id	sid	tid	token	lemma	pos
cana-1534	1	1	communications	communication	NOUN
cana-1534	1	2	on	on	ADP
cana-1534	1	3	applied	apply	VERB
cana-1534	1	4	nonlinear	nonlinear	ADJ
cana-1534	1	5	analysis	analysis	NOUN
cana-1534	1	6	issn	issn	NOUN
cana-1534	1	7	:	:	PUNCT
cana-1534	1	8	1074	1074	NUM
cana-1534	1	9	-	-	PUNCT
cana-1534	1	10	133x	133x	NUM
cana-1534	1	11	vol	vol	NOUN
cana-1534	1	12	31	31	NUM
cana-1534	1	13	no	no	NOUN
cana-1534	1	14	.	.	PUNCT
cana-1534	2	1	8s	8s	PROPN
cana-1534	2	2	(	(	PUNCT
cana-1534	2	3	2024	2024	NUM
cana-1534	2	4	)	)	PUNCT
cana-1534	2	5	421	421	NUM
cana-1534	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	2	7	some	some	DET
cana-1534	2	8	results	result	VERB
cana-1534	2	9	on	on	ADP
cana-1534	2	10	non	non	ADJ
cana-1534	2	11	-	-	ADJ
cana-1534	2	12	isolated	isolated	ADJ
cana-1534	2	13	resolving	resolving	NOUN
cana-1534	2	14	number	number	NOUN
cana-1534	2	15	p.	p.	NOUN
cana-1534	2	16	jeya	jeya	NOUN
cana-1534	2	17	bala	bala	PROPN
cana-1534	2	18	chitra1	chitra1	PROPN
cana-1534	2	19	,	,	PUNCT
cana-1534	2	20	selvam	selvam	ADJ
cana-1534	2	21	avadayappan2	avadayappan2	PROPN
cana-1534	2	22	,	,	PUNCT
cana-1534	2	23	m.	m.	NOUN
cana-1534	2	24	bhuvaneshwari3	bhuvaneshwari3	PROPN
cana-1534	3	1	1,2,3	1,2,3	NUM
cana-1534	3	2	research	research	NOUN
cana-1534	3	3	department	department	NOUN
cana-1534	3	4	of	of	ADP
cana-1534	3	5	mathematics	mathematics	PROPN
cana-1534	3	6	,	,	PUNCT
cana-1534	3	7	vhnsn	vhnsn	PROPN
cana-1534	3	8	college	college	PROPN
cana-1534	3	9	,	,	PUNCT
cana-1534	3	10	virudhunagar	virudhunagar	VERB
cana-1534	3	11	626	626	NUM
cana-1534	3	12	001	001	NUM
cana-1534	3	13	,	,	PUNCT
cana-1534	3	14	india	india	PROPN
cana-1534	3	15	.	.	PUNCT
cana-1534	4	1	durga1maths@gmail.com1	durga1maths@gmail.com1	PROPN
cana-1534	4	2	,	,	PUNCT
cana-1534	4	3	selvam−avadayappan@yahoo.co.in2	selvam−avadayappan@yahoo.co.in2	PROPN
cana-1534	4	4	,	,	PUNCT
cana-1534	4	5	bhuvaneshwari@vhnsnc.edu.in3	bhuvaneshwari@vhnsnc.edu.in3	PROPN
cana-1534	4	6	article	article	NOUN
cana-1534	4	7	history	history	NOUN
cana-1534	4	8	:	:	PUNCT
cana-1534	4	9	received	receive	VERB
cana-1534	4	10	:	:	PUNCT
cana-1534	4	11	06	06	NUM
cana-1534	4	12	-	-	SYM
cana-1534	4	13	05	05	NUM
cana-1534	4	14	-	-	PUNCT
cana-1534	4	15	2024	2024	NUM
cana-1534	4	16	revised	revise	VERB
cana-1534	4	17	:	:	PUNCT
cana-1534	4	18	25	25	NUM
cana-1534	4	19	-	-	PUNCT
cana-1534	4	20	06	06	NUM
cana-1534	4	21	-	-	PUNCT
cana-1534	4	22	2024	2024	NUM
cana-1534	4	23	accepted	accept	VERB
cana-1534	4	24	:	:	PUNCT
cana-1534	4	25	09	09	NUM
cana-1534	4	26	-	-	SYM
cana-1534	4	27	07	07	NUM
cana-1534	4	28	-	-	PUNCT
cana-1534	4	29	2024	2024	NUM
cana-1534	4	30	abstract	abstract	NOUN
cana-1534	4	31	:	:	PUNCT
cana-1534	5	1	connected	connect	VERB
cana-1534	5	2	graphs	graph	NOUN
cana-1534	5	3	g.	g.	PROPN
cana-1534	5	4	w	w	PROPN
cana-1534	6	1	=	=	PUNCT
cana-1534	6	2	{	{	PUNCT
cana-1534	6	3	w1	w1	NOUN
cana-1534	6	4	,	,	PUNCT
cana-1534	6	5	w2	w2	NOUN
cana-1534	6	6	,	,	PUNCT
cana-1534	6	7	...	...	PUNCT
cana-1534	6	8	,	,	PUNCT
cana-1534	6	9	wk	wk	PROPN
cana-1534	6	10	}	}	PUNCT
cana-1534	6	11	is	be	AUX
cana-1534	6	12	a	a	DET
cana-1534	6	13	subset	subset	NOUN
cana-1534	6	14	of	of	ADP
cana-1534	6	15	v	v	NOUN
cana-1534	6	16	with	with	ADP
cana-1534	6	17	a	a	DET
cana-1534	6	18	predetermined	predetermine	VERB
cana-1534	6	19	order	order	NOUN
cana-1534	6	20	.	.	PUNCT
cana-1534	7	1	the	the	DET
cana-1534	7	2	line	line	NOUN
cana-1534	7	3	vector	vector	NOUN
cana-1534	7	4	r(v|w	r(v|w	X
cana-1534	7	5	)	)	PUNCT
cana-1534	8	1	=	=	SYM
cana-1534	8	2	(	(	PUNCT
cana-1534	8	3	d(v	d(v	PROPN
cana-1534	8	4	,	,	PUNCT
cana-1534	8	5	w1	w1	NOUN
cana-1534	8	6	)	)	PUNCT
cana-1534	8	7	,	,	PUNCT
cana-1534	8	8	d(v	d(v	PROPN
cana-1534	8	9	,	,	PUNCT
cana-1534	8	10	w2	w2	NOUN
cana-1534	8	11	)	)	PUNCT
cana-1534	8	12	,	,	PUNCT
cana-1534	8	13	...	...	PUNCT
cana-1534	8	14	,	,	PUNCT
cana-1534	8	15	d(v	d(v	PROPN
cana-1534	8	16	,	,	PUNCT
cana-1534	8	17	wk	wk	NOUN
cana-1534	8	18	)	)	PUNCT
cana-1534	8	19	)	)	PUNCT
cana-1534	8	20	is	be	AUX
cana-1534	8	21	the	the	DET
cana-1534	8	22	measurement	measurement	NOUN
cana-1534	8	23	depicting	depict	VERB
cana-1534	8	24	v	v	NOUN
cana-1534	8	25	with	with	ADP
cana-1534	8	26	regards	regard	NOUN
cana-1534	8	27	to	to	ADP
cana-1534	8	28	w	w	NOUN
cana-1534	8	29	for	for	ADP
cana-1534	8	30	each	each	DET
cana-1534	8	31	v	v	NOUN
cana-1534	8	32	∈	∈	PROPN
cana-1534	8	33	v.	v.	ADP
cana-1534	8	34	if	if	SCONJ
cana-1534	8	35	v	v	NOUN
cana-1534	8	36	's	's	PART
cana-1534	8	37	vertex	vertex	NOUN
cana-1534	8	38	utilize	utilize	VERB
cana-1534	8	39	various	various	ADJ
cana-1534	8	40	metrics	metric	NOUN
cana-1534	8	41	,	,	PUNCT
cana-1534	8	42	w	w	NOUN
cana-1534	8	43	resolves	resolve	NOUN
cana-1534	8	44	g.	g.	VERB
cana-1534	8	45	their	their	PRON
cana-1534	8	46	fundamental	fundamental	ADJ
cana-1534	8	47	magnitude	magnitude	NOUN
cana-1534	8	48	,	,	PUNCT
cana-1534	8	49	dim(g	dim(g	PROPN
cana-1534	8	50	)	)	PUNCT
cana-1534	8	51	,	,	PUNCT
cana-1534	8	52	is	be	AUX
cana-1534	8	53	their	their	PRON
cana-1534	8	54	lowest	low	ADJ
cana-1534	8	55	cardinality	cardinality	NOUN
cana-1534	8	56	.	.	PUNCT
cana-1534	9	1	a	a	DET
cana-1534	9	2	resolved	resolve	VERB
cana-1534	9	3	set	set	NOUN
cana-1534	9	4	w	w	NOUN
cana-1534	9	5	is	be	AUX
cana-1534	9	6	nonisolated	nonisolate	VERB
cana-1534	9	7	if	if	SCONJ
cana-1534	9	8	its	its	PRON
cana-1534	9	9	influenced	influence	VERB
cana-1534	9	10	subsection	subsection	NOUN
cana-1534	9	11	⟨w	⟨w	X
cana-1534	9	12	⟩	⟩	PROPN
cana-1534	9	13	has	have	VERB
cana-1534	9	14	no	no	DET
cana-1534	9	15	single	single	ADJ
cana-1534	9	16	vertex	vertex	NOUN
cana-1534	9	17	.	.	PUNCT
cana-1534	10	1	the	the	DET
cana-1534	10	2	simplest	simple	ADJ
cana-1534	10	3	connection	connection	NOUN
cana-1534	10	4	of	of	ADP
cana-1534	10	5	a	a	DET
cana-1534	10	6	non	non	ADJ
cana-1534	10	7	-	-	ADJ
cana-1534	10	8	isolated	isolated	ADJ
cana-1534	10	9	resolved	resolve	VERB
cana-1534	10	10	set	set	NOUN
cana-1534	10	11	of	of	ADP
cana-1534	10	12	g	g	PROPN
cana-1534	10	13	is	be	AUX
cana-1534	10	14	nr	nr	PRON
cana-1534	10	15	.	.	PUNCT
cana-1534	11	1	an	an	DET
cana-1534	11	2	nr	nr	ADV
cana-1534	11	3	-	-	NOUN
cana-1534	11	4	set	set	NOUN
cana-1534	11	5	for	for	ADP
cana-1534	11	6	g	g	PROPN
cana-1534	11	7	is	be	AUX
cana-1534	11	8	a	a	DET
cana-1534	11	9	non	non	ADJ
cana-1534	11	10	-	-	ADJ
cana-1534	11	11	isolated	isolated	ADJ
cana-1534	11	12	resolution	resolution	NOUN
cana-1534	11	13	set	set	NOUN
cana-1534	11	14	of	of	ADP
cana-1534	11	15	cardinality	cardinality	NOUN
cana-1534	11	16	nr(g	nr(g	NUM
cana-1534	11	17	)	)	PUNCT
cana-1534	11	18	.	.	PUNCT
cana-1534	12	1	in	in	ADP
cana-1534	12	2	this	this	DET
cana-1534	12	3	study	study	NOUN
cana-1534	12	4	,	,	PUNCT
cana-1534	12	5	we	we	PRON
cana-1534	12	6	prove	prove	VERB
cana-1534	12	7	that	that	SCONJ
cana-1534	12	8	the	the	DET
cana-1534	12	9	chart	chart	NOUN
cana-1534	12	10	g	g	PROPN
cana-1534	12	11	has	have	VERB
cana-1534	12	12	a	a	DET
cana-1534	12	13	unique	unique	ADJ
cana-1534	12	14	nr	nr	NOUN
cana-1534	12	15	-	-	NOUN
cana-1534	12	16	set	set	NOUN
cana-1534	12	17	.	.	PUNCT
cana-1534	13	1	we	we	PRON
cana-1534	13	2	also	also	ADV
cana-1534	13	3	build	build	VERB
cana-1534	13	4	a	a	DET
cana-1534	13	5	2n	2n	NUM
cana-1534	13	6	-	-	PUNCT
cana-1534	13	7	vertex	vertex	NOUN
cana-1534	13	8	graph	graph	NOUN
cana-1534	13	9	g	g	NOUN
cana-1534	13	10	using	use	VERB
cana-1534	13	11	nr	nr	NOUN
cana-1534	13	12	-	-	PUNCT
cana-1534	13	13	set	set	VERB
cana-1534	13	14	.	.	PUNCT
cana-1534	14	1	w	w	ADP
cana-1534	14	2	which	which	PRON
cana-1534	14	3	means	mean	VERB
cana-1534	14	4	nr(g	nr(g	PRON
cana-1534	14	5	)	)	PUNCT
cana-1534	15	1	=	=	SYM
cana-1534	15	2	n	n	PROPN
cana-1534	15	3	and	and	CCONJ
cana-1534	15	4	r(vi|w	r(vi|w	VERB
cana-1534	15	5	)	)	PUNCT
cana-1534	15	6	=	=	SYM
cana-1534	15	7	(	(	PUNCT
cana-1534	15	8	1,	1,	NUM
cana-1534	15	9	...	...	SYM
cana-1534	15	10	2	2	NUM
cana-1534	15	11	,	,	PUNCT
cana-1534	15	12	1	1	NUM
cana-1534	15	13	)	)	PUNCT
cana-1534	15	14	,	,	PUNCT
cana-1534	15	15	where	where	SCONJ
cana-1534	15	16	2	2	NUM
cana-1534	15	17	is	be	AUX
cana-1534	15	18	in	in	ADP
cana-1534	15	19	the	the	DET
cana-1534	15	20	ith	ith	PROPN
cana-1534	15	21	location	location	NOUN
cana-1534	15	22	,	,	PUNCT
cana-1534	15	23	represents	represent	VERB
cana-1534	15	24	every	every	DET
cana-1534	15	25	vertex	vertex	NOUN
cana-1534	15	26	not	not	PART
cana-1534	15	27	in	in	ADP
cana-1534	15	28	w.	w.	PROPN
cana-1534	15	29	further	far	ADV
cana-1534	15	30	we	we	PRON
cana-1534	15	31	established	establish	VERB
cana-1534	15	32	the	the	DET
cana-1534	15	33	nr	nr	NOUN
cana-1534	15	34	-	-	NOUN
cana-1534	15	35	value	value	NOUN
cana-1534	15	36	for	for	ADP
cana-1534	15	37	the	the	DET
cana-1534	15	38	highly	highly	ADV
cana-1534	15	39	irregular	irregular	ADJ
cana-1534	15	40	graph	graph	NOUN
cana-1534	15	41	hn	hn	PROPN
cana-1534	15	42	,	,	PUNCT
cana-1534	15	43	n	n	PROPN
cana-1534	15	44	and	and	CCONJ
cana-1534	15	45	for	for	ADP
cana-1534	15	46	the	the	DET
cana-1534	15	47	wheel	wheel	NOUN
cana-1534	15	48	wn	wn	PROPN
cana-1534	15	49	.	.	PUNCT
cana-1534	16	1	also	also	ADV
cana-1534	16	2	we	we	PRON
cana-1534	16	3	determined	determine	VERB
cana-1534	16	4	the	the	DET
cana-1534	16	5	nr	nr	NOUN
cana-1534	16	6	-	-	NOUN
cana-1534	16	7	value	value	NOUN
cana-1534	16	8	for	for	ADP
cana-1534	16	9	corona	corona	NOUN
cana-1534	16	10	product	product	NOUN
cana-1534	16	11	of	of	ADP
cana-1534	16	12	some	some	DET
cana-1534	16	13	graphs	graph	NOUN
cana-1534	16	14	.	.	PUNCT
cana-1534	17	1	keywords	keyword	NOUN
cana-1534	17	2	:	:	PUNCT
cana-1534	17	3	resolution	resolution	NOUN
cana-1534	17	4	set	set	NOUN
cana-1534	17	5	,	,	PUNCT
cana-1534	17	6	metric	metric	ADJ
cana-1534	17	7	dimensions	dimension	NOUN
cana-1534	17	8	,	,	PUNCT
cana-1534	17	9	non	non	ADJ
cana-1534	17	10	-	-	ADJ
cana-1534	17	11	isolated	isolated	ADJ
cana-1534	17	12	resolution	resolution	NOUN
cana-1534	17	13	set	set	NOUN
cana-1534	17	14	,	,	PUNCT
cana-1534	17	15	number	number	NOUN
cana-1534	17	16	.	.	PUNCT
cana-1534	18	1	ams	am	NOUN
cana-1534	18	2	subject	subject	VERB
cana-1534	18	3	classification	classification	NOUN
cana-1534	18	4	code	code	NOUN
cana-1534	18	5	(	(	PUNCT
cana-1534	18	6	2010	2010	NUM
cana-1534	18	7	):	):	PUNCT
cana-1534	18	8	05c12	05c12	NOUN
cana-1534	18	9	1	1	NUM
cana-1534	18	10	introduction	introduction	NOUN
cana-1534	18	11	we	we	PRON
cana-1534	18	12	exclusively	exclusively	ADV
cana-1534	18	13	discuss	discuss	VERB
cana-1534	18	14	limited	limited	ADJ
cana-1534	18	15	,	,	PUNCT
cana-1534	18	16	simple	simple	ADJ
cana-1534	18	17	,	,	PUNCT
cana-1534	18	18	uncontrolled	uncontrolled	ADJ
cana-1534	18	19	,	,	PUNCT
cana-1534	18	20	connected	connected	ADJ
cana-1534	18	21	networks	network	NOUN
cana-1534	18	22	in	in	ADP
cana-1534	18	23	this	this	DET
cana-1534	18	24	study	study	NOUN
cana-1534	18	25	.	.	PUNCT
cana-1534	19	1	the	the	DET
cana-1534	19	2	vertex	vertex	NOUN
cana-1534	19	3	graph	graph	NOUN
cana-1534	19	4	g	g	PROPN
cana-1534	19	5	are	be	AUX
cana-1534	19	6	v	v	ADP
cana-1534	19	7	(	(	PUNCT
cana-1534	19	8	g	g	NOUN
cana-1534	19	9	)	)	PUNCT
cana-1534	19	10	edge	edge	NOUN
cana-1534	19	11	for	for	ADP
cana-1534	19	12	e(g	e(g	PROPN
cana-1534	19	13	)	)	PUNCT
cana-1534	19	14	.	.	PUNCT
cana-1534	20	1	see	see	VERB
cana-1534	21	1	[	[	X
cana-1534	21	2	6	6	NUM
cana-1534	21	3	]	]	PUNCT
cana-1534	21	4	for	for	ADP
cana-1534	21	5	fundamental	fundamental	ADJ
cana-1534	21	6	symbols	symbol	NOUN
cana-1534	21	7	and	and	CCONJ
cana-1534	21	8	nomenclature	nomenclature	NOUN
cana-1534	21	9	.	.	PUNCT
cana-1534	22	1	the	the	DET
cana-1534	22	2	distance	distance	NOUN
cana-1534	22	3	of	of	ADP
cana-1534	22	4	a	a	DET
cana-1534	22	5	shortest	short	ADJ
cana-1534	22	6	pathway	pathway	NOUN
cana-1534	22	7	among	among	ADP
cana-1534	22	8	the	the	DET
cana-1534	22	9	two	two	NUM
cana-1534	22	10	points	point	NOUN
cana-1534	22	11	is	be	AUX
cana-1534	22	12	d(u	d(u	PROPN
cana-1534	22	13	,	,	PUNCT
cana-1534	22	14	v	v	NOUN
cana-1534	22	15	)	)	PUNCT
cana-1534	22	16	.	.	PUNCT
cana-1534	23	1	the	the	DET
cana-1534	23	2	graph	graph	NOUN
cana-1534	23	3	g1&g2	g1&g2	PROPN
cana-1534	23	4	with	with	ADP
cana-1534	23	5	a	a	DET
cana-1534	23	6	single	single	ADJ
cana-1534	23	7	instance	instance	NOUN
cana-1534	23	8	of	of	ADP
cana-1534	23	9	g1	g1	PROPN
cana-1534	23	10	and	and	CCONJ
cana-1534	23	11	|v	|v	PROPN
cana-1534	23	12	(	(	PUNCT
cana-1534	23	13	g1)|	g1)|	PROPN
cana-1534	23	14	duplicates	duplicate	NOUN
cana-1534	23	15	of	of	ADP
cana-1534	23	16	g2	g2	PROPN
cana-1534	23	17	is	be	AUX
cana-1534	23	18	the	the	DET
cana-1534	23	19	corona	corona	NOUN
cana-1534	23	20	of	of	ADP
cana-1534	23	21	g1	g1	PROPN
cana-1534	23	22	and	and	CCONJ
cana-1534	23	23	g2	g2	PROPN
cana-1534	23	24	.	.	PUNCT
cana-1534	24	1	it	it	PRON
cana-1534	24	2	's	be	AUX
cana-1534	24	3	built	build	VERB
cana-1534	24	4	by	by	ADP
cana-1534	24	5	connecting	connect	VERB
cana-1534	24	6	each	each	DET
cana-1534	24	7	g2	g2	PROPN
cana-1534	24	8	node	node	NOUN
cana-1534	24	9	to	to	ADP
cana-1534	24	10	the	the	DET
cana-1534	24	11	ith	ith	PROPN
cana-1534	24	12	g1	g1	PROPN
cana-1534	24	13	vertex	vertex	NOUN
cana-1534	24	14	.	.	PUNCT
cana-1534	25	1	a	a	DET
cana-1534	25	2	wheel	wheel	NOUN
cana-1534	25	3	wn	wn	PROPN
cana-1534	25	4	network	network	NOUN
cana-1534	25	5	is	be	AUX
cana-1534	25	6	generated	generate	VERB
cana-1534	25	7	from	from	ADP
cana-1534	25	8	a	a	DET
cana-1534	25	9	single	single	ADJ
cana-1534	25	10	cycle	cycle	NOUN
cana-1534	25	11	cn	cn	PROPN
cana-1534	25	12	through	through	ADP
cana-1534	25	13	adding	add	VERB
cana-1534	25	14	a	a	DET
cana-1534	25	15	new	new	ADJ
cana-1534	25	16	vertex	vertex	NOUN
cana-1534	25	17	v	v	NOUN
cana-1534	25	18	and	and	CCONJ
cana-1534	25	19	attaching	attach	VERB
cana-1534	25	20	it	it	PRON
cana-1534	25	21	to	to	ADP
cana-1534	25	22	all	all	DET
cana-1534	25	23	the	the	DET
cana-1534	25	24	cycle	cycle	NOUN
cana-1534	25	25	's	's	PART
cana-1534	25	26	corners	corner	NOUN
cana-1534	25	27	.	.	PUNCT
cana-1534	26	1	the	the	DET
cana-1534	26	2	ribs	rib	NOUN
cana-1534	26	3	of	of	ADP
cana-1534	26	4	the	the	DET
cana-1534	26	5	wheels	wheel	NOUN
cana-1534	26	6	are	be	AUX
cana-1534	26	7	the	the	DET
cana-1534	26	8	newly	newly	ADV
cana-1534	26	9	added	add	VERB
cana-1534	26	10	connections	connection	NOUN
cana-1534	26	11	.	.	PUNCT
cana-1534	27	1	the	the	DET
cana-1534	27	2	graph	graph	NOUN
cana-1534	27	3	hn	hn	PROPN
cana-1534	27	4	,	,	PUNCT
cana-1534	27	5	n	n	PROPN
cana-1534	27	6	is	be	AUX
cana-1534	27	7	an	an	DET
cana-1534	27	8	irregular	irregular	ADJ
cana-1534	27	9	graph	graph	NOUN
cana-1534	27	10	defined	define	VERB
cana-1534	27	11	by	by	ADP
cana-1534	27	12	v	v	NOUN
cana-1534	27	13	(	(	PUNCT
cana-1534	27	14	hn	hn	PROPN
cana-1534	27	15	,	,	PUNCT
cana-1534	27	16	n	n	CCONJ
cana-1534	27	17	)	)	PUNCT
cana-1534	27	18	=	=	SYM
cana-1534	27	19	{	{	PUNCT
cana-1534	27	20	v1	v1	PROPN
cana-1534	27	21	,	,	PUNCT
cana-1534	27	22	v2	v2	PROPN
cana-1534	27	23	,	,	PUNCT
cana-1534	27	24	...	...	PUNCT
cana-1534	27	25	,	,	PUNCT
cana-1534	27	26	vn	vn	PROPN
cana-1534	27	27	;	;	PUNCT
cana-1534	27	28	u1	u1	PROPN
cana-1534	27	29	,	,	PUNCT
cana-1534	27	30	u2	u2	NOUN
cana-1534	27	31	,	,	PUNCT
cana-1534	27	32	...	...	PUNCT
cana-1534	27	33	,	,	PUNCT
cana-1534	27	34	un	un	ADJ
cana-1534	27	35	}	}	PUNCT
cana-1534	27	36	and	and	CCONJ
cana-1534	27	37	e(hn	e(hn	PROPN
cana-1534	27	38	,	,	PUNCT
cana-1534	27	39	n	n	CCONJ
cana-1534	27	40	)	)	PUNCT
cana-1534	27	41	=	=	PRON
cana-1534	27	42	{	{	PUNCT
cana-1534	27	43	viuj	viuj	ADJ
cana-1534	27	44	:	:	PUNCT
cana-1534	27	45	1	1	NUM
cana-1534	27	46	≤	≤	NUM
cana-1534	27	47	i	i	PRON
cana-1534	27	48	≤	≤	PROPN
cana-1534	27	49	n	n	CCONJ
cana-1534	27	50	,	,	PUNCT
cana-1534	27	51	n	n	CCONJ
cana-1534	27	52	−	−	PROPN
cana-1534	28	1	i	i	PRON
cana-1534	28	2	+	+	CCONJ
cana-1534	28	3	1	1	NUM
cana-1534	28	4	≤	≤	NUM
cana-1534	28	5	j	j	PROPN
cana-1534	28	6	≤	≤	PROPN
cana-1534	28	7	n	n	CCONJ
cana-1534	28	8	}	}	PUNCT
cana-1534	28	9	.	.	PUNCT
cana-1534	29	1	a	a	DET
cana-1534	29	2	graph	graph	NOUN
cana-1534	29	3	component	component	NOUN
cana-1534	29	4	a	a	DET
cana-1534	29	5	spanning	span	VERB
cana-1534	29	6	subsection	subsection	NOUN
cana-1534	29	7	with	with	ADP
cana-1534	29	8	the	the	DET
cana-1534	29	9	same	same	ADJ
cana-1534	29	10	vertex	vertex	NOUN
cana-1534	29	11	collection	collection	NOUN
cana-1534	29	12	as	as	SCONJ
cana-1534	29	13	g.	g.	PROPN
cana-1534	29	14	a	a	DET
cana-1534	29	15	k	k	NOUN
cana-1534	29	16	-	-	PUNCT
cana-1534	29	17	factor	factor	NOUN
cana-1534	29	18	is	be	AUX
cana-1534	29	19	a	a	DET
cana-1534	29	20	spanned	spanned	ADJ
cana-1534	29	21	k	k	ADJ
cana-1534	29	22	-	-	ADJ
cana-1534	29	23	regular	regular	ADJ
cana-1534	29	24	subsection	subsection	NOUN
cana-1534	29	25	.	.	PUNCT
cana-1534	30	1	in	in	ADP
cana-1534	30	2	particular	particular	ADJ
cana-1534	30	3	,	,	PUNCT
cana-1534	30	4	a	a	DET
cana-1534	30	5	1	1	NUM
cana-1534	30	6	-	-	PUNCT
cana-1534	30	7	aspect	aspect	NOUN
cana-1534	30	8	matches	match	NOUN
cana-1534	30	9	perfectly	perfectly	ADV
cana-1534	30	10	.	.	PUNCT
cana-1534	31	1	resolving	resolve	VERB
cana-1534	31	2	sets	set	NOUN
cana-1534	31	3	have	have	AUX
cana-1534	31	4	been	be	AUX
cana-1534	31	5	mentioned	mention	VERB
cana-1534	31	6	in	in	ADP
cana-1534	31	7	the	the	DET
cana-1534	31	8	literature	literature	NOUN
cana-1534	31	9	.	.	PUNCT
cana-1534	32	1	slater	slater	PROPN
cana-1534	32	2	established	establish	VERB
cana-1534	32	3	these	these	DET
cana-1534	32	4	concepts	concept	NOUN
cana-1534	32	5	in	in	ADP
cana-1534	32	6	[	[	X
cana-1534	32	7	18	18	NUM
cana-1534	32	8	]	]	PUNCT
cana-1534	32	9	and	and	CCONJ
cana-1534	32	10	[	[	X
cana-1534	32	11	19	19	NUM
cana-1534	32	12	]	]	PUNCT
cana-1534	32	13	and	and	CCONJ
cana-1534	32	14	utilized	utilize	VERB
cana-1534	32	15	finding	finding	NOUN
cana-1534	32	16	set	set	VERB
cana-1534	32	17	for	for	ADP
cana-1534	32	18	resolution	resolution	NOUN
cana-1534	32	19	set	set	NOUN
cana-1534	32	20	.	.	PUNCT
cana-1534	33	1	location	location	NOUN
cana-1534	33	2	number	number	NOUN
cana-1534	33	3	loc(g	loc(g	PROPN
cana-1534	33	4	)	)	PUNCT
cana-1534	33	5	is	be	AUX
cana-1534	33	6	the	the	DET
cana-1534	33	7	relationship	relationship	NOUN
cana-1534	33	8	g	g	NOUN
cana-1534	33	9	's	's	PART
cana-1534	33	10	minimal	minimal	ADJ
cana-1534	33	11	resolved	resolve	VERB
cana-1534	33	12	sets	set	NOUN
cana-1534	33	13	.	.	PUNCT
cana-1534	34	1	slater	slater	NOUN
cana-1534	35	1	[	[	X
cana-1534	35	2	10	10	NUM
cana-1534	35	3	,	,	PUNCT
cana-1534	35	4	11	11	NUM
cana-1534	35	5	,	,	PUNCT
cana-1534	35	6	12	12	NUM
cana-1534	35	7	]	]	PUNCT
cana-1534	35	8	describes	describe	VERB
cana-1534	35	9	a	a	DET
cana-1534	35	10	resolution	resolution	NOUN
cana-1534	35	11	set	set	VERB
cana-1534	35	12	as	as	ADP
cana-1534	35	13	an	an	DET
cana-1534	35	14	array	array	NOUN
cana-1534	35	15	w	w	ADP
cana-1534	35	16	of	of	ADP
cana-1534	35	17	vertices	vertex	NOUN
cana-1534	35	18	in	in	ADP
cana-1534	35	19	a	a	DET
cana-1534	35	20	data	data	NOUN
cana-1534	35	21	structure	structure	NOUN
cana-1534	35	22	g	g	PROPN
cana-1534	35	23	wherein	wherein	ADJ
cana-1534	35	24	distances	distance	NOUN
cana-1534	35	25	to	to	AUX
cana-1534	35	26	w	w	AUX
