id	sid	tid	token	lemma	pos
cana-369	1	1	c:/users	c:/user	NOUN
cana-369	1	2	/	/	SYM
cana-369	1	3	welcome	welcome	ADJ
cana-369	1	4	/	/	SYM
cana-369	1	5	desktop	desktop	NOUN
cana-369	1	6	/	/	SYM
cana-369	1	7	aip	aip	PROPN
cana-369	1	8	/	/	SYM
cana-369	1	9	some	some	PRON
cana-369	1	10	results	result	VERB
cana-369	1	11	on	on	ADP
cana-369	1	12	odd	odd	ADJ
cana-369	1	13	even	even	ADV
cana-369	1	14	congruence	congruence	ADJ
cana-369	1	15	labeling	labeling	NOUN
cana-369	1	16	of	of	ADP
cana-369	1	17	graphs	graph	NOUN
cana-369	1	18	/	/	SYM
cana-369	1	19	latex	latex	NOUN
cana-369	1	20	-	-	PUNCT
cana-369	1	21	oddeven	oddeven	VERB
cana-369	1	22	-	-	PUNCT
cana-369	1	23	labeling.dvi	labeling.dvi	NOUN
cana-369	1	24	communications	communication	NOUN
cana-369	1	25	on	on	ADP
cana-369	1	26	applied	apply	VERB
cana-369	1	27	nonlinear	nonlinear	ADJ
cana-369	1	28	analysis	analysis	NOUN
cana-369	1	29	issn	issn	NOUN
cana-369	1	30	:	:	PUNCT
cana-369	1	31	1074	1074	NUM
cana-369	1	32	-	-	PUNCT
cana-369	1	33	133x	133x	NUM
cana-369	1	34	vol	vol	NOUN
cana-369	1	35	31	31	NUM
cana-369	1	36	no	no	NOUN
cana-369	1	37	.	.	NOUN
cana-369	1	38	1	1	NUM
cana-369	1	39	(	(	PUNCT
cana-369	1	40	2024	2024	NUM
cana-369	1	41	)	)	PUNCT
cana-369	1	42	141	141	NUM
cana-369	1	43	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	2	1	some	some	PRON
cana-369	2	2	results	result	VERB
cana-369	2	3	on	on	ADP
cana-369	2	4	odd	odd	ADJ
cana-369	2	5	-	-	PUNCT
cana-369	2	6	even	even	ADV
cana-369	2	7	congruence	congruence	NOUN
cana-369	2	8	labeling	labeling	NOUN
cana-369	2	9	of	of	ADP
cana-369	2	10	graphs	graph	NOUN
cana-369	2	11	g.thamizhendhi	g.thamizhendhi	DET
cana-369	2	12	1	1	NUM
cana-369	2	13	and	and	CCONJ
cana-369	2	14	k.	k.	PROPN
cana-369	2	15	kanakambika	kanakambika	PROPN
cana-369	2	16	2	2	NUM
cana-369	2	17	*	*	SYM
cana-369	2	18	1department	1department	NUM
cana-369	2	19	of	of	ADP
cana-369	2	20	mathematics	mathematic	NOUN
cana-369	2	21	,	,	PUNCT
cana-369	2	22	sri	sri	PROPN
cana-369	2	23	vasavi	vasavi	PROPN
cana-369	2	24	college	college	PROPN
cana-369	2	25	,	,	PUNCT
cana-369	2	26	erode-638	erode-638	VERB
cana-369	2	27	316	316	NUM
cana-369	2	28	,	,	PUNCT
cana-369	2	29	tamilnadu	tamilnadu	NOUN
cana-369	2	30	,	,	PUNCT
cana-369	2	31	india	india	PROPN
cana-369	2	32	.	.	PUNCT
cana-369	3	1	gkthamil@gmail.com	gkthamil@gmail.com	X
cana-369	4	1	2*department	2*department	NOUN
cana-369	4	2	of	of	ADP
cana-369	4	3	mathematics	mathematic	NOUN
cana-369	4	4	,	,	PUNCT
cana-369	4	5	vellalar	vellalar	ADJ
cana-369	4	6	college	college	NOUN
cana-369	4	7	for	for	ADP
cana-369	4	8	women	woman	NOUN
cana-369	4	9	(	(	PUNCT
cana-369	4	10	autonomous	autonomous	ADJ
cana-369	4	11	)	)	PUNCT
cana-369	4	12	,	,	PUNCT
cana-369	4	13	erode-638	erode-638	VERB
cana-369	4	14	102	102	NUM
cana-369	4	15	,	,	PUNCT
cana-369	4	16	tamilnadu	tamilnadu	ADJ
cana-369	4	17	,	,	PUNCT
cana-369	4	18	india	india	PROPN
cana-369	4	19	.	.	PUNCT
cana-369	5	1	email:kkanakambikavel@gmail.com	email:kkanakambikavel@gmail.com	X
cana-369	5	2	article	article	NOUN
cana-369	5	3	history	history	NOUN
cana-369	5	4	:	:	PUNCT
cana-369	5	5	received	receive	VERB
cana-369	5	6	:	:	PUNCT
cana-369	5	7	05	05	NUM
cana-369	5	8	-	-	SYM
cana-369	5	9	10	10	NUM
cana-369	5	10	-	-	PUNCT
cana-369	5	11	2023	2023	NUM
cana-369	5	12	revised	revise	VERB
cana-369	5	13	:	:	PUNCT
cana-369	5	14	15	15	NUM
cana-369	5	15	-	-	SYM
cana-369	5	16	11	11	NUM
cana-369	5	17	-	-	SYM
cana-369	5	18	2023	2023	NUM
cana-369	5	19	accepted	accept	VERB
cana-369	5	20	:	:	PUNCT
cana-369	5	21	02	02	NUM
cana-369	5	22	-	-	SYM
cana-369	5	23	12	12	NUM
cana-369	5	24	-	-	PUNCT
cana-369	5	25	2023	2023	NUM
cana-369	5	26	abstract	abstract	NOUN
cana-369	5	27	:	:	PUNCT
cana-369	5	28	introduction	introduction	NOUN
cana-369	5	29	:	:	PUNCT
cana-369	5	30	labeling	labeling	NOUN
cana-369	5	31	of	of	ADP
cana-369	5	32	graphs	graph	NOUN
cana-369	5	33	has	have	AUX
cana-369	5	34	been	be	AUX
cana-369	5	35	introduced	introduce	VERB
cana-369	5	36	in	in	ADP
cana-369	5	37	1966	1966	NUM
cana-369	5	38	.	.	PUNCT
cana-369	6	1	assignment	assignment	NOUN
cana-369	6	2	of	of	ADP
cana-369	6	3	natural	natural	ADJ
cana-369	6	4	numbers	number	NOUN
cana-369	6	5	to	to	PART
cana-369	6	6	vertices	vertex	NOUN
cana-369	6	7	and/or	and/or	CCONJ
cana-369	6	8	edges	edge	NOUN
cana-369	6	9	is	be	AUX
cana-369	6	10	referred	refer	VERB
cana-369	6	11	as	as	ADP
cana-369	6	12	graph	graph	NOUN
cana-369	6	13	labeling	labeling	NOUN
cana-369	6	14	.	.	PUNCT
cana-369	7	1	inspired	inspire	VERB
cana-369	7	2	by	by	ADP
cana-369	7	3	the	the	DET
cana-369	7	4	ample	ample	ADJ
cana-369	7	5	application	application	NOUN
cana-369	7	6	of	of	ADP
cana-369	7	7	graph	graph	NOUN
cana-369	7	8	labeling	labeling	NOUN
cana-369	7	9	technique	technique	NOUN
cana-369	7	10	in	in	ADP
cana-369	7	11	real	real	ADJ
cana-369	7	12	life	life	NOUN
cana-369	7	13	problems	problem	NOUN
cana-369	7	14	,	,	PUNCT
cana-369	7	15	multifarious	multifarious	ADJ
cana-369	7	16	labeling	labeling	NOUN
cana-369	7	17	strategy	strategy	NOUN
cana-369	7	18	was	be	AUX
cana-369	7	19	adopted	adopt	VERB
cana-369	7	20	and	and	CCONJ
cana-369	7	21	investigated	investigate	VERB
cana-369	7	22	by	by	ADP
cana-369	7	23	many	many	ADJ
cana-369	7	24	researchers	researcher	NOUN
cana-369	7	25	.	.	PUNCT
cana-369	8	1	objectives	objective	NOUN
cana-369	8	2	:	:	PUNCT
cana-369	8	3	graph	graph	NOUN
cana-369	8	4	labeling	labeling	NOUN
cana-369	8	5	plays	play	VERB
cana-369	8	6	a	a	DET
cana-369	8	7	vital	vital	ADJ
cana-369	8	8	role	role	NOUN
cana-369	8	9	in	in	ADP
cana-369	8	10	various	various	ADJ
cana-369	8	11	fields	field	NOUN
cana-369	8	12	and	and	CCONJ
cana-369	8	13	can	can	AUX
cana-369	8	14	be	be	AUX
cana-369	8	15	implemented	implement	VERB
cana-369	8	16	in	in	ADP
cana-369	8	17	multitudinous	multitudinous	ADJ
cana-369	8	18	discipline	discipline	NOUN
cana-369	8	19	including	include	VERB
cana-369	8	20	coding	code	VERB
cana-369	8	21	theory	theory	NOUN
cana-369	8	22	,	,	PUNCT
cana-369	8	23	x	x	NOUN
cana-369	8	24	-	-	NOUN
cana-369	8	25	ray	ray	NOUN
cana-369	8	26	,	,	PUNCT
cana-369	8	27	psychology	psychology	NOUN
cana-369	8	28	,	,	PUNCT
cana-369	8	29	crystallography	crystallography	NOUN
cana-369	8	30	,	,	PUNCT
cana-369	8	31	circuit	circuit	NOUN
cana-369	8	32	design	design	NOUN
cana-369	8	33	,	,	PUNCT
cana-369	8	34	communication	communication	NOUN
cana-369	8	35	networks	network	NOUN
cana-369	8	36	,	,	PUNCT
cana-369	8	37	astronomy	astronomy	NOUN
cana-369	8	38	,	,	PUNCT
cana-369	8	39	radar	radar	NOUN
cana-369	8	40	,	,	PUNCT
cana-369	8	41	data	datum	NOUN
cana-369	8	42	security	security	NOUN
cana-369	8	43	,	,	PUNCT
cana-369	8	44	secret	secret	ADJ
cana-369	8	45	sharing	sharing	NOUN
cana-369	8	46	,	,	PUNCT
cana-369	8	47	data	datum	NOUN
cana-369	8	48	base	base	NOUN
cana-369	8	49	management	management	NOUN
cana-369	8	50	and	and	CCONJ
cana-369	8	51	so	so	ADV
cana-369	8	52	on	on	ADV
cana-369	8	53	.	.	PUNCT
cana-369	9	1	apart	apart	ADV
cana-369	9	2	from	from	ADP
cana-369	9	3	these	these	DET
cana-369	9	4	labeling	labeling	NOUN
cana-369	9	5	techniques	technique	NOUN
cana-369	9	6	serve	serve	VERB
cana-369	9	7	as	as	ADP
cana-369	9	8	a	a	DET
cana-369	9	9	model	model	NOUN
cana-369	9	10	to	to	PART
cana-369	9	11	understand	understand	VERB
cana-369	9	12	discrete	discrete	ADJ
cana-369	9	13	mathematical	mathematical	ADJ
cana-369	9	14	domains	domain	NOUN
cana-369	9	15	.	.	PUNCT
cana-369	10	1	methodology	methodology	NOUN
cana-369	10	2	:	:	PUNCT
cana-369	10	3	in	in	ADP
cana-369	10	4	this	this	DET
cana-369	10	5	paper	paper	NOUN
cana-369	10	6	,	,	PUNCT
cana-369	10	7	an	an	DET
cana-369	10	8	attempt	attempt	NOUN
cana-369	10	9	has	have	AUX
cana-369	10	10	been	be	AUX
cana-369	10	11	made	make	VERB
cana-369	10	12	to	to	PART
cana-369	10	13	introduce	introduce	VERB
cana-369	10	14	new	new	ADJ
cana-369	10	15	labeling	labeling	NOUN
cana-369	10	16	such	such	ADJ
cana-369	10	17	as	as	ADP
cana-369	10	18	o	o	PROPN
cana-369	10	19	dd	dd	VERB
cana-369	10	20	-	-	ADJ
cana-369	10	21	even	even	ADV
cana-369	10	22	congruence	congruence	ADJ
cana-369	10	23	labeling	labeling	NOUN
cana-369	10	24	.	.	PUNCT
cana-369	11	1	congruence	congruence	NOUN
cana-369	11	2	graph	graph	NOUN
cana-369	11	3	labeling	labeling	NOUN
cana-369	11	4	is	be	AUX
cana-369	11	5	an	an	DET
cana-369	11	6	allocation	allocation	NOUN
cana-369	11	7	of	of	ADP
cana-369	11	8	natural	natural	ADJ
cana-369	11	9	numbers	number	NOUN
cana-369	11	10	as	as	ADP
cana-369	11	11	labels	label	NOUN
cana-369	11	12	for	for	ADP
cana-369	11	13	the	the	DET
cana-369	11	14	edges	edge	NOUN
cana-369	11	15	and	and	CCONJ
cana-369	11	16	vertices	vertex	NOUN
cana-369	11	17	of	of	ADP
cana-369	11	18	a	a	DET
cana-369	11	19	graph	graph	NOUN
cana-369	11	20	based	base	VERB
cana-369	11	21	on	on	ADP
cana-369	11	22	modular	modular	ADJ
cana-369	11	23	arithmetic	arithmetic	ADJ
cana-369	11	24	property	property	NOUN
cana-369	11	25	.	.	PUNCT
cana-369	12	1	odd	odd	ADV
cana-369	12	2	-	-	PUNCT
cana-369	12	3	even	even	ADV
cana-369	12	4	congruence	congruence	NOUN
cana-369	12	5	labeling	labeling	NOUN
cana-369	12	6	is	be	AUX
cana-369	12	7	an	an	DET
cana-369	12	8	allocation	allocation	NOUN
cana-369	12	9	of	of	ADP
cana-369	12	10	odd	odd	ADJ
cana-369	12	11	integers	integer	NOUN
cana-369	12	12	to	to	PART
cana-369	12	13	vertices	vertex	NOUN
cana-369	12	14	and	and	CCONJ
cana-369	12	15	even	even	ADV
cana-369	12	16	integers	integer	NOUN
cana-369	12	17	to	to	PART
cana-369	12	18	edges	edge	NOUN
cana-369	12	19	in	in	ADP
cana-369	12	20	addition	addition	NOUN
cana-369	12	21	to	to	ADP
cana-369	12	22	congruence	congruence	NOUN
cana-369	12	23	graph	graph	NOUN
cana-369	12	24	labeling	labeling	NOUN
cana-369	12	25	.	.	PUNCT
cana-369	13	1	result	result	NOUN
cana-369	13	2	:	:	PUNCT
cana-369	13	3	the	the	DET
cana-369	13	4	suggested	suggest	VERB
cana-369	13	5	labeling	labeling	NOUN
cana-369	13	6	has	have	AUX
cana-369	13	7	been	be	AUX
cana-369	13	8	identified	identify	VERB
cana-369	13	9	on	on	ADP
cana-369	13	10	complete	complete	ADJ
cana-369	13	11	bipartite	bipartite	NOUN
cana-369	13	12	graph	graph	NOUN
cana-369	13	13	,	,	PUNCT
cana-369	13	14	comb	comb	NOUN
cana-369	13	15	graph	graph	NOUN
cana-369	13	16	and	and	CCONJ
cana-369	13	17	spliting	split	VERB
cana-369	13	18	graph	graph	NOUN
cana-369	13	19	of	of	ADP
cana-369	13	20	a	a	DET
cana-369	13	21	star	star	NOUN
cana-369	13	22	graph	graph	NOUN
cana-369	13	23	.	.	PUNCT
cana-369	14	1	further	far	ADV
cana-369	14	2	,	,	PUNCT
cana-369	14	3	it	it	PRON
cana-369	14	4	is	be	AUX
cana-369	14	5	proved	prove	VERB
cana-369	14	6	that	that	SCONJ
cana-369	14	7	graph	graph	NOUN
cana-369	14	8	acquired	acquire	VERB
cana-369	14	9	by	by	ADP
cana-369	14	10	connecting	connect	VERB
cana-369	14	11	two	two	NUM
cana-369	14	12	copies	copy	NOUN
cana-369	14	13	of	of	ADP
cana-369	14	14	even	even	ADV
cana-369	14	15	cycle	cycle	NOUN
cana-369	14	16	cr	cr	NOUN
cana-369	14	17	by	by	ADP
cana-369	14	18	a	a	DET
cana-369	14	19	path	path	NOUN
cana-369	14	20	pt	pt	NOUN
cana-369	14	21	,	,	PUNCT
cana-369	14	22	k1,t⊗p2	k1,t⊗p2	NOUN
cana-369	14	23	and	and	CCONJ
cana-369	14	24	d2	d2	PROPN
cana-369	14	25	(	(	PUNCT
cana-369	14	26	pt	pt	NOUN
cana-369	14	27	)	)	PUNCT
cana-369	14	28	are	be	AUX
cana-369	14	29	odd	odd	ADJ
cana-369	14	30	-	-	PUNCT
cana-369	14	31	even	even	ADV
cana-369	14	32	congurence	congurence	NOUN
cana-369	14	33	graph	graph	NOUN
cana-369	14	34	.	.	PUNCT
cana-369	15	1	keywords	keyword	NOUN
cana-369	15	2	:	:	PUNCT
cana-369	15	3	labeling	labeling	NOUN
cana-369	15	4	,	,	PUNCT
cana-369	15	5	congruence	congruence	NOUN
cana-369	15	6	labeling	labeling	NOUN
cana-369	15	7	,	,	PUNCT
cana-369	15	8	odd	odd	ADV
cana-369	15	9	-	-	PUNCT
cana-369	15	10	even	even	ADV
cana-369	15	11	congruence	congruence	ADJ
cana-369	15	12	labeling	labeling	NOUN
cana-369	15	13	,	,	PUNCT
cana-369	15	14	tensor	tensor	NOUN
cana-369	15	15	graph	graph	NOUN
cana-369	15	16	.	.	PUNCT
cana-369	16	1	1	1	X
cana-369	16	2	.	.	X
cana-369	16	3	introduction	introduction	NOUN
cana-369	16	4	the	the	DET
cana-369	16	5	relation	relation	NOUN
cana-369	16	6	between	between	ADP
cana-369	16	7	any	any	PRON
cana-369	16	8	of	of	ADP
cana-369	16	9	the	the	DET
cana-369	16	10	objects	object	NOUN
cana-369	16	11	in	in	ADP
cana-369	16	12	the	the	DET
