id	sid	tid	token	lemma	pos
cana-2401	1	1	communications	communication	NOUN
cana-2401	1	2	on	on	ADP
cana-2401	1	3	applied	apply	VERB
cana-2401	1	4	nonlinear	nonlinear	ADJ
cana-2401	1	5	analysis	analysis	NOUN
cana-2401	1	6	issn	issn	NOUN
cana-2401	1	7	:	:	PUNCT
cana-2401	1	8	1074	1074	NUM
cana-2401	1	9	-	-	PUNCT
cana-2401	1	10	133x	133x	NUM
cana-2401	1	11	vol	vol	NOUN
cana-2401	1	12	32	32	NUM
cana-2401	1	13	no	no	NOUN
cana-2401	1	14	.	.	PUNCT
cana-2401	2	1	2s	2s	NUM
cana-2401	2	2	(	(	PUNCT
cana-2401	2	3	2025	2025	NUM
cana-2401	2	4	)	)	PUNCT
cana-2401	2	5	296	296	NUM
cana-2401	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	2	7	even	even	ADV
cana-2401	2	8	vertex	vertex	NOUN
cana-2401	2	9	odd	odd	ADJ
cana-2401	2	10	edge	edge	NOUN
cana-2401	2	11	root	root	NOUN
cana-2401	2	12	square	square	ADJ
cana-2401	2	13	mean	mean	ADJ
cana-2401	2	14	labelling	labelling	NOUN
cana-2401	2	15	of	of	ADP
cana-2401	2	16	path	path	NOUN
cana-2401	2	17	related	relate	VERB
cana-2401	2	18	graphs	graph	NOUN
cana-2401	2	19	k.	k.	PROPN
cana-2401	2	20	n.	n.	PROPN
cana-2401	2	21	babu1	babu1	PROPN
cana-2401	2	22	,	,	PUNCT
cana-2401	2	23	s.	s.	PROPN
cana-2401	2	24	meenakshi2	meenakshi2	PROPN
cana-2401	2	25	1	1	NUM
cana-2401	2	26	research	research	NOUN
cana-2401	2	27	scholar	scholar	NOUN
cana-2401	2	28	,	,	PUNCT
cana-2401	2	29	department	department	NOUN
cana-2401	2	30	of	of	ADP
cana-2401	2	31	mathematics	mathematics	PROPN
cana-2401	2	32	,	,	PUNCT
cana-2401	2	33	vels	vels	PROPN
cana-2401	2	34	institute	institute	PROPN
cana-2401	2	35	of	of	ADP
cana-2401	2	36	science	science	NOUN
cana-2401	2	37	,	,	PUNCT
cana-2401	2	38	technology	technology	NOUN
cana-2401	2	39	and	and	CCONJ
cana-2401	2	40	advanced	advanced	ADJ
cana-2401	2	41	studies	study	NOUN
cana-2401	2	42	(	(	PUNCT
cana-2401	2	43	vistas	vista	NOUN
cana-2401	2	44	)	)	PUNCT
cana-2401	2	45	,	,	PUNCT
cana-2401	2	46	associate	associate	NOUN
cana-2401	2	47	professor	professor	NOUN
cana-2401	2	48	,	,	PUNCT
cana-2401	2	49	sri	sri	PROPN
cana-2401	2	50	malolan	malolan	PROPN
cana-2401	2	51	college	college	PROPN
cana-2401	2	52	of	of	ADP
cana-2401	2	53	arts	art	NOUN
cana-2401	2	54	and	and	CCONJ
cana-2401	2	55	science	science	NOUN
cana-2401	2	56	.	.	PUNCT
cana-2401	3	1	email	email	NOUN
cana-2401	3	2	:	:	PUNCT
cana-2401	3	3	babumalolan@gmail.com	babumalolan@gmail.com	X
cana-2401	3	4	2	2	NUM
cana-2401	3	5	research	research	NOUN
cana-2401	3	6	supervisor	supervisor	NOUN
cana-2401	3	7	,	,	PUNCT
cana-2401	3	8	vels	vels	PROPN
cana-2401	3	9	institute	institute	PROPN
cana-2401	3	10	of	of	ADP
cana-2401	3	11	science	science	NOUN
cana-2401	3	12	,	,	PUNCT
cana-2401	3	13	technology	technology	NOUN
cana-2401	3	14	and	and	CCONJ
cana-2401	3	15	advanced	advanced	ADJ
cana-2401	3	16	studies	study	NOUN
cana-2401	3	17	(	(	PUNCT
cana-2401	3	18	vistas	vista	NOUN
cana-2401	3	19	)	)	PUNCT
cana-2401	3	20	,	,	PUNCT
cana-2401	3	21	chennai	chennai	PROPN
cana-2401	3	22	,	,	PUNCT
cana-2401	3	23	india	india	PROPN
cana-2401	3	24	.	.	PUNCT
cana-2401	4	1	email	email	NOUN
cana-2401	4	2	:	:	PUNCT
cana-2401	4	3	meenakshikarthikeyan@yahoo.in	meenakshikarthikeyan@yahoo.in	ADV
cana-2401	4	4	article	article	NOUN
cana-2401	4	5	history	history	NOUN
cana-2401	4	6	:	:	PUNCT
cana-2401	4	7	received	receive	VERB
cana-2401	4	8	:	:	PUNCT
cana-2401	4	9	14	14	NUM
cana-2401	4	10	-	-	SYM
cana-2401	4	11	09	09	NUM
cana-2401	4	12	-	-	PUNCT
cana-2401	4	13	2024	2024	NUM
cana-2401	4	14	revised	revise	VERB
cana-2401	4	15	:	:	PUNCT
cana-2401	4	16	22	22	NUM
cana-2401	4	17	-	-	SYM
cana-2401	4	18	10	10	NUM
cana-2401	4	19	-	-	PUNCT
cana-2401	4	20	2024	2024	NUM
cana-2401	4	21	accepted	accept	VERB
cana-2401	4	22	:	:	PUNCT
cana-2401	4	23	02	02	NUM
cana-2401	4	24	-	-	SYM
cana-2401	4	25	11	11	NUM
cana-2401	4	26	-	-	PUNCT
cana-2401	4	27	2024	2024	NUM
cana-2401	4	28	abstract	abstract	NOUN
cana-2401	4	29	:	:	PUNCT
cana-2401	4	30	consider	consider	VERB
cana-2401	4	31	g	g	NOUN
cana-2401	4	32	be	be	AUX
cana-2401	4	33	a	a	DET
cana-2401	4	34	graph	graph	NOUN
cana-2401	4	35	with	with	ADP
cana-2401	4	36	p	p	NOUN
cana-2401	4	37	vertices	vertex	NOUN
cana-2401	4	38	and	and	CCONJ
cana-2401	4	39	q	q	NOUN
cana-2401	4	40	edges	edge	NOUN
cana-2401	4	41	.	.	PUNCT
cana-2401	5	1	a	a	DET
cana-2401	5	2	graph	graph	NOUN
cana-2401	5	3	g	g	NOUN
cana-2401	5	4	is	be	AUX
cana-2401	5	5	said	say	VERB
cana-2401	5	6	to	to	PART
cana-2401	5	7	be	be	AUX
cana-2401	5	8	even	even	ADV
cana-2401	5	9	vertex	vertex	NOUN
cana-2401	5	10	odd	odd	ADJ
cana-2401	5	11	edge	edge	NOUN
cana-2401	5	12	root	root	NOUN
cana-2401	5	13	square	square	ADJ
cana-2401	5	14	mean	mean	NOUN
cana-2401	5	15	labelling	labelling	NOUN
cana-2401	5	16	if	if	SCONJ
cana-2401	5	17	there	there	PRON
cana-2401	5	18	exist	exist	VERB
cana-2401	5	19	an	an	DET
cana-2401	5	20	injective	injective	ADJ
cana-2401	5	21	map	map	NOUN
cana-2401	6	1	f	f	X
cana-2401	6	2	:	:	PUNCT
cana-2401	6	3	v(g	v(g	NUM
cana-2401	6	4	)	)	PUNCT
cana-2401	6	5	→	→	SYM
cana-2401	6	6	{	{	PUNCT
cana-2401	6	7	0,1,2,3	0,1,2,3	NUM
cana-2401	6	8	,	,	PUNCT
cana-2401	6	9	…	…	PUNCT
cana-2401	6	10	,	,	PUNCT
cana-2401	6	11	2q	2q	X
cana-2401	6	12	}	}	PUNCT
cana-2401	6	13	such	such	ADJ
cana-2401	6	14	that	that	SCONJ
cana-2401	6	15	the	the	DET
cana-2401	6	16	induced	induced	ADJ
cana-2401	6	17	edge	edge	NOUN
cana-2401	6	18	labels	label	NOUN
cana-2401	6	19	are	be	AUX
cana-2401	6	20	odd	odd	ADJ
cana-2401	6	21	and	and	CCONJ
cana-2401	6	22	distinct	distinct	ADJ
cana-2401	6	23	which	which	PRON
cana-2401	6	24	can	can	AUX
cana-2401	6	25	be	be	AUX
cana-2401	6	26	obtained	obtain	VERB
cana-2401	6	27	by	by	ADP
cana-2401	6	28	f	f	PROPN
cana-2401	6	29	∗	∗	NOUN
cana-2401	6	30	(	(	PUNCT
cana-2401	6	31	e	e	NOUN
cana-2401	6	32	)	)	PUNCT
cana-2401	6	33	=	=	NOUN
cana-2401	7	1	⌈√	⌈√	X
cana-2401	7	2	f(u)2	f(u)2	PROPN
cana-2401	7	3	+	+	PROPN
cana-2401	7	4	f(v)2	f(v)2	PROPN
cana-2401	7	5	2	2	NUM
cana-2401	7	6	⌉	⌉	NOUN
cana-2401	7	7	or	or	CCONJ
cana-2401	7	8	⌊√	⌊√	PROPN
cana-2401	7	9	f(u)2	f(u)2	PROPN
cana-2401	7	10	+	+	PROPN
cana-2401	7	11	f(v)2	f(v)2	ADJ
cana-2401	7	12	2	2	NUM
cana-2401	7	13	⌋.	⌋.	NOUN
cana-2401	7	14	any	any	DET
cana-2401	7	15	graph	graph	NOUN
cana-2401	7	16	which	which	PRON
cana-2401	7	17	admits	admit	VERB
cana-2401	7	18	even	even	ADV
cana-2401	7	19	vertex	vertex	NOUN
cana-2401	7	20	odd	odd	ADJ
cana-2401	7	21	edge	edge	NOUN
cana-2401	7	22	root	root	NOUN
cana-2401	7	23	square	square	ADJ
cana-2401	7	24	mean	mean	NOUN
cana-2401	7	25	labelling	labelling	NOUN
cana-2401	7	26	then	then	ADV
cana-2401	7	27	it	it	PRON
cana-2401	7	28	is	be	AUX
cana-2401	7	29	called	call	VERB
cana-2401	7	30	as	as	ADP
cana-2401	7	31	even	even	ADV
cana-2401	7	32	vertex	vertex	NOUN
cana-2401	7	33	odd	odd	ADJ
cana-2401	7	34	edge	edge	NOUN
cana-2401	7	35	root	root	NOUN
cana-2401	7	36	square	square	ADJ
cana-2401	7	37	mean	mean	ADJ
cana-2401	7	38	labelling	labelling	NOUN
cana-2401	7	39	graph	graph	NOUN
cana-2401	7	40	.	.	PUNCT
cana-2401	8	1	keywords	keyword	NOUN
cana-2401	8	2	:	:	PUNCT
cana-2401	8	3	evoe	evoe	NOUN
cana-2401	8	4	–	–	PUNCT
cana-2401	8	5	rsml	rsml	NOUN
cana-2401	8	6	,	,	PUNCT
cana-2401	8	7	graph	graph	NOUN
cana-2401	8	8	,	,	PUNCT
cana-2401	8	9	ladder	ladder	NOUN
cana-2401	8	10	,	,	PUNCT
cana-2401	8	11	corona	corona	NOUN
cana-2401	8	12	graph	graph	NOUN
cana-2401	8	13	1.introduction	1.introduction	NUM
cana-2401	8	14	:	:	PUNCT
cana-2401	8	15	throughout	throughout	ADP
cana-2401	8	16	the	the	DET
cana-2401	8	17	work	work	NOUN
cana-2401	8	18	,	,	PUNCT
cana-2401	8	19	we	we	PRON
cana-2401	8	20	consider	consider	VERB
cana-2401	8	21	a	a	DET
cana-2401	8	22	simple	simple	ADJ
cana-2401	8	23	graph	graph	NOUN
cana-2401	8	24	.	.	PUNCT
cana-2401	9	1	the	the	DET
cana-2401	9	2	terminology	terminology	NOUN
cana-2401	9	3	and	and	CCONJ
cana-2401	9	4	the	the	DET
cana-2401	9	5	definitions	definition	NOUN
cana-2401	9	6	of	of	ADP
cana-2401	9	7	graph	graph	NOUN
cana-2401	9	8	were	be	AUX
cana-2401	9	9	followed	follow	VERB
cana-2401	9	10	in	in	ADP
cana-2401	9	11	[	[	X
cana-2401	9	12	1	1	NUM
cana-2401	9	13	]	]	PUNCT
cana-2401	9	14	.	.	PUNCT
cana-2401	10	1	for	for	ADP
cana-2401	10	2	a	a	DET
cana-2401	10	3	complete	complete	ADJ
cana-2401	10	4	analysis	analysis	NOUN
cana-2401	10	5	of	of	ADP
cana-2401	10	6	labeling	labeling	NOUN
cana-2401	10	7	studied	study	VERB
cana-2401	10	8	by	by	ADP
cana-2401	10	9	gallian	gallian	ADJ
cana-2401	10	10	[	[	X
cana-2401	10	11	1	1	NUM
cana-2401	10	12	]	]	PUNCT
cana-2401	10	13	.	.	PUNCT
cana-2401	11	1	in	in	ADP
cana-2401	11	2	1960	1960	NUM
cana-2401	11	3	’s	’s	PART
cana-2401	11	4	graph	graph	NOUN
cana-2401	11	5	labeling	labeling	NOUN
cana-2401	11	6	was	be	AUX
cana-2401	11	7	introduced	introduce	VERB
cana-2401	11	8	by	by	ADP
cana-2401	11	9	rosa	rosa	PROPN
cana-2401	11	10	.	.	PROPN
cana-2401	11	11	odd	odd	ADJ
cana-2401	11	12	vertex	vertex	NOUN
cana-2401	11	13	even	even	ADV
cana-2401	11	14	edge	edge	NOUN
cana-2401	11	15	root	root	NOUN
cana-2401	11	16	square	square	ADJ
cana-2401	11	17	mean	mean	NOUN
cana-2401	11	18	labeling	labeling	NOUN
cana-2401	11	19	discussed	discuss	VERB
cana-2401	11	20	in	in	ADP
cana-2401	11	21	[	[	X
cana-2401	11	22	5	5	NUM
cana-2401	11	23	]	]	PUNCT
cana-2401	11	24	which	which	PRON
cana-2401	11	25	motivated	motivate	VERB
cana-2401	11	26	to	to	PART
cana-2401	11	27	construct	construct	VERB
cana-2401	11	28	the	the	DET
cana-2401	11	29	above	above	ADJ
cana-2401	11	30	said	say	VERB
cana-2401	11	31	labeling	labeling	NOUN
cana-2401	11	32	pattern	pattern	NOUN
cana-2401	11	33	.	.	PUNCT
cana-2401	12	1	we	we	PRON
cana-2401	12	2	study	study	VERB
cana-2401	12	3	a	a	DET
cana-2401	12	4	few	few	ADJ
cana-2401	12	5	paths	path	NOUN
cana-2401	12	6	related	relate	VERB
cana-2401	12	7	graphs	graph	NOUN
cana-2401	12	8	which	which	PRON
cana-2401	12	9	satisfies	satisfy	VERB
cana-2401	12	10	the	the	DET
cana-2401	12	11	above	above	ADJ
cana-2401	12	12	pattern	pattern	NOUN
cana-2401	12	13	.	.	PUNCT
cana-2401	13	1	mean	mean	VERB
cana-2401	13	2	labeling	labeling	NOUN
cana-2401	13	3	,	,	PUNCT
cana-2401	13	4	root	root	NOUN
cana-2401	13	5	square	square	ADJ
cana-2401	13	6	mean	mean	NOUN
cana-2401	13	7	and	and	CCONJ
cana-2401	13	8	super	super	ADJ
cana-2401	13	9	root	root	PROPN
cana-2401	13	10	square	square	ADJ
cana-2401	13	11	mean	mean	NOUN
cana-2401	13	12	labeling	labeling	NOUN
cana-2401	13	13	were	be	AUX
cana-2401	13	14	discussed	discuss	VERB
cana-2401	13	15	in	in	ADP
cana-2401	13	16	[	[	X
cana-2401	13	17	4,6,7,8	4,6,7,8	NUM
cana-2401	13	18	]	]	PUNCT
cana-2401	13	19	.	.	PUNCT
cana-2401	14	1	2	2	NUM
cana-2401	14	2	preliminary	preliminary	NOUN
cana-2401	14	3	:	:	PUNCT
cana-2401	14	4	definition	definition	NOUN
cana-2401	14	5	2.1	2.1	NUM
cana-2401	14	6	[	[	SYM
cana-2401	14	7	5	5	NUM
cana-2401	14	8	]	]	PUNCT
cana-2401	14	9	a	a	DET
cana-2401	14	10	graph	graph	NOUN
cana-2401	14	11	g	g	NOUN
cana-2401	14	12	is	be	AUX
cana-2401	14	13	said	say	VERB
cana-2401	14	14	to	to	PART
cana-2401	14	15	be	be	AUX
cana-2401	14	16	odd	odd	ADJ
cana-2401	14	17	vertex	vertex	NOUN
cana-2401	14	18	even	even	ADV
cana-2401	14	19	edge	edge	NOUN
cana-2401	14	20	root	root	NOUN
cana-2401	14	21	square	square	ADJ
cana-2401	14	22	mean	mean	NOUN
cana-2401	14	23	labelling	labelling	NOUN
cana-2401	14	24	if	if	SCONJ
cana-2401	14	25	there	there	PRON
cana-2401	14	26	exist	exist	VERB
cana-2401	14	27	an	an	DET
cana-2401	14	28	injective	injective	ADJ
cana-2401	14	29	map	map	NOUN
cana-2401	15	1	f	f	X
cana-2401	15	2	:	:	PUNCT
cana-2401	15	3	v(g	v(g	NUM
cana-2401	15	4	)	)	PUNCT
cana-2401	15	5	→	→	SYM
cana-2401	15	6	{	{	PUNCT
cana-2401	15	7	1,2,3	1,2,3	NUM
cana-2401	15	8	,	,	PUNCT
cana-2401	15	9	…	…	PUNCT
cana-2401	15	10	,	,	PUNCT
cana-2401	15	11	2q	2q	X
cana-2401	15	12	+	+	NOUN
cana-2401	15	13	1	1	X
cana-2401	15	14	}	}	PUNCT
cana-2401	15	15	such	such	ADJ
cana-2401	15	16	that	that	SCONJ
cana-2401	15	17	the	the	DET
cana-2401	15	18	induced	induced	ADJ
cana-2401	15	19	edge	edge	NOUN
cana-2401	15	20	labels	label	NOUN
cana-2401	15	21	are	be	AUX
cana-2401	15	22	odd	odd	ADJ
cana-2401	15	23	and	and	CCONJ
cana-2401	15	24	distinct	distinct	ADJ
cana-2401	15	25	which	which	PRON
cana-2401	15	26	can	can	AUX
cana-2401	15	27	be	be	AUX
cana-2401	15	28	obtained	obtain	VERB
cana-2401	15	29	by	by	ADP
cana-2401	15	30	f	f	PROPN
cana-2401	15	31	∗	∗	NOUN
cana-2401	15	32	(	(	PUNCT
cana-2401	15	33	e	e	NOUN
cana-2401	15	34	)	)	PUNCT
cana-2401	15	35	=	=	PUNCT
cana-2401	16	1	⌈√f(u)2	⌈√f(u)2	PROPN
