB
cana-1260	67	4	nonlinear	nonlinear	ADJ
cana-1260	67	5	analysis	analysis	NOUN
cana-1260	67	6	issn	issn	NOUN
cana-1260	67	7	:	:	PUNCT
cana-1260	67	8	1074	1074	NUM
cana-1260	67	9	-	-	PUNCT
cana-1260	67	10	133x	133x	NUM
cana-1260	67	11	vol	vol	NOUN
cana-1260	67	12	31	31	NUM
cana-1260	67	13	no	no	NOUN
cana-1260	67	14	.	.	PUNCT
cana-1260	68	1	6s	6s	NUM
cana-1260	68	2	(	(	PUNCT
cana-1260	68	3	2024	2024	NUM
cana-1260	68	4	)	)	PUNCT
cana-1260	68	5	673	673	NUM
cana-1260	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	68	7	definition	definition	NOUN
cana-1260	68	8	1.4	1.4	NUM
cana-1260	68	9	.	.	PUNCT
cana-1260	69	1	line	line	NOUN
cana-1260	69	2	graph	graph	NOUN
cana-1260	69	3	:	:	PUNCT
cana-1260	69	4	line	line	NOUN
cana-1260	69	5	graph	graph	NOUN
cana-1260	69	6	of	of	ADP
cana-1260	69	7	𝐺	𝐺	PROPN
cana-1260	69	8	is	be	AUX
cana-1260	69	9	𝐿	𝐿	PROPN
cana-1260	69	10	(	(	PUNCT
cana-1260	69	11	𝐺	𝐺	NOUN
cana-1260	69	12	)	)	PUNCT
cana-1260	69	13	having	have	VERB
cana-1260	69	14	its	its	PRON
cana-1260	69	15	vertices	vertex	NOUN
cana-1260	69	16	are	be	AUX
cana-1260	69	17	converted	convert	VERB
cana-1260	69	18	as	as	SCONJ
cana-1260	69	19	the	the	DET
cana-1260	69	20	edges	edge	NOUN
cana-1260	69	21	of	of	ADP
cana-1260	69	22	original	original	ADJ
cana-1260	69	23	graph	graph	NOUN
cana-1260	69	24	𝐺	𝐺	NOUN
cana-1260	69	25	with	with	ADP
cana-1260	69	26	two	two	NUM
cana-1260	69	27	vertices	vertex	NOUN
cana-1260	69	28	of	of	ADP
cana-1260	69	29	graph	graph	NOUN
cana-1260	69	30	𝐿	𝐿	PROPN
cana-1260	69	31	(	(	PUNCT
cana-1260	69	32	𝐺	𝐺	NOUN
cana-1260	69	33	)	)	PUNCT
cana-1260	69	34	are	be	AUX
cana-1260	69	35	adjacent	adjacent	ADJ
cana-1260	69	36	for	for	ADP
cana-1260	69	37	the	the	DET
cana-1260	69	38	condition	condition	NOUN
cana-1260	69	39	if	if	SCONJ
cana-1260	69	40	corresponding	corresponding	ADJ
cana-1260	69	41	edges	edge	NOUN
cana-1260	69	42	share	share	VERB
cana-1260	69	43	a	a	DET
cana-1260	69	44	common	common	ADJ
cana-1260	69	45	vertex	vertex	NOUN
cana-1260	69	46	in	in	ADP
cana-1260	69	47	graph	graph	NOUN
cana-1260	69	48	𝐺.	𝐺.	NOUN
cana-1260	69	49	another	another	DET
cana-1260	69	50	definition	definition	NOUN
cana-1260	69	51	that	that	SCONJ
cana-1260	69	52	the	the	DET
cana-1260	69	53	vertices	vertex	NOUN
cana-1260	69	54	of	of	ADP
cana-1260	69	55	𝐿	𝐿	PROPN
cana-1260	69	56	(	(	PUNCT
cana-1260	69	57	𝐺	𝐺	NOUN
cana-1260	69	58	)	)	PUNCT
cana-1260	69	59	are	be	AUX
cana-1260	69	60	the	the	DET
cana-1260	69	61	edges	edge	NOUN
cana-1260	69	62	of	of	ADP
cana-1260	69	63	graph	graph	NOUN
cana-1260	69	64	𝐺	𝐺	PROPN
cana-1260	69	65	with	with	ADP
cana-1260	69	66	two	two	NUM
cana-1260	69	67	vertices	vertex	NOUN
cana-1260	69	68	are	be	AUX
cana-1260	69	69	adjacent	adjacent	ADJ
cana-1260	69	70	whenever	whenever	SCONJ
cana-1260	69	71	the	the	DET
cana-1260	69	72	corresponding	corresponding	ADJ
cana-1260	69	73	edges	edge	NOUN
cana-1260	69	74	of	of	ADP
cana-1260	69	75	𝐺	𝐺	PROPN
cana-1260	69	76	are	be	AUX
cana-1260	69	77	adjacent	adjacent	ADJ
cana-1260	69	78	.	.	PUNCT
cana-1260	70	1	in	in	ADP
cana-1260	70	2	any	any	DET
cana-1260	70	3	minimal	minimal	ADJ
cana-1260	70	4	dominating	dominating	NOUN
cana-1260	70	5	set	set	NOUN
cana-1260	70	6	𝑆	𝑆	PROPN
cana-1260	70	7	is	be	AUX
cana-1260	70	8	subset	subset	VERB
cana-1260	70	9	of	of	ADP
cana-1260	70	10	𝑉	𝑉	PROPN
cana-1260	70	11	(	(	PUNCT
cana-1260	70	12	𝐺	𝐺	NOUN
cana-1260	70	13	)	)	PUNCT
cana-1260	70	14	,	,	PUNCT
cana-1260	70	15	such	such	ADJ
cana-1260	70	16	that	that	DET
cana-1260	70	17	𝑁[𝑆	𝑁[𝑆	NOUN
cana-1260	70	18	]	]	X
cana-1260	71	1	=	=	SYM
cana-1260	71	2	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	71	3	)	)	PUNCT
cana-1260	71	4	and	and	CCONJ
cana-1260	71	5	the	the	DET
cana-1260	71	6	closed	closed	ADJ
cana-1260	71	7	neighborhood	neighborhood	NOUN
cana-1260	71	8	of	of	ADP
cana-1260	71	9	each	each	DET
cana-1260	71	10	𝑣	𝑣	PROPN
cana-1260	71	11	∈	∈	PROPN
cana-1260	71	12	𝑆	𝑆	PROPN
cana-1260	71	13	contains	contain	VERB
cana-1260	71	14	at	at	ADV
cana-1260	71	15	least	least	ADV
cana-1260	71	16	one	one	NUM
cana-1260	71	17	vertex	vertex	NOUN
cana-1260	71	18	and	and	CCONJ
cana-1260	71	19	which	which	PRON
cana-1260	71	20	is	be	AUX
cana-1260	71	21	not	not	PART
cana-1260	71	22	in	in	ADP
cana-1260	71	23	the	the	DET
cana-1260	71	24	closed	closed	ADJ
cana-1260	71	25	neighborhood	neighborhood	NOUN
cana-1260	71	26	of	of	ADP
cana-1260	71	27	any	any	DET
cana-1260	71	28	other	other	ADJ
cana-1260	71	29	member	member	NOUN
cana-1260	71	30	of	of	ADP
cana-1260	71	31	𝑆	𝑆	PROPN
cana-1260	71	32	.	.	PUNCT
cana-1260	72	1	(	(	PUNCT
cana-1260	72	2	if	if	SCONJ
cana-1260	72	3	this	this	PRON
cana-1260	72	4	is	be	AUX
cana-1260	72	5	not	not	PART
cana-1260	72	6	true	true	ADJ
cana-1260	72	7	for	for	ADP
cana-1260	72	8	some	some	DET
cana-1260	72	9	𝑣	𝑣	PRON
cana-1260	72	10	∈	∈	PROPN
cana-1260	72	11	𝑆	𝑆	PROPN
cana-1260	72	12	then	then	ADV
cana-1260	72	13	𝑆	𝑆	PROPN
cana-1260	72	14	−	−	PROPN
cana-1260	72	15	{	{	PUNCT
cana-1260	72	16	𝑣	𝑣	NOUN
cana-1260	72	17	}	}	PUNCT
cana-1260	72	18	is	be	AUX
cana-1260	72	19	a	a	DET
cana-1260	72	20	smaller	small	ADJ
cana-1260	72	21	dominating	dominating	NOUN
cana-1260	72	22	set	set	NOUN
cana-1260	72	23	,	,	PUNCT
cana-1260	72	24	contradicting	contradict	VERB
cana-1260	72	25	the	the	DET
cana-1260	72	26	minimality	minimality	NOUN
cana-1260	72	27	of	of	ADP
cana-1260	72	28	𝑆	𝑆	PROPN
cana-1260	72	29	)	)	PUNCT
cana-1260	72	30	.	.	PUNCT
cana-1260	73	1	consequently	consequently	ADV
cana-1260	73	2	,	,	PUNCT
cana-1260	73	3	every	every	DET
cana-1260	73	4	maximal	maximal	ADJ
cana-1260	73	5	irredundant	irredundant	NOUN
cana-1260	73	6	set	set	NOUN
cana-1260	73	7	is	be	AUX
cana-1260	73	8	likewise	likewise	ADV
cana-1260	73	9	a	a	DET
cana-1260	73	10	minimal	minimal	ADJ
cana-1260	73	11	dominant	dominant	ADJ
cana-1260	73	12	set	set	NOUN
cana-1260	73	13	.	.	PUNCT
cana-1260	74	1	hence	hence	ADV
cana-1260	74	2	𝑖𝑟	𝑖𝑟	PROPN
cana-1260	74	3	(	(	PUNCT
cana-1260	74	4	𝐺	𝐺	NOUN
cana-1260	74	5	)	)	PUNCT
cana-1260	74	6	≤	≤	NUM
cana-1260	74	7	𝛾(𝐺	𝛾(𝐺	NOUN
cana-1260	74	8	)	)	PUNCT
cana-1260	74	9	earlier	early	ADV
cana-1260	74	10	this	this	DET
cana-1260	74	11	result	result	NOUN
cana-1260	74	12	was	be	AUX
cana-1260	74	13	obtained	obtain	VERB
cana-1260	74	14	by	by	ADP
cana-1260	74	15	[	[	X
cana-1260	74	16	3	3	NUM
cana-1260	74	17	]	]	PUNCT
cana-1260	74	18	every	every	DET
cana-1260	74	19	minimal	minimal	ADJ
cana-1260	74	20	dominating	dominating	NOUN
cana-1260	74	21	set	set	NOUN
cana-1260	74	22	induces	induce	VERB
cana-1260	74	23	a	a	DET
cana-1260	74	24	minimum	minimum	ADJ
cana-1260	74	25	dominating	dominating	NOUN
cana-1260	74	26	function	function	NOUN
cana-1260	74	27	,	,	PUNCT
cana-1260	74	28	and	and	CCONJ
cana-1260	74	29	every	every	DET
cana-1260	74	30	minimal	minimal	ADJ
cana-1260	74	31	dominating	dominating	NOUN
cana-1260	74	32	function	function	NOUN
cana-1260	74	33	is	be	AUX
cana-1260	74	34	said	say	VERB
cana-1260	74	35	to	to	PART
cana-1260	74	36	be	be	AUX
cana-1260	74	37	a	a	DET
cana-1260	74	38	minimal	minimal	ADJ
cana-1260	74	39	fractional	fractional	ADJ
cana-1260	74	40	dominating	dominating	NOUN
cana-1260	74	41	function	function	NOUN
cana-1260	74	42	.	.	PUNCT
cana-1260	75	1	this	this	PRON
cana-1260	75	2	implies	imply	VERB
cana-1260	75	3	that	that	SCONJ
cana-1260	75	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	75	5	(	(	PUNCT
cana-1260	75	6	𝐺	𝐺	NOUN
cana-1260	75	7	)	)	PUNCT
cana-1260	75	8	≤	≤	NUM
cana-1260	75	9	𝛾(𝐺	𝛾(𝐺	NOUN
cana-1260	75	10	)	)	PUNCT
cana-1260	75	11	≤	≤	NUM
cana-1260	75	12	г(g	г(g	NOUN
cana-1260	75	13	)	)	PUNCT
cana-1260	75	14	≤	≤	NOUN
cana-1260	75	15	г𝑓	г𝑓	ADP
cana-1260	75	16	(	(	PUNCT
cana-1260	75	17	𝐺	𝐺	PROPN
cana-1260	75	18	)	)	PUNCT
cana-1260	75	19	.	.	PUNCT
cana-1260	76	1	some	some	DET
cana-1260	76	2	preliminary	preliminary	ADJ
cana-1260	76	3	known	know	VERB
cana-1260	76	4	results	result	NOUN
cana-1260	76	5	:	:	PUNCT
cana-1260	76	6	theorem	theorem	VERB
cana-1260	76	7	1.1	1.1	NUM
cana-1260	76	8	.	.	PUNCT
cana-1260	77	1	(	(	PUNCT
cana-1260	77	2	[	[	X
cana-1260	77	3	5	5	NUM
cana-1260	77	4	]	]	PUNCT
cana-1260	77	5	)	)	PUNCT
cana-1260	77	6	.	.	PUNCT
cana-1260	78	1	for	for	ADP
cana-1260	78	2	every	every	DET
cana-1260	78	3	graph	graph	NOUN
cana-1260	78	4	𝐺	𝐺	NOUN
cana-1260	78	5	we	we	PRON
cana-1260	78	6	have	have	VERB
cana-1260	78	7	⌈	⌈	SYM
cana-1260	78	8	𝑛	𝑛	PRON
cana-1260	78	9	1+∆(𝐺	1+∆(𝐺	NUM
cana-1260	78	10	)	)	PUNCT
cana-1260	78	11	⌉	⌉	ADP
cana-1260	78	12	≤	≤	NUM
cana-1260	78	13	𝛾	𝛾	X
cana-1260	78	14	(	(	PUNCT
cana-1260	78	15	𝐺	𝐺	NOUN
cana-1260	78	16	)	)	PUNCT
cana-1260	78	17	≤	≤	NOUN
cana-1260	78	18	𝑛	𝑛	PRON
cana-1260	78	19	−	−	NOUN
cana-1260	78	20	𝛥(𝐺	𝛥(𝐺	NUM
cana-1260	78	21	)	)	PUNCT
cana-1260	78	22	.	.	PUNCT
cana-1260	79	1	theorem	theorem	VERB
cana-1260	79	2	1.2	1.2	NUM
cana-1260	79	3	.	.	PUNCT
cana-1260	80	1	(	(	PUNCT
cana-1260	80	2	[	[	X
cana-1260	80	3	5	5	NUM
cana-1260	80	4	]	]	PUNCT
cana-1260	80	5	)	)	PUNCT
cana-1260	80	6	.	.	PUNCT
cana-1260	81	1	for	for	ADP
cana-1260	81	2	every	every	DET
cana-1260	81	3	graph	graph	NOUN
cana-1260	81	4	𝐺	𝐺	NOUN
cana-1260	81	5	we	we	PRON
cana-1260	81	6	have	have	VERB
cana-1260	81	7	𝑛	𝑛	DET
cana-1260	81	8	1+∆(𝐺	1+∆(𝐺	NUM
cana-1260	81	9	)	)	PUNCT
cana-1260	81	10	≤	≤	NOUN
cana-1260	81	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	81	12	(	(	PUNCT
cana-1260	81	13	𝐺	𝐺	NOUN
cana-1260	81	14	)	)	PUNCT
cana-1260	81	15	≤	≤	NOUN
cana-1260	81	16	𝑛	𝑛	DET
cana-1260	81	17	1+𝛿(𝐺	1+𝛿(𝐺	NUM
cana-1260	81	18	)	)	PUNCT
cana-1260	81	19	,	,	PUNCT
cana-1260	81	20	where	where	SCONJ
cana-1260	81	21	∆(𝐺	∆(𝐺	NOUN
cana-1260	81	22	)	)	PUNCT
cana-1260	82	1	𝑖𝑠	𝑖𝑠	CCONJ
cana-1260	82	2	noted	note	VERB
cana-1260	82	3	as	as	ADP
cana-1260	82	4	maximum	maximum	NOUN
cana-1260	82	5	degree	degree	NOUN
cana-1260	82	6	and	and	CCONJ
cana-1260	82	7	𝛿(𝐺	𝛿(𝐺	NUM
cana-1260	82	8	)	)	PUNCT
cana-1260	82	9	is	be	AUX
cana-1260	82	10	noted	note	VERB
cana-1260	82	11	as	as	ADP
cana-1260	82	12	minimum	minimum	NOUN
cana-1260	82	13	degree	degree	NOUN
cana-1260	82	14	of	of	ADP
cana-1260	82	15	graph	graph	NOUN
cana-1260	82	16	g.	g.	PROPN
cana-1260	82	17	theorem	theorem	VERB
cana-1260	82	18	1.3	1.3	NUM
cana-1260	82	19	.	.	PUNCT
cana-1260	83	1	(	(	PUNCT
cana-1260	83	2	[	[	X
cana-1260	83	3	5	5	NUM
cana-1260	83	4	]	]	PUNCT
cana-1260	83	5	)	)	PUNCT
cana-1260	83	6	.	.	PUNCT
cana-1260	84	1	for	for	ADP
cana-1260	84	2	any	any	DET
cana-1260	84	3	graph	graph	NOUN
cana-1260	84	4	𝐺	𝐺	NOUN
cana-1260	84	5	fractional	fractional	ADJ
cana-1260	84	6	domination	domination	NOUN
cana-1260	84	7	number	number	NOUN
cana-1260	84	8	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	84	9	(	(	PUNCT
cana-1260	84	10	𝐺	𝐺	NOUN
cana-1260	84	11	)	)	PUNCT
cana-1260	84	12	=	=	SYM
cana-1260	84	13	1	1	NUM
cana-1260	84	14	𝑖𝑓	𝑖𝑓	NUM
cana-1260	84	15	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1260	84	16	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
cana-1260	84	17	𝑖𝑓	𝑖𝑓	ADP
cana-1260	84	18	𝛥(𝐺	𝛥(𝐺	NUM
cana-1260	84	19	)	)	PUNCT
cana-1260	84	20	=	=	SYM
cana-1260	84	21	𝑛	𝑛	PRON
cana-1260	84	22	−	−	NUM
cana-1260	84	23	1	1	NUM
cana-1260	84	24	.	.	PUNCT
cana-1260	84	25	corollary	corollary	ADJ
cana-1260	84	26	1.1	1.1	NUM
cana-1260	84	27	.	.	PUNCT
cana-1260	85	1	(	(	PUNCT
cana-1260	85	2	[	[	X
cana-1260	85	3	5	5	NUM
cana-1260	85	4	]	]	PUNCT
cana-1260	85	5	)	)	PUNCT
cana-1260	85	6	.	.	PUNCT
cana-1260	86	1	let	let	VERB
cana-1260	86	2	𝐺	𝐺	PROPN
cana-1260	86	3	is	be	AUX
cana-1260	86	4	any	any	DET
cana-1260	86	5	simple	simple	ADJ
cana-1260	86	6	connected	connected	ADJ
cana-1260	86	7	graph	graph	NOUN
cana-1260	86	8	with	with	ADP
cana-1260	86	9	𝛥(𝐺	𝛥(𝐺	NUM
cana-1260	86	10	)	)	PUNCT
cana-1260	86	11	=	=	SYM
cana-1260	86	12	𝑛	𝑛	DET
cana-1260	86	13	−	−	NOUN
cana-1260	86	14	1	1	NUM
cana-1260	86	15	and	and	CCONJ
cana-1260	86	16	let	let	VERB
cana-1260	86	17	𝑓	𝑓	PRON
cana-1260	86	18	be	be	AUX
cana-1260	86	19	any	any	DET
cana-1260	86	20	fractional	fractional	ADJ
cana-1260	86	21	domination	domination	NOUN
cana-1260	86	22	function	function	NOUN
cana-1260	86	23	with	with	ADP
cana-1260	86	24	(	(	PUNCT
cana-1260	86	25	𝐺	𝐺	NOUN
cana-1260	86	26	)	)	PUNCT
cana-1260	86	27	=	=	SYM
cana-1260	87	1	|𝑓|	|𝑓|	PROPN
cana-1260	87	2	.	.	PUNCT
cana-1260	88	1	then	then	ADV
cana-1260	88	2	for	for	ADP
cana-1260	88	3	only	only	ADV
cana-1260	88	4	those	those	DET
cana-1260	88	5	vertices	vertex	NOUN
cana-1260	88	6	𝑣	𝑣	ADP
cana-1260	88	7	∈	∈	PROPN
cana-1260	88	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	88	9	)	)	PUNCT
cana-1260	88	10	with	with	ADP
cana-1260	88	11	𝑑𝑒𝑔	𝑑𝑒𝑔	PROPN
cana-1260	88	12	(	(	PUNCT
cana-1260	88	13	𝑣	𝑣	NOUN
cana-1260	88	14	)	)	PUNCT
cana-1260	88	15	=	=	SYM
cana-1260	88	16	𝑛	𝑛	DET
cana-1260	88	17	−	−	NUM
cana-1260	88	18	1	1	NUM
cana-1260	88	19	satisfy	satisfy	NOUN
cana-1260	88	20	𝑓(𝐺	𝑓(𝐺	PROPN
cana-1260	88	21	)	)	PUNCT
cana-1260	88	22	>	>	X
cana-1260	88	23	0	0	X
cana-1260	88	24	.	.	PUNCT
cana-1260	88	25	theorem	theorem	VERB
cana-1260	88	26	1.4	1.4	NUM
cana-1260	88	27	.	.	PUNCT
cana-1260	89	1	(	(	PUNCT
cana-1260	89	2	[	[	X
cana-1260	89	3	3	3	NUM
cana-1260	89	4	]	]	PUNCT
cana-1260	89	5	)	)	PUNCT
cana-1260	89	6	.	.	PUNCT
cana-1260	90	1	if	if	SCONJ
cana-1260	90	2	the	the	DET
cana-1260	90	3	graph	graph	NOUN
cana-1260	90	4	𝐺	𝐺	NOUN
cana-1260	90	5	is	be	AUX
cana-1260	90	6	r	r	NOUN
cana-1260	90	7	-	-	ADJ
cana-1260	90	8	regular	regular	NOUN
cana-1260	90	9	then	then	ADV
cana-1260	90	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	90	11	(	(	PUNCT
cana-1260	90	12	𝐺	𝐺	NOUN
cana-1260	90	13	)	)	PUNCT
cana-1260	90	14	=	=	PUNCT
cana-1260	90	15	𝑛	𝑛	DET
cana-1260	90	16	𝑟+1	𝑟+1	PROPN
cana-1260	90	17	.	.	PUNCT
cana-1260	91	1	in	in	ADP
cana-1260	91	2	section	section	NOUN
cana-1260	91	3	2	2	NUM
cana-1260	91	4	we	we	PRON
cana-1260	91	5	determine	determine	VERB
cana-1260	91	6	the	the	DET
cana-1260	91	7	fractional	fractional	ADJ
cana-1260	91	8	and	and	CCONJ
cana-1260	91	9	upper	upper	ADJ
cana-1260	91	10	fractional	fractional	ADJ
cana-1260	91	11	dominating	dominating	NOUN
cana-1260	91	12	number	number	NOUN
cana-1260	91	13	of	of	ADP
cana-1260	91	14	graph	graph	NOUN
cana-1260	91	15	𝐺	𝐺	PROPN
cana-1260	91	16	and	and	CCONJ
cana-1260	91	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	91	18	)	)	PUNCT
cana-1260	91	19	.	.	PUNCT
cana-1260	92	1	we	we	PRON
cana-1260	92	2	also	also	ADV
cana-1260	92	3	determine	determine	VERB
cana-1260	92	4	the	the	DET
cana-1260	92	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	92	6	(	(	PUNCT
cana-1260	92	7	𝐺	𝐺	NOUN
cana-1260	92	8	)	)	PUNCT
cana-1260	92	9	+	+	CCONJ
cana-1260	92	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	92	11	(	(	PUNCT
cana-1260	92	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	92	13	)	)	PUNCT
cana-1260	92	14	)	)	PUNCT
cana-1260	92	15	,	,	PUNCT
cana-1260	92	16	𝛾𝑓	𝛾𝑓	X
cana-1260	92	17	(	(	PUNCT
cana-1260	92	18	𝐺	𝐺	NOUN
cana-1260	92	19	)	)	PUNCT
cana-1260	92	20	∗	∗	NOUN
cana-1260	92	21	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	92	22	(	(	PUNCT
cana-1260	92	23	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	92	24	)	)	PUNCT
cana-1260	92	25	)	)	PUNCT
cana-1260	92	26	and	and	CCONJ
cana-1260	92	27	г𝑓	г𝑓	INTJ
cana-1260	92	28	(	(	PUNCT
cana-1260	92	29	𝐺	𝐺	NOUN
cana-1260	92	30	)	)	PUNCT
cana-1260	92	31	+	+	NUM
cana-1260	92	32	г𝑓	г𝑓	ADJ
cana-1260	92	33	(	(	PUNCT
cana-1260	92	34	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	92	35	)	)	PUNCT
cana-1260	92	36	)	)	PUNCT
cana-1260	92	37	,	,	PUNCT
cana-1260	92	38	г𝑓	г𝑓	PROPN
cana-1260	92	39	(	(	PUNCT
cana-1260	92	40	𝐺	𝐺	NOUN
cana-1260	92	41	)	)	PUNCT
cana-1260	92	42	∗	∗	NOUN
cana-1260	92	43	г𝑓	г𝑓	PROPN
cana-1260	92	44	(	(	PUNCT
cana-1260	92	45	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	92	46	)	)	PUNCT
cana-1260	92	47	)	)	PUNCT
cana-1260	92	48	.	.	PUNCT
cana-1260	93	1	we	we	PRON
cana-1260	93	2	have	have	AUX
cana-1260	93	3	taken	take	VERB
cana-1260	93	4	the	the	DET
cana-1260	93	5	standard	standard	ADJ
cana-1260	93	6	graphs	graph	NOUN
cana-1260	93	7	like	like	ADP
cana-1260	93	8	cycle	cycle	NOUN
cana-1260	93	9	,	,	PUNCT
cana-1260	93	10	complete	complete	ADJ
cana-1260	93	11	,	,	PUNCT
cana-1260	93	12	star	star	NOUN
cana-1260	93	13	,	,	PUNCT
cana-1260	93	14	bistar	bistar	PROPN
cana-1260	93	15	graph	graph	NOUN
cana-1260	93	16	,	,	PUNCT
cana-1260	93	17	wheel	wheel	NOUN
cana-1260	93	18	graph	graph	NOUN
cana-1260	93	19	,	,	PUNCT
cana-1260	93	20	cubic	cubic	ADJ
cana-1260	93	21	graph	graph	NOUN
cana-1260	93	22	,	,	PUNCT
cana-1260	93	23	cartesian	cartesian	ADJ
cana-1260	93	24	product	product	NOUN
cana-1260	93	25	graphs	graph	NOUN
cana-1260	93	26	.	.	PUNCT
cana-1260	94	1	in	in	ADP
cana-1260	94	2	section	section	NOUN
cana-1260	94	3	3	3	NUM
cana-1260	94	4	we	we	PRON
cana-1260	94	5	introduce	introduce	VERB
cana-1260	94	6	applications	application	NOUN
cana-1260	94	7	in	in	ADP
cana-1260	94	8	terms	term	NOUN
cana-1260	94	9	of	of	ADP
cana-1260	94	10	union	union	NOUN
cana-1260	94	11	and	and	CCONJ
cana-1260	94	12	join	join	NOUN
cana-1260	94	13	of	of	ADP
cana-1260	94	14	the	the	DET
cana-1260	94	15	considered	consider	VERB
cana-1260	94	16	graphs	graph	NOUN
cana-1260	94	17	.	.	PUNCT
cana-1260	95	1	2	2	X
cana-1260	95	2	.	.	X
cana-1260	95	3	relation	relation	NOUN
cana-1260	95	4	between	between	ADP
cana-1260	95	5	the	the	DET
cana-1260	95	6	fractional	fractional	ADJ
cana-1260	95	7	domination	domination	NOUN
cana-1260	95	8	number	number	NOUN
cana-1260	95	9	of	of	ADP
cana-1260	95	10	graph	graph	NOUN
cana-1260	95	11	and	and	CCONJ
cana-1260	95	12	their	their	PRON
cana-1260	95	13	line	line	NOUN
cana-1260	95	14	graph	graph	NOUN
cana-1260	95	15	2.1	2.1	NUM
cana-1260	95	16	cycle	cycle	NOUN
cana-1260	95	17	graph	graph	NOUN
cana-1260	95	18	(	(	PUNCT
cana-1260	95	19	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	95	20	)	)	PUNCT
cana-1260	95	21	simple	simple	ADJ
cana-1260	95	22	graph	graph	NOUN
cana-1260	95	23	𝐺	𝐺	PROPN
cana-1260	95	24	of	of	ADP
cana-1260	95	25	order	order	NOUN
cana-1260	95	26	𝑛	𝑛	ADP
cana-1260	95	27	where	where	SCONJ
cana-1260	95	28	(	(	PUNCT
cana-1260	95	29	𝑛	𝑛	DET
cana-1260	95	30	≥	≥	NOUN
cana-1260	95	31	3	3	NUM
cana-1260	95	32	)	)	PUNCT
cana-1260	95	33	and	and	CCONJ
cana-1260	95	34	number	number	NOUN
cana-1260	95	35	of	of	ADP
cana-1260	95	36	edges	edge	NOUN
cana-1260	95	37	𝑛	𝑛	PROPN
cana-1260	95	38	is	be	AUX
cana-1260	95	39	known	know	VERB
cana-1260	95	40	as	as	ADP
cana-1260	95	41	a	a	DET
cana-1260	95	42	cycle	cycle	NOUN
cana-1260	95	43	graph	graph	NOUN
cana-1260	95	44	.	.	PUNCT
cana-1260	96	1	every	every	DET
cana-1260	96	2	edge	edge	NOUN
cana-1260	96	3	in	in	ADP
cana-1260	96	4	the	the	DET
cana-1260	96	5	graph	graph	NOUN
cana-1260	96	6	creates	create	VERB
cana-1260	96	7	a	a	DET
cana-1260	96	8	cycle	cycle	NOUN
cana-1260	96	9	with	with	ADP
cana-1260	96	10	length	length	NOUN
cana-1260	96	11	𝑛	𝑛	PROPN
cana-1260	96	12	and	and	CCONJ
cana-1260	96	13	each	each	DET
cana-1260	96	14	vertex	vertex	NOUN
cana-1260	96	15	has	have	AUX
cana-1260	96	16	degree	degree	NOUN
cana-1260	96	17	two	two	NUM
cana-1260	96	18	.	.	PUNCT
cana-1260	97	1	the	the	DET
cana-1260	97	2	line	line	NOUN
cana-1260	97	3	graph	graph	NOUN
cana-1260	97	4	of	of	ADP
cana-1260	97	5	cycle	cycle	NOUN
cana-1260	97	6	graph	graph	NOUN
cana-1260	97	7	is	be	AUX
cana-1260	97	8	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	97	9	)	)	PUNCT
cana-1260	97	10	isomorphic	isomorphic	ADJ
cana-1260	97	11	to	to	ADP
cana-1260	97	12	its	its	PRON
cana-1260	97	13	cycle	cycle	NOUN
cana-1260	97	14	graph	graph	NOUN
cana-1260	97	15	𝐺	𝐺	PROPN
cana-1260	97	16	with	with	ADP
cana-1260	97	17	same	same	ADJ
cana-1260	97	18	vertex	vertex	NOUN
cana-1260	97	19	numbers	number	NOUN
cana-1260	97	20	and	and	CCONJ
cana-1260	97	21	are	be	AUX
cana-1260	97	22	adjacent	adjacent	ADJ
cana-1260	97	23	in	in	ADP
cana-1260	97	24	the	the	DET
cana-1260	97	25	provided	provide	VERB
cana-1260	97	26	graph	graph	NOUN
cana-1260	97	27	if	if	SCONJ
cana-1260	97	28	the	the	DET
cana-1260	97	29	corresponding	corresponding	ADJ
cana-1260	97	30	edges	edge	NOUN
cana-1260	97	31	share	share	VERB
cana-1260	97	32	a	a	DET
cana-1260	97	33	common	common	ADJ
cana-1260	97	34	vertex	vertex	NOUN
cana-1260	97	35	.	.	PUNCT
cana-1260	98	1	theorem	theorem	VERB
cana-1260	98	2	2.1.1	2.1.1	NUM
cana-1260	98	3	.	.	PUNCT
cana-1260	99	1	if	if	SCONJ
cana-1260	99	2	𝐺	𝐺	PROPN
cana-1260	99	3	is	be	AUX
cana-1260	99	4	the	the	DET
cana-1260	99	5	simple	simple	ADJ
cana-1260	99	6	connected	connected	ADJ
cana-1260	99	7	cycle	cycle	NOUN
cana-1260	99	8	graph	graph	NOUN
cana-1260	99	9	(	(	PUNCT
cana-1260	99	10	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	99	11	)	)	PUNCT
cana-1260	99	12	on	on	ADP
cana-1260	99	13	𝑛	𝑛	DET
cana-1260	99	14	vertices	vertex	NOUN
cana-1260	99	15	and	and	CCONJ
cana-1260	99	16	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	99	17	)	)	PUNCT
cana-1260	99	18	is	be	AUX
cana-1260	99	19	line	line	NOUN
cana-1260	99	20	graph	graph	NOUN
cana-1260	99	21	,	,	PUNCT
cana-1260	99	22	then	then	ADV
cana-1260	99	23	communications	communication	NOUN
cana-1260	99	24	on	on	ADP
cana-1260	99	25	applied	apply	VERB
cana-1260	99	26	nonlinear	nonlinear	ADJ
cana-1260	99	27	analysis	analysis	NOUN
cana-1260	99	28	issn	issn	NOUN
cana-1260	99	29	:	:	PUNCT
cana-1260	99	30	1074	1074	NUM
cana-1260	99	31	-	-	PUNCT
cana-1260	99	32	133x	133x	NUM
cana-1260	99	33	vol	vol	NOUN
cana-1260	99	34	31	31	NUM
cana-1260	99	35	no	no	NOUN
cana-1260	99	36	.	.	PUNCT
cana-1260	100	1	6s	6s	NUM
cana-1260	100	2	(	(	PUNCT
cana-1260	100	3	2024	2024	NUM
cana-1260	100	4	)	)	PUNCT
cana-1260	100	5	674	674	NUM
cana-1260	100	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	100	7	i	i	NOUN
cana-1260	100	8	)	)	PUNCT
cana-1260	100	9	2	2	NUM
cana-1260	100	10	≤	≤	NOUN
cana-1260	100	11	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	100	12	(	(	PUNCT
cana-1260	100	13	𝐺	𝐺	NOUN
cana-1260	100	14	)	)	PUNCT
cana-1260	101	1	+	+	CCONJ
cana-1260	101	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	101	3	(	(	PUNCT
cana-1260	101	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	101	5	)	)	PUNCT
cana-1260	101	6	)	)	PUNCT
cana-1260	101	7	≤	≤	NUM
cana-1260	102	1	2𝑛	2𝑛	PROPN
cana-1260	102	2	3	3	NUM
cana-1260	102	3	ii	ii	NOUN
cana-1260	102	4	)	)	PUNCT
cana-1260	102	5	1	1	NUM
cana-1260	102	6	≤	≤	NUM
cana-1260	102	7	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	102	8	(	(	PUNCT
cana-1260	102	9	𝐺	𝐺	NOUN
cana-1260	102	10	)	)	PUNCT
cana-1260	102	11	∗	∗	NOUN
cana-1260	102	12	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	102	13	(	(	PUNCT
cana-1260	102	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	102	15	)	)	PUNCT
cana-1260	102	16	)	)	PUNCT
cana-1260	102	17	≤	≤	NUM
cana-1260	102	18	𝑛	𝑛	PRON
cana-1260	102	19	2/9	2/9	NUM
cana-1260	102	20	iii	iii	NOUN
cana-1260	102	21	)	)	PUNCT
cana-1260	102	22	г𝑓	г𝑓	PROPN
cana-1260	102	23	(	(	PUNCT
cana-1260	102	24	𝐺	𝐺	NOUN
cana-1260	102	25	)	)	PUNCT
cana-1260	102	26	+	+	NUM
cana-1260	102	27	г𝑓	г𝑓	ADJ
cana-1260	102	28	(	(	PUNCT
cana-1260	102	29	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	102	30	)	)	PUNCT
cana-1260	102	31	)	)	PUNCT
cana-1260	103	1	=	=	SYM
cana-1260	103	2	𝑛	𝑛	PROPN
cana-1260	103	3	iv	iv	X
cana-1260	103	4	)	)	PUNCT
cana-1260	103	5	г𝑓	г𝑓	PROPN
cana-1260	103	6	(	(	PUNCT
cana-1260	103	7	𝐺	𝐺	NOUN
cana-1260	103	8	)	)	PUNCT
cana-1260	103	9	∗	∗	NOUN
cana-1260	103	10	г𝑓	г𝑓	PROPN
cana-1260	103	11	(	(	PUNCT
cana-1260	103	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	103	13	)	)	PUNCT
cana-1260	103	14	)	)	PUNCT
cana-1260	104	1	=	=	SYM
cana-1260	104	2	𝑛	𝑛	DET
cana-1260	104	3	2/4	2/4	NUM
cana-1260	104	4	proof	proof	NOUN
cana-1260	104	5	:	:	PUNCT
cana-1260	104	6	i	i	NOUN
cana-1260	104	7	)	)	PUNCT
cana-1260	104	8	and	and	CCONJ
cana-1260	104	9	ii	ii	X
cana-1260	104	10	)	)	PUNCT
cana-1260	104	11	with	with	ADP
cana-1260	104	12	the	the	DET
cana-1260	104	13	reference	reference	NOUN
cana-1260	104	14	of	of	ADP
cana-1260	104	15	previous	previous	ADJ
cana-1260	104	16	chapter	chapter	NOUN
cana-1260	104	17	for	for	ADP
cana-1260	104	18	cycle	cycle	NOUN
cana-1260	104	19	graph	graph	NOUN
cana-1260	104	20	we	we	PRON
cana-1260	104	21	have	have	VERB
cana-1260	104	22	fractional	fractional	ADJ
cana-1260	104	23	domination	domination	NOUN
cana-1260	104	24	number	number	NOUN
cana-1260	104	25	is	be	AUX
cana-1260	104	26	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	104	27	(	(	PUNCT
cana-1260	104	28	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	104	29	)	)	PUNCT
cana-1260	104	30	=	=	SYM
cana-1260	104	31	𝑛/3	𝑛/3	NUM
cana-1260	104	32	,	,	PUNCT
cana-1260	104	33	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1260	104	34	𝑛	𝑛	NOUN
cana-1260	104	35	=	=	SYM
cana-1260	104	36	3,4,5	3,4,5	NUM
cana-1260	104	37	,	,	PUNCT
cana-1260	104	38	…	…	PUNCT
cana-1260	104	39	𝑛.	𝑛.	NOUN
cana-1260	104	40	the	the	DET
cana-1260	104	41	line	line	NOUN
cana-1260	104	42	graph	graph	NOUN
cana-1260	104	43	of	of	ADP
cana-1260	104	44	cycle	cycle	NOUN
cana-1260	104	45	graph	graph	NOUN
cana-1260	104	46	is	be	AUX
cana-1260	104	47	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	104	48	)	)	PUNCT
cana-1260	104	49	isomorphic	isomorphic	ADJ
cana-1260	104	50	to	to	ADP
cana-1260	104	51	its	its	PRON
cana-1260	104	52	cycle	cycle	NOUN
cana-1260	104	53	graph	graph	NOUN
cana-1260	104	54	𝐺	𝐺	PROPN
cana-1260	104	55	with	with	ADP
cana-1260	104	56	same	same	ADJ
cana-1260	104	57	vertex	vertex	NOUN
cana-1260	104	58	numbers	number	NOUN
cana-1260	104	59	and	and	CCONJ
cana-1260	104	60	are	be	AUX
cana-1260	104	61	adjacent	adjacent	ADJ
cana-1260	104	62	in	in	ADP
cana-1260	104	63	the	the	DET
cana-1260	104	64	provided	provide	VERB
cana-1260	104	65	graph	graph	NOUN
cana-1260	104	66	if	if	SCONJ
cana-1260	104	67	the	the	DET
cana-1260	104	68	corresponding	corresponding	ADJ
cana-1260	104	69	edges	edge	NOUN
cana-1260	104	70	share	share	VERB
cana-1260	104	71	a	a	DET
cana-1260	104	72	common	common	ADJ
cana-1260	104	73	vertex	vertex	NOUN
cana-1260	104	74	.	.	PUNCT
cana-1260	105	1	therefore	therefore	ADV
cana-1260	105	2	,	,	PUNCT
cana-1260	105	3	for	for	ADP
cana-1260	105	4	line	line	NOUN
cana-1260	105	5	graph	graph	NOUN
cana-1260	105	6	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	105	7	)	)	PUNCT
cana-1260	105	8	the	the	DET
cana-1260	105	9	fractional	fractional	ADJ
cana-1260	105	10	dominating	dominating	NOUN
cana-1260	105	11	number	number	NOUN
cana-1260	105	12	is	be	AUX
cana-1260	105	13	𝛾𝑓	𝛾𝑓	X
cana-1260	105	14	(	(	PUNCT
cana-1260	105	15	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	105	16	)	)	PUNCT
cana-1260	105	17	)	)	PUNCT
cana-1260	106	1	=	=	SYM
cana-1260	106	2	𝑛/3	𝑛/3	NUM
cana-1260	106	3	.	.	PUNCT
cana-1260	107	1	for	for	ADP
cana-1260	107	2	the	the	DET
cana-1260	107	3	sum	sum	NOUN
cana-1260	107	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	107	5	(	(	PUNCT
cana-1260	107	6	𝐺	𝐺	NOUN
cana-1260	107	7	)	)	PUNCT
cana-1260	108	1	+	+	CCONJ
cana-1260	108	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	108	3	(	(	PUNCT
cana-1260	108	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	108	5	)	)	PUNCT
cana-1260	108	6	)	)	PUNCT
cana-1260	109	1	lower	lower	ADV
cana-1260	109	2	bound	bind	VERB
cana-1260	109	3	is	be	AUX
cana-1260	109	4	2	2	NUM
cana-1260	109	5	and	and	CCONJ
cana-1260	109	6	upper	upper	ADJ
cana-1260	109	7	bound	bind	VERB
cana-1260	109	8	is	be	AUX
cana-1260	109	9	2𝑛	2𝑛	PROPN
cana-1260	109	10	3	3	NUM
cana-1260	109	11	.	.	PUNCT
cana-1260	110	1	therefore	therefore	ADV
cana-1260	110	2	we	we	PRON
cana-1260	110	3	have	have	VERB
cana-1260	110	4	2	2	NUM
cana-1260	110	5	≤	≤	NUM
cana-1260	110	6	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	110	7	(	(	PUNCT
cana-1260	110	8	𝐺	𝐺	NOUN
cana-1260	110	9	)	)	PUNCT
cana-1260	110	10	+	+	CCONJ
cana-1260	110	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	110	12	(	(	PUNCT
cana-1260	110	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	110	14	)	)	PUNCT
cana-1260	110	15	)	)	PUNCT
cana-1260	110	16	≤	≤	NUM
cana-1260	111	1	2𝑛	2𝑛	NUM
cana-1260	111	2	3	3	NUM
cana-1260	111	3	and	and	CCONJ
cana-1260	111	4	1	1	NUM
cana-1260	111	5	≤	≤	NUM
cana-1260	111	6	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	111	7	(	(	PUNCT
cana-1260	111	8	𝐺	𝐺	NOUN
cana-1260	111	9	)	)	PUNCT
cana-1260	111	10	∗	∗	NOUN
cana-1260	111	11	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	111	12	(	(	PUNCT
cana-1260	111	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	111	14	)	)	PUNCT
cana-1260	111	15	)	)	PUNCT
cana-1260	111	16	≤	≤	PROPN
cana-1260	112	1	𝑛2/9	𝑛2/9	PROPN
cana-1260	112	2	,	,	PUNCT
cana-1260	112	3	𝑛	𝑛	PROPN
cana-1260	112	4	=	=	SYM
cana-1260	112	5	3,4,5	3,4,5	NUM
cana-1260	112	6	,	,	PUNCT
cana-1260	112	7	…	…	PUNCT
cana-1260	112	8	𝑛.	𝑛.	NOUN
cana-1260	112	9	iii	iii	NOUN
cana-1260	112	10	)	)	PUNCT
cana-1260	112	11	and	and	CCONJ
cana-1260	112	12	iv	iv	X
cana-1260	112	13	)	)	PUNCT
cana-1260	112	14	the	the	DET
cana-1260	112	15	upper	upper	ADJ
cana-1260	112	16	fractional	fractional	ADJ
cana-1260	112	17	domination	domination	NOUN
cana-1260	112	18	number	number	NOUN
cana-1260	112	19	of	of	ADP
cana-1260	112	20	cycle	cycle	NOUN
cana-1260	112	21	graph	graph	NOUN
cana-1260	113	1	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	113	2	is	be	AUX
cana-1260	113	3	г𝑓	г𝑓	DET
cana-1260	113	4	(	(	PUNCT
cana-1260	113	5	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	113	6	)	)	PUNCT
cana-1260	113	7	=	=	SYM
cana-1260	113	8	𝑛/2	𝑛/2	PROPN
cana-1260	113	9	is	be	AUX
cana-1260	113	10	observed	observe	VERB
cana-1260	113	11	by	by	ADP
cana-1260	113	12	assigning	assign	VERB
cana-1260	113	13	weight	weight	NOUN
cana-1260	113	14	(	(	PUNCT
cana-1260	113	15	½	½	NOUN
cana-1260	113	16	)	)	PUNCT
cana-1260	113	17	to	to	ADP
cana-1260	113	18	every	every	DET
cana-1260	113	19	vertex	vertex	NOUN
cana-1260	113	20	and	and	CCONJ
cana-1260	113	21	the	the	DET
cana-1260	113	22	condition	condition	NOUN
cana-1260	113	23	where	where	SCONJ
cana-1260	113	24	vertex	vertex	NOUN
cana-1260	113	25	𝑤	𝑤	ADP
cana-1260	113	26	∈	∈	PROPN
cana-1260	113	27	𝑁[𝑣	𝑁[𝑣	PROPN
cana-1260	113	28	]	]	X
cana-1260	113	29	such	such	ADJ
cana-1260	113	30	that∑	that∑	NOUN
cana-1260	113	31	𝑓(𝑣	𝑓(𝑣	VERB
cana-1260	113	32	)	)	PUNCT
cana-1260	113	33	⬚	⬚	PROPN
cana-1260	113	34	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	113	35	]	]	X
cana-1260	113	36	=	=	SYM
cana-1260	113	37	1	1	NUM
cana-1260	113	38	condition	condition	NOUN
cana-1260	113	39	is	be	AUX
cana-1260	113	40	satisfied	satisfied	ADJ
cana-1260	113	41	.	.	PUNCT
cana-1260	114	1	therefore	therefore	ADV
cana-1260	114	2	we	we	PRON
cana-1260	114	3	found	find	VERB
cana-1260	114	4	that	that	SCONJ
cana-1260	114	5	г𝑓	г𝑓	PROPN
cana-1260	114	6	(	(	PUNCT
cana-1260	114	7	𝐺	𝐺	NOUN
cana-1260	114	8	)	)	PUNCT
cana-1260	115	1	+	+	NUM
cana-1260	115	2	г𝑓	г𝑓	ADJ
cana-1260	115	3	(	(	PUNCT
cana-1260	115	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	115	5	)	)	PUNCT
cana-1260	115	6	)	)	PUNCT
cana-1260	116	1	=	=	SYM
cana-1260	116	2	𝑛	𝑛	PROPN
cana-1260	116	3	and	and	CCONJ
cana-1260	116	4	г𝑓	г𝑓	INTJ
cana-1260	116	5	(	(	PUNCT
cana-1260	116	6	𝐺	𝐺	NOUN
cana-1260	116	7	)	)	PUNCT
cana-1260	116	8	∗	∗	NOUN
cana-1260	116	9	г𝑓	г𝑓	PROPN
cana-1260	116	10	(	(	PUNCT
cana-1260	116	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	116	12	)	)	PUNCT
cana-1260	116	13	)	)	PUNCT
cana-1260	117	1	=	=	SYM
cana-1260	117	2	𝑛	𝑛	PRON
cana-1260	117	3	2/4	2/4	NUM
cana-1260	117	4	.	.	PUNCT
cana-1260	118	1	□	□	PUNCT
cana-1260	118	2	theorem	theorem	ADJ
cana-1260	118	3	2.1.2	2.1.2	NUM
cana-1260	118	4	let	let	VERB
cana-1260	118	5	graph	graph	NOUN
cana-1260	118	6	𝐺	𝐺	PROPN
cana-1260	118	7	is	be	AUX
cana-1260	118	8	𝑟-regular	𝑟-regular	NUM
cana-1260	118	9	of	of	ADP
cana-1260	118	10	order	order	NOUN
cana-1260	118	11	𝑛	𝑛	PRON
cana-1260	118	12	then	then	ADV
cana-1260	118	13	its	its	PRON
cana-1260	118	14	upper	upper	ADJ
cana-1260	118	15	fractional	fractional	ADJ
cana-1260	118	16	domination	domination	NOUN
cana-1260	118	17	number	number	NOUN
cana-1260	118	18	is	be	AUX
cana-1260	118	19	г𝑓	г𝑓	DET
cana-1260	118	20	(	(	PUNCT
cana-1260	118	21	𝐺	𝐺	NOUN
cana-1260	118	22	)	)	PUNCT
cana-1260	118	23	≤	≤	NOUN
cana-1260	119	1	𝑛/𝑟	𝑛/𝑟	NOUN
cana-1260	119	2	proof	proof	NOUN
cana-1260	119	3	:	:	PUNCT
cana-1260	119	4	theorem	theorem	VERB
cana-1260	119	5	1.4	1.4	NUM
cana-1260	119	6	.	.	PUNCT
cana-1260	120	1	gives	give	VERB
cana-1260	120	2	if	if	SCONJ
cana-1260	120	3	graph	graph	NOUN
cana-1260	120	4	𝐺	𝐺	PROPN
cana-1260	120	5	be	be	AUX
cana-1260	120	6	𝑟-regular	𝑟-regular	NUM
cana-1260	120	7	of	of	ADP
cana-1260	120	8	order	order	NOUN
cana-1260	120	9	𝑛	𝑛	NOUN
cana-1260	120	10	then	then	ADV
cana-1260	120	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	120	12	(	(	PUNCT
cana-1260	120	13	𝐺	𝐺	NOUN
cana-1260	120	14	)	)	PUNCT
cana-1260	120	15	=	=	PUNCT
cana-1260	120	16	𝑛	𝑛	PRON
cana-1260	120	17	𝑟+1	𝑟+1	NUM
cana-1260	120	18	.	.	PUNCT
cana-1260	121	1	the	the	DET
cana-1260	121	2	fractional	fractional	ADJ
cana-1260	121	3	dominating	dominating	NOUN
cana-1260	121	4	function	function	NOUN
cana-1260	121	5	𝑓	𝑓	PROPN
cana-1260	121	6	that	that	PRON
cana-1260	121	7	allocates	allocate	VERB
cana-1260	121	8	the	the	DET
cana-1260	121	9	weight	weight	NOUN
cana-1260	121	10	1/(𝑟	1/(𝑟	NUM
cana-1260	121	11	+	+	CCONJ
cana-1260	121	12	1	1	NUM
cana-1260	121	13	)	)	PUNCT
cana-1260	121	14	to	to	ADP
cana-1260	121	15	each	each	DET
cana-1260	121	16	vertex	vertex	NOUN
cana-1260	121	17	of	of	ADP
cana-1260	121	18	𝐺	𝐺	NOUN
cana-1260	121	19	so	so	SCONJ
cana-1260	121	20	that	that	SCONJ
cana-1260	121	21	function	function	NOUN
cana-1260	121	22	is	be	AUX
cana-1260	121	23	the	the	DET
cana-1260	121	24	minimal	minimal	ADJ
cana-1260	121	25	weight	weight	NOUN
cana-1260	121	26	fractional	fractional	ADJ
cana-1260	121	27	dominating	dominating	NOUN
cana-1260	121	28	function	function	NOUN
cana-1260	121	29	,	,	PUNCT
cana-1260	121	30	and	and	CCONJ
cana-1260	121	31	its	its	PRON
cana-1260	121	32	generalized	generalized	ADJ
cana-1260	121	33	form	form	NOUN
cana-1260	121	34	is	be	AUX
cana-1260	121	35	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	121	36	(	(	PUNCT
cana-1260	121	37	𝐺	𝐺	NOUN
cana-1260	121	38	)	)	PUNCT
cana-1260	121	39	=	=	PUNCT
cana-1260	121	40	𝑛	𝑛	DET
cana-1260	121	41	𝑟+1	𝑟+1	PROPN
cana-1260	121	42	.	.	PUNCT
cana-1260	122	1	so	so	ADV
cana-1260	122	2	an	an	DET
cana-1260	122	3	identical	identical	ADJ
cana-1260	122	4	argument	argument	NOUN
cana-1260	122	5	works	work	VERB
cana-1260	122	6	for	for	ADP
cana-1260	122	7	given	give	VERB
cana-1260	122	8	upper	upper	ADJ
cana-1260	122	9	fractional	fractional	ADJ
cana-1260	122	10	domination	domination	NOUN
cana-1260	122	11	number	number	NOUN
cana-1260	122	12	where	where	SCONJ
cana-1260	122	13	function	function	VERB
cana-1260	122	14	𝑓	𝑓	DET
cana-1260	122	15	allocating	allocate	VERB
cana-1260	122	16	weights	weight	NOUN
cana-1260	122	17	(	(	PUNCT
cana-1260	122	18	1/𝑟	1/𝑟	NUM
cana-1260	122	19	)	)	PUNCT
cana-1260	122	20	to	to	ADP
cana-1260	122	21	each	each	DET
cana-1260	122	22	vertex	vertex	NOUN
cana-1260	122	23	of	of	ADP
cana-1260	122	24	𝐺	𝐺	NOUN
cana-1260	122	25	so	so	SCONJ
cana-1260	122	26	that	that	SCONJ
cana-1260	122	27	г𝑓	г𝑓	PROPN
cana-1260	122	28	(	(	PUNCT
cana-1260	122	29	𝐺	𝐺	NOUN
cana-1260	122	30	)	)	PUNCT
cana-1260	122	31	is	be	AUX
cana-1260	122	32	maximum	maximum	ADJ
cana-1260	122	33	cardinality	cardinality	NOUN
cana-1260	122	34	of	of	ADP
cana-1260	122	35	function	function	NOUN
cana-1260	122	36	𝑓	𝑓	PRON
cana-1260	122	37	where	where	SCONJ
cana-1260	122	38	𝑓	𝑓	PRON
cana-1260	122	39	is	be	AUX
cana-1260	122	40	minimal	minimal	ADJ
cana-1260	122	41	fractional	fractional	ADJ
cana-1260	122	42	dominating	dominating	NOUN
cana-1260	122	43	function	function	NOUN
cana-1260	122	44	of	of	ADP
cana-1260	122	45	𝐺.	𝐺.	NOUN
cana-1260	122	46	its	its	PRON
cana-1260	122	47	generalized	generalized	ADJ
cana-1260	122	48	form	form	NOUN
cana-1260	122	49	is	be	AUX
cana-1260	122	50	given	give	VERB
cana-1260	122	51	by	by	ADP
cana-1260	122	52	г𝑓	г𝑓	PROPN
cana-1260	122	53	(	(	PUNCT
cana-1260	122	54	𝐺	𝐺	NOUN
cana-1260	122	55	)	)	PUNCT
cana-1260	122	56	≤	≤	NOUN
cana-1260	122	57	𝑛/𝑟.	𝑛/𝑟.	NOUN
cana-1260	122	58	let	let	VERB
cana-1260	122	59	𝑣	𝑣	PRON
cana-1260	122	60	∈	∈	PROPN
cana-1260	122	61	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	122	62	)	)	PUNCT
cana-1260	122	63	such	such	ADJ
cana-1260	122	64	that	that	SCONJ
cana-1260	122	65	𝑓(𝑁[𝑉	𝑓(𝑁[𝑉	PROPN
cana-1260	122	66	]	]	X
cana-1260	122	67	)	)	PUNCT
cana-1260	123	1	=	=	SYM
cana-1260	123	2	∑	∑	PUNCT
cana-1260	123	3	𝑓(𝑣	𝑓(𝑣	NUM
cana-1260	123	4	)	)	PUNCT
cana-1260	123	5	⬚	⬚	PROPN
cana-1260	123	6	𝑣∈𝑁[𝑣	𝑣∈𝑁[𝑣	PROPN
cana-1260	123	7	]	]	PUNCT
cana-1260	123	8	≥	≥	NUM
cana-1260	123	9	1	1	NUM
cana-1260	123	10	.	.	PUNCT
cana-1260	123	11	г𝑓	г𝑓	PROPN
cana-1260	123	12	(	(	PUNCT
cana-1260	123	13	𝐺	𝐺	NOUN
cana-1260	123	14	)	)	PUNCT
cana-1260	123	15	=	=	PUNCT
cana-1260	123	16	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1260	123	17	{	{	PUNCT
cana-1260	123	18	|𝑓|	|𝑓|	NOUN
cana-1260	123	19	:	:	PUNCT
cana-1260	123	20	𝑓	𝑓	DET
cana-1260	123	21	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	123	22	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	𝑚𝑖𝑛𝑖𝑚𝑎𝑙	NOUN
cana-1260	123	23	𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙	𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙	NOUN
cana-1260	123	24	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔	NOUN
cana-1260	123	25	𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛	𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛	ADV
cana-1260	123	26	𝑜𝑓	𝑜𝑓	ADP
cana-1260	123	27	𝐺	𝐺	PROPN
cana-1260	123	28	}	}	PUNCT
cana-1260	123	29	.	.	PUNCT
cana-1260	124	1	where	where	SCONJ
cana-1260	124	2	|𝑓|	|𝑓|	NOUN
cana-1260	124	3	=	=	NOUN
cana-1260	124	4	∑	∑	PUNCT
cana-1260	124	5	𝑓(𝑣	𝑓(𝑣	NUM
cana-1260	124	6	)	)	PUNCT
cana-1260	124	7	⬚	⬚	PROPN
cana-1260	124	8	𝑣∈𝑉	𝑣∈𝑉	PROPN
cana-1260	124	9	.	.	PUNCT
cana-1260	125	1	so	so	ADV
cana-1260	125	2	if	if	SCONJ
cana-1260	125	3	graph	graph	NOUN
cana-1260	125	4	𝐺	𝐺	PROPN
cana-1260	125	5	is	be	AUX
cana-1260	125	6	𝑟-regular	𝑟-regular	NUM
cana-1260	125	7	of	of	ADP
cana-1260	125	8	order	order	NOUN
cana-1260	125	9	𝑛	𝑛	PRON
cana-1260	125	10	then	then	ADV
cana-1260	125	11	its	its	PRON
cana-1260	125	12	upper	upper	ADJ
cana-1260	125	13	fractional	fractional	ADJ
cana-1260	125	14	domination	domination	NOUN
cana-1260	125	15	number	number	NOUN
cana-1260	125	16	is	be	AUX
cana-1260	125	17	г𝑓	г𝑓	DET
cana-1260	125	18	(	(	PUNCT
cana-1260	125	19	𝐺	𝐺	NOUN
cana-1260	125	20	)	)	PUNCT
cana-1260	125	21	≤	≤	NOUN
cana-1260	125	22	𝑛/𝑟	𝑛/𝑟	NOUN
cana-1260	125	23	.	.	PUNCT
cana-1260	126	1	□	□	PUNCT
cana-1260	126	2	2.2	2.2	NUM
cana-1260	126	3	complete	complete	ADJ
cana-1260	126	4	graph	graph	NOUN
cana-1260	126	5	(	(	PUNCT
cana-1260	126	6	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	126	7	)	)	PUNCT
cana-1260	126	8	a	a	DET
cana-1260	126	9	simple	simple	ADJ
cana-1260	126	10	connected	connected	ADJ
cana-1260	126	11	graph	graph	NOUN
cana-1260	126	12	𝐺	𝐺	PROPN
cana-1260	126	13	is	be	AUX
cana-1260	126	14	considered	consider	VERB
cana-1260	126	15	to	to	PART
cana-1260	126	16	complete	complete	VERB
cana-1260	126	17	if	if	SCONJ
cana-1260	126	18	every	every	DET
cana-1260	126	19	pair	pair	NOUN
cana-1260	126	20	of	of	ADP
cana-1260	126	21	vertices	vertex	NOUN
cana-1260	126	22	are	be	AUX
cana-1260	126	23	adjacent	adjacent	ADJ
cana-1260	126	24	.	.	PUNCT
cana-1260	127	1	a	a	DET
cana-1260	127	2	complete	complete	ADJ
cana-1260	127	3	graph	graph	NOUN
cana-1260	127	4	𝐺	𝐺	NOUN
cana-1260	127	5	is	be	AUX
cana-1260	127	6	simple	simple	ADJ
cana-1260	127	7	undirected	undirected	ADJ
cana-1260	127	8	graph	graph	NOUN
cana-1260	127	9	in	in	SCONJ
cana-1260	127	10	which	which	DET
cana-1260	127	11	pair	pair	NOUN
cana-1260	127	12	of	of	ADP
cana-1260	127	13	different	different	ADJ
cana-1260	127	14	vertices	vertex	NOUN
cana-1260	127	15	is	be	AUX
cana-1260	127	16	connected	connect	VERB
cana-1260	127	17	by	by	ADP
cana-1260	127	18	a	a	DET
cana-1260	127	19	unique	unique	ADJ
cana-1260	127	20	edge	edge	NOUN
cana-1260	127	21	.	.	PUNCT
cana-1260	128	1	in	in	ADP
cana-1260	128	2	the	the	DET
cana-1260	128	3	line	line	NOUN
cana-1260	128	4	graph	graph	NOUN
cana-1260	128	5	of	of	ADP
cana-1260	128	6	complete	complete	ADJ
cana-1260	128	7	graph	graph	NOUN
cana-1260	128	8	number	number	NOUN
cana-1260	128	9	of	of	ADP
cana-1260	128	10	vertices	vertex	NOUN
cana-1260	128	11	in	in	ADP
cana-1260	128	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	128	13	)	)	PUNCT
cana-1260	128	14	are	be	AUX
cana-1260	128	15	equal	equal	ADJ
cana-1260	128	16	in	in	ADP
cana-1260	128	17	number	number	NOUN
cana-1260	128	18	of	of	ADP
cana-1260	128	19	edges	edge	NOUN
cana-1260	128	20	in	in	ADP
cana-1260	128	21	graph	graph	NOUN
cana-1260	128	22	𝐺.	𝐺.	PROPN
cana-1260	128	23	https://en.wikipedia.org/wiki/simple_graph	https://en.wikipedia.org/wiki/simple_graph	NOUN
cana-1260	128	24	https://en.wikipedia.org/wiki/undirected_graph	https://en.wikipedia.org/wiki/undirected_graph	NOUN
cana-1260	128	25	https://en.wikipedia.org/wiki/vertex_(graph_theory	https://en.wikipedia.org/wiki/vertex_(graph_theory	NOUN
cana-1260	128	26	)	)	PUNCT
cana-1260	128	27	https://en.wikipedia.org/wiki/edge_(graph_theory	https://en.wikipedia.org/wiki/edge_(graph_theory	NOUN
cana-1260	128	28	)	)	PUNCT
cana-1260	128	29	communications	communication	NOUN
cana-1260	128	30	on	on	ADP
cana-1260	128	31	applied	apply	VERB
cana-1260	128	32	nonlinear	nonlinear	ADJ
cana-1260	128	33	analysis	analysis	NOUN
cana-1260	128	34	issn	issn	NOUN
cana-1260	128	35	:	:	PUNCT
cana-1260	128	36	1074	1074	NUM
cana-1260	128	37	-	-	PUNCT
cana-1260	128	38	133x	133x	NUM
cana-1260	128	39	vol	vol	NOUN
cana-1260	128	40	31	31	NUM
cana-1260	128	41	no	no	NOUN
cana-1260	128	42	.	.	PUNCT
cana-1260	129	1	6s	6s	NUM
cana-1260	129	2	(	(	PUNCT
cana-1260	129	3	2024	2024	NUM
cana-1260	129	4	)	)	PUNCT
cana-1260	129	5	675	675	NUM
cana-1260	129	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	129	7	theorem	theorem	VERB
cana-1260	129	8	2.2.1	2.2.1	NUM
cana-1260	129	9	.	.	PUNCT
cana-1260	130	1	let	let	VERB
cana-1260	130	2	graph	graph	NOUN
cana-1260	130	3	𝐺	𝐺	PROPN
cana-1260	130	4	is	be	AUX
cana-1260	130	5	a	a	DET
cana-1260	130	6	complete	complete	ADJ
cana-1260	130	7	on	on	ADP
cana-1260	130	8	𝑛	𝑛	DET
cana-1260	130	9	vertices	vertex	NOUN
cana-1260	130	10	and	and	CCONJ
cana-1260	130	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	130	12	)	)	PUNCT
cana-1260	130	13	is	be	AUX
cana-1260	130	14	its	its	PRON
cana-1260	130	15	line	line	NOUN
cana-1260	130	16	graph	graph	NOUN
cana-1260	130	17	on	on	ADP
cana-1260	130	18	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	130	19	)	)	PUNCT
cana-1260	130	20	2	2	NUM
cana-1260	130	21	vertices	vertex	NOUN
cana-1260	130	22	then	then	ADV
cana-1260	130	23	i	i	PRON
cana-1260	130	24	)	)	PUNCT
cana-1260	131	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	131	2	(	(	PUNCT
cana-1260	131	3	𝐺	𝐺	NOUN
cana-1260	131	4	)	)	PUNCT
cana-1260	132	1	+	+	CCONJ
cana-1260	132	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	132	3	(	(	PUNCT
cana-1260	132	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	132	5	)	)	PUNCT
cana-1260	132	6	)	)	PUNCT
cana-1260	132	7	=	=	SYM
cana-1260	133	1	1	1	NUM
cana-1260	133	2	+	+	CCONJ
cana-1260	133	3	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1260	133	4	−	−	PROPN
cana-1260	133	5	1)/2	1)/2	NUM
cana-1260	133	6	(	(	PUNCT
cana-1260	133	7	2𝑛	2𝑛	PROPN
cana-1260	133	8	−	−	PROPN
cana-1260	133	9	4	4	NUM
cana-1260	133	10	)	)	PUNCT
cana-1260	133	11	+	+	CCONJ
cana-1260	133	12	1	1	NUM
cana-1260	133	13	ii	ii	NOUN
cana-1260	133	14	)	)	PUNCT
cana-1260	133	15	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	133	16	(	(	PUNCT
cana-1260	133	17	𝐺	𝐺	NOUN
cana-1260	133	18	)	)	PUNCT
cana-1260	133	19	∗	∗	NOUN
cana-1260	133	20	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	133	21	(	(	PUNCT
cana-1260	133	22	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	133	23	)	)	PUNCT
cana-1260	133	24	)	)	PUNCT
cana-1260	134	1	=	=	SYM
cana-1260	134	2	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	134	3	)	)	PUNCT
cana-1260	134	4	2	2	NUM
cana-1260	134	5	(	(	PUNCT
cana-1260	134	6	2𝑛−4)+1	2𝑛−4)+1	NUM
cana-1260	134	7	iii	iii	NOUN
cana-1260	134	8	)	)	PUNCT
cana-1260	134	9	г𝑓	г𝑓	PROPN
cana-1260	134	10	(	(	PUNCT
cana-1260	134	11	𝐺	𝐺	NOUN
cana-1260	134	12	)	)	PUNCT
cana-1260	134	13	+	+	NUM
cana-1260	134	14	г𝑓	г𝑓	ADJ
cana-1260	134	15	(	(	PUNCT
cana-1260	134	16	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	134	17	)	)	PUNCT
cana-1260	134	18	)	)	PUNCT
cana-1260	135	1	=	=	SYM
cana-1260	135	2	1	1	NUM
cana-1260	136	1	+	+	CCONJ
cana-1260	136	2	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1260	136	3	−	−	PROPN
cana-1260	136	4	1)/2	1)/2	NUM
cana-1260	136	5	2𝑛	2𝑛	NOUN
cana-1260	136	6	−	−	PROPN
cana-1260	136	7	4	4	NUM
cana-1260	136	8	iv	iv	NOUN
cana-1260	136	9	)	)	PUNCT
cana-1260	136	10	г𝑓	г𝑓	PROPN
cana-1260	136	11	(	(	PUNCT
cana-1260	136	12	𝐺	𝐺	NOUN
cana-1260	136	13	)	)	PUNCT
cana-1260	136	14	∗	∗	NOUN
cana-1260	136	15	г𝑓	г𝑓	PROPN
cana-1260	136	16	(	(	PUNCT
cana-1260	136	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	136	18	)	)	PUNCT
cana-1260	136	19	)	)	PUNCT
cana-1260	137	1	=	=	PUNCT
cana-1260	137	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	137	3	2𝑛−4	2𝑛−4	NUM
cana-1260	137	4	proof	proof	NOUN
cana-1260	137	5	:	:	PUNCT
cana-1260	137	6	i	i	NOUN
cana-1260	137	7	)	)	PUNCT
cana-1260	137	8	and	and	CCONJ
cana-1260	137	9	ii	ii	X
cana-1260	137	10	)	)	PUNCT
cana-1260	137	11	if	if	SCONJ
cana-1260	137	12	graph	graph	NOUN
cana-1260	137	13	𝐺	𝐺	PROPN
cana-1260	137	14	is	be	AUX
cana-1260	137	15	the	the	DET
cana-1260	137	16	complete	complete	ADJ
cana-1260	137	17	then	then	ADV
cana-1260	137	18	all	all	DET
cana-1260	137	19	pair	pair	NOUN
cana-1260	137	20	of	of	ADP
cana-1260	137	21	other	other	ADJ
cana-1260	137	22	vertices	vertex	NOUN
cana-1260	137	23	are	be	AUX
cana-1260	137	24	connected	connect	VERB
cana-1260	137	25	by	by	ADP
cana-1260	137	26	a	a	DET
cana-1260	137	27	unique	unique	ADJ
cana-1260	137	28	edge	edge	NOUN
cana-1260	137	29	and	and	CCONJ
cana-1260	137	30	the	the	DET
cana-1260	137	31	fractional	fractional	ADJ
cana-1260	137	32	dominating	dominating	NOUN
