id	sid	tid	token	lemma	pos
cana-2520	1	1	communications	communication	NOUN
cana-2520	1	2	on	on	ADP
cana-2520	1	3	applied	apply	VERB
cana-2520	1	4	nonlinear	nonlinear	ADJ
cana-2520	1	5	analysis	analysis	NOUN
cana-2520	1	6	issn	issn	NOUN
cana-2520	1	7	:	:	PUNCT
cana-2520	1	8	1074	1074	NUM
cana-2520	1	9	-	-	PUNCT
cana-2520	1	10	133x	133x	NUM
cana-2520	1	11	vol	vol	NOUN
cana-2520	1	12	32	32	NUM
cana-2520	1	13	no	no	NOUN
cana-2520	1	14	.	.	PUNCT
cana-2520	2	1	2s	2s	NUM
cana-2520	2	2	(	(	PUNCT
cana-2520	2	3	2025	2025	NUM
cana-2520	2	4	)	)	PUNCT
cana-2520	2	5	592	592	NUM
cana-2520	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	2	7	prime	prime	ADJ
cana-2520	2	8	labeling	labeling	NOUN
cana-2520	2	9	of	of	ADP
cana-2520	2	10	bull	bull	NOUN
cana-2520	2	11	graph	graph	NOUN
cana-2520	2	12	dr	dr	PROPN
cana-2520	2	13	.	.	PROPN
cana-2520	2	14	m.	m.	NOUN
cana-2520	2	15	ganeshan	ganeshan	PROPN
cana-2520	2	16	assistant	assistant	NOUN
cana-2520	2	17	professor	professor	NOUN
cana-2520	2	18	,	,	PUNCT
cana-2520	2	19	pg	pg	NOUN
cana-2520	2	20	and	and	CCONJ
cana-2520	2	21	research	research	PROPN
cana-2520	2	22	department	department	PROPN
cana-2520	2	23	of	of	ADP
cana-2520	2	24	mathematics	mathematic	NOUN
cana-2520	2	25	,	,	PUNCT
cana-2520	2	26	agurchand	agurchand	VERB
cana-2520	2	27	manmull	manmull	ADJ
cana-2520	2	28	jain	jain	PROPN
cana-2520	2	29	college	college	PROPN
cana-2520	2	30	,	,	PUNCT
cana-2520	2	31	university	university	PROPN
cana-2520	2	32	of	of	ADP
cana-2520	2	33	madras	madras	PROPN
cana-2520	2	34	,	,	PUNCT
cana-2520	2	35	tamilnadu	tamilnadu	ADJ
cana-2520	2	36	,	,	PUNCT
cana-2520	2	37	chennai	chennai	PROPN
cana-2520	2	38	india	india	PROPN
cana-2520	2	39	.	.	PUNCT
cana-2520	3	1	email	email	NOUN
cana-2520	3	2	:	:	PUNCT
cana-2520	3	3	sivananthan21oct@gmail.com	sivananthan21oct@gmail.com	X
cana-2520	3	4	article	article	NOUN
cana-2520	3	5	history	history	NOUN
cana-2520	3	6	:	:	PUNCT
cana-2520	3	7	received	receive	VERB
cana-2520	3	8	:	:	PUNCT
cana-2520	3	9	27	27	NUM
cana-2520	3	10	-	-	SYM
cana-2520	3	11	09	09	NUM
cana-2520	3	12	-	-	PUNCT
cana-2520	3	13	2024	2024	NUM
cana-2520	3	14	revised	revise	VERB
cana-2520	3	15	:	:	PUNCT
cana-2520	3	16	02	02	NUM
cana-2520	3	17	-	-	SYM
cana-2520	3	18	11	11	NUM
cana-2520	3	19	-	-	PUNCT
cana-2520	3	20	2024	2024	NUM
cana-2520	3	21	accepted	accept	VERB
cana-2520	3	22	:	:	PUNCT
cana-2520	3	23	14	14	NUM
cana-2520	3	24	-	-	SYM
cana-2520	3	25	11	11	NUM
cana-2520	3	26	-	-	PUNCT
cana-2520	3	27	2024	2024	NUM
cana-2520	3	28	abstract	abstract	NOUN
cana-2520	3	29	:	:	PUNCT
cana-2520	3	30	let	let	VERB
cana-2520	3	31	g	g	PRON
cana-2520	3	32	be	be	AUX
cana-2520	3	33	a	a	DET
cana-2520	3	34	graph	graph	NOUN
cana-2520	3	35	.	.	PUNCT
cana-2520	4	1	a	a	DET
cana-2520	4	2	bijection	bijection	ADJ
cana-2520	4	3	f	f	NOUN
cana-2520	4	4	:	:	PUNCT
cana-2520	4	5	v	v	NOUN
cana-2520	4	6	→	→	SYM
cana-2520	4	7	{	{	PUNCT
cana-2520	4	8	1,2	1,2	NUM
cana-2520	4	9	,	,	PUNCT
cana-2520	4	10	…	…	PUNCT
cana-2520	4	11	.	.	PUNCT
cana-2520	4	12	.	.	PUNCT
cana-2520	5	1	|v|	|v|	X
cana-2520	5	2	}	}	PUNCT
cana-2520	5	3	is	be	AUX
cana-2520	5	4	called	call	VERB
cana-2520	5	5	a	a	DET
cana-2520	5	6	prime	prime	ADJ
cana-2520	5	7	labeling	labeling	NOUN
cana-2520	5	8	[	[	X
cana-2520	5	9	3	3	X
cana-2520	5	10	]	]	X
cana-2520	5	11	if	if	SCONJ
cana-2520	5	12	for	for	ADP
cana-2520	5	13	each	each	DET
cana-2520	5	14	edge	edge	NOUN
cana-2520	5	15	e	e	NOUN
cana-2520	5	16	=	=	NOUN
cana-2520	5	17	uv	uv	NOUN
cana-2520	5	18	in	in	ADP
cana-2520	5	19	e	e	NOUN
cana-2520	5	20	,	,	PUNCT
cana-2520	5	21	we	we	PRON
cana-2520	5	22	have	have	VERB
cana-2520	5	23	gcd	gcd	NOUN
cana-2520	5	24	{	{	PUNCT
cana-2520	5	25	f(u	f(u	PROPN
cana-2520	5	26	)	)	PUNCT
cana-2520	5	27	,	,	PUNCT
cana-2520	5	28	f(v	f(v	NOUN
cana-2520	5	29	)	)	PUNCT
cana-2520	5	30	}	}	PUNCT
cana-2520	5	31	=	=	SYM
cana-2520	6	1	1	1	X
cana-2520	6	2	.	.	PUNCT
cana-2520	6	3	a	a	DET
cana-2520	6	4	graph	graph	NOUN
cana-2520	6	5	that	that	PRON
cana-2520	6	6	admits	admit	VERB
cana-2520	6	7	a	a	DET
cana-2520	6	8	prime	prime	ADJ
cana-2520	6	9	labeling	labeling	NOUN
cana-2520	6	10	is	be	AUX
cana-2520	6	11	said	say	VERB
cana-2520	6	12	to	to	PART
cana-2520	6	13	be	be	AUX
cana-2520	6	14	a	a	DET
cana-2520	6	15	prime	prime	ADJ
cana-2520	6	16	graph	graph	NOUN
cana-2520	6	17	.	.	PUNCT
cana-2520	7	1	in	in	ADP
cana-2520	7	2	this	this	DET
cana-2520	7	3	paper	paper	NOUN
cana-2520	7	4	we	we	PRON
cana-2520	7	5	show	show	VERB
cana-2520	7	6	that	that	SCONJ
cana-2520	7	7	bull	bull	NOUN
cana-2520	7	8	graph	graph	NOUN
cana-2520	7	9	admits	admit	VERB
cana-2520	7	10	prime	prime	ADJ
cana-2520	7	11	labeling	labeling	NOUN
cana-2520	7	12	in	in	ADP
cana-2520	7	13	the	the	DET
cana-2520	7	14	context	context	NOUN
cana-2520	7	15	of	of	ADP
cana-2520	7	16	variety	variety	NOUN
cana-2520	7	17	graph	graph	NOUN
cana-2520	7	18	operations	operation	NOUN
cana-2520	7	19	namely	namely	ADV
cana-2520	7	20	duplication	duplication	NOUN
cana-2520	7	21	of	of	ADP
cana-2520	7	22	vertex	vertex	NOUN
cana-2520	7	23	,	,	PUNCT
cana-2520	7	24	fusion	fusion	NOUN
cana-2520	7	25	of	of	ADP
cana-2520	7	26	vertices	vertex	NOUN
cana-2520	7	27	and	and	CCONJ
cana-2520	7	28	switching	switch	VERB
cana-2520	7	29	in	in	ADP
cana-2520	7	30	bull	bull	NOUN
cana-2520	7	31	graph	graph	NOUN
cana-2520	7	32	.	.	PUNCT
cana-2520	8	1	keywords	keyword	NOUN
cana-2520	8	2	:	:	PUNCT
cana-2520	8	3	prime	prime	ADJ
cana-2520	8	4	labeling	labeling	NOUN
cana-2520	8	5	,	,	PUNCT
cana-2520	8	6	bull	bull	NOUN
cana-2520	8	7	graph	graph	NOUN
cana-2520	8	8	,	,	PUNCT
cana-2520	8	9	duplication	duplication	NOUN
cana-2520	8	10	,	,	PUNCT
cana-2520	8	11	fusion	fusion	NOUN
cana-2520	8	12	and	and	CCONJ
cana-2520	8	13	switching	switching	NOUN
cana-2520	8	14	.	.	PUNCT
cana-2520	9	1	1	1	X
cana-2520	9	2	.	.	X
cana-2520	9	3	introduction	introduction	NOUN
cana-2520	9	4	graph	graph	NOUN
cana-2520	9	5	labeling	labeling	NOUN
cana-2520	9	6	is	be	AUX
cana-2520	9	7	one	one	NUM
cana-2520	9	8	of	of	ADP
cana-2520	9	9	the	the	DET
cana-2520	9	10	stimulating	stimulate	VERB
cana-2520	9	11	areas	area	NOUN
cana-2520	9	12	with	with	ADP
cana-2520	9	13	plentiful	plentiful	ADJ
cana-2520	9	14	applications	application	NOUN
cana-2520	9	15	in	in	ADP
cana-2520	9	16	various	various	ADJ
cana-2520	9	17	fields	field	NOUN
cana-2520	9	18	.	.	PUNCT
cana-2520	10	1	in	in	ADP
cana-2520	10	2	this	this	DET
cana-2520	10	3	paper	paper	NOUN
cana-2520	10	4	we	we	PRON
cana-2520	10	5	consider	consider	VERB
cana-2520	10	6	simple	simple	ADJ
cana-2520	10	7	and	and	CCONJ
cana-2520	10	8	finite	finite	ADJ
cana-2520	10	9	graphs	graph	NOUN
cana-2520	10	10	only	only	ADV
cana-2520	10	11	.	.	PUNCT
cana-2520	11	1	the	the	DET
cana-2520	11	2	notion	notion	NOUN
cana-2520	11	3	of	of	ADP
cana-2520	11	4	prime	prime	ADJ
cana-2520	11	5	labeling	labeling	NOUN
cana-2520	11	6	was	be	AUX
cana-2520	11	7	introduced	introduce	VERB
cana-2520	11	8	by	by	ADP
cana-2520	11	9	roger	roger	PROPN
cana-2520	11	10	entringer	entringer	PROPN
cana-2520	11	11	and	and	CCONJ
cana-2520	11	12	was	be	AUX
cana-2520	11	13	discussed	discuss	VERB
cana-2520	11	14	in	in	ADP
cana-2520	11	15	a	a	DET
cana-2520	11	16	paper	paper	NOUN
cana-2520	11	17	by	by	ADP
cana-2520	11	18	a.	a.	NOUN
cana-2520	11	19	tout	tout	PROPN
cana-2520	11	20	(	(	PUNCT
cana-2520	11	21	1982	1982	NUM
cana-2520	11	22	p	p	NOUN
cana-2520	11	23	365	365	NUM
cana-2520	11	24	-	-	SYM
cana-2520	11	25	368	368	NUM
cana-2520	11	26	)	)	PUNCT
cana-2520	11	27	.	.	PUNCT
cana-2520	12	1	this	this	DET
cana-2520	12	2	paper	paper	NOUN
cana-2520	12	3	is	be	AUX
cana-2520	12	4	organized	organize	VERB
cana-2520	12	5	as	as	SCONJ
cana-2520	12	6	follows	follow	VERB
cana-2520	12	7	.	.	PUNCT
cana-2520	13	1	in	in	ADP
cana-2520	13	2	section	section	NOUN
cana-2520	13	3	2	2	NUM
cana-2520	13	4	we	we	PRON
cana-2520	13	5	provide	provide	VERB
cana-2520	13	6	the	the	DET
cana-2520	13	7	preliminary	preliminary	ADJ
cana-2520	13	8	definitions	definition	NOUN
cana-2520	13	9	.	.	PUNCT
cana-2520	14	1	in	in	ADP
cana-2520	14	2	section	section	NOUN
cana-2520	14	3	3	3	NUM
cana-2520	14	4	,	,	PUNCT
cana-2520	14	5	we	we	PRON
cana-2520	14	6	prove	prove	VERB
cana-2520	14	7	the	the	DET
cana-2520	14	8	main	main	ADJ
cana-2520	14	9	results	result	NOUN
cana-2520	14	10	of	of	ADP
cana-2520	14	11	the	the	DET
cana-2520	14	12	paper	paper	NOUN
cana-2520	14	13	,	,	PUNCT
cana-2520	14	14	where	where	SCONJ
cana-2520	14	15	we	we	PRON
cana-2520	14	16	prove	prove	VERB
cana-2520	14	17	the	the	DET
cana-2520	14	18	graph	graph	NOUN
cana-2520	14	19	obtained	obtain	VERB
cana-2520	14	20	by	by	ADP
cana-2520	14	21	duplicating	duplicate	VERB
cana-2520	14	22	arbitrary	arbitrary	ADJ
cana-2520	14	23	vertex	vertex	NOUN
cana-2520	14	24	of	of	ADP
cana-2520	14	25	bull	bull	NOUN
cana-2520	14	26	graph	graph	NOUN
cana-2520	14	27	is	be	AUX
cana-2520	14	28	a	a	DET
cana-2520	14	29	prime	prime	ADJ
cana-2520	14	30	graph	graph	NOUN
cana-2520	14	31	,	,	PUNCT
cana-2520	14	32	the	the	DET
cana-2520	14	33	graph	graph	NOUN
cana-2520	14	34	obtained	obtain	VERB
cana-2520	14	35	by	by	ADP
cana-2520	14	36	switching	switching	NOUN
cana-2520	14	37	of	of	ADP
cana-2520	14	38	any	any	DET
cana-2520	14	39	vertex	vertex	NOUN
cana-2520	14	40	in	in	ADP
cana-2520	14	41	a	a	DET
cana-2520	14	42	bull	bull	NOUN
cana-2520	14	43	graph	graph	NOUN
cana-2520	14	44	is	be	AUX
cana-2520	14	45	a	a	DET
cana-2520	14	46	prime	prime	ADJ
cana-2520	14	47	graph	graph	NOUN
cana-2520	14	48	and	and	CCONJ
cana-2520	14	49	we	we	PRON
cana-2520	14	50	also	also	ADV
cana-2520	14	51	prove	prove	VERB
cana-2520	14	52	that	that	SCONJ
cana-2520	14	53	in	in	ADP
cana-2520	14	54	a	a	DET
cana-2520	14	55	bull	bull	NOUN
cana-2520	14	56	graph	graph	NOUN
cana-2520	14	57	fusion	fusion	NOUN
cana-2520	14	58	of	of	ADP
cana-2520	14	59	any	any	DET
cana-2520	14	60	arbitrary	arbitrary	ADJ
cana-2520	14	61	vertex	vertex	NOUN
cana-2520	14	62	with	with	ADP
cana-2520	14	63	𝑣1	𝑣1	NOUN
cana-2520	14	64	produces	produce	VERB
cana-2520	14	65	a	a	DET
cana-2520	14	66	prime	prime	ADJ
cana-2520	14	67	graph	graph	NOUN
cana-2520	14	68	in	in	ADP
cana-2520	14	69	section	section	NOUN
cana-2520	14	70	4	4	NUM
cana-2520	14	71	,	,	PUNCT
cana-2520	14	72	we	we	PRON
cana-2520	14	73	conclude	conclude	VERB
cana-2520	14	74	the	the	DET
cana-2520	14	75	paper	paper	NOUN
cana-2520	14	76	and	and	CCONJ
cana-2520	14	77	also	also	ADV
cana-2520	14	78	provide	provide	VERB
cana-2520	14	79	the	the	DET
cana-2520	14	80	insight	insight	NOUN
cana-2520	14	81	for	for	ADP
cana-2520	14	82	future	future	ADJ
cana-2520	14	83	work	work	NOUN
cana-2520	14	84	.	.	PUNCT
cana-2520	15	1	for	for	SCONJ
cana-2520	15	2	number	number	NOUN
cana-2520	15	3	theory	theory	NOUN
cana-2520	15	4	concept	concept	NOUN
cana-2520	15	5	refer	refer	VERB
cana-2520	15	6	[	[	X
cana-2520	15	7	2	2	NUM
cana-2520	15	8	]	]	PUNCT
cana-2520	15	9	.	.	PUNCT
cana-2520	16	1	2.preliminary	2.preliminary	NUM
cana-2520	16	2	definitions	definition	NOUN
cana-2520	16	3	definition	definition	NOUN
cana-2520	17	1	[	[	X
cana-2520	17	2	7]-2.1	7]-2.1	NUM
cana-2520	17	3	.	.	PUNCT
cana-2520	17	4	duplication	duplication	NOUN
cana-2520	17	5	of	of	ADP
cana-2520	17	6	a	a	DET
cana-2520	17	7	vertex	vertex	NOUN
cana-2520	17	8	vi	vi	NOUN
cana-2520	17	9	of	of	ADP
cana-2520	17	10	a	a	DET
cana-2520	17	11	graph	graph	NOUN
cana-2520	17	12	g	g	NOUN
cana-2520	17	13	produces	produce	VERB
cana-2520	17	14	a	a	DET
cana-2520	17	15	new	new	ADJ
cana-2520	17	16	graph	graph	NOUN
cana-2520	17	17	g1	g1	NOUN
cana-2520	17	18	by	by	ADP
cana-2520	17	19	adding	add	VERB
cana-2520	17	20	a	a	DET
cana-2520	17	21	vertex	vertex	NOUN
cana-2520	17	22	vi	vi	NOUN
cana-2520	17	23	′	′	NOUN
cana-2520	17	24	with	with	ADP
cana-2520	17	25	n(vi	n(vi	NUM
cana-2520	17	26	′	′	NUM
cana-2520	17	27	)	)	PUNCT
cana-2520	17	28	=	=	SYM
cana-2520	17	29	n(vi	n(vi	PROPN
cana-2520	17	30	)	)	PUNCT
cana-2520	17	31	.	.	PUNCT
cana-2520	18	1	in	in	ADP
cana-2520	18	2	other	other	ADJ
cana-2520	18	3	words	word	NOUN
cana-2520	18	4	,	,	PUNCT
cana-2520	18	5	a	a	DET
cana-2520	18	6	vertex	vertex	NOUN
cana-2520	18	7	vi	vi	NOUN
cana-2520	18	8	′	′	NOUN
cana-2520	18	9	is	be	AUX
cana-2520	18	10	said	say	VERB
cana-2520	18	11	to	to	PART
cana-2520	18	12	be	be	AUX
cana-2520	18	13	a	a	DET
cana-2520	18	14	duplication	duplication	NOUN
cana-2520	18	15	of	of	ADP