cana-2401	16	2	+	+	NOUN
cana-2401	16	3	f(v)2	f(v)2	ADJ
cana-2401	16	4	2	2	NUM
cana-2401	16	5	⌉	⌉	NOUN
cana-2401	16	6	or	or	CCONJ
cana-2401	16	7	⌊√f(u)2	⌊√f(u)2	ADP
cana-2401	16	8	+	+	ADJ
cana-2401	16	9	f(v)2	f(v)2	PROPN
cana-2401	16	10	2	2	NUM
cana-2401	16	11	⌋	⌋	NOUN
cana-2401	16	12	.	.	PUNCT
cana-2401	17	1	any	any	DET
cana-2401	17	2	graph	graph	NOUN
cana-2401	17	3	which	which	PRON
cana-2401	17	4	admits	admit	VERB
cana-2401	17	5	even	even	ADV
cana-2401	17	6	vertex	vertex	NOUN
cana-2401	17	7	odd	odd	ADJ
cana-2401	17	8	edge	edge	NOUN
cana-2401	17	9	root	root	NOUN
cana-2401	17	10	square	square	ADJ
cana-2401	17	11	mean	mean	NOUN
cana-2401	17	12	labelling	labelling	NOUN
cana-2401	17	13	then	then	ADV
cana-2401	17	14	it	it	PRON
cana-2401	17	15	is	be	AUX
cana-2401	17	16	called	call	VERB
cana-2401	17	17	as	as	ADP
cana-2401	17	18	odd	odd	ADJ
cana-2401	17	19	vertex	vertex	NOUN
cana-2401	17	20	even	even	ADV
cana-2401	17	21	edge	edge	NOUN
cana-2401	17	22	root	root	NOUN
cana-2401	17	23	square	square	ADJ
cana-2401	17	24	mean	mean	ADJ
cana-2401	17	25	labelling	labelling	NOUN
cana-2401	17	26	graph	graph	NOUN
cana-2401	17	27	.	.	PUNCT
cana-2401	18	1	definition	definition	NOUN
cana-2401	18	2	2.2	2.2	NUM
cana-2401	18	3	a	a	DET
cana-2401	18	4	graph	graph	NOUN
cana-2401	18	5	g	g	NOUN
cana-2401	18	6	is	be	AUX
cana-2401	18	7	said	say	VERB
cana-2401	18	8	to	to	PART
cana-2401	18	9	be	be	AUX
cana-2401	18	10	even	even	ADV
cana-2401	18	11	vertex	vertex	NOUN
cana-2401	18	12	odd	odd	ADJ
cana-2401	18	13	edge	edge	NOUN
cana-2401	18	14	root	root	NOUN
cana-2401	18	15	square	square	ADJ
cana-2401	18	16	mean	mean	NOUN
cana-2401	18	17	labelling	labelling	NOUN
cana-2401	18	18	if	if	SCONJ
cana-2401	18	19	there	there	PRON
cana-2401	18	20	exist	exist	VERB
cana-2401	18	21	an	an	DET
cana-2401	18	22	injective	injective	ADJ
cana-2401	18	23	map	map	NOUN
cana-2401	19	1	f	f	X
cana-2401	19	2	:	:	PUNCT
cana-2401	19	3	v(g	v(g	NUM
cana-2401	19	4	)	)	PUNCT
cana-2401	19	5	→	→	SYM
cana-2401	19	6	{	{	PUNCT
cana-2401	19	7	0,1,2,3	0,1,2,3	NUM
cana-2401	19	8	,	,	PUNCT
cana-2401	19	9	…	…	PUNCT
cana-2401	19	10	,	,	PUNCT
cana-2401	19	11	2q	2q	X
cana-2401	19	12	}	}	PUNCT
cana-2401	19	13	such	such	ADJ
cana-2401	19	14	that	that	SCONJ
cana-2401	19	15	the	the	DET
cana-2401	19	16	induced	induced	ADJ
cana-2401	19	17	edge	edge	NOUN
cana-2401	19	18	labels	label	NOUN
cana-2401	19	19	are	be	AUX
cana-2401	19	20	odd	odd	ADJ
cana-2401	19	21	and	and	CCONJ
cana-2401	19	22	distinct	distinct	ADJ
cana-2401	19	23	which	which	PRON
cana-2401	19	24	can	can	AUX
cana-2401	19	25	be	be	AUX
cana-2401	19	26	obtained	obtain	VERB
cana-2401	19	27	by	by	ADP
cana-2401	19	28	f	f	PROPN
cana-2401	19	29	∗	∗	NOUN
cana-2401	19	30	(	(	PUNCT
cana-2401	19	31	e	e	NOUN
cana-2401	19	32	)	)	PUNCT
cana-2401	19	33	=	=	PUNCT
cana-2401	19	34	⌈√f(u)2	⌈√f(u)2	PROPN
cana-2401	19	35	+	+	NOUN
cana-2401	19	36	f(v)2	f(v)2	ADJ
cana-2401	19	37	2	2	NUM
cana-2401	19	38	⌉	⌉	NOUN
cana-2401	19	39	or	or	CCONJ
cana-2401	19	40	⌊√f(u)2	⌊√f(u)2	ADP
cana-2401	19	41	+	+	ADJ
cana-2401	19	42	f(v)2	f(v)2	PROPN
cana-2401	19	43	2	2	NUM
cana-2401	19	44	⌋	⌋	NOUN
cana-2401	19	45	.	.	PUNCT
cana-2401	20	1	any	any	DET
cana-2401	20	2	graph	graph	NOUN
cana-2401	20	3	which	which	PRON
cana-2401	20	4	admits	admit	VERB
cana-2401	20	5	even	even	ADV
cana-2401	20	6	vertex	vertex	NOUN
cana-2401	20	7	odd	odd	ADJ
cana-2401	20	8	edge	edge	NOUN
cana-2401	20	9	root	root	NOUN
cana-2401	20	10	square	square	ADJ
cana-2401	20	11	mean	mean	NOUN
cana-2401	20	12	labelling	labelling	NOUN
cana-2401	20	13	then	then	ADV
cana-2401	20	14	it	it	PRON
cana-2401	20	15	is	be	AUX
cana-2401	20	16	called	call	VERB
cana-2401	20	17	as	as	ADP
cana-2401	20	18	even	even	ADV
cana-2401	20	19	vertex	vertex	NOUN
cana-2401	20	20	odd	odd	ADJ
cana-2401	20	21	edge	edge	NOUN
cana-2401	20	22	root	root	NOUN
cana-2401	20	23	square	square	ADJ
cana-2401	20	24	mean	mean	ADJ
cana-2401	20	25	labelling	labelling	NOUN
cana-2401	20	26	graph	graph	NOUN
cana-2401	20	27	.	.	PUNCT
cana-2401	21	1	mailto:babumalolan@gmail.com	mailto:babumalolan@gmail.com	PROPN
cana-2401	21	2	communications	communication	NOUN
cana-2401	21	3	on	on	ADP
cana-2401	21	4	applied	apply	VERB
cana-2401	21	5	nonlinear	nonlinear	ADJ
cana-2401	21	6	analysis	analysis	NOUN
cana-2401	21	7	issn	issn	NOUN
cana-2401	21	8	:	:	PUNCT
cana-2401	21	9	1074	1074	NUM
cana-2401	21	10	-	-	PUNCT
cana-2401	21	11	133x	133x	NUM
cana-2401	21	12	vol	vol	NOUN
cana-2401	21	13	32	32	NUM
cana-2401	21	14	no	no	NOUN
cana-2401	21	15	.	.	PUNCT
cana-2401	22	1	2s	2s	NUM
cana-2401	22	2	(	(	PUNCT
cana-2401	22	3	2025	2025	NUM
cana-2401	22	4	)	)	PUNCT
cana-2401	22	5	297	297	NUM
cana-2401	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	22	7	definition	definition	NOUN
cana-2401	22	8	2.3	2.3	NUM
cana-2401	23	1	[	[	X
cana-2401	23	2	6	6	NUM
cana-2401	23	3	]	]	PUNCT
cana-2401	23	4	:	:	PUNCT
cana-2401	23	5	a	a	DET
cana-2401	23	6	graph	graph	NOUN
cana-2401	23	7	g	g	NOUN
cana-2401	23	8	with	with	ADP
cana-2401	23	9	p	p	NOUN
cana-2401	23	10	vertices	vertex	NOUN
cana-2401	23	11	and	and	CCONJ
cana-2401	23	12	q	q	NOUN
cana-2401	23	13	edges	edge	NOUN
cana-2401	23	14	,	,	PUNCT
cana-2401	23	15	is	be	AUX
cana-2401	23	16	a	a	DET
cana-2401	23	17	mean	mean	ADJ
cana-2401	23	18	graph	graph	NOUN
cana-2401	23	19	if	if	SCONJ
cana-2401	23	20	there	there	PRON
cana-2401	23	21	is	be	VERB
cana-2401	23	22	an	an	DET
cana-2401	23	23	injective	injective	ADJ
cana-2401	23	24	function	function	NOUN
cana-2401	23	25	f	f	NOUN
cana-2401	23	26	from	from	ADP
cana-2401	23	27	the	the	DET
cana-2401	23	28	vertex	vertex	NOUN
cana-2401	23	29	set	set	VERB
cana-2401	23	30	to	to	ADP
cana-2401	23	31	{	{	PUNCT
cana-2401	23	32	1	1	NUM
cana-2401	23	33	,	,	PUNCT
cana-2401	23	34	2	2	NUM
cana-2401	23	35	,	,	PUNCT
cana-2401	23	36	3	3	NUM
cana-2401	23	37	…	…	SYM
cana-2401	23	38	q	q	NOUN
cana-2401	23	39	}	}	PUNCT
cana-2401	23	40	when	when	SCONJ
cana-2401	23	41	each	each	DET
cana-2401	23	42	edge	edge	NOUN
cana-2401	23	43	uv	uv	NOUN
cana-2401	23	44	is	be	AUX
cana-2401	23	45	labelled	label	VERB
cana-2401	23	46	with	with	ADP
cana-2401	23	47	𝑓(𝑢)+𝑓(𝑣	𝑓(𝑢)+𝑓(𝑣	PROPN
cana-2401	23	48	)	)	PUNCT
cana-2401	23	49	2	2	NUM
cana-2401	23	50	if	if	SCONJ
cana-2401	23	51	f(u)+f(v	f(u)+f(v	VERB
cana-2401	23	52	)	)	PUNCT
cana-2401	23	53	is	be	AUX
cana-2401	23	54	even	even	ADV
cana-2401	23	55	and	and	CCONJ
cana-2401	23	56	𝑓(𝑢)+𝑓(𝑣)+1	𝑓(𝑢)+𝑓(𝑣)+1	NOUN
cana-2401	23	57	2	2	NUM
cana-2401	23	58	if	if	SCONJ
cana-2401	23	59	f(u)+f(v)+1	f(u)+f(v)+1	NOUN
cana-2401	23	60	is	be	AUX
cana-2401	23	61	odd	odd	ADJ
cana-2401	23	62	then	then	ADV
cana-2401	23	63	the	the	DET
cana-2401	23	64	labelling	labelling	NOUN
cana-2401	23	65	edges	edge	NOUN
cana-2401	23	66	are	be	AUX
cana-2401	23	67	distinct	distinct	ADJ
cana-2401	23	68	.	.	PUNCT
cana-2401	24	1	definition	definition	NOUN
cana-2401	24	2	2.4	2.4	NUM
cana-2401	25	1	[	[	X
cana-2401	25	2	4	4	NUM
cana-2401	25	3	]	]	PUNCT
cana-2401	25	4	:	:	PUNCT
cana-2401	25	5	a	a	DET
cana-2401	25	6	graph	graph	NOUN
cana-2401	25	7	g	g	NOUN
cana-2401	25	8	with	with	ADP
cana-2401	25	9	p	p	NOUN
cana-2401	25	10	vertices	vertex	NOUN
cana-2401	25	11	and	and	CCONJ
cana-2401	25	12	q	q	NOUN
cana-2401	25	13	edges	edge	NOUN
cana-2401	25	14	is	be	AUX
cana-2401	25	15	called	call	VERB
cana-2401	25	16	a	a	DET
cana-2401	25	17	root	root	NOUN
cana-2401	25	18	square	square	ADJ
cana-2401	25	19	mean	mean	NOUN
cana-2401	25	20	graph	graph	NOUN
cana-2401	25	21	if	if	SCONJ
cana-2401	25	22	is	be	AUX
cana-2401	25	23	possible	possible	ADJ
cana-2401	25	24	to	to	PART
cana-2401	25	25	label	label	VERB
cana-2401	25	26	the	the	DET
cana-2401	25	27	vertices	vertex	NOUN
cana-2401	25	28	v	v	NOUN
cana-2401	25	29	with	with	ADP
cana-2401	25	30	distinct	distinct	ADJ
cana-2401	25	31	labels	label	NOUN
cana-2401	25	32	f(x	f(x	PROPN
cana-2401	25	33	)	)	PUNCT
cana-2401	25	34	from	from	ADP
cana-2401	25	35	1	1	NUM
cana-2401	25	36	,	,	PUNCT
cana-2401	25	37	2	2	NUM
cana-2401	25	38	,	,	PUNCT
cana-2401	25	39	3	3	NUM
cana-2401	25	40	,	,	PUNCT
cana-2401	25	41	…	…	PUNCT
cana-2401	25	42	q+1	q+1	NUM
cana-2401	25	43	in	in	ADP
cana-2401	25	44	such	such	DET
cana-2401	25	45	a	a	DET
cana-2401	25	46	way	way	NOUN
cana-2401	25	47	that	that	SCONJ
cana-2401	25	48	when	when	SCONJ
cana-2401	25	49	each	each	DET
cana-2401	25	50	edge	edge	NOUN
cana-2401	25	51	e	e	X
cana-2401	25	52	=	=	NOUN
cana-2401	25	53	uv	uv	NOUN
cana-2401	25	54	is	be	AUX
cana-2401	25	55	labelled	label	VERB
cana-2401	25	56	with	with	ADP
cana-2401	25	57	f(e	f(e	PROPN
cana-2401	25	58	)	)	PUNCT
cana-2401	26	1	=	=	VERB
cana-2401	26	2	⌊√	⌊√	NOUN
cana-2401	26	3	(	(	PUNCT
cana-2401	26	4	f(u)2	f(u)2	NOUN
cana-2401	26	5	+	+	PROPN
cana-2401	26	6	f(v)2	f(v)2	ADJ
cana-2401	26	7	2	2	NUM
cana-2401	26	8	)	)	PUNCT
cana-2401	26	9	⌋	⌋	NOUN
cana-2401	26	10	or	or	CCONJ
cana-2401	26	11	⌈√	⌈√	NOUN
cana-2401	26	12	(	(	PUNCT
cana-2401	26	13	f(u)2	f(u)2	PROPN
cana-2401	26	14	+	+	PROPN
cana-2401	26	15	f(v)2	f(v)2	PROPN
cana-2401	26	16	2	2	NUM
cana-2401	26	17	)	)	PUNCT
cana-2401	26	18	⌉	⌉	VERB
cana-2401	26	19	then	then	ADV
cana-2401	26	20	each	each	DET
cana-2401	26	21	labels	label	NOUN
cana-2401	26	22	are	be	AUX
cana-2401	26	23	distinct	distinct	ADJ
cana-2401	26	24	.	.	PUNCT
cana-2401	27	1	definition	definition	NOUN
cana-2401	27	2	2.5	2.5	NUM
cana-2401	28	1	[	[	X
cana-2401	28	2	4	4	NUM
cana-2401	28	3	]	]	PUNCT
cana-2401	28	4	:	:	PUNCT
cana-2401	28	5	the	the	DET
cana-2401	28	6	corona	corona	NOUN
cana-2401	28	7	graph	graph	NOUN
cana-2401	28	8	is	be	AUX
cana-2401	28	9	obtained	obtain	VERB
cana-2401	28	10	by	by	ADP
cana-2401	28	11	taking	take	VERB
cana-2401	28	12	one	one	NUM
cana-2401	28	13	copy	copy	NOUN
cana-2401	28	14	of	of	ADP
cana-2401	28	15	path	path	NOUN
cana-2401	28	16	pn	pn	PROPN
cana-2401	28	17	and	and	CCONJ
cana-2401	28	18	n	n	PRON
cana-2401	28	19	copies	copy	NOUN
cana-2401	28	20	of	of	ADP
cana-2401	28	21	k1	k1	NOUN
cana-2401	28	22	then	then	ADV
cana-2401	28	23	joining	join	VERB
cana-2401	28	24	the	the	DET
cana-2401	28	25	ith	ith	PROPN
cana-2401	28	26	vertex	vertex	NOUN
cana-2401	28	27	of	of	ADP
cana-2401	28	28	pn	pn	PROPN
cana-2401	28	29	with	with	ADP
cana-2401	28	30	an	an	DET
cana-2401	28	31	edge	edge	NOUN
cana-2401	28	32	to	to	ADP
cana-2401	28	33	every	every	DET
cana-2401	28	34	vertex	vertex	NOUN
cana-2401	28	35	in	in	ADP
cana-2401	28	36	the	the	DET
cana-2401	28	37	ith	ith	PROPN
cana-2401	28	38	copy	copy	NOUN
cana-2401	28	39	of	of	ADP
cana-2401	28	40	k1	k1	NOUN
cana-2401	28	41	.	.	PUNCT
cana-2401	29	1	it	it	PRON
cana-2401	29	2	is	be	AUX
cana-2401	29	3	denoted	denote	VERB
cana-2401	29	4	by	by	ADP
cana-2401	29	5	pn	pn	PROPN
cana-2401	29	6	k1	k1	PROPN
cana-2401	29	7	.	.	PUNCT
cana-2401	30	1	definition	definition	NOUN
cana-2401	30	2	2.6	2.6	NUM
cana-2401	31	1	[	[	X
cana-2401	31	2	7	7	NUM
cana-2401	31	3	]	]	SYM
cana-2401	31	4	:	:	PUNCT
cana-2401	31	5	tw(pn	tw(pn	NOUN
cana-2401	31	6	)	)	PUNCT
cana-2401	31	7	is	be	AUX
cana-2401	31	8	a	a	DET
cana-2401	31	9	graph	graph	NOUN
cana-2401	31	10	which	which	PRON
cana-2401	31	11	is	be	AUX
cana-2401	31	12	obtained	obtain	VERB
cana-2401	31	13	from	from	ADP
cana-2401	31	14	a	a	DET
cana-2401	31	15	path	path	NOUN
cana-2401	31	16	by	by	ADP
cana-2401	31	17	identifying	identify	VERB
cana-2401	31	18	k1,2	k1,2	PROPN
cana-2401	31	19	to	to	ADP
cana-2401	31	20	all	all	DET
cana-2401	31	21	the	the	DET
cana-2401	31	22	vertices	vertex	NOUN
cana-2401	31	23	of	of	ADP
cana-2401	31	24	the	the	DET
cana-2401	31	25	path	path	NOUN
cana-2401	31	26	except	except	SCONJ
cana-2401	31	27	one	one	NUM
cana-2401	31	28	pendent	pendent	NOUN
cana-2401	31	29	vertex	vertex	NOUN
cana-2401	31	30	(	(	PUNCT
cana-2401	31	31	twing	twing	NOUN
cana-2401	31	32	graph	graph	NOUN
cana-2401	31	33	)	)	PUNCT
cana-2401	31	34	a	a	DET
cana-2401	31	35	path	path	NOUN
cana-2401	31	36	with	with	ADP
cana-2401	31	37	a	a	DET
cana-2401	31	38	least	least	ADJ
cana-2401	31	39	vertex	vertex	NOUN
cana-2401	31	40	is	be	AUX
cana-2401	31	41	connected	connect	VERB
cana-2401	31	42	and	and	CCONJ
cana-2401	31	43	has	have	VERB
cana-2401	31	44	two	two	NUM
cana-2401	31	45	terminal	terminal	ADJ
cana-2401	31	46	vertices	vertex	NOUN
cana-2401	31	47	while	while	SCONJ
cana-2401	31	48	all	all	DET
cana-2401	31	49	other	other	ADJ
cana-2401	31	50	vertices	vertex	NOUN
cana-2401	31	51	have	have	VERB
cana-2401	31	52	degree	degree	NOUN
cana-2401	31	53	2	2	NUM
cana-2401	31	54	.	.	PUNCT