cana-1534	35	27	precisely	precisely	ADV
cana-1534	35	28	define	define	VERB
cana-1534	35	29	every	every	DET
cana-1534	35	30	vertices	vertex	NOUN
cana-1534	35	31	.	.	PUNCT
cana-1534	36	1	harary	harary	NOUN
cana-1534	36	2	and	and	CCONJ
cana-1534	36	3	melter	melter	NOUN
cana-1534	37	1	[	[	X
cana-1534	37	2	9	9	NUM
cana-1534	37	3	]	]	PUNCT
cana-1534	37	4	separately	separately	ADV
cana-1534	37	5	found	find	VERB
cana-1534	37	6	location	location	NOUN
cana-1534	37	7	numbers	number	NOUN
cana-1534	37	8	but	but	CCONJ
cana-1534	37	9	called	call	VERB
cana-1534	37	10	them	they	PRON
cana-1534	37	11	metric	metric	ADJ
cana-1534	37	12	dimensions	dimension	NOUN
cana-1534	37	13	.	.	PUNCT
cana-1534	38	1	let	let	VERB
cana-1534	38	2	w	w	VERB
cana-1534	38	3	=	=	PUNCT
cana-1534	38	4	{	{	PUNCT
cana-1534	38	5	w1	w1	NOUN
cana-1534	38	6	,	,	PUNCT
cana-1534	38	7	…	…	PUNCT
cana-1534	38	8	wk	wk	ADP
cana-1534	38	9	}	}	PUNCT
cana-1534	38	10	be	be	AUX
cana-1534	38	11	a	a	DET
cana-1534	38	12	structured	structure	VERB
cana-1534	38	13	with	with	ADP
cana-1534	38	14	g	g	PROPN
cana-1534	38	15	&	&	CCONJ
cana-1534	38	16	v.	v.	PROPN
cana-1534	38	17	g	g	PROPN
cana-1534	38	18	's	's	PART
cana-1534	38	19	resolved	resolve	VERB
cana-1534	38	20	set	set	NOUN
cana-1534	38	21	is	be	AUX
cana-1534	38	22	w	w	NOUN
cana-1534	38	23	if	if	SCONJ
cana-1534	38	24	various	various	ADJ
cana-1534	38	25	edges	edge	NOUN
cana-1534	38	26	of	of	ADP
cana-1534	38	27	g	g	NOUN
cana-1534	38	28	have	have	VERB
cana-1534	38	29	different	different	ADJ
cana-1534	38	30	interpretations	interpretation	NOUN
cana-1534	38	31	in	in	ADP
cana-1534	38	32	w.	w.	NOUN
cana-1534	38	33	bases	basis	NOUN
cana-1534	38	34	for	for	ADP
cana-1534	38	35	g	g	PROPN
cana-1534	38	36	are	be	AUX
cana-1534	38	37	resolving	resolve	VERB
cana-1534	38	38	sets	set	NOUN
cana-1534	38	39	with	with	ADP
cana-1534	38	40	minimal	minimal	ADJ
cana-1534	38	41	cardinality	cardinality	NOUN
cana-1534	38	42	,	,	PUNCT
cana-1534	38	43	which	which	PRON
cana-1534	38	44	is	be	AUX
cana-1534	38	45	the	the	DET
cana-1534	38	46	metric	metric	ADJ
cana-1534	38	47	depth	depth	NOUN
cana-1534	38	48	of	of	ADP
cana-1534	38	49	g	g	PROPN
cana-1534	38	50	(	(	PUNCT
cana-1534	38	51	dim(g	dim(g	PROPN
cana-1534	38	52	)	)	PUNCT
cana-1534	38	53	)	)	PUNCT
cana-1534	38	54	.	.	PUNCT
cana-1534	39	1	resolved	resolve	VERB
cana-1534	39	2	sets	set	NOUN
cana-1534	39	3	have	have	VERB
cana-1534	39	4	uses	use	NOUN
cana-1534	39	5	for	for	ADP
cana-1534	39	6	coin	coin	NOUN
cana-1534	39	7	measuring	measuring	NOUN
cana-1534	39	8	,	,	PUNCT
cana-1534	39	9	discovering	discover	VERB
cana-1534	39	10	drugs	drug	NOUN
cana-1534	39	11	,	,	PUNCT
cana-1534	39	12	robot	robot	NOUN
cana-1534	39	13	communications	communication	NOUN
cana-1534	39	14	on	on	ADP
cana-1534	39	15	applied	apply	VERB
cana-1534	39	16	nonlinear	nonlinear	ADJ
cana-1534	39	17	analysis	analysis	NOUN
cana-1534	39	18	issn	issn	NOUN
cana-1534	39	19	:	:	PUNCT
cana-1534	39	20	1074	1074	NUM
cana-1534	39	21	-	-	PUNCT
cana-1534	39	22	133x	133x	NUM
cana-1534	39	23	vol	vol	NOUN
cana-1534	39	24	31	31	NUM
cana-1534	39	25	no	no	NOUN
cana-1534	39	26	.	.	PUNCT
cana-1534	40	1	8s	8s	PROPN
cana-1534	40	2	(	(	PUNCT
cana-1534	40	3	2024	2024	NUM
cana-1534	40	4	)	)	PUNCT
cana-1534	40	5	422	422	NUM
cana-1534	40	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	40	7	navigating	navigate	VERB
cana-1534	40	8	,	,	PUNCT
cana-1534	40	9	network	network	NOUN
cana-1534	40	10	exploration	exploration	NOUN
cana-1534	40	11	,	,	PUNCT
cana-1534	40	12	graph	graph	NOUN
cana-1534	40	13	participates	participate	VERB
cana-1534	40	14	,	,	PUNCT
cana-1534	40	15	and	and	CCONJ
cana-1534	40	16	mass	mass	ADJ
cana-1534	40	17	mind	mind	NOUN
cana-1534	40	18	game	game	NOUN
cana-1534	40	19	techniques	technique	NOUN
cana-1534	40	20	[	[	X
cana-1534	40	21	16	16	NUM
cana-1534	40	22	,	,	PUNCT
cana-1534	40	23	7	7	NUM
cana-1534	40	24	,	,	PUNCT
cana-1534	40	25	8	8	NUM
cana-1534	40	26	]	]	PUNCT
cana-1534	40	27	.	.	PUNCT
cana-1534	41	1	chartrand	chartrand	NOUN
cana-1534	41	2	and	and	CCONJ
cana-1534	41	3	zhang	zhang	PROPN
cana-1534	42	1	[	[	X
cana-1534	42	2	7	7	NUM
cana-1534	42	3	]	]	PUNCT
cana-1534	42	4	survey	survey	NOUN
cana-1534	42	5	measurement	measurement	NOUN
cana-1534	42	6	outcomes	outcome	NOUN
cana-1534	42	7	.	.	PUNCT
cana-1534	43	1	by	by	ADP
cana-1534	43	2	setting	set	VERB
cana-1534	43	3	requirements	requirement	NOUN
cana-1534	43	4	on	on	ADP
cana-1534	43	5	the	the	DET
cana-1534	43	6	smaller	small	ADJ
cana-1534	43	7	graphs	graph	NOUN
cana-1534	43	8	created	create	VERB
cana-1534	43	9	by	by	ADP
cana-1534	43	10	a	a	DET
cana-1534	43	11	resolved	resolve	VERB
cana-1534	43	12	set	set	NOUN
cana-1534	43	13	,	,	PUNCT
cana-1534	43	14	numerous	numerous	ADJ
cana-1534	43	15	designs	design	NOUN
cana-1534	43	16	have	have	AUX
cana-1534	43	17	been	be	AUX
cana-1534	43	18	examined	examine	VERB
cana-1534	43	19	.	.	PUNCT
cana-1534	44	1	these	these	DET
cana-1534	44	2	factors	factor	NOUN
cana-1534	44	3	are	be	AUX
cana-1534	44	4	well	well	ADV
cana-1534	44	5	-	-	PUNCT
cana-1534	44	6	studied	study	VERB
cana-1534	44	7	,	,	PUNCT
cana-1534	44	8	including	include	VERB
cana-1534	44	9	linked	link	VERB
cana-1534	44	10	and	and	CCONJ
cana-1534	44	11	independent	independent	ADJ
cana-1534	44	12	resolved	resolve	VERB
cana-1534	44	13	sets	set	NOUN
cana-1534	44	14	[	[	X
cana-1534	44	15	15	15	NUM
cana-1534	44	16	,	,	PUNCT
cana-1534	44	17	17	17	NUM
cana-1534	44	18	]	]	PUNCT
cana-1534	44	19	.	.	PUNCT
cana-1534	45	1	a	a	DET
cana-1534	45	2	resolved	resolve	VERB
cana-1534	45	3	set	set	NOUN
cana-1534	45	4	w	w	NOUN
cana-1534	45	5	of	of	ADP
cana-1534	45	6	g	g	PROPN
cana-1534	45	7	exists	exist	VERB
cana-1534	45	8	independently	independently	ADV
cana-1534	45	9	if	if	SCONJ
cana-1534	45	10	no	no	DET
cana-1534	45	11	two	two	NUM
cana-1534	45	12	vertex	vertex	NOUN
cana-1534	45	13	are	be	AUX
cana-1534	45	14	neighboring	neighbor	VERB
cana-1534	45	15	.	.	PUNCT
cana-1534	46	1	a	a	DET
cana-1534	46	2	resolution	resolution	NOUN
cana-1534	46	3	set	set	VERB
cana-1534	46	4	w	w	NOUN
cana-1534	46	5	of	of	ADP
cana-1534	46	6	g	g	PROPN
cana-1534	46	7	has	have	VERB
cana-1534	46	8	connections	connection	NOUN
cana-1534	46	9	if	if	SCONJ
cana-1534	46	10	its	its	PRON
cana-1534	46	11	induced	induced	ADJ
cana-1534	46	12	sub	sub	NOUN
cana-1534	46	13	network	network	NOUN
cana-1534	46	14	⟨w	⟨w	X
cana-1534	46	15	⟩	⟩	PROPN
cana-1534	46	16	is	be	AUX
cana-1534	46	17	a	a	DET
cana-1534	46	18	non	non	ADJ
cana-1534	46	19	-	-	ADJ
cana-1534	46	20	trivial	trivial	ADJ
cana-1534	46	21	linked	link	VERB
cana-1534	46	22	sub	sub	NOUN
cana-1534	46	23	graph	graph	NOUN
cana-1534	46	24	of	of	ADP
cana-1534	46	25	g.	g.	PROPN
cana-1534	46	26	as	as	ADP
cana-1534	46	27	in	in	ADP
cana-1534	46	28	[	[	X
cana-1534	46	29	13	13	NUM
cana-1534	46	30	]	]	PUNCT
cana-1534	46	31	,	,	PUNCT
cana-1534	46	32	a	a	DET
cana-1534	46	33	non	non	ADJ
cana-1534	46	34	-	-	ADJ
cana-1534	46	35	isolated	isolated	ADJ
cana-1534	46	36	resolve	resolve	NOUN
cana-1534	46	37	collection	collection	NOUN
cana-1534	46	38	was	be	AUX
cana-1534	46	39	developed	develop	VERB
cana-1534	46	40	.	.	PUNCT
cana-1534	47	1	a	a	DET
cana-1534	47	2	resolution	resolution	NOUN
cana-1534	47	3	setting	set	VERB
cana-1534	47	4	w	w	NOUN
cana-1534	47	5	of	of	ADP
cana-1534	47	6	g	g	NOUN
cana-1534	47	7	with	with	ADP
cana-1534	47	8	a	a	DET
cana-1534	47	9	minimum	minimum	NOUN
cana-1534	47	10	of	of	ADP
cana-1534	47	11	2	2	NUM
cana-1534	47	12	vertex	vertex	NOUN
cana-1534	47	13	is	be	AUX
cana-1534	47	14	declared	declare	VERB
cana-1534	47	15	non	non	ADJ
cana-1534	47	16	-	-	ADJ
cana-1534	47	17	isolated	isolated	ADJ
cana-1534	47	18	if	if	SCONJ
cana-1534	47	19	the	the	DET
cana-1534	47	20	resulting	result	VERB
cana-1534	47	21	subgraph	subgraph	NOUN
cana-1534	47	22	⟨w	⟨w	X
cana-1534	47	23	⟩	⟩	NOUN
cana-1534	47	24	contains	contain	VERB
cana-1534	47	25	no	no	DET
cana-1534	47	26	isolation	isolation	NOUN
cana-1534	47	27	vertex	vertex	NOUN
cana-1534	47	28	.	.	PUNCT
cana-1534	48	1	the	the	DET
cana-1534	48	2	smallest	small	ADJ
cana-1534	48	3	relationship	relationship	NOUN
cana-1534	48	4	of	of	ADP
cana-1534	48	5	a	a	DET
cana-1534	48	6	non	non	ADJ
cana-1534	48	7	-	-	ADJ
cana-1534	48	8	isolating	isolating	ADJ
cana-1534	48	9	resolution	resolution	NOUN
cana-1534	48	10	setting	set	VERB
cana-1534	48	11	graph	graph	NOUN
cana-1534	48	12	g	g	NOUN
cana-1534	48	13	is	be	AUX
cana-1534	48	14	termed	term	VERB
cana-1534	48	15	nr(g	nr(g	NUM
cana-1534	48	16	)	)	PUNCT
cana-1534	48	17	.	.	PUNCT
cana-1534	49	1	unisolated	unisolated	ADJ
cana-1534	49	2	resolving	resolving	NOUN
cana-1534	49	3	sets	set	NOUN
cana-1534	49	4	of	of	ADP
cana-1534	49	5	relationship	relationship	NOUN
cana-1534	49	6	nr(g	nr(g	NUM
cana-1534	49	7	)	)	PUNCT
cana-1534	49	8	are	be	AUX
cana-1534	49	9	termed	term	VERB
cana-1534	49	10	nr	nr	NOUN
cana-1534	49	11	-	-	NOUN
cana-1534	49	12	sets	set	NOUN
cana-1534	49	13	of	of	ADP
cana-1534	49	14	g.	g.	PROPN
cana-1534	49	15	in	in	ADP
cana-1534	49	16	[	[	X
cana-1534	49	17	13	13	NUM
cana-1534	49	18	]	]	PUNCT
cana-1534	49	19	,	,	PUNCT
cana-1534	49	20	the	the	DET
cana-1534	49	21	nr	nr	NOUN
cana-1534	49	22	-	-	PUNCT
cana-1534	49	23	value	value	NOUN
cana-1534	49	24	and	and	CCONJ
cana-1534	49	25	cartesian	cartesian	ADJ
cana-1534	49	26	products	product	NOUN
cana-1534	49	27	of	of	ADP
cana-1534	49	28	various	various	ADJ
cana-1534	49	29	graph	graph	NOUN
cana-1534	49	30	topologies	topology	NOUN
cana-1534	49	31	were	be	AUX
cana-1534	49	32	found	find	VERB
cana-1534	49	33	.	.	PUNCT
cana-1534	50	1	furthermore	furthermore	ADV
cana-1534	50	2	,	,	PUNCT
cana-1534	50	3	a	a	DET
cana-1534	50	4	line	line	NOUN
cana-1534	50	5	g	g	NOUN
cana-1534	50	6	of	of	ADP
cana-1534	50	7	rank	rank	NOUN
cana-1534	50	8	n	n	CCONJ
cana-1534	50	9	within	within	ADP
cana-1534	50	10	nr(g	nr(g	PRON
cana-1534	50	11	)	)	PUNCT
cana-1534	51	1	=	=	SYM
cana-1534	51	2	k	k	PROPN
cana-1534	51	3	created	create	VERB
cana-1534	51	4	for	for	ADP
cana-1534	51	5	any	any	DET
cana-1534	51	6	combination	combination	NOUN
cana-1534	51	7	of	of	ADP
cana-1534	51	8	k	k	PROPN
cana-1534	51	9	and	and	CCONJ
cana-1534	51	10	n	n	PROPN
cana-1534	51	11	with	with	ADP
cana-1534	51	12	2	2	NUM
cana-1534	51	13	≤	≤	NOUN
cana-1534	51	14	k	k	X
cana-1534	51	15	<	<	X
cana-1534	51	16	n	n	CCONJ
cana-1534	51	17	−	−	PROPN
cana-1534	51	18	1	1	NUM
cana-1534	51	19	.	.	PUNCT
cana-1534	52	1	in	in	ADP
cana-1534	52	2	[	[	X
cana-1534	52	3	1	1	NUM
cana-1534	52	4	]	]	PUNCT
cana-1534	52	5	,	,	PUNCT
cana-1534	52	6	the	the	DET
cana-1534	52	7	precise	precise	ADJ
cana-1534	52	8	nr(g	nr(g	NUM
cana-1534	52	9	)	)	PUNCT
cana-1534	52	10	networks	network	NOUN
cana-1534	52	11	and	and	CCONJ
cana-1534	52	12	standard	standard	ADJ
cana-1534	52	13	network	network	NOUN
cana-1534	52	14	subdivisions	subdivision	NOUN
cana-1534	52	15	are	be	AUX
cana-1534	52	16	given	give	VERB
cana-1534	52	17	.	.	PUNCT
cana-1534	53	1	additionally	additionally	ADV
cana-1534	53	2	,	,	PUNCT
cana-1534	53	3	[	[	X
cana-1534	53	4	2	2	NUM
cana-1534	53	5	]	]	PUNCT
cana-1534	53	6	discusses	discuss	VERB
cana-1534	53	7	the	the	DET
cana-1534	53	8	correlation	correlation	NOUN
cana-1534	53	9	among	among	ADP
cana-1534	53	10	nr	nr	NOUN
cana-1534	53	11	-	-	PUNCT
cana-1534	53	12	value	value	NOUN
cana-1534	53	13	and	and	CCONJ
cana-1534	53	14	factors	factor	NOUN
cana-1534	53	15	like	like	ADP
cana-1534	53	16	χ(g	χ(g	PROPN
cana-1534	53	17	)	)	PUNCT
cana-1534	53	18	&	&	CCONJ
cana-1534	53	19	∆(g	∆(g	PROPN
cana-1534	53	20	)	)	PUNCT
cana-1534	53	21	.	.	PUNCT
cana-1534	54	1	we	we	PRON
cana-1534	54	2	show	show	VERB
cana-1534	54	3	in	in	ADP
cana-1534	54	4	this	this	DET
cana-1534	54	5	work	work	NOUN
cana-1534	54	6	that	that	PRON
cana-1534	54	7	a	a	DET
cana-1534	54	8	graph	graph	NOUN
cana-1534	54	9	has	have	VERB
cana-1534	54	10	a	a	DET
cana-1534	54	11	unique	unique	ADJ
cana-1534	54	12	nr	nr	NOUN
cana-1534	54	13	-	-	NOUN
cana-1534	54	14	set	set	NOUN
cana-1534	54	15	.	.	PUNCT
cana-1534	55	1	furthermore	furthermore	ADV
cana-1534	55	2	,	,	PUNCT
cana-1534	55	3	we	we	PRON
cana-1534	55	4	built	build	VERB
cana-1534	55	5	a	a	DET
cana-1534	55	6	2n	2n	NUM
cana-1534	55	7	-	-	PUNCT
cana-1534	55	8	vertex	vertex	NOUN
cana-1534	55	9	graph	graph	NOUN
cana-1534	55	10	g	g	NOUN
cana-1534	55	11	with	with	ADP
cana-1534	55	12	nr	nr	NOUN
cana-1534	55	13	-	-	PUNCT
cana-1534	55	14	set	set	VERB
cana-1534	55	15	w.	w.	NOUN
cana-1534	55	16	such	such	ADJ
cana-1534	55	17	that	that	PRON
cana-1534	55	18	nr(g	nr(g	PUNCT
cana-1534	55	19	)	)	PUNCT
cana-1534	56	1	=	=	SYM
cana-1534	56	2	n	n	PROPN
cana-1534	56	3	and	and	CCONJ
cana-1534	56	4	the	the	DET
cana-1534	56	5	representation	representation	NOUN
cana-1534	56	6	of	of	ADP
cana-1534	56	7	each	each	DET
cana-1534	56	8	vertex	vertex	NOUN
cana-1534	56	9	not	not	PART
cana-1534	56	10	in	in	ADP
cana-1534	56	11	w	w	PROPN
cana-1534	56	12	is	be	AUX
cana-1534	56	13	r(vi|w	r(vi|w	VERB
cana-1534	56	14	)	)	PUNCT
cana-1534	56	15	=	=	PUNCT
cana-1534	56	16	(	(	PUNCT
cana-1534	56	17	1	1	NUM
cana-1534	56	18	,	,	PUNCT
cana-1534	56	19	2	2	NUM
cana-1534	56	20	,	,	PUNCT
cana-1534	56	21	..	..	PUNCT
cana-1534	57	1	1	1	NUM
cana-1534	57	2	,	,	PUNCT
cana-1534	57	3	.	.	PUNCT
cana-1534	58	1	,	,	PUNCT
cana-1534	58	2	1	1	NUM
cana-1534	58	3	)	)	PUNCT
cana-1534	58	4	.	.	PUNCT
cana-1534	59	1	also	also	ADV
cana-1534	59	2	we	we	PRON
cana-1534	59	3	established	establish	VERB
cana-1534	59	4	the	the	DET
cana-1534	59	5	nr	nr	NOUN
cana-1534	59	6	-	-	NOUN
cana-1534	59	7	value	value	NOUN
cana-1534	59	8	for	for	ADP
cana-1534	59	9	the	the	DET
cana-1534	59	10	highly	highly	ADV
cana-1534	59	11	irregular	irregular	ADJ
cana-1534	59	12	graph	graph	NOUN
cana-1534	59	13	hn	hn	PROPN
cana-1534	59	14	,	,	PUNCT
cana-1534	59	15	n	n	PROPN
cana-1534	59	16	and	and	CCONJ
cana-1534	59	17	for	for	ADP
cana-1534	59	18	the	the	DET
cana-1534	59	19	wheel	wheel	NOUN
cana-1534	59	20	wn	wn	PROPN
cana-1534	59	21	.	.	PROPN
cana-1534	59	22	and	and	CCONJ
cana-1534	59	23	also	also	ADV
cana-1534	59	24	the	the	DET
cana-1534	59	25	nrvalue	nrvalue	NOUN
cana-1534	59	26	for	for	ADP
cana-1534	59	27	corona	corona	NOUN
cana-1534	59	28	product	product	NOUN
cana-1534	59	29	of	of	ADP
cana-1534	59	30	some	some	DET
cana-1534	59	31	graphs	graph	NOUN
cana-1534	59	32	are	be	AUX
cana-1534	59	33	determined	determine	VERB
cana-1534	59	34	.	.	PUNCT
cana-1534	60	1	2	2	NUM
cana-1534	60	2	graphs	graph	NOUN
cana-1534	60	3	with	with	ADP
cana-1534	60	4	unique	unique	ADJ
cana-1534	60	5	nr	nr	NOUN
cana-1534	60	6	-	-	ADJ
cana-1534	60	7	set	set	NOUN
cana-1534	60	8	generally	generally	ADV
cana-1534	60	9	,	,	PUNCT
cana-1534	60	10	graphs	graph	NOUN
cana-1534	60	11	contain	contain	VERB
cana-1534	60	12	many	many	ADJ
cana-1534	60	13	nr	nr	NOUN
cana-1534	60	14	-	-	NOUN
cana-1534	60	15	sets	set	NOUN
cana-1534	60	16	.	.	PUNCT
cana-1534	61	1	this	this	DET
cana-1534	61	2	section	section	NOUN
cana-1534	61	3	demonstrates	demonstrate	VERB
cana-1534	61	4	that	that	SCONJ
cana-1534	61	5	a	a	DET
cana-1534	61	6	network	network	NOUN
cana-1534	61	7	with	with	ADP
cana-1534	61	8	a	a	DET
cana-1534	61	9	distinctive	distinctive	ADJ
cana-1534	61	10	nrset	nrset	NOUN
cana-1534	61	11	of	of	ADP
cana-1534	61	12	relationship	relationship	NOUN
cana-1534	61	13	n	n	PRON
cana-1534	61	14	occurs	occur	VERB
cana-1534	61	15	for	for	ADP
cana-1534	61	16	any	any	DET
cana-1534	61	17	integer	integer	NOUN
cana-1534	61	18	n	n	PRON
cana-1534	61	19	≥	≥	NOUN
cana-1534	61	20	2	2	NUM
cana-1534	61	21	.	.	PUNCT
cana-1534	61	22	theor	theor	PROPN
cana-1534	61	23	2.1	2.1	NUM
cana-1534	61	24	.	.	PUNCT
cana-1534	62	1	any	any	DET
cana-1534	62	2	linked	link	VERB
cana-1534	62	3	graph	graph	NOUN
cana-1534	62	4	g	g	ADP
cana-1534	62	5	having	have	VERB
cana-1534	62	6	order	order	NOUN
cana-1534	62	7	n	n	PRON
cana-1534	62	8	≥	≥	NOUN
cana-1534	62	9	2	2	NUM
cana-1534	62	10	has	have	VERB
cana-1534	62	11	a	a	DET
cana-1534	62	12	unique	unique	ADJ
cana-1534	62	13	nr	nr	NOUN
cana-1534	62	14	-	-	NOUN
cana-1534	62	15	set	set	NOUN
cana-1534	62	16	of	of	ADP
cana-1534	62	17	relationship	relationship	NOUN
cana-1534	62	18	n	n	CCONJ
cana-1534	62	19	,	,	PUNCT
cana-1534	62	20	resulting	result	VERB
cana-1534	62	21	in	in	ADP
cana-1534	62	22	an	an	DET
cana-1534	62	23	isotropic	isotropic	ADJ
cana-1534	62	24	graph	graph	NOUN
cana-1534	62	25	.	.	PUNCT
cana-1534	63	1	proof	proof	NOUN
cana-1534	63	2	.	.	PUNCT
cana-1534	64	1	let	let	VERB
cana-1534	64	2	g	g	NOUN
cana-1534	64	3	be	be	AUX
cana-1534	64	4	any	any	DET
cana-1534	64	5	connected	connected	ADJ
cana-1534	64	6	graph	graph	NOUN
cana-1534	64	7	on	on	ADP
cana-1534	64	8	n	n	PRON
cana-1534	64	9	≥	≥	NUM
cana-1534	64	10	2	2	NUM
cana-1534	64	11	vertices	vertex	NOUN
cana-1534	64	12	and	and	CCONJ
cana-1534	64	13	v	v	X
cana-1534	64	14	(	(	PUNCT
cana-1534	64	15	g	g	NOUN
cana-1534	64	16	)	)	PUNCT
cana-1534	64	17	=	=	SYM
cana-1534	64	18	{	{	PUNCT
cana-1534	64	19	v0	v0	NOUN
cana-1534	64	20	,	,	PUNCT
cana-1534	64	21	…	…	PUNCT
cana-1534	64	22	vn−1	vn−1	ADJ
cana-1534	64	23	}	}	PUNCT
cana-1534	64	24	.	.	PUNCT
cana-1534	65	1	let	let	VERB
cana-1534	65	2	g′	g′	NOUN
cana-1534	65	3	=	=	PUNCT
cana-1534	65	4	k2n−1	k2n−1	PROPN
cana-1534	65	5	with	with	ADP
cana-1534	65	6	the	the	DET
cana-1534	65	7	vertex	vertex	NOUN
cana-1534	65	8	set	set	VERB
cana-1534	65	9	v	v	NOUN
cana-1534	65	10	(	(	PUNCT
cana-1534	65	11	g′	g′	NOUN
cana-1534	65	12	)	)	PUNCT
cana-1534	66	1	=	=	NOUN
cana-1534	66	2	{	{	PUNCT
cana-1534	66	3	u0	u0	ADJ
cana-1534	66	4	,	,	PUNCT
cana-1534	66	5	u1	u1	NOUN
cana-1534	66	6	,	,	PUNCT
cana-1534	66	7	u2	u2	NOUN
cana-1534	66	8	,	,	PUNCT
cana-1534	66	9	...	...	PUNCT
cana-1534	66	10	,	,	PUNCT
cana-1534	66	11	u2n	u2n	PROPN
cana-1534	66	12	−2	−2	NOUN
cana-1534	66	13	}	}	PUNCT
cana-1534	66	14	.	.	PUNCT
cana-1534	67	1	a	a	DET
cana-1534	67	2	chart	chart	NOUN
cana-1534	67	3	h	h	NOUN
cana-1534	67	4	with	with	ADP
cana-1534	67	5	a	a	DET
cana-1534	67	6	complete	complete	ADJ
cana-1534	67	7	vertex	vertex	NOUN
cana-1534	67	8	v	v	NOUN
cana-1534	67	9	is	be	AUX
cana-1534	67	10	generated	generate	VERB
cana-1534	67	11	from	from	ADP
cana-1534	67	12	g	g	PROPN
cana-1534	67	13	and	and	CCONJ
cana-1534	67	14	g′	g′	NOUN
cana-1534	67	15	by	by	ADP
cana-1534	67	16	adding	add	VERB
cana-1534	67	17	edges	edge	NOUN
cana-1534	67	18	connecting	connect	VERB
cana-1534	67	19	v	v	ADP
cana-1534	67	20	(	(	PUNCT
cana-1534	67	21	g	g	NOUN
cana-1534	67	22	)	)	PUNCT
cana-1534	67	23	and	and	CCONJ
cana-1534	67	24	v	v	NOUN
cana-1534	67	25	(	(	PUNCT
cana-1534	67	26	g′	g′	NOUN
cana-1534	67	27	):	):	PUNCT
cana-1534	67	28	give	give	VERB
cana-1534	67	29	every	every	DET
cana-1534	67	30	number	number	NOUN
cana-1534	67	31	i	i	PRON
cana-1534	67	32	(	(	PUNCT
cana-1534	67	33	1	1	NUM
cana-1534	67	34	<	<	X
cana-1534	67	35	i	i	PROPN
cana-1534	67	36	≤	≤	NOUN
cana-1534	67	37	2n	2n	NUM
cana-1534	67	38	−	−	ADP
cana-1534	67	39	2	2	X
cana-1534	67	40	)	)	PUNCT
cana-1534	67	41	its	its	PRON
cana-1534	67	42	base	base	NOUN
cana-1534	67	43	2	2	NUM
cana-1534	67	44	(	(	PUNCT
cana-1534	67	45	binary	binary	ADJ
cana-1534	67	46	)	)	PUNCT
cana-1534	67	47	form	form	NOUN
cana-1534	67	48	.	.	PUNCT
cana-1534	68	1	every	every	DET
