id	sid	tid	token	lemma	pos
cana-512	1	1	communications	communication	NOUN
cana-512	1	2	on	on	ADP
cana-512	1	3	applied	apply	VERB
cana-512	1	4	nonlinear	nonlinear	ADJ
cana-512	1	5	analysis	analysis	NOUN
cana-512	1	6	issn	issn	NOUN
cana-512	1	7	:	:	PUNCT
cana-512	1	8	1074	1074	NUM
cana-512	1	9	-	-	PUNCT
cana-512	1	10	133x	133x	NUM
cana-512	1	11	vol	vol	NOUN
cana-512	1	12	31	31	NUM
cana-512	1	13	no	no	NOUN
cana-512	1	14	.	.	NOUN
cana-512	1	15	2	2	NUM
cana-512	1	16	(	(	PUNCT
cana-512	1	17	2024	2024	NUM
cana-512	1	18	)	)	PUNCT
cana-512	1	19	42	42	NUM
cana-512	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	1	21	on	on	ADP
cana-512	1	22	edge	edge	NOUN
cana-512	1	23	prime	prime	ADJ
cana-512	1	24	index	index	NOUN
cana-512	1	25	of	of	ADP
cana-512	1	26	a	a	DET
cana-512	1	27	graph	graph	NOUN
cana-512	1	28	janani	janani	NOUN
cana-512	1	29	r1	r1	PROPN
cana-512	1	30	*	*	PROPN
cana-512	1	31	,	,	PUNCT
cana-512	1	32	ramachandran	ramachandran	PROPN
cana-512	1	33	t2	t2	PROPN
cana-512	1	34	1	1	NUM
cana-512	1	35	*	*	PROPN
cana-512	1	36	department	department	NOUN
cana-512	1	37	of	of	ADP
cana-512	1	38	mathematics	mathematics	PROPN
cana-512	1	39	,	,	PUNCT
cana-512	1	40	ssm	ssm	PROPN
cana-512	1	41	institute	institute	PROPN
cana-512	1	42	of	of	ADP
cana-512	1	43	engineering	engineering	NOUN
cana-512	1	44	and	and	CCONJ
cana-512	1	45	technology	technology	NOUN
cana-512	1	46	,	,	PUNCT
cana-512	1	47	dindigul	dindigul	ADJ
cana-512	1	48	,	,	PUNCT
cana-512	1	49	tamilnadu	tamilnadu	ADJ
cana-512	1	50	,	,	PUNCT
cana-512	1	51	india	india	PROPN
cana-512	1	52	.	.	PROPN
cana-512	1	53	2	2	NUM
cana-512	1	54	department	department	NOUN
cana-512	1	55	of	of	ADP
cana-512	1	56	mathematics	mathematic	NOUN
cana-512	1	57	,	,	PUNCT
cana-512	1	58	mvm	mvm	PROPN
cana-512	1	59	government	government	NOUN
cana-512	1	60	arts	arts	PROPN
cana-512	1	61	college	college	PROPN
cana-512	1	62	(	(	PUNCT
cana-512	1	63	w	w	NOUN
cana-512	1	64	)	)	PUNCT
cana-512	1	65	,	,	PUNCT
cana-512	1	66	dindigul	dindigul	ADJ
cana-512	1	67	,	,	PUNCT
cana-512	1	68	tamilnadu	tamilnadu	ADJ
cana-512	1	69	,	,	PUNCT
cana-512	1	70	india	india	PROPN
cana-512	1	71	.	.	PUNCT
cana-512	2	1	*	*	PUNCT
cana-512	2	2	corresponding	corresponding	PROPN
cana-512	2	3	author(s	author(s	NOUN
cana-512	2	4	)	)	PUNCT
cana-512	2	5	.	.	PUNCT
cana-512	3	1	e	e	X
cana-512	3	2	-	-	PUNCT
cana-512	3	3	mail(s	mail(s	NOUN
cana-512	3	4	):	):	PUNCT
cana-512	3	5	raseja3@gmail.com	raseja3@gmail.com	PROPN
cana-512	3	6	;	;	PUNCT
cana-512	3	7	contributing	contribute	VERB
cana-512	3	8	authors	author	NOUN
cana-512	3	9	:	:	PUNCT
cana-512	3	10	yasrams@gmail.com	yasrams@gmail.com	X
cana-512	3	11	;	;	PUNCT
cana-512	3	12	article	article	NOUN
cana-512	3	13	history	history	NOUN
cana-512	3	14	:	:	PUNCT
cana-512	3	15	received	receive	VERB
cana-512	3	16	:	:	PUNCT
cana-512	3	17	20	20	NUM
cana-512	3	18	-	-	SYM
cana-512	3	19	01	01	NUM
cana-512	3	20	-	-	PUNCT
cana-512	3	21	2024	2024	NUM
cana-512	3	22	revised	revise	VERB
cana-512	3	23	:	:	PUNCT
cana-512	3	24	30	30	NUM
cana-512	3	25	-	-	SYM
cana-512	3	26	03	03	NUM
cana-512	3	27	-	-	PUNCT
cana-512	3	28	2024	2024	NUM
cana-512	3	29	accepted	accept	VERB
cana-512	3	30	:	:	PUNCT
cana-512	3	31	24	24	NUM
cana-512	3	32	-	-	PUNCT
cana-512	3	33	04	04	NUM
cana-512	3	34	-	-	PUNCT
cana-512	3	35	2024	2024	NUM
cana-512	3	36	abstract	abstract	NOUN
cana-512	3	37	:	:	PUNCT
cana-512	3	38	relatively	relatively	ADV
cana-512	3	39	prime	prime	ADJ
cana-512	3	40	edge	edge	NOUN
cana-512	3	41	labeling	labeling	NOUN
cana-512	3	42	extends	extend	VERB
cana-512	3	43	the	the	DET
cana-512	3	44	notion	notion	NOUN
cana-512	3	45	of	of	ADP
cana-512	3	46	prime	prime	ADJ
cana-512	3	47	labeling	labeling	NOUN
cana-512	3	48	by	by	ADP
cana-512	3	49	considering	consider	VERB
cana-512	3	50	edges	edge	NOUN
cana-512	3	51	.	.	PUNCT
cana-512	4	1	prime	prime	ADJ
cana-512	4	2	labeling	labeling	NOUN
cana-512	4	3	requires	require	VERB
cana-512	4	4	adjacent	adjacent	ADJ
cana-512	4	5	vertices	vertex	NOUN
cana-512	4	6	to	to	PART
cana-512	4	7	possess	possess	VERB
cana-512	4	8	relatively	relatively	ADV
cana-512	4	9	prime	prime	ADJ
cana-512	4	10	labels	label	NOUN
cana-512	4	11	,	,	PUNCT
cana-512	4	12	while	while	SCONJ
cana-512	4	13	relatively	relatively	ADV
cana-512	4	14	prime	prime	ADJ
cana-512	4	15	edge	edge	NOUN
cana-512	4	16	labeling	labeling	NOUN
cana-512	4	17	requires	require	VERB
cana-512	4	18	adjacent	adjacent	ADJ
cana-512	4	19	edges	edge	NOUN
cana-512	4	20	to	to	PART
cana-512	4	21	have	have	VERB
cana-512	4	22	relatively	relatively	ADV
cana-512	4	23	prime	prime	ADJ
cana-512	4	24	labels	label	NOUN
cana-512	4	25	.	.	PUNCT
cana-512	5	1	the	the	DET
cana-512	5	2	transformation	transformation	NOUN
cana-512	5	3	of	of	ADP
cana-512	5	4	a	a	DET
cana-512	5	5	coprime	coprime	ADJ
cana-512	5	6	edge	edge	NOUN
cana-512	5	7	-	-	PUNCT
cana-512	5	8	labeled	label	VERB
cana-512	5	9	graph	graph	NOUN
cana-512	5	10	into	into	ADP
cana-512	5	11	a	a	DET
cana-512	5	12	relatively	relatively	ADV
cana-512	5	13	prime	prime	ADJ
cana-512	5	14	edge	edge	NOUN
cana-512	5	15	-	-	PUNCT
cana-512	5	16	labeled	label	VERB
cana-512	5	17	graph	graph	NOUN
cana-512	5	18	introduces	introduce	VERB
cana-512	5	19	the	the	DET
cana-512	5	20	concept	concept	NOUN
cana-512	5	21	of	of	ADP
cana-512	5	22	edge	edge	ADJ
cana-512	5	23	prime	prime	ADJ
cana-512	5	24	index	index	NOUN
cana-512	5	25	(	(	PUNCT
cana-512	5	26	or	or	CCONJ
cana-512	5	27	relatively	relatively	ADV
cana-512	5	28	prime	prime	ADJ
cana-512	5	29	index	index	NOUN
cana-512	5	30	)	)	PUNCT
cana-512	5	31	.	.	PUNCT
cana-512	6	1	this	this	DET
cana-512	6	2	study	study	NOUN
cana-512	6	3	focuses	focus	VERB
cana-512	6	4	on	on	ADP
cana-512	6	5	cases	case	NOUN
cana-512	6	6	where	where	SCONJ
cana-512	6	7	a	a	DET
cana-512	6	8	coprime	coprime	ADJ
cana-512	6	9	edge	edge	NOUN
cana-512	6	10	-	-	PUNCT
cana-512	6	11	labeled	label	VERB
cana-512	6	12	graph	graph	NOUN
cana-512	6	13	can	can	AUX
cana-512	6	14	be	be	AUX
cana-512	6	15	converted	convert	VERB
cana-512	6	16	into	into	ADP
cana-512	6	17	a	a	DET
cana-512	6	18	relatively	relatively	ADV
cana-512	6	19	prime	prime	ADJ
cana-512	6	20	edge	edge	NOUN
cana-512	6	21	-	-	PUNCT
cana-512	6	22	labeled	label	VERB
cana-512	6	23	graph	graph	NOUN
cana-512	6	24	by	by	ADP
cana-512	6	25	removing	remove	VERB
cana-512	6	26	certain	certain	ADJ
cana-512	6	27	edges	edge	NOUN
cana-512	6	28	from	from	ADP
cana-512	6	29	graph	graph	NOUN
cana-512	6	30	g	g	NOUN
cana-512	6	31	,	,	PUNCT
cana-512	6	32	thereby	thereby	ADV
cana-512	6	33	establishing	establish	VERB
cana-512	6	34	the	the	DET
cana-512	6	35	concept	concept	NOUN
cana-512	6	36	of	of	ADP
cana-512	6	37	edge	edge	ADJ
cana-512	6	38	prime	prime	ADJ
cana-512	6	39	index	index	NOUN
cana-512	6	40	.	.	PUNCT
cana-512	7	1	finally	finally	ADV
cana-512	7	2	,	,	PUNCT
cana-512	7	3	the	the	DET
cana-512	7	4	edge	edge	NOUN
cana-512	7	5	prime	prime	ADJ
cana-512	7	6	index	index	NOUN
cana-512	7	7	of	of	ADP
cana-512	7	8	some	some	DET
cana-512	7	9	graphs	graph	NOUN
cana-512	7	10	are	be	AUX
cana-512	7	11	found	find	VERB
cana-512	7	12	.	.	PUNCT
cana-512	8	1	keywords	keyword	NOUN
cana-512	8	2	:	:	PUNCT
cana-512	8	3	prime	prime	ADJ
cana-512	8	4	labeling	labeling	NOUN
cana-512	8	5	,	,	PUNCT
cana-512	8	6	relatively	relatively	ADV
cana-512	8	7	prime	prime	ADJ
cana-512	8	8	edge	edge	NOUN
cana-512	8	9	labeling	labeling	NOUN
cana-512	8	10	,	,	PUNCT
cana-512	8	11	coprime	coprime	NOUN
cana-512	8	12	edge	edge	NOUN
cana-512	8	13	labeling	labeling	NOUN
cana-512	8	14	,	,	PUNCT
cana-512	8	15	prime	prime	ADJ
cana-512	8	16	index	index	NOUN
cana-512	8	17	,	,	PUNCT
cana-512	8	18	edge	edge	VERB
cana-512	8	19	prime	prime	ADJ
cana-512	8	20	index	index	NOUN
cana-512	8	21	.	.	PUNCT
cana-512	9	1	1	1	X
cana-512	9	2	.	.	X
cana-512	9	3	introduction	introduction	NOUN
cana-512	9	4	labeling	labeling	NOUN
cana-512	9	5	plays	play	VERB
cana-512	9	6	a	a	DET
cana-512	9	7	significant	significant	ADJ
cana-512	9	8	role	role	NOUN
cana-512	9	9	in	in	ADP
cana-512	9	10	the	the	DET
cana-512	9	11	field	field	NOUN
cana-512	9	12	of	of	ADP
cana-512	9	13	graph	graph	NOUN
cana-512	9	14	theory	theory	NOUN
cana-512	9	15	.	.	PUNCT
cana-512	10	1	to	to	PART
cana-512	10	2	meet	meet	VERB
cana-512	10	3	out	out	ADP
cana-512	10	4	the	the	DET
cana-512	10	5	current	current	ADJ
cana-512	10	6	needs	need	VERB
cana-512	10	7	different	different	ADJ
cana-512	10	8	types	type	NOUN
cana-512	10	9	of	of	ADP
cana-512	10	10	labeling	labeling	NOUN
cana-512	10	11	are	be	AUX
cana-512	10	12	emerging	emerge	VERB
cana-512	10	13	now	now	ADV
cana-512	10	14	a	a	DET
cana-512	10	15	days	day	NOUN
cana-512	10	16	(	(	PUNCT
cana-512	10	17	2	2	NUM
cana-512	10	18	,	,	PUNCT
cana-512	10	19	9	9	NUM
cana-512	10	20	)	)	PUNCT
cana-512	10	21	.	.	PUNCT
cana-512	11	1	one	one	NUM
cana-512	11	2	such	such	ADJ
cana-512	11	3	labeling	labeling	NOUN
cana-512	11	4	is	be	AUX
cana-512	11	5	prime	prime	ADJ
cana-512	11	6	labeling	labeling	NOUN
cana-512	11	7	.	.	PUNCT
cana-512	12	1	in	in	ADP
cana-512	12	2	prime	prime	ADJ
cana-512	12	3	labeling	labeling	NOUN
cana-512	12	4	,	,	PUNCT
cana-512	12	5	vertices	vertex	NOUN
cana-512	12	6	are	be	AUX
cana-512	12	7	labeled	label	VERB
cana-512	12	8	from	from	ADP
cana-512	12	9	1	1	NUM
cana-512	12	10	to	to	ADP
cana-512	12	11	n	n	CCONJ
cana-512	12	12	,	,	PUNCT
cana-512	12	13	with	with	ADP
cana-512	12	14	the	the	DET
cana-512	12	15	condition	condition	NOUN
cana-512	12	16	that	that	SCONJ
cana-512	12	17	any	any	DET
cana-512	12	18	two	two	NUM
cana-512	12	19	adjacent	adjacent	ADJ
cana-512	12	20	vertices	vertex	NOUN
cana-512	12	21	have	have	AUX
cana-512	12	22	relatively	relatively	ADV
cana-512	12	23	prime	prime	ADJ
cana-512	12	24	labels	label	NOUN
cana-512	12	25	(	(	PUNCT
cana-512	12	26	1	1	NUM
cana-512	12	27	)	)	PUNCT
cana-512	12	28	.	.	PUNCT
cana-512	13	1	from	from	ADP
cana-512	13	2	the	the	DET
cana-512	13	3	knowledge	knowledge	NOUN
cana-512	13	4	attained	attain	VERB
cana-512	13	5	from	from	ADP
cana-512	13	6	prime	prime	ADJ
cana-512	13	7	labeling	labeling	NOUN
cana-512	13	8	,	,	PUNCT
cana-512	13	9	relatively	relatively	ADV
cana-512	13	10	prime	prime	ADJ
cana-512	13	11	edge	edge	NOUN
cana-512	13	12	labeling	labeling	NOUN
cana-512	13	13	technique	technique	NOUN
cana-512	13	14	focuses	focus	VERB
cana-512	13	15	on	on	ADP
cana-512	13	16	labeling	label	VERB
cana-512	13	17	the	the	DET
cana-512	13	18	edges	edge	NOUN
cana-512	13	19	such	such	ADJ
cana-512	13	20	that	that	SCONJ
cana-512	13	21	adjacent	adjacent	ADJ
cana-512	13	22	edges	edge	NOUN
cana-512	13	23	have	have	VERB
cana-512	13	24	relatively	relatively	ADV
cana-512	13	25	prime	prime	ADJ
cana-512	13	26	labels	label	NOUN
cana-512	13	27	,	,	PUNCT
cana-512	13	28	unlike	unlike	ADP
cana-512	13	29	prime	prime	ADJ
cana-512	13	30	labeling	labeling	NOUN
cana-512	13	31	,	,	PUNCT
cana-512	13	32	which	which	PRON
cana-512	13	33	assigns	assign	VERB
cana-512	13	34	relatively	relatively	ADV
cana-512	13	35	prime	prime	ADJ
cana-512	13	36	labels	label	NOUN
cana-512	13	37	to	to	ADP
cana-512	13	38	adjacent	adjacent	ADJ
cana-512	13	39	vertices	vertex	NOUN
cana-512	13	40	.	.	PUNCT
cana-512	14	1	a	a	DET
cana-512	14	2	graph	graph	NOUN
cana-512	14	3	that	that	PRON
cana-512	14	4	allows	allow	VERB
cana-512	14	5	relatively	relatively	ADV
cana-512	14	6	prime	prime	ADJ
cana-512	14	7	edge	edge	NOUN
cana-512	14	8	labeling	labeling	NOUN
cana-512	14	9	is	be	AUX
cana-512	14	10	known	know	VERB
cana-512	14	11	as	as	ADP
cana-512	14	12	a	a	DET
cana-512	14	13	relatively	relatively	ADV
cana-512	14	14	prime	prime	ADJ
cana-512	14	15	edge	edge	NOUN
cana-512	14	16	-	-	PUNCT
cana-512	14	17	labeled	label	VERB
cana-512	14	18	graph	graph	NOUN
cana-512	14	19	.	.	PUNCT
cana-512	15	1	coprime	coprime	NOUN
cana-512	15	2	labeling	labeling	NOUN
cana-512	15	3	is	be	AUX
cana-512	15	4	another	another	DET
cana-512	15	5	labeling	labeling	NOUN
cana-512	15	6	technique	technique	NOUN
cana-512	15	7	that	that	PRON
cana-512	15	8	is	be	AUX
cana-512	15	9	derived	derive	VERB
cana-512	15	10	from	from	ADP
cana-512	15	11	prime	prime	ADJ
cana-512	15	12	labeling	labeling	NOUN
cana-512	15	13	(	(	PUNCT
cana-512	15	14	3	3	NUM
cana-512	15	15	,	,	PUNCT
cana-512	15	16	4	4	NUM
cana-512	15	17	)	)	PUNCT
cana-512	15	18	.	.	PUNCT
cana-512	16	1	in	in	ADP
cana-512	16	2	coprime	coprime	ADJ
cana-512	16	3	labeling	labeling	NOUN
cana-512	16	4	,	,	PUNCT
cana-512	16	5	the	the	DET
cana-512	16	6	labels	label	NOUN
cana-512	16	7	are	be	AUX
cana-512	16	8	not	not	PART
cana-512	16	9	limited	limit	VERB
cana-512	16	10	to	to	ADP
cana-512	16	11	1	1	NUM
cana-512	16	12	to	to	ADP
cana-512	16	13	n	n	PROPN
cana-512	16	14	as	as	ADP
cana-512	16	15	in	in	ADP
cana-512	16	16	prime	prime	ADJ
cana-512	16	17	labeling	labeling	NOUN
cana-512	16	18	.	.	PUNCT
cana-512	17	1	if	if	SCONJ
cana-512	17	2	the	the	DET
cana-512	17	3	vertices	vertex	NOUN
cana-512	17	4	are	be	AUX
cana-512	17	5	labeled	label	VERB
cana-512	17	6	from	from	ADP
cana-512	17	7	1	1	NUM
cana-512	17	8	to	to	ADP
cana-512	17	9	k	k	NOUN
cana-512	17	10	,	,	PUNCT
cana-512	17	11	then	then	ADV
cana-512	17	12	the	the	DET
cana-512	17	13	least	least	ADJ
cana-512	17	14	k	k	NOUN
cana-512	17	15	is	be	AUX
cana-512	17	16	called	call	VERB
cana-512	17	17	the	the	DET
cana-512	17	18	minimum	minimum	NOUN
cana-512	17	19	coprime	coprime	NOUN
cana-512	17	20	number	number	NOUN
cana-512	17	21	of	of	ADP
cana-512	17	22	g.	g.	PROPN
cana-512	17	23	inspired	inspire	VERB
cana-512	17	24	by	by	ADP
cana-512	17	25	the	the	DET
cana-512	17	26	above	above	ADJ
cana-512	17	27	research	research	NOUN
cana-512	17	28	,	,	PUNCT
cana-512	17	29	coprime	coprime	NOUN
cana-512	17	30	edge	edge	NOUN
cana-512	17	31	labeling	labeling	NOUN
cana-512	17	32	is	be	AUX
cana-512	17	33	employed	employ	VERB
cana-512	17	34	when	when	SCONJ
cana-512	17	35	a	a	DET
cana-512	17	36	graph	graph	NOUN
cana-512	17	37	does	do	AUX
cana-512	17	38	not	not	PART
cana-512	17	39	have	have	VERB
cana-512	17	40	a	a	DET
cana-512	17	41	relatively	relatively	ADV
cana-512	17	42	prime	prime	ADJ
cana-512	17	43	edge	edge	NOUN
cana-512	17	44	labeling	labeling	NOUN
cana-512	17	45	.	.	PUNCT
cana-512	18	1	in	in	ADP
cana-512	18	2	relatively	relatively	ADV
cana-512	18	3	prime	prime	ADJ
cana-512	18	4	edge	edge	NOUN
cana-512	18	5	labeling	labeling	NOUN
cana-512	18	6	,	,	PUNCT
cana-512	18	7	the	the	DET
cana-512	18	8	edges	edge	NOUN
cana-512	18	9	are	be	AUX
cana-512	18	10	labeled	label	VERB
cana-512	18	11	using	use	VERB
cana-512	18	12	numbers	number	NOUN
cana-512	18	13	from	from	ADP
cana-512	18	14	1	1	NUM
cana-512	18	15	to	to	ADP
cana-512	18	16	q.	q.	PROPN
cana-512	18	17	however	however	ADV
cana-512	18	18	,	,	PUNCT
cana-512	18	19	coprime	coprime	NOUN
cana-512	18	20	edge	edge	NOUN
cana-512	18	21	labeling	labeling	NOUN
cana-512	18	22	does	do	AUX
cana-512	18	23	not	not	PART
cana-512	18	24	have	have	VERB
cana-512	18	25	any	any	DET
cana-512	18	26	restrictions	restriction	NOUN
cana-512	18	27	on	on	ADP
cana-512	18	28	the	the	DET
cana-512	18	29	labels	label	NOUN
cana-512	18	30	used	use	VERB
cana-512	18	31	for	for	ADP
cana-512	18	32	the	the	DET
cana-512	18	33	edges	edge	NOUN
cana-512	18	34	.	.	PUNCT
cana-512	19	1	a	a	DET
cana-512	19	2	prime	prime	ADJ
cana-512	19	3	graph	graph	NOUN
cana-512	19	4	g	g	PROPN
cana-512	19	5	is	be	AUX
cana-512	19	6	a	a	DET
cana-512	19	7	bijection	bijection	NOUN
cana-512	19	8	f	f	PROPN
cana-512	19	9	∶	∶	NOUN
cana-512	19	10	v	v	NOUN
cana-512	19	11	→	→	SYM
cana-512	19	12	{	{	PUNCT
cana-512	19	13	1,2,3	1,2,3	NUM
cana-512	19	14	…	…	PUNCT
cana-512	19	15	.	.	PUNCT
cana-512	20	1	,	,	PUNCT
cana-512	20	2	p	p	X
cana-512	20	3	}	}	PUNCT
cana-512	20	4	such	such	ADJ
cana-512	20	5	that	that	SCONJ
cana-512	20	6	,	,	PUNCT
cana-512	20	7	for	for	ADP
cana-512	20	8	each	each	DET
cana-512	20	9	edge	edge	NOUN
cana-512	20	10	e	e	NOUN
cana-512	20	11	=	=	NOUN
cana-512	20	12	uv	uv	PROPN
cana-512	20	13			NOUN
cana-512	20	14	e	e	NOUN
cana-512	20	15	,	,	PUNCT
cana-512	20	16	we	we	PRON
cana-512	20	17	have	have	AUX
cana-512	20	18	gcd	gcd	VERB
cana-512	20	19	(	(	PUNCT
cana-512	20	20	f(u	f(u	PROPN
cana-512	20	21	)	)	PUNCT
cana-512	20	22	,	,	PUNCT
cana-512	20	23	f(v	f(v	NOUN
cana-512	20	24	)	)	PUNCT
cana-512	20	25	)	)	PUNCT
cana-512	21	1	=	=	SYM
cana-512	21	2	1	1	NUM
cana-512	21	3	(	(	PUNCT
cana-512	21	4	2	2	NUM
cana-512	21	5	)	)	PUNCT
cana-512	21	6	.	.	PUNCT
cana-512	22	1	communications	communication	NOUN
cana-512	22	2	on	on	ADP
cana-512	22	3	applied	apply	VERB
cana-512	22	4	nonlinear	nonlinear	ADJ
cana-512	22	5	analysis	analysis	NOUN
cana-512	22	6	issn	issn	NOUN
cana-512	22	7	:	:	PUNCT
cana-512	22	8	1074	1074	NUM
cana-512	22	9	-	-	PUNCT
cana-512	22	10	133x	133x	NUM
cana-512	22	11	vol	vol	NOUN
cana-512	22	12	31	31	NUM
cana-512	22	13	no	no	NOUN
cana-512	22	14	.	.	NOUN
cana-512	22	15	2	2	NUM
cana-512	22	16	(	(	PUNCT
cana-512	22	17	2024	2024	NUM
cana-512	22	18	)	)	PUNCT
cana-512	23	1	43	43	NUM
cana-512	23	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	23	3	definition	definition	NOUN
cana-512	23	4	1.1	1.1	NUM
cana-512	23	5	let	let	VERB
cana-512	23	6	g	g	NOUN
cana-512	23	7	=	=	SYM
cana-512	23	8	(	(	PUNCT
cana-512	23	9	v	v	NOUN
cana-512	23	10	,	,	PUNCT
cana-512	23	11	e	e	NOUN
cana-512	23	12	)	)	PUNCT
cana-512	23	13	be	be	AUX
cana-512	23	14	a	a	DET
cana-512	23	15	graph	graph	NOUN
cana-512	23	16	.	.	PUNCT
cana-512	24	1	a	a	DET
cana-512	24	2	bijection	bijection	NOUN
cana-512	24	3	𝑓	𝑓	DET
cana-512	24	4	∶	∶	NOUN
cana-512	24	5	𝐸	𝐸	NOUN
cana-512	24	6	→	→	SYM
cana-512	24	7	{	{	PUNCT
cana-512	24	8	1,2,3	1,2,3	NUM
cana-512	24	9	…	…	PUNCT