cana-2401	31	55	theorem	theorem	VERB
cana-2401	31	56	3.1	3.1	NUM
cana-2401	31	57	:	:	PUNCT
cana-2401	31	58	a	a	DET
cana-2401	31	59	path	path	NOUN
cana-2401	31	60	pn	pn	PROPN
cana-2401	31	61	is	be	AUX
cana-2401	31	62	an	an	DET
cana-2401	31	63	even	even	ADJ
cana-2401	31	64	vertex	vertex	NOUN
cana-2401	31	65	odd	odd	ADJ
cana-2401	31	66	edge	edge	NOUN
cana-2401	31	67	root	root	NOUN
cana-2401	31	68	square	square	ADJ
cana-2401	31	69	mean	mean	ADJ
cana-2401	31	70	labelling	labelling	NOUN
cana-2401	31	71	graph	graph	NOUN
cana-2401	31	72	.	.	PUNCT
cana-2401	32	1	proof	proof	NOUN
cana-2401	32	2	:	:	PUNCT
cana-2401	32	3	let	let	VERB
cana-2401	32	4	g	g	PRON
cana-2401	32	5	be	be	AUX
cana-2401	32	6	a	a	DET
cana-2401	32	7	path	path	NOUN
cana-2401	32	8	pn	pn	NOUN
cana-2401	32	9	,	,	PUNCT
cana-2401	32	10	where	where	SCONJ
cana-2401	32	11	v(g)=	v(g)=	PROPN
cana-2401	32	12	{	{	PUNCT
cana-2401	32	13	ui:1≤	ui:1≤	PROPN
cana-2401	32	14	𝑖	𝑖	PROPN
cana-2401	32	15	≤	≤	PROPN
cana-2401	32	16	𝑛	𝑛	PRON
cana-2401	32	17	}	}	PUNCT
cana-2401	32	18	and	and	CCONJ
cana-2401	32	19	e(g	e(g	NOUN
cana-2401	32	20	)	)	PUNCT
cana-2401	33	1	=	=	PRON
cana-2401	33	2	{	{	PUNCT
cana-2401	33	3	ui	ui	PROPN
cana-2401	33	4	,	,	PUNCT
cana-2401	33	5	ui+1	ui+1	PROPN
cana-2401	33	6	:	:	SYM
cana-2401	33	7	1≤	1≤	NUM
cana-2401	33	8	𝑖	𝑖	SYM
cana-2401	33	9	≤	≤	NUM
cana-2401	34	1	𝑛	𝑛	DET
cana-2401	34	2	−	−	PROPN
cana-2401	34	3	1	1	NUM
cana-2401	34	4	}	}	PUNCT
cana-2401	34	5	it	it	PRON
cana-2401	34	6	is	be	AUX
cana-2401	34	7	evident	evident	ADJ
cana-2401	34	8	that	that	SCONJ
cana-2401	34	9	g	g	PROPN
cana-2401	34	10	allows	allow	VERB
cana-2401	34	11	an	an	DET
cana-2401	34	12	even	even	ADJ
cana-2401	34	13	-	-	PUNCT
cana-2401	34	14	vertex	vertex	NOUN
cana-2401	34	15	odd	odd	ADJ
cana-2401	34	16	-	-	PUNCT
cana-2401	34	17	edge	edge	NOUN
cana-2401	34	18	root	root	NOUN
cana-2401	34	19	square	square	ADJ
cana-2401	34	20	mean	mean	NOUN
cana-2401	34	21	labeling	labeling	NOUN
cana-2401	34	22	,	,	PUNCT
cana-2401	34	23	as	as	SCONJ
cana-2401	34	24	labels	label	NOUN
cana-2401	34	25	can	can	AUX
cana-2401	34	26	be	be	AUX
cana-2401	34	27	assigned	assign	VERB
cana-2401	34	28	to	to	ADP
cana-2401	34	29	its	its	PRON
cana-2401	34	30	vertices	vertex	NOUN
cana-2401	34	31	and	and	CCONJ
cana-2401	34	32	edges	edge	NOUN
cana-2401	34	33	in	in	ADP
cana-2401	34	34	a	a	DET
cana-2401	34	35	sequential	sequential	ADJ
cana-2401	34	36	manner	manner	NOUN
cana-2401	34	37	.	.	PUNCT
cana-2401	35	1	hence	hence	ADV
cana-2401	35	2	,	,	PUNCT
cana-2401	35	3	g	g	PROPN
cana-2401	35	4	is	be	AUX
cana-2401	35	5	classified	classify	VERB
cana-2401	35	6	as	as	ADP
cana-2401	35	7	an	an	DET
cana-2401	35	8	even	even	ADJ
cana-2401	35	9	-	-	PUNCT
cana-2401	35	10	vertex	vertex	NOUN
cana-2401	35	11	odd	odd	ADJ
cana-2401	35	12	-	-	PUNCT
cana-2401	35	13	edge	edge	NOUN
cana-2401	35	14	root	root	NOUN
cana-2401	35	15	square	square	ADJ
cana-2401	35	16	mean	mean	NOUN
cana-2401	35	17	graph	graph	NOUN
cana-2401	35	18	.	.	PUNCT
cana-2401	36	1	illustration	illustration	NOUN
cana-2401	36	2	:	:	PUNCT
cana-2401	36	3	figure	figure	NOUN
cana-2401	36	4	1	1	NUM
cana-2401	36	5	shows	show	VERB
cana-2401	36	6	g	g	NOUN
cana-2401	36	7	=	=	NOUN
cana-2401	36	8	p7	p7	PROPN
cana-2401	36	9	is	be	AUX
cana-2401	36	10	even	even	ADV
cana-2401	36	11	vertex	vertex	NOUN
cana-2401	36	12	odd	odd	ADJ
cana-2401	36	13	edge	edge	NOUN
cana-2401	36	14	root	root	NOUN
cana-2401	36	15	square	square	ADJ
cana-2401	36	16	mean	mean	NOUN
cana-2401	36	17	labelling	labelling	NOUN
cana-2401	36	18	.	.	PUNCT
cana-2401	37	1	figure	figure	VERB
cana-2401	37	2	1	1	NUM
cana-2401	37	3	even	even	ADV
cana-2401	37	4	vertex	vertex	NOUN
cana-2401	37	5	odd	odd	ADJ
cana-2401	37	6	edge	edge	NOUN
cana-2401	37	7	root	root	NOUN
cana-2401	37	8	square	square	ADJ
cana-2401	37	9	mean	mean	ADJ
cana-2401	37	10	labelling	labelling	NOUN
cana-2401	37	11	of	of	ADP
cana-2401	37	12	p7	p7	PROPN
cana-2401	37	13	theorem	theorem	VERB
cana-2401	37	14	3.2	3.2	NUM
cana-2401	37	15	:	:	PUNCT
cana-2401	37	16	pn	pn	PROPN
cana-2401	37	17	ʘ	ʘ	PROPN
cana-2401	37	18	k1	k1	PROPN
cana-2401	37	19	is	be	AUX
cana-2401	37	20	an	an	DET
cana-2401	37	21	even	even	ADJ
cana-2401	37	22	vertex	vertex	NOUN
cana-2401	37	23	odd	odd	ADJ
cana-2401	37	24	edge	edge	NOUN
cana-2401	37	25	root	root	NOUN
cana-2401	37	26	square	square	ADJ
cana-2401	37	27	mean	mean	ADJ
cana-2401	37	28	labelling	labelling	NOUN
cana-2401	37	29	graph	graph	NOUN
cana-2401	37	30	.	.	PUNCT
cana-2401	38	1	proof	proof	NOUN
cana-2401	38	2	:	:	PUNCT
cana-2401	38	3	let	let	VERB
cana-2401	38	4	g	g	PROPN
cana-2401	38	5	=	=	VERB
cana-2401	38	6	pn	pn	PROPN
cana-2401	38	7	ʘ	ʘ	PROPN
cana-2401	38	8	k1	k1	PROPN
cana-2401	38	9	the	the	DET
cana-2401	38	10	vertex	vertex	NOUN
cana-2401	38	11	set	set	NOUN
cana-2401	38	12	and	and	CCONJ
cana-2401	38	13	edge	edge	NOUN
cana-2401	38	14	set	set	NOUN
cana-2401	38	15	of	of	ADP
cana-2401	38	16	g	g	PROPN
cana-2401	38	17	is	be	AUX
cana-2401	38	18	defined	define	VERB
cana-2401	38	19	as	as	SCONJ
cana-2401	38	20	follows	follow	VERB
cana-2401	38	21	;	;	PUNCT
cana-2401	38	22	v(g	v(g	NUM
cana-2401	38	23	)	)	PUNCT
cana-2401	38	24	=	=	SYM
cana-2401	38	25	{	{	PUNCT
cana-2401	38	26	ui	ui	NOUN
cana-2401	38	27	,	,	PUNCT
cana-2401	38	28	u′i	u′i	PROPN
cana-2401	38	29	;	;	PUNCT
cana-2401	38	30	1≤	1≤	NUM
cana-2401	38	31	𝑖	𝑖	SYM
cana-2401	38	32	≤	≤	NUM
cana-2401	38	33	𝑛	𝑛	PRON
cana-2401	38	34	}	}	PUNCT
cana-2401	38	35	and	and	CCONJ
cana-2401	38	36	e(g	e(g	NOUN
cana-2401	38	37	)	)	PUNCT
cana-2401	39	1	=	=	PRON
cana-2401	39	2	{	{	PUNCT
cana-2401	39	3	ui	ui	X
cana-2401	39	4	u′i	u′i	PROPN
cana-2401	39	5	;	;	PUNCT
cana-2401	39	6	1≤	1≤	NUM
cana-2401	39	7	𝑖	𝑖	SYM
cana-2401	39	8	≤	≤	NUM
cana-2401	39	9	𝑛	𝑛	PRON
cana-2401	39	10	and	and	CCONJ
cana-2401	39	11	ui	ui	PROPN
cana-2401	39	12	u′i+1	u′i+1	PROPN
cana-2401	39	13	;	;	PUNCT
cana-2401	39	14	1≤	1≤	X
cana-2401	39	15	𝑖	𝑖	SYM
cana-2401	39	16	≤	≤	PROPN
cana-2401	39	17	𝑛-1	𝑛-1	ADV
cana-2401	39	18	}	}	PUNCT
cana-2401	39	19	we	we	PRON
cana-2401	39	20	define	define	VERB
cana-2401	39	21	a	a	DET
cana-2401	39	22	map	map	NOUN
cana-2401	39	23	f	f	X
cana-2401	39	24	:	:	PUNCT
cana-2401	39	25	v(g	v(g	NUM
cana-2401	39	26	)	)	PUNCT
cana-2401	39	27	→	→	SYM
cana-2401	39	28	{	{	PUNCT
cana-2401	39	29	0,1	0,1	NUM
cana-2401	39	30	,	,	PUNCT
cana-2401	39	31	2	2	NUM
cana-2401	39	32	,	,	PUNCT
cana-2401	39	33	3	3	NUM
cana-2401	39	34	…	…	NUM
cana-2401	39	35	…	…	PUNCT
cana-2401	39	36	.2q	.2q	NUM
cana-2401	39	37	}	}	PUNCT
cana-2401	39	38	.	.	PUNCT
cana-2401	40	1	communications	communication	NOUN
cana-2401	40	2	on	on	ADP
cana-2401	40	3	applied	apply	VERB
cana-2401	40	4	nonlinear	nonlinear	ADJ
cana-2401	40	5	analysis	analysis	NOUN
cana-2401	40	6	issn	issn	NOUN
cana-2401	40	7	:	:	PUNCT
cana-2401	40	8	1074	1074	NUM
cana-2401	40	9	-	-	PUNCT
cana-2401	40	10	133x	133x	NUM
cana-2401	40	11	vol	vol	NOUN
cana-2401	40	12	32	32	NUM
cana-2401	40	13	no	no	NOUN
cana-2401	40	14	.	.	PUNCT
cana-2401	41	1	2s	2s	NUM
cana-2401	41	2	(	(	PUNCT
cana-2401	41	3	2025	2025	NUM
cana-2401	41	4	)	)	PUNCT
cana-2401	41	5	298	298	NUM
cana-2401	41	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	41	7	the	the	DET
cana-2401	41	8	labelling	labelling	NOUN
cana-2401	41	9	pattern	pattern	NOUN
cana-2401	41	10	of	of	ADP
cana-2401	41	11	vertices	vertex	NOUN
cana-2401	41	12	can	can	AUX
cana-2401	41	13	be	be	AUX
cana-2401	41	14	defined	define	VERB
cana-2401	41	15	as	as	SCONJ
cana-2401	41	16	follows	follow	VERB
cana-2401	41	17	;	;	PUNCT
cana-2401	41	18	f	f	PROPN
cana-2401	41	19	(	(	PUNCT
cana-2401	41	20	u2i−1	u2i−1	PROPN
cana-2401	41	21	)	)	PUNCT
cana-2401	41	22	=	=	PUNCT
cana-2401	41	23	8i	8i	NUM
cana-2401	41	24	–	–	PUNCT
cana-2401	41	25	6	6	NUM
cana-2401	41	26	and	and	CCONJ
cana-2401	41	27	f	f	PROPN
cana-2401	41	28	(	(	PUNCT
cana-2401	41	29	u2i	u2i	PROPN
cana-2401	41	30	)	)	PUNCT
cana-2401	41	31	=	=	PUNCT
cana-2401	41	32	8i	8i	NUM
cana-2401	41	33	–	–	PUNCT
cana-2401	41	34	4	4	NUM
cana-2401	41	35	;	;	PUNCT
cana-2401	41	36	1≤	1≤	NUM
cana-2401	41	37	𝑖	𝑖	SYM
cana-2401	41	38	≤	≤	NUM
cana-2401	41	39	𝑛	𝑛	DET
cana-2401	41	40	f	f	PROPN
cana-2401	41	41	(	(	PUNCT
cana-2401	41	42	u′2i−1	u′2i−1	PROPN
cana-2401	41	43	)	)	PUNCT
cana-2401	41	44	=	=	SYM
cana-2401	41	45	8(i-1	8(i-1	X
cana-2401	41	46	)	)	PUNCT
cana-2401	41	47	and	and	CCONJ
cana-2401	41	48	f	f	X
cana-2401	41	49	(	(	PUNCT
cana-2401	41	50	u′2i	u′2i	PROPN
cana-2401	41	51	)	)	PUNCT
cana-2401	41	52	=	=	SYM
cana-2401	41	53	8i-2	8i-2	NOUN
cana-2401	41	54	;	;	PUNCT
cana-2401	41	55	1≤	1≤	NUM
cana-2401	41	56	𝑖	𝑖	SYM
cana-2401	41	57	≤	≤	NOUN
cana-2401	41	58	𝑛	𝑛	PRON
cana-2401	41	59	then	then	ADV
cana-2401	41	60	the	the	DET
cana-2401	41	61	induced	induced	ADJ
cana-2401	41	62	edge	edge	NOUN
cana-2401	41	63	labels	label	NOUN
cana-2401	41	64	are	be	AUX
cana-2401	41	65	f	f	PROPN
cana-2401	41	66	′	′	NUM
cana-2401	41	67	(	(	PUNCT
cana-2401	41	68	uiu′i	uiu′i	ADJ
cana-2401	41	69	)	)	PUNCT
cana-2401	41	70	=	=	SYM
cana-2401	41	71	4i	4i	NOUN
cana-2401	41	72	–	–	PUNCT
cana-2401	41	73	3	3	NUM
cana-2401	41	74	;	;	PUNCT
cana-2401	41	75	1	1	NUM
cana-2401	41	76	≤	≤	NUM
cana-2401	41	77	i	i	PRON
cana-2401	41	78	≤	≤	NOUN
cana-2401	42	1	n	n	PRON
cana-2401	42	2	f	f	NOUN
cana-2401	42	3	′	′	NUM
cana-2401	42	4	(	(	PUNCT
cana-2401	42	5	uiu′i+1	uiu′i+1	PROPN
cana-2401	42	6	)	)	PUNCT
cana-2401	43	1	=	=	SYM
cana-2401	43	2	4i	4i	NOUN
cana-2401	43	3	1	1	NUM
cana-2401	43	4	;	;	PUNCT
cana-2401	43	5	1	1	NUM
cana-2401	43	6	≤	≤	NUM
cana-2401	43	7	i	i	PRON
cana-2401	43	8	≤	≤	ADJ
cana-2401	43	9	n	n	CCONJ
cana-2401	43	10	−	−	PROPN
cana-2401	43	11	1	1	NUM
cana-2401	43	12	thus	thus	ADV
cana-2401	43	13	g	g	PROPN
cana-2401	43	14	=	=	PUNCT
cana-2401	43	15	pn	pn	PROPN
cana-2401	43	16	ʘ	ʘ	PROPN
cana-2401	43	17	k1	k1	PROPN
cana-2401	43	18	admit	admit	VERB
cana-2401	43	19	even	even	ADV
cana-2401	43	20	vertex	vertex	NOUN
cana-2401	43	21	odd	odd	ADJ
cana-2401	43	22	edge	edge	NOUN
cana-2401	43	23	root	root	NOUN
cana-2401	43	24	square	square	ADJ
cana-2401	43	25	mean	mean	ADJ
cana-2401	43	26	labelling	labelling	NOUN
cana-2401	43	27	graph	graph	NOUN
cana-2401	43	28	.	.	PUNCT
cana-2401	44	1	illustration	illustration	NOUN
cana-2401	44	2	:	:	PUNCT
cana-2401	44	3	figure	figure	NOUN
cana-2401	44	4	2	2	NUM
cana-2401	44	5	illustrate	illustrate	VERB
cana-2401	44	6	the	the	DET
cana-2401	44	7	g	g	NOUN
cana-2401	44	8	=	=	PROPN
cana-2401	44	9	p4	p4	PROPN
cana-2401	44	10	ʘ	ʘ	PROPN
cana-2401	44	11	k1	k1	PROPN
cana-2401	44	12	is	be	AUX
cana-2401	44	13	even	even	ADV
cana-2401	44	14	vertex	vertex	NOUN
cana-2401	44	15	odd	odd	ADJ
cana-2401	44	16	edge	edge	NOUN
cana-2401	44	17	root	root	NOUN
cana-2401	44	18	square	square	ADJ
cana-2401	44	19	mean	mean	NOUN
cana-2401	44	20	labeling	labeling	NOUN
cana-2401	44	21	figure	figure	NOUN
cana-2401	44	22	2	2	NUM
cana-2401	44	23	g	g	NOUN
cana-2401	44	24	=	=	PROPN
cana-2401	44	25	p4	p4	PROPN
cana-2401	44	26	ʘ	ʘ	PROPN
cana-2401	44	27	k1	k1	PROPN
cana-2401	44	28	theorem	theorem	VERB
cana-2401	44	29	3.3	3.3	NUM
cana-2401	44	30	:	:	PUNCT
cana-2401	44	31	pn	pn	PROPN
cana-2401	44	32	ʘ	ʘ	PROPN
cana-2401	44	33	k1,2	k1,2	PROPN
cana-2401	44	34	is	be	AUX
cana-2401	44	35	an	an	DET
cana-2401	44	36	even	even	ADJ
cana-2401	44	37	vertex	vertex	NOUN
cana-2401	44	38	odd	odd	ADJ
cana-2401	44	39	edge	edge	NOUN
cana-2401	44	40	root	root	NOUN
cana-2401	44	41	square	square	ADJ
cana-2401	44	42	mean	mean	ADJ
cana-2401	44	43	labelling	labelling	NOUN
cana-2401	44	44	graph	graph	NOUN
cana-2401	44	45	.	.	PUNCT
cana-2401	45	1	proof	proof	NOUN
cana-2401	45	2	:	:	PUNCT
cana-2401	45	3	let	let	VERB
cana-2401	45	4	g	g	PROPN
cana-2401	45	5	=	=	VERB
cana-2401	45	6	pn	pn	PROPN
cana-2401	45	7	ʘ	ʘ	PROPN
cana-2401	45	8	k1,2	k1,2	PROPN
cana-2401	45	9	is	be	AUX
cana-2401	45	10	a	a	DET
cana-2401	45	11	graph	graph	NOUN
cana-2401	45	12	formed	form	VERB
cana-2401	45	13	by	by	ADP
cana-2401	45	14	attaching	attach	VERB
cana-2401	45	15	a	a	DET
cana-2401	45	16	k1,2	k1,2	NOUN
cana-2401	45	17	(	(	PUNCT
cana-2401	45	18	a	a	DET
cana-2401	45	19	star	star	NOUN
cana-2401	45	20	with	with	ADP
cana-2401	45	21	one	one	NUM
cana-2401	45	22	central	central	ADJ
cana-2401	45	23	vertex	vertex	NOUN
cana-2401	45	24	and	and	CCONJ
cana-2401	45	25	two	two	NUM
cana-2401	45	26	leaves	leave	NOUN
cana-2401	45	27	)	)	PUNCT
cana-2401	45	28	to	to	ADP
cana-2401	45	29	each	each	DET
cana-2401	45	30	vertex	vertex	NOUN
cana-2401	45	31	of	of	ADP
