id	sid	tid	token	lemma	pos
cana-410	1	1	communications	communication	NOUN
cana-410	1	2	on	on	ADP
cana-410	1	3	applied	apply	VERB
cana-410	1	4	nonlinear	nonlinear	ADJ
cana-410	1	5	analysis	analysis	NOUN
cana-410	1	6	issn	issn	NOUN
cana-410	1	7	:	:	PUNCT
cana-410	1	8	1074	1074	NUM
cana-410	1	9	-	-	PUNCT
cana-410	1	10	133x	133x	NUM
cana-410	1	11	vol	vol	NOUN
cana-410	1	12	31	31	NUM
cana-410	1	13	no	no	NOUN
cana-410	1	14	.	.	NOUN
cana-410	1	15	1	1	NUM
cana-410	1	16	(	(	PUNCT
cana-410	1	17	2024	2024	NUM
cana-410	1	18	)	)	PUNCT
cana-410	1	19	253	253	NUM
cana-410	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	1	21	a	a	DET
cana-410	1	22	study	study	NOUN
cana-410	1	23	on	on	ADP
cana-410	1	24	emergence	emergence	NOUN
cana-410	1	25	of	of	ADP
cana-410	1	26	clutch	clutch	ADJ
cana-410	1	27	graphs	graph	NOUN
cana-410	1	28	from	from	ADP
cana-410	1	29	cycle	cycle	NOUN
cana-410	1	30	graphs	graph	NOUN
cana-410	1	31	:	:	PUNCT
cana-410	1	32	a	a	DET
cana-410	1	33	comprehensive	comprehensive	ADJ
cana-410	1	34	analysis	analysis	NOUN
cana-410	1	35	tamilselvi	tamilselvi	NOUN
cana-410	1	36	v1	v1	PROPN
cana-410	1	37	,	,	PUNCT
cana-410	1	38	thamizhendhi	thamizhendhi	NOUN
cana-410	1	39	g2	g2	PROPN
cana-410	1	40	1	1	NUM
cana-410	1	41	assistant	assistant	NOUN
cana-410	1	42	professor	professor	NOUN
cana-410	1	43	of	of	ADP
cana-410	1	44	mathematics	mathematic	NOUN
cana-410	1	45	,	,	PUNCT
cana-410	1	46	vellalar	vellalar	ADJ
cana-410	1	47	college	college	NOUN
cana-410	1	48	for	for	ADP
cana-410	1	49	women(autonomous	women(autonomous	PROPN
cana-410	1	50	)	)	PUNCT
cana-410	1	51	,	,	PUNCT
cana-410	1	52	erode	erode	VERB
cana-410	1	53	,	,	PUNCT
cana-410	1	54	tamilnadu	tamilnadu	NOUN
cana-410	1	55	,	,	PUNCT
cana-410	1	56	india	india	PROPN
cana-410	1	57	,	,	PUNCT
cana-410	1	58	email	email	NOUN
cana-410	2	1	i	i	PROPN
cana-410	2	2	d	d	PROPN
cana-410	2	3	:	:	PUNCT
cana-410	2	4	tmlselvi18@gmail.com	tmlselvi18@gmail.com	X
cana-410	2	5	2	2	NUM
cana-410	2	6	assistant	assistant	NOUN
cana-410	2	7	professor	professor	NOUN
cana-410	2	8	of	of	ADP
cana-410	2	9	mathematics	mathematic	NOUN
cana-410	2	10	,	,	PUNCT
cana-410	2	11	sri	sri	PROPN
cana-410	2	12	vasavi	vasavi	PROPN
cana-410	2	13	college	college	PROPN
cana-410	2	14	,	,	PUNCT
cana-410	2	15	erode	erode	VERB
cana-410	2	16	,	,	PUNCT
cana-410	2	17	tamilnadu	tamilnadu	NOUN
cana-410	2	18	,	,	PUNCT
cana-410	2	19	india	india	PROPN
cana-410	2	20	,	,	PUNCT
cana-410	2	21	email	email	NOUN
cana-410	3	1	i	i	PROPN
cana-410	3	2	d	d	PROPN
cana-410	3	3	:	:	PUNCT
cana-410	3	4	gkthamil@gmail.com	gkthamil@gmail.com	X
cana-410	3	5	article	article	NOUN
cana-410	3	6	history	history	NOUN
cana-410	3	7	:	:	PUNCT
cana-410	3	8	received	receive	VERB
cana-410	3	9	:	:	PUNCT
cana-410	3	10	28	28	NUM
cana-410	3	11	-	-	SYM
cana-410	3	12	10	10	NUM
cana-410	3	13	-	-	PUNCT
cana-410	3	14	2023	2023	NUM
cana-410	3	15	revised	revise	VERB
cana-410	3	16	:	:	PUNCT
cana-410	3	17	09	09	NUM
cana-410	3	18	-	-	SYM
cana-410	3	19	12	12	NUM
cana-410	3	20	-	-	PUNCT
cana-410	3	21	2023	2023	NUM
cana-410	3	22	accepted	accept	VERB
cana-410	3	23	:	:	PUNCT
cana-410	3	24	24	24	NUM
cana-410	3	25	-	-	SYM
cana-410	3	26	12	12	NUM
cana-410	3	27	-	-	PUNCT
cana-410	3	28	2023	2023	NUM
cana-410	3	29	abstract	abstract	NOUN
cana-410	3	30	:	:	PUNCT
cana-410	3	31	the	the	DET
cana-410	3	32	clutch	clutch	NOUN
cana-410	3	33	is	be	AUX
cana-410	3	34	one	one	NUM
cana-410	3	35	of	of	ADP
cana-410	3	36	the	the	DET
cana-410	3	37	substantial	substantial	ADJ
cana-410	3	38	devices	device	NOUN
cana-410	3	39	for	for	ADP
cana-410	3	40	constructing	construct	VERB
cana-410	3	41	vehicles	vehicle	NOUN
cana-410	3	42	in	in	ADP
cana-410	3	43	automobile	automobile	NOUN
cana-410	3	44	engineering	engineering	NOUN
cana-410	3	45	.	.	PUNCT
cana-410	4	1	in	in	ADP
cana-410	4	2	terms	term	NOUN
cana-410	4	3	of	of	ADP
cana-410	4	4	network	network	NOUN
cana-410	4	5	toughness	toughness	NOUN
cana-410	4	6	against	against	ADP
cana-410	4	7	failures	failure	NOUN
cana-410	4	8	or	or	CCONJ
cana-410	4	9	disruptions	disruption	NOUN
cana-410	4	10	,	,	PUNCT
cana-410	4	11	the	the	DET
cana-410	4	12	clutchbased	clutchbase	VERB
cana-410	4	13	graph	graph	NOUN
cana-410	4	14	can	can	AUX
cana-410	4	15	be	be	AUX
cana-410	4	16	used	use	VERB
cana-410	4	17	to	to	PART
cana-410	4	18	model	model	VERB
cana-410	4	19	insufficiency	insufficiency	NOUN
cana-410	4	20	of	of	ADP
cana-410	4	21	primary	primary	ADJ
cana-410	4	22	pathways	pathway	NOUN
cana-410	4	23	and	and	CCONJ
cana-410	4	24	the	the	DET
cana-410	4	25	availability	availability	NOUN
cana-410	4	26	of	of	ADP
cana-410	4	27	alternative	alternative	ADJ
cana-410	4	28	routes	route	NOUN
cana-410	4	29	in	in	ADP
cana-410	4	30	communication	communication	NOUN
cana-410	4	31	networks	network	NOUN
cana-410	4	32	.	.	PUNCT
cana-410	5	1	in	in	ADP
cana-410	5	2	this	this	DET
cana-410	5	3	paper	paper	NOUN
cana-410	5	4	,	,	PUNCT
cana-410	5	5	the	the	DET
cana-410	5	6	identification	identification	NOUN
cana-410	5	7	of	of	ADP
cana-410	5	8	the	the	DET
cana-410	5	9	clutch	clutch	NOUN
cana-410	5	10	graph	graph	NOUN
cana-410	5	11	cl3n(g	cl3n(g	NOUN
cana-410	5	12	)	)	PUNCT
cana-410	5	13	from	from	ADP
cana-410	5	14	the	the	DET
cana-410	5	15	cycle	cycle	NOUN
cana-410	5	16	graph	graph	NOUN
cana-410	5	17	cn(g	cn(g	X
cana-410	5	18	)	)	PUNCT
cana-410	5	19	has	have	AUX
cana-410	5	20	been	be	AUX
cana-410	5	21	proposed	propose	VERB
cana-410	5	22	.	.	PUNCT
cana-410	6	1	a	a	DET
cana-410	6	2	clutch	clutch	ADJ
cana-410	6	3	graph	graph	NOUN
cana-410	6	4	cl3n(g	cl3n(g	NOUN
cana-410	6	5	)	)	PUNCT
cana-410	6	6	generating	generate	VERB
cana-410	6	7	from	from	ADP
cana-410	6	8	a	a	DET
cana-410	6	9	cycle	cycle	NOUN
cana-410	6	10	graph	graph	NOUN
cana-410	6	11	with	with	ADP
cana-410	6	12	3n	3n	NUM
cana-410	6	13	vertices	vertex	NOUN
cana-410	6	14	and	and	CCONJ
cana-410	6	15	4n	4n	ADJ
cana-410	6	16	edges	edge	NOUN
cana-410	6	17	(	(	PUNCT
cana-410	6	18	n	n	CCONJ
cana-410	6	19	≥	≥	NOUN
cana-410	6	20	4	4	NUM
cana-410	6	21	and	and	CCONJ
cana-410	6	22	even	even	ADV
cana-410	6	23	)	)	PUNCT
cana-410	6	24	.	.	PUNCT
cana-410	7	1	the	the	DET
cana-410	7	2	notions	notion	NOUN
cana-410	7	3	of	of	ADP
cana-410	7	4	degree	degree	NOUN
cana-410	7	5	,	,	PUNCT
cana-410	7	6	girth	girth	NOUN
cana-410	7	7	,	,	PUNCT
cana-410	7	8	and	and	CCONJ
cana-410	7	9	chromatic	chromatic	ADJ
cana-410	7	10	number	number	NOUN
cana-410	7	11	of	of	ADP
cana-410	7	12	the	the	DET
cana-410	7	13	clutch	clutch	NOUN
cana-410	7	14	graph	graph	NOUN
cana-410	7	15	have	have	AUX
cana-410	7	16	been	be	AUX
cana-410	7	17	discussed	discuss	VERB
cana-410	7	18	.	.	PUNCT
cana-410	8	1	further	far	ADV
cana-410	8	2	,	,	PUNCT
cana-410	8	3	the	the	DET
cana-410	8	4	existence	existence	NOUN
cana-410	8	5	of	of	ADP
cana-410	8	6	the	the	DET
cana-410	8	7	bipartite	bipartite	NOUN
cana-410	8	8	and	and	CCONJ
cana-410	8	9	hamiltonian	hamiltonian	ADJ
cana-410	8	10	graphs	graph	NOUN
cana-410	8	11	on	on	ADP
cana-410	8	12	the	the	DET
cana-410	8	13	clutch	clutch	NOUN
cana-410	8	14	graph	graph	NOUN
cana-410	8	15	has	have	AUX
cana-410	8	16	been	be	AUX
cana-410	8	17	examined	examine	VERB
cana-410	8	18	.	.	PUNCT
cana-410	9	1	keywords	keyword	NOUN
cana-410	9	2	:	:	PUNCT
cana-410	9	3	clutch	clutch	NOUN
cana-410	9	4	graph	graph	NOUN
cana-410	9	5	,	,	PUNCT
cana-410	9	6	cycle	cycle	NOUN
cana-410	9	7	graph	graph	NOUN
cana-410	9	8	,	,	PUNCT
cana-410	9	9	spanning	span	VERB
cana-410	9	10	tree	tree	NOUN
cana-410	9	11	,	,	PUNCT
cana-410	9	12	bipartite	bipartite	ADJ
cana-410	9	13	,	,	PUNCT
cana-410	9	14	hamiltonian	hamiltonian	NOUN
cana-410	9	15	.	.	PUNCT
cana-410	10	1	1	1	X
cana-410	10	2	.	.	X
cana-410	10	3	introduction	introduction	NOUN
cana-410	10	4	graph	graph	NOUN
cana-410	10	5	theory	theory	NOUN
cana-410	10	6	explores	explore	VERB
cana-410	10	7	connections	connection	NOUN
cana-410	10	8	between	between	ADP
cana-410	10	9	vertices	vertex	NOUN
cana-410	10	10	and	and	CCONJ
cana-410	10	11	edges	edge	NOUN
cana-410	10	12	,	,	PUNCT
cana-410	10	13	offering	offer	VERB
cana-410	10	14	a	a	DET
cana-410	10	15	way	way	NOUN
cana-410	10	16	to	to	PART
cana-410	10	17	understand	understand	VERB
cana-410	10	18	and	and	CCONJ
cana-410	10	19	analyze	analyze	VERB
cana-410	10	20	various	various	ADJ
cana-410	10	21	real	real	ADJ
cana-410	10	22	-	-	PUNCT
cana-410	10	23	world	world	NOUN
cana-410	10	24	systems	system	NOUN
cana-410	10	25	[	[	X
cana-410	10	26	6	6	NUM
cana-410	10	27	]	]	PUNCT
cana-410	10	28	.	.	PUNCT
cana-410	11	1	a	a	DET
cana-410	11	2	cycle	cycle	NOUN
cana-410	11	3	in	in	ADP
cana-410	11	4	graph	graph	NOUN
cana-410	11	5	theory	theory	NOUN
cana-410	11	6	is	be	AUX
cana-410	11	7	a	a	DET
cana-410	11	8	closed	closed	ADJ
cana-410	11	9	path	path	NOUN
cana-410	11	10	that	that	PRON
cana-410	11	11	starts	start	VERB
cana-410	11	12	and	and	CCONJ
cana-410	11	13	ends	end	VERB
cana-410	11	14	at	at	ADP
cana-410	11	15	the	the	DET
cana-410	11	16	same	same	ADJ
cana-410	11	17	vertex	vertex	NOUN
cana-410	11	18	.	.	PUNCT
cana-410	12	1	a	a	DET
cana-410	12	2	cycle	cycle	NOUN
cana-410	12	3	graph	graph	NOUN
cana-410	12	4	is	be	AUX
cana-410	12	5	a	a	DET
cana-410	12	6	graph	graph	NOUN
cana-410	12	7	composed	compose	VERB
cana-410	12	8	of	of	ADP
cana-410	12	9	a	a	DET
cana-410	12	10	single	single	ADJ
cana-410	12	11	cycle	cycle	NOUN
cana-410	12	12	,	,	PUNCT
cana-410	12	13	forming	form	VERB
cana-410	12	14	a	a	DET
cana-410	12	15	circular	circular	ADJ
cana-410	12	16	structure	structure	NOUN
cana-410	12	17	[	[	X
cana-410	12	18	7	7	NUM
cana-410	12	19	]	]	PUNCT
cana-410	12	20	.	.	PUNCT
cana-410	13	1	a	a	DET
cana-410	13	2	clutch	clutch	NOUN
cana-410	13	3	is	be	AUX
cana-410	13	4	a	a	DET
cana-410	13	5	mechanical	mechanical	ADJ
cana-410	13	6	device	device	NOUN
cana-410	13	7	that	that	PRON
cana-410	13	8	connects	connect	VERB
cana-410	13	9	or	or	CCONJ
cana-410	13	10	disconnects	disconnect	VERB
cana-410	13	11	power	power	NOUN
cana-410	13	12	transmission	transmission	NOUN
cana-410	13	13	,	,	PUNCT
cana-410	13	14	enabling	enable	VERB
cana-410	13	15	smooth	smooth	ADJ
cana-410	13	16	control	control	NOUN
cana-410	13	17	over	over	ADP
cana-410	13	18	energy	energy	NOUN
cana-410	13	19	transfer	transfer	NOUN
cana-410	13	20	[	[	X
cana-410	13	21	8	8	NUM
cana-410	13	22	]	]	PUNCT
cana-410	13	23	.	.	PUNCT
cana-410	14	1	in	in	ADP
cana-410	14	2	this	this	DET
cana-410	14	3	way	way	NOUN
cana-410	14	4	,	,	PUNCT
cana-410	14	5	the	the	DET
cana-410	14	6	authors	author	NOUN
cana-410	14	7	motivated	motivate	VERB
cana-410	14	8	to	to	PART
cana-410	14	9	define	define	VERB
cana-410	14	10	clutch	clutch	NOUN
cana-410	14	11	based	base	VERB
cana-410	14	12	graph	graph	NOUN
cana-410	14	13	that	that	PRON
cana-410	14	14	is	be	AUX
cana-410	14	15	used	use	VERB
cana-410	14	16	to	to	PART
cana-410	14	17	design	design	VERB
cana-410	14	18	networks	network	NOUN
cana-410	14	19	.	.	PUNCT
cana-410	15	1	clutch	clutch	NOUN
cana-410	15	2	graph	graph	NOUN
cana-410	15	3	based	base	VERB
cana-410	15	4	network	network	NOUN
cana-410	15	5	models	model	NOUN
cana-410	15	6	effective	effective	ADJ
cana-410	15	7	in	in	ADP
cana-410	15	8	identifying	identify	VERB
cana-410	15	9	the	the	DET
cana-410	15	10	smooth	smooth	ADJ
cana-410	15	11	control	control	NOUN
cana-410	15	12	of	of	ADP
cana-410	15	13	data	datum	NOUN
cana-410	15	14	transmission	transmission	NOUN
cana-410	15	15	between	between	ADP
cana-410	15	16	entities	entity	NOUN
cana-410	15	17	.	.	PUNCT
cana-410	16	1	the	the	DET
cana-410	16	2	clutch	clutch	ADJ
cana-410	16	3	graph	graph	NOUN
cana-410	16	4	inherits	inherit	VERB
cana-410	16	5	the	the	DET
cana-410	16	6	properties	property	NOUN
cana-410	16	7	of	of	ADP
cana-410	16	8	bipartite	bipartite	NOUN
cana-410	16	9	and	and	CCONJ
cana-410	16	10	hamiltonian	hamiltonian	ADJ
cana-410	16	11	graphs	graph	NOUN
cana-410	16	12	.	.	PUNCT
cana-410	17	1	the	the	DET
cana-410	17	2	chromatic	chromatic	ADJ
cana-410	17	3	number	number	NOUN
cana-410	17	4	and	and	CCONJ
cana-410	17	5	girth	girth	NOUN
cana-410	17	6	have	have	AUX
cana-410	17	7	been	be	AUX
cana-410	17	8	explored	explore	VERB
cana-410	17	9	with	with	ADP
cana-410	17	10	appropriate	appropriate	ADJ
cana-410	17	11	illustration	illustration	NOUN
cana-410	17	12	.	.	PUNCT
cana-410	18	1	2	2	X
cana-410	18	2	.	.	X
cana-410	18	3	preliminaries	preliminary	NOUN
cana-410	18	4	definition	definition	NOUN
cana-410	18	5	2.1:[1	2.1:[1	NUM
cana-410	18	6	]	]	PUNCT
cana-410	18	7	the	the	DET
cana-410	18	8	cycle	cycle	NOUN
cana-410	18	9	cn	cn	PROPN
cana-410	18	10	,	,	PUNCT
cana-410	18	11	n	n	CCONJ
cana-410	18	12	≥	≥	NOUN
cana-410	18	13	3	3	NUM
cana-410	18	14	,	,	PUNCT
cana-410	18	15	made	make	VERB
cana-410	18	16	up	up	ADP
cana-410	18	17	of	of	ADP
cana-410	18	18	n	n	PRON
cana-410	18	19	vertices	vertex	NOUN
cana-410	18	20	c1	c1	PROPN
cana-410	18	21	,	,	PUNCT
cana-410	18	22	c2	c2	PROPN
cana-410	18	23	,	,	PUNCT
cana-410	18	24	.	.	PUNCT
cana-410	18	25	.	.	PUNCT
cana-410	19	1	.	.	PUNCT
cana-410	20	1	,	,	PUNCT
cana-410	20	2	cn	cn	PROPN
cana-410	20	3	and	and	CCONJ
cana-410	20	4	edges	edge	VERB
cana-410	20	5	{	{	PUNCT
cana-410	20	6	c1	c1	NOUN
cana-410	20	7	,	,	PUNCT
cana-410	20	8	c2},{c2	c2},{c2	PROPN
cana-410	20	9	,	,	PUNCT
cana-410	20	10	c3	c3	NOUN
cana-410	20	11	}	}	PUNCT
cana-410	20	12	,	,	PUNCT
cana-410	20	13	.	.	PUNCT
cana-410	20	14	.	.	PUNCT
cana-410	20	15	.	.	PUNCT
cana-410	21	1	,	,	PUNCT
cana-410	21	2	{	{	PUNCT
cana-410	21	3	cn−1	cn−1	PROPN
cana-410	21	4	,	,	PUNCT
cana-410	21	5	cn},{cn	cn},{cn	PROPN
cana-410	21	6	,	,	PUNCT
cana-410	21	7	c1	c1	PROPN
cana-410	21	8	}	}	PUNCT
cana-410	21	9	.	.	PUNCT
cana-410	22	1	the	the	DET
cana-410	22	2	cycles	cycle	NOUN
cana-410	22	3	c3	c3	PROPN
cana-410	22	4	and	and	CCONJ
cana-410	22	5	c4	c4	NOUN
cana-410	22	6	are	be	AUX
cana-410	22	7	shown	show	VERB
cana-410	22	8	in	in	ADP
cana-410	22	9	figure	figure	NOUN
cana-410	22	10	1	1	NUM
cana-410	22	11	and	and	CCONJ
cana-410	22	12	figure	figure	VERB
cana-410	22	13	2	2	NUM
cana-410	22	14	.	.	PUNCT
cana-410	23	1	mailto:tmlselvi18@gmail.com	mailto:tmlselvi18@gmail.com	PROPN
cana-410	23	2	mailto:gkthamil@gmail.com	mailto:gkthamil@gmail.com	PROPN
cana-410	23	3	communications	communication	NOUN
cana-410	23	4	on	on	ADP
cana-410	23	5	applied	apply	VERB
cana-410	23	6	nonlinear	nonlinear	ADJ
cana-410	23	7	analysis	analysis	NOUN
cana-410	23	8	issn	issn	NOUN
cana-410	23	9	:	:	PUNCT
cana-410	23	10	1074	1074	NUM
cana-410	23	11	-	-	PUNCT
cana-410	23	12	133x	133x	NUM
cana-410	23	13	vol	vol	NOUN
cana-410	23	14	31	31	NUM
cana-410	23	15	no	no	NOUN
cana-410	23	16	.	.	NOUN
cana-410	23	17	1	1	NUM
cana-410	23	18	(	(	PUNCT
cana-410	23	19	2024	2024	NUM
cana-410	23	20	)	)	PUNCT
cana-410	23	21	254	254	NUM
cana-410	23	22	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	23	23	definition	definition	NOUN
cana-410	23	24	2.2:[1	2.2:[1	NUM
cana-410	23	25	]	]	PUNCT
cana-410	23	26	a	a	DET
cana-410	23	27	graph	graph	NOUN
cana-410	23	28	with	with	ADP
cana-410	23	29	a	a	DET
cana-410	23	30	single	single	ADJ
cana-410	23	31	cycle	cycle	NOUN
cana-410	23	32	(	(	PUNCT
cana-410	23	33	at	at	ADP
cana-410	23	34	least	least	ADJ
cana-410	23	35	3	3	NUM
cana-410	23	36	vertices	vertex	NOUN
cana-410	23	37	)	)	PUNCT
cana-410	23	38	connected	connect	VERB
cana-410	23	39	in	in	ADP
cana-410	23	40	a	a	DET
cana-410	23	41	closed	closed	ADJ
cana-410	23	42	chain	chain	NOUN
cana-410	23	43	is	be	AUX
cana-410	23	44	referred	refer	VERB
cana-410	23	45	to	to	ADP
cana-410	23	46	as	as	ADP
cana-410	23	47	a	a	DET
cana-410	23	48	cycle	cycle	NOUN
cana-410	23	49	graph	graph	NOUN
cana-410	23	50	cn	cn	ADJ
cana-410	23	51	or	or	CCONJ
cana-410	23	52	circular	circular	ADJ
cana-410	23	53	graph	graph	NOUN
cana-410	23	54	.	.	PUNCT
cana-410	24	1	in	in	ADP
cana-410	24	2	cn	cn	PROPN
cana-410	24	3	,	,	PUNCT
cana-410	24	4	the	the	DET
cana-410	24	5	number	number	NOUN
cana-410	24	6	of	of	ADP
cana-410	24	7	edges	edge	NOUN
cana-410	24	8	is	be	AUX
cana-410	24	9	equal	equal	ADJ
cana-410	24	10	to	to	ADP
cana-410	24	11	the	the	DET
cana-410	24	12	number	number	NOUN
cana-410	24	13	of	of	ADP
cana-410	24	14	vertices	vertex	NOUN
cana-410	24	15	,	,	PUNCT
cana-410	24	16	and	and	CCONJ
cana-410	24	17	each	each	DET
cana-410	24	18	vertex	vertex	NOUN
cana-410	24	19	has	have	VERB
cana-410	24	20	degree	degree	NOUN
cana-410	24	21	2	2	NUM
cana-410	24	22	.	.	PUNCT
cana-410	25	1	definition	definition	NOUN
cana-410	25	2	2.3:[2	2.3:[2	NUM
cana-410	25	3	]	]	PUNCT
cana-410	25	4	the	the	DET
cana-410	25	5	number	number	NOUN
cana-410	25	6	of	of	ADP
cana-410	25	7	edges	edge	NOUN
cana-410	25	8	that	that	PRON
cana-410	25	9	are	be	AUX
cana-410	25	10	incident	incident	NOUN
cana-410	25	11	on	on	ADP
cana-410	25	12	a	a	DET
cana-410	25	13	vertex	vertex	NOUN
cana-410	25	14	is	be	AUX
cana-410	25	15	the	the	DET
cana-410	25	16	degree	degree	NOUN
cana-410	25	17	of	of	ADP
cana-410	25	18	the	the	DET
cana-410	25	19	graph	graph	NOUN
cana-410	25	20	.	.	PUNCT
cana-410	26	1	definition	definition	NOUN
cana-410	26	2	2.4:[5	2.4:[5	NUM
cana-410	26	3	]	]	PUNCT
cana-410	26	4	the	the	DET
cana-410	26	5	length	length	NOUN
cana-410	26	6	of	of	ADP
cana-410	26	7	shortest	short	ADJ
cana-410	26	8	cycle	cycle	NOUN
cana-410	26	9	in	in	ADP
cana-410	26	10	the	the	DET
cana-410	26	11	graph	graph	NOUN
cana-410	26	12	is	be	AUX
cana-410	26	13	said	say	VERB
cana-410	26	14	to	to	PART
cana-410	26	15	be	be	AUX
cana-410	26	16	girth	girth	ADJ
cana-410	26	17	.	.	PUNCT
cana-410	27	1	definition	definition	NOUN
cana-410	27	2	2.5:[1	2.5:[1	NUM
cana-410	27	3	]	]	PUNCT
cana-410	27	4	the	the	DET
cana-410	27	5	least	least	ADJ
cana-410	27	6	number	number	NOUN
cana-410	27	7	of	of	ADP
cana-410	27	8	colors	color	NOUN
cana-410	27	9	that	that	PRON
cana-410	27	10	allows	allow	VERB
cana-410	27	11	to	to	PART
cana-410	27	12	be	be	AUX
cana-410	27	13	colored	color	VERB
cana-410	27	14	differently	differently	ADV
cana-410	27	15	for	for	ADP
cana-410	27	16	the	the	DET
cana-410	27	17	adjacent	adjacent	ADJ
cana-410	27	18	vertices	vertex	NOUN
cana-410	27	19	is	be	AUX
cana-410	27	20	said	say	VERB
cana-410	27	21	to	to	PART
cana-410	27	22	be	be	AUX
cana-410	27	23	chromatic	chromatic	ADJ
cana-410	27	24	number	number	NOUN
cana-410	27	25	.	.	PUNCT
cana-410	28	1	3	3	X
cana-410	28	2	.	.	X
cana-410	28	3	main	main	ADJ
cana-410	28	4	results	result	NOUN
cana-410	28	5	definition	definition	NOUN
cana-410	28	6	3.1	3.1	NUM
cana-410	28	7	:	:	PUNCT
