id	sid	tid	token	lemma	pos
cana-3685	1	1	communications	communication	NOUN
cana-3685	1	2	on	on	ADP
cana-3685	1	3	applied	apply	VERB
cana-3685	1	4	nonlinear	nonlinear	ADJ
cana-3685	1	5	analysis	analysis	NOUN
cana-3685	1	6	issn	issn	NOUN
cana-3685	1	7	:	:	PUNCT
cana-3685	1	8	1074	1074	NUM
cana-3685	1	9	-	-	PUNCT
cana-3685	1	10	133x	133x	NUM
cana-3685	1	11	vol	vol	NOUN
cana-3685	1	12	32	32	NUM
cana-3685	1	13	no	no	NOUN
cana-3685	1	14	.	.	PUNCT
cana-3685	2	1	8s	8s	PROPN
cana-3685	2	2	(	(	PUNCT
cana-3685	2	3	2025	2025	NUM
cana-3685	2	4	)	)	PUNCT
cana-3685	2	5	414	414	NUM
cana-3685	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3685	2	7	binomial	binomial	ADJ
cana-3685	2	8	labeling	labeling	NOUN
cana-3685	2	9	of	of	ADP
cana-3685	2	10	intersection	intersection	NOUN
cana-3685	2	11	mirror	mirror	NOUN
cana-3685	2	12	lagoon	lagoon	NOUN
cana-3685	2	13	step	step	NOUN
cana-3685	2	14	graph	graph	NOUN
cana-3685	2	15	and	and	CCONJ
cana-3685	2	16	intersection	intersection	NOUN
cana-3685	2	17	opposite	opposite	ADJ
cana-3685	2	18	lagoon	lagoon	NOUN
cana-3685	2	19	step	step	NOUN
cana-3685	2	20	graph	graph	NOUN
cana-3685	2	21	with	with	ADP
cana-3685	2	22	python	python	NOUN
cana-3685	2	23	implementation	implementation	NOUN
cana-3685	2	24	1j	1j	NUM
cana-3685	2	25	.	.	PUNCT
cana-3685	3	1	vasanthi	vasanthi	NOUN
cana-3685	3	2	,	,	PUNCT
cana-3685	3	3	2n	2n	NUM
cana-3685	3	4	.	.	PUNCT
cana-3685	4	1	ramya	ramya	PROPN
cana-3685	4	2	1research	1research	NUM
cana-3685	4	3	scholar	scholar	NOUN
cana-3685	4	4	,	,	PUNCT
cana-3685	4	5	department	department	NOUN
cana-3685	4	6	of	of	ADP
cana-3685	4	7	mathematics	mathematics	PROPN
cana-3685	4	8	,	,	PUNCT
cana-3685	4	9	bharath	bharath	PROPN
cana-3685	4	10	institute	institute	PROPN
cana-3685	4	11	of	of	ADP
cana-3685	4	12	higher	high	ADJ
cana-3685	4	13	education	education	NOUN
cana-3685	4	14	&	&	CCONJ
cana-3685	4	15	research	research	PROPN
cana-3685	4	16	,	,	PUNCT
cana-3685	4	17	chennai	chennai	PROPN
cana-3685	4	18	,	,	PUNCT
cana-3685	4	19	tamil	tamil	PROPN
cana-3685	4	20	nadu	nadu	PROPN
cana-3685	4	21	,	,	PUNCT
cana-3685	4	22	india	india	PROPN
cana-3685	4	23	.	.	PUNCT
cana-3685	5	1	2professor	2professor	NUM
cana-3685	5	2	,	,	PUNCT
cana-3685	5	3	department	department	NOUN
cana-3685	5	4	of	of	ADP
cana-3685	5	5	mathematics	mathematics	PROPN
cana-3685	5	6	,	,	PUNCT
cana-3685	5	7	bharath	bharath	PROPN
cana-3685	5	8	institute	institute	PROPN
cana-3685	5	9	of	of	ADP
cana-3685	5	10	higher	high	ADJ
cana-3685	5	11	education	education	NOUN
cana-3685	5	12	&	&	CCONJ
cana-3685	5	13	research	research	PROPN
cana-3685	5	14	,	,	PUNCT
cana-3685	5	15	chennai	chennai	PROPN
cana-3685	5	16	,	,	PUNCT
cana-3685	5	17	tamil	tamil	PROPN
cana-3685	5	18	nadu	nadu	PROPN
cana-3685	5	19	,	,	PUNCT
cana-3685	5	20	india	india	PROPN
cana-3685	5	21	.	.	PUNCT
cana-3685	6	1	article	article	PROPN
cana-3685	6	2	history	history	NOUN
cana-3685	6	3	:	:	PUNCT
cana-3685	6	4	received	receive	VERB
cana-3685	6	5	:	:	PUNCT
cana-3685	6	6	27	27	NUM
cana-3685	6	7	-	-	SYM
cana-3685	6	8	10	10	NUM
cana-3685	6	9	-	-	PUNCT
cana-3685	6	10	2024	2024	NUM
cana-3685	6	11	revised:01	revised:01	VERB
cana-3685	6	12	-	-	PUNCT
cana-3685	6	13	12	12	NUM
cana-3685	6	14	-	-	PUNCT
cana-3685	6	15	2024	2024	NUM
cana-3685	6	16	accepted:28	accepted:28	NUM
cana-3685	6	17	-	-	PUNCT
cana-3685	6	18	12	12	NUM
cana-3685	6	19	-	-	PUNCT
cana-3685	6	20	2024	2024	NUM
cana-3685	6	21	abstract	abstract	NOUN
cana-3685	6	22	:	:	PUNCT
cana-3685	6	23	assume	assume	VERB
cana-3685	6	24	g	g	PROPN
cana-3685	6	25	=	=	SYM
cana-3685	6	26	{	{	PUNCT
cana-3685	6	27	v	v	NOUN
cana-3685	6	28	,	,	PUNCT
cana-3685	6	29	e	e	NOUN
cana-3685	6	30	}	}	PUNCT
cana-3685	6	31	be	be	AUX
cana-3685	6	32	the	the	DET
cana-3685	6	33	graph	graph	NOUN
cana-3685	6	34	with	with	ADP
cana-3685	6	35	n	n	ADP
cana-3685	6	36	vertices	vertex	NOUN
cana-3685	6	37	and	and	CCONJ
cana-3685	6	38	e	e	NOUN
cana-3685	6	39	edges	edge	NOUN
cana-3685	6	40	.	.	PUNCT
cana-3685	7	1	the	the	DET
cana-3685	7	2	binomial	binomial	ADJ
cana-3685	7	3	labeling	labeling	NOUN
cana-3685	7	4	classification	classification	NOUN
cana-3685	7	5	has	have	AUX
cana-3685	7	6	been	be	AUX
cana-3685	7	7	used	use	VERB
cana-3685	7	8	for	for	ADP
cana-3685	7	9	intersection	intersection	NOUN
cana-3685	7	10	mirror	mirror	NOUN
cana-3685	7	11	lagoon	lagoon	NOUN
cana-3685	7	12	step	step	NOUN
cana-3685	7	13	graph	graph	NOUN
cana-3685	7	14	i	i	PRON
cana-3685	7	15	m	m	VERB
cana-3685	7	16	lnsm	lnsm	NOUN
cana-3685	7	17	and	and	CCONJ
cana-3685	7	18	intersection	intersection	NOUN
cana-3685	7	19	opposite	opposite	ADJ
cana-3685	7	20	lagoon	lagoon	NOUN
cana-3685	7	21	step	step	NOUN
cana-3685	7	22	graph	graph	NOUN
cana-3685	7	23	io	io	X
cana-3685	7	24	ln	ln	ADV
cana-3685	7	25	sm	sm	INTJ
cana-3685	7	26	in	in	ADP
cana-3685	7	27	this	this	DET
cana-3685	7	28	paper	paper	NOUN
cana-3685	7	29	.	.	PUNCT
cana-3685	8	1	the	the	DET
cana-3685	8	2	concepts	concept	NOUN
cana-3685	8	3	stated	state	VERB
cana-3685	8	4	below	below	ADV
cana-3685	8	5	in	in	ADP
cana-3685	8	6	the	the	DET
cana-3685	8	7	paper	paper	NOUN
cana-3685	8	8	have	have	AUX
cana-3685	8	9	been	be	AUX
cana-3685	8	10	implemented	implement	VERB
cana-3685	8	11	using	use	VERB
cana-3685	8	12	python	python	NOUN
cana-3685	8	13	coding	coding	NOUN
cana-3685	8	14	.	.	PUNCT
cana-3685	9	1	keywords	keyword	NOUN
cana-3685	9	2	:	:	PUNCT
cana-3685	9	3	intersection	intersection	NOUN
cana-3685	9	4	mirror	mirror	NOUN
cana-3685	9	5	lagoon	lagoon	NOUN
cana-3685	9	6	step	step	NOUN
cana-3685	9	7	graph	graph	NOUN
cana-3685	9	8	,	,	PUNCT
cana-3685	9	9	intersection	intersection	NOUN
cana-3685	9	10	opposite	opposite	ADJ
cana-3685	9	11	lagoon	lagoon	NOUN
cana-3685	9	12	step	step	NOUN
cana-3685	9	13	graph	graph	NOUN
cana-3685	9	14	,	,	PUNCT
cana-3685	9	15	binomial	binomial	ADJ
cana-3685	9	16	labeling	labeling	NOUN
cana-3685	9	17	.	.	PUNCT
cana-3685	10	1	1	1	X
cana-3685	10	2	.	.	X
cana-3685	10	3	introduction	introduction	NOUN
cana-3685	10	4	rosa	rosa	PROPN
cana-3685	10	5	[	[	X
cana-3685	10	6	1	1	NUM
cana-3685	10	7	]	]	PUNCT
cana-3685	10	8	first	first	ADV
cana-3685	10	9	proposed	propose	VERB
cana-3685	10	10	the	the	DET
cana-3685	10	11	idea	idea	NOUN
cana-3685	10	12	of	of	ADP
cana-3685	10	13	graph	graph	NOUN
cana-3685	10	14	labeling	labeling	NOUN
cana-3685	10	15	in	in	ADP
cana-3685	10	16	1967	1967	NUM
cana-3685	10	17	.	.	PUNCT
cana-3685	11	1	gallian	gallian	ADJ
cana-3685	12	1	[	[	X
cana-3685	12	2	2	2	X
cana-3685	12	3	]	]	PUNCT
cana-3685	12	4	updates	update	VERB
cana-3685	12	5	a	a	DET
cana-3685	12	6	dynamic	dynamic	ADJ
cana-3685	12	7	overview	overview	NOUN
cana-3685	12	8	of	of	ADP
cana-3685	12	9	graph	graph	NOUN
cana-3685	12	10	labeling	labeling	NOUN
cana-3685	12	11	techniques	technique	NOUN
