id	sid	tid	token	lemma	pos
cana-1005	1	1	communications	communication	NOUN
cana-1005	1	2	on	on	ADP
cana-1005	1	3	applied	apply	VERB
cana-1005	1	4	nonlinear	nonlinear	ADJ
cana-1005	1	5	analysis	analysis	NOUN
cana-1005	1	6	issn	issn	NOUN
cana-1005	1	7	:	:	PUNCT
cana-1005	1	8	1074	1074	NUM
cana-1005	1	9	-	-	PUNCT
cana-1005	1	10	133x	133x	NUM
cana-1005	1	11	vol	vol	NOUN
cana-1005	1	12	31	31	NUM
cana-1005	1	13	no	no	NOUN
cana-1005	1	14	.	.	PUNCT
cana-1005	2	1	5s	5s	NUM
cana-1005	2	2	(	(	PUNCT
cana-1005	2	3	2024	2024	NUM
cana-1005	2	4	)	)	PUNCT
cana-1005	2	5	110	110	NUM
cana-1005	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1005	2	7	extended	extend	VERB
cana-1005	2	8	reverse	reverse	NOUN
cana-1005	2	9	r	r	NOUN
cana-1005	2	10	degrees	degree	NOUN
cana-1005	2	11	of	of	ADP
cana-1005	2	12	vertices	vertex	NOUN
cana-1005	2	13	and	and	CCONJ
cana-1005	2	14	extended	extend	VERB
cana-1005	2	15	reverse	reverse	NOUN
cana-1005	2	16	r	r	NOUN
cana-1005	2	17	indices	index	NOUN
cana-1005	2	18	of	of	ADP
cana-1005	2	19	graphs	graph	NOUN
cana-1005	2	20	1	1	NUM
cana-1005	2	21	t.	t.	NOUN
cana-1005	2	22	lavanya	lavanya	NOUN
cana-1005	2	23	,	,	PUNCT
cana-1005	2	24	2k.a	2k.a	NUM
cana-1005	2	25	.	.	PUNCT
cana-1005	3	1	venkatesh	venkatesh	PROPN
cana-1005	3	2	,	,	PUNCT
cana-1005	3	3	3d	3d	NUM
cana-1005	3	4	.	.	PUNCT
cana-1005	4	1	amsaveni	amsaveni	PROPN
cana-1005	4	2	1,3department	1,3department	NUM
cana-1005	4	3	of	of	ADP
cana-1005	4	4	mathematics	mathematic	NOUN
cana-1005	4	5	,	,	PUNCT
cana-1005	4	6	1bharat	1bharat	NUM
cana-1005	4	7	ratna	ratna	PROPN
cana-1005	4	8	puratchi	puratchi	PROPN
cana-1005	4	9	thalaivar	thalaivar	PROPN
cana-1005	4	10	dr	dr	PROPN
cana-1005	4	11	.	.	PROPN
cana-1005	4	12	m.g.r	m.g.r	PROPN
cana-1005	4	13	.	.	PUNCT
cana-1005	5	1	government	government	NOUN
cana-1005	5	2	arts	art	NOUN
cana-1005	5	3	and	and	CCONJ
cana-1005	5	4	science	science	NOUN
cana-1005	5	5	,	,	PUNCT
cana-1005	5	6	palacode-636808	palacode-636808	VERB
cana-1005	5	7	2myanmar	2myanmar	NUM
cana-1005	5	8	institute	institute	NOUN
cana-1005	5	9	of	of	ADP
cana-1005	5	10	information	information	NOUN
cana-1005	5	11	technology	technology	PROPN
cana-1005	5	12	,	,	PUNCT
cana-1005	5	13	myanmar	myanmar	PROPN
cana-1005	5	14	.	.	PUNCT
cana-1005	6	1	3sri	3sri	NUM
cana-1005	6	2	sarada	sarada	PROPN
cana-1005	6	3	college	college	PROPN
cana-1005	6	4	for	for	ADP
cana-1005	6	5	women(autonomous	women(autonomous	PROPN
cana-1005	6	6	)	)	PUNCT
cana-1005	6	7	,	,	PUNCT
cana-1005	6	8	salem-636016	salem-636016	NOUN
cana-1005	6	9	.	.	PUNCT
cana-1005	7	1	email	email	NOUN
cana-1005	7	2	:	:	PUNCT
cana-1005	8	1	1tplaly@gmail.com	1tplaly@gmail.com	NUM
cana-1005	8	2	,	,	PUNCT
cana-1005	8	3	2prof.kavenatesh@gmail.com	2prof.kavenatesh@gmail.com	NUM
cana-1005	8	4	and	and	CCONJ
cana-1005	8	5	3d_amsaveni@rediffmail.com	3d_amsaveni@rediffmail.com	NUM
cana-1005	8	6	article	article	NOUN
cana-1005	8	7	history	history	NOUN
cana-1005	8	8	:	:	PUNCT
cana-1005	8	9	received	receive	VERB
cana-1005	8	10	:	:	PUNCT
cana-1005	8	11	18	18	NUM
cana-1005	8	12	-	-	SYM
cana-1005	8	13	05	05	NUM
cana-1005	8	14	-	-	PUNCT
cana-1005	8	15	2024	2024	NUM
cana-1005	8	16	revised	revise	VERB
cana-1005	8	17	:	:	PUNCT
cana-1005	8	18	14	14	NUM
cana-1005	8	19	-	-	SYM
cana-1005	8	20	06	06	NUM
cana-1005	8	21	-	-	PUNCT
cana-1005	8	22	2024	2024	NUM
cana-1005	8	23	accepted	accept	VERB
cana-1005	8	24	:	:	PUNCT
cana-1005	8	25	30	30	NUM
cana-1005	8	26	-	-	SYM
cana-1005	8	27	06	06	NUM
cana-1005	8	28	-	-	PUNCT
cana-1005	8	29	2024	2024	NUM
cana-1005	8	30	abstract	abstract	NOUN
cana-1005	8	31	a	a	DET
cana-1005	8	32	topological	topological	ADJ
cana-1005	8	33	representation	representation	NOUN
cana-1005	8	34	of	of	ADP
cana-1005	8	35	a	a	DET
cana-1005	8	36	molecule	molecule	NOUN
cana-1005	8	37	is	be	AUX
cana-1005	8	38	called	call	VERB
cana-1005	8	39	molecular	molecular	ADJ
cana-1005	8	40	graph	graph	NOUN
cana-1005	8	41	.	.	PUNCT
cana-1005	9	1	a	a	DET
cana-1005	9	2	molecular	molecular	ADJ
cana-1005	9	3	graph	graph	NOUN
cana-1005	9	4	is	be	AUX
cana-1005	9	5	a	a	DET
cana-1005	9	6	collection	collection	NOUN
cana-1005	9	7	of	of	ADP
cana-1005	9	8	points	point	NOUN
cana-1005	9	9	representing	represent	VERB
cana-1005	9	10	the	the	DET
cana-1005	9	11	atoms	atom	NOUN
cana-1005	9	12	in	in	ADP
cana-1005	9	13	the	the	DET
cana-1005	9	14	molecule	molecule	NOUN
cana-1005	9	15	and	and	CCONJ
cana-1005	9	16	set	set	NOUN
cana-1005	9	17	of	of	ADP
cana-1005	9	18	lines	line	NOUN
cana-1005	9	19	represent	represent	VERB
cana-1005	9	20	the	the	DET
cana-1005	9	21	covalent	covalent	ADJ
cana-1005	9	22	bonds	bond	NOUN
cana-1005	9	23	.	.	PUNCT
cana-1005	10	1	topological	topological	ADJ
cana-1005	10	2	indices	index	NOUN
cana-1005	10	3	gather	gather	VERB
cana-1005	10	4	data	datum	NOUN
cana-1005	10	5	from	from	ADP
cana-1005	10	6	the	the	DET
cana-1005	10	7	graph	graph	NOUN
cana-1005	10	8	of	of	ADP
cana-1005	10	9	molecule	molecule	NOUN
cana-1005	10	10	and	and	CCONJ
cana-1005	10	11	help	help	VERB
cana-1005	10	12	to	to	PART
cana-1005	10	13	foresee	foresee	VERB
cana-1005	10	14	properties	property	NOUN
cana-1005	10	15	of	of	ADP
cana-1005	10	16	the	the	DET
cana-1005	10	17	concealing	conceal	VERB
cana-1005	10	18	molecule	molecule	NOUN
cana-1005	10	19	.	.	PUNCT
cana-1005	11	1	all	all	DET
cana-1005	11	2	the	the	DET
cana-1005	11	3	degree	degree	NOUN
cana-1005	11	4	based	base	VERB
cana-1005	11	5	topological	topological	ADJ
cana-1005	11	6	indices	index	NOUN
cana-1005	11	7	have	have	AUX
cana-1005	11	8	been	be	AUX
cana-1005	11	9	defined	define	VERB
cana-1005	11	10	through	through	ADP
cana-1005	11	11	classical	classical	ADJ
cana-1005	11	12	degree	degree	NOUN
cana-1005	11	13	concept	concept	NOUN
cana-1005	11	14	.	.	PUNCT
cana-1005	12	1	in	in	ADP
cana-1005	12	2	this	this	DET
cana-1005	12	3	paper	paper	NOUN
cana-1005	12	4	,	,	PUNCT
cana-1005	12	5	we	we	PRON
cana-1005	12	6	define	define	VERB
cana-1005	12	7	a	a	DET
cana-1005	12	8	novel	novel	ADJ
cana-1005	12	9	degree	degree	NOUN
cana-1005	12	10	concept	concept	NOUN
cana-1005	12	11	for	for	ADP
cana-1005	12	12	a	a	DET
cana-1005	12	13	vertex	vertex	NOUN
cana-1005	12	14	of	of	ADP
cana-1005	12	15	a	a	DET
cana-1005	12	16	simple	simple	ADJ
cana-1005	12	17	connected	connected	ADJ
cana-1005	12	18	graph	graph	NOUN
cana-1005	12	19	:	:	PUNCT
cana-1005	12	20	extended	extended	ADJ
cana-1005	12	21	reverse	reverse	NOUN
cana-1005	12	22	r	r	NOUN
cana-1005	12	23	degree	degree	NOUN
cana-1005	12	24	and	and	CCONJ
cana-1005	12	25	also	also	ADV
cana-1005	12	26	,	,	PUNCT
cana-1005	12	27	we	we	PRON
cana-1005	12	28	define	define	VERB
cana-1005	12	29	extended	extended	ADJ
cana-1005	12	30	reverse	reverse	NOUN
cana-1005	12	31	r	r	NOUN
cana-1005	12	32	indices	index	NOUN
cana-1005	12	33	of	of	ADP
cana-1005	12	34	a	a	DET
cana-1005	12	35	simple	simple	ADJ
cana-1005	12	36	connected	connected	ADJ
cana-1005	12	37	graph	graph	NOUN
cana-1005	12	38	by	by	ADP
cana-1005	12	39	using	use	VERB
cana-1005	12	40	the	the	DET
cana-1005	12	41	extended	extended	ADJ
cana-1005	12	42	reverse	reverse	NOUN
cana-1005	12	43	r	r	NOUN
cana-1005	12	44	degree	degree	NOUN
cana-1005	12	45	concept	concept	NOUN
cana-1005	12	46	.	.	PUNCT
cana-1005	13	1	we	we	PRON
cana-1005	13	2	compute	compute	VERB
cana-1005	13	3	the	the	DET
cana-1005	13	4	extended	extended	ADJ
cana-1005	13	5	reverse	reverse	NOUN
cana-1005	13	6	r	r	NOUN
cana-1005	13	7	indices	index	NOUN
cana-1005	13	8	using	use	VERB
cana-1005	13	9	the	the	DET
cana-1005	13	10	above	above	ADJ
cana-1005	13	11	contemporary	contemporary	ADJ
cana-1005	13	12	degree	degree	NOUN
cana-1005	13	13	concept	concept	NOUN
cana-1005	13	14	for	for	ADP
cana-1005	13	15	well	well	ADV
cana-1005	13	16	-	-	PUNCT
cana-1005	13	17	known	know	VERB
cana-1005	13	18	simple	simple	ADJ
cana-1005	13	19	connected	connected	ADJ
cana-1005	13	20	graphs	graph	NOUN
cana-1005	13	21	such	such	ADJ
cana-1005	13	22	as	as	ADP
cana-1005	13	23	complete	complete	ADJ
cana-1005	13	24	bipartite	bipartite	NOUN
cana-1005	13	25	graph	graph	NOUN
cana-1005	13	26	,	,	PUNCT
cana-1005	13	27	wheel	wheel	NOUN
cana-1005	13	28	graph	graph	NOUN
cana-1005	13	29	,	,	PUNCT
cana-1005	13	30	generalized	generalized	ADJ
cana-1005	13	31	peterson	peterson	NOUN
cana-1005	13	32	graph	graph	NOUN
cana-1005	13	33	,	,	PUNCT
cana-1005	13	34	crown	crown	NOUN
cana-1005	13	35	graph	graph	NOUN
cana-1005	13	36	,	,	PUNCT
cana-1005	13	37	double	double	ADJ
cana-1005	13	38	star	star	NOUN
cana-1005	13	39	graph	graph	NOUN
cana-1005	13	40	,	,	PUNCT
cana-1005	13	41	and	and	CCONJ
cana-1005	13	42	windmill	windmill	NOUN
cana-1005	13	43	graph	graph	NOUN
cana-1005	13	44	.	.	PUNCT
cana-1005	14	1	keywords	keyword	NOUN
cana-1005	14	2	:	:	PUNCT
cana-1005	14	3	reverse	reverse	NOUN
cana-1005	14	4	degree	degree	NOUN
cana-1005	14	5	,	,	PUNCT
cana-1005	14	6	topological	topological	ADJ
cana-1005	14	7	indices	index	NOUN
cana-1005	14	8	,	,	PUNCT
cana-1005	14	9	extended	extend	VERB
cana-1005	14	10	reverse	reverse	NOUN
cana-1005	14	11	r	r	NOUN
cana-1005	14	12	degree	degree	NOUN
cana-1005	14	13	,	,	PUNCT
cana-1005	14	14	extended	extend	VERB
cana-1005	14	15	reverse	reverse	NOUN
cana-1005	14	16	r	r	NOUN
cana-1005	14	17	indices	index	NOUN
cana-1005	14	18	.	.	PUNCT
cana-1005	15	1	ams	am	NOUN
cana-1005	15	2	mathematics	mathematics	PROPN
cana-1005	15	3	subject	subject	ADJ
cana-1005	15	4	classification	classification	NOUN
cana-1005	15	5	(	(	PUNCT
cana-1005	15	6	2020	2020	NUM
cana-1005	15	7	):	):	PUNCT
cana-1005	15	8	05c09	05c09	NUM
cana-1005	15	9	,	,	PUNCT
cana-1005	15	10	05c07	05c07	NOUN
cana-1005	15	11	,	,	PUNCT
cana-1005	15	12	05c31,05c38	05c31,05c38	NOUN
cana-1005	15	13	.	.	PROPN
cana-1005	16	1	1	1	NUM
cana-1005	16	2	.	.	X
cana-1005	16	3	introduction	introduction	NOUN
cana-1005	16	4	a	a	DET
cana-1005	16	5	topological	topological	ADJ
cana-1005	16	6	index	index	NOUN
cana-1005	16	7	is	be	AUX
cana-1005	16	8	a	a	DET
cana-1005	16	9	mathematical	mathematical	ADJ
cana-1005	16	10	invariant	invariant	NOUN
cana-1005	16	11	that	that	PRON
cana-1005	16	12	characterize	characterize	VERB
cana-1005	16	13	the	the	DET
cana-1005	16	14	chemical	chemical	NOUN
cana-1005	16	15	properties	property	NOUN
cana-1005	16	16	of	of	ADP
cana-1005	16	17	a	a	DET
cana-1005	16	18	molecule	molecule	NOUN
cana-1005	16	19	.	.	PUNCT
cana-1005	17	1	these	these	DET
cana-1005	17	2	indices	index	NOUN
cana-1005	17	3	are	be	AUX
cana-1005	17	4	used	use	VERB
cana-1005	17	5	in	in	ADP
cana-1005	17	6	quantitative	quantitative	ADJ
cana-1005	17	7	structure	structure	NOUN
cana-1005	17	8	property	property	NOUN
cana-1005	17	9	relations	relation	NOUN
cana-1005	17	10	(	(	PUNCT
cana-1005	17	11	qspr	qspr	NOUN
cana-1005	17	12	)	)	PUNCT
cana-1005	17	13	research	research	NOUN
cana-1005	17	14	.	.	PUNCT
cana-1005	18	1	topological	topological	ADJ
cana-1005	18	2	indices	index	NOUN
cana-1005	18	3	are	be	AUX
cana-1005	18	4	important	important	ADJ
cana-1005	18	5	tools	tool	NOUN
cana-1005	18	6	for	for	ADP
cana-1005	18	7	analyzing	analyze	VERB
cana-1005	18	8	some	some	DET
cana-1005	18	9	physicochemical	physicochemical	ADJ
cana-1005	18	10	properties	property	NOUN
cana-1005	18	11	of	of	ADP
cana-1005	18	12	molecules	molecule	NOUN
cana-1005	18	13	without	without	ADP
cana-1005	18	14	performing	perform	VERB
cana-1005	18	15	any	any	DET
cana-1005	18	16	experiment	experiment	NOUN
cana-1005	18	17	.	.	PUNCT
cana-1005	19	1	the	the	DET
cana-1005	19	2	wiener	wiener	NOUN
cana-1005	19	3	index	index	NOUN
cana-1005	19	4	w(g	w(g	PROPN
cana-1005	19	5	)	)	PUNCT
cana-1005	19	6	is	be	AUX
cana-1005	19	7	a	a	DET
cana-1005	19	8	distance	distance	NOUN
cana-1005	19	9	-	-	PUNCT
cana-1005	19	10	based	base	VERB
cana-1005	19	11	topological	topological	ADJ
cana-1005	19	12	invariant	invariant	ADJ
cana-1005	19	13	much	much	ADV
cana-1005	19	14	used	use	VERB
cana-1005	19	15	in	in	ADP
cana-1005	19	16	the	the	DET
cana-1005	19	17	study	study	NOUN
cana-1005	19	18	of	of	ADP
cana-1005	19	19	the	the	DET
cana-1005	19	20	structure	structure	NOUN
cana-1005	19	21	-	-	PUNCT
cana-1005	19	22	property	property	NOUN
cana-1005	19	23	and	and	CCONJ
cana-1005	19	24	the	the	DET
cana-1005	19	25	structure	structure	NOUN
cana-1005	19	26	-	-	PUNCT
cana-1005	19	27	activity	activity	NOUN
cana-1005	19	28	relationships	relationship	NOUN
cana-1005	19	29	of	of	ADP
cana-1005	19	30	various	various	ADJ
cana-1005	19	31	classes	class	NOUN
cana-1005	19	32	of	of	ADP
cana-1005	19	33	biochemically	biochemically	ADV
cana-1005	19	34	interesting	interesting	ADJ
cana-1005	19	35	compounds	compound	NOUN
cana-1005	19	36	,	,	PUNCT
cana-1005	19	37	which	which	PRON
cana-1005	19	38	is	be	AUX
cana-1005	19	39	introduced	introduce	VERB
cana-1005	19	40	in	in	ADP
cana-1005	19	41	1947	1947	NUM
cana-1005	19	42	for	for	ADP
cana-1005	19	43	prognosticating	prognosticate	VERB
cana-1005	19	44	boiling	boiling	NOUN
cana-1005	19	45	points	point	NOUN
cana-1005	19	46	by	by	ADP
cana-1005	19	47	harold	harold	PROPN
cana-1005	19	48	wiener	wiener	NOUN
cana-1005	19	49	[	[	X
cana-1005	19	50	16	16	NUM
cana-1005	19	51	]	]	PUNCT
cana-1005	19	52	.	.	PUNCT
cana-1005	20	1	as	as	ADP
cana-1005	20	2	of	of	ADP
cana-1005	20	3	now	now	ADV
cana-1005	20	4	,	,	PUNCT
cana-1005	20	5	myriad	myriad	ADJ
cana-1005	20	6	“	"	PUNCT
cana-1005	20	7	molecular	molecular	ADJ
cana-1005	20	8	descriptors	descriptor	NOUN
cana-1005	20	9	”	"	PUNCT
cana-1005	20	10	are	be	AUX
cana-1005	20	11	being	be	AUX
cana-1005	20	12	put	put	VERB
cana-1005	20	13	forwarded	forward	VERB
cana-1005	20	14	.	.	PUNCT
cana-1005	21	1	recently	recently	ADV
cana-1005	21	2	,	,	PUNCT
cana-1005	21	3	degree	degree	NOUN
cana-1005	21	4	based	base	VERB
cana-1005	21	5	topological	topological	ADJ
cana-1005	21	6	indices	index	NOUN
cana-1005	21	7	are	be	AUX
cana-1005	21	8	also	also	ADV
cana-1005	21	9	formed	form	VERB
cana-1005	21	10	a	a	DET
cana-1005	21	11	good	good	ADJ
cana-1005	21	12	correlation	correlation	NOUN
cana-1005	21	13	with	with	ADP
cana-1005	21	14	chemical	chemical	ADJ
cana-1005	21	15	properties	property	NOUN
cana-1005	21	16	of	of	ADP
cana-1005	21	17	a	a	DET
cana-1005	21	18	molecule	molecule	NOUN
cana-1005	21	19	.	.	PUNCT
cana-1005	22	1	some	some	DET
cana-1005	22	2	well	well	ADV
cana-1005	22	3	-	-	PUNCT
cana-1005	22	4	known	know	VERB
cana-1005	22	5	degree	degree	NOUN
cana-1005	22	6	based	base	VERB
cana-1005	22	7	topological	topological	ADJ
cana-1005	22	8	indices	index	NOUN
cana-1005	22	9	are	be	AUX
cana-1005	22	10	randic	randic	ADJ
cana-1005	22	11	index	index	NOUN
cana-1005	22	12	,	,	PUNCT
cana-1005	22	13	first	first	ADJ
cana-1005	22	14	and	and	CCONJ
cana-1005	22	15	second	second	ADJ
cana-1005	22	16	zagreb	zagreb	PROPN
cana-1005	22	17	indices	index	NOUN
cana-1005	22	18	,	,	PUNCT
cana-1005	22	19	reformulated	reformulate	VERB
cana-1005	22	20	first	first	ADV
cana-1005	22	21	and	and	CCONJ
cana-1005	22	22	second	second	ADJ
cana-1005	22	23	zagreb	zagreb	PROPN
cana-1005	22	24	indices	index	NOUN
cana-1005	22	25	,	,	PUNCT
cana-1005	22	26	atom	atom	NOUN
cana-1005	22	27	-	-	PUNCT
cana-1005	22	28	bond	bond	NOUN
cana-1005	22	29	connectivity	connectivity	NOUN
cana-1005	22	30	index	index	NOUN
cana-1005	22	31	,	,	PUNCT
cana-1005	22	32	augmented	augment	VERB
cana-1005	22	33	zagreb	zagreb	PROPN
cana-1005	22	34	index	index	PROPN
cana-1005	22	35	,	,	PUNCT
cana-1005	22	36	harmonic	harmonic	ADJ
cana-1005	22	37	index	index	NOUN
cana-1005	22	38	,	,	PUNCT
cana-1005	22	39	geometric	geometric	ADJ