cana-2520	18	16	vi	vi	NOUN
cana-2520	18	17	if	if	SCONJ
cana-2520	18	18	all	all	DET
cana-2520	18	19	the	the	DET
cana-2520	18	20	vertices	vertex	NOUN
cana-2520	18	21	adjacent	adjacent	ADJ
cana-2520	18	22	to	to	ADP
cana-2520	18	23	vi	vi	PROPN
cana-2520	18	24	are	be	AUX
cana-2520	18	25	now	now	ADV
cana-2520	18	26	adjacent	adjacent	ADJ
cana-2520	18	27	to	to	PART
cana-2520	18	28	vi	vi	VERB
cana-2520	18	29	′	′	NUM
cana-2520	18	30	also	also	ADV
cana-2520	18	31	.	.	PUNCT
cana-2520	19	1	definition	definition	NOUN
cana-2520	19	2	[	[	X
cana-2520	19	3	7]-2.2	7]-2.2	X
cana-2520	19	4	.	.	PUNCT
cana-2520	20	1	let	let	VERB
cana-2520	20	2	u	u	PRON
cana-2520	20	3	and	and	CCONJ
cana-2520	20	4	v	v	NOUN
cana-2520	20	5	be	be	AUX
cana-2520	20	6	two	two	NUM
cana-2520	20	7	distinct	distinct	ADJ
cana-2520	20	8	vertices	vertex	NOUN
cana-2520	20	9	of	of	ADP
cana-2520	20	10	a	a	DET
cana-2520	20	11	graph	graph	NOUN
cana-2520	20	12	g.	g.	NOUN
cana-2520	20	13	a	a	DET
cana-2520	20	14	new	new	ADJ
cana-2520	20	15	graph	graph	NOUN
cana-2520	20	16	g1	g1	PROPN
cana-2520	20	17	is	be	AUX
cana-2520	20	18	constructed	construct	VERB
cana-2520	20	19	by	by	ADP
cana-2520	20	20	fusing	fuse	VERB
cana-2520	20	21	two	two	NUM
cana-2520	20	22	vertices	vertex	NOUN
cana-2520	20	23	u	u	NOUN
cana-2520	20	24	and	and	CCONJ
cana-2520	20	25	v	v	NUM
cana-2520	20	26	by	by	ADP
cana-2520	20	27	a	a	DET
cana-2520	20	28	single	single	ADJ
cana-2520	20	29	vertex	vertex	NOUN
cana-2520	20	30	w	w	ADP
cana-2520	20	31	such	such	ADJ
cana-2520	20	32	that	that	SCONJ
cana-2520	20	33	every	every	DET
cana-2520	20	34	edge	edge	NOUN
cana-2520	20	35	incident	incident	NOUN
cana-2520	20	36	to	to	ADP
cana-2520	20	37	u	u	NOUN
cana-2520	20	38	and	and	CCONJ
cana-2520	20	39	v	v	NOUN
cana-2520	20	40	is	be	AUX
cana-2520	20	41	now	now	ADV
cana-2520	20	42	incident	incident	NOUN
cana-2520	20	43	with	with	ADP
cana-2520	20	44	w	w	NOUN
cana-2520	20	45	in	in	ADP
cana-2520	20	46	g1	g1	NOUN
cana-2520	20	47	.	.	PUNCT
cana-2520	21	1	definition	definition	NOUN
cana-2520	21	2	[	[	X
cana-2520	21	3	7	7	NUM
cana-2520	21	4	]	]	SYM
cana-2520	21	5	-2.3	-2.3	NOUN
cana-2520	21	6	.	.	PUNCT
cana-2520	22	1	a	a	DET
cana-2520	22	2	vertex	vertex	NOUN
cana-2520	22	3	switching	switch	VERB
cana-2520	22	4	gu	gu	NOUN
cana-2520	22	5	in	in	ADP
cana-2520	22	6	a	a	DET
cana-2520	22	7	graph	graph	NOUN
cana-2520	22	8	g	g	NOUN
cana-2520	22	9	is	be	AUX
cana-2520	22	10	obtained	obtain	VERB
cana-2520	22	11	by	by	ADP
cana-2520	22	12	taking	take	VERB
cana-2520	22	13	a	a	DET
cana-2520	22	14	vertex	vertex	NOUN
cana-2520	22	15	u	u	NOUN
cana-2520	22	16	of	of	ADP
cana-2520	22	17	g	g	NOUN
cana-2520	22	18	,	,	PUNCT
cana-2520	22	19	removing	remove	VERB
cana-2520	22	20	all	all	DET
cana-2520	22	21	the	the	DET
cana-2520	22	22	edges	edge	NOUN
cana-2520	22	23	incident	incident	NOUN
cana-2520	22	24	to	to	ADP
cana-2520	22	25	u	u	NOUN
cana-2520	22	26	and	and	CCONJ
cana-2520	22	27	adding	add	VERB
cana-2520	22	28	edges	edge	NOUN
cana-2520	22	29	joining	join	VERB
cana-2520	22	30	u	u	NOUN
cana-2520	22	31	to	to	ADP
cana-2520	22	32	every	every	DET
cana-2520	22	33	non	non	ADJ
cana-2520	22	34	-	-	ADJ
cana-2520	22	35	adjacent	adjacent	ADJ
cana-2520	22	36	vertex	vertex	NOUN
cana-2520	22	37	of	of	ADP
cana-2520	22	38	u	u	PROPN
cana-2520	22	39	in	in	ADP
cana-2520	22	40	g.	g.	PROPN
cana-2520	22	41	definition	definition	NOUN
cana-2520	22	42	[	[	X
cana-2520	22	43	5	5	NUM
cana-2520	22	44	]	]	PUNCT
cana-2520	22	45	,	,	PUNCT
cana-2520	22	46	-2.4	-2.4	PROPN
cana-2520	22	47	.	.	PUNCT
cana-2520	23	1	the	the	DET
cana-2520	23	2	bull	bull	NOUN
cana-2520	23	3	graph	graph	NOUN
cana-2520	23	4	is	be	AUX
cana-2520	23	5	a	a	DET
cana-2520	23	6	graph	graph	NOUN
cana-2520	23	7	with	with	ADP
cana-2520	23	8	5	5	NUM
cana-2520	23	9	vertices	vertex	NOUN
cana-2520	23	10	and	and	CCONJ
cana-2520	23	11	5	5	NUM
cana-2520	23	12	edges	edge	NOUN
cana-2520	23	13	consisting	consist	VERB
cana-2520	23	14	of	of	ADP
cana-2520	23	15	a	a	DET
cana-2520	23	16	triangle	triangle	NOUN
cana-2520	23	17	with	with	ADP
cana-2520	23	18	two	two	NUM
cana-2520	23	19	disjoint	disjoint	ADJ
cana-2520	23	20	pendant	pendant	ADJ
cana-2520	23	21	edges	edge	NOUN
cana-2520	23	22	.	.	PUNCT
cana-2520	24	1	3.main	3.main	NUM
cana-2520	24	2	results	result	VERB
cana-2520	24	3	theorem-3.1	theorem-3.1	PROPN
cana-2520	24	4	.	.	PUNCT
cana-2520	25	1	the	the	DET
cana-2520	25	2	graph	graph	NOUN
cana-2520	25	3	obtained	obtain	VERB
cana-2520	25	4	by	by	ADP
cana-2520	25	5	duplicating	duplicate	VERB
cana-2520	25	6	arbitrary	arbitrary	ADJ
cana-2520	25	7	vertex	vertex	NOUN
cana-2520	25	8	of	of	ADP
cana-2520	25	9	bull	bull	NOUN
cana-2520	25	10	graph	graph	NOUN
cana-2520	25	11	is	be	AUX
cana-2520	25	12	a	a	DET
cana-2520	25	13	prime	prime	ADJ
cana-2520	25	14	graph	graph	NOUN
cana-2520	25	15	.	.	PUNCT
cana-2520	26	1	communications	communication	NOUN
cana-2520	26	2	on	on	ADP
cana-2520	26	3	applied	apply	VERB
cana-2520	26	4	nonlinear	nonlinear	ADJ
cana-2520	26	5	analysis	analysis	NOUN
cana-2520	26	6	issn	issn	NOUN
cana-2520	26	7	:	:	PUNCT
cana-2520	26	8	1074	1074	NUM
cana-2520	26	9	-	-	PUNCT
cana-2520	26	10	133x	133x	NUM
cana-2520	26	11	vol	vol	NOUN
cana-2520	26	12	32	32	NUM
cana-2520	26	13	no	no	NOUN
cana-2520	26	14	.	.	PUNCT
cana-2520	27	1	2s	2s	NUM
cana-2520	27	2	(	(	PUNCT
cana-2520	27	3	2025	2025	NUM
cana-2520	27	4	)	)	PUNCT
cana-2520	27	5	593	593	NUM
cana-2520	27	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	27	7	proof	proof	NOUN
cana-2520	27	8	:	:	PUNCT
cana-2520	27	9	figure	figure	NOUN
cana-2520	27	10	1	1	NUM
cana-2520	27	11	.	.	PUNCT
cana-2520	27	12	bull	bull	NOUN
cana-2520	27	13	graph	graph	NOUN
cana-2520	27	14	case-1	case-1	PROPN
cana-2520	27	15	.	.	PUNCT
cana-2520	28	1	duplication	duplication	NOUN
cana-2520	28	2	of	of	ADP
cana-2520	28	3	the	the	DET
cana-2520	28	4	vertex	vertex	NOUN
cana-2520	28	5	𝑣1	𝑣1	NOUN
cana-2520	28	6	let	let	VERB
cana-2520	28	7	𝐺1	𝐺1	NOUN
cana-2520	28	8	be	be	AUX
cana-2520	28	9	the	the	DET
cana-2520	28	10	graph	graph	NOUN
cana-2520	28	11	obtained	obtain	VERB
cana-2520	28	12	by	by	ADP
cana-2520	28	13	duplicating	duplicate	VERB
cana-2520	28	14	the	the	DET
cana-2520	28	15	vertex	vertex	NOUN
cana-2520	28	16	𝑣1	𝑣1	NOUN
cana-2520	28	17	define	define	VERB
cana-2520	28	18	ℬ	ℬ	X
cana-2520	28	19	:	:	PUNCT
cana-2520	28	20	𝑉	𝑉	PROPN
cana-2520	28	21	(	(	PUNCT
cana-2520	28	22	𝐺1	𝐺1	NOUN
cana-2520	28	23	)	)	PUNCT
cana-2520	28	24	→	→	SYM
cana-2520	28	25	{	{	PUNCT
cana-2520	28	26	1,2,3	1,2,3	NUM
cana-2520	28	27	,	,	PUNCT
cana-2520	28	28	…	…	PUNCT
cana-2520	28	29	.	.	PUNCT
cana-2520	28	30	.	.	PUNCT
cana-2520	29	1	,	,	PUNCT
cana-2520	29	2	6	6	X
cana-2520	29	3	}	}	PUNCT
cana-2520	29	4	by	by	ADP
cana-2520	29	5	ℬ(𝑣𝑖	ℬ(𝑣𝑖	ADJ
cana-2520	29	6	)	)	PUNCT
cana-2520	29	7	=	=	PUNCT
cana-2520	30	1	𝑖	𝑖	PROPN
cana-2520	31	1	+	+	NOUN
cana-2520	31	2	1	1	NUM
cana-2520	31	3	,	,	PUNCT
cana-2520	31	4	1	1	NUM
cana-2520	31	5	≤	≤	NUM
cana-2520	31	6	𝑖	𝑖	SYM
cana-2520	31	7	≤	≤	NOUN
cana-2520	31	8	5	5	NUM
cana-2520	31	9	and	and	CCONJ
cana-2520	31	10	ℬ(𝑣1	ℬ(𝑣1	ADJ
cana-2520	31	11	′	′	NUM
cana-2520	31	12	)	)	PUNCT
cana-2520	32	1	=	=	SYM
cana-2520	32	2	1	1	NUM
cana-2520	32	3	evidently	evidently	ADV
cana-2520	32	4	all	all	DET
cana-2520	32	5	the	the	DET
cana-2520	32	6	vertex	vertex	NOUN
cana-2520	32	7	labels	label	NOUN
cana-2520	32	8	are	be	AUX
cana-2520	32	9	distinct	distinct	ADJ
cana-2520	32	10	for	for	ADP
cana-2520	32	11	edges	edge	NOUN
cana-2520	32	12	in	in	ADP
cana-2520	32	13	𝐺1	𝐺1	NOUN
cana-2520	32	14	g.c.d	g.c.d	NOUN
cana-2520	32	15	(	(	PUNCT
cana-2520	32	16	ℬ(𝑣𝑖	ℬ(𝑣𝑖	PROPN
cana-2520	32	17	)	)	PUNCT
cana-2520	32	18	,	,	PUNCT
cana-2520	32	19	ℬ(𝑣𝑖+1	ℬ(𝑣𝑖+1	PROPN
cana-2520	32	20	)	)	PUNCT
cana-2520	32	21	)	)	PUNCT
cana-2520	33	1	=	=	PUNCT
cana-2520	33	2	1	1	NUM
cana-2520	33	3	,	,	PUNCT
cana-2520	33	4	1	1	NUM
cana-2520	33	5	≤	≤	NUM
cana-2520	33	6	𝑖	𝑖	SYM
cana-2520	33	7	≤	≤	NOUN
cana-2520	33	8	4	4	NUM
cana-2520	33	9	g.c.d	g.c.d	NOUN
cana-2520	33	10	(	(	PUNCT
cana-2520	33	11	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	33	12	)	)	PUNCT
cana-2520	33	13	,	,	PUNCT
cana-2520	33	14	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	33	15	)	)	PUNCT
cana-2520	33	16	)	)	PUNCT
cana-2520	34	1	=	=	SYM
cana-2520	34	2	1	1	NUM
cana-2520	34	3	g.c.d	g.c.d	NOUN
cana-2520	34	4	(	(	PUNCT
cana-2520	34	5	ℬ(𝑣1	ℬ(𝑣1	PROPN
cana-2520	34	6	′	′	NUM
cana-2520	34	7	)	)	PUNCT
cana-2520	34	8	,	,	PUNCT
cana-2520	34	9	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	34	10	)	)	PUNCT
cana-2520	34	11	)	)	PUNCT
cana-2520	34	12	=	=	SYM
cana-2520	35	1	1	1	NUM
cana-2520	35	2	clearly	clearly	ADV
cana-2520	35	3	ℬ	ℬ	PROPN
cana-2520	35	4	is	be	AUX
cana-2520	35	5	a	a	DET
cana-2520	35	6	prime	prime	ADJ
cana-2520	35	7	labeling	labeling	NOUN
cana-2520	35	8	on	on	ADP
cana-2520	35	9	𝐺1.hence	𝐺1.hence	NOUN
cana-2520	35	10	𝐺1	𝐺1	NOUN
cana-2520	35	11	is	be	AUX
cana-2520	35	12	a	a	DET
cana-2520	35	13	prime	prime	ADJ
cana-2520	35	14	graph	graph	NOUN
cana-2520	35	15	.	.	PUNCT
cana-2520	36	1	figure	figure	NOUN
cana-2520	36	2	2	2	NUM
cana-2520	36	3	.	.	PUNCT
cana-2520	36	4	prime	prime	ADJ
cana-2520	36	5	labeling	labeling	NOUN
cana-2520	36	6	of	of	ADP
cana-2520	36	7	duplication	duplication	NOUN
cana-2520	36	8	of	of	ADP
cana-2520	36	9	vertex	vertex	NOUN
cana-2520	36	10	𝑣1	𝑣1	NOUN
cana-2520	36	11	in	in	ADP
cana-2520	36	12	bull	bull	NOUN
cana-2520	36	13	graph	graph	NOUN
cana-2520	36	14	case-2	case-2	PROPN
cana-2520	36	15	.	.	PUNCT
cana-2520	37	1	duplication	duplication	NOUN
cana-2520	37	2	of	of	ADP
cana-2520	37	3	the	the	DET
cana-2520	37	4	vertex	vertex	NOUN
cana-2520	37	5	𝑣2	𝑣2	PRON
cana-2520	37	6	let	let	VERB
cana-2520	37	7	𝐺2	𝐺2	NOUN
cana-2520	37	8	be	be	AUX
cana-2520	37	9	the	the	DET
cana-2520	37	10	graph	graph	NOUN
cana-2520	37	11	obtained	obtain	VERB
cana-2520	37	12	by	by	ADP
cana-2520	37	13	duplicating	duplicate	VERB
cana-2520	37	14	the	the	DET
cana-2520	37	15	vertex	vertex	NOUN
cana-2520	37	16	𝑣2	𝑣2	NOUN
cana-2520	37	17	define	define	VERB
cana-2520	37	18	ℬ	ℬ	X
cana-2520	37	19	:	:	PUNCT
cana-2520	37	20	𝑉	𝑉	PROPN
cana-2520	37	21	(	(	PUNCT
cana-2520	37	22	𝐺2	𝐺2	ADJ
cana-2520	37	23	)	)	PUNCT
cana-2520	37	24	→	→	SYM
cana-2520	37	25	{	{	PUNCT
cana-2520	37	26	1,2,3	1,2,3	NUM
cana-2520	37	27	,	,	PUNCT
cana-2520	37	28	…	…	PUNCT
cana-2520	37	29	.	.	PUNCT
cana-2520	37	30	.	.	PUNCT
cana-2520	38	1	,	,	PUNCT
cana-2520	38	2	6	6	X
cana-2520	38	3	}	}	PUNCT
cana-2520	38	4	by	by	ADP
cana-2520	38	5	ℬ(𝑣𝑖	ℬ(𝑣𝑖	ADJ
cana-2520	38	6	)	)	PUNCT
cana-2520	38	7	=	=	PUNCT
cana-2520	39	1	𝑖	𝑖	PROPN
cana-2520	40	1	+	+	NOUN
cana-2520	40	2	1	1	NUM
cana-2520	40	3	,	,	PUNCT
cana-2520	40	4	1	1	NUM
cana-2520	40	5	≤	≤	NUM
cana-2520	40	6	𝑖	𝑖	SYM
cana-2520	40	7	≤	≤	NOUN
cana-2520	40	8	5	5	NUM
cana-2520	40	9	and	and	CCONJ
cana-2520	40	10	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	40	11	′	′	NUM
cana-2520	40	12	)	)	PUNCT
cana-2520	41	1	=	=	SYM
cana-2520	41	2	1	1	NUM
cana-2520	41	3	communications	communication	NOUN
cana-2520	41	4	on	on	ADP
cana-2520	41	5	applied	apply	VERB
cana-2520	41	6	nonlinear	nonlinear	ADJ
cana-2520	41	7	analysis	analysis	NOUN
cana-2520	41	8	issn	issn	NOUN
cana-2520	41	9	:	:	PUNCT
cana-2520	41	10	1074	1074	NUM
cana-2520	41	11	-	-	PUNCT
cana-2520	41	12	133x	133x	NUM
cana-2520	41	13	vol	vol	NOUN
cana-2520	41	14	32	32	NUM
cana-2520	41	15	no	no	NOUN
cana-2520	41	16	.	.	PUNCT
cana-2520	42	1	2s	2s	NUM
cana-2520	42	2	(	(	PUNCT
cana-2520	42	3	2025	2025	NUM
cana-2520	42	4	)	)	PUNCT
cana-2520	42	5	594	594	NUM
cana-2520	42	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	42	7	clearly	clearly	ADV
cana-2520	42	8	all	all	DET
cana-2520	42	9	the	the	DET
cana-2520	42	10	vertex	vertex	NOUN
cana-2520	42	11	labels	label	NOUN
cana-2520	42	12	are	be	AUX
cana-2520	42	13	distinct	distinct	ADJ
cana-2520	42	14	for	for	ADP
cana-2520	42	15	edges	edge	NOUN
cana-2520	42	16	in	in	ADP
cana-2520	42	17	𝐺2	𝐺2	ADJ
cana-2520	42	18	g.c.d	g.c.d	NOUN
cana-2520	42	19	(	(	PUNCT
cana-2520	42	20	ℬ(𝑣𝑖	ℬ(𝑣𝑖	PROPN
cana-2520	42	21	)	)	PUNCT
cana-2520	42	22	,	,	PUNCT
cana-2520	42	23	ℬ(𝑣𝑖+1	ℬ(𝑣𝑖+1	PROPN
cana-2520	42	24	)	)	PUNCT
cana-2520	42	25	)	)	PUNCT
cana-2520	43	1	=	=	PUNCT
cana-2520	43	2	1	1	NUM
cana-2520	43	3	,	,	PUNCT
cana-2520	43	4	1	1	NUM
cana-2520	43	5	≤	≤	NUM
cana-2520	43	6	𝑖	𝑖	SYM
cana-2520	43	7	≤	≤	NOUN
