id	sid	tid	token	lemma	pos
cana-1239	1	1	communications	communication	NOUN
cana-1239	1	2	on	on	ADP
cana-1239	1	3	applied	apply	VERB
cana-1239	1	4	nonlinear	nonlinear	ADJ
cana-1239	1	5	analysis	analysis	NOUN
cana-1239	1	6	issn	issn	NOUN
cana-1239	1	7	:	:	PUNCT
cana-1239	1	8	1074	1074	NUM
cana-1239	1	9	-	-	PUNCT
cana-1239	1	10	133x	133x	NUM
cana-1239	1	11	vol	vol	NOUN
cana-1239	1	12	31	31	NUM
cana-1239	1	13	no	no	NOUN
cana-1239	1	14	.	.	PUNCT
cana-1239	2	1	6s	6s	NUM
cana-1239	2	2	(	(	PUNCT
cana-1239	2	3	2024	2024	NUM
cana-1239	2	4	)	)	PUNCT
cana-1239	2	5	489	489	NUM
cana-1239	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	3	2	(	(	PUNCT
cana-1239	3	3	𝑺	𝑺	NOUN
cana-1239	3	4	,	,	PUNCT
cana-1239	3	5	𝒅)magic	𝒅)magic	ADJ
cana-1239	3	6	labeling	labeling	NOUN
cana-1239	3	7	of	of	ADP
cana-1239	3	8	some	some	DET
cana-1239	3	9	cycle	cycle	NOUN
cana-1239	3	10	related	relate	VERB
cana-1239	3	11	graphs	graph	NOUN
cana-1239	3	12	p.	p.	NOUN
cana-1239	3	13	sumathi	sumathi	ADJ
cana-1239	3	14	1	1	NUM
cana-1239	3	15	,	,	PUNCT
cana-1239	3	16	p.	p.	NOUN
cana-1239	3	17	mala2	mala2	NOUN
cana-1239	4	1	1department	1department	NUM
cana-1239	4	2	of	of	ADP
cana-1239	4	3	mathematics	mathematic	NOUN
cana-1239	4	4	,	,	PUNCT
cana-1239	4	5	c.	c.	PROPN
cana-1239	4	6	kandaswami	kandaswami	PROPN
cana-1239	4	7	college	college	PROPN
cana-1239	4	8	for	for	ADP
cana-1239	4	9	men	man	NOUN
cana-1239	4	10	,	,	PUNCT
cana-1239	4	11	anna	anna	PROPN
cana-1239	4	12	nagar	nagar	PROPN
cana-1239	4	13	,	,	PUNCT
cana-1239	4	14	chennai-102	chennai-102	NOUN
cana-1239	4	15	.	.	PUNCT
cana-1239	5	1	2department	2department	NUM
cana-1239	5	2	of	of	ADP
cana-1239	5	3	mathematics	mathematic	NOUN
cana-1239	5	4	,	,	PUNCT
cana-1239	5	5	st	st	PROPN
cana-1239	5	6	thomas	thomas	PROPN
cana-1239	5	7	college	college	PROPN
cana-1239	5	8	of	of	ADP
cana-1239	5	9	arts	art	NOUN
cana-1239	5	10	and	and	CCONJ
cana-1239	5	11	science	science	NOUN
cana-1239	5	12	,	,	PUNCT
cana-1239	5	13	koyambedu	koyambedu	PROPN
cana-1239	5	14	,	,	PUNCT
cana-1239	5	15	chennai-107	chennai-107	NOUN
cana-1239	5	16	.	.	PUNCT
cana-1239	6	1	1sumathipaul@gmail.com	1sumathipaul@gmail.com	NUM
cana-1239	6	2	,	,	PUNCT
cana-1239	6	3	2mala.vedha@gmail.com	2mala.vedha@gmail.com	NUM
cana-1239	6	4	article	article	NOUN
cana-1239	6	5	history	history	NOUN
cana-1239	6	6	:	:	PUNCT
cana-1239	6	7	received	receive	VERB
cana-1239	6	8	:	:	PUNCT
cana-1239	6	9	20	20	NUM
cana-1239	6	10	-	-	SYM
cana-1239	6	11	06	06	NUM
cana-1239	6	12	-	-	PUNCT
cana-1239	6	13	2024	2024	NUM
cana-1239	6	14	revised	revise	VERB
cana-1239	6	15	:	:	PUNCT
cana-1239	6	16	21	21	NUM
cana-1239	6	17	-	-	SYM
cana-1239	6	18	07	07	NUM
cana-1239	6	19	-	-	PUNCT
cana-1239	6	20	2024	2024	NUM
cana-1239	6	21	accepted	accept	VERB
cana-1239	6	22	:	:	PUNCT
cana-1239	6	23	09	09	NUM
cana-1239	6	24	-	-	SYM
cana-1239	6	25	08	08	NUM
cana-1239	6	26	-	-	PUNCT
cana-1239	6	27	2024	2024	NUM
cana-1239	6	28	abstract	abstract	NOUN
cana-1239	6	29	:	:	PUNCT
cana-1239	6	30	objective	objective	NOUN
cana-1239	6	31	:	:	PUNCT
cana-1239	6	32	to	to	PART
cana-1239	6	33	examine	examine	VERB
cana-1239	6	34	the	the	DET
cana-1239	6	35	existence	existence	NOUN
cana-1239	6	36	of	of	ADP
cana-1239	6	37	(	(	PUNCT
cana-1239	6	38	s	s	X
cana-1239	6	39	,	,	PUNCT
cana-1239	6	40	d	d	NOUN
cana-1239	6	41	)	)	PUNCT
cana-1239	6	42	magic	magic	ADJ
cana-1239	6	43	labeling	labeling	NOUN
cana-1239	6	44	on	on	ADP
cana-1239	6	45	cycle	cycle	NOUN
cana-1239	6	46	related	relate	VERB
cana-1239	6	47	graphs	graph	NOUN
cana-1239	6	48	.	.	PUNCT
cana-1239	7	1	methods	method	NOUN
cana-1239	7	2	:	:	PUNCT
cana-1239	7	3	let	let	VERB
cana-1239	7	4	g	g	PROPN
cana-1239	7	5	(	(	PUNCT
cana-1239	7	6	p	p	X
cana-1239	7	7	,	,	PUNCT
cana-1239	7	8	q	q	NOUN
cana-1239	7	9	)	)	PUNCT
cana-1239	7	10	be	be	AUX
cana-1239	7	11	a	a	DET
cana-1239	7	12	simple	simple	ADJ
cana-1239	7	13	,	,	PUNCT
cana-1239	7	14	non	non	ADJ
cana-1239	7	15	-	-	ADJ
cana-1239	7	16	trivial	trivial	ADJ
cana-1239	7	17	,	,	PUNCT
cana-1239	7	18	connected	connect	VERB
cana-1239	7	19	,	,	PUNCT
cana-1239	7	20	undirected	undirected	ADJ
cana-1239	7	21	graph	graph	NOUN
cana-1239	7	22	with	with	ADP
cana-1239	7	23	p	p	NOUN
cana-1239	7	24	vertices	vertex	NOUN
cana-1239	7	25	and	and	CCONJ
cana-1239	7	26	q	q	NOUN
cana-1239	7	27	edges	edge	NOUN
cana-1239	7	28	.	.	PUNCT
cana-1239	8	1	let	let	VERB
cana-1239	8	2	f	f	PRON
cana-1239	8	3	:	:	PUNCT
cana-1239	8	4	v(g)→{s	v(g)→{s	ADJ
cana-1239	8	5	,	,	PUNCT
cana-1239	8	6	s+d	s+d	ADJ
cana-1239	8	7	,	,	PUNCT
cana-1239	8	8	s+2d,	s+2d,	NOUN
cana-1239	8	9	..	..	PUNCT
cana-1239	8	10	s+(q+1)d	s+(q+1)d	PROPN
cana-1239	8	11	}	}	PUNCT
cana-1239	8	12	and	and	CCONJ
cana-1239	8	13	g	g	NOUN
cana-1239	8	14	:	:	PUNCT
cana-1239	8	15	e(g)→{d,2d,3d	e(g)→{d,2d,3d	PROPN
cana-1239	8	16	…	…	SYM
cana-1239	8	17	2(q-1)d	2(q-1)d	NUM
cana-1239	8	18	}	}	PUNCT
cana-1239	8	19	be	be	AUX
cana-1239	8	20	an	an	DET
cana-1239	8	21	injective	injective	ADJ
cana-1239	8	22	function	function	NOUN
cana-1239	8	23	.	.	PUNCT
cana-1239	9	1	then	then	ADV
cana-1239	9	2	,	,	PUNCT
cana-1239	9	3	for	for	ADP
cana-1239	9	4	any	any	DET
cana-1239	9	5	u	u	NOUN
cana-1239	9	6	,	,	PUNCT
cana-1239	9	7	v∈v(g	v∈v(g	NOUN
cana-1239	9	8	)	)	PUNCT
cana-1239	9	9	and	and	CCONJ
cana-1239	9	10	uv∈e(g),f(u)+g(uv)+f(v	uv∈e(g),f(u)+g(uv)+f(v	ADJ
cana-1239	9	11	)	)	PUNCT
cana-1239	9	12	is	be	AUX
cana-1239	9	13	a	a	DET
cana-1239	9	14	constant	constant	ADJ
cana-1239	9	15	,	,	PUNCT
cana-1239	9	16	and	and	CCONJ
cana-1239	9	17	the	the	DET
cana-1239	9	18	function	function	NOUN
cana-1239	9	19	f	f	PROPN
cana-1239	9	20	is	be	AUX
cana-1239	9	21	said	say	VERB
cana-1239	9	22	to	to	PART
cana-1239	9	23	be	be	AUX
cana-1239	9	24	(	(	PUNCT
cana-1239	9	25	s	s	X
cana-1239	9	26	,	,	PUNCT
cana-1239	9	27	d	d	NOUN
cana-1239	9	28	)	)	PUNCT
cana-1239	9	29	magic	magic	ADJ
cana-1239	9	30	labeling	labeling	NOUN
cana-1239	9	31	.	.	PUNCT
cana-1239	10	1	if	if	SCONJ
cana-1239	10	2	a	a	DET
cana-1239	10	3	graph	graph	NOUN
cana-1239	10	4	g	g	PROPN
cana-1239	10	5	admits	admit	VERB
cana-1239	10	6	(	(	PUNCT
cana-1239	10	7	s	s	X
cana-1239	10	8	,	,	PUNCT
cana-1239	10	9	d	d	NOUN
cana-1239	10	10	)	)	PUNCT
cana-1239	10	11	magic	magic	ADJ
cana-1239	10	12	labeling	labeling	NOUN
cana-1239	10	13	,	,	PUNCT
cana-1239	10	14	then	then	ADV
cana-1239	10	15	it	it	PRON
cana-1239	10	16	is	be	AUX
cana-1239	10	17	referred	refer	VERB
cana-1239	10	18	to	to	ADP
cana-1239	10	19	as	as	ADP
cana-1239	10	20	a	a	DET
cana-1239	10	21	(	(	PUNCT
cana-1239	10	22	s	s	NOUN
cana-1239	10	23	,	,	PUNCT
cana-1239	10	24	d	d	NOUN
cana-1239	10	25	)	)	PUNCT
cana-1239	10	26	magic	magic	ADJ
cana-1239	10	27	graph	graph	NOUN
cana-1239	10	28	.	.	PUNCT
cana-1239	11	1	findings	finding	NOUN
cana-1239	11	2	:	:	PUNCT
cana-1239	11	3	in	in	ADP
cana-1239	11	4	this	this	DET
cana-1239	11	5	paper	paper	NOUN
cana-1239	11	6	the	the	DET
cana-1239	11	7	existence	existence	NOUN
cana-1239	11	8	of	of	ADP
cana-1239	11	9	(	(	PUNCT
cana-1239	11	10	s	s	X
cana-1239	11	11	,	,	PUNCT
cana-1239	11	12	d	d	NOUN
cana-1239	11	13	)	)	PUNCT
cana-1239	11	14	magic	magic	ADJ
cana-1239	11	15	labeling	labeling	NOUN
cana-1239	11	16	in	in	ADP
cana-1239	11	17	some	some	DET
cana-1239	11	18	cycle	cycle	NOUN
cana-1239	11	19	related	relate	VERB
cana-1239	11	20	graphs	graph	NOUN
cana-1239	11	21	such	such	ADJ
cana-1239	11	22	as	as	ADP
cana-1239	11	23	a	a	DET
cana-1239	11	24	cycle	cycle	NOUN
cana-1239	11	25	graph	graph	NOUN
cana-1239	11	26	c_(n⊙k_(1,m	c_(n⊙k_(1,m	PROPN
cana-1239	11	27	)	)	PUNCT
cana-1239	11	28	)	)	PUNCT
cana-1239	11	29	graph	graph	NOUN
cana-1239	11	30	,	,	PUNCT
cana-1239	11	31	n	n	CCONJ
cana-1239	11	32	-sunlet	-sunlet	NOUN
cana-1239	11	33	graph	graph	NOUN
cana-1239	11	34	,	,	PUNCT
cana-1239	11	35	friendship	friendship	NOUN
cana-1239	11	36	graph	graph	NOUN
cana-1239	11	37	flower	flower	NOUN
cana-1239	11	38	graph	graph	NOUN
cana-1239	11	39	and	and	CCONJ
cana-1239	11	40	wheel	wheel	NOUN
cana-1239	11	41	graph	graph	NOUN
cana-1239	11	42	were	be	AUX
cana-1239	11	43	found	find	VERB
cana-1239	11	44	.	.	PUNCT
cana-1239	12	1	novelty	novelty	NOUN
cana-1239	12	2	:	:	PUNCT
cana-1239	12	3	the	the	DET
cana-1239	12	4	labeling	labeling	NOUN
cana-1239	12	5	of	of	ADP
cana-1239	12	6	the	the	DET
cana-1239	12	7	vertices	vertex	NOUN
cana-1239	12	8	and	and	CCONJ
cana-1239	12	9	edges	edge	NOUN
cana-1239	12	10	is	be	AUX
cana-1239	12	11	done	do	VERB
cana-1239	12	12	mathematically	mathematically	ADV
cana-1239	12	13	,	,	PUNCT
cana-1239	12	14	and	and	CCONJ
cana-1239	12	15	this	this	PRON
cana-1239	12	16	leads	lead	VERB
cana-1239	12	17	to	to	ADP
cana-1239	12	18	the	the	DET
cana-1239	12	19	creation	creation	NOUN
cana-1239	12	20	of	of	ADP
cana-1239	12	21	a	a	DET
cana-1239	12	22	new	new	ADJ
cana-1239	12	23	labeling	labeling	NOUN
cana-1239	12	24	known	know	VERB
cana-1239	12	25	as	as	ADP
cana-1239	12	26	(	(	PUNCT
cana-1239	12	27	s	s	X
cana-1239	12	28	,	,	PUNCT
cana-1239	12	29	d	d	NOUN
cana-1239	12	30	)	)	PUNCT
cana-1239	12	31	magic	magic	ADJ
cana-1239	12	32	labeling	labeling	NOUN
cana-1239	12	33	.	.	PUNCT
cana-1239	13	1	keywords	keyword	NOUN
cana-1239	13	2	:	:	PUNCT
cana-1239	13	3	cycle	cycle	NOUN
cana-1239	13	4	graph	graph	NOUN
cana-1239	13	5	,	,	PUNCT
cana-1239	13	6	c_(n⊙k_(1,m	c_(n⊙k_(1,m	PROPN
cana-1239	13	7	)	)	PUNCT
cana-1239	13	8	)	)	PUNCT
cana-1239	13	9	graph	graph	NOUN
cana-1239	13	10	,	,	PUNCT
cana-1239	13	11	n	n	CCONJ
cana-1239	13	12	-	-	PUNCT
cana-1239	13	13	sunlet	sunlet	NOUN
cana-1239	13	14	graph	graph	NOUN
cana-1239	13	15	,	,	PUNCT
cana-1239	13	16	friendship	friendship	NOUN
cana-1239	13	17	graph	graph	NOUN
cana-1239	13	18	,	,	PUNCT
cana-1239	13	19	flower	flower	NOUN
cana-1239	13	20	graph	graph	NOUN
cana-1239	13	21	and	and	CCONJ
cana-1239	13	22	wheel	wheel	NOUN
cana-1239	13	23	graph	graph	NOUN
cana-1239	13	24	.	.	PUNCT
cana-1239	14	1	1	1	X
cana-1239	14	2	.	.	X
cana-1239	14	3	introduction	introduction	NOUN
cana-1239	15	1	[	[	X
cana-1239	15	2	1,3	1,3	NUM
cana-1239	15	3	]	]	X
cana-1239	15	4	the	the	DET
cana-1239	15	5	graphs	graph	NOUN
cana-1239	15	6	that	that	PRON
cana-1239	15	7	are	be	AUX
cana-1239	15	8	being	be	AUX
cana-1239	15	9	studied	study	VERB
cana-1239	15	10	are	be	AUX
cana-1239	15	11	simple	simple	ADJ
cana-1239	15	12	,	,	PUNCT
cana-1239	15	13	undirected	undirected	ADJ
cana-1239	15	14	,	,	PUNCT
cana-1239	15	15	finite	finite	ADJ
cana-1239	15	16	,	,	PUNCT
cana-1239	15	17	and	and	CCONJ
cana-1239	15	18	non	non	ADJ
cana-1239	15	19	-	-	ADJ
cana-1239	15	20	trivial	trivial	ADJ
cana-1239	15	21	.	.	PUNCT
cana-1239	16	1	graph	graph	NOUN
cana-1239	16	2	labeling	labeling	NOUN
cana-1239	16	3	has	have	AUX
cana-1239	16	4	advanced	advance	VERB
cana-1239	16	5	significantly	significantly	ADV
cana-1239	16	6	according	accord	VERB
cana-1239	16	7	to	to	ADP
cana-1239	16	8	graph	graph	NOUN
cana-1239	16	9	theory	theory	NOUN
cana-1239	16	10	,	,	PUNCT
cana-1239	16	11	which	which	PRON
cana-1239	16	12	has	have	VERB
cana-1239	16	13	a	a	DET
cana-1239	16	14	wide	wide	ADJ
cana-1239	16	15	range	range	NOUN
cana-1239	16	16	of	of	ADP
cana-1239	16	17	uses	use	NOUN
cana-1239	16	18	.	.	PUNCT
cana-1239	17	1	in	in	ADP
cana-1239	17	2	1963	1963	NUM
cana-1239	17	3	,	,	PUNCT
cana-1239	17	4	sedl´ȧcek	sedl´ȧcek	NUM
cana-1239	17	5	developed	develop	VERB
cana-1239	17	6	the	the	DET
cana-1239	17	7	first	first	ADJ
cana-1239	17	8	magic	magic	ADJ
cana-1239	17	9	-	-	PUNCT
cana-1239	17	10	type	type	NOUN
cana-1239	17	11	labeling	labeling	NOUN
cana-1239	17	12	.	.	PUNCT
cana-1239	18	1	he	he	PRON
cana-1239	18	2	gave	give	VERB
cana-1239	18	3	a	a	DET
cana-1239	18	4	graph	graph	NOUN
cana-1239	18	5	's	's	PART
cana-1239	18	6	edges	edge	NOUN
cana-1239	18	7	real	real	ADJ
cana-1239	18	8	numbers	number	NOUN
cana-1239	18	9	and	and	CCONJ
cana-1239	18	10	mandated	mandate	VERB
cana-1239	18	11	that	that	SCONJ
cana-1239	18	12	the	the	DET
cana-1239	18	13	total	total	NOUN
cana-1239	18	14	of	of	ADP
cana-1239	18	15	all	all	DET
cana-1239	18	16	the	the	DET
cana-1239	18	17	labels	label	NOUN
cana-1239	18	18	on	on	ADP
cana-1239	18	19	all	all	DET
cana-1239	18	20	the	the	DET
cana-1239	18	21	edges	edge	NOUN
cana-1239	18	22	that	that	PRON
cana-1239	18	23	are	be	AUX
cana-1239	18	24	incident	incident	NOUN
cana-1239	18	25	to	to	ADP
cana-1239	18	26	a	a	DET
cana-1239	18	27	vertex	vertex	NOUN
cana-1239	18	28	must	must	AUX
cana-1239	18	29	remain	remain	VERB
cana-1239	18	30	constant	constant	ADJ
cana-1239	18	31	.	.	PUNCT
cana-1239	19	1	[	[	X
cana-1239	19	2	5	5	NUM
cana-1239	19	3	]	]	X
cana-1239	19	4	hegde	hegde	PROPN
cana-1239	19	5	sm	sm	PROPN
cana-1239	19	6	,	,	PUNCT
cana-1239	19	7	shetty	shetty	PROPN
cana-1239	19	8	s.	s.	PROPN
cana-1239	19	9	have	have	AUX
cana-1239	19	10	introduced	introduce	VERB
cana-1239	19	11	(	(	PUNCT
cana-1239	19	12	𝑘	𝑘	X
cana-1239	19	13	,	,	PUNCT
cana-1239	19	14	𝑑	𝑑	NOUN
cana-1239	19	15	)	)	PUNCT
cana-1239	19	16	arithmetic	arithmetic	ADJ
cana-1239	19	17	labeling	labeling	NOUN
cana-1239	19	18	of	of	ADP
cana-1239	19	19	graphs	graph	NOUN
cana-1239	19	20	.	.	PUNCT
cana-1239	20	1	we	we	PRON
cana-1239	20	2	have	have	AUX
cana-1239	20	3	showed	show	VERB
cana-1239	20	4	(	(	PUNCT
cana-1239	20	5	𝑠	𝑠	INTJ
cana-1239	20	6	,	,	PUNCT
cana-1239	20	7	𝑑).magic	𝑑).magic	ADJ
cana-1239	20	8	labeling	labeling	NOUN
cana-1239	20	9	for	for	ADP
cana-1239	20	10	the	the	DET
cana-1239	20	11	path	path	NOUN
cana-1239	20	12	,	,	PUNCT
cana-1239	20	13	star	star	NOUN
cana-1239	20	14	,	,	PUNCT
cana-1239	20	15	and	and	CCONJ
cana-1239	20	16	a	a	DET
cana-1239	20	17	few	few	ADJ
cana-1239	20	18	standard	standard	ADJ
cana-1239	20	19	graphs	graph	NOUN
cana-1239	20	20	.	.	PUNCT
cana-1239	21	1	this	this	DET
cana-1239	21	2	work	work	NOUN
cana-1239	21	3	established	establish	VERB
cana-1239	21	4	the	the	DET
cana-1239	21	5	existence	existence	NOUN
cana-1239	21	6	of	of	ADP
cana-1239	21	7	(	(	PUNCT
cana-1239	21	8	𝑠	𝑠	PROPN
cana-1239	21	9	,	,	PUNCT
cana-1239	21	10	𝑑	𝑑	NOUN
cana-1239	21	11	)	)	PUNCT
cana-1239	21	12	magic	magic	ADJ
cana-1239	21	13	labeling	labeling	NOUN
cana-1239	21	14	in	in	ADP
cana-1239	21	15	certain	certain	ADJ
cana-1239	21	16	cycle	cycle	NOUN
cana-1239	21	17	-	-	PUNCT
cana-1239	21	18	related	relate	VERB
cana-1239	21	19	graphs	graph	NOUN
cana-1239	21	20	.	.	PUNCT
cana-1239	22	1	[	[	X
cana-1239	22	2	10	10	NUM
cana-1239	22	3	]	]	PUNCT
cana-1239	22	4	let	let	VERB
cana-1239	22	5	𝐺	𝐺	PROPN
cana-1239	22	6	(	(	PUNCT
cana-1239	22	7	𝑝	𝑝	PROPN
cana-1239	22	8	,	,	PUNCT
cana-1239	22	9	𝑞	𝑞	NOUN
cana-1239	22	10	)	)	PUNCT
cana-1239	22	11	be	be	VERB
cana-1239	22	12	a	a	DET
cana-1239	22	13	simple	simple	ADJ
cana-1239	22	14	,	,	PUNCT
cana-1239	22	15	non	non	ADJ
cana-1239	22	16	-	-	ADJ
cana-1239	22	17	trivial	trivial	ADJ
cana-1239	22	18	,	,	PUNCT
cana-1239	22	19	connected	connect	VERB
cana-1239	22	20	,	,	PUNCT
cana-1239	22	21	undirected	undirected	ADJ
cana-1239	22	22	graph	graph	NOUN
cana-1239	22	23	with	with	ADP
cana-1239	22	24	p	p	NOUN
cana-1239	22	25	vertices	vertex	NOUN