cana-369	16	13	real	real	ADJ
cana-369	16	14	world	world	NOUN
cana-369	16	15	can	can	AUX
cana-369	16	16	be	be	AUX
cana-369	16	17	represented	represent	VERB
cana-369	16	18	as	as	ADP
cana-369	16	19	graphs	graph	NOUN
cana-369	16	20	,	,	PUNCT
cana-369	16	21	whose	whose	DET
cana-369	16	22	structures	structure	NOUN
cana-369	16	23	,	,	PUNCT
cana-369	16	24	properties	property	NOUN
cana-369	16	25	,	,	PUNCT
cana-369	16	26	interrelations	interrelation	NOUN
cana-369	16	27	and	and	CCONJ
cana-369	16	28	correlations	correlation	NOUN
cana-369	16	29	are	be	AUX
cana-369	16	30	interpreted	interpret	VERB
cana-369	16	31	in	in	ADP
cana-369	16	32	graph	graph	NOUN
cana-369	16	33	theory	theory	NOUN
cana-369	16	34	.	.	PUNCT
cana-369	17	1	the	the	DET
cana-369	17	2	graph	graph	NOUN
cana-369	17	3	comprises	comprise	NOUN
cana-369	17	4	of	of	ADP
cana-369	17	5	dots	dot	NOUN
cana-369	17	6	associated	associate	VERB
cana-369	17	7	by	by	ADP
cana-369	17	8	lines	line	NOUN
cana-369	17	9	which	which	PRON
cana-369	17	10	are	be	AUX
cana-369	17	11	referred	refer	VERB
cana-369	17	12	as	as	ADP
cana-369	17	13	vertices	vertex	NOUN
cana-369	17	14	and	and	CCONJ
cana-369	17	15	edges	edge	NOUN
cana-369	17	16	respectively	respectively	ADV
cana-369	17	17	.	.	PUNCT
cana-369	18	1	it	it	PRON
cana-369	18	2	aids	aid	VERB
cana-369	18	3	the	the	DET
cana-369	18	4	researchers	researcher	NOUN
cana-369	18	5	to	to	PART
cana-369	18	6	frame	frame	VERB
cana-369	18	7	the	the	DET
cana-369	18	8	hypothesis	hypothesis	NOUN
cana-369	18	9	and	and	CCONJ
cana-369	18	10	establish	establish	VERB
cana-369	18	11	the	the	DET
cana-369	18	12	solution	solution	NOUN
cana-369	18	13	certainly	certainly	ADV
cana-369	18	14	.	.	PUNCT
cana-369	19	1	graph	graph	NOUN
cana-369	19	2	labeling	labeling	NOUN
cana-369	19	3	was	be	AUX
cana-369	19	4	procured	procure	VERB
cana-369	19	5	as	as	ADP
cana-369	19	6	the	the	DET
cana-369	19	7	most	most	ADV
cana-369	19	8	prominent	prominent	ADJ
cana-369	19	9	one	one	NOUN
cana-369	19	10	among	among	ADP
cana-369	19	11	the	the	DET
cana-369	19	12	multifarious	multifarious	ADJ
cana-369	19	13	conception	conception	NOUN
cana-369	19	14	of	of	ADP
cana-369	19	15	graph	graph	NOUN
cana-369	19	16	theory	theory	NOUN
cana-369	19	17	to	to	PART
cana-369	19	18	design	design	VERB
cana-369	19	19	the	the	DET
cana-369	19	20	graphical	graphical	ADJ
cana-369	19	21	model	model	NOUN
cana-369	19	22	of	of	ADP
cana-369	19	23	the	the	DET
cana-369	19	24	real	real	ADJ
cana-369	19	25	life	life	NOUN
cana-369	19	26	situations	situation	NOUN
cana-369	19	27	.	.	PUNCT
cana-369	20	1	in	in	ADP
cana-369	20	2	the	the	DET
cana-369	20	3	middle	middle	NOUN
cana-369	20	4	of	of	ADP
cana-369	20	5	19th	19th	ADJ
cana-369	20	6	century	century	NOUN
cana-369	20	7	,	,	PUNCT
cana-369	20	8	a.rosa	a.rosa	VERB
cana-369	20	9	introduced	introduce	VERB
cana-369	20	10	graph	graph	NOUN
cana-369	20	11	labeling[8	labeling[8	NOUN
cana-369	20	12	]	]	PUNCT
cana-369	20	13	.	.	PUNCT
cana-369	21	1	the	the	DET
cana-369	21	2	vertices	vertex	NOUN
cana-369	21	3	and	and	CCONJ
cana-369	21	4	edges	edge	NOUN
cana-369	21	5	of	of	ADP
cana-369	21	6	the	the	DET
cana-369	21	7	graphs	graph	NOUN
cana-369	21	8	are	be	AUX
cana-369	21	9	labeled	label	VERB
cana-369	21	10	with	with	ADP
cana-369	21	11	natural	natural	ADJ
cana-369	21	12	numbers	number	NOUN
cana-369	21	13	with	with	ADP
cana-369	21	14	certain	certain	ADJ
cana-369	21	15	constraints	constraint	NOUN
cana-369	21	16	in	in	ADP
cana-369	21	17	order	order	NOUN
cana-369	21	18	to	to	PART
cana-369	21	19	discriminate	discriminate	VERB
cana-369	21	20	individually	individually	ADV
cana-369	21	21	.	.	PUNCT
cana-369	22	1	inspired	inspire	VERB
cana-369	22	2	by	by	ADP
cana-369	22	3	its	its	PRON
cana-369	22	4	comprehensive	comprehensive	ADJ
cana-369	22	5	application	application	NOUN
cana-369	22	6	in	in	ADP
cana-369	22	7	modeling	model	VERB
cana-369	22	8	all	all	DET
cana-369	22	9	the	the	DET
cana-369	22	10	circumstances	circumstance	NOUN
cana-369	22	11	,	,	PUNCT
cana-369	22	12	numerous	numerous	ADJ
cana-369	22	13	labeling	labeling	NOUN
cana-369	22	14	technique	technique	NOUN
cana-369	22	15	has	have	AUX
cana-369	22	16	been	be	AUX
cana-369	22	17	proposed	propose	VERB
cana-369	22	18	and	and	CCONJ
cana-369	22	19	overworked	overwork	VERB
cana-369	22	20	by	by	ADP
cana-369	22	21	several	several	ADJ
cana-369	22	22	researchers	researcher	NOUN
cana-369	22	23	.	.	PUNCT
cana-369	23	1	in	in	ADP
cana-369	23	2	this	this	DET
cana-369	23	3	paper	paper	NOUN
cana-369	23	4	,	,	PUNCT
cana-369	23	5	a	a	DET
cana-369	23	6	new	new	ADJ
cana-369	23	7	labeling	labeling	NOUN
cana-369	23	8	procedure	procedure	NOUN
cana-369	23	9	named	name	VERB
cana-369	23	10	odd	odd	ADJ
cana-369	23	11	-	-	PUNCT
cana-369	23	12	even	even	ADV
cana-369	23	13	congruence	congruence	NOUN
cana-369	23	14	labeling	labeling	NOUN
cana-369	23	15	is	be	AUX
cana-369	23	16	established	establish	VERB
cana-369	23	17	based	base	VERB
cana-369	23	18	on	on	ADP
cana-369	23	19	modular	modular	ADJ
cana-369	23	20	division	division	NOUN
cana-369	23	21	.	.	PUNCT
cana-369	24	1	a	a	DET
cana-369	24	2	graph	graph	NOUN
cana-369	24	3	g	g	NOUN
cana-369	24	4	is	be	AUX
cana-369	24	5	identified	identify	VERB
cana-369	24	6	as	as	ADP
cana-369	24	7	odd	odd	ADJ
cana-369	24	8	-	-	PUNCT
cana-369	24	9	even	even	ADV
cana-369	24	10	congruence	congruence	NOUN
cana-369	24	11	graph	graph	NOUN
cana-369	24	12	,	,	PUNCT
cana-369	24	13	if	if	SCONJ
cana-369	24	14	its	its	PRON
cana-369	24	15	vertex	vertex	NOUN
cana-369	24	16	set	set	NOUN
cana-369	24	17	and	and	CCONJ
cana-369	24	18	edge	edge	NOUN
cana-369	24	19	set	set	VERB
cana-369	24	20	are	be	AUX
cana-369	24	21	tagged	tag	VERB
cana-369	24	22	with	with	ADP
cana-369	24	23	distinct	distinct	ADJ
cana-369	24	24	odd	odd	ADJ
cana-369	24	25	and	and	CCONJ
cana-369	24	26	even	even	ADV
cana-369	24	27	integers	integer	NOUN
cana-369	24	28	respectively	respectively	ADV
cana-369	24	29	,	,	PUNCT
cana-369	24	30	further	further	ADJ
cana-369	24	31	communications	communication	NOUN
cana-369	24	32	on	on	ADP
cana-369	24	33	applied	apply	VERB
cana-369	24	34	nonlinear	nonlinear	ADJ
cana-369	24	35	analysis	analysis	NOUN
cana-369	24	36	issn	issn	NOUN
cana-369	24	37	:	:	PUNCT
cana-369	24	38	1074	1074	NUM
cana-369	24	39	-	-	PUNCT
cana-369	24	40	133x	133x	NUM
cana-369	24	41	vol	vol	NOUN
cana-369	24	42	31	31	NUM
cana-369	24	43	no	no	NOUN
cana-369	24	44	.	.	NOUN
cana-369	24	45	1	1	NUM
cana-369	24	46	(	(	PUNCT
cana-369	24	47	2024	2024	NUM
cana-369	24	48	)	)	PUNCT
cana-369	24	49	142	142	NUM
cana-369	24	50	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	24	51	f	f	PROPN
cana-369	24	52	(	(	PUNCT
cana-369	24	53	sp	sp	NOUN
cana-369	24	54	)	)	PUNCT
cana-369	24	55	≡	≡	PROPN
cana-369	24	56	f	f	PROPN
cana-369	24	57	(	(	PUNCT
cana-369	24	58	sq)(mod	sq)(mod	NUM
cana-369	24	59	g	g	PROPN
cana-369	24	60	(	(	PUNCT
cana-369	24	61	w	w	NOUN
cana-369	24	62	)	)	PUNCT
cana-369	24	63	)	)	PUNCT
cana-369	24	64	,	,	PUNCT
cana-369	24	65	sp	sp	ADP
cana-369	24	66	and	and	CCONJ
cana-369	24	67	sq	sq	NOUN
cana-369	24	68	are	be	AUX
cana-369	24	69	adjacent	adjacent	ADJ
cana-369	24	70	vertices	vertex	NOUN
cana-369	24	71	in	in	ADP
cana-369	24	72	g.	g.	PROPN
cana-369	24	73	this	this	DET
cana-369	24	74	paper	paper	NOUN
cana-369	24	75	is	be	AUX
cana-369	24	76	devoted	devote	VERB
cana-369	24	77	for	for	ADP
cana-369	24	78	investigating	investigate	VERB
cana-369	24	79	the	the	DET
cana-369	24	80	existence	existence	NOUN
cana-369	24	81	of	of	ADP
cana-369	24	82	odd	odd	ADV
cana-369	24	83	-	-	PUNCT
cana-369	24	84	even	even	ADV
cana-369	24	85	congruence	congruence	NOUN
cana-369	24	86	graphs	graph	NOUN
cana-369	24	87	of	of	ADP
cana-369	24	88	complete	complete	ADJ
cana-369	24	89	bipartite	bipartite	NOUN
cana-369	24	90	graph	graph	NOUN
cana-369	24	91	,	,	PUNCT
cana-369	24	92	comb	comb	NOUN
cana-369	24	93	graph	graph	NOUN
cana-369	24	94	and	and	CCONJ
cana-369	24	95	spliting	split	VERB
cana-369	24	96	graph	graph	NOUN
cana-369	24	97	of	of	ADP
cana-369	24	98	a	a	DET
cana-369	24	99	star	star	NOUN
cana-369	24	100	graph	graph	NOUN
cana-369	24	101	.	.	PUNCT
cana-369	25	1	in	in	ADP
cana-369	25	2	addition	addition	NOUN
cana-369	25	3	,	,	PUNCT
cana-369	25	4	graph	graph	NOUN
cana-369	25	5	acquired	acquire	VERB
cana-369	25	6	by	by	ADP
cana-369	25	7	connecting	connect	VERB
cana-369	25	8	two	two	NUM
cana-369	25	9	copies	copy	NOUN
cana-369	25	10	of	of	ADP
cana-369	25	11	even	even	ADV
cana-369	25	12	cycle	cycle	NOUN
cana-369	25	13	cr	cr	NOUN
cana-369	25	14	by	by	ADP
cana-369	25	15	a	a	DET
cana-369	25	16	path	path	NOUN
cana-369	25	17	pt	pt	PROPN
cana-369	25	18	,	,	PUNCT
cana-369	25	19	the	the	DET
cana-369	25	20	tensor	tensor	NOUN
cana-369	25	21	product	product	NOUN
cana-369	25	22	of	of	ADP
cana-369	25	23	k1,t	k1,t	PROPN
cana-369	25	24	&	&	CCONJ
cana-369	25	25	p2	p2	PROPN
cana-369	25	26	and	and	CCONJ
cana-369	25	27	the	the	DET
cana-369	25	28	shadow	shadow	NOUN
cana-369	25	29	graph	graph	NOUN
cana-369	25	30	of	of	ADP
cana-369	25	31	the	the	DET
cana-369	25	32	path	path	NOUN
cana-369	25	33	pt	pt	NOUN
cana-369	25	34	are	be	AUX
cana-369	25	35	proved	prove	VERB
cana-369	25	36	as	as	ADP
cana-369	25	37	odd	odd	ADJ
cana-369	25	38	-	-	PUNCT
cana-369	25	39	even	even	ADV
cana-369	25	40	congruence	congruence	NOUN
cana-369	25	41	graph	graph	NOUN
cana-369	25	42	.	.	PUNCT
cana-369	26	1	2	2	X
cana-369	26	2	.	.	X
cana-369	26	3	preliminaries	preliminary	NOUN
cana-369	26	4	definition	definition	NOUN
cana-369	26	5	2.1[2	2.1[2	NUM
cana-369	26	6	]	]	X
cana-369	26	7	bipartite	bipartite	PROPN
cana-369	26	8	graph	graph	NOUN
cana-369	26	9	g	g	PROPN
cana-369	27	1	[	[	X
cana-369	27	2	x	x	PROPN
cana-369	27	3	,	,	PUNCT
cana-369	27	4	y	y	PROPN
cana-369	27	5	]	]	PUNCT
cana-369	27	6	is	be	AUX
cana-369	27	7	recognized	recognize	VERB
cana-369	27	8	as	as	ADP
cana-369	27	9	complete	complete	ADJ
cana-369	27	10	bipartite	bipartite	NOUN
cana-369	27	11	graph	graph	NOUN
cana-369	27	12	kr	kr	PROPN
cana-369	27	13	,	,	PUNCT
cana-369	27	14	t	t	PROPN
cana-369	27	15	,	,	PUNCT
cana-369	27	16	the	the	DET
cana-369	27	17	edges	edge	NOUN
cana-369	27	18	occur	occur	VERB
cana-369	27	19	in	in	ADP
cana-369	27	20	between	between	ADP
cana-369	27	21	every	every	DET
cana-369	27	22	distinct	distinct	ADJ
cana-369	27	23	pair	pair	NOUN
cana-369	27	24	of	of	ADP
cana-369	27	25	spf	spf	PROPN
cana-369	27	26	q	q	PROPN
cana-369	27	27	such	such	ADJ
cana-369	27	28	that	that	PRON
cana-369	27	29	sp	sp	ADP
cana-369	27	30	∈	∈	PROPN
cana-369	27	31	x	x	X
cana-369	27	32	and	and	CCONJ
cana-369	27	33	f	f	PROPN
cana-369	27	34	q	q	PROPN
cana-369	27	35	∈	∈	PROPN
cana-369	27	36	y	y	PROPN
cana-369	27	37	.	.	PUNCT
cana-369	28	1	definition	definition	NOUN
cana-369	28	2	2.2[4	2.2[4	NOUN
cana-369	28	3	]	]	PUNCT
cana-369	28	4	comb	comb	NOUN
cana-369	28	5	graph	graph	NOUN
cana-369	28	6	pt	pt	PROPN
cana-369	28	7	⊙	⊙	PROPN
cana-369	28	8	k1	k1	PROPN
cana-369	28	9	is	be	AUX
cana-369	28	10	constructed	construct	VERB
cana-369	28	11	by	by	ADP
cana-369	28	12	introducing	introduce	VERB
cana-369	28	13	single	single	ADJ
cana-369	28	14	pendent	pendent	ADJ
cana-369	28	15	edge	edge	NOUN
cana-369	28	16	to	to	PART
cana-369	28	17	link	link	VERB
cana-369	28	18	all	all	DET
cana-369	28	19	the	the	DET
cana-369	28	20	vertices	vertex	NOUN
cana-369	28	21	in	in	ADP
cana-369	28	22	pt	pt	PROPN
cana-369	28	23	.	.	PROPN
cana-369	28	24	definition	definition	NOUN
cana-369	28	25	2.3[7	2.3[7	NUM
cana-369	28	26	]	]	PUNCT
cana-369	28	27	in	in	ADP
cana-369	28	28	a	a	DET
cana-369	28	29	graph	graph	NOUN
cana-369	28	30	g(v	g(v	NOUN
cana-369	28	31	,	,	PUNCT
cana-369	28	32	e	e	NOUN
cana-369	28	33	)	)	PUNCT
cana-369	28	34	,	,	PUNCT
cana-369	28	35	if	if	SCONJ
cana-369	28	36	the	the	DET
cana-369	28	37	edge	edge	NOUN
cana-369	28	38	set	set	NOUN
cana-369	28	39	is	be	AUX
cana-369	28	40	e	e	NOUN
cana-369	28	41	=	=	PUNCT
cana-369	28	42	{	{	PUNCT
cana-369	28	43	spf	spf	PROPN
cana-369	28	44	/	/	SYM
cana-369	28	45	sp	sp	ADP
cana-369	28	46	f	f	PROPN
cana-369	28	47	∈	∈	PROPN
cana-369	28	48	v	v	PROPN
cana-369	28	49	&	&	CCONJ
cana-369	28	50	sp	sp	ADP
cana-369	28	51	/=	/=	PROPN
cana-369	29	1	f	f	X
cana-369	29	2	}	}	PUNCT
cana-369	29	3	and	and	CCONJ
cana-369	29	4	f	f	PROPN
cana-369	29	5	is	be	AUX