cana-1005	22	40	-	-	PUNCT
cana-1005	22	41	arithmetic	arithmetic	ADJ
cana-1005	22	42	index	index	NOUN
cana-1005	22	43	,	,	PUNCT
cana-1005	22	44	sum	sum	NOUN
cana-1005	22	45	connectivity	connectivity	NOUN
cana-1005	22	46	index	index	NOUN
cana-1005	22	47	are	be	AUX
cana-1005	22	48	studied	study	VERB
cana-1005	22	49	in	in	ADP
cana-1005	22	50	[	[	PUNCT
cana-1005	22	51	1	1	NUM
cana-1005	22	52	-	-	SYM
cana-1005	22	53	9	9	NUM
cana-1005	22	54	]	]	PUNCT
cana-1005	22	55	,	,	PUNCT
cana-1005	22	56	[	[	X
cana-1005	22	57	11	11	NUM
cana-1005	22	58	]	]	PUNCT
cana-1005	22	59	,	,	PUNCT
cana-1005	22	60	[	[	X
cana-1005	22	61	12	12	NUM
cana-1005	22	62	]	]	PUNCT
cana-1005	22	63	,	,	PUNCT
cana-1005	22	64	[	[	X
cana-1005	22	65	15	15	NUM
cana-1005	22	66	]	]	PUNCT
cana-1005	22	67	,	,	PUNCT
cana-1005	22	68	[	[	X
cana-1005	22	69	18	18	NUM
cana-1005	22	70	]	]	PUNCT
cana-1005	22	71	and	and	CCONJ
cana-1005	22	72	[	[	X
cana-1005	22	73	19	19	NUM
cana-1005	22	74	]	]	PUNCT
cana-1005	22	75	.	.	PUNCT
cana-1005	23	1	the	the	DET
cana-1005	23	2	comparative	comparative	ADJ
cana-1005	23	3	testing	testing	NOUN
cana-1005	23	4	of	of	ADP
cana-1005	23	5	these	these	DET
cana-1005	23	6	well	well	ADV
cana-1005	23	7	-	-	PUNCT
cana-1005	23	8	known	know	VERB
cana-1005	23	9	degree	degree	NOUN
cana-1005	23	10	based	base	VERB
cana-1005	23	11	topological	topological	ADJ
cana-1005	23	12	indices	index	NOUN
cana-1005	23	13	were	be	AUX
cana-1005	23	14	given	give	VERB
cana-1005	23	15	in	in	ADP
cana-1005	23	16	[	[	X
cana-1005	23	17	10	10	NUM
cana-1005	23	18	]	]	PUNCT
cana-1005	23	19	.	.	PUNCT
cana-1005	24	1	the	the	DET
cana-1005	24	2	concept	concept	NOUN
cana-1005	24	3	of	of	ADP
cana-1005	24	4	r	r	NOUN
cana-1005	24	5	degree	degree	NOUN
cana-1005	24	6	of	of	ADP
cana-1005	24	7	a	a	DET
cana-1005	24	8	vertex	vertex	NOUN
cana-1005	24	9	communications	communication	NOUN
cana-1005	24	10	on	on	ADP
cana-1005	24	11	applied	apply	VERB
cana-1005	24	12	nonlinear	nonlinear	ADJ
cana-1005	24	13	analysis	analysis	NOUN
cana-1005	24	14	issn	issn	NOUN
cana-1005	24	15	:	:	PUNCT
cana-1005	24	16	1074	1074	NUM
cana-1005	24	17	-	-	PUNCT
cana-1005	24	18	133x	133x	NUM
cana-1005	24	19	vol	vol	NOUN
cana-1005	24	20	31	31	NUM
cana-1005	24	21	no	no	NOUN
cana-1005	24	22	.	.	PUNCT
cana-1005	25	1	5s	5s	NUM
cana-1005	25	2	(	(	PUNCT
cana-1005	25	3	2024	2024	NUM
cana-1005	25	4	)	)	PUNCT
cana-1005	25	5	111	111	NUM
cana-1005	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1005	25	7	and	and	CCONJ
cana-1005	25	8	r	r	NOUN
cana-1005	25	9	index	index	NOUN
cana-1005	25	10	of	of	ADP
cana-1005	25	11	a	a	DET
cana-1005	25	12	graph	graph	NOUN
cana-1005	25	13	were	be	AUX
cana-1005	25	14	introduced	introduce	VERB
cana-1005	25	15	by	by	ADP
cana-1005	25	16	siileyman	siileyman	NOUN
cana-1005	25	17	ediz[17	ediz[17	PROPN
cana-1005	25	18	]	]	PUNCT
cana-1005	25	19	.	.	PUNCT
cana-1005	26	1	in	in	ADP
cana-1005	26	2	this	this	DET
cana-1005	26	3	paper	paper	NOUN
cana-1005	26	4	,	,	PUNCT
cana-1005	26	5	the	the	DET
cana-1005	26	6	extended	extended	ADJ
cana-1005	26	7	reverse	reverse	NOUN
cana-1005	26	8	ℛ	ℛ	PROPN
cana-1005	26	9	indices	index	NOUN
cana-1005	26	10	for	for	ADP
cana-1005	26	11	well	well	ADV
cana-1005	26	12	-	-	PUNCT
cana-1005	26	13	known	know	VERB
cana-1005	26	14	simple	simple	ADJ
cana-1005	26	15	connected	connected	ADJ
cana-1005	26	16	graphs	graph	NOUN
cana-1005	26	17	such	such	ADJ
cana-1005	26	18	as	as	ADP
cana-1005	26	19	complete	complete	ADJ
cana-1005	26	20	bipartite	bipartite	NOUN
cana-1005	26	21	graph	graph	NOUN
cana-1005	26	22	,	,	PUNCT
cana-1005	26	23	wheel	wheel	NOUN
cana-1005	26	24	graph	graph	NOUN
cana-1005	26	25	,	,	PUNCT
cana-1005	26	26	generalized	generalized	ADJ
cana-1005	26	27	peterson	peterson	NOUN
cana-1005	26	28	graph	graph	NOUN
cana-1005	26	29	,	,	PUNCT
cana-1005	26	30	crown	crown	NOUN
cana-1005	26	31	graph	graph	NOUN
cana-1005	26	32	,	,	PUNCT
cana-1005	26	33	double	double	ADJ
cana-1005	26	34	star	star	NOUN
cana-1005	26	35	graph	graph	NOUN
cana-1005	26	36	and	and	CCONJ
cana-1005	26	37	windmill	windmill	NOUN
cana-1005	26	38	graph	graph	NOUN
cana-1005	26	39	are	be	AUX
cana-1005	26	40	obtained	obtain	VERB
cana-1005	26	41	.	.	PUNCT
cana-1005	27	1	throughout	throughout	ADP
cana-1005	27	2	this	this	DET
cana-1005	27	3	paper	paper	NOUN
cana-1005	27	4	only	only	ADV
cana-1005	27	5	simple	simple	ADJ
cana-1005	27	6	connected	connected	ADJ
cana-1005	27	7	graphs	graph	NOUN
cana-1005	27	8	were	be	AUX
cana-1005	27	9	considered	consider	VERB
cana-1005	27	10	,	,	PUNCT
cana-1005	27	11	that	that	PRON
cana-1005	27	12	is	be	AUX
cana-1005	27	13	connected	connect	VERB
cana-1005	27	14	graphs	graph	NOUN
cana-1005	27	15	without	without	ADP
cana-1005	27	16	self	self	NOUN
cana-1005	27	17	-	-	PUNCT
cana-1005	27	18	loops	loop	NOUN
cana-1005	27	19	and	and	CCONJ
cana-1005	27	20	parallel	parallel	ADJ
cana-1005	27	21	edges	edge	NOUN
cana-1005	27	22	.	.	PUNCT
cana-1005	28	1	2	2	X
cana-1005	28	2	.	.	NUM
cana-1005	28	3	extended	extend	VERB
cana-1005	28	4	reverse	reverse	NOUN
cana-1005	28	5	r	r	NOUN
cana-1005	28	6	indices	indice	VERB
cana-1005	28	7	the	the	DET
cana-1005	28	8	graph	graph	NOUN
cana-1005	28	9	g	g	PROPN
cana-1005	28	10	=	=	SYM
cana-1005	28	11	(	(	PUNCT
cana-1005	28	12	v	v	NOUN
cana-1005	28	13	,	,	PUNCT
cana-1005	28	14	e	e	NOUN
cana-1005	28	15	)	)	PUNCT
cana-1005	28	16	=	=	SYM
cana-1005	28	17	(	(	PUNCT
cana-1005	28	18	v(g),e(g	v(g),e(g	NOUN
cana-1005	28	19	)	)	PUNCT
cana-1005	28	20	)	)	PUNCT
cana-1005	29	1	have	have	VERB
cana-1005	29	2	the	the	DET
cana-1005	29	3	set	set	NOUN
cana-1005	29	4	of	of	ADP
cana-1005	29	5	all	all	DET
cana-1005	29	6	vertices	vertex	NOUN
cana-1005	29	7	v(g	v(g	NOUN
cana-1005	29	8	)	)	PUNCT
cana-1005	29	9	and	and	CCONJ
cana-1005	29	10	the	the	DET
cana-1005	29	11	set	set	NOUN
cana-1005	29	12	of	of	ADP
cana-1005	29	13	all	all	DET
cana-1005	29	14	edges	edge	NOUN
cana-1005	29	15	e(g	e(g	PROPN
cana-1005	29	16	)	)	PUNCT
cana-1005	29	17	respectively	respectively	ADV
cana-1005	29	18	.	.	PUNCT
cana-1005	30	1	the	the	DET
cana-1005	30	2	degree	degree	NOUN
cana-1005	30	3	of	of	ADP
cana-1005	30	4	the	the	DET
cana-1005	30	5	vertex	vertex	NOUN
cana-1005	30	6	v	v	NOUN
cana-1005	30	7	is	be	AUX
cana-1005	30	8	defined	define	VERB
cana-1005	30	9	as	as	ADP
cana-1005	30	10	the	the	DET
cana-1005	30	11	number	number	NOUN
cana-1005	30	12	of	of	ADP
cana-1005	30	13	edges	edge	NOUN
cana-1005	30	14	incident	incident	NOUN
cana-1005	30	15	with	with	ADP
cana-1005	30	16	v	v	NOUN
cana-1005	30	17	and	and	CCONJ
cana-1005	30	18	denoted	denote	VERB
cana-1005	30	19	by	by	ADP
cana-1005	30	20	d(v	d(v	PROPN
cana-1005	30	21	)	)	PUNCT
cana-1005	30	22	.	.	PUNCT
cana-1005	31	1	the	the	DET
cana-1005	31	2	set	set	NOUN
cana-1005	31	3	of	of	ADP
cana-1005	31	4	all	all	DET
cana-1005	31	5	vertices	vertex	NOUN
cana-1005	31	6	which	which	PRON
cana-1005	31	7	are	be	AUX
cana-1005	31	8	adjacent	adjacent	ADJ
cana-1005	31	9	to	to	ADP
cana-1005	31	10	v	v	PROPN
cana-1005	31	11	is	be	AUX
cana-1005	31	12	called	call	VERB
cana-1005	31	13	the	the	DET
cana-1005	31	14	neighborhood	neighborhood	NOUN
cana-1005	31	15	of	of	ADP
cana-1005	31	16	v	v	NOUN
cana-1005	31	17	and	and	CCONJ
cana-1005	31	18	it	it	PRON
cana-1005	31	19	is	be	AUX
cana-1005	31	20	denoted	denote	VERB
cana-1005	31	21	by	by	ADP
cana-1005	31	22	n(v	n(v	PROPN
cana-1005	31	23	)	)	PUNCT
cana-1005	31	24	.	.	PUNCT
cana-1005	32	1	the	the	DET
cana-1005	32	2	reverse	reverse	ADJ
cana-1005	32	3	degree	degree	NOUN
cana-1005	32	4	of	of	ADP
cana-1005	32	5	a	a	DET
cana-1005	32	6	vertex	vertex	NOUN
cana-1005	32	7	v	v	NOUN
cana-1005	32	8	is	be	AUX
cana-1005	32	9	rdv	rdv	NOUN
cana-1005	32	10	=	=	SYM
cana-1005	32	11	𝛥	𝛥	PROPN
cana-1005	32	12	−	−	PROPN
cana-1005	32	13	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1005	32	14	)	)	PUNCT
cana-1005	33	1	+	+	CCONJ
cana-1005	33	2	1	1	NUM
cana-1005	33	3	where	where	SCONJ
cana-1005	33	4	𝛥	𝛥	PROPN
cana-1005	33	5	is	be	AUX
cana-1005	33	6	maximum	maximum	ADJ
cana-1005	33	7	degree	degree	NOUN
cana-1005	33	8	of	of	ADP
cana-1005	33	9	g.	g.	PROPN
cana-1005	33	10	definition:2.1	definition:2.1	VERB
cana-1005	33	11	the	the	DET
cana-1005	33	12	reverse	reverse	ADJ
cana-1005	33	13	sum	sum	NOUN
cana-1005	33	14	degree	degree	NOUN
cana-1005	33	15	of	of	ADP
cana-1005	33	16	v	v	NOUN
cana-1005	33	17	is	be	AUX
cana-1005	33	18	defined	define	VERB
cana-1005	33	19	as	as	ADP
cana-1005	33	20	𝑅𝑆𝑉	𝑅𝑆𝑉	NOUN
cana-1005	33	21	=	=	SYM
cana-1005	33	22	∑	∑	PUNCT
cana-1005	33	23	𝑅𝑑𝑢𝑢∈𝑁(𝑣	𝑅𝑑𝑢𝑢∈𝑁(𝑣	PROPN
cana-1005	33	24	)	)	PUNCT
cana-1005	33	25	and	and	CCONJ
cana-1005	33	26	the	the	DET
cana-1005	33	27	reverse	reverse	ADJ
cana-1005	33	28	multiplication	multiplication	NOUN
cana-1005	33	29	degree	degree	NOUN
cana-1005	33	30	of	of	ADP
cana-1005	33	31	a	a	DET
cana-1005	33	32	vertex	vertex	NOUN
cana-1005	33	33	v	v	NOUN
cana-1005	33	34	is	be	AUX
cana-1005	33	35	defined	define	VERB
cana-1005	33	36	as	as	ADP
cana-1005	33	37	𝑅𝑀𝑣	𝑅𝑀𝑣	NOUN
cana-1005	33	38	=	=	SYM
cana-1005	33	39	∏	∏	NUM
cana-1005	33	40	𝑅𝑑𝑢𝑢∈𝑁(𝑣	𝑅𝑑𝑢𝑢∈𝑁(𝑣	PROPN
cana-1005	33	41	)	)	PUNCT
cana-1005	33	42	.	.	PUNCT
cana-1005	34	1	definition:2.2	definition:2.2	VERB
cana-1005	34	2	the	the	DET
cana-1005	34	3	extended	extended	ADJ
cana-1005	34	4	reverse	reverse	ADJ
cana-1005	34	5	ℛ	ℛ	NOUN
cana-1005	34	6	degree	degree	NOUN
cana-1005	34	7	of	of	ADP
cana-1005	34	8	a	a	DET
cana-1005	34	9	vertex	vertex	NOUN
cana-1005	34	10	v	v	NOUN
cana-1005	34	11	of	of	ADP
cana-1005	34	12	a	a	DET
cana-1005	34	13	simple	simple	ADJ
cana-1005	34	14	connected	connected	ADJ
cana-1005	34	15	graph	graph	NOUN
cana-1005	34	16	g	g	NOUN
cana-1005	34	17	is	be	AUX
cana-1005	34	18	defined	define	VERB
cana-1005	34	19	as	as	ADP
cana-1005	34	20	𝐸𝑅𝑟(𝑣	𝐸𝑅𝑟(𝑣	NOUN
cana-1005	34	21	)	)	PUNCT
cana-1005	34	22	=	=	SYM
cana-1005	34	23	𝑅𝑆𝑉	𝑅𝑆𝑉	NOUN
cana-1005	34	24	+	+	CCONJ
cana-1005	34	25	𝑅𝑀𝑣.	𝑅𝑀𝑣.	PUNCT
cana-1005	34	26	definition:2.3	definition:2.3	NOUN
cana-1005	34	27	let	let	VERB
cana-1005	34	28	𝐺	𝐺	PROPN
cana-1005	34	29	=	=	SYM
cana-1005	34	30	(	(	PUNCT
cana-1005	34	31	𝑉	𝑉	PROPN
cana-1005	34	32	,	,	PUNCT
cana-1005	34	33	𝐸	𝐸	PROPN
cana-1005	34	34	)	)	PUNCT
cana-1005	34	35	be	be	VERB
cana-1005	34	36	a	a	DET
cana-1005	34	37	graph	graph	NOUN
cana-1005	34	38	.	.	PUNCT
cana-1005	35	1	(	(	PUNCT
cana-1005	35	2	a	a	X
cana-1005	35	3	)	)	PUNCT
cana-1005	35	4	the	the	DET
cana-1005	35	5	extended	extended	ADJ
cana-1005	35	6	reverse	reverse	ADJ
cana-1005	35	7	first	first	ADJ
cana-1005	35	8	index	index	NOUN
cana-1005	35	9	of	of	ADP
cana-1005	35	10	a	a	DET
cana-1005	35	11	simple	simple	ADJ
cana-1005	35	12	connected	connected	ADJ
cana-1005	35	13	graph	graph	NOUN
cana-1005	35	14	g	g	PROPN
cana-1005	35	15	defined	define	VERB
cana-1005	35	16	as	as	ADP
cana-1005	35	17	𝐸𝑅𝑟	𝐸𝑅𝑟	NOUN
cana-1005	35	18	1(𝐺	1(𝐺	NOUN
cana-1005	35	19	)	)	PUNCT
cana-1005	35	20	=	=	PUNCT
cana-1005	36	1	∑	∑	PUNCT
cana-1005	37	1	[	[	X
cana-1005	37	2	𝐸𝑅𝑟(𝑣)]2	𝐸𝑅𝑟(𝑣)]2	NOUN
cana-1005	37	3	𝑣∈𝑉(𝐺	𝑣∈𝑉(𝐺	NOUN
cana-1005	37	4	)	)	PUNCT
cana-1005	37	5	.	.	PUNCT
cana-1005	38	1	(	(	PUNCT
cana-1005	38	2	b)the	b)the	DET
cana-1005	38	3	extended	extended	ADJ
cana-1005	38	4	reverse	reverse	ADJ
cana-1005	38	5	second	second	ADJ
cana-1005	38	6	index	index	NOUN
cana-1005	38	7	of	of	ADP
cana-1005	38	8	a	a	DET
cana-1005	38	9	simple	simple	ADJ
cana-1005	38	10	connected	connected	ADJ
cana-1005	38	11	graph	graph	NOUN
cana-1005	38	12	g	g	PROPN
cana-1005	38	13	defined	define	VERB
cana-1005	38	14	as	as	ADP
cana-1005	38	15	𝐸𝑅𝑟	𝐸𝑅𝑟	NOUN
cana-1005	38	16	2(𝐺	2(𝐺	NOUN
cana-1005	38	17	)	)	PUNCT
cana-1005	38	18	=	=	PUNCT
cana-1005	38	19	∑	∑	PUNCT
cana-1005	38	20	[	[	X
cana-1005	38	21	𝐸𝑅𝑟(𝑢)𝐸𝑅𝑟(𝑣)]<𝑢𝑣>∈𝐸(𝐺	𝐸𝑅𝑟(𝑢)𝐸𝑅𝑟(𝑣)]<𝑢𝑣>∈𝐸(𝐺	X
cana-1005	38	22	)	)	PUNCT
cana-1005	38	23	.	.	PUNCT
cana-1005	39	1	(	(	PUNCT
cana-1005	39	2	b	b	X
cana-1005	39	3	)	)	PUNCT
cana-1005	39	4	the	the	DET
cana-1005	39	5	extended	extend	VERB
cana-1005	39	6	reverse	reverse	ADJ
cana-1005	39	7	third	third	ADJ
cana-1005	39	8	index	index	NOUN
cana-1005	39	9	of	of	ADP
cana-1005	39	10	a	a	DET
cana-1005	39	11	simple	simple	ADJ
cana-1005	39	12	connected	connected	ADJ
cana-1005	39	13	graph	graph	NOUN
cana-1005	39	14	g	g	PROPN
cana-1005	39	15	defined	define	VERB
cana-1005	39	16	as	as	ADP
cana-1005	39	17	𝐸𝑅𝑟	𝐸𝑅𝑟	NOUN
cana-1005	39	18	3(𝐺	3(𝐺	NUM
cana-1005	39	19	)	)	PUNCT
cana-1005	39	20	=	=	PUNCT
cana-1005	40	1	∑	∑	PUNCT
cana-1005	41	1	[	[	X
cana-1005	41	2	𝐸𝑅𝑟(𝑢	𝐸𝑅𝑟(𝑢	X
cana-1005	41	3	)	)	PUNCT
cana-1005	42	1	+	+	PUNCT
cana-1005	43	1	𝐸𝑅𝑟(𝑣)]<𝑢𝑣>∈𝐸(𝐺	𝐸𝑅𝑟(𝑣)]<𝑢𝑣>∈𝐸(𝐺	NOUN
cana-1005	43	2	)	)	PUNCT
cana-1005	43	3	.	.	PUNCT
cana-1005	44	1	hence	hence	ADV
cana-1005	44	2	,	,	PUNCT
cana-1005	44	3	the	the	DET
cana-1005	44	4	extended	extended	ADJ
cana-1005	44	5	reverse	reverse	NOUN
cana-1005	44	6	ℛ	ℛ	PROPN
cana-1005	44	7	indices	index	NOUN
cana-1005	44	8	are	be	AUX
cana-1005	44	9	topological	topological	ADJ
cana-1005	44	10	indices	index	NOUN
cana-1005	44	11	.	.	PUNCT
cana-1005	45	1	3	3	X
cana-1005	45	2	.	.	NUM
cana-1005	45	3	extended	extend	VERB
cana-1005	45	4	reverse	reverse	ADJ
cana-1005	45	5	ℛ	ℛ	PROPN
cana-1005	45	6	indices	index	NOUN
cana-1005	45	7	of	of	ADP
cana-1005	45	8	some	some	DET
cana-1005	45	9	graphs	graph	NOUN
cana-1005	45	10	in	in	ADP
cana-1005	45	11	this	this	DET
cana-1005	45	12	section	section	NOUN
cana-1005	45	13	,	,	PUNCT
cana-1005	45	14	the	the	DET
cana-1005	45	15	complete	complete	ADJ
cana-1005	45	16	bipartite	bipartite	NOUN
cana-1005	45	17	graph	graph	NOUN
cana-1005	45	18	,	,	PUNCT
cana-1005	45	19	wheel	wheel	NOUN
cana-1005	45	20	graph	graph	NOUN
cana-1005	45	21	,	,	PUNCT
cana-1005	45	22	generalized	generalized	ADJ
cana-1005	45	23	peterson	peterson	NOUN
cana-1005	45	24	graph	graph	NOUN
cana-1005	45	25	,	,	PUNCT
cana-1005	45	26	crown	crown	NOUN
cana-1005	45	27	graph	graph	NOUN
cana-1005	45	28	,	,	PUNCT
cana-1005	45	29	double	double	ADJ
cana-1005	45	30	star	star	NOUN
cana-1005	45	31	graph	graph	NOUN
cana-1005	45	32	and	and	CCONJ
cana-1005	45	33	windmill	windmill	NOUN
cana-1005	45	34	graph	graph	NOUN
cana-1005	45	35	are	be	AUX
cana-1005	45	36	characterized	characterize	VERB
cana-1005	45	37	using	use	VERB
cana-1005	45	38	the	the	DET
cana-1005	45	39	extended	extended	ADJ
cana-1005	45	40	reverse	reverse	ADJ
cana-1005	45	41	ℛ	ℛ	PROPN
cana-1005	45	42	indices	index	NOUN
cana-1005	45	43	.	.	PUNCT
cana-1005	46	1	theorem	theorem	VERB
cana-1005	46	2	3.1	3.1	NUM
cana-1005	46	3	if	if	SCONJ
cana-1005	46	4	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	46	5	is	be	AUX
cana-1005	46	6	the	the	DET
cana-1005	46	7	complete	complete	ADJ
cana-1005	46	8	bipartite	bipartite	NOUN
cana-1005	46	9	graph	graph	NOUN
cana-1005	46	10	with	with	ADP
cana-1005	46	11	n+m	n+m	NUM
cana-1005	46	12	vertices	vertex	NOUN
cana-1005	46	13	,	,	PUNCT
cana-1005	46	14	n	n	CCONJ
cana-1005	46	15	>	>	X
cana-1005	46	16	m	m	PROPN
cana-1005	46	17	and	and	CCONJ
cana-1005	46	18	mn	mn	PROPN
cana-1005	46	19	edges	edge	NOUN
cana-1005	46	20	then	then	ADV
cana-1005	46	21	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	46	22	1	1	NUM
cana-1005	46	23	(	(	PUNCT
cana-1005	46	24	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	46	25	)	)	PUNCT
cana-1005	46	26	=	=	PUNCT
cana-1005	47	1	𝑛[𝑚	𝑛[𝑚	VERB
cana-1005	47	2	+	+	NUM
cana-1005	47	3	1]2	1]2	NUM
cana-1005	47	4	+	+	CCONJ
cana-1005	47	5	𝑚[(𝑛	𝑚[(𝑛	ADJ
cana-1005	47	6	−	−	PROPN
cana-1005	47	7	𝑚	𝑚	NOUN
cana-1005	47	8	+	+	NOUN
cana-1005	47	9	1)(𝑛	1)(𝑛	NUM