cana-1239	22	26	and	and	CCONJ
cana-1239	22	27	q	q	NOUN
cana-1239	22	28	edges	edge	NOUN
cana-1239	22	29	.	.	PUNCT
cana-1239	23	1	consider	consider	VERB
cana-1239	23	2	the	the	DET
cana-1239	23	3	following	follow	VERB
cana-1239	23	4	:	:	PUNCT
cana-1239	23	5	𝑓	𝑓	X
cana-1239	23	6	:	:	PUNCT
cana-1239	23	7	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1239	23	8	)	)	PUNCT
cana-1239	23	9	→	→	SYM
cana-1239	23	10	{	{	PUNCT
cana-1239	23	11	𝑠	𝑠	PROPN
cana-1239	23	12	,	,	PUNCT
cana-1239	23	13	𝑠	𝑠	PROPN
cana-1239	23	14	+	+	PROPN
cana-1239	23	15	𝑑	𝑑	PROPN
cana-1239	23	16	,	,	PUNCT
cana-1239	23	17	𝑠	𝑠	PROPN
cana-1239	23	18	+	+	NOUN
cana-1239	23	19	2𝑑	2𝑑	NUM
cana-1239	23	20	,	,	PUNCT
cana-1239	23	21	.	.	PUNCT
cana-1239	23	22	.	.	PUNCT
cana-1239	24	1	𝑠	𝑠	PROPN
cana-1239	25	1	+	+	CCONJ
cana-1239	25	2	(	(	PUNCT
cana-1239	25	3	𝑞	𝑞	X
cana-1239	25	4	+	+	NOUN
cana-1239	25	5	1)𝑑	1)𝑑	NUM
cana-1239	25	6	}	}	PUNCT
cana-1239	25	7	and	and	CCONJ
cana-1239	25	8	𝑔	𝑔	ADJ
cana-1239	25	9	:	:	PUNCT
cana-1239	25	10	𝐸(𝐺	𝐸(𝐺	NUM
cana-1239	25	11	)	)	PUNCT
cana-1239	25	12	→	→	SYM
cana-1239	25	13	{	{	PUNCT
cana-1239	25	14	𝑑	𝑑	NOUN
cana-1239	25	15	,	,	PUNCT
cana-1239	25	16	2𝑑	2𝑑	NUM
cana-1239	25	17	,	,	PUNCT
cana-1239	25	18	3𝑑	3𝑑	NUM
cana-1239	25	19	…	…	PUNCT
cana-1239	25	20	2(𝑞	2(𝑞	NUM
cana-1239	25	21	−	−	NOUN
cana-1239	25	22	1)𝑑	1)𝑑	NUM
cana-1239	25	23	}	}	PUNCT
cana-1239	25	24	be	be	AUX
cana-1239	25	25	an	an	DET
cana-1239	25	26	injective	injective	ADJ
cana-1239	25	27	function	function	NOUN
cana-1239	25	28	.	.	PUNCT
cana-1239	26	1	then	then	ADV
cana-1239	26	2	,	,	PUNCT
cana-1239	26	3	for	for	ADP
cana-1239	26	4	any	any	DET
cana-1239	26	5	𝑢	𝑢	NOUN
cana-1239	26	6	,	,	PUNCT
cana-1239	26	7	𝑣	𝑣	PRON
cana-1239	26	8	∈	∈	PROPN
cana-1239	26	9	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1239	26	10	)	)	PUNCT
cana-1239	26	11	and	and	CCONJ
cana-1239	26	12	𝑢𝑣	𝑢𝑣	PROPN
cana-1239	26	13	∈	∈	PROPN
cana-1239	26	14	𝐸(𝐺	𝐸(𝐺	NOUN
cana-1239	26	15	)	)	PUNCT
cana-1239	26	16	,	,	PUNCT
cana-1239	26	17	𝑓(𝑢	𝑓(𝑢	PROPN
cana-1239	26	18	)	)	PUNCT
cana-1239	26	19	+	+	NUM
cana-1239	26	20	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-1239	26	21	)	)	PUNCT
cana-1239	27	1	+	+	CCONJ
cana-1239	27	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1239	27	3	)	)	PUNCT
cana-1239	27	4	is	be	AUX
cana-1239	27	5	a	a	DET
cana-1239	27	6	constant	constant	ADJ
cana-1239	27	7	,	,	PUNCT
cana-1239	27	8	and	and	CCONJ
cana-1239	27	9	the	the	DET
cana-1239	27	10	function	function	NOUN
cana-1239	27	11	f	f	PROPN
cana-1239	27	12	is	be	AUX
cana-1239	27	13	said	say	VERB
cana-1239	27	14	to	to	PART
cana-1239	27	15	be	be	AUX
cana-1239	27	16	(	(	PUNCT
cana-1239	27	17	𝑠	𝑠	PROPN
cana-1239	27	18	,	,	PUNCT
cana-1239	27	19	𝑑	𝑑	NOUN
cana-1239	27	20	)	)	PUNCT
cana-1239	27	21	magic	magic	ADJ
cana-1239	27	22	labeling	labeling	NOUN
cana-1239	27	23	.	.	PUNCT
cana-1239	28	1	if	if	SCONJ
cana-1239	28	2	a	a	DET
cana-1239	28	3	graph	graph	NOUN
cana-1239	28	4	g	g	PROPN
cana-1239	28	5	admits	admit	VERB
cana-1239	28	6	(	(	PUNCT
cana-1239	28	7	𝑠	𝑠	PROPN
cana-1239	28	8	,	,	PUNCT
cana-1239	28	9	𝑑	𝑑	NOUN
cana-1239	28	10	)	)	PUNCT
cana-1239	28	11	magic	magic	ADJ
cana-1239	28	12	labeling	labeling	NOUN
cana-1239	28	13	,	,	PUNCT
cana-1239	28	14	then	then	ADV
cana-1239	28	15	it	it	PRON
cana-1239	28	16	is	be	AUX
cana-1239	28	17	referred	refer	VERB
cana-1239	28	18	to	to	ADP
cana-1239	28	19	as	as	ADP
cana-1239	28	20	a	a	DET
cana-1239	28	21	(	(	PUNCT
cana-1239	28	22	𝑠	𝑠	PROPN
cana-1239	28	23	,	,	PUNCT
cana-1239	28	24	𝑑	𝑑	NOUN
cana-1239	28	25	)	)	PUNCT
cana-1239	28	26	magic	magic	ADJ
cana-1239	28	27	graph	graph	NOUN
cana-1239	28	28	.	.	PUNCT
cana-1239	29	1	several	several	ADJ
cana-1239	29	2	studies	study	NOUN
cana-1239	29	3	contribute	contribute	VERB
cana-1239	29	4	to	to	ADP
cana-1239	29	5	graph	graph	NOUN
cana-1239	29	6	theory	theory	NOUN
cana-1239	29	7	:	:	PUNCT
cana-1239	29	8	[	[	X
cana-1239	29	9	2	2	X
cana-1239	29	10	]	]	PUNCT
cana-1239	29	11	farida	farida	NOUN
cana-1239	29	12	et	et	PROPN
cana-1239	29	13	al	al	PROPN
cana-1239	29	14	.	.	PROPN
cana-1239	29	15	explore	explore	VERB
cana-1239	29	16	graph	graph	NOUN
cana-1239	29	17	labeling	labeling	NOUN
cana-1239	29	18	,	,	PUNCT
cana-1239	29	19	emphasizing	emphasize	VERB
cana-1239	29	20	magic	magic	ADJ
cana-1239	29	21	covering	covering	NOUN
cana-1239	29	22	and	and	CCONJ
cana-1239	29	23	edge	edge	VERB
cana-1239	29	24	super	super	ADJ
cana-1239	29	25	magic	magic	ADJ
cana-1239	29	26	labeling	labeling	NOUN
cana-1239	29	27	,	,	PUNCT
cana-1239	29	28	revealing	reveal	VERB
cana-1239	29	29	practical	practical	ADJ
cana-1239	29	30	applications	application	NOUN
cana-1239	29	31	in	in	ADP
cana-1239	29	32	secret	secret	ADJ
cana-1239	29	33	sharing	sharing	NOUN
cana-1239	29	34	and	and	CCONJ
cana-1239	29	35	ruler	ruler	NOUN
cana-1239	29	36	models	model	NOUN
cana-1239	29	37	.	.	PUNCT
cana-1239	30	1	[	[	X
cana-1239	30	2	4,6	4,6	X
cana-1239	30	3	]	]	X
cana-1239	30	4	ghodasara	ghodasara	PROPN
cana-1239	30	5	g.v	g.v	PROPN
cana-1239	30	6	.	.	PROPN
cana-1239	30	7	et	et	PROPN
cana-1239	30	8	al	al	PROPN
cana-1239	30	9	.	.	PROPN
cana-1239	30	10	and	and	CCONJ
cana-1239	30	11	baskar	baskar	NOUN
cana-1239	30	12	babujee	babujee	NOUN
cana-1239	30	13	j.et.al	j.et.al	PROPN
cana-1239	30	14	.	.	PUNCT
cana-1239	31	1	introduce	introduce	VERB
cana-1239	31	2	prime	prime	ADJ
cana-1239	31	3	cordial	cordial	ADJ
cana-1239	31	4	labeling	labeling	NOUN
cana-1239	31	5	,	,	PUNCT
cana-1239	31	6	communications	communication	NOUN
cana-1239	31	7	on	on	ADP
cana-1239	31	8	applied	apply	VERB
cana-1239	31	9	nonlinear	nonlinear	ADJ
cana-1239	31	10	analysis	analysis	NOUN
cana-1239	31	11	issn	issn	NOUN
cana-1239	31	12	:	:	PUNCT
cana-1239	31	13	1074	1074	NUM
cana-1239	31	14	-	-	PUNCT
cana-1239	31	15	133x	133x	NUM
cana-1239	31	16	vol	vol	NOUN
cana-1239	31	17	31	31	NUM
cana-1239	31	18	no	no	NOUN
cana-1239	31	19	.	.	PUNCT
cana-1239	32	1	6s	6s	NUM
cana-1239	32	2	(	(	PUNCT
cana-1239	32	3	2024	2024	NUM
cana-1239	32	4	)	)	PUNCT
cana-1239	32	5	490	490	NUM
cana-1239	32	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	32	7	presenting	present	VERB
cana-1239	32	8	innovative	innovative	ADJ
cana-1239	32	9	constructions	construction	NOUN
cana-1239	32	10	and	and	CCONJ
cana-1239	32	11	advancements	advancement	NOUN
cana-1239	32	12	in	in	ADP
cana-1239	32	13	the	the	DET
cana-1239	32	14	field	field	NOUN
cana-1239	32	15	.	.	PUNCT
cana-1239	33	1	[	[	X
cana-1239	33	2	7	7	X
cana-1239	33	3	]	]	PUNCT
cana-1239	33	4	k.	k.	PROPN
cana-1239	33	5	kavitha	kavitha	PROPN
cana-1239	33	6	et	et	PROPN
cana-1239	34	1	al	al	PROPN
cana-1239	34	2	.	.	PROPN
cana-1239	34	3	define	define	PROPN
cana-1239	34	4	group	group	NOUN
cana-1239	34	5	magic	magic	ADJ
cana-1239	34	6	labeling	labeling	NOUN
cana-1239	34	7	for	for	ADP
cana-1239	34	8	connected	connected	ADJ
cana-1239	34	9	graphs	graph	NOUN
cana-1239	34	10	,	,	PUNCT
cana-1239	34	11	focusing	focus	VERB
cana-1239	34	12	on	on	ADP
cana-1239	34	13	specific	specific	ADJ
cana-1239	34	14	structures	structure	NOUN
cana-1239	34	15	like	like	ADP
cana-1239	34	16	cycles	cycle	NOUN
cana-1239	34	17	with	with	ADP
cana-1239	34	18	a	a	DET
cana-1239	34	19	common	common	ADJ
cana-1239	34	20	vertex	vertex	NOUN
cana-1239	34	21	and	and	CCONJ
cana-1239	34	22	chains	chain	NOUN
cana-1239	34	23	of	of	ADP
cana-1239	34	24	even	even	ADV
cana-1239	34	25	cycles	cycle	NOUN
cana-1239	34	26	.	.	PUNCT
cana-1239	35	1	[	[	X
cana-1239	35	2	8	8	X
cana-1239	35	3	]	]	PUNCT
cana-1239	35	4	s.	s.	PROPN
cana-1239	35	5	k.	k.	PROPN
cana-1239	35	6	vaidya	vaidya	PROPN
cana-1239	35	7	et	et	PROPN
cana-1239	35	8	al	al	PROPN
cana-1239	35	9	.	.	PROPN
cana-1239	35	10	introduce	introduce	VERB
cana-1239	35	11	prime	prime	ADJ
cana-1239	35	12	cordial	cordial	ADJ
cana-1239	35	13	labeling	labeling	NOUN
cana-1239	35	14	based	base	VERB
cana-1239	35	15	on	on	ADP
cana-1239	35	16	gcd	gcd	NOUN
cana-1239	35	17	properties	property	NOUN
cana-1239	35	18	,	,	PUNCT
cana-1239	35	19	demonstrating	demonstrate	VERB
cana-1239	35	20	its	its	PRON
cana-1239	35	21	applicability	applicability	NOUN
cana-1239	35	22	to	to	ADP
cana-1239	35	23	various	various	ADJ
cana-1239	35	24	graph	graph	NOUN
cana-1239	35	25	structures	structure	NOUN
cana-1239	35	26	such	such	ADJ
cana-1239	35	27	as	as	ADP
cana-1239	35	28	gear	gear	NOUN
cana-1239	35	29	graphs	graph	NOUN
cana-1239	35	30	,	,	PUNCT
cana-1239	35	31	helms	helm	NOUN
cana-1239	35	32	,	,	PUNCT
cana-1239	35	33	closed	closed	ADJ
cana-1239	35	34	helms	helm	NOUN
cana-1239	35	35	,	,	PUNCT
cana-1239	35	36	and	and	CCONJ
cana-1239	35	37	flower	flower	NOUN
cana-1239	35	38	graphs	graph	NOUN
cana-1239	35	39	.	.	PUNCT
cana-1239	36	1	2	2	X
cana-1239	36	2	.	.	X
cana-1239	36	3	methodology	methodology	NOUN
cana-1239	36	4	definition	definition	NOUN
cana-1239	36	5	2.1	2.1	NUM
cana-1239	36	6	:	:	PUNCT
cana-1239	36	7	cycle	cycle	NOUN
cana-1239	36	8	graph	graph	NOUN
cana-1239	36	9	is	be	AUX
cana-1239	36	10	a	a	DET
cana-1239	36	11	graph	graph	NOUN
cana-1239	36	12	of	of	ADP
cana-1239	36	13	n	n	NOUN
cana-1239	36	14	vertices	vertex	NOUN
cana-1239	36	15	with	with	ADP
cana-1239	36	16	exactly	exactly	ADV
cana-1239	36	17	n	n	PRON
cana-1239	36	18	edges	edge	NOUN
cana-1239	36	19	.	.	PUNCT
cana-1239	37	1	cycle	cycle	NOUN
cana-1239	37	2	graph	graph	NOUN
cana-1239	37	3	contains	contain	VERB
cana-1239	37	4	cycle	cycle	NOUN
cana-1239	37	5	of	of	ADP
cana-1239	37	6	length	length	NOUN
cana-1239	37	7	n	n	PRON
cana-1239	37	8	definition	definition	NOUN
cana-1239	37	9	2.2	2.2	NUM
cana-1239	37	10	:	:	PUNCT
cana-1239	38	1	[	[	X
cana-1239	38	2	9	9	X
cana-1239	38	3	]	]	X
cana-1239	38	4	the	the	DET
cana-1239	38	5	graph	graph	NOUN
cana-1239	38	6	𝐶𝑛	𝐶𝑛	PROPN
cana-1239	38	7	⊙𝐾1,𝑚	⊙𝐾1,𝑚	PROPN
cana-1239	38	8	is	be	AUX
cana-1239	38	9	obtained	obtain	VERB
cana-1239	38	10	by	by	ADP
cana-1239	38	11	attaching	attach	VERB
cana-1239	38	12	m	m	PROPN
cana-1239	38	13	leaves	leave	NOUN
cana-1239	38	14	to	to	ADP
cana-1239	38	15	each	each	DET
cana-1239	38	16	vertex	vertex	NOUN
cana-1239	38	17	of	of	ADP
cana-1239	38	18	the	the	DET
cana-1239	38	19	cycle	cycle	NOUN
cana-1239	38	20	𝐶𝑛	𝐶𝑛	PROPN
cana-1239	38	21	definition	definition	NOUN
cana-1239	38	22	2.3	2.3	NUM
cana-1239	38	23	:	:	PUNCT
cana-1239	38	24	the	the	DET
cana-1239	38	25	sunlet	sunlet	NOUN
cana-1239	38	26	graph	graph	NOUN
cana-1239	38	27	,	,	PUNCT
cana-1239	38	28	represented	represent	VERB
cana-1239	38	29	by	by	ADP
cana-1239	38	30	𝑆𝑛	𝑆𝑛	PROPN
cana-1239	38	31	,	,	PUNCT
cana-1239	38	32	is	be	AUX
cana-1239	38	33	a	a	DET
cana-1239	38	34	graph	graph	NOUN
cana-1239	38	35	with	with	ADP
cana-1239	38	36	two	two	NUM
cana-1239	38	37	n	n	PRON
cana-1239	38	38	vertices	vertex	NOUN
cana-1239	38	39	that	that	PRON
cana-1239	38	40	is	be	AUX
cana-1239	38	41	created	create	VERB
cana-1239	38	42	by	by	ADP
cana-1239	38	43	joining	join	VERB
cana-1239	38	44	n	n	CCONJ
cana-1239	38	45	−	−	PROPN
cana-1239	38	46	pendant	pendant	ADJ
cana-1239	38	47	edges	edge	NOUN
cana-1239	38	48	to	to	ADP
cana-1239	38	49	the	the	DET
cana-1239	38	50	cycle	cycle	NOUN
cana-1239	38	51	𝐶𝑛.	𝐶𝑛.	NOUN
cana-1239	38	52	definition	definition	NOUN
cana-1239	38	53	2.4	2.4	NUM
cana-1239	38	54	:	:	PUNCT
cana-1239	38	55	a	a	DET
cana-1239	38	56	graph	graph	NOUN
cana-1239	38	57	called	call	VERB
cana-1239	38	58	friendship	friendship	NOUN
cana-1239	38	59	,	,	PUNCT
cana-1239	38	60	denoted	denote	VERB
cana-1239	38	61	by	by	ADP
cana-1239	38	62	𝐹𝑛	𝐹𝑛	PROPN
cana-1239	38	63	is	be	AUX
cana-1239	38	64	constructed	construct	VERB
cana-1239	38	65	by	by	ADP
cana-1239	38	66	n	n	PRON
cana-1239	38	67	triangles	triangle	NOUN
cana-1239	38	68	that	that	PRON
cana-1239	38	69	share	share	VERB
cana-1239	38	70	a	a	DET
cana-1239	38	71	vertex	vertex	NOUN
cana-1239	38	72	.	.	PUNCT
cana-1239	39	1	definition	definition	NOUN
cana-1239	39	2	2.5	2.5	NUM
cana-1239	39	3	:	:	PUNCT
cana-1239	39	4	a	a	DET
cana-1239	39	5	flower	flower	NOUN
cana-1239	39	6	graph	graph	NOUN
cana-1239	39	7	,	,	PUNCT
cana-1239	39	8	or	or	CCONJ
cana-1239	39	9	𝐹𝑙𝑛	𝐹𝑙𝑛	PROPN
cana-1239	39	10	,	,	PUNCT
cana-1239	39	11	is	be	AUX
cana-1239	39	12	a	a	DET
cana-1239	39	13	graph	graph	NOUN
cana-1239	39	14	that	that	PRON
cana-1239	39	15	is	be	AUX
cana-1239	39	16	created	create	VERB
cana-1239	39	17	by	by	ADP
cana-1239	39	18	connecting	connect	VERB
cana-1239	39	19	each	each	DET
cana-1239	39	20	pendent	pendent	NOUN
cana-1239	39	21	to	to	ADP
cana-1239	39	22	the	the	DET
cana-1239	39	23	helm	helm	NOUN
cana-1239	39	24	's	's	PART
cana-1239	39	25	central	central	ADJ
cana-1239	39	26	vertex	vertex	NOUN
cana-1239	39	27	.	.	PUNCT
cana-1239	40	1	definition	definition	NOUN
cana-1239	40	2	2.6	2.6	NUM
cana-1239	40	3	:	:	PUNCT
cana-1239	40	4	the	the	DET
cana-1239	40	5	graph	graph	NOUN
cana-1239	40	6	𝐶𝑛+	𝐶𝑛+	PROPN
cana-1239	40	7	𝐾1.denoted	𝐾1.denote	VERB
cana-1239	40	8	by	by	ADP
cana-1239	40	9	𝑊𝑛	𝑊𝑛	PROPN
cana-1239	40	10	is	be	AUX
cana-1239	40	11	called	call	VERB
cana-1239	40	12	wheel	wheel	NOUN
cana-1239	40	13	graph	graph	NOUN
cana-1239	40	14	,	,	PUNCT
cana-1239	40	15	n	n	PRON
cana-1239	40	16	is	be	AUX
cana-1239	40	17	a	a	DET
cana-1239	40	18	number	number	NOUN
cana-1239	40	19	of	of	ADP
cana-1239	40	20	vertices	vertex	NOUN
cana-1239	40	21	in	in	ADP
cana-1239	40	22	the	the	DET
cana-1239	40	23	cycle	cycle	NOUN
cana-1239	40	24	.	.	PUNCT
cana-1239	41	1	3	3	X
cana-1239	41	2	.	.	NOUN
cana-1239	41	3	results	result	NOUN
cana-1239	41	4	and	and	CCONJ
cana-1239	41	5	discussion	discussion	NOUN
cana-1239	41	6	theorem	theorem	VERB
cana-1239	41	7	3.1	3.1	NUM
cana-1239	41	8	the	the	DET
cana-1239	41	9	cycle	cycle	NOUN
cana-1239	41	10	graph	graph	NOUN
cana-1239	41	11	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	41	12	is	be	AUX
cana-1239	41	13	(	(	PUNCT
cana-1239	41	14	𝑠	𝑠	PROPN
cana-1239	41	15	,	,	PUNCT
cana-1239	41	16	𝑑	𝑑	NOUN
cana-1239	41	17	)	)	PUNCT
cana-1239	41	18	magic	magic	ADJ
cana-1239	41	19	labeling	labeling	NOUN
cana-1239	41	20	proof	proof	NOUN
cana-1239	41	21	:	:	PUNCT
cana-1239	41	22	let	let	VERB
cana-1239	41	23	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	41	24	be	be	AUX
cana-1239	41	25	the	the	DET
cana-1239	41	26	cycle	cycle	NOUN
cana-1239	41	27	graph	graph	NOUN
cana-1239	41	28	.	.	PUNCT
cana-1239	41	29	|𝑉(𝐶𝜂)|	|𝑉(𝐶𝜂)|	PUNCT
cana-1239	42	1	=	=	PUNCT
cana-1239	42	2	|𝐸(𝐶𝜂)|=	|𝐸(𝐶𝜂)|=	PROPN
cana-1239	42	3	η	η	PROPN
cana-1239	42	4	.	.	PROPN
cana-1239	42	5	let	let	VERB
cana-1239	42	6	𝑉(𝐶𝜂	𝑉(𝐶𝜂	NUM
cana-1239	42	7	)	)	PUNCT
cana-1239	42	8	=	=	SYM
cana-1239	42	9	{	{	PUNCT
cana-1239	42	10	𝑣𝑙̇	𝑣𝑙̇	NOUN
cana-1239	42	11	;	;	PUNCT
cana-1239	42	12	1	1	NUM
cana-1239	42	13	≤	≤	NOUN
cana-1239	42	14	𝑙̇	𝑙̇	VERB
cana-1239	42	15	≤	≤	NUM
cana-1239	42	16	𝜂	𝜂	X
cana-1239	42	17	}	}	PUNCT
cana-1239	42	18	and	and	CCONJ
cana-1239	42	19	𝐸(𝐶𝜂	𝐸(𝐶𝜂	NUM
cana-1239	42	20	)	)	PUNCT
cana-1239	43	1	=	=	PRON
cana-1239	43	2	{	{	PUNCT
cana-1239	43	3	𝑣𝑙̇𝑣𝑙̇+1	𝑣𝑙̇𝑣𝑙̇+1	NOUN
cana-1239	43	4	:	:	PUNCT
cana-1239	43	5	1	1	NUM
cana-1239	43	6	≤	≤	NOUN
cana-1239	43	7	𝑙̇	𝑙̇	VERB