cana-2520	43	8	4	4	NUM
cana-2520	43	9	g.c.d	g.c.d	NOUN
cana-2520	43	10	(	(	PUNCT
cana-2520	43	11	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	43	12	)	)	PUNCT
cana-2520	43	13	,	,	PUNCT
cana-2520	43	14	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	43	15	)	)	PUNCT
cana-2520	43	16	)	)	PUNCT
cana-2520	44	1	=	=	SYM
cana-2520	44	2	1	1	NUM
cana-2520	44	3	g.c.d	g.c.d	NOUN
cana-2520	44	4	(	(	PUNCT
cana-2520	44	5	ℬ(𝑣2	ℬ(𝑣2	INTJ
cana-2520	44	6	′	′	NUM
cana-2520	44	7	)	)	PUNCT
cana-2520	44	8	,	,	PUNCT
cana-2520	44	9	ℬ(𝑣1	ℬ(𝑣1	PROPN
cana-2520	44	10	)	)	PUNCT
cana-2520	44	11	)	)	PUNCT
cana-2520	45	1	=	=	SYM
cana-2520	45	2	1	1	NUM
cana-2520	45	3	g.c.d	g.c.d	NOUN
cana-2520	45	4	(	(	PUNCT
cana-2520	45	5	ℬ(𝑣2	ℬ(𝑣2	INTJ
cana-2520	45	6	′	′	NUM
cana-2520	45	7	)	)	PUNCT
cana-2520	45	8	,	,	PUNCT
cana-2520	45	9	ℬ(𝑣3	ℬ(𝑣3	NOUN
cana-2520	45	10	)	)	PUNCT
cana-2520	45	11	)	)	PUNCT
cana-2520	45	12	=	=	SYM
cana-2520	45	13	1	1	NUM
cana-2520	45	14	g.c.d	g.c.d	NOUN
cana-2520	45	15	(	(	PUNCT
cana-2520	45	16	ℬ(𝑣2	ℬ(𝑣2	INTJ
cana-2520	45	17	′	′	NUM
cana-2520	45	18	)	)	PUNCT
cana-2520	45	19	,	,	PUNCT
cana-2520	45	20	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	45	21	)	)	PUNCT
cana-2520	45	22	)	)	PUNCT
cana-2520	45	23	=	=	SYM
cana-2520	46	1	1	1	NUM
cana-2520	46	2	therefore	therefore	ADV
cana-2520	46	3	ℬ	ℬ	PROPN
cana-2520	46	4	is	be	AUX
cana-2520	46	5	a	a	DET
cana-2520	46	6	prime	prime	ADJ
cana-2520	46	7	labeling	labeling	NOUN
cana-2520	46	8	on	on	ADP
cana-2520	46	9	𝐺2	𝐺2	NOUN
cana-2520	46	10	.	.	PUNCT
cana-2520	47	1	hence	hence	ADV
cana-2520	47	2	𝐺2	𝐺2	ADV
cana-2520	47	3	is	be	AUX
cana-2520	47	4	a	a	DET
cana-2520	47	5	prime	prime	ADJ
cana-2520	47	6	graph	graph	NOUN
cana-2520	47	7	.	.	PUNCT
cana-2520	48	1	figure	figure	NOUN
cana-2520	48	2	3	3	NUM
cana-2520	48	3	.	.	PUNCT
cana-2520	48	4	prime	prime	ADJ
cana-2520	48	5	labeling	labeling	NOUN
cana-2520	48	6	of	of	ADP
cana-2520	48	7	duplication	duplication	NOUN
cana-2520	48	8	of	of	ADP
cana-2520	48	9	vertex	vertex	NOUN
cana-2520	48	10	𝑣2	𝑣2	PROPN
cana-2520	48	11	in	in	ADP
cana-2520	48	12	bull	bull	NOUN
cana-2520	48	13	graph	graph	NOUN
cana-2520	48	14	case-3	case-3	PROPN
cana-2520	48	15	.	.	PUNCT
cana-2520	49	1	duplication	duplication	NOUN
cana-2520	49	2	of	of	ADP
cana-2520	49	3	the	the	DET
cana-2520	49	4	vertex	vertex	NOUN
cana-2520	49	5	𝑣3	𝑣3	PROPN
cana-2520	49	6	let	let	VERB
cana-2520	49	7	𝐺3	𝐺3	PROPN
cana-2520	49	8	be	be	AUX
cana-2520	49	9	the	the	DET
cana-2520	49	10	graph	graph	NOUN
cana-2520	49	11	obtained	obtain	VERB
cana-2520	49	12	by	by	ADP
cana-2520	49	13	duplicating	duplicate	VERB
cana-2520	49	14	the	the	DET
cana-2520	49	15	vertex	vertex	NOUN
cana-2520	49	16	𝑣3	𝑣3	NOUN
cana-2520	49	17	define	define	VERB
cana-2520	49	18	ℬ	ℬ	NOUN
cana-2520	49	19	:	:	PUNCT
cana-2520	49	20	𝑉	𝑉	PROPN
cana-2520	49	21	(	(	PUNCT
cana-2520	49	22	𝐺3	𝐺3	PROPN
cana-2520	49	23	)	)	PUNCT
cana-2520	49	24	→	→	SYM
cana-2520	49	25	{	{	PUNCT
cana-2520	49	26	1,2,3	1,2,3	NUM
cana-2520	49	27	,	,	PUNCT
cana-2520	49	28	…	…	PUNCT
cana-2520	49	29	.	.	PUNCT
cana-2520	49	30	.	.	PUNCT
cana-2520	50	1	,	,	PUNCT
cana-2520	50	2	6	6	X
cana-2520	50	3	}	}	PUNCT
cana-2520	50	4	by	by	ADP
cana-2520	50	5	ℬ(𝑣𝑖	ℬ(𝑣𝑖	ADJ
cana-2520	50	6	)	)	PUNCT
cana-2520	50	7	=	=	PUNCT
cana-2520	51	1	𝑖	𝑖	PROPN
cana-2520	52	1	+	+	NOUN
cana-2520	52	2	1	1	NUM
cana-2520	52	3	,	,	PUNCT
cana-2520	52	4	1	1	NUM
cana-2520	52	5	≤	≤	NUM
cana-2520	52	6	𝑖	𝑖	SYM
cana-2520	52	7	≤	≤	NOUN
cana-2520	52	8	5	5	NUM
cana-2520	52	9	and	and	CCONJ
cana-2520	52	10	ℬ(𝑣3	ℬ(𝑣3	X
cana-2520	52	11	′	′	NUM
cana-2520	52	12	)	)	PUNCT
cana-2520	53	1	=	=	SYM
cana-2520	53	2	1	1	NUM
cana-2520	53	3	clearly	clearly	ADV
cana-2520	53	4	all	all	DET
cana-2520	53	5	the	the	DET
cana-2520	53	6	vertex	vertex	NOUN
cana-2520	53	7	labels	label	NOUN
cana-2520	53	8	are	be	AUX
cana-2520	53	9	distinct	distinct	ADJ
cana-2520	53	10	for	for	ADP
cana-2520	53	11	edges	edge	NOUN
cana-2520	53	12	in	in	ADP
cana-2520	53	13	𝐺3	𝐺3	PRON
cana-2520	53	14	g.c.d	g.c.d	NOUN
cana-2520	53	15	(	(	PUNCT
cana-2520	53	16	ℬ(𝑣𝑖	ℬ(𝑣𝑖	PROPN
cana-2520	53	17	)	)	PUNCT
cana-2520	53	18	,	,	PUNCT
cana-2520	53	19	ℬ(𝑣𝑖+1	ℬ(𝑣𝑖+1	PROPN
cana-2520	53	20	)	)	PUNCT
cana-2520	53	21	)	)	PUNCT
cana-2520	54	1	=	=	PUNCT
cana-2520	54	2	1	1	NUM
cana-2520	54	3	,	,	PUNCT
cana-2520	54	4	1	1	NUM
cana-2520	54	5	≤	≤	NUM
cana-2520	54	6	𝑖	𝑖	SYM
cana-2520	54	7	≤	≤	NOUN
cana-2520	54	8	4	4	NUM
cana-2520	54	9	g.c.d	g.c.d	NOUN
cana-2520	54	10	(	(	PUNCT
cana-2520	54	11	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	54	12	)	)	PUNCT
cana-2520	54	13	,	,	PUNCT
cana-2520	54	14	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	54	15	)	)	PUNCT
cana-2520	54	16	)	)	PUNCT
cana-2520	55	1	=	=	SYM
cana-2520	55	2	1	1	NUM
cana-2520	55	3	g.c.d	g.c.d	NOUN
cana-2520	55	4	(	(	PUNCT
cana-2520	55	5	ℬ(𝑣3	ℬ(𝑣3	PROPN
cana-2520	55	6	′	′	NUM
cana-2520	55	7	)	)	PUNCT
cana-2520	55	8	,	,	PUNCT
cana-2520	55	9	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	55	10	)	)	PUNCT
cana-2520	55	11	)	)	PUNCT
cana-2520	55	12	=	=	SYM
cana-2520	55	13	1	1	NUM
cana-2520	55	14	g.c.d	g.c.d	NOUN
cana-2520	55	15	(	(	PUNCT
cana-2520	55	16	ℬ(𝑣3	ℬ(𝑣3	PROPN
cana-2520	55	17	′	′	NUM
cana-2520	55	18	)	)	PUNCT
cana-2520	55	19	,	,	PUNCT
cana-2520	55	20	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	55	21	)	)	PUNCT
cana-2520	55	22	)	)	PUNCT
cana-2520	55	23	=	=	SYM
cana-2520	55	24	1	1	NUM
cana-2520	55	25	thus	thus	ADV
cana-2520	55	26	ℬ	ℬ	NOUN
cana-2520	55	27	is	be	AUX
cana-2520	55	28	a	a	DET
cana-2520	55	29	prime	prime	ADJ
cana-2520	55	30	labeling	labeling	NOUN
cana-2520	55	31	on	on	ADP
cana-2520	55	32	𝐺3	𝐺3	PROPN
cana-2520	55	33	.	.	PUNCT
cana-2520	56	1	hence	hence	ADV
cana-2520	56	2	𝐺3	𝐺3	PROPN
cana-2520	56	3	is	be	AUX
cana-2520	56	4	a	a	DET
cana-2520	56	5	prime	prime	ADJ
cana-2520	56	6	graph	graph	NOUN
cana-2520	56	7	.	.	PUNCT
cana-2520	57	1	communications	communication	NOUN
cana-2520	57	2	on	on	ADP
cana-2520	57	3	applied	apply	VERB
cana-2520	57	4	nonlinear	nonlinear	ADJ
cana-2520	57	5	analysis	analysis	NOUN
cana-2520	57	6	issn	issn	NOUN
cana-2520	57	7	:	:	PUNCT
cana-2520	57	8	1074	1074	NUM
cana-2520	57	9	-	-	PUNCT
cana-2520	57	10	133x	133x	NUM
cana-2520	57	11	vol	vol	NOUN
cana-2520	57	12	32	32	NUM
cana-2520	57	13	no	no	NOUN
cana-2520	57	14	.	.	PUNCT
cana-2520	58	1	2s	2s	NUM
cana-2520	58	2	(	(	PUNCT
cana-2520	58	3	2025	2025	NUM
cana-2520	58	4	)	)	PUNCT
cana-2520	58	5	595	595	NUM
cana-2520	58	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	58	7	figure	figure	NOUN
cana-2520	58	8	4	4	NUM
cana-2520	58	9	.	.	PUNCT
cana-2520	58	10	prime	prime	ADJ
cana-2520	58	11	labeling	labeling	NOUN
cana-2520	58	12	of	of	ADP
cana-2520	58	13	duplication	duplication	NOUN
cana-2520	58	14	of	of	ADP
cana-2520	58	15	vertex	vertex	NOUN
cana-2520	58	16	𝑣3	𝑣3	NOUN
cana-2520	58	17	in	in	ADP
cana-2520	58	18	bull	bull	NOUN
cana-2520	58	19	graph	graph	NOUN
cana-2520	58	20	case-4	case-4	PROPN
cana-2520	58	21	.	.	PUNCT
cana-2520	58	22	duplication	duplication	NOUN
cana-2520	58	23	of	of	ADP
cana-2520	58	24	the	the	DET
cana-2520	58	25	vertex	vertex	NOUN
cana-2520	58	26	𝑣4	𝑣4	NOUN
cana-2520	58	27	let	let	VERB
cana-2520	58	28	𝐺4	𝐺4	NOUN
cana-2520	58	29	be	be	AUX
cana-2520	58	30	the	the	DET
cana-2520	58	31	graph	graph	NOUN
cana-2520	58	32	obtained	obtain	VERB
cana-2520	58	33	by	by	ADP
cana-2520	58	34	duplicating	duplicate	VERB
cana-2520	58	35	the	the	DET
cana-2520	58	36	vertex	vertex	NOUN
cana-2520	58	37	𝑣4	𝑣4	NOUN
cana-2520	58	38	define	define	VERB
cana-2520	58	39	ℬ	ℬ	NOUN
cana-2520	58	40	:	:	PUNCT
cana-2520	58	41	𝑉	𝑉	PROPN
cana-2520	58	42	(	(	PUNCT
cana-2520	58	43	𝐺4	𝐺4	PROPN
cana-2520	58	44	)	)	PUNCT
cana-2520	58	45	→	→	SYM
cana-2520	58	46	{	{	PUNCT
cana-2520	58	47	1,2,3	1,2,3	NUM
cana-2520	58	48	,	,	PUNCT
cana-2520	58	49	…	…	PUNCT
cana-2520	58	50	.	.	PUNCT
cana-2520	58	51	.	.	PUNCT
cana-2520	59	1	,	,	PUNCT
cana-2520	59	2	6	6	X
cana-2520	59	3	}	}	PUNCT
cana-2520	59	4	by	by	ADP
cana-2520	59	5	ℬ(𝑣𝑖	ℬ(𝑣𝑖	ADJ
cana-2520	59	6	)	)	PUNCT
cana-2520	59	7	=	=	PUNCT
cana-2520	60	1	𝑖	𝑖	PROPN
cana-2520	61	1	+	+	NOUN
cana-2520	61	2	1	1	NUM
cana-2520	61	3	,	,	PUNCT
cana-2520	61	4	1	1	NUM
cana-2520	61	5	≤	≤	NUM
cana-2520	61	6	𝑖	𝑖	SYM
cana-2520	61	7	≤	≤	NOUN
cana-2520	61	8	5	5	NUM
cana-2520	61	9	and	and	CCONJ
cana-2520	61	10	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	61	11	′	′	NUM
cana-2520	61	12	)	)	PUNCT
cana-2520	62	1	=	=	SYM
cana-2520	62	2	1	1	NUM
cana-2520	62	3	obviously	obviously	ADV
cana-2520	62	4	all	all	DET
cana-2520	62	5	the	the	DET
cana-2520	62	6	vertex	vertex	NOUN
cana-2520	62	7	labels	label	NOUN
cana-2520	62	8	are	be	AUX
cana-2520	62	9	distinct	distinct	ADJ
cana-2520	62	10	for	for	ADP
cana-2520	62	11	edges	edge	NOUN
cana-2520	62	12	in	in	ADP
cana-2520	62	13	𝐺4	𝐺4	PROPN
cana-2520	62	14	g.c.d	g.c.d	NOUN
cana-2520	62	15	(	(	PUNCT
cana-2520	62	16	ℬ(𝑣𝑖	ℬ(𝑣𝑖	PROPN
cana-2520	62	17	)	)	PUNCT
cana-2520	62	18	,	,	PUNCT
cana-2520	62	19	ℬ(𝑣𝑖+1	ℬ(𝑣𝑖+1	PROPN
cana-2520	62	20	)	)	PUNCT
cana-2520	62	21	)	)	PUNCT
cana-2520	63	1	=	=	PUNCT
cana-2520	63	2	1	1	NUM
cana-2520	63	3	,	,	PUNCT
cana-2520	63	4	1	1	NUM
cana-2520	63	5	≤	≤	NUM
cana-2520	63	6	𝑖	𝑖	SYM
cana-2520	63	7	≤	≤	NOUN
cana-2520	63	8	4	4	NUM
cana-2520	63	9	g.c.d	g.c.d	NOUN
cana-2520	63	10	(	(	PUNCT
cana-2520	63	11	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	63	12	)	)	PUNCT
cana-2520	63	13	,	,	PUNCT
cana-2520	63	14	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	63	15	)	)	PUNCT
cana-2520	63	16	)	)	PUNCT
cana-2520	64	1	=	=	SYM
cana-2520	64	2	1	1	NUM
cana-2520	64	3	g.c.d	g.c.d	NOUN
cana-2520	64	4	(	(	PUNCT
cana-2520	64	5	ℬ(𝑣4	ℬ(𝑣4	PROPN
cana-2520	64	6	′	′	NUM
cana-2520	64	7	)	)	PUNCT
cana-2520	64	8	,	,	PUNCT
cana-2520	64	9	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	64	10	)	)	PUNCT
cana-2520	64	11	)	)	PUNCT
cana-2520	64	12	=	=	SYM
cana-2520	64	13	1	1	NUM
cana-2520	64	14	g.c.d	g.c.d	NOUN
cana-2520	64	15	(	(	PUNCT
cana-2520	64	16	ℬ(𝑣4	ℬ(𝑣4	PROPN
cana-2520	64	17	′	′	NUM
cana-2520	64	18	)	)	PUNCT
cana-2520	64	19	,	,	PUNCT
cana-2520	64	20	ℬ(𝑣3	ℬ(𝑣3	NOUN
cana-2520	64	21	)	)	PUNCT
cana-2520	64	22	)	)	PUNCT
cana-2520	64	23	=	=	SYM
cana-2520	64	24	1	1	NUM
cana-2520	64	25	g.c.d	g.c.d	NOUN
cana-2520	64	26	(	(	PUNCT
cana-2520	64	27	ℬ(𝑣4	ℬ(𝑣4	PROPN
cana-2520	64	28	′	′	NUM
cana-2520	64	29	)	)	PUNCT
cana-2520	64	30	,	,	PUNCT
cana-2520	64	31	ℬ(𝑣5	ℬ(𝑣5	PROPN
cana-2520	64	32	)	)	PUNCT
cana-2520	64	33	)	)	PUNCT
cana-2520	64	34	=	=	PUNCT
cana-2520	65	1	1	1	NUM
cana-2520	65	2	clearly	clearly	ADV
cana-2520	65	3	ℬ	ℬ	PROPN
cana-2520	65	4	is	be	AUX
cana-2520	65	5	a	a	DET
cana-2520	65	6	prime	prime	ADJ
cana-2520	65	7	labeling	labeling	NOUN
cana-2520	65	8	on	on	ADP
cana-2520	65	9	𝐺4	𝐺4	PROPN
cana-2520	65	10	.	.	PUNCT
cana-2520	66	1	hence	hence	ADV
cana-2520	66	2	𝐺4	𝐺4	PROPN
cana-2520	66	3	is	be	AUX
cana-2520	66	4	a	a	DET
cana-2520	66	5	prime	prime	ADJ
cana-2520	66	6	graph	graph	NOUN
cana-2520	66	7	.	.	PUNCT
cana-2520	67	1	figure	figure	NOUN
cana-2520	67	2	5	5	NUM
cana-2520	67	3	.	.	PUNCT
cana-2520	67	4	prime	prime	ADJ
cana-2520	67	5	labeling	labeling	NOUN
cana-2520	67	6	of	of	ADP
cana-2520	67	7	duplication	duplication	NOUN
cana-2520	67	8	of	of	ADP
cana-2520	67	9	vertex	vertex	NOUN
cana-2520	67	10	𝑣4	𝑣4	NOUN
cana-2520	67	11	in	in	ADP
cana-2520	67	12	bull	bull	NOUN
cana-2520	67	13	graph	graph	NOUN
cana-2520	67	14	case-5	case-5	NOUN
cana-2520	67	15	.	.	NOUN
cana-2520	67	16	duplication	duplication	NOUN
cana-2520	67	17	of	of	ADP
cana-2520	67	18	the	the	DET
cana-2520	67	19	vertex	vertex	NOUN
cana-2520	67	20	𝑣5	𝑣5	NOUN
cana-2520	67	21	communications	communication	NOUN
cana-2520	67	22	on	on	ADP
cana-2520	67	23	applied	apply	VERB
cana-2520	67	24	nonlinear	nonlinear	ADJ
cana-2520	67	25	analysis	analysis	NOUN
cana-2520	67	26	issn	issn	NOUN
cana-2520	67	27	:	:	PUNCT
cana-2520	67	28	1074	1074	NUM
cana-2520	67	29	-	-	PUNCT
cana-2520	67	30	133x	133x	NUM
cana-2520	67	31	vol	vol	NOUN
cana-2520	67	32	32	32	NUM
cana-2520	67	33	no	no	NOUN
cana-2520	67	34	.	.	PUNCT
cana-2520	68	1	2s	2s	NUM
cana-2520	68	2	(	(	PUNCT
cana-2520	68	3	2025	2025	NUM
cana-2520	68	4	)	)	PUNCT
cana-2520	68	5	596	596	NUM
cana-2520	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	68	7	let	let	VERB
cana-2520	68	8	𝐺5	𝐺5	NOUN
cana-2520	68	9	be	be	AUX
cana-2520	68	10	the	the	DET