cana-410	28	8	a	a	DET
cana-410	28	9	clutch	clutch	ADJ
cana-410	28	10	graph	graph	NOUN
cana-410	28	11	cl3n(g	cl3n(g	NOUN
cana-410	28	12	)	)	PUNCT
cana-410	29	1	=	=	SYM
cana-410	29	2	(	(	PUNCT
cana-410	29	3	v	v	NOUN
cana-410	29	4	,	,	PUNCT
cana-410	29	5	e	e	NOUN
cana-410	29	6	)	)	PUNCT
cana-410	29	7	is	be	AUX
cana-410	29	8	a	a	DET
cana-410	29	9	type	type	NOUN
cana-410	29	10	of	of	ADP
cana-410	29	11	graph	graph	NOUN
cana-410	29	12	that	that	PRON
cana-410	29	13	can	can	AUX
cana-410	29	14	be	be	AUX
cana-410	29	15	de	de	VERB
cana-410	29	16	rived	rive	VERB
cana-410	29	17	from	from	ADP
cana-410	29	18	the	the	DET
cana-410	29	19	cycle	cycle	NOUN
cana-410	29	20	graph	graph	NOUN
cana-410	29	21	.	.	PUNCT
cana-410	30	1	the	the	DET
cana-410	30	2	construction	construction	NOUN
cana-410	30	3	of	of	ADP
cana-410	30	4	a	a	DET
cana-410	30	5	clutch	clutch	ADJ
cana-410	30	6	graph	graph	NOUN
cana-410	30	7	cl3n(g	cl3n(g	NOUN
cana-410	30	8	)	)	PUNCT
cana-410	30	9	involves	involve	VERB
cana-410	30	10	specific	specific	ADJ
cana-410	30	11	steps	step	NOUN
cana-410	30	12	as	as	SCONJ
cana-410	30	13	described	describe	VERB
cana-410	30	14	below	below	ADV
cana-410	30	15	.	.	PUNCT
cana-410	31	1	•	•	NUM
cana-410	31	2	start	start	VERB
cana-410	31	3	with	with	ADP
cana-410	31	4	a	a	DET
cana-410	31	5	cycle	cycle	NOUN
cana-410	31	6	graph	graph	NOUN
cana-410	31	7	cn(g	cn(g	X
cana-410	31	8	)	)	PUNCT
cana-410	32	1	=	=	SYM
cana-410	32	2	(	(	PUNCT
cana-410	32	3	vc	vc	PROPN
cana-410	32	4	,	,	PUNCT
cana-410	32	5	ec	ec	PROPN
cana-410	32	6	)	)	PUNCT
cana-410	32	7	,	,	PUNCT
cana-410	32	8	with	with	ADP
cana-410	32	9	vertex	vertex	NOUN
cana-410	32	10	set	set	VERB
cana-410	32	11	vc	vc	PROPN
cana-410	32	12	=	=	SYM
cana-410	32	13	{	{	PUNCT
cana-410	32	14	c1	c1	PROPN
cana-410	32	15	,	,	PUNCT
cana-410	32	16	c2	c2	PROPN
cana-410	32	17	,	,	PUNCT
cana-410	32	18	.	.	PUNCT
cana-410	32	19	.	.	PUNCT
cana-410	33	1	.	.	PUNCT
cana-410	34	1	,	,	PUNCT
cana-410	34	2	cn},where	cn},where	INTJ
cana-410	34	3	(	(	PUNCT
cana-410	34	4	n	n	X
cana-410	34	5	≥	≥	NUM
cana-410	34	6	4	4	NUM
cana-410	34	7	,	,	PUNCT
cana-410	34	8	even	even	ADV
cana-410	34	9	)	)	PUNCT
cana-410	34	10	is	be	AUX
cana-410	34	11	the	the	DET
cana-410	34	12	number	number	NOUN
cana-410	34	13	of	of	ADP
cana-410	34	14	vertices	vertex	NOUN
cana-410	34	15	and	and	CCONJ
cana-410	34	16	edge	edge	NOUN
cana-410	34	17	set	set	NOUN
cana-410	34	18	,	,	PUNCT
cana-410	34	19	ec	ec	PROPN
cana-410	34	20	=	=	SYM
cana-410	34	21	{	{	PUNCT
cana-410	34	22	(	(	PUNCT
cana-410	34	23	ci	ci	NOUN
cana-410	34	24	,	,	PUNCT
cana-410	34	25	ci+1)|i	ci+1)|i	NOUN
cana-410	34	26	∈	∈	PROPN
cana-410	34	27	{	{	PUNCT
cana-410	34	28	1	1	NUM
cana-410	34	29	,	,	PUNCT
cana-410	34	30	2	2	NUM
cana-410	34	31	,	,	PUNCT
cana-410	34	32	.	.	PUNCT
cana-410	34	33	.	.	PUNCT
cana-410	35	1	.	.	PUNCT
cana-410	36	1	,	,	PUNCT
cana-410	36	2	n	n	CCONJ
cana-410	36	3	−	−	PROPN
cana-410	36	4	1	1	NUM
cana-410	36	5	}	}	PUNCT
cana-410	36	6	}	}	PUNCT
cana-410	36	7	∪	∪	X
cana-410	36	8	(	(	PUNCT
cana-410	36	9	cn	cn	PROPN
cana-410	36	10	,	,	PUNCT
cana-410	36	11	c1	c1	PROPN
cana-410	36	12	)	)	PUNCT
cana-410	36	13	•	•	NOUN
cana-410	36	14	add	add	VERB
cana-410	36	15	a	a	DET
cana-410	36	16	set	set	NOUN
cana-410	36	17	of	of	ADP
cana-410	36	18	vertices	vertex	NOUN
cana-410	36	19	,	,	PUNCT
cana-410	36	20	vp	vp	X
cana-410	36	21	=	=	SYM
cana-410	36	22	{	{	PUNCT
cana-410	36	23	p1	p1	PROPN
cana-410	36	24	,	,	PUNCT
cana-410	36	25	p2	p2	NOUN
cana-410	36	26	,	,	PUNCT
cana-410	36	27	.	.	PUNCT
cana-410	36	28	.	.	PUNCT
cana-410	37	1	.	.	PUNCT
cana-410	38	1	,	,	PUNCT
cana-410	38	2	pn	pn	X
cana-410	38	3	}	}	PUNCT
cana-410	38	4	to	to	ADP
cana-410	38	5	the	the	DET
cana-410	38	6	existing	exist	VERB
cana-410	38	7	vertex	vertex	NOUN
cana-410	38	8	set	set	VERB
cana-410	38	9	vc	vc	PROPN
cana-410	38	10	•	•	NUM
cana-410	38	11	establish	establish	VERB
cana-410	38	12	the	the	DET
cana-410	38	13	edge	edge	NOUN
cana-410	38	14	set	set	NOUN
cana-410	38	15	,	,	PUNCT
cana-410	38	16	ecp	ecp	NOUN
cana-410	38	17	by	by	ADP
cana-410	38	18	joining	join	VERB
cana-410	38	19	each	each	DET
cana-410	38	20	vertex	vertex	NOUN
cana-410	38	21	ci	ci	NOUN
cana-410	38	22	with	with	ADP
cana-410	38	23	its	its	PRON
cana-410	38	24	corresponding	corresponding	ADJ
cana-410	38	25	pi	pi	NOUN
cana-410	38	26	,	,	PUNCT
cana-410	38	27	ecp	ecp	NOUN
cana-410	38	28	=	=	PRON
cana-410	38	29	{	{	PUNCT
cana-410	38	30	(	(	PUNCT
cana-410	38	31	ci	ci	NOUN
cana-410	38	32	,	,	PUNCT
cana-410	38	33	pi)|i	pi)|i	NOUN
cana-410	38	34	∈	∈	PROPN
cana-410	38	35	{	{	PUNCT
cana-410	38	36	1	1	NUM
cana-410	38	37	,	,	PUNCT
cana-410	38	38	2	2	NUM
cana-410	38	39	,	,	PUNCT
cana-410	38	40	.	.	PUNCT
cana-410	38	41	.	.	PUNCT
cana-410	39	1	.	.	PUNCT
cana-410	40	1	,	,	PUNCT
cana-410	40	2	n	n	CCONJ
cana-410	40	3	}	}	PUNCT
cana-410	40	4	}	}	PUNCT
cana-410	40	5	•	•	NOUN
cana-410	40	6	form	form	VERB
cana-410	40	7	an	an	DET
cana-410	40	8	edge	edge	NOUN
cana-410	40	9	set	set	NOUN
cana-410	40	10	,	,	PUNCT
cana-410	40	11	ep	ep	PROPN
cana-410	40	12	=	=	SYM
cana-410	40	13	{	{	PUNCT
cana-410	40	14	(	(	PUNCT
cana-410	40	15	pi	pi	NOUN
cana-410	40	16	,	,	PUNCT
cana-410	40	17	pi+1)|i	pi+1)|i	NOUN
cana-410	40	18	∈	∈	NOUN
cana-410	40	19	{	{	PUNCT
cana-410	40	20	1	1	NUM
cana-410	40	21	,	,	PUNCT
cana-410	40	22	3	3	NUM
cana-410	40	23	,	,	PUNCT
cana-410	40	24	5	5	NUM
cana-410	40	25	,	,	PUNCT
cana-410	40	26	.	.	PUNCT
cana-410	40	27	.	.	PUNCT
cana-410	41	1	.	.	PUNCT
cana-410	42	1	,	,	PUNCT
cana-410	42	2	n	n	CCONJ
cana-410	42	3	−	−	PROPN
cana-410	42	4	1	1	NUM
cana-410	42	5	}	}	PUNCT
cana-410	42	6	}	}	PUNCT
cana-410	42	7	•	•	X
cana-410	42	8	create	create	VERB
cana-410	42	9	another	another	DET
cana-410	42	10	set	set	NOUN
cana-410	42	11	of	of	ADP
cana-410	42	12	vertices	vertex	NOUN
cana-410	42	13	,	,	PUNCT
cana-410	42	14	vq	vq	NOUN
cana-410	42	15	=	=	SYM
cana-410	42	16	{	{	PUNCT
cana-410	42	17	q1	q1	PROPN
cana-410	42	18	,	,	PUNCT
cana-410	42	19	q2	q2	NOUN
cana-410	42	20	,	,	PUNCT
cana-410	42	21	.	.	PUNCT
cana-410	42	22	.	.	PUNCT
cana-410	43	1	.	.	PUNCT
cana-410	44	1	,	,	PUNCT
cana-410	44	2	qn	qn	VERB
cana-410	44	3	}	}	PUNCT
cana-410	44	4	and	and	CCONJ
cana-410	44	5	add	add	VERB
cana-410	44	6	to	to	ADP
cana-410	44	7	the	the	DET
cana-410	44	8	vertex	vertex	NOUN
cana-410	44	9	set	set	NOUN
cana-410	44	10	vp	vp	PROPN
cana-410	44	11	.	.	PUNCT
cana-410	45	1	the	the	DET
cana-410	45	2	resulting	result	VERB
cana-410	45	3	vertex	vertex	NOUN
cana-410	45	4	set	set	NOUN
cana-410	45	5	be	be	AUX
cana-410	45	6	vcpq	vcpq	NOUN
cana-410	45	7	=	=	SYM
cana-410	45	8	{	{	PUNCT
cana-410	45	9	c1	c1	PROPN
cana-410	45	10	,	,	PUNCT
cana-410	45	11	c2	c2	PROPN
cana-410	45	12	,	,	PUNCT
cana-410	45	13	.	.	PUNCT
cana-410	45	14	.	.	PUNCT
cana-410	46	1	.	.	PUNCT
cana-410	47	1	,	,	PUNCT
cana-410	47	2	cn	cn	PROPN
cana-410	47	3	,	,	PUNCT
cana-410	47	4	p1	p1	NOUN
cana-410	47	5	,	,	PUNCT
cana-410	47	6	p2	p2	NOUN
cana-410	47	7	,	,	PUNCT
cana-410	47	8	.	.	PUNCT
cana-410	47	9	.	.	PUNCT
cana-410	47	10	.	.	PUNCT
cana-410	48	1	,	,	PUNCT
cana-410	48	2	pn	pn	PROPN
cana-410	48	3	,	,	PUNCT
cana-410	48	4	q1	q1	PROPN
cana-410	48	5	,	,	PUNCT
cana-410	48	6	q2	q2	NOUN
cana-410	48	7	,	,	PUNCT
cana-410	48	8	.	.	PUNCT
cana-410	48	9	.	.	PUNCT
cana-410	49	1	.	.	PUNCT
cana-410	50	1	,	,	PUNCT
cana-410	50	2	qn	qn	AUX
cana-410	50	3	}	}	PUNCT
cana-410	50	4	•	•	NUM
cana-410	50	5	built	build	VERB
cana-410	50	6	an	an	DET
cana-410	50	7	edge	edge	NOUN
cana-410	50	8	set	set	VERB
cana-410	50	9	epq	epq	NOUN
cana-410	50	10	=	=	SYM
cana-410	50	11	{	{	PUNCT
cana-410	50	12	(	(	PUNCT
cana-410	50	13	pi	pi	NOUN
cana-410	50	14	,	,	PUNCT
cana-410	50	15	qi)|i	qi)|i	NOUN
cana-410	50	16	∈	∈	PROPN
cana-410	50	17	{	{	PUNCT
cana-410	50	18	1	1	NUM
cana-410	50	19	,	,	PUNCT
cana-410	50	20	2	2	NUM
cana-410	50	21	,	,	PUNCT
cana-410	50	22	.	.	PUNCT
cana-410	50	23	.	.	PUNCT
cana-410	51	1	.	.	PUNCT
cana-410	52	1	,	,	PUNCT
cana-410	52	2	n	n	CCONJ
cana-410	52	3	}	}	PUNCT
cana-410	52	4	}	}	PUNCT
cana-410	52	5	connecting	connect	VERB
cana-410	52	6	corresponding	corresponding	ADJ
cana-410	52	7	vertices	vertex	NOUN
cana-410	52	8	from	from	ADP
cana-410	52	9	vp	vp	PROPN
cana-410	52	10	and	and	CCONJ
cana-410	52	11	vq	vq	PROPN
cana-410	52	12	•	•	NUM
cana-410	52	13	finally	finally	ADV
cana-410	52	14	,	,	PUNCT
cana-410	52	15	create	create	VERB
cana-410	52	16	an	an	DET
cana-410	52	17	edge	edge	NOUN
cana-410	52	18	set	set	NOUN
cana-410	52	19	,	,	PUNCT
cana-410	52	20	eq	eq	NOUN
cana-410	52	21	=	=	SYM
cana-410	52	22	{	{	PUNCT
cana-410	52	23	(	(	PUNCT
cana-410	52	24	qi	qi	PROPN
cana-410	52	25	,	,	PUNCT
cana-410	52	26	qi+1)|i	qi+1)|i	NOUN
cana-410	52	27	∈	∈	PROPN
cana-410	52	28	{	{	PUNCT
cana-410	52	29	2	2	NUM
cana-410	52	30	,	,	PUNCT
cana-410	52	31	4	4	NUM
cana-410	52	32	,	,	PUNCT
cana-410	52	33	.	.	PUNCT
cana-410	52	34	.	.	PUNCT
cana-410	53	1	.	.	PUNCT
cana-410	54	1	,	,	PUNCT
cana-410	54	2	n	n	CCONJ
cana-410	54	3	}	}	PUNCT
cana-410	54	4	}	}	PUNCT
cana-410	54	5	∪	∪	NOUN
cana-410	54	6	(	(	PUNCT
cana-410	54	7	qn	qn	INTJ
cana-410	54	8	,	,	PUNCT
cana-410	54	9	q1	q1	PROPN
cana-410	54	10	)	)	PUNCT
cana-410	54	11	}	}	PUNCT
cana-410	54	12	by	by	ADP
cana-410	54	13	following	follow	VERB
cana-410	54	14	these	these	DET
cana-410	54	15	steps	step	NOUN
cana-410	54	16	,	,	PUNCT
cana-410	54	17	the	the	DET
cana-410	54	18	clutch	clutch	ADJ
cana-410	54	19	graph	graph	NOUN
cana-410	54	20	cl3n(g	cl3n(g	NOUN
cana-410	54	21	)	)	PUNCT
cana-410	54	22	have	have	VERB
cana-410	54	23	the	the	DET
cana-410	54	24	vertex	vertex	NOUN
cana-410	54	25	set	set	VERB
cana-410	54	26	v	v	NOUN
cana-410	54	27	=	=	SYM
cana-410	54	28	vc	vc	NOUN
cana-410	54	29	∪	∪	X
cana-410	54	30	vp	vp	PROPN
cana-410	54	31	∪	∪	X
cana-410	54	32	vq	vq	NOUN
cana-410	54	33	and	and	CCONJ
cana-410	54	34	edge	edge	VERB
cana-410	54	35	set	set	VERB
cana-410	54	36	e	e	NOUN
cana-410	54	37	=	=	SYM
cana-410	54	38	ec	ec	PROPN
cana-410	54	39	∪	∪	VERB
cana-410	54	40	ecp	ecp	PROPN
cana-410	54	41	∪	∪	ADP
cana-410	54	42	ep	ep	PROPN
cana-410	54	43	∪	∪	ADP
cana-410	54	44	epq	epq	PROPN
cana-410	54	45	∪	∪	NOUN
cana-410	54	46	eq	eq	NOUN
cana-410	54	47	.	.	PUNCT
cana-410	55	1	the	the	DET
cana-410	55	2	resulting	result	VERB
cana-410	55	3	clutch	clutch	ADJ
cana-410	55	4	graph	graph	NOUN
cana-410	55	5	is	be	AUX
cana-410	55	6	characterized	characterize	VERB
cana-410	55	7	by	by	ADP
cana-410	55	8	with	with	ADP
cana-410	55	9	|v	|v	PROPN
cana-410	56	1	|	|	NOUN
cana-410	56	2	=	=	SYM
cana-410	57	1	3n	3n	NUM
cana-410	57	2	vertices	vertex	NOUN
cana-410	57	3	and	and	CCONJ
cana-410	57	4	|e|	|e|	PRON
cana-410	57	5	=	=	SYM
cana-410	57	6	4n	4n	ADJ
cana-410	57	7	edges	edge	NOUN
cana-410	57	8	(	(	PUNCT
cana-410	57	9	n	n	CCONJ
cana-410	57	10	≥	≥	NOUN
cana-410	57	11	4	4	NUM
cana-410	57	12	&	&	CCONJ
cana-410	57	13	even	even	ADV
cana-410	57	14	)	)	PUNCT
cana-410	57	15	.	.	PUNCT
cana-410	58	1	example	example	NOUN
cana-410	58	2	:	:	PUNCT
cana-410	58	3	the	the	DET
cana-410	58	4	clutch	clutch	NOUN
cana-410	58	5	graph	graph	NOUN
cana-410	58	6	cl18(g	cl18(g	PROPN
cana-410	58	7	)	)	PUNCT
cana-410	58	8	in	in	ADP
cana-410	58	9	fig	fig	NOUN
cana-410	58	10	:	:	PUNCT
cana-410	58	11	3	3	NUM
cana-410	58	12	and	and	CCONJ
cana-410	58	13	cl24(g	cl24(g	NOUN
cana-410	58	14	)	)	PUNCT
cana-410	58	15	in	in	ADP
cana-410	58	16	fig	fig	NOUN
cana-410	58	17	:	:	PUNCT
cana-410	58	18	4	4	NUM
cana-410	58	19	which	which	PRON
cana-410	58	20	have	have	AUX
cana-410	58	21	been	be	AUX
cana-410	58	22	constructed	construct	VERB
cana-410	58	23	from	from	ADP
cana-410	58	24	the	the	DET
cana-410	58	25	cycle	cycle	NOUN
cana-410	58	26	graph	graph	NOUN
cana-410	58	27	c6(g	c6(g	PROPN
cana-410	58	28	)	)	PUNCT
cana-410	58	29	and	and	CCONJ
cana-410	58	30	c8(g	c8(g	NOUN
cana-410	58	31	)	)	PUNCT
cana-410	58	32	respectively	respectively	ADV
cana-410	58	33	.	.	PUNCT
cana-410	59	1	c4	c4	NOUN
cana-410	59	2	c3	c3	PROPN
cana-410	59	3	c3	c3	PROPN
cana-410	59	4	c1	c1	PROPN
cana-410	59	5	c2	c2	PROPN
cana-410	59	6	c1	c1	PROPN
cana-410	59	7	c2	c2	PROPN
cana-410	59	8	figure	figure	VERB
cana-410	59	9	1	1	NUM
cana-410	59	10	:	:	PUNCT
cana-410	59	11	c3	c3	PROPN
cana-410	59	12	(	(	PUNCT
cana-410	59	13	g	g	NOUN
cana-410	59	14	)	)	PUNCT
cana-410	59	15	figure	figure	NOUN
cana-410	59	16	2	2	NUM
cana-410	59	17	:	:	PUNCT
cana-410	59	18	c4	c4	NOUN
cana-410	59	19	(	(	PUNCT
cana-410	59	20	g	g	NOUN
cana-410	59	21	)	)	PUNCT
cana-410	59	22	communications	communication	NOUN
cana-410	59	23	on	on	ADP
cana-410	59	24	applied	apply	VERB
cana-410	59	25	nonlinear	nonlinear	ADJ
cana-410	59	26	analysis	analysis	NOUN
cana-410	59	27	issn	issn	NOUN
cana-410	59	28	:	:	PUNCT
cana-410	59	29	1074	1074	NUM
cana-410	59	30	-	-	PUNCT
cana-410	59	31	133x	133x	NUM
cana-410	59	32	vol	vol	NOUN
cana-410	59	33	31	31	NUM
cana-410	59	34	no	no	NOUN
cana-410	59	35	.	.	NOUN
cana-410	59	36	1	1	NUM
cana-410	59	37	(	(	PUNCT
cana-410	59	38	2024	2024	NUM
cana-410	59	39	)	)	PUNCT
cana-410	59	40	255	255	NUM
cana-410	59	41	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	59	42	figure	figure	NOUN
cana-410	59	43	3	3	NUM
cana-410	59	44	:	:	PUNCT
cana-410	59	45	cl18	cl18	PROPN
cana-410	59	46	(	(	PUNCT
cana-410	59	47	g	g	NOUN
cana-410	59	48	)	)	PUNCT
cana-410	59	49	figure	figure	NOUN
cana-410	59	50	4	4	NUM
cana-410	59	51	:	:	PUNCT
cana-410	59	52	cl24	cl24	PROPN
cana-410	59	53	(	(	PUNCT
cana-410	59	54	g	g	NOUN
cana-410	59	55	)	)	PUNCT
cana-410	59	56	definition	definition	NOUN
cana-410	59	57	3.2	3.2	NUM
cana-410	59	58	:	:	PUNCT
cana-410	59	59	the	the	DET
cana-410	59	60	degree	degree	NOUN
cana-410	59	61	of	of	ADP
cana-410	59	62	the	the	DET
cana-410	59	63	clutch	clutch	NOUN
cana-410	59	64	graph	graph	NOUN
cana-410	59	65	is	be	AUX
cana-410	59	66	represented	represent	VERB
cana-410	59	67	by	by	ADP
cana-410	59	68	deg	deg	PROPN
cana-410	59	69	cl3n(g	cl3n(g	PROPN
cana-410	59	70	)	)	PUNCT
cana-410	59	71	.	.	PUNCT
cana-410	60	1	the	the	DET
cana-410	60	2	degree	degree	NOUN
cana-410	60	3	of	of	ADP
cana-410	60	4	the	the	DET
cana-410	60	5	vertices	vertex	NOUN
cana-410	60	6	ci	ci	NOUN
cana-410	60	7	in	in	ADP
cana-410	60	8	vc	vc	PROPN
cana-410	60	9	and	and	CCONJ
cana-410	60	10	pi	pi	NOUN
cana-410	60	11	in	in	ADP
cana-410	60	12	vp	vp	PROPN
cana-410	60	13	of	of	ADP
cana-410	60	14	the	the	DET
cana-410	60	15	clutch	clutch	NOUN
cana-410	60	16	graph	graph	NOUN
cana-410	60	17	are	be	AUX
cana-410	60	18	denoted	denote	VERB
cana-410	60	19	by	by	ADP
cana-410	60	20	s.	s.	PROPN
cana-410	60	21	the	the	DET
cana-410	60	22	degree	degree	NOUN
cana-410	60	23	of	of	ADP
cana-410	60	24	the	the	DET
cana-410	60	25	vertices	vertex	NOUN
cana-410	60	26	qi	qi	PROPN
cana-410	60	27	in	in	ADP
cana-410	60	28	vq	vq	PROPN
cana-410	60	29	is	be	AUX
cana-410	60	30	denoted	denote	VERB
cana-410	60	31	by	by	ADP
cana-410	60	32	t.	t.	PROPN
cana-410	60	33	𝑑𝑒𝑔𝑐𝑖	𝑑𝑒𝑔𝑐𝑖	PROPN
cana-410	60	34	∈	∈	PROPN
cana-410	61	1	𝑉𝑐	𝑉𝑐	PROPN
cana-410	61	2	𝐶𝑙3𝑛(𝑐𝑖	𝐶𝑙3𝑛(𝑐𝑖	PROPN
cana-410	61	3	)	)	PUNCT
cana-410	61	4	=	=	SYM
cana-410	61	5	s	s	NOUN
cana-410	61	6	=	=	SYM
cana-410	61	7	3	3	NUM
cana-410	61	8	𝑑𝑒𝑔𝑝𝑖	𝑑𝑒𝑔𝑝𝑖	NOUN
cana-410	61	9	∈	∈	NOUN
cana-410	62	1	𝑉𝑝	𝑉𝑝	ADJ
cana-410	62	2	𝐶𝑙3𝑛(𝑝𝑖	𝐶𝑙3𝑛(𝑝𝑖	NOUN
cana-410	62	3	)	)	PUNCT
cana-410	62	4	=	=	SYM
cana-410	62	5	s	s	NOUN
cana-410	62	6	=	=	SYM
cana-410	62	7	3	3	NUM
cana-410	62	8	,	,	PUNCT
cana-410	62	9	and	and	CCONJ
cana-410	62	10	𝑑𝑒𝑔𝑞𝑖	𝑑𝑒𝑔𝑞𝑖	VERB
cana-410	62	11	∈	∈	PROPN
cana-410	62	12	𝑉𝑞	𝑉𝑞	PROPN
cana-410	62	13	𝐶𝑙3𝑛(𝑞𝑖	𝐶𝑙3𝑛(𝑞𝑖	NOUN
cana-410	62	14	)	)	PUNCT
cana-410	62	15	=	=	SYM
cana-410	62	16	t	t	NOUN
cana-410	62	17	=	=	SYM
cana-410	62	18	2	2	NUM
cana-410	62	19	definition	definition	NOUN
cana-410	62	20	3.3	3.3	NUM
cana-410	62	21	:	:	PUNCT
cana-410	62	22	the	the	DET
cana-410	62	23	girth	girth	NOUN
cana-410	62	24	g	g	NOUN
cana-410	62	25	of	of	ADP
cana-410	62	26	a	a	DET
cana-410	62	27	clutch	clutch	NOUN
cana-410	62	28	graph	graph	NOUN
cana-410	62	29	(	(	PUNCT
cana-410	62	30	cl3n(g	cl3n(g	NOUN
cana-410	62	31	)	)	PUNCT
cana-410	62	32	)	)	PUNCT
cana-410	62	33	is	be	AUX
cana-410	62	34	always	always	ADV
cana-410	62	35	4	4	NUM
cana-410	62	36	.	.	PUNCT
cana-410	63	1	g	g	NOUN
cana-410	63	2	=	=	PRON
cana-410	63	3	{	{	PUNCT
cana-410	63	4	(	(	PUNCT
cana-410	63	5	𝑐2𝑖−𝑖	𝑐2𝑖−𝑖	NUM
cana-410	63	6	,	,	PUNCT
cana-410	63	7	𝑝2𝑖−1	𝑝2𝑖−1	PROPN
cana-410	63	8	,	,	PUNCT
cana-410	63	9	𝑝2𝑖	𝑝2𝑖	NOUN
cana-410	63	10	,	,	PUNCT
cana-410	63	11	𝑐2𝑖)|𝑖	𝑐2𝑖)|𝑖	PROPN
cana-410	63	12	∈	∈	PROPN
cana-410	63	13	1	1	NUM
cana-410	63	14	,	,	PUNCT
cana-410	63	15	2	2	NUM
cana-410	63	16	,	,	PUNCT
cana-410	63	17	.	.	PUNCT
cana-410	63	18	.	.	PUNCT
cana-410	64	1	.	.	PUNCT
cana-410	65	1	,	,	PUNCT
cana-410	65	2	𝑛	𝑛	PRON
cana-410	65	3	2	2	NUM
cana-410	65	4	}	}	PUNCT
cana-410	65	5	definition	definition	NOUN
cana-410	65	6	3.4	3.4	NUM
cana-410	65	7	:	:	PUNCT
cana-410	65	8	the	the	DET
cana-410	65	9	chromatic	chromatic	ADJ
cana-410	65	10	number	number	NOUN
cana-410	65	11	for	for	ADP
cana-410	65	12	clutch	clutch	ADJ
cana-410	65	13	graph	graph	NOUN
cana-410	65	14	χ(cl3n(g	χ(cl3n(g	PROPN
cana-410	65	15	)	)	PUNCT
cana-410	65	16	)	)	PUNCT
cana-410	65	17	is	be	AUX
cana-410	65	18	2	2	NUM
cana-410	65	19	.	.	NOUN
cana-410	65	20	𝜒(𝐶𝑙3𝑛(𝐺	𝜒(𝐶𝑙3𝑛(𝐺	NUM
cana-410	65	21	)	)	PUNCT
cana-410	65	22	)	)	PUNCT
cana-410	66	1	=	=	PRON
cana-410	66	2	{	{	PUNCT
cana-410	66	3	1	1	NUM
cana-410	66	4	𝑖𝑓	𝑖𝑓	SYM
cana-410	66	5	{	{	PUNCT
cana-410	66	6	(	(	PUNCT
cana-410	66	7	𝑐2𝑖−1	𝑐2𝑖−1	PROPN
cana-410	66	8	,	,	PUNCT
cana-410	66	9	𝑝2𝑖	𝑝2𝑖	NOUN
cana-410	66	10	,	,	PUNCT
cana-410	66	11	𝑞2𝑖−1)|𝑖	𝑞2𝑖−1)|𝑖	PROPN
cana-410	66	12	∈	∈	PROPN
cana-410	66	13	{	{	PUNCT
cana-410	66	14	1	1	NUM
cana-410	66	15	,	,	PUNCT
cana-410	66	16	2	2	NUM
cana-410	66	17	,	,	PUNCT
cana-410	66	18	.	.	PUNCT
cana-410	66	19	.	.	PUNCT
cana-410	67	1	.	.	PUNCT
cana-410	68	1	,	,	PUNCT
cana-410	68	2	𝑛	𝑛	DET
cana-410	68	3	2	2	NUM
cana-410	68	4	}	}	SYM
cana-410	68	5	2	2	NUM
cana-410	68	6	𝑖𝑓	𝑖𝑓	NOUN
cana-410	68	7	{	{	PUNCT
cana-410	68	8	(	(	PUNCT
cana-410	68	9	𝑐2𝑖	𝑐2𝑖	ADJ
cana-410	68	10	,	,	PUNCT
cana-410	68	11	𝑝2𝑖−𝑖	𝑝2𝑖−𝑖	ADJ
cana-410	68	12	,	,	PUNCT
cana-410	68	13	𝑞2𝑖)|𝑖	𝑞2𝑖)|𝑖	NOUN
cana-410	68	14	∈	∈	NOUN
cana-410	68	15	{	{	PUNCT
cana-410	68	16	1	1	NUM
cana-410	68	17	,	,	PUNCT
cana-410	68	18	2	2	NUM
cana-410	68	19	,	,	PUNCT
cana-410	68	20	.	.	PUNCT
cana-410	68	21	.	.	PUNCT
cana-410	69	1	.	.	PUNCT
cana-410	70	1	,	,	PUNCT
cana-410	70	2	𝑛	𝑛	PRON
cana-410	70	3	2	2	X
cana-410	70	4	}	}	PUNCT
cana-410	70	5	here	here	ADV