cana-1239	43	8	≤	≤	NUM
cana-1239	43	9	𝜂	𝜂	NOUN
cana-1239	43	10	−	−	NUM
cana-1239	43	11	1	1	NUM
cana-1239	43	12	}	}	PUNCT
cana-1239	43	13	∪	∪	ADJ
cana-1239	43	14	{	{	PUNCT
cana-1239	43	15	𝑣1𝑣𝜂	𝑣1𝑣𝜂	NOUN
cana-1239	43	16	}	}	PUNCT
cana-1239	43	17	define	define	VERB
cana-1239	43	18	the	the	DET
cana-1239	43	19	function	function	NOUN
cana-1239	43	20	f	f	PROPN
cana-1239	43	21	from	from	ADP
cana-1239	43	22	the	the	DET
cana-1239	43	23	vertex	vertex	NOUN
cana-1239	43	24	set	set	VERB
cana-1239	43	25	to	to	ADP
cana-1239	43	26	{	{	PUNCT
cana-1239	43	27	{	{	PUNCT
cana-1239	43	28	𝑠	𝑠	PROPN
cana-1239	43	29	,	,	PUNCT
cana-1239	43	30	𝑠	𝑠	PROPN
cana-1239	43	31	+	+	PROPN
cana-1239	43	32	𝑑	𝑑	PROPN
cana-1239	43	33	,	,	PUNCT
cana-1239	43	34	𝑠	𝑠	PROPN
cana-1239	43	35	+	+	NUM
cana-1239	43	36	2𝑑	2𝑑	NUM
cana-1239	43	37	…	…	PUNCT
cana-1239	43	38	.	.	PUNCT
cana-1239	44	1	𝑠	𝑠	INTJ
cana-1239	45	1	+	+	CCONJ
cana-1239	45	2	(	(	PUNCT
cana-1239	45	3	|𝐸|	|𝐸|	NOUN
cana-1239	45	4	+	+	X
cana-1239	45	5	1	1	NUM
cana-1239	45	6	)	)	PUNCT
cana-1239	45	7	}	}	PUNCT
cana-1239	45	8	,	,	PUNCT
cana-1239	45	9	𝑔	𝑔	NOUN
cana-1239	45	10	:	:	PUNCT
cana-1239	45	11	𝐸(𝐺	𝐸(𝐺	ADJ
cana-1239	45	12	)	)	PUNCT
cana-1239	45	13	→	→	SYM
cana-1239	45	14	{	{	PUNCT
cana-1239	45	15	𝑑	𝑑	NOUN
cana-1239	45	16	,	,	PUNCT
cana-1239	45	17	2𝑑	2𝑑	NUM
cana-1239	45	18	,	,	PUNCT
cana-1239	45	19	3𝑑	3𝑑	NUM
cana-1239	45	20	…	…	PUNCT
cana-1239	45	21	.	.	PUNCT
cana-1239	46	1	.2(|𝐸|	.2(|𝐸|	X
cana-1239	47	1	−	−	PROPN
cana-1239	47	2	1)𝑑	1)𝑑	NUM
cana-1239	47	3	}	}	PUNCT
cana-1239	47	4	labeling	labeling	NOUN
cana-1239	47	5	of	of	ADP
cana-1239	47	6	vertices	vertex	NOUN
cana-1239	47	7	cases	case	NOUN
cana-1239	47	8	value	value	NOUN
cana-1239	47	9	of	of	ADP
cana-1239	47	10	𝑙	𝑙	X
cana-1239	47	11	̇	̇	VERB
cana-1239	47	12	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	NOUN
cana-1239	47	13	)	)	PUNCT
cana-1239	47	14	η	η	PROPN
cana-1239	47	15	is	be	AUX
cana-1239	47	16	even	even	ADV
cana-1239	47	17	𝑙̇	𝑙̇	ADJ
cana-1239	47	18	=	=	PUNCT
cana-1239	47	19	1	1	NUM
cana-1239	47	20	s	s	NOUN
cana-1239	47	21	2	2	NUM
cana-1239	47	22	≤	≤	NOUN
cana-1239	47	23	𝑙̇	𝑙̇	VERB
cana-1239	47	24	≤	≤	NUM
cana-1239	47	25	𝜂	𝜂	DET
cana-1239	47	26	2	2	NUM
cana-1239	47	27	+	+	SYM
cana-1239	47	28	1	1	NUM
cana-1239	47	29	𝑠	𝑠	NOUN
cana-1239	47	30	+	+	CCONJ
cana-1239	47	31	(	(	PUNCT
cana-1239	47	32	2𝑙̇	2𝑙̇	NUM
cana-1239	47	33	−	−	NOUN
cana-1239	47	34	3)𝑑	3)𝑑	NUM
cana-1239	47	35	𝜂	𝜂	NOUN
cana-1239	47	36	2	2	NUM
cana-1239	47	37	+	+	CCONJ
cana-1239	47	38	2	2	NUM
cana-1239	47	39	≤	≤	NOUN
cana-1239	47	40	𝑙̇	𝑙̇	VERB
cana-1239	47	41	≤	≤	NUM
cana-1239	47	42	𝜂	𝜂	X
cana-1239	47	43	𝑠	𝑠	PROPN
cana-1239	48	1	+	+	CCONJ
cana-1239	48	2	(	(	PUNCT
cana-1239	48	3	18	18	NUM
cana-1239	48	4	−	−	PROPN
cana-1239	48	5	2𝑙)̇𝑑	2𝑙)̇𝑑	NUM
cana-1239	48	6	η	η	PROPN
cana-1239	48	7	is	be	AUX
cana-1239	48	8	odd	odd	ADJ
cana-1239	48	9	1	1	NUM
cana-1239	48	10	≤	≤	NOUN
cana-1239	48	11	𝑙̇	𝑙̇	VERB
cana-1239	48	12	≤	≤	NUM
cana-1239	48	13	𝜂	𝜂	X
cana-1239	48	14	𝑠	𝑠	PROPN
cana-1239	48	15	+	+	CCONJ
cana-1239	48	16	(	(	PUNCT
cana-1239	48	17	𝑙̇	𝑙̇	VERB
cana-1239	48	18	−	−	NUM
cana-1239	48	19	1)𝑑	1)𝑑	NUM
cana-1239	48	20	communications	communication	NOUN
cana-1239	48	21	on	on	ADP
cana-1239	48	22	applied	apply	VERB
cana-1239	48	23	nonlinear	nonlinear	ADJ
cana-1239	48	24	analysis	analysis	NOUN
cana-1239	48	25	issn	issn	NOUN
cana-1239	48	26	:	:	PUNCT
cana-1239	48	27	1074	1074	NUM
cana-1239	48	28	-	-	PUNCT
cana-1239	48	29	133x	133x	NUM
cana-1239	48	30	vol	vol	NOUN
cana-1239	48	31	31	31	NUM
cana-1239	48	32	no	no	NOUN
cana-1239	48	33	.	.	PUNCT
cana-1239	49	1	6s	6s	NUM
cana-1239	49	2	(	(	PUNCT
cana-1239	49	3	2024	2024	NUM
cana-1239	49	4	)	)	PUNCT
cana-1239	49	5	491	491	NUM
cana-1239	49	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	49	7	table	table	NOUN
cana-1239	49	8	1	1	NUM
cana-1239	49	9	:	:	PUNCT
cana-1239	49	10	labeling	labeling	NOUN
cana-1239	49	11	of	of	ADP
cana-1239	49	12	vertices	vertex	NOUN
cana-1239	49	13	for	for	ADP
cana-1239	49	14	the	the	DET
cana-1239	49	15	graph	graph	NOUN
cana-1239	49	16	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	49	17	labeling	labeling	NOUN
cana-1239	49	18	of	of	ADP
cana-1239	49	19	edges	edge	NOUN
cana-1239	49	20	cases	case	NOUN
cana-1239	49	21	value	value	NOUN
cana-1239	49	22	of	of	ADP
cana-1239	49	23	𝑙	𝑙	PRON
cana-1239	49	24	̇	̇	PROPN
cana-1239	49	25	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	49	26	)	)	PUNCT
cana-1239	49	27	𝑔(𝑣𝑙̇𝑣1	𝑔(𝑣𝑙̇𝑣1	PROPN
cana-1239	49	28	)	)	PUNCT
cana-1239	49	29	η	η	PROPN
cana-1239	49	30	is	be	AUX
cana-1239	49	31	even	even	ADV
cana-1239	49	32	𝑙̇	𝑙̇	ADJ
cana-1239	50	1	=	=	PUNCT
cana-1239	50	2	𝜂	𝜂	NOUN
cana-1239	50	3	2𝑠	2𝑠	NOUN
cana-1239	50	4	+	+	CCONJ
cana-1239	50	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	50	6	−	−	PROPN
cana-1239	50	7	1)𝑑	1)𝑑	NUM
cana-1239	50	8	−	−	PROPN
cana-1239	50	9	(	(	PUNCT
cana-1239	50	10	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	50	11	)	)	PUNCT
cana-1239	51	1	+	+	CCONJ
cana-1239	51	2	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	51	3	)	)	PUNCT
cana-1239	51	4	)	)	PUNCT
cana-1239	51	5	1	1	NUM
cana-1239	51	6	≤	≤	NOUN
cana-1239	51	7	𝑙̇	𝑙̇	VERB
cana-1239	51	8	≤	≤	NUM
cana-1239	52	1	𝜂	𝜂	NOUN
cana-1239	52	2	−	−	PROPN
cana-1239	52	3	1	1	NUM
cana-1239	52	4	2𝑠	2𝑠	NOUN
cana-1239	52	5	+	+	CCONJ
cana-1239	52	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	52	7	−	−	PROPN
cana-1239	52	8	1)𝑑	1)𝑑	NUM
cana-1239	52	9	−	−	PROPN
cana-1239	52	10	(	(	PUNCT
cana-1239	52	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	52	12	)	)	PUNCT
cana-1239	52	13	+	+	NUM
cana-1239	52	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	52	15	)	)	PUNCT
cana-1239	52	16	)	)	PUNCT
cana-1239	53	1	η	η	PROPN
cana-1239	53	2	is	be	AUX
cana-1239	53	3	odd	odd	ADJ
cana-1239	53	4	𝑙̇	𝑙̇	NOUN
cana-1239	53	5	=	=	PUNCT
cana-1239	53	6	𝜂	𝜂	NOUN
cana-1239	53	7	2𝑠	2𝑠	NOUN
cana-1239	53	8	+	+	CCONJ
cana-1239	53	9	2(|𝐸|	2(|𝐸|	NUM
cana-1239	53	10	−	−	PROPN
cana-1239	53	11	1)𝑑	1)𝑑	NUM
cana-1239	53	12	−	−	PROPN
cana-1239	53	13	(	(	PUNCT
cana-1239	53	14	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	53	15	)	)	PUNCT
cana-1239	54	1	+	+	CCONJ
cana-1239	54	2	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	54	3	)	)	PUNCT
cana-1239	54	4	)	)	PUNCT
cana-1239	54	5	1	1	NUM
cana-1239	54	6	≤	≤	NOUN
cana-1239	54	7	𝑙̇	𝑙̇	VERB
cana-1239	54	8	≤	≤	NUM
cana-1239	55	1	𝜂	𝜂	NOUN
cana-1239	55	2	−	−	PROPN
cana-1239	55	3	1	1	NUM
cana-1239	55	4	2𝑠	2𝑠	NOUN
cana-1239	55	5	+	+	CCONJ
cana-1239	55	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	55	7	−	−	PROPN
cana-1239	55	8	1)𝑑	1)𝑑	NUM
cana-1239	55	9	−	−	PROPN
cana-1239	55	10	(	(	PUNCT
cana-1239	55	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	55	12	)	)	PUNCT
cana-1239	55	13	+	+	NUM
cana-1239	55	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	55	15	)	)	PUNCT
cana-1239	55	16	)	)	PUNCT
cana-1239	55	17	table	table	NOUN
cana-1239	55	18	2	2	NUM
cana-1239	55	19	:	:	PUNCT
cana-1239	55	20	labeling	labeling	NOUN
cana-1239	55	21	of	of	ADP
cana-1239	55	22	edges	edge	NOUN
cana-1239	55	23	for	for	ADP
cana-1239	55	24	the	the	DET
cana-1239	55	25	graph	graph	NOUN
cana-1239	55	26	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	55	27	therefore	therefore	ADV
cana-1239	55	28	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	ADV
cana-1239	55	29	)	)	PUNCT
cana-1239	56	1	+	+	NUM
cana-1239	56	2	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	56	3	)	)	PUNCT
cana-1239	57	1	+	+	NUM
cana-1239	57	2	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	57	3	)	)	PUNCT
cana-1239	57	4	,	,	PUNCT
cana-1239	57	5	𝑓	𝑓	PROPN
cana-1239	57	6	(	(	PUNCT
cana-1239	57	7	𝑣1	𝑣1	PROPN
cana-1239	57	8	)	)	PUNCT
cana-1239	57	9	+	+	X
cana-1239	57	10	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	57	11	)	)	PUNCT
cana-1239	58	1	+	+	CCONJ
cana-1239	58	2	𝑔(𝑣1𝑣𝑙̇	𝑔(𝑣1𝑣𝑙̇	ADV
cana-1239	58	3	)	)	PUNCT
cana-1239	58	4	are	be	AUX
cana-1239	58	5	constant	constant	ADJ
cana-1239	58	6	equals	equal	VERB
cana-1239	58	7	to	to	ADP
cana-1239	58	8	2(𝑠	2(𝑠	NUM
cana-1239	58	9	+	+	CCONJ
cana-1239	58	10	(	(	PUNCT
cana-1239	58	11	|𝐸|	|𝐸|	NOUN
cana-1239	58	12	−	−	PROPN
cana-1239	58	13	1)𝑑	1)𝑑	NUM
cana-1239	58	14	)	)	PUNCT
cana-1239	58	15	.	.	PUNCT
cana-1239	59	1	hence	hence	ADV
cana-1239	59	2	the	the	DET
cana-1239	59	3	cycle	cycle	NOUN
cana-1239	59	4	graph	graph	NOUN
cana-1239	59	5	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	59	6	admits	admit	VERB
cana-1239	59	7	(	(	PUNCT
cana-1239	59	8	𝑠	𝑠	PROPN
cana-1239	59	9	,	,	PUNCT
cana-1239	59	10	𝑑	𝑑	NOUN
cana-1239	59	11	)	)	PUNCT
cana-1239	59	12	magic	magic	ADJ
cana-1239	59	13	labeling	labeling	NOUN
cana-1239	59	14	.	.	PUNCT
cana-1239	60	1	theorem	theorem	VERB
cana-1239	60	2	3.2	3.2	NUM
cana-1239	60	3	the	the	DET
cana-1239	60	4	graph	graph	NOUN
cana-1239	60	5	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	60	6	⊙𝐾1,𝑚is	⊙𝐾1,𝑚is	SYM
cana-1239	60	7	(	(	PUNCT
cana-1239	60	8	𝑠	𝑠	PROPN
cana-1239	60	9	,	,	PUNCT
cana-1239	60	10	𝑑	𝑑	NOUN
cana-1239	60	11	)	)	PUNCT
cana-1239	60	12	magic	magic	ADJ
cana-1239	60	13	labeling	labeling	NOUN
cana-1239	60	14	proof	proof	NOUN
cana-1239	60	15	:	:	PUNCT
cana-1239	60	16	let	let	VERB
cana-1239	60	17	𝑉(𝐶𝑛	𝑉(𝐶𝑛	NOUN
cana-1239	60	18	⊙𝐾1,𝑚	⊙𝐾1,𝑚	ADJ
cana-1239	60	19	)	)	PUNCT
cana-1239	61	1	=	=	PRON
cana-1239	61	2	{	{	PUNCT
cana-1239	61	3	𝑣𝑖	𝑣𝑖	X
cana-1239	61	4	:1	:1	PUNCT
cana-1239	61	5	≤	≤	NUM
cana-1239	61	6	𝑙̇	𝑙̇	VERB
cana-1239	61	7	≤	≤	NUM
cana-1239	61	8	𝜂	𝜂	X
cana-1239	61	9	}	}	PUNCT
cana-1239	61	10	∪	∪	NOUN
cana-1239	61	11	(	(	PUNCT
cana-1239	61	12	𝑢	𝑢	PART
cana-1239	61	13	𝑙̇	𝑙̇	VERB
cana-1239	61	14	𝑗	𝑗	PROPN
cana-1239	61	15	∶	∶	NOUN
cana-1239	61	16	1	1	NUM
cana-1239	61	17	≤	≤	NOUN
cana-1239	61	18	𝑙̇	𝑙̇	VERB
cana-1239	61	19	≤	≤	NUM
cana-1239	61	20	𝜂	𝜂	NOUN
cana-1239	61	21	,	,	PUNCT
cana-1239	61	22	1	1	NUM
cana-1239	61	23	≤	≤	NOUN
cana-1239	61	24	𝑙̇	𝑙̇	NOUN
cana-1239	61	25	≤m	≤m	PROPN
cana-1239	61	26	}	}	PUNCT
cana-1239	61	27	and	and	CCONJ
cana-1239	61	28	𝐸(𝐶𝑛	𝐸(𝐶𝑛	PROPN
cana-1239	61	29	⊙𝐾1,𝑚	⊙𝐾1,𝑚	NOUN
cana-1239	61	30	)	)	PUNCT
cana-1239	61	31	=	=	PRON
cana-1239	61	32	{	{	PUNCT
cana-1239	61	33	𝑣𝑙̇	𝑣𝑙̇	X
cana-1239	61	34	𝑣𝑙̇+1:1	𝑣𝑙̇+1:1	PROPN
cana-1239	61	35	≤	≤	NUM
cana-1239	61	36	𝑙̇	𝑙̇	VERB
cana-1239	61	37	≤	≤	NUM
cana-1239	62	1	𝜂	𝜂	NOUN
cana-1239	62	2	−	−	NUM
cana-1239	62	3	1	1	NUM
cana-1239	62	4	}	}	PUNCT
cana-1239	62	5	∪	∪	ADJ
cana-1239	62	6	{	{	PUNCT
cana-1239	62	7	𝑣𝜂𝑣1	𝑣𝜂𝑣1	NOUN
cana-1239	62	8	}	}	PUNCT
cana-1239	62	9	∪	∪	NOUN
cana-1239	62	10	(	(	PUNCT
cana-1239	62	11	𝑣𝑙̇𝑢𝑙̇	𝑣𝑙̇𝑢𝑙̇	NOUN
cana-1239	62	12	𝑗	𝑗	NOUN
cana-1239	62	13	∶	∶	NOUN
cana-1239	62	14	1	1	NUM
cana-1239	62	15	≤	≤	NOUN
cana-1239	62	16	𝑙̇	𝑙̇	VERB
cana-1239	62	17	≤	≤	NUM
cana-1239	62	18	𝜂	𝜂	NOUN
cana-1239	62	19	,	,	PUNCT
cana-1239	62	20	1	1	NUM
cana-1239	62	21	≤	≤	NOUN
cana-1239	62	22	𝑙̇	𝑙̇	NOUN
cana-1239	62	23	≤m	≤m	PROPN
cana-1239	62	24	}	}	PUNCT
cana-1239	62	25	|𝑉(𝐶𝑛	|𝑉(𝐶𝑛	NUM
cana-1239	62	26	⊙𝐾1,𝑚)|=|𝐸(𝐶𝑛	⊙𝐾1,𝑚)|=|𝐸(𝐶𝑛	PROPN
cana-1239	62	27	⊙𝐾1,𝑚)|=	⊙𝐾1,𝑚)|=	PROPN
cana-1239	62	28	η+ηm	η+ηm	PROPN
cana-1239	62	29	.	.	PROPN
cana-1239	62	30	define	define	VERB
cana-1239	62	31	the	the	DET
cana-1239	62	32	function	function	NOUN
cana-1239	62	33	f	f	PROPN
cana-1239	62	34	from	from	ADP
cana-1239	62	35	the	the	DET
cana-1239	62	36	vertex	vertex	NOUN
cana-1239	62	37	set	set	VERB
cana-1239	62	38	to	to	ADP
cana-1239	62	39	{	{	PUNCT
cana-1239	62	40	𝑠	𝑠	PROPN
cana-1239	62	41	,	,	PUNCT
cana-1239	62	42	𝑠	𝑠	PROPN
cana-1239	62	43	+	+	PROPN
cana-1239	62	44	𝑑	𝑑	PROPN
cana-1239	62	45	,	,	PUNCT
cana-1239	62	46	𝑠	𝑠	PROPN
cana-1239	63	1	+	+	NUM
cana-1239	63	2	2𝑑	2𝑑	NUM
cana-1239	63	3	…	…	PUNCT
cana-1239	63	4	.	.	PUNCT
cana-1239	64	1	𝑠	𝑠	INTJ
cana-1239	65	1	+	+	CCONJ
cana-1239	65	2	(	(	PUNCT
cana-1239	65	3	|𝐸|	|𝐸|	NOUN
cana-1239	65	4	+	+	X
cana-1239	65	5	1	1	NUM
cana-1239	65	6	)	)	PUNCT
cana-1239	65	7	}	}	PUNCT
cana-1239	65	8	,	,	PUNCT
cana-1239	65	9	𝑔	𝑔	NOUN
cana-1239	65	10	:	:	PUNCT
cana-1239	65	11	𝐸(𝐺	𝐸(𝐺	ADJ
cana-1239	65	12	)	)	PUNCT
cana-1239	65	13	→	→	SYM
cana-1239	65	14	{	{	PUNCT
cana-1239	65	15	𝑑	𝑑	NOUN
cana-1239	65	16	,	,	PUNCT
cana-1239	65	17	2𝑑	2𝑑	NUM
cana-1239	65	18	,	,	PUNCT
cana-1239	65	19	3𝑑	3𝑑	NUM
cana-1239	65	20	…	…	PUNCT
cana-1239	65	21	.	.	PUNCT
cana-1239	66	1	.2(|𝐸|	.2(|𝐸|	X
cana-1239	67	1	−	−	PROPN
cana-1239	67	2	1)𝑑	1)𝑑	NUM
cana-1239	67	3	}	}	PUNCT
cana-1239	67	4	labeling	labeling	NOUN
cana-1239	67	5	of	of	ADP
cana-1239	67	6	vertices	vertex	NOUN
cana-1239	67	7	cases	case	NOUN
cana-1239	67	8	value	value	NOUN
cana-1239	67	9	of	of	ADP
cana-1239	67	10	𝑙	𝑙	X
cana-1239	67	11	̇	̇	VERB
cana-1239	67	12	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	NOUN
cana-1239	67	13	)	)	PUNCT
cana-1239	67	14	𝑓(𝑢	𝑓(𝑢	PROPN
cana-1239	67	15	𝑙̇	𝑙̇	VERB
cana-1239	67	16	𝑗	𝑗	INTJ
cana-1239	67	17	)	)	PUNCT
cana-1239	67	18	𝑓(𝑢𝜂	𝑓(𝑢𝜂	PROPN
cana-1239	67	19	𝑗	𝑗	NOUN
cana-1239	67	20	)	)	PUNCT
cana-1239	67	21	η	η	PROPN
cana-1239	67	22	is	be	AUX
cana-1239	67	23	even	even	ADV
cana-1239	67	24	1	1	NUM
cana-1239	67	25	≤	≤	NOUN
cana-1239	67	26	𝑙̇	𝑙̇	VERB
cana-1239	67	27	≤	≤	NUM
cana-1239	68	1	𝜂	𝜂	NOUN
cana-1239	68	2	−	−	NUM
cana-1239	68	3	1	1	NUM
cana-1239	68	4	𝑠	𝑠	PROPN
cana-1239	68	5	+	+	CCONJ
cana-1239	69	1	[	[	X
cana-1239	69	2	(	(	PUNCT
cana-1239	69	3	𝑙̇	𝑙̇	INTJ
cana-1239	69	4	−	−	PROPN
cana-1239	69	5	1)(𝑚	1)(𝑚	NUM
cana-1239	69	6	+	+	CCONJ
cana-1239	69	7	1)]𝑑	1)]𝑑	ADJ
cana-1239	69	8	𝑙̇	𝑙̇	NOUN
cana-1239	69	9	=	=	SYM
cana-1239	69	10	𝜂	𝜂	X
cana-1239	69	11	𝑠	𝑠	NOUN
cana-1239	69	12	+	+	CCONJ
cana-1239	70	1	[	[	X
cana-1239	70	2	(	(	PUNCT
cana-1239	70	3	𝑙̇	𝑙̇	INTJ
cana-1239	70	4	−	−	PROPN
cana-1239	70	5	1)(𝑚	1)(𝑚	NUM
cana-1239	70	6	+	+	CCONJ
cana-1239	70	7	1	1	NUM
cana-1239	70	8	)	)	PUNCT
cana-1239	70	9	+	+	CCONJ
cana-1239	70	10	1]𝑑	1]𝑑	NUM
cana-1239	70	11	1	1	NUM
cana-1239	70	12	≤	≤	NOUN
cana-1239	70	13	𝑙̇	𝑙̇	VERB
cana-1239	70	14	≤	≤	NUM
cana-1239	71	1	𝜂	𝜂	NOUN
cana-1239	71	2	−	−	PROPN
cana-1239	71	3	1	1	NUM
cana-1239	71	4	,	,	PUNCT
cana-1239	71	5	1	1	NUM
cana-1239	71	6	≤	≤	NUM
cana-1239	71	7	𝑗	𝑗	PRON
cana-1239	71	8	≤	≤	NUM
cana-1239	71	9	𝑚	𝑚	X
cana-1239	71	10	𝑠	𝑠	NOUN
cana-1239	71	11	+	+	X
cana-1239	71	12	[	[	X
cana-1239	71	13	(	(	PUNCT
cana-1239	71	14	𝑙̇	𝑙̇	INTJ
cana-1239	71	15	−	−	PROPN
cana-1239	71	16	1)(𝑚	1)(𝑚	NUM
cana-1239	71	17	+	+	CCONJ
cana-1239	71	18	1	1	NUM
cana-1239	71	19	)	)	PUNCT
cana-1239	71	20	+	+	CCONJ
cana-1239	71	21	𝑗]𝑑	𝑗]𝑑	NOUN
cana-1239	71	22	communications	communication	NOUN
cana-1239	71	23	on	on	ADP
cana-1239	71	24	applied	apply	VERB
cana-1239	71	25	nonlinear	nonlinear	ADJ
cana-1239	71	26	analysis	analysis	NOUN
cana-1239	71	27	issn	issn	NOUN
cana-1239	71	28	:	:	PUNCT
cana-1239	71	29	1074	1074	NUM
cana-1239	71	30	-	-	PUNCT
cana-1239	71	31	133x	133x	NUM
cana-1239	71	32	vol	vol	NOUN
cana-1239	71	33	31	31	NUM
cana-1239	71	34	no	no	NOUN
cana-1239	71	35	.	.	PUNCT
cana-1239	72	1	6s	6s	NUM
cana-1239	72	2	(	(	PUNCT
cana-1239	72	3	2024	2024	NUM
cana-1239	72	4	)	)	PUNCT
cana-1239	72	5	492	492	NUM
