id	sid	tid	token	lemma	pos
cana-2086	1	1	communications	communication	NOUN
cana-2086	1	2	on	on	ADP
cana-2086	1	3	applied	apply	VERB
cana-2086	1	4	nonlinear	nonlinear	ADJ
cana-2086	1	5	analysis	analysis	NOUN
cana-2086	1	6	issn	issn	NOUN
cana-2086	1	7	:	:	PUNCT
cana-2086	1	8	1074	1074	NUM
cana-2086	1	9	-	-	PUNCT
cana-2086	1	10	133x	133x	NUM
cana-2086	1	11	vol	vol	NOUN
cana-2086	1	12	32	32	NUM
cana-2086	1	13	no	no	NOUN
cana-2086	1	14	.	.	PUNCT
cana-2086	2	1	1s	1s	NUM
cana-2086	2	2	(	(	PUNCT
cana-2086	2	3	2025	2025	NUM
cana-2086	2	4	)	)	PUNCT
cana-2086	2	5	31	31	NUM
cana-2086	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	3	2	elegant	elegant	ADJ
cana-2086	3	3	fuzzy	fuzzy	ADJ
cana-2086	3	4	labeling	labeling	NOUN
cana-2086	3	5	of	of	ADP
cana-2086	3	6	graphs	graph	NOUN
cana-2086	3	7	nithya	nithya	PROPN
cana-2086	3	8	sree	sree	PROPN
cana-2086	3	9	a1	a1	PROPN
cana-2086	3	10	,	,	PUNCT
cana-2086	3	11	*	*	PROPN
cana-2086	3	12	,	,	PUNCT
cana-2086	3	13	mahavir	mahavir	PROPN
cana-2086	3	14	b2	b2	PROPN
cana-2086	3	15	and	and	CCONJ
cana-2086	3	16	paulraj	paulraj	PROPN
cana-2086	3	17	m	m	PROPN
cana-2086	3	18	s3	s3	PROPN
cana-2086	3	19	1,2,3pg	1,2,3pg	NOUN
cana-2086	3	20	and	and	CCONJ
cana-2086	3	21	research	research	PROPN
cana-2086	3	22	department	department	PROPN
cana-2086	3	23	of	of	ADP
cana-2086	3	24	mathematics	mathematic	NOUN
cana-2086	3	25	,	,	PUNCT
cana-2086	3	26	agurchand	agurchand	VERB
cana-2086	3	27	manmull	manmull	ADJ
cana-2086	3	28	jain	jain	PROPN
cana-2086	3	29	college	college	PROPN
cana-2086	3	30	,	,	PUNCT
cana-2086	3	31	university	university	PROPN
cana-2086	3	32	of	of	ADP
cana-2086	3	33	madras	madras	PROPN
cana-2086	3	34	,	,	PUNCT
cana-2086	3	35	chennai	chennai	PROPN
cana-2086	3	36	,	,	PUNCT
cana-2086	3	37	india	india	PROPN
cana-2086	3	38	.	.	PUNCT
cana-2086	4	1	e	e	X
cana-2086	4	2	-	-	NOUN
cana-2086	4	3	mail	mail	NOUN
cana-2086	4	4	:	:	PUNCT
cana-2086	4	5	*	*	PUNCT
cana-2086	4	6	ananthanarayanan.nithya@gmail.com	ananthanarayanan.nithya@gmail.com	X
cana-2086	4	7	(	(	PUNCT
cana-2086	4	8	corresponding	corresponding	ADJ
cana-2086	4	9	author	author	NOUN
cana-2086	4	10	)	)	PUNCT
cana-2086	4	11	;	;	PUNCT
cana-2086	4	12	mahavirb@gmail.com	mahavirb@gmail.com	X
cana-2086	4	13	;	;	PUNCT
cana-2086	4	14	mspaulraj65@gmail.com	mspaulraj65@gmail.com	X
cana-2086	4	15	article	article	NOUN
cana-2086	4	16	history	history	NOUN
cana-2086	4	17	:	:	PUNCT
cana-2086	4	18	received	receive	VERB
cana-2086	4	19	:	:	PUNCT
cana-2086	4	20	05	05	NUM
cana-2086	4	21	-	-	SYM
cana-2086	4	22	08	08	NUM
cana-2086	4	23	-	-	PUNCT
cana-2086	4	24	2024	2024	NUM
cana-2086	4	25	revised	revise	VERB
cana-2086	4	26	:	:	PUNCT
cana-2086	4	27	25	25	NUM
cana-2086	4	28	-	-	PUNCT
cana-2086	4	29	09	09	NUM
cana-2086	4	30	-	-	PUNCT
cana-2086	4	31	2024	2024	NUM
cana-2086	4	32	accepted	accept	VERB
cana-2086	4	33	:	:	PUNCT
cana-2086	4	34	07	07	NUM
cana-2086	4	35	-	-	SYM
cana-2086	4	36	10	10	NUM
cana-2086	4	37	-	-	PUNCT
cana-2086	4	38	2024	2024	NUM
cana-2086	4	39	abstract	abstract	NOUN
cana-2086	4	40	:	:	PUNCT
cana-2086	4	41	fuzzy	fuzzy	ADJ
cana-2086	4	42	graphs	graph	NOUN
cana-2086	4	43	and	and	CCONJ
cana-2086	4	44	fuzzy	fuzzy	ADJ
cana-2086	4	45	labelings	labeling	NOUN
cana-2086	4	46	are	be	AUX
cana-2086	4	47	crucial	crucial	ADJ
cana-2086	4	48	in	in	ADP
cana-2086	4	49	modeling	model	VERB
cana-2086	4	50	real	real	ADJ
cana-2086	4	51	-	-	PUNCT
cana-2086	4	52	time	time	NOUN
cana-2086	4	53	systems	system	NOUN
cana-2086	4	54	by	by	ADP
cana-2086	4	55	accommodating	accommodate	VERB
cana-2086	4	56	varying	vary	VERB
cana-2086	4	57	degrees	degree	NOUN
cana-2086	4	58	of	of	ADP
cana-2086	4	59	information	information	NOUN
cana-2086	4	60	precision	precision	NOUN
cana-2086	4	61	.	.	PUNCT
cana-2086	5	1	in	in	ADP
cana-2086	5	2	this	this	DET
cana-2086	5	3	paper	paper	NOUN
cana-2086	5	4	,	,	PUNCT
cana-2086	5	5	we	we	PRON
cana-2086	5	6	introduce	introduce	VERB
cana-2086	5	7	a	a	DET
cana-2086	5	8	novel	novel	ADJ
cana-2086	5	9	fuzzy	fuzzy	ADJ
cana-2086	5	10	labeling	labeling	NOUN
cana-2086	5	11	called	call	VERB
cana-2086	5	12	elegant	elegant	ADJ
cana-2086	5	13	fuzzy	fuzzy	ADJ
cana-2086	5	14	labeling	labeling	NOUN
cana-2086	5	15	.	.	PUNCT
cana-2086	6	1	we	we	PRON
cana-2086	6	2	prove	prove	VERB
cana-2086	6	3	that	that	SCONJ
cana-2086	6	4	a	a	DET
cana-2086	6	5	simple	simple	ADJ
cana-2086	6	6	graph	graph	NOUN
cana-2086	6	7	admits	admit	VERB
cana-2086	6	8	elegant	elegant	ADJ
cana-2086	6	9	fuzzy	fuzzy	ADJ
cana-2086	6	10	labeling	labeling	NOUN
cana-2086	6	11	if	if	SCONJ
cana-2086	6	12	and	and	CCONJ
cana-2086	6	13	only	only	ADV
cana-2086	6	14	if	if	SCONJ
cana-2086	6	15	it	it	PRON
cana-2086	6	16	admits	admit	VERB
cana-2086	6	17	elegant	elegant	ADJ
cana-2086	6	18	labeling	labeling	NOUN
cana-2086	6	19	.	.	PUNCT
cana-2086	7	1	furthermore	furthermore	ADV
cana-2086	7	2	,	,	PUNCT
cana-2086	7	3	we	we	PRON
cana-2086	7	4	prove	prove	VERB
cana-2086	7	5	that	that	SCONJ
cana-2086	7	6	while	while	SCONJ
cana-2086	7	7	a	a	DET
cana-2086	7	8	simple	simple	ADJ
cana-2086	7	9	graph	graph	NOUN
cana-2086	7	10	that	that	PRON
cana-2086	7	11	admits	admit	VERB
cana-2086	7	12	elegant	elegant	ADJ
cana-2086	7	13	fuzzy	fuzzy	ADJ
cana-2086	7	14	labeling	labeling	NOUN
cana-2086	7	15	also	also	ADV
cana-2086	7	16	admits	admit	VERB
cana-2086	7	17	fuzzy	fuzzy	ADJ
cana-2086	7	18	labeling	labeling	NOUN
cana-2086	7	19	,	,	PUNCT
cana-2086	7	20	but	but	CCONJ
cana-2086	7	21	not	not	PART
cana-2086	7	22	conversely	conversely	ADV
cana-2086	7	23	.	.	PUNCT
cana-2086	8	1	we	we	PRON
cana-2086	8	2	identify	identify	VERB
cana-2086	8	3	specific	specific	ADJ
cana-2086	8	4	classes	class	NOUN
cana-2086	8	5	of	of	ADP
cana-2086	8	6	simple	simple	ADJ
cana-2086	8	7	graphs	graph	NOUN
cana-2086	8	8	that	that	PRON
cana-2086	8	9	admit	admit	VERB
cana-2086	8	10	elegant	elegant	ADJ
cana-2086	8	11	fuzzy	fuzzy	ADJ
cana-2086	8	12	labeling	labeling	NOUN
cana-2086	8	13	and	and	CCONJ
cana-2086	8	14	explore	explore	VERB
cana-2086	8	15	practical	practical	ADJ
cana-2086	8	16	application	application	NOUN
cana-2086	8	17	of	of	ADP
cana-2086	8	18	this	this	DET
cana-2086	8	19	new	new	ADJ
cana-2086	8	20	labeling	labeling	NOUN
cana-2086	8	21	approach	approach	NOUN
cana-2086	8	22	.	.	PUNCT
cana-2086	9	1	keywords	keyword	NOUN
cana-2086	9	2	:	:	PUNCT
cana-2086	9	3	elegant	elegant	ADJ
cana-2086	9	4	labeling	labeling	NOUN
cana-2086	9	5	,	,	PUNCT
cana-2086	9	6	fuzzy	fuzzy	ADJ
cana-2086	9	7	graph	graph	NOUN
cana-2086	9	8	,	,	PUNCT
cana-2086	9	9	fuzzy	fuzzy	ADJ
cana-2086	9	10	labeling	labeling	NOUN
cana-2086	9	11	,	,	PUNCT
cana-2086	9	12	elegant	elegant	ADJ
cana-2086	9	13	fuzzy	fuzzy	ADJ
cana-2086	9	14	labeling	labeling	NOUN
cana-2086	9	15	.	.	PUNCT
cana-2086	10	1	mathematics	mathematic	NOUN
cana-2086	10	2	subject	subject	ADJ
cana-2086	10	3	classification	classification	NOUN
cana-2086	10	4	:	:	PUNCT
cana-2086	10	5	05c78	05c78	NUM
cana-2086	10	6	,	,	PUNCT
cana-2086	10	7	05c72	05c72	NOUN
cana-2086	10	8	.	.	PUNCT
cana-2086	11	1	1	1	X
cana-2086	11	2	.	.	X
cana-2086	11	3	introduction	introduction	NOUN
cana-2086	11	4	graph	graph	NOUN
cana-2086	11	5	labeling	labeling	NOUN
cana-2086	11	6	refers	refer	VERB
cana-2086	11	7	to	to	ADP
cana-2086	11	8	assigning	assign	VERB
cana-2086	11	9	labels	label	NOUN
cana-2086	11	10	to	to	ADP
cana-2086	11	11	vertices	vertex	NOUN
cana-2086	11	12	,	,	PUNCT
cana-2086	11	13	edges	edge	NOUN
cana-2086	11	14	,	,	PUNCT
cana-2086	11	15	or	or	CCONJ
cana-2086	11	16	both	both	PRON
cana-2086	11	17	,	,	PUNCT
cana-2086	11	18	of	of	ADP
cana-2086	11	19	a	a	DET
cana-2086	11	20	graph	graph	NOUN
cana-2086	11	21	while	while	SCONJ
cana-2086	11	22	adhering	adhere	VERB
cana-2086	11	23	to	to	ADP
cana-2086	11	24	specific	specific	ADJ
cana-2086	11	25	conditions	condition	NOUN
cana-2086	11	26	.	.	PUNCT
cana-2086	12	1	graph	graph	NOUN
cana-2086	12	2	labeling	labeling	NOUN
cana-2086	12	3	trace	trace	NOUN
cana-2086	12	4	their	their	PRON
cana-2086	12	5	origin	origin	NOUN
cana-2086	12	6	back	back	ADV
cana-2086	12	7	to	to	ADP
cana-2086	12	8	the	the	DET
cana-2086	12	9	one	one	NOUN
cana-2086	12	10	introduced	introduce	VERB
cana-2086	12	11	by	by	ADP
cana-2086	12	12	a.	a.	PROPN
cana-2086	12	13	rosa	rosa	PROPN
cana-2086	13	1	[	[	X
cana-2086	13	2	20	20	NUM
cana-2086	13	3	]	]	PUNCT
cana-2086	13	4	in	in	ADP
cana-2086	13	5	1967	1967	NUM
cana-2086	13	6	called	call	VERB
cana-2086	13	7	β	β	NOUN
cana-2086	13	8	-	-	NOUN
cana-2086	13	9	labeling	labeling	NOUN
cana-2086	13	10	.	.	PUNCT
cana-2086	14	1	s.	s.	PROPN
cana-2086	14	2	w.	w.	PROPN
cana-2086	14	3	golomb	golomb	PROPN
cana-2086	15	1	[	[	X
cana-2086	15	2	9	9	NUM
cana-2086	15	3	]	]	PUNCT
cana-2086	15	4	renamed	rename	VERB
cana-2086	15	5	β	β	NOUN
cana-2086	15	6	-	-	NOUN
cana-2086	15	7	labeling	labeling	NOUN
cana-2086	15	8	as	as	ADP
cana-2086	15	9	graceful	graceful	ADJ
cana-2086	15	10	labeling	labeling	NOUN
cana-2086	15	11	.	.	PUNCT
cana-2086	16	1	gallian	gallian	ADJ
cana-2086	17	1	[	[	X
cana-2086	17	2	6	6	NUM
cana-2086	17	3	]	]	PUNCT
cana-2086	17	4	in	in	ADP
cana-2086	17	5	his	his	PRON
cana-2086	17	6	survey	survey	NOUN
cana-2086	17	7	paper	paper	NOUN
cana-2086	17	8	has	have	AUX
cana-2086	17	9	given	give	VERB
cana-2086	17	10	an	an	DET
cana-2086	17	11	extensive	extensive	ADJ
cana-2086	17	12	account	account	NOUN
cana-2086	17	13	of	of	ADP
cana-2086	17	14	various	various	ADJ
cana-2086	17	15	types	type	NOUN
cana-2086	17	16	of	of	ADP
cana-2086	17	17	graph	graph	NOUN
cana-2086	17	18	labelings	labeling	NOUN
cana-2086	17	19	.	.	PUNCT
cana-2086	18	1	the	the	DET
cana-2086	18	2	graph	graph	NOUN
cana-2086	18	3	labeling	labeling	NOUN
cana-2086	18	4	has	have	VERB
cana-2086	18	5	wide	wide	ADJ
cana-2086	18	6	range	range	NOUN
cana-2086	18	7	of	of	ADP
cana-2086	18	8	applications	application	NOUN
cana-2086	18	9	such	such	ADJ
cana-2086	18	10	as	as	ADP
cana-2086	18	11	x	x	NOUN
cana-2086	18	12	-	-	NOUN
cana-2086	18	13	ray	ray	NOUN
cana-2086	18	14	crystallography	crystallography	NOUN
cana-2086	18	15	,	,	PUNCT
cana-2086	18	16	coding	code	VERB
cana-2086	18	17	theory	theory	NOUN
cana-2086	18	18	,	,	PUNCT
cana-2086	18	19	radar	radar	NOUN
cana-2086	18	20	astronomy	astronomy	NOUN
cana-2086	18	21	,	,	PUNCT
cana-2086	18	22	network	network	NOUN
cana-2086	18	23	design	design	NOUN
cana-2086	18	24	,	,	PUNCT
cana-2086	18	25	circuit	circuit	NOUN
cana-2086	18	26	design	design	NOUN
cana-2086	18	27	,	,	PUNCT
cana-2086	18	28	etc	etc	X
cana-2086	18	29	.	.	X
cana-2086	18	30	in	in	ADP
cana-2086	18	31	real	real	ADJ
cana-2086	18	32	life	life	NOUN
cana-2086	18	33	,	,	PUNCT
cana-2086	18	34	we	we	PRON
cana-2086	18	35	encounter	encounter	VERB
cana-2086	18	36	many	many	ADJ
cana-2086	18	37	problems	problem	NOUN
cana-2086	18	38	that	that	PRON
cana-2086	18	39	are	be	AUX
cana-2086	18	40	imprecise	imprecise	ADJ
cana-2086	18	41	and	and	CCONJ
cana-2086	18	42	ambiguous	ambiguous	ADJ
cana-2086	18	43	.	.	PUNCT
cana-2086	19	1	to	to	PART
cana-2086	19	2	deal	deal	VERB
cana-2086	19	3	with	with	ADP
cana-2086	19	4	such	such	ADJ
cana-2086	19	5	problems	problem	NOUN
cana-2086	19	6	zadeh	zadeh	PROPN
cana-2086	20	1	[	[	X
cana-2086	20	2	30	30	NUM
cana-2086	20	3	]	]	PUNCT
cana-2086	20	4	introduced	introduce	VERB
cana-2086	20	5	the	the	DET
cana-2086	20	6	concept	concept	NOUN
cana-2086	20	7	,	,	PUNCT
cana-2086	20	8	fuzzy	fuzzy	ADJ
cana-2086	20	9	sets	set	NOUN
cana-2086	20	10	.	.	PUNCT
cana-2086	21	1	since	since	SCONJ
cana-2086	21	2	the	the	DET
cana-2086	21	3	graphs	graph	NOUN
cana-2086	21	4	are	be	AUX
cana-2086	21	5	used	use	VERB
cana-2086	21	6	to	to	PART
cana-2086	21	7	model	model	VERB
cana-2086	21	8	many	many	ADJ
cana-2086	21	9	real	real	ADJ
cana-2086	21	10	world	world	NOUN
cana-2086	21	11	problems	problem	NOUN
cana-2086	21	12	,	,	PUNCT
cana-2086	21	13	fuzzy	fuzzy	ADJ
cana-2086	21	14	graph	graph	NOUN
cana-2086	21	15	models	model	NOUN
cana-2086	21	16	are	be	AUX
cana-2086	21	17	required	require	VERB
cana-2086	21	18	to	to	PART
cana-2086	21	19	represent	represent	VERB
cana-2086	21	20	the	the	DET
cana-2086	21	21	vagueness	vagueness	NOUN
cana-2086	21	22	in	in	ADP
cana-2086	21	23	the	the	DET
cana-2086	21	24	objects	object	NOUN
cana-2086	21	25	and	and	CCONJ
cana-2086	21	26	the	the	DET
cana-2086	21	27	vagueness	vagueness	NOUN
cana-2086	21	28	in	in	ADP
cana-2086	21	29	the	the	DET
cana-2086	21	30	relationship	relationship	NOUN
cana-2086	21	31	between	between	ADP
cana-2086	21	32	them	they	PRON
cana-2086	21	33	.	.	PUNCT
cana-2086	22	1	the	the	DET
cana-2086	22	2	first	first	ADJ
cana-2086	22	3	definition	definition	NOUN
cana-2086	22	4	of	of	ADP
cana-2086	22	5	fuzzy	fuzzy	ADJ
cana-2086	22	6	graphs	graph	NOUN
cana-2086	22	7	was	be	AUX
cana-2086	22	8	given	give	VERB
cana-2086	22	9	by	by	ADP
cana-2086	22	10	kaufman	kaufman	PROPN
cana-2086	22	11	[	[	X
cana-2086	22	12	11	11	NUM
cana-2086	22	13	]	]	PUNCT
cana-2086	22	14	that	that	PRON
cana-2086	22	15	was	be	AUX
cana-2086	22	16	based	base	VERB
cana-2086	22	17	on	on	ADP
cana-2086	22	18	zadeh	zadeh	PROPN
cana-2086	22	19	’s	’s	PART
cana-2086	22	20	fuzzy	fuzzy	ADJ
cana-2086	22	21	relations	relation	NOUN
cana-2086	22	22	.	.	PUNCT
cana-2086	23	1	later	later	ADV
cana-2086	23	2	,	,	PUNCT
cana-2086	23	3	rosenfield	rosenfield	VERB
cana-2086	23	4	[	[	X
cana-2086	23	5	21	21	NUM
cana-2086	23	6	]	]	PUNCT
cana-2086	23	7	introduced	introduce	VERB
cana-2086	23	8	the	the	DET
cana-2086	23	9	basic	basic	ADJ
cana-2086	23	10	graph	graph	NOUN
cana-2086	23	11	theoretic	theoretic	ADJ
cana-2086	23	12	concepts	concept	NOUN
cana-2086	23	13	such	such	ADJ
cana-2086	23	14	as	as	ADP
cana-2086	23	15	bridges	bridge	NOUN
cana-2086	23	16	,	,	PUNCT
cana-2086	23	17	paths	path	NOUN
cana-2086	23	18	,	,	PUNCT
cana-2086	23	19	cycles	cycle	NOUN
cana-2086	23	20	,	,	PUNCT
cana-2086	23	21	trees	tree	NOUN
cana-2086	23	22	and	and	CCONJ
cana-2086	23	23	connectedness	connectedness	NOUN
cana-2086	23	24	in	in	ADP
cana-2086	23	25	the	the	DET
cana-2086	23	26	fuzzy	fuzzy	ADJ
cana-2086	23	27	setting	setting	NOUN
cana-2086	23	28	and	and	CCONJ
cana-2086	23	29	established	establish	VERB
cana-2086	23	30	some	some	PRON
cana-2086	23	31	of	of	ADP
cana-2086	23	32	their	their	PRON
cana-2086	23	33	properties	property	NOUN
cana-2086	23	34	.	.	PUNCT
cana-2086	24	1	fuzzy	fuzzy	ADJ
cana-2086	24	2	graphs	graph	NOUN
cana-2086	24	3	are	be	AUX
cana-2086	24	4	highly	highly	ADV
cana-2086	24	5	effective	effective	ADJ
cana-2086	24	6	in	in	ADP
cana-2086	24	7	modeling	model	VERB
cana-2086	24	8	realtime	realtime	NOUN
cana-2086	24	9	systems	system	NOUN
cana-2086	24	10	,	,	PUNCT
cana-2086	24	11	where	where	SCONJ
cana-2086	24	12	varying	varying	ADJ
cana-2086	24	13	levels	level	NOUN
cana-2086	24	14	of	of	ADP
cana-2086	24	15	imprecision	imprecision	NOUN
cana-2086	24	16	are	be	AUX
cana-2086	24	17	inherently	inherently	ADV
cana-2086	24	18	present	present	ADJ
cana-2086	24	19	.	.	PUNCT
cana-2086	25	1	gani	gani	PROPN
cana-2086	25	2	et	et	PROPN
cana-2086	25	3	al	al	PROPN
cana-2086	25	4	.	.	PUNCT
cana-2086	26	1	[	[	X
cana-2086	26	2	7,15	7,15	X
cana-2086	26	3	]	]	PUNCT
cana-2086	26	4	introduced	introduce	VERB
cana-2086	26	5	the	the	DET
cana-2086	26	6	concept	concept	NOUN
cana-2086	26	7	of	of	ADP
cana-2086	26	8	fuzzy	fuzzy	ADJ
cana-2086	26	9	labeling	labeling	NOUN
cana-2086	26	10	graphs	graph	NOUN
cana-2086	26	11	and	and	CCONJ
cana-2086	26	12	studied	study	VERB
cana-2086	26	13	their	their	PRON
cana-2086	26	14	properties	property	NOUN
cana-2086	26	15	.	.	PUNCT
cana-2086	27	1	fuzzy	fuzzy	ADJ
cana-2086	27	2	labeling	labeling	NOUN
cana-2086	27	3	is	be	AUX
cana-2086	27	4	more	more	ADV
cana-2086	27	5	appropriate	appropriate	ADJ
cana-2086	27	6	than	than	ADP
cana-2086	27	7	the	the	DET
cana-2086	27	8	classical	classical	ADJ
cana-2086	27	9	labeling	labeling	NOUN
cana-2086	27	10	for	for	ADP
cana-2086	27	11	many	many	ADJ
cana-2086	27	12	real	real	ADJ
cana-2086	27	13	-	-	PUNCT
cana-2086	27	14	world	world	NOUN
cana-2086	27	15	problems	problem	NOUN
cana-2086	27	16	.	.	PUNCT
cana-2086	28	1	several	several	ADJ
cana-2086	28	2	studies	study	NOUN
cana-2086	28	3	have	have	AUX
cana-2086	28	4	explored	explore	VERB
cana-2086	28	5	different	different	ADJ
cana-2086	28	6	aspects	aspect	NOUN
cana-2086	28	7	of	of	ADP
cana-2086	28	8	fuzzy	fuzzy	ADJ
cana-2086	28	9	graphs	graph	NOUN
cana-2086	28	10	,	,	PUNCT
cana-2086	28	11	highlighting	highlight	VERB
cana-2086	28	12	their	their	PRON
cana-2086	28	13	versatility	versatility	NOUN
cana-2086	28	14	and	and	CCONJ
cana-2086	28	15	utility	utility	NOUN
cana-2086	28	16	.	.	PUNCT
cana-2086	29	1	for	for	ADP
cana-2086	29	2	instance	instance	NOUN
cana-2086	29	3	,	,	PUNCT
cana-2086	29	4	fathalian	fathalian	PROPN
cana-2086	29	5	et	et	PROPN
cana-2086	29	6	al	al	PROPN
cana-2086	29	7	.	.	PUNCT
cana-2086	30	1	[	[	X
cana-2086	30	2	5	5	NUM
cana-2086	30	3	]	]	PUNCT
cana-2086	30	4	studied	study	VERB
cana-2086	30	5	fuzzy	fuzzy	ADJ
cana-2086	30	6	magic	magic	ADJ
cana-2086	30	7	labeling	labeling	NOUN
cana-2086	30	8	on	on	ADP
cana-2086	30	9	simple	simple	ADJ
cana-2086	30	10	graphs	graph	NOUN
cana-2086	30	11	.	.	PUNCT
cana-2086	31	1	giri	giri	PROPN
cana-2086	31	2	et	et	PROPN
cana-2086	31	3	al	al	PROPN
cana-2086	31	4	.	.	PUNCT
cana-2086	32	1	[	[	X
cana-2086	32	2	8	8	NUM
cana-2086	32	3	]	]	PUNCT
cana-2086	32	4	analyzed	analyze	VERB
cana-2086	32	5	fermatean	fermatean	ADJ
cana-2086	32	6	fuzzy	fuzzy	ADJ
cana-2086	32	7	graphs	graph	NOUN
cana-2086	32	8	.	.	PUNCT
cana-2086	33	1	kosari	kosari	PROPN
cana-2086	33	2	et	et	PROPN
cana-2086	33	3	al	al	PROPN
cana-2086	33	4	.	.	PUNCT
cana-2086	34	1	[	[	X
cana-2086	34	2	13	13	NUM
cana-2086	34	3	]	]	PUNCT
cana-2086	34	4	studied	study	VERB
cana-2086	34	5	perfectly	perfectly	ADV
cana-2086	34	6	regular	regular	ADJ
cana-2086	34	7	fuzzy	fuzzy	ADJ
cana-2086	34	8	graphs	graph	NOUN
cana-2086	34	9	.	.	PUNCT
cana-2086	35	1	selvarasu	selvarasu	PROPN
cana-2086	35	2	and	and	CCONJ
cana-2086	35	3	murugan	murugan	PROPN
cana-2086	36	1	[	[	X
cana-2086	36	2	23	23	NUM
cana-2086	36	3	]	]	PUNCT
cana-2086	36	4	studied	study	VERB
cana-2086	36	5	fuzzy	fuzzy	ADJ
cana-2086	36	6	antimagic	antimagic	ADJ
cana-2086	36	7	labeling	labeling	NOUN
cana-2086	36	8	in	in	ADP
cana-2086	36	9	graphs	graph	NOUN
cana-2086	36	10	.	.	PUNCT
cana-2086	37	1	shanmugapriya	shanmugapriya	PROPN
cana-2086	37	2	and	and	CCONJ
cana-2086	37	3	hemalatha	hemalatha	NOUN
cana-2086	38	1	[	[	X
cana-2086	38	2	24	24	NUM
cana-2086	38	3	]	]	PUNCT
cana-2086	38	4	examined	examine	VERB
cana-2086	38	5	fuzzy	fuzzy	ADJ
cana-2086	38	6	vertex	vertex	NOUN
cana-2086	38	7	magic	magic	ADJ
cana-2086	38	8	labeling	labeling	NOUN
cana-2086	38	9	.	.	PUNCT
cana-2086	39	1	borzooei	borzooei	PROPN
cana-2086	39	2	and	and	CCONJ
cana-2086	39	3	rashmanlou	rashmanlou	VERB
cana-2086	40	1	[	[	X
cana-2086	40	2	1	1	X
cana-2086	40	3	]	]	PUNCT
cana-2086	40	4	investigated	investigate	VERB
cana-2086	40	5	cayley	cayley	ADJ
cana-2086	40	6	interval	interval	NOUN
cana-2086	40	7	-	-	PUNCT
cana-2086	40	8	valued	value	VERB
cana-2086	40	9	fuzzy	fuzzy	ADJ
cana-2086	40	10	graphs	graph	NOUN
cana-2086	40	11	.	.	PUNCT
cana-2086	41	1	shoaib	shoaib	PROPN
cana-2086	41	2	et	et	PROPN
cana-2086	41	3	al	al	PROPN
cana-2086	41	4	.	.	PUNCT
cana-2086	42	1	[	[	X
cana-2086	42	2	29	29	NUM
cana-2086	42	3	]	]	PUNCT
cana-2086	42	4	studied	study	VERB
cana-2086	42	5	pythagorean	pythagorean	PROPN
cana-2086	42	6	fuzzy	fuzzy	ADJ
cana-2086	42	7	graphs	graph	NOUN
cana-2086	42	8	.	.	PUNCT
cana-2086	43	1	further	far	ADV
cana-2086	43	2	extending	extend	VERB
cana-2086	43	3	the	the	DET
cana-2086	43	4	field	field	NOUN
cana-2086	43	5	,	,	PUNCT
cana-2086	43	6	shi	shi	PROPN
cana-2086	43	7	et	et	PROPN
cana-2086	43	8	al	al	PROPN
cana-2086	43	9	.	.	PUNCT
cana-2086	44	1	[	[	X
cana-2086	44	2	28	28	NUM
cana-2086	44	3	]	]	PUNCT
cana-2086	44	4	expanded	expand	VERB
cana-2086	44	5	the	the	DET
cana-2086	44	6	concept	concept	NOUN
cana-2086	44	7	of	of	ADP
cana-2086	44	8	energy	energy	NOUN
cana-2086	44	9	on	on	ADP
cana-2086	44	10	the	the	DET
cana-2086	44	11	picture	picture	NOUN
cana-2086	44	12	fuzzy	fuzzy	ADJ
cana-2086	44	13	graphs	graph	NOUN
cana-2086	44	14	.	.	PUNCT
cana-2086	45	1	rao	rao	NOUN
cana-2086	45	2	et	et	PROPN
cana-2086	45	3	al	al	PROPN
cana-2086	45	4	.	.	PUNCT
cana-2086	46	1	[	[	X
cana-2086	46	2	22	22	NUM
cana-2086	46	3	]	]	PUNCT
cana-2086	46	4	introduced	introduce	VERB
cana-2086	46	5	intuitionistic	intuitionistic	ADJ
cana-2086	46	6	fuzzy	fuzzy	ADJ
cana-2086	46	7	trees	tree	NOUN
cana-2086	46	8	.	.	PUNCT
cana-2086	47	1	moreover	moreover	ADV
cana-2086	47	2	,	,	PUNCT
cana-2086	47	3	studies	study	NOUN
cana-2086	47	4	on	on	ADP
cana-2086	47	5	vague	vague	ADJ
cana-2086	47	6	graphs	graph	NOUN
cana-2086	47	7	,	,	PUNCT
cana-2086	47	8	an	an	DET
cana-2086	47	9	extension	extension	NOUN
cana-2086	47	10	of	of	ADP
cana-2086	47	11	fuzzy	fuzzy	ADJ
cana-2086	47	12	graphs	graph	NOUN
cana-2086	47	13	,	,	PUNCT
cana-2086	47	14	have	have	AUX
cana-2086	47	15	been	be	AUX
cana-2086	47	16	conducted	conduct	VERB
cana-2086	47	17	by	by	ADP
cana-2086	47	18	kosari	kosari	PROPN
cana-2086	47	19	et	et	PROPN
cana-2086	47	20	al	al	PROPN
cana-2086	47	21	.	.	PUNCT
cana-2086	48	1	[	[	X
cana-2086	48	2	12	12	NUM
cana-2086	48	3	]	]	PUNCT
cana-2086	48	4	,	,	PUNCT
cana-2086	48	5	rashmanlou	rashmanlou	PROPN
cana-2086	48	6	et	et	PROPN
cana-2086	48	7	al	al	PROPN
cana-2086	48	8	.	.	PUNCT
cana-2086	49	1	[	[	X
cana-2086	49	2	19	19	NUM
cana-2086	49	3	]	]	PUNCT
cana-2086	49	4	,	,	PUNCT
cana-2086	49	5	and	and	CCONJ
cana-2086	49	6	shao	shao	PROPN
cana-2086	49	7	et	et	PROPN
cana-2086	49	8	al	al	PROPN
cana-2086	49	9	.	.	PUNCT
cana-2086	50	1	[	[	X
cana-2086	50	2	25,26	25,26	X
cana-2086	50	3	]	]	X
cana-2086	50	4	.	.	PUNCT
cana-2086	51	1	mailto:ananthanarayanan.nithya@gmail.com	mailto:ananthanarayanan.nithya@gmail.com	X
cana-2086	52	1	mailto:mahavirb@gmail.com	mailto:mahavirb@gmail.com	PROPN
cana-2086	52	2	mailto:mspaulraj65@gmail.com	mailto:mspaulraj65@gmail.com	X
cana-2086	52	3	communications	communication	NOUN
cana-2086	52	4	on	on	ADP
cana-2086	52	5	applied	apply	VERB
cana-2086	52	6	nonlinear	nonlinear	ADJ
cana-2086	52	7	analysis	analysis	NOUN
cana-2086	52	8	issn	issn	NOUN
cana-2086	52	9	:	:	PUNCT
cana-2086	52	10	1074	1074	NUM
cana-2086	52	11	-	-	PUNCT
cana-2086	52	12	133x	133x	NUM
cana-2086	52	13	vol	vol	NOUN
cana-2086	52	14	32	32	NUM
cana-2086	52	15	no	no	NOUN
cana-2086	52	16	.	.	PUNCT
cana-2086	53	1	1s	1s	NUM
cana-2086	53	2	(	(	PUNCT
cana-2086	53	3	2025	2025	NUM
cana-2086	53	4	)	)	PUNCT
cana-2086	53	5	32	32	NUM
cana-2086	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	53	7	in	in	ADP
cana-2086	53	8	1980	1980	NUM
cana-2086	53	9	,	,	PUNCT
cana-2086	53	10	harmonious	harmonious	ADJ
cana-2086	53	11	graphs	graph	NOUN
cana-2086	53	12	were	be	AUX
cana-2086	53	13	introduced	introduce	VERB
cana-2086	53	14	by	by	ADP
cana-2086	53	15	graham	graham	PROPN
cana-2086	53	16	and	and	CCONJ
cana-2086	53	17	sloane	sloane	NOUN
cana-2086	53	18	[	[	X
cana-2086	53	19	10	10	NUM
cana-2086	53	20	]	]	PUNCT
cana-2086	53	21	.	.	PUNCT
cana-2086	54	1	they	they	PRON
cana-2086	54	2	defined	define	VERB
cana-2086	54	3	a	a	DET
cana-2086	54	4	graph	graph	NOUN
cana-2086	54	5	𝐺	𝐺	NOUN
cana-2086	54	6	with	with	ADP
cana-2086	54	7	𝑞	𝑞	PRON
cana-2086	54	8	edges	edge	NOUN
cana-2086	54	9	to	to	PART
cana-2086	54	10	be	be	AUX
cana-2086	54	11	harmonious	harmonious	ADJ
cana-2086	54	12	if	if	SCONJ
cana-2086	54	13	there	there	PRON
cana-2086	54	14	is	be	VERB
cana-2086	54	15	an	an	DET
cana-2086	54	16	injection	injection	NOUN
cana-2086	54	17	𝑓	𝑓	ADP
cana-2086	54	18	from	from	ADP
cana-2086	54	19	the	the	DET
cana-2086	54	20	vertices	vertex	NOUN
cana-2086	54	21	of	of	ADP
cana-2086	54	22	𝐺	𝐺	PROPN
cana-2086	54	23	to	to	ADP
cana-2086	54	24	the	the	DET
cana-2086	54	25	group	group	NOUN
cana-2086	54	26	of	of	ADP
cana-2086	54	27	integers	integer	NOUN
cana-2086	54	28	modulo	modulo	VERB
cana-2086	54	29	𝑞	𝑞	PRON
cana-2086	54	30	such	such	ADJ
cana-2086	54	31	that	that	SCONJ
cana-2086	54	32	when	when	SCONJ
cana-2086	54	33	each	each	DET
cana-2086	54	34	edge	edge	NOUN
cana-2086	54	35	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	54	36	is	be	AUX
cana-2086	54	37	assigned	assign	VERB
cana-2086	54	38	the	the	DET
cana-2086	54	39	label	label	NOUN
cana-2086	54	40	(	(	PUNCT
cana-2086	54	41	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	54	42	)	)	PUNCT
cana-2086	54	43	+	+	CCONJ
cana-2086	54	44	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	54	45	)	)	PUNCT
cana-2086	54	46	)	)	PUNCT
cana-2086	54	47	𝑚𝑜𝑑	𝑚𝑜𝑑	PROPN
cana-2086	55	1	𝑞	𝑞	PROPN
cana-2086	55	2	,	,	PUNCT
cana-2086	55	3	the	the	DET
cana-2086	55	4	resulting	result	VERB
cana-2086	55	5	edge	edge	NOUN
cana-2086	55	6	labels	label	NOUN
cana-2086	55	7	are	be	AUX
cana-2086	55	8	distinct	distinct	ADJ
cana-2086	55	9	.	.	PUNCT
cana-2086	56	1	elegant	elegant	ADJ
cana-2086	56	2	labeling	labeling	NOUN
cana-2086	56	3	,	,	PUNCT
cana-2086	56	4	a	a	DET
cana-2086	56	5	variation	variation	NOUN
cana-2086	56	6	of	of	ADP
cana-2086	56	7	harmonious	harmonious	ADJ
cana-2086	56	8	labeling	labeling	NOUN
cana-2086	56	9	was	be	AUX
cana-2086	56	10	defined	define	VERB
cana-2086	56	11	by	by	ADP
cana-2086	56	12	chang	chang	PROPN
cana-2086	56	13	et	et	PROPN
cana-2086	56	14	al	al	PROPN
cana-2086	56	15	.	.	PUNCT
cana-2086	57	1	[	[	X
cana-2086	57	2	3	3	X
cana-2086	57	3	]	]	PUNCT
cana-2086	57	4	in	in	ADP
cana-2086	57	5	1981	1981	NUM
cana-2086	57	6	.	.	PUNCT
cana-2086	58	1	they	they	PRON
cana-2086	58	2	defined	define	VERB
cana-2086	58	3	,	,	PUNCT
cana-2086	58	4	elegant	elegant	ADJ
cana-2086	58	5	labeling	labeling	NOUN
cana-2086	58	6	𝑓	𝑓	PRON
cana-2086	58	7	of	of	ADP
cana-2086	58	8	a	a	DET
cana-2086	58	9	graph	graph	NOUN
cana-2086	58	10	𝐺	𝐺	NOUN
cana-2086	58	11	with	with	ADP
cana-2086	58	12	𝑞	𝑞	PRON
cana-2086	58	13	edges	edge	NOUN
cana-2086	58	14	as	as	ADP
cana-2086	58	15	an	an	DET
cana-2086	58	16	injective	injective	ADJ
cana-2086	58	17	function	function	NOUN
cana-2086	58	18	from	from	ADP
cana-2086	58	19	the	the	DET
cana-2086	58	20	vertices	vertex	NOUN
cana-2086	58	21	of	of	ADP
cana-2086	58	22	𝐺	𝐺	PROPN
cana-2086	58	23	to	to	ADP
cana-2086	58	24	the	the	DET
cana-2086	58	25	set	set	NOUN
cana-2086	58	26	{	{	PUNCT
cana-2086	58	27	0,1	0,1	NUM
cana-2086	58	28	,	,	PUNCT
cana-2086	58	29	…	…	PUNCT
cana-2086	58	30	,	,	PUNCT
cana-2086	58	31	𝑞	𝑞	X
cana-2086	58	32	}	}	PUNCT
cana-2086	58	33	such	such	ADJ
cana-2086	58	34	that	that	SCONJ
cana-2086	58	35	when	when	SCONJ
cana-2086	58	36	each	each	DET
cana-2086	58	37	edge	edge	NOUN
cana-2086	58	38	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	58	39	is	be	AUX
cana-2086	58	40	assigned	assign	VERB
cana-2086	58	41	the	the	DET
cana-2086	58	42	label	label	NOUN
cana-2086	58	43	(	(	PUNCT
cana-2086	58	44	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	58	45	)	)	PUNCT
cana-2086	59	1	+	+	CCONJ
cana-2086	59	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	59	3	)	)	PUNCT
cana-2086	59	4	)	)	PUNCT
cana-2086	59	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	59	6	(	(	PUNCT
cana-2086	59	7	𝑞	𝑞	X
cana-2086	59	8	+	+	NOUN
cana-2086	59	9	1	1	NUM
cana-2086	59	10	)	)	PUNCT
cana-2086	59	11	,	,	PUNCT
cana-2086	59	12	the	the	DET
cana-2086	59	13	resulting	result	VERB
cana-2086	59	14	edge	edge	NOUN
cana-2086	59	15	labels	label	NOUN
cana-2086	59	16	are	be	AUX
cana-2086	59	17	distinct	distinct	ADJ
cana-2086	59	18	and	and	CCONJ
cana-2086	59	19	nonzero	nonzero	NOUN
cana-2086	59	20	.	.	PUNCT
cana-2086	60	1	cahit	cahit	PROPN
cana-2086	61	1	[	[	X
cana-2086	61	2	2	2	NUM
cana-2086	61	3	]	]	PUNCT
cana-2086	61	4	,	,	PUNCT
cana-2086	61	5	elumalai	elumalai	NOUN
cana-2086	61	6	and	and	CCONJ
cana-2086	61	7	sethuraman	sethuraman	NOUN
cana-2086	62	1	[	[	X
cana-2086	62	2	4	4	NUM
cana-2086	62	3	]	]	PUNCT
cana-2086	62	4	,	,	PUNCT
cana-2086	62	5	maulidia	maulidia	NOUN
cana-2086	62	6	and	and	CCONJ
cana-2086	62	7	purwanto	purwanto	ADP
cana-2086	62	8	[	[	X
cana-2086	62	9	14	14	NUM
cana-2086	62	10	]	]	PUNCT
cana-2086	62	11	,	,	PUNCT
cana-2086	62	12	parwati	parwati	NOUN
cana-2086	62	13	and	and	CCONJ
cana-2086	62	14	purwanto	purwanto	ADP
cana-2086	62	15	[	[	X
cana-2086	62	16	16	16	NUM
cana-2086	62	17	]	]	PUNCT
cana-2086	62	18	,	,	PUNCT
cana-2086	62	19	prihandini	prihandini	PROPN
cana-2086	62	20	et	et	PROPN
cana-2086	62	21	al	al	PROPN
cana-2086	62	22	.	.	PUNCT
cana-2086	63	1	[	[	X
cana-2086	63	2	17,18	17,18	X
cana-2086	63	3	]	]	X
cana-2086	63	4	and	and	CCONJ
cana-2086	63	5	,	,	PUNCT
cana-2086	63	6	sherly	sherly	NOUN
cana-2086	63	7	and	and	CCONJ
cana-2086	63	8	purwanto	purwanto	ADP
cana-2086	63	9	[	[	X
cana-2086	63	10	27	27	NUM
cana-2086	63	11	]	]	PUNCT
cana-2086	63	12	have	have	AUX
cana-2086	63	13	produced	produce	VERB
cana-2086	63	14	several	several	ADJ
cana-2086	63	15	results	result	NOUN
cana-2086	63	16	on	on	ADP
cana-2086	63	17	elegant	elegant	ADJ
cana-2086	63	18	graphs	graph	NOUN
cana-2086	63	19	.	.	PUNCT
cana-2086	64	1	motivated	motivate	VERB
cana-2086	64	2	by	by	ADP
cana-2086	64	3	the	the	DET
cana-2086	64	4	need	need	NOUN
cana-2086	64	5	to	to	PART
cana-2086	64	6	address	address	VERB
cana-2086	64	7	uncertainties	uncertainty	NOUN
cana-2086	64	8	and	and	CCONJ
cana-2086	64	9	impreciseness	impreciseness	ADJ
cana-2086	64	10	in	in	ADP
cana-2086	64	11	problems	problem	NOUN
cana-2086	64	12	more	more	ADV
cana-2086	64	13	effectively	effectively	ADV
cana-2086	64	14	,	,	PUNCT
cana-2086	64	15	in	in	ADP
cana-2086	64	16	this	this	DET
cana-2086	64	17	paper	paper	NOUN
cana-2086	64	18	,	,	PUNCT
cana-2086	64	19	we	we	PRON
cana-2086	64	20	introduce	introduce	VERB
cana-2086	64	21	a	a	DET
cana-2086	64	22	new	new	ADJ
cana-2086	64	23	type	type	NOUN
cana-2086	64	24	of	of	ADP
cana-2086	64	25	fuzzy	fuzzy	ADJ
cana-2086	64	26	labeling	labeling	NOUN
cana-2086	64	27	called	call	VERB
cana-2086	64	28	the	the	DET
cana-2086	64	29	elegant	elegant	ADJ
cana-2086	64	30	fuzzy	fuzzy	ADJ
cana-2086	64	31	labeling	labeling	NOUN
cana-2086	64	32	.	.	PUNCT
cana-2086	65	1	we	we	PRON
cana-2086	65	2	prove	prove	VERB
cana-2086	65	3	that	that	SCONJ
cana-2086	65	4	if	if	SCONJ
cana-2086	65	5	a	a	DET
cana-2086	65	6	simple	simple	ADJ
cana-2086	65	7	graph	graph	NOUN
cana-2086	65	8	admits	admit	VERB
cana-2086	65	9	elegant	elegant	ADJ
cana-2086	65	10	fuzzy	fuzzy	ADJ
cana-2086	65	11	labeling	labeling	NOUN
cana-2086	65	12	,	,	PUNCT
cana-2086	65	13	then	then	ADV
cana-2086	65	14	it	it	PRON
cana-2086	65	15	admits	admit	VERB
cana-2086	65	16	elegant	elegant	ADJ
cana-2086	65	17	labeling	labeling	NOUN
cana-2086	65	18	and	and	CCONJ
cana-2086	65	19	fuzzy	fuzzy	ADJ
cana-2086	65	20	labeling	labeling	NOUN
cana-2086	65	21	,	,	PUNCT
cana-2086	65	22	and	and	CCONJ
cana-2086	65	23	also	also	ADV
cana-2086	65	24	prove	prove	VERB
cana-2086	65	25	that	that	SCONJ
cana-2086	65	26	if	if	SCONJ
cana-2086	65	27	a	a	DET
cana-2086	65	28	simple	simple	ADJ
cana-2086	65	29	graph	graph	NOUN
cana-2086	65	30	admits	admit	VERB
cana-2086	65	31	elegant	elegant	ADJ
cana-2086	65	32	labeling	labeling	NOUN
cana-2086	65	33	,	,	PUNCT
cana-2086	65	34	then	then	ADV
cana-2086	65	35	it	it	PRON
cana-2086	65	36	admits	admit	VERB
cana-2086	65	37	elegant	elegant	ADJ
cana-2086	65	38	fuzzy	fuzzy	ADJ
cana-2086	65	39	labeling	labeling	NOUN
cana-2086	65	40	.	.	PUNCT
cana-2086	66	1	we	we	PRON
cana-2086	66	2	prove	prove	VERB
cana-2086	66	3	that	that	SCONJ
cana-2086	66	4	the	the	DET