cana-512	24	10	…	…	PUNCT
cana-512	24	11	𝑞	𝑞	X
cana-512	24	12	}	}	PUNCT
cana-512	24	13	is	be	AUX
cana-512	24	14	called	call	VERB
cana-512	24	15	relatively	relatively	ADV
cana-512	24	16	prime	prime	ADJ
cana-512	24	17	edge	edge	NOUN
cana-512	24	18	labeling	labeling	NOUN
cana-512	24	19	if	if	SCONJ
cana-512	24	20	,	,	PUNCT
cana-512	24	21	for	for	ADP
cana-512	24	22	each	each	DET
cana-512	24	23	vertex	vertex	NOUN
cana-512	24	24	v	v	ADP
cana-512	24	25			NOUN
cana-512	24	26	v(g	v(g	PROPN
cana-512	24	27	)	)	PUNCT
cana-512	24	28	,	,	PUNCT
cana-512	24	29	the	the	DET
cana-512	24	30	labels	label	NOUN
cana-512	24	31	of	of	ADP
cana-512	24	32	the	the	DET
cana-512	24	33	edge	edge	NOUN
cana-512	24	34	’s	’s	PART
cana-512	24	35	incident	incident	NOUN
cana-512	24	36	on	on	ADP
cana-512	24	37	𝑣	𝑣	PROPN
cana-512	24	38	are	be	AUX
cana-512	24	39	pairwise	pairwise	NOUN
cana-512	24	40	relatively	relatively	ADV
cana-512	24	41	prime	prime	ADJ
cana-512	24	42	.	.	PUNCT
cana-512	25	1	a	a	DET
cana-512	25	2	graph	graph	NOUN
cana-512	25	3	that	that	PRON
cana-512	25	4	admits	admit	VERB
cana-512	25	5	a	a	DET
cana-512	25	6	relatively	relatively	ADV
cana-512	25	7	prime	prime	ADJ
cana-512	25	8	edge	edge	NOUN
cana-512	25	9	labeling	labeling	NOUN
cana-512	25	10	is	be	AUX
cana-512	25	11	called	call	VERB
cana-512	25	12	a	a	DET
cana-512	25	13	relatively	relatively	ADV
cana-512	25	14	prime	prime	ADJ
cana-512	25	15	edge	edge	NOUN
cana-512	25	16	labeled	label	VERB
cana-512	25	17	graph	graph	NOUN
cana-512	25	18	.	.	PUNCT
cana-512	26	1	(	(	PUNCT
cana-512	26	2	8)	8)	NUM
cana-512	26	3	in	in	ADP
cana-512	26	4	other	other	ADJ
cana-512	26	5	words	word	NOUN
cana-512	26	6	,	,	PUNCT
cana-512	26	7	a	a	DET
cana-512	26	8	graph	graph	NOUN
cana-512	26	9	with	with	ADP
cana-512	26	10	𝑝	𝑝	NOUN
cana-512	26	11	vertices	vertex	NOUN
cana-512	26	12	and	and	CCONJ
cana-512	26	13	𝑞	𝑞	PRON
cana-512	26	14	edges	edge	NOUN
cana-512	26	15	are	be	AUX
cana-512	26	16	said	say	VERB
cana-512	26	17	to	to	PART
cana-512	26	18	be	be	AUX
cana-512	26	19	a	a	DET
cana-512	26	20	relatively	relatively	ADV
cana-512	26	21	prime	prime	ADJ
cana-512	26	22	edge	edge	NOUN
cana-512	26	23	labeled	label	VERB
cana-512	26	24	graph	graph	NOUN
cana-512	26	25	,	,	PUNCT
cana-512	26	26	if	if	SCONJ
cana-512	26	27	the	the	DET
cana-512	26	28	edges	edge	NOUN
cana-512	26	29	are	be	AUX
cana-512	26	30	labeled	label	VERB
cana-512	26	31	with	with	ADP
cana-512	26	32	the	the	DET
cana-512	26	33	first	first	ADJ
cana-512	26	34	q	q	ADJ
cana-512	26	35	natural	natural	ADJ
cana-512	26	36	numbers	number	NOUN
cana-512	26	37	with	with	ADP
cana-512	26	38	the	the	DET
cana-512	26	39	condition	condition	NOUN
cana-512	26	40	that	that	SCONJ
cana-512	26	41	any	any	DET
cana-512	26	42	two	two	NUM
cana-512	26	43	adjacent	adjacent	ADJ
cana-512	26	44	edges	edge	NOUN
cana-512	26	45	have	have	AUX
cana-512	26	46	relatively	relatively	ADV
cana-512	26	47	prime	prime	ADJ
cana-512	26	48	labels	label	NOUN
cana-512	26	49	.	.	PUNCT
cana-512	27	1	considering	consider	VERB
cana-512	27	2	,	,	PUNCT
cana-512	27	3	edge	edge	NOUN
cana-512	27	4	incident	incident	NOUN
cana-512	27	5	on	on	ADP
cana-512	27	6	pendent	pendent	ADJ
cana-512	27	7	vertex	vertex	NOUN
cana-512	27	8	as	as	ADP
cana-512	27	9	relatively	relatively	ADV
cana-512	27	10	prime	prime	ADJ
cana-512	27	11	.	.	PUNCT
cana-512	28	1	(	(	PUNCT
cana-512	28	2	8)	8)	NUM
cana-512	28	3	definition	definition	NOUN
cana-512	28	4	1.2	1.2	NUM
cana-512	28	5	for	for	ADP
cana-512	28	6	a	a	DET
cana-512	28	7	graph	graph	NOUN
cana-512	28	8	,	,	PUNCT
cana-512	28	9	g	g	NOUN
cana-512	28	10	=	=	PUNCT
cana-512	28	11	(	(	PUNCT
cana-512	28	12	p	p	X
cana-512	28	13	,	,	PUNCT
cana-512	28	14	q	q	NOUN
cana-512	28	15	)	)	PUNCT
cana-512	28	16	,	,	PUNCT
cana-512	28	17	coprime	coprime	NOUN
cana-512	28	18	edge	edge	NOUN
cana-512	28	19	labeling	labeling	NOUN
cana-512	28	20	is	be	AUX
cana-512	28	21	defined	define	VERB
cana-512	28	22	to	to	PART
cana-512	28	23	be	be	AUX
cana-512	28	24	a	a	DET
cana-512	28	25	bijection	bijection	NOUN
cana-512	28	26	𝑓	𝑓	PRON
cana-512	28	27	:	:	PUNCT
cana-512	28	28	𝐸	𝐸	PROPN
cana-512	28	29	→	→	SYM
cana-512	28	30	{	{	PUNCT
cana-512	28	31	1	1	NUM
cana-512	28	32	,	,	PUNCT
cana-512	28	33	2	2	NUM
cana-512	28	34	,	,	PUNCT
cana-512	28	35	.	.	PUNCT
cana-512	28	36	.	.	PUNCT
cana-512	29	1	.	.	PUNCT
cana-512	30	1	,	,	PUNCT
cana-512	30	2	𝑘	𝑘	X
cana-512	30	3	}	}	PUNCT
cana-512	30	4	such	such	ADJ
cana-512	30	5	that	that	SCONJ
cana-512	30	6	,	,	PUNCT
cana-512	30	7	for	for	ADP
cana-512	30	8	𝑘	𝑘	DET
cana-512	30	9	≥	≥	NOUN
cana-512	30	10	𝑞	𝑞	NOUN
cana-512	30	11	,	,	PUNCT
cana-512	30	12	and	and	CCONJ
cana-512	30	13	for	for	ADP
cana-512	30	14	each	each	DET
cana-512	30	15	vertex	vertex	NOUN
cana-512	30	16	v	v	ADP
cana-512	30	17			PROPN
cana-512	30	18	v	v	NOUN
cana-512	30	19	,	,	PUNCT
cana-512	30	20	the	the	DET
cana-512	30	21	labels	label	NOUN
cana-512	30	22	of	of	ADP
cana-512	30	23	the	the	DET
cana-512	30	24	edge	edge	NOUN
cana-512	30	25	’s	’s	PART
cana-512	30	26	incident	incident	NOUN
cana-512	30	27	on	on	ADP
cana-512	30	28	𝑣	𝑣	PROPN
cana-512	30	29	are	be	AUX
cana-512	30	30	pairwise	pairwise	NOUN
cana-512	30	31	relatively	relatively	ADV
cana-512	30	32	prime	prime	ADJ
cana-512	30	33	.	.	PUNCT
cana-512	31	1	(	(	PUNCT
cana-512	31	2	10	10	NUM
cana-512	31	3	)	)	PUNCT
cana-512	31	4	the	the	DET
cana-512	31	5	minimum	minimum	ADJ
cana-512	31	6	value	value	NOUN
cana-512	31	7	of	of	ADP
cana-512	31	8	𝑘	𝑘	NOUN
cana-512	31	9	,	,	PUNCT
cana-512	31	10	for	for	ADP
cana-512	31	11	which	which	PRON
cana-512	31	12	g	g	NOUN
cana-512	31	13	is	be	AUX
cana-512	31	14	coprime	coprime	ADJ
cana-512	31	15	edge	edge	NOUN
cana-512	31	16	labeling	labeling	NOUN
cana-512	31	17	is	be	AUX
cana-512	31	18	called	call	VERB
cana-512	31	19	as	as	ADP
cana-512	31	20	minimum	minimum	NOUN
cana-512	31	21	coprime	coprime	NOUN
cana-512	31	22	edge	edge	NOUN
cana-512	31	23	labeling	labeling	NOUN
cana-512	31	24	,	,	PUNCT
cana-512	31	25	with	with	ADP
cana-512	31	26	minimum	minimum	NOUN
cana-512	31	27	coprime	coprime	NOUN
cana-512	31	28	edge	edge	NOUN
cana-512	31	29	number	number	NOUN
cana-512	31	30	,	,	PUNCT
cana-512	31	31	𝑝𝜏𝐸(𝐺	𝑝𝜏𝐸(𝐺	NOUN
cana-512	31	32	)	)	PUNCT
cana-512	31	33	=	=	SYM
cana-512	32	1	𝑘.	𝑘.	NOUN
cana-512	32	2	definition	definition	NOUN
cana-512	32	3	1.3	1.3	NUM
cana-512	32	4	for	for	ADP
cana-512	32	5	a	a	DET
cana-512	32	6	coprime	coprime	ADJ
cana-512	32	7	graph	graph	NOUN
cana-512	32	8	g	g	NOUN
cana-512	32	9	,	,	PUNCT
cana-512	32	10	the	the	DET
cana-512	32	11	prime	prime	ADJ
cana-512	32	12	index	index	NOUN
cana-512	32	13	is	be	AUX
cana-512	32	14	the	the	DET
cana-512	32	15	least	least	ADJ
cana-512	32	16	number	number	NOUN
cana-512	32	17	of	of	ADP
cana-512	32	18	edges	edge	NOUN
cana-512	32	19	removed	remove	VERB
cana-512	32	20	from	from	ADP
cana-512	32	21	g	g	PROPN
cana-512	32	22	to	to	PART
cana-512	32	23	form	form	VERB
cana-512	32	24	a	a	DET
cana-512	32	25	prime	prime	ADJ
cana-512	32	26	graph	graph	NOUN
cana-512	32	27	g	g	PROPN
cana-512	32	28	*	*	PUNCT
cana-512	32	29	.	.	PUNCT
cana-512	32	30	and	and	CCONJ
cana-512	32	31	is	be	AUX
cana-512	32	32	denoted	denote	VERB
cana-512	32	33	by	by	ADP
cana-512	32	34	,	,	PUNCT
cana-512	32	35	ε(g	ε(g	PROPN
cana-512	32	36	)	)	PUNCT
cana-512	32	37	.	.	PUNCT
cana-512	33	1	in	in	ADP
cana-512	33	2	other	other	ADJ
cana-512	33	3	words	word	NOUN
cana-512	33	4	,	,	PUNCT
cana-512	33	5	(	(	PUNCT
cana-512	33	6	10	10	NUM
cana-512	33	7	)	)	PUNCT
cana-512	33	8	ε(g	ε(g	NOUN
cana-512	33	9	)	)	PUNCT
cana-512	33	10	=	=	SYM
cana-512	33	11	min	min	NOUN
cana-512	33	12	{	{	PUNCT
cana-512	33	13	|	|	ADV
cana-512	33	14	e(h	e(h	PROPN
cana-512	33	15	)	)	PUNCT
cana-512	34	1	|	|	ADV
cana-512	34	2	∶	∶	NOUN
cana-512	34	3	h	h	NOUN
cana-512	34	4	⊆	⊆	NUM
cana-512	34	5	g	g	NOUN
cana-512	34	6	and	and	CCONJ
cana-512	34	7	g	g	PROPN
cana-512	34	8	−	−	PROPN
cana-512	34	9	e(h	e(h	PROPN
cana-512	34	10	)	)	PUNCT
cana-512	34	11	is	be	AUX
cana-512	34	12	prime	prime	ADJ
cana-512	34	13	}	}	PUNCT
cana-512	34	14	2	2	NUM
cana-512	34	15	.	.	NOUN
cana-512	34	16	edge	edge	NOUN
cana-512	34	17	prime	prime	ADJ
cana-512	34	18	index	index	NOUN
cana-512	34	19	in	in	ADP
cana-512	34	20	this	this	DET
cana-512	34	21	section	section	NOUN
cana-512	34	22	,	,	PUNCT
cana-512	34	23	the	the	DET
cana-512	34	24	formal	formal	ADJ
cana-512	34	25	definition	definition	NOUN
cana-512	34	26	of	of	ADP
cana-512	34	27	edge	edge	ADJ
cana-512	34	28	prime	prime	ADJ
cana-512	34	29	index	index	NOUN
cana-512	34	30	is	be	AUX
cana-512	34	31	defined	define	VERB
cana-512	34	32	with	with	ADP
cana-512	34	33	an	an	DET
cana-512	34	34	appropriate	appropriate	ADJ
cana-512	34	35	example	example	NOUN
cana-512	34	36	.	.	PUNCT
cana-512	35	1	2.1	2.1	NUM
cana-512	35	2	.	.	PUNCT
cana-512	36	1	definition	definition	NOUN
cana-512	36	2	edge	edge	VERB
cana-512	36	3	prime	prime	ADJ
cana-512	36	4	index	index	NOUN
cana-512	36	5	:	:	PUNCT
cana-512	36	6	let	let	VERB
cana-512	36	7	g	g	PRON
cana-512	36	8	be	be	AUX
cana-512	36	9	a	a	DET
cana-512	36	10	coprime	coprime	ADJ
cana-512	36	11	edge	edge	NOUN
cana-512	36	12	labeled	label	VERB
cana-512	36	13	graph	graph	NOUN
cana-512	36	14	.	.	PUNCT
cana-512	37	1	edge	edge	NOUN
cana-512	37	2	prime	prime	ADJ
cana-512	37	3	index	index	NOUN
cana-512	37	4	𝜀𝑟(𝐺	𝜀𝑟(𝐺	NOUN
cana-512	37	5	)	)	PUNCT
cana-512	37	6	is	be	AUX
cana-512	37	7	defined	define	VERB
cana-512	37	8	to	to	PART
cana-512	37	9	be	be	AUX
cana-512	37	10	the	the	DET
cana-512	37	11	minimum	minimum	ADJ
cana-512	37	12	number	number	NOUN
cana-512	37	13	of	of	ADP
cana-512	37	14	edges	edge	NOUN
cana-512	37	15	removed	remove	VERB
cana-512	37	16	from	from	ADP
cana-512	37	17	g	g	PROPN
cana-512	37	18	to	to	PART
cana-512	37	19	form	form	VERB
cana-512	37	20	a	a	DET
cana-512	37	21	relatively	relatively	ADV
cana-512	37	22	prime	prime	ADJ
cana-512	37	23	edge	edge	NOUN
cana-512	37	24	labeled	label	VERB
cana-512	37	25	graph	graph	NOUN
cana-512	37	26	𝐺∗.	𝐺∗.	PROPN
cana-512	37	27	in	in	ADP
cana-512	37	28	other	other	ADJ
cana-512	37	29	words	word	NOUN
cana-512	37	30	,	,	PUNCT
cana-512	37	31	𝜀𝑟(𝐺	𝜀𝑟(𝐺	ADV
cana-512	37	32	)	)	PUNCT
cana-512	38	1	=	=	SYM
cana-512	38	2	𝑚𝑖𝑛{|	𝑚𝑖𝑛{|	NUM
cana-512	38	3	𝐸(𝐻	𝐸(𝐻	NOUN
cana-512	38	4	)	)	PUNCT
cana-512	39	1	|	|	ADV
cana-512	39	2	∶	∶	VERB
cana-512	39	3	𝐻	𝐻	PROPN
cana-512	39	4	⊆	⊆	PROPN
cana-512	39	5	𝐺	𝐺	PROPN
cana-512	39	6	&	&	CCONJ
cana-512	39	7	𝐺	𝐺	PROPN
cana-512	39	8	−	−	PROPN
cana-512	39	9	𝐸(𝐻	𝐸(𝐻	PROPN
cana-512	39	10	)	)	PUNCT
cana-512	40	1	𝑖𝑠	𝑖𝑠	VERB
cana-512	40	2	𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦	𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦	PROPN
cana-512	40	3	𝑝𝑟𝑖𝑚𝑒	𝑝𝑟𝑖𝑚𝑒	NOUN
cana-512	40	4	𝑒𝑑𝑔𝑒	𝑒𝑑𝑔𝑒	PROPN
cana-512	40	5	𝑙𝑎𝑏𝑒𝑙𝑒𝑑	𝑙𝑎𝑏𝑒𝑙𝑒𝑑	PROPN
cana-512	40	6	𝑔𝑟𝑎𝑝ℎ	𝑔𝑟𝑎𝑝ℎ	PROPN
cana-512	40	7	}	}	PUNCT
cana-512	40	8	2.2.illustration	2.2.illustration	NUM
cana-512	40	9	for	for	ADP
cana-512	40	10	a	a	DET
cana-512	40	11	complete	complete	ADJ
cana-512	40	12	graph	graph	NOUN
cana-512	40	13	𝐾4	𝐾4	NOUN
cana-512	40	14	,	,	PUNCT
cana-512	40	15	the	the	DET
cana-512	40	16	edge	edge	NOUN
cana-512	40	17	prime	prime	ADJ
cana-512	40	18	index	index	NOUN
cana-512	40	19	is	be	AUX
cana-512	40	20	explained	explain	VERB
cana-512	40	21	in	in	ADP
cana-512	40	22	the	the	DET
cana-512	40	23	given	give	VERB
cana-512	40	24	figure	figure	NOUN
cana-512	40	25	1	1	NUM
cana-512	40	26	.	.	PUNCT
cana-512	41	1	that	that	PRON
cana-512	41	2	is	is	ADV
cana-512	41	3	,	,	PUNCT
cana-512	41	4	by	by	ADP
cana-512	41	5	removing	remove	VERB
cana-512	41	6	an	an	DET
cana-512	41	7	edge	edge	NOUN
cana-512	41	8	from	from	ADP
cana-512	41	9	𝐾4	𝐾4	NOUN
cana-512	41	10	,	,	PUNCT
cana-512	41	11	it	it	PRON
cana-512	41	12	becomes	become	VERB
cana-512	41	13	a	a	DET
cana-512	41	14	relatively	relatively	ADV
cana-512	41	15	prime	prime	ADJ
cana-512	41	16	edge	edge	NOUN
cana-512	41	17	labeled	label	VERB
cana-512	41	18	graph	graph	NOUN
cana-512	41	19	.	.	PUNCT
cana-512	42	1	communications	communication	NOUN
cana-512	42	2	on	on	ADP
cana-512	42	3	applied	apply	VERB
cana-512	42	4	nonlinear	nonlinear	ADJ
cana-512	42	5	analysis	analysis	NOUN
cana-512	42	6	issn	issn	NOUN
cana-512	42	7	:	:	PUNCT
cana-512	42	8	1074	1074	NUM
cana-512	42	9	-	-	PUNCT
cana-512	42	10	133x	133x	NUM
cana-512	42	11	vol	vol	NOUN
cana-512	42	12	31	31	NUM
cana-512	42	13	no	no	NOUN
cana-512	42	14	.	.	NOUN
cana-512	42	15	2	2	NUM
cana-512	42	16	(	(	PUNCT
cana-512	42	17	2024	2024	NUM
cana-512	42	18	)	)	PUNCT
cana-512	42	19	44	44	NUM
cana-512	42	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	42	21	figure	figure	NOUN
cana-512	42	22	1	1	NUM
cana-512	42	23	:	:	PUNCT
cana-512	42	24	εr(k4	εr(k4	NOUN
cana-512	42	25	)	)	PUNCT
cana-512	42	26	=	=	SYM
cana-512	42	27	1	1	NUM
cana-512	42	28	3	3	NUM
cana-512	42	29	.	.	PUNCT
cana-512	42	30	relation	relation	NOUN
cana-512	42	31	with	with	ADP
cana-512	42	32	other	other	ADJ
cana-512	42	33	parameters	parameter	NOUN
cana-512	42	34	the	the	DET
cana-512	42	35	next	next	ADJ
cana-512	42	36	theorem	theorem	NOUN
cana-512	42	37	helps	help	VERB
cana-512	42	38	to	to	PART
cana-512	42	39	find	find	VERB
cana-512	42	40	the	the	DET
cana-512	42	41	upper	upper	ADJ
cana-512	42	42	bound	bound	NOUN
cana-512	42	43	of	of	ADP
cana-512	42	44	the	the	DET
cana-512	42	45	edge	edge	NOUN
cana-512	42	46	prime	prime	ADJ
cana-512	42	47	index	index	NOUN
cana-512	42	48	of	of	ADP
cana-512	42	49	a	a	DET
cana-512	42	50	graph	graph	NOUN
cana-512	42	51	having	have	VERB
cana-512	42	52	a	a	DET
cana-512	42	53	hamiltonian	hamiltonian	ADJ
cana-512	42	54	circuit	circuit	NOUN
cana-512	42	55	(	(	PUNCT
cana-512	42	56	7	7	NUM
cana-512	42	57	)	)	PUNCT
cana-512	42	58	.	.	PUNCT
cana-512	43	1	theorem	theorem	VERB
cana-512	43	2	3.1	3.1	NUM
cana-512	43	3	for	for	ADP
cana-512	43	4	a	a	DET
cana-512	43	5	graph	graph	NOUN
cana-512	43	6	𝐺	𝐺	NOUN
cana-512	43	7	=	=	SYM
cana-512	43	8	(	(	PUNCT
cana-512	43	9	𝑝	𝑝	PROPN
cana-512	43	10	,	,	PUNCT
cana-512	43	11	𝑞	𝑞	NOUN
cana-512	43	12	)	)	PUNCT
cana-512	43	13	which	which	PRON
cana-512	43	14	contains	contain	VERB
cana-512	43	15	a	a	DET
cana-512	43	16	hamiltonian	hamiltonian	ADJ
cana-512	43	17	circuit	circuit	NOUN
cana-512	43	18	of	of	ADP
cana-512	43	19	length	length	NOUN
cana-512	43	20	k	k	PROPN
cana-512	43	21	,	,	PUNCT
cana-512	43	22	then	then	ADV
cana-512	43	23	εr(g	εr(g	PUNCT
cana-512	43	24	)	)	PUNCT
cana-512	43	25	≤	≤	NOUN
cana-512	44	1	𝑞	𝑞	ADP
cana-512	44	2	−	−	PROPN
cana-512	44	3	𝑘.	𝑘.	ADJ
cana-512	44	4	proof	proof	NOUN
cana-512	44	5	.	.	PUNCT
cana-512	45	1	suppose	suppose	VERB
cana-512	45	2	,	,	PUNCT
cana-512	45	3	εr(g	εr(g	PUNCT
cana-512	45	4	)	)	PUNCT
cana-512	45	5	>	>	X
cana-512	46	1	𝑞	𝑞	X
cana-512	46	2	−	−	X
cana-512	46	3	𝑘	𝑘	PROPN
cana-512	46	4	,	,	PUNCT
cana-512	46	5	where	where	SCONJ
cana-512	46	6	k	k	PROPN
cana-512	46	7	is	be	AUX
cana-512	46	8	the	the	DET
cana-512	46	9	length	length	NOUN
cana-512	46	10	of	of	ADP
cana-512	46	11	the	the	DET
cana-512	46	12	hamiltonian	hamiltonian	ADJ
cana-512	46	13	circuit	circuit	NOUN
cana-512	46	14	and	and	CCONJ
cana-512	46	15	q	q	NOUN
cana-512	46	16	is	be	AUX
cana-512	46	17	the	the	DET
cana-512	46	18	number	number	NOUN
cana-512	46	19	of	of	ADP
cana-512	46	20	edges	edge	NOUN
cana-512	46	21	.	.	PUNCT
cana-512	47	1	let	let	VERB
cana-512	47	2	𝑣1	𝑣1	PROPN
cana-512	47	3	,	,	PUNCT
cana-512	47	4	𝑣2	𝑣2	PROPN
cana-512	47	5	,	,	PUNCT
cana-512	47	6	⋯	⋯	PROPN
cana-512	47	7	,	,	PUNCT
cana-512	47	8	𝑣𝑝	𝑣𝑝	NOUN
cana-512	47	9	be	be	AUX
cana-512	47	10	the	the	DET
cana-512	47	11	p	p	NOUN
cana-512	47	12	vertices	vertex	NOUN
cana-512	47	13	.	.	PUNCT
cana-512	48	1	as	as	SCONJ
cana-512	48	2	g	g	PROPN
cana-512	48	3	contains	contain	VERB
cana-512	48	4	a	a	DET
cana-512	48	5	hamiltonian	hamiltonian	ADJ
cana-512	48	6	circuit	circuit	NOUN
cana-512	48	7	of	of	ADP
cana-512	48	8	length	length	NOUN
cana-512	48	9	k	k	PROPN
cana-512	48	10	,	,	PUNCT
cana-512	48	11	say	say	VERB
cana-512	48	12	𝑣1	𝑣1	PROPN
cana-512	48	13	,	,	PUNCT
cana-512	48	14	𝑣2	𝑣2	PROPN
cana-512	48	15	,	,	PUNCT
cana-512	48	16	⋯	⋯	PROPN
cana-512	48	17	,	,	PUNCT
cana-512	48	18	𝑣𝑘	𝑣𝑘	ADV
cana-512	48	19	then	then	ADV
cana-512	48	20	label	label	VERB
cana-512	48	21	the	the	DET
cana-512	48	22	edges	edge	NOUN
cana-512	48	23	of	of	ADP
cana-512	48	24	the	the	DET
cana-512	48	25	hamiltonian	hamiltonian	ADJ
cana-512	48	26	circuit	circuit	NOUN
cana-512	48	27	in	in	ADP
cana-512	48	28	such	such	DET
cana-512	48	29	a	a	DET
cana-512	48	30	way	way	NOUN
cana-512	48	31	that	that	SCONJ
cana-512	48	32	,	,	PUNCT
cana-512	48	33	l(vivi+1	l(vivi+1	NOUN
cana-512	48	34	)	)	PUNCT
cana-512	49	1	=	=	VERB
cana-512	49	2	i	i	PRON
cana-512	49	3	for	for	ADP
cana-512	49	4	𝑖	𝑖	X
cana-512	49	5	=	=	SYM
cana-512	49	6	1	1	NUM
cana-512	49	7	,	,	PUNCT
cana-512	49	8	2	2	NUM
cana-512	49	9	,	,	PUNCT
cana-512	49	10	…	…	PUNCT
cana-512	49	11	,	,	PUNCT
cana-512	49	12	𝑘	𝑘	PRON
cana-512	49	13	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-512	49	14	𝑣𝑘+1	𝑣𝑘+1	ADJ
cana-512	49	15	=	=	SYM
cana-512	49	16	𝑣1	𝑣1	PROPN
cana-512	49	17	.	.	PUNCT
cana-512	50	1	since	since	SCONJ
cana-512	50	2	the	the	DET
cana-512	50	3	prime	prime	ADJ
cana-512	50	4	index	index	NOUN
cana-512	50	5	is	be	AUX
cana-512	50	6	greater	great	ADJ
cana-512	50	7	than	than	ADP
cana-512	50	8	𝑞	𝑞	PROPN
cana-512	50	9	−	−	PROPN
cana-512	50	10	𝑘	𝑘	X
cana-512	50	11	,	,	PUNCT
cana-512	50	12	which	which	PRON
cana-512	50	13	is	be	AUX
cana-512	50	14	the	the	DET
cana-512	50	15	contradiction	contradiction	NOUN
cana-512	50	16	to	to	ADP
cana-512	50	17	the	the	DET
cana-512	50	18	above	above	ADJ