cana-3685	12	12	on	on	ADP
cana-3685	12	13	a	a	DET
cana-3685	12	14	regular	regular	ADJ
cana-3685	12	15	basis	basis	NOUN
cana-3685	12	16	,	,	PUNCT
cana-3685	12	17	and	and	CCONJ
cana-3685	12	18	the	the	DET
cana-3685	12	19	electronic	electronic	ADJ
cana-3685	12	20	journal	journal	NOUN
cana-3685	12	21	of	of	ADP
cana-3685	12	22	combinatory	combinatory	NOUN
cana-3685	12	23	publishes	publish	VERB
cana-3685	12	24	it	it	PRON
cana-3685	12	25	.	.	PUNCT
cana-3685	13	1	we	we	PRON
cana-3685	13	2	adhere	adhere	VERB
cana-3685	13	3	to	to	ADP
cana-3685	13	4	bondy	bondy	PROPN
cana-3685	13	5	and	and	CCONJ
cana-3685	13	6	murthy	murthy	PROPN
cana-3685	13	7	's	's	PART
cana-3685	13	8	[	[	X
cana-3685	13	9	3	3	NUM
cana-3685	13	10	]	]	X
cana-3685	13	11	notation	notation	NOUN
cana-3685	13	12	and	and	CCONJ
cana-3685	13	13	nomenclature	nomenclature	NOUN
cana-3685	13	14	for	for	ADP
cana-3685	13	15	several	several	ADJ
cana-3685	13	16	aspects	aspect	NOUN
cana-3685	13	17	of	of	ADP
cana-3685	13	18	graph	graph	NOUN
cana-3685	13	19	theory	theory	NOUN
cana-3685	13	20	.	.	PUNCT
cana-3685	14	1	binomial	binomial	ADJ
cana-3685	14	2	labeling	labeling	NOUN
cana-3685	14	3	introduced	introduce	VERB
cana-3685	14	4	by	by	ADP
cana-3685	14	5	chandrakala	chandrakala	PROPN
cana-3685	14	6	and	and	CCONJ
cana-3685	14	7	sekar	sekar	PROPN
cana-3685	14	8	[	[	X
cana-3685	14	9	4	4	NUM
cana-3685	14	10	]	]	X
cana-3685	14	11	et.al	et.al	NOUN
cana-3685	14	12	.	.	PUNCT
cana-3685	15	1	the	the	DET
cana-3685	15	2	binomial	binomial	ADJ
cana-3685	15	3	labeling	labeling	NOUN
cana-3685	15	4	for	for	ADP
cana-3685	15	5	intersection	intersection	NOUN
cana-3685	15	6	mirror	mirror	NOUN
cana-3685	15	7	lagoon	lagoon	NOUN
cana-3685	15	8	step	step	NOUN
cana-3685	15	9	graph	graph	NOUN
cana-3685	15	10	i	i	PRON
cana-3685	15	11	m	m	VERB
cana-3685	15	12	ln	ln	ADJ
cana-3685	15	13	sm	sm	NOUN
cana-3685	15	14	and	and	CCONJ
cana-3685	15	15	the	the	DET
cana-3685	15	16	intersection	intersection	NOUN
cana-3685	15	17	opposite	opposite	ADJ
cana-3685	15	18	lagoon	lagoon	NOUN
cana-3685	15	19	step	step	NOUN
cana-3685	15	20	graph	graph	NOUN
cana-3685	16	1	io	io	PROPN
cana-3685	16	2	ln	ln	NOUN
cana-3685	16	3	sm	sm	PROPN
cana-3685	16	4	are	be	AUX
cana-3685	16	5	constructed	construct	VERB
cana-3685	16	6	here	here	ADV
cana-3685	16	7	using	use	VERB
cana-3685	16	8	python	python	PROPN
cana-3685	16	9	code	code	NOUN
cana-3685	17	1	[	[	X
cana-3685	17	2	5][6	5][6	NUM
cana-3685	17	3	]	]	X
cana-3685	17	4	.	.	PUNCT
cana-3685	18	1	2	2	X
cana-3685	18	2	.	.	X
cana-3685	18	3	preliminaries	preliminary	NOUN
cana-3685	18	4	definition	definition	NOUN
cana-3685	18	5	1	1	NUM
cana-3685	18	6	:	:	PUNCT
cana-3685	18	7	a	a	DET
cana-3685	18	8	lagoon	lagoon	NOUN
cana-3685	18	9	step	step	NOUN
cana-3685	18	10	graph	graph	NOUN
cana-3685	18	11	ln	ln	ADV
cana-3685	18	12	sm	sm	PROPN
cana-3685	18	13	is	be	AUX
cana-3685	18	14	obtained	obtain	VERB
cana-3685	18	15	from	from	ADP
cana-3685	18	16	lagoon	lagoon	NOUN
cana-3685	18	17	ln	ln	NOUN
cana-3685	18	18	(	(	PUNCT
cana-3685	18	19	l	l	NOUN
cana-3685	18	20	shape	shape	NOUN
cana-3685	18	21	)	)	PUNCT
cana-3685	18	22	graph	graph	NOUN
cana-3685	18	23	with	with	ADP
cana-3685	18	24	vertices	vertex	NOUN
cana-3685	18	25	n	n	CCONJ
cana-3685	18	26	(	(	PUNCT
cana-3685	18	27	n≥3	n≥3	NOUN
cana-3685	18	28	,	,	PUNCT
cana-3685	18	29	must	must	AUX
cana-3685	18	30	be	be	AUX
cana-3685	18	31	an	an	DET
cana-3685	18	32	odd	odd	ADJ
cana-3685	18	33	)	)	PUNCT
cana-3685	18	34	joining	join	VERB
cana-3685	18	35	with	with	ADP
cana-3685	18	36	step	step	NOUN
cana-3685	18	37	graph	graph	NOUN
cana-3685	18	38	sm	sm	INTJ
cana-3685	18	39	(	(	PUNCT
cana-3685	18	40	m≥1,m	m≥1,m	PROPN
cana-3685	18	41	is	be	AUX
cana-3685	18	42	number	number	NOUN
cana-3685	18	43	of	of	ADP
cana-3685	18	44	steps	step	NOUN
cana-3685	18	45	with	with	ADP
cana-3685	18	46	m	m	PROPN
cana-3685	18	47	=(	=(	NOUN
cana-3685	18	48	n-1)/2	n-1)/2	PROPN
cana-3685	18	49	,	,	PUNCT
cana-3685	18	50	the	the	DET
cana-3685	18	51	graph	graph	NOUN
cana-3685	18	52	consists	consist	VERB
cana-3685	18	53	of	of	ADP
cana-3685	18	54	n+2m-1	n+2m-1	NOUN
cana-3685	18	55	vertices	vertex	NOUN
cana-3685	18	56	and	and	CCONJ
cana-3685	18	57	n+2m-1	n+2m-1	NOUN
cana-3685	18	58	edges	edge	NOUN
cana-3685	18	59	and	and	CCONJ
cana-3685	18	60	is	be	AUX
cana-3685	18	61	described	describe	VERB
cana-3685	18	62	below	below	ADP
cana-3685	18	63	figure	figure	NOUN
cana-3685	18	64	1	1	NUM
cana-3685	18	65	:	:	PUNCT
cana-3685	18	66	l5s2	l5s2	PROPN
cana-3685	18	67	&	&	CCONJ
cana-3685	18	68	l7s3	l7s3	ADJ
cana-3685	18	69	definition	definition	NOUN
cana-3685	18	70	2	2	NUM
cana-3685	18	71	:	:	PUNCT
cana-3685	18	72	intersection	intersection	NOUN
cana-3685	18	73	mirror	mirror	NOUN
cana-3685	18	74	lagoon	lagoon	NOUN
cana-3685	18	75	step	step	NOUN
cana-3685	18	76	graph	graph	NOUN
cana-3685	19	1	i	i	PRON
cana-3685	19	2	m	m	VERB
cana-3685	19	3	ln	ln	ADJ
cana-3685	19	4	sm	sm	PROPN
cana-3685	19	5	obtained	obtain	VERB
cana-3685	19	6	by	by	ADP
cana-3685	19	7	attaching	attach	VERB
cana-3685	19	8	lagoon	lagoon	NOUN
cana-3685	19	9	step	step	NOUN
cana-3685	19	10	graph	graph	NOUN
cana-3685	19	11	with	with	ADP
cana-3685	19	12	mirror	mirror	NOUN
cana-3685	19	13	image	image	NOUN
cana-3685	19	14	lagoon	lagoon	NOUN
cana-3685	19	15	step	step	NOUN
cana-3685	19	16	graph	graph	NOUN
cana-3685	19	17	consists	consist	VERB
cana-3685	19	18	of	of	ADP
cana-3685	19	19	3n+m-4	3n+m-4	NUM
cana-3685	19	20	vertices	vertex	NOUN
cana-3685	19	21	and	and	CCONJ
cana-3685	19	22	3n+m-3	3n+m-3	NUM
cana-3685	19	23	edges	edge	NOUN
cana-3685	19	24	(	(	PUNCT
cana-3685	19	25	with	with	ADP
cana-3685	19	26	n	n	PRON
cana-3685	19	27	≥	≥	NOUN
cana-3685	19	28	3	3	NUM
cana-3685	19	29	must	must	AUX
cana-3685	19	30	be	be	AUX
cana-3685	19	31	an	an	DET
cana-3685	19	32	odd	odd	ADJ
cana-3685	19	33	and	and	CCONJ
cana-3685	19	34	m	m	VERB
cana-3685	19	35	=	=	SYM
cana-3685	19	36	(	(	PUNCT
cana-3685	19	37	n-1)/2	n-1)/2	PROPN
cana-3685	19	38	,	,	PUNCT
cana-3685	19	39	m	m	VERB
cana-3685	19	40	≥	≥	NOUN
cana-3685	19	41	1	1	NUM
cana-3685	19	42	)	)	PUNCT
cana-3685	19	43	described	describe	VERB
cana-3685	19	44	below	below	ADP
cana-3685	19	45	communications	communication	NOUN
cana-3685	19	46	on	on	ADP
cana-3685	19	47	applied	apply	VERB
cana-3685	19	48	nonlinear	nonlinear	ADJ
cana-3685	19	49	analysis	analysis	NOUN
cana-3685	19	50	issn	issn	NOUN
cana-3685	19	51	:	:	PUNCT
cana-3685	19	52	1074	1074	NUM
cana-3685	19	53	-	-	PUNCT
cana-3685	19	54	133x	133x	NUM
cana-3685	19	55	vol	vol	NOUN
cana-3685	19	56	32	32	NUM
cana-3685	19	57	no	no	NOUN
cana-3685	19	58	.	.	PUNCT
cana-3685	20	1	8s	8s	PROPN
cana-3685	20	2	(	(	PUNCT
cana-3685	20	3	2025	2025	NUM
cana-3685	20	4	)	)	PUNCT
cana-3685	20	5	415	415	NUM