cana-369	29	6	a	a	DET
cana-369	29	7	fixed	fix	VERB
cana-369	29	8	vertex	vertex	NOUN
cana-369	29	9	then	then	ADV
cana-369	29	10	g	g	PROPN
cana-369	29	11	is	be	AUX
cana-369	29	12	stated	state	VERB
cana-369	29	13	as	as	ADP
cana-369	29	14	a	a	DET
cana-369	29	15	star	star	NOUN
cana-369	29	16	graph	graph	NOUN
cana-369	29	17	,	,	PUNCT
cana-369	29	18	it	it	PRON
cana-369	29	19	is	be	AUX
cana-369	29	20	denoted	denote	VERB
cana-369	29	21	by	by	ADP
cana-369	29	22	st	st	PROPN
cana-369	29	23	.	.	PROPN
cana-369	29	24	definition	definition	NOUN
cana-369	29	25	2.4[5	2.4[5	NUM
cana-369	29	26	]	]	PUNCT
cana-369	29	27	the	the	DET
cana-369	29	28	spliting	split	VERB
cana-369	29	29	graph	graph	NOUN
cana-369	29	30	,	,	PUNCT
cana-369	29	31	spl(g	spl(g	PROPN
cana-369	29	32	)	)	PUNCT
cana-369	29	33	is	be	AUX
cana-369	29	34	arrived	arrive	VERB
cana-369	29	35	by	by	ADP
cana-369	29	36	introducing	introduce	VERB
cana-369	29	37	a	a	DET
cana-369	29	38	new	new	ADJ
cana-369	29	39	vertex	vertex	NOUN
cana-369	29	40	fp	fp	X
cana-369	29	41	for	for	ADP
cana-369	29	42	each	each	DET
cana-369	29	43	existing	exist	VERB
cana-369	29	44	vertex	vertex	NOUN
cana-369	29	45	sp	sp	ADP
cana-369	29	46	moreover	moreover	ADV
cana-369	29	47	n	n	CCONJ
cana-369	29	48	(	(	PUNCT
cana-369	29	49	sp	sp	NOUN
cana-369	29	50	)	)	PUNCT
cana-369	29	51	=	=	SYM
cana-369	29	52	n	n	CCONJ
cana-369	29	53	(	(	PUNCT
cana-369	29	54	fp	fp	NOUN
cana-369	29	55	)	)	PUNCT
cana-369	29	56	,	,	PUNCT
cana-369	29	57	where	where	SCONJ
cana-369	29	58	n	n	X
cana-369	29	59	(	(	PUNCT
cana-369	29	60	sp	sp	NOUN
cana-369	29	61	)	)	PUNCT
cana-369	29	62	is	be	AUX
cana-369	29	63	the	the	DET
cana-369	29	64	neighborhood	neighborhood	NOUN
cana-369	29	65	of	of	ADP
cana-369	29	66	sp	sp	NOUN
cana-369	29	67	.	.	NOUN
cana-369	29	68	definition	definition	NOUN
cana-369	29	69	2.5[11	2.5[11	NUM
cana-369	29	70	]	]	PUNCT
cana-369	29	71	tensor	tensor	NOUN
cana-369	29	72	product	product	NOUN
cana-369	29	73	g1	g1	PROPN
cana-369	29	74	⊗	⊗	PROPN
cana-369	29	75	g2	g2	PROPN
cana-369	29	76	of	of	ADP
cana-369	29	77	g1	g1	PROPN
cana-369	29	78	and	and	CCONJ
cana-369	29	79	g2	g2	PROPN
cana-369	29	80	is	be	AUX
cana-369	29	81	a	a	DET
cana-369	29	82	graph	graph	NOUN
cana-369	29	83	whose	whose	DET
cana-369	29	84	vertices	vertex	NOUN
cana-369	29	85	and	and	CCONJ
cana-369	29	86	edges	edge	VERB
cana-369	29	87	a	a	DET
cana-369	29	88	re	re	NOUN
cana-369	29	89	v	v	NOUN
cana-369	29	90	(	(	PUNCT
cana-369	29	91	g1	g1	PROPN
cana-369	29	92	⊗	⊗	PROPN
cana-369	29	93	g2	g2	PROPN
cana-369	29	94	)	)	PUNCT
cana-369	30	1	=	=	SYM
cana-369	30	2	v	v	X
cana-369	30	3	(	(	PUNCT
cana-369	30	4	g1	g1	PROPN
cana-369	30	5	)	)	PUNCT
cana-369	30	6	×	×	NOUN
cana-369	30	7	v	v	NOUN
cana-369	30	8	(	(	PUNCT
cana-369	30	9	g2	g2	PROPN
cana-369	30	10	)	)	PUNCT
cana-369	30	11	and	and	CCONJ
cana-369	30	12	e(g1	e(g1	ADJ
cana-369	30	13	⊗	⊗	PROPN
cana-369	30	14	g2	g2	PROPN
cana-369	30	15	)	)	PUNCT
cana-369	30	16	=	=	PRON
cana-369	30	17	{	{	PUNCT
cana-369	30	18	(	(	PUNCT
cana-369	30	19	s1	s1	NOUN
cana-369	30	20	,	,	PUNCT
cana-369	30	21	s2)(s3	s2)(s3	PROPN
cana-369	30	22	,	,	PUNCT
cana-369	30	23	s	s	VERB
cana-369	30	24	4)|s1s3	4)|s1s3	NUM
cana-369	30	25	∈	∈	NOUN
cana-369	30	26	e(g1	e(g1	NOUN
cana-369	30	27	)	)	PUNCT
cana-369	30	28	and	and	CCONJ
cana-369	30	29	s2s4	s2s4	PROPN
cana-369	30	30	∈	∈	PROPN
cana-369	30	31	e(g2	e(g2	ADV
cana-369	30	32	)	)	PUNCT
cana-369	30	33	}	}	PUNCT
cana-369	30	34	.	.	PUNCT
cana-369	31	1	definition	definition	NOUN
cana-369	31	2	2.6[12	2.6[12	NUM
cana-369	31	3	]	]	PUNCT
cana-369	31	4	shadow	shadow	NOUN
cana-369	31	5	graph	graph	NOUN
cana-369	31	6	d2	d2	PROPN
cana-369	31	7	(	(	PUNCT
cana-369	31	8	g	g	NOUN
cana-369	31	9	)	)	PUNCT
cana-369	31	10	is	be	AUX
cana-369	31	11	constituted	constitute	VERB
cana-369	31	12	by	by	ADP
cana-369	31	13	picking	pick	VERB
cana-369	31	14	𝐺′and	𝐺′and	PRON
cana-369	31	15	𝐺′′alike	𝐺′′alike	PROPN
cana-369	31	16	g	g	NOUN
cana-369	31	17	and	and	CCONJ
cana-369	31	18	introduce	introduce	VERB
cana-369	31	19	edges	edge	NOUN
cana-369	31	20	in	in	ADP
cana-369	31	21	between	between	ADP
cana-369	31	22	𝑠′∈	𝑠′∈	PROPN
cana-369	31	23	𝐺′and	𝐺′and	ADP
cana-369	31	24	𝑠′′∈	𝑠′′∈	PROPN
cana-369	31	25	𝐺′′where	𝐺′′where	PROPN
cana-369	31	26	𝑠′′	𝑠′′	ADJ
cana-369	31	27	is	be	AUX
cana-369	31	28	the	the	DET
cana-369	31	29	neighbors	neighbor	NOUN
cana-369	31	30	of	of	ADP
cana-369	31	31	parallel	parallel	ADJ
cana-369	31	32	vertex	vertex	NOUN
cana-369	31	33	of	of	ADP
cana-369	31	34	𝑠′.	𝑠′.	NOUN
cana-369	31	35	definition	definition	NOUN
cana-369	31	36	2.7[13	2.7[13	NUM
cana-369	31	37	]	]	X
cana-369	31	38	a	a	DET
cana-369	31	39	bijection	bijection	ADJ
cana-369	31	40	h	h	NOUN
cana-369	31	41	:	:	PUNCT
cana-369	31	42	v	v	X
cana-369	31	43	→	→	SYM
cana-369	31	44	{	{	PUNCT
cana-369	31	45	1	1	NUM
cana-369	31	46	,	,	PUNCT
cana-369	31	47	2	2	NUM
cana-369	31	48	,	,	PUNCT
cana-369	31	49	....	....	PUNCT
cana-369	32	1	d	d	X
cana-369	32	2	}	}	PUNCT
cana-369	32	3	and	and	CCONJ
cana-369	32	4	k	k	NOUN
cana-369	32	5	:	:	PUNCT
cana-369	32	6	e	e	X
cana-369	32	7	→	→	PUNCT
cana-369	32	8	{	{	PUNCT
cana-369	32	9	1	1	NUM
cana-369	32	10	,	,	PUNCT
cana-369	32	11	2,	2,	NUM
cana-369	32	12	…	…	NUM
cana-369	32	13	.d	.d	ADJ
cana-369	32	14	−	−	NOUN
cana-369	32	15	1}of	1}of	NOUN
cana-369	32	16	g	g	NOUN
cana-369	32	17	is	be	AUX
cana-369	32	18	claimed	claim	VERB
cana-369	32	19	as	as	ADP
cana-369	32	20	congruence	congruence	NOUN
cana-369	32	21	graph	graph	NOUN
cana-369	32	22	,	,	PUNCT
cana-369	32	23	if	if	SCONJ
cana-369	32	24	h	h	PROPN
cana-369	32	25	(	(	PUNCT
cana-369	32	26	sp	sp	NOUN
cana-369	32	27	)	)	PUNCT
cana-369	32	28	≡	≡	PROPN
cana-369	32	29	h	h	NOUN
cana-369	32	30	(	(	PUNCT
cana-369	32	31	sq)(mod	sq)(mod	X
cana-369	32	32	k	k	X
cana-369	32	33	(	(	PUNCT
cana-369	32	34	wp	wp	PROPN
cana-369	32	35	)	)	PUNCT
cana-369	32	36	)	)	PUNCT
cana-369	32	37	,	,	PUNCT
cana-369	32	38	where	where	SCONJ
cana-369	32	39	d	d	PROPN
cana-369	32	40	=	=	SYM
cana-369	32	41	min	min	PROPN
cana-369	32	42	{	{	PUNCT
cana-369	32	43	2	2	NUM
cana-369	32	44	|v	|v	ADP
cana-369	32	45	|	|	ADV
cana-369	32	46	,	,	PUNCT
cana-369	32	47	2	2	NUM
cana-369	32	48	|e	|e	NOUN
cana-369	32	49	|	|	ADV
cana-369	32	50	}	}	PUNCT
cana-369	32	51	.	.	PUNCT
cana-369	33	1	definition	definition	NOUN
cana-369	33	2	2.8[14	2.8[14	NUM
cana-369	33	3	]	]	PUNCT
cana-369	33	4	a	a	DET
cana-369	33	5	bijection	bijection	ADJ
cana-369	33	6	h	h	NOUN
cana-369	33	7	:	:	PUNCT
cana-369	33	8	v	v	X
cana-369	33	9	→	→	SYM
cana-369	33	10	{	{	PUNCT
cana-369	33	11	1	1	NUM
cana-369	33	12	,	,	PUNCT
cana-369	33	13	2	2	NUM
cana-369	33	14	,	,	PUNCT
cana-369	33	15	....	....	PUNCT
cana-369	33	16	2d+1	2d+1	NUM
cana-369	33	17	}	}	PUNCT
cana-369	33	18	and	and	CCONJ
cana-369	33	19	k	k	NOUN
cana-369	33	20	:	:	PUNCT
cana-369	33	21	e	e	X
cana-369	33	22	→	→	PUNCT
cana-369	33	23	{	{	PUNCT
cana-369	33	24	2,4,	2,4,	PROPN
cana-369	33	25	…	…	SYM
cana-369	33	26	.2d	.2d	NOUN
cana-369	33	27	}	}	PUNCT
cana-369	33	28	of	of	ADP
cana-369	33	29	g	g	PROPN
cana-369	33	30	is	be	AUX
cana-369	33	31	referred	refer	VERB
cana-369	33	32	as	as	ADP
cana-369	33	33	odd	odd	ADJ
cana-369	33	34	-	-	PUNCT
cana-369	33	35	even	even	ADV
cana-369	33	36	congruence	congruence	NOUN
cana-369	33	37	graph	graph	NOUN
cana-369	33	38	,	,	PUNCT
cana-369	33	39	if	if	SCONJ
cana-369	33	40	h	h	PROPN
cana-369	33	41	(	(	PUNCT
cana-369	33	42	sp	sp	NOUN
cana-369	33	43	)	)	PUNCT
cana-369	33	44	≡	≡	PROPN
cana-369	33	45	h	h	NOUN
cana-369	33	46	(	(	PUNCT
cana-369	33	47	sq)(mod	sq)(mod	X
cana-369	33	48	k	k	X
cana-369	33	49	(	(	PUNCT
cana-369	33	50	wp	wp	PROPN
cana-369	33	51	)	)	PUNCT
cana-369	33	52	)	)	PUNCT
cana-369	33	53	,	,	PUNCT
cana-369	33	54	where	where	SCONJ
cana-369	33	55	d	d	PROPN
cana-369	33	56	=	=	SYM
cana-369	33	57	min	min	PROPN
cana-369	33	58	{	{	PUNCT
cana-369	33	59	2	2	NUM
cana-369	33	60	|v	|v	ADP
cana-369	33	61	|	|	ADV
cana-369	33	62	,	,	PUNCT
cana-369	33	63	2	2	NUM
cana-369	33	64	|e	|e	NOUN
cana-369	33	65	|	|	ADV
cana-369	33	66	}	}	PUNCT
cana-369	33	67	.	.	PUNCT
cana-369	34	1	3	3	X
cana-369	34	2	.	.	X
cana-369	34	3	main	main	ADJ
cana-369	34	4	results	result	NOUN
cana-369	34	5	in	in	ADP
cana-369	34	6	this	this	DET
cana-369	34	7	section	section	NOUN
cana-369	34	8	,	,	PUNCT
cana-369	34	9	simple	simple	ADJ
cana-369	34	10	finite	finite	NOUN
cana-369	34	11	connected	connect	VERB
cana-369	34	12	graph	graph	NOUN
cana-369	34	13	g	g	PROPN
cana-369	34	14	=	=	SYM
cana-369	34	15	(	(	PUNCT
cana-369	34	16	v	v	NOUN
cana-369	34	17	,	,	PUNCT
cana-369	34	18	e	e	NOUN
cana-369	34	19	)	)	PUNCT
cana-369	34	20	with	with	ADP
cana-369	34	21	|v	|v	PROPN
cana-369	34	22	|	|	NOUN
cana-369	34	23	=	=	SYM
cana-369	34	24	r	r	NOUN
cana-369	34	25	and	and	CCONJ
cana-369	34	26	|e|	|e|	PROPN
cana-369	34	27	=	=	PUNCT
cana-369	34	28	t	t	PROPN
cana-369	34	29	were	be	AUX
cana-369	34	30	considered	consider	VERB
cana-369	34	31	and	and	CCONJ
cana-369	34	32	proved	prove	VERB
cana-369	34	33	that	that	SCONJ
cana-369	34	34	it	it	PRON
cana-369	34	35	admits	admit	VERB
cana-369	34	36	odd	odd	ADJ
cana-369	34	37	-	-	PUNCT
cana-369	34	38	even	even	ADV
cana-369	34	39	congruence	congruence	ADJ
cana-369	34	40	labeling	labeling	NOUN
cana-369	34	41	.	.	PUNCT
cana-369	35	1	theorem	theorem	VERB
cana-369	35	2	3.1	3.1	NUM
cana-369	35	3	every	every	DET
cana-369	35	4	kr	kr	PROPN
cana-369	35	5	,	,	PUNCT
cana-369	35	6	t	t	PROPN
cana-369	35	7	is	be	AUX
cana-369	35	8	odd	odd	ADV
cana-369	35	9	-	-	PUNCT
cana-369	35	10	even	even	ADV
cana-369	35	11	congruence	congruence	NOUN
cana-369	35	12	graph	graph	NOUN
cana-369	35	13	for	for	ADP
cana-369	35	14	r	r	NOUN
cana-369	35	15	≥	≥	NUM
cana-369	35	16	1	1	NUM
cana-369	35	17	,	,	PUNCT
cana-369	35	18	t	t	PROPN
cana-369	35	19	≥	≥	NUM
cana-369	35	20	1	1	NUM
cana-369	35	21	.	.	PUNCT
cana-369	36	1	proof	proof	NOUN
cana-369	36	2	:	:	PUNCT
cana-369	36	3	consider	consider	VERB
cana-369	36	4	,	,	PUNCT
cana-369	36	5	km	km	PROPN
cana-369	36	6	,	,	PUNCT
cana-369	36	7	n	n	CCONJ
cana-369	36	8	with	with	ADP
cana-369	36	9	|v	|v	PROPN
cana-369	36	10	|	|	NOUN
cana-369	37	1	=	=	SYM
cana-369	37	2	r	r	NOUN
cana-369	37	3	+	+	NOUN
cana-369	37	4	t	t	PROPN
cana-369	37	5	and	and	CCONJ
cana-369	37	6	|e|	|e|	PROPN
cana-369	37	7	=	=	PROPN
cana-369	37	8	rt	rt	PROPN
cana-369	37	9	.	.	PUNCT
cana-369	38	1	d	d	X
cana-369	38	2	=	=	SYM
cana-369	38	3	min	min	PROPN
cana-369	38	4	{	{	PUNCT
cana-369	38	5	2	2	NUM
cana-369	38	6	|v	|v	ADP
cana-369	39	1	|	|	ADV
cana-369	39	2	,	,	PUNCT
cana-369	39	3	2	2	NUM
cana-369	39	4	|e	|e	NOUN
cana-369	39	5	|	|	ADV
cana-369	39	6	}	}	PUNCT
cana-369	39	7	for	for	ADP
cana-369	39	8	g	g	NOUN
cana-369	39	9	=	=	SYM
cana-369	39	10	kr	kr	PROPN
cana-369	39	11	,	,	PUNCT
cana-369	39	12	t	t	PROPN
cana-369	39	13	we	we	PRON
cana-369	39	14	have	have	VERB
cana-369	39	15	,	,	PUNCT
cana-369	39	16	d	d	X
cana-369	39	17	=	=	SYM
cana-369	39	18	min	min	PROPN
cana-369	39	19	{	{	PUNCT
cana-369	39	20	2	2	NUM
cana-369	39	21	(	(	PUNCT
cana-369	39	22	r	r	NOUN
cana-369	39	23	+	+	PROPN
cana-369	39	24	t	t	PROPN
cana-369	39	25	)	)	PUNCT
cana-369	39	26	,	,	PUNCT
cana-369	39	27	2rt	2rt	NOUN
cana-369	39	28	}	}	PUNCT
cana-369	39	29	=	=	SYM
cana-369	39	30	2	2	NUM
cana-369	39	31	(	(	PUNCT
cana-369	39	32	r	r	NOUN
cana-369	39	33	+	+	SYM
cana-369	39	34	t	t	NOUN
cana-369	39	35	)	)	PUNCT
cana-369	39	36	there	there	PRON
cana-369	39	37	exist	exist	VERB
cana-369	39	38	two	two	NUM
cana-369	39	39	independent	independent	ADJ
cana-369	39	40	and	and	CCONJ
cana-369	39	41	disjoint	disjoint	ADJ
cana-369	39	42	vertex	vertex	NOUN
cana-369	39	43	sets	set	NOUN
cana-369	39	44	such	such	ADJ
cana-369	39	45	as	as	ADP
cana-369	39	46	v1	v1	NOUN
cana-369	39	47	=	=	SYM
cana-369	39	48	{	{	PUNCT
cana-369	39	49	s1	s1	NOUN
cana-369	39	50	,	,	PUNCT
cana-369	39	51	s2,	s2,	NOUN
cana-369	39	52	…	…	SYM
cana-369	39	53	sr	sr	NOUN
cana-369	39	54	}	}	PUNCT
cana-369	39	55	and	and	CCONJ
cana-369	39	56	v2	v2	NOUN
cana-369	39	57	=	=	SYM
cana-369	39	58	{	{	PUNCT
cana-369	39	59	f1	f1	NOUN
cana-369	39	60	,	,	PUNCT
cana-369	39	61	f2	f2	PROPN
cana-369	39	62	,	,	PUNCT
cana-369	39	63	....	....	PUNCT
cana-369	39	64	ft	ft	X
cana-369	39	65	}	}	PUNCT
cana-369	39	66	.	.	PUNCT
cana-369	40	1	the	the	DET
cana-369	40	2	edge	edge	NOUN
cana-369	40	3	set	set	VERB