cana-1005	47	10	+	+	CCONJ
cana-1005	47	11	(	(	PUNCT
cana-1005	47	12	𝑛	𝑛	PRON
cana-1005	47	13	−	−	PROPN
cana-1005	47	14	𝑚	𝑚	PROPN
cana-1005	48	1	+	+	NUM
cana-1005	48	2	1)𝑛−1)]2	1)𝑛−1)]2	NUM
cana-1005	48	3	,	,	PUNCT
cana-1005	48	4	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	48	5	2	2	NUM
cana-1005	48	6	(	(	PUNCT
cana-1005	48	7	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	48	8	)	)	PUNCT
cana-1005	48	9	=	=	PUNCT
cana-1005	49	1	𝑚𝑛{[𝑚	𝑚𝑛{[𝑚	NOUN
cana-1005	49	2	+	+	NOUN
cana-1005	49	3	1][(𝑛	1][(𝑛	NUM
cana-1005	49	4	−	−	NOUN
cana-1005	49	5	𝑚	𝑚	PROPN
cana-1005	50	1	+	+	NOUN
cana-1005	50	2	1)(𝑛	1)(𝑛	NUM
cana-1005	50	3	+	+	CCONJ
cana-1005	50	4	(	(	PUNCT
cana-1005	50	5	𝑛	𝑛	PRON
cana-1005	50	6	−	−	PROPN
cana-1005	50	7	𝑚	𝑚	NOUN
cana-1005	50	8	+	+	NOUN
cana-1005	50	9	1)𝑛−1	1)𝑛−1	NUM
cana-1005	50	10	)	)	PUNCT
cana-1005	50	11	]	]	PUNCT
cana-1005	50	12	}	}	PUNCT
cana-1005	50	13	,	,	PUNCT
cana-1005	50	14	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	50	15	3	3	NUM
cana-1005	50	16	(	(	PUNCT
cana-1005	50	17	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	50	18	)	)	PUNCT
cana-1005	50	19	=	=	PUNCT
cana-1005	51	1	𝑚𝑛{[𝑚	𝑚𝑛{[𝑚	NOUN
cana-1005	51	2	+	+	NOUN
cana-1005	51	3	1	1	NUM
cana-1005	51	4	]	]	PUNCT
cana-1005	51	5	+	+	CCONJ
cana-1005	52	1	[	[	X
cana-1005	52	2	(	(	PUNCT
cana-1005	52	3	𝑛	𝑛	PRON
cana-1005	52	4	−	−	PROPN
cana-1005	52	5	𝑚	𝑚	PROPN
cana-1005	52	6	+	+	NOUN
cana-1005	52	7	1)(𝑛	1)(𝑛	NUM
cana-1005	52	8	+	+	CCONJ
cana-1005	52	9	(	(	PUNCT
cana-1005	52	10	𝑛	𝑛	DET
cana-1005	52	11	−	−	PROPN
cana-1005	52	12	𝑚	𝑚	NOUN
cana-1005	52	13	+	+	NOUN
cana-1005	52	14	1)𝑛−1	1)𝑛−1	NUM
cana-1005	52	15	)	)	PUNCT
cana-1005	52	16	]	]	PUNCT
cana-1005	52	17	}	}	PUNCT
cana-1005	52	18	.	.	PUNCT
cana-1005	53	1	communications	communication	NOUN
cana-1005	53	2	on	on	ADP
cana-1005	53	3	applied	apply	VERB
cana-1005	53	4	nonlinear	nonlinear	ADJ
cana-1005	53	5	analysis	analysis	NOUN
cana-1005	53	6	issn	issn	NOUN
cana-1005	53	7	:	:	PUNCT
cana-1005	53	8	1074	1074	NUM
cana-1005	53	9	-	-	PUNCT
cana-1005	53	10	133x	133x	NUM
cana-1005	53	11	vol	vol	NOUN
cana-1005	53	12	31	31	NUM
cana-1005	53	13	no	no	NOUN
cana-1005	53	14	.	.	PUNCT
cana-1005	54	1	5s	5s	NUM
cana-1005	54	2	(	(	PUNCT
cana-1005	54	3	2024	2024	NUM
cana-1005	54	4	)	)	PUNCT
cana-1005	54	5	112	112	NUM
cana-1005	54	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1005	54	7	proof	proof	NOUN
cana-1005	54	8	:	:	PUNCT
cana-1005	54	9	let	let	VERB
cana-1005	54	10	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	54	11	be	be	AUX
cana-1005	54	12	a	a	DET
cana-1005	54	13	complete	complete	ADJ
cana-1005	54	14	bipartite	bipartite	NOUN
cana-1005	54	15	graph	graph	NOUN
cana-1005	54	16	(	(	PUNCT
cana-1005	54	17	𝑉𝑖	𝑉𝑖	PROPN
cana-1005	54	18	,	,	PUNCT
cana-1005	54	19	𝑉𝑗	𝑉𝑗	PROPN
cana-1005	54	20	,	,	PUNCT
cana-1005	54	21	𝐸	𝐸	PROPN
cana-1005	54	22	)	)	PUNCT
cana-1005	54	23	.	.	PUNCT
cana-1005	55	1	the	the	DET
cana-1005	55	2	{	{	PUNCT
cana-1005	55	3	𝑢1	𝑢1	PROPN
cana-1005	55	4	,	,	PUNCT
cana-1005	55	5	𝑢2	𝑢2	PROPN
cana-1005	55	6	,	,	PUNCT
cana-1005	55	7	…	…	PUNCT
cana-1005	55	8	,	,	PUNCT
cana-1005	55	9	𝑢𝑚	𝑢𝑚	ADP
cana-1005	55	10	}	}	PUNCT
cana-1005	55	11	⊂	⊂	PROPN
cana-1005	55	12	𝑉𝑖	𝑉𝑖	PROPN
cana-1005	55	13	and	and	CCONJ
cana-1005	55	14	{	{	PUNCT
cana-1005	55	15	𝑣1	𝑣1	PROPN
cana-1005	55	16	,	,	PUNCT
cana-1005	55	17	𝑣2	𝑣2	PROPN
cana-1005	55	18	,	,	PUNCT
cana-1005	55	19	…	…	PUNCT
cana-1005	55	20	,	,	PUNCT
cana-1005	55	21	𝑣𝑛	𝑣𝑛	NOUN
cana-1005	55	22	}	}	PUNCT
cana-1005	55	23	⊂	⊂	PROPN
cana-1005	56	1	𝑉𝑗	𝑉𝑗	PROPN
cana-1005	56	2	are	be	AUX
cana-1005	56	3	any	any	DET
cana-1005	56	4	two	two	NUM
cana-1005	56	5	sets	set	NOUN
cana-1005	56	6	of	of	ADP
cana-1005	56	7	vertices	vertex	NOUN
cana-1005	56	8	.	.	PUNCT
cana-1005	57	1	here	here	ADV
cana-1005	57	2	,	,	PUNCT
cana-1005	57	3	(	(	PUNCT
cana-1005	57	4	𝑢𝑖,𝑢𝑗	𝑢𝑖,𝑢𝑗	NOUN
cana-1005	57	5	)	)	PUNCT
cana-1005	57	6	,	,	PUNCT
cana-1005	57	7	𝑖	𝑖	X
cana-1005	58	1	=	=	NOUN
cana-1005	58	2	1,2,3	1,2,3	NUM
cana-1005	58	3	,	,	PUNCT
cana-1005	58	4	…	…	PUNCT
cana-1005	58	5	,	,	PUNCT
cana-1005	58	6	𝑚	𝑚	NOUN
cana-1005	58	7	,	,	PUNCT
cana-1005	58	8	𝑗	𝑗	NOUN
cana-1005	58	9	=	=	NOUN
cana-1005	58	10	1,2,3	1,2,3	NUM
cana-1005	58	11	,	,	PUNCT
cana-1005	58	12	…	…	PUNCT
cana-1005	58	13	,	,	PUNCT
cana-1005	58	14	𝑛	𝑛	PROPN
cana-1005	58	15	are	be	AUX
cana-1005	58	16	edges	edge	NOUN
cana-1005	58	17	in	in	ADP
cana-1005	58	18	e.	e.	PROPN
cana-1005	59	1	a	a	DET
cana-1005	59	2	complete	complete	ADJ
cana-1005	59	3	bipartite	bipartite	NOUN
cana-1005	59	4	graph	graph	NOUN
cana-1005	59	5	with	with	ADP
cana-1005	59	6	partitions	partition	NOUN
cana-1005	59	7	of	of	ADP
cana-1005	59	8	size	size	NOUN
cana-1005	59	9	|𝑉𝑖|	|𝑉𝑖|	PROPN
cana-1005	59	10	=	=	SYM
cana-1005	59	11	𝑚	𝑚	PROPN
cana-1005	59	12	,	,	PUNCT
cana-1005	59	13	|𝑉𝑗|	|𝑉𝑗|	PROPN
cana-1005	59	14	=	=	SYM
cana-1005	59	15	𝑛	𝑛	NOUN
cana-1005	59	16	,	,	PUNCT
cana-1005	59	17	|𝑉(𝐾𝑚,𝑛)|	|𝑉(𝐾𝑚,𝑛)|	X
cana-1005	59	18	=	=	PUNCT
cana-1005	59	19	𝑚	𝑚	PROPN
cana-1005	59	20	+	+	NOUN
cana-1005	59	21	𝑛	𝑛	NOUN
cana-1005	59	22	and	and	CCONJ
cana-1005	59	23	|𝐸(𝐾𝑚,𝑛)|	|𝐸(𝐾𝑚,𝑛)|	ADV
cana-1005	60	1	=	=	SYM
cana-1005	60	2	𝑚𝑛	𝑚𝑛	NOUN
cana-1005	60	3	then	then	ADV
cana-1005	60	4	the	the	DET
cana-1005	60	5	reverse	reverse	ADJ
cana-1005	60	6	vertex	vertex	NOUN
cana-1005	60	7	degree	degree	NOUN
cana-1005	60	8	ℜ𝑑𝑢𝑖	ℜ𝑑𝑢𝑖	PROPN
cana-1005	60	9	=	=	SYM
cana-1005	60	10	1	1	NUM
cana-1005	60	11	,	,	PUNCT
cana-1005	60	12	ℜ𝑑𝑣𝑗	ℜ𝑑𝑣𝑗	PROPN
cana-1005	60	13	=	=	PUNCT
cana-1005	60	14	𝑛	𝑛	DET
cana-1005	60	15	−	−	NOUN
cana-1005	60	16	𝑚	𝑚	PROPN
cana-1005	60	17	+	+	PROPN
cana-1005	60	18	1	1	NUM
cana-1005	60	19	,	,	PUNCT
cana-1005	60	20	the	the	DET
cana-1005	60	21	reverse	reverse	ADJ
cana-1005	60	22	sum	sum	NOUN
cana-1005	60	23	degree	degree	NOUN
cana-1005	60	24	ℜ𝑆𝑢𝑖	ℜ𝑆𝑢𝑖	NOUN
cana-1005	60	25	=	=	PUNCT
cana-1005	61	1	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1005	61	2	−	−	NOUN
cana-1005	61	3	𝑚	𝑚	NOUN
cana-1005	61	4	+	+	PROPN
cana-1005	61	5	1	1	NUM
cana-1005	61	6	)	)	PUNCT
cana-1005	61	7	,	,	PUNCT
cana-1005	61	8	ℜ𝑆𝑣𝑗	ℜ𝑆𝑣𝑗	PROPN
cana-1005	61	9	=	=	PUNCT
cana-1005	61	10	𝑚	𝑚	PROPN
cana-1005	61	11	and	and	CCONJ
cana-1005	61	12	the	the	DET
cana-1005	61	13	reverse	reverse	ADJ
cana-1005	61	14	multiplication	multiplication	NOUN
cana-1005	61	15	degree	degree	NOUN
cana-1005	61	16	ℜ𝑀𝑢𝑖	ℜ𝑀𝑢𝑖	PROPN
cana-1005	61	17	=	=	PUNCT
cana-1005	61	18	(	(	PUNCT
cana-1005	61	19	𝑛	𝑛	PRON
cana-1005	61	20	−	−	NOUN
cana-1005	61	21	𝑚	𝑚	PROPN
cana-1005	61	22	+	+	SYM
cana-1005	61	23	1)𝑛	1)𝑛	NUM
cana-1005	61	24	,	,	PUNCT
cana-1005	61	25	ℜ𝑀𝑣𝑗	ℜ𝑀𝑣𝑗	PROPN
cana-1005	61	26	=	=	SYM
cana-1005	62	1	1	1	X
cana-1005	62	2	.	.	PUNCT
cana-1005	62	3	then	then	ADV
cana-1005	62	4	the	the	DET
cana-1005	62	5	extended	extended	ADJ
cana-1005	62	6	reverse	reverse	ADJ
cana-1005	62	7	ℛ	ℛ	PROPN
cana-1005	62	8	degrees	degree	NOUN
cana-1005	62	9	are	be	AUX
cana-1005	62	10	𝔼ℜℛ𝑢𝑖	𝔼ℜℛ𝑢𝑖	NOUN
cana-1005	62	11	1	1	NUM
cana-1005	62	12	=	=	SYM
cana-1005	62	13	(	(	PUNCT
cana-1005	62	14	𝑛	𝑛	PRON
cana-1005	62	15	−	−	NOUN
cana-1005	62	16	𝑚	𝑚	NOUN
cana-1005	62	17	+	+	CCONJ
cana-1005	62	18	1)[(𝑛	1)[(𝑛	NUM
cana-1005	62	19	+	+	CCONJ
cana-1005	62	20	(	(	PUNCT
cana-1005	62	21	𝑛	𝑛	PRON
cana-1005	62	22	−	−	PROPN
cana-1005	62	23	𝑚	𝑚	PROPN
cana-1005	63	1	+	+	NOUN
cana-1005	63	2	1)𝑛−1)]2	1)𝑛−1)]2	NUM
cana-1005	63	3	,	,	PUNCT
cana-1005	63	4	𝔼ℜℛ𝑣𝑗	𝔼ℜℛ𝑣𝑗	PROPN
cana-1005	63	5	1	1	NUM
cana-1005	63	6	=	=	SYM
cana-1005	63	7	𝑚	𝑚	PROPN
cana-1005	63	8	+	+	NOUN
cana-1005	63	9	1	1	NUM
cana-1005	63	10	.	.	PUNCT
cana-1005	64	1	hence	hence	ADV
cana-1005	64	2	,	,	PUNCT
cana-1005	64	3	the	the	DET
cana-1005	64	4	extended	extended	ADJ
cana-1005	64	5	reverse	reverse	ADJ
cana-1005	64	6	ℛ	ℛ	PROPN
cana-1005	64	7	topological	topological	ADJ
cana-1005	64	8	indices	index	NOUN
cana-1005	64	9	of	of	ADP
cana-1005	64	10	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	64	11	are	be	AUX
cana-1005	64	12	𝔼ℜℛ	𝔼ℜℛ	SYM
cana-1005	64	13	1	1	NUM
cana-1005	64	14	(	(	PUNCT
cana-1005	64	15	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	64	16	)	)	PUNCT
cana-1005	64	17	=	=	PUNCT
cana-1005	65	1	𝑛[𝑚	𝑛[𝑚	VERB
cana-1005	65	2	+	+	NUM
cana-1005	65	3	1]2	1]2	NUM
cana-1005	65	4	+	+	CCONJ
cana-1005	65	5	𝑚[(𝑛	𝑚[(𝑛	ADJ
cana-1005	65	6	−	−	PROPN
cana-1005	65	7	𝑚	𝑚	NOUN
cana-1005	65	8	+	+	NOUN
cana-1005	65	9	1)(𝑛	1)(𝑛	NUM
cana-1005	65	10	+	+	CCONJ
cana-1005	65	11	(	(	PUNCT
cana-1005	65	12	𝑛	𝑛	PRON
cana-1005	65	13	−	−	PROPN
cana-1005	65	14	𝑚	𝑚	PROPN
cana-1005	66	1	+	+	NUM
cana-1005	66	2	1)𝑛−1)]2	1)𝑛−1)]2	NUM
cana-1005	66	3	,	,	PUNCT
cana-1005	66	4	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	66	5	2	2	NUM
cana-1005	66	6	(	(	PUNCT
cana-1005	66	7	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	66	8	)	)	PUNCT
cana-1005	66	9	=	=	PUNCT
cana-1005	67	1	𝑚𝑛{[𝑚	𝑚𝑛{[𝑚	NOUN
cana-1005	67	2	+	+	NOUN
cana-1005	67	3	1][(𝑛	1][(𝑛	NUM
cana-1005	67	4	−	−	NOUN
cana-1005	67	5	𝑚	𝑚	PROPN
cana-1005	68	1	+	+	NOUN
cana-1005	68	2	1)(𝑛	1)(𝑛	NUM
cana-1005	68	3	+	+	CCONJ
cana-1005	68	4	(	(	PUNCT
cana-1005	68	5	𝑛	𝑛	PRON
cana-1005	68	6	−	−	PROPN
cana-1005	68	7	𝑚	𝑚	NOUN
cana-1005	68	8	+	+	NOUN
cana-1005	68	9	1)𝑛−1	1)𝑛−1	NUM
cana-1005	68	10	)	)	PUNCT
cana-1005	68	11	]	]	PUNCT
cana-1005	68	12	}	}	PUNCT
cana-1005	68	13	,	,	PUNCT
cana-1005	68	14	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	68	15	3	3	NUM
cana-1005	68	16	(	(	PUNCT
cana-1005	68	17	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-1005	68	18	)	)	PUNCT
cana-1005	68	19	=	=	PUNCT
cana-1005	69	1	𝑚𝑛{[𝑚	𝑚𝑛{[𝑚	NOUN
cana-1005	69	2	+	+	NOUN
cana-1005	69	3	1	1	NUM
cana-1005	69	4	]	]	PUNCT
cana-1005	69	5	+	+	CCONJ
cana-1005	70	1	[	[	X
cana-1005	70	2	(	(	PUNCT
cana-1005	70	3	𝑛	𝑛	PRON
cana-1005	70	4	−	−	PROPN
cana-1005	70	5	𝑚	𝑚	PROPN
cana-1005	70	6	+	+	NOUN
cana-1005	70	7	1)(𝑛	1)(𝑛	NUM
cana-1005	70	8	+	+	CCONJ
cana-1005	70	9	(	(	PUNCT
cana-1005	70	10	𝑛	𝑛	DET
cana-1005	70	11	−	−	PROPN
cana-1005	70	12	𝑚	𝑚	NOUN
cana-1005	70	13	+	+	NOUN
cana-1005	70	14	1)𝑛−1	1)𝑛−1	NUM
cana-1005	70	15	)	)	PUNCT
cana-1005	70	16	]	]	PUNCT
cana-1005	70	17	}	}	PUNCT
cana-1005	70	18	.	.	PUNCT
cana-1005	71	1	theorem	theorem	ADJ
cana-1005	71	2	3.2	3.2	NUM
cana-1005	71	3	if	if	SCONJ
cana-1005	71	4	(	(	PUNCT
cana-1005	71	5	𝑊𝑛	𝑊𝑛	NOUN
cana-1005	71	6	)	)	PUNCT
cana-1005	71	7	is	be	AUX
cana-1005	71	8	the	the	DET
cana-1005	71	9	wheel	wheel	NOUN
cana-1005	71	10	graph	graph	NOUN
cana-1005	71	11	with	with	ADP
cana-1005	71	12	n	n	ADP
cana-1005	71	13	vertices	vertex	NOUN
cana-1005	71	14	n	n	PRON
cana-1005	71	15	≥	≥	NOUN
cana-1005	71	16	4	4	NUM
cana-1005	71	17	then	then	ADV
cana-1005	71	18	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	71	19	1	1	NUM
cana-1005	71	20	(	(	PUNCT
cana-1005	71	21	𝑊𝑛	𝑊𝑛	PROPN
cana-1005	71	22	)	)	PUNCT
cana-1005	71	23	=	=	PUNCT
cana-1005	72	1	[	[	X
cana-1005	72	2	(	(	PUNCT
cana-1005	72	3	𝑛	𝑛	PRON
cana-1005	72	4	−	−	NOUN
cana-1005	72	5	1)(𝑛	1)(𝑛	NUM
cana-1005	72	6	−	−	NOUN
cana-1005	72	7	3	3	NUM
cana-1005	72	8	)	)	PUNCT
cana-1005	72	9	+	+	CCONJ
cana-1005	72	10	(	(	PUNCT
cana-1005	72	11	𝑛	𝑛	DET
cana-1005	72	12	−	−	PROPN
cana-1005	72	13	3)𝑛−1]2	3)𝑛−1]2	PROPN
cana-1005	72	14	+	+	CCONJ
cana-1005	72	15	(	(	PUNCT
cana-1005	72	16	𝑛	𝑛	PRON
cana-1005	72	17	−	−	NOUN
cana-1005	72	18	1)(𝑛	1)(𝑛	NUM
cana-1005	72	19	−	−	PROPN
cana-1005	72	20	2)4	2)4	NUM
cana-1005	72	21	,	,	PUNCT
cana-1005	72	22	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	72	23	2	2	NUM
cana-1005	72	24	(	(	PUNCT
cana-1005	72	25	𝑊𝑛	𝑊𝑛	PROPN
cana-1005	72	26	)	)	PUNCT
cana-1005	72	27	=	=	SYM
cana-1005	72	28	(	(	PUNCT
cana-1005	72	29	𝑛	𝑛	PRON
cana-1005	72	30	−	−	NUM
cana-1005	72	31	1)[{(𝑛	1)[{(𝑛	NUM
cana-1005	72	32	−	−	NOUN
cana-1005	72	33	1)(𝑛	1)(𝑛	NUM
cana-1005	72	34	−	−	NOUN
cana-1005	72	35	3	3	NUM
cana-1005	72	36	)	)	PUNCT
cana-1005	72	37	+	+	CCONJ
cana-1005	72	38	(	(	PUNCT
cana-1005	72	39	𝑛	𝑛	PRON
cana-1005	72	40	−	−	PROPN
cana-1005	72	41	3)𝑛−1][𝑛	3)𝑛−1][𝑛	NUM
cana-1005	72	42	−	−	PROPN
cana-1005	72	43	2]2	2]2	NUM
cana-1005	72	44	}	}	PUNCT
cana-1005	72	45	+	+	CCONJ
cana-1005	72	46	(	(	PUNCT
cana-1005	72	47	𝑛	𝑛	DET
cana-1005	72	48	−	−	PROPN
cana-1005	72	49	2)4	2)4	NUM
cana-1005	72	50	,	,	PUNCT
cana-1005	72	51	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	72	52	3	3	NUM
cana-1005	72	53	(	(	PUNCT
cana-1005	72	54	𝑊𝑛	𝑊𝑛	PROPN
cana-1005	72	55	)	)	PUNCT
cana-1005	72	56	=	=	SYM
cana-1005	72	57	(	(	PUNCT
cana-1005	72	58	𝑛	𝑛	PRON
cana-1005	72	59	−	−	NOUN
cana-1005	72	60	1)(𝑛	1)(𝑛	NUM
cana-1005	72	61	−	−	PROPN
cana-1005	72	62	3)𝑛−1	3)𝑛−1	NUM
cana-1005	72	63	+	+	NUM
cana-1005	72	64	4𝑛3	4𝑛3	NUM
cana-1005	72	65	−	−	NOUN
cana-1005	73	1	20𝑛2	20𝑛2	NUM
cana-1005	73	2	+	+	NUM
cana-1005	73	3	31𝑛	31𝑛	NOUN
cana-1005	73	4	−	−	PROPN
cana-1005	73	5	15	15	NUM
cana-1005	73	6	.	.	PUNCT
cana-1005	74	1	communications	communication	NOUN
cana-1005	74	2	on	on	ADP
cana-1005	74	3	applied	apply	VERB
cana-1005	74	4	nonlinear	nonlinear	ADJ
cana-1005	74	5	analysis	analysis	NOUN
cana-1005	74	6	issn	issn	NOUN
cana-1005	74	7	:	:	PUNCT
cana-1005	74	8	1074	1074	NUM
cana-1005	74	9	-	-	PUNCT
cana-1005	74	10	133x	133x	NUM
cana-1005	74	11	vol	vol	NOUN
cana-1005	74	12	31	31	NUM
cana-1005	74	13	no	no	NOUN
cana-1005	74	14	.	.	PUNCT
cana-1005	75	1	5s	5s	NUM
cana-1005	75	2	(	(	PUNCT
cana-1005	75	3	2024	2024	NUM
cana-1005	75	4	)	)	PUNCT
cana-1005	75	5	113	113	NUM
cana-1005	75	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1005	75	7	proof	proof	NOUN
cana-1005	75	8	:	:	PUNCT
cana-1005	75	9	a	a	DET
cana-1005	75	10	wheel	wheel	NOUN
cana-1005	75	11	graph	graph	NOUN
cana-1005	75	12	wn	wn	NOUN
cana-1005	75	13	with	with	ADP
cana-1005	75	14	n	n	ADP
cana-1005	75	15	vertices	vertex	NOUN
cana-1005	75	16	and	and	CCONJ
cana-1005	75	17	2n-2	2n-2	PRON
cana-1005	75	18	edges	edge	NOUN
cana-1005	75	19	is	be	AUX
cana-1005	75	20	obtained	obtain	VERB
cana-1005	75	21	by	by	ADP
cana-1005	75	22	connecting	connect	VERB
cana-1005	75	23	a	a	DET
cana-1005	75	24	single	single	ADJ
cana-1005	75	25	vertex	vertex	NOUN
cana-1005	75	26	to	to	ADP
cana-1005	75	27	a	a	DET
cana-1005	75	28	vertices	vertex	NOUN
cana-1005	75	29	of	of	ADP
cana-1005	75	30	a	a	DET
cana-1005	75	31	cycle	cycle	NOUN
cana-1005	75	32	of	of	ADP
cana-1005	75	33	length	length	NOUN
cana-1005	75	34	n-1	n-1	PROPN
cana-1005	75	35	.	.	PUNCT
cana-1005	76	1	the	the	DET
cana-1005	76	2	set	set	NOUN
cana-1005	76	3	of	of	ADP
cana-1005	76	4	vertices	vertex	NOUN
cana-1005	76	5	{	{	PUNCT
cana-1005	76	6	𝑣1	𝑣1	PROPN
cana-1005	76	7	,	,	PUNCT
cana-1005	76	8	𝑣2	𝑣2	PROPN
cana-1005	76	9	,	,	PUNCT
cana-1005	76	10	𝑣3	𝑣3	ADJ
cana-1005	76	11	,	,	PUNCT
cana-1005	76	12	.	.	PUNCT
cana-1005	76	13	.	.	PUNCT
cana-1005	76	14	.	.	PUNCT
cana-1005	77	1	,	,	PUNCT
cana-1005	77	2	𝑣𝑛	𝑣𝑛	NOUN
cana-1005	77	3	}	}	PUNCT
cana-1005	77	4	can	can	AUX
cana-1005	77	5	be	be	AUX
cana-1005	77	6	classified	classify	VERB
cana-1005	77	7	into	into	ADP
cana-1005	77	8	two	two	NUM
cana-1005	77	9	sets	set	NOUN
cana-1005	77	10	of	of	ADP
cana-1005	77	11	vertices	vertex	NOUN
cana-1005	77	12	such	such	ADJ
cana-1005	77	13	that	that	DET
cana-1005	77	14	𝑣1	𝑣1	NOUN
cana-1005	77	15	and	and	CCONJ
cana-1005	77	16	{	{	PUNCT
cana-1005	77	17	𝑣𝑗	𝑣𝑗	ADP
cana-1005	77	18	,	,	PUNCT
cana-1005	77	19	𝑗	𝑗	NOUN
cana-1005	77	20	=	=	SYM
cana-1005	77	21	1,2	1,2	NUM
cana-1005	77	22	,	,	PUNCT
cana-1005	77	23	…	…	PUNCT
cana-1005	77	24	,	,	PUNCT
cana-1005	77	25	𝑛	𝑛	PROPN
cana-1005	77	26	}	}	PUNCT
cana-1005	77	27	.	.	PUNCT
cana-1005	78	1	then	then	ADV