cana-410	70	6	1,2	1,2	NUM
cana-410	70	7	represents	represent	VERB
cana-410	70	8	the	the	DET
cana-410	70	9	colors	color	NOUN
cana-410	70	10	assign	assign	VERB
cana-410	70	11	to	to	ADP
cana-410	70	12	the	the	DET
cana-410	70	13	vertices	vertex	NOUN
cana-410	70	14	of	of	ADP
cana-410	70	15	the	the	DET
cana-410	70	16	clutch	clutch	NOUN
cana-410	70	17	graph	graph	NOUN
cana-410	70	18	.	.	PUNCT
cana-410	71	1	4	4	X
cana-410	71	2	.	.	X
cana-410	72	1	some	some	DET
cana-410	72	2	properties	property	NOUN
cana-410	72	3	of	of	ADP
cana-410	72	4	clutch	clutch	ADJ
cana-410	72	5	graph	graph	NOUN
cana-410	72	6	theorem	theorem	VERB
cana-410	72	7	4.1	4.1	NUM
cana-410	72	8	for	for	ADP
cana-410	72	9	every	every	DET
cana-410	72	10	clutch	clutch	NOUN
cana-410	72	11	graph	graph	NOUN
cana-410	72	12	,	,	PUNCT
cana-410	72	13	the	the	DET
cana-410	72	14	sum	sum	NOUN
cana-410	72	15	of	of	ADP
cana-410	72	16	the	the	DET
cana-410	72	17	degrees	degree	NOUN
cana-410	72	18	of	of	ADP
cana-410	72	19	vertices	vertex	NOUN
cana-410	72	20	is	be	AUX
cana-410	72	21	equal	equal	ADJ
cana-410	72	22	to	to	ADP
cana-410	72	23	twice	twice	DET
cana-410	72	24	the	the	DET
cana-410	72	25	number	number	NOUN
cana-410	72	26	of	of	ADP
cana-410	72	27	edges	edge	NOUN
cana-410	72	28	.	.	PUNCT
cana-410	73	1	proof	proof	NOUN
cana-410	73	2	let	let	VERB
cana-410	73	3	us	we	PRON
cana-410	73	4	consider	consider	VERB
cana-410	73	5	a	a	DET
cana-410	73	6	clutch	clutch	ADJ
cana-410	73	7	graph	graph	NOUN
cana-410	73	8	cl3n(g	cl3n(g	NOUN
cana-410	73	9	)	)	PUNCT
cana-410	73	10	with	with	ADP
cana-410	73	11	vertex	vertex	NOUN
cana-410	73	12	set	set	VERB
cana-410	73	13	vcpq	vcpq	NOUN
cana-410	73	14	where	where	SCONJ
cana-410	73	15	vcpq	vcpq	NOUN
cana-410	73	16	=(	=(	NOUN
cana-410	73	17	vc∪vp∪vq	vc∪vp∪vq	PROPN
cana-410	73	18	)	)	PUNCT
cana-410	73	19	.	.	PUNCT
cana-410	74	1	each	each	DET
cana-410	74	2	vertex	vertex	NOUN
cana-410	74	3	ci	ci	NOUN
cana-410	74	4	in	in	ADP
cana-410	74	5	vc	vc	PROPN
cana-410	74	6	and	and	CCONJ
cana-410	74	7	pi	pi	NOUN
cana-410	74	8	in	in	ADP
cana-410	74	9	vp	vp	PROPN
cana-410	74	10	have	have	VERB
cana-410	74	11	degrees	degree	NOUN
cana-410	74	12	3	3	NUM
cana-410	74	13	and	and	CCONJ
cana-410	74	14	each	each	DET
cana-410	74	15	vertex	vertex	NOUN
cana-410	74	16	qi	qi	PROPN
cana-410	74	17	in	in	ADP
cana-410	74	18	vq	vq	PROPN
cana-410	74	19	have	have	VERB
cana-410	74	20	degrees	degree	NOUN
cana-410	74	21	2	2	NUM
cana-410	74	22	.	.	PUNCT
cana-410	74	23	∑	∑	PUNCT
cana-410	74	24	𝑑𝑒𝑔(𝑐𝑖	𝑑𝑒𝑔(𝑐𝑖	NOUN
cana-410	74	25	)	)	PUNCT
cana-410	74	26	𝑐𝑖	𝑐𝑖	NOUN
cana-410	74	27	∈	∈	NOUN
cana-410	75	1	𝑉𝑐	𝑉𝑐	PROPN
cana-410	75	2	+	+	ADJ
cana-410	75	3	∑	∑	ADJ
cana-410	75	4	𝑑𝑒𝑔(𝑝𝑖	𝑑𝑒𝑔(𝑝𝑖	NOUN
cana-410	75	5	)	)	PUNCT
cana-410	75	6	𝑝𝑖	𝑝𝑖	PART
cana-410	75	7	∈	∈	NOUN
cana-410	75	8	𝑉𝑝	𝑉𝑝	PROPN
cana-410	75	9	+	+	CCONJ
cana-410	75	10	∑	∑	NUM
cana-410	75	11	𝑑𝑒𝑔(𝑞𝑖	𝑑𝑒𝑔(𝑞𝑖	NOUN
cana-410	75	12	)	)	PUNCT
cana-410	75	13	𝑞𝑖	𝑞𝑖	PRON
cana-410	75	14	∈	∈	PROPN
cana-410	76	1	𝑉𝑞	𝑉𝑞	PROPN
cana-410	76	2	=	=	NOUN
cana-410	76	3	3n	3n	NUM
cana-410	76	4	+	+	NUM
cana-410	76	5	3n	3n	NUM
cana-410	76	6	+	+	CCONJ
cana-410	76	7	2n	2n	NUM
cana-410	76	8	=	=	SYM
cana-410	76	9	8n	8n	NOUN
cana-410	76	10	=	=	SYM
cana-410	76	11	2(4n	2(4n	NUM
cana-410	76	12	)	)	PUNCT
cana-410	76	13	=	=	SYM
cana-410	76	14	2e	2e	NOUN
cana-410	76	15	.	.	PUNCT
cana-410	77	1	example	example	NOUN
cana-410	77	2	:	:	PUNCT
cana-410	77	3	in	in	ADP
cana-410	77	4	figure	figure	NOUN
cana-410	77	5	3	3	NUM
cana-410	77	6	,	,	PUNCT
cana-410	77	7	the	the	DET
cana-410	77	8	clutch	clutch	ADJ
cana-410	77	9	graph	graph	NOUN
cana-410	77	10	cl18	cl18	PROPN
cana-410	77	11	(	(	PUNCT
cana-410	77	12	g	g	NOUN
cana-410	77	13	)	)	PUNCT
cana-410	77	14	with	with	ADP
cana-410	77	15	n	n	NOUN
cana-410	77	16	=	=	SYM
cana-410	77	17	6	6	NUM
cana-410	77	18	vertices	vertex	NOUN
cana-410	77	19	.	.	PUNCT
cana-410	78	1	communications	communication	NOUN
cana-410	78	2	on	on	ADP
cana-410	78	3	applied	apply	VERB
cana-410	78	4	nonlinear	nonlinear	ADJ
cana-410	78	5	analysis	analysis	NOUN
cana-410	78	6	issn	issn	NOUN
cana-410	78	7	:	:	PUNCT
cana-410	78	8	1074	1074	NUM
cana-410	78	9	-	-	PUNCT
cana-410	78	10	133x	133x	NUM
cana-410	78	11	vol	vol	NOUN
cana-410	78	12	31	31	NUM
cana-410	78	13	no	no	NOUN
cana-410	78	14	.	.	NOUN
cana-410	78	15	1	1	NUM
cana-410	78	16	(	(	PUNCT
cana-410	78	17	2024	2024	NUM
cana-410	78	18	)	)	PUNCT
cana-410	78	19	256	256	NUM
cana-410	78	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	78	21	∑	∑	PUNCT
cana-410	78	22	𝑑𝑒𝑔(𝑐𝑖	𝑑𝑒𝑔(𝑐𝑖	NOUN
cana-410	78	23	)	)	PUNCT
cana-410	78	24	𝑐𝑖	𝑐𝑖	NOUN
cana-410	78	25	∈	∈	NOUN
cana-410	79	1	𝑉𝑐	𝑉𝑐	PROPN
cana-410	79	2	+	+	ADJ
cana-410	79	3	∑	∑	ADJ
cana-410	79	4	𝑑𝑒𝑔(𝑝𝑖	𝑑𝑒𝑔(𝑝𝑖	NOUN
cana-410	79	5	)	)	PUNCT
cana-410	79	6	𝑝𝑖	𝑝𝑖	PART
cana-410	79	7	∈	∈	NOUN
cana-410	79	8	𝑉𝑝	𝑉𝑝	PROPN
cana-410	79	9	+	+	CCONJ
cana-410	79	10	∑	∑	NUM
cana-410	79	11	𝑑𝑒𝑔(𝑞𝑖	𝑑𝑒𝑔(𝑞𝑖	NOUN
cana-410	79	12	)	)	PUNCT
cana-410	79	13	𝑞𝑖	𝑞𝑖	PRON
cana-410	79	14	∈	∈	NOUN
cana-410	80	1	𝑉𝑞	𝑉𝑞	PROPN
cana-410	80	2	=	=	NOUN
cana-410	80	3	=	=	SYM
cana-410	80	4	18	18	NUM
cana-410	80	5	+	+	NUM
cana-410	80	6	18	18	NUM
cana-410	80	7	+	+	SYM
cana-410	80	8	12	12	NUM
cana-410	80	9	=	=	SYM
cana-410	80	10	48	48	NUM
cana-410	80	11	=	=	SYM
cana-410	80	12	2(24	2(24	NUM
cana-410	80	13	)	)	PUNCT
cana-410	80	14	=	=	SYM
cana-410	80	15	2e	2e	NOUN
cana-410	80	16	.	.	PUNCT
cana-410	81	1	theorem	theorem	VERB
cana-410	81	2	4.2	4.2	NUM
cana-410	81	3	let	let	VERB
cana-410	81	4	cl3n	cl3n	NOUN
cana-410	81	5	(	(	PUNCT
cana-410	81	6	g	g	NOUN
cana-410	81	7	)	)	PUNCT
cana-410	81	8	be	be	AUX
cana-410	81	9	a	a	DET
cana-410	81	10	clutch	clutch	ADJ
cana-410	81	11	graph	graph	NOUN
cana-410	81	12	with	with	ADP
cana-410	81	13	vertices	vertex	NOUN
cana-410	81	14	of	of	ADP
cana-410	81	15	degrees	degree	NOUN
cana-410	81	16	s	s	PART
cana-410	81	17	(	(	PUNCT
cana-410	81	18	or	or	CCONJ
cana-410	81	19	)	)	PUNCT
cana-410	81	20	t	t	PROPN
cana-410	81	21	,	,	PUNCT
cana-410	81	22	then	then	ADV
cana-410	81	23	show	show	VERB
cana-410	81	24	that	that	SCONJ
cana-410	81	25	n(2s	n(2s	VERB
cana-410	81	26	+	+	SYM
cana-410	81	27	t	t	NOUN
cana-410	81	28	)	)	PUNCT
cana-410	81	29	=	=	SYM
cana-410	81	30	2	2	NUM
cana-410	81	31	∈	∈	PROPN
cana-410	81	32	,	,	PUNCT
cana-410	81	33	n	n	X
cana-410	81	34	is	be	AUX
cana-410	81	35	the	the	DET
cana-410	81	36	number	number	NOUN
cana-410	81	37	of	of	ADP
cana-410	81	38	vertices	vertex	NOUN
cana-410	81	39	in	in	ADP
cana-410	81	40	the	the	DET
cana-410	81	41	cycle	cycle	NOUN
cana-410	81	42	graph	graph	NOUN
cana-410	81	43	.	.	PUNCT
cana-410	82	1	proof	proof	NOUN
cana-410	82	2	given	give	VERB
cana-410	82	3	that	that	DET
cana-410	82	4	cl3n	cl3n	NOUN
cana-410	82	5	(	(	PUNCT
cana-410	82	6	g	g	NOUN
cana-410	82	7	)	)	PUNCT
cana-410	82	8	is	be	AUX
cana-410	82	9	a	a	DET
cana-410	82	10	clutch	clutch	ADJ
cana-410	82	11	graph	graph	NOUN
cana-410	82	12	whose	whose	DET
cana-410	82	13	vertices	vertex	NOUN
cana-410	82	14	have	have	VERB
cana-410	82	15	degree	degree	NOUN
cana-410	82	16	s	s	NOUN
cana-410	82	17	or	or	CCONJ
cana-410	82	18	t.	t.	NOUN
cana-410	82	19	the	the	DET
cana-410	82	20	vertices	vertex	NOUN
cana-410	82	21	in	in	ADP
cana-410	82	22	vc	vc	PROPN
cana-410	82	23	and	and	CCONJ
cana-410	82	24	vp	vp	PROPN
cana-410	82	25	have	have	AUX
cana-410	82	26	degrees	degree	NOUN
cana-410	82	27	s	s	PART
cana-410	82	28	and	and	CCONJ
cana-410	82	29	vq	vq	PROPN
cana-410	82	30	have	have	AUX
cana-410	82	31	degrees	degree	NOUN
cana-410	82	32	t	t	NOUN
cana-410	82	33	respectively	respectively	ADV
cana-410	82	34	.	.	PUNCT
cana-410	83	1	by	by	ADP
cana-410	83	2	theorem	theorem	NOUN
cana-410	83	3	[	[	PUNCT
cana-410	83	4	4.1	4.1	NUM
cana-410	83	5	]	]	PUNCT
cana-410	83	6	,	,	PUNCT
cana-410	83	7	given	give	VERB
cana-410	83	8	that	that	SCONJ
cana-410	83	9	,	,	PUNCT
cana-410	83	10	∑	∑	ADV
cana-410	83	11	deg(ci	deg(ci	X
cana-410	83	12	)	)	PUNCT
cana-410	83	13	ci	ci	NOUN
cana-410	83	14	∈	∈	PROPN
cana-410	83	15	vc	vc	PROPN
cana-410	83	16	+	+	CCONJ
cana-410	83	17	∑	∑	PROPN
cana-410	83	18	deg(pi	deg(pi	NOUN
cana-410	83	19	)	)	PUNCT
cana-410	83	20	pi	pi	NOUN
cana-410	83	21	∈	∈	PROPN
cana-410	83	22	vp	vp	X
cana-410	83	23	+	+	CCONJ
cana-410	83	24	∑	∑	PART
cana-410	83	25	deg(qi	deg(qi	NOUN
cana-410	83	26	)	)	PUNCT
cana-410	83	27	qi	qi	PROPN
cana-410	83	28	∈	∈	PROPN
cana-410	83	29	vq	vq	PROPN
cana-410	83	30	=	=	SYM
cana-410	83	31	2e	2e	NUM
cana-410	83	32	𝑛𝑠	𝑛𝑠	VERB
cana-410	83	33	+	+	CCONJ
cana-410	83	34	𝑛𝑠	𝑛𝑠	VERB
cana-410	83	35	+	+	NOUN
cana-410	83	36	𝑛𝑡	𝑛𝑡	NOUN
cana-410	83	37	=	=	SYM
cana-410	83	38	2	2	NUM
cana-410	83	39	∈	∈	NOUN
cana-410	83	40	𝑛(2𝑠	𝑛(2𝑠	NOUN
cana-410	83	41	+	+	NUM
cana-410	83	42	𝑡	𝑡	X
cana-410	83	43	)	)	PUNCT
cana-410	83	44	=	=	SYM
cana-410	83	45	2	2	NUM
cana-410	83	46	∈	∈	NOUN
cana-410	83	47	hence	hence	ADV
cana-410	83	48	proved	prove	VERB
cana-410	83	49	theorem	theorem	VERB
cana-410	83	50	4.3	4.3	NUM
cana-410	83	51	every	every	DET
cana-410	83	52	clutch	clutch	ADJ
cana-410	83	53	graph	graph	NOUN
cana-410	83	54	cl3n	cl3n	NOUN
cana-410	83	55	(	(	PUNCT
cana-410	83	56	g	g	NOUN
cana-410	83	57	)	)	PUNCT
cana-410	83	58	is	be	AUX
cana-410	83	59	constructed	construct	VERB
cana-410	83	60	with	with	ADP
cana-410	83	61	2n	2n	NUM
cana-410	83	62	vertices	vertex	NOUN
cana-410	83	63	having	have	VERB
cana-410	83	64	of	of	ADP
cana-410	83	65	odd	odd	ADJ
cana-410	83	66	degree	degree	NOUN
cana-410	83	67	in	in	ADP
cana-410	83	68	vc	vc	PROPN
cana-410	83	69	and	and	CCONJ
cana-410	83	70	vp	vp	PROPN
cana-410	83	71	,	,	PUNCT
cana-410	83	72	and	and	CCONJ
cana-410	83	73	n	n	DET
cana-410	83	74	vertices	vertex	NOUN
cana-410	83	75	of	of	ADP
cana-410	83	76	even	even	ADJ
cana-410	83	77	degree	degree	NOUN
cana-410	83	78	in	in	ADP
cana-410	83	79	vq	vq	NOUN
cana-410	83	80	,	,	PUNCT
cana-410	83	81	results	result	NOUN
cana-410	83	82	in	in	ADP
cana-410	83	83	a	a	DET
cana-410	83	84	connected	connected	ADJ
cana-410	83	85	graph	graph	NOUN
cana-410	83	86	.	.	PUNCT
cana-410	84	1	proof	proof	NOUN
cana-410	84	2	given	give	VERB
cana-410	84	3	that	that	PRON
cana-410	84	4	s	s	NOUN
cana-410	84	5	=	=	SYM
cana-410	84	6	3	3	NUM
cana-410	84	7	be	be	AUX
cana-410	84	8	the	the	DET
cana-410	84	9	degree	degree	NOUN
cana-410	84	10	of	of	ADP
cana-410	84	11	each	each	DET
cana-410	84	12	vertex	vertex	NOUN
cana-410	84	13	ci	ci	NOUN
cana-410	84	14	in	in	ADP
cana-410	84	15	vc	vc	PROPN
cana-410	84	16	and	and	CCONJ
cana-410	84	17	each	each	DET
cana-410	84	18	vertex	vertex	NOUN
cana-410	84	19	pi	pi	NOUN
cana-410	84	20	in	in	ADP
cana-410	84	21	vp	vp	PROPN
cana-410	84	22	.	.	PUNCT
cana-410	85	1	then	then	ADV
cana-410	85	2	total	total	ADJ
cana-410	85	3	sum	sum	NOUN
cana-410	85	4	of	of	ADP
cana-410	85	5	odd	odd	ADJ
cana-410	85	6	degrees	degree	NOUN
cana-410	85	7	for	for	ADP
cana-410	85	8	these	these	DET
cana-410	85	9	2n	2n	NUM
cana-410	85	10	vertices	vertex	NOUN
cana-410	85	11	is	be	AUX
cana-410	85	12	∑	∑	X
cana-410	85	13	deg(𝑐𝑖	deg(𝑐𝑖	NOUN
cana-410	85	14	)	)	PUNCT
cana-410	86	1	+	+	CCONJ
cana-410	86	2	2𝑛	2𝑛	PROPN
cana-410	86	3	𝑖=1	𝑖=1	PUNCT
cana-410	86	4	∑	∑	PUNCT
cana-410	86	5	deg	deg	PROPN
cana-410	86	6	(	(	PUNCT
cana-410	86	7	𝑝𝑖	𝑝𝑖	NOUN
cana-410	86	8	)	)	PUNCT
cana-410	86	9	2𝑛	2𝑛	NOUN
cana-410	87	1	𝑖=1	𝑖=1	PUNCT
cana-410	87	2	=	=	PUNCT
cana-410	88	1	2𝑛.	2𝑛.	NUM
cana-410	88	2	𝑠	𝑠	X
cana-410	88	3	=	=	SYM
cana-410	88	4	6𝑛	6𝑛	NOUN
cana-410	88	5	also	also	ADV
cana-410	88	6	t	t	NOUN
cana-410	88	7	=	=	SYM
cana-410	88	8	2	2	NUM
cana-410	88	9	be	be	AUX
cana-410	88	10	the	the	DET
cana-410	88	11	degree	degree	NOUN
cana-410	88	12	of	of	ADP
cana-410	88	13	each	each	DET
cana-410	88	14	vertex	vertex	NOUN
cana-410	88	15	qi	qi	PROPN
cana-410	88	16	in	in	ADP
cana-410	88	17	vq	vq	PROPN
cana-410	88	18	.	.	PUNCT
cana-410	89	1	the	the	DET
cana-410	89	2	total	total	ADJ
cana-410	89	3	sum	sum	NOUN
cana-410	89	4	of	of	ADP
cana-410	89	5	even	even	ADJ
cana-410	89	6	degrees	degree	NOUN
cana-410	89	7	for	for	ADP
cana-410	89	8	these	these	DET
cana-410	89	9	n	n	PRON
cana-410	89	10	vertices	vertex	NOUN
cana-410	89	11	is	be	AUX
cana-410	89	12	∑	∑	ADV
cana-410	89	13	deg(𝑞𝑖	deg(𝑞𝑖	NOUN
cana-410	89	14	)	)	PUNCT
cana-410	90	1	=	=	NOUN
cana-410	90	2	𝑛.	𝑛.	NOUN
cana-410	90	3	𝑡	𝑡	X
cana-410	90	4	=	=	NOUN
cana-410	90	5	2𝑛.𝑛	2𝑛.𝑛	NUM
cana-410	90	6	𝑖=1	𝑖=1	PUNCT
cana-410	90	7	the	the	DET
cana-410	90	8	total	total	ADJ
cana-410	90	9	sum	sum	NOUN
cana-410	90	10	of	of	ADP
cana-410	90	11	degrees	degree	NOUN
cana-410	90	12	for	for	ADP
cana-410	90	13	all	all	DET
cana-410	90	14	vertices	vertex	NOUN
cana-410	90	15	is	be	AUX
cana-410	90	16	6n	6n	NUM
cana-410	90	17	+	+	CCONJ
cana-410	90	18	2n	2n	NUM
cana-410	90	19	=	=	NOUN
cana-410	90	20	8n	8n	NOUN
cana-410	90	21	.	.	PUNCT
cana-410	91	1	according	accord	VERB
cana-410	91	2	to	to	ADP
cana-410	91	3	the	the	DET
cana-410	91	4	handshaking	handshake	VERB
cana-410	91	5	lemma	lemma	NOUN
cana-410	91	6	,	,	PUNCT
cana-410	91	7	in	in	ADP
cana-410	91	8	a	a	DET
cana-410	91	9	graph	graph	NOUN
cana-410	91	10	∑	∑	VERB
cana-410	91	11	deg	deg	VERB
cana-410	91	12	𝐶𝐼𝑣𝜖𝑉	𝐶𝐼𝑣𝜖𝑉	PROPN
cana-410	91	13	3n(g)=2.|e|	3n(g)=2.|e|	NUM
cana-410	91	14	=	=	SYM
cana-410	91	15	2.4n	2.4n	NUM
cana-410	91	16	∑	∑	PUNCT
cana-410	91	17	deg	deg	VERB
cana-410	91	18	𝐶𝐼𝑣∈𝑉	𝐶𝐼𝑣∈𝑉	ADV
cana-410	91	19	3n	3n	NUM
cana-410	91	20	(	(	PUNCT
cana-410	91	21	g	g	NOUN
cana-410	91	22	)	)	PUNCT
cana-410	91	23	=	=	NOUN
cana-410	91	24	8n	8n	NOUN
cana-410	91	25	the	the	DET
cana-410	91	26	fact	fact	NOUN
cana-410	91	27	that	that	SCONJ
cana-410	91	28	∑	∑	PUNCT
cana-410	91	29	deg	deg	VERB
cana-410	91	30	𝐶𝐼𝑣∈𝑉	𝐶𝐼𝑣∈𝑉	ADV
cana-410	91	31	3n	3n	NUM
cana-410	91	32	(	(	PUNCT
cana-410	91	33	g	g	NOUN
cana-410	91	34	)	)	PUNCT
cana-410	92	1	=	=	NOUN
cana-410	92	2	8n	8n	NOUN
cana-410	92	3	implies	imply	VERB
cana-410	92	4	|𝐸|	|𝐸|	NOUN
cana-410	92	5	=	=	SYM
cana-410	92	6	1	1	NUM
cana-410	92	7	2	2	NUM
cana-410	92	8	∑	∑	ADP
cana-410	92	9	deg𝐶𝐼𝑣∈𝑉	deg𝐶𝐼𝑣∈𝑉	VERB
cana-410	92	10	3n	3n	NUM
cana-410	92	11	(	(	PUNCT
cana-410	92	12	g	g	NOUN
cana-410	92	13	)	)	PUNCT
cana-410	92	14	=	=	SYM
cana-410	92	15	4𝑛	4𝑛	NOUN
cana-410	92	16	edges	edge	VERB
cana-410	92	17	.	.	PUNCT
cana-410	93	1	this	this	PRON
cana-410	93	2	ensures	ensure	VERB
cana-410	93	3	that	that	SCONJ
cana-410	93	4	every	every	DET
cana-410	93	5	vertex	vertex	NOUN
cana-410	93	6	is	be	AUX
cana-410	93	7	incident	incident	NOUN
cana-410	93	8	to	to	ADP
cana-410	93	9	at	at	ADV
cana-410	93	10	least	least	ADV
cana-410	93	11	one	one	NUM
cana-410	93	12	edge	edge	NOUN
cana-410	93	13	,	,	PUNCT
cana-410	93	14	establishing	establish	VERB
cana-410	93	15	connectivity	connectivity	NOUN
cana-410	93	16	in	in	ADP
cana-410	93	17	the	the	DET
cana-410	93	18	clutch	clutch	NOUN
cana-410	93	19	graph	graph	NOUN
cana-410	93	20	.	.	PUNCT
cana-410	94	1	theorem	theorem	VERB
cana-410	94	2	4.4	4.4	NUM
cana-410	94	3	for	for	ADP
cana-410	94	4	every	every	DET
cana-410	94	5	clutch	clutch	NOUN
cana-410	94	6	graph	graph	NOUN
cana-410	94	7	,	,	PUNCT
cana-410	94	8	χ	χ	X
cana-410	95	1	+	+	X
cana-410	95	2	g	g	NOUN
cana-410	95	3	≤	≤	NUM
cana-410	95	4	n	n	NOUN
cana-410	95	5	+	+	CCONJ
cana-410	95	6	2	2	NUM
cana-410	95	7	,	,	PUNCT
cana-410	95	8	where	where	SCONJ
cana-410	95	9	χ	χ	NOUN
cana-410	95	10	is	be	AUX
cana-410	95	11	chromatic	chromatic	ADJ
cana-410	95	12	number	number	NOUN
cana-410	95	13	,	,	PUNCT
cana-410	95	14	g	g	PROPN
cana-410	95	15	is	be	AUX
cana-410	95	16	girth	girth	ADJ
cana-410	95	17	and	and	CCONJ
cana-410	95	18	n	n	ADV
cana-410	95	19	is	be	AUX
cana-410	95	20	number	number	NOUN
cana-410	95	21	of	of	ADP
cana-410	95	22	vertices	vertex	NOUN
cana-410	95	23	.	.	PUNCT
cana-410	96	1	communications	communication	NOUN
cana-410	96	2	on	on	ADP
cana-410	96	3	applied	apply	VERB
cana-410	96	4	nonlinear	nonlinear	ADJ
cana-410	96	5	analysis	analysis	NOUN
cana-410	96	6	issn	issn	NOUN
cana-410	96	7	:	:	PUNCT
cana-410	96	8	1074	1074	NUM
cana-410	96	9	-	-	PUNCT
cana-410	96	10	133x	133x	NUM
cana-410	96	11	vol	vol	NOUN
cana-410	96	12	31	31	NUM
cana-410	96	13	no	no	NOUN
cana-410	96	14	.	.	NOUN
cana-410	96	15	1	1	NUM
cana-410	96	16	(	(	PUNCT
cana-410	96	17	2024	2024	NUM
cana-410	96	18	)	)	PUNCT
cana-410	96	19	257	257	NUM
cana-410	96	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	96	21	example	example	NOUN
cana-410	96	22	:	:	PUNCT
cana-410	96	23	in	in	ADP
cana-410	96	24	fig	fig	NOUN
cana-410	96	25	:	:	PUNCT
cana-410	96	26	3	3	NUM
cana-410	96	27	,	,	PUNCT
cana-410	96	28	the	the	DET
cana-410	96	29	clutch	clutch	ADJ
cana-410	96	30	graph	graph	NOUN
cana-410	96	31	cl18	cl18	PROPN
cana-410	96	32	(	(	PUNCT
cana-410	96	33	g	g	NOUN
cana-410	96	34	)	)	PUNCT
cana-410	96	35	with	with	ADP
cana-410	96	36	n	n	NOUN
cana-410	96	37	=	=	NOUN
cana-410	96	38	6	6	NUM
cana-410	96	39	vertices	vertex	NOUN
cana-410	96	40	,	,	PUNCT
cana-410	96	41	resulting	result	VERB
cana-410	96	42	in	in	ADP
cana-410	96	43	3n=18	3n=18	NUM
cana-410	96	44	vertices	vertex	NOUN
cana-410	96	45	.	.	PUNCT
cana-410	97	1	for	for	ADP
cana-410	97	2	every	every	DET
cana-410	97	3	clutch	clutch	NOUN
cana-410	97	4	graph	graph	NOUN
cana-410	97	5	,	,	PUNCT
cana-410	97	6	the	the	DET
cana-410	97	7	chromatic	chromatic	ADJ
cana-410	97	8	number	number	NOUN
cana-410	97	9	χ	χ	NOUN
cana-410	97	10	is	be	AUX
cana-410	97	11	2	2	NUM
cana-410	97	12	,	,	PUNCT
cana-410	97	13	and	and	CCONJ
cana-410	97	14	the	the	DET
cana-410	97	15	girth	girth	NOUN
cana-410	97	16	g	g	PROPN
cana-410	97	17	is	be	AUX
cana-410	97	18	4	4	NUM
cana-410	97	19	.	.	PUNCT
cana-410	98	1	therefore	therefore	ADV
cana-410	98	2	,	,	PUNCT
cana-410	98	3	𝜒	𝜒	X
cana-410	98	4	+	+	X
cana-410	98	5	𝑔	𝑔	X
cana-410	98	6	=	=	SYM
cana-410	98	7	2	2	NUM
cana-410	98	8	+	+	CCONJ
cana-410	98	9	4	4	NUM
cana-410	98	10	=	=	SYM
cana-410	98	11	6	6	NUM
cana-410	98	12	𝑛	𝑛	NOUN
cana-410	98	13	+	+	CCONJ
cana-410	98	14	2	2	NUM
cana-410	98	15	=	=	SYM
cana-410	98	16	6	6	NUM
cana-410	98	17	+	+	SYM
cana-410	98	18	2	2	NUM
cana-410	98	19	=	=	SYM
cana-410	98	20	8	8	NUM
cana-410	98	21	the	the	DET
cana-410	98	22	example	example	NOUN
cana-410	98	23	satisfies	satisfy	VERB
cana-410	98	24	the	the	DET
cana-410	98	25	inequality	inequality	NOUN
cana-410	98	26	χ	χ	PROPN
cana-410	99	1	+	+	X
cana-410	99	2	g	g	NOUN
cana-410	99	3	≤	≤	NUM
cana-410	99	4	n	n	CCONJ
cana-410	99	5	+	+	NUM
cana-410	99	6	2	2	NUM
cana-410	99	7	,	,	PUNCT
cana-410	99	8	illustrating	illustrate	VERB
cana-410	99	9	the	the	DET
cana-410	99	10	validity	validity	NOUN