cana-2086	66	5	path	path	NOUN
cana-2086	66	6	graphs	graph	VERB
cana-2086	66	7	𝑃𝑛	𝑃𝑛	NOUN
cana-2086	66	8	,	,	PUNCT
cana-2086	66	9	𝑛	𝑛	PRON
cana-2086	66	10	≠	≠	PROPN
cana-2086	66	11	4	4	NUM
cana-2086	66	12	and	and	CCONJ
cana-2086	66	13	cycles	cycle	NOUN
cana-2086	66	14	𝐶𝑛	𝐶𝑛	PROPN
cana-2086	66	15	,	,	PUNCT
cana-2086	66	16	𝑛	𝑛	PRON
cana-2086	66	17	≡	≡	PROPN
cana-2086	66	18	0,3	0,3	NUM
cana-2086	66	19	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	66	20	4	4	NUM
cana-2086	66	21	,	,	PUNCT
cana-2086	66	22	admit	admit	VERB
cana-2086	66	23	elegant	elegant	ADJ
cana-2086	66	24	fuzzy	fuzzy	ADJ
cana-2086	66	25	labeling	labeling	NOUN
cana-2086	66	26	.	.	PUNCT
cana-2086	67	1	the	the	DET
cana-2086	67	2	line	line	NOUN
cana-2086	67	3	graph	graph	NOUN
cana-2086	67	4	𝐿(𝐺	𝐿(𝐺	PROPN
cana-2086	67	5	)	)	PUNCT
cana-2086	67	6	of	of	ADP
cana-2086	67	7	a	a	DET
cana-2086	67	8	simple	simple	ADJ
cana-2086	67	9	graph	graph	NOUN
cana-2086	67	10	𝐺	𝐺	NOUN
cana-2086	67	11	is	be	AUX
cana-2086	67	12	the	the	DET
cana-2086	67	13	graph	graph	NOUN
cana-2086	67	14	with	with	ADP
cana-2086	67	15	edges	edge	NOUN
cana-2086	67	16	of	of	ADP
cana-2086	67	17	𝐺	𝐺	PROPN
cana-2086	67	18	as	as	ADP
cana-2086	67	19	its	its	PRON
cana-2086	67	20	vertices	vertex	NOUN
cana-2086	67	21	,	,	PUNCT
cana-2086	67	22	where	where	SCONJ
cana-2086	67	23	two	two	NUM
cana-2086	67	24	vertices	vertex	NOUN
cana-2086	67	25	are	be	AUX
cana-2086	67	26	adjacent	adjacent	ADJ
cana-2086	67	27	in	in	ADP
cana-2086	67	28	𝐿(𝐺	𝐿(𝐺	PROPN
cana-2086	67	29	)	)	PUNCT
cana-2086	68	1	if	if	SCONJ
cana-2086	68	2	and	and	CCONJ
cana-2086	68	3	only	only	ADV
cana-2086	68	4	if	if	SCONJ
cana-2086	68	5	the	the	DET
cana-2086	68	6	corresponding	corresponding	ADJ
cana-2086	68	7	edges	edge	NOUN
cana-2086	68	8	are	be	AUX
cana-2086	68	9	incident	incident	NOUN
cana-2086	68	10	in	in	ADP
cana-2086	68	11	𝐺.	𝐺.	NOUN
cana-2086	68	12	we	we	PRON
cana-2086	68	13	prove	prove	VERB
cana-2086	68	14	that	that	SCONJ
cana-2086	68	15	𝐿(𝑃𝑛	𝐿(𝑃𝑛	PROPN
cana-2086	68	16	)	)	PUNCT
cana-2086	68	17	,	,	PUNCT
cana-2086	68	18	where	where	SCONJ
cana-2086	68	19	𝑛	𝑛	DET
cana-2086	68	20	≠	≠	PROPN
cana-2086	68	21	5	5	NUM
cana-2086	68	22	and	and	CCONJ
cana-2086	68	23	𝐿(𝐶𝑛	𝐿(𝐶𝑛	PRON
cana-2086	68	24	)	)	PUNCT
cana-2086	68	25	,	,	PUNCT
cana-2086	68	26	where	where	SCONJ
cana-2086	68	27	𝑛	𝑛	DET
cana-2086	68	28	≡	≡	PROPN
cana-2086	68	29	0,3	0,3	NUM
cana-2086	68	30	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	68	31	4	4	NUM
cana-2086	68	32	,	,	PUNCT
cana-2086	68	33	admit	admit	VERB
cana-2086	68	34	elegant	elegant	ADJ
cana-2086	68	35	fuzzy	fuzzy	ADJ
cana-2086	68	36	labeling	labeling	NOUN
cana-2086	68	37	.	.	PUNCT
cana-2086	69	1	we	we	PRON
cana-2086	69	2	also	also	ADV
cana-2086	69	3	provide	provide	VERB
cana-2086	69	4	an	an	DET
cana-2086	69	5	application	application	NOUN
cana-2086	69	6	of	of	ADP
cana-2086	69	7	elegant	elegant	ADJ
cana-2086	69	8	fuzzy	fuzzy	ADJ
cana-2086	69	9	labeling	labeling	NOUN
cana-2086	69	10	.	.	PUNCT
cana-2086	70	1	if	if	SCONJ
cana-2086	70	2	𝐺	𝐺	PROPN
cana-2086	70	3	is	be	AUX
cana-2086	70	4	a	a	DET
cana-2086	70	5	graph	graph	NOUN
cana-2086	70	6	,	,	PUNCT
cana-2086	70	7	we	we	PRON
cana-2086	70	8	denote	denote	VERB
cana-2086	70	9	the	the	DET
cana-2086	70	10	vertex	vertex	NOUN
cana-2086	70	11	set	set	NOUN
cana-2086	70	12	of	of	ADP
cana-2086	70	13	𝐺	𝐺	PROPN
cana-2086	70	14	by	by	ADP
cana-2086	70	15	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	70	16	)	)	PUNCT
cana-2086	70	17	and	and	CCONJ
cana-2086	70	18	the	the	DET
cana-2086	70	19	edge	edge	NOUN
cana-2086	70	20	set	set	NOUN
cana-2086	70	21	of	of	ADP
cana-2086	70	22	𝐺	𝐺	PROPN
cana-2086	70	23	by	by	ADP
cana-2086	70	24	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	70	25	)	)	PUNCT
cana-2086	70	26	.	.	PUNCT
cana-2086	71	1	2	2	X
cana-2086	71	2	.	.	X
cana-2086	71	3	preliminaries	preliminary	NOUN
cana-2086	71	4	definition	definition	NOUN
cana-2086	71	5	2.1	2.1	NUM
cana-2086	71	6	.	.	PUNCT
cana-2086	72	1	[	[	X
cana-2086	72	2	3	3	NUM
cana-2086	72	3	]	]	X
cana-2086	72	4	elegant	elegant	ADJ
cana-2086	72	5	labeling	labeling	NOUN
cana-2086	72	6	𝑓	𝑓	PRON
cana-2086	72	7	of	of	ADP
cana-2086	72	8	a	a	DET
cana-2086	72	9	graph	graph	NOUN
cana-2086	72	10	𝐺	𝐺	NOUN
cana-2086	72	11	with	with	ADP
cana-2086	72	12	𝑞	𝑞	PRON
cana-2086	72	13	edges	edge	NOUN
cana-2086	72	14	is	be	AUX
cana-2086	72	15	an	an	DET
cana-2086	72	16	injective	injective	ADJ
cana-2086	72	17	function	function	NOUN
cana-2086	72	18	from	from	ADP
cana-2086	72	19	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	72	20	)	)	PUNCT
cana-2086	72	21	into	into	ADP
cana-2086	72	22	the	the	DET
cana-2086	72	23	set	set	NOUN
cana-2086	72	24	{	{	PUNCT
cana-2086	72	25	0	0	NUM
cana-2086	72	26	,	,	PUNCT
cana-2086	72	27	1	1	NUM
cana-2086	72	28	,	,	PUNCT
cana-2086	72	29	…	…	PUNCT
cana-2086	72	30	,	,	PUNCT
cana-2086	72	31	𝑞	𝑞	X
cana-2086	72	32	}	}	PUNCT
cana-2086	72	33	such	such	ADJ
cana-2086	72	34	that	that	SCONJ
cana-2086	72	35	the	the	DET
cana-2086	72	36	function	function	NOUN
cana-2086	72	37	𝑔	𝑔	PROPN
cana-2086	72	38	from	from	ADP
cana-2086	72	39	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	72	40	)	)	PUNCT
cana-2086	72	41	into	into	ADP
cana-2086	72	42	the	the	DET
cana-2086	72	43	set	set	NOUN
cana-2086	72	44	{	{	PUNCT
cana-2086	72	45	1	1	NUM
cana-2086	72	46	,	,	PUNCT
cana-2086	72	47	2	2	NUM
cana-2086	72	48	,	,	PUNCT
cana-2086	72	49	…	…	PUNCT
cana-2086	72	50	,	,	PUNCT
cana-2086	72	51	𝑞	𝑞	X
cana-2086	72	52	}	}	PUNCT
cana-2086	72	53	defined	define	VERB
cana-2086	72	54	as	as	ADP
cana-2086	72	55	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-2086	72	56	)	)	PUNCT
cana-2086	72	57	=	=	PUNCT
cana-2086	72	58	(	(	PUNCT
cana-2086	72	59	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	72	60	)	)	PUNCT
cana-2086	73	1	+	+	CCONJ
cana-2086	73	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	73	3	)	)	PUNCT
cana-2086	73	4	)	)	PUNCT
cana-2086	73	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	73	6	(	(	PUNCT
cana-2086	73	7	𝑞	𝑞	NOUN
cana-2086	73	8	+	+	NOUN
cana-2086	73	9	1	1	NUM
cana-2086	73	10	)	)	PUNCT
cana-2086	73	11	for	for	ADP
cana-2086	73	12	every	every	DET
cana-2086	73	13	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	73	14	in	in	ADP
cana-2086	73	15	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	73	16	)	)	PUNCT
cana-2086	73	17	is	be	AUX
cana-2086	73	18	injective	injective	ADJ
cana-2086	73	19	.	.	PUNCT
cana-2086	73	20	example	example	NOUN
cana-2086	74	1	2.2	2.2	NUM
cana-2086	74	2	.	.	PUNCT
cana-2086	74	3	fig	fig	NOUN
cana-2086	74	4	.	.	PUNCT
cana-2086	75	1	1	1	NUM
cana-2086	75	2	illustrates	illustrate	VERB
cana-2086	75	3	an	an	DET
cana-2086	75	4	elegant	elegant	ADJ
cana-2086	75	5	graph	graph	NOUN
cana-2086	75	6	𝐺	𝐺	NOUN
cana-2086	75	7	with	with	ADP
cana-2086	75	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	75	9	)	)	PUNCT
cana-2086	75	10	=	=	SYM
cana-2086	75	11	{	{	PUNCT
cana-2086	75	12	𝑣1	𝑣1	PROPN
cana-2086	75	13	,	,	PUNCT
cana-2086	75	14	𝑣2	𝑣2	PROPN
cana-2086	75	15	,	,	PUNCT
cana-2086	75	16	𝑣3	𝑣3	ADJ
cana-2086	75	17	}	}	PUNCT
cana-2086	75	18	and	and	CCONJ
cana-2086	75	19	𝐸(𝐺	𝐸(𝐺	NUM
cana-2086	75	20	)	)	PUNCT
cana-2086	75	21	=	=	SYM
cana-2086	75	22	{	{	PUNCT
cana-2086	75	23	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-2086	75	24	,	,	PUNCT
cana-2086	75	25	𝑣2𝑣3	𝑣2𝑣3	ADV
cana-2086	75	26	,	,	PUNCT
cana-2086	75	27	𝑣1𝑣3	𝑣1𝑣3	ADV
cana-2086	75	28	}	}	PUNCT
cana-2086	75	29	.	.	PUNCT
cana-2086	76	1	an	an	DET
cana-2086	76	2	elegant	elegant	ADJ
cana-2086	76	3	labeling	labeling	NOUN
cana-2086	76	4	𝑓	𝑓	PRON
cana-2086	76	5	:	:	PUNCT
cana-2086	76	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	76	7	)	)	PUNCT
cana-2086	76	8	→	→	SYM
cana-2086	76	9	{	{	PUNCT
cana-2086	76	10	0	0	NUM
cana-2086	76	11	,	,	PUNCT
cana-2086	76	12	1	1	NUM
cana-2086	76	13	,	,	PUNCT
cana-2086	76	14	2	2	NUM
cana-2086	76	15	,	,	PUNCT
cana-2086	76	16	3	3	NUM
cana-2086	76	17	}	}	PUNCT
cana-2086	76	18	of	of	ADP
cana-2086	76	19	𝐺	𝐺	PROPN
cana-2086	76	20	is	be	AUX
cana-2086	76	21	defined	define	VERB
cana-2086	76	22	as	as	ADP
cana-2086	76	23	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-2086	76	24	)	)	PUNCT
cana-2086	76	25	=	=	SYM
cana-2086	76	26	0	0	NUM
cana-2086	76	27	,	,	PUNCT
cana-2086	76	28	𝑓(𝑣2	𝑓(𝑣2	NOUN
cana-2086	76	29	)	)	PUNCT
cana-2086	76	30	=	=	SYM
cana-2086	76	31	1	1	NUM
cana-2086	76	32	,	,	PUNCT
cana-2086	76	33	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	76	34	)	)	PUNCT
cana-2086	77	1	=	=	SYM
cana-2086	77	2	2	2	X
cana-2086	77	3	.	.	X
cana-2086	77	4	hence	hence	ADV
cana-2086	77	5	the	the	DET
cana-2086	77	6	edge	edge	NOUN
cana-2086	77	7	labels	label	NOUN
cana-2086	77	8	are	be	AUX
cana-2086	77	9	𝑔(𝑣1𝑣2	𝑔(𝑣1𝑣2	NOUN
cana-2086	77	10	)	)	PUNCT
cana-2086	77	11	=	=	SYM
cana-2086	77	12	1	1	NUM
cana-2086	77	13	,	,	PUNCT
cana-2086	77	14	𝑔(𝑣2𝑣3	𝑔(𝑣2𝑣3	NOUN
cana-2086	77	15	)	)	PUNCT
cana-2086	77	16	=	=	SYM
cana-2086	77	17	3	3	NUM
cana-2086	77	18	,	,	PUNCT
cana-2086	77	19	𝑔(𝑣1𝑣3	𝑔(𝑣1𝑣3	NOUN
cana-2086	77	20	)	)	PUNCT
cana-2086	77	21	=	=	SYM
cana-2086	77	22	2	2	X
cana-2086	77	23	.	.	X
cana-2086	77	24	definition	definition	NOUN
cana-2086	77	25	2.3	2.3	NUM
cana-2086	77	26	.	.	PUNCT
cana-2086	78	1	[	[	X
cana-2086	78	2	21	21	NUM
cana-2086	78	3	]	]	X
cana-2086	78	4	a	a	DET
cana-2086	78	5	graph	graph	NOUN
cana-2086	78	6	𝐺	𝐺	NOUN
cana-2086	78	7	=	=	SYM
cana-2086	78	8	(	(	PUNCT
cana-2086	78	9	𝜇	𝜇	X
cana-2086	78	10	,	,	PUNCT
cana-2086	78	11	𝜌	𝜌	X
cana-2086	78	12	)	)	PUNCT
cana-2086	78	13	with	with	ADP
cana-2086	78	14	vertex	vertex	NOUN
cana-2086	78	15	set	set	VERB
cana-2086	78	16	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	78	17	)	)	PUNCT
cana-2086	78	18	,	,	PUNCT
cana-2086	78	19	edge	edge	VERB
cana-2086	78	20	set	set	VERB
cana-2086	78	21	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	78	22	)	)	PUNCT
cana-2086	78	23	and	and	CCONJ
cana-2086	78	24	a	a	DET
cana-2086	78	25	pair	pair	NOUN
cana-2086	78	26	of	of	ADP
cana-2086	78	27	functions	function	NOUN
cana-2086	78	28	𝜇	𝜇	ADP
cana-2086	78	29	:	:	PUNCT
cana-2086	78	30	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	78	31	)	)	PUNCT
cana-2086	78	32	→	→	PUNCT
cana-2086	79	1	[	[	X
cana-2086	79	2	0,1	0,1	NUM
cana-2086	79	3	]	]	PUNCT
cana-2086	79	4	and	and	CCONJ
cana-2086	79	5	𝜌	𝜌	ADP
cana-2086	79	6	:	:	PUNCT
cana-2086	79	7	𝐸(𝐺	𝐸(𝐺	X
cana-2086	79	8	)	)	PUNCT
cana-2086	79	9	→	→	PUNCT
cana-2086	80	1	[	[	X
cana-2086	80	2	0,1	0,1	NUM
cana-2086	80	3	]	]	PUNCT
cana-2086	80	4	is	be	AUX
cana-2086	80	5	called	call	VERB
cana-2086	80	6	fuzzy	fuzzy	ADJ
cana-2086	80	7	graph	graph	NOUN
cana-2086	80	8	if	if	SCONJ
cana-2086	80	9	for	for	ADP
cana-2086	80	10	every	every	DET
cana-2086	80	11	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	80	12	in	in	ADP
cana-2086	80	13	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	80	14	)	)	PUNCT
cana-2086	80	15	,	,	PUNCT
cana-2086	80	16	𝜌(𝑢𝑣	𝜌(𝑢𝑣	NOUN
cana-2086	80	17	)	)	PUNCT
cana-2086	80	18	≤	≤	NUM
cana-2086	81	1	𝜇(𝑢	𝜇(𝑢	NOUN
cana-2086	81	2	)	)	PUNCT
cana-2086	81	3	∧	∧	NOUN
cana-2086	81	4	𝜇(𝑣	𝜇(𝑣	NOUN
cana-2086	81	5	)	)	PUNCT
cana-2086	82	1	(=	(=	NOUN
cana-2086	82	2	𝑚𝑖𝑛{𝜇(𝑢	𝑚𝑖𝑛{𝜇(𝑢	NOUN
cana-2086	82	3	)	)	PUNCT
cana-2086	82	4	,	,	PUNCT
cana-2086	82	5	𝜇(𝑣	𝜇(𝑣	PROPN
cana-2086	82	6	)	)	PUNCT
cana-2086	82	7	}	}	PUNCT
cana-2086	82	8	)	)	PUNCT
cana-2086	82	9	.	.	PUNCT
cana-2086	83	1	example	example	NOUN
cana-2086	84	1	2.4	2.4	NUM
cana-2086	84	2	.	.	PUNCT
cana-2086	84	3	fig	fig	NOUN
cana-2086	84	4	.	.	PUNCT
cana-2086	85	1	2	2	NUM
cana-2086	85	2	illustrates	illustrate	VERB
cana-2086	85	3	a	a	DET
cana-2086	85	4	fuzzy	fuzzy	ADJ
cana-2086	85	5	graph	graph	NOUN
cana-2086	85	6	g	g	NOUN
cana-2086	85	7	with	with	ADP
cana-2086	85	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	85	9	)	)	PUNCT
cana-2086	85	10	=	=	SYM
cana-2086	85	11	{	{	PUNCT
cana-2086	85	12	𝑣1	𝑣1	PROPN
cana-2086	85	13	,	,	PUNCT
cana-2086	85	14	𝑣2	𝑣2	PROPN
cana-2086	85	15	,	,	PUNCT
cana-2086	85	16	𝑣3	𝑣3	ADJ
cana-2086	85	17	,	,	PUNCT
cana-2086	85	18	𝑣4	𝑣4	NOUN
cana-2086	85	19	}	}	PUNCT
cana-2086	85	20	and	and	CCONJ
cana-2086	85	21	𝐸(𝐺	𝐸(𝐺	NUM
cana-2086	85	22	)	)	PUNCT
cana-2086	85	23	=	=	SYM
cana-2086	85	24	{	{	PUNCT
cana-2086	85	25	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-2086	85	26	,	,	PUNCT
cana-2086	85	27	𝑣2𝑣3	𝑣2𝑣3	ADV
cana-2086	85	28	,	,	PUNCT
cana-2086	85	29	𝑣3𝑣4	𝑣3𝑣4	ADP
cana-2086	85	30	,	,	PUNCT
cana-2086	85	31	𝑣1𝑣4	𝑣1𝑣4	NOUN
cana-2086	85	32	}	}	PUNCT
cana-2086	85	33	.	.	PUNCT
cana-2086	86	1	a	a	DET
cana-2086	86	2	function	function	NOUN
cana-2086	86	3	𝜇	𝜇	ADP
cana-2086	86	4	:	:	PUNCT
cana-2086	86	5	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	86	6	)	)	PUNCT
cana-2086	86	7	→	→	PUNCT
cana-2086	87	1	[	[	X
cana-2086	87	2	0,1	0,1	NUM
cana-2086	87	3	]	]	PUNCT
cana-2086	87	4	is	be	AUX
cana-2086	87	5	defined	define	VERB
cana-2086	87	6	as	as	ADP
cana-2086	87	7	𝜇(𝑣1	𝜇(𝑣1	NOUN
cana-2086	87	8	)	)	PUNCT
cana-2086	87	9	=	=	SYM
cana-2086	87	10	0.5	0.5	NUM
cana-2086	87	11	,	,	PUNCT
cana-2086	87	12	𝜇(𝑣2	𝜇(𝑣2	NOUN
cana-2086	87	13	)	)	PUNCT
cana-2086	87	14	=	=	SYM
cana-2086	87	15	0.3	0.3	NUM
cana-2086	87	16	,	,	PUNCT
cana-2086	87	17	𝜇(𝑣3	𝜇(𝑣3	NOUN
cana-2086	87	18	)	)	PUNCT
cana-2086	87	19	=	=	SYM
cana-2086	87	20	0.4	0.4	NUM
cana-2086	87	21	,	,	PUNCT
cana-2086	87	22	𝜇(𝑣4	𝜇(𝑣4	NOUN
cana-2086	87	23	)	)	PUNCT
cana-2086	87	24	=	=	SYM
cana-2086	87	25	0.6	0.6	NUM
cana-2086	87	26	,	,	PUNCT
cana-2086	87	27	and	and	CCONJ
cana-2086	87	28	𝜌	𝜌	ADP
cana-2086	87	29	:	:	PUNCT
cana-2086	87	30	𝐸(𝐺	𝐸(𝐺	X
cana-2086	87	31	)	)	PUNCT
cana-2086	87	32	→	→	PUNCT
cana-2086	88	1	[	[	X
cana-2086	88	2	0,1	0,1	NUM
cana-2086	88	3	]	]	PUNCT
cana-2086	88	4	is	be	AUX
cana-2086	88	5	defined	define	VERB
cana-2086	88	6	as	as	ADP
cana-2086	88	7	𝜌(𝑣1𝑣2	𝜌(𝑣1𝑣2	ADJ
cana-2086	88	8	)	)	PUNCT
cana-2086	88	9	=	=	SYM
cana-2086	88	10	0.2	0.2	NUM
cana-2086	88	11	,	,	PUNCT
cana-2086	88	12	𝜌(𝑣2𝑣3	𝜌(𝑣2𝑣3	NOUN
cana-2086	88	13	)	)	PUNCT
cana-2086	88	14	=	=	SYM
cana-2086	88	15	0.3	0.3	NUM
cana-2086	88	16	,	,	PUNCT
cana-2086	88	17	𝜌(𝑣3𝑣4	𝜌(𝑣3𝑣4	NOUN
cana-2086	88	18	)	)	PUNCT
cana-2086	88	19	=	=	SYM
cana-2086	88	20	0.4	0.4	NUM
cana-2086	88	21	,	,	PUNCT
cana-2086	88	22	𝜌(𝑣1𝑣4	𝜌(𝑣1𝑣4	ADJ
cana-2086	88	23	)	)	PUNCT
cana-2086	88	24	=	=	SYM
cana-2086	88	25	0.1	0.1	NUM
cana-2086	88	26	.	.	PUNCT
cana-2086	89	1	definition	definition	NOUN
cana-2086	89	2	2.5	2.5	NUM
cana-2086	89	3	.	.	PUNCT
cana-2086	90	1	[	[	X
cana-2086	90	2	7	7	X
cana-2086	90	3	]	]	X
cana-2086	90	4	a	a	DET
cana-2086	90	5	fuzzy	fuzzy	ADJ
cana-2086	90	6	graph	graph	NOUN
cana-2086	90	7	𝐺	𝐺	NOUN
cana-2086	90	8	=	=	PUNCT
cana-2086	90	9	(	(	PUNCT
cana-2086	90	10	𝜇	𝜇	X
cana-2086	90	11	,	,	PUNCT
cana-2086	90	12	𝜌	𝜌	X
cana-2086	90	13	)	)	PUNCT
cana-2086	90	14	is	be	AUX
cana-2086	90	15	said	say	VERB
cana-2086	90	16	to	to	PART
cana-2086	90	17	be	be	AUX
cana-2086	90	18	a	a	DET
cana-2086	90	19	fuzzy	fuzzy	ADJ
cana-2086	90	20	labeling	labeling	NOUN
cana-2086	90	21	graph	graph	NOUN
cana-2086	90	22	,	,	PUNCT
cana-2086	90	23	if	if	SCONJ
cana-2086	90	24	𝜇	𝜇	X
cana-2086	90	25	:	:	PUNCT
cana-2086	90	26	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	90	27	)	)	PUNCT
cana-2086	90	28	→	→	PUNCT
cana-2086	91	1	[	[	X
cana-2086	91	2	0,1	0,1	NUM
cana-2086	91	3	]	]	PUNCT
cana-2086	91	4	and	and	CCONJ
cana-2086	91	5	𝜌	𝜌	ADP
cana-2086	91	6	:	:	PUNCT
cana-2086	91	7	𝐸(𝐺	𝐸(𝐺	X
cana-2086	91	8	)	)	PUNCT
cana-2086	91	9	→	→	PUNCT
cana-2086	92	1	[	[	X
cana-2086	92	2	0,1	0,1	NUM
cana-2086	92	3	]	]	PUNCT
cana-2086	92	4	are	be	AUX
cana-2086	92	5	injective	injective	ADJ
cana-2086	92	6	and	and	CCONJ
cana-2086	92	7	for	for	ADP
cana-2086	92	8	every	every	DET
cana-2086	92	9	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	92	10	in	in	ADP
cana-2086	92	11	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	92	12	)	)	PUNCT
cana-2086	92	13	,	,	PUNCT
cana-2086	92	14	𝜌(𝑢𝑣	𝜌(𝑢𝑣	ADP
cana-2086	92	15	)	)	PUNCT
cana-2086	92	16	<	<	X
cana-2086	92	17	𝜇(𝑢	𝜇(𝑢	PROPN
cana-2086	92	18	)	)	PUNCT
cana-2086	92	19	∧	∧	PROPN
cana-2086	92	20	𝜇(𝑣	𝜇(𝑣	PROPN
cana-2086	92	21	)	)	PUNCT
cana-2086	92	22	.	.	PUNCT
cana-2086	93	1	example	example	NOUN
cana-2086	94	1	2.6	2.6	NUM
cana-2086	94	2	.	.	PUNCT
cana-2086	94	3	fig	fig	NOUN
cana-2086	94	4	.	.	PUNCT
cana-2086	95	1	3	3	NUM
cana-2086	95	2	illustrates	illustrate	VERB
cana-2086	95	3	a	a	DET
cana-2086	95	4	fuzzy	fuzzy	ADJ
cana-2086	95	5	labeling	labeling	NOUN
cana-2086	95	6	graph	graph	NOUN
cana-2086	95	7	𝐺	𝐺	PROPN
cana-2086	95	8	with	with	ADP
cana-2086	95	9	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	95	10	)	)	PUNCT
cana-2086	95	11	=	=	SYM
cana-2086	95	12	{	{	PUNCT
cana-2086	95	13	𝑣1	𝑣1	PROPN
cana-2086	95	14	,	,	PUNCT
cana-2086	95	15	𝑣2	𝑣2	PROPN
cana-2086	95	16	,	,	PUNCT
cana-2086	95	17	𝑣3	𝑣3	ADJ
cana-2086	95	18	,	,	PUNCT
cana-2086	95	19	𝑣4	𝑣4	NOUN
cana-2086	95	20	}	}	PUNCT
cana-2086	95	21	and	and	CCONJ
cana-2086	95	22	𝐸(𝐺	𝐸(𝐺	NUM
cana-2086	95	23	)	)	PUNCT
cana-2086	95	24	=	=	SYM
cana-2086	95	25	{	{	PUNCT
cana-2086	95	26	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-2086	95	27	,	,	PUNCT
cana-2086	95	28	𝑣2𝑣3	𝑣2𝑣3	ADV
cana-2086	95	29	,	,	PUNCT
cana-2086	95	30	𝑣3𝑣4	𝑣3𝑣4	ADJ
cana-2086	95	31	,	,	PUNCT
cana-2086	95	32	𝑣2𝑣4	𝑣2𝑣4	ADP
cana-2086	95	33	}	}	PUNCT
cana-2086	95	34	.	.	PUNCT
cana-2086	96	1	a	a	DET
cana-2086	96	2	fuzzy	fuzzy	ADJ
cana-2086	96	3	labeling	labeling	NOUN
cana-2086	96	4	𝜇	𝜇	ADP
cana-2086	96	5	:	:	PUNCT
cana-2086	96	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	96	7	)	)	PUNCT
cana-2086	96	8	→	→	PUNCT
cana-2086	97	1	[	[	X
cana-2086	97	2	0,1	0,1	NUM
cana-2086	97	3	]	]	PUNCT
cana-2086	97	4	and	and	CCONJ
cana-2086	97	5	𝜌	𝜌	ADP
cana-2086	97	6	:	:	PUNCT
cana-2086	97	7	𝐸(𝐺	𝐸(𝐺	X
cana-2086	97	8	)	)	PUNCT
cana-2086	97	9	→	→	PUNCT
cana-2086	98	1	[	[	X
cana-2086	98	2	0,1	0,1	NUM
cana-2086	98	3	]	]	PUNCT
cana-2086	98	4	are	be	AUX
cana-2086	98	5	defined	define	VERB
cana-2086	98	6	as	as	SCONJ
cana-2086	98	7	follows	follow	VERB
cana-2086	98	8	:	:	PUNCT
cana-2086	98	9	𝜇(𝑣1	𝜇(𝑣1	ADJ
cana-2086	98	10	)	)	PUNCT
cana-2086	98	11	=	=	SYM
cana-2086	98	12	0.5	0.5	NUM
cana-2086	98	13	,	,	PUNCT
cana-2086	98	14	𝜇(𝑣2	𝜇(𝑣2	NOUN
cana-2086	98	15	)	)	PUNCT
cana-2086	98	16	=	=	SYM
cana-2086	98	17	0.6	0.6	NUM
cana-2086	98	18	,	,	PUNCT
cana-2086	98	19	𝜇(𝑣3	𝜇(𝑣3	NOUN
cana-2086	98	20	)	)	PUNCT
cana-2086	98	21	=	=	SYM
cana-2086	98	22	0.7	0.7	NUM
cana-2086	98	23	,	,	PUNCT
cana-2086	98	24	𝜇(𝑣4	𝜇(𝑣4	NOUN
cana-2086	98	25	)	)	PUNCT
cana-2086	98	26	=	=	SYM
cana-2086	99	1	0.8	0.8	NUM
cana-2086	99	2	,	,	PUNCT
cana-2086	99	3	𝜌(𝑣1𝑣2	𝜌(𝑣1𝑣2	ADJ
cana-2086	99	4	)	)	PUNCT
cana-2086	99	5	=	=	SYM
cana-2086	99	6	0.1	0.1	NUM
cana-2086	99	7	,	,	PUNCT
cana-2086	99	8	𝜌(𝑣2𝑣3	𝜌(𝑣2𝑣3	NOUN
cana-2086	99	9	)	)	PUNCT
cana-2086	99	10	=	=	SYM
cana-2086	99	11	0.2	0.2	NUM
cana-2086	99	12	,	,	PUNCT
cana-2086	99	13	𝜌(𝑣3𝑣4	𝜌(𝑣3𝑣4	NOUN
cana-2086	99	14	)	)	PUNCT
cana-2086	99	15	=	=	PUNCT
cana-2086	99	16	0.3	0.3	NUM
cana-2086	99	17	,	,	PUNCT
cana-2086	99	18	𝜌(𝑣2𝑣4	𝜌(𝑣2𝑣4	NOUN
cana-2086	99	19	)	)	PUNCT
cana-2086	99	20	=	=	SYM
cana-2086	99	21	0.4	0.4	NUM
cana-2086	99	22	.	.	PUNCT
cana-2086	100	1	communications	communication	NOUN
cana-2086	100	2	on	on	ADP
cana-2086	100	3	applied	apply	VERB
cana-2086	100	4	nonlinear	nonlinear	ADJ
cana-2086	100	5	analysis	analysis	NOUN
cana-2086	100	6	issn	issn	NOUN
cana-2086	100	7	:	:	PUNCT
cana-2086	100	8	1074	1074	NUM
cana-2086	100	9	-	-	PUNCT
cana-2086	100	10	133x	133x	NUM
cana-2086	100	11	vol	vol	NOUN
cana-2086	100	12	32	32	NUM
cana-2086	100	13	no	no	NOUN
cana-2086	100	14	.	.	PUNCT
cana-2086	101	1	1s	1s	NUM
cana-2086	101	2	(	(	PUNCT
cana-2086	101	3	2025	2025	NUM
cana-2086	101	4	)	)	PUNCT
cana-2086	101	5	33	33	NUM
cana-2086	101	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	101	7	definition	definition	NOUN
cana-2086	101	8	2.7	2.7	NUM
cana-2086	101	9	.	.	PUNCT
cana-2086	102	1	let	let	VERB
cana-2086	102	2	𝐺	𝐺	PRON
cana-2086	102	3	be	be	AUX
cana-2086	102	4	a	a	DET
cana-2086	102	5	(	(	PUNCT
cana-2086	102	6	𝑝	𝑝	PROPN
cana-2086	102	7	,	,	PUNCT
cana-2086	102	8	𝑞	𝑞	NOUN
cana-2086	102	9	)	)	PUNCT
cana-2086	102	10	graph	graph	NOUN
cana-2086	102	11	.	.	PUNCT
cana-2086	103	1	let	let	VERB
cana-2086	103	2	𝑓	𝑓	PRON
cana-2086	103	3	be	be	AUX
cana-2086	103	4	an	an	DET
cana-2086	103	5	injective	injective	ADJ
cana-2086	103	6	function	function	NOUN
cana-2086	103	7	from	from	ADP
cana-2086	103	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	103	9	)	)	PUNCT
cana-2086	103	10	into	into	ADP
cana-2086	103	11	the	the	DET
cana-2086	103	12	set	set	NOUN
cana-2086	103	13	{	{	PUNCT
cana-2086	103	14	ℎ′	ℎ′	NOUN
cana-2086	103	15	,	,	PUNCT
cana-2086	103	16	ℎ′	ℎ′	ADJ
cana-2086	104	1	+	+	ADV
cana-2086	104	2	1	1	NUM
cana-2086	104	3	,	,	PUNCT
cana-2086	104	4	.	.	PUNCT
cana-2086	104	5	.	.	PUNCT
cana-2086	105	1	.	.	PUNCT
cana-2086	106	1	,	,	PUNCT
cana-2086	106	2	ℎ′	ℎ′	ADJ
cana-2086	106	3	+	+	PUNCT
cana-2086	106	4	𝑞	𝑞	X
cana-2086	106	5	−	−	PROPN
cana-2086	106	6	1	1	NUM
cana-2086	106	7	,	,	PUNCT
cana-2086	106	8	ℎ′	ℎ′	ADJ
cana-2086	106	9	+	+	PUNCT
cana-2086	106	10	𝑞	𝑞	X
cana-2086	106	11	}	}	PUNCT
cana-2086	106	12	where	where	SCONJ
cana-2086	106	13	ℎ′	ℎ′	ADJ
cana-2086	106	14	is	be	AUX
cana-2086	106	15	a	a	DET
cana-2086	106	16	suitable	suitable	ADJ
cana-2086	106	17	constant	constant	ADJ
cana-2086	106	18	,	,	PUNCT
cana-2086	106	19	and	and	CCONJ
cana-2086	106	20	𝑔	𝑔	PROPN
cana-2086	106	21	be	be	AUX
cana-2086	106	22	an	an	DET
cana-2086	106	23	injective	injective	ADJ
cana-2086	106	24	function	function	NOUN
cana-2086	106	25	from	from	ADP
cana-2086	106	26	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	106	27	)	)	PUNCT
cana-2086	106	28	into	into	ADP
cana-2086	106	29	the	the	DET
cana-2086	106	30	set	set	NOUN
cana-2086	106	31	{	{	PUNCT
cana-2086	106	32	0,1	0,1	NUM
cana-2086	106	33	,	,	PUNCT
cana-2086	106	34	…	…	PUNCT
cana-2086	106	35	,	,	PUNCT
cana-2086	106	36	𝑞	𝑞	X
cana-2086	106	37	}	}	PUNCT
cana-2086	106	38	defined	define	VERB
cana-2086	106	39	as	as	ADP
cana-2086	106	40	,	,	PUNCT
cana-2086	106	41	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-2086	106	42	)	)	PUNCT
cana-2086	106	43	=	=	PUNCT
cana-2086	106	44	(	(	PUNCT
cana-2086	106	45	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	106	46	)	)	PUNCT
cana-2086	107	1	+	+	CCONJ
cana-2086	107	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	107	3	)	)	PUNCT
cana-2086	107	4	)	)	PUNCT
cana-2086	107	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	107	6	(	(	PUNCT
cana-2086	107	7	𝑞	𝑞	NOUN
cana-2086	107	8	+	+	NOUN
cana-2086	107	9	1	1	NUM
cana-2086	107	10	)	)	PUNCT
cana-2086	107	11	for	for	ADP
cana-2086	107	12	every	every	DET
cana-2086	107	13	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	107	14	in	in	ADP
cana-2086	107	15	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	107	16	)	)	PUNCT
cana-2086	107	17	.	.	PUNCT
cana-2086	108	1	define	define	VERB
cana-2086	108	2	vertex	vertex	NOUN
cana-2086	108	3	labelling	labelling	NOUN
cana-2086	108	4	𝜇	𝜇	ADP
cana-2086	108	5	:	:	PUNCT
cana-2086	108	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	108	7	)	)	PUNCT
cana-2086	108	8	→	→	PUNCT
cana-2086	109	1	[	[	X
cana-2086	109	2	0,1	0,1	NUM
cana-2086	109	3	]	]	PUNCT
cana-2086	109	4	as	as	ADP
cana-2086	109	5	𝜇(𝑣	𝜇(𝑣	NOUN
cana-2086	109	6	)	)	PUNCT
cana-2086	109	7	=	=	SYM
cana-2086	109	8	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	109	9	)	)	PUNCT
cana-2086	109	10	.	.	PUNCT
cana-2086	110	1	ℎ	ℎ	PROPN
cana-2086	110	2	and	and	CCONJ
cana-2086	110	3	the	the	DET
cana-2086	110	4	edge	edge	NOUN
cana-2086	110	5	labeling	label	VERB
cana-2086	110	6	𝜌	𝜌	ADP
cana-2086	110	7	:	:	PUNCT
cana-2086	110	8	𝐸(𝐺	𝐸(𝐺	X
cana-2086	110	9	)	)	PUNCT
cana-2086	110	10	→	→	PUNCT
cana-2086	111	1	[	[	X
cana-2086	111	2	0,1	0,1	NUM
cana-2086	111	3	]	]	PUNCT
cana-2086	111	4	as	as	ADP
cana-2086	111	5	𝜌(𝑢𝑣	𝜌(𝑢𝑣	NOUN
cana-2086	111	6	)	)	PUNCT
cana-2086	111	7	=	=	SYM
cana-2086	111	8	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	111	9	)	)	PUNCT
cana-2086	111	10	.	.	PUNCT
cana-2086	112	1	ℎ	ℎ	PROPN
cana-2086	112	2	,	,	PUNCT
cana-2086	112	3	where	where	SCONJ
cana-2086	112	4	ℎ	ℎ	PROPN
cana-2086	112	5	≤	≤	NOUN
cana-2086	112	6	1	1	NUM
cana-2086	112	7	ℎ′+𝑞	ℎ′+𝑞	ADV
cana-2086	112	8	.	.	PUNCT
cana-2086	113	1	if	if	SCONJ
cana-2086	113	2	the	the	DET
cana-2086	113	3	edge	edge	NOUN
cana-2086	113	4	labels	label	NOUN
cana-2086	113	5	are	be	AUX
cana-2086	113	6	distinct	distinct	ADJ
cana-2086	113	7	and	and	CCONJ
cana-2086	113	8	nonzero	nonzero	NOUN
cana-2086	113	9	,	,	PUNCT
cana-2086	113	10	and	and	CCONJ
cana-2086	113	11	if	if	SCONJ
cana-2086	113	12	the	the	DET
cana-2086	113	13	condition	condition	NOUN
cana-2086	113	14	𝜌(𝑢𝑣	𝜌(𝑢𝑣	VERB
cana-2086	113	15	)	)	PUNCT
cana-2086	113	16	<	<	X
cana-2086	113	17	𝜇(𝑢	𝜇(𝑢	PROPN
cana-2086	113	18	)	)	PUNCT
cana-2086	113	19	∧	∧	PROPN
cana-2086	113	20	𝜇(𝑣	𝜇(𝑣	NOUN
cana-2086	113	21	)	)	PUNCT
cana-2086	113	22	is	be	AUX
cana-2086	113	23	satisfied	satisfied	ADJ
cana-2086	113	24	for	for	ADP
cana-2086	113	25	all	all	DET
cana-2086	113	26	𝑢𝑣	𝑢𝑣	PROPN
cana-2086	113	27	∈	∈	PROPN
cana-2086	113	28	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	113	29	)	)	PUNCT
cana-2086	113	30	,	,	PUNCT
cana-2086	113	31	then	then	ADV
cana-2086	113	32	𝐺	𝐺	PROPN
cana-2086	113	33	is	be	AUX
cana-2086	113	34	said	say	VERB
cana-2086	113	35	to	to	PART
cana-2086	113	36	be	be	AUX
cana-2086	113	37	an	an	DET
cana-2086	113	38	elegant	elegant	ADJ
cana-2086	113	39	fuzzy	fuzzy	ADJ
cana-2086	113	40	labeling	labeling	NOUN
cana-2086	113	41	graph	graph	NOUN
cana-2086	113	42	and	and	CCONJ
cana-2086	113	43	(	(	PUNCT
cana-2086	113	44	𝜇	𝜇	ADP
cana-2086	113	45	,	,	PUNCT
cana-2086	113	46	𝜌	𝜌	X
cana-2086	113	47	)	)	PUNCT
cana-2086	113	48	is	be	AUX
cana-2086	113	49	an	an	DET
cana-2086	113	50	elegant	elegant	ADJ
cana-2086	113	51	fuzzy	fuzzy	ADJ
cana-2086	113	52	labeling	labeling	NOUN
cana-2086	113	53	of	of	ADP
cana-2086	113	54	𝐺.	𝐺.	NOUN
cana-2086	113	55	in	in	ADP
cana-2086	113	56	the	the	DET
cana-2086	113	57	definition	definition	NOUN
cana-2086	113	58	2.7	2.7	NUM
cana-2086	113	59	,	,	PUNCT
cana-2086	113	60	if	if	SCONJ
cana-2086	113	61	ℎ′	ℎ′	ADJ
cana-2086	113	62	=	=	SYM
cana-2086	113	63	0	0	NUM
cana-2086	113	64	,	,	PUNCT
cana-2086	113	65	then	then	ADV
cana-2086	113	66	𝑓	𝑓	PRON
cana-2086	113	67	is	be	AUX
cana-2086	113	68	an	an	DET
cana-2086	113	69	elegant	elegant	ADJ
cana-2086	113	70	labeling	labeling	NOUN
cana-2086	113	71	.	.	PUNCT
cana-2086	114	1	suppose	suppose	VERB
cana-2086	114	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	114	3	)	)	PUNCT
cana-2086	114	4	=	=	SYM
cana-2086	114	5	0	0	NUM
cana-2086	114	6	for	for	ADP
cana-2086	114	7	some	some	DET
cana-2086	114	8	𝑣	𝑣	PRON
cana-2086	114	9	∈	∈	PROPN
cana-2086	114	10	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	114	11	)	)	PUNCT
cana-2086	114	12	and	and	CCONJ
cana-2086	114	13	𝑢𝑣	𝑢𝑣	PROPN
cana-2086	114	14	∈	∈	PROPN
cana-2086	114	15	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	114	16	)	)	PUNCT
cana-2086	114	17	then	then	ADV
cana-2086	114	18	,	,	PUNCT
cana-2086	114	19	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	114	20	)	)	PUNCT
cana-2086	114	21	=	=	PUNCT
cana-2086	114	22	(	(	PUNCT
cana-2086	114	23	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	114	24	)	)	PUNCT
cana-2086	115	1	+	+	CCONJ
cana-2086	115	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	115	3	)	)	PUNCT
cana-2086	115	4	)	)	PUNCT
cana-2086	115	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	115	6	(	(	PUNCT
cana-2086	115	7	𝑞	𝑞	X
cana-2086	115	8	+	+	NOUN
cana-2086	115	9	1	1	NUM
cana-2086	115	10	)	)	PUNCT
cana-2086	115	11	,	,	PUNCT
cana-2086	115	12	where	where	SCONJ
cana-2086	115	13	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	115	14	)	)	PUNCT
cana-2086	115	15	=	=	SYM
cana-2086	115	16	0	0	NUM
cana-2086	115	17	and	and	CCONJ
cana-2086	115	18	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	115	19	)	)	PUNCT
cana-2086	115	20	≠	≠	PROPN
cana-2086	115	21	0	0	NUM
cana-2086	115	22	,	,	PUNCT
cana-2086	115	23	simplifying	simplify	VERB
cana-2086	115	24	to	to	ADP
cana-2086	115	25	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	115	26	)	)	PUNCT
cana-2086	115	27	=	=	SYM
cana-2086	115	28	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	115	29	)	)	PUNCT
cana-2086	115	30	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	115	31	(	(	PUNCT
cana-2086	115	32	𝑞	𝑞	X
cana-2086	115	33	+	+	NOUN
cana-2086	115	34	1	1	NUM
cana-2086	115	35	)	)	PUNCT
cana-2086	115	36	.	.	PUNCT
cana-2086	116	1	according	accord	VERB
cana-2086	116	2	to	to	ADP
cana-2086	116	3	definition	definition	NOUN
cana-2086	116	4	2.7	2.7	NUM
cana-2086	116	5	,	,	PUNCT
cana-2086	116	6	0	0	NUM
cana-2086	116	7	<	<	X
cana-2086	116	8	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	116	9	)	)	PUNCT
cana-2086	116	10	<	<	X
cana-2086	116	11	𝑞	𝑞	X
cana-2086	116	12	+	+	NOUN
cana-2086	116	13	1	1	NUM
cana-2086	116	14	,	,	PUNCT
cana-2086	116	15	hence	hence	ADV
cana-2086	116	16	we	we	PRON
cana-2086	116	17	have	have	VERB
cana-2086	116	18	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-2086	116	19	)	)	PUNCT
cana-2086	116	20	>	>	X
cana-2086	116	21	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	116	22	)	)	PUNCT
cana-2086	116	23	∧	∧	PROPN
cana-2086	116	24	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	116	25	)	)	PUNCT
cana-2086	116	26	.	.	PUNCT
cana-2086	117	1	similarly	similarly	ADV
cana-2086	117	2	,	,	PUNCT
cana-2086	117	3	suppose	suppose	VERB
cana-2086	117	4	𝑓(𝑣′	𝑓(𝑣′	ADJ
cana-2086	117	5	)	)	PUNCT
cana-2086	117	6	=	=	SYM
cana-2086	117	7	1	1	NUM
cana-2086	117	8	for	for	ADP
cana-2086	117	9	some	some	DET
cana-2086	117	10	𝑣′	𝑣′	NUM
cana-2086	117	11	∈	∈	PROPN
cana-2086	117	12	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	117	13	)	)	PUNCT
cana-2086	117	14	and	and	CCONJ
cana-2086	117	15	𝑢′𝑣′	𝑢′𝑣′	VERB
cana-2086	117	16	∈	∈	PROPN
cana-2086	117	17	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	117	18	)	)	PUNCT
cana-2086	117	19	,	,	PUNCT
cana-2086	117	20	we	we	PRON
cana-2086	117	21	get	get	VERB
cana-2086	117	22	𝑔(𝑢′𝑣′	𝑔(𝑢′𝑣′	ADJ
cana-2086	117	23	)	)	PUNCT
cana-2086	117	24	≥	≥	NOUN
cana-2086	117	25	𝑓(𝑢′	𝑓(𝑢′	VERB
cana-2086	117	26	)	)	PUNCT
cana-2086	117	27	∧	∧	NOUN
cana-2086	117	28	𝑓(𝑣′	𝑓(𝑣′	NOUN
cana-2086	117	29	)	)	PUNCT
cana-2086	117	30	.	.	PUNCT
cana-2086	118	1	since	since	SCONJ
cana-2086	118	2	𝑔	𝑔	NOUN
cana-2086	118	3	:	:	PUNCT
cana-2086	118	4	𝐸(𝐺	𝐸(𝐺	NUM
cana-2086	118	5	)	)	PUNCT
cana-2086	118	6	→	→	SYM
cana-2086	118	7	{	{	PUNCT
cana-2086	118	8	1	1	NUM
cana-2086	118	9	,	,	PUNCT
cana-2086	118	10	2	2	NUM
cana-2086	118	11	,	,	PUNCT
cana-2086	118	12	…	…	PUNCT
cana-2086	118	13	,	,	PUNCT
cana-2086	118	14	𝑞	𝑞	X
cana-2086	118	15	}	}	PUNCT
cana-2086	118	16	is	be	AUX
cana-2086	118	17	injective	injective	ADJ
cana-2086	118	18	,	,	PUNCT
cana-2086	118	19	by	by	ADP
cana-2086	118	20	the	the	DET
cana-2086	118	21	definition	definition	NOUN
cana-2086	118	22	2.7	2.7	NUM
cana-2086	118	23	,	,	PUNCT
cana-2086	118	24	for	for	ADP
cana-2086	118	25	some	some	DET
cana-2086	118	26	𝑢′′𝑣′′	𝑢′′𝑣′′	NOUN
cana-2086	118	27	∈	∈	PROPN
cana-2086	118	28	𝐺	𝐺	PROPN
cana-2086	118	29	,	,	PUNCT
cana-2086	118	30	𝑔(𝑢′′𝑣′′	𝑔(𝑢′′𝑣′′	NOUN
cana-2086	118	31	)	)	PUNCT
cana-2086	118	32	=	=	PUNCT
cana-2086	119	1	𝑞	𝑞	X
cana-2086	119	2	which	which	PRON
cana-2086	119	3	implies	imply	VERB
cana-2086	119	4	,	,	PUNCT