cana-3685	20	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3685	20	7	figure	figure	NOUN
cana-3685	20	8	2	2	NUM
cana-3685	20	9	:	:	PUNCT
cana-3685	20	10	i	i	PRON
cana-3685	20	11	m	m	VERB
cana-3685	20	12	l7s3	l7s3	VERB
cana-3685	20	13	definition	definition	NOUN
cana-3685	20	14	3	3	NUM
cana-3685	20	15	:	:	PUNCT
cana-3685	20	16	intersection	intersection	NOUN
cana-3685	20	17	opposite	opposite	ADJ
cana-3685	20	18	lagoon	lagoon	NOUN
cana-3685	20	19	step	step	NOUN
cana-3685	20	20	graph	graph	NOUN
cana-3685	20	21	io	io	PROPN
cana-3685	20	22	lns	lns	PROPN
cana-3685	20	23	m	m	AUX
cana-3685	20	24	obtained	obtain	VERB
cana-3685	20	25	by	by	ADP
cana-3685	20	26	attaching	attach	VERB
cana-3685	20	27	lagoon	lagoon	NOUN
cana-3685	20	28	step	step	NOUN
cana-3685	20	29	graph	graph	NOUN
cana-3685	20	30	with	with	ADP
cana-3685	20	31	opposite	opposite	ADJ
cana-3685	20	32	direction	direction	NOUN
cana-3685	20	33	of	of	ADP
cana-3685	20	34	lagoon	lagoon	NOUN
cana-3685	20	35	step	step	NOUN
cana-3685	20	36	graph	graph	NOUN
cana-3685	20	37	consists	consist	VERB
cana-3685	20	38	of	of	ADP
cana-3685	20	39	vertices	vertex	NOUN
cana-3685	20	40	2n+4m-4	2n+4m-4	PROPN
cana-3685	20	41	vertices	vertex	NOUN
cana-3685	20	42	and	and	CCONJ
cana-3685	20	43	edges	edge	VERB
cana-3685	21	1	2n+4m-3(with	2n+4m-3(with	NUM
cana-3685	21	2	n	n	PRON
cana-3685	21	3	≥	≥	NUM
cana-3685	21	4	3	3	NUM
cana-3685	21	5	must	must	AUX
cana-3685	21	6	be	be	AUX
cana-3685	21	7	an	an	DET
cana-3685	21	8	odd	odd	ADJ
cana-3685	21	9	and	and	CCONJ
cana-3685	21	10	m	m	VERB
cana-3685	21	11	=	=	SYM
cana-3685	21	12	(	(	PUNCT
cana-3685	21	13	n-1)/2	n-1)/2	PROPN
cana-3685	21	14	,	,	PUNCT
cana-3685	21	15	m	m	VERB
cana-3685	21	16	≥	≥	NOUN
cana-3685	21	17	1	1	NUM
cana-3685	21	18	)	)	PUNCT
cana-3685	21	19	described	describe	VERB
cana-3685	21	20	below	below	ADP
cana-3685	21	21	figure	figure	NOUN
cana-3685	21	22	3	3	NUM
cana-3685	21	23	:	:	PUNCT
cana-3685	21	24	io	io	X
cana-3685	21	25	l7s3	l7s3	PROPN
cana-3685	21	26	definition	definition	NOUN
cana-3685	21	27	4	4	NUM
cana-3685	21	28	:	:	PUNCT
cana-3685	21	29	for	for	ADP
cana-3685	21	30	positive	positive	ADJ
cana-3685	21	31	integers	integer	NOUN
cana-3685	21	32	n	n	NOUN
cana-3685	21	33	and	and	CCONJ
cana-3685	21	34	r	r	PRON
cana-3685	21	35	such	such	ADJ
cana-3685	21	36	that	that	DET
cana-3685	21	37	0	0	NUM
cana-3685	21	38	≤	≤	NUM
cana-3685	21	39	r	r	NOUN
cana-3685	21	40	≤	≤	NUM
cana-3685	21	41	n	n	CCONJ
cana-3685	21	42	,	,	PUNCT
cana-3685	21	43	then	then	ADV
cana-3685	21	44	binomial	binomial	ADJ
cana-3685	21	45	coefficient	coefficient	NOUN
cana-3685	21	46	ncr	ncr	PROPN
cana-3685	21	47	=	=	PROPN
cana-3685	21	48	n!/(r!(n	n!/(r!(n	PROPN
cana-3685	21	49	-	-	PUNCT
cana-3685	21	50	r	r	NOUN
cana-3685	21	51	)	)	PUNCT
cana-3685	21	52	!	!	PUNCT
cana-3685	21	53	)	)	PUNCT
cana-3685	22	1	is	be	AUX
cana-3685	22	2	always	always	ADV
cana-3685	22	3	an	an	DET
cana-3685	22	4	integer	integer	NOUN
cana-3685	22	5	.	.	PUNCT
cana-3685	23	1	definition	definition	NOUN
cana-3685	23	2	5	5	NUM
cana-3685	23	3	:	:	PUNCT
cana-3685	23	4	consider	consider	VERB
cana-3685	23	5	g	g	NOUN
cana-3685	23	6	=	=	SYM
cana-3685	23	7	(	(	PUNCT
cana-3685	23	8	p	p	X
cana-3685	23	9	,	,	PUNCT
cana-3685	23	10	q	q	NOUN
cana-3685	23	11	)	)	PUNCT
cana-3685	23	12	be	be	AUX
cana-3685	23	13	a	a	DET
cana-3685	23	14	graph	graph	NOUN
cana-3685	23	15	.	.	PUNCT
cana-3685	24	1	if	if	SCONJ
cana-3685	24	2	the	the	DET
cana-3685	24	3	graph	graph	NOUN
cana-3685	24	4	is	be	AUX
cana-3685	24	5	an	an	DET
cana-3685	24	6	injective	injective	ADJ
cana-3685	24	7	map	map	NOUN
cana-3685	24	8	f	f	X
cana-3685	24	9	:	:	PUNCT
cana-3685	24	10	v(g	v(g	NUM
cana-3685	24	11	)	)	PUNCT
cana-3685	24	12	→	→	SYM
cana-3685	24	13	{	{	PUNCT
cana-3685	24	14	1	1	NUM
cana-3685	24	15	,	,	PUNCT
cana-3685	24	16	2	2	NUM
cana-3685	24	17	,	,	PUNCT
cana-3685	24	18	3	3	NUM
cana-3685	24	19	,	,	PUNCT
cana-3685	24	20	…	…	PUNCT
cana-3685	24	21	q+1	q+1	X
cana-3685	24	22	}	}	PUNCT
cana-3685	24	23	where	where	SCONJ
cana-3685	24	24	the	the	DET
cana-3685	24	25	induced	induced	ADJ
cana-3685	24	26	f	f	NOUN
cana-3685	24	27	*	*	NOUN
cana-3685	24	28	:	:	PUNCT
cana-3685	24	29	e(g	e(g	NOUN
cana-3685	24	30	)	)	PUNCT
cana-3685	24	31	→	→	PUNCT
cana-3685	24	32	n	n	X
cana-3685	24	33	is	be	AUX
cana-3685	24	34	provided	provide	VERB
cana-3685	24	35	by	by	ADP
cana-3685	24	36	,	,	PUNCT
cana-3685	24	37	then	then	ADV
cana-3685	24	38	g	g	PROPN
cana-3685	24	39	has	have	VERB
cana-3685	24	40	a	a	DET
cana-3685	24	41	binomial	binomial	ADJ
cana-3685	24	42	labeling	labeling	NOUN
cana-3685	24	43	by	by	ADP
cana-3685	24	44	f	f	PROPN
cana-3685	24	45	*	*	PUNCT
cana-3685	24	46	(	(	PUNCT
cana-3685	24	47	u	u	NOUN
cana-3685	24	48	v	v	NOUN
cana-3685	24	49	)	)	PUNCT
cana-3685	24	50	=	=	PUNCT
cana-3685	25	1	m	m	VERB
cana-3685	25	2	cm	cm	NOUN
cana-3685	25	3	=	=	SYM
cana-3685	25	4	m!/(m!(m	m!/(m!(m	NUM
cana-3685	25	5	-	-	PUNCT
cana-3685	25	6	m	m	NOUN
cana-3685	25	7	)	)	PUNCT
cana-3685	25	8	!	!	PUNCT
cana-3685	25	9	)	)	PUNCT
cana-3685	26	1	where	where	SCONJ
cana-3685	26	2	m	m	NOUN
cana-3685	26	3	=	=	SYM
cana-3685	26	4	max	max	PROPN
cana-3685	26	5	{	{	PUNCT
cana-3685	26	6	f(u	f(u	PROPN
cana-3685	26	7	)	)	PUNCT
cana-3685	26	8	,	,	PUNCT
cana-3685	26	9	f(v	f(v	NOUN
cana-3685	26	10	)	)	PUNCT
cana-3685	26	11	}	}	PUNCT
cana-3685	26	12	,	,	PUNCT
cana-3685	26	13	m	m	NOUN
cana-3685	26	14	=	=	NOUN
cana-3685	26	15	min	min	X
cana-3685	26	16	{	{	PUNCT
cana-3685	26	17	f(u	f(u	PROPN
cana-3685	26	18	)	)	PUNCT
cana-3685	26	19	,	,	PUNCT
cana-3685	26	20	f(v)}assigns	f(v)}assign	VERB
cana-3685	26	21	distinct	distinct	ADJ
cana-3685	26	22	labels	label	NOUN
cana-3685	26	23	for	for	ADP
cana-3685	26	24	the	the	DET
cana-3685	26	25	edges	edge	NOUN
cana-3685	26	26	.	.	PUNCT
cana-3685	27	1	definition	definition	NOUN
cana-3685	27	2	6	6	NUM
cana-3685	27	3	:	:	PUNCT
cana-3685	27	4	the	the	DET
cana-3685	27	5	term	term	NOUN
cana-3685	27	6	"	"	PUNCT
cana-3685	27	7	binomial	binomial	ADJ
cana-3685	27	8	graph	graph	NOUN
cana-3685	27	9	"	"	PUNCT
cana-3685	27	10	refers	refer	VERB
cana-3685	27	11	to	to	ADP
cana-3685	27	12	a	a	DET
cana-3685	27	13	graph	graph	NOUN
cana-3685	27	14	that	that	PRON
cana-3685	27	15	has	have	VERB
cana-3685	27	16	the	the	DET
cana-3685	27	17	binomial	binomial	ADJ
cana-3685	27	18	labeling	labeling	NOUN
cana-3685	27	19	.	.	PUNCT
cana-3685	28	1	3	3	X
cana-3685	28	2	.	.	X
cana-3685	28	3	main	main	ADJ
cana-3685	28	4	results	result	NOUN
cana-3685	28	5	3	3	NUM
cana-3685	28	6	.	.	SYM
cana-3685	28	7	1	1	NUM
cana-3685	28	8	theorem	theorem	NOUN
cana-3685	28	9	:	:	PUNCT
cana-3685	28	10	intersection	intersection	NOUN
cana-3685	28	11	mirror	mirror	NOUN
cana-3685	28	12	lagoon	lagoon	NOUN
cana-3685	28	13	step	step	NOUN
cana-3685	28	14	graph	graph	NOUN