cana-512	50	19	labeling	labeling	NOUN
cana-512	50	20	.	.	PUNCT
cana-512	51	1	hence	hence	ADV
cana-512	51	2	the	the	DET
cana-512	51	3	maximum	maximum	ADJ
cana-512	51	4	number	number	NOUN
cana-512	51	5	of	of	ADP
cana-512	51	6	edges	edge	NOUN
cana-512	51	7	to	to	PART
cana-512	51	8	be	be	AUX
cana-512	51	9	removed	remove	VERB
cana-512	51	10	from	from	ADP
cana-512	51	11	g	g	NOUN
cana-512	51	12	is	be	AUX
cana-512	51	13	less	less	ADJ
cana-512	51	14	than	than	ADP
cana-512	51	15	or	or	CCONJ
cana-512	51	16	equal	equal	ADJ
cana-512	51	17	to	to	ADP
cana-512	51	18	𝑞	𝑞	PROPN
cana-512	51	19	−	−	PROPN
cana-512	51	20	𝑘	𝑘	PROPN
cana-512	51	21	.	.	PUNCT
cana-512	52	1	corollary	corollary	ADJ
cana-512	52	2	3.2	3.2	NUM
cana-512	52	3	for	for	ADP
cana-512	52	4	a	a	DET
cana-512	52	5	complete	complete	ADJ
cana-512	52	6	graph	graph	NOUN
cana-512	52	7	𝐾4	𝐾4	NOUN
cana-512	52	8	,	,	PUNCT
cana-512	52	9	εr(𝐾4	εr(𝐾4	PROPN
cana-512	52	10	)	)	PUNCT
cana-512	53	1	≤	≤	NUM
cana-512	53	2	2	2	NUM
cana-512	53	3	.	.	PUNCT
cana-512	53	4	proof	proof	NOUN
cana-512	53	5	.	.	PUNCT
cana-512	54	1	by	by	ADP
cana-512	54	2	the	the	DET
cana-512	54	3	above	above	ADJ
cana-512	54	4	theorem	theorem	NOUN
cana-512	54	5	,	,	PUNCT
cana-512	54	6	figure	figure	NOUN
cana-512	54	7	2	2	NUM
cana-512	54	8	shows	show	VERB
cana-512	54	9	that	that	SCONJ
cana-512	54	10	,	,	PUNCT
cana-512	54	11	𝐾4	𝐾4	NOUN
cana-512	54	12	contains	contain	VERB
cana-512	54	13	a	a	DET
cana-512	54	14	hamiltonian	hamiltonian	ADJ
cana-512	54	15	circuit	circuit	NOUN
cana-512	54	16	of	of	ADP
cana-512	54	17	length	length	NOUN
cana-512	54	18	4	4	NUM
cana-512	54	19	and	and	CCONJ
cana-512	54	20	the	the	DET
cana-512	54	21	number	number	NOUN
cana-512	54	22	of	of	ADP
cana-512	54	23	edges	edge	NOUN
cana-512	54	24	in	in	ADP
cana-512	54	25	𝐾4	𝐾4	NOUN
cana-512	54	26	is	be	AUX
cana-512	54	27	6	6	NUM
cana-512	54	28	.	.	PUNCT
cana-512	54	29	hence	hence	ADV
cana-512	54	30	εr(g	εr(g	PUNCT
cana-512	54	31	)	)	PUNCT
cana-512	54	32	≤	≤	ADV
cana-512	55	1	6	6	NUM
cana-512	55	2	−	−	NOUN
cana-512	55	3	4	4	NUM
cana-512	55	4	=	=	SYM
cana-512	55	5	2	2	NUM
cana-512	55	6	.	.	PUNCT
cana-512	55	7	communications	communication	NOUN
cana-512	55	8	on	on	ADP
cana-512	55	9	applied	apply	VERB
cana-512	55	10	nonlinear	nonlinear	ADJ
cana-512	55	11	analysis	analysis	NOUN
cana-512	55	12	issn	issn	NOUN
cana-512	55	13	:	:	PUNCT
cana-512	55	14	1074	1074	NUM
cana-512	55	15	-	-	PUNCT
cana-512	55	16	133x	133x	NUM
cana-512	55	17	vol	vol	NOUN
cana-512	55	18	31	31	NUM
cana-512	55	19	no	no	NOUN
cana-512	55	20	.	.	NOUN
cana-512	55	21	2	2	NUM
cana-512	55	22	(	(	PUNCT
cana-512	55	23	2024	2024	NUM
cana-512	55	24	)	)	PUNCT
cana-512	55	25	45	45	NUM
cana-512	55	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	55	27	figure	figure	NOUN
cana-512	55	28	2	2	NUM
cana-512	55	29	:	:	PUNCT
cana-512	55	30	complete	complete	ADJ
cana-512	55	31	graph	graph	NOUN
cana-512	55	32	with	with	ADP
cana-512	55	33	4	4	NUM
cana-512	55	34	vertices	vertex	NOUN
cana-512	55	35	4	4	NUM
cana-512	55	36	.	.	NOUN
cana-512	55	37	edge	edge	NOUN
cana-512	55	38	prime	prime	ADJ
cana-512	55	39	index	index	NOUN
cana-512	55	40	of	of	ADP
cana-512	55	41	some	some	DET
cana-512	55	42	class	class	NOUN
cana-512	55	43	of	of	ADP
cana-512	55	44	graphs	graph	NOUN
cana-512	55	45	in	in	ADP
cana-512	55	46	this	this	DET
cana-512	55	47	section	section	NOUN
cana-512	55	48	,	,	PUNCT
cana-512	55	49	edge	edge	NOUN
cana-512	55	50	prime	prime	ADJ
cana-512	55	51	index	index	NOUN
cana-512	55	52	is	be	AUX
cana-512	55	53	found	find	VERB
cana-512	55	54	for	for	ADP
cana-512	55	55	specific	specific	ADJ
cana-512	55	56	classes	class	NOUN
cana-512	55	57	of	of	ADP
cana-512	55	58	graphs	graph	NOUN
cana-512	55	59	,	,	PUNCT
cana-512	55	60	namely	namely	ADV
cana-512	55	61	the	the	DET
cana-512	55	62	complete	complete	ADJ
cana-512	55	63	graph	graph	NOUN
cana-512	55	64	,	,	PUNCT
cana-512	55	65	the	the	DET
cana-512	55	66	corona	corona	NOUN
cana-512	55	67	product	product	NOUN
cana-512	55	68	of	of	ADP
cana-512	55	69	graphs	graph	NOUN
cana-512	55	70	and	and	CCONJ
cana-512	55	71	so	so	ADV
cana-512	55	72	on	on	ADP
cana-512	55	73	(	(	PUNCT
cana-512	55	74	7	7	NUM
cana-512	55	75	)	)	PUNCT
cana-512	55	76	.	.	PUNCT
cana-512	56	1	4.1	4.1	NUM
cana-512	56	2	.	.	PUNCT
cana-512	56	3	corona	corona	NOUN
cana-512	56	4	product	product	NOUN
cana-512	56	5	of	of	ADP
cana-512	56	6	graph	graph	NOUN
cana-512	56	7	in	in	ADP
cana-512	56	8	the	the	DET
cana-512	56	9	following	following	NOUN
cana-512	56	10	theorem	theorem	NOUN
cana-512	56	11	,	,	PUNCT
cana-512	56	12	the	the	DET
cana-512	56	13	relatively	relatively	ADV
cana-512	56	14	prime	prime	ADJ
cana-512	56	15	index	index	NOUN
cana-512	56	16	of	of	ADP
cana-512	56	17	the	the	DET
cana-512	56	18	corona	corona	NOUN
cana-512	56	19	product	product	NOUN
cana-512	56	20	of	of	ADP
cana-512	56	21	kn	kn	PROPN
cana-512	56	22	and	and	CCONJ
cana-512	56	23	k1	k1	PROPN
cana-512	56	24	is	be	AUX
cana-512	56	25	determined	determine	VERB
cana-512	56	26	.	.	PUNCT
cana-512	57	1	the	the	DET
cana-512	57	2	corona	corona	NOUN
cana-512	57	3	product	product	NOUN
cana-512	57	4	of	of	ADP
cana-512	57	5	two	two	NUM
cana-512	57	6	graphs	graph	NOUN
cana-512	57	7	g	g	NOUN
cana-512	57	8	and	and	CCONJ
cana-512	57	9	h	h	NOUN
cana-512	57	10	is	be	AUX
cana-512	57	11	defined	define	VERB
cana-512	57	12	as	as	SCONJ
cana-512	57	13	the	the	DET
cana-512	57	14	graph	graph	NOUN
cana-512	57	15	obtained	obtain	VERB
cana-512	57	16	by	by	ADP
cana-512	57	17	taking	take	VERB
cana-512	57	18	one	one	NUM
cana-512	57	19	copy	copy	NOUN
cana-512	57	20	of	of	ADP
cana-512	57	21	g	g	PROPN
cana-512	57	22	and	and	CCONJ
cana-512	57	23	|v(g)|	|v(g)|	ADJ
cana-512	57	24	copies	copy	NOUN
cana-512	57	25	of	of	ADP
cana-512	57	26	h	h	NOUN
cana-512	57	27	and	and	CCONJ
cana-512	57	28	joining	join	VERB
cana-512	57	29	the	the	DET
cana-512	57	30	ith	ith	PROPN
cana-512	57	31	vertex	vertex	NOUN
cana-512	57	32	of	of	ADP
cana-512	57	33	g	g	NOUN
cana-512	57	34	to	to	ADP
cana-512	57	35	every	every	DET
cana-512	57	36	vertex	vertex	NOUN
cana-512	57	37	in	in	ADP
cana-512	57	38	the	the	DET
cana-512	57	39	ith	ith	PROPN
cana-512	57	40	copy	copy	NOUN
cana-512	57	41	of	of	ADP
cana-512	57	42	h.	h.	PROPN
cana-512	57	43	theorem	theorem	VERB
cana-512	57	44	4.1	4.1	NUM
cana-512	57	45	for	for	ADP
cana-512	57	46	a	a	DET
cana-512	57	47	graph	graph	NOUN
cana-512	57	48	kn	kn	PROPN
cana-512	57	49	⊙	⊙	PROPN
cana-512	57	50	k1	k1	PROPN
cana-512	57	51	,	,	PUNCT
cana-512	57	52	𝜀𝑟(kn	𝜀𝑟(kn	PROPN
cana-512	57	53	⊙	⊙	PROPN
cana-512	57	54	k1	k1	PROPN
cana-512	57	55	)	)	PUNCT
cana-512	58	1	=	=	PRON
cana-512	58	2	{	{	PUNCT
cana-512	58	3	𝑛(𝑛−3	𝑛(𝑛−3	PROPN
cana-512	58	4	)	)	PUNCT
cana-512	58	5	2	2	NUM
cana-512	58	6	,	,	PUNCT
cana-512	58	7	𝑖𝑓	𝑖𝑓	NUM
cana-512	58	8	2𝑛	2𝑛	PROPN
cana-512	58	9	+	+	CCONJ
cana-512	58	10	1	1	NUM
cana-512	58	11	≡	≡	PROPN
cana-512	58	12	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	NOUN
cana-512	58	13	3	3	NUM
cana-512	58	14	)	)	PUNCT
cana-512	58	15	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	58	16	)	)	PUNCT
cana-512	58	17	2	2	NUM
cana-512	58	18	−	−	NOUN
cana-512	58	19	1	1	NUM
cana-512	58	20	,	,	PUNCT
cana-512	58	21	𝑖𝑓	𝑖𝑓	NUM
cana-512	58	22	2𝑛	2𝑛	NOUN
cana-512	58	23	+	+	CCONJ
cana-512	58	24	1	1	NUM
cana-512	58	25	≢	≢	NUM
cana-512	58	26	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-512	58	27	3	3	NUM
cana-512	58	28	)	)	PUNCT
cana-512	58	29	proof	proof	NOUN
cana-512	58	30	let	let	VERB
cana-512	58	31	g	g	PROPN
cana-512	58	32	=	=	PROPN
cana-512	58	33	kn	kn	PROPN
cana-512	58	34	⊙	⊙	PROPN
cana-512	58	35	k1	k1	PROPN
cana-512	58	36	be	be	AUX
cana-512	58	37	the	the	DET
cana-512	58	38	graph	graph	NOUN
cana-512	58	39	with	with	ADP
cana-512	58	40	2n	2n	ADJ
cana-512	58	41	vertices	vertex	NOUN
cana-512	58	42	and	and	CCONJ
cana-512	58	43	𝑛(𝑛+1	𝑛(𝑛+1	NUM
cana-512	58	44	)	)	PUNCT
cana-512	58	45	2	2	NUM
cana-512	58	46	edges	edge	NOUN
cana-512	58	47	and	and	CCONJ
cana-512	58	48	let	let	VERB
cana-512	58	49	𝑣1	𝑣1	PROPN
cana-512	58	50	,	,	PUNCT
cana-512	58	51	𝑣2	𝑣2	PROPN
cana-512	58	52	,	,	PUNCT
cana-512	58	53	⋯	⋯	PROPN
cana-512	58	54	⋯	⋯	PROPN
cana-512	58	55	,	,	PUNCT
cana-512	58	56	𝑣𝑛	𝑣𝑛	PROPN
cana-512	58	57	,	,	PUNCT
cana-512	58	58	𝑢1	𝑢1	NOUN
cana-512	58	59	,	,	PUNCT
cana-512	58	60	𝑢2	𝑢2	PROPN
cana-512	58	61	,	,	PUNCT
cana-512	58	62	⋯	⋯	PROPN
cana-512	58	63	⋯	⋯	PROPN
cana-512	58	64	,	,	PUNCT
cana-512	58	65	𝑢𝑛	𝑢𝑛	NOUN
cana-512	58	66	be	be	VERB
cana-512	58	67	the	the	DET
cana-512	58	68	vertices	vertex	NOUN
cana-512	58	69	of	of	ADP
cana-512	58	70	kn	kn	PROPN
cana-512	58	71	⊙	⊙	PROPN
cana-512	58	72	k1	k1	PROPN
cana-512	58	73	.	.	PUNCT
cana-512	59	1	case	case	NOUN
cana-512	59	2	1	1	NUM
cana-512	59	3	:	:	PUNCT
cana-512	59	4	for	for	ADP
cana-512	59	5	2𝑛	2𝑛	PROPN
cana-512	59	6	+	+	CCONJ
cana-512	59	7	1	1	NUM
cana-512	59	8	≡	≡	PROPN
cana-512	59	9	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	NOUN
cana-512	59	10	3	3	NUM
cana-512	59	11	)	)	PUNCT
cana-512	59	12	.	.	PUNCT
cana-512	60	1	it	it	PRON
cana-512	60	2	is	be	AUX
cana-512	60	3	enough	enough	ADJ
cana-512	60	4	to	to	PART
cana-512	60	5	prove	prove	VERB
cana-512	60	6	that	that	SCONJ
cana-512	60	7	,	,	PUNCT
cana-512	60	8	the	the	DET
cana-512	60	9	removal	removal	NOUN
cana-512	60	10	of	of	ADP
cana-512	60	11	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	60	12	)	)	PUNCT
cana-512	60	13	2	2	NUM
cana-512	60	14	edges	edge	NOUN
cana-512	60	15	results	result	NOUN
cana-512	60	16	in	in	ADP
cana-512	60	17	a	a	DET
cana-512	60	18	relatively	relatively	ADV
cana-512	60	19	prime	prime	ADJ
cana-512	60	20	edge	edge	NOUN
cana-512	60	21	labeled	label	VERB
cana-512	60	22	graph	graph	NOUN
cana-512	60	23	.	.	PUNCT
cana-512	61	1	suppose	suppose	VERB
cana-512	61	2	the	the	DET
cana-512	61	3	removal	removal	NOUN
cana-512	61	4	of	of	ADP
cana-512	61	5	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	61	6	)	)	PUNCT
cana-512	61	7	2	2	NUM
cana-512	61	8	−	−	NOUN
cana-512	61	9	1	1	NUM
cana-512	61	10	edges	edge	NOUN
cana-512	61	11	in	in	ADP
cana-512	61	12	kn	kn	PROPN
cana-512	61	13	⊙	⊙	PROPN
cana-512	61	14	k1	k1	PROPN
cana-512	61	15	results	result	NOUN
cana-512	61	16	in	in	ADP
cana-512	61	17	a	a	DET
cana-512	61	18	relatively	relatively	ADV
cana-512	61	19	prime	prime	ADJ
cana-512	61	20	edge	edge	NOUN
cana-512	61	21	labeled	label	VERB
cana-512	61	22	graph	graph	NOUN
cana-512	61	23	.	.	PUNCT
cana-512	62	1	that	that	PRON
cana-512	62	2	is	is	ADV
cana-512	62	3	,	,	PUNCT
cana-512	62	4	remaining	remain	VERB
cana-512	62	5	𝑛(𝑛+1	𝑛(𝑛+1	NUM
cana-512	62	6	)	)	PUNCT
cana-512	62	7	2	2	NUM
cana-512	62	8	−	−	PROPN
cana-512	62	9	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	62	10	)	)	PUNCT
cana-512	62	11	2	2	NUM
cana-512	63	1	+	+	SYM
cana-512	63	2	1	1	NUM
cana-512	63	3	=	=	SYM
cana-512	63	4	2𝑛	2𝑛	NOUN
cana-512	63	5	+	+	CCONJ
cana-512	63	6	1	1	NUM
cana-512	63	7	edges	edge	NOUN
cana-512	63	8	of	of	ADP
cana-512	63	9	kn	kn	PROPN
cana-512	63	10	can	can	AUX
cana-512	63	11	be	be	AUX
cana-512	63	12	labeled	label	VERB
cana-512	63	13	from	from	ADP
cana-512	63	14	1	1	NUM
cana-512	63	15	to	to	ADP
cana-512	63	16	2𝑛	2𝑛	PROPN
cana-512	63	17	+	+	CCONJ
cana-512	64	1	1	1	X
cana-512	64	2	.	.	X
cana-512	64	3	hence	hence	ADV
cana-512	64	4	by	by	ADP
cana-512	64	5	removing	remove	VERB
cana-512	64	6	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	64	7	)	)	PUNCT
cana-512	64	8	2	2	NUM
cana-512	64	9	−	−	SYM
cana-512	64	10	1	1	NUM
cana-512	64	11	interior	interior	ADJ
cana-512	64	12	edges	edge	NOUN
cana-512	64	13	of	of	ADP
cana-512	64	14	kn	kn	PROPN
cana-512	64	15	⊙	⊙	PROPN
cana-512	64	16	k1	k1	PROPN
cana-512	64	17	,	,	PUNCT
cana-512	64	18	the	the	DET
cana-512	64	19	resultant	resultant	NOUN
cana-512	64	20	graph	graph	NOUN
cana-512	64	21	will	will	AUX
cana-512	64	22	be	be	AUX
cana-512	64	23	of	of	ADP
cana-512	64	24	the	the	DET
cana-512	64	25	form	form	NOUN
cana-512	64	26	cn	cn	VERB
cana-512	64	27	with	with	ADP
cana-512	64	28	n	n	NOUN
cana-512	64	29	edges	edge	NOUN
cana-512	64	30	,	,	PUNCT
cana-512	64	31	n	n	DET
cana-512	64	32	pendent	pendent	NOUN
cana-512	64	33	vertices	vertex	NOUN
cana-512	64	34	(	(	PUNCT
cana-512	64	35	𝑢1	𝑢1	PROPN
cana-512	64	36	,	,	PUNCT
cana-512	64	37	𝑢2	𝑢2	PROPN
cana-512	64	38	,	,	PUNCT
cana-512	64	39	⋯	⋯	PROPN
cana-512	64	40	⋯	⋯	PROPN
cana-512	64	41	,	,	PUNCT
cana-512	64	42	𝑢𝑛	𝑢𝑛	NOUN
cana-512	64	43	)	)	PUNCT
cana-512	64	44	connecting	connect	VERB
cana-512	64	45	to	to	ADP
cana-512	64	46	the	the	DET
cana-512	64	47	each	each	DET
cana-512	64	48	vertex	vertex	NOUN
cana-512	64	49	of	of	ADP
cana-512	64	50	cn	cn	PROPN
cana-512	64	51	and	and	CCONJ
cana-512	64	52	an	an	DET
cana-512	64	53	edge	edge	NOUN
cana-512	64	54	connecting	connect	VERB
cana-512	64	55	any	any	DET
cana-512	64	56	two	two	NUM
cana-512	64	57	non	non	ADJ
cana-512	64	58	-	-	ADJ
cana-512	64	59	adjacent	adjacent	ADJ
cana-512	64	60	vertices	vertex	NOUN
cana-512	64	61	of	of	ADP
cana-512	64	62	cn	cn	PROPN
cana-512	64	63	.	.	PUNCT
cana-512	65	1	each	each	DET
cana-512	65	2	vertex	vertex	NOUN
cana-512	65	3	𝑣1	𝑣1	PROPN
cana-512	65	4	,	,	PUNCT
cana-512	65	5	𝑣2	𝑣2	PROPN
cana-512	65	6	,	,	PUNCT
cana-512	65	7	⋯	⋯	PROPN
cana-512	65	8	⋯	⋯	PROPN
cana-512	65	9	,	,	PUNCT
cana-512	65	10	𝑣𝑛	𝑣𝑛	PROPN
cana-512	65	11	of	of	ADP
cana-512	65	12	cn	cn	PROPN
cana-512	65	13	is	be	AUX
cana-512	65	14	of	of	ADP
cana-512	65	15	degree	degree	NOUN
cana-512	65	16	3	3	NUM
cana-512	65	17	.	.	PUNCT
cana-512	66	1	by	by	ADP
cana-512	66	2	labeling	label	VERB
cana-512	66	3	the	the	DET
cana-512	66	4	edges	edge	NOUN
cana-512	66	5	of	of	ADP
cana-512	66	6	cn	cn	PROPN
cana-512	66	7	with	with	ADP
cana-512	66	8	1	1	NUM
cana-512	66	9	,	,	PUNCT
cana-512	66	10	3	3	NUM
cana-512	66	11	,	,	PUNCT
cana-512	66	12	5	5	NUM
cana-512	66	13	,	,	PUNCT
cana-512	66	14	…	…	PUNCT
cana-512	66	15	.	.	PUNCT
cana-512	67	1	2𝑛	2𝑛	NOUN
cana-512	68	1	−	−	NOUN
cana-512	68	2	1	1	NUM
cana-512	68	3	,	,	PUNCT
cana-512	68	4	each	each	DET
cana-512	68	5	edge	edge	NOUN
cana-512	68	6	incident	incident	NOUN
cana-512	68	7	on	on	ADP
cana-512	68	8	the	the	DET
cana-512	68	9	pendant	pendant	ADJ
cana-512	68	10	vertices	vertex	NOUN
cana-512	68	11	is	be	AUX
cana-512	68	12	labeled	label	VERB
cana-512	68	13	with	with	ADP
cana-512	68	14	2	2	NUM
cana-512	68	15	,	,	PUNCT
cana-512	68	16	4	4	NUM
cana-512	68	17	,	,	PUNCT
cana-512	68	18	6	6	NUM
cana-512	68	19	,	,	PUNCT
cana-512	68	20	…	…	PUNCT
cana-512	68	21	,	,	PUNCT
cana-512	68	22	2𝑛	2𝑛	PROPN
cana-512	68	23	and	and	CCONJ
cana-512	68	24	an	an	DET
cana-512	68	25	edge	edge	NOUN
cana-512	68	26	connecting	connect	VERB
cana-512	68	27	any	any	DET
cana-512	68	28	two	two	NUM
cana-512	68	29	non	non	ADJ
cana-512	68	30	-	-	ADJ
cana-512	68	31	adjacent	adjacent	ADJ
cana-512	68	32	vertices	vertex	NOUN
cana-512	68	33	of	of	ADP
cana-512	68	34	cn	cn	PROPN
cana-512	68	35	is	be	AUX
cana-512	68	36	labeled	label	VERB
cana-512	68	37	with	with	ADP
cana-512	68	38	communications	communication	NOUN
cana-512	68	39	on	on	ADP
cana-512	68	40	applied	apply	VERB
cana-512	68	41	nonlinear	nonlinear	ADJ
cana-512	68	42	analysis	analysis	NOUN
cana-512	68	43	issn	issn	NOUN
cana-512	68	44	:	:	PUNCT
cana-512	68	45	1074	1074	NUM
cana-512	68	46	-	-	PUNCT
cana-512	68	47	133x	133x	NUM
cana-512	68	48	vol	vol	NOUN
cana-512	68	49	31	31	NUM
cana-512	68	50	no	no	NOUN
cana-512	68	51	.	.	NOUN
cana-512	68	52	2	2	NUM
cana-512	68	53	(	(	PUNCT
cana-512	68	54	2024	2024	NUM
cana-512	68	55	)	)	PUNCT
cana-512	68	56	46	46	NUM
cana-512	68	57	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	68	58	2𝑛	2𝑛	NOUN
cana-512	69	1	+	+	CCONJ
cana-512	69	2	1	1	X
cana-512	69	3	.	.	X
cana-512	69	4	as	as	ADP
cana-512	69	5	2𝑛	2𝑛	PROPN
cana-512	69	6	+	+	CCONJ
cana-512	69	7	1	1	NUM
cana-512	69	8	≡	≡	PROPN
cana-512	69	9	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	NOUN
cana-512	69	10	3	3	NUM
cana-512	69	11	)	)	PUNCT
cana-512	69	12	,	,	PUNCT
cana-512	69	13	that	that	PRON
cana-512	69	14	is	be	AUX
cana-512	69	15	2𝑛	2𝑛	PROPN
cana-512	69	16	+	+	CCONJ
cana-512	69	17	1	1	NUM
cana-512	69	18	=	=	SYM
cana-512	69	19	3𝑚	3𝑚	NOUN
cana-512	69	20	,	,	PUNCT
cana-512	69	21	the	the	DET
cana-512	69	22	label	label	NOUN
cana-512	69	23	incident	incident	NOUN
cana-512	69	24	on	on	ADP
cana-512	69	25	the	the	DET
cana-512	69	26	vertices	vertex	NOUN
cana-512	69	27	of	of	ADP
cana-512	69	28	the	the	DET
cana-512	69	29	edge	edge	NOUN
cana-512	69	30	with	with	ADP
cana-512	69	31	label	label	NOUN
cana-512	69	32	2𝑛	2𝑛	NOUN
cana-512	69	33	+	+	CCONJ
cana-512	69	34	1	1	NUM
cana-512	69	35	fails	fail	VERB
cana-512	69	36	to	to	PART
cana-512	69	37	be	be	AUX
cana-512	69	38	relatively	relatively	ADV
cana-512	69	39	prime	prime	ADJ
cana-512	69	40	.	.	PUNCT
cana-512	70	1	case	case	NOUN
cana-512	70	2	2	2	NUM
cana-512	70	3	:	:	PUNCT
cana-512	70	4	for	for	ADP
cana-512	70	5	2𝑛	2𝑛	PROPN
cana-512	70	6	+	+	CCONJ
cana-512	70	7	1	1	NUM
cana-512	70	8	≢	≢	NUM
cana-512	70	9	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-512	70	10	3	3	NUM
cana-512	70	11	)	)	PUNCT
cana-512	70	12	.	.	PUNCT
cana-512	71	1	it	it	PRON
cana-512	71	2	is	be	AUX
cana-512	71	3	enough	enough	ADJ
cana-512	71	4	to	to	PART
cana-512	71	5	prove	prove	VERB
cana-512	71	6	that	that	SCONJ
cana-512	71	7	,	,	PUNCT
cana-512	71	8	the	the	DET
cana-512	71	9	removal	removal	NOUN
cana-512	71	10	of	of	ADP
cana-512	71	11	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	71	12	)	)	PUNCT
cana-512	71	13	2	2	NUM
cana-512	71	14	−	−	NOUN
cana-512	71	15	1	1	NUM
cana-512	71	16	edges	edge	NOUN
cana-512	71	17	results	result	NOUN
cana-512	71	18	in	in	ADP
cana-512	71	19	a	a	DET