cana-1260	137	33	number	number	NOUN
cana-1260	137	34	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	137	35	(	(	PUNCT
cana-1260	137	36	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	137	37	)	)	PUNCT
cana-1260	137	38	=	=	PUNCT
cana-1260	138	1	1	1	X
cana-1260	138	2	.	.	X
cana-1260	139	1	any	any	DET
cana-1260	139	2	set	set	NOUN
cana-1260	139	3	containing	contain	VERB
cana-1260	139	4	a	a	DET
cana-1260	139	5	single	single	ADJ
cana-1260	139	6	vertex	vertex	NOUN
cana-1260	139	7	is	be	AUX
cana-1260	139	8	a	a	DET
cana-1260	139	9	dominating	dominating	NOUN
cana-1260	139	10	set	set	VERB
cana-1260	139	11	in	in	ADP
cana-1260	139	12	𝐾𝑛.	𝐾𝑛.	PROPN
cana-1260	139	13	so	so	ADJ
cana-1260	139	14	𝛾	𝛾	PROPN
cana-1260	139	15	(	(	PUNCT
cana-1260	139	16	𝐺	𝐺	NOUN
cana-1260	139	17	)	)	PUNCT
cana-1260	139	18	=	=	SYM
cana-1260	140	1	1	1	X
cana-1260	140	2	.	.	X
cana-1260	140	3	for	for	ADP
cana-1260	140	4	any	any	DET
cana-1260	140	5	graph	graph	NOUN
cana-1260	140	6	𝐺	𝐺	NOUN
cana-1260	140	7	we	we	PRON
cana-1260	140	8	have	have	VERB
cana-1260	140	9	𝛾𝑓	𝛾𝑓	VERB
cana-1260	140	10	(	(	PUNCT
cana-1260	140	11	𝐺	𝐺	NOUN
cana-1260	140	12	)	)	PUNCT
cana-1260	140	13	≤	≤	NOUN
cana-1260	140	14	𝛾	𝛾	X
cana-1260	140	15	(	(	PUNCT
cana-1260	140	16	𝐺	𝐺	NOUN
cana-1260	140	17	)	)	PUNCT
cana-1260	140	18	.	.	PUNCT
cana-1260	141	1	the	the	DET
cana-1260	141	2	graph	graph	NOUN
cana-1260	141	3	𝐺	𝐺	NOUN
cana-1260	141	4	is	be	AUX
cana-1260	141	5	a	a	DET
cana-1260	141	6	complete	complete	ADJ
cana-1260	141	7	on	on	ADP
cana-1260	141	8	𝑛	𝑛	DET
cana-1260	141	9	vertices	vertex	NOUN
cana-1260	141	10	as	as	ADP
cana-1260	141	11	𝑛	𝑛	PRON
cana-1260	141	12	−	−	NUM
cana-1260	141	13	1	1	NUM
cana-1260	141	14	regular	regular	ADJ
cana-1260	141	15	graph	graph	NOUN
cana-1260	141	16	we	we	PRON
cana-1260	141	17	have	have	VERB
cana-1260	141	18	г𝑓	г𝑓	DET
cana-1260	141	19	(	(	PUNCT
cana-1260	141	20	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	141	21	)	)	PUNCT
cana-1260	141	22	≤	≤	NUM
cana-1260	141	23	𝑛	𝑛	DET
cana-1260	141	24	𝑛−1	𝑛−1	PROPN
cana-1260	141	25	.	.	PUNCT
cana-1260	142	1	the	the	DET
cana-1260	142	2	line	line	NOUN
cana-1260	142	3	graph	graph	NOUN
cana-1260	142	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	142	5	)	)	PUNCT
cana-1260	142	6	is	be	AUX
cana-1260	142	7	having	have	VERB
cana-1260	142	8	its	its	PRON
cana-1260	142	9	vertices	vertex	NOUN
cana-1260	142	10	are	be	AUX
cana-1260	142	11	nothing	nothing	PRON
cana-1260	142	12	but	but	SCONJ
cana-1260	142	13	edges	edge	NOUN
cana-1260	142	14	of	of	ADP
cana-1260	142	15	𝐺	𝐺	PROPN
cana-1260	142	16	with	with	ADP
cana-1260	142	17	two	two	NUM
cana-1260	142	18	vertices	vertex	NOUN
cana-1260	142	19	of	of	ADP
cana-1260	142	20	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	142	21	)	)	PUNCT
cana-1260	142	22	are	be	AUX
cana-1260	142	23	adjacent	adjacent	ADJ
cana-1260	142	24	if	if	SCONJ
cana-1260	142	25	common	common	ADJ
cana-1260	142	26	vertex	vertex	NOUN
cana-1260	142	27	shared	share	VERB
cana-1260	142	28	by	by	ADP
cana-1260	142	29	corresponding	correspond	VERB
cana-1260	142	30	edges	edge	NOUN
cana-1260	142	31	in	in	ADP
cana-1260	142	32	𝐺.	𝐺.	NOUN
cana-1260	142	33	in	in	ADP
cana-1260	142	34	the	the	DET
cana-1260	142	35	line	line	NOUN
cana-1260	142	36	graph	graph	NOUN
cana-1260	142	37	of	of	ADP
cana-1260	142	38	complete	complete	ADJ
cana-1260	142	39	graph	graph	NOUN
cana-1260	142	40	number	number	NOUN
cana-1260	142	41	of	of	ADP
cana-1260	142	42	vertices	vertex	NOUN
cana-1260	142	43	in	in	ADP
cana-1260	142	44	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	142	45	)	)	PUNCT
cana-1260	142	46	are	be	AUX
cana-1260	142	47	equal	equal	ADJ
cana-1260	142	48	in	in	ADP
cana-1260	142	49	number	number	NOUN
cana-1260	142	50	of	of	ADP
cana-1260	142	51	edges	edge	NOUN
cana-1260	142	52	in	in	ADP
cana-1260	142	53	graph	graph	NOUN
cana-1260	142	54	𝐺.	𝐺.	NOUN
cana-1260	142	55	(	(	PUNCT
cana-1260	142	56	see	see	VERB
cana-1260	142	57	the	the	DET
cana-1260	142	58	figure	figure	NOUN
cana-1260	142	59	2.2.1a	2.2.1a	NOUN
cana-1260	142	60	and	and	CCONJ
cana-1260	142	61	2.2.1b	2.2.1b	NUM
cana-1260	142	62	)	)	PUNCT
cana-1260	142	63	where	where	SCONJ
cana-1260	142	64	𝐾4	𝐾4	NOUN
cana-1260	142	65	is	be	AUX
cana-1260	142	66	complete	complete	ADJ
cana-1260	142	67	graph	graph	NOUN
cana-1260	142	68	with	with	ADP
cana-1260	142	69	4	4	NUM
cana-1260	142	70	vertices	vertex	NOUN
cana-1260	142	71	and	and	CCONJ
cana-1260	142	72	6	6	NUM
cana-1260	142	73	edges	edge	NOUN
cana-1260	142	74	so	so	ADV
cana-1260	142	75	in	in	ADP
cana-1260	142	76	its	its	PRON
cana-1260	142	77	line	line	NOUN
cana-1260	142	78	graph	graph	NOUN
cana-1260	142	79	of	of	ADP
cana-1260	142	80	𝐾4	𝐾4	NOUN
cana-1260	142	81	we	we	PRON
cana-1260	142	82	have	have	VERB
cana-1260	142	83	6	6	NUM
cana-1260	142	84	vertices	vertex	NOUN
cana-1260	142	85	with	with	ADP
cana-1260	142	86	each	each	DET
cana-1260	142	87	vertex	vertex	NOUN
cana-1260	142	88	of	of	ADP
cana-1260	142	89	degree	degree	NOUN
cana-1260	142	90	4	4	NUM
cana-1260	142	91	.	.	PUNCT
cana-1260	143	1	so	so	ADV
cana-1260	143	2	𝐿(𝐾4	𝐿(𝐾4	NOUN
cana-1260	143	3	)	)	PUNCT
cana-1260	143	4	is	be	AUX
cana-1260	143	5	4	4	NUM
cana-1260	143	6	-	-	PUNCT
cana-1260	143	7	regular	regular	ADJ
cana-1260	143	8	graph	graph	NOUN
cana-1260	143	9	.	.	PUNCT
cana-1260	144	1	in	in	ADP
cana-1260	144	2	line	line	NOUN
cana-1260	144	3	graph	graph	NOUN
cana-1260	144	4	of	of	ADP
cana-1260	144	5	complete	complete	ADJ
cana-1260	144	6	graph	graph	NOUN
cana-1260	144	7	𝐾4	𝐾4	NOUN
cana-1260	144	8	we	we	PRON
cana-1260	144	9	have	have	VERB
cana-1260	144	10	every	every	DET
cana-1260	144	11	two	two	NUM
cana-1260	144	12	nonadjacent	nonadjacent	ADJ
cana-1260	144	13	vertices	vertex	NOUN
cana-1260	144	14	are	be	AUX
cana-1260	144	15	mutually	mutually	ADV
cana-1260	144	16	adjacent	adjacent	ADJ
cana-1260	144	17	to	to	ADP
cana-1260	144	18	exactly	exactly	ADV
cana-1260	144	19	four	four	NUM
cana-1260	144	20	vertices	vertex	NOUN
cana-1260	144	21	.	.	PUNCT
cana-1260	145	1	(	(	PUNCT
cana-1260	145	2	a	a	X
cana-1260	145	3	)	)	PUNCT
cana-1260	145	4	complete	complete	ADJ
cana-1260	145	5	graph	graph	NOUN
cana-1260	145	6	𝐾4	𝐾4	NOUN
cana-1260	145	7	(	(	PUNCT
cana-1260	145	8	b	b	NOUN
cana-1260	145	9	)	)	PUNCT
cana-1260	145	10	line	line	NOUN
cana-1260	145	11	graph	graph	NOUN
cana-1260	145	12	of	of	ADP
cana-1260	145	13	complete	complete	ADJ
cana-1260	145	14	graph	graph	NOUN
cana-1260	145	15	𝐾4	𝐾4	NOUN
cana-1260	145	16	figure	figure	NOUN
cana-1260	145	17	2.2.1	2.2.1	NUM
cana-1260	145	18	similarly	similarly	ADV
cana-1260	145	19	,	,	PUNCT
cana-1260	145	20	𝐾5	𝐾5	PROPN
cana-1260	145	21	is	be	AUX
cana-1260	145	22	complete	complete	ADJ
cana-1260	145	23	with	with	ADP
cana-1260	145	24	5	5	NUM
cana-1260	145	25	vertices	vertex	NOUN
cana-1260	145	26	and	and	CCONJ
cana-1260	145	27	10	10	NUM
cana-1260	145	28	edges	edge	NOUN
cana-1260	145	29	so	so	ADV
cana-1260	145	30	in	in	ADP
cana-1260	145	31	the	the	DET
cana-1260	145	32	line	line	NOUN
cana-1260	145	33	graph	graph	NOUN
cana-1260	145	34	of	of	ADP
cana-1260	145	35	𝐾5	𝐾5	NOUN
cana-1260	145	36	we	we	PRON
cana-1260	145	37	get	get	VERB
cana-1260	145	38	10	10	NUM
cana-1260	145	39	vertices	vertex	NOUN
cana-1260	145	40	with	with	ADP
cana-1260	145	41	each	each	DET
cana-1260	145	42	vertex	vertex	NOUN
cana-1260	145	43	of	of	ADP
cana-1260	145	44	degree	degree	NOUN
cana-1260	145	45	6	6	NUM
cana-1260	145	46	.	.	PUNCT
cana-1260	146	1	for	for	ADP
cana-1260	146	2	the	the	DET
cana-1260	146	3	generalized	generalized	ADJ
cana-1260	146	4	result	result	NOUN
cana-1260	146	5	we	we	PRON
cana-1260	146	6	have	have	VERB
cana-1260	146	7	the	the	DET
cana-1260	146	8	sequence	sequence	NOUN
cana-1260	146	9	of	of	ADP
cana-1260	146	10	complete	complete	ADJ
cana-1260	146	11	graph	graph	NOUN
cana-1260	146	12	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	146	13	with	with	ADP
cana-1260	146	14	set	set	NOUN
cana-1260	146	15	of	of	ADP
cana-1260	146	16	vertices	vertex	NOUN
cana-1260	146	17	as	as	ADP
cana-1260	146	18	{	{	PUNCT
cana-1260	146	19	1,2,3,4,5,6,7	1,2,3,4,5,6,7	NUM
cana-1260	146	20	,	,	PUNCT
cana-1260	146	21	…	…	PUNCT
cana-1260	146	22	𝑛	𝑛	NOUN
cana-1260	146	23	}	}	PUNCT
cana-1260	146	24	.	.	PUNCT
cana-1260	147	1	so	so	ADV
cana-1260	147	2	for	for	ADP
cana-1260	147	3	complete	complete	ADJ
cana-1260	147	4	graph	graph	NOUN
cana-1260	147	5	on	on	ADP
cana-1260	147	6	set	set	NOUN
cana-1260	147	7	of	of	ADP
cana-1260	147	8	vertices	vertex	NOUN
cana-1260	147	9	𝑛	𝑛	VERB
cana-1260	147	10	=	=	PUNCT
cana-1260	147	11	{	{	PUNCT
cana-1260	147	12	2,3,4,5	2,3,4,5	NUM
cana-1260	147	13	,	,	PUNCT
cana-1260	147	14	…	…	PUNCT
cana-1260	147	15	,	,	PUNCT
cana-1260	147	16	𝑛	𝑛	PROPN
cana-1260	147	17	}	}	PUNCT
cana-1260	147	18	the	the	DET
cana-1260	147	19	sequence	sequence	NOUN
cana-1260	147	20	formed	form	VERB
cana-1260	147	21	for	for	ADP
cana-1260	147	22	graphs	graph	NOUN
cana-1260	147	23	with	with	ADP
cana-1260	147	24	number	number	NOUN
cana-1260	147	25	of	of	ADP
cana-1260	147	26	edges	edge	NOUN
cana-1260	147	27	{	{	PUNCT
cana-1260	147	28	1,3,6,10,15,21,28	1,3,6,10,15,21,28	NUM
cana-1260	147	29	,	,	PUNCT
cana-1260	147	30	…	…	PUNCT
cana-1260	147	31	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	147	32	)	)	PUNCT
cana-1260	147	33	2	2	NUM
cana-1260	147	34	}	}	PUNCT
cana-1260	147	35	,	,	PUNCT
cana-1260	147	36	is	be	AUX
cana-1260	147	37	triangular	triangular	NOUN
cana-1260	147	38	number	number	NOUN
cana-1260	147	39	sequence	sequence	NOUN
cana-1260	147	40	.	.	PUNCT
cana-1260	148	1	for	for	ADP
cana-1260	148	2	𝐺	𝐺	PROPN
cana-1260	148	3	is	be	AUX
cana-1260	148	4	complete	complete	ADJ
cana-1260	148	5	graph	graph	NOUN
cana-1260	148	6	with	with	ADP
cana-1260	148	7	𝑛	𝑛	DET
cana-1260	148	8	vertices	vertex	NOUN
cana-1260	148	9	as	as	ADP
cana-1260	148	10	𝑛	𝑛	PRON
cana-1260	148	11	−	−	PROPN
cana-1260	148	12	1	1	NUM
cana-1260	148	13	regular	regular	ADJ
cana-1260	148	14	connected	connected	ADJ
cana-1260	148	15	graph	graph	NOUN
cana-1260	148	16	where	where	SCONJ
cana-1260	148	17	𝑛	𝑛	ADJ
cana-1260	148	18	=	=	PUNCT
cana-1260	148	19	{	{	PUNCT
cana-1260	148	20	2,3,4,5,6	2,3,4,5,6	NUM
cana-1260	148	21	,	,	PUNCT
cana-1260	148	22	…	…	PUNCT
cana-1260	148	23	}	}	PUNCT
cana-1260	148	24	,	,	PUNCT
cana-1260	148	25	the	the	DET
cana-1260	148	26	line	line	NOUN
cana-1260	148	27	graph	graph	NOUN
cana-1260	148	28	of	of	ADP
cana-1260	148	29	complete	complete	ADJ
cana-1260	148	30	graph	graph	NOUN
cana-1260	148	31	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	148	32	)	)	PUNCT
cana-1260	148	33	on	on	ADP
cana-1260	148	34	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	148	35	)	)	PUNCT
cana-1260	148	36	2	2	NUM
cana-1260	148	37	vertices	vertex	NOUN
cana-1260	148	38	are	be	AUX
cana-1260	148	39	(	(	PUNCT
cana-1260	148	40	2𝑛	2𝑛	NOUN
cana-1260	148	41	−	−	PROPN
cana-1260	148	42	4	4	X
cana-1260	148	43	)	)	PUNCT
cana-1260	148	44	regular	regular	ADJ
cana-1260	148	45	connected	connected	ADJ
cana-1260	148	46	graph	graph	NOUN
cana-1260	148	47	.	.	PUNCT
cana-1260	149	1	line	line	NOUN
cana-1260	149	2	graph	graph	NOUN
cana-1260	149	3	having	have	VERB
cana-1260	149	4	each	each	DET
cana-1260	149	5	vertex	vertex	NOUN
cana-1260	149	6	with	with	ADP
cana-1260	149	7	degree	degree	NOUN
cana-1260	149	8	sequence	sequence	NOUN
cana-1260	149	9	is	be	AUX
cana-1260	149	10	{	{	PUNCT
cana-1260	149	11	0,2,4,6,8,10,	0,2,4,6,8,10,	NOUN
cana-1260	149	12	…	…	PUNCT
cana-1260	149	13	2𝑛	2𝑛	NOUN
cana-1260	149	14	−	−	PROPN
cana-1260	149	15	4	4	NUM
cana-1260	149	16	}	}	PUNCT
cana-1260	149	17	.	.	PUNCT
cana-1260	150	1	so	so	ADV
cana-1260	150	2	https://en.wikipedia.org/wiki/vertex_(graph_theory	https://en.wikipedia.org/wiki/vertex_(graph_theory	NOUN
cana-1260	150	3	)	)	PUNCT
cana-1260	150	4	https://en.wikipedia.org/wiki/edge_(graph_theory	https://en.wikipedia.org/wiki/edge_(graph_theory	NOUN
cana-1260	150	5	)	)	PUNCT
cana-1260	150	6	communications	communication	NOUN
cana-1260	150	7	on	on	ADP
cana-1260	150	8	applied	apply	VERB
cana-1260	150	9	nonlinear	nonlinear	ADJ
cana-1260	150	10	analysis	analysis	NOUN
cana-1260	150	11	issn	issn	NOUN
cana-1260	150	12	:	:	PUNCT
cana-1260	150	13	1074	1074	NUM
cana-1260	150	14	-	-	PUNCT
cana-1260	150	15	133x	133x	NUM
cana-1260	150	16	vol	vol	NOUN
cana-1260	150	17	31	31	NUM
cana-1260	150	18	no	no	NOUN
cana-1260	150	19	.	.	PUNCT
cana-1260	151	1	6s	6s	NUM
cana-1260	151	2	(	(	PUNCT
cana-1260	151	3	2024	2024	NUM
cana-1260	151	4	)	)	PUNCT
cana-1260	151	5	676	676	NUM
cana-1260	151	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	152	1	we	we	PRON
cana-1260	152	2	have	have	VERB
cana-1260	152	3	𝛾𝑓	𝛾𝑓	VERB
cana-1260	152	4	(	(	PUNCT
cana-1260	152	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	152	6	)	)	PUNCT
cana-1260	152	7	)	)	PUNCT
cana-1260	153	1	=	=	SYM
cana-1260	153	2	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	153	3	)	)	PUNCT
cana-1260	153	4	2	2	NUM
cana-1260	153	5	(	(	PUNCT
cana-1260	153	6	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	153	7	hence	hence	ADV
cana-1260	153	8	for	for	ADP
cana-1260	153	9	the	the	DET
cana-1260	153	10	sum	sum	NOUN
cana-1260	153	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	153	12	(	(	PUNCT
cana-1260	153	13	𝐺	𝐺	NOUN
cana-1260	153	14	)	)	PUNCT
cana-1260	154	1	+	+	CCONJ
cana-1260	154	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	154	3	(	(	PUNCT
cana-1260	154	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	154	5	)	)	PUNCT
cana-1260	154	6	)	)	PUNCT
cana-1260	154	7	=	=	SYM
cana-1260	155	1	1	1	NUM
cana-1260	155	2	+	+	SYM
cana-1260	155	3	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	155	4	)	)	PUNCT
cana-1260	155	5	2	2	NUM
cana-1260	155	6	(	(	PUNCT
cana-1260	155	7	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	155	8	and	and	CCONJ
cana-1260	155	9	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	155	10	(	(	PUNCT
cana-1260	155	11	𝐺	𝐺	NOUN
cana-1260	155	12	)	)	PUNCT
cana-1260	155	13	∗	∗	NOUN
cana-1260	155	14	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	155	15	(	(	PUNCT
cana-1260	155	16	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	155	17	)	)	PUNCT
cana-1260	155	18	)	)	PUNCT
cana-1260	156	1	=	=	SYM
cana-1260	156	2	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	156	3	)	)	PUNCT
cana-1260	156	4	2	2	NUM
cana-1260	156	5	(	(	PUNCT
cana-1260	156	6	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	156	7	.	.	PUNCT
cana-1260	157	1	iii	iii	X
cana-1260	157	2	)	)	PUNCT
cana-1260	157	3	and	and	CCONJ
cana-1260	157	4	iv	iv	X
cana-1260	157	5	)	)	PUNCT
cana-1260	157	6	the	the	DET
cana-1260	157	7	condition	condition	NOUN
cana-1260	157	8	satisfies	satisfy	VERB
cana-1260	157	9	for	for	ADP
cana-1260	157	10	the	the	DET
cana-1260	157	11	complete	complete	ADJ
cana-1260	157	12	graph	graph	NOUN
cana-1260	157	13	on	on	ADP
cana-1260	157	14	𝑛	𝑛	DET
cana-1260	157	15	vertices	vertex	NOUN
cana-1260	157	16	as	as	ADP
cana-1260	157	17	(	(	PUNCT
cana-1260	157	18	𝑛	𝑛	PRON
cana-1260	157	19	−	−	PROPN
cana-1260	157	20	1	1	NUM
cana-1260	157	21	)	)	PUNCT
cana-1260	157	22	regular	regular	ADJ
cana-1260	157	23	graph	graph	NOUN
cana-1260	157	24	where	where	SCONJ
cana-1260	157	25	each	each	DET
cana-1260	157	26	vertex	vertex	NOUN
cana-1260	157	27	𝑤	𝑤	ADP
cana-1260	157	28	∈	∈	PROPN
cana-1260	157	29	𝑁[𝑉	𝑁[𝑉	NOUN
cana-1260	157	30	]	]	PUNCT
cana-1260	157	31	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	X
cana-1260	157	32	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
cana-1260	157	33	∑	∑	PUNCT
cana-1260	157	34	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	157	35	)	)	PUNCT
cana-1260	157	36	⬚	⬚	PROPN
cana-1260	157	37	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	157	38	]	]	X
cana-1260	157	39	=	=	SYM
cana-1260	157	40	1	1	NUM
cana-1260	157	41	and	and	CCONJ
cana-1260	157	42	it	it	PRON
cana-1260	157	43	is	be	AUX
cana-1260	157	44	known	know	VERB
cana-1260	157	45	that	that	SCONJ
cana-1260	157	46	г𝑓	г𝑓	X
cana-1260	157	47	(	(	PUNCT
cana-1260	157	48	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	157	49	)	)	PUNCT
cana-1260	157	50	=	=	NOUN
cana-1260	157	51	1	1	X
cana-1260	157	52	.	.	PUNCT
cana-1260	158	1	so	so	ADV
cana-1260	158	2	if	if	SCONJ
cana-1260	158	3	graph	graph	NOUN
cana-1260	158	4	𝐺	𝐺	NOUN
cana-1260	158	5	is	be	AUX
cana-1260	158	6	complete	complete	ADJ
cana-1260	158	7	and	and	CCONJ
cana-1260	158	8	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	158	9	)	)	PUNCT
cana-1260	158	10	is	be	AUX
cana-1260	158	11	its	its	PRON
cana-1260	158	12	line	line	NOUN
cana-1260	158	13	graph	graph	NOUN
cana-1260	158	14	then	then	ADV
cana-1260	158	15	upper	upper	ADJ
cana-1260	158	16	fractional	fractional	ADJ
cana-1260	158	17	dominating	dominating	NOUN
cana-1260	158	18	number	number	NOUN
cana-1260	158	19	of	of	ADP
cana-1260	158	20	line	line	NOUN
cana-1260	158	21	graph	graph	NOUN
cana-1260	158	22	is	be	AUX
cana-1260	158	23	г𝑓	г𝑓	DET
cana-1260	158	24	(	(	PUNCT
cana-1260	158	25	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	158	26	)	)	PUNCT
cana-1260	158	27	)	)	PUNCT
cana-1260	159	1	=	=	PUNCT
cana-1260	159	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	159	3	2𝑛−4	2𝑛−4	NUM
cana-1260	159	4	and	and	CCONJ
cana-1260	159	5	for	for	ADP
cana-1260	159	6	the	the	DET
cana-1260	159	7	sum	sum	NOUN
cana-1260	159	8	we	we	PRON
cana-1260	159	9	get	get	VERB
cana-1260	159	10	г𝑓	г𝑓	DET
cana-1260	159	11	(	(	PUNCT
cana-1260	159	12	𝐺	𝐺	NOUN
cana-1260	159	13	)	)	PUNCT
cana-1260	160	1	+	+	NUM
cana-1260	160	2	г𝑓	г𝑓	ADJ
cana-1260	160	3	(	(	PUNCT
cana-1260	160	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	160	5	)	)	PUNCT
cana-1260	160	6	)	)	PUNCT
cana-1260	160	7	=	=	SYM
cana-1260	161	1	1	1	NUM
cana-1260	161	2	+	+	NUM
cana-1260	161	3	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	161	4	2𝑛−4	2𝑛−4	NUM
cana-1260	161	5	and	and	CCONJ
cana-1260	161	6	г𝑓	г𝑓	PROPN
cana-1260	161	7	(	(	PUNCT
cana-1260	161	8	𝐺	𝐺	NOUN
cana-1260	161	9	)	)	PUNCT
cana-1260	161	10	∗	∗	NOUN
cana-1260	161	11	г𝑓	г𝑓	PROPN
cana-1260	161	12	(	(	PUNCT
cana-1260	161	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	161	14	)	)	PUNCT
cana-1260	161	15	)	)	PUNCT
cana-1260	162	1	=	=	SYM
cana-1260	162	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	162	3	(	(	PUNCT
cana-1260	162	4	2𝑛−4	2𝑛−4	NUM
cana-1260	162	5	)	)	PUNCT
cana-1260	162	6	.	.	PUNCT
cana-1260	163	1	□	□	PUNCT
cana-1260	163	2	number	number	NOUN
cana-1260	163	3	of	of	ADP
cana-1260	163	4	vertices	vertex	NOUN
cana-1260	163	5	number	number	NOUN
cana-1260	163	6	of	of	ADP
cana-1260	163	7	edges	edge	NOUN
cana-1260	163	8	graph	graph	NOUN
cana-1260	163	9	with	with	ADP
cana-1260	163	10	each	each	DET
cana-1260	163	11	vertex	vertex	NOUN
cana-1260	163	12	of	of	ADP
cana-1260	163	13	degree	degree	NOUN
cana-1260	163	14	line	line	NOUN
cana-1260	163	15	graph	graph	NOUN
cana-1260	163	16	with	with	ADP
cana-1260	163	17	each	each	DET
cana-1260	163	18	vertex	vertex	NOUN
cana-1260	163	19	of	of	ADP
cana-1260	163	20	degree	degree	NOUN
cana-1260	163	21	2	2	NUM
cana-1260	163	22	1	1	NUM
cana-1260	163	23	1	1	NUM
cana-1260	163	24	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	163	25	(	(	PUNCT
cana-1260	163	26	𝐺	𝐺	NOUN
cana-1260	163	27	)	)	PUNCT
cana-1260	163	28	=	=	SYM
cana-1260	163	29	1	1	NUM
cana-1260	163	30	0	0	NUM
cana-1260	163	31	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	163	32	(	(	PUNCT
cana-1260	163	33	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	163	34	)	)	PUNCT
cana-1260	163	35	)	)	PUNCT
cana-1260	164	1	=	=	SYM
cana-1260	164	2	1	1	NUM
cana-1260	164	3	3	3	NUM
cana-1260	164	4	3	3	NUM
cana-1260	164	5	2	2	NUM
cana-1260	164	6	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	164	7	(	(	PUNCT
cana-1260	164	8	𝐺	𝐺	NOUN
cana-1260	164	9	)	)	PUNCT
cana-1260	164	10	=	=	SYM
cana-1260	164	11	1	1	NUM
cana-1260	164	12	2	2	NUM
cana-1260	164	13	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	164	14	(	(	PUNCT
cana-1260	164	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	164	16	)	)	PUNCT
cana-1260	164	17	)	)	PUNCT
cana-1260	165	1	=	=	SYM
cana-1260	166	1	1	1	NUM
cana-1260	166	2	4	4	NUM
cana-1260	166	3	6	6	NUM
cana-1260	166	4	3	3	NUM
cana-1260	166	5	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	166	6	(	(	PUNCT
cana-1260	166	7	𝐺	𝐺	NOUN
cana-1260	166	8	)	)	PUNCT
cana-1260	166	9	=	=	SYM
cana-1260	166	10	1	1	NUM
cana-1260	166	11	4	4	NUM
cana-1260	166	12	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	166	13	(	(	PUNCT
cana-1260	166	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	166	15	)	)	PUNCT
cana-1260	166	16	)	)	PUNCT
cana-1260	166	17	=	=	PUNCT
cana-1260	167	1	1.2	1.2	NUM
cana-1260	167	2	5	5	NUM
cana-1260	167	3	10	10	NUM
cana-1260	167	4	4	4	NUM
cana-1260	167	5	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	167	6	(	(	PUNCT
cana-1260	167	7	𝐺	𝐺	NOUN
cana-1260	167	8	)	)	PUNCT
cana-1260	167	9	=	=	SYM
cana-1260	167	10	1	1	NUM
cana-1260	167	11	6	6	NUM
cana-1260	167	12	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	167	13	(	(	PUNCT
cana-1260	167	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	167	15	)	)	PUNCT
cana-1260	167	16	)	)	PUNCT
cana-1260	168	1	=	=	PUNCT
cana-1260	169	1	1.42	1.42	NUM
cana-1260	169	2	6	6	NUM
cana-1260	169	3	15	15	NUM
cana-1260	169	4	5	5	NUM
cana-1260	169	5	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	169	6	(	(	PUNCT
cana-1260	169	7	𝐺	𝐺	NOUN
cana-1260	169	8	)	)	PUNCT
cana-1260	169	9	=	=	SYM
cana-1260	169	10	1	1	NUM
cana-1260	169	11	8	8	NUM
cana-1260	169	12	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	169	13	(	(	PUNCT
cana-1260	169	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	169	15	)	)	PUNCT
cana-1260	169	16	)	)	PUNCT
cana-1260	170	1	=	=	PUNCT
cana-1260	171	1	1.66	1.66	NUM
cana-1260	171	2	…	…	PUNCT
cana-1260	171	3	…	…	PUNCT
cana-1260	171	4	…	…	PUNCT
cana-1260	171	5	…	…	PUNCT
cana-1260	171	6	…	…	PUNCT
cana-1260	171	7	…	…	PUNCT
cana-1260	172	1	𝑛	𝑛	PRON
cana-1260	172	2	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1260	172	3	−	−	PROPN
cana-1260	172	4	1	1	NUM
cana-1260	172	5	)	)	SYM
cana-1260	172	6	2	2	NUM
cana-1260	172	7	𝑛	𝑛	PRON
cana-1260	172	8	−	−	PROPN
cana-1260	172	9	1	1	NUM
cana-1260	172	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	172	11	(	(	PUNCT
cana-1260	172	12	𝐺	𝐺	NOUN
cana-1260	172	13	)	)	PUNCT
cana-1260	172	14	=	=	SYM
cana-1260	172	15	1	1	NUM
cana-1260	172	16	(	(	PUNCT
cana-1260	172	17	2𝑛	2𝑛	NOUN
cana-1260	172	18	−	−	PROPN
cana-1260	172	19	4	4	X
cana-1260	172	20	)	)	PUNCT
cana-1260	172	21	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1260	172	22	−	−	PROPN
cana-1260	172	23	1	1	NUM
cana-1260	172	24	)	)	SYM
cana-1260	172	25	2	2	NUM
cana-1260	172	26	(	(	PUNCT
cana-1260	172	27	2𝑛	2𝑛	NOUN
cana-1260	172	28	−	−	PROPN
cana-1260	172	29	4	4	NUM
cana-1260	172	30	)	)	PUNCT
cana-1260	172	31	+	+	CCONJ
cana-1260	172	32	1	1	NUM
cana-1260	172	33	table	table	NOUN
cana-1260	172	34	2.2.1	2.2.1	NUM
cana-1260	172	35	2.3	2.3	NUM
cana-1260	172	36	star	star	NOUN
cana-1260	172	37	graph	graph	NOUN
cana-1260	172	38	(	(	PUNCT
cana-1260	172	39	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	172	40	)	)	PUNCT
cana-1260	172	41	star	star	NOUN
cana-1260	172	42	graph	graph	NOUN
cana-1260	172	43	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	172	44	is	be	AUX
cana-1260	172	45	n	n	PRON
cana-1260	172	46	o	o	NOUN
cana-1260	172	47	t	t	NOUN
cana-1260	173	1	i	i	INTJ
cana-1260	173	2	c	c	NOUN
cana-1260	173	3	e	e	PROPN
cana-1260	173	4	d	d	X
cana-1260	173	5	a	a	DET
cana-1260	173	6	s	s	X
cana-1260	173	7	complete	complete	ADJ
cana-1260	173	8	bipartite	bipartite	NOUN
cana-1260	173	9	graph	graph	NOUN
cana-1260	173	10	l	l	NOUN
cana-1260	174	1	i	i	NOUN
cana-1260	174	2	k	k	PROPN
cana-1260	174	3	e	e	X
cana-1260	174	4	(	(	PUNCT
cana-1260	174	5	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	174	6	)	)	PUNCT
cana-1260	174	7	,	,	PUNCT
cana-1260	174	8	an	an	DET
cana-1260	174	9	interior	interior	ADJ
cana-1260	174	10	node	node	NOUN
cana-1260	174	11	serves	serve	VERB
cana-1260	174	12	as	as	ADP
cana-1260	174	13	the	the	DET
cana-1260	174	14	central	central	ADJ
cana-1260	174	15	vertex	vertex	NOUN
cana-1260	174	16	of	of	ADP
cana-1260	174	17	a	a	DET
cana-1260	174	18	tree	tree	NOUN
cana-1260	174	19	containg	contae	VERB
cana-1260	174	20	𝑛	𝑛	PRON
cana-1260	174	21	leaves	leave	NOUN
cana-1260	174	22	.	.	PUNCT
cana-1260	175	1	theorem	theorem	ADJ
cana-1260	175	2	2.3.1	2.3.1	NUM
cana-1260	175	3	let	let	VERB
cana-1260	175	4	𝐺	𝐺	PROPN
cana-1260	175	5	is	be	AUX
cana-1260	175	6	the	the	DET
cana-1260	175	7	star	star	NOUN
cana-1260	175	8	graph	graph	NOUN
cana-1260	175	9	of	of	ADP
cana-1260	175	10	order	order	NOUN
cana-1260	175	11	𝑛	𝑛	PRON
cana-1260	175	12	and	and	CCONJ
cana-1260	175	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	175	14	)	)	PUNCT
cana-1260	175	15	is	be	AUX
cana-1260	175	16	its	its	PRON
cana-1260	175	17	line	line	NOUN
cana-1260	175	18	graph	graph	NOUN
cana-1260	175	19	then	then	ADV
cana-1260	175	20	i	i	PRON
cana-1260	175	21	)	)	PUNCT
cana-1260	175	22	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	175	23	(	(	PUNCT
cana-1260	175	24	𝐺	𝐺	NOUN
cana-1260	175	25	)	)	PUNCT
cana-1260	176	1	+	+	CCONJ
cana-1260	176	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	176	3	(	(	PUNCT
cana-1260	176	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	176	5	)	)	PUNCT
cana-1260	176	6	)	)	PUNCT
cana-1260	176	7	=	=	SYM
cana-1260	176	8	2	2	NUM
cana-1260	176	9	ii	ii	NOUN
cana-1260	176	10	)	)	PUNCT
cana-1260	176	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	176	12	(	(	PUNCT
cana-1260	176	13	𝐺	𝐺	NOUN
cana-1260	176	14	)	)	PUNCT
cana-1260	176	15	∗	∗	NOUN
cana-1260	176	16	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	176	17	(	(	PUNCT
cana-1260	176	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	176	19	)	)	PUNCT
cana-1260	176	20	)	)	PUNCT
cana-1260	177	1	=	=	SYM
cana-1260	177	2	1	1	NUM
cana-1260	177	3	iii	iii	NOUN
cana-1260	177	4	)	)	PUNCT
cana-1260	177	5	г𝑓	г𝑓	PROPN
cana-1260	177	6	(	(	PUNCT
cana-1260	177	7	𝐺	𝐺	NOUN
cana-1260	177	8	)	)	PUNCT
cana-1260	177	9	+	+	NUM
cana-1260	177	10	г𝑓	г𝑓	ADJ
cana-1260	177	11	(	(	PUNCT
cana-1260	177	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	177	13	)	)	PUNCT
cana-1260	177	14	)	)	PUNCT
cana-1260	177	15	≤	≤	NOUN
cana-1260	178	1	𝑛	𝑛	DET
cana-1260	178	2	+	+	CCONJ
cana-1260	178	3	𝑛	𝑛	PRON
cana-1260	178	4	−	−	PROPN
cana-1260	178	5	1	1	NUM
cana-1260	178	6	𝑛	𝑛	DET
cana-1260	178	7	−	−	NUM
cana-1260	178	8	2	2	NUM
cana-1260	178	9	iv	iv	NOUN
cana-1260	178	10	)	)	PUNCT
cana-1260	178	11	г𝑓	г𝑓	PROPN
cana-1260	178	12	(	(	PUNCT
cana-1260	178	13	𝐺	𝐺	NOUN
cana-1260	178	14	)	)	PUNCT
cana-1260	178	15	∗	∗	NOUN
cana-1260	178	16	г𝑓	г𝑓	PROPN
cana-1260	178	17	(	(	PUNCT
cana-1260	178	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	178	19	)	)	PUNCT
cana-1260	178	20	)	)	PUNCT
cana-1260	179	1	≤	≤	NUM
cana-1260	179	2	𝑛2−𝑛	𝑛2−𝑛	NOUN
cana-1260	179	3	𝑛−2	𝑛−2	PROPN
cana-1260	179	4	proof	proof	NOUN
cana-1260	179	5	:	:	PUNCT
cana-1260	179	6	i	i	NOUN
cana-1260	179	7	)	)	PUNCT
cana-1260	179	8	and	and	CCONJ
cana-1260	179	9	ii	ii	X
cana-1260	179	10	)	)	PUNCT
cana-1260	179	11	the	the	DET
cana-1260	179	12	term	term	NOUN
cana-1260	179	13	star	star	NOUN
cana-1260	179	14	graph	graph	NOUN
cana-1260	179	15	refers	refer	VERB
cana-1260	179	16	to	to	PART
cana-1260	179	17	be	be	AUX
cana-1260	179	18	complete	complete	ADJ
cana-1260	179	19	bipartite	bipartite	NOUN
cana-1260	179	20	graph	graph	NOUN
cana-1260	179	21	in	in	ADP
cana-1260	179	22	which	which	PRON
cana-1260	179	23	each	each	DET
cana-1260	179	24	vertex	vertex	NOUN
cana-1260	179	25	in	in	ADP
cana-1260	179	26	the	the	DET
cana-1260	179	27	graph	graph	NOUN
cana-1260	179	28	belongs	belong	VERB
cana-1260	179	29	to	to	ADP
cana-1260	179	30	one	one	NUM
cana-1260	179	31	set	set	VERB
cana-1260	179	32	and	and	CCONJ
cana-1260	179	33	other	other	ADJ
cana-1260	179	34	vertices	vertex	NOUN
cana-1260	179	35	to	to	ADP
cana-1260	179	36	the	the	DET
cana-1260	179	37	other	other	ADJ
cana-1260	179	38	set	set	NOUN
cana-1260	179	39	.	.	PUNCT
cana-1260	180	1	one	one	NUM
cana-1260	180	2	central	central	ADJ
cana-1260	180	3	vertex	vertex	NOUN
cana-1260	180	4	is	be	AUX
cana-1260	180	5	adjacent	adjacent	ADJ
cana-1260	180	6	to	to	ADP
cana-1260	180	7	every	every	DET
cana-1260	180	8	other	other	ADJ
cana-1260	180	9	vertex	vertex	NOUN
cana-1260	180	10	.	.	PUNCT
cana-1260	181	1	so	so	ADV
cana-1260	181	2	,	,	PUNCT
cana-1260	181	3	if	if	SCONJ
cana-1260	181	4	𝐺	𝐺	PROPN
cana-1260	181	5	be	be	VERB
cana-1260	181	6	the	the	DET
cana-1260	181	7	star	star	NOUN
cana-1260	181	8	graph	graph	NOUN
cana-1260	181	9	(	(	PUNCT
cana-1260	181	10	𝐾1,5	𝐾1,5	NOUN
cana-1260	181	11	)	)	PUNCT
cana-1260	181	12	then	then	ADV
cana-1260	181	13	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	181	14	(	(	PUNCT
cana-1260	181	15	𝐺	𝐺	NOUN
cana-1260	181	16	)	)	PUNCT
cana-1260	181	17	=	=	SYM
cana-1260	181	18	1	1	NUM
cana-1260	181	19	see	see	VERB
cana-1260	181	20	the	the	DET
cana-1260	181	21	figure	figure	NOUN
cana-1260	181	22	2.3.1a	2.3.1a	NUM
cana-1260	181	23	)	)	PUNCT
cana-1260	181	24	.	.	PUNCT
cana-1260	182	1	the	the	DET
cana-1260	182	2	line	line	NOUN
cana-1260	182	3	graph	graph	NOUN
cana-1260	182	4	of	of	ADP
cana-1260	182	5	(	(	PUNCT
cana-1260	182	6	𝐾1,5	𝐾1,5	NOUN
cana-1260	182	7	)	)	PUNCT
cana-1260	182	8	is	be	AUX
cana-1260	182	9	complete	complete	ADJ
cana-1260	182	10	graph	graph	NOUN
cana-1260	182	11	see	see	VERB
cana-1260	182	12	the	the	DET
cana-1260	182	13	figure	figure	NOUN
cana-1260	182	14	2.3.1b	2.3.1b	NUM
cana-1260	182	15	)	)	PUNCT
cana-1260	182	16	and	and	CCONJ
cana-1260	182	17	hence	hence	ADV
cana-1260	182	18	the	the	DET
cana-1260	182	19	fractional	fractional	ADJ
cana-1260	182	20	domination	domination	NOUN
cana-1260	182	21	number	number	NOUN
cana-1260	182	22	is	be	AUX
cana-1260	182	23	one	one	NUM
cana-1260	182	24	i.e.	i.e.	X
cana-1260	182	25	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	182	26	(	(	PUNCT
cana-1260	182	27	𝐺	𝐺	NOUN
cana-1260	182	28	)	)	PUNCT
cana-1260	182	29	=	=	SYM
cana-1260	183	1	1	1	X
cana-1260	183	2	.	.	PUNCT
cana-1260	183	3	therefore	therefore	ADV
cana-1260	183	4	,	,	PUNCT
cana-1260	183	5	star	star	NOUN
cana-1260	183	6	graph	graph	NOUN
cana-1260	183	7	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	183	8	is	be	AUX
cana-1260	183	9	o	o	PROPN
cana-1260	183	10	b	b	X
cana-1260	183	11	s	s	X
cana-1260	183	12	e	e	NOUN
cana-1260	183	13	r	r	NOUN
cana-1260	183	14	v	v	NOUN
cana-1260	183	15	e	e	X
cana-1260	183	16	d	d	X
cana-1260	183	17	a	a	DET
cana-1260	183	18	s	s	X
cana-1260	183	19	complete	complete	ADJ
cana-1260	183	20	bipartite	bipartite	NOUN
cana-1260	183	21	graph	graph	NOUN
cana-1260	183	22	communications	communication	NOUN
cana-1260	183	23	on	on	ADP
cana-1260	183	24	applied	apply	VERB
cana-1260	183	25	nonlinear	nonlinear	ADJ
cana-1260	183	26	analysis	analysis	NOUN
cana-1260	183	27	issn	issn	NOUN
cana-1260	183	28	:	:	PUNCT
cana-1260	183	29	1074	1074	NUM
cana-1260	183	30	-	-	PUNCT
cana-1260	183	31	133x	133x	NUM
cana-1260	183	32	vol	vol	NOUN
cana-1260	183	33	31	31	NUM
cana-1260	183	34	no	no	NOUN
cana-1260	183	35	.	.	PUNCT
cana-1260	184	1	6s	6s	NUM
cana-1260	184	2	(	(	PUNCT
cana-1260	184	3	2024	2024	NUM
cana-1260	184	4	)	)	PUNCT
cana-1260	184	5	677	677	NUM
cana-1260	184	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	185	1	l	l	NOUN
cana-1260	186	1	i	i	VERB
cana-1260	186	2	k	k	PROPN
cana-1260	186	3	e	e	X
cana-1260	186	4	(	(	PUNCT
cana-1260	186	5	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	186	6	)	)	PUNCT
cana-1260	186	7	where	where	SCONJ
cana-1260	186	8	𝑛	𝑛	ADV
cana-1260	186	9	=	=	SYM
cana-1260	186	10	{	{	PUNCT
cana-1260	186	11	3,4,5	3,4,5	NUM
cana-1260	186	12	,	,	PUNCT
cana-1260	186	13	…	…	PUNCT
cana-1260	186	14	}	}	PUNCT
cana-1260	186	15	and	and	CCONJ
cana-1260	186	16	line	line	NOUN
cana-1260	186	17	graphs	graph	NOUN
cana-1260	186	18	are	be	AUX
cana-1260	186	19	all	all	PRON
cana-1260	186	20	complete	complete	ADJ
cana-1260	186	21	graph	graph	NOUN
cana-1260	186	22	with	with	ADP
cana-1260	186	23	𝑛	𝑛	DET
cana-1260	186	24	vertices	vertex	NOUN
cana-1260	186	25	where	where	SCONJ
cana-1260	186	26	𝑛	𝑛	VERB
cana-1260	186	27	=	=	PUNCT
cana-1260	186	28	{	{	PUNCT
cana-1260	186	29	3,4,5,6	3,4,5,6	NUM
cana-1260	186	30	,	,	PUNCT
cana-1260	186	31	…	…	PUNCT
cana-1260	186	32	}	}	PUNCT
cana-1260	186	33	so	so	SCONJ
cana-1260	186	34	the	the	DET
cana-1260	186	35	general	general	ADJ
cana-1260	186	36	result	result	NOUN
cana-1260	186	37	for	for	ADP
cana-1260	186	38	the	the	DET
cana-1260	186	39	sum	sum	NOUN
cana-1260	186	40	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	186	41	(	(	PUNCT
cana-1260	186	42	𝐺	𝐺	NOUN
cana-1260	186	43	)	)	PUNCT
cana-1260	187	1	+	+	CCONJ
cana-1260	187	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	187	3	(	(	PUNCT
cana-1260	187	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	187	5	)	)	PUNCT
cana-1260	187	6	)	)	PUNCT
cana-1260	187	7	=	=	SYM
cana-1260	187	8	2	2	NUM
cana-1260	187	9	and	and	CCONJ
cana-1260	187	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	187	11	(	(	PUNCT
cana-1260	187	12	𝐺	𝐺	NOUN
cana-1260	187	13	)	)	PUNCT
cana-1260	187	14	∗	∗	NOUN
cana-1260	187	15	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	187	16	(	(	PUNCT
cana-1260	187	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	187	18	)	)	PUNCT
cana-1260	187	19	)	)	PUNCT
cana-1260	188	1	=	=	PUNCT
cana-1260	188	2	1	1	X
cana-1260	188	3	.	.	PUNCT
cana-1260	188	4	(	(	PUNCT
cana-1260	188	5	a	a	X
cana-1260	188	6	)	)	PUNCT
cana-1260	188	7	star	star	NOUN
cana-1260	188	8	graph	graph	NOUN
cana-1260	188	9	𝐾1,5	𝐾1,5	PROPN
cana-1260	188	10	(	(	PUNCT
cana-1260	188	11	b	b	NOUN
cana-1260	188	12	)	)	PUNCT
cana-1260	188	13	line	line	NOUN
cana-1260	188	14	graph	graph	NOUN
cana-1260	188	15	of	of	ADP
cana-1260	188	16	star	star	NOUN
cana-1260	188	17	graph	graph	NOUN
cana-1260	188	18	𝐾1,5	𝐾1,5	PROPN
cana-1260	188	19	figure	figure	NOUN
cana-1260	188	20	2.3.1	2.3.1	NUM
cana-1260	188	21	iii	iii	NOUN
cana-1260	188	22	)	)	PUNCT
cana-1260	188	23	and	and	CCONJ
cana-1260	188	24	iv	iv	X
cana-1260	188	25	)	)	PUNCT
cana-1260	188	26	as	as	ADP
cana-1260	188	27	per	per	ADP
cana-1260	188	28	observations	observation	NOUN
cana-1260	188	29	the	the	DET
cana-1260	188	30	upper	upper	ADJ
cana-1260	188	31	fractional	fractional	ADJ
cana-1260	188	32	dominating	dominating	NOUN
cana-1260	188	33	number	number	NOUN
cana-1260	188	34	of	of	ADP
cana-1260	188	35	star	star	NOUN
cana-1260	188	36	graph	graph	NOUN
cana-1260	188	37	(	(	PUNCT
cana-1260	188	38	𝐾1,5	𝐾1,5	NOUN
cana-1260	188	39	)	)	PUNCT
cana-1260	188	40	is	be	AUX
cana-1260	188	41	5	5	NUM
cana-1260	188	42	with	with	ADP
cana-1260	188	43	considered	consider	VERB
cana-1260	188	44	closed	closed	ADJ
cana-1260	188	45	neighborhood	neighborhood	NOUN
cana-1260	188	46	of	of	ADP
cana-1260	188	47	every	every	DET
cana-1260	188	48	vertex	vertex	NOUN
cana-1260	188	49	satisfying	satisfy	VERB
cana-1260	188	50	the	the	DET
cana-1260	188	51	condition	condition	NOUN
cana-1260	188	52	where	where	SCONJ
cana-1260	188	53	the	the	DET
cana-1260	188	54	vertex	vertex	NOUN
cana-1260	188	55	𝑤	𝑤	ADP
cana-1260	188	56	∈	∈	NOUN
cana-1260	188	57	𝑁(𝑣	𝑁(𝑣	NOUN
cana-1260	188	58	)	)	PUNCT
cana-1260	188	59	such	such	ADJ
cana-1260	188	60	that	that	SCONJ
cana-1260	188	61	∑	∑	PUNCT
cana-1260	188	62	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	188	63	)	)	PUNCT
cana-1260	188	64	⬚	⬚	PROPN
cana-1260	188	65	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	188	66	]	]	X
cana-1260	188	67	=	=	SYM
cana-1260	188	68	1	1	X
cana-1260	188	69	.	.	PUNCT
cana-1260	188	70	by	by	ADP
cana-1260	188	71	assigning	assign	VERB
cana-1260	188	72	weight	weight	NOUN
cana-1260	188	73	zero	zero	NUM
cana-1260	188	74	to	to	ADP
cana-1260	188	75	the	the	DET
cana-1260	188	76	central	central	ADJ
cana-1260	188	77	vertex	vertex	NOUN
cana-1260	188	78	and	and	CCONJ
cana-1260	188	79	weight	weight	NOUN
cana-1260	188	80	one	one	NUM
cana-1260	188	81	to	to	ADP
cana-1260	188	82	remaining	remain	VERB
cana-1260	188	83	vertices	vertex	NOUN
cana-1260	188	84	.	.	PUNCT
cana-1260	189	1	upper	upper	ADJ
cana-1260	189	2	fractional	fractional	ADJ
cana-1260	189	3	dominating	dominating	NOUN
cana-1260	189	4	number	number	NOUN
cana-1260	189	5	of	of	ADP
cana-1260	189	6	line	line	NOUN
cana-1260	189	7	graph	graph	NOUN
cana-1260	189	8	of	of	ADP
cana-1260	189	9	𝐾1,5	𝐾1,5	NOUN
cana-1260	189	10	is	be	AUX
cana-1260	189	11	one	one	NUM
cana-1260	189	12	.	.	PUNCT
cana-1260	190	1	a	a	DET
cana-1260	190	2	star	star	NOUN
cana-1260	190	3	graph	graph	NOUN
cana-1260	190	4	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	190	5	is	be	AUX
cana-1260	190	6	the	the	DET
cana-1260	190	7	complete	complete	ADJ
cana-1260	190	8	bipartite	bipartite	NOUN
cana-1260	190	9	graph	graph	NOUN
cana-1260	190	10	like	like	ADP
cana-1260	190	11	(	(	PUNCT
cana-1260	190	12	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	190	13	)	)	PUNCT
cana-1260	190	14	where	where	SCONJ
cana-1260	190	15	𝑛	𝑛	PRON
cana-1260	190	16	=	=	SYM
cana-1260	190	17	3,4,5	3,4,5	NUM
cana-1260	190	18	,	,	PUNCT
cana-1260	190	19	…	…	PUNCT
cana-1260	190	20	we	we	PRON
cana-1260	190	21	have	have	VERB
cana-1260	190	22	complete	complete	ADJ
cana-1260	190	23	graph	graph	NOUN
cana-1260	190	24	on	on	ADP
cana-1260	190	25	𝑛	𝑛	DET
cana-1260	190	26	vertices	vertex	NOUN
cana-1260	190	27	where	where	SCONJ
cana-1260	190	28	𝑛	𝑛	VERB
cana-1260	190	29	=	=	PUNCT
cana-1260	190	30	{	{	PUNCT
cana-1260	190	31	3,4,5,6	3,4,5,6	NUM
cana-1260	190	32	,	,	PUNCT
cana-1260	190	33	…	…	PUNCT
cana-1260	190	34	}	}	PUNCT
cana-1260	190	35	so	so	ADV
cana-1260	190	36	the	the	DET
cana-1260	190	37	generalized	generalized	ADJ
cana-1260	190	38	result	result	NOUN
cana-1260	190	39	for	for	SCONJ
cana-1260	190	40	𝐺	𝐺	PROPN
cana-1260	190	41	be	be	AUX
cana-1260	190	42	the	the	DET
cana-1260	190	43	star	star	NOUN
cana-1260	190	44	graph	graph	NOUN
cana-1260	190	45	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	190	46	on	on	ADP
cana-1260	190	47	n	n	CCONJ
cana-1260	190	48	vertices	vertex	NOUN
cana-1260	190	49	and	and	CCONJ
cana-1260	190	50	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	190	51	)	)	PUNCT
cana-1260	190	52	be	be	VERB
cana-1260	190	53	its	its	PRON
cana-1260	190	54	line	line	NOUN
cana-1260	190	55	graph	graph	NOUN
cana-1260	190	56	on	on	ADP
cana-1260	190	57	𝑛	𝑛	DET
cana-1260	190	58	−	−	NUM
cana-1260	190	59	1	1	NUM
cana-1260	190	60	vertices	vertex	NOUN
cana-1260	190	61	with	with	ADP
cana-1260	190	62	𝑛	𝑛	DET
cana-1260	190	63	−	−	NUM
cana-1260	190	64	2	2	NUM
cana-1260	190	65	regular	regular	ADJ
cana-1260	190	66	graph	graph	NOUN
cana-1260	190	67	then	then	ADV
cana-1260	190	68	the	the	DET
cana-1260	190	69	sum	sum	NOUN
cana-1260	190	70	г𝑓	г𝑓	PROPN
cana-1260	190	71	(	(	PUNCT
cana-1260	190	72	𝐺	𝐺	NOUN
cana-1260	190	73	)	)	PUNCT
cana-1260	191	1	+	+	NUM
cana-1260	191	2	г𝑓	г𝑓	ADJ
cana-1260	191	3	(	(	PUNCT
cana-1260	191	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	191	5	)	)	PUNCT
cana-1260	191	6	)	)	PUNCT
cana-1260	191	7	≤	≤	NOUN
cana-1260	191	8	𝑛	𝑛	DET
cana-1260	191	9	+	+	CCONJ
cana-1260	191	10	𝑛−1	𝑛−1	PROPN
cana-1260	191	11	𝑛−2	𝑛−2	NOUN
cana-1260	191	12	and	and	CCONJ
cana-1260	191	13	product	product	NOUN
cana-1260	191	14	г𝑓	г𝑓	PROPN
cana-1260	191	15	(	(	PUNCT
cana-1260	191	16	𝐺	𝐺	NOUN
cana-1260	191	17	)	)	PUNCT
cana-1260	191	18	∗	∗	NOUN
cana-1260	191	19	г𝑓	г𝑓	PROPN
cana-1260	191	20	(	(	PUNCT
cana-1260	191	21	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	191	22	)	)	PUNCT
cana-1260	191	23	)	)	PUNCT
cana-1260	191	24	≤	≤	NUM
cana-1260	192	1	𝑛2−𝑛	𝑛2−𝑛	PROPN
cana-1260	192	2	𝑛−2	𝑛−2	PROPN
cana-1260	192	3	.	.	PUNCT
cana-1260	193	1	□	□	PUNCT
cana-1260	193	2	2.4	2.4	NUM
cana-1260	193	3	bi	bi	ADJ
cana-1260	193	4	-	-	ADJ
cana-1260	193	5	star	star	NOUN
cana-1260	193	6	graph	graph	NOUN
cana-1260	193	7	the	the	DET
cana-1260	193	8	bi	bi	NOUN
cana-1260	193	9	-s	-s	INTJ
cana-1260	194	1	ta	ta	ADP
cana-1260	194	2	r	r	NOUN
cana-1260	194	3	g	g	PROPN
cana-1260	194	4	raph	raph	NOUN
cana-1260	195	1	i	i	PRON
cana-1260	195	2	s	s	VERB
cana-1260	195	3	two	two	NUM
cana-1260	195	4	copies	copy	NOUN
cana-1260	195	5	of	of	ADP
cana-1260	195	6	the	the	DET
cana-1260	195	7	star	star	NOUN
cana-1260	195	8	graph	graph	NOUN
cana-1260	195	9	or	or	CCONJ
cana-1260	195	10	two	two	NUM
cana-1260	195	11	copies	copy	NOUN
cana-1260	195	12	of	of	ADP
cana-1260	195	13	(	(	PUNCT
cana-1260	195	14	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	195	15	)	)	PUNCT
cana-1260	195	16	with	with	ADP
cana-1260	195	17	a	a	DET
cana-1260	195	18	tree	tree	NOUN
cana-1260	195	19	's	's	PART
cana-1260	195	20	core	core	NOUN
cana-1260	195	21	vertex	vertex	NOUN
cana-1260	195	22	being	be	AUX
cana-1260	195	23	an	an	DET
cana-1260	195	24	internal	internal	ADJ
cana-1260	195	25	node	node	NOUN
cana-1260	195	26	and	and	CCONJ
cana-1260	195	27	𝑛	𝑛	DET
cana-1260	195	28	leaves	leave	NOUN
cana-1260	195	29	.	.	PUNCT
cana-1260	196	1	theorem	theorem	ADJ
cana-1260	196	2	2.4.1	2.4.1	NUM
cana-1260	196	3	.	.	PUNCT
cana-1260	197	1	if	if	SCONJ
cana-1260	197	2	simple	simple	ADJ
cana-1260	197	3	graph	graph	NOUN
cana-1260	197	4	𝐺	𝐺	PROPN
cana-1260	197	5	is	be	AUX
cana-1260	197	6	the	the	DET
cana-1260	197	7	bi	bi	NOUN
cana-1260	197	8	-	-	NOUN
cana-1260	197	9	star	star	NOUN
cana-1260	197	10	of	of	ADP
cana-1260	197	11	𝑛	𝑛	PROPN
cana-1260	197	12	vertices	vertex	NOUN
cana-1260	197	13	and	and	CCONJ
cana-1260	197	14	graph	graph	NOUN
cana-1260	197	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	197	16	)	)	PUNCT
cana-1260	197	17	be	be	AUX
cana-1260	197	18	its	its	PRON
cana-1260	197	19	line	line	NOUN
cana-1260	197	20	graph	graph	NOUN
cana-1260	197	21	then	then	ADV
cana-1260	197	22	i	i	PRON
cana-1260	197	23	)	)	PUNCT
cana-1260	197	24	𝛾	𝛾	ADP
cana-1260	197	25	𝑓	𝑓	PRON
cana-1260	197	26	(	(	PUNCT
cana-1260	197	27	𝐺	𝐺	NOUN
cana-1260	197	28	)	)	PUNCT
cana-1260	197	29	+	+	CCONJ
cana-1260	197	30	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	197	31	(	(	PUNCT
cana-1260	197	32	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	197	33	)	)	PUNCT
cana-1260	197	34	)	)	PUNCT
cana-1260	198	1	=	=	SYM
cana-1260	198	2	3	3	NUM
cana-1260	198	3	ii	ii	NOUN
cana-1260	198	4	)	)	PUNCT
cana-1260	198	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	198	6	(	(	PUNCT
cana-1260	198	7	𝐺	𝐺	NOUN
cana-1260	198	8	)	)	PUNCT
cana-1260	198	9	∗	∗	NOUN
cana-1260	198	10	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	198	11	(	(	PUNCT
cana-1260	198	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	198	13	)	)	PUNCT
cana-1260	198	14	)	)	PUNCT
cana-1260	199	1	=	=	SYM
cana-1260	199	2	2	2	NUM
cana-1260	199	3	iii	iii	NOUN
cana-1260	199	4	)	)	PUNCT
cana-1260	199	5	г𝑓	г𝑓	PROPN
cana-1260	199	6	(	(	PUNCT
cana-1260	199	7	𝐺	𝐺	NOUN
cana-1260	199	8	)	)	PUNCT
cana-1260	199	9	+	+	NUM
cana-1260	199	10	г𝑓	г𝑓	ADJ
cana-1260	199	11	(	(	PUNCT
cana-1260	199	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	199	13	)	)	PUNCT
cana-1260	199	14	)	)	PUNCT
cana-1260	200	1	=	=	SYM
cana-1260	201	1	2𝑛	2𝑛	PROPN
cana-1260	201	2	+	+	CCONJ
cana-1260	201	3	2	2	NUM
cana-1260	201	4	iv	iv	NUM
cana-1260	201	5	)	)	PUNCT
cana-1260	201	6	г𝑓	г𝑓	PROPN
cana-1260	201	7	(	(	PUNCT
cana-1260	201	8	𝐺	𝐺	NOUN
cana-1260	201	9	)	)	PUNCT
cana-1260	201	10	∗	∗	NOUN
cana-1260	201	11	г𝑓	г𝑓	PROPN
cana-1260	201	12	(	(	PUNCT
cana-1260	201	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	201	14	)	)	PUNCT
cana-1260	201	15	)	)	PUNCT
cana-1260	202	1	=	=	PUNCT
cana-1260	203	1	4𝑛	4𝑛	NOUN
cana-1260	203	2	proof	proof	NOUN
cana-1260	203	3	i	i	PRON
cana-1260	203	4	)	)	PUNCT
cana-1260	203	5	and	and	CCONJ
cana-1260	203	6	ii	ii	NUM
cana-1260	203	7	):	):	PUNCT
cana-1260	203	8	(	(	PUNCT
cana-1260	203	9	[	[	X
cana-1260	203	10	11	11	NUM
cana-1260	203	11	]	]	SYM
cana-1260	203	12	)	)	PUNCT
cana-1260	203	13	a	a	DET
cana-1260	203	14	bi	bi	ADJ
cana-1260	203	15	-	-	ADJ
cana-1260	203	16	star	star	ADJ
cana-1260	203	17	graph	graph	NOUN
cana-1260	203	18	is	be	AUX
cana-1260	203	19	created	create	VERB
cana-1260	203	20	by	by	ADP
cana-1260	203	21	combining	combine	VERB
cana-1260	203	22	the	the	DET
cana-1260	203	23	apex	apex	NOUN
cana-1260	203	24	vertices	vertice	VERB
cana-1260	203	25	with	with	ADP
cana-1260	203	26	two	two	NUM
cana-1260	203	27	copies	copy	NOUN
cana-1260	203	28	of	of	ADP
cana-1260	203	29	a	a	DET
cana-1260	203	30	star	star	NOUN
cana-1260	203	31	graph	graph	NOUN
cana-1260	203	32	.	.	PUNCT
cana-1260	204	1	the	the	DET
cana-1260	204	2	set	set	NOUN
cana-1260	204	3	of	of	ADP
cana-1260	204	4	vertices	vertex	NOUN
cana-1260	204	5	of	of	ADP
cana-1260	204	6	𝐵𝑛,𝑛	𝐵𝑛,𝑛	PROPN
cana-1260	204	7	,	,	PUNCT
cana-1260	204	8	𝑉	𝑉	PROPN
cana-1260	204	9	(	(	PUNCT
cana-1260	204	10	𝐵𝑛,𝑛	𝐵𝑛,𝑛	PROPN
cana-1260	204	11	)	)	PUNCT
cana-1260	204	12	=	=	PRON
cana-1260	204	13	{	{	PUNCT
cana-1260	204	14	𝑢	𝑢	X
cana-1260	204	15	,	,	PUNCT
cana-1260	204	16	𝑣	𝑣	NOUN
cana-1260	204	17	,	,	PUNCT
cana-1260	204	18	𝑢𝑖	𝑢𝑖	INTJ
cana-1260	204	19	,	,	PUNCT
cana-1260	204	20	𝑣𝑖	𝑣𝑖	ADV
cana-1260	204	21	|	|	ADV
cana-1260	204	22	1	1	NUM
cana-1260	204	23	≤	≤	NOUN
cana-1260	204	24	𝑖	𝑖	SYM
cana-1260	204	25	≤	≤	NUM
cana-1260	204	26	𝑛	𝑛	PROPN
cana-1260	204	27	}	}	PUNCT
cana-1260	204	28	,	,	PUNCT
cana-1260	204	29	where	where	SCONJ
cana-1260	204	30	𝑢	𝑢	X
cana-1260	204	31	,	,	PUNCT
cana-1260	204	32	𝑣	𝑣	PRON
cana-1260	204	33	are	be	AUX
cana-1260	204	34	center	center	NOUN
cana-1260	204	35	vertices	vertex	NOUN
cana-1260	204	36	and	and	CCONJ
cana-1260	204	37	𝑢𝑖	𝑢𝑖	NOUN
cana-1260	204	38	,	,	PUNCT
cana-1260	204	39	𝑣𝑖	𝑣𝑖	NOUN
cana-1260	204	40	are	be	AUX
cana-1260	204	41	all	all	DET
cana-1260	204	42	pendent	pendent	ADJ
cana-1260	204	43	vertices	vertex	NOUN
cana-1260	204	44	.	.	PUNCT
cana-1260	205	1	the	the	DET
cana-1260	205	2	set	set	NOUN
cana-1260	205	3	of	of	ADP
cana-1260	205	4	edges	edge	NOUN
cana-1260	205	5	of	of	ADP
cana-1260	205	6	𝐵𝑛,𝑛	𝐵𝑛,𝑛	PROPN
cana-1260	205	7	is	be	AUX
cana-1260	205	8	𝐸	𝐸	PROPN
cana-1260	205	9	(	(	PUNCT
cana-1260	205	10	𝐵𝑛,𝑛	𝐵𝑛,𝑛	PROPN
cana-1260	205	11	)	)	PUNCT
cana-1260	205	12	=	=	PRON
cana-1260	205	13	{	{	PUNCT
cana-1260	205	14	𝑢𝑣	𝑢𝑣	NOUN
cana-1260	205	15	,	,	PUNCT
cana-1260	205	16	𝑢𝑢𝑖	𝑢𝑢𝑖	PROPN
cana-1260	205	17	,	,	PUNCT
cana-1260	205	18	𝑣𝑣𝑖	𝑣𝑣𝑖	NOUN
cana-1260	205	19	|	|	ADV
cana-1260	205	20	1	1	NUM
cana-1260	205	21	≤	≤	NOUN
cana-1260	205	22	𝑖	𝑖	SYM