cana-1005	78	2	the	the	DET
cana-1005	78	3	reverse	reverse	ADJ
cana-1005	78	4	vertex	vertex	NOUN
cana-1005	78	5	degree	degree	NOUN
cana-1005	78	6	ℜ𝑑𝑣1	ℜ𝑑𝑣1	PROPN
cana-1005	78	7	=	=	SYM
cana-1005	78	8	1	1	NUM
cana-1005	78	9	,	,	PUNCT
cana-1005	78	10	ℜ𝑑𝑣𝑗	ℜ𝑑𝑣𝑗	PROPN
cana-1005	78	11	=	=	PUNCT
cana-1005	78	12	𝑛	𝑛	PRON
cana-1005	78	13	−	−	NOUN
cana-1005	78	14	3	3	X
cana-1005	78	15	.	.	PUNCT
cana-1005	79	1	the	the	DET
cana-1005	79	2	reverse	reverse	ADJ
cana-1005	79	3	sum	sum	NOUN
cana-1005	79	4	degree	degree	NOUN
cana-1005	79	5	ℜ𝑆𝑣1	ℜ𝑆𝑣1	X
cana-1005	79	6	=	=	PUNCT
cana-1005	79	7	(	(	PUNCT
cana-1005	79	8	𝑛	𝑛	PRON
cana-1005	79	9	−	−	NOUN
cana-1005	79	10	1)(𝑛	1)(𝑛	NUM
cana-1005	79	11	−	−	NOUN
cana-1005	79	12	3	3	NUM
cana-1005	79	13	)	)	PUNCT
cana-1005	79	14	,	,	PUNCT
cana-1005	79	15	ℜ𝑆𝑣𝑗	ℜ𝑆𝑣𝑗	PROPN
cana-1005	79	16	=	=	SYM
cana-1005	79	17	2𝑛	2𝑛	PROPN
cana-1005	79	18	−	−	NUM
cana-1005	79	19	5	5	NUM
cana-1005	79	20	and	and	CCONJ
cana-1005	79	21	the	the	DET
cana-1005	79	22	reverse	reverse	ADJ
cana-1005	79	23	multiplication	multiplication	NOUN
cana-1005	79	24	degree	degree	NOUN
cana-1005	79	25	ℜ𝑀𝑣1	ℜ𝑀𝑣1	PROPN
cana-1005	79	26	=	=	PUNCT
cana-1005	79	27	(	(	PUNCT
cana-1005	79	28	𝑛	𝑛	PROPN
cana-1005	79	29	−	−	PROPN
cana-1005	79	30	3)𝑛−1	3)𝑛−1	NUM
cana-1005	79	31	,	,	PUNCT
cana-1005	79	32	ℜ𝑀𝑣𝑗	ℜ𝑀𝑣𝑗	PROPN
cana-1005	79	33	=	=	SYM
cana-1005	79	34	(	(	PUNCT
cana-1005	79	35	𝑛	𝑛	PRON
cana-1005	79	36	−	−	NOUN
cana-1005	79	37	3)2	3)2	NUM
cana-1005	79	38	.	.	PUNCT
cana-1005	80	1	then	then	ADV
cana-1005	80	2	the	the	DET
cana-1005	80	3	extended	extended	ADJ
cana-1005	80	4	reverse	reverse	ADJ
cana-1005	80	5	ℛ	ℛ	PROPN
cana-1005	80	6	degrees	degree	NOUN
cana-1005	80	7	are	be	AUX
cana-1005	80	8	𝔼ℜℛ𝑣1	𝔼ℜℛ𝑣1	NOUN
cana-1005	80	9	⬚	⬚	PROPN
cana-1005	80	10	=	=	PUNCT
cana-1005	80	11	(	(	PUNCT
cana-1005	80	12	𝑛	𝑛	PRON
cana-1005	80	13	−	−	NOUN
cana-1005	80	14	1)(𝑛	1)(𝑛	NUM
cana-1005	80	15	−	−	NOUN
cana-1005	80	16	3	3	NUM
cana-1005	80	17	)	)	PUNCT
cana-1005	81	1	+	+	CCONJ
cana-1005	81	2	(	(	PUNCT
cana-1005	81	3	𝑛	𝑛	PRON
cana-1005	81	4	−	−	PROPN
cana-1005	81	5	3)𝑛−1	3)𝑛−1	NUM
cana-1005	81	6	,	,	PUNCT
cana-1005	81	7	𝔼ℜℛ𝑣𝑗	𝔼ℜℛ𝑣𝑗	PROPN
cana-1005	81	8	⬚	⬚	PROPN
cana-1005	81	9	=	=	PUNCT
cana-1005	81	10	(	(	PUNCT
cana-1005	81	11	𝑛	𝑛	PRON
cana-1005	81	12	−	−	PROPN
cana-1005	81	13	2)2	2)2	NUM
cana-1005	81	14	.	.	PUNCT
cana-1005	82	1	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	82	2	1	1	NUM
cana-1005	82	3	(	(	PUNCT
cana-1005	82	4	𝑊𝑛	𝑊𝑛	PROPN
cana-1005	82	5	)	)	PUNCT
cana-1005	82	6	=	=	PUNCT
cana-1005	83	1	[	[	X
cana-1005	83	2	(	(	PUNCT
cana-1005	83	3	𝑛	𝑛	PRON
cana-1005	83	4	−	−	NOUN
cana-1005	83	5	1)(𝑛	1)(𝑛	NUM
cana-1005	83	6	−	−	NOUN
cana-1005	83	7	3	3	NUM
cana-1005	83	8	)	)	PUNCT
cana-1005	83	9	+	+	CCONJ
cana-1005	83	10	(	(	PUNCT
cana-1005	83	11	𝑛	𝑛	DET
cana-1005	83	12	−	−	PROPN
cana-1005	83	13	3)𝑛−1]2	3)𝑛−1]2	PROPN
cana-1005	83	14	+	+	CCONJ
cana-1005	83	15	(	(	PUNCT
cana-1005	83	16	𝑛	𝑛	PRON
cana-1005	83	17	−	−	NOUN
cana-1005	83	18	1)(𝑛	1)(𝑛	NUM
cana-1005	83	19	−	−	PROPN
cana-1005	83	20	2)4	2)4	NUM
cana-1005	83	21	,	,	PUNCT
cana-1005	83	22	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	83	23	2	2	NUM
cana-1005	83	24	(	(	PUNCT
cana-1005	83	25	𝑊𝑛	𝑊𝑛	PROPN
cana-1005	83	26	)	)	PUNCT
cana-1005	83	27	=	=	SYM
cana-1005	83	28	(	(	PUNCT
cana-1005	83	29	𝑛	𝑛	PRON
cana-1005	83	30	−	−	NUM
cana-1005	83	31	1)[{(𝑛	1)[{(𝑛	NUM
cana-1005	83	32	−	−	NOUN
cana-1005	83	33	1)(𝑛	1)(𝑛	NUM
cana-1005	83	34	−	−	NOUN
cana-1005	83	35	3	3	NUM
cana-1005	83	36	)	)	PUNCT
cana-1005	83	37	+	+	CCONJ
cana-1005	83	38	(	(	PUNCT
cana-1005	83	39	𝑛	𝑛	PRON
cana-1005	83	40	−	−	PROPN
cana-1005	83	41	3)𝑛−1][𝑛	3)𝑛−1][𝑛	NUM
cana-1005	83	42	−	−	PROPN
cana-1005	83	43	2]2	2]2	NUM
cana-1005	83	44	}	}	PUNCT
cana-1005	83	45	+	+	CCONJ
cana-1005	83	46	(	(	PUNCT
cana-1005	83	47	𝑛	𝑛	DET
cana-1005	83	48	−	−	PROPN
cana-1005	83	49	2)4	2)4	NUM
cana-1005	83	50	,	,	PUNCT
cana-1005	83	51	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	83	52	3	3	NUM
cana-1005	83	53	(	(	PUNCT
cana-1005	83	54	𝑊𝑛	𝑊𝑛	PROPN
cana-1005	83	55	)	)	PUNCT
cana-1005	83	56	=	=	SYM
cana-1005	83	57	(	(	PUNCT
cana-1005	83	58	𝑛	𝑛	PRON
cana-1005	83	59	−	−	NOUN
cana-1005	83	60	1)(𝑛	1)(𝑛	NUM
cana-1005	83	61	−	−	PROPN
cana-1005	83	62	3)𝑛−1	3)𝑛−1	NUM
cana-1005	83	63	+	+	NUM
cana-1005	83	64	4𝑛3	4𝑛3	NUM
cana-1005	83	65	−	−	NOUN
cana-1005	84	1	20𝑛2	20𝑛2	NUM
cana-1005	84	2	+	+	NUM
cana-1005	84	3	31𝑛	31𝑛	NOUN
cana-1005	84	4	−	−	PROPN
cana-1005	84	5	15	15	NUM
cana-1005	84	6	.	.	PUNCT
cana-1005	85	1	theorem	theorem	VERB
cana-1005	85	2	3.3	3.3	NUM
cana-1005	85	3	if	if	SCONJ
cana-1005	85	4	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	PRON
cana-1005	85	5	be	be	AUX
cana-1005	85	6	the	the	DET
cana-1005	85	7	generalized	generalized	ADJ
cana-1005	85	8	peterson	peterson	NOUN
cana-1005	85	9	graph	graph	NOUN
cana-1005	85	10	with	with	ADP
cana-1005	85	11	n	n	NUM
cana-1005	85	12	≥	≥	NOUN
cana-1005	85	13	3	3	NUM
cana-1005	85	14	and	and	CCONJ
cana-1005	85	15	1	1	NUM
cana-1005	85	16	≤	≤	NOUN
cana-1005	85	17	𝑘	𝑘	DET
cana-1005	85	18	≤	≤	ADJ
cana-1005	85	19	⌊	⌊	PROPN
cana-1005	85	20	(	(	PUNCT
cana-1005	85	21	𝑛−1	𝑛−1	PROPN
cana-1005	85	22	)	)	PUNCT
cana-1005	85	23	2	2	NUM
cana-1005	85	24	⌋	⌋	NOUN
cana-1005	85	25	then	then	ADV
cana-1005	85	26	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	85	27	1	1	NUM
cana-1005	85	28	(	(	PUNCT
cana-1005	85	29	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	NOUN
cana-1005	85	30	)	)	PUNCT
cana-1005	85	31	=	=	SYM
cana-1005	85	32	2𝑛(𝑛	2𝑛(𝑛	PROPN
cana-1005	85	33	+	+	CCONJ
cana-1005	85	34	1)2	1)2	NUM
cana-1005	85	35	,	,	PUNCT
cana-1005	85	36	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	85	37	2	2	NUM
cana-1005	85	38	(	(	PUNCT
cana-1005	85	39	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	NOUN
cana-1005	85	40	)	)	PUNCT
cana-1005	85	41	=	=	SYM
cana-1005	85	42	3𝑛(𝑛	3𝑛(𝑛	NUM
cana-1005	85	43	+	+	CCONJ
cana-1005	85	44	1)2	1)2	NUM
cana-1005	85	45	,	,	PUNCT
cana-1005	85	46	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	85	47	3	3	NUM
cana-1005	85	48	(	(	PUNCT
cana-1005	85	49	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	NOUN
cana-1005	85	50	)	)	PUNCT
cana-1005	85	51	=	=	SYM
cana-1005	86	1	6𝑛(𝑛	6𝑛(𝑛	NUM
cana-1005	87	1	+	+	CCONJ
cana-1005	87	2	1)2	1)2	NUM
cana-1005	87	3	.	.	PUNCT
cana-1005	88	1	proof	proof	NOUN
cana-1005	88	2	:	:	PUNCT
cana-1005	88	3	the	the	DET
cana-1005	88	4	vertex	vertex	NOUN
cana-1005	88	5	and	and	CCONJ
cana-1005	88	6	edge	edge	NOUN
cana-1005	88	7	cardinality	cardinality	NOUN
cana-1005	88	8	of	of	ADP
cana-1005	88	9	generalized	generalized	ADJ
cana-1005	88	10	peterson	peterson	NOUN
cana-1005	88	11	graph	graph	NOUN
cana-1005	88	12	is	be	AUX
cana-1005	88	13	|𝑉(𝐺𝑃𝑛,𝑘)|	|𝑉(𝐺𝑃𝑛,𝑘)|	PROPN
cana-1005	88	14	=	=	SYM
cana-1005	88	15	2𝑛	2𝑛	NUM
cana-1005	88	16	,	,	PUNCT
cana-1005	88	17	|𝐸(𝐺𝑃𝑛,𝑘)|	|𝐸(𝐺𝑃𝑛,𝑘)|	PROPN
cana-1005	88	18	=	=	SYM
cana-1005	88	19	3𝑛	3𝑛	NUM
cana-1005	88	20	respectively	respectively	ADV
cana-1005	88	21	.	.	PUNCT
cana-1005	89	1	the	the	DET
cana-1005	89	2	reverse	reverse	ADJ
cana-1005	89	3	vertex	vertex	NOUN
cana-1005	89	4	degree	degree	NOUN
cana-1005	89	5	degree	degree	NOUN
cana-1005	90	1	ℜ𝑑𝑣	ℜ𝑑𝑣	NOUN
cana-1005	90	2	=	=	SYM
cana-1005	90	3	1	1	X
cana-1005	90	4	.	.	PUNCT
cana-1005	91	1	the	the	DET
cana-1005	91	2	reverse	reverse	ADJ
cana-1005	91	3	sum	sum	NOUN
cana-1005	91	4	degree	degree	NOUN
cana-1005	91	5	ℜ𝑆𝑣	ℜ𝑆𝑣	NOUN
cana-1005	91	6	=	=	SYM
cana-1005	91	7	𝑛	𝑛	PROPN
cana-1005	91	8	and	and	CCONJ
cana-1005	91	9	the	the	DET
cana-1005	91	10	reverse	reverse	ADJ
cana-1005	91	11	multiplication	multiplication	NOUN
cana-1005	91	12	degree	degree	NOUN
cana-1005	91	13	ℜ𝑀𝑣1	ℜ𝑀𝑣1	PROPN
cana-1005	91	14	=	=	SYM
cana-1005	91	15	1	1	X
cana-1005	91	16	.	.	PUNCT
cana-1005	92	1	then	then	ADV
cana-1005	92	2	the	the	DET
cana-1005	92	3	extended	extended	ADJ
cana-1005	92	4	reverse	reverse	ADJ
cana-1005	92	5	ℛ	ℛ	NOUN
cana-1005	92	6	degree	degree	NOUN
cana-1005	92	7	𝔼ℜ𝑣	𝔼ℜ𝑣	NOUN
cana-1005	92	8	⬚	⬚	PROPN
cana-1005	92	9	=	=	SYM
cana-1005	92	10	(	(	PUNCT
cana-1005	92	11	𝑛	𝑛	PROPN
cana-1005	92	12	+	+	NOUN
cana-1005	92	13	1	1	NUM
cana-1005	92	14	)	)	PUNCT
cana-1005	92	15	.	.	PUNCT
cana-1005	93	1	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	93	2	1	1	NUM
cana-1005	93	3	(	(	PUNCT
cana-1005	93	4	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	NOUN
cana-1005	93	5	)	)	PUNCT
cana-1005	93	6	=	=	SYM
cana-1005	93	7	2𝑛(𝑛	2𝑛(𝑛	PROPN
cana-1005	93	8	+	+	CCONJ
cana-1005	93	9	1)2	1)2	NUM
cana-1005	93	10	,	,	PUNCT
cana-1005	93	11	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	93	12	2	2	NUM
cana-1005	93	13	(	(	PUNCT
cana-1005	93	14	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	NOUN
cana-1005	93	15	)	)	PUNCT
cana-1005	93	16	=	=	SYM
cana-1005	93	17	3𝑛(𝑛	3𝑛(𝑛	NUM
cana-1005	93	18	+	+	CCONJ
cana-1005	93	19	1)2	1)2	NUM
cana-1005	93	20	,	,	PUNCT
cana-1005	93	21	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	93	22	3	3	NUM
cana-1005	93	23	(	(	PUNCT
cana-1005	93	24	𝐺𝑃𝑛,𝑘	𝐺𝑃𝑛,𝑘	NOUN
cana-1005	93	25	)	)	PUNCT
cana-1005	93	26	=	=	SYM
cana-1005	93	27	6𝑛(𝑛	6𝑛(𝑛	NUM
cana-1005	94	1	+	+	CCONJ
cana-1005	94	2	1)2	1)2	NUM
cana-1005	94	3	.	.	PUNCT
cana-1005	95	1	communications	communication	NOUN
cana-1005	95	2	on	on	ADP
cana-1005	95	3	applied	apply	VERB
cana-1005	95	4	nonlinear	nonlinear	ADJ
cana-1005	95	5	analysis	analysis	NOUN
cana-1005	95	6	issn	issn	NOUN
cana-1005	95	7	:	:	PUNCT
cana-1005	95	8	1074	1074	NUM
cana-1005	95	9	-	-	PUNCT
cana-1005	95	10	133x	133x	NUM
cana-1005	95	11	vol	vol	NOUN
cana-1005	95	12	31	31	NUM
cana-1005	95	13	no	no	NOUN
cana-1005	95	14	.	.	PUNCT
cana-1005	96	1	5s	5s	NUM
cana-1005	96	2	(	(	PUNCT
cana-1005	96	3	2024	2024	NUM
cana-1005	96	4	)	)	PUNCT
cana-1005	96	5	114	114	NUM
cana-1005	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1005	96	7	theorem	theorem	VERB
cana-1005	96	8	3.4	3.4	NUM
cana-1005	96	9	if	if	SCONJ
cana-1005	96	10	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	96	11	is	be	AUX
cana-1005	96	12	the	the	DET
cana-1005	96	13	crown	crown	NOUN
cana-1005	96	14	graph	graph	NOUN
cana-1005	96	15	with	with	ADP
cana-1005	96	16	2n	2n	ADJ
cana-1005	96	17	vertices	vertex	NOUN
cana-1005	96	18	and	and	CCONJ
cana-1005	96	19	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1005	96	20	−	−	PROPN
cana-1005	96	21	1	1	NUM
cana-1005	96	22	)	)	PUNCT
cana-1005	96	23	edges	edge	NOUN
cana-1005	96	24	then	then	ADV
cana-1005	96	25	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	96	26	1	1	NUM
cana-1005	96	27	(	(	PUNCT
cana-1005	96	28	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	96	29	)	)	PUNCT
cana-1005	96	30	=	=	SYM
cana-1005	97	1	2𝑛3	2𝑛3	NUM
cana-1005	97	2	,	,	PUNCT
cana-1005	97	3	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	97	4	2	2	NUM
cana-1005	97	5	(	(	PUNCT
cana-1005	97	6	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	97	7	)	)	PUNCT
cana-1005	97	8	=	=	PUNCT
cana-1005	98	1	𝑛3(𝑛	𝑛3(𝑛	NOUN
cana-1005	98	2	−	−	NOUN
cana-1005	98	3	1	1	NUM
cana-1005	98	4	)	)	PUNCT
cana-1005	98	5	,	,	PUNCT
cana-1005	98	6	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	98	7	3	3	NUM
cana-1005	98	8	(	(	PUNCT
cana-1005	98	9	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	98	10	)	)	PUNCT
cana-1005	98	11	=	=	NOUN
cana-1005	98	12	2𝑛2(𝑛	2𝑛2(𝑛	NUM
cana-1005	98	13	−	−	NOUN
cana-1005	98	14	1	1	NUM
cana-1005	98	15	)	)	PUNCT
cana-1005	98	16	.	.	PUNCT
cana-1005	99	1	proof	proof	NOUN
cana-1005	99	2	:	:	PUNCT
cana-1005	99	3	the	the	DET
cana-1005	99	4	crown	crown	NOUN
cana-1005	99	5	graph	graph	NOUN
cana-1005	99	6	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	99	7	is	be	AUX
cana-1005	99	8	a	a	DET
cana-1005	99	9	graph	graph	NOUN
cana-1005	99	10	whose	whose	DET
cana-1005	99	11	vertices	vertex	NOUN
cana-1005	99	12	can	can	AUX
cana-1005	99	13	be	be	AUX
cana-1005	99	14	subdivided	subdivide	VERB
cana-1005	99	15	into	into	ADP
cana-1005	99	16	two	two	NUM
cana-1005	99	17	sets	set	NOUN
cana-1005	99	18	of	of	ADP
cana-1005	99	19	vertices	vertex	NOUN
cana-1005	99	20	{	{	PUNCT
cana-1005	99	21	𝑢1	𝑢1	PROPN
cana-1005	99	22	,	,	PUNCT
cana-1005	99	23	𝑢2	𝑢2	PROPN
cana-1005	99	24	,	,	PUNCT
cana-1005	99	25	𝑢3	𝑢3	PROPN
cana-1005	99	26	,	,	PUNCT
cana-1005	99	27	.	.	PUNCT
cana-1005	99	28	.	.	PUNCT
cana-1005	100	1	.	.	PUNCT
cana-1005	101	1	,	,	PUNCT
cana-1005	101	2	𝑢𝑛	𝑢𝑛	X
cana-1005	101	3	}	}	PUNCT
cana-1005	101	4	and	and	CCONJ
cana-1005	101	5	{	{	PUNCT
cana-1005	101	6	𝑣1	𝑣1	PROPN
cana-1005	101	7	,	,	PUNCT
cana-1005	101	8	𝑣2	𝑣2	PROPN
cana-1005	101	9	,	,	PUNCT
cana-1005	101	10	𝑣3	𝑣3	ADJ
cana-1005	101	11	,	,	PUNCT
cana-1005	101	12	.	.	PUNCT
cana-1005	101	13	.	.	PUNCT
cana-1005	102	1	.	.	PUNCT
cana-1005	103	1	,	,	PUNCT
cana-1005	103	2	𝑣𝑛	𝑣𝑛	ADP
cana-1005	103	3	}	}	PUNCT
cana-1005	103	4	as	as	ADP
cana-1005	103	5	𝑣	𝑣	PROPN
cana-1005	103	6	and	and	CCONJ
cana-1005	103	7	with	with	ADP
cana-1005	103	8	an	an	DET
cana-1005	103	9	edge	edge	NOUN
cana-1005	103	10	from	from	ADP
cana-1005	103	11	𝑢𝑖	𝑢𝑖	NOUN
cana-1005	103	12	to	to	ADP
cana-1005	103	13	𝑣𝑗	𝑣𝑗	ADP
cana-1005	103	14	whenever	whenever	SCONJ
cana-1005	103	15	𝑖	𝑖	DET
cana-1005	103	16	≠	≠	PROPN
cana-1005	103	17	𝑗.	𝑗.	NOUN
cana-1005	103	18	a	a	DET
cana-1005	103	19	size	size	NOUN
cana-1005	103	20	of	of	ADP
cana-1005	103	21	crown	crown	NOUN
cana-1005	103	22	graph	graph	NOUN
cana-1005	103	23	is	be	AUX
cana-1005	103	24	|𝑉(𝑆𝑛)|	|𝑉(𝑆𝑛)|	ADV
cana-1005	103	25	=	=	SYM
cana-1005	103	26	2𝑛	2𝑛	NUM
cana-1005	103	27	,	,	PUNCT
cana-1005	103	28	|𝐸(𝑆𝑛)|	|𝐸(𝑆𝑛)|	X
cana-1005	103	29	=	=	PUNCT
cana-1005	104	1	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1005	104	2	−	−	NOUN
cana-1005	104	3	1	1	NUM
cana-1005	104	4	)	)	PUNCT
cana-1005	104	5	.	.	PUNCT
cana-1005	105	1	then	then	ADV
cana-1005	105	2	the	the	DET
cana-1005	105	3	reverse	reverse	ADJ
cana-1005	105	4	vertex	vertex	NOUN
cana-1005	105	5	degree	degree	NOUN
cana-1005	105	6	degree	degree	NOUN
cana-1005	105	7	ℜ𝑑𝑣	ℜ𝑑𝑣	NOUN
cana-1005	105	8	=	=	SYM
cana-1005	105	9	1	1	X
cana-1005	105	10	.	.	PUNCT
cana-1005	106	1	the	the	DET
cana-1005	106	2	reverse	reverse	ADJ
cana-1005	106	3	sum	sum	NOUN
cana-1005	106	4	degree	degree	NOUN
cana-1005	106	5	ℜ𝑆𝑣	ℜ𝑆𝑣	NOUN
cana-1005	106	6	=	=	PUNCT
cana-1005	106	7	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1005	106	8	−	−	NOUN
cana-1005	106	9	1	1	NUM
cana-1005	106	10	)	)	PUNCT
cana-1005	106	11	and	and	CCONJ
cana-1005	106	12	the	the	DET
cana-1005	106	13	reverse	reverse	ADJ
cana-1005	106	14	multiplication	multiplication	NOUN
cana-1005	106	15	degree	degree	NOUN
cana-1005	106	16	ℜ𝑀𝑣1	ℜ𝑀𝑣1	PROPN
cana-1005	106	17	=	=	NOUN
cana-1005	106	18	1	1	X
cana-1005	106	19	.	.	PUNCT
cana-1005	107	1	the	the	DET
cana-1005	107	2	extended	extended	ADJ
cana-1005	107	3	reverse	reverse	ADJ
cana-1005	107	4	ℛ	ℛ	NOUN
cana-1005	107	5	degree	degree	NOUN
cana-1005	107	6	𝔼ℜ𝑅𝑣	𝔼ℜ𝑅𝑣	NOUN
cana-1005	107	7	⬚	⬚	NOUN
cana-1005	107	8	=	=	SYM
cana-1005	107	9	𝑛.	𝑛.	NOUN
cana-1005	107	10	hence	hence	ADV
cana-1005	107	11	,	,	PUNCT
cana-1005	107	12	the	the	DET
cana-1005	107	13	extended	extended	ADJ
cana-1005	107	14	reverse	reverse	ADJ
cana-1005	107	15	ℛ	ℛ	PROPN
cana-1005	107	16	topological	topological	ADJ
cana-1005	107	17	indices	index	NOUN
cana-1005	107	18	of	of	ADP
cana-1005	107	19	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	107	20	are	be	AUX
cana-1005	107	21	𝔼ℜℛ	𝔼ℜℛ	SYM
cana-1005	107	22	1	1	NUM
cana-1005	107	23	(	(	PUNCT