cana-1534	68	2	such	such	ADJ
cana-1534	68	3	i	i	PRON
cana-1534	68	4	may	may	AUX
cana-1534	68	5	be	be	AUX
cana-1534	68	6	described	describe	VERB
cana-1534	68	7	as	as	ADP
cana-1534	68	8	a	a	DET
cana-1534	68	9	series	series	NOUN
cana-1534	68	10	of	of	ADP
cana-1534	68	11	k	k	PROPN
cana-1534	68	12	co	co	NOUN
cana-1534	68	13	-	-	NOUN
cana-1534	68	14	ordinates	ordinate	NOUN
cana-1534	68	15	,	,	PUNCT
cana-1534	68	16	or	or	CCONJ
cana-1534	68	17	k	k	NOUN
cana-1534	68	18	-	-	NOUN
cana-1534	68	19	vectors	vector	NOUN
cana-1534	68	20	,	,	PUNCT
cana-1534	68	21	in	in	ADP
cana-1534	68	22	which	which	PRON
cana-1534	68	23	the	the	DET
cana-1534	68	24	rightmost	rightmost	NOUN
cana-1534	68	25	co	co	NOUN
cana-1534	68	26	-	-	NOUN
cana-1534	68	27	ordinate	ordinate	NOUN
cana-1534	68	28	is	be	AUX
cana-1534	68	29	an	an	DET
cana-1534	68	30	integer	integer	NOUN
cana-1534	68	31	(	(	PUNCT
cana-1534	68	32	either	either	CCONJ
cana-1534	68	33	0	0	NUM
cana-1534	68	34	or	or	CCONJ
cana-1534	68	35	1	1	NUM
cana-1534	68	36	)	)	PUNCT
cana-1534	68	37	at	at	ADP
cana-1534	68	38	the	the	DET
cana-1534	68	39	20	20	NUM
cana-1534	68	40	status	status	NOUN
cana-1534	68	41	,	,	PUNCT
cana-1534	68	42	the	the	DET
cana-1534	68	43	value	value	NOUN
cana-1534	68	44	to	to	ADP
cana-1534	68	45	its	its	PRON
cana-1534	68	46	immediately	immediately	ADV
cana-1534	68	47	left	leave	VERB
cana-1534	68	48	is	be	AUX
cana-1534	68	49	the	the	DET
cana-1534	68	50	21	21	NUM
cana-1534	68	51	status	status	NOUN
cana-1534	68	52	,	,	PUNCT
cana-1534	68	53	etc	etc	X
cana-1534	68	54	.	.	X
cana-1534	69	1	for	for	ADP
cana-1534	69	2	integers	integer	NOUN
cana-1534	69	3	i	i	PRON
cana-1534	69	4	,	,	PUNCT
cana-1534	69	5	j	j	PROPN
cana-1534	69	6	with	with	ADP
cana-1534	69	7	0	0	NUM
cana-1534	69	8	≤	≤	NUM
cana-1534	69	9	j	j	PROPN
cana-1534	69	10	≤	≤	PROPN
cana-1534	69	11	n	n	CCONJ
cana-1534	69	12	−	−	PROPN
cana-1534	69	13	1	1	NUM
cana-1534	69	14	and	and	CCONJ
cana-1534	69	15	0	0	NUM
cana-1534	69	16	≤	≤	NUM
cana-1534	69	17	i	i	PRON
cana-1534	69	18	≤	≤	NOUN
cana-1534	69	19	2n	2n	NUM
cana-1534	69	20	−	−	NOUN
cana-1534	69	21	2	2	NUM
cana-1534	69	22	,	,	PUNCT
cana-1534	69	23	we	we	PRON
cana-1534	69	24	combine	combine	VERB
cana-1534	69	25	ui	ui	PROPN
cana-1534	69	26	and	and	CCONJ
cana-1534	69	27	vj	vj	INTJ
cana-1534	69	28	if	if	SCONJ
cana-1534	70	1	and	and	CCONJ
cana-1534	70	2	only	only	ADV
cana-1534	70	3	if	if	SCONJ
cana-1534	70	4	i	i	PRON
cana-1534	70	5	's	be	AUX
cana-1534	70	6	(	(	PUNCT
cana-1534	70	7	2j	2j	NUM
cana-1534	70	8	)	)	PUNCT
cana-1534	70	9	th	th	PROPN
cana-1534	70	10	binary	binary	PROPN
cana-1534	70	11	code	code	NOUN
cana-1534	70	12	value	value	NOUN
cana-1534	70	13	is	be	AUX
cana-1534	70	14	1	1	NUM
cana-1534	70	15	.	.	PUNCT
cana-1534	70	16	first	first	ADV
cana-1534	70	17	let	let	VERB
cana-1534	70	18	us	we	PRON
cana-1534	70	19	prove	prove	VERB
cana-1534	70	20	that	that	PRON
cana-1534	70	21	nr(h	nr(h	NOUN
cana-1534	70	22	)	)	PUNCT
cana-1534	70	23	=	=	SYM
cana-1534	70	24	n.	n.	NOUN
cana-1534	70	25	communications	communication	NOUN
cana-1534	70	26	on	on	ADP
cana-1534	70	27	applied	apply	VERB
cana-1534	70	28	nonlinear	nonlinear	ADJ
cana-1534	70	29	analysis	analysis	NOUN
cana-1534	70	30	issn	issn	NOUN
cana-1534	70	31	:	:	PUNCT
cana-1534	70	32	1074	1074	NUM
cana-1534	70	33	-	-	PUNCT
cana-1534	70	34	133x	133x	NUM
cana-1534	70	35	vol	vol	NOUN
cana-1534	70	36	31	31	NUM
cana-1534	70	37	no	no	NOUN
cana-1534	70	38	.	.	PUNCT
cana-1534	71	1	8s	8s	PROPN
cana-1534	71	2	(	(	PUNCT
cana-1534	71	3	2024	2024	NUM
cana-1534	71	4	)	)	PUNCT
cana-1534	71	5	423	423	NUM
cana-1534	71	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	71	7	let	let	VERB
cana-1534	71	8	w	w	PROPN
cana-1534	71	9	=	=	SYM
cana-1534	71	10	v	v	ADJ
cana-1534	71	11	(	(	PUNCT
cana-1534	71	12	g	g	NOUN
cana-1534	71	13	)	)	PUNCT
cana-1534	71	14	.	.	PUNCT
cana-1534	72	1	else	else	ADV
cana-1534	72	2	r(v|w	r(v|w	VERB
cana-1534	72	3	)	)	PUNCT
cana-1534	73	1	=	=	PUNCT
cana-1534	73	2	(	(	PUNCT
cana-1534	73	3	1	1	NUM
cana-1534	73	4	,	,	PUNCT
cana-1534	73	5	1	1	NUM
cana-1534	73	6	,	,	PUNCT
cana-1534	73	7	...	...	PUNCT
cana-1534	73	8	,	,	PUNCT
cana-1534	73	9	1	1	X
cana-1534	73	10	)	)	PUNCT
cana-1534	73	11	and	and	CCONJ
cana-1534	73	12	r(ui|w	r(ui|w	VERB
cana-1534	73	13	)	)	PUNCT
cana-1534	73	14	=	=	PUNCT
cana-1534	74	1	(	(	PUNCT
cana-1534	74	2	2	2	NUM
cana-1534	74	3	−	−	NOUN
cana-1534	74	4	bn−1	bn−1	ADJ
cana-1534	74	5	,	,	PUNCT
cana-1534	74	6	2	2	NUM
cana-1534	74	7	−	−	NOUN
cana-1534	74	8	bn−2	bn−2	PROPN
cana-1534	74	9	,	,	PUNCT
cana-1534	74	10	...	...	PUNCT
cana-1534	74	11	,	,	PUNCT
cana-1534	74	12	2	2	NUM
cana-1534	74	13	−	−	NOUN
cana-1534	74	14	b0	b0	NOUN
cana-1534	74	15	)	)	PUNCT
cana-1534	74	16	.	.	PUNCT
cana-1534	75	1	0	0	PUNCT
cana-1534	76	1	≤	≤	NUM
cana-1534	76	2	i	i	PRON
cana-1534	76	3	≤	≤	NOUN
cana-1534	76	4	2n	2n	NUM
cana-1534	76	5	−	−	NOUN
cana-1534	76	6	2	2	X
cana-1534	76	7	.	.	PUNCT
cana-1534	76	8	also	also	ADV
cana-1534	76	9	⟨w	⟨w	VERB
cana-1534	76	10	⟩	⟩	NOUN
cana-1534	76	11	=	=	NOUN
cana-1534	76	12	∼	∼	NOUN
cana-1534	76	13	g.	g.	NOUN
cana-1534	76	14	w	w	NOUN
cana-1534	76	15	resolves	resolve	NOUN
cana-1534	76	16	h	h	NOUN
cana-1534	76	17	non	non	ADJ
cana-1534	76	18	-	-	ADJ
cana-1534	76	19	isolatedly	isolatedly	ADV
cana-1534	76	20	.	.	PUNCT
cana-1534	77	1	therefore	therefore	ADV
cana-1534	77	2	,	,	PUNCT
cana-1534	77	3	nr(h	nr(h	NOUN
cana-1534	77	4	)	)	PUNCT
cana-1534	77	5	≤	≤	NOUN
cana-1534	77	6	n.	n.	NOUN
cana-1534	77	7	just	just	ADV
cana-1534	77	8	show	show	VERB
cana-1534	77	9	that	that	PRON
cana-1534	77	10	nr(h	nr(h	NOUN
cana-1534	77	11	)	)	PUNCT
cana-1534	77	12	>	>	X
cana-1534	78	1	n.	n.	PROPN
cana-1534	78	2	non	non	ADJ
cana-1534	78	3	-	-	ADJ
cana-1534	78	4	isolated	isolated	ADJ
cana-1534	78	5	h	h	NOUN
cana-1534	78	6	-	-	PUNCT
cana-1534	78	7	resolved	resolve	VERB
cana-1534	78	8	sets	set	NOUN
cana-1534	78	9	may	may	AUX
cana-1534	78	10	be	be	AUX
cana-1534	78	11	w.	w.	NOUN
cana-1534	78	12	if	if	SCONJ
cana-1534	78	13	vj	vj	INTJ
cana-1534	78	14	∈/	∈/	PROPN
cana-1534	78	15	w	w	PROPN
cana-1534	78	16	for	for	ADP
cana-1534	78	17	some	some	DET
cana-1534	78	18	j	j	NOUN
cana-1534	78	19	,	,	PUNCT
cana-1534	78	20	0	0	NUM
cana-1534	78	21	≤	≤	NUM
cana-1534	78	22	j	j	PROPN
cana-1534	78	23	≤	≤	PROPN
cana-1534	78	24	n	n	CCONJ
cana-1534	78	25	−	−	PROPN
cana-1534	78	26	1	1	NUM
cana-1534	78	27	,	,	PUNCT
cana-1534	78	28	else	else	ADV
cana-1534	78	29	r(vj|w	r(vj|w	PROPN
cana-1534	78	30	)	)	PUNCT
cana-1534	78	31	=	=	PUNCT
cana-1534	78	32	r(ui|w	r(ui|w	VERB
cana-1534	78	33	)	)	PUNCT
cana-1534	78	34	for	for	ADP
cana-1534	78	35	some	some	DET
cana-1534	78	36	i	i	PROPN
cana-1534	78	37	,	,	PUNCT
cana-1534	78	38	0	0	NUM
cana-1534	78	39	≤	≤	NUM
cana-1534	79	1	i	i	PRON
cana-1534	79	2	≤	≤	NOUN
cana-1534	79	3	2n	2n	NUM
cana-1534	79	4	−	−	ADP
cana-1534	79	5	2	2	NUM
cana-1534	79	6	,	,	PUNCT
cana-1534	79	7	which	which	PRON
cana-1534	79	8	is	be	AUX
cana-1534	79	9	a	a	DET
cana-1534	79	10	contradiction	contradiction	NOUN
cana-1534	79	11	.	.	PUNCT
cana-1534	80	1	therefore	therefore	ADV
cana-1534	80	2	,	,	PUNCT
cana-1534	80	3	w	w	PROPN
cana-1534	80	4	must	must	AUX
cana-1534	80	5	contain	contain	VERB
cana-1534	80	6	all	all	DET
cana-1534	80	7	the	the	DET
cana-1534	80	8	vertices	vertex	NOUN
cana-1534	80	9	of	of	ADP
cana-1534	80	10	g.	g.	NOUN
cana-1534	80	11	hence	hence	ADV
cana-1534	80	12	|w	|w	PROPN
cana-1534	81	1	|	|	ADV
cana-1534	81	2	≥	≥	NOUN
cana-1534	81	3	n.	n.	NOUN
cana-1534	81	4	thus	thus	ADV
cana-1534	81	5	nr(h	nr(h	NOUN
cana-1534	81	6	)	)	PUNCT
cana-1534	81	7	=	=	VERB
cana-1534	82	1	n.	n.	NOUN
cana-1534	82	2	now	now	ADV
cana-1534	82	3	we	we	PRON
cana-1534	82	4	have	have	VERB
cana-1534	82	5	to	to	PART
cana-1534	82	6	show	show	VERB
cana-1534	82	7	that	that	SCONJ
cana-1534	82	8	there	there	PRON
cana-1534	82	9	is	be	VERB
cana-1534	82	10	no	no	DET
cana-1534	82	11	other	other	ADJ
cana-1534	82	12	nr	nr	ADV
cana-1534	82	13	-	-	PUNCT
cana-1534	82	14	set	set	VERB
cana-1534	82	15	for	for	ADP
cana-1534	82	16	h.	h.	NOUN
cana-1534	82	17	let	let	VERB
cana-1534	82	18	vj	vj	INTJ
cana-1534	82	19	be	be	AUX
cana-1534	82	20	the	the	DET
cana-1534	82	21	vertex	vertex	NOUN
cana-1534	82	22	which	which	PRON
cana-1534	82	23	is	be	AUX
cana-1534	82	24	in	in	ADP
cana-1534	82	25	w	w	PROPN
cana-1534	82	26	but	but	CCONJ
cana-1534	82	27	not	not	PART
cana-1534	82	28	in	in	ADP
cana-1534	82	29	w	w	NOUN
cana-1534	82	30	′	′	NUM
cana-1534	82	31	for	for	ADP
cana-1534	82	32	some	some	DET
cana-1534	82	33	j	j	PROPN
cana-1534	82	34	,	,	PUNCT
cana-1534	82	35	0	0	NUM
cana-1534	82	36	≤	≤	NUM
cana-1534	82	37	j	j	PROPN
cana-1534	82	38	≤	≤	PROPN
cana-1534	82	39	n	n	CCONJ
cana-1534	82	40	−	−	PROPN
cana-1534	82	41	1	1	NUM
cana-1534	82	42	.	.	PUNCT
cana-1534	82	43	else	else	ADV
cana-1534	83	1	w	w	ADP
cana-1534	83	2	′	′	NUM
cana-1534	83	3	must	must	AUX
cana-1534	83	4	contain	contain	VERB
cana-1534	83	5	one	one	NUM
cana-1534	83	6	vertex	vertex	NOUN
cana-1534	83	7	from	from	ADP
cana-1534	83	8	each	each	PRON
cana-1534	83	9	of	of	ADP
cana-1534	83	10	{	{	PUNCT
cana-1534	83	11	us	us	PROPN
cana-1534	83	12	,	,	PUNCT
cana-1534	83	13	ut	ut	PROPN
cana-1534	83	14	}	}	PUNCT
cana-1534	83	15	with	with	ADP
cana-1534	83	16	t	t	PROPN
cana-1534	83	17	−	−	PROPN
cana-1534	83	18	s	s	PART
cana-1534	83	19	=	=	NOUN
cana-1534	83	20	2j	2j	NOUN
cana-1534	83	21	for	for	ADP
cana-1534	83	22	all	all	PRON
cana-1534	83	23	0	0	NUM
cana-1534	83	24	≤	≤	NUM
cana-1534	83	25	s	s	PART
cana-1534	83	26	<	<	X
cana-1534	83	27	t	t	X
cana-1534	83	28	≤	≤	PUNCT
cana-1534	83	29	2n	2n	NUM
cana-1534	84	1	−	−	ADP
cana-1534	84	2	2	2	X
cana-1534	84	3	.	.	PUNCT
cana-1534	84	4	therefore	therefore	ADV
cana-1534	84	5	|w	|w	ADJ
cana-1534	84	6	′|	′|	NUM
cana-1534	84	7	≥	≥	NOUN
cana-1534	84	8	2n	2n	NUM
cana-1534	84	9	−	−	NOUN
cana-1534	85	1	1	1	X
cana-1534	85	2	.	.	PUNCT
cana-1534	86	1	the	the	DET
cana-1534	86	2	following	follow	VERB
cana-1534	86	3	theorem	theorem	NOUN
cana-1534	86	4	constructs	construct	VERB
cana-1534	86	5	a	a	DET
cana-1534	86	6	graph	graph	NOUN
cana-1534	86	7	g	g	NOUN
cana-1534	86	8	on	on	ADP
cana-1534	86	9	2n	2n	NUM
cana-1534	86	10	triangles	triangle	NOUN
cana-1534	86	11	with	with	ADP
cana-1534	86	12	a	a	DET
cana-1534	86	13	nr	nr	ADV
cana-1534	86	14	-	-	ADJ
cana-1534	86	15	set	set	VERB
cana-1534	86	16	w	w	ADP
cana-1534	86	17	such	such	ADJ
cana-1534	86	18	that	that	PRON
cana-1534	86	19	nr(g	nr(g	PUNCT
cana-1534	86	20	)	)	PUNCT
cana-1534	87	1	=	=	SYM
cana-1534	87	2	n	n	PROPN
cana-1534	87	3	and	and	CCONJ
cana-1534	87	4	the	the	DET
cana-1534	87	5	value	value	NOUN
cana-1534	87	6	of	of	ADP
cana-1534	87	7	every	every	DET
cana-1534	87	8	vertices	vertex	NOUN
cana-1534	87	9	is	be	AUX
cana-1534	87	10	nr	nr	PRON
cana-1534	87	11	.	.	PUNCT
cana-1534	88	1	w	w	PROPN
cana-1534	88	2	is	be	AUX
cana-1534	88	3	r(vi|w	r(vi|w	VERB
cana-1534	88	4	)	)	PUNCT
cana-1534	88	5	=	=	PUNCT
cana-1534	88	6	(	(	PUNCT
cana-1534	88	7	1	1	NUM
cana-1534	88	8	,	,	PUNCT
cana-1534	88	9	,	,	PUNCT
cana-1534	88	10	2	2	NUM
cana-1534	88	11	,	,	PUNCT
cana-1534	88	12	…	…	PUNCT
cana-1534	88	13	1	1	NUM
cana-1534	88	14	,	,	PUNCT
cana-1534	88	15	...	...	PUNCT
cana-1534	88	16	,	,	PUNCT
cana-1534	88	17	1	1	X
cana-1534	88	18	)	)	PUNCT
cana-1534	88	19	.	.	PUNCT
cana-1534	89	1	theorem	theorem	VERB
cana-1534	89	2	2.2	2.2	NUM
cana-1534	89	3	.	.	PUNCT
cana-1534	90	1	there	there	PRON
cana-1534	90	2	is	be	VERB
cana-1534	90	3	a	a	DET
cana-1534	90	4	chart	chart	NOUN
cana-1534	90	5	g	g	NOUN
cana-1534	90	6	on	on	ADP
cana-1534	90	7	2n	2n	NUM
cana-1534	90	8	vertex	vertex	NOUN
cana-1534	90	9	with	with	ADP
cana-1534	90	10	a	a	DET
cana-1534	90	11	nr	nr	ADV
cana-1534	90	12	-	-	PUNCT
cana-1534	90	13	set	set	VERB
cana-1534	90	14	w	w	NOUN
cana-1534	90	15	,	,	PUNCT
cana-1534	90	16	nr(g	nr(g	NUM
cana-1534	90	17	)	)	PUNCT
cana-1534	90	18	=	=	SYM
cana-1534	90	19	n	n	CCONJ
cana-1534	90	20	,	,	PUNCT
cana-1534	90	21	and	and	CCONJ
cana-1534	90	22	r(vi|w	r(vi|w	VERB
cana-1534	90	23	)	)	PUNCT
cana-1534	90	24	=	=	SYM
cana-1534	90	25	(	(	PUNCT
cana-1534	90	26	1	1	NUM
cana-1534	90	27	,	,	PUNCT
cana-1534	90	28	1,	1,	NUM
cana-1534	90	29	..	..	SYM
cana-1534	90	30	2	2	NUM
cana-1534	90	31	…	…	SYM
cana-1534	90	32	1	1	NUM
cana-1534	90	33	..	..	PUNCT
cana-1534	90	34	,	,	PUNCT
cana-1534	90	35	1	1	NUM
cana-1534	90	36	,	,	PUNCT
cana-1534	90	37	...	...	PUNCT
cana-1534	90	38	,	,	PUNCT
cana-1534	90	39	1	1	X
cana-1534	90	40	)	)	PUNCT
cana-1534	90	41	for	for	ADP
cana-1534	90	42	each	each	DET
cana-1534	90	43	significant	significant	ADJ
cana-1534	90	44	integers	integer	NOUN
cana-1534	90	45	n	n	PRON
cana-1534	90	46	≥	≥	NOUN
cana-1534	90	47	2	2	NUM
cana-1534	90	48	.	.	PUNCT
cana-1534	90	49	proof	proof	NOUN
cana-1534	90	50	.	.	PUNCT
cana-1534	91	1	let	let	VERB
cana-1534	91	2	g	g	NOUN
cana-1534	91	3	=	=	PUNCT
cana-1534	91	4	(	(	PUNCT
cana-1534	91	5	kn	kn	PROPN
cana-1534	91	6	+	+	CCONJ
cana-1534	91	7	kn)/1	kn)/1	PROPN
cana-1534	91	8	−	−	PROPN
cana-1534	91	9	factor	factor	NOUN
cana-1534	91	10	and	and	CCONJ
cana-1534	91	11	v	v	NOUN
cana-1534	91	12	(	(	PUNCT
cana-1534	91	13	g	g	NOUN
cana-1534	91	14	)	)	PUNCT
cana-1534	91	15	=	=	SYM
cana-1534	91	16	v1	v1	VERB
cana-1534	91	17	∪	∪	NOUN
cana-1534	91	18	v2	v2	PROPN
cana-1534	91	19	wherein	wherein	ADJ
cana-1534	91	20	v1	v1	NOUN
cana-1534	91	21	and	and	CCONJ
cana-1534	91	22	v2	v2	NOUN
cana-1534	91	23	=	=	SYM
cana-1534	91	24	{	{	PUNCT
cana-1534	91	25	v′	v′	NOUN
cana-1534	91	26	…	…	PUNCT
cana-1534	91	27	.v′	.v′	PUNCT
cana-1534	91	28	}	}	PUNCT
cana-1534	91	29	.	.	PUNCT
cana-1534	92	1	communications	communication	NOUN
cana-1534	92	2	on	on	ADP
cana-1534	92	3	applied	apply	VERB
cana-1534	92	4	nonlinear	nonlinear	ADJ
cana-1534	92	5	analysis	analysis	NOUN
cana-1534	92	6	issn	issn	NOUN
cana-1534	92	7	:	:	PUNCT
cana-1534	92	8	1074	1074	NUM
cana-1534	92	9	-	-	PUNCT
cana-1534	92	10	133x	133x	NUM
cana-1534	92	11	vol	vol	NOUN
cana-1534	92	12	31	31	NUM
cana-1534	92	13	no	no	NOUN
cana-1534	92	14	.	.	PUNCT
cana-1534	93	1	8s	8s	PROPN
cana-1534	93	2	(	(	PUNCT
cana-1534	93	3	2024	2024	NUM
cana-1534	93	4	)	)	PUNCT
cana-1534	93	5	424	424	NUM
cana-1534	93	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	93	7	let	let	VERB
cana-1534	93	8	w	w	VERB
cana-1534	93	9	=	=	PUNCT
cana-1534	93	10	{	{	PUNCT
cana-1534	93	11	v1	v1	PROPN
cana-1534	93	12	,	,	PUNCT
cana-1534	93	13	v2	v2	PROPN
cana-1534	93	14	,	,	PUNCT
cana-1534	93	15	...	...	PUNCT
cana-1534	93	16	,	,	PUNCT
cana-1534	93	17	vn	vn	PROPN
cana-1534	93	18	}	}	PUNCT
cana-1534	93	19	.	.	PUNCT
cana-1534	94	1	else	else	ADV
cana-1534	94	2	r(v′|w	r(v′|w	NOUN
cana-1534	94	3	)	)	PUNCT
cana-1534	94	4	=	=	PUNCT
cana-1534	94	5	(	(	PUNCT
cana-1534	94	6	1	1	NUM
cana-1534	94	7	,	,	PUNCT
cana-1534	94	8	2	2	NUM
cana-1534	94	9	,	,	PUNCT
cana-1534	94	10	..	..	PUNCT
cana-1534	94	11	,	,	PUNCT
cana-1534	94	12	1	1	X
cana-1534	94	13	)	)	PUNCT
cana-1534	94	14	supposed	suppose	VERB
cana-1534	94	15	that	that	SCONJ
cana-1534	94	16	w	w	ADP
cana-1534	94	17	⊆	⊆	NUM
cana-1534	94	18	v1	v1	NOUN
cana-1534	94	19	,	,	PUNCT
cana-1534	94	20	else	else	ADV
cana-1534	94	21	if	if	SCONJ
cana-1534	94	22	vi	vi	PROPN
cana-1534	94	23	∈/	∈/	PROPN
cana-1534	94	24	w	w	PROPN
cana-1534	94	25	for	for	ADP
cana-1534	94	26	any	any	DET
cana-1534	94	27	i	i	NOUN
cana-1534	94	28	,	,	PUNCT
cana-1534	94	29	1	1	NUM
cana-1534	94	30	≤	≤	NUM
cana-1534	94	31	i	i	PRON
cana-1534	94	32	≤	≤	PROPN
cana-1534	94	33	n	n	CCONJ
cana-1534	94	34	,	,	PUNCT
cana-1534	94	35	else	else	ADV
cana-1534	94	36	r(vi|w	r(vi|w	VERB
cana-1534	94	37	)	)	PUNCT
cana-1534	94	38	=	=	SYM
cana-1534	94	39	r(v′|w	r(v′|w	NOUN
cana-1534	94	40	)	)	PUNCT
cana-1534	94	41	=	=	PUNCT
cana-1534	95	1	(	(	PUNCT
cana-1534	95	2	1	1	NUM
cana-1534	95	3	,	,	PUNCT
cana-1534	95	4	…	…	PUNCT
cana-1534	95	5	1	1	NUM
cana-1534	95	6	)	)	PUNCT
cana-1534	95	7	,	,	PUNCT
cana-1534	95	8	a	a	DET
cana-1534	95	9	contradiction	contradiction	NOUN
cana-1534	95	10	.	.	PUNCT
cana-1534	96	1	w	w	NOUN
cana-1534	96	2	must	must	AUX
cana-1534	96	3	contain	contain	VERB
cana-1534	96	4	all	all	DET
cana-1534	96	5	vi	vi	NOUN
cana-1534	96	6	’s	’s	NOUN
cana-1534	96	7	,	,	PUNCT
cana-1534	96	8	1	1	NUM
cana-1534	96	9	≤	≤	NUM
cana-1534	96	10	i	i	PRON
cana-1534	96	11	≤	≤	ADJ
cana-1534	97	1	n.	n.	NOUN
cana-1534	97	2	hence	hence	ADV
cana-1534	97	3	|w	|w	PROPN
cana-1534	97	4	|	|	ADV
cana-1534	97	5	≥	≥	PUNCT
cana-1534	97	6	n.	n.	NOUN
cana-1534	97	7	similarly	similarly	ADV
cana-1534	97	8	if	if	SCONJ
cana-1534	97	9	w	w	PROPN
cana-1534	97	10	⊆	⊆	NUM
cana-1534	97	11	v2	v2	NOUN
cana-1534	97	12	,	,	PUNCT
cana-1534	97	13	else	else	ADV
cana-1534	97	14	|w	|w	NOUN
cana-1534	98	1	|	|	ADV
cana-1534	98	2	≥	≥	PUNCT
cana-1534	98	3	n.	n.	PROPN
cana-1534	98	4	now	now	ADV
cana-1534	98	5	assume	assume	VERB
cana-1534	98	6	that	that	SCONJ
cana-1534	98	7	w	w	ADJ
cana-1534	98	8	⊆	⊆	NUM
cana-1534	98	9	v1	v1	NOUN
cana-1534	98	10	∪	∪	NOUN
cana-1534	98	11	v2	v2	NOUN
cana-1534	98	12	.	.	PUNCT
cana-1534	99	1	if	if	SCONJ
cana-1534	99	2	vi	vi	PROPN
cana-1534	99	3	,	,	PUNCT
cana-1534	99	4	v′	v′	X
cana-1534	99	5	∈/	∈/	PROPN
cana-1534	99	6	w	w	PROPN
cana-1534	99	7	,	,	PUNCT
cana-1534	99	8	1	1	NUM
cana-1534	99	9	≤	≤	NUM
cana-1534	99	10	i	i	PRON
cana-1534	99	11	≤	≤	PROPN
cana-1534	99	12	n	n	CCONJ
cana-1534	99	13	,	,	PUNCT
cana-1534	99	14	else	else	ADV
cana-1534	99	15	r(vi|w	r(vi|w	VERB
cana-1534	99	16	)	)	PUNCT
cana-1534	99	17	=	=	SYM
cana-1534	99	18	r(v′|w	r(v′|w	NOUN
cana-1534	99	19	)	)	PUNCT
cana-1534	99	20	=	=	PUNCT
cana-1534	99	21	(	(	PUNCT
cana-1534	99	22	1	1	NUM
cana-1534	99	23	,	,	PUNCT
cana-1534	99	24	...	...	PUNCT
cana-1534	99	25	,	,	PUNCT
cana-1534	99	26	1	1	X
cana-1534	99	27	)	)	PUNCT
cana-1534	99	28	,	,	PUNCT
cana-1534	99	29	a	a	DET
cana-1534	99	30	contradiction	contradiction	NOUN
cana-1534	99	31	.	.	PUNCT
cana-1534	100	1	else	else	ADV
cana-1534	101	1	w	w	PROPN
cana-1534	101	2	must	must	AUX
cana-1534	101	3	contain	contain	VERB
cana-1534	101	4	either	either	CCONJ
cana-1534	101	5	vi	vi	NOUN
cana-1534	101	6	or	or	CCONJ
cana-1534	101	7	v′for	v′for	ADP
cana-1534	101	8	all	all	PRON
cana-1534	101	9	i	i	PRON
cana-1534	101	10	,	,	PUNCT
cana-1534	101	11	1	1	NUM
cana-1534	101	12	≤	≤	NUM
cana-1534	101	13	i	i	PRON
cana-1534	101	14	≤	≤	ADJ
cana-1534	101	15	n.	n.	NOUN
cana-1534	101	16	hence	hence	ADV
cana-1534	101	17	|w	|w	PROPN
cana-1534	102	1	|	|	ADV
cana-1534	102	2	≥	≥	NOUN
cana-1534	102	3	n.	n.	NOUN
cana-1534	102	4	corollary	corollary	ADJ
cana-1534	102	5	2.3	2.3	NUM
cana-1534	102	6	states	state	NOUN
cana-1534	102	7	that	that	SCONJ
cana-1534	102	8	no	no	DET
cana-1534	102	9	graph	graph	NOUN
cana-1534	102	10	g	g	NOUN
cana-1534	102	11	fulfilling	fulfil	VERB
cana-1534	102	12	the	the	DET
cana-1534	102	13	characteristics	characteristic	NOUN
cana-1534	102	14	of	of	ADP
cana-1534	102	15	the	the	DET
cana-1534	102	16	foregoing	foregoing	NOUN
cana-1534	102	17	exists	exist	VERB
cana-1534	102	18	for	for	ADP
cana-1534	102	19	positives	positive	NOUN
cana-1534	102	20	integers	integer	NOUN
cana-1534	102	21	n	n	PRON
cana-1534	102	22	≥	≥	NUM
cana-1534	102	23	2	2	NUM
cana-1534	102	24	and	and	CCONJ
cana-1534	102	25	k	k	PROPN
cana-1534	102	26	≥	≥	NUM
cana-1534	102	27	3	3	NUM
cana-1534	102	28	.	.	PUNCT