cana-1239	73	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	73	2	labeling	labeling	NOUN
cana-1239	73	3	of	of	ADP
cana-1239	73	4	edges	edge	NOUN
cana-1239	73	5	cases	case	NOUN
cana-1239	73	6	value	value	NOUN
cana-1239	73	7	of	of	ADP
cana-1239	73	8	𝑙	𝑙	PRON
cana-1239	73	9	̇	̇	PROPN
cana-1239	73	10	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	73	11	)	)	PUNCT
cana-1239	73	12	𝑔(𝑣1𝑣𝑙̇	𝑔(𝑣1𝑣𝑙̇	ADV
cana-1239	73	13	)	)	PUNCT
cana-1239	73	14	𝑔(𝑣𝑙̇𝑢𝑙̇	𝑔(𝑣𝑙̇𝑢𝑙̇	PROPN
cana-1239	73	15	𝑗	𝑗	PROPN
cana-1239	73	16	)	)	PUNCT
cana-1239	73	17	η	η	PROPN
cana-1239	73	18	is	be	AUX
cana-1239	73	19	even	even	ADV
cana-1239	73	20	1	1	NUM
cana-1239	73	21	≤	≤	NOUN
cana-1239	73	22	𝑙̇	𝑙̇	VERB
cana-1239	73	23	≤	≤	NUM
cana-1239	74	1	𝜂	𝜂	NOUN
cana-1239	74	2	−	−	PROPN
cana-1239	74	3	1	1	NUM
cana-1239	74	4	2𝑠	2𝑠	NOUN
cana-1239	74	5	+	+	CCONJ
cana-1239	74	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	74	7	−	−	PROPN
cana-1239	74	8	1)𝑑	1)𝑑	NUM
cana-1239	74	9	−	−	PROPN
cana-1239	74	10	(	(	PUNCT
cana-1239	74	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	74	12	)	)	PUNCT
cana-1239	74	13	+	+	NUM
cana-1239	74	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	74	15	)	)	PUNCT
cana-1239	74	16	)	)	PUNCT
cana-1239	75	1	𝑙̇	𝑙̇	NOUN
cana-1239	76	1	=	=	PUNCT
cana-1239	76	2	𝜂	𝜂	NOUN
cana-1239	76	3	2𝑠	2𝑠	NOUN
cana-1239	76	4	+	+	CCONJ
cana-1239	76	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	76	6	−	−	PROPN
cana-1239	76	7	1)𝑑	1)𝑑	NUM
cana-1239	76	8	−	−	PROPN
cana-1239	76	9	(	(	PUNCT
cana-1239	76	10	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	76	11	)	)	PUNCT
cana-1239	76	12	+	+	CCONJ
cana-1239	76	13	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	76	14	)	)	PUNCT
cana-1239	76	15	)	)	PUNCT
cana-1239	76	16	1	1	NUM
cana-1239	76	17	≤	≤	NOUN
cana-1239	76	18	𝑙̇	𝑙̇	VERB
cana-1239	76	19	≤	≤	NUM
cana-1239	76	20	𝜂	𝜂	NOUN
cana-1239	76	21	,	,	PUNCT
cana-1239	76	22	1	1	NUM
cana-1239	76	23	≤	≤	NUM
cana-1239	76	24	𝑗	𝑗	PRON
cana-1239	76	25	≤	≤	NOUN
cana-1239	76	26	𝑚	𝑚	PRON
cana-1239	76	27	2𝑠	2𝑠	NOUN
cana-1239	76	28	+	+	CCONJ
cana-1239	76	29	2(|𝐸|	2(|𝐸|	NUM
cana-1239	76	30	−	−	PROPN
cana-1239	76	31	1)𝑑	1)𝑑	NUM
cana-1239	76	32	−	−	PROPN
cana-1239	76	33	(	(	PUNCT
cana-1239	76	34	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	76	35	)	)	PUNCT
cana-1239	77	1	+	+	CCONJ
cana-1239	77	2	𝑓(𝑢	𝑓(𝑢	PROPN
cana-1239	77	3	𝑙̇	𝑙̇	NOUN
cana-1239	77	4	𝑗	𝑗	INTJ
cana-1239	77	5	)	)	PUNCT
cana-1239	77	6	)	)	PUNCT
cana-1239	78	1	η	η	PROPN
cana-1239	78	2	is	be	AUX
cana-1239	78	3	odd	odd	ADJ
cana-1239	78	4	1	1	NUM
cana-1239	78	5	≤	≤	NOUN
cana-1239	78	6	𝑙̇	𝑙̇	VERB
cana-1239	78	7	≤	≤	NUM
cana-1239	78	8	𝜂	𝜂	NOUN
cana-1239	78	9	,	,	PUNCT
cana-1239	78	10	1	1	NUM
cana-1239	78	11	≤	≤	NUM
cana-1239	78	12	𝑗	𝑗	PRON
cana-1239	78	13	≤	≤	NOUN
cana-1239	78	14	𝑚	𝑚	PRON
cana-1239	78	15	2𝑠	2𝑠	NOUN
cana-1239	78	16	+	+	CCONJ
cana-1239	78	17	2(|𝐸|	2(|𝐸|	NUM
cana-1239	78	18	−	−	PROPN
cana-1239	78	19	1)𝑑	1)𝑑	NUM
cana-1239	78	20	−(𝑓(𝑣𝑖	−(𝑓(𝑣𝑖	NOUN
cana-1239	78	21	)	)	PUNCT
cana-1239	79	1	+	+	PUNCT
cana-1239	79	2	𝑓(𝑢𝑖	𝑓(𝑢𝑖	X
cana-1239	79	3	𝑗	𝑗	NOUN
cana-1239	79	4	)	)	PUNCT
cana-1239	79	5	)	)	PUNCT
cana-1239	79	6	1	1	NUM
cana-1239	79	7	≤	≤	NOUN
cana-1239	79	8	𝑙̇	𝑙̇	VERB
cana-1239	79	9	≤	≤	NUM
cana-1239	80	1	𝜂	𝜂	NOUN
cana-1239	80	2	−	−	PROPN
cana-1239	80	3	1	1	NUM
cana-1239	80	4	2𝑠	2𝑠	NOUN
cana-1239	80	5	+	+	CCONJ
cana-1239	80	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	80	7	−	−	PROPN
cana-1239	80	8	1)𝑑	1)𝑑	NUM
cana-1239	80	9	−	−	PROPN
cana-1239	80	10	(	(	PUNCT
cana-1239	80	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	80	12	)	)	PUNCT
cana-1239	80	13	+	+	NUM
cana-1239	80	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	80	15	)	)	PUNCT
cana-1239	80	16	)	)	PUNCT
cana-1239	81	1	𝑙̇	𝑙̇	NOUN
cana-1239	82	1	=	=	PUNCT
cana-1239	82	2	𝜂	𝜂	NOUN
cana-1239	82	3	2𝑠	2𝑠	NOUN
cana-1239	82	4	+	+	CCONJ
cana-1239	82	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	82	6	−	−	NOUN
cana-1239	82	7	1	1	NUM
cana-1239	82	8	)	)	PUNCT
cana-1239	82	9	−	−	PROPN
cana-1239	82	10	(	(	PUNCT
cana-1239	82	11	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	82	12	)	)	PUNCT
cana-1239	82	13	+	+	CCONJ
cana-1239	82	14	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	82	15	)	)	PUNCT
cana-1239	82	16	)	)	PUNCT
cana-1239	82	17	table	table	NOUN
cana-1239	82	18	4	4	NUM
cana-1239	82	19	:	:	PUNCT
cana-1239	82	20	labeling	labeling	NOUN
cana-1239	82	21	of	of	ADP
cana-1239	82	22	edges	edge	NOUN
cana-1239	82	23	of	of	ADP
cana-1239	82	24	the	the	DET
cana-1239	82	25	graph	graph	NOUN
cana-1239	82	26	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	82	27	⊙𝐾1,𝑚	⊙𝐾1,𝑚	NOUN
cana-1239	82	28	1	1	NUM
cana-1239	82	29	≤	≤	NUM
cana-1239	82	30	𝑗	𝑗	PRON
cana-1239	82	31	≤	≤	NUM
cana-1239	82	32	𝑚	𝑚	X
cana-1239	82	33	𝑠	𝑠	NOUN
cana-1239	82	34	+	+	X
cana-1239	83	1	[	[	X
cana-1239	83	2	(	(	PUNCT
cana-1239	83	3	𝜂	𝜂	NOUN
cana-1239	83	4	−	−	PROPN
cana-1239	83	5	1)(𝑚	1)(𝑚	NUM
cana-1239	83	6	+	+	CCONJ
cana-1239	83	7	1	1	NUM
cana-1239	83	8	)	)	PUNCT
cana-1239	84	1	+	+	CCONJ
cana-1239	84	2	𝑗]𝑑	𝑗]𝑑	X
cana-1239	84	3	η	η	PROPN
cana-1239	84	4	is	be	AUX
cana-1239	84	5	odd	odd	ADJ
cana-1239	84	6	1	1	NUM
cana-1239	84	7	≤	≤	NOUN
cana-1239	84	8	𝑙̇	𝑙̇	VERB
cana-1239	84	9	≤	≤	NUM
cana-1239	84	10	𝜂	𝜂	X
cana-1239	84	11	𝑠	𝑠	NOUN
cana-1239	85	1	+	+	CCONJ
cana-1239	85	2	[	[	X
cana-1239	85	3	(	(	PUNCT
cana-1239	85	4	𝑙̇	𝑙̇	INTJ
cana-1239	85	5	−	−	PROPN
cana-1239	85	6	1)(𝑚	1)(𝑚	NUM
cana-1239	85	7	+	+	CCONJ
cana-1239	85	8	1)]𝑑	1)]𝑑	NUM
cana-1239	85	9	1	1	NUM
cana-1239	85	10	≤	≤	NUM
cana-1239	85	11	𝑙̇	𝑙̇	VERB
cana-1239	85	12	≤	≤	NUM
cana-1239	85	13	𝜂	𝜂	DET
cana-1239	85	14	1	1	NUM
cana-1239	85	15	≤	≤	NUM
cana-1239	85	16	𝑗	𝑗	PRON
cana-1239	85	17	≤	≤	NUM
cana-1239	85	18	𝑚	𝑚	X
cana-1239	85	19	𝑠	𝑠	NOUN
cana-1239	86	1	+	+	X
cana-1239	87	1	[	[	X
cana-1239	87	2	(	(	PUNCT
cana-1239	87	3	𝑙̇	𝑙̇	INTJ
cana-1239	87	4	−	−	PROPN
cana-1239	87	5	1)(𝑚	1)(𝑚	NUM
cana-1239	87	6	+	+	CCONJ
cana-1239	87	7	1	1	NUM
cana-1239	87	8	)	)	PUNCT
cana-1239	87	9	+	+	CCONJ
cana-1239	87	10	𝑗]𝑑	𝑗]𝑑	PRON
cana-1239	87	11	table	table	NOUN
cana-1239	87	12	3	3	NUM
cana-1239	87	13	:	:	PUNCT
cana-1239	87	14	labeling	labeling	NOUN
cana-1239	87	15	of	of	ADP
cana-1239	87	16	vertices	vertex	NOUN
cana-1239	87	17	of	of	ADP
cana-1239	87	18	the	the	DET
cana-1239	87	19	graph	graph	NOUN
cana-1239	87	20	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	87	21	⊙𝐾1,𝑚	⊙𝐾1,𝑚	ADJ
cana-1239	87	22	communications	communication	NOUN
cana-1239	87	23	on	on	ADP
cana-1239	87	24	applied	apply	VERB
cana-1239	87	25	nonlinear	nonlinear	ADJ
cana-1239	87	26	analysis	analysis	NOUN
cana-1239	87	27	issn	issn	NOUN
cana-1239	87	28	:	:	PUNCT
cana-1239	87	29	1074	1074	NUM
cana-1239	87	30	-	-	PUNCT
cana-1239	87	31	133x	133x	NUM
cana-1239	87	32	vol	vol	NOUN
cana-1239	87	33	31	31	NUM
cana-1239	87	34	no	no	NOUN
cana-1239	87	35	.	.	PUNCT
cana-1239	88	1	6s	6s	NUM
cana-1239	88	2	(	(	PUNCT
cana-1239	88	3	2024	2024	NUM
cana-1239	88	4	)	)	PUNCT
cana-1239	88	5	493	493	NUM
cana-1239	88	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	88	7	therefore	therefore	ADV
cana-1239	88	8	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	88	9	)	)	PUNCT
cana-1239	89	1	+	+	CCONJ
cana-1239	89	2	𝑓(𝑢	𝑓(𝑢	PROPN
cana-1239	89	3	𝑙̇	𝑙̇	NOUN
cana-1239	89	4	𝑗	𝑗	INTJ
cana-1239	89	5	)	)	PUNCT
cana-1239	90	1	+	+	VERB
cana-1239	90	2	𝑔(𝑣𝑙̇𝑢𝑙̇	𝑔(𝑣𝑙̇𝑢𝑙̇	ADJ
cana-1239	90	3	𝑗	𝑗	NOUN
cana-1239	90	4	)	)	PUNCT
cana-1239	90	5	,	,	PUNCT
cana-1239	90	6	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	90	7	)	)	PUNCT
cana-1239	90	8	+	+	NUM
cana-1239	90	9	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	90	10	)	)	PUNCT
cana-1239	91	1	+	+	NUM
cana-1239	91	2	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	91	3	)	)	PUNCT
cana-1239	91	4	,	,	PUNCT
cana-1239	91	5	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	91	6	)	)	PUNCT
cana-1239	91	7	+	+	X
cana-1239	91	8	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	91	9	)	)	PUNCT
cana-1239	92	1	+	+	CCONJ
cana-1239	92	2	𝑔(𝑣1𝑣𝑙̇	𝑔(𝑣1𝑣𝑙̇	ADV
cana-1239	92	3	)	)	PUNCT
cana-1239	92	4	are	be	AUX
cana-1239	92	5	constant	constant	ADJ
cana-1239	92	6	equals	equal	VERB
cana-1239	92	7	to	to	ADP
cana-1239	92	8	2(𝑠	2(𝑠	NUM
cana-1239	92	9	+	+	CCONJ
cana-1239	92	10	(	(	PUNCT
cana-1239	92	11	|𝐸|	|𝐸|	NOUN
cana-1239	92	12	−	−	PROPN
cana-1239	92	13	1)𝑑	1)𝑑	NUM
cana-1239	92	14	)	)	PUNCT
cana-1239	92	15	.hence	.hence	ADP
cana-1239	92	16	the	the	DET
cana-1239	92	17	𝐶𝜂	𝐶𝜂	PROPN
cana-1239	92	18	⊙𝐾1,𝑚	⊙𝐾1,𝑚	ADJ
cana-1239	92	19	graph	graph	NOUN
cana-1239	92	20	admits	admit	VERB
cana-1239	92	21	(	(	PUNCT
cana-1239	92	22	𝑠	𝑠	PROPN
cana-1239	92	23	,	,	PUNCT
cana-1239	92	24	𝑑	𝑑	NOUN
cana-1239	92	25	)	)	PUNCT
cana-1239	92	26	magic	magic	ADJ
cana-1239	92	27	labeling	labeling	NOUN
cana-1239	92	28	.	.	PUNCT
cana-1239	93	1	theorem	theorem	VERB
cana-1239	93	2	3.3	3.3	NUM
cana-1239	93	3	the	the	DET
cana-1239	93	4	η	η	PROPN
cana-1239	93	5	-	-	ADJ
cana-1239	93	6	sunlet	sunlet	NOUN
cana-1239	93	7	graph	graph	NOUN
cana-1239	93	8	𝑆𝜂	𝑆𝜂	PROPN
cana-1239	93	9	is	be	AUX
cana-1239	93	10	a	a	DET
cana-1239	93	11	(	(	PUNCT
cana-1239	93	12	𝑠	𝑠	PROPN
cana-1239	93	13	,	,	PUNCT
cana-1239	93	14	𝑑	𝑑	NOUN
cana-1239	93	15	)	)	PUNCT
cana-1239	93	16	magic	magic	ADJ
cana-1239	93	17	labeling	labeling	NOUN
cana-1239	93	18	.	.	PUNCT
cana-1239	94	1	proof	proof	NOUN
cana-1239	94	2	:	:	PUNCT
cana-1239	94	3	let	let	VERB
cana-1239	94	4	g	g	PRON
cana-1239	94	5	be	be	AUX
cana-1239	94	6	a	a	DET
cana-1239	94	7	𝑆𝜂	𝑆𝜂	PROPN
cana-1239	94	8	graph	graph	NOUN
cana-1239	94	9	.	.	PUNCT
cana-1239	95	1	𝑉(𝑆𝜂	𝑉(𝑆𝜂	NOUN
cana-1239	95	2	)	)	PUNCT
cana-1239	95	3	=	=	PRON
cana-1239	95	4	{	{	PUNCT
cana-1239	95	5	𝑢𝑙̇	𝑢𝑙̇	ADV
cana-1239	95	6	}	}	PUNCT
cana-1239	95	7	∪	∪	X
cana-1239	95	8	{	{	PUNCT
cana-1239	95	9	𝑣𝑙̇	𝑣𝑙̇	ADV
cana-1239	95	10	}	}	PUNCT
cana-1239	95	11	;	;	PUNCT
cana-1239	95	12	1	1	NUM
cana-1239	95	13	≤	≤	NOUN
cana-1239	95	14	𝑙̇	𝑙̇	VERB
cana-1239	95	15	≤	≤	NUM
cana-1239	95	16	𝜂	𝜂	NOUN
cana-1239	95	17	and	and	CCONJ
cana-1239	95	18	𝐸(𝑆𝜂	𝐸(𝑆𝜂	NOUN
cana-1239	95	19	)	)	PUNCT
cana-1239	96	1	=	=	PRON
cana-1239	96	2	{	{	PUNCT
cana-1239	96	3	(	(	PUNCT
cana-1239	96	4	𝑢𝑙̇𝑣𝑙̇	𝑢𝑙̇𝑣𝑙̇	X
cana-1239	96	5	)	)	PUNCT
cana-1239	96	6	;	;	PUNCT
cana-1239	96	7	1	1	NUM
cana-1239	96	8	≤	≤	NOUN
cana-1239	96	9	𝑙̇	𝑙̇	VERB
cana-1239	96	10	≤	≤	NUM
cana-1239	96	11	η}∪	η}∪	X
cana-1239	96	12	(	(	PUNCT
cana-1239	96	13	𝑣𝑙̇𝑣𝑙̇+1	𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	96	14	)	)	PUNCT
cana-1239	96	15	;	;	PUNCT
cana-1239	96	16	1	1	NUM
cana-1239	96	17	≤	≤	NOUN
cana-1239	96	18	𝑙̇	𝑙̇	VERB
cana-1239	96	19	≤	≤	PROPN
cana-1239	96	20	η-1}∪	η-1}∪	PROPN
cana-1239	96	21	{	{	PUNCT
cana-1239	96	22	𝑣1𝑣𝑙̇	𝑣1𝑣𝑙̇	PROPN
cana-1239	96	23	)	)	PUNCT
cana-1239	96	24	;	;	PUNCT
cana-1239	96	25	𝑙̇	𝑙̇	PROPN
cana-1239	96	26	=	=	SYM
cana-1239	96	27	η	η	NOUN
cana-1239	96	28	}	}	PUNCT
cana-1239	96	29	.	.	PUNCT
cana-1239	97	1	define	define	VERB
cana-1239	97	2	the	the	DET
cana-1239	97	3	function	function	NOUN
cana-1239	97	4	f	f	PROPN
cana-1239	97	5	from	from	ADP
cana-1239	97	6	the	the	DET
cana-1239	97	7	vertex	vertex	NOUN
cana-1239	97	8	set	set	VERB
cana-1239	97	9	to	to	ADP
cana-1239	97	10	{	{	PUNCT
cana-1239	97	11	𝑠	𝑠	PROPN
cana-1239	97	12	,	,	PUNCT
cana-1239	97	13	𝑠	𝑠	PROPN
cana-1239	97	14	+	+	PROPN
cana-1239	97	15	𝑑	𝑑	PROPN
cana-1239	97	16	,	,	PUNCT
cana-1239	97	17	𝑠	𝑠	PROPN
cana-1239	98	1	+	+	NUM
cana-1239	98	2	2𝑑	2𝑑	NUM
cana-1239	98	3	…	…	PUNCT
cana-1239	98	4	.	.	PUNCT
cana-1239	99	1	𝑠	𝑠	INTJ
cana-1239	100	1	+	+	CCONJ
cana-1239	100	2	(	(	PUNCT
cana-1239	100	3	|𝐸|	|𝐸|	NOUN
cana-1239	100	4	+	+	X
cana-1239	100	5	1	1	NUM
cana-1239	100	6	)	)	PUNCT
cana-1239	100	7	}	}	PUNCT
cana-1239	100	8	,	,	PUNCT
cana-1239	100	9	𝑔	𝑔	NOUN
cana-1239	100	10	:	:	PUNCT
cana-1239	100	11	𝐸(𝐺	𝐸(𝐺	ADJ
cana-1239	100	12	)	)	PUNCT
cana-1239	100	13	→	→	SYM
cana-1239	100	14	{	{	PUNCT
cana-1239	100	15	𝑑	𝑑	NOUN
cana-1239	100	16	,	,	PUNCT
cana-1239	100	17	2𝑑	2𝑑	NUM
cana-1239	100	18	,	,	PUNCT
cana-1239	100	19	3𝑑	3𝑑	NUM
cana-1239	100	20	…	…	PUNCT
cana-1239	100	21	.	.	PUNCT
cana-1239	101	1	.2(|𝐸|	.2(|𝐸|	X
cana-1239	102	1	−	−	PROPN
cana-1239	102	2	1)𝑑	1)𝑑	NUM
cana-1239	102	3	}	}	PUNCT
cana-1239	102	4	labeling	labeling	NOUN
cana-1239	102	5	of	of	ADP
cana-1239	102	6	vertices	vertex	NOUN
cana-1239	102	7	cases	case	NOUN
cana-1239	102	8	value	value	NOUN
cana-1239	102	9	of	of	ADP
cana-1239	102	10	𝑙	𝑙	X
cana-1239	102	11	̇	̇	VERB
cana-1239	102	12	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	102	13	)	)	PUNCT
cana-1239	102	14	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	102	15	)	)	PUNCT
cana-1239	102	16	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	102	17	)	)	PUNCT
cana-1239	102	18	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	102	19	)	)	PUNCT
cana-1239	103	1	η	η	PROPN
cana-1239	103	2	is	be	AUX
cana-1239	103	3	odd	odd	ADJ
cana-1239	103	4	𝑙̇	𝑙̇	NOUN
cana-1239	104	1	=	=	PUNCT
cana-1239	104	2	0	0	NUM
cana-1239	104	3	s	s	PART
cana-1239	104	4	𝑠	𝑠	NOUN
cana-1239	105	1	+	+	CCONJ
cana-1239	105	2	𝑑	𝑑	PROPN
cana-1239	105	3	1	1	NUM
cana-1239	105	4	≤	≤	NOUN
cana-1239	105	5	𝑙̇	𝑙̇	VERB
cana-1239	105	6	≤	≤	NUM
cana-1239	106	1	𝜂	𝜂	NOUN
cana-1239	106	2	−	−	NUM
cana-1239	106	3	1	1	NUM
cana-1239	106	4	𝑠	𝑠	PROPN
cana-1239	106	5	+	+	NOUN
cana-1239	106	6	2𝑙	2𝑙	NOUN
cana-1239	106	7	�	�	PROPN
cana-1239	106	8	̇	̇	PROPN
cana-1239	106	9	�	�	PROPN
cana-1239	106	10	𝑠	𝑠	PROPN
cana-1239	106	11	+	+	X
cana-1239	106	12	(	(	PUNCT
cana-1239	106	13	2𝑙̇	2𝑙̇	NUM
cana-1239	106	14	+	+	SYM
cana-1239	106	15	1)𝑑	1)𝑑	NUM
cana-1239	106	16	η	η	PROPN
cana-1239	106	17	is	be	AUX
cana-1239	106	18	even	even	ADV
cana-1239	106	19	𝑙̇	𝑙̇	ADJ
cana-1239	107	1	=	=	NOUN
cana-1239	107	2	0	0	NUM
cana-1239	107	3	s	s	PART
cana-1239	107	4	𝑠	𝑠	NOUN
cana-1239	107	5	+	+	CCONJ
cana-1239	107	6	𝑑	𝑑	PROPN
cana-1239	107	7	𝑙̇	𝑙̇	NOUN
cana-1239	108	1	=	=	PUNCT
cana-1239	108	2	𝜂	𝜂	X
cana-1239	108	3	𝑠	𝑠	PROPN
cana-1239	108	4	+	+	CCONJ
cana-1239	108	5	(	(	PUNCT
cana-1239	108	6	2(𝜂	2(𝜂	NUM
cana-1239	108	7	−	−	NOUN
cana-1239	108	8	1	1	NUM
cana-1239	108	9	)	)	PUNCT
cana-1239	108	10	+	+	SYM
cana-1239	108	11	1)𝑑	1)𝑑	NUM
cana-1239	109	1	𝑠	𝑠	X
cana-1239	109	2	+	+	CCONJ
cana-1239	109	3	(	(	PUNCT
cana-1239	109	4	2(𝜂	2(𝜂	NUM
cana-1239	109	5	−	−	NOUN
cana-1239	109	6	1))𝑑	1))𝑑	NUM
cana-1239	109	7	1	1	NUM
cana-1239	109	8	≤	≤	NOUN
cana-1239	109	9	𝑙̇	𝑙̇	VERB
cana-1239	109	10	≤	≤	NUM
cana-1239	110	1	𝜂	𝜂	NOUN
cana-1239	110	2	−	−	NUM
cana-1239	110	3	2	2	NUM
cana-1239	110	4	−	−	NOUN
cana-1239	111	1	𝑠	𝑠	PROPN
cana-1239	112	1	+	+	CCONJ
cana-1239	113	1	(	(	PUNCT
cana-1239	113	2	2𝑙̇	2𝑙̇	NUM
cana-1239	113	3	+	+	CCONJ
cana-1239	113	4	1)𝑑	1)𝑑	NUM
cana-1239	113	5	table	table	NOUN
cana-1239	113	6	5	5	NUM
cana-1239	113	7	:	:	PUNCT
cana-1239	113	8	labeling	labeling	NOUN
cana-1239	113	9	of	of	ADP
cana-1239	113	10	vertices	vertex	NOUN
cana-1239	113	11	of	of	ADP
cana-1239	113	12	the	the	DET
cana-1239	113	13	graph	graph	NOUN