cana-369	40	4	be	be	AUX
cana-369	40	5	e	e	NOUN
cana-369	40	6	=	=	X
cana-369	40	7	{	{	PUNCT
cana-369	40	8	w1	w1	NOUN
cana-369	40	9	,	,	PUNCT
cana-369	40	10	w2,	w2,	X
cana-369	40	11	…	…	SYM
cana-369	40	12	wrt	wrt	X
cana-369	40	13	}	}	PUNCT
cana-369	40	14	.	.	PUNCT
cana-369	41	1	where	where	SCONJ
cana-369	41	2	w1	w1	NOUN
cana-369	41	3	=	=	SYM
cana-369	41	4	s1f1	s1f1	PROPN
cana-369	41	5	,	,	PUNCT
cana-369	41	6	w2	w2	NOUN
cana-369	41	7	=	=	SYM
cana-369	41	8	s2f1	s2f1	PROPN
cana-369	41	9	,	,	PUNCT
cana-369	41	10	.......	.......	PUNCT
cana-369	41	11	,	,	PUNCT
cana-369	41	12	wm	wm	PROPN
cana-369	41	13	=	=	SYM
cana-369	41	14	srf1	srf1	PROPN
cana-369	41	15	,	,	PUNCT
cana-369	41	16	...	...	PUNCT
cana-369	41	17	,	,	PUNCT
cana-369	41	18	er+1	er+1	PROPN
cana-369	41	19	=	=	SYM
cana-369	41	20	s1f2	s1f2	PROPN
cana-369	41	21	,	,	PUNCT
cana-369	41	22	…	…	PUNCT
cana-369	41	23	.	.	PUNCT
cana-369	41	24	,	,	PUNCT
cana-369	41	25	ert	ert	NOUN
cana-369	41	26	=	=	ADJ
cana-369	41	27	srft	srft	NOUN
cana-369	41	28	.	.	PUNCT
cana-369	42	1	bijection	bijection	NOUN
cana-369	42	2	h	h	NOUN
cana-369	42	3	:	:	PUNCT
cana-369	42	4	v	v	X
cana-369	42	5	(	(	PUNCT
cana-369	42	6	g	g	NOUN
cana-369	42	7	)	)	PUNCT
cana-369	42	8	→	→	SYM
cana-369	42	9	{	{	PUNCT
cana-369	42	10	1	1	NUM
cana-369	42	11	,	,	PUNCT
cana-369	42	12	3	3	NUM
cana-369	42	13	,	,	PUNCT
cana-369	42	14	…	…	PUNCT
cana-369	42	15	.	.	NUM
cana-369	42	16	,	,	PUNCT
cana-369	42	17	4r	4r	NOUN
cana-369	42	18	+	+	CCONJ
cana-369	42	19	4	4	NUM
cana-369	42	20	t	t	NOUN
cana-369	42	21	+	+	NOUN
cana-369	42	22	1	1	NUM
cana-369	42	23	}	}	PUNCT
cana-369	42	24	and	and	CCONJ
cana-369	42	25	k	k	NOUN
cana-369	42	26	:	:	PUNCT
cana-369	43	1	e	e	X
cana-369	43	2	(	(	PUNCT
cana-369	43	3	g	g	NOUN
cana-369	43	4	)	)	PUNCT
cana-369	43	5	→	→	SYM
cana-369	43	6	{	{	PUNCT
cana-369	43	7	2	2	NUM
cana-369	43	8	,	,	PUNCT
cana-369	43	9	4	4	NUM
cana-369	43	10	,	,	PUNCT
cana-369	43	11	…	…	PUNCT
cana-369	43	12	.	.	PUNCT
cana-369	43	13	,	,	PUNCT
cana-369	43	14	4	4	NUM
cana-369	43	15	m	m	NOUN
cana-369	43	16	+	+	NOUN
cana-369	43	17	4n	4n	X
cana-369	43	18	}	}	PUNCT
cana-369	43	19	is	be	AUX
cana-369	43	20	defined	define	VERB
cana-369	43	21	as	as	ADP
cana-369	43	22	v1	v1	NOUN
cana-369	43	23	:	:	PUNCT
cana-369	43	24	h	h	NOUN
cana-369	43	25	(	(	PUNCT
cana-369	43	26	sp	sp	NOUN
cana-369	43	27	)	)	PUNCT
cana-369	43	28	=	=	SYM
cana-369	43	29	2p	2p	NOUN
cana-369	43	30	−	−	NOUN
cana-369	43	31	1	1	NUM
cana-369	43	32	,	,	PUNCT
cana-369	43	33	for	for	ADP
cana-369	43	34	all	all	DET
cana-369	43	35	p	p	NOUN
cana-369	43	36	=	=	NOUN
cana-369	43	37	1	1	NUM
cana-369	43	38	to	to	PART
cana-369	43	39	r	r	NOUN
cana-369	43	40	v2	v2	NOUN
cana-369	43	41	:	:	PUNCT
cana-369	43	42	h	h	NOUN
cana-369	43	43	(	(	PUNCT
cana-369	43	44	f	f	X
cana-369	43	45	q	q	NOUN
cana-369	43	46	)	)	PUNCT
cana-369	44	1	=	=	SYM
cana-369	44	2	2r	2r	NUM
cana-369	44	3	(	(	PUNCT
cana-369	44	4	t	t	NOUN
cana-369	44	5	+	+	CCONJ
cana-369	44	6	1	1	NUM
cana-369	44	7	)	)	PUNCT
cana-369	44	8	–	–	PUNCT
cana-369	45	1	2r	2r	NUM
cana-369	45	2	q	q	X
cana-369	46	1	+	+	NUM
cana-369	46	2	1	1	NUM
cana-369	46	3	,	,	PUNCT
cana-369	46	4	for	for	ADP
cana-369	46	5	all	all	DET
cana-369	46	6	q	q	NOUN
cana-369	46	7	=	=	SYM
cana-369	46	8	1	1	NUM
cana-369	46	9	to	to	ADP
cana-369	46	10	t	t	PROPN
cana-369	46	11	and	and	CCONJ
cana-369	46	12	k	k	PROPN
cana-369	46	13	(	(	PUNCT
cana-369	46	14	wl	wl	PROPN
cana-369	46	15	)	)	PUNCT
cana-369	46	16	=	=	PUNCT
cana-369	47	1	2rt	2rt	NOUN
cana-369	47	2	−	−	NOUN
cana-369	47	3	2	2	NUM
cana-369	47	4	(	(	PUNCT
cana-369	47	5	l	l	NOUN
cana-369	47	6	−	−	NOUN
cana-369	47	7	1	1	NUM
cana-369	47	8	)	)	PUNCT
cana-369	47	9	,	,	PUNCT
cana-369	47	10	for	for	ADP
cana-369	47	11	all	all	DET
cana-369	47	12	l	l	NOUN
cana-369	47	13	=	=	SYM
cana-369	47	14	1	1	NUM
cana-369	47	15	to	to	PART
cana-369	47	16	rt	rt	VERB
cana-369	47	17	to	to	PART
cana-369	47	18	prove	prove	VERB
cana-369	47	19	the	the	DET
cana-369	47	20	existence	existence	NOUN
cana-369	47	21	of	of	ADP
cana-369	47	22	odd	odd	ADV
cana-369	47	23	-	-	PUNCT
cana-369	47	24	even	even	ADV
cana-369	47	25	congruence	congruence	NOUN
cana-369	47	26	labeling	labeling	NOUN
cana-369	47	27	,	,	PUNCT
cana-369	47	28	consider	consider	VERB
cana-369	47	29	communications	communication	NOUN
cana-369	47	30	on	on	ADP
cana-369	47	31	applied	apply	VERB
cana-369	47	32	nonlinear	nonlinear	ADJ
cana-369	47	33	analysis	analysis	NOUN
cana-369	47	34	issn	issn	NOUN
cana-369	47	35	:	:	PUNCT
cana-369	47	36	1074	1074	NUM
cana-369	47	37	-	-	PUNCT
cana-369	47	38	133x	133x	NUM
cana-369	47	39	vol	vol	NOUN
cana-369	47	40	31	31	NUM
cana-369	47	41	no	no	NOUN
cana-369	47	42	.	.	NOUN
cana-369	47	43	1	1	NUM
cana-369	47	44	(	(	PUNCT
cana-369	47	45	2024	2024	NUM
cana-369	47	46	)	)	PUNCT
cana-369	47	47	143	143	NUM
cana-369	47	48	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	47	49	10	10	NUM
cana-369	47	50	26	26	NUM
cana-369	47	51	24	24	NUM
cana-369	47	52	30	30	NUM
cana-369	47	53	28	28	NUM
cana-369	47	54	22	22	NUM
cana-369	47	55	10	10	NUM
cana-369	47	56	20	20	NUM
cana-369	47	57	16	16	NUM
cana-369	47	58	14	14	NUM
cana-369	47	59	12	12	NUM
cana-369	47	60	18	18	NUM
cana-369	47	61	h	h	NOUN
cana-369	47	62	(	(	PUNCT
cana-369	47	63	sp	sp	NOUN
cana-369	47	64	)	)	PUNCT
cana-369	47	65	−	−	NOUN
cana-369	47	66	h	h	NOUN
cana-369	47	67	(	(	PUNCT
cana-369	47	68	f	f	NOUN
cana-369	47	69	q	q	ADJ
cana-369	47	70	)	)	PUNCT
cana-369	47	71	≡	≡	PROPN
cana-369	47	72	(	(	PUNCT
cana-369	47	73	mod	mod	PROPN
cana-369	47	74	k	k	PROPN
cana-369	47	75	(	(	PUNCT
cana-369	47	76	wp	wp	PROPN
cana-369	47	77	)	)	PUNCT
cana-369	47	78	)	)	PUNCT
cana-369	48	1	(	(	PUNCT
cana-369	48	2	2p	2p	NUM
cana-369	48	3	−	−	NOUN
cana-369	48	4	1	1	NUM
cana-369	48	5	)	)	PUNCT
cana-369	48	6	−	−	PROPN
cana-369	48	7	(	(	PUNCT
cana-369	48	8	2rt	2rt	NOUN
cana-369	48	9	+	+	CCONJ
cana-369	48	10	2r	2r	NUM
cana-369	48	11	–	–	PUNCT
cana-369	48	12	2r	2r	NUM
cana-369	48	13	q	q	NOUN
cana-369	49	1	+	+	NUM
cana-369	49	2	1	1	NUM
cana-369	49	3	)	)	PUNCT
cana-369	49	4	≡	≡	PROPN
cana-369	49	5	(	(	PUNCT
cana-369	49	6	mod	mod	PROPN
cana-369	49	7	(	(	PUNCT
cana-369	49	8	2rt	2rt	ADJ
cana-369	49	9	−	−	NOUN
cana-369	49	10	2l	2l	NOUN
cana-369	49	11	+	+	CCONJ
cana-369	49	12	2	2	NUM
cana-369	49	13	)	)	PUNCT
cana-369	49	14	)	)	PUNCT
cana-369	50	1	(	(	PUNCT
cana-369	50	2	2rt	2rt	NOUN
cana-369	50	3	−	−	NOUN
cana-369	50	4	2l	2l	NOUN
cana-369	50	5	+	+	CCONJ
cana-369	50	6	2	2	X
cana-369	50	7	)	)	PUNCT
cana-369	50	8	divides	divide	NOUN
cana-369	50	9	(	(	PUNCT
cana-369	50	10	2p	2p	NUM
cana-369	50	11	−	−	NOUN
cana-369	50	12	1	1	NUM
cana-369	50	13	–	–	PUNCT
cana-369	50	14	2rt	2rt	NOUN
cana-369	50	15	–	–	PUNCT
cana-369	50	16	2r	2r	NUM
cana-369	51	1	+	+	CCONJ
cana-369	51	2	2r	2r	NUM
cana-369	51	3	q	q	NOUN
cana-369	51	4	−	−	NOUN
cana-369	51	5	1	1	NUM
cana-369	51	6	)	)	PUNCT
cana-369	51	7	for	for	ADP
cana-369	51	8	all	all	DET
cana-369	51	9	p	p	NOUN
cana-369	51	10	and	and	CCONJ
cana-369	51	11	q.	q.	PROPN
cana-369	51	12	hence	hence	ADV
cana-369	51	13	,	,	PUNCT
cana-369	51	14	every	every	DET
cana-369	51	15	complete	complete	ADJ
cana-369	51	16	bipartite	bipartite	NOUN
cana-369	51	17	graph	graph	NOUN
cana-369	51	18	is	be	AUX
cana-369	51	19	odd	odd	ADJ
cana-369	51	20	-	-	PUNCT
cana-369	51	21	even	even	ADV
cana-369	51	22	congruence	congruence	NOUN
cana-369	51	23	graph	graph	NOUN
cana-369	51	24	.	.	PUNCT
cana-369	51	25	example	example	NOUN
cana-369	51	26	3.2	3.2	NUM
cana-369	51	27	consider	consider	VERB
cana-369	51	28	the	the	DET
cana-369	51	29	graph	graph	NOUN
cana-369	51	30	g	g	NOUN
cana-369	51	31	=	=	SYM
cana-369	51	32	k3,2	k3,2	PROPN
cana-369	51	33	with	with	ADP
cana-369	51	34	|v1	|v1	PROPN
cana-369	51	35	|	|	ADV
cana-369	51	36	=	=	SYM
cana-369	51	37	3	3	NUM
cana-369	51	38	and	and	CCONJ
cana-369	51	39	|v2	|v2	NOUN
cana-369	52	1	|	|	ADV
cana-369	52	2	=	=	NOUN
cana-369	52	3	2	2	NUM
cana-369	52	4	.	.	NOUN
cana-369	52	5	13	13	NUM
cana-369	52	6	7	7	NUM
cana-369	52	7	12	12	NUM
cana-369	52	8	2	2	NUM
cana-369	52	9	1	1	NUM
cana-369	52	10	3	3	NUM
cana-369	52	11	5	5	NUM
cana-369	52	12	figure	figure	NOUN
cana-369	52	13	1	1	NUM
cana-369	52	14	k3,2	k3,2	NOUN
cana-369	52	15	the	the	DET
cana-369	52	16	odd	odd	ADV
cana-369	52	17	-	-	PUNCT
cana-369	52	18	even	even	ADV
cana-369	52	19	congruence	congruence	NOUN
cana-369	52	20	labeling	labeling	NOUN
cana-369	52	21	of	of	ADP
cana-369	52	22	complete	complete	ADJ
cana-369	52	23	bipartite	bipartite	NOUN
cana-369	52	24	graph	graph	NOUN
cana-369	52	25	is	be	AUX
cana-369	52	26	revealed	reveal	VERB
cana-369	52	27	in	in	ADP
cana-369	52	28	figure	figure	NOUN
cana-369	52	29	1	1	NUM
cana-369	52	30	.	.	PUNCT
cana-369	52	31	suppose	suppose	VERB
cana-369	52	32	g	g	PROPN
cana-369	52	33	=	=	SYM
cana-369	52	34	k4,4	k4,4	PROPN
cana-369	52	35	with	with	ADP
cana-369	52	36	|v1	|v1	PROPN
cana-369	52	37	|	|	ADV
cana-369	52	38	=	=	NOUN
cana-369	52	39	4	4	NUM
cana-369	52	40	and	and	CCONJ
cana-369	52	41	|v2	|v2	NOUN
cana-369	53	1	|	|	ADV
cana-369	53	2	=	=	NOUN
cana-369	53	3	4	4	NUM
cana-369	53	4	.	.	NOUN
cana-369	53	5	33	33	NUM
cana-369	53	6	25	25	NUM
cana-369	53	7	17	17	NUM
cana-369	53	8	9	9	NUM
cana-369	53	9	32	32	NUM
cana-369	53	10	2	2	NUM
cana-369	53	11	1	1	NUM
cana-369	53	12	3	3	NUM
cana-369	53	13	5	5	NUM
cana-369	53	14	7	7	NUM
cana-369	53	15	figure	figure	NOUN
cana-369	53	16	2	2	NUM
cana-369	53	17	k4,4	k4,4	PROPN
cana-369	53	18	figure	figure	NOUN
cana-369	53	19	2	2	NUM
cana-369	53	20	,	,	PUNCT
cana-369	53	21	shows	show	VERB
cana-369	53	22	that	that	SCONJ
cana-369	53	23	k4,4	k4,4	PROPN
cana-369	53	24	receives	receive	VERB
cana-369	53	25	odd	odd	ADV
cana-369	53	26	-	-	PUNCT
cana-369	53	27	even	even	ADV
cana-369	53	28	congruence	congruence	NOUN
cana-369	53	29	labeling	labeling	NOUN
cana-369	53	30	theorem	theorem	VERB
cana-369	53	31	3.3	3.3	NUM
cana-369	53	32	comb	comb	NOUN
cana-369	53	33	graph	graph	NOUN
cana-369	53	34	pt	pt	PROPN
cana-369	53	35	⊙	⊙	PROPN
cana-369	53	36	k1	k1	PROPN
cana-369	53	37	is	be	AUX
cana-369	53	38	odd	odd	ADJ
cana-369	53	39	-	-	PUNCT
cana-369	53	40	even	even	ADV
cana-369	53	41	congruence	congruence	NOUN
cana-369	53	42	graph	graph	NOUN
cana-369	53	43	,	,	PUNCT
cana-369	53	44	t	t	PROPN
cana-369	53	45	≥	≥	NUM
cana-369	53	46	1	1	NUM
cana-369	53	47	.	.	PUNCT
cana-369	54	1	proof	proof	NOUN
cana-369	54	2	:	:	PUNCT
cana-369	54	3	let	let	VERB
cana-369	54	4	g	g	PROPN
cana-369	54	5	=	=	PUNCT
cana-369	54	6	pt	pt	PROPN
cana-369	54	7	⊙	⊙	PROPN
cana-369	54	8	k1	k1	PROPN
cana-369	54	9	with	with	ADP
cana-369	54	10	|v	|v	PROPN
cana-369	55	1	|	|	NOUN
cana-369	55	2	=	=	SYM
cana-369	55	3	2	2	NUM
cana-369	55	4	t	t	NOUN
cana-369	55	5	and	and	CCONJ
cana-369	55	6	|e|	|e|	NOUN
cana-369	55	7	=	=	PUNCT
cana-369	55	8	(	(	PUNCT
cana-369	55	9	2	2	NUM
cana-369	55	10	t	t	NOUN
cana-369	55	11	−	−	NOUN
cana-369	55	12	1	1	NUM
cana-369	55	13	)	)	PUNCT
cana-369	55	14	.	.	PUNCT
cana-369	56	1	then	then	ADV
cana-369	56	2	d	d	X
cana-369	56	3	=	=	SYM
cana-369	56	4	min	min	PROPN
cana-369	56	5	(	(	PUNCT
cana-369	56	6	2	2	NUM
cana-369	56	7	(	(	PUNCT
cana-369	56	8	2	2	NUM
cana-369	56	9	t	t	NOUN
cana-369	56	10	)	)	PUNCT
cana-369	56	11	,	,	PUNCT
cana-369	56	12	2	2	NUM
cana-369	56	13	(	(	PUNCT
cana-369	56	14	2	2	NUM
cana-369	56	15	t	t	NOUN
cana-369	56	16	−	−	NOUN
cana-369	56	17	1	1	NUM
cana-369	56	18	)	)	PUNCT
cana-369	56	19	)	)	PUNCT
cana-369	57	1	=	=	PUNCT
cana-369	57	2	4	4	NUM
cana-369	57	3	t	t	NOUN
cana-369	57	4	−	−	NUM
cana-369	57	5	2	2	NUM
cana-369	57	6	let	let	VERB
cana-369	57	7	{	{	PUNCT
cana-369	57	8	sp/1	sp/1	VERB
cana-369	57	9	≤	≤	NOUN
cana-369	57	10	p	p	PROPN
cana-369	57	11	≤	≤	PROPN
cana-369	57	12	t	t	PROPN
cana-369	57	13	}	}	PUNCT
cana-369	57	14	and	and	CCONJ
cana-369	57	15	{	{	PUNCT
cana-369	57	16	fp/1	fp/1	NOUN
cana-369	57	17	≤	≤	PROPN
cana-369	57	18	p	p	PROPN
cana-369	57	19	≤	≤	PROPN
cana-369	57	20	t	t	PROPN
cana-369	57	21	}	}	PUNCT
cana-369	57	22	are	be	AUX
cana-369	57	23	vertex	vertex	NOUN
cana-369	57	24	sets	set	NOUN
cana-369	57	25	,	,	PUNCT
cana-369	57	26	here	here	ADV
cana-369	57	27	fp	fp	NOUN
cana-369	57	28	represents	represent	VERB
cana-369	57	29	pendent	pendent	ADJ
cana-369	57	30	vertices	vertex	NOUN
cana-369	57	31	.	.	PUNCT
cana-369	58	1	further	far	ADV
cana-369	58	2	,	,	PUNCT