cana-1534	103	1	proof	proof	NOUN
cana-1534	103	2	.	.	PUNCT
cana-1534	104	1	let	let	VERB
cana-1534	104	2	n	n	PRON
cana-1534	104	3	≥	≥	NOUN
cana-1534	104	4	2	2	NUM
cana-1534	104	5	,	,	PUNCT
cana-1534	104	6	k	k	X
cana-1534	104	7	≥	≥	NUM
cana-1534	104	8	3	3	X
cana-1534	104	9	.	.	PUNCT
cana-1534	104	10	consider	consider	VERB
cana-1534	104	11	g	g	NOUN
cana-1534	104	12	on	on	ADP
cana-1534	104	13	2n	2n	NUM
cana-1534	104	14	vertex	vertex	NOUN
cana-1534	104	15	with	with	ADP
cana-1534	104	16	a	a	DET
cana-1534	104	17	nr	nr	ADV
cana-1534	104	18	-	-	ADJ
cana-1534	104	19	set	set	VERB
cana-1534	104	20	w	w	ADP
cana-1534	104	21	such	such	ADJ
cana-1534	104	22	that	that	PRON
cana-1534	104	23	nr(g	nr(g	PUNCT
cana-1534	104	24	)	)	PUNCT
cana-1534	104	25	=	=	SYM
cana-1534	104	26	n	n	PROPN
cana-1534	104	27	and	and	CCONJ
cana-1534	104	28	r(vi|w	r(vi|w	VERB
cana-1534	104	29	)	)	PUNCT
cana-1534	104	30	=	=	SYM
cana-1534	104	31	(	(	PUNCT
cana-1534	104	32	1	1	NUM
cana-1534	104	33	,	,	PUNCT
cana-1534	104	34	1	1	NUM
cana-1534	104	35	,	,	PUNCT
cana-1534	104	36	...	...	PUNCT
cana-1534	104	37	,	,	PUNCT
cana-1534	104	38	1	1	X
cana-1534	104	39	,	,	PUNCT
cana-1534	104	40	k	k	NOUN
cana-1534	104	41	,	,	PUNCT
cana-1534	104	42	1	1	NUM
cana-1534	104	43	,	,	PUNCT
cana-1534	104	44	1	1	NUM
cana-1534	104	45	,	,	PUNCT
cana-1534	104	46	...	...	PUNCT
cana-1534	104	47	,	,	PUNCT
cana-1534	104	48	1	1	X
cana-1534	104	49	)	)	PUNCT
cana-1534	104	50	where	where	SCONJ
cana-1534	104	51	k	k	PROPN
cana-1534	104	52	occurs	occur	VERB
cana-1534	104	53	in	in	ADP
cana-1534	104	54	the	the	DET
cana-1534	104	55	ith	ith	PROPN
cana-1534	104	56	location	location	NOUN
cana-1534	104	57	.	.	PUNCT
cana-1534	105	1	let	let	VERB
cana-1534	105	2	v	v	X
cana-1534	105	3	(	(	PUNCT
cana-1534	105	4	g	g	NOUN
cana-1534	105	5	)	)	PUNCT
cana-1534	105	6	=	=	SYM
cana-1534	105	7	{	{	PUNCT
cana-1534	105	8	u1	u1	NOUN
cana-1534	105	9	,	,	PUNCT
cana-1534	105	10	,	,	PUNCT
cana-1534	105	11	…	…	PUNCT
cana-1534	105	12	un	un	ADJ
cana-1534	105	13	;	;	PUNCT
cana-1534	105	14	v1	v1	NOUN
cana-1534	105	15	,	,	PUNCT
cana-1534	105	16	v1	v1	NOUN
cana-1534	105	17	...	...	PUNCT
cana-1534	105	18	,	,	PUNCT
cana-1534	105	19	vn	vn	PROPN
cana-1534	105	20	}	}	PUNCT
cana-1534	105	21	and	and	CCONJ
cana-1534	105	22	w	w	NOUN
cana-1534	105	23	=	=	SYM
cana-1534	105	24	{	{	PUNCT
cana-1534	105	25	u1	u1	NOUN
cana-1534	105	26	,	,	PUNCT
cana-1534	105	27	u2	u2	PROPN
cana-1534	105	28	,	,	PUNCT
cana-1534	105	29	...	...	PUNCT
cana-1534	105	30	,	,	PUNCT
cana-1534	105	31	un	un	PROPN
cana-1534	105	32	}	}	PUNCT
cana-1534	105	33	.	.	PUNCT
cana-1534	106	1	now	now	ADV
cana-1534	106	2	by	by	ADP
cana-1534	106	3	our	our	PRON
cana-1534	106	4	assumption	assumption	NOUN
cana-1534	106	5	,	,	PUNCT
cana-1534	106	6	d(v1	d(v1	NOUN
cana-1534	106	7	,	,	PUNCT
cana-1534	106	8	u2	u2	PROPN
cana-1534	106	9	)	)	PUNCT
cana-1534	106	10	=	=	SYM
cana-1534	106	11	1	1	X
cana-1534	106	12	.	.	PUNCT
cana-1534	106	13	therefore	therefore	ADV
cana-1534	106	14	,	,	PUNCT
cana-1534	106	15	r(v1|w	r(v1|w	X
cana-1534	106	16	)	)	PUNCT
cana-1534	106	17	=	=	PUNCT
cana-1534	107	1	(	(	PUNCT
cana-1534	107	2	2	2	NUM
cana-1534	107	3	,	,	PUNCT
cana-1534	107	4	1	1	NUM
cana-1534	107	5	,	,	PUNCT
cana-1534	107	6	1	1	NUM
cana-1534	107	7	,	,	PUNCT
cana-1534	107	8	...	...	PUNCT
cana-1534	107	9	,	,	PUNCT
cana-1534	107	10	1	1	X
cana-1534	107	11	)	)	PUNCT
cana-1534	107	12	,	,	PUNCT
cana-1534	107	13	a	a	DET
cana-1534	107	14	contradiction	contradiction	NOUN
cana-1534	107	15	.	.	PUNCT
cana-1534	108	1	3	3	NUM
cana-1534	108	2	nr	nr	NOUN
cana-1534	108	3	-	-	NOUN
cana-1534	108	4	values	value	NOUN
cana-1534	108	5	for	for	ADP
cana-1534	108	6	the	the	DET
cana-1534	108	7	graphs	graph	NOUN
cana-1534	108	8	hn	hn	PROPN
cana-1534	108	9	,	,	PUNCT
cana-1534	108	10	n	n	PROPN
cana-1534	108	11	and	and	CCONJ
cana-1534	108	12	wn	wn	PROPN
cana-1534	108	13	the	the	DET
cana-1534	108	14	non	non	ADJ
cana-1534	108	15	-	-	ADJ
cana-1534	108	16	isolated	isolated	ADJ
cana-1534	108	17	resolution	resolution	NOUN
cana-1534	108	18	factor	factor	NOUN
cana-1534	108	19	for	for	ADP
cana-1534	108	20	extremely	extremely	ADV
cana-1534	108	21	erratic	erratic	ADJ
cana-1534	108	22	graph	graph	NOUN
cana-1534	108	23	hn	hn	PROPN
cana-1534	108	24	,	,	PUNCT
cana-1534	108	25	n	n	PROPN
cana-1534	108	26	and	and	CCONJ
cana-1534	108	27	for	for	ADP
cana-1534	108	28	the	the	DET
cana-1534	108	29	wheel	wheel	PROPN
cana-1534	108	30	wn	wn	PROPN
cana-1534	108	31	.	.	PUNCT
cana-1534	108	32	communications	communication	NOUN
cana-1534	108	33	on	on	ADP
cana-1534	108	34	applied	apply	VERB
cana-1534	108	35	nonlinear	nonlinear	ADJ
cana-1534	108	36	analysis	analysis	NOUN
cana-1534	108	37	issn	issn	NOUN
cana-1534	108	38	:	:	PUNCT
cana-1534	108	39	1074	1074	NUM
cana-1534	108	40	-	-	PUNCT
cana-1534	108	41	133x	133x	NUM
cana-1534	108	42	vol	vol	NOUN
cana-1534	108	43	31	31	NUM
cana-1534	108	44	no	no	NOUN
cana-1534	108	45	.	.	PUNCT
cana-1534	109	1	8s	8s	PROPN
cana-1534	109	2	(	(	PUNCT
cana-1534	109	3	2024	2024	NUM
cana-1534	109	4	)	)	PUNCT
cana-1534	109	5	425	425	NUM
cana-1534	109	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	109	7	theorem	theorem	VERB
cana-1534	109	8	3.1	3.1	NUM
cana-1534	109	9	.	.	PUNCT
cana-1534	110	1	for	for	ADP
cana-1534	110	2	any	any	DET
cana-1534	110	3	positive	positive	ADJ
cana-1534	110	4	integer	integer	NOUN
cana-1534	110	5	n	n	PRON
cana-1534	110	6	≥	≥	NOUN
cana-1534	110	7	4	4	NUM
cana-1534	110	8	,	,	PUNCT
cana-1534	110	9	nr(hn	nr(hn	PROPN
cana-1534	110	10	,	,	PUNCT
cana-1534	110	11	n	n	CCONJ
cana-1534	110	12	)	)	PUNCT
cana-1534	110	13	=	=	SYM
cana-1534	110	14	n	n	CCONJ
cana-1534	110	15	−	−	NOUN
cana-1534	110	16	1	1	NUM
cana-1534	110	17	.	.	PUNCT
cana-1534	111	1	proof	proof	NOUN
cana-1534	111	2	.	.	PUNCT
cana-1534	112	1	let	let	VERB
cana-1534	112	2	v	v	X
cana-1534	112	3	(	(	PUNCT
cana-1534	112	4	hn	hn	PROPN
cana-1534	112	5	,	,	PUNCT
cana-1534	112	6	n	n	CCONJ
cana-1534	112	7	)	)	PUNCT
cana-1534	112	8	=	=	SYM
cana-1534	112	9	{	{	PUNCT
cana-1534	112	10	v1	v1	PROPN
cana-1534	112	11	,	,	PUNCT
cana-1534	112	12	v2	v2	PROPN
cana-1534	112	13	,	,	PUNCT
cana-1534	112	14	...	...	PUNCT
cana-1534	112	15	,	,	PUNCT
cana-1534	112	16	vn	vn	PROPN
cana-1534	112	17	;	;	PUNCT
cana-1534	112	18	u1	u1	PROPN
cana-1534	112	19	,	,	PUNCT
cana-1534	112	20	u2	u2	NOUN
cana-1534	112	21	,	,	PUNCT
cana-1534	112	22	...	...	PUNCT
cana-1534	112	23	,	,	PUNCT
cana-1534	112	24	un	un	ADJ
cana-1534	112	25	}	}	PUNCT
cana-1534	112	26	and	and	CCONJ
cana-1534	112	27	e(hn	e(hn	PROPN
cana-1534	112	28	,	,	PUNCT
cana-1534	112	29	n	n	CCONJ
cana-1534	112	30	)	)	PUNCT
cana-1534	113	1	=	=	PRON
cana-1534	113	2	{	{	PUNCT
cana-1534	113	3	viuj	viuj	ADJ
cana-1534	113	4	:	:	PUNCT
cana-1534	113	5	1	1	NUM
cana-1534	113	6	≤	≤	NUM
cana-1534	113	7	i	i	PRON
cana-1534	113	8	≤	≤	PROPN
cana-1534	113	9	n	n	CCONJ
cana-1534	113	10	,	,	PUNCT
cana-1534	113	11	n	n	CCONJ
cana-1534	113	12	−	−	PROPN
cana-1534	114	1	i	i	PRON
cana-1534	114	2	+	+	CCONJ
cana-1534	114	3	1	1	NUM
cana-1534	114	4	≤	≤	NUM
cana-1534	114	5	j	j	PROPN
cana-1534	114	6	≤	≤	PROPN
cana-1534	114	7	n	n	CCONJ
cana-1534	114	8	}	}	PUNCT
cana-1534	114	9	.	.	PUNCT
cana-1534	115	1	let	let	VERB
cana-1534	115	2	w	w	VERB
cana-1534	115	3	=	=	PRON
cana-1534	115	4	{	{	PUNCT
cana-1534	115	5	un−1	un−1	PROPN
cana-1534	115	6	,	,	PUNCT
cana-1534	115	7	un−2	un−2	PROPN
cana-1534	115	8	;	;	PUNCT
cana-1534	115	9	v3	v3	PROPN
cana-1534	115	10	,	,	PUNCT
cana-1534	115	11	v4	v4	PROPN
cana-1534	115	12	,	,	PUNCT
cana-1534	115	13	...	...	PUNCT
cana-1534	115	14	,	,	PUNCT
cana-1534	115	15	vn−1	vn−1	ADJ
cana-1534	115	16	}	}	PUNCT
cana-1534	115	17	.	.	PUNCT
cana-1534	116	1	else	else	PROPN
cana-1534	116	2	r(v1|w	r(v1|w	PROPN
cana-1534	116	3	)	)	PUNCT
cana-1534	116	4	=	=	PUNCT
cana-1534	116	5	(	(	PUNCT
cana-1534	116	6	3	3	NUM
cana-1534	116	7	,	,	PUNCT
cana-1534	116	8	2	2	NUM
cana-1534	116	9	)	)	PUNCT
cana-1534	116	10	,	,	PUNCT
cana-1534	116	11	r(v2|w	r(v2|w	PROPN
cana-1534	116	12	)	)	PUNCT
cana-1534	116	13	=	=	PUNCT
cana-1534	116	14	(	(	PUNCT
cana-1534	116	15	1	1	NUM
cana-1534	116	16	,	,	PUNCT
cana-1534	116	17	3	3	NUM
cana-1534	116	18	,	,	PUNCT
cana-1534	116	19	2	2	NUM
cana-1534	116	20	,	,	PUNCT
cana-1534	116	21	...	...	PUNCT
cana-1534	116	22	2	2	NUM
cana-1534	116	23	)	)	PUNCT
cana-1534	116	24	,	,	PUNCT
cana-1534	116	25	r(un|w	r(un|w	PROPN
cana-1534	116	26	)	)	PUNCT
cana-1534	116	27	=	=	PUNCT
cana-1534	116	28	(	(	PUNCT
cana-1534	116	29	2	2	NUM
cana-1534	116	30	,	,	PUNCT
cana-1534	116	31	2	2	NUM
cana-1534	116	32	,	,	PUNCT
cana-1534	116	33	1	1	NUM
cana-1534	116	34	,	,	PUNCT
cana-1534	116	35	1	1	NUM
cana-1534	116	36	,	,	PUNCT
cana-1534	116	37	...	...	PUNCT
cana-1534	116	38	,	,	PUNCT
cana-1534	116	39	1	1	X
cana-1534	116	40	)	)	PUNCT
cana-1534	116	41	,	,	PUNCT
cana-1534	116	42	r(u1|w	r(u1|w	PROPN
cana-1534	116	43	)	)	PUNCT
cana-1534	116	44	=	=	PUNCT
cana-1534	116	45	(	(	PUNCT
cana-1534	116	46	2	2	NUM
cana-1534	116	47	,	,	PUNCT
cana-1534	116	48	2	2	NUM
cana-1534	116	49	,	,	PUNCT
cana-1534	116	50	3	3	NUM
cana-1534	116	51	,	,	PUNCT
cana-1534	116	52	3	3	NUM
cana-1534	116	53	,	,	PUNCT
cana-1534	116	54	...	...	PUNCT
cana-1534	116	55	,	,	PUNCT
cana-1534	116	56	3	3	X
cana-1534	116	57	)	)	PUNCT
cana-1534	116	58	and	and	CCONJ
cana-1534	116	59	r(uj|w	r(uj|w	NUM
cana-1534	116	60	)	)	PUNCT
cana-1534	116	61	=	=	PUNCT
cana-1534	116	62	(	(	PUNCT
cana-1534	116	63	2	2	NUM
cana-1534	116	64	,	,	PUNCT
cana-1534	116	65	2	2	NUM
cana-1534	116	66	,	,	PUNCT
cana-1534	116	67	3	3	NUM
cana-1534	116	68	,	,	PUNCT
cana-1534	116	69	3	3	NUM
cana-1534	116	70	,	,	PUNCT
cana-1534	116	71	...	...	PUNCT
cana-1534	116	72	,	,	PUNCT
cana-1534	117	1	3,˛¸	3,˛¸	NUM
cana-1534	117	2	x	x	SYM
cana-1534	118	1	[	[	X
cana-1534	118	2	n−(j+2)]times	n−(j+2)]time	NOUN
cana-1534	118	3	1	1	NUM
cana-1534	118	4	,	,	PUNCT
cana-1534	118	5	1	1	NUM
cana-1534	118	6	,	,	PUNCT
cana-1534	118	7	...	...	PUNCT
cana-1534	118	8	,	,	PUNCT
cana-1534	118	9	1	1	X
cana-1534	118	10	)	)	PUNCT
cana-1534	118	11	where	where	SCONJ
cana-1534	118	12	2	2	NUM
cana-1534	118	13	≤	≤	NUM
cana-1534	118	14	j	j	PROPN
cana-1534	118	15	≤	≤	PROPN
cana-1534	118	16	n	n	CCONJ
cana-1534	118	17	−	−	PROPN
cana-1534	118	18	3	3	NUM
cana-1534	118	19	.	.	PUNCT
cana-1534	118	20	also	also	ADV
cana-1534	118	21	⟨w	⟨w	X
cana-1534	118	22	is	be	AUX
cana-1534	118	23	isomorphic	isomorphic	ADJ
cana-1534	118	24	to	to	ADP
cana-1534	118	25	k	k	PROPN
cana-1534	118	26	−.hence	−.hence	PUNCT
cana-1534	118	27	nr(hn	nr(hn	PROPN
cana-1534	118	28	,	,	PUNCT
cana-1534	118	29	n	n	CCONJ
cana-1534	118	30	)	)	PUNCT
cana-1534	118	31	≤	≤	NOUN
cana-1534	118	32	n	n	CCONJ
cana-1534	118	33	−	−	PROPN
cana-1534	118	34	1	1	NUM
cana-1534	118	35	.	.	PUNCT
cana-1534	119	1	it	it	PRON
cana-1534	119	2	's	be	AUX
cana-1534	119	3	sufficient	sufficient	ADJ
cana-1534	119	4	to	to	PART
cana-1534	119	5	show	show	VERB
cana-1534	119	6	that	that	SCONJ
cana-1534	119	7	nr(hn	nr(hn	PROPN
cana-1534	119	8	,	,	PUNCT
cana-1534	119	9	n	n	CCONJ
cana-1534	119	10	)	)	PUNCT
cana-1534	119	11	>	>	X
cana-1534	120	1	n	n	CCONJ
cana-1534	121	1	−	−	NOUN
cana-1534	121	2	1	1	NUM
cana-1534	121	3	.	.	PUNCT
cana-1534	122	1	use	use	VERB
cana-1534	122	2	any	any	DET
cana-1534	122	3	non	non	ADJ
cana-1534	122	4	-	-	ADJ
cana-1534	122	5	isolated	isolated	ADJ
cana-1534	122	6	resolution	resolution	NOUN
cana-1534	122	7	set	set	VERB
cana-1534	122	8	for	for	ADP
cana-1534	122	9	hn	hn	PROPN
cana-1534	122	10	,	,	PUNCT
cana-1534	122	11	n.	n.	NOUN
cana-1534	122	12	if	if	SCONJ
cana-1534	122	13	w	w	NOUN
cana-1534	122	14	contains	contain	VERB
cana-1534	122	15	only	only	ADV
cana-1534	122	16	vi	vi	NOUN
cana-1534	122	17	’s	’s	NOUN
cana-1534	122	18	or	or	CCONJ
cana-1534	122	19	ui	ui	NOUN
cana-1534	122	20	’s	’s	ADV
cana-1534	122	21	,	,	PUNCT
cana-1534	122	22	else	else	ADV
cana-1534	122	23	⟨w	⟨w	X
cana-1534	122	24	⟩	⟩	NOUN
cana-1534	122	25	is	be	AUX
cana-1534	122	26	a	a	DET
cana-1534	122	27	null	null	ADJ
cana-1534	122	28	graph	graph	NOUN
cana-1534	122	29	,	,	PUNCT
cana-1534	122	30	a	a	DET
cana-1534	122	31	contradiction	contradiction	NOUN
cana-1534	122	32	.	.	PUNCT
cana-1534	123	1	suppose	suppose	VERB
cana-1534	123	2	that	that	SCONJ
cana-1534	123	3	vn	vn	PROPN
cana-1534	123	4	∈	∈	PROPN
cana-1534	123	5	w	w	PROPN
cana-1534	123	6	.	.	PUNCT
cana-1534	124	1	if	if	SCONJ
cana-1534	124	2	u1	u1	PROPN
cana-1534	124	3	,	,	PUNCT
cana-1534	124	4	un	un	PROPN
cana-1534	124	5	belongs	belong	VERB
cana-1534	124	6	to	to	ADP
cana-1534	124	7	w	w	PROPN
cana-1534	124	8	,	,	PUNCT
cana-1534	124	9	else	else	ADJ
cana-1534	124	10	removal	removal	NOUN
cana-1534	124	11	of	of	ADP
cana-1534	124	12	them	they	PRON
cana-1534	124	13	from	from	ADP
cana-1534	124	14	w	w	PROPN
cana-1534	124	15	will	will	AUX
cana-1534	124	16	result	result	VERB
cana-1534	124	17	again	again	ADV
cana-1534	124	18	a	a	DET
cana-1534	124	19	non	non	ADJ
cana-1534	124	20	-	-	ADJ
cana-1534	124	21	isolated	isolated	ADJ
cana-1534	124	22	set	set	NOUN
cana-1534	124	23	.	.	PUNCT
cana-1534	125	1	thus	thus	ADV
cana-1534	125	2	u1	u1	VERB
cana-1534	125	3	and	and	CCONJ
cana-1534	125	4	un	un	PROPN
cana-1534	125	5	need	need	AUX
cana-1534	125	6	not	not	PART
cana-1534	125	7	be	be	AUX
cana-1534	125	8	in	in	ADP
cana-1534	125	9	w.	w.	PROPN
cana-1534	125	10	if	if	SCONJ
cana-1534	125	11	vi	vi	PROPN
cana-1534	125	12	∈/	∈/	PROPN
cana-1534	125	13	w	w	PROPN
cana-1534	125	14	for	for	ADP
cana-1534	125	15	all	all	DET
cana-1534	125	16	i	i	PRON
cana-1534	125	17	,	,	PUNCT
cana-1534	125	18	1	1	NUM
cana-1534	125	19	≤	≤	NUM
cana-1534	125	20	i	i	PRON
cana-1534	125	21	≤	≤	NOUN
cana-1534	125	22	n	n	CCONJ
cana-1534	125	23	−	−	PROPN
cana-1534	125	24	1	1	NUM
cana-1534	125	25	,	,	PUNCT
cana-1534	125	26	else	else	ADV
cana-1534	125	27	r(u1|w	r(u1|w	NOUN
cana-1534	125	28	)	)	PUNCT
cana-1534	126	1	=	=	SYM
cana-1534	126	2	r(un|w	r(un|w	PROPN
cana-1534	126	3	)	)	PUNCT
cana-1534	126	4	.	.	PUNCT
cana-1534	127	1	hence	hence	ADV
cana-1534	127	2	at	at	ADV
cana-1534	127	3	least	least	ADV
cana-1534	127	4	one	one	NUM
cana-1534	127	5	vi	vi	NOUN
cana-1534	127	6	,	,	PUNCT
cana-1534	127	7	1	1	NUM
cana-1534	127	8	≤	≤	NUM
cana-1534	127	9	i	i	PRON
cana-1534	127	10	≤	≤	NOUN
cana-1534	128	1	n	n	CCONJ
cana-1534	128	2	−	−	PROPN
cana-1534	128	3	1	1	NUM
cana-1534	128	4	must	must	AUX
cana-1534	128	5	be	be	AUX
cana-1534	128	6	in	in	ADP
cana-1534	128	7	w.	w.	NOUN
cana-1534	128	8	now	now	ADV
cana-1534	128	9	if	if	SCONJ
cana-1534	128	10	uj	uj	PROPN
cana-1534	128	11	∈/	∈/	PROPN
cana-1534	128	12	w	w	PROPN
cana-1534	128	13	for	for	ADP
cana-1534	128	14	some	some	DET
cana-1534	128	15	j	j	NOUN
cana-1534	128	16	,	,	PUNCT
cana-1534	128	17	communications	communication	NOUN
cana-1534	128	18	on	on	ADP
cana-1534	128	19	applied	apply	VERB
cana-1534	128	20	nonlinear	nonlinear	ADJ
cana-1534	128	21	analysis	analysis	NOUN
cana-1534	128	22	issn	issn	NOUN
cana-1534	128	23	:	:	PUNCT
cana-1534	128	24	1074	1074	NUM
cana-1534	128	25	-	-	PUNCT
cana-1534	128	26	133x	133x	NUM
cana-1534	128	27	vol	vol	NOUN
cana-1534	128	28	31	31	NUM
cana-1534	128	29	no	no	NOUN
cana-1534	128	30	.	.	PUNCT
cana-1534	129	1	8s	8s	PROPN
cana-1534	129	2	(	(	PUNCT
cana-1534	129	3	2024	2024	NUM
cana-1534	129	4	)	)	PUNCT
cana-1534	129	5	426	426	NUM
cana-1534	129	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	129	7	2	2	NUM
cana-1534	129	8	≤	≤	NUM
cana-1534	129	9	j	j	PROPN
cana-1534	129	10	≤	≤	PROPN
cana-1534	129	11	n	n	CCONJ
cana-1534	129	12	−	−	PROPN
cana-1534	129	13	1	1	NUM
cana-1534	129	14	,	,	PUNCT
cana-1534	129	15	else	else	ADV
cana-1534	129	16	for	for	ADP
cana-1534	129	17	all	all	DET
cana-1534	129	18	2	2	NUM
cana-1534	129	19	≤	≤	NUM
cana-1534	129	20	j	j	PROPN
cana-1534	129	21	≤	≤	PROPN
cana-1534	129	22	n	n	CCONJ
cana-1534	129	23	−	−	PROPN
cana-1534	130	1	i	i	PRON
cana-1534	130	2	,	,	PUNCT
cana-1534	130	3	r	r	NOUN
cana-1534	130	4	(	(	PUNCT
cana-1534	130	5	uj|w	uj|w	INTJ
cana-1534	130	6	)	)	PUNCT
cana-1534	131	1	=	=	PUNCT
cana-1534	131	2	r(u1|w	r(u1|w	PROPN
cana-1534	131	3	)	)	PUNCT
cana-1534	131	4	and	and	CCONJ
cana-1534	131	5	for	for	ADP
cana-1534	131	6	all	all	DET
cana-1534	131	7	n	n	PRON
cana-1534	131	8	−	−	NOUN
cana-1534	132	1	i	i	PRON
cana-1534	132	2	−	−	PROPN
cana-1534	132	3	1	1	NUM
cana-1534	132	4	≤	≤	NUM
cana-1534	132	5	j	j	PROPN
cana-1534	132	6	≤	≤	PROPN
cana-1534	132	7	n	n	CCONJ
cana-1534	132	8	−	−	PROPN
cana-1534	132	9	1	1	NUM
cana-1534	132	10	,	,	PUNCT
cana-1534	132	11	r(uj|w	r(uj|w	NOUN
cana-1534	132	12	)	)	PUNCT
cana-1534	133	1	=	=	SYM
cana-1534	133	2	r(un|w	r(un|w	PROPN
cana-1534	133	3	)	)	PUNCT
cana-1534	133	4	,	,	PUNCT
cana-1534	133	5	a	a	DET
cana-1534	133	6	contradiction	contradiction	NOUN
cana-1534	133	7	.	.	PUNCT
cana-1534	134	1	therefore	therefore	ADV
cana-1534	134	2	,	,	PUNCT
cana-1534	134	3	all	all	PRON
cana-1534	134	4	uj	uj	PROPN
cana-1534	134	5	’s	’s	NOUN
cana-1534	134	6	,	,	PUNCT
cana-1534	134	7	where	where	SCONJ
cana-1534	134	8	2	2	NUM
cana-1534	134	9	≤	≤	NUM
cana-1534	134	10	j	j	PROPN
cana-1534	134	11	≤	≤	PROPN
cana-1534	134	12	n	n	CCONJ
cana-1534	134	13	−	−	PROPN
cana-1534	134	14	1	1	NUM
cana-1534	134	15	must	must	AUX
cana-1534	134	16	be	be	AUX
cana-1534	134	17	in	in	ADP
cana-1534	134	18	w.	w.	NOUN
cana-1534	134	19	hence	hence	ADV
cana-1534	134	20	|w	|w	PROPN
cana-1534	135	1	|	|	ADV
cana-1534	135	2	≥	≥	NOUN
cana-1534	135	3	1	1	NUM
cana-1534	135	4	+	+	CCONJ
cana-1534	135	5	1	1	NUM
cana-1534	135	6	+	+	NUM
cana-1534	135	7	n	n	CCONJ
cana-1534	135	8	−	−	NUM
cana-1534	135	9	3	3	NUM
cana-1534	135	10	=	=	SYM
cana-1534	135	11	n	n	CCONJ
cana-1534	135	12	−	−	NOUN
cana-1534	135	13	1	1	NUM
cana-1534	135	14	.	.	PUNCT
cana-1534	136	1	thus	thus	ADV
cana-1534	136	2	|w	|w	ADJ
cana-1534	136	3	|	|	NOUN
cana-1534	136	4	≥	≥	NOUN
cana-1534	136	5	n	n	CCONJ
cana-1534	136	6	−	−	PROPN
cana-1534	136	7	1	1	NUM
cana-1534	136	8	.	.	PUNCT
cana-1534	137	1	similarly	similarly	ADV
cana-1534	137	2	,	,	PUNCT
cana-1534	137	3	if	if	SCONJ
cana-1534	137	4	un	un	PROPN
cana-1534	137	5	∈	∈	PROPN
cana-1534	137	6	w	w	PROPN
cana-1534	137	7	,	,	PUNCT
cana-1534	137	8	else	else	ADV
cana-1534	137	9	|w	|w	ADJ
cana-1534	137	10	|	|	ADV
cana-1534	137	11	≥	≥	NOUN
cana-1534	137	12	n	n	CCONJ
cana-1534	137	13	−	−	PROPN
cana-1534	137	14	1	1	NUM
cana-1534	137	15	.	.	PUNCT
cana-1534	137	16	now	now	ADV
cana-1534	137	17	let	let	VERB
cana-1534	137	18	vn	vn	PROPN
cana-1534	137	19	∈/	∈/	PROPN
cana-1534	137	20	w.	w.	PROPN
cana-1534	137	21	suppose	suppose	VERB
cana-1534	137	22	v1	v1	PROPN
cana-1534	137	23	∈	∈	PROPN
cana-1534	137	24	w	w	PROPN
cana-1534	137	25	,	,	PUNCT
cana-1534	137	26	else	else	ADV
cana-1534	137	27	un	un	PROPN
cana-1534	137	28	must	must	AUX
cana-1534	137	29	be	be	AUX
cana-1534	137	30	in	in	ADP
cana-1534	137	31	w	w	PROPN
cana-1534	137	32	,	,	PUNCT
cana-1534	137	33	since	since	SCONJ
cana-1534	137	34	n	n	PROPN
cana-1534	137	35	(	(	PUNCT
cana-1534	137	36	v1	v1	NOUN
cana-1534	137	37	)	)	PUNCT
cana-1534	137	38	=	=	PUNCT
cana-1534	137	39	{	{	PUNCT
cana-1534	137	40	un	un	PROPN
cana-1534	137	41	}	}	PUNCT
cana-1534	137	42	.	.	PUNCT
cana-1534	138	1	hence	hence	ADV
cana-1534	138	2	as	as	SCONJ
cana-1534	138	3	discussed	discuss	VERB
cana-1534	138	4	above	above	ADP
cana-1534	138	5	|w	|w	ADJ