cana-2520	68	11	graph	graph	NOUN
cana-2520	68	12	obtained	obtain	VERB
cana-2520	68	13	by	by	ADP
cana-2520	68	14	duplicating	duplicate	VERB
cana-2520	68	15	the	the	DET
cana-2520	68	16	vertex	vertex	NOUN
cana-2520	68	17	𝑣5	𝑣5	NOUN
cana-2520	68	18	define	define	VERB
cana-2520	68	19	ℬ	ℬ	X
cana-2520	68	20	:	:	PUNCT
cana-2520	68	21	𝑉	𝑉	PROPN
cana-2520	68	22	(	(	PUNCT
cana-2520	68	23	𝐺5	𝐺5	NOUN
cana-2520	68	24	)	)	PUNCT
cana-2520	68	25	→	→	SYM
cana-2520	68	26	{	{	PUNCT
cana-2520	68	27	1,2,3	1,2,3	NUM
cana-2520	68	28	,	,	PUNCT
cana-2520	68	29	…	…	PUNCT
cana-2520	68	30	.	.	PUNCT
cana-2520	68	31	.	.	PUNCT
cana-2520	69	1	,	,	PUNCT
cana-2520	69	2	6	6	X
cana-2520	69	3	}	}	PUNCT
cana-2520	69	4	by	by	ADP
cana-2520	69	5	ℬ(𝑣𝑖	ℬ(𝑣𝑖	ADJ
cana-2520	69	6	)	)	PUNCT
cana-2520	69	7	=	=	PUNCT
cana-2520	70	1	𝑖	𝑖	PROPN
cana-2520	71	1	+	+	NOUN
cana-2520	71	2	1	1	NUM
cana-2520	71	3	,	,	PUNCT
cana-2520	71	4	1	1	NUM
cana-2520	71	5	≤	≤	NUM
cana-2520	71	6	𝑖	𝑖	SYM
cana-2520	71	7	≤	≤	NOUN
cana-2520	71	8	5	5	NUM
cana-2520	71	9	and	and	CCONJ
cana-2520	71	10	ℬ(𝑣5	ℬ(𝑣5	PROPN
cana-2520	71	11	′	′	NUM
cana-2520	71	12	)	)	PUNCT
cana-2520	72	1	=	=	PUNCT
cana-2520	72	2	1	1	NUM
cana-2520	72	3	evidently	evidently	ADV
cana-2520	72	4	all	all	DET
cana-2520	72	5	the	the	DET
cana-2520	72	6	vertex	vertex	NOUN
cana-2520	72	7	labels	label	NOUN
cana-2520	72	8	are	be	AUX
cana-2520	72	9	distinct	distinct	ADJ
cana-2520	72	10	for	for	ADP
cana-2520	72	11	edges	edge	NOUN
cana-2520	72	12	in	in	ADP
cana-2520	72	13	𝐺5	𝐺5	NOUN
cana-2520	72	14	g.c.d	g.c.d	NOUN
cana-2520	72	15	(	(	PUNCT
cana-2520	72	16	ℬ(𝑣𝑖	ℬ(𝑣𝑖	PROPN
cana-2520	72	17	)	)	PUNCT
cana-2520	72	18	,	,	PUNCT
cana-2520	72	19	ℬ(𝑣𝑖+1	ℬ(𝑣𝑖+1	PROPN
cana-2520	72	20	)	)	PUNCT
cana-2520	72	21	)	)	PUNCT
cana-2520	73	1	=	=	PUNCT
cana-2520	73	2	1	1	NUM
cana-2520	73	3	,	,	PUNCT
cana-2520	73	4	1	1	NUM
cana-2520	73	5	≤	≤	NUM
cana-2520	73	6	𝑖	𝑖	SYM
cana-2520	73	7	≤	≤	NOUN
cana-2520	73	8	4	4	NUM
cana-2520	73	9	g.c.d	g.c.d	NOUN
cana-2520	73	10	(	(	PUNCT
cana-2520	73	11	ℬ(𝑣2	ℬ(𝑣2	NOUN
cana-2520	73	12	)	)	PUNCT
cana-2520	73	13	,	,	PUNCT
cana-2520	73	14	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	73	15	)	)	PUNCT
cana-2520	73	16	)	)	PUNCT
cana-2520	74	1	=	=	SYM
cana-2520	74	2	1	1	NUM
cana-2520	74	3	g.c.d	g.c.d	NOUN
cana-2520	74	4	(	(	PUNCT
cana-2520	74	5	ℬ(𝑣5	ℬ(𝑣5	PROPN
cana-2520	74	6	′	′	NUM
cana-2520	74	7	)	)	PUNCT
cana-2520	74	8	,	,	PUNCT
cana-2520	74	9	ℬ(𝑣4	ℬ(𝑣4	NOUN
cana-2520	74	10	)	)	PUNCT
cana-2520	74	11	)	)	PUNCT
cana-2520	74	12	=	=	SYM
cana-2520	75	1	1	1	NUM
cana-2520	75	2	clearly	clearly	ADV
cana-2520	75	3	ℬ	ℬ	PROPN
cana-2520	75	4	is	be	AUX
cana-2520	75	5	a	a	DET
cana-2520	75	6	prime	prime	ADJ
cana-2520	75	7	labeling	labeling	NOUN
cana-2520	75	8	on	on	ADP
cana-2520	75	9	𝐺5	𝐺5	NOUN
cana-2520	75	10	.	.	PUNCT
cana-2520	76	1	hence	hence	ADV
cana-2520	76	2	𝐺5	𝐺5	PROPN
cana-2520	76	3	is	be	AUX
cana-2520	76	4	a	a	DET
cana-2520	76	5	prime	prime	ADJ
cana-2520	76	6	graph	graph	NOUN
cana-2520	76	7	.	.	PUNCT
cana-2520	77	1	figure	figure	NOUN
cana-2520	77	2	6	6	NUM
cana-2520	77	3	.	.	PUNCT
cana-2520	77	4	prime	prime	ADJ
cana-2520	77	5	labeling	labeling	NOUN
cana-2520	77	6	of	of	ADP
cana-2520	77	7	duplication	duplication	NOUN
cana-2520	77	8	of	of	ADP
cana-2520	77	9	vertex	vertex	NOUN
cana-2520	77	10	𝑣5	𝑣5	NOUN
cana-2520	77	11	of	of	ADP
cana-2520	77	12	bull	bull	NOUN
cana-2520	77	13	graph	graph	NOUN
cana-2520	77	14	thus	thus	ADV
cana-2520	77	15	,	,	PUNCT
cana-2520	77	16	in	in	ADP
cana-2520	77	17	all	all	DET
cana-2520	77	18	the	the	DET
cana-2520	77	19	cases	case	NOUN
cana-2520	77	20	the	the	DET
cana-2520	77	21	graph	graph	NOUN
cana-2520	77	22	obtained	obtain	VERB
cana-2520	77	23	by	by	ADP
cana-2520	77	24	duplication	duplication	NOUN
cana-2520	77	25	of	of	ADP
cana-2520	77	26	any	any	DET
cana-2520	77	27	arbitrary	arbitrary	ADJ
cana-2520	77	28	vertex	vertex	NOUN
cana-2520	77	29	of	of	ADP
cana-2520	77	30	bull	bull	NOUN
cana-2520	77	31	graph	graph	NOUN
cana-2520	77	32	is	be	AUX
cana-2520	77	33	a	a	DET
cana-2520	77	34	prime	prime	ADJ
cana-2520	77	35	graph	graph	NOUN
cana-2520	77	36	.	.	PUNCT
cana-2520	77	37	theorem-3.2	theorem-3.2	NOUN
cana-2520	77	38	.	.	PUNCT
cana-2520	78	1	the	the	DET
cana-2520	78	2	graph	graph	NOUN
cana-2520	78	3	obtained	obtain	VERB
cana-2520	78	4	by	by	ADP
cana-2520	78	5	switching	switching	NOUN
cana-2520	78	6	of	of	ADP
cana-2520	78	7	any	any	DET
cana-2520	78	8	vertex	vertex	NOUN
cana-2520	78	9	in	in	ADP
cana-2520	78	10	a	a	DET
cana-2520	78	11	bull	bull	NOUN
cana-2520	78	12	graph	graph	NOUN
cana-2520	78	13	is	be	AUX
cana-2520	78	14	a	a	DET
cana-2520	78	15	prime	prime	ADJ
cana-2520	78	16	graph	graph	NOUN
cana-2520	78	17	.	.	PUNCT
cana-2520	79	1	proof	proof	NOUN
cana-2520	79	2	.	.	PUNCT
cana-2520	80	1	figure	figure	VERB
cana-2520	80	2	7	7	NUM
cana-2520	80	3	.	.	PUNCT
cana-2520	80	4	bull	bull	NOUN
cana-2520	80	5	graph	graph	NOUN
cana-2520	80	6	communications	communication	NOUN
cana-2520	80	7	on	on	ADP
cana-2520	80	8	applied	apply	VERB
cana-2520	80	9	nonlinear	nonlinear	ADJ
cana-2520	80	10	analysis	analysis	NOUN
cana-2520	80	11	issn	issn	NOUN
cana-2520	80	12	:	:	PUNCT
cana-2520	80	13	1074	1074	NUM
cana-2520	80	14	-	-	PUNCT
cana-2520	80	15	133x	133x	NUM
cana-2520	80	16	vol	vol	NOUN
cana-2520	80	17	32	32	NUM
cana-2520	80	18	no	no	NOUN
cana-2520	80	19	.	.	PUNCT
cana-2520	81	1	2s	2s	NUM
cana-2520	81	2	(	(	PUNCT
cana-2520	81	3	2025	2025	NUM
cana-2520	81	4	)	)	PUNCT
cana-2520	81	5	597	597	NUM
cana-2520	81	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	81	7	case-1	case-1	NUM
cana-2520	81	8	.	.	PUNCT
cana-2520	82	1	switching	switch	VERB
cana-2520	82	2	the	the	DET
cana-2520	82	3	vertex	vertex	NOUN
cana-2520	82	4	𝑣1	𝑣1	NOUN
cana-2520	82	5	let	let	VERB
cana-2520	82	6	𝐺1	𝐺1	NOUN
cana-2520	82	7	be	be	AUX
cana-2520	82	8	the	the	DET
cana-2520	82	9	graph	graph	NOUN
cana-2520	82	10	obtained	obtain	VERB
cana-2520	82	11	by	by	ADP
cana-2520	82	12	switching	switch	VERB
cana-2520	82	13	the	the	DET
cana-2520	82	14	vertex	vertex	NOUN
cana-2520	82	15	𝑣1	𝑣1	NOUN
cana-2520	82	16	define	define	VERB
cana-2520	82	17	℘	℘	PROPN
cana-2520	82	18	:	:	PUNCT
cana-2520	82	19	𝑉	𝑉	PROPN
cana-2520	82	20	(	(	PUNCT
cana-2520	82	21	𝐺1	𝐺1	NOUN
cana-2520	82	22	)	)	PUNCT
cana-2520	82	23	→	→	SYM
cana-2520	82	24	{	{	PUNCT
cana-2520	82	25	1,2,3	1,2,3	NUM
cana-2520	82	26	,	,	PUNCT
cana-2520	82	27	…	…	PUNCT
cana-2520	82	28	.	.	PUNCT
cana-2520	82	29	.	.	PUNCT
cana-2520	83	1	,	,	PUNCT
cana-2520	83	2	5	5	X
cana-2520	83	3	}	}	PUNCT
cana-2520	83	4	by	by	ADP
cana-2520	83	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	83	6	)	)	PUNCT
cana-2520	83	7	=	=	SYM
cana-2520	83	8	1	1	NUM
cana-2520	83	9	,	,	PUNCT
cana-2520	83	10	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	83	11	)	)	PUNCT
cana-2520	83	12	=	=	SYM
cana-2520	83	13	5	5	NUM
cana-2520	83	14	,	,	PUNCT
cana-2520	83	15	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	83	16	)	)	PUNCT
cana-2520	83	17	=	=	SYM
cana-2520	84	1	4	4	NUM
cana-2520	84	2	,	,	PUNCT
cana-2520	84	3	℘(𝑣4	℘(𝑣4	ADJ
cana-2520	84	4	)	)	PUNCT
cana-2520	84	5	=	=	SYM
cana-2520	84	6	3	3	NUM
cana-2520	84	7	,	,	PUNCT
cana-2520	84	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	84	9	)	)	PUNCT
cana-2520	85	1	=	=	SYM
cana-2520	85	2	2	2	NUM
cana-2520	85	3	evidently	evidently	ADV
cana-2520	85	4	all	all	DET
cana-2520	85	5	the	the	DET
cana-2520	85	6	vertex	vertex	NOUN
cana-2520	85	7	labels	label	NOUN
cana-2520	85	8	are	be	AUX
cana-2520	85	9	distinct	distinct	ADJ
cana-2520	85	10	for	for	ADP
cana-2520	85	11	edges	edge	NOUN
cana-2520	85	12	in	in	ADP
cana-2520	85	13	𝐺1	𝐺1	NOUN
cana-2520	85	14	g.c.d	g.c.d	NOUN
cana-2520	85	15	(	(	PUNCT
cana-2520	85	16	℘(𝑣𝑖	℘(𝑣𝑖	PROPN
cana-2520	85	17	)	)	PUNCT
cana-2520	85	18	,	,	PUNCT
cana-2520	85	19	℘(𝑣𝑖+1	℘(𝑣𝑖+1	PROPN
cana-2520	85	20	)	)	PUNCT
cana-2520	85	21	)	)	PUNCT
cana-2520	86	1	=	=	SYM
cana-2520	86	2	1	1	NUM
cana-2520	86	3	,	,	PUNCT
cana-2520	86	4	2	2	NUM
cana-2520	86	5	≤	≤	NOUN
cana-2520	86	6	𝑖	𝑖	SYM
cana-2520	86	7	≤	≤	NOUN
cana-2520	86	8	4	4	NUM
cana-2520	86	9	g.c.d	g.c.d	NOUN
cana-2520	86	10	(	(	PUNCT
cana-2520	86	11	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	86	12	)	)	PUNCT
cana-2520	86	13	,	,	PUNCT
cana-2520	86	14	℘(𝑣4	℘(𝑣4	NOUN
cana-2520	86	15	)	)	PUNCT
cana-2520	86	16	)	)	PUNCT
cana-2520	87	1	=	=	SYM
cana-2520	87	2	1	1	NUM
cana-2520	87	3	g.c.d	g.c.d	NOUN
cana-2520	87	4	(	(	PUNCT
cana-2520	87	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	87	6	)	)	PUNCT
cana-2520	87	7	,	,	PUNCT
cana-2520	87	8	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	87	9	)	)	PUNCT
cana-2520	87	10	)	)	PUNCT
cana-2520	88	1	=	=	SYM
cana-2520	88	2	1	1	NUM
cana-2520	88	3	g.c.d	g.c.d	NOUN
cana-2520	88	4	(	(	PUNCT
cana-2520	88	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	88	6	)	)	PUNCT
cana-2520	88	7	,	,	PUNCT
cana-2520	88	8	℘(𝑣4	℘(𝑣4	NOUN
cana-2520	88	9	)	)	PUNCT
cana-2520	88	10	)	)	PUNCT
cana-2520	89	1	=	=	SYM
cana-2520	89	2	1	1	NUM
cana-2520	89	3	g.c.d	g.c.d	NOUN
cana-2520	89	4	(	(	PUNCT
cana-2520	89	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	89	6	)	)	PUNCT
cana-2520	89	7	,	,	PUNCT
cana-2520	89	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	89	9	)	)	PUNCT
cana-2520	89	10	)	)	PUNCT
cana-2520	90	1	=	=	SYM
cana-2520	90	2	1	1	NUM
cana-2520	90	3	thus	thus	ADV
cana-2520	90	4	℘	℘	PROPN
cana-2520	90	5	is	be	AUX
cana-2520	90	6	a	a	DET
cana-2520	90	7	prime	prime	ADJ
cana-2520	90	8	labeling	labeling	NOUN
cana-2520	90	9	on	on	ADP
cana-2520	90	10	𝐺1	𝐺1	NOUN
cana-2520	90	11	.	.	PUNCT
cana-2520	91	1	hence	hence	ADV
cana-2520	91	2	𝐺1	𝐺1	PROPN
cana-2520	91	3	is	be	AUX
cana-2520	91	4	a	a	DET
cana-2520	91	5	prime	prime	ADJ
cana-2520	91	6	graph	graph	NOUN
cana-2520	91	7	.	.	PUNCT
cana-2520	92	1	figure	figure	NOUN
cana-2520	92	2	8	8	NUM
cana-2520	92	3	.	.	PUNCT
cana-2520	93	1	prime	prime	ADJ
cana-2520	93	2	labeling	labeling	NOUN
cana-2520	93	3	of	of	ADP
cana-2520	93	4	switching	switching	NOUN
cana-2520	93	5	of	of	ADP
cana-2520	93	6	vertex	vertex	NOUN
cana-2520	93	7	𝑣1	𝑣1	NOUN
cana-2520	93	8	in	in	ADP
cana-2520	93	9	bull	bull	NOUN
cana-2520	93	10	graph	graph	NOUN
cana-2520	93	11	case-2	case-2	NOUN
cana-2520	93	12	.	.	PUNCT
cana-2520	94	1	switching	switch	VERB
cana-2520	94	2	the	the	DET
cana-2520	94	3	vertex	vertex	NOUN
cana-2520	94	4	𝑣2	𝑣2	PRON
cana-2520	94	5	let	let	VERB
cana-2520	94	6	𝐺2	𝐺2	NOUN
cana-2520	94	7	be	be	AUX
cana-2520	94	8	the	the	DET
cana-2520	94	9	graph	graph	NOUN
cana-2520	94	10	obtained	obtain	VERB
cana-2520	94	11	by	by	ADP
cana-2520	94	12	switching	switch	VERB
cana-2520	94	13	the	the	DET
cana-2520	94	14	vertex	vertex	NOUN
cana-2520	94	15	𝑣2	𝑣2	PRON
cana-2520	94	16	define	define	VERB
cana-2520	94	17	℘	℘	PROPN
cana-2520	94	18	:	:	PUNCT
cana-2520	94	19	𝑉	𝑉	PROPN
cana-2520	94	20	(	(	PUNCT
cana-2520	94	21	𝐺2	𝐺2	ADJ
cana-2520	94	22	)	)	PUNCT
cana-2520	94	23	→	→	SYM
cana-2520	94	24	{	{	PUNCT
cana-2520	94	25	1,2,3	1,2,3	NUM
cana-2520	94	26	,	,	PUNCT
cana-2520	94	27	…	…	PUNCT
cana-2520	94	28	.	.	PUNCT
cana-2520	94	29	.	.	PUNCT
cana-2520	95	1	,	,	PUNCT
cana-2520	95	2	5	5	X
cana-2520	95	3	}	}	PUNCT
cana-2520	95	4	by	by	ADP
cana-2520	95	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	95	6	)	)	PUNCT
cana-2520	95	7	=	=	SYM
cana-2520	95	8	1	1	NUM
cana-2520	95	9	,	,	PUNCT
cana-2520	95	10	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	95	11	)	)	PUNCT
cana-2520	95	12	=	=	SYM
cana-2520	95	13	5	5	NUM
cana-2520	95	14	,	,	PUNCT
cana-2520	95	15	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	95	16	)	)	PUNCT
cana-2520	95	17	=	=	SYM
cana-2520	96	1	4	4	NUM
cana-2520	96	2	,	,	PUNCT
cana-2520	96	3	℘(𝑣4	℘(𝑣4	ADJ
cana-2520	96	4	)	)	PUNCT
cana-2520	96	5	=	=	SYM
cana-2520	96	6	3	3	NUM
cana-2520	96	7	,	,	PUNCT
cana-2520	96	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	96	9	)	)	PUNCT
cana-2520	97	1	=	=	SYM
cana-2520	97	2	2	2	NUM
cana-2520	97	3	clearly	clearly	ADV
cana-2520	97	4	all	all	DET
cana-2520	97	5	the	the	DET
cana-2520	97	6	vertex	vertex	NOUN
cana-2520	97	7	labels	label	NOUN
cana-2520	97	8	are	be	AUX
cana-2520	97	9	distinct	distinct	ADJ
cana-2520	97	10	for	for	ADP
cana-2520	97	11	edges	edge	NOUN
cana-2520	97	12	in	in	ADP
cana-2520	97	13	𝐺2	𝐺2	ADJ
cana-2520	97	14	g.c.d	g.c.d	NOUN
cana-2520	97	15	(	(	PUNCT
cana-2520	97	16	℘(𝑣𝑖	℘(𝑣𝑖	PROPN