cana-512	71	20	relatively	relatively	ADV
cana-512	71	21	prime	prime	ADJ
cana-512	71	22	edge	edge	NOUN
cana-512	71	23	labeled	label	VERB
cana-512	71	24	graph	graph	NOUN
cana-512	71	25	.	.	PUNCT
cana-512	72	1	suppose	suppose	VERB
cana-512	72	2	the	the	DET
cana-512	72	3	removal	removal	NOUN
cana-512	72	4	of	of	ADP
cana-512	72	5	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	72	6	)	)	PUNCT
cana-512	72	7	2	2	NUM
cana-512	72	8	−	−	SYM
cana-512	72	9	2	2	NUM
cana-512	72	10	edges	edge	NOUN
cana-512	72	11	in	in	ADP
cana-512	72	12	kn	kn	PROPN
cana-512	72	13	⊙	⊙	PROPN
cana-512	72	14	k1	k1	PROPN
cana-512	72	15	results	result	NOUN
cana-512	72	16	in	in	ADP
cana-512	72	17	a	a	DET
cana-512	72	18	relatively	relatively	ADV
cana-512	72	19	prime	prime	ADJ
cana-512	72	20	edge	edge	NOUN
cana-512	72	21	labeled	label	VERB
cana-512	72	22	graph	graph	NOUN
cana-512	72	23	.	.	PUNCT
cana-512	73	1	that	that	PRON
cana-512	73	2	is	is	ADV
cana-512	73	3	,	,	PUNCT
cana-512	73	4	remaining	remain	VERB
cana-512	73	5	𝑛(𝑛+1	𝑛(𝑛+1	NUM
cana-512	73	6	)	)	PUNCT
cana-512	73	7	2	2	NUM
cana-512	73	8	−	−	PROPN
cana-512	73	9	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	73	10	)	)	PUNCT
cana-512	73	11	2	2	NUM
cana-512	74	1	+	+	SYM
cana-512	74	2	2	2	NUM
cana-512	74	3	=	=	SYM
cana-512	74	4	2𝑛	2𝑛	NOUN
cana-512	74	5	+	+	CCONJ
cana-512	74	6	2	2	NUM
cana-512	74	7	edges	edge	NOUN
cana-512	74	8	of	of	ADP
cana-512	74	9	kn	kn	PROPN
cana-512	74	10	can	can	AUX
cana-512	74	11	be	be	AUX
cana-512	74	12	labeled	label	VERB
cana-512	74	13	from	from	ADP
cana-512	74	14	1	1	NUM
cana-512	74	15	to	to	ADP
cana-512	74	16	2𝑛	2𝑛	NOUN
cana-512	74	17	+	+	CCONJ
cana-512	75	1	2	2	X
cana-512	75	2	.	.	X
cana-512	75	3	hence	hence	ADV
cana-512	75	4	by	by	ADP
cana-512	75	5	removing	remove	VERB
cana-512	75	6	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	75	7	)	)	PUNCT
cana-512	75	8	2	2	NUM
cana-512	75	9	−	−	NUM
cana-512	75	10	2	2	NUM
cana-512	75	11	interior	interior	ADJ
cana-512	75	12	edges	edge	NOUN
cana-512	75	13	of	of	ADP
cana-512	75	14	kn	kn	PROPN
cana-512	75	15	⊙	⊙	PROPN
cana-512	75	16	k1	k1	PROPN
cana-512	75	17	,	,	PUNCT
cana-512	75	18	the	the	DET
cana-512	75	19	resultant	resultant	NOUN
cana-512	75	20	graph	graph	NOUN
cana-512	75	21	will	will	AUX
cana-512	75	22	be	be	AUX
cana-512	75	23	of	of	ADP
cana-512	75	24	the	the	DET
cana-512	75	25	form	form	NOUN
cana-512	75	26	cn	cn	VERB
cana-512	75	27	with	with	ADP
cana-512	75	28	n	n	NOUN
cana-512	75	29	edges	edge	NOUN
cana-512	75	30	,	,	PUNCT
cana-512	75	31	n	n	DET
cana-512	75	32	pendent	pendent	NOUN
cana-512	75	33	vertices	vertex	NOUN
cana-512	75	34	(	(	PUNCT
cana-512	75	35	𝑢1	𝑢1	PROPN
cana-512	75	36	,	,	PUNCT
cana-512	75	37	𝑢2	𝑢2	PROPN
cana-512	75	38	,	,	PUNCT
cana-512	75	39	⋯	⋯	PROPN
cana-512	75	40	⋯	⋯	PROPN
cana-512	75	41	,	,	PUNCT
cana-512	75	42	𝑢𝑛	𝑢𝑛	NOUN
cana-512	75	43	)	)	PUNCT
cana-512	75	44	connecting	connect	VERB
cana-512	75	45	to	to	ADP
cana-512	75	46	the	the	DET
cana-512	75	47	each	each	DET
cana-512	75	48	vertex	vertex	NOUN
cana-512	75	49	of	of	ADP
cana-512	75	50	cn	cn	PROPN
cana-512	75	51	and	and	CCONJ
cana-512	75	52	two	two	NUM
cana-512	75	53	edges	edge	NOUN
cana-512	75	54	connecting	connect	VERB
cana-512	75	55	any	any	DET
cana-512	75	56	two	two	NUM
cana-512	75	57	non	non	ADJ
cana-512	75	58	-	-	ADJ
cana-512	75	59	adjacent	adjacent	ADJ
cana-512	75	60	vertices	vertex	NOUN
cana-512	75	61	of	of	ADP
cana-512	75	62	cn	cn	PROPN
cana-512	75	63	.	.	PUNCT
cana-512	76	1	each	each	DET
cana-512	76	2	vertex	vertex	NOUN
cana-512	76	3	𝑣1	𝑣1	PROPN
cana-512	76	4	,	,	PUNCT
cana-512	76	5	𝑣2	𝑣2	PROPN
cana-512	76	6	,	,	PUNCT
cana-512	76	7	⋯	⋯	PROPN
cana-512	76	8	⋯	⋯	PROPN
cana-512	76	9	,	,	PUNCT
cana-512	76	10	𝑣𝑛	𝑣𝑛	PROPN
cana-512	76	11	of	of	ADP
cana-512	76	12	cn	cn	PROPN
cana-512	76	13	is	be	AUX
cana-512	76	14	of	of	ADP
cana-512	76	15	degree	degree	NOUN
cana-512	76	16	3	3	NUM
cana-512	76	17	.	.	PUNCT
cana-512	77	1	by	by	ADP
cana-512	77	2	labeling	label	VERB
cana-512	77	3	the	the	DET
cana-512	77	4	edges	edge	NOUN
cana-512	77	5	of	of	ADP
cana-512	77	6	cn	cn	PROPN
cana-512	77	7	with	with	ADP
cana-512	77	8	1	1	NUM
cana-512	77	9	,	,	PUNCT
cana-512	77	10	3	3	NUM
cana-512	77	11	,	,	PUNCT
cana-512	77	12	5	5	NUM
cana-512	77	13	,	,	PUNCT
cana-512	77	14	…	…	PUNCT
cana-512	77	15	.	.	PUNCT
cana-512	78	1	2𝑛	2𝑛	NOUN
cana-512	79	1	−	−	NOUN
cana-512	79	2	1	1	NUM
cana-512	79	3	,	,	PUNCT
cana-512	79	4	each	each	DET
cana-512	79	5	edge	edge	NOUN
cana-512	79	6	incident	incident	NOUN
cana-512	79	7	on	on	ADP
cana-512	79	8	the	the	DET
cana-512	79	9	pendant	pendant	ADJ
cana-512	79	10	vertices	vertex	NOUN
cana-512	79	11	is	be	AUX
cana-512	79	12	labeled	label	VERB
cana-512	79	13	with	with	ADP
cana-512	79	14	2	2	NUM
cana-512	79	15	,	,	PUNCT
cana-512	79	16	4	4	NUM
cana-512	79	17	,	,	PUNCT
cana-512	79	18	6	6	NUM
cana-512	79	19	,	,	PUNCT
cana-512	79	20	…	…	PUNCT
cana-512	79	21	,	,	PUNCT
cana-512	79	22	2𝑛	2𝑛	PROPN
cana-512	79	23	and	and	CCONJ
cana-512	79	24	the	the	DET
cana-512	79	25	edge	edge	NOUN
cana-512	79	26	connecting	connect	VERB
cana-512	79	27	any	any	DET
cana-512	79	28	two	two	NUM
cana-512	79	29	non	non	ADJ
cana-512	79	30	-	-	ADJ
cana-512	79	31	adjacent	adjacent	ADJ
cana-512	79	32	vertices	vertex	NOUN
cana-512	79	33	of	of	ADP
cana-512	79	34	cn	cn	PROPN
cana-512	79	35	is	be	AUX
cana-512	79	36	labeled	label	VERB
cana-512	79	37	with	with	ADP
cana-512	79	38	2𝑛	2𝑛	PROPN
cana-512	80	1	+	+	CCONJ
cana-512	80	2	1	1	NUM
cana-512	80	3	,	,	PUNCT
cana-512	80	4	2𝑛	2𝑛	NOUN
cana-512	80	5	+	+	CCONJ
cana-512	80	6	2	2	X
cana-512	80	7	.	.	X
cana-512	81	1	as	as	ADP
cana-512	81	2	2𝑛	2𝑛	PROPN
cana-512	81	3	+	+	CCONJ
cana-512	81	4	1	1	NUM
cana-512	81	5	≢	≢	NUM
cana-512	81	6	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-512	81	7	3	3	NUM
cana-512	81	8	)	)	PUNCT
cana-512	81	9	,	,	PUNCT
cana-512	81	10	the	the	DET
cana-512	81	11	label	label	NOUN
cana-512	81	12	incident	incident	NOUN
cana-512	81	13	on	on	ADP
cana-512	81	14	the	the	DET
cana-512	81	15	vertices	vertex	NOUN
cana-512	81	16	of	of	ADP
cana-512	81	17	the	the	DET
cana-512	81	18	edge	edge	NOUN
cana-512	81	19	with	with	ADP
cana-512	81	20	label	label	NOUN
cana-512	81	21	2𝑛	2𝑛	NOUN
cana-512	82	1	+	+	CCONJ
cana-512	82	2	2	2	NUM
cana-512	82	3	fails	fail	VERB
cana-512	82	4	to	to	PART
cana-512	82	5	be	be	AUX
cana-512	82	6	relatively	relatively	ADV
cana-512	82	7	prime	prime	ADJ
cana-512	82	8	.	.	PUNCT
cana-512	83	1	illustration	illustration	NOUN
cana-512	83	2	as	as	ADP
cana-512	83	3	an	an	DET
cana-512	83	4	illustration	illustration	NOUN
cana-512	83	5	of	of	ADP
cana-512	83	6	above	above	ADP
cana-512	83	7	theorem	theorem	ADJ
cana-512	83	8	,	,	PUNCT
cana-512	83	9	edge	edge	ADJ
cana-512	83	10	prime	prime	ADJ
cana-512	83	11	index	index	NOUN
cana-512	83	12	of	of	ADP
cana-512	83	13	,	,	PUNCT
cana-512	83	14	k4	k4	PROPN
cana-512	83	15	⊙	⊙	PROPN
cana-512	83	16	k1	k1	PROPN
cana-512	83	17	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-512	83	18	,	,	PUNCT
cana-512	83	19	k5	k5	PROPN
cana-512	83	20	⊙	⊙	PROPN
cana-512	83	21	k1	k1	PROPN
cana-512	83	22	is	be	AUX
cana-512	83	23	given	give	VERB
cana-512	83	24	in	in	ADP
cana-512	83	25	figure	figure	NOUN
cana-512	83	26	–	–	PUNCT
cana-512	83	27	3	3	NUM
cana-512	83	28	,	,	PUNCT
cana-512	83	29	4	4	NUM
cana-512	83	30	respectively	respectively	ADV
cana-512	83	31	.	.	PUNCT
cana-512	84	1	for	for	ADP
cana-512	84	2	2𝑛	2𝑛	PROPN
cana-512	84	3	+	+	CCONJ
cana-512	84	4	1	1	NUM
cana-512	84	5	≡	≡	PROPN
cana-512	84	6	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	NOUN
cana-512	84	7	3	3	NUM
cana-512	84	8	)	)	PUNCT
cana-512	84	9	,	,	PUNCT
cana-512	84	10	consider	consider	VERB
cana-512	84	11	n	n	NOUN
cana-512	84	12	=	=	SYM
cana-512	84	13	4	4	NUM
cana-512	84	14	that	that	PRON
cana-512	84	15	is	be	AUX
cana-512	84	16	,	,	PUNCT
cana-512	84	17	k4	k4	PROPN
cana-512	84	18	⊙	⊙	PROPN
cana-512	84	19	k1	k1	PROPN
cana-512	84	20	,	,	PUNCT
cana-512	84	21	the	the	DET
cana-512	84	22	edge	edge	NOUN
cana-512	84	23	prime	prime	ADJ
cana-512	84	24	index	index	NOUN
cana-512	84	25	is	be	AUX
cana-512	84	26	𝜀𝑟(k4	𝜀𝑟(k4	PROPN
cana-512	84	27	⊙	⊙	PROPN
cana-512	84	28	k1	k1	PROPN
cana-512	84	29	)	)	PUNCT
cana-512	84	30	=	=	SYM
cana-512	84	31	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	84	32	)	)	PUNCT
cana-512	84	33	2	2	NUM
cana-512	84	34	=	=	SYM
cana-512	84	35	4	4	NUM
cana-512	84	36	2	2	NUM
cana-512	84	37	=	=	SYM
cana-512	84	38	2	2	NUM
cana-512	84	39	,	,	PUNCT
cana-512	84	40	which	which	PRON
cana-512	84	41	is	be	AUX
cana-512	84	42	illustrated	illustrate	VERB
cana-512	84	43	in	in	ADP
cana-512	84	44	figure	figure	NOUN
cana-512	84	45	3	3	NUM
cana-512	84	46	.	.	PUNCT
cana-512	84	47	figure	figure	VERB
cana-512	84	48	3	3	NUM
cana-512	84	49	:	:	PUNCT
cana-512	84	50	edge	edge	VERB
cana-512	84	51	prime	prime	ADJ
cana-512	84	52	index	index	NOUN
cana-512	84	53	𝜀𝑟(k4	𝜀𝑟(k4	PROPN
cana-512	84	54	⊙	⊙	PROPN
cana-512	84	55	k1	k1	PROPN
cana-512	84	56	)	)	PUNCT
cana-512	84	57	=	=	SYM
cana-512	84	58	2	2	NUM
cana-512	84	59	for	for	ADP
cana-512	84	60	2𝑛	2𝑛	PROPN
cana-512	84	61	+	+	CCONJ
cana-512	84	62	1	1	NUM
cana-512	84	63	≢	≢	NUM
cana-512	84	64	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-512	84	65	3	3	NUM
cana-512	84	66	)	)	PUNCT
cana-512	84	67	,	,	PUNCT
cana-512	84	68	consider	consider	VERB
cana-512	84	69	n	n	NOUN
cana-512	84	70	=	=	SYM
cana-512	84	71	5	5	NUM
cana-512	84	72	,	,	PUNCT
cana-512	84	73	that	that	PRON
cana-512	84	74	is	be	AUX
cana-512	84	75	k5	k5	PROPN
cana-512	84	76	⊙	⊙	PROPN
cana-512	84	77	k1	k1	PROPN
cana-512	84	78	.	.	PUNCT
cana-512	84	79	.	.	PUNCT
cana-512	85	1	communications	communication	NOUN
cana-512	85	2	on	on	ADP
cana-512	85	3	applied	apply	VERB
cana-512	85	4	nonlinear	nonlinear	ADJ
cana-512	85	5	analysis	analysis	NOUN
cana-512	85	6	issn	issn	NOUN
cana-512	85	7	:	:	PUNCT
cana-512	85	8	1074	1074	NUM
cana-512	85	9	-	-	PUNCT
cana-512	85	10	133x	133x	NUM
cana-512	85	11	vol	vol	NOUN
cana-512	85	12	31	31	NUM
cana-512	85	13	no	no	NOUN
cana-512	85	14	.	.	NOUN
cana-512	85	15	2	2	NUM
cana-512	85	16	(	(	PUNCT
cana-512	85	17	2024	2024	NUM
cana-512	85	18	)	)	PUNCT
cana-512	85	19	47	47	NUM
cana-512	85	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	85	21	the	the	DET
cana-512	85	22	edge	edge	NOUN
cana-512	85	23	prime	prime	ADJ
cana-512	85	24	index	index	NOUN
cana-512	85	25	of	of	ADP
cana-512	85	26	k5	k5	PROPN
cana-512	85	27	⊙	⊙	PROPN
cana-512	85	28	k1	k1	PROPN
cana-512	85	29	.	.	PROPN
cana-512	86	1	is	be	AUX
cana-512	86	2	𝜀𝑟(k5	𝜀𝑟(k5	PROPN
cana-512	86	3	⊙	⊙	PROPN
cana-512	86	4	k1	k1	PROPN
cana-512	86	5	)	)	PUNCT
cana-512	86	6	=	=	SYM
cana-512	86	7	𝑛(𝑛−3	𝑛(𝑛−3	NOUN
cana-512	86	8	)	)	PUNCT
cana-512	86	9	2	2	NUM
cana-512	86	10	−	−	NOUN
cana-512	86	11	1	1	NUM
cana-512	86	12	=	=	SYM
cana-512	86	13	5×2	5×2	NUM
cana-512	86	14	2	2	NUM
cana-512	86	15	−	−	NOUN
cana-512	86	16	1	1	NUM
cana-512	86	17	=	=	SYM
cana-512	86	18	4	4	NUM
cana-512	86	19	,	,	PUNCT
cana-512	86	20	which	which	PRON
cana-512	86	21	is	be	AUX
cana-512	86	22	illustrated	illustrate	VERB
cana-512	86	23	in	in	ADP
cana-512	86	24	figure	figure	NOUN
cana-512	86	25	4	4	NUM
cana-512	86	26	.	.	PUNCT
cana-512	86	27	figure	figure	VERB
cana-512	86	28	4	4	NUM
cana-512	86	29	:	:	PUNCT
cana-512	86	30	edge	edge	VERB
cana-512	86	31	prime	prime	ADJ
cana-512	86	32	index	index	NOUN
cana-512	86	33	𝜀𝑟(k5	𝜀𝑟(k5	PROPN
cana-512	86	34	⊙	⊙	PROPN
cana-512	86	35	k1	k1	PROPN
cana-512	86	36	)	)	PUNCT
cana-512	87	1	=	=	PUNCT
cana-512	87	2	4	4	NUM
cana-512	87	3	4.2	4.2	NUM
cana-512	87	4	.	.	PUNCT
cana-512	88	1	complete	complete	ADJ
cana-512	88	2	bipartite	bipartite	PROPN
cana-512	88	3	graph	graph	NOUN
cana-512	88	4	the	the	DET
cana-512	88	5	following	follow	VERB
cana-512	88	6	theorem	theorem	NOUN
cana-512	88	7	finds	find	VERB
cana-512	88	8	the	the	DET
cana-512	88	9	edge	edge	NOUN
cana-512	88	10	prime	prime	ADJ
cana-512	88	11	index	index	NOUN
cana-512	88	12	of	of	ADP
cana-512	88	13	the	the	DET
cana-512	88	14	complete	complete	ADJ
cana-512	88	15	bipartite	bipartite	PROPN
cana-512	88	16	graph	graph	NOUN
cana-512	88	17	𝐾2,𝑡	𝐾2,𝑡	PROPN
cana-512	88	18	and	and	CCONJ
cana-512	88	19	𝐾3,𝑡	𝐾3,𝑡	PROPN
cana-512	88	20	with	with	ADP
cana-512	88	21	proper	proper	ADJ
cana-512	88	22	illustration	illustration	NOUN
cana-512	88	23	(	(	PUNCT
cana-512	88	24	7	7	NUM
cana-512	88	25	)	)	PUNCT
cana-512	88	26	.	.	PUNCT
cana-512	89	1	theorem	theorem	VERB
cana-512	89	2	4.2	4.2	NUM
cana-512	89	3	for	for	ADP
cana-512	89	4	a	a	DET
cana-512	89	5	graph	graph	NOUN
cana-512	89	6	g	g	PROPN
cana-512	89	7	=	=	PROPN
cana-512	89	8	𝐾2,𝑡	𝐾2,𝑡	PROPN
cana-512	89	9	,	,	PUNCT
cana-512	89	10	𝑡	𝑡	X
cana-512	89	11	>	>	X
cana-512	89	12	2	2	NUM
cana-512	89	13	then	then	ADV
cana-512	89	14	,	,	PUNCT
cana-512	89	15	εr(g	εr(g	PUNCT
cana-512	89	16	)	)	PUNCT
cana-512	90	1	=	=	NOUN
cana-512	90	2	2𝑡	2𝑡	NOUN
cana-512	90	3	−	−	NOUN
cana-512	90	4	5	5	X
cana-512	90	5	.	.	PUNCT
cana-512	91	1	proof	proof	NOUN
cana-512	91	2	.	.	PUNCT
cana-512	92	1	we	we	PRON
cana-512	92	2	know	know	VERB
cana-512	92	3	that	that	SCONJ
cana-512	92	4	,	,	PUNCT
cana-512	92	5	𝐺	𝐺	PROPN
cana-512	92	6	=	=	PUNCT
cana-512	92	7	𝐾2,𝑡	𝐾2,𝑡	PROPN
cana-512	92	8	is	be	AUX
cana-512	92	9	not	not	PART
cana-512	92	10	rpel	rpel	NOUN
cana-512	92	11	graph	graph	NOUN
cana-512	92	12	.	.	PUNCT
cana-512	93	1	now	now	ADV
cana-512	93	2	,	,	PUNCT
cana-512	93	3	it	it	PRON
cana-512	93	4	is	be	AUX
cana-512	93	5	enough	enough	ADJ
cana-512	93	6	to	to	PART
cana-512	93	7	find	find	VERB
cana-512	93	8	the	the	DET
cana-512	93	9	minimum	minimum	ADJ
cana-512	93	10	number	number	NOUN
cana-512	93	11	of	of	ADP
cana-512	93	12	edges	edge	NOUN
cana-512	93	13	to	to	PART
cana-512	93	14	be	be	AUX
cana-512	93	15	removed	remove	VERB
cana-512	93	16	from	from	ADP
cana-512	93	17	g	g	PROPN
cana-512	93	18	to	to	PART
cana-512	93	19	make	make	VERB
cana-512	93	20	g	g	NOUN
cana-512	93	21	as	as	ADP
cana-512	93	22	a	a	DET
cana-512	93	23	rpel	rpel	NOUN
cana-512	93	24	.	.	PUNCT
cana-512	94	1	the	the	DET
cana-512	94	2	number	number	NOUN
cana-512	94	3	of	of	ADP
cana-512	94	4	vertices	vertex	NOUN
cana-512	94	5	and	and	CCONJ
cana-512	94	6	edges	edge	NOUN
cana-512	94	7	in	in	ADP
cana-512	94	8	𝐺	𝐺	PROPN
cana-512	94	9	=	=	PUNCT
cana-512	94	10	𝐾2,𝑡	𝐾2,𝑡	PROPN
cana-512	94	11	is	be	AUX
cana-512	94	12	𝑡	𝑡	PROPN
cana-512	94	13	+	+	PROPN
cana-512	94	14	2	2	NUM
cana-512	94	15	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-512	94	16	2𝑡.	2𝑡.	NUM
cana-512	94	17	label	label	NOUN
cana-512	94	18	the	the	DET
cana-512	94	19	edges	edge	NOUN
cana-512	94	20	of	of	ADP
cana-512	94	21	g	g	NOUN
cana-512	94	22	in	in	ADP
cana-512	94	23	such	such	DET
cana-512	94	24	a	a	DET
cana-512	94	25	way	way	NOUN
cana-512	94	26	that	that	SCONJ
cana-512	94	27	,	,	PUNCT
cana-512	94	28	𝐿(𝑢1𝑣1	𝐿(𝑢1𝑣1	PROPN
cana-512	94	29	)	)	PUNCT
cana-512	94	30	=	=	SYM
cana-512	94	31	1	1	NUM
cana-512	94	32	,	,	PUNCT
cana-512	94	33	𝐿(𝑢1𝑣2	𝐿(𝑢1𝑣2	NOUN
cana-512	94	34	)	)	PUNCT
cana-512	94	35	=	=	SYM
cana-512	94	36	2	2	NUM
cana-512	94	37	,	,	PUNCT
cana-512	94	38	𝐿(𝑢1𝑣3	𝐿(𝑢1𝑣3	ADJ
cana-512	94	39	)	)	PUNCT
cana-512	94	40	=	=	SYM
cana-512	94	41	3	3	NUM
cana-512	94	42	,	,	PUNCT
cana-512	94	43	𝐿(𝑢1𝑣4	𝐿(𝑢1𝑣4	NOUN
cana-512	94	44	)	)	PUNCT
cana-512	94	45	=	=	SYM
cana-512	94	46	5	5	NUM
cana-512	94	47	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-512	94	48	𝐿(𝑢2𝑣1	𝐿(𝑢2𝑣1	NOUN
cana-512	94	49	)	)	PUNCT
cana-512	94	50	=	=	SYM
cana-512	94	51	4	4	NUM
cana-512	94	52	as	as	ADP
cana-512	94	53	in	in	ADP
cana-512	94	54	figure	figure	NOUN
cana-512	94	55	5	5	NUM
cana-512	94	56	.	.	PUNCT
cana-512	94	57	communications	communication	NOUN
cana-512	94	58	on	on	ADP
cana-512	94	59	applied	apply	VERB
cana-512	94	60	nonlinear	nonlinear	ADJ
cana-512	94	61	analysis	analysis	NOUN
cana-512	94	62	issn	issn	NOUN
cana-512	94	63	:	:	PUNCT
cana-512	94	64	1074	1074	NUM
cana-512	94	65	-	-	PUNCT
cana-512	94	66	133x	133x	NUM
cana-512	94	67	vol	vol	NOUN
cana-512	94	68	31	31	NUM
cana-512	94	69	no	no	NOUN
cana-512	94	70	.	.	NOUN
cana-512	94	71	2	2	NUM
cana-512	94	72	(	(	PUNCT
cana-512	94	73	2024	2024	NUM
cana-512	94	74	)	)	PUNCT
cana-512	94	75	48	48	NUM
cana-512	94	76	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-512	94	77	figure	figure	NOUN
cana-512	94	78	5	5	NUM
cana-512	94	79	edge	edge	NOUN
cana-512	94	80	prime	prime	ADJ
cana-512	94	81	index	index	NOUN
cana-512	94	82	illustration	illustration	NOUN
cana-512	94	83	also	also	ADV
cana-512	94	84	,	,	PUNCT
cana-512	94	85	the	the	DET
cana-512	94	86	label	label	NOUN
cana-512	94	87	6	6	NUM
cana-512	94	88	can	can	AUX
cana-512	94	89	not	not	PART
cana-512	94	90	be	be	AUX
cana-512	94	91	labeled	label	VERB
cana-512	94	92	in	in	ADP
cana-512	94	93	any	any	PRON
cana-512	94	94	of	of	ADP
cana-512	94	95	the	the	DET
cana-512	94	96	edge	edge	NOUN
cana-512	94	97	’s	’s	PART
cana-512	94	98	incident	incident	NOUN
cana-512	94	99	on	on	ADP
cana-512	94	100	𝑢1	𝑢1	PROPN
cana-512	94	101	and	and	CCONJ
cana-512	94	102	𝑢2	𝑢2	PROPN
cana-512	94	103	.	.	PUNCT
cana-512	95	1	thus	thus	ADV
cana-512	95	2	,	,	PUNCT
cana-512	95	3	it	it	PRON
cana-512	95	4	is	be	AUX
cana-512	95	5	needed	need	VERB
cana-512	95	6	to	to	PART
cana-512	95	7	remove	remove	VERB
cana-512	95	8	2𝑡	2𝑡	NOUN
cana-512	95	9	−	−	ADP
cana-512	95	10	5	5	NUM
cana-512	95	11	edges	edge	VERB
cana-512	95	12	from	from	ADP
cana-512	95	13	g	g	NOUN
cana-512	95	14	to	to	PART
cana-512	95	15	make	make	VERB
cana-512	95	16	it	it	PRON
cana-512	95	17	as	as	ADP
cana-512	95	18	a	a	DET
cana-512	95	19	relatively	relatively	ADV
cana-512	95	20	prime	prime	ADJ
cana-512	95	21	edge	edge	NOUN