cana-1534	138	6	|	|	NOUN
cana-1534	138	7	≥	≥	NOUN
cana-1534	138	8	n	n	CCONJ
cana-1534	138	9	−	−	PROPN
cana-1534	138	10	1	1	NUM
cana-1534	138	11	.	.	PUNCT
cana-1534	139	1	similarly	similarly	ADV
cana-1534	139	2	,	,	PUNCT
cana-1534	139	3	if	if	SCONJ
cana-1534	139	4	u1	u1	VERB
cana-1534	139	5	∈	∈	PROPN
cana-1534	139	6	w	w	PROPN
cana-1534	139	7	,	,	PUNCT
cana-1534	139	8	else	else	ADV
cana-1534	139	9	|w	|w	ADJ
cana-1534	139	10	|	|	ADV
cana-1534	139	11	≥	≥	NOUN
cana-1534	139	12	n	n	CCONJ
cana-1534	139	13	−	−	PROPN
cana-1534	139	14	1	1	NUM
cana-1534	139	15	.	.	PUNCT
cana-1534	139	16	now	now	ADV
cana-1534	139	17	let	let	VERB
cana-1534	139	18	vn	vn	PROPN
cana-1534	139	19	,	,	PUNCT
cana-1534	139	20	v1	v1	NOUN
cana-1534	139	21	,	,	PUNCT
cana-1534	139	22	u1	u1	NOUN
cana-1534	139	23	,	,	PUNCT
cana-1534	139	24	un	un	PROPN
cana-1534	139	25	are	be	AUX
cana-1534	139	26	all	all	ADV
cana-1534	139	27	not	not	PART
cana-1534	139	28	in	in	ADP
cana-1534	139	29	w	w	PROPN
cana-1534	139	30	.	.	PUNCT
cana-1534	140	1	if	if	SCONJ
cana-1534	140	2	v2	v2	PROPN
cana-1534	140	3	,	,	PUNCT
cana-1534	140	4	un−1	un−1	PROPN
cana-1534	140	5	∈/	∈/	PROPN
cana-1534	140	6	w	w	PROPN
cana-1534	140	7	,	,	PUNCT
cana-1534	140	8	else	else	ADV
cana-1534	140	9	r(v1|w	r(v1|w	PROPN
cana-1534	140	10	)	)	PUNCT
cana-1534	140	11	=	=	SYM
cana-1534	140	12	r(v2|w	r(v2|w	PROPN
cana-1534	140	13	)	)	PUNCT
cana-1534	140	14	and	and	CCONJ
cana-1534	140	15	r(un|w	r(un|w	PROPN
cana-1534	140	16	)	)	PUNCT
cana-1534	141	1	=	=	PUNCT
cana-1534	142	1	r(un−1|w	r(un−1|w	NOUN
cana-1534	142	2	be	be	VERB
cana-1534	142	3	contract	contract	NOUN
cana-1534	142	4	.	.	PUNCT
cana-1534	143	1	if	if	SCONJ
cana-1534	143	2	v2	v2	VERB
cana-1534	143	3	∈	∈	PROPN
cana-1534	143	4	w	w	NOUN
cana-1534	143	5	,	,	PUNCT
cana-1534	143	6	else	else	ADV
cana-1534	143	7	r(un|w	r(un|w	PROPN
cana-1534	143	8	)	)	PUNCT
cana-1534	144	1	=	=	SYM
cana-1534	144	2	r(un−1|w	r(un−1|w	NOUN
cana-1534	144	3	)	)	PUNCT
cana-1534	144	4	,	,	PUNCT
cana-1534	144	5	representation	representation	NOUN
cana-1534	144	6	of	of	ADP
cana-1534	144	7	the	the	DET
cana-1534	144	8	vertices	vertex	NOUN
cana-1534	144	9	v1	v1	NOUN
cana-1534	144	10	,	,	PUNCT
cana-1534	144	11	u2	u2	NOUN
cana-1534	144	12	and	and	CCONJ
cana-1534	144	13	un	un	PROPN
cana-1534	144	14	.	.	PROPN
cana-1534	145	1	thus	thus	ADV
cana-1534	145	2	un−1	un−1	PROPN
cana-1534	145	3	∈	∈	PROPN
cana-1534	145	4	w.	w.	NOUN
cana-1534	145	5	now	now	ADV
cana-1534	145	6	r(v2|w	r(v2|w	PROPN
cana-1534	145	7	)	)	PUNCT
cana-1534	146	1	=	=	SYM
cana-1534	146	2	r(v3|w	r(v3|w	PROPN
cana-1534	146	3	)	)	PUNCT
cana-1534	146	4	and	and	CCONJ
cana-1534	146	5	r(un|w	r(un|w	PROPN
cana-1534	146	6	)	)	PUNCT
cana-1534	146	7	=	=	PUNCT
cana-1534	146	8	r(un−2|w	r(un−2|w	NOUN
cana-1534	146	9	)	)	PUNCT
cana-1534	146	10	and	and	CCONJ
cana-1534	146	11	un−2	un−2	VERB
cana-1534	146	12	.	.	PROPN
cana-1534	146	13	and	and	CCONJ
cana-1534	146	14	un	un	PROPN
cana-1534	146	15	,	,	PUNCT
cana-1534	146	16	un−2	un−2	PROPN
cana-1534	146	17	.	.	PUNCT
cana-1534	146	18	communications	communication	NOUN
cana-1534	146	19	on	on	ADP
cana-1534	146	20	applied	apply	VERB
cana-1534	146	21	nonlinear	nonlinear	ADJ
cana-1534	146	22	analysis	analysis	NOUN
cana-1534	146	23	issn	issn	NOUN
cana-1534	146	24	:	:	PUNCT
cana-1534	146	25	1074	1074	NUM
cana-1534	146	26	-	-	PUNCT
cana-1534	146	27	133x	133x	NUM
cana-1534	146	28	vol	vol	NOUN
cana-1534	146	29	31	31	NUM
cana-1534	146	30	no	no	NOUN
cana-1534	146	31	.	.	PUNCT
cana-1534	147	1	8s	8s	PROPN
cana-1534	147	2	(	(	PUNCT
cana-1534	147	3	2024	2024	NUM
cana-1534	147	4	)	)	PUNCT
cana-1534	147	5	427	427	NUM
cana-1534	147	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	147	7	hence	hence	ADV
cana-1534	147	8	un−2	un−2	VERB
cana-1534	147	9	∈	∈	PROPN
cana-1534	147	10	w	w	NOUN
cana-1534	147	11	.	.	PUNCT
cana-1534	148	1	if	if	SCONJ
cana-1534	148	2	vi	vi	PROPN
cana-1534	148	3	∈/	∈/	PROPN
cana-1534	148	4	w	w	PROPN
cana-1534	148	5	for	for	ADP
cana-1534	148	6	some	some	DET
cana-1534	148	7	i	i	PRON
cana-1534	148	8	,	,	PUNCT
cana-1534	148	9	3	3	NUM
cana-1534	148	10	≤	≤	NUM
cana-1534	148	11	i	i	PRON
cana-1534	148	12	≤	≤	NOUN
cana-1534	148	13	n	n	CCONJ
cana-1534	148	14	−	−	PROPN
cana-1534	148	15	1	1	NUM
cana-1534	148	16	,	,	PUNCT
cana-1534	148	17	else	else	ADV
cana-1534	148	18	r(vi|w	r(vi|w	VERB
cana-1534	148	19	)	)	PUNCT
cana-1534	148	20	=	=	SYM
cana-1534	148	21	r(vi+1	r(vi+1	ADJ
cana-1534	148	22	|w	|w	NOUN
cana-1534	148	23	)	)	PUNCT
cana-1534	148	24	unless	unless	SCONJ
cana-1534	148	25	un−i	un−i	NOUN
cana-1534	148	26	∈	∈	PROPN
cana-1534	148	27	w	w	NOUN
cana-1534	148	28	.	.	PUNCT
cana-1534	149	1	thus	thus	ADV
cana-1534	149	2	either	either	CCONJ
cana-1534	149	3	vi	vi	NOUN
cana-1534	149	4	or	or	CCONJ
cana-1534	149	5	un−i	un−i	NOUN
cana-1534	149	6	belongs	belong	VERB
cana-1534	149	7	to	to	ADP
cana-1534	149	8	w	w	PROPN
cana-1534	149	9	for	for	ADP
cana-1534	149	10	all	all	DET
cana-1534	149	11	i	i	PRON
cana-1534	149	12	,	,	PUNCT
cana-1534	149	13	3	3	NUM
cana-1534	149	14	≤	≤	NUM
cana-1534	149	15	i	i	PRON
cana-1534	149	16	≤	≤	NOUN
cana-1534	149	17	n	n	CCONJ
cana-1534	149	18	−	−	PROPN
cana-1534	149	19	1	1	NUM
cana-1534	149	20	.	.	PUNCT
cana-1534	150	1	therefore	therefore	ADV
cana-1534	150	2	,	,	PUNCT
cana-1534	150	3	|w	|w	ADJ
cana-1534	150	4	|	|	ADV
cana-1534	150	5	≥	≥	NOUN
cana-1534	150	6	2	2	NUM
cana-1534	150	7	+	+	CCONJ
cana-1534	150	8	n	n	CCONJ
cana-1534	150	9	−	−	NUM
cana-1534	150	10	3	3	NUM
cana-1534	150	11	=	=	SYM
cana-1534	150	12	n	n	CCONJ
cana-1534	150	13	−	−	NOUN
cana-1534	150	14	1	1	NUM
cana-1534	150	15	.	.	PUNCT
cana-1534	151	1	hence	hence	ADV
cana-1534	151	2	|w	|w	ADJ
cana-1534	151	3	|	|	ADV
cana-1534	151	4	≥	≥	NOUN
cana-1534	151	5	n	n	CCONJ
cana-1534	151	6	−	−	PROPN
cana-1534	152	1	1	1	NUM
cana-1534	152	2	.	.	PUNCT
cana-1534	153	1	next	next	ADV
cana-1534	153	2	we	we	PRON
cana-1534	153	3	find	find	VERB
cana-1534	153	4	the	the	DET
cana-1534	153	5	nrvalue	nrvalue	NOUN
cana-1534	153	6	for	for	ADP
cana-1534	153	7	the	the	DET
cana-1534	153	8	wheel	wheel	NOUN
cana-1534	153	9	wn	wn	PROPN
cana-1534	153	10	.	.	PUNCT
cana-1534	154	1	it	it	PRON
cana-1534	154	2	can	can	AUX
cana-1534	154	3	be	be	AUX
cana-1534	154	4	easily	easily	ADV
cana-1534	154	5	verify	verify	VERB
cana-1534	154	6	that	that	SCONJ
cana-1534	154	7	,	,	PUNCT
cana-1534	154	8	nr(w3	nr(w3	PROPN
cana-1534	154	9	)	)	PUNCT
cana-1534	154	10	=	=	SYM
cana-1534	154	11	3	3	NUM
cana-1534	154	12	,	,	PUNCT
cana-1534	154	13	nr(w4	nr(w4	NOUN
cana-1534	154	14	)	)	PUNCT
cana-1534	154	15	=	=	SYM
cana-1534	154	16	2	2	NUM
cana-1534	154	17	,	,	PUNCT
cana-1534	154	18	nr(w5	nr(w5	NUM
cana-1534	154	19	)	)	PUNCT
cana-1534	154	20	=	=	SYM
cana-1534	154	21	2	2	NUM
cana-1534	154	22	and	and	CCONJ
cana-1534	154	23	nr(w9	nr(w9	NOUN
cana-1534	154	24	)	)	PUNCT
cana-1534	154	25	=	=	SYM
cana-1534	154	26	4	4	X
cana-1534	154	27	.	.	NOUN
cana-1534	154	28	for	for	ADP
cana-1534	154	29	n	n	NUM
cana-1534	154	30	≥	≥	NUM
cana-1534	154	31	6	6	NUM
cana-1534	154	32	and	and	CCONJ
cana-1534	154	33	n	n	PRON
cana-1534	154	34	=/	=/	NOUN
cana-1534	154	35	9	9	NUM
cana-1534	154	36	,	,	PUNCT
cana-1534	154	37	we	we	PRON
cana-1534	154	38	define	define	VERB
cana-1534	154	39	a	a	DET
cana-1534	154	40	formula	formula	NOUN
cana-1534	154	41	for	for	ADP
cana-1534	154	42	the	the	DET
cana-1534	154	43	wheel	wheel	NOUN
cana-1534	154	44	wn	wn	NOUN
cana-1534	154	45	in	in	ADP
cana-1534	154	46	the	the	DET
cana-1534	154	47	following	follow	VERB
cana-1534	154	48	theorem	theorem	PROPN
cana-1534	154	49	.	.	PUNCT
cana-1534	154	50	theorem	theorem	PROPN
cana-1534	154	51	3.2	3.2	NUM
cana-1534	154	52	.	.	PUNCT
cana-1534	155	1	for	for	ADP
cana-1534	155	2	any	any	DET
cana-1534	155	3	integer	integer	NOUN
cana-1534	155	4	with	with	ADP
cana-1534	155	5	positive	positive	ADJ
cana-1534	155	6	n	n	CCONJ
cana-1534	155	7	≥	≥	NUM
cana-1534	155	8	6	6	NUM
cana-1534	155	9	and	and	CCONJ
cana-1534	155	10	n	n	CCONJ
cana-1534	155	11	≠	≠	PROPN
cana-1534	155	12	9	9	NUM
cana-1534	155	13	:	:	PUNCT
cana-1534	155	14	proof	proof	NOUN
cana-1534	155	15	.	.	PUNCT
cana-1534	156	1	let	let	VERB
cana-1534	156	2	v	v	X
cana-1534	156	3	(	(	PUNCT
cana-1534	156	4	wn	wn	PROPN
cana-1534	156	5	)	)	PUNCT
cana-1534	156	6	=	=	SYM
cana-1534	156	7	{	{	PUNCT
cana-1534	156	8	v	v	NOUN
cana-1534	156	9	,	,	PUNCT
cana-1534	156	10	v1	v1	NOUN
cana-1534	156	11	,	,	PUNCT
cana-1534	156	12	v2	v2	PROPN
cana-1534	156	13	,	,	PUNCT
cana-1534	156	14	...	...	PUNCT
cana-1534	156	15	,	,	PUNCT
cana-1534	156	16	vn	vn	PROPN
cana-1534	156	17	}	}	PUNCT
cana-1534	156	18	.	.	PUNCT
cana-1534	157	1	case1	case1	PROPN
cana-1534	157	2	.	.	PUNCT
cana-1534	158	1	let	let	VERB
cana-1534	158	2	n	n	PRON
cana-1534	158	3	≡	≡	PROPN
cana-1534	158	4	0	0	NUM
cana-1534	158	5	,	,	PUNCT
cana-1534	158	6	1	1	NUM
cana-1534	158	7	(	(	PUNCT
cana-1534	158	8	modulo	modulo	NOUN
cana-1534	158	9	5	5	NUM
cana-1534	158	10	)	)	PUNCT
cana-1534	158	11	.	.	PUNCT
cana-1534	159	1	take	take	VERB
cana-1534	159	2	w=	w=	PROPN
cana-1534	159	3	{	{	PUNCT
cana-1534	159	4	v	v	NOUN
cana-1534	159	5	,	,	PUNCT
cana-1534	159	6	v1	v1	NOUN
cana-1534	159	7	,	,	PUNCT
cana-1534	159	8	v5	v5	NOUN
cana-1534	159	9	,	,	PUNCT
cana-1534	159	10	v5i+2	v5i+2	ADJ
cana-1534	159	11	,	,	PUNCT
cana-1534	159	12	v5(i+1	v5(i+1	NOUN
cana-1534	159	13	)	)	PUNCT
cana-1534	159	14	:	:	PUNCT
cana-1534	159	15	1	1	NUM
cana-1534	159	16	≤	≤	NUM
cana-1534	159	17	i	i	X
cana-1534	159	18	≤	≤	ADV
cana-1534	159	19	6	6	NUM
cana-1534	159	20	}	}	PUNCT
cana-1534	159	21	.	.	PUNCT
cana-1534	160	1	else	else	ADV
cana-1534	160	2	for	for	ADP
cana-1534	160	3	n≡	n≡	NOUN
cana-1534	160	4	(	(	PUNCT
cana-1534	160	5	modulo	modulo	PROPN
cana-1534	160	6	5	5	NUM
cana-1534	160	7	)	)	PUNCT
cana-1534	160	8	,	,	PUNCT
cana-1534	160	9	∣w∣	∣w∣	NOUN
cana-1534	160	10	=	=	SYM
cana-1534	160	11	2𝑛	2𝑛	PROPN
cana-1534	160	12	5	5	NUM
cana-1534	160	13	+1	+1	NOUN
cana-1534	160	14	and	and	CCONJ
cana-1534	160	15	for	for	ADP
cana-1534	160	16	n≡	n≡	ADP
cana-1534	160	17	1	1	NUM
cana-1534	160	18	(	(	PUNCT
cana-1534	160	19	modulo	modulo	NOUN
cana-1534	160	20	5	5	NUM
cana-1534	160	21	)	)	PUNCT
cana-1534	160	22	,	,	PUNCT
cana-1534	160	23	∣w∣	∣w∣	NOUN
cana-1534	160	24	=	=	SYM
cana-1534	160	25	2𝑛	2𝑛	PROPN
cana-1534	160	26	5	5	NUM
cana-1534	160	27	and	and	CCONJ
cana-1534	160	28	r(v|w)=	r(v|w)=	PROPN
cana-1534	160	29	(	(	PUNCT
cana-1534	160	30	1,1,2,2,	1,1,2,2,	NUM
cana-1534	160	31	…	…	SYM
cana-1534	160	32	,2,1	,2,1	NOUN
cana-1534	160	33	)	)	PUNCT
cana-1534	160	34	where	where	SCONJ
cana-1534	160	35	1	1	NUM
cana-1534	160	36	appears	appear	VERB
cana-1534	160	37	in	in	ADP
cana-1534	160	38	the	the	DET
cana-1534	160	39	first	first	ADJ
cana-1534	160	40	,	,	PUNCT
cana-1534	160	41	second	second	ADJ
cana-1534	160	42	,	,	PUNCT
cana-1534	160	43	and	and	CCONJ
cana-1534	160	44	2𝑛	2𝑛	NUM
cana-1534	160	45	5	5	NUM
cana-1534	160	46	𝑡ℎ	𝑡ℎ	NOUN
cana-1534	160	47	places	place	NOUN
cana-1534	160	48	.	.	PUNCT
cana-1534	161	1	case	case	NOUN
cana-1534	161	2	2	2	X
cana-1534	161	3	.	.	PUNCT
cana-1534	162	1	let	let	VERB
cana-1534	162	2	n≡2,4	n≡2,4	PROPN
cana-1534	162	3	(	(	PUNCT
cana-1534	162	4	modulo	modulo	NOUN
cana-1534	162	5	5	5	X
cana-1534	162	6	)	)	PUNCT
cana-1534	162	7	take	take	VERB
cana-1534	162	8	w	w	NOUN
cana-1534	162	9	=	=	PUNCT
cana-1534	162	10	{	{	PUNCT
cana-1534	162	11	v	v	NOUN
cana-1534	162	12	,	,	PUNCT
cana-1534	162	13	v1	v1	NOUN
cana-1534	162	14	,	,	PUNCT
cana-1534	162	15	v5	v5	NOUN
cana-1534	162	16	,	,	PUNCT
cana-1534	162	17	v5i+2	v5i+2	ADJ
cana-1534	162	18	,	,	PUNCT
cana-1534	162	19	v5(i+1	v5(i+1	NOUN
cana-1534	162	20	)	)	PUNCT
cana-1534	162	21	,	,	PUNCT
cana-1534	162	22	vn	vn	X
cana-1534	162	23	:	:	PUNCT
cana-1534	162	24	1	1	NUM
cana-1534	162	25	≤	≤	NUM
cana-1534	162	26	i	i	PRON
cana-1534	162	27	≤	≤	ADV
cana-1534	162	28	6	6	NUM
cana-1534	162	29	.	.	PUNCT
cana-1534	163	1	else	else	ADV
cana-1534	163	2	|w	|w	ADJ
cana-1534	163	3	|	|	NOUN
cana-1534	163	4	=	=	SYM
cana-1534	163	5	2𝑛	2𝑛	PROPN
cana-1534	163	6	5	5	NUM
cana-1534	163	7	+1	+1	NOUN
cana-1534	163	8	and	and	CCONJ
cana-1534	163	9	r(vn-1|w	r(vn-1|w	NOUN
cana-1534	163	10	)	)	PUNCT
cana-1534	163	11	=	=	PUNCT
cana-1534	163	12	(	(	PUNCT
cana-1534	163	13	1,2,2,	1,2,2,	NUM
cana-1534	163	14	…	…	SYM
cana-1534	163	15	,2,1,1	,2,1,1	NOUN
cana-1534	163	16	)	)	PUNCT
cana-1534	163	17	where	where	SCONJ
cana-1534	163	18	1	1	NUM
cana-1534	163	19	appears	appear	VERB
cana-1534	163	20	in	in	ADP
cana-1534	163	21	the	the	DET
cana-1534	163	22	first	first	ADJ
cana-1534	163	23	.	.	PUNCT
cana-1534	164	1	2𝑛	2𝑛	NOUN
cana-1534	164	2	5	5	NUM
cana-1534	164	3	−	−	PROPN
cana-1534	164	4	𝑡ℎ	𝑡ℎ	NOUN
cana-1534	164	5	,	,	PUNCT
cana-1534	164	6	and	and	CCONJ
cana-1534	164	7	2𝑛	2𝑛	NUM
cana-1534	164	8	5	5	NUM
cana-1534	164	9	+	+	CCONJ
cana-1534	164	10	1	1	NUM
cana-1534	164	11	−	−	NOUN
cana-1534	164	12	𝑡ℎ	𝑡ℎ	NOUN
cana-1534	164	13	places	place	NOUN
cana-1534	164	14	.	.	PUNCT
cana-1534	165	1	communications	communication	NOUN
cana-1534	165	2	on	on	ADP
cana-1534	165	3	applied	apply	VERB
cana-1534	165	4	nonlinear	nonlinear	ADJ
cana-1534	165	5	analysis	analysis	NOUN
cana-1534	165	6	issn	issn	NOUN
cana-1534	165	7	:	:	PUNCT
cana-1534	165	8	1074	1074	NUM
cana-1534	165	9	-	-	PUNCT
cana-1534	165	10	133x	133x	NUM
cana-1534	165	11	vol	vol	NOUN
cana-1534	165	12	31	31	NUM
cana-1534	165	13	no	no	NOUN
cana-1534	165	14	.	.	PUNCT
cana-1534	166	1	8s	8s	PROPN
cana-1534	166	2	(	(	PUNCT
cana-1534	166	3	2024	2024	NUM
cana-1534	166	4	)	)	PUNCT
cana-1534	166	5	428	428	NUM
cana-1534	166	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	166	7	case	case	NOUN
cana-1534	166	8	2	2	X
cana-1534	166	9	.	.	X
cana-1534	166	10	let	let	VERB
cana-1534	166	11	𝑛≡3	𝑛≡3	PROPN
cana-1534	166	12	(	(	PUNCT
cana-1534	166	13	modulo	modulo	NOUN
cana-1534	166	14	5	5	X
cana-1534	166	15	)	)	PUNCT
cana-1534	166	16	take	take	VERB
cana-1534	166	17	w	w	NOUN
cana-1534	166	18	=	=	PUNCT
cana-1534	166	19	{	{	PUNCT
cana-1534	166	20	v	v	NOUN
cana-1534	166	21	,	,	PUNCT
cana-1534	166	22	v1	v1	NOUN
cana-1534	166	23	,	,	PUNCT
cana-1534	166	24	v5	v5	NOUN
cana-1534	166	25	,	,	PUNCT
cana-1534	166	26	v5i+2	v5i+2	ADJ
cana-1534	166	27	,	,	PUNCT
cana-1534	166	28	v5(i+1	v5(i+1	NOUN
cana-1534	166	29	)	)	PUNCT
cana-1534	166	30	,	,	PUNCT
cana-1534	166	31	vn	vn	X
cana-1534	166	32	:	:	PUNCT
cana-1534	166	33	1	1	NUM
cana-1534	166	34	≤	≤	NUM
cana-1534	166	35	i	i	PRON
cana-1534	166	36	≤	≤	ADV
cana-1534	166	37	6	6	NUM
cana-1534	166	38	.	.	PUNCT
cana-1534	167	1	else	else	ADV
cana-1534	167	2	|w	|w	ADJ
cana-1534	167	3	|	|	NOUN
cana-1534	167	4	=	=	SYM
cana-1534	167	5	2𝑛	2𝑛	PROPN
cana-1534	167	6	5	5	NUM
cana-1534	167	7	and	and	CCONJ
cana-1534	167	8	r(vn-1|w	r(vn-1|w	NOUN
cana-1534	167	9	)	)	PUNCT
cana-1534	167	10	=	=	SYM
cana-1534	167	11	(	(	PUNCT
cana-1534	167	12	1,2,2,	1,2,2,	NUM
cana-1534	167	13	…	…	SYM
cana-1534	167	14	,2,1,1	,2,1,1	NOUN
cana-1534	167	15	)	)	PUNCT
cana-1534	167	16	where	where	SCONJ
cana-1534	167	17	1	1	NUM
cana-1534	167	18	appears	appear	VERB
cana-1534	167	19	in	in	ADP
cana-1534	167	20	the	the	DET
cana-1534	167	21	first	first	ADJ
cana-1534	167	22	.	.	PUNCT
cana-1534	168	1	2𝑛	2𝑛	NOUN
cana-1534	168	2	5	5	NUM
cana-1534	168	3	−	−	PROPN
cana-1534	168	4	𝑡ℎ	𝑡ℎ	NOUN
cana-1534	168	5	,	,	PUNCT
cana-1534	168	6	and	and	CCONJ
cana-1534	168	7	2𝑛	2𝑛	NUM
cana-1534	168	8	5	5	NUM
cana-1534	169	1	−	−	NOUN
cana-1534	169	2	1	1	NUM
cana-1534	169	3	−	−	NOUN
cana-1534	169	4	𝑡ℎ	𝑡ℎ	NOUN
cana-1534	169	5	places	place	NOUN
cana-1534	169	6	,	,	PUNCT
cana-1534	169	7	and	and	CCONJ
cana-1534	169	8	r(vw)=(1,1,2,2,	r(vw)=(1,1,2,2,	PROPN
cana-1534	169	9	…	…	SYM
cana-1534	169	10	,2,1	,2,1	PROPN
cana-1534	169	11	)	)	PUNCT
cana-1534	169	12	,	,	PUNCT
cana-1534	169	13	where	where	SCONJ
cana-1534	169	14	1	1	NUM
cana-1534	169	15	appears	appear	VERB
cana-1534	169	16	in	in	ADP
cana-1534	169	17	the	the	DET
cana-1534	169	18	first	first	ADJ
cana-1534	169	19	,	,	PUNCT
cana-1534	169	20	second	second	ADJ
cana-1534	169	21	,	,	PUNCT
cana-1534	169	22	and	and	CCONJ
cana-1534	169	23	2𝑛	2𝑛	NUM
cana-1534	169	24	5	5	NUM
cana-1534	169	25	−	−	PROPN
cana-1534	169	26	𝑡ℎ	𝑡ℎ	NOUN
cana-1534	169	27	places	place	NOUN
cana-1534	169	28	.	.	PUNCT
cana-1534	170	1	in	in	ADP
cana-1534	170	2	all	all	DET
cana-1534	170	3	the	the	DET
cana-1534	170	4	above	above	ADJ
cana-1534	170	5	cases	case	NOUN
cana-1534	170	6	,	,	PUNCT
cana-1534	170	7	r(v2|w	r(v2|w	PROPN
cana-1534	170	8	)	)	PUNCT
cana-1534	170	9	=	=	PUNCT
cana-1534	171	1	(	(	PUNCT
cana-1534	171	2	1	1	NUM
cana-1534	171	3	,	,	PUNCT
cana-1534	171	4	,	,	PUNCT
cana-1534	171	5	...	...	PUNCT
cana-1534	171	6	,	,	PUNCT
cana-1534	171	7	2	2	X
cana-1534	171	8	)	)	PUNCT
cana-1534	171	9	,	,	PUNCT
cana-1534	171	10	r(v3|w	r(v3|w	PROPN
cana-1534	171	11	)	)	PUNCT
cana-1534	171	12	=	=	PUNCT
cana-1534	172	1	(	(	PUNCT
cana-1534	172	2	1	1	NUM
cana-1534	172	3	,	,	PUNCT
cana-1534	172	4	,	,	PUNCT
cana-1534	172	5	...	...	PUNCT
cana-1534	172	6	,	,	PUNCT
cana-1534	172	7	2	2	X
cana-1534	172	8	)	)	PUNCT
cana-1534	172	9	,	,	PUNCT
cana-1534	172	10	r(v4|w	r(v4|w	NOUN
cana-1534	172	11	)	)	PUNCT
cana-1534	172	12	=	=	PUNCT
cana-1534	172	13	(	(	PUNCT
cana-1534	172	14	1	1	NUM
cana-1534	172	15	,	,	PUNCT
cana-1534	172	16	2,1	2,1	NUM
cana-1534	172	17	,	,	PUNCT
cana-1534	172	18	2	2	NUM
cana-1534	172	19	,	,	PUNCT
cana-1534	172	20	2	2	NUM
cana-1534	172	21	,	,	PUNCT
cana-1534	172	22	...	...	PUNCT
cana-1534	172	23	,	,	PUNCT
cana-1534	172	24	2	2	X
cana-1534	172	25	)	)	PUNCT
cana-1534	172	26	,	,	PUNCT
cana-1534	172	27	r(v5i+1|w	r(v5i+1|w	PROPN
cana-1534	172	28	)	)	PUNCT
cana-1534	172	29	where	where	SCONJ
cana-1534	172	30	1	1	NUM
cana-1534	172	31	appears	appear	VERB
cana-1534	172	32	in	in	ADP
cana-1534	172	33	the	the	DET
cana-1534	172	34	first	first	ADJ
cana-1534	172	35	and	and	CCONJ
cana-1534	172	36	(	(	PUNCT
cana-1534	172	37	5i)th	5i)th	NUM
cana-1534	172	38	and	and	CCONJ
cana-1534	172	39	(	(	PUNCT
cana-1534	172	40	5i	5i	NUM
cana-1534	172	41	+	+	SYM
cana-1534	172	42	2)th	2)th	NUM
cana-1534	172	43	places	place	NOUN
cana-1534	172	44	,	,	PUNCT
cana-1534	172	45	r(v5i+3|w	r(v5i+3|w	NOUN
cana-1534	172	46	)	)	PUNCT
cana-1534	172	47	=	=	PUNCT
cana-1534	172	48	(	(	PUNCT
cana-1534	172	49	1	1	NUM
cana-1534	172	50	,	,	PUNCT
cana-1534	172	51	2	2	NUM
cana-1534	172	52	,	,	PUNCT
cana-1534	172	53	2	2	NUM
cana-1534	172	54	,	,	PUNCT
cana-1534	172	55	...	...	PUNCT
cana-1534	172	56	,	,	PUNCT
cana-1534	172	57	2	2	NUM
cana-1534	172	58	,	,	PUNCT
cana-1534	172	59	1	1	NUM
cana-1534	172	60	,	,	PUNCT
cana-1534	172	61	2	2	NUM
cana-1534	172	62	,	,	PUNCT
cana-1534	172	63	2	2	NUM
cana-1534	172	64	,	,	PUNCT
cana-1534	172	65	...	...	PUNCT
cana-1534	172	66	,	,	PUNCT
cana-1534	172	67	2	2	X
cana-1534	172	68	)	)	PUNCT
cana-1534	172	69	where	where	SCONJ
cana-1534	172	70	1	1	NUM
cana-1534	172	71	appears	appear	VERB
cana-1534	172	72	in	in	ADP
cana-1534	172	73	the	the	DET
cana-1534	172	74	first	first	ADJ
cana-1534	172	75	and	and	CCONJ
cana-1534	172	76	(	(	PUNCT
cana-1534	172	77	5i	5i	NUM
cana-1534	172	78	+	+	SYM
cana-1534	172	79	2)th	2)th	NUM
cana-1534	172	80	places	place	NOUN
cana-1534	172	81	and	and	CCONJ
cana-1534	172	82	r(v5i+4|w	r(v5i+4|w	NOUN
cana-1534	172	83	)	)	PUNCT