cana-1260	205	23	≤	≤	NUM
cana-1260	205	24	𝑛	𝑛	NOUN
cana-1260	205	25	}	}	PUNCT
cana-1260	205	26	.	.	PUNCT
cana-1260	206	1	so	so	ADV
cana-1260	206	2	,	,	PUNCT
cana-1260	206	3	|𝑉	|𝑉	X
cana-1260	206	4	(	(	PUNCT
cana-1260	206	5	𝐵𝑛,𝑛	𝐵𝑛,𝑛	PROPN
cana-1260	206	6	)	)	PUNCT
cana-1260	206	7	|	|	ADV
cana-1260	206	8	=	=	SYM
cana-1260	206	9	2𝑛	2𝑛	PROPN
cana-1260	207	1	+	+	CCONJ
cana-1260	207	2	2	2	NUM
cana-1260	207	3	and	and	CCONJ
cana-1260	207	4	|𝐸	|𝐸	NOUN
cana-1260	207	5	(	(	PUNCT
cana-1260	207	6	𝐵𝑛,𝑛	𝐵𝑛,𝑛	PROPN
cana-1260	207	7	)	)	PUNCT
cana-1260	207	8	|	|	ADV
cana-1260	207	9	=	=	SYM
cana-1260	207	10	2𝑛	2𝑛	PROPN
cana-1260	208	1	+	+	CCONJ
cana-1260	208	2	1	1	X
cana-1260	208	3	.	.	X
cana-1260	208	4	the	the	DET
cana-1260	208	5	fractional	fractional	ADJ
cana-1260	208	6	dominating	dominating	NOUN
cana-1260	208	7	number	number	NOUN
cana-1260	208	8	of	of	ADP
cana-1260	208	9	bistar	bistar	ADJ
cana-1260	208	10	graph	graph	NOUN
cana-1260	208	11	will	will	AUX
cana-1260	208	12	be	be	AUX
cana-1260	208	13	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	208	14	(	(	PUNCT
cana-1260	208	15	𝐺	𝐺	NOUN
cana-1260	208	16	)	)	PUNCT
cana-1260	208	17	=	=	SYM
cana-1260	208	18	2	2	NUM
cana-1260	208	19	by	by	ADP
cana-1260	208	20	assigning	assign	VERB
cana-1260	208	21	weight	weight	NOUN
cana-1260	208	22	one	one	NUM
cana-1260	208	23	to	to	ADP
cana-1260	208	24	central	central	ADJ
cana-1260	208	25	vertices	vertex	NOUN
cana-1260	208	26	and	and	CCONJ
cana-1260	208	27	weight	weight	NOUN
cana-1260	208	28	zero	zero	NUM
cana-1260	208	29	to	to	ADP
cana-1260	208	30	remaining	remain	VERB
cana-1260	208	31	vertices	vertex	NOUN
cana-1260	208	32	(	(	PUNCT
cana-1260	208	33	see	see	VERB
cana-1260	208	34	the	the	DET
cana-1260	208	35	figure	figure	NOUN
cana-1260	208	36	2.4.1a	2.4.1a	PROPN
cana-1260	208	37	)	)	PUNCT
cana-1260	208	38	.	.	PUNCT
cana-1260	209	1	the	the	DET
cana-1260	209	2	given	give	VERB
cana-1260	209	3	line	line	NOUN
cana-1260	209	4	graph	graph	NOUN
cana-1260	209	5	of	of	ADP
cana-1260	209	6	bi	bi	NOUN
cana-1260	209	7	-	-	NOUN
cana-1260	209	8	star	star	NOUN
cana-1260	209	9	(	(	PUNCT
cana-1260	209	10	𝐾1,3	𝐾1,3	NOUN
cana-1260	209	11	)	)	PUNCT
cana-1260	209	12	is	be	AUX
cana-1260	209	13	connected	connect	VERB
cana-1260	209	14	graph	graph	NOUN
cana-1260	209	15	with	with	ADP
cana-1260	209	16	two	two	NUM
cana-1260	209	17	communications	communication	NOUN
cana-1260	209	18	on	on	ADP
cana-1260	209	19	applied	apply	VERB
cana-1260	209	20	nonlinear	nonlinear	ADJ
cana-1260	209	21	analysis	analysis	NOUN
cana-1260	209	22	issn	issn	NOUN
cana-1260	209	23	:	:	PUNCT
cana-1260	209	24	1074	1074	NUM
cana-1260	209	25	-	-	PUNCT
cana-1260	209	26	133x	133x	NUM
cana-1260	209	27	vol	vol	NOUN
cana-1260	209	28	31	31	NUM
cana-1260	209	29	no	no	NOUN
cana-1260	209	30	.	.	PUNCT
cana-1260	210	1	6s	6s	NUM
cana-1260	210	2	(	(	PUNCT
cana-1260	210	3	2024	2024	NUM
cana-1260	210	4	)	)	PUNCT
cana-1260	210	5	678	678	NUM
cana-1260	210	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	210	7	copies	copy	NOUN
cana-1260	210	8	of	of	ADP
cana-1260	210	9	complete	complete	ADJ
cana-1260	210	10	graph	graph	NOUN
cana-1260	210	11	connected	connect	VERB
cana-1260	210	12	by	by	ADP
cana-1260	210	13	one	one	NUM
cana-1260	210	14	point	point	NOUN
cana-1260	210	15	so	so	SCONJ
cana-1260	210	16	this	this	DET
cana-1260	210	17	graph	graph	NOUN
cana-1260	210	18	is	be	AUX
cana-1260	210	19	one	one	NUM
cana-1260	210	20	point	point	NOUN
cana-1260	210	21	connectivity	connectivity	NOUN
cana-1260	210	22	graph	graph	NOUN
cana-1260	210	23	(	(	PUNCT
cana-1260	210	24	see	see	VERB
cana-1260	210	25	the	the	DET
cana-1260	210	26	figure	figure	NOUN
cana-1260	210	27	2.4.1b	2.4.1b	PROPN
cana-1260	210	28	)	)	PUNCT
cana-1260	210	29	.	.	PUNCT
cana-1260	211	1	a	a	DET
cana-1260	211	2	vertex	vertex	NOUN
cana-1260	211	3	connected	connect	VERB
cana-1260	211	4	graph	graph	NOUN
cana-1260	211	5	is	be	AUX
cana-1260	211	6	one	one	NUM
cana-1260	211	7	with	with	ADP
cana-1260	211	8	a	a	DET
cana-1260	211	9	minimal	minimal	ADJ
cana-1260	211	10	number	number	NOUN
cana-1260	211	11	of	of	ADP
cana-1260	211	12	vertices	vertex	NOUN
cana-1260	211	13	whose	whose	DET
cana-1260	211	14	removal	removal	NOUN
cana-1260	211	15	which	which	PRON
cana-1260	211	16	results	result	VERB
cana-1260	211	17	in	in	ADP
cana-1260	211	18	a	a	DET
cana-1260	211	19	disconnected	disconnected	ADJ
cana-1260	211	20	graph	graph	NOUN
cana-1260	211	21	.	.	PUNCT
cana-1260	212	1	we	we	PRON
cana-1260	212	2	get	get	VERB
cana-1260	212	3	line	line	NOUN
cana-1260	212	4	graph	graph	NOUN
cana-1260	212	5	of	of	ADP
cana-1260	212	6	bi	bi	ADJ
cana-1260	212	7	-	-	ADJ
cana-1260	212	8	star	star	ADJ
cana-1260	212	9	graph	graph	NOUN
cana-1260	212	10	is	be	AUX
cana-1260	212	11	one	one	NUM
cana-1260	212	12	connected	connected	ADJ
cana-1260	212	13	graph	graph	NOUN
cana-1260	212	14	therefore	therefore	ADV
cana-1260	212	15	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	212	16	(	(	PUNCT
cana-1260	212	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	212	18	)	)	PUNCT
cana-1260	212	19	)	)	PUNCT
cana-1260	213	1	=	=	PUNCT
cana-1260	213	2	1	1	X
cana-1260	213	3	.	.	PUNCT
cana-1260	214	1	in	in	ADP
cana-1260	214	2	general	general	ADJ
cana-1260	214	3	for	for	ADP
cana-1260	214	4	bi	bi	ADJ
cana-1260	214	5	-	-	ADJ
cana-1260	214	6	star	star	ADJ
cana-1260	214	7	graph	graph	NOUN
cana-1260	214	8	or	or	CCONJ
cana-1260	214	9	two	two	NUM
cana-1260	214	10	copies	copy	NOUN
cana-1260	214	11	of	of	ADP
cana-1260	214	12	(	(	PUNCT
cana-1260	214	13	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	214	14	)	)	PUNCT
cana-1260	214	15	where	where	SCONJ
cana-1260	214	16	𝑛	𝑛	ADV
cana-1260	214	17	=	=	PRON
cana-1260	214	18	{	{	PUNCT
cana-1260	214	19	3,4,5,6,7	3,4,5,6,7	NUM
cana-1260	214	20	,	,	PUNCT
cana-1260	214	21	…	…	PUNCT
cana-1260	214	22	}	}	PUNCT
cana-1260	214	23	we	we	PRON
cana-1260	214	24	get	get	VERB
cana-1260	214	25	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	214	26	(	(	PUNCT
cana-1260	214	27	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	214	28	)	)	PUNCT
cana-1260	215	1	=	=	SYM
cana-1260	215	2	2	2	NUM
cana-1260	215	3	and	and	CCONJ
cana-1260	215	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	215	5	(	(	PUNCT
cana-1260	215	6	𝐿(𝐾1,𝑛	𝐿(𝐾1,𝑛	NOUN
cana-1260	215	7	)	)	PUNCT
cana-1260	215	8	)	)	PUNCT
cana-1260	216	1	=	=	PUNCT
cana-1260	216	2	1	1	X
cana-1260	216	3	.	.	X
cana-1260	216	4	for	for	ADP
cana-1260	216	5	the	the	DET
cana-1260	216	6	sum	sum	NOUN
cana-1260	216	7	we	we	PRON
cana-1260	216	8	have	have	VERB
cana-1260	216	9	𝛾𝑓	𝛾𝑓	VERB
cana-1260	216	10	(	(	PUNCT
cana-1260	216	11	𝐺	𝐺	NOUN
cana-1260	216	12	)	)	PUNCT
cana-1260	217	1	+	+	CCONJ
cana-1260	217	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	217	3	(	(	PUNCT
cana-1260	217	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	217	5	)	)	PUNCT
cana-1260	217	6	)	)	PUNCT
cana-1260	218	1	=	=	SYM
cana-1260	218	2	3	3	NUM
cana-1260	218	3	and	and	CCONJ
cana-1260	218	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	218	5	(	(	PUNCT
cana-1260	218	6	𝐺	𝐺	NOUN
cana-1260	218	7	)	)	PUNCT
cana-1260	218	8	∗	∗	NOUN
cana-1260	218	9	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	218	10	(	(	PUNCT
cana-1260	218	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	218	12	)	)	PUNCT
cana-1260	218	13	)	)	PUNCT
cana-1260	219	1	=	=	SYM
cana-1260	219	2	2	2	X
cana-1260	219	3	.	.	PUNCT
cana-1260	219	4	(	(	PUNCT
cana-1260	219	5	a	a	X
cana-1260	219	6	)	)	PUNCT
cana-1260	219	7	bi	bi	ADJ
cana-1260	219	8	-	-	ADJ
cana-1260	219	9	star	star	ADJ
cana-1260	219	10	graph	graph	NOUN
cana-1260	219	11	𝐾(1,3	𝐾(1,3	PROPN
cana-1260	219	12	)	)	PUNCT
cana-1260	219	13	(	(	PUNCT
cana-1260	219	14	b	b	X
cana-1260	219	15	)	)	PUNCT
cana-1260	219	16	line	line	NOUN
cana-1260	219	17	graph	graph	NOUN
cana-1260	219	18	of	of	ADP
cana-1260	219	19	bi	bi	ADJ
cana-1260	219	20	-	-	ADJ
cana-1260	219	21	star	star	ADJ
cana-1260	219	22	graph	graph	NOUN
cana-1260	219	23	𝐾(1,3	𝐾(1,3	NOUN
cana-1260	219	24	)	)	PUNCT
cana-1260	219	25	figure	figure	NOUN
cana-1260	219	26	2.4.1	2.4.1	NUM
cana-1260	219	27	iii	iii	NOUN
cana-1260	219	28	)	)	PUNCT
cana-1260	219	29	and	and	CCONJ
cana-1260	219	30	iv	iv	X
cana-1260	219	31	)	)	PUNCT
cana-1260	219	32	the	the	DET
cana-1260	219	33	upper	upper	ADJ
cana-1260	219	34	fractional	fractional	ADJ
cana-1260	219	35	dominating	dominating	NOUN
cana-1260	219	36	number	number	NOUN
cana-1260	219	37	of	of	ADP
cana-1260	219	38	(	(	PUNCT
cana-1260	219	39	𝐾1,3	𝐾1,3	NOUN
cana-1260	219	40	)	)	PUNCT
cana-1260	219	41	is	be	AUX
cana-1260	219	42	г𝑓	г𝑓	PRON
cana-1260	219	43	(	(	PUNCT
cana-1260	219	44	(	(	PUNCT
cana-1260	219	45	𝐾1,3	𝐾1,3	NOUN
cana-1260	219	46	)	)	PUNCT
cana-1260	219	47	)	)	PUNCT
cana-1260	220	1	=	=	PUNCT
cana-1260	220	2	6	6	NUM
cana-1260	220	3	with	with	ADP
cana-1260	220	4	consideration	consideration	NOUN
cana-1260	220	5	of	of	ADP
cana-1260	220	6	each	each	DET
cana-1260	220	7	vertex	vertex	NOUN
cana-1260	220	8	's	's	PART
cana-1260	220	9	closed	closed	ADJ
cana-1260	220	10	neighborhood	neighborhood	NOUN
cana-1260	220	11	that	that	PRON
cana-1260	220	12	satisfies	satisfy	VERB
cana-1260	220	13	the	the	DET
cana-1260	220	14	requirement	requirement	NOUN
cana-1260	220	15	where	where	SCONJ
cana-1260	220	16	the	the	DET
cana-1260	220	17	vertex	vertex	NOUN
cana-1260	220	18	𝑤	𝑤	ADP
cana-1260	220	19	∈	∈	PROPN
cana-1260	220	20	𝑁[𝑉	𝑁[𝑉	NOUN
cana-1260	220	21	]	]	PUNCT
cana-1260	220	22	such	such	ADJ
cana-1260	220	23	that	that	SCONJ
cana-1260	220	24	∑	∑	PUNCT
cana-1260	220	25	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	220	26	)	)	PUNCT
cana-1260	220	27	⬚	⬚	PROPN
cana-1260	220	28	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	220	29	]	]	X
cana-1260	220	30	=	=	SYM
cana-1260	220	31	1	1	NUM
cana-1260	220	32	by	by	ADP
cana-1260	220	33	assigning	assign	VERB
cana-1260	220	34	weight	weight	NOUN
cana-1260	220	35	zero	zero	NUM
cana-1260	220	36	to	to	ADP
cana-1260	220	37	the	the	DET
cana-1260	220	38	central	central	ADJ
cana-1260	220	39	vertices	vertex	NOUN
cana-1260	220	40	and	and	CCONJ
cana-1260	220	41	weight	weight	NOUN
cana-1260	220	42	one	one	NUM
cana-1260	220	43	to	to	ADP
cana-1260	220	44	remaining	remain	VERB
cana-1260	220	45	vertices	vertex	NOUN
cana-1260	220	46	.	.	PUNCT
cana-1260	221	1	the	the	DET
cana-1260	221	2	upper	upper	ADJ
cana-1260	221	3	fractional	fractional	ADJ
cana-1260	221	4	dominating	dominating	NOUN
cana-1260	221	5	number	number	NOUN
cana-1260	221	6	of	of	ADP
cana-1260	221	7	line	line	NOUN
cana-1260	221	8	graph	graph	NOUN
cana-1260	221	9	of	of	ADP
cana-1260	221	10	(	(	PUNCT
cana-1260	221	11	𝐾1,3	𝐾1,3	NOUN
cana-1260	221	12	)	)	PUNCT
cana-1260	221	13	is	be	AUX
cana-1260	221	14	г𝑓	г𝑓	PRON
cana-1260	221	15	(	(	PUNCT
cana-1260	221	16	𝐿(𝐾1,3	𝐿(𝐾1,3	PROPN
cana-1260	221	17	)	)	PUNCT
cana-1260	221	18	)	)	PUNCT
cana-1260	222	1	=	=	SYM
cana-1260	222	2	2	2	NUM
cana-1260	222	3	by	by	ADP
cana-1260	222	4	assigning	assign	VERB
cana-1260	222	5	weight	weight	NOUN
cana-1260	222	6	one	one	NUM
cana-1260	222	7	to	to	ADP
cana-1260	222	8	vertex	vertex	NOUN
cana-1260	222	9	𝑒1	𝑒1	NOUN
cana-1260	222	10	and	and	CCONJ
cana-1260	222	11	𝑒5	𝑒5	NOUN
cana-1260	222	12	for	for	ADP
cana-1260	222	13	remaining	remain	VERB
cana-1260	222	14	vertices	vertex	NOUN
cana-1260	222	15	weight	weight	NOUN
cana-1260	222	16	zero	zero	NUM
cana-1260	222	17	so	so	SCONJ
cana-1260	222	18	that	that	SCONJ
cana-1260	222	19	adjacency	adjacency	PROPN
cana-1260	222	20	relation	relation	NOUN
cana-1260	222	21	will	will	AUX
cana-1260	222	22	satisfied	satisfied	VERB
cana-1260	222	23	.	.	PUNCT
cana-1260	223	1	in	in	ADP
cana-1260	223	2	general	general	ADJ
cana-1260	223	3	for	for	ADP
cana-1260	223	4	bi	bi	ADJ
cana-1260	223	5	-	-	ADJ
cana-1260	223	6	star	star	ADJ
cana-1260	223	7	graph	graph	NOUN
cana-1260	223	8	or	or	CCONJ
cana-1260	223	9	two	two	NUM
cana-1260	223	10	copies	copy	NOUN
cana-1260	223	11	of	of	ADP
cana-1260	223	12	(	(	PUNCT
cana-1260	223	13	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	223	14	)	)	PUNCT
cana-1260	223	15	where	where	SCONJ
cana-1260	223	16	𝑛	𝑛	ADV
cana-1260	223	17	=	=	PRON
cana-1260	223	18	{	{	PUNCT
cana-1260	223	19	3,4,5,6,7	3,4,5,6,7	NUM
cana-1260	223	20	…	…	PUNCT
cana-1260	223	21	}	}	PUNCT
cana-1260	223	22	we	we	PRON
cana-1260	223	23	get	get	VERB
cana-1260	223	24	result	result	NOUN
cana-1260	223	25	for	for	SCONJ
cana-1260	223	26	𝐺	𝐺	PROPN
cana-1260	223	27	be	be	AUX
cana-1260	223	28	the	the	DET
cana-1260	223	29	bi	bi	ADJ
cana-1260	223	30	-	-	ADJ
cana-1260	223	31	star	star	ADJ
cana-1260	223	32	graph	graph	NOUN
cana-1260	223	33	(	(	PUNCT
cana-1260	223	34	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	223	35	)	)	PUNCT
cana-1260	223	36	on	on	ADP
cana-1260	223	37	𝑛	𝑛	DET
cana-1260	223	38	vertices	vertex	NOUN
cana-1260	223	39	its	its	PRON
cana-1260	223	40	г𝑓	г𝑓	X
cana-1260	223	41	(	(	PUNCT
cana-1260	223	42	(	(	PUNCT
cana-1260	223	43	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	223	44	)	)	PUNCT
cana-1260	223	45	)	)	PUNCT
cana-1260	224	1	=	=	SYM
cana-1260	224	2	2𝑛	2𝑛	NOUN
cana-1260	224	3	and	and	CCONJ
cana-1260	224	4	line	line	NOUN
cana-1260	224	5	graph	graph	NOUN
cana-1260	224	6	of	of	ADP
cana-1260	224	7	bi	bi	ADJ
cana-1260	224	8	-	-	ADJ
cana-1260	224	9	star	star	ADJ
cana-1260	224	10	graph	graph	NOUN
cana-1260	224	11	(	(	PUNCT
cana-1260	224	12	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	224	13	)	)	PUNCT
cana-1260	224	14	on	on	ADP
cana-1260	224	15	𝑛	𝑛	PRON
cana-1260	224	16	−	−	NOUN
cana-1260	224	17	1	1	NUM
cana-1260	224	18	vertices	vertice	VERB
cana-1260	224	19	its	its	PRON
cana-1260	224	20	г𝑓	г𝑓	X
cana-1260	224	21	(	(	PUNCT
cana-1260	224	22	𝐿(𝐾1,𝑛	𝐿(𝐾1,𝑛	NOUN
cana-1260	224	23	)	)	PUNCT
cana-1260	224	24	)	)	PUNCT
cana-1260	225	1	=	=	SYM
cana-1260	225	2	2	2	NUM
cana-1260	225	3	then	then	ADV
cana-1260	225	4	for	for	ADP
cana-1260	225	5	sum	sum	NOUN
cana-1260	225	6	г𝑓	г𝑓	PROPN
cana-1260	225	7	(	(	PUNCT
cana-1260	225	8	𝐺	𝐺	NOUN
cana-1260	225	9	)	)	PUNCT
cana-1260	226	1	+	+	NUM
cana-1260	226	2	г𝑓	г𝑓	ADJ
cana-1260	226	3	(	(	PUNCT
cana-1260	226	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	226	5	)	)	PUNCT
cana-1260	226	6	)	)	PUNCT
cana-1260	227	1	=	=	SYM
cana-1260	228	1	2𝑛	2𝑛	PROPN
cana-1260	229	1	+	+	CCONJ
cana-1260	229	2	2	2	NUM
cana-1260	229	3	and	and	CCONJ
cana-1260	229	4	г𝑓	г𝑓	ADJ
cana-1260	229	5	(	(	PUNCT
cana-1260	229	6	𝐺	𝐺	NOUN
cana-1260	229	7	)	)	PUNCT
cana-1260	229	8	∗	∗	NOUN
cana-1260	229	9	г𝑓	г𝑓	PROPN
cana-1260	229	10	(	(	PUNCT
cana-1260	229	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	229	12	)	)	PUNCT
cana-1260	229	13	)	)	PUNCT
cana-1260	230	1	=	=	SYM
cana-1260	230	2	4𝑛.	4𝑛.	NUM
cana-1260	230	3	□	□	PUNCT
cana-1260	230	4	2.5	2.5	NUM
cana-1260	230	5	wheel	wheel	NOUN
cana-1260	230	6	graph	graph	NOUN
cana-1260	230	7	(	(	PUNCT
cana-1260	230	8	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	230	9	)	)	PUNCT
cana-1260	230	10	to	to	PART
cana-1260	230	11	create	create	VERB
cana-1260	230	12	a	a	DET
cana-1260	230	13	wheel	wheel	NOUN
cana-1260	230	14	graph	graph	NOUN
cana-1260	230	15	a	a	DET
cana-1260	230	16	cycle	cycle	NOUN
cana-1260	230	17	graph	graph	NOUN
cana-1260	230	18	𝐶𝑛−1	𝐶𝑛−1	NOUN
cana-1260	230	19	,	,	PUNCT
cana-1260	230	20	is	be	AUX
cana-1260	230	21	used	use	VERB
cana-1260	230	22	,	,	PUNCT
cana-1260	230	23	by	by	ADP
cana-1260	230	24	adding	add	VERB
cana-1260	230	25	one	one	NUM
cana-1260	230	26	vertex	vertex	NOUN
cana-1260	230	27	or	or	CCONJ
cana-1260	230	28	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	230	29	=	=	PUNCT
cana-1260	230	30	𝐶𝑛−1	𝐶𝑛−1	PROPN
cana-1260	230	31	+	+	CCONJ
cana-1260	230	32	𝐾1	𝐾1	NOUN
cana-1260	230	33	.	.	PUNCT
cana-1260	231	1	every	every	DET
cana-1260	231	2	vertex	vertex	NOUN
cana-1260	231	3	is	be	AUX
cana-1260	231	4	linked	link	VERB
cana-1260	231	5	to	to	ADP
cana-1260	231	6	every	every	DET
cana-1260	231	7	other	other	ADJ
cana-1260	231	8	vertex	vertex	NOUN
cana-1260	231	9	,	,	PUNCT
cana-1260	231	10	forming	form	VERB
cana-1260	231	11	a	a	DET
cana-1260	231	12	hub	hub	NOUN
cana-1260	231	13	of	of	ADP
cana-1260	231	14	𝐶𝑛−1	𝐶𝑛−1	NOUN
cana-1260	231	15	.	.	PUNCT
cana-1260	232	1	the	the	DET
cana-1260	232	2	line	line	NOUN
cana-1260	232	3	graph	graph	NOUN
cana-1260	232	4	of	of	ADP
cana-1260	232	5	wheel	wheel	NOUN
cana-1260	232	6	graph	graph	NOUN
cana-1260	232	7	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	232	8	)	)	PUNCT
cana-1260	232	9	contains	contain	VERB
cana-1260	232	10	a	a	DET
cana-1260	232	11	complete	complete	ADJ
cana-1260	232	12	graph	graph	NOUN
cana-1260	232	13	𝐾𝑛−1	𝐾𝑛−1	NOUN
cana-1260	232	14	with	with	ADP
cana-1260	232	15	cycle	cycle	NOUN
cana-1260	232	16	of	of	ADP
cana-1260	232	17	𝑛	𝑛	PRON
cana-1260	232	18	−	−	NUM
cana-1260	232	19	1	1	NUM
cana-1260	232	20	vertices	vertex	NOUN
cana-1260	232	21	where	where	SCONJ
cana-1260	232	22	each	each	DET
cana-1260	232	23	vertex	vertex	NOUN
cana-1260	232	24	of	of	ADP
cana-1260	232	25	the	the	DET
cana-1260	232	26	cycle	cycle	NOUN
cana-1260	232	27	is	be	AUX
cana-1260	232	28	connected	connect	VERB
cana-1260	232	29	to	to	ADP
cana-1260	232	30	two	two	NUM
cana-1260	232	31	vertices	vertex	NOUN
cana-1260	232	32	of	of	ADP
cana-1260	232	33	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-1260	232	34	.	.	PUNCT
cana-1260	233	1	theorem	theorem	VERB
cana-1260	233	2	2.5.1	2.5.1	NUM
cana-1260	233	3	if	if	SCONJ
cana-1260	233	4	𝐺	𝐺	PROPN
cana-1260	233	5	is	be	AUX
cana-1260	233	6	wheel	wheel	NOUN
cana-1260	233	7	graph	graph	NOUN
cana-1260	233	8	on	on	ADP
cana-1260	233	9	𝑛	𝑛	DET
cana-1260	233	10	vertices	vertex	NOUN
cana-1260	233	11	with	with	ADP
cana-1260	233	12	size	size	NOUN
cana-1260	233	13	𝑚	𝑚	PROPN
cana-1260	233	14	and	and	CCONJ
cana-1260	233	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	233	16	)	)	PUNCT
cana-1260	233	17	be	be	AUX
cana-1260	233	18	its	its	PRON
cana-1260	233	19	line	line	NOUN
cana-1260	233	20	graph	graph	NOUN
cana-1260	233	21	of	of	ADP
cana-1260	233	22	𝑚	𝑚	PROPN
cana-1260	233	23	vertices	vertex	NOUN
cana-1260	233	24	then	then	ADV
cana-1260	233	25	𝑖	𝑖	X
cana-1260	233	26	)	)	PUNCT
cana-1260	234	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	234	2	(	(	PUNCT
cana-1260	234	3	𝐺	𝐺	NOUN
cana-1260	234	4	)	)	PUNCT
cana-1260	235	1	+	+	CCONJ
cana-1260	235	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	235	3	(	(	PUNCT
cana-1260	235	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	235	5	)	)	PUNCT
cana-1260	235	6	)	)	PUNCT
cana-1260	235	7	=	=	SYM
cana-1260	236	1	1	1	NUM
cana-1260	236	2	+	+	NUM
cana-1260	236	3	𝑚	𝑚	PRON
cana-1260	236	4	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	236	5	,	,	PUNCT
cana-1260	236	6	where	where	SCONJ
cana-1260	236	7	𝛿	𝛿	ADJ
cana-1260	236	8	(	(	PUNCT
cana-1260	236	9	𝐺	𝐺	NOUN
cana-1260	236	10	)	)	PUNCT
cana-1260	236	11	is	be	AUX
cana-1260	236	12	minimum	minimum	NOUN
cana-1260	236	13	degree	degree	NOUN
cana-1260	236	14	of	of	ADP
cana-1260	236	15	𝐿	𝐿	PROPN
cana-1260	236	16	(	(	PUNCT
cana-1260	236	17	𝐺	𝐺	NOUN
cana-1260	236	18	)	)	PUNCT
cana-1260	236	19	and	and	CCONJ
cana-1260	236	20	𝑖𝑖	𝑖𝑖	NOUN
cana-1260	236	21	)	)	PUNCT
cana-1260	236	22	г𝑓	г𝑓	PROPN
cana-1260	236	23	(	(	PUNCT
cana-1260	236	24	𝐺	𝐺	NOUN
cana-1260	236	25	)	)	PUNCT
cana-1260	237	1	+	+	NUM
cana-1260	237	2	г𝑓	г𝑓	ADJ
cana-1260	237	3	(	(	PUNCT
cana-1260	237	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	237	5	)	)	PUNCT
cana-1260	237	6	)	)	PUNCT
cana-1260	237	7	≤	≤	NUM
cana-1260	238	1	2	2	NUM
cana-1260	238	2	+	+	CCONJ
cana-1260	238	3	𝑛−1	𝑛−1	NUM
cana-1260	238	4	2	2	NUM
cana-1260	238	5	proof	proof	NOUN
cana-1260	238	6	i	i	PRON
cana-1260	238	7	):	):	PUNCT
cana-1260	238	8	[	[	X
cana-1260	238	9	11	11	NUM
cana-1260	238	10	]	]	PUNCT
cana-1260	238	11	a	a	DET
cana-1260	238	12	wheel	wheel	NOUN
cana-1260	238	13	graph	graph	NOUN
cana-1260	238	14	of	of	ADP
cana-1260	238	15	𝑛	𝑛	DET
cana-1260	238	16	vertices	vertex	NOUN
cana-1260	238	17	contains	contain	VERB
cana-1260	238	18	a	a	DET
cana-1260	238	19	cycle	cycle	NOUN
cana-1260	238	20	graph	graph	NOUN
cana-1260	238	21	of	of	ADP
cana-1260	238	22	length	length	NOUN
cana-1260	238	23	𝑛	𝑛	DET
cana-1260	238	24	−	−	PROPN
cana-1260	238	25	1	1	NUM
cana-1260	238	26	and	and	CCONJ
cana-1260	238	27	all	all	DET
cana-1260	238	28	the	the	DET
cana-1260	238	29	vertices	vertex	NOUN
cana-1260	238	30	of	of	ADP
cana-1260	238	31	the	the	DET
cana-1260	238	32	cycle	cycle	NOUN
cana-1260	238	33	are	be	AUX
cana-1260	238	34	connected	connect	VERB
cana-1260	238	35	to	to	ADP
cana-1260	238	36	a	a	DET
cana-1260	238	37	single	single	ADJ
cana-1260	238	38	central	central	ADJ
cana-1260	238	39	vertex	vertex	NOUN
cana-1260	238	40	.	.	PUNCT
cana-1260	239	1	thus	thus	ADV
cana-1260	239	2	,	,	PUNCT
cana-1260	239	3	wheel	wheel	NOUN
cana-1260	239	4	graph	graph	NOUN
cana-1260	239	5	on	on	ADP
cana-1260	239	6	𝑛	𝑛	DET
cana-1260	239	7	vertices	vertex	NOUN
cana-1260	239	8	denoted	denote	VERB
cana-1260	239	9	by	by	ADP
cana-1260	239	10	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	239	11	for	for	ADP
cana-1260	239	12	𝑛	𝑛	NOUN
cana-1260	239	13	>	>	SYM
cana-1260	239	14	3	3	NUM
cana-1260	239	15	is	be	AUX
cana-1260	239	16	acquired	acquire	VERB
cana-1260	239	17	by	by	ADP
cana-1260	239	18	connecting	connect	VERB
cana-1260	239	19	a	a	DET
cana-1260	239	20	vertex	vertex	NOUN
cana-1260	239	21	to	to	ADP
cana-1260	239	22	every	every	DET
cana-1260	239	23	vertex	vertex	NOUN
cana-1260	239	24	of	of	ADP
cana-1260	239	25	cycle	cycle	NOUN
cana-1260	239	26	graph	graph	NOUN
cana-1260	239	27	.	.	PUNCT
cana-1260	240	1	the	the	DET
cana-1260	240	2	fractional	fractional	ADJ
cana-1260	240	3	dominating	dominating	NOUN
cana-1260	240	4	communications	communication	NOUN
cana-1260	240	5	on	on	ADP
cana-1260	240	6	applied	apply	VERB
cana-1260	240	7	nonlinear	nonlinear	ADJ
cana-1260	240	8	analysis	analysis	NOUN
cana-1260	240	9	issn	issn	NOUN
cana-1260	240	10	:	:	PUNCT
cana-1260	240	11	1074	1074	NUM
cana-1260	240	12	-	-	PUNCT
cana-1260	240	13	133x	133x	NUM
cana-1260	240	14	vol	vol	NOUN
cana-1260	240	15	31	31	NUM
cana-1260	240	16	no	no	NOUN
cana-1260	240	17	.	.	PUNCT
cana-1260	241	1	6s	6s	NUM
cana-1260	241	2	(	(	PUNCT
cana-1260	241	3	2024	2024	NUM
cana-1260	241	4	)	)	PUNCT
cana-1260	241	5	679	679	NUM
cana-1260	241	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	241	7	number	number	NOUN
cana-1260	241	8	of	of	ADP
cana-1260	241	9	wheel	wheel	NOUN
cana-1260	241	10	graph	graph	NOUN
cana-1260	241	11	will	will	AUX
cana-1260	241	12	be	be	AUX
cana-1260	241	13	𝛾𝑓	𝛾𝑓	ADP
cana-1260	241	14	(	(	PUNCT
cana-1260	241	15	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	241	16	)	)	PUNCT
cana-1260	241	17	=	=	SYM
cana-1260	241	18	1	1	X
cana-1260	241	19	.	.	NOUN
cana-1260	241	20	wheel	wheel	NOUN
cana-1260	241	21	graph	graph	NOUN
cana-1260	241	22	of	of	ADP
cana-1260	241	23	𝑛	𝑛	DET
cana-1260	241	24	vertices	vertex	NOUN
cana-1260	241	25	contains	contain	VERB
cana-1260	241	26	𝑚	𝑚	ADP
cana-1260	241	27	edges	edge	NOUN
cana-1260	241	28	so	so	SCONJ
cana-1260	241	29	the	the	DET
cana-1260	241	30	line	line	NOUN
cana-1260	241	31	graph	graph	NOUN
cana-1260	241	32	of	of	ADP
cana-1260	241	33	wheel	wheel	NOUN
cana-1260	241	34	graph	graph	NOUN
cana-1260	241	35	𝐿	𝐿	PROPN
cana-1260	241	36	(	(	PUNCT
cana-1260	241	37	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	241	38	)	)	PUNCT
cana-1260	241	39	is	be	AUX
cana-1260	241	40	a	a	DET
cana-1260	241	41	graph	graph	NOUN
cana-1260	241	42	in	in	ADP
cana-1260	241	43	which	which	PRON
cana-1260	241	44	vertices	vertice	VERB
cana-1260	241	45	corresponds	correspond	VERB
cana-1260	241	46	to	to	ADP
cana-1260	241	47	the	the	DET
cana-1260	241	48	edges	edge	NOUN
cana-1260	241	49	of	of	ADP
cana-1260	241	50	the	the	DET
cana-1260	241	51	original	original	ADJ
cana-1260	241	52	wheel	wheel	NOUN
cana-1260	241	53	graph	graph	NOUN
cana-1260	241	54	so	so	SCONJ
cana-1260	241	55	it	it	PRON
cana-1260	241	56	contains	contain	VERB
cana-1260	241	57	𝑚	𝑚	ADP
cana-1260	241	58	vertices	vertex	NOUN
cana-1260	241	59	with	with	ADP
cana-1260	241	60	a	a	DET
cana-1260	241	61	complete	complete	ADJ
cana-1260	241	62	graph	graph	NOUN
cana-1260	241	63	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-1260	241	64	and	and	CCONJ
cana-1260	241	65	the	the	DET
cana-1260	241	66	cycle	cycle	NOUN
cana-1260	241	67	of	of	ADP
cana-1260	241	68	𝑛	𝑛	PRON
cana-1260	241	69	−	−	NUM
cana-1260	241	70	1	1	NUM
cana-1260	241	71	vertices	vertex	NOUN
cana-1260	241	72	where	where	SCONJ
cana-1260	241	73	every	every	DET
cana-1260	241	74	vertex	vertex	NOUN
cana-1260	241	75	of	of	ADP
cana-1260	241	76	the	the	DET
cana-1260	241	77	cycle	cycle	NOUN
cana-1260	241	78	is	be	AUX
cana-1260	241	79	adjacent	adjacent	ADJ
cana-1260	241	80	to	to	ADP
cana-1260	241	81	other	other	ADJ
cana-1260	241	82	two	two	NUM
cana-1260	241	83	vertices	vertex	NOUN
cana-1260	241	84	of	of	ADP
cana-1260	241	85	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-1260	241	86	.	.	PUNCT
cana-1260	242	1	the	the	DET
cana-1260	242	2	max	max	PROPN
cana-1260	242	3	degree	degree	NOUN
cana-1260	242	4	of	of	ADP
cana-1260	242	5	line	line	NOUN
cana-1260	242	6	graph	graph	NOUN
cana-1260	242	7	𝛥	𝛥	PROPN
cana-1260	242	8	(	(	PUNCT
cana-1260	242	9	𝐺	𝐺	NOUN
cana-1260	242	10	)	)	PUNCT
cana-1260	242	11	=	=	SYM
cana-1260	242	12	𝑛	𝑛	PROPN
cana-1260	242	13	and	and	CCONJ
cana-1260	242	14	min	min	NOUN
cana-1260	242	15	degree	degree	NOUN
cana-1260	242	16	of	of	ADP
cana-1260	242	17	line	line	NOUN
cana-1260	242	18	graph	graph	NOUN
cana-1260	242	19	𝛿	𝛿	PROPN
cana-1260	242	20	(	(	PUNCT
cana-1260	242	21	𝐺	𝐺	NOUN
cana-1260	242	22	)	)	PUNCT
cana-1260	242	23	=	=	PUNCT
cana-1260	242	24	4	4	X
cana-1260	242	25	.	.	X
cana-1260	243	1	for	for	ADP
cana-1260	243	2	example	example	NOUN
cana-1260	244	1	i	i	PRON
cana-1260	244	2	f	f	PROPN
cana-1260	244	3	wheel	wheel	NOUN
cana-1260	244	4	graph	graph	NOUN
cana-1260	244	5	𝐺	𝐺	PROPN
cana-1260	244	6	=	=	PUNCT
cana-1260	244	7	𝑊4	𝑊4	PROPN
cana-1260	244	8	with	with	ADP
cana-1260	244	9	number	number	NOUN
cana-1260	244	10	of	of	ADP
cana-1260	244	11	vertices	vertex	NOUN
cana-1260	244	12	𝑛	𝑛	PROPN
cana-1260	244	13	=	=	SYM
cana-1260	244	14	4	4	NUM
cana-1260	244	15	and	and	CCONJ
cana-1260	244	16	number	number	NOUN
cana-1260	244	17	of	of	ADP
cana-1260	244	18	edges	edge	NOUN
cana-1260	244	19	𝑚	𝑚	X
cana-1260	244	20	=	=	SYM
cana-1260	244	21	6	6	NUM
cana-1260	244	22	then	then	ADV
cana-1260	244	23	line	line	NOUN
cana-1260	244	24	graph	graph	NOUN
cana-1260	244	25	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	244	26	)	)	PUNCT
cana-1260	244	27	contains	contain	VERB
cana-1260	244	28	𝑚	𝑚	NOUN
cana-1260	244	29	=	=	SYM
cana-1260	244	30	6	6	NUM
cana-1260	244	31	vertices	vertex	NOUN
cana-1260	244	32	with	with	ADP
cana-1260	244	33	a	a	DET
cana-1260	244	34	complete	complete	ADJ
cana-1260	244	35	graph	graph	NOUN
cana-1260	244	36	𝐾4	𝐾4	NOUN
cana-1260	244	37	-	-	PUNCT
cana-1260	244	38	1	1	NUM
cana-1260	244	39	and	and	CCONJ
cana-1260	244	40	the	the	DET
cana-1260	244	41	cycle	cycle	NOUN
cana-1260	244	42	of	of	ADP
cana-1260	244	43	𝑛	𝑛	DET
cana-1260	244	44	−	−	PROPN
cana-1260	244	45	1	1	NUM
cana-1260	244	46	=	=	SYM
cana-1260	244	47	3	3	NUM
cana-1260	244	48	vertices	vertex	NOUN
cana-1260	244	49	where	where	SCONJ
cana-1260	244	50	every	every	DET
cana-1260	244	51	vertex	vertex	NOUN
cana-1260	244	52	of	of	ADP
cana-1260	244	53	the	the	DET
cana-1260	244	54	cycle	cycle	NOUN
cana-1260	244	55	is	be	AUX
cana-1260	244	56	adjacent	adjacent	ADJ
cana-1260	244	57	to	to	ADP
cana-1260	244	58	two	two	NUM
cana-1260	244	59	vertices	vertex	NOUN
cana-1260	244	60	of	of	ADP
cana-1260	244	61	𝐾4	𝐾4	NOUN
cana-1260	244	62	-	-	PUNCT
cana-1260	244	63	1	1	NUM
cana-1260	244	64	.	.	PUNCT
cana-1260	245	1	we	we	PRON
cana-1260	245	2	know	know	VERB
cana-1260	245	3	that	that	SCONJ
cana-1260	245	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	245	5	(	(	PUNCT
cana-1260	245	6	𝑊4	𝑊4	PROPN
cana-1260	245	7	)	)	PUNCT
cana-1260	245	8	=	=	SYM
cana-1260	245	9	1	1	NUM
cana-1260	245	10	and	and	CCONJ
cana-1260	245	11	the	the	DET
cana-1260	245	12	line	line	NOUN
cana-1260	245	13	graph	graph	NOUN
cana-1260	245	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	245	15	)	)	PUNCT
cana-1260	246	1	where	where	SCONJ
cana-1260	246	2	𝐺	𝐺	NOUN
cana-1260	246	3	=	=	X
cana-1260	246	4	𝑊4	𝑊4	PROPN
cana-1260	246	5	is	be	AUX
cana-1260	246	6	of	of	ADP
cana-1260	246	7	order	order	NOUN
cana-1260	246	8	𝑚	𝑚	NOUN
cana-1260	246	9	=	=	SYM
cana-1260	246	10	6	6	NUM
cana-1260	246	11	and	and	CCONJ
cana-1260	246	12	𝛿(𝐺	𝛿(𝐺	NUM
cana-1260	246	13	)	)	PUNCT
cana-1260	247	1	=	=	SYM
cana-1260	247	2	4	4	NUM
cana-1260	247	3	we	we	PRON
cana-1260	247	4	get	get	VERB
cana-1260	247	5	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	247	6	(	(	PUNCT
cana-1260	247	7	𝐿(𝑊4	𝐿(𝑊4	ADJ
cana-1260	247	8	)	)	PUNCT
cana-1260	247	9	)	)	PUNCT
cana-1260	248	1	=	=	PUNCT
cana-1260	248	2	𝑚	𝑚	PROPN
cana-1260	248	3	𝛿(𝐺)+1	𝛿(𝐺)+1	PROPN
cana-1260	248	4	≤	≤	NOUN
cana-1260	248	5	6/5	6/5	NUM
cana-1260	248	6	by	by	ADP
cana-1260	248	7	assigning	assign	VERB
cana-1260	248	8	weight	weight	NOUN
cana-1260	248	9	(	(	PUNCT
cana-1260	248	10	1/5	1/5	NUM
cana-1260	248	11	)	)	PUNCT
cana-1260	248	12	for	for	ADP
cana-1260	248	13	each	each	DET
cana-1260	248	14	vertex	vertex	NOUN
cana-1260	248	15	of	of	ADP
cana-1260	248	16	line	line	NOUN
cana-1260	248	17	graph	graph	NOUN
cana-1260	248	18	.	.	PUNCT
cana-1260	249	1	if	if	SCONJ
cana-1260	249	2	wheel	wheel	NOUN
cana-1260	249	3	graph	graph	NOUN
cana-1260	249	4	𝐺	𝐺	PROPN
cana-1260	249	5	=	=	PUNCT
cana-1260	249	6	𝑊5	𝑊5	PROPN
cana-1260	249	7	is	be	AUX
cana-1260	249	8	of	of	ADP
cana-1260	249	9	order	order	NOUN
cana-1260	249	10	𝑛	𝑛	NOUN
cana-1260	249	11	=	=	SYM
cana-1260	249	12	5	5	NUM
cana-1260	249	13	and	and	CCONJ
cana-1260	249	14	size	size	NOUN
cana-1260	249	15	𝑚	𝑚	X
cana-1260	249	16	=	=	SYM
cana-1260	249	17	8	8	NUM
cana-1260	249	18	then	then	ADV
cana-1260	249	19	line	line	NOUN
cana-1260	249	20	graph	graph	NOUN
cana-1260	249	21	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	249	22	)	)	PUNCT
cana-1260	249	23	contains	contain	VERB
cana-1260	249	24	𝑚	𝑚	NOUN
cana-1260	249	25	=	=	SYM
cana-1260	249	26	8	8	NUM
cana-1260	249	27	vertices	vertex	NOUN
cana-1260	249	28	with	with	ADP
cana-1260	249	29	a	a	DET
cana-1260	249	30	complete	complete	ADJ
cana-1260	249	31	graph	graph	NOUN
cana-1260	249	32	𝐾5	𝐾5	PROPN
cana-1260	249	33	-	-	PUNCT
cana-1260	249	34	1	1	NUM
cana-1260	249	35	and	and	CCONJ
cana-1260	249	36	cycle	cycle	NOUN
cana-1260	249	37	of	of	ADP
cana-1260	249	38	𝑛	𝑛	DET
cana-1260	249	39	−	−	PROPN
cana-1260	249	40	1	1	NUM
cana-1260	249	41	=	=	SYM
cana-1260	249	42	4	4	NUM
cana-1260	249	43	vertices	vertex	NOUN
cana-1260	249	44	where	where	SCONJ
cana-1260	249	45	each	each	DET
cana-1260	249	46	vertex	vertex	NOUN
cana-1260	249	47	of	of	ADP
cana-1260	249	48	cycle	cycle	NOUN
cana-1260	249	49	is	be	AUX
cana-1260	249	50	connected	connect	VERB
cana-1260	249	51	to	to	ADP
cana-1260	249	52	two	two	NUM
cana-1260	249	53	vertices	vertex	NOUN
cana-1260	249	54	of	of	ADP
cana-1260	249	55	𝐾5	𝐾5	NOUN
cana-1260	249	56	-	-	PUNCT
cana-1260	249	57	1	1	NUM
cana-1260	249	58	.	.	PUNCT
cana-1260	250	1	we	we	PRON
cana-1260	250	2	found	find	VERB
cana-1260	250	3	that	that	SCONJ
cana-1260	250	4	if	if	SCONJ
cana-1260	250	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	250	6	(	(	PUNCT
cana-1260	250	7	𝑊5	𝑊5	X
cana-1260	250	8	)	)	PUNCT
cana-1260	250	9	=	=	SYM
cana-1260	250	10	1	1	NUM
cana-1260	250	11	and	and	CCONJ
cana-1260	250	12	its	its	PRON
cana-1260	250	13	line	line	NOUN
cana-1260	250	14	graph	graph	NOUN
cana-1260	250	15	of	of	ADP
cana-1260	250	16	order	order	NOUN
cana-1260	250	17	𝑚	𝑚	NOUN
cana-1260	250	18	=	=	SYM
cana-1260	250	19	8	8	NUM
cana-1260	250	20	and	and	CCONJ
cana-1260	250	21	𝛿(𝐺	𝛿(𝐺	NUM
cana-1260	250	22	)	)	PUNCT
cana-1260	250	23	=	=	SYM
cana-1260	251	1	4	4	NUM
cana-1260	251	2	then	then	ADV
cana-1260	251	3	𝛾𝑓	𝛾𝑓	X
cana-1260	251	4	(	(	PUNCT
cana-1260	251	5	𝐿(𝑊5	𝐿(𝑊5	NOUN
cana-1260	251	6	)	)	PUNCT
cana-1260	251	7	)	)	PUNCT
cana-1260	252	1	=	=	PUNCT
cana-1260	253	1	𝑚	𝑚	PROPN
cana-1260	253	2	𝛿(𝐺)+1	𝛿(𝐺)+1	PROPN
cana-1260	253	3	≤	≤	NUM
cana-1260	253	4	8/5	8/5	NUM
cana-1260	253	5	by	by	ADP
cana-1260	253	6	assigning	assign	VERB
cana-1260	253	7	weight	weight	NOUN
cana-1260	253	8	(	(	PUNCT
cana-1260	253	9	1/5	1/5	NUM
cana-1260	253	10	)	)	PUNCT
cana-1260	253	11	for	for	ADP
cana-1260	253	12	each	each	DET
cana-1260	253	13	vertex	vertex	NOUN
cana-1260	253	14	of	of	ADP
cana-1260	253	15	line	line	NOUN
cana-1260	253	16	graph	graph	NOUN
cana-1260	253	17	.	.	PUNCT
cana-1260	254	1	if	if	SCONJ
cana-1260	254	2	wheel	wheel	NOUN
cana-1260	254	3	graph	graph	NOUN
cana-1260	254	4	𝐺	𝐺	PROPN
cana-1260	254	5	=	=	PUNCT
cana-1260	254	6	𝑊6	𝑊6	PROPN
cana-1260	254	7	is	be	AUX
cana-1260	254	8	of	of	ADP
cana-1260	254	9	order	order	NOUN
cana-1260	254	10	𝑛	𝑛	NOUN
cana-1260	254	11	=	=	SYM
cana-1260	254	12	6	6	NUM
cana-1260	254	13	and	and	CCONJ
cana-1260	254	14	size	size	NOUN
cana-1260	254	15	𝑚	𝑚	X
cana-1260	254	16	=	=	SYM
cana-1260	254	17	10	10	NUM
cana-1260	254	18	(	(	PUNCT
cana-1260	254	19	see	see	VERB
cana-1260	254	20	the	the	DET
cana-1260	254	21	figure	figure	NOUN
cana-1260	254	22	2.5.1a	2.5.1a	NUM
cana-1260	254	23	)	)	PUNCT
cana-1260	254	24	then	then	ADV
cana-1260	254	25	line	line	NOUN
cana-1260	254	26	graph	graph	NOUN
cana-1260	254	27	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	254	28	)	)	PUNCT
cana-1260	255	1	contains	contain	VERB
cana-1260	255	2	𝑚	𝑚	NOUN
cana-1260	255	3	=	=	SYM
cana-1260	255	4	10	10	NUM
cana-1260	255	5	vertices	vertex	NOUN
cana-1260	255	6	with	with	ADP
cana-1260	255	7	complete	complete	ADJ
cana-1260	255	8	graph	graph	NOUN
cana-1260	255	9	𝐾6	𝐾6	NOUN
cana-1260	255	10	-	-	PUNCT
cana-1260	255	11	1	1	NUM
cana-1260	255	12	and	and	CCONJ
cana-1260	255	13	the	the	DET
cana-1260	255	14	cycle	cycle	NOUN
cana-1260	255	15	of	of	ADP
cana-1260	255	16	𝑛	𝑛	DET
cana-1260	255	17	−	−	PROPN
cana-1260	255	18	1	1	NUM
cana-1260	255	19	=	=	SYM
cana-1260	255	20	5	5	NUM
cana-1260	255	21	vertices	vertex	NOUN
cana-1260	255	22	where	where	SCONJ
cana-1260	255	23	each	each	DET
cana-1260	255	24	vertex	vertex	NOUN
cana-1260	255	25	of	of	ADP
cana-1260	255	26	cycle	cycle	NOUN
cana-1260	255	27	is	be	AUX
cana-1260	255	28	connected	connect	VERB
cana-1260	255	29	to	to	ADP
cana-1260	255	30	two	two	NUM
cana-1260	255	31	vertices	vertex	NOUN
cana-1260	255	32	of	of	ADP
cana-1260	255	33	𝐾6	𝐾6	NOUN
cana-1260	255	34	-	-	PUNCT
cana-1260	255	35	1	1	NUM
cana-1260	255	36	(	(	PUNCT
cana-1260	255	37	see	see	VERB
cana-1260	255	38	the	the	DET
cana-1260	255	39	figure	figure	NOUN
cana-1260	255	40	2.5.1b	2.5.1b	NUM
cana-1260	255	41	)	)	PUNCT
cana-1260	255	42	.	.	PUNCT
cana-1260	256	1	we	we	PRON
cana-1260	256	2	found	find	VERB
cana-1260	256	3	if	if	SCONJ
cana-1260	256	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	256	5	(	(	PUNCT
cana-1260	256	6	𝑊6	𝑊6	PROPN
cana-1260	256	7	)	)	PUNCT
cana-1260	256	8	=	=	SYM
cana-1260	256	9	1	1	NUM
cana-1260	256	10	its	its	PRON
cana-1260	256	11	line	line	NOUN
cana-1260	256	12	graph	graph	NOUN
cana-1260	256	13	of	of	ADP
cana-1260	256	14	order	order	NOUN
cana-1260	256	15	𝑚	𝑚	NOUN
cana-1260	256	16	=	=	SYM
cana-1260	256	17	10	10	NUM
cana-1260	256	18	and	and	CCONJ
cana-1260	256	19	𝛿(𝐺	𝛿(𝐺	NUM
cana-1260	256	20	)	)	PUNCT
cana-1260	256	21	=	=	SYM
cana-1260	257	1	4	4	NUM
cana-1260	257	2	then	then	ADV
cana-1260	257	3	we	we	PRON
cana-1260	257	4	get	get	VERB
cana-1260	257	5	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	257	6	(	(	PUNCT
cana-1260	257	7	𝐿(𝑊6	𝐿(𝑊6	ADJ
cana-1260	257	8	)	)	PUNCT
cana-1260	257	9	)	)	PUNCT
cana-1260	258	1	≤	≤	NOUN
cana-1260	258	2	𝑚	𝑚	ADP
cana-1260	258	3	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	258	4	=	=	SYM
cana-1260	258	5	2	2	NUM
cana-1260	258	6	hence	hence	ADV
cana-1260	258	7	,	,	PUNCT
cana-1260	258	8	we	we	PRON
cana-1260	258	9	get	get	VERB
cana-1260	258	10	the	the	DET
cana-1260	258	11	generalized	generalized	ADJ
cana-1260	258	12	result	result	NOUN
cana-1260	258	13	for	for	ADP
cana-1260	258	14	the	the	DET
cana-1260	258	15	sum	sum	NOUN
cana-1260	258	16	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	258	17	(	(	PUNCT
cana-1260	258	18	𝐺	𝐺	NOUN
cana-1260	258	19	)	)	PUNCT
cana-1260	259	1	+	+	CCONJ
cana-1260	259	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	259	3	(	(	PUNCT
cana-1260	259	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	259	5	)	)	PUNCT
cana-1260	259	6	)	)	PUNCT
cana-1260	259	7	≤	≤	ADV
cana-1260	259	8	1	1	NUM
cana-1260	259	9	+	+	NUM
cana-1260	259	10	𝑚	𝑚	ADP
cana-1260	259	11	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	259	12	.	.	PUNCT
cana-1260	260	1	(	(	PUNCT
cana-1260	260	2	a	a	X
cana-1260	260	3	)	)	PUNCT
cana-1260	260	4	wheel	wheel	NOUN
cana-1260	260	5	graph	graph	NOUN
cana-1260	260	6	w6	w6	PROPN
cana-1260	260	7	(	(	PUNCT
cana-1260	260	8	b	b	NOUN
cana-1260	260	9	)	)	PUNCT
cana-1260	260	10	line	line	NOUN
cana-1260	260	11	graph	graph	NOUN
cana-1260	260	12	of	of	ADP
cana-1260	260	13	wheel	wheel	NOUN
cana-1260	260	14	graph	graph	NOUN
cana-1260	260	15	w6	w6	PROPN
cana-1260	260	16	figure	figure	NOUN
cana-1260	260	17	2.5.1	2.5.1	NUM
cana-1260	260	18	ii	ii	NOUN
cana-1260	260	19	)	)	PUNCT
cana-1260	260	20	upper	upper	ADJ
cana-1260	260	21	fractional	fractional	ADJ
cana-1260	260	22	dominating	dominating	NOUN
cana-1260	260	23	number	number	NOUN
cana-1260	260	24	of	of	ADP
cana-1260	260	25	wheel	wheel	NOUN
cana-1260	260	26	graph	graph	NOUN
cana-1260	261	1	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	261	2	is	be	AUX
cana-1260	261	3	г𝑓	г𝑓	PRON
cana-1260	261	4	(	(	PUNCT
cana-1260	261	5	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	261	6	)	)	PUNCT
cana-1260	261	7	=	=	SYM
cana-1260	261	8	2	2	NUM
cana-1260	261	9	with	with	ADP
cana-1260	261	10	consideration	consideration	NOUN
cana-1260	261	11	of	of	ADP
cana-1260	261	12	the	the	DET
cana-1260	261	13	closed	closed	ADJ
cana-1260	261	14	neighborhood	neighborhood	NOUN
cana-1260	261	15	of	of	ADP
cana-1260	261	16	each	each	DET
cana-1260	261	17	vertex	vertex	NOUN
cana-1260	261	18	satisfying	satisfy	VERB
cana-1260	261	19	the	the	DET
cana-1260	261	20	condition	condition	NOUN
cana-1260	261	21	where	where	SCONJ
cana-1260	261	22	the	the	DET
cana-1260	261	23	vertex	vertex	NOUN
cana-1260	261	24	𝑤	𝑤	ADP
cana-1260	261	25	∈	∈	PROPN
cana-1260	261	26	𝑁[𝑉	𝑁[𝑉	NOUN
cana-1260	261	27	]	]	PUNCT
cana-1260	261	28	such	such	ADJ
cana-1260	261	29	that	that	SCONJ
cana-1260	261	30	∑	∑	PUNCT
cana-1260	261	31	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	261	32	)	)	PUNCT
cana-1260	261	33	⬚	⬚	PROPN
cana-1260	261	34	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	261	35	]	]	X
cana-1260	261	36	=	=	SYM
cana-1260	261	37	1	1	X
cana-1260	261	38	.	.	X
cana-1260	261	39	for	for	ADP
cana-1260	261	40	upper	upper	ADJ
cana-1260	261	41	fractional	fractional	ADJ
cana-1260	261	42	dominating	dominating	NOUN
cana-1260	261	43	number	number	NOUN
cana-1260	261	44	of	of	ADP
cana-1260	261	45	line	line	NOUN
cana-1260	261	46	graph	graph	NOUN
cana-1260	261	47	of	of	ADP
cana-1260	261	48	wheel	wheel	NOUN
cana-1260	261	49	graph	graph	NOUN
cana-1260	261	50	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	261	51	is	be	AUX
cana-1260	261	52	г𝑓(𝐿(𝑊𝑛	г𝑓(𝐿(𝑊𝑛	NUM
cana-1260	261	53	)	)	PUNCT
cana-1260	261	54	)	)	PUNCT
cana-1260	261	55	≤	≤	NUM
cana-1260	262	1	𝑛−1	𝑛−1	NUM
cana-1260	262	2	2	2	NUM
cana-1260	262	3	as	as	SCONJ
cana-1260	262	4	we	we	PRON
cana-1260	262	5	have	have	VERB
cana-1260	262	6	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	262	7	=	=	PUNCT
cana-1260	262	8	𝐶𝑛−1	𝐶𝑛−1	PROPN
cana-1260	262	9	+	+	CCONJ
cana-1260	262	10	𝐾1	𝐾1	NOUN
cana-1260	262	11	and	and	CCONJ
cana-1260	262	12	the	the	DET
cana-1260	262	13	line	line	NOUN
cana-1260	262	14	graph	graph	NOUN
cana-1260	262	15	of	of	ADP
cana-1260	262	16	wheel	wheel	NOUN
cana-1260	262	17	graph	graph	NOUN
cana-1260	262	18	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	262	19	)	)	PUNCT
cana-1260	262	20	contains	contain	VERB
cana-1260	262	21	a	a	DET
cana-1260	262	22	complete	complete	ADJ
cana-1260	262	23	graph	graph	NOUN
cana-1260	262	24	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-1260	262	25	with	with	ADP
cana-1260	262	26	the	the	DET
cana-1260	262	27	cycle	cycle	NOUN
cana-1260	262	28	of	of	ADP
cana-1260	262	29	𝑛	𝑛	PRON
cana-1260	262	30	−	−	NUM
cana-1260	262	31	1	1	NUM
cana-1260	262	32	vertices	vertex	NOUN
cana-1260	262	33	where	where	SCONJ
cana-1260	262	34	each	each	DET
cana-1260	262	35	vertex	vertex	NOUN
cana-1260	262	36	of	of	ADP
cana-1260	262	37	the	the	DET
cana-1260	262	38	cycle	cycle	NOUN
cana-1260	262	39	is	be	AUX
cana-1260	262	40	adjacent	adjacent	ADJ
cana-1260	262	41	to	to	ADP
cana-1260	262	42	two	two	NUM
cana-1260	262	43	vertices	vertex	NOUN
cana-1260	262	44	of	of	ADP
cana-1260	262	45	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-1260	262	46	and	and	CCONJ
cana-1260	262	47	each	each	DET
cana-1260	262	48	vertex	vertex	NOUN
cana-1260	262	49	in	in	ADP
cana-1260	262	50	cycle	cycle	NOUN
cana-1260	262	51	having	have	VERB
cana-1260	262	52	degree	degree	NOUN
cana-1260	262	53	four	four	NUM
cana-1260	262	54	.	.	PUNCT
cana-1260	263	1	if	if	SCONJ
cana-1260	263	2	we	we	PRON
cana-1260	263	3	assign	assign	VERB
cana-1260	263	4	the	the	DET
cana-1260	263	5	weights	weight	NOUN
cana-1260	263	6	(	(	PUNCT
cana-1260	263	7	½	½	NOUN
cana-1260	263	8	)	)	PUNCT
cana-1260	263	9	for	for	ADP
cana-1260	263	10	each	each	DET
cana-1260	263	11	vertex	vertex	NOUN
cana-1260	263	12	of	of	ADP
cana-1260	263	13	complete	complete	ADJ
cana-1260	263	14	graph	graph	NOUN
cana-1260	263	15	𝐾𝑛−1	𝐾𝑛−1	PROPN
cana-1260	263	16	and	and	CCONJ
cana-1260	263	17	each	each	DET
cana-1260	263	18	vertex	vertex	NOUN
cana-1260	263	19	in	in	ADP
cana-1260	263	20	cycle	cycle	NOUN
cana-1260	263	21	having	have	VERB
cana-1260	263	22	weight	weight	NOUN
cana-1260	263	23	zero	zero	NUM
cana-1260	263	24	will	will	AUX
cana-1260	263	25	get	get	VERB
cana-1260	263	26	the	the	DET
cana-1260	263	27	result	result	NOUN
cana-1260	263	28	г𝑓	г𝑓	ADP
cana-1260	263	29	(	(	PUNCT
cana-1260	263	30	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	263	31	)	)	PUNCT
cana-1260	263	32	)	)	PUNCT
cana-1260	264	1	≤	≤	NUM
cana-1260	265	1	𝑛−1	𝑛−1	NUM
cana-1260	265	2	2	2	NUM
cana-1260	265	3	and	and	CCONJ
cana-1260	265	4	hence	hence	ADV
cana-1260	265	5	result	result	VERB
cana-1260	265	6	for	for	ADP
cana-1260	265	7	the	the	DET
cana-1260	265	8	sum	sum	NOUN
cana-1260	265	9	г𝑓	г𝑓	PROPN
cana-1260	265	10	(	(	PUNCT
cana-1260	265	11	𝐺	𝐺	NOUN
cana-1260	265	12	)	)	PUNCT
cana-1260	265	13	+	+	NUM
cana-1260	265	14	г𝑓	г𝑓	ADJ
cana-1260	265	15	(	(	PUNCT
cana-1260	265	16	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	265	17	)	)	PUNCT
cana-1260	265	18	)	)	PUNCT
cana-1260	265	19	≤	≤	NUM
cana-1260	265	20	2	2	NUM
cana-1260	265	21	+	+	CCONJ
cana-1260	265	22	𝑛−1	𝑛−1	NUM
cana-1260	265	23	2	2	NUM
cana-1260	265	24	.	.	PUNCT
cana-1260	266	1	communications	communication	NOUN
cana-1260	266	2	on	on	ADP
cana-1260	266	3	applied	apply	VERB
cana-1260	266	4	nonlinear	nonlinear	ADJ
cana-1260	266	5	analysis	analysis	NOUN
cana-1260	266	6	issn	issn	NOUN
cana-1260	266	7	:	:	PUNCT
cana-1260	266	8	1074	1074	NUM
cana-1260	266	9	-	-	PUNCT
cana-1260	266	10	133x	133x	NUM
cana-1260	266	11	vol	vol	NOUN
cana-1260	266	12	31	31	NUM
cana-1260	266	13	no	no	NOUN
cana-1260	266	14	.	.	PUNCT
cana-1260	267	1	6s	6s	NUM
cana-1260	267	2	(	(	PUNCT
cana-1260	267	3	2024	2024	NUM
cana-1260	267	4	)	)	PUNCT
cana-1260	267	5	680	680	NUM
cana-1260	267	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1260	267	7	2.6	2.6	NUM
cana-1260	267	8	cubic	cubic	ADJ
cana-1260	267	9	graph	graph	NOUN
cana-1260	267	10	the	the	DET
cana-1260	267	11	regular	regular	ADJ
cana-1260	267	12	graph	graph	NOUN
cana-1260	267	13	with	with	ADP
cana-1260	267	14	degree	degree	NOUN
cana-1260	267	15	three	three	NUM
cana-1260	267	16	is	be	AUX
cana-1260	267	17	known	know	VERB
cana-1260	267	18	as	as	ADP
cana-1260	267	19	cubic	cubic	ADJ
cana-1260	267	20	graph	graph	NOUN
cana-1260	267	21	.	.	PUNCT
cana-1260	268	1	trivalent	trivalent	NOUN
cana-1260	268	2	graphs	graph	NOUN
cana-1260	268	3	are	be	AUX
cana-1260	268	4	another	another	DET
cana-1260	268	5	name	name	NOUN
cana-1260	268	6	for	for	ADP
cana-1260	268	7	cubic	cubic	ADJ
cana-1260	268	8	graphs	graph	NOUN
cana-1260	268	9	.	.	PUNCT
cana-1260	269	1	the	the	DET
cana-1260	269	2	cubic	cubic	ADJ
cana-1260	269	3	graphs	graph	NOUN
cana-1260	269	4	with	with	ADP
cana-1260	269	5	𝑛	𝑛	DET
cana-1260	269	6	vertices	vertex	NOUN
cana-1260	269	7	exist	exist	VERB
cana-1260	269	8	only	only	ADV
cana-1260	269	9	for	for	ADP
cana-1260	269	10	even	even	ADV
cana-1260	269	11	𝑛	𝑛	ADJ
cana-1260	269	12	whereas	whereas	SCONJ
cana-1260	269	13	the	the	DET
cana-1260	269	14	condition	condition	NOUN
cana-1260	269	15	for	for	ADP
cana-1260	269	16	a	a	DET
cana-1260	269	17	graph	graph	NOUN
cana-1260	269	18	is	be	AUX
cana-1260	269	19	to	to	PART
cana-1260	269	20	be	be	AUX
cana-1260	269	21	cubic	cubic	ADJ
cana-1260	269	22	is	be	AUX
cana-1260	269	23	given	give	VERB
cana-1260	269	24	by	by	ADP
cana-1260	269	25	𝑚/𝑛	𝑚/𝑛	ADJ
cana-1260	269	26	=	=	SYM
cana-1260	269	27	3/2	3/2	NUM
cana-1260	269	28	,	,	PUNCT
cana-1260	269	29	with	with	ADP
cana-1260	269	30	m	m	PROPN
cana-1260	269	31	as	as	ADP
cana-1260	269	32	size	size	NOUN
cana-1260	269	33	of	of	ADP
cana-1260	269	34	graph	graph	NOUN
cana-1260	269	35	and	and	CCONJ
cana-1260	269	36	𝑛	𝑛	PRON
cana-1260	269	37	be	be	AUX
cana-1260	269	38	the	the	DET
cana-1260	269	39	order	order	NOUN
cana-1260	269	40	of	of	ADP
cana-1260	269	41	graph	graph	NOUN
cana-1260	269	42	.	.	PUNCT
cana-1260	270	1	theorem	theorem	VERB
cana-1260	270	2	2.6.1	2.6.1	NUM
cana-1260	270	3	let	let	VERB
cana-1260	270	4	graph	graph	NOUN
cana-1260	270	5	𝐺	𝐺	PROPN
cana-1260	270	6	is	be	AUX
cana-1260	270	7	the	the	DET
cana-1260	270	8	connected	connected	ADJ
cana-1260	270	9	cubic	cubic	ADJ
cana-1260	270	10	graph	graph	NOUN
cana-1260	270	11	or	or	CCONJ
cana-1260	270	12	trivalent	trivalent	NOUN
cana-1260	270	13	graph	graph	NOUN
cana-1260	270	14	with	with	ADP
cana-1260	270	15	𝑛	𝑛	PROPN
cana-1260	270	16	vertices	vertex	NOUN
cana-1260	270	17	and	and	CCONJ
cana-1260	270	18	𝑚	𝑚	ADP
cana-1260	270	19	edges	edge	NOUN
cana-1260	270	20	and	and	CCONJ