cana-2520	97	17	)	)	PUNCT
cana-2520	97	18	,	,	PUNCT
cana-2520	97	19	℘(𝑣𝑖+1	℘(𝑣𝑖+1	PROPN
cana-2520	97	20	)	)	PUNCT
cana-2520	97	21	)	)	PUNCT
cana-2520	98	1	=	=	SYM
cana-2520	98	2	1	1	NUM
cana-2520	98	3	,	,	PUNCT
cana-2520	98	4	3	3	NUM
cana-2520	98	5	≤	≤	NUM
cana-2520	98	6	𝑖	𝑖	SYM
cana-2520	98	7	≤	≤	NOUN
cana-2520	98	8	4	4	NUM
cana-2520	98	9	g.c.d	g.c.d	NOUN
cana-2520	98	10	(	(	PUNCT
cana-2520	98	11	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	98	12	)	)	PUNCT
cana-2520	98	13	,	,	PUNCT
cana-2520	98	14	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	98	15	)	)	PUNCT
cana-2520	98	16	)	)	PUNCT
cana-2520	99	1	=	=	SYM
cana-2520	99	2	1	1	NUM
cana-2520	99	3	hence	hence	ADV
cana-2520	99	4	℘	℘	PROPN
cana-2520	99	5	is	be	AUX
cana-2520	99	6	a	a	DET
cana-2520	99	7	prime	prime	ADJ
cana-2520	99	8	labeling	labeling	NOUN
cana-2520	99	9	on	on	ADP
cana-2520	99	10	𝐺2	𝐺2	NOUN
cana-2520	99	11	.	.	PUNCT
cana-2520	100	1	thus	thus	ADV
cana-2520	100	2	𝐺2	𝐺2	ADV
cana-2520	100	3	is	be	AUX
cana-2520	100	4	a	a	DET
cana-2520	100	5	prime	prime	ADJ
cana-2520	100	6	graph	graph	NOUN
cana-2520	100	7	.	.	PUNCT
cana-2520	101	1	communications	communication	NOUN
cana-2520	101	2	on	on	ADP
cana-2520	101	3	applied	apply	VERB
cana-2520	101	4	nonlinear	nonlinear	ADJ
cana-2520	101	5	analysis	analysis	NOUN
cana-2520	101	6	issn	issn	NOUN
cana-2520	101	7	:	:	PUNCT
cana-2520	101	8	1074	1074	NUM
cana-2520	101	9	-	-	PUNCT
cana-2520	101	10	133x	133x	NUM
cana-2520	101	11	vol	vol	NOUN
cana-2520	101	12	32	32	NUM
cana-2520	101	13	no	no	NOUN
cana-2520	101	14	.	.	PUNCT
cana-2520	102	1	2s	2s	NUM
cana-2520	102	2	(	(	PUNCT
cana-2520	102	3	2025	2025	NUM
cana-2520	102	4	)	)	PUNCT
cana-2520	102	5	598	598	NUM
cana-2520	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	102	7	figure	figure	NOUN
cana-2520	102	8	9	9	NUM
cana-2520	102	9	.	.	PUNCT
cana-2520	102	10	prime	prime	ADJ
cana-2520	102	11	labeling	labeling	NOUN
cana-2520	102	12	of	of	ADP
cana-2520	102	13	switching	switching	NOUN
cana-2520	102	14	of	of	ADP
cana-2520	102	15	vertex	vertex	NOUN
cana-2520	102	16	𝑣2	𝑣2	PROPN
cana-2520	102	17	in	in	ADP
cana-2520	102	18	bull	bull	NOUN
cana-2520	102	19	graph	graph	NOUN
cana-2520	102	20	case-3	case-3	PROPN
cana-2520	102	21	.	.	PUNCT
cana-2520	103	1	switching	switch	VERB
cana-2520	103	2	the	the	DET
cana-2520	103	3	vertex	vertex	NOUN
cana-2520	103	4	𝑣3	𝑣3	PROPN
cana-2520	103	5	let	let	VERB
cana-2520	103	6	𝐺3	𝐺3	PROPN
cana-2520	103	7	be	be	AUX
cana-2520	103	8	the	the	DET
cana-2520	103	9	graph	graph	NOUN
cana-2520	103	10	obtained	obtain	VERB
cana-2520	103	11	by	by	ADP
cana-2520	103	12	switching	switch	VERB
cana-2520	103	13	the	the	DET
cana-2520	103	14	vertex	vertex	NOUN
cana-2520	103	15	𝑣3	𝑣3	NOUN
cana-2520	103	16	define	define	VERB
cana-2520	103	17	℘	℘	PROPN
cana-2520	103	18	:	:	PUNCT
cana-2520	103	19	𝑉	𝑉	PROPN
cana-2520	103	20	(	(	PUNCT
cana-2520	103	21	𝐺3	𝐺3	PROPN
cana-2520	103	22	)	)	PUNCT
cana-2520	103	23	→	→	SYM
cana-2520	103	24	{	{	PUNCT
cana-2520	103	25	1,2,3	1,2,3	NUM
cana-2520	103	26	,	,	PUNCT
cana-2520	103	27	…	…	PUNCT
cana-2520	103	28	.	.	PUNCT
cana-2520	103	29	.	.	PUNCT
cana-2520	104	1	,	,	PUNCT
cana-2520	104	2	5	5	X
cana-2520	104	3	}	}	PUNCT
cana-2520	104	4	by	by	ADP
cana-2520	104	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	104	6	)	)	PUNCT
cana-2520	104	7	=	=	SYM
cana-2520	104	8	5	5	NUM
cana-2520	104	9	,	,	PUNCT
cana-2520	104	10	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	104	11	)	)	PUNCT
cana-2520	104	12	=	=	SYM
cana-2520	105	1	4	4	NUM
cana-2520	105	2	,	,	PUNCT
cana-2520	105	3	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	105	4	)	)	PUNCT
cana-2520	105	5	=	=	SYM
cana-2520	105	6	1	1	NUM
cana-2520	105	7	,	,	PUNCT
cana-2520	105	8	℘(𝑣4	℘(𝑣4	ADJ
cana-2520	105	9	)	)	PUNCT
cana-2520	106	1	=	=	SYM
cana-2520	106	2	3	3	NUM
cana-2520	106	3	,	,	PUNCT
cana-2520	106	4	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	106	5	)	)	PUNCT
cana-2520	107	1	=	=	SYM
cana-2520	107	2	2	2	NUM
cana-2520	107	3	visibly	visibly	ADV
cana-2520	107	4	all	all	DET
cana-2520	107	5	the	the	DET
cana-2520	107	6	vertex	vertex	NOUN
cana-2520	107	7	labels	label	NOUN
cana-2520	107	8	are	be	AUX
cana-2520	107	9	distinct	distinct	ADJ
cana-2520	107	10	for	for	ADP
cana-2520	107	11	edges	edge	NOUN
cana-2520	107	12	in	in	ADP
cana-2520	107	13	𝐺3	𝐺3	ADJ
cana-2520	107	14	g.c.d	g.c.d	NOUN
cana-2520	107	15	(	(	PUNCT
cana-2520	107	16	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	107	17	)	)	PUNCT
cana-2520	107	18	,	,	PUNCT
cana-2520	107	19	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	107	20	)	)	PUNCT
cana-2520	107	21	)	)	PUNCT
cana-2520	108	1	=	=	SYM
cana-2520	108	2	1	1	NUM
cana-2520	108	3	g.c.d	g.c.d	NOUN
cana-2520	108	4	(	(	PUNCT
cana-2520	108	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	108	6	)	)	PUNCT
cana-2520	108	7	,	,	PUNCT
cana-2520	108	8	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	108	9	)	)	PUNCT
cana-2520	108	10	)	)	PUNCT
cana-2520	109	1	=	=	SYM
cana-2520	109	2	1	1	NUM
cana-2520	109	3	g.c.d	g.c.d	NOUN
cana-2520	109	4	(	(	PUNCT
cana-2520	109	5	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	109	6	)	)	PUNCT
cana-2520	109	7	,	,	PUNCT
cana-2520	109	8	℘(𝑣4	℘(𝑣4	NOUN
cana-2520	109	9	)	)	PUNCT
cana-2520	109	10	)	)	PUNCT
cana-2520	110	1	=	=	SYM
cana-2520	110	2	1	1	NUM
cana-2520	110	3	g.c.d	g.c.d	NOUN
cana-2520	110	4	(	(	PUNCT
cana-2520	110	5	℘(𝑣4	℘(𝑣4	ADJ
cana-2520	110	6	)	)	PUNCT
cana-2520	110	7	,	,	PUNCT
cana-2520	110	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	110	9	)	)	PUNCT
cana-2520	110	10	)	)	PUNCT
cana-2520	111	1	=	=	SYM
cana-2520	111	2	1	1	NUM
cana-2520	111	3	g.c.d	g.c.d	NOUN
cana-2520	111	4	(	(	PUNCT
cana-2520	111	5	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	111	6	)	)	PUNCT
cana-2520	111	7	,	,	PUNCT
cana-2520	111	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	111	9	)	)	PUNCT
cana-2520	111	10	)	)	PUNCT
cana-2520	112	1	=	=	SYM
cana-2520	112	2	1	1	NUM
cana-2520	112	3	therefore	therefore	ADV
cana-2520	112	4	℘	℘	PROPN
cana-2520	112	5	is	be	AUX
cana-2520	112	6	a	a	DET
cana-2520	112	7	prime	prime	ADJ
cana-2520	112	8	labeling	labeling	NOUN
cana-2520	112	9	on	on	ADP
cana-2520	112	10	𝐺3	𝐺3	PROPN
cana-2520	112	11	.	.	PUNCT
cana-2520	113	1	hence	hence	ADV
cana-2520	113	2	𝐺3	𝐺3	PROPN
cana-2520	113	3	is	be	AUX
cana-2520	113	4	a	a	DET
cana-2520	113	5	prime	prime	ADJ
cana-2520	113	6	graph	graph	NOUN
cana-2520	113	7	.	.	PUNCT
cana-2520	114	1	figure	figure	NOUN
cana-2520	114	2	10	10	NUM
cana-2520	114	3	.	.	PUNCT
cana-2520	115	1	prime	prime	ADJ
cana-2520	115	2	labeling	labeling	NOUN
cana-2520	115	3	of	of	ADP
cana-2520	115	4	switching	switching	NOUN
cana-2520	115	5	of	of	ADP
cana-2520	115	6	vertex	vertex	NOUN
cana-2520	115	7	𝑣3	𝑣3	NOUN
cana-2520	115	8	in	in	ADP
cana-2520	115	9	bull	bull	NOUN
cana-2520	115	10	graph	graph	NOUN
cana-2520	115	11	case-4	case-4	NOUN
cana-2520	115	12	.	.	PUNCT
cana-2520	116	1	switching	switch	VERB
cana-2520	116	2	the	the	DET
cana-2520	116	3	vertex	vertex	NOUN
cana-2520	116	4	𝑣4	𝑣4	NOUN
cana-2520	116	5	let	let	VERB
cana-2520	116	6	𝐺4	𝐺4	NOUN
cana-2520	116	7	be	be	AUX
cana-2520	116	8	the	the	DET
cana-2520	116	9	graph	graph	NOUN
cana-2520	116	10	obtained	obtain	VERB
cana-2520	116	11	by	by	ADP
cana-2520	116	12	switching	switch	VERB
cana-2520	116	13	the	the	DET
cana-2520	116	14	vertex	vertex	NOUN
cana-2520	116	15	𝑣4	𝑣4	NOUN
cana-2520	116	16	define	define	VERB
cana-2520	116	17	℘	℘	PROPN
cana-2520	116	18	:	:	PUNCT
cana-2520	116	19	𝑉	𝑉	PROPN
cana-2520	116	20	(	(	PUNCT
cana-2520	116	21	𝐺4	𝐺4	PROPN
cana-2520	116	22	)	)	PUNCT
cana-2520	116	23	→	→	SYM
cana-2520	116	24	{	{	PUNCT
cana-2520	116	25	1,2,3	1,2,3	NUM
cana-2520	116	26	,	,	PUNCT
cana-2520	116	27	…	…	PUNCT
cana-2520	116	28	.	.	PUNCT
cana-2520	116	29	.	.	PUNCT
cana-2520	117	1	,	,	PUNCT
cana-2520	117	2	5	5	X
cana-2520	117	3	}	}	PUNCT
cana-2520	117	4	by	by	ADP
cana-2520	117	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	117	6	)	)	PUNCT
cana-2520	117	7	=	=	SYM
cana-2520	117	8	1	1	NUM
cana-2520	117	9	,	,	PUNCT
cana-2520	117	10	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	117	11	)	)	PUNCT
cana-2520	117	12	=	=	SYM
cana-2520	117	13	5	5	NUM
cana-2520	117	14	,	,	PUNCT
cana-2520	117	15	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	117	16	)	)	PUNCT
cana-2520	117	17	=	=	SYM
cana-2520	118	1	4	4	NUM
cana-2520	118	2	,	,	PUNCT
cana-2520	118	3	℘(𝑣4	℘(𝑣4	ADJ
cana-2520	118	4	)	)	PUNCT
cana-2520	118	5	=	=	SYM
cana-2520	118	6	3	3	NUM
cana-2520	118	7	,	,	PUNCT
cana-2520	118	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	118	9	)	)	PUNCT
cana-2520	119	1	=	=	SYM
cana-2520	119	2	2	2	NUM
cana-2520	119	3	clearly	clearly	ADV
cana-2520	119	4	all	all	DET
cana-2520	119	5	the	the	DET
cana-2520	119	6	vertex	vertex	NOUN
cana-2520	119	7	labels	label	NOUN
cana-2520	119	8	are	be	AUX
cana-2520	119	9	distinct	distinct	ADJ
cana-2520	119	10	communications	communication	NOUN
cana-2520	119	11	on	on	ADP
cana-2520	119	12	applied	apply	VERB
cana-2520	119	13	nonlinear	nonlinear	ADJ
cana-2520	119	14	analysis	analysis	NOUN
cana-2520	119	15	issn	issn	NOUN
cana-2520	119	16	:	:	PUNCT
cana-2520	119	17	1074	1074	NUM
cana-2520	119	18	-	-	PUNCT
cana-2520	119	19	133x	133x	NUM
cana-2520	119	20	vol	vol	NOUN
cana-2520	119	21	32	32	NUM
cana-2520	119	22	no	no	NOUN
cana-2520	119	23	.	.	PUNCT
cana-2520	120	1	2s	2s	NUM
cana-2520	120	2	(	(	PUNCT
cana-2520	120	3	2025	2025	NUM
cana-2520	120	4	)	)	PUNCT
cana-2520	120	5	599	599	NUM
cana-2520	120	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	120	7	for	for	ADP
cana-2520	120	8	edges	edge	NOUN
cana-2520	120	9	in	in	ADP
cana-2520	120	10	𝐺4	𝐺4	PROPN
cana-2520	120	11	g.c.d	g.c.d	NOUN
cana-2520	120	12	(	(	PUNCT
cana-2520	120	13	℘(𝑣𝑖	℘(𝑣𝑖	PROPN
cana-2520	120	14	)	)	PUNCT
cana-2520	120	15	,	,	PUNCT
cana-2520	120	16	℘(𝑣𝑖+1	℘(𝑣𝑖+1	PROPN
cana-2520	120	17	)	)	PUNCT
cana-2520	120	18	)	)	PUNCT
cana-2520	121	1	=	=	SYM
cana-2520	121	2	1	1	NUM
cana-2520	121	3	,	,	PUNCT
cana-2520	121	4	1	1	NUM
cana-2520	121	5	≤	≤	NUM
cana-2520	121	6	𝑖	𝑖	SYM
cana-2520	121	7	≤	≤	NOUN
cana-2520	121	8	2	2	NUM
cana-2520	121	9	g.c.d	g.c.d	NOUN
cana-2520	121	10	(	(	PUNCT
cana-2520	121	11	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	121	12	)	)	PUNCT
cana-2520	121	13	,	,	PUNCT
cana-2520	121	14	℘(𝑣4	℘(𝑣4	NOUN
cana-2520	121	15	)	)	PUNCT
cana-2520	121	16	)	)	PUNCT
cana-2520	122	1	=	=	SYM
cana-2520	122	2	1	1	NUM
cana-2520	122	3	hence	hence	ADV
cana-2520	122	4	℘	℘	PROPN
cana-2520	122	5	is	be	AUX
cana-2520	122	6	a	a	DET
cana-2520	122	7	prime	prime	ADJ
cana-2520	122	8	labeling	labeling	NOUN
cana-2520	122	9	on	on	ADP
cana-2520	122	10	𝐺4	𝐺4	PROPN
cana-2520	122	11	.	.	PUNCT
cana-2520	123	1	therefore	therefore	ADV
cana-2520	123	2	𝐺4	𝐺4	PROPN
cana-2520	123	3	is	be	AUX
cana-2520	123	4	a	a	DET
cana-2520	123	5	prime	prime	ADJ
cana-2520	123	6	graph	graph	NOUN
cana-2520	123	7	figure	figure	NOUN
cana-2520	123	8	11	11	NUM
cana-2520	123	9	.	.	PUNCT
cana-2520	124	1	prime	prime	ADJ
cana-2520	124	2	labeling	labeling	NOUN
cana-2520	124	3	of	of	ADP
cana-2520	124	4	switching	switching	NOUN
cana-2520	124	5	of	of	ADP
cana-2520	124	6	vertex	vertex	NOUN
cana-2520	124	7	𝑣4	𝑣4	NOUN
cana-2520	124	8	in	in	ADP
cana-2520	124	9	bull	bull	NOUN
cana-2520	124	10	graph	graph	NOUN
cana-2520	124	11	case-5	case-5	NOUN
cana-2520	124	12	.	.	PUNCT
cana-2520	125	1	switching	switch	VERB
cana-2520	125	2	the	the	DET
cana-2520	125	3	vertex	vertex	NOUN
cana-2520	125	4	𝑣5	𝑣5	NOUN
cana-2520	125	5	let	let	VERB
cana-2520	125	6	𝐺5	𝐺5	NOUN
cana-2520	125	7	be	be	AUX
cana-2520	125	8	the	the	DET
cana-2520	125	9	graph	graph	NOUN
cana-2520	125	10	obtained	obtain	VERB
cana-2520	125	11	by	by	ADP
cana-2520	125	12	switching	switch	VERB
cana-2520	125	13	the	the	DET
cana-2520	125	14	vertex	vertex	NOUN
cana-2520	125	15	𝑣5	𝑣5	NOUN
cana-2520	125	16	define	define	VERB
cana-2520	125	17	℘	℘	PROPN
cana-2520	125	18	:	:	PUNCT
cana-2520	125	19	𝑉	𝑉	PROPN
cana-2520	125	20	(	(	PUNCT
cana-2520	125	21	𝐺5	𝐺5	NOUN
cana-2520	125	22	)	)	PUNCT
cana-2520	125	23	→	→	SYM
cana-2520	125	24	{	{	PUNCT
cana-2520	125	25	1,2,3	1,2,3	NUM
cana-2520	125	26	,	,	PUNCT
cana-2520	125	27	…	…	PUNCT
cana-2520	125	28	.	.	PUNCT
cana-2520	125	29	.	.	PUNCT
cana-2520	126	1	,	,	PUNCT
cana-2520	126	2	5	5	X
cana-2520	126	3	}	}	PUNCT