cana-410	99	11	of	of	ADP
cana-410	99	12	the	the	DET
cana-410	99	13	theorem	theorem	NOUN
cana-410	99	14	for	for	ADP
cana-410	99	15	this	this	DET
cana-410	99	16	clutch	clutch	ADJ
cana-410	99	17	graph	graph	NOUN
cana-410	99	18	with	with	ADP
cana-410	99	19	𝑛	𝑛	PROPN
cana-410	99	20	=	=	SYM
cana-410	99	21	6	6	NUM
cana-410	99	22	.	.	PUNCT
cana-410	99	23	theorem	theorem	VERB
cana-410	99	24	4.5	4.5	NUM
cana-410	99	25	in	in	ADP
cana-410	99	26	clutch	clutch	ADJ
cana-410	99	27	graph	graph	NOUN
cana-410	99	28	cl3n	cl3n	NOUN
cana-410	99	29	(	(	PUNCT
cana-410	99	30	g	g	NOUN
cana-410	99	31	)	)	PUNCT
cana-410	99	32	,	,	PUNCT
cana-410	99	33	the	the	DET
cana-410	99	34	number	number	NOUN
cana-410	99	35	of	of	ADP
cana-410	99	36	vertices	vertex	NOUN
cana-410	99	37	with	with	ADP
cana-410	99	38	odd	odd	ADJ
cana-410	99	39	degree	degree	NOUN
cana-410	99	40	is	be	AUX
cana-410	99	41	even	even	ADV
cana-410	99	42	.	.	PUNCT
cana-410	100	1	proof	proof	NOUN
cana-410	100	2	the	the	DET
cana-410	100	3	clutch	clutch	ADJ
cana-410	100	4	graph	graph	NOUN
cana-410	100	5	cl3n	cl3n	NOUN
cana-410	100	6	(	(	PUNCT
cana-410	100	7	g	g	NOUN
cana-410	100	8	)	)	PUNCT
cana-410	100	9	has	have	VERB
cana-410	100	10	3n	3n	NUM
cana-410	100	11	vertices	vertex	NOUN
cana-410	100	12	.	.	PUNCT
cana-410	101	1	since	since	SCONJ
cana-410	101	2	,	,	PUNCT
cana-410	101	3	the	the	DET
cana-410	101	4	sum	sum	NOUN
cana-410	101	5	of	of	ADP
cana-410	101	6	degrees	degree	NOUN
cana-410	101	7	of	of	ADP
cana-410	101	8	all	all	DET
cana-410	101	9	vertices	vertex	NOUN
cana-410	101	10	in	in	ADP
cana-410	101	11	the	the	DET
cana-410	101	12	clutch	clutch	NOUN
cana-410	101	13	graph	graph	NOUN
cana-410	101	14	is	be	AUX
cana-410	101	15	2e	2e	NUM
cana-410	101	16	,	,	PUNCT
cana-410	101	17	e	e	X
cana-410	101	18	is	be	AUX
cana-410	101	19	the	the	DET
cana-410	101	20	number	number	NOUN
cana-410	101	21	of	of	ADP
cana-410	101	22	edges	edge	NOUN
cana-410	101	23	.	.	PUNCT
cana-410	102	1	∑	∑	PUNCT
cana-410	102	2	deg(𝑣𝑖	deg(𝑣𝑖	VERB
cana-410	102	3	)	)	PUNCT
cana-410	102	4	𝑛	𝑛	PRON
cana-410	102	5	𝑖=1	𝑖=1	PUNCT
cana-410	102	6	=	=	SYM
cana-410	102	7	2e	2e	NUM
cana-410	102	8	,	,	PUNCT
cana-410	102	9	vi	vi	PROPN
cana-410	102	10	∈	∈	PROPN
cana-410	103	1	v	v	NOUN
cana-410	103	2	=	=	SYM
cana-410	103	3	vc	vc	PROPN
cana-410	103	4	∪	∪	ADJ
cana-410	103	5	vp	vp	PROPN
cana-410	103	6	∪	∪	X
cana-410	103	7	vq	vq	PROPN
cana-410	103	8	∑	∑	NOUN
cana-410	103	9	𝑑𝑒𝑔(𝑐𝑖	𝑑𝑒𝑔(𝑐𝑖	NOUN
cana-410	103	10	)	)	PUNCT
cana-410	103	11	𝑐𝑖	𝑐𝑖	NOUN
cana-410	103	12	∈	∈	NOUN
cana-410	104	1	𝑉𝑐	𝑉𝑐	PROPN
cana-410	104	2	+	+	ADJ
cana-410	104	3	∑	∑	ADJ
cana-410	104	4	𝑑𝑒𝑔(𝑝𝑖	𝑑𝑒𝑔(𝑝𝑖	NOUN
cana-410	104	5	)	)	PUNCT
cana-410	104	6	𝑝𝑖	𝑝𝑖	PART
cana-410	104	7	∈	∈	NOUN
cana-410	104	8	𝑉𝑝	𝑉𝑝	PROPN
cana-410	104	9	+	+	CCONJ
cana-410	104	10	∑	∑	NUM
cana-410	104	11	𝑑𝑒𝑔(𝑞𝑖	𝑑𝑒𝑔(𝑞𝑖	NOUN
cana-410	104	12	)	)	PUNCT
cana-410	104	13	𝑞𝑖	𝑞𝑖	PRON
cana-410	104	14	∈	∈	PROPN
cana-410	105	1	𝑉𝑞	𝑉𝑞	PROPN
cana-410	105	2	=	=	PROPN
cana-410	105	3	2e	2e	PROPN
cana-410	105	4	in	in	ADP
cana-410	105	5	the	the	DET
cana-410	105	6	clutch	clutch	NOUN
cana-410	105	7	graph	graph	NOUN
cana-410	105	8	,	,	PUNCT
cana-410	105	9	each	each	DET
cana-410	105	10	vertex	vertex	NOUN
cana-410	105	11	in	in	ADP
cana-410	105	12	vc	vc	PROPN
cana-410	105	13	and	and	CCONJ
cana-410	105	14	vp	vp	PROPN
cana-410	105	15	with	with	ADP
cana-410	105	16	degree	degree	NOUN
cana-410	105	17	3	3	NUM
cana-410	105	18	,	,	PUNCT
cana-410	105	19	and	and	CCONJ
cana-410	105	20	each	each	DET
cana-410	105	21	vertex	vertex	NOUN
cana-410	105	22	in	in	ADP
cana-410	105	23	vq	vq	NOUN
cana-410	105	24	with	with	ADP
cana-410	105	25	degree	degree	NOUN
cana-410	105	26	2	2	NUM
cana-410	105	27	.	.	PUNCT
cana-410	105	28	let	let	VERB
cana-410	105	29	n	n	PRON
cana-410	105	30	be	be	AUX
cana-410	105	31	the	the	DET
cana-410	105	32	number	number	NOUN
cana-410	105	33	of	of	ADP
cana-410	105	34	vertices	vertex	NOUN
cana-410	105	35	of	of	ADP
cana-410	105	36	the	the	DET
cana-410	105	37	vertex	vertex	NOUN
cana-410	105	38	set	set	VERB
cana-410	105	39	vc	vc	PROPN
cana-410	105	40	,	,	PUNCT
cana-410	105	41	vp	vp	PROPN
cana-410	105	42	and	and	CCONJ
cana-410	105	43	vq	vq	NOUN
cana-410	105	44	respectively	respectively	ADV
cana-410	105	45	.	.	PUNCT
cana-410	106	1	3n	3n	NUM
cana-410	106	2	+	+	NUM
cana-410	106	3	3n	3n	NUM
cana-410	106	4	+	+	CCONJ
cana-410	106	5	2n	2n	NUM
cana-410	106	6	=	=	SYM
cana-410	106	7	2e	2e	NOUN
cana-410	106	8	8n	8n	NOUN
cana-410	106	9	=	=	SYM
cana-410	106	10	2e	2e	NOUN
cana-410	106	11	here	here	ADV
cana-410	106	12	the	the	DET
cana-410	106	13	sum	sum	NOUN
cana-410	106	14	of	of	ADP
cana-410	106	15	degrees	degree	NOUN
cana-410	106	16	is	be	AUX
cana-410	106	17	even	even	ADV
cana-410	106	18	,	,	PUNCT
cana-410	106	19	and	and	CCONJ
cana-410	106	20	the	the	DET
cana-410	106	21	degrees	degree	NOUN
cana-410	106	22	in	in	ADP
cana-410	106	23	vq	vq	NOUN
cana-410	106	24	are	be	AUX
cana-410	106	25	all	all	ADV
cana-410	106	26	even	even	ADV
cana-410	106	27	,	,	PUNCT
cana-410	106	28	then	then	ADV
cana-410	106	29	the	the	DET
cana-410	106	30	sum	sum	NOUN
cana-410	106	31	of	of	ADP
cana-410	106	32	degrees	degree	NOUN
cana-410	106	33	in	in	ADP
cana-410	106	34	vc	vc	PROPN
cana-410	106	35	and	and	CCONJ
cana-410	106	36	vp	vp	PROPN
cana-410	106	37	(	(	PUNCT
cana-410	106	38	which	which	PRON
cana-410	106	39	are	be	AUX
cana-410	106	40	all	all	PRON
cana-410	106	41	odd	odd	ADJ
cana-410	106	42	)	)	PUNCT
cana-410	106	43	must	must	AUX
cana-410	106	44	be	be	AUX
cana-410	106	45	even	even	ADV
cana-410	106	46	.	.	PUNCT
cana-410	107	1	it	it	PRON
cana-410	107	2	follows	follow	VERB
cana-410	107	3	that	that	SCONJ
cana-410	107	4	,	,	PUNCT
cana-410	107	5	the	the	DET
cana-410	107	6	number	number	NOUN
cana-410	107	7	of	of	ADP
cana-410	107	8	vertices	vertex	NOUN
cana-410	107	9	with	with	ADP
cana-410	107	10	odd	odd	ADJ
cana-410	107	11	degree	degree	NOUN
cana-410	107	12	is	be	AUX
cana-410	107	13	even	even	ADV
cana-410	107	14	.	.	PUNCT
cana-410	108	1	theorem	theorem	VERB
cana-410	108	2	4.6	4.6	NUM
cana-410	108	3	every	every	DET
cana-410	108	4	clutch	clutch	ADJ
cana-410	108	5	graph	graph	NOUN
cana-410	108	6	cl3n	cl3n	NOUN
cana-410	108	7	(	(	PUNCT
cana-410	108	8	g	g	NOUN
cana-410	108	9	)	)	PUNCT
cana-410	108	10	has	have	VERB
cana-410	108	11	78[4n	78[4n	NUM
cana-410	108	12	−	−	NOUN
cana-410	108	13	k]k+1	k]k+1	NOUN
cana-410	108	14	spanning	span	VERB
cana-410	108	15	trees	tree	NOUN
cana-410	108	16	.	.	PUNCT
cana-410	109	1	proof	proof	NOUN
cana-410	109	2	step	step	NOUN
cana-410	109	3	1	1	NUM
cana-410	109	4	:	:	PUNCT
cana-410	109	5	consider	consider	VERB
cana-410	109	6	the	the	DET
cana-410	109	7	clutch	clutch	ADJ
cana-410	109	8	graph	graph	NOUN
cana-410	109	9	cl3n	cl3n	NOUN
cana-410	109	10	(	(	PUNCT
cana-410	109	11	g	g	NOUN
cana-410	109	12	)	)	PUNCT
cana-410	109	13	with	with	ADP
cana-410	109	14	the	the	DET
cana-410	109	15	vertex	vertex	NOUN
cana-410	109	16	set	set	VERB
cana-410	109	17	v	v	NOUN
cana-410	109	18	=	=	SYM
cana-410	109	19	vc	vc	NOUN
cana-410	109	20	∪	∪	X
cana-410	109	21	vp	vp	PROPN
cana-410	109	22	∪	∪	X
cana-410	109	23	vq	vq	NOUN
cana-410	109	24	and	and	CCONJ
cana-410	109	25	edge	edge	NOUN
cana-410	109	26	sets	set	NOUN
cana-410	109	27	e	e	NOUN
cana-410	109	28	=	=	SYM
cana-410	109	29	ec	ec	PROPN
cana-410	109	30	∪	∪	VERB
cana-410	109	31	ecp	ecp	PROPN
cana-410	109	32	∪	∪	ADP
cana-410	109	33	ep	ep	PROPN
cana-410	109	34	∪	∪	ADP
cana-410	109	35	epq	epq	PROPN
cana-410	109	36	∪	∪	NOUN
cana-410	109	37	eq	eq	NOUN
cana-410	109	38	.	.	PUNCT
cana-410	110	1	step	step	NOUN
cana-410	110	2	2	2	NUM
cana-410	110	3	:	:	PUNCT
cana-410	110	4	the	the	DET
cana-410	110	5	degrees	degree	NOUN
cana-410	110	6	of	of	ADP
cana-410	110	7	the	the	DET
cana-410	110	8	vertices	vertex	NOUN
cana-410	110	9	in	in	ADP
cana-410	110	10	v	v	NUM
cana-410	110	11	are	be	AUX
cana-410	110	12	𝑑𝑒𝑔(𝑣𝑖	𝑑𝑒𝑔(𝑣𝑖	NUM
cana-410	110	13	)	)	PUNCT
cana-410	111	1	=	=	PRON
cana-410	111	2	{	{	PUNCT
cana-410	111	3	3	3	NUM
cana-410	111	4	,	,	PUNCT
cana-410	111	5	𝑖𝑓	𝑖𝑓	NUM
cana-410	111	6	𝑣𝑖	𝑣𝑖	PRON
cana-410	111	7	∈	∈	PROPN
cana-410	112	1	𝑉𝑐	𝑉𝑐	PRON
cana-410	112	2	∪	∪	VERB
cana-410	112	3	𝑉𝑝	𝑉𝑝	PROPN
cana-410	112	4	2	2	NUM
cana-410	112	5	,	,	PUNCT
cana-410	112	6	𝑖𝑓	𝑖𝑓	ADP
cana-410	112	7	𝑣𝑖	𝑣𝑖	ADP
cana-410	112	8	∈	∈	PROPN
cana-410	113	1	𝑉𝑞	𝑉𝑞	PROPN
cana-410	113	2	then	then	ADV
cana-410	113	3	,	,	PUNCT
cana-410	113	4	d(cl3n	d(cl3n	PROPN
cana-410	113	5	(	(	PUNCT
cana-410	113	6	g	g	NOUN
cana-410	113	7	)	)	PUNCT
cana-410	113	8	)	)	PUNCT
cana-410	113	9	is	be	AUX
cana-410	113	10	the	the	DET
cana-410	113	11	diagonal	diagonal	ADJ
cana-410	113	12	matrix	matrix	NOUN
cana-410	113	13	of	of	ADP
cana-410	113	14	vertex	vertex	NOUN
cana-410	113	15	degrees	degree	NOUN
cana-410	113	16	which	which	PRON
cana-410	113	17	is	be	AUX
cana-410	113	18	given	give	VERB
cana-410	113	19	by	by	ADP
cana-410	113	20	d(cl3n	d(cl3n	NOUN
cana-410	113	21	(	(	PUNCT
cana-410	113	22	g	g	NOUN
cana-410	113	23	)	)	PUNCT
cana-410	113	24	)	)	PUNCT
cana-410	114	1	=	=	PUNCT
cana-410	114	2	[	[	PUNCT
cana-410	114	3	𝐷𝑐𝑐	𝐷𝑐𝑐	NOUN
cana-410	114	4	0	0	NUM
cana-410	114	5	0	0	NUM
cana-410	114	6	0	0	X
cana-410	115	1	𝐷𝑝𝑝	𝐷𝑝𝑝	ADJ
cana-410	115	2	0	0	NUM
cana-410	115	3	0	0	NUM
cana-410	115	4	0	0	NUM
cana-410	115	5	𝐷𝑞𝑞	𝐷𝑞𝑞	NOUN
cana-410	115	6	]	]	PUNCT
cana-410	115	7	step	step	NOUN
cana-410	115	8	3	3	NUM
cana-410	115	9	:	:	PUNCT
cana-410	115	10	form	form	VERB
cana-410	115	11	an	an	DET
cana-410	115	12	adjacency	adjacency	NOUN
cana-410	115	13	matrix	matrix	NOUN
cana-410	115	14	based	base	VERB
cana-410	115	15	on	on	ADP
cana-410	115	16	the	the	DET
cana-410	115	17	edges	edge	NOUN
cana-410	115	18	of	of	ADP
cana-410	115	19	the	the	DET
cana-410	115	20	clutch	clutch	NOUN
cana-410	115	21	graph	graph	NOUN
cana-410	115	22	𝐴(𝐶𝑙3𝑛(𝐺	𝐴(𝐶𝑙3𝑛(𝐺	NUM
cana-410	115	23	)	)	PUNCT
cana-410	115	24	)	)	PUNCT
cana-410	116	1	=	=	PRON
cana-410	116	2	{	{	PUNCT
cana-410	116	3	1	1	NUM
cana-410	116	4	,	,	PUNCT
cana-410	116	5	if	if	SCONJ
cana-410	116	6	there	there	PRON
cana-410	116	7	is	be	VERB
cana-410	116	8	an	an	DET
cana-410	116	9	edge	edge	NOUN
cana-410	116	10	between	between	ADP
cana-410	116	11	vertices	vertex	NOUN
cana-410	116	12	𝑖	𝑖	X
cana-410	116	13	and	and	CCONJ
cana-410	116	14	𝑗	𝑗	PROPN
cana-410	116	15	0	0	NUM
cana-410	116	16	,	,	PUNCT
cana-410	116	17	otherwise	otherwise	ADV
cana-410	116	18	i	i	PRON
cana-410	116	19	communications	communication	VERB
cana-410	116	20	on	on	ADP
cana-410	116	21	applied	apply	VERB
cana-410	116	22	nonlinear	nonlinear	ADJ
cana-410	116	23	analysis	analysis	NOUN
cana-410	116	24	issn	issn	NOUN
cana-410	116	25	:	:	PUNCT
cana-410	116	26	1074	1074	NUM
cana-410	116	27	-	-	PUNCT
cana-410	116	28	133x	133x	NUM
cana-410	116	29	vol	vol	NOUN
cana-410	116	30	31	31	NUM
cana-410	116	31	no	no	NOUN
cana-410	116	32	.	.	NOUN
cana-410	116	33	1	1	NUM
cana-410	116	34	(	(	PUNCT
cana-410	116	35	2024	2024	NUM
cana-410	116	36	)	)	PUNCT
cana-410	116	37	258	258	NUM
cana-410	116	38	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	116	39	𝐴(𝐶𝑙3𝑛(𝐺	𝐴(𝐶𝑙3𝑛(𝐺	NUM
cana-410	116	40	)	)	PUNCT
cana-410	116	41	)	)	PUNCT
cana-410	117	1	=	=	PUNCT
cana-410	117	2	[	[	PUNCT
cana-410	117	3	𝐴𝑐𝑐	𝐴𝑐𝑐	PROPN
cana-410	117	4	𝐴𝑐𝑝	𝐴𝑐𝑝	PROPN
cana-410	117	5	0	0	PUNCT
cana-410	118	1	𝐴𝑐𝑝	𝐴𝑐𝑝	PROPN
cana-410	118	2	𝐴𝑝𝑝	𝐴𝑝𝑝	PROPN
cana-410	118	3	𝐴𝑝𝑞	𝐴𝑝𝑞	PROPN
cana-410	118	4	0	0	NUM
cana-410	118	5	𝐴𝑃𝑞	𝐴𝑃𝑞	NOUN
cana-410	118	6	𝐴𝑞𝑞	𝐴𝑞𝑞	PROPN
cana-410	118	7	]	]	PUNCT
cana-410	118	8	step	step	NOUN
cana-410	118	9	4	4	NUM
cana-410	118	10	:	:	PUNCT
cana-410	118	11	the	the	DET
cana-410	118	12	laplacian	laplacian	ADJ
cana-410	118	13	matrix	matrix	NOUN
cana-410	118	14	is	be	AUX
cana-410	118	15	defined	define	VERB
cana-410	118	16	as	as	ADP
cana-410	118	17	l(cl3n	l(cl3n	X
cana-410	118	18	(	(	PUNCT
cana-410	118	19	g	g	NOUN
cana-410	118	20	)	)	PUNCT
cana-410	118	21	)	)	PUNCT
cana-410	119	1	=	=	PUNCT
cana-410	119	2	d(cl3n	d(cl3n	X
cana-410	119	3	(	(	PUNCT
cana-410	119	4	g	g	NOUN
cana-410	119	5	)	)	PUNCT
cana-410	119	6	)	)	PUNCT
cana-410	119	7	a(cl3n	a(cl3n	PUNCT
cana-410	119	8	(	(	PUNCT
cana-410	119	9	g	g	NOUN
cana-410	119	10	)	)	PUNCT
cana-410	119	11	)	)	PUNCT
cana-410	119	12	.	.	PUNCT
cana-410	120	1	so	so	ADV
cana-410	120	2	𝐿(𝐶𝑙3𝑛(𝐺	𝐿(𝐶𝑙3𝑛(𝐺	NUM
cana-410	120	3	)	)	PUNCT
cana-410	120	4	)	)	PUNCT
cana-410	121	1	=	=	PUNCT
cana-410	121	2	[	[	PUNCT
cana-410	121	3	𝐷𝑐𝑐	𝐷𝑐𝑐	PROPN
cana-410	121	4	−	−	PROPN
cana-410	121	5	𝐴𝑐𝑐	𝐴𝑐𝑐	PROPN
cana-410	121	6	−	−	PROPN
cana-410	121	7	𝐴𝑐𝑝	𝐴𝑐𝑝	PROPN
cana-410	121	8	0	0	SYM
cana-410	121	9	−𝐴𝑐𝑝	−𝐴𝑐𝑝	NOUN
cana-410	121	10	𝐷𝑝𝑝	𝐷𝑝𝑝	NOUN
cana-410	121	11	−	−	PROPN
cana-410	121	12	𝐴𝑝𝑝	𝐴𝑝𝑝	PROPN
cana-410	121	13	−	−	PROPN
cana-410	122	1	𝐴𝑝𝑞	𝐴𝑝𝑞	PROPN
cana-410	122	2	0	0	NUM
cana-410	123	1	−	−	NOUN
cana-410	123	2	𝐴𝑃𝑞	𝐴𝑃𝑞	NOUN
cana-410	123	3	𝐷𝑞𝑞	𝐷𝑞𝑞	PROPN
cana-410	123	4	−	−	PROPN
cana-410	123	5	𝐴𝑞𝑞	𝐴𝑞𝑞	PROPN
cana-410	123	6	]	]	PUNCT
cana-410	123	7	then	then	ADV
cana-410	123	8	,	,	PUNCT
cana-410	123	9	by	by	ADP
cana-410	123	10	kirchoff	kirchoff	PROPN
cana-410	123	11	’s	’s	PART
cana-410	123	12	theorem	theorem	PROPN
cana-410	123	13	,	,	PUNCT
cana-410	123	14	the	the	DET
cana-410	123	15	number	number	NOUN
cana-410	123	16	of	of	ADP
cana-410	123	17	distinct	distinct	ADJ
cana-410	123	18	spanning	span	VERB
cana-410	123	19	tree	tree	NOUN
cana-410	123	20	of	of	ADP
cana-410	123	21	the	the	DET
cana-410	123	22	graph	graph	NOUN
cana-410	123	23	is	be	AUX
cana-410	123	24	equal	equal	ADJ
cana-410	123	25	to	to	ADP
cana-410	123	26	any	any	DET
cana-410	123	27	cofactors	cofactor	NOUN
cana-410	123	28	of	of	ADP
cana-410	123	29	its	its	PRON
cana-410	123	30	laplacian	laplacian	ADJ
cana-410	123	31	matrix	matrix	NOUN
cana-410	123	32	.	.	PUNCT
cana-410	124	1	hence	hence	ADV
cana-410	124	2	,	,	PUNCT
cana-410	124	3	the	the	DET
cana-410	124	4	cofactor	cofactor	NOUN
cana-410	124	5	of	of	ADP
cana-410	124	6	l(cl3n	l(cl3n	X
cana-410	124	7	(	(	PUNCT
cana-410	124	8	g	g	NOUN
cana-410	124	9	)	)	PUNCT
cana-410	124	10	)	)	PUNCT
cana-410	124	11	is	be	AUX
cana-410	124	12	calculated	calculate	VERB
cana-410	124	13	by	by	ADP
cana-410	124	14	using	use	VERB
cana-410	124	15	(	(	PUNCT
cana-410	124	16	−1)i+j	−1)i+j	PUNCT
cana-410	124	17	·	·	PUNCT
cana-410	124	18	det(l	det(l	PROPN
cana-410	124	19	)	)	PUNCT
cana-410	124	20	that	that	PRON
cana-410	124	21	remains	remain	VERB
cana-410	124	22	after	after	ADP
cana-410	124	23	deleting	delete	VERB
cana-410	124	24	ith	ith	PROPN
cana-410	124	25	row	row	NOUN
cana-410	124	26	and	and	CCONJ
cana-410	124	27	jth	jth	PROPN
cana-410	124	28	column	column	PROPN
cana-410	124	29	.	.	PUNCT
cana-410	125	1	therefore	therefore	ADV
cana-410	125	2	,	,	PUNCT
cana-410	125	3	the	the	DET
cana-410	125	4	clutch	clutch	NOUN
cana-410	125	5	graph	graph	NOUN
cana-410	125	6	has	have	VERB
cana-410	125	7	n	n	CCONJ
cana-410	125	8	distinct	distinct	ADJ
cana-410	125	9	spanning	span	VERB
cana-410	125	10	trees	tree	NOUN
cana-410	125	11	.	.	PUNCT
cana-410	126	1	example	example	NOUN
cana-410	126	2	:	:	PUNCT
cana-410	126	3	let	let	VERB
cana-410	126	4	’s	’s	PRON
cana-410	126	5	find	find	VERB
cana-410	126	6	the	the	DET
cana-410	126	7	cofactors	cofactor	NOUN
cana-410	126	8	of	of	ADP
cana-410	126	9	laplacian	laplacian	ADJ
cana-410	126	10	matrix	matrix	NOUN
cana-410	126	11	for	for	ADP
cana-410	126	12	the	the	DET
cana-410	126	13	clutch	clutch	NOUN
cana-410	126	14	graph	graph	NOUN
cana-410	126	15	cl12	cl12	PROPN
cana-410	126	16	(	(	PUNCT
cana-410	126	17	g	g	NOUN
cana-410	126	18	)	)	PUNCT
cana-410	126	19	using	use	VERB
cana-410	126	20	jupyter	jupyter	ADJ
cana-410	126	21	notebook	notebook	NOUN
cana-410	126	22	software	software	NOUN
cana-410	126	23	.	.	PUNCT
cana-410	127	1	start	start	VERB
cana-410	127	2	with	with	ADP
cana-410	127	3	the	the	DET
cana-410	127	4	construction	construction	NOUN
cana-410	127	5	of	of	ADP
cana-410	127	6	cl12	cl12	PROPN
cana-410	127	7	(	(	PUNCT
cana-410	127	8	g	g	NOUN
cana-410	127	9	)	)	PUNCT
cana-410	127	10	.	.	PUNCT
cana-410	128	1	import	import	NOUN
cana-410	128	2	matplotlib.pyplot	matplotlib.pyplot	PROPN
cana-410	128	3	as	as	ADP
cana-410	128	4	plt	plt	NOUN
cana-410	128	5	import	import	NOUN
cana-410	128	6	numpy	numpy	NOUN
cana-410	128	7	as	as	ADP
cana-410	128	8	np	np	NOUN
cana-410	128	9	#	#	SYM
cana-410	128	10	points	point	NOUN
cana-410	128	11	points=	points=	NOUN
cana-410	128	12	{	{	PUNCT
cana-410	128	13	’	'	PUNCT
cana-410	128	14	c1	c1	NOUN
cana-410	128	15	’	'	PUNCT
cana-410	128	16	:	:	PUNCT
cana-410	128	17	(	(	PUNCT
cana-410	128	18	1.48	1.48	NUM
cana-410	128	19	,	,	PUNCT
cana-410	128	20	0.41	0.41	NUM
cana-410	128	21	)	)	PUNCT
cana-410	128	22	,	,	PUNCT
cana-410	128	23	’	'	PUNCT
cana-410	128	24	c2	c2	PROPN
cana-410	128	25	’	'	PUNCT
cana-410	128	26	:	:	PUNCT
cana-410	128	27	(	(	PUNCT
cana-410	128	28	2.44	2.44	NUM
cana-410	128	29	,	,	PUNCT
cana-410	128	30	0.41	0.41	NUM
cana-410	128	31	)	)	PUNCT
cana-410	128	32	,	,	PUNCT
cana-410	128	33	’	'	PUNCT
cana-410	128	34	c3	c3	PROPN
cana-410	128	35	’	'	PUNCT
cana-410	128	36	:	:	PUNCT
cana-410	128	37	(	(	PUNCT
cana-410	128	38	2.46	2.46	NUM
cana-410	128	39	,	,	PUNCT
cana-410	128	40	-0.41	-0.41	NUM
cana-410	128	41	)	)	PUNCT
cana-410	128	42	,	,	PUNCT
cana-410	128	43	’	'	PUNCT
cana-410	128	44	c4	c4	NOUN
cana-410	128	45	’	'	PUNCT
cana-410	128	46	:	:	PUNCT
cana-410	128	47	(	(	PUNCT
cana-410	128	48	1.46	1.46	NUM
cana-410	128	49	,	,	PUNCT
cana-410	128	50	-0.43	-0.43	NUM
cana-410	128	51	)	)	PUNCT
cana-410	128	52	,	,	PUNCT
cana-410	128	53	’	'	PUNCT
cana-410	128	54	p1	p1	PROPN
cana-410	128	55	’	'	PUNCT
cana-410	128	56	:	:	PUNCT
cana-410	128	57	(	(	PUNCT
cana-410	128	58	0.88	0.88	NUM
cana-410	128	59	,	,	PUNCT
cana-410	128	60	0.79	0.79	NUM
cana-410	128	61	)	)	PUNCT
cana-410	128	62	,	,	PUNCT
cana-410	128	63	’	'	PUNCT
cana-410	128	64	q2	q2	NOUN
cana-410	128	65	’	'	PUNCT
cana-410	128	66	:	:	PUNCT
cana-410	128	67	(	(	PUNCT
cana-410	128	68	3.46	3.46	NUM
cana-410	128	69	,	,	PUNCT
cana-410	128	70	1.39	1.39	NUM
cana-410	128	71	)	)	PUNCT
cana-410	128	72	,	,	PUNCT
cana-410	128	73	’	'	PUNCT
cana-410	128	74	p3	p3	PROPN
cana-410	128	75	’	'	PUNCT
cana-410	128	76	:	:	PUNCT
cana-410	128	77	(	(	PUNCT
cana-410	128	78	2.86	2.86	NUM
cana-410	128	79	,	,	PUNCT
cana-410	128	80	-0.81	-0.81	NUM
cana-410	128	81	)	)	PUNCT
cana-410	128	82	,	,	PUNCT
cana-410	128	83	’	'	PUNCT
cana-410	128	84	p4	p4	ADJ
cana-410	128	85	’	'	PUNCT
cana-410	128	86	:	:	PUNCT
cana-410	128	87	(	(	PUNCT
cana-410	128	88	0.84	0.84	NUM