cana-2401	45	32	the	the	DET
cana-2401	45	33	path	path	NOUN
cana-2401	46	1	pn	pn	PROPN
cana-2401	46	2	.	.	PROPN
cana-2401	46	3	let	let	VERB
cana-2401	46	4	v(g	v(g	NUM
cana-2401	46	5	)	)	PUNCT
cana-2401	46	6	=	=	PRON
cana-2401	46	7	{	{	PUNCT
cana-2401	46	8	ui	ui	NOUN
cana-2401	46	9	,	,	PUNCT
cana-2401	46	10	u′i	u′i	PROPN
cana-2401	46	11	,	,	PUNCT
cana-2401	46	12	u′′i	u′′i	PROPN
cana-2401	46	13	;	;	PUNCT
cana-2401	47	1	1≤	1≤	NUM
cana-2401	47	2	𝑖	𝑖	SYM
cana-2401	47	3	≤	≤	NUM
cana-2401	47	4	𝑛	𝑛	PRON
cana-2401	47	5	}	}	PUNCT
cana-2401	47	6	and	and	CCONJ
cana-2401	47	7	e(g	e(g	NOUN
cana-2401	47	8	)	)	PUNCT
cana-2401	48	1	=	=	PRON
cana-2401	48	2	{	{	PUNCT
cana-2401	48	3	ui	ui	PROPN
cana-2401	48	4	u′i	u′i	INTJ
cana-2401	48	5	,	,	PUNCT
cana-2401	48	6	ui	ui	PROPN
cana-2401	48	7	u′′	u′′	PROPN
cana-2401	49	1	i	i	PRON
cana-2401	49	2	:	:	PUNCT
cana-2401	49	3	1≤	1≤	X
cana-2401	49	4	𝑖	𝑖	SYM
cana-2401	49	5	≤	≤	NUM
cana-2401	49	6	𝑛	𝑛	PRON
cana-2401	49	7	and	and	CCONJ
cana-2401	49	8	ui	ui	PROPN
cana-2401	49	9	u′i+1	u′i+1	NUM
cana-2401	49	10	:	:	PUNCT
cana-2401	49	11	1≤	1≤	NUM
cana-2401	49	12	𝑖	𝑖	SYM
cana-2401	49	13	≤	≤	NOUN
cana-2401	49	14	𝑛-1	𝑛-1	ADV
cana-2401	49	15	}	}	PUNCT
cana-2401	49	16	this	this	PRON
cana-2401	49	17	describes	describe	VERB
cana-2401	49	18	the	the	DET
cana-2401	49	19	standard	standard	ADJ
cana-2401	49	20	labeling	labeling	NOUN
cana-2401	49	21	of	of	ADP
cana-2401	49	22	the	the	DET
cana-2401	49	23	vertices	vertex	NOUN
cana-2401	49	24	and	and	CCONJ
cana-2401	49	25	edges	edge	NOUN
cana-2401	49	26	of	of	ADP
cana-2401	49	27	g.	g.	NOUN
cana-2401	49	28	we	we	PRON
cana-2401	49	29	define	define	VERB
cana-2401	49	30	a	a	DET
cana-2401	49	31	map	map	NOUN
cana-2401	49	32	f	f	NOUN
cana-2401	49	33	:	:	PUNCT
cana-2401	49	34	𝑉(𝐺	𝑉(𝐺	NOUN
cana-2401	49	35	)	)	PUNCT
cana-2401	49	36	→	→	SYM
cana-2401	49	37	{	{	PUNCT
cana-2401	49	38	0	0	NUM
cana-2401	49	39	,	,	PUNCT
cana-2401	49	40	1	1	NUM
cana-2401	49	41	,	,	PUNCT
cana-2401	49	42	2	2	NUM
cana-2401	49	43	,	,	PUNCT
cana-2401	49	44	…	…	PUNCT
cana-2401	49	45	…	…	SYM
cana-2401	49	46	2q	2q	NUM
cana-2401	49	47	}	}	PUNCT
cana-2401	49	48	by	by	ADP
cana-2401	49	49	f(𝑢𝑖	f(𝑢𝑖	NOUN
cana-2401	49	50	)	)	PUNCT
cana-2401	49	51	=	=	SYM
cana-2401	49	52	6𝑖	6𝑖	NOUN
cana-2401	49	53	–	–	PUNCT
cana-2401	49	54	4	4	NUM
cana-2401	49	55	∶	∶	NOUN
cana-2401	49	56	𝑖	𝑖	NOUN
cana-2401	49	57	≤	≤	NOUN
cana-2401	49	58	𝑖	𝑖	SYM
cana-2401	49	59	≤	≤	NUM
cana-2401	50	1	𝑛	𝑛	DET
cana-2401	50	2	f	f	X
cana-2401	50	3	(	(	PUNCT
cana-2401	50	4	u′i	u′i	ADJ
cana-2401	50	5	)	)	PUNCT
cana-2401	50	6	=	=	SYM
cana-2401	50	7	6(i-1	6(i-1	NUM
cana-2401	50	8	)	)	PUNCT
cana-2401	50	9	and	and	CCONJ
cana-2401	50	10	f	f	PROPN
cana-2401	50	11	(	(	PUNCT
cana-2401	50	12	u′′i	u′′i	PROPN
cana-2401	50	13	)	)	PUNCT
cana-2401	51	1	=	=	SYM
cana-2401	52	1	6i	6i	NUM
cana-2401	52	2	2	2	NUM
cana-2401	52	3	:	:	SYM
cana-2401	52	4	1≤	1≤	NUM
cana-2401	52	5	𝑖	𝑖	SYM
cana-2401	52	6	≤	≤	NOUN
cana-2401	52	7	𝑛	𝑛	PRON
cana-2401	52	8	it	it	PRON
cana-2401	52	9	is	be	AUX
cana-2401	52	10	evident	evident	ADJ
cana-2401	52	11	that	that	SCONJ
cana-2401	52	12	the	the	DET
cana-2401	52	13	included	include	VERB
cana-2401	52	14	edge	edge	NOUN
cana-2401	52	15	labels	label	NOUN
cana-2401	52	16	are	be	AUX
cana-2401	52	17	both	both	PRON
cana-2401	52	18	odd	odd	ADJ
cana-2401	52	19	and	and	CCONJ
cana-2401	52	20	distinct	distinct	ADJ
cana-2401	52	21	.	.	PUNCT
cana-2401	53	1	hence	hence	ADV
cana-2401	53	2	g	g	PROPN
cana-2401	53	3	=	=	PUNCT
cana-2401	53	4	pn	pn	PROPN
cana-2401	53	5	ʘ	ʘ	PROPN
cana-2401	53	6	k1,2	k1,2	PROPN
cana-2401	53	7	is	be	AUX
cana-2401	53	8	an	an	DET
cana-2401	53	9	even	even	ADJ
cana-2401	53	10	vertex	vertex	NOUN
cana-2401	53	11	odd	odd	ADJ
cana-2401	53	12	edge	edge	NOUN
cana-2401	53	13	root	root	NOUN
cana-2401	53	14	square	square	ADJ
cana-2401	53	15	mean	mean	ADJ
cana-2401	53	16	labelling	labelling	NOUN
cana-2401	53	17	.	.	PUNCT
cana-2401	54	1	illustration	illustration	NOUN
cana-2401	54	2	:	:	PUNCT
cana-2401	54	3	figure	figure	NOUN
cana-2401	54	4	3	3	NUM
cana-2401	54	5	illustrate	illustrate	VERB
cana-2401	54	6	the	the	DET
cana-2401	54	7	g	g	NOUN
cana-2401	54	8	=	=	PROPN
cana-2401	54	9	p4	p4	PROPN
cana-2401	54	10	ʘ	ʘ	PROPN
cana-2401	54	11	k1,2	k1,2	PROPN
cana-2401	54	12	is	be	AUX
cana-2401	54	13	even	even	ADV
cana-2401	54	14	vertex	vertex	NOUN
cana-2401	54	15	odd	odd	ADJ
cana-2401	54	16	edge	edge	NOUN
cana-2401	54	17	root	root	NOUN
cana-2401	54	18	square	square	ADJ
cana-2401	54	19	mean	mean	NOUN
cana-2401	54	20	labeling	labeling	NOUN
cana-2401	54	21	communications	communication	NOUN
cana-2401	54	22	on	on	ADP
cana-2401	54	23	applied	apply	VERB
cana-2401	54	24	nonlinear	nonlinear	ADJ
cana-2401	54	25	analysis	analysis	NOUN
cana-2401	54	26	issn	issn	NOUN
cana-2401	54	27	:	:	PUNCT
cana-2401	54	28	1074	1074	NUM
cana-2401	54	29	-	-	PUNCT
cana-2401	54	30	133x	133x	NUM
cana-2401	54	31	vol	vol	NOUN
cana-2401	54	32	32	32	NUM
cana-2401	54	33	no	no	NOUN
cana-2401	54	34	.	.	PUNCT
cana-2401	55	1	2s	2s	NUM
cana-2401	55	2	(	(	PUNCT
cana-2401	55	3	2025	2025	NUM
cana-2401	55	4	)	)	PUNCT
cana-2401	55	5	299	299	NUM
cana-2401	55	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	55	7	figure	figure	NOUN
cana-2401	55	8	3	3	NUM
cana-2401	55	9	g	g	NOUN
cana-2401	55	10	=	=	PROPN
cana-2401	55	11	p4	p4	PROPN
cana-2401	55	12	ʘ	ʘ	PROPN
cana-2401	55	13	k1,2	k1,2	PROPN
cana-2401	55	14	theorem	theorem	VERB
cana-2401	55	15	3.4	3.4	NUM
cana-2401	55	16	:	:	PUNCT
cana-2401	55	17	the	the	DET
cana-2401	55	18	graph	graph	NOUN
cana-2401	55	19	sln	sln	NOUN
cana-2401	55	20	is	be	AUX
cana-2401	55	21	an	an	DET
cana-2401	55	22	even	even	ADJ
cana-2401	55	23	vertex	vertex	NOUN
cana-2401	55	24	odd	odd	ADJ
cana-2401	55	25	edge	edge	NOUN
cana-2401	55	26	mean	mean	PROPN
cana-2401	55	27	square	square	ADJ
cana-2401	55	28	labelling	labelling	NOUN
cana-2401	55	29	graph	graph	NOUN
cana-2401	55	30	.	.	PUNCT
cana-2401	56	1	proof	proof	NOUN
cana-2401	56	2	:	:	PUNCT
cana-2401	56	3	let	let	VERB
cana-2401	56	4	g	g	PROPN
cana-2401	56	5	=	=	PROPN
cana-2401	56	6	sln	sln	PROPN
cana-2401	56	7	let	let	VERB
cana-2401	56	8	the	the	DET
cana-2401	56	9	vertex	vertex	NOUN
cana-2401	56	10	set	set	VERB
cana-2401	56	11	v(g	v(g	PROPN
cana-2401	56	12	)	)	PUNCT
cana-2401	56	13	=	=	SYM
cana-2401	56	14	{	{	PUNCT
cana-2401	56	15	𝑢𝑖	𝑢𝑖	NOUN
cana-2401	56	16	,	,	PUNCT
cana-2401	56	17	𝑣𝑖	𝑣𝑖	ADP
cana-2401	56	18	∶	∶	NOUN
cana-2401	56	19	1	1	NUM
cana-2401	56	20	≤	≤	NOUN
cana-2401	56	21	𝑖	𝑖	SYM
cana-2401	56	22	≤	≤	NUM
cana-2401	56	23	𝑛	𝑛	PRON
cana-2401	56	24	}	}	PUNCT
cana-2401	56	25	and	and	CCONJ
cana-2401	56	26	edge	edge	VERB
cana-2401	56	27	set	set	ADJ
cana-2401	56	28	e	e	NOUN
cana-2401	56	29	(	(	PUNCT
cana-2401	56	30	g	g	NOUN
cana-2401	56	31	)	)	PUNCT
cana-2401	56	32	=	=	PRON
cana-2401	56	33	{	{	PUNCT
cana-2401	57	1	ui	ui	INTJ
cana-2401	57	2	ui	ui	PROPN
cana-2401	58	1	+	+	CCONJ
cana-2401	58	2	1	1	NUM
cana-2401	58	3	∶	∶	NOUN
cana-2401	58	4	1	1	NUM
cana-2401	58	5	≤	≤	NOUN
cana-2401	58	6	i	i	PRON
cana-2401	58	7	≤	≤	ADJ
cana-2401	58	8	n	n	CCONJ
cana-2401	58	9	−	−	PROPN
cana-2401	58	10	1	1	NUM
cana-2401	58	11	vi	vi	NOUN
cana-2401	58	12	vi	vi	NOUN
cana-2401	59	1	+	+	CCONJ
cana-2401	59	2	1	1	NUM
cana-2401	59	3	∶	∶	NOUN
cana-2401	59	4	1	1	NUM
cana-2401	59	5	≤	≤	NOUN
cana-2401	59	6	i	i	PRON
cana-2401	59	7	≤	≤	NOUN
cana-2401	59	8	n	n	CCONJ
cana-2401	59	9	−	−	PROPN
cana-2401	59	10	1	1	NUM
cana-2401	59	11	vi+2	vi+2	NUM
cana-2401	59	12	vi	vi	NOUN
cana-2401	59	13	∶	∶	NOUN
cana-2401	59	14	2	2	NUM
cana-2401	59	15	≤	≤	NOUN
cana-2401	59	16	i	i	PRON
cana-2401	59	17	≤	≤	NOUN
cana-2401	60	1	n	n	CCONJ
cana-2401	60	2	−	−	PROPN
cana-2401	60	3	1	1	NUM
cana-2401	60	4	define	define	VERB
cana-2401	60	5	a	a	DET
cana-2401	60	6	map	map	NOUN
cana-2401	60	7	f	f	X
cana-2401	60	8	:	:	PUNCT
cana-2401	60	9	v(g	v(g	NUM
cana-2401	60	10	)	)	PUNCT
cana-2401	60	11	→	→	SYM
cana-2401	60	12	{	{	PUNCT
cana-2401	60	13	0,1	0,1	NUM
cana-2401	60	14	,	,	PUNCT
cana-2401	60	15	2	2	NUM
cana-2401	60	16	,	,	PUNCT
cana-2401	60	17	…	…	PUNCT
cana-2401	60	18	…	…	SYM
cana-2401	60	19	2q	2q	NUM
cana-2401	60	20	}	}	PUNCT
cana-2401	60	21	by	by	ADP
cana-2401	60	22	f	f	PROPN
cana-2401	60	23	(	(	PUNCT
cana-2401	60	24	u1	u1	PROPN
cana-2401	60	25	)	)	PUNCT
cana-2401	60	26	=	=	SYM
cana-2401	60	27	0	0	NUM
cana-2401	60	28	,	,	PUNCT
cana-2401	60	29	f(𝑣𝑛	f(𝑣𝑛	X
cana-2401	60	30	)	)	PUNCT
cana-2401	60	31	=	=	SYM
cana-2401	60	32	2𝑞	2𝑞	NOUN
cana-2401	60	33	,	,	PUNCT
cana-2401	60	34	further	further	ADJ
cana-2401	60	35	f	f	X
cana-2401	60	36	(	(	PUNCT
cana-2401	60	37	ui	ui	PROPN
cana-2401	60	38	)	)	PUNCT
cana-2401	60	39	=	=	SYM
cana-2401	60	40	6𝑖	6𝑖	NOUN
cana-2401	60	41	–	–	PUNCT
cana-2401	60	42	10	10	NUM
cana-2401	60	43	∶	∶	NOUN
cana-2401	60	44	2	2	NUM
cana-2401	60	45	≤	≤	NOUN
cana-2401	60	46	𝑖	𝑖	SYM
cana-2401	60	47	≤	≤	NUM
cana-2401	60	48	𝑛	𝑛	PROPN
cana-2401	60	49	,	,	PUNCT
cana-2401	60	50	f	f	PROPN
cana-2401	60	51	(	(	PUNCT
cana-2401	60	52	vi	vi	NOUN
cana-2401	60	53	)	)	PUNCT
cana-2401	60	54	=	=	SYM
cana-2401	60	55	6𝑖	6𝑖	NOUN
cana-2401	60	56	–	–	PUNCT
cana-2401	60	57	2	2	NUM
cana-2401	60	58	∶	∶	NOUN
cana-2401	60	59	1	1	NUM
cana-2401	60	60	≤	≤	NOUN
cana-2401	60	61	𝑖	𝑖	SYM
cana-2401	60	62	≤	≤	NUM
cana-2401	60	63	𝑛	𝑛	PRON
cana-2401	60	64	−	−	PROPN
cana-2401	60	65	1	1	NUM
cana-2401	60	66	then	then	ADV
cana-2401	60	67	the	the	DET
cana-2401	60	68	induced	induced	ADJ
cana-2401	60	69	edge	edge	NOUN
cana-2401	60	70	labels	label	NOUN
cana-2401	60	71	are	be	AUX
cana-2401	60	72	defined	define	VERB
cana-2401	60	73	as	as	SCONJ
cana-2401	60	74	follows	follow	VERB
cana-2401	60	75	;	;	PUNCT
cana-2401	60	76	𝑓′	𝑓′	X
cana-2401	60	77	(	(	PUNCT
cana-2401	60	78	vivi+1	vivi+1	NOUN
cana-2401	60	79	)	)	PUNCT
cana-2401	60	80	=	=	SYM
cana-2401	60	81	6𝑖	6𝑖	NOUN
cana-2401	61	1	+	+	CCONJ
cana-2401	61	2	1∶	1∶	NUM
cana-2401	61	3	1	1	NUM
cana-2401	61	4	≤	≤	NUM
cana-2401	61	5	𝑖	𝑖	SYM
cana-2401	61	6	≤	≤	NUM
cana-2401	61	7	𝑛	𝑛	PRON
cana-2401	61	8	−	−	PROPN
cana-2401	61	9	2	2	NUM
cana-2401	61	10	,	,	PUNCT
cana-2401	61	11	𝑓′	𝑓′	X
cana-2401	61	12	(	(	PUNCT
cana-2401	61	13	vn−1vn	vn−1vn	NUM
cana-2401	61	14	)	)	PUNCT
cana-2401	61	15	=	=	SYM
cana-2401	61	16	2𝑞	2𝑞	NUM
cana-2401	61	17	–	–	PUNCT
cana-2401	61	18	1	1	NUM
cana-2401	61	19	∶	∶	NOUN
cana-2401	61	20	1	1	NUM
cana-2401	61	21	≤	≤	NOUN
cana-2401	61	22	𝑖	𝑖	SYM
cana-2401	61	23	≤	≤	NUM
cana-2401	61	24	𝑛	𝑛	PRON
cana-2401	61	25	−	−	PROPN
cana-2401	61	26	2	2	NUM
cana-2401	61	27	and	and	CCONJ
cana-2401	61	28	𝑓′	𝑓′	NUM
cana-2401	61	29	(	(	PUNCT
cana-2401	61	30	ui+1vi	ui+1vi	NOUN
cana-2401	61	31	)	)	PUNCT
cana-2401	61	32	=	=	SYM
cana-2401	61	33	6𝑖	6𝑖	NOUN
cana-2401	61	34	−	−	NOUN
cana-2401	61	35	3∶	3∶	NUM
cana-2401	61	36	1	1	NUM
cana-2401	61	37	≤	≤	NUM
cana-2401	61	38	𝑖	𝑖	SYM
cana-2401	61	39	≤	≤	NUM
cana-2401	61	40	𝑛	𝑛	PRON
cana-2401	61	41	−	−	NOUN
cana-2401	61	42	1	1	NUM
cana-2401	61	43	.	.	PUNCT
cana-2401	62	1	hence	hence	ADV
cana-2401	62	2	,	,	PUNCT
cana-2401	62	3	sln	sln	PROPN
cana-2401	62	4	admits	admit	VERB
cana-2401	62	5	even	even	ADV
cana-2401	62	6	vertex	vertex	NOUN
cana-2401	62	7	odd	odd	ADJ
cana-2401	62	8	edge	edge	NOUN
cana-2401	62	9	mean	mean	PROPN
cana-2401	62	10	square	square	ADJ
cana-2401	62	11	labelling	labelling	NOUN
cana-2401	62	12	graph	graph	NOUN
cana-2401	62	13	.	.	PUNCT
cana-2401	63	1	illustration	illustration	NOUN
cana-2401	63	2	:	:	PUNCT
cana-2401	63	3	figure	figure	NOUN
cana-2401	63	4	4	4	NUM
cana-2401	63	5	illustrate	illustrate	VERB
cana-2401	63	6	the	the	DET
cana-2401	63	7	g	g	PROPN
cana-2401	63	8	=	=	PROPN
cana-2401	63	9	sl6	sl6	PROPN
cana-2401	63	10	is	be	AUX
cana-2401	63	11	even	even	ADV
cana-2401	63	12	vertex	vertex	NOUN
cana-2401	63	13	odd	odd	ADJ
cana-2401	63	14	edge	edge	NOUN
cana-2401	63	15	root	root	NOUN
cana-2401	63	16	square	square	ADJ
cana-2401	63	17	mean	mean	NOUN
cana-2401	63	18	labeling	labeling	NOUN
cana-2401	63	19	figure	figure	NOUN
cana-2401	63	20	4	4	NUM
cana-2401	63	21	g	g	NOUN
cana-2401	63	22	=	=	SYM
cana-2401	63	23	sl6	sl6	PROPN
cana-2401	63	24	theorem	theorem	VERB
cana-2401	63	25	3.5:[5	3.5:[5	NOUN
cana-2401	63	26	]	]	PUNCT
cana-2401	63	27	let	let	VERB
cana-2401	63	28	gp	gp	NOUN
cana-2401	63	29	be	be	AUX
cana-2401	63	30	a	a	DET
cana-2401	63	31	graph	graph	NOUN
cana-2401	63	32	formed	form	VERB
cana-2401	63	33	by	by	ADP
cana-2401	63	34	attaching	attach	VERB
cana-2401	63	35	a	a	DET
cana-2401	63	36	pendant	pendant	ADJ
cana-2401	63	37	edge	edge	NOUN