cana-369	58	3	wp	wp	PROPN
cana-369	58	4	=	=	PRON
cana-369	58	5	{	{	PUNCT
cana-369	58	6	spsp+1/1	spsp+1/1	NOUN
cana-369	58	7	≤	≤	NUM
cana-369	58	8	p	p	NOUN
cana-369	58	9	≤	≤	PROPN
cana-369	58	10	t	t	NOUN
cana-369	58	11	−	−	NOUN
cana-369	58	12	1	1	NUM
cana-369	58	13	}	}	PUNCT
cana-369	58	14	are	be	AUX
cana-369	58	15	edges	edge	NOUN
cana-369	58	16	of	of	ADP
cana-369	58	17	pt	pt	NOUN
cana-369	58	18	and	and	CCONJ
cana-369	58	19	edges	edge	NOUN
cana-369	58	20	adjacent	adjacent	ADJ
cana-369	58	21	to	to	ADP
cana-369	58	22	fp	fp	PROPN
cana-369	58	23	are	be	AUX
cana-369	58	24	denoted	denote	VERB
cana-369	58	25	as	as	ADP
cana-369	58	26	{	{	PUNCT
cana-369	58	27	ep	ep	PROPN
cana-369	58	28	=	=	PROPN
cana-369	58	29	spfq/1	spfq/1	PROPN
cana-369	58	30	≤	≤	PUNCT
cana-369	58	31	p	p	X
cana-369	58	32	≤	≤	NUM
cana-369	58	33	t	t	PROPN
cana-369	58	34	}	}	PUNCT
cana-369	58	35	.	.	PUNCT
cana-369	59	1	define	define	VERB
cana-369	59	2	the	the	DET
cana-369	59	3	bijection	bijection	ADJ
cana-369	59	4	h	h	NOUN
cana-369	59	5	:	:	PUNCT
cana-369	59	6	v	v	X
cana-369	59	7	→	→	SYM
cana-369	59	8	{	{	PUNCT
cana-369	59	9	1	1	NUM
cana-369	59	10	,	,	PUNCT
cana-369	59	11	3	3	NUM
cana-369	59	12	,	,	PUNCT
cana-369	59	13	...	...	PUNCT
cana-369	59	14	,	,	PUNCT
cana-369	59	15	8	8	NUM
cana-369	59	16	t	t	NOUN
cana-369	59	17	−	−	NOUN
cana-369	59	18	3	3	NUM
cana-369	59	19	}	}	PUNCT
cana-369	59	20	as	as	SCONJ
cana-369	59	21	communications	communication	NOUN
cana-369	59	22	on	on	ADP
cana-369	59	23	applied	apply	VERB
cana-369	59	24	nonlinear	nonlinear	ADJ
cana-369	59	25	analysis	analysis	NOUN
cana-369	59	26	issn	issn	NOUN
cana-369	59	27	:	:	PUNCT
cana-369	59	28	1074	1074	NUM
cana-369	59	29	-	-	PUNCT
cana-369	59	30	133x	133x	NUM
cana-369	59	31	vol	vol	NOUN
cana-369	59	32	31	31	NUM
cana-369	59	33	no	no	NOUN
cana-369	59	34	.	.	NOUN
cana-369	59	35	1	1	NUM
cana-369	59	36	(	(	PUNCT
cana-369	59	37	2024	2024	NUM
cana-369	59	38	)	)	PUNCT
cana-369	59	39	144	144	NUM
cana-369	59	40	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	59	41	h	h	NOUN
cana-369	59	42	(	(	PUNCT
cana-369	59	43	s2p−1	s2p−1	PROPN
cana-369	59	44	)	)	PUNCT
cana-369	59	45	=	=	SYM
cana-369	59	46	4	4	NUM
cana-369	59	47	(	(	PUNCT
cana-369	59	48	t	t	NOUN
cana-369	59	49	−	−	PROPN
cana-369	59	50	p	p	NOUN
cana-369	59	51	)	)	PUNCT
cana-369	59	52	+	+	CCONJ
cana-369	59	53	3	3	NUM
cana-369	59	54	h	h	NOUN
cana-369	59	55	(	(	PUNCT
cana-369	59	56	s2p	s2p	X
cana-369	59	57	)	)	PUNCT
cana-369	59	58	=	=	SYM
cana-369	59	59	4p	4p	NOUN
cana-369	59	60	−	−	NOUN
cana-369	59	61	1	1	NUM
cana-369	59	62	h	h	NOUN
cana-369	59	63	(	(	PUNCT
cana-369	59	64	f2p−1	f2p−1	PROPN
cana-369	59	65	)	)	PUNCT
cana-369	60	1	=	=	SYM
cana-369	60	2	4p	4p	NUM
cana-369	60	3	–	–	PUNCT
cana-369	60	4	3	3	NUM
cana-369	60	5	h	h	NOUN
cana-369	60	6	(	(	PUNCT
cana-369	60	7	f2p	f2p	NOUN
cana-369	60	8	)	)	PUNCT
cana-369	60	9	=	=	SYM
cana-369	60	10	4	4	NUM
cana-369	60	11	(	(	PUNCT
cana-369	60	12	t	t	NOUN
cana-369	60	13	−	−	PROPN
cana-369	60	14	p	p	NOUN
cana-369	60	15	)	)	PUNCT
cana-369	60	16	+	+	CCONJ
cana-369	60	17	1	1	NUM
cana-369	60	18	where	where	SCONJ
cana-369	60	19	p	p	NOUN
cana-369	60	20	=	=	NOUN
cana-369	60	21	1	1	NUM
cana-369	60	22	to	to	ADP
cana-369	60	23	(	(	PUNCT
cana-369	60	24	t/2	t/2	NUM
cana-369	60	25	)	)	PUNCT
cana-369	60	26	if	if	SCONJ
cana-369	60	27	t	t	PROPN
cana-369	60	28	is	be	AUX
cana-369	60	29	even	even	ADV
cana-369	60	30	p	p	NOUN
cana-369	60	31	=	=	NOUN
cana-369	60	32	1	1	NUM
cana-369	60	33	to	to	PART
cana-369	60	34	(	(	PUNCT
cana-369	60	35	t+1)/2	t+1)/2	VERB
cana-369	60	36	if	if	SCONJ
cana-369	60	37	t	t	PROPN
cana-369	60	38	is	be	AUX
cana-369	60	39	odd	odd	ADJ
cana-369	60	40	the	the	DET
cana-369	60	41	edge	edge	NOUN
cana-369	60	42	labeling	labeling	NOUN
cana-369	60	43	k	k	X
cana-369	60	44	:	:	PUNCT
cana-369	60	45	e	e	X
cana-369	60	46	(	(	PUNCT
cana-369	60	47	g	g	NOUN
cana-369	60	48	)	)	PUNCT
cana-369	60	49	→	→	SYM
cana-369	60	50	{	{	PUNCT
cana-369	60	51	2	2	NUM
cana-369	60	52	,	,	PUNCT
cana-369	60	53	4	4	NUM
cana-369	60	54	,	,	PUNCT
cana-369	60	55	...	...	PUNCT
cana-369	60	56	,	,	PUNCT
cana-369	60	57	8	8	NUM
cana-369	60	58	t	t	NOUN
cana-369	60	59	−	−	NOUN
cana-369	60	60	4	4	NUM
cana-369	60	61	}	}	PUNCT
cana-369	60	62	is	be	AUX
cana-369	60	63	defined	define	VERB
cana-369	60	64	as	as	ADP
cana-369	60	65	k	k	PROPN
cana-369	60	66	(	(	PUNCT
cana-369	60	67	wp	wp	X
cana-369	60	68	)	)	PUNCT
cana-369	60	69	=	=	SYM
cana-369	61	1	4	4	NUM
cana-369	61	2	(	(	PUNCT
cana-369	61	3	t	t	NOUN
cana-369	61	4	−	−	PROPN
cana-369	61	5	p	p	NOUN
cana-369	61	6	)	)	PUNCT
cana-369	61	7	,	,	PUNCT
cana-369	61	8	p	p	NOUN
cana-369	61	9	=	=	NOUN
cana-369	61	10	1	1	NUM
cana-369	61	11	to	to	ADP
cana-369	61	12	t	t	PROPN
cana-369	61	13	−	−	PROPN
cana-369	61	14	1	1	NUM
cana-369	61	15	k	k	X
cana-369	61	16	(	(	PUNCT
cana-369	61	17	ep	ep	PROPN
cana-369	61	18	)	)	PUNCT
cana-369	61	19	=	=	SYM
cana-369	61	20	4	4	NUM
cana-369	61	21	(	(	PUNCT
cana-369	61	22	t	t	NOUN
cana-369	61	23	−	−	PROPN
cana-369	61	24	p	p	NOUN
cana-369	61	25	)	)	PUNCT
cana-369	61	26	+	+	CCONJ
cana-369	61	27	2	2	NUM
cana-369	61	28	,	,	PUNCT
cana-369	61	29	p	p	NOUN
cana-369	61	30	=	=	NOUN
cana-369	61	31	1	1	NUM
cana-369	61	32	to	to	ADP
cana-369	61	33	t	t	PROPN
cana-369	61	34	clearly	clearly	ADV
cana-369	61	35	,	,	PUNCT
cana-369	61	36	k	k	PROPN
cana-369	61	37	(	(	PUNCT
cana-369	61	38	wi	wi	PROPN
cana-369	61	39	)	)	PUNCT
cana-369	61	40	divides	divide	VERB
cana-369	61	41	(	(	PUNCT
cana-369	61	42	h	h	NOUN
cana-369	61	43	(	(	PUNCT
cana-369	61	44	s2p−1	s2p−1	PROPN
cana-369	61	45	)	)	PUNCT
cana-369	61	46	−	−	PROPN
cana-369	61	47	h	h	NOUN
cana-369	61	48	(	(	PUNCT
cana-369	61	49	s2p	s2p	NOUN
cana-369	61	50	)	)	PUNCT
cana-369	61	51	)	)	PUNCT
cana-369	61	52	and	and	CCONJ
cana-369	61	53	k	k	PROPN
cana-369	61	54	(	(	PUNCT
cana-369	61	55	ep	ep	NOUN
cana-369	61	56	)	)	PUNCT
cana-369	61	57	divides	divide	VERB
cana-369	61	58	(	(	PUNCT
cana-369	61	59	h	h	NOUN
cana-369	61	60	(	(	PUNCT
cana-369	61	61	sp	sp	NOUN
cana-369	61	62	)	)	PUNCT
cana-369	61	63	−	−	NOUN
cana-369	62	1	h	h	NOUN
cana-369	62	2	(	(	PUNCT
cana-369	62	3	fp	fp	ADJ
cana-369	62	4	)	)	PUNCT
cana-369	62	5	)	)	PUNCT
cana-369	63	1	hence	hence	ADV
cana-369	63	2	,	,	PUNCT
cana-369	63	3	the	the	DET
cana-369	63	4	comb	comb	NOUN
cana-369	63	5	graph	graph	NOUN
cana-369	63	6	pt	pt	PROPN
cana-369	63	7	⊙	⊙	PROPN
cana-369	63	8	k1	k1	PROPN
cana-369	63	9	is	be	AUX
cana-369	63	10	odd	odd	ADJ
cana-369	63	11	-	-	PUNCT
cana-369	63	12	even	even	ADV
cana-369	63	13	congruence	congruence	NOUN
cana-369	63	14	graph	graph	NOUN
cana-369	63	15	.	.	PUNCT
cana-369	63	16	example	example	NOUN
cana-369	63	17	3.4	3.4	NUM
cana-369	63	18	consider	consider	VERB
cana-369	63	19	a	a	DET
cana-369	63	20	graph	graph	NOUN
cana-369	63	21	g	g	NOUN
cana-369	63	22	=	=	SYM
cana-369	63	23	pt	pt	PROPN
cana-369	63	24	⊙	⊙	PROPN
cana-369	63	25	k1	k1	PROPN
cana-369	63	26	with	with	ADP
cana-369	63	27	t	t	PROPN
cana-369	63	28	=	=	SYM
cana-369	63	29	10	10	NUM
cana-369	63	30	.	.	PUNCT
cana-369	64	1	figure	figure	VERB
cana-369	64	2	3	3	NUM
cana-369	64	3	p10	p10	PROPN
cana-369	64	4	⊙	⊙	PROPN
cana-369	64	5	k1	k1	PROPN
cana-369	64	6	figure	figure	NOUN
cana-369	64	7	3	3	NUM
cana-369	64	8	exhibits	exhibit	VERB
cana-369	64	9	the	the	DET
cana-369	64	10	odd	odd	ADV
cana-369	64	11	-	-	PUNCT
cana-369	64	12	even	even	ADV
cana-369	64	13	congruence	congruence	NOUN
cana-369	64	14	labeling	labeling	NOUN
cana-369	64	15	of	of	ADP
cana-369	64	16	the	the	DET
cana-369	64	17	comb	comb	NOUN
cana-369	64	18	graph	graph	NOUN
cana-369	64	19	p10	p10	PROPN
cana-369	64	20	⊙	⊙	PROPN
cana-369	64	21	k1	k1	PROPN
cana-369	64	22	.	.	PUNCT
cana-369	65	1	theorem	theorem	VERB
cana-369	65	2	3.5	3.5	NUM
cana-369	65	3	spliting	split	VERB
cana-369	65	4	graph	graph	NOUN
cana-369	65	5	of	of	ADP
cana-369	65	6	a	a	DET
cana-369	65	7	star	star	NOUN
cana-369	65	8	graph	graph	NOUN
cana-369	65	9	st	st	PROPN
cana-369	65	10	is	be	AUX
cana-369	65	11	odd	odd	ADJ
cana-369	65	12	-	-	PUNCT
cana-369	65	13	even	even	ADV
cana-369	65	14	congruence	congruence	NOUN
cana-369	65	15	graph	graph	NOUN
cana-369	65	16	.	.	PUNCT
cana-369	66	1	communications	communication	NOUN
cana-369	66	2	on	on	ADP
cana-369	66	3	applied	apply	VERB
cana-369	66	4	nonlinear	nonlinear	ADJ
cana-369	66	5	analysis	analysis	NOUN
cana-369	66	6	issn	issn	NOUN
cana-369	66	7	:	:	PUNCT
cana-369	66	8	1074	1074	NUM
cana-369	66	9	-	-	PUNCT
cana-369	66	10	133x	133x	NUM
cana-369	66	11	vol	vol	NOUN
cana-369	66	12	31	31	NUM
cana-369	66	13	no	no	NOUN
cana-369	66	14	.	.	NOUN
cana-369	66	15	1	1	NUM
cana-369	66	16	(	(	PUNCT
cana-369	66	17	2024	2024	NUM
cana-369	66	18	)	)	PUNCT
cana-369	66	19	145	145	NUM
cana-369	66	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	66	21	48	48	NUM
cana-369	66	22	20	20	NUM
cana-369	66	23	44	44	NUM
cana-369	66	24	40	40	NUM
cana-369	66	25	36	36	NUM
cana-369	66	26	32	32	NUM
cana-369	66	27	28	28	NUM
cana-369	67	1	b	b	X
cana-369	67	2	47	47	NUM
cana-369	67	3	b	b	NUM
cana-369	67	4	43	43	NUM
cana-369	67	5	b	b	NOUN
cana-369	67	6	39	39	NUM
cana-369	67	7	b	b	PROPN
cana-369	67	8	b	b	PROPN
cana-369	67	9	35	35	NUM
cana-369	67	10	24	24	NUM
cana-369	67	11	b	b	SYM
cana-369	67	12	31	31	NUM
cana-369	67	13	26	26	NUM
cana-369	67	14	b	b	NUM
cana-369	67	15	27	27	NUM
cana-369	67	16	46	46	NUM
cana-369	67	17	42	42	NUM
cana-369	67	18	38	38	NUM
cana-369	67	19	30	30	NUM
cana-369	67	20	34	34	NUM
cana-369	67	21	50	50	NUM
cana-369	67	22	22	22	NUM
cana-369	67	23	18	18	NUM
cana-369	67	24	8	8	NUM
cana-369	67	25	10	10	NUM
cana-369	67	26	12	12	NUM
cana-369	67	27	14	14	NUM
cana-369	67	28	16	16	NUM
cana-369	67	29	proof	proof	NOUN
cana-369	67	30	:	:	PUNCT
cana-369	67	31	suppose	suppose	VERB
cana-369	67	32	g	g	PROPN
cana-369	67	33	=	=	SYM
cana-369	67	34	spl	spl	PROPN
cana-369	67	35	(	(	PUNCT
cana-369	67	36	st	st	PROPN
cana-369	67	37	)	)	PUNCT
cana-369	67	38	is	be	AUX
cana-369	67	39	splitting	split	VERB
cana-369	67	40	graph	graph	NOUN
cana-369	67	41	with	with	ADP
cana-369	67	42	|v	|v	PROPN
cana-369	67	43	|	|	NOUN
cana-369	67	44	=	=	SYM
cana-369	67	45	2	2	NUM
cana-369	67	46	t	t	NOUN
cana-369	67	47	+	+	CCONJ
cana-369	67	48	2	2	NUM
cana-369	67	49	and	and	CCONJ
cana-369	67	50	|e|	|e|	NOUN
cana-369	67	51	=	=	SYM
cana-369	67	52	3	3	NUM
cana-369	67	53	t.	t.	NOUN
cana-369	67	54	the	the	DET
cana-369	67	55	vertex	vertex	NOUN
cana-369	67	56	set	set	NOUN
cana-369	67	57	be	be	AUX
cana-369	67	58	v	v	ADP
cana-369	67	59	(	(	PUNCT
cana-369	67	60	g	g	NOUN
cana-369	67	61	)	)	PUNCT
cana-369	67	62	=	=	PUNCT
cana-369	67	63	{	{	PUNCT
cana-369	67	64	s	s	PROPN
cana-369	67	65	,	,	PUNCT
cana-369	67	66	s1	s1	NOUN
cana-369	67	67	,	,	PUNCT
cana-369	67	68	s2	s2	PROPN
cana-369	67	69	,	,	PUNCT
cana-369	67	70	...	...	PUNCT
cana-369	67	71	st	st	X
cana-369	67	72	}	}	PUNCT
cana-369	67	73	∪	∪	X
cana-369	67	74	{	{	PUNCT
cana-369	67	75	f	f	PROPN
cana-369	67	76	,	,	PUNCT
cana-369	67	77	f1	f1	NOUN
cana-369	67	78	,	,	PUNCT
cana-369	67	79	f2	f2	PROPN
cana-369	67	80	,	,	PUNCT
cana-369	67	81	...	...	PUNCT
cana-369	67	82	ft	ft	X
cana-369	67	83	}	}	PUNCT
cana-369	67	84	where	where	SCONJ
cana-369	67	85	,	,	PUNCT
cana-369	67	86	s	s	X
cana-369	67	87	,	,	PUNCT
cana-369	67	88	s1	s1	NOUN
cana-369	67	89	,	,	PUNCT
cana-369	67	90	s2	s2	PROPN
cana-369	67	91	,	,	PUNCT
cana-369	67	92	...	...	PUNCT
cana-369	67	93	st	st	PROPN
cana-369	67	94	is	be	AUX
cana-369	67	95	vertex	vertex	NOUN
cana-369	67	96	set	set	NOUN
cana-369	67	97	of	of	ADP
cana-369	67	98	star	star	NOUN
cana-369	67	99	graph	graph	NOUN
cana-369	67	100	and	and	CCONJ
cana-369	67	101	s	s	NOUN
cana-369	67	102	is	be	AUX
cana-369	67	103	apex	apex	NOUN
cana-369	67	104	vertex	vertex	NOUN
cana-369	67	105	f	f	PROPN
cana-369	67	106	,	,	PUNCT
cana-369	67	107	f1	f1	NOUN
cana-369	67	108	,	,	PUNCT
cana-369	67	109	f2	f2	PROPN
cana-369	67	110	,	,	PUNCT
cana-369	67	111	...	...	PUNCT
cana-369	67	112	ft	ft	PRON
cana-369	67	113	are	be	AUX
cana-369	67	114	the	the	DET
cana-369	67	115	vertices	vertex	NOUN
cana-369	67	116	added	add	VERB
cana-369	67	117	to	to	PART
cana-369	67	118	form	form	VERB
cana-369	67	119	g.	g.	PROPN
cana-369	67	120	also	also	ADV
cana-369	67	121	the	the	DET
cana-369	67	122	edge	edge	NOUN
cana-369	67	123	set	set	VERB
cana-369	67	124	be	be	AUX
cana-369	67	125	e	e	NOUN