cana-410	128	89	,	,	PUNCT
cana-410	128	90	-0.79	-0.79	NOUN
cana-410	128	91	)	)	PUNCT
cana-410	128	92	,	,	PUNCT
cana-410	128	93	’	'	PUNCT
cana-410	128	94	q1	q1	PROPN
cana-410	128	95	’	'	PUNCT
cana-410	128	96	:	:	PUNCT
cana-410	128	97	(	(	PUNCT
cana-410	128	98	0.08	0.08	NUM
cana-410	128	99	,	,	PUNCT
cana-410	128	100	1.37	1.37	NUM
cana-410	128	101	)	)	PUNCT
cana-410	128	102	,	,	PUNCT
cana-410	128	103	’	'	PUNCT
cana-410	128	104	q3	q3	PROPN
cana-410	128	105	’	'	PUNCT
cana-410	128	106	:	:	PUNCT
cana-410	128	107	(	(	PUNCT
cana-410	128	108	3.46	3.46	NUM
cana-410	128	109	,	,	PUNCT
cana-410	128	110	1.37	1.37	NUM
cana-410	128	111	)	)	PUNCT
cana-410	128	112	,	,	PUNCT
cana-410	128	113	’	'	PUNCT
cana-410	128	114	p2	p2	X
cana-410	128	115	’	'	PUNCT
cana-410	128	116	:	:	PUNCT
cana-410	128	117	(	(	PUNCT
cana-410	128	118	2.88	2.88	NUM
cana-410	128	119	,	,	PUNCT
cana-410	128	120	0.81	0.81	NUM
cana-410	128	121	)	)	PUNCT
cana-410	128	122	,	,	PUNCT
cana-410	128	123	’	'	PUNCT
cana-410	128	124	q4	q4	PROPN
cana-410	128	125	’	'	PUNCT
cana-410	128	126	:	:	PUNCT
cana-410	128	127	(	(	PUNCT
cana-410	128	128	0.06	0.06	NUM
cana-410	128	129	,	,	PUNCT
cana-410	128	130	-1.39	-1.39	NUM
cana-410	128	131	)	)	PUNCT
cana-410	128	132	}	}	PUNCT
cana-410	128	133	#	#	NOUN
cana-410	128	134	plot	plot	NOUN
cana-410	128	135	points	point	NOUN
cana-410	128	136	for	for	ADP
cana-410	128	137	point	point	NOUN
cana-410	128	138	,	,	PUNCT
cana-410	128	139	coordinates	coordinate	NOUN
cana-410	128	140	in	in	ADP
cana-410	128	141	points.items	points.item	NOUN
cana-410	128	142	(	(	PUNCT
cana-410	128	143	):	):	PUNCT
cana-410	128	144	plt.scatter(*coordinates	plt.scatter(*coordinate	NOUN
cana-410	128	145	,	,	PUNCT
cana-410	128	146	label	label	NOUN
cana-410	128	147	=	=	NOUN
cana-410	128	148	point	point	NOUN
cana-410	128	149	)	)	PUNCT
cana-410	128	150	#	#	NOUN
cana-410	128	151	connect	connect	NOUN
cana-410	128	152	points	point	NOUN
cana-410	128	153	with	with	ADP
cana-410	128	154	lines	line	NOUN
cana-410	128	155	lines	line	NOUN
cana-410	128	156	=	=	PUNCT
cana-410	129	1	[	[	PUNCT
cana-410	129	2	communications	communication	NOUN
cana-410	129	3	on	on	ADP
cana-410	129	4	applied	apply	VERB
cana-410	129	5	nonlinear	nonlinear	ADJ
cana-410	129	6	analysis	analysis	NOUN
cana-410	129	7	issn	issn	NOUN
cana-410	129	8	:	:	PUNCT
cana-410	129	9	1074	1074	NUM
cana-410	129	10	-	-	PUNCT
cana-410	129	11	133x	133x	NUM
cana-410	129	12	vol	vol	NOUN
cana-410	129	13	31	31	NUM
cana-410	129	14	no	no	NOUN
cana-410	129	15	.	.	NOUN
cana-410	129	16	1	1	NUM
cana-410	129	17	(	(	PUNCT
cana-410	129	18	2024	2024	NUM
cana-410	129	19	)	)	PUNCT
cana-410	129	20	259	259	NUM
cana-410	129	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	129	22	(	(	PUNCT
cana-410	129	23	‘	'	PUNCT
cana-410	129	24	c1’,’c2’,’c3’,’c4’,’c1	c1’,’c2’,’c3’,’c4’,’c1	NOUN
cana-410	129	25	’	'	PUNCT
cana-410	129	26	)	)	PUNCT
cana-410	129	27	,	,	PUNCT
cana-410	129	28	(	(	PUNCT
cana-410	129	29	’	'	PUNCT
cana-410	129	30	c1	c1	PROPN
cana-410	129	31	’	'	PUNCT
cana-410	129	32	,	,	PUNCT
cana-410	129	33	’	'	PUNCT
cana-410	129	34	p1	p1	PROPN
cana-410	129	35	’	'	PUNCT
cana-410	129	36	)	)	PUNCT
cana-410	129	37	,	,	PUNCT
cana-410	129	38	(	(	PUNCT
cana-410	129	39	’	'	PUNCT
cana-410	129	40	c2	c2	PROPN
cana-410	129	41	’	'	PUNCT
cana-410	129	42	,	,	PUNCT
cana-410	129	43	’	'	PUNCT
cana-410	129	44	p2	p2	X
cana-410	129	45	’	'	PUNCT
cana-410	129	46	)	)	PUNCT
cana-410	129	47	,	,	PUNCT
cana-410	129	48	(	(	PUNCT
cana-410	129	49	’	'	PUNCT
cana-410	129	50	c3	c3	PROPN
cana-410	129	51	’	'	PUNCT
cana-410	129	52	,	,	PUNCT
cana-410	129	53	’	'	PUNCT
cana-410	129	54	p3	p3	PROPN
cana-410	129	55	’	'	PUNCT
cana-410	129	56	)	)	PUNCT
cana-410	129	57	,	,	PUNCT
cana-410	129	58	(	(	PUNCT
cana-410	129	59	’	'	PUNCT
cana-410	129	60	c4	c4	NOUN
cana-410	129	61	’	'	PUNCT
cana-410	129	62	,	,	PUNCT
cana-410	129	63	’	'	PUNCT
cana-410	129	64	p4	p4	ADJ
cana-410	129	65	’	'	PUNCT
cana-410	129	66	)	)	PUNCT
cana-410	129	67	,	,	PUNCT
cana-410	129	68	(	(	PUNCT
cana-410	129	69	’	'	PUNCT
cana-410	129	70	p1	p1	PROPN
cana-410	129	71	’	'	PUNCT
cana-410	129	72	,	,	PUNCT
cana-410	129	73	’	'	PUNCT
cana-410	129	74	p2	p2	X
cana-410	129	75	’	'	PUNCT
cana-410	129	76	)	)	PUNCT
cana-410	129	77	,	,	PUNCT
cana-410	129	78	(	(	PUNCT
cana-410	129	79	’	'	PUNCT
cana-410	129	80	p3	p3	PROPN
cana-410	129	81	’	'	PUNCT
cana-410	129	82	,	,	PUNCT
cana-410	129	83	’	'	PUNCT
cana-410	129	84	p4	p4	ADJ
cana-410	129	85	’	'	PUNCT
cana-410	129	86	)	)	PUNCT
cana-410	129	87	,	,	PUNCT
cana-410	129	88	(	(	PUNCT
cana-410	129	89	’	'	PUNCT
cana-410	129	90	p1	p1	PROPN
cana-410	129	91	’	'	PUNCT
cana-410	129	92	,	,	PUNCT
cana-410	129	93	’	'	PUNCT
cana-410	129	94	q1	q1	PROPN
cana-410	129	95	’	'	PUNCT
cana-410	129	96	)	)	PUNCT
cana-410	129	97	,	,	PUNCT
cana-410	129	98	(	(	PUNCT
cana-410	129	99	’	'	PUNCT
cana-410	129	100	p2	p2	PROPN
cana-410	129	101	’	'	PUNCT
cana-410	129	102	,	,	PUNCT
cana-410	129	103	’	'	PUNCT
cana-410	129	104	q2	q2	NOUN
cana-410	129	105	’	'	PUNCT
cana-410	129	106	)	)	PUNCT
cana-410	129	107	,	,	PUNCT
cana-410	129	108	(	(	PUNCT
cana-410	129	109	’	'	PUNCT
cana-410	129	110	p3	p3	PROPN
cana-410	129	111	’	'	PUNCT
cana-410	129	112	,	,	PUNCT
cana-410	129	113	’	'	PUNCT
cana-410	129	114	q3	q3	NOUN
cana-410	129	115	’	'	PUNCT
cana-410	129	116	)	)	PUNCT
cana-410	129	117	,	,	PUNCT
cana-410	129	118	(	(	PUNCT
cana-410	129	119	’	'	PUNCT
cana-410	129	120	p4	p4	ADJ
cana-410	129	121	’	'	PUNCT
cana-410	129	122	,	,	PUNCT
cana-410	129	123	’	'	PUNCT
cana-410	129	124	q4	q4	PROPN
cana-410	129	125	’	'	PUNCT
cana-410	129	126	)	)	PUNCT
cana-410	129	127	,	,	PUNCT
cana-410	129	128	(	(	PUNCT
cana-410	129	129	’	'	PUNCT
cana-410	129	130	q2	q2	NOUN
cana-410	129	131	’	'	PUNCT
cana-410	129	132	,	,	PUNCT
cana-410	129	133	’	'	PUNCT
cana-410	129	134	q3	q3	NOUN
cana-410	129	135	’	'	PUNCT
cana-410	129	136	)	)	PUNCT
cana-410	129	137	,	,	PUNCT
cana-410	129	138	(	(	PUNCT
cana-410	129	139	’	'	PUNCT
cana-410	129	140	q4	q4	PROPN
cana-410	129	141	’	'	PUNCT
cana-410	129	142	,	,	PUNCT
cana-410	129	143	’	'	PUNCT
cana-410	129	144	q1	q1	PROPN
cana-410	129	145	’	'	PUNCT
cana-410	129	146	)	)	PUNCT
cana-410	129	147	]	]	PUNCT
cana-410	129	148	for	for	ADP
cana-410	129	149	line	line	NOUN
cana-410	129	150	in	in	ADP
cana-410	129	151	lines	line	NOUN
cana-410	129	152	:	:	PUNCT
cana-410	129	153	plt.plot(*zip(*[points[point	plt.plot(*zip(*[points[point	NOUN
cana-410	129	154	]	]	PUNCT
cana-410	129	155	for	for	ADP
cana-410	129	156	point	point	NOUN
cana-410	129	157	in	in	ADP
cana-410	129	158	line	line	NOUN
cana-410	129	159	]	]	PUNCT
cana-410	129	160	)	)	PUNCT
cana-410	129	161	)	)	PUNCT
cana-410	129	162	#	#	NOUN
cana-410	129	163	label	label	NOUN
cana-410	129	164	points	point	NOUN
cana-410	129	165	for	for	ADP
cana-410	129	166	point	point	NOUN
cana-410	129	167	,	,	PUNCT
cana-410	129	168	coordinates	coordinate	NOUN
cana-410	129	169	in	in	ADP
cana-410	129	170	points.items	points.item	NOUN
cana-410	129	171	(	(	PUNCT
cana-410	129	172	):	):	PUNCT
cana-410	129	173	plt.annotate(point	plt.annotate(point	NOUN
cana-410	129	174	,	,	PUNCT
cana-410	129	175	coordinates	coordinate	NOUN
cana-410	129	176	,	,	PUNCT
cana-410	129	177	textcoords="offset	textcoords="offset	PROPN
cana-410	129	178	points",xytext=(0	points",xytext=(0	ADV
cana-410	129	179	,	,	PUNCT
cana-410	129	180	5	5	NUM
cana-410	129	181	)	)	PUNCT
cana-410	129	182	,	,	PUNCT
cana-410	129	183	ha=’center	ha=’center	PROPN
cana-410	129	184	’	'	PUNCT
cana-410	129	185	)	)	PUNCT
cana-410	129	186	#	#	NOUN
cana-410	129	187	add	add	VERB
cana-410	129	188	title	title	NOUN
cana-410	129	189	below	below	ADP
cana-410	129	190	the	the	DET
cana-410	129	191	graph	graph	NOUN
cana-410	129	192	fig.text(0.5	fig.text(0.5	NOUN
cana-410	129	193	,	,	PUNCT
cana-410	129	194	0.02	0.02	NUM
cana-410	129	195	,	,	PUNCT
cana-410	129	196	’	'	PUNCT
cana-410	129	197	graph	graph	NOUN
cana-410	129	198	diagram	diagram	NOUN
cana-410	129	199	’	'	PUNCT
cana-410	129	200	,	,	PUNCT
cana-410	129	201	ha=’center	ha=’center	PROPN
cana-410	129	202	’	'	PUNCT
cana-410	129	203	,	,	PUNCT
cana-410	129	204	fontsize=12	fontsize=12	PROPN
cana-410	129	205	)	)	PUNCT
cana-410	129	206	plt.legend	plt.legend	PROPN
cana-410	129	207	(	(	PUNCT
cana-410	129	208	)	)	PUNCT
cana-410	129	209	plt.grid(true	plt.grid(true	NOUN
cana-410	129	210	)	)	PUNCT
cana-410	129	211	plt.show	plt.show	PRON
cana-410	129	212	(	(	PUNCT
cana-410	129	213	)	)	PUNCT
cana-410	129	214	figure	figure	NOUN
cana-410	129	215	5	5	NUM
cana-410	129	216	:	:	PUNCT
cana-410	130	1	cl12	cl12	PROPN
cana-410	130	2	(	(	PUNCT
cana-410	130	3	g	g	NOUN
cana-410	130	4	)	)	PUNCT
cana-410	130	5	first	first	ADV
cana-410	130	6	create	create	VERB
cana-410	130	7	a	a	DET
cana-410	130	8	diagonal	diagonal	ADJ
cana-410	130	9	matrix	matrix	NOUN
cana-410	130	10	d(cl3n(g	d(cl3n(g	NOUN
cana-410	130	11	)	)	PUNCT
cana-410	130	12	)	)	PUNCT
cana-410	130	13	.	.	PUNCT
cana-410	131	1	import	import	NOUN
cana-410	131	2	networkx	networkx	PROPN
cana-410	131	3	as	as	ADP
cana-410	131	4	nx	nx	PROPN
cana-410	131	5	import	import	NOUN
cana-410	131	6	numpy	numpy	NOUN
cana-410	131	7	as	as	SCONJ
cana-410	131	8	np	np	NOUN
cana-410	131	9	#	#	NOUN
cana-410	131	10	create	create	VERB
cana-410	131	11	a	a	DET
cana-410	131	12	graph	graph	NOUN
cana-410	131	13	g	g	ADP
cana-410	131	14	=	=	SYM
cana-410	131	15	nx.graph	nx.graph	X
cana-410	131	16	(	(	PUNCT
cana-410	131	17	)	)	PUNCT
cana-410	131	18	edges	edge	NOUN
cana-410	131	19	=	=	PUNCT
cana-410	132	1	[	[	X
cana-410	132	2	(	(	PUNCT
cana-410	132	3	1	1	NUM
cana-410	132	4	,	,	PUNCT
cana-410	132	5	2	2	NUM
cana-410	132	6	)	)	PUNCT
cana-410	132	7	,	,	PUNCT
cana-410	132	8	(	(	PUNCT
cana-410	132	9	2	2	NUM
cana-410	132	10	,	,	PUNCT
cana-410	132	11	3	3	NUM
cana-410	132	12	)	)	PUNCT
cana-410	132	13	,	,	PUNCT
cana-410	132	14	(	(	PUNCT
cana-410	132	15	3	3	NUM
cana-410	132	16	,	,	PUNCT
cana-410	132	17	4	4	NUM
cana-410	132	18	)	)	PUNCT
cana-410	132	19	,	,	PUNCT
cana-410	132	20	(	(	PUNCT
cana-410	132	21	4	4	NUM
cana-410	132	22	,	,	PUNCT
cana-410	132	23	1	1	NUM
cana-410	132	24	)	)	PUNCT
cana-410	132	25	,	,	PUNCT
cana-410	132	26	(	(	PUNCT
cana-410	132	27	1	1	NUM
cana-410	132	28	,	,	PUNCT
cana-410	132	29	5	5	NUM
cana-410	132	30	)	)	PUNCT
cana-410	132	31	,	,	PUNCT
cana-410	132	32	(	(	PUNCT
cana-410	132	33	2	2	NUM
cana-410	132	34	,	,	PUNCT
cana-410	132	35	6	6	NUM
cana-410	132	36	)	)	PUNCT
cana-410	132	37	,	,	PUNCT
cana-410	132	38	(	(	PUNCT
cana-410	132	39	3	3	NUM
cana-410	132	40	,	,	PUNCT
cana-410	132	41	7	7	NUM
cana-410	132	42	)	)	PUNCT
cana-410	132	43	,	,	PUNCT
cana-410	132	44	(	(	PUNCT
cana-410	132	45	4	4	NUM
cana-410	132	46	,	,	PUNCT
cana-410	132	47	8)	8)	NUM
cana-410	132	48	,	,	PUNCT
cana-410	132	49	(	(	PUNCT
cana-410	132	50	5	5	NUM
cana-410	132	51	,	,	PUNCT
cana-410	132	52	6	6	NUM
cana-410	132	53	)	)	PUNCT
cana-410	132	54	,	,	PUNCT
cana-410	132	55	(	(	PUNCT
cana-410	132	56	7	7	NUM
cana-410	132	57	,	,	PUNCT
cana-410	132	58	8)	8)	NUM
cana-410	132	59	,	,	PUNCT
cana-410	132	60	(	(	PUNCT
cana-410	132	61	5	5	NUM
cana-410	132	62	,	,	PUNCT
cana-410	132	63	9	9	NUM
cana-410	132	64	)	)	PUNCT
cana-410	132	65	,	,	PUNCT
cana-410	132	66	communications	communication	NOUN
cana-410	132	67	on	on	ADP
cana-410	132	68	applied	apply	VERB
cana-410	132	69	nonlinear	nonlinear	ADJ
cana-410	132	70	analysis	analysis	NOUN
cana-410	132	71	issn	issn	NOUN
cana-410	132	72	:	:	PUNCT
cana-410	132	73	1074	1074	NUM
cana-410	132	74	-	-	PUNCT
cana-410	132	75	133x	133x	NUM
cana-410	132	76	vol	vol	NOUN
cana-410	132	77	31	31	NUM
cana-410	132	78	no	no	NOUN
cana-410	132	79	.	.	NOUN
cana-410	132	80	1	1	NUM
cana-410	132	81	(	(	PUNCT
cana-410	132	82	2024	2024	NUM
cana-410	132	83	)	)	PUNCT
cana-410	132	84	260	260	NUM
cana-410	132	85	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	132	86	(	(	PUNCT
cana-410	132	87	6	6	NUM
cana-410	132	88	,	,	PUNCT
cana-410	132	89	10	10	NUM
cana-410	132	90	)	)	PUNCT
cana-410	132	91	,	,	PUNCT
cana-410	132	92	(	(	PUNCT
cana-410	132	93	7	7	NUM
cana-410	132	94	,	,	PUNCT
cana-410	132	95	11	11	NUM
cana-410	132	96	)	)	PUNCT
cana-410	132	97	,	,	PUNCT
cana-410	132	98	(	(	PUNCT
cana-410	132	99	8	8	NUM
cana-410	132	100	,	,	PUNCT
cana-410	132	101	12	12	NUM
cana-410	132	102	)	)	PUNCT
cana-410	132	103	,	,	PUNCT
cana-410	132	104	(	(	PUNCT
cana-410	132	105	10	10	NUM
cana-410	132	106	,	,	PUNCT
cana-410	132	107	11	11	NUM
cana-410	132	108	)	)	PUNCT
cana-410	132	109	,	,	PUNCT
cana-410	132	110	(	(	PUNCT
cana-410	132	111	12	12	NUM
cana-410	132	112	,	,	PUNCT
cana-410	132	113	9	9	NUM
cana-410	132	114	)	)	PUNCT
cana-410	132	115	]	]	PUNCT
cana-410	133	1	g.add_edges_from(edges	g.add_edges_from(edge	NOUN
cana-410	133	2	)	)	PUNCT
cana-410	133	3	#	#	NOUN
cana-410	133	4	get	get	VERB
cana-410	133	5	the	the	DET
cana-410	133	6	nodes	node	NOUN
cana-410	133	7	from	from	ADP
cana-410	133	8	the	the	DET
cana-410	133	9	edges	edge	NOUN
cana-410	133	10	nodes	node	NOUN
cana-410	133	11	=	=	SYM
cana-410	133	12	set(node	set(node	CCONJ
cana-410	133	13	for	for	ADP
cana-410	133	14	edge	edge	NOUN
cana-410	133	15	in	in	ADP
cana-410	133	16	edges	edge	NOUN
cana-410	133	17	for	for	ADP
cana-410	133	18	node	node	NOUN
cana-410	133	19	in	in	ADP
cana-410	133	20	edge	edge	NOUN
cana-410	133	21	)	)	PUNCT
cana-410	133	22	#	#	NOUN
cana-410	133	23	create	create	VERB
cana-410	133	24	a	a	DET
cana-410	133	25	diagonal	diagonal	ADJ
cana-410	133	26	matrix	matrix	NOUN
cana-410	133	27	with	with	ADP
cana-410	133	28	zeros	zero	NOUN
cana-410	133	29	diagonal_matrix	diagonal_matrix	NOUN
cana-410	133	30	=	=	SYM
cana-410	133	31	np.zeros((len(nodes	np.zeros((len(node	NOUN
cana-410	133	32	)	)	PUNCT
cana-410	133	33	,	,	PUNCT
cana-410	133	34	len(nodes	len(node	NOUN
cana-410	133	35	)	)	PUNCT
cana-410	133	36	)	)	PUNCT
cana-410	133	37	)	)	PUNCT
cana-410	134	1	#	#	NOUN
cana-410	134	2	assign	assign	NOUN
cana-410	134	3	diagonal	diagonal	ADJ
cana-410	134	4	values	value	NOUN
cana-410	134	5	for	for	ADP
cana-410	134	6	i	i	PRON
cana-410	134	7	,	,	PUNCT
cana-410	134	8	node	node	NOUN
cana-410	134	9	in	in	ADP
cana-410	134	10	enumerate(nodes	enumerate(node	NOUN
cana-410	134	11	):	):	PUNCT
cana-410	134	12	diagonal_matrix[i	diagonal_matrix[i	PROPN
cana-410	134	13	,	,	PUNCT
cana-410	134	14	i	i	PRON
cana-410	134	15	]	]	X
cana-410	134	16	=	=	PUNCT
cana-410	134	17	g.degree(node	g.degree(node	ADJ
cana-410	134	18	)	)	PUNCT
cana-410	134	19	#	#	NOUN
cana-410	134	20	assign	assign	VERB
cana-410	134	21	a	a	DET
cana-410	134	22	name	name	NOUN
cana-410	134	23	to	to	ADP
cana-410	134	24	the	the	DET
cana-410	134	25	matrix	matrix	NOUN
cana-410	134	26	d_cl3n	d_cl3n	NOUN
cana-410	134	27	=	=	PUNCT
cana-410	134	28	diagonal_matrix	diagonal_matrix	NOUN
cana-410	134	29	#	#	NOUN
cana-410	134	30	print	print	NOUN
cana-410	134	31	the	the	DET
cana-410	134	32	diagonal	diagonal	ADJ
cana-410	134	33	matrix	matrix	NOUN
cana-410	134	34	with	with	ADP
cana-410	134	35	the	the	DET
cana-410	134	36	assigned	assign	VERB
cana-410	134	37	name	name	NOUN
cana-410	134	38	print("d(cl_3n	print("d(cl_3n	NOUN
cana-410	134	39	):	):	PUNCT
cana-410	134	40	"	"	PUNCT
cana-410	134	41	)	)	PUNCT
cana-410	134	42	print(d_cl3n	print(d_cl3n	NOUN
cana-410	134	43	)	)	PUNCT
cana-410	134	44	which	which	PRON
cana-410	134	45	display	display	VERB
cana-410	134	46	the	the	DET
cana-410	134	47	output	output	NOUN
cana-410	134	48	as	as	SCONJ
cana-410	134	49	create	create	VERB
cana-410	134	50	an	an	DET
cana-410	134	51	adjacency	adjacency	NOUN
cana-410	134	52	matrix	matrix	NOUN
cana-410	134	53	a(cl3n(g	a(cl3n(g	PROPN
cana-410	134	54	)	)	PUNCT
cana-410	134	55	)	)	PUNCT
cana-410	134	56	using	use	VERB
cana-410	134	57	import	import	NOUN
cana-410	134	58	networkx	networkx	NOUN
cana-410	134	59	as	as	ADP
cana-410	134	60	nx	nx	PROPN
cana-410	134	61	import	import	NOUN
cana-410	134	62	numpy	numpy	NOUN
cana-410	134	63	as	as	SCONJ
cana-410	134	64	np	np	NOUN
cana-410	134	65	#	#	NOUN
cana-410	134	66	create	create	VERB
cana-410	134	67	a	a	DET
cana-410	134	68	graph	graph	NOUN
cana-410	134	69	g	g	ADP
cana-410	134	70	=	=	SYM
cana-410	134	71	nx.graph	nx.graph	X
cana-410	134	72	(	(	PUNCT
cana-410	134	73	)	)	PUNCT
cana-410	134	74	g.add	g.add	PROPN
cana-410	134	75	edges	edge	NOUN
cana-410	134	76	from([(1	from([(1	PROPN
cana-410	134	77	,	,	PUNCT
cana-410	134	78	2	2	NUM
cana-410	134	79	)	)	PUNCT
cana-410	134	80	,	,	PUNCT
cana-410	134	81	(	(	PUNCT
cana-410	134	82	2	2	NUM
cana-410	134	83	,	,	PUNCT
cana-410	134	84	3	3	NUM
cana-410	134	85	)	)	PUNCT
cana-410	134	86	,	,	PUNCT
cana-410	134	87	(	(	PUNCT
cana-410	134	88	3	3	NUM
cana-410	134	89	,	,	PUNCT
cana-410	134	90	4	4	NUM
cana-410	134	91	)	)	PUNCT
cana-410	134	92	,	,	PUNCT
cana-410	134	93	(	(	PUNCT
cana-410	134	94	4	4	NUM
cana-410	134	95	,	,	PUNCT
cana-410	134	96	1	1	NUM
cana-410	134	97	)	)	PUNCT
cana-410	134	98	,	,	PUNCT
cana-410	134	99	(	(	PUNCT
cana-410	134	100	1	1	NUM
cana-410	134	101	,	,	PUNCT
cana-410	134	102	5	5	NUM
cana-410	134	103	)	)	PUNCT
cana-410	134	104	,	,	PUNCT
cana-410	134	105	(	(	PUNCT
cana-410	134	106	2	2	NUM
cana-410	134	107	,	,	PUNCT
cana-410	134	108	6	6	NUM
cana-410	134	109	)	)	PUNCT
cana-410	134	110	,	,	PUNCT
cana-410	134	111	(	(	PUNCT
cana-410	134	112	3	3	NUM
cana-410	134	113	,	,	PUNCT
cana-410	134	114	7	7	NUM
cana-410	134	115	)	)	PUNCT
cana-410	134	116	,	,	PUNCT
cana-410	134	117	(	(	PUNCT
cana-410	134	118	4	4	NUM
cana-410	134	119	,	,	PUNCT
cana-410	134	120	8)	8)	NUM
cana-410	134	121	,	,	PUNCT
cana-410	134	122	(	(	PUNCT
cana-410	134	123	5	5	NUM
cana-410	134	124	,	,	PUNCT
cana-410	134	125	6	6	NUM
cana-410	134	126	)	)	PUNCT
cana-410	134	127	,	,	PUNCT
cana-410	134	128	(	(	PUNCT
cana-410	134	129	7	7	NUM
cana-410	134	130	,	,	PUNCT
cana-410	134	131	8)	8)	NUM
cana-410	134	132	,	,	PUNCT
cana-410	134	133	(	(	PUNCT
cana-410	134	134	5	5	NUM
cana-410	134	135	,	,	PUNCT
cana-410	134	136	9	9	NUM
cana-410	134	137	)	)	PUNCT
cana-410	134	138	,	,	PUNCT
cana-410	134	139	(	(	PUNCT
cana-410	134	140	6	6	NUM
cana-410	134	141	,	,	PUNCT
cana-410	134	142	10	10	NUM
cana-410	134	143	)	)	PUNCT
cana-410	134	144	,	,	PUNCT
cana-410	134	145	(	(	PUNCT
cana-410	134	146	7	7	NUM
cana-410	134	147	,	,	PUNCT
cana-410	134	148	11	11	NUM
cana-410	134	149	)	)	PUNCT
cana-410	134	150	,	,	PUNCT
cana-410	134	151	(	(	PUNCT
cana-410	134	152	8	8	NUM
cana-410	134	153	,	,	PUNCT
cana-410	134	154	12	12	NUM
cana-410	134	155	)	)	PUNCT
cana-410	134	156	,	,	PUNCT