cana-2401	63	38	to	to	ADP
cana-2401	63	39	each	each	DET
cana-2401	63	40	internal	internal	ADJ
cana-2401	63	41	vertex	vertex	NOUN
cana-2401	63	42	of	of	ADP
cana-2401	63	43	a	a	DET
cana-2401	63	44	path	path	NOUN
cana-2401	63	45	.	.	PUNCT
cana-2401	64	1	this	this	DET
cana-2401	64	2	graph	graph	NOUN
cana-2401	64	3	gp	gp	NOUN
cana-2401	64	4	is	be	AUX
cana-2401	64	5	an	an	DET
cana-2401	64	6	even	even	ADJ
cana-2401	64	7	vertex	vertex	NOUN
cana-2401	64	8	odd	odd	ADJ
cana-2401	64	9	edge	edge	NOUN
cana-2401	64	10	root	root	NOUN
cana-2401	64	11	mean	mean	VERB
cana-2401	64	12	square	square	ADJ
cana-2401	64	13	labelling	labelling	NOUN
cana-2401	64	14	graph	graph	NOUN
cana-2401	64	15	.	.	PUNCT
cana-2401	65	1	proof	proof	NOUN
cana-2401	65	2	:	:	PUNCT
cana-2401	65	3	let	let	VERB
cana-2401	65	4	v(g	v(g	NUM
cana-2401	65	5	)	)	PUNCT
cana-2401	65	6	=	=	PRON
cana-2401	65	7	{	{	PUNCT
cana-2401	65	8	ui	ui	NOUN
cana-2401	65	9	:	:	PUNCT
cana-2401	65	10	1	1	NUM
cana-2401	65	11	≤	≤	NUM
cana-2401	65	12	i	i	PRON
cana-2401	65	13	≤	≤	ADJ
cana-2401	65	14	n	n	CCONJ
cana-2401	65	15	and	and	CCONJ
cana-2401	65	16	u′i	u′i	ADJ
cana-2401	65	17	:	:	PUNCT
cana-2401	65	18	2	2	NUM
cana-2401	65	19	≤	≤	NUM
cana-2401	65	20	i	i	PRON
cana-2401	65	21	≤	≤	ADJ
cana-2401	66	1	n	n	CCONJ
cana-2401	66	2	−	−	PROPN
cana-2401	66	3	1	1	NUM
cana-2401	66	4	and	and	CCONJ
cana-2401	66	5	communications	communication	NOUN
cana-2401	66	6	on	on	ADP
cana-2401	66	7	applied	apply	VERB
cana-2401	66	8	nonlinear	nonlinear	ADJ
cana-2401	66	9	analysis	analysis	NOUN
cana-2401	66	10	issn	issn	NOUN
cana-2401	66	11	:	:	PUNCT
cana-2401	66	12	1074	1074	NUM
cana-2401	66	13	-	-	PUNCT
cana-2401	66	14	133x	133x	NUM
cana-2401	66	15	vol	vol	NOUN
cana-2401	66	16	32	32	NUM
cana-2401	66	17	no	no	NOUN
cana-2401	66	18	.	.	PUNCT
cana-2401	67	1	2s	2s	NUM
cana-2401	67	2	(	(	PUNCT
cana-2401	67	3	2025	2025	NUM
cana-2401	67	4	)	)	PUNCT
cana-2401	67	5	300	300	NUM
cana-2401	67	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	67	7	e(g	e(g	PROPN
cana-2401	67	8	)	)	PUNCT
cana-2401	68	1	=	=	PRON
cana-2401	68	2	{	{	PUNCT
cana-2401	68	3	ui	ui	PROPN
cana-2401	68	4	ui+1	ui+1	PROPN
cana-2401	68	5	:	:	PUNCT
cana-2401	68	6	1	1	NUM
cana-2401	68	7	≤	≤	NUM
cana-2401	68	8	i	i	PRON
cana-2401	68	9	≤	≤	ADJ
cana-2401	68	10	n	n	CCONJ
cana-2401	68	11	−	−	PROPN
cana-2401	68	12	2	2	NUM
cana-2401	68	13	ui	ui	PROPN
cana-2401	68	14	u′i	u′i	ADJ
cana-2401	68	15	∶	∶	NOUN
cana-2401	69	1	i	i	X
cana-2401	69	2	=	=	NOUN
cana-2401	69	3	2,3,4,5	2,3,4,5	NUM
cana-2401	69	4	.	.	PUNCT
cana-2401	69	5	.	.	PUNCT
cana-2401	69	6	.	.	PUNCT
cana-2401	69	7	.	.	PUNCT
cana-2401	70	1	n	n	CCONJ
cana-2401	70	2	−	−	PROPN
cana-2401	70	3	2	2	NUM
cana-2401	70	4	be	be	AUX
cana-2401	70	5	the	the	DET
cana-2401	70	6	vertex	vertex	NOUN
cana-2401	70	7	set	set	NOUN
cana-2401	70	8	and	and	CCONJ
cana-2401	70	9	edge	edge	NOUN
cana-2401	70	10	set	set	NOUN
cana-2401	70	11	of	of	ADP
cana-2401	70	12	g	g	NOUN
cana-2401	70	13	respectively	respectively	ADV
cana-2401	70	14	.	.	PUNCT
cana-2401	71	1	define	define	VERB
cana-2401	71	2	a	a	DET
cana-2401	71	3	map	map	NOUN
cana-2401	71	4	f	f	X
cana-2401	71	5	:	:	PUNCT
cana-2401	71	6	v(g	v(g	NUM
cana-2401	71	7	)	)	PUNCT
cana-2401	71	8	→	→	SYM
cana-2401	71	9	{	{	PUNCT
cana-2401	71	10	0,2,4	0,2,4	NOUN
cana-2401	71	11	,	,	PUNCT
cana-2401	71	12	.	.	PUNCT
cana-2401	71	13	.	.	PUNCT
cana-2401	71	14	.	.	PUNCT
cana-2401	72	1	2q	2q	NUM
cana-2401	72	2	}	}	PUNCT
cana-2401	72	3	by	by	ADP
cana-2401	72	4	f(u	f(u	PROPN
cana-2401	72	5	)	)	PUNCT
cana-2401	72	6	where	where	SCONJ
cana-2401	72	7	f(u	f(u	PROPN
cana-2401	72	8	)	)	PUNCT
cana-2401	72	9	is	be	AUX
cana-2401	72	10	determined	determine	VERB
cana-2401	72	11	based	base	VERB
cana-2401	72	12	on	on	ADP
cana-2401	72	13	the	the	DET
cana-2401	72	14	following	follow	VERB
cana-2401	72	15	two	two	NUM
cana-2401	72	16	cases	case	NOUN
cana-2401	72	17	.	.	PUNCT
cana-2401	73	1	case	case	NOUN
cana-2401	73	2	1	1	NUM
cana-2401	73	3	:	:	PUNCT
cana-2401	73	4	for	for	ADP
cana-2401	73	5	n	n	PRON
cana-2401	73	6	is	be	AUX
cana-2401	73	7	odd	odd	ADJ
cana-2401	73	8	f	f	X
cana-2401	73	9	(	(	PUNCT
cana-2401	73	10	ui	ui	PROPN
cana-2401	73	11	)	)	PUNCT
cana-2401	73	12	=	=	SYM
cana-2401	73	13	4(i	4(i	NUM
cana-2401	73	14	-1	-1	NOUN
cana-2401	73	15	)	)	PUNCT
cana-2401	73	16	and	and	CCONJ
cana-2401	74	1	f	f	PROPN
cana-2401	74	2	(	(	PUNCT
cana-2401	74	3	u′i	u′i	ADJ
cana-2401	74	4	)	)	PUNCT
cana-2401	74	5	=	=	SYM
cana-2401	74	6	4i-6	4i-6	PROPN
cana-2401	74	7	:	:	PUNCT
cana-2401	74	8	i	i	NOUN
cana-2401	74	9	=	=	NOUN
cana-2401	74	10	3	3	NUM
cana-2401	74	11	,	,	PUNCT
cana-2401	74	12	5	5	NUM
cana-2401	74	13	,	,	PUNCT
cana-2401	74	14	…	…	PUNCT
cana-2401	74	15	f	f	X
cana-2401	74	16	(	(	PUNCT
cana-2401	74	17	ui	ui	NOUN
cana-2401	74	18	)	)	PUNCT
cana-2401	74	19	=	=	SYM
cana-2401	74	20	4i	4i	NOUN
cana-2401	75	1	-6	-6	CCONJ
cana-2401	75	2	and	and	CCONJ
cana-2401	75	3	f	f	PROPN
cana-2401	75	4	(	(	PUNCT
cana-2401	75	5	u′i	u′i	ADJ
cana-2401	75	6	)	)	PUNCT
cana-2401	75	7	=	=	SYM
cana-2401	75	8	4(i-1	4(i-1	NUM
cana-2401	75	9	):	):	PUNCT
cana-2401	75	10	i	i	PRON
cana-2401	75	11	=	=	NOUN
cana-2401	75	12	2	2	NUM
cana-2401	75	13	,	,	PUNCT
cana-2401	75	14	4	4	NUM
cana-2401	75	15	,	,	PUNCT
cana-2401	75	16	…	…	PUNCT
cana-2401	75	17	the	the	DET
cana-2401	75	18	labels	label	NOUN
cana-2401	75	19	of	of	ADP
cana-2401	75	20	the	the	DET
cana-2401	75	21	edges	edge	NOUN
cana-2401	75	22	are	be	AUX
cana-2401	75	23	then	then	ADV
cana-2401	75	24	defined	define	VERB
cana-2401	75	25	as	as	SCONJ
cana-2401	75	26	follows	follow	VERB
cana-2401	75	27	f′(ui	f′(ui	PROPN
cana-2401	75	28	u′i	u′i	ADJ
cana-2401	75	29	)	)	PUNCT
cana-2401	75	30	=	=	SYM
cana-2401	75	31	4i	4i	NOUN
cana-2401	75	32	–	–	PUNCT
cana-2401	75	33	5	5	NUM
cana-2401	75	34	,	,	PUNCT
cana-2401	75	35	i	i	PRON
cana-2401	75	36	=	=	NOUN
cana-2401	75	37	2	2	NUM
cana-2401	75	38	,	,	PUNCT
cana-2401	75	39	4	4	NUM
cana-2401	75	40	,	,	PUNCT
cana-2401	75	41	…	…	PUNCT
cana-2401	75	42	.	.	PUNCT
cana-2401	76	1	n-1	n-1	PROPN
cana-2401	76	2	and	and	CCONJ
cana-2401	76	3	f′(ui	f′(ui	PROPN
cana-2401	76	4	ui+1	ui+1	NOUN
cana-2401	76	5	)	)	PUNCT
cana-2401	77	1	=	=	SYM
cana-2401	77	2	4i	4i	NOUN
cana-2401	77	3	–	–	PUNCT
cana-2401	77	4	3	3	NUM
cana-2401	77	5	,	,	PUNCT
cana-2401	77	6	i	i	PRON
cana-2401	77	7	=	=	NOUN
cana-2401	77	8	1	1	NUM
cana-2401	77	9	,	,	PUNCT
cana-2401	77	10	2	2	NUM
cana-2401	77	11	,	,	PUNCT
cana-2401	77	12	…	…	PUNCT
cana-2401	77	13	.n-1	.n-1	PROPN
cana-2401	77	14	case	case	NOUN
cana-2401	77	15	2	2	NUM
cana-2401	77	16	:	:	PUNCT
cana-2401	77	17	for	for	ADP
cana-2401	77	18	n	n	PRON
cana-2401	77	19	is	be	AUX
cana-2401	77	20	even	even	ADV
cana-2401	77	21	f	f	PROPN
cana-2401	77	22	(	(	PUNCT
cana-2401	77	23	ui	ui	PROPN
cana-2401	77	24	)	)	PUNCT
cana-2401	77	25	=	=	SYM
cana-2401	78	1	4i	4i	NOUN
cana-2401	78	2	-5	-5	PUNCT
cana-2401	78	3	and	and	CCONJ
cana-2401	78	4	f	f	X
cana-2401	78	5	(	(	PUNCT
cana-2401	78	6	u′i	u′i	ADJ
cana-2401	79	1	)	)	PUNCT
cana-2401	79	2	=	=	SYM
cana-2401	79	3	4i-6	4i-6	PROPN
cana-2401	79	4	:	:	PUNCT
cana-2401	79	5	i	i	NOUN
cana-2401	79	6	=	=	NOUN
cana-2401	79	7	3	3	NUM
cana-2401	79	8	,	,	PUNCT
cana-2401	79	9	5	5	NUM
cana-2401	79	10	,	,	PUNCT
cana-2401	79	11	…	…	PUNCT
cana-2401	79	12	f	f	X
cana-2401	79	13	(	(	PUNCT
cana-2401	79	14	ui	ui	NOUN
cana-2401	79	15	)	)	PUNCT
cana-2401	79	16	=	=	SYM
cana-2401	79	17	4i	4i	NOUN
cana-2401	80	1	-6	-6	CCONJ
cana-2401	80	2	and	and	CCONJ
cana-2401	80	3	f	f	PROPN
cana-2401	80	4	(	(	PUNCT
cana-2401	80	5	u′i	u′i	ADJ
cana-2401	80	6	)	)	PUNCT
cana-2401	80	7	=	=	SYM
cana-2401	80	8	4(i-1	4(i-1	NUM
cana-2401	80	9	):	):	PUNCT
cana-2401	80	10	i	i	PRON
cana-2401	80	11	=	=	NOUN
cana-2401	80	12	2	2	NUM
cana-2401	80	13	,	,	PUNCT
cana-2401	80	14	4	4	NUM
cana-2401	80	15	,	,	PUNCT
cana-2401	80	16	…	…	PUNCT
cana-2401	80	17	then	then	ADV
cana-2401	80	18	the	the	DET
cana-2401	80	19	induced	induced	ADJ
cana-2401	80	20	edge	edge	NOUN
cana-2401	80	21	labels	label	NOUN
cana-2401	80	22	are	be	AUX
cana-2401	80	23	defined	define	VERB
cana-2401	80	24	as	as	SCONJ
cana-2401	80	25	follows	follow	VERB
cana-2401	80	26	:	:	PUNCT
cana-2401	80	27	f′(𝐮𝐢	f′(𝐮𝐢	VERB
cana-2401	80	28	𝐮′𝐢	𝐮′𝐢	X
cana-2401	80	29	)	)	PUNCT
cana-2401	81	1	=	=	SYM
cana-2401	81	2	4i	4i	NOUN
cana-2401	81	3	–	–	PUNCT
cana-2401	81	4	5	5	NUM
cana-2401	81	5	,	,	PUNCT
cana-2401	81	6	i	i	PRON
cana-2401	81	7	=	=	NOUN
cana-2401	81	8	2	2	NUM
cana-2401	81	9	,	,	PUNCT
cana-2401	81	10	4,	4,	ADJ
cana-2401	81	11	…	…	PUNCT
cana-2401	81	12	.n-1	.n-1	PUNCT
cana-2401	81	13	and	and	CCONJ
cana-2401	81	14	f′(𝐮𝐢	f′(𝐮𝐢	VERB
cana-2401	81	15	𝐮𝐢+𝟏	𝐮𝐢+𝟏	X
cana-2401	81	16	)	)	PUNCT
cana-2401	82	1	=	=	SYM
cana-2401	82	2	4i	4i	NOUN
cana-2401	82	3	–	–	PUNCT
cana-2401	82	4	3	3	NUM
cana-2401	82	5	,	,	PUNCT
cana-2401	82	6	i	i	PRON
cana-2401	82	7	=	=	NOUN
cana-2401	82	8	1	1	NUM
cana-2401	82	9	,	,	PUNCT
cana-2401	82	10	2	2	NUM
cana-2401	82	11	,	,	PUNCT
cana-2401	82	12	…	…	PUNCT
cana-2401	82	13	.n-1	.n-1	ADV
cana-2401	83	1	hence	hence	ADV
cana-2401	83	2	f	f	PROPN
cana-2401	83	3	is	be	AUX
cana-2401	83	4	an	an	DET
cana-2401	83	5	even	even	ADJ
cana-2401	83	6	vertex	vertex	NOUN
cana-2401	83	7	odd	odd	ADJ
cana-2401	83	8	edge	edge	NOUN
cana-2401	83	9	root	root	NOUN
cana-2401	83	10	square	square	ADJ
cana-2401	83	11	mean	mean	ADJ
cana-2401	83	12	labelling	labelling	NOUN
cana-2401	83	13	of	of	ADP
cana-2401	83	14	graph	graph	NOUN
cana-2401	83	15	g.	g.	PROPN
cana-2401	83	16	therefore	therefore	ADV
cana-2401	83	17	,	,	PUNCT
cana-2401	83	18	g	g	PROPN
cana-2401	83	19	is	be	AUX
cana-2401	83	20	classified	classify	VERB
cana-2401	83	21	as	as	ADP
cana-2401	83	22	an	an	DET
cana-2401	83	23	even	even	ADJ
cana-2401	83	24	vertex	vertex	NOUN
cana-2401	83	25	odd	odd	ADJ
cana-2401	83	26	edge	edge	NOUN
cana-2401	83	27	root	root	NOUN
cana-2401	83	28	square	square	ADJ
cana-2401	83	29	mean	mean	ADJ
cana-2401	83	30	labelling	labelling	NOUN
cana-2401	83	31	of	of	ADP
cana-2401	83	32	graph	graph	NOUN
cana-2401	83	33	with	with	ADP
cana-2401	83	34	respect	respect	NOUN
cana-2401	83	35	to	to	ADP
cana-2401	83	36	f.	f.	PROPN
cana-2401	83	37	theorem	theorem	PROPN
cana-2401	83	38	3.6	3.6	NUM
cana-2401	83	39	:	:	PUNCT
cana-2401	83	40	tw(pn	tw(pn	NOUN
cana-2401	83	41	)	)	PUNCT
cana-2401	83	42	is	be	AUX
cana-2401	83	43	an	an	DET
cana-2401	83	44	even	even	ADJ
cana-2401	83	45	vertex	vertex	NOUN
cana-2401	83	46	odd	odd	ADJ
cana-2401	83	47	edge	edge	NOUN
cana-2401	83	48	root	root	NOUN
cana-2401	83	49	square	square	ADJ
cana-2401	83	50	mean	mean	ADJ
cana-2401	83	51	labelling	labelling	NOUN
cana-2401	83	52	of	of	ADP
cana-2401	83	53	graph	graph	NOUN
cana-2401	83	54	.	.	PUNCT
cana-2401	84	1	proof	proof	NOUN
cana-2401	84	2	:	:	PUNCT
cana-2401	84	3	let	let	VERB
cana-2401	84	4	g	g	PROPN
cana-2401	84	5	=	=	PUNCT
cana-2401	84	6	tw(pn	tw(pn	NOUN
cana-2401	84	7	)	)	PUNCT
cana-2401	84	8	v(g	v(g	ADJ
cana-2401	84	9	)	)	PUNCT
cana-2401	84	10	=	=	PRON
cana-2401	84	11	{	{	PUNCT
cana-2401	84	12	ui	ui	NOUN
cana-2401	84	13	:	:	PUNCT
cana-2401	84	14	1	1	NUM
cana-2401	84	15	≤	≤	NUM
cana-2401	84	16	i	i	PRON
cana-2401	84	17	≤	≤	ADJ
cana-2401	84	18	n	n	CCONJ
cana-2401	84	19	and	and	CCONJ
cana-2401	84	20	u′i	u′i	ADJ
cana-2401	84	21	and	and	CCONJ
cana-2401	84	22	u′′i	u′′i	PROPN
cana-2401	84	23	∶	∶	NOUN
cana-2401	84	24	2	2	NUM
cana-2401	84	25	≤	≤	NOUN
cana-2401	85	1	i	i	PRON
cana-2401	85	2	≤	≤	NOUN
cana-2401	85	3	n	n	PRON
cana-2401	85	4	e(g	e(g	NOUN
cana-2401	85	5	)	)	PUNCT
cana-2401	86	1	=	=	PRON
cana-2401	86	2	{	{	PUNCT
cana-2401	86	3	uiui+1	uiui+1	NOUN
cana-2401	86	4	:	:	PUNCT
cana-2401	86	5	1	1	NUM
cana-2401	86	6	≤	≤	NUM
cana-2401	86	7	i	i	PRON
cana-2401	86	8	≤	≤	ADJ
cana-2401	86	9	n	n	CCONJ
cana-2401	86	10	−	−	PROPN
cana-2401	86	11	1	1	NUM
cana-2401	86	12	and	and	CCONJ
cana-2401	86	13	uiu′i	uiu′i	ADJ
cana-2401	86	14	and	and	CCONJ
cana-2401	86	15	uiu′′	uiu′′	PROPN
cana-2401	86	16	i	i	PRON
cana-2401	86	17	∶	∶	VERB
cana-2401	86	18	2	2	NUM
cana-2401	86	19	≤	≤	NUM
cana-2401	87	1	i	i	PRON
cana-2401	87	2	≤	≤	NOUN
cana-2401	88	1	n	n	CCONJ
cana-2401	88	2	be	be	VERB
cana-2401	88	3	the	the	DET
cana-2401	88	4	vertex	vertex	NOUN
cana-2401	88	5	set	set	NOUN
cana-2401	88	6	and	and	CCONJ
cana-2401	88	7	edge	edge	NOUN
cana-2401	88	8	set	set	NOUN
cana-2401	88	9	of	of	ADP
cana-2401	88	10	g	g	NOUN
cana-2401	88	11	respectively	respectively	ADV
cana-2401	88	12	.	.	PUNCT
cana-2401	89	1	we	we	PRON