cana-369	67	126	(	(	PUNCT
cana-369	67	127	g	g	NOUN
cana-369	67	128	)	)	PUNCT
cana-369	67	129	=	=	SYM
cana-369	67	130	{	{	PUNCT
cana-369	67	131	w1	w1	NOUN
cana-369	67	132	,	,	PUNCT
cana-369	67	133	w2	w2	NOUN
cana-369	67	134	,	,	PUNCT
cana-369	67	135	...	...	PUNCT
cana-369	67	136	,	,	PUNCT
cana-369	67	137	wt	wt	PROPN
cana-369	67	138	,	,	PUNCT
cana-369	67	139	wt+1	wt+1	PROPN
cana-369	67	140	,	,	PUNCT
cana-369	67	141	....	....	PUNCT
cana-369	67	142	,	,	PUNCT
cana-369	67	143	w2	w2	PROPN
cana-369	67	144	t	t	PROPN
cana-369	67	145	,	,	PUNCT
cana-369	67	146	w2t+1	w2t+1	PROPN
cana-369	67	147	,	,	PUNCT
cana-369	67	148	…	…	PUNCT
cana-369	67	149	,	,	PUNCT
cana-369	67	150	w3	w3	PROPN
cana-369	67	151	t	t	PROPN
cana-369	67	152	}	}	PUNCT
cana-369	67	153	here	here	ADV
cana-369	67	154	,	,	PUNCT
cana-369	67	155	w1	w1	NOUN
cana-369	67	156	,	,	PUNCT
cana-369	67	157	w2	w2	NOUN
cana-369	67	158	,	,	PUNCT
cana-369	67	159	...	...	PUNCT
cana-369	67	160	,	,	PUNCT
cana-369	67	161	wt	wt	PROPN
cana-369	67	162	are	be	AUX
cana-369	67	163	the	the	DET
cana-369	67	164	edges	edge	NOUN
cana-369	67	165	of	of	ADP
cana-369	67	166	sn	sn	PROPN
cana-369	67	167	,	,	PUNCT
cana-369	67	168	wt+1	wt+1	PROPN
cana-369	67	169	,	,	PUNCT
cana-369	67	170	....	....	PUNCT
cana-369	67	171	,	,	PUNCT
cana-369	67	172	w2	w2	PROPN
cana-369	67	173	t	t	PROPN
cana-369	67	174	are	be	AUX
cana-369	67	175	edges	edge	NOUN
cana-369	67	176	adjacent	adjacent	ADJ
cana-369	67	177	to	to	ADP
cana-369	67	178	f	f	PROPN
cana-369	67	179	and	and	CCONJ
cana-369	67	180	sp	sp	ADP
cana-369	67	181	and	and	CCONJ
cana-369	67	182	w2t+1	w2t+1	PROPN
cana-369	67	183	,	,	PUNCT
cana-369	67	184	…	…	PUNCT
cana-369	67	185	..	..	PUNCT
cana-369	67	186	,	,	PUNCT
cana-369	67	187	w3	w3	PROPN
cana-369	67	188	t	t	PROPN
cana-369	67	189	are	be	AUX
cana-369	67	190	edges	edge	NOUN
cana-369	67	191	adjacent	adjacent	ADJ
cana-369	67	192	to	to	ADP
cana-369	67	193	s	s	PROPN
cana-369	67	194	and	and	CCONJ
cana-369	67	195	fp	fp	INTJ
cana-369	67	196	.	.	PUNCT
cana-369	68	1	now	now	ADV
cana-369	68	2	,	,	PUNCT
cana-369	68	3	d	d	PROPN
cana-369	68	4	=	=	SYM
cana-369	68	5	min	min	PROPN
cana-369	68	6	(	(	PUNCT
cana-369	68	7	4	4	NUM
cana-369	68	8	t	t	NOUN
cana-369	68	9	+	+	CCONJ
cana-369	68	10	4	4	NUM
cana-369	68	11	,	,	PUNCT
cana-369	68	12	6	6	NUM
cana-369	68	13	t	t	NOUN
cana-369	68	14	)	)	PUNCT
cana-369	68	15	=	=	PUNCT
cana-369	69	1	4	4	NUM
cana-369	69	2	t	t	NOUN
cana-369	69	3	+	+	NOUN
cana-369	69	4	4	4	NUM
cana-369	69	5	then	then	ADV
cana-369	69	6	h	h	NOUN
cana-369	69	7	:	:	PUNCT
cana-369	69	8	v	v	X
cana-369	69	9	(	(	PUNCT
cana-369	69	10	g	g	NOUN
cana-369	69	11	)	)	PUNCT
cana-369	69	12	→	→	SYM
cana-369	69	13	{	{	PUNCT
cana-369	69	14	1	1	NUM
cana-369	69	15	,	,	PUNCT
cana-369	69	16	3	3	NUM
cana-369	69	17	,	,	PUNCT
cana-369	69	18	…	…	PUNCT
cana-369	69	19	.	.	NUM
cana-369	69	20	,	,	PUNCT
cana-369	69	21	8	8	NUM
cana-369	69	22	t	t	NOUN
cana-369	69	23	+	+	NOUN
cana-369	69	24	9	9	NUM
cana-369	69	25	}	}	PUNCT
cana-369	69	26	and	and	CCONJ
cana-369	69	27	k	k	NOUN
cana-369	69	28	:	:	PUNCT
cana-369	70	1	e	e	X
cana-369	70	2	(	(	PUNCT
cana-369	70	3	g	g	NOUN
cana-369	70	4	)	)	PUNCT
cana-369	70	5	→	→	SYM
cana-369	70	6	{	{	PUNCT
cana-369	70	7	2	2	NUM
cana-369	70	8	,	,	PUNCT
cana-369	70	9	4	4	NUM
cana-369	70	10	,	,	PUNCT
cana-369	70	11	…	…	PUNCT
cana-369	70	12	.	.	NUM
cana-369	70	13	,	,	PUNCT
cana-369	70	14	8	8	NUM
cana-369	70	15	t	t	NOUN
cana-369	70	16	+	+	NOUN
cana-369	70	17	8	8	NUM
cana-369	70	18	}	}	PUNCT
cana-369	70	19	are	be	AUX
cana-369	70	20	assigned	assign	VERB
cana-369	70	21	as	as	ADP
cana-369	70	22	h	h	NOUN
cana-369	70	23	(	(	PUNCT
cana-369	70	24	sp	sp	NOUN
cana-369	70	25	)	)	PUNCT
cana-369	70	26	=	=	SYM
cana-369	70	27	6	6	NUM
cana-369	70	28	t	t	NOUN
cana-369	70	29	–	–	PUNCT
cana-369	70	30	4p	4p	NOUN
cana-369	70	31	+	+	CCONJ
cana-369	70	32	7	7	NUM
cana-369	70	33	,	,	PUNCT
cana-369	70	34	p	p	NOUN
cana-369	70	35	=	=	NOUN
cana-369	70	36	1	1	NUM
cana-369	70	37	to	to	ADP
cana-369	70	38	t	t	PROPN
cana-369	70	39	h	h	PROPN
cana-369	70	40	(	(	PUNCT
cana-369	70	41	s	s	X
cana-369	70	42	)	)	PUNCT
cana-369	70	43	=	=	SYM
cana-369	70	44	1	1	NUM
cana-369	70	45	h	h	NOUN
cana-369	70	46	(	(	PUNCT
cana-369	70	47	f	f	X
cana-369	70	48	)	)	PUNCT
cana-369	70	49	=	=	SYM
cana-369	70	50	3	3	NUM
cana-369	70	51	h	h	NOUN
cana-369	70	52	(	(	PUNCT
cana-369	70	53	fp	fp	X
cana-369	70	54	)	)	PUNCT
cana-369	70	55	=	=	NOUN
cana-369	70	56	2p	2p	NOUN
cana-369	70	57	+	+	CCONJ
cana-369	70	58	3	3	NUM
cana-369	70	59	,	,	PUNCT
cana-369	70	60	p	p	NOUN
cana-369	70	61	=	=	NOUN
cana-369	70	62	1	1	NUM
cana-369	70	63	to	to	ADP
cana-369	70	64	t	t	PROPN
cana-369	70	65	𝑘	𝑘	PROPN
cana-369	70	66	(	(	PUNCT
cana-369	70	67	𝑤𝑞	𝑤𝑞	ADP
cana-369	70	68	)	)	PUNCT
cana-369	70	69	=	=	SYM
cana-369	70	70	{	{	PUNCT
cana-369	70	71	7𝑡	7𝑡	NOUN
cana-369	70	72	−	−	PROPN
cana-369	70	73	4𝑞	4𝑞	NOUN
cana-369	70	74	+	+	CCONJ
cana-369	70	75	2	2	NUM
cana-369	70	76	;	;	PUNCT
cana-369	70	77	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-369	70	78	𝑞	𝑞	X
cana-369	70	79	=	=	SYM
cana-369	70	80	1	1	NUM
cana-369	70	81	𝑡𝑜	𝑡𝑜	PROPN
cana-369	70	82	𝑡	𝑡	PROPN
cana-369	70	83	10	10	NUM
cana-369	70	84	t	t	NOUN
cana-369	70	85	−	−	NOUN
cana-369	70	86	4q	4q	NOUN
cana-369	70	87	+	+	CCONJ
cana-369	70	88	4	4	NUM
cana-369	70	89	;	;	PUNCT
cana-369	70	90	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-369	70	91	q	q	NOUN
cana-369	70	92	=	=	PUNCT
cana-369	70	93	t	t	PROPN
cana-369	70	94	+	+	CCONJ
cana-369	70	95	1	1	NUM
cana-369	70	96	𝑡𝑜	𝑡𝑜	NOUN
cana-369	70	97	2	2	NUM
cana-369	70	98	t	t	NOUN
cana-369	70	99	2q	2q	NUM
cana-369	70	100	−	−	PROPN
cana-369	70	101	4	4	NUM
cana-369	70	102	t	t	NOUN
cana-369	70	103	+	+	CCONJ
cana-369	70	104	2	2	NUM
cana-369	70	105	;	;	PUNCT
cana-369	70	106	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-369	70	107	q	q	NOUN
cana-369	70	108	=	=	SYM
cana-369	70	109	2	2	NUM
cana-369	70	110	t	t	NOUN
cana-369	70	111	+	+	NOUN
cana-369	70	112	1	1	NUM
cana-369	70	113	𝑡𝑜	𝑡𝑜	NOUN
cana-369	70	114	3	3	NUM
cana-369	70	115	t	t	NOUN
cana-369	70	116	the	the	DET
cana-369	70	117	above	above	ADJ
cana-369	70	118	labeling	labeling	NOUN
cana-369	70	119	construction	construction	NOUN
cana-369	70	120	satisfies	satisfie	NOUN
cana-369	70	121	h	h	X
cana-369	70	122	(	(	PUNCT
cana-369	70	123	fp	fp	X
cana-369	70	124	)	)	PUNCT
cana-369	70	125	≡	≡	PROPN
cana-369	70	126	h	h	PROPN
cana-369	70	127	(	(	PUNCT
cana-369	70	128	s)(mod	s)(mod	PROPN
cana-369	70	129	k	k	PROPN
cana-369	70	130	(	(	PUNCT
cana-369	70	131	w	w	PROPN
cana-369	70	132	q	q	NOUN
cana-369	70	133	)	)	PUNCT
cana-369	70	134	)	)	PUNCT
cana-369	70	135	for	for	ADP
cana-369	70	136	every	every	DET
cana-369	70	137	edge	edge	NOUN
cana-369	70	138	of	of	ADP
cana-369	70	139	g.	g.	PROPN
cana-369	70	140	hence	hence	ADV
cana-369	70	141	,	,	PUNCT
cana-369	70	142	spliting	split	VERB
cana-369	70	143	graph	graph	NOUN
cana-369	70	144	of	of	ADP
cana-369	70	145	a	a	DET
cana-369	70	146	star	star	NOUN
cana-369	70	147	graph	graph	NOUN
cana-369	70	148	is	be	AUX
cana-369	70	149	odd	odd	ADJ
cana-369	70	150	-	-	PUNCT
cana-369	70	151	even	even	ADV
cana-369	70	152	congruence	congruence	NOUN
cana-369	70	153	graph	graph	NOUN
cana-369	70	154	.	.	PUNCT
cana-369	70	155	example	example	NOUN
cana-369	70	156	3.6	3.6	NUM
cana-369	70	157	consider	consider	VERB
cana-369	70	158	the	the	DET
cana-369	70	159	graph	graph	NOUN
cana-369	70	160	g	g	PROPN
cana-369	70	161	=	=	PROPN
cana-369	70	162	spl	spl	PROPN
cana-369	70	163	(	(	PUNCT
cana-369	70	164	st	st	PROPN
cana-369	70	165	)	)	PUNCT
cana-369	70	166	with	with	ADP
cana-369	70	167	t	t	NOUN
cana-369	70	168	=	=	SYM
cana-369	70	169	8	8	NUM
cana-369	70	170	3	3	NUM
cana-369	70	171	51	51	NUM
cana-369	70	172	23	23	NUM
cana-369	70	173	5	5	NUM
cana-369	70	174	7	7	NUM
cana-369	70	175	9	9	NUM
cana-369	70	176	11	11	NUM
cana-369	70	177	13	13	NUM
cana-369	70	178	15	15	NUM
cana-369	70	179	17	17	NUM
cana-369	70	180	19	19	NUM
cana-369	70	181	figure	figure	NOUN
cana-369	70	182	4	4	NUM
cana-369	70	183	spl	spl	PROPN
cana-369	70	184	(	(	PUNCT
cana-369	70	185	s8	s8	PROPN
cana-369	70	186	)	)	PUNCT
cana-369	70	187	the	the	DET
cana-369	70	188	given	give	VERB
cana-369	70	189	g	g	PROPN
cana-369	70	190	=	=	SYM
cana-369	70	191	spl	spl	PROPN
cana-369	70	192	(	(	PUNCT
cana-369	70	193	s8	s8	PROPN
cana-369	70	194	)	)	PUNCT
cana-369	70	195	admits	admit	VERB
cana-369	70	196	odd	odd	ADJ
cana-369	70	197	-	-	PUNCT
cana-369	70	198	even	even	ADV
cana-369	70	199	congruence	congruence	NOUN
cana-369	70	200	labeling	labeling	NOUN
cana-369	70	201	and	and	CCONJ
cana-369	70	202	it	it	PRON
cana-369	70	203	depicts	depict	VERB
cana-369	70	204	in	in	ADP
cana-369	70	205	figure	figure	NOUN
cana-369	70	206	–	–	PUNCT
cana-369	70	207	4	4	NUM
cana-369	70	208	communications	communication	NOUN
cana-369	70	209	on	on	ADP
cana-369	70	210	applied	apply	VERB
cana-369	70	211	nonlinear	nonlinear	ADJ
cana-369	70	212	analysis	analysis	NOUN
cana-369	70	213	issn	issn	NOUN
cana-369	70	214	:	:	PUNCT
cana-369	70	215	1074	1074	NUM
cana-369	70	216	-	-	PUNCT
cana-369	70	217	133x	133x	NUM
cana-369	70	218	vol	vol	NOUN
cana-369	70	219	31	31	NUM
cana-369	70	220	no	no	NOUN
cana-369	70	221	.	.	NOUN
cana-369	70	222	1	1	NUM
cana-369	70	223	(	(	PUNCT
cana-369	70	224	2024	2024	NUM
cana-369	70	225	)	)	PUNCT
cana-369	70	226	146	146	NUM
cana-369	70	227	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	70	228	theorem	theorem	VERB
cana-369	70	229	3.7	3.7	NUM
cana-369	70	230	the	the	DET
cana-369	70	231	graph	graph	NOUN
cana-369	70	232	acquired	acquire	VERB
cana-369	70	233	by	by	ADP
cana-369	70	234	connecting	connect	VERB
cana-369	70	235	two	two	NUM
cana-369	70	236	copies	copy	NOUN
cana-369	70	237	of	of	ADP
cana-369	70	238	even	even	ADV
cana-369	70	239	cycle	cycle	NOUN
cana-369	70	240	cr	cr	PROPN
cana-369	70	241	by	by	ADP
cana-369	70	242	pn	pn	PROPN
cana-369	70	243	is	be	AUX
cana-369	70	244	odd	odd	ADJ
cana-369	70	245	-	-	PUNCT
cana-369	70	246	even	even	ADV
cana-369	70	247	congruence	congruence	NOUN
cana-369	70	248	graph	graph	NOUN
cana-369	70	249	.	.	PUNCT
cana-369	71	1	proof	proof	NOUN
cana-369	71	2	:	:	PUNCT
cana-369	71	3	suppose	suppose	VERB
cana-369	71	4	g	g	PROPN
cana-369	71	5	is	be	AUX
cana-369	71	6	acquired	acquire	VERB
cana-369	71	7	by	by	ADP
cana-369	71	8	connecting	connect	VERB
cana-369	71	9	two	two	NUM
cana-369	71	10	copies	copy	NOUN
cana-369	71	11	of	of	ADP
cana-369	71	12	even	even	ADV
cana-369	71	13	cycle	cycle	NOUN
cana-369	71	14	cr	cr	PROPN
cana-369	71	15	by	by	ADP
cana-369	71	16	pt	pt	PROPN
cana-369	71	17	,	,	PUNCT
cana-369	71	18	with	with	ADP
cana-369	71	19	|v	|v	PROPN
cana-369	71	20	|	|	NOUN
cana-369	72	1	=	=	SYM
cana-369	72	2	2r	2r	NUM
cana-369	73	1	+	+	CCONJ
cana-369	73	2	t	t	PROPN
cana-369	73	3	−	−	NUM
cana-369	73	4	2	2	NUM
cana-369	73	5	and	and	CCONJ
cana-369	73	6	|e|	|e|	NOUN
cana-369	73	7	=	=	SYM
cana-369	73	8	2r	2r	NUM
cana-369	74	1	+	+	CCONJ
cana-369	74	2	t	t	PROPN
cana-369	74	3	−	−	NUM
cana-369	74	4	1	1	NUM
cana-369	74	5	edges	edge	NOUN
cana-369	74	6	.	.	PUNCT
cana-369	75	1	let	let	VERB
cana-369	75	2	s1	s1	NOUN
cana-369	75	3	,	,	PUNCT
cana-369	75	4	s2	s2	PROPN
cana-369	75	5	,	,	PUNCT
cana-369	75	6	...	...	PUNCT
cana-369	75	7	sr	sr	PROPN
cana-369	75	8	,	,	PUNCT
cana-369	75	9	sr+1	sr+1	PROPN
cana-369	75	10	,	,	PUNCT
cana-369	75	11	sr+2	sr+2	NUM
cana-369	75	12	,	,	PUNCT
cana-369	75	13	…	…	NUM
cana-369	75	14	,	,	PUNCT
cana-369	75	15	s2r+t−3	s2r+t−3	NOUN
cana-369	75	16	,	,	PUNCT
cana-369	75	17	s2r+t−2	s2r+t−2	NOUN
cana-369	75	18	be	be	AUX
cana-369	75	19	the	the	DET
cana-369	75	20	vertices	vertex	NOUN
cana-369	75	21	of	of	ADP
cana-369	75	22	g.	g.	PROPN
cana-369	75	23	the	the	DET
cana-369	75	24	path	path	NOUN
cana-369	75	25	s1	s1	PROPN
cana-369	75	26	to	to	PART
cana-369	75	27	s2r+t−2	s2r+t−2	VERB
cana-369	75	28	form	form	NOUN
cana-369	75	29	a	a	DET
cana-369	75	30	spanning	span	VERB
cana-369	75	31	path	path	NOUN
cana-369	75	32	in	in	ADP
cana-369	75	33	g.	g.	PROPN
cana-369	75	34	the	the	DET
cana-369	75	35	vertex	vertex	NOUN
cana-369	75	36	sm	sm	INTJ
cana-369	75	37	and	and	CCONJ
cana-369	75	38	s[r+(t−2])+1	s[r+(t−2])+1	NOUN
cana-369	75	39	are	be	AUX
cana-369	75	40	the	the	DET
cana-369	75	41	common	common	ADJ
cana-369	75	42	vertex	vertex	NOUN