cana-410	134	157	(	(	PUNCT
cana-410	134	158	10	10	NUM
cana-410	134	159	,	,	PUNCT
cana-410	134	160	11	11	NUM
cana-410	134	161	)	)	PUNCT
cana-410	134	162	,	,	PUNCT
cana-410	134	163	(	(	PUNCT
cana-410	134	164	12	12	NUM
cana-410	134	165	,	,	PUNCT
cana-410	134	166	9)])\\	9)])\\	PROPN
cana-410	134	167	#	#	NOUN
cana-410	134	168	obtain	obtain	VERB
cana-410	134	169	the	the	DET
cana-410	134	170	adjacency	adjacency	NOUN
cana-410	134	171	matrix	matrix	NOUN
cana-410	134	172	adjacency	adjacency	NOUN
cana-410	134	173	matrix	matrix	NOUN
cana-410	134	174	=	=	SYM
cana-410	134	175	nx.adjacency	nx.adjacency	SYM
cana-410	134	176	matrix(g).todense	matrix(g).todense	NOUN
cana-410	134	177	(	(	PUNCT
cana-410	134	178	)	)	PUNCT
cana-410	134	179	#	#	NOUN
cana-410	134	180	assign	assign	VERB
cana-410	134	181	a	a	DET
cana-410	134	182	name	name	NOUN
cana-410	134	183	to	to	ADP
cana-410	134	184	the	the	DET
cana-410	134	185	matrix	matrix	NOUN
cana-410	134	186	communications	communication	NOUN
cana-410	134	187	on	on	ADP
cana-410	134	188	applied	apply	VERB
cana-410	134	189	nonlinear	nonlinear	ADJ
cana-410	134	190	analysis	analysis	NOUN
cana-410	134	191	issn	issn	NOUN
cana-410	134	192	:	:	PUNCT
cana-410	134	193	1074	1074	NUM
cana-410	134	194	-	-	PUNCT
cana-410	134	195	133x	133x	NUM
cana-410	134	196	vol	vol	NOUN
cana-410	134	197	31	31	NUM
cana-410	134	198	no	no	NOUN
cana-410	134	199	.	.	NOUN
cana-410	134	200	1	1	NUM
cana-410	134	201	(	(	PUNCT
cana-410	134	202	2024	2024	NUM
cana-410	134	203	)	)	PUNCT
cana-410	134	204	261	261	NUM
cana-410	134	205	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	134	206	a(cl_{3n}(g	a(cl_{3n}(g	PROPN
cana-410	134	207	)	)	PUNCT
cana-410	134	208	)	)	PUNCT
cana-410	134	209	=	=	PUNCT
cana-410	135	1	$	$	SYM
cana-410	135	2	adjacency	adjacency	NOUN
cana-410	135	3	matrix	matrix	NOUN
cana-410	135	4	$	$	SYM
cana-410	135	5	#	#	NOUN
cana-410	135	6	print	print	NOUN
cana-410	135	7	the	the	DET
cana-410	135	8	adjacency	adjacency	NOUN
cana-410	135	9	matrix	matrix	NOUN
cana-410	135	10	with	with	ADP
cana-410	135	11	the	the	DET
cana-410	135	12	assigned	assign	VERB
cana-410	135	13	name	name	NOUN
cana-410	135	14	print(f"a(cl_{3n}(g))):\n{a(cl_{3n}(g	print(f"a(cl_{3n}(g))):\n{a(cl_{3n}(g	NOUN
cana-410	135	15	)	)	PUNCT
cana-410	135	16	)	)	PUNCT
cana-410	135	17	"	"	PUNCT
cana-410	135	18	}	}	PUNCT
cana-410	135	19	results	result	VERB
cana-410	135	20	in	in	ADP
cana-410	135	21	next	next	ADJ
cana-410	135	22	,	,	PUNCT
cana-410	135	23	the	the	DET
cana-410	135	24	laplacian	laplacian	ADJ
cana-410	135	25	matrix	matrix	NOUN
cana-410	135	26	is	be	AUX
cana-410	135	27	given	give	VERB
cana-410	135	28	by	by	ADP
cana-410	135	29	l(cl3n	l(cl3n	X
cana-410	135	30	(	(	PUNCT
cana-410	135	31	g	g	NOUN
cana-410	135	32	)	)	PUNCT
cana-410	135	33	)	)	PUNCT
cana-410	136	1	=	=	PUNCT
cana-410	136	2	d(cl3n	d(cl3n	X
cana-410	136	3	(	(	PUNCT
cana-410	136	4	g	g	NOUN
cana-410	136	5	)	)	PUNCT
cana-410	136	6	)	)	PUNCT
cana-410	137	1	−	−	NOUN
cana-410	137	2	a(cl3n	a(cl3n	PUNCT
cana-410	137	3	(	(	PUNCT
cana-410	137	4	g	g	NOUN
cana-410	137	5	)	)	PUNCT
cana-410	137	6	)	)	PUNCT
cana-410	137	7	.	.	PUNCT
cana-410	138	1	import	import	NOUN
cana-410	138	2	networkx	networkx	PROPN
cana-410	138	3	as	as	ADP
cana-410	138	4	nx	nx	PROPN
cana-410	138	5	import	import	NOUN
cana-410	138	6	numpy	numpy	NOUN
cana-410	138	7	as	as	SCONJ
cana-410	138	8	np	np	NOUN
cana-410	138	9	#	#	NOUN
cana-410	138	10	create	create	VERB
cana-410	138	11	a	a	DET
cana-410	138	12	graph	graph	NOUN
cana-410	138	13	g	g	ADP
cana-410	138	14	=	=	SYM
cana-410	138	15	nx.graph	nx.graph	X
cana-410	138	16	(	(	PUNCT
cana-410	138	17	)	)	PUNCT
cana-410	138	18	edges	edge	NOUN
cana-410	138	19	=	=	PUNCT
cana-410	139	1	[	[	X
cana-410	139	2	(	(	PUNCT
cana-410	139	3	1	1	NUM
cana-410	139	4	,	,	PUNCT
cana-410	139	5	2	2	NUM
cana-410	139	6	)	)	PUNCT
cana-410	139	7	,	,	PUNCT
cana-410	139	8	(	(	PUNCT
cana-410	139	9	2	2	NUM
cana-410	139	10	,	,	PUNCT
cana-410	139	11	3	3	NUM
cana-410	139	12	)	)	PUNCT
cana-410	139	13	,	,	PUNCT
cana-410	139	14	(	(	PUNCT
cana-410	139	15	3	3	NUM
cana-410	139	16	,	,	PUNCT
cana-410	139	17	4	4	NUM
cana-410	139	18	)	)	PUNCT
cana-410	139	19	,	,	PUNCT
cana-410	139	20	(	(	PUNCT
cana-410	139	21	4	4	NUM
cana-410	139	22	,	,	PUNCT
cana-410	139	23	1	1	NUM
cana-410	139	24	)	)	PUNCT
cana-410	139	25	,	,	PUNCT
cana-410	139	26	(	(	PUNCT
cana-410	139	27	1	1	NUM
cana-410	139	28	,	,	PUNCT
cana-410	139	29	5	5	NUM
cana-410	139	30	)	)	PUNCT
cana-410	139	31	,	,	PUNCT
cana-410	139	32	(	(	PUNCT
cana-410	139	33	2	2	NUM
cana-410	139	34	,	,	PUNCT
cana-410	139	35	6	6	NUM
cana-410	139	36	)	)	PUNCT
cana-410	139	37	,	,	PUNCT
cana-410	139	38	(	(	PUNCT
cana-410	139	39	3	3	NUM
cana-410	139	40	,	,	PUNCT
cana-410	139	41	7	7	NUM
cana-410	139	42	)	)	PUNCT
cana-410	139	43	,	,	PUNCT
cana-410	139	44	(	(	PUNCT
cana-410	139	45	4	4	NUM
cana-410	139	46	,	,	PUNCT
cana-410	139	47	8)	8)	NUM
cana-410	139	48	,	,	PUNCT
cana-410	139	49	(	(	PUNCT
cana-410	139	50	5	5	NUM
cana-410	139	51	,	,	PUNCT
cana-410	139	52	6	6	NUM
cana-410	139	53	)	)	PUNCT
cana-410	139	54	,	,	PUNCT
cana-410	139	55	(	(	PUNCT
cana-410	139	56	7	7	NUM
cana-410	139	57	,	,	PUNCT
cana-410	139	58	8)	8)	NUM
cana-410	139	59	,	,	PUNCT
cana-410	139	60	(	(	PUNCT
cana-410	139	61	5	5	NUM
cana-410	139	62	,	,	PUNCT
cana-410	139	63	9	9	NUM
cana-410	139	64	)	)	PUNCT
cana-410	139	65	,	,	PUNCT
cana-410	139	66	(	(	PUNCT
cana-410	139	67	6	6	NUM
cana-410	139	68	,	,	PUNCT
cana-410	139	69	10	10	NUM
cana-410	139	70	)	)	PUNCT
cana-410	139	71	,	,	PUNCT
cana-410	139	72	(	(	PUNCT
cana-410	139	73	7	7	NUM
cana-410	139	74	,	,	PUNCT
cana-410	139	75	11	11	NUM
cana-410	139	76	)	)	PUNCT
cana-410	139	77	,	,	PUNCT
cana-410	139	78	(	(	PUNCT
cana-410	139	79	8	8	NUM
cana-410	139	80	,	,	PUNCT
cana-410	139	81	12	12	NUM
cana-410	139	82	)	)	PUNCT
cana-410	139	83	,	,	PUNCT
cana-410	139	84	(	(	PUNCT
cana-410	139	85	10	10	NUM
cana-410	139	86	,	,	PUNCT
cana-410	139	87	11	11	NUM
cana-410	139	88	)	)	PUNCT
cana-410	139	89	,	,	PUNCT
cana-410	139	90	(	(	PUNCT
cana-410	139	91	12	12	NUM
cana-410	139	92	,	,	PUNCT
cana-410	139	93	9	9	NUM
cana-410	139	94	)	)	PUNCT
cana-410	139	95	]	]	PUNCT
cana-410	139	96	g.add_edges_from(edges	g.add_edges_from(edge	NOUN
cana-410	139	97	)	)	PUNCT
cana-410	139	98	#	#	NOUN
cana-410	139	99	get	get	VERB
cana-410	139	100	the	the	DET
cana-410	139	101	nodes	node	NOUN
cana-410	139	102	from	from	ADP
cana-410	139	103	the	the	DET
cana-410	139	104	edges	edge	NOUN
cana-410	139	105	nodes	node	NOUN
cana-410	139	106	=	=	SYM
cana-410	139	107	set(node	set(node	CCONJ
cana-410	139	108	for	for	ADP
cana-410	139	109	edge	edge	NOUN
cana-410	139	110	in	in	ADP
cana-410	139	111	edges	edge	NOUN
cana-410	139	112	for	for	ADP
cana-410	139	113	node	node	NOUN
cana-410	139	114	in	in	ADP
cana-410	139	115	edge	edge	NOUN
cana-410	139	116	)	)	PUNCT
cana-410	139	117	#	#	NOUN
cana-410	139	118	create	create	VERB
cana-410	139	119	the	the	DET
cana-410	139	120	adjacency	adjacency	NOUN
cana-410	139	121	matrix	matrix	NOUN
cana-410	139	122	a_cl3n	a_cl3n	NOUN
cana-410	139	123	=	=	SYM
cana-410	139	124	nx.adjacency_matrix(g).todense	nx.adjacency_matrix(g).todense	NOUN
cana-410	139	125	(	(	PUNCT
cana-410	139	126	)	)	PUNCT
cana-410	139	127	#	#	NOUN
cana-410	139	128	create	create	VERB
cana-410	139	129	the	the	DET
cana-410	139	130	diagonal	diagonal	ADJ
cana-410	139	131	matrix	matrix	NOUN
cana-410	139	132	d_cl3n	d_cl3n	NOUN
cana-410	139	133	=	=	SYM
cana-410	139	134	np.diag([g.degree(node	np.diag([g.degree(node	PROPN
cana-410	139	135	)	)	PUNCT
cana-410	139	136	for	for	ADP
cana-410	139	137	node	node	NOUN
cana-410	139	138	in	in	ADP
cana-410	139	139	nodes	node	NOUN
cana-410	139	140	]	]	PUNCT
cana-410	139	141	)	)	PUNCT
cana-410	139	142	#	#	NOUN
cana-410	139	143	calculate	calculate	NOUN
cana-410	139	144	the	the	DET
cana-410	139	145	laplacian	laplacian	ADJ
cana-410	139	146	matrix	matrix	NOUN
cana-410	139	147	l_cl3n	l_cl3n	NOUN
cana-410	139	148	=	=	PUNCT
cana-410	139	149	d_cl3n	d_cl3n	VERB
cana-410	139	150	a_cl3n	a_cl3n	NOUN
cana-410	139	151	#	#	NOUN
cana-410	139	152	print	print	NOUN
cana-410	139	153	the	the	DET
cana-410	139	154	laplacian	laplacian	ADJ
cana-410	139	155	matrix	matrix	NOUN
cana-410	139	156	print("l(cl_3n	print("l(cl_3n	NOUN
cana-410	139	157	):	):	PUNCT
cana-410	139	158	"	"	PUNCT
cana-410	139	159	)	)	PUNCT
cana-410	139	160	print(l_cl3n	print(l_cl3n	NOUN
cana-410	139	161	)	)	PUNCT
cana-410	139	162	communications	communication	NOUN
cana-410	139	163	on	on	ADP
cana-410	139	164	applied	apply	VERB
cana-410	139	165	nonlinear	nonlinear	ADJ
cana-410	139	166	analysis	analysis	NOUN
cana-410	139	167	issn	issn	NOUN
cana-410	139	168	:	:	PUNCT
cana-410	139	169	1074	1074	NUM
cana-410	139	170	-	-	PUNCT
cana-410	139	171	133x	133x	NUM
cana-410	139	172	vol	vol	NOUN
cana-410	139	173	31	31	NUM
cana-410	139	174	no	no	NOUN
cana-410	139	175	.	.	NOUN
cana-410	139	176	1	1	NUM
cana-410	139	177	(	(	PUNCT
cana-410	139	178	2024	2024	NUM
cana-410	139	179	)	)	PUNCT
cana-410	139	180	262	262	NUM
cana-410	139	181	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	139	182	now	now	ADV
cana-410	139	183	to	to	PART
cana-410	139	184	find	find	VERB
cana-410	139	185	the	the	DET
cana-410	139	186	cofactor	cofactor	NOUN
cana-410	139	187	of	of	ADP
cana-410	139	188	laplacian	laplacian	ADJ
cana-410	139	189	matrix	matrix	NOUN
cana-410	139	190	import	import	NOUN
cana-410	139	191	numpy	numpy	NOUN
cana-410	139	192	as	as	SCONJ
cana-410	139	193	np	np	NOUN
cana-410	139	194	#	#	NOUN
cana-410	139	195	define	define	VERB
cana-410	139	196	the	the	DET
cana-410	139	197	laplacian	laplacian	ADJ
cana-410	139	198	matrix	matrix	NOUN
cana-410	139	199	l	l	NOUN
cana-410	139	200	l	l	NOUN
cana-410	140	1	=	=	SYM
cana-410	140	2	np.array	np.array	X
cana-410	140	3	(	(	PUNCT
cana-410	140	4	[	[	PUNCT
cana-410	140	5	[	[	X
cana-410	140	6	3	3	NUM
cana-410	140	7	,	,	PUNCT
cana-410	140	8	-1	-1	ADJ
cana-410	140	9	,	,	PUNCT
cana-410	140	10	0	0	NUM
cana-410	140	11	,	,	PUNCT
cana-410	140	12	-1	-1	ADJ
cana-410	140	13	,	,	PUNCT
cana-410	140	14	-1	-1	INTJ
cana-410	140	15	,	,	PUNCT
cana-410	140	16	0	0	NUM
cana-410	140	17	,	,	PUNCT
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cana-410	140	28	0	0	NUM
cana-410	140	29	]	]	PUNCT
cana-410	140	30	,	,	PUNCT
cana-410	140	31	[	[	X
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cana-410	140	36	-1	-1	ADJ
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cana-410	140	55	]	]	PUNCT
cana-410	140	56	,	,	PUNCT
cana-410	140	57	[	[	X
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cana-410	140	82	,	,	PUNCT
cana-410	140	83	[	[	X
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cana-410	140	104	0	0	NUM
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cana-410	140	107	]	]	PUNCT
cana-410	140	108	,	,	PUNCT
cana-410	140	109	[	[	X
cana-410	140	110	-1	-1	X
cana-410	140	111	,	,	PUNCT
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cana-410	140	133	]	]	PUNCT
cana-410	140	134	,	,	PUNCT
cana-410	140	135	[	[	X
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cana-410	140	139	,	,	PUNCT
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cana-410	140	145	,	,	PUNCT
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cana-410	140	158	0	0	NUM
cana-410	140	159	]	]	PUNCT
cana-410	140	160	,	,	PUNCT
cana-410	140	161	[	[	X
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cana-410	140	187	[	[	X
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cana-410	140	237	]	]	PUNCT
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cana-410	140	306	-1	-1	ADJ
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cana-410	140	308	-1	-1	INTJ
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cana-410	140	314	2	2	NUM
cana-410	140	315	]	]	PUNCT
cana-410	140	316	]	]	PUNCT
cana-410	140	317	)	)	PUNCT
cana-410	140	318	#	#	NOUN
cana-410	140	319	function	function	NOUN
cana-410	140	320	to	to	PART
cana-410	140	321	find	find	VERB
cana-410	140	322	the	the	DET
cana-410	140	323	cofactor	cofactor	NOUN
cana-410	140	324	of	of	ADP
cana-410	140	325	a	a	DET
cana-410	140	326	matrix	matrix	NOUN
cana-410	140	327	def	def	PROPN
cana-410	140	328	cofactor(matrix	cofactor(matrix	PROPN
cana-410	140	329	,	,	PUNCT
cana-410	140	330	row	row	NOUN
cana-410	140	331	,	,	PUNCT
cana-410	140	332	col	col	NOUN
cana-410	140	333	):	):	PUNCT
cana-410	140	334	minor_matrix	minor_matrix	X
cana-410	140	335	=	=	SYM
cana-410	140	336	np.delete(np.delete(matrix	np.delete(np.delete(matrix	PROPN
cana-410	140	337	,	,	PUNCT
cana-410	140	338	row	row	NOUN
cana-410	140	339	,	,	PUNCT
cana-410	140	340	axis=0	axis=0	PROPN
cana-410	140	341	)	)	PUNCT
cana-410	140	342	,	,	PUNCT
cana-410	140	343	col	col	PROPN
cana-410	140	344	,	,	PUNCT
cana-410	140	345	axis=1	axis=1	PROPN
cana-410	140	346	)	)	PUNCT
cana-410	140	347	sign	sign	NOUN
cana-410	140	348	=	=	SYM
cana-410	140	349	(	(	PUNCT
cana-410	140	350	-1	-1	INTJ
cana-410	140	351	)	)	PUNCT
cana-410	140	352	*	*	PUNCT
cana-410	140	353	*	*	PUNCT
cana-410	140	354	(	(	PUNCT
cana-410	140	355	row	row	NOUN
cana-410	140	356	+	+	CCONJ
cana-410	140	357	col	col	NOUN
cana-410	140	358	)	)	PUNCT
cana-410	140	359	return	return	NOUN
cana-410	140	360	sign	sign	NOUN
cana-410	140	361	*	*	SYM
cana-410	140	362	np.linalg.det(minor_matrix	np.linalg.det(minor_matrix	ADJ
cana-410	140	363	)	)	PUNCT
cana-410	140	364	#	#	NOUN
cana-410	140	365	calculate	calculate	VERB
cana-410	140	366	the	the	DET
cana-410	140	367	cofactor	cofactor	NOUN
cana-410	140	368	c12	c12	NOUN
cana-410	140	369	row_index	row_index	NOUN
cana-410	140	370	=	=	NOUN
cana-410	140	371	0	0	NUM
cana-410	140	372	col_index	col_index	NOUN
cana-410	140	373	=	=	SYM
cana-410	140	374	1	1	NUM
cana-410	140	375	cofactor_c12	cofactor_c12	NOUN
cana-410	140	376	=	=	SYM
cana-410	140	377	cofactor(l	cofactor(l	PROPN
cana-410	140	378	,	,	PUNCT
cana-410	140	379	row_index	row_index	NOUN
cana-410	140	380	,	,	PUNCT
cana-410	140	381	col_index	col_index	NOUN
cana-410	140	382	)	)	PUNCT
cana-410	140	383	communications	communication	NOUN
cana-410	140	384	on	on	ADP
cana-410	140	385	applied	apply	VERB
cana-410	140	386	nonlinear	nonlinear	ADJ
cana-410	140	387	analysis	analysis	NOUN
cana-410	140	388	issn	issn	NOUN
cana-410	140	389	:	:	PUNCT
cana-410	140	390	1074	1074	NUM
cana-410	140	391	-	-	PUNCT
cana-410	140	392	133x	133x	NUM
cana-410	140	393	vol	vol	NOUN
cana-410	140	394	31	31	NUM
cana-410	140	395	no	no	NOUN
cana-410	140	396	.	.	NOUN
cana-410	140	397	1	1	NUM
cana-410	140	398	(	(	PUNCT
cana-410	140	399	2024	2024	NUM
cana-410	140	400	)	)	PUNCT
cana-410	140	401	263	263	NUM
cana-410	140	402	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-410	140	403	2	2	NUM
cana-410	140	404	print(f"cofactor	print(f"cofactor	NOUN
cana-410	140	405	c12	c12	NOUN
cana-410	140	406	:	:	PUNCT
cana-410	140	407	{	{	PUNCT
cana-410	140	408	cofactor_c12	cofactor_c12	NOUN
cana-410	140	409	}	}	PUNCT
cana-410	140	410	"	"	PUNCT
cana-410	140	411	)	)	PUNCT
cana-410	140	412	finally	finally	ADV
cana-410	140	413	get	get	VERB
cana-410	140	414	the	the	DET
cana-410	140	415	output	output	NOUN
cana-410	140	416	c12	c12	NOUN
cana-410	140	417	:	:	PUNCT
cana-410	140	418	1248	1248	NUM
cana-410	140	419	,	,	PUNCT
cana-410	140	420	which	which	PRON
cana-410	140	421	is	be	AUX
cana-410	140	422	one	one	NUM
cana-410	140	423	of	of	ADP
cana-410	140	424	the	the	DET
cana-410	140	425	cofactor	cofactor	NOUN
cana-410	140	426	of	of	ADP
cana-410	140	427	the	the	DET
cana-410	140	428	laplacian	laplacian	ADJ
cana-410	140	429	matrix	matrix	NOUN
cana-410	140	430	.	.	PUNCT
cana-410	141	1	hence	hence	ADV
cana-410	141	2	,	,	PUNCT
cana-410	141	3	for	for	ADP
cana-410	141	4	n	n	NOUN
cana-410	141	5	=	=	SYM
cana-410	141	6	4	4	NUM
cana-410	141	7	in	in	ADP
cana-410	141	8	the	the	DET
cana-410	141	9	clutch	clutch	NOUN
cana-410	141	10	graph	graph	NOUN
cana-410	141	11	cl3n(g	cl3n(g	PROPN
cana-410	141	12	)	)	PUNCT
cana-410	141	13	,	,	PUNCT
cana-410	141	14	the	the	DET
cana-410	141	15	number	number	NOUN
cana-410	141	16	of	of	ADP
cana-410	141	17	spanning	span	VERB
cana-410	141	18	trees	tree	NOUN
cana-410	141	19	is	be	AUX
cana-410	141	20	1248	1248	NUM
cana-410	141	21	.	.	PUNCT
cana-410	142	1	following	follow	VERB
cana-410	142	2	the	the	DET
cana-410	142	3	same	same	ADJ
cana-410	142	4	approach	approach	NOUN
cana-410	142	5	,	,	PUNCT
cana-410	142	6	we	we	PRON
cana-410	142	7	found	find	VERB
cana-410	142	8	41262	41262	NUM
cana-410	142	9	spanning	span	VERB
cana-410	142	10	trees	tree	NOUN
cana-410	142	11	for	for	ADP
cana-410	142	12	n	n	NOUN
cana-410	142	13	=	=	SYM
cana-410	142	14	6	6	NUM
cana-410	142	15	and	and	CCONJ
cana-410	142	16	210240	210240	NUM
cana-410	142	17	for	for	ADP
cana-410	142	18	n	n	NOUN
cana-410	142	19	=	=	SYM
cana-410	142	20	8	8	NUM
cana-410	142	21	.	.	PUNCT
cana-410	143	1	based	base	VERB
cana-410	143	2	on	on	ADP
cana-410	143	3	these	these	DET
cana-410	143	4	results	result	NOUN
cana-410	143	5	,	,	PUNCT
cana-410	143	6	we	we	PRON
cana-410	143	7	derived	derive	VERB
cana-410	143	8	a	a	DET
cana-410	143	9	general	general	ADJ
cana-410	143	10	formula	formula	NOUN
cana-410	143	11	for	for	ADP
cana-410	143	12	calculating	calculate	VERB
cana-410	143	13	the	the	DET
cana-410	143	14	number	number	NOUN
cana-410	143	15	of	of	ADP
cana-410	143	16	spanning	span	VERB
cana-410	143	17	trees	tree	NOUN
cana-410	143	18	in	in	ADP
cana-410	143	19	the	the	DET
cana-410	143	20	clutch	clutch	NOUN
cana-410	143	21	graph	graph	NOUN
cana-410	143	22	cl3n(g	cl3n(g	NOUN
cana-410	143	23	)	)	PUNCT
cana-410	143	24	as	as	ADP
cana-410	143	25	78[4n	78[4n	NUM
cana-410	143	26	−	−	PROPN
cana-410	143	27	k]k+1	k]k+1	NOUN
cana-410	143	28	,	,	PUNCT
cana-410	143	29	where	where	SCONJ
cana-410	143	30	k	k	PROPN
cana-410	143	31	∈	∈	PROPN
cana-410	143	32	{	{	PUNCT
cana-410	143	33	0	0	NUM
cana-410	143	34	,	,	PUNCT
cana-410	143	35	1	1	NUM
cana-410	143	36	,	,	PUNCT
cana-410	143	37	2	2	NUM
cana-410	143	38	,	,	PUNCT
cana-410	143	39	.	.	PUNCT
cana-410	143	40	.	.	PUNCT
cana-410	143	41	.	.	PUNCT
cana-410	143	42	}	}	PUNCT
cana-410	143	43	.	.	PUNCT
cana-410	144	1	therefore	therefore	ADV
cana-410	144	2	,	,	PUNCT
cana-410	144	3	the	the	DET
cana-410	144	4	total	total	ADJ
cana-410	144	5	number	number	NOUN
cana-410	144	6	of	of	ADP
cana-410	144	7	spanning	span	VERB
cana-410	144	8	trees	tree	NOUN
cana-410	144	9	in	in	ADP
cana-410	144	10	the	the	DET
cana-410	144	11	clutch	clutch	NOUN