cana-2086	119	5	𝑔(𝑢′′𝑣′′	𝑔(𝑢′′𝑣′′	NOUN
cana-2086	119	6	)	)	PUNCT
cana-2086	119	7	>	>	X
cana-2086	119	8	𝑓(𝑢′′	𝑓(𝑢′′	PROPN
cana-2086	119	9	)	)	PUNCT
cana-2086	119	10	∧	∧	PROPN
cana-2086	119	11	𝑓(𝑣′′	𝑓(𝑣′′	NOUN
cana-2086	119	12	)	)	PUNCT
cana-2086	119	13	.	.	PUNCT
cana-2086	120	1	however	however	ADV
cana-2086	120	2	,	,	PUNCT
cana-2086	120	3	we	we	PRON
cana-2086	120	4	must	must	AUX
cana-2086	120	5	have	have	VERB
cana-2086	120	6	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	120	7	)	)	PUNCT
cana-2086	120	8	<	<	X
cana-2086	120	9	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	120	10	)	)	PUNCT
cana-2086	120	11	∧	∧	PROPN
cana-2086	120	12	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	120	13	)	)	PUNCT
cana-2086	120	14	for	for	ADP
cana-2086	120	15	all	all	DET
cana-2086	120	16	𝑢𝑣	𝑢𝑣	PROPN
cana-2086	120	17	∈	∈	PROPN
cana-2086	120	18	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	120	19	)	)	PUNCT
cana-2086	120	20	to	to	PART
cana-2086	120	21	attain	attain	VERB
cana-2086	120	22	𝜌(𝑢𝑣	𝜌(𝑢𝑣	ADP
cana-2086	120	23	)	)	PUNCT
cana-2086	120	24	<	<	X
cana-2086	120	25	μ(𝑢	μ(𝑢	PROPN
cana-2086	120	26	)	)	PUNCT
cana-2086	120	27	∧	∧	PROPN
cana-2086	120	28	μ(𝑣	μ(𝑣	PROPN
cana-2086	120	29	)	)	PUNCT
cana-2086	120	30	for	for	ADP
cana-2086	120	31	all	all	DET
cana-2086	120	32	𝑢𝑣	𝑢𝑣	PROPN
cana-2086	120	33	∈	∈	PROPN
cana-2086	120	34	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	120	35	)	)	PUNCT
cana-2086	120	36	.	.	PUNCT
cana-2086	121	1	thus	thus	ADV
cana-2086	121	2	,	,	PUNCT
cana-2086	121	3	we	we	PRON
cana-2086	121	4	have	have	AUX
cana-2086	121	5	translated	translate	VERB
cana-2086	121	6	the	the	DET
cana-2086	121	7	set	set	NOUN
cana-2086	121	8	{	{	PUNCT
cana-2086	121	9	0,1	0,1	NUM
cana-2086	121	10	,	,	PUNCT
cana-2086	121	11	…	…	PUNCT
cana-2086	121	12	,	,	PUNCT
cana-2086	121	13	𝑞	𝑞	X
cana-2086	121	14	}	}	PUNCT
cana-2086	121	15	to	to	ADP
cana-2086	121	16	{	{	PUNCT
cana-2086	121	17	ℎ′	ℎ′	NOUN
cana-2086	121	18	,	,	PUNCT
cana-2086	121	19	ℎ′	ℎ′	ADJ
cana-2086	121	20	+	+	NOUN
cana-2086	121	21	1	1	NUM
cana-2086	121	22	,	,	PUNCT
cana-2086	121	23	…	…	PUNCT
cana-2086	121	24	,	,	PUNCT
cana-2086	121	25	ℎ′	ℎ′	ADJ
cana-2086	121	26	+	+	CCONJ
cana-2086	121	27	𝑞	𝑞	X
cana-2086	121	28	}	}	PUNCT
cana-2086	121	29	,	,	PUNCT
cana-2086	121	30	where	where	SCONJ
cana-2086	121	31	ℎ′	ℎ′	ADJ
cana-2086	121	32	is	be	AUX
cana-2086	121	33	the	the	DET
cana-2086	121	34	suitable	suitable	ADJ
cana-2086	121	35	translation	translation	NOUN
cana-2086	121	36	constant	constant	ADJ
cana-2086	121	37	.	.	PUNCT
cana-2086	122	1	to	to	PART
cana-2086	122	2	define	define	VERB
cana-2086	122	3	labeling	labeling	NOUN
cana-2086	122	4	functions	function	NOUN
cana-2086	122	5	of	of	ADP
cana-2086	122	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	122	7	)	)	PUNCT
cana-2086	122	8	and	and	CCONJ
cana-2086	122	9	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	122	10	)	)	PUNCT
cana-2086	122	11	into	into	ADP
cana-2086	122	12	[	[	X
cana-2086	122	13	0,1	0,1	NUM
cana-2086	122	14	]	]	PUNCT
cana-2086	122	15	,	,	PUNCT
cana-2086	122	16	we	we	PRON
cana-2086	122	17	have	have	AUX
cana-2086	122	18	performed	perform	VERB
cana-2086	122	19	contraction	contraction	NOUN
cana-2086	122	20	with	with	ADP
cana-2086	122	21	scaling	scale	VERB
cana-2086	122	22	factor	factor	NOUN
cana-2086	122	23	ℎ.	ℎ.	NOUN
cana-2086	122	24	3	3	X
cana-2086	122	25	.	.	PUNCT
cana-2086	122	26	elegant	elegant	ADJ
cana-2086	122	27	fuzzy	fuzzy	ADJ
cana-2086	122	28	labeling	labeling	NOUN
cana-2086	122	29	in	in	ADP
cana-2086	122	30	the	the	DET
cana-2086	122	31	sequel	sequel	NOUN
cana-2086	122	32	we	we	PRON
cana-2086	122	33	take	take	VERB
cana-2086	122	34	ℎ′	ℎ′	NOUN
cana-2086	122	35	=	=	PUNCT
cana-2086	123	1	𝑞	𝑞	X
cana-2086	123	2	+	+	NOUN
cana-2086	123	3	1	1	NUM
cana-2086	123	4	and	and	CCONJ
cana-2086	123	5	ℎ	ℎ	X
cana-2086	123	6	=	=	SYM
cana-2086	123	7	1	1	NUM
cana-2086	123	8	2𝑞+1	2𝑞+1	NUM
cana-2086	123	9	.	.	PUNCT
cana-2086	124	1	so	so	ADV
cana-2086	124	2	we	we	PRON
cana-2086	124	3	consider	consider	VERB
cana-2086	124	4	the	the	DET
cana-2086	124	5	following	follow	VERB
cana-2086	124	6	definition	definition	NOUN
cana-2086	124	7	of	of	ADP
cana-2086	124	8	elegant	elegant	ADJ
cana-2086	124	9	fuzzy	fuzzy	ADJ
cana-2086	124	10	labeling	labeling	NOUN
cana-2086	124	11	of	of	ADP
cana-2086	124	12	graphs	graph	NOUN
cana-2086	124	13	,	,	PUNCT
cana-2086	124	14	and	and	CCONJ
cana-2086	124	15	obtain	obtain	VERB
cana-2086	124	16	the	the	DET
cana-2086	124	17	results	result	NOUN
cana-2086	124	18	and	and	CCONJ
cana-2086	124	19	prove	prove	VERB
cana-2086	124	20	that	that	SCONJ
cana-2086	124	21	certain	certain	ADJ
cana-2086	124	22	classes	class	NOUN
cana-2086	124	23	of	of	ADP
cana-2086	124	24	graphs	graph	NOUN
cana-2086	124	25	admit	admit	VERB
cana-2086	124	26	elegant	elegant	ADJ
cana-2086	124	27	fuzzy	fuzzy	ADJ
cana-2086	124	28	labeling	labeling	NOUN
cana-2086	124	29	.	.	PUNCT
cana-2086	125	1	definition	definition	NOUN
cana-2086	125	2	3.1	3.1	NUM
cana-2086	125	3	.	.	PUNCT
cana-2086	126	1	let	let	VERB
cana-2086	126	2	𝐺	𝐺	PRON
cana-2086	126	3	be	be	AUX
cana-2086	126	4	a	a	DET
cana-2086	126	5	(	(	PUNCT
cana-2086	126	6	𝑝	𝑝	PROPN
cana-2086	126	7	,	,	PUNCT
cana-2086	126	8	𝑞	𝑞	NOUN
cana-2086	126	9	)	)	PUNCT
cana-2086	126	10	graph	graph	NOUN
cana-2086	126	11	.	.	PUNCT
cana-2086	127	1	let	let	VERB
cana-2086	127	2	𝑓	𝑓	PRON
cana-2086	127	3	be	be	AUX
cana-2086	127	4	an	an	DET
cana-2086	127	5	injective	injective	ADJ
cana-2086	127	6	function	function	NOUN
cana-2086	127	7	from	from	ADP
cana-2086	127	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	127	9	)	)	PUNCT
cana-2086	127	10	into	into	ADP
cana-2086	127	11	the	the	DET
cana-2086	127	12	set	set	NOUN
cana-2086	127	13	{	{	PUNCT
cana-2086	127	14	𝑞	𝑞	X
cana-2086	127	15	+	+	PROPN
cana-2086	127	16	1	1	NUM
cana-2086	127	17	,	,	PUNCT
cana-2086	127	18	𝑞	𝑞	X
cana-2086	127	19	+	+	NOUN
cana-2086	127	20	2	2	NUM
cana-2086	127	21	,	,	PUNCT
cana-2086	127	22	…	…	PUNCT
cana-2086	127	23	,	,	PUNCT
cana-2086	127	24	2𝑞	2𝑞	NOUN
cana-2086	127	25	+	+	CCONJ
cana-2086	127	26	1	1	X
cana-2086	127	27	}	}	PUNCT
cana-2086	127	28	and	and	CCONJ
cana-2086	127	29	𝑔	𝑔	PROPN
cana-2086	127	30	be	be	AUX
cana-2086	127	31	an	an	DET
cana-2086	127	32	injective	injective	ADJ
cana-2086	127	33	function	function	NOUN
cana-2086	127	34	from	from	ADP
cana-2086	127	35	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	127	36	)	)	PUNCT
cana-2086	127	37	into	into	ADP
cana-2086	127	38	the	the	DET
cana-2086	127	39	set	set	NOUN
cana-2086	127	40	{	{	PUNCT
cana-2086	127	41	0,1	0,1	NUM
cana-2086	127	42	,	,	PUNCT
cana-2086	127	43	…	…	PUNCT
cana-2086	127	44	,	,	PUNCT
cana-2086	127	45	𝑞	𝑞	X
cana-2086	127	46	}	}	PUNCT
cana-2086	127	47	defined	define	VERB
cana-2086	127	48	as	as	ADP
cana-2086	127	49	,	,	PUNCT
cana-2086	127	50	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-2086	127	51	)	)	PUNCT
cana-2086	127	52	=	=	PUNCT
cana-2086	127	53	(	(	PUNCT
cana-2086	127	54	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	127	55	)	)	PUNCT
cana-2086	128	1	+	+	CCONJ
cana-2086	128	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	128	3	)	)	PUNCT
cana-2086	128	4	)	)	PUNCT
cana-2086	128	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	128	6	(	(	PUNCT
cana-2086	128	7	𝑞	𝑞	NOUN
cana-2086	128	8	+	+	NOUN
cana-2086	128	9	1	1	NUM
cana-2086	128	10	)	)	PUNCT
cana-2086	128	11	for	for	ADP
cana-2086	128	12	every	every	DET
cana-2086	128	13	𝑢𝑣	𝑢𝑣	NOUN
cana-2086	128	14	in	in	ADP
cana-2086	128	15	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	128	16	)	)	PUNCT
cana-2086	128	17	.	.	PUNCT
cana-2086	129	1	define	define	VERB
cana-2086	129	2	vertex	vertex	NOUN
cana-2086	129	3	labeling	labeling	NOUN
cana-2086	129	4	𝜇	𝜇	ADP
cana-2086	129	5	:	:	PUNCT
cana-2086	129	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	129	7	)	)	PUNCT
cana-2086	129	8	→	→	PUNCT
cana-2086	130	1	[	[	X
cana-2086	130	2	0,1	0,1	NUM
cana-2086	130	3	]	]	PUNCT
cana-2086	130	4	as	as	ADP
cana-2086	130	5	𝜇(𝑣	𝜇(𝑣	NOUN
cana-2086	130	6	)	)	PUNCT
cana-2086	130	7	=	=	SYM
cana-2086	130	8	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	130	9	)	)	PUNCT
cana-2086	130	10	.	.	PUNCT
cana-2086	131	1	ℎ	ℎ	PROPN
cana-2086	131	2	and	and	CCONJ
cana-2086	131	3	the	the	DET
cana-2086	131	4	edge	edge	NOUN
cana-2086	131	5	labeling	label	VERB
cana-2086	131	6	𝜌	𝜌	ADP
cana-2086	131	7	:	:	PUNCT
cana-2086	131	8	𝐸(𝐺	𝐸(𝐺	X
cana-2086	131	9	)	)	PUNCT
cana-2086	131	10	→	→	PUNCT
cana-2086	132	1	[	[	X
cana-2086	132	2	0,1	0,1	NUM
cana-2086	132	3	]	]	PUNCT
cana-2086	132	4	as	as	ADP
cana-2086	132	5	𝜌(𝑢𝑣	𝜌(𝑢𝑣	NOUN
cana-2086	132	6	)	)	PUNCT
cana-2086	132	7	=	=	SYM
cana-2086	132	8	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	132	9	)	)	PUNCT
cana-2086	132	10	.	.	PUNCT
cana-2086	133	1	ℎ	ℎ	PROPN
cana-2086	133	2	,	,	PUNCT
cana-2086	133	3	where	where	SCONJ
cana-2086	133	4	ℎ	ℎ	X
cana-2086	133	5	=	=	SYM
cana-2086	133	6	1	1	NUM
cana-2086	133	7	2𝑞+1	2𝑞+1	NUM
cana-2086	133	8	.	.	PUNCT
cana-2086	134	1	if	if	SCONJ
cana-2086	134	2	the	the	DET
cana-2086	134	3	edge	edge	NOUN
cana-2086	134	4	labels	label	NOUN
cana-2086	134	5	are	be	AUX
cana-2086	134	6	distinct	distinct	ADJ
cana-2086	134	7	and	and	CCONJ
cana-2086	134	8	nonzero	nonzero	NOUN
cana-2086	134	9	,	,	PUNCT
cana-2086	134	10	and	and	CCONJ
cana-2086	134	11	if	if	SCONJ
cana-2086	134	12	the	the	DET
cana-2086	134	13	condition	condition	NOUN
cana-2086	134	14	𝜌(𝑢𝑣	𝜌(𝑢𝑣	VERB
cana-2086	134	15	)	)	PUNCT
cana-2086	134	16	<	<	X
cana-2086	134	17	𝜇(𝑢	𝜇(𝑢	PROPN
cana-2086	134	18	)	)	PUNCT
cana-2086	134	19	∧	∧	PROPN
cana-2086	134	20	𝜇(𝑣	𝜇(𝑣	NOUN
cana-2086	134	21	)	)	PUNCT
cana-2086	134	22	is	be	AUX
cana-2086	134	23	satisfied	satisfied	ADJ
cana-2086	134	24	for	for	ADP
cana-2086	134	25	all	all	DET
cana-2086	134	26	𝑢𝑣	𝑢𝑣	PROPN
cana-2086	134	27	∈	∈	PROPN
cana-2086	134	28	𝐸(𝐺	𝐸(𝐺	NOUN
cana-2086	134	29	)	)	PUNCT
cana-2086	134	30	,	,	PUNCT
cana-2086	134	31	then	then	ADV
cana-2086	134	32	𝐺	𝐺	PROPN
cana-2086	134	33	is	be	AUX
cana-2086	134	34	said	say	VERB
cana-2086	134	35	to	to	PART
cana-2086	134	36	be	be	AUX
cana-2086	134	37	an	an	DET
cana-2086	134	38	elegant	elegant	ADJ
cana-2086	134	39	fuzzy	fuzzy	ADJ
cana-2086	134	40	labeling	labeling	NOUN
cana-2086	134	41	graph	graph	NOUN
cana-2086	134	42	and	and	CCONJ
cana-2086	134	43	(	(	PUNCT
cana-2086	134	44	𝜇	𝜇	ADP
cana-2086	134	45	,	,	PUNCT
cana-2086	134	46	𝜌	𝜌	X
cana-2086	134	47	)	)	PUNCT
cana-2086	134	48	is	be	AUX
cana-2086	134	49	an	an	DET
cana-2086	134	50	elegant	elegant	ADJ
cana-2086	134	51	fuzzy	fuzzy	ADJ
cana-2086	134	52	labeling	labeling	NOUN
cana-2086	134	53	of	of	ADP
cana-2086	134	54	𝐺.	𝐺.	NOUN
cana-2086	134	55	example	example	NOUN
cana-2086	134	56	3.2	3.2	NUM
cana-2086	134	57	.	.	PUNCT
cana-2086	135	1	fig	fig	NOUN
cana-2086	135	2	.	.	PUNCT
cana-2086	136	1	4	4	NUM
cana-2086	136	2	illustrates	illustrate	VERB
cana-2086	136	3	an	an	DET
cana-2086	136	4	elegant	elegant	ADJ
cana-2086	136	5	fuzzy	fuzzy	ADJ
cana-2086	136	6	labeling	labeling	NOUN
cana-2086	136	7	graph	graph	NOUN
cana-2086	136	8	𝐺	𝐺	PROPN
cana-2086	136	9	,	,	PUNCT
cana-2086	136	10	where	where	SCONJ
cana-2086	136	11	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	136	12	)	)	PUNCT
cana-2086	136	13	=	=	SYM
cana-2086	136	14	{	{	PUNCT
cana-2086	136	15	𝑣1	𝑣1	PROPN
cana-2086	136	16	,	,	PUNCT
cana-2086	136	17	𝑣2	𝑣2	PROPN
cana-2086	136	18	,	,	PUNCT
cana-2086	136	19	𝑣3	𝑣3	ADJ
cana-2086	136	20	,	,	PUNCT
cana-2086	136	21	𝑣4	𝑣4	NOUN
cana-2086	136	22	}	}	PUNCT
cana-2086	136	23	and	and	CCONJ
cana-2086	136	24	𝐸(𝐺	𝐸(𝐺	NUM
cana-2086	136	25	)	)	PUNCT
cana-2086	136	26	=	=	SYM
cana-2086	136	27	{	{	PUNCT
cana-2086	136	28	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-2086	136	29	,	,	PUNCT
cana-2086	136	30	 	 	SPACE
cana-2086	136	31	𝑣1𝑣3	𝑣1𝑣3	ADP
cana-2086	136	32	,	,	PUNCT
cana-2086	136	33	 	 	SPACE
cana-2086	136	34	𝑣1𝑣4	𝑣1𝑣4	VERB
cana-2086	136	35	,	,	PUNCT
cana-2086	136	36	 	 	SPACE
cana-2086	136	37	𝑣2𝑣3	𝑣2𝑣3	ADV
cana-2086	136	38	,	,	PUNCT
cana-2086	136	39	 	 	SPACE
cana-2086	136	40	𝑣2𝑣4	𝑣2𝑣4	PROPN
cana-2086	136	41	,	,	PUNCT
cana-2086	136	42	 	 	SPACE
cana-2086	136	43	𝑣3𝑣4	𝑣3𝑣4	ADP
cana-2086	136	44	}	}	PUNCT
cana-2086	136	45	.	.	PUNCT
cana-2086	137	1	the	the	DET
cana-2086	137	2	function	function	NOUN
cana-2086	137	3	𝑓	𝑓	PROPN
cana-2086	137	4	is	be	AUX
cana-2086	137	5	defined	define	VERB
cana-2086	137	6	as	as	SCONJ
cana-2086	137	7	follows	follow	VERB
cana-2086	137	8	:	:	PUNCT
cana-2086	137	9	𝑓(𝑣1	𝑓(𝑣1	X
cana-2086	137	10	)	)	PUNCT
cana-2086	137	11	=	=	SYM
cana-2086	137	12	7	7	NUM
cana-2086	137	13	,	,	PUNCT
cana-2086	137	14	𝑓(𝑣2	𝑓(𝑣2	NOUN
cana-2086	137	15	)	)	PUNCT
cana-2086	137	16	=	=	SYM
cana-2086	137	17	8	8	NUM
cana-2086	137	18	,	,	PUNCT
cana-2086	137	19	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	137	20	)	)	PUNCT
cana-2086	137	21	=	=	SYM
cana-2086	137	22	9	9	NUM
cana-2086	137	23	,	,	PUNCT
cana-2086	137	24	𝑓(𝑣4	𝑓(𝑣4	NOUN
cana-2086	137	25	)	)	PUNCT
cana-2086	137	26	=	=	SYM
cana-2086	137	27	11	11	NUM
cana-2086	137	28	.	.	PUNCT
cana-2086	138	1	consequently	consequently	ADV
cana-2086	138	2	,	,	PUNCT
cana-2086	138	3	𝑔(𝑣1𝑣2	𝑔(𝑣1𝑣2	PROPN
cana-2086	138	4	)	)	PUNCT
cana-2086	138	5	=	=	SYM
cana-2086	138	6	1	1	NUM
cana-2086	138	7	,	,	PUNCT
cana-2086	138	8	𝑔(𝑣1𝑣3	𝑔(𝑣1𝑣3	NOUN
cana-2086	138	9	)	)	PUNCT
cana-2086	138	10	=	=	SYM
cana-2086	138	11	2	2	NUM
cana-2086	138	12	,	,	PUNCT
cana-2086	138	13	𝑔(𝑣1𝑣4	𝑔(𝑣1𝑣4	NOUN
cana-2086	138	14	)	)	PUNCT
cana-2086	138	15	=	=	SYM
cana-2086	138	16	4	4	NUM
cana-2086	138	17	,	,	PUNCT
cana-2086	138	18	𝑔(𝑣2𝑣3	𝑔(𝑣2𝑣3	NOUN
cana-2086	138	19	)	)	PUNCT
cana-2086	138	20	=	=	SYM
cana-2086	138	21	3	3	NUM
cana-2086	138	22	,	,	PUNCT
cana-2086	138	23	𝑔(𝑣2𝑣4	𝑔(𝑣2𝑣4	NOUN
cana-2086	138	24	)	)	PUNCT
cana-2086	138	25	=	=	SYM
cana-2086	138	26	5	5	NUM
cana-2086	138	27	,	,	PUNCT
cana-2086	138	28	𝑔(𝑣3𝑣4	𝑔(𝑣3𝑣4	ADJ
cana-2086	138	29	)	)	PUNCT
cana-2086	138	30	=	=	SYM
cana-2086	138	31	6	6	NUM
cana-2086	138	32	.	.	PUNCT
cana-2086	139	1	therefore	therefore	ADV
cana-2086	139	2	,	,	PUNCT
cana-2086	139	3	an	an	DET
cana-2086	139	4	elegant	elegant	ADJ
cana-2086	139	5	fuzzy	fuzzy	ADJ
cana-2086	139	6	labeling	labeling	NOUN
cana-2086	139	7	(	(	PUNCT
cana-2086	139	8	𝜇	𝜇	ADP
cana-2086	139	9	,	,	PUNCT
cana-2086	139	10	𝜌	𝜌	X
cana-2086	139	11	)	)	PUNCT
cana-2086	139	12	is	be	AUX
cana-2086	139	13	given	give	VERB
cana-2086	139	14	as	as	SCONJ
cana-2086	139	15	communications	communication	NOUN
cana-2086	139	16	on	on	ADP
cana-2086	139	17	applied	apply	VERB
cana-2086	139	18	nonlinear	nonlinear	ADJ
cana-2086	139	19	analysis	analysis	NOUN
cana-2086	139	20	issn	issn	NOUN
cana-2086	139	21	:	:	PUNCT
cana-2086	139	22	1074	1074	NUM
cana-2086	139	23	-	-	PUNCT
cana-2086	139	24	133x	133x	NUM
cana-2086	139	25	vol	vol	NOUN
cana-2086	139	26	32	32	NUM
cana-2086	139	27	no	no	NOUN
cana-2086	139	28	.	.	PUNCT
cana-2086	140	1	1s	1s	NUM
cana-2086	140	2	(	(	PUNCT
cana-2086	140	3	2025	2025	NUM
cana-2086	140	4	)	)	PUNCT
cana-2086	140	5	34	34	NUM
cana-2086	140	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	140	7	follows	follow	VERB
cana-2086	140	8	:	:	PUNCT
cana-2086	140	9	𝜇(𝑣1	𝜇(𝑣1	ADJ
cana-2086	140	10	)	)	PUNCT
cana-2086	140	11	=	=	SYM
cana-2086	140	12	0.54	0.54	NUM
cana-2086	140	13	,	,	PUNCT
cana-2086	140	14	𝜇(𝑣2	𝜇(𝑣2	NOUN
cana-2086	140	15	)	)	PUNCT
cana-2086	140	16	=	=	SYM
cana-2086	140	17	0.62	0.62	NUM
cana-2086	140	18	,	,	PUNCT
cana-2086	140	19	𝜇(𝑣3	𝜇(𝑣3	NOUN
cana-2086	140	20	)	)	PUNCT
cana-2086	140	21	=	=	SYM
cana-2086	140	22	0.69	0.69	NUM
cana-2086	140	23	,	,	PUNCT
cana-2086	140	24	𝜇(𝑣4	𝜇(𝑣4	NOUN
cana-2086	140	25	)	)	PUNCT
cana-2086	140	26	=	=	SYM
cana-2086	140	27	0.85	0.85	NUM
cana-2086	140	28	,	,	PUNCT
cana-2086	140	29	𝜌(𝑣1𝑣2	𝜌(𝑣1𝑣2	ADJ
cana-2086	140	30	)	)	PUNCT
cana-2086	140	31	=	=	SYM
cana-2086	140	32	0.077	0.077	NUM
cana-2086	140	33	,	,	PUNCT
cana-2086	140	34	𝜌(𝑣1𝑣3	𝜌(𝑣1𝑣3	X
cana-2086	140	35	)	)	PUNCT
cana-2086	140	36	=	=	SYM
cana-2086	141	1	0.15	0.15	NUM
cana-2086	141	2	,	,	PUNCT
cana-2086	141	3	𝜌(𝑣1𝑣4	𝜌(𝑣1𝑣4	ADJ
cana-2086	141	4	)	)	PUNCT
cana-2086	141	5	=	=	SYM
cana-2086	141	6	0.31	0.31	NUM
cana-2086	141	7	,	,	PUNCT
cana-2086	141	8	𝜌(𝑣2𝑣3	𝜌(𝑣2𝑣3	NOUN
cana-2086	141	9	)	)	PUNCT
cana-2086	141	10	=	=	SYM
cana-2086	141	11	0.23	0.23	NUM
cana-2086	141	12	,	,	PUNCT
cana-2086	141	13	𝜌(𝑣2𝑣4	𝜌(𝑣2𝑣4	NOUN
cana-2086	141	14	)	)	PUNCT
cana-2086	141	15	=	=	SYM
cana-2086	141	16	0.38	0.38	NUM
cana-2086	141	17	,	,	PUNCT
cana-2086	141	18	𝜌(𝑣3𝑣4	𝜌(𝑣3𝑣4	NOUN
cana-2086	141	19	)	)	PUNCT
cana-2086	141	20	=	=	SYM
cana-2086	141	21	0.46	0.46	NUM
cana-2086	141	22	.	.	PUNCT
cana-2086	141	23	3.1	3.1	NUM
cana-2086	141	24	.	.	PUNCT
cana-2086	142	1	general	general	ADJ
cana-2086	142	2	results	result	NOUN
cana-2086	142	3	on	on	ADP
cana-2086	142	4	elegant	elegant	ADJ
cana-2086	142	5	fuzzy	fuzzy	ADJ
cana-2086	142	6	labeling	labeling	NOUN
cana-2086	142	7	of	of	ADP
cana-2086	142	8	simple	simple	ADJ
cana-2086	142	9	graphs	graph	NOUN
cana-2086	142	10	proposition	proposition	NOUN
cana-2086	142	11	3.1.1	3.1.1	NUM
cana-2086	142	12	.	.	PUNCT
cana-2086	143	1	a	a	DET
cana-2086	143	2	simple	simple	ADJ
cana-2086	143	3	graph	graph	NOUN
cana-2086	143	4	𝐺	𝐺	PROPN
cana-2086	143	5	admits	admit	VERB
cana-2086	143	6	elegant	elegant	ADJ
cana-2086	143	7	fuzzy	fuzzy	ADJ
cana-2086	143	8	labeling	labeling	NOUN
cana-2086	143	9	if	if	SCONJ
cana-2086	143	10	and	and	CCONJ
cana-2086	143	11	only	only	ADV
cana-2086	143	12	if	if	SCONJ
cana-2086	143	13	it	it	PRON
cana-2086	143	14	admits	admit	VERB
cana-2086	143	15	elegant	elegant	ADJ
cana-2086	143	16	labeling	labeling	NOUN
cana-2086	143	17	.	.	PUNCT
cana-2086	144	1	proof	proof	NOUN
cana-2086	144	2	.	.	PUNCT
cana-2086	145	1	let	let	VERB
cana-2086	145	2	𝐺	𝐺	PRON
cana-2086	145	3	be	be	AUX
cana-2086	145	4	a	a	DET
cana-2086	145	5	simple	simple	ADJ
cana-2086	145	6	graph	graph	NOUN
cana-2086	145	7	with	with	ADP
cana-2086	145	8	𝑝	𝑝	NOUN
cana-2086	145	9	vertices	vertex	NOUN
cana-2086	145	10	and	and	CCONJ
cana-2086	145	11	𝑞	𝑞	NOUN
cana-2086	145	12	edges	edge	NOUN
cana-2086	145	13	.	.	PUNCT
cana-2086	146	1	let	let	VERB
cana-2086	146	2	𝐺	𝐺	PROPN
cana-2086	146	3	admit	admit	VERB
cana-2086	146	4	elegant	elegant	ADJ
cana-2086	146	5	fuzzy	fuzzy	ADJ
cana-2086	146	6	labeling	labeling	NOUN
cana-2086	146	7	.	.	PUNCT
cana-2086	147	1	by	by	ADP
cana-2086	147	2	definition	definition	NOUN
cana-2086	147	3	3.1	3.1	NUM
cana-2086	147	4	,	,	PUNCT
cana-2086	147	5	𝐺	𝐺	PROPN
cana-2086	147	6	admits	admit	VERB
cana-2086	147	7	elegant	elegant	ADJ
cana-2086	147	8	labeling	labeling	NOUN
cana-2086	147	9	.	.	PUNCT
cana-2086	148	1	conversely	conversely	ADV
cana-2086	148	2	,	,	PUNCT
cana-2086	148	3	let	let	VERB
cana-2086	148	4	𝐺	𝐺	PRON
cana-2086	148	5	admit	admit	VERB
cana-2086	148	6	elegant	elegant	ADJ
cana-2086	148	7	labeling	labeling	NOUN
cana-2086	148	8	.	.	PUNCT
cana-2086	149	1	by	by	ADP
cana-2086	149	2	definition	definition	NOUN
cana-2086	149	3	2.1	2.1	NUM
cana-2086	149	4	,	,	PUNCT
cana-2086	149	5	there	there	PRON
cana-2086	149	6	exist	exist	VERB
cana-2086	149	7	an	an	DET
cana-2086	149	8	injective	injective	ADJ
cana-2086	149	9	function	function	NOUN
cana-2086	149	10	𝑓	𝑓	NOUN
cana-2086	149	11	:	:	PUNCT
cana-2086	149	12	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	149	13	)	)	PUNCT
cana-2086	149	14	→	→	SYM
cana-2086	149	15	{	{	PUNCT
cana-2086	149	16	0,1	0,1	NUM
cana-2086	149	17	,	,	PUNCT
cana-2086	149	18	…	…	PUNCT
cana-2086	149	19	,	,	PUNCT
cana-2086	149	20	𝑞	𝑞	X
cana-2086	149	21	}	}	PUNCT
cana-2086	149	22	and	and	CCONJ
cana-2086	149	23	an	an	DET
cana-2086	149	24	injective	injective	ADJ
cana-2086	149	25	function	function	NOUN
cana-2086	149	26	𝑔	𝑔	NOUN
cana-2086	149	27	:	:	PUNCT
cana-2086	149	28	𝐸(𝐺	𝐸(𝐺	PROPN
cana-2086	149	29	)	)	PUNCT
cana-2086	149	30	→	→	SYM
cana-2086	149	31	{	{	PUNCT
cana-2086	149	32	1	1	NUM
cana-2086	149	33	,	,	PUNCT
cana-2086	149	34	…	…	PUNCT
cana-2086	149	35	,	,	PUNCT
cana-2086	149	36	𝑞	𝑞	X
cana-2086	149	37	}	}	PUNCT
cana-2086	149	38	defined	define	VERB
cana-2086	149	39	as	as	ADP
cana-2086	149	40	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-2086	149	41	)	)	PUNCT
cana-2086	149	42	=	=	PUNCT
cana-2086	149	43	(	(	PUNCT
cana-2086	149	44	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	149	45	)	)	PUNCT
cana-2086	150	1	+	+	CCONJ
cana-2086	150	2	𝑓(𝑣	𝑓(𝑣	NOUN
cana-2086	150	3	)	)	PUNCT
cana-2086	150	4	)	)	PUNCT
cana-2086	150	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	150	6	(	(	PUNCT
cana-2086	150	7	𝑞	𝑞	X
cana-2086	150	8	+	+	NOUN
cana-2086	150	9	1	1	NUM
cana-2086	150	10	)	)	PUNCT
cana-2086	150	11	.	.	PUNCT
cana-2086	151	1	now	now	ADV
cana-2086	151	2	translate	translate	VERB
cana-2086	151	3	the	the	DET
cana-2086	151	4	function	function	NOUN
cana-2086	151	5	𝑓	𝑓	ADP
cana-2086	151	6	by	by	X
cana-2086	151	7	(	(	PUNCT
cana-2086	151	8	𝑞	𝑞	X
cana-2086	151	9	+	+	NOUN
cana-2086	151	10	1	1	NUM
cana-2086	151	11	)	)	PUNCT
cana-2086	151	12	,	,	PUNCT
cana-2086	151	13	so	so	SCONJ
cana-2086	151	14	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	151	15	)	)	PUNCT
cana-2086	151	16	increases	increase	VERB
cana-2086	151	17	by	by	ADP
cana-2086	151	18	𝑞	𝑞	PROPN
cana-2086	151	19	+	+	PROPN
cana-2086	151	20	1	1	NUM
cana-2086	151	21	and	and	CCONJ
cana-2086	151	22	𝑔(𝑢𝑣	𝑔(𝑢𝑣	NOUN
cana-2086	151	23	)	)	PUNCT
cana-2086	151	24	remains	remain	VERB
cana-2086	151	25	unchanged	unchanged	ADJ
cana-2086	151	26	.	.	PUNCT
cana-2086	152	1	now	now	ADV
cana-2086	152	2	define	define	VERB
cana-2086	152	3	vertex	vertex	NOUN
cana-2086	152	4	labeling	labeling	NOUN
cana-2086	152	5	𝜇	𝜇	ADP
cana-2086	152	6	:	:	PUNCT
cana-2086	152	7	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2086	152	8	)	)	PUNCT
cana-2086	152	9	→	→	PUNCT
cana-2086	153	1	[	[	X
cana-2086	153	2	0,1	0,1	NUM
cana-2086	153	3	]	]	PUNCT
cana-2086	153	4	as	as	ADP
cana-2086	153	5	𝜇(𝑢	𝜇(𝑢	PROPN
cana-2086	153	6	)	)	PUNCT
cana-2086	153	7	=	=	SYM
cana-2086	153	8	𝑓(𝑢	𝑓(𝑢	PROPN
cana-2086	153	9	)	)	PUNCT
cana-2086	153	10	.	.	PUNCT
cana-2086	154	1	ℎ	ℎ	NOUN
cana-2086	154	2	and	and	CCONJ
cana-2086	154	3	edge	edge	VERB
cana-2086	154	4	labeling	labeling	NOUN
cana-2086	154	5	𝜌	𝜌	NOUN
cana-2086	154	6	:	:	PUNCT
cana-2086	154	7	𝐸(𝐺	𝐸(𝐺	X
cana-2086	154	8	)	)	PUNCT
cana-2086	154	9	→	→	PUNCT
cana-2086	155	1	[	[	X
cana-2086	155	2	0,1	0,1	NUM
cana-2086	155	3	]	]	PUNCT
cana-2086	155	4	as	as	ADP
cana-2086	155	5	𝜌(𝑢𝑣	𝜌(𝑢𝑣	NOUN
cana-2086	155	6	)	)	PUNCT
cana-2086	155	7	=	=	SYM
cana-2086	155	8	𝑔(𝑢𝑣	𝑔(𝑢𝑣	PROPN
cana-2086	155	9	)	)	PUNCT
cana-2086	155	10	.	.	PUNCT
cana-2086	156	1	ℎ	ℎ	PROPN
cana-2086	156	2	,	,	PUNCT
cana-2086	156	3	where	where	SCONJ
cana-2086	156	4	ℎ	ℎ	X
cana-2086	156	5	=	=	SYM
cana-2086	156	6	1	1	NUM
cana-2086	156	7	2𝑞+1	2𝑞+1	NUM
cana-2086	156	8	.	.	PUNCT
cana-2086	157	1	the	the	DET
cana-2086	157	2	functions	function	NOUN
cana-2086	157	3	f	f	PROPN
cana-2086	157	4	and	and	CCONJ
cana-2086	157	5	g	g	PROPN
cana-2086	157	6	are	be	AUX
cana-2086	157	7	injective	injective	ADJ
cana-2086	157	8	.	.	PUNCT
cana-2086	158	1	hence	hence	ADV
cana-2086	158	2	,	,	PUNCT
cana-2086	158	3	the	the	DET
cana-2086	158	4	vertex	vertex	NOUN
cana-2086	158	5	labels	label	NOUN
cana-2086	158	6	and	and	CCONJ
cana-2086	158	7	edge	edge	NOUN
cana-2086	158	8	labels	label	NOUN
cana-2086	158	9	are	be	AUX
cana-2086	158	10	distinct	distinct	ADJ
cana-2086	158	11	and	and	CCONJ
cana-2086	158	12	nonzero	nonzero	NOUN
cana-2086	158	13	.	.	PUNCT
cana-2086	159	1	since	since	SCONJ
cana-2086	159	2	𝑚𝑖𝑛(𝑓(𝑢	𝑚𝑖𝑛(𝑓(𝑢	NOUN
cana-2086	159	3	)	)	PUNCT
cana-2086	159	4	)	)	PUNCT
cana-2086	160	1	=	=	PUNCT
cana-2086	161	1	𝑞	𝑞	X
cana-2086	161	2	+	+	CCONJ
cana-2086	161	3	1	1	NUM
cana-2086	161	4	and	and	CCONJ
cana-2086	161	5	𝑚𝑎𝑥(𝑔(𝑢𝑣	𝑚𝑎𝑥(𝑔(𝑢𝑣	ADJ
cana-2086	161	6	)	)	PUNCT
cana-2086	161	7	)	)	PUNCT
cana-2086	162	1	=	=	SYM
cana-2086	162	2	𝑞	𝑞	NOUN
cana-2086	162	3	,	,	PUNCT
cana-2086	162	4	edge	edge	NOUN
cana-2086	162	5	labels	label	NOUN
cana-2086	162	6	are	be	AUX
cana-2086	162	7	less	less	ADJ
cana-2086	162	8	than	than	ADP
cana-2086	162	9	the	the	DET
cana-2086	162	10	minimum	minimum	NOUN
cana-2086	162	11	of	of	ADP
cana-2086	162	12	their	their	PRON
cana-2086	162	13	respective	respective	ADJ
cana-2086	162	14	endpoints	endpoint	NOUN
cana-2086	162	15	labels	label	NOUN
cana-2086	162	16	.	.	PUNCT
cana-2086	163	1	therefore	therefore	ADV
cana-2086	163	2	(	(	PUNCT
cana-2086	163	3	𝜇	𝜇	X
cana-2086	163	4	,	,	PUNCT
cana-2086	163	5	𝜌	𝜌	X
cana-2086	163	6	)	)	PUNCT
cana-2086	163	7	is	be	AUX
cana-2086	163	8	an	an	DET
cana-2086	163	9	elegant	elegant	ADJ
cana-2086	163	10	fuzzy	fuzzy	ADJ
cana-2086	163	11	labeling	labeling	NOUN
cana-2086	163	12	of	of	ADP
cana-2086	163	13	𝐺.	𝐺.	NOUN
cana-2086	163	14	□	□	PUNCT
cana-2086	163	15	proposition	proposition	NOUN
cana-2086	163	16	3.1.2	3.1.2	NOUN
cana-2086	163	17	.	.	PUNCT
cana-2086	164	1	if	if	SCONJ
cana-2086	164	2	a	a	DET
cana-2086	164	3	simple	simple	ADJ
cana-2086	164	4	graph	graph	NOUN
cana-2086	164	5	𝐺	𝐺	PROPN
cana-2086	164	6	admits	admit	VERB
cana-2086	164	7	elegant	elegant	ADJ
cana-2086	164	8	fuzzy	fuzzy	ADJ
cana-2086	164	9	labeling	labeling	NOUN
cana-2086	164	10	then	then	ADV
cana-2086	164	11	it	it	PRON
cana-2086	164	12	admits	admit	VERB
cana-2086	164	13	fuzzy	fuzzy	ADJ
cana-2086	164	14	labeling	labeling	NOUN
cana-2086	164	15	,	,	PUNCT
cana-2086	164	16	but	but	CCONJ
cana-2086	164	17	not	not	PART
cana-2086	164	18	conversely	conversely	ADV
cana-2086	164	19	.	.	PUNCT
cana-2086	165	1	proof	proof	NOUN
cana-2086	165	2	.	.	PUNCT
cana-2086	166	1	let	let	VERB
cana-2086	166	2	𝐺	𝐺	PRON
cana-2086	166	3	be	be	AUX
cana-2086	166	4	a	a	DET
cana-2086	166	5	simple	simple	ADJ
cana-2086	166	6	graph	graph	NOUN
cana-2086	166	7	with	with	ADP
cana-2086	166	8	𝑝	𝑝	NOUN
cana-2086	166	9	vertices	vertex	NOUN
cana-2086	166	10	and	and	CCONJ
cana-2086	166	11	𝑞	𝑞	NOUN
cana-2086	166	12	edges	edge	NOUN
cana-2086	166	13	.	.	PUNCT
cana-2086	167	1	let	let	VERB
cana-2086	167	2	𝐺	𝐺	PROPN
cana-2086	167	3	admit	admit	VERB
cana-2086	167	4	elegant	elegant	ADJ
cana-2086	167	5	fuzzy	fuzzy	ADJ
cana-2086	167	6	labeling	labeling	NOUN
cana-2086	167	7	.	.	PUNCT
cana-2086	168	1	by	by	ADP
cana-2086	168	2	definition	definition	NOUN
cana-2086	168	3	3.1	3.1	NUM
cana-2086	168	4	,	,	PUNCT
cana-2086	168	5	𝐺	𝐺	PROPN
cana-2086	168	6	admits	admit	VERB
cana-2086	168	7	fuzzy	fuzzy	ADJ
cana-2086	168	8	labeling	labeling	NOUN
cana-2086	168	9	.	.	PUNCT
cana-2086	169	1	conversely	conversely	ADV
cana-2086	169	2	,	,	PUNCT
cana-2086	169	3	if	if	SCONJ
cana-2086	169	4	𝐺	𝐺	PROPN
cana-2086	169	5	admits	admit	VERB
cana-2086	169	6	fuzzy	fuzzy	ADJ
cana-2086	169	7	labeling	labeling	NOUN
cana-2086	169	8	then	then	ADV
cana-2086	169	9	it	it	PRON
cana-2086	169	10	is	be	AUX
cana-2086	169	11	not	not	PART
cana-2086	169	12	necessary	necessary	ADJ
cana-2086	169	13	that	that	SCONJ
cana-2086	169	14	𝐺	𝐺	PROPN
cana-2086	169	15	admits	admit	VERB
cana-2086	169	16	elegant	elegant	ADJ
cana-2086	169	17	fuzzy	fuzzy	ADJ
cana-2086	169	18	labeling	labeling	NOUN
cana-2086	169	19	.	.	PUNCT
cana-2086	170	1	for	for	ADP
cana-2086	170	2	instance	instance	NOUN
cana-2086	170	3	consider	consider	VERB
cana-2086	170	4	the	the	DET
cana-2086	170	5	graph	graph	NOUN
cana-2086	170	6	𝑃4	𝑃4	NOUN
cana-2086	170	7	with	with	ADP
cana-2086	170	8	vertex	vertex	NOUN
cana-2086	170	9	set	set	VERB
cana-2086	170	10	𝑉(𝑃4	𝑉(𝑃4	PUNCT
cana-2086	170	11	)	)	PUNCT
cana-2086	170	12	=	=	PRON
cana-2086	170	13	{	{	PUNCT
cana-2086	170	14	𝑣1	𝑣1	PROPN
cana-2086	170	15	,	,	PUNCT
cana-2086	170	16	𝑣2	𝑣2	PROPN
cana-2086	170	17	,	,	PUNCT
cana-2086	170	18	𝑣3	𝑣3	ADJ
cana-2086	170	19	,	,	PUNCT
cana-2086	170	20	𝑣4	𝑣4	NOUN
cana-2086	170	21	}	}	PUNCT
cana-2086	170	22	and	and	CCONJ
cana-2086	170	23	edge	edge	VERB
cana-2086	170	24	set	set	VERB
cana-2086	170	25	𝐸(𝑃4	𝐸(𝑃4	NOUN
cana-2086	170	26	)	)	PUNCT
cana-2086	170	27	=	=	SYM
cana-2086	170	28	{	{	PUNCT
cana-2086	170	29	𝑣𝑖𝑣𝑖+1	𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	170	30	,	,	PUNCT
cana-2086	170	31	1	1	NUM
cana-2086	170	32	≤	≤	NUM
cana-2086	170	33	𝑖	𝑖	SYM
cana-2086	170	34	≤	≤	NOUN
cana-2086	170	35	3	3	NUM
cana-2086	170	36	}	}	PUNCT
cana-2086	170	37	.	.	PUNCT
cana-2086	171	1	define	define	VERB
cana-2086	171	2	vertex	vertex	NOUN
cana-2086	171	3	labeling	labeling	NOUN
cana-2086	171	4	𝜇	𝜇	ADP
cana-2086	171	5	:	:	PUNCT
cana-2086	171	6	𝑉(𝑃4	𝑉(𝑃4	ADJ
cana-2086	171	7	)	)	PUNCT
cana-2086	171	8	→	→	PUNCT
cana-2086	172	1	[	[	X
cana-2086	172	2	0,1	0,1	NUM
cana-2086	172	3	]	]	PUNCT
cana-2086	172	4	as	as	ADP
cana-2086	172	5	𝜇(𝑣𝑖	𝜇(𝑣𝑖	NUM
cana-2086	172	6	)	)	PUNCT
cana-2086	173	1	=	=	PUNCT
cana-2086	174	1	𝑖+3	𝑖+3	DET
cana-2086	174	2	10	10	NUM
cana-2086	174	3	and	and	CCONJ
cana-2086	174	4	edge	edge	VERB
cana-2086	174	5	labeling	labeling	NOUN
cana-2086	174	6	𝜌	𝜌	ADP
cana-2086	174	7	:	:	PUNCT
cana-2086	174	8	𝐸(𝑃4	𝐸(𝑃4	NOUN
cana-2086	174	9	)	)	PUNCT
cana-2086	174	10	→	→	PUNCT
cana-2086	175	1	[	[	X
cana-2086	175	2	0,1	0,1	NUM
cana-2086	175	3	]	]	PUNCT
cana-2086	175	4	as	as	ADP
cana-2086	175	5	𝜌(𝑣𝑖𝑣𝑖+1	𝜌(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	175	6	)	)	PUNCT
cana-2086	175	7	=	=	PUNCT
cana-2086	175	8	𝑖	𝑖	SYM
cana-2086	175	9	10	10	NUM
cana-2086	175	10	.	.	PUNCT
cana-2086	176	1	by	by	ADP
cana-2086	176	2	definition	definition	NOUN
cana-2086	176	3	2.5	2.5	NUM
cana-2086	176	4	,	,	PUNCT
cana-2086	176	5	𝑃4	𝑃4	PROPN
cana-2086	176	6	admits	admit	VERB
cana-2086	176	7	fuzzy	fuzzy	ADJ
cana-2086	176	8	labeling	labeling	NOUN
cana-2086	176	9	.	.	PUNCT
cana-2086	177	1	but	but	CCONJ
cana-2086	177	2	𝑃4	𝑃4	NOUN
cana-2086	177	3	is	be	AUX
cana-2086	177	4	not	not	PART
cana-2086	177	5	elegant	elegant	ADJ
cana-2086	177	6	[	[	X
cana-2086	177	7	2	2	NUM
cana-2086	177	8	]	]	PUNCT
cana-2086	177	9	.	.	PUNCT
cana-2086	178	1	□	□	PUNCT
cana-2086	178	2	3.2	3.2	NUM
cana-2086	178	3	.	.	PUNCT
cana-2086	179	1	elegant	elegant	ADJ
cana-2086	179	2	fuzzy	fuzzy	ADJ
cana-2086	179	3	labeling	labeling	NOUN
cana-2086	179	4	of	of	ADP
cana-2086	179	5	certain	certain	ADJ
cana-2086	179	6	classes	class	NOUN
cana-2086	179	7	of	of	ADP
cana-2086	179	8	simple	simple	ADJ
cana-2086	179	9	graphs	graph	NOUN
cana-2086	179	10	theorem	theorem	VERB
cana-2086	179	11	3.2.1	3.2.1	NUM
cana-2086	179	12	.	.	PUNCT
cana-2086	180	1	path	path	NOUN
cana-2086	180	2	graphs	graph	NOUN
cana-2086	181	1	𝑃𝑛	𝑃𝑛	PROPN
cana-2086	181	2	admit	admit	VERB
cana-2086	181	3	elegant	elegant	ADJ
cana-2086	181	4	fuzzy	fuzzy	ADJ
cana-2086	181	5	labeling	labeling	NOUN
cana-2086	181	6	except	except	SCONJ
cana-2086	181	7	for	for	ADP
cana-2086	181	8	𝑛	𝑛	PROPN
cana-2086	181	9	=	=	SYM
cana-2086	181	10	4	4	X
cana-2086	181	11	.	.	PUNCT
cana-2086	181	12	proof	proof	NOUN
cana-2086	181	13	.	.	PUNCT
cana-2086	182	1	let	let	VERB
cana-2086	182	2	𝑃𝑛	𝑃𝑛	NOUN
cana-2086	182	3	be	be	AUX
cana-2086	182	4	a	a	DET
cana-2086	182	5	path	path	NOUN
cana-2086	182	6	with	with	ADP
cana-2086	182	7	vertex	vertex	NOUN
cana-2086	182	8	set	set	VERB
cana-2086	182	9	𝑉(𝑃𝑛	𝑉(𝑃𝑛	NOUN
cana-2086	182	10	)	)	PUNCT
cana-2086	183	1	=	=	PRON
cana-2086	183	2	{	{	PUNCT
cana-2086	183	3	𝑣1	𝑣1	PROPN
cana-2086	183	4	,	,	PUNCT
cana-2086	183	5	𝑣2	𝑣2	PROPN
cana-2086	183	6	,	,	PUNCT
cana-2086	183	7	…	…	PUNCT
cana-2086	183	8	,	,	PUNCT
cana-2086	183	9	𝑣𝑛	𝑣𝑛	NOUN
cana-2086	183	10	}	}	PUNCT
cana-2086	183	11	and	and	CCONJ
cana-2086	183	12	edge	edge	VERB
cana-2086	183	13	set	set	VERB
cana-2086	183	14	𝐸(𝑃𝑛	𝐸(𝑃𝑛	PROPN
cana-2086	183	15	)	)	PUNCT
cana-2086	183	16	=	=	PRON