cana-369	75	43	of	of	ADP
cana-369	75	44	the	the	DET
cana-369	75	45	first	first	ADJ
cana-369	75	46	and	and	CCONJ
cana-369	75	47	second	second	ADJ
cana-369	75	48	cr	cr	PROPN
cana-369	75	49	&	&	CCONJ
cana-369	75	50	pt	pt	PROPN
cana-369	75	51	respectively	respectively	ADV
cana-369	75	52	.	.	PUNCT
cana-369	76	1	d	d	NOUN
cana-369	76	2	=	=	SYM
cana-369	76	3	min	min	PROPN
cana-369	76	4	(	(	PUNCT
cana-369	76	5	2	2	NUM
cana-369	76	6	(	(	PUNCT
cana-369	76	7	2r	2r	NUM
cana-369	77	1	+	+	NUM
cana-369	77	2	t	t	NOUN
cana-369	77	3	−	−	PROPN
cana-369	77	4	2	2	NUM
cana-369	77	5	)	)	PUNCT
cana-369	77	6	,	,	PUNCT
cana-369	77	7	2	2	NUM
cana-369	77	8	(	(	PUNCT
cana-369	77	9	2r	2r	NUM
cana-369	78	1	+	+	NUM
cana-369	78	2	t	t	NOUN
cana-369	78	3	−	−	NUM
cana-369	78	4	1	1	NUM
cana-369	78	5	)	)	PUNCT
cana-369	78	6	)	)	PUNCT
cana-369	79	1	=	=	SYM
cana-369	79	2	2	2	NUM
cana-369	79	3	(	(	PUNCT
cana-369	79	4	2r	2r	NUM
cana-369	79	5	+	+	NUM
cana-369	79	6	t	t	PROPN
cana-369	79	7	−	−	NOUN
cana-369	79	8	2	2	NUM
cana-369	79	9	)	)	PUNCT
cana-369	79	10	define	define	VERB
cana-369	79	11	h	h	NOUN
cana-369	79	12	:	:	PUNCT
cana-369	79	13	v	v	X
cana-369	79	14	(	(	PUNCT
cana-369	79	15	g	g	NOUN
cana-369	79	16	)	)	PUNCT
cana-369	79	17	→	→	SYM
cana-369	79	18	{	{	PUNCT
cana-369	79	19	1	1	NUM
cana-369	79	20	,	,	PUNCT
cana-369	79	21	3,	3,	ADJ
cana-369	79	22	…	…	NUM
cana-369	79	23	,(8r	,(8r	PUNCT
cana-369	79	24	+	+	PUNCT
cana-369	79	25	4	4	NUM
cana-369	79	26	t	t	NOUN
cana-369	79	27	−	−	NUM
cana-369	79	28	4	4	NUM
cana-369	79	29	)	)	PUNCT
cana-369	79	30	}	}	PUNCT
cana-369	79	31	as	as	ADP
cana-369	79	32	following	follow	VERB
cana-369	79	33	for	for	ADP
cana-369	79	34	1	1	NUM
cana-369	79	35	≤	≤	NOUN
cana-369	79	36	p	p	NOUN
cana-369	79	37	≤	≤	ADJ
cana-369	79	38	r	r	NOUN
cana-369	79	39	−	−	NOUN
cana-369	79	40	1	1	NUM
cana-369	79	41	ℎ(𝑠𝑝	ℎ(𝑠𝑝	PROPN
cana-369	79	42	)	)	PUNCT
cana-369	79	43	=	=	SYM
cana-369	79	44	{	{	PUNCT
cana-369	79	45	𝑝	𝑝	NOUN
cana-369	79	46	;	;	PUNCT
cana-369	79	47	𝑝	𝑝	PROPN
cana-369	79	48	𝑖𝑠	𝑖𝑠	NOUN
cana-369	79	49	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-369	80	1	𝑑	𝑑	PROPN
cana-369	80	2	+	+	ADJ
cana-369	80	3	7	7	NUM
cana-369	80	4	−	−	PROPN
cana-369	80	5	𝑝	𝑝	NOUN
cana-369	80	6	;	;	PUNCT
cana-369	80	7	𝑝	𝑝	PROPN
cana-369	80	8	𝑖𝑠	𝑖𝑠	NOUN
cana-369	80	9	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-369	80	10	for	for	ADP
cana-369	80	11	r	r	NOUN
cana-369	80	12	≤	≤	NOUN
cana-369	80	13	p	p	NOUN
cana-369	80	14	≤	≤	NOUN
cana-369	80	15	(	(	PUNCT
cana-369	80	16	3r)/2+t-1	3r)/2+t-1	NUM
cana-369	80	17	ℎ(𝑠𝑝	ℎ(𝑠𝑝	NOUN
cana-369	80	18	)	)	PUNCT
cana-369	80	19	=	=	PRON
cana-369	80	20	{	{	PUNCT
cana-369	80	21	𝑑	𝑑	PROPN
cana-369	80	22	+	+	NUM
cana-369	80	23	8	8	NUM
cana-369	80	24	−	−	PROPN
cana-369	80	25	𝑝	𝑝	NOUN
cana-369	80	26	;	;	PUNCT
cana-369	80	27	𝑝	𝑝	PROPN
cana-369	80	28	𝑖𝑠	𝑖𝑠	NOUN
cana-369	80	29	𝑜𝑑𝑑	𝑜𝑑𝑑	PROPN
cana-369	80	30	𝑝	𝑝	ADP
cana-369	81	1	+	+	ADV
cana-369	81	2	1	1	NUM
cana-369	81	3	;	;	PUNCT
cana-369	81	4	𝑝	𝑝	ADP
cana-369	81	5	𝑖𝑠	𝑖𝑠	NOUN
cana-369	81	6	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-369	81	7	for	for	ADP
cana-369	81	8	(	(	PUNCT
cana-369	81	9	3r)/2+t	3r)/2+t	NUM
cana-369	81	10	≤	≤	NOUN
cana-369	81	11	p	p	NOUN
cana-369	81	12	≤	≤	NUM
cana-369	81	13	2r+t-2	2r+t-2	NUM
cana-369	81	14	ℎ(𝑠𝑝	ℎ(𝑠𝑝	NOUN
cana-369	81	15	)	)	PUNCT
cana-369	81	16	=	=	PRON
cana-369	81	17	{	{	PUNCT
cana-369	81	18	𝑑	𝑑	PROPN
cana-369	81	19	+	+	PROPN
cana-369	81	20	6	6	NUM
cana-369	81	21	−	−	PROPN
cana-369	81	22	𝑝	𝑝	NOUN
cana-369	81	23	;	;	PUNCT
cana-369	81	24	𝑝	𝑝	PROPN
cana-369	81	25	𝑖𝑠	𝑖𝑠	NOUN
cana-369	81	26	𝑜𝑑𝑑	𝑜𝑑𝑑	PROPN
cana-369	81	27	𝑝	𝑝	NOUN
cana-369	82	1	+	+	CCONJ
cana-369	82	2	3	3	NUM
cana-369	82	3	;	;	PUNCT
cana-369	82	4	𝑝	𝑝	ADP
cana-369	82	5	𝑖𝑠	𝑖𝑠	NOUN
cana-369	83	1	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	NOUN
cana-369	83	2	then	then	ADV
cana-369	83	3	the	the	DET
cana-369	83	4	edges	edge	NOUN
cana-369	83	5	of	of	ADP
cana-369	83	6	g	g	NOUN
cana-369	83	7	are	be	AUX
cana-369	83	8	labeled	label	VERB
cana-369	83	9	as	as	ADP
cana-369	83	10	k	k	PROPN
cana-369	83	11	(	(	PUNCT
cana-369	83	12	wp	wp	X
cana-369	83	13	)	)	PUNCT
cana-369	83	14	=	=	PUNCT
cana-369	84	1	|	|	INTJ
cana-369	84	2	h	h	NOUN
cana-369	84	3	(	(	PUNCT
cana-369	84	4	sp	sp	NOUN
cana-369	84	5	)	)	PUNCT
cana-369	84	6	−	−	NOUN
cana-369	84	7	h	h	NOUN
cana-369	84	8	(	(	PUNCT
cana-369	84	9	sq	sq	ADJ
cana-369	84	10	)	)	PUNCT
cana-369	84	11	|	|	ADV
cana-369	84	12	obviously	obviously	ADV
cana-369	84	13	,	,	PUNCT
cana-369	84	14	h(sp	h(sp	X
cana-369	84	15	)	)	PUNCT
cana-369	84	16	satisfies	satisfy	VERB
cana-369	84	17	modulo	modulo	NOUN
cana-369	84	18	division	division	NOUN
cana-369	84	19	by	by	ADP
cana-369	84	20	k	k	PROPN
cana-369	84	21	(	(	PUNCT
cana-369	84	22	wp	wp	PROPN
cana-369	84	23	)	)	PUNCT
cana-369	84	24	.	.	PUNCT
cana-369	85	1	thus	thus	ADV
cana-369	85	2	,	,	PUNCT
cana-369	85	3	the	the	DET
cana-369	85	4	graph	graph	NOUN
cana-369	85	5	acquired	acquire	VERB
cana-369	85	6	by	by	ADP
cana-369	85	7	connecting	connect	VERB
cana-369	85	8	two	two	NUM
cana-369	85	9	copies	copy	NOUN
cana-369	85	10	of	of	ADP
cana-369	85	11	even	even	ADV
cana-369	85	12	cycle	cycle	NOUN
cana-369	85	13	cr	cr	NOUN
cana-369	85	14	by	by	ADP
cana-369	85	15	a	a	DET
cana-369	85	16	path	path	NOUN
cana-369	85	17	pt	pt	NOUN
cana-369	85	18	is	be	AUX
cana-369	85	19	odd	odd	ADJ
cana-369	85	20	-	-	PUNCT
cana-369	85	21	even	even	ADV
cana-369	85	22	congruence	congruence	NOUN
cana-369	85	23	graph	graph	NOUN
cana-369	85	24	.	.	PUNCT
cana-369	85	25	example	example	NOUN
cana-369	85	26	3.8	3.8	NUM
cana-369	85	27	let	let	VERB
cana-369	85	28	g	g	NOUN
cana-369	85	29	be	be	AUX
cana-369	85	30	a	a	DET
cana-369	85	31	graph	graph	NOUN
cana-369	85	32	acquired	acquire	VERB
cana-369	85	33	by	by	ADP
cana-369	85	34	connecting	connect	VERB
cana-369	85	35	two	two	NUM
cana-369	85	36	copies	copy	NOUN
cana-369	85	37	of	of	ADP
cana-369	85	38	even	even	ADV
cana-369	85	39	cycle	cycle	NOUN
cana-369	85	40	c10	c10	VERB
cana-369	85	41	by	by	ADP
cana-369	85	42	a	a	DET
cana-369	85	43	path	path	NOUN
cana-369	85	44	p5	p5	ADJ
cana-369	85	45	figure	figure	NOUN
cana-369	85	46	5	5	NUM
cana-369	85	47	figure	figure	NOUN
cana-369	85	48	5	5	NUM
cana-369	85	49	reveals	reveal	VERB
cana-369	85	50	that	that	SCONJ
cana-369	85	51	the	the	DET
cana-369	85	52	graph	graph	NOUN
cana-369	85	53	acquired	acquire	VERB
cana-369	85	54	by	by	ADP
cana-369	85	55	connecting	connect	VERB
cana-369	85	56	two	two	NUM
cana-369	85	57	copies	copy	NOUN
cana-369	85	58	of	of	ADP
cana-369	85	59	even	even	ADV
cana-369	85	60	cycle	cycle	NOUN
cana-369	85	61	c10	c10	VERB
cana-369	85	62	by	by	ADP
cana-369	85	63	a	a	DET
cana-369	85	64	path	path	NOUN
cana-369	85	65	p5	p5	NOUN
cana-369	85	66	is	be	AUX
cana-369	85	67	an	an	DET
cana-369	85	68	odd	odd	ADJ
cana-369	85	69	-	-	PUNCT
cana-369	85	70	even	even	ADV
cana-369	85	71	congruence	congruence	NOUN
cana-369	85	72	graph	graph	NOUN
cana-369	85	73	communications	communication	NOUN
cana-369	85	74	on	on	ADP
cana-369	85	75	applied	apply	VERB
cana-369	85	76	nonlinear	nonlinear	ADJ
cana-369	85	77	analysis	analysis	NOUN
cana-369	85	78	issn	issn	NOUN
cana-369	85	79	:	:	PUNCT
cana-369	85	80	1074	1074	NUM
cana-369	85	81	-	-	PUNCT
cana-369	85	82	133x	133x	NUM
cana-369	85	83	vol	vol	NOUN
cana-369	85	84	31	31	NUM
cana-369	85	85	no	no	NOUN
cana-369	85	86	.	.	NOUN
cana-369	85	87	1	1	NUM
cana-369	85	88	(	(	PUNCT
cana-369	85	89	2024	2024	NUM
cana-369	85	90	)	)	PUNCT
cana-369	85	91	147	147	NUM
cana-369	85	92	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	85	93	theorem	theorem	VERB
cana-369	85	94	3.9	3.9	NUM
cana-369	85	95	k1,t	k1,t	PROPN
cana-369	85	96	⊗	⊗	PROPN
cana-369	85	97	p2	p2	PROPN
cana-369	85	98	is	be	AUX
cana-369	85	99	odd	odd	ADJ
cana-369	85	100	-	-	PUNCT
cana-369	85	101	even	even	ADV
cana-369	85	102	congruence	congruence	NOUN
cana-369	85	103	graph	graph	NOUN
cana-369	85	104	.	.	PUNCT
cana-369	86	1	proof	proof	NOUN
cana-369	86	2	:	:	PUNCT
cana-369	86	3	the	the	DET
cana-369	86	4	tensor	tensor	NOUN
cana-369	86	5	product	product	NOUN
cana-369	86	6	of	of	ADP
cana-369	86	7	star	star	NOUN
cana-369	86	8	graph	graph	NOUN
cana-369	86	9	k1,t	k1,t	PROPN
cana-369	86	10	and	and	CCONJ
cana-369	86	11	path	path	NOUN
cana-369	86	12	p2	p2	PROPN
cana-369	86	13	is	be	AUX
cana-369	86	14	denoted	denote	VERB
cana-369	86	15	as	as	ADP
cana-369	86	16	g	g	PROPN
cana-369	86	17	=	=	PROPN
cana-369	86	18	k1,t	k1,t	PROPN
cana-369	86	19	⊗	⊗	PROPN
cana-369	86	20	p2	p2	PROPN
cana-369	86	21	with	with	ADP
cana-369	86	22	|v	|v	PROPN
cana-369	86	23	|	|	NOUN
cana-369	86	24	=	=	SYM
cana-369	86	25	2	2	NUM
cana-369	86	26	t	t	NOUN
cana-369	86	27	+	+	CCONJ
cana-369	86	28	2	2	NUM
cana-369	86	29	and	and	CCONJ
cana-369	86	30	|e|	|e|	NOUN
cana-369	86	31	=	=	SYM
cana-369	86	32	2	2	NUM
cana-369	86	33	t.	t.	NOUN
cana-369	86	34	let	let	VERB
cana-369	86	35	s1	s1	NOUN
cana-369	86	36	,	,	PUNCT
cana-369	86	37	s2,	s2,	NOUN
cana-369	86	38	…	…	SYM
cana-369	86	39	..	..	SYM
cana-369	86	40	st+1	st+1	NUM
cana-369	86	41	are	be	AUX
cana-369	86	42	vertex	vertex	NOUN
cana-369	86	43	set	set	NOUN
cana-369	86	44	of	of	ADP
cana-369	86	45	k1,t	k1,t	PROPN
cana-369	86	46	,	,	PUNCT
cana-369	86	47	s1	s1	PROPN
cana-369	86	48	is	be	AUX
cana-369	86	49	apex	apex	NOUN
cana-369	86	50	vertex	vertex	NOUN
cana-369	86	51	and	and	CCONJ
cana-369	86	52	f1	f1	NOUN
cana-369	86	53	,	,	PUNCT
cana-369	86	54	f2	f2	PROPN
cana-369	86	55	is	be	AUX
cana-369	86	56	vertex	vertex	NOUN
cana-369	86	57	set	set	NOUN
cana-369	86	58	of	of	ADP
cana-369	86	59	p2	p2	PROPN
cana-369	86	60	.	.	PUNCT
cana-369	87	1	v	v	X
cana-369	87	2	(	(	PUNCT
cana-369	87	3	g	g	NOUN
cana-369	87	4	)	)	PUNCT
cana-369	87	5	=	=	SYM
cana-369	87	6	(	(	PUNCT
cana-369	87	7	s1	s1	NOUN
cana-369	87	8	,	,	PUNCT
cana-369	87	9	f1),(s2	f1),(s2	NOUN
cana-369	87	10	,	,	PUNCT
cana-369	87	11	f1	f1	NOUN
cana-369	87	12	)	)	PUNCT
cana-369	87	13	,	,	PUNCT
cana-369	87	14	......	......	PUNCT
cana-369	87	15	,	,	PUNCT
cana-369	87	16	(	(	PUNCT
cana-369	87	17	st+1	st+1	NOUN
cana-369	87	18	,	,	PUNCT
cana-369	87	19	f1),(s1	f1),(s1	ADJ
cana-369	87	20	,	,	PUNCT
cana-369	87	21	f2),(s2	f2),(s2	PROPN
cana-369	87	22	,	,	PUNCT
cana-369	87	23	f2	f2	PROPN
cana-369	87	24	)	)	PUNCT
cana-369	87	25	,	,	PUNCT
cana-369	87	26	.......	.......	PUNCT
cana-369	87	27	,	,	PUNCT
cana-369	87	28	(	(	PUNCT
cana-369	87	29	st+1	st+1	NOUN
cana-369	87	30	,	,	PUNCT
cana-369	87	31	f2	f2	PROPN
cana-369	87	32	)	)	PUNCT
cana-369	87	33	e	e	NOUN
cana-369	87	34	(	(	PUNCT
cana-369	87	35	g	g	NOUN
cana-369	87	36	)	)	PUNCT
cana-369	87	37	=	=	SYM
cana-369	87	38	e1	e1	PROPN
cana-369	87	39	,	,	PUNCT
cana-369	87	40	e2	e2	PROPN
cana-369	87	41	,	,	PUNCT
cana-369	87	42	.....	.....	PUNCT
cana-369	87	43	,	,	PUNCT
cana-369	87	44	et	et	NOUN
cana-369	87	45	,	,	PUNCT
cana-369	87	46	et+1	et+1	PROPN
cana-369	87	47	,	,	PUNCT
cana-369	87	48	.....	.....	PUNCT
cana-369	87	49	,	,	PUNCT
cana-369	87	50	e2	e2	PROPN
cana-369	87	51	t	t	PROPN
cana-369	87	52	where	where	SCONJ
cana-369	87	53	,	,	PUNCT
cana-369	87	54	w1	w1	NOUN
cana-369	87	55	,	,	PUNCT
cana-369	87	56	w2	w2	NOUN
cana-369	87	57	,	,	PUNCT
cana-369	87	58	.......	.......	PUNCT
cana-369	87	59	,	,	PUNCT
cana-369	87	60	wn	wn	PROPN
cana-369	87	61	are	be	AUX
cana-369	87	62	the	the	DET
cana-369	87	63	edges	edge	NOUN
cana-369	87	64	adjacent	adjacent	ADJ
cana-369	87	65	with	with	ADP
cana-369	87	66	the	the	DET
cana-369	87	67	vertex	vertex	NOUN
cana-369	87	68	(	(	PUNCT
cana-369	87	69	sp	sp	NOUN
cana-369	87	70	,	,	PUNCT
cana-369	87	71	f1	f1	NOUN
cana-369	87	72	)	)	PUNCT
cana-369	87	73	and	and	CCONJ
cana-369	87	74	wt+1	wt+1	PROPN
cana-369	87	75	,	,	PUNCT
cana-369	87	76	wt+2	wt+2	PROPN
cana-369	87	77	,	,	PUNCT
cana-369	87	78	........	........	PUNCT
cana-369	87	79	,	,	PUNCT
cana-369	87	80	w2	w2	PROPN
cana-369	87	81	t	t	PROPN
cana-369	87	82	are	be	AUX
cana-369	87	83	the	the	DET
cana-369	87	84	edges	edge	NOUN
cana-369	87	85	adjacent	adjacent	ADJ
cana-369	87	86	with	with	ADP
cana-369	87	87	the	the	DET
cana-369	87	88	vertex	vertex	NOUN
cana-369	87	89	(	(	PUNCT
cana-369	87	90	sq	sq	ADJ
cana-369	87	91	,	,	PUNCT
cana-369	87	92	f2	f2	PROPN
cana-369	87	93	)	)	PUNCT
cana-369	87	94	d	d	PROPN
cana-369	87	95	=	=	SYM
cana-369	87	96	min	min	PROPN