cana-2401	89	2	define	define	VERB
cana-2401	89	3	a	a	DET
cana-2401	89	4	map	map	NOUN
cana-2401	89	5	f	f	X
cana-2401	89	6	:	:	PUNCT
cana-2401	89	7	v(g	v(g	NUM
cana-2401	89	8	)	)	PUNCT
cana-2401	89	9	→	→	PUNCT
cana-2401	89	10	0,2,4	0,2,4	NUM
cana-2401	89	11	,	,	PUNCT
cana-2401	89	12	…	…	PUNCT
cana-2401	89	13	2q	2q	NUM
cana-2401	89	14	by	by	ADP
cana-2401	89	15	f(u	f(u	PROPN
cana-2401	89	16	)	)	PUNCT
cana-2401	89	17	=	=	SYM
cana-2401	90	1	0	0	NUM
cana-2401	90	2	;	;	PUNCT
cana-2401	90	3	f	f	X
cana-2401	90	4	(	(	PUNCT
cana-2401	90	5	u′i	u′i	SYM
cana-2401	90	6	)	)	PUNCT
cana-2401	90	7	=	=	SYM
cana-2401	90	8	2(i	2(i	NUM
cana-2401	91	1	+	+	CCONJ
cana-2401	91	2	j	j	PROPN
cana-2401	91	3	-1	-1	PUNCT
cana-2401	91	4	)	)	PUNCT
cana-2401	92	1	i	i	PRON
cana-2401	92	2	=	=	NOUN
cana-2401	92	3	2	2	NUM
cana-2401	92	4	and	and	CCONJ
cana-2401	92	5	j	j	NOUN
cana-2401	92	6	=	=	SYM
cana-2401	92	7	1	1	NUM
cana-2401	92	8	,	,	PUNCT
cana-2401	92	9	2	2	NUM
cana-2401	92	10	;	;	PUNCT
cana-2401	92	11	f	f	X
cana-2401	92	12	(	(	PUNCT
cana-2401	92	13	u′i	u′i	SYM
cana-2401	92	14	)	)	PUNCT
cana-2401	92	15	=	=	SYM
cana-2401	92	16	6(i-1	6(i-1	NUM
cana-2401	92	17	)	)	PUNCT
cana-2401	92	18	,	,	PUNCT
cana-2401	92	19	i	i	NOUN
cana-2401	92	20	=	=	NOUN
cana-2401	92	21	3,4	3,4	NUM
cana-2401	92	22	,	,	PUNCT
cana-2401	92	23	…	…	PUNCT
cana-2401	92	24	n	n	CCONJ
cana-2401	92	25	;	;	PUNCT
cana-2401	92	26	f	f	PROPN
cana-2401	92	27	(	(	PUNCT
cana-2401	92	28	u′′i	u′′i	PROPN
cana-2401	92	29	)	)	PUNCT
cana-2401	93	1	=	=	SYM
cana-2401	93	2	6i-10	6i-10	NUM
cana-2401	93	3	,	,	PUNCT
cana-2401	93	4	i	i	PRON
cana-2401	93	5	=	=	NOUN
cana-2401	93	6	3	3	NUM
cana-2401	93	7	,	,	PUNCT
cana-2401	93	8	4,	4,	ADJ
cana-2401	93	9	…	…	SYM
cana-2401	93	10	n	n	PRON
cana-2401	93	11	then	then	ADV
cana-2401	93	12	the	the	DET
cana-2401	93	13	induced	induced	ADJ
cana-2401	93	14	edge	edge	NOUN
cana-2401	93	15	labels	label	NOUN
cana-2401	93	16	are	be	AUX
cana-2401	93	17	defined	define	VERB
cana-2401	93	18	as	as	SCONJ
cana-2401	93	19	follows	follow	VERB
cana-2401	93	20	;	;	PUNCT
cana-2401	93	21	f′(ui	f′(ui	PROPN
cana-2401	93	22	u′i	u′i	ADJ
cana-2401	93	23	)	)	PUNCT
cana-2401	93	24	=	=	SYM
cana-2401	93	25	6i	6i	NUM
cana-2401	93	26	–	–	PUNCT
cana-2401	93	27	9	9	NUM
cana-2401	93	28	,	,	PUNCT
cana-2401	93	29	f′(ui	f′(ui	PROPN
cana-2401	93	30	u′′i	u′′i	PROPN
cana-2401	93	31	)	)	PUNCT
cana-2401	94	1	=	=	SYM
cana-2401	94	2	6i	6i	NUM
cana-2401	94	3	–	–	PUNCT
cana-2401	94	4	7	7	NUM
cana-2401	94	5	,	,	PUNCT
cana-2401	94	6	i	i	PRON
cana-2401	94	7	=	=	NOUN
cana-2401	94	8	2,4	2,4	NUM
cana-2401	94	9	,	,	PUNCT
cana-2401	94	10	…	…	PUNCT
cana-2401	94	11	,	,	PUNCT
cana-2401	94	12	n-1	n-1	NOUN
cana-2401	94	13	and	and	CCONJ
cana-2401	94	14	f′(ui	f′(ui	PROPN
cana-2401	94	15	ui+1	ui+1	NOUN
cana-2401	94	16	)	)	PUNCT
cana-2401	95	1	=	=	SYM
cana-2401	95	2	6i-5	6i-5	NOUN
cana-2401	95	3	;	;	PUNCT
cana-2401	95	4	i	i	PRON
cana-2401	95	5	=	=	NOUN
cana-2401	95	6	1	1	NUM
cana-2401	95	7	,	,	PUNCT
cana-2401	95	8	2,	2,	NUM
cana-2401	95	9	…	…	NUM
cana-2401	95	10	n-1	n-1	NOUN
cana-2401	95	11	hence	hence	ADV
cana-2401	95	12	g	g	PROPN
cana-2401	95	13	is	be	AUX
cana-2401	95	14	an	an	DET
cana-2401	95	15	even	even	ADJ
cana-2401	95	16	vertex	vertex	NOUN
cana-2401	95	17	odd	odd	ADJ
cana-2401	95	18	edge	edge	NOUN
cana-2401	95	19	root	root	NOUN
cana-2401	95	20	square	square	ADJ
cana-2401	95	21	mean	mean	ADJ
cana-2401	95	22	labelling	labelling	NOUN
cana-2401	95	23	of	of	ADP
cana-2401	95	24	graph	graph	NOUN
cana-2401	95	25	since	since	SCONJ
cana-2401	95	26	f	f	PROPN
cana-2401	95	27	is	be	AUX
cana-2401	95	28	an	an	DET
cana-2401	95	29	even	even	ADJ
cana-2401	95	30	vertex	vertex	NOUN
cana-2401	95	31	odd	odd	ADJ
cana-2401	95	32	edge	edge	NOUN
cana-2401	95	33	root	root	NOUN
cana-2401	95	34	square	square	ADJ
cana-2401	95	35	mean	mean	ADJ
cana-2401	95	36	labelling	labelling	NOUN
cana-2401	95	37	of	of	ADP
cana-2401	95	38	graph	graph	NOUN
cana-2401	95	39	g.	g.	NOUN
cana-2401	95	40	communications	communication	NOUN
cana-2401	95	41	on	on	ADP
cana-2401	95	42	applied	apply	VERB
cana-2401	95	43	nonlinear	nonlinear	ADJ
cana-2401	95	44	analysis	analysis	NOUN
cana-2401	95	45	issn	issn	NOUN
cana-2401	95	46	:	:	PUNCT
cana-2401	95	47	1074	1074	NUM
cana-2401	95	48	-	-	PUNCT
cana-2401	95	49	133x	133x	NUM
cana-2401	95	50	vol	vol	NOUN
cana-2401	95	51	32	32	NUM
cana-2401	95	52	no	no	NOUN
cana-2401	95	53	.	.	PUNCT
cana-2401	96	1	2s	2s	NUM
cana-2401	96	2	(	(	PUNCT
cana-2401	96	3	2025	2025	NUM
cana-2401	96	4	)	)	PUNCT
cana-2401	96	5	301	301	NUM
cana-2401	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	96	7	illustration	illustration	NOUN
cana-2401	96	8	:	:	PUNCT
cana-2401	96	9	figure	figure	NOUN
cana-2401	96	10	5	5	NUM
cana-2401	96	11	illustrate	illustrate	VERB
cana-2401	96	12	the	the	DET
cana-2401	96	13	gp	gp	NOUN
cana-2401	96	14	on	on	ADP
cana-2401	96	15	6	6	NUM
cana-2401	96	16	vertices	vertex	NOUN
cana-2401	96	17	is	be	AUX
cana-2401	96	18	even	even	ADV
cana-2401	96	19	vertex	vertex	NOUN
cana-2401	96	20	odd	odd	ADJ
cana-2401	96	21	edge	edge	NOUN
cana-2401	96	22	root	root	NOUN
cana-2401	96	23	square	square	ADJ
cana-2401	96	24	mean	mean	NOUN
cana-2401	96	25	labeling	labeling	NOUN
cana-2401	96	26	figure	figure	NOUN
cana-2401	96	27	5	5	NUM
cana-2401	96	28	gp	gp	NOUN
cana-2401	96	29	on	on	ADP
cana-2401	96	30	6	6	NUM
cana-2401	96	31	vertices	vertex	NOUN
cana-2401	96	32	theorem	theorem	VERB
cana-2401	96	33	3.5	3.5	NUM
cana-2401	96	34	:	:	PUNCT
cana-2401	96	35	k1,n	k1,n	PROPN
cana-2401	96	36	an	an	DET
cana-2401	96	37	even	even	ADJ
cana-2401	96	38	vertex	vertex	NOUN
cana-2401	96	39	odd	odd	ADJ
cana-2401	96	40	edge	edge	NOUN
cana-2401	96	41	mean	mean	PROPN
cana-2401	96	42	square	square	ADJ
cana-2401	96	43	labelling	labelling	NOUN
cana-2401	96	44	graph	graph	NOUN
cana-2401	96	45	.	.	PUNCT
cana-2401	97	1	if	if	SCONJ
cana-2401	97	2	n	n	PRON
cana-2401	97	3	≤	≤	ADV
cana-2401	97	4	3	3	NUM
cana-2401	97	5	.	.	PUNCT
cana-2401	98	1	proof	proof	NOUN
cana-2401	98	2	:	:	PUNCT
cana-2401	98	3	let	let	VERB
cana-2401	98	4	the	the	DET
cana-2401	98	5	vertex	vertex	NOUN
cana-2401	98	6	and	and	CCONJ
cana-2401	98	7	edge	edge	NOUN
cana-2401	98	8	sets	set	NOUN
cana-2401	98	9	of	of	ADP
cana-2401	98	10	k1,n	k1,n	PROPN
cana-2401	98	11	be	be	AUX
cana-2401	98	12	defined	define	VERB
cana-2401	98	13	as	as	ADP
cana-2401	98	14	{	{	PUNCT
cana-2401	98	15	u	u	NOUN
cana-2401	98	16	,	,	PUNCT
cana-2401	98	17	v1	v1	NOUN
cana-2401	98	18	,	,	PUNCT
cana-2401	98	19	v2	v2	PROPN
cana-2401	98	20	,	,	PUNCT
cana-2401	98	21	…	…	PUNCT
cana-2401	98	22	,	,	PUNCT
cana-2401	98	23	vn}and	vn}and	CCONJ
cana-2401	98	24	{	{	PUNCT
cana-2401	98	25	e1	e1	PROPN
cana-2401	98	26	,	,	PUNCT
cana-2401	98	27	e2	e2	PROPN
cana-2401	98	28	,	,	PUNCT
cana-2401	98	29	…	…	PUNCT
cana-2401	98	30	,	,	PUNCT
cana-2401	98	31	en	en	ADP
cana-2401	98	32	}	}	PUNCT
cana-2401	98	33	respectively	respectively	ADV
cana-2401	98	34	.	.	PUNCT
cana-2401	99	1	for	for	ADP
cana-2401	99	2	n	n	PRON
cana-2401	99	3	≤	≤	NOUN
cana-2401	99	4	3	3	NUM
cana-2401	99	5	,	,	PUNCT
cana-2401	99	6	it	it	PRON
cana-2401	99	7	exhibits	exhibit	VERB
cana-2401	99	8	an	an	DET
cana-2401	99	9	odd	odd	ADJ
cana-2401	99	10	vertex	vertex	NOUN
cana-2401	99	11	even	even	ADV
cana-2401	99	12	edge	edge	NOUN
cana-2401	99	13	mean	mean	PROPN
cana-2401	99	14	square	square	ADJ
cana-2401	99	15	labelling	labelling	NOUN
cana-2401	99	16	graph	graph	NOUN
cana-2401	99	17	.	.	PUNCT
cana-2401	100	1	this	this	PRON
cana-2401	100	2	demonstrates	demonstrate	VERB
cana-2401	100	3	the	the	DET
cana-2401	100	4	figures	figure	NOUN
cana-2401	100	5	that	that	PRON
cana-2401	100	6	are	be	AUX
cana-2401	100	7	admitted	admit	VERB
cana-2401	100	8	as	as	ADP
cana-2401	100	9	an	an	DET
cana-2401	100	10	odd	odd	ADJ
cana-2401	100	11	vertex	vertex	NOUN
cana-2401	100	12	even	even	ADV
cana-2401	100	13	edge	edge	NOUN
cana-2401	100	14	mean	mean	PROPN
cana-2401	100	15	square	square	ADJ
cana-2401	100	16	labelling	labelling	NOUN
cana-2401	100	17	graph	graph	NOUN
cana-2401	100	18	.	.	PUNCT
cana-2401	101	1	if	if	SCONJ
cana-2401	101	2	n	n	NOUN
cana-2401	101	3	=	=	SYM
cana-2401	101	4	4	4	NUM
cana-2401	101	5	k	k	NOUN
cana-2401	101	6	1	1	NUM
cana-2401	101	7	,	,	PUNCT
cana-2401	101	8	4	4	NUM
cana-2401	101	9	is	be	AUX
cana-2401	101	10	not	not	PART
cana-2401	101	11	even	even	ADV
cana-2401	101	12	vertex	vertex	NOUN
cana-2401	101	13	and	and	CCONJ
cana-2401	101	14	odd	odd	ADJ
cana-2401	101	15	edge	edge	NOUN
cana-2401	101	16	labelling	labelling	NOUN
cana-2401	101	17	graph	graph	NOUN
cana-2401	101	18	since	since	SCONJ
cana-2401	101	19	the	the	DET
cana-2401	101	20	graph	graph	NOUN
cana-2401	101	21	depicts	depict	VERB
cana-2401	101	22	that	that	SCONJ
cana-2401	101	23	the	the	DET
cana-2401	101	24	edge	edge	NOUN
cana-2401	101	25	label	label	NOUN
cana-2401	101	26	5	5	NUM
cana-2401	101	27	repeated	repeat	VERB
cana-2401	101	28	two	two	NUM
cana-2401	101	29	times	time	NOUN
cana-2401	101	30	if	if	SCONJ
cana-2401	101	31	n	n	NOUN
cana-2401	101	32	=	=	SYM
cana-2401	101	33	5	5	NUM
cana-2401	101	34	communications	communication	NOUN
cana-2401	101	35	on	on	ADP
cana-2401	101	36	applied	apply	VERB
cana-2401	101	37	nonlinear	nonlinear	ADJ
cana-2401	101	38	analysis	analysis	NOUN
cana-2401	101	39	issn	issn	NOUN
cana-2401	101	40	:	:	PUNCT
cana-2401	101	41	1074	1074	NUM
cana-2401	101	42	-	-	PUNCT
cana-2401	101	43	133x	133x	NUM
cana-2401	101	44	vol	vol	NOUN
cana-2401	101	45	32	32	NUM
cana-2401	101	46	no	no	NOUN
cana-2401	101	47	.	.	PUNCT
cana-2401	102	1	2s	2s	NUM
cana-2401	102	2	(	(	PUNCT
cana-2401	102	3	2025	2025	NUM
cana-2401	102	4	)	)	PUNCT
cana-2401	102	5	302	302	NUM
cana-2401	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	102	7	since	since	SCONJ
cana-2401	102	8	the	the	DET
cana-2401	102	9	graph	graph	NOUN
cana-2401	102	10	depicts	depict	VERB
cana-2401	102	11	that	that	SCONJ
cana-2401	102	12	the	the	DET
cana-2401	102	13	edge	edge	NOUN
cana-2401	102	14	label	label	NOUN
cana-2401	102	15	5	5	NUM
cana-2401	102	16	repeated	repeat	VERB
cana-2401	102	17	two	two	NUM
cana-2401	102	18	times	time	NOUN
cana-2401	102	19	in	in	ADP
cana-2401	102	20	general	general	ADJ
cana-2401	102	21	,	,	PUNCT
cana-2401	102	22	n	n	CCONJ
cana-2401	102	23	>	>	SYM
cana-2401	102	24	3	3	NUM
cana-2401	102	25	case	case	NOUN
cana-2401	102	26	(	(	PUNCT
cana-2401	102	27	i	i	NOUN
cana-2401	102	28	)	)	PUNCT
cana-2401	102	29	let	let	VERB
cana-2401	102	30	f(u	f(u	PROPN
cana-2401	102	31	)	)	PUNCT
cana-2401	102	32	=	=	SYM
cana-2401	103	1	0	0	NUM
cana-2401	103	2	,	,	PUNCT
cana-2401	103	3	then	then	ADV
cana-2401	103	4	f	f	PROPN
cana-2401	103	5	(	(	PUNCT
cana-2401	103	6	ui	ui	PROPN
cana-2401	103	7	)	)	PUNCT
cana-2401	103	8	=	=	PUNCT
cana-2401	103	9	{	{	PUNCT
cana-2401	103	10	2	2	NUM
cana-2401	103	11	,	,	PUNCT
cana-2401	103	12	4	4	NUM
cana-2401	103	13	,	,	PUNCT
cana-2401	103	14	6	6	NUM
cana-2401	103	15	,	,	PUNCT
cana-2401	103	16	…	…	PUNCT
cana-2401	103	17	,	,	PUNCT
cana-2401	103	18	2q	2q	NUM
cana-2401	103	19	}	}	PUNCT
cana-2401	103	20	in	in	ADP
cana-2401	103	21	this	this	DET
cana-2401	103	22	case	case	NOUN
cana-2401	103	23	,	,	PUNCT
cana-2401	103	24	(	(	PUNCT
cana-2401	103	25	0,6	0,6	NUM
cana-2401	103	26	)	)	PUNCT
cana-2401	103	27	,	,	PUNCT
cana-2401	103	28	(	(	PUNCT
cana-2401	103	29	0	0	NUM
cana-2401	103	30	,	,	PUNCT
cana-2401	103	31	8)	8)	NUM
cana-2401	103	32	,	,	PUNCT
cana-2401	103	33	will	will	AUX
cana-2401	103	34	get	get	VERB
cana-2401	103	35	same	same	ADJ
cana-2401	103	36	labels	label	NOUN
cana-2401	103	37	,	,	PUNCT
cana-2401	103	38	which	which	PRON
cana-2401	103	39	is	be	AUX
cana-2401	103	40	contradiction	contradiction	NOUN
cana-2401	103	41	to	to	ADP
cana-2401	103	42	the	the	DET
cana-2401	103	43	definition	definition	NOUN
cana-2401	103	44	.	.	PUNCT
cana-2401	104	1	case	case	NOUN
cana-2401	104	2	(	(	PUNCT
cana-2401	104	3	ii	ii	NOUN
cana-2401	104	4	)	)	PUNCT
cana-2401	104	5	let	let	VERB
cana-2401	104	6	f(u	f(u	PROPN
cana-2401	104	7	)	)	PUNCT
cana-2401	105	1	=	=	SYM
cana-2401	105	2	2q	2q	NOUN
cana-2401	105	3	,	,	PUNCT
cana-2401	105	4	then	then	ADV
cana-2401	105	5	no	no	DET
cana-2401	105	6	edge	edge	NOUN
cana-2401	105	7	can	can	AUX
cana-2401	105	8	get	get	VERB
cana-2401	105	9	the	the	DET
cana-2401	105	10	label	label	NOUN
cana-2401	105	11	1	1	NUM
cana-2401	105	12	.	.	PUNCT
cana-2401	106	1	since	since	SCONJ
cana-2401	106	2	(	(	PUNCT
cana-2401	106	3	0	0	NUM
cana-2401	106	4	,	,	PUNCT