cana-512	95	22	labeled	label	VERB
cana-512	95	23	graph	graph	NOUN
cana-512	95	24	.	.	PUNCT
cana-512	96	1	hence	hence	ADV
cana-512	96	2	,	,	PUNCT
cana-512	96	3	εr(g	εr(g	PUNCT
cana-512	96	4	)	)	PUNCT
cana-512	97	1	=	=	NOUN
cana-512	97	2	2𝑡	2𝑡	NOUN
cana-512	97	3	−	−	NOUN
cana-512	97	4	5	5	X
cana-512	97	5	.	.	PUNCT
cana-512	97	6	theorem	theorem	VERB
cana-512	97	7	4.3	4.3	NUM
cana-512	97	8	.	.	PUNCT
cana-512	98	1	for	for	ADP
cana-512	98	2	a	a	DET
cana-512	98	3	graph	graph	NOUN
cana-512	98	4	g	g	PROPN
cana-512	98	5	=	=	PUNCT
cana-512	98	6	𝐾3,𝑡	𝐾3,𝑡	PROPN
cana-512	98	7	,	,	PUNCT
cana-512	98	8	𝑡	𝑡	X
cana-512	98	9	>	>	X
cana-512	98	10	3	3	NUM
cana-512	98	11	,	,	PUNCT
cana-512	98	12	then	then	ADV
cana-512	98	13	,	,	PUNCT
cana-512	98	14	εr(g	εr(g	PUNCT
cana-512	98	15	)	)	PUNCT
cana-512	99	1	=	=	SYM
cana-512	99	2	3𝑡	3𝑡	NUM
cana-512	99	3	−	−	NOUN
cana-512	99	4	7	7	NUM
cana-512	99	5	.	.	PUNCT
cana-512	99	6	proof	proof	NOUN
cana-512	99	7	.	.	PUNCT
cana-512	100	1	since	since	SCONJ
cana-512	100	2	𝐺	𝐺	PROPN
cana-512	100	3	=	=	PRON
cana-512	100	4	𝐾3,𝑡	𝐾3,𝑡	PROPN
cana-512	100	5	is	be	AUX
cana-512	100	6	not	not	PART
cana-512	100	7	rpel	rpel	NOUN
cana-512	100	8	graph	graph	NOUN
cana-512	100	9	.	.	PUNCT
cana-512	101	1	it	it	PRON
cana-512	101	2	is	be	AUX
cana-512	101	3	enough	enough	ADJ
cana-512	101	4	to	to	PART
cana-512	101	5	find	find	VERB
cana-512	101	6	the	the	DET
cana-512	101	7	minimum	minimum	ADJ
cana-512	101	8	number	number	NOUN
cana-512	101	9	of	of	ADP
cana-512	101	10	edges	edge	NOUN
cana-512	101	11	to	to	PART
cana-512	101	12	be	be	AUX
cana-512	101	13	removed	remove	VERB
cana-512	101	14	from	from	ADP
cana-512	101	15	g	g	PROPN
cana-512	101	16	to	to	PART
cana-512	101	17	make	make	VERB
cana-512	101	18	it	it	PRON
cana-512	101	19	as	as	ADP
cana-512	101	20	a	a	DET
cana-512	101	21	rpel	rpel	NOUN
cana-512	101	22	graph	graph	NOUN
cana-512	101	23	.	.	PUNCT
cana-512	102	1	the	the	DET
cana-512	102	2	number	number	NOUN
cana-512	102	3	of	of	ADP
cana-512	102	4	vertices	vertex	NOUN
cana-512	102	5	and	and	CCONJ
cana-512	102	6	edges	edge	NOUN
cana-512	102	7	in	in	ADP
cana-512	102	8	𝐺	𝐺	PROPN
cana-512	102	9	=	=	PUNCT
cana-512	102	10	𝐾3,𝑡	𝐾3,𝑡	PROPN
cana-512	102	11	is	be	AUX
cana-512	102	12	𝑡	𝑡	PROPN
cana-512	102	13	+	+	NOUN
cana-512	102	14	3	3	NUM
cana-512	102	15	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-512	102	16	3𝑡.	3𝑡.	NUM
cana-512	102	17	label	label	NOUN
cana-512	102	18	the	the	DET
cana-512	102	19	edges	edge	NOUN
cana-512	102	20	of	of	ADP
cana-512	102	21	g	g	NOUN
cana-512	102	22	in	in	ADP
cana-512	102	23	such	such	DET
cana-512	102	24	a	a	DET
cana-512	102	25	way	way	NOUN
cana-512	102	26	that	that	SCONJ
cana-512	102	27	,	,	PUNCT
cana-512	102	28	𝐿(𝑢1𝑣1	𝐿(𝑢1𝑣1	PROPN
cana-512	102	29	)	)	PUNCT
cana-512	102	30	=	=	SYM
cana-512	102	31	1	1	NUM
cana-512	102	32	,	,	PUNCT
cana-512	102	33	𝐿(𝑢1𝑣2	𝐿(𝑢1𝑣2	NOUN
cana-512	102	34	)	)	PUNCT
cana-512	102	35	=	=	SYM
cana-512	102	36	2	2	NUM
cana-512	102	37	,	,	PUNCT
cana-512	102	38	𝐿(𝑢1𝑣3	𝐿(𝑢1𝑣3	ADJ
cana-512	102	39	)	)	PUNCT
cana-512	102	40	=	=	SYM
cana-512	102	41	3	3	NUM
cana-512	102	42	,	,	PUNCT
cana-512	102	43	𝐿(𝑢1𝑣4	𝐿(𝑢1𝑣4	NOUN
cana-512	102	44	)	)	PUNCT
cana-512	102	45	=	=	SYM
cana-512	102	46	5	5	NUM
cana-512	102	47	,	,	PUNCT
cana-512	102	48	𝐿(𝑢2𝑣1	𝐿(𝑢2𝑣1	NOUN
cana-512	102	49	)	)	PUNCT
cana-512	102	50	=	=	SYM
cana-512	102	51	4	4	NUM
cana-512	102	52	,	,	PUNCT
cana-512	102	53	𝐿(𝑢2𝑣2	𝐿(𝑢2𝑣2	NOUN
cana-512	102	54	)	)	PUNCT
cana-512	102	55	=	=	SYM
cana-512	102	56	7	7	NUM
cana-512	102	57	and	and	CCONJ
cana-512	102	58	,	,	PUNCT
cana-512	102	59	𝐿(𝑢3𝑣4	𝐿(𝑢3𝑣4	NUM
cana-512	102	60	)	)	PUNCT
cana-512	102	61	=	=	SYM
cana-512	102	62	6	6	NUM
cana-512	102	63	as	as	ADP
cana-512	102	64	in	in	ADP
cana-512	102	65	figure	figure	NOUN
cana-512	102	66	6	6	NUM
cana-512	102	67	.	.	PUNCT
cana-512	103	1	figure	figure	VERB
cana-512	103	2	6	6	NUM
cana-512	103	3	edge	edge	NOUN
cana-512	103	4	prime	prime	ADJ
cana-512	103	5	index	index	NOUN
cana-512	103	6	communications	communication	NOUN
cana-512	103	7	on	on	ADP
cana-512	103	8	applied	apply	VERB
cana-512	103	9	nonlinear	nonlinear	ADJ
cana-512	103	10	analysis	analysis	NOUN
cana-512	103	11	issn	issn	NOUN
cana-512	103	12	:	:	PUNCT
cana-512	103	13	1074	1074	NUM
cana-512	103	14	-	-	PUNCT
cana-512	103	15	133x	133x	NUM
cana-512	103	16	vol	vol	NOUN
cana-512	103	17	31	31	NUM
cana-512	103	18	no	no	NOUN
cana-512	103	19	.	.	NOUN
cana-512	103	20	2	2	NUM
cana-512	103	21	(	(	PUNCT
cana-512	103	22	2024	2024	NUM
cana-512	103	23	)	)	PUNCT
cana-512	103	24	49	49	NUM
cana-512	103	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	103	26	also	also	ADV
cana-512	103	27	,	,	PUNCT
cana-512	103	28	the	the	DET
cana-512	103	29	label	label	NOUN
cana-512	103	30	8	8	NUM
cana-512	103	31	can	can	AUX
cana-512	103	32	not	not	PART
cana-512	103	33	be	be	AUX
cana-512	103	34	labeled	label	VERB
cana-512	103	35	in	in	ADP
cana-512	103	36	any	any	PRON
cana-512	103	37	of	of	ADP
cana-512	103	38	the	the	DET
cana-512	103	39	edge	edge	NOUN
cana-512	103	40	’s	’s	PART
cana-512	103	41	incident	incident	NOUN
cana-512	103	42	on	on	ADP
cana-512	103	43	𝑢1	𝑢1	PROPN
cana-512	103	44	,	,	PUNCT
cana-512	103	45	𝑢2	𝑢2	PROPN
cana-512	103	46	and	and	CCONJ
cana-512	103	47	𝑢3	𝑢3	PROPN
cana-512	103	48	.	.	PUNCT
cana-512	104	1	thus	thus	ADV
cana-512	104	2	,	,	PUNCT
cana-512	104	3	it	it	PRON
cana-512	104	4	is	be	AUX
cana-512	104	5	needed	need	VERB
cana-512	104	6	to	to	PART
cana-512	104	7	remove	remove	VERB
cana-512	104	8	3𝑡	3𝑡	NOUN
cana-512	104	9	−	−	PROPN
cana-512	104	10	7	7	NUM
cana-512	104	11	edges	edge	NOUN
cana-512	104	12	from	from	ADP
cana-512	104	13	g	g	NOUN
cana-512	104	14	to	to	PART
cana-512	104	15	make	make	VERB
cana-512	104	16	it	it	PRON
cana-512	104	17	as	as	ADP
cana-512	104	18	a	a	DET
cana-512	104	19	relatively	relatively	ADV
cana-512	104	20	prime	prime	ADJ
cana-512	104	21	edge	edge	NOUN
cana-512	104	22	labeled	label	VERB
cana-512	104	23	graph	graph	NOUN
cana-512	104	24	.	.	PUNCT
cana-512	105	1	hence	hence	ADV
cana-512	105	2	,	,	PUNCT
cana-512	105	3	εr(g	εr(g	PUNCT
cana-512	105	4	)	)	PUNCT
cana-512	106	1	=	=	SYM
cana-512	106	2	3𝑡	3𝑡	NUM
cana-512	106	3	−	−	PROPN
cana-512	106	4	7	7	NUM
cana-512	106	5	.	.	X
cana-512	106	6	4.3	4.3	NUM
cana-512	106	7	.	.	PUNCT
cana-512	107	1	generalized	generalized	ADJ
cana-512	107	2	petersen	petersen	PROPN
cana-512	107	3	graph	graph	NOUN
cana-512	107	4	the	the	DET
cana-512	107	5	next	next	ADJ
cana-512	107	6	theorem	theorem	NOUN
cana-512	107	7	finds	find	VERB
cana-512	107	8	the	the	DET
cana-512	107	9	edge	edge	NOUN
cana-512	107	10	prime	prime	ADJ
cana-512	107	11	index	index	NOUN
cana-512	107	12	of	of	ADP
cana-512	107	13	the	the	DET
cana-512	107	14	generalized	generalized	ADJ
cana-512	107	15	petersen	petersen	NOUN
cana-512	107	16	graph	graph	NOUN
cana-512	107	17	g	g	PROPN
cana-512	107	18	=	=	PUNCT
cana-512	107	19	p(n	p(n	PROPN
cana-512	107	20	,	,	PUNCT
cana-512	107	21	k	k	NOUN
cana-512	107	22	)	)	PUNCT
cana-512	107	23	.	.	PUNCT
cana-512	108	1	the	the	DET
cana-512	108	2	generalized	generalized	ADJ
cana-512	108	3	petersen	petersen	NOUN
cana-512	108	4	graphs	graph	NOUN
cana-512	108	5	p(n	p(n	PROPN
cana-512	108	6	,	,	PUNCT
cana-512	108	7	k	k	NOUN
cana-512	108	8	)	)	PUNCT
cana-512	108	9	with	with	ADP
cana-512	108	10	n	n	PRON
cana-512	108	11	≥	≥	NOUN
cana-512	108	12	3	3	NUM
cana-512	108	13	and	and	CCONJ
cana-512	108	14	1	1	NUM
cana-512	108	15	≤	≤	NUM
cana-512	108	16	k	k	NOUN
cana-512	108	17	≤	≤	NOUN
cana-512	108	18	n	n	DET
cana-512	108	19	2	2	NUM
cana-512	108	20	are	be	AUX
cana-512	108	21	defined	define	VERB
cana-512	108	22	to	to	PART
cana-512	108	23	be	be	AUX
cana-512	108	24	a	a	DET
cana-512	108	25	graph	graph	NOUN
cana-512	108	26	with	with	ADP
cana-512	108	27	v(p(n	v(p(n	PROPN
cana-512	108	28	,	,	PUNCT
cana-512	108	29	k	k	NOUN
cana-512	108	30	)	)	PUNCT
cana-512	108	31	)	)	PUNCT
cana-512	109	1	=	=	PRON
cana-512	109	2	{	{	PUNCT
cana-512	109	3	uj	uj	PROPN
cana-512	109	4	,	,	PUNCT
cana-512	109	5	vj	vj	PROPN
cana-512	109	6	∶	∶	NOUN
cana-512	109	7	1	1	NUM
cana-512	109	8	≤	≤	NUM
cana-512	109	9	j	j	PROPN
cana-512	109	10	≤	≤	PROPN
cana-512	109	11	n	n	CCONJ
cana-512	109	12	}	}	PUNCT
cana-512	109	13	and	and	CCONJ
cana-512	109	14	e(p(n	e(p(n	PROPN
cana-512	109	15	,	,	PUNCT
cana-512	109	16	k	k	NOUN
cana-512	109	17	)	)	PUNCT
cana-512	109	18	)	)	PUNCT
cana-512	110	1	=	=	PRON
cana-512	110	2	{	{	PUNCT
cana-512	110	3	v1vj+1	v1vj+1	X
cana-512	110	4	,	,	PUNCT
cana-512	110	5	vjuj	vjuj	VERB
cana-512	110	6	,	,	PUNCT
cana-512	110	7	ujuj+k	ujuj+k	VERB
cana-512	110	8	∶	∶	NOUN
cana-512	110	9	1	1	NUM
cana-512	110	10	≤	≤	NUM
cana-512	110	11	j	j	PROPN
cana-512	110	12	≤	≤	PROPN
cana-512	110	13	n	n	CCONJ
cana-512	110	14	,	,	PUNCT
cana-512	110	15	subscript	subscript	NOUN
cana-512	110	16	mod	mod	PROPN
cana-512	110	17	n	n	CCONJ
cana-512	110	18	}	}	PUNCT
cana-512	110	19	.	.	PUNCT
cana-512	111	1	(	(	PUNCT
cana-512	111	2	5,6	5,6	NUM
cana-512	111	3	)	)	PUNCT
cana-512	111	4	theorem	theorem	VERB
cana-512	111	5	4.4	4.4	NUM
cana-512	111	6	.	.	PUNCT
cana-512	112	1	for	for	ADP
cana-512	112	2	even	even	ADV
cana-512	112	3	n	n	CCONJ
cana-512	112	4	,	,	PUNCT
cana-512	112	5	the	the	DET
cana-512	112	6	generalized	generalized	ADJ
cana-512	112	7	petersen	petersen	NOUN
cana-512	112	8	graph	graph	NOUN
cana-512	112	9	g	g	PROPN
cana-512	112	10	=	=	PUNCT
cana-512	112	11	p(n	p(n	PROPN
cana-512	112	12	,	,	PUNCT
cana-512	112	13	2	2	NUM
cana-512	112	14	)	)	PUNCT
cana-512	112	15	then	then	ADV
cana-512	112	16	,	,	PUNCT
cana-512	112	17	εr(g	εr(g	PUNCT
cana-512	112	18	)	)	PUNCT
cana-512	112	19	=	=	SYM
cana-512	112	20	𝑛	𝑛	PRON
cana-512	112	21	−	−	NUM
cana-512	112	22	1	1	X
cana-512	112	23	.	.	PUNCT
cana-512	112	24	proof	proof	NOUN
cana-512	112	25	the	the	DET
cana-512	112	26	vertices	vertex	NOUN
cana-512	112	27	of	of	ADP
cana-512	112	28	p(n	p(n	PROPN
cana-512	112	29	,	,	PUNCT
cana-512	112	30	2	2	NUM
cana-512	112	31	)	)	PUNCT
cana-512	112	32	are	be	AUX
cana-512	112	33	{	{	PUNCT
cana-512	112	34	𝑣1	𝑣1	PROPN
cana-512	112	35	,	,	PUNCT
cana-512	112	36	𝑣2	𝑣2	PROPN
cana-512	112	37	…	…	PUNCT
cana-512	112	38	.	.	PUNCT
cana-512	113	1	𝑣𝑛	𝑣𝑛	PROPN
cana-512	113	2	,	,	PUNCT
cana-512	113	3	𝑢1	𝑢1	NOUN
cana-512	113	4	,	,	PUNCT
cana-512	113	5	𝑢2	𝑢2	PROPN
cana-512	113	6	,	,	PUNCT
cana-512	113	7	…	…	PUNCT
cana-512	113	8	,	,	PUNCT
cana-512	113	9	𝑢𝑛	𝑢𝑛	NOUN
cana-512	113	10	}	}	PUNCT
cana-512	113	11	,	,	PUNCT
cana-512	113	12	where	where	SCONJ
cana-512	113	13	𝑣1	𝑣1	NOUN
cana-512	113	14	,	,	PUNCT
cana-512	113	15	𝑣2	𝑣2	PROPN
cana-512	113	16	…	…	PUNCT
cana-512	113	17	.	.	PUNCT
cana-512	113	18	𝑣𝑛	𝑣𝑛	PROPN
cana-512	113	19	represents	represent	VERB
cana-512	113	20	the	the	DET
cana-512	113	21	outer	outer	ADJ
cana-512	113	22	vertices	vertex	NOUN
cana-512	113	23	and	and	CCONJ
cana-512	113	24	𝑢1	𝑢1	NOUN
cana-512	113	25	,	,	PUNCT
cana-512	113	26	𝑢2	𝑢2	PROPN
cana-512	113	27	,	,	PUNCT
cana-512	113	28	…	…	PUNCT
cana-512	113	29	,	,	PUNCT
cana-512	113	30	𝑢𝑛	𝑢𝑛	NOUN
cana-512	113	31	represents	represent	VERB
cana-512	113	32	the	the	DET
cana-512	113	33	inner	inner	ADJ
cana-512	113	34	vertices	vertex	NOUN
cana-512	113	35	and	and	CCONJ
cana-512	113	36	the	the	DET
cana-512	113	37	edges	edge	NOUN
cana-512	113	38	of	of	ADP
cana-512	113	39	p(n	p(n	PROPN
cana-512	113	40	,	,	PUNCT
cana-512	113	41	2	2	NUM
cana-512	113	42	)	)	PUNCT
cana-512	113	43	are	be	AUX
cana-512	113	44	{	{	PUNCT
cana-512	113	45	𝑣1𝑣𝑖+1	𝑣1𝑣𝑖+1	PROPN
cana-512	113	46	,	,	PUNCT
cana-512	113	47	𝑣𝑖	𝑣𝑖	ADP
cana-512	113	48	𝑢𝑖	𝑢𝑖	NOUN
cana-512	113	49	,	,	PUNCT
cana-512	113	50	𝑢𝑖𝑢𝑖+2	𝑢𝑖𝑢𝑖+2	NOUN
cana-512	113	51	:	:	PUNCT
cana-512	113	52	1	1	NUM
cana-512	113	53	≤	≤	NUM
cana-512	113	54	𝑖	𝑖	SYM
cana-512	113	55	≤	≤	NUM
cana-512	113	56	𝑛	𝑛	PRON
cana-512	113	57	,	,	PUNCT
cana-512	113	58	subscript	subscript	NOUN
cana-512	113	59	mod	mod	NOUN
cana-512	113	60	𝑛	𝑛	X
cana-512	113	61	}	}	PUNCT
cana-512	113	62	.	.	PUNCT
cana-512	114	1	thus	thus	ADV
cana-512	114	2	,	,	PUNCT
cana-512	114	3	there	there	PRON
cana-512	114	4	are	be	VERB
cana-512	114	5	2n	2n	NUM
cana-512	114	6	vertices	vertex	NOUN
cana-512	114	7	and	and	CCONJ
cana-512	114	8	3n	3n	NUM
cana-512	114	9	edges	edge	NOUN
cana-512	114	10	in	in	ADP
cana-512	114	11	p(n	p(n	NOUN
cana-512	114	12	,	,	PUNCT
cana-512	114	13	2	2	NUM
cana-512	114	14	)	)	PUNCT
cana-512	114	15	.	.	PUNCT
cana-512	115	1	as	as	ADP
cana-512	115	2	the	the	DET
cana-512	115	3	gcd(n	gcd(n	NOUN
cana-512	115	4	,	,	PUNCT
cana-512	115	5	2	2	NUM
cana-512	115	6	)	)	PUNCT
cana-512	115	7	=	=	SYM
cana-512	115	8	2	2	NUM
cana-512	115	9	,	,	PUNCT
cana-512	115	10	for	for	ADP
cana-512	115	11	even	even	ADV
cana-512	115	12	n	n	CCONJ
cana-512	115	13	,	,	PUNCT
cana-512	115	14	there	there	PRON
cana-512	115	15	exists	exist	VERB
cana-512	115	16	2	2	NUM
cana-512	115	17	disjoint	disjoint	NOUN
cana-512	115	18	inner	inner	ADJ
cana-512	115	19	cycles	cycle	NOUN
cana-512	115	20	of	of	ADP
cana-512	115	21	length	length	NOUN
cana-512	115	22	𝑛	𝑛	ADP
cana-512	115	23	2	2	NUM
cana-512	115	24	each	each	PRON
cana-512	115	25	and	and	CCONJ
cana-512	115	26	there	there	PRON
cana-512	115	27	exists	exist	VERB
cana-512	115	28	an	an	DET
cana-512	115	29	outer	outer	ADJ
cana-512	115	30	cycle	cycle	NOUN
cana-512	115	31	{	{	PUNCT
cana-512	115	32	𝑣1	𝑣1	NOUN
cana-512	115	33	𝑣2	𝑣2	PROPN
cana-512	115	34	…	…	PUNCT
cana-512	115	35	.	.	PUNCT
cana-512	116	1	𝑣𝑛𝑣1	𝑣𝑛𝑣1	PROPN
cana-512	116	2	}	}	PUNCT
cana-512	116	3	of	of	ADP
cana-512	116	4	length	length	NOUN
cana-512	116	5	n.	n.	PROPN
cana-512	116	6	now	now	ADV
cana-512	116	7	,	,	PUNCT
cana-512	116	8	label	label	VERB
cana-512	116	9	the	the	DET
cana-512	116	10	edges	edge	NOUN
cana-512	116	11	of	of	ADP
cana-512	116	12	p(n	p(n	PROPN
cana-512	116	13	,	,	PUNCT
cana-512	116	14	2	2	NUM
cana-512	116	15	)	)	PUNCT
cana-512	116	16	in	in	ADP
cana-512	116	17	such	such	DET
cana-512	116	18	a	a	DET
cana-512	116	19	way	way	NOUN
cana-512	116	20	that	that	PRON
cana-512	116	21	,	,	PUNCT
cana-512	116	22	the	the	DET
cana-512	116	23	two	two	NUM
cana-512	116	24	inner	inner	ADJ
cana-512	116	25	cycles	cycle	NOUN
cana-512	116	26	receive	receive	VERB
cana-512	116	27	the	the	DET
cana-512	116	28	label	label	NOUN
cana-512	116	29	{	{	PUNCT
cana-512	116	30	1	1	NUM
cana-512	116	31	,	,	PUNCT
cana-512	116	32	2	2	NUM
cana-512	116	33	,	,	PUNCT
cana-512	116	34	3	3	NUM
cana-512	116	35	,	,	PUNCT
cana-512	116	36	…	…	PUNCT
cana-512	116	37	.	.	PUNCT
cana-512	117	1	,	,	PUNCT
cana-512	117	2	𝑛	𝑛	PRON
cana-512	117	3	2	2	NUM
cana-512	117	4	}	}	PUNCT
cana-512	117	5	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-512	117	6	{	{	PUNCT
cana-512	117	7	𝑛+2	𝑛+2	NOUN
cana-512	117	8	2	2	NUM
cana-512	117	9	,	,	PUNCT
cana-512	117	10	…	…	PUNCT
cana-512	117	11	.	.	PUNCT
cana-512	117	12	,	,	PUNCT
cana-512	117	13	𝑛	𝑛	X
cana-512	117	14	}	}	PUNCT
cana-512	117	15	and	and	CCONJ
cana-512	117	16	the	the	DET
cana-512	117	17	outer	outer	ADJ
cana-512	117	18	cycle	cycle	NOUN
cana-512	117	19	with	with	ADP
cana-512	117	20	{	{	PUNCT
cana-512	117	21	𝑛	𝑛	PROPN
cana-512	117	22	+	+	NUM
cana-512	117	23	1	1	NUM
cana-512	117	24	,	,	PUNCT
cana-512	117	25	𝑛	𝑛	PRON
cana-512	117	26	+	+	NOUN
cana-512	117	27	2	2	NUM
cana-512	117	28	,	,	PUNCT
cana-512	117	29	…	…	PUNCT
cana-512	117	30	.	.	PUNCT
cana-512	117	31	.	.	PUNCT
cana-512	118	1	2𝑛	2𝑛	NUM
cana-512	118	2	}	}	PUNCT
cana-512	118	3	.	.	PUNCT
cana-512	119	1	as	as	ADP
cana-512	119	2	2𝑛	2𝑛	PROPN
cana-512	119	3	+	+	CCONJ
cana-512	119	4	1	1	NUM
cana-512	119	5	is	be	AUX
cana-512	119	6	odd	odd	ADJ
cana-512	119	7	,	,	PUNCT
cana-512	119	8	label	label	VERB
cana-512	119	9	any	any	PRON
cana-512	119	10	of	of	ADP
cana-512	119	11	the	the	DET
cana-512	119	12	edge	edge	NOUN
cana-512	119	13	{	{	PUNCT
cana-512	119	14	𝑣𝑖	𝑣𝑖	NOUN
cana-512	119	15	𝑢𝑖	𝑢𝑖	NOUN
cana-512	119	16	}	}	PUNCT
cana-512	119	17	as	as	ADP
cana-512	119	18	2𝑛	2𝑛	PROPN
cana-512	119	19	+	+	CCONJ
cana-512	119	20	1	1	X
cana-512	119	21	.	.	PUNCT
cana-512	119	22	also	also	ADV
cana-512	119	23	,	,	PUNCT
cana-512	119	24	2𝑛	2𝑛	PROPN
cana-512	119	25	+	+	CCONJ
cana-512	119	26	2	2	NUM
cana-512	119	27	can	can	AUX
cana-512	119	28	not	not	PART
cana-512	119	29	be	be	AUX
cana-512	119	30	labeled	label	VERB
cana-512	119	31	on	on	ADP
cana-512	119	32	any	any	DET
cana-512	119	33	edge	edge	NOUN
cana-512	119	34	because	because	SCONJ
cana-512	119	35	2𝑛	2𝑛	PROPN
cana-512	119	36	+	+	CCONJ
cana-512	119	37	2	2	NUM
cana-512	119	38	is	be	AUX
cana-512	119	39	an	an	DET
cana-512	119	40	even	even	ADJ
cana-512	119	41	number	number	NOUN
cana-512	119	42	that	that	PRON
cana-512	119	43	violates	violate	VERB
cana-512	119	44	the	the	DET
cana-512	119	45	relative	relative	ADJ
cana-512	119	46	prime	prime	ADJ
cana-512	119	47	property	property	NOUN
cana-512	119	48	.	.	PUNCT
cana-512	120	1	thus	thus	ADV
cana-512	120	2	,	,	PUNCT
cana-512	120	3	the	the	DET
cana-512	120	4	relatively	relatively	ADV
cana-512	120	5	prime	prime	ADJ
cana-512	120	6	index	index	NOUN
cana-512	120	7	of	of	ADP
cana-512	120	8	p(n	p(n	PROPN
cana-512	120	9	,	,	PUNCT
cana-512	120	10	2	2	NUM
cana-512	120	11	)	)	PUNCT
cana-512	120	12	for	for	ADP
cana-512	120	13	even	even	ADV
cana-512	120	14	n	n	CCONJ
cana-512	120	15	,	,	PUNCT
cana-512	120	16	is	be	AUX
cana-512	120	17	εr(p(n	εr(p(n	NUM
cana-512	120	18	,	,	PUNCT
cana-512	120	19	2	2	NUM
cana-512	120	20	)	)	PUNCT
cana-512	120	21	)	)	PUNCT
cana-512	121	1	=	=	SYM
cana-512	121	2	3𝑛	3𝑛	NUM
cana-512	121	3	−	−	PROPN
cana-512	121	4	2𝑛	2𝑛	NOUN
cana-512	121	5	−	−	PROPN
cana-512	121	6	1	1	NUM
cana-512	121	7	=	=	SYM
cana-512	121	8	𝑛	𝑛	PRON
cana-512	121	9	−	−	NOUN
cana-512	121	10	1	1	NUM