cana-1260	270	21	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	270	22	)	)	PUNCT
cana-1260	270	23	be	be	VERB
cana-1260	270	24	its	its	PRON
cana-1260	270	25	line	line	NOUN
cana-1260	270	26	graph	graph	NOUN
cana-1260	270	27	with	with	ADP
cana-1260	270	28	order	order	NOUN
cana-1260	270	29	𝑚	𝑚	X
cana-1260	270	30	then	then	ADV
cana-1260	270	31	i	i	PRON
cana-1260	270	32	)	)	PUNCT
cana-1260	271	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	271	2	(	(	PUNCT
cana-1260	271	3	𝐺	𝐺	NOUN
cana-1260	271	4	)	)	PUNCT
cana-1260	272	1	+	+	CCONJ
cana-1260	272	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	272	3	(	(	PUNCT
cana-1260	272	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	272	5	)	)	PUNCT
cana-1260	272	6	)	)	PUNCT
cana-1260	272	7	=	=	PUNCT
cana-1260	273	1	5𝑛+4𝑚	5𝑛+4𝑚	NUM
cana-1260	273	2	20	20	NUM
cana-1260	273	3	ii	ii	NOUN
cana-1260	273	4	)	)	PUNCT
cana-1260	273	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	273	6	(	(	PUNCT
cana-1260	273	7	𝐺	𝐺	NOUN
cana-1260	273	8	)	)	PUNCT
cana-1260	273	9	∗	∗	NOUN
cana-1260	273	10	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	273	11	(	(	PUNCT
cana-1260	273	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	273	13	)	)	PUNCT
cana-1260	273	14	)	)	PUNCT
cana-1260	274	1	=	=	PUNCT
cana-1260	274	2	𝑚𝑛	𝑚𝑛	NUM
cana-1260	274	3	20	20	NUM
cana-1260	274	4	iii	iii	NOUN
cana-1260	274	5	)	)	PUNCT
cana-1260	274	6	г𝑓	г𝑓	PROPN
cana-1260	274	7	(	(	PUNCT
cana-1260	274	8	𝐺	𝐺	NOUN
cana-1260	274	9	)	)	PUNCT
cana-1260	274	10	+	+	NUM
cana-1260	274	11	г𝑓	г𝑓	ADJ
cana-1260	274	12	(	(	PUNCT
cana-1260	274	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	274	14	)	)	PUNCT
cana-1260	274	15	)	)	PUNCT
cana-1260	274	16	≤	≤	NUM
cana-1260	275	1	4𝑛+3𝑚	4𝑛+3𝑚	NUM
cana-1260	275	2	12	12	NUM
cana-1260	275	3	iv	iv	NOUN
cana-1260	275	4	)	)	PUNCT
cana-1260	275	5	г𝑓	г𝑓	PROPN
cana-1260	275	6	(	(	PUNCT
cana-1260	275	7	𝐺	𝐺	NOUN
cana-1260	275	8	)	)	PUNCT
cana-1260	275	9	∗	∗	NOUN
cana-1260	275	10	г𝑓	г𝑓	PROPN
cana-1260	275	11	(	(	PUNCT
cana-1260	275	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	275	13	)	)	PUNCT
cana-1260	275	14	)	)	PUNCT
cana-1260	275	15	≤	≤	NUM
cana-1260	276	1	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	276	2	12	12	NUM
cana-1260	276	3	proof	proof	NOUN
cana-1260	276	4	i	i	PRON
cana-1260	276	5	)	)	PUNCT
cana-1260	276	6	and	and	CCONJ
cana-1260	276	7	ii	ii	PROPN
cana-1260	276	8	):	):	PUNCT
cana-1260	276	9	a	a	DET
cana-1260	276	10	graph	graph	NOUN
cana-1260	276	11	with	with	ADP
cana-1260	276	12	degree	degree	NOUN
cana-1260	276	13	three	three	NUM
cana-1260	276	14	at	at	ADP
cana-1260	276	15	every	every	DET
cana-1260	276	16	vertex	vertex	NOUN
cana-1260	276	17	is	be	AUX
cana-1260	276	18	called	call	VERB
cana-1260	276	19	a	a	DET
cana-1260	276	20	cubic	cubic	ADJ
cana-1260	276	21	graph	graph	NOUN
cana-1260	276	22	or	or	CCONJ
cana-1260	276	23	3	3	NUM
cana-1260	276	24	-	-	PUNCT
cana-1260	276	25	regular	regular	ADJ
cana-1260	276	26	graph	graph	NOUN
cana-1260	276	27	.	.	PUNCT
cana-1260	277	1	cubic	cubic	ADJ
cana-1260	277	2	graph	graph	NOUN
cana-1260	277	3	are	be	AUX
cana-1260	277	4	also	also	ADV
cana-1260	277	5	called	call	VERB
cana-1260	277	6	trivalent	trivalent	NOUN
cana-1260	277	7	graph	graph	NOUN
cana-1260	277	8	.	.	PUNCT
cana-1260	278	1	cubic	cubic	ADJ
cana-1260	278	2	graphs	graph	NOUN
cana-1260	278	3	on	on	ADP
cana-1260	278	4	𝑛	𝑛	DET
cana-1260	278	5	vertices	vertex	NOUN
cana-1260	278	6	exist	exist	VERB
cana-1260	278	7	only	only	ADV
cana-1260	278	8	for	for	ADP
cana-1260	278	9	even	even	ADV
cana-1260	278	10	𝑛.	𝑛.	NOUN
cana-1260	278	11	for	for	ADP
cana-1260	278	12	example	example	NOUN
cana-1260	278	13	if	if	SCONJ
cana-1260	278	14	𝐺	𝐺	PROPN
cana-1260	278	15	be	be	VERB
cana-1260	278	16	the	the	DET
cana-1260	278	17	cubic	cubic	ADJ
cana-1260	278	18	graph	graph	NOUN
cana-1260	278	19	with	with	ADP
cana-1260	278	20	𝑛	𝑛	NOUN
cana-1260	278	21	=	=	SYM
cana-1260	278	22	6	6	NUM
cana-1260	278	23	vertices	vertex	NOUN
cana-1260	278	24	or	or	CCONJ
cana-1260	278	25	3	3	NUM
cana-1260	278	26	-	-	PUNCT
cana-1260	278	27	regular	regular	ADJ
cana-1260	278	28	graph	graph	NOUN
cana-1260	278	29	with	with	ADP
cana-1260	278	30	size	size	NOUN
cana-1260	278	31	𝑚	𝑚	NOUN
cana-1260	278	32	=	=	SYM
cana-1260	278	33	9	9	NUM
cana-1260	278	34	then	then	ADV
cana-1260	278	35	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	278	36	(	(	PUNCT
cana-1260	278	37	𝐺	𝐺	NOUN
cana-1260	278	38	)	)	PUNCT
cana-1260	278	39	=	=	SYM
cana-1260	278	40	𝑛	𝑛	PRON
cana-1260	278	41	3	3	NUM
cana-1260	278	42	+	+	NOUN
cana-1260	278	43	1	1	NUM
cana-1260	278	44	=	=	SYM
cana-1260	278	45	6/4	6/4	NUM
cana-1260	278	46	by	by	ADP
cana-1260	278	47	the	the	DET
cana-1260	278	48	definition	definition	NOUN
cana-1260	278	49	of	of	ADP
cana-1260	278	50	fractional	fractional	ADJ
cana-1260	278	51	domination	domination	NOUN
cana-1260	278	52	number	number	NOUN
cana-1260	278	53	(	(	PUNCT
cana-1260	278	54	see	see	VERB
cana-1260	278	55	the	the	DET
cana-1260	278	56	figure	figure	NOUN
cana-1260	278	57	2.6.1a	2.6.1a	NUM
cana-1260	278	58	)	)	PUNCT
cana-1260	278	59	.	.	PUNCT
cana-1260	279	1	its	its	PRON
cana-1260	279	2	line	line	NOUN
cana-1260	279	3	graph	graph	NOUN
cana-1260	279	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	279	5	)	)	PUNCT
cana-1260	279	6	has	have	VERB
cana-1260	279	7	𝑚	𝑚	NOUN
cana-1260	279	8	=	=	SYM
cana-1260	279	9	9	9	NUM
cana-1260	279	10	vertices	vertex	NOUN
cana-1260	279	11	as	as	ADP
cana-1260	279	12	4	4	NUM
cana-1260	279	13	-	-	PUNCT
cana-1260	279	14	regular	regular	ADJ
cana-1260	279	15	graph	graph	NOUN
cana-1260	279	16	so	so	SCONJ
cana-1260	279	17	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	279	18	(	(	PUNCT
cana-1260	279	19	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	279	20	)	)	PUNCT
cana-1260	279	21	)	)	PUNCT
cana-1260	280	1	=	=	PUNCT
cana-1260	280	2	𝑚	𝑚	ADP
cana-1260	280	3	4	4	NUM
cana-1260	280	4	+	+	SYM
cana-1260	280	5	1	1	NUM
cana-1260	280	6	=	=	SYM
cana-1260	280	7	9/5	9/5	NUM
cana-1260	280	8	(	(	PUNCT
cana-1260	280	9	see	see	VERB
cana-1260	280	10	the	the	DET
cana-1260	280	11	figure	figure	NOUN
cana-1260	280	12	2.6.1b	2.6.1b	PROPN
cana-1260	280	13	)	)	PUNCT
cana-1260	280	14	.	.	PUNCT
cana-1260	281	1	the	the	DET
cana-1260	281	2	cubic	cubic	ADJ
cana-1260	281	3	graph	graph	NOUN
cana-1260	281	4	with	with	ADP
cana-1260	281	5	𝑛	𝑛	PROPN
cana-1260	281	6	=	=	SYM
cana-1260	281	7	4	4	NUM
cana-1260	281	8	called	call	VERB
cana-1260	281	9	tetrahedral	tetrahedral	ADJ
cana-1260	281	10	graph	graph	NOUN
cana-1260	281	11	,	,	PUNCT
cana-1260	281	12	the	the	DET
cana-1260	281	13	cubic	cubic	ADJ
cana-1260	281	14	graph	graph	NOUN
cana-1260	281	15	with	with	ADP
cana-1260	281	16	𝑛	𝑛	PROPN
cana-1260	281	17	=	=	SYM
cana-1260	281	18	8	8	NUM
cana-1260	281	19	called	call	VERB
cana-1260	281	20	cubical	cubical	ADJ
cana-1260	281	21	graph	graph	NOUN
cana-1260	281	22	,	,	PUNCT
cana-1260	281	23	for	for	ADP
cana-1260	281	24	all	all	DET
cana-1260	281	25	such	such	ADJ
cana-1260	281	26	connected	connected	ADJ
cana-1260	281	27	cubic	cubic	ADJ
cana-1260	281	28	graphs	graph	NOUN
cana-1260	281	29	which	which	PRON
cana-1260	281	30	are	be	AUX
cana-1260	281	31	3	3	NUM
cana-1260	281	32	-	-	PUNCT
cana-1260	281	33	regular	regular	ADJ
cana-1260	281	34	graphs	graph	NOUN
cana-1260	281	35	of	of	ADP
cana-1260	281	36	order	order	NOUN
cana-1260	281	37	𝑛	𝑛	VERB
cana-1260	281	38	=	=	PUNCT
cana-1260	281	39	{	{	PUNCT
cana-1260	281	40	4,6,8	4,6,8	NUM
cana-1260	281	41	,	,	PUNCT
cana-1260	281	42	…	…	PUNCT
cana-1260	281	43	}	}	PUNCT
cana-1260	281	44	we	we	PRON
cana-1260	281	45	get	get	VERB
cana-1260	281	46	the	the	DET
cana-1260	281	47	generalized	generalized	ADJ
cana-1260	281	48	result	result	NOUN
cana-1260	281	49	as	as	ADP
cana-1260	281	50	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	281	51	(	(	PUNCT
cana-1260	281	52	𝐺	𝐺	NOUN
cana-1260	281	53	)	)	PUNCT
cana-1260	281	54	=	=	SYM
cana-1260	282	1	𝑛	𝑛	PRON
cana-1260	282	2	𝑟+1	𝑟+1	NUM
cana-1260	282	3	where	where	SCONJ
cana-1260	282	4	𝑟	𝑟	AUX
cana-1260	282	5	=	=	SYM
cana-1260	282	6	3	3	X
cana-1260	282	7	.	.	NOUN
cana-1260	282	8	for	for	ADP
cana-1260	282	9	its	its	PRON
cana-1260	282	10	line	line	NOUN
cana-1260	282	11	graph	graph	NOUN
cana-1260	282	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	282	13	)	)	PUNCT
cana-1260	282	14	of	of	ADP
cana-1260	282	15	order	order	NOUN
cana-1260	282	16	𝑚	𝑚	NOUN
cana-1260	282	17	with	with	ADP
cana-1260	282	18	(	(	PUNCT
cana-1260	282	19	𝑟	𝑟	X
cana-1260	282	20	+	+	NOUN
cana-1260	282	21	1	1	X
cana-1260	282	22	)	)	PUNCT
cana-1260	282	23	regular	regular	ADJ
cana-1260	282	24	graph	graph	NOUN
cana-1260	282	25	where	where	SCONJ
cana-1260	282	26	𝑟	𝑟	NOUN
cana-1260	282	27	=	=	SYM
cana-1260	282	28	3	3	NUM
cana-1260	282	29	,	,	PUNCT
cana-1260	282	30	we	we	PRON
cana-1260	282	31	have	have	VERB
cana-1260	282	32	𝛾𝑓	𝛾𝑓	VERB
cana-1260	282	33	(	(	PUNCT
cana-1260	282	34	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	282	35	)	)	PUNCT
cana-1260	282	36	)	)	PUNCT
cana-1260	283	1	=	=	PUNCT
cana-1260	283	2	𝑚	𝑚	X
cana-1260	283	3	𝑟+2	𝑟+2	NUM
cana-1260	283	4	so	so	ADV
cana-1260	283	5	for	for	ADP
cana-1260	283	6	the	the	DET
cana-1260	283	7	sum	sum	NOUN
cana-1260	283	8	we	we	PRON
cana-1260	283	9	get	get	VERB
cana-1260	283	10	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	283	11	(	(	PUNCT
cana-1260	283	12	𝐺	𝐺	NOUN
cana-1260	283	13	)	)	PUNCT
cana-1260	284	1	+	+	CCONJ
cana-1260	284	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	284	3	(	(	PUNCT
cana-1260	284	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	284	5	)	)	PUNCT
cana-1260	284	6	)	)	PUNCT
cana-1260	285	1	=	=	PUNCT
cana-1260	285	2	𝑛	𝑛	PRON
cana-1260	285	3	𝑟+1	𝑟+1	NUM
cana-1260	285	4	+	+	ADP
cana-1260	285	5	𝑚	𝑚	X
cana-1260	285	6	𝑟+2	𝑟+2	NUM
cana-1260	285	7	if	if	SCONJ
cana-1260	285	8	𝑟	𝑟	NOUN
cana-1260	285	9	=	=	SYM
cana-1260	285	10	3	3	NUM
cana-1260	285	11	then	then	ADV
cana-1260	285	12	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	285	13	(	(	PUNCT
cana-1260	285	14	𝐺	𝐺	NOUN
cana-1260	285	15	)	)	PUNCT
cana-1260	286	1	+	+	CCONJ
cana-1260	286	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	286	3	(	(	PUNCT
cana-1260	286	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	286	5	)	)	PUNCT
cana-1260	286	6	)	)	PUNCT
cana-1260	287	1	=	=	SYM
cana-1260	287	2	𝑛	𝑛	PRON
cana-1260	287	3	4	4	NUM
cana-1260	287	4	+	+	CCONJ
cana-1260	287	5	𝑚	𝑚	PROPN
cana-1260	287	6	5	5	NUM
cana-1260	287	7	=	=	SYM
cana-1260	287	8	5𝑛+4𝑚	5𝑛+4𝑚	NUM
cana-1260	287	9	20	20	NUM
cana-1260	287	10	and	and	CCONJ
cana-1260	287	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	287	12	(	(	PUNCT
cana-1260	287	13	𝐺	𝐺	NOUN
cana-1260	287	14	)	)	PUNCT
cana-1260	287	15	∗	∗	NOUN
cana-1260	287	16	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	287	17	(	(	PUNCT
cana-1260	287	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	287	19	)	)	PUNCT
cana-1260	287	20	)	)	PUNCT
cana-1260	288	1	=	=	SYM
cana-1260	288	2	𝑛	𝑛	PRON
cana-1260	288	3	4	4	NUM
cana-1260	288	4	∗	∗	NOUN
cana-1260	288	5	𝑚	𝑚	ADP
cana-1260	288	6	5	5	NUM
cana-1260	288	7	=	=	SYM
cana-1260	288	8	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	288	9	20	20	NUM
cana-1260	288	10	.	.	PUNCT
cana-1260	289	1	(	(	PUNCT
cana-1260	289	2	a	a	X
cana-1260	289	3	)	)	PUNCT
cana-1260	289	4	cubic	cubic	ADJ
cana-1260	289	5	graph	graph	NOUN
cana-1260	289	6	of	of	ADP
cana-1260	289	7	6	6	NUM
cana-1260	289	8	vertices	vertex	NOUN
cana-1260	289	9	(	(	PUNCT
cana-1260	289	10	b	b	NOUN
cana-1260	289	11	)	)	PUNCT
cana-1260	289	12	line	line	NOUN
cana-1260	289	13	graph	graph	NOUN
cana-1260	289	14	of	of	ADP
cana-1260	289	15	3	3	NUM
cana-1260	289	16	-	-	PUNCT
cana-1260	289	17	prism	prism	NOUN
cana-1260	289	18	graph	graph	NOUN
cana-1260	289	19	figure	figure	NOUN
cana-1260	289	20	2.6.1	2.6.1	NUM
cana-1260	289	21	communications	communication	NOUN
cana-1260	289	22	on	on	ADP
cana-1260	289	23	applied	apply	VERB
cana-1260	289	24	nonlinear	nonlinear	ADJ
cana-1260	289	25	analysis	analysis	NOUN
cana-1260	289	26	issn	issn	NOUN
cana-1260	289	27	:	:	PUNCT
cana-1260	289	28	1074	1074	NUM
cana-1260	289	29	-	-	PUNCT
cana-1260	289	30	133x	133x	NUM
cana-1260	289	31	vol	vol	NOUN
cana-1260	289	32	31	31	NUM
cana-1260	289	33	no	no	NOUN
cana-1260	289	34	.	.	PUNCT
cana-1260	290	1	6s	6s	NUM
cana-1260	290	2	(	(	PUNCT
cana-1260	290	3	2024	2024	NUM
cana-1260	290	4	)	)	PUNCT
cana-1260	290	5	681	681	NUM
cana-1260	290	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	290	7	iii	iii	NOUN
cana-1260	290	8	)	)	PUNCT
cana-1260	290	9	and	and	CCONJ
cana-1260	290	10	iv	iv	X
cana-1260	290	11	)	)	PUNCT
cana-1260	290	12	for	for	ADP
cana-1260	290	13	upper	upper	ADJ
cana-1260	290	14	fractional	fractional	ADJ
cana-1260	290	15	domination	domination	NOUN
cana-1260	290	16	number	number	NOUN
cana-1260	290	17	we	we	PRON
cana-1260	290	18	used	use	VERB
cana-1260	290	19	result	result	NOUN
cana-1260	290	20	2.1.2	2.1.2	NUM
cana-1260	290	21	let	let	VERB
cana-1260	290	22	𝐺	𝐺	PROPN
cana-1260	290	23	is	be	AUX
cana-1260	290	24	𝑟-regular	𝑟-regular	ADJ
cana-1260	290	25	graph	graph	NOUN
cana-1260	290	26	of	of	ADP
cana-1260	290	27	n	n	NOUN
cana-1260	290	28	vertices	vertex	NOUN
cana-1260	290	29	and	and	CCONJ
cana-1260	290	30	m	m	PRON
cana-1260	290	31	edges	edge	NOUN
cana-1260	290	32	then	then	ADV
cana-1260	290	33	г𝑓	г𝑓	X
cana-1260	290	34	(	(	PUNCT
cana-1260	290	35	𝐺	𝐺	NOUN
cana-1260	290	36	)	)	PUNCT
cana-1260	290	37	≤	≤	NOUN
cana-1260	290	38	𝑛/𝑟.	𝑛/𝑟.	NOUN
cana-1260	290	39	so	so	SCONJ
cana-1260	290	40	the	the	DET
cana-1260	290	41	generalized	generalized	ADJ
cana-1260	290	42	result	result	NOUN
cana-1260	290	43	for	for	ADP
cana-1260	290	44	connected	connected	ADJ
cana-1260	290	45	cubic	cubic	ADJ
cana-1260	290	46	graphs	graph	NOUN
cana-1260	290	47	with	with	ADP
cana-1260	290	48	degree	degree	NOUN
cana-1260	290	49	three	three	NUM
cana-1260	290	50	at	at	ADP
cana-1260	290	51	every	every	DET
cana-1260	290	52	vertex	vertex	NOUN
cana-1260	290	53	of	of	ADP
cana-1260	290	54	order	order	NOUN
cana-1260	290	55	𝑛	𝑛	VERB
cana-1260	290	56	=	=	PUNCT
cana-1260	290	57	{	{	PUNCT
cana-1260	290	58	4,6,8	4,6,8	NUM
cana-1260	290	59	,	,	PUNCT
cana-1260	290	60	…	…	PUNCT
cana-1260	290	61	}	}	PUNCT
cana-1260	290	62	and	and	CCONJ
cana-1260	290	63	size	size	NOUN
cana-1260	290	64	𝑚	𝑚	PROPN
cana-1260	290	65	for	for	ADP
cana-1260	290	66	𝑟	𝑟	NOUN
cana-1260	290	67	=	=	SYM
cana-1260	290	68	3	3	NUM
cana-1260	290	69	,	,	PUNCT
cana-1260	290	70	we	we	PRON
cana-1260	290	71	get	get	VERB
cana-1260	290	72	г𝑓	г𝑓	DET
cana-1260	290	73	(	(	PUNCT
cana-1260	290	74	𝐺	𝐺	NOUN
cana-1260	290	75	)	)	PUNCT
cana-1260	290	76	≤	≤	NUM
cana-1260	290	77	𝑛	𝑛	DET
cana-1260	290	78	3	3	NUM
cana-1260	290	79	and	and	CCONJ
cana-1260	290	80	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	290	81	)	)	PUNCT
cana-1260	290	82	is	be	AUX
cana-1260	290	83	(	(	PUNCT
cana-1260	290	84	𝑟	𝑟	X
cana-1260	290	85	+	+	CCONJ
cana-1260	290	86	1	1	X
cana-1260	290	87	)	)	PUNCT
cana-1260	290	88	regular	regular	ADJ
cana-1260	290	89	graph	graph	NOUN
cana-1260	290	90	so	so	SCONJ
cana-1260	290	91	г𝑓	г𝑓	ADJ
cana-1260	290	92	(	(	PUNCT
cana-1260	290	93	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	290	94	)	)	PUNCT
cana-1260	290	95	)	)	PUNCT
cana-1260	291	1	≤	≤	NOUN
cana-1260	292	1	𝑚	𝑚	X
cana-1260	292	2	𝑟+1	𝑟+1	PROPN
cana-1260	292	3	we	we	PRON
cana-1260	292	4	get	get	VERB
cana-1260	292	5	г𝑓	г𝑓	PRON
cana-1260	292	6	(	(	PUNCT
cana-1260	292	7	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	292	8	)	)	PUNCT
cana-1260	292	9	)	)	PUNCT
cana-1260	293	1	≤	≤	NOUN
cana-1260	293	2	𝑚	𝑚	ADP
cana-1260	293	3	4	4	NUM
cana-1260	293	4	.	.	PUNCT
cana-1260	294	1	so	so	ADV
cana-1260	294	2	sum	sum	VERB
cana-1260	294	3	г𝑓	г𝑓	PROPN
cana-1260	294	4	(	(	PUNCT
cana-1260	294	5	𝐺	𝐺	NOUN
cana-1260	294	6	)	)	PUNCT
cana-1260	295	1	+	+	NUM
cana-1260	295	2	г𝑓	г𝑓	ADJ
cana-1260	295	3	(	(	PUNCT
cana-1260	295	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	295	5	)	)	PUNCT
cana-1260	295	6	)	)	PUNCT
cana-1260	295	7	≤	≤	NUM
cana-1260	295	8	𝑛	𝑛	DET
cana-1260	295	9	3	3	NUM
cana-1260	295	10	+	+	CCONJ
cana-1260	295	11	𝑚	𝑚	PROPN
cana-1260	295	12	4	4	NUM
cana-1260	295	13	=	=	SYM
cana-1260	295	14	4𝑛+3𝑚	4𝑛+3𝑚	NUM
cana-1260	295	15	12	12	NUM
cana-1260	296	1	and	and	CCONJ
cana-1260	296	2	г𝑓	г𝑓	ADJ
cana-1260	296	3	(	(	PUNCT
cana-1260	296	4	𝐺	𝐺	NOUN
cana-1260	296	5	)	)	PUNCT
cana-1260	297	1	∗	∗	NOUN
cana-1260	297	2	г𝑓	г𝑓	PROPN
cana-1260	297	3	(	(	PUNCT
cana-1260	297	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	297	5	)	)	PUNCT
cana-1260	297	6	)	)	PUNCT
cana-1260	298	1	≤	≤	NUM
cana-1260	298	2	𝑛	𝑛	DET
cana-1260	298	3	3	3	NUM
cana-1260	298	4	∗	∗	NOUN
cana-1260	298	5	𝑚	𝑚	ADP
cana-1260	298	6	4	4	NUM
cana-1260	298	7	=	=	SYM
cana-1260	298	8	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	298	9	12	12	NUM
cana-1260	298	10	.	.	PUNCT
cana-1260	299	1	□	□	PUNCT
cana-1260	299	2	2.7	2.7	NUM
cana-1260	299	3	cartesian	cartesian	ADJ
cana-1260	299	4	product	product	NOUN
cana-1260	299	5	of	of	ADP
cana-1260	299	6	graph	graph	NOUN
cana-1260	299	7	(	(	PUNCT
cana-1260	299	8	𝑲𝟐	𝑲𝟐	NOUN
cana-1260	299	9	×	×	NOUN
cana-1260	299	10	𝑷𝒏	𝑷𝒏	PROPN
cana-1260	299	11	)	)	PUNCT
cana-1260	299	12	if	if	SCONJ
cana-1260	299	13	we	we	PRON
cana-1260	299	14	take	take	VERB
cana-1260	299	15	path	path	NOUN
cana-1260	299	16	graph	graph	NOUN
cana-1260	299	17	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	299	18	and	and	CCONJ
cana-1260	299	19	complete	complete	ADJ
cana-1260	299	20	graph	graph	NOUN
cana-1260	299	21	𝐾2	𝐾2	NOUN
cana-1260	299	22	then	then	ADV
cana-1260	299	23	cartesian	cartesian	ADJ
cana-1260	299	24	product	product	NOUN
cana-1260	299	25	graph	graph	NOUN
cana-1260	299	26	by	by	ADP
cana-1260	299	27	𝐾2	𝐾2	ADJ
cana-1260	299	28	×	×	PROPN
cana-1260	299	29	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	299	30	.	.	PUNCT
cana-1260	300	1	for	for	ADP
cana-1260	300	2	the	the	DET
cana-1260	300	3	complete	complete	ADJ
cana-1260	300	4	graph	graph	NOUN
cana-1260	300	5	like	like	ADP
cana-1260	300	6	𝐾2	𝐾2	NOUN
cana-1260	300	7	we	we	PRON
cana-1260	300	8	have	have	VERB
cana-1260	300	9	the	the	DET
cana-1260	300	10	set	set	NOUN
cana-1260	300	11	of	of	ADP
cana-1260	300	12	vertices	vertex	NOUN
cana-1260	300	13	𝑉(𝐾2	𝑉(𝐾2	PART
cana-1260	300	14	)	)	PUNCT
cana-1260	300	15	=	=	SYM
cana-1260	300	16	{	{	PUNCT
cana-1260	300	17	𝑘1	𝑘1	PROPN
cana-1260	300	18	,	,	PUNCT
cana-1260	300	19	𝑘2	𝑘2	PROPN
cana-1260	300	20	}	}	PUNCT
cana-1260	300	21	and	and	CCONJ
cana-1260	300	22	the	the	DET
cana-1260	300	23	set	set	NOUN
cana-1260	300	24	of	of	ADP
cana-1260	300	25	vertices	vertex	NOUN
cana-1260	300	26	of	of	ADP
cana-1260	300	27	path	path	NOUN
cana-1260	300	28	graph	graph	NOUN
cana-1260	300	29	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	300	30	is	be	AUX
cana-1260	300	31	{	{	PUNCT
cana-1260	300	32	𝑝1	𝑝1	NOUN
cana-1260	300	33	,	,	PUNCT
cana-1260	300	34	𝑝2,	𝑝2,	NOUN
cana-1260	300	35	…	…	SYM
cana-1260	300	36	,𝑝𝑛	,𝑝𝑛	NOUN
cana-1260	300	37	}	}	PUNCT
cana-1260	300	38	respectively	respectively	ADV
cana-1260	300	39	.	.	PUNCT
cana-1260	301	1	the	the	DET
cana-1260	301	2	set	set	NOUN
cana-1260	301	3	of	of	ADP
cana-1260	301	4	verticesvfor	verticesvfor	ADP
cana-1260	301	5	cartesian	cartesian	ADJ
cana-1260	301	6	product	product	NOUN
cana-1260	301	7	of	of	ADP
cana-1260	301	8	𝐾2	𝐾2	ADJ
cana-1260	301	9	×	×	PROPN
cana-1260	301	10	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	301	11	is	be	AUX
cana-1260	301	12	𝑉	𝑉	PROPN
cana-1260	301	13	(	(	PUNCT
cana-1260	301	14	𝐾2	𝐾2	ADJ
cana-1260	301	15	×	×	PROPN
cana-1260	301	16	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	301	17	)	)	PUNCT
cana-1260	302	1	=	=	SYM
cana-1260	302	2	𝑉	𝑉	PROPN
cana-1260	302	3	(	(	PUNCT
cana-1260	302	4	𝐾2	𝐾2	PROPN
cana-1260	302	5	)	)	PUNCT
cana-1260	302	6	×	×	PROPN
cana-1260	302	7	𝑉	𝑉	PROPN
cana-1260	302	8	(	(	PUNCT
cana-1260	302	9	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	302	10	)	)	PUNCT
cana-1260	302	11	and	and	CCONJ
cana-1260	302	12	𝑉(𝐾2	𝑉(𝐾2	X
cana-1260	302	13	)	)	PUNCT
cana-1260	302	14	×	×	PROPN
cana-1260	302	15	𝑉	𝑉	PROPN
cana-1260	302	16	(	(	PUNCT
cana-1260	302	17	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	302	18	)	)	PUNCT
cana-1260	302	19	=	=	PRON
cana-1260	302	20	{	{	PUNCT
cana-1260	302	21	(	(	PUNCT
cana-1260	302	22	𝑘𝑖	𝑘𝑖	NOUN
cana-1260	302	23	,	,	PUNCT
cana-1260	302	24	𝑝𝑗	𝑝𝑗	NOUN
cana-1260	302	25	)	)	PUNCT
cana-1260	302	26	|	|	ADV
cana-1260	302	27	𝑘𝑖	𝑘𝑖	NOUN
cana-1260	302	28	∈	∈	NOUN
cana-1260	302	29	𝑉(𝐾2	𝑉(𝐾2	NOUN
cana-1260	302	30	)	)	PUNCT
cana-1260	302	31	,	,	PUNCT
cana-1260	302	32	𝑝𝑗	𝑝𝑗	PROPN
cana-1260	302	33	∈	∈	PROPN
cana-1260	302	34	𝑉	𝑉	PROPN
cana-1260	302	35	(	(	PUNCT
cana-1260	302	36	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	302	37	)	)	PUNCT
cana-1260	302	38	}	}	PUNCT
cana-1260	302	39	and	and	CCONJ
cana-1260	302	40	‘	'	PUNCT
cana-1260	302	41	𝑒	𝑒	X
cana-1260	302	42	’	'	PUNCT
cana-1260	302	43	be	be	VERB
cana-1260	302	44	the	the	DET
cana-1260	302	45	edge	edge	NOUN
cana-1260	302	46	for	for	ADP
cana-1260	302	47	cartesian	cartesian	ADJ
cana-1260	302	48	product	product	NOUN
cana-1260	302	49	𝐾2	𝐾2	ADJ
cana-1260	302	50	×	×	NOUN
cana-1260	303	1	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	303	2	iff	iff	VERB
cana-1260	303	3	𝑒	𝑒	ADP
cana-1260	303	4	=	=	X
cana-1260	303	5	<	<	X
cana-1260	303	6	(	(	PUNCT
cana-1260	303	7	𝑘𝑖	𝑘𝑖	NOUN
cana-1260	303	8	,	,	PUNCT
cana-1260	303	9	𝑝𝑗	𝑝𝑗	NOUN
cana-1260	303	10	)	)	PUNCT
cana-1260	303	11	,	,	PUNCT
cana-1260	303	12	(	(	PUNCT
cana-1260	303	13	𝑘𝑟	𝑘𝑟	INTJ
cana-1260	303	14	,	,	PUNCT
cana-1260	303	15	𝑝𝑠	𝑝𝑠	CCONJ
cana-1260	303	16	)	)	PUNCT
cana-1260	303	17	>	>	X
cana-1260	303	18	where	where	SCONJ
cana-1260	303	19	the	the	DET
cana-1260	303	20	condition	condition	NOUN
cana-1260	303	21	holds	hold	VERB
cana-1260	303	22	as	as	ADP
cana-1260	303	23	𝑖	𝑖	NOUN
cana-1260	303	24	)	)	PUNCT
cana-1260	303	25	𝑖	𝑖	NOUN
cana-1260	303	26	=	=	SYM
cana-1260	303	27	𝑟	𝑟	NOUN
cana-1260	303	28	and	and	CCONJ
cana-1260	303	29	𝑝𝑗	𝑝𝑗	PROPN
cana-1260	303	30	𝑝𝑠	𝑝𝑠	ADV
cana-1260	303	31	∈	∈	PROPN
cana-1260	303	32	𝐸𝑑𝑔𝑒	𝐸𝑑𝑔𝑒	PROPN
cana-1260	303	33	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-1260	303	34	𝑜𝑓	𝑜𝑓	X
cana-1260	303	35	(	(	PUNCT
cana-1260	303	36	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	303	37	)	)	PUNCT
cana-1260	303	38	,	,	PUNCT
cana-1260	303	39	𝑖𝑖	𝑖𝑖	X
cana-1260	303	40	)	)	PUNCT
cana-1260	303	41	𝑗	𝑗	NOUN
cana-1260	303	42	=	=	SYM
cana-1260	303	43	𝑠	𝑠	PROPN
cana-1260	303	44	and	and	CCONJ
cana-1260	303	45	𝑝𝑗	𝑝𝑗	NOUN
cana-1260	303	46	𝑝𝑠	𝑝𝑠	ADV
cana-1260	303	47	∈	∈	PROPN
cana-1260	303	48	𝐸𝑑𝑔𝑒	𝐸𝑑𝑔𝑒	PROPN
cana-1260	303	49	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-1260	303	50	𝑜𝑓	𝑜𝑓	X
cana-1260	303	51	(	(	PUNCT
cana-1260	303	52	𝐾2	𝐾2	PROPN
cana-1260	303	53	)	)	PUNCT
cana-1260	303	54	.	.	PUNCT
cana-1260	304	1	we	we	PRON
cana-1260	304	2	need	need	VERB
cana-1260	304	3	the	the	DET
cana-1260	304	4	following	follow	VERB
cana-1260	304	5	theorems	theorem	NOUN
cana-1260	304	6	on	on	ADP
cana-1260	304	7	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	304	8	(	(	PUNCT
cana-1260	304	9	𝐾2	𝐾2	ADJ
cana-1260	304	10	×	×	PROPN
cana-1260	304	11	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	304	12	)	)	PUNCT
cana-1260	304	13	.	.	PUNCT
cana-1260	305	1	theorem	theorem	VERB
cana-1260	305	2	2.7.1	2.7.1	NUM
cana-1260	305	3	(	(	PUNCT
cana-1260	305	4	[	[	X
cana-1260	305	5	10,11	10,11	NOUN
cana-1260	305	6	]	]	PUNCT
cana-1260	305	7	)	)	PUNCT
cana-1260	305	8	if	if	SCONJ
cana-1260	305	9	𝐺	𝐺	PROPN
cana-1260	305	10	is	be	AUX
cana-1260	305	11	the	the	DET
cana-1260	305	12	cartesian	cartesian	ADJ
cana-1260	305	13	product	product	NOUN
cana-1260	305	14	graph	graph	NOUN
cana-1260	305	15	(	(	PUNCT
cana-1260	305	16	𝐾2	𝐾2	ADJ
cana-1260	305	17	×	×	PROPN
cana-1260	305	18	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	305	19	)	)	PUNCT
cana-1260	305	20	for	for	ADP
cana-1260	305	21	𝑛	𝑛	PROPN
cana-1260	305	22	>	>	SYM
cana-1260	305	23	1	1	NUM
cana-1260	305	24	then	then	ADV
cana-1260	305	25	𝛾𝑓	𝛾𝑓	X
cana-1260	305	26	(	(	PUNCT
cana-1260	305	27	𝐾2	𝐾2	ADJ
cana-1260	305	28	×	×	PROPN
cana-1260	305	29	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	305	30	)	)	PUNCT
cana-1260	306	1	=	=	PRON
cana-1260	306	2	{	{	PUNCT
cana-1260	306	3	𝑛	𝑛	PROPN
cana-1260	306	4	+	+	NUM
cana-1260	306	5	1	1	NUM
cana-1260	306	6	2	2	NUM
cana-1260	306	7	,	,	PUNCT
cana-1260	306	8	𝑖𝑓	𝑖𝑓	ADP
cana-1260	306	9	𝑛	𝑛	DET
cana-1260	306	10	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	306	11	𝑜𝑑𝑑	𝑜𝑑𝑑	PROPN
cana-1260	306	12	𝑛2	𝑛2	NOUN
cana-1260	306	13	+	+	CCONJ
cana-1260	306	14	2𝑛	2𝑛	PROPN
cana-1260	306	15	2(𝑛	2(𝑛	NUM
cana-1260	306	16	+	+	CCONJ
cana-1260	306	17	1	1	NUM
cana-1260	306	18	)	)	PUNCT
cana-1260	306	19	,	,	PUNCT
cana-1260	306	20	𝑖𝑓	𝑖𝑓	ADP
cana-1260	306	21	𝑛	𝑛	DET
cana-1260	306	22	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	306	23	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-1260	306	24	theorem	theorem	VERB
cana-1260	306	25	2.7.2	2.7.2	NUM
cana-1260	306	26	if	if	SCONJ
cana-1260	306	27	𝐺	𝐺	PROPN
cana-1260	306	28	is	be	AUX
cana-1260	306	29	the	the	DET
cana-1260	306	30	cartesian	cartesian	ADJ
cana-1260	306	31	product	product	NOUN
cana-1260	306	32	(	(	PUNCT
cana-1260	306	33	𝐾2	𝐾2	ADJ
cana-1260	306	34	×	×	PROPN
cana-1260	306	35	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	306	36	)	)	PUNCT
cana-1260	306	37	of	of	ADP
cana-1260	306	38	order	order	NOUN
cana-1260	306	39	2𝑛	2𝑛	NUM
cana-1260	306	40	for	for	ADP
cana-1260	306	41	𝑛	𝑛	ADJ
cana-1260	306	42	>	>	SYM
cana-1260	306	43	1	1	NUM
cana-1260	306	44	and	and	CCONJ
cana-1260	306	45	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	306	46	)	)	PUNCT
cana-1260	306	47	be	be	VERB
cana-1260	306	48	its	its	PRON
cana-1260	306	49	line	line	NOUN
cana-1260	306	50	graph	graph	NOUN
cana-1260	306	51	of	of	ADP
cana-1260	306	52	order	order	NOUN
cana-1260	306	53	3𝑛	3𝑛	NUM
cana-1260	306	54	−	−	PROPN
cana-1260	306	55	2	2	NUM
cana-1260	306	56	then	then	ADV
cana-1260	306	57	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	306	58	(	(	PUNCT
cana-1260	306	59	𝐿(𝐾2	𝐿(𝐾2	PRON
cana-1260	306	60	×	×	NOUN
cana-1260	306	61	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	306	62	)	)	PUNCT
cana-1260	306	63	)	)	PUNCT
cana-1260	307	1	=	=	SYM
cana-1260	307	2	2𝑛	2𝑛	PROPN
cana-1260	307	3	3	3	NUM
cana-1260	307	4	.	.	PUNCT
cana-1260	308	1	proof	proof	NOUN
cana-1260	308	2	:	:	PUNCT
cana-1260	308	3	in	in	ADP
cana-1260	308	4	cartesian	cartesian	ADJ
cana-1260	308	5	product	product	NOUN
cana-1260	308	6	graph	graph	NOUN
cana-1260	308	7	(	(	PUNCT
cana-1260	308	8	𝐾2	𝐾2	ADJ
cana-1260	308	9	×	×	PROPN
cana-1260	308	10	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	308	11	)	)	PUNCT
cana-1260	308	12	for	for	ADP
cana-1260	308	13	𝑛	𝑛	PROPN
cana-1260	308	14	>	>	SYM
cana-1260	308	15	1	1	NUM
cana-1260	308	16	,	,	PUNCT
cana-1260	308	17	we	we	PRON
cana-1260	308	18	noted	note	VERB
cana-1260	308	19	that	that	SCONJ
cana-1260	308	20	𝐾2	𝐾2	NOUN
cana-1260	308	21	and	and	CCONJ
cana-1260	308	22	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	308	23	both	both	PRON
cana-1260	308	24	are	be	AUX
cana-1260	308	25	one	one	NUM
cana-1260	308	26	distance	distance	NOUN
cana-1260	308	27	graph	graph	NOUN
cana-1260	308	28	and	and	CCONJ
cana-1260	308	29	the	the	DET
cana-1260	308	30	cartesian	cartesian	ADJ
cana-1260	308	31	product	product	NOUN
cana-1260	308	32	of	of	ADP
cana-1260	308	33	the	the	DET
cana-1260	308	34	two	two	NUM
cana-1260	308	35	unit	unit	NOUN
cana-1260	308	36	distance	distance	NOUN
cana-1260	308	37	graphs	graph	NOUN
cana-1260	308	38	is	be	AUX
cana-1260	308	39	also	also	ADV
cana-1260	308	40	unit	unit	NOUN
cana-1260	308	41	distance	distance	NOUN
cana-1260	308	42	graph	graph	NOUN
cana-1260	308	43	.	.	PUNCT
cana-1260	309	1	let	let	VERB
cana-1260	309	2	𝐺1	𝐺1	NOUN
cana-1260	309	3	=	=	SYM
cana-1260	309	4	(	(	PUNCT
cana-1260	309	5	𝐾2	𝐾2	PROPN
cana-1260	309	6	×	×	PROPN
cana-1260	309	7	𝑃2	𝑃2	NOUN
cana-1260	309	8	)	)	PUNCT
cana-1260	309	9	is	be	AUX
cana-1260	309	10	the	the	DET
cana-1260	309	11	cartesian	cartesian	ADJ
cana-1260	309	12	product	product	NOUN
cana-1260	309	13	of	of	ADP
cana-1260	309	14	𝐾2	𝐾2	PROPN
cana-1260	309	15	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1260	309	16	𝑃2	𝑃2	NOUN
cana-1260	309	17	we	we	PRON
cana-1260	309	18	can	can	AUX
cana-1260	309	19	see	see	VERB
cana-1260	309	20	that	that	SCONJ
cana-1260	309	21	it	it	PRON
cana-1260	309	22	is	be	AUX
cana-1260	309	23	cycle	cycle	NOUN
cana-1260	309	24	graph	graph	NOUN
cana-1260	309	25	of	of	ADP
cana-1260	309	26	order	order	NOUN
cana-1260	309	27	4	4	NUM
cana-1260	309	28	so	so	ADV
cana-1260	309	29	its	its	PRON
cana-1260	309	30	fractional	fractional	ADJ
cana-1260	309	31	domination	domination	NOUN
cana-1260	309	32	number	number	NOUN
cana-1260	309	33	is	be	AUX
cana-1260	309	34	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	309	35	(	(	PUNCT
cana-1260	309	36	𝐾2	𝐾2	PROPN
cana-1260	309	37	×	×	PROPN
cana-1260	309	38	𝑃2	𝑃2	NOUN
cana-1260	309	39	)	)	PUNCT
cana-1260	310	1	=	=	PRON
cana-1260	310	2	(	(	PUNCT
cana-1260	310	3	2)2	2)2	NUM
cana-1260	310	4	+	+	NOUN
cana-1260	310	5	2(2	2(2	NUM
cana-1260	310	6	)	)	PUNCT
cana-1260	310	7	2(2	2(2	NUM
cana-1260	311	1	+	+	SYM
cana-1260	311	2	1	1	NUM
cana-1260	311	3	)	)	PUNCT
cana-1260	311	4	=	=	SYM
cana-1260	311	5	4/3	4/3	NUM
cana-1260	311	6	and	and	CCONJ
cana-1260	311	7	its	its	PRON
cana-1260	311	8	line	line	NOUN
cana-1260	311	9	graph	graph	NOUN
cana-1260	311	10	𝐿(𝐺1	𝐿(𝐺1	NOUN
cana-1260	311	11	)	)	PUNCT
cana-1260	311	12	is	be	AUX
cana-1260	311	13	isomorphic	isomorphic	ADJ
cana-1260	311	14	to	to	ADP
cana-1260	311	15	original	original	ADJ
cana-1260	311	16	graph	graph	NOUN
cana-1260	311	17	𝐺1	𝐺1	NOUN
cana-1260	311	18	=	=	SYM
cana-1260	311	19	(	(	PUNCT
cana-1260	311	20	𝐾2	𝐾2	PROPN
cana-1260	311	21	×	×	PROPN
cana-1260	311	22	𝑃2	𝑃2	NOUN
cana-1260	311	23	)	)	PUNCT
cana-1260	311	24	of	of	ADP
cana-1260	311	25	order	order	NOUN
cana-1260	311	26	3(2	3(2	NUM
cana-1260	311	27	)	)	PUNCT
cana-1260	312	1	−	−	PROPN
cana-1260	312	2	2	2	NUM
cana-1260	312	3	=	=	SYM
cana-1260	312	4	4	4	NUM
cana-1260	312	5	so	so	ADV
cana-1260	312	6	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	312	7	(	(	PUNCT
cana-1260	312	8	𝐿(𝐾2	𝐿(𝐾2	PROPN
cana-1260	312	9	×	×	NOUN
cana-1260	312	10	𝑃2	𝑃2	NOUN
cana-1260	312	11	)	)	PUNCT
cana-1260	312	12	)	)	PUNCT
cana-1260	313	1	=	=	SYM
cana-1260	313	2	2(2	2(2	NUM
cana-1260	313	3	)	)	PUNCT
cana-1260	313	4	3	3	NUM
cana-1260	313	5	=	=	SYM
cana-1260	313	6	4/3	4/3	NUM
cana-1260	313	7	.	.	PUNCT
cana-1260	314	1	let	let	VERB
cana-1260	314	2	𝐺2	𝐺2	NOUN
cana-1260	314	3	=	=	PUNCT
cana-1260	314	4	(	(	PUNCT
cana-1260	314	5	𝐾2	𝐾2	ADJ
cana-1260	314	6	×	×	PROPN
cana-1260	314	7	𝑃3	𝑃3	NOUN
cana-1260	314	8	)	)	PUNCT
cana-1260	314	9	be	be	VERB
cana-1260	314	10	the	the	DET
cana-1260	314	11	cartesian	cartesian	ADJ
cana-1260	314	12	product	product	NOUN
cana-1260	314	13	of	of	ADP
cana-1260	314	14	𝐾2	𝐾2	NOUN
cana-1260	314	15	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1260	314	16	𝑃3	𝑃3	NOUN
cana-1260	314	17	we	we	PRON
cana-1260	314	18	can	can	AUX
cana-1260	314	19	see	see	VERB
cana-1260	314	20	that	that	DET
cana-1260	314	21	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	314	22	(	(	PUNCT
cana-1260	314	23	𝐾2	𝐾2	ADJ
cana-1260	314	24	×	×	NOUN
cana-1260	314	25	𝑃3	𝑃3	NOUN
cana-1260	314	26	)	)	PUNCT
cana-1260	314	27	=	=	PUNCT
cana-1260	315	1	3	3	NUM
cana-1260	315	2	+	+	SYM
cana-1260	315	3	1	1	NUM
cana-1260	315	4	2	2	NUM
cana-1260	315	5	=	=	SYM
cana-1260	315	6	2	2	NUM
cana-1260	315	7	.	.	X
cana-1260	315	8	its	its	PRON
cana-1260	315	9	line	line	NOUN
cana-1260	315	10	graph	graph	NOUN
cana-1260	315	11	𝐿(𝐺2	𝐿(𝐺2	PROPN
cana-1260	315	12	)	)	PUNCT
cana-1260	315	13	is	be	AUX
cana-1260	315	14	of	of	ADP
cana-1260	315	15	order	order	NOUN
cana-1260	315	16	3(3	3(3	NUM
cana-1260	315	17	)	)	PUNCT
cana-1260	315	18	−	−	PROPN
cana-1260	315	19	2	2	NUM
cana-1260	315	20	=	=	SYM
cana-1260	315	21	7	7	NUM
cana-1260	315	22	so	so	ADV
cana-1260	315	23	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	315	24	(	(	PUNCT
cana-1260	315	25	𝐿(𝐾2	𝐿(𝐾2	PROPN
cana-1260	315	26	×	×	NOUN
cana-1260	315	27	𝑃3	𝑃3	NOUN
cana-1260	315	28	)	)	PUNCT
cana-1260	315	29	)	)	PUNCT
cana-1260	316	1	=	=	SYM
cana-1260	316	2	2(3	2(3	NUM
cana-1260	316	3	)	)	PUNCT
cana-1260	316	4	3	3	NUM
cana-1260	316	5	=	=	SYM
cana-1260	316	6	2	2	NUM
cana-1260	316	7	by	by	ADP
cana-1260	316	8	assigning	assign	VERB
cana-1260	316	9	each	each	DET
cana-1260	316	10	vertex	vertex	NOUN
cana-1260	316	11	with	with	ADP
cana-1260	316	12	weight	weight	NOUN
cana-1260	316	13	(	(	PUNCT
cana-1260	316	14	1/3	1/3	NUM
cana-1260	316	15	)	)	PUNCT
cana-1260	316	16	except	except	SCONJ
cana-1260	316	17	central	central	ADJ
cana-1260	316	18	vertex	vertex	NOUN
cana-1260	316	19	.	.	PUNCT
cana-1260	317	1	for	for	ADP
cana-1260	317	2	central	central	ADJ
cana-1260	317	3	(	(	PUNCT
cana-1260	317	4	hub	hub	NOUN
cana-1260	317	5	)	)	PUNCT
cana-1260	317	6	vertex	vertex	NOUN
cana-1260	317	7	we	we	PRON
cana-1260	317	8	can	can	AUX
cana-1260	317	9	assign	assign	VERB
cana-1260	317	10	weight	weight	NOUN
cana-1260	317	11	zero	zero	NUM
cana-1260	317	12	so	so	SCONJ
cana-1260	317	13	that	that	SCONJ
cana-1260	317	14	adjacency	adjacency	PROPN
cana-1260	317	15	relation	relation	NOUN
cana-1260	317	16	will	will	AUX
cana-1260	317	17	satisfied	satisfied	VERB
cana-1260	317	18	.	.	PUNCT
cana-1260	318	1	similarly	similarly	ADV
cana-1260	318	2	if	if	SCONJ
cana-1260	318	3	𝐺3	𝐺3	PRON
cana-1260	318	4	=	=	PUNCT
cana-1260	318	5	𝐾2	𝐾2	ADJ
cana-1260	318	6	×	×	NOUN
cana-1260	318	7	𝑃4	𝑃4	NOUN
cana-1260	318	8	be	be	VERB
cana-1260	318	9	the	the	DET
cana-1260	318	10	cartesian	cartesian	ADJ
cana-1260	318	11	product	product	NOUN
cana-1260	318	12	of	of	ADP
cana-1260	318	13	𝐾2	𝐾2	NOUN
cana-1260	318	14	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1260	318	15	𝑃4	𝑃4	PROPN
cana-1260	318	16	is	be	AUX
cana-1260	318	17	graph	graph	NOUN
cana-1260	318	18	of	of	ADP
cana-1260	318	19	8	8	NUM
cana-1260	318	20	vertices	vertex	NOUN
cana-1260	318	21	with	with	ADP
cana-1260	318	22	{	{	PUNCT
cana-1260	318	23	𝑣2	𝑣2	NOUN
cana-1260	318	24	,	,	PUNCT
cana-1260	318	25	𝑣3	𝑣3	ADJ
cana-1260	318	26	,	,	PUNCT
cana-1260	318	27	𝑣6	𝑣6	PROPN
cana-1260	318	28	,	,	PUNCT
cana-1260	318	29	𝑣7	𝑣7	NOUN
cana-1260	318	30	}	}	PUNCT
cana-1260	318	31	having	have	VERB
cana-1260	318	32	weights	weight	NOUN
cana-1260	318	33	(	(	PUNCT
cana-1260	318	34	1/5	1/5	NUM
cana-1260	318	35	)	)	PUNCT
cana-1260	318	36	and	and	CCONJ
cana-1260	318	37	{	{	PUNCT
cana-1260	318	38	𝑣1	𝑣1	NOUN
cana-1260	318	39	,	,	PUNCT
cana-1260	318	40	𝑣4	𝑣4	NOUN
cana-1260	318	41	,	,	PUNCT
cana-1260	318	42	𝑣5	𝑣5	NOUN
cana-1260	318	43	,	,	PUNCT
cana-1260	318	44	𝑣8	𝑣8	PROPN
cana-1260	318	45	}	}	PUNCT
cana-1260	318	46	having	have	VERB
cana-1260	318	47	weights	weight	NOUN
cana-1260	318	48	(	(	PUNCT
cana-1260	318	49	2/5	2/5	NUM
cana-1260	318	50	)	)	PUNCT
cana-1260	319	1	so	so	ADV
cana-1260	319	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	319	3	(	(	PUNCT
cana-1260	319	4	𝐾2	𝐾2	PROPN
cana-1260	319	5	×	×	PROPN
cana-1260	319	6	𝑃4	𝑃4	NOUN
cana-1260	319	7	)	)	PUNCT
cana-1260	320	1	=	=	PUNCT
cana-1260	320	2	communications	communication	NOUN
cana-1260	320	3	on	on	ADP
cana-1260	320	4	applied	apply	VERB
cana-1260	320	5	nonlinear	nonlinear	ADJ
cana-1260	320	6	analysis	analysis	NOUN
cana-1260	320	7	issn	issn	NOUN
cana-1260	320	8	:	:	PUNCT
cana-1260	320	9	1074	1074	NUM
cana-1260	320	10	-	-	PUNCT
cana-1260	320	11	133x	133x	NUM
cana-1260	320	12	vol	vol	NOUN
cana-1260	320	13	31	31	NUM
cana-1260	320	14	no	no	NOUN
cana-1260	320	15	.	.	PUNCT
cana-1260	321	1	6s	6s	NUM
cana-1260	321	2	(	(	PUNCT
cana-1260	321	3	2024	2024	NUM
cana-1260	321	4	)	)	PUNCT
cana-1260	321	5	682	682	NUM
cana-1260	321	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	321	7	42	42	NUM
cana-1260	321	8	+	+	NOUN
cana-1260	321	9	2(4	2(4	NUM
cana-1260	321	10	)	)	PUNCT
cana-1260	321	11	2(4	2(4	NUM
cana-1260	322	1	+	+	PROPN
cana-1260	322	2	1	1	NUM
cana-1260	322	3	)	)	PUNCT
cana-1260	322	4	=	=	NOUN
cana-1260	322	5	12	12	NUM
cana-1260	322	6	5	5	NUM
cana-1260	322	7	=	=	SYM
cana-1260	322	8	2.4	2.4	NUM
cana-1260	322	9	(	(	PUNCT
cana-1260	322	10	see	see	VERB
cana-1260	322	11	the	the	DET
cana-1260	322	12	figure	figure	NOUN
cana-1260	322	13	2.7.2a	2.7.2a	NUM
cana-1260	322	14	)	)	PUNCT
cana-1260	322	15	.	.	PUNCT
cana-1260	323	1	(	(	PUNCT
cana-1260	323	2	a	a	X
cana-1260	323	3	)	)	PUNCT
cana-1260	323	4	cartesian	cartesian	ADJ
cana-1260	323	5	product	product	NOUN
cana-1260	323	6	of	of	ADP
cana-1260	323	7	(	(	PUNCT
cana-1260	323	8	𝐾2	𝐾2	PROPN
cana-1260	323	9	×	×	PROPN
cana-1260	323	10	𝑃4	𝑃4	NOUN
cana-1260	323	11	)	)	PUNCT
cana-1260	323	12	(	(	PUNCT
cana-1260	323	13	b	b	X
cana-1260	323	14	)	)	PUNCT
cana-1260	323	15	its	its	PRON
cana-1260	323	16	line	line	NOUN
cana-1260	323	17	graph	graph	NOUN
cana-1260	323	18	(	(	PUNCT
cana-1260	323	19	𝐾2	𝐾2	ADJ
cana-1260	323	20	×	×	PROPN
cana-1260	323	21	𝑃4	𝑃4	NOUN
cana-1260	323	22	)	)	PUNCT
cana-1260	323	23	figure	figure	VERB
cana-1260	323	24	2.7.2	2.7.2	NUM
cana-1260	323	25	its	its	PRON
cana-1260	323	26	line	line	NOUN
cana-1260	323	27	graph	graph	NOUN
cana-1260	323	28	𝐿(𝐺3	𝐿(𝐺3	PUNCT
cana-1260	323	29	)	)	PUNCT
cana-1260	323	30	is	be	AUX
cana-1260	323	31	of	of	ADP
cana-1260	323	32	order	order	NOUN
cana-1260	323	33	3(4	3(4	NUM
cana-1260	323	34	)	)	PUNCT
cana-1260	324	1	−	−	PROPN
cana-1260	325	1	2	2	NUM
cana-1260	325	2	=	=	SYM
cana-1260	325	3	10	10	NUM
cana-1260	325	4	so	so	ADV
cana-1260	325	5	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	325	6	(	(	PUNCT
cana-1260	325	7	𝐿(𝐾2	𝐿(𝐾2	PROPN
cana-1260	325	8	×	×	NOUN
cana-1260	325	9	𝑃4	𝑃4	NOUN
cana-1260	325	10	)	)	PUNCT
cana-1260	325	11	)	)	PUNCT
cana-1260	326	1	=	=	SYM
cana-1260	326	2	2(4	2(4	NUM
cana-1260	326	3	)	)	PUNCT
cana-1260	326	4	3	3	NUM
cana-1260	326	5	=	=	NOUN
cana-1260	326	6	8/3	8/3	NUM
cana-1260	326	7	with	with	ADP
cana-1260	326	8	vertices	vertex	NOUN
cana-1260	326	9	{	{	PUNCT
cana-1260	326	10	𝑒1	𝑒1	NOUN
cana-1260	326	11	,	,	PUNCT
cana-1260	326	12	𝑒2	𝑒2	PROPN
cana-1260	326	13	,	,	PUNCT
cana-1260	326	14	𝑒3	𝑒3	NOUN
cana-1260	326	15	,	,	PUNCT
cana-1260	326	16	𝑒4	𝑒4	PROPN
cana-1260	326	17	,	,	PUNCT
cana-1260	326	18	𝑒5	𝑒5	PROPN
cana-1260	326	19	,	,	PUNCT
cana-1260	326	20	𝑒6	𝑒6	NOUN
cana-1260	326	21	,	,	PUNCT
cana-1260	326	22	𝑒7	𝑒7	NOUN
cana-1260	326	23	,	,	PUNCT
cana-1260	326	24	𝑒10	𝑒10	NOUN
cana-1260	326	25	,	,	PUNCT
cana-1260	326	26	}	}	PUNCT
cana-1260	326	27	having	have	VERB
cana-1260	326	28	weights	weight	NOUN
cana-1260	326	29	(	(	PUNCT
cana-1260	326	30	1/3	1/3	NUM
cana-1260	326	31	)	)	PUNCT
cana-1260	326	32	and	and	CCONJ
cana-1260	326	33	vertices	vertice	VERB
cana-1260	326	34	{	{	PUNCT
cana-1260	326	35	𝑒8	𝑒8	NOUN
cana-1260	326	36	,	,	PUNCT
cana-1260	326	37	𝑒9	𝑒9	NOUN
cana-1260	326	38	}	}	PUNCT
cana-1260	326	39	having	have	VERB
cana-1260	326	40	weights	weight	NOUN
cana-1260	326	41	zero	zero	NUM
cana-1260	326	42	.	.	PUNCT
cana-1260	327	1	(	(	PUNCT
cana-1260	327	2	see	see	VERB
cana-1260	327	3	the	the	DET
cana-1260	327	4	figure	figure	NOUN
cana-1260	327	5	2.7.2b	2.7.2b	NOUN
cana-1260	327	6	)	)	PUNCT
cana-1260	327	7	.	.	PUNCT
cana-1260	328	1	these	these	PRON
cana-1260	328	2	are	be	AUX
cana-1260	328	3	minimal	minimal	ADJ
cana-1260	328	4	fractional	fractional	ADJ
cana-1260	328	5	domination	domination	NOUN
cana-1260	328	6	numbers	number	NOUN
cana-1260	328	7	by	by	ADP
cana-1260	328	8	considering	consider	VERB
cana-1260	328	9	the	the	DET
cana-1260	328	10	closed	closed	ADJ
cana-1260	328	11	neighborhood	neighborhood	NOUN
cana-1260	328	12	of	of	ADP
cana-1260	328	13	each	each	DET
cana-1260	328	14	vertex	vertex	NOUN
cana-1260	328	15	satisfying	satisfy	VERB
cana-1260	328	16	the	the	DET
cana-1260	328	17	condition	condition	NOUN
cana-1260	328	18	where	where	SCONJ
cana-1260	328	19	the	the	DET
cana-1260	328	20	vertex	vertex	NOUN
cana-1260	328	21	𝑤	𝑤	ADP
cana-1260	328	22	∈	∈	PROPN
cana-1260	328	23	𝑁[𝑉	𝑁[𝑉	NOUN
cana-1260	328	24	]	]	PUNCT
cana-1260	328	25	such	such	ADJ
cana-1260	328	26	that	that	SCONJ
cana-1260	328	27	∑	∑	PUNCT
cana-1260	328	28	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	328	29	)	)	PUNCT
cana-1260	328	30	⬚	⬚	PROPN
cana-1260	328	31	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	328	32	]	]	X
cana-1260	328	33	=	=	SYM
cana-1260	328	34	1	1	X
cana-1260	328	35	.	.	X
cana-1260	328	36	hence	hence	ADV
cana-1260	328	37	in	in	ADP
cana-1260	328	38	general	general	ADJ
cana-1260	328	39	form	form	NOUN
cana-1260	328	40	by	by	ADP
cana-1260	328	41	theorem	theorem	NOUN
cana-1260	328	42	2.7.1	2.7.1	NUM
cana-1260	328	43	we	we	PRON
cana-1260	328	44	get	get	VERB
cana-1260	328	45	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	328	46	(	(	PUNCT
cana-1260	328	47	𝐾2	𝐾2	ADJ
cana-1260	328	48	×	×	PROPN
cana-1260	328	49	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	328	50	)	)	PUNCT
cana-1260	329	1	=	=	PUNCT
cana-1260	329	2	𝑛+1	𝑛+1	ADP
cana-1260	329	3	2	2	NUM
cana-1260	329	4	,	,	PUNCT
cana-1260	329	5	𝑖𝑓	𝑖𝑓	ADP
cana-1260	329	6	𝑛	𝑛	DET
cana-1260	329	7	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	329	8	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-1260	329	9	and	and	CCONJ
cana-1260	329	10	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	329	11	(	(	PUNCT
cana-1260	329	12	𝐾2	𝐾2	ADJ
cana-1260	329	13	×	×	PROPN
cana-1260	329	14	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	329	15	)	)	PUNCT
cana-1260	329	16	=	=	SYM
cana-1260	329	17	𝑛2	𝑛2	NOUN
cana-1260	329	18	+	+	PROPN
cana-1260	329	19	2𝑛	2𝑛	PROPN
cana-1260	329	20	2(𝑛+1	2(𝑛+1	NUM
cana-1260	329	21	)	)	PUNCT
cana-1260	329	22	,	,	PUNCT
cana-1260	329	23	𝑖𝑓	𝑖𝑓	ADP
cana-1260	329	24	𝑛	𝑛	PRON
cana-1260	329	25	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	329	26	𝑒𝑣𝑒𝑛.	𝑒𝑣𝑒𝑛.	NOUN
cana-1260	329	27	therefore	therefore	ADV
cana-1260	329	28	in	in	ADP
cana-1260	329	29	its	its	PRON
cana-1260	329	30	line	line	NOUN
cana-1260	329	31	graph	graph	NOUN
cana-1260	329	32	l(𝐾2	l(𝐾2	PROPN
cana-1260	329	33	×	×	NOUN
cana-1260	329	34	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	329	35	)	)	PUNCT
cana-1260	329	36	with	with	ADP
cana-1260	329	37	3𝑛	3𝑛	NUM
cana-1260	329	38	−	−	NOUN
cana-1260	329	39	2	2	NUM
cana-1260	329	40	vertices	vertex	NOUN
cana-1260	329	41	for	for	ADP
cana-1260	329	42	𝑛	𝑛	PROPN
cana-1260	329	43	>	>	SYM
cana-1260	329	44	1	1	NUM
cana-1260	329	45	we	we	PRON
cana-1260	329	46	have	have	AUX
cana-1260	329	47	𝛾𝑓	𝛾𝑓	VERB
cana-1260	329	48	(	(	PUNCT
cana-1260	329	49	𝐿(𝐾2	𝐿(𝐾2	VERB
cana-1260	329	50	×	×	NOUN
cana-1260	329	51	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	329	52	)	)	PUNCT
cana-1260	329	53	)	)	PUNCT
cana-1260	330	1	=	=	SYM
cana-1260	330	2	2𝑛	2𝑛	PROPN
cana-1260	330	3	3	3	NUM
cana-1260	330	4	.	.	PUNCT
cana-1260	331	1	□	□	PUNCT
cana-1260	331	2	2.8	2.8	NUM
cana-1260	331	3	cartesian	cartesian	ADJ
cana-1260	331	4	product	product	NOUN
cana-1260	331	5	graph	graph	NOUN
cana-1260	331	6	(	(	PUNCT
cana-1260	331	7	𝑲𝟑	𝑲𝟑	ADJ
cana-1260	331	8	×	×	X
cana-1260	331	9	𝑷𝒏	𝑷𝒏	PROPN
cana-1260	331	10	)	)	PUNCT
cana-1260	331	11	if	if	SCONJ
cana-1260	331	12	we	we	PRON
cana-1260	331	13	take	take	VERB
cana-1260	331	14	the	the	DET
cana-1260	331	15	path	path	NOUN
cana-1260	331	16	graph	graph	NOUN
cana-1260	331	17	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	331	18	and	and	CCONJ
cana-1260	331	19	complete	complete	ADJ
cana-1260	331	20	graph	graph	NOUN
cana-1260	331	21	𝐾3	𝐾3	NOUN
cana-1260	331	22	then	then	ADV
cana-1260	331	23	cartesian	cartesian	ADJ
cana-1260	331	24	product	product	NOUN
cana-1260	331	25	graph	graph	NOUN
cana-1260	331	26	represented	represent	VERB
cana-1260	331	27	by	by	ADP
cana-1260	331	28	𝐾3	𝐾3	NOUN
cana-1260	331	29	×	×	PROPN
cana-1260	331	30	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	331	31	.	.	PUNCT
cana-1260	332	1	for	for	ADP
cana-1260	332	2	the	the	DET
cana-1260	332	3	three	three	NUM
cana-1260	332	4	vertex	vertex	NOUN
cana-1260	332	5	complete	complete	ADJ
cana-1260	332	6	graph	graph	NOUN
cana-1260	332	7	𝐾3	𝐾3	PROPN
cana-1260	332	8	,	,	PUNCT
cana-1260	332	9	the	the	DET
cana-1260	332	10	vertex	vertex	NOUN
cana-1260	332	11	set	set	NOUN
cana-1260	332	12	𝑉(𝐾3	𝑉(𝐾3	NOUN
cana-1260	332	13	)	)	PUNCT
cana-1260	332	14	=	=	SYM
cana-1260	332	15	{	{	PUNCT
cana-1260	332	16	𝑘1	𝑘1	PROPN
cana-1260	332	17	,	,	PUNCT
cana-1260	332	18	𝑘2	𝑘2	PROPN
cana-1260	332	19	,	,	PUNCT
cana-1260	332	20	𝑘3	𝑘3	PROPN
cana-1260	332	21	}	}	PUNCT
cana-1260	332	22	and	and	CCONJ
cana-1260	332	23	the	the	DET
cana-1260	332	24	vertex	vertex	NOUN
cana-1260	332	25	set	set	VERB
cana-1260	332	26	for	for	ADP
cana-1260	332	27	path	path	NOUN
cana-1260	332	28	graph	graph	NOUN
cana-1260	332	29	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	332	30	is	be	AUX
cana-1260	332	31	{	{	PUNCT
cana-1260	332	32	𝑝1	𝑝1	NOUN
cana-1260	332	33	,	,	PUNCT
cana-1260	332	34	𝑝2,	𝑝2,	NOUN
cana-1260	332	35	…	…	SYM
cana-1260	332	36	𝑝𝑛	𝑝𝑛	NOUN
cana-1260	332	37	}	}	PUNCT
cana-1260	332	38	resp	resp	NOUN
cana-1260	332	39	.	.	PUNCT
cana-1260	333	1	the	the	DET
cana-1260	333	2	vertex	vertex	NOUN
cana-1260	333	3	set	set	VERB
cana-1260	333	4	for	for	ADP
cana-1260	333	5	cartesian	cartesian	ADJ
cana-1260	333	6	product	product	NOUN
cana-1260	333	7	of	of	ADP
cana-1260	333	8	(	(	PUNCT
cana-1260	333	9	𝐾3	𝐾3	VERB
cana-1260	333	10	×	×	PROPN
cana-1260	333	11	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	333	12	)	)	PUNCT
cana-1260	333	13	is	be	AUX
cana-1260	333	14	𝑉	𝑉	PROPN
cana-1260	333	15	(	(	PUNCT
cana-1260	333	16	𝐾3	𝐾3	VERB
cana-1260	333	17	×	×	PROPN
cana-1260	333	18	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	333	19	)	)	PUNCT
cana-1260	333	20	=	=	SYM
cana-1260	333	21	𝑉(𝐾3	𝑉(𝐾3	NOUN
cana-1260	333	22	)	)	PUNCT
cana-1260	333	23	×	×	PROPN
cana-1260	333	24	𝑉	𝑉	PROPN
cana-1260	333	25	(	(	PUNCT
cana-1260	333	26	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	333	27	)	)	PUNCT
cana-1260	333	28	and	and	CCONJ
cana-1260	333	29	𝑉(𝐾3	𝑉(𝐾3	NOUN
cana-1260	333	30	)	)	PUNCT
cana-1260	333	31	×	×	PROPN
cana-1260	333	32	𝑉	𝑉	PROPN
cana-1260	333	33	(	(	PUNCT
cana-1260	333	34	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	333	35	)	)	PUNCT
cana-1260	333	36	=	=	PRON
cana-1260	333	37	{	{	PUNCT
cana-1260	333	38	(	(	PUNCT
cana-1260	333	39	𝑘𝑖	𝑘𝑖	INTJ