cana-3685	28	15	i	i	PRON
cana-3685	28	16	m	m	VERB
cana-3685	28	17	ln	ln	ADJ
cana-3685	28	18	sm	sm	PROPN
cana-3685	28	19	is	be	AUX
cana-3685	28	20	a	a	DET
cana-3685	28	21	binomial	binomial	ADJ
cana-3685	28	22	graph	graph	NOUN
cana-3685	28	23	.	.	PUNCT
cana-3685	29	1	proof	proof	NOUN
cana-3685	29	2	:	:	PUNCT
cana-3685	29	3	consider	consider	VERB
cana-3685	29	4	the	the	DET
cana-3685	29	5	graph	graph	NOUN
cana-3685	29	6	g	g	NOUN
cana-3685	29	7	be	be	AUX
cana-3685	29	8	the	the	DET
cana-3685	29	9	intersection	intersection	NOUN
cana-3685	29	10	mirror	mirror	NOUN
cana-3685	29	11	lagoon	lagoon	NOUN
cana-3685	29	12	step	step	NOUN
cana-3685	29	13	graph	graph	NOUN
cana-3685	29	14	i	i	PRON
cana-3685	29	15	m	m	VERB
cana-3685	29	16	ln	ln	ADJ
cana-3685	29	17	sm	sm	ADV
cana-3685	29	18	with	with	ADP
cana-3685	29	19	3n+m-4	3n+m-4	NUM
cana-3685	29	20	vertices	vertex	NOUN
cana-3685	29	21	and	and	CCONJ
cana-3685	29	22	3n+m-3	3n+m-3	NUM
cana-3685	29	23	edges	edge	VERB
cana-3685	29	24	when	when	SCONJ
cana-3685	29	25	n	n	X
cana-3685	29	26	=	=	SYM
cana-3685	29	27	3	3	NUM
cana-3685	29	28	,	,	PUNCT
cana-3685	29	29	5	5	NUM
cana-3685	29	30	,	,	PUNCT
cana-3685	29	31	7	7	NUM
cana-3685	29	32	,	,	PUNCT
cana-3685	29	33	9	9	NUM
cana-3685	29	34	…	…	PUNCT
cana-3685	29	35	then	then	ADV
cana-3685	29	36	m	m	VERB
cana-3685	29	37	=	=	ADJ
cana-3685	29	38	1	1	NUM
cana-3685	29	39	,	,	PUNCT
cana-3685	29	40	2	2	NUM
cana-3685	29	41	,	,	PUNCT
cana-3685	29	42	3	3	NUM
cana-3685	29	43	…	…	PUNCT
cana-3685	29	44	respectively	respectively	ADV
cana-3685	29	45	.	.	PUNCT
cana-3685	30	1	case	case	NOUN
cana-3685	31	1	i	i	PRON
cana-3685	31	2	:	:	PUNCT
cana-3685	31	3	when	when	SCONJ
cana-3685	31	4	n	n	X
cana-3685	31	5	=	=	SYM
cana-3685	31	6	5	5	NUM
cana-3685	31	7	,	,	PUNCT
cana-3685	31	8	7	7	NUM
cana-3685	31	9	,	,	PUNCT
cana-3685	31	10	9	9	NUM
cana-3685	31	11	…	…	PUNCT
cana-3685	31	12	then	then	ADV
cana-3685	31	13	m	m	VERB
cana-3685	31	14	=	=	ADJ
cana-3685	31	15	2	2	NUM
cana-3685	31	16	,	,	PUNCT
cana-3685	31	17	3,4	3,4	NUM
cana-3685	31	18	…	…	PUNCT
cana-3685	31	19	respectively	respectively	ADV
cana-3685	31	20	.	.	PUNCT
cana-3685	32	1	let	let	VERB
cana-3685	32	2	vi	vi	NOUN
cana-3685	32	3	be	be	AUX
cana-3685	32	4	the	the	DET
cana-3685	32	5	outer	outer	ADJ
cana-3685	32	6	vertices	vertex	NOUN
cana-3685	32	7	starts	start	VERB
cana-3685	32	8	from	from	ADP
cana-3685	32	9	left	left	ADJ
cana-3685	32	10	step	step	NOUN
cana-3685	32	11	corner	corner	NOUN
cana-3685	32	12	to	to	ADP
cana-3685	32	13	right	right	ADJ
cana-3685	32	14	step	step	NOUN
cana-3685	32	15	corner	corner	NOUN
cana-3685	32	16	,	,	PUNCT
cana-3685	32	17	w	w	VERB
cana-3685	32	18	i	i	PRON
cana-3685	32	19	be	be	VERB
cana-3685	32	20	the	the	DET
cana-3685	32	21	vertical	vertical	ADJ
cana-3685	32	22	inner	inner	ADJ
cana-3685	32	23	vertices	vertex	NOUN
cana-3685	32	24	and	and	CCONJ
cana-3685	32	25	xi	xi	AUX
cana-3685	32	26	be	be	AUX
cana-3685	32	27	the	the	DET
cana-3685	32	28	base	base	NOUN
cana-3685	32	29	vertices	vertex	NOUN
cana-3685	32	30	.	.	PUNCT
cana-3685	33	1	classify	classify	VERB
cana-3685	33	2	f	f	NOUN
cana-3685	33	3	:	:	PUNCT
cana-3685	33	4	v	v	NOUN
cana-3685	33	5	(	(	PUNCT
cana-3685	33	6	g	g	NOUN
cana-3685	33	7	)	)	PUNCT
cana-3685	33	8	→	→	SYM
cana-3685	33	9	{	{	PUNCT
cana-3685	33	10	1	1	NUM
cana-3685	33	11	,	,	PUNCT
cana-3685	33	12	2	2	NUM
cana-3685	33	13	,	,	PUNCT
cana-3685	33	14	3	3	NUM
cana-3685	33	15	…	…	SYM
cana-3685	33	16	3n+m-2	3n+m-2	NOUN
cana-3685	33	17	}	}	PUNCT
cana-3685	33	18	is	be	AUX
cana-3685	33	19	given	give	VERB
cana-3685	33	20	below	below	ADP
cana-3685	33	21	f(vi	f(vi	NOUN
cana-3685	33	22	)	)	PUNCT
cana-3685	34	1	=	=	SYM
cana-3685	35	1	i	i	PROPN
cana-3685	35	2	,	,	PUNCT
cana-3685	35	3	i	i	PRON
cana-3685	35	4	=	=	NOUN
cana-3685	35	5	1	1	NUM
cana-3685	35	6	,	,	PUNCT
cana-3685	35	7	2	2	NUM
cana-3685	35	8	…	…	PUNCT
cana-3685	35	9	n+2m-2	n+2m-2	NOUN
cana-3685	35	10	,	,	PUNCT
cana-3685	35	11	when	when	SCONJ
cana-3685	35	12	n	n	X
cana-3685	35	13	=	=	SYM
cana-3685	35	14	5	5	NUM
cana-3685	35	15	,	,	PUNCT
cana-3685	35	16	7	7	NUM
cana-3685	35	17	,	,	PUNCT
cana-3685	35	18	9	9	NUM
cana-3685	35	19	…	…	PUNCT
cana-3685	35	20	,	,	PUNCT
cana-3685	35	21	then	then	ADV
cana-3685	35	22	m	m	VERB
cana-3685	35	23	=	=	ADJ
cana-3685	35	24	2	2	NUM
cana-3685	35	25	,	,	PUNCT
cana-3685	35	26	3	3	NUM
cana-3685	35	27	,	,	PUNCT
cana-3685	35	28	4	4	NUM
cana-3685	35	29	…	…	PUNCT
cana-3685	35	30	respectively	respectively	ADV
cana-3685	35	31	,	,	PUNCT
cana-3685	35	32	f(wi	f(wi	NOUN
cana-3685	35	33	)	)	PUNCT
cana-3685	35	34	=	=	PUNCT
cana-3685	36	1	2n+2m+j-1	2n+2m+j-1	PROPN
cana-3685	36	2	,	,	PUNCT
cana-3685	36	3	which	which	PRON
cana-3685	36	4	can	can	AUX
cana-3685	36	5	be	be	AUX
cana-3685	36	6	expressed	express	VERB
cana-3685	36	7	as	as	ADP
cana-3685	36	8	i	i	PRON
cana-3685	36	9	n	n	VERB
cana-3685	36	10	m	m	VERB
cana-3685	36	11	j	j	NOUN
cana-3685	36	12	1	1	NUM
cana-3685	36	13	5	5	NUM
cana-3685	36	14	2	2	NUM
cana-3685	36	15	0	0	NUM
cana-3685	36	16	1	1	NUM
cana-3685	36	17	,	,	PUNCT
cana-3685	36	18	2	2	NUM
cana-3685	36	19	7	7	NUM
cana-3685	36	20	3	3	NUM
cana-3685	36	21	0	0	NUM
cana-3685	36	22	,	,	PUNCT
cana-3685	36	23	1	1	NUM
cana-3685	36	24	1	1	NUM
cana-3685	36	25	,	,	PUNCT
cana-3685	36	26	2	2	NUM
cana-3685	36	27	,	,	PUNCT
cana-3685	36	28	3	3	NUM
cana-3685	36	29	9	9	NUM
cana-3685	36	30	4	4	NUM
cana-3685	36	31	0	0	NUM
cana-3685	36	32	,	,	PUNCT
cana-3685	36	33	1	1	NUM
cana-3685	36	34	,	,	PUNCT
cana-3685	36	35	2	2	NUM
cana-3685	36	36	communications	communication	NOUN
cana-3685	36	37	on	on	ADP
cana-3685	36	38	applied	apply	VERB
cana-3685	36	39	nonlinear	nonlinear	ADJ
cana-3685	36	40	analysis	analysis	NOUN
cana-3685	36	41	issn	issn	NOUN
cana-3685	36	42	:	:	PUNCT
cana-3685	36	43	1074	1074	NUM
cana-3685	36	44	-	-	PUNCT
cana-3685	36	45	133x	133x	NUM
cana-3685	36	46	vol	vol	NOUN
cana-3685	36	47	32	32	NUM
cana-3685	36	48	no	no	NOUN
cana-3685	36	49	.	.	PUNCT
cana-3685	37	1	8s	8s	PROPN
cana-3685	37	2	(	(	PUNCT
cana-3685	37	3	2025	2025	NUM
cana-3685	37	4	)	)	PUNCT
cana-3685	37	5	416	416	NUM
cana-3685	37	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3685	37	7	.	.	PUNCT
cana-3685	37	8	.	.	PUNCT
cana-3685	37	9	.	.	PUNCT
cana-3685	37	10	.	.	PUNCT
cana-3685	37	11	.	.	PUNCT
cana-3685	37	12	.	.	PUNCT
cana-3685	37	13	.	.	PUNCT
cana-3685	37	14	.	.	PUNCT
cana-3685	37	15	.	.	PUNCT
cana-3685	37	16	.	.	PUNCT
cana-3685	37	17	.	.	PUNCT
cana-3685	37	18	.	.	PUNCT
cana-3685	38	1	respectively	respectively	ADV
cana-3685	38	2	.	.	PUNCT
cana-3685	39	1	f(xi	f(xi	X
cana-3685	39	2	)	)	PUNCT
cana-3685	39	3	=	=	SYM
cana-3685	40	1	n+2m+i-2	n+2m+i-2	X
cana-3685	40	2	which	which	PRON
cana-3685	40	3	can	can	AUX
cana-3685	40	4	be	be	AUX
cana-3685	40	5	expressed	express	VERB
cana-3685	40	6	as	as	ADP