cana-1005	107	24	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	107	25	)	)	PUNCT
cana-1005	107	26	=	=	SYM
cana-1005	107	27	2𝑛3	2𝑛3	NUM
cana-1005	107	28	,	,	PUNCT
cana-1005	107	29	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	107	30	2	2	NUM
cana-1005	107	31	(	(	PUNCT
cana-1005	107	32	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	107	33	)	)	PUNCT
cana-1005	107	34	=	=	PUNCT
cana-1005	108	1	𝑛3(𝑛	𝑛3(𝑛	NOUN
cana-1005	108	2	−	−	NOUN
cana-1005	108	3	1	1	NUM
cana-1005	108	4	)	)	PUNCT
cana-1005	108	5	,	,	PUNCT
cana-1005	108	6	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	108	7	3	3	NUM
cana-1005	108	8	(	(	PUNCT
cana-1005	108	9	𝑆𝑛	𝑆𝑛	PROPN
cana-1005	108	10	)	)	PUNCT
cana-1005	108	11	=	=	NOUN
cana-1005	108	12	2𝑛2(𝑛	2𝑛2(𝑛	NUM
cana-1005	108	13	−	−	NOUN
cana-1005	108	14	1	1	NUM
cana-1005	108	15	)	)	PUNCT
cana-1005	108	16	.	.	PUNCT
cana-1005	109	1	theorem	theorem	VERB
cana-1005	109	2	3.5	3.5	NUM
cana-1005	109	3	if	if	SCONJ
cana-1005	109	4	sm	sm	PROPN
cana-1005	109	5	,	,	PUNCT
cana-1005	109	6	n	n	X
cana-1005	109	7	is	be	AUX
cana-1005	109	8	the	the	DET
cana-1005	109	9	double	double	ADJ
cana-1005	109	10	star	star	NOUN
cana-1005	109	11	graph	graph	NOUN
cana-1005	109	12	with	with	ADP
cana-1005	109	13	n+m+2	n+m+2	PROPN
cana-1005	109	14	vertices	vertex	NOUN
cana-1005	109	15	(	(	PUNCT
cana-1005	109	16	n	n	CCONJ
cana-1005	109	17	>	>	X
cana-1005	109	18	m	m	PROPN
cana-1005	109	19	)	)	PUNCT
cana-1005	109	20	and	and	CCONJ
cana-1005	109	21	n+m+1edges	n+m+1edge	NOUN
cana-1005	109	22	then	then	ADV
cana-1005	109	23	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	109	24	1	1	NUM
cana-1005	109	25	(	(	PUNCT
cana-1005	109	26	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	109	27	)	)	PUNCT
cana-1005	109	28	=	=	PRON
cana-1005	110	1	{	{	PUNCT
cana-1005	110	2	(	(	PUNCT
cana-1005	110	3	𝑛2	𝑛2	NOUN
cana-1005	110	4	−	−	PROPN
cana-1005	110	5	𝑚	𝑚	PROPN
cana-1005	110	6	+	+	SYM
cana-1005	110	7	1)𝑛(𝑚−1	1)𝑛(𝑚−1	NUM
cana-1005	110	8	)	)	PUNCT
cana-1005	111	1	+	+	CCONJ
cana-1005	111	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	111	3	−	−	NOUN
cana-1005	111	4	1	1	NUM
cana-1005	111	5	)	)	PUNCT
cana-1005	111	6	+	+	CCONJ
cana-1005	111	7	1	1	NUM
cana-1005	111	8	}	}	SYM
cana-1005	111	9	2	2	NUM
cana-1005	111	10	{	{	PUNCT
cana-1005	111	11	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	111	12	−	−	ADP
cana-1005	111	13	𝑚	𝑚	NOUN
cana-1005	111	14	+	+	NOUN
cana-1005	111	15	1	1	NUM
cana-1005	111	16	)	)	PUNCT
cana-1005	111	17	+	+	NUM
cana-1005	111	18	𝑛2	𝑛2	NOUN
cana-1005	111	19	−	−	PROPN
cana-1005	111	20	𝑚	𝑚	NOUN
cana-1005	111	21	+	+	NUM
cana-1005	111	22	1}2	1}2	NUM
cana-1005	111	23	+	+	CCONJ
cana-1005	111	24	4(𝑚	4(𝑚	NUM
cana-1005	111	25	−	−	NUM
cana-1005	111	26	1){𝑛	1){𝑛	ADV
cana-1005	111	27	−	−	ADP
cana-1005	111	28	𝑚	𝑚	NOUN
cana-1005	112	1	+	+	NUM
cana-1005	112	2	1}2	1}2	NUM
cana-1005	112	3	+	+	CCONJ
cana-1005	112	4	4𝑛	4𝑛	NOUN
cana-1005	112	5	−	−	NOUN
cana-1005	112	6	4	4	NUM
cana-1005	112	7	,	,	PUNCT
cana-1005	112	8	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	112	9	2	2	NUM
cana-1005	112	10	(	(	PUNCT
cana-1005	112	11	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	112	12	)	)	PUNCT
cana-1005	112	13	=	=	PUNCT
cana-1005	113	1	2(𝑚	2(𝑚	NUM
cana-1005	113	2	−	−	NUM
cana-1005	113	3	1){(𝑛(𝑚−1	1){(𝑛(𝑚−1	NUM
cana-1005	113	4	)	)	PUNCT
cana-1005	114	1	+	+	CCONJ
cana-1005	114	2	(	(	PUNCT
cana-1005	114	3	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	114	4	−	−	NOUN
cana-1005	114	5	1	1	NUM
cana-1005	114	6	)	)	PUNCT
cana-1005	114	7	+	+	CCONJ
cana-1005	114	8	1))(𝑛	1))(𝑛	NUM
cana-1005	114	9	−	−	NOUN
cana-1005	114	10	𝑚	𝑚	NOUN
cana-1005	114	11	+	+	PROPN
cana-1005	114	12	1	1	NUM
cana-1005	114	13	)	)	PUNCT
cana-1005	114	14	}	}	PUNCT
cana-1005	115	1	+	+	CCONJ
cana-1005	115	2	2(𝑛	2(𝑛	NUM
cana-1005	115	3	−	−	NOUN
cana-1005	115	4	1	1	NUM
cana-1005	115	5	)	)	PUNCT
cana-1005	115	6	(	(	PUNCT
cana-1005	115	7	𝑛2	𝑛2	NOUN
cana-1005	115	8	−	−	PROPN
cana-1005	115	9	𝑚	𝑚	PROPN
cana-1005	115	10	+	+	PROPN
cana-1005	115	11	1	1	NUM
cana-1005	115	12	+	+	CCONJ
cana-1005	115	13	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	115	14	−	−	ADP
cana-1005	115	15	𝑚	𝑚	NOUN
cana-1005	115	16	+	+	NOUN
cana-1005	115	17	1	1	NUM
cana-1005	115	18	)	)	PUNCT
cana-1005	115	19	)	)	PUNCT
cana-1005	116	1	+	+	CCONJ
cana-1005	116	2	(	(	PUNCT
cana-1005	116	3	𝑛(𝑚−1	𝑛(𝑚−1	NOUN
cana-1005	116	4	)	)	PUNCT
cana-1005	116	5	+	+	NUM
cana-1005	116	6	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	116	7	−	−	NOUN
cana-1005	116	8	1	1	NUM
cana-1005	116	9	)	)	PUNCT
cana-1005	116	10	+	+	CCONJ
cana-1005	116	11	1)(𝑛𝑛−1(𝑛	1)(𝑛𝑛−1(𝑛	NUM
cana-1005	116	12	−	−	NOUN
cana-1005	116	13	𝑚	𝑚	NOUN
cana-1005	116	14	+	+	NOUN
cana-1005	116	15	1	1	NUM
cana-1005	116	16	)	)	PUNCT
cana-1005	116	17	+	+	NUM
cana-1005	116	18	𝑛2	𝑛2	NOUN
cana-1005	116	19	−	−	PROPN
cana-1005	116	20	𝑚	𝑚	PROPN
cana-1005	116	21	+	+	PROPN
cana-1005	116	22	1	1	NUM
cana-1005	116	23	)	)	PUNCT
cana-1005	116	24	,	,	PUNCT
cana-1005	116	25	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	116	26	3	3	NUM
cana-1005	116	27	(	(	PUNCT
cana-1005	116	28	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	116	29	)	)	PUNCT
cana-1005	116	30	=	=	PUNCT
cana-1005	116	31	(	(	PUNCT
cana-1005	116	32	𝑚	𝑚	PROPN
cana-1005	116	33	−	−	PROPN
cana-1005	116	34	1){𝑛(𝑚−1	1){𝑛(𝑚−1	NUM
cana-1005	116	35	)	)	PUNCT
cana-1005	116	36	+	+	CCONJ
cana-1005	116	37	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	116	38	−	−	NOUN
cana-1005	116	39	1	1	NUM
cana-1005	116	40	)	)	PUNCT
cana-1005	116	41	+	+	CCONJ
cana-1005	116	42	1	1	NUM
cana-1005	116	43	+	+	SYM
cana-1005	116	44	2(𝑛	2(𝑛	NUM
cana-1005	116	45	−	−	NOUN
cana-1005	116	46	𝑚	𝑚	NOUN
cana-1005	116	47	+	+	NOUN
cana-1005	116	48	1	1	NUM
cana-1005	116	49	)	)	PUNCT
cana-1005	116	50	}	}	PUNCT
cana-1005	117	1	+	+	CCONJ
cana-1005	117	2	(	(	PUNCT
cana-1005	117	3	𝑛	𝑛	PRON
cana-1005	117	4	−	−	PROPN
cana-1005	117	5	1){(𝑛2	1){(𝑛2	NOUN
cana-1005	117	6	−	−	NOUN
cana-1005	118	1	𝑚	𝑚	PROPN
cana-1005	118	2	+	+	NOUN
cana-1005	118	3	3	3	NUM
cana-1005	118	4	+	+	CCONJ
cana-1005	118	5	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	118	6	−	−	ADP
cana-1005	118	7	𝑚	𝑚	NOUN
cana-1005	118	8	+	+	NOUN
cana-1005	118	9	1	1	NUM
cana-1005	118	10	)	)	PUNCT
cana-1005	118	11	)	)	PUNCT
cana-1005	119	1	+	+	CCONJ
cana-1005	119	2	2	2	X
cana-1005	119	3	}	}	PUNCT
cana-1005	119	4	+	+	CCONJ
cana-1005	119	5	(	(	PUNCT
cana-1005	119	6	𝑛(𝑚−1	𝑛(𝑚−1	NOUN
cana-1005	119	7	)	)	PUNCT
cana-1005	119	8	+	+	NUM
cana-1005	119	9	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	119	10	−	−	NOUN
cana-1005	119	11	1	1	NUM
cana-1005	119	12	)	)	PUNCT
cana-1005	119	13	+	+	CCONJ
cana-1005	119	14	1	1	X
cana-1005	119	15	)	)	PUNCT
cana-1005	119	16	+	+	CCONJ
cana-1005	119	17	(	(	PUNCT
cana-1005	119	18	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	119	19	−	−	ADP
cana-1005	119	20	𝑚	𝑚	NOUN
cana-1005	119	21	+	+	NOUN
cana-1005	119	22	1	1	NUM
cana-1005	119	23	)	)	PUNCT
cana-1005	119	24	+	+	NUM
cana-1005	119	25	𝑛2	𝑛2	NOUN
cana-1005	119	26	−	−	PROPN
cana-1005	119	27	𝑚	𝑚	PROPN
cana-1005	119	28	+	+	PROPN
cana-1005	119	29	1	1	NUM
cana-1005	119	30	)	)	PUNCT
cana-1005	119	31	.	.	PUNCT
cana-1005	120	1	communications	communication	NOUN
cana-1005	120	2	on	on	ADP
cana-1005	120	3	applied	apply	VERB
cana-1005	120	4	nonlinear	nonlinear	ADJ
cana-1005	120	5	analysis	analysis	NOUN
cana-1005	120	6	issn	issn	NOUN
cana-1005	120	7	:	:	PUNCT
cana-1005	120	8	1074	1074	NUM
cana-1005	120	9	-	-	PUNCT
cana-1005	120	10	133x	133x	NUM
cana-1005	120	11	vol	vol	NOUN
cana-1005	120	12	31	31	NUM
cana-1005	120	13	no	no	NOUN
cana-1005	120	14	.	.	PUNCT
cana-1005	121	1	5s	5s	NUM
cana-1005	121	2	(	(	PUNCT
cana-1005	121	3	2024	2024	NUM
cana-1005	121	4	)	)	PUNCT
cana-1005	121	5	115	115	NUM
cana-1005	122	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1005	122	2	proof	proof	NOUN
cana-1005	122	3	:	:	PUNCT
cana-1005	122	4	the	the	DET
cana-1005	122	5	double	double	ADJ
cana-1005	122	6	star	star	NOUN
cana-1005	122	7	graph	graph	NOUN
cana-1005	122	8	sm	sm	PROPN
cana-1005	122	9	,	,	PUNCT
cana-1005	122	10	n	n	PROPN
cana-1005	122	11	n	n	CCONJ
cana-1005	122	12	,	,	PUNCT
cana-1005	122	13	m	m	VERB
cana-1005	122	14	≥	≥	NOUN
cana-1005	122	15	2	2	NUM
cana-1005	122	16	and	and	CCONJ
cana-1005	122	17	(	(	PUNCT
cana-1005	122	18	n	n	CCONJ
cana-1005	122	19	>	>	X
cana-1005	122	20	m	m	PROPN
cana-1005	122	21	)	)	PUNCT
cana-1005	122	22	.	.	PUNCT
cana-1005	123	1	here	here	ADV
cana-1005	123	2	,	,	PUNCT
cana-1005	123	3	|𝑉(𝑆𝑚,𝑛)|	|𝑉(𝑆𝑚,𝑛)|	ADV
cana-1005	123	4	=	=	SYM
cana-1005	123	5	𝑛	𝑛	PROPN
cana-1005	123	6	+	+	NUM
cana-1005	123	7	𝑚	𝑚	X
cana-1005	123	8	+	+	ADJ
cana-1005	123	9	2	2	NUM
cana-1005	123	10	,	,	PUNCT
cana-1005	123	11	|𝐸(𝑆𝑚,𝑛)|	|𝐸(𝑆𝑚,𝑛)|	ADV
cana-1005	123	12	=	=	PUNCT
cana-1005	123	13	𝑛	𝑛	DET
cana-1005	123	14	−	−	NOUN
cana-1005	123	15	𝑚	𝑚	NOUN
cana-1005	123	16	+	+	NOUN
cana-1005	123	17	1	1	X
cana-1005	123	18	.	.	PUNCT
cana-1005	124	1	then	then	ADV
cana-1005	124	2	,	,	PUNCT
cana-1005	124	3	reverse	reverse	VERB
cana-1005	124	4	vertex	vertex	NOUN
cana-1005	124	5	degrees	degree	NOUN
cana-1005	124	6	are	be	AUX
cana-1005	124	7	ℜ𝑑𝑉𝐼	ℜ𝑑𝑉𝐼	PROPN
cana-1005	124	8	=	=	SYM
cana-1005	124	9	𝑛	𝑛	DET
cana-1005	124	10	−	−	NOUN
cana-1005	124	11	𝑚	𝑚	NOUN
cana-1005	124	12	+	+	PROPN
cana-1005	124	13	1	1	NUM
cana-1005	124	14	,	,	PUNCT
cana-1005	124	15	ℜ𝑑𝑉𝐼𝐼	ℜ𝑑𝑉𝐼𝐼	NOUN
cana-1005	124	16	=	=	SYM
cana-1005	124	17	𝑛	𝑛	PROPN
cana-1005	124	18	and	and	CCONJ
cana-1005	124	19	ℜ𝑑𝑉𝐼𝐼𝐼	ℜ𝑑𝑉𝐼𝐼𝐼	NOUN
cana-1005	124	20	=	=	SYM
cana-1005	124	21	1	1	NUM
cana-1005	124	22	where	where	SCONJ
cana-1005	124	23	𝑉𝐼	𝑉𝐼	PROPN
cana-1005	124	24	is	be	AUX
cana-1005	124	25	the	the	DET
cana-1005	124	26	central	central	ADJ
cana-1005	124	27	vertex	vertex	NOUN
cana-1005	124	28	of	of	ADP
cana-1005	124	29	m	m	PROPN
cana-1005	124	30	star	star	NOUN
cana-1005	124	31	,	,	PUNCT
cana-1005	124	32	is	be	AUX
cana-1005	124	33	the	the	DET
cana-1005	124	34	pendent	pendent	ADJ
cana-1005	124	35	vertex	vertex	NOUN
cana-1005	124	36	of	of	ADP
cana-1005	124	37	m	m	PROPN
cana-1005	124	38	star	star	NOUN
cana-1005	124	39	,	,	PUNCT
cana-1005	124	40	is	be	AUX
cana-1005	124	41	the	the	DET
cana-1005	124	42	central	central	ADJ
cana-1005	124	43	vertex	vertex	NOUN
cana-1005	124	44	of	of	ADP
cana-1005	124	45	n	n	DET
cana-1005	124	46	star	star	NOUN
cana-1005	124	47	and	and	CCONJ
cana-1005	124	48	is	be	AUX
cana-1005	124	49	the	the	DET
cana-1005	124	50	pendent	pendent	ADJ
cana-1005	124	51	vertex	vertex	NOUN
cana-1005	124	52	of	of	ADP
cana-1005	124	53	n	n	PROPN
cana-1005	124	54	star	star	NOUN
cana-1005	124	55	graph	graph	NOUN
cana-1005	124	56	.	.	PUNCT
cana-1005	125	1	the	the	DET
cana-1005	125	2	reverse	reverse	ADJ
cana-1005	125	3	sum	sum	NOUN
cana-1005	125	4	degrees	degree	NOUN
cana-1005	125	5	are	be	AUX
cana-1005	125	6	ℜ𝑆𝑉𝐼	ℜ𝑆𝑉𝐼	PROPN
cana-1005	125	7	=	=	PUNCT
cana-1005	126	1	𝑚𝑛	𝑚𝑛	PROPN
cana-1005	126	2	−	−	PROPN
cana-1005	127	1	𝑛	𝑛	PROPN
cana-1005	128	1	+	+	NOUN
cana-1005	128	2	1	1	NUM
cana-1005	128	3	,	,	PUNCT
cana-1005	128	4	ℜ𝑆𝑉𝐼𝐼	ℜ𝑆𝑉𝐼𝐼	ADJ
cana-1005	128	5	=	=	SYM
cana-1005	128	6	𝑛	𝑛	DET
cana-1005	128	7	−	−	NOUN
cana-1005	128	8	𝑚	𝑚	NOUN
cana-1005	128	9	+	+	NOUN
cana-1005	128	10	1	1	NUM
cana-1005	128	11	,	,	PUNCT
cana-1005	128	12	ℜ𝑆𝑉𝐼𝐼𝐼	ℜ𝑆𝑉𝐼𝐼𝐼	NOUN
cana-1005	128	13	=	=	PUNCT
cana-1005	128	14	𝑛2	𝑛2	NOUN
cana-1005	128	15	−	−	PROPN
cana-1005	128	16	𝑚	𝑚	PROPN
cana-1005	128	17	+	+	PROPN
cana-1005	128	18	1	1	NUM
cana-1005	128	19	,	,	PUNCT
cana-1005	128	20	ℜ𝑆𝑉𝐼𝑉	ℜ𝑆𝑉𝐼𝑉	ADJ
cana-1005	128	21	=	=	SYM
cana-1005	128	22	1	1	NUM
cana-1005	128	23	and	and	CCONJ
cana-1005	128	24	the	the	DET
cana-1005	128	25	reverse	reverse	ADJ
cana-1005	128	26	multiplication	multiplication	NOUN
cana-1005	128	27	degree	degree	NOUN
cana-1005	128	28	ℜ𝑀𝑉𝐼	ℜ𝑀𝑉𝐼	PROPN
cana-1005	128	29	=	=	SYM
cana-1005	128	30	(	(	PUNCT
cana-1005	128	31	𝑛2	𝑛2	NOUN
cana-1005	128	32	−	−	PROPN
cana-1005	128	33	𝑚	𝑚	PROPN
cana-1005	128	34	+	+	SYM
cana-1005	128	35	1)𝑛𝑚−1	1)𝑛𝑚−1	NUM
cana-1005	128	36	,	,	PUNCT
cana-1005	128	37	ℜ𝑀𝑉𝐼𝐼	ℜ𝑀𝑉𝐼𝐼	ADJ
cana-1005	128	38	=	=	SYM
cana-1005	128	39	𝑛	𝑛	PRON
cana-1005	128	40	−	−	NOUN
cana-1005	128	41	𝑚	𝑚	NOUN
cana-1005	128	42	+	+	NOUN
cana-1005	128	43	1	1	NUM
cana-1005	128	44	,	,	PUNCT
cana-1005	128	45	ℜ𝑀𝑉𝐼𝐼𝐼	ℜ𝑀𝑉𝐼𝐼𝐼	NOUN
cana-1005	128	46	=	=	SYM
cana-1005	128	47	(	(	PUNCT
cana-1005	128	48	𝑛	𝑛	PRON
cana-1005	128	49	−	−	NOUN
cana-1005	128	50	𝑚	𝑚	PROPN
cana-1005	128	51	+	+	CCONJ
cana-1005	128	52	1)𝑛𝑛−1	1)𝑛𝑛−1	NOUN
cana-1005	128	53	,	,	PUNCT
cana-1005	128	54	ℜ𝑀𝑉𝐼𝑉	ℜ𝑀𝑉𝐼𝑉	PROPN
cana-1005	128	55	=	=	SYM
cana-1005	129	1	1	1	X
cana-1005	129	2	.	.	PUNCT
cana-1005	129	3	then	then	ADV
cana-1005	129	4	the	the	DET
cana-1005	129	5	extended	extended	ADJ
cana-1005	129	6	reverse	reverse	ADJ
cana-1005	129	7	ℛ	ℛ	PROPN
cana-1005	129	8	degrees	degree	NOUN
cana-1005	129	9	are	be	AUX
cana-1005	129	10	𝔼ℜℛ𝑣𝐼	𝔼ℜℛ𝑣𝐼	NOUN
cana-1005	129	11	⬚	⬚	PROPN
cana-1005	129	12	=	=	SYM
cana-1005	129	13	(	(	PUNCT
cana-1005	129	14	𝑛2	𝑛2	NOUN
cana-1005	129	15	−	−	PROPN
cana-1005	129	16	𝑚	𝑚	PROPN
cana-1005	129	17	+	+	SYM
cana-1005	129	18	1)𝑛(𝑚−1	1)𝑛(𝑚−1	NUM
cana-1005	129	19	)	)	PUNCT
cana-1005	130	1	+	+	CCONJ
cana-1005	130	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	130	3	−	−	NOUN
cana-1005	130	4	1	1	NUM
cana-1005	130	5	)	)	PUNCT
cana-1005	130	6	+	+	NUM
cana-1005	130	7	1	1	NUM
cana-1005	130	8	,	,	PUNCT
cana-1005	130	9	𝔼ℜℛ𝑉𝐼𝐼	𝔼ℜℛ𝑉𝐼𝐼	ADJ
cana-1005	130	10	⬚	⬚	PROPN
cana-1005	130	11	=	=	SYM
cana-1005	130	12	2(𝑛	2(𝑛	NUM
cana-1005	130	13	−	−	NOUN
cana-1005	130	14	𝑚	𝑚	NOUN
cana-1005	130	15	+	+	PROPN
cana-1005	130	16	1	1	NUM
cana-1005	130	17	)	)	PUNCT
cana-1005	130	18	,	,	PUNCT
cana-1005	130	19	𝔼ℜℛ𝑉𝐼𝐼𝐼	𝔼ℜℛ𝑉𝐼𝐼𝐼	PROPN
cana-1005	130	20	⬚	⬚	NOUN
cana-1005	130	21	=	=	SYM
cana-1005	130	22	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	130	23	−	−	ADP
cana-1005	130	24	𝑚	𝑚	NOUN
cana-1005	130	25	+	+	NOUN
cana-1005	130	26	1	1	NUM
cana-1005	130	27	)	)	PUNCT
cana-1005	130	28	+	+	NUM
cana-1005	130	29	𝑛2	𝑛2	NOUN
cana-1005	130	30	−	−	PROPN
cana-1005	130	31	𝑚	𝑚	PROPN
cana-1005	130	32	+	+	PROPN
cana-1005	130	33	1	1	NUM
cana-1005	130	34	,	,	PUNCT
cana-1005	130	35	𝔼ℜℛ𝑉𝐼𝑉	𝔼ℜℛ𝑉𝐼𝑉	PROPN
cana-1005	130	36	⬚	⬚	PROPN
cana-1005	130	37	=	=	SYM
cana-1005	130	38	2	2	X
cana-1005	130	39	.	.	X
cana-1005	130	40	hence	hence	ADV
cana-1005	130	41	,	,	PUNCT
cana-1005	130	42	the	the	DET
cana-1005	130	43	extended	extended	ADJ
cana-1005	130	44	reverse	reverse	ADJ
cana-1005	130	45	ℛ	ℛ	PROPN
cana-1005	130	46	topological	topological	ADJ
cana-1005	130	47	indices	index	NOUN
cana-1005	130	48	of	of	ADP
cana-1005	130	49	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	130	50	are	be	AUX
cana-1005	130	51	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	130	52	1	1	NUM
cana-1005	130	53	(	(	PUNCT
cana-1005	130	54	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	130	55	)	)	PUNCT
cana-1005	130	56	=	=	PRON
cana-1005	130	57	{	{	PUNCT
cana-1005	130	58	(	(	PUNCT
cana-1005	130	59	𝑛2	𝑛2	NOUN
cana-1005	130	60	−	−	PROPN
cana-1005	130	61	𝑚	𝑚	PROPN
cana-1005	130	62	+	+	SYM
cana-1005	130	63	1)𝑛(𝑚−1	1)𝑛(𝑚−1	NUM
cana-1005	130	64	)	)	PUNCT
cana-1005	131	1	+	+	CCONJ
cana-1005	131	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	131	3	−	−	NOUN
cana-1005	131	4	1	1	NUM
cana-1005	131	5	)	)	PUNCT
cana-1005	131	6	+	+	CCONJ
cana-1005	131	7	1	1	NUM
cana-1005	131	8	}	}	SYM
cana-1005	131	9	2	2	NUM
cana-1005	131	10	{	{	PUNCT
cana-1005	131	11	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	131	12	−	−	ADP