cana-1260	333	40	,	,	PUNCT
cana-1260	333	41	𝑝𝑗	𝑝𝑗	NOUN
cana-1260	333	42	)	)	PUNCT
cana-1260	333	43	|	|	ADV
cana-1260	333	44	𝑘𝑖	𝑘𝑖	NOUN
cana-1260	333	45	∈	∈	PROPN
cana-1260	333	46	𝑉(𝐾3	𝑉(𝐾3	NOUN
cana-1260	333	47	)	)	PUNCT
cana-1260	333	48	,	,	PUNCT
cana-1260	333	49	𝑝𝑗	𝑝𝑗	PROPN
cana-1260	333	50	∈	∈	PROPN
cana-1260	333	51	𝑉	𝑉	PROPN
cana-1260	333	52	(	(	PUNCT
cana-1260	333	53	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	333	54	)	)	PUNCT
cana-1260	333	55	}	}	PUNCT
cana-1260	333	56	and	and	CCONJ
cana-1260	333	57	‘	'	PUNCT
cana-1260	333	58	𝑒	𝑒	X
cana-1260	333	59	’	'	PUNCT
cana-1260	333	60	be	be	VERB
cana-1260	333	61	the	the	DET
cana-1260	333	62	edge	edge	NOUN
cana-1260	333	63	for	for	ADP
cana-1260	333	64	cartesian	cartesian	ADJ
cana-1260	333	65	product	product	NOUN
cana-1260	334	1	(	(	PUNCT
cana-1260	334	2	𝐾3	𝐾3	VERB
cana-1260	334	3	×	×	PROPN
cana-1260	334	4	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	334	5	)	)	PUNCT
cana-1260	334	6	iff	iff	PROPN
cana-1260	334	7	𝑒	𝑒	PROPN
cana-1260	334	8	=	=	X
cana-1260	334	9	<	<	X
cana-1260	334	10	(	(	PUNCT
cana-1260	334	11	𝑘𝑖	𝑘𝑖	INTJ
cana-1260	334	12	,	,	PUNCT
cana-1260	334	13	𝑝𝑗	𝑝𝑗	PROPN
cana-1260	334	14	)	)	PUNCT
cana-1260	334	15	,	,	PUNCT
cana-1260	334	16	(	(	PUNCT
cana-1260	334	17	𝑘𝑟	𝑘𝑟	INTJ
cana-1260	334	18	,	,	PUNCT
cana-1260	334	19	𝑝𝑠	𝑝𝑠	CCONJ
cana-1260	334	20	)	)	PUNCT
cana-1260	334	21	>	>	X
cana-1260	334	22	where	where	SCONJ
cana-1260	334	23	the	the	DET
cana-1260	334	24	condition	condition	NOUN
cana-1260	334	25	holds	hold	VERB
cana-1260	334	26	as	as	ADP
cana-1260	334	27	𝑖	𝑖	NOUN
cana-1260	334	28	)	)	PUNCT
cana-1260	334	29	𝑖	𝑖	NOUN
cana-1260	334	30	=	=	SYM
cana-1260	334	31	𝑟	𝑟	NOUN
cana-1260	334	32	and	and	CCONJ
cana-1260	334	33	𝑝𝑗	𝑝𝑗	PROPN
cana-1260	334	34	𝑝𝑠	𝑝𝑠	ADV
cana-1260	334	35	∈	∈	PROPN
cana-1260	335	1	𝐸𝑑𝑔𝑒	𝐸𝑑𝑔𝑒	PROPN
cana-1260	335	2	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-1260	335	3	𝑜𝑓	𝑜𝑓	X
cana-1260	335	4	(	(	PUNCT
cana-1260	335	5	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	335	6	)	)	PUNCT
cana-1260	335	7	,	,	PUNCT
cana-1260	335	8	𝑖𝑖	𝑖𝑖	X
cana-1260	335	9	)	)	PUNCT
cana-1260	335	10	𝑗	𝑗	NOUN
cana-1260	335	11	=	=	SYM
cana-1260	335	12	𝑠	𝑠	PROPN
cana-1260	335	13	and	and	CCONJ
cana-1260	335	14	𝑝𝑗	𝑝𝑗	NOUN
cana-1260	335	15	𝑝𝑠	𝑝𝑠	ADV
cana-1260	335	16	∈	∈	PROPN
cana-1260	336	1	𝐸𝑑𝑔𝑒	𝐸𝑑𝑔𝑒	PROPN
cana-1260	336	2	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-1260	336	3	𝑜𝑓	𝑜𝑓	X
cana-1260	336	4	(	(	PUNCT
cana-1260	336	5	𝐾3	𝐾3	PROPN
cana-1260	336	6	)	)	PUNCT
cana-1260	336	7	.	.	PUNCT
cana-1260	337	1	theorem	theorem	NOUN
cana-1260	337	2	2.8.1	2.8.1	NUM
cana-1260	337	3	let	let	VERB
cana-1260	337	4	g	g	PROPN
cana-1260	337	5	is	be	AUX
cana-1260	337	6	the	the	DET
cana-1260	337	7	cartesian	cartesian	ADJ
cana-1260	337	8	product	product	NOUN
cana-1260	337	9	graph	graph	NOUN
cana-1260	337	10	(	(	PUNCT
cana-1260	337	11	𝐾3	𝐾3	VERB
cana-1260	337	12	×	×	PROPN
cana-1260	337	13	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	337	14	)	)	PUNCT
cana-1260	337	15	of	of	ADP
cana-1260	337	16	order	order	NOUN
cana-1260	337	17	3𝑛	3𝑛	NUM
cana-1260	337	18	for	for	ADP
cana-1260	337	19	𝑛	𝑛	PRON
cana-1260	337	20	≥	≥	NUM
cana-1260	337	21	3	3	NUM
cana-1260	337	22	and	and	CCONJ
cana-1260	337	23	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	337	24	)	)	PUNCT
cana-1260	337	25	be	be	VERB
cana-1260	337	26	its	its	PRON
cana-1260	337	27	line	line	NOUN
cana-1260	337	28	graph	graph	NOUN
cana-1260	337	29	of	of	ADP
cana-1260	337	30	order	order	NOUN
cana-1260	337	31	(	(	PUNCT
cana-1260	337	32	6𝑛	6𝑛	NOUN
cana-1260	337	33	−	−	NOUN
cana-1260	337	34	3	3	NUM
cana-1260	337	35	)	)	PUNCT
cana-1260	337	36	.	.	PUNCT
cana-1260	338	1	then	then	ADV
cana-1260	338	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	338	3	(	(	PUNCT
cana-1260	338	4	𝐾3	𝐾3	VERB
cana-1260	338	5	×	×	PROPN
cana-1260	338	6	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	338	7	)	)	PUNCT
cana-1260	338	8	=	=	SYM
cana-1260	339	1	3𝑛+2	3𝑛+2	NUM
cana-1260	339	2	5	5	NUM
cana-1260	339	3	and	and	CCONJ
cana-1260	339	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	339	5	(	(	PUNCT
cana-1260	339	6	𝐿(𝐾3	𝐿(𝐾3	ADV
cana-1260	339	7	×	×	NOUN
cana-1260	339	8	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	339	9	)	)	PUNCT
cana-1260	339	10	)	)	PUNCT
cana-1260	340	1	=	=	SYM
cana-1260	340	2	3(2𝑛+1	3(2𝑛+1	X
cana-1260	340	3	)	)	PUNCT
cana-1260	340	4	7	7	NUM
cana-1260	340	5	.	.	PUNCT
cana-1260	341	1	proof	proof	NOUN
cana-1260	341	2	:	:	PUNCT
cana-1260	341	3	for	for	ADP
cana-1260	341	4	the	the	DET
cana-1260	341	5	complete	complete	ADJ
cana-1260	341	6	graph	graph	NOUN
cana-1260	341	7	𝐾3	𝐾3	NOUN
cana-1260	341	8	and	and	CCONJ
cana-1260	341	9	path	path	NOUN
cana-1260	341	10	graph	graph	NOUN
cana-1260	341	11	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	341	12	,	,	PUNCT
cana-1260	341	13	we	we	PRON
cana-1260	341	14	have	have	VERB
cana-1260	341	15	graph	graph	NOUN
cana-1260	341	16	of	of	ADP
cana-1260	341	17	cartesian	cartesian	ADJ
cana-1260	341	18	product	product	NOUN
cana-1260	341	19	denoted	denote	VERB
cana-1260	341	20	by	by	ADP
cana-1260	341	21	(	(	PUNCT
cana-1260	341	22	𝐾3	𝐾3	VERB
cana-1260	341	23	×	×	PROPN
cana-1260	341	24	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	341	25	)	)	PUNCT
cana-1260	341	26	of	of	ADP
cana-1260	341	27	order	order	NOUN
cana-1260	341	28	3𝑛	3𝑛	NUM
cana-1260	341	29	for	for	ADP
cana-1260	341	30	𝑛	𝑛	PRON
cana-1260	341	31	≥	≥	NUM
cana-1260	341	32	3	3	NUM
cana-1260	341	33	.	.	PUNCT
cana-1260	342	1	let	let	VERB
cana-1260	342	2	𝐺1	𝐺1	NOUN
cana-1260	342	3	=	=	SYM
cana-1260	342	4	(	(	PUNCT
cana-1260	342	5	𝐾3	𝐾3	VERB
cana-1260	342	6	×	×	PROPN
cana-1260	342	7	𝑃3	𝑃3	NOUN
cana-1260	342	8	)	)	PUNCT
cana-1260	342	9	be	be	VERB
cana-1260	342	10	the	the	DET
cana-1260	342	11	cartesian	cartesian	ADJ
cana-1260	342	12	product	product	NOUN
cana-1260	342	13	of	of	ADP
cana-1260	342	14	𝐾3	𝐾3	NOUN
cana-1260	342	15	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1260	342	16	𝑃3	𝑃3	NOUN
cana-1260	342	17	we	we	PRON
cana-1260	342	18	can	can	AUX
cana-1260	342	19	see	see	VERB
cana-1260	342	20	that	that	SCONJ
cana-1260	342	21	it	it	PRON
cana-1260	342	22	is	be	AUX
cana-1260	342	23	graph	graph	NOUN
cana-1260	342	24	of	of	ADP
cana-1260	342	25	order	order	NOUN
cana-1260	342	26	9	9	NUM
cana-1260	342	27	so	so	ADV
cana-1260	342	28	its	its	PRON
cana-1260	342	29	fractional	fractional	ADJ
cana-1260	342	30	domination	domination	NOUN
cana-1260	342	31	number	number	NOUN
cana-1260	342	32	is	be	AUX
cana-1260	342	33	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	342	34	(	(	PUNCT
cana-1260	342	35	𝐾3	𝐾3	VERB
cana-1260	342	36	×	×	NOUN
cana-1260	342	37	𝑃3	𝑃3	NOUN
cana-1260	342	38	)	)	PUNCT
cana-1260	342	39	=	=	SYM
cana-1260	342	40	3(3)+2	3(3)+2	NUM
cana-1260	342	41	5	5	NUM
cana-1260	342	42	=	=	SYM
cana-1260	342	43	11/5	11/5	NUM
cana-1260	342	44	by	by	ADP
cana-1260	342	45	assigning	assign	VERB
cana-1260	342	46	weight	weight	NOUN
cana-1260	342	47	(	(	PUNCT
cana-1260	342	48	1/5	1/5	NUM
cana-1260	342	49	)	)	PUNCT
cana-1260	342	50	to	to	ADP
cana-1260	342	51	every	every	DET
cana-1260	342	52	vertex	vertex	NOUN
cana-1260	342	53	of	of	ADP
cana-1260	342	54	the	the	DET
cana-1260	342	55	set	set	NOUN
cana-1260	342	56	{	{	PUNCT
cana-1260	342	57	𝑣1	𝑣1	PROPN
cana-1260	342	58	,	,	PUNCT
cana-1260	342	59	𝑣2	𝑣2	PROPN
cana-1260	342	60	,	,	PUNCT
cana-1260	342	61	𝑣3	𝑣3	ADJ
cana-1260	342	62	,	,	PUNCT
cana-1260	342	63	𝑣5	𝑣5	NOUN
cana-1260	342	64	,	,	PUNCT
cana-1260	342	65	𝑣6	𝑣6	NOUN
cana-1260	342	66	,	,	PUNCT
cana-1260	342	67	𝑣7	𝑣7	ADJ
cana-1260	342	68	,	,	PUNCT
cana-1260	342	69	𝑣9	𝑣9	PROPN
cana-1260	342	70	,	,	PUNCT
cana-1260	342	71	}	}	PUNCT
cana-1260	342	72	and	and	CCONJ
cana-1260	342	73	weight	weight	NOUN
cana-1260	342	74	(	(	PUNCT
cana-1260	342	75	2/5	2/5	NUM
cana-1260	342	76	)	)	PUNCT
cana-1260	342	77	to	to	ADP
cana-1260	342	78	every	every	DET
cana-1260	342	79	vertex	vertex	NOUN
cana-1260	342	80	of	of	ADP
cana-1260	342	81	the	the	DET
cana-1260	342	82	set	set	NOUN
cana-1260	342	83	{	{	PUNCT
cana-1260	342	84	𝑣4	𝑣4	NOUN
cana-1260	342	85	,	,	PUNCT
cana-1260	342	86	𝑣8	𝑣8	NOUN
cana-1260	342	87	,	,	PUNCT
cana-1260	342	88	}	}	PUNCT
cana-1260	342	89	.	.	PUNCT
cana-1260	343	1	for	for	ADP
cana-1260	343	2	𝐺	𝐺	PROPN
cana-1260	343	3	is	be	AUX
cana-1260	343	4	cartesian	cartesian	ADJ
cana-1260	343	5	product	product	NOUN
cana-1260	343	6	graph	graph	NOUN
cana-1260	343	7	(	(	PUNCT
cana-1260	343	8	𝐾3	𝐾3	VERB
cana-1260	343	9	×	×	PROPN
cana-1260	343	10	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	343	11	)	)	PUNCT
cana-1260	343	12	of	of	ADP
cana-1260	343	13	order	order	NOUN
cana-1260	343	14	3𝑛	3𝑛	NUM
cana-1260	343	15	for	for	ADP
cana-1260	343	16	𝑛	𝑛	PRON
cana-1260	343	17	≥	≥	NUM
cana-1260	343	18	3	3	NUM
cana-1260	343	19	then	then	ADV
cana-1260	343	20	for	for	ADP
cana-1260	343	21	fractional	fractional	ADJ
cana-1260	343	22	domination	domination	NOUN
cana-1260	343	23	numbers	number	NOUN
cana-1260	343	24	we	we	PRON
cana-1260	343	25	found	find	VERB
cana-1260	343	26	sequence	sequence	NOUN
cana-1260	343	27	of	of	ADP
cana-1260	343	28	numbers	number	NOUN
cana-1260	343	29	as	as	ADP
cana-1260	343	30	{	{	PUNCT
cana-1260	343	31	11/5	11/5	NUM
cana-1260	343	32	,	,	PUNCT
cana-1260	343	33	14/5	14/5	NUM
cana-1260	343	34	,	,	PUNCT
cana-1260	343	35	17/5	17/5	NUM
cana-1260	343	36	,	,	PUNCT
cana-1260	343	37	20/5	20/5	NUM
cana-1260	343	38	,	,	PUNCT
cana-1260	343	39	…	…	PUNCT
cana-1260	343	40	3𝑛+2	3𝑛+2	NUM
cana-1260	343	41	5	5	NUM
cana-1260	343	42	}	}	PUNCT
cana-1260	343	43	so	so	SCONJ
cana-1260	343	44	that	that	SCONJ
cana-1260	343	45	in	in	ADP
cana-1260	343	46	general	general	ADJ
cana-1260	343	47	𝛾𝑓	𝛾𝑓	X
cana-1260	343	48	(	(	PUNCT
cana-1260	343	49	𝐾3	𝐾3	VERB
cana-1260	343	50	×	×	PROPN
cana-1260	343	51	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	343	52	)	)	PUNCT
cana-1260	344	1	=	=	SYM
cana-1260	344	2	3𝑛+2	3𝑛+2	NUM
cana-1260	344	3	5	5	NUM
cana-1260	344	4	see	see	VERB
cana-1260	344	5	the	the	DET
cana-1260	344	6	figure	figure	NOUN
cana-1260	344	7	2.8.1a	2.8.1a	NOUN
cana-1260	344	8	)	)	PUNCT
cana-1260	344	9	.	.	PUNCT
cana-1260	345	1	communications	communication	NOUN
cana-1260	345	2	on	on	ADP
cana-1260	345	3	applied	apply	VERB
cana-1260	345	4	nonlinear	nonlinear	ADJ
cana-1260	345	5	analysis	analysis	NOUN
cana-1260	345	6	issn	issn	NOUN
cana-1260	345	7	:	:	PUNCT
cana-1260	345	8	1074	1074	NUM
cana-1260	345	9	-	-	PUNCT
cana-1260	345	10	133x	133x	NUM
cana-1260	345	11	vol	vol	NOUN
cana-1260	345	12	31	31	NUM
cana-1260	345	13	no	no	NOUN
cana-1260	345	14	.	.	PUNCT
cana-1260	346	1	6s	6s	NUM
cana-1260	346	2	(	(	PUNCT
cana-1260	346	3	2024	2024	NUM
cana-1260	346	4	)	)	PUNCT
cana-1260	346	5	683	683	NUM
cana-1260	346	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	346	7	in	in	ADP
cana-1260	346	8	contradiction	contradiction	NOUN
cana-1260	346	9	if	if	SCONJ
cana-1260	346	10	we	we	PRON
cana-1260	346	11	have	have	AUX
cana-1260	346	12	𝛾𝑓	𝛾𝑓	VERB
cana-1260	346	13	(	(	PUNCT
cana-1260	346	14	𝐾3	𝐾3	VERB
cana-1260	346	15	×	×	PROPN
cana-1260	346	16	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	346	17	)	)	PUNCT
cana-1260	347	1	=	=	SYM
cana-1260	347	2	𝑘	𝑘	PROPN
cana-1260	347	3	and	and	CCONJ
cana-1260	347	4	𝑘	𝑘	X
cana-1260	347	5	<	<	X
cana-1260	347	6	3𝑛+2	3𝑛+2	NUM
cana-1260	347	7	5	5	NUM
cana-1260	347	8	then	then	ADV
cana-1260	347	9	adjacency	adjacency	PROPN
cana-1260	347	10	relation	relation	NOUN
cana-1260	347	11	is	be	AUX
cana-1260	347	12	not	not	PART
cana-1260	347	13	satisfied	satisfied	ADJ
cana-1260	347	14	with	with	ADP
cana-1260	347	15	∑	∑	ADV
cana-1260	347	16	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1260	347	17	)	)	PUNCT
cana-1260	347	18	⬚	⬚	PROPN
cana-1260	347	19	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	347	20	]	]	X
cana-1260	347	21	=	=	SYM
cana-1260	347	22	1	1	NUM
cana-1260	347	23	therefore	therefore	ADV
cana-1260	347	24	there	there	PRON
cana-1260	347	25	does	do	AUX
cana-1260	347	26	not	not	PART
cana-1260	347	27	exist	exist	VERB
cana-1260	347	28	such	such	ADJ
cana-1260	347	29	𝑘	𝑘	ADP
cana-1260	347	30	where	where	SCONJ
cana-1260	347	31	𝑘	𝑘	PRON
cana-1260	347	32	<	<	X
cana-1260	347	33	3𝑛+2	3𝑛+2	PROPN
cana-1260	347	34	5	5	NUM
cana-1260	347	35	.	.	PUNCT
cana-1260	348	1	hence	hence	ADV
cana-1260	348	2	with	with	ADP
cana-1260	348	3	minimal	minimal	ADJ
cana-1260	348	4	weights	weight	NOUN
cana-1260	348	5	we	we	PRON
cana-1260	348	6	have	have	VERB
cana-1260	348	7	𝛾𝑓	𝛾𝑓	VERB
cana-1260	348	8	(	(	PUNCT
cana-1260	348	9	𝐾3	𝐾3	VERB
cana-1260	348	10	×	×	PROPN
cana-1260	348	11	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	348	12	)	)	PUNCT
cana-1260	348	13	=	=	PUNCT
cana-1260	349	1	3𝑛+2	3𝑛+2	NUM
cana-1260	349	2	5	5	NUM
cana-1260	349	3	.	.	PUNCT
cana-1260	350	1	(	(	PUNCT
cana-1260	350	2	a	a	X
cana-1260	350	3	)	)	PUNCT
cana-1260	350	4	cartesian	cartesian	ADJ
cana-1260	350	5	product	product	NOUN
cana-1260	350	6	of	of	ADP
cana-1260	350	7	(	(	PUNCT
cana-1260	350	8	𝑲𝟑	𝑲𝟑	ADJ
cana-1260	350	9	×	×	NOUN
cana-1260	350	10	𝑷𝟑	𝑷𝟑	NOUN
cana-1260	350	11	)	)	PUNCT
cana-1260	350	12	figure	figure	NOUN
cana-1260	350	13	2.8.1	2.8.1	NUM
cana-1260	350	14	in	in	ADP
cana-1260	350	15	its	its	PRON
cana-1260	350	16	line	line	NOUN
cana-1260	350	17	graph	graph	NOUN
cana-1260	350	18	𝐿(𝐺1	𝐿(𝐺1	NOUN
cana-1260	350	19	)	)	PUNCT
cana-1260	350	20	=	=	PUNCT
cana-1260	351	1	𝐿(𝐾3	𝐿(𝐾3	NOUN
cana-1260	351	2	×	×	NOUN
cana-1260	351	3	𝑃3	𝑃3	NOUN
cana-1260	351	4	)	)	PUNCT
cana-1260	351	5	of	of	ADP
cana-1260	351	6	order	order	NOUN
cana-1260	351	7	6(3	6(3	NUM
cana-1260	351	8	)	)	PUNCT
cana-1260	351	9	−	−	NOUN
cana-1260	351	10	3	3	NUM
cana-1260	351	11	=	=	SYM
cana-1260	351	12	15	15	NUM
cana-1260	351	13	we	we	PRON
cana-1260	351	14	get	get	VERB
cana-1260	351	15	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	351	16	(	(	PUNCT
cana-1260	351	17	𝐿(𝐾3	𝐿(𝐾3	ADV
cana-1260	351	18	×	×	NOUN
cana-1260	351	19	𝑃3	𝑃3	NOUN
cana-1260	351	20	)	)	PUNCT
cana-1260	351	21	)	)	PUNCT
cana-1260	352	1	=	=	PUNCT
cana-1260	352	2	3[2(3)+1	3[2(3)+1	NUM
cana-1260	352	3	]	]	SYM
cana-1260	352	4	7	7	NUM
cana-1260	352	5	=	=	SYM
cana-1260	352	6	21	21	NUM
cana-1260	352	7	7	7	NUM
cana-1260	352	8	=	=	SYM
cana-1260	352	9	3	3	X
cana-1260	352	10	.	.	PUNCT
cana-1260	352	11	by	by	ADP
cana-1260	352	12	assigning	assign	VERB
cana-1260	352	13	weight	weight	NOUN
cana-1260	352	14	(	(	PUNCT
cana-1260	352	15	1/7	1/7	NUM
cana-1260	352	16	)	)	PUNCT
cana-1260	352	17	to	to	ADP
cana-1260	352	18	every	every	DET
cana-1260	352	19	vertex	vertex	NOUN
cana-1260	352	20	of	of	ADP
cana-1260	352	21	the	the	DET
cana-1260	352	22	set	set	NOUN
cana-1260	352	23	{	{	PUNCT
cana-1260	352	24	𝑒1	𝑒1	NOUN
cana-1260	352	25	,	,	PUNCT
cana-1260	352	26	𝑒3	𝑒3	NOUN
cana-1260	352	27	,	,	PUNCT
cana-1260	352	28	𝑒5	𝑒5	NOUN
cana-1260	352	29	,	,	PUNCT
cana-1260	352	30	𝑒6	𝑒6	NOUN
cana-1260	352	31	,	,	PUNCT
cana-1260	352	32	𝑒9	𝑒9	NOUN
cana-1260	352	33	,	,	PUNCT
cana-1260	352	34	𝑒10	𝑒10	NOUN
cana-1260	352	35	,	,	PUNCT
cana-1260	352	36	,	,	PUNCT
cana-1260	352	37	𝑒12	𝑒12	PROPN
cana-1260	352	38	,	,	PUNCT
cana-1260	352	39	𝑒14	𝑒14	ADJ
cana-1260	352	40	,	,	PUNCT
cana-1260	352	41	𝑒15	𝑒15	NOUN
cana-1260	352	42	}	}	PUNCT
cana-1260	352	43	and	and	CCONJ
cana-1260	352	44	weight	weight	NOUN
cana-1260	352	45	(	(	PUNCT
cana-1260	352	46	2/7	2/7	NUM
cana-1260	352	47	)	)	PUNCT
cana-1260	352	48	to	to	ADP
cana-1260	352	49	every	every	DET
cana-1260	352	50	vertex	vertex	NOUN
cana-1260	352	51	of	of	ADP
cana-1260	352	52	the	the	DET
cana-1260	352	53	set	set	NOUN
cana-1260	352	54	{	{	PUNCT
cana-1260	352	55	𝑒2	𝑒2	NOUN
cana-1260	352	56	,	,	PUNCT
cana-1260	352	57	𝑒4	𝑒4	PROPN
cana-1260	352	58	,	,	PUNCT
cana-1260	352	59	𝑒7	𝑒7	NOUN
cana-1260	352	60	,	,	PUNCT
cana-1260	352	61	𝑒8	𝑒8	PROPN
cana-1260	352	62	𝑒11	𝑒11	NOUN
cana-1260	352	63	,	,	PUNCT
cana-1260	352	64	𝑒13	𝑒13	PROPN
cana-1260	352	65	}	}	PUNCT
cana-1260	352	66	see	see	VERB
cana-1260	352	67	the	the	DET
cana-1260	352	68	figure	figure	NOUN
cana-1260	352	69	2.8.1b	2.8.1b	NUM
cana-1260	352	70	)	)	PUNCT
cana-1260	352	71	.	.	PUNCT
cana-1260	353	1	(	(	PUNCT
cana-1260	353	2	b	b	X
cana-1260	353	3	)	)	PUNCT
cana-1260	353	4	line	line	NOUN
cana-1260	353	5	graph	graph	NOUN
cana-1260	353	6	of	of	ADP
cana-1260	353	7	cartesian	cartesian	ADJ
cana-1260	353	8	product	product	NOUN
cana-1260	353	9	of	of	ADP
cana-1260	353	10	(	(	PUNCT
cana-1260	353	11	𝑲𝟑	𝑲𝟑	ADJ
cana-1260	353	12	×	×	NOUN
cana-1260	353	13	𝑷𝟑	𝑷𝟑	NOUN
cana-1260	353	14	)	)	PUNCT
cana-1260	353	15	figure	figure	NOUN
cana-1260	353	16	2.8.1	2.8.1	NUM
cana-1260	353	17	let	let	VERB
cana-1260	353	18	𝐺2	𝐺2	NOUN
cana-1260	353	19	=	=	PUNCT
cana-1260	353	20	(	(	PUNCT
cana-1260	353	21	𝐾3	𝐾3	VERB
cana-1260	353	22	×	×	PROPN
cana-1260	353	23	𝑃4	𝑃4	NOUN
cana-1260	353	24	)	)	PUNCT
cana-1260	353	25	be	be	VERB
cana-1260	353	26	the	the	DET
cana-1260	353	27	cartesian	cartesian	ADJ
cana-1260	353	28	product	product	NOUN
cana-1260	353	29	of	of	ADP
cana-1260	353	30	𝐾3	𝐾3	NOUN
cana-1260	353	31	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1260	353	32	𝑃4	𝑃4	NOUN
cana-1260	353	33	we	we	PRON
cana-1260	353	34	can	can	AUX
cana-1260	353	35	see	see	VERB
cana-1260	353	36	that	that	SCONJ
cana-1260	353	37	it	it	PRON
cana-1260	353	38	is	be	AUX
cana-1260	353	39	graph	graph	NOUN
cana-1260	353	40	of	of	ADP
cana-1260	353	41	order	order	NOUN
cana-1260	353	42	12	12	NUM
cana-1260	353	43	so	so	ADV
cana-1260	353	44	its	its	PRON
cana-1260	353	45	fractional	fractional	ADJ
cana-1260	353	46	domination	domination	NOUN
cana-1260	353	47	number	number	NOUN
cana-1260	353	48	is	be	AUX
cana-1260	353	49	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	353	50	(	(	PUNCT
cana-1260	353	51	𝐾3	𝐾3	VERB
cana-1260	353	52	×	×	PROPN
cana-1260	353	53	𝑃4	𝑃4	NOUN
cana-1260	353	54	)	)	PUNCT
cana-1260	353	55	=	=	PUNCT
cana-1260	354	1	3(4)+2	3(4)+2	NUM
cana-1260	354	2	5	5	NUM
cana-1260	354	3	=	=	SYM
cana-1260	354	4	14/5	14/5	NUM
cana-1260	354	5	by	by	ADP
cana-1260	354	6	assigning	assign	VERB
cana-1260	354	7	weight	weight	NOUN
cana-1260	354	8	(	(	PUNCT
cana-1260	354	9	1/5	1/5	NUM
cana-1260	354	10	)	)	PUNCT
cana-1260	354	11	to	to	ADP
cana-1260	354	12	10	10	NUM
cana-1260	354	13	number	number	NOUN
cana-1260	354	14	of	of	ADP
cana-1260	354	15	vertices	vertex	NOUN
cana-1260	354	16	and	and	CCONJ
cana-1260	354	17	weight	weight	NOUN
cana-1260	354	18	(	(	PUNCT
cana-1260	354	19	2/5	2/5	NUM
cana-1260	354	20	)	)	PUNCT
cana-1260	354	21	to	to	ADP
cana-1260	354	22	other	other	ADJ
cana-1260	354	23	two	two	NUM
cana-1260	354	24	vertices	vertex	NOUN
cana-1260	354	25	.	.	PUNCT
cana-1260	355	1	in	in	ADP
cana-1260	355	2	its	its	PRON
cana-1260	355	3	line	line	NOUN
cana-1260	355	4	graph	graph	NOUN
cana-1260	355	5	𝐿(𝐺2	𝐿(𝐺2	PROPN
cana-1260	355	6	)	)	PUNCT
cana-1260	355	7	=	=	PUNCT
cana-1260	356	1	𝐿(𝐾3	𝐿(𝐾3	ADJ
cana-1260	356	2	×	×	NOUN
cana-1260	356	3	𝑃4	𝑃4	NOUN
cana-1260	356	4	)	)	PUNCT
cana-1260	356	5	of	of	ADP
cana-1260	356	6	order	order	NOUN
cana-1260	356	7	[	[	X
cana-1260	356	8	6(4	6(4	NOUN
cana-1260	356	9	)	)	PUNCT
cana-1260	356	10	−	−	NOUN
cana-1260	357	1	3	3	X
cana-1260	357	2	]	]	X
cana-1260	357	3	=	=	SYM
cana-1260	357	4	21	21	NUM
cana-1260	357	5	so	so	ADV
cana-1260	357	6	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	357	7	(	(	PUNCT
cana-1260	357	8	𝐿(𝐾3	𝐿(𝐾3	ADV
cana-1260	357	9	×	×	NOUN
cana-1260	357	10	𝑃4	𝑃4	NOUN
cana-1260	357	11	)	)	PUNCT
cana-1260	357	12	)	)	PUNCT
cana-1260	358	1	=	=	PUNCT
cana-1260	358	2	3[2(4)+1	3[2(4)+1	NUM
cana-1260	358	3	]	]	X
cana-1260	358	4	7	7	NUM
cana-1260	358	5	=	=	SYM
cana-1260	358	6	27	27	NUM
cana-1260	358	7	7	7	NUM
cana-1260	358	8	by	by	ADP
cana-1260	358	9	assigning	assign	VERB
cana-1260	358	10	weight	weight	NOUN
cana-1260	358	11	(	(	PUNCT
cana-1260	358	12	1/7	1/7	NUM
cana-1260	358	13	)	)	PUNCT
cana-1260	358	14	to	to	ADP
cana-1260	358	15	15	15	NUM
cana-1260	358	16	number	number	NOUN
cana-1260	358	17	of	of	ADP
cana-1260	358	18	vertices	vertex	NOUN
cana-1260	358	19	and	and	CCONJ
cana-1260	358	20	weight	weight	NOUN
cana-1260	358	21	(	(	PUNCT
cana-1260	358	22	2/7	2/7	NUM
cana-1260	358	23	)	)	PUNCT
cana-1260	358	24	to	to	ADP
cana-1260	358	25	six	six	NUM
cana-1260	358	26	number	number	NOUN
cana-1260	358	27	of	of	ADP
cana-1260	358	28	vertices	vertex	NOUN
cana-1260	358	29	.	.	PUNCT
cana-1260	359	1	these	these	PRON
cana-1260	359	2	are	be	AUX
cana-1260	359	3	minimal	minimal	ADJ
cana-1260	359	4	fractional	fractional	ADJ
cana-1260	359	5	domination	domination	NOUN
cana-1260	359	6	numbers	number	NOUN
cana-1260	359	7	by	by	ADP
cana-1260	359	8	definition	definition	NOUN
cana-1260	359	9	satisfying	satisfy	VERB
cana-1260	359	10	the	the	DET
cana-1260	359	11	condition	condition	NOUN
cana-1260	359	12	where	where	SCONJ
cana-1260	359	13	the	the	DET
cana-1260	359	14	vertex	vertex	NOUN
cana-1260	359	15	𝑤	𝑤	ADP
cana-1260	359	16	∈	∈	PROPN
cana-1260	359	17	𝑁[𝑉	𝑁[𝑉	NOUN
cana-1260	359	18	]	]	PUNCT
cana-1260	359	19	such	such	ADJ
cana-1260	359	20	that	that	SCONJ
cana-1260	359	21	∑	∑	PUNCT
cana-1260	359	22	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1260	359	23	)	)	PUNCT
cana-1260	359	24	⬚	⬚	PROPN
cana-1260	359	25	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	359	26	]	]	X
cana-1260	359	27	=	=	SYM
cana-1260	359	28	1	1	X
cana-1260	359	29	.	.	PUNCT
cana-1260	360	1	hence	hence	ADV
cana-1260	360	2	if	if	SCONJ
cana-1260	360	3	graph	graph	NOUN
cana-1260	360	4	𝐺	𝐺	PROPN
cana-1260	360	5	is	be	AUX
cana-1260	360	6	the	the	DET
cana-1260	360	7	cartesian	cartesian	ADJ
cana-1260	360	8	product	product	NOUN
cana-1260	360	9	(	(	PUNCT
cana-1260	360	10	𝐾3	𝐾3	VERB
cana-1260	360	11	×	×	PROPN
cana-1260	360	12	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	360	13	)	)	PUNCT
cana-1260	360	14	of	of	ADP
cana-1260	360	15	order	order	NOUN
cana-1260	360	16	3𝑛	3𝑛	NUM
cana-1260	360	17	for	for	ADP
cana-1260	360	18	𝑛	𝑛	PRON
cana-1260	360	19	≥	≥	NUM
cana-1260	360	20	3	3	NUM
cana-1260	360	21	then	then	ADV
cana-1260	360	22	for	for	SCONJ
cana-1260	360	23	the	the	DET
cana-1260	360	24	fractional	fractional	ADJ
cana-1260	360	25	communications	communication	NOUN
cana-1260	360	26	on	on	ADP
cana-1260	360	27	applied	apply	VERB
cana-1260	360	28	nonlinear	nonlinear	ADJ
cana-1260	360	29	analysis	analysis	NOUN
cana-1260	360	30	issn	issn	NOUN
cana-1260	360	31	:	:	PUNCT
cana-1260	360	32	1074	1074	NUM
cana-1260	360	33	-	-	PUNCT
cana-1260	360	34	133x	133x	NUM
cana-1260	360	35	vol	vol	NOUN
cana-1260	360	36	31	31	NUM
cana-1260	360	37	no	no	NOUN
cana-1260	360	38	.	.	PUNCT
cana-1260	361	1	6s	6s	NUM
cana-1260	361	2	(	(	PUNCT
cana-1260	361	3	2024	2024	NUM
cana-1260	361	4	)	)	PUNCT
cana-1260	361	5	684	684	NUM
cana-1260	361	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	361	7	domination	domination	NOUN
cana-1260	361	8	numbers	number	NOUN
cana-1260	361	9	we	we	PRON
cana-1260	361	10	found	find	VERB
cana-1260	361	11	sequence	sequence	NOUN
cana-1260	361	12	of	of	ADP
cana-1260	361	13	numbers	number	NOUN
cana-1260	361	14	as	as	ADP
cana-1260	361	15	{	{	PUNCT
cana-1260	361	16	11/5	11/5	NUM
cana-1260	361	17	,	,	PUNCT
cana-1260	361	18	14/5	14/5	NUM
cana-1260	361	19	,	,	PUNCT
cana-1260	361	20	17/5	17/5	NUM
cana-1260	361	21	,	,	PUNCT
cana-1260	361	22	20/5	20/5	NUM
cana-1260	361	23	,	,	PUNCT
cana-1260	361	24	…	…	PUNCT
cana-1260	361	25	3𝑛+2	3𝑛+2	NUM
cana-1260	361	26	5	5	NUM
cana-1260	361	27	}	}	PUNCT
cana-1260	361	28	and	and	CCONJ
cana-1260	361	29	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	361	30	)	)	PUNCT
cana-1260	361	31	be	be	VERB
cana-1260	361	32	its	its	PRON
cana-1260	361	33	line	line	NOUN
cana-1260	361	34	graph	graph	NOUN
cana-1260	361	35	of	of	ADP
cana-1260	361	36	order	order	NOUN
cana-1260	361	37	(	(	PUNCT
cana-1260	361	38	6𝑛	6𝑛	NOUN
cana-1260	361	39	−	−	NOUN
cana-1260	361	40	3	3	NUM
cana-1260	361	41	)	)	PUNCT
cana-1260	361	42	then	then	ADV
cana-1260	361	43	fractional	fractional	ADJ
cana-1260	361	44	domination	domination	NOUN
cana-1260	361	45	numbers	number	NOUN
cana-1260	361	46	formed	form	VERB
cana-1260	361	47	by	by	ADP
cana-1260	361	48	sequence	sequence	NOUN
cana-1260	361	49	{	{	PUNCT
cana-1260	361	50	21/7	21/7	NUM
cana-1260	361	51	,	,	PUNCT
cana-1260	361	52	27/7	27/7	NUM
cana-1260	361	53	,	,	PUNCT
cana-1260	361	54	33/7	33/7	NUM
cana-1260	361	55	,	,	PUNCT
cana-1260	361	56	39/7	39/7	NUM
cana-1260	361	57	,	,	PUNCT
cana-1260	361	58	…	…	PUNCT
cana-1260	361	59	3(2𝑛+1	3(2𝑛+1	NUM
cana-1260	361	60	)	)	PUNCT
cana-1260	361	61	7	7	NUM
cana-1260	361	62	}	}	PUNCT
cana-1260	361	63	therefore	therefore	ADV
cana-1260	361	64	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	361	65	(	(	PUNCT
cana-1260	361	66	𝐾3	𝐾3	VERB
cana-1260	361	67	×	×	PROPN
cana-1260	361	68	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	361	69	)	)	PUNCT
cana-1260	361	70	=	=	SYM
cana-1260	362	1	3𝑛+2	3𝑛+2	NUM
cana-1260	362	2	5	5	NUM
cana-1260	362	3	and	and	CCONJ
cana-1260	362	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	362	5	(	(	PUNCT
cana-1260	362	6	𝐿(𝐾3	𝐿(𝐾3	ADV
cana-1260	362	7	×	×	NOUN
cana-1260	362	8	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	362	9	)	)	PUNCT
cana-1260	362	10	)	)	PUNCT
cana-1260	363	1	=	=	SYM
cana-1260	363	2	3(2𝑛+1	3(2𝑛+1	X
cana-1260	363	3	)	)	PUNCT
cana-1260	363	4	7	7	NUM
cana-1260	363	5	.	.	PUNCT
cana-1260	364	1	in	in	ADP
cana-1260	364	2	contradiction	contradiction	NOUN
cana-1260	364	3	if	if	SCONJ
cana-1260	364	4	we	we	PRON
cana-1260	364	5	have	have	AUX
cana-1260	364	6	𝛾𝑓	𝛾𝑓	VERB
cana-1260	364	7	(	(	PUNCT
cana-1260	364	8	𝐿(𝐾3	𝐿(𝐾3	ADV
cana-1260	364	9	×	×	NOUN
cana-1260	364	10	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	364	11	)	)	PUNCT
cana-1260	364	12	)	)	PUNCT
cana-1260	365	1	=	=	PUNCT
cana-1260	365	2	𝑘	𝑘	PROPN
cana-1260	365	3	and	and	CCONJ
cana-1260	365	4	𝑘	𝑘	DET
cana-1260	365	5	<	<	X
cana-1260	365	6	3(2𝑛+1	3(2𝑛+1	NUM
cana-1260	365	7	)	)	PUNCT
cana-1260	365	8	7	7	NUM
cana-1260	365	9	then	then	ADV
cana-1260	365	10	adjacency	adjacency	PROPN
cana-1260	365	11	relation	relation	NOUN
cana-1260	365	12	is	be	AUX
cana-1260	365	13	not	not	PART
cana-1260	365	14	satisfied	satisfied	ADJ
cana-1260	365	15	with	with	ADP
cana-1260	365	16	∑	∑	ADV
cana-1260	365	17	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1260	365	18	)	)	PUNCT
cana-1260	365	19	⬚	⬚	PROPN
cana-1260	365	20	𝑣∈𝑁[𝑤	𝑣∈𝑁[𝑤	PROPN
cana-1260	365	21	]	]	X
cana-1260	365	22	=	=	SYM
cana-1260	365	23	1	1	NUM
cana-1260	365	24	therefore	therefore	ADV
cana-1260	365	25	there	there	PRON
cana-1260	365	26	does	do	AUX
cana-1260	365	27	not	not	PART
cana-1260	365	28	exist	exist	VERB
cana-1260	365	29	such	such	ADJ
cana-1260	365	30	𝑘	𝑘	ADP
cana-1260	365	31	where	where	SCONJ
cana-1260	365	32	𝑘	𝑘	PRON
cana-1260	365	33	<	<	X
cana-1260	365	34	3(2𝑛+1	3(2𝑛+1	NUM
cana-1260	365	35	)	)	PUNCT
cana-1260	365	36	7	7	NUM
cana-1260	365	37	.	.	PUNCT
cana-1260	366	1	hence	hence	ADV
cana-1260	366	2	with	with	ADP
cana-1260	366	3	minimal	minimal	ADJ
cana-1260	366	4	weights	weight	NOUN
cana-1260	366	5	we	we	PRON
cana-1260	366	6	have	have	VERB
cana-1260	366	7	𝛾𝑓	𝛾𝑓	VERB
cana-1260	366	8	(	(	PUNCT
cana-1260	366	9	𝐿(𝐾3	𝐿(𝐾3	ADV
cana-1260	366	10	×	×	NOUN
cana-1260	366	11	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	366	12	)	)	PUNCT
cana-1260	366	13	)	)	PUNCT
cana-1260	367	1	=	=	SYM
cana-1260	367	2	3(2𝑛+1	3(2𝑛+1	X
cana-1260	367	3	)	)	PUNCT
cana-1260	367	4	7	7	NUM
cana-1260	367	5	.	.	PUNCT
cana-1260	368	1	□	□	PUNCT
cana-1260	368	2	2.9	2.9	NUM
cana-1260	368	3	the	the	DET
cana-1260	368	4	cartesian	cartesian	ADJ
cana-1260	368	5	product	product	NOUN
cana-1260	368	6	of	of	ADP
cana-1260	368	7	two	two	NUM
cana-1260	368	8	cycle	cycle	NOUN
cana-1260	368	9	graphs	graph	NOUN
cana-1260	368	10	𝑪𝒎	𝑪𝒎	PROPN
cana-1260	368	11	×	×	PROPN
cana-1260	368	12	𝑪𝒏	𝑪𝒏	PROPN
cana-1260	368	13	cycle	cycle	NOUN
cana-1260	368	14	graph	graph	NOUN
cana-1260	368	15	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	368	16	and	and	CCONJ
cana-1260	368	17	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	368	18	taken	take	VERB
cana-1260	368	19	to	to	PART
cana-1260	368	20	form	form	VERB
cana-1260	368	21	cartesian	cartesian	ADJ
cana-1260	368	22	product	product	NOUN
cana-1260	368	23	and	and	CCONJ
cana-1260	368	24	is	be	AUX
cana-1260	368	25	denoted	denote	VERB
cana-1260	368	26	by	by	ADP
cana-1260	368	27	(	(	PUNCT
cana-1260	368	28	𝐶𝑚	𝐶𝑚	PRON
cana-1260	368	29	×	×	NOUN
cana-1260	368	30	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	368	31	)	)	PUNCT
cana-1260	368	32	whose	whose	DET
cana-1260	368	33	vertex	vertex	NOUN
cana-1260	368	34	set	set	NOUN
cana-1260	368	35	is	be	AUX
cana-1260	368	36	𝑉	𝑉	PROPN
cana-1260	368	37	(	(	PUNCT
cana-1260	368	38	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	368	39	)	)	PUNCT
cana-1260	368	40	×	×	PROPN
cana-1260	368	41	𝑉	𝑉	PROPN
cana-1260	368	42	(	(	PUNCT
cana-1260	368	43	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	368	44	)	)	PUNCT
cana-1260	368	45	where	where	SCONJ
cana-1260	368	46	𝑢	𝑢	AUX
cana-1260	368	47	=	=	SYM
cana-1260	368	48	(	(	PUNCT
cana-1260	368	49	𝑢1	𝑢1	PROPN
cana-1260	368	50	,	,	PUNCT
cana-1260	368	51	𝑢2	𝑢2	PROPN
cana-1260	368	52	)	)	PUNCT
cana-1260	368	53	and	and	CCONJ
cana-1260	368	54	𝑣	𝑣	X
cana-1260	368	55	=	=	PUNCT
cana-1260	368	56	(	(	PUNCT
cana-1260	368	57	𝑣1	𝑣1	PROPN
cana-1260	368	58	,	,	PUNCT
cana-1260	368	59	𝑣2	𝑣2	PROPN
cana-1260	368	60	)	)	PUNCT
cana-1260	368	61	are	be	AUX
cana-1260	368	62	connected	connect	VERB
cana-1260	368	63	if	if	SCONJ
cana-1260	368	64	𝑢1	𝑢1	PROPN
cana-1260	368	65	=	=	SYM
cana-1260	368	66	𝑣1	𝑣1	PROPN
cana-1260	368	67	in	in	ADP
cana-1260	368	68	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	368	69	and	and	CCONJ
cana-1260	368	70	𝑢2	𝑢2	PROPN
cana-1260	368	71	is	be	AUX
cana-1260	368	72	adjacent	adjacent	ADJ
cana-1260	368	73	to	to	ADP
cana-1260	368	74	𝑣2	𝑣2	PROPN
cana-1260	368	75	in	in	ADP
cana-1260	368	76	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	368	77	or	or	CCONJ
cana-1260	368	78	𝑢1	𝑢1	PROPN
cana-1260	368	79	is	be	AUX
cana-1260	368	80	adjacent	adjacent	ADJ
cana-1260	368	81	to	to	PART
cana-1260	368	82	𝑣1	𝑣1	VERB
cana-1260	368	83	in	in	ADP
cana-1260	368	84	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	368	85	and	and	CCONJ
cana-1260	368	86	𝑢2	𝑢2	PROPN
cana-1260	368	87	=	=	PROPN
cana-1260	368	88	𝑣2	𝑣2	PROPN
cana-1260	368	89	in	in	ADP
cana-1260	368	90	𝐶𝑛.	𝐶𝑛.	PROPN
cana-1260	368	91	it	it	PRON
cana-1260	368	92	is	be	AUX
cana-1260	368	93	also	also	ADV
cana-1260	368	94	known	know	VERB
cana-1260	368	95	as	as	SCONJ
cana-1260	368	96	torus	torus	NOUN
cana-1260	368	97	grid	grid	NOUN
cana-1260	368	98	graph	graph	NOUN
cana-1260	368	99	𝑇𝑚,𝑛	𝑇𝑚,𝑛	PROPN
cana-1260	368	100	formed	form	VERB
cana-1260	368	101	by	by	ADP
cana-1260	368	102	cartesian	cartesian	ADJ
cana-1260	368	103	product	product	NOUN
cana-1260	368	104	of	of	ADP
cana-1260	368	105	two	two	NUM
cana-1260	368	106	cycle	cycle	NOUN
cana-1260	368	107	graphs	graph	NOUN
cana-1260	368	108	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	368	109	and	and	CCONJ
cana-1260	368	110	𝐶𝑛.	𝐶𝑛.	PROPN
cana-1260	368	111	theorem	theorem	NOUN
cana-1260	368	112	2.9.1	2.9.1	NUM
cana-1260	368	113	let	let	VERB
cana-1260	368	114	cycle	cycle	NOUN
cana-1260	368	115	graphs	graph	NOUN
cana-1260	369	1	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	369	2	and	and	CCONJ
cana-1260	369	3	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	369	4	are	be	AUX
cana-1260	369	5	two	two	NUM
cana-1260	369	6	graphs	graph	NOUN
cana-1260	369	7	with	with	ADP
cana-1260	369	8	𝑚	𝑚	PROPN
cana-1260	369	9	and	and	CCONJ
cana-1260	369	10	𝑛	𝑛	DET
cana-1260	369	11	vertices	vertex	NOUN
cana-1260	369	12	respectively	respectively	ADV
cana-1260	369	13	and	and	CCONJ
cana-1260	369	14	let	let	VERB
cana-1260	369	15	𝐺	𝐺	NOUN
cana-1260	369	16	=	=	PUNCT
cana-1260	370	1	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	370	2	×	×	NOUN
cana-1260	370	3	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	370	4	is	be	AUX
cana-1260	370	5	cartesian	cartesian	ADJ
cana-1260	370	6	product	product	NOUN
cana-1260	370	7	of	of	ADP
cana-1260	370	8	two	two	NUM
cana-1260	370	9	cycle	cycle	NOUN
cana-1260	370	10	graphs	graph	NOUN
cana-1260	370	11	.	.	PUNCT
cana-1260	371	1	let	let	VERB
cana-1260	371	2	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	371	3	)	)	PUNCT
cana-1260	371	4	be	be	AUX
cana-1260	371	5	its	its	PRON
cana-1260	371	6	line	line	NOUN
cana-1260	371	7	graph	graph	NOUN
cana-1260	371	8	then	then	ADV
cana-1260	371	9	𝑖	𝑖	NUM
cana-1260	371	10	)	)	PUNCT
cana-1260	372	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	372	2	(	(	PUNCT
cana-1260	372	3	𝐺	𝐺	NOUN
cana-1260	372	4	)	)	PUNCT
cana-1260	373	1	+	+	CCONJ
cana-1260	373	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	373	3	(	(	PUNCT
cana-1260	373	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	373	5	)	)	PUNCT
cana-1260	373	6	)	)	PUNCT
cana-1260	374	1	=	=	PUNCT
cana-1260	374	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	374	3	5	5	NUM
cana-1260	374	4	+	+	CCONJ
cana-1260	374	5	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	374	6	7	7	NUM
cana-1260	374	7	𝑖𝑖	𝑖𝑖	NOUN
cana-1260	374	8	)	)	PUNCT
cana-1260	374	9	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	374	10	(	(	PUNCT
cana-1260	374	11	𝐺	𝐺	NOUN
cana-1260	374	12	)	)	PUNCT
cana-1260	374	13	∗	∗	NOUN
cana-1260	374	14	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	374	15	(	(	PUNCT
cana-1260	374	16	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	374	17	)	)	PUNCT
cana-1260	374	18	)	)	PUNCT
cana-1260	375	1	=	=	PUNCT
cana-1260	375	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	375	3	5	5	NUM
cana-1260	375	4	∗	∗	NOUN
cana-1260	375	5	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	375	6	7	7	NUM
cana-1260	375	7	,	,	PUNCT
cana-1260	375	8	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
cana-1260	375	9	)	)	PUNCT
cana-1260	375	10	г𝑓	г𝑓	PROPN
cana-1260	375	11	(	(	PUNCT
cana-1260	375	12	𝐺	𝐺	NOUN
cana-1260	375	13	)	)	PUNCT
cana-1260	375	14	+	+	NUM
cana-1260	375	15	г𝑓	г𝑓	ADJ
cana-1260	375	16	(	(	PUNCT
cana-1260	375	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	375	18	)	)	PUNCT
cana-1260	375	19	)	)	PUNCT
cana-1260	376	1	≤	≤	NUM
cana-1260	376	2	𝑚𝑛	𝑚𝑛	ADP
cana-1260	377	1	4	4	NUM
cana-1260	377	2	+	+	CCONJ
cana-1260	377	3	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	377	4	3	3	NUM
cana-1260	377	5	𝑖𝑣	𝑖𝑣	NOUN
cana-1260	377	6	)	)	PUNCT
cana-1260	377	7	г𝑓	г𝑓	PROPN
cana-1260	377	8	(	(	PUNCT
cana-1260	377	9	𝐺	𝐺	NOUN
cana-1260	377	10	)	)	PUNCT
cana-1260	377	11	∗	∗	NOUN
cana-1260	377	12	г𝑓	г𝑓	PROPN
cana-1260	377	13	(	(	PUNCT
cana-1260	377	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	377	15	)	)	PUNCT
cana-1260	377	16	)	)	PUNCT
cana-1260	377	17	≤	≤	NUM
cana-1260	378	1	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	378	2	4	4	NUM
cana-1260	378	3	∗	∗	NOUN
cana-1260	378	4	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	378	5	3	3	NUM
cana-1260	378	6	.	.	PUNCT
cana-1260	379	1	proof	proof	NOUN
cana-1260	379	2	i	i	PRON
cana-1260	379	3	)	)	PUNCT
cana-1260	379	4	and	and	CCONJ
cana-1260	379	5	ii	ii	NUM
cana-1260	379	6	):	):	PUNCT
cana-1260	379	7	we	we	PRON
cana-1260	379	8	have	have	AUX
cana-1260	379	9	taken	take	VERB
cana-1260	379	10	reference	reference	NOUN
cana-1260	379	11	to	to	ADP
cana-1260	379	12	[	[	X
cana-1260	379	13	7	7	NUM
cana-1260	379	14	]	]	PUNCT
cana-1260	379	15	,	,	PUNCT
cana-1260	379	16	in	in	ADP
cana-1260	379	17	graphs	graph	NOUN
cana-1260	379	18	of	of	ADP
cana-1260	379	19	cartesian	cartesian	ADJ
cana-1260	379	20	product	product	NOUN
cana-1260	380	1	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	380	2	×	×	NOUN
cana-1260	380	3	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	380	4	)	)	PUNCT
cana-1260	380	5	the	the	DET
cana-1260	380	6	set	set	NOUN
cana-1260	380	7	of	of	ADP
cana-1260	380	8	vertices	vertex	NOUN
cana-1260	380	9	of	of	ADP
cana-1260	380	10	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	380	11	will	will	AUX
cana-1260	380	12	be	be	AUX
cana-1260	380	13	𝑉	𝑉	PROPN
cana-1260	380	14	=	=	SYM
cana-1260	380	15	{	{	PUNCT
cana-1260	380	16	𝑣1	𝑣1	NOUN
cana-1260	380	17	,	,	PUNCT
cana-1260	380	18	𝑣2	𝑣2	NOUN
cana-1260	380	19	,	,	PUNCT
cana-1260	380	20	𝑣3	𝑣3	PROPN
cana-1260	380	21	,	,	PUNCT
cana-1260	380	22	…	…	PUNCT
cana-1260	380	23	,	,	PUNCT
cana-1260	380	24	𝑣𝑚	𝑣𝑚	VERB
cana-1260	380	25	}	}	PUNCT
cana-1260	380	26	and	and	CCONJ
cana-1260	380	27	the	the	DET
cana-1260	380	28	set	set	NOUN
cana-1260	380	29	of	of	ADP
cana-1260	380	30	vertices	vertex	NOUN
cana-1260	380	31	of	of	ADP
cana-1260	380	32	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	380	33	will	will	AUX
cana-1260	380	34	be	be	AUX
cana-1260	380	35	𝑈	𝑈	PROPN
cana-1260	380	36	=	=	SYM
cana-1260	380	37	{	{	PUNCT
cana-1260	380	38	𝑢1	𝑢1	PROPN
cana-1260	380	39	,	,	PUNCT
cana-1260	380	40	𝑢2	𝑢2	PROPN
cana-1260	380	41	,	,	PUNCT
cana-1260	380	42	𝑢3	𝑢3	PROPN
cana-1260	380	43	,	,	PUNCT
cana-1260	380	44	…	…	PUNCT
cana-1260	380	45	,	,	PUNCT
cana-1260	380	46	𝑢𝑚	𝑢𝑚	INTJ
cana-1260	380	47	}	}	PUNCT
cana-1260	380	48	the	the	DET
cana-1260	380	49	graph	graph	NOUN
cana-1260	380	50	𝐺	𝐺	PROPN
cana-1260	380	51	has	have	VERB
cana-1260	380	52	order	order	NOUN
cana-1260	380	53	𝑚𝑛	𝑚𝑛	PRON
cana-1260	380	54	and	and	CCONJ
cana-1260	380	55	size	size	NOUN
cana-1260	380	56	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	380	57	with	with	ADP
cana-1260	380	58	each	each	DET
cana-1260	380	59	vertex	vertex	NOUN
cana-1260	380	60	has	have	AUX
cana-1260	380	61	degree	degree	NOUN
cana-1260	380	62	4	4	NUM
cana-1260	380	63	or	or	CCONJ
cana-1260	380	64	4regular	4regular	NUM
cana-1260	380	65	graph	graph	NOUN
cana-1260	380	66	.	.	PUNCT
cana-1260	381	1	the	the	DET
cana-1260	381	2	line	line	NOUN
cana-1260	381	3	graph	graph	NOUN
cana-1260	381	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	381	5	)	)	PUNCT
cana-1260	381	6	has	have	VERB
cana-1260	381	7	order	order	NOUN
cana-1260	381	8	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	381	9	with	with	ADP
cana-1260	381	10	each	each	DET
cana-1260	381	11	vertex	vertex	NOUN
cana-1260	381	12	has	have	AUX
cana-1260	381	13	degree	degree	NOUN
cana-1260	381	14	6	6	NUM
cana-1260	381	15	or	or	CCONJ
cana-1260	381	16	6	6	NUM
cana-1260	381	17	-	-	PUNCT
cana-1260	381	18	regular	regular	ADJ
cana-1260	381	19	graph	graph	NOUN
cana-1260	381	20	.	.	PUNCT
cana-1260	382	1	for	for	ADP
cana-1260	382	2	fractional	fractional	ADJ
cana-1260	382	3	domination	domination	NOUN
cana-1260	382	4	number	number	NOUN
cana-1260	382	5	we	we	PRON
cana-1260	382	6	used	use	VERB
cana-1260	382	7	the	the	DET
cana-1260	382	8	theorem	theorem	NOUN
cana-1260	382	9	1.9	1.9	NUM
cana-1260	382	10	for	for	ADP
cana-1260	382	11	following	follow	VERB
cana-1260	382	12	cases	case	NOUN
cana-1260	382	13	:	:	PUNCT
cana-1260	382	14	case	case	NOUN
cana-1260	382	15	1	1	NUM
cana-1260	382	16	:	:	PUNCT
cana-1260	382	17	when	when	SCONJ
cana-1260	382	18	𝑚	𝑚	PROPN
cana-1260	382	19	and	and	CCONJ
cana-1260	382	20	𝑛	𝑛	PROPN
cana-1260	382	21	are	be	AUX
cana-1260	382	22	even	even	ADV
cana-1260	382	23	and	and	CCONJ
cana-1260	382	24	distinct	distinct	ADJ
cana-1260	382	25	if	if	SCONJ
cana-1260	382	26	𝑚	𝑚	NOUN
cana-1260	382	27	=	=	SYM
cana-1260	382	28	4	4	NUM
cana-1260	382	29	,	,	PUNCT
cana-1260	382	30	𝑛	𝑛	NOUN
cana-1260	382	31	=	=	NUM
cana-1260	382	32	6	6	NUM
cana-1260	382	33	then	then	ADV
cana-1260	382	34	the	the	DET
cana-1260	382	35	graph	graph	NOUN
cana-1260	382	36	𝐺	𝐺	NOUN
cana-1260	382	37	=	=	PUNCT
cana-1260	382	38	(	(	PUNCT
cana-1260	382	39	𝐶𝑚	𝐶𝑚	PROPN
cana-1260	382	40	×	×	PROPN
cana-1260	382	41	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	382	42	)	)	PUNCT
cana-1260	382	43	has	have	VERB
cana-1260	382	44	order	order	NOUN
cana-1260	382	45	24	24	NUM
cana-1260	382	46	and	and	CCONJ
cana-1260	382	47	size	size	NOUN
cana-1260	382	48	48	48	NUM
cana-1260	382	49	with	with	ADP
cana-1260	382	50	each	each	DET
cana-1260	382	51	vertex	vertex	NOUN
cana-1260	382	52	has	have	VERB
cana-1260	382	53	degree	degree	NOUN
cana-1260	382	54	4	4	NUM
cana-1260	382	55	.	.	PUNCT
cana-1260	383	1	hence	hence	ADV
cana-1260	383	2	we	we	PRON
cana-1260	383	3	have	have	VERB
cana-1260	383	4	𝛾𝑓	𝛾𝑓	VERB
cana-1260	383	5	(	(	PUNCT
cana-1260	383	6	𝐶4	𝐶4	PROPN
cana-1260	383	7	×	×	NOUN
cana-1260	383	8	𝐶6	𝐶6	X
cana-1260	383	9	)	)	PUNCT
cana-1260	384	1	=	=	SYM
cana-1260	384	2	24/5	24/5	NUM
cana-1260	384	3	.	.	PUNCT
cana-1260	385	1	the	the	DET
cana-1260	385	2	line	line	NOUN
cana-1260	385	3	graph	graph	NOUN
cana-1260	385	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	385	5	)	)	PUNCT
cana-1260	385	6	has	have	VERB
cana-1260	385	7	48	48	NUM
cana-1260	385	8	number	number	NOUN
cana-1260	385	9	of	of	ADP
cana-1260	385	10	vertices	vertex	NOUN
cana-1260	385	11	with	with	ADP
cana-1260	385	12	each	each	DET
cana-1260	385	13	vertex	vertex	NOUN
cana-1260	385	14	has	have	AUX
cana-1260	385	15	degree	degree	NOUN
cana-1260	385	16	6	6	NUM
cana-1260	385	17	.	.	PUNCT
cana-1260	386	1	therefore	therefore	ADV
cana-1260	386	2	we	we	PRON
cana-1260	386	3	get	get	VERB
cana-1260	386	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	386	5	(	(	PUNCT
cana-1260	386	6	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	386	7	)	)	PUNCT
cana-1260	386	8	)	)	PUNCT
cana-1260	387	1	=	=	PRON
cana-1260	387	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	387	3	(	(	PUNCT
cana-1260	387	4	𝐶4	𝐶4	PROPN
cana-1260	387	5	×	×	NOUN
cana-1260	387	6	𝐶6	𝐶6	X
cana-1260	387	7	)	)	PUNCT
cana-1260	387	8	=	=	SYM
cana-1260	388	1	48/7	48/7	PROPN
cana-1260	388	2	.	.	PUNCT
cana-1260	388	3	case	case	NOUN
cana-1260	388	4	2	2	NUM
cana-1260	388	5	:	:	PUNCT
cana-1260	388	6	when	when	SCONJ
cana-1260	388	7	𝑚	𝑚	PROPN
cana-1260	388	8	is	be	AUX
cana-1260	388	9	even	even	ADV
cana-1260	388	10	and	and	CCONJ
cana-1260	388	11	𝑛	𝑛	PROPN
cana-1260	388	12	is	be	AUX
cana-1260	388	13	odd	odd	ADJ
cana-1260	388	14	if	if	SCONJ
cana-1260	388	15	𝑚	𝑚	X
cana-1260	388	16	=	=	SYM
cana-1260	388	17	4	4	NUM
cana-1260	388	18	,	,	PUNCT
cana-1260	388	19	𝑛	𝑛	NOUN
cana-1260	388	20	=	=	SYM
cana-1260	388	21	3	3	NUM
cana-1260	388	22	then	then	ADV
cana-1260	388	23	the	the	DET
cana-1260	388	24	graph	graph	NOUN
cana-1260	388	25	𝐺	𝐺	PROPN
cana-1260	388	26	=	=	PUNCT
cana-1260	388	27	(	(	PUNCT
cana-1260	388	28	𝐶4	𝐶4	PROPN
cana-1260	388	29	×	×	PROPN
cana-1260	388	30	𝐶3	𝐶3	NOUN
cana-1260	388	31	)	)	PUNCT
cana-1260	388	32	for	for	ADP
cana-1260	388	33	example	example	NOUN
cana-1260	388	34	circulant	circulant	NOUN
cana-1260	388	35	graph	graph	NOUN
cana-1260	388	36	has	have	VERB
cana-1260	388	37	order	order	NOUN
cana-1260	388	38	12	12	NUM
cana-1260	388	39	and	and	CCONJ
cana-1260	388	40	size	size	NOUN
cana-1260	388	41	24	24	NUM
cana-1260	388	42	with	with	ADP
cana-1260	388	43	every	every	DET
cana-1260	388	44	vertex	vertex	NOUN
cana-1260	388	45	has	have	VERB
cana-1260	388	46	degree	degree	NOUN
cana-1260	388	47	4	4	NUM
cana-1260	388	48	.	.	PUNCT
cana-1260	389	1	hence	hence	ADV
cana-1260	389	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	389	3	(	(	PUNCT
cana-1260	389	4	𝐶4	𝐶4	PROPN
cana-1260	389	5	×	×	PROPN
cana-1260	389	6	𝐶3	𝐶3	NOUN
cana-1260	389	7	)	)	PUNCT
cana-1260	389	8	=	=	SYM
cana-1260	389	9	12/5	12/5	PROPN
cana-1260	389	10	.	.	PUNCT
cana-1260	390	1	the	the	DET
cana-1260	390	2	line	line	NOUN
cana-1260	390	3	graph	graph	NOUN
cana-1260	390	4	𝐺	𝐺	PROPN
cana-1260	390	5	has	have	VERB
cana-1260	390	6	order	order	NOUN
cana-1260	390	7	24	24	NUM
cana-1260	390	8	with	with	ADP
cana-1260	390	9	each	each	DET
cana-1260	390	10	vertex	vertex	NOUN
cana-1260	390	11	has	have	AUX
cana-1260	390	12	degree	degree	NOUN
cana-1260	390	13	6	6	NUM
cana-1260	390	14	.	.	PUNCT
cana-1260	391	1	therefore	therefore	ADV
cana-1260	391	2	we	we	PRON
cana-1260	391	3	get	get	VERB
cana-1260	391	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	391	5	(	(	PUNCT
cana-1260	391	6	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	391	7	)	)	PUNCT
cana-1260	391	8	)	)	PUNCT
cana-1260	392	1	=	=	PRON
cana-1260	392	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	392	3	(	(	PUNCT
cana-1260	392	4	𝐶4	𝐶4	PROPN
cana-1260	392	5	×	×	PROPN
cana-1260	392	6	𝐶3	𝐶3	NOUN
cana-1260	392	7	)	)	PUNCT
cana-1260	392	8	=	=	SYM
cana-1260	392	9	24/7	24/7	NOUN
cana-1260	392	10	.	.	PUNCT
cana-1260	393	1	case	case	NOUN
cana-1260	393	2	3	3	NUM
cana-1260	393	3	:	:	PUNCT
cana-1260	393	4	when	when	SCONJ
cana-1260	393	5	𝑚	𝑚	PROPN
cana-1260	393	6	and	and	CCONJ
cana-1260	393	7	𝑛	𝑛	PROPN
cana-1260	393	8	are	be	AUX
cana-1260	393	9	odd	odd	ADJ
cana-1260	393	10	and	and	CCONJ
cana-1260	393	11	equal	equal	ADJ
cana-1260	393	12	if	if	SCONJ
cana-1260	393	13	𝑚	𝑚	PROPN
cana-1260	393	14	=	=	SYM
cana-1260	393	15	3	3	NUM
cana-1260	393	16	,	,	PUNCT
cana-1260	393	17	𝑛	𝑛	NOUN
cana-1260	393	18	=	=	SYM
cana-1260	393	19	3	3	NUM
cana-1260	393	20	then	then	ADV
cana-1260	394	1	the	the	DET
cana-1260	394	2	graph	graph	NOUN
cana-1260	394	3	𝐺	𝐺	NOUN
cana-1260	394	4	=	=	PUNCT
cana-1260	394	5	𝐶3	𝐶3	PROPN
cana-1260	394	6	×	×	PROPN
cana-1260	394	7	𝐶3	𝐶3	NOUN
cana-1260	394	8	for	for	ADP
cana-1260	394	9	example	example	NOUN
cana-1260	394	10	generalized	generalize	VERB
cana-1260	394	11	quadrangle	quadrangle	NOUN
cana-1260	394	12	graph	graph	NOUN
cana-1260	394	13	has	have	VERB
cana-1260	394	14	order	order	NOUN
cana-1260	394	15	9	9	NUM
cana-1260	394	16	and	and	CCONJ
cana-1260	394	17	size	size	NOUN
cana-1260	394	18	18	18	NUM
cana-1260	394	19	with	with	ADP
cana-1260	394	20	each	each	DET
cana-1260	394	21	vertex	vertex	NOUN
cana-1260	394	22	has	have	VERB
cana-1260	394	23	degree	degree	NOUN
cana-1260	394	24	4	4	NUM
cana-1260	394	25	.	.	PUNCT
cana-1260	395	1	therefore	therefore	ADV
cana-1260	395	2	we	we	PRON
cana-1260	395	3	get	get	VERB
cana-1260	395	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	395	5	(	(	PUNCT
cana-1260	395	6	𝐶3	𝐶3	PROPN
cana-1260	395	7	×	×	PROPN
cana-1260	395	8	𝐶3	𝐶3	NOUN
cana-1260	395	9	)	)	PUNCT
cana-1260	396	1	=	=	SYM
cana-1260	396	2	9/5	9/5	NUM
cana-1260	396	3	.	.	PUNCT
cana-1260	397	1	the	the	DET
cana-1260	397	2	line	line	NOUN
cana-1260	397	3	graph	graph	NOUN
cana-1260	397	4	(	(	PUNCT
cana-1260	397	5	𝐺	𝐺	NOUN
cana-1260	397	6	)	)	PUNCT
cana-1260	397	7	has	have	VERB
cana-1260	397	8	order	order	NOUN
cana-1260	397	9	18	18	NUM
cana-1260	397	10	with	with	ADP
cana-1260	397	11	every	every	DET
cana-1260	397	12	vertex	vertex	NOUN