cana-2401	106	5	2q	2q	NUM
cana-2401	106	6	)	)	PUNCT
cana-2401	106	7	=	=	SYM
cana-2401	106	8	√2(q)2	√2(q)2	NOUN
cana-2401	106	9	=	=	PUNCT
cana-2401	106	10	√2	√2	NOUN
cana-2401	106	11	q	q	PROPN
cana-2401	106	12	,	,	PUNCT
cana-2401	106	13	which	which	PRON
cana-2401	106	14	is	be	AUX
cana-2401	106	15	not	not	PART
cana-2401	106	16	possible	possible	ADJ
cana-2401	106	17	.	.	PUNCT
cana-2401	107	1	case	case	NOUN
cana-2401	107	2	(	(	PUNCT
cana-2401	107	3	iii	iii	NOUN
cana-2401	107	4	)	)	PUNCT
cana-2401	107	5	if	if	SCONJ
cana-2401	107	6	f(u	f(u	PROPN
cana-2401	107	7	)	)	PUNCT
cana-2401	107	8	=	=	PRON
cana-2401	107	9	{	{	PUNCT
cana-2401	107	10	2	2	NUM
cana-2401	107	11	or	or	CCONJ
cana-2401	107	12	4	4	NUM
cana-2401	107	13	or	or	CCONJ
cana-2401	107	14	6	6	NUM
cana-2401	107	15	or	or	CCONJ
cana-2401	107	16	…	…	PUNCT
cana-2401	107	17	.	.	NOUN
cana-2401	107	18	2(q-1	2(q-1	NUM
cana-2401	107	19	)	)	PUNCT
cana-2401	107	20	}	}	PUNCT
cana-2401	107	21	,	,	PUNCT
cana-2401	107	22	then	then	ADV
cana-2401	107	23	we	we	PRON
cana-2401	107	24	get	get	VERB
cana-2401	107	25	the	the	DET
cana-2401	107	26	repetition	repetition	NOUN
cana-2401	107	27	of	of	ADP
cana-2401	107	28	odd	odd	ADJ
cana-2401	107	29	edge	edge	NOUN
cana-2401	107	30	labelling	labelling	NOUN
cana-2401	107	31	.	.	PUNCT
cana-2401	108	1	so	so	ADV
cana-2401	108	2	,	,	PUNCT
cana-2401	108	3	clearly	clearly	ADV
cana-2401	108	4	one	one	PRON
cana-2401	108	5	can	can	AUX
cana-2401	108	6	check	check	VERB
cana-2401	108	7	labelling	labelling	NOUN
cana-2401	108	8	of	of	ADP
cana-2401	108	9	a	a	DET
cana-2401	108	10	graph	graph	NOUN
cana-2401	108	11	.	.	PUNCT
cana-2401	109	1	theorem	theorem	VERB
cana-2401	109	2	3.6	3.6	NUM
cana-2401	109	3	a	a	DET
cana-2401	109	4	subdivision	subdivision	NOUN
cana-2401	109	5	of	of	ADP
cana-2401	109	6	pn	pn	PROPN
cana-2401	109	7	ʘ	ʘ	PROPN
cana-2401	109	8	k1	k1	PROPN
cana-2401	109	9	is	be	AUX
cana-2401	109	10	an	an	DET
cana-2401	109	11	even	even	ADJ
cana-2401	109	12	vertex	vertex	NOUN
cana-2401	109	13	odd	odd	ADJ
cana-2401	109	14	edge	edge	NOUN
cana-2401	109	15	root	root	NOUN
cana-2401	109	16	square	square	ADJ
cana-2401	109	17	mean	mean	ADJ
cana-2401	109	18	labelling	labelling	NOUN
cana-2401	109	19	graph	graph	NOUN
cana-2401	109	20	.	.	PUNCT
cana-2401	110	1	proof	proof	NOUN
cana-2401	110	2	:	:	PUNCT
cana-2401	110	3	the	the	DET
cana-2401	110	4	vertex	vertex	NOUN
cana-2401	110	5	set	set	NOUN
cana-2401	110	6	and	and	CCONJ
cana-2401	110	7	edge	edge	NOUN
cana-2401	110	8	set	set	NOUN
cana-2401	110	9	of	of	ADP
cana-2401	110	10	,	,	PUNCT
cana-2401	110	11	g	g	PROPN
cana-2401	110	12	,	,	PUNCT
cana-2401	110	13	is	be	AUX
cana-2401	110	14	defined	define	VERB
cana-2401	110	15	as	as	ADP
cana-2401	110	16	follows	follow	VERB
cana-2401	110	17	;	;	PUNCT
cana-2401	110	18	v(g	v(g	NUM
cana-2401	110	19	)	)	PUNCT
cana-2401	110	20	=	=	SYM
cana-2401	110	21	{	{	PUNCT
cana-2401	110	22	ui	ui	NOUN
cana-2401	110	23	;	;	PUNCT
cana-2401	110	24	1≤	1≤	NUM
cana-2401	110	25	𝑖	𝑖	SYM
cana-2401	110	26	≤	≤	NUM
cana-2401	110	27	𝑛	𝑛	PRON
cana-2401	110	28	}	}	PUNCT
cana-2401	110	29	∪	∪	ADJ
cana-2401	110	30	{	{	PUNCT
cana-2401	110	31	vi	vi	NOUN
cana-2401	110	32	1	1	NUM
cana-2401	110	33	≤	≤	NUM
cana-2401	110	34	i	i	PRON
cana-2401	110	35	≤	≤	NOUN
cana-2401	110	36	n	n	CCONJ
cana-2401	110	37	−	−	PROPN
cana-2401	110	38	1	1	NUM
cana-2401	110	39	}	}	PUNCT
cana-2401	110	40	∪	∪	ADJ
cana-2401	110	41	{	{	PUNCT
cana-2401	110	42	ui	ui	NOUN
cana-2401	110	43	,	,	PUNCT
cana-2401	110	44	u′′i	u′′i	PROPN
cana-2401	110	45	1	1	NUM
cana-2401	110	46	≤	≤	NUM
cana-2401	110	47	i	i	PRON
cana-2401	110	48	≤	≤	NOUN
cana-2401	110	49	n	n	CCONJ
cana-2401	110	50	}	}	PUNCT
cana-2401	110	51	and	and	CCONJ
cana-2401	110	52	e(g	e(g	NOUN
cana-2401	110	53	)	)	PUNCT
cana-2401	111	1	=	=	NOUN
cana-2401	111	2	{	{	PUNCT
cana-2401	111	3	{	{	PUNCT
cana-2401	111	4	ui	ui	PROPN
cana-2401	111	5	u′i	u′i	ADJ
cana-2401	111	6	}	}	PUNCT
cana-2401	111	7	∪	∪	X
cana-2401	111	8	{	{	PUNCT
cana-2401	111	9	ui	ui	NOUN
cana-2401	111	10	,	,	PUNCT
cana-2401	111	11	u′′i	u′′i	PROPN
cana-2401	111	12	}	}	PUNCT
cana-2401	111	13	∪	∪	VERB
cana-2401	111	14	{	{	PUNCT
cana-2401	111	15	ui	ui	NOUN
cana-2401	111	16	vi	vi	PROPN
cana-2401	111	17	}	}	PUNCT
cana-2401	111	18	∪	∪	X
cana-2401	111	19	{	{	PUNCT
cana-2401	111	20	ui	ui	NOUN
cana-2401	111	21	vi	vi	PROPN
cana-2401	112	1	+	+	CCONJ
cana-2401	112	2	1	1	NUM
cana-2401	112	3	}	}	PUNCT
cana-2401	112	4	:	:	PUNCT
cana-2401	112	5	1≤	1≤	X
cana-2401	112	6	𝑖	𝑖	SYM
cana-2401	112	7	≤	≤	NOUN
cana-2401	112	8	𝑛-1	𝑛-1	ADV
cana-2401	112	9	}	}	PUNCT
cana-2401	112	10	we	we	PRON
cana-2401	112	11	define	define	VERB
cana-2401	112	12	a	a	DET
cana-2401	112	13	map	map	NOUN
cana-2401	112	14	f	f	X
cana-2401	112	15	:	:	PUNCT
cana-2401	112	16	v(g	v(g	NUM
cana-2401	112	17	)	)	PUNCT
cana-2401	112	18	→	→	SYM
cana-2401	112	19	{	{	PUNCT
cana-2401	112	20	0,1	0,1	NUM
cana-2401	112	21	,	,	PUNCT
cana-2401	112	22	2	2	NUM
cana-2401	112	23	,	,	PUNCT
cana-2401	112	24	3,	3,	ADJ
cana-2401	112	25	…	…	SYM
cana-2401	112	26	2q	2q	NUM
cana-2401	112	27	}	}	PUNCT
cana-2401	112	28	.	.	PUNCT
cana-2401	113	1	the	the	DET
cana-2401	113	2	labelling	labelling	NOUN
cana-2401	113	3	pattern	pattern	NOUN
cana-2401	113	4	of	of	ADP
cana-2401	113	5	vertices	vertex	NOUN
cana-2401	113	6	can	can	AUX
cana-2401	113	7	be	be	AUX
cana-2401	113	8	defined	define	VERB
cana-2401	113	9	as	as	SCONJ
cana-2401	113	10	follows	follow	VERB
cana-2401	113	11	;	;	PUNCT
cana-2401	113	12	f	f	PROPN
cana-2401	113	13	(	(	PUNCT
cana-2401	113	14	ui	ui	PROPN
cana-2401	113	15	)	)	PUNCT
cana-2401	114	1	=	=	NOUN
cana-2401	114	2	4i	4i	NOUN
cana-2401	114	3	;	;	PUNCT
cana-2401	115	1	i	i	NOUN
cana-2401	115	2	=	=	NOUN
cana-2401	115	3	1,2	1,2	NUM
cana-2401	115	4	f	f	X
cana-2401	115	5	(	(	PUNCT
cana-2401	115	6	ui	ui	PROPN
cana-2401	115	7	)	)	PUNCT
cana-2401	115	8	=	=	PUNCT
cana-2401	115	9	8i	8i	NUM
cana-2401	115	10	–	–	PUNCT
cana-2401	115	11	6	6	NUM
cana-2401	115	12	;	;	PUNCT
cana-2401	115	13	i	i	PRON
cana-2401	115	14	=	=	NOUN
cana-2401	115	15	3,4,5	3,4,5	NUM
cana-2401	115	16	…	…	NUM
cana-2401	115	17	,	,	PUNCT
cana-2401	115	18	n-1	n-1	NOUN
cana-2401	115	19	f(vi	f(vi	NOUN
cana-2401	115	20	)	)	PUNCT
cana-2401	115	21	=	=	SYM
cana-2401	115	22	6i	6i	NUM
cana-2401	115	23	;	;	PUNCT
cana-2401	115	24	i	i	NOUN
cana-2401	115	25	=	=	NOUN
cana-2401	115	26	1,2	1,2	NUM
cana-2401	115	27	f	f	X
cana-2401	115	28	(	(	PUNCT
cana-2401	115	29	vi	vi	NOUN
cana-2401	115	30	)	)	PUNCT
cana-2401	115	31	=	=	PUNCT
cana-2401	115	32	8i	8i	NUM
cana-2401	115	33	–	–	PUNCT
cana-2401	115	34	2	2	NUM
cana-2401	115	35	;	;	PUNCT
cana-2401	115	36	i	i	PROPN
cana-2401	115	37	=	=	NOUN
cana-2401	115	38	3,4,5	3,4,5	NUM
cana-2401	115	39	…	…	NUM
cana-2401	115	40	,	,	PUNCT
cana-2401	115	41	n	n	PRON
cana-2401	115	42	f	f	X
cana-2401	115	43	(	(	PUNCT
cana-2401	115	44	u′i	u′i	SYM
cana-2401	115	45	)	)	PUNCT
cana-2401	115	46	=	=	PUNCT
cana-2401	115	47	8i	8i	NUM
cana-2401	115	48	-6	-6	NOUN
cana-2401	115	49	;	;	PUNCT
cana-2401	115	50	i	i	NOUN
cana-2401	115	51	=	=	NOUN
cana-2401	115	52	1,2	1,2	NUM
cana-2401	115	53	f(u′i	f(u′i	ADJ
cana-2401	115	54	)	)	PUNCT
cana-2401	115	55	=	=	SYM
cana-2401	115	56	8(i	8(i	NUM
cana-2401	115	57	–	–	PUNCT
cana-2401	115	58	8)	8)	NUM
cana-2401	115	59	;	;	PUNCT
cana-2401	115	60	i	i	PROPN
cana-2401	115	61	=	=	NOUN
cana-2401	115	62	3,4,5	3,4,5	NUM
cana-2401	115	63	…	…	PUNCT
cana-2401	115	64	,	,	PUNCT
cana-2401	115	65	n-1	n-1	PROPN
cana-2401	115	66	f	f	PROPN
cana-2401	115	67	(	(	PUNCT
cana-2401	115	68	u′′i	u′′i	PROPN
cana-2401	115	69	)	)	PUNCT
cana-2401	115	70	=	=	SYM
cana-2401	115	71	14(i-1	14(i-1	NUM
cana-2401	115	72	)	)	PUNCT
cana-2401	115	73	;	;	PUNCT
cana-2401	115	74	i	i	NOUN
cana-2401	115	75	=	=	NOUN
cana-2401	115	76	1,2	1,2	NUM
cana-2401	115	77	f	f	X
cana-2401	115	78	(	(	PUNCT
cana-2401	115	79	u′′i	u′′i	PROPN
cana-2401	115	80	)	)	PUNCT
cana-2401	115	81	=	=	PUNCT
cana-2401	115	82	8i	8i	NUM
cana-2401	115	83	4	4	NUM
cana-2401	115	84	;	;	PUNCT
cana-2401	115	85	i	i	PRON
cana-2401	115	86	=	=	PUNCT
cana-2401	115	87	3,4,5	3,4,5	NUM
cana-2401	115	88	…	…	NUM
cana-2401	115	89	,	,	PUNCT
cana-2401	115	90	n-1	n-1	NOUN
cana-2401	115	91	and	and	CCONJ
cana-2401	115	92	f	f	PROPN
cana-2401	115	93	(	(	PUNCT
cana-2401	115	94	un	un	PROPN
cana-2401	115	95	)	)	PUNCT
cana-2401	116	1	=	=	PUNCT
cana-2401	117	1	2(q	2(q	NUM
cana-2401	117	2	−	−	NUM
cana-2401	117	3	1	1	NUM
cana-2401	117	4	)	)	PUNCT
cana-2401	117	5	communications	communication	NOUN
cana-2401	117	6	on	on	ADP
cana-2401	117	7	applied	apply	VERB
cana-2401	117	8	nonlinear	nonlinear	ADJ
cana-2401	117	9	analysis	analysis	NOUN
cana-2401	117	10	issn	issn	NOUN
cana-2401	117	11	:	:	PUNCT
cana-2401	117	12	1074	1074	NUM
cana-2401	117	13	-	-	PUNCT
cana-2401	117	14	133x	133x	NUM
cana-2401	117	15	vol	vol	NOUN
cana-2401	117	16	32	32	NUM
cana-2401	117	17	no	no	NOUN
cana-2401	117	18	.	.	PUNCT
cana-2401	118	1	2s	2s	NUM
cana-2401	118	2	(	(	PUNCT
cana-2401	118	3	2025	2025	NUM
cana-2401	118	4	)	)	PUNCT
cana-2401	118	5	303	303	NUM
cana-2401	118	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2401	118	7	then	then	ADV
cana-2401	118	8	the	the	DET
cana-2401	118	9	induced	induced	ADJ
cana-2401	118	10	edge	edge	NOUN
cana-2401	118	11	labels	label	NOUN
cana-2401	118	12	are	be	AUX
cana-2401	118	13	f	f	PROPN
cana-2401	118	14	′	′	NUM
cana-2401	118	15	(	(	PUNCT
cana-2401	118	16	uivi	uivi	NOUN
cana-2401	118	17	)	)	PUNCT
cana-2401	118	18	=	=	SYM
cana-2401	119	1	6i	6i	NUM
cana-2401	119	2	–	–	PUNCT
cana-2401	119	3	1	1	NUM
cana-2401	119	4	;	;	PUNCT
cana-2401	119	5	i	i	NOUN
cana-2401	119	6	=	=	NOUN
cana-2401	120	1	1,2	1,2	NUM
cana-2401	120	2	f	f	NOUN
cana-2401	120	3	′	′	NUM
cana-2401	120	4	(	(	PUNCT
cana-2401	120	5	uivi	uivi	NOUN
cana-2401	120	6	)	)	PUNCT
cana-2401	120	7	=	=	PUNCT
cana-2401	120	8	8i	8i	NUM
cana-2401	120	9	–	–	PUNCT
cana-2401	120	10	3	3	NUM
cana-2401	120	11	;	;	PUNCT
cana-2401	120	12	i	i	PROPN
cana-2401	120	13	=	=	NOUN
cana-2401	120	14	3,4,5	3,4,5	NUM
cana-2401	120	15	…	…	PUNCT
cana-2401	120	16	,	,	PUNCT
cana-2401	120	17	n-1	n-1	VERB
cana-2401	120	18	f	f	PROPN
cana-2401	120	19	′(viui+1	′(viui+1	PROPN
cana-2401	120	20	)	)	PUNCT
cana-2401	120	21	=	=	PUNCT
cana-2401	120	22	8i	8i	NUM
cana-2401	120	23	–	–	PUNCT
cana-2401	120	24	1	1	NUM
cana-2401	120	25	;	;	PUNCT
cana-2401	120	26	1≤	1≤	NUM
cana-2401	120	27	𝑖	𝑖	SYM
cana-2401	120	28	≤	≤	PROPN
cana-2401	120	29	𝑛-1	𝑛-1	ADV
cana-2401	121	1	f	f	PROPN
cana-2401	122	1	′	′	NUM
cana-2401	122	2	(	(	PUNCT
cana-2401	122	3	uiu′i	uiu′i	PROPN
cana-2401	122	4	)	)	PUNCT
cana-2401	122	5	=	=	SYM
cana-2401	122	6	6i	6i	NUM
cana-2401	122	7	–	–	PUNCT
cana-2401	122	8	3	3	NUM
cana-2401	122	9	;	;	PUNCT
cana-2401	122	10	i	i	NOUN
cana-2401	122	11	=	=	NOUN
cana-2401	122	12	1,2	1,2	NUM
cana-2401	122	13	f	f	NOUN
cana-2401	122	14	′	′	NUM
cana-2401	122	15	(	(	PUNCT
cana-2401	122	16	uiu′i	uiu′i	PROPN
cana-2401	122	17	)	)	PUNCT
cana-2401	122	18	=	=	PUNCT
cana-2401	122	19	8i	8i	NUM
cana-2401	122	20	–	–	PUNCT
cana-2401	122	21	7	7	NUM
cana-2401	122	22	;	;	PUNCT
cana-2401	122	23	i	i	PROPN
cana-2401	122	24	=	=	NOUN
cana-2401	122	25	3,4,5	3,4,5	NUM
cana-2401	122	26	…	…	PUNCT
cana-2401	122	27	,	,	PUNCT
cana-2401	122	28	n-1	n-1	NOUN
cana-2401	123	1	f	f	PROPN
cana-2401	123	2	′	′	NUM
cana-2401	123	3	(	(	PUNCT
cana-2401	123	4	u′iu′′i	u′iu′′i	PROPN
cana-2401	123	5	)	)	PUNCT
cana-2401	124	1	=	=	NOUN
cana-2401	124	2	12i	12i	NOUN
cana-2401	124	3	–	–	PUNCT
cana-2401	124	4	11	11	NUM
cana-2401	124	5	;	;	PUNCT
cana-2401	124	6	i	i	PRON
cana-2401	124	7	=	=	NOUN
cana-2401	125	1	1,2	1,2	NUM
cana-2401	125	2	f	f	NOUN
cana-2401	125	3	′	′	NUM
cana-2401	125	4	(	(	PUNCT
cana-2401	125	5	u′iu′′i	u′iu′′i	PROPN
cana-2401	125	6	)	)	PUNCT
cana-2401	125	7	=	=	PUNCT
cana-2401	125	8	8i	8i	NUM
cana-2401	125	9	–	–	PUNCT
cana-2401	125	10	5	5	NUM
cana-2401	125	11	;	;	PUNCT
cana-2401	125	12	i	i	PROPN
cana-2401	125	13	=	=	NOUN
cana-2401	125	14	3,4,5	3,4,5	NUM
cana-2401	125	15	…	…	NUM
cana-2401	125	16	,	,	PUNCT
cana-2401	125	17	n-1	n-1	VERB
cana-2401	125	18	thus	thus	ADV
cana-2401	125	19	,	,	PUNCT
cana-2401	125	20	subdivision	subdivision	NOUN
cana-2401	125	21	of	of	ADP
cana-2401	125	22	g	g	PROPN
cana-2401	125	23	=	=	PROPN
cana-2401	125	24	pn	pn	PROPN
cana-2401	125	25	ʘ	ʘ	PROPN
cana-2401	125	26	k1	k1	PROPN
cana-2401	125	27	admit	admit	VERB
cana-2401	125	28	even	even	ADV
cana-2401	125	29	vertex	vertex	NOUN
cana-2401	125	30	odd	odd	ADJ
cana-2401	125	31	edge	edge	NOUN