cana-2086	183	17	{	{	PUNCT
cana-2086	183	18	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-2086	183	19	,	,	PUNCT
cana-2086	183	20	𝑣2𝑣3	𝑣2𝑣3	PROPN
cana-2086	183	21	,	,	PUNCT
cana-2086	183	22	…	…	PUNCT
cana-2086	183	23	,	,	PUNCT
cana-2086	183	24	𝑣𝑛−1𝑣𝑛	𝑣𝑛−1𝑣𝑛	NUM
cana-2086	183	25	}	}	PUNCT
cana-2086	183	26	.	.	PUNCT
cana-2086	184	1	let	let	VERB
cana-2086	184	2	𝑓	𝑓	PRON
cana-2086	184	3	be	be	AUX
cana-2086	184	4	an	an	DET
cana-2086	184	5	injective	injective	ADJ
cana-2086	184	6	function	function	NOUN
cana-2086	184	7	from	from	ADP
cana-2086	184	8	𝑉(𝑃𝑛	𝑉(𝑃𝑛	NOUN
cana-2086	184	9	)	)	PUNCT
cana-2086	184	10	into	into	ADP
cana-2086	184	11	the	the	DET
cana-2086	184	12	set	set	NOUN
cana-2086	184	13	{	{	PUNCT
cana-2086	184	14	𝑛	𝑛	PROPN
cana-2086	184	15	,	,	PUNCT
cana-2086	184	16	𝑛	𝑛	PROPN
cana-2086	184	17	+	+	NOUN
cana-2086	184	18	1	1	NUM
cana-2086	184	19	,	,	PUNCT
cana-2086	184	20	…	…	PUNCT
cana-2086	184	21	,	,	PUNCT
cana-2086	184	22	2𝑛	2𝑛	PROPN
cana-2086	184	23	−	−	PROPN
cana-2086	184	24	1	1	X
cana-2086	184	25	}	}	PUNCT
cana-2086	184	26	and	and	CCONJ
cana-2086	184	27	𝑔	𝑔	PROPN
cana-2086	184	28	be	be	AUX
cana-2086	184	29	a	a	DET
cana-2086	184	30	function	function	NOUN
cana-2086	184	31	from	from	ADP
cana-2086	184	32	𝐸(𝑃𝑛	𝐸(𝑃𝑛	PROPN
cana-2086	184	33	)	)	PUNCT
cana-2086	184	34	into	into	ADP
cana-2086	184	35	the	the	DET
cana-2086	184	36	set	set	NOUN
cana-2086	184	37	{	{	PUNCT
cana-2086	184	38	0,1	0,1	NUM
cana-2086	184	39	,	,	PUNCT
cana-2086	184	40	…	…	PUNCT
cana-2086	184	41	,	,	PUNCT
cana-2086	185	1	𝑛	𝑛	DET
cana-2086	185	2	−	−	PROPN
cana-2086	185	3	1	1	NUM
cana-2086	185	4	}	}	PUNCT
cana-2086	185	5	defined	define	VERB
cana-2086	185	6	as	as	ADP
cana-2086	185	7	,	,	PUNCT
cana-2086	185	8	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	185	9	)	)	PUNCT
cana-2086	185	10	=	=	PUNCT
cana-2086	185	11	(	(	PUNCT
cana-2086	185	12	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	185	13	)	)	PUNCT
cana-2086	185	14	+	+	CCONJ
cana-2086	185	15	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	185	16	)	)	PUNCT
cana-2086	185	17	)	)	PUNCT
cana-2086	185	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	185	19	𝑛	𝑛	NOUN
cana-2086	185	20	for	for	ADP
cana-2086	185	21	1	1	NUM
cana-2086	185	22	≤	≤	NUM
cana-2086	185	23	𝑖	𝑖	SYM
cana-2086	185	24	≤	≤	NUM
cana-2086	185	25	𝑛	𝑛	PRON
cana-2086	185	26	−	−	NUM
cana-2086	185	27	1	1	NUM
cana-2086	185	28	.	.	PUNCT
cana-2086	185	29	case	case	NOUN
cana-2086	185	30	1	1	NUM
cana-2086	185	31	:	:	PUNCT
cana-2086	185	32	for	for	ADP
cana-2086	185	33	𝑃2𝑚+1	𝑃2𝑚+1	PROPN
cana-2086	185	34	,	,	PUNCT
cana-2086	185	35	𝑚	𝑚	X
cana-2086	185	36	=	=	SYM
cana-2086	185	37	1	1	NUM
cana-2086	185	38	,	,	PUNCT
cana-2086	185	39	2	2	NUM
cana-2086	185	40	,	,	PUNCT
cana-2086	185	41	3	3	NUM
cana-2086	185	42	,	,	PUNCT
cana-2086	185	43	…	…	PUNCT
cana-2086	185	44	,	,	PUNCT
cana-2086	185	45	𝑓	𝑓	X
cana-2086	185	46	:	:	PUNCT
cana-2086	185	47	𝑉(𝑃2𝑚+1	𝑉(𝑃2𝑚+1	NUM
cana-2086	185	48	)	)	PUNCT
cana-2086	185	49	→	→	SYM
cana-2086	185	50	{	{	PUNCT
cana-2086	185	51	2𝑚	2𝑚	NOUN
cana-2086	185	52	+	+	CCONJ
cana-2086	185	53	1	1	NUM
cana-2086	185	54	,	,	PUNCT
cana-2086	185	55	2𝑚	2𝑚	NOUN
cana-2086	185	56	+	+	CCONJ
cana-2086	185	57	2	2	NUM
cana-2086	185	58	,	,	PUNCT
cana-2086	185	59	2𝑚	2𝑚	NOUN
cana-2086	185	60	+	+	CCONJ
cana-2086	185	61	3	3	NUM
cana-2086	185	62	,	,	PUNCT
cana-2086	185	63	…	…	PUNCT
cana-2086	185	64	,	,	PUNCT
cana-2086	185	65	4𝑚	4𝑚	NUM
cana-2086	185	66	+	+	CCONJ
cana-2086	185	67	1	1	X
cana-2086	185	68	}	}	PUNCT
cana-2086	185	69	is	be	AUX
cana-2086	185	70	defined	define	VERB
cana-2086	185	71	as	as	ADP
cana-2086	185	72	,	,	PUNCT
cana-2086	185	73	communications	communication	NOUN
cana-2086	185	74	on	on	ADP
cana-2086	185	75	applied	apply	VERB
cana-2086	185	76	nonlinear	nonlinear	ADJ
cana-2086	185	77	analysis	analysis	NOUN
cana-2086	185	78	issn	issn	NOUN
cana-2086	185	79	:	:	PUNCT
cana-2086	185	80	1074	1074	NUM
cana-2086	185	81	-	-	PUNCT
cana-2086	185	82	133x	133x	NUM
cana-2086	185	83	vol	vol	NOUN
cana-2086	185	84	32	32	NUM
cana-2086	186	1	no	no	NOUN
cana-2086	186	2	.	.	PUNCT
cana-2086	187	1	1s	1s	NUM
cana-2086	187	2	(	(	PUNCT
cana-2086	187	3	2025	2025	NUM
cana-2086	187	4	)	)	PUNCT
cana-2086	187	5	35	35	NUM
cana-2086	187	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	187	7	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	187	8	)	)	PUNCT
cana-2086	187	9	=	=	PRON
cana-2086	187	10	{	{	PUNCT
cana-2086	187	11	3𝑚	3𝑚	NUM
cana-2086	188	1	+	+	NOUN
cana-2086	188	2	1	1	NUM
cana-2086	188	3	+	+	CCONJ
cana-2086	188	4	𝑖	𝑖	SYM
cana-2086	188	5	1	1	NUM
cana-2086	188	6	≤	≤	NUM
cana-2086	188	7	𝑖	𝑖	SYM
cana-2086	188	8	≤	≤	NOUN
cana-2086	188	9	𝑚	𝑚	PRON
cana-2086	188	10	2𝑚	2𝑚	NOUN
cana-2086	188	11	+	+	CCONJ
cana-2086	188	12	1	1	NUM
cana-2086	188	13	𝑖	𝑖	NOUN
cana-2086	188	14	=	=	PUNCT
cana-2086	188	15	𝑚	𝑚	PROPN
cana-2086	188	16	+	+	ADJ
cana-2086	188	17	1	1	NUM
cana-2086	188	18	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	188	19	)	)	PUNCT
cana-2086	188	20	+	+	CCONJ
cana-2086	188	21	1	1	NUM
cana-2086	188	22	𝑚	𝑚	NOUN
cana-2086	188	23	+	+	NOUN
cana-2086	188	24	2	2	NUM
cana-2086	188	25	≤	≤	NOUN
cana-2086	188	26	𝑖	𝑖	PRON
cana-2086	188	27	≤	≤	NOUN
cana-2086	188	28	2𝑚	2𝑚	NOUN
cana-2086	188	29	+	+	CCONJ
cana-2086	188	30	1	1	X
cana-2086	188	31	.	.	X
cana-2086	189	1	for	for	ADP
cana-2086	189	2	1	1	NUM
cana-2086	189	3	≤	≤	NUM
cana-2086	189	4	𝑖	𝑖	SYM
cana-2086	189	5	≤	≤	NOUN
cana-2086	189	6	𝑚	𝑚	ADP
cana-2086	189	7	−	−	PROPN
cana-2086	189	8	1	1	NUM
cana-2086	189	9	,	,	PUNCT
cana-2086	189	10	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	189	11	)	)	PUNCT
cana-2086	189	12	=	=	PUNCT
cana-2086	189	13	(	(	PUNCT
cana-2086	189	14	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	189	15	)	)	PUNCT
cana-2086	189	16	+	+	CCONJ
cana-2086	189	17	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	189	18	)	)	PUNCT
cana-2086	189	19	)	)	PUNCT
cana-2086	189	20	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	189	21	(	(	PUNCT
cana-2086	189	22	2𝑚	2𝑚	NOUN
cana-2086	189	23	+	+	CCONJ
cana-2086	189	24	1	1	X
cana-2086	189	25	)	)	PUNCT
cana-2086	189	26	=	=	SYM
cana-2086	189	27	(	(	PUNCT
cana-2086	189	28	3𝑚	3𝑚	NUM
cana-2086	189	29	+	+	NOUN
cana-2086	189	30	1	1	NUM
cana-2086	189	31	+	+	NUM
cana-2086	189	32	𝑖	𝑖	SYM
cana-2086	189	33	+	+	NOUN
cana-2086	189	34	3𝑚	3𝑚	NUM
cana-2086	189	35	+	+	NOUN
cana-2086	189	36	1	1	NUM
cana-2086	189	37	+	+	NUM
cana-2086	189	38	𝑖	𝑖	SYM
cana-2086	189	39	+	+	NOUN
cana-2086	189	40	1	1	NUM
cana-2086	189	41	)	)	PUNCT
cana-2086	189	42	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	189	43	(	(	PUNCT
cana-2086	189	44	2𝑚	2𝑚	NOUN
cana-2086	189	45	+	+	CCONJ
cana-2086	189	46	1	1	X
cana-2086	189	47	)	)	PUNCT
cana-2086	189	48	=	=	SYM
cana-2086	189	49	(	(	PUNCT
cana-2086	189	50	3(2𝑚	3(2𝑚	NUM
cana-2086	189	51	+	+	PROPN
cana-2086	189	52	1	1	NUM
cana-2086	189	53	)	)	PUNCT
cana-2086	189	54	+	+	NUM
cana-2086	189	55	2𝑖	2𝑖	NUM
cana-2086	189	56	)	)	PUNCT
cana-2086	189	57	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	189	58	(	(	PUNCT
cana-2086	189	59	2𝑚	2𝑚	NOUN
cana-2086	189	60	+	+	CCONJ
cana-2086	189	61	1	1	X
cana-2086	189	62	)	)	PUNCT
cana-2086	189	63	=	=	SYM
cana-2086	189	64	2𝑖.	2𝑖.	NUM
cana-2086	189	65	for	for	ADP
cana-2086	189	66	𝑖	𝑖	PROPN
cana-2086	189	67	=	=	SYM
cana-2086	189	68	𝑚	𝑚	PROPN
cana-2086	189	69	,	,	PUNCT
cana-2086	189	70	𝑔(𝑣𝑚𝑣𝑚+1	𝑔(𝑣𝑚𝑣𝑚+1	PROPN
cana-2086	189	71	)	)	PUNCT
cana-2086	189	72	=	=	SYM
cana-2086	189	73	(	(	PUNCT
cana-2086	189	74	𝑓(𝑣𝑚	𝑓(𝑣𝑚	NOUN
cana-2086	189	75	)	)	PUNCT
cana-2086	189	76	+	+	CCONJ
cana-2086	189	77	𝑓(𝑣𝑚+1	𝑓(𝑣𝑚+1	NUM
cana-2086	189	78	)	)	PUNCT
cana-2086	189	79	)	)	PUNCT
cana-2086	189	80	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	189	81	(	(	PUNCT
cana-2086	189	82	2𝑚	2𝑚	NOUN
cana-2086	189	83	+	+	CCONJ
cana-2086	189	84	1	1	X
cana-2086	189	85	)	)	PUNCT
cana-2086	189	86	=	=	SYM
cana-2086	189	87	(	(	PUNCT
cana-2086	189	88	3𝑚	3𝑚	NUM
cana-2086	189	89	+	+	NOUN
cana-2086	189	90	1	1	NUM
cana-2086	190	1	+	+	NOUN
cana-2086	190	2	𝑚	𝑚	X
cana-2086	190	3	+	+	NOUN
cana-2086	190	4	2𝑚+	2𝑚+	NUM
cana-2086	190	5	1	1	NUM
cana-2086	190	6	)	)	PUNCT
cana-2086	190	7	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	190	8	(	(	PUNCT
cana-2086	190	9	2𝑚	2𝑚	NOUN
cana-2086	190	10	+	+	CCONJ
cana-2086	190	11	1	1	X
cana-2086	190	12	)	)	PUNCT
cana-2086	190	13	=	=	SYM
cana-2086	190	14	2𝑚.	2𝑚.	NUM
cana-2086	190	15	for	for	ADP
cana-2086	190	16	𝑚	𝑚	PROPN
cana-2086	190	17	+	+	PROPN
cana-2086	190	18	1	1	NUM
cana-2086	190	19	≤	≤	NUM
cana-2086	190	20	𝑖	𝑖	PRON
cana-2086	190	21	≤	≤	NOUN
cana-2086	190	22	2𝑚	2𝑚	NOUN
cana-2086	190	23	,	,	PUNCT
cana-2086	190	24	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	190	25	)	)	PUNCT
cana-2086	190	26	=	=	PUNCT
cana-2086	190	27	(	(	PUNCT
cana-2086	190	28	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	190	29	)	)	PUNCT
cana-2086	190	30	+	+	CCONJ
cana-2086	190	31	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	190	32	)	)	PUNCT
cana-2086	190	33	)	)	PUNCT
cana-2086	190	34	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	190	35	(	(	PUNCT
cana-2086	190	36	2𝑚	2𝑚	NOUN
cana-2086	190	37	+	+	CCONJ
cana-2086	190	38	1	1	X
cana-2086	190	39	)	)	PUNCT
cana-2086	190	40	=	=	NOUN
cana-2086	190	41	(	(	PUNCT
cana-2086	190	42	2𝑓(𝑣𝑖	2𝑓(𝑣𝑖	NUM
cana-2086	190	43	)	)	PUNCT
cana-2086	191	1	+	+	CCONJ
cana-2086	191	2	1	1	X
cana-2086	191	3	)	)	PUNCT
cana-2086	191	4	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	191	5	(	(	PUNCT
cana-2086	191	6	2𝑚	2𝑚	NOUN
cana-2086	191	7	+	+	CCONJ
cana-2086	191	8	1	1	X
cana-2086	191	9	)	)	PUNCT
cana-2086	191	10	=	=	SYM
cana-2086	191	11	(	(	PUNCT
cana-2086	191	12	2𝑓(𝑣𝑖−1	2𝑓(𝑣𝑖−1	NUM
cana-2086	191	13	)	)	PUNCT
cana-2086	191	14	+	+	CCONJ
cana-2086	191	15	3	3	X
cana-2086	191	16	)	)	PUNCT
cana-2086	191	17	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	191	18	(	(	PUNCT
cana-2086	191	19	2𝑚	2𝑚	NOUN
cana-2086	191	20	+	+	CCONJ
cana-2086	191	21	1	1	NUM
cana-2086	191	22	)	)	PUNCT
cana-2086	191	23	⋮	⋮	NOUN
cana-2086	191	24	=	=	SYM
cana-2086	191	25	(	(	PUNCT
cana-2086	191	26	2𝑓(𝑣𝑖−𝑗	2𝑓(𝑣𝑖−𝑗	NUM
cana-2086	191	27	)	)	PUNCT
cana-2086	192	1	+	+	NUM
cana-2086	192	2	2𝑗	2𝑗	NUM
cana-2086	192	3	+	+	CCONJ
cana-2086	192	4	1	1	X
cana-2086	192	5	)	)	PUNCT
cana-2086	192	6	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	192	7	(	(	PUNCT
cana-2086	192	8	2𝑚	2𝑚	NOUN
cana-2086	192	9	+	+	CCONJ
cana-2086	192	10	1	1	NUM
cana-2086	192	11	)	)	PUNCT
cana-2086	192	12	,	,	PUNCT
cana-2086	192	13	𝑗	𝑗	NOUN
cana-2086	193	1	=	=	SYM
cana-2086	193	2	0	0	NUM
cana-2086	193	3	,	,	PUNCT
cana-2086	193	4	1	1	NUM
cana-2086	193	5	,	,	PUNCT
cana-2086	193	6	2	2	NUM
cana-2086	193	7	,	,	PUNCT
cana-2086	193	8	…	…	PUNCT
cana-2086	193	9	,	,	PUNCT
cana-2086	193	10	𝑖	𝑖	PRON
cana-2086	193	11	−	−	NOUN
cana-2086	193	12	𝑚	𝑚	INTJ
cana-2086	193	13	−	−	PROPN
cana-2086	193	14	1	1	NUM
cana-2086	193	15	,	,	PUNCT
cana-2086	193	16	and	and	CCONJ
cana-2086	193	17	when	when	SCONJ
cana-2086	193	18	𝑗	𝑗	X
cana-2086	193	19	=	=	SYM
cana-2086	193	20	𝑖	𝑖	SYM
cana-2086	193	21	−𝑚	−𝑚	NOUN
cana-2086	193	22	−	−	PROPN
cana-2086	193	23	1	1	NUM
cana-2086	193	24	,	,	PUNCT
cana-2086	193	25	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	193	26	)	)	PUNCT
cana-2086	193	27	=	=	PUNCT
cana-2086	193	28	(	(	PUNCT
cana-2086	193	29	2𝑓(𝑣𝑚+1	2𝑓(𝑣𝑚+1	NUM
cana-2086	193	30	)	)	PUNCT
cana-2086	194	1	+	+	CCONJ
cana-2086	195	1	2(𝑖	2(𝑖	NUM
cana-2086	195	2	−	−	NOUN
cana-2086	196	1	𝑚	𝑚	ADP
cana-2086	196	2	−	−	PROPN
cana-2086	196	3	1	1	NUM
cana-2086	196	4	)	)	PUNCT
cana-2086	196	5	+	+	CCONJ
cana-2086	196	6	1	1	X
cana-2086	196	7	)	)	PUNCT
cana-2086	196	8	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	196	9	(	(	PUNCT
cana-2086	196	10	2𝑚	2𝑚	NOUN
cana-2086	196	11	+	+	CCONJ
cana-2086	196	12	1	1	X
cana-2086	196	13	)	)	PUNCT
cana-2086	196	14	=	=	NOUN
cana-2086	197	1	(	(	PUNCT
cana-2086	197	2	2(2𝑚	2(2𝑚	NUM
cana-2086	197	3	+	+	CCONJ
cana-2086	197	4	1	1	X
cana-2086	197	5	)	)	PUNCT
cana-2086	198	1	+	+	CCONJ
cana-2086	198	2	2(𝑖	2(𝑖	NUM
cana-2086	198	3	−	−	NOUN
cana-2086	198	4	𝑚	𝑚	ADP
cana-2086	198	5	−	−	PROPN
cana-2086	198	6	1	1	NUM
cana-2086	198	7	)	)	PUNCT
cana-2086	198	8	+	+	CCONJ
cana-2086	198	9	1)𝑚𝑜𝑑(2𝑚	1)𝑚𝑜𝑑(2𝑚	NUM
cana-2086	198	10	+	+	SYM
cana-2086	198	11	1	1	NUM
cana-2086	198	12	)	)	PUNCT
cana-2086	198	13	=	=	SYM
cana-2086	199	1	2(𝑖	2(𝑖	NUM
cana-2086	199	2	−	−	NOUN
cana-2086	200	1	𝑚	𝑚	INTJ
cana-2086	200	2	−	−	PROPN
cana-2086	200	3	1	1	NUM
cana-2086	200	4	)	)	PUNCT
cana-2086	200	5	+	+	NOUN
cana-2086	200	6	1	1	X
cana-2086	200	7	.	.	X
cana-2086	200	8	hence	hence	ADV
cana-2086	200	9	,	,	PUNCT
cana-2086	200	10	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	200	11	)	)	PUNCT
cana-2086	200	12	=	=	NOUN
cana-2086	200	13	{	{	PUNCT
cana-2086	200	14	2𝑖	2𝑖	NOUN
cana-2086	200	15	1	1	NUM
cana-2086	200	16	≤	≤	NOUN
cana-2086	200	17	𝑖	𝑖	SYM
cana-2086	200	18	≤	≤	NOUN
cana-2086	200	19	𝑚	𝑚	ADP
cana-2086	200	20	2(𝑖	2(𝑖	NUM
cana-2086	200	21	−	−	NOUN
cana-2086	200	22	𝑚	𝑚	ADP
cana-2086	200	23	−	−	PROPN
cana-2086	200	24	1	1	NUM
cana-2086	200	25	)	)	PUNCT
cana-2086	200	26	+	+	CCONJ
cana-2086	200	27	1	1	NUM
cana-2086	200	28	𝑚	𝑚	NOUN
cana-2086	200	29	+	+	NOUN
cana-2086	200	30	1	1	NUM
cana-2086	200	31	≤	≤	NUM
cana-2086	200	32	𝑖	𝑖	PRON
cana-2086	200	33	≤	≤	ADJ
cana-2086	200	34	2𝑚.	2𝑚.	NUM
cana-2086	200	35	case	case	NOUN
cana-2086	200	36	2	2	NUM
cana-2086	200	37	:	:	PUNCT
cana-2086	200	38	for	for	ADP
cana-2086	200	39	𝑃2𝑚	𝑃2𝑚	PROPN
cana-2086	200	40	,	,	PUNCT
cana-2086	200	41	𝑚	𝑚	X
cana-2086	200	42	=	=	SYM
cana-2086	200	43	3	3	NUM
cana-2086	200	44	,	,	PUNCT
cana-2086	200	45	5	5	NUM
cana-2086	200	46	,	,	PUNCT
cana-2086	200	47	7	7	NUM
cana-2086	200	48	,	,	PUNCT
cana-2086	200	49	…	…	PUNCT
cana-2086	200	50	,	,	PUNCT
cana-2086	200	51	𝑓	𝑓	X
cana-2086	200	52	:	:	PUNCT
cana-2086	200	53	𝑉(𝑃2𝑚	𝑉(𝑃2𝑚	NUM
cana-2086	200	54	)	)	PUNCT
cana-2086	200	55	→	→	SYM
cana-2086	200	56	{	{	PUNCT
cana-2086	200	57	2𝑚	2𝑚	NOUN
cana-2086	200	58	,	,	PUNCT
cana-2086	200	59	2𝑚	2𝑚	NOUN
cana-2086	200	60	+	+	CCONJ
cana-2086	200	61	1	1	NUM
cana-2086	200	62	,	,	PUNCT
cana-2086	200	63	2𝑚	2𝑚	NOUN
cana-2086	200	64	+	+	CCONJ
cana-2086	200	65	2	2	NUM
cana-2086	200	66	,	,	PUNCT
cana-2086	200	67	…	…	PUNCT
cana-2086	200	68	,	,	PUNCT
cana-2086	200	69	4𝑚	4𝑚	NUM
cana-2086	201	1	−	−	NOUN
cana-2086	201	2	1	1	NUM
cana-2086	201	3	}	}	PUNCT
cana-2086	201	4	is	be	AUX
cana-2086	201	5	defined	define	VERB
cana-2086	201	6	as	as	ADP
cana-2086	201	7	,	,	PUNCT
cana-2086	201	8	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	201	9	)	)	PUNCT
cana-2086	201	10	=	=	SYM
cana-2086	201	11	{	{	PUNCT
cana-2086	201	12	2𝑚	2𝑚	NOUN
cana-2086	201	13	+	+	CCONJ
cana-2086	201	14	1	1	NUM
cana-2086	201	15	𝑖	𝑖	SYM
cana-2086	201	16	=	=	NOUN
cana-2086	201	17	1	1	NUM
cana-2086	201	18	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	201	19	)	)	PUNCT
cana-2086	201	20	+	+	CCONJ
cana-2086	201	21	2	2	NUM
cana-2086	201	22	2	2	NUM
cana-2086	201	23	≤	≤	NOUN
cana-2086	201	24	𝑖	𝑖	PUNCT
cana-2086	201	25	≤	≤	NUM
cana-2086	202	1	⌊	⌊	VERB
cana-2086	202	2	𝑚	𝑚	ADP
cana-2086	202	3	2	2	NUM
cana-2086	202	4	⌋	⌋	NOUN
cana-2086	202	5	3𝑚	3𝑚	NUM
cana-2086	202	6	𝑖	𝑖	NOUN
cana-2086	202	7	=	=	PUNCT
cana-2086	202	8	⌊	⌊	VERB
cana-2086	202	9	𝑚	𝑚	ADP
cana-2086	202	10	2	2	NUM
cana-2086	202	11	⌋	⌋	NOUN
cana-2086	202	12	+	+	CCONJ
cana-2086	202	13	1	1	NUM
cana-2086	202	14	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	202	15	)	)	PUNCT
cana-2086	202	16	+	+	CCONJ
cana-2086	202	17	1	1	NUM
cana-2086	202	18	⌊	⌊	NOUN
cana-2086	202	19	𝑚	𝑚	ADP
cana-2086	202	20	2	2	NUM
cana-2086	202	21	⌋	⌋	NOUN
cana-2086	202	22	+	+	CCONJ
cana-2086	202	23	2	2	NUM
cana-2086	202	24	≤	≤	NUM
cana-2086	202	25	𝑖	𝑖	NOUN
cana-2086	202	26	≤	≤	NOUN
cana-2086	202	27	𝑚	𝑚	X
cana-2086	203	1	+	+	CCONJ
cana-2086	203	2	⌊	⌊	PROPN
cana-2086	203	3	𝑚	𝑚	ADP
cana-2086	203	4	2	2	NUM
cana-2086	203	5	⌋	⌋	NOUN
cana-2086	203	6	2𝑚	2𝑚	NOUN
cana-2086	203	7	𝑖	𝑖	NOUN
cana-2086	203	8	=	=	SYM
cana-2086	203	9	𝑚	𝑚	PROPN
cana-2086	204	1	+	+	CCONJ
cana-2086	204	2	⌊	⌊	PROPN
cana-2086	204	3	𝑚	𝑚	ADP
cana-2086	204	4	2	2	NUM
cana-2086	204	5	⌋	⌋	NOUN
cana-2086	204	6	+	+	CCONJ
cana-2086	204	7	1	1	NUM
cana-2086	204	8	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	204	9	)	)	PUNCT
cana-2086	204	10	+	+	CCONJ
cana-2086	204	11	2	2	NUM
cana-2086	204	12	𝑚	𝑚	X
cana-2086	204	13	+	+	CCONJ
cana-2086	204	14	⌊	⌊	PROPN
cana-2086	204	15	𝑚	𝑚	ADP
cana-2086	204	16	2	2	NUM
cana-2086	204	17	⌋	⌋	NOUN
cana-2086	204	18	+	+	CCONJ
cana-2086	204	19	2	2	NUM
cana-2086	204	20	≤	≤	NUM
cana-2086	204	21	𝑖	𝑖	PRON
cana-2086	204	22	≤	≤	ADJ
cana-2086	204	23	2𝑚.	2𝑚.	NUM
cana-2086	204	24	communications	communication	NOUN
cana-2086	204	25	on	on	ADP
cana-2086	204	26	applied	apply	VERB
cana-2086	204	27	nonlinear	nonlinear	ADJ
cana-2086	204	28	analysis	analysis	NOUN
cana-2086	204	29	issn	issn	NOUN
cana-2086	204	30	:	:	PUNCT
cana-2086	204	31	1074	1074	NUM
cana-2086	204	32	-	-	PUNCT
cana-2086	204	33	133x	133x	NUM
cana-2086	204	34	vol	vol	NOUN
cana-2086	204	35	32	32	NUM
cana-2086	205	1	no	no	NOUN
cana-2086	205	2	.	.	PUNCT
cana-2086	206	1	1s	1s	NUM
cana-2086	206	2	(	(	PUNCT
cana-2086	206	3	2025	2025	NUM
cana-2086	206	4	)	)	PUNCT
cana-2086	206	5	36	36	NUM
cana-2086	206	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	206	7	following	follow	VERB
cana-2086	206	8	a	a	DET
cana-2086	206	9	similar	similar	ADJ
cana-2086	206	10	approach	approach	NOUN
cana-2086	206	11	as	as	ADP
cana-2086	206	12	in	in	ADP
cana-2086	206	13	case	case	NOUN
cana-2086	206	14	1	1	NUM
cana-2086	206	15	,	,	PUNCT
cana-2086	206	16	we	we	PRON
cana-2086	206	17	obtain	obtain	VERB
cana-2086	206	18	,	,	PUNCT
cana-2086	206	19	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	206	20	)	)	PUNCT
cana-2086	206	21	=	=	PRON
cana-2086	206	22	{	{	PUNCT
cana-2086	206	23	4𝑖	4𝑖	NOUN
cana-2086	206	24	1	1	NUM
cana-2086	206	25	≤	≤	NOUN
cana-2086	206	26	𝑖	𝑖	PUNCT
cana-2086	206	27	≤	≤	NUM
cana-2086	207	1	⌊	⌊	VERB
cana-2086	207	2	𝑚	𝑚	ADP
cana-2086	207	3	2	2	NUM
cana-2086	207	4	⌋	⌋	NOUN
cana-2086	207	5	2	2	NUM
cana-2086	207	6	(	(	PUNCT
cana-2086	207	7	𝑖	𝑖	SYM
cana-2086	207	8	−	−	NOUN
cana-2086	207	9	⌊	⌊	VERB
cana-2086	207	10	𝑚	𝑚	ADP
cana-2086	207	11	2	2	NUM
cana-2086	207	12	⌋	⌋	NOUN
cana-2086	207	13	−	−	NOUN
cana-2086	207	14	1	1	NUM
cana-2086	207	15	)	)	PUNCT
cana-2086	207	16	+	+	CCONJ
cana-2086	207	17	1	1	NUM
cana-2086	207	18	⌊	⌊	NOUN
cana-2086	207	19	𝑚	𝑚	ADP
cana-2086	207	20	2	2	NUM
cana-2086	207	21	⌋	⌋	NOUN
cana-2086	207	22	+	+	CCONJ
cana-2086	207	23	1	1	NUM
cana-2086	207	24	≤	≤	NUM
cana-2086	207	25	𝑖	𝑖	NOUN
cana-2086	207	26	≤	≤	NOUN
cana-2086	207	27	𝑚	𝑚	X
cana-2086	208	1	+	+	CCONJ
cana-2086	208	2	⌊	⌊	PROPN
cana-2086	208	3	𝑚	𝑚	ADP
cana-2086	208	4	2	2	NUM
cana-2086	208	5	⌋	⌋	NOUN
cana-2086	208	6	−	−	NOUN
cana-2086	208	7	1	1	NUM
cana-2086	208	8	2𝑚	2𝑚	NOUN
cana-2086	208	9	−	−	NOUN
cana-2086	208	10	1	1	NUM
cana-2086	208	11	𝑖	𝑖	NOUN
cana-2086	208	12	=	=	PUNCT
cana-2086	208	13	𝑚	𝑚	PROPN
cana-2086	209	1	+	+	CCONJ
cana-2086	209	2	⌊	⌊	PROPN
cana-2086	209	3	𝑚	𝑚	ADP
cana-2086	209	4	2	2	NUM
cana-2086	209	5	⌋	⌋	NOUN
cana-2086	209	6	4	4	NUM
cana-2086	209	7	(	(	PUNCT
cana-2086	209	8	𝑖	𝑖	NOUN
cana-2086	209	9	−	−	NOUN
cana-2086	209	10	𝑚	𝑚	INTJ
cana-2086	209	11	−	−	PROPN
cana-2086	209	12	⌊	⌊	PROPN
cana-2086	209	13	𝑚	𝑚	ADP
cana-2086	209	14	2	2	NUM
cana-2086	209	15	⌋	⌋	NOUN
cana-2086	209	16	−	−	NOUN
cana-2086	209	17	1	1	NUM
cana-2086	209	18	)	)	PUNCT
cana-2086	209	19	+	+	CCONJ
cana-2086	209	20	2	2	NUM
cana-2086	209	21	𝑚	𝑚	X
cana-2086	209	22	+	+	CCONJ
cana-2086	209	23	⌊	⌊	PROPN
cana-2086	209	24	𝑚	𝑚	ADP
cana-2086	209	25	2	2	NUM
cana-2086	209	26	⌋	⌋	NOUN
cana-2086	209	27	+	+	CCONJ
cana-2086	209	28	1	1	NUM
cana-2086	209	29	≤	≤	NUM
cana-2086	209	30	𝑖	𝑖	PRON
cana-2086	209	31	≤	≤	NOUN
cana-2086	209	32	2𝑚	2𝑚	NOUN
cana-2086	209	33	−	−	NOUN
cana-2086	209	34	1	1	NUM
cana-2086	209	35	.	.	PUNCT
cana-2086	209	36	case	case	NOUN
cana-2086	209	37	3	3	NUM
cana-2086	209	38	:	:	PUNCT
cana-2086	209	39	for	for	ADP
cana-2086	209	40	𝑃4𝑚	𝑃4𝑚	PROPN
cana-2086	209	41	,	,	PUNCT
cana-2086	209	42	𝑚	𝑚	X
cana-2086	209	43	=	=	SYM
cana-2086	209	44	3	3	NUM
cana-2086	209	45	,	,	PUNCT
cana-2086	209	46	5	5	NUM
cana-2086	209	47	,	,	PUNCT
cana-2086	209	48	7	7	NUM
cana-2086	209	49	,	,	PUNCT
cana-2086	209	50	…	…	PUNCT
cana-2086	209	51	,	,	PUNCT
cana-2086	209	52	𝑓	𝑓	X
cana-2086	209	53	:	:	PUNCT
cana-2086	209	54	𝑉(𝑃4𝑚	𝑉(𝑃4𝑚	NOUN
cana-2086	209	55	)	)	PUNCT
cana-2086	209	56	→	→	SYM
cana-2086	209	57	{	{	PUNCT
cana-2086	209	58	4𝑚	4𝑚	NUM
cana-2086	209	59	,	,	PUNCT
cana-2086	209	60	4𝑚	4𝑚	NOUN
cana-2086	209	61	+	+	CCONJ
cana-2086	209	62	1	1	NUM
cana-2086	209	63	,	,	PUNCT
cana-2086	209	64	4𝑚	4𝑚	NOUN
cana-2086	209	65	+	+	CCONJ
cana-2086	209	66	2	2	NUM
cana-2086	209	67	,	,	PUNCT
cana-2086	209	68	…	…	PUNCT
cana-2086	209	69	,	,	PUNCT
cana-2086	209	70	8𝑚	8𝑚	NUM
cana-2086	209	71	−	−	NOUN
cana-2086	209	72	1	1	NUM
cana-2086	209	73	}	}	PUNCT
cana-2086	209	74	is	be	AUX
cana-2086	209	75	defined	define	VERB
cana-2086	209	76	as	as	ADP
cana-2086	209	77	,	,	PUNCT
cana-2086	209	78	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	209	79	)	)	PUNCT
cana-2086	209	80	=	=	SYM
cana-2086	209	81	{	{	PUNCT
cana-2086	209	82	4𝑚	4𝑚	NOUN
cana-2086	209	83	𝑖	𝑖	SYM
cana-2086	209	84	=	=	NOUN
cana-2086	209	85	1	1	NUM
cana-2086	209	86	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	209	87	)	)	PUNCT
cana-2086	209	88	+	+	CCONJ
cana-2086	209	89	2	2	NUM
cana-2086	209	90	2	2	NUM
cana-2086	209	91	≤	≤	NOUN
cana-2086	209	92	𝑖	𝑖	SYM
cana-2086	209	93	≤	≤	NOUN
cana-2086	209	94	𝑚	𝑚	ADP
cana-2086	209	95	4𝑚	4𝑚	NOUN
cana-2086	209	96	+	+	CCONJ
cana-2086	209	97	1	1	NUM
cana-2086	209	98	𝑖	𝑖	X
cana-2086	209	99	=	=	PUNCT
cana-2086	209	100	𝑚	𝑚	PROPN
cana-2086	209	101	+	+	ADJ
cana-2086	209	102	1	1	NUM
cana-2086	209	103	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	209	104	)	)	PUNCT
cana-2086	209	105	+	+	CCONJ
cana-2086	209	106	2	2	NUM
cana-2086	209	107	𝑚	𝑚	NOUN
cana-2086	209	108	+	+	NOUN
cana-2086	209	109	2	2	NUM
cana-2086	209	110	≤	≤	NUM
cana-2086	209	111	𝑖	𝑖	NOUN
cana-2086	209	112	≤	≤	NOUN
cana-2086	209	113	𝑚	𝑚	X
cana-2086	210	1	+	+	CCONJ
cana-2086	210	2	⌊	⌊	PROPN
cana-2086	210	3	𝑚	𝑚	ADP
cana-2086	210	4	2	2	NUM
cana-2086	210	5	⌋	⌋	NOUN
cana-2086	210	6	7𝑚	7𝑚	NUM
cana-2086	210	7	𝑖	𝑖	SYM
cana-2086	210	8	=	=	SYM
cana-2086	210	9	𝑚	𝑚	PROPN
cana-2086	211	1	+	+	CCONJ
cana-2086	211	2	⌊	⌊	PROPN
cana-2086	211	3	𝑚	𝑚	ADP
cana-2086	211	4	2	2	NUM
cana-2086	211	5	⌋	⌋	NOUN
cana-2086	211	6	+	+	CCONJ
cana-2086	211	7	1	1	NUM
cana-2086	211	8	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	211	9	)	)	PUNCT
cana-2086	211	10	+	+	CCONJ
cana-2086	211	11	1	1	NUM
cana-2086	211	12	𝑚	𝑚	X
cana-2086	211	13	+	+	CCONJ
cana-2086	211	14	⌊	⌊	PROPN
cana-2086	211	15	𝑚	𝑚	ADP
cana-2086	211	16	2	2	NUM
cana-2086	211	17	⌋	⌋	NOUN
cana-2086	211	18	+	+	CCONJ
cana-2086	211	19	2	2	NUM
cana-2086	211	20	≤	≤	NOUN
cana-2086	211	21	𝑖	𝑖	PRON
cana-2086	211	22	≤	≤	NOUN
cana-2086	211	23	2𝑚	2𝑚	NOUN
cana-2086	211	24	+	+	CCONJ
cana-2086	211	25	⌊	⌊	VERB
cana-2086	211	26	𝑚	𝑚	ADP
cana-2086	211	27	2	2	NUM
cana-2086	211	28	⌋	⌋	NUM
cana-2086	211	29	6𝑚	6𝑚	PRON
cana-2086	211	30	−	−	PROPN
cana-2086	212	1	1	1	NUM
cana-2086	212	2	𝑖	𝑖	NOUN
cana-2086	212	3	=	=	NOUN
cana-2086	212	4	2𝑚	2𝑚	NOUN
cana-2086	213	1	+	+	CCONJ
cana-2086	213	2	⌊	⌊	VERB
cana-2086	213	3	𝑚	𝑚	ADP
cana-2086	213	4	2	2	NUM
cana-2086	213	5	⌋	⌋	NOUN
cana-2086	213	6	+	+	CCONJ
cana-2086	213	7	1	1	NUM
cana-2086	213	8	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	213	9	)	)	PUNCT
cana-2086	213	10	−	−	ADP
cana-2086	213	11	2	2	NUM
cana-2086	213	12	2𝑚	2𝑚	NOUN
cana-2086	213	13	+	+	CCONJ
cana-2086	213	14	⌊	⌊	VERB
cana-2086	213	15	𝑚	𝑚	ADP
cana-2086	213	16	2	2	NUM
cana-2086	213	17	⌋	⌋	NOUN
cana-2086	213	18	+	+	CCONJ
cana-2086	213	19	2	2	NUM
cana-2086	213	20	≤	≤	NUM
cana-2086	213	21	𝑖	𝑖	SYM
cana-2086	213	22	≤	≤	NUM
cana-2086	214	1	3𝑚	3𝑚	NUM
cana-2086	214	2	7𝑚	7𝑚	NUM
cana-2086	214	3	−	−	NOUN
cana-2086	214	4	1	1	NUM
cana-2086	214	5	𝑖	𝑖	SYM
cana-2086	214	6	=	=	NOUN
cana-2086	214	7	3𝑚	3𝑚	NOUN
cana-2086	214	8	+	+	CCONJ
cana-2086	214	9	1	1	NUM
cana-2086	214	10	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	214	11	)	)	PUNCT
cana-2086	214	12	−	−	ADP
cana-2086	214	13	1	1	NUM
cana-2086	214	14	3𝑚	3𝑚	NUM
cana-2086	214	15	+	+	CCONJ
cana-2086	214	16	2	2	NUM
cana-2086	214	17	≤	≤	NUM
cana-2086	214	18	𝑖	𝑖	PRON
cana-2086	214	19	≤	≤	NOUN
cana-2086	214	20	4𝑚.	4𝑚.	NUM
cana-2086	214	21	following	follow	VERB
cana-2086	214	22	a	a	DET
cana-2086	214	23	similar	similar	ADJ
cana-2086	214	24	approach	approach	NOUN
cana-2086	214	25	as	as	ADP
cana-2086	214	26	in	in	ADP
cana-2086	214	27	case	case	NOUN
cana-2086	214	28	1	1	NUM
cana-2086	214	29	,	,	PUNCT
cana-2086	214	30	we	we	PRON
cana-2086	214	31	obtain	obtain	VERB
cana-2086	214	32	,	,	PUNCT
cana-2086	214	33	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	214	34	)	)	PUNCT
cana-2086	214	35	=	=	SYM
cana-2086	214	36	{	{	PUNCT
cana-2086	214	37	4𝑖	4𝑖	NOUN
cana-2086	214	38	−	−	NOUN
cana-2086	214	39	2	2	NUM
cana-2086	214	40	1	1	NUM
cana-2086	214	41	≤	≤	NUM
cana-2086	214	42	𝑖	𝑖	SYM
cana-2086	214	43	≤	≤	NOUN
cana-2086	214	44	𝑚	𝑚	ADP
cana-2086	214	45	−	−	NUM
cana-2086	214	46	1	1	NUM
cana-2086	214	47	2𝑚	2𝑚	NOUN
cana-2086	214	48	−	−	NOUN
cana-2086	214	49	1	1	NUM
cana-2086	214	50	𝑖	𝑖	NOUN
cana-2086	214	51	=	=	PUNCT
cana-2086	214	52	𝑚	𝑚	X
cana-2086	214	53	4𝑖	4𝑖	NOUN
cana-2086	214	54	−	−	PROPN
cana-2086	214	55	4𝑚	4𝑚	NOUN
cana-2086	214	56	𝑚	𝑚	X
cana-2086	214	57	+	+	PROPN
cana-2086	214	58	1	1	NUM
cana-2086	214	59	≤	≤	NUM
cana-2086	214	60	𝑖	𝑖	ADP
cana-2086	214	61	≤	≤	NOUN
cana-2086	214	62	𝑚	𝑚	X
cana-2086	215	1	+	+	CCONJ
cana-2086	215	2	⌊	⌊	PROPN
cana-2086	215	3	𝑚	𝑚	ADP
cana-2086	215	4	2	2	NUM
cana-2086	215	5	⌋	⌋	NOUN
cana-2086	215	6	−	−	PROPN
cana-2086	215	7	1	1	NUM
cana-2086	215	8	4𝑚	4𝑚	NOUN
cana-2086	215	9	−	−	NOUN
cana-2086	215	10	2	2	NUM
cana-2086	215	11	𝑖	𝑖	SYM
cana-2086	215	12	=	=	SYM
cana-2086	215	13	𝑚	𝑚	PROPN
cana-2086	216	1	+	+	CCONJ
cana-2086	216	2	⌊	⌊	PROPN
cana-2086	216	3	𝑚	𝑚	ADP
cana-2086	216	4	2	2	NUM
cana-2086	216	5	⌋	⌋	NOUN
cana-2086	216	6	2𝑖	2𝑖	NOUN
cana-2086	216	7	−	−	PROPN
cana-2086	217	1	𝑚	𝑚	NOUN
cana-2086	217	2	𝑚	𝑚	X
cana-2086	218	1	+	+	CCONJ
cana-2086	218	2	⌊	⌊	PROPN
cana-2086	218	3	𝑚	𝑚	ADP
cana-2086	218	4	2	2	NUM
cana-2086	218	5	⌋	⌋	NOUN
cana-2086	218	6	+	+	CCONJ
cana-2086	218	7	1	1	NUM
cana-2086	218	8	≤	≤	NUM
cana-2086	218	9	𝑖	𝑖	PRON
cana-2086	218	10	≤	≤	NOUN
cana-2086	218	11	2𝑚	2𝑚	NOUN
cana-2086	218	12	+	+	CCONJ
cana-2086	218	13	⌊	⌊	VERB
cana-2086	218	14	𝑚	𝑚	ADP
cana-2086	218	15	2	2	NUM
cana-2086	218	16	⌋	⌋	NOUN
cana-2086	218	17	−	−	NOUN
cana-2086	218	18	1	1	NUM
cana-2086	218	19	2𝑚	2𝑚	NOUN
cana-2086	218	20	−	−	NOUN
cana-2086	218	21	2	2	NUM
cana-2086	218	22	𝑖	𝑖	NOUN
cana-2086	218	23	=	=	NOUN
cana-2086	218	24	2𝑚	2𝑚	NOUN
cana-2086	218	25	+	+	CCONJ
cana-2086	218	26	⌊	⌊	VERB
cana-2086	218	27	𝑚	𝑚	ADP
cana-2086	218	28	2	2	NUM
cana-2086	218	29	⌋	⌋	NOUN
cana-2086	218	30	14𝑚	14𝑚	NOUN
cana-2086	218	31	−	−	PROPN
cana-2086	218	32	4𝑖	4𝑖	NOUN
cana-2086	218	33	−	−	NOUN
cana-2086	218	34	2	2	NUM
cana-2086	218	35	2𝑚	2𝑚	NOUN
cana-2086	218	36	+	+	CCONJ
cana-2086	218	37	⌊	⌊	VERB
cana-2086	218	38	𝑚	𝑚	ADP
cana-2086	218	39	2	2	NUM
cana-2086	218	40	⌋	⌋	NOUN
cana-2086	218	41	+	+	CCONJ
cana-2086	218	42	1	1	NUM
cana-2086	218	43	≤	≤	NUM
cana-2086	218	44	𝑖	𝑖	SYM
cana-2086	218	45	≤	≤	NOUN
cana-2086	219	1	3𝑚	3𝑚	NUM
cana-2086	219	2	−	−	PROPN
cana-2086	219	3	1	1	NUM
cana-2086	219	4	4𝑚	4𝑚	NOUN
cana-2086	219	5	−	−	NOUN
cana-2086	219	6	1	1	NUM
cana-2086	219	7	𝑖	𝑖	SYM
cana-2086	219	8	=	=	NOUN
cana-2086	219	9	3𝑚	3𝑚	NUM
cana-2086	219	10	8𝑚	8𝑚	NUM
cana-2086	219	11	−	−	PROPN
cana-2086	219	12	2𝑖	2𝑖	NOUN
cana-2086	219	13	−	−	NOUN
cana-2086	219	14	1	1	NUM
cana-2086	219	15	3𝑚	3𝑚	NUM
cana-2086	219	16	+	+	CCONJ
cana-2086	219	17	1	1	NUM
cana-2086	219	18	≤	≤	NUM
cana-2086	219	19	𝑖	𝑖	PRON
cana-2086	219	20	≤	≤	NOUN
cana-2086	219	21	4𝑚	4𝑚	NOUN
cana-2086	219	22	−	−	NOUN
cana-2086	219	23	1	1	X
cana-2086	219	24	.	.	X
cana-2086	219	25	case	case	NOUN
cana-2086	219	26	4	4	NUM
cana-2086	219	27	:	:	PUNCT
cana-2086	219	28	for	for	ADP
cana-2086	219	29	𝑃8𝑚	𝑃8𝑚	PROPN
cana-2086	219	30	,	,	PUNCT
cana-2086	219	31	𝑚	𝑚	X
cana-2086	219	32	=	=	SYM
cana-2086	219	33	1	1	NUM
cana-2086	219	34	,	,	PUNCT
cana-2086	219	35	2	2	NUM
cana-2086	219	36	,	,	PUNCT
cana-2086	219	37	3	3	NUM
cana-2086	219	38	,	,	PUNCT
cana-2086	219	39	…	…	PUNCT
cana-2086	219	40	,	,	PUNCT
cana-2086	219	41	𝑓	𝑓	X
cana-2086	219	42	:	:	PUNCT
cana-2086	219	43	𝑉(𝑃8𝑚	𝑉(𝑃8𝑚	PROPN
cana-2086	219	44	)	)	PUNCT
cana-2086	219	45	→	→	SYM
cana-2086	219	46	{	{	PUNCT
cana-2086	219	47	8𝑚	8𝑚	NUM
cana-2086	219	48	,	,	PUNCT
cana-2086	219	49	8𝑚	8𝑚	NUM
cana-2086	219	50	+	+	CCONJ
cana-2086	219	51	1	1	NUM
cana-2086	219	52	,	,	PUNCT
cana-2086	219	53	8𝑚	8𝑚	NUM
cana-2086	219	54	+	+	CCONJ
cana-2086	219	55	2	2	NUM
cana-2086	219	56	,	,	PUNCT
cana-2086	219	57	…	…	PUNCT
cana-2086	219	58	,	,	PUNCT
cana-2086	219	59	16𝑚	16𝑚	NUM
cana-2086	219	60	−	−	PROPN
cana-2086	219	61	1	1	NUM
cana-2086	219	62	}	}	PUNCT
cana-2086	219	63	is	be	AUX
cana-2086	219	64	defined	define	VERB
cana-2086	219	65	as	as	ADP
cana-2086	219	66	,	,	PUNCT
cana-2086	219	67	communications	communication	NOUN
cana-2086	219	68	on	on	ADP
cana-2086	219	69	applied	apply	VERB
cana-2086	219	70	nonlinear	nonlinear	ADJ
cana-2086	219	71	analysis	analysis	NOUN
cana-2086	219	72	issn	issn	NOUN
cana-2086	219	73	:	:	PUNCT
cana-2086	219	74	1074	1074	NUM
cana-2086	219	75	-	-	PUNCT
cana-2086	219	76	133x	133x	NUM
cana-2086	219	77	vol	vol	NOUN
cana-2086	219	78	32	32	NUM
cana-2086	219	79	no	no	NOUN
cana-2086	219	80	.	.	PUNCT
cana-2086	220	1	1s	1s	NUM
cana-2086	220	2	(	(	PUNCT
cana-2086	220	3	2025	2025	NUM
cana-2086	220	4	)	)	PUNCT
cana-2086	220	5	37	37	NUM
cana-2086	220	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	220	7	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	220	8	)	)	PUNCT
cana-2086	220	9	=	=	SYM
cana-2086	220	10	{	{	PUNCT
cana-2086	220	11	14𝑚	14𝑚	NOUN
cana-2086	220	12	−	−	PROPN
cana-2086	220	13	1	1	NUM
cana-2086	220	14	𝑖	𝑖	NOUN
cana-2086	220	15	=	=	NOUN
cana-2086	220	16	1	1	NUM
cana-2086	220	17	10𝑚	10𝑚	ADP
cana-2086	220	18	−	−	PROPN
cana-2086	220	19	2	2	NUM
cana-2086	220	20	𝑖	𝑖	NOUN
cana-2086	220	21	=	=	SYM
cana-2086	220	22	2	2	NUM
cana-2086	220	23	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	220	24	)	)	PUNCT
cana-2086	220	25	−	−	PROPN
cana-2086	221	1	2	2	NUM
cana-2086	221	2	3	3	NUM
cana-2086	221	3	≤	≤	NOUN
cana-2086	221	4	𝑖	𝑖	PRON
cana-2086	221	5	≤	≤	NOUN
cana-2086	221	6	2𝑚	2𝑚	NOUN
cana-2086	221	7	12𝑚	12𝑚	NOUN
cana-2086	221	8	−	−	PROPN
cana-2086	221	9	1	1	NUM
cana-2086	221	10	𝑖	𝑖	NOUN
cana-2086	221	11	=	=	NOUN
cana-2086	221	12	2𝑚	2𝑚	NOUN
cana-2086	221	13	+	+	CCONJ
cana-2086	221	14	1	1	NUM
cana-2086	221	15	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	221	16	)	)	PUNCT
cana-2086	221	17	−	−	ADP
cana-2086	221	18	2	2	NUM
cana-2086	221	19	2𝑚	2𝑚	NOUN
cana-2086	221	20	+	+	CCONJ
cana-2086	221	21	2	2	NUM
cana-2086	221	22	≤	≤	NUM
cana-2086	221	23	𝑖	𝑖	PRON
cana-2086	221	24	≤	≤	NOUN
cana-2086	221	25	4𝑚	4𝑚	NUM
cana-2086	221	26	16𝑚	16𝑚	NOUN
cana-2086	222	1	−	−	ADP
cana-2086	222	2	2	2	NUM
cana-2086	222	3	𝑖	𝑖	SYM
cana-2086	222	4	=	=	SYM
cana-2086	222	5	4𝑚	4𝑚	ADJ
cana-2086	223	1	+	+	CCONJ
cana-2086	223	2	1	1	NUM
cana-2086	223	3	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	223	4	)	)	PUNCT