cana-1260	397	13	has	have	VERB
cana-1260	397	14	degree	degree	NOUN
cana-1260	397	15	communications	communication	NOUN
cana-1260	397	16	on	on	ADP
cana-1260	397	17	applied	apply	VERB
cana-1260	397	18	nonlinear	nonlinear	ADJ
cana-1260	397	19	analysis	analysis	NOUN
cana-1260	397	20	issn	issn	NOUN
cana-1260	397	21	:	:	PUNCT
cana-1260	397	22	1074	1074	NUM
cana-1260	397	23	-	-	PUNCT
cana-1260	397	24	133x	133x	NUM
cana-1260	397	25	vol	vol	NOUN
cana-1260	397	26	31	31	NUM
cana-1260	397	27	no	no	NOUN
cana-1260	397	28	.	.	PUNCT
cana-1260	398	1	6s	6s	NUM
cana-1260	398	2	(	(	PUNCT
cana-1260	398	3	2024	2024	NUM
cana-1260	398	4	)	)	PUNCT
cana-1260	398	5	685	685	NUM
cana-1260	398	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	398	7	6	6	NUM
cana-1260	398	8	.	.	PUNCT
cana-1260	398	9	therefore	therefore	ADV
cana-1260	398	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	398	11	(	(	PUNCT
cana-1260	398	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	398	13	)	)	PUNCT
cana-1260	398	14	)	)	PUNCT
cana-1260	399	1	=	=	PRON
cana-1260	399	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	399	3	(	(	PUNCT
cana-1260	399	4	𝐶4	𝐶4	PROPN
cana-1260	399	5	×	×	PROPN
cana-1260	399	6	𝐶3	𝐶3	NOUN
cana-1260	399	7	)	)	PUNCT
cana-1260	400	1	=	=	SYM
cana-1260	400	2	18/7	18/7	NUM
cana-1260	400	3	.	.	PUNCT
cana-1260	400	4	case	case	NOUN
cana-1260	400	5	4	4	NUM
cana-1260	400	6	:	:	PUNCT
cana-1260	400	7	when	when	SCONJ
cana-1260	400	8	𝑚	𝑚	PROPN
cana-1260	400	9	and	and	CCONJ
cana-1260	400	10	𝑛	𝑛	PROPN
cana-1260	400	11	are	be	AUX
cana-1260	400	12	even	even	ADV
cana-1260	400	13	and	and	CCONJ
cana-1260	400	14	equal	equal	ADJ
cana-1260	400	15	if	if	SCONJ
cana-1260	400	16	𝑚	𝑚	NOUN
cana-1260	400	17	=	=	SYM
cana-1260	400	18	4	4	NUM
cana-1260	400	19	,	,	PUNCT
cana-1260	400	20	𝑛	𝑛	NOUN
cana-1260	400	21	=	=	NOUN
cana-1260	400	22	4	4	NUM
cana-1260	400	23	then	then	ADV
cana-1260	400	24	the	the	DET
cana-1260	400	25	graph	graph	NOUN
cana-1260	400	26	𝐺	𝐺	NOUN
cana-1260	400	27	=	=	NOUN
cana-1260	400	28	𝐶4	𝐶4	ADJ
cana-1260	400	29	×	×	PROPN
cana-1260	400	30	𝐶4	𝐶4	NOUN
cana-1260	400	31	for	for	ADP
cana-1260	400	32	example	example	NOUN
cana-1260	400	33	tesseract	tesseract	NOUN
cana-1260	400	34	graph	graph	NOUN
cana-1260	400	35	has	have	VERB
cana-1260	400	36	order	order	NOUN
cana-1260	400	37	16	16	NUM
cana-1260	400	38	and	and	CCONJ
cana-1260	400	39	size	size	NOUN
cana-1260	400	40	32	32	NUM
cana-1260	400	41	with	with	ADP
cana-1260	400	42	each	each	DET
cana-1260	400	43	vertex	vertex	NOUN
cana-1260	400	44	has	have	VERB
cana-1260	400	45	degree	degree	NOUN
cana-1260	400	46	4	4	NUM
cana-1260	400	47	.	.	PUNCT
cana-1260	401	1	hence	hence	ADV
cana-1260	401	2	we	we	PRON
cana-1260	401	3	get	get	VERB
cana-1260	401	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	401	5	(	(	PUNCT
cana-1260	401	6	𝐶4	𝐶4	PROPN
cana-1260	401	7	×	×	PROPN
cana-1260	401	8	𝐶4	𝐶4	NOUN
cana-1260	401	9	)	)	PUNCT
cana-1260	401	10	=	=	PUNCT
cana-1260	402	1	16	16	NUM
cana-1260	402	2	5	5	NUM
cana-1260	402	3	.	.	PUNCT
cana-1260	403	1	the	the	DET
cana-1260	403	2	line	line	NOUN
cana-1260	403	3	graph	graph	NOUN
cana-1260	403	4	(	(	PUNCT
cana-1260	403	5	𝐺	𝐺	NOUN
cana-1260	403	6	)	)	PUNCT
cana-1260	403	7	has	have	VERB
cana-1260	403	8	order	order	NOUN
cana-1260	403	9	32	32	NUM
cana-1260	403	10	with	with	ADP
cana-1260	403	11	every	every	DET
cana-1260	403	12	vertex	vertex	NOUN
cana-1260	403	13	has	have	AUX
cana-1260	403	14	degree	degree	NOUN
cana-1260	403	15	6	6	NUM
cana-1260	403	16	.	.	PUNCT
cana-1260	404	1	therefore	therefore	ADV
cana-1260	404	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	404	3	(	(	PUNCT
cana-1260	404	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	404	5	)	)	PUNCT
cana-1260	404	6	)	)	PUNCT
cana-1260	405	1	=	=	PRON
cana-1260	405	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	405	3	(	(	PUNCT
cana-1260	405	4	𝐶4	𝐶4	PROPN
cana-1260	405	5	×	×	PROPN
cana-1260	405	6	𝐶4	𝐶4	NOUN
cana-1260	405	7	)	)	PUNCT
cana-1260	405	8	=	=	PUNCT
cana-1260	406	1	32/7	32/7	ADJ
cana-1260	406	2	.	.	PUNCT
cana-1260	406	3	case	case	NOUN
cana-1260	406	4	5	5	NUM
cana-1260	406	5	:	:	PUNCT
cana-1260	406	6	when	when	SCONJ
cana-1260	406	7	𝑚	𝑚	PROPN
cana-1260	406	8	and	and	CCONJ
cana-1260	406	9	𝑛	𝑛	DET
cana-1260	406	10	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
cana-1260	406	11	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-1260	406	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1260	406	13	distinct	distinct	ADJ
cana-1260	406	14	if	if	SCONJ
cana-1260	406	15	𝑚	𝑚	X
cana-1260	406	16	=	=	SYM
cana-1260	406	17	3	3	NUM
cana-1260	406	18	,	,	PUNCT
cana-1260	406	19	𝑛	𝑛	NOUN
cana-1260	406	20	=	=	SYM
cana-1260	406	21	5	5	NUM
cana-1260	406	22	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
cana-1260	406	23	𝑡ℎ𝑒	𝑡ℎ𝑒	PROPN
cana-1260	406	24	𝑔𝑟𝑎𝑝ℎ	𝑔𝑟𝑎𝑝ℎ	PROPN
cana-1260	406	25	𝐺	𝐺	PROPN
cana-1260	406	26	=	=	PUNCT
cana-1260	406	27	𝐶3	𝐶3	PROPN
cana-1260	406	28	×	×	PROPN
cana-1260	406	29	𝐶5	𝐶5	VERB
cana-1260	406	30	ℎ𝑎𝑠	ℎ𝑎𝑠	NOUN
cana-1260	406	31	15	15	NUM
cana-1260	406	32	𝑛𝑢𝑚𝑏𝑒𝑟	𝑛𝑢𝑚𝑏𝑒𝑟	NOUN
cana-1260	406	33	of	of	ADP
cana-1260	406	34	vertices	vertex	NOUN
cana-1260	406	35	and	and	CCONJ
cana-1260	406	36	size	size	NOUN
cana-1260	406	37	30	30	NUM
cana-1260	406	38	with	with	ADP
cana-1260	406	39	every	every	DET
cana-1260	406	40	vertex	vertex	NOUN
cana-1260	406	41	has	have	VERB
cana-1260	406	42	degree	degree	NOUN
cana-1260	406	43	4	4	NUM
cana-1260	406	44	.	.	PUNCT
cana-1260	406	45	hence	hence	ADV
cana-1260	406	46	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	406	47	(	(	PUNCT
cana-1260	406	48	𝐶3	𝐶3	PROPN
cana-1260	406	49	×	×	PROPN
cana-1260	406	50	𝐶5	𝐶5	PROPN
cana-1260	406	51	)	)	PUNCT
cana-1260	406	52	=	=	SYM
cana-1260	407	1	15	15	NUM
cana-1260	407	2	5	5	NUM
cana-1260	407	3	=	=	SYM
cana-1260	407	4	3	3	NUM
cana-1260	407	5	.	.	PUNCT
cana-1260	408	1	the	the	DET
cana-1260	408	2	line	line	NOUN
cana-1260	408	3	graph	graph	NOUN
cana-1260	408	4	(	(	PUNCT
cana-1260	408	5	𝐺	𝐺	NOUN
cana-1260	408	6	)	)	PUNCT
cana-1260	408	7	has	have	VERB
cana-1260	408	8	30	30	NUM
cana-1260	408	9	number	number	NOUN
cana-1260	408	10	of	of	ADP
cana-1260	408	11	vertices	vertex	NOUN
cana-1260	408	12	with	with	ADP
cana-1260	408	13	each	each	DET
cana-1260	408	14	vertex	vertex	NOUN
cana-1260	408	15	has	have	AUX
cana-1260	408	16	degree	degree	NOUN
cana-1260	408	17	6	6	NUM
cana-1260	408	18	.	.	PUNCT
cana-1260	409	1	therefore	therefore	ADV
cana-1260	409	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	409	3	(	(	PUNCT
cana-1260	409	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	409	5	)	)	PUNCT
cana-1260	409	6	)	)	PUNCT
cana-1260	410	1	=	=	PRON
cana-1260	410	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	410	3	(	(	PUNCT
cana-1260	410	4	𝐶3	𝐶3	PROPN
cana-1260	410	5	×	×	PROPN
cana-1260	410	6	𝐶5	𝐶5	ADV
cana-1260	410	7	)	)	PUNCT
cana-1260	410	8	=	=	SYM
cana-1260	411	1	30/7	30/7	NOUN
cana-1260	411	2	.	.	PUNCT
cana-1260	411	3	hence	hence	ADV
cana-1260	411	4	in	in	ADP
cana-1260	411	5	gene	gene	NOUN
cana-1260	411	6	ra	ra	PROPN
cana-1260	411	7	l	l	NOUN
cana-1260	412	1	i	i	PRON
cana-1260	412	2	zed	zed	VERB
cana-1260	413	1	i	i	PRON
cana-1260	413	2	f	f	PROPN
cana-1260	413	3	𝐺	𝐺	PROPN
cana-1260	413	4	=	=	PUNCT
cana-1260	414	1	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	414	2	×	×	NOUN
cana-1260	414	3	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	414	4	be	be	AUX
cana-1260	414	5	4	4	NUM
cana-1260	414	6	-	-	PUNCT
cana-1260	414	7	regular	regular	ADJ
cana-1260	414	8	graph	graph	NOUN
cana-1260	414	9	with	with	ADP
cana-1260	414	10	order	order	NOUN
cana-1260	414	11	𝑚𝑛	𝑚𝑛	PRON
cana-1260	414	12	and	and	CCONJ
cana-1260	414	13	graph	graph	VERB
cana-1260	414	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	414	15	)	)	PUNCT
cana-1260	414	16	be	be	VERB
cana-1260	414	17	6	6	NUM
cana-1260	414	18	-	-	PUNCT
cana-1260	414	19	regular	regular	ADJ
cana-1260	414	20	graph	graph	NOUN
cana-1260	414	21	of	of	ADP
cana-1260	414	22	order	order	NOUN
cana-1260	414	23	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	415	1	then	then	ADV
cana-1260	415	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	415	3	(	(	PUNCT
cana-1260	415	4	𝐺	𝐺	NOUN
cana-1260	415	5	)	)	PUNCT
cana-1260	415	6	+	+	CCONJ
cana-1260	415	7	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	415	8	(	(	PUNCT
cana-1260	415	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	415	10	)	)	PUNCT
cana-1260	415	11	)	)	PUNCT
cana-1260	416	1	=	=	PUNCT
cana-1260	416	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	416	3	5	5	NUM
cana-1260	416	4	+	+	CCONJ
cana-1260	416	5	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	416	6	7	7	NUM
cana-1260	416	7	and	and	CCONJ
cana-1260	416	8	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	416	9	(	(	PUNCT
cana-1260	416	10	𝐺	𝐺	NOUN
cana-1260	416	11	)	)	PUNCT
cana-1260	416	12	∗	∗	NOUN
cana-1260	416	13	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	416	14	(	(	PUNCT
cana-1260	416	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	416	16	)	)	PUNCT
cana-1260	416	17	)	)	PUNCT
cana-1260	417	1	=	=	PUNCT
cana-1260	417	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	417	3	5	5	NUM
cana-1260	417	4	∗	∗	NOUN
cana-1260	417	5	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	417	6	7	7	NUM
cana-1260	417	7	.	.	PUNCT
cana-1260	417	8	example	example	NOUN
cana-1260	418	1	2.9.1	2.9.1	NUM
cana-1260	418	2	when	when	SCONJ
cana-1260	418	3	𝑚	𝑚	PROPN
cana-1260	418	4	=	=	SYM
cana-1260	418	5	3	3	NUM
cana-1260	418	6	,	,	PUNCT
cana-1260	418	7	𝑛	𝑛	NOUN
cana-1260	418	8	=	=	SYM
cana-1260	418	9	4	4	NUM
cana-1260	418	10	the	the	DET
cana-1260	418	11	cartesian	cartesian	ADJ
cana-1260	418	12	product	product	NOUN
cana-1260	418	13	graph	graph	NOUN
cana-1260	418	14	𝐺	𝐺	PROPN
cana-1260	418	15	=	=	PROPN
cana-1260	418	16	𝐶3	𝐶3	PROPN
cana-1260	418	17	×	×	PROPN
cana-1260	418	18	𝐶4	𝐶4	NOUN
cana-1260	418	19	given	give	VERB
cana-1260	418	20	in	in	ADP
cana-1260	418	21	figure	figure	NOUN
cana-1260	418	22	2.9.1a	2.9.1a	PROPN
cana-1260	418	23	)	)	PUNCT
cana-1260	418	24	.	.	PUNCT
cana-1260	419	1	so	so	ADV
cana-1260	419	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	419	3	(	(	PUNCT
cana-1260	419	4	𝐶3	𝐶3	PROPN
cana-1260	419	5	×	×	PROPN
cana-1260	419	6	𝐶4	𝐶4	NOUN
cana-1260	419	7	)	)	PUNCT
cana-1260	420	1	=	=	SYM
cana-1260	420	2	12/5	12/5	PROPN
cana-1260	420	3	.	.	PUNCT
cana-1260	421	1	(	(	PUNCT
cana-1260	421	2	a	a	X
cana-1260	421	3	)	)	PUNCT
cana-1260	421	4	cartesian	cartesian	ADJ
cana-1260	421	5	product	product	NOUN
cana-1260	421	6	graph	graph	NOUN
cana-1260	421	7	𝐺	𝐺	PROPN
cana-1260	421	8	=	=	PROPN
cana-1260	421	9	𝐶3	𝐶3	PROPN
cana-1260	421	10	×	×	PROPN
cana-1260	421	11	𝐶4	𝐶4	NOUN
cana-1260	421	12	(	(	PUNCT
cana-1260	421	13	a	a	NOUN
cana-1260	421	14	)	)	PUNCT
cana-1260	421	15	line	line	NOUN
cana-1260	421	16	graph	graph	NOUN
cana-1260	421	17	of	of	ADP
cana-1260	421	18	cartesian	cartesian	ADJ
cana-1260	421	19	product	product	NOUN
cana-1260	421	20	𝐺	𝐺	NOUN
cana-1260	421	21	=	=	PUNCT
cana-1260	421	22	𝐶3	𝐶3	PROPN
cana-1260	421	23	×	×	PROPN
cana-1260	421	24	𝐶4	𝐶4	NOUN
cana-1260	421	25	figure	figure	NOUN
cana-1260	421	26	2.9.1	2.9.1	NUM
cana-1260	421	27	the	the	DET
cana-1260	421	28	line	line	NOUN
cana-1260	421	29	graph	graph	NOUN
cana-1260	421	30	of	of	ADP
cana-1260	421	31	cartesian	cartesian	ADJ
cana-1260	421	32	product	product	NOUN
cana-1260	421	33	𝐺	𝐺	NOUN
cana-1260	421	34	=	=	PUNCT
cana-1260	421	35	𝐶3	𝐶3	PROPN
cana-1260	421	36	×	×	PROPN
cana-1260	421	37	𝐶4	𝐶4	NOUN
cana-1260	421	38	given	give	VERB
cana-1260	421	39	in	in	ADP
cana-1260	421	40	figure	figure	NOUN
cana-1260	421	41	2.9.1b	2.9.1b	PROPN
cana-1260	421	42	)	)	PUNCT
cana-1260	421	43	the	the	DET
cana-1260	421	44	line	line	NOUN
cana-1260	421	45	graph	graph	NOUN
cana-1260	421	46	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	421	47	)	)	PUNCT
cana-1260	421	48	has	have	VERB
cana-1260	421	49	24	24	NUM
cana-1260	421	50	vertices	vertex	NOUN
cana-1260	421	51	with	with	ADP
cana-1260	421	52	each	each	DET
cana-1260	421	53	vertex	vertex	NOUN
cana-1260	421	54	has	have	AUX
cana-1260	421	55	degree	degree	NOUN
cana-1260	421	56	6	6	NUM
cana-1260	421	57	.	.	PUNCT
cana-1260	422	1	therefore	therefore	ADV
cana-1260	422	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	422	3	(	(	PUNCT
cana-1260	422	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	422	5	)	)	PUNCT
cana-1260	422	6	)	)	PUNCT
cana-1260	423	1	=	=	PRON
cana-1260	423	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	423	3	(	(	PUNCT
cana-1260	423	4	𝐶3	𝐶3	PROPN
cana-1260	423	5	×	×	PROPN
cana-1260	423	6	𝐶4	𝐶4	NOUN
cana-1260	423	7	)	)	PUNCT
cana-1260	423	8	=	=	SYM
cana-1260	423	9	24/7	24/7	X
cana-1260	423	10	.	.	PUNCT
cana-1260	423	11	iii	iii	X
cana-1260	423	12	)	)	PUNCT
cana-1260	423	13	and	and	CCONJ
cana-1260	423	14	iv	iv	X
cana-1260	423	15	)	)	PUNCT
cana-1260	423	16	the	the	DET
cana-1260	423	17	theorem	theorem	ADJ
cana-1260	423	18	2.1.2	2.1.2	NUM
cana-1260	423	19	give	give	VERB
cana-1260	423	20	𝐺	𝐺	NOUN
cana-1260	423	21	=	=	PUNCT
cana-1260	424	1	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	424	2	×	×	NOUN
cana-1260	424	3	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	424	4	be	be	AUX
cana-1260	424	5	4	4	NUM
cana-1260	424	6	-	-	PUNCT
cana-1260	424	7	regular	regular	ADJ
cana-1260	424	8	graph	graph	NOUN
cana-1260	424	9	of	of	ADP
cana-1260	424	10	order	order	NOUN
cana-1260	425	1	𝑚𝑛	𝑚𝑛	PRON
cana-1260	425	2	then	then	ADV
cana-1260	425	3	its	its	PRON
cana-1260	425	4	upper	upper	ADJ
cana-1260	425	5	fractional	fractional	ADJ
cana-1260	425	6	domination	domination	NOUN
cana-1260	425	7	number	number	NOUN
cana-1260	425	8	is	be	AUX
cana-1260	425	9	г𝑓	г𝑓	DET
cana-1260	425	10	(	(	PUNCT
cana-1260	425	11	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	425	12	×	×	NOUN
cana-1260	425	13	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	425	14	)	)	PUNCT
cana-1260	425	15	≤	≤	ADJ
cana-1260	425	16	𝑚𝑛/𝑟	𝑚𝑛/𝑟	NOUN
cana-1260	425	17	and	and	CCONJ
cana-1260	425	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	425	19	)	)	PUNCT
cana-1260	425	20	be	be	VERB
cana-1260	425	21	6	6	NUM
cana-1260	425	22	-	-	PUNCT
cana-1260	425	23	regular	regular	ADJ
cana-1260	425	24	graph	graph	NOUN
cana-1260	425	25	of	of	ADP
cana-1260	425	26	order	order	NOUN
cana-1260	426	1	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	427	1	then	then	ADV
cana-1260	427	2	г𝑓	г𝑓	X
cana-1260	427	3	(	(	PUNCT
cana-1260	427	4	𝐿(𝐶𝑚	𝐿(𝐶𝑚	PROPN
cana-1260	427	5	×	×	PROPN
cana-1260	427	6	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	427	7	)	)	PUNCT
cana-1260	427	8	)	)	PUNCT
cana-1260	428	1	≤	≤	NUM
cana-1260	428	2	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	429	1	𝑟+2	𝑟+2	PUNCT
cana-1260	429	2	hence	hence	ADV
cana-1260	429	3	г𝑓	г𝑓	PROPN
cana-1260	429	4	(	(	PUNCT
cana-1260	429	5	𝐺	𝐺	NOUN
cana-1260	429	6	)	)	PUNCT
cana-1260	430	1	+	+	NUM
cana-1260	430	2	г𝑓	г𝑓	ADJ
cana-1260	430	3	(	(	PUNCT
cana-1260	430	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	430	5	)	)	PUNCT
cana-1260	430	6	)	)	PUNCT
cana-1260	430	7	≤	≤	NUM
cana-1260	430	8	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	431	1	𝑟	𝑟	NOUN
cana-1260	431	2	+	+	X
cana-1260	431	3	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	431	4	𝑟+2	𝑟+2	NUM
cana-1260	431	5	and	and	CCONJ
cana-1260	431	6	г𝑓	г𝑓	PROPN
cana-1260	431	7	(	(	PUNCT
cana-1260	431	8	𝐺	𝐺	NOUN
cana-1260	431	9	)	)	PUNCT
cana-1260	432	1	∗	∗	NOUN
cana-1260	432	2	г𝑓	г𝑓	PROPN
cana-1260	432	3	(	(	PUNCT
cana-1260	432	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	432	5	)	)	PUNCT
cana-1260	432	6	)	)	PUNCT
cana-1260	433	1	≤	≤	NUM
cana-1260	433	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	433	3	4	4	NUM
cana-1260	433	4	∗	∗	NOUN
cana-1260	433	5	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	433	6	3	3	NUM
cana-1260	433	7	.	.	PUNCT
cana-1260	434	1	□	□	PUNCT
cana-1260	434	2	communications	communication	NOUN
cana-1260	434	3	on	on	ADP
cana-1260	434	4	applied	apply	VERB
cana-1260	434	5	nonlinear	nonlinear	ADJ
cana-1260	434	6	analysis	analysis	NOUN
cana-1260	434	7	issn	issn	NOUN
cana-1260	434	8	:	:	PUNCT
cana-1260	434	9	1074	1074	NUM
cana-1260	434	10	-	-	PUNCT
cana-1260	434	11	133x	133x	NUM
cana-1260	434	12	vol	vol	NOUN
cana-1260	434	13	31	31	NUM
cana-1260	434	14	no	no	NOUN
cana-1260	434	15	.	.	PUNCT
cana-1260	435	1	6s	6s	NUM
cana-1260	435	2	(	(	PUNCT
cana-1260	435	3	2024	2024	NUM
cana-1260	435	4	)	)	PUNCT
cana-1260	435	5	686	686	NUM
cana-1260	435	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1260	435	7	3	3	NUM
cana-1260	435	8	applications	application	NOUN
cana-1260	435	9	join	join	VERB
cana-1260	435	10	and	and	CCONJ
cana-1260	435	11	union	union	VERB
cana-1260	435	12	the	the	DET
cana-1260	435	13	first	first	ADJ
cana-1260	435	14	graph	graph	NOUN
cana-1260	435	15	𝐺	𝐺	NOUN
cana-1260	435	16	and	and	CCONJ
cana-1260	435	17	second	second	ADJ
cana-1260	435	18	graph	graph	NOUN
cana-1260	435	19	𝐻	𝐻	PROPN
cana-1260	435	20	have	have	VERB
cana-1260	435	21	two	two	NUM
cana-1260	435	22	set	set	NOUN
cana-1260	435	23	of	of	ADP
cana-1260	435	24	vertices	vertex	NOUN
cana-1260	435	25	as	as	ADP
cana-1260	435	26	𝑉1	𝑉1	NOUN
cana-1260	435	27	and	and	CCONJ
cana-1260	435	28	𝑉2	𝑉2	NOUN
cana-1260	435	29	which	which	PRON
cana-1260	435	30	lack	lack	VERB
cana-1260	435	31	coherence	coherence	NOUN
cana-1260	435	32	or	or	CCONJ
cana-1260	435	33	disjoint	disjoint	NOUN
cana-1260	435	34	whereas	whereas	SCONJ
cana-1260	435	35	the	the	DET
cana-1260	435	36	edge	edge	NOUN
cana-1260	435	37	sets	set	NOUN
cana-1260	435	38	are	be	AUX
cana-1260	435	39	𝑋1	𝑋1	NOUN
cana-1260	435	40	and	and	CCONJ
cana-1260	435	41	𝑋2	𝑋2	VERB
cana-1260	435	42	respectively	respectively	ADV
cana-1260	435	43	.	.	PUNCT
cana-1260	436	1	the	the	DET
cana-1260	436	2	union	union	NOUN
cana-1260	436	3	(	(	PUNCT
cana-1260	436	4	𝐺	𝐺	PROPN
cana-1260	436	5	𝘜	𝘜	PROPN
cana-1260	436	6	𝐻	𝐻	PROPN
cana-1260	436	7	)	)	PUNCT
cana-1260	436	8	represented	represent	VERB
cana-1260	436	9	as	as	ADP
cana-1260	436	10	graphs	graph	NOUN
cana-1260	436	11	such	such	ADJ
cana-1260	436	12	that	that	DET
cana-1260	436	13	𝑉	𝑉	PROPN
cana-1260	436	14	=	=	PUNCT
cana-1260	436	15	(	(	PUNCT
cana-1260	436	16	𝑉1	𝑉1	PROPN
cana-1260	436	17	ս	ս	NOUN
cana-1260	436	18	𝑉2	𝑉2	PROPN
cana-1260	436	19	)	)	PUNCT
cana-1260	436	20	and	and	CCONJ
cana-1260	436	21	𝑋	𝑋	PROPN
cana-1260	436	22	=	=	SYM
cana-1260	436	23	(	(	PUNCT
cana-1260	436	24	𝑋1	𝑋1	PROPN
cana-1260	436	25	ս	ս	X
cana-1260	436	26	𝑋2	𝑋2	VERB
cana-1260	436	27	)	)	PUNCT
cana-1260	436	28	if	if	SCONJ
cana-1260	436	29	graph	graph	NOUN
cana-1260	436	30	𝐺	𝐺	PROPN
cana-1260	436	31	has	have	VERB
cana-1260	436	32	order	order	NOUN
cana-1260	436	33	𝑛1	𝑛1	NOUN
cana-1260	436	34	and	and	CCONJ
cana-1260	436	35	size	size	NOUN
cana-1260	436	36	𝑚1	𝑚1	NOUN
cana-1260	436	37	,	,	PUNCT
cana-1260	436	38	graph	graph	NOUN
cana-1260	436	39	𝐻	𝐻	PROPN
cana-1260	436	40	has	have	VERB
cana-1260	436	41	order	order	NOUN
cana-1260	436	42	𝑛2	𝑛2	NOUN
cana-1260	436	43	and	and	CCONJ
cana-1260	436	44	size	size	NOUN
cana-1260	436	45	𝑚2	𝑚2	NOUN
cana-1260	436	46	then	then	ADV
cana-1260	436	47	for	for	ADP
cana-1260	436	48	union	union	NOUN
cana-1260	436	49	𝐺	𝐺	NOUN
cana-1260	436	50	𝘜	𝘜	NOUN
cana-1260	436	51	𝐻	𝐻	NOUN
cana-1260	436	52	we	we	PRON
cana-1260	436	53	have	have	VERB
cana-1260	436	54	𝑛1	𝑛1	ADJ
cana-1260	436	55	+	+	CCONJ
cana-1260	436	56	𝑛2	𝑛2	NOUN
cana-1260	436	57	vertices	vertex	NOUN
cana-1260	436	58	and	and	CCONJ
cana-1260	436	59	𝑚1	𝑚1	NOUN
cana-1260	436	60	+	+	CCONJ
cana-1260	436	61	𝑚2	𝑚2	NOUN
cana-1260	436	62	edges	edge	NOUN
cana-1260	436	63	.	.	PUNCT
cana-1260	437	1	so	so	ADV
cana-1260	437	2	it	it	PRON
cana-1260	437	3	is	be	AUX
cana-1260	437	4	recognized	recognize	VERB
cana-1260	437	5	that	that	SCONJ
cana-1260	437	6	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	437	7	(	(	PUNCT
cana-1260	437	8	𝐺	𝐺	NOUN
cana-1260	437	9	𝘜	𝘜	PROPN
cana-1260	437	10	𝐻	𝐻	PROPN
cana-1260	437	11	)	)	PUNCT
cana-1260	438	1	=	=	SYM
cana-1260	438	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	438	3	(	(	PUNCT
cana-1260	438	4	𝐺	𝐺	NOUN
cana-1260	438	5	)	)	PUNCT
cana-1260	439	1	+	+	CCONJ
cana-1260	439	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	439	3	(	(	PUNCT
cana-1260	439	4	𝐻	𝐻	PROPN
cana-1260	439	5	)	)	PUNCT
cana-1260	439	6	and	and	CCONJ
cana-1260	439	7	г𝑓	г𝑓	INTJ
cana-1260	439	8	(	(	PUNCT
cana-1260	439	9	𝐺	𝐺	NOUN
cana-1260	439	10	𝘜	𝘜	PROPN
cana-1260	439	11	𝐻	𝐻	PROPN
cana-1260	439	12	)	)	PUNCT
cana-1260	440	1	=	=	NOUN
cana-1260	440	2	г𝑓	г𝑓	PROPN
cana-1260	440	3	(	(	PUNCT
cana-1260	440	4	𝐺	𝐺	NOUN
cana-1260	440	5	)	)	PUNCT
cana-1260	440	6	+	+	NUM
cana-1260	440	7	г𝑓	г𝑓	ADJ
cana-1260	440	8	(	(	PUNCT
cana-1260	440	9	𝐻	𝐻	PROPN
cana-1260	440	10	)	)	PUNCT
cana-1260	440	11	here	here	ADV
cana-1260	440	12	we	we	PRON
cana-1260	440	13	have	have	AUX
cana-1260	440	14	taken	take	VERB
cana-1260	440	15	graph	graph	NOUN
cana-1260	440	16	(	(	PUNCT
cana-1260	440	17	𝐶3	𝐶3	PROPN
cana-1260	440	18	𝘜	𝘜	PROPN
cana-1260	440	19	𝑃3	𝑃3	NOUN
cana-1260	440	20	)	)	PUNCT
cana-1260	441	1	so	so	ADV
cana-1260	441	2	immediately	immediately	ADV
cana-1260	441	3	we	we	PRON
cana-1260	441	4	have	have	AUX
cana-1260	441	5	𝛾𝑓	𝛾𝑓	VERB
cana-1260	441	6	(	(	PUNCT
cana-1260	441	7	𝐶3	𝐶3	PROPN
cana-1260	441	8	𝘜	𝘜	PROPN
cana-1260	441	9	𝑃3	𝑃3	NOUN
cana-1260	441	10	)	)	PUNCT
cana-1260	442	1	=	=	SYM
cana-1260	442	2	𝛾𝑓	𝛾𝑓	X
cana-1260	442	3	(	(	PUNCT
cana-1260	442	4	𝐶3	𝐶3	PROPN
cana-1260	442	5	)	)	PUNCT
cana-1260	442	6	+	+	NUM
cana-1260	442	7	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	442	8	(	(	PUNCT
cana-1260	442	9	𝑃3	𝑃3	NOUN
cana-1260	442	10	)	)	PUNCT
cana-1260	442	11	and	and	CCONJ
cana-1260	442	12	г𝑓	г𝑓	X
cana-1260	442	13	(	(	PUNCT
cana-1260	442	14	𝐶3	𝐶3	PROPN
cana-1260	442	15	𝘜	𝘜	PROPN
cana-1260	442	16	𝑃3	𝑃3	NOUN
cana-1260	442	17	)	)	PUNCT
cana-1260	442	18	=	=	SYM
cana-1260	442	19	г𝑓	г𝑓	PROPN
cana-1260	442	20	(	(	PUNCT
cana-1260	442	21	𝐶3	𝐶3	PROPN
cana-1260	442	22	)	)	PUNCT
cana-1260	442	23	+	+	NUM
cana-1260	442	24	г𝑓	г𝑓	PRON
cana-1260	442	25	(	(	PUNCT
cana-1260	442	26	𝑃3	𝑃3	NOUN
cana-1260	442	27	)	)	PUNCT
cana-1260	442	28	.	.	PUNCT
cana-1260	443	1	(	(	PUNCT
cana-1260	443	2	see	see	VERB
cana-1260	443	3	figure	figure	NOUN
cana-1260	443	4	3a	3a	NUM
cana-1260	443	5	)	)	PUNCT
cana-1260	443	6	.	.	PUNCT
cana-1260	444	1	the	the	DET
cana-1260	444	2	join	join	NOUN
cana-1260	444	3	for	for	ADP
cana-1260	444	4	graph	graph	NOUN
cana-1260	444	5	g	g	PROPN
cana-1260	444	6	and	and	CCONJ
cana-1260	444	7	h	h	NOUN
cana-1260	444	8	denoted	denote	VERB
cana-1260	444	9	by	by	ADP
cana-1260	444	10	(	(	PUNCT
cana-1260	444	11	𝐺	𝐺	PROPN
cana-1260	444	12	+	+	CCONJ
cana-1260	444	13	𝐻	𝐻	PROPN
cana-1260	444	14	)	)	PUNCT
cana-1260	444	15	has	have	VERB
cana-1260	444	16	vertex	vertex	NOUN
cana-1260	444	17	set	set	VERB
cana-1260	444	18	v	v	NOUN
cana-1260	444	19	(	(	PUNCT
cana-1260	444	20	𝐺	𝐺	NOUN
cana-1260	444	21	+	+	CCONJ
cana-1260	444	22	𝐻	𝐻	PROPN
cana-1260	444	23	)	)	PUNCT
cana-1260	444	24	=	=	SYM
cana-1260	444	25	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	444	26	)	)	PUNCT
cana-1260	444	27	∪	∪	NOUN
cana-1260	444	28	𝑉(𝐻	𝑉(𝐻	NOUN
cana-1260	444	29	)	)	PUNCT
cana-1260	444	30	and	and	CCONJ
cana-1260	444	31	edge	edge	NOUN
cana-1260	444	32	set	set	VERB
cana-1260	444	33	𝐸(𝐺	𝐸(𝐺	NOUN
cana-1260	444	34	+	+	CCONJ
cana-1260	444	35	𝐻	𝐻	PROPN
cana-1260	444	36	)	)	PUNCT
cana-1260	444	37	=	=	SYM
cana-1260	444	38	𝐸(𝐺	𝐸(𝐺	X
cana-1260	444	39	)	)	PUNCT
cana-1260	444	40	∪	∪	PROPN
cana-1260	444	41	𝐸(𝐻	𝐸(𝐻	PROPN
cana-1260	444	42	)	)	PUNCT
cana-1260	444	43	∪	∪	X
cana-1260	444	44	{	{	PUNCT
cana-1260	444	45	(	(	PUNCT
cana-1260	444	46	𝑔ℎ)|𝑔	𝑔ℎ)|𝑔	ADJ
cana-1260	444	47	∈	∈	NOUN
cana-1260	444	48	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1260	444	49	)	)	PUNCT
cana-1260	444	50	,	,	PUNCT
cana-1260	444	51	ℎ	ℎ	PROPN
cana-1260	444	52	∈	∈	PROPN
cana-1260	444	53	𝑉(𝐻	𝑉(𝐻	NOUN
cana-1260	444	54	)	)	PUNCT
cana-1260	444	55	}	}	PUNCT
cana-1260	444	56	.	.	PUNCT
cana-1260	445	1	if	if	SCONJ
cana-1260	445	2	graph	graph	NOUN
cana-1260	445	3	𝐺	𝐺	PROPN
cana-1260	445	4	has	have	VERB
cana-1260	445	5	order	order	NOUN
cana-1260	445	6	𝑛1	𝑛1	NOUN
cana-1260	445	7	and	and	CCONJ
cana-1260	445	8	size	size	NOUN
cana-1260	445	9	𝑚1	𝑚1	NOUN
cana-1260	445	10	,	,	PUNCT
cana-1260	445	11	graph	graph	NOUN
cana-1260	445	12	𝐻	𝐻	PROPN
cana-1260	445	13	has	have	VERB
cana-1260	445	14	order	order	NOUN
cana-1260	445	15	𝑛2	𝑛2	NOUN
cana-1260	445	16	and	and	CCONJ
cana-1260	445	17	𝑚2	𝑚2	NOUN
cana-1260	445	18	number	number	NOUN
cana-1260	445	19	of	of	ADP
cana-1260	445	20	edges	edge	NOUN
cana-1260	445	21	then	then	ADV
cana-1260	445	22	for	for	ADP
cana-1260	445	23	join	join	NOUN
cana-1260	445	24	𝐺	𝐺	PROPN
cana-1260	445	25	+	+	CCONJ
cana-1260	445	26	𝐻	𝐻	NOUN
cana-1260	446	1	we	we	PRON
cana-1260	446	2	have	have	VERB
cana-1260	446	3	𝑛1	𝑛1	ADJ
cana-1260	446	4	+	+	CCONJ
cana-1260	446	5	𝑛2	𝑛2	NOUN
cana-1260	446	6	vertices	vertex	NOUN
cana-1260	446	7	and	and	CCONJ
cana-1260	446	8	(	(	PUNCT
cana-1260	446	9	𝑚1	𝑚1	NOUN
cana-1260	446	10	+	+	CCONJ
cana-1260	446	11	𝑚2	𝑚2	NOUN
cana-1260	446	12	+	+	CCONJ
cana-1260	446	13	𝑛1𝑛2	𝑛1𝑛2	NOUN
cana-1260	446	14	)	)	PUNCT
cana-1260	446	15	number	number	NOUN
cana-1260	446	16	of	of	ADP
cana-1260	446	17	edges	edge	NOUN
cana-1260	446	18	.	.	PUNCT
cana-1260	447	1	here	here	ADV
cana-1260	447	2	we	we	PRON
cana-1260	447	3	consider	consider	VERB
cana-1260	447	4	the	the	DET
cana-1260	447	5	graph	graph	NOUN
cana-1260	447	6	(	(	PUNCT
cana-1260	447	7	𝐶3	𝐶3	NOUN
cana-1260	447	8	+	+	NUM
cana-1260	447	9	𝑃3	𝑃3	NOUN
cana-1260	447	10	)	)	PUNCT
cana-1260	447	11	(	(	PUNCT
cana-1260	447	12	see	see	VERB
cana-1260	447	13	figure	figure	NOUN
cana-1260	447	14	3b	3b	NUM
cana-1260	447	15	)	)	PUNCT
cana-1260	447	16	.	.	PUNCT
cana-1260	448	1	(	(	PUNCT
cana-1260	448	2	a	a	X
cana-1260	448	3	)	)	PUNCT
cana-1260	448	4	graph	graph	NOUN
cana-1260	448	5	𝐶3	𝐶3	PROPN
cana-1260	448	6	u	u	PROPN
cana-1260	448	7	𝑃3	𝑃3	NOUN
cana-1260	448	8	(	(	PUNCT
cana-1260	448	9	b	b	NOUN
cana-1260	448	10	)	)	PUNCT
cana-1260	448	11	graph	graph	NOUN
cana-1260	448	12	𝐶3	𝐶3	ADJ
cana-1260	448	13	+	+	NUM
cana-1260	448	14	𝑃3	𝑃3	NOUN
cana-1260	448	15	figure	figure	NOUN
cana-1260	448	16	3	3	NUM
cana-1260	448	17	we	we	PRON
cana-1260	448	18	will	will	AUX
cana-1260	448	19	now	now	ADV
cana-1260	448	20	state	state	VERB
cana-1260	448	21	several	several	ADJ
cana-1260	448	22	results	result	NOUN
cana-1260	448	23	involving	involve	VERB
cana-1260	448	24	fractional	fractional	ADJ
cana-1260	448	25	domination	domination	NOUN
cana-1260	448	26	of	of	ADP
cana-1260	448	27	graphs	graph	NOUN
cana-1260	448	28	.	.	PUNCT
cana-1260	449	1	clearly	clearly	ADV
cana-1260	449	2	,	,	PUNCT
cana-1260	449	3	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	449	4	(	(	PUNCT
cana-1260	449	5	𝐺	𝐺	PROPN
cana-1260	449	6	+	+	CCONJ
cana-1260	449	7	𝐻	𝐻	PROPN
cana-1260	449	8	)	)	PUNCT
cana-1260	449	9	≤	≤	NUM
cana-1260	449	10	min	min	NOUN
cana-1260	449	11	{	{	PUNCT
cana-1260	449	12	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	449	13	(	(	PUNCT
cana-1260	449	14	𝐺	𝐺	NOUN
cana-1260	449	15	)	)	PUNCT
cana-1260	449	16	,	,	PUNCT
cana-1260	449	17	𝛾𝑓	𝛾𝑓	X
cana-1260	449	18	(	(	PUNCT
cana-1260	449	19	𝐻	𝐻	NOUN
cana-1260	449	20	)	)	PUNCT
cana-1260	449	21	}	}	PUNCT
cana-1260	449	22	and	and	CCONJ
cana-1260	449	23	г𝑓	г𝑓	INTJ
cana-1260	449	24	(	(	PUNCT
cana-1260	449	25	𝐺	𝐺	PROPN
cana-1260	449	26	+	+	CCONJ
cana-1260	449	27	𝐻	𝐻	PROPN
cana-1260	449	28	)	)	PUNCT
cana-1260	449	29	=	=	SYM
cana-1260	449	30	max	max	PROPN
cana-1260	449	31	{	{	PUNCT
cana-1260	449	32	г𝑓	г𝑓	X
cana-1260	449	33	(	(	PUNCT
cana-1260	449	34	𝐺	𝐺	PROPN
cana-1260	449	35	)	)	PUNCT
cana-1260	449	36	,	,	PUNCT
cana-1260	449	37	г𝑓	г𝑓	X
cana-1260	449	38	(	(	PUNCT
cana-1260	449	39	𝐻	𝐻	PROPN
cana-1260	449	40	)	)	PUNCT
cana-1260	449	41	}	}	PUNCT
cana-1260	449	42	.	.	PUNCT
cana-1260	450	1	we	we	PRON
cana-1260	450	2	have	have	VERB
cana-1260	450	3	the	the	DET
cana-1260	450	4	following	follow	VERB
cana-1260	450	5	theorems	theorem	NOUN
cana-1260	450	6	on	on	ADP
cana-1260	450	7	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	450	8	(	(	PUNCT
cana-1260	450	9	𝐺	𝐺	PROPN
cana-1260	450	10	+	+	CCONJ
cana-1260	450	11	𝐻	𝐻	PROPN
cana-1260	450	12	)	)	PUNCT
cana-1260	450	13	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1260	450	14	г𝑓	г𝑓	NOUN
cana-1260	450	15	(	(	PUNCT
cana-1260	450	16	𝐺	𝐺	PROPN
cana-1260	450	17	+	+	CCONJ
cana-1260	450	18	𝐻	𝐻	PROPN
cana-1260	450	19	)	)	PUNCT
cana-1260	450	20	.	.	PUNCT
cana-1260	451	1	theorem	theorem	VERB
cana-1260	451	2	3.1	3.1	NUM
cana-1260	451	3	(	(	PUNCT
cana-1260	451	4	[	[	X
cana-1260	451	5	4	4	NUM
cana-1260	451	6	]	]	NUM
cana-1260	451	7	)	)	PUNCT
cana-1260	451	8	.	.	PUNCT
cana-1260	452	1	given	give	VERB
cana-1260	452	2	graphs	graph	NOUN
cana-1260	452	3	𝐺	𝐺	PROPN
cana-1260	452	4	and	and	CCONJ
cana-1260	452	5	𝐻	𝐻	PROPN
cana-1260	452	6	,	,	PUNCT
cana-1260	452	7	we	we	PRON
cana-1260	452	8	have	have	AUX
cana-1260	452	9	𝛾𝑓	𝛾𝑓	VERB
cana-1260	452	10	(	(	PUNCT
cana-1260	452	11	𝐺	𝐺	PROPN
cana-1260	452	12	+	+	CCONJ
cana-1260	452	13	𝐻	𝐻	PROPN
cana-1260	452	14	)	)	PUNCT
cana-1260	452	15	=	=	NOUN
cana-1260	452	16	{	{	PUNCT
cana-1260	452	17	1	1	NUM
cana-1260	452	18	𝑖𝑓	𝑖𝑓	NUM
cana-1260	452	19	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	452	20	(	(	PUNCT
cana-1260	452	21	𝐺	𝐺	NOUN
cana-1260	452	22	)	)	PUNCT
cana-1260	452	23	=	=	SYM
cana-1260	452	24	1	1	NUM
cana-1260	452	25	𝑜𝑟	𝑜𝑟	PRON
cana-1260	452	26	𝛾𝑓	𝛾𝑓	X
cana-1260	452	27	(	(	PUNCT
cana-1260	452	28	𝐻	𝐻	PROPN
cana-1260	452	29	)	)	PUNCT
cana-1260	452	30	=	=	SYM
cana-1260	452	31	1	1	NUM
cana-1260	452	32	,	,	PUNCT
cana-1260	452	33	2	2	NUM
cana-1260	452	34	−	−	NOUN
cana-1260	452	35	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	452	36	(	(	PUNCT
cana-1260	452	37	𝐺)+	𝐺)+	NOUN
cana-1260	452	38	𝛾𝑓	𝛾𝑓	X
cana-1260	452	39	(	(	PUNCT
cana-1260	452	40	𝐻	𝐻	PROPN
cana-1260	452	41	)	)	PUNCT
cana-1260	452	42	−	−	PROPN
cana-1260	452	43	2	2	NUM
cana-1260	452	44	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	452	45	(	(	PUNCT
cana-1260	452	46	𝐺)∗	𝐺)∗	NOUN
cana-1260	452	47	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	452	48	(	(	PUNCT
cana-1260	452	49	𝐻	𝐻	PROPN
cana-1260	452	50	)	)	PUNCT
cana-1260	452	51	−	−	PROPN
cana-1260	452	52	1	1	NUM
cana-1260	452	53	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	452	54	.	.	PUNCT
cana-1260	453	1	and	and	CCONJ
cana-1260	453	2	г𝑓	г𝑓	INTJ
cana-1260	453	3	(	(	PUNCT
cana-1260	453	4	𝐺	𝐺	PROPN
cana-1260	453	5	+	+	CCONJ
cana-1260	453	6	𝐻	𝐻	PROPN
cana-1260	453	7	)	)	PUNCT
cana-1260	453	8	=	=	SYM
cana-1260	453	9	2	2	NUM
cana-1260	453	10	−	−	NOUN
cana-1260	453	11	г𝑓	г𝑓	NOUN
cana-1260	453	12	(	(	PUNCT
cana-1260	453	13	𝐺)+	𝐺)+	NOUN
cana-1260	453	14	г𝑓	г𝑓	X
cana-1260	453	15	(	(	PUNCT
cana-1260	453	16	𝐻	𝐻	PROPN
cana-1260	453	17	)	)	PUNCT
cana-1260	453	18	−	−	PROPN
cana-1260	453	19	2	2	NUM
cana-1260	453	20	г𝑓	г𝑓	NOUN
cana-1260	453	21	(	(	PUNCT
cana-1260	453	22	𝐺)∗	𝐺)∗	NOUN
cana-1260	453	23	г𝑓	г𝑓	PART
cana-1260	453	24	(	(	PUNCT
cana-1260	453	25	𝐻	𝐻	PROPN
cana-1260	453	26	)	)	PUNCT
cana-1260	453	27	−	−	PROPN
cana-1260	453	28	1	1	NUM
cana-1260	453	29	.	.	PUNCT
cana-1260	454	1	theorem	theorem	VERB
cana-1260	454	2	3.2	3.2	NUM
cana-1260	454	3	(	(	PUNCT
cana-1260	454	4	[	[	X
cana-1260	454	5	4	4	NUM
cana-1260	454	6	]	]	NUM
cana-1260	454	7	)	)	PUNCT
cana-1260	454	8	.	.	PUNCT
cana-1260	455	1	for	for	ADP
cana-1260	455	2	graph	graph	NOUN
cana-1260	455	3	𝐺	𝐺	NOUN
cana-1260	455	4	and	and	CCONJ
cana-1260	455	5	graph	graph	NOUN
cana-1260	455	6	𝐻	𝐻	PROPN
cana-1260	455	7	we	we	PRON
cana-1260	455	8	have	have	AUX
cana-1260	455	9	using	use	VERB
cana-1260	455	10	1/∞	1/∞	NUM
cana-1260	455	11	=	=	SYM
cana-1260	455	12	0	0	NUM
cana-1260	455	13	,	,	PUNCT
cana-1260	455	14	𝛾𝑓	𝛾𝑓	X
cana-1260	455	15	(	(	PUNCT
cana-1260	455	16	𝐺	𝐺	PROPN
cana-1260	455	17	+	+	CCONJ
cana-1260	455	18	𝐻	𝐻	PROPN
cana-1260	455	19	)	)	PUNCT
cana-1260	455	20	=	=	SYM
cana-1260	455	21	2	2	NUM
cana-1260	455	22	−	−	NOUN
cana-1260	455	23	{	{	PUNCT
cana-1260	455	24	1/	1/	NUM
cana-1260	455	25	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	455	26	(	(	PUNCT
cana-1260	455	27	𝐻	𝐻	PROPN
cana-1260	455	28	)	)	PUNCT
cana-1260	455	29	,	,	PUNCT
cana-1260	455	30	𝑖𝑓	𝑖𝑓	ADP
cana-1260	456	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	456	2	(	(	PUNCT
cana-1260	456	3	𝐺	𝐺	NOUN
cana-1260	456	4	)	)	PUNCT
cana-1260	456	5	=	=	SYM
cana-1260	457	1	∞	∞	PROPN
cana-1260	457	2	1/	1/	NUM
cana-1260	457	3	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	457	4	(	(	PUNCT
cana-1260	457	5	𝐺	𝐺	PROPN
cana-1260	457	6	)	)	PUNCT
cana-1260	457	7	,	,	PUNCT
cana-1260	457	8	𝑖𝑓	𝑖𝑓	ADP
cana-1260	458	1	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	458	2	(	(	PUNCT
cana-1260	458	3	𝐻	𝐻	PROPN
cana-1260	458	4	)	)	PUNCT
cana-1260	458	5	=	=	SYM
cana-1260	458	6	∞	∞	NUM
cana-1260	459	1	г𝑓	г𝑓	NOUN
cana-1260	459	2	(	(	PUNCT
cana-1260	459	3	𝐺)+	𝐺)+	NOUN
cana-1260	459	4	г𝑓	г𝑓	X
cana-1260	459	5	(	(	PUNCT
cana-1260	459	6	𝐻	𝐻	PROPN
cana-1260	459	7	)	)	PUNCT
cana-1260	459	8	−	−	PROPN
cana-1260	459	9	2	2	NUM
cana-1260	459	10	г𝑓	г𝑓	NOUN
cana-1260	459	11	(	(	PUNCT
cana-1260	459	12	𝐺)∗	𝐺)∗	NOUN
cana-1260	459	13	г𝑓	г𝑓	PART
cana-1260	459	14	(	(	PUNCT
cana-1260	459	15	𝐻	𝐻	PROPN
cana-1260	459	16	)	)	PUNCT
cana-1260	459	17	−	−	PROPN
cana-1260	459	18	1	1	NUM
cana-1260	459	19	,	,	PUNCT
cana-1260	459	20	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	459	21	communications	communication	NOUN
cana-1260	459	22	on	on	ADP
cana-1260	459	23	applied	apply	VERB
cana-1260	459	24	nonlinear	nonlinear	ADJ
cana-1260	459	25	analysis	analysis	NOUN
cana-1260	459	26	issn	issn	NOUN
cana-1260	459	27	:	:	PUNCT
cana-1260	459	28	1074	1074	NUM
cana-1260	459	29	-	-	PUNCT
cana-1260	459	30	133x	133x	NUM
cana-1260	459	31	vol	vol	NOUN
cana-1260	459	32	31	31	NUM
cana-1260	459	33	no	no	NOUN
cana-1260	459	34	.	.	PUNCT
cana-1260	460	1	6s	6s	NUM
cana-1260	460	2	(	(	PUNCT
cana-1260	460	3	2024	2024	NUM
cana-1260	460	4	)	)	PUNCT
cana-1260	460	5	687	687	NUM
cana-1260	460	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	460	7	(	(	PUNCT
cana-1260	460	8	figure	figure	NOUN
cana-1260	460	9	3a	3a	NUM
cana-1260	460	10	)	)	PUNCT
cana-1260	460	11	as	as	ADP
cana-1260	460	12	union	union	NOUN
cana-1260	460	13	for	for	ADP
cana-1260	460	14	3	3	NUM
cana-1260	460	15	vertex	vertex	NOUN
cana-1260	460	16	cycle	cycle	NOUN
cana-1260	460	17	and	and	CCONJ
cana-1260	460	18	3	3	NUM
cana-1260	460	19	vertex	vertex	NOUN
cana-1260	460	20	path	path	NOUN
cana-1260	460	21	graph	graph	NOUN
cana-1260	460	22	and	and	CCONJ
cana-1260	460	23	(	(	PUNCT
cana-1260	460	24	figure	figure	NOUN
cana-1260	460	25	3b	3b	NUM
cana-1260	460	26	)	)	PUNCT
cana-1260	460	27	as	as	SCONJ
cana-1260	460	28	join	join	VERB
cana-1260	460	29	for	for	ADP
cana-1260	460	30	the	the	DET
cana-1260	460	31	same	same	ADJ
cana-1260	460	32	.	.	PUNCT
cana-1260	461	1	so	so	ADV
cana-1260	461	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	461	3	(	(	PUNCT
cana-1260	461	4	𝐶3	𝐶3	PROPN
cana-1260	461	5	)	)	PUNCT
cana-1260	461	6	=	=	SYM
cana-1260	461	7	1	1	NUM
cana-1260	461	8	whereas	whereas	SCONJ
cana-1260	461	9	г𝑓	г𝑓	X
cana-1260	461	10	(	(	PUNCT
cana-1260	461	11	𝐶3	𝐶3	PROPN
cana-1260	461	12	)	)	PUNCT
cana-1260	461	13	=	=	SYM
cana-1260	461	14	3/2	3/2	NUM
cana-1260	461	15	,	,	PUNCT
cana-1260	461	16	𝛾𝑓	𝛾𝑓	ADJ
cana-1260	461	17	(	(	PUNCT
cana-1260	461	18	𝑃3	𝑃3	NOUN
cana-1260	461	19	)	)	PUNCT
cana-1260	461	20	=	=	SYM
cana-1260	461	21	1	1	NUM
cana-1260	461	22	and	and	CCONJ
cana-1260	461	23	г𝑓	г𝑓	NOUN
cana-1260	461	24	(	(	PUNCT
cana-1260	461	25	𝑃3	𝑃3	NOUN
cana-1260	461	26	)	)	PUNCT
cana-1260	461	27	=	=	SYM
cana-1260	461	28	3/2	3/2	NUM
cana-1260	461	29	,	,	PUNCT
cana-1260	461	30	same	same	ADJ
cana-1260	461	31	concept	concept	NOUN
cana-1260	461	32	for	for	ADP
cana-1260	461	33	5	5	NUM
cana-1260	461	34	vertex	vertex	NOUN
cana-1260	461	35	cycle	cycle	NOUN
cana-1260	461	36	graph	graph	NOUN
cana-1260	461	37	and	and	CCONJ
cana-1260	461	38	4	4	NUM
cana-1260	461	39	vertex	vertex	NOUN
cana-1260	461	40	path	path	NOUN
cana-1260	461	41	graph	graph	NOUN
cana-1260	461	42	,	,	PUNCT
cana-1260	461	43	as	as	ADP
cana-1260	461	44	per	per	ADP
cana-1260	461	45	the	the	DET
cana-1260	461	46	join	join	NOUN
cana-1260	461	47	and	and	CCONJ
cana-1260	461	48	union	union	NOUN
cana-1260	461	49	we	we	PRON
cana-1260	461	50	get	get	VERB
cana-1260	461	51	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	461	52	(	(	PUNCT
cana-1260	461	53	𝐶5	𝐶5	PROPN
cana-1260	461	54	)	)	PUNCT
cana-1260	461	55	=	=	SYM
cana-1260	461	56	5/3	5/3	NUM
cana-1260	461	57	and	and	CCONJ
cana-1260	461	58	г𝑓	г𝑓	NOUN
cana-1260	461	59	(	(	PUNCT
cana-1260	461	60	𝐶5	𝐶5	PROPN
cana-1260	461	61	)	)	PUNCT
cana-1260	461	62	=	=	SYM
cana-1260	461	63	5/2	5/2	NUM
cana-1260	461	64	whereas	whereas	SCONJ
cana-1260	461	65	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	461	66	(	(	PUNCT
cana-1260	461	67	𝑃4	𝑃4	PROPN
cana-1260	461	68	)	)	PUNCT
cana-1260	461	69	=	=	SYM
cana-1260	461	70	2	2	NUM
cana-1260	461	71	and	and	CCONJ
cana-1260	461	72	г𝑓	г𝑓	ADJ
cana-1260	461	73	(	(	PUNCT
cana-1260	461	74	𝑃4	𝑃4	PROPN
cana-1260	461	75	)	)	PUNCT
cana-1260	461	76	=	=	SYM
cana-1260	461	77	4/2	4/2	NUM
cana-1260	461	78	.	.	PUNCT
cana-1260	461	79	therefore	therefore	ADV
cana-1260	461	80	theorem	theorem	VERB
cana-1260	461	81	on	on	ADP
cana-1260	461	82	union	union	NOUN
cana-1260	461	83	gives	give	VERB
cana-1260	461	84	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	461	85	(	(	PUNCT
cana-1260	461	86	𝐶5	𝐶5	NOUN
cana-1260	461	87	𝘜	𝘜	PROPN
cana-1260	461	88	𝑃4	𝑃4	NOUN
cana-1260	461	89	)	)	PUNCT
cana-1260	462	1	=	=	PUNCT
cana-1260	462	2	5/3	5/3	NUM
cana-1260	462	3	+	+	CCONJ
cana-1260	462	4	2	2	NUM
cana-1260	462	5	=	=	SYM
cana-1260	462	6	11/3	11/3	NUM
cana-1260	462	7	and	and	CCONJ
cana-1260	462	8	г𝑓	г𝑓	X
cana-1260	462	9	(	(	PUNCT
cana-1260	462	10	𝐶5	𝐶5	PROPN
cana-1260	462	11	𝘜	𝘜	PROPN
cana-1260	462	12	𝑃4	𝑃4	NOUN
cana-1260	462	13	)	)	PUNCT
cana-1260	462	14	=	=	PUNCT
cana-1260	463	1	5/2	5/2	NUM
cana-1260	463	2	+	+	CCONJ
cana-1260	463	3	2	2	NUM
cana-1260	463	4	=	=	SYM
cana-1260	463	5	9/2	9/2	NUM
cana-1260	463	6	.	.	PUNCT
cana-1260	464	1	theorem	theorem	VERB
cana-1260	464	2	on	on	ADP
cana-1260	464	3	join	join	NOUN
cana-1260	464	4	gives	gives	AUX
cana-1260	464	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	464	6	(	(	PUNCT
cana-1260	464	7	𝐶5	𝐶5	NOUN
cana-1260	464	8	+	+	CCONJ
cana-1260	464	9	𝑃4	𝑃4	NOUN
cana-1260	464	10	)	)	PUNCT
cana-1260	464	11	=	=	SYM
cana-1260	464	12	2	2	NUM
cana-1260	464	13	−	−	NOUN
cana-1260	464	14	(	(	PUNCT
cana-1260	464	15	5/3)+2−2	5/3)+2−2	NUM
cana-1260	464	16	(	(	PUNCT
cana-1260	464	17	5/3)∗2−1	5/3)∗2−1	NUM
cana-1260	464	18	=	=	SYM
cana-1260	464	19	9/7	9/7	NUM
cana-1260	464	20	and	and	CCONJ
cana-1260	464	21	г𝑓	г𝑓	NOUN
cana-1260	464	22	(	(	PUNCT
cana-1260	464	23	𝐶5	𝐶5	NOUN
cana-1260	464	24	+	+	CCONJ
cana-1260	464	25	𝑃4	𝑃4	NOUN
cana-1260	464	26	)	)	PUNCT
cana-1260	464	27	=	=	SYM
cana-1260	464	28	2	2	NUM
cana-1260	464	29	−	−	NOUN
cana-1260	464	30	(	(	PUNCT
cana-1260	464	31	5/2)+2−2	5/2)+2−2	NUM
cana-1260	464	32	(	(	PUNCT
cana-1260	464	33	5/2)∗2−1	5/2)∗2−1	NUM
cana-1260	464	34	=	=	SYM
cana-1260	464	35	11/8	11/8	NUM
cana-1260	464	36	.	.	PUNCT
cana-1260	464	37	theorem	theorem	VERB
cana-1260	464	38	3.3	3.3	NUM
cana-1260	464	39	let	let	VERB
cana-1260	464	40	𝐺	𝐺	PROPN
cana-1260	464	41	is	be	AUX
cana-1260	464	42	cycle	cycle	NOUN
cana-1260	464	43	graph	graph	NOUN
cana-1260	464	44	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	464	45	and	and	CCONJ
cana-1260	464	46	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	464	47	)	)	PUNCT
cana-1260	464	48	is	be	AUX
cana-1260	464	49	line	line	NOUN
cana-1260	464	50	graph	graph	NOUN
cana-1260	464	51	,	,	PUNCT
cana-1260	464	52	we	we	PRON
cana-1260	464	53	have	have	AUX
cana-1260	464	54	𝛾𝑓	𝛾𝑓	VERB
cana-1260	464	55	(	(	PUNCT
cana-1260	464	56	𝐺	𝐺	PROPN
cana-1260	464	57	+	+	CCONJ
cana-1260	464	58	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	464	59	)	)	PUNCT
cana-1260	464	60	)	)	PUNCT
cana-1260	465	1	=	=	PRON
cana-1260	465	2	{	{	PUNCT
cana-1260	465	3	1	1	NUM
cana-1260	465	4	𝑖𝑓	𝑖𝑓	NUM
cana-1260	466	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	466	2	(	(	PUNCT
cana-1260	466	3	𝐺	𝐺	NOUN
cana-1260	466	4	)	)	PUNCT
cana-1260	466	5	=	=	SYM
cana-1260	466	6	1	1	NUM
cana-1260	466	7	𝑜𝑟	𝑜𝑟	PRON
cana-1260	466	8	𝛾𝑓	𝛾𝑓	X
cana-1260	466	9	(	(	PUNCT
cana-1260	466	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	466	11	)	)	PUNCT
cana-1260	466	12	)	)	PUNCT
cana-1260	466	13	=	=	PUNCT
cana-1260	467	1	1	1	NUM
cana-1260	467	2	,	,	PUNCT
cana-1260	467	3	2	2	NUM
cana-1260	467	4	−	−	PROPN
cana-1260	467	5	2𝑛	2𝑛	PROPN
cana-1260	467	6	3	3	NUM
cana-1260	467	7	−	−	PROPN
cana-1260	467	8	2	2	NUM
cana-1260	467	9	𝑛2	𝑛2	NOUN
cana-1260	467	10	9	9	NUM
cana-1260	467	11	−	−	NOUN
cana-1260	467	12	1	1	NUM
cana-1260	467	13	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	467	14	and	and	CCONJ
cana-1260	467	15	г𝑓	г𝑓	NOUN
cana-1260	467	16	(	(	PUNCT
cana-1260	467	17	𝐺	𝐺	PROPN
cana-1260	467	18	+	+	CCONJ
cana-1260	467	19	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	467	20	)	)	PUNCT
cana-1260	467	21	)	)	PUNCT
cana-1260	468	1	=	=	SYM
cana-1260	468	2	2	2	NUM
cana-1260	468	3	−	−	NOUN
cana-1260	468	4	𝑛	𝑛	DET
cana-1260	468	5	−	−	NUM
cana-1260	468	6	2	2	NUM
cana-1260	468	7	𝑛2	𝑛2	NOUN
cana-1260	468	8	4	4	NUM
cana-1260	468	9	−	−	NOUN
cana-1260	468	10	1	1	NUM
cana-1260	468	11	.	.	PUNCT
cana-1260	469	1	proof	proof	NOUN
cana-1260	469	2	.	.	PUNCT
cana-1260	470	1	theorem	theorem	VERB
cana-1260	470	2	2.1.1	2.1.1	NUM
cana-1260	470	3	and	and	CCONJ
cana-1260	470	4	theorem	theorem	VERB
cana-1260	470	5	3.1	3.1	NUM
cana-1260	470	6	gives	give	VERB
cana-1260	470	7	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	470	8	(	(	PUNCT
cana-1260	470	9	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	470	10	)	)	PUNCT
cana-1260	470	11	=	=	NOUN
cana-1260	470	12	𝑛/3	𝑛/3	NUM
cana-1260	470	13	and	and	CCONJ
cana-1260	470	14	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	470	15	(	(	PUNCT
cana-1260	470	16	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	470	17	)	)	PUNCT
cana-1260	470	18	)	)	PUNCT
cana-1260	471	1	=	=	SYM
cana-1260	471	2	𝑛/3	𝑛/3	NUM
cana-1260	471	3	so	so	ADV
cana-1260	471	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	471	5	(	(	PUNCT
cana-1260	471	6	𝐺	𝐺	NOUN
cana-1260	471	7	)	)	PUNCT
cana-1260	471	8	+	+	CCONJ
cana-1260	471	9	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	471	10	(	(	PUNCT
cana-1260	471	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	471	12	)	)	PUNCT
cana-1260	471	13	)	)	PUNCT
cana-1260	472	1	=	=	SYM
cana-1260	472	2	2𝑛	2𝑛	NOUN
cana-1260	472	3	3	3	NUM
cana-1260	472	4	and	and	CCONJ
cana-1260	472	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	472	6	(	(	PUNCT
cana-1260	472	7	𝐺	𝐺	NOUN
cana-1260	472	8	)	)	PUNCT
cana-1260	472	9	∗	∗	NOUN
cana-1260	472	10	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	472	11	(	(	PUNCT
cana-1260	472	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	472	13	)	)	PUNCT
cana-1260	472	14	)	)	PUNCT
cana-1260	473	1	=	=	NOUN
cana-1260	473	2	𝑛2	𝑛2	NOUN
cana-1260	473	3	9	9	NUM
cana-1260	473	4	hence	hence	ADV
cana-1260	473	5	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	474	1	(	(	PUNCT
cana-1260	474	2	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	474	3	+	+	NOUN
cana-1260	474	4	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	474	5	)	)	PUNCT
cana-1260	474	6	)	)	PUNCT
cana-1260	475	1	=	=	SYM
cana-1260	475	2	1	1	NUM
cana-1260	475	3	𝑖𝑓	𝑖𝑓	NUM
cana-1260	475	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	475	5	(	(	PUNCT
cana-1260	475	6	𝐺	𝐺	NOUN
cana-1260	475	7	)	)	PUNCT
cana-1260	475	8	=	=	SYM
cana-1260	475	9	1	1	NUM
cana-1260	475	10	𝑜𝑟	𝑜𝑟	PRON
cana-1260	475	11	𝛾𝑓	𝛾𝑓	X
cana-1260	475	12	(	(	PUNCT
cana-1260	475	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	475	14	)	)	PUNCT
cana-1260	475	15	)	)	PUNCT
cana-1260	476	1	=	=	SYM
cana-1260	476	2	1	1	NUM
cana-1260	476	3	otherwise	otherwise	ADV
cana-1260	476	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	476	5	(	(	PUNCT
cana-1260	476	6	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	476	7	+	+	NOUN
cana-1260	476	8	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	476	9	)	)	PUNCT
cana-1260	476	10	)	)	PUNCT
cana-1260	477	1	=	=	SYM
cana-1260	477	2	2	2	NUM
cana-1260	477	3	−	−	PROPN
cana-1260	477	4	2𝑛	2𝑛	NOUN
cana-1260	477	5	3	3	NUM
cana-1260	477	6	−	−	PROPN
cana-1260	477	7	2	2	NUM
cana-1260	477	8	𝑛2	𝑛2	NOUN
cana-1260	477	9	9	9	NUM
cana-1260	477	10	−	−	NOUN
cana-1260	477	11	1	1	NUM
cana-1260	477	12	=	=	SYM
cana-1260	477	13	2	2	NUM
cana-1260	477	14	−	−	PROPN
cana-1260	477	15	6𝑛−18	6𝑛−18	PRON
cana-1260	477	16	𝑛2−9	𝑛2−9	VERB
cana-1260	477	17	.	.	PUNCT
cana-1260	478	1	similarly	similarly	ADV
cana-1260	478	2	г𝑓	г𝑓	X
cana-1260	478	3	(	(	PUNCT
cana-1260	478	4	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	478	5	)	)	PUNCT
cana-1260	478	6	=	=	SYM
cana-1260	478	7	𝑛/2	𝑛/2	PROPN
cana-1260	478	8	and	and	CCONJ
cana-1260	478	9	г𝑓	г𝑓	X
cana-1260	478	10	(	(	PUNCT
cana-1260	478	11	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	478	12	)	)	PUNCT
cana-1260	478	13	)	)	PUNCT
cana-1260	479	1	=	=	SYM
cana-1260	479	2	𝑛/2	𝑛/2	PROPN
cana-1260	480	1	so	so	ADV
cana-1260	480	2	г𝑓	г𝑓	ADP
cana-1260	480	3	(	(	PUNCT
cana-1260	480	4	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	480	5	)	)	PUNCT
cana-1260	481	1	+	+	NUM
cana-1260	481	2	г𝑓	г𝑓	X
cana-1260	481	3	(	(	PUNCT
cana-1260	481	4	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	481	5	)	)	PUNCT
cana-1260	481	6	)	)	PUNCT
cana-1260	482	1	=	=	SYM
cana-1260	482	2	𝑛	𝑛	PROPN
cana-1260	482	3	and	and	CCONJ
cana-1260	482	4	г𝑓	г𝑓	X
cana-1260	482	5	(	(	PUNCT
cana-1260	482	6	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	482	7	)	)	PUNCT
cana-1260	482	8	∗	∗	NOUN
cana-1260	482	9	г𝑓	г𝑓	NOUN
cana-1260	482	10	(	(	PUNCT
cana-1260	482	11	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	482	12	)	)	PUNCT
cana-1260	482	13	)	)	PUNCT
cana-1260	483	1	=	=	NOUN
cana-1260	483	2	𝑛2	𝑛2	NOUN
cana-1260	483	3	4	4	NUM
cana-1260	483	4	hence	hence	ADV
cana-1260	483	5	г𝑓	г𝑓	NOUN
cana-1260	483	6	(	(	PUNCT