cana-512	121	11	.	.	PUNCT
cana-512	122	1	hence	hence	ADV
cana-512	122	2	the	the	DET
cana-512	122	3	proof	proof	NOUN
cana-512	122	4	.	.	PUNCT
cana-512	123	1	illustration	illustration	NOUN
cana-512	123	2	:	:	PUNCT
cana-512	123	3	figure	figure	NOUN
cana-512	123	4	7	7	NUM
cana-512	123	5	shows	show	VERB
cana-512	123	6	the	the	DET
cana-512	123	7	illustration	illustration	NOUN
cana-512	123	8	of	of	ADP
cana-512	123	9	the	the	DET
cana-512	123	10	above	above	ADJ
cana-512	123	11	theorem	theorem	PROPN
cana-512	123	12	.	.	PUNCT
cana-512	123	13	consider	consider	VERB
cana-512	123	14	a	a	DET
cana-512	123	15	generalized	generalized	ADJ
cana-512	123	16	petersen	petersen	NOUN
cana-512	123	17	graph	graph	NOUN
cana-512	123	18	with	with	ADP
cana-512	123	19	16	16	NUM
cana-512	123	20	vertices	vertex	NOUN
cana-512	123	21	,	,	PUNCT
cana-512	123	22	p(8,2	p(8,2	NOUN
cana-512	123	23	)	)	PUNCT
cana-512	123	24	.	.	PUNCT
cana-512	124	1	then	then	ADV
cana-512	124	2	the	the	DET
cana-512	124	3	edge	edge	NOUN
cana-512	124	4	prime	prime	ADJ
cana-512	124	5	index	index	NOUN
cana-512	124	6	is	be	AUX
cana-512	124	7	given	give	VERB
cana-512	124	8	by	by	ADP
cana-512	124	9	,	,	PUNCT
cana-512	124	10	εr(p(8	εr(p(8	NUM
cana-512	124	11	,	,	PUNCT
cana-512	124	12	2	2	NUM
cana-512	124	13	)	)	PUNCT
cana-512	124	14	)	)	PUNCT
cana-512	125	1	=	=	SYM
cana-512	126	1	𝑛	𝑛	DET
cana-512	126	2	−	−	NUM
cana-512	126	3	1	1	NUM
cana-512	126	4	=	=	SYM
cana-512	126	5	8	8	NUM
cana-512	126	6	−	−	NOUN
cana-512	126	7	1	1	NUM
cana-512	126	8	=	=	SYM
cana-512	126	9	7	7	NUM
cana-512	126	10	communications	communication	NOUN
cana-512	126	11	on	on	ADP
cana-512	126	12	applied	apply	VERB
cana-512	126	13	nonlinear	nonlinear	ADJ
cana-512	126	14	analysis	analysis	NOUN
cana-512	126	15	issn	issn	NOUN
cana-512	126	16	:	:	PUNCT
cana-512	126	17	1074	1074	NUM
cana-512	126	18	-	-	PUNCT
cana-512	126	19	133x	133x	NUM
cana-512	126	20	vol	vol	NOUN
cana-512	126	21	31	31	NUM
cana-512	126	22	no	no	NOUN
cana-512	126	23	.	.	NOUN
cana-512	126	24	2	2	NUM
cana-512	126	25	(	(	PUNCT
cana-512	126	26	2024	2024	NUM
cana-512	126	27	)	)	PUNCT
cana-512	126	28	50	50	NUM
cana-512	126	29	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	126	30	figure	figure	NOUN
cana-512	126	31	7	7	NUM
cana-512	126	32	:	:	SYM
cana-512	126	33	εr(p(8	εr(p(8	NUM
cana-512	126	34	,	,	PUNCT
cana-512	126	35	2	2	NUM
cana-512	126	36	)	)	PUNCT
cana-512	126	37	)	)	PUNCT
cana-512	127	1	=	=	SYM
cana-512	127	2	7	7	NUM
cana-512	127	3	theorem	theorem	VERB
cana-512	127	4	4.5	4.5	NUM
cana-512	127	5	.	.	PUNCT
cana-512	128	1	for	for	ADP
cana-512	128	2	even	even	ADV
cana-512	128	3	n	n	CCONJ
cana-512	128	4	,	,	PUNCT
cana-512	128	5	k	k	PROPN
cana-512	128	6	,	,	PUNCT
cana-512	128	7	the	the	DET
cana-512	128	8	generalized	generalized	ADJ
cana-512	128	9	petersen	petersen	NOUN
cana-512	128	10	graph	graph	NOUN
cana-512	128	11	g	g	PROPN
cana-512	128	12	=	=	PUNCT
cana-512	128	13	p(n	p(n	PROPN
cana-512	128	14	,	,	PUNCT
cana-512	128	15	k	k	NOUN
cana-512	128	16	)	)	PUNCT
cana-512	128	17	then	then	ADV
cana-512	128	18	,	,	PUNCT
cana-512	128	19	εr(g	εr(g	PUNCT
cana-512	128	20	)	)	PUNCT
cana-512	128	21	=	=	SYM
cana-512	128	22	𝑛	𝑛	PRON
cana-512	128	23	−	−	NUM
cana-512	128	24	1	1	NUM
cana-512	128	25	.	.	PUNCT
cana-512	129	1	proof	proof	NOUN
cana-512	129	2	:	:	PUNCT
cana-512	129	3	the	the	DET
cana-512	129	4	vertices	vertex	NOUN
cana-512	129	5	of	of	ADP
cana-512	129	6	p(n	p(n	PROPN
cana-512	129	7	,	,	PUNCT
cana-512	129	8	k	k	NOUN
cana-512	129	9	)	)	PUNCT
cana-512	129	10	are	be	AUX
cana-512	129	11	{	{	PUNCT
cana-512	129	12	𝑣1	𝑣1	PROPN
cana-512	129	13	,	,	PUNCT
cana-512	129	14	𝑣2	𝑣2	PROPN
cana-512	129	15	…	…	PUNCT
cana-512	129	16	.	.	PUNCT
cana-512	130	1	𝑣𝑛	𝑣𝑛	PROPN
cana-512	130	2	,	,	PUNCT
cana-512	130	3	𝑢1	𝑢1	NOUN
cana-512	130	4	,	,	PUNCT
cana-512	130	5	𝑢2	𝑢2	PROPN
cana-512	130	6	,	,	PUNCT
cana-512	130	7	…	…	PUNCT
cana-512	130	8	,	,	PUNCT
cana-512	130	9	𝑢𝑛	𝑢𝑛	NOUN
cana-512	130	10	}	}	PUNCT
cana-512	130	11	,	,	PUNCT
cana-512	130	12	where	where	SCONJ
cana-512	130	13	𝑣1	𝑣1	NOUN
cana-512	130	14	,	,	PUNCT
cana-512	130	15	𝑣2	𝑣2	PROPN
cana-512	130	16	…	…	PUNCT
cana-512	130	17	.	.	PUNCT
cana-512	130	18	𝑣𝑛	𝑣𝑛	PROPN
cana-512	130	19	represents	represent	VERB
cana-512	130	20	the	the	DET
cana-512	130	21	outer	outer	ADJ
cana-512	130	22	vertices	vertex	NOUN
cana-512	130	23	and	and	CCONJ
cana-512	130	24	𝑢1	𝑢1	NOUN
cana-512	130	25	,	,	PUNCT
cana-512	130	26	𝑢2	𝑢2	PROPN
cana-512	130	27	,	,	PUNCT
cana-512	130	28	…	…	PUNCT
cana-512	130	29	,	,	PUNCT
cana-512	130	30	𝑢𝑛	𝑢𝑛	NOUN
cana-512	130	31	represents	represent	VERB
cana-512	130	32	the	the	DET
cana-512	130	33	inner	inner	ADJ
cana-512	130	34	vertices	vertex	NOUN
cana-512	130	35	and	and	CCONJ
cana-512	130	36	the	the	DET
cana-512	130	37	edges	edge	NOUN
cana-512	130	38	of	of	ADP
cana-512	130	39	p(n	p(n	PROPN
cana-512	130	40	,	,	PUNCT
cana-512	130	41	k	k	NOUN
cana-512	130	42	)	)	PUNCT
cana-512	130	43	are	be	AUX
cana-512	130	44	{	{	PUNCT
cana-512	130	45	𝑣1𝑣𝑖+1	𝑣1𝑣𝑖+1	PROPN
cana-512	130	46	,	,	PUNCT
cana-512	130	47	𝑣𝑖	𝑣𝑖	ADP
cana-512	130	48	𝑢𝑖	𝑢𝑖	NOUN
cana-512	130	49	,	,	PUNCT
cana-512	130	50	𝑢𝑖𝑢𝑖+𝑘	𝑢𝑖𝑢𝑖+𝑘	INTJ
cana-512	130	51	:	:	PUNCT
cana-512	130	52	1	1	NUM
cana-512	130	53	≤	≤	NUM
cana-512	130	54	𝑖	𝑖	SYM
cana-512	130	55	≤	≤	NUM
cana-512	130	56	𝑛	𝑛	PRON
cana-512	130	57	,	,	PUNCT
cana-512	130	58	subscript	subscript	NOUN
cana-512	130	59	mod	mod	NOUN
cana-512	130	60	𝑛	𝑛	PROPN
cana-512	130	61	}	}	PUNCT
cana-512	130	62	.	.	PUNCT
cana-512	131	1	we	we	PRON
cana-512	131	2	know	know	VERB
cana-512	131	3	that	that	SCONJ
cana-512	131	4	,	,	PUNCT
cana-512	131	5	there	there	PRON
cana-512	131	6	exist	exist	VERB
cana-512	131	7	gcd(n	gcd(n	NOUN
cana-512	131	8	,	,	PUNCT
cana-512	131	9	k	k	NOUN
cana-512	131	10	)	)	PUNCT
cana-512	131	11	=	=	SYM
cana-512	132	1	d	d	ADP
cana-512	132	2	inner	inner	ADJ
cana-512	132	3	cycles	cycle	NOUN
cana-512	132	4	of	of	ADP
cana-512	132	5	length	length	NOUN
cana-512	132	6	𝑛	𝑛	ADP
cana-512	132	7	𝑘	𝑘	NOUN
cana-512	132	8	each	each	PRON
cana-512	132	9	.	.	PUNCT
cana-512	133	1	label	label	VERB
cana-512	133	2	the	the	DET
cana-512	133	3	edges	edge	NOUN
cana-512	133	4	of	of	ADP
cana-512	133	5	p(n	p(n	PROPN
cana-512	133	6	,	,	PUNCT
cana-512	133	7	k	k	NOUN
cana-512	133	8	)	)	PUNCT
cana-512	133	9	in	in	ADP
cana-512	133	10	such	such	DET
cana-512	133	11	a	a	DET
cana-512	133	12	way	way	NOUN
cana-512	133	13	that	that	PRON
cana-512	133	14	,	,	PUNCT
cana-512	133	15	the	the	DET
cana-512	133	16	d	d	PROPN
cana-512	133	17	inner	inner	ADJ
cana-512	133	18	cycles	cycle	NOUN
cana-512	133	19	receive	receive	VERB
cana-512	133	20	the	the	DET
cana-512	133	21	label	label	NOUN
cana-512	133	22	{	{	PUNCT
cana-512	133	23	1	1	NUM
cana-512	133	24	,	,	PUNCT
cana-512	133	25	2	2	NUM
cana-512	133	26	,	,	PUNCT
cana-512	133	27	3	3	NUM
cana-512	133	28	,	,	PUNCT
cana-512	133	29	…	…	PUNCT
cana-512	133	30	.	.	PUNCT
cana-512	134	1	,	,	PUNCT
cana-512	134	2	𝑛	𝑛	X
cana-512	134	3	}	}	PUNCT
cana-512	134	4	consecutively	consecutively	ADV
cana-512	134	5	and	and	CCONJ
cana-512	134	6	the	the	DET
cana-512	134	7	outer	outer	ADJ
cana-512	134	8	cycle	cycle	NOUN
cana-512	134	9	with	with	ADP
cana-512	134	10	{	{	PUNCT
cana-512	134	11	𝑛	𝑛	PROPN
cana-512	134	12	+	+	NUM
cana-512	134	13	1	1	NUM
cana-512	134	14	,	,	PUNCT
cana-512	134	15	𝑛	𝑛	PRON
cana-512	134	16	+	+	NOUN
cana-512	134	17	2	2	NUM
cana-512	134	18	,	,	PUNCT
cana-512	134	19	…	…	PUNCT
cana-512	134	20	.	.	PUNCT
cana-512	134	21	.	.	PUNCT
cana-512	135	1	2𝑛	2𝑛	NUM
cana-512	135	2	}	}	PUNCT
cana-512	135	3	.	.	PUNCT
cana-512	136	1	as	as	ADP
cana-512	136	2	2𝑛	2𝑛	PROPN
cana-512	136	3	+	+	CCONJ
cana-512	136	4	1	1	NUM
cana-512	136	5	is	be	AUX
cana-512	136	6	odd	odd	ADJ
cana-512	136	7	,	,	PUNCT
cana-512	136	8	label	label	VERB
cana-512	136	9	any	any	PRON
cana-512	136	10	of	of	ADP
cana-512	136	11	the	the	DET
cana-512	136	12	edge	edge	NOUN
cana-512	136	13	{	{	PUNCT
cana-512	136	14	𝑣𝑖	𝑣𝑖	NOUN
cana-512	136	15	𝑢𝑖	𝑢𝑖	NOUN
cana-512	136	16	}	}	PUNCT
cana-512	136	17	as	as	ADP
cana-512	136	18	2𝑛	2𝑛	PROPN
cana-512	136	19	+	+	CCONJ
cana-512	136	20	1	1	X
cana-512	136	21	.	.	PUNCT
cana-512	136	22	also	also	ADV
cana-512	136	23	,	,	PUNCT
cana-512	136	24	2𝑛	2𝑛	PROPN
cana-512	136	25	+	+	CCONJ
cana-512	136	26	2	2	NUM
cana-512	136	27	can	can	AUX
cana-512	136	28	not	not	PART
cana-512	136	29	be	be	AUX
cana-512	136	30	labeled	label	VERB
cana-512	136	31	on	on	ADP
cana-512	136	32	any	any	DET
cana-512	136	33	edge	edge	NOUN
cana-512	136	34	because	because	SCONJ
cana-512	136	35	2𝑛	2𝑛	PROPN
cana-512	136	36	+	+	CCONJ
cana-512	136	37	2	2	NUM
cana-512	136	38	is	be	AUX
cana-512	136	39	an	an	DET
cana-512	136	40	even	even	ADJ
cana-512	136	41	number	number	NOUN
cana-512	136	42	that	that	PRON
cana-512	136	43	violates	violate	VERB
cana-512	136	44	the	the	DET
cana-512	136	45	relative	relative	ADJ
cana-512	136	46	prime	prime	ADJ
cana-512	136	47	property	property	NOUN
cana-512	136	48	.	.	PUNCT
cana-512	137	1	thus	thus	ADV
cana-512	137	2	,	,	PUNCT
cana-512	137	3	the	the	DET
cana-512	137	4	edge	edge	NOUN
cana-512	137	5	prime	prime	ADJ
cana-512	137	6	index	index	NOUN
cana-512	137	7	of	of	ADP
cana-512	137	8	p(n	p(n	PROPN
cana-512	137	9	,	,	PUNCT
cana-512	137	10	k	k	NOUN
cana-512	137	11	)	)	PUNCT
cana-512	137	12	for	for	ADP
cana-512	137	13	even	even	ADV
cana-512	137	14	n	n	CCONJ
cana-512	137	15	,	,	PUNCT
cana-512	137	16	k	k	X
cana-512	137	17	,	,	PUNCT
cana-512	137	18	is	be	AUX
cana-512	137	19	εr(p(n	εr(p(n	PROPN
cana-512	137	20	,	,	PUNCT
cana-512	137	21	k	k	NOUN
cana-512	137	22	)	)	PUNCT
cana-512	137	23	)	)	PUNCT
cana-512	138	1	=	=	SYM
cana-512	138	2	3𝑛	3𝑛	NUM
cana-512	138	3	−	−	PROPN
cana-512	138	4	2𝑛	2𝑛	NOUN
cana-512	138	5	−	−	PROPN
cana-512	138	6	1	1	NUM
cana-512	138	7	=	=	SYM
cana-512	138	8	𝑛	𝑛	PRON
cana-512	138	9	−	−	NOUN
cana-512	138	10	1	1	NUM
cana-512	138	11	.	.	PUNCT
cana-512	139	1	hence	hence	ADV
cana-512	139	2	the	the	DET
cana-512	139	3	proof	proof	NOUN
cana-512	139	4	.	.	PUNCT
cana-512	140	1	4.4	4.4	NUM
cana-512	140	2	.	.	PUNCT
cana-512	141	1	ladder	ladder	NOUN
cana-512	141	2	graph	graph	VERB
cana-512	141	3	the	the	DET
cana-512	141	4	ladder	ladder	NOUN
cana-512	141	5	graph	graph	NOUN
cana-512	141	6	l(n	l(n	PROPN
cana-512	141	7	)	)	PUNCT
cana-512	141	8	consists	consist	VERB
cana-512	141	9	of	of	ADP
cana-512	141	10	two	two	NUM
cana-512	141	11	parallel	parallel	ADJ
cana-512	141	12	paths	path	NOUN
cana-512	141	13	,	,	PUNCT
cana-512	141	14	each	each	PRON
cana-512	141	15	with	with	ADP
cana-512	141	16	n	n	ADP
cana-512	141	17	vertices	vertex	NOUN
cana-512	141	18	,	,	PUNCT
cana-512	141	19	and	and	CCONJ
cana-512	141	20	these	these	DET
cana-512	141	21	paths	path	NOUN
cana-512	141	22	are	be	AUX
cana-512	141	23	connected	connect	VERB
cana-512	141	24	by	by	ADP
cana-512	141	25	n-1	n-1	ADJ
cana-512	141	26	edges	edge	NOUN
cana-512	141	27	known	know	VERB
cana-512	141	28	as	as	ADP
cana-512	141	29	"	"	PUNCT
cana-512	141	30	rails	rail	NOUN
cana-512	141	31	.	.	PUNCT
cana-512	141	32	"	"	PUNCT
cana-512	142	1	the	the	DET
cana-512	142	2	first	first	ADJ
cana-512	142	3	and	and	CCONJ
cana-512	142	4	last	last	ADJ
cana-512	142	5	vertices	vertex	NOUN
cana-512	142	6	of	of	ADP
cana-512	142	7	each	each	DET
cana-512	142	8	path	path	NOUN
cana-512	142	9	are	be	AUX
cana-512	142	10	also	also	ADV
cana-512	142	11	connected	connect	VERB
cana-512	142	12	by	by	ADP
cana-512	142	13	an	an	DET
cana-512	142	14	additional	additional	ADJ
cana-512	142	15	edge	edge	NOUN
cana-512	142	16	called	call	VERB
cana-512	142	17	the	the	DET
cana-512	142	18	"	"	PUNCT
cana-512	142	19	rung	rung	NOUN
cana-512	142	20	.	.	PUNCT
cana-512	142	21	"	"	PUNCT
cana-512	143	1	therefore	therefore	ADV
cana-512	143	2	,	,	PUNCT
cana-512	143	3	a	a	DET
cana-512	143	4	ladder	ladder	NOUN
cana-512	143	5	graph	graph	NOUN
cana-512	143	6	l(n	l(n	NOUN
cana-512	143	7	)	)	PUNCT
cana-512	143	8	has	have	VERB
cana-512	143	9	a	a	DET
cana-512	143	10	total	total	NOUN
cana-512	143	11	of	of	ADP
cana-512	143	12	2n	2n	NUM
cana-512	143	13	vertices	vertex	NOUN
cana-512	143	14	and	and	CCONJ
cana-512	143	15	3n-2	3n-2	NUM
cana-512	143	16	edges	edge	NOUN
cana-512	143	17	.	.	PUNCT
cana-512	144	1	in	in	ADP
cana-512	144	2	this	this	DET
cana-512	144	3	section	section	NOUN
cana-512	144	4	,	,	PUNCT
cana-512	144	5	the	the	DET
cana-512	144	6	edge	edge	NOUN
cana-512	144	7	prime	prime	ADJ
cana-512	144	8	index	index	NOUN
cana-512	144	9	of	of	ADP
cana-512	144	10	the	the	DET
cana-512	144	11	ladder	ladder	NOUN
cana-512	144	12	graph	graph	NOUN
cana-512	144	13	l(n	l(n	PROPN
cana-512	144	14	)	)	PUNCT
cana-512	144	15	is	be	AUX
cana-512	144	16	discussed	discuss	VERB
cana-512	144	17	.	.	PUNCT
cana-512	145	1	communications	communication	NOUN
cana-512	145	2	on	on	ADP
cana-512	145	3	applied	apply	VERB
cana-512	145	4	nonlinear	nonlinear	ADJ
cana-512	145	5	analysis	analysis	NOUN
cana-512	145	6	issn	issn	NOUN
cana-512	145	7	:	:	PUNCT
cana-512	145	8	1074	1074	NUM
cana-512	145	9	-	-	PUNCT
cana-512	145	10	133x	133x	NUM
cana-512	145	11	vol	vol	NOUN
cana-512	145	12	31	31	NUM
cana-512	145	13	no	no	NOUN
cana-512	145	14	.	.	NOUN
cana-512	145	15	2	2	NUM
cana-512	145	16	(	(	PUNCT
cana-512	145	17	2024	2024	NUM
cana-512	145	18	)	)	PUNCT
cana-512	145	19	51	51	NUM
cana-512	145	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	145	21	theorem	theorem	VERB
cana-512	145	22	4.6	4.6	NUM
cana-512	145	23	.	.	PUNCT
cana-512	146	1	for	for	ADP
cana-512	146	2	a	a	DET
cana-512	146	3	ladder	ladder	NOUN
cana-512	146	4	graph	graph	NOUN
cana-512	146	5	g	g	PROPN
cana-512	146	6	=	=	SYM
cana-512	146	7	𝐿𝑛	𝐿𝑛	PROPN
cana-512	146	8	,	,	PUNCT
cana-512	146	9	𝑛	𝑛	PROPN
cana-512	146	10	>	>	X
cana-512	146	11	3	3	NUM
cana-512	146	12	then	then	ADV
cana-512	146	13	,	,	PUNCT
cana-512	146	14	εr(g	εr(g	PUNCT
cana-512	146	15	)	)	PUNCT
cana-512	146	16	=	=	SYM
cana-512	146	17	𝑛	𝑛	PRON
cana-512	146	18	−	−	NOUN
cana-512	146	19	3	3	X
cana-512	146	20	.	.	X
cana-512	146	21	proof	proof	NOUN
cana-512	146	22	:	:	PUNCT
cana-512	146	23	let	let	VERB
cana-512	146	24	the	the	DET
cana-512	146	25	vertices	vertex	NOUN
cana-512	146	26	in	in	ADP
cana-512	146	27	the	the	DET
cana-512	146	28	ladder	ladder	NOUN
cana-512	146	29	graph	graph	NOUN
cana-512	146	30	are	be	AUX
cana-512	146	31	{	{	PUNCT
cana-512	146	32	𝑣1	𝑣1	PROPN
cana-512	146	33	,	,	PUNCT
cana-512	146	34	𝑣2	𝑣2	PROPN
cana-512	146	35	…	…	PUNCT
cana-512	146	36	.	.	PUNCT
cana-512	147	1	𝑣𝑛	𝑣𝑛	PROPN
cana-512	147	2	,	,	PUNCT
cana-512	147	3	𝑢1	𝑢1	NOUN
cana-512	147	4	,	,	PUNCT
cana-512	147	5	𝑢2	𝑢2	PROPN
cana-512	147	6	,	,	PUNCT
cana-512	147	7	…	…	PUNCT
cana-512	147	8	,	,	PUNCT
cana-512	147	9	𝑢𝑛	𝑢𝑛	NOUN
cana-512	147	10	}	}	PUNCT
cana-512	147	11	.	.	PUNCT
cana-512	148	1	then	then	ADV
cana-512	148	2	by	by	ADP
cana-512	148	3	the	the	DET
cana-512	148	4	definition	definition	NOUN
cana-512	148	5	of	of	ADP
cana-512	148	6	ladder	ladder	NOUN
cana-512	148	7	graph	graph	NOUN
cana-512	148	8	,	,	PUNCT
cana-512	148	9	there	there	PRON
cana-512	148	10	are	be	VERB
cana-512	148	11	2n	2n	NUM
cana-512	148	12	and	and	CCONJ
cana-512	148	13	3n-2	3n-2	NUM
cana-512	148	14	vertices	vertex	NOUN
cana-512	148	15	and	and	CCONJ
cana-512	148	16	edges	edge	NOUN
cana-512	148	17	respectively	respectively	ADV
cana-512	148	18	.	.	PUNCT
cana-512	149	1	now	now	ADV
cana-512	149	2	,	,	PUNCT
cana-512	149	3	it	it	PRON
cana-512	149	4	is	be	AUX
cana-512	149	5	enough	enough	ADJ
cana-512	149	6	to	to	PART
cana-512	149	7	find	find	VERB
cana-512	149	8	the	the	DET
cana-512	149	9	minimum	minimum	ADJ
cana-512	149	10	number	number	NOUN
cana-512	149	11	of	of	ADP
cana-512	149	12	edges	edge	NOUN
cana-512	149	13	to	to	PART
cana-512	149	14	be	be	AUX
cana-512	149	15	removed	remove	VERB
cana-512	149	16	from	from	ADP
cana-512	149	17	the	the	DET
cana-512	149	18	ladder	ladder	NOUN
cana-512	149	19	graph	graph	NOUN
cana-512	149	20	to	to	PART
cana-512	149	21	make	make	VERB
cana-512	149	22	it	it	PRON
cana-512	149	23	as	as	ADP
cana-512	149	24	a	a	DET
cana-512	149	25	relatively	relatively	ADV
cana-512	149	26	prime	prime	ADJ
cana-512	149	27	edge	edge	NOUN
cana-512	149	28	labeled	label	VERB
cana-512	149	29	graph	graph	NOUN
cana-512	149	30	.	.	PUNCT
cana-512	150	1	the	the	DET
cana-512	150	2	maximal	maximal	ADJ
cana-512	150	3	cycle	cycle	NOUN
cana-512	150	4	formed	form	VERB
cana-512	150	5	by	by	ADP
cana-512	150	6	the	the	DET
cana-512	150	7	vertices	vertex	NOUN
cana-512	150	8	of	of	ADP
cana-512	150	9	the	the	DET
cana-512	150	10	ladder	ladder	NOUN
cana-512	150	11	graph	graph	NOUN
cana-512	150	12	is	be	AUX
cana-512	150	13	𝑣1	𝑣1	NOUN
cana-512	150	14	,	,	PUNCT
cana-512	150	15	𝑣2	𝑣2	NOUN
cana-512	150	16	…	…	PUNCT
cana-512	150	17	.	.	PUNCT
cana-512	151	1	𝑣𝑛	𝑣𝑛	ADP
cana-512	151	2	,	,	PUNCT
cana-512	151	3	𝑢𝑛	𝑢𝑛	NOUN
cana-512	151	4	,	,	PUNCT
cana-512	151	5	𝑢𝑛−1	𝑢𝑛−1	NOUN
cana-512	151	6	,	,	PUNCT
cana-512	151	7	…	…	PUNCT
cana-512	151	8	,	,	PUNCT
cana-512	151	9	𝑢1	𝑢1	PROPN
cana-512	151	10	,	,	PUNCT
cana-512	151	11	,	,	PUNCT
cana-512	151	12	𝑣1	𝑣1	PROPN
cana-512	151	13	.	.	PUNCT
cana-512	152	1	now	now	ADV
cana-512	152	2	,	,	PUNCT
cana-512	152	3	label	label	VERB
cana-512	152	4	the	the	DET
cana-512	152	5	edges	edge	NOUN
cana-512	152	6	of	of	ADP
cana-512	152	7	g	g	NOUN
cana-512	152	8	in	in	ADP
cana-512	152	9	such	such	DET
cana-512	152	10	a	a	DET
cana-512	152	11	way	way	NOUN
cana-512	152	12	that	that	PRON
cana-512	152	13	,	,	PUNCT
cana-512	152	14	the	the	DET
cana-512	152	15	maximal	maximal	ADJ
cana-512	152	16	cycle	cycle	NOUN
cana-512	152	17	is	be	AUX
cana-512	152	18	labeled	label	VERB
cana-512	152	19	with	with	ADP
cana-512	152	20	{	{	PUNCT