cana-3685	40	7	n	n	ADP
cana-3685	40	8	m	m	NOUN
cana-3685	40	9	i	i	ADV
cana-3685	40	10	5	5	NUM
cana-3685	40	11	2	2	NUM
cana-3685	40	12	1	1	NUM
cana-3685	40	13	,	,	PUNCT
cana-3685	40	14	2	2	NUM
cana-3685	40	15	,	,	PUNCT
cana-3685	40	16	3	3	NUM
cana-3685	40	17	,	,	PUNCT
cana-3685	40	18	4	4	NUM
cana-3685	40	19	,	,	PUNCT
cana-3685	40	20	5	5	NUM
cana-3685	40	21	7	7	NUM
cana-3685	40	22	3	3	NUM
cana-3685	40	23	1	1	NUM
cana-3685	40	24	,	,	PUNCT
cana-3685	40	25	2	2	NUM
cana-3685	40	26	,	,	PUNCT
cana-3685	40	27	3	3	NUM
cana-3685	40	28	,	,	PUNCT
cana-3685	40	29	4	4	NUM
cana-3685	40	30	,	,	PUNCT
cana-3685	40	31	5	5	NUM
cana-3685	40	32	,	,	PUNCT
cana-3685	40	33	6	6	NUM
cana-3685	40	34	,	,	PUNCT
cana-3685	40	35	7	7	NUM
cana-3685	40	36	9	9	NUM
cana-3685	40	37	4	4	NUM
cana-3685	40	38	1	1	NUM
cana-3685	40	39	,	,	PUNCT
cana-3685	40	40	2	2	NUM
cana-3685	40	41	,	,	PUNCT
cana-3685	40	42	3	3	NUM
cana-3685	40	43	,	,	PUNCT
cana-3685	40	44	4	4	NUM
cana-3685	40	45	,	,	PUNCT
cana-3685	40	46	5	5	NUM
cana-3685	40	47	,	,	PUNCT
cana-3685	40	48	6	6	NUM
cana-3685	40	49	,	,	PUNCT
cana-3685	40	50	7	7	NUM
cana-3685	40	51	,	,	PUNCT
cana-3685	40	52	8	8	NUM
cana-3685	40	53	,	,	PUNCT
cana-3685	40	54	9	9	NUM
cana-3685	40	55	.	.	PUNCT
cana-3685	40	56	.	.	PUNCT
cana-3685	40	57	.	.	PUNCT
cana-3685	40	58	.	.	PUNCT
cana-3685	40	59	.	.	PUNCT
cana-3685	40	60	.	.	PUNCT
cana-3685	40	61	.	.	PUNCT
cana-3685	40	62	.	.	PUNCT
cana-3685	40	63	.	.	PUNCT
cana-3685	41	1	respectively	respectively	ADV
cana-3685	41	2	.	.	PUNCT
cana-3685	42	1	the	the	DET
cana-3685	42	2	equation	equation	NOUN
cana-3685	42	3	f	f	X
cana-3685	42	4	*	*	PUNCT
cana-3685	42	5	(	(	PUNCT
cana-3685	42	6	e	e	NOUN
cana-3685	42	7	=	=	NOUN
cana-3685	42	8	uv	uv	NOUN
cana-3685	42	9	)	)	PUNCT
cana-3685	42	10	=	=	SYM
cana-3685	42	11	r	r	NOUN
cana-3685	42	12	cr	cr	NOUN
cana-3685	42	13	=	=	PUNCT
cana-3685	42	14	r!/(r!(r	r!/(r!(r	NOUN
cana-3685	42	15	-	-	PUNCT
cana-3685	42	16	r	r	NOUN
cana-3685	42	17	)	)	PUNCT
cana-3685	42	18	!	!	PUNCT
cana-3685	42	19	)	)	PUNCT
cana-3685	43	1	yields	yield	VERB
cana-3685	43	2	f	f	NOUN
cana-3685	43	3	*	*	NOUN
cana-3685	43	4	:	:	PUNCT
cana-3685	43	5	e(g	e(g	NOUN
cana-3685	43	6	)	)	PUNCT
cana-3685	43	7	→	→	SYM
cana-3685	43	8	n	n	CCONJ
cana-3685	43	9	,	,	PUNCT
cana-3685	43	10	wherein	wherein	SCONJ
cana-3685	43	11	r	r	NOUN
cana-3685	43	12	=	=	SYM
cana-3685	43	13	max{f(u	max{f(u	PROPN
cana-3685	43	14	)	)	PUNCT
cana-3685	43	15	,	,	PUNCT
cana-3685	43	16	f(v	f(v	PROPN
cana-3685	43	17	)	)	PUNCT
cana-3685	43	18	}	}	PUNCT
cana-3685	43	19	and	and	CCONJ
cana-3685	43	20	r	r	NOUN
cana-3685	43	21	=	=	SYM
cana-3685	43	22	min	min	NOUN
cana-3685	43	23	{	{	PUNCT
cana-3685	43	24	f(u	f(u	PROPN
cana-3685	43	25	)	)	PUNCT
cana-3685	43	26	,	,	PUNCT
cana-3685	43	27	f(v	f(v	PROPN
cana-3685	43	28	)	)	PUNCT
cana-3685	43	29	}	}	PUNCT
cana-3685	43	30	for	for	ADP
cana-3685	43	31	every	every	DET
cana-3685	43	32	uv	uv	NOUN
cana-3685	43	33	ϵ	ϵ	PROPN
cana-3685	43	34	e(g	e(g	PROPN
cana-3685	43	35	)	)	PUNCT
cana-3685	43	36	.every	.every	PUNCT
cana-3685	43	37	edge	edge	VERB
cana-3685	43	38	in	in	ADP
cana-3685	43	39	this	this	DET
cana-3685	43	40	case	case	NOUN
cana-3685	43	41	has	have	VERB
cana-3685	43	42	a	a	DET
cana-3685	43	43	unique	unique	ADJ
cana-3685	43	44	label	label	NOUN
cana-3685	43	45	.	.	PUNCT
cana-3685	44	1	figure	figure	NOUN
cana-3685	44	2	4	4	NUM
cana-3685	44	3	:	:	PUNCT
cana-3685	44	4	binomial	binomial	ADJ
cana-3685	44	5	graph	graph	NOUN
cana-3685	44	6	iml5s2	iml5s2	ADJ
cana-3685	44	7	figure	figure	NOUN
cana-3685	44	8	5	5	NUM
cana-3685	44	9	:	:	PUNCT
cana-3685	44	10	binomial	binomial	ADJ
cana-3685	44	11	graph	graph	NOUN
cana-3685	44	12	i	i	PRON
cana-3685	44	13	m	m	VERB
cana-3685	44	14	l7s3	l7s3	ADV
cana-3685	44	15	special	special	ADJ
cana-3685	44	16	case	case	NOUN
cana-3685	45	1	i	i	PRON
cana-3685	45	2	m	m	VERB
cana-3685	45	3	l3s1	l3s1	VERB
cana-3685	45	4	:	:	PUNCT
cana-3685	45	5	consider	consider	VERB
cana-3685	45	6	v1	v1	NOUN
cana-3685	45	7	,	,	PUNCT
cana-3685	45	8	v2	v2	PROPN
cana-3685	45	9	,	,	PUNCT
cana-3685	45	10	v3	v3	PROPN
cana-3685	45	11	,	,	PUNCT
cana-3685	45	12	v4	v4	PROPN
cana-3685	45	13	,	,	PUNCT
cana-3685	45	14	v5	v5	PROPN
cana-3685	45	15	and	and	CCONJ
cana-3685	45	16	v6	v6	NOUN
cana-3685	45	17	be	be	VERB
cana-3685	45	18	the	the	DET
cana-3685	45	19	six	six	NUM
cana-3685	45	20	vertices	vertex	NOUN
cana-3685	45	21	of	of	ADP
cana-3685	45	22	i	i	PRON
cana-3685	45	23	m	m	VERB
cana-3685	45	24	l3s1	l3s1	X
cana-3685	45	25	.	.	PUNCT
cana-3685	46	1	vertex	vertex	NOUN
cana-3685	46	2	labeling	labeling	NOUN
cana-3685	46	3	are	be	AUX
cana-3685	46	4	f(vi	f(vi	NOUN
cana-3685	46	5	)	)	PUNCT
cana-3685	47	1	=	=	SYM
cana-3685	47	2	i	i	PROPN
cana-3685	47	3	,	,	PUNCT
cana-3685	47	4	i	i	PRON
cana-3685	47	5	=	=	NOUN
cana-3685	47	6	1	1	NUM
cana-3685	47	7	,	,	PUNCT
cana-3685	47	8	2	2	NUM
cana-3685	47	9	,	,	PUNCT
cana-3685	47	10	3	3	NUM
cana-3685	47	11	,	,	PUNCT
cana-3685	47	12	4	4	NUM
cana-3685	47	13	,	,	PUNCT
cana-3685	47	14	5	5	NUM
cana-3685	47	15	and	and	CCONJ
cana-3685	47	16	f(vi	f(vi	NOUN
cana-3685	47	17	)	)	PUNCT
cana-3685	47	18	=	=	PUNCT
cana-3685	48	1	i+1	i+1	NUM
cana-3685	48	2	,	,	PUNCT
cana-3685	48	3	i	i	PRON
cana-3685	48	4	=	=	NOUN
cana-3685	48	5	6	6	X
cana-3685	48	6	.	.	X
cana-3685	48	7	figure	figure	VERB
cana-3685	48	8	6	6	NUM
cana-3685	48	9	:	:	PUNCT
cana-3685	48	10	binomial	binomial	ADJ
cana-3685	48	11	graph	graph	NOUN
cana-3685	48	12	iml3s1	iml3s1	NOUN
cana-3685	48	13	every	every	DET
cana-3685	48	14	edge	edge	NOUN
cana-3685	48	15	in	in	ADP
cana-3685	48	16	this	this	DET
cana-3685	48	17	case	case	NOUN
cana-3685	48	18	has	have	VERB
cana-3685	48	19	a	a	DET
cana-3685	48	20	unique	unique	ADJ
cana-3685	48	21	label	label	NOUN
cana-3685	48	22	.	.	PUNCT
cana-3685	49	1	imlnsm	imlnsm	NOUN
cana-3685	49	2	is	be	AUX
cana-3685	49	3	a	a	DET
cana-3685	49	4	binomial	binomial	ADJ
cana-3685	49	5	graph	graph	NOUN
cana-3685	49	6	as	as	ADP
cana-3685	49	7	a	a	DET
cana-3685	49	8	result	result	NOUN
cana-3685	49	9	.	.	PUNCT
cana-3685	50	1	3	3	X
cana-3685	50	2	.	.	SYM
cana-3685	50	3	2	2	NUM
cana-3685	50	4	theorem	theorem	NOUN
cana-3685	50	5	:	:	PUNCT
cana-3685	50	6	intersection	intersection	NOUN
cana-3685	50	7	opposite	opposite	ADJ
cana-3685	50	8	lagoon	lagoon	NOUN
cana-3685	50	9	step	step	NOUN
cana-3685	50	10	graph	graph	NOUN
cana-3685	50	11	io	io	X
cana-3685	50	12	lnsm	lnsm	PROPN
cana-3685	50	13	is	be	AUX
cana-3685	50	14	a	a	DET
cana-3685	50	15	binomial	binomial	ADJ
cana-3685	50	16	graph	graph	NOUN
cana-3685	50	17	.	.	PUNCT
cana-3685	51	1	proof	proof	NOUN
cana-3685	51	2	:	:	PUNCT
cana-3685	51	3	consider	consider	VERB
cana-3685	51	4	the	the	DET
cana-3685	51	5	graph	graph	NOUN
cana-3685	51	6	g	g	NOUN
cana-3685	51	7	be	be	AUX