cana-1534	173	1	=	=	PUNCT
cana-1534	173	2	(	(	PUNCT
cana-1534	173	3	1	1	NUM
cana-1534	173	4	,	,	PUNCT
cana-1534	173	5	2	2	NUM
cana-1534	173	6	,	,	PUNCT
cana-1534	173	7	2	2	NUM
cana-1534	173	8	,	,	PUNCT
cana-1534	173	9	...	...	PUNCT
cana-1534	173	10	,	,	PUNCT
cana-1534	173	11	2	2	NUM
cana-1534	173	12	,	,	PUNCT
cana-1534	173	13	1	1	NUM
cana-1534	173	14	,	,	PUNCT
cana-1534	173	15	2	2	NUM
cana-1534	173	16	,	,	PUNCT
cana-1534	173	17	2	2	NUM
cana-1534	173	18	,	,	PUNCT
cana-1534	173	19	...	...	PUNCT
cana-1534	173	20	,	,	PUNCT
cana-1534	173	21	2	2	X
cana-1534	173	22	)	)	PUNCT
cana-1534	173	23	where	where	SCONJ
cana-1534	173	24	1	1	NUM
cana-1534	173	25	appears	appear	VERB
cana-1534	173	26	in	in	ADP
cana-1534	173	27	the	the	DET
cana-1534	173	28	first	first	ADJ
cana-1534	173	29	and	and	CCONJ
cana-1534	173	30	(	(	PUNCT
cana-1534	173	31	5i)th	5i)th	NUM
cana-1534	173	32	places	place	NOUN
cana-1534	173	33	,	,	PUNCT
cana-1534	173	34	1	1	NUM
cana-1534	173	35	≤	≤	NUM
cana-1534	173	36	i	i	PRON
cana-1534	173	37	≤	≤	NOUN
cana-1534	173	38	6	6	NUM
cana-1534	173	39	,	,	PUNCT
cana-1534	173	40	n	n	CCONJ
cana-1534	173	41	,	,	PUNCT
cana-1534	173	42	.	.	PUNCT
cana-1534	174	1	also	also	ADV
cana-1534	174	2	for	for	ADP
cana-1534	174	3	n	n	X
cana-1534	174	4	≡	≡	PROPN
cana-1534	174	5	1	1	NUM
cana-1534	174	6	,	,	PUNCT
cana-1534	174	7	3	3	NUM
cana-1534	174	8	(	(	PUNCT
cana-1534	174	9	modulo	modulo	NOUN
cana-1534	174	10	5	5	NUM
cana-1534	174	11	)	)	PUNCT
cana-1534	174	12	,	,	PUNCT
cana-1534	174	13	⟨w	⟨w	X
cana-1534	174	14	⟩	⟩	NOUN
cana-1534	174	15	is	be	AUX
cana-1534	174	16	isomorphic	isomorphic	ADJ
cana-1534	174	17	to	to	PART
cana-1534	174	18	k1,⌊	k1,⌊	VERB
cana-1534	174	19	2n	2n	NUM
cana-1534	174	20	⌋	⌋	NOUN
cana-1534	174	21	and	and	CCONJ
cana-1534	174	22	for	for	ADP
cana-1534	174	23	n	n	X
cana-1534	174	24	≡	≡	PROPN
cana-1534	174	25	0	0	NUM
cana-1534	174	26	,	,	PUNCT
cana-1534	174	27	2	2	NUM
cana-1534	174	28	,	,	PUNCT
cana-1534	174	29	4	4	NUM
cana-1534	174	30	(	(	PUNCT
cana-1534	174	31	modulo	modulo	NOUN
cana-1534	174	32	5	5	NUM
cana-1534	174	33	)	)	PUNCT
cana-1534	174	34	,	,	PUNCT
cana-1534	174	35	⟨w	⟨w	X
cana-1534	174	36	⟩	⟩	NOUN
cana-1534	174	37	is	be	AUX
cana-1534	174	38	isomorphic	isomorphic	ADJ
cana-1534	174	39	to	to	PART
cana-1534	174	40	k1,⌈2n	k1,⌈2n	VERB
cana-1534	174	41	⌉.	⌉.	ADV
cana-1534	174	42	hence	hence	ADV
cana-1534	174	43	nr(wn	nr(wn	NOUN
cana-1534	174	44	)	)	PUNCT
cana-1534	174	45	≤	≤	NOUN
cana-1534	174	46	5	5	NUM
cana-1534	175	1	+	+	CCONJ
cana-1534	175	2	1	1	NUM
cana-1534	175	3	,	,	PUNCT
cana-1534	175	4	if	if	SCONJ
cana-1534	175	5	n	n	PRON
cana-1534	175	6	≡	≡	PROPN
cana-1534	175	7	0	0	NUM
cana-1534	175	8	,	,	PUNCT
cana-1534	175	9	2	2	NUM
cana-1534	175	10	,	,	PUNCT
cana-1534	175	11	4	4	NUM
cana-1534	175	12	(	(	PUNCT
cana-1534	175	13	modulo	modulo	NOUN
cana-1534	175	14	,	,	PUNCT
cana-1534	175	15	2n	2n	NUM
cana-1534	175	16	,	,	PUNCT
cana-1534	175	17	5	5	NUM
cana-1534	175	18	)	)	PUNCT
cana-1534	175	19	and	and	CCONJ
cana-1534	175	20	nr(wn	nr(wn	PROPN
cana-1534	175	21	)	)	PUNCT
cana-1534	175	22	≤	≤	NUM
cana-1534	175	23	5	5	NUM
cana-1534	175	24	,	,	PUNCT
cana-1534	175	25	if	if	SCONJ
cana-1534	175	26	n	n	PRON
cana-1534	175	27	≡	≡	PROPN
cana-1534	175	28	1	1	NUM
cana-1534	175	29	,	,	PUNCT
cana-1534	175	30	3	3	NUM
cana-1534	175	31	(	(	PUNCT
cana-1534	175	32	modulo	modulo	NOUN
cana-1534	175	33	5	5	NUM
cana-1534	175	34	)	)	PUNCT
cana-1534	175	35	.	.	PUNCT
cana-1534	176	1	shows	show	VERB
cana-1534	176	2	n	n	NOUN
cana-1534	176	3	=	=	SYM
cana-1534	176	4	6	6	NUM
cana-1534	176	5	,	,	PUNCT
cana-1534	176	6	7	7	NUM
cana-1534	176	7	,	,	PUNCT
cana-1534	176	8	8	8	NUM
cana-1534	176	9	,	,	PUNCT
cana-1534	176	10	and	and	CCONJ
cana-1534	176	11	14	14	NUM
cana-1534	176	12	examples	example	NOUN
cana-1534	176	13	.	.	PUNCT
cana-1534	177	1	vertices	vertex	NOUN
cana-1534	177	2	in	in	ADP
cana-1534	177	3	non	non	ADJ
cana-1534	177	4	-	-	ADJ
cana-1534	177	5	isolated	isolated	ADJ
cana-1534	177	6	resolution	resolution	NOUN
cana-1534	177	7	sets	set	NOUN
cana-1534	177	8	are	be	AUX
cana-1534	177	9	expanded	expand	VERB
cana-1534	177	10	in	in	ADP
cana-1534	177	11	the	the	DET
cana-1534	177	12	graph	graph	NOUN
cana-1534	177	13	.	.	PUNCT
cana-1534	178	1	consider	consider	VERB
cana-1534	178	2	the	the	DET
cana-1534	178	3	vertex	vertex	NOUN
cana-1534	178	4	vi	vi	NOUN
cana-1534	178	5	,	,	PUNCT
cana-1534	178	6	1	1	NUM
cana-1534	178	7	≤	≤	NUM
cana-1534	178	8	i	i	PRON
cana-1534	178	9	≤	≤	ADJ
cana-1534	178	10	n.	n.	NOUN
cana-1534	179	1	if	if	SCONJ
cana-1534	179	2	vi	vi	VERB
cana-1534	179	3	∈	∈	PROPN
cana-1534	179	4	w	w	NOUN
cana-1534	179	5	,	,	PUNCT
cana-1534	179	6	else	else	ADV
cana-1534	179	7	one	one	NUM
cana-1534	179	8	among	among	ADP
cana-1534	179	9	vi+2	vi+2	NUM
cana-1534	179	10	,	,	PUNCT
cana-1534	179	11	vi+3	vi+3	X
cana-1534	179	12	,	,	PUNCT
cana-1534	179	13	vi+4	vi+4	X
cana-1534	179	14	∈	∈	PROPN
cana-1534	179	15	w.	w.	NOUN
cana-1534	179	16	since	since	SCONJ
cana-1534	179	17	vi+4	vi+4	PROPN
cana-1534	179	18	∈	∈	PROPN
cana-1534	179	19	w	w	PROPN
cana-1534	179	20	,	,	PUNCT
cana-1534	179	21	r(vi+5|w	r(vi+5|w	PROPN
cana-1534	179	22	)	)	PUNCT
cana-1534	179	23	=	=	SYM
cana-1534	179	24	r(vi+3|w	r(vi+3|w	PROPN
cana-1534	179	25	)	)	PUNCT
cana-1534	179	26	for	for	ADP
cana-1534	179	27	any	any	DET
cana-1534	179	28	w	w	NOUN
cana-1534	179	29	unless	unless	SCONJ
cana-1534	179	30	vi+6	vi+6	PROPN
cana-1534	179	31	∈	∈	PROPN
cana-1534	179	32	w	w	PROPN
cana-1534	179	33	.	.	PUNCT
cana-1534	180	1	communications	communication	NOUN
cana-1534	180	2	on	on	ADP
cana-1534	180	3	applied	apply	VERB
cana-1534	180	4	nonlinear	nonlinear	ADJ
cana-1534	180	5	analysis	analysis	NOUN
cana-1534	180	6	issn	issn	NOUN
cana-1534	180	7	:	:	PUNCT
cana-1534	180	8	1074	1074	NUM
cana-1534	180	9	-	-	PUNCT
cana-1534	180	10	133x	133x	NUM
cana-1534	180	11	vol	vol	NOUN
cana-1534	180	12	31	31	NUM
cana-1534	180	13	no	no	NOUN
cana-1534	180	14	.	.	PUNCT
cana-1534	181	1	8s	8s	PROPN
cana-1534	181	2	(	(	PUNCT
cana-1534	181	3	2024	2024	NUM
cana-1534	181	4	)	)	PUNCT
cana-1534	181	5	429	429	NUM
cana-1534	181	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	182	1	so	so	ADV
cana-1534	182	2	we	we	PRON
cana-1534	182	3	conclude	conclude	VERB
cana-1534	182	4	that	that	SCONJ
cana-1534	182	5	vi+6	vi+6	PROPN
cana-1534	182	6	∈	∈	PROPN
cana-1534	182	7	w	w	PROPN
cana-1534	182	8	.	.	PUNCT
cana-1534	183	1	already	already	ADV
cana-1534	183	2	r(vi+5|w	r(vi+5|w	ADJ
cana-1534	183	3	)	)	PUNCT
cana-1534	183	4	/=	/=	PUNCT
cana-1534	184	1	r(vi+7|w	r(vi+7|w	PROPN
cana-1534	184	2	)	)	PUNCT
cana-1534	184	3	hence	hence	ADV
cana-1534	184	4	vi+9	vi+9	ADP
cana-1534	184	5	∈	∈	PROPN
cana-1534	184	6	w	w	ADP
cana-1534	184	7	otherwise	otherwise	ADV
cana-1534	184	8	r(vi+2|w	r(vi+2|w	NOUN
cana-1534	184	9	)	)	PUNCT
cana-1534	184	10	=	=	SYM
cana-1534	184	11	r(vi+8|w	r(vi+8|w	NOUN
cana-1534	184	12	)	)	PUNCT
cana-1534	184	13	.	.	PUNCT
cana-1534	185	1	now	now	ADV
cana-1534	185	2	,	,	PUNCT
cana-1534	185	3	the	the	DET
cana-1534	185	4	vertex	vertex	NOUN
cana-1534	185	5	v	v	NOUN
cana-1534	185	6	must	must	AUX
cana-1534	185	7	belong	belong	VERB
cana-1534	185	8	to	to	ADP
cana-1534	185	9	w	w	ADP
cana-1534	185	10	otherwise	otherwise	ADV
cana-1534	185	11	⟨w	⟨w	X
cana-1534	185	12	⟩	⟩	NOUN
cana-1534	185	13	will	will	AUX
cana-1534	185	14	contain	contain	VERB
cana-1534	185	15	the	the	DET
cana-1534	185	16	isolated	isolated	ADJ
cana-1534	185	17	vertices	vertex	NOUN
cana-1534	185	18	.	.	PUNCT
cana-1534	186	1	therefore	therefore	ADV
cana-1534	186	2	,	,	PUNCT
cana-1534	186	3	any	any	DET
cana-1534	186	4	non	non	ADJ
cana-1534	186	5	-	-	ADJ
cana-1534	186	6	isolated	isolated	ADJ
cana-1534	186	7	resolved	resolve	VERB
cana-1534	186	8	sets	set	NOUN
cana-1534	186	9	it	it	PRON
cana-1534	186	10	contain	contain	VERB
cana-1534	186	11	2𝑛	2𝑛	NUM
cana-1534	186	12	5	5	NUM
cana-1534	186	13	vertices	vertex	NOUN
cana-1534	186	14	for	for	ADP
cana-1534	186	15	n	n	X
cana-1534	186	16	≡	≡	PROPN
cana-1534	186	17	1	1	NUM
cana-1534	186	18	,	,	PUNCT
cana-1534	186	19	3	3	NUM
cana-1534	186	20	(	(	PUNCT
cana-1534	186	21	modulo	modulo	NOUN
cana-1534	186	22	5	5	NUM
cana-1534	186	23	)	)	PUNCT
cana-1534	186	24	and	and	CCONJ
cana-1534	186	25	2𝑛	2𝑛	NUM
cana-1534	186	26	5	5	NUM
cana-1534	186	27	+	+	CCONJ
cana-1534	186	28	1	1	NUM
cana-1534	186	29	vertices	vertex	NOUN
cana-1534	186	30	for	for	ADP
cana-1534	186	31	n	n	PRON
cana-1534	186	32	≡	≡	PROPN
cana-1534	186	33	0	0	NUM
cana-1534	186	34	,	,	PUNCT
cana-1534	186	35	2	2	NUM
cana-1534	186	36	,	,	PUNCT
cana-1534	186	37	4	4	NUM
cana-1534	186	38	(	(	PUNCT
cana-1534	186	39	modulo	modulo	NOUN
cana-1534	186	40	5	5	NUM
cana-1534	186	41	)	)	PUNCT
cana-1534	186	42	.	.	PUNCT
cana-1534	187	1	hence	hence	ADV
cana-1534	187	2	|w|	|w|	VERB
cana-1534	187	3	≥	≥	NUM
cana-1534	187	4	0	0	NUM
cana-1534	187	5	,	,	PUNCT
cana-1534	187	6	2	2	NUM
cana-1534	187	7	,	,	PUNCT
cana-1534	187	8	4	4	NUM
cana-1534	187	9	(	(	PUNCT
cana-1534	187	10	modulo	modulo	NOUN
cana-1534	187	11	5	5	NUM
cana-1534	187	12	)	)	PUNCT
cana-1534	187	13	.	.	PUNCT
cana-1534	188	1	2𝑛	2𝑛	NOUN
cana-1534	188	2	5	5	NUM
cana-1534	188	3	5	5	NUM
cana-1534	188	4	,	,	PUNCT
cana-1534	188	5	if	if	SCONJ
cana-1534	188	6	n	n	PRON
cana-1534	188	7	≡	≡	PROPN
cana-1534	188	8	1	1	NUM
cana-1534	188	9	,	,	PUNCT
cana-1534	188	10	3	3	NUM
cana-1534	188	11	(	(	PUNCT
cana-1534	188	12	modulo	modulo	NOUN
cana-1534	188	13	5	5	NUM
cana-1534	188	14	)	)	PUNCT
cana-1534	188	15	and	and	CCONJ
cana-1534	188	16	|w|	|w|	VERB
cana-1534	188	17	≥	≥	NUM
cana-1534	188	18	2𝑛	2𝑛	PROPN
cana-1534	188	19	5	5	NUM
cana-1534	189	1	+	+	CCONJ
cana-1534	189	2	1	1	NUM
cana-1534	189	3	,	,	PUNCT
cana-1534	189	4	if	if	SCONJ
cana-1534	189	5	n	n	PRON
cana-1534	189	6	≡	≡	PROPN
cana-1534	189	7	2𝑛	2𝑛	PROPN
cana-1534	189	8	5	5	NUM
cana-1534	189	9	thus	thus	ADV
cana-1534	189	10	we	we	PRON
cana-1534	189	11	conclude	conclude	VERB
cana-1534	189	12	that	that	SCONJ
cana-1534	189	13	nr(wn	nr(wn	PROPN
cana-1534	189	14	,	,	PUNCT
cana-1534	189	15	+	+	NOUN
cana-1534	190	1	1	1	NUM
cana-1534	190	2	,	,	PUNCT
cana-1534	190	3	if	if	SCONJ
cana-1534	190	4	n	n	PRON
cana-1534	190	5	≡	≡	PROPN
cana-1534	190	6	0	0	NUM
cana-1534	190	7	,	,	PUNCT
cana-1534	190	8	2	2	NUM
cana-1534	190	9	,	,	PUNCT
cana-1534	190	10	4	4	NUM
cana-1534	190	11	(	(	PUNCT
cana-1534	190	12	modulo	modulo	NOUN
cana-1534	190	13	5	5	NUM
cana-1534	190	14	)	)	PUNCT
cana-1534	190	15	.	.	PUNCT
cana-1534	190	16	)	)	PUNCT
cana-1534	191	1	=	=	SYM
cana-1534	191	2	2𝑛	2𝑛	NUM
cana-1534	191	3	5	5	NUM
cana-1534	191	4	,	,	PUNCT
cana-1534	191	5	if	if	SCONJ
cana-1534	191	6	n	n	PRON
cana-1534	191	7	≡	≡	PROPN
cana-1534	191	8	1	1	NUM
cana-1534	191	9	,	,	PUNCT
cana-1534	191	10	3	3	NUM
cana-1534	191	11	(	(	PUNCT
cana-1534	191	12	modulo	modulo	NOUN
cana-1534	191	13	5	5	NUM
cana-1534	191	14	)	)	PUNCT
cana-1534	191	15	and	and	CCONJ
cana-1534	191	16	nr(w	nr(w	ADV
cana-1534	191	17	)	)	PUNCT
cana-1534	192	1	=	=	NOUN
cana-1534	192	2	nr	nr	NOUN
cana-1534	192	3	-	-	NOUN
cana-1534	192	4	value	value	NOUN
cana-1534	192	5	of	of	ADP
cana-1534	192	6	g	g	NOUN
cana-1534	192	7	◦	◦	NOUN
cana-1534	192	8	km	km	NOUN
cana-1534	192	9	,	,	PUNCT
cana-1534	192	10	g	g	NOUN
cana-1534	192	11	◦	◦	NOUN
cana-1534	192	12	km	km	NOUN
cana-1534	192	13	and	and	CCONJ
cana-1534	192	14	g	g	ADP
cana-1534	192	15	◦	◦	NOUN
cana-1534	192	16	k1	k1	NOUN
cana-1534	192	17	,	,	PUNCT
cana-1534	192	18	m	m	VERB
cana-1534	192	19	for	for	ADP
cana-1534	192	20	m	m	PROPN
cana-1534	192	21	≥	≥	NOUN
cana-1534	192	22	2	2	NUM
cana-1534	192	23	.	.	PUNCT
cana-1534	192	24	theorem	theorem	VERB
cana-1534	192	25	3.3	3.3	NUM
cana-1534	192	26	.	.	PUNCT
cana-1534	193	1	consider	consider	VERB
cana-1534	193	2	g	g	NOUN
cana-1534	193	3	,	,	PUNCT
cana-1534	193	4	a	a	DET
cana-1534	193	5	graph	graph	NOUN
cana-1534	193	6	that	that	PRON
cana-1534	193	7	is	be	AUX
cana-1534	193	8	connected	connect	VERB
cana-1534	193	9	with	with	ADP
cana-1534	193	10	order	order	NOUN
cana-1534	193	11	n	n	PRON
cana-1534	193	12	≥	≥	NOUN
cana-1534	193	13	2	2	NUM
cana-1534	193	14	.	.	PUNCT
cana-1534	193	15	else	else	ADV
cana-1534	193	16	,	,	PUNCT
cana-1534	193	17	for	for	ADP
cana-1534	193	18	each	each	DET
cana-1534	193	19	significant	significant	ADJ
cana-1534	193	20	integer	integer	NOUN
cana-1534	193	21	m	m	PROPN
cana-1534	193	22	≥	≥	NOUN
cana-1534	193	23	3	3	NUM
cana-1534	193	24	,	,	PUNCT
cana-1534	193	25	nr(g	nr(g	ADV
cana-1534	193	26	◦	◦	NOUN
cana-1534	193	27	km	km	NOUN
cana-1534	193	28	)	)	PUNCT
cana-1534	193	29	=	=	PUNCT
cana-1534	193	30	nm	nm	X
cana-1534	193	31	.	.	PUNCT
cana-1534	193	32	proof	proof	NOUN
cana-1534	193	33	.	.	PUNCT
cana-1534	194	1	let	let	VERB
cana-1534	194	2	h	h	NOUN
cana-1534	194	3	=	=	SYM
cana-1534	194	4	g	g	PROPN
cana-1534	194	5	◦	◦	NOUN
cana-1534	194	6	km	km	NOUN
cana-1534	194	7	and	and	CCONJ
cana-1534	194	8	v	v	NOUN
cana-1534	194	9	(	(	PUNCT
cana-1534	194	10	h	h	NOUN
cana-1534	194	11	)	)	PUNCT
cana-1534	194	12	=	=	SYM
cana-1534	194	13	{	{	PUNCT
cana-1534	194	14	vi	vi	PROPN
cana-1534	194	15	;	;	PUNCT
cana-1534	194	16	ui1	ui1	ADJ
cana-1534	194	17	,	,	PUNCT
cana-1534	194	18	ui2	ui2	INTJ
cana-1534	194	19	,	,	PUNCT
cana-1534	194	20	...	...	PUNCT
cana-1534	194	21	,	,	PUNCT
cana-1534	194	22	uim	uim	X
cana-1534	194	23	:	:	PUNCT
cana-1534	194	24	1	1	NUM
cana-1534	194	25	≤	≤	NUM
cana-1534	194	26	i	i	PRON
cana-1534	194	27	≤	≤	NOUN
cana-1534	194	28	n	n	CCONJ
cana-1534	194	29	}	}	PUNCT
cana-1534	194	30	where	where	SCONJ
cana-1534	194	31	v	v	X
cana-1534	194	32	(	(	PUNCT
cana-1534	194	33	g	g	NOUN
cana-1534	194	34	)	)	PUNCT
cana-1534	194	35	=	=	PRON
cana-1534	194	36	{	{	PUNCT
cana-1534	194	37	vi	vi	NOUN
cana-1534	194	38	:	:	PUNCT
cana-1534	194	39	1	1	NUM
cana-1534	194	40	≤	≤	NUM
cana-1534	194	41	i	i	PRON
cana-1534	194	42	≤	≤	NOUN
cana-1534	194	43	n	n	CCONJ
cana-1534	194	44	}	}	PUNCT
cana-1534	194	45	and	and	CCONJ
cana-1534	194	46	ui	ui	NOUN
cana-1534	194	47	,	,	PUNCT
cana-1534	194	48	ui	ui	PROPN
cana-1534	194	49	,	,	PUNCT
cana-1534	194	50	...	...	PUNCT
cana-1534	194	51	,	,	PUNCT
cana-1534	194	52	ui	ui	PROPN
cana-1534	194	53	are	be	AUX
cana-1534	194	54	the	the	DET
cana-1534	194	55	vertices	vertex	NOUN
cana-1534	194	56	in	in	ADP
cana-1534	194	57	the	the	DET
cana-1534	194	58	ith	ith	PROPN
cana-1534	194	59	copy	copy	NOUN
cana-1534	194	60	of	of	ADP
cana-1534	194	61	km	km	PROPN
cana-1534	194	62	,	,	PUNCT
cana-1534	194	63	1	1	NUM
cana-1534	194	64	≤	≤	NUM
cana-1534	194	65	i	i	PRON
cana-1534	194	66	≤	≤	ADJ
cana-1534	194	67	n.	n.	NOUN
cana-1534	194	68	let	let	VERB
cana-1534	195	1	w	w	VERB
cana-1534	195	2	=	=	PRON
cana-1534	195	3	{	{	PUNCT
cana-1534	195	4	vi	vi	PROPN
cana-1534	195	5	,	,	PUNCT
cana-1534	195	6	uij	uij	PRON
cana-1534	195	7	:	:	PUNCT
cana-1534	195	8	1	1	NUM
cana-1534	195	9	≤	≤	NUM
cana-1534	195	10	i	i	PRON
cana-1534	195	11	≤	≤	PROPN
cana-1534	195	12	n	n	CCONJ
cana-1534	195	13	,	,	PUNCT
cana-1534	195	14	1	1	NUM
cana-1534	195	15	≤	≤	NUM
cana-1534	195	16	j	j	PROPN
cana-1534	195	17	≤	≤	NUM
cana-1534	195	18	m	m	VERB
cana-1534	195	19	−	−	NOUN
cana-1534	195	20	1	1	NUM
cana-1534	195	21	}	}	PUNCT
cana-1534	195	22	.	.	PUNCT
cana-1534	196	1	else	else	ADV
cana-1534	196	2	vi	vi	PROPN
cana-1534	196	3	is	be	AUX
cana-1534	196	4	the	the	DET
cana-1534	196	5	only	only	ADJ
cana-1534	196	6	vertices	vertex	NOUN
cana-1534	196	7	which	which	PRON
cana-1534	196	8	is	be	AUX
cana-1534	196	9	at	at	ADP
cana-1534	196	10	a	a	DET
cana-1534	196	11	distance	distance	NOUN
cana-1534	196	12	1	1	NUM
cana-1534	196	13	from	from	ADP
cana-1534	196	14	uim	uim	PROPN
cana-1534	196	15	,	,	PUNCT
cana-1534	196	16	for	for	ADP
cana-1534	196	17	all	all	DET
cana-1534	196	18	1	1	NUM
cana-1534	196	19	≤	≤	NUM
cana-1534	196	20	i	i	PRON
cana-1534	196	21	≤	≤	ADJ
cana-1534	196	22	n.	n.	NOUN
cana-1534	196	23	hence	hence	ADV
cana-1534	196	24	the	the	DET
cana-1534	196	25	representation	representation	NOUN
cana-1534	196	26	of	of	ADP
cana-1534	196	27	all	all	DET
cana-1534	196	28	ui	ui	PROPN
cana-1534	196	29	,	,	PUNCT
cana-1534	196	30	1	1	NUM
cana-1534	196	31	≤	≤	NUM
cana-1534	196	32	i	i	PRON
cana-1534	196	33	≤	≤	NOUN
cana-1534	196	34	n	n	PRON
cana-1534	196	35	differs	differ	VERB
cana-1534	196	36	at	at	ADP
cana-1534	196	37	least	least	ADJ
cana-1534	196	38	in	in	ADP
cana-1534	196	39	the	the	DET
cana-1534	196	40	ith	ith	PROPN
cana-1534	196	41	place	place	NOUN
cana-1534	196	42	.	.	PUNCT
cana-1534	197	1	nr(h	nr(h	NOUN
cana-1534	197	2	)	)	PUNCT
cana-1534	197	3	≤	≤	NUM
cana-1534	197	4	nm	nm	NOUN
cana-1534	197	5	.	.	PUNCT
cana-1534	198	1	communications	communication	NOUN
cana-1534	198	2	on	on	ADP
cana-1534	198	3	applied	apply	VERB
cana-1534	198	4	nonlinear	nonlinear	ADJ
cana-1534	198	5	analysis	analysis	NOUN
cana-1534	198	6	issn	issn	NOUN
cana-1534	198	7	:	:	PUNCT
cana-1534	198	8	1074	1074	NUM
cana-1534	198	9	-	-	PUNCT
cana-1534	198	10	133x	133x	NUM
cana-1534	198	11	vol	vol	NOUN
cana-1534	198	12	31	31	NUM
cana-1534	198	13	no	no	NOUN
cana-1534	198	14	.	.	PUNCT
cana-1534	199	1	8s	8s	PROPN
cana-1534	199	2	(	(	PUNCT
cana-1534	199	3	2024	2024	NUM
cana-1534	199	4	)	)	PUNCT
cana-1534	199	5	430	430	NUM
cana-1534	199	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	199	7	if	if	SCONJ
cana-1534	199	8	uij	uij	PRON
cana-1534	199	9	,	,	PUNCT
cana-1534	199	10	uik	uik	NOUN
cana-1534	199	11	∈/	∈/	PROPN
cana-1534	199	12	w	w	PROPN
cana-1534	199	13	for	for	ADP
cana-1534	199	14	some	some	DET
cana-1534	199	15	i	i	PROPN
cana-1534	199	16	,	,	PUNCT
cana-1534	199	17	j	j	PROPN
cana-1534	199	18	,	,	PUNCT
cana-1534	199	19	k	k	PROPN
cana-1534	199	20	such	such	ADJ
cana-1534	199	21	that	that	SCONJ
cana-1534	199	22	1	1	NUM
cana-1534	199	23	≤	≤	NUM
cana-1534	199	24	i	i	PRON
cana-1534	199	25	≤	≤	ADJ
cana-1534	199	26	n	n	CCONJ
cana-1534	199	27	and	and	CCONJ
cana-1534	199	28	1	1	NUM
cana-1534	199	29	≤	≤	NUM
cana-1534	200	1	j	j	NOUN
cana-1534	200	2	/=	/=	PROPN
cana-1534	201	1	k	k	PROPN
cana-1534	201	2	≤	≤	PROPN
cana-1534	201	3	m	m	PROPN
cana-1534	201	4	,	,	PUNCT
cana-1534	201	5	else	else	ADV
cana-1534	201	6	r(uij	r(uij	PROPN
cana-1534	201	7	|w	|w	NOUN
cana-1534	201	8	)	)	PUNCT
cana-1534	201	9	=	=	PUNCT
cana-1534	201	10	r(uik	r(uik	NOUN
cana-1534	201	11	|w	|w	NOUN
cana-1534	201	12	)	)	PUNCT
cana-1534	201	13	..	..	PUNCT
cana-1534	202	1	also	also	ADV
cana-1534	202	2	w	w	PROPN
cana-1534	202	3	must	must	AUX
cana-1534	202	4	contain	contain	VERB
cana-1534	202	5	all	all	DET
cana-1534	202	6	vi	vi	NOUN
cana-1534	202	7	’s	’s	NOUN
cana-1534	202	8	,	,	PUNCT
cana-1534	202	9	1	1	NUM
cana-1534	202	10	≤	≤	NUM
cana-1534	202	11	i	i	PRON
cana-1534	202	12	≤	≤	PROPN
cana-1534	202	13	n	n	CCONJ
cana-1534	202	14	,	,	PUNCT
cana-1534	202	15	otherwise	otherwise	ADV
cana-1534	202	16	⟨w	⟨w	VERB
cana-1534	202	17	⟩	⟩	NOUN
cana-1534	202	18	will	will	AUX
cana-1534	202	19	contain	contain	VERB
cana-1534	202	20	ui1	ui1	ADV
cana-1534	202	21	as	as	ADP
cana-1534	202	22	an	an	DET
cana-1534	202	23	isolated	isolated	ADJ
cana-1534	202	24	vertex	vertex	NOUN
cana-1534	202	25	.	.	PUNCT
cana-1534	203	1	thus	thus	ADV
cana-1534	203	2	|w	|w	ADJ
cana-1534	203	3	|	|	ADV
cana-1534	203	4	≥	≥	NOUN
cana-1534	203	5	nm	nm	NOUN
cana-1534	203	6	.	.	PUNCT
cana-1534	204	1	hence	hence	ADV