cana-2520	126	4	by	by	ADP
cana-2520	126	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	126	6	)	)	PUNCT
cana-2520	126	7	=	=	SYM
cana-2520	126	8	2	2	NUM
cana-2520	126	9	,	,	PUNCT
cana-2520	126	10	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	126	11	)	)	PUNCT
cana-2520	126	12	=	=	SYM
cana-2520	126	13	3	3	NUM
cana-2520	126	14	,	,	PUNCT
cana-2520	126	15	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	126	16	)	)	PUNCT
cana-2520	126	17	=	=	SYM
cana-2520	126	18	4	4	NUM
cana-2520	126	19	,	,	PUNCT
cana-2520	126	20	℘(𝑣4	℘(𝑣4	ADJ
cana-2520	126	21	)	)	PUNCT
cana-2520	126	22	=	=	SYM
cana-2520	126	23	5	5	NUM
cana-2520	126	24	,	,	PUNCT
cana-2520	126	25	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	126	26	)	)	PUNCT
cana-2520	127	1	=	=	SYM
cana-2520	127	2	1	1	NUM
cana-2520	127	3	visibly	visibly	ADV
cana-2520	127	4	all	all	DET
cana-2520	127	5	the	the	DET
cana-2520	127	6	vertex	vertex	NOUN
cana-2520	127	7	labels	label	NOUN
cana-2520	127	8	are	be	AUX
cana-2520	127	9	distinct	distinct	ADJ
cana-2520	127	10	for	for	ADP
cana-2520	127	11	edges	edge	NOUN
cana-2520	127	12	in	in	ADP
cana-2520	127	13	𝐺5	𝐺5	NOUN
cana-2520	127	14	g.c.d	g.c.d	NOUN
cana-2520	127	15	(	(	PUNCT
cana-2520	127	16	℘(𝑣𝑖	℘(𝑣𝑖	PROPN
cana-2520	127	17	)	)	PUNCT
cana-2520	127	18	,	,	PUNCT
cana-2520	127	19	℘(𝑣𝑖+1	℘(𝑣𝑖+1	PROPN
cana-2520	127	20	)	)	PUNCT
cana-2520	127	21	)	)	PUNCT
cana-2520	128	1	=	=	SYM
cana-2520	128	2	1	1	NUM
cana-2520	128	3	,	,	PUNCT
cana-2520	128	4	1	1	NUM
cana-2520	128	5	≤	≤	NUM
cana-2520	128	6	𝑖	𝑖	SYM
cana-2520	128	7	≤	≤	NOUN
cana-2520	128	8	3	3	NUM
cana-2520	128	9	g.c.d	g.c.d	NOUN
cana-2520	128	10	(	(	PUNCT
cana-2520	128	11	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	128	12	)	)	PUNCT
cana-2520	128	13	,	,	PUNCT
cana-2520	128	14	℘(𝑣4	℘(𝑣4	NOUN
cana-2520	128	15	)	)	PUNCT
cana-2520	128	16	)	)	PUNCT
cana-2520	129	1	=	=	SYM
cana-2520	129	2	1	1	NUM
cana-2520	129	3	g.c.d	g.c.d	NOUN
cana-2520	129	4	(	(	PUNCT
cana-2520	129	5	℘(𝑣1	℘(𝑣1	ADJ
cana-2520	129	6	)	)	PUNCT
cana-2520	129	7	,	,	PUNCT
cana-2520	129	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	129	9	)	)	PUNCT
cana-2520	129	10	)	)	PUNCT
cana-2520	130	1	=	=	SYM
cana-2520	130	2	1	1	NUM
cana-2520	130	3	g.c.d	g.c.d	NOUN
cana-2520	130	4	(	(	PUNCT
cana-2520	130	5	℘(𝑣2	℘(𝑣2	PROPN
cana-2520	130	6	)	)	PUNCT
cana-2520	130	7	,	,	PUNCT
cana-2520	130	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	130	9	)	)	PUNCT
cana-2520	130	10	)	)	PUNCT
cana-2520	131	1	=	=	SYM
cana-2520	131	2	1	1	NUM
cana-2520	131	3	g.c.d	g.c.d	NOUN
cana-2520	131	4	(	(	PUNCT
cana-2520	131	5	℘(𝑣3	℘(𝑣3	NOUN
cana-2520	131	6	)	)	PUNCT
cana-2520	131	7	,	,	PUNCT
cana-2520	131	8	℘(𝑣5	℘(𝑣5	NOUN
cana-2520	131	9	)	)	PUNCT
cana-2520	131	10	)	)	PUNCT
cana-2520	132	1	=	=	SYM
cana-2520	132	2	1	1	NUM
cana-2520	132	3	hence	hence	ADV
cana-2520	132	4	℘	℘	PROPN
cana-2520	132	5	is	be	AUX
cana-2520	132	6	a	a	DET
cana-2520	132	7	prime	prime	ADJ
cana-2520	132	8	labeling	labeling	NOUN
cana-2520	132	9	on	on	ADP
cana-2520	132	10	𝐺5	𝐺5	NOUN
cana-2520	132	11	.	.	PUNCT
cana-2520	133	1	so	so	ADV
cana-2520	133	2	𝐺5	𝐺5	NOUN
cana-2520	133	3	is	be	AUX
cana-2520	133	4	a	a	DET
cana-2520	133	5	prime	prime	ADJ
cana-2520	133	6	graph	graph	NOUN
cana-2520	133	7	figure	figure	NOUN
cana-2520	133	8	12	12	NUM
cana-2520	133	9	.	.	PUNCT
cana-2520	134	1	prime	prime	ADJ
cana-2520	134	2	labeling	labeling	NOUN
cana-2520	134	3	of	of	ADP
cana-2520	134	4	switching	switching	NOUN
cana-2520	134	5	of	of	ADP
cana-2520	134	6	vertex	vertex	NOUN
cana-2520	134	7	𝑣5	𝑣5	NOUN
cana-2520	134	8	in	in	ADP
cana-2520	134	9	bull	bull	NOUN
cana-2520	134	10	graph	graph	NOUN
cana-2520	134	11	communications	communication	NOUN
cana-2520	134	12	on	on	ADP
cana-2520	134	13	applied	apply	VERB
cana-2520	134	14	nonlinear	nonlinear	ADJ
cana-2520	134	15	analysis	analysis	NOUN
cana-2520	134	16	issn	issn	NOUN
cana-2520	134	17	:	:	PUNCT
cana-2520	134	18	1074	1074	NUM
cana-2520	134	19	-	-	PUNCT
cana-2520	134	20	133x	133x	NUM
cana-2520	134	21	vol	vol	NOUN
cana-2520	134	22	32	32	NUM
cana-2520	134	23	no	no	NOUN
cana-2520	134	24	.	.	PUNCT
cana-2520	135	1	2s	2s	NUM
cana-2520	135	2	(	(	PUNCT
cana-2520	135	3	2025	2025	NUM
cana-2520	135	4	)	)	PUNCT
cana-2520	135	5	600	600	NUM
cana-2520	135	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	135	7	thus	thus	ADV
cana-2520	135	8	,	,	PUNCT
cana-2520	135	9	in	in	ADP
cana-2520	135	10	all	all	DET
cana-2520	135	11	the	the	DET
cana-2520	135	12	cases	case	NOUN
cana-2520	135	13	the	the	DET
cana-2520	135	14	graph	graph	NOUN
cana-2520	135	15	obtained	obtain	VERB
cana-2520	135	16	by	by	ADP
cana-2520	135	17	switching	switching	NOUN
cana-2520	135	18	of	of	ADP
cana-2520	135	19	any	any	DET
cana-2520	135	20	arbitrary	arbitrary	ADJ
cana-2520	135	21	vertex	vertex	NOUN
cana-2520	135	22	of	of	ADP
cana-2520	135	23	bull	bull	NOUN
cana-2520	135	24	graph	graph	NOUN
cana-2520	135	25	is	be	AUX
cana-2520	135	26	a	a	DET
cana-2520	135	27	prime	prime	ADJ
cana-2520	135	28	graph	graph	NOUN
cana-2520	135	29	.	.	PUNCT
cana-2520	136	1	theorem-3.3	theorem-3.3	NOUN
cana-2520	136	2	.	.	PUNCT
cana-2520	137	1	in	in	ADP
cana-2520	137	2	a	a	DET
cana-2520	137	3	bull	bull	NOUN
cana-2520	137	4	graph	graph	NOUN
cana-2520	137	5	fusion	fusion	NOUN
cana-2520	137	6	of	of	ADP
cana-2520	137	7	any	any	DET
cana-2520	137	8	arbitrary	arbitrary	ADJ
cana-2520	137	9	vertex	vertex	NOUN
cana-2520	137	10	with	with	ADP
cana-2520	137	11	𝑣1	𝑣1	NOUN
cana-2520	137	12	produces	produce	VERB
cana-2520	137	13	a	a	DET
cana-2520	137	14	prime	prime	ADJ
cana-2520	137	15	graph	graph	NOUN
cana-2520	137	16	.	.	PUNCT
cana-2520	138	1	proof	proof	NOUN
cana-2520	138	2	.	.	PUNCT
cana-2520	139	1	figure	figure	VERB
cana-2520	139	2	13	13	NUM
cana-2520	139	3	.	.	PUNCT
cana-2520	140	1	bull	bull	NOUN
cana-2520	140	2	graph	graph	NOUN
cana-2520	140	3	case-1	case-1	PROPN
cana-2520	140	4	.	.	PUNCT
cana-2520	140	5	fusion	fusion	NOUN
cana-2520	140	6	of	of	ADP
cana-2520	140	7	𝑣2	𝑣2	PROPN
cana-2520	140	8	with	with	ADP
cana-2520	140	9	𝑣1	𝑣1	PROPN
cana-2520	140	10	let	let	VERB
cana-2520	140	11	𝐺1	𝐺1	NOUN
cana-2520	140	12	be	be	AUX
cana-2520	140	13	the	the	DET
cana-2520	140	14	graph	graph	NOUN
cana-2520	140	15	obtained	obtain	VERB
cana-2520	140	16	by	by	ADP
cana-2520	140	17	fusion	fusion	NOUN
cana-2520	140	18	of	of	ADP
cana-2520	140	19	𝑣2	𝑣2	NUM
cana-2520	140	20	with	with	ADP
cana-2520	140	21	𝑣1	𝑣1	PROPN
cana-2520	140	22	define	define	VERB
cana-2520	140	23	𝒰	𝒰	PROPN
cana-2520	140	24	:	:	PUNCT
cana-2520	140	25	𝑉	𝑉	PROPN
cana-2520	140	26	(	(	PUNCT
cana-2520	140	27	𝐺1	𝐺1	NOUN
cana-2520	140	28	)	)	PUNCT
cana-2520	140	29	→	→	SYM
cana-2520	140	30	{	{	PUNCT
cana-2520	140	31	1,2,3,4	1,2,3,4	NUM
cana-2520	140	32	}	}	PUNCT
cana-2520	140	33	by	by	ADP
cana-2520	140	34	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	140	35	=	=	SYM
cana-2520	140	36	𝑣2	𝑣2	PROPN
cana-2520	140	37	)	)	PUNCT
cana-2520	140	38	=	=	SYM
cana-2520	140	39	1	1	NUM
cana-2520	140	40	,	,	PUNCT
cana-2520	140	41	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	140	42	)	)	PUNCT
cana-2520	140	43	=	=	SYM
cana-2520	140	44	2	2	NUM
cana-2520	140	45	,	,	PUNCT
cana-2520	140	46	𝒰(𝑣4	𝒰(𝑣4	PROPN
cana-2520	140	47	)	)	PUNCT
cana-2520	140	48	=	=	SYM
cana-2520	140	49	3	3	NUM
cana-2520	140	50	,	,	PUNCT
cana-2520	140	51	𝒰(𝑣5	𝒰(𝑣5	PROPN
cana-2520	140	52	)	)	PUNCT
cana-2520	140	53	=	=	SYM
cana-2520	140	54	4	4	NUM
cana-2520	140	55	evidently	evidently	ADV
cana-2520	140	56	all	all	DET
cana-2520	140	57	the	the	DET
cana-2520	140	58	vertex	vertex	NOUN
cana-2520	140	59	labels	label	NOUN
cana-2520	140	60	are	be	AUX
cana-2520	140	61	distinct	distinct	ADJ
cana-2520	140	62	for	for	ADP
cana-2520	140	63	edges	edge	NOUN
cana-2520	140	64	in	in	ADP
cana-2520	140	65	𝐺1	𝐺1	NOUN
cana-2520	140	66	g.c.d	g.c.d	NOUN
cana-2520	140	67	(	(	PUNCT
cana-2520	140	68	𝒰(𝑣𝑖	𝒰(𝑣𝑖	NOUN
cana-2520	140	69	)	)	PUNCT
cana-2520	140	70	,	,	PUNCT
cana-2520	140	71	𝒰(𝑣𝑖+1	𝒰(𝑣𝑖+1	PROPN
cana-2520	140	72	)	)	PUNCT
cana-2520	140	73	)	)	PUNCT
cana-2520	141	1	=	=	SYM
cana-2520	141	2	1	1	NUM
cana-2520	141	3	,	,	PUNCT
cana-2520	141	4	3	3	NUM
cana-2520	141	5	≤	≤	NUM
cana-2520	141	6	𝑖	𝑖	SYM
cana-2520	141	7	≤	≤	NOUN
cana-2520	141	8	4	4	NUM
cana-2520	141	9	g.c.d	g.c.d	NOUN
cana-2520	141	10	(	(	PUNCT
cana-2520	141	11	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	141	12	=	=	SYM
cana-2520	141	13	𝑣2	𝑣2	PROPN
cana-2520	141	14	)	)	PUNCT
cana-2520	141	15	,	,	PUNCT
cana-2520	141	16	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	141	17	)	)	PUNCT
cana-2520	141	18	)	)	PUNCT
cana-2520	142	1	=	=	SYM
cana-2520	142	2	1	1	NUM
cana-2520	142	3	g.c.d	g.c.d	NOUN
cana-2520	142	4	(	(	PUNCT
cana-2520	142	5	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	142	6	=	=	SYM
cana-2520	142	7	𝑣2	𝑣2	PROPN
cana-2520	142	8	)	)	PUNCT
cana-2520	142	9	,	,	PUNCT
cana-2520	142	10	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	142	11	)	)	PUNCT
cana-2520	142	12	)	)	PUNCT
cana-2520	143	1	=	=	SYM
cana-2520	144	1	1	1	NUM
cana-2520	144	2	hence	hence	ADV
cana-2520	144	3	𝒰	𝒰	PROPN
cana-2520	144	4	is	be	AUX
cana-2520	144	5	a	a	DET
cana-2520	144	6	prime	prime	ADJ
cana-2520	144	7	labeling	labeling	NOUN
cana-2520	144	8	on	on	ADP
cana-2520	144	9	𝐺1	𝐺1	NOUN
cana-2520	144	10	.	.	PUNCT
cana-2520	145	1	so	so	ADV
cana-2520	145	2	𝐺1	𝐺1	PROPN
cana-2520	145	3	is	be	AUX
cana-2520	145	4	a	a	DET
cana-2520	145	5	prime	prime	ADJ
cana-2520	145	6	graph	graph	NOUN
cana-2520	145	7	figure	figure	NOUN
cana-2520	145	8	14	14	NUM
cana-2520	145	9	.	.	PUNCT
cana-2520	146	1	prime	prime	ADJ
cana-2520	146	2	labeling	labeling	NOUN
cana-2520	146	3	of	of	ADP
cana-2520	146	4	fusion	fusion	NOUN
cana-2520	146	5	of	of	ADP
cana-2520	146	6	vertices	vertex	NOUN
cana-2520	146	7	𝑣2	𝑣2	PROPN
cana-2520	146	8	with	with	ADP
cana-2520	146	9	𝑣1	𝑣1	PROPN
cana-2520	146	10	in	in	ADP
cana-2520	146	11	bull	bull	NOUN
cana-2520	146	12	graph	graph	NOUN
cana-2520	146	13	case-2	case-2	NOUN
cana-2520	146	14	.	.	NOUN
cana-2520	146	15	fusion	fusion	NOUN
cana-2520	146	16	of	of	ADP
cana-2520	146	17	𝑣3	𝑣3	NOUN
cana-2520	146	18	with	with	ADP
cana-2520	146	19	𝑣1	𝑣1	NOUN
cana-2520	146	20	communications	communication	NOUN
cana-2520	146	21	on	on	ADP
cana-2520	146	22	applied	apply	VERB
cana-2520	146	23	nonlinear	nonlinear	ADJ
cana-2520	146	24	analysis	analysis	NOUN
cana-2520	146	25	issn	issn	NOUN
cana-2520	146	26	:	:	PUNCT
cana-2520	146	27	1074	1074	NUM
cana-2520	146	28	-	-	PUNCT
cana-2520	146	29	133x	133x	NUM
cana-2520	146	30	vol	vol	NOUN
cana-2520	146	31	32	32	NUM
cana-2520	146	32	no	no	NOUN
cana-2520	146	33	.	.	PUNCT
cana-2520	147	1	2s	2s	NUM
cana-2520	147	2	(	(	PUNCT
cana-2520	147	3	2025	2025	NUM
cana-2520	147	4	)	)	PUNCT
cana-2520	147	5	601	601	NUM
cana-2520	147	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	147	7	let	let	VERB
cana-2520	147	8	𝐺2	𝐺2	NOUN
cana-2520	147	9	be	be	AUX
cana-2520	147	10	the	the	DET
cana-2520	147	11	graph	graph	NOUN
cana-2520	147	12	obtained	obtain	VERB
cana-2520	147	13	by	by	ADP
cana-2520	147	14	fusion	fusion	NOUN
cana-2520	147	15	of	of	ADP
cana-2520	147	16	𝑣3	𝑣3	NOUN
cana-2520	147	17	with	with	ADP
cana-2520	147	18	𝑣1	𝑣1	NOUN
cana-2520	147	19	define	define	VERB
cana-2520	147	20	𝒰	𝒰	PROPN
cana-2520	147	21	:	:	PUNCT
cana-2520	147	22	𝑉	𝑉	PROPN
cana-2520	147	23	(	(	PUNCT
cana-2520	147	24	𝐺1	𝐺1	NOUN
cana-2520	147	25	)	)	PUNCT
cana-2520	147	26	→	→	SYM
cana-2520	147	27	{	{	PUNCT
cana-2520	147	28	1,2,3,4	1,2,3,4	NUM
cana-2520	147	29	}	}	PUNCT
cana-2520	147	30	by	by	ADP
cana-2520	147	31	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	147	32	=	=	SYM
cana-2520	147	33	𝑣3	𝑣3	ADJ
cana-2520	147	34	)	)	PUNCT
cana-2520	147	35	=	=	SYM
cana-2520	147	36	1	1	NUM
cana-2520	147	37	,	,	PUNCT
cana-2520	147	38	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	147	39	)	)	PUNCT
cana-2520	147	40	=	=	SYM
cana-2520	147	41	2	2	NUM
cana-2520	147	42	,	,	PUNCT
cana-2520	147	43	𝒰(𝑣4	𝒰(𝑣4	PROPN
cana-2520	147	44	)	)	PUNCT
cana-2520	147	45	=	=	SYM
cana-2520	147	46	3	3	NUM
cana-2520	147	47	,	,	PUNCT
cana-2520	147	48	𝒰(𝑣5	𝒰(𝑣5	PROPN
cana-2520	147	49	)	)	PUNCT
cana-2520	147	50	=	=	SYM
cana-2520	148	1	4	4	NUM
cana-2520	148	2	clearly	clearly	ADV
cana-2520	148	3	all	all	DET
cana-2520	148	4	the	the	DET
cana-2520	148	5	vertex	vertex	NOUN
cana-2520	148	6	labels	label	NOUN
cana-2520	148	7	are	be	AUX