cana-1005	131	13	𝑚	𝑚	NOUN
cana-1005	131	14	+	+	NOUN
cana-1005	131	15	1	1	NUM
cana-1005	131	16	)	)	PUNCT
cana-1005	131	17	+	+	NUM
cana-1005	131	18	𝑛2	𝑛2	NOUN
cana-1005	131	19	−	−	PROPN
cana-1005	131	20	𝑚	𝑚	NOUN
cana-1005	131	21	+	+	NUM
cana-1005	131	22	1}2	1}2	NUM
cana-1005	131	23	+	+	CCONJ
cana-1005	131	24	4(𝑚	4(𝑚	NUM
cana-1005	131	25	−	−	NUM
cana-1005	131	26	1){𝑛	1){𝑛	ADV
cana-1005	131	27	−	−	ADP
cana-1005	131	28	𝑚	𝑚	NOUN
cana-1005	132	1	+	+	NUM
cana-1005	132	2	1}2	1}2	NUM
cana-1005	132	3	+	+	CCONJ
cana-1005	132	4	4𝑛	4𝑛	NOUN
cana-1005	132	5	−	−	NOUN
cana-1005	132	6	4	4	NUM
cana-1005	132	7	,	,	PUNCT
cana-1005	132	8	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	132	9	2	2	NUM
cana-1005	132	10	(	(	PUNCT
cana-1005	132	11	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	132	12	)	)	PUNCT
cana-1005	132	13	=	=	PUNCT
cana-1005	133	1	2(𝑚	2(𝑚	NUM
cana-1005	133	2	−	−	NUM
cana-1005	133	3	1){(𝑛(𝑚−1	1){(𝑛(𝑚−1	NUM
cana-1005	133	4	)	)	PUNCT
cana-1005	134	1	+	+	CCONJ
cana-1005	134	2	(	(	PUNCT
cana-1005	134	3	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	134	4	−	−	NOUN
cana-1005	134	5	1	1	NUM
cana-1005	134	6	)	)	PUNCT
cana-1005	134	7	+	+	CCONJ
cana-1005	134	8	1))(𝑛	1))(𝑛	NUM
cana-1005	134	9	−	−	NOUN
cana-1005	134	10	𝑚	𝑚	NOUN
cana-1005	134	11	+	+	PROPN
cana-1005	134	12	1	1	NUM
cana-1005	134	13	)	)	PUNCT
cana-1005	134	14	}	}	PUNCT
cana-1005	135	1	+	+	CCONJ
cana-1005	135	2	2(𝑛	2(𝑛	NUM
cana-1005	135	3	−	−	NOUN
cana-1005	135	4	1)(𝑛2	1)(𝑛2	NUM
cana-1005	135	5	−	−	NOUN
cana-1005	135	6	𝑚	𝑚	PROPN
cana-1005	135	7	+	+	NOUN
cana-1005	135	8	1	1	NUM
cana-1005	136	1	+	+	CCONJ
cana-1005	136	2	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	136	3	−	−	ADP
cana-1005	136	4	𝑚	𝑚	NOUN
cana-1005	136	5	+	+	NOUN
cana-1005	136	6	1	1	NUM
cana-1005	136	7	)	)	PUNCT
cana-1005	136	8	)	)	PUNCT
cana-1005	137	1	+	+	CCONJ
cana-1005	137	2	(	(	PUNCT
cana-1005	137	3	𝑛(𝑚−1	𝑛(𝑚−1	NOUN
cana-1005	137	4	)	)	PUNCT
cana-1005	137	5	+	+	NUM
cana-1005	137	6	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	137	7	−	−	NOUN
cana-1005	137	8	1	1	NUM
cana-1005	137	9	)	)	PUNCT
cana-1005	137	10	+	+	CCONJ
cana-1005	137	11	1)(𝑛𝑛−1(𝑛	1)(𝑛𝑛−1(𝑛	NUM
cana-1005	137	12	−	−	NOUN
cana-1005	137	13	𝑚	𝑚	NOUN
cana-1005	137	14	+	+	NOUN
cana-1005	137	15	1	1	NUM
cana-1005	137	16	)	)	PUNCT
cana-1005	137	17	+	+	NUM
cana-1005	137	18	𝑛2	𝑛2	NOUN
cana-1005	137	19	−	−	PROPN
cana-1005	137	20	𝑚	𝑚	PROPN
cana-1005	137	21	+	+	PROPN
cana-1005	137	22	1	1	NUM
cana-1005	137	23	)	)	PUNCT
cana-1005	137	24	,	,	PUNCT
cana-1005	137	25	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	137	26	3	3	NUM
cana-1005	137	27	(	(	PUNCT
cana-1005	137	28	𝑆𝑚,𝑛	𝑆𝑚,𝑛	NOUN
cana-1005	137	29	)	)	PUNCT
cana-1005	137	30	=	=	PUNCT
cana-1005	137	31	(	(	PUNCT
cana-1005	137	32	𝑚	𝑚	PROPN
cana-1005	137	33	−	−	PROPN
cana-1005	137	34	1){𝑛(𝑚−1	1){𝑛(𝑚−1	NUM
cana-1005	137	35	)	)	PUNCT
cana-1005	137	36	+	+	CCONJ
cana-1005	137	37	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	137	38	−	−	NOUN
cana-1005	137	39	1	1	NUM
cana-1005	137	40	)	)	PUNCT
cana-1005	137	41	+	+	CCONJ
cana-1005	137	42	1	1	NUM
cana-1005	137	43	+	+	SYM
cana-1005	137	44	2(𝑛	2(𝑛	NUM
cana-1005	137	45	−	−	NOUN
cana-1005	137	46	𝑚	𝑚	NOUN
cana-1005	137	47	+	+	NOUN
cana-1005	137	48	1	1	NUM
cana-1005	137	49	)	)	PUNCT
cana-1005	137	50	}	}	PUNCT
cana-1005	138	1	+	+	CCONJ
cana-1005	138	2	(	(	PUNCT
cana-1005	138	3	𝑛	𝑛	PRON
cana-1005	138	4	−	−	PROPN
cana-1005	138	5	1){(𝑛2	1){(𝑛2	NOUN
cana-1005	138	6	−	−	NOUN
cana-1005	139	1	𝑚	𝑚	PROPN
cana-1005	139	2	+	+	NOUN
cana-1005	139	3	3	3	NUM
cana-1005	139	4	+	+	CCONJ
cana-1005	139	5	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	139	6	−	−	ADP
cana-1005	139	7	𝑚	𝑚	NOUN
cana-1005	139	8	+	+	NOUN
cana-1005	139	9	1	1	NUM
cana-1005	139	10	)	)	PUNCT
cana-1005	139	11	)	)	PUNCT
cana-1005	140	1	+	+	CCONJ
cana-1005	140	2	2	2	X
cana-1005	140	3	}	}	PUNCT
cana-1005	140	4	+	+	CCONJ
cana-1005	140	5	(	(	PUNCT
cana-1005	140	6	𝑛(𝑚−1	𝑛(𝑚−1	NOUN
cana-1005	140	7	)	)	PUNCT
cana-1005	140	8	+	+	NUM
cana-1005	140	9	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	140	10	−	−	NOUN
cana-1005	140	11	1	1	NUM
cana-1005	140	12	)	)	PUNCT
cana-1005	140	13	+	+	CCONJ
cana-1005	140	14	1	1	X
cana-1005	140	15	)	)	PUNCT
cana-1005	140	16	+	+	CCONJ
cana-1005	140	17	(	(	PUNCT
cana-1005	140	18	𝑛𝑛−1(𝑛	𝑛𝑛−1(𝑛	ADJ
cana-1005	140	19	−	−	ADP
cana-1005	140	20	𝑚	𝑚	NOUN
cana-1005	140	21	+	+	NOUN
cana-1005	140	22	1	1	NUM
cana-1005	140	23	)	)	PUNCT
cana-1005	140	24	+	+	NUM
cana-1005	140	25	𝑛2	𝑛2	NOUN
cana-1005	140	26	−	−	PROPN
cana-1005	140	27	𝑚	𝑚	PROPN
cana-1005	140	28	+	+	PROPN
cana-1005	140	29	1	1	NUM
cana-1005	140	30	)	)	PUNCT
cana-1005	140	31	.	.	PUNCT
cana-1005	141	1	theorem	theorem	VERB
cana-1005	141	2	3.6	3.6	NUM
cana-1005	141	3	if	if	SCONJ
cana-1005	141	4	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	141	5	(	(	PUNCT
cana-1005	141	6	𝑛	𝑛	NOUN
cana-1005	141	7	)	)	PUNCT
cana-1005	141	8	is	be	AUX
cana-1005	141	9	the	the	DET
cana-1005	141	10	windmill	windmill	NOUN
cana-1005	141	11	graph	graph	NOUN
cana-1005	141	12	with	with	ADP
cana-1005	141	13	m	m	PROPN
cana-1005	141	14	,	,	PUNCT
cana-1005	142	1	n	n	PRON
cana-1005	142	2	≥	≥	NOUN
cana-1005	142	3	2	2	NUM
cana-1005	142	4	then	then	ADV
cana-1005	142	5	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	142	6	1	1	NUM
cana-1005	142	7	(	(	PUNCT
cana-1005	142	8	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	142	9	(	(	PUNCT
cana-1005	142	10	𝑛	𝑛	NOUN
cana-1005	142	11	)	)	PUNCT
cana-1005	142	12	)	)	PUNCT
cana-1005	143	1	=	=	PRON
cana-1005	143	2	{	{	PUNCT
cana-1005	143	3	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	143	4	−	−	PROPN
cana-1005	143	5	1)[(𝑚	1)[(𝑚	NUM
cana-1005	143	6	−	−	NOUN
cana-1005	143	7	1)(𝑛	1)(𝑛	NUM
cana-1005	143	8	−	−	NOUN
cana-1005	143	9	1	1	NUM
cana-1005	143	10	)	)	PUNCT
cana-1005	143	11	+	+	CCONJ
cana-1005	143	12	1	1	X
cana-1005	143	13	]	]	PUNCT
cana-1005	143	14	+	+	CCONJ
cana-1005	143	15	[	[	X
cana-1005	143	16	(	(	PUNCT
cana-1005	143	17	𝑚	𝑚	PROPN
cana-1005	143	18	−	−	PROPN
cana-1005	143	19	1)(𝑛	1)(𝑛	NUM
cana-1005	143	20	−	−	NOUN
cana-1005	143	21	1	1	NUM
cana-1005	143	22	)	)	PUNCT
cana-1005	143	23	+	+	CCONJ
cana-1005	143	24	1]𝑛(𝑚−1	1]𝑛(𝑚−1	X
cana-1005	143	25	)	)	PUNCT
cana-1005	143	26	}	}	PUNCT
cana-1005	143	27	2	2	NUM
cana-1005	144	1	+	+	NUM
cana-1005	144	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	144	3	−	−	PROPN
cana-1005	144	4	1)[2(𝑚	1)[2(𝑚	NUM
cana-1005	144	5	−	−	NOUN
cana-1005	144	6	1)(𝑛	1)(𝑛	NUM
cana-1005	144	7	−	−	NOUN
cana-1005	144	8	1	1	NUM
cana-1005	144	9	)	)	PUNCT
cana-1005	144	10	+	+	CCONJ
cana-1005	144	11	3]2	3]2	NOUN
cana-1005	144	12	,	,	PUNCT
cana-1005	144	13	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	144	14	2	2	NUM
cana-1005	144	15	(	(	PUNCT
cana-1005	144	16	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	144	17	(	(	PUNCT
cana-1005	144	18	𝑛	𝑛	NOUN
cana-1005	144	19	)	)	PUNCT
cana-1005	144	20	)	)	PUNCT
cana-1005	145	1	=	=	PUNCT
cana-1005	145	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	145	3	−	−	NOUN
cana-1005	145	4	1){(𝑛(𝑚	1){(𝑛(𝑚	ADJ
cana-1005	145	5	−	−	PROPN
cana-1005	145	6	1)[(𝑚	1)[(𝑚	NUM
cana-1005	145	7	−	−	NOUN
cana-1005	145	8	1)(𝑛	1)(𝑛	NUM
cana-1005	145	9	−	−	NOUN
cana-1005	145	10	1	1	NUM
cana-1005	145	11	)	)	PUNCT
cana-1005	145	12	+	+	CCONJ
cana-1005	145	13	1	1	X
cana-1005	145	14	]	]	PUNCT
cana-1005	145	15	+	+	CCONJ
cana-1005	146	1	[	[	X
cana-1005	146	2	(	(	PUNCT
cana-1005	146	3	𝑚	𝑚	PROPN
cana-1005	146	4	−	−	PROPN
cana-1005	146	5	1)(𝑛	1)(𝑛	NUM
cana-1005	146	6	−	−	NOUN
cana-1005	146	7	1	1	NUM
cana-1005	146	8	)	)	PUNCT
cana-1005	146	9	+	+	NUM
cana-1005	146	10	1]𝑛(𝑚−1	1]𝑛(𝑚−1	NUM
cana-1005	146	11	)	)	PUNCT
cana-1005	146	12	)	)	PUNCT
cana-1005	147	1	(	(	PUNCT
cana-1005	147	2	2(𝑚	2(𝑚	NUM
cana-1005	147	3	−	−	NOUN
cana-1005	147	4	1)(𝑛	1)(𝑛	NUM
cana-1005	147	5	−	−	NOUN
cana-1005	147	6	1	1	NUM
cana-1005	147	7	)	)	PUNCT
cana-1005	147	8	+	+	CCONJ
cana-1005	147	9	3	3	NUM
cana-1005	147	10	)	)	PUNCT
cana-1005	147	11	}	}	PUNCT
cana-1005	147	12	+	+	PUNCT
cana-1005	148	1	[	[	X
cana-1005	148	2	2(𝑚	2(𝑚	NUM
cana-1005	148	3	−	−	NOUN
cana-1005	148	4	1)(𝑛	1)(𝑛	NUM
cana-1005	148	5	−	−	NOUN
cana-1005	148	6	1	1	NUM
cana-1005	148	7	)	)	PUNCT
cana-1005	148	8	+	+	CCONJ
cana-1005	148	9	3]2	3]2	NOUN
cana-1005	148	10	,	,	PUNCT
cana-1005	148	11	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	148	12	3	3	NUM
cana-1005	148	13	(	(	PUNCT
cana-1005	148	14	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	148	15	(	(	PUNCT
cana-1005	148	16	𝑛	𝑛	NOUN
cana-1005	148	17	)	)	PUNCT
cana-1005	148	18	)	)	PUNCT
cana-1005	149	1	=	=	PUNCT
cana-1005	149	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	149	3	−	−	NOUN
cana-1005	149	4	1){(𝑛(𝑚	1){(𝑛(𝑚	ADJ
cana-1005	149	5	−	−	PROPN
cana-1005	149	6	1)[(𝑚	1)[(𝑚	NUM
cana-1005	149	7	−	−	NOUN
cana-1005	149	8	1)(𝑛	1)(𝑛	NUM
cana-1005	149	9	−	−	NOUN
cana-1005	149	10	1	1	NUM
cana-1005	149	11	)	)	PUNCT
cana-1005	149	12	+	+	CCONJ
cana-1005	149	13	1	1	X
cana-1005	149	14	]	]	PUNCT
cana-1005	149	15	+	+	CCONJ
cana-1005	150	1	[	[	X
cana-1005	150	2	(	(	PUNCT
cana-1005	150	3	𝑚	𝑚	PROPN
cana-1005	150	4	−	−	PROPN
cana-1005	150	5	1)(𝑛	1)(𝑛	NUM
cana-1005	150	6	−	−	NOUN
cana-1005	150	7	1	1	NUM
cana-1005	150	8	)	)	PUNCT
cana-1005	150	9	+	+	NUM
cana-1005	150	10	1]𝑛(𝑚−1	1]𝑛(𝑚−1	NUM
cana-1005	150	11	)	)	PUNCT
cana-1005	150	12	)	)	PUNCT
cana-1005	151	1	+	+	CCONJ
cana-1005	151	2	(	(	PUNCT
cana-1005	151	3	2(𝑚	2(𝑚	NUM
cana-1005	151	4	−	−	NOUN
cana-1005	151	5	1)(𝑛	1)(𝑛	NUM
cana-1005	151	6	−	−	NOUN
cana-1005	151	7	1	1	NUM
cana-1005	151	8	)	)	PUNCT
cana-1005	151	9	+	+	CCONJ
cana-1005	151	10	3	3	NUM
cana-1005	151	11	)	)	PUNCT
cana-1005	151	12	}	}	PUNCT
cana-1005	151	13	+	+	PUNCT
cana-1005	152	1	[	[	X
cana-1005	152	2	2(𝑚	2(𝑚	NUM
cana-1005	152	3	−	−	NOUN
cana-1005	152	4	1)(𝑛	1)(𝑛	NUM
cana-1005	152	5	−	−	NOUN
cana-1005	152	6	1	1	NUM
cana-1005	152	7	)	)	PUNCT
cana-1005	152	8	+	+	CCONJ
cana-1005	152	9	3]2	3]2	NOUN
cana-1005	152	10	.	.	PUNCT
cana-1005	153	1	communications	communication	NOUN
cana-1005	153	2	on	on	ADP
cana-1005	153	3	applied	apply	VERB
cana-1005	153	4	nonlinear	nonlinear	ADJ
cana-1005	153	5	analysis	analysis	NOUN
cana-1005	153	6	issn	issn	NOUN
cana-1005	153	7	:	:	PUNCT
cana-1005	153	8	1074	1074	NUM
cana-1005	153	9	-	-	PUNCT
cana-1005	153	10	133x	133x	NUM
cana-1005	153	11	vol	vol	NOUN
cana-1005	153	12	31	31	NUM
cana-1005	153	13	no	no	NOUN
cana-1005	153	14	.	.	PUNCT
cana-1005	154	1	5s	5s	NUM
cana-1005	154	2	(	(	PUNCT
cana-1005	154	3	2024	2024	NUM
cana-1005	154	4	)	)	PUNCT
cana-1005	154	5	116	116	NUM
cana-1005	155	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1005	155	2	proof	proof	NOUN
cana-1005	155	3	:	:	PUNCT
cana-1005	155	4	the	the	DET
cana-1005	155	5	windmill	windmill	NOUN
cana-1005	155	6	graph	graph	NOUN
cana-1005	155	7	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	155	8	(	(	PUNCT
cana-1005	155	9	𝑛	𝑛	NOUN
cana-1005	155	10	)	)	PUNCT
cana-1005	155	11	is	be	AUX
cana-1005	155	12	an	an	DET
cana-1005	155	13	undirected	undirected	ADJ
cana-1005	155	14	graph	graph	NOUN
cana-1005	155	15	constructed	construct	VERB
cana-1005	155	16	for	for	ADP
cana-1005	155	17	𝑚	𝑚	PROPN
cana-1005	155	18	,	,	PUNCT
cana-1005	155	19	𝑛	𝑛	DET
cana-1005	155	20	≥	≥	NOUN
cana-1005	155	21	2	2	NUM
cana-1005	155	22	by	by	ADP
cana-1005	155	23	joining	join	VERB
cana-1005	155	24	n	n	PRON
cana-1005	155	25	copies	copy	NOUN
cana-1005	155	26	of	of	ADP
cana-1005	155	27	the	the	DET
cana-1005	155	28	complete	complete	ADJ
cana-1005	155	29	graph	graph	NOUN
cana-1005	155	30	m	m	VERB
cana-1005	155	31	at	at	ADP
cana-1005	155	32	a	a	DET
cana-1005	155	33	shared	share	VERB
cana-1005	155	34	universal	universal	ADJ
cana-1005	155	35	vertex	vertex	NOUN
cana-1005	155	36	and	and	CCONJ
cana-1005	155	37	|𝑉(𝑊𝑚	|𝑉(𝑊𝑚	X
cana-1005	155	38	(	(	PUNCT
cana-1005	155	39	𝑛	𝑛	NOUN
cana-1005	155	40	)	)	PUNCT
cana-1005	155	41	)	)	PUNCT
cana-1005	156	1	|	|	ADV
cana-1005	156	2	=	=	SYM
cana-1005	156	3	(	(	PUNCT
cana-1005	156	4	𝑚	𝑚	PROPN
cana-1005	156	5	−	−	PROPN
cana-1005	156	6	1)𝑛	1)𝑛	NUM
cana-1005	157	1	+	+	CCONJ
cana-1005	157	2	1	1	NUM
cana-1005	157	3	,	,	PUNCT
cana-1005	157	4	|𝐸(𝑊𝑚	|𝐸(𝑊𝑚	PROPN
cana-1005	157	5	(	(	PUNCT
cana-1005	157	6	𝑛	𝑛	NOUN
cana-1005	157	7	)	)	PUNCT
cana-1005	157	8	)	)	PUNCT
cana-1005	158	1	|	|	ADV
cana-1005	158	2	=	=	SYM
cana-1005	158	3	𝑚𝑛(𝑚−1	𝑚𝑛(𝑚−1	PROPN
cana-1005	158	4	)	)	PUNCT
cana-1005	158	5	2	2	NUM
cana-1005	158	6	.	.	PUNCT
cana-1005	159	1	the	the	DET
cana-1005	159	2	reverse	reverse	ADJ
cana-1005	159	3	vertex	vertex	NOUN
cana-1005	159	4	degrees	degree	NOUN
cana-1005	159	5	are	be	AUX
cana-1005	159	6	ℜ𝑑𝑉𝐼	ℜ𝑑𝑉𝐼	PROPN
cana-1005	159	7	=	=	SYM
cana-1005	159	8	1	1	NUM
cana-1005	159	9	,	,	PUNCT
cana-1005	159	10	ℜ𝑑𝑉𝐼𝐼	ℜ𝑑𝑉𝐼𝐼	NOUN
cana-1005	159	11	=	=	SYM
cana-1005	159	12	(	(	PUNCT
cana-1005	159	13	𝑚	𝑚	PROPN
cana-1005	159	14	−	−	PROPN
cana-1005	159	15	1)(𝑛	1)(𝑛	NUM
cana-1005	159	16	−	−	NOUN
cana-1005	159	17	1	1	NUM
cana-1005	159	18	)	)	PUNCT
cana-1005	159	19	+	+	CCONJ
cana-1005	159	20	1	1	NUM
cana-1005	159	21	where	where	SCONJ
cana-1005	159	22	𝑉𝐼	𝑉𝐼	PROPN
cana-1005	159	23	is	be	AUX
cana-1005	159	24	the	the	DET
cana-1005	159	25	central	central	ADJ
cana-1005	159	26	vertex	vertex	NOUN
cana-1005	159	27	,	,	PUNCT
cana-1005	159	28	𝑉𝐼𝐼	𝑉𝐼𝐼	PROPN
cana-1005	159	29	is	be	AUX
cana-1005	159	30	the	the	DET
cana-1005	159	31	vertices	vertex	NOUN
cana-1005	159	32	which	which	PRON
cana-1005	159	33	are	be	AUX
cana-1005	159	34	all	all	ADV
cana-1005	159	35	adjacent	adjacent	ADJ
cana-1005	159	36	to	to	ADP
cana-1005	159	37	the	the	DET
cana-1005	159	38	central	central	ADJ
cana-1005	159	39	vertex	vertex	NOUN
cana-1005	159	40	.	.	PUNCT
cana-1005	160	1	the	the	DET
cana-1005	160	2	reverse	reverse	ADJ
cana-1005	160	3	sum	sum	NOUN
cana-1005	160	4	degrees	degree	NOUN
cana-1005	160	5	are	be	AUX
cana-1005	160	6	ℜ𝑆𝑉𝐼	ℜ𝑆𝑉𝐼	PROPN
cana-1005	160	7	=	=	PUNCT
cana-1005	160	8	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	160	9	−	−	PROPN
cana-1005	160	10	1)[(𝑚	1)[(𝑚	NUM
cana-1005	160	11	−	−	NOUN
cana-1005	160	12	1)(𝑛	1)(𝑛	NUM
cana-1005	160	13	−	−	NOUN
cana-1005	160	14	1	1	NUM
cana-1005	160	15	)	)	PUNCT
cana-1005	160	16	+	+	CCONJ
cana-1005	160	17	1	1	NUM
cana-1005	160	18	]	]	PUNCT
cana-1005	160	19	,	,	PUNCT
cana-1005	160	20	ℜ𝑆𝑉𝐼𝐼	ℜ𝑆𝑉𝐼𝐼	PROPN
cana-1005	160	21	=	=	SYM
cana-1005	160	22	(	(	PUNCT
cana-1005	160	23	𝑚	𝑚	PROPN
cana-1005	160	24	−	−	PROPN
cana-1005	160	25	1)(𝑛	1)(𝑛	NUM
cana-1005	160	26	−	−	NOUN
cana-1005	160	27	1	1	NUM
cana-1005	160	28	)	)	PUNCT
cana-1005	160	29	+	+	CCONJ
cana-1005	160	30	2	2	NUM
cana-1005	160	31	and	and	CCONJ
cana-1005	160	32	the	the	DET
cana-1005	160	33	reverse	reverse	ADJ
cana-1005	160	34	multiplicative	multiplicative	ADJ
cana-1005	160	35	degrees	degree	NOUN
cana-1005	160	36	are	be	AUX
cana-1005	160	37	ℜ𝑀𝑉𝐼	ℜ𝑀𝑉𝐼	NOUN
cana-1005	160	38	=	=	PUNCT
cana-1005	161	1	[	[	X
cana-1005	161	2	(	(	PUNCT
cana-1005	161	3	𝑚	𝑚	PROPN
cana-1005	161	4	−	−	PROPN
cana-1005	161	5	1)(𝑛	1)(𝑛	NUM
cana-1005	161	6	−	−	NOUN
cana-1005	161	7	1	1	NUM
cana-1005	161	8	)	)	PUNCT
cana-1005	161	9	+	+	CCONJ
cana-1005	161	10	1]𝑛(𝑚−1	1]𝑛(𝑚−1	NUM
cana-1005	161	11	)	)	PUNCT
cana-1005	161	12	,	,	PUNCT
cana-1005	161	13	ℜ𝑀𝑉𝐼𝐼	ℜ𝑀𝑉𝐼𝐼	PROPN
cana-1005	161	14	=	=	SYM
cana-1005	161	15	(	(	PUNCT
cana-1005	161	16	𝑚	𝑚	PROPN
cana-1005	161	17	−	−	PROPN
cana-1005	161	18	1)(𝑛	1)(𝑛	NUM
cana-1005	161	19	−	−	NOUN
cana-1005	161	20	1	1	NUM
cana-1005	161	21	)	)	PUNCT