cana-1239	113	14	𝑆𝜂	𝑆𝜂	PROPN
cana-1239	113	15	labeling	labeling	NOUN
cana-1239	113	16	of	of	ADP
cana-1239	113	17	edges	edge	NOUN
cana-1239	113	18	cases	case	NOUN
cana-1239	113	19	value	value	NOUN
cana-1239	113	20	of	of	ADP
cana-1239	113	21	𝑙	𝑙	X
cana-1239	113	22	̇	̇	VERB
cana-1239	113	23	𝑔(𝑢𝑙̇𝑣𝑙̇	𝑔(𝑢𝑙̇𝑣𝑙̇	ADV
cana-1239	113	24	)	)	PUNCT
cana-1239	113	25	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	113	26	)	)	PUNCT
cana-1239	113	27	𝑔(𝑣1𝑣𝑙̇	𝑔(𝑣1𝑣𝑙̇	ADV
cana-1239	113	28	)	)	PUNCT
cana-1239	113	29	η	η	PROPN
cana-1239	113	30	is	be	AUX
cana-1239	113	31	odd	odd	ADJ
cana-1239	113	32	1	1	NUM
cana-1239	113	33	≤	≤	NOUN
cana-1239	113	34	𝑙̇	𝑙̇	VERB
cana-1239	113	35	≤	≤	NUM
cana-1239	113	36	𝜂	𝜂	PRON
cana-1239	113	37	2𝑠	2𝑠	NOUN
cana-1239	113	38	+	+	CCONJ
cana-1239	113	39	2(|𝐸|	2(|𝐸|	NUM
cana-1239	113	40	−	−	PROPN
cana-1239	113	41	1)𝑑	1)𝑑	NUM
cana-1239	113	42	−	−	PROPN
cana-1239	113	43	(	(	PUNCT
cana-1239	113	44	𝑓(𝑢𝑖	𝑓(𝑢𝑖	X
cana-1239	113	45	)	)	PUNCT
cana-1239	113	46	+	+	CCONJ
cana-1239	113	47	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-1239	113	48	)	)	PUNCT
cana-1239	113	49	)	)	PUNCT
cana-1239	113	50	1	1	NUM
cana-1239	113	51	≤	≤	NOUN
cana-1239	113	52	𝑙̇	𝑙̇	VERB
cana-1239	113	53	≤	≤	NUM
cana-1239	114	1	𝜂	𝜂	NOUN
cana-1239	114	2	−	−	PROPN
cana-1239	114	3	1	1	NUM
cana-1239	114	4	2𝑠	2𝑠	NOUN
cana-1239	114	5	+	+	CCONJ
cana-1239	114	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	114	7	−	−	PROPN
cana-1239	114	8	1)𝑑	1)𝑑	NUM
cana-1239	114	9	−	−	PROPN
cana-1239	114	10	(	(	PUNCT
cana-1239	114	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	114	12	)	)	PUNCT
cana-1239	114	13	+	+	NUM
cana-1239	114	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	114	15	)	)	PUNCT
cana-1239	114	16	)	)	PUNCT
cana-1239	115	1	𝑙̇	𝑙̇	NOUN
cana-1239	116	1	=	=	PUNCT
cana-1239	116	2	𝜂	𝜂	NOUN
cana-1239	116	3	2𝑠	2𝑠	NOUN
cana-1239	116	4	+	+	CCONJ
cana-1239	116	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	116	6	−	−	PROPN
cana-1239	116	7	1)𝑑	1)𝑑	NUM
cana-1239	116	8	−(𝑓(𝑣1	−(𝑓(𝑣1	NOUN
cana-1239	116	9	)	)	PUNCT
cana-1239	117	1	+	+	CCONJ
cana-1239	117	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	117	3	)	)	PUNCT
cana-1239	117	4	)	)	PUNCT
cana-1239	117	5	η	η	PROPN
cana-1239	117	6	is	be	AUX
cana-1239	117	7	even	even	ADV
cana-1239	117	8	1	1	NUM
cana-1239	117	9	≤	≤	NOUN
cana-1239	117	10	𝑙̇	𝑙̇	VERB
cana-1239	117	11	≤	≤	NUM
cana-1239	117	12	𝜂	𝜂	PRON
cana-1239	117	13	2𝑠	2𝑠	NOUN
cana-1239	117	14	+	+	CCONJ
cana-1239	117	15	2(|𝐸|	2(|𝐸|	NUM
cana-1239	117	16	−	−	PROPN
cana-1239	117	17	1)𝑑	1)𝑑	NUM
cana-1239	118	1	−	−	NOUN
cana-1239	118	2	(	(	PUNCT
cana-1239	118	3	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	118	4	)	)	PUNCT
cana-1239	119	1	+	+	CCONJ
cana-1239	119	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	119	3	)	)	PUNCT
cana-1239	119	4	)	)	PUNCT
cana-1239	119	5	1	1	NUM
cana-1239	119	6	≤	≤	NOUN
cana-1239	119	7	𝑙̇	𝑙̇	VERB
cana-1239	119	8	≤	≤	NUM
cana-1239	120	1	𝜂	𝜂	NOUN
cana-1239	120	2	−	−	PROPN
cana-1239	120	3	1	1	NUM
cana-1239	120	4	2𝑠	2𝑠	NOUN
cana-1239	120	5	+	+	CCONJ
cana-1239	120	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	120	7	−	−	PROPN
cana-1239	120	8	1)𝑑	1)𝑑	NUM
cana-1239	120	9	−	−	PROPN
cana-1239	120	10	(	(	PUNCT
cana-1239	120	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	120	12	)	)	PUNCT
cana-1239	120	13	+	+	NUM
cana-1239	120	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	120	15	)	)	PUNCT
cana-1239	120	16	)	)	PUNCT
cana-1239	120	17	communications	communication	NOUN
cana-1239	120	18	on	on	ADP
cana-1239	120	19	applied	apply	VERB
cana-1239	120	20	nonlinear	nonlinear	ADJ
cana-1239	120	21	analysis	analysis	NOUN
cana-1239	120	22	issn	issn	NOUN
cana-1239	120	23	:	:	PUNCT
cana-1239	120	24	1074	1074	NUM
cana-1239	120	25	-	-	PUNCT
cana-1239	120	26	133x	133x	NUM
cana-1239	120	27	vol	vol	NOUN
cana-1239	120	28	31	31	NUM
cana-1239	120	29	no	no	NOUN
cana-1239	120	30	.	.	PUNCT
cana-1239	121	1	6s	6s	NUM
cana-1239	121	2	(	(	PUNCT
cana-1239	121	3	2024	2024	NUM
cana-1239	121	4	)	)	PUNCT
cana-1239	121	5	494	494	NUM
cana-1239	121	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	121	7	𝑙̇	𝑙̇	NOUN
cana-1239	122	1	=	=	PUNCT
cana-1239	122	2	𝜂	𝜂	NOUN
cana-1239	122	3	2𝑠	2𝑠	NOUN
cana-1239	122	4	+	+	CCONJ
cana-1239	122	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	122	6	−	−	PROPN
cana-1239	122	7	1)𝑑	1)𝑑	NUM
cana-1239	122	8	−	−	PROPN
cana-1239	122	9	(	(	PUNCT
cana-1239	122	10	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	122	11	)	)	PUNCT
cana-1239	122	12	+	+	CCONJ
cana-1239	122	13	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	122	14	)	)	PUNCT
cana-1239	122	15	)	)	PUNCT
cana-1239	122	16	table	table	NOUN
cana-1239	122	17	6	6	NUM
cana-1239	122	18	:	:	PUNCT
cana-1239	122	19	labeling	labeling	NOUN
cana-1239	122	20	of	of	ADP
cana-1239	122	21	edges	edge	NOUN
cana-1239	122	22	of	of	ADP
cana-1239	122	23	the	the	DET
cana-1239	122	24	graph	graph	NOUN
cana-1239	122	25	𝑆𝜂	𝑆𝜂	PROPN
cana-1239	122	26	therefore𝑓(𝑢𝑙̇	therefore𝑓(𝑢𝑙̇	ADJ
cana-1239	122	27	)	)	PUNCT
cana-1239	122	28	+	+	X
cana-1239	122	29	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	122	30	)	)	PUNCT
cana-1239	123	1	+	+	CCONJ
cana-1239	123	2	𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑢𝑙̇	𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑢𝑙̇	ADJ
cana-1239	123	3	)	)	PUNCT
cana-1239	124	1	+	+	CCONJ
cana-1239	124	2	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	124	3	)	)	PUNCT
cana-1239	125	1	+	+	SYM
cana-1239	125	2	𝑔(𝑢𝑙̇𝑢𝑙̇+1	𝑔(𝑢𝑙̇𝑢𝑙̇+1	NOUN
cana-1239	125	3	)	)	PUNCT
cana-1239	125	4	are	be	AUX
cana-1239	125	5	constant	constant	ADJ
cana-1239	125	6	equals	equal	VERB
cana-1239	125	7	to	to	ADP
cana-1239	125	8	2(𝑠	2(𝑠	NUM
cana-1239	125	9	+	+	CCONJ
cana-1239	125	10	(	(	PUNCT
cana-1239	125	11	|𝐸|	|𝐸|	NOUN
cana-1239	125	12	−	−	PROPN
cana-1239	125	13	1)𝑑).hence	1)𝑑).hence	NUM
cana-1239	125	14	the	the	DET
cana-1239	125	15	η	η	ADJ
cana-1239	125	16	-	-	ADJ
cana-1239	125	17	sunlet	sunlet	NOUN
cana-1239	125	18	graph	graph	NOUN
cana-1239	125	19	𝑆𝜂	𝑆𝜂	PROPN
cana-1239	125	20	admits	admit	VERB
cana-1239	125	21	(	(	PUNCT
cana-1239	125	22	𝑠	𝑠	PROPN
cana-1239	125	23	,	,	PUNCT
cana-1239	125	24	𝑑	𝑑	NOUN
cana-1239	125	25	)	)	PUNCT
cana-1239	125	26	magic	magic	ADJ
cana-1239	125	27	labeling	labeling	NOUN
cana-1239	125	28	.	.	PUNCT
cana-1239	126	1	theorem	theorem	VERB
cana-1239	126	2	3.4	3.4	NUM
cana-1239	126	3	the	the	DET
cana-1239	126	4	friendship	friendship	NOUN
cana-1239	126	5	graph	graph	NOUN
cana-1239	126	6	𝐹𝜂	𝐹𝜂	PROPN
cana-1239	126	7	is	be	AUX
cana-1239	126	8	a	a	DET
cana-1239	126	9	(	(	PUNCT
cana-1239	126	10	𝑠	𝑠	PROPN
cana-1239	126	11	,	,	PUNCT
cana-1239	126	12	𝑑	𝑑	NOUN
cana-1239	126	13	)	)	PUNCT
cana-1239	126	14	magic	magic	ADJ
cana-1239	126	15	labeling	labeling	NOUN
cana-1239	126	16	for	for	ADP
cana-1239	126	17	η≥	η≥	PROPN
cana-1239	126	18	3	3	X
cana-1239	126	19	.	.	X
cana-1239	127	1	proof	proof	NOUN
cana-1239	127	2	:	:	PUNCT
cana-1239	127	3	let	let	VERB
cana-1239	127	4	𝐹𝜂	𝐹𝜂	PROPN
cana-1239	127	5	be	be	AUX
cana-1239	127	6	a	a	DET
cana-1239	127	7	friendship	friendship	NOUN
cana-1239	127	8	graph	graph	NOUN
cana-1239	127	9	.	.	PUNCT
cana-1239	128	1	v	v	X
cana-1239	128	2	(	(	PUNCT
cana-1239	128	3	𝐹𝜂)={{𝑢𝑙̇	𝐹𝜂)={{𝑢𝑙̇	PROPN
cana-1239	128	4	}	}	PUNCT
cana-1239	128	5	;	;	PUNCT
cana-1239	128	6	1	1	NUM
cana-1239	128	7	≤	≤	NOUN
cana-1239	128	8	𝑙̇	𝑙̇	VERB
cana-1239	128	9	≤	≤	NUM
cana-1239	128	10	𝜂	𝜂	X
cana-1239	128	11	}	}	PUNCT
cana-1239	128	12	∪	∪	X
cana-1239	128	13	{	{	PUNCT
cana-1239	128	14	𝑣𝑙̇	𝑣𝑙̇	NOUN
cana-1239	128	15	}	}	PUNCT
cana-1239	128	16	;	;	PUNCT
cana-1239	128	17	1	1	NUM
cana-1239	128	18	≤	≤	NOUN
cana-1239	128	19	𝑙̇	𝑙̇	VERB
cana-1239	128	20	≤	≤	NUM
cana-1239	128	21	𝜂	𝜂	X
cana-1239	128	22	}	}	PUNCT
cana-1239	128	23	∪	∪	X
cana-1239	128	24	{	{	PUNCT
cana-1239	128	25	𝑣′	𝑣′	PROPN
cana-1239	128	26	}	}	PUNCT
cana-1239	128	27	e	e	X
cana-1239	128	28	(	(	PUNCT
cana-1239	128	29	𝐹𝜂)={(𝑢𝑙̇𝑣𝑙̇	𝐹𝜂)={(𝑢𝑙̇𝑣𝑙̇	ADV
cana-1239	128	30	)	)	PUNCT
cana-1239	128	31	;	;	PUNCT
cana-1239	128	32	1	1	NUM
cana-1239	128	33	≤	≤	NOUN
cana-1239	128	34	𝑙̇	𝑙̇	VERB
cana-1239	128	35	≤	≤	NUM
cana-1239	128	36	𝜂	𝜂	X
cana-1239	128	37	}	}	PUNCT
cana-1239	128	38	∪	∪	X
cana-1239	128	39	{	{	PUNCT
cana-1239	128	40	(	(	PUNCT
cana-1239	128	41	𝑣′𝑢𝑙̇	𝑣′𝑢𝑙̇	ADJ
cana-1239	128	42	)	)	PUNCT
cana-1239	128	43	;	;	PUNCT
cana-1239	128	44	1	1	NUM
cana-1239	128	45	≤	≤	NOUN
cana-1239	128	46	𝑙̇	𝑙̇	VERB
cana-1239	128	47	≤	≤	NUM
cana-1239	128	48	𝜂	𝜂	X
cana-1239	128	49	}	}	PUNCT
cana-1239	128	50	∪	∪	ADJ
cana-1239	128	51	(	(	PUNCT
cana-1239	128	52	𝑣′𝑣𝑙̇	𝑣′𝑣𝑙̇	NOUN
cana-1239	128	53	)	)	PUNCT
cana-1239	128	54	;	;	PUNCT
cana-1239	128	55	1	1	NUM
cana-1239	128	56	≤	≤	NOUN
cana-1239	128	57	𝑙̇	𝑙̇	VERB
cana-1239	128	58	≤	≤	ADJ
cana-1239	128	59	𝜂	𝜂	NOUN
cana-1239	128	60	here	here	ADV
cana-1239	128	61	|v(𝐹𝜂)|=2𝜂	|v(𝐹𝜂)|=2𝜂	PROPN
cana-1239	129	1	+	+	CCONJ
cana-1239	129	2	1,|e(𝐹𝜂)|=3𝜂	1,|e(𝐹𝜂)|=3𝜂	NOUN
cana-1239	129	3	.define	.define	NOUN
cana-1239	129	4	the	the	DET
cana-1239	129	5	function	function	NOUN
cana-1239	129	6	f	f	PROPN
cana-1239	129	7	from	from	ADP
cana-1239	129	8	the	the	DET
cana-1239	129	9	vertex	vertex	NOUN
cana-1239	129	10	set	set	VERB
cana-1239	129	11	to	to	ADP
cana-1239	129	12	{	{	PUNCT
cana-1239	129	13	𝑠	𝑠	PROPN
cana-1239	129	14	,	,	PUNCT
cana-1239	129	15	𝑠	𝑠	PROPN
cana-1239	129	16	+	+	PROPN
cana-1239	129	17	𝑑	𝑑	PROPN
cana-1239	129	18	,	,	PUNCT
cana-1239	129	19	𝑠	𝑠	PROPN
cana-1239	129	20	+	+	NUM
cana-1239	129	21	2𝑑	2𝑑	NUM
cana-1239	129	22	…	…	PUNCT
cana-1239	129	23	.	.	PUNCT
cana-1239	130	1	𝑠	𝑠	INTJ
cana-1239	131	1	+	+	CCONJ
cana-1239	131	2	(	(	PUNCT
cana-1239	131	3	|𝐸|	|𝐸|	NOUN
cana-1239	131	4	+	+	X
cana-1239	131	5	1	1	NUM
cana-1239	131	6	)	)	PUNCT
cana-1239	131	7	}	}	PUNCT
cana-1239	131	8	,	,	PUNCT
cana-1239	131	9	𝑔	𝑔	NOUN
cana-1239	131	10	:	:	PUNCT
cana-1239	131	11	𝐸(𝐺	𝐸(𝐺	ADJ
cana-1239	131	12	)	)	PUNCT
cana-1239	131	13	→	→	SYM
cana-1239	131	14	{	{	PUNCT
cana-1239	131	15	𝑑	𝑑	NOUN
cana-1239	131	16	,	,	PUNCT
cana-1239	131	17	2𝑑	2𝑑	NUM
cana-1239	131	18	,	,	PUNCT
cana-1239	131	19	3𝑑	3𝑑	NUM
cana-1239	131	20	…	…	PUNCT
cana-1239	131	21	.	.	PUNCT
cana-1239	132	1	.2(|𝐸|	.2(|𝐸|	X
cana-1239	133	1	−	−	PROPN
cana-1239	133	2	1)𝑑	1)𝑑	NUM
cana-1239	133	3	}	}	PUNCT
cana-1239	133	4	labeling	labeling	NOUN
cana-1239	133	5	of	of	ADP
cana-1239	133	6	vertices	vertex	NOUN
cana-1239	133	7	𝑓(𝑣′)=𝑢𝜂−1	𝑓(𝑣′)=𝑢𝜂−1	ADJ
cana-1239	133	8	+	+	CCONJ
cana-1239	133	9	𝜂𝑑	𝜂𝑑	ADP
cana-1239	133	10	value	value	NOUN
cana-1239	133	11	𝑙	𝑙	X
cana-1239	133	12	̇	̇	VERB
cana-1239	133	13	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	133	14	)	)	PUNCT
cana-1239	133	15	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	133	16	)	)	PUNCT
cana-1239	133	17	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	133	18	)	)	PUNCT
cana-1239	133	19	𝑙̇	𝑙̇	NOUN
cana-1239	134	1	=	=	NOUN
cana-1239	134	2	1	1	NUM
cana-1239	134	3	𝑠	𝑠	INTJ
cana-1239	134	4	+	+	CCONJ
cana-1239	134	5	(	(	PUNCT
cana-1239	134	6	𝜂	𝜂	PROPN
cana-1239	134	7	+	+	NUM
cana-1239	134	8	1)𝑑	1)𝑑	NUM
cana-1239	134	9	𝑙̇	𝑙̇	NOUN
cana-1239	134	10	=	=	SYM
cana-1239	134	11	𝜂	𝜂	X
cana-1239	134	12	𝑠	𝑠	PROPN
cana-1239	134	13	+	+	CCONJ
cana-1239	134	14	𝜂𝑑	𝜂𝑑	ADP
cana-1239	134	15	𝑠	𝑠	PROPN
cana-1239	134	16	1	1	NUM
cana-1239	134	17	≤	≤	NOUN
cana-1239	134	18	𝑙̇	𝑙̇	VERB
cana-1239	134	19	≤	≤	NUM
cana-1239	135	1	𝜂	𝜂	NOUN
cana-1239	135	2	−	−	NUM
cana-1239	135	3	1	1	NUM
cana-1239	135	4	𝑠	𝑠	PROPN
cana-1239	135	5	+	+	PROPN
cana-1239	135	6	𝑙	𝑙	PROPN
cana-1239	135	7	�	�	PROPN
cana-1239	135	8	̇	̇	VERB
cana-1239	135	9	�	�	PROPN
cana-1239	135	10	1	1	NUM
cana-1239	135	11	≤	≤	NOUN
cana-1239	135	12	𝑙̇	𝑙̇	VERB
cana-1239	135	13	≤	≤	NUM
cana-1239	136	1	𝜂	𝜂	NOUN
cana-1239	136	2	−	−	NUM
cana-1239	136	3	2	2	NUM
cana-1239	136	4	𝑢𝑙̇	𝑢𝑙̇	ADP
cana-1239	136	5	+	+	CCONJ
cana-1239	136	6	𝑑	𝑑	NOUN
cana-1239	136	7	table	table	NOUN
cana-1239	136	8	7	7	NUM
cana-1239	136	9	:	:	PUNCT
cana-1239	136	10	labeling	labeling	NOUN
cana-1239	136	11	of	of	ADP
cana-1239	136	12	vertices	vertex	NOUN
cana-1239	136	13	of	of	ADP
cana-1239	136	14	the	the	DET
cana-1239	136	15	graph	graph	NOUN
cana-1239	136	16	𝐹𝜂	𝐹𝜂	PROPN
cana-1239	136	17	labeling	labeling	NOUN
cana-1239	136	18	of	of	ADP
cana-1239	136	19	edges	edge	NOUN
cana-1239	136	20	value	value	NOUN
cana-1239	136	21	𝑙	𝑙	X
cana-1239	136	22	̇	̇	VERB
cana-1239	136	23	𝑔(𝑢𝑙̇𝑣𝑙̇	𝑔(𝑢𝑙̇𝑣𝑙̇	ADV
cana-1239	136	24	)	)	PUNCT
cana-1239	136	25	𝑔(𝑣′𝑢𝑙̇	𝑔(𝑣′𝑢𝑙̇	NUM
cana-1239	136	26	)	)	PUNCT
cana-1239	136	27	𝑔(𝑣′𝑣𝑙̇	𝑔(𝑣′𝑣𝑙̇	NOUN
cana-1239	136	28	)	)	PUNCT
cana-1239	136	29	1	1	NUM
cana-1239	136	30	≤	≤	NOUN
cana-1239	136	31	𝑙̇	𝑙̇	VERB
cana-1239	136	32	≤	≤	NUM
cana-1239	136	33	𝜂	𝜂	PRON
cana-1239	136	34	2𝑠	2𝑠	NOUN
cana-1239	136	35	+	+	CCONJ
cana-1239	136	36	2(|𝐸|	2(|𝐸|	NUM
cana-1239	136	37	−	−	PROPN
cana-1239	136	38	1)𝑑	1)𝑑	NUM
cana-1239	136	39	−(𝑓(𝑢𝑙̇	−(𝑓(𝑢𝑙̇	NOUN
cana-1239	136	40	)	)	PUNCT
cana-1239	136	41	+	+	CCONJ
cana-1239	137	1	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	137	2	)	)	PUNCT
cana-1239	137	3	)	)	PUNCT
cana-1239	138	1	2𝑠	2𝑠	NOUN
cana-1239	139	1	+	+	CCONJ
cana-1239	139	2	2(|𝐸|	2(|𝐸|	NUM
cana-1239	139	3	−	−	PROPN
cana-1239	139	4	1)𝑑	1)𝑑	NUM
cana-1239	139	5	−	−	PROPN
cana-1239	139	6	(	(	PUNCT
cana-1239	139	7	𝑓(𝑣′	𝑓(𝑣′	NOUN
cana-1239	139	8	)	)	PUNCT
cana-1239	139	9	+	+	NUM
cana-1239	139	10	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	139	11	)	)	PUNCT
cana-1239	139	12	)	)	PUNCT
cana-1239	140	1	2𝑠	2𝑠	NOUN
cana-1239	141	1	+	+	CCONJ
cana-1239	141	2	2(|𝐸|	2(|𝐸|	NUM
cana-1239	141	3	−	−	NUM
cana-1239	141	4	1)𝑑	1)𝑑	NUM
cana-1239	141	5	−(𝑓(𝑣′	−(𝑓(𝑣′	NOUN
cana-1239	141	6	)	)	PUNCT
cana-1239	141	7	+	+	X
cana-1239	141	8	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	141	9	)	)	PUNCT
cana-1239	141	10	)	)	PUNCT
cana-1239	141	11	table	table	NOUN
cana-1239	141	12	8	8	NUM
cana-1239	141	13	:	:	PUNCT
cana-1239	141	14	labeling	labeling	NOUN
cana-1239	141	15	of	of	ADP
cana-1239	141	16	edges	edge	NOUN
cana-1239	141	17	of	of	ADP
cana-1239	141	18	the	the	DET
cana-1239	141	19	graph	graph	NOUN
cana-1239	141	20	𝐹𝜂	𝐹𝜂	PROPN
cana-1239	141	21	therefore	therefore	ADV
cana-1239	141	22	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	141	23	)	)	PUNCT
cana-1239	142	1	+	+	CCONJ
cana-1239	142	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	142	3	)	)	PUNCT
cana-1239	142	4	+	+	NUM
cana-1239	142	5	𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑣′	𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑣′	NOUN
cana-1239	142	6	)	)	PUNCT
cana-1239	143	1	+	+	CCONJ
cana-1239	143	2	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	X
cana-1239	143	3	)	)	PUNCT
cana-1239	144	1	+	+	NUM
cana-1239	144	2	𝑔(𝑣′𝑢𝑙̇	𝑔(𝑣′𝑢𝑙̇	NUM
cana-1239	144	3	)	)	PUNCT
cana-1239	144	4	and	and	CCONJ
cana-1239	144	5	𝑓(𝑣′	𝑓(𝑣′	NOUN
cana-1239	144	6	)	)	PUNCT
cana-1239	144	7	+	+	X
cana-1239	144	8	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	144	9	)	)	PUNCT
cana-1239	145	1	+	+	PUNCT
cana-1239	145	2	𝑔(𝑣′𝑣𝑙̇	𝑔(𝑣′𝑣𝑙̇	NOUN