cana-2520	148	8	distinct	distinct	ADJ
cana-2520	148	9	for	for	ADP
cana-2520	148	10	edges	edge	NOUN
cana-2520	148	11	in	in	ADP
cana-2520	148	12	𝐺2	𝐺2	ADJ
cana-2520	148	13	g.c.d	g.c.d	NOUN
cana-2520	148	14	(	(	PUNCT
cana-2520	148	15	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	148	16	=	=	SYM
cana-2520	148	17	𝑣3	𝑣3	ADJ
cana-2520	148	18	)	)	PUNCT
cana-2520	148	19	,	,	PUNCT
cana-2520	148	20	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	148	21	)	)	PUNCT
cana-2520	148	22	)	)	PUNCT
cana-2520	149	1	=	=	SYM
cana-2520	149	2	1	1	NUM
cana-2520	149	3	g.c.d	g.c.d	NOUN
cana-2520	149	4	(	(	PUNCT
cana-2520	149	5	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	149	6	=	=	SYM
cana-2520	149	7	𝑣3	𝑣3	ADJ
cana-2520	149	8	)	)	PUNCT
cana-2520	149	9	,	,	PUNCT
cana-2520	149	10	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	149	11	)	)	PUNCT
cana-2520	149	12	)	)	PUNCT
cana-2520	150	1	=	=	SYM
cana-2520	150	2	1	1	NUM
cana-2520	150	3	g.c.d	g.c.d	NOUN
cana-2520	150	4	(	(	PUNCT
cana-2520	150	5	𝒰(𝑣2	𝒰(𝑣2	PROPN
cana-2520	150	6	)	)	PUNCT
cana-2520	150	7	,	,	PUNCT
cana-2520	150	8	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	150	9	)	)	PUNCT
cana-2520	150	10	)	)	PUNCT
cana-2520	151	1	=	=	SYM
cana-2520	151	2	1	1	NUM
cana-2520	151	3	g.c.d	g.c.d	NOUN
cana-2520	151	4	(	(	PUNCT
cana-2520	151	5	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	151	6	)	)	PUNCT
cana-2520	151	7	,	,	PUNCT
cana-2520	151	8	𝒰(𝑣5	𝒰(𝑣5	PROPN
cana-2520	151	9	)	)	PUNCT
cana-2520	151	10	)	)	PUNCT
cana-2520	152	1	=	=	SYM
cana-2520	152	2	1	1	NUM
cana-2520	152	3	hence	hence	ADV
cana-2520	152	4	𝒰	𝒰	PROPN
cana-2520	152	5	is	be	AUX
cana-2520	152	6	a	a	DET
cana-2520	152	7	prime	prime	ADJ
cana-2520	152	8	labeling	labeling	NOUN
cana-2520	152	9	on	on	ADP
cana-2520	152	10	𝐺2	𝐺2	NOUN
cana-2520	152	11	.	.	PUNCT
cana-2520	153	1	therefore	therefore	ADV
cana-2520	153	2	𝐺2	𝐺2	ADV
cana-2520	153	3	is	be	AUX
cana-2520	153	4	a	a	DET
cana-2520	153	5	prime	prime	ADJ
cana-2520	153	6	graph	graph	NOUN
cana-2520	153	7	figure	figure	NOUN
cana-2520	153	8	15	15	NUM
cana-2520	153	9	.	.	PUNCT
cana-2520	154	1	prime	prime	ADJ
cana-2520	154	2	labeling	labeling	NOUN
cana-2520	154	3	of	of	ADP
cana-2520	154	4	fusion	fusion	NOUN
cana-2520	154	5	of	of	ADP
cana-2520	154	6	vertices	vertex	NOUN
cana-2520	154	7	𝑣3	𝑣3	ADJ
cana-2520	154	8	with	with	ADP
cana-2520	154	9	𝑣1	𝑣1	PROPN
cana-2520	154	10	in	in	ADP
cana-2520	154	11	bull	bull	NOUN
cana-2520	154	12	graph	graph	NOUN
cana-2520	154	13	case-3	case-3	PROPN
cana-2520	154	14	.	.	NOUN
cana-2520	154	15	fusion	fusion	NOUN
cana-2520	154	16	of	of	ADP
cana-2520	154	17	𝑣4	𝑣4	NOUN
cana-2520	154	18	with	with	ADP
cana-2520	154	19	𝑣1	𝑣1	PROPN
cana-2520	154	20	let	let	VERB
cana-2520	154	21	𝐺3	𝐺3	PROPN
cana-2520	154	22	be	be	AUX
cana-2520	154	23	the	the	DET
cana-2520	154	24	graph	graph	NOUN
cana-2520	154	25	obtained	obtain	VERB
cana-2520	154	26	by	by	ADP
cana-2520	154	27	fusion	fusion	NOUN
cana-2520	154	28	of	of	ADP
cana-2520	154	29	𝑣4	𝑣4	NOUN
cana-2520	154	30	with	with	ADP
cana-2520	154	31	𝑣1	𝑣1	NOUN
cana-2520	154	32	define	define	VERB
cana-2520	154	33	𝒰	𝒰	PROPN
cana-2520	154	34	:	:	PUNCT
cana-2520	154	35	𝑉	𝑉	PROPN
cana-2520	154	36	(	(	PUNCT
cana-2520	154	37	𝐺1	𝐺1	NOUN
cana-2520	154	38	)	)	PUNCT
cana-2520	154	39	→	→	SYM
cana-2520	154	40	{	{	PUNCT
cana-2520	154	41	1,2,3,4	1,2,3,4	NUM
cana-2520	154	42	}	}	PUNCT
cana-2520	154	43	by	by	ADP
cana-2520	154	44	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	154	45	=	=	SYM
cana-2520	154	46	𝑣4	𝑣4	NOUN
cana-2520	154	47	)	)	PUNCT
cana-2520	154	48	=	=	SYM
cana-2520	154	49	1	1	NUM
cana-2520	154	50	,	,	PUNCT
cana-2520	154	51	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	154	52	)	)	PUNCT
cana-2520	154	53	=	=	SYM
cana-2520	154	54	2	2	NUM
cana-2520	154	55	,	,	PUNCT
cana-2520	154	56	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	154	57	)	)	PUNCT
cana-2520	154	58	=	=	SYM
cana-2520	154	59	3	3	NUM
cana-2520	154	60	,	,	PUNCT
cana-2520	154	61	𝒰(𝑣5	𝒰(𝑣5	PROPN
cana-2520	154	62	)	)	PUNCT
cana-2520	154	63	=	=	SYM
cana-2520	155	1	4	4	NUM
cana-2520	155	2	evidently	evidently	ADV
cana-2520	155	3	all	all	DET
cana-2520	155	4	the	the	DET
cana-2520	155	5	vertex	vertex	NOUN
cana-2520	155	6	labels	label	NOUN
cana-2520	155	7	are	be	AUX
cana-2520	155	8	distinct	distinct	ADJ
cana-2520	155	9	for	for	ADP
cana-2520	155	10	edges	edge	NOUN
cana-2520	155	11	in	in	ADP
cana-2520	155	12	𝐺3	𝐺3	ADJ
cana-2520	155	13	g.c.d	g.c.d	NOUN
cana-2520	155	14	(	(	PUNCT
cana-2520	155	15	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	155	16	=	=	SYM
cana-2520	155	17	𝑣4	𝑣4	NOUN
cana-2520	155	18	)	)	PUNCT
cana-2520	155	19	,	,	PUNCT
cana-2520	155	20	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	155	21	)	)	PUNCT
cana-2520	155	22	)	)	PUNCT
cana-2520	155	23	=	=	SYM
cana-2520	155	24	1	1	NUM
cana-2520	155	25	g.c.d	g.c.d	NOUN
cana-2520	155	26	(	(	PUNCT
cana-2520	155	27	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	155	28	=	=	SYM
cana-2520	155	29	𝑣4	𝑣4	NOUN
cana-2520	155	30	)	)	PUNCT
cana-2520	155	31	,	,	PUNCT
cana-2520	155	32	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	155	33	)	)	PUNCT
cana-2520	155	34	)	)	PUNCT
cana-2520	156	1	=	=	SYM
cana-2520	156	2	1	1	NUM
cana-2520	156	3	g.c.d	g.c.d	NOUN
cana-2520	156	4	(	(	PUNCT
cana-2520	156	5	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	156	6	=	=	SYM
cana-2520	156	7	𝑣4	𝑣4	NOUN
cana-2520	156	8	)	)	PUNCT
cana-2520	156	9	,	,	PUNCT
cana-2520	156	10	𝒰(𝑣5	𝒰(𝑣5	PROPN
cana-2520	156	11	)	)	PUNCT
cana-2520	156	12	)	)	PUNCT
cana-2520	157	1	=	=	SYM
cana-2520	157	2	1	1	NUM
cana-2520	157	3	g.c.d	g.c.d	NOUN
cana-2520	157	4	(	(	PUNCT
cana-2520	157	5	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	157	6	)	)	PUNCT
cana-2520	157	7	,	,	PUNCT
cana-2520	157	8	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	157	9	)	)	PUNCT
cana-2520	157	10	)	)	PUNCT
cana-2520	157	11	=	=	SYM
cana-2520	158	1	1	1	NUM
cana-2520	158	2	hence	hence	ADV
cana-2520	158	3	𝒰	𝒰	PROPN
cana-2520	158	4	is	be	AUX
cana-2520	158	5	a	a	DET
cana-2520	158	6	prime	prime	ADJ
cana-2520	158	7	labeling	labeling	NOUN
cana-2520	158	8	on	on	ADP
cana-2520	158	9	𝐺3	𝐺3	PROPN
cana-2520	158	10	.	.	PUNCT
cana-2520	159	1	so	so	ADV
cana-2520	159	2	𝐺3	𝐺3	PROPN
cana-2520	159	3	is	be	AUX
cana-2520	159	4	a	a	DET
cana-2520	159	5	prime	prime	ADJ
cana-2520	159	6	graph	graph	NOUN
cana-2520	159	7	communications	communication	NOUN
cana-2520	159	8	on	on	ADP
cana-2520	159	9	applied	apply	VERB
cana-2520	159	10	nonlinear	nonlinear	ADJ
cana-2520	159	11	analysis	analysis	NOUN
cana-2520	159	12	issn	issn	NOUN
cana-2520	159	13	:	:	PUNCT
cana-2520	159	14	1074	1074	NUM
cana-2520	159	15	-	-	PUNCT
cana-2520	159	16	133x	133x	NUM
cana-2520	159	17	vol	vol	NOUN
cana-2520	159	18	32	32	NUM
cana-2520	159	19	no	no	NOUN
cana-2520	159	20	.	.	PUNCT
cana-2520	160	1	2s	2s	NUM
cana-2520	160	2	(	(	PUNCT
cana-2520	160	3	2025	2025	NUM
cana-2520	160	4	)	)	PUNCT
cana-2520	160	5	602	602	NUM
cana-2520	161	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	161	2	figure	figure	NOUN
cana-2520	161	3	16	16	NUM
cana-2520	161	4	.	.	PUNCT
cana-2520	162	1	prime	prime	ADJ
cana-2520	162	2	labeling	labeling	NOUN
cana-2520	162	3	of	of	ADP
cana-2520	162	4	fusion	fusion	NOUN
cana-2520	162	5	of	of	ADP
cana-2520	162	6	vertices	vertex	NOUN
cana-2520	162	7	𝑣4	𝑣4	NOUN
cana-2520	162	8	with	with	ADP
cana-2520	162	9	𝑣1	𝑣1	PROPN
cana-2520	162	10	in	in	ADP
cana-2520	162	11	bull	bull	NOUN
cana-2520	162	12	graph	graph	NOUN
cana-2520	162	13	case-4	case-4	PROPN
cana-2520	162	14	.	.	PUNCT
cana-2520	162	15	fusion	fusion	NOUN
cana-2520	162	16	of	of	ADP
cana-2520	162	17	𝑣5	𝑣5	NOUN
cana-2520	162	18	with	with	ADP
cana-2520	162	19	𝑣1	𝑣1	PROPN
cana-2520	162	20	let	let	VERB
cana-2520	162	21	𝐺4	𝐺4	PROPN
cana-2520	162	22	be	be	AUX
cana-2520	162	23	the	the	DET
cana-2520	162	24	graph	graph	NOUN
cana-2520	162	25	obtained	obtain	VERB
cana-2520	162	26	by	by	ADP
cana-2520	162	27	fusion	fusion	NOUN
cana-2520	162	28	of	of	ADP
cana-2520	162	29	𝑣5	𝑣5	NOUN
cana-2520	162	30	with	with	ADP
cana-2520	162	31	𝑣1	𝑣1	PROPN
cana-2520	162	32	define	define	VERB
cana-2520	162	33	𝒰	𝒰	PROPN
cana-2520	162	34	:	:	PUNCT
cana-2520	162	35	𝑉	𝑉	PROPN
cana-2520	162	36	(	(	PUNCT
cana-2520	162	37	𝐺1	𝐺1	NOUN
cana-2520	162	38	)	)	PUNCT
cana-2520	162	39	→	→	SYM
cana-2520	162	40	{	{	PUNCT
cana-2520	162	41	1,2,3,4	1,2,3,4	NUM
cana-2520	162	42	}	}	PUNCT
cana-2520	162	43	by	by	ADP
cana-2520	162	44	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	162	45	=	=	PUNCT
cana-2520	162	46	𝑣5	𝑣5	NOUN
cana-2520	162	47	)	)	PUNCT
cana-2520	162	48	=	=	SYM
cana-2520	163	1	4	4	X
cana-2520	163	2	,	,	PUNCT
cana-2520	163	3	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	163	4	)	)	PUNCT
cana-2520	163	5	=	=	SYM
cana-2520	163	6	3	3	NUM
cana-2520	163	7	,	,	PUNCT
cana-2520	163	8	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	163	9	)	)	PUNCT
cana-2520	163	10	=	=	SYM
cana-2520	163	11	2	2	NUM
cana-2520	163	12	,	,	PUNCT
cana-2520	163	13	𝒰(𝑣4	𝒰(𝑣4	PROPN
cana-2520	163	14	)	)	PUNCT
cana-2520	163	15	=	=	SYM
cana-2520	164	1	1	1	NUM
cana-2520	164	2	obviously	obviously	ADV
cana-2520	164	3	all	all	DET
cana-2520	164	4	the	the	DET
cana-2520	164	5	vertex	vertex	NOUN
cana-2520	164	6	labels	label	NOUN
cana-2520	164	7	are	be	AUX
cana-2520	164	8	distinct	distinct	ADJ
cana-2520	164	9	for	for	ADP
cana-2520	164	10	edges	edge	NOUN
cana-2520	164	11	in	in	ADP
cana-2520	164	12	𝐺4	𝐺4	PROPN
cana-2520	164	13	g.c.d	g.c.d	NOUN
cana-2520	164	14	(	(	PUNCT
cana-2520	164	15	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	164	16	)	)	PUNCT
cana-2520	164	17	,	,	PUNCT
cana-2520	164	18	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	164	19	)	)	PUNCT
cana-2520	164	20	)	)	PUNCT
cana-2520	165	1	=	=	SYM
cana-2520	165	2	1	1	NUM
cana-2520	165	3	g.c.d	g.c.d	NOUN
cana-2520	165	4	(	(	PUNCT
cana-2520	165	5	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	165	6	)	)	PUNCT
cana-2520	165	7	,	,	PUNCT
cana-2520	165	8	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	165	9	)	)	PUNCT
cana-2520	165	10	)	)	PUNCT
cana-2520	166	1	=	=	SYM
cana-2520	166	2	1	1	NUM
cana-2520	166	3	g.c.d	g.c.d	NOUN
cana-2520	166	4	(	(	PUNCT
cana-2520	166	5	𝒰(𝑣3	𝒰(𝑣3	NOUN
cana-2520	166	6	)	)	PUNCT
cana-2520	166	7	,	,	PUNCT
cana-2520	166	8	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	166	9	)	)	PUNCT
cana-2520	166	10	)	)	PUNCT
cana-2520	167	1	=	=	SYM
cana-2520	167	2	1	1	NUM
cana-2520	167	3	g.c.d	g.c.d	NOUN
cana-2520	167	4	(	(	PUNCT
cana-2520	167	5	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	167	6	=	=	PROPN
cana-2520	167	7	𝑣5	𝑣5	NOUN
cana-2520	167	8	)	)	PUNCT
cana-2520	167	9	,	,	PUNCT
cana-2520	167	10	𝒰(𝑣2	𝒰(𝑣2	NOUN
cana-2520	167	11	)	)	PUNCT
cana-2520	167	12	)	)	PUNCT
cana-2520	167	13	=	=	SYM
cana-2520	167	14	1	1	NUM
cana-2520	167	15	g.c.d	g.c.d	NOUN
cana-2520	167	16	(	(	PUNCT
cana-2520	167	17	𝒰(𝑣1	𝒰(𝑣1	PROPN
cana-2520	167	18	=	=	PROPN
cana-2520	167	19	𝑣5	𝑣5	NOUN
cana-2520	167	20	)	)	PUNCT
cana-2520	167	21	,	,	PUNCT
cana-2520	167	22	𝒰(𝑣4	𝒰(𝑣4	NOUN
cana-2520	167	23	)	)	PUNCT
cana-2520	167	24	)	)	PUNCT
cana-2520	167	25	=	=	SYM
cana-2520	168	1	1	1	NUM
cana-2520	168	2	thus	thus	ADV
cana-2520	168	3	𝒰	𝒰	PROPN
cana-2520	168	4	is	be	AUX
cana-2520	168	5	a	a	DET
cana-2520	168	6	prime	prime	ADJ
cana-2520	168	7	labeling	labeling	NOUN
cana-2520	168	8	on	on	ADP
cana-2520	168	9	𝐺4	𝐺4	PROPN
cana-2520	168	10	.	.	PUNCT
cana-2520	169	1	hence	hence	ADV
cana-2520	169	2	𝐺4	𝐺4	PROPN
cana-2520	169	3	is	be	AUX
cana-2520	169	4	a	a	DET
cana-2520	169	5	prime	prime	ADJ
cana-2520	169	6	graph	graph	NOUN
cana-2520	169	7	.	.	PUNCT
cana-2520	170	1	figure	figure	NOUN
cana-2520	170	2	17	17	NUM
cana-2520	170	3	.	.	PUNCT
cana-2520	171	1	prime	prime	ADJ
cana-2520	171	2	labeling	labeling	NOUN
cana-2520	171	3	of	of	ADP
cana-2520	171	4	fusion	fusion	NOUN
cana-2520	171	5	of	of	ADP
cana-2520	171	6	vertices	vertex	NOUN
cana-2520	171	7	𝑣5	𝑣5	NOUN
cana-2520	171	8	with	with	ADP
cana-2520	171	9	𝑣1	𝑣1	PROPN
cana-2520	171	10	in	in	ADP
cana-2520	171	11	bull	bull	NOUN
cana-2520	171	12	graph	graph	NOUN
cana-2520	171	13	communications	communication	NOUN
cana-2520	171	14	on	on	ADP
cana-2520	171	15	applied	apply	VERB
cana-2520	171	16	nonlinear	nonlinear	ADJ
cana-2520	171	17	analysis	analysis	NOUN