cana-2086	223	5	−	−	PROPN
cana-2086	223	6	2	2	NUM
cana-2086	223	7	4𝑚	4𝑚	NOUN
cana-2086	223	8	+	+	CCONJ
cana-2086	223	9	2	2	NUM
cana-2086	223	10	≤	≤	NUM
cana-2086	223	11	𝑖	𝑖	PUNCT
cana-2086	223	12	≤	≤	NOUN
cana-2086	224	1	6𝑚	6𝑚	PRON
cana-2086	224	2	+	+	CCONJ
cana-2086	224	3	1	1	NUM
cana-2086	224	4	16𝑚	16𝑚	NUM
cana-2086	224	5	−	−	PROPN
cana-2086	224	6	1	1	NUM
cana-2086	224	7	𝑖	𝑖	SYM
cana-2086	224	8	=	=	PUNCT
cana-2086	224	9	6𝑚	6𝑚	NOUN
cana-2086	224	10	+	+	CCONJ
cana-2086	224	11	2	2	NUM
cana-2086	224	12	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	224	13	)	)	PUNCT
cana-2086	224	14	−	−	PROPN
cana-2086	224	15	2	2	NUM
cana-2086	225	1	6𝑚	6𝑚	PRON
cana-2086	225	2	+	+	CCONJ
cana-2086	225	3	3	3	NUM
cana-2086	225	4	≤	≤	NUM
cana-2086	225	5	𝑖	𝑖	SYM
cana-2086	225	6	≤	≤	NOUN
cana-2086	225	7	8𝑚.	8𝑚.	NUM
cana-2086	225	8	following	follow	VERB
cana-2086	225	9	a	a	DET
cana-2086	225	10	similar	similar	ADJ
cana-2086	225	11	approach	approach	NOUN
cana-2086	225	12	as	as	ADP
cana-2086	225	13	in	in	ADP
cana-2086	225	14	case	case	NOUN
cana-2086	225	15	1	1	NUM
cana-2086	225	16	,	,	PUNCT
cana-2086	225	17	we	we	PRON
cana-2086	225	18	obtain	obtain	VERB
cana-2086	225	19	,	,	PUNCT
cana-2086	225	20	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	225	21	)	)	PUNCT
cana-2086	225	22	=	=	SYM
cana-2086	225	23	{	{	PUNCT
cana-2086	225	24	8𝑚	8𝑚	NUM
cana-2086	226	1	−	−	NOUN
cana-2086	226	2	3	3	NUM
cana-2086	226	3	𝑖	𝑖	SYM
cana-2086	226	4	=	=	NOUN
cana-2086	226	5	1	1	NUM
cana-2086	226	6	8𝑚	8𝑚	NUM
cana-2086	226	7	−	−	PROPN
cana-2086	226	8	2𝑖	2𝑖	NOUN
cana-2086	226	9	−	−	NOUN
cana-2086	226	10	1	1	NUM
cana-2086	226	11	2	2	NUM
cana-2086	226	12	≤	≤	NOUN
cana-2086	226	13	𝑖	𝑖	PRON
cana-2086	226	14	≤	≤	NOUN
cana-2086	226	15	2𝑚	2𝑚	NOUN
cana-2086	226	16	−	−	PROPN
cana-2086	226	17	1	1	NUM
cana-2086	226	18	4𝑚	4𝑚	NOUN
cana-2086	226	19	−	−	NOUN
cana-2086	226	20	1	1	NUM
cana-2086	226	21	𝑖	𝑖	NOUN
cana-2086	226	22	=	=	NOUN
cana-2086	226	23	2𝑚	2𝑚	NUM
cana-2086	226	24	16𝑚	16𝑚	NUM
cana-2086	226	25	−	−	PROPN
cana-2086	226	26	4𝑖	4𝑖	ADJ
cana-2086	226	27	2𝑚	2𝑚	NOUN
cana-2086	226	28	+	+	CCONJ
cana-2086	226	29	1	1	NUM
cana-2086	226	30	≤	≤	NUM
cana-2086	226	31	𝑖	𝑖	PRON
cana-2086	226	32	≤	≤	NOUN
cana-2086	226	33	4𝑚	4𝑚	NOUN
cana-2086	226	34	−	−	NUM
cana-2086	226	35	1	1	NUM
cana-2086	226	36	8𝑚	8𝑚	NOUN
cana-2086	226	37	−	−	NOUN
cana-2086	226	38	1	1	NUM
cana-2086	226	39	𝑖	𝑖	SYM
cana-2086	226	40	=	=	SYM
cana-2086	226	41	4𝑚	4𝑚	PROPN
cana-2086	226	42	24𝑚	24𝑚	PUNCT
cana-2086	226	43	−	−	PROPN
cana-2086	226	44	(	(	PUNCT
cana-2086	226	45	4𝑖	4𝑖	NOUN
cana-2086	226	46	+	+	CCONJ
cana-2086	226	47	2	2	X
cana-2086	226	48	)	)	PUNCT
cana-2086	226	49	4𝑚	4𝑚	NOUN
cana-2086	227	1	+	+	CCONJ
cana-2086	227	2	1	1	NUM
cana-2086	227	3	≤	≤	NUM
cana-2086	227	4	𝑖	𝑖	PUNCT
cana-2086	227	5	≤	≤	NOUN
cana-2086	228	1	6𝑚	6𝑚	PRON
cana-2086	229	1	−	−	NOUN
cana-2086	229	2	1	1	NUM
cana-2086	229	3	32𝑚	32𝑚	NOUN
cana-2086	229	4	−	−	PROPN
cana-2086	229	5	(	(	PUNCT
cana-2086	229	6	4𝑖	4𝑖	NOUN
cana-2086	229	7	+	+	CCONJ
cana-2086	229	8	2	2	X
cana-2086	229	9	)	)	PUNCT
cana-2086	229	10	𝑖	𝑖	NOUN
cana-2086	230	1	=	=	PUNCT
cana-2086	230	2	6𝑚	6𝑚	PRON
cana-2086	230	3	4𝑚	4𝑚	VERB
cana-2086	230	4	−	−	ADP
cana-2086	230	5	3	3	NUM
cana-2086	230	6	𝑖	𝑖	SYM
cana-2086	230	7	=	=	PUNCT
cana-2086	231	1	6𝑚	6𝑚	NOUN
cana-2086	231	2	+	+	CCONJ
cana-2086	231	3	1	1	NUM
cana-2086	231	4	16𝑚	16𝑚	NUM
cana-2086	231	5	−	−	NOUN
cana-2086	231	6	2𝑖	2𝑖	NOUN
cana-2086	231	7	−	−	NOUN
cana-2086	231	8	1	1	NUM
cana-2086	232	1	6𝑚	6𝑚	PRON
cana-2086	232	2	+	+	CCONJ
cana-2086	232	3	2	2	NUM
cana-2086	232	4	≤	≤	NUM
cana-2086	232	5	𝑖	𝑖	PRON
cana-2086	232	6	≤	≤	NUM
cana-2086	232	7	8𝑚	8𝑚	NOUN
cana-2086	232	8	−	−	NOUN
cana-2086	233	1	1	1	X
cana-2086	233	2	.	.	PUNCT
cana-2086	234	1	now	now	ADV
cana-2086	234	2	we	we	PRON
cana-2086	234	3	have	have	VERB
cana-2086	234	4	ℎ	ℎ	NOUN
cana-2086	234	5	=	=	SYM
cana-2086	234	6	1	1	NUM
cana-2086	234	7	2𝑛−1	2𝑛−1	NUM
cana-2086	234	8	.	.	PUNCT
cana-2086	235	1	the	the	DET
cana-2086	235	2	vertex	vertex	NOUN
cana-2086	235	3	labeling	labeling	NOUN
cana-2086	235	4	𝜇	𝜇	ADP
cana-2086	235	5	:	:	PUNCT
cana-2086	235	6	𝑉(𝑃𝑛	𝑉(𝑃𝑛	NUM
cana-2086	235	7	)	)	PUNCT
cana-2086	235	8	→	→	PUNCT
cana-2086	236	1	[	[	X
cana-2086	236	2	0,1	0,1	NUM
cana-2086	236	3	]	]	PUNCT
cana-2086	236	4	is	be	AUX
cana-2086	236	5	defined	define	VERB
cana-2086	236	6	as	as	ADP
cana-2086	236	7	,	,	PUNCT
cana-2086	236	8	𝜇(𝑣𝑖	𝜇(𝑣𝑖	NOUN
cana-2086	236	9	)	)	PUNCT
cana-2086	236	10	=	=	PUNCT
cana-2086	236	11	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	236	12	)	)	PUNCT
cana-2086	236	13	.	.	PUNCT
cana-2086	237	1	1	1	NUM
cana-2086	237	2	2𝑛−1	2𝑛−1	NUM
cana-2086	237	3	for	for	ADP
cana-2086	237	4	every	every	DET
cana-2086	237	5	1	1	NUM
cana-2086	237	6	≤	≤	NUM
cana-2086	237	7	𝑖	𝑖	SYM
cana-2086	237	8	≤	≤	NOUN
cana-2086	237	9	𝑛	𝑛	ADP
cana-2086	237	10	and	and	CCONJ
cana-2086	237	11	edge	edge	VERB
cana-2086	237	12	labeling	labeling	NOUN
cana-2086	237	13	𝜌	𝜌	ADP
cana-2086	237	14	:	:	PUNCT
cana-2086	237	15	𝐸(𝑃𝑛	𝐸(𝑃𝑛	NUM
cana-2086	237	16	)	)	PUNCT
cana-2086	237	17	→	→	PUNCT
cana-2086	238	1	[	[	X
cana-2086	238	2	0,1	0,1	NUM
cana-2086	238	3	]	]	PUNCT
cana-2086	238	4	is	be	AUX
cana-2086	238	5	defined	define	VERB
cana-2086	238	6	as	as	ADP
cana-2086	238	7	,	,	PUNCT
cana-2086	238	8	𝜌(𝑣𝑖𝑣𝑖+1	𝜌(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	238	9	)	)	PUNCT
cana-2086	238	10	=	=	SYM
cana-2086	238	11	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	238	12	)	)	PUNCT
cana-2086	238	13	.	.	PUNCT
cana-2086	239	1	1	1	NUM
cana-2086	239	2	2𝑛−1	2𝑛−1	NUM
cana-2086	239	3	for	for	ADP
cana-2086	239	4	every	every	DET
cana-2086	239	5	1	1	NUM
cana-2086	239	6	≤	≤	NUM
cana-2086	239	7	𝑖	𝑖	SYM
cana-2086	239	8	≤	≤	NUM
cana-2086	239	9	𝑛	𝑛	PRON
cana-2086	239	10	−	−	NOUN
cana-2086	239	11	1	1	NUM
cana-2086	239	12	.	.	PUNCT
cana-2086	240	1	it	it	PRON
cana-2086	240	2	can	can	AUX
cana-2086	240	3	be	be	AUX
cana-2086	240	4	verified	verify	VERB
cana-2086	240	5	that	that	SCONJ
cana-2086	240	6	in	in	ADP
cana-2086	240	7	all	all	DET
cana-2086	240	8	the	the	DET
cana-2086	240	9	cases	case	NOUN
cana-2086	240	10	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	NOUN
cana-2086	240	11	)	)	PUNCT
cana-2086	240	12	are	be	AUX
cana-2086	240	13	distinct	distinct	ADJ
cana-2086	240	14	and	and	CCONJ
cana-2086	240	15	nonzero	nonzero	NOUN
cana-2086	240	16	,	,	PUNCT
cana-2086	240	17	therefore	therefore	ADV
cana-2086	240	18	𝜌(𝑣𝑖𝑣𝑖+1	𝜌(𝑣𝑖𝑣𝑖+1	VERB
cana-2086	240	19	)	)	PUNCT
cana-2086	240	20	are	be	AUX
cana-2086	240	21	distinct	distinct	ADJ
cana-2086	240	22	and	and	CCONJ
cana-2086	240	23	nonzero	nonzero	NOUN
cana-2086	240	24	.	.	PUNCT
cana-2086	241	1	further	further	ADJ
cana-2086	241	2	max{𝑔(𝑣𝑖𝑣𝑖+1	max{𝑔(𝑣𝑖𝑣𝑖+1	NOUN
cana-2086	241	3	)	)	PUNCT
cana-2086	241	4	}	}	PUNCT
cana-2086	242	1	=	=	SYM
cana-2086	242	2	𝑛	𝑛	PRON
cana-2086	242	3	−	−	NUM
cana-2086	242	4	1	1	NUM
cana-2086	242	5	and	and	CCONJ
cana-2086	242	6	min	min	NOUN
cana-2086	242	7	{	{	PUNCT
cana-2086	242	8	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	242	9	)	)	PUNCT
cana-2086	242	10	}	}	PUNCT
cana-2086	242	11	=	=	SYM
cana-2086	242	12	𝑛.	𝑛.	NOUN
cana-2086	242	13	therefore	therefore	ADV
cana-2086	242	14	,	,	PUNCT
cana-2086	242	15	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	242	16	)	)	PUNCT
cana-2086	242	17	<	<	X
cana-2086	242	18	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	242	19	)	)	PUNCT
cana-2086	242	20	∧	∧	PROPN
cana-2086	242	21	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	242	22	)	)	PUNCT
cana-2086	242	23	for	for	ADP
cana-2086	242	24	every	every	DET
cana-2086	242	25	1	1	NUM
cana-2086	242	26	≤	≤	NUM
cana-2086	242	27	𝑖	𝑖	SYM
cana-2086	242	28	≤	≤	NUM
cana-2086	242	29	𝑛	𝑛	PRON
cana-2086	242	30	−	−	NOUN
cana-2086	242	31	1	1	NUM
cana-2086	242	32	.	.	PUNCT
cana-2086	242	33	hence	hence	ADV
cana-2086	242	34	for	for	ADP
cana-2086	242	35	every	every	DET
cana-2086	242	36	1	1	NUM
cana-2086	242	37	≤	≤	NUM
cana-2086	242	38	𝑖	𝑖	SYM
cana-2086	242	39	≤	≤	NUM
cana-2086	242	40	𝑛	𝑛	PRON
cana-2086	242	41	−	−	PROPN
cana-2086	242	42	1	1	NUM
cana-2086	242	43	,	,	PUNCT
cana-2086	242	44	𝜌(𝑣𝑖𝑣𝑖+1	𝜌(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	242	45	)	)	PUNCT
cana-2086	242	46	=	=	SYM
cana-2086	242	47	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	242	48	)	)	PUNCT
cana-2086	242	49	.	.	PUNCT
cana-2086	243	1	1	1	NUM
cana-2086	243	2	2𝑛−1	2𝑛−1	NUM
cana-2086	243	3	<	<	X
cana-2086	243	4	(	(	PUNCT
cana-2086	243	5	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	243	6	)	)	PUNCT
cana-2086	243	7	∧	∧	PROPN
cana-2086	243	8	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	243	9	)	)	PUNCT
cana-2086	243	10	)	)	PUNCT
cana-2086	243	11	.	.	PUNCT
cana-2086	244	1	1	1	NUM
cana-2086	244	2	2𝑛−1	2𝑛−1	NUM
cana-2086	244	3	<	<	X
cana-2086	244	4	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	244	5	)	)	PUNCT
cana-2086	244	6	.	.	PUNCT
cana-2086	245	1	1	1	NUM
cana-2086	245	2	2𝑛−1	2𝑛−1	NUM
cana-2086	245	3	∧	∧	PROPN
cana-2086	245	4	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	245	5	)	)	PUNCT
cana-2086	245	6	.	.	PUNCT
cana-2086	246	1	1	1	NUM
cana-2086	246	2	2𝑛−1	2𝑛−1	NUM
cana-2086	246	3	<	<	X
cana-2086	246	4	𝜇(𝑣𝑖	𝜇(𝑣𝑖	NOUN
cana-2086	246	5	)	)	PUNCT
cana-2086	246	6	∧	∧	NOUN
cana-2086	246	7	𝜇(𝑣𝑖+1	𝜇(𝑣𝑖+1	NUM
cana-2086	246	8	)	)	PUNCT
cana-2086	246	9	.	.	PUNCT
cana-2086	247	1	this	this	PRON
cana-2086	247	2	implies	imply	VERB
cana-2086	247	3	that	that	SCONJ
cana-2086	247	4	,	,	PUNCT
cana-2086	247	5	the	the	DET
cana-2086	247	6	edge	edge	NOUN
cana-2086	247	7	labels	label	NOUN
cana-2086	247	8	are	be	AUX
cana-2086	247	9	less	less	ADJ
cana-2086	247	10	than	than	ADP
cana-2086	247	11	the	the	DET
cana-2086	247	12	minimum	minimum	NOUN
cana-2086	247	13	of	of	ADP
cana-2086	247	14	their	their	PRON
cana-2086	247	15	respective	respective	ADJ
cana-2086	247	16	endpoints	endpoint	NOUN
cana-2086	247	17	labels	label	NOUN
cana-2086	247	18	.	.	PUNCT
cana-2086	248	1	consequently	consequently	ADV
cana-2086	248	2	,	,	PUNCT
cana-2086	248	3	𝑃𝑛	𝑃𝑛	PROPN
cana-2086	248	4	,	,	PUNCT
cana-2086	248	5	𝑛	𝑛	DET
cana-2086	248	6	≠	≠	PROPN
cana-2086	248	7	4	4	NUM
cana-2086	248	8	are	be	AUX
cana-2086	248	9	elegant	elegant	ADJ
cana-2086	248	10	fuzzy	fuzzy	ADJ
cana-2086	248	11	labeling	labeling	NOUN
cana-2086	248	12	graphs	graph	NOUN
cana-2086	248	13	and	and	CCONJ
cana-2086	248	14	(	(	PUNCT
cana-2086	248	15	𝜇	𝜇	ADP
cana-2086	248	16	,	,	PUNCT
cana-2086	248	17	𝜌	𝜌	X
cana-2086	248	18	)	)	PUNCT
cana-2086	248	19	is	be	AUX
cana-2086	248	20	an	an	DET
cana-2086	248	21	elegant	elegant	ADJ
cana-2086	248	22	fuzzy	fuzzy	ADJ
cana-2086	248	23	labeling	labeling	NOUN
cana-2086	248	24	of	of	ADP
cana-2086	248	25	𝑃𝑛	𝑃𝑛	PROPN
cana-2086	248	26	,	,	PUNCT
cana-2086	248	27	𝑛	𝑛	DET
cana-2086	248	28	≠	≠	PROPN
cana-2086	248	29	4	4	NUM
cana-2086	248	30	.	.	PUNCT
cana-2086	248	31	□	□	PUNCT
cana-2086	248	32	theorem	theorem	ADJ
cana-2086	248	33	3.2.2	3.2.2	NUM
cana-2086	248	34	.	.	PUNCT
cana-2086	248	35	line	line	NOUN
cana-2086	248	36	graph	graph	NOUN
cana-2086	248	37	of	of	ADP
cana-2086	248	38	path	path	NOUN
cana-2086	248	39	graphs	graph	NOUN
cana-2086	248	40	𝐿(𝑃𝑛	𝐿(𝑃𝑛	NOUN
cana-2086	248	41	)	)	PUNCT
cana-2086	248	42	admit	admit	VERB
cana-2086	248	43	elegant	elegant	ADJ
cana-2086	248	44	fuzzy	fuzzy	ADJ
cana-2086	248	45	labeling	labeling	NOUN
cana-2086	248	46	except	except	SCONJ
cana-2086	248	47	for	for	ADP
cana-2086	248	48	𝑛	𝑛	PROPN
cana-2086	248	49	=	=	SYM
cana-2086	248	50	5	5	NUM
cana-2086	248	51	.	.	PUNCT
cana-2086	248	52	proof	proof	NOUN
cana-2086	248	53	.	.	PUNCT
cana-2086	249	1	by	by	ADP
cana-2086	249	2	theorem	theorem	NOUN
cana-2086	249	3	3.2.1	3.2.1	NUM
cana-2086	249	4	,	,	PUNCT
cana-2086	249	5	𝑃𝑛	𝑃𝑛	PROPN
cana-2086	249	6	admit	admit	VERB
cana-2086	249	7	elegant	elegant	ADJ
cana-2086	249	8	fuzzy	fuzzy	ADJ
cana-2086	249	9	labeling	labeling	NOUN
cana-2086	249	10	except	except	SCONJ
cana-2086	249	11	for	for	ADP
cana-2086	249	12	𝑛	𝑛	PROPN
cana-2086	249	13	=	=	SYM
cana-2086	249	14	4	4	NUM
cana-2086	249	15	.	.	PUNCT
cana-2086	249	16	𝐿(𝑃𝑛	𝐿(𝑃𝑛	PROPN
cana-2086	249	17	)	)	PUNCT
cana-2086	249	18	≡	≡	PROPN
cana-2086	249	19	𝑃𝑛−1	𝑃𝑛−1	PROPN
cana-2086	249	20	.	.	PUNCT
cana-2086	250	1	therefore	therefore	ADV
cana-2086	250	2	𝐿(𝑃𝑛	𝐿(𝑃𝑛	PROPN
cana-2086	250	3	)	)	PUNCT
cana-2086	250	4	admit	admit	VERB
cana-2086	250	5	elegant	elegant	ADJ
cana-2086	250	6	fuzzy	fuzzy	ADJ
cana-2086	250	7	labeling	labeling	NOUN
cana-2086	250	8	except	except	SCONJ
cana-2086	250	9	for	for	ADP
cana-2086	250	10	𝑛	𝑛	NOUN
cana-2086	250	11	=	=	SYM
cana-2086	250	12	5	5	NUM
cana-2086	250	13	.	.	PUNCT
cana-2086	251	1	□	□	PUNCT
cana-2086	251	2	example	example	NOUN
cana-2086	251	3	3.2.3	3.2.3	X
cana-2086	251	4	.	.	PUNCT
cana-2086	252	1	consider	consider	VERB
cana-2086	252	2	a	a	DET
cana-2086	252	3	path	path	NOUN
cana-2086	252	4	graph	graph	NOUN
cana-2086	252	5	𝑃5	𝑃5	NOUN
cana-2086	252	6	in	in	ADP
cana-2086	252	7	fig	fig	NOUN
cana-2086	252	8	.	.	PUNCT
cana-2086	253	1	5	5	X
cana-2086	253	2	.	.	PUNCT
cana-2086	253	3	by	by	ADP
cana-2086	253	4	theorem	theorem	ADJ
cana-2086	253	5	3.2.1	3.2.1	NUM
cana-2086	253	6	,	,	PUNCT
cana-2086	253	7	a	a	DET
cana-2086	253	8	function	function	NOUN
cana-2086	253	9	𝑓	𝑓	NOUN
cana-2086	253	10	:	:	PUNCT
cana-2086	253	11	𝑉(𝑃5	𝑉(𝑃5	NOUN
cana-2086	253	12	)	)	PUNCT
cana-2086	253	13	→	→	SYM
cana-2086	253	14	{	{	PUNCT
cana-2086	253	15	5	5	NUM
cana-2086	253	16	,	,	PUNCT
cana-2086	253	17	6	6	NUM
cana-2086	253	18	,	,	PUNCT
cana-2086	253	19	7	7	NUM
cana-2086	253	20	,	,	PUNCT
cana-2086	253	21	8	8	NUM
cana-2086	253	22	,	,	PUNCT
cana-2086	253	23	9	9	NUM
cana-2086	253	24	}	}	PUNCT
cana-2086	253	25	is	be	AUX
cana-2086	253	26	defined	define	VERB
cana-2086	253	27	as	as	SCONJ
cana-2086	253	28	follows	follow	VERB
cana-2086	253	29	:	:	PUNCT
cana-2086	253	30	𝑓(𝑣1	𝑓(𝑣1	ADJ
cana-2086	253	31	)	)	PUNCT
cana-2086	253	32	=	=	SYM
cana-2086	253	33	8	8	NUM
cana-2086	253	34	,	,	PUNCT
cana-2086	253	35	𝑓(𝑣2	𝑓(𝑣2	NOUN
cana-2086	253	36	)	)	PUNCT
cana-2086	253	37	=	=	SYM
cana-2086	253	38	9	9	NUM
cana-2086	253	39	,	,	PUNCT
cana-2086	253	40	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	253	41	)	)	PUNCT
cana-2086	253	42	=	=	SYM
cana-2086	253	43	5	5	NUM
cana-2086	253	44	,	,	PUNCT
cana-2086	253	45	𝑓(𝑣4	𝑓(𝑣4	NOUN
cana-2086	253	46	)	)	PUNCT
cana-2086	253	47	=	=	SYM
cana-2086	253	48	6	6	NUM
cana-2086	253	49	,	,	PUNCT
cana-2086	253	50	𝑓(𝑣5	𝑓(𝑣5	NOUN
cana-2086	253	51	)	)	PUNCT
cana-2086	254	1	=	=	PUNCT
cana-2086	254	2	7	7	X
cana-2086	254	3	.	.	PUNCT
cana-2086	254	4	consequently	consequently	ADV
cana-2086	254	5	,	,	PUNCT
cana-2086	254	6	𝑔(𝑣1𝑣2	𝑔(𝑣1𝑣2	PROPN
cana-2086	254	7	)	)	PUNCT
cana-2086	254	8	=	=	SYM
cana-2086	254	9	2	2	NUM
cana-2086	254	10	,	,	PUNCT
cana-2086	254	11	𝑔(𝑣2𝑣3	𝑔(𝑣2𝑣3	NOUN
cana-2086	254	12	)	)	PUNCT
cana-2086	254	13	=	=	SYM
cana-2086	254	14	4	4	NUM
cana-2086	254	15	,	,	PUNCT
cana-2086	254	16	𝑔(𝑣3𝑣4	𝑔(𝑣3𝑣4	ADJ
cana-2086	254	17	)	)	PUNCT
cana-2086	254	18	=	=	SYM
cana-2086	254	19	1	1	NUM
cana-2086	254	20	,	,	PUNCT
cana-2086	254	21	𝑔(𝑣4𝑣5	𝑔(𝑣4𝑣5	NOUN
cana-2086	254	22	)	)	PUNCT
cana-2086	254	23	=	=	SYM
cana-2086	255	1	3	3	X
cana-2086	255	2	.	.	X
cana-2086	255	3	hence	hence	ADV
cana-2086	255	4	an	an	DET
cana-2086	255	5	elegant	elegant	ADJ
cana-2086	255	6	fuzzy	fuzzy	ADJ
cana-2086	255	7	labeling	labeling	NOUN
cana-2086	255	8	(	(	PUNCT
cana-2086	255	9	𝜇	𝜇	ADP
cana-2086	255	10	,	,	PUNCT
cana-2086	255	11	𝜌	𝜌	X
cana-2086	255	12	)	)	PUNCT
cana-2086	255	13	of	of	ADP
cana-2086	255	14	𝑃5	𝑃5	NOUN
cana-2086	255	15	is	be	AUX
cana-2086	255	16	given	give	VERB
cana-2086	255	17	as	as	SCONJ
cana-2086	255	18	follows	follow	VERB
cana-2086	255	19	:	:	PUNCT
cana-2086	255	20	𝜇(𝑣1	𝜇(𝑣1	ADJ
cana-2086	255	21	)	)	PUNCT
cana-2086	255	22	=	=	SYM
cana-2086	255	23	0.89	0.89	NUM
cana-2086	255	24	,	,	PUNCT
cana-2086	255	25	𝜇(𝑣2	𝜇(𝑣2	NOUN
cana-2086	255	26	)	)	PUNCT
cana-2086	255	27	=	=	SYM
cana-2086	255	28	1	1	NUM
cana-2086	255	29	,	,	PUNCT
cana-2086	255	30	𝜇(𝑣3	𝜇(𝑣3	NOUN
cana-2086	255	31	)	)	PUNCT
cana-2086	255	32	=	=	PUNCT
cana-2086	255	33	0.56	0.56	NUM
cana-2086	255	34	,	,	PUNCT
cana-2086	255	35	𝜇(𝑣4	𝜇(𝑣4	NOUN
cana-2086	255	36	)	)	PUNCT
cana-2086	255	37	=	=	SYM
cana-2086	255	38	0.67	0.67	NUM
cana-2086	255	39	,	,	PUNCT
cana-2086	255	40	𝜇(𝑣5	𝜇(𝑣5	NOUN
cana-2086	255	41	)	)	PUNCT
cana-2086	255	42	=	=	SYM
cana-2086	255	43	0.78	0.78	NUM
cana-2086	255	44	,	,	PUNCT
cana-2086	255	45	𝜌(𝑣1𝑣2	𝜌(𝑣1𝑣2	ADJ
cana-2086	255	46	)	)	PUNCT
cana-2086	255	47	=	=	SYM
cana-2086	255	48	0.22	0.22	NUM
cana-2086	255	49	,	,	PUNCT
cana-2086	255	50	𝜌(𝑣2𝑣3	𝜌(𝑣2𝑣3	NOUN
cana-2086	255	51	)	)	PUNCT
cana-2086	255	52	=	=	SYM
cana-2086	255	53	0.44	0.44	NUM
cana-2086	255	54	,	,	PUNCT
cana-2086	255	55	𝜌(𝑣3𝑣4	𝜌(𝑣3𝑣4	NOUN
cana-2086	255	56	)	)	PUNCT
cana-2086	255	57	=	=	SYM
cana-2086	255	58	0.11	0.11	NUM
cana-2086	255	59	,	,	PUNCT
cana-2086	255	60	𝜌(𝑣4𝑣5	𝜌(𝑣4𝑣5	NOUN
cana-2086	255	61	)	)	PUNCT
cana-2086	255	62	=	=	SYM
cana-2086	255	63	0.33	0.33	NUM
cana-2086	255	64	.	.	PUNCT
cana-2086	256	1	communications	communication	NOUN
cana-2086	256	2	on	on	ADP
cana-2086	256	3	applied	apply	VERB
cana-2086	256	4	nonlinear	nonlinear	ADJ
cana-2086	256	5	analysis	analysis	NOUN
cana-2086	256	6	issn	issn	NOUN
cana-2086	256	7	:	:	PUNCT
cana-2086	256	8	1074	1074	NUM
cana-2086	256	9	-	-	PUNCT
cana-2086	256	10	133x	133x	NUM
cana-2086	256	11	vol	vol	NOUN
cana-2086	256	12	32	32	NUM
cana-2086	256	13	no	no	NOUN
cana-2086	256	14	.	.	PUNCT
cana-2086	257	1	1s	1s	NUM
cana-2086	257	2	(	(	PUNCT
cana-2086	257	3	2025	2025	NUM
cana-2086	257	4	)	)	PUNCT
cana-2086	257	5	38	38	NUM
cana-2086	257	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	257	7	theorem	theorem	VERB
cana-2086	257	8	3.2.4	3.2.4	NUM
cana-2086	257	9	.	.	PUNCT
cana-2086	258	1	cycles	cycle	NOUN
cana-2086	259	1	𝐶𝑛	𝐶𝑛	NOUN
cana-2086	259	2	admit	admit	VERB
cana-2086	259	3	elegant	elegant	ADJ
cana-2086	259	4	fuzzy	fuzzy	ADJ
cana-2086	259	5	labeling	labeling	NOUN
cana-2086	259	6	for	for	ADP
cana-2086	259	7	𝑛	𝑛	DET
cana-2086	259	8	≡	≡	PROPN
cana-2086	259	9	0	0	NUM
cana-2086	259	10	,	,	PUNCT
cana-2086	259	11	3	3	NUM
cana-2086	259	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	259	13	4	4	NUM
cana-2086	259	14	.	.	PUNCT
cana-2086	260	1	proof	proof	NOUN
cana-2086	260	2	.	.	PUNCT
cana-2086	261	1	let	let	VERB
cana-2086	261	2	𝐶𝑛	𝐶𝑛	PART
cana-2086	261	3	be	be	AUX
cana-2086	261	4	a	a	DET
cana-2086	261	5	cycle	cycle	NOUN
cana-2086	261	6	with	with	ADP
cana-2086	261	7	vertex	vertex	NOUN
cana-2086	261	8	set	set	VERB
cana-2086	261	9	𝑉(𝐶𝑛	𝑉(𝐶𝑛	NUM
cana-2086	261	10	)	)	PUNCT
cana-2086	262	1	=	=	PRON
cana-2086	262	2	{	{	PUNCT
cana-2086	262	3	𝑣1	𝑣1	PROPN
cana-2086	262	4	,	,	PUNCT
cana-2086	262	5	𝑣2	𝑣2	PROPN
cana-2086	262	6	,	,	PUNCT
cana-2086	262	7	…	…	PUNCT
cana-2086	262	8	,	,	PUNCT
cana-2086	262	9	𝑣𝑛	𝑣𝑛	NOUN
cana-2086	262	10	}	}	PUNCT
cana-2086	262	11	and	and	CCONJ
cana-2086	262	12	edge	edge	VERB
cana-2086	262	13	set	set	VERB
cana-2086	262	14	𝐸(𝐶𝑛	𝐸(𝐶𝑛	NOUN
cana-2086	262	15	)	)	PUNCT
cana-2086	262	16	=	=	PRON
cana-2086	262	17	{	{	PUNCT
cana-2086	262	18	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-2086	262	19	,	,	PUNCT
cana-2086	262	20	𝑣2𝑣3	𝑣2𝑣3	PROPN
cana-2086	262	21	,	,	PUNCT
cana-2086	262	22	…	…	PUNCT
cana-2086	262	23	,	,	PUNCT
cana-2086	262	24	𝑣𝑛−1𝑣𝑛	𝑣𝑛−1𝑣𝑛	NUM
cana-2086	262	25	,	,	PUNCT
cana-2086	262	26	𝑣𝑛𝑣1	𝑣𝑛𝑣1	PROPN
cana-2086	262	27	}	}	PUNCT
cana-2086	262	28	.	.	PUNCT
cana-2086	263	1	let	let	VERB
cana-2086	263	2	𝑓	𝑓	PRON
cana-2086	263	3	be	be	AUX
cana-2086	263	4	an	an	DET
cana-2086	263	5	injective	injective	ADJ
cana-2086	263	6	function	function	NOUN
cana-2086	263	7	from	from	ADP
cana-2086	263	8	𝑉(𝐶𝑛	𝑉(𝐶𝑛	NUM
cana-2086	263	9	)	)	PUNCT
cana-2086	263	10	into	into	ADP
cana-2086	263	11	the	the	DET
cana-2086	263	12	set	set	NOUN
cana-2086	263	13	{	{	PUNCT
cana-2086	263	14	𝑛	𝑛	PROPN
cana-2086	263	15	+	+	NUM
cana-2086	263	16	1	1	NUM
cana-2086	263	17	,	,	PUNCT
cana-2086	263	18	𝑛	𝑛	PROPN
cana-2086	263	19	+	+	NOUN
cana-2086	263	20	2	2	NUM
cana-2086	263	21	,	,	PUNCT
cana-2086	263	22	…	…	PUNCT
cana-2086	263	23	,	,	PUNCT
cana-2086	263	24	2𝑛	2𝑛	PROPN
cana-2086	264	1	+	+	CCONJ
cana-2086	264	2	1	1	X
cana-2086	264	3	}	}	PUNCT
cana-2086	264	4	and	and	CCONJ
cana-2086	264	5	𝑔	𝑔	PROPN
cana-2086	264	6	be	be	AUX
cana-2086	264	7	a	a	DET
cana-2086	264	8	function	function	NOUN
cana-2086	264	9	from	from	ADP
cana-2086	264	10	𝐸(𝐶𝑛	𝐸(𝐶𝑛	PROPN
cana-2086	264	11	)	)	PUNCT
cana-2086	264	12	into	into	ADP
cana-2086	264	13	the	the	DET
cana-2086	264	14	set	set	NOUN
cana-2086	264	15	{	{	PUNCT
cana-2086	264	16	0,1	0,1	NUM
cana-2086	264	17	,	,	PUNCT
cana-2086	264	18	…	…	PUNCT
cana-2086	264	19	,	,	PUNCT
cana-2086	264	20	𝑛	𝑛	PROPN
cana-2086	264	21	}	}	PUNCT
cana-2086	264	22	defined	define	VERB
cana-2086	264	23	as	as	ADP
cana-2086	264	24	,	,	PUNCT
cana-2086	264	25	𝑔(𝑣𝑛𝑣1	𝑔(𝑣𝑛𝑣1	NUM
cana-2086	264	26	)	)	PUNCT
cana-2086	264	27	=	=	SYM
cana-2086	264	28	(	(	PUNCT
cana-2086	264	29	𝑓(𝑣𝑛	𝑓(𝑣𝑛	X
cana-2086	264	30	)	)	PUNCT
cana-2086	264	31	+	+	CCONJ
cana-2086	264	32	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-2086	264	33	)	)	PUNCT
cana-2086	264	34	)	)	PUNCT
cana-2086	265	1	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	265	2	(	(	PUNCT
cana-2086	265	3	𝑛	𝑛	PROPN
cana-2086	265	4	+	+	ADJ
cana-2086	265	5	1	1	NUM
cana-2086	265	6	)	)	PUNCT
cana-2086	265	7	and	and	CCONJ
cana-2086	265	8	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	NOUN
cana-2086	265	9	)	)	PUNCT
cana-2086	265	10	=	=	NOUN
cana-2086	265	11	(	(	PUNCT
cana-2086	265	12	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	265	13	)	)	PUNCT
cana-2086	265	14	+	+	CCONJ
cana-2086	265	15	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	265	16	)	)	PUNCT
cana-2086	265	17	)	)	PUNCT
cana-2086	265	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	265	19	(	(	PUNCT
cana-2086	265	20	𝑛	𝑛	PROPN
cana-2086	265	21	+	+	NOUN
cana-2086	265	22	1	1	NUM
cana-2086	265	23	)	)	PUNCT
cana-2086	265	24	for	for	ADP
cana-2086	265	25	1	1	NUM
cana-2086	265	26	≤	≤	NUM
cana-2086	265	27	𝑖	𝑖	SYM
cana-2086	265	28	≤	≤	NUM
cana-2086	265	29	𝑛	𝑛	PRON
cana-2086	265	30	−	−	NUM
cana-2086	265	31	1	1	NUM
cana-2086	265	32	.	.	PUNCT
cana-2086	265	33	case	case	NOUN
cana-2086	265	34	1	1	NUM
cana-2086	265	35	:	:	PUNCT
cana-2086	265	36	for	for	SCONJ
cana-2086	265	37	𝐶2𝑚+1	𝐶2𝑚+1	PROPN
cana-2086	265	38	,	,	PUNCT
cana-2086	265	39	𝑚	𝑚	X
cana-2086	265	40	=	=	SYM
cana-2086	265	41	1	1	NUM
cana-2086	265	42	,	,	PUNCT
cana-2086	265	43	3	3	NUM
cana-2086	265	44	,	,	PUNCT
cana-2086	265	45	5	5	NUM
cana-2086	265	46	,	,	PUNCT
cana-2086	265	47	…	…	PUNCT
cana-2086	265	48	,	,	PUNCT
cana-2086	265	49	𝑓	𝑓	X
cana-2086	265	50	:	:	PUNCT
cana-2086	265	51	𝑉(𝐶2𝑚+1	𝑉(𝐶2𝑚+1	NUM
cana-2086	265	52	)	)	PUNCT
cana-2086	265	53	→	→	SYM
cana-2086	265	54	{	{	PUNCT
cana-2086	265	55	2𝑚	2𝑚	NOUN
cana-2086	265	56	+	+	CCONJ
cana-2086	265	57	2	2	NUM
cana-2086	265	58	,	,	PUNCT
cana-2086	265	59	2𝑚	2𝑚	NOUN
cana-2086	265	60	+	+	CCONJ
cana-2086	265	61	3	3	NUM
cana-2086	265	62	,	,	PUNCT
cana-2086	265	63	2𝑚	2𝑚	NOUN
cana-2086	265	64	+	+	CCONJ
cana-2086	265	65	4	4	NUM
cana-2086	265	66	,	,	PUNCT
cana-2086	265	67	…	…	PUNCT
cana-2086	265	68	,	,	PUNCT
cana-2086	265	69	4𝑚	4𝑚	NUM
cana-2086	265	70	+	+	CCONJ
cana-2086	265	71	3	3	X
cana-2086	265	72	}	}	PUNCT
cana-2086	265	73	is	be	AUX
cana-2086	265	74	defined	define	VERB
cana-2086	265	75	as	as	ADP
cana-2086	265	76	,	,	PUNCT
cana-2086	265	77	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	265	78	)	)	PUNCT
cana-2086	265	79	=	=	SYM
cana-2086	265	80	{	{	PUNCT
cana-2086	265	81	2𝑚	2𝑚	NOUN
cana-2086	265	82	+	+	CCONJ
cana-2086	265	83	2	2	NUM
cana-2086	265	84	𝑖	𝑖	SYM
cana-2086	265	85	=	=	SYM
cana-2086	265	86	1	1	NUM
cana-2086	265	87	3𝑚	3𝑚	NOUN
cana-2086	266	1	+	+	CCONJ
cana-2086	266	2	1	1	NUM
cana-2086	266	3	+	+	CCONJ
cana-2086	266	4	𝑖	𝑖	SYM
cana-2086	266	5	2	2	NUM
cana-2086	266	6	≤	≤	NOUN
cana-2086	266	7	𝑖	𝑖	SYM
cana-2086	266	8	≤	≤	NUM
cana-2086	266	9	3	3	NUM
cana-2086	266	10	2𝑚	2𝑚	NOUN
cana-2086	266	11	+	+	CCONJ
cana-2086	266	12	3	3	NUM
cana-2086	266	13	𝑖	𝑖	NOUN
cana-2086	266	14	=	=	NOUN
cana-2086	266	15	4	4	NUM
cana-2086	266	16	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	266	17	)	)	PUNCT
cana-2086	266	18	+	+	CCONJ
cana-2086	266	19	1	1	NUM
cana-2086	266	20	5	5	NUM
cana-2086	266	21	≤	≤	NOUN
cana-2086	266	22	𝑖	𝑖	SYM
cana-2086	266	23	≤	≤	NOUN
cana-2086	266	24	𝑚	𝑚	ADP
cana-2086	266	25	+	+	ADJ
cana-2086	266	26	2	2	NUM
cana-2086	266	27	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	266	28	)	)	PUNCT
cana-2086	266	29	+	+	CCONJ
cana-2086	266	30	1	1	NUM
cana-2086	266	31	𝑖	𝑖	X
cana-2086	266	32	=	=	PUNCT
cana-2086	266	33	𝑚	𝑚	PROPN
cana-2086	266	34	+	+	PROPN
cana-2086	266	35	3	3	NUM
cana-2086	266	36	𝑓(𝑣𝑖−3	𝑓(𝑣𝑖−3	NOUN
cana-2086	266	37	)	)	PUNCT
cana-2086	267	1	+	+	CCONJ
cana-2086	267	2	2	2	NUM
cana-2086	267	3	𝑖	𝑖	NOUN
cana-2086	267	4	=	=	PUNCT
cana-2086	267	5	𝑚	𝑚	PROPN
cana-2086	267	6	+	+	NUM
cana-2086	267	7	4	4	NUM
cana-2086	267	8	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	267	9	)	)	PUNCT
cana-2086	268	1	+	+	CCONJ
cana-2086	268	2	1	1	NUM
cana-2086	268	3	𝑚	𝑚	NOUN
cana-2086	268	4	+	+	NUM
cana-2086	268	5	5	5	NUM
cana-2086	268	6	≤	≤	NUM
cana-2086	268	7	𝑖	𝑖	PRON
cana-2086	268	8	≤	≤	NOUN
cana-2086	268	9	2𝑚	2𝑚	NOUN
cana-2086	268	10	+	+	CCONJ
cana-2086	268	11	1	1	X
cana-2086	268	12	.	.	X
cana-2086	269	1	for	for	ADP
cana-2086	269	2	𝑖	𝑖	PRON
cana-2086	269	3	=	=	SYM
cana-2086	269	4	1	1	NUM
cana-2086	269	5	,	,	PUNCT
cana-2086	269	6	𝑔(𝑣1𝑣2	𝑔(𝑣1𝑣2	PROPN
cana-2086	269	7	)	)	PUNCT
cana-2086	269	8	=	=	PUNCT
cana-2086	269	9	(	(	PUNCT
cana-2086	269	10	2𝑚	2𝑚	NOUN
cana-2086	269	11	+	+	CCONJ
cana-2086	269	12	2	2	NUM
cana-2086	269	13	+	+	NUM
cana-2086	269	14	3𝑚	3𝑚	NUM
cana-2086	269	15	+	+	CCONJ
cana-2086	269	16	3	3	NUM
cana-2086	269	17	)	)	PUNCT
cana-2086	269	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	269	19	(	(	PUNCT
cana-2086	269	20	2𝑚	2𝑚	NOUN
cana-2086	269	21	+	+	CCONJ
cana-2086	269	22	2	2	X
cana-2086	269	23	)	)	PUNCT
cana-2086	269	24	=	=	NOUN
cana-2086	269	25	(	(	PUNCT
cana-2086	269	26	5𝑚	5𝑚	NOUN
cana-2086	269	27	+	+	CCONJ
cana-2086	269	28	5	5	NUM
cana-2086	269	29	)	)	PUNCT
cana-2086	269	30	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	269	31	(	(	PUNCT
cana-2086	269	32	2𝑚	2𝑚	NOUN
cana-2086	269	33	+	+	CCONJ
cana-2086	269	34	2	2	X
cana-2086	269	35	)	)	PUNCT
cana-2086	269	36	=	=	PUNCT
cana-2086	269	37	𝑚	𝑚	PROPN
cana-2086	269	38	+	+	NOUN
cana-2086	269	39	1	1	X
cana-2086	269	40	.	.	X
cana-2086	269	41	for	for	ADP
cana-2086	269	42	𝑖	𝑖	PRON
cana-2086	269	43	=	=	SYM
cana-2086	269	44	2	2	NUM
cana-2086	269	45	,	,	PUNCT
cana-2086	269	46	𝑔(𝑣2𝑣3	𝑔(𝑣2𝑣3	NOUN
cana-2086	269	47	)	)	PUNCT
cana-2086	269	48	=	=	SYM
cana-2086	270	1	(	(	PUNCT
cana-2086	270	2	3𝑚	3𝑚	NUM
cana-2086	270	3	+	+	NUM
cana-2086	270	4	3	3	NUM
cana-2086	270	5	+	+	NUM
cana-2086	270	6	3𝑚	3𝑚	NUM
cana-2086	270	7	+	+	NOUN
cana-2086	270	8	4	4	NUM
cana-2086	270	9	)	)	PUNCT
cana-2086	270	10	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	270	11	(	(	PUNCT
cana-2086	270	12	2𝑚	2𝑚	NOUN
cana-2086	270	13	+	+	CCONJ
cana-2086	270	14	2	2	X
cana-2086	270	15	)	)	PUNCT
cana-2086	270	16	=	=	NOUN
cana-2086	270	17	(	(	PUNCT
cana-2086	270	18	6𝑚	6𝑚	NOUN
cana-2086	270	19	+	+	CCONJ
cana-2086	270	20	7	7	X
cana-2086	270	21	)	)	PUNCT
cana-2086	270	22	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	270	23	(	(	PUNCT
cana-2086	270	24	2𝑚	2𝑚	NOUN
cana-2086	270	25	+	+	CCONJ
cana-2086	270	26	2	2	X
cana-2086	270	27	)	)	PUNCT
cana-2086	270	28	=	=	SYM
cana-2086	270	29	1	1	X
cana-2086	270	30	.	.	X
cana-2086	271	1	for	for	ADP
cana-2086	271	2	𝑖	𝑖	PRON
cana-2086	271	3	=	=	SYM
cana-2086	271	4	3	3	NUM
cana-2086	271	5	,	,	PUNCT
cana-2086	271	6	𝑔(𝑣3𝑣4	𝑔(𝑣3𝑣4	ADJ
cana-2086	271	7	)	)	PUNCT
cana-2086	271	8	=	=	SYM
cana-2086	271	9	(	(	PUNCT
cana-2086	271	10	3𝑚	3𝑚	NUM
cana-2086	271	11	+	+	NUM
cana-2086	271	12	4	4	NUM
cana-2086	271	13	+	+	NUM
cana-2086	271	14	2𝑚	2𝑚	NOUN
cana-2086	271	15	+	+	CCONJ
cana-2086	271	16	3	3	X
cana-2086	271	17	)	)	PUNCT
cana-2086	271	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	271	19	(	(	PUNCT
cana-2086	271	20	2𝑚	2𝑚	NOUN
cana-2086	271	21	+	+	CCONJ
cana-2086	271	22	2	2	X
cana-2086	271	23	)	)	PUNCT
cana-2086	271	24	=	=	NOUN
cana-2086	271	25	(	(	PUNCT
cana-2086	271	26	5𝑚	5𝑚	NOUN
cana-2086	271	27	+	+	CCONJ
cana-2086	271	28	7	7	NUM
cana-2086	271	29	)	)	PUNCT
cana-2086	271	30	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	271	31	(	(	PUNCT
cana-2086	271	32	2𝑚	2𝑚	NOUN
cana-2086	271	33	+	+	CCONJ
cana-2086	271	34	2	2	X
cana-2086	271	35	)	)	PUNCT
cana-2086	271	36	=	=	PUNCT
cana-2086	271	37	𝑚	𝑚	PROPN
cana-2086	271	38	+	+	NOUN
cana-2086	271	39	3	3	X
cana-2086	271	40	.	.	X
cana-2086	271	41	for	for	ADP
cana-2086	271	42	4	4	NUM
cana-2086	271	43	≤	≤	NOUN
cana-2086	271	44	𝑖	𝑖	SYM
cana-2086	271	45	≤	≤	NOUN
cana-2086	271	46	𝑚	𝑚	ADP
cana-2086	271	47	+	+	PROPN
cana-2086	271	48	1	1	NUM
cana-2086	271	49	,	,	PUNCT
cana-2086	271	50	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	271	51	)	)	PUNCT
cana-2086	271	52	=	=	PUNCT
cana-2086	271	53	(	(	PUNCT
cana-2086	271	54	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	271	55	)	)	PUNCT
cana-2086	271	56	+	+	CCONJ
cana-2086	271	57	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	271	58	)	)	PUNCT
cana-2086	271	59	)	)	PUNCT
cana-2086	271	60	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	271	61	(	(	PUNCT
cana-2086	271	62	2𝑚	2𝑚	NOUN
cana-2086	271	63	+	+	CCONJ
cana-2086	271	64	2	2	X
cana-2086	271	65	)	)	PUNCT
cana-2086	271	66	=	=	SYM
cana-2086	271	67	(	(	PUNCT
cana-2086	271	68	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	271	69	)	)	PUNCT
cana-2086	271	70	+	+	CCONJ
cana-2086	271	71	1	1	NUM
cana-2086	271	72	+	+	NUM
cana-2086	271	73	𝑓(𝑣𝑖+1−2	𝑓(𝑣𝑖+1−2	NOUN
cana-2086	271	74	)	)	PUNCT
cana-2086	271	75	+	+	CCONJ
cana-2086	271	76	1	1	X
cana-2086	271	77	)	)	PUNCT
cana-2086	271	78	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	271	79	(	(	PUNCT
cana-2086	271	80	2𝑚	2𝑚	NOUN
cana-2086	271	81	+	+	CCONJ
cana-2086	271	82	2	2	X
cana-2086	271	83	)	)	PUNCT
cana-2086	271	84	=	=	NOUN
cana-2086	271	85	(	(	PUNCT
cana-2086	271	86	𝑓(𝑣𝑖−4	𝑓(𝑣𝑖−4	NOUN
cana-2086	271	87	)	)	PUNCT
cana-2086	271	88	+	+	CCONJ
cana-2086	271	89	2	2	NUM
cana-2086	271	90	+	+	NUM
cana-2086	271	91	𝑓(𝑣𝑖+1−4	𝑓(𝑣𝑖+1−4	NOUN
cana-2086	271	92	)	)	PUNCT
cana-2086	271	93	+	+	CCONJ
cana-2086	271	94	2	2	X
cana-2086	271	95	)	)	PUNCT
cana-2086	271	96	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	271	97	(	(	PUNCT
cana-2086	271	98	2𝑚	2𝑚	NOUN
cana-2086	271	99	+	+	CCONJ
cana-2086	271	100	2	2	NUM
cana-2086	271	101	)	)	PUNCT
cana-2086	271	102	⋮	⋮	NOUN
cana-2086	271	103	subcase	subcase	VERB
cana-2086	271	104	1	1	NUM
cana-2086	271	105	:	:	PUNCT
cana-2086	271	106	when	when	SCONJ
cana-2086	271	107	4	4	NUM
cana-2086	271	108	≤	≤	NUM
cana-2086	271	109	𝑖	𝑖	SYM
cana-2086	271	110	≤	≤	NOUN
cana-2086	271	111	𝑚	𝑚	ADP
cana-2086	271	112	+	+	CCONJ
cana-2086	271	113	1	1	NUM
cana-2086	271	114	and	and	CCONJ
cana-2086	271	115	odd	odd	ADJ
cana-2086	271	116	,	,	PUNCT
cana-2086	271	117	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	271	118	)	)	PUNCT
cana-2086	271	119	=	=	SYM
cana-2086	271	120	(	(	PUNCT
cana-2086	271	121	𝑓(𝑣𝑖−𝑗	𝑓(𝑣𝑖−𝑗	PROPN
cana-2086	271	122	)	)	PUNCT
cana-2086	272	1	+	+	CCONJ
cana-2086	272	2	𝑗	𝑗	PROPN
cana-2086	272	3	2	2	NUM
cana-2086	272	4	+	+	NUM