cana-1534	204	2	nr(h	nr(h	NOUN
cana-1534	204	3	)	)	PUNCT
cana-1534	204	4	≥	≥	NUM
cana-1534	204	5	nm	nm	PROPN
cana-1534	204	6	.	.	PUNCT
cana-1534	205	1	therefore	therefore	ADV
cana-1534	205	2	,	,	PUNCT
cana-1534	205	3	nr(h	nr(h	NOUN
cana-1534	205	4	)	)	PUNCT
cana-1534	205	5	=	=	SYM
cana-1534	205	6	nm	nm	X
cana-1534	205	7	.	.	PUNCT
cana-1534	205	8	theorem	theorem	VERB
cana-1534	205	9	3.3	3.3	NUM
cana-1534	205	10	.	.	PUNCT
cana-1534	206	1	if	if	SCONJ
cana-1534	206	2	g	g	PROPN
cana-1534	206	3	is	be	AUX
cana-1534	206	4	a	a	DET
cana-1534	206	5	connected	connected	ADJ
cana-1534	206	6	graph	graph	NOUN
cana-1534	206	7	with	with	ADP
cana-1534	206	8	at	at	ADV
cana-1534	206	9	least	least	ADJ
cana-1534	206	10	2	2	NUM
cana-1534	206	11	vertices	vertex	NOUN
cana-1534	206	12	,	,	PUNCT
cana-1534	206	13	and	and	CCONJ
cana-1534	206	14	you	you	PRON
cana-1534	206	15	create	create	VERB
cana-1534	206	16	a	a	DET
cana-1534	206	17	new	new	ADJ
cana-1534	206	18	graph	graph	NOUN
cana-1534	206	19	by	by	ADP
cana-1534	206	20	combining	combine	VERB
cana-1534	206	21	g	g	NOUN
cana-1534	206	22	with	with	ADP
cana-1534	206	23	multiple	multiple	ADJ
cana-1534	206	24	copies	copy	NOUN
cana-1534	206	25	of	of	ADP
cana-1534	206	26	a	a	DET
cana-1534	206	27	smaller	small	ADJ
cana-1534	206	28	graph	graph	NOUN
cana-1534	206	29	km	km	NOUN
cana-1534	206	30	(	(	PUNCT
cana-1534	206	31	where	where	SCONJ
cana-1534	206	32	m	m	NOUN
cana-1534	206	33	is	be	AUX
cana-1534	206	34	a	a	DET
cana-1534	206	35	positive	positive	ADJ
cana-1534	206	36	integer	integer	NOUN
cana-1534	206	37	greater	great	ADJ
cana-1534	206	38	than	than	ADP
cana-1534	206	39	or	or	CCONJ
cana-1534	206	40	equal	equal	ADJ
cana-1534	206	41	to	to	ADP
cana-1534	206	42	3	3	NUM
cana-1534	206	43	)	)	PUNCT
cana-1534	206	44	,	,	PUNCT
cana-1534	206	45	else	else	ADV
cana-1534	206	46	the	the	DET
cana-1534	206	47	minimum	minimum	ADJ
cana-1534	206	48	size	size	NOUN
cana-1534	206	49	of	of	ADP
cana-1534	206	50	a	a	DET
cana-1534	206	51	non	non	ADJ
cana-1534	206	52	-	-	ADJ
cana-1534	206	53	isolated	isolated	ADJ
cana-1534	206	54	resolving	resolving	NOUN
cana-1534	206	55	set	set	VERB
cana-1534	206	56	for	for	ADP
cana-1534	206	57	this	this	DET
cana-1534	206	58	new	new	ADJ
cana-1534	206	59	graph	graph	NOUN
cana-1534	206	60	is	be	AUX
cana-1534	206	61	g	g	PROPN
cana-1534	206	62	×	×	PROPN
cana-1534	206	63	n×m	n×m	PROPN
cana-1534	206	64	.	.	PUNCT
cana-1534	207	1	proof	proof	NOUN
cana-1534	207	2	.	.	PUNCT
cana-1534	208	1	consider	consider	VERB
cana-1534	208	2	the	the	DET
cana-1534	208	3	new	new	ADJ
cana-1534	208	4	graph	graph	NOUN
cana-1534	208	5	obtained	obtain	VERB
cana-1534	208	6	by	by	ADP
cana-1534	208	7	adding	add	VERB
cana-1534	208	8	to	to	ADP
cana-1534	208	9	g	g	PROPN
cana-1534	208	10	copies	copy	NOUN
cana-1534	208	11	of	of	ADP
cana-1534	208	12	k	k	PROPN
cana-1534	208	13	m.	m.	NOUN
cana-1534	208	14	in	in	ADP
cana-1534	208	15	this	this	DET
cana-1534	208	16	graph	graph	NOUN
cana-1534	208	17	,	,	PUNCT
cana-1534	208	18	each	each	DET
cana-1534	208	19	vertex	vertex	NOUN
cana-1534	208	20	of	of	ADP
cana-1534	208	21	g	g	PROPN
cana-1534	208	22	is	be	AUX
cana-1534	208	23	adjacent	adjacent	ADJ
cana-1534	208	24	to	to	ADP
cana-1534	208	25	every	every	DET
cana-1534	208	26	vertex	vertex	NOUN
cana-1534	208	27	in	in	ADP
cana-1534	208	28	one	one	NUM
cana-1534	208	29	copy	copy	NOUN
cana-1534	208	30	of	of	ADP
cana-1534	208	31	k	k	PROPN
cana-1534	208	32	m.	m.	NOUN
cana-1534	208	33	we	we	PRON
cana-1534	208	34	will	will	AUX
cana-1534	208	35	exhibit	exhibit	VERB
cana-1534	208	36	a	a	DET
cana-1534	208	37	set	set	NOUN
cana-1534	208	38	of	of	ADP
cana-1534	208	39	vertices	vertex	NOUN
cana-1534	208	40	that	that	PRON
cana-1534	208	41	consists	consist	VERB
cana-1534	208	42	of	of	ADP
cana-1534	208	43	one	one	NUM
cana-1534	208	44	vertex	vertex	NOUN
cana-1534	208	45	of	of	ADP
cana-1534	208	46	g	g	NOUN
cana-1534	208	47	and	and	CCONJ
cana-1534	208	48	all	all	DET
cana-1534	208	49	but	but	SCONJ
cana-1534	208	50	one	one	NUM
cana-1534	208	51	vertex	vertex	NOUN
cana-1534	208	52	in	in	ADP
cana-1534	208	53	each	each	DET
cana-1534	208	54	copy	copy	NOUN
cana-1534	208	55	of	of	ADP
cana-1534	208	56	k	k	PROPN
cana-1534	208	57	m	m	PROPN
cana-1534	208	58	.	.	PUNCT
cana-1534	209	1	this	this	DET
cana-1534	209	2	set	set	NOUN
cana-1534	209	3	is	be	AUX
cana-1534	209	4	large	large	ADJ
cana-1534	209	5	enough	enough	ADV
cana-1534	209	6	to	to	PART
cana-1534	209	7	distinguish	distinguish	VERB
cana-1534	209	8	all	all	DET
cana-1534	209	9	other	other	ADJ
cana-1534	209	10	vertices	vertex	NOUN
cana-1534	209	11	in	in	ADP
cana-1534	209	12	the	the	DET
cana-1534	209	13	graph	graph	NOUN
cana-1534	209	14	and	and	CCONJ
cana-1534	209	15	therefore	therefore	ADV
cana-1534	209	16	resolves	resolve	VERB
cana-1534	209	17	the	the	DET
cana-1534	209	18	graph	graph	NOUN
cana-1534	209	19	.	.	PUNCT
cana-1534	210	1	let	let	VERB
cana-1534	210	2	us	we	PRON
cana-1534	210	3	now	now	ADV
cana-1534	210	4	try	try	VERB
cana-1534	210	5	to	to	PART
cana-1534	210	6	show	show	VERB
cana-1534	210	7	that	that	SCONJ
cana-1534	210	8	this	this	DET
cana-1534	210	9	set	set	NOUN
cana-1534	210	10	is	be	AUX
cana-1534	210	11	the	the	DET
cana-1534	210	12	smallest	small	ADJ
cana-1534	210	13	possible	possible	ADJ
cana-1534	210	14	.	.	PUNCT
cana-1534	211	1	consider	consider	VERB
cana-1534	211	2	any	any	DET
cana-1534	211	3	other	other	ADJ
cana-1534	211	4	resolving	resolving	NOUN
cana-1534	211	5	set	set	NOUN
cana-1534	211	6	.	.	PUNCT
cana-1534	212	1	suppose	suppose	VERB
cana-1534	212	2	that	that	SCONJ
cana-1534	212	3	the	the	DET
cana-1534	212	4	set	set	NOUN
cana-1534	212	5	does	do	AUX
cana-1534	212	6	not	not	PART
cana-1534	212	7	contain	contain	VERB
cana-1534	212	8	enough	enough	ADJ
cana-1534	212	9	vertices	vertex	NOUN
cana-1534	212	10	from	from	ADP
cana-1534	212	11	each	each	DET
cana-1534	212	12	copy	copy	NOUN
cana-1534	212	13	of	of	ADP
cana-1534	212	14	k	k	PROPN
cana-1534	212	15	m.	m.	NOUN
cana-1534	212	16	,	,	PUNCT
cana-1534	212	17	some	some	DET
cana-1534	212	18	vertices	vertex	NOUN
cana-1534	212	19	of	of	ADP
cana-1534	212	20	that	that	DET
cana-1534	212	21	copy	copy	NOUN
cana-1534	212	22	would	would	AUX
cana-1534	212	23	be	be	AUX
cana-1534	212	24	indistinguishable	indistinguishable	ADJ
cana-1534	212	25	,	,	PUNCT
cana-1534	212	26	contradicting	contradict	VERB
cana-1534	212	27	the	the	DET
cana-1534	212	28	defining	define	VERB
cana-1534	212	29	property	property	NOUN
cana-1534	212	30	of	of	ADP
cana-1534	212	31	a	a	DET
cana-1534	212	32	resolving	resolving	NOUN
cana-1534	212	33	set	set	NOUN
cana-1534	212	34	.	.	PUNCT
cana-1534	213	1	the	the	DET
cana-1534	213	2	set	set	NOUN
cana-1534	213	3	must	must	AUX
cana-1534	213	4	therefore	therefore	ADV
cana-1534	213	5	contain	contain	VERB
cana-1534	213	6	at	at	ADP
cana-1534	213	7	least	least	ADJ
cana-1534	213	8	m×n	m×n	ADJ
cana-1534	213	9	vertices	vertex	NOUN
cana-1534	213	10	,	,	PUNCT
cana-1534	213	11	so	so	SCONJ
cana-1534	213	12	the	the	DET
cana-1534	213	13	minimum	minimum	ADJ
cana-1534	213	14	possible	possible	ADJ
cana-1534	213	15	size	size	NOUN
cana-1534	213	16	is	be	AUX
cana-1534	213	17	exactly	exactly	ADV
cana-1534	213	18	that	that	PRON
cana-1534	213	19	.	.	PUNCT
cana-1534	214	1	theorem	theorem	VERB
cana-1534	214	2	3.4	3.4	NUM
cana-1534	214	3	.	.	PUNCT
cana-1534	215	1	if	if	SCONJ
cana-1534	215	2	g	g	PROPN
cana-1534	215	3	is	be	AUX
cana-1534	215	4	a	a	DET
cana-1534	215	5	connected	connected	ADJ
cana-1534	215	6	graph	graph	NOUN
cana-1534	215	7	with	with	ADP
cana-1534	215	8	at	at	ADV
cana-1534	215	9	least	least	ADJ
cana-1534	215	10	2	2	NUM
cana-1534	215	11	vertices	vertex	NOUN
cana-1534	215	12	,	,	PUNCT
cana-1534	215	13	and	and	CCONJ
cana-1534	215	14	you	you	PRON
cana-1534	215	15	make	make	VERB
cana-1534	215	16	a	a	DET
cana-1534	215	17	new	new	ADJ
cana-1534	215	18	graph	graph	NOUN
cana-1534	215	19	by	by	ADP
cana-1534	215	20	combining	combine	VERB
cana-1534	215	21	g	g	NOUN
cana-1534	215	22	with	with	ADP
cana-1534	215	23	multiple	multiple	ADJ
cana-1534	215	24	copies	copy	NOUN
cana-1534	215	25	of	of	ADP
cana-1534	215	26	a	a	DET
cana-1534	215	27	smaller	small	ADJ
cana-1534	215	28	graph	graph	NOUN
cana-1534	215	29	k	k	X
cana-1534	215	30	(	(	PUNCT
cana-1534	215	31	where	where	SCONJ
cana-1534	215	32	,	,	PUNCT
cana-1534	215	33	if	if	SCONJ
cana-1534	215	34	m	m	NOUN
cana-1534	215	35	is	be	AUX
cana-1534	215	36	a	a	DET
cana-1534	215	37	positive	positive	ADJ
cana-1534	215	38	integer	integer	NOUN
cana-1534	215	39	greater	great	ADJ
cana-1534	215	40	or	or	CCONJ
cana-1534	215	41	equal	equal	ADJ
cana-1534	215	42	than	than	ADP
cana-1534	215	43	3	3	NUM
cana-1534	215	44	,	,	PUNCT
cana-1534	215	45	else	else	ADV
cana-1534	215	46	the	the	DET
cana-1534	215	47	minimum	minimum	ADJ
cana-1534	215	48	cardinality	cardinality	NOUN
cana-1534	215	49	of	of	ADP
cana-1534	215	50	a	a	DET
cana-1534	215	51	non	non	ADJ
cana-1534	215	52	-	-	ADJ
cana-1534	215	53	isolated	isolated	ADJ
cana-1534	215	54	resolving	resolving	NOUN
cana-1534	215	55	set	set	VERB
cana-1534	215	56	for	for	ADP
cana-1534	215	57	this	this	DET
cana-1534	215	58	new	new	ADJ
cana-1534	215	59	graph	graph	NOUN
cana-1534	215	60	is	be	AUX
cana-1534	215	61	nn×(m−1	nn×(m−1	NOUN
cana-1534	215	62	)	)	PUNCT
cana-1534	215	63	.	.	PUNCT
cana-1534	216	1	proof	proof	NOUN
cana-1534	216	2	.	.	PUNCT
cana-1534	217	1	in	in	ADP
cana-1534	217	2	this	this	DET
cana-1534	217	3	case	case	NOUN
cana-1534	217	4	we	we	PRON
cana-1534	217	5	will	will	AUX
cana-1534	217	6	deal	deal	VERB
cana-1534	217	7	with	with	ADP
cana-1534	217	8	another	another	DET
cana-1534	217	9	set	set	NOUN
cana-1534	217	10	of	of	ADP
cana-1534	217	11	vertices	vertex	NOUN
cana-1534	217	12	,	,	PUNCT
cana-1534	217	13	which	which	PRON
cana-1534	217	14	contains	contain	VERB
cana-1534	217	15	all	all	PRON
cana-1534	217	16	but	but	CCONJ
cana-1534	217	17	one	one	NUM
cana-1534	217	18	vertices	vertex	NOUN
cana-1534	217	19	from	from	ADP
cana-1534	217	20	each	each	DET
cana-1534	217	21	copy	copy	NOUN
cana-1534	217	22	of	of	ADP
cana-1534	217	23	k.	k.	PROPN
cana-1534	217	24	this	this	DET
cana-1534	217	25	set	set	NOUN
cana-1534	217	26	still	still	ADV
cana-1534	217	27	can	can	AUX
cana-1534	217	28	distinguish	distinguish	VERB
cana-1534	217	29	all	all	DET
cana-1534	217	30	vertices	vertex	NOUN
cana-1534	217	31	in	in	ADP
cana-1534	217	32	the	the	DET
cana-1534	217	33	graph	graph	NOUN
cana-1534	217	34	and	and	CCONJ
cana-1534	217	35	resolve	resolve	VERB
cana-1534	217	36	it	it	PRON
cana-1534	217	37	.	.	PUNCT
cana-1534	218	1	to	to	PART
cana-1534	218	2	see	see	VERB
cana-1534	218	3	this	this	DET
cana-1534	218	4	set	set	NOUN
cana-1534	218	5	is	be	AUX
cana-1534	218	6	minimal	minimal	ADJ
cana-1534	218	7	,	,	PUNCT
cana-1534	218	8	consider	consider	VERB
cana-1534	218	9	any	any	DET
cana-1534	218	10	other	other	ADJ
cana-1534	218	11	resolving	resolve	VERB
cana-1534	218	12	set	set	VERB
cana-1534	218	13	.	.	PUNCT
cana-1534	219	1	communications	communication	NOUN
cana-1534	219	2	on	on	ADP
cana-1534	219	3	applied	apply	VERB
cana-1534	219	4	nonlinear	nonlinear	ADJ
cana-1534	219	5	analysis	analysis	NOUN
cana-1534	219	6	issn	issn	NOUN
cana-1534	219	7	:	:	PUNCT
cana-1534	219	8	1074	1074	NUM
cana-1534	219	9	-	-	PUNCT
cana-1534	219	10	133x	133x	NUM
cana-1534	219	11	vol	vol	NOUN
cana-1534	219	12	31	31	NUM
cana-1534	219	13	no	no	NOUN
cana-1534	219	14	.	.	PUNCT
cana-1534	220	1	8s	8s	PROPN
cana-1534	220	2	(	(	PUNCT
cana-1534	220	3	2024	2024	NUM
cana-1534	220	4	)	)	PUNCT
cana-1534	220	5	431	431	NUM
cana-1534	220	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	220	7	if	if	SCONJ
cana-1534	220	8	it	it	PRON
cana-1534	220	9	contains	contain	VERB
cana-1534	220	10	too	too	ADV
cana-1534	220	11	few	few	ADJ
cana-1534	220	12	vertices	vertex	NOUN
cana-1534	220	13	from	from	ADP
cana-1534	220	14	each	each	DET
cana-1534	220	15	copy	copy	NOUN
cana-1534	220	16	of	of	ADP
cana-1534	220	17	k	k	PROPN
cana-1534	220	18	m	m	PROPN
cana-1534	220	19	,	,	PUNCT
cana-1534	220	20	else	else	ADV
cana-1534	220	21	it	it	PRON
cana-1534	220	22	wo	will	AUX
cana-1534	220	23	n't	not	PART
cana-1534	220	24	be	be	AUX
cana-1534	220	25	able	able	ADJ
cana-1534	220	26	to	to	PART
cana-1534	220	27	distinguish	distinguish	VERB
cana-1534	220	28	some	some	DET
cana-1534	220	29	pairs	pair	NOUN
cana-1534	220	30	of	of	ADP
cana-1534	220	31	vertices	vertex	NOUN
cana-1534	220	32	.	.	PUNCT
cana-1534	221	1	hence	hence	ADV
cana-1534	221	2	the	the	DET
cana-1534	221	3	set	set	NOUN
cana-1534	221	4	must	must	AUX
cana-1534	221	5	contain	contain	VERB
cana-1534	221	6	at	at	ADV
cana-1534	221	7	least	least	ADJ
cana-1534	221	8	(	(	PUNCT
cana-1534	221	9	m−1)nn×(m−1	m−1)nn×(m−1	PROPN
cana-1534	221	10	)	)	PUNCT
cana-1534	221	11	vertices	vertex	NOUN
cana-1534	221	12	,	,	PUNCT
cana-1534	221	13	which	which	PRON
cana-1534	221	14	shows	show	VERB
cana-1534	221	15	this	this	PRON
cana-1534	221	16	is	be	AUX
cana-1534	221	17	the	the	DET
cana-1534	221	18	minimum	minimum	ADJ
cana-1534	221	19	possible	possible	ADJ
cana-1534	221	20	size	size	NOUN
cana-1534	221	21	.	.	PUNCT
cana-1534	222	1	theorem	theorem	VERB
cana-1534	222	2	3.5	3.5	NUM
cana-1534	222	3	.	.	PUNCT
cana-1534	223	1	if	if	SCONJ
cana-1534	223	2	let	let	VERB
cana-1534	223	3	g	g	PRON
cana-1534	223	4	be	be	AUX
cana-1534	223	5	a	a	DET
cana-1534	223	6	connected	connected	ADJ
cana-1534	223	7	graph	graph	NOUN
cana-1534	223	8	with	with	ADP
cana-1534	223	9	at	at	ADV
cana-1534	223	10	least	least	ADJ
cana-1534	223	11	2	2	NUM
cana-1534	223	12	vertices	vertex	NOUN
cana-1534	223	13	,	,	PUNCT
cana-1534	223	14	and	and	CCONJ
cana-1534	223	15	combine	combine	VERB
cana-1534	223	16	it	it	PRON
cana-1534	223	17	with	with	ADP
cana-1534	223	18	multiple	multiple	ADJ
cana-1534	223	19	copies	copy	NOUN
cana-1534	223	20	of	of	ADP
cana-1534	223	21	a	a	DET
cana-1534	223	22	star	star	NOUN
cana-1534	223	23	graph	graph	NOUN
cana-1534	223	24	.	.	PUNCT
cana-1534	224	1	a	a	DET
cana-1534	224	2	star	star	NOUN
cana-1534	224	3	graph	graph	NOUN
cana-1534	224	4	stands	stand	VERB
cana-1534	224	5	for	for	ADP
cana-1534	224	6	a	a	DET
cana-1534	224	7	special	special	ADJ
cana-1534	224	8	kind	kind	NOUN
cana-1534	224	9	of	of	ADP
cana-1534	224	10	small	small	ADJ
cana-1534	224	11	graph	graph	NOUN
cana-1534	224	12	where	where	SCONJ
cana-1534	224	13	one	one	NUM
cana-1534	224	14	central	central	ADJ
cana-1534	224	15	vertex	vertex	NOUN
cana-1534	224	16	is	be	AUX
cana-1534	224	17	connected	connect	VERB
cana-1534	224	18	to	to	ADP
cana-1534	224	19	all	all	DET
cana-1534	224	20	others	other	NOUN
cana-1534	224	21	.	.	PUNCT
cana-1534	225	1	the	the	DET
cana-1534	225	2	minimum	minimum	ADJ
cana-1534	225	3	size	size	NOUN
cana-1534	225	4	of	of	ADP
cana-1534	225	5	a	a	DET
cana-1534	225	6	non	non	ADJ
cana-1534	225	7	-	-	ADJ
cana-1534	225	8	isolated	isolated	ADJ
cana-1534	225	9	resolving	resolving	NOUN
cana-1534	225	10	set	set	VERB
cana-1534	225	11	in	in	ADP
cana-1534	225	12	this	this	DET
cana-1534	225	13	new	new	ADJ
cana-1534	225	14	graph	graph	NOUN
cana-1534	225	15	would	would	AUX
cana-1534	225	16	be	be	AUX
cana-1534	225	17	equal	equal	ADJ
cana-1534	225	18	to	to	ADP
cana-1534	225	19	the	the	DET
cana-1534	225	20	total	total	ADJ
cana-1534	225	21	number	number	NOUN
cana-1534	225	22	of	of	ADP
cana-1534	225	23	vertices	vertex	NOUN
cana-1534	225	24	of	of	ADP
cana-1534	225	25	the	the	DET
cana-1534	225	26	combined	combine	VERB
cana-1534	225	27	graph	graph	NOUN
cana-1534	225	28	.	.	PUNCT
cana-1534	226	1	proof	proof	NOUN
cana-1534	226	2	.	.	PUNCT
cana-1534	227	1	in	in	ADP
cana-1534	227	2	such	such	DET
cana-1534	227	3	a	a	DET
cana-1534	227	4	graph	graph	NOUN
cana-1534	227	5	,	,	PUNCT
cana-1534	227	6	each	each	DET
cana-1534	227	7	vertex	vertex	NOUN
cana-1534	227	8	of	of	ADP
cana-1534	227	9	g	g	PROPN
cana-1534	227	10	is	be	AUX
cana-1534	227	11	connected	connect	VERB
cana-1534	227	12	to	to	ADP
cana-1534	227	13	the	the	DET
cana-1534	227	14	center	center	NOUN
cana-1534	227	15	of	of	ADP
cana-1534	227	16	a	a	DET
cana-1534	227	17	star	star	NOUN
cana-1534	227	18	graph	graph	NOUN
cana-1534	227	19	.	.	PUNCT
cana-1534	228	1	the	the	DET
cana-1534	228	2	resolving	resolving	NOUN
cana-1534	228	3	set	set	NOUN
cana-1534	228	4	should	should	AUX
cana-1534	228	5	contain	contain	VERB
cana-1534	228	6	enough	enough	ADJ
cana-1534	228	7	vertices	vertex	NOUN
cana-1534	228	8	from	from	ADP
cana-1534	228	9	each	each	DET
cana-1534	228	10	star	star	NOUN
cana-1534	228	11	so	so	SCONJ
cana-1534	228	12	that	that	SCONJ
cana-1534	228	13	all	all	DET
cana-1534	228	14	vertices	vertex	NOUN
cana-1534	228	15	are	be	AUX
cana-1534	228	16	distinguishable	distinguishable	ADJ
cana-1534	228	17	.	.	PUNCT
cana-1534	229	1	if	if	SCONJ
cana-1534	229	2	there	there	PRON
cana-1534	229	3	are	be	VERB
cana-1534	229	4	n't	not	PART
cana-1534	229	5	enough	enough	ADJ
cana-1534	229	6	vertices	vertex	NOUN
cana-1534	229	7	else	else	ADV
cana-1534	229	8	some	some	DET
cana-1534	229	9	vertices	vertex	NOUN
cana-1534	229	10	might	might	AUX
cana-1534	229	11	not	not	PART
cana-1534	229	12	be	be	AUX
cana-1534	229	13	distinguishable	distinguishable	ADJ
cana-1534	229	14	.	.	PUNCT
cana-1534	230	1	thus	thus	ADV
cana-1534	230	2	the	the	DET
cana-1534	230	3	set	set	NOUN
cana-1534	230	4	will	will	AUX
cana-1534	230	5	have	have	VERB
cana-1534	230	6	at	at	ADP
cana-1534	230	7	least	least	ADJ
cana-1534	230	8	the	the	DET
cana-1534	230	9	total	total	ADJ
cana-1534	230	10	number	number	NOUN
cana-1534	230	11	of	of	ADP
cana-1534	230	12	vertices	vertex	NOUN
cana-1534	230	13	in	in	ADP
cana-1534	230	14	the	the	DET
cana-1534	230	15	graph	graph	NOUN
cana-1534	230	16	confirming	confirm	VERB
cana-1534	230	17	that	that	SCONJ
cana-1534	230	18	this	this	PRON
cana-1534	230	19	is	be	AUX
cana-1534	230	20	the	the	DET
cana-1534	230	21	minimum	minimum	ADJ
cana-1534	230	22	size	size	NOUN
cana-1534	230	23	.	.	PUNCT
cana-1534	231	1	referances	referance	NOUN
cana-1534	231	2	[	[	X
cana-1534	231	3	1	1	X
cana-1534	231	4	]	]	X
cana-1534	231	5	selvam	selvam	PROPN
cana-1534	231	6	avadayappan	avadayappan	PROPN
cana-1534	231	7	,	,	PUNCT
cana-1534	231	8	m.	m.	NOUN
cana-1534	231	9	bhuvaneshwari	bhuvaneshwari	PROPN
cana-1534	231	10	,	,	PUNCT
cana-1534	231	11	p.	p.	NOUN
cana-1534	231	12	jeya	jeya	PROPN
cana-1534	231	13	bala	bala	PROPN
cana-1534	231	14	chitra	chitra	PROPN
cana-1534	231	15	,	,	PUNCT
cana-1534	231	16	more	more	ADJ
cana-1534	231	17	results	result	NOUN
cana-1534	231	18	on	on	ADP
cana-1534	231	19	non	non	ADJ
cana-1534	231	20	-	-	ADJ
cana-1534	231	21	isolated	isolated	ADJ
cana-1534	231	22	resolving	resolving	NOUN
cana-1534	231	23	number	number	NOUN
cana-1534	231	24	of	of	ADP
cana-1534	231	25	a	a	DET
cana-1534	231	26	graph	graph	NOUN
cana-1534	231	27	,	,	PUNCT
cana-1534	231	28	international	international	ADJ
cana-1534	231	29	journal	journal	NOUN
cana-1534	231	30	of	of	ADP
cana-1534	231	31	applied	apply	VERB
cana-1534	231	32	and	and	CCONJ
cana-1534	231	33	advanced	advanced	ADJ
cana-1534	231	34	scientific	scientific	ADJ
cana-1534	231	35	research	research	NOUN
cana-1534	231	36	,	,	PUNCT
cana-1534	231	37	special	special	ADJ
cana-1534	231	38	issue	issue	NOUN
cana-1534	231	39	(	(	PUNCT