cana-1239	145	3	)	)	PUNCT
cana-1239	145	4	are	be	AUX
cana-1239	145	5	constant	constant	ADJ
cana-1239	145	6	equals	equal	VERB
cana-1239	145	7	to	to	ADP
cana-1239	145	8	2(𝑠	2(𝑠	NUM
cana-1239	145	9	+	+	CCONJ
cana-1239	145	10	(	(	PUNCT
cana-1239	145	11	|𝐸|	|𝐸|	NOUN
cana-1239	145	12	−	−	PROPN
cana-1239	145	13	1)𝑑).hence	1)𝑑).hence	NUM
cana-1239	145	14	the	the	DET
cana-1239	145	15	friendship	friendship	NOUN
cana-1239	145	16	graph	graph	NOUN
cana-1239	145	17	𝐹𝜂	𝐹𝜂	PROPN
cana-1239	145	18	admits	admit	VERB
cana-1239	145	19	(	(	PUNCT
cana-1239	145	20	𝑠	𝑠	PROPN
cana-1239	145	21	,	,	PUNCT
cana-1239	145	22	𝑑	𝑑	NOUN
cana-1239	145	23	)	)	PUNCT
cana-1239	145	24	magic	magic	ADJ
cana-1239	145	25	labeling	labeling	NOUN
cana-1239	145	26	.	.	PUNCT
cana-1239	146	1	theorem	theorem	VERB
cana-1239	146	2	3.5	3.5	NUM
cana-1239	146	3	the	the	DET
cana-1239	146	4	flower	flower	NOUN
cana-1239	146	5	graph	graph	NOUN
cana-1239	146	6	𝐹𝑙𝜂	𝐹𝑙𝜂	PROPN
cana-1239	146	7	is	be	AUX
cana-1239	146	8	a	a	DET
cana-1239	146	9	(	(	PUNCT
cana-1239	146	10	𝑠	𝑠	PROPN
cana-1239	146	11	,	,	PUNCT
cana-1239	146	12	𝑑	𝑑	NOUN
cana-1239	146	13	)	)	PUNCT
cana-1239	146	14	magic	magic	ADJ
cana-1239	146	15	labeling	labeling	NOUN
cana-1239	146	16	.	.	PUNCT
cana-1239	147	1	proof	proof	NOUN
cana-1239	147	2	:	:	PUNCT
cana-1239	147	3	let	let	VERB
cana-1239	147	4	𝐺	𝐺	PRON
cana-1239	147	5	be	be	AUX
cana-1239	147	6	a	a	DET
cana-1239	147	7	graph	graph	NOUN
cana-1239	147	8	𝐹𝑙𝜂	𝐹𝑙𝜂	ADJ
cana-1239	147	9	communications	communication	NOUN
cana-1239	147	10	on	on	ADP
cana-1239	147	11	applied	apply	VERB
cana-1239	147	12	nonlinear	nonlinear	ADJ
cana-1239	147	13	analysis	analysis	NOUN
cana-1239	147	14	issn	issn	NOUN
cana-1239	147	15	:	:	PUNCT
cana-1239	147	16	1074	1074	NUM
cana-1239	147	17	-	-	PUNCT
cana-1239	147	18	133x	133x	NUM
cana-1239	147	19	vol	vol	NOUN
cana-1239	147	20	31	31	NUM
cana-1239	147	21	no	no	NOUN
cana-1239	147	22	.	.	PUNCT
cana-1239	148	1	6s	6s	NUM
cana-1239	148	2	(	(	PUNCT
cana-1239	148	3	2024	2024	NUM
cana-1239	148	4	)	)	PUNCT
cana-1239	148	5	495	495	NUM
cana-1239	148	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	148	7	|𝑉(𝐺)|=2𝜂	|𝑉(𝐺)|=2𝜂	NOUN
cana-1239	148	8	+	+	CCONJ
cana-1239	148	9	1	1	NUM
cana-1239	148	10	,	,	PUNCT
cana-1239	148	11	|𝐸(𝐺)|=4𝜂	|𝐸(𝐺)|=4𝜂	VERB
cana-1239	148	12	𝑉(𝐺)=	𝑉(𝐺)=	NOUN
cana-1239	148	13	{	{	PUNCT
cana-1239	148	14	𝑢𝑙̇	𝑢𝑙̇	X
cana-1239	148	15	;	;	PUNCT
cana-1239	148	16	1	1	NUM
cana-1239	148	17	≤	≤	NOUN
cana-1239	148	18	𝑙̇	𝑙̇	VERB
cana-1239	148	19	≤	≤	NUM
cana-1239	148	20	𝜂	𝜂	X
cana-1239	148	21	}	}	PUNCT
cana-1239	148	22	∪	∪	NOUN
cana-1239	148	23	{	{	PUNCT
cana-1239	148	24	𝑣𝑙̇	𝑣𝑙̇	PROPN
cana-1239	148	25	;	;	PUNCT
cana-1239	148	26	1	1	NUM
cana-1239	148	27	≤	≤	NOUN
cana-1239	148	28	𝑙̇	𝑙̇	VERB
cana-1239	148	29	≤	≤	NUM
cana-1239	148	30	𝜂	𝜂	X
cana-1239	148	31	}	}	PUNCT
cana-1239	148	32	∪	∪	ADJ
cana-1239	148	33	{	{	PUNCT
cana-1239	148	34	𝑣	𝑣	NOUN
cana-1239	148	35	}	}	PUNCT
cana-1239	148	36	and	and	CCONJ
cana-1239	148	37	𝐸(𝐺)=	𝐸(𝐺)=	NOUN
cana-1239	148	38	{	{	PUNCT
cana-1239	148	39	𝑢𝑙̇𝑢𝑙̇+1:1≤	𝑢𝑙̇𝑢𝑙̇+1:1≤	NOUN
cana-1239	148	40	𝑖	𝑖	X
cana-1239	148	41	≤η-1}∪	≤η-1}∪	X
cana-1239	148	42	{	{	PUNCT
cana-1239	148	43	𝑢1𝑢𝜂	𝑢1𝑢𝜂	PUNCT
cana-1239	148	44	}	}	PUNCT
cana-1239	148	45	∪	∪	NOUN
cana-1239	148	46	{	{	PUNCT
cana-1239	148	47	𝑢𝑙̇𝑣𝑙̇	𝑢𝑙̇𝑣𝑙̇	X
cana-1239	148	48	;	;	PUNCT
cana-1239	148	49	1	1	NUM
cana-1239	148	50	≤	≤	NOUN
cana-1239	148	51	𝑙̇	𝑙̇	VERB
cana-1239	148	52	≤	≤	NUM
cana-1239	148	53	𝜂	𝜂	X
cana-1239	148	54	}	}	PUNCT
cana-1239	148	55	∪	∪	ADJ
cana-1239	148	56	{	{	PUNCT
cana-1239	148	57	𝑣𝑢𝑙̇	𝑣𝑢𝑙̇	ADJ
cana-1239	148	58	;	;	PUNCT
cana-1239	148	59	≤	≤	NOUN
cana-1239	148	60	𝑙̇	𝑙̇	VERB
cana-1239	148	61	≤	≤	NUM
cana-1239	148	62	𝜂	𝜂	X
cana-1239	148	63	}	}	PUNCT
cana-1239	148	64	∪	∪	X
cana-1239	148	65	{	{	PUNCT
cana-1239	148	66	𝑣𝑣𝑙̇	𝑣𝑣𝑙̇	ADV
cana-1239	148	67	;	;	PUNCT
cana-1239	148	68	≤	≤	NOUN
cana-1239	148	69	𝑙̇	𝑙̇	VERB
cana-1239	148	70	≤	≤	NUM
cana-1239	148	71	𝜂	𝜂	X
cana-1239	148	72	}	}	PUNCT
cana-1239	148	73	define	define	VERB
cana-1239	148	74	the	the	DET
cana-1239	148	75	function	function	NOUN
cana-1239	148	76	f	f	PROPN
cana-1239	148	77	from	from	ADP
cana-1239	148	78	the	the	DET
cana-1239	148	79	vertex	vertex	NOUN
cana-1239	148	80	set	set	VERB
cana-1239	148	81	to	to	ADP
cana-1239	148	82	{	{	PUNCT
cana-1239	148	83	{	{	PUNCT
cana-1239	148	84	𝑠	𝑠	PROPN
cana-1239	148	85	,	,	PUNCT
cana-1239	148	86	𝑠	𝑠	PROPN
cana-1239	148	87	+	+	PROPN
cana-1239	148	88	𝑑	𝑑	PROPN
cana-1239	148	89	,	,	PUNCT
cana-1239	148	90	𝑠	𝑠	PROPN
cana-1239	148	91	+	+	NUM
cana-1239	148	92	2𝑑	2𝑑	NUM
cana-1239	148	93	…	…	PUNCT
cana-1239	148	94	.	.	PUNCT
cana-1239	149	1	𝑠	𝑠	INTJ
cana-1239	150	1	+	+	CCONJ
cana-1239	150	2	(	(	PUNCT
cana-1239	150	3	|𝐸|	|𝐸|	NOUN
cana-1239	150	4	+	+	X
cana-1239	150	5	1	1	NUM
cana-1239	150	6	)	)	PUNCT
cana-1239	150	7	}	}	PUNCT
cana-1239	150	8	,	,	PUNCT
cana-1239	150	9	𝑔	𝑔	NOUN
cana-1239	150	10	:	:	PUNCT
cana-1239	150	11	𝐸(𝐺	𝐸(𝐺	ADJ
cana-1239	150	12	)	)	PUNCT
cana-1239	150	13	→	→	SYM
cana-1239	150	14	{	{	PUNCT
cana-1239	150	15	𝑑	𝑑	NOUN
cana-1239	150	16	,	,	PUNCT
cana-1239	150	17	2𝑑	2𝑑	NUM
cana-1239	150	18	,	,	PUNCT
cana-1239	150	19	3𝑑	3𝑑	NUM
cana-1239	150	20	…	…	PUNCT
cana-1239	150	21	.	.	PUNCT
cana-1239	151	1	.2(|𝐸|	.2(|𝐸|	X
cana-1239	152	1	−	−	PROPN
cana-1239	152	2	1)𝑑	1)𝑑	NUM
cana-1239	152	3	}	}	PUNCT
cana-1239	152	4	labeling	labeling	NOUN
cana-1239	152	5	of	of	ADP
cana-1239	152	6	vertices	vertex	NOUN
cana-1239	152	7	cases	case	NOUN
cana-1239	152	8	value	value	NOUN
cana-1239	152	9	of	of	ADP
cana-1239	152	10	𝑙	𝑙	X
cana-1239	152	11	̇	̇	VERB
cana-1239	152	12	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	152	13	)	)	PUNCT
cana-1239	152	14	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	NOUN
cana-1239	152	15	)	)	PUNCT
cana-1239	152	16	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	152	17	)	)	PUNCT
cana-1239	152	18	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	152	19	)	)	PUNCT
cana-1239	153	1	η	η	PROPN
cana-1239	153	2	is	be	AUX
cana-1239	153	3	odd	odd	ADJ
cana-1239	153	4	f(v)=s+(4η-2)d	f(v)=s+(4η-2)d	NOUN
cana-1239	153	5	𝑙̇	𝑙̇	NOUN
cana-1239	154	1	=	=	NOUN
cana-1239	154	2	0	0	NUM
cana-1239	155	1	𝑠	𝑠	INTJ
cana-1239	155	2	𝑠	𝑠	PROPN
cana-1239	156	1	+	+	CCONJ
cana-1239	156	2	𝑑	𝑑	PROPN
cana-1239	156	3	1	1	NUM
cana-1239	156	4	≤	≤	NOUN
cana-1239	156	5	𝑙̇	𝑙̇	VERB
cana-1239	156	6	≤	≤	NUM
cana-1239	157	1	𝜂	𝜂	NOUN
cana-1239	157	2	−	−	NUM
cana-1239	157	3	1	1	NUM
cana-1239	157	4	𝑠	𝑠	PROPN
cana-1239	157	5	+	+	NOUN
cana-1239	157	6	2𝑙	2𝑙	NOUN
cana-1239	157	7	�	�	PROPN
cana-1239	157	8	̇	̇	PROPN
cana-1239	157	9	�	�	PROPN
cana-1239	157	10	𝑠	𝑠	PROPN
cana-1239	157	11	+	+	X
cana-1239	157	12	(	(	PUNCT
cana-1239	157	13	2𝑙̇	2𝑙̇	NUM
cana-1239	157	14	+	+	SYM
cana-1239	157	15	1)𝑑	1)𝑑	NUM
cana-1239	157	16	η	η	PROPN
cana-1239	157	17	is	be	AUX
cana-1239	157	18	even	even	ADV
cana-1239	157	19	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1239	157	20	)	)	PUNCT
cana-1239	157	21	=	=	SYM
cana-1239	157	22	𝑠	𝑠	PROPN
cana-1239	158	1	+	+	CCONJ
cana-1239	158	2	(	(	PUNCT
cana-1239	158	3	𝜂	𝜂	X
cana-1239	158	4	−	−	PROPN
cana-1239	158	5	2)𝑑	2)𝑑	NOUN
cana-1239	158	6	𝑙̇	𝑙̇	NOUN
cana-1239	159	1	=	=	NOUN
cana-1239	159	2	0	0	NUM
cana-1239	159	3	𝑠	𝑠	INTJ
cana-1239	159	4	𝑠	𝑠	PROPN
cana-1239	160	1	+	+	CCONJ
cana-1239	160	2	𝑑	𝑑	PROPN
cana-1239	160	3	1	1	NUM
cana-1239	160	4	≤	≤	NOUN
cana-1239	160	5	𝑙̇	𝑙̇	VERB
cana-1239	160	6	≤	≤	NUM
cana-1239	161	1	𝜂	𝜂	NOUN
cana-1239	161	2	−	−	NUM
cana-1239	161	3	2	2	NUM
cana-1239	161	4	𝑠	𝑠	PROPN
cana-1239	161	5	+	+	NOUN
cana-1239	161	6	2𝑙	2𝑙	NOUN
cana-1239	161	7	�	�	PROPN
cana-1239	161	8	̇	̇	PROPN
cana-1239	161	9	�	�	PROPN
cana-1239	161	10	𝑠	𝑠	PROPN
cana-1239	161	11	+	+	X
cana-1239	161	12	(	(	PUNCT
cana-1239	161	13	2𝑙̇	2𝑙̇	NUM
cana-1239	161	14	+	+	SYM
cana-1239	161	15	1)𝑑	1)𝑑	NUM
cana-1239	161	16	𝑙̇	𝑙̇	NOUN
cana-1239	161	17	=	=	SYM
cana-1239	162	1	𝜂	𝜂	NOUN
cana-1239	162	2	𝑢𝜂−1	𝑢𝜂−1	PROPN
cana-1239	163	1	+	+	CCONJ
cana-1239	163	2	𝑑	𝑑	PROPN
cana-1239	163	3	𝑣𝜂−1	𝑣𝜂−1	NOUN
cana-1239	163	4	+	+	CCONJ
cana-1239	163	5	3𝑑	3𝑑	NUM
cana-1239	163	6	table	table	NOUN
cana-1239	163	7	9	9	NUM
cana-1239	163	8	:	:	PUNCT
cana-1239	163	9	labeling	labeling	NOUN
cana-1239	163	10	of	of	ADP
cana-1239	163	11	vertices	vertex	NOUN
cana-1239	163	12	of	of	ADP
cana-1239	163	13	the	the	DET
cana-1239	163	14	graph	graph	NOUN
cana-1239	163	15	𝐹𝑙𝜂	𝐹𝑙𝜂	ADJ
cana-1239	163	16	labeling	labeling	NOUN
cana-1239	163	17	of	of	ADP
cana-1239	163	18	edges	edge	NOUN
cana-1239	163	19	cases	case	NOUN
cana-1239	163	20	value	value	NOUN
cana-1239	163	21	of	of	ADP
cana-1239	163	22	𝑙	𝑙	PRON
cana-1239	163	23	̇	̇	ADJ
cana-1239	163	24	𝑔(𝑢𝑙̇𝑢𝑙̇+1	𝑔(𝑢𝑙̇𝑢𝑙̇+1	NOUN
cana-1239	163	25	)	)	PUNCT
cana-1239	163	26	𝑔(𝑢𝑙̇𝑣𝑙̇	𝑔(𝑢𝑙̇𝑣𝑙̇	ADV
cana-1239	163	27	)	)	PUNCT
cana-1239	163	28	𝑔(𝑢𝑙̇𝑢1	𝑔(𝑢𝑙̇𝑢1	NOUN
cana-1239	163	29	)	)	PUNCT
cana-1239	163	30	𝑔(𝑣𝑢𝑙̇	𝑔(𝑣𝑢𝑙̇	NOUN
cana-1239	163	31	)	)	PUNCT
cana-1239	163	32	𝑔(𝑣𝑣𝑙̇	𝑔(𝑣𝑣𝑙̇	ADV
cana-1239	163	33	)	)	PUNCT
cana-1239	163	34	η	η	PROPN
cana-1239	163	35	is	be	AUX
cana-1239	163	36	odd	odd	ADJ
cana-1239	163	37	1	1	NUM
cana-1239	163	38	≤	≤	NOUN
cana-1239	163	39	𝑙̇	𝑙̇	VERB
cana-1239	163	40	≤	≤	NUM
cana-1239	164	1	𝜂	𝜂	NOUN
cana-1239	164	2	−	−	PROPN
cana-1239	164	3	1	1	NUM
cana-1239	164	4	2𝑠	2𝑠	NOUN
cana-1239	164	5	+	+	CCONJ
cana-1239	164	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	164	7	−	−	PROPN
cana-1239	164	8	1)𝑑	1)𝑑	NUM
cana-1239	164	9	−	−	NOUN
cana-1239	164	10	(	(	PUNCT
cana-1239	164	11	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	164	12	)	)	PUNCT
cana-1239	165	1	+	+	CCONJ
cana-1239	165	2	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	165	3	)	)	PUNCT
cana-1239	165	4	1	1	NUM
cana-1239	165	5	≤	≤	NOUN
cana-1239	165	6	𝑙̇	𝑙̇	VERB
cana-1239	165	7	≤	≤	NUM
cana-1239	165	8	𝜂	𝜂	PRON
cana-1239	165	9	2𝑠	2𝑠	NOUN
cana-1239	165	10	+	+	CCONJ
cana-1239	165	11	2(|𝐸|	2(|𝐸|	NUM
cana-1239	165	12	−	−	PROPN
cana-1239	165	13	1)𝑑	1)𝑑	NUM
cana-1239	166	1	−	−	NOUN
cana-1239	166	2	(	(	PUNCT
cana-1239	166	3	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	166	4	)	)	PUNCT
cana-1239	167	1	+	+	CCONJ
cana-1239	167	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	167	3	)	)	PUNCT
cana-1239	167	4	)	)	PUNCT
cana-1239	167	5	2𝑠	2𝑠	NOUN
cana-1239	168	1	+	+	CCONJ
cana-1239	168	2	2(|𝐸|	2(|𝐸|	NUM
cana-1239	168	3	−	−	PROPN
cana-1239	168	4	1)𝑑	1)𝑑	NUM
cana-1239	168	5	−	−	NOUN
cana-1239	168	6	(	(	PUNCT
cana-1239	168	7	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1239	168	8	)	)	PUNCT
cana-1239	168	9	+	+	NUM
cana-1239	168	10	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	168	11	)	)	PUNCT
cana-1239	168	12	)	)	PUNCT
cana-1239	169	1	2𝑠	2𝑠	NOUN
cana-1239	170	1	+	+	CCONJ
cana-1239	170	2	2(|𝐸|	2(|𝐸|	NUM
cana-1239	170	3	−	−	PROPN
cana-1239	170	4	1)𝑑	1)𝑑	NUM
cana-1239	170	5	−	−	NOUN
cana-1239	170	6	(	(	PUNCT
cana-1239	170	7	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1239	170	8	)	)	PUNCT
cana-1239	170	9	+	+	X
cana-1239	170	10	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	170	11	)	)	PUNCT
cana-1239	170	12	)	)	PUNCT
cana-1239	171	1	𝑙̇	𝑙̇	NOUN
cana-1239	172	1	=	=	PUNCT
cana-1239	172	2	𝜂	𝜂	NOUN
cana-1239	172	3	2𝑠	2𝑠	NOUN
cana-1239	172	4	+	+	CCONJ
cana-1239	172	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	172	6	−	−	PROPN
cana-1239	172	7	1)𝑑	1)𝑑	NUM
cana-1239	172	8	−	−	NOUN
cana-1239	172	9	(	(	PUNCT
cana-1239	172	10	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	172	11	)	)	PUNCT
cana-1239	172	12	+	+	NUM
cana-1239	172	13	𝑓(𝑢1	𝑓(𝑢1	NOUN
cana-1239	172	14	)	)	PUNCT
cana-1239	172	15	)	)	PUNCT
cana-1239	172	16	communications	communication	NOUN
cana-1239	172	17	on	on	ADP
cana-1239	172	18	applied	apply	VERB
cana-1239	172	19	nonlinear	nonlinear	ADJ
cana-1239	172	20	analysis	analysis	NOUN
cana-1239	172	21	issn	issn	NOUN
cana-1239	172	22	:	:	PUNCT
cana-1239	172	23	1074	1074	NUM
cana-1239	172	24	-	-	PUNCT
cana-1239	172	25	133x	133x	NUM
cana-1239	172	26	vol	vol	NOUN
cana-1239	172	27	31	31	NUM
cana-1239	172	28	no	no	NOUN
cana-1239	172	29	.	.	PUNCT
cana-1239	173	1	6s	6s	NUM
cana-1239	173	2	(	(	PUNCT
cana-1239	173	3	2024	2024	NUM
cana-1239	173	4	)	)	PUNCT
cana-1239	173	5	496	496	NUM
cana-1239	173	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	173	7	η	η	PROPN
cana-1239	173	8	is	be	AUX
cana-1239	173	9	even	even	ADV
cana-1239	173	10	1	1	NUM
cana-1239	173	11	≤	≤	NOUN
cana-1239	173	12	𝑙̇	𝑙̇	VERB
cana-1239	173	13	≤	≤	NUM
cana-1239	174	1	𝜂	𝜂	NOUN
cana-1239	174	2	−	−	PROPN
cana-1239	174	3	1	1	NUM
cana-1239	174	4	2𝑠	2𝑠	NOUN
cana-1239	174	5	+	+	CCONJ
cana-1239	174	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	174	7	−	−	PROPN
cana-1239	174	8	1)𝑑	1)𝑑	NUM
cana-1239	174	9	−	−	NOUN
cana-1239	174	10	(	(	PUNCT
cana-1239	174	11	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	174	12	)	)	PUNCT
cana-1239	175	1	+	+	CCONJ
cana-1239	175	2	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	175	3	)	)	PUNCT
cana-1239	175	4	1	1	NUM
cana-1239	175	5	≤	≤	NOUN
cana-1239	175	6	𝑙̇	𝑙̇	VERB
cana-1239	175	7	≤	≤	NUM
cana-1239	175	8	𝜂	𝜂	PRON
cana-1239	175	9	2𝑠	2𝑠	NOUN
cana-1239	175	10	+	+	CCONJ
cana-1239	175	11	2(|𝐸|	2(|𝐸|	NUM
cana-1239	175	12	−	−	PROPN
cana-1239	175	13	1)𝑑	1)𝑑	NUM
cana-1239	176	1	−	−	NOUN
cana-1239	176	2	(	(	PUNCT
cana-1239	176	3	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	176	4	)	)	PUNCT
cana-1239	177	1	+	+	CCONJ
cana-1239	177	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	177	3	)	)	PUNCT
cana-1239	177	4	)	)	PUNCT
cana-1239	177	5	2𝑠	2𝑠	NOUN
cana-1239	178	1	+	+	CCONJ
cana-1239	178	2	2(|𝐸|	2(|𝐸|	NUM
cana-1239	178	3	−	−	PROPN
cana-1239	178	4	1)𝑑	1)𝑑	NUM
cana-1239	178	5	−	−	NOUN
cana-1239	178	6	(	(	PUNCT
cana-1239	178	7	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1239	178	8	)	)	PUNCT
cana-1239	178	9	+	+	NUM
cana-1239	178	10	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	178	11	)	)	PUNCT
cana-1239	178	12	)	)	PUNCT
cana-1239	179	1	2𝑠	2𝑠	NOUN
cana-1239	180	1	+	+	CCONJ
cana-1239	180	2	2(|𝐸|	2(|𝐸|	NUM
cana-1239	180	3	−	−	PROPN
cana-1239	180	4	1)𝑑	1)𝑑	NUM
cana-1239	180	5	−	−	NOUN
cana-1239	180	6	(	(	PUNCT
cana-1239	180	7	𝑓(𝑣	𝑓(𝑣	PROPN
cana-1239	180	8	)	)	PUNCT
cana-1239	180	9	+	+	X
cana-1239	180	10	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	180	11	)	)	PUNCT
cana-1239	180	12	)	)	PUNCT
cana-1239	181	1	𝑙̇	𝑙̇	NOUN
cana-1239	182	1	=	=	PUNCT
cana-1239	182	2	η	η	X
cana-1239	182	3	2𝑠	2𝑠	NOUN
cana-1239	182	4	+	+	CCONJ
cana-1239	182	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	182	6	−	−	PROPN
cana-1239	182	7	1)𝑑	1)𝑑	NUM
cana-1239	182	8	−	−	NOUN
cana-1239	182	9	(	(	PUNCT
cana-1239	182	10	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	182	11	)	)	PUNCT
cana-1239	182	12	+	+	NUM
cana-1239	182	13	𝑓(𝑢1	𝑓(𝑢1	NOUN
cana-1239	182	14	)	)	PUNCT
cana-1239	182	15	)	)	PUNCT
cana-1239	182	16	table	table	NOUN
cana-1239	182	17	10	10	NUM
cana-1239	182	18	labeling	labeling	NOUN
cana-1239	182	19	of	of	ADP
cana-1239	182	20	edges	edge	NOUN
cana-1239	182	21	of	of	ADP
cana-1239	182	22	the	the	DET