cana-2086	272	5	𝑓(𝑣𝑖+1−𝑗	𝑓(𝑣𝑖+1−𝑗	NOUN
cana-2086	272	6	)	)	PUNCT
cana-2086	273	1	+	+	CCONJ
cana-2086	273	2	𝑗	𝑗	PROPN
cana-2086	273	3	2	2	X
cana-2086	273	4	)	)	PUNCT
cana-2086	273	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	273	6	(	(	PUNCT
cana-2086	273	7	2𝑚	2𝑚	NOUN
cana-2086	273	8	+	+	CCONJ
cana-2086	273	9	2	2	NUM
cana-2086	273	10	)	)	PUNCT
cana-2086	273	11	,	,	PUNCT
cana-2086	273	12	𝑗	𝑗	NOUN
cana-2086	273	13	=	=	SYM
cana-2086	273	14	2	2	NUM
cana-2086	273	15	,	,	PUNCT
cana-2086	273	16	4	4	NUM
cana-2086	273	17	,	,	PUNCT
cana-2086	273	18	…	…	PUNCT
cana-2086	273	19	,	,	PUNCT
cana-2086	273	20	𝑖	𝑖	PRON
cana-2086	273	21	−	−	PROPN
cana-2086	273	22	3	3	NUM
cana-2086	273	23	,	,	PUNCT
cana-2086	273	24	and	and	CCONJ
cana-2086	273	25	when	when	SCONJ
cana-2086	273	26	𝑗	𝑗	X
cana-2086	273	27	=	=	SYM
cana-2086	273	28	𝑖	𝑖	SYM
cana-2086	273	29	−	−	PROPN
cana-2086	273	30	3	3	NUM
cana-2086	273	31	,	,	PUNCT
cana-2086	273	32	communications	communication	NOUN
cana-2086	273	33	on	on	ADP
cana-2086	273	34	applied	apply	VERB
cana-2086	273	35	nonlinear	nonlinear	ADJ
cana-2086	273	36	analysis	analysis	NOUN
cana-2086	273	37	issn	issn	NOUN
cana-2086	273	38	:	:	PUNCT
cana-2086	273	39	1074	1074	NUM
cana-2086	273	40	-	-	PUNCT
cana-2086	273	41	133x	133x	NUM
cana-2086	273	42	vol	vol	NOUN
cana-2086	273	43	32	32	NUM
cana-2086	274	1	no	no	NOUN
cana-2086	274	2	.	.	PUNCT
cana-2086	275	1	1s	1s	NUM
cana-2086	275	2	(	(	PUNCT
cana-2086	275	3	2025	2025	NUM
cana-2086	275	4	)	)	PUNCT
cana-2086	275	5	39	39	NUM
cana-2086	275	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	275	7	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	275	8	)	)	PUNCT
cana-2086	275	9	=	=	PUNCT
cana-2086	275	10	(	(	PUNCT
cana-2086	275	11	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	275	12	)	)	PUNCT
cana-2086	275	13	+	+	CCONJ
cana-2086	275	14	𝑓(𝑣4	𝑓(𝑣4	ADJ
cana-2086	275	15	)	)	PUNCT
cana-2086	276	1	+	+	CCONJ
cana-2086	276	2	2	2	NUM
cana-2086	276	3	(	(	PUNCT
cana-2086	276	4	𝑖	𝑖	NOUN
cana-2086	276	5	−	−	PROPN
cana-2086	276	6	3	3	NUM
cana-2086	276	7	2	2	NUM
cana-2086	276	8	)	)	PUNCT
cana-2086	276	9	)	)	PUNCT
cana-2086	276	10	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	276	11	(	(	PUNCT
cana-2086	276	12	2𝑚	2𝑚	NOUN
cana-2086	276	13	+	+	CCONJ
cana-2086	276	14	2	2	X
cana-2086	276	15	)	)	PUNCT
cana-2086	276	16	=	=	SYM
cana-2086	277	1	(	(	PUNCT
cana-2086	277	2	3𝑚	3𝑚	NUM
cana-2086	277	3	+	+	NOUN
cana-2086	277	4	4	4	NUM
cana-2086	277	5	+	+	NUM
cana-2086	277	6	2𝑚	2𝑚	NOUN
cana-2086	277	7	+	+	CCONJ
cana-2086	277	8	3	3	NUM
cana-2086	277	9	+	+	NOUN
cana-2086	277	10	𝑖	𝑖	NOUN
cana-2086	277	11	−	−	NOUN
cana-2086	277	12	3	3	NUM
cana-2086	277	13	)	)	PUNCT
cana-2086	277	14	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	277	15	(	(	PUNCT
cana-2086	277	16	2𝑚	2𝑚	NOUN
cana-2086	277	17	+	+	CCONJ
cana-2086	277	18	2	2	X
cana-2086	277	19	)	)	PUNCT
cana-2086	277	20	=	=	NOUN
cana-2086	277	21	(	(	PUNCT
cana-2086	277	22	2(2𝑚	2(2𝑚	NUM
cana-2086	277	23	+	+	CCONJ
cana-2086	277	24	2	2	X
cana-2086	277	25	)	)	PUNCT
cana-2086	277	26	+	+	NOUN
cana-2086	277	27	𝑚	𝑚	X
cana-2086	277	28	+	+	NOUN
cana-2086	277	29	𝑖	𝑖	X
cana-2086	277	30	)	)	PUNCT
cana-2086	277	31	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	277	32	(	(	PUNCT
cana-2086	277	33	2𝑚	2𝑚	NOUN
cana-2086	277	34	+	+	CCONJ
cana-2086	277	35	2	2	X
cana-2086	277	36	)	)	PUNCT
cana-2086	277	37	=	=	PUNCT
cana-2086	277	38	𝑚	𝑚	PROPN
cana-2086	277	39	+	+	CCONJ
cana-2086	277	40	𝑖.	𝑖.	ADJ
cana-2086	277	41	subcase	subcase	NOUN
cana-2086	277	42	2	2	NUM
cana-2086	277	43	:	:	PUNCT
cana-2086	277	44	when	when	SCONJ
cana-2086	277	45	4	4	NUM
cana-2086	277	46	≤	≤	NUM
cana-2086	277	47	𝑖	𝑖	SYM
cana-2086	277	48	≤	≤	NOUN
cana-2086	277	49	𝑚	𝑚	ADP
cana-2086	277	50	+	+	ADP
cana-2086	277	51	1	1	NUM
cana-2086	277	52	and	and	CCONJ
cana-2086	277	53	even	even	ADV
cana-2086	277	54	,	,	PUNCT
cana-2086	277	55	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	277	56	)	)	PUNCT
cana-2086	277	57	=	=	SYM
cana-2086	277	58	(	(	PUNCT
cana-2086	277	59	𝑓(𝑣𝑖−𝑗	𝑓(𝑣𝑖−𝑗	PROPN
cana-2086	277	60	)	)	PUNCT
cana-2086	278	1	+	+	CCONJ
cana-2086	278	2	𝑗	𝑗	PROPN
cana-2086	278	3	2	2	NUM
cana-2086	278	4	+	+	NUM
cana-2086	278	5	𝑓(𝑣𝑖+1−𝑗	𝑓(𝑣𝑖+1−𝑗	NOUN
cana-2086	278	6	)	)	PUNCT
cana-2086	279	1	+	+	CCONJ
cana-2086	279	2	𝑗	𝑗	PROPN
cana-2086	279	3	2	2	X
cana-2086	279	4	)	)	PUNCT
cana-2086	279	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	279	6	(	(	PUNCT
cana-2086	279	7	2𝑚	2𝑚	NOUN
cana-2086	279	8	+	+	CCONJ
cana-2086	279	9	2	2	NUM
cana-2086	279	10	)	)	PUNCT
cana-2086	279	11	,	,	PUNCT
cana-2086	279	12	𝑗	𝑗	NOUN
cana-2086	279	13	=	=	SYM
cana-2086	279	14	0	0	NUM
cana-2086	279	15	,	,	PUNCT
cana-2086	279	16	2	2	NUM
cana-2086	279	17	,	,	PUNCT
cana-2086	279	18	4	4	NUM
cana-2086	279	19	,	,	PUNCT
cana-2086	279	20	…	…	PUNCT
cana-2086	279	21	,	,	PUNCT
cana-2086	279	22	𝑖	𝑖	PRON
cana-2086	279	23	−	−	PROPN
cana-2086	279	24	4	4	NUM
cana-2086	279	25	,	,	PUNCT
cana-2086	279	26	and	and	CCONJ
cana-2086	279	27	when	when	SCONJ
cana-2086	279	28	𝑗	𝑗	X
cana-2086	279	29	=	=	SYM
cana-2086	279	30	𝑖	𝑖	SYM
cana-2086	279	31	−	−	PROPN
cana-2086	279	32	4	4	NUM
cana-2086	279	33	,	,	PUNCT
cana-2086	279	34	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	279	35	)	)	PUNCT
cana-2086	279	36	=	=	PUNCT
cana-2086	279	37	(	(	PUNCT
cana-2086	279	38	𝑓(𝑣4	𝑓(𝑣4	NOUN
cana-2086	279	39	)	)	PUNCT
cana-2086	279	40	+	+	CCONJ
cana-2086	279	41	𝑓(𝑣5	𝑓(𝑣5	X
cana-2086	279	42	)	)	PUNCT
cana-2086	280	1	+	+	CCONJ
cana-2086	280	2	2	2	NUM
cana-2086	280	3	(	(	PUNCT
cana-2086	280	4	𝑖	𝑖	NOUN
cana-2086	280	5	−	−	PROPN
cana-2086	280	6	4	4	NUM
cana-2086	280	7	2	2	NUM
cana-2086	280	8	)	)	PUNCT
cana-2086	280	9	)	)	PUNCT
cana-2086	280	10	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	280	11	(	(	PUNCT
cana-2086	280	12	2𝑚	2𝑚	NOUN
cana-2086	280	13	+	+	CCONJ
cana-2086	280	14	2	2	X
cana-2086	280	15	)	)	PUNCT
cana-2086	280	16	=	=	NOUN
cana-2086	280	17	(	(	PUNCT
cana-2086	280	18	𝑓(𝑣4	𝑓(𝑣4	NOUN
cana-2086	280	19	)	)	PUNCT
cana-2086	281	1	+	+	CCONJ
cana-2086	281	2	𝑓(𝑣3	𝑓(𝑣3	VERB
cana-2086	281	3	)	)	PUNCT
cana-2086	281	4	+	+	CCONJ
cana-2086	281	5	1	1	NUM
cana-2086	281	6	+	+	NUM
cana-2086	281	7	2	2	NUM
cana-2086	281	8	(	(	PUNCT
cana-2086	281	9	𝑖−4	𝑖−4	PROPN
cana-2086	281	10	2	2	NUM
cana-2086	281	11	)	)	PUNCT
cana-2086	281	12	)	)	PUNCT
cana-2086	281	13	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	281	14	(	(	PUNCT
cana-2086	281	15	2𝑚	2𝑚	NOUN
cana-2086	281	16	+	+	CCONJ
cana-2086	281	17	2	2	X
cana-2086	281	18	)	)	PUNCT
cana-2086	281	19	=	=	NOUN
cana-2086	281	20	(	(	PUNCT
cana-2086	281	21	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	281	22	)	)	PUNCT
cana-2086	281	23	+	+	CCONJ
cana-2086	281	24	𝑓(𝑣4	𝑓(𝑣4	ADJ
cana-2086	281	25	)	)	PUNCT
cana-2086	282	1	+	+	CCONJ
cana-2086	282	2	2	2	NUM
cana-2086	282	3	(	(	PUNCT
cana-2086	282	4	𝑖−3	𝑖−3	NOUN
cana-2086	282	5	2	2	NUM
cana-2086	282	6	)	)	PUNCT
cana-2086	282	7	)	)	PUNCT
cana-2086	282	8	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	282	9	(	(	PUNCT
cana-2086	282	10	2𝑚	2𝑚	NOUN
cana-2086	282	11	+	+	CCONJ
cana-2086	282	12	2	2	X
cana-2086	282	13	)	)	PUNCT
cana-2086	282	14	=	=	SYM
cana-2086	283	1	(	(	PUNCT
cana-2086	283	2	3𝑚	3𝑚	NUM
cana-2086	283	3	+	+	NOUN
cana-2086	283	4	4	4	NUM
cana-2086	283	5	+	+	NUM
cana-2086	283	6	2𝑚	2𝑚	NOUN
cana-2086	283	7	+	+	CCONJ
cana-2086	283	8	3	3	NUM
cana-2086	283	9	+	+	NOUN
cana-2086	283	10	𝑖	𝑖	NOUN
cana-2086	283	11	−	−	NOUN
cana-2086	283	12	3	3	NUM
cana-2086	283	13	)	)	PUNCT
cana-2086	283	14	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	283	15	(	(	PUNCT
cana-2086	283	16	2𝑚	2𝑚	NOUN
cana-2086	283	17	+	+	CCONJ
cana-2086	283	18	2	2	X
cana-2086	283	19	)	)	PUNCT
cana-2086	283	20	=	=	NOUN
cana-2086	283	21	(	(	PUNCT
cana-2086	283	22	2(2𝑚	2(2𝑚	NUM
cana-2086	283	23	+	+	CCONJ
cana-2086	283	24	2	2	X
cana-2086	283	25	)	)	PUNCT
cana-2086	283	26	+	+	NOUN
cana-2086	283	27	𝑚	𝑚	X
cana-2086	283	28	+	+	NOUN
cana-2086	283	29	𝑖	𝑖	X
cana-2086	283	30	)	)	PUNCT
cana-2086	283	31	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	283	32	(	(	PUNCT
cana-2086	283	33	2𝑚	2𝑚	NOUN
cana-2086	283	34	+	+	CCONJ
cana-2086	283	35	2	2	X
cana-2086	283	36	)	)	PUNCT
cana-2086	283	37	=	=	PUNCT
cana-2086	283	38	𝑚	𝑚	PROPN
cana-2086	283	39	+	+	NOUN
cana-2086	283	40	𝑖	𝑖	X
cana-2086	283	41	.	.	PUNCT
cana-2086	284	1	for	for	ADP
cana-2086	284	2	𝑖	𝑖	PRON
cana-2086	284	3	=	=	SYM
cana-2086	284	4	𝑚	𝑚	PROPN
cana-2086	284	5	+	+	PROPN
cana-2086	284	6	2	2	NUM
cana-2086	284	7	,	,	PUNCT
cana-2086	284	8	𝑔(𝑣𝑚+2𝑣𝑚+3	𝑔(𝑣𝑚+2𝑣𝑚+3	NOUN
cana-2086	284	9	)	)	PUNCT
cana-2086	284	10	=	=	PUNCT
cana-2086	284	11	(	(	PUNCT
cana-2086	284	12	𝑓(𝑣𝑚+2	𝑓(𝑣𝑚+2	PROPN
cana-2086	284	13	)	)	PUNCT
cana-2086	284	14	+	+	CCONJ
cana-2086	284	15	𝑓(𝑣𝑚+3	𝑓(𝑣𝑚+3	NOUN
cana-2086	284	16	)	)	PUNCT
cana-2086	284	17	)	)	PUNCT
cana-2086	284	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	284	19	(	(	PUNCT
cana-2086	284	20	2𝑚	2𝑚	NOUN
cana-2086	284	21	+	+	CCONJ
cana-2086	284	22	2	2	X
cana-2086	284	23	)	)	PUNCT
cana-2086	284	24	=	=	NOUN
cana-2086	284	25	(	(	PUNCT
cana-2086	284	26	2𝑓(𝑣𝑚+2	2𝑓(𝑣𝑚+2	NUM
cana-2086	284	27	)	)	PUNCT
cana-2086	284	28	+	+	NOUN
cana-2086	284	29	1	1	X
cana-2086	284	30	)	)	PUNCT
cana-2086	284	31	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	284	32	(	(	PUNCT
cana-2086	284	33	2𝑚	2𝑚	NOUN
cana-2086	284	34	+	+	CCONJ
cana-2086	284	35	2	2	X
cana-2086	284	36	)	)	PUNCT
cana-2086	284	37	=	=	NOUN
cana-2086	284	38	(	(	PUNCT
cana-2086	284	39	2𝑓(𝑣𝑚	2𝑓(𝑣𝑚	NUM
cana-2086	284	40	)	)	PUNCT
cana-2086	284	41	+	+	CCONJ
cana-2086	284	42	2	2	NUM
cana-2086	284	43	+	+	CCONJ
cana-2086	284	44	1	1	NUM
cana-2086	284	45	)	)	PUNCT
cana-2086	284	46	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	284	47	(	(	PUNCT
cana-2086	284	48	2𝑚	2𝑚	NOUN
cana-2086	284	49	+	+	CCONJ
cana-2086	284	50	2	2	X
cana-2086	284	51	)	)	PUNCT
cana-2086	284	52	=	=	SYM
cana-2086	284	53	(	(	PUNCT
cana-2086	284	54	2𝑓(𝑣𝑚−2	2𝑓(𝑣𝑚−2	NUM
cana-2086	284	55	)	)	PUNCT
cana-2086	285	1	+	+	CCONJ
cana-2086	285	2	4	4	NUM
cana-2086	285	3	+	+	SYM
cana-2086	285	4	1	1	NUM
cana-2086	285	5	)	)	PUNCT
cana-2086	285	6	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	285	7	(	(	PUNCT
cana-2086	285	8	2𝑚	2𝑚	NOUN
cana-2086	285	9	+	+	CCONJ
cana-2086	285	10	2	2	NUM
cana-2086	285	11	)	)	PUNCT
cana-2086	285	12	⋮	⋮	NOUN
cana-2086	285	13	=	=	SYM
cana-2086	285	14	(	(	PUNCT
cana-2086	285	15	2𝑓(𝑣(𝑚+2)−(𝑚+2−3	2𝑓(𝑣(𝑚+2)−(𝑚+2−3	NOUN
cana-2086	285	16	)	)	PUNCT
cana-2086	285	17	)	)	PUNCT
cana-2086	286	1	+	+	CCONJ
cana-2086	286	2	𝑚	𝑚	X
cana-2086	286	3	+	+	ADJ
cana-2086	286	4	2	2	NUM
cana-2086	286	5	−	−	NOUN
cana-2086	286	6	3	3	NUM
cana-2086	286	7	+	+	CCONJ
cana-2086	286	8	1	1	NUM
cana-2086	286	9	)	)	PUNCT
cana-2086	286	10	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	11	(	(	PUNCT
cana-2086	286	12	2𝑚	2𝑚	NOUN
cana-2086	286	13	+	+	CCONJ
cana-2086	286	14	2	2	X
cana-2086	286	15	)	)	PUNCT
cana-2086	286	16	=	=	SYM
cana-2086	286	17	(	(	PUNCT
cana-2086	286	18	2𝑓(𝑣3	2𝑓(𝑣3	NUM
cana-2086	286	19	)	)	PUNCT
cana-2086	286	20	+	+	CCONJ
cana-2086	286	21	𝑚	𝑚	X
cana-2086	286	22	+	+	ADJ
cana-2086	286	23	2	2	NUM
cana-2086	286	24	−	−	NUM
cana-2086	286	25	2	2	NUM
cana-2086	286	26	)	)	PUNCT
cana-2086	286	27	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	28	(	(	PUNCT
cana-2086	286	29	2𝑚	2𝑚	NOUN
cana-2086	286	30	+	+	CCONJ
cana-2086	286	31	2	2	X
cana-2086	286	32	)	)	PUNCT
cana-2086	286	33	=	=	NOUN
cana-2086	286	34	(	(	PUNCT
cana-2086	286	35	6𝑚	6𝑚	NOUN
cana-2086	286	36	+	+	CCONJ
cana-2086	286	37	6	6	NUM
cana-2086	286	38	+	+	SYM
cana-2086	286	39	2	2	NUM
cana-2086	286	40	+	+	CCONJ
cana-2086	286	41	𝑚	𝑚	NOUN
cana-2086	286	42	)	)	PUNCT
cana-2086	286	43	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	44	(	(	PUNCT
cana-2086	286	45	2𝑚	2𝑚	NOUN
cana-2086	286	46	+	+	CCONJ
cana-2086	286	47	2	2	X
cana-2086	286	48	)	)	PUNCT
cana-2086	286	49	=	=	PUNCT
cana-2086	286	50	𝑚	𝑚	PROPN
cana-2086	286	51	+	+	NOUN
cana-2086	286	52	2	2	X
cana-2086	286	53	.	.	X
cana-2086	286	54	for	for	ADP
cana-2086	286	55	𝑖	𝑖	PRON
cana-2086	286	56	=	=	SYM
cana-2086	286	57	𝑚	𝑚	PROPN
cana-2086	286	58	+	+	NUM
cana-2086	286	59	3	3	NUM
cana-2086	286	60	,	,	PUNCT
cana-2086	286	61	𝑔(𝑣𝑚+3𝑣𝑚+4	𝑔(𝑣𝑚+3𝑣𝑚+4	NOUN
cana-2086	286	62	)	)	PUNCT
cana-2086	286	63	=	=	SYM
cana-2086	286	64	(	(	PUNCT
cana-2086	286	65	𝑓(𝑣𝑚+3	𝑓(𝑣𝑚+3	NOUN
cana-2086	286	66	)	)	PUNCT
cana-2086	286	67	+	+	NUM
cana-2086	286	68	𝑓(𝑣𝑚+4	𝑓(𝑣𝑚+4	NOUN
cana-2086	286	69	)	)	PUNCT
cana-2086	286	70	)	)	PUNCT
cana-2086	286	71	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	72	(	(	PUNCT
cana-2086	286	73	2𝑚	2𝑚	NOUN
cana-2086	286	74	+	+	CCONJ
cana-2086	286	75	2	2	X
cana-2086	286	76	)	)	PUNCT
cana-2086	286	77	=	=	SYM
cana-2086	286	78	(	(	PUNCT
cana-2086	286	79	𝑓(𝑣𝑚+2	𝑓(𝑣𝑚+2	PROPN
cana-2086	286	80	)	)	PUNCT
cana-2086	286	81	+	+	CCONJ
cana-2086	286	82	1	1	NUM
cana-2086	286	83	+	+	CCONJ
cana-2086	286	84	𝑓(𝑣𝑚+1	𝑓(𝑣𝑚+1	NUM
cana-2086	286	85	)	)	PUNCT
cana-2086	286	86	+	+	CCONJ
cana-2086	286	87	2	2	X
cana-2086	286	88	)	)	PUNCT
cana-2086	286	89	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	90	(	(	PUNCT
cana-2086	286	91	2𝑚	2𝑚	NOUN
cana-2086	286	92	+	+	CCONJ
cana-2086	286	93	2	2	X
cana-2086	286	94	)	)	PUNCT
cana-2086	286	95	=	=	SYM
cana-2086	286	96	(	(	PUNCT
cana-2086	286	97	𝑓(𝑣𝑚	𝑓(𝑣𝑚	NOUN
cana-2086	286	98	)	)	PUNCT
cana-2086	286	99	+	+	CCONJ
cana-2086	286	100	1	1	NUM
cana-2086	286	101	+	+	NUM
cana-2086	286	102	𝑓(𝑣𝑚−1	𝑓(𝑣𝑚−1	NOUN
cana-2086	286	103	)	)	PUNCT
cana-2086	286	104	+	+	CCONJ
cana-2086	286	105	1	1	NUM
cana-2086	286	106	+	+	NUM
cana-2086	286	107	3	3	NUM
cana-2086	286	108	)	)	PUNCT
cana-2086	286	109	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	110	(	(	PUNCT
cana-2086	286	111	2𝑚	2𝑚	NOUN
cana-2086	286	112	+	+	CCONJ
cana-2086	286	113	2	2	X
cana-2086	286	114	)	)	PUNCT
cana-2086	286	115	=	=	NOUN
cana-2086	286	116	(	(	PUNCT
cana-2086	286	117	𝑓(𝑣𝑚−2	𝑓(𝑣𝑚−2	NOUN
cana-2086	286	118	)	)	PUNCT
cana-2086	286	119	+	+	CCONJ
cana-2086	286	120	2	2	NUM
cana-2086	286	121	+	+	CCONJ
cana-2086	286	122	𝑓(𝑣𝑚−3	𝑓(𝑣𝑚−3	NOUN
cana-2086	286	123	)	)	PUNCT
cana-2086	286	124	+	+	CCONJ
cana-2086	286	125	2	2	NUM
cana-2086	286	126	+	+	NUM
cana-2086	286	127	3	3	NUM
cana-2086	286	128	)	)	PUNCT
cana-2086	286	129	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	130	(	(	PUNCT
cana-2086	286	131	2𝑚	2𝑚	NOUN
cana-2086	286	132	+	+	CCONJ
cana-2086	286	133	2	2	NUM
cana-2086	286	134	)	)	PUNCT
cana-2086	286	135	⋮	⋮	NOUN
cana-2086	286	136	=	=	SYM
cana-2086	286	137	(	(	PUNCT
cana-2086	286	138	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	286	139	)	)	PUNCT
cana-2086	286	140	+	+	CCONJ
cana-2086	286	141	𝑚−1	𝑚−1	PROPN
cana-2086	286	142	2	2	NUM
cana-2086	286	143	+	+	CCONJ
cana-2086	286	144	𝑓(𝑣4	𝑓(𝑣4	ADJ
cana-2086	286	145	)	)	PUNCT
cana-2086	286	146	+	+	CCONJ
cana-2086	286	147	𝑚−3	𝑚−3	NOUN
cana-2086	286	148	2	2	NUM
cana-2086	286	149	+	+	NUM
cana-2086	286	150	3	3	NUM
cana-2086	286	151	)	)	PUNCT
cana-2086	286	152	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	286	153	(	(	PUNCT
cana-2086	286	154	2𝑚	2𝑚	NOUN
cana-2086	286	155	+	+	CCONJ
cana-2086	286	156	2	2	X
cana-2086	286	157	)	)	PUNCT
cana-2086	286	158	=	=	SYM
cana-2086	287	1	(	(	PUNCT
cana-2086	287	2	3𝑚	3𝑚	NUM
cana-2086	287	3	+	+	NUM
cana-2086	287	4	4	4	NUM
cana-2086	287	5	+	+	NUM
cana-2086	287	6	2𝑚	2𝑚	NOUN
cana-2086	287	7	+	+	CCONJ
cana-2086	287	8	3	3	NUM
cana-2086	287	9	+	+	CCONJ
cana-2086	287	10	𝑚−1	𝑚−1	PROPN
cana-2086	287	11	2	2	NUM
cana-2086	287	12	+	+	CCONJ
cana-2086	287	13	𝑚−3	𝑚−3	NOUN
cana-2086	287	14	2	2	NUM
cana-2086	287	15	+	+	NOUN
cana-2086	287	16	3)𝑚𝑜𝑑	3)𝑚𝑜𝑑	NUM
cana-2086	287	17	(	(	PUNCT
cana-2086	287	18	2𝑚	2𝑚	NOUN
cana-2086	287	19	+	+	CCONJ
cana-2086	287	20	2	2	X
cana-2086	287	21	)	)	PUNCT
cana-2086	287	22	=	=	NOUN
cana-2086	287	23	(	(	PUNCT
cana-2086	287	24	5𝑚	5𝑚	NUM
cana-2086	287	25	+	+	CCONJ
cana-2086	287	26	10	10	NUM
cana-2086	287	27	+	+	NOUN
cana-2086	287	28	𝑚	𝑚	NOUN
cana-2086	287	29	−	−	NUM
cana-2086	287	30	2	2	NUM
cana-2086	287	31	)	)	PUNCT
cana-2086	287	32	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	287	33	(	(	PUNCT
cana-2086	287	34	2𝑚	2𝑚	NOUN
cana-2086	287	35	+	+	CCONJ
cana-2086	287	36	2	2	X
cana-2086	287	37	)	)	PUNCT
cana-2086	287	38	=	=	SYM
cana-2086	287	39	2	2	X
cana-2086	287	40	.	.	X
cana-2086	287	41	communications	communication	NOUN
cana-2086	287	42	on	on	ADP
cana-2086	287	43	applied	apply	VERB
cana-2086	287	44	nonlinear	nonlinear	ADJ
cana-2086	287	45	analysis	analysis	NOUN
cana-2086	287	46	issn	issn	NOUN
cana-2086	287	47	:	:	PUNCT
cana-2086	287	48	1074	1074	NUM
cana-2086	287	49	-	-	PUNCT
cana-2086	287	50	133x	133x	NUM
cana-2086	287	51	vol	vol	NOUN
cana-2086	287	52	32	32	NUM
cana-2086	287	53	no	no	NOUN
cana-2086	287	54	.	.	PUNCT
cana-2086	288	1	1s	1s	NUM
cana-2086	288	2	(	(	PUNCT
cana-2086	288	3	2025	2025	NUM
cana-2086	288	4	)	)	PUNCT
cana-2086	288	5	40	40	NUM
cana-2086	288	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	288	7	for	for	ADP
cana-2086	288	8	𝑚	𝑚	PROPN
cana-2086	288	9	+	+	PROPN
cana-2086	288	10	4	4	NUM
cana-2086	288	11	≤	≤	NUM
cana-2086	288	12	𝑖	𝑖	PRON
cana-2086	288	13	≤	≤	NOUN
cana-2086	288	14	2𝑚	2𝑚	NOUN
cana-2086	288	15	,	,	PUNCT
cana-2086	288	16	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	288	17	)	)	PUNCT
cana-2086	288	18	=	=	PUNCT
cana-2086	288	19	(	(	PUNCT
cana-2086	288	20	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	288	21	)	)	PUNCT
cana-2086	288	22	+	+	CCONJ
cana-2086	288	23	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	288	24	)	)	PUNCT
cana-2086	288	25	)	)	PUNCT
cana-2086	288	26	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	288	27	(	(	PUNCT
cana-2086	288	28	2𝑚	2𝑚	NOUN
cana-2086	288	29	+	+	CCONJ
cana-2086	288	30	2	2	X
cana-2086	288	31	)	)	PUNCT
cana-2086	288	32	=	=	SYM
cana-2086	288	33	(	(	PUNCT
cana-2086	288	34	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	288	35	)	)	PUNCT
cana-2086	288	36	+	+	CCONJ
cana-2086	288	37	1	1	NUM
cana-2086	288	38	+	+	NUM
cana-2086	288	39	𝑓(𝑣𝑖+1−2	𝑓(𝑣𝑖+1−2	NOUN
cana-2086	288	40	)	)	PUNCT
cana-2086	288	41	+	+	CCONJ
cana-2086	288	42	1	1	X
cana-2086	288	43	)	)	PUNCT
cana-2086	288	44	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	288	45	(	(	PUNCT
cana-2086	288	46	2𝑚	2𝑚	NOUN
cana-2086	288	47	+	+	CCONJ
cana-2086	288	48	2	2	X
cana-2086	288	49	)	)	PUNCT
cana-2086	288	50	=	=	NOUN
cana-2086	288	51	(	(	PUNCT
cana-2086	288	52	𝑓(𝑣𝑖−4	𝑓(𝑣𝑖−4	NOUN
cana-2086	288	53	)	)	PUNCT
cana-2086	288	54	+	+	CCONJ
cana-2086	288	55	2	2	NUM
cana-2086	288	56	+	+	NUM
cana-2086	288	57	𝑓(𝑣𝑖+1−4	𝑓(𝑣𝑖+1−4	NOUN
cana-2086	288	58	)	)	PUNCT
cana-2086	288	59	+	+	CCONJ
cana-2086	288	60	2	2	X
cana-2086	288	61	)	)	PUNCT
cana-2086	288	62	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	288	63	(	(	PUNCT
cana-2086	288	64	2𝑚	2𝑚	NOUN
cana-2086	288	65	+	+	CCONJ
cana-2086	288	66	2	2	NUM
cana-2086	288	67	)	)	PUNCT
cana-2086	288	68	⋮	⋮	NOUN
cana-2086	288	69	subcase	subcase	VERB
cana-2086	288	70	1	1	NUM
cana-2086	288	71	:	:	PUNCT
cana-2086	288	72	when	when	SCONJ
cana-2086	288	73	𝑚	𝑚	X
cana-2086	288	74	+	+	CCONJ
cana-2086	288	75	4	4	NUM
cana-2086	288	76	≤	≤	NUM
cana-2086	288	77	𝑖	𝑖	NUM
cana-2086	288	78	≤	≤	NOUN
cana-2086	288	79	2𝑚	2𝑚	NOUN
cana-2086	288	80	and	and	CCONJ
cana-2086	288	81	even	even	ADV
cana-2086	288	82	,	,	PUNCT
cana-2086	288	83	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	288	84	)	)	PUNCT
cana-2086	288	85	=	=	SYM
cana-2086	288	86	(	(	PUNCT
cana-2086	288	87	𝑓(𝑣𝑖−𝑗	𝑓(𝑣𝑖−𝑗	PROPN
cana-2086	288	88	)	)	PUNCT
cana-2086	289	1	+	+	CCONJ
cana-2086	289	2	𝑗	𝑗	PROPN
cana-2086	289	3	2	2	NUM
cana-2086	289	4	+	+	NUM
cana-2086	289	5	𝑓(𝑣𝑖+1−𝑗	𝑓(𝑣𝑖+1−𝑗	NOUN
cana-2086	289	6	)	)	PUNCT
cana-2086	290	1	+	+	CCONJ
cana-2086	290	2	𝑗	𝑗	PROPN
cana-2086	290	3	2	2	NUM
cana-2086	290	4	)	)	PUNCT
cana-2086	290	5	𝑚𝑜𝑑(2𝑚	𝑚𝑜𝑑(2𝑚	PROPN
cana-2086	291	1	+	+	NUM
cana-2086	291	2	2	2	NUM
cana-2086	291	3	)	)	PUNCT
cana-2086	291	4	,	,	PUNCT
cana-2086	291	5	𝑗	𝑗	NOUN
cana-2086	291	6	=	=	SYM
cana-2086	291	7	2	2	NUM
cana-2086	291	8	,	,	PUNCT
cana-2086	291	9	4	4	NUM
cana-2086	291	10	,	,	PUNCT
cana-2086	291	11	…	…	PUNCT
cana-2086	291	12	,	,	PUNCT
cana-2086	291	13	𝑖	𝑖	PRON
cana-2086	291	14	−	−	NOUN
cana-2086	291	15	𝑚	𝑚	INTJ
cana-2086	291	16	−	−	PROPN
cana-2086	291	17	3	3	NUM
cana-2086	291	18	,	,	PUNCT
cana-2086	291	19	and	and	CCONJ
cana-2086	291	20	when	when	SCONJ
cana-2086	291	21	𝑗	𝑗	X
cana-2086	291	22	=	=	SYM
cana-2086	291	23	𝑖	𝑖	SYM
cana-2086	291	24	−𝑚	−𝑚	NOUN
cana-2086	291	25	−	−	PROPN
cana-2086	291	26	3	3	NUM
cana-2086	291	27	,	,	PUNCT
cana-2086	291	28	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	291	29	)	)	PUNCT
cana-2086	291	30	=	=	SYM
cana-2086	291	31	(	(	PUNCT
cana-2086	291	32	𝑓(𝑣𝑚+3	𝑓(𝑣𝑚+3	NOUN
cana-2086	291	33	)	)	PUNCT
cana-2086	291	34	+	+	NUM
cana-2086	291	35	𝑓(𝑣𝑚+4	𝑓(𝑣𝑚+4	NOUN
cana-2086	291	36	)	)	PUNCT
cana-2086	291	37	+	+	CCONJ
cana-2086	291	38	(	(	PUNCT
cana-2086	291	39	𝑖	𝑖	SYM
cana-2086	291	40	−	−	NOUN
cana-2086	291	41	𝑚	𝑚	NOUN
cana-2086	291	42	−	−	PROPN
cana-2086	291	43	3	3	NUM
cana-2086	291	44	)	)	PUNCT
cana-2086	291	45	)	)	PUNCT
cana-2086	291	46	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	291	47	(	(	PUNCT
cana-2086	291	48	2𝑚	2𝑚	NOUN
cana-2086	291	49	+	+	CCONJ
cana-2086	291	50	2	2	X
cana-2086	291	51	)	)	PUNCT
cana-2086	291	52	=	=	NOUN
cana-2086	291	53	(	(	PUNCT
cana-2086	292	1	6𝑚	6𝑚	NOUN
cana-2086	292	2	+	+	CCONJ
cana-2086	292	3	6	6	NUM
cana-2086	292	4	+	+	SYM
cana-2086	292	5	2	2	NUM
cana-2086	292	6	+	+	CCONJ
cana-2086	292	7	𝑖	𝑖	PROPN
cana-2086	292	8	−𝑚	−𝑚	NOUN
cana-2086	292	9	−	−	NOUN
cana-2086	292	10	3	3	NUM
cana-2086	292	11	)	)	PUNCT
cana-2086	292	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	292	13	(	(	PUNCT
cana-2086	292	14	2𝑚	2𝑚	NOUN
cana-2086	292	15	+	+	CCONJ
cana-2086	292	16	2	2	X
cana-2086	292	17	)	)	PUNCT
cana-2086	292	18	=	=	PUNCT
cana-2086	293	1	𝑖	𝑖	PRON
cana-2086	293	2	−	−	NOUN
cana-2086	293	3	𝑚	𝑚	INTJ
cana-2086	293	4	−	−	PROPN
cana-2086	293	5	1	1	NUM
cana-2086	293	6	.	.	PUNCT
cana-2086	293	7	subcase	subcase	PROPN
cana-2086	293	8	2	2	NUM
cana-2086	293	9	:	:	PUNCT
cana-2086	293	10	when	when	SCONJ
cana-2086	293	11	𝑚	𝑚	X
cana-2086	293	12	+	+	CCONJ
cana-2086	293	13	4	4	NUM
cana-2086	293	14	≤	≤	NUM
cana-2086	293	15	𝑖	𝑖	NUM
cana-2086	293	16	≤	≤	NOUN
cana-2086	293	17	2𝑚	2𝑚	NOUN
cana-2086	293	18	and	and	CCONJ
cana-2086	293	19	odd	odd	ADJ
cana-2086	293	20	,	,	PUNCT
cana-2086	293	21	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	293	22	)	)	PUNCT
cana-2086	293	23	=	=	SYM
cana-2086	293	24	(	(	PUNCT
cana-2086	293	25	𝑓(𝑣𝑖−𝑗	𝑓(𝑣𝑖−𝑗	PROPN
cana-2086	293	26	)	)	PUNCT
cana-2086	294	1	+	+	CCONJ
cana-2086	294	2	𝑗	𝑗	PROPN
cana-2086	294	3	2	2	NUM
cana-2086	294	4	+	+	NUM
cana-2086	294	5	𝑓(𝑣𝑖+1−𝑗	𝑓(𝑣𝑖+1−𝑗	NOUN
cana-2086	294	6	)	)	PUNCT
cana-2086	295	1	+	+	CCONJ
cana-2086	295	2	𝑗	𝑗	PROPN
cana-2086	295	3	2	2	X
cana-2086	295	4	)	)	PUNCT
cana-2086	295	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	295	6	(	(	PUNCT
cana-2086	295	7	2𝑚	2𝑚	NOUN
cana-2086	295	8	+	+	CCONJ
cana-2086	295	9	2	2	NUM
cana-2086	295	10	)	)	PUNCT
cana-2086	295	11	,	,	PUNCT
cana-2086	295	12	𝑗	𝑗	NOUN
cana-2086	295	13	=	=	SYM
cana-2086	295	14	0	0	NUM
cana-2086	295	15	,	,	PUNCT
cana-2086	295	16	2	2	NUM
cana-2086	295	17	,	,	PUNCT
cana-2086	295	18	4	4	NUM
cana-2086	295	19	,	,	PUNCT
cana-2086	295	20	…	…	PUNCT
cana-2086	295	21	,	,	PUNCT
cana-2086	296	1	𝑖	𝑖	PRON
cana-2086	296	2	−	−	NOUN
cana-2086	296	3	𝑚	𝑚	ADP
cana-2086	296	4	−	−	PROPN
cana-2086	296	5	4	4	NUM
cana-2086	296	6	,	,	PUNCT
cana-2086	296	7	and	and	CCONJ
cana-2086	296	8	when	when	SCONJ
cana-2086	296	9	𝑗	𝑗	X
cana-2086	296	10	=	=	SYM
cana-2086	296	11	𝑖	𝑖	SYM
cana-2086	296	12	−𝑚	−𝑚	NOUN
cana-2086	296	13	−	−	PROPN
cana-2086	296	14	4	4	NUM
cana-2086	296	15	,	,	PUNCT
cana-2086	296	16	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	296	17	)	)	PUNCT
cana-2086	296	18	=	=	SYM
cana-2086	296	19	(	(	PUNCT
cana-2086	296	20	𝑓(𝑣𝑚+4	𝑓(𝑣𝑚+4	X
cana-2086	296	21	)	)	PUNCT
cana-2086	296	22	+	+	CCONJ
cana-2086	296	23	𝑓(𝑣𝑚+5	𝑓(𝑣𝑚+5	NOUN
cana-2086	296	24	)	)	PUNCT
cana-2086	296	25	+	+	CCONJ
cana-2086	296	26	(	(	PUNCT
cana-2086	296	27	𝑖	𝑖	SYM
cana-2086	296	28	−	−	NOUN
cana-2086	296	29	𝑚	𝑚	NOUN
cana-2086	296	30	−	−	PROPN
cana-2086	296	31	4	4	NUM
cana-2086	296	32	)	)	PUNCT
cana-2086	296	33	)	)	PUNCT
cana-2086	296	34	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	296	35	(	(	PUNCT
cana-2086	296	36	2𝑚	2𝑚	NOUN
cana-2086	296	37	+	+	CCONJ
cana-2086	296	38	2	2	X
cana-2086	296	39	)	)	PUNCT
cana-2086	296	40	=	=	SYM
cana-2086	296	41	(	(	PUNCT
cana-2086	296	42	𝑓(𝑣𝑚+4	𝑓(𝑣𝑚+4	X
cana-2086	296	43	)	)	PUNCT
cana-2086	296	44	+	+	NUM
cana-2086	296	45	𝑓(𝑣𝑚+3	𝑓(𝑣𝑚+3	NOUN
cana-2086	296	46	)	)	PUNCT
cana-2086	297	1	+	+	CCONJ
cana-2086	297	2	1	1	NUM
cana-2086	297	3	+	+	CCONJ
cana-2086	297	4	(	(	PUNCT
cana-2086	297	5	𝑖	𝑖	SYM
cana-2086	297	6	−𝑚	−𝑚	NOUN
cana-2086	297	7	−	−	NOUN
cana-2086	297	8	4	4	NUM
cana-2086	297	9	)	)	PUNCT
cana-2086	297	10	)	)	PUNCT
cana-2086	297	11	𝑚𝑜𝑑(2𝑚	𝑚𝑜𝑑(2𝑚	PROPN
cana-2086	298	1	+	+	CCONJ
cana-2086	298	2	2	2	X
cana-2086	298	3	)	)	PUNCT
cana-2086	298	4	=	=	SYM
cana-2086	298	5	(	(	PUNCT
cana-2086	298	6	𝑓(𝑣𝑚+3	𝑓(𝑣𝑚+3	NOUN
cana-2086	298	7	)	)	PUNCT
cana-2086	298	8	+	+	NUM
cana-2086	298	9	𝑓(𝑣𝑚+4	𝑓(𝑣𝑚+4	NOUN
cana-2086	298	10	)	)	PUNCT
cana-2086	299	1	+	+	CCONJ
cana-2086	299	2	(	(	PUNCT
cana-2086	299	3	𝑖	𝑖	SYM
cana-2086	299	4	−	−	NOUN
cana-2086	299	5	𝑚	𝑚	NOUN
cana-2086	299	6	−	−	PROPN
cana-2086	299	7	3	3	NUM
cana-2086	299	8	)	)	PUNCT
cana-2086	299	9	)	)	PUNCT
cana-2086	299	10	𝑚𝑜𝑑(2𝑚	𝑚𝑜𝑑(2𝑚	PROPN
cana-2086	300	1	+	+	CCONJ
cana-2086	300	2	2	2	X
cana-2086	300	3	)	)	PUNCT
cana-2086	300	4	=	=	NOUN
cana-2086	300	5	(	(	PUNCT
cana-2086	300	6	6𝑚	6𝑚	NOUN
cana-2086	300	7	+	+	CCONJ
cana-2086	300	8	6	6	NUM
cana-2086	300	9	+	+	SYM
cana-2086	300	10	2	2	NUM
cana-2086	300	11	+	+	CCONJ
cana-2086	300	12	𝑖	𝑖	PROPN
cana-2086	300	13	−𝑚	−𝑚	NOUN
cana-2086	301	1	−	−	NOUN
cana-2086	301	2	3)𝑚𝑜𝑑(2𝑚	3)𝑚𝑜𝑑(2𝑚	NUM
cana-2086	302	1	+	+	CCONJ
cana-2086	302	2	2	2	NUM
cana-2086	302	3	)	)	PUNCT
cana-2086	302	4	=	=	PUNCT
cana-2086	303	1	𝑖	𝑖	PRON
cana-2086	303	2	−	−	NOUN
cana-2086	303	3	𝑚	𝑚	INTJ
cana-2086	303	4	−	−	PROPN
cana-2086	303	5	1	1	NUM
cana-2086	303	6	.	.	PUNCT
cana-2086	304	1	now	now	ADV
cana-2086	304	2	,	,	PUNCT
cana-2086	304	3	𝑔(𝑣2𝑚+1𝑣1	𝑔(𝑣2𝑚+1𝑣1	PROPN
cana-2086	304	4	)	)	PUNCT
cana-2086	305	1	=	=	PRON
cana-2086	305	2	(	(	PUNCT
cana-2086	305	3	𝑓(𝑣2𝑚+1	𝑓(𝑣2𝑚+1	NOUN
cana-2086	305	4	)	)	PUNCT
cana-2086	306	1	+	+	CCONJ
cana-2086	306	2	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-2086	306	3	)	)	PUNCT
cana-2086	306	4	)	)	PUNCT
cana-2086	306	5	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	306	6	(	(	PUNCT
cana-2086	306	7	2𝑚	2𝑚	NOUN
cana-2086	306	8	+	+	CCONJ
cana-2086	306	9	2	2	X
cana-2086	306	10	)	)	PUNCT
cana-2086	306	11	=	=	SYM
cana-2086	306	12	(	(	PUNCT
cana-2086	306	13	𝑓(𝑣2𝑚+1−2	𝑓(𝑣2𝑚+1−2	X
cana-2086	306	14	)	)	PUNCT
cana-2086	306	15	+	+	CCONJ
cana-2086	306	16	1	1	NUM
cana-2086	306	17	+	+	NUM
cana-2086	306	18	2𝑚	2𝑚	NUM
cana-2086	306	19	+	+	CCONJ
cana-2086	306	20	2	2	NUM
cana-2086	306	21	)	)	PUNCT
cana-2086	306	22	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	306	23	(	(	PUNCT
cana-2086	306	24	2𝑚	2𝑚	NOUN
cana-2086	306	25	+	+	CCONJ
cana-2086	306	26	2	2	X
cana-2086	306	27	)	)	PUNCT
cana-2086	306	28	=	=	NOUN
cana-2086	306	29	(	(	PUNCT
cana-2086	306	30	𝑓(𝑣2𝑚+1−4	𝑓(𝑣2𝑚+1−4	PROPN
cana-2086	306	31	)	)	PUNCT
cana-2086	306	32	+	+	CCONJ
cana-2086	306	33	2	2	NUM
cana-2086	306	34	+	+	NUM
cana-2086	306	35	2𝑚	2𝑚	NOUN
cana-2086	306	36	+	+	CCONJ
cana-2086	306	37	2	2	NUM
cana-2086	306	38	)	)	PUNCT
cana-2086	306	39	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	306	40	(	(	PUNCT
cana-2086	306	41	2𝑚	2𝑚	NOUN
cana-2086	306	42	+	+	CCONJ
cana-2086	306	43	2	2	NUM
cana-2086	306	44	)	)	PUNCT
cana-2086	306	45	⋮	⋮	NOUN
cana-2086	306	46	=	=	SYM
cana-2086	306	47	(	(	PUNCT
cana-2086	306	48	𝑓(𝑣𝑚+4	𝑓(𝑣𝑚+4	X
cana-2086	306	49	)	)	PUNCT
cana-2086	307	1	+	+	CCONJ
cana-2086	307	2	2𝑚+1−𝑚−4	2𝑚+1−𝑚−4	NOUN
cana-2086	307	3	2	2	NUM
cana-2086	307	4	+	+	CCONJ
cana-2086	307	5	2𝑚+	2𝑚+	NUM
cana-2086	307	6	2	2	NUM
cana-2086	307	7	)	)	PUNCT
cana-2086	307	8	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	307	9	(	(	PUNCT
cana-2086	307	10	2𝑚	2𝑚	NOUN
cana-2086	307	11	+	+	CCONJ
cana-2086	307	12	2	2	X
cana-2086	307	13	)	)	PUNCT
cana-2086	307	14	=	=	SYM
cana-2086	307	15	(	(	PUNCT
cana-2086	307	16	𝑓(𝑣𝑚+4−3	𝑓(𝑣𝑚+4−3	NOUN
cana-2086	307	17	)	)	PUNCT
cana-2086	307	18	+	+	CCONJ
cana-2086	307	19	2	2	NUM
cana-2086	307	20	+	+	CCONJ
cana-2086	307	21	𝑚−3	𝑚−3	PROPN
cana-2086	307	22	2	2	NUM
cana-2086	307	23	+	+	NUM
cana-2086	307	24	2𝑚	2𝑚	NOUN
cana-2086	307	25	+	+	CCONJ
cana-2086	307	26	2)𝑚𝑜𝑑	2)𝑚𝑜𝑑	NUM
cana-2086	307	27	(	(	PUNCT
cana-2086	307	28	2𝑚	2𝑚	NOUN
cana-2086	307	29	+	+	CCONJ
cana-2086	307	30	2	2	X
cana-2086	307	31	)	)	PUNCT
cana-2086	307	32	=	=	NOUN
cana-2086	307	33	(	(	PUNCT
cana-2086	307	34	𝑓(𝑣𝑚+1	𝑓(𝑣𝑚+1	PROPN
cana-2086	307	35	)	)	PUNCT
cana-2086	307	36	+	+	CCONJ
cana-2086	307	37	𝑚−3	𝑚−3	NOUN
cana-2086	307	38	2	2	NUM
cana-2086	307	39	+	+	NUM
cana-2086	307	40	2𝑚+	2𝑚+	NUM
cana-2086	307	41	4)𝑚𝑜𝑑(2𝑚	4)𝑚𝑜𝑑(2𝑚	NUM
cana-2086	307	42	+	+	CCONJ
cana-2086	307	43	2	2	X
cana-2086	307	44	)	)	PUNCT
cana-2086	307	45	=	=	SYM
cana-2086	307	46	(	(	PUNCT
cana-2086	307	47	𝑓(𝑣𝑚+1−2	𝑓(𝑣𝑚+1−2	PROPN
cana-2086	307	48	)	)	PUNCT
cana-2086	307	49	+	+	CCONJ
cana-2086	307	50	1	1	NUM
cana-2086	307	51	+	+	CCONJ
cana-2086	307	52	𝑚−3	𝑚−3	PROPN
cana-2086	307	53	2	2	NUM
cana-2086	307	54	+	+	NUM
cana-2086	307	55	2𝑚	2𝑚	NOUN
cana-2086	307	56	+	+	CCONJ
cana-2086	307	57	4)𝑚𝑜𝑑(2𝑚	4)𝑚𝑜𝑑(2𝑚	NUM
cana-2086	307	58	+	+	CCONJ
cana-2086	307	59	2	2	NUM
cana-2086	307	60	)	)	PUNCT
cana-2086	307	61	⋮	⋮	NOUN
cana-2086	307	62	=	=	SYM
cana-2086	307	63	(	(	PUNCT
cana-2086	307	64	𝑓(𝑣𝑚+1−𝑚−1	𝑓(𝑣𝑚+1−𝑚−1	ADJ
cana-2086	307	65	+	+	NOUN
cana-2086	307	66	4	4	NUM
cana-2086	307	67	)	)	PUNCT
cana-2086	307	68	+	+	CCONJ
cana-2086	307	69	𝑚−3	𝑚−3	NOUN
cana-2086	307	70	2	2	NUM
cana-2086	307	71	+	+	NUM
cana-2086	307	72	𝑚−3	𝑚−3	NOUN
cana-2086	307	73	2	2	NUM
cana-2086	307	74	+	+	NUM
cana-2086	307	75	2𝑚	2𝑚	NOUN
cana-2086	307	76	+	+	CCONJ
cana-2086	307	77	4)𝑚𝑜𝑑(2𝑚	4)𝑚𝑜𝑑(2𝑚	NUM
cana-2086	307	78	+	+	CCONJ
cana-2086	307	79	2	2	X
cana-2086	307	80	)	)	PUNCT
cana-2086	307	81	=	=	NOUN
cana-2086	307	82	(	(	PUNCT
cana-2086	307	83	𝑓(𝑣4	𝑓(𝑣4	NOUN
cana-2086	307	84	)	)	PUNCT
cana-2086	307	85	+	+	CCONJ
cana-2086	307	86	𝑚−3	𝑚−3	NOUN
cana-2086	307	87	2	2	NUM
cana-2086	307	88	+	+	CCONJ
cana-2086	307	89	𝑚−3	𝑚−3	NOUN
cana-2086	307	90	2	2	NUM
cana-2086	307	91	+	+	NUM
cana-2086	307	92	2𝑚+	2𝑚+	NUM
cana-2086	307	93	4	4	NUM
cana-2086	307	94	)	)	PUNCT
cana-2086	307	95	𝑚𝑜𝑑(2𝑚	𝑚𝑜𝑑(2𝑚	PROPN
cana-2086	307	96	+	+	CCONJ
cana-2086	307	97	2	2	X
cana-2086	307	98	)	)	PUNCT
cana-2086	307	99	=	=	NOUN
cana-2086	307	100	(	(	PUNCT
cana-2086	307	101	2𝑚	2𝑚	NOUN
cana-2086	307	102	+	+	CCONJ
cana-2086	307	103	3	3	NUM
cana-2086	307	104	+	+	NOUN
cana-2086	307	105	𝑚	𝑚	ADP
cana-2086	307	106	−	−	NUM
cana-2086	307	107	3	3	NUM
cana-2086	307	108	+	+	NOUN
cana-2086	307	109	2𝑚	2𝑚	NOUN
cana-2086	307	110	+	+	CCONJ
cana-2086	307	111	4	4	NUM
cana-2086	307	112	)	)	PUNCT
cana-2086	307	113	𝑚𝑜𝑑(2𝑚	𝑚𝑜𝑑(2𝑚	ADV
cana-2086	307	114	+	+	CCONJ
cana-2086	307	115	2	2	X
cana-2086	307	116	)	)	PUNCT
cana-2086	307	117	communications	communication	NOUN
cana-2086	307	118	on	on	ADP
cana-2086	307	119	applied	apply	VERB
cana-2086	307	120	nonlinear	nonlinear	ADJ