cana-1534	231	40	2017	2017	NUM
cana-1534	231	41	)	)	PUNCT
cana-1534	231	42	49	49	NUM
cana-1534	231	43	-	-	SYM
cana-1534	231	44	53	53	NUM
cana-1534	231	45	.	.	PUNCT
cana-1534	232	1	[	[	X
cana-1534	232	2	2	2	NUM
cana-1534	232	3	]	]	PUNCT
cana-1534	232	4	selvam	selvam	PROPN
cana-1534	232	5	avadayappan	avadayappan	PROPN
cana-1534	232	6	,	,	PUNCT
cana-1534	232	7	m.	m.	NOUN
cana-1534	232	8	bhuvaneshwari	bhuvaneshwari	PROPN
cana-1534	232	9	,	,	PUNCT
cana-1534	232	10	p.	p.	NOUN
cana-1534	232	11	jeya	jeya	PROPN
cana-1534	232	12	bala	bala	PROPN
cana-1534	232	13	chitra	chitra	PROPN
cana-1534	232	14	,	,	PUNCT
cana-1534	232	15	non	non	ADJ
cana-1534	232	16	-	-	ADJ
cana-1534	232	17	isolated	isolated	ADJ
cana-1534	232	18	resolving	resolving	NOUN
cana-1534	232	19	number	number	NOUN
cana-1534	232	20	of	of	ADP
cana-1534	232	21	a	a	DET
cana-1534	232	22	graph	graph	NOUN
cana-1534	232	23	,	,	PUNCT
cana-1534	232	24	global	global	ADJ
cana-1534	232	25	journal	journal	NOUN
cana-1534	232	26	of	of	ADP
cana-1534	232	27	pure	pure	ADJ
cana-1534	232	28	and	and	CCONJ
cana-1534	232	29	applied	applied	ADJ
cana-1534	232	30	mathematics	mathematic	NOUN
cana-1534	232	31	,	,	PUNCT
cana-1534	232	32	vol.13	vol.13	PROPN
cana-1534	232	33	,	,	PUNCT
cana-1534	232	34	no.5	no.5	PROPN
cana-1534	232	35	(	(	PUNCT
cana-1534	232	36	2017	2017	NUM
cana-1534	232	37	)	)	PUNCT
cana-1534	232	38	336	336	NUM
cana-1534	232	39	-	-	SYM
cana-1534	232	40	346	346	NUM
cana-1534	232	41	.	.	PUNCT
cana-1534	233	1	[	[	X
cana-1534	233	2	3	3	NUM
cana-1534	233	3	]	]	X
cana-1534	233	4	selvam	selvam	PROPN
cana-1534	233	5	avadayappan	avadayappan	PROPN
cana-1534	233	6	,	,	PUNCT
cana-1534	233	7	m.	m.	NOUN
cana-1534	233	8	bhuvaneshwari	bhuvaneshwari	PROPN
cana-1534	233	9	,	,	PUNCT
cana-1534	233	10	p.	p.	NOUN
cana-1534	233	11	jeya	jeya	PROPN
cana-1534	233	12	bala	bala	PROPN
cana-1534	233	13	chitra	chitra	PROPN
cana-1534	233	14	,	,	PUNCT
cana-1534	233	15	non	non	ADJ
cana-1534	233	16	-	-	ADJ
cana-1534	233	17	isolated	isolated	ADJ
cana-1534	233	18	resolving	resolving	NOUN
cana-1534	233	19	number	number	NOUN
cana-1534	233	20	for	for	ADP
cana-1534	233	21	some	some	DET
cana-1534	233	22	splitting	splitting	NOUN
cana-1534	233	23	graphs	graph	NOUN
cana-1534	233	24	,	,	PUNCT
cana-1534	233	25	international	international	ADJ
cana-1534	233	26	j.	j.	PROPN
cana-1534	233	27	math	math	PROPN
cana-1534	233	28	.	.	PUNCT
cana-1534	234	1	combin	combin	NOUN
cana-1534	234	2	.	.	PUNCT
cana-1534	234	3	,	,	PUNCT
cana-1534	234	4	special	special	ADJ
cana-1534	234	5	issue	issue	NOUN
cana-1534	234	6	1	1	NUM
cana-1534	234	7	(	(	PUNCT
cana-1534	234	8	2018	2018	NUM
cana-1534	234	9	)	)	PUNCT
cana-1534	234	10	9	9	NUM
cana-1534	234	11	-	-	SYM
cana-1534	234	12	18	18	NUM
cana-1534	234	13	.	.	PUNCT
cana-1534	235	1	[	[	X
cana-1534	235	2	4	4	X
cana-1534	235	3	]	]	PUNCT
cana-1534	235	4	z.	z.	PROPN
cana-1534	235	5	beuliova	beuliova	PROPN
cana-1534	235	6	,	,	PUNCT
cana-1534	235	7	f.	f.	PROPN
cana-1534	235	8	eberhard	eberhard	PROPN
cana-1534	235	9	,	,	PUNCT
cana-1534	235	10	t.	t.	PROPN
cana-1534	235	11	erlebach	erlebach	PROPN
cana-1534	235	12	,	,	PUNCT
cana-1534	235	13	a.	a.	NOUN
cana-1534	235	14	hall	hall	PROPN
cana-1534	235	15	,	,	PUNCT
cana-1534	235	16	m.	m.	NOUN
cana-1534	235	17	hoffman	hoffman	PROPN
cana-1534	235	18	,	,	PUNCT
cana-1534	235	19	m.	m.	NOUN
cana-1534	235	20	mihalak	mihalak	PROPN
cana-1534	235	21	,	,	PUNCT
cana-1534	235	22	l.	l.	PROPN
cana-1534	235	23	ram	ram	PROPN
cana-1534	235	24	,	,	PUNCT
cana-1534	235	25	network	network	NOUN
cana-1534	235	26	discovery	discovery	NOUN
cana-1534	235	27	and	and	CCONJ
cana-1534	235	28	verification	verification	NOUN
cana-1534	235	29	,	,	PUNCT
cana-1534	235	30	ieee	ieee	PROPN
cana-1534	235	31	j.	j.	PROPN
cana-1534	235	32	sel	sel	PROPN
cana-1534	235	33	.	.	PUNCT
cana-1534	235	34	areas	area	NOUN
cana-1534	235	35	commun	commun	PROPN
cana-1534	235	36	.	.	PROPN
cana-1534	235	37	,	,	PUNCT
cana-1534	235	38	24	24	NUM
cana-1534	235	39	(	(	PUNCT
cana-1534	235	40	2006	2006	NUM
cana-1534	235	41	)	)	PUNCT
cana-1534	235	42	21682181	21682181	NUM
cana-1534	235	43	.	.	PUNCT
cana-1534	236	1	[	[	X
cana-1534	236	2	5	5	X
cana-1534	236	3	]	]	PUNCT
cana-1534	236	4	g.	g.	PROPN
cana-1534	236	5	chartrand	chartrand	PROPN
cana-1534	236	6	,	,	PUNCT
cana-1534	236	7	l.	l.	PROPN
cana-1534	236	8	eroh	eroh	PROPN
cana-1534	236	9	,	,	PUNCT
cana-1534	236	10	m.	m.	PROPN
cana-1534	236	11	johnson	johnson	PROPN
cana-1534	236	12	,	,	PUNCT
cana-1534	236	13	o.r	o.r	PROPN
cana-1534	236	14	.	.	PROPN
cana-1534	236	15	oellermann	oellermann	PROPN
cana-1534	236	16	,	,	PUNCT
cana-1534	236	17	resolvability	resolvability	NOUN
cana-1534	236	18	in	in	ADP
cana-1534	236	19	graphs	graph	NOUN
cana-1534	236	20	and	and	CCONJ
cana-1534	236	21	the	the	DET
cana-1534	236	22	metric	metric	ADJ
cana-1534	236	23	dimension	dimension	NOUN
cana-1534	236	24	of	of	ADP
cana-1534	236	25	a	a	DET
cana-1534	236	26	graph	graph	NOUN
cana-1534	236	27	,	,	PUNCT
cana-1534	236	28	discrete	discrete	ADJ
cana-1534	236	29	appl	appl	NOUN
cana-1534	236	30	.	.	PUNCT
cana-1534	236	31	math	math	PROPN
cana-1534	236	32	.	.	PUNCT
cana-1534	236	33	,	,	PUNCT
cana-1534	236	34	105	105	NUM
cana-1534	236	35	(	(	PUNCT
cana-1534	236	36	2000	2000	NUM
cana-1534	236	37	)	)	PUNCT
cana-1534	236	38	99	99	NUM
cana-1534	236	39	-	-	SYM
cana-1534	236	40	113	113	NUM
cana-1534	236	41	.	.	PUNCT
cana-1534	237	1	[	[	X
cana-1534	237	2	6	6	NUM
cana-1534	237	3	]	]	PUNCT
cana-1534	237	4	g.	g.	PROPN
cana-1534	237	5	chartrand	chartrand	PROPN
cana-1534	237	6	and	and	CCONJ
cana-1534	237	7	l.	l.	PROPN
cana-1534	237	8	lesniak	lesniak	PROPN
cana-1534	237	9	,	,	PUNCT
cana-1534	237	10	graphs	graph	NOUN
cana-1534	237	11	and	and	CCONJ
cana-1534	237	12	digraphs	digraph	NOUN
cana-1534	237	13	,	,	PUNCT
cana-1534	237	14	3rd	3rd	ADJ
cana-1534	237	15	ed	ed	NOUN
cana-1534	237	16	.	.	PROPN
cana-1534	237	17	,	,	PUNCT
cana-1534	237	18	chapman	chapman	PROPN
cana-1534	237	19	and	and	CCONJ
cana-1534	237	20	hall	hall	PROPN
cana-1534	237	21	,	,	PUNCT
cana-1534	237	22	london	london	PROPN
cana-1534	237	23	,	,	PUNCT
cana-1534	237	24	(	(	PUNCT
cana-1534	237	25	1996	1996	NUM
cana-1534	237	26	)	)	PUNCT
cana-1534	237	27	.	.	PUNCT
cana-1534	238	1	[	[	X
cana-1534	238	2	7	7	X
cana-1534	238	3	]	]	X
cana-1534	238	4	g.	g.	PROPN
cana-1534	238	5	chartrand	chartrand	PROPN
cana-1534	238	6	,	,	PUNCT
cana-1534	238	7	p.	p.	PROPN
cana-1534	238	8	zhang	zhang	PROPN
cana-1534	238	9	,	,	PUNCT
cana-1534	238	10	the	the	DET
cana-1534	238	11	theory	theory	NOUN
cana-1534	238	12	and	and	CCONJ
cana-1534	238	13	application	application	NOUN
cana-1534	238	14	of	of	ADP
cana-1534	238	15	resolvability	resolvability	NOUN
cana-1534	238	16	in	in	ADP
cana-1534	238	17	graphs	graph	NOUN
cana-1534	238	18	:	:	PUNCT
cana-1534	238	19	a	a	DET
cana-1534	238	20	survey	survey	NOUN
cana-1534	238	21	,	,	PUNCT
cana-1534	238	22	congr	congr	NOUN
cana-1534	238	23	.	.	PUNCT
cana-1534	239	1	numer	numer	PROPN
cana-1534	239	2	.	.	PROPN
cana-1534	240	1	,	,	PUNCT
cana-1534	240	2	160	160	NUM
cana-1534	240	3	(	(	PUNCT
cana-1534	240	4	2003	2003	NUM
cana-1534	240	5	)	)	PUNCT
cana-1534	240	6	47	47	NUM
cana-1534	240	7	-	-	SYM
cana-1534	240	8	68	68	NUM
cana-1534	240	9	.	.	PUNCT
cana-1534	241	1	[	[	X
cana-1534	241	2	8	8	NUM
cana-1534	241	3	]	]	X
cana-1534	241	4	v.	v.	ADP
cana-1534	241	5	chavatal	chavatal	ADJ
cana-1534	241	6	,	,	PUNCT
cana-1534	241	7	mastermind	mastermind	NOUN
cana-1534	241	8	,	,	PUNCT
cana-1534	241	9	combinatorica	combinatorica	PROPN
cana-1534	241	10	,	,	PUNCT
cana-1534	241	11	3	3	NUM
cana-1534	241	12	(	(	PUNCT
cana-1534	241	13	1983	1983	NUM
cana-1534	241	14	)	)	PUNCT
cana-1534	241	15	325	325	NUM
cana-1534	241	16	-	-	SYM
cana-1534	241	17	329	329	NUM
cana-1534	241	18	.	.	PUNCT
cana-1534	242	1	[	[	X
cana-1534	242	2	9	9	NUM
cana-1534	242	3	]	]	PUNCT
cana-1534	242	4	f.	f.	PROPN
cana-1534	242	5	harary	harary	PROPN
cana-1534	242	6	,	,	PUNCT
cana-1534	242	7	r.a	r.a	PROPN
cana-1534	242	8	.	.	NOUN
cana-1534	242	9	melter	melter	NOUN
cana-1534	242	10	,	,	PUNCT
cana-1534	242	11	on	on	ADP
cana-1534	242	12	the	the	DET
cana-1534	242	13	metric	metric	ADJ
cana-1534	242	14	dimension	dimension	NOUN
cana-1534	242	15	of	of	ADP
cana-1534	242	16	a	a	DET
cana-1534	242	17	graph	graph	NOUN
cana-1534	242	18	,	,	PUNCT
cana-1534	242	19	ars	ar	NOUN
cana-1534	242	20	.	.	PUNCT
cana-1534	243	1	combin	combin	NOUN
cana-1534	243	2	.	.	PUNCT
cana-1534	243	3	,	,	PUNCT
cana-1534	243	4	2	2	NUM
cana-1534	243	5	(	(	PUNCT
cana-1534	243	6	1976	1976	NUM
cana-1534	243	7	)	)	PUNCT
cana-1534	243	8	191	191	NUM
cana-1534	243	9	-	-	SYM
cana-1534	243	10	195	195	NUM
cana-1534	243	11	.	.	PUNCT
cana-1534	244	1	[	[	X
cana-1534	244	2	10	10	NUM
cana-1534	244	3	]	]	X
cana-1534	244	4	b.l	b.l	PROPN
cana-1534	244	5	hulme	hulme	PROPN
cana-1534	244	6	,	,	PUNCT
cana-1534	244	7	a.w	a.w	PROPN
cana-1534	244	8	shiver	shiver	NOUN
cana-1534	244	9	and	and	CCONJ
cana-1534	244	10	p.j	p.j	PROPN
cana-1534	244	11	.	.	PROPN
cana-1534	244	12	slater	slater	PROPN
cana-1534	244	13	,	,	PUNCT
cana-1534	244	14	fire	fire	NOUN
cana-1534	244	15	:	:	PUNCT
cana-1534	244	16	a	a	DET
cana-1534	244	17	subroutine	subroutine	NOUN
cana-1534	244	18	for	for	ADP
cana-1534	244	19	fire	fire	NOUN
cana-1534	244	20	protection	protection	NOUN
cana-1534	244	21	and	and	CCONJ
cana-1534	244	22	network	network	NOUN
cana-1534	244	23	analysis	analysis	NOUN
cana-1534	244	24	,	,	PUNCT
cana-1534	244	25	sand	sand	NOUN
cana-1534	244	26	811261	811261	NUM
cana-1534	244	27	,	,	PUNCT
cana-1534	244	28	sandia	sandia	PROPN
cana-1534	244	29	national	national	ADJ
cana-1534	244	30	laboratories	laboratory	NOUN
cana-1534	244	31	,	,	PUNCT
cana-1534	244	32	albuquerque	albuquerque	NOUN
cana-1534	244	33	,	,	PUNCT
cana-1534	244	34	nm(1981	nm(1981	PROPN
cana-1534	244	35	)	)	PUNCT
cana-1534	244	36	.	.	PUNCT
cana-1534	245	1	[	[	X
cana-1534	245	2	11	11	NUM
cana-1534	245	3	]	]	X
cana-1534	245	4	b.l	b.l	PROPN
cana-1534	245	5	hulme	hulme	PROPN
cana-1534	245	6	,	,	PUNCT
cana-1534	245	7	a.w	a.w	PROPN
cana-1534	245	8	shiver	shiver	NOUN
cana-1534	245	9	and	and	CCONJ
cana-1534	245	10	p.j	p.j	PROPN
cana-1534	245	11	.	.	PROPN
cana-1534	245	12	slater	slater	PROPN
cana-1534	245	13	,	,	PUNCT
cana-1534	245	14	computing	compute	VERB
cana-1534	245	15	minimum	minimum	NOUN
cana-1534	245	16	cast	cast	NOUN
cana-1534	245	17	fire	fire	NOUN
cana-1534	245	18	protection	protection	NOUN
cana-1534	245	19	,	,	PUNCT
cana-1534	245	20	sand	sand	VERB
cana-1534	245	21	82	82	NUM
cana-1534	245	22	-	-	PUNCT
cana-1534	245	23	0809	0809	NUM
cana-1534	245	24	,	,	PUNCT
cana-1534	245	25	sandia	sandia	PROPN
cana-1534	245	26	national	national	ADJ
cana-1534	245	27	laboratories	laboratory	NOUN
cana-1534	245	28	,	,	PUNCT
cana-1534	245	29	albuquerque	albuquerque	NOUN
cana-1534	245	30	,	,	PUNCT
cana-1534	245	31	nm(1982	nm(1982	NUM
cana-1534	245	32	)	)	PUNCT
cana-1534	245	33	.	.	PUNCT
cana-1534	246	1	[	[	X
cana-1534	246	2	12	12	NUM
cana-1534	246	3	]	]	X
cana-1534	246	4	b.l	b.l	PROPN
cana-1534	246	5	hulme	hulme	PROPN
cana-1534	246	6	,	,	PUNCT
cana-1534	246	7	a.w	a.w	PROPN
cana-1534	246	8	shiver	shiver	NOUN
cana-1534	246	9	and	and	CCONJ
cana-1534	246	10	p.j	p.j	PROPN
cana-1534	246	11	.	.	PROPN
cana-1534	246	12	slater	slater	PROPN
cana-1534	246	13	,	,	PUNCT
cana-1534	246	14	a	a	DET
cana-1534	246	15	boolean	boolean	ADJ
cana-1534	246	16	algebraic	algebraic	ADJ
cana-1534	246	17	analysis	analysis	NOUN
cana-1534	246	18	of	of	ADP
cana-1534	246	19	fire	fire	NOUN
cana-1534	246	20	protection	protection	NOUN
cana-1534	246	21	,	,	PUNCT
cana-1534	246	22	annals	annal	NOUN
cana-1534	246	23	of	of	ADP
cana-1534	246	24	discrete	discrete	ADJ
cana-1534	246	25	mathematics	mathematic	NOUN
cana-1534	246	26	,	,	PUNCT
cana-1534	246	27	algebraic	algebraic	ADJ
cana-1534	246	28	structure	structure	NOUN
cana-1534	246	29	in	in	ADP
cana-1534	246	30	operations	operation	NOUN
cana-1534	246	31	research	research	NOUN
cana-1534	246	32	(	(	PUNCT
cana-1534	246	33	1984	1984	NUM
cana-1534	246	34	)	)	PUNCT
cana-1534	246	35	215	215	NUM
cana-1534	246	36	-	-	SYM
cana-1534	246	37	228	228	NUM
cana-1534	246	38	.	.	PUNCT
cana-1534	247	1	[	[	X
cana-1534	247	2	13	13	NUM
cana-1534	247	3	]	]	PUNCT
cana-1534	247	4	p.	p.	NOUN
cana-1534	247	5	jeya	jeya	PROPN
cana-1534	247	6	bala	bala	PROPN
cana-1534	247	7	chitra	chitra	PROPN
cana-1534	247	8	,	,	PUNCT
cana-1534	247	9	s.	s.	PROPN
cana-1534	247	10	arumugam	arumugam	PROPN
cana-1534	247	11	,	,	PUNCT
cana-1534	247	12	resolving	resolve	VERB
cana-1534	247	13	sets	set	NOUN
cana-1534	247	14	without	without	ADP
cana-1534	247	15	isolated	isolated	ADJ
cana-1534	247	16	vertices	vertex	NOUN
cana-1534	247	17	,	,	PUNCT
cana-1534	247	18	procedia	procedia	NOUN
cana-1534	247	19	computer	computer	NOUN
cana-1534	247	20	science	science	NOUN
cana-1534	247	21	,	,	PUNCT
cana-1534	247	22	74	74	NUM
cana-1534	247	23	(	(	PUNCT
cana-1534	247	24	2015	2015	NUM
cana-1534	247	25	)	)	PUNCT
cana-1534	247	26	38	38	NUM
cana-1534	247	27	-	-	SYM
cana-1534	247	28	42	42	NUM
cana-1534	247	29	.	.	PUNCT
cana-1534	248	1	[	[	X
cana-1534	248	2	14	14	NUM
cana-1534	248	3	]	]	X
cana-1534	248	4	s.	s.	PROPN
cana-1534	248	5	khuller	khuller	PROPN
cana-1534	248	6	,	,	PUNCT
cana-1534	248	7	b.	b.	PROPN
cana-1534	248	8	raghavachari	raghavachari	PROPN
cana-1534	248	9	,	,	PUNCT
cana-1534	248	10	a.	a.	NOUN
cana-1534	248	11	rosenfield	rosenfield	PROPN
cana-1534	248	12	,	,	PUNCT
cana-1534	248	13	landmarks	landmark	NOUN
cana-1534	248	14	in	in	ADP
cana-1534	248	15	graphs	graph	NOUN
cana-1534	248	16	,	,	PUNCT
cana-1534	248	17	discrete	discrete	ADJ
cana-1534	248	18	appl	appl	NOUN
cana-1534	248	19	.	.	PUNCT
cana-1534	248	20	math	math	PROPN
cana-1534	248	21	.	.	PUNCT
cana-1534	249	1	,	,	PUNCT
cana-1534	249	2	70	70	NUM
cana-1534	249	3	(	(	PUNCT
cana-1534	249	4	1996	1996	NUM
cana-1534	249	5	)	)	PUNCT
cana-1534	249	6	217	217	NUM
cana-1534	249	7	-	-	SYM
cana-1534	249	8	229	229	NUM
cana-1534	249	9	.	.	PUNCT
cana-1534	250	1	communications	communication	NOUN
cana-1534	250	2	on	on	ADP
cana-1534	250	3	applied	apply	VERB
cana-1534	250	4	nonlinear	nonlinear	ADJ
cana-1534	250	5	analysis	analysis	NOUN
cana-1534	250	6	issn	issn	NOUN
cana-1534	250	7	:	:	PUNCT
cana-1534	250	8	1074	1074	NUM
cana-1534	250	9	-	-	PUNCT
cana-1534	250	10	133x	133x	NUM
cana-1534	250	11	vol	vol	NOUN
cana-1534	250	12	31	31	NUM
cana-1534	250	13	no	no	NOUN
cana-1534	250	14	.	.	PUNCT
cana-1534	251	1	8s	8s	PROPN
cana-1534	251	2	(	(	PUNCT
cana-1534	251	3	2024	2024	NUM
cana-1534	251	4	)	)	PUNCT
cana-1534	251	5	432	432	NUM
cana-1534	251	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1534	252	1	[	[	X
cana-1534	252	2	15	15	X
cana-1534	252	3	]	]	X
cana-1534	252	4	v.	v.	ADP
cana-1534	252	5	saenpholphat	saenpholphat	PROPN
cana-1534	252	6	and	and	CCONJ
cana-1534	252	7	ping	ping	PROPN
cana-1534	252	8	zhang	zhang	PROPN
cana-1534	252	9	,	,	PUNCT
cana-1534	252	10	connected	connected	ADJ
cana-1534	252	11	resolvability	resolvability	NOUN
cana-1534	252	12	of	of	ADP
cana-1534	252	13	graphs	graph	NOUN
cana-1534	252	14	,	,	PUNCT
cana-1534	252	15	czechoslovak	czechoslovak	ADJ
cana-1534	252	16	mathematical	mathematical	ADJ
cana-1534	252	17	journal	journal	NOUN
cana-1534	252	18	,	,	PUNCT
cana-1534	252	19	53	53	NUM
cana-1534	252	20	(	(	PUNCT
cana-1534	252	21	2003	2003	NUM
cana-1534	252	22	)	)	PUNCT
cana-1534	252	23	,	,	PUNCT
cana-1534	252	24	827	827	NUM
cana-1534	252	25	-	-	SYM
cana-1534	252	26	840	840	NUM
cana-1534	252	27	.	.	PUNCT
cana-1534	253	1	[	[	X
cana-1534	253	2	16	16	NUM
cana-1534	253	3	]	]	PUNCT
cana-1534	253	4	a.	a.	NOUN
cana-1534	253	5	sebo	sebo	NOUN
cana-1534	253	6	,	,	PUNCT
cana-1534	253	7	e.	e.	PROPN
cana-1534	253	8	tannier	tannier	PROPN
cana-1534	253	9	,	,	PUNCT
cana-1534	253	10	on	on	ADP
cana-1534	253	11	metric	metric	ADJ
cana-1534	253	12	generators	generator	NOUN
cana-1534	253	13	of	of	ADP
cana-1534	253	14	graphs	graph	NOUN
cana-1534	253	15	,	,	PUNCT
cana-1534	253	16	math	math	NOUN
cana-1534	253	17	.	.	PUNCT
cana-1534	254	1	oper	oper	PROPN
cana-1534	254	2	.	.	PUNCT
cana-1534	254	3	res	re	NOUN
cana-1534	254	4	,	,	PUNCT
cana-1534	254	5	29	29	NUM
cana-1534	254	6	(	(	PUNCT
cana-1534	254	7	2004	2004	NUM
cana-1534	254	8	)	)	PUNCT
cana-1534	254	9	383	383	NUM
cana-1534	254	10	-	-	SYM
cana-1534	254	11	393	393	NUM
cana-1534	254	12	.	.	PUNCT
cana-1534	255	1	[	[	X
cana-1534	255	2	17	17	NUM
cana-1534	255	3	]	]	SYM
cana-1534	255	4	shahida	shahida	NOUN
cana-1534	255	5	.	.	PUNCT
cana-1534	255	6	a.	a.	PROPN
cana-1534	255	7	t	t	PROPN
cana-1534	255	8	,	,	PUNCT
cana-1534	255	9	m.	m.	NOUN
cana-1534	255	10	s.	s.	PROPN
cana-1534	255	11	sunitha	sunitha	PROPN
cana-1534	255	12	,	,	PUNCT
cana-1534	255	13	on	on	ADP
cana-1534	255	14	the	the	DET
cana-1534	255	15	metric	metric	ADJ
cana-1534	255	16	dimension	dimension	NOUN
cana-1534	255	17	of	of	ADP
cana-1534	255	18	joins	join	NOUN
cana-1534	255	19	of	of	ADP
cana-1534	255	20	two	two	NUM
cana-1534	255	21	graphs	graph	NOUN
cana-1534	255	22	,	,	PUNCT
cana-1534	255	23	international	international	ADJ
cana-1534	255	24	journal	journal	NOUN
cana-1534	255	25	of	of	ADP
cana-1534	255	26	scientific	scientific	ADJ
cana-1534	255	27	and	and	CCONJ
cana-1534	255	28	research	research	NOUN
cana-1534	255	29	,	,	PUNCT
cana-1534	255	30	vol	vol	NOUN
cana-1534	255	31	.	.	PROPN
cana-1534	255	32	5	5	NUM
cana-1534	255	33	,	,	PUNCT
cana-1534	255	34	9	9	NUM
cana-1534	255	35	(	(	PUNCT
cana-1534	255	36	2014	2014	NUM
cana-1534	255	37	)	)	PUNCT
cana-1534	255	38	2229	2229	NUM
cana-1534	255	39	-	-	SYM
cana-1534	255	40	5518	5518	NUM
cana-1534	255	41	.	.	PUNCT
cana-1534	256	1	[	[	X
cana-1534	256	2	18	18	NUM
cana-1534	256	3	]	]	X
cana-1534	256	4	p.j	p.j	PROPN
cana-1534	256	5	.	.	PROPN
cana-1534	256	6	slater	slater	PROPN
cana-1534	256	7	,	,	PUNCT
cana-1534	256	8	leaves	leave	NOUN
cana-1534	256	9	of	of	ADP
cana-1534	256	10	trees	tree	NOUN
cana-1534	256	11	,	,	PUNCT
cana-1534	256	12	congr	congr	NOUN
cana-1534	256	13	.	.	PUNCT
cana-1534	257	1	numer	numer	PROPN
cana-1534	257	2	.	.	PROPN
cana-1534	258	1	,	,	PUNCT
cana-1534	258	2	14	14	NUM
cana-1534	258	3	(	(	PUNCT
cana-1534	258	4	1975	1975	NUM
cana-1534	258	5	)	)	PUNCT
cana-1534	258	6	549	549	NUM
cana-1534	258	7	-	-	SYM
cana-1534	258	8	559	559	NUM
cana-1534	258	9	.	.	PUNCT
cana-1534	259	1	[	[	X
cana-1534	259	2	19	19	NUM
cana-1534	259	3	]	]	X
cana-1534	259	4	p.j	p.j	PROPN
cana-1534	259	5	.	.	PROPN
cana-1534	259	6	slater	slater	PROPN
cana-1534	259	7	:	:	PUNCT
cana-1534	259	8	dominating	dominating	NOUN
cana-1534	259	9	and	and	CCONJ
cana-1534	259	10	reference	reference	NOUN
cana-1534	259	11	sets	set	NOUN
cana-1534	259	12	in	in	ADP
cana-1534	259	13	graphs	graph	NOUN
cana-1534	259	14	,	,	PUNCT
cana-1534	259	15	j.	j.	PROPN
cana-1534	259	16	math	math	PROPN
cana-1534	259	17	.	.	PUNCT
cana-1534	260	1	phys	phy	NOUN
cana-1534	260	2	.	.	PUNCT
cana-1534	261	1	sci	sci	PROPN
cana-1534	261	2	.	.	PROPN
cana-1534	261	3	,	,	PUNCT
cana-1534	261	4	22	22	NUM
cana-1534	261	5	(	(	PUNCT
cana-1534	261	6	1988	1988	NUM
cana-1534	261	7	)	)	PUNCT
cana-1534	261	8	445	445	NUM
cana-1534	261	9	-	-	SYM
cana-1534	261	10	455	455	NUM
cana-1534	261	11	.	.	PUNCT