cana-410	144	12	graph	graph	NOUN
cana-410	144	13	cl3n(g	cl3n(g	NOUN
cana-410	144	14	)	)	PUNCT
cana-410	144	15	can	can	AUX
cana-410	144	16	be	be	AUX
cana-410	144	17	expressed	express	VERB
cana-410	144	18	by	by	ADP
cana-410	144	19	the	the	DET
cana-410	144	20	formula	formula	NOUN
cana-410	144	21	78[4n	78[4n	NUM
cana-410	144	22	−	−	PROPN
cana-410	144	23	k]k+1	k]k+1	NOUN
cana-410	144	24	.	.	PUNCT
cana-410	145	1	theorem	theorem	VERB
cana-410	145	2	4.7	4.7	NUM
cana-410	145	3	every	every	DET
cana-410	145	4	clutch	clutch	ADJ
cana-410	145	5	graph	graph	NOUN
cana-410	145	6	cl3n	cl3n	NOUN
cana-410	145	7	(	(	PUNCT
cana-410	145	8	g	g	NOUN
cana-410	145	9	)	)	PUNCT
cana-410	145	10	is	be	AUX
cana-410	145	11	bipartite	bipartite	ADJ
cana-410	145	12	.	.	PUNCT
cana-410	146	1	proof	proof	NOUN
cana-410	146	2	the	the	DET
cana-410	146	3	clutch	clutch	NOUN
cana-410	146	4	graph	graph	NOUN
cana-410	146	5	has	have	VERB
cana-410	146	6	three	three	NUM
cana-410	146	7	sets	set	NOUN
cana-410	146	8	of	of	ADP
cana-410	146	9	vertices	vertex	NOUN
cana-410	146	10	:	:	PUNCT
cana-410	146	11	vc	vc	INTJ
cana-410	146	12	(	(	PUNCT
cana-410	146	13	cycle	cycle	NOUN
cana-410	146	14	vertices	vertex	NOUN
cana-410	146	15	)	)	PUNCT
cana-410	146	16	,	,	PUNCT
cana-410	146	17	vp	vp	X
cana-410	146	18	(	(	PUNCT
cana-410	146	19	vertices	vertex	NOUN
cana-410	146	20	introduced	introduce	VERB
cana-410	146	21	to	to	ADP
cana-410	146	22	the	the	DET
cana-410	146	23	cycle	cycle	NOUN
cana-410	146	24	)	)	PUNCT
cana-410	146	25	,	,	PUNCT
cana-410	146	26	and	and	CCONJ
cana-410	146	27	vq	vq	PROPN
cana-410	146	28	(	(	PUNCT
cana-410	146	29	vertices	vertex	NOUN
cana-410	146	30	introduced	introduce	VERB
cana-410	146	31	to	to	ADP
cana-410	146	32	vp	vp	PROPN
cana-410	146	33	)	)	PUNCT
cana-410	146	34	.	.	PUNCT
cana-410	147	1	now	now	ADV
cana-410	147	2	,	,	PUNCT
cana-410	147	3	consider	consider	VERB
cana-410	147	4	two	two	NUM
cana-410	147	5	disjoint	disjoint	NOUN
cana-410	147	6	sets	set	NOUN
cana-410	147	7	v1	v1	VERB
cana-410	147	8	and	and	CCONJ
cana-410	147	9	v2	v2	VERB
cana-410	147	10	such	such	ADJ
cana-410	147	11	that	that	DET
cana-410	147	12	v1	v1	NOUN
cana-410	147	13	∩	∩	ADJ
cana-410	147	14	v2	v2	NOUN
cana-410	147	15	=	=	SYM
cana-410	147	16	∅	∅	NOUN
cana-410	147	17	and	and	CCONJ
cana-410	147	18	v1	v1	VERB
cana-410	147	19	∪	∪	ADJ
cana-410	147	20	v2	v2	NOUN
cana-410	147	21	=	=	PUNCT
cana-410	147	22	v.	v.	PROPN
cana-410	147	23	partitioned	partition	VERB
cana-410	147	24	the	the	DET
cana-410	147	25	vertex	vertex	NOUN
cana-410	147	26	sets	set	NOUN
cana-410	147	27	vc	vc	PROPN
cana-410	147	28	,	,	PUNCT
cana-410	147	29	vp	vp	PROPN
cana-410	147	30	,	,	PUNCT
cana-410	147	31	vq	vq	NOUN
cana-410	147	32	in	in	ADP
cana-410	147	33	cl3n	cl3n	NOUN
cana-410	147	34	(	(	PUNCT
cana-410	147	35	g	g	NOUN
cana-410	147	36	)	)	PUNCT
cana-410	147	37	such	such	ADJ
cana-410	147	38	that	that	SCONJ
cana-410	147	39	,	,	PUNCT
cana-410	147	40	each	each	DET
cana-410	147	41	edge	edge	NOUN
cana-410	147	42	(	(	PUNCT
cana-410	147	43	vi	vi	NOUN
cana-410	147	44	,	,	PUNCT
cana-410	147	45	vj	vj	NOUN
cana-410	147	46	)	)	PUNCT
cana-410	147	47	in	in	ADP
cana-410	147	48	the	the	DET
cana-410	147	49	clutch	clutch	NOUN
cana-410	147	50	graph	graph	NOUN
cana-410	147	51	cl3n	cl3n	NOUN
cana-410	147	52	(	(	PUNCT
cana-410	147	53	g	g	NOUN
cana-410	147	54	)	)	PUNCT
cana-410	147	55	is	be	AUX
cana-410	147	56	of	of	ADP
cana-410	147	57	the	the	DET
cana-410	147	58	form	form	NOUN
cana-410	147	59	if	if	SCONJ
cana-410	147	60	vi	vi	NOUN
cana-410	147	61	∈	∈	PROPN
cana-410	147	62	v1	v1	NOUN
cana-410	147	63	,	,	PUNCT
cana-410	147	64	then	then	ADV
cana-410	147	65	vj	vj	PROPN
cana-410	147	66	must	must	AUX
cana-410	147	67	be	be	AUX
cana-410	147	68	in	in	ADP
cana-410	147	69	v2	v2	PROPN
cana-410	147	70	if	if	SCONJ
cana-410	147	71	vj	vj	INTJ
cana-410	147	72	∈	∈	PROPN
cana-410	147	73	v2	v2	PROPN
cana-410	147	74	,	,	PUNCT
cana-410	147	75	then	then	ADV
cana-410	147	76	vi	vi	PROPN
cana-410	147	77	must	must	AUX
cana-410	147	78	be	be	AUX
cana-410	147	79	in	in	ADP
cana-410	147	80	v1	v1	VERB
cana-410	147	81	the	the	DET
cana-410	147	82	partition	partition	NOUN
cana-410	147	83	of	of	ADP
cana-410	147	84	the	the	DET
cana-410	147	85	vertex	vertex	NOUN
cana-410	147	86	set	set	NOUN
cana-410	147	87	is	be	AUX
cana-410	147	88	v1	v1	NOUN
cana-410	147	89	=	=	SYM
cana-410	147	90	{	{	PUNCT
cana-410	147	91	(	(	PUNCT
cana-410	147	92	c2i−1	c2i−1	PROPN
cana-410	147	93	,	,	PUNCT
cana-410	147	94	p2i	p2i	PROPN
cana-410	147	95	,	,	PUNCT
cana-410	147	96	q2i−1	q2i−1	PROPN
cana-410	147	97	)	)	PUNCT
cana-410	148	1	|	|	ADV
cana-410	148	2	i	i	PRON
cana-410	148	3	∈	∈	PROPN
cana-410	148	4	{	{	PUNCT
cana-410	148	5	1	1	NUM
cana-410	148	6	,	,	PUNCT
cana-410	148	7	2	2	NUM
cana-410	148	8	,	,	PUNCT
cana-410	148	9	.	.	PUNCT
cana-410	148	10	.	.	PUNCT
cana-410	148	11	.	.	PUNCT
cana-410	149	1	.	.	PUNCT
cana-410	149	2	,	,	PUNCT
cana-410	149	3	𝑛	𝑛	PRON
cana-410	149	4	2	2	NUM
cana-410	149	5	}	}	PUNCT
cana-410	149	6	}	}	PUNCT
cana-410	149	7	v2	v2	NOUN
cana-410	149	8	=	=	SYM
cana-410	149	9	{	{	PUNCT
cana-410	149	10	(	(	PUNCT
cana-410	149	11	c2i	c2i	PROPN
cana-410	149	12	,	,	PUNCT
cana-410	149	13	p2i−1	p2i−1	PROPN
cana-410	149	14	,	,	PUNCT
cana-410	149	15	q2i	q2i	NOUN
cana-410	149	16	)	)	PUNCT
cana-410	150	1	|	|	ADV
cana-410	150	2	i	i	PRON
cana-410	150	3	∈	∈	PROPN
cana-410	150	4	{	{	PUNCT
cana-410	150	5	1	1	NUM
cana-410	150	6	,	,	PUNCT
cana-410	150	7	2	2	NUM
cana-410	150	8	,	,	PUNCT
cana-410	150	9	.	.	PUNCT
cana-410	150	10	.	.	PUNCT
cana-410	150	11	.	.	PUNCT
cana-410	151	1	,	,	PUNCT
cana-410	151	2	𝑛	𝑛	PRON
cana-410	151	3	2	2	NUM
cana-410	151	4	}	}	PUNCT
cana-410	151	5	}	}	PUNCT
cana-410	151	6	the	the	DET
cana-410	151	7	condition	condition	NOUN
cana-410	151	8	holds	hold	VERB
cana-410	151	9	for	for	ADP
cana-410	151	10	every	every	DET
cana-410	151	11	edge	edge	NOUN
cana-410	151	12	in	in	ADP
cana-410	151	13	the	the	DET
cana-410	151	14	clutch	clutch	NOUN
cana-410	151	15	graph	graph	NOUN
cana-410	151	16	.	.	PUNCT
cana-410	152	1	the	the	DET
cana-410	152	2	graph	graph	NOUN
cana-410	152	3	is	be	AUX
cana-410	152	4	bipartite	bipartite	PROPN
cana-410	152	5	iff	iff	PROPN
cana-410	152	6	it	it	PRON
cana-410	152	7	has	have	VERB
cana-410	152	8	no	no	DET
cana-410	152	9	odd	odd	ADJ
cana-410	152	10	cycles	cycle	NOUN
cana-410	152	11	.	.	PUNCT
cana-410	153	1	based	base	VERB
cana-410	153	2	on	on	ADP
cana-410	153	3	the	the	DET
cana-410	153	4	results	result	NOUN
cana-410	153	5	,	,	PUNCT
cana-410	153	6	the	the	DET
cana-410	153	7	clutch	clutch	ADJ
cana-410	153	8	graph	graph	NOUN
cana-410	153	9	has	have	VERB
cana-410	153	10	no	no	DET
cana-410	153	11	odd	odd	ADJ
cana-410	153	12	cycles	cycle	NOUN
cana-410	153	13	.	.	PUNCT
cana-410	154	1	so	so	ADV
cana-410	154	2	,	,	PUNCT
cana-410	154	3	it	it	PRON
cana-410	154	4	is	be	AUX
cana-410	154	5	proved	prove	VERB
cana-410	154	6	that	that	SCONJ
cana-410	154	7	every	every	DET
cana-410	154	8	clutch	clutch	ADJ
cana-410	154	9	graph	graph	NOUN
cana-410	154	10	is	be	AUX
cana-410	154	11	bipartite	bipartite	ADJ
cana-410	154	12	.	.	PUNCT
cana-410	155	1	theorem	theorem	VERB
cana-410	155	2	4.8	4.8	NUM
cana-410	155	3	every	every	DET
cana-410	155	4	clutch	clutch	ADJ
cana-410	155	5	graph	graph	NOUN
cana-410	155	6	cl3n	cl3n	NOUN
cana-410	155	7	(	(	PUNCT
cana-410	155	8	g	g	NOUN
cana-410	155	9	)	)	PUNCT
cana-410	155	10	,	,	PUNCT
cana-410	155	11	(	(	PUNCT
cana-410	155	12	n	n	X
cana-410	155	13	≥	≥	NOUN
cana-410	155	14	4	4	NUM
cana-410	155	15	,	,	PUNCT
cana-410	155	16	even	even	ADV
cana-410	155	17	)	)	PUNCT
cana-410	155	18	is	be	AUX
cana-410	155	19	hamiltonian	hamiltonian	ADJ
cana-410	155	20	.	.	PUNCT
cana-410	156	1	proof	proof	NOUN
cana-410	156	2	consider	consider	VERB
cana-410	156	3	the	the	DET
cana-410	156	4	clutch	clutch	ADJ
cana-410	156	5	graph	graph	NOUN
cana-410	156	6	cl3n	cl3n	NOUN
cana-410	156	7	(	(	PUNCT
cana-410	156	8	g	g	NOUN
cana-410	156	9	)	)	PUNCT
cana-410	156	10	with	with	ADP
cana-410	156	11	vertex	vertex	NOUN
cana-410	156	12	set	set	VERB
cana-410	156	13	v	v	NOUN
cana-410	156	14	and	and	CCONJ
cana-410	156	15	edge	edge	NOUN
cana-410	156	16	set	set	VERB
cana-410	156	17	e.	e.	PROPN
cana-410	156	18	let	let	VERB
cana-410	156	19	us	we	PRON
cana-410	156	20	decompose	decompose	VERB
cana-410	156	21	the	the	DET
cana-410	156	22	clutch	clutch	NOUN
cana-410	156	23	graph	graph	NOUN
cana-410	156	24	into	into	ADP
cana-410	156	25	two	two	NUM
cana-410	156	26	components	component	NOUN
cana-410	156	27	as	as	ADP
cana-410	156	28	g1	g1	PROPN
cana-410	156	29	representing	represent	VERB
cana-410	156	30	the	the	DET
cana-410	156	31	cycle	cycle	NOUN
cana-410	156	32	graph	graph	NOUN
cana-410	156	33	cn(g	cn(g	X
cana-410	156	34	)	)	PUNCT
cana-410	156	35	and	and	CCONJ
cana-410	156	36	g2	g2	PROPN
cana-410	156	37	representing	represent	VERB
cana-410	156	38	the	the	DET
cana-410	156	39	additional	additional	ADJ
cana-410	156	40	edges	edge	NOUN
cana-410	156	41	connecting	connect	VERB
cana-410	156	42	vp	vp	PROPN
cana-410	156	43	and	and	CCONJ
cana-410	156	44	vq	vq	PROPN
cana-410	156	45	.	.	PUNCT
cana-410	156	46	g1	g1	PROPN
cana-410	156	47	=	=	SYM
cana-410	156	48	(	(	PUNCT
cana-410	156	49	vc	vc	INTJ
cana-410	156	50	,	,	PUNCT
cana-410	156	51	ec	ec	PROPN
cana-410	156	52	)	)	PUNCT
cana-410	156	53	g2	g2	PROPN
cana-410	156	54	=	=	PRON
cana-410	156	55	(	(	PUNCT
cana-410	156	56	vp	vp	PROPN
cana-410	156	57	∪	∪	PROPN
cana-410	156	58	vq	vq	PROPN
cana-410	156	59	,	,	PUNCT
cana-410	156	60	ecp	ecp	NOUN
cana-410	156	61	∪	∪	ADP
cana-410	156	62	epq	epq	NOUN
cana-410	156	63	∪	∪	NOUN
cana-410	156	64	eq	eq	NOUN
cana-410	156	65	)	)	PUNCT
cana-410	156	66	since	since	SCONJ
cana-410	156	67	g1	g1	PROPN
cana-410	156	68	is	be	AUX
cana-410	156	69	a	a	DET
cana-410	156	70	cycle	cycle	NOUN
cana-410	156	71	graph	graph	NOUN
cana-410	156	72	,	,	PUNCT
cana-410	156	73	there	there	PRON
cana-410	156	74	exists	exist	VERB
cana-410	156	75	a	a	DET
cana-410	156	76	hamiltonian	hamiltonian	ADJ
cana-410	156	77	cycle	cycle	NOUN
cana-410	156	78	h1	h1	NOUN
cana-410	156	79	.	.	PUNCT
cana-410	157	1	further	far	ADV
cana-410	157	2	,	,	PUNCT
cana-410	157	3	connecting	connect	VERB
cana-410	157	4	the	the	DET
cana-410	157	5	edges	edge	NOUN
cana-410	157	6	vp	vp	PROPN
cana-410	157	7	and	and	CCONJ
cana-410	157	8	vq	vq	PROPN
cana-410	157	9	form	form	VERB
cana-410	157	10	a	a	DET
cana-410	157	11	hamiltonian	hamiltonian	ADJ
cana-410	157	12	path	path	NOUN
cana-410	157	13	p2	p2	NOUN
cana-410	157	14	.	.	PUNCT
cana-410	158	1	h1	h1	PROPN
cana-410	158	2	=(	=(	PROPN
cana-410	158	3	ci1	ci1	PROPN
cana-410	158	4	,	,	PUNCT
cana-410	158	5	𝑐𝑖2	𝑐𝑖2	NOUN
cana-410	158	6	,	,	PUNCT
cana-410	158	7	…	…	PUNCT
cana-410	158	8	…	…	PUNCT
cana-410	158	9	.	.	NUM
cana-410	158	10	,	,	PUNCT
cana-410	158	11	cin	cin	PROPN
cana-410	158	12	,	,	PUNCT
cana-410	158	13	ci1	ci1	PROPN
cana-410	158	14	)	)	PUNCT
cana-410	158	15	p2	p2	PROPN
cana-410	158	16	=	=	SYM
cana-410	158	17	(	(	PUNCT
cana-410	158	18	pj1	pj1	NOUN
cana-410	158	19	,	,	PUNCT
cana-410	158	20	qj1	qj1	PROPN
cana-410	158	21	,	,	PUNCT
cana-410	158	22	pj2	pj2	NOUN
cana-410	158	23	,	,	PUNCT
cana-410	158	24	qj2,	qj2,	NOUN
cana-410	158	25	…	…	X
cana-410	158	26	…	…	PUNCT
cana-410	158	27	…	…	PUNCT
cana-410	158	28	,pjn	,pjn	NOUN
cana-410	158	29	,	,	PUNCT
cana-410	158	30	qjn	qjn	ADJ
cana-410	158	31	)	)	PUNCT
cana-410	158	32	communications	communication	NOUN
cana-410	158	33	on	on	ADP
cana-410	158	34	applied	apply	VERB
cana-410	158	35	nonlinear	nonlinear	ADJ
cana-410	158	36	analysis	analysis	NOUN
cana-410	158	37	issn	issn	NOUN
cana-410	158	38	:	:	PUNCT
cana-410	158	39	1074	1074	NUM
cana-410	158	40	-	-	PUNCT
cana-410	158	41	133x	133x	NUM
cana-410	158	42	vol	vol	NOUN
cana-410	158	43	31	31	NUM
cana-410	158	44	no	no	NOUN
cana-410	158	45	.	.	NOUN
cana-410	158	46	1	1	NUM
cana-410	158	47	(	(	PUNCT
cana-410	158	48	2024	2024	NUM
cana-410	158	49	)	)	PUNCT
cana-410	158	50	264	264	NUM
cana-410	158	51	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	158	52	combine	combine	VERB
cana-410	158	53	h1	h1	NOUN
cana-410	158	54	and	and	CCONJ
cana-410	158	55	p2	p2	NOUN
cana-410	158	56	to	to	PART
cana-410	158	57	form	form	VERB
cana-410	158	58	a	a	DET
cana-410	158	59	hamiltonian	hamiltonian	ADJ
cana-410	158	60	cycle	cycle	NOUN
cana-410	158	61	h	h	NOUN
cana-410	158	62	for	for	ADP
cana-410	158	63	cl3n	cl3n	NOUN
cana-410	158	64	(	(	PUNCT
cana-410	158	65	g	g	NOUN
cana-410	158	66	)	)	PUNCT
cana-410	158	67	.	.	PUNCT
cana-410	159	1	h	h	NOUN
cana-410	159	2	=	=	PRON
cana-410	159	3	(	(	PUNCT
cana-410	159	4	ci1	ci1	PROPN
cana-410	159	5	,	,	PUNCT
cana-410	159	6	ci2	ci2	PROPN
cana-410	159	7	,	,	PUNCT
cana-410	159	8	...	...	PUNCT
cana-410	159	9	,	,	PUNCT
cana-410	159	10	cin	cin	PROPN
cana-410	159	11	,	,	PUNCT
cana-410	159	12	ci1	ci1	PROPN
cana-410	159	13	,	,	PUNCT
cana-410	159	14	pj1	pj1	NOUN
cana-410	159	15	,	,	PUNCT
cana-410	159	16	qj1	qj1	PROPN
cana-410	159	17	,	,	PUNCT
cana-410	159	18	pj2	pj2	NOUN
cana-410	159	19	,	,	PUNCT
cana-410	159	20	qj2	qj2	INTJ
cana-410	159	21	,	,	PUNCT
cana-410	159	22	...	...	PUNCT
cana-410	159	23	,	,	PUNCT
cana-410	159	24	pjn	pjn	ADJ
cana-410	159	25	,	,	PUNCT
cana-410	159	26	qjn	qjn	ADJ
cana-410	159	27	)	)	PUNCT
cana-410	159	28	.	.	PUNCT
cana-410	160	1	as	as	SCONJ
cana-410	160	2	it	it	PRON
cana-410	160	3	concluded	conclude	VERB
cana-410	160	4	that	that	SCONJ
cana-410	160	5	every	every	DET
cana-410	160	6	clutch	clutch	ADJ
cana-410	160	7	graph	graph	NOUN
cana-410	160	8	with	with	ADP
cana-410	160	9	n	n	CCONJ
cana-410	160	10	vertices	vertex	NOUN
cana-410	160	11	(	(	PUNCT
cana-410	160	12	n	n	CCONJ
cana-410	160	13	≥	≥	NOUN
cana-410	160	14	4	4	NUM
cana-410	160	15	,	,	PUNCT
cana-410	160	16	even	even	ADV
cana-410	160	17	)	)	PUNCT
cana-410	160	18	has	have	VERB
cana-410	160	19	a	a	DET
cana-410	160	20	hamiltonian	hamiltonian	ADJ
cana-410	160	21	cycle	cycle	NOUN
cana-410	160	22	.	.	PUNCT
cana-410	161	1	let	let	VERB
cana-410	161	2	’s	’s	NOUN
cana-410	161	3	consider	consider	VERB
cana-410	161	4	the	the	DET
cana-410	161	5	case	case	NOUN
cana-410	161	6	where	where	SCONJ
cana-410	161	7	n	n	PROPN
cana-410	161	8	=	=	SYM
cana-410	161	9	6	6	NUM
cana-410	161	10	and	and	CCONJ
cana-410	161	11	construct	construct	VERB
cana-410	161	12	the	the	DET
cana-410	161	13	clutch	clutch	NOUN
cana-410	161	14	graph	graph	NOUN
cana-410	161	15	cl18	cl18	PROPN
cana-410	161	16	(	(	PUNCT
cana-410	161	17	g	g	NOUN
cana-410	161	18	)	)	PUNCT
cana-410	161	19	.	.	PUNCT
cana-410	162	1	the	the	DET
cana-410	162	2	cycle	cycle	NOUN
cana-410	162	3	graph	graph	NOUN
cana-410	162	4	with	with	ADP
cana-410	162	5	n	n	NOUN
cana-410	162	6	=	=	NUM
cana-410	162	7	6	6	NUM
cana-410	162	8	has	have	VERB
cana-410	162	9	vertices	vertex	NOUN
cana-410	162	10	c1	c1	PROPN
cana-410	162	11	,	,	PUNCT
cana-410	162	12	c2	c2	PROPN
cana-410	162	13	,	,	PUNCT
cana-410	162	14	c3	c3	PROPN
cana-410	162	15	,	,	PUNCT
cana-410	162	16	c4	c4	NOUN
cana-410	162	17	,	,	PUNCT
cana-410	162	18	c5	c5	PROPN
cana-410	162	19	,	,	PUNCT
cana-410	162	20	c6	c6	PROPN
cana-410	162	21	and	and	CCONJ
cana-410	162	22	edges	edge	NOUN
cana-410	162	23	:	:	PUNCT
cana-410	162	24	cn	cn	PROPN
cana-410	162	25	=	=	SYM
cana-410	162	26	(	(	PUNCT
cana-410	162	27	c1	c1	PROPN
cana-410	162	28	,	,	PUNCT
cana-410	162	29	c2	c2	PROPN
cana-410	162	30	)	)	PUNCT
cana-410	162	31	,	,	PUNCT
cana-410	162	32	(	(	PUNCT
cana-410	162	33	c2	c2	PROPN
cana-410	162	34	,	,	PUNCT
cana-410	162	35	c3	c3	PROPN
cana-410	162	36	)	)	PUNCT
cana-410	162	37	,	,	PUNCT
cana-410	162	38	(	(	PUNCT
cana-410	162	39	c3	c3	PROPN
cana-410	162	40	,	,	PUNCT
cana-410	162	41	c4	c4	NOUN
cana-410	162	42	)	)	PUNCT
cana-410	162	43	,	,	PUNCT
cana-410	162	44	(	(	PUNCT
cana-410	162	45	c4	c4	NOUN
cana-410	162	46	,	,	PUNCT
cana-410	162	47	c5	c5	PROPN
cana-410	162	48	)	)	PUNCT
cana-410	162	49	,	,	PUNCT
cana-410	162	50	(	(	PUNCT
cana-410	162	51	c5	c5	PROPN
cana-410	162	52	,	,	PUNCT
cana-410	162	53	c6	c6	PROPN
cana-410	162	54	)	)	PUNCT
cana-410	162	55	,	,	PUNCT
cana-410	162	56	(	(	PUNCT
cana-410	162	57	c6	c6	PROPN
cana-410	162	58	,	,	PUNCT
cana-410	162	59	c1	c1	PROPN
cana-410	162	60	)	)	PUNCT
cana-410	162	61	•	•	ADP
cana-410	162	62	add	add	VERB
cana-410	162	63	vertices	vertex	NOUN
cana-410	162	64	vc	vc	PROPN
cana-410	162	65	=	=	SYM
cana-410	162	66	{	{	PUNCT
cana-410	162	67	c1	c1	PROPN
cana-410	162	68	,	,	PUNCT
cana-410	162	69	c2	c2	PROPN
cana-410	162	70	,	,	PUNCT
cana-410	162	71	c3	c3	PROPN
cana-410	162	72	,	,	PUNCT
cana-410	162	73	c4	c4	NOUN
cana-410	162	74	,	,	PUNCT
cana-410	162	75	c5	c5	PROPN
cana-410	162	76	,	,	PUNCT
cana-410	162	77	c6	c6	PROPN
cana-410	162	78	}	}	PUNCT
cana-410	162	79	,	,	PUNCT
cana-410	162	80	vp	vp	PROPN
cana-410	162	81	=	=	SYM
cana-410	162	82	{	{	PUNCT
cana-410	162	83	p1	p1	PROPN
cana-410	162	84	,	,	PUNCT
cana-410	162	85	p2	p2	NOUN
cana-410	162	86	,	,	PUNCT
cana-410	162	87	p3	p3	NOUN
cana-410	162	88	,	,	PUNCT
cana-410	162	89	p4	p4	ADJ
cana-410	162	90	,	,	PUNCT
cana-410	162	91	p5	p5	ADJ
cana-410	162	92	,	,	PUNCT
cana-410	162	93	p6	p6	PROPN
cana-410	162	94	}	}	PUNCT
cana-410	162	95	,	,	PUNCT
cana-410	162	96	and	and	CCONJ
cana-410	162	97	vq	vq	PROPN
cana-410	162	98	=	=	SYM
cana-410	162	99	{	{	PUNCT
cana-410	162	100	q1	q1	PROPN
cana-410	162	101	,	,	PUNCT
cana-410	162	102	q2	q2	NOUN
cana-410	162	103	,	,	PUNCT
cana-410	162	104	q3	q3	PROPN
cana-410	162	105	,	,	PUNCT
cana-410	162	106	q4	q4	PROPN
cana-410	162	107	,	,	PUNCT
cana-410	162	108	q5	q5	PROPN
cana-410	162	109	,	,	PUNCT
cana-410	162	110	q6	q6	PROPN
cana-410	162	111	}	}	PUNCT
cana-410	162	112	.	.	PUNCT
cana-410	163	1	•	•	NUM
cana-410	163	2	connect	connect	VERB
cana-410	163	3	each	each	DET
cana-410	163	4	ci	ci	NOUN
cana-410	163	5	to	to	ADP
cana-410	163	6	its	its	PRON
cana-410	163	7	corresponding	corresponding	ADJ
cana-410	163	8	pi	pi	NOUN
cana-410	163	9	:	:	PUNCT
cana-410	163	10	ecp	ecp	NOUN
cana-410	163	11	=	=	PRON
cana-410	163	12	{	{	PUNCT
cana-410	163	13	(	(	PUNCT
cana-410	163	14	c1	c1	NOUN
cana-410	163	15	,	,	PUNCT
cana-410	163	16	p1	p1	PROPN
cana-410	163	17	)	)	PUNCT
cana-410	163	18	,	,	PUNCT
cana-410	163	19	(	(	PUNCT
cana-410	163	20	c2	c2	PROPN
cana-410	163	21	,	,	PUNCT
cana-410	163	22	p2	p2	PROPN
cana-410	163	23	)	)	PUNCT
cana-410	163	24	,	,	PUNCT
cana-410	163	25	.	.	PUNCT
cana-410	163	26	.	.	PUNCT
cana-410	164	1	.	.	PUNCT
cana-410	165	1	,	,	PUNCT
cana-410	165	2	(	(	PUNCT
cana-410	165	3	c6	c6	PROPN