cana-2086	307	121	analysis	analysis	NOUN
cana-2086	307	122	issn	issn	NOUN
cana-2086	307	123	:	:	PUNCT
cana-2086	307	124	1074	1074	NUM
cana-2086	307	125	-	-	PUNCT
cana-2086	307	126	133x	133x	NUM
cana-2086	307	127	vol	vol	NOUN
cana-2086	307	128	32	32	NUM
cana-2086	307	129	no	no	NOUN
cana-2086	307	130	.	.	PUNCT
cana-2086	308	1	1s	1s	NUM
cana-2086	308	2	(	(	PUNCT
cana-2086	308	3	2025	2025	NUM
cana-2086	308	4	)	)	PUNCT
cana-2086	308	5	41	41	NUM
cana-2086	309	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	309	2	=	=	SYM
cana-2086	309	3	(	(	PUNCT
cana-2086	309	4	4𝑚	4𝑚	NOUN
cana-2086	309	5	+	+	CCONJ
cana-2086	309	6	4	4	NUM
cana-2086	309	7	+	+	NUM
cana-2086	309	8	𝑚	𝑚	NOUN
cana-2086	309	9	)	)	PUNCT
cana-2086	309	10	𝑚𝑜𝑑(2𝑚	𝑚𝑜𝑑(2𝑚	PROPN
cana-2086	310	1	+	+	CCONJ
cana-2086	310	2	2	2	X
cana-2086	310	3	)	)	PUNCT
cana-2086	310	4	=	=	VERB
cana-2086	310	5	𝑚.	𝑚.	ADV
cana-2086	310	6	hence	hence	ADV
cana-2086	310	7	,	,	PUNCT
cana-2086	310	8	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	310	9	)	)	PUNCT
cana-2086	310	10	=	=	NOUN
cana-2086	310	11	{	{	PUNCT
cana-2086	310	12	𝑚	𝑚	X
cana-2086	310	13	+	+	NOUN
cana-2086	310	14	1	1	NUM
cana-2086	310	15	𝑖	𝑖	NOUN
cana-2086	310	16	=	=	NOUN
cana-2086	310	17	1	1	NUM
cana-2086	310	18	1	1	NUM
cana-2086	310	19	𝑖	𝑖	NOUN
cana-2086	310	20	=	=	NOUN
cana-2086	310	21	2	2	NUM
cana-2086	310	22	𝑚	𝑚	NOUN
cana-2086	310	23	+	+	NOUN
cana-2086	310	24	3	3	NUM
cana-2086	310	25	𝑖	𝑖	NOUN
cana-2086	310	26	=	=	NOUN
cana-2086	310	27	3	3	NUM
cana-2086	310	28	𝑚	𝑚	NOUN
cana-2086	310	29	+	+	NOUN
cana-2086	310	30	𝑖	𝑖	SYM
cana-2086	310	31	4	4	NUM
cana-2086	310	32	≤	≤	NUM
cana-2086	310	33	𝑖	𝑖	SYM
cana-2086	310	34	≤	≤	NOUN
cana-2086	310	35	𝑚	𝑚	ADP
cana-2086	310	36	+	+	PROPN
cana-2086	310	37	1	1	NUM
cana-2086	310	38	𝑚	𝑚	NOUN
cana-2086	310	39	+	+	NOUN
cana-2086	310	40	2	2	NUM
cana-2086	310	41	𝑖	𝑖	NOUN
cana-2086	310	42	=	=	PUNCT
cana-2086	310	43	𝑚	𝑚	PROPN
cana-2086	310	44	+	+	NOUN
cana-2086	310	45	2	2	NUM
cana-2086	310	46	2	2	NUM
cana-2086	310	47	𝑖	𝑖	NOUN
cana-2086	310	48	=	=	PUNCT
cana-2086	310	49	𝑚	𝑚	PROPN
cana-2086	310	50	+	+	NOUN
cana-2086	310	51	3	3	NUM
cana-2086	310	52	𝑖	𝑖	NOUN
cana-2086	310	53	−𝑚	−𝑚	NOUN
cana-2086	310	54	−	−	NOUN
cana-2086	310	55	1	1	NUM
cana-2086	310	56	𝑚	𝑚	PRON
cana-2086	310	57	+	+	CCONJ
cana-2086	310	58	4	4	NUM
cana-2086	310	59	≤	≤	NUM
cana-2086	310	60	𝑖	𝑖	PRON
cana-2086	310	61	≤	≤	NOUN
cana-2086	310	62	2𝑚.	2𝑚.	NUM
cana-2086	310	63	𝑔(𝑣2𝑚+1𝑣1	𝑔(𝑣2𝑚+1𝑣1	NUM
cana-2086	310	64	)	)	PUNCT
cana-2086	311	1	=	=	PUNCT
cana-2086	311	2	𝑚	𝑚	PRON
cana-2086	311	3	case	case	NOUN
cana-2086	311	4	2	2	NUM
cana-2086	311	5	:	:	PUNCT
cana-2086	311	6	for	for	ADP
cana-2086	311	7	𝐶4𝑚	𝐶4𝑚	PROPN
cana-2086	311	8	,	,	PUNCT
cana-2086	311	9	𝑚	𝑚	X
cana-2086	311	10	=	=	SYM
cana-2086	311	11	1	1	NUM
cana-2086	311	12	,	,	PUNCT
cana-2086	311	13	2	2	NUM
cana-2086	311	14	,	,	PUNCT
cana-2086	311	15	3	3	NUM
cana-2086	311	16	,	,	PUNCT
cana-2086	311	17	…	…	PUNCT
cana-2086	311	18	,	,	PUNCT
cana-2086	311	19	𝑓	𝑓	X
cana-2086	311	20	:	:	PUNCT
cana-2086	311	21	𝑉(𝐶4𝑚	𝑉(𝐶4𝑚	NUM
cana-2086	311	22	)	)	PUNCT
cana-2086	311	23	→	→	SYM
cana-2086	311	24	{	{	PUNCT
cana-2086	311	25	4𝑚	4𝑚	NOUN
cana-2086	311	26	+	+	CCONJ
cana-2086	311	27	1	1	NUM
cana-2086	311	28	,	,	PUNCT
cana-2086	311	29	4𝑚	4𝑚	NOUN
cana-2086	311	30	+	+	CCONJ
cana-2086	311	31	2	2	NUM
cana-2086	311	32	,	,	PUNCT
cana-2086	311	33	4𝑚	4𝑚	NOUN
cana-2086	311	34	+	+	CCONJ
cana-2086	311	35	3	3	NUM
cana-2086	311	36	,	,	PUNCT
cana-2086	311	37	…	…	PUNCT
cana-2086	311	38	,	,	PUNCT
cana-2086	311	39	8𝑚	8𝑚	NUM
cana-2086	311	40	+	+	CCONJ
cana-2086	311	41	1	1	X
cana-2086	311	42	}	}	PUNCT
cana-2086	311	43	is	be	AUX
cana-2086	311	44	defined	define	VERB
cana-2086	311	45	as	as	ADP
cana-2086	311	46	,	,	PUNCT
cana-2086	311	47	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	311	48	)	)	PUNCT
cana-2086	311	49	=	=	SYM
cana-2086	311	50	{	{	PUNCT
cana-2086	311	51	4𝑚	4𝑚	NOUN
cana-2086	312	1	+	+	CCONJ
cana-2086	312	2	2	2	NUM
cana-2086	312	3	𝑖	𝑖	SYM
cana-2086	312	4	=	=	NOUN
cana-2086	312	5	1	1	NUM
cana-2086	312	6	8𝑚	8𝑚	NOUN
cana-2086	312	7	+	+	CCONJ
cana-2086	312	8	𝑖	𝑖	SYM
cana-2086	312	9	−	−	NOUN
cana-2086	312	10	2	2	NUM
cana-2086	312	11	2	2	NUM
cana-2086	312	12	≤	≤	NOUN
cana-2086	312	13	𝑖	𝑖	SYM
cana-2086	312	14	≤	≤	NUM
cana-2086	312	15	3	3	NUM
cana-2086	312	16	𝑓(𝑣𝑖−2	𝑓(𝑣𝑖−2	NUM
cana-2086	312	17	)	)	PUNCT
cana-2086	312	18	−	−	ADP
cana-2086	312	19	2	2	NUM
cana-2086	312	20	4	4	NUM
cana-2086	312	21	≤	≤	NOUN
cana-2086	312	22	𝑖	𝑖	PRON
cana-2086	312	23	≤	≤	NOUN
cana-2086	312	24	2𝑚	2𝑚	NOUN
cana-2086	312	25	+	+	CCONJ
cana-2086	312	26	1	1	NUM
cana-2086	312	27	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	312	28	)	)	PUNCT
cana-2086	312	29	−	−	ADP
cana-2086	313	1	2	2	NUM
cana-2086	313	2	𝑖	𝑖	NOUN
cana-2086	313	3	=	=	NOUN
cana-2086	313	4	2𝑚	2𝑚	NOUN
cana-2086	313	5	+	+	CCONJ
cana-2086	313	6	2	2	NUM
cana-2086	313	7	𝑓(𝑣𝑖−1	𝑓(𝑣𝑖−1	NUM
cana-2086	313	8	)	)	PUNCT
cana-2086	313	9	−	−	ADP
cana-2086	313	10	1	1	NUM
cana-2086	313	11	2𝑚	2𝑚	NOUN
cana-2086	313	12	+	+	CCONJ
cana-2086	313	13	3	3	NUM
cana-2086	313	14	≤	≤	NUM
cana-2086	313	15	𝑖	𝑖	PRON
cana-2086	313	16	≤	≤	NOUN
cana-2086	313	17	4𝑚.	4𝑚.	NUM
cana-2086	313	18	following	follow	VERB
cana-2086	313	19	a	a	DET
cana-2086	313	20	similar	similar	ADJ
cana-2086	313	21	approach	approach	NOUN
cana-2086	313	22	as	as	ADP
cana-2086	313	23	in	in	ADP
cana-2086	313	24	case	case	NOUN
cana-2086	313	25	1	1	NUM
cana-2086	313	26	,	,	PUNCT
cana-2086	313	27	we	we	PRON
cana-2086	313	28	obtain	obtain	VERB
cana-2086	313	29	,	,	PUNCT
cana-2086	313	30	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	313	31	)	)	PUNCT
cana-2086	313	32	=	=	SYM
cana-2086	313	33	{	{	PUNCT
cana-2086	313	34	4𝑚	4𝑚	NOUN
cana-2086	313	35	𝑖	𝑖	SYM
cana-2086	314	1	=	=	SYM
cana-2086	314	2	1	1	NUM
cana-2086	314	3	4𝑚	4𝑚	NUM
cana-2086	314	4	−	−	NOUN
cana-2086	314	5	2	2	NUM
cana-2086	314	6	𝑖	𝑖	NOUN
cana-2086	314	7	=	=	SYM
cana-2086	314	8	2	2	NUM
cana-2086	314	9	4𝑚	4𝑚	NOUN
cana-2086	315	1	+	+	CCONJ
cana-2086	315	2	2	2	NUM
cana-2086	315	3	−	−	NOUN
cana-2086	315	4	2𝑖	2𝑖	NOUN
cana-2086	315	5	3	3	NUM
cana-2086	315	6	≤	≤	NOUN
cana-2086	315	7	𝑖	𝑖	PRON
cana-2086	315	8	≤	≤	NOUN
cana-2086	315	9	2𝑚	2𝑚	NOUN
cana-2086	316	1	1	1	NUM
cana-2086	316	2	𝑖	𝑖	NOUN
cana-2086	316	3	=	=	NOUN
cana-2086	316	4	2𝑚	2𝑚	NOUN
cana-2086	316	5	+	+	CCONJ
cana-2086	316	6	1	1	NUM
cana-2086	316	7	8𝑚	8𝑚	NUM
cana-2086	316	8	+	+	CCONJ
cana-2086	317	1	3	3	NUM
cana-2086	317	2	−	−	NOUN
cana-2086	317	3	2𝑖	2𝑖	NOUN
cana-2086	317	4	2𝑚	2𝑚	NOUN
cana-2086	317	5	+	+	CCONJ
cana-2086	317	6	2	2	NUM
cana-2086	317	7	≤	≤	NUM
cana-2086	317	8	𝑖	𝑖	PRON
cana-2086	317	9	≤	≤	NOUN
cana-2086	317	10	4𝑚	4𝑚	NOUN
cana-2086	317	11	−	−	NOUN
cana-2086	318	1	1	1	X
cana-2086	318	2	.	.	PUNCT
cana-2086	318	3	𝑔(𝑣4𝑚𝑣1	𝑔(𝑣4𝑚𝑣1	X
cana-2086	318	4	)	)	PUNCT
cana-2086	319	1	=	=	SYM
cana-2086	319	2	3	3	X
cana-2086	319	3	.	.	PUNCT
cana-2086	320	1	now	now	ADV
cana-2086	320	2	we	we	PRON
cana-2086	320	3	have	have	VERB
cana-2086	320	4	ℎ	ℎ	X
cana-2086	320	5	=	=	SYM
cana-2086	320	6	1	1	NUM
cana-2086	320	7	2𝑛+1	2𝑛+1	NUM
cana-2086	320	8	.	.	PUNCT
cana-2086	321	1	the	the	DET
cana-2086	321	2	vertex	vertex	NOUN
cana-2086	321	3	labeling	labeling	NOUN
cana-2086	321	4	𝜇	𝜇	ADP
cana-2086	321	5	:	:	PUNCT
cana-2086	321	6	𝑉(𝐶𝑛	𝑉(𝐶𝑛	NUM
cana-2086	321	7	)	)	PUNCT
cana-2086	321	8	→	→	PUNCT
cana-2086	322	1	[	[	X
cana-2086	322	2	0,1	0,1	NUM
cana-2086	322	3	]	]	PUNCT
cana-2086	322	4	is	be	AUX
cana-2086	322	5	defined	define	VERB
cana-2086	322	6	as	as	ADP
cana-2086	322	7	,	,	PUNCT
cana-2086	322	8	𝜇(𝑣𝑖	𝜇(𝑣𝑖	NOUN
cana-2086	322	9	)	)	PUNCT
cana-2086	322	10	=	=	PUNCT
cana-2086	322	11	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	322	12	)	)	PUNCT
cana-2086	322	13	.	.	PUNCT
cana-2086	323	1	1	1	NUM
cana-2086	323	2	2𝑛+1	2𝑛+1	NUM
cana-2086	323	3	for	for	ADP
cana-2086	323	4	every	every	DET
cana-2086	323	5	1	1	NUM
cana-2086	323	6	≤	≤	NUM
cana-2086	323	7	𝑖	𝑖	SYM
cana-2086	323	8	≤	≤	NOUN
cana-2086	323	9	𝑛	𝑛	PRON
cana-2086	323	10	and	and	CCONJ
cana-2086	323	11	the	the	DET
cana-2086	323	12	edge	edge	NOUN
cana-2086	323	13	labeling	labeling	NOUN
cana-2086	323	14	𝜌	𝜌	ADP
cana-2086	323	15	:	:	PUNCT
cana-2086	323	16	𝐸(𝐶𝑛	𝐸(𝐶𝑛	NUM
cana-2086	323	17	)	)	PUNCT
cana-2086	323	18	→	→	PUNCT
cana-2086	324	1	[	[	X
cana-2086	324	2	0,1	0,1	NUM
cana-2086	324	3	]	]	PUNCT
cana-2086	324	4	is	be	AUX
cana-2086	324	5	defined	define	VERB
cana-2086	324	6	as	as	ADP
cana-2086	324	7	,	,	PUNCT
cana-2086	324	8	𝜌(𝑣𝑖𝑣𝑖+1	𝜌(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	324	9	)	)	PUNCT
cana-2086	324	10	=	=	SYM
cana-2086	324	11	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	324	12	)	)	PUNCT
cana-2086	324	13	.	.	PUNCT
cana-2086	325	1	1	1	NUM
cana-2086	325	2	2𝑛+1	2𝑛+1	NUM
cana-2086	325	3	for	for	ADP
cana-2086	325	4	every	every	DET
cana-2086	325	5	1	1	NUM
cana-2086	325	6	≤	≤	NUM
cana-2086	325	7	𝑖	𝑖	SYM
cana-2086	325	8	≤	≤	NUM
cana-2086	325	9	𝑛	𝑛	PRON
cana-2086	325	10	−	−	PROPN
cana-2086	325	11	1	1	NUM
cana-2086	325	12	and	and	CCONJ
cana-2086	325	13	𝜌(𝑣𝑛𝑣1	𝜌(𝑣𝑛𝑣1	NUM
cana-2086	325	14	)	)	PUNCT
cana-2086	325	15	=	=	SYM
cana-2086	326	1	𝑔(𝑣𝑛𝑣1	𝑔(𝑣𝑛𝑣1	NOUN
cana-2086	326	2	)	)	PUNCT
cana-2086	326	3	.	.	PUNCT
cana-2086	327	1	1	1	NUM
cana-2086	327	2	2𝑛+1	2𝑛+1	NUM
cana-2086	327	3	.	.	PUNCT
cana-2086	328	1	it	it	PRON
cana-2086	328	2	can	can	AUX
cana-2086	328	3	be	be	AUX
cana-2086	328	4	verified	verify	VERB
cana-2086	328	5	that	that	SCONJ
cana-2086	328	6	in	in	ADP
cana-2086	328	7	all	all	DET
cana-2086	328	8	the	the	DET
cana-2086	328	9	cases	case	NOUN
cana-2086	328	10	the	the	DET
cana-2086	328	11	elements	element	NOUN
cana-2086	328	12	of	of	ADP
cana-2086	328	13	the	the	DET
cana-2086	328	14	set	set	NOUN
cana-2086	328	15	{	{	PUNCT
cana-2086	328	16	𝑔(𝑣𝑛𝑣1	𝑔(𝑣𝑛𝑣1	NOUN
cana-2086	328	17	)	)	PUNCT
cana-2086	328	18	,	,	PUNCT
cana-2086	328	19	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	328	20	)	)	PUNCT
cana-2086	328	21	,	,	PUNCT
cana-2086	328	22	1	1	NUM
cana-2086	328	23	≤	≤	NUM
cana-2086	328	24	𝑖	𝑖	SYM
cana-2086	328	25	≤	≤	NUM
cana-2086	329	1	𝑛	𝑛	PRON
cana-2086	329	2	−	−	NUM
cana-2086	329	3	1	1	NUM
cana-2086	329	4	}	}	PUNCT
cana-2086	329	5	are	be	AUX
cana-2086	329	6	distinct	distinct	ADJ
cana-2086	329	7	and	and	CCONJ
cana-2086	329	8	nonzero	nonzero	NOUN
cana-2086	329	9	.	.	PUNCT
cana-2086	330	1	therefore	therefore	ADV
cana-2086	330	2	,	,	PUNCT
cana-2086	330	3	the	the	DET
cana-2086	330	4	edge	edge	NOUN
cana-2086	330	5	labels	label	NOUN
cana-2086	330	6	are	be	AUX
cana-2086	330	7	distinct	distinct	ADJ
cana-2086	330	8	and	and	CCONJ
cana-2086	330	9	non	non	ADJ
cana-2086	330	10	zero	zero	NUM
cana-2086	330	11	.	.	PUNCT
cana-2086	331	1	further	further	PROPN
cana-2086	331	2	max	max	PROPN
cana-2086	331	3	{	{	PUNCT
cana-2086	331	4	𝑔(𝑣𝑛𝑣1	𝑔(𝑣𝑛𝑣1	NOUN
cana-2086	331	5	)	)	PUNCT
cana-2086	331	6	,	,	PUNCT
cana-2086	331	7	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	PROPN
cana-2086	331	8	)	)	PUNCT
cana-2086	331	9	,	,	PUNCT
cana-2086	331	10	1	1	NUM
cana-2086	331	11	≤	≤	NUM
cana-2086	331	12	𝑖	𝑖	SYM
cana-2086	331	13	≤	≤	NUM
cana-2086	331	14	𝑛	𝑛	DET
cana-2086	331	15	−	−	PROPN
cana-2086	331	16	1	1	NUM
cana-2086	331	17	}	}	PUNCT
cana-2086	331	18	=	=	SYM
cana-2086	331	19	𝑛	𝑛	PROPN
cana-2086	331	20	and	and	CCONJ
cana-2086	331	21	min	min	ADJ
cana-2086	331	22	{	{	PUNCT
cana-2086	331	23	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	331	24	)	)	PUNCT
cana-2086	331	25	}	}	PUNCT
cana-2086	331	26	=	=	SYM
cana-2086	331	27	𝑛	𝑛	PROPN
cana-2086	331	28	+	+	NOUN
cana-2086	331	29	1	1	X
cana-2086	331	30	.	.	PUNCT
cana-2086	331	31	therefore	therefore	ADV
cana-2086	331	32	,	,	PUNCT
cana-2086	331	33	𝑔(𝑣𝑛𝑣1	𝑔(𝑣𝑛𝑣1	NOUN
cana-2086	331	34	)	)	PUNCT
cana-2086	331	35	<	<	X
cana-2086	331	36	𝑓(𝑣𝑛	𝑓(𝑣𝑛	X
cana-2086	331	37	)	)	PUNCT
cana-2086	331	38	∧	∧	PROPN
cana-2086	331	39	𝑓(𝑣1	𝑓(𝑣1	NOUN
cana-2086	331	40	)	)	PUNCT
cana-2086	331	41	and	and	CCONJ
cana-2086	331	42	𝑔(𝑣𝑖𝑣𝑖+1	𝑔(𝑣𝑖𝑣𝑖+1	NOUN
cana-2086	331	43	)	)	PUNCT
cana-2086	331	44	<	<	X
cana-2086	331	45	𝑓(𝑣𝑖	𝑓(𝑣𝑖	NOUN
cana-2086	331	46	)	)	PUNCT
cana-2086	331	47	∧	∧	PROPN
cana-2086	331	48	𝑓(𝑣𝑖+1	𝑓(𝑣𝑖+1	NOUN
cana-2086	331	49	)	)	PUNCT
cana-2086	331	50	for	for	ADP
cana-2086	331	51	every	every	DET
cana-2086	331	52	1	1	NUM
cana-2086	331	53	≤	≤	NUM
cana-2086	331	54	𝑖	𝑖	SYM
cana-2086	331	55	≤	≤	NUM
cana-2086	331	56	𝑛	𝑛	PRON
cana-2086	331	57	−	−	NOUN
cana-2086	331	58	1	1	NUM
cana-2086	331	59	.	.	PUNCT
cana-2086	332	1	hence	hence	ADV
cana-2086	332	2	the	the	DET
cana-2086	332	3	edge	edge	NOUN
cana-2086	332	4	labels	label	NOUN
cana-2086	332	5	are	be	AUX
cana-2086	332	6	less	less	ADJ
cana-2086	332	7	than	than	ADP
cana-2086	332	8	the	the	DET
cana-2086	332	9	minimum	minimum	NOUN
cana-2086	332	10	of	of	ADP
cana-2086	332	11	their	their	PRON
cana-2086	332	12	respective	respective	ADJ
cana-2086	332	13	endpoints	endpoint	NOUN
cana-2086	332	14	labels	label	NOUN
cana-2086	332	15	.	.	PUNCT
cana-2086	333	1	consequently	consequently	ADV
cana-2086	333	2	,	,	PUNCT
cana-2086	333	3	𝐶𝑛	𝐶𝑛	PROPN
cana-2086	333	4	for	for	ADP
cana-2086	333	5	𝑛	𝑛	DET
cana-2086	333	6	≡	≡	PROPN
cana-2086	333	7	0,3	0,3	NUM
cana-2086	333	8	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	333	9	4	4	NUM
cana-2086	333	10	are	be	AUX
cana-2086	333	11	elegant	elegant	ADJ
cana-2086	333	12	fuzzy	fuzzy	ADJ
cana-2086	333	13	labeling	labeling	NOUN
cana-2086	333	14	graphs	graph	NOUN
cana-2086	333	15	and	and	CCONJ
cana-2086	333	16	(	(	PUNCT
cana-2086	333	17	𝜇	𝜇	ADP
cana-2086	333	18	,	,	PUNCT
cana-2086	333	19	𝜌	𝜌	X
cana-2086	333	20	)	)	PUNCT
cana-2086	333	21	is	be	AUX
cana-2086	333	22	an	an	DET
cana-2086	333	23	elegant	elegant	ADJ
cana-2086	333	24	fuzzy	fuzzy	ADJ
cana-2086	333	25	labeling	labeling	NOUN
cana-2086	333	26	of	of	ADP
cana-2086	333	27	𝐶𝑛	𝐶𝑛	PROPN
cana-2086	333	28	for	for	ADP
cana-2086	333	29	𝑛	𝑛	DET
cana-2086	333	30	≡	≡	PROPN
cana-2086	333	31	0,3	0,3	NUM
cana-2086	333	32	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	333	33	4	4	NUM
cana-2086	333	34	.	.	PUNCT
cana-2086	333	35	□	□	PUNCT
cana-2086	333	36	remark	remark	NOUN
cana-2086	333	37	3.2.5	3.2.5	NUM
cana-2086	333	38	.	.	PUNCT
cana-2086	334	1	cycles	cycle	NOUN
cana-2086	335	1	𝐶𝑛	𝐶𝑛	PROPN
cana-2086	335	2	where	where	SCONJ
cana-2086	335	3	𝑛	𝑛	DET
cana-2086	335	4	≡	≡	PROPN
cana-2086	335	5	1	1	NUM
cana-2086	335	6	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	335	7	4	4	NUM
cana-2086	335	8	are	be	AUX
cana-2086	335	9	not	not	PART
cana-2086	335	10	elegant	elegant	ADJ
cana-2086	335	11	[	[	X
cana-2086	335	12	3	3	NUM
cana-2086	335	13	]	]	PUNCT
cana-2086	335	14	,	,	PUNCT
cana-2086	335	15	therefore	therefore	ADV
cana-2086	335	16	it	it	PRON
cana-2086	335	17	does	do	AUX
cana-2086	335	18	not	not	PART
cana-2086	335	19	admit	admit	VERB
cana-2086	335	20	elegant	elegant	ADJ
cana-2086	335	21	fuzzy	fuzzy	ADJ
cana-2086	335	22	labeling	labeling	NOUN
cana-2086	335	23	.	.	PUNCT
cana-2086	336	1	communications	communication	NOUN
cana-2086	336	2	on	on	ADP
cana-2086	336	3	applied	apply	VERB
cana-2086	336	4	nonlinear	nonlinear	ADJ
cana-2086	336	5	analysis	analysis	NOUN
cana-2086	336	6	issn	issn	NOUN
cana-2086	336	7	:	:	PUNCT
cana-2086	336	8	1074	1074	NUM
cana-2086	336	9	-	-	PUNCT
cana-2086	336	10	133x	133x	NUM
cana-2086	336	11	vol	vol	NOUN
cana-2086	336	12	32	32	NUM
cana-2086	336	13	no	no	NOUN
cana-2086	336	14	.	.	PUNCT
cana-2086	337	1	1s	1s	NUM
cana-2086	337	2	(	(	PUNCT
cana-2086	337	3	2025	2025	NUM
cana-2086	337	4	)	)	PUNCT
cana-2086	337	5	42	42	NUM
cana-2086	337	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2086	337	7	note	note	NOUN
cana-2086	337	8	.	.	PUNCT
cana-2086	338	1	the	the	DET
cana-2086	338	2	case	case	NOUN
cana-2086	338	3	for	for	ADP
cana-2086	338	4	cycles	cycle	NOUN
cana-2086	338	5	𝐶𝑛	𝐶𝑛	PROPN
cana-2086	338	6	where	where	SCONJ
cana-2086	338	7	𝑛	𝑛	DET
cana-2086	338	8	≡	≡	PROPN
cana-2086	338	9	2	2	NUM
cana-2086	338	10	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	338	11	4	4	NUM
cana-2086	338	12	remains	remain	VERB
cana-2086	338	13	an	an	DET
cana-2086	338	14	open	open	ADJ
cana-2086	338	15	question	question	NOUN
cana-2086	338	16	.	.	PUNCT
cana-2086	339	1	future	future	ADJ
cana-2086	339	2	work	work	NOUN
cana-2086	339	3	could	could	AUX
cana-2086	339	4	investigate	investigate	VERB
cana-2086	339	5	whether	whether	SCONJ
cana-2086	339	6	this	this	DET
cana-2086	339	7	class	class	NOUN
cana-2086	339	8	of	of	ADP
cana-2086	339	9	cycles	cycle	NOUN
cana-2086	339	10	can	can	AUX
cana-2086	339	11	admit	admit	VERB
cana-2086	339	12	elegant	elegant	ADJ
cana-2086	339	13	fuzzy	fuzzy	ADJ
cana-2086	339	14	labeling	labeling	NOUN
cana-2086	339	15	or	or	CCONJ
cana-2086	339	16	if	if	SCONJ
cana-2086	339	17	modifications	modification	NOUN
cana-2086	339	18	to	to	ADP
cana-2086	339	19	the	the	DET
cana-2086	339	20	labeling	labeling	NOUN
cana-2086	339	21	scheme	scheme	NOUN
cana-2086	339	22	are	be	AUX
cana-2086	339	23	required	require	VERB
cana-2086	339	24	to	to	PART
cana-2086	339	25	accommodate	accommodate	VERB
cana-2086	339	26	these	these	DET
cana-2086	339	27	structures	structure	NOUN
cana-2086	339	28	.	.	PUNCT
cana-2086	340	1	theorem	theorem	VERB
cana-2086	340	2	3.2.6	3.2.6	NUM
cana-2086	340	3	.	.	PUNCT
cana-2086	340	4	line	line	NOUN
cana-2086	340	5	graph	graph	NOUN
cana-2086	340	6	of	of	ADP
cana-2086	340	7	cycles	cycle	NOUN
cana-2086	340	8	𝐿(𝐶𝑛	𝐿(𝐶𝑛	PRON
cana-2086	340	9	)	)	PUNCT
cana-2086	340	10	admit	admit	VERB
cana-2086	340	11	elegant	elegant	ADJ
cana-2086	340	12	fuzzy	fuzzy	ADJ
cana-2086	340	13	labeling	labeling	NOUN
cana-2086	340	14	when	when	SCONJ
cana-2086	340	15	𝑛	𝑛	DET
cana-2086	340	16	≡	≡	PROPN
cana-2086	340	17	0,3	0,3	NUM
cana-2086	340	18	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	340	19	4	4	NUM
cana-2086	340	20	.	.	PUNCT
cana-2086	341	1	proof	proof	NOUN
cana-2086	341	2	.	.	PUNCT
cana-2086	342	1	by	by	ADP
cana-2086	342	2	theorem	theorem	NOUN
cana-2086	342	3	3.2.4	3.2.4	NUM
cana-2086	342	4	,	,	PUNCT
cana-2086	342	5	𝐶𝑛	𝐶𝑛	PROPN
cana-2086	342	6	admit	admit	VERB
cana-2086	342	7	elegant	elegant	ADJ
cana-2086	342	8	fuzzy	fuzzy	ADJ
cana-2086	342	9	labeling	labeling	NOUN
cana-2086	342	10	when	when	SCONJ
cana-2086	342	11	𝑛	𝑛	DET
cana-2086	342	12	≡	≡	PROPN
cana-2086	342	13	0,3	0,3	NUM
cana-2086	342	14	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	342	15	4	4	NUM
cana-2086	342	16	.	.	PUNCT
cana-2086	343	1	𝐿(𝐶𝑛	𝐿(𝐶𝑛	PRON
cana-2086	343	2	)	)	PUNCT
cana-2086	343	3	≡	≡	PROPN
cana-2086	343	4	𝐶𝑛.	𝐶𝑛.	PROPN
cana-2086	343	5	therefore	therefore	ADV
cana-2086	343	6	𝐿(𝐶𝑛	𝐿(𝐶𝑛	PRON
cana-2086	343	7	)	)	PUNCT
cana-2086	343	8	admit	admit	VERB
cana-2086	343	9	elegant	elegant	ADJ
cana-2086	343	10	fuzzy	fuzzy	ADJ
cana-2086	343	11	labeling	labeling	NOUN
cana-2086	343	12	when	when	SCONJ
cana-2086	343	13	𝑛	𝑛	DET
cana-2086	343	14	≡	≡	PROPN
cana-2086	343	15	0,3	0,3	NUM
cana-2086	343	16	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-2086	343	17	4	4	NUM
cana-2086	343	18	.	.	PUNCT
cana-2086	344	1	□	□	PUNCT
cana-2086	344	2	example	example	NOUN
cana-2086	344	3	3.2.7	3.2.7	NUM
cana-2086	344	4	.	.	PUNCT
cana-2086	344	5	consider	consider	VERB
cana-2086	344	6	a	a	DET
cana-2086	344	7	cycle	cycle	NOUN
cana-2086	344	8	𝐶8	𝐶8	NOUN
cana-2086	344	9	in	in	ADP
cana-2086	344	10	fig.6	fig.6	PROPN
cana-2086	344	11	.	.	PUNCT
cana-2086	345	1	by	by	ADP
cana-2086	345	2	theorem	theorem	NOUN
cana-2086	345	3	3.2.4	3.2.4	NUM
cana-2086	345	4	,	,	PUNCT
cana-2086	345	5	the	the	DET
cana-2086	345	6	function	function	NOUN
cana-2086	345	7	𝑓	𝑓	NOUN
cana-2086	345	8	:	:	PUNCT
cana-2086	345	9	𝑉(𝐶8	𝑉(𝐶8	NUM
cana-2086	345	10	)	)	PUNCT
cana-2086	345	11	→	→	SYM
cana-2086	345	12	{	{	PUNCT
cana-2086	345	13	9	9	NUM
cana-2086	345	14	,	,	PUNCT
cana-2086	345	15	10	10	NUM
cana-2086	345	16	,	,	PUNCT
cana-2086	345	17	11	11	NUM
cana-2086	345	18	,	,	PUNCT
cana-2086	345	19	12	12	NUM
cana-2086	345	20	,	,	PUNCT
cana-2086	345	21	13,14	13,14	NUM
cana-2086	345	22	,	,	PUNCT
cana-2086	345	23	15	15	NUM
cana-2086	345	24	,	,	PUNCT
cana-2086	345	25	16	16	NUM
cana-2086	345	26	,	,	PUNCT
cana-2086	345	27	17	17	NUM
cana-2086	345	28	}	}	PUNCT
cana-2086	345	29	is	be	AUX
cana-2086	345	30	defined	define	VERB
cana-2086	345	31	as	as	SCONJ
cana-2086	345	32	follows	follow	VERB
cana-2086	345	33	:	:	PUNCT
cana-2086	345	34	𝑓(𝑣1	𝑓(𝑣1	ADJ
cana-2086	345	35	)	)	PUNCT
cana-2086	345	36	=	=	SYM
cana-2086	345	37	10	10	NUM
cana-2086	345	38	,	,	PUNCT
cana-2086	345	39	𝑓(𝑣2	𝑓(𝑣2	NOUN
cana-2086	345	40	)	)	PUNCT
cana-2086	345	41	=	=	SYM
cana-2086	345	42	16	16	NUM
cana-2086	345	43	,	,	PUNCT
cana-2086	345	44	𝑓(𝑣3	𝑓(𝑣3	ADJ
cana-2086	345	45	)	)	PUNCT
cana-2086	345	46	=	=	SYM
cana-2086	345	47	17	17	NUM
cana-2086	345	48	,	,	PUNCT
cana-2086	345	49	𝑓(𝑣4	𝑓(𝑣4	NOUN
cana-2086	345	50	)	)	PUNCT
cana-2086	345	51	=	=	SYM
cana-2086	345	52	14	14	NUM
cana-2086	345	53	,	,	PUNCT
cana-2086	345	54	𝑓(𝑣5	𝑓(𝑣5	NOUN
cana-2086	345	55	)	)	PUNCT
cana-2086	346	1	=	=	SYM
cana-2086	346	2	15	15	NUM
cana-2086	346	3	,	,	PUNCT
cana-2086	346	4	𝑓(𝑣6	𝑓(𝑣6	ADJ
cana-2086	346	5	)	)	PUNCT
cana-2086	346	6	=	=	SYM
cana-2086	346	7	13	13	NUM
cana-2086	346	8	,	,	PUNCT
cana-2086	346	9	𝑓(𝑣7	𝑓(𝑣7	NOUN
cana-2086	346	10	)	)	PUNCT
cana-2086	346	11	=	=	SYM
cana-2086	346	12	12	12	NUM
cana-2086	346	13	,	,	PUNCT
cana-2086	346	14	𝑓(𝑣8	𝑓(𝑣8	X
cana-2086	346	15	)	)	PUNCT
cana-2086	346	16	=	=	SYM
cana-2086	347	1	11	11	NUM
cana-2086	347	2	.	.	PUNCT
cana-2086	348	1	consequently	consequently	ADV
cana-2086	348	2	,	,	PUNCT
cana-2086	348	3	𝑔(𝑣1𝑣2	𝑔(𝑣1𝑣2	PROPN
cana-2086	348	4	)	)	PUNCT
cana-2086	348	5	=	=	SYM
cana-2086	348	6	8	8	NUM
cana-2086	348	7	,	,	PUNCT
cana-2086	348	8	𝑔(𝑣2𝑣3	𝑔(𝑣2𝑣3	NOUN
cana-2086	348	9	)	)	PUNCT
cana-2086	348	10	=	=	SYM
cana-2086	348	11	6	6	NUM
cana-2086	348	12	,	,	PUNCT
cana-2086	348	13	𝑔(𝑣3𝑣4	𝑔(𝑣3𝑣4	ADJ
cana-2086	348	14	)	)	PUNCT
cana-2086	348	15	=	=	SYM
cana-2086	348	16	4	4	NUM
cana-2086	348	17	,	,	PUNCT
cana-2086	348	18	𝑔(𝑣4𝑣5	𝑔(𝑣4𝑣5	NOUN
cana-2086	348	19	)	)	PUNCT
cana-2086	348	20	=	=	SYM
cana-2086	348	21	2	2	NUM
cana-2086	348	22	,	,	PUNCT
cana-2086	348	23	𝑔(𝑣5𝑣6	𝑔(𝑣5𝑣6	NOUN
cana-2086	348	24	)	)	PUNCT
cana-2086	348	25	=	=	SYM
cana-2086	349	1	1	1	NUM
cana-2086	349	2	,	,	PUNCT
cana-2086	349	3	𝑔(𝑣6𝑣7	𝑔(𝑣6𝑣7	NOUN
cana-2086	349	4	)	)	PUNCT
cana-2086	349	5	=	=	SYM
cana-2086	349	6	7	7	NUM
cana-2086	349	7	,	,	PUNCT
cana-2086	349	8	𝑔(𝑣7𝑣8	𝑔(𝑣7𝑣8	NOUN
cana-2086	349	9	)	)	PUNCT
cana-2086	349	10	=	=	SYM
cana-2086	349	11	5	5	NUM
cana-2086	349	12	,	,	PUNCT
cana-2086	349	13	𝑔(𝑣8𝑣1	𝑔(𝑣8𝑣1	CCONJ
cana-2086	349	14	)	)	PUNCT
cana-2086	349	15	=	=	SYM
cana-2086	350	1	3	3	X
cana-2086	350	2	.	.	X
cana-2086	351	1	hence	hence	ADV
cana-2086	351	2	,	,	PUNCT
cana-2086	351	3	an	an	DET
cana-2086	351	4	elegant	elegant	ADJ
cana-2086	351	5	fuzzy	fuzzy	ADJ
cana-2086	351	6	labeling	labeling	NOUN
cana-2086	351	7	(	(	PUNCT
cana-2086	351	8	𝜇	𝜇	ADP
cana-2086	351	9	,	,	PUNCT
cana-2086	351	10	𝜌	𝜌	X
cana-2086	351	11	)	)	PUNCT
cana-2086	351	12	of	of	ADP
cana-2086	351	13	𝐶8	𝐶8	NOUN
cana-2086	351	14	is	be	AUX
cana-2086	351	15	given	give	VERB
cana-2086	351	16	as	as	SCONJ
cana-2086	351	17	follows	follow	VERB
cana-2086	351	18	:	:	PUNCT
cana-2086	351	19	𝜇(𝑣1	𝜇(𝑣1	ADJ
cana-2086	351	20	)	)	PUNCT
cana-2086	351	21	=	=	SYM
cana-2086	351	22	0.59	0.59	NUM
cana-2086	351	23	,	,	PUNCT
cana-2086	351	24	𝜇(𝑣2	𝜇(𝑣2	NOUN
cana-2086	351	25	)	)	PUNCT
cana-2086	351	26	=	=	SYM
cana-2086	351	27	0.94	0.94	NUM
cana-2086	351	28	,	,	PUNCT
cana-2086	351	29	𝜇(𝑣3	𝜇(𝑣3	NOUN
cana-2086	351	30	)	)	PUNCT
cana-2086	351	31	=	=	SYM
cana-2086	351	32	1	1	NUM
cana-2086	351	33	,	,	PUNCT
cana-2086	351	34	𝜇(𝑣4	𝜇(𝑣4	NOUN
cana-2086	351	35	)	)	PUNCT
cana-2086	351	36	=	=	SYM
cana-2086	351	37	0.82	0.82	NUM
cana-2086	351	38	,	,	PUNCT
cana-2086	351	39	𝜇(𝑣5	𝜇(𝑣5	NOUN
cana-2086	351	40	)	)	PUNCT
cana-2086	351	41	=	=	SYM
cana-2086	351	42	0.88	0.88	NUM
cana-2086	351	43	,	,	PUNCT
cana-2086	351	44	𝜇(𝑣6	𝜇(𝑣6	NOUN
cana-2086	351	45	)	)	PUNCT
cana-2086	351	46	=	=	SYM
cana-2086	351	47	0.76	0.76	NUM
cana-2086	351	48	,	,	PUNCT
cana-2086	351	49	𝜇(𝑣7	𝜇(𝑣7	NOUN
cana-2086	351	50	)	)	PUNCT
cana-2086	351	51	=	=	SYM
cana-2086	351	52	0.71	0.71	NUM
cana-2086	351	53	,	,	PUNCT
cana-2086	351	54	𝜇(𝑣8	𝜇(𝑣8	NOUN
cana-2086	351	55	)	)	PUNCT
cana-2086	351	56	=	=	SYM
cana-2086	351	57	0.65	0.65	NUM
cana-2086	351	58	,	,	PUNCT
cana-2086	351	59	𝜌(𝑣1𝑣2	𝜌(𝑣1𝑣2	ADJ
cana-2086	351	60	)	)	PUNCT
cana-2086	351	61	=	=	SYM
cana-2086	351	62	0.47	0.47	NUM
cana-2086	351	63	,	,	PUNCT
cana-2086	351	64	𝜌(𝑣2𝑣3	𝜌(𝑣2𝑣3	NOUN
cana-2086	351	65	)	)	PUNCT
cana-2086	351	66	=	=	SYM
cana-2086	351	67	0.35	0.35	NUM
cana-2086	351	68	,	,	PUNCT
cana-2086	351	69	𝜌(𝑣3𝑣4	𝜌(𝑣3𝑣4	NOUN
cana-2086	351	70	)	)	PUNCT
cana-2086	351	71	=	=	NOUN
cana-2086	351	72	0.24	0.24	NUM
cana-2086	351	73	,	,	PUNCT
cana-2086	351	74	𝜌(𝑣4𝑣5	𝜌(𝑣4𝑣5	NOUN
cana-2086	351	75	)	)	PUNCT
cana-2086	351	76	=	=	SYM
cana-2086	351	77	0.12	0.12	NUM
cana-2086	351	78	,	,	PUNCT
cana-2086	351	79	𝜌(𝑣5𝑣6	𝜌(𝑣5𝑣6	NOUN
cana-2086	351	80	)	)	PUNCT
cana-2086	351	81	=	=	SYM
cana-2086	351	82	0.06	0.06	NUM
cana-2086	351	83	,	,	PUNCT
cana-2086	351	84	𝜌(𝑣6𝑣7	𝜌(𝑣6𝑣7	NUM
cana-2086	351	85	)	)	PUNCT
cana-2086	351	86	=	=	SYM
cana-2086	351	87	0.41	0.41	NUM
cana-2086	351	88	,	,	PUNCT
cana-2086	351	89	𝜌(𝑣7𝑣8	𝜌(𝑣7𝑣8	PROPN
cana-2086	351	90	)	)	PUNCT
cana-2086	351	91	=	=	SYM
cana-2086	351	92	0.29	0.29	NUM
cana-2086	351	93	,	,	PUNCT
cana-2086	351	94	𝜌(𝑣8𝑣1	𝜌(𝑣8𝑣1	NUM
cana-2086	351	95	)	)	PUNCT
cana-2086	351	96	=	=	SYM
cana-2086	351	97	0.18	0.18	NUM
cana-2086	351	98	.	.	NOUN
cana-2086	352	1	4	4	NUM
cana-2086	352	2	.	.	X
cana-2086	352	3	application	application	NOUN
cana-2086	352	4	consider	consider	VERB
cana-2086	352	5	a	a	DET
cana-2086	352	6	transit	transit	NOUN
cana-2086	352	7	network	network	NOUN
cana-2086	352	8	as	as	ADP
cana-2086	352	9	an	an	DET
cana-2086	352	10	elegant	elegant	ADJ
cana-2086	352	11	fuzzy	fuzzy	ADJ
cana-2086	352	12	labeling	labeling	NOUN
cana-2086	352	13	graph	graph	NOUN
cana-2086	352	14	,	,	PUNCT
cana-2086	352	15	where	where	SCONJ
cana-2086	352	16	vertices	vertex	NOUN
cana-2086	352	17	represent	represent	VERB
cana-2086	352	18	stations	station	NOUN
cana-2086	352	19	and	and	CCONJ
cana-2086	352	20	edges	edge	NOUN
cana-2086	352	21	represent	represent	VERB
cana-2086	352	22	routes	route	NOUN
cana-2086	352	23	.	.	PUNCT
cana-2086	353	1	vertex	vertex	NOUN
cana-2086	353	2	labels	label	NOUN
cana-2086	353	3	denote	denote	VERB
cana-2086	353	4	passenger	passenger	NOUN
cana-2086	353	5	capacity	capacity	NOUN
cana-2086	353	6	at	at	ADP
cana-2086	353	7	each	each	DET
cana-2086	353	8	station	station	NOUN
cana-2086	353	9	,	,	PUNCT
cana-2086	353	10	while	while	SCONJ
cana-2086	353	11	edge	edge	NOUN
cana-2086	353	12	labels	label	NOUN
cana-2086	353	13	represent	represent	VERB
cana-2086	353	14	the	the	DET
cana-2086	353	15	passenger	passenger	NOUN
cana-2086	353	16	flow	flow	NOUN
cana-2086	353	17	along	along	ADP
cana-2086	353	18	the	the	DET
cana-2086	353	19	routes	route	NOUN
cana-2086	353	20	.	.	PUNCT
cana-2086	354	1	the	the	DET
cana-2086	354	2	condition	condition	NOUN
cana-2086	354	3	of	of	ADP
cana-2086	354	4	elegant	elegant	ADJ
cana-2086	354	5	fuzzy	fuzzy	ADJ
cana-2086	354	6	labeling	labeling	NOUN
cana-2086	354	7	,	,	PUNCT
cana-2086	354	8	that	that	SCONJ
cana-2086	354	9	labels	label	NOUN
cana-2086	354	10	are	be	AUX
cana-2086	354	11	distinct	distinct	ADJ
cana-2086	354	12	and	and	CCONJ
cana-2086	354	13	non	non	ADJ
cana-2086	354	14	-	-	ADJ
cana-2086	354	15	zero	zero	NUM
cana-2086	354	16	,	,	PUNCT
cana-2086	354	17	guarantees	guarantee	VERB
cana-2086	354	18	that	that	SCONJ
cana-2086	354	19	every	every	DET
cana-2086	354	20	station	station	NOUN
cana-2086	354	21	and	and	CCONJ
cana-2086	354	22	route	route	NOUN
cana-2086	354	23	is	be	AUX
cana-2086	354	24	utilized	utilize	VERB
cana-2086	354	25	effectively	effectively	ADV
cana-2086	354	26	.	.	PUNCT
cana-2086	355	1	the	the	DET
cana-2086	355	2	condition	condition	NOUN
cana-2086	355	3	that	that	PRON
cana-2086	355	4	edge	edge	NOUN
cana-2086	355	5	labels	label	NOUN
cana-2086	355	6	are	be	AUX
cana-2086	355	7	less	less	ADJ
cana-2086	355	8	than	than	SCONJ
cana-2086	355	9	their	their	PRON
cana-2086	355	10	endpoints	endpoint	NOUN
cana-2086	355	11	labels	label	NOUN
cana-2086	355	12	ensures	ensure	VERB
cana-2086	355	13	that	that	SCONJ
cana-2086	355	14	the	the	DET
cana-2086	355	15	passenger	passenger	NOUN
cana-2086	355	16	flow	flow	NOUN
cana-2086	355	17	along	along	ADP
cana-2086	355	18	the	the	DET
cana-2086	355	19	route	route	NOUN
cana-2086	355	20	does	do	AUX
cana-2086	355	21	not	not	PART
cana-2086	355	22	exceed	exceed	VERB
cana-2086	355	23	the	the	DET
cana-2086	355	24	station	station	NOUN
cana-2086	355	25	capacity	capacity	NOUN
cana-2086	355	26	,	,	PUNCT
cana-2086	355	27	thus	thus	ADV
cana-2086	355	28	preventing	prevent	VERB
cana-2086	355	29	congestion	congestion	NOUN
cana-2086	355	30	at	at	ADP
cana-2086	355	31	stations	station	NOUN
cana-2086	355	32	and	and	CCONJ
cana-2086	355	33	along	along	ADP
cana-2086	355	34	the	the	DET
cana-2086	355	35	routes	route	NOUN
cana-2086	355	36	.	.	PUNCT
cana-2086	356	1	this	this	DET
cana-2086	356	2	approach	approach	NOUN
cana-2086	356	3	enables	enable	VERB
cana-2086	356	4	judicious	judicious	ADJ
cana-2086	356	5	allocation	allocation	NOUN
cana-2086	356	6	of	of	ADP
cana-2086	356	7	resources	resource	NOUN
cana-2086	356	8	,	,	PUNCT
cana-2086	356	9	improves	improve	VERB
cana-2086	356	10	routing	routing	NOUN
cana-2086	356	11	and	and	CCONJ
cana-2086	356	12	scheduling	scheduling	NOUN
cana-2086	356	13	,	,	PUNCT
cana-2086	356	14	and	and	CCONJ
cana-2086	356	15	ensures	ensure	VERB
cana-2086	356	16	smooth	smooth	ADJ
cana-2086	356	17	movement	movement	NOUN
cana-2086	356	18	of	of	ADP
cana-2086	356	19	passengers	passenger	NOUN
cana-2086	356	20	.	.	PUNCT
cana-2086	357	1	5	5	X
cana-2086	357	2	.	.	X
cana-2086	357	3	conclusion	conclusion	NOUN
cana-2086	357	4	in	in	ADP
cana-2086	357	5	this	this	DET
cana-2086	357	6	paper	paper	NOUN
cana-2086	357	7	,	,	PUNCT
cana-2086	357	8	we	we	PRON
cana-2086	357	9	have	have	AUX
cana-2086	357	10	introduced	introduce	VERB
cana-2086	357	11	a	a	DET
cana-2086	357	12	new	new	ADJ
cana-2086	357	13	type	type	NOUN
cana-2086	357	14	of	of	ADP
cana-2086	357	15	fuzzy	fuzzy	ADJ
cana-2086	357	16	labeling	labeling	NOUN
cana-2086	357	17	called	call	VERB
cana-2086	357	18	elegant	elegant	ADJ
cana-2086	357	19	fuzzy	fuzzy	ADJ
cana-2086	357	20	labeling	labeling	NOUN
cana-2086	357	21	.	.	PUNCT
cana-2086	358	1	we	we	PRON
cana-2086	358	2	proved	prove	VERB
cana-2086	358	3	that	that	SCONJ
cana-2086	358	4	a	a	DET
cana-2086	358	5	simple	simple	ADJ
cana-2086	358	6	graph	graph	NOUN
cana-2086	358	7	admits	admit	VERB
cana-2086	358	8	elegant	elegant	ADJ
cana-2086	358	9	fuzzy	fuzzy	ADJ
cana-2086	358	10	labeling	labeling	NOUN
cana-2086	358	11	if	if	SCONJ
cana-2086	358	12	and	and	CCONJ
cana-2086	358	13	only	only	ADV
cana-2086	358	14	if	if	SCONJ
cana-2086	358	15	it	it	PRON
cana-2086	358	16	admits	admit	VERB
cana-2086	358	17	elegant	elegant	ADJ
cana-2086	358	18	labeling	labeling	NOUN
cana-2086	358	19	.	.	PUNCT
cana-2086	359	1	additionally	additionally	ADV
cana-2086	359	2	,	,	PUNCT
cana-2086	359	3	we	we	PRON
cana-2086	359	4	showed	show	VERB
cana-2086	359	5	that	that	SCONJ
cana-2086	359	6	while	while	SCONJ
cana-2086	359	7	a	a	DET
cana-2086	359	8	simple	simple	ADJ
cana-2086	359	9	graph	graph	NOUN
cana-2086	359	10	admitting	admit	VERB
cana-2086	359	11	elegant	elegant	ADJ