cana-512	152	21	2	2	NUM
cana-512	152	22	,	,	PUNCT
cana-512	152	23	3	3	NUM
cana-512	152	24	,	,	PUNCT
cana-512	152	25	…	…	PUNCT
cana-512	152	26	.	.	PUNCT
cana-512	153	1	,	,	PUNCT
cana-512	153	2	2𝑛	2𝑛	PROPN
cana-512	153	3	+	+	CCONJ
cana-512	153	4	1	1	X
cana-512	153	5	}	}	PUNCT
cana-512	153	6	and	and	CCONJ
cana-512	153	7	the	the	DET
cana-512	153	8	edge	edge	NOUN
cana-512	153	9	𝑢2𝑣2	𝑢2𝑣2	NOUN
cana-512	153	10	with	with	ADP
cana-512	153	11	the	the	DET
cana-512	153	12	label	label	NOUN
cana-512	153	13	1	1	NUM
cana-512	153	14	as	as	SCONJ
cana-512	153	15	shown	show	VERB
cana-512	153	16	in	in	ADP
cana-512	153	17	figure	figure	NOUN
cana-512	153	18	8	8	NUM
cana-512	153	19	.	.	PUNCT
cana-512	153	20	figure	figure	NOUN
cana-512	153	21	8	8	NUM
cana-512	153	22	:	:	PUNCT
cana-512	153	23	εr(g	εr(g	PUNCT
cana-512	153	24	)	)	PUNCT
cana-512	154	1	=	=	SYM
cana-512	154	2	n	n	CCONJ
cana-512	154	3	−	−	NOUN
cana-512	154	4	3	3	NUM
cana-512	154	5	also	also	ADV
cana-512	154	6	,	,	PUNCT
cana-512	154	7	2n+2	2n+2	PROPN
cana-512	154	8	is	be	AUX
cana-512	154	9	an	an	DET
cana-512	154	10	even	even	ADJ
cana-512	154	11	number	number	NOUN
cana-512	154	12	and	and	CCONJ
cana-512	154	13	to	to	PART
cana-512	154	14	make	make	VERB
cana-512	154	15	the	the	DET
cana-512	154	16	edges	edge	NOUN
cana-512	154	17	incident	incident	NOUN
cana-512	154	18	of	of	ADP
cana-512	154	19	each	each	DET
cana-512	154	20	vertex	vertex	NOUN
cana-512	154	21	as	as	ADP
cana-512	154	22	pairwise	pairwise	NOUN
cana-512	154	23	relatively	relatively	ADV
cana-512	154	24	prime	prime	ADJ
cana-512	154	25	,	,	PUNCT
cana-512	154	26	it	it	PRON
cana-512	154	27	can	can	AUX
cana-512	154	28	not	not	PART
cana-512	154	29	be	be	AUX
cana-512	154	30	labeled	label	VERB
cana-512	154	31	in	in	ADP
cana-512	154	32	any	any	PRON
cana-512	154	33	of	of	ADP
cana-512	154	34	the	the	DET
cana-512	154	35	edges	edge	NOUN
cana-512	154	36	.	.	PUNCT
cana-512	155	1	thus	thus	ADV
cana-512	155	2	,	,	PUNCT
cana-512	155	3	εr(g	εr(g	PUNCT
cana-512	155	4	)	)	PUNCT
cana-512	156	1	=	=	SYM
cana-512	156	2	3𝑛	3𝑛	NUM
cana-512	156	3	−	−	PROPN
cana-512	156	4	2	2	NUM
cana-512	156	5	−	−	PROPN
cana-512	156	6	2𝑛	2𝑛	NOUN
cana-512	156	7	−	−	PROPN
cana-512	156	8	1	1	NUM
cana-512	156	9	=	=	SYM
cana-512	156	10	𝑛	𝑛	PRON
cana-512	156	11	−	−	NOUN
cana-512	157	1	3	3	X
cana-512	157	2	.	.	PUNCT
cana-512	158	1	hence	hence	ADV
cana-512	158	2	the	the	DET
cana-512	158	3	proof	proof	NOUN
cana-512	158	4	.	.	PUNCT
cana-512	159	1	4.5	4.5	NUM
cana-512	159	2	.	.	PUNCT
cana-512	160	1	powers	power	NOUN
cana-512	160	2	of	of	ADP
cana-512	160	3	paths	path	NOUN
cana-512	160	4	and	and	CCONJ
cana-512	160	5	cycle	cycle	NOUN
cana-512	160	6	for	for	ADP
cana-512	160	7	the	the	DET
cana-512	160	8	powers	power	NOUN
cana-512	160	9	of	of	ADP
cana-512	160	10	path	path	NOUN
cana-512	160	11	and	and	CCONJ
cana-512	160	12	cycle	cycle	NOUN
cana-512	160	13	graph	graph	NOUN
cana-512	160	14	,	,	PUNCT
cana-512	160	15	the	the	DET
cana-512	160	16	edge	edge	NOUN
cana-512	160	17	prime	prime	ADJ
cana-512	160	18	index	index	NOUN
cana-512	160	19	is	be	AUX
cana-512	160	20	determined	determine	VERB
cana-512	160	21	in	in	ADP
cana-512	160	22	this	this	DET
cana-512	160	23	section	section	NOUN
cana-512	160	24	.	.	PUNCT
cana-512	161	1	the	the	DET
cana-512	161	2	m	m	PROPN
cana-512	161	3	-	-	PUNCT
cana-512	161	4	th	th	VERB
cana-512	161	5	power	power	NOUN
cana-512	161	6	of	of	ADP
cana-512	161	7	a	a	DET
cana-512	161	8	simple	simple	ADJ
cana-512	161	9	graph	graph	NOUN
cana-512	161	10	g	g	NOUN
cana-512	161	11	can	can	AUX
cana-512	161	12	be	be	AUX
cana-512	161	13	defined	define	VERB
cana-512	161	14	as	as	ADP
cana-512	161	15	the	the	DET
cana-512	161	16	graph	graph	NOUN
cana-512	161	17	𝐺𝑚	𝐺𝑚	PROPN
cana-512	161	18	,	,	PUNCT
cana-512	161	19	where	where	SCONJ
cana-512	161	20	the	the	DET
cana-512	161	21	set	set	NOUN
cana-512	161	22	of	of	ADP
cana-512	161	23	vertices	vertex	NOUN
cana-512	161	24	in	in	ADP
cana-512	161	25	𝐺𝑚	𝐺𝑚	PROPN
cana-512	161	26	is	be	AUX
cana-512	161	27	the	the	DET
cana-512	161	28	same	same	ADJ
cana-512	161	29	communications	communication	NOUN
cana-512	161	30	on	on	ADP
cana-512	161	31	applied	apply	VERB
cana-512	161	32	nonlinear	nonlinear	ADJ
cana-512	161	33	analysis	analysis	NOUN
cana-512	161	34	issn	issn	NOUN
cana-512	161	35	:	:	PUNCT
cana-512	161	36	1074	1074	NUM
cana-512	161	37	-	-	PUNCT
cana-512	161	38	133x	133x	NUM
cana-512	161	39	vol	vol	NOUN
cana-512	161	40	31	31	NUM
cana-512	161	41	no	no	NOUN
cana-512	161	42	.	.	NOUN
cana-512	161	43	2	2	NUM
cana-512	161	44	(	(	PUNCT
cana-512	161	45	2024	2024	NUM
cana-512	161	46	)	)	PUNCT
cana-512	161	47	52	52	NUM
cana-512	161	48	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	161	49	as	as	ADP
cana-512	161	50	that	that	PRON
cana-512	161	51	in	in	ADP
cana-512	161	52	g.	g.	PROPN
cana-512	161	53	in	in	ADP
cana-512	161	54	𝐺𝑚	𝐺𝑚	PROPN
cana-512	161	55	,	,	PUNCT
cana-512	161	56	two	two	NUM
cana-512	161	57	vertices	vertex	NOUN
cana-512	161	58	are	be	AUX
cana-512	161	59	adjacent	adjacent	ADJ
cana-512	161	60	if	if	SCONJ
cana-512	161	61	and	and	CCONJ
cana-512	161	62	only	only	ADV
cana-512	161	63	if	if	SCONJ
cana-512	161	64	their	their	PRON
cana-512	161	65	distance	distance	NOUN
cana-512	161	66	in	in	ADP
cana-512	161	67	g	g	PROPN
cana-512	161	68	is	be	AUX
cana-512	161	69	not	not	PART
cana-512	161	70	more	more	ADJ
cana-512	161	71	than	than	ADP
cana-512	161	72	m.	m.	NOUN
cana-512	161	73	that	that	PRON
cana-512	161	74	is	be	AUX
cana-512	161	75	,	,	PUNCT
cana-512	161	76	their	their	PRON
cana-512	161	77	distance	distance	NOUN
cana-512	161	78	in	in	ADP
cana-512	161	79	g	g	PROPN
cana-512	161	80	is	be	AUX
cana-512	161	81	at	at	ADP
cana-512	161	82	most	most	ADJ
cana-512	161	83	𝑚.	𝑚.	ADJ
cana-512	161	84	theorem	theorem	ADJ
cana-512	161	85	4.7	4.7	NUM
cana-512	161	86	.	.	PUNCT
cana-512	162	1	for	for	ADP
cana-512	162	2	even	even	ADV
cana-512	162	3	n	n	CCONJ
cana-512	162	4	,	,	PUNCT
cana-512	162	5	the	the	DET
cana-512	162	6	power	power	NOUN
cana-512	162	7	graph	graph	NOUN
cana-512	162	8	of	of	ADP
cana-512	162	9	a	a	DET
cana-512	162	10	path	path	NOUN
cana-512	162	11	g	g	NOUN
cana-512	162	12	=	=	PUNCT
cana-512	162	13	p𝑛	p𝑛	PROPN
cana-512	162	14	2	2	NUM
cana-512	162	15	,	,	PUNCT
cana-512	162	16	then	then	ADV
cana-512	162	17	,	,	PUNCT
cana-512	162	18	εr(g	εr(g	PUNCT
cana-512	162	19	)	)	PUNCT
cana-512	162	20	=	=	PRON
cana-512	162	21	{	{	PUNCT
cana-512	162	22	𝑛	𝑛	PRON
cana-512	162	23	−	−	NUM
cana-512	162	24	3	3	NUM
cana-512	162	25	𝑖𝑓	𝑖𝑓	ADP
cana-512	162	26	𝑛	𝑛	PRON
cana-512	162	27	𝑖𝑠	𝑖𝑠	NOUN
cana-512	162	28	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-512	162	29	𝑛	𝑛	DET
cana-512	162	30	−	−	PROPN
cana-512	162	31	4	4	NUM
cana-512	162	32	𝑖𝑓	𝑖𝑓	ADP
cana-512	162	33	𝑛	𝑛	DET
cana-512	162	34	𝑖𝑠	𝑖𝑠	NOUN
cana-512	162	35	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-512	162	36	proof	proof	NOUN
cana-512	162	37	:	:	PUNCT
cana-512	162	38	let	let	VERB
cana-512	162	39	the	the	DET
cana-512	162	40	number	number	NOUN
cana-512	162	41	of	of	ADP
cana-512	162	42	vertices	vertex	NOUN
cana-512	162	43	and	and	CCONJ
cana-512	162	44	edges	edge	NOUN
cana-512	162	45	in	in	ADP
cana-512	162	46	𝑃𝑛	𝑃𝑛	PROPN
cana-512	162	47	2	2	NUM
cana-512	162	48	are	be	AUX
cana-512	162	49	𝑛	𝑛	PRON
cana-512	162	50	and	and	CCONJ
cana-512	162	51	2𝑛	2𝑛	PROPN
cana-512	162	52	−	−	PROPN
cana-512	162	53	3	3	NUM
cana-512	162	54	respectively	respectively	ADV
cana-512	162	55	.	.	PUNCT
cana-512	163	1	from	from	ADP
cana-512	163	2	the	the	DET
cana-512	163	3	definition	definition	NOUN
cana-512	163	4	of	of	ADP
cana-512	163	5	𝑃𝑛	𝑃𝑛	PROPN
cana-512	163	6	2	2	NUM
cana-512	163	7	,	,	PUNCT
cana-512	163	8	there	there	PRON
cana-512	163	9	exists	exist	VERB
cana-512	163	10	a	a	DET
cana-512	163	11	maximal	maximal	ADJ
cana-512	163	12	cycle	cycle	NOUN
cana-512	163	13	of	of	ADP
cana-512	163	14	length	length	NOUN
cana-512	163	15	n.	n.	PROPN
cana-512	163	16	now	now	ADV
cana-512	163	17	,	,	PUNCT
cana-512	163	18	label	label	VERB
cana-512	163	19	the	the	DET
cana-512	163	20	maximal	maximal	ADJ
cana-512	163	21	cycle	cycle	NOUN
cana-512	163	22	with	with	ADP
cana-512	163	23	{	{	PUNCT
cana-512	163	24	1	1	NUM
cana-512	163	25	,	,	PUNCT
cana-512	163	26	2	2	NUM
cana-512	163	27	,	,	PUNCT
cana-512	163	28	3	3	NUM
cana-512	163	29	,	,	PUNCT
cana-512	163	30	…	…	PUNCT
cana-512	163	31	.	.	PUNCT
cana-512	163	32	.	.	PUNCT
cana-512	164	1	.	.	PUNCT
cana-512	165	1	,	,	PUNCT
cana-512	165	2	𝑛	𝑛	ADJ
cana-512	165	3	}	}	PUNCT
cana-512	165	4	.	.	PUNCT
cana-512	166	1	case	case	NOUN
cana-512	166	2	1	1	NUM
cana-512	166	3	:	:	PUNCT
cana-512	166	4	n	n	PRON
cana-512	166	5	is	be	AUX
cana-512	166	6	odd	odd	ADJ
cana-512	166	7	as	as	SCONJ
cana-512	166	8	n	n	NUM
cana-512	166	9	is	be	AUX
cana-512	166	10	odd	odd	ADJ
cana-512	166	11	,	,	PUNCT
cana-512	166	12	𝑛	𝑛	PRON
cana-512	166	13	+	+	NOUN
cana-512	166	14	1	1	NUM
cana-512	166	15	is	be	AUX
cana-512	166	16	even	even	ADV
cana-512	166	17	.	.	PUNCT
cana-512	167	1	hence	hence	ADV
cana-512	167	2	it	it	PRON
cana-512	167	3	is	be	AUX
cana-512	167	4	not	not	PART
cana-512	167	5	possible	possible	ADJ
cana-512	167	6	to	to	PART
cana-512	167	7	label	label	VERB
cana-512	167	8	𝑛	𝑛	DET
cana-512	167	9	+	+	NOUN
cana-512	167	10	1	1	NUM
cana-512	167	11	in	in	ADP
cana-512	167	12	any	any	PRON
cana-512	167	13	of	of	ADP
cana-512	167	14	the	the	DET
cana-512	167	15	remaining	remain	VERB
cana-512	167	16	edges	edge	NOUN
cana-512	167	17	.	.	PUNCT
cana-512	168	1	because	because	SCONJ
cana-512	168	2	the	the	DET
cana-512	168	3	remaining	remain	VERB
cana-512	168	4	edges	edge	NOUN
cana-512	168	5	associated	associate	VERB
cana-512	168	6	with	with	ADP
cana-512	168	7	a	a	DET
cana-512	168	8	vertex	vertex	NOUN
cana-512	168	9	already	already	ADV
cana-512	168	10	contains	contain	VERB
cana-512	168	11	the	the	DET
cana-512	168	12	label	label	NOUN
cana-512	168	13	of	of	ADP
cana-512	168	14	even	even	ADV
cana-512	168	15	multiple	multiple	ADJ
cana-512	168	16	.	.	PUNCT
cana-512	169	1	thus	thus	ADV
cana-512	169	2	,	,	PUNCT
cana-512	169	3	by	by	ADP
cana-512	169	4	labeling	label	VERB
cana-512	169	5	𝑛	𝑛	PRON
cana-512	169	6	+	+	NOUN
cana-512	169	7	1	1	NUM
cana-512	169	8	will	will	AUX
cana-512	169	9	violate	violate	VERB
cana-512	169	10	the	the	DET
cana-512	169	11	relatively	relatively	ADV
cana-512	169	12	prime	prime	ADJ
cana-512	169	13	property	property	NOUN
cana-512	169	14	.	.	PUNCT
cana-512	170	1	therefore	therefore	ADV
cana-512	170	2	,	,	PUNCT
cana-512	170	3	it	it	PRON
cana-512	170	4	is	be	AUX
cana-512	170	5	necessary	necessary	ADJ
cana-512	170	6	to	to	PART
cana-512	170	7	remove	remove	VERB
cana-512	170	8	2𝑛	2𝑛	PROPN
cana-512	170	9	−	−	PROPN
cana-512	170	10	3	3	NUM
cana-512	170	11	–	–	PUNCT
cana-512	170	12	𝑛	𝑛	NOUN
cana-512	170	13	=	=	SYM
cana-512	170	14	𝑛	𝑛	DET
cana-512	170	15	−	−	NUM
cana-512	170	16	3	3	NUM
cana-512	170	17	edges	edge	NOUN
cana-512	170	18	from	from	ADP
cana-512	170	19	p𝑛	p𝑛	PROPN
cana-512	170	20	2	2	NUM
cana-512	170	21	to	to	PART
cana-512	170	22	make	make	VERB
cana-512	170	23	it	it	PRON
cana-512	170	24	as	as	ADP
cana-512	170	25	a	a	DET
cana-512	170	26	relatively	relatively	ADV
cana-512	170	27	prime	prime	ADJ
cana-512	170	28	labeled	label	VERB
cana-512	170	29	graph	graph	NOUN
cana-512	170	30	.	.	PUNCT
cana-512	171	1	that	that	PRON
cana-512	171	2	is	be	AUX
cana-512	171	3	,	,	PUNCT
cana-512	171	4	εr(g	εr(g	PUNCT
cana-512	171	5	)	)	PUNCT
cana-512	172	1	=	=	SYM
cana-512	172	2	𝑛	𝑛	PRON
cana-512	172	3	−	−	NOUN
cana-512	172	4	3	3	NUM
cana-512	172	5	.	.	PUNCT
cana-512	172	6	(	(	PUNCT
cana-512	172	7	example	example	NOUN
cana-512	172	8	for	for	ADP
cana-512	172	9	𝑛	𝑛	NOUN
cana-512	172	10	=	=	SYM
cana-512	172	11	7	7	NUM
cana-512	172	12	is	be	AUX
cana-512	172	13	illustrated	illustrate	VERB
cana-512	172	14	in	in	ADP
cana-512	172	15	figure	figure	NOUN
cana-512	172	16	9	9	NUM
cana-512	172	17	)	)	PUNCT
cana-512	172	18	figure	figure	NOUN
cana-512	172	19	9	9	NUM
cana-512	172	20	:	:	PUNCT
cana-512	172	21	εr(g	εr(g	PUNCT
cana-512	172	22	)	)	PUNCT
cana-512	173	1	=	=	SYM
cana-512	173	2	n	n	CCONJ
cana-512	173	3	−	−	NUM
cana-512	173	4	3	3	NUM
cana-512	173	5	=	=	SYM
cana-512	173	6	4	4	NUM
cana-512	173	7	case	case	NOUN
cana-512	173	8	2	2	NUM
cana-512	173	9	:	:	PUNCT
cana-512	173	10	n	n	PRON
cana-512	173	11	is	be	AUX
cana-512	173	12	even	even	ADV
cana-512	173	13	as	as	SCONJ
cana-512	173	14	n	n	NUM
cana-512	173	15	is	be	AUX
cana-512	173	16	even	even	ADV
cana-512	173	17	,	,	PUNCT
cana-512	173	18	𝑛	𝑛	PRON
cana-512	173	19	+	+	NOUN
cana-512	173	20	1	1	NUM
cana-512	173	21	is	be	AUX
cana-512	173	22	odd	odd	ADJ
cana-512	173	23	.	.	PUNCT
cana-512	174	1	hence	hence	ADV
cana-512	174	2	it	it	PRON
cana-512	174	3	is	be	AUX
cana-512	174	4	possible	possible	ADJ
cana-512	174	5	to	to	PART
cana-512	174	6	label	label	VERB
cana-512	174	7	𝑛	𝑛	DET
cana-512	174	8	+	+	NOUN
cana-512	174	9	1	1	NUM
cana-512	174	10	in	in	ADP
cana-512	174	11	any	any	PRON
cana-512	174	12	of	of	ADP
cana-512	174	13	the	the	DET
cana-512	174	14	remaining	remain	VERB
cana-512	174	15	edges	edge	NOUN
cana-512	174	16	.	.	PUNCT
cana-512	175	1	and	and	CCONJ
cana-512	175	2	,	,	PUNCT
cana-512	175	3	it	it	PRON
cana-512	175	4	is	be	AUX
cana-512	175	5	not	not	PART
cana-512	175	6	possible	possible	ADJ
cana-512	175	7	to	to	PART
cana-512	175	8	label	label	VERB
cana-512	175	9	𝑛	𝑛	DET
cana-512	175	10	+	+	NUM
cana-512	175	11	2	2	NUM
cana-512	175	12	in	in	ADP
cana-512	175	13	the	the	DET
cana-512	175	14	remaining	remain	VERB
cana-512	175	15	edges	edge	NOUN
cana-512	175	16	.	.	PUNCT
cana-512	176	1	therefore	therefore	ADV
cana-512	176	2	,	,	PUNCT
cana-512	176	3	it	it	PRON
cana-512	176	4	is	be	AUX
cana-512	176	5	necessary	necessary	ADJ
cana-512	176	6	to	to	PART
cana-512	176	7	remove	remove	VERB
cana-512	176	8	2𝑛	2𝑛	PROPN
cana-512	176	9	−	−	PROPN
cana-512	176	10	3	3	NUM
cana-512	176	11	–	–	PUNCT
cana-512	176	12	𝑛	𝑛	NOUN
cana-512	176	13	−	−	PROPN
cana-512	176	14	1	1	NUM
cana-512	176	15	=	=	SYM
cana-512	176	16	𝑛	𝑛	DET
cana-512	176	17	−	−	NUM
cana-512	176	18	4	4	NUM
cana-512	176	19	edges	edge	NOUN
cana-512	176	20	from	from	ADP
cana-512	176	21	p𝑛	p𝑛	PROPN
cana-512	176	22	2	2	NUM
cana-512	176	23	to	to	PART
cana-512	176	24	make	make	VERB
cana-512	176	25	it	it	PRON
cana-512	176	26	as	as	ADP
cana-512	176	27	a	a	DET
cana-512	176	28	relatively	relatively	ADV
cana-512	176	29	prime	prime	ADJ
cana-512	176	30	labeled	label	VERB
cana-512	176	31	graph	graph	NOUN
cana-512	176	32	.	.	PUNCT
cana-512	177	1	that	that	PRON
cana-512	177	2	is	be	AUX
cana-512	177	3	,	,	PUNCT
cana-512	177	4	εr(g	εr(g	PUNCT
cana-512	177	5	)	)	PUNCT
cana-512	178	1	=	=	SYM
cana-512	178	2	𝑛	𝑛	PRON
cana-512	178	3	−	−	NUM
cana-512	178	4	4	4	NUM
cana-512	178	5	.	.	PUNCT
cana-512	179	1	(	(	PUNCT
cana-512	179	2	example	example	NOUN
cana-512	179	3	for	for	ADP
cana-512	179	4	𝑛	𝑛	PROPN
cana-512	179	5	=	=	SYM
cana-512	179	6	8	8	NUM
cana-512	179	7	is	be	AUX
cana-512	179	8	illustrated	illustrate	VERB
cana-512	179	9	in	in	ADP
cana-512	179	10	figure	figure	NOUN
cana-512	179	11	10	10	NUM
cana-512	179	12	)	)	PUNCT
cana-512	179	13	figure	figure	NOUN
cana-512	179	14	10	10	NUM
cana-512	179	15	:	:	PUNCT
cana-512	179	16	εr(g	εr(g	PUNCT
cana-512	179	17	)	)	PUNCT
cana-512	180	1	=	=	SYM
cana-512	180	2	n	n	CCONJ
cana-512	180	3	−	−	NUM
cana-512	180	4	4	4	NUM
cana-512	180	5	=	=	SYM
cana-512	180	6	4	4	NUM
cana-512	180	7	communications	communication	NOUN
cana-512	180	8	on	on	ADP
cana-512	180	9	applied	apply	VERB
cana-512	180	10	nonlinear	nonlinear	ADJ
cana-512	180	11	analysis	analysis	NOUN
cana-512	180	12	issn	issn	NOUN
cana-512	180	13	:	:	PUNCT
cana-512	180	14	1074	1074	NUM
cana-512	180	15	-	-	PUNCT
cana-512	180	16	133x	133x	NUM
cana-512	180	17	vol	vol	NOUN
cana-512	180	18	31	31	NUM
cana-512	180	19	no	no	NOUN
cana-512	180	20	.	.	NOUN
cana-512	180	21	2	2	NUM
cana-512	180	22	(	(	PUNCT
cana-512	180	23	2024	2024	NUM
cana-512	180	24	)	)	PUNCT
cana-512	180	25	53	53	NUM
cana-512	180	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	180	27	theorem	theorem	VERB
cana-512	180	28	4.8	4.8	NUM
cana-512	180	29	.	.	PUNCT
cana-512	181	1	for	for	ADP
cana-512	181	2	a	a	DET
cana-512	181	3	power	power	NOUN
cana-512	181	4	graph	graph	NOUN
cana-512	181	5	of	of	ADP
cana-512	181	6	a	a	DET
cana-512	181	7	cycle	cycle	NOUN
cana-512	181	8	g	g	NOUN
cana-512	181	9	=	=	SYM
cana-512	181	10	c𝑛	c𝑛	PROPN
cana-512	181	11	2	2	NUM
cana-512	181	12	,	,	PUNCT
cana-512	181	13	then	then	ADV
cana-512	181	14	,	,	PUNCT
cana-512	181	15	εr(g	εr(g	PUNCT
cana-512	181	16	)	)	PUNCT
cana-512	182	1	=	=	PRON
cana-512	182	2	{	{	PUNCT
cana-512	182	3	𝑛	𝑛	INTJ
cana-512	182	4	𝑖𝑓	𝑖𝑓	NOUN
cana-512	182	5	𝑛	𝑛	PRON
cana-512	182	6	𝑖𝑠	𝑖𝑠	NOUN
cana-512	182	7	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-512	183	1	𝑛	𝑛	PRON
cana-512	183	2	−	−	PROPN
cana-512	183	3	1	1	NUM
cana-512	183	4	,	,	PUNCT
cana-512	183	5	𝑖𝑓	𝑖𝑓	ADP
cana-512	183	6	𝑛	𝑛	DET
cana-512	183	7	𝑖𝑠	𝑖𝑠	NOUN
cana-512	183	8	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-512	183	9	proof	proof	NOUN
cana-512	183	10	:	:	PUNCT
cana-512	183	11	let	let	VERB
cana-512	183	12	the	the	DET
cana-512	183	13	number	number	NOUN
cana-512	183	14	of	of	ADP
cana-512	183	15	vertices	vertex	NOUN
cana-512	183	16	and	and	CCONJ
cana-512	183	17	edges	edge	NOUN
cana-512	183	18	in	in	ADP
cana-512	183	19	𝐶𝑛	𝐶𝑛	PROPN
cana-512	183	20	2	2	NUM
cana-512	183	21	are	be	AUX
cana-512	183	22	𝑛	𝑛	PRON
cana-512	183	23	and	and	CCONJ
cana-512	183	24	2𝑛	2𝑛	PROPN
cana-512	183	25	respectively	respectively	ADV
cana-512	183	26	.	.	PUNCT
cana-512	184	1	from	from	ADP
cana-512	184	2	the	the	DET