cana-2520	171	18	issn	issn	NOUN
cana-2520	171	19	:	:	PUNCT
cana-2520	171	20	1074	1074	NUM
cana-2520	171	21	-	-	PUNCT
cana-2520	171	22	133x	133x	NUM
cana-2520	171	23	vol	vol	NOUN
cana-2520	171	24	32	32	NUM
cana-2520	171	25	no	no	NOUN
cana-2520	171	26	.	.	PUNCT
cana-2520	172	1	2s	2s	NUM
cana-2520	172	2	(	(	PUNCT
cana-2520	172	3	2025	2025	NUM
cana-2520	172	4	)	)	PUNCT
cana-2520	172	5	603	603	NUM
cana-2520	172	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2520	172	7	thus	thus	ADV
cana-2520	172	8	,	,	PUNCT
cana-2520	172	9	in	in	ADP
cana-2520	172	10	all	all	DET
cana-2520	172	11	the	the	DET
cana-2520	172	12	cases	case	NOUN
cana-2520	172	13	the	the	DET
cana-2520	172	14	graph	graph	NOUN
cana-2520	172	15	obtained	obtain	VERB
cana-2520	172	16	by	by	ADP
cana-2520	172	17	fusion	fusion	NOUN
cana-2520	172	18	of	of	ADP
cana-2520	172	19	any	any	DET
cana-2520	172	20	arbitrary	arbitrary	ADJ
cana-2520	172	21	vertex	vertex	NOUN
cana-2520	172	22	to	to	ADP
cana-2520	172	23	𝑣1	𝑣1	NOUN
cana-2520	172	24	of	of	ADP
cana-2520	172	25	bull	bull	NOUN
cana-2520	172	26	graph	graph	NOUN
cana-2520	172	27	is	be	AUX
cana-2520	172	28	a	a	DET
cana-2520	172	29	prime	prime	ADJ
cana-2520	172	30	graph	graph	NOUN
cana-2520	172	31	.	.	PUNCT
cana-2520	173	1	4.conclusion	4.conclusion	NOUN
cana-2520	173	2	and	and	CCONJ
cana-2520	173	3	future	future	ADJ
cana-2520	173	4	work	work	NOUN
cana-2520	173	5	in	in	ADP
cana-2520	173	6	this	this	DET
cana-2520	173	7	paper	paper	NOUN
cana-2520	173	8	we	we	PRON
cana-2520	173	9	have	have	AUX
cana-2520	173	10	proved	prove	VERB
cana-2520	173	11	that	that	SCONJ
cana-2520	173	12	bull	bull	NOUN
cana-2520	173	13	graph	graph	NOUN
cana-2520	173	14	admits	admit	VERB
cana-2520	173	15	prime	prime	ADJ
cana-2520	173	16	labeling	labeling	NOUN
cana-2520	173	17	in	in	ADP
cana-2520	173	18	the	the	DET
cana-2520	173	19	context	context	NOUN
cana-2520	173	20	of	of	ADP
cana-2520	173	21	graph	graph	NOUN
cana-2520	173	22	operations	operation	NOUN
cana-2520	173	23	namely	namely	ADV
cana-2520	173	24	duplication	duplication	NOUN
cana-2520	173	25	,	,	PUNCT
cana-2520	173	26	fusion	fusion	NOUN
cana-2520	173	27	and	and	CCONJ
cana-2520	173	28	switching	switching	NOUN
cana-2520	173	29	.	.	PUNCT
cana-2520	174	1	there	there	PRON
cana-2520	174	2	exist	exist	VERB
cana-2520	174	3	many	many	ADJ
cana-2520	174	4	such	such	ADJ
cana-2520	174	5	graphs	graph	NOUN
cana-2520	174	6	that	that	PRON
cana-2520	174	7	admit	admit	VERB
cana-2520	174	8	prime	prime	ADJ
cana-2520	174	9	labeling	labeling	NOUN
cana-2520	174	10	.	.	PUNCT
cana-2520	175	1	an	an	DET
cana-2520	175	2	investigation	investigation	NOUN
cana-2520	175	3	to	to	PART
cana-2520	175	4	identify	identify	VERB
cana-2520	175	5	such	such	ADJ
cana-2520	175	6	graphs	graph	NOUN
cana-2520	175	7	can	can	AUX
cana-2520	175	8	be	be	AUX
cana-2520	175	9	considered	consider	VERB
cana-2520	175	10	as	as	ADP
cana-2520	175	11	future	future	ADJ
cana-2520	175	12	work	work	NOUN
cana-2520	175	13	.	.	PUNCT
cana-2520	176	1	refrences	refrence	VERB
cana-2520	177	1	[	[	X
cana-2520	177	2	1	1	NUM
cana-2520	177	3	]	]	X
cana-2520	177	4	ashokkumar	ashokkumar	X
cana-2520	177	5	.	.	PUNCT
cana-2520	178	1	s	s	PART
cana-2520	178	2	,	,	PUNCT
cana-2520	178	3	and	and	CCONJ
cana-2520	178	4	maragathavalli	maragathavalli	NOUN
cana-2520	178	5	.	.	PUNCT
cana-2520	179	1	s,(2015	s,(2015	ADV
cana-2520	179	2	)	)	PUNCT
cana-2520	179	3	,	,	PUNCT
cana-2520	179	4	prime	prime	ADJ
cana-2520	179	5	labeling	labeling	NOUN
cana-2520	179	6	of	of	ADP
cana-2520	179	7	some	some	DET
cana-2520	179	8	special	special	ADJ
cana-2520	179	9	graphs	graph	NOUN
cana-2520	179	10	,	,	PUNCT
cana-2520	179	11	vol	vol	NOUN
cana-2520	179	12	.	.	PROPN
cana-2520	179	13	11	11	NUM
cana-2520	179	14	,	,	PUNCT
cana-2520	179	15	no.1	no.1	NUM
cana-2520	179	16	,	,	PUNCT
cana-2520	179	17	isor	isor	NOUN
cana-2520	179	18	journal	journal	PROPN
cana-2520	179	19	of	of	ADP
cana-2520	179	20	mathematics	mathematic	NOUN
cana-2520	179	21	,	,	PUNCT
cana-2520	179	22	pp	pp	ADJ
cana-2520	179	23	.	.	PUNCT
cana-2520	180	1	01	01	NUM
cana-2520	180	2	-	-	SYM
cana-2520	180	3	05	05	NUM
cana-2520	180	4	.	.	PUNCT
cana-2520	181	1	[	[	X
cana-2520	181	2	2	2	NUM
cana-2520	181	3	]	]	SYM
cana-2520	181	4	burton	burton	PROPN
cana-2520	181	5	,	,	PUNCT
cana-2520	181	6	d.	d.	PROPN
cana-2520	181	7	m	m	PROPN
cana-2520	181	8	(	(	PUNCT
cana-2520	181	9	1980	1980	NUM
cana-2520	181	10	)	)	PUNCT
cana-2520	181	11	elementary	elementary	ADJ
cana-2520	181	12	number	number	NOUN
cana-2520	181	13	theory	theory	NOUN
cana-2520	181	14	,	,	PUNCT
cana-2520	181	15	second	second	ADJ
cana-2520	181	16	edition	edition	NOUN
cana-2520	181	17	,	,	PUNCT
cana-2520	181	18	wm	wm	PROPN
cana-2520	181	19	.	.	PROPN
cana-2520	181	20	c.	c.	PROPN
cana-2520	181	21	brown	brown	PROPN
cana-2520	181	22	company	company	NOUN
cana-2520	181	23	publishers	publisher	NOUN
cana-2520	181	24	,	,	PUNCT
cana-2520	181	25	[	[	X
cana-2520	181	26	3	3	NUM
cana-2520	181	27	]	]	X
cana-2520	181	28	fu	fu	ADJ
cana-2520	181	29	,	,	PUNCT
cana-2520	181	30	h	h	PROPN
cana-2520	181	31	,	,	PUNCT
cana-2520	181	32	huang	huang	PROPN
cana-2520	181	33	,	,	PUNCT
cana-2520	181	34	k.	k.	PROPN
cana-2520	181	35	(	(	PUNCT
cana-2520	181	36	1994	1994	NUM
cana-2520	181	37	)	)	PUNCT
cana-2520	181	38	.	.	PUNCT
cana-2520	182	1	on	on	ADP
cana-2520	182	2	prime	prime	ADJ
cana-2520	182	3	labelings	labeling	NOUN
cana-2520	182	4	.	.	PUNCT
cana-2520	183	1	discrete	discrete	ADJ
cana-2520	183	2	math	math	NOUN
cana-2520	183	3	.	.	PUNCT
cana-2520	184	1	127(1–3	127(1–3	NUM
cana-2520	184	2	):	):	PUNCT
cana-2520	184	3	181–186	181–186	NUM
cana-2520	184	4	.	.	PUNCT
cana-2520	185	1	[	[	X
cana-2520	185	2	4	4	NUM
cana-2520	185	3	]	]	SYM
cana-2520	185	4	gallian	gallian	NOUN
cana-2520	185	5	,	,	PUNCT
cana-2520	185	6	j.	j.	PROPN
cana-2520	185	7	(	(	PUNCT
cana-2520	185	8	2019	2019	NUM
cana-2520	185	9	)	)	PUNCT
cana-2520	185	10	.	.	PUNCT
cana-2520	186	1	a	a	DET
cana-2520	186	2	dynamic	dynamic	ADJ
cana-2520	186	3	survey	survey	NOUN
cana-2520	186	4	of	of	ADP
cana-2520	186	5	graph	graph	NOUN
cana-2520	186	6	labeling	labeling	NOUN
cana-2520	186	7	.	.	PUNCT
cana-2520	187	1	electron	electron	PROPN
cana-2520	187	2	.	.	PUNCT
cana-2520	188	1	j.	j.	PROPN
cana-2520	188	2	combin.https://www.combinatorics	combin.https://www.combinatorics	PROPN
cana-2520	188	3	.	.	PUNCT
cana-2520	189	1	org/	org/	PRON
cana-2520	189	2	ojs	ojs	PROPN
cana-2520	189	3	/	/	SYM
cana-2520	189	4	index.php	index.php	VERB
cana-2520	189	5	/	/	NOUN
cana-2520	189	6	eljc	eljc	NOUN
cana-2520	189	7	/	/	SYM
cana-2520	189	8	article	article	NOUN
cana-2520	189	9	/	/	SYM
cana-2520	189	10	viewfile	viewfile	ADJ
cana-2520	189	11	/	/	SYM
cana-2520	189	12	ds6	ds6	NOUN
cana-2520	189	13	/	/	SYM
cana-2520	189	14	pdf	pdf	NOUN
cana-2520	190	1	[	[	X
cana-2520	190	2	5	5	NUM
cana-2520	190	3	]	]	PUNCT
cana-2520	190	4	ganeshan	ganeshan	NOUN
cana-2520	190	5	m	m	PROPN
cana-2520	190	6	(	(	PUNCT
cana-2520	190	7	2022	2022	NUM
cana-2520	190	8	)	)	PUNCT
cana-2520	190	9	,	,	PUNCT
cana-2520	190	10	sum	sum	VERB
cana-2520	190	11	divisor	divisor	NOUN
cana-2520	190	12	cordial	cordial	ADJ
cana-2520	190	13	labeling	labeling	NOUN
cana-2520	190	14	of	of	ADP
cana-2520	190	15	almost	almost	ADV
cana-2520	190	16	complete	complete	ADJ
cana-2520	190	17	bipartite	bipartite	NOUN
cana-2520	190	18	graph	graph	NOUN
cana-2520	190	19	and	and	CCONJ
cana-2520	190	20	bull	bull	NOUN
cana-2520	190	21	graph	graph	NOUN
cana-2520	190	22	,	,	PUNCT
cana-2520	190	23	a	a	DET
cana-2520	190	24	journal	journal	NOUN
cana-2520	190	25	of	of	ADP
cana-2520	190	26	composition	composition	NOUN
cana-2520	190	27	theory	theory	NOUN
cana-2520	190	28	,	,	PUNCT
cana-2520	190	29	issn	issn	PROPN
cana-2520	190	30	:	:	PUNCT
cana-2520	190	31	0731	0731	NUM
cana-2520	190	32	-	-	SYM
cana-2520	190	33	6755	6755	NUM
cana-2520	190	34	,	,	PUNCT
cana-2520	190	35	volume	volume	NOUN
cana-2520	190	36	xv	xv	PROPN
cana-2520	190	37	,	,	PUNCT
cana-2520	190	38	issue	issue	NOUN
cana-2520	190	39	viii	viii	NOUN
cana-2520	190	40	,	,	PUNCT
cana-2520	190	41	pp.120	pp.120	NOUN
cana-2520	190	42	-	-	PUNCT
cana-2520	190	43	124	124	NUM
cana-2520	190	44	.	.	PUNCT
cana-2520	191	1	[	[	X
cana-2520	191	2	6	6	NUM
cana-2520	191	3	]	]	X
cana-2520	191	4	keerthi	keerthi	PROPN
cana-2520	191	5	kamal	kamal	PROPN
cana-2520	191	6	adusumilli	adusumilli	PROPN
cana-2520	191	7	,	,	PUNCT
cana-2520	191	8	member	member	NOUN
cana-2520	191	9	,	,	PUNCT
cana-2520	191	10	iaeng	iaeng	NOUN
cana-2520	191	11	,	,	PUNCT
cana-2520	191	12	odd	odd	ADJ
cana-2520	191	13	even	even	ADV
cana-2520	191	14	based	base	VERB
cana-2520	191	15	cryptography,36:1	cryptography,36:1	PROPN
cana-2520	191	16	,	,	PUNCT
cana-2520	191	17	ijam_36_1_12	ijam_36_1_12	ADP
cana-2520	191	18	advance	advance	VERB
cana-2520	191	19	online	online	ADV
cana-2520	191	20	publication:1	publication:1	PROPN
cana-2520	191	21	february2007	february2007	PROPN
cana-2520	191	22	,	,	PUNCT
cana-2520	191	23	issn	issn	PROPN
cana-2520	191	24	:	:	PUNCT
cana-2520	191	25	19929986	19929986	NUM
cana-2520	191	26	(	(	PUNCT
cana-2520	191	27	online	online	ADJ
cana-2520	191	28	version	version	PROPN
cana-2520	191	29	)	)	PUNCT
cana-2520	191	30	;	;	PUNCT
cana-2520	192	1	[	[	X
cana-2520	192	2	7	7	X
cana-2520	192	3	]	]	X
cana-2520	192	4	meena	meena	PROPN
cana-2520	192	5	.	.	PUNCT
cana-2520	192	6	s	s	PART
cana-2520	192	7	and	and	CCONJ
cana-2520	192	8	vaithilingam	vaithilingam	ADV
cana-2520	192	9	.	.	PUNCT
cana-2520	193	1	k	k	X
cana-2520	193	2	(	(	PUNCT
cana-2520	193	3	2012	2012	NUM
cana-2520	193	4	)	)	PUNCT
cana-2520	193	5	,	,	PUNCT
cana-2520	193	6	prime	prime	ADJ
cana-2520	193	7	labeling	labeling	NOUN
cana-2520	193	8	for	for	ADP
cana-2520	193	9	some	some	DET
cana-2520	193	10	fan	fan	NOUN
cana-2520	193	11	related	relate	VERB
cana-2520	193	12	graphs	graph	NOUN
cana-2520	193	13	,	,	PUNCT
cana-2520	193	14	international	international	ADJ
cana-2520	193	15	journal	journal	NOUN
cana-2520	193	16	of	of	ADP
cana-2520	193	17	engineering	engineering	PROPN
cana-2520	193	18	research	research	NOUN
cana-2520	193	19	&	&	CCONJ
cana-2520	193	20	technology	technology	PROPN
cana-2520	193	21	(	(	PUNCT
cana-2520	193	22	ijert	ijert	NOUN
cana-2520	193	23	)	)	PUNCT
cana-2520	193	24	vol	vol	NOUN
cana-2520	193	25	.	.	PROPN
cana-2520	193	26	1	1	NUM
cana-2520	193	27	,	,	PUNCT
cana-2520	193	28	issue	issue	NOUN
cana-2520	193	29	9	9	NUM
cana-2520	193	30	.	.	PUNCT
cana-2520	194	1	[	[	X
cana-2520	194	2	8	8	NUM
cana-2520	194	3	]	]	SYM
cana-2520	194	4	tout	tout	X
cana-2520	194	5	,	,	PUNCT
cana-2520	194	6	a	a	DET
cana-2520	194	7	,	,	PUNCT
cana-2520	194	8	.dabboucy	.dabboucy	PROPN
cana-2520	194	9	,	,	PUNCT
cana-2520	194	10	a.n	a.n	PROPN
cana-2520	194	11	and	and	CCONJ
cana-2520	194	12	howalla	howalla	NOUN
cana-2520	194	13	k.	k.	PROPN
cana-2520	194	14	(	(	PUNCT
cana-2520	194	15	1982)prime	1982)prime	NUM
cana-2520	194	16	labeling	labeling	NOUN
cana-2520	194	17	of	of	ADP
cana-2520	194	18	graphs	graph	NOUN
cana-2520	194	19	.	.	PUNCT
cana-2520	195	1	nat.acad.sci	nat.acad.sci	PRON
cana-2520	195	2	letter	letter	VERB
cana-2520	195	3	11	11	NUM
cana-2520	195	4	365	365	NUM
cana-2520	195	5	-	-	SYM
cana-2520	195	6	368	368	NUM
cana-2520	195	7	.	.	PUNCT
cana-2520	196	1	[	[	X
cana-2520	196	2	9	9	NUM
cana-2520	196	3	]	]	X
cana-2520	196	4	vaidya	vaidya	PROPN
cana-2520	196	5	,	,	PUNCT
cana-2520	196	6	s.	s.	PROPN
cana-2520	196	7	k	k	PROPN
cana-2520	196	8	and	and	CCONJ
cana-2520	196	9	kanmani	kanmani	PROPN
cana-2520	196	10	,	,	PUNCT
cana-2520	196	11	k.k	k.k	PROPN
cana-2520	196	12	(	(	PUNCT
cana-2520	196	13	2010	2010	NUM
cana-2520	196	14	)	)	PUNCT
cana-2520	196	15	,	,	PUNCT
cana-2520	196	16	prime	prime	ADJ
cana-2520	196	17	labeling	labeling	NOUN
cana-2520	196	18	for	for	ADP
cana-2520	196	19	some	some	DET
cana-2520	196	20	cycle	cycle	NOUN
cana-2520	196	21	related	relate	VERB
cana-2520	196	22	graphs	graph	NOUN
cana-2520	196	23	,	,	PUNCT
cana-2520	196	24	journals	journal	NOUN
cana-2520	196	25	of	of	ADP
cana-2520	196	26	mathematics	mathematics	PROPN
cana-2520	196	27	research	research	PROPN
cana-2520	196	28	vol.2	vol.2	PROPN
cana-2520	196	29	.	.	PUNCT
cana-2520	197	1	no.2	no.2	PROPN
cana-2520	197	2	.	.	PROPN
cana-2520	197	3	,	,	PUNCT
cana-2520	197	4	98	98	NUM
cana-2520	197	5	-	-	SYM
cana-2520	197	6	104	104	NUM
cana-2520	197	7	.	.	PUNCT