cana-1005	161	22	+	+	CCONJ
cana-1005	161	23	1	1	X
cana-1005	161	24	.	.	X
cana-1005	161	25	then	then	ADV
cana-1005	161	26	the	the	DET
cana-1005	161	27	extended	extended	ADJ
cana-1005	161	28	reverse	reverse	ADJ
cana-1005	161	29	ℛ	ℛ	PROPN
cana-1005	161	30	degrees	degree	NOUN
cana-1005	161	31	are	be	AUX
cana-1005	161	32	𝔼ℜℛ𝑣𝐼	𝔼ℜℛ𝑣𝐼	NOUN
cana-1005	161	33	⬚	⬚	NOUN
cana-1005	161	34	=	=	PUNCT
cana-1005	161	35	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	161	36	−	−	PROPN
cana-1005	161	37	1)[(𝑚	1)[(𝑚	NUM
cana-1005	161	38	−	−	NOUN
cana-1005	161	39	1)(𝑛	1)(𝑛	NUM
cana-1005	162	1	−	−	NOUN
cana-1005	162	2	1	1	NUM
cana-1005	162	3	)	)	PUNCT
cana-1005	162	4	+	+	CCONJ
cana-1005	162	5	1	1	X
cana-1005	162	6	]	]	PUNCT
cana-1005	162	7	+	+	CCONJ
cana-1005	163	1	[	[	X
cana-1005	163	2	(	(	PUNCT
cana-1005	163	3	𝑚	𝑚	PROPN
cana-1005	163	4	−	−	PROPN
cana-1005	163	5	1)(𝑛	1)(𝑛	NUM
cana-1005	163	6	−	−	NOUN
cana-1005	163	7	1	1	NUM
cana-1005	163	8	)	)	PUNCT
cana-1005	163	9	+	+	CCONJ
cana-1005	163	10	1]𝑛(𝑚−1	1]𝑛(𝑚−1	NUM
cana-1005	163	11	)	)	PUNCT
cana-1005	163	12	,	,	PUNCT
cana-1005	163	13	𝔼ℜℛ𝑉𝐼𝐼	𝔼ℜℛ𝑉𝐼𝐼	PROPN
cana-1005	163	14	⬚	⬚	PROPN
cana-1005	163	15	=	=	SYM
cana-1005	163	16	2(𝑚	2(𝑚	NUM
cana-1005	163	17	−	−	PROPN
cana-1005	163	18	1)(𝑛	1)(𝑛	NUM
cana-1005	163	19	−	−	NOUN
cana-1005	163	20	1	1	NUM
cana-1005	163	21	)	)	PUNCT
cana-1005	163	22	+	+	CCONJ
cana-1005	163	23	3	3	X
cana-1005	163	24	.	.	X
cana-1005	163	25	hence	hence	ADV
cana-1005	163	26	,	,	PUNCT
cana-1005	163	27	the	the	DET
cana-1005	163	28	extended	extended	ADJ
cana-1005	163	29	reverse	reverse	ADJ
cana-1005	163	30	ℛ	ℛ	PROPN
cana-1005	163	31	topological	topological	ADJ
cana-1005	163	32	indices	index	NOUN
cana-1005	163	33	of	of	ADP
cana-1005	163	34	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	163	35	(	(	PUNCT
cana-1005	163	36	𝑛	𝑛	NOUN
cana-1005	163	37	)	)	PUNCT
cana-1005	163	38	are	be	AUX
cana-1005	163	39	𝔼ℜℛ	𝔼ℜℛ	SYM
cana-1005	163	40	1	1	NUM
cana-1005	163	41	(	(	PUNCT
cana-1005	163	42	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	163	43	(	(	PUNCT
cana-1005	163	44	𝑛	𝑛	NOUN
cana-1005	163	45	)	)	PUNCT
cana-1005	163	46	)	)	PUNCT
cana-1005	164	1	=	=	PRON
cana-1005	164	2	{	{	PUNCT
cana-1005	164	3	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	164	4	−	−	PROPN
cana-1005	164	5	1)[(𝑚	1)[(𝑚	NUM
cana-1005	164	6	−	−	NOUN
cana-1005	164	7	1)(𝑛	1)(𝑛	NUM
cana-1005	164	8	−	−	NOUN
cana-1005	164	9	1	1	NUM
cana-1005	164	10	)	)	PUNCT
cana-1005	164	11	+	+	CCONJ
cana-1005	164	12	1	1	X
cana-1005	164	13	]	]	PUNCT
cana-1005	164	14	+	+	CCONJ
cana-1005	164	15	[	[	X
cana-1005	164	16	(	(	PUNCT
cana-1005	164	17	𝑚	𝑚	PROPN
cana-1005	164	18	−	−	PROPN
cana-1005	164	19	1)(𝑛	1)(𝑛	NUM
cana-1005	164	20	−	−	NOUN
cana-1005	164	21	1	1	NUM
cana-1005	164	22	)	)	PUNCT
cana-1005	164	23	+	+	CCONJ
cana-1005	164	24	1]𝑛(𝑚−1	1]𝑛(𝑚−1	X
cana-1005	164	25	)	)	PUNCT
cana-1005	164	26	}	}	PUNCT
cana-1005	164	27	2	2	NUM
cana-1005	165	1	+	+	NUM
cana-1005	165	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	165	3	−	−	PROPN
cana-1005	165	4	1)[2(𝑚	1)[2(𝑚	NUM
cana-1005	165	5	−	−	NOUN
cana-1005	165	6	1)(𝑛	1)(𝑛	NUM
cana-1005	165	7	−	−	NOUN
cana-1005	165	8	1	1	NUM
cana-1005	165	9	)	)	PUNCT
cana-1005	165	10	+	+	CCONJ
cana-1005	165	11	3]2	3]2	NOUN
cana-1005	165	12	,	,	PUNCT
cana-1005	165	13	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	165	14	2	2	NUM
cana-1005	165	15	(	(	PUNCT
cana-1005	165	16	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	165	17	(	(	PUNCT
cana-1005	165	18	𝑛	𝑛	NOUN
cana-1005	165	19	)	)	PUNCT
cana-1005	165	20	)	)	PUNCT
cana-1005	166	1	=	=	PUNCT
cana-1005	166	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	166	3	−	−	NOUN
cana-1005	166	4	1){(𝑛(𝑚	1){(𝑛(𝑚	ADJ
cana-1005	166	5	−	−	PROPN
cana-1005	166	6	1)[(𝑚	1)[(𝑚	NUM
cana-1005	166	7	−	−	NOUN
cana-1005	166	8	1)(𝑛	1)(𝑛	NUM
cana-1005	166	9	−	−	NOUN
cana-1005	166	10	1	1	NUM
cana-1005	166	11	)	)	PUNCT
cana-1005	166	12	+	+	CCONJ
cana-1005	166	13	1	1	X
cana-1005	166	14	]	]	PUNCT
cana-1005	166	15	+	+	CCONJ
cana-1005	167	1	[	[	X
cana-1005	167	2	(	(	PUNCT
cana-1005	167	3	𝑚	𝑚	PROPN
cana-1005	167	4	−	−	PROPN
cana-1005	167	5	1)(𝑛	1)(𝑛	NUM
cana-1005	167	6	−	−	NOUN
cana-1005	167	7	1	1	NUM
cana-1005	167	8	)	)	PUNCT
cana-1005	167	9	+	+	NUM
cana-1005	167	10	1]𝑛(𝑚−1	1]𝑛(𝑚−1	NUM
cana-1005	167	11	)	)	PUNCT
cana-1005	167	12	)	)	PUNCT
cana-1005	168	1	(	(	PUNCT
cana-1005	168	2	2(𝑚	2(𝑚	NUM
cana-1005	168	3	−	−	NOUN
cana-1005	168	4	1)(𝑛	1)(𝑛	NUM
cana-1005	168	5	−	−	NOUN
cana-1005	168	6	1	1	NUM
cana-1005	168	7	)	)	PUNCT
cana-1005	168	8	+	+	CCONJ
cana-1005	168	9	3	3	NUM
cana-1005	168	10	)	)	PUNCT
cana-1005	168	11	}	}	PUNCT
cana-1005	168	12	+	+	PUNCT
cana-1005	169	1	[	[	X
cana-1005	169	2	2(𝑚	2(𝑚	NUM
cana-1005	169	3	−	−	NOUN
cana-1005	169	4	1)(𝑛	1)(𝑛	NUM
cana-1005	169	5	−	−	NOUN
cana-1005	169	6	1	1	NUM
cana-1005	169	7	)	)	PUNCT
cana-1005	169	8	+	+	CCONJ
cana-1005	169	9	3]2	3]2	NOUN
cana-1005	169	10	,	,	PUNCT
cana-1005	169	11	𝔼ℜℛ	𝔼ℜℛ	PROPN
cana-1005	169	12	3	3	NUM
cana-1005	169	13	(	(	PUNCT
cana-1005	169	14	𝑊𝑚	𝑊𝑚	PROPN
cana-1005	169	15	(	(	PUNCT
cana-1005	169	16	𝑛	𝑛	NOUN
cana-1005	169	17	)	)	PUNCT
cana-1005	169	18	)	)	PUNCT
cana-1005	170	1	=	=	PUNCT
cana-1005	170	2	𝑛(𝑚	𝑛(𝑚	NOUN
cana-1005	170	3	−	−	NOUN
cana-1005	170	4	1){(𝑛(𝑚	1){(𝑛(𝑚	ADJ
cana-1005	170	5	−	−	PROPN
cana-1005	170	6	1)[(𝑚	1)[(𝑚	NUM
cana-1005	170	7	−	−	NOUN
cana-1005	170	8	1)(𝑛	1)(𝑛	NUM
cana-1005	170	9	−	−	NOUN
cana-1005	170	10	1	1	NUM
cana-1005	170	11	)	)	PUNCT
cana-1005	170	12	+	+	CCONJ
cana-1005	170	13	1	1	X
cana-1005	170	14	]	]	PUNCT
cana-1005	170	15	+	+	CCONJ
cana-1005	171	1	[	[	X
cana-1005	171	2	(	(	PUNCT
cana-1005	171	3	𝑚	𝑚	PROPN
cana-1005	171	4	−	−	PROPN
cana-1005	171	5	1)(𝑛	1)(𝑛	NUM
cana-1005	171	6	−	−	NOUN
cana-1005	171	7	1	1	NUM
cana-1005	171	8	)	)	PUNCT
cana-1005	171	9	+	+	NUM
cana-1005	171	10	1]𝑛(𝑚−1	1]𝑛(𝑚−1	NUM
cana-1005	171	11	)	)	PUNCT
cana-1005	171	12	)	)	PUNCT
cana-1005	172	1	+	+	CCONJ
cana-1005	172	2	(	(	PUNCT
cana-1005	172	3	2(𝑚	2(𝑚	NUM
cana-1005	172	4	−	−	NOUN
cana-1005	172	5	1)(𝑛	1)(𝑛	NUM
cana-1005	172	6	−	−	NOUN
cana-1005	172	7	1	1	NUM
cana-1005	172	8	)	)	PUNCT
cana-1005	172	9	+	+	CCONJ
cana-1005	172	10	3	3	NUM
cana-1005	172	11	)	)	PUNCT
cana-1005	172	12	}	}	PUNCT
cana-1005	172	13	+	+	PUNCT
cana-1005	173	1	[	[	X
cana-1005	173	2	2(𝑚	2(𝑚	NUM
cana-1005	173	3	−	−	NOUN
cana-1005	173	4	1)(𝑛	1)(𝑛	NUM
cana-1005	173	5	−	−	NOUN
cana-1005	173	6	1	1	NUM
cana-1005	173	7	)	)	PUNCT
cana-1005	173	8	+	+	CCONJ
cana-1005	173	9	3]2	3]2	NOUN
cana-1005	173	10	.	.	PUNCT
cana-1005	173	11	4	4	NUM
cana-1005	173	12	.	.	X
cana-1005	173	13	conclusion	conclusion	NOUN
cana-1005	173	14	in	in	ADP
cana-1005	173	15	this	this	DET
cana-1005	173	16	paper	paper	NOUN
cana-1005	173	17	,	,	PUNCT
cana-1005	173	18	the	the	DET
cana-1005	173	19	brand	brand	NOUN
cana-1005	173	20	new	new	ADJ
cana-1005	173	21	degree	degree	NOUN
cana-1005	173	22	based	base	VERB
cana-1005	173	23	topological	topological	ADJ
cana-1005	173	24	indices	index	NOUN
cana-1005	173	25	such	such	ADJ
cana-1005	173	26	as	as	ADP
cana-1005	173	27	extended	extended	ADJ
cana-1005	173	28	reverse	reverse	NOUN
cana-1005	173	29	r	r	NOUN
cana-1005	173	30	indices	index	NOUN
cana-1005	173	31	are	be	AUX
cana-1005	173	32	elucidated	elucidate	VERB
cana-1005	173	33	using	use	VERB
cana-1005	173	34	the	the	DET
cana-1005	173	35	reverse	reverse	ADJ
cana-1005	173	36	sum	sum	NOUN
cana-1005	173	37	degree	degree	NOUN
cana-1005	173	38	,	,	PUNCT
cana-1005	173	39	reverse	reverse	VERB
cana-1005	173	40	multiplicative	multiplicative	ADJ
cana-1005	173	41	degree	degree	NOUN
cana-1005	173	42	and	and	CCONJ
cana-1005	173	43	extended	extend	VERB
cana-1005	173	44	reverse	reverse	NOUN
cana-1005	173	45	r	r	NOUN
cana-1005	173	46	degree	degree	NOUN
cana-1005	173	47	.	.	PUNCT
cana-1005	174	1	make	make	VERB
cana-1005	174	2	use	use	NOUN
cana-1005	174	3	of	of	ADP
cana-1005	174	4	,	,	PUNCT
cana-1005	174	5	extended	extend	VERB
cana-1005	174	6	reverse	reverse	NOUN
cana-1005	174	7	r	r	NOUN
cana-1005	174	8	indices	index	NOUN
cana-1005	174	9	for	for	ADP
cana-1005	174	10	the	the	DET
cana-1005	174	11	complete	complete	ADJ
cana-1005	174	12	bipartite	bipartite	NOUN
cana-1005	174	13	graph	graph	NOUN
cana-1005	174	14	,	,	PUNCT
cana-1005	174	15	wheel	wheel	NOUN
cana-1005	174	16	graph	graph	NOUN
cana-1005	174	17	,	,	PUNCT
cana-1005	174	18	generalized	generalized	ADJ
cana-1005	174	19	peterson	peterson	NOUN
cana-1005	174	20	graph	graph	NOUN
cana-1005	174	21	,	,	PUNCT
cana-1005	174	22	crown	crown	NOUN
cana-1005	174	23	graph	graph	NOUN
cana-1005	174	24	,	,	PUNCT
cana-1005	174	25	double	double	ADJ
cana-1005	174	26	star	star	NOUN
cana-1005	174	27	graph	graph	NOUN
cana-1005	174	28	and	and	CCONJ
cana-1005	174	29	windmill	windmill	NOUN
cana-1005	174	30	graph	graph	NOUN
cana-1005	174	31	are	be	AUX
cana-1005	174	32	structurally	structurally	ADV
cana-1005	174	33	communications	communication	NOUN
cana-1005	174	34	on	on	ADP
cana-1005	174	35	applied	apply	VERB
cana-1005	174	36	nonlinear	nonlinear	ADJ
cana-1005	174	37	analysis	analysis	NOUN
cana-1005	174	38	issn	issn	NOUN
cana-1005	174	39	:	:	PUNCT
cana-1005	174	40	1074	1074	NUM
cana-1005	174	41	-	-	PUNCT
cana-1005	174	42	133x	133x	NUM
cana-1005	174	43	vol	vol	NOUN
cana-1005	174	44	31	31	NUM
cana-1005	174	45	no	no	NOUN
cana-1005	174	46	.	.	PUNCT
cana-1005	175	1	5s	5s	NUM
cana-1005	175	2	(	(	PUNCT
cana-1005	175	3	2024	2024	NUM
cana-1005	175	4	)	)	PUNCT
cana-1005	175	5	117	117	NUM
cana-1005	175	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1005	175	7	characterized	characterized	ADJ
cana-1005	175	8	.	.	PUNCT
cana-1005	176	1	this	this	DET
cana-1005	176	2	study	study	NOUN
cana-1005	176	3	may	may	AUX
cana-1005	176	4	be	be	AUX
cana-1005	176	5	regarded	regard	VERB
cana-1005	176	6	as	as	ADP
cana-1005	176	7	an	an	DET
cana-1005	176	8	introduction	introduction	NOUN
cana-1005	176	9	to	to	ADP
cana-1005	176	10	the	the	DET
cana-1005	176	11	topic	topic	NOUN
cana-1005	176	12	and	and	CCONJ
cana-1005	176	13	may	may	AUX
cana-1005	176	14	seek	seek	VERB
cana-1005	176	15	to	to	PART
cana-1005	176	16	identify	identify	VERB
cana-1005	176	17	further	further	ADJ
cana-1005	176	18	recourse	recourse	NOUN
cana-1005	176	19	for	for	ADP
cana-1005	176	20	simple	simple	ADJ
cana-1005	176	21	connected	connected	ADJ
cana-1005	176	22	graphs	graph	NOUN
cana-1005	176	23	.	.	PUNCT
cana-1005	177	1	also	also	ADV
cana-1005	177	2	one	one	PRON
cana-1005	177	3	could	could	AUX
cana-1005	177	4	concentrate	concentrate	VERB
cana-1005	177	5	on	on	ADP
cana-1005	177	6	chemical	chemical	NOUN
cana-1005	177	7	graphs	graph	NOUN
cana-1005	177	8	.	.	PUNCT
cana-1005	178	1	acknowledgement	acknowledgement	NOUN
cana-1005	178	2	.	.	PUNCT
cana-1005	179	1	this	this	DET
cana-1005	179	2	research	research	NOUN
cana-1005	179	3	work	work	NOUN
cana-1005	179	4	funded	fund	VERB
cana-1005	179	5	by	by	ADP
cana-1005	179	6	‘	'	PUNCT
cana-1005	179	7	university	university	NOUN
cana-1005	179	8	grand	grand	ADJ
cana-1005	179	9	commission(ugc	commission(ugc	PROPN
cana-1005	179	10	)	)	PUNCT
cana-1005	179	11	’	'	PUNCT
cana-1005	179	12	india	india	PROPN
cana-1005	179	13	under	under	ADP
cana-1005	179	14	the	the	DET
cana-1005	179	15	rajiv	rajiv	PROPN
cana-1005	179	16	gandhi	gandhi	PROPN
cana-1005	179	17	national	national	ADJ
cana-1005	179	18	fellowship	fellowship	NOUN
cana-1005	179	19	(	(	PUNCT
cana-1005	179	20	rgnf	rgnf	NOUN
cana-1005	179	21	)	)	PUNCT
cana-1005	179	22	scheme	scheme	NOUN
cana-1005	179	23	and	and	CCONJ
cana-1005	179	24	the	the	DET
cana-1005	179	25	scheme	scheme	NOUN
cana-1005	179	26	no	no	NOUN
cana-1005	179	27	:	:	PUNCT
cana-1005	179	28	f117.1/201617	f117.1/201617	ADJ
cana-1005	179	29	/	/	SYM
cana-1005	179	30	rgnf201517sctam27633/(saiii	rgnf201517sctam27633/(saiii	PROPN
cana-1005	179	31	/	/	SYM
cana-1005	179	32	website	website	NOUN
cana-1005	179	33	)	)	PUNCT
cana-1005	179	34	.	.	PUNCT
cana-1005	180	1	references	reference	NOUN
cana-1005	180	2	[	[	X
cana-1005	180	3	1	1	NUM
cana-1005	180	4	]	]	X
cana-1005	180	5	das	das	PROPN
cana-1005	180	6	,	,	PUNCT
cana-1005	180	7	k.c	k.c	PROPN
cana-1005	180	8	.	.	PROPN
cana-1005	180	9	;	;	PUNCT
cana-1005	180	10	xu	xu	PROPN
cana-1005	180	11	k.	k.	PROPN
cana-1005	180	12	;	;	PUNCT
cana-1005	180	13	nam	nam	PROPN
cana-1005	180	14	,	,	PUNCT
cana-1005	180	15	j.	j.	PROPN
cana-1005	180	16	:	:	PUNCT
cana-1005	180	17	on	on	ADP
cana-1005	180	18	zagreb	zagreb	PROPN
cana-1005	180	19	indices	index	NOUN
cana-1005	180	20	of	of	ADP
cana-1005	180	21	graphs	graph	NOUN
cana-1005	180	22	,	,	PUNCT
cana-1005	180	23	front	front	NOUN
cana-1005	180	24	.	.	PUNCT
cana-1005	181	1	math	math	NOUN
cana-1005	181	2	.	.	PUNCT
cana-1005	182	1	china	china	PROPN
cana-1005	182	2	,	,	PUNCT
cana-1005	182	3	10	10	NUM
cana-1005	182	4	562	562	NUM
cana-1005	182	5	-	-	SYM
cana-1005	182	6	582	582	NUM
cana-1005	182	7	(	(	PUNCT
cana-1005	182	8	2015	2015	NUM
cana-1005	182	9	)	)	PUNCT
cana-1005	182	10	.	.	PUNCT
cana-1005	183	1	[	[	X
cana-1005	183	2	2	2	NUM
cana-1005	183	3	]	]	PUNCT
cana-1005	183	4	durgia	durgia	NOUN
cana-1005	183	5	,	,	PUNCT
cana-1005	183	6	b.s	b.s	PROPN
cana-1005	183	7	.	.	PROPN
cana-1005	183	8	;	;	PUNCT
cana-1005	183	9	mekkalikeb	mekkalikeb	PROPN
cana-1005	183	10	,	,	PUNCT
cana-1005	183	11	s.m.;and	s.m.;and	X
cana-1005	183	12	ramane	ramane	PROPN
cana-1005	183	13	,	,	PUNCT
cana-1005	183	14	h.s	h.s	PROPN
cana-1005	183	15	.	.	PROPN
cana-1005	183	16	:	:	PUNCT
cana-1005	183	17	on	on	ADP
cana-1005	183	18	the	the	DET
cana-1005	183	19	zagreb	zagreb	PROPN
cana-1005	183	20	indices	index	NOUN
cana-1005	183	21	of	of	ADP
cana-1005	183	22	semi	semi	ADJ
cana-1005	183	23	total	total	ADJ
cana-1005	183	24	point	point	NOUN
cana-1005	183	25	graphs	graph	NOUN
cana-1005	183	26	of	of	ADP
cana-1005	183	27	some	some	DET
cana-1005	183	28	graphs	graph	NOUN
cana-1005	183	29	handed	hand	VERB
cana-1005	183	30	,	,	PUNCT
cana-1005	183	31	annals	annal	NOUN
cana-1005	183	32	of	of	ADP
cana-1005	183	33	pure	pure	ADJ
cana-1005	183	34	and	and	CCONJ
cana-1005	183	35	applied	applied	ADJ
cana-1005	183	36	mathematics	mathematic	NOUN
cana-1005	183	37	,	,	PUNCT
cana-1005	183	38	12(1	12(1	NUM
cana-1005	183	39	)	)	PUNCT
cana-1005	183	40	49	49	NUM
cana-1005	183	41	-	-	SYM
cana-1005	183	42	57(2016	57(2016	NUM
cana-1005	183	43	)	)	PUNCT
cana-1005	183	44	.	.	PUNCT
cana-1005	184	1	[	[	X
cana-1005	184	2	3	3	X
cana-1005	184	3	]	]	X
cana-1005	184	4	estrada	estrada	PROPN
cana-1005	184	5	,	,	PUNCT
cana-1005	184	6	e.	e.	PROPN
cana-1005	184	7	;	;	PUNCT
cana-1005	184	8	torres	torre	NOUN
cana-1005	184	9	,	,	PUNCT
cana-1005	184	10	l.	l.	PROPN
cana-1005	184	11	;	;	PUNCT
cana-1005	184	12	rodríguez	rodríguez	PROPN
cana-1005	184	13	l.	l.	PROPN
cana-1005	184	14	;	;	PUNCT
cana-1005	184	15	and	and	CCONJ
cana-1005	184	16	gutman	gutman	NOUN
cana-1005	184	17	,	,	PUNCT
cana-1005	184	18	i.	i.	PROPN
cana-1005	184	19	:	:	PUNCT
cana-1005	184	20	indian	indian	PROPN
cana-1005	184	21	j.	j.	PROPN
cana-1005	184	22	chem	chem	PROPN
cana-1005	184	23	.	.	PUNCT
cana-1005	184	24	,	,	PUNCT
cana-1005	184	25	37a	37a	NOUN
cana-1005	184	26	849(1998	849(1998	NUM
cana-1005	184	27	)	)	PUNCT
cana-1005	184	28	.	.	PUNCT
cana-1005	185	1	[	[	X
cana-1005	185	2	4	4	NUM
cana-1005	185	3	]	]	X
cana-1005	185	4	gao	gao	PROPN
cana-1005	185	5	,	,	PUNCT
cana-1005	185	6	w.	w.	PROPN
cana-1005	185	7	;	;	PUNCT
cana-1005	185	8	liang	liang	PROPN
cana-1005	185	9	,	,	PUNCT
cana-1005	185	10	l.	l.	PROPN
cana-1005	185	11	;	;	PUNCT