cana-512	184	3	definition	definition	NOUN
cana-512	184	4	of	of	ADP
cana-512	184	5	𝐶𝑛	𝐶𝑛	PROPN
cana-512	184	6	2	2	NUM
cana-512	184	7	,	,	PUNCT
cana-512	184	8	there	there	PRON
cana-512	184	9	exists	exist	VERB
cana-512	184	10	a	a	DET
cana-512	184	11	cycle	cycle	NOUN
cana-512	184	12	of	of	ADP
cana-512	184	13	length	length	NOUN
cana-512	184	14	n.	n.	PROPN
cana-512	184	15	now	now	ADV
cana-512	184	16	,	,	PUNCT
cana-512	184	17	label	label	VERB
cana-512	184	18	the	the	DET
cana-512	184	19	cycle	cycle	NOUN
cana-512	184	20	with	with	ADP
cana-512	184	21	{	{	PUNCT
cana-512	184	22	1	1	NUM
cana-512	184	23	,	,	PUNCT
cana-512	184	24	2	2	NUM
cana-512	184	25	,	,	PUNCT
cana-512	184	26	3	3	NUM
cana-512	184	27	,	,	PUNCT
cana-512	184	28	…	…	PUNCT
cana-512	184	29	.	.	PUNCT
cana-512	184	30	.	.	PUNCT
cana-512	185	1	.	.	PUNCT
cana-512	186	1	,	,	PUNCT
cana-512	186	2	𝑛	𝑛	ADJ
cana-512	186	3	}	}	PUNCT
cana-512	186	4	.	.	PUNCT
cana-512	187	1	case	case	NOUN
cana-512	187	2	1	1	NUM
cana-512	187	3	:	:	PUNCT
cana-512	187	4	n	n	PRON
cana-512	187	5	is	be	AUX
cana-512	187	6	odd	odd	ADJ
cana-512	187	7	as	as	SCONJ
cana-512	187	8	n	n	NUM
cana-512	187	9	is	be	AUX
cana-512	187	10	odd	odd	ADJ
cana-512	187	11	,	,	PUNCT
cana-512	187	12	𝑛	𝑛	PRON
cana-512	187	13	+	+	NOUN
cana-512	187	14	1	1	NUM
cana-512	187	15	is	be	AUX
cana-512	187	16	even	even	ADV
cana-512	187	17	.	.	PUNCT
cana-512	188	1	hence	hence	ADV
cana-512	188	2	it	it	PRON
cana-512	188	3	is	be	AUX
cana-512	188	4	not	not	PART
cana-512	188	5	possible	possible	ADJ
cana-512	188	6	to	to	PART
cana-512	188	7	label	label	VERB
cana-512	188	8	𝑛	𝑛	DET
cana-512	188	9	+	+	NOUN
cana-512	188	10	1	1	NUM
cana-512	188	11	in	in	ADP
cana-512	188	12	any	any	PRON
cana-512	188	13	of	of	ADP
cana-512	188	14	the	the	DET
cana-512	188	15	remaining	remain	VERB
cana-512	188	16	edges	edge	NOUN
cana-512	188	17	.	.	PUNCT
cana-512	189	1	because	because	SCONJ
cana-512	189	2	the	the	DET
cana-512	189	3	remaining	remain	VERB
cana-512	189	4	edges	edge	NOUN
cana-512	189	5	associated	associate	VERB
cana-512	189	6	with	with	ADP
cana-512	189	7	a	a	DET
cana-512	189	8	vertex	vertex	NOUN
cana-512	189	9	already	already	ADV
cana-512	189	10	contains	contain	VERB
cana-512	189	11	the	the	DET
cana-512	189	12	label	label	NOUN
cana-512	189	13	of	of	ADP
cana-512	189	14	even	even	ADV
cana-512	189	15	multiple	multiple	ADJ
cana-512	189	16	.	.	PUNCT
cana-512	190	1	thus	thus	ADV
cana-512	190	2	,	,	PUNCT
cana-512	190	3	by	by	ADP
cana-512	190	4	labeling	label	VERB
cana-512	190	5	𝑛	𝑛	PRON
cana-512	190	6	+	+	NOUN
cana-512	190	7	1	1	NUM
cana-512	190	8	will	will	AUX
cana-512	190	9	violate	violate	VERB
cana-512	190	10	the	the	DET
cana-512	190	11	relatively	relatively	ADV
cana-512	190	12	prime	prime	ADJ
cana-512	190	13	property	property	NOUN
cana-512	190	14	.	.	PUNCT
cana-512	191	1	therefore	therefore	ADV
cana-512	191	2	,	,	PUNCT
cana-512	191	3	it	it	PRON
cana-512	191	4	is	be	AUX
cana-512	191	5	necessary	necessary	ADJ
cana-512	191	6	to	to	PART
cana-512	191	7	remove	remove	VERB
cana-512	191	8	2𝑛	2𝑛	PROPN
cana-512	191	9	–	–	PUNCT
cana-512	191	10	𝑛	𝑛	NOUN
cana-512	191	11	=	=	SYM
cana-512	191	12	𝑛	𝑛	PROPN
cana-512	191	13	edges	edge	NOUN
cana-512	191	14	from	from	ADP
cana-512	191	15	c𝑛	c𝑛	PROPN
cana-512	191	16	2	2	NUM
cana-512	191	17	to	to	PART
cana-512	191	18	make	make	VERB
cana-512	191	19	it	it	PRON
cana-512	191	20	as	as	ADP
cana-512	191	21	a	a	DET
cana-512	191	22	relatively	relatively	ADV
cana-512	191	23	prime	prime	ADJ
cana-512	191	24	labeled	label	VERB
cana-512	191	25	graph	graph	NOUN
cana-512	191	26	.	.	PUNCT
cana-512	192	1	that	that	PRON
cana-512	192	2	is	be	AUX
cana-512	192	3	,	,	PUNCT
cana-512	192	4	εr(g	εr(g	PUNCT
cana-512	192	5	)	)	PUNCT
cana-512	193	1	=	=	SYM
cana-512	193	2	𝑛	𝑛	PROPN
cana-512	193	3	(	(	PUNCT
cana-512	193	4	example	example	NOUN
cana-512	193	5	for	for	ADP
cana-512	193	6	𝑛	𝑛	NOUN
cana-512	193	7	=	=	SYM
cana-512	193	8	5	5	NUM
cana-512	193	9	is	be	AUX
cana-512	193	10	illustrated	illustrate	VERB
cana-512	193	11	in	in	ADP
cana-512	193	12	figure	figure	NOUN
cana-512	193	13	11	11	NUM
cana-512	193	14	)	)	PUNCT
cana-512	193	15	figure	figure	NOUN
cana-512	193	16	11	11	NUM
cana-512	193	17	:	:	PUNCT
cana-512	193	18	εr(g	εr(g	PUNCT
cana-512	193	19	)	)	PUNCT
cana-512	194	1	=	=	SYM
cana-512	194	2	n	n	NOUN
cana-512	194	3	=	=	SYM
cana-512	194	4	5	5	NUM
cana-512	194	5	figure	figure	NOUN
cana-512	194	6	12	12	NUM
cana-512	194	7	:	:	PUNCT
cana-512	194	8	εr(g	εr(g	PUNCT
cana-512	194	9	)	)	PUNCT
cana-512	194	10	=	=	SYM
cana-512	195	1	n	n	CCONJ
cana-512	195	2	−	−	NUM
cana-512	195	3	1	1	NUM
cana-512	195	4	=	=	SYM
cana-512	195	5	5	5	NUM
cana-512	195	6	case	case	NOUN
cana-512	195	7	2	2	NUM
cana-512	195	8	:	:	PUNCT
cana-512	195	9	n	n	PRON
cana-512	195	10	is	be	AUX
cana-512	195	11	even	even	ADV
cana-512	195	12	as	as	SCONJ
cana-512	195	13	n	n	NUM
cana-512	195	14	is	be	AUX
cana-512	195	15	even	even	ADV
cana-512	195	16	,	,	PUNCT
cana-512	195	17	𝑛	𝑛	PRON
cana-512	195	18	+	+	NOUN
cana-512	195	19	1	1	NUM
cana-512	195	20	is	be	AUX
cana-512	195	21	odd	odd	ADJ
cana-512	195	22	.	.	PUNCT
cana-512	196	1	hence	hence	ADV
cana-512	196	2	it	it	PRON
cana-512	196	3	is	be	AUX
cana-512	196	4	possible	possible	ADJ
cana-512	196	5	to	to	PART
cana-512	196	6	label	label	VERB
cana-512	196	7	𝑛	𝑛	DET
cana-512	196	8	+	+	NOUN
cana-512	196	9	1	1	NUM
cana-512	196	10	in	in	ADP
cana-512	196	11	any	any	PRON
cana-512	196	12	of	of	ADP
cana-512	196	13	the	the	DET
cana-512	196	14	remaining	remain	VERB
cana-512	196	15	edges	edge	NOUN
cana-512	196	16	.	.	PUNCT
cana-512	197	1	and	and	CCONJ
cana-512	197	2	,	,	PUNCT
cana-512	197	3	it	it	PRON
cana-512	197	4	is	be	AUX
cana-512	197	5	not	not	PART
cana-512	197	6	possible	possible	ADJ
cana-512	197	7	to	to	PART
cana-512	197	8	label	label	VERB
cana-512	197	9	𝑛	𝑛	DET
cana-512	197	10	+	+	NUM
cana-512	197	11	2	2	NUM
cana-512	197	12	in	in	ADP
cana-512	197	13	the	the	DET
cana-512	197	14	remaining	remain	VERB
cana-512	197	15	edges	edge	NOUN
cana-512	197	16	.	.	PUNCT
cana-512	198	1	therefore	therefore	ADV
cana-512	198	2	,	,	PUNCT
cana-512	198	3	it	it	PRON
cana-512	198	4	is	be	AUX
cana-512	198	5	necessary	necessary	ADJ
cana-512	198	6	to	to	PART
cana-512	198	7	remove	remove	VERB
cana-512	198	8	2𝑛	2𝑛	PROPN
cana-512	198	9	–	–	PUNCT
cana-512	198	10	𝑛	𝑛	DET
cana-512	198	11	−	−	NUM
cana-512	198	12	1	1	NUM
cana-512	198	13	=	=	SYM
cana-512	198	14	𝑛	𝑛	PRON
cana-512	198	15	−	−	NUM
cana-512	198	16	1	1	NUM
cana-512	198	17	edges	edge	NOUN
cana-512	198	18	from	from	ADP
cana-512	198	19	c𝑛	c𝑛	PROPN
cana-512	198	20	2	2	NUM
cana-512	198	21	to	to	PART
cana-512	198	22	make	make	VERB
cana-512	198	23	it	it	PRON
cana-512	198	24	as	as	ADP
cana-512	198	25	a	a	DET
cana-512	198	26	relatively	relatively	ADV
cana-512	198	27	prime	prime	ADJ
cana-512	198	28	labeled	label	VERB
cana-512	198	29	graph	graph	NOUN
cana-512	198	30	.	.	PUNCT
cana-512	199	1	that	that	PRON
cana-512	199	2	is	be	AUX
cana-512	199	3	,	,	PUNCT
cana-512	199	4	εr(g	εr(g	PUNCT
cana-512	199	5	)	)	PUNCT
cana-512	200	1	=	=	SYM
cana-512	200	2	𝑛	𝑛	PRON
cana-512	200	3	−	−	NUM
cana-512	200	4	1	1	NUM
cana-512	200	5	.	.	PUNCT
cana-512	201	1	(	(	PUNCT
cana-512	201	2	example	example	NOUN
cana-512	201	3	for	for	ADP
cana-512	201	4	𝑛	𝑛	NOUN
cana-512	201	5	=	=	SYM
cana-512	201	6	6	6	NUM
cana-512	201	7	is	be	AUX
cana-512	201	8	illustrated	illustrate	VERB
cana-512	201	9	in	in	ADP
cana-512	201	10	figure	figure	NOUN
cana-512	201	11	12	12	NUM
cana-512	201	12	)	)	PUNCT
cana-512	201	13	references	reference	NOUN
cana-512	201	14	[	[	X
cana-512	201	15	1	1	NUM
cana-512	201	16	]	]	PUNCT
cana-512	201	17	a.	a.	NOUN
cana-512	201	18	tout	tout	PROPN
cana-512	201	19	,	,	PUNCT
cana-512	201	20	a.	a.	PROPN
cana-512	201	21	n.	n.	PROPN
cana-512	201	22	dabboucy	dabboucy	PROPN
cana-512	201	23	,	,	PUNCT
cana-512	201	24	k.	k.	NOUN
cana-512	201	25	howalla	howalla	PROPN
cana-512	201	26	,	,	PUNCT
cana-512	201	27	1982	1982	NUM
cana-512	201	28	,	,	PUNCT
cana-512	201	29	prime	prime	ADJ
cana-512	201	30	labeling	labeling	NOUN
cana-512	201	31	of	of	ADP
cana-512	201	32	graphs	graph	NOUN
cana-512	201	33	,	,	PUNCT
cana-512	201	34	nat	nat	PROPN
cana-512	201	35	.	.	PUNCT
cana-512	202	1	acad	acad	PROPN
cana-512	202	2	.	.	PUNCT
cana-512	203	1	sci	sci	PROPN
cana-512	203	2	.	.	PUNCT
cana-512	203	3	letters	letter	NOUN
cana-512	203	4	,	,	PUNCT
cana-512	203	5	11	11	NUM
cana-512	203	6	,	,	PUNCT
cana-512	203	7	365	365	NUM
cana-512	203	8	-	-	SYM
cana-512	203	9	368	368	NUM
cana-512	203	10	.	.	PUNCT
cana-512	204	1	[	[	X
cana-512	204	2	2	2	X
cana-512	204	3	]	]	PUNCT
cana-512	204	4	j.	j.	PROPN
cana-512	204	5	a.	a.	PROPN
cana-512	204	6	gallian	gallian	PROPN
cana-512	204	7	,	,	PUNCT
cana-512	204	8	2018	2018	NUM
cana-512	204	9	.	.	PUNCT
cana-512	205	1	a	a	DET
cana-512	205	2	dynamic	dynamic	ADJ
cana-512	205	3	survey	survey	NOUN
cana-512	205	4	of	of	ADP
cana-512	205	5	graph	graph	NOUN
cana-512	205	6	labeling	labeling	NOUN
cana-512	205	7	,	,	PUNCT
cana-512	205	8	electronic	electronic	ADJ
cana-512	205	9	journal	journal	NOUN
cana-512	205	10	of	of	ADP
cana-512	205	11	combinatorics	combinatorics	PROPN
cana-512	205	12	,	,	PUNCT
cana-512	205	13	vol	vol	NOUN
cana-512	205	14	.	.	PROPN
cana-512	205	15	1	1	NUM
cana-512	205	16	.	.	PUNCT
cana-512	206	1	[	[	X
cana-512	206	2	3	3	X
cana-512	206	3	]	]	PUNCT
cana-512	206	4	j.	j.	PROPN
cana-512	206	5	asplund	asplund	PROPN
cana-512	206	6	,	,	PUNCT
cana-512	206	7	n.	n.	PROPN
cana-512	206	8	bradley	bradley	PROPN
cana-512	206	9	fox	fox	PROPN
cana-512	206	10	,	,	PUNCT
cana-512	206	11	2017	2017	NUM
cana-512	206	12	,	,	PUNCT
cana-512	206	13	minimum	minimum	NOUN
cana-512	206	14	coprime	coprime	NOUN
cana-512	206	15	labelings	labeling	NOUN
cana-512	206	16	for	for	ADP
cana-512	206	17	operations	operation	NOUN
cana-512	206	18	on	on	ADP
cana-512	206	19	graphs	graph	NOUN
cana-512	206	20	,	,	PUNCT
cana-512	206	21	integers	integer	NOUN
cana-512	206	22	,	,	PUNCT
cana-512	206	23	pp	pp	ADJ
cana-512	206	24	.	.	PUNCT
cana-512	206	25	1–19	1–19	NOUN
cana-512	206	26	.	.	PUNCT
cana-512	207	1	[	[	X
cana-512	207	2	4	4	NUM
cana-512	207	3	]	]	PUNCT
cana-512	207	4	a.	a.	NOUN
cana-512	207	5	h.	h.	PROPN
cana-512	207	6	berliner	berliner	PROPN
cana-512	207	7	,	,	PUNCT
cana-512	207	8	n.	n.	PROPN
cana-512	207	9	dean	dean	PROPN
cana-512	207	10	,	,	PUNCT
cana-512	207	11	j.	j.	PROPN
cana-512	207	12	hook	hook	PROPN
cana-512	207	13	,	,	PUNCT
cana-512	207	14	a.	a.	NOUN
cana-512	207	15	marr	marr	PROPN
cana-512	207	16	,	,	PUNCT
cana-512	207	17	a.	a.	NOUN
cana-512	207	18	mbirika	mbirika	PROPN
cana-512	207	19	,	,	PUNCT
cana-512	207	20	c.	c.	PROPN
cana-512	207	21	d.	d.	PROPN
cana-512	207	22	mcbee	mcbee	PROPN
cana-512	207	23	,	,	PUNCT
cana-512	207	24	2016	2016	NUM
cana-512	207	25	,	,	PUNCT
cana-512	207	26	coprime	coprime	NOUN
cana-512	207	27	and	and	CCONJ
cana-512	207	28	prime	prime	ADJ
cana-512	207	29	labelings	labeling	NOUN
cana-512	207	30	of	of	ADP
cana-512	207	31	graphs	graph	NOUN
cana-512	207	32	,	,	PUNCT
cana-512	207	33	j.	j.	PROPN
cana-512	207	34	integer	integer	PROPN
cana-512	207	35	seq	seq	PROPN
cana-512	207	36	.	.	PROPN
cana-512	207	37	,	,	PUNCT
cana-512	207	38	19(5	19(5	NUM
cana-512	207	39	)	)	PUNCT
cana-512	207	40	.	.	PUNCT
cana-512	208	1	[	[	X
cana-512	208	2	5	5	X
cana-512	208	3	]	]	PUNCT
cana-512	208	4	k.	k.	PROPN
cana-512	208	5	m.	m.	PROPN
cana-512	208	6	m.	m.	PROPN
cana-512	208	7	haque	haque	PROPN
cana-512	208	8	,	,	PUNCT
cana-512	208	9	l.	l.	PROPN
cana-512	208	10	xiaohui	xiaohui	PROPN
cana-512	208	11	,	,	PUNCT
cana-512	208	12	y.	y.	PROPN
cana-512	208	13	yuansheng	yuansheng	PROPN
cana-512	208	14	,	,	PUNCT
cana-512	208	15	z.	z.	PROPN
cana-512	208	16	pingzhong	pingzhong	PROPN
cana-512	208	17	,	,	PUNCT
cana-512	208	18	2010	2010	NUM
cana-512	208	19	.	.	PUNCT
cana-512	209	1	on	on	ADP
cana-512	209	2	the	the	DET
cana-512	209	3	prime	prime	ADJ
cana-512	209	4	labeling	labeling	NOUN
cana-512	209	5	of	of	ADP
cana-512	209	6	generalized	generalized	ADJ
cana-512	209	7	petersen	petersen	NOUN
cana-512	209	8	graph	graph	NOUN
cana-512	209	9	p	p	PROPN
cana-512	209	10	(	(	PUNCT
cana-512	209	11	n	n	CCONJ
cana-512	209	12	,	,	PUNCT
cana-512	209	13	1	1	NUM
cana-512	209	14	)	)	PUNCT
cana-512	209	15	.	.	PUNCT
cana-512	210	1	utilitas	utilitas	PROPN
cana-512	210	2	mathematica	mathematica	PROPN
cana-512	210	3	,	,	PUNCT
cana-512	210	4	83	83	NUM
cana-512	210	5	.	.	PUNCT
cana-512	211	1	communications	communication	NOUN
cana-512	211	2	on	on	ADP
cana-512	211	3	applied	apply	VERB
cana-512	211	4	nonlinear	nonlinear	ADJ
cana-512	211	5	analysis	analysis	NOUN
cana-512	211	6	issn	issn	NOUN
cana-512	211	7	:	:	PUNCT
cana-512	211	8	1074	1074	NUM
cana-512	211	9	-	-	PUNCT
cana-512	211	10	133x	133x	NUM
cana-512	211	11	vol	vol	NOUN
cana-512	211	12	31	31	NUM
cana-512	211	13	no	no	NOUN
cana-512	211	14	.	.	NOUN
cana-512	211	15	2	2	NUM
cana-512	211	16	(	(	PUNCT
cana-512	211	17	2024	2024	NUM
cana-512	211	18	)	)	PUNCT
cana-512	211	19	54	54	NUM
cana-512	211	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-512	212	1	[	[	X
cana-512	212	2	6	6	NUM
cana-512	212	3	]	]	X
cana-512	212	4	u.	u.	PROPN
cana-512	212	5	m.	m.	PROPN
cana-512	212	6	prajapati	prajapati	PROPN
cana-512	212	7	,	,	PUNCT
cana-512	212	8	s.	s.	PROPN
cana-512	212	9	j.	j.	PROPN
cana-512	212	10	gajjar	gajjar	PROPN
cana-512	212	11	.	.	PUNCT
cana-512	213	1	2015	2015	NUM
cana-512	213	2	,	,	PUNCT
cana-512	213	3	prime	prime	ADJ
cana-512	213	4	labeling	labeling	NOUN
cana-512	213	5	of	of	ADP
cana-512	213	6	generalized	generalized	ADJ
cana-512	213	7	petersen	petersen	NOUN
cana-512	213	8	graph	graph	NOUN
cana-512	213	9	,	,	PUNCT
cana-512	213	10	internat	internat	PROPN
cana-512	213	11	.	.	PUNCT
cana-512	214	1	j.	j.	PROPN
cana-512	214	2	math	math	PROPN
cana-512	214	3	.	.	PUNCT
cana-512	215	1	and	and	CCONJ
cana-512	215	2	soft	soft	ADJ
cana-512	215	3	comput	comput	NOUN
cana-512	215	4	5	5	NUM
cana-512	215	5	:	:	SYM
cana-512	215	6	65	65	NUM
cana-512	215	7	-	-	SYM
cana-512	215	8	71	71	NUM
cana-512	215	9	.	.	PUNCT
cana-512	216	1	[	[	X
cana-512	216	2	7	7	X
cana-512	216	3	]	]	X
cana-512	216	4	n	n	DET
cana-512	216	5	deo	deo	NOUN
cana-512	216	6	,	,	PUNCT
cana-512	216	7	1974	1974	NUM
cana-512	216	8	.	.	PUNCT
cana-512	217	1	graph	graph	NOUN
cana-512	217	2	theory	theory	NOUN
cana-512	217	3	with	with	ADP
cana-512	217	4	applications	application	NOUN
cana-512	217	5	to	to	ADP
cana-512	217	6	engineering	engineering	NOUN
cana-512	217	7	and	and	CCONJ
cana-512	217	8	computer	computer	NOUN
cana-512	217	9	science	science	NOUN
cana-512	217	10	,	,	PUNCT
cana-512	217	11	courier	courier	PROPN
cana-512	217	12	dover	dover	PROPN
cana-512	217	13	publications	publication	NOUN
cana-512	217	14	,	,	PUNCT
cana-512	217	15	2017	2017	NUM
cana-512	217	16	.	.	PUNCT
cana-512	218	1	[	[	X
cana-512	218	2	8	8	NUM
cana-512	218	3	]	]	X
cana-512	218	4	r.	r.	PROPN
cana-512	218	5	janani	janani	PROPN
cana-512	218	6	,	,	PUNCT
cana-512	218	7	and	and	CCONJ
cana-512	218	8	t.	t.	PROPN
cana-512	218	9	ramachandran	ramachandran	PROPN
cana-512	218	10	,	,	PUNCT
cana-512	218	11	on	on	ADP
cana-512	218	12	relatively	relatively	ADV
cana-512	218	13	prime	prime	ADJ
cana-512	218	14	edge	edge	NOUN
cana-512	218	15	labeling	labeling	NOUN
cana-512	218	16	of	of	ADP
cana-512	218	17	graphs	graph	NOUN
cana-512	218	18	,	,	PUNCT
cana-512	218	19	engineering	engineering	NOUN
cana-512	218	20	letters	letter	NOUN
cana-512	218	21	.	.	PUNCT
cana-512	219	1	30(2	30(2	NUM
cana-512	219	2	)	)	PUNCT
cana-512	219	3	,	,	PUNCT
cana-512	219	4	659665	659665	NUM
cana-512	219	5	(	(	PUNCT
cana-512	219	6	2022	2022	NUM
cana-512	219	7	)	)	PUNCT
cana-512	219	8	.	.	PUNCT
cana-512	220	1	[	[	X
cana-512	220	2	9	9	NUM
cana-512	220	3	]	]	X
cana-512	220	4	janani	janani	NOUN
cana-512	220	5	r	r	PROPN
cana-512	220	6	,	,	PUNCT
cana-512	220	7	ramachandran	ramachandran	PROPN
cana-512	220	8	t	t	PROPN
cana-512	220	9	,	,	PUNCT
cana-512	220	10	applications	application	NOUN
cana-512	220	11	of	of	ADP
cana-512	220	12	labeling	labeling	NOUN
cana-512	220	13	in	in	ADP
cana-512	220	14	hypergraph	hypergraph	NOUN
cana-512	220	15	,	,	PUNCT
cana-512	220	16	international	international	ADJ
cana-512	220	17	research	research	NOUN
cana-512	220	18	journal	journal	NOUN
cana-512	220	19	of	of	ADP
cana-512	220	20	multidisciplinary	multidisciplinary	ADJ
cana-512	220	21	scope	scope	NOUN
cana-512	220	22	,	,	PUNCT
cana-512	220	23	5	5	NUM
cana-512	220	24	(	(	PUNCT
cana-512	220	25	1	1	NUM
cana-512	220	26	)	)	PUNCT
cana-512	220	27	,	,	PUNCT
cana-512	220	28	379386	379386	NUM
cana-512	220	29	.	.	PUNCT
cana-512	221	1	[	[	X
cana-512	221	2	10	10	NUM
cana-512	221	3	]	]	X
cana-512	221	4	janani	janani	PROPN
cana-512	221	5	,	,	PUNCT
cana-512	221	6	r.	r.	PROPN
cana-512	221	7	,	,	PUNCT
cana-512	221	8	and	and	CCONJ
cana-512	221	9	t.	t.	PROPN
cana-512	221	10	ramachandran	ramachandran	PROPN
cana-512	221	11	,	,	PUNCT
cana-512	221	12	on	on	ADP
cana-512	221	13	prime	prime	ADJ
cana-512	221	14	index	index	NOUN
cana-512	221	15	of	of	ADP
cana-512	221	16	a	a	DET
cana-512	221	17	graph	graph	NOUN
cana-512	221	18	.	.	PUNCT
cana-512	222	1	ratio	ratio	PROPN
cana-512	222	2	mathematica	mathematica	PROPN
cana-512	222	3	48	48	NUM
cana-512	222	4	(	(	PUNCT
cana-512	222	5	2023	2023	NUM
cana-512	222	6	)	)	PUNCT
cana-512	222	7	.	.	PUNCT
cana-512	223	1	doi	doi	NOUN
cana-512	223	2	:	:	PUNCT
cana-512	223	3	http://dx.doi.org/10.23755/rm.v48i0.1315	http://dx.doi.org/10.23755/rm.v48i0.1315	PROPN
cana-512	223	4	http://dx.doi.org/10.23755/rm.v48i0.1315	http://dx.doi.org/10.23755/rm.v48i0.1315	PROPN