cana-2086	359	12	fuzzy	fuzzy	ADJ
cana-2086	359	13	labeling	labeling	NOUN
cana-2086	359	14	will	will	AUX
cana-2086	359	15	also	also	ADV
cana-2086	359	16	admit	admit	VERB
cana-2086	359	17	fuzzy	fuzzy	ADJ
cana-2086	359	18	labeling	labeling	NOUN
cana-2086	359	19	,	,	PUNCT
cana-2086	359	20	the	the	DET
cana-2086	359	21	converse	converse	NOUN
cana-2086	359	22	is	be	AUX
cana-2086	359	23	not	not	PART
cana-2086	359	24	necessarily	necessarily	ADV
cana-2086	359	25	true	true	ADJ
cana-2086	359	26	.	.	PUNCT
cana-2086	360	1	we	we	PRON
cana-2086	360	2	have	have	AUX
cana-2086	360	3	investigated	investigate	VERB
cana-2086	360	4	certain	certain	ADJ
cana-2086	360	5	classes	class	NOUN
cana-2086	360	6	of	of	ADP
cana-2086	360	7	simple	simple	ADJ
cana-2086	360	8	graphs	graph	NOUN
cana-2086	360	9	to	to	PART
cana-2086	360	10	determine	determine	VERB
cana-2086	360	11	if	if	SCONJ
cana-2086	360	12	they	they	PRON
cana-2086	360	13	admit	admit	VERB
cana-2086	360	14	elegant	elegant	ADJ
cana-2086	360	15	fuzzy	fuzzy	ADJ
cana-2086	360	16	labeling	labeling	NOUN
cana-2086	360	17	and	and	CCONJ
cana-2086	360	18	provided	provide	VERB
cana-2086	360	19	an	an	DET
cana-2086	360	20	application	application	NOUN
cana-2086	360	21	using	use	VERB
cana-2086	360	22	this	this	DET
cana-2086	360	23	labeling	labeling	NOUN
cana-2086	360	24	method	method	NOUN
cana-2086	360	25	.	.	PUNCT
cana-2086	361	1	we	we	PRON
cana-2086	361	2	employed	employ	VERB
cana-2086	361	3	translation	translation	NOUN
cana-2086	361	4	and	and	CCONJ
cana-2086	361	5	contraction	contraction	NOUN
cana-2086	361	6	of	of	ADP
cana-2086	361	7	mapping	mapping	NOUN
cana-2086	361	8	to	to	PART
cana-2086	361	9	achieve	achieve	VERB
cana-2086	361	10	the	the	DET
cana-2086	361	11	labeling	labeling	NOUN
cana-2086	361	12	.	.	PUNCT
cana-2086	362	1	for	for	ADP
cana-2086	362	2	future	future	ADJ
cana-2086	362	3	work	work	NOUN
cana-2086	362	4	,	,	PUNCT
cana-2086	362	5	we	we	PRON
cana-2086	362	6	plan	plan	VERB
cana-2086	362	7	to	to	PART
cana-2086	362	8	explore	explore	VERB
cana-2086	362	9	elegant	elegant	ADJ
cana-2086	362	10	fuzzy	fuzzy	ADJ
cana-2086	362	11	labeling	labeling	NOUN
cana-2086	362	12	on	on	ADP
cana-2086	362	13	other	other	ADJ
cana-2086	362	14	graph	graph	NOUN
cana-2086	362	15	families	family	NOUN
cana-2086	362	16	,	,	PUNCT
cana-2086	362	17	investigate	investigate	VERB
cana-2086	362	18	whether	whether	SCONJ
cana-2086	362	19	communications	communication	NOUN
cana-2086	362	20	on	on	ADP
cana-2086	362	21	applied	apply	VERB
cana-2086	362	22	nonlinear	nonlinear	ADJ
cana-2086	362	23	analysis	analysis	NOUN
cana-2086	362	24	issn	issn	NOUN
cana-2086	362	25	:	:	PUNCT
cana-2086	362	26	1074	1074	NUM
cana-2086	362	27	-	-	PUNCT
cana-2086	362	28	133x	133x	NUM
cana-2086	362	29	vol	vol	NOUN
cana-2086	362	30	32	32	NUM
cana-2086	362	31	no	no	NOUN
cana-2086	362	32	.	.	PUNCT
cana-2086	363	1	1s	1s	NUM
cana-2086	363	2	(	(	PUNCT
cana-2086	363	3	2025	2025	NUM
cana-2086	363	4	)	)	PUNCT
cana-2086	363	5	43	43	NUM
cana-2086	364	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2086	364	2	different	different	ADJ
cana-2086	364	3	translation	translation	NOUN
cana-2086	364	4	constants	constant	NOUN
cana-2086	364	5	and	and	CCONJ
cana-2086	364	6	contraction	contraction	NOUN
cana-2086	364	7	scales	scale	NOUN
cana-2086	364	8	can	can	AUX
cana-2086	364	9	be	be	AUX
cana-2086	364	10	used	use	VERB
cana-2086	364	11	to	to	PART
cana-2086	364	12	define	define	VERB
cana-2086	364	13	elegant	elegant	ADJ
cana-2086	364	14	fuzzy	fuzzy	ADJ
cana-2086	364	15	labeling	labeling	NOUN
cana-2086	364	16	and	and	CCONJ
cana-2086	364	17	extend	extend	VERB
cana-2086	364	18	the	the	DET
cana-2086	364	19	concept	concept	NOUN
cana-2086	364	20	to	to	ADP
cana-2086	364	21	various	various	ADJ
cana-2086	364	22	extensions	extension	NOUN
cana-2086	364	23	of	of	ADP
cana-2086	364	24	fuzzy	fuzzy	ADJ
cana-2086	364	25	graphs	graph	NOUN
cana-2086	364	26	.	.	PUNCT
cana-2086	365	1	declarations	declaration	NOUN
cana-2086	365	2	conflicts	conflict	NOUN
cana-2086	365	3	of	of	ADP
cana-2086	365	4	interest	interest	NOUN
cana-2086	365	5	:	:	PUNCT
cana-2086	365	6	none	none	NOUN
cana-2086	365	7	of	of	ADP
cana-2086	365	8	the	the	DET
cana-2086	365	9	authors	author	NOUN
cana-2086	365	10	have	have	VERB
cana-2086	365	11	any	any	DET
cana-2086	365	12	conflict	conflict	NOUN
cana-2086	365	13	of	of	ADP
cana-2086	365	14	interest	interest	NOUN
cana-2086	365	15	.	.	PUNCT
cana-2086	366	1	references	reference	NOUN
cana-2086	366	2	[	[	X
cana-2086	366	3	1	1	NUM
cana-2086	366	4	]	]	X
cana-2086	366	5	borzooei	borzooei	PROPN
cana-2086	366	6	,	,	PUNCT
cana-2086	366	7	r.	r.	PROPN
cana-2086	366	8	a.	a.	PROPN
cana-2086	366	9	,	,	PUNCT
cana-2086	366	10	rashmanlou	rashmanlou	PROPN
cana-2086	366	11	,	,	PUNCT
cana-2086	366	12	h.	h.	PROPN
cana-2086	366	13	:	:	PUNCT
cana-2086	366	14	cayley	cayley	ADJ
cana-2086	366	15	interval	interval	NOUN
cana-2086	366	16	-	-	PUNCT
cana-2086	366	17	valued	value	VERB
cana-2086	366	18	fuzzy	fuzzy	ADJ
cana-2086	366	19	graphs	graph	NOUN
cana-2086	366	20	.	.	PUNCT
cana-2086	367	1	upb	upb	ADJ
cana-2086	367	2	scientific	scientific	ADJ
cana-2086	367	3	bulletin	bulletin	NOUN
cana-2086	367	4	,	,	PUNCT
cana-2086	367	5	series	series	PROPN
cana-2086	367	6	a	a	PRON
cana-2086	367	7	:	:	PUNCT
cana-2086	367	8	applied	applied	ADJ
cana-2086	367	9	mathematics	mathematic	NOUN
cana-2086	367	10	and	and	CCONJ
cana-2086	367	11	physics	physics	NOUN
cana-2086	367	12	78	78	NUM
cana-2086	367	13	,	,	PUNCT
cana-2086	367	14	no	no	INTJ
cana-2086	367	15	.	.	NOUN
cana-2086	368	1	3	3	NUM
cana-2086	368	2	:	:	PUNCT
cana-2086	368	3	83	83	NUM
cana-2086	368	4	-	-	SYM
cana-2086	368	5	94	94	NUM
cana-2086	368	6	(	(	PUNCT
cana-2086	368	7	2016	2016	NUM
cana-2086	368	8	)	)	PUNCT
cana-2086	368	9	.	.	PUNCT
cana-2086	369	1	[	[	X
cana-2086	369	2	2	2	NUM
cana-2086	369	3	]	]	X
cana-2086	369	4	cahit	cahit	ADJ
cana-2086	369	5	,	,	PUNCT
cana-2086	369	6	ibrahim	ibrahim	PROPN
cana-2086	369	7	.	.	PUNCT
cana-2086	369	8	:	:	PUNCT
cana-2086	370	1	elegant	elegant	ADJ
cana-2086	370	2	valuation	valuation	NOUN
cana-2086	370	3	of	of	ADP
cana-2086	370	4	the	the	DET
cana-2086	370	5	paths	path	NOUN
cana-2086	370	6	.	.	PUNCT
cana-2086	371	1	ars	ar	VERB
cana-2086	371	2	combinatoria	combinatoria	PROPN
cana-2086	371	3	16	16	NUM
cana-2086	371	4	(	(	PUNCT
cana-2086	371	5	1983	1983	NUM
cana-2086	371	6	)	)	PUNCT
cana-2086	371	7	.	.	PUNCT
cana-2086	372	1	[	[	X
cana-2086	372	2	3	3	X
cana-2086	372	3	]	]	X
cana-2086	372	4	chang	chang	PROPN
cana-2086	372	5	,	,	PUNCT
cana-2086	372	6	g.	g.	PROPN
cana-2086	372	7	j.	j.	PROPN
cana-2086	372	8	,	,	PUNCT
cana-2086	372	9	hue	hue	PROPN
cana-2086	372	10	,	,	PUNCT
cana-2086	372	11	d.	d.	PROPN
cana-2086	372	12	f.	f.	PROPN
cana-2086	372	13	,	,	PUNCT
cana-2086	372	14	rogers	rogers	PROPN
cana-2086	372	15	,	,	PUNCT
cana-2086	372	16	d.	d.	PROPN
cana-2086	372	17	g.	g.	PROPN
cana-2086	372	18	:	:	PUNCT
cana-2086	372	19	additive	additive	ADJ
cana-2086	372	20	variations	variation	NOUN
cana-2086	372	21	on	on	ADP
cana-2086	372	22	a	a	DET
cana-2086	372	23	graceful	graceful	ADJ
cana-2086	372	24	theme	theme	NOUN
cana-2086	372	25	:	:	PUNCT
cana-2086	372	26	some	some	DET
cana-2086	372	27	results	result	NOUN
cana-2086	372	28	on	on	ADP
cana-2086	372	29	harmonious	harmonious	ADJ
cana-2086	372	30	and	and	CCONJ
cana-2086	372	31	other	other	ADJ
cana-2086	372	32	related	related	ADJ
cana-2086	372	33	graphs	graph	NOUN
cana-2086	372	34	.	.	PUNCT
cana-2086	373	1	congr	congr	NOUN
cana-2086	373	2	.	.	PUNCT
cana-2086	374	1	numer	numer	PROPN
cana-2086	374	2	.	.	PROPN
cana-2086	375	1	32	32	NUM
cana-2086	375	2	,	,	PUNCT
cana-2086	375	3	181	181	NUM
cana-2086	375	4	-	-	SYM
cana-2086	375	5	197	197	NUM
cana-2086	375	6	(	(	PUNCT
cana-2086	375	7	1981	1981	NUM
cana-2086	375	8	)	)	PUNCT
cana-2086	375	9	.	.	PUNCT
cana-2086	376	1	[	[	X
cana-2086	376	2	4	4	NUM
cana-2086	376	3	]	]	X
cana-2086	376	4	elumalai	elumalai	NOUN
cana-2086	376	5	,	,	PUNCT
cana-2086	376	6	a.	a.	NOUN
cana-2086	376	7	,	,	PUNCT
cana-2086	376	8	sethuraman	sethuraman	NOUN
cana-2086	376	9	,	,	PUNCT
cana-2086	376	10	g.	g.	PROPN
cana-2086	376	11	:	:	PUNCT
cana-2086	376	12	elegant	elegant	ADJ
cana-2086	376	13	labeled	label	VERB
cana-2086	376	14	graphs	graph	NOUN
cana-2086	376	15	.	.	PUNCT
cana-2086	377	1	journal	journal	NOUN
cana-2086	377	2	of	of	ADP
cana-2086	377	3	informatics	informatic	NOUN
cana-2086	377	4	and	and	CCONJ
cana-2086	377	5	mathematical	mathematical	ADJ
cana-2086	377	6	sciences	science	NOUN
cana-2086	377	7	2	2	NUM
cana-2086	377	8	(	(	PUNCT
cana-2086	377	9	1	1	NUM
cana-2086	377	10	):	):	PUNCT
cana-2086	377	11	45	45	NUM
cana-2086	377	12	-	-	SYM
cana-2086	377	13	49	49	NUM
cana-2086	377	14	(	(	PUNCT
cana-2086	377	15	2010	2010	NUM
cana-2086	377	16	)	)	PUNCT
cana-2086	377	17	.	.	PUNCT
cana-2086	378	1	[	[	X
cana-2086	378	2	5	5	NUM
cana-2086	378	3	]	]	PUNCT
cana-2086	378	4	fathalian	fathalian	PROPN
cana-2086	378	5	,	,	PUNCT
cana-2086	378	6	m.	m.	NOUN
cana-2086	378	7	,	,	PUNCT
cana-2086	378	8	borzooei	borzooei	PROPN
cana-2086	378	9	,	,	PUNCT
cana-2086	378	10	r.	r.	PROPN
cana-2086	378	11	a.	a.	PROPN
cana-2086	378	12	,	,	PUNCT
cana-2086	378	13	hamidi	hamidi	NOUN
cana-2086	378	14	,	,	PUNCT
cana-2086	378	15	m.	m.	NOUN
cana-2086	378	16	:	:	PUNCT
cana-2086	378	17	fuzzy	fuzzy	ADJ
cana-2086	378	18	magic	magic	ADJ
cana-2086	378	19	labeling	labeling	NOUN
cana-2086	378	20	of	of	ADP
cana-2086	378	21	simple	simple	ADJ
cana-2086	378	22	graphs	graph	NOUN
cana-2086	378	23	.	.	PUNCT
cana-2086	379	1	journal	journal	NOUN
cana-2086	379	2	of	of	ADP
cana-2086	379	3	applied	apply	VERB
cana-2086	379	4	mathematics	mathematic	NOUN
cana-2086	379	5	and	and	CCONJ
cana-2086	379	6	computing	computing	NOUN
cana-2086	379	7	60	60	NUM
cana-2086	379	8	,	,	PUNCT
cana-2086	379	9	369	369	NUM
cana-2086	379	10	-	-	SYM
cana-2086	379	11	385	385	NUM
cana-2086	379	12	(	(	PUNCT
cana-2086	379	13	2019	2019	NUM
cana-2086	379	14	)	)	PUNCT
cana-2086	379	15	.	.	PUNCT
cana-2086	380	1	https://doi.org/10.1007/s12190-018-01218-x	https://doi.org/10.1007/s12190-018-01218-x	PROPN
cana-2086	380	2	[	[	X
cana-2086	380	3	6	6	NUM
cana-2086	380	4	]	]	PUNCT
cana-2086	380	5	gallian	gallian	NOUN
cana-2086	380	6	,	,	PUNCT
cana-2086	380	7	j.	j.	PROPN
cana-2086	380	8	a.	a.	PROPN
cana-2086	380	9	:	:	PUNCT
cana-2086	380	10	a	a	DET
cana-2086	380	11	dynamic	dynamic	ADJ
cana-2086	380	12	survey	survey	NOUN
cana-2086	380	13	of	of	ADP
cana-2086	380	14	graph	graph	NOUN
cana-2086	380	15	labeling	labeling	NOUN
cana-2086	380	16	.	.	PUNCT
cana-2086	381	1	electronic	electronic	ADJ
cana-2086	381	2	journal	journal	NOUN
cana-2086	381	3	of	of	ADP
cana-2086	381	4	combinatorics	combinatoric	NOUN
cana-2086	381	5	1(dynamic	1(dynamic	NUM
cana-2086	381	6	surveys	survey	NOUN
cana-2086	381	7	)	)	PUNCT
cana-2086	381	8	,	,	PUNCT
cana-2086	381	9	ds6	ds6	NOUN
cana-2086	381	10	(	(	PUNCT
cana-2086	381	11	2022	2022	NUM
cana-2086	381	12	)	)	PUNCT
cana-2086	381	13	.	.	PUNCT
cana-2086	382	1	https://doi.org/10.37236/27	https://doi.org/10.37236/27	PROPN
cana-2086	383	1	[	[	X
cana-2086	383	2	7	7	NUM
cana-2086	383	3	]	]	X
cana-2086	383	4	gani	gani	X
cana-2086	383	5	,	,	PUNCT
cana-2086	383	6	a.	a.	NOUN
cana-2086	383	7	n.	n.	NOUN
cana-2086	383	8	,	,	PUNCT
cana-2086	383	9	subahashini	subahashini	PROPN
cana-2086	383	10	,	,	PUNCT
cana-2086	383	11	d.	d.	PROPN
cana-2086	383	12	r.	r.	PROPN
cana-2086	383	13	:	:	PUNCT
cana-2086	383	14	properties	property	NOUN
cana-2086	383	15	of	of	ADP
cana-2086	383	16	fuzzy	fuzzy	ADJ
cana-2086	383	17	labeling	labeling	NOUN
cana-2086	383	18	graph	graph	NOUN
cana-2086	383	19	.	.	PUNCT
cana-2086	384	1	applied	apply	VERB
cana-2086	384	2	mathematical	mathematical	ADJ
cana-2086	384	3	sciences	science	NOUN
cana-2086	384	4	6(70	6(70	NUM
cana-2086	384	5	)	)	PUNCT
cana-2086	384	6	,	,	PUNCT
cana-2086	384	7	34613466	34613466	NUM
cana-2086	384	8	(	(	PUNCT
cana-2086	384	9	2012	2012	NUM
cana-2086	384	10	)	)	PUNCT
cana-2086	384	11	.	.	PUNCT
cana-2086	385	1	[	[	X
cana-2086	385	2	8	8	NUM
cana-2086	385	3	]	]	X
cana-2086	385	4	giri	giri	PROPN
cana-2086	385	5	,	,	PUNCT
cana-2086	385	6	p.	p.	NOUN
cana-2086	385	7	,	,	PUNCT
cana-2086	385	8	amanathulla	amanathulla	PROPN
cana-2086	385	9	,	,	PUNCT
cana-2086	385	10	s.	s.	PROPN
cana-2086	385	11	,	,	PUNCT
cana-2086	385	12	das	das	PROPN
cana-2086	385	13	,	,	PUNCT
cana-2086	385	14	k.m	k.m	PROPN
cana-2086	385	15	.	.	PROPN
cana-2086	385	16	:	:	PUNCT
cana-2086	385	17	an	an	DET
cana-2086	385	18	analysis	analysis	NOUN
cana-2086	385	19	of	of	ADP
cana-2086	385	20	fermatean	fermatean	ADJ
cana-2086	385	21	fuzzy	fuzzy	ADJ
cana-2086	385	22	graph	graph	NOUN
cana-2086	385	23	and	and	CCONJ
cana-2086	385	24	its	its	PRON
cana-2086	385	25	application	application	NOUN
cana-2086	385	26	in	in	ADP
cana-2086	385	27	a	a	DET
cana-2086	385	28	car	car	NOUN
cana-2086	385	29	company	company	NOUN
cana-2086	385	30	.	.	PUNCT
cana-2086	386	1	j.	j.	PROPN
cana-2086	386	2	appl	appl	PROPN
cana-2086	386	3	.	.	PROPN
cana-2086	386	4	math	math	PROPN
cana-2086	386	5	.	.	PUNCT
cana-2086	387	1	comput	comput	NOUN
cana-2086	387	2	.	.	PUNCT
cana-2086	388	1	(	(	PUNCT
cana-2086	388	2	2024	2024	NUM
cana-2086	388	3	)	)	PUNCT
cana-2086	388	4	.	.	PUNCT
cana-2086	389	1	https://doi.org/10.1007/s12190-024-02094-4	https://doi.org/10.1007/s12190-024-02094-4	PRON
cana-2086	390	1	[	[	X
cana-2086	390	2	9	9	NUM
cana-2086	390	3	]	]	PUNCT
cana-2086	390	4	golomb	golomb	NOUN
cana-2086	390	5	,	,	PUNCT
cana-2086	390	6	s.	s.	PROPN
cana-2086	390	7	w.	w.	PROPN
cana-2086	390	8	:	:	PUNCT
cana-2086	390	9	how	how	SCONJ
cana-2086	390	10	to	to	PART
cana-2086	390	11	number	number	VERB
cana-2086	390	12	a	a	DET
cana-2086	390	13	graph	graph	NOUN
cana-2086	390	14	.	.	PUNCT
cana-2086	391	1	graph	graph	NOUN
cana-2086	391	2	theory	theory	NOUN
cana-2086	391	3	and	and	CCONJ
cana-2086	391	4	computing	computing	NOUN
cana-2086	391	5	,	,	PUNCT
cana-2086	391	6	r.	r.	PROPN
cana-2086	391	7	c.	c.	PROPN
cana-2086	391	8	read	read	PROPN
cana-2086	391	9	,	,	PUNCT
cana-2086	391	10	ed	ed	NOUN
cana-2086	391	11	.	.	PROPN
cana-2086	391	12	,	,	PUNCT
cana-2086	391	13	academic	academic	ADJ
cana-2086	391	14	press	press	NOUN
cana-2086	391	15	,	,	PUNCT
cana-2086	391	16	new	new	PROPN
cana-2086	391	17	york	york	PROPN
cana-2086	391	18	,	,	PUNCT
cana-2086	391	19	23	23	NUM
cana-2086	391	20	-	-	SYM
cana-2086	391	21	37	37	NUM
cana-2086	391	22	(	(	PUNCT
cana-2086	391	23	1972	1972	NUM
cana-2086	391	24	)	)	PUNCT
cana-2086	391	25	.	.	PUNCT
cana-2086	392	1	[	[	X
cana-2086	392	2	10	10	NUM
cana-2086	392	3	]	]	X
cana-2086	392	4	graham	graham	PROPN
cana-2086	392	5	,	,	PUNCT
cana-2086	392	6	r.	r.	PROPN
cana-2086	392	7	l.	l.	PROPN
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cana-2086	393	7	1(4	1(4	NUM
cana-2086	393	8	)	)	PUNCT
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cana-2086	393	13	(	(	PUNCT
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cana-2086	393	15	)	)	PUNCT
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cana-2086	394	4	]	]	X
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cana-2086	394	8	:	:	PUNCT
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cana-2086	394	10	a	a	DET
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cana-2086	395	10	)	)	PUNCT
cana-2086	395	11	.	.	PUNCT
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cana-2086	396	3	]	]	PUNCT
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cana-2086	396	5	,	,	PUNCT
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cana-2086	396	9	,	,	PUNCT
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cana-2086	396	11	,	,	PUNCT
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cana-2086	396	17	,	,	PUNCT
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cana-2086	398	3	]	]	SYM
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cana-2086	398	5	,	,	PUNCT
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cana-2086	398	11	,	,	PUNCT
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cana-2086	398	13	,	,	PUNCT
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cana-2086	398	23	:	:	PUNCT
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cana-2086	398	37	.	.	PUNCT
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cana-2086	399	11	(	(	PUNCT
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cana-2086	399	13	)	)	PUNCT
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cana-2086	400	3	]	]	SYM
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cana-2086	400	10	.	.	PUNCT
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cana-2086	400	20	.	.	PUNCT
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cana-2086	401	4	:	:	PUNCT
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cana-2086	401	7	(	(	PUNCT
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cana-2086	401	9	.	.	PUNCT
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cana-2086	401	11	,	,	PUNCT
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cana-2086	401	15	,	,	PUNCT
cana-2086	401	16	p.	p.	NOUN
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cana-2086	401	18	)	)	PUNCT
cana-2086	401	19	.	.	PUNCT
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cana-2086	402	3	(	(	PUNCT
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cana-2086	402	5	)	)	PUNCT
cana-2086	402	6	.	.	PUNCT
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cana-2086	404	17	:	:	PUNCT
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cana-2086	405	7	.	.	PUNCT
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cana-2086	407	1	(	(	PUNCT
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cana-2086	407	3	)	)	PUNCT
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cana-2086	411	12	)	)	PUNCT
cana-2086	411	13	.	.	PUNCT
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cana-2086	414	4	]	]	PUNCT
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cana-2086	414	27	:	:	PUNCT
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cana-2086	414	33	.	.	PUNCT
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cana-2086	415	7	.	.	PUNCT
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cana-2086	415	9	.	.	PUNCT
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cana-2086	417	1	(	(	PUNCT
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cana-2086	417	3	)	)	PUNCT
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cana-2086	419	3	]	]	PUNCT
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cana-2086	419	8	,	,	PUNCT
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cana-2086	419	29	:	:	PUNCT
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cana-2086	419	36	)	)	PUNCT
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cana-2086	420	21	a	a	DET
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cana-2086	420	26	.	.	PUNCT
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cana-2086	421	7	:	:	SYM
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cana-2086	421	9	(	(	PUNCT
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cana-2086	421	11	)	)	PUNCT
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cana-2086	422	1	https://doi.org/10.1186/s40064-016-2892-z	https://doi.org/10.1186/s40064-016-2892-z	NOUN
cana-2086	422	2	[	[	PUNCT
cana-2086	422	3	20	20	NUM
cana-2086	422	4	]	]	X
cana-2086	422	5	rosa	rosa	PROPN
cana-2086	422	6	,	,	PUNCT
cana-2086	422	7	a.	a.	NOUN
cana-2086	422	8	:	:	PUNCT
cana-2086	422	9	on	on	ADP
cana-2086	422	10	certain	certain	ADJ
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cana-2086	422	12	of	of	ADP
cana-2086	422	13	the	the	DET
cana-2086	422	14	vertices	vertex	NOUN
cana-2086	422	15	of	of	ADP
cana-2086	422	16	a	a	DET
cana-2086	422	17	graph	graph	NOUN
cana-2086	422	18	,	,	PUNCT
cana-2086	422	19	theory	theory	NOUN
cana-2086	422	20	of	of	ADP
cana-2086	422	21	graphs	graph	NOUN
cana-2086	422	22	(	(	PUNCT
cana-2086	422	23	internat	internat	PROPN
cana-2086	422	24	.	.	PUNCT
cana-2086	423	1	symposium	symposium	PROPN
cana-2086	423	2	,	,	PUNCT
cana-2086	423	3	rome	rome	PROPN
cana-2086	423	4	,	,	PUNCT
cana-2086	423	5	1966	1966	NUM
cana-2086	423	6	)	)	PUNCT
cana-2086	423	7	.	.	PUNCT
cana-2086	424	1	[	[	X
cana-2086	424	2	21	21	NUM
cana-2086	424	3	]	]	PUNCT
cana-2086	424	4	rosenfield	rosenfield	VERB
cana-2086	424	5	,	,	PUNCT
cana-2086	424	6	a.	a.	NOUN
cana-2086	424	7	:	:	PUNCT
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cana-2086	424	9	graphs	graph	NOUN
cana-2086	424	10	.	.	PUNCT
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cana-2086	425	2	sets	set	NOUN
cana-2086	425	3	and	and	CCONJ
cana-2086	425	4	their	their	PRON
cana-2086	425	5	applications	application	NOUN
cana-2086	425	6	to	to	PART
cana-2086	425	7	cognitive	cognitive	VERB
cana-2086	425	8	and	and	CCONJ
cana-2086	425	9	decision	decision	NOUN
cana-2086	425	10	processes	process	NOUN
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cana-2086	425	12	pp	pp	ADP
cana-2086	425	13	.	.	PUNCT
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cana-2086	426	2	-	-	SYM
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cana-2086	426	7	,	,	PUNCT
cana-2086	426	8	new	new	PROPN
cana-2086	426	9	york	york	PROPN
cana-2086	426	10	(	(	PUNCT
cana-2086	426	11	1975	1975	NUM
cana-2086	426	12	)	)	PUNCT
cana-2086	426	13	.	.	PUNCT
cana-2086	427	1	https://doi.org/10.1016/b978-0-12-775260-0.50008-6	https://doi.org/10.1016/b978-0-12-775260-0.50008-6	PROPN
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cana-2086	428	3	]	]	X
cana-2086	428	4	rao	rao	PROPN
cana-2086	428	5	,	,	PUNCT
cana-2086	428	6	y.	y.	PROPN
cana-2086	428	7	,	,	PUNCT
cana-2086	428	8	kosari	kosari	X
cana-2086	428	9	,	,	PUNCT
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cana-2086	428	14	z.	z.	PROPN
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cana-2086	428	16	talebi	talebi	ADV
cana-2086	428	17	,	,	PUNCT
cana-2086	428	18	a.	a.	NOUN
cana-2086	428	19	a.	a.	PROPN
cana-2086	428	20	,	,	PUNCT
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cana-2086	428	22	,	,	PUNCT
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cana-2086	428	28	:	:	PUNCT
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cana-2086	428	34	trees	tree	NOUN
cana-2086	428	35	with	with	ADP
cana-2086	428	36	applications	application	NOUN
cana-2086	428	37	.	.	PUNCT
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cana-2086	429	8	:	:	SYM
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cana-2086	429	12	(	(	PUNCT
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cana-2086	429	15	.	.	PUNCT
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cana-2086	431	3	]	]	X
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cana-2086	431	5	,	,	PUNCT
cana-2086	431	6	s.	s.	PROPN
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cana-2086	432	7	11(2	11(2	X
cana-2086	432	8	)	)	PUNCT
cana-2086	432	9	(	(	PUNCT
cana-2086	432	10	2024	2024	NUM
cana-2086	432	11	)	)	PUNCT
cana-2086	432	12	.	.	PUNCT
cana-2086	433	1	https://doi.org/10.18280/mmep.110222	https://doi.org/10.18280/mmep.110222	NOUN
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cana-2086	433	5	https://epubs.siam.org/doi/10.1137/0601045	https://epubs.siam.org/doi/10.1137/0601045	PROPN
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cana-2086	433	21	:	:	PUNCT
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cana-2086	433	23	-	-	PUNCT
cana-2086	433	24	133x	133x	NUM
cana-2086	433	25	vol	vol	NOUN
cana-2086	433	26	32	32	NUM
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cana-2086	433	28	.	.	PUNCT
cana-2086	434	1	1s	1s	NUM
cana-2086	434	2	(	(	PUNCT
cana-2086	434	3	2025	2025	NUM
cana-2086	434	4	)	)	PUNCT
cana-2086	434	5	44	44	NUM
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cana-2086	435	1	[	[	X
cana-2086	435	2	24	24	NUM
cana-2086	435	3	]	]	SYM
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cana-2086	435	5	,	,	PUNCT
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cana-2086	435	7	,	,	PUNCT
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cana-2086	435	9	,	,	PUNCT
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cana-2086	435	11	k.	k.	PROPN
cana-2086	436	1	:	:	PUNCT
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cana-2086	437	4	vertex	vertex	NOUN
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cana-2086	437	7	.	.	PUNCT
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cana-2086	438	2	of	of	ADP
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cana-2086	438	5	conference	conference	NOUN
cana-2086	438	6	on	on	ADP
cana-2086	438	7	mathematical	mathematical	ADJ
cana-2086	438	8	modeling	modeling	NOUN
cana-2086	438	9	and	and	CCONJ
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cana-2086	438	11	science	science	NOUN
cana-2086	438	12	:	:	PUNCT
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cana-2086	439	2	-	-	SYM
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cana-2086	439	5	.	.	PUNCT
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cana-2086	440	5	)	)	PUNCT
cana-2086	440	6	.	.	PUNCT
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cana-2086	441	17	,	,	PUNCT
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cana-2086	441	19	:	:	PUNCT
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cana-2086	442	5	:	:	SYM
cana-2086	442	6	357	357	NUM
cana-2086	442	7	(	(	PUNCT
cana-2086	442	8	2020	2020	NUM
cana-2086	442	9	)	)	PUNCT
cana-2086	442	10	.	.	PUNCT
cana-2086	443	1	https://doi.org/10.3389/fphy.2020.00357	https://doi.org/10.3389/fphy.2020.00357	PROPN
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cana-2086	443	3	26	26	NUM
cana-2086	443	4	]	]	X
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cana-2086	443	8	,	,	PUNCT
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cana-2086	443	10	,	,	PUNCT
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cana-2086	443	16	,	,	PUNCT
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cana-2086	443	22	,	,	PUNCT
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cana-2086	445	11	:	:	PUNCT
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cana-2086	447	13	.	.	PUNCT
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cana-2086	448	3	(	(	PUNCT
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cana-2086	448	6	.	.	PUNCT
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cana-2086	450	3	]	]	X
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cana-2086	450	7	,	,	PUNCT
cana-2086	450	8	kosari	kosari	PROPN
cana-2086	450	9	,	,	PUNCT
cana-2086	450	10	s.	s.	PROPN
cana-2086	450	11	,	,	PUNCT
cana-2086	450	12	talebi	talebi	ADV
cana-2086	450	13	,	,	PUNCT
cana-2086	450	14	a.	a.	NOUN
cana-2086	450	15	a	a	NOUN
cana-2086	450	16	,	,	PUNCT
cana-2086	450	17	sadati	sadati	PROPN
cana-2086	450	18	,	,	PUNCT
cana-2086	450	19	s.h	s.h	PROPN
cana-2086	450	20	.	.	PROPN
cana-2086	450	21	,	,	PUNCT
cana-2086	450	22	rashmanlou	rashmanlou	PROPN
cana-2086	450	23	,	,	PUNCT
cana-2086	450	24	h.	h.	NOUN
cana-2086	450	25	:	:	PUNCT
cana-2086	450	26	investigation	investigation	NOUN
cana-2086	450	27	of	of	ADP
cana-2086	450	28	the	the	DET
cana-2086	450	29	main	main	ADJ
cana-2086	450	30	energies	energy	NOUN
cana-2086	450	31	of	of	ADP
cana-2086	450	32	picture	picture	NOUN
cana-2086	450	33	fuzzy	fuzzy	ADJ
cana-2086	450	34	graph	graph	NOUN
cana-2086	450	35	and	and	CCONJ
cana-2086	450	36	its	its	PRON
cana-2086	450	37	applications	application	NOUN
cana-2086	450	38	.	.	PUNCT
cana-2086	451	1	international	international	ADJ
cana-2086	451	2	journal	journal	NOUN
cana-2086	451	3	of	of	ADP
cana-2086	451	4	computational	computational	ADJ
cana-2086	451	5	intelligence	intelligence	NOUN
cana-2086	451	6	systems	system	NOUN
cana-2086	451	7	15	15	NUM
cana-2086	451	8	,	,	PUNCT
cana-2086	451	9	no	no	INTJ
cana-2086	451	10	.	.	NOUN
cana-2086	451	11	1	1	NUM
cana-2086	451	12	:	:	SYM
cana-2086	451	13	31	31	NUM
cana-2086	451	14	(	(	PUNCT
cana-2086	451	15	2022	2022	NUM
cana-2086	451	16	)	)	PUNCT
cana-2086	451	17	.	.	PUNCT
cana-2086	452	1	https://doi.org/10.1007/s44196-022-00086-5	https://doi.org/10.1007/s44196-022-00086-5	NUM
cana-2086	453	1	[	[	X
cana-2086	453	2	29	29	NUM
cana-2086	453	3	]	]	X
cana-2086	453	4	shoaib	shoaib	PROPN
cana-2086	453	5	,	,	PUNCT
cana-2086	453	6	m.	m.	NOUN
cana-2086	453	7	,	,	PUNCT
cana-2086	453	8	kosari	kosari	X
cana-2086	453	9	,	,	PUNCT
cana-2086	453	10	s.	s.	PROPN
cana-2086	453	11	,	,	PUNCT
cana-2086	453	12	rashmanlou	rashmanlou	PROPN
cana-2086	453	13	,	,	PUNCT
cana-2086	453	14	h.	h.	PROPN
cana-2086	453	15	,	,	PUNCT
cana-2086	453	16	aslam	aslam	PROPN
cana-2086	453	17	malik	malik	PROPN
cana-2086	453	18	,	,	PUNCT
cana-2086	453	19	m.	m.	NOUN
cana-2086	453	20	,	,	PUNCT
cana-2086	453	21	rao	rao	PROPN
cana-2086	453	22	,	,	PUNCT
cana-2086	453	23	y.	y.	PROPN
cana-2086	453	24	,	,	PUNCT
cana-2086	453	25	talebi	talebi	PRON
cana-2086	453	26	,	,	PUNCT
cana-2086	453	27	y.	y.	NOUN
cana-2086	453	28	,	,	PUNCT
cana-2086	453	29	mofidnakhaei	mofidnakhaei	NOUN
cana-2086	453	30	,	,	PUNCT
cana-2086	453	31	f.	f.	PROPN
cana-2086	453	32	:	:	PUNCT
cana-2086	453	33	notion	notion	NOUN
cana-2086	453	34	of	of	ADP
cana-2086	453	35	complex	complex	ADJ
cana-2086	453	36	pythagorean	pythagorean	ADJ
cana-2086	453	37	fuzzy	fuzzy	ADJ
cana-2086	453	38	graph	graph	NOUN
cana-2086	453	39	with	with	ADP
cana-2086	453	40	properties	property	NOUN
cana-2086	453	41	and	and	CCONJ
cana-2086	453	42	application	application	NOUN
cana-2086	453	43	.	.	PUNCT
cana-2086	454	1	journal	journal	NOUN
cana-2086	454	2	of	of	ADP
cana-2086	454	3	multiple	multiple	ADV
cana-2086	454	4	-	-	PUNCT
cana-2086	454	5	valued	value	VERB
cana-2086	454	6	logic	logic	NOUN
cana-2086	454	7	&	&	CCONJ
cana-2086	454	8	soft	soft	ADJ
cana-2086	454	9	computing	compute	VERB
cana-2086	454	10	34	34	NUM
cana-2086	454	11	(	(	PUNCT
cana-2086	454	12	2020	2020	NUM
cana-2086	454	13	)	)	PUNCT
cana-2086	454	14	.	.	PUNCT
cana-2086	455	1	[	[	X
cana-2086	455	2	30	30	NUM
cana-2086	455	3	]	]	X
cana-2086	455	4	zadeh	zadeh	PROPN
cana-2086	455	5	,	,	PUNCT
cana-2086	455	6	l.	l.	PROPN
cana-2086	455	7	a.	a.	PROPN
cana-2086	455	8	:	:	PUNCT
cana-2086	455	9	fuzzy	fuzzy	ADJ
cana-2086	455	10	sets	set	NOUN
cana-2086	455	11	.	.	PUNCT
cana-2086	456	1	information	information	NOUN
cana-2086	456	2	and	and	CCONJ
cana-2086	456	3	control	control	NOUN
cana-2086	456	4	,	,	PUNCT
cana-2086	456	5	8(3	8(3	NUM
cana-2086	456	6	)	)	PUNCT
cana-2086	456	7	,	,	PUNCT
cana-2086	456	8	338	338	NUM
cana-2086	456	9	-	-	SYM
cana-2086	456	10	353	353	NUM
cana-2086	456	11	(	(	PUNCT
cana-2086	456	12	1965	1965	NUM
cana-2086	456	13	)	)	PUNCT
cana-2086	456	14	.	.	PUNCT
cana-2086	457	1	https://doi.org/10.1016/s00199958(65)90241-x	https://doi.org/10.1016/s00199958(65)90241-x	PRON
cana-2086	457	2	https://doi.org/10.1007/978-981-33-4389-4_53	https://doi.org/10.1007/978-981-33-4389-4_53	VERB
cana-2086	457	3	https://doi.org/10.3389/fphy.2020.00357	https://doi.org/10.3389/fphy.2020.00357	PROPN
cana-2086	457	4	https://doi.org/10.1063/5.0099955	https://doi.org/10.1063/5.0099955	PROPN
cana-2086	457	5	https://doi.org/10.1007/s44196-022-00086-5	https://doi.org/10.1007/s44196-022-00086-5	NUM
cana-2086	457	6	https://doi.org/10.1016/s0019-9958(65)90241-x	https://doi.org/10.1016/s0019-9958(65)90241-x	PROPN
cana-2086	457	7	https://doi.org/10.1016/s0019-9958(65)90241-x	https://doi.org/10.1016/s0019-9958(65)90241-x	NOUN