cana-1005	185	12	and	and	CCONJ
cana-1005	185	13	chen	chen	PROPN
cana-1005	185	14	,	,	PUNCT
cana-1005	185	15	y.	y.	NOUN
cana-1005	185	16	:	:	PUNCT
cana-1005	185	17	on	on	ADP
cana-1005	185	18	second	second	ADJ
cana-1005	185	19	geometric	geometric	ADJ
cana-1005	185	20	-	-	PUNCT
cana-1005	185	21	arithmetic	arithmetic	ADJ
cana-1005	185	22	index	index	NOUN
cana-1005	185	23	and	and	CCONJ
cana-1005	185	24	co	co	ADJ
cana-1005	185	25	-	-	ADJ
cana-1005	185	26	pi	pi	ADJ
cana-1005	185	27	index	index	NOUN
cana-1005	185	28	of	of	ADP
cana-1005	185	29	special	special	ADJ
cana-1005	185	30	chemical	chemical	NOUN
cana-1005	185	31	molecular	molecular	ADJ
cana-1005	185	32	structures	structure	NOUN
cana-1005	185	33	,	,	PUNCT
cana-1005	185	34	annals	annal	NOUN
cana-1005	185	35	of	of	ADP
cana-1005	185	36	pure	pure	ADJ
cana-1005	185	37	and	and	CCONJ
cana-1005	185	38	applied	apply	VERB
cana-1005	185	39	mathematics,13(1	mathematics,13(1	PROPN
cana-1005	185	40	)	)	PUNCT
cana-1005	185	41	99	99	NUM
cana-1005	185	42	-	-	SYM
cana-1005	185	43	117	117	NUM
cana-1005	185	44	(	(	PUNCT
cana-1005	185	45	2017	2017	NUM
cana-1005	185	46	)	)	PUNCT
cana-1005	185	47	.	.	PUNCT
cana-1005	186	1	[	[	X
cana-1005	186	2	5	5	NUM
cana-1005	186	3	]	]	X
cana-1005	186	4	gutman	gutman	NOUN
cana-1005	186	5	,	,	PUNCT
cana-1005	186	6	i.	i.	PROPN
cana-1005	186	7	;	;	PUNCT
cana-1005	186	8	and	and	CCONJ
cana-1005	186	9	trinajstic	trinajstic	ADJ
cana-1005	186	10	,	,	PUNCT
cana-1005	186	11	n.	n.	NOUN
cana-1005	186	12	:	:	PUNCT
cana-1005	186	13	graph	graph	NOUN
cana-1005	186	14	theory	theory	NOUN
cana-1005	186	15	and	and	CCONJ
cana-1005	186	16	molecular	molecular	ADJ
cana-1005	186	17	orbitals	orbital	NOUN
cana-1005	186	18	,	,	PUNCT
cana-1005	186	19	total	total	ADJ
cana-1005	186	20	pi	pi	NOUN
cana-1005	186	21	-	-	PUNCT
cana-1005	186	22	electron	electron	NOUN
cana-1005	186	23	energy	energy	NOUN
cana-1005	186	24	of	of	ADP
cana-1005	186	25	alternant	alternant	ADJ
cana-1005	186	26	hydrocarbons	hydrocarbon	NOUN
cana-1005	186	27	,	,	PUNCT
cana-1005	186	28	chemical	chemical	NOUN
cana-1005	186	29	physics	physics	NOUN
cana-1005	186	30	letters	letter	NOUN
cana-1005	186	31	,	,	PUNCT
cana-1005	186	32	17	17	NUM
cana-1005	186	33	535	535	NUM
cana-1005	186	34	-	-	PUNCT
cana-1005	186	35	538(1972	538(1972	NUM
cana-1005	186	36	)	)	PUNCT
cana-1005	186	37	.	.	PUNCT
cana-1005	187	1	[	[	X
cana-1005	187	2	6	6	NUM
cana-1005	187	3	]	]	X
cana-1005	187	4	gutman	gutman	NOUN
cana-1005	187	5	,	,	PUNCT
cana-1005	187	6	i.	i.	PROPN
cana-1005	187	7	;	;	PUNCT
cana-1005	187	8	ruščić	ruščić	PROPN
cana-1005	187	9	,	,	PUNCT
cana-1005	187	10	b.	b.	PROPN
cana-1005	187	11	;	;	PUNCT
cana-1005	187	12	trinajstić	trinajstić	PROPN
cana-1005	187	13	,	,	PUNCT
cana-1005	187	14	n.	n.	PROPN
cana-1005	187	15	;	;	PUNCT
cana-1005	187	16	wilcox	wilcox	PROPN
cana-1005	187	17	,	,	PUNCT
cana-1005	187	18	c.n	c.n	PROPN
cana-1005	187	19	.	.	PROPN
cana-1005	187	20	:	:	PUNCT
cana-1005	187	21	graph	graph	NOUN
cana-1005	187	22	theory	theory	NOUN
cana-1005	187	23	and	and	CCONJ
cana-1005	187	24	molecular	molecular	ADJ
cana-1005	187	25	orbitals	orbital	NOUN
cana-1005	187	26	xii	xii	NOUN
cana-1005	187	27	,	,	PUNCT
cana-1005	187	28	acyclic	acyclic	ADJ
cana-1005	187	29	polyenes	polyene	NOUN
cana-1005	187	30	,	,	PUNCT
cana-1005	187	31	j.chem	j.chem	NOUN
cana-1005	187	32	.	.	PUNCT
cana-1005	188	1	phys	phy	NOUN
cana-1005	188	2	.	.	PUNCT
cana-1005	188	3	,	,	PUNCT
cana-1005	188	4	62	62	NUM
cana-1005	188	5	3399	3399	NUM
cana-1005	188	6	-	-	SYM
cana-1005	188	7	3405(1975	3405(1975	NUM
cana-1005	188	8	)	)	PUNCT
cana-1005	188	9	.	.	PUNCT
cana-1005	189	1	[	[	X
cana-1005	189	2	7	7	X
cana-1005	189	3	]	]	X
cana-1005	189	4	gutman	gutman	NOUN
cana-1005	189	5	,	,	PUNCT
cana-1005	189	6	i.	i.	PROPN
cana-1005	189	7	and	and	CCONJ
cana-1005	189	8	trinajstić	trinajstić	PROPN
cana-1005	189	9	,	,	PUNCT
cana-1005	189	10	n.	n.	NOUN
cana-1005	189	11	:	:	PUNCT
cana-1005	189	12	graph	graph	NOUN
cana-1005	189	13	theory	theory	NOUN
cana-1005	189	14	and	and	CCONJ
cana-1005	189	15	molecular	molecular	ADJ
cana-1005	189	16	orbitals	orbital	NOUN
cana-1005	189	17	.	.	PUNCT
cana-1005	190	1	xv	xv	PROPN
cana-1005	190	2	.	.	PUNCT
cana-1005	191	1	the	the	DET
cana-1005	191	2	huckle	huckle	NOUN
cana-1005	191	3	rule	rule	NOUN
cana-1005	191	4	,	,	PUNCT
cana-1005	191	5	the	the	DET
cana-1005	191	6	journal	journal	NOUN
cana-1005	191	7	of	of	ADP
cana-1005	191	8	chemical	chemical	PROPN
cana-1005	191	9	physics	physics	PROPN
cana-1005	191	10	,	,	PUNCT
cana-1005	191	11	64	64	NUM
cana-1005	191	12	(	(	PUNCT
cana-1005	191	13	1976	1976	NUM
cana-1005	191	14	)	)	PUNCT
cana-1005	191	15	4921	4921	NUM
cana-1005	191	16	.	.	PUNCT
cana-1005	192	1	[	[	X
cana-1005	192	2	8	8	NUM
cana-1005	192	3	]	]	X
cana-1005	192	4	ilić	ilić	ADJ
cana-1005	192	5	.a	.a	NOUN
cana-1005	192	6	,	,	PUNCT
cana-1005	192	7	note	note	VERB
cana-1005	192	8	on	on	ADP
cana-1005	192	9	the	the	DET
cana-1005	192	10	harmonic	harmonic	ADJ
cana-1005	192	11	index	index	NOUN
cana-1005	192	12	of	of	ADP
cana-1005	192	13	a	a	DET
cana-1005	192	14	graph	graph	NOUN
cana-1005	192	15	,	,	PUNCT
cana-1005	192	16	ars	ar	VERB
cana-1005	192	17	combin	combin	NOUN
cana-1005	192	18	.	.	PUNCT
cana-1005	192	19	,	,	PUNCT
cana-1005	192	20	128	128	NUM
cana-1005	192	21	295–299(2016	295–299(2016	NUM
cana-1005	192	22	)	)	PUNCT
cana-1005	192	23	.	.	PUNCT
cana-1005	193	1	[	[	X
cana-1005	193	2	9	9	NUM
cana-1005	193	3	]	]	PUNCT
cana-1005	193	4	ivan	ivan	PROPN
cana-1005	193	5	gutman	gutman	PROPN
cana-1005	193	6	,	,	PUNCT
cana-1005	193	7	degree	degree	NOUN
cana-1005	193	8	-	-	PUNCT
cana-1005	193	9	based	base	VERB
cana-1005	193	10	topological	topological	ADJ
cana-1005	193	11	indices	index	NOUN
cana-1005	193	12	,	,	PUNCT
cana-1005	193	13	croat	croat	NOUN
cana-1005	193	14	.	.	PUNCT
cana-1005	194	1	chem	chem	PROPN
cana-1005	194	2	.	.	PUNCT
cana-1005	195	1	acta	acta	PROPN
cana-1005	195	2	,	,	PUNCT
cana-1005	195	3	86	86	NUM
cana-1005	195	4	(	(	PUNCT
cana-1005	195	5	4	4	NUM
cana-1005	195	6	)	)	PUNCT
cana-1005	195	7	351–361(2013	351–361(2013	NUM
cana-1005	195	8	)	)	PUNCT
cana-1005	195	9	.	.	PUNCT
cana-1005	196	1	[	[	X
cana-1005	196	2	10	10	NUM
cana-1005	196	3	]	]	X
cana-1005	196	4	li	li	PROPN
cana-1005	196	5	,	,	PUNCT
cana-1005	196	6	j.	j.	PROPN
cana-1005	196	7	;	;	PUNCT
cana-1005	196	8	lv	lv	PROPN
cana-1005	196	9	,	,	PUNCT
cana-1005	196	10	j.b	j.b	PROPN
cana-1005	196	11	.	.	PROPN
cana-1005	196	12	and	and	CCONJ
cana-1005	196	13	liu	liu	PROPN
cana-1005	196	14	,	,	PUNCT
cana-1005	196	15	y.	y.	PROPN
cana-1005	196	16	:	:	PUNCT
cana-1005	196	17	the	the	DET
cana-1005	196	18	harmonic	harmonic	ADJ
cana-1005	196	19	index	index	NOUN
cana-1005	196	20	of	of	ADP
cana-1005	196	21	some	some	DET
cana-1005	196	22	graphs	graph	NOUN
cana-1005	196	23	,	,	PUNCT
cana-1005	196	24	bull	bull	NOUN
cana-1005	196	25	.	.	PUNCT
cana-1005	197	1	malays	malays	PROPN
cana-1005	197	2	.	.	PUNCT
cana-1005	198	1	math	math	NOUN
cana-1005	198	2	.	.	PUNCT
cana-1005	199	1	sci	sci	PROPN
cana-1005	199	2	soc	soc	PROPN
cana-1005	199	3	.	.	PUNCT
cana-1005	199	4	,39	,39	PUNCT
cana-1005	199	5	331–340(2016	331–340(2016	NUM
cana-1005	199	6	)	)	PUNCT
cana-1005	199	7	.	.	PUNCT
cana-1005	200	1	[	[	X
cana-1005	200	2	11	11	NUM
cana-1005	200	3	]	]	X
cana-1005	200	4	milicevic	milicevic	ADJ
cana-1005	200	5	,	,	PUNCT
cana-1005	200	6	a.	a.	NOUN
cana-1005	200	7	;	;	PUNCT
cana-1005	200	8	niiolic	niiolic	PROPN
cana-1005	200	9	,	,	PUNCT
cana-1005	200	10	s.	s.	PROPN
cana-1005	200	11	and	and	CCONJ
cana-1005	200	12	trinajstic	trinajstic	ADJ
cana-1005	200	13	,	,	PUNCT
cana-1005	200	14	n.	n.	NOUN
cana-1005	200	15	:	:	PUNCT
cana-1005	200	16	on	on	ADP
cana-1005	200	17	reformulated	reformulate	VERB
cana-1005	200	18	zagreb	zagreb	PROPN
cana-1005	200	19	indices	index	NOUN
cana-1005	200	20	,	,	PUNCT
cana-1005	200	21	mo	mo	PROPN
cana-1005	200	22	..	..	PROPN
cana-1005	200	23	divers	diver	NOUN
cana-1005	200	24	..	..	PUNCT
cana-1005	200	25	8	8	NUM
cana-1005	200	26	393	393	NUM
cana-1005	200	27	-	-	NUM
cana-1005	200	28	399(2000	399(2000	NUM
cana-1005	200	29	)	)	PUNCT
cana-1005	200	30	.	.	PUNCT
cana-1005	201	1	[	[	X
cana-1005	201	2	12	12	NUM
cana-1005	201	3	]	]	PUNCT
cana-1005	201	4	randić	randić	NOUN
cana-1005	201	5	,	,	PUNCT
cana-1005	201	6	m.	m.	NOUN
cana-1005	201	7	:	:	PUNCT
cana-1005	201	8	characterization	characterization	NOUN
cana-1005	201	9	of	of	ADP
cana-1005	201	10	molecular	molecular	ADJ
cana-1005	201	11	branching	branching	NOUN
cana-1005	201	12	,	,	PUNCT
cana-1005	201	13	j.	j.	PROPN
cana-1005	201	14	am	am	PROPN
cana-1005	201	15	.	.	PUNCT
cana-1005	202	1	chem	chem	NOUN
cana-1005	202	2	.	.	PUNCT
cana-1005	203	1	soc	soc	PROPN
cana-1005	203	2	.	.	PUNCT
cana-1005	204	1	,	,	PUNCT
cana-1005	204	2	97,6609	97,6609	PROPN
cana-1005	204	3	-	-	SYM
cana-1005	204	4	6615	6615	NUM
cana-1005	204	5	(	(	PUNCT
cana-1005	204	6	1975	1975	NUM
cana-1005	204	7	)	)	PUNCT
cana-1005	204	8	.	.	PUNCT
cana-1005	205	1	[	[	X
cana-1005	205	2	13	13	NUM
cana-1005	205	3	]	]	SYM
cana-1005	205	4	siileyman	siileyman	NOUN
cana-1005	205	5	,	,	PUNCT
cana-1005	205	6	e.	e.	PROPN
cana-1005	205	7	:	:	PUNCT
cana-1005	205	8	on	on	ADP
cana-1005	205	9	r	r	NOUN
cana-1005	205	10	degrees	degree	NOUN
cana-1005	205	11	of	of	ADP
cana-1005	205	12	vertices	vertex	NOUN
cana-1005	205	13	and	and	CCONJ
cana-1005	205	14	r	r	NOUN
cana-1005	205	15	indices	index	NOUN
cana-1005	205	16	of	of	ADP
cana-1005	205	17	graphs	graph	NOUN
cana-1005	205	18	,	,	PUNCT
cana-1005	205	19	international	international	ADJ
cana-1005	205	20	journal	journal	NOUN
cana-1005	205	21	of	of	ADP
cana-1005	205	22	advanced	advanced	ADJ
cana-1005	205	23	chemistry	chemistry	NOUN
cana-1005	205	24	,	,	PUNCT
cana-1005	205	25	5(2	5(2	NUM
cana-1005	205	26	)	)	PUNCT
cana-1005	205	27	70	70	NUM
cana-1005	205	28	-	-	SYM
cana-1005	205	29	72(2017	72(2017	NUM
cana-1005	205	30	)	)	PUNCT
cana-1005	205	31	.	.	PUNCT
cana-1005	206	1	[	[	X
cana-1005	206	2	14	14	NUM
cana-1005	206	3	]	]	X
cana-1005	206	4	sumathi	sumathi	ADV
cana-1005	206	5	,	,	PUNCT
cana-1005	206	6	a.	a.	NOUN
cana-1005	206	7	:	:	PUNCT
cana-1005	206	8	r	r	NOUN
cana-1005	206	9	index	index	NOUN
cana-1005	206	10	of	of	ADP
cana-1005	206	11	some	some	DET
cana-1005	206	12	graphs	graph	NOUN
cana-1005	206	13	,	,	PUNCT
cana-1005	206	14	annals	annal	NOUN
cana-1005	206	15	of	of	ADP
cana-1005	206	16	pure	pure	ADJ
cana-1005	206	17	andappliedmathematics	andappliedmathematic	NOUN
cana-1005	206	18	,	,	PUNCT
cana-1005	206	19	vol.16,no.1	vol.16,no.1	VERB
cana-1005	206	20	,	,	PUNCT
cana-1005	206	21	63	63	NUM
cana-1005	206	22	-	-	SYM
cana-1005	206	23	67	67	NUM
cana-1005	206	24	(	(	PUNCT
cana-1005	206	25	2018	2018	NUM
cana-1005	206	26	)	)	PUNCT
cana-1005	206	27	.	.	PUNCT
cana-1005	207	1	[	[	X
cana-1005	207	2	15	15	NUM
cana-1005	207	3	]	]	X
cana-1005	207	4	vukičević	vukičević	NOUN
cana-1005	207	5	,	,	PUNCT
cana-1005	207	6	d.	d.	PROPN
cana-1005	207	7	and	and	CCONJ
cana-1005	207	8	furtula	furtula	PROPN
cana-1005	207	9	,	,	PUNCT
cana-1005	207	10	b.	b.	PROPN
cana-1005	207	11	:	:	PUNCT
cana-1005	207	12	topological	topological	ADJ
cana-1005	207	13	index	index	NOUN
cana-1005	207	14	based	base	VERB
cana-1005	207	15	on	on	ADP
cana-1005	207	16	the	the	DET
cana-1005	207	17	ratios	ratio	NOUN
cana-1005	207	18	of	of	ADP
cana-1005	207	19	geometrical	geometrical	ADJ
cana-1005	207	20	and	and	CCONJ
cana-1005	207	21	arithmetical	arithmetical	ADJ
cana-1005	207	22	means	mean	NOUN
cana-1005	207	23	of	of	ADP
cana-1005	207	24	end	end	NOUN
cana-1005	207	25	-	-	PUNCT
cana-1005	207	26	vertex	vertex	NOUN
cana-1005	207	27	degrees	degree	NOUN
cana-1005	207	28	of	of	ADP
cana-1005	207	29	edges	edge	NOUN
cana-1005	207	30	,	,	PUNCT
cana-1005	207	31	j.	j.	PROPN
cana-1005	207	32	math	math	PROPN
cana-1005	207	33	.	.	PUNCT
cana-1005	208	1	chem	chem	PROPN
cana-1005	208	2	.	.	PUNCT
cana-1005	208	3	,	,	PUNCT
cana-1005	208	4	46,1369	46,1369	PROPN
cana-1005	208	5	-	-	PUNCT
cana-1005	208	6	1376((2009	1376((2009	NUM
cana-1005	208	7	)	)	PUNCT
cana-1005	208	8	)	)	PUNCT
cana-1005	208	9	.	.	PUNCT
cana-1005	209	1	[	[	X
cana-1005	209	2	16	16	NUM
cana-1005	209	3	]	]	PUNCT
cana-1005	209	4	wiener	wiener	NOUN
cana-1005	209	5	,	,	PUNCT
cana-1005	209	6	h.	h.	NOUN
cana-1005	209	7	:	:	PUNCT
cana-1005	209	8	structural	structural	ADJ
cana-1005	209	9	determination	determination	NOUN
cana-1005	209	10	of	of	ADP
cana-1005	209	11	paraffin	paraffin	NOUN
cana-1005	209	12	boiling	boiling	NOUN
cana-1005	209	13	points	point	NOUN
cana-1005	209	14	,	,	PUNCT
cana-1005	209	15	j.	j.	PROPN
cana-1005	209	16	am	be	AUX
cana-1005	209	17	chem	chem	NOUN
cana-1005	209	18	.	.	PUNCT
cana-1005	210	1	soc	soc	PROPN
cana-1005	210	2	.	.	PUNCT
cana-1005	211	1	,	,	PUNCT
cana-1005	211	2	6,17	6,17	NOUN
cana-1005	211	3	-	-	SYM
cana-1005	211	4	20	20	NUM
cana-1005	211	5	(	(	PUNCT
cana-1005	211	6	1947	1947	NUM
cana-1005	211	7	)	)	PUNCT
cana-1005	211	8	.	.	PUNCT
cana-1005	212	1	[	[	X
cana-1005	212	2	17	17	NUM
cana-1005	212	3	]	]	X
cana-1005	212	4	zhong	zhong	PROPN
cana-1005	212	5	,	,	PUNCT
cana-1005	212	6	l.	l.	PROPN
cana-1005	212	7	:	:	PUNCT
cana-1005	212	8	the	the	DET
cana-1005	212	9	harmonic	harmonic	ADJ
cana-1005	212	10	index	index	NOUN
cana-1005	212	11	for	for	ADP
cana-1005	212	12	graphs	graph	NOUN
cana-1005	212	13	,	,	PUNCT
cana-1005	212	14	applied	apply	VERB
cana-1005	212	15	mathematics	mathematics	NOUN
cana-1005	212	16	letters	letter	NOUN
cana-1005	212	17	,	,	PUNCT
cana-1005	212	18	25	25	NUM
cana-1005	212	19	,	,	PUNCT
cana-1005	212	20	561	561	NUM
cana-1005	212	21	-	-	SYM
cana-1005	212	22	566	566	NUM
cana-1005	212	23	(	(	PUNCT
cana-1005	212	24	2012	2012	NUM
cana-1005	212	25	)	)	PUNCT
cana-1005	212	26	.	.	PUNCT
cana-1005	213	1	[	[	X
cana-1005	213	2	18	18	NUM
cana-1005	213	3	]	]	X
cana-1005	213	4	zhong	zhong	PROPN
cana-1005	213	5	,	,	PUNCT
cana-1005	213	6	l.	l.	PROPN
cana-1005	213	7	:	:	PUNCT
cana-1005	213	8	the	the	DET
cana-1005	213	9	harmonic	harmonic	ADJ
cana-1005	213	10	index	index	NOUN
cana-1005	213	11	on	on	ADP
cana-1005	213	12	unicyclic	unicyclic	ADJ
cana-1005	213	13	graphs	graph	NOUN
cana-1005	213	14	,	,	PUNCT
cana-1005	213	15	ars	ar	VERB
cana-1005	213	16	combin	combin	NOUN
cana-1005	213	17	.	.	PUNCT
cana-1005	213	18	,	,	PUNCT
cana-1005	213	19	104	104	NUM
cana-1005	213	20	,	,	PUNCT
cana-1005	213	21	261	261	NUM
cana-1005	213	22	-	-	PUNCT
cana-1005	213	23	269(2012	269(2012	NUM
cana-1005	213	24	)	)	PUNCT
cana-1005	213	25	.	.	PUNCT
cana-1005	214	1	[	[	X
cana-1005	214	2	19	19	NUM
cana-1005	214	3	]	]	X
cana-1005	214	4	zhou	zhou	PROPN
cana-1005	214	5	,	,	PUNCT
cana-1005	214	6	b.	b.	PROPN
cana-1005	214	7	and	and	CCONJ
cana-1005	214	8	trinajstić	trinajstić	PROPN
cana-1005	214	9	,	,	PUNCT
cana-1005	214	10	n.	n.	NOUN
cana-1005	214	11	:	:	PUNCT
cana-1005	214	12	on	on	ADP
cana-1005	214	13	a	a	DET
cana-1005	214	14	novel	novel	ADJ
cana-1005	214	15	connectivity	connectivity	NOUN
cana-1005	214	16	index	index	NOUN
cana-1005	214	17	,	,	PUNCT
cana-1005	214	18	j.	j.	PROPN
cana-1005	214	19	math	math	PROPN
cana-1005	214	20	.	.	PUNCT
cana-1005	214	21	chem	chem	PROPN
cana-1005	214	22	.	.	PUNCT
cana-1005	214	23	,	,	PUNCT
cana-1005	214	24	46	46	NUM
cana-1005	214	25	,	,	PUNCT
cana-1005	214	26	1252	1252	NUM
cana-1005	214	27	-	-	SYM
cana-1005	214	28	1270	1270	NUM
cana-1005	214	29	(	(	PUNCT
cana-1005	214	30	2009	2009	NUM
cana-1005	214	31	)	)	PUNCT
cana-1005	214	32	.	.	PUNCT