cana-1260	483	7	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	483	8	+	+	NOUN
cana-1260	483	9	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	483	10	)	)	PUNCT
cana-1260	483	11	)	)	PUNCT
cana-1260	484	1	=	=	SYM
cana-1260	484	2	2	2	NUM
cana-1260	484	3	−	−	NOUN
cana-1260	484	4	𝑛	𝑛	DET
cana-1260	484	5	−	−	NUM
cana-1260	484	6	2	2	NUM
cana-1260	484	7	𝑛2	𝑛2	NOUN
cana-1260	484	8	4	4	NUM
cana-1260	484	9	−	−	NOUN
cana-1260	484	10	1	1	NUM
cana-1260	484	11	=	=	SYM
cana-1260	484	12	2	2	NUM
cana-1260	484	13	−	−	NOUN
cana-1260	484	14	4𝑛	4𝑛	NOUN
cana-1260	484	15	−	−	NOUN
cana-1260	484	16	8	8	NUM
cana-1260	484	17	𝑛2−4	𝑛2−4	PROPN
cana-1260	484	18	.	.	PUNCT
cana-1260	485	1	□	□	PUNCT
cana-1260	485	2	theorem	theorem	VERB
cana-1260	485	3	3.4	3.4	NUM
cana-1260	485	4	for	for	ADP
cana-1260	485	5	the	the	DET
cana-1260	485	6	graph	graph	NOUN
cana-1260	485	7	𝐺	𝐺	PROPN
cana-1260	485	8	as	as	ADP
cana-1260	485	9	complete	complete	ADJ
cana-1260	485	10	graph	graph	NOUN
cana-1260	485	11	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	485	12	and	and	CCONJ
cana-1260	485	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	485	14	)	)	PUNCT
cana-1260	485	15	be	be	VERB
cana-1260	485	16	its	its	PRON
cana-1260	485	17	line	line	NOUN
cana-1260	485	18	graph	graph	NOUN
cana-1260	485	19	,	,	PUNCT
cana-1260	485	20	we	we	PRON
cana-1260	485	21	have	have	AUX
cana-1260	485	22	𝛾𝑓	𝛾𝑓	VERB
cana-1260	485	23	(	(	PUNCT
cana-1260	485	24	𝐺	𝐺	PROPN
cana-1260	485	25	+	+	CCONJ
cana-1260	485	26	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	485	27	)	)	PUNCT
cana-1260	485	28	)	)	PUNCT
cana-1260	486	1	=	=	PRON
cana-1260	486	2	{	{	PUNCT
cana-1260	486	3	1	1	NUM
cana-1260	486	4	𝑖𝑓	𝑖𝑓	NUM
cana-1260	487	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	487	2	(	(	PUNCT
cana-1260	487	3	𝐺	𝐺	NOUN
cana-1260	487	4	)	)	PUNCT
cana-1260	487	5	=	=	SYM
cana-1260	487	6	1	1	NUM
cana-1260	487	7	𝑜𝑟	𝑜𝑟	PRON
cana-1260	487	8	𝛾𝑓	𝛾𝑓	X
cana-1260	487	9	(	(	PUNCT
cana-1260	487	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	487	11	)	)	PUNCT
cana-1260	487	12	)	)	PUNCT
cana-1260	487	13	=	=	PUNCT
cana-1260	488	1	1	1	NUM
cana-1260	488	2	,	,	PUNCT
cana-1260	488	3	2	2	NUM
cana-1260	488	4	−	−	NOUN
cana-1260	488	5	1	1	NUM
cana-1260	488	6	+	+	NUM
cana-1260	488	7	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	488	8	(	(	PUNCT
cana-1260	488	9	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	488	10	−	−	NOUN
cana-1260	488	11	2	2	NUM
cana-1260	488	12	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	488	13	(	(	PUNCT
cana-1260	488	14	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	488	15	−	−	NOUN
cana-1260	488	16	1	1	NUM
cana-1260	488	17	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	488	18	and	and	CCONJ
cana-1260	488	19	г𝑓	г𝑓	NOUN
cana-1260	488	20	(	(	PUNCT
cana-1260	488	21	𝐺	𝐺	PROPN
cana-1260	488	22	+	+	CCONJ
cana-1260	488	23	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	488	24	)	)	PUNCT
cana-1260	488	25	)	)	PUNCT
cana-1260	489	1	=	=	SYM
cana-1260	489	2	2	2	NUM
cana-1260	489	3	−	−	NUM
cana-1260	489	4	1	1	NUM
cana-1260	489	5	+	+	NUM
cana-1260	489	6	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	489	7	2𝑛−4	2𝑛−4	NUM
cana-1260	489	8	−	−	NUM
cana-1260	489	9	2	2	NUM
cana-1260	489	10	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	489	11	2𝑛−4	2𝑛−4	NUM
cana-1260	489	12	−	−	NOUN
cana-1260	489	13	1	1	NUM
cana-1260	489	14	.	.	PUNCT
cana-1260	490	1	proof	proof	NOUN
cana-1260	490	2	:	:	PUNCT
cana-1260	490	3	theorem	theorem	VERB
cana-1260	490	4	2.2.1	2.2.1	NUM
cana-1260	490	5	and	and	CCONJ
cana-1260	490	6	theorem	theorem	VERB
cana-1260	490	7	3.1	3.1	NUM
cana-1260	490	8	gives	give	VERB
cana-1260	490	9	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	490	10	(	(	PUNCT
cana-1260	490	11	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	490	12	)	)	PUNCT
cana-1260	490	13	=	=	SYM
cana-1260	490	14	1	1	NUM
cana-1260	490	15	and	and	CCONJ
cana-1260	490	16	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	490	17	(	(	PUNCT
cana-1260	490	18	𝐿(𝐾𝑛	𝐿(𝐾𝑛	NOUN
cana-1260	490	19	)	)	PUNCT
cana-1260	490	20	)	)	PUNCT
cana-1260	491	1	=	=	SYM
cana-1260	491	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	491	3	(	(	PUNCT
cana-1260	491	4	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	491	5	so	so	ADV
cana-1260	491	6	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	491	7	(	(	PUNCT
cana-1260	491	8	𝐺	𝐺	NOUN
cana-1260	491	9	)	)	PUNCT
cana-1260	491	10	+	+	CCONJ
cana-1260	491	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	491	12	(	(	PUNCT
cana-1260	491	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	491	14	)	)	PUNCT
cana-1260	491	15	)	)	PUNCT
cana-1260	492	1	=	=	SYM
cana-1260	492	2	1	1	NUM
cana-1260	492	3	+	+	NUM
cana-1260	492	4	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	492	5	(	(	PUNCT
cana-1260	492	6	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	492	7	and	and	CCONJ
cana-1260	492	8	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	492	9	(	(	PUNCT
cana-1260	492	10	𝐺	𝐺	NOUN
cana-1260	492	11	)	)	PUNCT
cana-1260	492	12	∗	∗	NOUN
cana-1260	492	13	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	492	14	(	(	PUNCT
cana-1260	492	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	492	16	)	)	PUNCT
cana-1260	492	17	)	)	PUNCT
cana-1260	493	1	=	=	SYM
cana-1260	493	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	493	3	(	(	PUNCT
cana-1260	493	4	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	493	5	.	.	PUNCT
cana-1260	494	1	hence	hence	ADV
cana-1260	494	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	494	3	(	(	PUNCT
cana-1260	494	4	𝐾𝑛	𝐾𝑛	NOUN
cana-1260	494	5	+	+	CCONJ
cana-1260	494	6	𝐿(𝐾𝑛	𝐿(𝐾𝑛	NOUN
cana-1260	494	7	)	)	PUNCT
cana-1260	494	8	)	)	PUNCT
cana-1260	495	1	=	=	PUNCT
cana-1260	495	2	1	1	NUM
cana-1260	495	3	,	,	PUNCT
cana-1260	495	4	𝑖𝑓	𝑖𝑓	ADV
cana-1260	495	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	495	6	(	(	PUNCT
cana-1260	495	7	𝐺	𝐺	NOUN
cana-1260	495	8	)	)	PUNCT
cana-1260	495	9	=	=	SYM
cana-1260	496	1	1	1	NUM
cana-1260	496	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	496	3	𝛾𝑓	𝛾𝑓	X
cana-1260	496	4	(	(	PUNCT
cana-1260	496	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	496	6	)	)	PUNCT
cana-1260	496	7	)	)	PUNCT
cana-1260	497	1	=	=	SYM
cana-1260	497	2	1	1	NUM
cana-1260	497	3	,	,	PUNCT
cana-1260	497	4	otherwise	otherwise	ADV
cana-1260	497	5	communications	communication	NOUN
cana-1260	497	6	on	on	ADP
cana-1260	497	7	applied	apply	VERB
cana-1260	497	8	nonlinear	nonlinear	ADJ
cana-1260	497	9	analysis	analysis	NOUN
cana-1260	497	10	issn	issn	NOUN
cana-1260	497	11	:	:	PUNCT
cana-1260	497	12	1074	1074	NUM
cana-1260	497	13	-	-	PUNCT
cana-1260	497	14	133x	133x	NUM
cana-1260	497	15	vol	vol	NOUN
cana-1260	497	16	31	31	NUM
cana-1260	497	17	no	no	NOUN
cana-1260	497	18	.	.	PUNCT
cana-1260	498	1	6s	6s	NUM
cana-1260	498	2	(	(	PUNCT
cana-1260	498	3	2024	2024	NUM
cana-1260	498	4	)	)	PUNCT
cana-1260	498	5	688	688	NUM
cana-1260	498	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	498	7	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	498	8	(	(	PUNCT
cana-1260	498	9	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	498	10	+	+	CCONJ
cana-1260	498	11	𝐿(𝐾𝑛	𝐿(𝐾𝑛	NOUN
cana-1260	498	12	)	)	PUNCT
cana-1260	498	13	)	)	PUNCT
cana-1260	499	1	=	=	SYM
cana-1260	499	2	2	2	NUM
cana-1260	499	3	−	−	NUM
cana-1260	499	4	1	1	NUM
cana-1260	499	5	+	+	NUM
cana-1260	499	6	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	499	7	(	(	PUNCT
cana-1260	499	8	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	499	9	−	−	NOUN
cana-1260	499	10	2	2	NUM
cana-1260	499	11	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	499	12	(	(	PUNCT
cana-1260	499	13	2𝑛−4)+1	2𝑛−4)+1	NOUN
cana-1260	499	14	−	−	NOUN
cana-1260	499	15	1	1	NUM
cana-1260	499	16	.	.	PUNCT
cana-1260	500	1	similarly	similarly	ADV
cana-1260	500	2	г𝑓	г𝑓	X
cana-1260	500	3	(	(	PUNCT
cana-1260	500	4	𝐾𝑛	𝐾𝑛	PROPN
cana-1260	500	5	)	)	PUNCT
cana-1260	500	6	=	=	SYM
cana-1260	500	7	1	1	NUM
cana-1260	500	8	and	and	CCONJ
cana-1260	500	9	г𝑓	г𝑓	INTJ
cana-1260	500	10	(	(	PUNCT
cana-1260	500	11	𝐿(𝐾𝑛	𝐿(𝐾𝑛	PROPN
cana-1260	500	12	)	)	PUNCT
cana-1260	500	13	)	)	PUNCT
cana-1260	501	1	=	=	PUNCT
cana-1260	501	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	501	3	2𝑛−4	2𝑛−4	NUM
cana-1260	501	4	so	so	ADV
cana-1260	501	5	г𝑓	г𝑓	PROPN
cana-1260	501	6	(	(	PUNCT
cana-1260	501	7	𝐺	𝐺	NOUN
cana-1260	501	8	)	)	PUNCT
cana-1260	502	1	+	+	NUM
cana-1260	502	2	г𝑓	г𝑓	ADJ
cana-1260	502	3	(	(	PUNCT
cana-1260	502	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	502	5	)	)	PUNCT
cana-1260	502	6	)	)	PUNCT
cana-1260	502	7	=	=	SYM
cana-1260	503	1	1	1	NUM
cana-1260	503	2	+	+	NUM
cana-1260	503	3	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	503	4	2𝑛−4	2𝑛−4	NUM
cana-1260	503	5	and	and	CCONJ
cana-1260	503	6	г𝑓	г𝑓	PROPN
cana-1260	503	7	(	(	PUNCT
cana-1260	503	8	𝐺	𝐺	NOUN
cana-1260	503	9	)	)	PUNCT
cana-1260	503	10	∗	∗	NOUN
cana-1260	503	11	г𝑓	г𝑓	PROPN
cana-1260	503	12	(	(	PUNCT
cana-1260	503	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	503	14	)	)	PUNCT
cana-1260	503	15	)	)	PUNCT
cana-1260	504	1	=	=	PUNCT
cana-1260	504	2	𝑛(𝑛−1)/2	𝑛(𝑛−1)/2	PROPN
cana-1260	504	3	2𝑛−4	2𝑛−4	NUM
cana-1260	504	4	hence	hence	ADV
cana-1260	504	5	г𝑓	г𝑓	X
cana-1260	504	6	(	(	PUNCT
cana-1260	504	7	𝐾𝑛	𝐾𝑛	NOUN
cana-1260	504	8	+	+	CCONJ
cana-1260	504	9	𝐿(𝐾𝑛	𝐿(𝐾𝑛	NOUN
cana-1260	504	10	)	)	PUNCT
cana-1260	504	11	)	)	PUNCT
cana-1260	505	1	=	=	SYM
cana-1260	506	1	2	2	NUM
cana-1260	506	2	−	−	NUM
cana-1260	506	3	1	1	NUM
cana-1260	506	4	+	+	SYM
cana-1260	506	5	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	506	6	)	)	PUNCT
cana-1260	506	7	2	2	NUM
cana-1260	506	8	2𝑛−4	2𝑛−4	NUM
cana-1260	506	9	−	−	PROPN
cana-1260	506	10	2	2	NUM
cana-1260	506	11	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-1260	506	12	)	)	PUNCT
cana-1260	506	13	2	2	NUM
cana-1260	506	14	2𝑛−4	2𝑛−4	NUM
cana-1260	506	15	−	−	NOUN
cana-1260	506	16	1	1	NUM
cana-1260	506	17	.	.	PUNCT
cana-1260	506	18	□	□	PUNCT
cana-1260	506	19	theorem	theorem	ADJ
cana-1260	506	20	3.5	3.5	NUM
cana-1260	506	21	let	let	VERB
cana-1260	506	22	𝐺	𝐺	PROPN
cana-1260	506	23	is	be	AUX
cana-1260	506	24	the	the	DET
cana-1260	506	25	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	506	26	as	as	ADP
cana-1260	506	27	star	star	NOUN
cana-1260	506	28	graph	graph	NOUN
cana-1260	506	29	,	,	PUNCT
cana-1260	506	30	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	506	31	)	)	PUNCT
cana-1260	506	32	be	be	AUX
cana-1260	506	33	its	its	PRON
cana-1260	506	34	line	line	NOUN
cana-1260	506	35	graph	graph	NOUN
cana-1260	506	36	,	,	PUNCT
cana-1260	506	37	we	we	PRON
cana-1260	506	38	have	have	AUX
cana-1260	506	39	𝛾𝑓	𝛾𝑓	VERB
cana-1260	506	40	(	(	PUNCT
cana-1260	506	41	𝐺	𝐺	PROPN
cana-1260	506	42	+	+	CCONJ
cana-1260	506	43	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	506	44	)	)	PUNCT
cana-1260	506	45	)	)	PUNCT
cana-1260	507	1	=	=	PRON
cana-1260	507	2	{	{	PUNCT
cana-1260	507	3	1	1	NUM
cana-1260	507	4	𝑖𝑓	𝑖𝑓	NUM
cana-1260	508	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	508	2	(	(	PUNCT
cana-1260	508	3	𝐺	𝐺	NOUN
cana-1260	508	4	)	)	PUNCT
cana-1260	508	5	=	=	SYM
cana-1260	508	6	1	1	NUM
cana-1260	508	7	𝑜𝑟	𝑜𝑟	PRON
cana-1260	508	8	𝛾𝑓	𝛾𝑓	X
cana-1260	508	9	(	(	PUNCT
cana-1260	508	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	508	11	)	)	PUNCT
cana-1260	508	12	)	)	PUNCT
cana-1260	508	13	=	=	PUNCT
cana-1260	509	1	1	1	NUM
cana-1260	509	2	,	,	PUNCT
cana-1260	509	3	2	2	NUM
cana-1260	509	4	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	509	5	and	and	CCONJ
cana-1260	509	6	г𝑓	г𝑓	NOUN
cana-1260	509	7	(	(	PUNCT
cana-1260	509	8	𝐺	𝐺	PROPN
cana-1260	509	9	+	+	CCONJ
cana-1260	509	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	509	11	)	)	PUNCT
cana-1260	509	12	)	)	PUNCT
cana-1260	509	13	≤	≤	ADV
cana-1260	510	1	2	2	NUM
cana-1260	510	2	−	−	NOUN
cana-1260	510	3	𝑛	𝑛	PROPN
cana-1260	510	4	+	+	CCONJ
cana-1260	510	5	𝑛−1	𝑛−1	NUM
cana-1260	510	6	𝑛−2	𝑛−2	NOUN
cana-1260	510	7	−	−	NOUN
cana-1260	510	8	2	2	NUM
cana-1260	510	9	𝑛2−𝑛	𝑛2−𝑛	NOUN
cana-1260	510	10	𝑛−2	𝑛−2	NOUN
cana-1260	510	11	−	−	NOUN
cana-1260	510	12	1	1	NUM
cana-1260	510	13	.	.	PUNCT
cana-1260	511	1	proof	proof	NOUN
cana-1260	511	2	:	:	PUNCT
cana-1260	511	3	theorem	theorem	VERB
cana-1260	511	4	2.3.1	2.3.1	NUM
cana-1260	511	5	and	and	CCONJ
cana-1260	511	6	theorem	theorem	VERB
cana-1260	511	7	3.1	3.1	NUM
cana-1260	511	8	gives	give	VERB
cana-1260	511	9	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	511	10	(	(	PUNCT
cana-1260	511	11	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	511	12	)	)	PUNCT
cana-1260	511	13	=	=	SYM
cana-1260	511	14	1	1	NUM
cana-1260	511	15	and	and	CCONJ
cana-1260	511	16	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	511	17	(	(	PUNCT
cana-1260	511	18	𝐿(𝑆𝑛	𝐿(𝑆𝑛	PROPN
cana-1260	511	19	)	)	PUNCT
cana-1260	511	20	)	)	PUNCT
cana-1260	512	1	=	=	SYM
cana-1260	512	2	1	1	NUM
cana-1260	512	3	so	so	ADV
cana-1260	512	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	512	5	(	(	PUNCT
cana-1260	512	6	𝐺	𝐺	NOUN
cana-1260	512	7	)	)	PUNCT
cana-1260	513	1	+	+	CCONJ
cana-1260	513	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	513	3	(	(	PUNCT
cana-1260	513	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	513	5	)	)	PUNCT
cana-1260	513	6	)	)	PUNCT
cana-1260	513	7	=	=	SYM
cana-1260	513	8	2	2	NUM
cana-1260	513	9	and	and	CCONJ
cana-1260	513	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	513	11	(	(	PUNCT
cana-1260	513	12	𝐺	𝐺	NOUN
cana-1260	513	13	)	)	PUNCT
cana-1260	513	14	∗	∗	NOUN
cana-1260	513	15	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	513	16	(	(	PUNCT
cana-1260	513	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	513	18	)	)	PUNCT
cana-1260	513	19	)	)	PUNCT
cana-1260	514	1	=	=	SYM
cana-1260	514	2	1	1	NUM
cana-1260	514	3	hence	hence	ADV
cana-1260	514	4	𝛾𝑓	𝛾𝑓	X
cana-1260	514	5	(	(	PUNCT
cana-1260	514	6	𝑆𝑛	𝑆𝑛	NOUN
cana-1260	514	7	+	+	CCONJ
cana-1260	514	8	𝐿(𝑆𝑛	𝐿(𝑆𝑛	NOUN
cana-1260	514	9	)	)	PUNCT
cana-1260	514	10	)	)	PUNCT
cana-1260	515	1	=	=	PUNCT
cana-1260	515	2	1	1	NUM
cana-1260	515	3	if	if	SCONJ
cana-1260	515	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	515	5	(	(	PUNCT
cana-1260	515	6	𝐺	𝐺	NOUN
cana-1260	515	7	)	)	PUNCT
cana-1260	515	8	=	=	SYM
cana-1260	516	1	1	1	NUM
cana-1260	516	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	516	3	𝛾𝑓	𝛾𝑓	X
cana-1260	516	4	(	(	PUNCT
cana-1260	516	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	516	6	)	)	PUNCT
cana-1260	516	7	)	)	PUNCT
cana-1260	517	1	=	=	SYM
cana-1260	517	2	1	1	NUM
cana-1260	517	3	otherwise	otherwise	ADV
cana-1260	517	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	517	5	(	(	PUNCT
cana-1260	517	6	𝑆𝑛	𝑆𝑛	NOUN
cana-1260	517	7	+	+	CCONJ
cana-1260	517	8	𝐿(𝑆𝑛	𝐿(𝑆𝑛	NOUN
cana-1260	517	9	)	)	PUNCT
cana-1260	517	10	)	)	PUNCT
cana-1260	518	1	=	=	SYM
cana-1260	518	2	2	2	X
cana-1260	518	3	.	.	X
cana-1260	518	4	similarly	similarly	ADV
cana-1260	518	5	г𝑓	г𝑓	X
cana-1260	518	6	(	(	PUNCT
cana-1260	518	7	𝑆𝑛	𝑆𝑛	PROPN
cana-1260	518	8	)	)	PUNCT
cana-1260	518	9	=	=	SYM
cana-1260	518	10	𝑛	𝑛	PROPN
cana-1260	518	11	and	and	CCONJ
cana-1260	518	12	г𝑓	г𝑓	X
cana-1260	518	13	(	(	PUNCT
cana-1260	518	14	𝐿(𝑆𝑛	𝐿(𝑆𝑛	PROPN
cana-1260	518	15	)	)	PUNCT
cana-1260	518	16	)	)	PUNCT
cana-1260	519	1	=	=	PUNCT
cana-1260	519	2	𝑛−1	𝑛−1	NUM
cana-1260	519	3	𝑛−2	𝑛−2	NOUN
cana-1260	519	4	so	so	ADV
cana-1260	519	5	г𝑓	г𝑓	ADP
cana-1260	519	6	(	(	PUNCT
cana-1260	519	7	𝐺	𝐺	NOUN
cana-1260	519	8	)	)	PUNCT
cana-1260	519	9	+	+	NUM
cana-1260	519	10	г𝑓	г𝑓	ADJ
cana-1260	519	11	(	(	PUNCT
cana-1260	519	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	519	13	)	)	PUNCT
cana-1260	519	14	)	)	PUNCT
cana-1260	519	15	≤	≤	NOUN
cana-1260	520	1	𝑛	𝑛	DET
cana-1260	520	2	+	+	CCONJ
cana-1260	520	3	𝑛−1	𝑛−1	PROPN
cana-1260	520	4	𝑛−2	𝑛−2	PROPN
cana-1260	520	5	and	and	CCONJ
cana-1260	520	6	г𝑓	г𝑓	NOUN
cana-1260	520	7	(	(	PUNCT
cana-1260	520	8	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	520	9	)	)	PUNCT
cana-1260	520	10	∗	∗	NOUN
cana-1260	520	11	г𝑓	г𝑓	NOUN
cana-1260	520	12	(	(	PUNCT
cana-1260	520	13	𝐿(𝐶𝑛	𝐿(𝐶𝑛	NUM
cana-1260	520	14	)	)	PUNCT
cana-1260	520	15	)	)	PUNCT
cana-1260	520	16	≤	≤	NOUN
cana-1260	521	1	𝑛2−𝑛	𝑛2−𝑛	NOUN
cana-1260	521	2	𝑛−2	𝑛−2	PROPN
cana-1260	521	3	hence	hence	ADV
cana-1260	521	4	г𝑓	г𝑓	NOUN
cana-1260	521	5	(	(	PUNCT
cana-1260	521	6	𝑆𝑛	𝑆𝑛	NOUN
cana-1260	521	7	+	+	CCONJ
cana-1260	521	8	𝐿(𝑆𝑛	𝐿(𝑆𝑛	NOUN
cana-1260	521	9	)	)	PUNCT
cana-1260	521	10	)	)	PUNCT
cana-1260	522	1	≤	≤	ADV
cana-1260	522	2	2	2	NUM
cana-1260	522	3	−	−	NOUN
cana-1260	522	4	𝑛	𝑛	PROPN
cana-1260	522	5	+	+	CCONJ
cana-1260	522	6	𝑛−1	𝑛−1	NUM
cana-1260	522	7	𝑛−2	𝑛−2	NOUN
cana-1260	522	8	−	−	NOUN
cana-1260	522	9	2	2	NUM
cana-1260	522	10	𝑛2−𝑛	𝑛2−𝑛	NOUN
cana-1260	522	11	𝑛−2	𝑛−2	NOUN
cana-1260	522	12	−	−	NOUN
cana-1260	522	13	1	1	NUM
cana-1260	522	14	.	.	PUNCT
cana-1260	523	1	□	□	PUNCT
cana-1260	523	2	theorem	theorem	ADJ
cana-1260	523	3	3.6	3.6	NUM
cana-1260	523	4	let	let	VERB
cana-1260	523	5	𝐺	𝐺	PROPN
cana-1260	523	6	is	be	AUX
cana-1260	523	7	the	the	DET
cana-1260	523	8	graph	graph	NOUN
cana-1260	523	9	(	(	PUNCT
cana-1260	523	10	𝐾1,𝑛	𝐾1,𝑛	PROPN
cana-1260	523	11	)	)	PUNCT
cana-1260	523	12	a	a	DET
cana-1260	523	13	as	as	ADP
cana-1260	523	14	bistar	bistar	NOUN
cana-1260	523	15	graph	graph	NOUN
cana-1260	523	16	and	and	CCONJ
cana-1260	523	17	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	523	18	)	)	PUNCT
cana-1260	523	19	be	be	VERB
cana-1260	523	20	its	its	PRON
cana-1260	523	21	line	line	NOUN
cana-1260	523	22	graph	graph	NOUN
cana-1260	523	23	,	,	PUNCT
cana-1260	523	24	we	we	PRON
cana-1260	523	25	have	have	AUX
cana-1260	523	26	𝛾𝑓	𝛾𝑓	VERB
cana-1260	523	27	(	(	PUNCT
cana-1260	523	28	𝐺	𝐺	PROPN
cana-1260	523	29	+	+	CCONJ
cana-1260	523	30	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	523	31	)	)	PUNCT
cana-1260	523	32	)	)	PUNCT
cana-1260	524	1	=	=	PUNCT
cana-1260	524	2	1	1	NUM
cana-1260	524	3	if	if	SCONJ
cana-1260	524	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	524	5	(	(	PUNCT
cana-1260	524	6	𝐺	𝐺	NOUN
cana-1260	524	7	)	)	PUNCT
cana-1260	524	8	=	=	SYM
cana-1260	525	1	1	1	NUM
cana-1260	525	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	525	3	𝛾𝑓	𝛾𝑓	X
cana-1260	525	4	(	(	PUNCT
cana-1260	525	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	525	6	)	)	PUNCT
cana-1260	525	7	)	)	PUNCT
cana-1260	526	1	=	=	SYM
cana-1260	526	2	1	1	NUM
cana-1260	526	3	and	and	CCONJ
cana-1260	526	4	г𝑓	г𝑓	X
cana-1260	526	5	(	(	PUNCT
cana-1260	526	6	𝐺	𝐺	PROPN
cana-1260	526	7	+	+	CCONJ
cana-1260	526	8	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	526	9	)	)	PUNCT
cana-1260	526	10	)	)	PUNCT
cana-1260	527	1	=	=	SYM
cana-1260	527	2	2	2	NUM
cana-1260	527	3	−	−	PROPN
cana-1260	527	4	2𝑛	2𝑛	PROPN
cana-1260	527	5	4𝑛−1	4𝑛−1	NOUN
cana-1260	527	6	.	.	PUNCT
cana-1260	528	1	proof	proof	NOUN
cana-1260	528	2	:	:	PUNCT
cana-1260	528	3	theorem	theorem	VERB
cana-1260	528	4	2.4.1	2.4.1	NUM
cana-1260	528	5	and	and	CCONJ
cana-1260	528	6	theorem	theorem	VERB
cana-1260	528	7	3.1	3.1	NUM
cana-1260	528	8	gives	give	VERB
cana-1260	528	9	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	528	10	(	(	PUNCT
cana-1260	528	11	𝐺	𝐺	NOUN
cana-1260	528	12	)	)	PUNCT
cana-1260	528	13	=	=	SYM
cana-1260	528	14	2	2	NUM
cana-1260	528	15	and	and	CCONJ
cana-1260	528	16	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	528	17	(	(	PUNCT
cana-1260	528	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	528	19	)	)	PUNCT
cana-1260	528	20	)	)	PUNCT
cana-1260	529	1	=	=	SYM
cana-1260	529	2	1	1	NUM
cana-1260	529	3	so	so	ADV
cana-1260	529	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	529	5	(	(	PUNCT
cana-1260	529	6	𝐺	𝐺	NOUN
cana-1260	529	7	)	)	PUNCT
cana-1260	530	1	+	+	CCONJ
cana-1260	530	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	530	3	(	(	PUNCT
cana-1260	530	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	530	5	)	)	PUNCT
cana-1260	530	6	)	)	PUNCT
cana-1260	531	1	=	=	SYM
cana-1260	531	2	3	3	NUM
cana-1260	531	3	and	and	CCONJ
cana-1260	531	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	531	5	(	(	PUNCT
cana-1260	531	6	𝐺	𝐺	NOUN
cana-1260	531	7	)	)	PUNCT
cana-1260	531	8	∗	∗	NOUN
cana-1260	531	9	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	531	10	(	(	PUNCT
cana-1260	531	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	531	12	)	)	PUNCT
cana-1260	531	13	)	)	PUNCT
cana-1260	532	1	=	=	SYM
cana-1260	532	2	2	2	NUM
cana-1260	532	3	hence	hence	ADV
cana-1260	532	4	𝛾𝑓	𝛾𝑓	VERB
cana-1260	532	5	(	(	PUNCT
cana-1260	532	6	𝐺	𝐺	PROPN
cana-1260	532	7	+	+	CCONJ
cana-1260	532	8	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	532	9	)	)	PUNCT
cana-1260	532	10	)	)	PUNCT
cana-1260	533	1	=	=	PUNCT
cana-1260	533	2	1	1	NUM
cana-1260	533	3	if	if	SCONJ
cana-1260	533	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	533	5	(	(	PUNCT
cana-1260	533	6	𝐺	𝐺	NOUN
cana-1260	533	7	)	)	PUNCT
cana-1260	533	8	=	=	SYM
cana-1260	534	1	1	1	NUM
cana-1260	534	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	534	3	𝛾𝑓	𝛾𝑓	X
cana-1260	534	4	(	(	PUNCT
cana-1260	534	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	534	6	)	)	PUNCT
cana-1260	534	7	)	)	PUNCT
cana-1260	535	1	=	=	PUNCT
cana-1260	535	2	1	1	X
cana-1260	535	3	.	.	X
cana-1260	535	4	similarly	similarly	ADV
cana-1260	535	5	г𝑓	г𝑓	X
cana-1260	535	6	(	(	PUNCT
cana-1260	535	7	𝐺	𝐺	NOUN
cana-1260	535	8	)	)	PUNCT
cana-1260	535	9	=	=	SYM
cana-1260	535	10	2𝑛	2𝑛	PROPN
cana-1260	535	11	and	and	CCONJ
cana-1260	535	12	г𝑓	г𝑓	PROPN
cana-1260	535	13	(	(	PUNCT
cana-1260	535	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	535	15	)	)	PUNCT
cana-1260	535	16	)	)	PUNCT
cana-1260	536	1	=	=	SYM
cana-1260	536	2	2	2	NUM
cana-1260	536	3	so	so	ADV
cana-1260	536	4	г𝑓	г𝑓	ADP
cana-1260	536	5	(	(	PUNCT
cana-1260	536	6	𝐺	𝐺	NOUN
cana-1260	536	7	)	)	PUNCT
cana-1260	536	8	+	+	NUM
cana-1260	536	9	г𝑓	г𝑓	ADJ
cana-1260	536	10	(	(	PUNCT
cana-1260	536	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	536	12	)	)	PUNCT
cana-1260	536	13	)	)	PUNCT
cana-1260	537	1	=	=	SYM
cana-1260	538	1	2𝑛	2𝑛	PROPN
cana-1260	539	1	+	+	CCONJ
cana-1260	539	2	2and	2and	NUM
cana-1260	539	3	г𝑓	г𝑓	NOUN
cana-1260	539	4	(	(	PUNCT
cana-1260	539	5	𝐺	𝐺	NOUN
cana-1260	539	6	)	)	PUNCT
cana-1260	539	7	∗	∗	NOUN
cana-1260	539	8	г𝑓	г𝑓	PROPN
cana-1260	539	9	(	(	PUNCT
cana-1260	539	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	539	11	)	)	PUNCT
cana-1260	539	12	)	)	PUNCT
cana-1260	540	1	=	=	PUNCT
cana-1260	541	1	4𝑛	4𝑛	NOUN
cana-1260	541	2	,	,	PUNCT
cana-1260	541	3	г𝑓	г𝑓	X
cana-1260	541	4	(	(	PUNCT
cana-1260	541	5	𝐺	𝐺	PROPN
cana-1260	541	6	+	+	CCONJ
cana-1260	541	7	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	541	8	)	)	PUNCT
cana-1260	541	9	)	)	PUNCT
cana-1260	542	1	≤	≤	ADV
cana-1260	542	2	2	2	NUM
cana-1260	542	3	−	−	PROPN
cana-1260	542	4	2𝑛	2𝑛	PROPN
cana-1260	542	5	4𝑛	4𝑛	PROPN
cana-1260	542	6	−	−	PROPN
cana-1260	543	1	1	1	NUM
cana-1260	543	2	.	.	PUNCT
cana-1260	544	1	□	□	PUNCT
cana-1260	544	2	theorem	theorem	VERB
cana-1260	544	3	3.7	3.7	NUM
cana-1260	544	4	for	for	ADP
cana-1260	544	5	the	the	DET
cana-1260	544	6	graph	graph	NOUN
cana-1260	544	7	𝐺	𝐺	NOUN
cana-1260	544	8	be	be	AUX
cana-1260	544	9	the	the	DET
cana-1260	544	10	wheel	wheel	NOUN
cana-1260	544	11	graph	graph	NOUN
cana-1260	544	12	on	on	ADP
cana-1260	544	13	𝑛	𝑛	DET
cana-1260	544	14	vertices	vertex	NOUN
cana-1260	544	15	with	with	ADP
cana-1260	544	16	size	size	NOUN
cana-1260	544	17	𝑚	𝑚	PROPN
cana-1260	544	18	and	and	CCONJ
cana-1260	544	19	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	544	20	)	)	PUNCT
cana-1260	544	21	be	be	AUX
cana-1260	544	22	its	its	PRON
cana-1260	544	23	line	line	NOUN
cana-1260	544	24	graph	graph	NOUN
cana-1260	544	25	,	,	PUNCT
cana-1260	544	26	where	where	SCONJ
cana-1260	544	27	𝛿(𝐺	𝛿(𝐺	NUM
cana-1260	544	28	)	)	PUNCT
cana-1260	544	29	is	be	AUX
cana-1260	544	30	minimum	minimum	NOUN
cana-1260	544	31	degree	degree	NOUN
cana-1260	544	32	of	of	ADP
cana-1260	544	33	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	544	34	)	)	PUNCT
cana-1260	544	35	we	we	PRON
cana-1260	544	36	have	have	AUX
cana-1260	544	37	𝛾𝑓	𝛾𝑓	VERB
cana-1260	544	38	(	(	PUNCT
cana-1260	544	39	𝐺	𝐺	PROPN
cana-1260	544	40	+	+	X
cana-1260	544	41	𝐿(𝐺))=	𝐿(𝐺))=	PUNCT
cana-1260	544	42	{	{	PUNCT
cana-1260	544	43	1	1	NUM
cana-1260	544	44	𝑖𝑓	𝑖𝑓	NUM
cana-1260	544	45	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	544	46	(	(	PUNCT
cana-1260	544	47	𝐺	𝐺	NOUN
cana-1260	544	48	)	)	PUNCT
cana-1260	544	49	=	=	SYM
cana-1260	544	50	1	1	NUM
cana-1260	544	51	𝑜𝑟	𝑜𝑟	PRON
cana-1260	544	52	𝛾𝑓	𝛾𝑓	X
cana-1260	544	53	(	(	PUNCT
cana-1260	544	54	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	544	55	)	)	PUNCT
cana-1260	544	56	)	)	PUNCT
cana-1260	545	1	=	=	PUNCT
cana-1260	545	2	1	1	NUM
cana-1260	545	3	,	,	PUNCT
cana-1260	545	4	2	2	NUM
cana-1260	545	5	−	−	NOUN
cana-1260	545	6	1	1	NUM
cana-1260	545	7	+	+	NUM
cana-1260	545	8	𝑚	𝑚	PROPN
cana-1260	545	9	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	545	10	−	−	PROPN
cana-1260	545	11	2	2	NUM
cana-1260	545	12	𝑚	𝑚	NOUN
cana-1260	545	13	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	545	14	−	−	PROPN
cana-1260	545	15	1	1	NUM
cana-1260	545	16	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	545	17	and	and	CCONJ
cana-1260	545	18	г𝑓	г𝑓	NOUN
cana-1260	545	19	(	(	PUNCT
cana-1260	545	20	𝐺	𝐺	PROPN
cana-1260	545	21	+	+	CCONJ
cana-1260	545	22	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	545	23	)	)	PUNCT
cana-1260	545	24	)	)	PUNCT
cana-1260	545	25	≤	≤	ADV
cana-1260	545	26	2	2	NUM
cana-1260	545	27	−	−	NUM
cana-1260	545	28	2	2	NUM
cana-1260	545	29	+	+	CCONJ
cana-1260	545	30	𝑛−1	𝑛−1	NUM
cana-1260	545	31	2	2	NUM
cana-1260	545	32	−	−	NOUN
cana-1260	545	33	2	2	NUM
cana-1260	545	34	𝑛−2	𝑛−2	NOUN
cana-1260	545	35	.	.	PUNCT
cana-1260	546	1	communications	communication	NOUN
cana-1260	546	2	on	on	ADP
cana-1260	546	3	applied	apply	VERB
cana-1260	546	4	nonlinear	nonlinear	ADJ
cana-1260	546	5	analysis	analysis	NOUN
cana-1260	546	6	issn	issn	NOUN
cana-1260	546	7	:	:	PUNCT
cana-1260	546	8	1074	1074	NUM
cana-1260	546	9	-	-	PUNCT
cana-1260	546	10	133x	133x	NUM
cana-1260	546	11	vol	vol	NOUN
cana-1260	546	12	31	31	NUM
cana-1260	546	13	no	no	NOUN
cana-1260	546	14	.	.	PUNCT
cana-1260	547	1	6s	6s	NUM
cana-1260	547	2	(	(	PUNCT
cana-1260	547	3	2024	2024	NUM
cana-1260	547	4	)	)	PUNCT
cana-1260	547	5	689	689	NUM
cana-1260	548	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	548	2	proof	proof	NOUN
cana-1260	548	3	:	:	PUNCT
cana-1260	548	4	theorem	theorem	VERB
cana-1260	548	5	2.5.1	2.5.1	NUM
cana-1260	548	6	and	and	CCONJ
cana-1260	548	7	theorem	theorem	VERB
cana-1260	548	8	3.1	3.1	NUM
cana-1260	548	9	give	give	NOUN
cana-1260	548	10	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	548	11	(	(	PUNCT
cana-1260	548	12	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	548	13	)	)	PUNCT
cana-1260	548	14	=	=	SYM
cana-1260	548	15	1	1	NUM
cana-1260	548	16	and	and	CCONJ
cana-1260	548	17	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	548	18	(	(	PUNCT
cana-1260	548	19	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	548	20	)	)	PUNCT
cana-1260	548	21	)	)	PUNCT
cana-1260	549	1	=	=	PUNCT
cana-1260	549	2	𝑚	𝑚	X
cana-1260	549	3	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	549	4	so	so	ADV
cana-1260	549	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	549	6	(	(	PUNCT
cana-1260	549	7	𝐺	𝐺	NOUN
cana-1260	549	8	)	)	PUNCT
cana-1260	550	1	+	+	CCONJ
cana-1260	550	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	550	3	(	(	PUNCT
cana-1260	550	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	550	5	)	)	PUNCT
cana-1260	550	6	)	)	PUNCT
cana-1260	550	7	=	=	SYM
cana-1260	551	1	1	1	NUM
cana-1260	551	2	+	+	NUM
cana-1260	551	3	𝑚	𝑚	ADP
cana-1260	551	4	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	551	5	and	and	CCONJ
cana-1260	551	6	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	551	7	(	(	PUNCT
cana-1260	551	8	𝐺	𝐺	NOUN
cana-1260	551	9	)	)	PUNCT
cana-1260	551	10	∗	∗	NOUN
cana-1260	551	11	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	551	12	(	(	PUNCT
cana-1260	551	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	551	14	)	)	PUNCT
cana-1260	551	15	)	)	PUNCT
cana-1260	552	1	=	=	PUNCT
cana-1260	552	2	𝑚	𝑚	X
cana-1260	552	3	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	552	4	hence	hence	ADV
cana-1260	552	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	553	1	(	(	PUNCT
cana-1260	553	2	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	553	3	+	+	NUM
cana-1260	553	4	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	553	5	)	)	PUNCT
cana-1260	553	6	)	)	PUNCT
cana-1260	554	1	=	=	PUNCT
cana-1260	554	2	1	1	NUM
cana-1260	554	3	if	if	SCONJ
cana-1260	554	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	554	5	(	(	PUNCT
cana-1260	554	6	𝐺	𝐺	NOUN
cana-1260	554	7	)	)	PUNCT
cana-1260	554	8	=	=	SYM
cana-1260	555	1	1	1	NUM
cana-1260	555	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	555	3	𝛾𝑓	𝛾𝑓	X
cana-1260	555	4	(	(	PUNCT
cana-1260	555	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	555	6	)	)	PUNCT
cana-1260	555	7	)	)	PUNCT
cana-1260	556	1	=	=	SYM
cana-1260	556	2	1	1	NUM
cana-1260	556	3	,	,	PUNCT
cana-1260	556	4	otherwise	otherwise	ADV
cana-1260	556	5	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	556	6	(	(	PUNCT
cana-1260	556	7	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	556	8	+	+	NUM
cana-1260	556	9	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	556	10	)	)	PUNCT
cana-1260	556	11	)	)	PUNCT
cana-1260	557	1	=	=	SYM
cana-1260	558	1	2	2	NUM
cana-1260	558	2	−	−	NOUN
cana-1260	558	3	1	1	NUM
cana-1260	558	4	+	+	NUM
cana-1260	558	5	𝑚	𝑚	PROPN
cana-1260	558	6	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	558	7	−	−	PROPN
cana-1260	558	8	2	2	NUM
cana-1260	558	9	𝑚	𝑚	NOUN
cana-1260	558	10	𝛿(𝐺)+1	𝛿(𝐺)+1	NOUN
cana-1260	558	11	−	−	PROPN
cana-1260	558	12	1	1	NUM
cana-1260	558	13	similarly	similarly	ADV
cana-1260	558	14	г𝑓	г𝑓	X
cana-1260	558	15	(	(	PUNCT
cana-1260	558	16	𝑊𝑛	𝑊𝑛	PROPN
cana-1260	558	17	)	)	PUNCT
cana-1260	558	18	=	=	SYM
cana-1260	558	19	2	2	NUM
cana-1260	558	20	and	and	CCONJ
cana-1260	558	21	г𝑓	г𝑓	NOUN
cana-1260	558	22	(	(	PUNCT
cana-1260	558	23	𝐿(𝑊𝑛	𝐿(𝑊𝑛	NOUN
cana-1260	558	24	)	)	PUNCT
cana-1260	558	25	)	)	PUNCT
cana-1260	559	1	=	=	SYM
cana-1260	559	2	𝑛	𝑛	DET
cana-1260	559	3	−	−	NUM
cana-1260	559	4	1/2	1/2	NUM
cana-1260	560	1	so	so	ADV
cana-1260	560	2	г𝑓	г𝑓	ADP
cana-1260	560	3	(	(	PUNCT
cana-1260	560	4	𝐺	𝐺	NOUN
cana-1260	560	5	)	)	PUNCT
cana-1260	560	6	+	+	NUM
cana-1260	560	7	г𝑓	г𝑓	ADJ
cana-1260	560	8	(	(	PUNCT
cana-1260	560	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	560	10	)	)	PUNCT
cana-1260	560	11	)	)	PUNCT
cana-1260	560	12	≤	≤	NUM
cana-1260	560	13	2	2	NUM
cana-1260	560	14	+	+	CCONJ
cana-1260	560	15	𝑛−1	𝑛−1	NUM
cana-1260	560	16	2	2	NUM
cana-1260	560	17	and	and	CCONJ
cana-1260	560	18	г𝑓	г𝑓	ADJ
cana-1260	560	19	(	(	PUNCT
cana-1260	560	20	𝐺	𝐺	NOUN
cana-1260	560	21	)	)	PUNCT
cana-1260	560	22	∗	∗	NOUN
cana-1260	560	23	г𝑓	г𝑓	PROPN
cana-1260	560	24	(	(	PUNCT
cana-1260	560	25	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	560	26	)	)	PUNCT
cana-1260	560	27	)	)	PUNCT
cana-1260	560	28	≤	≤	NUM
cana-1260	561	1	𝑛	𝑛	DET
cana-1260	561	2	−	−	NUM
cana-1260	561	3	1	1	NUM
cana-1260	561	4	hence	hence	ADV
cana-1260	561	5	г𝑓	г𝑓	NOUN
cana-1260	561	6	(	(	PUNCT
cana-1260	561	7	𝐺	𝐺	PROPN
cana-1260	561	8	+	+	CCONJ
cana-1260	561	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	561	10	)	)	PUNCT
cana-1260	561	11	)	)	PUNCT
cana-1260	561	12	≤	≤	ADV
cana-1260	561	13	2	2	NUM
cana-1260	561	14	−	−	NUM
cana-1260	561	15	2	2	NUM
cana-1260	561	16	+	+	CCONJ
cana-1260	561	17	𝑛−1	𝑛−1	NUM
cana-1260	561	18	2	2	NUM
cana-1260	561	19	−	−	NUM
cana-1260	561	20	2	2	NUM
cana-1260	561	21	𝑛−2	𝑛−2	NOUN
cana-1260	561	22	.	.	PUNCT
cana-1260	561	23	□	□	PUNCT
cana-1260	561	24	theorem	theorem	VERB
cana-1260	561	25	3.8	3.8	NUM
cana-1260	561	26	if	if	SCONJ
cana-1260	561	27	graph	graph	NOUN
cana-1260	561	28	𝐺	𝐺	PROPN
cana-1260	561	29	be	be	AUX
cana-1260	561	30	the	the	DET
cana-1260	561	31	connected	connected	ADJ
cana-1260	561	32	cubic	cubic	ADJ
cana-1260	561	33	graph	graph	NOUN
cana-1260	561	34	or	or	CCONJ
cana-1260	561	35	trivalent	trivalent	NOUN
cana-1260	561	36	graph	graph	NOUN
cana-1260	561	37	with	with	ADP
cana-1260	561	38	order	order	NOUN
cana-1260	561	39	n	n	CCONJ
cana-1260	561	40	,	,	PUNCT
cana-1260	561	41	size	size	NOUN
cana-1260	561	42	m	m	PROPN
cana-1260	561	43	and	and	CCONJ
cana-1260	561	44	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	561	45	)	)	PUNCT
cana-1260	561	46	be	be	AUX
cana-1260	561	47	its	its	PRON
cana-1260	561	48	line	line	NOUN
cana-1260	561	49	graph	graph	NOUN
cana-1260	561	50	of	of	ADP
cana-1260	561	51	order	order	NOUN
cana-1260	561	52	m	m	VERB
cana-1260	562	1	we	we	PRON
cana-1260	562	2	have	have	AUX
cana-1260	562	3	𝛾𝑓	𝛾𝑓	VERB
cana-1260	562	4	(	(	PUNCT
cana-1260	562	5	𝐺	𝐺	NOUN
cana-1260	562	6	+	+	X
cana-1260	562	7	𝐿(𝐺)=	𝐿(𝐺)=	NOUN
cana-1260	562	8	{	{	PUNCT
cana-1260	562	9	1	1	NUM
cana-1260	562	10	𝑖𝑓	𝑖𝑓	NOUN
cana-1260	562	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	562	12	(	(	PUNCT
cana-1260	562	13	𝐺	𝐺	NOUN
cana-1260	562	14	)	)	PUNCT
cana-1260	562	15	=	=	SYM
cana-1260	562	16	1	1	NUM
cana-1260	562	17	𝑜𝑟	𝑜𝑟	PRON
cana-1260	562	18	𝛾𝑓	𝛾𝑓	X
cana-1260	562	19	(	(	PUNCT
cana-1260	562	20	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	562	21	)	)	PUNCT
cana-1260	562	22	)	)	PUNCT
cana-1260	563	1	=	=	PUNCT
cana-1260	563	2	1	1	NUM
cana-1260	563	3	,	,	PUNCT
cana-1260	563	4	2	2	NUM
cana-1260	563	5	−	−	NOUN
cana-1260	563	6	5𝑛+4𝑚	5𝑛+4𝑚	NUM
cana-1260	563	7	20	20	NUM
cana-1260	563	8	−	−	NUM
cana-1260	563	9	2	2	NUM
cana-1260	563	10	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	563	11	20	20	NUM
cana-1260	563	12	−	−	NOUN
cana-1260	563	13	1	1	NUM
cana-1260	563	14	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	563	15	and	and	CCONJ
cana-1260	563	16	г𝑓	г𝑓	NOUN
cana-1260	563	17	(	(	PUNCT
cana-1260	563	18	𝐺	𝐺	PROPN
cana-1260	563	19	+	+	CCONJ
cana-1260	563	20	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	563	21	)	)	PUNCT
cana-1260	563	22	)	)	PUNCT
cana-1260	563	23	≤	≤	ADV
cana-1260	564	1	2	2	NUM
cana-1260	564	2	−	−	NOUN
cana-1260	564	3	4𝑛+3𝑚	4𝑛+3𝑚	NUM
cana-1260	564	4	12	12	NUM
cana-1260	564	5	−	−	NUM
cana-1260	564	6	2	2	NUM
cana-1260	564	7	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	564	8	12	12	NUM
cana-1260	564	9	−	−	NOUN
cana-1260	564	10	1	1	NUM
cana-1260	564	11	.	.	PUNCT
cana-1260	565	1	proof	proof	NOUN
cana-1260	565	2	:	:	PUNCT
cana-1260	565	3	theorem	theorem	VERB
cana-1260	565	4	2.6.1	2.6.1	NUM
cana-1260	565	5	and	and	CCONJ
cana-1260	565	6	theorem	theorem	VERB
cana-1260	565	7	3.1	3.1	NUM
cana-1260	565	8	gives	give	VERB
cana-1260	565	9	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	565	10	(	(	PUNCT
cana-1260	565	11	𝐺	𝐺	NOUN
cana-1260	565	12	)	)	PUNCT
cana-1260	565	13	=	=	SYM
cana-1260	565	14	𝑛/4	𝑛/4	NUM
cana-1260	565	15	and	and	CCONJ
cana-1260	565	16	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	565	17	(	(	PUNCT
cana-1260	565	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	565	19	)	)	PUNCT
cana-1260	565	20	)	)	PUNCT
cana-1260	566	1	=	=	PRON
cana-1260	566	2	𝑚/5	𝑚/5	PUNCT
cana-1260	566	3	so	so	ADV
cana-1260	566	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	566	5	(	(	PUNCT
cana-1260	566	6	𝐺	𝐺	NOUN
cana-1260	566	7	)	)	PUNCT
cana-1260	567	1	+	+	CCONJ
cana-1260	567	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	567	3	(	(	PUNCT
cana-1260	567	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	567	5	)	)	PUNCT
cana-1260	567	6	)	)	PUNCT
cana-1260	568	1	=	=	SYM
cana-1260	568	2	𝑛	𝑛	PRON
cana-1260	568	3	4	4	NUM
cana-1260	568	4	+	+	CCONJ
cana-1260	568	5	𝑚	𝑚	PROPN
cana-1260	568	6	5	5	NUM
cana-1260	568	7	=	=	SYM
cana-1260	568	8	5𝑛+4𝑚	5𝑛+4𝑚	NUM
cana-1260	568	9	20	20	NUM
cana-1260	568	10	and	and	CCONJ
cana-1260	568	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	568	12	(	(	PUNCT
cana-1260	568	13	𝐺	𝐺	NOUN
cana-1260	568	14	)	)	PUNCT
cana-1260	568	15	∗	∗	NOUN
cana-1260	568	16	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	568	17	(	(	PUNCT
cana-1260	568	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	568	19	)	)	PUNCT
cana-1260	568	20	)	)	PUNCT
cana-1260	569	1	=	=	SYM
cana-1260	569	2	𝑛	𝑛	PRON
cana-1260	569	3	4	4	NUM
cana-1260	569	4	∗	∗	NOUN
cana-1260	569	5	𝑚	𝑚	ADP
cana-1260	569	6	5	5	NUM
cana-1260	569	7	=	=	SYM
cana-1260	569	8	𝑚𝑛/20	𝑚𝑛/20	PROPN
cana-1260	569	9	hence	hence	ADV
cana-1260	569	10	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	569	11	(	(	PUNCT
cana-1260	569	12	𝐺	𝐺	PROPN
cana-1260	569	13	+	+	CCONJ
cana-1260	569	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	569	15	)	)	PUNCT
cana-1260	569	16	)	)	PUNCT
cana-1260	570	1	=	=	PUNCT
cana-1260	570	2	1	1	NUM
cana-1260	570	3	if	if	SCONJ
cana-1260	570	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	570	5	(	(	PUNCT
cana-1260	570	6	𝐺	𝐺	NOUN
cana-1260	570	7	)	)	PUNCT
cana-1260	570	8	=	=	SYM
cana-1260	571	1	1	1	NUM
cana-1260	571	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	571	3	𝛾𝑓	𝛾𝑓	X
cana-1260	571	4	(	(	PUNCT
cana-1260	571	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	571	6	)	)	PUNCT
cana-1260	571	7	)	)	PUNCT
cana-1260	572	1	=	=	SYM
cana-1260	572	2	1	1	NUM
cana-1260	572	3	otherwise	otherwise	ADV
cana-1260	572	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	572	5	(	(	PUNCT
cana-1260	572	6	𝐺	𝐺	PROPN
cana-1260	572	7	+	+	CCONJ
cana-1260	572	8	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	572	9	)	)	PUNCT
cana-1260	572	10	)	)	PUNCT
cana-1260	573	1	=	=	SYM
cana-1260	573	2	2	2	NUM
cana-1260	573	3	−	−	NUM
cana-1260	573	4	5𝑛+4𝑚	5𝑛+4𝑚	NUM
cana-1260	573	5	20	20	NUM
cana-1260	573	6	−	−	NUM
cana-1260	573	7	2	2	NUM
cana-1260	573	8	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	573	9	20	20	NUM
cana-1260	573	10	−	−	NOUN
cana-1260	573	11	1	1	NUM
cana-1260	573	12	.	.	PUNCT
cana-1260	574	1	similarly	similarly	ADV
cana-1260	574	2	г𝑓	г𝑓	X
cana-1260	574	3	(	(	PUNCT
cana-1260	574	4	𝐺	𝐺	NOUN
cana-1260	574	5	)	)	PUNCT
cana-1260	574	6	=	=	NOUN
cana-1260	574	7	𝑛/3	𝑛/3	NUM
cana-1260	574	8	and	and	CCONJ
cana-1260	574	9	г𝑓	г𝑓	PROPN
cana-1260	574	10	(	(	PUNCT
cana-1260	574	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	574	12	)	)	PUNCT
cana-1260	574	13	)	)	PUNCT
cana-1260	575	1	=	=	PUNCT
cana-1260	575	2	𝑚/4	𝑚/4	NOUN
cana-1260	575	3	.	.	PUNCT
cana-1260	576	1	so	so	ADV
cana-1260	576	2	г𝑓	г𝑓	INTJ
cana-1260	576	3	(	(	PUNCT
cana-1260	576	4	𝐺	𝐺	NOUN
cana-1260	576	5	)	)	PUNCT
cana-1260	576	6	+	+	NUM
cana-1260	576	7	г𝑓	г𝑓	ADJ
cana-1260	576	8	(	(	PUNCT
cana-1260	576	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	576	10	)	)	PUNCT
cana-1260	576	11	)	)	PUNCT
cana-1260	577	1	≤	≤	NUM
cana-1260	577	2	𝑛	𝑛	DET
cana-1260	577	3	3	3	NUM
cana-1260	577	4	+	+	CCONJ
cana-1260	578	1	𝑚	𝑚	PROPN
cana-1260	578	2	4	4	NUM
cana-1260	578	3	=	=	SYM
cana-1260	578	4	4𝑛+3𝑚	4𝑛+3𝑚	NUM
cana-1260	578	5	12	12	NUM
cana-1260	578	6	and	and	CCONJ
cana-1260	578	7	г𝑓	г𝑓	ADJ
cana-1260	578	8	(	(	PUNCT
cana-1260	578	9	𝐺	𝐺	NOUN
cana-1260	578	10	)	)	PUNCT
cana-1260	578	11	∗	∗	NOUN
cana-1260	578	12	г𝑓	г𝑓	PROPN
cana-1260	578	13	(	(	PUNCT
cana-1260	578	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	578	15	)	)	PUNCT
cana-1260	578	16	)	)	PUNCT
cana-1260	578	17	≤	≤	NOUN
cana-1260	579	1	𝑚𝑛/12	𝑚𝑛/12	PUNCT
cana-1260	579	2	hence	hence	ADV
cana-1260	579	3	г𝑓	г𝑓	X
cana-1260	579	4	(	(	PUNCT
cana-1260	579	5	𝐺	𝐺	PROPN
cana-1260	579	6	+	+	CCONJ
cana-1260	579	7	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	579	8	)	)	PUNCT
cana-1260	579	9	)	)	PUNCT
cana-1260	580	1	≤	≤	ADV
cana-1260	580	2	2	2	NUM
cana-1260	580	3	−	−	NOUN
cana-1260	580	4	4𝑛+3𝑚	4𝑛+3𝑚	NUM
cana-1260	580	5	12	12	NUM
cana-1260	580	6	−	−	NUM
cana-1260	580	7	2	2	NUM
cana-1260	580	8	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	580	9	12	12	NUM
cana-1260	580	10	−	−	NOUN
cana-1260	580	11	1	1	NUM
cana-1260	580	12	.	.	PUNCT
cana-1260	581	1	□	□	PUNCT
cana-1260	581	2	theorem	theorem	ADJ
cana-1260	581	3	3.9	3.9	NUM
cana-1260	581	4	let	let	VERB
cana-1260	581	5	𝐺	𝐺	PROPN
cana-1260	581	6	is	be	AUX
cana-1260	581	7	the	the	DET
cana-1260	581	8	cartesian	cartesian	ADJ
cana-1260	581	9	product	product	NOUN
cana-1260	581	10	graph	graph	NOUN
cana-1260	581	11	(	(	PUNCT
cana-1260	581	12	𝐾2	𝐾2	ADJ
cana-1260	581	13	×	×	PROPN
cana-1260	581	14	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	581	15	)	)	PUNCT
cana-1260	581	16	for	for	ADP
cana-1260	581	17	𝑛	𝑛	PROPN
cana-1260	581	18	>	>	SYM
cana-1260	581	19	1	1	NUM
cana-1260	581	20	and	and	CCONJ
cana-1260	581	21	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	581	22	)	)	PUNCT
cana-1260	581	23	be	be	VERB
cana-1260	581	24	line	line	NOUN
cana-1260	581	25	graph	graph	NOUN
cana-1260	581	26	of	of	ADP
cana-1260	581	27	order	order	NOUN
cana-1260	581	28	3𝑛	3𝑛	NUM
cana-1260	581	29	−	−	NOUN
cana-1260	581	30	2	2	NUM
cana-1260	581	31	we	we	PRON
cana-1260	581	32	have	have	AUX
cana-1260	581	33	𝛾𝑓	𝛾𝑓	VERB
cana-1260	581	34	(	(	PUNCT
cana-1260	581	35	𝐺	𝐺	PROPN
cana-1260	581	36	+	+	CCONJ
cana-1260	581	37	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	581	38	)	)	PUNCT
cana-1260	581	39	)	)	PUNCT
cana-1260	582	1	=	=	PRON
cana-1260	582	2	{	{	PUNCT
cana-1260	582	3	2	2	NUM
cana-1260	582	4	−	−	NOUN
cana-1260	582	5	𝑛	𝑛	PROPN
cana-1260	582	6	+	+	NOUN
cana-1260	582	7	1	1	NUM
cana-1260	582	8	2	2	NUM
cana-1260	582	9	+	+	NUM
cana-1260	582	10	2𝑛	2𝑛	PROPN
cana-1260	582	11	3	3	NUM
cana-1260	582	12	−	−	PROPN
cana-1260	582	13	2	2	NUM
cana-1260	582	14	𝑛	𝑛	NOUN
cana-1260	582	15	+	+	NOUN
cana-1260	582	16	1	1	NUM
cana-1260	582	17	2	2	NUM
cana-1260	582	18	∗	∗	NOUN
cana-1260	582	19	2𝑛	2𝑛	NOUN
cana-1260	582	20	3	3	NUM
cana-1260	582	21	−	−	PROPN
cana-1260	582	22	1	1	NUM
cana-1260	582	23	𝑖𝑓	𝑖𝑓	NUM
cana-1260	582	24	𝑛	𝑛	PRON
cana-1260	582	25	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	582	26	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-1260	582	27	2	2	NUM
cana-1260	582	28	−	−	NOUN
cana-1260	582	29	𝑛2	𝑛2	NOUN
cana-1260	582	30	+	+	CCONJ
cana-1260	582	31	2𝑛	2𝑛	PROPN
cana-1260	582	32	2(𝑛	2(𝑛	NUM
cana-1260	582	33	+	+	CCONJ
cana-1260	582	34	1	1	NUM
cana-1260	582	35	)	)	PUNCT
cana-1260	582	36	+	+	NUM
cana-1260	582	37	2𝑛	2𝑛	PROPN
cana-1260	582	38	3	3	NUM
cana-1260	582	39	−	−	PROPN
cana-1260	582	40	2	2	NUM
cana-1260	582	41	𝑛2	𝑛2	NOUN
cana-1260	582	42	+	+	CCONJ
cana-1260	582	43	2𝑛	2𝑛	NOUN
cana-1260	582	44	2(𝑛	2(𝑛	NUM
cana-1260	582	45	+	+	CCONJ
cana-1260	582	46	1	1	X
cana-1260	582	47	)	)	PUNCT
cana-1260	582	48	∗	∗	NOUN
cana-1260	582	49	2𝑛	2𝑛	NUM
cana-1260	582	50	3	3	NUM
cana-1260	582	51	−	−	PROPN
cana-1260	582	52	1	1	NUM
cana-1260	582	53	𝑖𝑓	𝑖𝑓	NUM
cana-1260	582	54	𝑛	𝑛	PRON
cana-1260	582	55	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	582	56	𝑒𝑣𝑒𝑛.	𝑒𝑣𝑒𝑛.	NOUN
cana-1260	582	57	proof	proof	NOUN
cana-1260	582	58	:	:	PUNCT
cana-1260	582	59	theorem	theorem	VERB
cana-1260	582	60	2.7.1	2.7.1	NUM
cana-1260	582	61	and	and	CCONJ
cana-1260	582	62	theorem	theorem	VERB
cana-1260	582	63	3.1	3.1	NUM
cana-1260	582	64	gives	give	VERB
cana-1260	582	65	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	582	66	(	(	PUNCT
cana-1260	582	67	𝐾2	𝐾2	ADJ
cana-1260	582	68	×	×	PROPN
cana-1260	582	69	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	582	70	)	)	PUNCT
cana-1260	583	1	=	=	PUNCT
cana-1260	583	2	𝑛+1	𝑛+1	ADP
cana-1260	583	3	2	2	NUM
cana-1260	583	4	,	,	PUNCT
cana-1260	583	5	𝑖𝑓	𝑖𝑓	ADP
cana-1260	583	6	𝑛	𝑛	DET
cana-1260	583	7	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	583	8	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-1260	583	9	and	and	CCONJ
cana-1260	583	10	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	583	11	(	(	PUNCT
cana-1260	583	12	𝐾2	𝐾2	ADJ
cana-1260	583	13	×	×	PROPN
cana-1260	583	14	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	583	15	)	)	PUNCT
cana-1260	583	16	=	=	SYM
cana-1260	583	17	𝑛2	𝑛2	NOUN
cana-1260	583	18	+	+	PROPN
cana-1260	583	19	2𝑛	2𝑛	PROPN
cana-1260	583	20	2(𝑛+1	2(𝑛+1	NUM
cana-1260	583	21	)	)	PUNCT
cana-1260	583	22	,	,	PUNCT
cana-1260	583	23	𝑖𝑓	𝑖𝑓	ADP
cana-1260	583	24	𝑛	𝑛	PRON
cana-1260	583	25	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	583	26	𝑒𝑣𝑒𝑛.	𝑒𝑣𝑒𝑛.	VERB
cana-1260	583	27	the	the	DET
cana-1260	583	28	theorem	theorem	NOUN
cana-1260	583	29	2.7.2	2.7.2	NUM
cana-1260	583	30	gives	give	VERB
cana-1260	583	31	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	583	32	(	(	PUNCT
cana-1260	583	33	𝐿(𝐾2	𝐿(𝐾2	PRON
cana-1260	583	34	×	×	NOUN
cana-1260	583	35	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	583	36	)	)	PUNCT
cana-1260	583	37	)	)	PUNCT
cana-1260	584	1	=	=	SYM
cana-1260	584	2	2𝑛	2𝑛	PROPN
cana-1260	584	3	3	3	NUM
cana-1260	584	4	.	.	PUNCT
cana-1260	585	1	so	so	ADV
cana-1260	585	2	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	585	3	(	(	PUNCT
cana-1260	585	4	𝐺	𝐺	NOUN
cana-1260	585	5	)	)	PUNCT
cana-1260	586	1	+	+	CCONJ
cana-1260	586	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	586	3	(	(	PUNCT
cana-1260	586	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	586	5	)	)	PUNCT
cana-1260	586	6	)	)	PUNCT
cana-1260	587	1	=	=	PUNCT
cana-1260	587	2	𝑛+1	𝑛+1	ADP
cana-1260	587	3	2	2	NUM
cana-1260	587	4	+	+	CCONJ
cana-1260	587	5	2𝑛	2𝑛	PROPN
cana-1260	587	6	3	3	NUM
cana-1260	587	7	if	if	SCONJ
cana-1260	587	8	n	n	NOUN
cana-1260	587	9	is	be	AUX
cana-1260	587	10	odd	odd	ADJ
cana-1260	587	11	and	and	CCONJ
cana-1260	587	12	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	587	13	(	(	PUNCT
cana-1260	587	14	𝐺	𝐺	NOUN
cana-1260	587	15	)	)	PUNCT
cana-1260	588	1	+	+	CCONJ
cana-1260	588	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	588	3	(	(	PUNCT
cana-1260	588	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	588	5	)	)	PUNCT
cana-1260	588	6	)	)	PUNCT
cana-1260	588	7	=	=	PUNCT
cana-1260	588	8	𝑛2	𝑛2	PROPN
cana-1260	588	9	+	+	PROPN
cana-1260	588	10	2𝑛	2𝑛	PROPN
cana-1260	588	11	2(𝑛+1	2(𝑛+1	NUM
cana-1260	588	12	)	)	PUNCT
cana-1260	589	1	+	+	NUM
cana-1260	589	2	2𝑛	2𝑛	PROPN
cana-1260	589	3	3	3	NUM
cana-1260	589	4	𝑖𝑓	𝑖𝑓	ADP
cana-1260	589	5	𝑛	𝑛	DET
cana-1260	589	6	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	589	7	𝑒𝑣𝑒𝑛.	𝑒𝑣𝑒𝑛.	NOUN