cana-410	165	4	,	,	PUNCT
cana-410	165	5	p6	p6	PROPN
cana-410	165	6	)	)	PUNCT
cana-410	165	7	}	}	PUNCT
cana-410	165	8	.	.	PUNCT
cana-410	166	1	•	•	NUM
cana-410	166	2	connect	connect	VERB
cana-410	166	3	pi	pi	NOUN
cana-410	166	4	to	to	ADP
cana-410	166	5	qi	qi	PROPN
cana-410	166	6	:	:	PUNCT
cana-410	166	7	epq	epq	NOUN
cana-410	166	8	=	=	SYM
cana-410	166	9	{	{	PUNCT
cana-410	166	10	(	(	PUNCT
cana-410	166	11	p1	p1	PROPN
cana-410	166	12	,	,	PUNCT
cana-410	166	13	q1	q1	PROPN
cana-410	166	14	)	)	PUNCT
cana-410	166	15	,	,	PUNCT
cana-410	166	16	(	(	PUNCT
cana-410	166	17	p2	p2	X
cana-410	166	18	,	,	PUNCT
cana-410	166	19	q2	q2	NOUN
cana-410	166	20	)	)	PUNCT
cana-410	166	21	,	,	PUNCT
cana-410	166	22	.	.	PUNCT
cana-410	166	23	.	.	PUNCT
cana-410	166	24	.	.	PUNCT
cana-410	167	1	,	,	PUNCT
cana-410	167	2	(	(	PUNCT
cana-410	167	3	p6	p6	PROPN
cana-410	167	4	,	,	PUNCT
cana-410	167	5	q6	q6	NOUN
cana-410	167	6	)	)	PUNCT
cana-410	167	7	}	}	PUNCT
cana-410	167	8	.	.	PUNCT
cana-410	168	1	•	•	NOUN
cana-410	168	2	connect	connect	VERB
cana-410	168	3	qi	qi	PROPN
cana-410	168	4	to	to	ADP
cana-410	168	5	qi+1	qi+1	PROPN
cana-410	168	6	(	(	PUNCT
cana-410	168	7	with	with	ADP
cana-410	168	8	q6	q6	PROPN
cana-410	168	9	connecting	connect	VERB
cana-410	168	10	to	to	PART
cana-410	168	11	q1	q1	VERB
cana-410	168	12	):	):	PUNCT
cana-410	168	13	eq	eq	NOUN
cana-410	168	14	=	=	SYM
cana-410	168	15	{	{	PUNCT
cana-410	168	16	(	(	PUNCT
cana-410	168	17	q2	q2	NOUN
cana-410	168	18	,	,	PUNCT
cana-410	168	19	q3	q3	PROPN
cana-410	168	20	)	)	PUNCT
cana-410	168	21	,	,	PUNCT
cana-410	168	22	(	(	PUNCT
cana-410	168	23	q4	q4	PROPN
cana-410	168	24	,	,	PUNCT
cana-410	168	25	q5	q5	PROPN
cana-410	168	26	)	)	PUNCT
cana-410	168	27	,	,	PUNCT
cana-410	168	28	(	(	PUNCT
cana-410	168	29	q6	q6	PROPN
cana-410	168	30	,	,	PUNCT
cana-410	168	31	q1	q1	PROPN
cana-410	168	32	)	)	PUNCT
cana-410	168	33	}	}	PUNCT
cana-410	168	34	.	.	PUNCT
cana-410	169	1	•	•	NUM
cana-410	169	2	connect	connect	VERB
cana-410	169	3	pi	pi	NOUN
cana-410	169	4	to	to	ADP
cana-410	169	5	pi+1	pi+1	NUM
cana-410	169	6	(	(	PUNCT
cana-410	169	7	with	with	ADP
cana-410	169	8	p6	p6	PROPN
cana-410	169	9	connecting	connect	VERB
cana-410	169	10	to	to	ADP
cana-410	169	11	p1	p1	PROPN
cana-410	169	12	):	):	PUNCT
cana-410	169	13	ep	ep	PROPN
cana-410	169	14	=	=	SYM
cana-410	169	15	{	{	PUNCT
cana-410	169	16	(	(	PUNCT
cana-410	169	17	p1	p1	NOUN
cana-410	169	18	,	,	PUNCT
cana-410	169	19	p2	p2	PROPN
cana-410	169	20	)	)	PUNCT
cana-410	169	21	,	,	PUNCT
cana-410	169	22	(	(	PUNCT
cana-410	169	23	p3	p3	PROPN
cana-410	169	24	,	,	PUNCT
cana-410	169	25	p4	p4	ADJ
cana-410	169	26	)	)	PUNCT
cana-410	169	27	,	,	PUNCT
cana-410	169	28	(	(	PUNCT
cana-410	169	29	p5	p5	ADJ
cana-410	169	30	,	,	PUNCT
cana-410	169	31	p6	p6	PROPN
cana-410	169	32	)	)	PUNCT
cana-410	169	33	}	}	PUNCT
cana-410	169	34	.	.	PUNCT
cana-410	170	1	a	a	DET
cana-410	170	2	hamiltonian	hamiltonian	ADJ
cana-410	170	3	cycle	cycle	NOUN
cana-410	170	4	can	can	AUX
cana-410	170	5	be	be	AUX
cana-410	170	6	traversed	traverse	VERB
cana-410	170	7	as	as	ADP
cana-410	170	8	c1	c1	PROPN
cana-410	170	9	→	→	SYM
cana-410	170	10	p1	p1	PROPN
cana-410	170	11	→	→	SYM
cana-410	170	12	q1	q1	PROPN
cana-410	170	13	→	→	SYM
cana-410	170	14	c2	c2	PROPN
cana-410	170	15	→	→	SYM
cana-410	170	16	p2	p2	PROPN
cana-410	170	17	→	→	SYM
cana-410	170	18	q2	q2	NOUN
cana-410	170	19	→	→	SYM
cana-410	170	20	c3	c3	PROPN
cana-410	170	21	→	→	SYM
cana-410	170	22	p3	p3	PROPN
cana-410	170	23	→	→	SYM
cana-410	170	24	q3	q3	PROPN
cana-410	170	25	→	→	SYM
cana-410	170	26	c4→	c4→	PRON
cana-410	170	27	p4	p4	PROPN
cana-410	170	28	→	→	SYM
cana-410	170	29	q4	q4	PROPN
cana-410	170	30	→	→	SYM
cana-410	170	31	c5	c5	PROPN
cana-410	170	32	→	→	PUNCT
cana-410	170	33	p5	p5	PROPN
cana-410	170	34	→	→	SYM
cana-410	170	35	q5	q5	PROPN
cana-410	170	36	→	→	SYM
cana-410	170	37	c6	c6	PROPN
cana-410	170	38	→	→	SYM
cana-410	170	39	p6	p6	PROPN
cana-410	170	40	→	→	SYM
cana-410	170	41	q6	q6	PROPN
cana-410	170	42	→	→	SYM
cana-410	170	43	c1	c1	PROPN
cana-410	170	44	conclusion	conclusion	NOUN
cana-410	170	45	this	this	DET
cana-410	170	46	paper	paper	NOUN
cana-410	170	47	introduces	introduce	VERB
cana-410	170	48	the	the	DET
cana-410	170	49	concept	concept	NOUN
cana-410	170	50	of	of	ADP
cana-410	170	51	clutch	clutch	ADJ
cana-410	170	52	graphs	graph	NOUN
cana-410	170	53	from	from	ADP
cana-410	170	54	cycle	cycle	NOUN
cana-410	170	55	graphs	graph	NOUN
cana-410	170	56	.	.	PUNCT
cana-410	171	1	the	the	DET
cana-410	171	2	clutch	clutch	ADJ
cana-410	171	3	graph	graph	NOUN
cana-410	171	4	properties	property	NOUN
cana-410	171	5	,	,	PUNCT
cana-410	171	6	such	such	ADJ
cana-410	171	7	as	as	ADP
cana-410	171	8	degree	degree	NOUN
cana-410	171	9	,	,	PUNCT
cana-410	171	10	girth	girth	NOUN
cana-410	171	11	,	,	PUNCT
cana-410	171	12	and	and	CCONJ
cana-410	171	13	chromatic	chromatic	ADJ
cana-410	171	14	number	number	NOUN
cana-410	171	15	have	have	AUX
cana-410	171	16	been	be	AUX
cana-410	171	17	discussed	discuss	VERB
cana-410	171	18	.	.	PUNCT
cana-410	172	1	theorems	theorem	NOUN
cana-410	172	2	are	be	AUX
cana-410	172	3	presented	present	VERB
cana-410	172	4	to	to	PART
cana-410	172	5	illustrate	illustrate	VERB
cana-410	172	6	the	the	DET
cana-410	172	7	sum	sum	NOUN
cana-410	172	8	of	of	ADP
cana-410	172	9	the	the	DET
cana-410	172	10	degrees	degree	NOUN
cana-410	172	11	and	and	CCONJ
cana-410	172	12	number	number	NOUN
cana-410	172	13	of	of	ADP
cana-410	172	14	distinct	distinct	ADJ
cana-410	172	15	spanning	span	VERB
cana-410	172	16	trees	tree	NOUN
cana-410	172	17	in	in	ADP
cana-410	172	18	the	the	DET
cana-410	172	19	clutch	clutch	NOUN
cana-410	172	20	graph	graph	NOUN
cana-410	172	21	.	.	PUNCT
cana-410	173	1	moreover	moreover	ADV
cana-410	173	2	,	,	PUNCT
cana-410	173	3	the	the	DET
cana-410	173	4	paper	paper	NOUN
cana-410	173	5	establishes	establish	VERB
cana-410	173	6	the	the	DET
cana-410	173	7	existence	existence	NOUN
cana-410	173	8	of	of	ADP
cana-410	173	9	bipartite	bipartite	NOUN
cana-410	173	10	and	and	CCONJ
cana-410	173	11	hamiltonian	hamiltonian	NOUN
cana-410	173	12	in	in	ADP
cana-410	173	13	the	the	DET
cana-410	173	14	clutch	clutch	NOUN
cana-410	173	15	graph	graph	NOUN
cana-410	173	16	.	.	PUNCT
cana-410	174	1	furthermore	furthermore	ADV
cana-410	174	2	,	,	PUNCT
cana-410	174	3	the	the	DET
cana-410	174	4	author	author	NOUN
cana-410	174	5	plans	plan	VERB
cana-410	174	6	to	to	PART
cana-410	174	7	apply	apply	VERB
cana-410	174	8	these	these	DET
cana-410	174	9	concepts	concept	NOUN
cana-410	174	10	to	to	PART
cana-410	174	11	network	network	VERB
cana-410	174	12	analysis	analysis	NOUN
cana-410	174	13	.	.	PUNCT
cana-410	175	1	references	reference	NOUN
cana-410	175	2	[	[	X
cana-410	175	3	1	1	NUM
cana-410	175	4	]	]	PUNCT
cana-410	175	5	vasudev.c	vasudev.c	ADV
cana-410	175	6	,	,	PUNCT
cana-410	175	7	graph	graph	NOUN
cana-410	175	8	theory	theory	NOUN
cana-410	175	9	and	and	CCONJ
cana-410	175	10	applications	application	NOUN
cana-410	175	11	,	,	PUNCT
cana-410	175	12	new	new	ADJ
cana-410	175	13	age	age	NOUN
cana-410	175	14	international	international	NOUN
cana-410	175	15	(	(	PUNCT
cana-410	175	16	p)ltd	p)ltd	PROPN
cana-410	175	17	,	,	PUNCT
cana-410	175	18	(	(	PUNCT
cana-410	175	19	2009	2009	NUM
cana-410	175	20	)	)	PUNCT
cana-410	175	21	.	.	PUNCT
cana-410	176	1	[	[	X
cana-410	176	2	2	2	NUM
cana-410	176	3	]	]	X
cana-410	176	4	bondy	bondy	PROPN
cana-410	176	5	.	.	PUNCT
cana-410	177	1	j.	j.	PROPN
cana-410	177	2	a	a	PROPN
cana-410	177	3	and	and	CCONJ
cana-410	177	4	murty	murty	NOUN
cana-410	177	5	.	.	PUNCT
cana-410	178	1	u.	u.	PROPN
cana-410	178	2	s.r	s.r	PROPN
cana-410	178	3	,	,	PUNCT
cana-410	178	4	graph	graph	NOUN
cana-410	178	5	theory	theory	NOUN
cana-410	178	6	,	,	PUNCT
cana-410	178	7	springer	springer	NOUN
cana-410	178	8	international	international	PROPN
cana-410	178	9	edition	edition	PROPN
cana-410	178	10	,	,	PUNCT
cana-410	178	11	(	(	PUNCT
cana-410	178	12	2008	2008	NUM
cana-410	178	13	)	)	PUNCT
cana-410	178	14	.	.	PUNCT
cana-410	179	1	[	[	X
cana-410	179	2	3	3	NUM
cana-410	179	3	]	]	PUNCT
cana-410	179	4	harary	harary	NOUN
cana-410	179	5	.	.	PUNCT
cana-410	180	1	f	f	X
cana-410	180	2	,	,	PUNCT
cana-410	180	3	graph	graph	NOUN
cana-410	180	4	theory	theory	NOUN
cana-410	180	5	,	,	PUNCT
cana-410	180	6	addison	addison	PROPN
cana-410	180	7	-	-	PUNCT
cana-410	180	8	wesley	wesley	PROPN
cana-410	180	9	,	,	PUNCT
cana-410	180	10	reading	reading	NOUN
cana-410	180	11	,	,	PUNCT
cana-410	180	12	mass	mass	PROPN
cana-410	180	13	,	,	PUNCT
cana-410	180	14	(	(	PUNCT
cana-410	180	15	1969	1969	NUM
cana-410	180	16	)	)	PUNCT
cana-410	180	17	.	.	PUNCT
cana-410	181	1	[	[	X
cana-410	181	2	4	4	X
cana-410	181	3	]	]	X
cana-410	181	4	kamran	kamran	PROPN
cana-410	181	5	azhar	azhar	PROPN
cana-410	181	6	,	,	PUNCT
cana-410	181	7	sohail	sohail	PROPN
cana-410	181	8	zafar	zafar	PROPN
cana-410	181	9	,	,	PUNCT
cana-410	181	10	agha	agha	PROPN
cana-410	181	11	kashif	kashif	PROPN
cana-410	181	12	,	,	PUNCT
cana-410	181	13	amer	amer	PROPN
cana-410	181	14	aljaedi	aljaedi	PROPN
cana-410	181	15	,	,	PUNCT
cana-410	181	16	umar	umar	PROPN
cana-410	181	17	albalawi	albalawi	PROPN
cana-410	181	18	,	,	PUNCT
cana-410	181	19	the	the	DET
cana-410	181	20	application	application	NOUN
cana-410	181	21	of	of	ADP
cana-410	181	22	fault	fault	NOUN
cana-410	181	23	-	-	PUNCT
cana-410	181	24	tolerant	tolerant	ADJ
cana-410	181	25	partition	partition	NOUN
cana-410	181	26	resolvability	resolvability	NOUN
cana-410	181	27	in	in	ADP
cana-410	181	28	cycle	cycle	NOUN
cana-410	181	29	-	-	PUNCT
cana-410	181	30	related	relate	VERB
cana-410	181	31	graphs	graph	NOUN
cana-410	181	32	,	,	PUNCT
cana-410	181	33	applied	apply	VERB
cana-410	181	34	science	science	NOUN
cana-410	181	35	(	(	PUNCT
cana-410	181	36	2022	2022	NUM
cana-410	181	37	)	)	PUNCT
cana-410	181	38	,	,	PUNCT
cana-410	181	39	12(19	12(19	NUM
cana-410	181	40	)	)	PUNCT
cana-410	181	41	,	,	PUNCT
cana-410	181	42	9558	9558	NUM
cana-410	181	43	.	.	PUNCT
cana-410	182	1	[	[	X
cana-410	182	2	5	5	X
cana-410	182	3	]	]	X
cana-410	182	4	jonathan	jonathan	PROPN
cana-410	182	5	l.	l.	PROPN
cana-410	182	6	gross	gross	PROPN
cana-410	182	7	,	,	PUNCT
cana-410	182	8	jay	jay	PROPN
cana-410	182	9	yellen	yellen	PROPN
cana-410	182	10	,	,	PUNCT
cana-410	182	11	mark	mark	PROPN
cana-410	182	12	anderson	anderson	PROPN
cana-410	182	13	,	,	PUNCT
cana-410	182	14	graph	graph	NOUN
cana-410	182	15	theory	theory	NOUN
cana-410	182	16	and	and	CCONJ
cana-410	182	17	its	its	PRON
cana-410	182	18	applications	application	NOUN
cana-410	182	19	,	,	PUNCT
cana-410	182	20	crc	crc	NOUN
cana-410	182	21	press	press	NOUN
cana-410	182	22	,	,	PUNCT
cana-410	182	23	taylor	taylor	PROPN
cana-410	182	24	francis	francis	PROPN
cana-410	182	25	group	group	PROPN
cana-410	182	26	,	,	PUNCT
cana-410	182	27	mass	mass	PROPN
cana-410	182	28	,	,	PUNCT
cana-410	182	29	(	(	PUNCT
cana-410	182	30	2019	2019	NUM
cana-410	182	31	)	)	PUNCT
cana-410	182	32	.	.	PUNCT
cana-410	183	1	[	[	X
cana-410	183	2	6	6	NUM
cana-410	183	3	]	]	SYM
cana-410	183	4	badwaik	badwaik	PROPN
cana-410	183	5	jyoti	jyoti	PROPN
cana-410	183	6	s.	s.	PROPN
cana-410	183	7	recent	recent	ADJ
cana-410	183	8	advances	advance	NOUN
cana-410	183	9	in	in	ADP
cana-410	183	10	graph	graph	NOUN
cana-410	183	11	theory	theory	NOUN
cana-410	183	12	and	and	CCONJ
cana-410	183	13	its	its	PRON
cana-410	183	14	applications	application	NOUN
cana-410	183	15	,	,	PUNCT
cana-410	183	16	international	international	ADJ
cana-410	183	17	journal	journal	NOUN
cana-410	183	18	of	of	ADP
cana-410	183	19	scientific	scientific	ADJ
cana-410	183	20	research	research	NOUN
cana-410	183	21	in	in	ADP
cana-410	183	22	science	science	NOUN
cana-410	183	23	,	,	PUNCT
cana-410	183	24	engineering	engineering	NOUN
cana-410	183	25	and	and	CCONJ
cana-410	183	26	technology	technology	NOUN
cana-410	183	27	,	,	PUNCT
cana-410	183	28	february	february	PROPN
cana-410	183	29	,	,	PUNCT
cana-410	183	30	(	(	PUNCT
cana-410	183	31	2020	2020	NUM
cana-410	183	32	)	)	PUNCT
cana-410	183	33	,	,	PUNCT
cana-410	183	34	special	special	ADJ
cana-410	183	35	issue	issue	NOUN
cana-410	183	36	a7	a7	PROPN
cana-410	183	37	(	(	PUNCT
cana-410	183	38	533	533	NUM
cana-410	183	39	-	-	SYM
cana-410	183	40	538	538	NUM
cana-410	183	41	)	)	PUNCT
cana-410	183	42	.	.	PUNCT
cana-410	184	1	[	[	X
cana-410	184	2	7	7	X
cana-410	184	3	]	]	X
cana-410	184	4	jayesh	jayesh	NOUN
cana-410	184	5	kudase1	kudase1	PROPN
cana-410	184	6	,	,	PUNCT
cana-410	184	7	priyanka	priyanka	PROPN
cana-410	184	8	bane	bane	PROPN
cana-410	184	9	,	,	PUNCT
cana-410	184	10	a	a	DET
cana-410	184	11	brief	brief	ADJ
cana-410	184	12	study	study	NOUN
cana-410	184	13	of	of	ADP
cana-410	184	14	graph	graph	NOUN
cana-410	184	15	data	datum	NOUN
cana-410	184	16	structure	structure	NOUN
cana-410	184	17	,	,	PUNCT
cana-410	184	18	international	international	ADJ
cana-410	184	19	journal	journal	NOUN
cana-410	184	20	of	of	ADP
cana-410	184	21	advanced	advanced	ADJ
cana-410	184	22	research	research	NOUN
cana-410	184	23	in	in	ADP
cana-410	184	24	computer	computer	NOUN
cana-410	184	25	and	and	CCONJ
cana-410	184	26	communication	communication	NOUN
cana-410	184	27	engineering	engineering	NOUN
cana-410	184	28	,	,	PUNCT
cana-410	184	29	5(6	5(6	NUM
cana-410	184	30	)	)	PUNCT
cana-410	184	31	,	,	PUNCT
cana-410	184	32	(	(	PUNCT
cana-410	184	33	2016	2016	NUM
cana-410	184	34	)	)	PUNCT
cana-410	184	35	,	,	PUNCT
cana-410	184	36	issn	issn	PROPN
cana-410	184	37	(	(	PUNCT
cana-410	184	38	print	print	NOUN
cana-410	184	39	)	)	PUNCT
cana-410	184	40	2319	2319	NUM
cana-410	184	41	-	-	SYM
cana-410	184	42	5940	5940	NUM
cana-410	184	43	.	.	PUNCT
cana-410	185	1	[	[	X
cana-410	185	2	8	8	NUM
cana-410	185	3	]	]	X
cana-410	185	4	sagar	sagar	PROPN
cana-410	185	5	jay	jay	PROPN
cana-410	185	6	bhoite	bhoite	PROPN
cana-410	185	7	,	,	PUNCT
cana-410	185	8	design	design	NOUN
cana-410	185	9	and	and	CCONJ
cana-410	185	10	analysis	analysis	NOUN
cana-410	185	11	of	of	ADP
cana-410	185	12	single	single	ADJ
cana-410	185	13	plate	plate	NOUN
cana-410	185	14	friction	friction	NOUN
cana-410	185	15	clutch	clutch	NOUN
cana-410	185	16	,	,	PUNCT
cana-410	185	17	international	international	ADJ
cana-410	185	18	journal	journal	NOUN
cana-410	185	19	communications	communication	NOUN
cana-410	185	20	on	on	ADP
cana-410	185	21	applied	apply	VERB
cana-410	185	22	nonlinear	nonlinear	ADJ
cana-410	185	23	analysis	analysis	NOUN
cana-410	185	24	issn	issn	NOUN
cana-410	185	25	:	:	PUNCT
cana-410	185	26	1074	1074	NUM
cana-410	185	27	-	-	PUNCT
cana-410	185	28	133x	133x	NUM
cana-410	185	29	vol	vol	NOUN
cana-410	185	30	31	31	NUM
cana-410	185	31	no	no	NOUN
cana-410	185	32	.	.	NOUN
cana-410	185	33	1	1	NUM
cana-410	185	34	(	(	PUNCT
cana-410	185	35	2024	2024	NUM
cana-410	185	36	)	)	PUNCT
cana-410	185	37	265	265	NUM
cana-410	185	38	https://internationalpubls.com	https://internationalpubls.com	X
cana-410	185	39	of	of	ADP
cana-410	185	40	innovative	innovative	ADJ
cana-410	185	41	research	research	NOUN
cana-410	185	42	in	in	ADP
cana-410	185	43	technology	technology	NOUN
cana-410	185	44	,	,	PUNCT
cana-410	185	45	february	february	PROPN
cana-410	185	46	(	(	PUNCT
cana-410	185	47	2022	2022	NUM
cana-410	185	48	)	)	PUNCT
cana-410	185	49	,	,	PUNCT
cana-410	185	50	8(9	8(9	NUM
cana-410	185	51	)	)	PUNCT
cana-410	185	52	,	,	PUNCT
cana-410	185	53	issn	issn	PROPN
cana-410	185	54	:	:	PUNCT
cana-410	185	55	2349	2349	NUM
cana-410	185	56	-	-	SYM
cana-410	185	57	6002	6002	NUM
cana-410	185	58	.	.	PUNCT
cana-410	186	1	[	[	X
cana-410	186	2	9	9	NUM
cana-410	186	3	]	]	X
cana-410	186	4	abhishek	abhishek	PROPN
cana-410	186	5	chowdhary	chowdhary	PROPN
cana-410	186	6	,	,	PUNCT
cana-410	186	7	anupam	anupam	PROPN
cana-410	186	8	kumar	kumar	PROPN
cana-410	186	9	,	,	PUNCT
cana-410	186	10	sanket	sanket	PROPN
cana-410	186	11	kumar	kumar	PROPN
cana-410	186	12	singh	singh	PROPN
cana-410	186	13	,	,	PUNCT
cana-410	186	14	design	design	NOUN
cana-410	186	15	and	and	CCONJ
cana-410	186	16	analysis	analysis	NOUN
cana-410	186	17	of	of	ADP
cana-410	186	18	an	an	DET
cana-410	186	19	electro	electro	NOUN
cana-410	186	20	-	-	PUNCT
cana-410	186	21	magnetic	magnetic	ADJ
cana-410	186	22	clutch	clutch	NOUN
cana-410	186	23	,	,	PUNCT
cana-410	186	24	international	international	ADJ
cana-410	186	25	journal	journal	NOUN
cana-410	186	26	of	of	ADP
cana-410	186	27	progressive	progressive	ADJ
cana-410	186	28	research	research	NOUN
cana-410	186	29	in	in	ADP
cana-410	186	30	science	science	NOUN
cana-410	186	31	and	and	CCONJ
cana-410	186	32	engineering	engineering	NOUN
cana-410	186	33	,	,	PUNCT
cana-410	186	34	june	june	PROPN
cana-410	186	35	(	(	PUNCT
cana-410	186	36	2020	2020	NUM
cana-410	186	37	)	)	PUNCT
cana-410	186	38	,	,	PUNCT
cana-410	186	39	1(3	1(3	NUM
cana-410	186	40	)	)	PUNCT
cana-410	186	41	,	,	PUNCT
cana-410	186	42	(	(	PUNCT
cana-410	186	43	89	89	NUM
cana-410	186	44	-	-	SYM
cana-410	186	45	95	95	NUM
cana-410	186	46	)	)	PUNCT
cana-410	186	47	.	.	PUNCT
cana-410	187	1	[	[	X
cana-410	187	2	10	10	NUM
cana-410	187	3	]	]	X
cana-410	187	4	manuel	manuel	PROPN
cana-410	187	5	tentarelli	tentarelli	PROPN
cana-410	187	6	,	,	PUNCT
cana-410	187	7	stefano	stefano	PROPN
cana-410	187	8	cantelli	cantelli	PROPN
cana-410	187	9	,	,	PUNCT
cana-410	187	10	silvio	silvio	NOUN
cana-410	187	11	sorrentino	sorrentino	PROPN
cana-410	187	12	,	,	PUNCT
cana-410	187	13	alessandro	alessandro	X
cana-410	187	14	de	de	X
cana-410	187	15	felice	felice	PROPN
cana-410	187	16	,	,	PUNCT
cana-410	187	17	a	a	DET
cana-410	187	18	new	new	ADJ
cana-410	187	19	approach	approach	NOUN
cana-410	187	20	to	to	ADP
cana-410	187	21	the	the	DET
cana-410	187	22	study	study	NOUN
cana-410	187	23	and	and	CCONJ
cana-410	187	24	prevention	prevention	NOUN
cana-410	187	25	of	of	ADP
cana-410	187	26	the	the	DET
cana-410	187	27	clutch	clutch	NOUN
cana-410	187	28	judder	judder	NOUN
cana-410	187	29	,	,	PUNCT
cana-410	187	30	the	the	DET
cana-410	187	31	american	american	ADJ
cana-410	187	32	society	society	NOUN
cana-410	187	33	of	of	ADP
cana-410	187	34	mechanical	mechanical	ADJ
cana-410	187	35	engineers	engineer	NOUN
cana-410	187	36	,	,	PUNCT
cana-410	187	37	february	february	PROPN
cana-410	187	38	(	(	PUNCT
cana-410	187	39	2023	2023	NUM
cana-410	187	40	)	)	PUNCT
cana-410	187	41	,	,	PUNCT
cana-410	187	42	1(3	1(3	NUM
cana-410	187	43	)	)	PUNCT
cana-410	187	44	.	.	PUNCT