cana-1239	182	23	graph	graph	NOUN
cana-1239	182	24	𝐹𝑙𝜂	𝐹𝑙𝜂	PROPN
cana-1239	182	25	therefore𝑓(𝑢𝑙̇	therefore𝑓(𝑢𝑙̇	NUM
cana-1239	182	26	)	)	PUNCT
cana-1239	182	27	+	+	X
cana-1239	182	28	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	182	29	)	)	PUNCT
cana-1239	183	1	+	+	CCONJ
cana-1239	183	2	𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑢𝑙̇	𝑔(𝑢𝑙̇𝑣𝑙̇),𝑓(𝑢𝑙̇	ADJ
cana-1239	183	3	)	)	PUNCT
cana-1239	184	1	+	+	CCONJ
cana-1239	184	2	𝑓(𝑢𝑙̇+1	𝑓(𝑢𝑙̇+1	NUM
cana-1239	184	3	)	)	PUNCT
cana-1239	185	1	+	+	CCONJ
cana-1239	185	2	𝑔(𝑢𝑙̇𝑢𝑙̇+1	𝑔(𝑢𝑙̇𝑢𝑙̇+1	NOUN
cana-1239	185	3	)	)	PUNCT
cana-1239	185	4	,	,	PUNCT
cana-1239	185	5	(	(	PUNCT
cana-1239	185	6	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	NOUN
cana-1239	185	7	)	)	PUNCT
cana-1239	185	8	+	+	NUM
cana-1239	185	9	𝑓(𝑢1	𝑓(𝑢1	NOUN
cana-1239	185	10	)	)	PUNCT
cana-1239	185	11	+	+	CCONJ
cana-1239	185	12	𝑔(𝑢𝑙̇𝑢1	𝑔(𝑢𝑙̇𝑢1	NOUN
cana-1239	185	13	)	)	PUNCT
cana-1239	185	14	,	,	PUNCT
cana-1239	185	15	𝑓(𝑣	𝑓(𝑣	X
cana-1239	185	16	)	)	PUNCT
cana-1239	185	17	+	+	NUM
cana-1239	185	18	𝑓(𝑢𝑙̇	𝑓(𝑢𝑙̇	ADV
cana-1239	185	19	)	)	PUNCT
cana-1239	186	1	+	+	CCONJ
cana-1239	186	2	𝑔(𝑣𝑢𝑙̇	𝑔(𝑣𝑢𝑙̇	NOUN
cana-1239	186	3	)	)	PUNCT
cana-1239	186	4	and	and	CCONJ
cana-1239	186	5	𝑓(𝑣	𝑓(𝑣	NOUN
cana-1239	186	6	)	)	PUNCT
cana-1239	186	7	+	+	X
cana-1239	186	8	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	186	9	)	)	PUNCT
cana-1239	187	1	+	+	CCONJ
cana-1239	187	2	𝑔(𝑣𝑣𝑙̇	𝑔(𝑣𝑣𝑙̇	NOUN
cana-1239	187	3	)	)	PUNCT
cana-1239	187	4	are	be	AUX
cana-1239	187	5	constant	constant	ADJ
cana-1239	187	6	equals	equal	VERB
cana-1239	187	7	to	to	ADP
cana-1239	187	8	2(𝑠	2(𝑠	NUM
cana-1239	187	9	+	+	CCONJ
cana-1239	187	10	(	(	PUNCT
cana-1239	187	11	|𝐸|	|𝐸|	NOUN
cana-1239	187	12	−	−	PROPN
cana-1239	187	13	1)𝑑).hence	1)𝑑).hence	PRON
cana-1239	187	14	the	the	DET
cana-1239	187	15	flower	flower	NOUN
cana-1239	187	16	graph	graph	NOUN
cana-1239	187	17	𝐹𝑙𝜂	𝐹𝑙𝜂	PROPN
cana-1239	187	18	admits	admit	VERB
cana-1239	187	19	(	(	PUNCT
cana-1239	187	20	𝑠	𝑠	PROPN
cana-1239	187	21	,	,	PUNCT
cana-1239	187	22	𝑑	𝑑	NOUN
cana-1239	187	23	)	)	PUNCT
cana-1239	187	24	magic	magic	ADJ
cana-1239	187	25	labeling	labeling	NOUN
cana-1239	187	26	.	.	PUNCT
cana-1239	188	1	figure1	figure1	X
cana-1239	188	2	:	:	PUNCT
cana-1239	189	1	flower	flower	NOUN
cana-1239	189	2	graph	graph	NOUN
cana-1239	189	3	𝑭𝒍𝟕	𝑭𝒍𝟕	NOUN
cana-1239	189	4	theorem	theorem	VERB
cana-1239	189	5	3.6	3.6	NUM
cana-1239	189	6	the	the	DET
cana-1239	189	7	wheel	wheel	NOUN
cana-1239	189	8	graph	graph	NOUN
cana-1239	189	9	𝑊𝜂	𝑊𝜂	PROPN
cana-1239	189	10	is	be	AUX
cana-1239	189	11	a	a	DET
cana-1239	189	12	(	(	PUNCT
cana-1239	189	13	𝑠	𝑠	PROPN
cana-1239	189	14	,	,	PUNCT
cana-1239	189	15	𝑑	𝑑	NOUN
cana-1239	189	16	)	)	PUNCT
cana-1239	189	17	magic	magic	ADJ
cana-1239	189	18	labeling	labeling	NOUN
cana-1239	189	19	η≥	η≥	VERB
cana-1239	189	20	4	4	NUM
cana-1239	189	21	communications	communication	NOUN
cana-1239	189	22	on	on	ADP
cana-1239	189	23	applied	apply	VERB
cana-1239	189	24	nonlinear	nonlinear	ADJ
cana-1239	189	25	analysis	analysis	NOUN
cana-1239	189	26	issn	issn	NOUN
cana-1239	189	27	:	:	PUNCT
cana-1239	189	28	1074	1074	NUM
cana-1239	189	29	-	-	PUNCT
cana-1239	189	30	133x	133x	NUM
cana-1239	189	31	vol	vol	NOUN
cana-1239	189	32	31	31	NUM
cana-1239	189	33	no	no	NOUN
cana-1239	189	34	.	.	PUNCT
cana-1239	190	1	6s	6s	NUM
cana-1239	190	2	(	(	PUNCT
cana-1239	190	3	2024	2024	NUM
cana-1239	190	4	)	)	PUNCT
cana-1239	190	5	497	497	NUM
cana-1239	190	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	190	7	proof	proof	NOUN
cana-1239	190	8	:	:	PUNCT
cana-1239	190	9	let	let	VERB
cana-1239	190	10	g	g	NOUN
cana-1239	190	11	=	=	SYM
cana-1239	190	12	𝑊𝜂	𝑊𝜂	PROPN
cana-1239	190	13	.	.	PUNCT
cana-1239	191	1	|𝑉(𝑊𝜂)|=𝜂	|𝑉(𝑊𝜂)|=𝜂	NOUN
cana-1239	191	2	+	+	CCONJ
cana-1239	191	3	1	1	NUM
cana-1239	191	4	,	,	PUNCT
cana-1239	191	5	|𝐸(𝑊𝜂)|=2𝜂.	|𝐸(𝑊𝜂)|=2𝜂.	PRON
cana-1239	191	6	define	define	VERB
cana-1239	191	7	the	the	DET
cana-1239	191	8	function	function	NOUN
cana-1239	191	9	f	f	PROPN
cana-1239	191	10	from	from	ADP
cana-1239	191	11	the	the	DET
cana-1239	191	12	vertex	vertex	NOUN
cana-1239	191	13	set	set	VERB
cana-1239	191	14	to	to	ADP
cana-1239	191	15	{	{	PUNCT
cana-1239	191	16	{	{	PUNCT
cana-1239	191	17	𝑠	𝑠	PROPN
cana-1239	191	18	,	,	PUNCT
cana-1239	191	19	𝑠	𝑠	PROPN
cana-1239	191	20	+	+	PROPN
cana-1239	191	21	𝑑	𝑑	PROPN
cana-1239	191	22	,	,	PUNCT
cana-1239	191	23	𝑠	𝑠	PROPN
cana-1239	191	24	+	+	NUM
cana-1239	191	25	2𝑑	2𝑑	NUM
cana-1239	191	26	…	…	PUNCT
cana-1239	191	27	.	.	PUNCT
cana-1239	192	1	𝑠	𝑠	INTJ
cana-1239	193	1	+	+	CCONJ
cana-1239	193	2	(	(	PUNCT
cana-1239	193	3	|𝐸|	|𝐸|	NOUN
cana-1239	193	4	+	+	X
cana-1239	193	5	1	1	NUM
cana-1239	193	6	)	)	PUNCT
cana-1239	193	7	}	}	PUNCT
cana-1239	193	8	,	,	PUNCT
cana-1239	193	9	𝑔	𝑔	NOUN
cana-1239	193	10	:	:	PUNCT
cana-1239	193	11	𝐸(𝐺	𝐸(𝐺	ADJ
cana-1239	193	12	)	)	PUNCT
cana-1239	193	13	→	→	SYM
cana-1239	193	14	{	{	PUNCT
cana-1239	193	15	𝑑	𝑑	NOUN
cana-1239	193	16	,	,	PUNCT
cana-1239	193	17	2𝑑	2𝑑	NUM
cana-1239	193	18	,	,	PUNCT
cana-1239	193	19	3𝑑	3𝑑	NUM
cana-1239	193	20	…	…	PUNCT
cana-1239	193	21	.	.	PUNCT
cana-1239	194	1	.2(|𝐸|	.2(|𝐸|	X
cana-1239	195	1	−	−	PROPN
cana-1239	195	2	1)𝑑	1)𝑑	NUM
cana-1239	195	3	}	}	PUNCT
cana-1239	195	4	labeling	labeling	NOUN
cana-1239	195	5	of	of	ADP
cana-1239	195	6	vertices	vertex	NOUN
cana-1239	195	7	cases	case	NOUN
cana-1239	195	8	value	value	NOUN
cana-1239	195	9	of	of	ADP
cana-1239	195	10	𝑙	𝑙	PROPN
cana-1239	195	11	̇	̇	VERB
cana-1239	195	12	f	f	PROPN
cana-1239	195	13	(	(	PUNCT
cana-1239	195	14	𝑣𝑙̇	𝑣𝑙̇	X
cana-1239	195	15	)	)	PUNCT
cana-1239	195	16	f(𝑣𝜂+1−𝑙̇	f(𝑣𝜂+1−𝑙̇	NOUN
cana-1239	195	17	)	)	PUNCT
cana-1239	195	18	η	η	PROPN
cana-1239	195	19	is	be	AUX
cana-1239	195	20	odd	odd	ADJ
cana-1239	195	21	1	1	NUM
cana-1239	195	22	≤	≤	NOUN
cana-1239	195	23	𝑙̇	𝑙̇	VERB
cana-1239	195	24	≤	≤	NUM
cana-1239	195	25	𝜂	𝜂	X
cana-1239	195	26	𝑠	𝑠	PROPN
cana-1239	195	27	+	+	CCONJ
cana-1239	195	28	(	(	PUNCT
cana-1239	195	29	𝑙̇	𝑙̇	VERB
cana-1239	195	30	−	−	NUM
cana-1239	195	31	1)𝑑	1)𝑑	NUM
cana-1239	195	32	𝑙̇	𝑙̇	NOUN
cana-1239	195	33	=	=	SYM
cana-1239	195	34	𝜂	𝜂	NOUN
cana-1239	195	35	+	+	NUM
cana-1239	195	36	1	1	NUM
cana-1239	195	37	𝑠	𝑠	PROPN
cana-1239	195	38	+	+	CCONJ
cana-1239	195	39	2(𝜂	2(𝜂	NUM
cana-1239	195	40	−	−	NUM
cana-1239	195	41	1)𝑑	1)𝑑	NUM
cana-1239	195	42	η	η	PROPN
cana-1239	195	43	is	be	AUX
cana-1239	195	44	even	even	ADV
cana-1239	195	45	𝑙̇	𝑙̇	ADJ
cana-1239	195	46	=	=	SYM
cana-1239	196	1	1	1	NUM
cana-1239	196	2	𝑠	𝑠	INTJ
cana-1239	196	3	𝑙̇	𝑙̇	NOUN
cana-1239	196	4	=	=	SYM
cana-1239	196	5	𝜂	𝜂	NOUN
cana-1239	197	1	+	+	NUM
cana-1239	197	2	1	1	NUM
cana-1239	197	3	𝑠	𝑠	PROPN
cana-1239	197	4	+	+	CCONJ
cana-1239	197	5	2(𝜂	2(𝜂	NUM
cana-1239	197	6	−	−	NUM
cana-1239	197	7	1)𝑑	1)𝑑	NUM
cana-1239	197	8	2	2	NUM
cana-1239	197	9	≤	≤	NOUN
cana-1239	197	10	𝑙̇	𝑙̇	VERB
cana-1239	197	11	≤	≤	NUM
cana-1239	197	12	𝜂	𝜂	NOUN
cana-1239	198	1	+	+	CCONJ
cana-1239	198	2	2	2	NUM
cana-1239	198	3	2	2	NUM
cana-1239	198	4	𝑠	𝑠	NOUN
cana-1239	198	5	+	+	CCONJ
cana-1239	198	6	(	(	PUNCT
cana-1239	198	7	2𝑙̇	2𝑙̇	NUM
cana-1239	198	8	−	−	NOUN
cana-1239	198	9	3)𝑑	3)𝑑	NUM
cana-1239	198	10	1	1	NUM
cana-1239	198	11	≤	≤	NOUN
cana-1239	198	12	𝑙̇	𝑙̇	VERB
cana-1239	198	13	≤	≤	NUM
cana-1239	198	14	𝜂	𝜂	NOUN
cana-1239	198	15	−	−	NUM
cana-1239	198	16	2	2	NUM
cana-1239	198	17	2	2	NUM
cana-1239	198	18	𝑠	𝑠	NOUN
cana-1239	198	19	+	+	NOUN
cana-1239	198	20	2𝑙	2𝑙	NOUN
cana-1239	198	21	�	�	PROPN
cana-1239	198	22	̇	̇	NOUN
cana-1239	198	23	�	�	PROPN
cana-1239	198	24	table	table	NOUN
cana-1239	198	25	11	11	NUM
cana-1239	198	26	:	:	PUNCT
cana-1239	198	27	labeling	labeling	NOUN
cana-1239	198	28	of	of	ADP
cana-1239	198	29	vertices	vertex	NOUN
cana-1239	198	30	of	of	ADP
cana-1239	198	31	the	the	DET
cana-1239	198	32	graph	graph	NOUN
cana-1239	198	33	𝑊𝜂	𝑊𝜂	PROPN
cana-1239	198	34	labeling	labeling	NOUN
cana-1239	198	35	of	of	ADP
cana-1239	198	36	edges	edge	NOUN
cana-1239	198	37	cases	case	NOUN
cana-1239	198	38	value	value	NOUN
cana-1239	198	39	of	of	ADP
cana-1239	198	40	𝑙	𝑙	PRON
cana-1239	198	41	̇	̇	PROPN
cana-1239	198	42	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	198	43	)	)	PUNCT
cana-1239	198	44	𝑔(𝑣𝑙̇𝑣1	𝑔(𝑣𝑙̇𝑣1	NOUN
cana-1239	198	45	)	)	PUNCT
cana-1239	198	46	𝑔(𝑣𝜂+1𝑣𝑙̇	𝑔(𝑣𝜂+1𝑣𝑙̇	ADV
cana-1239	198	47	)	)	PUNCT
cana-1239	198	48	η	η	PROPN
cana-1239	198	49	is	be	AUX
cana-1239	198	50	odd	odd	ADJ
cana-1239	198	51	𝑙̇	𝑙̇	NOUN
cana-1239	198	52	=	=	PUNCT
cana-1239	198	53	𝜂	𝜂	NOUN
cana-1239	198	54	2𝑠	2𝑠	NOUN
cana-1239	198	55	+	+	CCONJ
cana-1239	198	56	2(|𝐸|	2(|𝐸|	NUM
cana-1239	198	57	−	−	PROPN
cana-1239	198	58	1)𝑑	1)𝑑	NUM
cana-1239	198	59	−	−	PROPN
cana-1239	198	60	(	(	PUNCT
cana-1239	198	61	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	198	62	)	)	PUNCT
cana-1239	199	1	+	+	CCONJ
cana-1239	199	2	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-1239	199	3	)	)	PUNCT
cana-1239	199	4	)	)	PUNCT
cana-1239	199	5	1	1	NUM
cana-1239	199	6	≤	≤	NOUN
cana-1239	199	7	𝑙̇	𝑙̇	VERB
cana-1239	199	8	≤	≤	NUM
cana-1239	199	9	𝜂	𝜂	PRON
cana-1239	199	10	2𝑠	2𝑠	NOUN
cana-1239	199	11	+	+	CCONJ
cana-1239	199	12	2(|𝐸|	2(|𝐸|	NUM
cana-1239	199	13	−	−	PROPN
cana-1239	199	14	1)𝑑	1)𝑑	NUM
cana-1239	200	1	−	−	NOUN
cana-1239	200	2	(	(	PUNCT
cana-1239	200	3	𝑓(𝑣𝜂+1	𝑓(𝑣𝜂+1	NOUN
cana-1239	200	4	)	)	PUNCT
cana-1239	201	1	+	+	CCONJ
cana-1239	201	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	201	3	)	)	PUNCT
cana-1239	201	4	)	)	PUNCT
cana-1239	201	5	1	1	NUM
cana-1239	201	6	≤	≤	NOUN
cana-1239	201	7	𝑙̇	𝑙̇	VERB
cana-1239	201	8	≤	≤	NUM
cana-1239	202	1	𝜂	𝜂	NOUN
cana-1239	202	2	−	−	PROPN
cana-1239	202	3	1	1	NUM
cana-1239	202	4	2𝑠	2𝑠	NOUN
cana-1239	202	5	+	+	CCONJ
cana-1239	202	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	202	7	−	−	PROPN
cana-1239	202	8	1)𝑑	1)𝑑	NUM
cana-1239	202	9	−	−	PROPN
cana-1239	202	10	(	(	PUNCT
cana-1239	202	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	202	12	)	)	PUNCT
cana-1239	202	13	+	+	NUM
cana-1239	202	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	202	15	)	)	PUNCT
cana-1239	202	16	)	)	PUNCT
cana-1239	203	1	η	η	PROPN
cana-1239	203	2	is	be	AUX
cana-1239	203	3	even	even	ADV
cana-1239	203	4	1	1	NUM
cana-1239	203	5	≤	≤	NOUN
cana-1239	203	6	𝑙̇	𝑙̇	VERB
cana-1239	203	7	≤	≤	NUM
cana-1239	204	1	𝜂	𝜂	NOUN
cana-1239	204	2	−	−	PROPN
cana-1239	204	3	1	1	NUM
cana-1239	204	4	2𝑠	2𝑠	NOUN
cana-1239	204	5	+	+	CCONJ
cana-1239	204	6	2(|𝐸|	2(|𝐸|	NUM
cana-1239	204	7	−	−	PROPN
cana-1239	204	8	1)𝑑	1)𝑑	NUM
cana-1239	204	9	−	−	PROPN
cana-1239	204	10	(	(	PUNCT
cana-1239	204	11	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	204	12	)	)	PUNCT
cana-1239	204	13	+	+	NUM
cana-1239	204	14	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	204	15	)	)	PUNCT
cana-1239	204	16	)	)	PUNCT
cana-1239	204	17	1	1	NUM
cana-1239	204	18	≤	≤	NOUN
cana-1239	204	19	𝑙̇	𝑙̇	VERB
cana-1239	204	20	≤	≤	NUM
cana-1239	204	21	𝜂	𝜂	PRON
cana-1239	204	22	2𝑠	2𝑠	NOUN
cana-1239	204	23	+	+	CCONJ
cana-1239	204	24	2(|𝐸|	2(|𝐸|	NUM
cana-1239	204	25	−	−	PROPN
cana-1239	204	26	1)𝑑	1)𝑑	NUM
cana-1239	205	1	−	−	NOUN
cana-1239	205	2	(	(	PUNCT
cana-1239	205	3	𝑓(𝑣𝜂+1	𝑓(𝑣𝜂+1	NOUN
cana-1239	205	4	)	)	PUNCT
cana-1239	206	1	+	+	CCONJ
cana-1239	206	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	206	3	)	)	PUNCT
cana-1239	206	4	)	)	PUNCT
cana-1239	207	1	𝑙̇	𝑙̇	NOUN
cana-1239	208	1	=	=	PUNCT
cana-1239	208	2	𝜂	𝜂	NOUN
cana-1239	208	3	2𝑠	2𝑠	NOUN
cana-1239	208	4	+	+	CCONJ
cana-1239	208	5	2(|𝐸|	2(|𝐸|	NUM
cana-1239	208	6	−	−	PROPN
cana-1239	208	7	1)𝑑	1)𝑑	NUM
cana-1239	208	8	−	−	PROPN
cana-1239	208	9	(	(	PUNCT
cana-1239	208	10	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	208	11	)	)	PUNCT
cana-1239	209	1	+	+	CCONJ
cana-1239	209	2	𝑓(𝑣1	𝑓(𝑣1	ADJ
cana-1239	209	3	)	)	PUNCT
cana-1239	209	4	)	)	PUNCT
cana-1239	209	5	table	table	NOUN
cana-1239	209	6	12	12	NUM
cana-1239	209	7	:	:	PUNCT
cana-1239	209	8	labeling	labeling	NOUN
cana-1239	209	9	of	of	ADP
cana-1239	209	10	edges	edge	NOUN
cana-1239	209	11	of	of	ADP
cana-1239	209	12	the	the	DET
cana-1239	209	13	graph	graph	NOUN
cana-1239	209	14	𝑊𝜂	𝑊𝜂	PROPN
cana-1239	209	15	therefore𝑓(𝑣𝑙̇	therefore𝑓(𝑣𝑙̇	NUM
cana-1239	209	16	)	)	PUNCT
cana-1239	210	1	+	+	CCONJ
cana-1239	210	2	𝑓(𝑣𝑙̇+1	𝑓(𝑣𝑙̇+1	NUM
cana-1239	210	3	)	)	PUNCT
cana-1239	211	1	+	+	NUM
cana-1239	211	2	𝑔(𝑣𝑙̇𝑣𝑙̇+1	𝑔(𝑣𝑙̇𝑣𝑙̇+1	PROPN
cana-1239	211	3	)	)	PUNCT
cana-1239	211	4	,	,	PUNCT
cana-1239	211	5	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	211	6	)	)	PUNCT
cana-1239	212	1	+	+	CCONJ
cana-1239	212	2	𝑓(𝑣1	𝑓(𝑣1	ADJ
cana-1239	212	3	)	)	PUNCT
cana-1239	212	4	+	+	CCONJ
cana-1239	212	5	𝑔(𝑣𝑙̇𝑣1)and𝑓(𝑣𝜂+1	𝑔(𝑣𝑙̇𝑣1)and𝑓(𝑣𝜂+1	ADJ
cana-1239	212	6	)	)	PUNCT
cana-1239	213	1	+	+	X
cana-1239	213	2	𝑓(𝑣𝑙̇	𝑓(𝑣𝑙̇	X
cana-1239	213	3	)	)	PUNCT
cana-1239	214	1	+	+	CCONJ
cana-1239	214	2	𝑔(𝑣𝜂+1𝑣𝑙̇	𝑔(𝑣𝜂+1𝑣𝑙̇	ADV
cana-1239	214	3	)	)	PUNCT
cana-1239	214	4	are	be	AUX
cana-1239	214	5	constant	constant	ADJ
cana-1239	214	6	equals	equal	VERB
cana-1239	214	7	to	to	ADP
cana-1239	214	8	2(𝑠	2(𝑠	NUM
cana-1239	214	9	+	+	CCONJ
cana-1239	214	10	(	(	PUNCT
cana-1239	214	11	|𝐸|	|𝐸|	NOUN
cana-1239	214	12	−	−	PROPN
cana-1239	214	13	1)𝑑	1)𝑑	NUM
cana-1239	214	14	)	)	PUNCT
cana-1239	214	15	.	.	PUNCT
cana-1239	215	1	hence	hence	ADV
cana-1239	215	2	the	the	DET
cana-1239	215	3	wheel	wheel	NOUN
cana-1239	215	4	graph	graph	NOUN
cana-1239	215	5	𝑊𝜂	𝑊𝜂	PROPN
cana-1239	215	6	admits	admit	VERB
cana-1239	215	7	(	(	PUNCT
cana-1239	215	8	𝑠	𝑠	PROPN
cana-1239	215	9	,	,	PUNCT
cana-1239	215	10	𝑑	𝑑	NOUN
cana-1239	215	11	)	)	PUNCT
cana-1239	215	12	magic	magic	ADJ
cana-1239	215	13	labeling	labeling	NOUN
cana-1239	215	14	.	.	PUNCT
cana-1239	216	1	communications	communication	NOUN
cana-1239	216	2	on	on	ADP
cana-1239	216	3	applied	apply	VERB
cana-1239	216	4	nonlinear	nonlinear	ADJ
cana-1239	216	5	analysis	analysis	NOUN
cana-1239	216	6	issn	issn	NOUN
cana-1239	216	7	:	:	PUNCT
cana-1239	216	8	1074	1074	NUM
cana-1239	216	9	-	-	PUNCT
cana-1239	216	10	133x	133x	NUM
cana-1239	216	11	vol	vol	NOUN
cana-1239	216	12	31	31	NUM
cana-1239	216	13	no	no	NOUN
cana-1239	216	14	.	.	PUNCT
cana-1239	217	1	6s	6s	NUM
cana-1239	217	2	(	(	PUNCT
cana-1239	217	3	2024	2024	NUM
cana-1239	217	4	)	)	PUNCT
cana-1239	217	5	498	498	NUM
cana-1239	218	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1239	218	2	figure	figure	NOUN
cana-1239	218	3	2	2	NUM
cana-1239	218	4	:	:	PUNCT
cana-1239	218	5	wheel	wheel	NOUN
cana-1239	218	6	graph	graph	NOUN
cana-1239	218	7	𝑾𝟔	𝑾𝟔	PROPN
cana-1239	218	8	4	4	NUM
cana-1239	218	9	.	.	PUNCT
cana-1239	218	10	conclusion	conclusion	VERB
cana-1239	218	11	the	the	DET
cana-1239	218	12	discovery	discovery	NOUN
cana-1239	218	13	of	of	ADP
cana-1239	218	14	the	the	DET
cana-1239	218	15	(	(	PUNCT
cana-1239	218	16	s	s	PROPN
cana-1239	218	17	,	,	PUNCT
cana-1239	218	18	d	d	NOUN
cana-1239	218	19	)	)	PUNCT
cana-1239	218	20	magic	magic	ADJ
cana-1239	218	21	labeling	labeling	NOUN
cana-1239	218	22	for	for	ADP
cana-1239	218	23	diverse	diverse	ADJ
cana-1239	218	24	cycle	cycle	NOUN