cana-3685	51	8	the	the	DET
cana-3685	51	9	intersection	intersection	NOUN
cana-3685	51	10	opposite	opposite	ADJ
cana-3685	51	11	lagoon	lagoon	NOUN
cana-3685	51	12	step	step	NOUN
cana-3685	51	13	graph	graph	NOUN
cana-3685	51	14	with	with	ADP
cana-3685	51	15	2n+4m-4	2n+4m-4	PROPN
cana-3685	51	16	vertices	vertex	NOUN
cana-3685	51	17	and	and	CCONJ
cana-3685	51	18	2n+4m-3	2n+4m-3	NUM
cana-3685	51	19	edges	edge	NOUN
cana-3685	51	20	when	when	SCONJ
cana-3685	51	21	n	n	X
cana-3685	51	22	=	=	SYM
cana-3685	51	23	3	3	NUM
cana-3685	51	24	,	,	PUNCT
cana-3685	51	25	5	5	NUM
cana-3685	51	26	,	,	PUNCT
cana-3685	51	27	7	7	NUM
cana-3685	51	28	…	…	NUM
cana-3685	51	29	,	,	PUNCT
cana-3685	51	30	then	then	ADV
cana-3685	51	31	m	m	VERB
cana-3685	51	32	=	=	NOUN
cana-3685	51	33	1	1	NUM
cana-3685	51	34	,	,	PUNCT
cana-3685	51	35	2	2	NUM
cana-3685	51	36	,	,	PUNCT
cana-3685	51	37	3	3	NUM
cana-3685	51	38	,	,	PUNCT
cana-3685	51	39	4	4	NUM
cana-3685	51	40	…	…	PUNCT
cana-3685	51	41	respectively	respectively	ADV
cana-3685	51	42	.	.	PUNCT
cana-3685	52	1	case	case	NOUN
cana-3685	53	1	i	i	PRON
cana-3685	53	2	:	:	PUNCT
cana-3685	53	3	when	when	SCONJ
cana-3685	53	4	n	n	X
cana-3685	53	5	=	=	SYM
cana-3685	53	6	5	5	NUM
cana-3685	53	7	,	,	PUNCT
cana-3685	53	8	7	7	NUM
cana-3685	53	9	,	,	PUNCT
cana-3685	53	10	9	9	NUM
cana-3685	53	11	…	…	PUNCT
cana-3685	53	12	then	then	ADV
cana-3685	53	13	m	m	VERB
cana-3685	53	14	=	=	ADJ
cana-3685	53	15	2	2	NUM
cana-3685	53	16	,	,	PUNCT
cana-3685	53	17	3	3	NUM
cana-3685	53	18	,	,	PUNCT
cana-3685	53	19	4	4	NUM
cana-3685	53	20	…	…	PUNCT
cana-3685	53	21	respectively	respectively	ADV
cana-3685	53	22	.	.	PUNCT
cana-3685	54	1	let	let	VERB
cana-3685	54	2	vi	vi	NOUN
cana-3685	54	3	be	be	AUX
cana-3685	54	4	the	the	DET
cana-3685	54	5	outer	outer	ADJ
cana-3685	54	6	vertices	vertex	NOUN
cana-3685	54	7	except	except	SCONJ
cana-3685	54	8	last	last	ADJ
cana-3685	54	9	vertex	vertex	NOUN
cana-3685	54	10	vi1	vi1	NOUN
cana-3685	54	11	where	where	SCONJ
cana-3685	54	12	vi1	vi1	PROPN
cana-3685	54	13	be	be	AUX
cana-3685	54	14	the	the	DET
cana-3685	54	15	last	last	ADJ
cana-3685	54	16	vertex	vertex	NOUN
cana-3685	54	17	.	.	PUNCT
cana-3685	55	1	communications	communication	NOUN
cana-3685	55	2	on	on	ADP
cana-3685	55	3	applied	apply	VERB
cana-3685	55	4	nonlinear	nonlinear	ADJ
cana-3685	55	5	analysis	analysis	NOUN
cana-3685	55	6	issn	issn	NOUN
cana-3685	55	7	:	:	PUNCT
cana-3685	55	8	1074	1074	NUM
cana-3685	55	9	-	-	PUNCT
cana-3685	55	10	133x	133x	NUM
cana-3685	55	11	vol	vol	NOUN
cana-3685	55	12	32	32	NUM
cana-3685	55	13	no	no	NOUN
cana-3685	55	14	.	.	PUNCT
cana-3685	56	1	8s	8s	PROPN
cana-3685	56	2	(	(	PUNCT
cana-3685	56	3	2025	2025	NUM
cana-3685	56	4	)	)	PUNCT
cana-3685	56	5	417	417	NUM
cana-3685	56	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3685	56	7	classify	classify	VERB
cana-3685	56	8	f	f	X
cana-3685	56	9	:	:	PUNCT
cana-3685	56	10	v(g	v(g	NUM
cana-3685	56	11	)	)	PUNCT
cana-3685	56	12	→	→	SYM
cana-3685	56	13	{	{	PUNCT
cana-3685	56	14	1	1	NUM
cana-3685	56	15	,	,	PUNCT
cana-3685	56	16	2,3	2,3	NUM
cana-3685	56	17	…	…	PUNCT
cana-3685	56	18	2n+4m-2	2n+4m-2	NOUN
cana-3685	56	19	}	}	PUNCT
cana-3685	56	20	as	as	SCONJ
cana-3685	56	21	follows	follow	VERB
cana-3685	56	22	f(vi	f(vi	NOUN
cana-3685	56	23	)	)	PUNCT
cana-3685	56	24	=	=	SYM
cana-3685	57	1	i	i	PROPN
cana-3685	57	2	,	,	PUNCT
cana-3685	57	3	i	i	PRON
cana-3685	57	4	=	=	NOUN
cana-3685	57	5	1	1	NUM
cana-3685	57	6	,	,	PUNCT
cana-3685	57	7	2,3	2,3	NUM
cana-3685	57	8	…	…	SYM
cana-3685	57	9	2n+4m-5	2n+4m-5	NUM
cana-3685	57	10	when	when	SCONJ
cana-3685	57	11	n	n	X
cana-3685	57	12	=	=	SYM
cana-3685	57	13	5	5	NUM
cana-3685	57	14	,	,	PUNCT
cana-3685	57	15	7	7	NUM
cana-3685	57	16	,	,	PUNCT
cana-3685	57	17	9	9	NUM
cana-3685	57	18	…	…	PUNCT
cana-3685	57	19	then	then	ADV
cana-3685	57	20	m	m	VERB
cana-3685	57	21	=	=	ADJ
cana-3685	57	22	2	2	NUM
cana-3685	57	23	,	,	PUNCT
cana-3685	57	24	3	3	NUM
cana-3685	57	25	,	,	PUNCT
cana-3685	57	26	4	4	NUM
cana-3685	57	27	…	…	PUNCT
cana-3685	57	28	respectively	respectively	ADV
cana-3685	57	29	,	,	PUNCT
cana-3685	57	30	f(vi1	f(vi1	NOUN
cana-3685	57	31	)	)	PUNCT
cana-3685	57	32	)	)	PUNCT
cana-3685	58	1	=	=	PUNCT
cana-3685	59	1	i	i	PRON
cana-3685	59	2	+	+	NOUN
cana-3685	59	3	1	1	NUM
cana-3685	59	4	,	,	PUNCT
cana-3685	59	5	where	where	SCONJ
cana-3685	59	6	i	i	PRON
cana-3685	59	7	=	=	NOUN
cana-3685	59	8	2n+4m-4	2n+4m-4	NUM
cana-3685	59	9	when	when	SCONJ
cana-3685	59	10	n	n	X
cana-3685	59	11	=	=	SYM
cana-3685	59	12	5	5	NUM
cana-3685	59	13	,	,	PUNCT
cana-3685	59	14	7	7	NUM
cana-3685	59	15	,	,	PUNCT
cana-3685	59	16	9	9	NUM
cana-3685	59	17	…	…	PUNCT
cana-3685	59	18	then	then	ADV
cana-3685	59	19	m	m	VERB
cana-3685	59	20	=	=	ADJ
cana-3685	59	21	2	2	NUM
cana-3685	59	22	,	,	PUNCT
cana-3685	59	23	3	3	NUM
cana-3685	59	24	,	,	PUNCT
cana-3685	59	25	4	4	NUM
cana-3685	59	26	…	…	PUNCT
cana-3685	59	27	respectively	respectively	ADV
cana-3685	59	28	.	.	PUNCT
cana-3685	60	1	f	f	X
cana-3685	60	2	*	*	PUNCT
cana-3685	60	3	:	:	PUNCT
cana-3685	60	4	e(g	e(g	NOUN
cana-3685	60	5	)	)	PUNCT
cana-3685	60	6	→	→	SYM
cana-3685	60	7	n	n	X
cana-3685	60	8	is	be	AUX
cana-3685	60	9	given	give	VERB
cana-3685	60	10	by	by	ADP
cana-3685	60	11	f	f	PROPN
cana-3685	60	12	*	*	PUNCT
cana-3685	60	13	(	(	PUNCT
cana-3685	60	14	e	e	NOUN
cana-3685	60	15	=	=	NOUN
cana-3685	60	16	uv	uv	NOUN
cana-3685	60	17	)	)	PUNCT
cana-3685	60	18	=	=	SYM
cana-3685	60	19	r	r	NOUN
cana-3685	60	20	cr	cr	NOUN
cana-3685	60	21	=	=	PUNCT
cana-3685	60	22	r!/(r!(r	r!/(r!(r	NOUN
cana-3685	60	23	-	-	PUNCT
cana-3685	60	24	r	r	NOUN
cana-3685	60	25	)	)	PUNCT
cana-3685	60	26	!	!	PUNCT
cana-3685	60	27	)	)	PUNCT
cana-3685	61	1	for	for	ADP
cana-3685	61	2	all	all	DET
cana-3685	61	3	uv	uv	NOUN
cana-3685	61	4	ϵ	ϵ	PROPN
cana-3685	61	5	e(g	e(g	PROPN
cana-3685	61	6	)	)	PUNCT
cana-3685	61	7	where	where	SCONJ
cana-3685	61	8	r	r	NOUN
cana-3685	61	9	=	=	SYM
cana-3685	61	10	max	max	PROPN
cana-3685	61	11	{	{	PUNCT
cana-3685	61	12	f(u	f(u	PROPN
cana-3685	61	13	)	)	PUNCT
cana-3685	61	14	,	,	PUNCT
cana-3685	61	15	f(v	f(v	PROPN
cana-3685	61	16	)	)	PUNCT
cana-3685	61	17	}	}	PUNCT
cana-3685	61	18	and	and	CCONJ
cana-3685	61	19	r	r	NOUN
cana-3685	61	20	=	=	SYM
cana-3685	61	21	min	min	PROPN
cana-3685	61	22	{	{	PUNCT
cana-3685	61	23	f(u	f(u	PROPN
cana-3685	61	24	)	)	PUNCT
cana-3685	61	25	,	,	PUNCT
cana-3685	61	26	f(v)}.here	f(v)}.here	AUX
cana-3685	61	27	all	all	DET
cana-3685	61	28	the	the	DET
cana-3685	61	29	edges	edge	NOUN
cana-3685	61	30	have	have	VERB
cana-3685	61	31	distinct	distinct	ADJ
cana-3685	61	32	labels	label	NOUN
cana-3685	61	33	.	.	PUNCT
cana-3685	62	1	figure	figure	VERB
cana-3685	62	2	7	7	NUM
cana-3685	62	3	:	:	PUNCT
cana-3685	62	4	binomial	binomial	ADJ
cana-3685	62	5	graph	graph	NOUN