cana-1260	589	8	hence	hence	ADV
cana-1260	589	9	communications	communication	NOUN
cana-1260	589	10	on	on	ADP
cana-1260	589	11	applied	apply	VERB
cana-1260	589	12	nonlinear	nonlinear	ADJ
cana-1260	589	13	analysis	analysis	NOUN
cana-1260	589	14	issn	issn	NOUN
cana-1260	589	15	:	:	PUNCT
cana-1260	589	16	1074	1074	NUM
cana-1260	589	17	-	-	PUNCT
cana-1260	589	18	133x	133x	NUM
cana-1260	589	19	vol	vol	NOUN
cana-1260	589	20	31	31	NUM
cana-1260	589	21	no	no	NOUN
cana-1260	589	22	.	.	PUNCT
cana-1260	590	1	6s	6s	NUM
cana-1260	590	2	(	(	PUNCT
cana-1260	590	3	2024	2024	NUM
cana-1260	590	4	)	)	PUNCT
cana-1260	590	5	690	690	NUM
cana-1260	590	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1260	591	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	591	2	(	(	PUNCT
cana-1260	591	3	𝐺	𝐺	PROPN
cana-1260	591	4	+	+	CCONJ
cana-1260	591	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	591	6	)	)	PUNCT
cana-1260	591	7	)	)	PUNCT
cana-1260	592	1	=	=	SYM
cana-1260	592	2	2	2	NUM
cana-1260	592	3	−	−	NOUN
cana-1260	592	4	𝑛+1	𝑛+1	ADP
cana-1260	592	5	2	2	NUM
cana-1260	592	6	+	+	CCONJ
cana-1260	592	7	2𝑛	2𝑛	PROPN
cana-1260	592	8	3	3	NUM
cana-1260	592	9	−	−	PROPN
cana-1260	592	10	2	2	NUM
cana-1260	592	11	𝑛+1	𝑛+1	ADP
cana-1260	592	12	2	2	NUM
cana-1260	592	13	2𝑛	2𝑛	PROPN
cana-1260	592	14	3	3	NUM
cana-1260	592	15	−	−	PROPN
cana-1260	592	16	1	1	NUM
cana-1260	592	17	𝑖𝑓	𝑖𝑓	NOUN
cana-1260	592	18	𝑛	𝑛	PRON
cana-1260	592	19	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	592	20	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-1260	592	21	and	and	CCONJ
cana-1260	592	22	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	592	23	(	(	PUNCT
cana-1260	592	24	𝐺	𝐺	PROPN
cana-1260	592	25	+	+	CCONJ
cana-1260	592	26	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	592	27	)	)	PUNCT
cana-1260	592	28	)	)	PUNCT
cana-1260	593	1	=	=	SYM
cana-1260	593	2	2	2	NUM
cana-1260	593	3	−	−	NOUN
cana-1260	593	4	𝑛2	𝑛2	NOUN
cana-1260	593	5	+	+	PROPN
cana-1260	593	6	2𝑛	2𝑛	PROPN
cana-1260	593	7	2(𝑛+1	2(𝑛+1	NUM
cana-1260	593	8	)	)	PUNCT
cana-1260	594	1	+	+	NUM
cana-1260	594	2	2𝑛	2𝑛	PROPN
cana-1260	594	3	3	3	NUM
cana-1260	594	4	−	−	PROPN
cana-1260	594	5	2	2	NUM
cana-1260	594	6	𝑛2	𝑛2	NOUN
cana-1260	594	7	+	+	PROPN
cana-1260	594	8	2𝑛	2𝑛	PROPN
cana-1260	594	9	2(𝑛+1	2(𝑛+1	NUM
cana-1260	594	10	)	)	PUNCT
cana-1260	594	11	∗	∗	NOUN
cana-1260	594	12	2𝑛	2𝑛	NUM
cana-1260	594	13	3	3	NUM
cana-1260	594	14	−	−	PROPN
cana-1260	594	15	1	1	NUM
cana-1260	594	16	𝑖𝑓	𝑖𝑓	NOUN
cana-1260	594	17	𝑛	𝑛	PRON
cana-1260	594	18	𝑖𝑠	𝑖𝑠	NOUN
cana-1260	594	19	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-1260	594	20	.	.	PUNCT
cana-1260	595	1	□	□	PUNCT
cana-1260	595	2	theorem	theorem	VERB
cana-1260	595	3	3.10	3.10	NUM
cana-1260	595	4	let	let	VERB
cana-1260	595	5	g	g	PROPN
cana-1260	595	6	is	be	AUX
cana-1260	595	7	the	the	DET
cana-1260	595	8	cartesian	cartesian	ADJ
cana-1260	595	9	product	product	NOUN
cana-1260	595	10	graph	graph	NOUN
cana-1260	595	11	(	(	PUNCT
cana-1260	595	12	𝐾3	𝐾3	VERB
cana-1260	595	13	×	×	PROPN
cana-1260	595	14	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	595	15	)	)	PUNCT
cana-1260	595	16	of	of	ADP
cana-1260	595	17	order	order	NOUN
cana-1260	595	18	3𝑛	3𝑛	NUM
cana-1260	595	19	for	for	ADP
cana-1260	595	20	𝑛	𝑛	PRON
cana-1260	595	21	≥	≥	NUM
cana-1260	595	22	3	3	NUM
cana-1260	595	23	and	and	CCONJ
cana-1260	595	24	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	595	25	)	)	PUNCT
cana-1260	595	26	is	be	AUX
cana-1260	595	27	its	its	PRON
cana-1260	595	28	line	line	NOUN
cana-1260	595	29	graph	graph	NOUN
cana-1260	595	30	of	of	ADP
cana-1260	595	31	order	order	NOUN
cana-1260	595	32	(	(	PUNCT
cana-1260	595	33	6𝑛	6𝑛	NOUN
cana-1260	595	34	−	−	NOUN
cana-1260	595	35	3	3	NUM
cana-1260	595	36	)	)	PUNCT
cana-1260	595	37	then	then	ADV
cana-1260	595	38	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	595	39	(	(	PUNCT
cana-1260	595	40	𝐺	𝐺	PROPN
cana-1260	595	41	+	+	CCONJ
cana-1260	595	42	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	595	43	)	)	PUNCT
cana-1260	595	44	)	)	PUNCT
cana-1260	596	1	=	=	PRON
cana-1260	596	2	{	{	PUNCT
cana-1260	596	3	1	1	NUM
cana-1260	596	4	𝑖𝑓	𝑖𝑓	NUM
cana-1260	597	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	597	2	(	(	PUNCT
cana-1260	597	3	𝐺	𝐺	NOUN
cana-1260	597	4	)	)	PUNCT
cana-1260	597	5	=	=	SYM
cana-1260	597	6	1	1	NUM
cana-1260	597	7	𝑜𝑟	𝑜𝑟	PRON
cana-1260	597	8	𝛾𝑓	𝛾𝑓	X
cana-1260	597	9	(	(	PUNCT
cana-1260	597	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	597	11	)	)	PUNCT
cana-1260	597	12	)	)	PUNCT
cana-1260	597	13	=	=	PUNCT
cana-1260	598	1	1	1	NUM
cana-1260	598	2	,	,	PUNCT
cana-1260	598	3	2	2	NUM
cana-1260	598	4	−	−	NOUN
cana-1260	598	5	3𝑛	3𝑛	NUM
cana-1260	598	6	+	+	CCONJ
cana-1260	598	7	2	2	NUM
cana-1260	598	8	5	5	NUM
cana-1260	598	9	+	+	NUM
cana-1260	598	10	3(2𝑛	3(2𝑛	NUM
cana-1260	598	11	+	+	CCONJ
cana-1260	598	12	1	1	NUM
cana-1260	598	13	)	)	PUNCT
cana-1260	598	14	7	7	NUM
cana-1260	598	15	−	−	NUM
cana-1260	598	16	2	2	NUM
cana-1260	598	17	3𝑛	3𝑛	NUM
cana-1260	598	18	+	+	CCONJ
cana-1260	598	19	2	2	NUM
cana-1260	598	20	5	5	NUM
cana-1260	598	21	∗	∗	NOUN
cana-1260	598	22	3(2𝑛	3(2𝑛	NUM
cana-1260	598	23	+	+	CCONJ
cana-1260	598	24	1	1	NUM
cana-1260	598	25	)	)	PUNCT
cana-1260	598	26	7	7	NUM
cana-1260	598	27	−	−	SYM
cana-1260	598	28	1	1	NUM
cana-1260	598	29	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	598	30	proof	proof	NOUN
cana-1260	598	31	:	:	PUNCT
cana-1260	598	32	theorem	theorem	ADJ
cana-1260	598	33	2.8.1	2.8.1	NUM
cana-1260	598	34	and	and	CCONJ
cana-1260	598	35	theorem	theorem	VERB
cana-1260	598	36	3.1	3.1	NUM
cana-1260	598	37	gives	give	VERB
cana-1260	598	38	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	598	39	(	(	PUNCT
cana-1260	598	40	𝐾3	𝐾3	VERB
cana-1260	598	41	×	×	PROPN
cana-1260	598	42	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	598	43	)	)	PUNCT
cana-1260	598	44	=	=	SYM
cana-1260	599	1	3𝑛+2	3𝑛+2	NUM
cana-1260	599	2	5	5	NUM
cana-1260	599	3	and	and	CCONJ
cana-1260	599	4	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	599	5	(	(	PUNCT
cana-1260	599	6	𝐿(𝐾2	𝐿(𝐾2	PRON
cana-1260	599	7	×	×	NOUN
cana-1260	599	8	𝑃𝑛	𝑃𝑛	NOUN
cana-1260	599	9	)	)	PUNCT
cana-1260	599	10	)	)	PUNCT
cana-1260	600	1	=	=	SYM
cana-1260	600	2	3(2𝑛+1	3(2𝑛+1	X
cana-1260	600	3	)	)	PUNCT
cana-1260	600	4	7	7	NUM
cana-1260	601	1	so	so	ADV
cana-1260	601	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	601	3	(	(	PUNCT
cana-1260	601	4	𝐺	𝐺	NOUN
cana-1260	601	5	)	)	PUNCT
cana-1260	602	1	+	+	CCONJ
cana-1260	602	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	602	3	(	(	PUNCT
cana-1260	602	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	602	5	)	)	PUNCT
cana-1260	602	6	)	)	PUNCT
cana-1260	603	1	=	=	PUNCT
cana-1260	604	1	3𝑛+2	3𝑛+2	NUM
cana-1260	604	2	5	5	NUM
cana-1260	604	3	+	+	CCONJ
cana-1260	604	4	3(2𝑛+1	3(2𝑛+1	NUM
cana-1260	604	5	)	)	PUNCT
cana-1260	604	6	7	7	NUM
cana-1260	604	7	and	and	CCONJ
cana-1260	604	8	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	604	9	(	(	PUNCT
cana-1260	604	10	𝐺	𝐺	NOUN
cana-1260	604	11	)	)	PUNCT
cana-1260	604	12	∗	∗	NOUN
cana-1260	604	13	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	604	14	(	(	PUNCT
cana-1260	604	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	604	16	)	)	PUNCT
cana-1260	604	17	)	)	PUNCT
cana-1260	605	1	=	=	PUNCT
cana-1260	605	2	3𝑛+2	3𝑛+2	NUM
cana-1260	605	3	5	5	NUM
cana-1260	605	4	∗	∗	NOUN
cana-1260	605	5	3(2𝑛+1	3(2𝑛+1	NUM
cana-1260	605	6	)	)	PUNCT
cana-1260	605	7	7	7	NUM
cana-1260	605	8	hence	hence	ADV
cana-1260	605	9	𝛾𝑓	𝛾𝑓	VERB
cana-1260	605	10	(	(	PUNCT
cana-1260	605	11	𝐺	𝐺	PROPN
cana-1260	605	12	+	+	CCONJ
cana-1260	605	13	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	605	14	)	)	PUNCT
cana-1260	605	15	)	)	PUNCT
cana-1260	606	1	=	=	SYM
cana-1260	607	1	1	1	NUM
cana-1260	607	2	𝑖𝑓	𝑖𝑓	NUM
cana-1260	607	3	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	607	4	(	(	PUNCT
cana-1260	607	5	𝐺	𝐺	NOUN
cana-1260	607	6	)	)	PUNCT
cana-1260	607	7	=	=	SYM
cana-1260	607	8	1	1	NUM
cana-1260	607	9	𝑜𝑟	𝑜𝑟	PRON
cana-1260	607	10	𝛾𝑓	𝛾𝑓	X
cana-1260	607	11	(	(	PUNCT
cana-1260	607	12	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	607	13	)	)	PUNCT
cana-1260	607	14	)	)	PUNCT
cana-1260	607	15	=	=	SYM
cana-1260	607	16	1	1	NUM
cana-1260	607	17	and	and	CCONJ
cana-1260	607	18	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	607	19	(	(	PUNCT
cana-1260	607	20	𝐺	𝐺	PROPN
cana-1260	607	21	+	+	CCONJ
cana-1260	607	22	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	607	23	)	)	PUNCT
cana-1260	607	24	)	)	PUNCT
cana-1260	608	1	=	=	SYM
cana-1260	608	2	2	2	NUM
cana-1260	608	3	−	−	NOUN
cana-1260	608	4	3𝑛+2	3𝑛+2	NUM
cana-1260	608	5	5	5	NUM
cana-1260	608	6	+	+	CCONJ
cana-1260	608	7	3(2𝑛+1	3(2𝑛+1	X
cana-1260	608	8	)	)	PUNCT
cana-1260	608	9	7	7	NUM
cana-1260	608	10	−	−	SYM
cana-1260	608	11	2	2	NUM
cana-1260	608	12	3𝑛+2	3𝑛+2	NUM
cana-1260	608	13	5	5	NUM
cana-1260	608	14	∗	∗	NOUN
cana-1260	608	15	3(2𝑛+1	3(2𝑛+1	NUM
cana-1260	608	16	)	)	PUNCT
cana-1260	608	17	7	7	NUM
cana-1260	608	18	−	−	NOUN
cana-1260	608	19	1	1	NUM
cana-1260	608	20	otherwise	otherwise	ADV
cana-1260	608	21	.	.	PUNCT
cana-1260	609	1	□	□	PUNCT
cana-1260	609	2	theorem	theorem	VERB
cana-1260	609	3	3.11	3.11	NUM
cana-1260	609	4	let	let	VERB
cana-1260	610	1	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	610	2	and	and	CCONJ
cana-1260	610	3	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	610	4	are	be	AUX
cana-1260	610	5	the	the	DET
cana-1260	610	6	two	two	NUM
cana-1260	610	7	cycle	cycle	NOUN
cana-1260	610	8	graphs	graph	NOUN
cana-1260	610	9	with	with	ADP
cana-1260	610	10	𝑚	𝑚	PROPN
cana-1260	610	11	and	and	CCONJ
cana-1260	610	12	𝑛	𝑛	DET
cana-1260	610	13	vertices	vertex	NOUN
cana-1260	610	14	respectively	respectively	ADV
cana-1260	610	15	and	and	CCONJ
cana-1260	610	16	let	let	VERB
cana-1260	610	17	𝐺	𝐺	NOUN
cana-1260	610	18	=	=	PUNCT
cana-1260	611	1	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	611	2	×	×	NOUN
cana-1260	611	3	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	611	4	be	be	AUX
cana-1260	611	5	the	the	DET
cana-1260	611	6	cartesian	cartesian	ADJ
cana-1260	611	7	product	product	NOUN
cana-1260	611	8	.	.	PUNCT
cana-1260	612	1	let	let	VERB
cana-1260	612	2	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	612	3	)	)	PUNCT
cana-1260	612	4	be	be	AUX
cana-1260	612	5	its	its	PRON
cana-1260	612	6	line	line	NOUN
cana-1260	612	7	graph	graph	NOUN
cana-1260	612	8	we	we	PRON
cana-1260	612	9	have	have	VERB
cana-1260	612	10	𝛾𝑓	𝛾𝑓	VERB
cana-1260	612	11	(	(	PUNCT
cana-1260	612	12	𝐺	𝐺	PROPN
cana-1260	612	13	+	+	CCONJ
cana-1260	612	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	612	15	)	)	PUNCT
cana-1260	612	16	)	)	PUNCT
cana-1260	613	1	=	=	PRON
cana-1260	613	2	{	{	PUNCT
cana-1260	613	3	1	1	NUM
cana-1260	613	4	𝑖𝑓	𝑖𝑓	NUM
cana-1260	614	1	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	614	2	(	(	PUNCT
cana-1260	614	3	𝐺	𝐺	NOUN
cana-1260	614	4	)	)	PUNCT
cana-1260	614	5	=	=	SYM
cana-1260	614	6	1	1	NUM
cana-1260	614	7	𝑜𝑟	𝑜𝑟	PRON
cana-1260	614	8	𝛾𝑓	𝛾𝑓	X
cana-1260	614	9	(	(	PUNCT
cana-1260	614	10	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	614	11	)	)	PUNCT
cana-1260	614	12	)	)	PUNCT
cana-1260	614	13	=	=	PUNCT
cana-1260	615	1	1	1	NUM
cana-1260	615	2	,	,	PUNCT
cana-1260	615	3	2	2	NUM
cana-1260	615	4	−	−	NOUN
cana-1260	615	5	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	615	6	5	5	NUM
cana-1260	615	7	+	+	CCONJ
cana-1260	615	8	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	615	9	7	7	NUM
cana-1260	615	10	−	−	NOUN
cana-1260	615	11	2	2	NUM
cana-1260	615	12	𝑚𝑛	𝑚𝑛	SYM
cana-1260	615	13	5	5	NUM
cana-1260	615	14	∗	∗	NOUN
cana-1260	615	15	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	615	16	7	7	NUM
cana-1260	615	17	−	−	SYM
cana-1260	615	18	1	1	NUM
cana-1260	615	19	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1260	615	20	and	and	CCONJ
cana-1260	615	21	г𝑓	г𝑓	NOUN
cana-1260	615	22	(	(	PUNCT
cana-1260	615	23	𝐺	𝐺	PROPN
cana-1260	615	24	+	+	CCONJ
cana-1260	615	25	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	615	26	)	)	PUNCT
cana-1260	615	27	)	)	PUNCT
cana-1260	615	28	≤	≤	ADV
cana-1260	616	1	2	2	NUM
cana-1260	616	2	−	−	NOUN
cana-1260	616	3	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	616	4	4	4	NUM
cana-1260	616	5	+	+	CCONJ
cana-1260	616	6	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	616	7	3	3	NUM
cana-1260	616	8	−	−	NOUN
cana-1260	616	9	2	2	NUM
cana-1260	616	10	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	616	11	4	4	NUM
cana-1260	616	12	∗	∗	NOUN
cana-1260	616	13	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	616	14	3	3	NUM
cana-1260	616	15	−	−	NOUN
cana-1260	616	16	1	1	NUM
cana-1260	616	17	proof	proof	NOUN
cana-1260	616	18	:	:	PUNCT
cana-1260	616	19	theorem	theorem	VERB
cana-1260	616	20	2.9.1	2.9.1	NUM
cana-1260	616	21	and	and	CCONJ
cana-1260	616	22	theorem	theorem	VERB
cana-1260	616	23	3.1	3.1	NUM
cana-1260	616	24	gives	give	VERB
cana-1260	616	25	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	616	26	(	(	PUNCT
cana-1260	616	27	𝐺	𝐺	NOUN
cana-1260	616	28	)	)	PUNCT
cana-1260	616	29	=	=	SYM
cana-1260	617	1	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	617	2	5	5	NUM
cana-1260	617	3	and	and	CCONJ
cana-1260	617	4	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	617	5	(	(	PUNCT
cana-1260	617	6	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	617	7	)	)	PUNCT
cana-1260	617	8	)	)	PUNCT
cana-1260	618	1	=	=	PUNCT
cana-1260	618	2	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	618	3	7	7	NUM
cana-1260	618	4	where	where	SCONJ
cana-1260	618	5	𝐺	𝐺	NOUN
cana-1260	618	6	=	=	PUNCT
cana-1260	619	1	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	619	2	×	×	NOUN
cana-1260	619	3	𝐶𝑛	𝐶𝑛	NOUN
cana-1260	619	4	observed	observe	VERB
cana-1260	619	5	as	as	ADP
cana-1260	619	6	cartesian	cartesian	ADJ
cana-1260	619	7	product	product	NOUN
cana-1260	619	8	cycle	cycle	NOUN
cana-1260	619	9	graphs	graph	NOUN
cana-1260	619	10	so	so	SCONJ
cana-1260	619	11	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	619	12	(	(	PUNCT
cana-1260	619	13	𝐺	𝐺	NOUN
cana-1260	619	14	)	)	PUNCT
cana-1260	619	15	+	+	CCONJ
cana-1260	619	16	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	619	17	(	(	PUNCT
cana-1260	619	18	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	619	19	)	)	PUNCT
cana-1260	619	20	)	)	PUNCT
cana-1260	620	1	=	=	PUNCT
cana-1260	620	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	620	3	5	5	NUM
cana-1260	620	4	+	+	CCONJ
cana-1260	620	5	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	620	6	7	7	NUM
cana-1260	620	7	and	and	CCONJ
cana-1260	620	8	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	620	9	(	(	PUNCT
cana-1260	620	10	𝐺	𝐺	NOUN
cana-1260	620	11	)	)	PUNCT
cana-1260	620	12	∗	∗	NOUN
cana-1260	620	13	𝛾𝑓	𝛾𝑓	NOUN
cana-1260	620	14	(	(	PUNCT
cana-1260	620	15	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	620	16	)	)	PUNCT
cana-1260	620	17	)	)	PUNCT
cana-1260	621	1	=	=	PUNCT
cana-1260	621	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	621	3	5	5	NUM
cana-1260	622	1	+	+	CCONJ
cana-1260	622	2	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	622	3	7	7	NUM
cana-1260	622	4	hence	hence	ADV
cana-1260	622	5	𝛾𝑓	𝛾𝑓	VERB
cana-1260	622	6	(	(	PUNCT
cana-1260	622	7	𝐺	𝐺	PROPN
cana-1260	622	8	+	+	CCONJ
cana-1260	622	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	622	10	)	)	PUNCT
cana-1260	622	11	)	)	PUNCT
cana-1260	623	1	=	=	PUNCT
cana-1260	623	2	1	1	NUM
cana-1260	623	3	,	,	PUNCT
cana-1260	623	4	𝑖𝑓	𝑖𝑓	ADV
cana-1260	623	5	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	623	6	(	(	PUNCT
cana-1260	623	7	𝐺	𝐺	NOUN
cana-1260	623	8	)	)	PUNCT
cana-1260	623	9	=	=	SYM
cana-1260	624	1	1	1	NUM
cana-1260	624	2	𝑜𝑟	𝑜𝑟	PRON
cana-1260	624	3	𝛾𝑓	𝛾𝑓	X
cana-1260	624	4	(	(	PUNCT
cana-1260	624	5	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	624	6	)	)	PUNCT
cana-1260	624	7	)	)	PUNCT
cana-1260	625	1	=	=	SYM
cana-1260	625	2	1	1	NUM
cana-1260	625	3	,	,	PUNCT
cana-1260	625	4	otherwise	otherwise	ADV
cana-1260	625	5	𝛾𝑓	𝛾𝑓	INTJ
cana-1260	625	6	(	(	PUNCT
cana-1260	625	7	𝐺	𝐺	PROPN
cana-1260	625	8	+	+	CCONJ
cana-1260	625	9	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	625	10	)	)	PUNCT
cana-1260	625	11	)	)	PUNCT
cana-1260	626	1	=	=	SYM
cana-1260	626	2	2	2	NUM
cana-1260	626	3	−	−	NOUN
cana-1260	626	4	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	626	5	5	5	NUM
cana-1260	626	6	+	+	CCONJ
cana-1260	626	7	2𝑚𝑛	2𝑚𝑛	ADJ
cana-1260	626	8	7	7	NUM
cana-1260	626	9	−	−	NOUN
cana-1260	626	10	2	2	NUM
cana-1260	626	11	𝑚𝑛	𝑚𝑛	SYM
cana-1260	626	12	5	5	NUM
cana-1260	626	13	∗	∗	NOUN
cana-1260	626	14	2𝑚𝑛	2𝑚𝑛	NOUN
cana-1260	626	15	7	7	NUM
cana-1260	626	16	−	−	NOUN
cana-1260	626	17	1	1	NUM
cana-1260	626	18	.	.	PUNCT
cana-1260	627	1	similarly	similarly	ADV
cana-1260	627	2	г𝑓	г𝑓	X
cana-1260	627	3	(	(	PUNCT
cana-1260	627	4	𝐺	𝐺	NOUN
cana-1260	627	5	)	)	PUNCT
cana-1260	627	6	≤	≤	NOUN
cana-1260	628	1	𝑚𝑛	𝑚𝑛	ADP
cana-1260	628	2	4	4	NUM
cana-1260	628	3	and	and	CCONJ
cana-1260	628	4	г𝑓	г𝑓	NOUN
cana-1260	628	5	(	(	PUNCT
cana-1260	628	6	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	628	7	)	)	PUNCT
cana-1260	628	8	)	)	PUNCT
cana-1260	629	1	≤	≤	NUM
cana-1260	630	1	𝑚𝑛	𝑚𝑛	ADP
cana-1260	630	2	3	3	NUM
cana-1260	630	3	so	so	ADV
cana-1260	630	4	г𝑓	г𝑓	PROPN
cana-1260	630	5	(	(	PUNCT
cana-1260	630	6	𝐺	𝐺	NOUN
cana-1260	630	7	)	)	PUNCT
cana-1260	630	8	+	+	NUM
cana-1260	630	9	г𝑓	г𝑓	ADJ
cana-1260	630	10	(	(	PUNCT
cana-1260	630	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	630	12	)	)	PUNCT
cana-1260	630	13	)	)	PUNCT
cana-1260	630	14	≤	≤	NUM
cana-1260	631	1	𝑚𝑛	𝑚𝑛	ADP
cana-1260	631	2	4	4	NUM
cana-1260	631	3	+	+	CCONJ
cana-1260	631	4	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	631	5	3	3	NUM
cana-1260	631	6	and	and	CCONJ
cana-1260	631	7	г𝑓	г𝑓	PROPN
cana-1260	631	8	(	(	PUNCT
cana-1260	631	9	𝐺	𝐺	NOUN
cana-1260	631	10	)	)	PUNCT
cana-1260	631	11	∗	∗	NOUN
cana-1260	631	12	г𝑓	г𝑓	PROPN
cana-1260	631	13	(	(	PUNCT
cana-1260	631	14	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	631	15	)	)	PUNCT
cana-1260	631	16	)	)	PUNCT
cana-1260	631	17	≤	≤	NUM
cana-1260	632	1	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	632	2	4	4	NUM
cana-1260	632	3	∗	∗	NOUN
cana-1260	632	4	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	632	5	3	3	NUM
cana-1260	632	6	hence	hence	ADV
cana-1260	632	7	г𝑓	г𝑓	PROPN
cana-1260	632	8	(	(	PUNCT
cana-1260	632	9	𝐺	𝐺	PROPN
cana-1260	632	10	+	+	CCONJ
cana-1260	632	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	632	12	)	)	PUNCT
cana-1260	632	13	)	)	PUNCT
cana-1260	632	14	≤	≤	ADV
cana-1260	633	1	2	2	NUM
cana-1260	633	2	−	−	NOUN
cana-1260	633	3	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	633	4	4	4	NUM
cana-1260	633	5	+	+	CCONJ
cana-1260	633	6	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	633	7	3	3	NUM
cana-1260	633	8	−	−	NOUN
cana-1260	633	9	2	2	NUM
cana-1260	633	10	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	633	11	4	4	NUM
cana-1260	633	12	∗	∗	NOUN
cana-1260	633	13	𝑚𝑛	𝑚𝑛	NOUN
cana-1260	633	14	3	3	NUM
cana-1260	633	15	−	−	NOUN
cana-1260	633	16	1	1	NUM
cana-1260	633	17	.	.	PUNCT
cana-1260	634	1	4	4	X
cana-1260	634	2	.	.	X
cana-1260	634	3	applications	application	NOUN
cana-1260	634	4	•	•	NUM
cana-1260	634	5	transportation	transportation	NOUN
cana-1260	634	6	networks	network	NOUN
cana-1260	634	7	:	:	PUNCT
cana-1260	634	8	graph	graph	NOUN
cana-1260	634	9	theory	theory	NOUN
cana-1260	634	10	helps	help	VERB
cana-1260	634	11	in	in	ADP
cana-1260	634	12	optimizing	optimize	VERB
cana-1260	634	13	transportation	transportation	NOUN
cana-1260	634	14	networks	network	NOUN
cana-1260	634	15	,	,	PUNCT
cana-1260	634	16	such	such	ADJ
cana-1260	634	17	as	as	ADP
cana-1260	634	18	road	road	NOUN
cana-1260	634	19	networks	network	NOUN
cana-1260	634	20	or	or	CCONJ
cana-1260	634	21	flight	flight	NOUN
cana-1260	634	22	routes	route	NOUN
cana-1260	634	23	.	.	PUNCT
cana-1260	635	1	line	line	NOUN
cana-1260	635	2	graphs	graph	NOUN
cana-1260	635	3	can	can	AUX
cana-1260	635	4	represent	represent	VERB
cana-1260	635	5	these	these	DET
cana-1260	635	6	networks	network	NOUN
cana-1260	635	7	where	where	SCONJ
cana-1260	635	8	nodes	node	NOUN
cana-1260	635	9	represent	represent	VERB
cana-1260	635	10	intersections	intersection	NOUN
cana-1260	635	11	or	or	CCONJ
cana-1260	635	12	airports	airport	NOUN
cana-1260	635	13	,	,	PUNCT
cana-1260	635	14	and	and	CCONJ
cana-1260	635	15	edges	edge	NOUN
cana-1260	635	16	represent	represent	VERB
cana-1260	635	17	roads	road	NOUN
cana-1260	635	18	or	or	CCONJ
cana-1260	635	19	flight	flight	NOUN
cana-1260	635	20	paths	path	NOUN
cana-1260	635	21	.	.	PUNCT
cana-1260	636	1	communications	communication	NOUN
cana-1260	636	2	on	on	ADP
cana-1260	636	3	applied	apply	VERB
cana-1260	636	4	nonlinear	nonlinear	ADJ
cana-1260	636	5	analysis	analysis	NOUN
cana-1260	636	6	issn	issn	NOUN
cana-1260	636	7	:	:	PUNCT
cana-1260	636	8	1074	1074	NUM
cana-1260	636	9	-	-	PUNCT
cana-1260	636	10	133x	133x	NUM
cana-1260	636	11	vol	vol	NOUN
cana-1260	636	12	31	31	NUM
cana-1260	636	13	no	no	NOUN
cana-1260	636	14	.	.	PUNCT
cana-1260	637	1	6s	6s	NUM
cana-1260	637	2	(	(	PUNCT
cana-1260	637	3	2024	2024	NUM
cana-1260	637	4	)	)	PUNCT
cana-1260	637	5	691	691	NUM
cana-1260	637	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1260	637	7	•	•	NOUN
cana-1260	637	8	communication	communication	NOUN
cana-1260	637	9	networks	network	NOUN
cana-1260	637	10	:	:	PUNCT
cana-1260	637	11	graph	graph	NOUN
cana-1260	637	12	theory	theory	NOUN
cana-1260	637	13	is	be	AUX
cana-1260	637	14	used	use	VERB
cana-1260	637	15	in	in	ADP
cana-1260	637	16	designing	design	VERB
cana-1260	637	17	and	and	CCONJ
cana-1260	637	18	optimizing	optimize	VERB
cana-1260	637	19	communication	communication	NOUN
cana-1260	637	20	networks	network	NOUN
cana-1260	637	21	,	,	PUNCT
cana-1260	637	22	such	such	ADJ
cana-1260	637	23	as	as	ADP
cana-1260	637	24	telephone	telephone	NOUN
cana-1260	637	25	networks	network	NOUN
cana-1260	637	26	or	or	CCONJ
cana-1260	637	27	internet	internet	NOUN
cana-1260	637	28	routing	routing	NOUN
cana-1260	637	29	.	.	PUNCT
cana-1260	638	1	line	line	NOUN
cana-1260	638	2	graphs	graph	NOUN
cana-1260	638	3	can	can	AUX
cana-1260	638	4	represent	represent	VERB
cana-1260	638	5	these	these	DET
cana-1260	638	6	networks	network	NOUN
cana-1260	638	7	where	where	SCONJ
cana-1260	638	8	nodes	node	NOUN
cana-1260	638	9	are	be	AUX
cana-1260	638	10	switching	switch	VERB
cana-1260	638	11	points	point	NOUN
cana-1260	638	12	or	or	CCONJ
cana-1260	638	13	routers	router	NOUN
cana-1260	638	14	,	,	PUNCT
cana-1260	638	15	and	and	CCONJ
cana-1260	638	16	edges	edge	NOUN
cana-1260	638	17	represent	represent	VERB
cana-1260	638	18	communication	communication	NOUN
cana-1260	638	19	links	link	NOUN
cana-1260	638	20	.	.	PUNCT
cana-1260	639	1	•	•	NUM
cana-1260	639	2	computational	computational	ADJ
cana-1260	639	3	biology	biology	NOUN
cana-1260	639	4	:	:	PUNCT
cana-1260	640	1	[	[	X
cana-1260	640	2	13	13	NUM
cana-1260	640	3	]	]	PUNCT
cana-1260	640	4	here	here	ADV
cana-1260	640	5	we	we	PRON
cana-1260	640	6	have	have	AUX
cana-1260	640	7	used	use	VERB
cana-1260	640	8	these	these	DET
cana-1260	640	9	concepts	concept	NOUN
cana-1260	640	10	in	in	ADP
cana-1260	640	11	computational	computational	ADJ
cana-1260	640	12	biological	biological	ADJ
cana-1260	640	13	systems	system	NOUN
cana-1260	640	14	such	such	ADJ
cana-1260	640	15	as	as	ADP
cana-1260	640	16	gene	gene	NOUN
cana-1260	640	17	regulatory	regulatory	ADJ
cana-1260	640	18	networks	network	NOUN
cana-1260	640	19	,	,	PUNCT
cana-1260	640	20	protein	protein	NOUN
cana-1260	640	21	-	-	PUNCT
cana-1260	640	22	protein	protein	NOUN
cana-1260	640	23	interaction	interaction	NOUN
cana-1260	640	24	networks	network	NOUN
cana-1260	640	25	and	and	CCONJ
cana-1260	640	26	healthcare	healthcare	NOUN
cana-1260	640	27	network	network	NOUN
cana-1260	640	28	optimization	optimization	NOUN
cana-1260	640	29	.	.	PUNCT
cana-1260	641	1	line	line	NOUN
cana-1260	641	2	graphs	graph	NOUN
cana-1260	641	3	can	can	AUX
cana-1260	641	4	represent	represent	VERB
cana-1260	641	5	these	these	DET
cana-1260	641	6	networks	network	NOUN
cana-1260	641	7	where	where	SCONJ
cana-1260	641	8	nodes	node	NOUN
cana-1260	641	9	are	be	AUX
cana-1260	641	10	biological	biological	ADJ
cana-1260	641	11	entities	entity	NOUN
cana-1260	641	12	and	and	CCONJ
cana-1260	641	13	edges	edge	NOUN
cana-1260	641	14	represent	represent	VERB
cana-1260	641	15	interactions	interaction	NOUN
cana-1260	641	16	or	or	CCONJ
cana-1260	641	17	biochemical	biochemical	ADJ
cana-1260	641	18	reactions	reaction	NOUN
cana-1260	641	19	.	.	PUNCT
cana-1260	642	1	5	5	X
cana-1260	642	2	.	.	X
cana-1260	642	3	conclusion	conclusion	NOUN
cana-1260	642	4	we	we	PRON
cana-1260	642	5	have	have	AUX
cana-1260	642	6	obtained	obtain	VERB
cana-1260	642	7	bounds	bound	NOUN
cana-1260	642	8	like	like	ADP
cana-1260	642	9	lower	low	ADJ
cana-1260	642	10	and	and	CCONJ
cana-1260	642	11	upper	upper	ADJ
cana-1260	642	12	for	for	ADP
cana-1260	642	13	sum	sum	NOUN
cana-1260	642	14	of	of	ADP
cana-1260	642	15	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	642	16	(	(	PUNCT
cana-1260	642	17	𝐺	𝐺	NOUN
cana-1260	642	18	)	)	PUNCT
cana-1260	643	1	+	+	CCONJ
cana-1260	643	2	𝛾𝑓	𝛾𝑓	PROPN
cana-1260	643	3	(	(	PUNCT
cana-1260	643	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	643	5	)	)	PUNCT
cana-1260	643	6	)	)	PUNCT
cana-1260	643	7	and	and	CCONJ
cana-1260	643	8	exploring	explore	VERB
cana-1260	643	9	the	the	DET
cana-1260	643	10	graphs	graph	NOUN
cana-1260	643	11	for	for	ADP
cana-1260	643	12	upper	upper	ADJ
cana-1260	643	13	fractional	fractional	ADJ
cana-1260	643	14	domination	domination	NOUN
cana-1260	643	15	number	number	NOUN
cana-1260	643	16	including	include	VERB
cana-1260	643	17	cycle	cycle	NOUN
cana-1260	643	18	,	,	PUNCT
cana-1260	643	19	complete	complete	ADJ
cana-1260	643	20	,	,	PUNCT
cana-1260	643	21	star	star	NOUN
cana-1260	643	22	,	,	PUNCT
cana-1260	643	23	bi	bi	NOUN
cana-1260	643	24	-	-	NOUN
cana-1260	643	25	star	star	NOUN
cana-1260	643	26	,	,	PUNCT
cana-1260	643	27	wheel	wheel	NOUN
cana-1260	643	28	,	,	PUNCT
cana-1260	643	29	cubic	cubic	ADJ
cana-1260	643	30	graph	graph	NOUN
cana-1260	643	31	,	,	PUNCT
cana-1260	643	32	graph	graph	NOUN
cana-1260	643	33	of	of	ADP
cana-1260	643	34	cartesian	cartesian	ADJ
cana-1260	643	35	product	product	NOUN
cana-1260	643	36	like	like	ADP
cana-1260	643	37	(	(	PUNCT
cana-1260	643	38	𝐾2	𝐾2	ADJ
cana-1260	643	39	×	×	PROPN
cana-1260	643	40	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	643	41	)	)	PUNCT
cana-1260	643	42	,	,	PUNCT
cana-1260	643	43	(	(	PUNCT
cana-1260	643	44	𝐾3	𝐾3	VERB
cana-1260	643	45	×	×	PROPN
cana-1260	643	46	𝑃𝑛	𝑃𝑛	PROPN
cana-1260	643	47	)	)	PUNCT
cana-1260	643	48	and	and	CCONJ
cana-1260	643	49	(	(	PUNCT
cana-1260	643	50	𝐶𝑚	𝐶𝑚	NOUN
cana-1260	643	51	×	×	PROPN
cana-1260	643	52	𝐶𝑛	𝐶𝑛	PROPN
cana-1260	643	53	)	)	PUNCT
cana-1260	643	54	.	.	PUNCT
cana-1260	644	1	for	for	ADP
cana-1260	644	2	these	these	DET
cana-1260	644	3	graph	graph	NOUN
cana-1260	644	4	classes	class	NOUN
cana-1260	644	5	,	,	PUNCT
cana-1260	644	6	the	the	DET
cana-1260	644	7	fractional	fractional	ADJ
cana-1260	644	8	domination	domination	NOUN
cana-1260	644	9	number	number	NOUN
cana-1260	644	10	of	of	ADP
cana-1260	644	11	𝐿(𝐺	𝐿(𝐺	PROPN
cana-1260	644	12	)	)	PUNCT
cana-1260	644	13	can	can	AUX
cana-1260	644	14	be	be	AUX
cana-1260	644	15	related	relate	VERB
cana-1260	644	16	to	to	ADP
cana-1260	644	17	the	the	DET
cana-1260	644	18	fractional	fractional	ADJ
cana-1260	644	19	domination	domination	NOUN
cana-1260	644	20	number	number	NOUN
cana-1260	644	21	of	of	ADP
cana-1260	644	22	𝐺.	𝐺.	NOUN
cana-1260	644	23	our	our	PRON
cana-1260	644	24	results	result	NOUN
cana-1260	644	25	show	show	VERB
cana-1260	644	26	that	that	SCONJ
cana-1260	644	27	there	there	PRON
cana-1260	644	28	is	be	VERB
cana-1260	644	29	a	a	DET
cana-1260	644	30	correlation	correlation	NOUN
cana-1260	644	31	between	between	ADP
cana-1260	644	32	the	the	DET
cana-1260	644	33	sum	sum	NOUN
cana-1260	644	34	of	of	ADP
cana-1260	644	35	fractional	fractional	ADJ
cana-1260	644	36	domination	domination	NOUN
cana-1260	644	37	number	number	NOUN
cana-1260	644	38	of	of	ADP
cana-1260	644	39	a	a	DET
cana-1260	644	40	graph	graph	NOUN
cana-1260	644	41	and	and	CCONJ
cana-1260	644	42	its	its	PRON
cana-1260	644	43	line	line	NOUN
cana-1260	644	44	graph	graph	NOUN
cana-1260	644	45	.	.	PUNCT
cana-1260	645	1	parameters	parameter	NOUN
cana-1260	645	2	related	relate	VERB
cana-1260	645	3	to	to	ADP
cana-1260	645	4	fractional	fractional	ADJ
cana-1260	645	5	domination	domination	NOUN
cana-1260	645	6	in	in	ADP
cana-1260	645	7	line	line	NOUN
cana-1260	645	8	graphs	graph	NOUN
cana-1260	645	9	towards	towards	ADP
cana-1260	645	10	generalization	generalization	NOUN
cana-1260	645	11	.	.	PUNCT
cana-1260	646	1	references	reference	NOUN
cana-1260	646	2	[	[	X
cana-1260	646	3	1	1	X
cana-1260	646	4	]	]	PUNCT
cana-1260	646	5	s.	s.	PROPN
cana-1260	646	6	arumugam	arumugam	PROPN
cana-1260	646	7	,	,	PUNCT
cana-1260	646	8	v.	v.	PROPN
cana-1260	646	9	mathew	mathew	PROPN
cana-1260	646	10	,	,	PUNCT
cana-1260	646	11	and	and	CCONJ
cana-1260	646	12	k.	k.	PROPN
cana-1260	646	13	karuppasamy	karuppasamy	PROPN
cana-1260	646	14	,	,	PUNCT
cana-1260	646	15	fractional	fractional	ADJ
cana-1260	646	16	distance	distance	NOUN
cana-1260	646	17	domination	domination	NOUN
cana-1260	646	18	in	in	ADP
cana-1260	646	19	graphs	graph	NOUN
cana-1260	646	20	,	,	PUNCT
cana-1260	646	21	discussiones	discussione	NOUN
cana-1260	646	22	mathematicae	mathematicae	PROPN
cana-1260	646	23	graph	graph	NOUN
cana-1260	646	24	theory	theory	NOUN
cana-1260	646	25	,	,	PUNCT
cana-1260	646	26	32	32	NUM
cana-1260	646	27	(	(	PUNCT
cana-1260	646	28	2012	2012	NUM
cana-1260	646	29	)	)	PUNCT
cana-1260	646	30	,	,	PUNCT
cana-1260	646	31	449–459	449–459	NUM
cana-1260	646	32	.	.	PUNCT
cana-1260	647	1	doi	doi	NOUN
cana-1260	647	2	:	:	PUNCT
cana-1260	647	3	/10.7151	/10.7151	ADJ
cana-1260	647	4	/	/	SYM
cana-1260	647	5	dmgt.1609	dmgt.1609	PROPN
cana-1260	647	6	.	.	PUNCT
cana-1260	648	1	[	[	X
cana-1260	648	2	2	2	X
cana-1260	648	3	]	]	X
cana-1260	648	4	g.	g.	PROPN
cana-1260	648	5	a.	a.	PROPN
cana-1260	648	6	cheston	cheston	PROPN
cana-1260	648	7	,	,	PUNCT
cana-1260	648	8	g.	g.	PROPN
cana-1260	648	9	fricke	fricke	PROPN
cana-1260	648	10	,	,	PUNCT
cana-1260	648	11	s.t	s.t	PROPN
cana-1260	648	12	.	.	PROPN
cana-1260	648	13	hedetniemi	hedetniemi	PROPN
cana-1260	648	14	,	,	PUNCT
cana-1260	648	15	and	and	CCONJ
cana-1260	648	16	d.	d.	PROPN
cana-1260	648	17	p.	p.	PROPN
cana-1260	648	18	jacobs	jacobs	PROPN
cana-1260	648	19	,	,	PUNCT
cana-1260	648	20	on	on	ADP
cana-1260	648	21	the	the	DET
cana-1260	648	22	computational	computational	ADJ
cana-1260	648	23	complexity	complexity	NOUN
cana-1260	648	24	of	of	ADP
cana-1260	648	25	upper	upper	ADJ
cana-1260	648	26	fractional	fractional	ADJ
cana-1260	648	27	domination	domination	NOUN
cana-1260	648	28	,	,	PUNCT
cana-1260	648	29	discrete	discrete	ADJ
cana-1260	648	30	applied	apply	VERB
cana-1260	648	31	mathematics	mathematic	NOUN
cana-1260	648	32	,	,	PUNCT
cana-1260	648	33	27	27	NUM
cana-1260	648	34	(	(	PUNCT
cana-1260	648	35	1990	1990	NUM
cana-1260	648	36	)	)	PUNCT
cana-1260	648	37	,	,	PUNCT
cana-1260	648	38	195	195	NUM
cana-1260	648	39	-	-	SYM
cana-1260	648	40	20	20	NUM
cana-1260	648	41	.	.	PUNCT
cana-1260	649	1	doi	doi	NOUN
cana-1260	649	2	:	:	PUNCT
cana-1260	649	3	/10.1016/0166	/10.1016/0166	ADJ
cana-1260	649	4	-	-	PUNCT
cana-1260	649	5	218x(90)90065	218x(90)90065	NUM
cana-1260	649	6	-	-	PUNCT
cana-1260	649	7	k.	k.	NOUN
cana-1260	650	1	[	[	X
cana-1260	650	2	3	3	X
cana-1260	650	3	]	]	X
cana-1260	650	4	g.	g.	PROPN
cana-1260	650	5	s.	s.	PROPN
cana-1260	650	6	domke	domke	PROPN
cana-1260	650	7	,	,	PUNCT
cana-1260	650	8	s.t	s.t	PROPN
cana-1260	650	9	.	.	PROPN
cana-1260	650	10	hedetniemi	hedetniemi	PROPN
cana-1260	650	11	,	,	PUNCT
cana-1260	650	12	and	and	CCONJ
cana-1260	650	13	r.	r.	PROPN
cana-1260	650	14	c.	c.	PROPN
cana-1260	650	15	laskar	laskar	PROPN
cana-1260	650	16	,	,	PUNCT
cana-1260	650	17	fractional	fractional	ADJ
cana-1260	650	18	packings	packing	NOUN
cana-1260	650	19	,	,	PUNCT
cana-1260	650	20	coverings	covering	NOUN
cana-1260	650	21	and	and	CCONJ
cana-1260	650	22	irredundance	irredundance	NOUN
cana-1260	650	23	in	in	ADP
cana-1260	650	24	graphs	graph	NOUN
cana-1260	650	25	,	,	PUNCT
cana-1260	650	26	congressus	congressus	PROPN
cana-1260	650	27	numerantium	numerantium	PROPN
cana-1260	650	28	.	.	PUNCT
cana-1260	651	1	published	publish	VERB
cana-1260	651	2	(	(	PUNCT
cana-1260	651	3	1988	1988	NUM
cana-1260	651	4	)	)	PUNCT
cana-1260	651	5	,	,	PUNCT
cana-1260	651	6	66	66	NUM
cana-1260	651	7	,	,	PUNCT
cana-1260	651	8	227	227	NUM
cana-1260	651	9	-	-	SYM
cana-1260	651	10	238	238	NUM
cana-1260	651	11	.	.	PUNCT
cana-1260	652	1	[	[	X
cana-1260	652	2	4	4	NUM
cana-1260	652	3	]	]	X
cana-1260	652	4	d.c	d.c	PROPN
cana-1260	652	5	.	.	PROPN
cana-1260	652	6	fisher	fisher	PROPN
cana-1260	652	7	,	,	PUNCT
cana-1260	652	8	fractional	fractional	ADJ
cana-1260	652	9	domination	domination	NOUN
cana-1260	652	10	and	and	CCONJ
cana-1260	652	11	fractional	fractional	ADJ
cana-1260	652	12	total	total	ADJ
cana-1260	652	13	domination	domination	NOUN
cana-1260	652	14	of	of	ADP
cana-1260	652	15	graph	graph	NOUN
cana-1260	652	16	complements	complement	NOUN
cana-1260	652	17	,	,	PUNCT
cana-1260	652	18	discrete	discrete	ADJ
cana-1260	652	19	applied	apply	VERB
cana-1260	652	20	mathematics	mathematic	NOUN
cana-1260	652	21	,	,	PUNCT
cana-1260	652	22	122	122	NUM
cana-1260	652	23	(	(	PUNCT
cana-1260	652	24	2002	2002	NUM
cana-1260	652	25	)	)	PUNCT
cana-1260	652	26	,	,	PUNCT
cana-1260	652	27	283	283	NUM
cana-1260	652	28	-	-	SYM
cana-1260	652	29	291	291	NUM
cana-1260	652	30	,	,	PUNCT
cana-1260	652	31	doi	doi	NOUN
cana-1260	652	32	:	:	PUNCT
cana-1260	652	33	/10.1016	/10.1016	PROPN
cana-1260	652	34	/	/	SYM
cana-1260	652	35	s0166	s0166	NOUN
cana-1260	652	36	-	-	PUNCT
cana-1260	652	37	218x(01)00305	218x(01)00305	NUM
cana-1260	652	38	-	-	PUNCT
cana-1260	652	39	5	5	NUM
cana-1260	652	40	.	.	PUNCT
cana-1260	653	1	[	[	X
cana-1260	653	2	5	5	X
cana-1260	653	3	]	]	X
cana-1260	653	4	t.w	t.w	PROPN
cana-1260	653	5	.	.	PROPN
cana-1260	653	6	haynes	haynes	PROPN
cana-1260	653	7	s.t	s.t	PROPN
cana-1260	653	8	.	.	PROPN
cana-1260	653	9	hedetniemi	hedetniemi	PROPN
cana-1260	653	10	and	and	CCONJ
cana-1260	653	11	p.j	p.j	PROPN
cana-1260	653	12	.	.	PROPN
cana-1260	653	13	slater	slater	PROPN
cana-1260	653	14	,	,	PUNCT
cana-1260	653	15	fundamentals	fundamental	NOUN
cana-1260	653	16	of	of	ADP
cana-1260	653	17	domination	domination	NOUN
cana-1260	653	18	in	in	ADP
cana-1260	653	19	graphs	graph	NOUN
cana-1260	653	20	,	,	PUNCT
cana-1260	653	21	marcel	marcel	PROPN
cana-1260	653	22	dekker	dekker	PROPN
cana-1260	653	23	,	,	PUNCT
cana-1260	653	24	inc	inc	PROPN
cana-1260	653	25	.	.	PROPN
cana-1260	653	26	,	,	PUNCT
cana-1260	653	27	new	new	PROPN
cana-1260	653	28	york	york	PROPN
cana-1260	653	29	,	,	PUNCT
cana-1260	653	30	(	(	PUNCT
cana-1260	653	31	1998	1998	NUM
cana-1260	653	32	)	)	PUNCT
cana-1260	653	33	.	.	PUNCT
cana-1260	654	1	[	[	X
cana-1260	654	2	6	6	NUM
cana-1260	654	3	]	]	X
cana-1260	654	4	t.w	t.w	PROPN
cana-1260	654	5	.	.	PROPN
cana-1260	654	6	haynes	haynes	PROPN
cana-1260	654	7	,	,	PUNCT
cana-1260	654	8	s.t	s.t	PROPN
cana-1260	654	9	.	.	PROPN
cana-1260	654	10	hedetniemi	hedetniemi	ADV
cana-1260	654	11	,	,	PUNCT
cana-1260	654	12	and	and	CCONJ
cana-1260	654	13	p.j	p.j	PROPN
cana-1260	654	14	.	.	PROPN
cana-1260	654	15	slater	slater	PROPN
cana-1260	654	16	,	,	PUNCT
cana-1260	654	17	domination	domination	NOUN
cana-1260	654	18	in	in	ADP
cana-1260	654	19	graphs	graph	NOUN
cana-1260	654	20	advanced	advanced	ADJ
cana-1260	654	21	topics	topic	NOUN
cana-1260	654	22	,	,	PUNCT
cana-1260	654	23	pure	pure	ADJ
cana-1260	654	24	and	and	CCONJ
cana-1260	654	25	applied	applied	ADJ
cana-1260	654	26	mathematics	mathematic	NOUN
cana-1260	654	27	,	,	PUNCT
cana-1260	654	28	crc	crc	PROPN
cana-1260	654	29	press	press	PROPN
cana-1260	654	30	taylor	taylor	PROPN
cana-1260	654	31	&	&	CCONJ
cana-1260	654	32	francis	francis	PROPN
cana-1260	654	33	group	group	PROPN
cana-1260	654	34	,	,	PUNCT
cana-1260	654	35	marcel	marcel	PROPN
cana-1260	654	36	dekker	dekker	PROPN
cana-1260	654	37	,	,	PUNCT
cana-1260	654	38	inc	inc	PROPN
cana-1260	654	39	.	.	PROPN
cana-1260	654	40	,	,	PUNCT
cana-1260	654	41	new	new	PROPN
cana-1260	654	42	york	york	PROPN
cana-1260	654	43	,	,	PUNCT
cana-1260	654	44	(	(	PUNCT
cana-1260	654	45	1998	1998	NUM
cana-1260	654	46	)	)	PUNCT
cana-1260	654	47	.	.	PUNCT
cana-1260	655	1	[	[	X
cana-1260	655	2	7	7	X
cana-1260	655	3	]	]	X
cana-1260	655	4	frank	frank	PROPN
cana-1260	655	5	harary	harary	NOUN
cana-1260	655	6	,	,	PUNCT
cana-1260	655	7	graph	graph	NOUN
cana-1260	655	8	theory	theory	NOUN
cana-1260	655	9	,	,	PUNCT
cana-1260	655	10	addison	addison	PROPN
cana-1260	655	11	-	-	PUNCT
cana-1260	655	12	wesley	wesley	PROPN
cana-1260	655	13	publishing	publishing	PROPN
cana-1260	655	14	company	company	PROPN
cana-1260	655	15	inc	inc	PROPN
cana-1260	655	16	,	,	PUNCT
cana-1260	655	17	(	(	PUNCT
cana-1260	655	18	1969	1969	NUM
cana-1260	655	19	)	)	PUNCT
cana-1260	655	20	.	.	PUNCT
cana-1260	656	1	[	[	X
cana-1260	656	2	8	8	X
cana-1260	656	3	]	]	X
cana-1260	656	4	e.	e.	PROPN
cana-1260	656	5	murugan	murugan	PROPN
cana-1260	656	6	,	,	PUNCT
cana-1260	656	7	and	and	CCONJ
cana-1260	656	8	j.	j.	PROPN
cana-1260	656	9	p.	p.	PROPN
cana-1260	656	10	joseph	joseph	PROPN
cana-1260	656	11	,	,	PUNCT
cana-1260	656	12	on	on	ADP
cana-1260	656	13	the	the	DET
cana-1260	656	14	domination	domination	NOUN
cana-1260	656	15	number	number	NOUN
cana-1260	656	16	of	of	ADP
cana-1260	656	17	graph	graph	NOUN
cana-1260	656	18	and	and	CCONJ
cana-1260	656	19	its	its	PRON
cana-1260	656	20	line	line	NOUN
cana-1260	656	21	graph	graph	NOUN
cana-1260	656	22	,	,	PUNCT
cana-1260	656	23	international	international	ADJ
cana-1260	656	24	journal	journal	NOUN
cana-1260	656	25	of	of	ADP
cana-1260	656	26	mathematical	mathematical	ADJ
cana-1260	656	27	combinatorics	combinatoric	NOUN
cana-1260	656	28	,	,	PUNCT
cana-1260	656	29	issn	issn	PROPN
cana-1260	656	30	1937	1937	NUM
cana-1260	656	31	-	-	SYM
cana-1260	656	32	1055	1055	NUM
cana-1260	656	33	,	,	PUNCT
cana-1260	656	34	1	1	NUM
cana-1260	656	35	(	(	PUNCT
cana-1260	656	36	2018	2018	NUM
cana-1260	656	37	)	)	PUNCT
cana-1260	656	38	,	,	PUNCT
cana-1260	656	39	170	170	NUM
cana-1260	656	40	-	-	SYM
cana-1260	656	41	181	181	NUM
cana-1260	656	42	.	.	PUNCT
cana-1260	657	1	[	[	X
cana-1260	657	2	9	9	NUM
cana-1260	657	3	]	]	X
cana-1260	657	4	e.	e.	PROPN
cana-1260	657	5	murugan	murugan	PROPN
cana-1260	657	6	and	and	CCONJ
cana-1260	657	7	j.	j.	PROPN
cana-1260	657	8	p.	p.	PROPN
cana-1260	657	9	joseph	joseph	PROPN
cana-1260	657	10	,	,	PUNCT
cana-1260	657	11	further	further	ADJ
cana-1260	657	12	results	result	NOUN
cana-1260	657	13	on	on	ADP
cana-1260	657	14	domination	domination	NOUN
cana-1260	657	15	number	number	NOUN
cana-1260	657	16	of	of	ADP
cana-1260	657	17	a	a	DET
cana-1260	657	18	graph	graph	NOUN
cana-1260	657	19	and	and	CCONJ
cana-1260	657	20	its	its	PRON
cana-1260	657	21	line	line	NOUN
cana-1260	657	22	graph	graph	NOUN
cana-1260	657	23	,	,	PUNCT
cana-1260	657	24	global	global	ADJ
cana-1260	657	25	journal	journal	NOUN
cana-1260	657	26	of	of	ADP
cana-1260	657	27	engineering	engineering	NOUN
cana-1260	657	28	science	science	NOUN
cana-1260	657	29	and	and	CCONJ
cana-1260	657	30	researches	research	NOUN
cana-1260	657	31	,	,	PUNCT
cana-1260	657	32	issn	issn	PROPN
cana-1260	657	33	2348	2348	NUM
cana-1260	657	34	–	–	PUNCT
cana-1260	657	35	8034	8034	NUM
cana-1260	657	36	,	,	PUNCT
cana-1260	657	37	6	6	NUM
cana-1260	657	38	(	(	PUNCT
cana-1260	657	39	4	4	NUM
cana-1260	657	40	)	)	PUNCT
cana-1260	657	41	,	,	PUNCT
cana-1260	657	42	(	(	PUNCT
cana-1260	657	43	2019	2019	NUM
cana-1260	657	44	)	)	PUNCT
cana-1260	657	45	,	,	PUNCT
cana-1260	657	46	222	222	NUM
cana-1260	657	47	-	-	SYM
cana-1260	657	48	229	229	NUM
cana-1260	657	49	,	,	PUNCT
cana-1260	657	50	doi	doi	NOUN
cana-1260	657	51	:	:	PUNCT
cana-1260	657	52	/10.5281	/10.5281	PUNCT
cana-1260	657	53	/	/	SYM
cana-1260	658	1	zenodo.2649083	zenodo.2649083	NOUN
cana-1260	658	2	.	.	PUNCT
cana-1260	659	1	[	[	X
cana-1260	659	2	10	10	NUM
cana-1260	659	3	]	]	X
cana-1260	659	4	e.	e.	PROPN
cana-1260	659	5	r.	r.	PROPN
cana-1260	659	6	scheinerman	scheinerman	PROPN
cana-1260	659	7	,	,	PUNCT
cana-1260	659	8	and	and	CCONJ
cana-1260	659	9	d.h	d.h	PROPN
cana-1260	659	10	.	.	PROPN
cana-1260	659	11	ullman	ullman	PROPN
cana-1260	659	12	,	,	PUNCT
cana-1260	659	13	fractional	fractional	ADJ
cana-1260	659	14	graph	graph	NOUN
cana-1260	659	15	theory	theory	NOUN
cana-1260	659	16	,	,	PUNCT
cana-1260	659	17	centre	centre	NOUN
cana-1260	659	18	national	national	PROPN
cana-1260	659	19	de	de	X
cana-1260	659	20	la	la	X
cana-1260	659	21	recherche	recherche	X
cana-1260	659	22	scientifique	scientifique	PROPN
cana-1260	659	23	paris	paris	PROPN
cana-1260	659	24	,	,	PUNCT
cana-1260	659	25	france	france	PROPN
cana-1260	659	26	,	,	PUNCT
cana-1260	659	27	john	john	PROPN
cana-1260	659	28	wiley	wiley	PROPN
cana-1260	659	29	&	&	CCONJ
cana-1260	659	30	sons	son	NOUN
cana-1260	659	31	,	,	PUNCT
cana-1260	659	32	(	(	PUNCT
cana-1260	659	33	2008	2008	NUM
cana-1260	659	34	)	)	PUNCT
cana-1260	659	35	.	.	PUNCT
cana-1260	660	1	[	[	X
cana-1260	660	2	11	11	NUM
cana-1260	660	3	]	]	PUNCT
cana-1260	660	4	m.	m.	NOUN
cana-1260	660	5	sarada	sarada	PROPN
cana-1260	660	6	,	,	PUNCT
cana-1260	660	7	r.	r.	PROPN
cana-1260	660	8	jain	jain	PROPN
cana-1260	660	9	,	,	PUNCT
cana-1260	660	10	and	and	CCONJ
cana-1260	660	11	g.	g.	PROPN
cana-1260	660	12	mundhe	mundhe	PROPN
cana-1260	660	13	,	,	PUNCT
cana-1260	660	14	bounds	bound	VERB
cana-1260	660	15	for	for	ADP
cana-1260	660	16	fractional	fractional	ADJ
cana-1260	660	17	domination	domination	NOUN
cana-1260	660	18	number	number	NOUN
cana-1260	660	19	of	of	ADP
cana-1260	660	20	some	some	DET
cana-1260	660	21	graphs	graph	NOUN
cana-1260	660	22	and	and	CCONJ
cana-1260	660	23	their	their	PRON
cana-1260	660	24	dual	dual	ADJ
cana-1260	660	25	graphs	graph	NOUN
cana-1260	660	26	,	,	PUNCT
cana-1260	660	27	indian	indian	ADJ
cana-1260	660	28	journal	journal	NOUN
cana-1260	660	29	of	of	ADP
cana-1260	660	30	natural	natural	ADJ
cana-1260	660	31	sciences	science	NOUN
cana-1260	660	32	,	,	PUNCT
cana-1260	660	33	14	14	NUM
cana-1260	660	34	(	(	PUNCT
cana-1260	660	35	80	80	NUM
cana-1260	660	36	)	)	PUNCT
cana-1260	660	37	,	,	PUNCT
cana-1260	660	38	(	(	PUNCT
cana-1260	660	39	2023	2023	NUM
cana-1260	660	40	)	)	PUNCT
cana-1260	660	41	,	,	PUNCT
cana-1260	660	42	61661	61661	NUM
cana-1260	660	43	-	-	SYM
cana-1260	660	44	61670	61670	NUM
cana-1260	660	45	.	.	PUNCT
cana-1260	661	1	issn	issn	PROPN
cana-1260	661	2	:	:	PUNCT
cana-1260	661	3	0976	0976	NUM
cana-1260	661	4	–	–	PUNCT
cana-1260	661	5	0997	0997	NUM
cana-1260	661	6	.	.	PUNCT
cana-1260	662	1	[	[	X
cana-1260	662	2	12	12	NUM
cana-1260	662	3	]	]	PUNCT
cana-1260	662	4	p.	p.	NOUN
cana-1260	662	5	s.	s.	PROPN
cana-1260	662	6	gholap	gholap	PROPN
cana-1260	662	7	and	and	CCONJ
cana-1260	662	8	v.	v.	PROPN
cana-1260	662	9	e.	e.	PROPN
cana-1260	662	10	nikumbh	nikumbh	PROPN
cana-1260	662	11	,	,	PUNCT
cana-1260	662	12	topological	topological	ADJ
cana-1260	662	13	spaces	space	NOUN
cana-1260	662	14	generated	generate	VERB
cana-1260	662	15	by	by	ADP
cana-1260	662	16	graph	graph	NOUN
cana-1260	662	17	,	,	PUNCT
cana-1260	662	18	jñānābha	jñānābha	NOUN
cana-1260	662	19	,	,	PUNCT
cana-1260	662	20	52(1	52(1	NOUN
cana-1260	662	21	)	)	PUNCT
cana-1260	662	22	(	(	PUNCT
cana-1260	662	23	2022	2022	NUM
cana-1260	662	24	)	)	PUNCT
cana-1260	662	25	,	,	PUNCT
cana-1260	662	26	1	1	NUM
cana-1260	662	27	-	-	SYM
cana-1260	662	28	7	7	NUM
cana-1260	662	29	.	.	PUNCT
cana-1260	663	1	[	[	X
cana-1260	663	2	13	13	NUM
cana-1260	663	3	]	]	PUNCT
cana-1260	663	4	m.	m.	NOUN
cana-1260	663	5	sarada	sarada	PROPN
cana-1260	663	6	,	,	PUNCT
cana-1260	663	7	r.	r.	PROPN
cana-1260	663	8	jain	jain	PROPN
cana-1260	663	9	,	,	PUNCT
cana-1260	663	10	and	and	CCONJ
cana-1260	663	11	g.	g.	PROPN
cana-1260	663	12	mundhe	mundhe	PROPN
cana-1260	663	13	,	,	PUNCT
cana-1260	663	14	applications	application	NOUN
cana-1260	663	15	of	of	ADP
cana-1260	663	16	the	the	DET
cana-1260	663	17	fractional	fractional	ADJ
cana-1260	663	18	domination	domination	NOUN
cana-1260	663	19	in	in	ADP
cana-1260	663	20	computational	computational	ADJ
cana-1260	663	21	biology	biology	NOUN
cana-1260	663	22	using	use	VERB
cana-1260	663	23	lpp	lpp	PROPN
cana-1260	663	24	formulation	formulation	NOUN
cana-1260	663	25	,	,	PUNCT
cana-1260	663	26	african	african	ADJ
cana-1260	663	27	journal	journal	NOUN
cana-1260	663	28	of	of	ADP
cana-1260	663	29	biological	biological	ADJ
cana-1260	663	30	sciences	science	NOUN
cana-1260	663	31	,	,	PUNCT
cana-1260	663	32	06	06	NUM
cana-1260	663	33	(	(	PUNCT
cana-1260	663	34	05	05	NUM
cana-1260	663	35	)	)	PUNCT
cana-1260	663	36	,	,	PUNCT
cana-1260	663	37	(	(	PUNCT
cana-1260	663	38	2024	2024	NUM
cana-1260	663	39	)	)	PUNCT
cana-1260	663	40	,	,	PUNCT
cana-1260	663	41	issn	issn	PROPN
cana-1260	663	42	:	:	PUNCT
cana-1260	663	43	2663	2663	NUM
cana-1260	663	44	-	-	SYM
cana-1260	663	45	2187	2187	NUM
cana-1260	663	46	,	,	PUNCT
cana-1260	663	47	4341	4341	NUM
cana-1260	663	48	-	-	SYM
cana-1260	663	49	4358	4358	NUM
cana-1260	663	50	.	.	PUNCT
cana-1260	664	1	https://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=utf8&field-author=teresa+w.+haynes&search-alias=stripbooks	https://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=utf8&field-author=teresa+w.+haynes&search-alias=stripbook	VERB
cana-1260	664	2	https://www.amazon.in/s/ref=dp_byline_sr_book_2?ie=utf8&field-author=stephen+hedetniemi&search-alias=stripbooks	https://www.amazon.in/s/ref=dp_byline_sr_book_2?ie=utf8&field-author=stephen+hedetniemi&search-alias=stripbook	NOUN
cana-1260	664	3	https://www.amazon.in/s/ref=dp_byline_sr_book_3?ie=utf8&field-author=peter+slater&search-alias=stripbooks	https://www.amazon.in/s/ref=dp_byline_sr_book_3?ie=utf8&field-author=peter+slater&search-alias=stripbook	NOUN