cana-1239	218	25	graphs	graph	NOUN
cana-1239	218	26	underscores	underscore	VERB
cana-1239	218	27	the	the	DET
cana-1239	218	28	versatility	versatility	NOUN
cana-1239	218	29	and	and	CCONJ
cana-1239	218	30	applicability	applicability	NOUN
cana-1239	218	31	of	of	ADP
cana-1239	218	32	this	this	DET
cana-1239	218	33	labeling	labeling	NOUN
cana-1239	218	34	scheme	scheme	NOUN
cana-1239	218	35	across	across	ADP
cana-1239	218	36	different	different	ADJ
cana-1239	218	37	graph	graph	NOUN
cana-1239	218	38	structures	structure	NOUN
cana-1239	218	39	.	.	PUNCT
cana-1239	219	1	this	this	DET
cana-1239	219	2	finding	find	VERB
cana-1239	219	3	not	not	PART
cana-1239	219	4	only	only	ADV
cana-1239	219	5	expands	expand	VERB
cana-1239	219	6	our	our	PRON
cana-1239	219	7	understanding	understanding	NOUN
cana-1239	219	8	of	of	ADP
cana-1239	219	9	labeling	labeling	NOUN
cana-1239	219	10	techniques	technique	NOUN
cana-1239	219	11	but	but	CCONJ
cana-1239	219	12	also	also	ADV
cana-1239	219	13	opens	open	VERB
cana-1239	219	14	up	up	ADP
cana-1239	219	15	possibilities	possibility	NOUN
cana-1239	219	16	for	for	ADP
cana-1239	219	17	further	further	ADJ
cana-1239	219	18	exploration	exploration	NOUN
cana-1239	219	19	in	in	ADP
cana-1239	219	20	graph	graph	NOUN
cana-1239	219	21	theory	theory	NOUN
cana-1239	219	22	and	and	CCONJ
cana-1239	219	23	its	its	PRON
cana-1239	219	24	applications	application	NOUN
cana-1239	219	25	.	.	PUNCT
cana-1239	220	1	moreover	moreover	ADV
cana-1239	220	2	,	,	PUNCT
cana-1239	220	3	the	the	DET
cana-1239	220	4	ability	ability	NOUN
cana-1239	220	5	to	to	PART
cana-1239	220	6	apply	apply	VERB
cana-1239	220	7	this	this	DET
cana-1239	220	8	labeling	labeling	NOUN
cana-1239	220	9	to	to	ADP
cana-1239	220	10	various	various	ADJ
cana-1239	220	11	cycle	cycle	NOUN
cana-1239	220	12	graphs	graph	NOUN
cana-1239	220	13	suggests	suggest	VERB
cana-1239	220	14	potential	potential	ADJ
cana-1239	220	15	advancements	advancement	NOUN
cana-1239	220	16	in	in	ADP
cana-1239	220	17	areas	area	NOUN
cana-1239	220	18	such	such	ADJ
cana-1239	220	19	as	as	ADP
cana-1239	220	20	network	network	NOUN
cana-1239	220	21	design	design	NOUN
cana-1239	220	22	,	,	PUNCT
cana-1239	220	23	communication	communication	NOUN
cana-1239	220	24	systems	system	NOUN
cana-1239	220	25	,	,	PUNCT
cana-1239	220	26	and	and	CCONJ
cana-1239	220	27	algorithm	algorithm	NOUN
cana-1239	220	28	development	development	NOUN
cana-1239	220	29	.	.	PUNCT
cana-1239	221	1	further	further	ADJ
cana-1239	221	2	research	research	NOUN
cana-1239	221	3	could	could	AUX
cana-1239	221	4	focus	focus	VERB
cana-1239	221	5	on	on	ADP
cana-1239	221	6	refining	refine	VERB
cana-1239	221	7	the	the	DET
cana-1239	221	8	(	(	PUNCT
cana-1239	221	9	s	s	PROPN
cana-1239	221	10	,	,	PUNCT
cana-1239	221	11	d	d	NOUN
cana-1239	221	12	)	)	PUNCT
cana-1239	221	13	magic	magic	ADJ
cana-1239	221	14	labeling	labeling	NOUN
cana-1239	221	15	method	method	NOUN
cana-1239	221	16	or	or	CCONJ
cana-1239	221	17	exploring	explore	VERB
cana-1239	221	18	its	its	PRON
cana-1239	221	19	implications	implication	NOUN
cana-1239	221	20	in	in	ADP
cana-1239	221	21	other	other	ADJ
cana-1239	221	22	graph	graph	NOUN
cana-1239	221	23	theoretic	theoretic	ADJ
cana-1239	221	24	contexts	contexts	NOUN
cana-1239	221	25	.	.	PUNCT
cana-1239	222	1	future	future	ADJ
cana-1239	222	2	research	research	NOUN
cana-1239	222	3	will	will	AUX
cana-1239	222	4	examine	examine	VERB
cana-1239	222	5	the	the	DET
cana-1239	222	6	(	(	PUNCT
cana-1239	222	7	𝑠	𝑠	PROPN
cana-1239	222	8	,	,	PUNCT
cana-1239	222	9	𝑑	𝑑	NOUN
cana-1239	222	10	)	)	PUNCT
cana-1239	222	11	magic	magic	ADJ
cana-1239	222	12	labeling	labeling	NOUN
cana-1239	222	13	of	of	ADP
cana-1239	222	14	additional	additional	ADJ
cana-1239	222	15	graphs	graph	NOUN
cana-1239	222	16	and	and	CCONJ
cana-1239	222	17	some	some	DET
cana-1239	222	18	graph	graph	NOUN
cana-1239	222	19	families	family	NOUN
cana-1239	222	20	.	.	PUNCT
cana-1239	223	1	references	reference	NOUN
cana-1239	223	2	[	[	X
cana-1239	223	3	1	1	NUM
cana-1239	223	4	]	]	PUNCT
cana-1239	223	5	.	.	PUNCT
cana-1239	224	1	f.	f.	PROPN
cana-1239	224	2	harary	harary	PROPN
cana-1239	224	3	,	,	PUNCT
cana-1239	224	4	graph	graph	NOUN
cana-1239	224	5	theory	theory	NOUN
cana-1239	224	6	,	,	PUNCT
cana-1239	224	7	addision	addision	NOUN
cana-1239	224	8	-	-	PUNCT
cana-1239	224	9	wesley	wesley	PROPN
cana-1239	224	10	,	,	PUNCT
cana-1239	224	11	reading	reading	NOUN
cana-1239	224	12	,	,	PUNCT
cana-1239	224	13	ma	ma	PROPN
cana-1239	224	14	,	,	PUNCT
cana-1239	224	15	1969	1969	NUM
cana-1239	224	16	.	.	PUNCT
cana-1239	225	1	[	[	X
cana-1239	225	2	2	2	NUM
cana-1239	225	3	]	]	PUNCT
cana-1239	225	4	.	.	PUNCT
cana-1239	226	1	farida	farida	PROPN
cana-1239	226	2	,	,	PUNCT
cana-1239	226	3	a.	a.	NOUN
cana-1239	226	4	,	,	PUNCT
cana-1239	226	5	indah	indah	PROPN
cana-1239	226	6	,	,	PUNCT
cana-1239	226	7	r.	r.	PROPN
cana-1239	226	8	p.	p.	PROPN
cana-1239	226	9	,	,	PUNCT
cana-1239	226	10	&	&	CCONJ
cana-1239	226	11	sudibyo	sudibyo	NOUN
cana-1239	226	12	,	,	PUNCT
cana-1239	226	13	n.	n.	NOUN
cana-1239	226	14	a.	a.	NOUN
cana-1239	226	15	magic	magic	NOUN
cana-1239	226	16	covering	covering	NOUN
cana-1239	226	17	and	and	CCONJ
cana-1239	226	18	edge	edge	NOUN
cana-1239	226	19	magic	magic	ADJ
cana-1239	226	20	labelling	labelling	NOUN
cana-1239	226	21	and	and	CCONJ
cana-1239	226	22	its	its	PRON
cana-1239	226	23	application	application	NOUN
cana-1239	226	24	.	.	PUNCT
cana-1239	227	1	in	in	ADP
cana-1239	227	2	journal	journal	PROPN
cana-1239	227	3	of	of	ADP
cana-1239	227	4	physics	physics	PROPN
cana-1239	227	5	:	:	PUNCT
cana-1239	227	6	conference	conference	NOUN
cana-1239	227	7	series	series	PROPN
cana-1239	227	8	vol	vol	NOUN
cana-1239	227	9	.	.	PROPN
cana-1239	227	10	1657	1657	NUM
cana-1239	227	11	,	,	PUNCT
cana-1239	227	12	no	no	INTJ
cana-1239	227	13	.	.	NOUN
cana-1239	227	14	1	1	NUM
cana-1239	227	15	,	,	PUNCT
cana-1239	227	16	p.	p.	NOUN
cana-1239	227	17	012051	012051	NUM
cana-1239	227	18	,	,	PUNCT
cana-1239	227	19	(	(	PUNCT
cana-1239	227	20	2020	2020	NUM
cana-1239	227	21	)	)	PUNCT
cana-1239	227	22	.	.	PUNCT
cana-1239	228	1	iop	iop	NOUN
cana-1239	228	2	publishing	publishing	NOUN
cana-1239	228	3	.	.	PUNCT
cana-1239	229	1	[	[	X
cana-1239	229	2	3	3	NUM
cana-1239	229	3	]	]	PUNCT
cana-1239	229	4	.	.	PUNCT
cana-1239	230	1	gallian	gallian	PROPN
cana-1239	230	2	ja	ja	PROPN
cana-1239	230	3	.	.	PUNCT
cana-1239	231	1	a	a	DET
cana-1239	231	2	dynamic	dynamic	ADJ
cana-1239	231	3	survey	survey	NOUN
cana-1239	231	4	of	of	ADP
cana-1239	231	5	graph	graph	NOUN
cana-1239	231	6	labeling	labeling	NOUN
cana-1239	231	7	.	.	PUNCT
cana-1239	232	1	the	the	DET
cana-1239	232	2	electronic	electronic	ADJ
cana-1239	232	3	journal	journal	NOUN
cana-1239	232	4	of	of	ADP
cana-1239	232	5	combinatorics	combinatoric	NOUN
cana-1239	232	6	.	.	PUNCT
cana-1239	232	7	2022	2022	NUM
cana-1239	232	8	.	.	PUNCT
cana-1239	232	9	available	available	ADJ
cana-1239	232	10	from	from	ADP
cana-1239	232	11	:	:	PUNCT
cana-1239	232	12	http://www.combinatorics.org	http://www.combinatorics.org	PUNCT
cana-1239	233	1	[	[	X
cana-1239	233	2	4	4	NUM
cana-1239	233	3	]	]	PUNCT
cana-1239	233	4	.	.	PUNCT
cana-1239	234	1	ghodasara	ghodasara	PROPN
cana-1239	234	2	,	,	PUNCT
cana-1239	234	3	g.	g.	PROPN
cana-1239	234	4	v.	v.	PROPN
cana-1239	234	5	,	,	PUNCT
cana-1239	234	6	&	&	CCONJ
cana-1239	234	7	jena	jena	PROPN
cana-1239	234	8	,	,	PUNCT
cana-1239	234	9	j.	j.	PROPN
cana-1239	234	10	p	p	PROPN
cana-1239	234	11	..	..	PUNCT
cana-1239	234	12	prime	prime	ADJ
cana-1239	234	13	cordial	cordial	ADJ
cana-1239	234	14	labeling	labeling	NOUN
cana-1239	234	15	of	of	ADP
cana-1239	234	16	some	some	DET
cana-1239	234	17	special	special	ADJ
cana-1239	234	18	graph	graph	NOUN
cana-1239	234	19	families	family	NOUN
cana-1239	234	20	.	.	PUNCT
cana-1239	235	1	international	international	ADJ
cana-1239	235	2	journal	journal	PROPN
cana-1239	235	3	of	of	ADP
cana-1239	235	4	mathematics	mathematic	NOUN
cana-1239	235	5	and	and	CCONJ
cana-1239	235	6	soft	soft	ADJ
cana-1239	235	7	computing	computing	NOUN
cana-1239	235	8	,	,	PUNCT
cana-1239	235	9	4(2	4(2	NUM
cana-1239	235	10	)	)	PUNCT
cana-1239	235	11	,	,	PUNCT
cana-1239	235	12	41	41	NUM
cana-1239	235	13	-	-	SYM
cana-1239	235	14	48	48	NUM
cana-1239	235	15	,	,	PUNCT
cana-1239	235	16	(	(	PUNCT
cana-1239	235	17	2014	2014	NUM
cana-1239	235	18	)	)	PUNCT
cana-1239	235	19	.	.	PUNCT
cana-1239	236	1	[	[	X
cana-1239	236	2	5	5	NUM
cana-1239	236	3	]	]	PUNCT
cana-1239	236	4	.	.	PUNCT
cana-1239	237	1	hegde	hegde	PROPN
cana-1239	237	2	sm	sm	PROPN
cana-1239	237	3	,	,	PUNCT
cana-1239	237	4	shetty	shetty	PROPN
cana-1239	237	5	s.	s.	PROPN
cana-1239	237	6	on	on	ADP
cana-1239	237	7	arithmetic	arithmetic	ADJ
cana-1239	237	8	graphs	graph	NOUN
cana-1239	237	9	.	.	PUNCT
cana-1239	238	1	[	[	X
cana-1239	238	2	6	6	NUM
cana-1239	238	3	]	]	PUNCT
cana-1239	238	4	.	.	PUNCT
cana-1239	239	1	j.	j.	PROPN
cana-1239	239	2	baskar	baskar	PROPN
cana-1239	239	3	babujee	babujee	NOUN
cana-1239	239	4	and	and	CCONJ
cana-1239	239	5	s.	s.	PROPN
cana-1239	239	6	babitha	babitha	PROPN
cana-1239	239	7	,	,	PUNCT
cana-1239	239	8	prime	prime	ADJ
cana-1239	239	9	cordial	cordial	ADJ
cana-1239	239	10	labeling	labeling	NOUN
cana-1239	239	11	on	on	ADP
cana-1239	239	12	graphs	graph	NOUN
cana-1239	239	13	,	,	PUNCT
cana-1239	239	14	international	international	ADJ
cana-1239	239	15	journal	journal	NOUN
cana-1239	239	16	of	of	ADP
cana-1239	239	17	mathematical	mathematical	ADJ
cana-1239	239	18	sciences	science	NOUN
cana-1239	239	19	,	,	PUNCT
cana-1239	239	20	7(1)(2013	7(1)(2013	NUM
cana-1239	239	21	)	)	PUNCT
cana-1239	239	22	.	.	PUNCT
cana-1239	240	1	[	[	X
cana-1239	240	2	7	7	NUM
cana-1239	240	3	]	]	PUNCT
cana-1239	240	4	.	.	PUNCT
cana-1239	241	1	kavitha	kavitha	PROPN
cana-1239	241	2	k	k	PROPN
cana-1239	241	3	,	,	PUNCT
cana-1239	241	4	thirusangu	thirusangu	PROPN
cana-1239	241	5	k.	k.	PROPN
cana-1239	241	6	group	group	PROPN
cana-1239	241	7	magic	magic	PROPN
cana-1239	241	8	labeling	labeling	NOUN
cana-1239	241	9	of	of	ADP
cana-1239	241	10	multiple	multiple	ADJ
cana-1239	241	11	cycles	cycle	NOUN
cana-1239	241	12	.	.	PUNCT
cana-1239	242	1	international	international	ADJ
cana-1239	242	2	journal	journal	PROPN
cana-1239	242	3	of	of	ADP
cana-1239	242	4	computer	computer	NOUN
cana-1239	242	5	applications	application	NOUN
cana-1239	242	6	.	.	PUNCT
cana-1239	243	1	jan	jan	PROPN
cana-1239	243	2	1;59	1;59	NUM
cana-1239	243	3	,	,	PUNCT
cana-1239	243	4	12	12	NUM
cana-1239	243	5	(	(	PUNCT
cana-1239	243	6	2012	2012	NUM
cana-1239	243	7	)	)	PUNCT
cana-1239	243	8	.	.	PUNCT
cana-1239	244	1	[	[	X
cana-1239	244	2	8	8	NUM
cana-1239	244	3	]	]	PUNCT
cana-1239	244	4	.	.	PUNCT
cana-1239	245	1	s.	s.	PROPN
cana-1239	245	2	k.	k.	PROPN
cana-1239	245	3	vaidya	vaidya	PROPN
cana-1239	245	4	and	and	CCONJ
cana-1239	245	5	p.	p.	PROPN
cana-1239	245	6	l.	l.	PROPN
cana-1239	245	7	vihol	vihol	PROPN
cana-1239	245	8	,	,	PUNCT
cana-1239	245	9	prime	prime	ADJ
cana-1239	245	10	cordial	cordial	ADJ
cana-1239	245	11	labeling	labeling	NOUN
cana-1239	245	12	for	for	ADP
cana-1239	245	13	some	some	DET
cana-1239	245	14	cycle	cycle	NOUN
cana-1239	245	15	related	relate	VERB
cana-1239	245	16	graphs	graph	NOUN
cana-1239	245	17	,	,	PUNCT
cana-1239	245	18	international	international	ADJ
cana-1239	245	19	journal	journal	NOUN
cana-1239	245	20	of	of	ADP
cana-1239	245	21	open	open	ADJ
cana-1239	245	22	problems	problem	NOUN
cana-1239	245	23	in	in	ADP
cana-1239	245	24	computure	computure	NOUN
cana-1239	245	25	science	science	NOUN
cana-1239	245	26	mathematics	mathematic	NOUN
cana-1239	245	27	,	,	PUNCT
cana-1239	245	28	3(5	3(5	NUM
cana-1239	245	29	)	)	PUNCT
cana-1239	245	30	,	,	PUNCT
cana-1239	245	31	223	223	NUM
cana-1239	245	32	–	–	PUNCT
cana-1239	245	33	232	232	NUM
cana-1239	245	34	,	,	PUNCT
cana-1239	245	35	(	(	PUNCT
cana-1239	245	36	2010	2010	NUM
cana-1239	245	37	)	)	PUNCT
cana-1239	245	38	.	.	PUNCT
cana-1239	246	1	[	[	X
cana-1239	246	2	9	9	NUM
cana-1239	246	3	]	]	PUNCT
cana-1239	246	4	.	.	PUNCT
cana-1239	247	1	sumathi	sumathi	PROPN
cana-1239	247	2	p	p	X
cana-1239	247	3	,	,	PUNCT
cana-1239	247	4	kumar	kumar	PROPN
cana-1239	247	5	js	js	PROPN
cana-1239	247	6	.	.	PUNCT
cana-1239	247	7	fuzzy	fuzzy	ADJ
cana-1239	247	8	quotient-3	quotient-3	NUM
cana-1239	247	9	cordial	cordial	ADJ
cana-1239	247	10	labeling	labeling	NOUN
cana-1239	247	11	on	on	ADP
cana-1239	247	12	some	some	DET
cana-1239	247	13	unicyclic	unicyclic	ADJ
cana-1239	247	14	graphs	graph	NOUN
cana-1239	247	15	with	with	ADP
cana-1239	247	16	pendant	pendant	ADJ
cana-1239	247	17	edges	edge	NOUN
cana-1239	247	18	.	.	PUNCT
cana-1239	248	1	in	in	ADP
cana-1239	248	2	aip	aip	PROPN
cana-1239	248	3	conference	conference	NOUN
cana-1239	248	4	proceedings	proceeding	NOUN
cana-1239	248	5	2023	2023	NUM
cana-1239	248	6	may	may	AUX
cana-1239	248	7	24	24	NUM
cana-1239	248	8	(	(	PUNCT
cana-1239	248	9	vol	vol	NOUN
cana-1239	248	10	.	.	NOUN
cana-1239	248	11	2718	2718	NUM
cana-1239	248	12	,	,	PUNCT
cana-1239	248	13	no	no	INTJ
cana-1239	248	14	.	.	NOUN
cana-1239	248	15	1	1	NUM
cana-1239	248	16	)	)	PUNCT
cana-1239	248	17	.	.	PUNCT
cana-1239	249	1	aip	aip	PROPN
cana-1239	249	2	publishing	publishing	PROPN
cana-1239	249	3	.	.	PUNCT
cana-1239	250	1	aip	aip	PROPN
cana-1239	250	2	conf	conf	PROPN
cana-1239	250	3	.	.	PUNCT
cana-1239	251	1	proc	proc	PROPN
cana-1239	251	2	.	.	PUNCT
cana-1239	252	1	2718	2718	NUM
cana-1239	252	2	,	,	PUNCT
cana-1239	252	3	020007	020007	NUM
cana-1239	252	4	(	(	PUNCT
cana-1239	252	5	2023	2023	NUM
cana-1239	252	6	)	)	PUNCT
cana-1239	252	7	.	.	PUNCT
cana-1239	253	1	https://doi.org/10.1063/5.0140418	https://doi.org/10.1063/5.0140418	PROPN
cana-1239	254	1	[	[	X
cana-1239	254	2	10	10	NUM
cana-1239	254	3	]	]	PUNCT
cana-1239	254	4	.	.	PUNCT
cana-1239	255	1	sumathi	sumathi	PROPN
cana-1239	255	2	p	p	X
cana-1239	255	3	,	,	PUNCT
cana-1239	255	4	mala	mala	PROPN
cana-1239	255	5	p.	p.	PROPN
cana-1239	255	6	(	(	PUNCT
cana-1239	255	7	s	s	PROPN
cana-1239	255	8	,	,	PUNCT
cana-1239	255	9	d	d	NOUN
cana-1239	255	10	)	)	PUNCT
cana-1239	255	11	magic	magic	ADJ
cana-1239	255	12	labeling	labeling	NOUN
cana-1239	255	13	of	of	ADP
cana-1239	255	14	some	some	DET
cana-1239	255	15	ladder	ladder	NOUN
cana-1239	255	16	graphs	graph	NOUN
cana-1239	255	17	.	.	PUNCT
cana-1239	256	1	ine3s	ine3	NOUN
cana-1239	256	2	web	web	NOUN
cana-1239	256	3	of	of	ADP
cana-1239	256	4	conferences	conference	NOUN
cana-1239	256	5	vol	vol	NOUN
cana-1239	256	6	.	.	PUNCT
cana-1239	257	1	376	376	NUM
cana-1239	257	2	(	(	PUNCT
cana-1239	257	3	2023	2023	NUM
cana-1239	257	4	)	)	PUNCT
cana-1239	257	5	.	.	PUNCT
cana-1239	258	1	edp	edp	PROPN
cana-1239	258	2	sciences	sciences	PROPN
cana-1239	258	3	.	.	PUNCT
cana-1239	259	1	http://www.combinatorics.org/	http://www.combinatorics.org/	PROPN
cana-1239	259	2	http://www.combinatorics.org/	http://www.combinatorics.org/	ADJ
cana-1239	259	3	https://doi.org/10.1063/5.0140418	https://doi.org/10.1063/5.0140418	NOUN