cana-3685	62	6	iol5s2	iol5s2	ADJ
cana-3685	62	7	figure	figure	NOUN
cana-3685	62	8	8	8	NUM
cana-3685	62	9	:	:	PUNCT
cana-3685	62	10	binomial	binomial	ADJ
cana-3685	62	11	graph	graph	NOUN
cana-3685	62	12	iol7s3	iol7s3	DET
cana-3685	62	13	special	special	ADJ
cana-3685	62	14	case	case	NOUN
cana-3685	62	15	io	io	X
cana-3685	62	16	l3s1	l3s1	NOUN
cana-3685	62	17	:	:	PUNCT
cana-3685	62	18	let	let	VERB
cana-3685	62	19	v1	v1	NOUN
cana-3685	62	20	,	,	PUNCT
cana-3685	62	21	v2	v2	PROPN
cana-3685	62	22	,	,	PUNCT
cana-3685	62	23	v3	v3	PROPN
cana-3685	62	24	,	,	PUNCT
cana-3685	62	25	v4	v4	PROPN
cana-3685	62	26	,	,	PUNCT
cana-3685	62	27	v5	v5	PROPN
cana-3685	62	28	and	and	CCONJ
cana-3685	62	29	v6	v6	NOUN
cana-3685	62	30	be	be	VERB
cana-3685	62	31	the	the	DET
cana-3685	62	32	six	six	NUM
cana-3685	62	33	vertices	vertex	NOUN
cana-3685	62	34	of	of	ADP
cana-3685	62	35	io	io	X
cana-3685	62	36	l3s1	l3s1	PROPN
cana-3685	62	37	.	.	PUNCT
cana-3685	63	1	vertex	vertex	NOUN
cana-3685	63	2	labelings	labeling	NOUN
cana-3685	63	3	are	be	AUX
cana-3685	63	4	f(vi	f(vi	NOUN
cana-3685	63	5	)	)	PUNCT
cana-3685	64	1	=	=	SYM
cana-3685	64	2	i	i	PROPN
cana-3685	64	3	,	,	PUNCT
cana-3685	64	4	i	i	PRON
cana-3685	64	5	=	=	NOUN
cana-3685	64	6	1	1	NUM
cana-3685	64	7	,	,	PUNCT
cana-3685	64	8	2	2	NUM
cana-3685	64	9	,	,	PUNCT
cana-3685	64	10	3	3	NUM
cana-3685	64	11	,	,	PUNCT
cana-3685	64	12	4	4	NUM
cana-3685	64	13	,	,	PUNCT
cana-3685	64	14	5	5	NUM
cana-3685	64	15	and	and	CCONJ
cana-3685	64	16	f(vi	f(vi	NOUN
cana-3685	64	17	)	)	PUNCT
cana-3685	64	18	=	=	PUNCT
cana-3685	65	1	i+1	i+1	NUM
cana-3685	65	2	,	,	PUNCT
cana-3685	65	3	i	i	PRON
cana-3685	65	4	=	=	NOUN
cana-3685	65	5	6	6	X
cana-3685	65	6	.	.	PUNCT
cana-3685	65	7	figure	figure	NOUN
cana-3685	65	8	9	9	NUM
cana-3685	65	9	:	:	PUNCT
cana-3685	65	10	binomial	binomial	ADJ
cana-3685	65	11	graph	graph	NOUN
cana-3685	65	12	iol3s1	iol3s1	NOUN
cana-3685	65	13	every	every	DET
cana-3685	65	14	edge	edge	NOUN
cana-3685	65	15	in	in	ADP
cana-3685	65	16	this	this	DET
cana-3685	65	17	case	case	NOUN
cana-3685	65	18	has	have	VERB
cana-3685	65	19	a	a	DET
cana-3685	65	20	unique	unique	ADJ
cana-3685	65	21	label	label	NOUN
cana-3685	65	22	.	.	PUNCT
cana-3685	66	1	iolnsm	iolnsm	PROPN
cana-3685	66	2	is	be	AUX
cana-3685	66	3	a	a	DET
cana-3685	66	4	binomial	binomial	ADJ
cana-3685	66	5	graph	graph	NOUN
cana-3685	66	6	as	as	ADP
cana-3685	66	7	a	a	DET
cana-3685	66	8	result	result	NOUN
cana-3685	66	9	.	.	PUNCT
cana-3685	67	1	4	4	X
cana-3685	67	2	.	.	X
cana-3685	67	3	verification	verification	NOUN
cana-3685	67	4	using	use	VERB
cana-3685	67	5	python	python	NOUN
cana-3685	67	6	coding	code	VERB
cana-3685	67	7	4	4	NUM
cana-3685	67	8	.	.	SYM
cana-3685	67	9	1	1	NUM
cana-3685	67	10	verification	verification	NOUN
cana-3685	67	11	of	of	ADP
cana-3685	67	12	the	the	DET
cana-3685	67	13	binomial	binomial	NOUN
cana-3685	67	14	of	of	ADP
cana-3685	67	15	i	i	PROPN
cana-3685	67	16	m	m	PROPN
cana-3685	67	17	l9	l9	PROPN
cana-3685	67	18	s4	s4	PROPN
cana-3685	67	19	and	and	CCONJ
cana-3685	67	20	i	i	PROPN
cana-3685	67	21	m	m	PROPN
cana-3685	67	22	l11	l11	PROPN
cana-3685	67	23	s5	s5	PROPN
cana-3685	67	24	using	use	VERB
cana-3685	67	25	python	python	NOUN
cana-3685	67	26	implementation	implementation	NOUN
cana-3685	67	27	communications	communication	NOUN
cana-3685	67	28	on	on	ADP
cana-3685	67	29	applied	apply	VERB
cana-3685	67	30	nonlinear	nonlinear	ADJ
cana-3685	67	31	analysis	analysis	NOUN
cana-3685	67	32	issn	issn	NOUN
cana-3685	67	33	:	:	PUNCT
cana-3685	67	34	1074	1074	NUM
cana-3685	67	35	-	-	PUNCT
cana-3685	67	36	133x	133x	NUM
cana-3685	67	37	vol	vol	NOUN
cana-3685	67	38	32	32	NUM
cana-3685	67	39	no	no	NOUN
cana-3685	67	40	.	.	PUNCT
cana-3685	68	1	8s	8s	PROPN
cana-3685	68	2	(	(	PUNCT
cana-3685	68	3	2025	2025	NUM
cana-3685	68	4	)	)	PUNCT
cana-3685	68	5	418	418	NUM
cana-3685	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3685	68	7	4	4	NUM
cana-3685	68	8	.	.	SYM
cana-3685	68	9	2	2	NUM
cana-3685	68	10	verification	verification	NOUN
cana-3685	68	11	of	of	ADP
cana-3685	68	12	the	the	DET
cana-3685	68	13	binomial	binomial	NOUN
cana-3685	68	14	of	of	ADP
cana-3685	68	15	io	io	PROPN
cana-3685	68	16	l9	l9	PROPN
cana-3685	68	17	s4	s4	PROPN
cana-3685	68	18	and	and	CCONJ
cana-3685	68	19	io	io	PROPN
cana-3685	68	20	l11	l11	PROPN
cana-3685	68	21	s5	s5	PROPN
cana-3685	68	22	using	use	VERB
cana-3685	68	23	python	python	NOUN
cana-3685	68	24	implementation	implementation	NOUN
cana-3685	68	25	communications	communication	NOUN
cana-3685	68	26	on	on	ADP
cana-3685	68	27	applied	apply	VERB
cana-3685	68	28	nonlinear	nonlinear	ADJ
cana-3685	68	29	analysis	analysis	NOUN
cana-3685	68	30	issn	issn	NOUN
cana-3685	68	31	:	:	PUNCT
cana-3685	68	32	1074	1074	NUM
cana-3685	68	33	-	-	PUNCT
cana-3685	68	34	133x	133x	NUM
cana-3685	68	35	vol	vol	NOUN
cana-3685	68	36	32	32	NUM
cana-3685	68	37	no	no	NOUN
cana-3685	68	38	.	.	PUNCT
cana-3685	69	1	8s	8s	PROPN
cana-3685	69	2	(	(	PUNCT
cana-3685	69	3	2025	2025	NUM
cana-3685	69	4	)	)	PUNCT
cana-3685	69	5	419	419	NUM
cana-3685	70	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3685	70	2	5	5	X
cana-3685	70	3	.	.	PUNCT
cana-3685	70	4	conclusion	conclusion	VERB
cana-3685	70	5	the	the	DET
cana-3685	70	6	demonstration	demonstration	NOUN
cana-3685	70	7	of	of	ADP
cana-3685	70	8	graphs	graph	NOUN
cana-3685	70	9	like	like	ADP
cana-3685	70	10	intersection	intersection	NOUN
cana-3685	70	11	mirror	mirror	NOUN
cana-3685	70	12	lagoon	lagoon	NOUN
cana-3685	70	13	step	step	NOUN
cana-3685	70	14	graph	graph	NOUN
cana-3685	70	15	and	and	CCONJ
cana-3685	70	16	intersection	intersection	NOUN
cana-3685	70	17	opposite	opposite	ADJ
cana-3685	70	18	lagoon	lagoon	NOUN
cana-3685	70	19	step	step	NOUN
cana-3685	70	20	graph	graph	NOUN
cana-3685	70	21	exist	exist	VERB
cana-3685	70	22	when	when	SCONJ
cana-3685	70	23	it	it	PRON
cana-3685	70	24	comes	come	VERB
cana-3685	70	25	to	to	ADP
cana-3685	70	26	binomial	binomial	ADJ
cana-3685	70	27	labeling	labeling	NOUN
cana-3685	70	28	.	.	PUNCT
cana-3685	71	1	this	this	DET
cana-3685	71	2	paper	paper	NOUN
cana-3685	71	3	also	also	ADV
cana-3685	71	4	discusses	discuss	VERB
cana-3685	71	5	the	the	DET
cana-3685	71	6	limitations	limitation	NOUN
cana-3685	71	7	and	and	CCONJ
cana-3685	71	8	possible	possible	ADJ
cana-3685	71	9	uses	use	NOUN
cana-3685	71	10	of	of	ADP
cana-3685	71	11	this	this	DET
cana-3685	71	12	type	type	NOUN
cana-3685	71	13	of	of	ADP
cana-3685	71	14	labeling	labeling	NOUN
cana-3685	71	15	.	.	PUNCT
cana-3685	72	1	python	python	NOUN
cana-3685	72	2	coding	code	VERB
cana-3685	72	3	for	for	ADP
cana-3685	72	4	binomial	binomial	ADJ
cana-3685	72	5	labeling	labeling	NOUN
cana-3685	72	6	has	have	AUX
cana-3685	72	7	been	be	AUX
cana-3685	72	8	implemented	implement	VERB