cana-2401	125	32	root	root	NOUN
cana-2401	125	33	square	square	ADJ
cana-2401	125	34	mean	mean	ADJ
cana-2401	125	35	labelling	labelling	NOUN
cana-2401	125	36	graph	graph	NOUN
cana-2401	125	37	.	.	PUNCT
cana-2401	126	1	illustration	illustration	NOUN
cana-2401	126	2	:	:	PUNCT
cana-2401	126	3	figure	figure	NOUN
cana-2401	126	4	6	6	NUM
cana-2401	126	5	illustrate	illustrate	VERB
cana-2401	126	6	a	a	DET
cana-2401	126	7	subdivision	subdivision	NOUN
cana-2401	126	8	of	of	ADP
cana-2401	126	9	pn	pn	PROPN
cana-2401	126	10	ʘ	ʘ	PROPN
cana-2401	126	11	k1	k1	PROPN
cana-2401	126	12	on	on	ADP
cana-2401	126	13	8	8	NUM
cana-2401	126	14	vertices	vertex	NOUN
cana-2401	126	15	is	be	AUX
cana-2401	126	16	an	an	DET
cana-2401	126	17	even	even	ADJ
cana-2401	126	18	vertex	vertex	NOUN
cana-2401	126	19	odd	odd	ADJ
cana-2401	126	20	edge	edge	NOUN
cana-2401	126	21	root	root	NOUN
cana-2401	126	22	square	square	ADJ
cana-2401	126	23	mean	mean	NOUN
cana-2401	126	24	labeling	labeling	NOUN
cana-2401	126	25	figure	figure	NOUN
cana-2401	126	26	6	6	NUM
cana-2401	126	27	pn	pn	PROPN
cana-2401	126	28	ʘ	ʘ	PROPN
cana-2401	126	29	k1	k1	PROPN
cana-2401	126	30	on	on	ADP
cana-2401	126	31	8	8	NUM
cana-2401	126	32	vertices	vertex	NOUN
cana-2401	126	33	conclusion	conclusion	NOUN
cana-2401	126	34	:	:	PUNCT
cana-2401	126	35	in	in	ADP
cana-2401	126	36	conclusion	conclusion	NOUN
cana-2401	126	37	,	,	PUNCT
cana-2401	126	38	this	this	DET
cana-2401	126	39	paper	paper	NOUN
cana-2401	126	40	presents	present	VERB
cana-2401	126	41	a	a	DET
cana-2401	126	42	new	new	ADJ
cana-2401	126	43	labeling	labeling	NOUN
cana-2401	126	44	pattern	pattern	NOUN
cana-2401	126	45	based	base	VERB
cana-2401	126	46	on	on	ADP
cana-2401	126	47	the	the	DET
cana-2401	126	48	root	root	NOUN
cana-2401	126	49	square	square	ADJ
cana-2401	126	50	mean	mean	NOUN
cana-2401	126	51	,	,	PUNCT
cana-2401	126	52	which	which	PRON
cana-2401	126	53	has	have	AUX
cana-2401	126	54	been	be	AUX
cana-2401	126	55	successfully	successfully	ADV
cana-2401	126	56	applied	apply	VERB
cana-2401	126	57	to	to	ADP
cana-2401	126	58	path	path	NOUN
cana-2401	126	59	-	-	PUNCT
cana-2401	126	60	related	relate	VERB
cana-2401	126	61	graphs	graph	NOUN
cana-2401	126	62	,	,	PUNCT
cana-2401	126	63	the	the	DET
cana-2401	126	64	subdivision	subdivision	NOUN
cana-2401	126	65	of	of	ADP
cana-2401	126	66	path	path	NOUN
cana-2401	126	67	corona	corona	NOUN
cana-2401	126	68	graphs	graph	NOUN
cana-2401	126	69	,	,	PUNCT
cana-2401	126	70	and	and	CCONJ
cana-2401	126	71	star	star	NOUN
cana-2401	126	72	graphs	graph	NOUN
cana-2401	126	73	.	.	PUNCT
cana-2401	127	1	the	the	DET
cana-2401	127	2	results	result	NOUN
cana-2401	127	3	demonstrate	demonstrate	VERB
cana-2401	127	4	the	the	DET
cana-2401	127	5	effectiveness	effectiveness	NOUN
cana-2401	127	6	of	of	ADP
cana-2401	127	7	this	this	DET
cana-2401	127	8	approach	approach	NOUN
cana-2401	127	9	in	in	ADP
cana-2401	127	10	enhancing	enhance	VERB
cana-2401	127	11	graph	graph	NOUN
cana-2401	127	12	labeling	labeling	NOUN
cana-2401	127	13	techniques	technique	NOUN
cana-2401	127	14	and	and	CCONJ
cana-2401	127	15	offer	offer	VERB
cana-2401	127	16	potential	potential	NOUN
cana-2401	127	17	for	for	ADP
cana-2401	127	18	further	further	ADJ
cana-2401	127	19	research	research	NOUN
cana-2401	127	20	in	in	ADP
cana-2401	127	21	related	related	ADJ
cana-2401	127	22	areas	area	NOUN
cana-2401	127	23	.	.	PUNCT
cana-2401	128	1	references	reference	NOUN
cana-2401	128	2	[	[	X
cana-2401	128	3	1	1	NUM
cana-2401	128	4	]	]	X
cana-2401	128	5	j.a.gallian	j.a.gallian	ADJ
cana-2401	128	6	,	,	PUNCT
cana-2401	128	7	‘	'	PUNCT
cana-2401	128	8	a	a	DET
cana-2401	128	9	dynamic	dynamic	ADJ
cana-2401	128	10	survey	survey	NOUN
cana-2401	128	11	of	of	ADP
cana-2401	128	12	graphs	graph	NOUN
cana-2401	128	13	labelling’,the	labelling’,the	DET
cana-2401	128	14	electronic	electronic	ADJ
cana-2401	128	15	journal	journal	NOUN
cana-2401	128	16	of	of	ADP
cana-2401	128	17	combinatorics,18	combinatorics,18	NOUN
cana-2401	128	18	(	(	PUNCT
cana-2401	128	19	2015	2015	NUM
cana-2401	128	20	)	)	PUNCT
cana-2401	128	21	.	.	PUNCT
cana-2401	129	1	[	[	X
cana-2401	129	2	2	2	NUM
cana-2401	129	3	]	]	PUNCT
cana-2401	129	4	b.gayathri	b.gayathri	NOUN
cana-2401	129	5	and	and	CCONJ
cana-2401	129	6	r.gobi	r.gobi	NOUN
cana-2401	129	7	,	,	PUNCT
cana-2401	129	8	‘	'	PUNCT
cana-2401	129	9	(	(	PUNCT
cana-2401	129	10	k	k	X
cana-2401	129	11	,	,	PUNCT
cana-2401	129	12	d	d	NOUN
cana-2401	129	13	)	)	PUNCT
cana-2401	129	14	even	even	ADV
cana-2401	129	15	mean	mean	VERB
cana-2401	129	16	labelling	labelling	NOUN
cana-2401	129	17	of	of	ADP
cana-2401	129	18	corona	corona	NOUN
cana-2401	129	19	graphs	graph	NOUN
cana-2401	129	20	,	,	PUNCT
cana-2401	129	21	international	international	ADJ
cana-2401	129	22	journal	journal	NOUN
cana-2401	129	23	of	of	ADP
cana-2401	129	24	mathematics	mathematic	NOUN
cana-2401	129	25	and	and	CCONJ
cana-2401	129	26	soft	soft	ADJ
cana-2401	129	27	computing	computing	NOUN
cana-2401	129	28	,	,	PUNCT
cana-2401	129	29	vol.(1	vol.(1	PROPN
cana-2401	129	30	)	)	PUNCT
cana-2401	129	31	,	,	PUNCT
cana-2401	129	32	17	17	NUM
cana-2401	129	33	-	-	SYM
cana-2401	129	34	23	23	NUM
cana-2401	129	35	,	,	PUNCT
cana-2401	129	36	aug	aug	PROPN
cana-2401	129	37	(	(	PUNCT
cana-2401	129	38	2011	2011	NUM
cana-2401	129	39	)	)	PUNCT
cana-2401	129	40	.	.	PUNCT
cana-2401	130	1	[	[	X
cana-2401	130	2	3	3	X
cana-2401	130	3	]	]	PUNCT
cana-2401	130	4	b.gayathri	b.gayathri	NOUN
cana-2401	130	5	and	and	CCONJ
cana-2401	130	6	r.gobi	r.gobi	NOUN
cana-2401	130	7	,	,	PUNCT
cana-2401	130	8	‘	'	PUNCT
cana-2401	130	9	necessary	necessary	ADJ
cana-2401	130	10	condition	condition	NOUN
cana-2401	130	11	for	for	ADP
cana-2401	130	12	mean	mean	ADJ
cana-2401	130	13	labelling	labelling	NOUN
cana-2401	130	14	,	,	PUNCT
cana-2401	130	15	international	international	ADJ
cana-2401	130	16	journal	journal	NOUN
cana-2401	130	17	of	of	ADP
cana-2401	130	18	engineering	engineering	NOUN
cana-2401	130	19	science	science	NOUN
cana-2401	130	20	,	,	PUNCT
cana-2401	130	21	advanced	advanced	ADJ
cana-2401	130	22	computing	computing	NOUN
cana-2401	130	23	and	and	CCONJ
cana-2401	130	24	bio	bio	PROPN
cana-2401	130	25	technology	technology	PROPN
cana-2401	130	26	’	'	PUNCT
cana-2401	130	27	,	,	PUNCT
cana-2401	130	28	vol	vol	NOUN
cana-2401	130	29	.	.	PUNCT
cana-2401	131	1	4(3	4(3	NUM
cana-2401	131	2	)	)	PUNCT
cana-2401	131	3	,	,	PUNCT
cana-2401	131	4	43	43	NUM
cana-2401	131	5	-	-	SYM
cana-2401	131	6	52	52	NUM
cana-2401	131	7	,	,	PUNCT
cana-2401	131	8	july	july	PROPN
cana-2401	131	9	-	-	PUNCT
cana-2401	131	10	sep	sep	PROPN
cana-2401	131	11	.	.	PUNCT
cana-2401	132	1	(	(	PUNCT
cana-2401	132	2	2013	2013	NUM
cana-2401	132	3	)	)	PUNCT
cana-2401	132	4	.	.	PUNCT
cana-2401	133	1	[	[	X
cana-2401	133	2	4	4	X
cana-2401	133	3	]	]	X
cana-2401	133	4	r.ramdani	r.ramdani	X
cana-2401	133	5	,	,	PUNCT
cana-2401	133	6	a.y	a.y	NOUN
cana-2401	133	7	fanisa	fanisa	NOUN
cana-2401	133	8	,	,	PUNCT
cana-2401	133	9	s.	s.	PROPN
cana-2401	133	10	gumilar	gumilar	PROPN
cana-2401	133	11	,	,	PUNCT
cana-2401	133	12	s	s	NOUN
cana-2401	133	13	sabini	sabini	NOUN
cana-2401	133	14	,	,	PUNCT
cana-2401	133	15	‘	'	PUNCT
cana-2401	133	16	root	root	NOUN
cana-2401	133	17	square	square	ADJ
cana-2401	133	18	mean	mean	ADJ
cana-2401	133	19	labelling	labelling	NOUN
cana-2401	133	20	of	of	ADP
cana-2401	133	21	some	some	DET
cana-2401	133	22	graphs	graph	NOUN
cana-2401	133	23	obtained	obtain	VERB
cana-2401	133	24	from	from	ADP
cana-2401	133	25	paths	path	NOUN
cana-2401	133	26	‘	'	PUNCT
cana-2401	133	27	journal	journal	NOUN
cana-2401	133	28	of	of	ADP
cana-2401	133	29	physics	physics	PROPN
cana-2401	133	30	conference	conference	PROPN
cana-2401	133	31	series	series	NOUN
cana-2401	133	32	(	(	PUNCT
cana-2401	133	33	2021	2021	NUM
cana-2401	133	34	)	)	PUNCT
cana-2401	133	35	.	.	PUNCT
cana-2401	134	1	[	[	X
cana-2401	134	2	5	5	X
cana-2401	134	3	]	]	PUNCT
cana-2401	134	4	v.	v.	ADP
cana-2401	134	5	senthilkumar	senthilkumar	PROPN
cana-2401	134	6	,	,	PUNCT
cana-2401	134	7	k.	k.	PROPN
cana-2401	134	8	venkatesan	venkatesan	PROPN
cana-2401	134	9	,	,	PUNCT
cana-2401	134	10	“	"	PUNCT
cana-2401	134	11	odd	odd	ADJ
cana-2401	134	12	vertex	vertex	NOUN
cana-2401	134	13	even	even	ADV
cana-2401	134	14	edge	edge	NOUN
cana-2401	134	15	root	root	NOUN
cana-2401	134	16	square	square	ADJ
cana-2401	134	17	mean	mean	ADJ
cana-2401	134	18	labelling	labelling	NOUN
cana-2401	134	19	graphs	graph	NOUN
cana-2401	134	20	”	"	PUNCT
cana-2401	134	21	journal	journal	NOUN
cana-2401	134	22	of	of	ADP
cana-2401	134	23	interdisciplinary	interdisciplinary	ADJ
cana-2401	134	24	mathematics	mathematic	NOUN
cana-2401	134	25	,	,	PUNCT
cana-2401	134	26	vol	vol	NOUN
cana-2401	134	27	.	.	PROPN
cana-2401	134	28	27	27	NUM
cana-2401	134	29	(	(	PUNCT
cana-2401	134	30	2024	2024	NUM
cana-2401	134	31	)	)	PUNCT
cana-2401	134	32	,	,	PUNCT
cana-2401	134	33	no	no	INTJ
cana-2401	134	34	.	.	NOUN
cana-2401	134	35	5	5	NUM
cana-2401	134	36	,	,	PUNCT
cana-2401	134	37	pp	pp	ADJ
cana-2401	134	38	.	.	PUNCT
cana-2401	135	1	1001–1008	1001–1008	NUM
cana-2401	135	2	[	[	X
cana-2401	135	3	6	6	NUM
cana-2401	135	4	]	]	X
cana-2401	135	5	s.somasundaram	s.somasundaram	NOUN
cana-2401	135	6	and	and	CCONJ
cana-2401	135	7	r.ponraj	r.ponraj	NOUN
cana-2401	135	8	,	,	PUNCT
cana-2401	135	9	‘	'	PUNCT
cana-2401	135	10	mean	mean	ADJ
cana-2401	135	11	labelling	labelling	NOUN
cana-2401	135	12	of	of	ADP
cana-2401	135	13	graphs	graph	NOUN
cana-2401	135	14	’	'	PUNCT
cana-2401	135	15	,	,	PUNCT
cana-2401	135	16	national	national	PROPN
cana-2401	135	17	academy	academy	PROPN
cana-2401	135	18	science	science	PROPN
cana-2401	135	19	letters	letter	NOUN
cana-2401	135	20	,	,	PUNCT
cana-2401	135	21	26	26	NUM
cana-2401	135	22	,	,	PUNCT
cana-2401	135	23	210	210	NUM
cana-2401	135	24	-	-	SYM
cana-2401	135	25	213	213	NUM
cana-2401	135	26	(	(	PUNCT
cana-2401	135	27	2003	2003	NUM
cana-2401	135	28	)	)	PUNCT
cana-2401	135	29	.	.	PUNCT
cana-2401	136	1	[	[	X
cana-2401	136	2	7	7	X
cana-2401	136	3	]	]	PUNCT
cana-2401	136	4	s.	s.	PROPN
cana-2401	136	5	sandiya	sandiya	PROPN
cana-2401	136	6	,	,	PUNCT
cana-2401	136	7	s.	s.	PROPN
cana-2401	136	8	somasundaram	somasundaram	PROPN
cana-2401	136	9	and	and	CCONJ
cana-2401	136	10	s.arun,‘root	s.arun,‘root	NOUN
cana-2401	136	11	square	square	ADJ
cana-2401	136	12	mean	mean	NOUN
cana-2401	136	13	labelling	labelling	NOUN
cana-2401	136	14	graphs	graph	NOUN
cana-2401	136	15	’	'	PUNCT
cana-2401	136	16	internal	internal	ADJ
cana-2401	136	17	journal	journal	PROPN
cana-2401	136	18	contemporary	contemporary	PROPN
cana-2401	136	19	math	math	PROPN
cana-2401	136	20	.	.	PUNCT
cana-2401	137	1	science	science	NOUN
cana-2401	137	2	,	,	PUNCT
cana-2401	137	3	vol.9(14	vol.9(14	NOUN
cana-2401	137	4	)	)	PUNCT
cana-2401	137	5	,	,	PUNCT
cana-2401	137	6	page	page	NOUN
cana-2401	137	7	no	no	NOUN
cana-2401	137	8	.	.	NOUN
cana-2401	137	9	667	667	NUM
cana-2401	137	10	-	-	SYM
cana-2401	137	11	676	676	NUM
cana-2401	137	12	(	(	PUNCT
cana-2401	137	13	2014	2014	NUM
cana-2401	137	14	)	)	PUNCT
cana-2401	137	15	.	.	PUNCT
cana-2401	138	1	[	[	X
cana-2401	138	2	8	8	X
cana-2401	138	3	]	]	X
cana-2401	138	4	k.	k.	PROPN
cana-2401	138	5	thirugnasambandam	thirugnasambandam	PROPN
cana-2401	138	6	and	and	CCONJ
cana-2401	138	7	k.	k.	PROPN
cana-2401	138	8	venkatesan	venkatesan	PROPN
cana-2401	138	9	,	,	PUNCT
cana-2401	138	10	’	'	PUNCT
cana-2401	138	11	super	super	ADJ
cana-2401	138	12	root	root	PROPN
cana-2401	138	13	square	square	ADJ
cana-2401	138	14	mean	mean	ADJ
cana-2401	138	15	labelling	labelling	NOUN
cana-2401	138	16	of	of	ADP
cana-2401	138	17	graphs	graph	NOUN
cana-2401	138	18	’	'	PUNCT
cana-2401	138	19	,	,	PUNCT
cana-2401	138	20	international	international	ADJ
cana-2401	138	21	journal	journal	NOUN
cana-2401	138	22	of	of	ADP
cana-2401	138	23	mathematics	mathematic	NOUN
cana-2401	138	24	and	and	CCONJ
cana-2401	138	25	soft	soft	ADJ
cana-2401	138	26	computing	computing	NOUN
cana-2401	138	27	,	,	PUNCT
cana-2401	138	28	vol	vol	NOUN
cana-2401	138	29	.	.	PUNCT
cana-2401	138	30	5(2	5(2	NUM
cana-2401	138	31	)	)	PUNCT
cana-2401	138	32	,	,	PUNCT
cana-2401	138	33	page	page	NOUN
cana-2401	138	34	no	no	INTJ
cana-2401	138	35	.	.	NOUN
cana-2401	138	36	189	189	NUM
cana-2401	138	37	-	-	SYM
cana-2401	138	38	198	198	NUM
cana-2401	138	39	(	(	PUNCT
cana-2401	138	40	2025	2025	NUM
cana-2401	138	41	)	)	PUNCT
cana-2401	138	42	.	.	PUNCT