cana-3685	72	9	for	for	ADP
cana-3685	72	10	the	the	DET
cana-3685	72	11	above	above	ADJ
cana-3685	72	12	graphs	graph	NOUN
cana-3685	72	13	.	.	PUNCT
cana-3685	73	1	in	in	ADP
cana-3685	73	2	the	the	DET
cana-3685	73	3	near	near	ADJ
cana-3685	73	4	future	future	NOUN
cana-3685	73	5	,	,	PUNCT
cana-3685	73	6	more	more	ADJ
cana-3685	73	7	research	research	NOUN
cana-3685	73	8	under	under	ADP
cana-3685	73	9	the	the	DET
cana-3685	73	10	lagoon	lagoon	NOUN
cana-3685	73	11	step	step	NOUN
cana-3685	73	12	graph	graph	NOUN
cana-3685	73	13	using	use	VERB
cana-3685	73	14	various	various	ADJ
cana-3685	73	15	labeling	labeling	NOUN
cana-3685	73	16	strategies	strategy	NOUN
cana-3685	73	17	is	be	AUX
cana-3685	73	18	needed	need	VERB
cana-3685	73	19	.	.	PUNCT
cana-3685	74	1	references	reference	NOUN
cana-3685	74	2	[	[	X
cana-3685	74	3	1	1	NUM
cana-3685	74	4	]	]	PUNCT
cana-3685	74	5	a.	a.	NOUN
cana-3685	74	6	rosa	rosa	PROPN
cana-3685	74	7	,	,	PUNCT
cana-3685	74	8	on	on	ADP
cana-3685	74	9	certain	certain	ADJ
cana-3685	74	10	valuations	valuation	NOUN
cana-3685	74	11	of	of	ADP
cana-3685	74	12	the	the	DET
cana-3685	74	13	vertices	vertex	NOUN
cana-3685	74	14	of	of	ADP
cana-3685	74	15	a	a	DET
cana-3685	74	16	graph	graph	NOUN
cana-3685	74	17	,	,	PUNCT
cana-3685	74	18	in	in	ADP
cana-3685	74	19	theory	theory	NOUN
cana-3685	74	20	of	of	ADP
cana-3685	74	21	graphs	graph	NOUN
cana-3685	74	22	(	(	PUNCT
cana-3685	74	23	internat	internat	NOUN
cana-3685	74	24	.	.	PUNCT
cana-3685	75	1	sympos	sympos	PROPN
cana-3685	75	2	.	.	PUNCT
cana-3685	76	1	rome	rome	PROPN
cana-3685	76	2	.	.	PUNCT
cana-3685	77	1	(	(	PUNCT
cana-3685	77	2	1966	1966	NUM
cana-3685	77	3	)	)	PUNCT
cana-3685	77	4	(	(	PUNCT
cana-3685	77	5	1967	1967	NUM
cana-3685	77	6	)	)	PUNCT
cana-3685	77	7	,	,	PUNCT
cana-3685	77	8	349–359	349–359	NUM
cana-3685	77	9	,	,	PUNCT
cana-3685	77	10	gordan	gordan	PROPN
cana-3685	77	11	and	and	CCONJ
cana-3685	77	12	breach	breach	PROPN
cana-3685	77	13	.	.	PUNCT
cana-3685	78	1	newyork	newyork	PROPN
cana-3685	78	2	.	.	PUNCT
cana-3685	79	1	dunod	dunod	PROPN
cana-3685	79	2	.	.	PUNCT
cana-3685	80	1	paris	paris	PROPN
cana-3685	80	2	.	.	PUNCT
cana-3685	81	1	[	[	X
cana-3685	81	2	2	2	X
cana-3685	81	3	]	]	X
cana-3685	81	4	j.a	j.a	PROPN
cana-3685	81	5	.	.	PROPN
cana-3685	81	6	gallian	gallian	PROPN
cana-3685	81	7	,	,	PUNCT
cana-3685	81	8	a	a	DET
cana-3685	81	9	dynamic	dynamic	ADJ
cana-3685	81	10	survey	survey	NOUN
cana-3685	81	11	of	of	ADP
cana-3685	81	12	graph	graph	NOUN
cana-3685	81	13	labeling	labeling	NOUN
cana-3685	81	14	,	,	PUNCT
cana-3685	81	15	electronic	electronic	ADJ
cana-3685	81	16	journal	journal	NOUN
cana-3685	81	17	of	of	ADP
cana-3685	81	18	combinatorics	combinatoric	NOUN
cana-3685	81	19	,	,	PUNCT
cana-3685	81	20	17	17	NUM
cana-3685	81	21	(	(	PUNCT
cana-3685	81	22	ds6	ds6	NOUN
cana-3685	81	23	)	)	PUNCT
cana-3685	81	24	(	(	PUNCT
cana-3685	81	25	2016	2016	NUM
cana-3685	81	26	)	)	PUNCT
cana-3685	81	27	.	.	PUNCT
cana-3685	82	1	[	[	X
cana-3685	82	2	3	3	X
cana-3685	82	3	]	]	X
cana-3685	82	4	bondy	bondy	X
cana-3685	82	5	.j.a	.j.a	PROPN
cana-3685	82	6	and	and	CCONJ
cana-3685	82	7	murthy	murthy	PROPN
cana-3685	82	8	u.s.r	u.s.r	NOUN
cana-3685	82	9	,	,	PUNCT
cana-3685	82	10	“	"	PUNCT
cana-3685	82	11	graph	graph	NOUN
cana-3685	82	12	theory	theory	NOUN
cana-3685	82	13	and	and	CCONJ
cana-3685	82	14	application	application	NOUN
cana-3685	82	15	”	"	PUNCT
cana-3685	82	16	(	(	PUNCT
cana-3685	82	17	north	north	NOUN
cana-3685	82	18	holland	holland	PROPN
cana-3685	82	19	)	)	PUNCT
cana-3685	82	20	.	.	PUNCT
cana-3685	83	1	new	new	PROPN
cana-3685	83	2	york	york	PROPN
cana-3685	83	3	(	(	PUNCT
cana-3685	83	4	1976	1976	NUM
cana-3685	83	5	)	)	PUNCT
cana-3685	83	6	.	.	PUNCT
cana-3685	84	1	[	[	X
cana-3685	84	2	4].s	4].s	NUM
cana-3685	84	3	.	.	PUNCT
cana-3685	84	4	chandrakala	chandrakala	PROPN
cana-3685	84	5	and	and	CCONJ
cana-3685	84	6	c.sekar	c.sekar	ADJ
cana-3685	84	7	,	,	PUNCT
cana-3685	84	8	“	"	PUNCT
cana-3685	84	9	binomial	binomial	ADJ
cana-3685	84	10	labeling	labeling	NOUN
cana-3685	84	11	of	of	ADP
cana-3685	84	12	some	some	DET
cana-3685	84	13	graphs”,international	graphs”,international	ADJ
cana-3685	84	14	journal	journal	NOUN
cana-3685	84	15	for	for	ADP
cana-3685	84	16	research	research	NOUN
cana-3685	84	17	in	in	ADP
cana-3685	84	18	engineering	engineering	NOUN
cana-3685	84	19	application	application	NOUN
cana-3685	84	20	&	&	CCONJ
cana-3685	84	21	management	management	PROPN
cana-3685	84	22	issn	issn	PROPN
cana-3685	84	23	:	:	PUNCT
cana-3685	84	24	2454	2454	NUM
cana-3685	84	25	-	-	SYM
cana-3685	84	26	9150	9150	NUM
cana-3685	84	27	,	,	PUNCT
cana-3685	84	28	vol.04,issue	vol.04,issue	NUM
cana-3685	84	29	.	.	PUNCT
cana-3685	85	1	08	08	NUM
cana-3685	85	2	,	,	PUNCT
cana-3685	85	3	nov	nov	PROPN
cana-3685	85	4	2018	2018	NUM
cana-3685	85	5	.	.	PUNCT
cana-3685	86	1	[	[	X
cana-3685	86	2	5	5	X
cana-3685	86	3	]	]	X
cana-3685	86	4	d.	d.	PROPN
cana-3685	86	5	amuthavalli	amuthavalli	PROPN
cana-3685	86	6	,	,	PUNCT
cana-3685	86	7	o.	o.	PROPN
cana-3685	86	8	v.	v.	PROPN
cana-3685	86	9	shanmuga	shanmuga	PROPN
cana-3685	86	10	sundaram	sundaram	PROPN
cana-3685	86	11	,	,	PUNCT
cana-3685	86	12	super	super	PROPN
cana-3685	86	13	fibonacci	fibonacci	PROPN
cana-3685	86	14	graceful	graceful	ADJ
cana-3685	86	15	anti	anti	ADJ
cana-3685	86	16	–	–	PUNCT
cana-3685	86	17	magic	magic	ADJ
cana-3685	86	18	labeling	labeling	NOUN
cana-3685	86	19	for	for	ADP
cana-3685	86	20	flower	flower	NOUN
cana-3685	86	21	graphs	graph	NOUN
cana-3685	86	22	and	and	CCONJ
cana-3685	86	23	python	python	NOUN
cana-3685	86	24	coding	coding	NOUN
cana-3685	86	25	,	,	PUNCT
cana-3685	86	26	tuijin	tuijin	PROPN
cana-3685	86	27	jishu	jishu	PROPN
cana-3685	86	28	/	/	SYM
cana-3685	86	29	journal	journal	NOUN
cana-3685	86	30	of	of	ADP
cana-3685	86	31	propulsion	propulsion	NOUN
cana-3685	86	32	technology	technology	NOUN
cana-3685	86	33	,	,	PUNCT
cana-3685	86	34	vol	vol	NOUN
cana-3685	86	35	.	.	PROPN
cana-3685	86	36	44	44	NUM
cana-3685	86	37	no	no	NOUN
cana-3685	86	38	.	.	PUNCT
cana-3685	87	1	3,2023	3,2023	NUM
cana-3685	87	2	.	.	PUNCT
cana-3685	88	1	[	[	X
cana-3685	88	2	6	6	NUM
cana-3685	88	3	]	]	X
cana-3685	88	4	j.a.gadhiya	j.a.gadhiya	NOUN
cana-3685	88	5	and	and	CCONJ
cana-3685	88	6	r.solanki	r.solanki	NOUN
cana-3685	88	7	,	,	PUNCT
cana-3685	88	8	“	"	PUNCT
cana-3685	88	9	labeling	labeling	NOUN
cana-3685	88	10	of	of	ADP
cana-3685	88	11	some	some	DET
cana-3685	88	12	graphs	graph	NOUN
cana-3685	88	13	using	use	VERB
cana-3685	88	14	python	python	NOUN
cana-3685	88	15	programming	programming	NOUN
cana-3685	88	16	,	,	PUNCT
cana-3685	88	17	”	"	PUNCT
cana-3685	88	18	vol.20,no.17,pp.2288	vol.20,no.17,pp.2288	ADJ
cana-3685	88	19	2294,2022	2294,2022	NOUN
cana-3685	88	20	.	.	PUNCT