cana-369	87	97	(	(	PUNCT
cana-369	87	98	2	2	NUM
cana-369	87	99	(	(	PUNCT
cana-369	87	100	2	2	NUM
cana-369	87	101	t	t	NOUN
cana-369	87	102	+	+	NOUN
cana-369	87	103	2	2	NUM
cana-369	87	104	)	)	PUNCT
cana-369	87	105	,	,	PUNCT
cana-369	87	106	2	2	NUM
cana-369	87	107	(	(	PUNCT
cana-369	87	108	2	2	NUM
cana-369	87	109	t	t	NOUN
cana-369	87	110	)	)	PUNCT
cana-369	87	111	)	)	PUNCT
cana-369	88	1	=	=	PUNCT
cana-369	88	2	4	4	NUM
cana-369	88	3	t	t	NOUN
cana-369	88	4	label	label	NOUN
cana-369	88	5	the	the	DET
cana-369	88	6	vertices	vertex	NOUN
cana-369	88	7	h	h	NOUN
cana-369	88	8	:	:	PUNCT
cana-369	88	9	v	v	X
cana-369	88	10	(	(	PUNCT
cana-369	88	11	g	g	NOUN
cana-369	88	12	)	)	PUNCT
cana-369	88	13	→	→	SYM
cana-369	88	14	{	{	PUNCT
cana-369	88	15	1	1	NUM
cana-369	88	16	,	,	PUNCT
cana-369	88	17	3	3	NUM
cana-369	88	18	,	,	PUNCT
cana-369	88	19	......	......	PUNCT
cana-369	88	20	8	8	NUM
cana-369	88	21	t	t	NOUN
cana-369	88	22	+	+	CCONJ
cana-369	88	23	1	1	NUM
cana-369	88	24	}	}	PUNCT
cana-369	88	25	and	and	CCONJ
cana-369	88	26	edges	edge	VERB
cana-369	88	27	k	k	X
cana-369	88	28	:	:	PUNCT
cana-369	88	29	e	e	X
cana-369	88	30	(	(	PUNCT
cana-369	88	31	g	g	NOUN
cana-369	88	32	)	)	PUNCT
cana-369	88	33	→	→	SYM
cana-369	88	34	{	{	PUNCT
cana-369	88	35	2	2	NUM
cana-369	88	36	,	,	PUNCT
cana-369	88	37	4,	4,	ADJ
cana-369	88	38	…	…	SYM
cana-369	88	39	.,8	.,8	NOUN
cana-369	88	40	t	t	PROPN
cana-369	88	41	}	}	PUNCT
cana-369	88	42	are	be	AUX
cana-369	88	43	labeled	label	VERB
cana-369	88	44	in	in	ADP
cana-369	88	45	the	the	DET
cana-369	88	46	following	following	ADJ
cana-369	88	47	way	way	NOUN
cana-369	88	48	h	h	PROPN
cana-369	88	49	(	(	PUNCT
cana-369	88	50	sp	sp	NOUN
cana-369	88	51	,	,	PUNCT
cana-369	88	52	f1	f1	NOUN
cana-369	88	53	)	)	PUNCT
cana-369	88	54	=	=	SYM
cana-369	89	1	2p	2p	NUM
cana-369	89	2	−	−	NOUN
cana-369	89	3	1	1	NUM
cana-369	89	4	for	for	ADP
cana-369	89	5	1	1	NUM
cana-369	89	6	≤	≤	NOUN
cana-369	89	7	p	p	PROPN
cana-369	89	8	≤	≤	PROPN
cana-369	89	9	t	t	NOUN
cana-369	89	10	+	+	CCONJ
cana-369	89	11	1	1	NUM
cana-369	89	12	h	h	NOUN
cana-369	89	13	(	(	PUNCT
cana-369	89	14	sq	sq	PROPN
cana-369	89	15	,	,	PUNCT
cana-369	89	16	f2	f2	PROPN
cana-369	89	17	)	)	PUNCT
cana-369	89	18	=	=	SYM
cana-369	89	19	2	2	NUM
cana-369	89	20	(	(	PUNCT
cana-369	89	21	t	t	NOUN
cana-369	89	22	+	+	CCONJ
cana-369	89	23	q	q	X
cana-369	89	24	)	)	PUNCT
cana-369	89	25	+	+	CCONJ
cana-369	89	26	1	1	NUM
cana-369	89	27	for	for	ADP
cana-369	89	28	1	1	NUM
cana-369	89	29	≤	≤	NUM
cana-369	89	30	q	q	PROPN
cana-369	89	31	≤	≤	PROPN
cana-369	89	32	t	t	NOUN
cana-369	89	33	+	+	CCONJ
cana-369	89	34	1	1	NUM
cana-369	89	35	𝑘(𝑤𝑝	𝑘(𝑤𝑝	NOUN
cana-369	89	36	)	)	PUNCT
cana-369	89	37	=	=	PRON
cana-369	89	38	{	{	PUNCT
cana-369	89	39	3𝑡	3𝑡	NOUN
cana-369	89	40	+	+	CCONJ
cana-369	89	41	2(𝑝	2(𝑝	NUM
cana-369	89	42	−	−	NUM
cana-369	89	43	1	1	NUM
cana-369	89	44	)	)	PUNCT
cana-369	89	45	;	;	PUNCT
cana-369	89	46	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-369	89	47	𝑝	𝑝	X
cana-369	89	48	=	=	SYM
cana-369	89	49	1	1	NUM
cana-369	89	50	𝑡𝑜	𝑡𝑜	PROPN
cana-369	89	51	𝑡	𝑡	VERB
cana-369	89	52	4𝑡	4𝑡	NUM
cana-369	89	53	−	−	NOUN
cana-369	89	54	2𝑝	2𝑝	NOUN
cana-369	89	55	+	+	CCONJ
cana-369	89	56	2	2	NUM
cana-369	89	57	;	;	PUNCT
cana-369	89	58	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-369	89	59	𝑝	𝑝	X
cana-369	89	60	=	=	SYM
cana-369	89	61	𝑡	𝑡	PROPN
cana-369	89	62	+	+	NOUN
cana-369	89	63	1	1	NUM
cana-369	89	64	𝑡𝑜	𝑡𝑜	NOUN
cana-369	89	65	2𝑡	2𝑡	NOUN
cana-369	89	66	evidently	evidently	ADV
cana-369	89	67	,	,	PUNCT
cana-369	89	68	2	2	NUM
cana-369	89	69	(	(	PUNCT
cana-369	89	70	t	t	NOUN
cana-369	89	71	+	+	CCONJ
cana-369	89	72	q	q	X
cana-369	89	73	)	)	PUNCT
cana-369	90	1	+	+	CCONJ
cana-369	90	2	1	1	NUM
cana-369	90	3	–	–	PUNCT
cana-369	90	4	2p	2p	NUM
cana-369	90	5	+	+	SYM
cana-369	90	6	1	1	NUM
cana-369	90	7	≡	≡	PROPN
cana-369	90	8	(	(	PUNCT
cana-369	90	9	mod	mod	X
cana-369	90	10	(	(	PUNCT
cana-369	90	11	3	3	NUM
cana-369	90	12	t	t	NOUN
cana-369	90	13	+	+	NUM
cana-369	90	14	2p	2p	NUM
cana-369	90	15	−	−	NOUN
cana-369	90	16	2	2	NUM
cana-369	90	17	)	)	PUNCT
cana-369	90	18	)	)	PUNCT
cana-369	90	19	i.e	i.e	X
cana-369	90	20	,	,	PUNCT
cana-369	90	21	(	(	PUNCT
cana-369	90	22	3	3	NUM
cana-369	90	23	t	t	NOUN
cana-369	90	24	+	+	NUM
cana-369	90	25	2p	2p	NUM
cana-369	90	26	−	−	NOUN
cana-369	90	27	2	2	NUM
cana-369	90	28	)	)	PUNCT
cana-369	90	29	divides	divide	NOUN
cana-369	90	30	(	(	PUNCT
cana-369	90	31	2	2	NUM
cana-369	90	32	(	(	PUNCT
cana-369	90	33	t	t	NOUN
cana-369	90	34	+	+	CCONJ
cana-369	90	35	q	q	NOUN
cana-369	90	36	)	)	PUNCT
cana-369	90	37	–	–	PUNCT
cana-369	90	38	2p	2p	NUM
cana-369	90	39	+	+	CCONJ
cana-369	90	40	2	2	NUM
cana-369	90	41	)	)	PUNCT
cana-369	90	42	hence	hence	ADV
cana-369	90	43	,	,	PUNCT
cana-369	90	44	k1,t	k1,t	PROPN
cana-369	90	45	⊗	⊗	PROPN
cana-369	90	46	p2	p2	PROPN
cana-369	90	47	was	be	AUX
cana-369	90	48	an	an	DET
cana-369	90	49	odd	odd	ADJ
cana-369	90	50	-	-	PUNCT
cana-369	90	51	even	even	ADV
cana-369	90	52	congruence	congruence	NOUN
cana-369	90	53	graph	graph	NOUN
cana-369	90	54	.	.	PUNCT
cana-369	90	55	example	example	NOUN
cana-369	90	56	3.10	3.10	NUM
cana-369	90	57	let	let	VERB
cana-369	90	58	g	g	NOUN
cana-369	90	59	=	=	PROPN
cana-369	90	60	k1,4	k1,4	PROPN
cana-369	90	61	⊗	⊗	PROPN
cana-369	90	62	p2	p2	PROPN
cana-369	90	63	,	,	PUNCT
cana-369	90	64	t	t	NOUN
cana-369	90	65	=	=	SYM
cana-369	90	66	4	4	NUM
cana-369	90	67	1	1	NUM
cana-369	90	68	3	3	NUM
cana-369	90	69	5	5	NUM
cana-369	90	70	7	7	NUM
cana-369	90	71	9	9	NUM
cana-369	90	72	11	11	NUM
cana-369	90	73	13	13	NUM
cana-369	90	74	15	15	NUM
cana-369	90	75	17	17	NUM
cana-369	90	76	19	19	NUM
cana-369	90	77	figure	figure	NOUN
cana-369	90	78	6	6	NUM
cana-369	90	79	k1,4	k1,4	ADV
cana-369	90	80	⊗	⊗	PROPN
cana-369	90	81	p2	p2	PROPN
cana-369	90	82	figure	figure	NOUN
cana-369	90	83	6	6	NUM
cana-369	90	84	represents	represent	VERB
cana-369	90	85	the	the	DET
cana-369	90	86	odd	odd	ADV
cana-369	90	87	-	-	PUNCT
cana-369	90	88	even	even	ADV
cana-369	90	89	congruence	congruence	NOUN
cana-369	90	90	labeling	labeling	NOUN
cana-369	90	91	of	of	ADP
cana-369	90	92	k1,4	k1,4	PROPN
cana-369	90	93	⊗	⊗	PROPN
cana-369	90	94	p2	p2	PROPN
cana-369	90	95	theorem	theorem	VERB
cana-369	90	96	3.11	3.11	NUM
cana-369	90	97	the	the	DET
cana-369	90	98	graph	graph	NOUN
cana-369	90	99	d2	d2	PROPN
cana-369	90	100	(	(	PUNCT
cana-369	90	101	pt	pt	NOUN
cana-369	90	102	)	)	PUNCT
cana-369	90	103	is	be	AUX
cana-369	90	104	odd	odd	ADJ
cana-369	90	105	-	-	PUNCT
cana-369	90	106	even	even	ADV
cana-369	90	107	congruence	congruence	NOUN
cana-369	90	108	graph	graph	NOUN
cana-369	90	109	.	.	PUNCT
cana-369	91	1	proof	proof	NOUN
cana-369	91	2	:	:	PUNCT
cana-369	91	3	suppose	suppose	VERB
cana-369	91	4	g	g	PROPN
cana-369	91	5	=	=	PROPN
cana-369	91	6	d2	d2	PROPN
cana-369	91	7	(	(	PUNCT
cana-369	91	8	pt	pt	NOUN
cana-369	91	9	)	)	PUNCT
cana-369	91	10	is	be	AUX
cana-369	91	11	the	the	DET
cana-369	91	12	shadow	shadow	NOUN
cana-369	91	13	graph	graph	NOUN
cana-369	91	14	of	of	ADP
cana-369	91	15	the	the	DET
cana-369	91	16	path	path	NOUN
cana-369	91	17	pt	pt	NOUN
cana-369	91	18	with	with	ADP
cana-369	91	19	|v	|v	PROPN
cana-369	92	1	|	|	NOUN
cana-369	92	2	=	=	SYM
cana-369	92	3	2	2	NUM
cana-369	92	4	t	t	NOUN
cana-369	92	5	and	and	CCONJ
cana-369	92	6	|e|	|e|	NOUN
cana-369	92	7	=	=	SYM
cana-369	92	8	4	4	NUM
cana-369	92	9	(	(	PUNCT
cana-369	92	10	t	t	NOUN
cana-369	92	11	−	−	PROPN
cana-369	92	12	1	1	NUM
cana-369	92	13	)	)	PUNCT
cana-369	92	14	.	.	PUNCT
cana-369	93	1	let	let	VERB
cana-369	93	2	s1	s1	NOUN
cana-369	93	3	,	,	PUNCT
cana-369	93	4	s2	s2	PROPN
cana-369	93	5	,	,	PUNCT
cana-369	93	6	....	....	PUNCT
cana-369	93	7	st	st	PROPN
cana-369	93	8	be	be	AUX
cana-369	93	9	the	the	DET
cana-369	93	10	vertices	vertex	NOUN
cana-369	93	11	of	of	ADP
cana-369	93	12	first	first	ADJ
cana-369	93	13	pt	pt	PROPN
cana-369	93	14	and	and	CCONJ
cana-369	93	15	f1	f1	NOUN
cana-369	93	16	,	,	PUNCT
cana-369	93	17	f2,	f2,	ADJ
cana-369	93	18	…	…	SYM
cana-369	93	19	.,ft	.,ft	NUM
cana-369	93	20	are	be	AUX
cana-369	93	21	vertices	vertex	NOUN
cana-369	93	22	of	of	ADP
cana-369	93	23	second	second	ADJ
cana-369	93	24	path	path	NOUN
cana-369	93	25	pt	pt	NOUN
cana-369	93	26	.	.	PROPN
cana-369	93	27	here	here	ADV
cana-369	94	1	,	,	PUNCT
cana-369	94	2	d	d	PROPN
cana-369	94	3	=	=	SYM
cana-369	94	4	min	min	PROPN
cana-369	94	5	(	(	PUNCT
cana-369	94	6	2	2	NUM
cana-369	94	7	(	(	PUNCT
cana-369	94	8	2	2	NUM
cana-369	94	9	t	t	NOUN
cana-369	94	10	)	)	PUNCT
cana-369	94	11	,	,	PUNCT
cana-369	94	12	2	2	NUM
cana-369	94	13	(	(	PUNCT
cana-369	94	14	4	4	NUM
cana-369	94	15	(	(	PUNCT
cana-369	94	16	t	t	NOUN
cana-369	94	17	−	−	PROPN
cana-369	94	18	1	1	NUM
cana-369	94	19	)	)	PUNCT
cana-369	94	20	)	)	PUNCT
cana-369	94	21	)	)	PUNCT
cana-369	95	1	12	12	NUM
cana-369	95	2	14	14	NUM
cana-369	95	3	16	16	NUM
cana-369	95	4	18	18	NUM
cana-369	95	5	communications	communication	NOUN
cana-369	95	6	on	on	ADP
cana-369	95	7	applied	apply	VERB
cana-369	95	8	nonlinear	nonlinear	ADJ
cana-369	95	9	analysis	analysis	NOUN
cana-369	95	10	issn	issn	NOUN
cana-369	95	11	:	:	PUNCT
cana-369	95	12	1074	1074	NUM
cana-369	95	13	-	-	PUNCT
cana-369	95	14	133x	133x	NUM
cana-369	95	15	vol	vol	NOUN
cana-369	95	16	31	31	NUM
cana-369	95	17	no	no	NOUN
cana-369	95	18	.	.	NOUN
cana-369	95	19	1	1	NUM
cana-369	95	20	(	(	PUNCT
cana-369	95	21	2024	2024	NUM
cana-369	95	22	)	)	PUNCT
cana-369	95	23	148	148	NUM
cana-369	95	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	95	25	=	=	SYM
cana-369	95	26	4	4	NUM
cana-369	95	27	t	t	NOUN
cana-369	95	28	the	the	DET
cana-369	95	29	vertices	vertex	NOUN
cana-369	95	30	h	h	NOUN
cana-369	95	31	:	:	PUNCT
cana-369	96	1	v	v	X
cana-369	96	2	(	(	PUNCT
cana-369	96	3	g	g	NOUN
cana-369	96	4	)	)	PUNCT
cana-369	96	5	→	→	SYM
cana-369	96	6	{	{	PUNCT
cana-369	96	7	1	1	NUM
cana-369	96	8	,	,	PUNCT
cana-369	96	9	3	3	NUM
cana-369	96	10	,	,	PUNCT
cana-369	96	11	......	......	PUNCT
cana-369	96	12	8	8	NUM
cana-369	96	13	t	t	NOUN
cana-369	96	14	+	+	CCONJ
cana-369	96	15	1	1	NUM
cana-369	96	16	}	}	PUNCT
cana-369	96	17	and	and	CCONJ
cana-369	96	18	edges	edge	VERB
cana-369	96	19	k	k	X
cana-369	96	20	:	:	PUNCT
cana-369	96	21	e	e	X
cana-369	96	22	(	(	PUNCT
cana-369	96	23	g	g	NOUN
cana-369	96	24	)	)	PUNCT
cana-369	96	25	→	→	SYM
cana-369	96	26	{	{	PUNCT
cana-369	96	27	2	2	NUM
cana-369	96	28	,	,	PUNCT
cana-369	96	29	4,	4,	ADJ
cana-369	96	30	…	…	SYM
cana-369	96	31	…	…	SYM
cana-369	96	32	8	8	NUM
cana-369	96	33	t	t	PROPN
cana-369	96	34	}	}	PUNCT
cana-369	96	35	are	be	AUX
cana-369	96	36	labeled	label	VERB
cana-369	96	37	as	as	SCONJ
cana-369	96	38	given	give	VERB
cana-369	96	39	below	below	ADP
cana-369	96	40	ℎ(𝑠𝑝	ℎ(𝑠𝑝	NOUN
cana-369	96	41	)	)	PUNCT
cana-369	96	42	=	=	PRON
cana-369	96	43	{	{	PUNCT
cana-369	96	44	4𝑝	4𝑝	PROPN
cana-369	97	1	−	−	PROPN
cana-369	97	2	3	3	NUM
cana-369	97	3	;	;	PUNCT
cana-369	97	4	𝑝	𝑝	PROPN
cana-369	97	5	𝑖𝑠	𝑖𝑠	NOUN
cana-369	97	6	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-369	97	7	8𝑡	8𝑡	NUM
cana-369	97	8	−	−	NOUN
cana-369	98	1	4𝑝	4𝑝	PROPN
cana-369	99	1	+	+	CCONJ
cana-369	99	2	1	1	NUM
cana-369	99	3	;	;	PUNCT
cana-369	99	4	𝑝	𝑝	NOUN
cana-369	99	5	𝑖𝑠	𝑖𝑠	NOUN
cana-369	99	6	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-369	99	7	ℎ(𝑓𝑝	ℎ(𝑓𝑝	NOUN
cana-369	99	8	)	)	PUNCT
cana-369	99	9	=	=	PRON
cana-369	99	10	{	{	PUNCT
cana-369	99	11	4𝑝	4𝑝	PROPN
cana-369	99	12	−	−	PROPN
cana-369	99	13	1	1	NUM
cana-369	99	14	;	;	PUNCT
cana-369	99	15	𝑝	𝑝	PROPN
cana-369	99	16	𝑖𝑠	𝑖𝑠	NOUN
cana-369	99	17	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-369	99	18	8𝑡	8𝑡	NUM
cana-369	99	19	−	−	NOUN
cana-369	99	20	4𝑝	4𝑝	PROPN
cana-369	99	21	−	−	PROPN
cana-369	99	22	3	3	NUM
cana-369	99	23	;	;	PUNCT
cana-369	99	24	𝑝	𝑝	ADP
cana-369	99	25	𝑖𝑠	𝑖𝑠	NOUN
cana-369	99	26	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	VERB
cana-369	99	27	the	the	DET
cana-369	99	28	edge	edge	NOUN
cana-369	99	29	wp	wp	NOUN
cana-369	100	1	=	=	PUNCT
cana-369	101	1	sf	sf	NOUN
cana-369	101	2	are	be	AUX
cana-369	101	3	labeled	label	VERB
cana-369	101	4	as	as	SCONJ
cana-369	101	5	follows	follow	VERB
cana-369	101	6	k	k	PROPN
cana-369	101	7	(	(	PUNCT
cana-369	101	8	wp	wp	X
cana-369	101	9	)	)	PUNCT
cana-369	101	10	=	=	PUNCT
cana-369	102	1	|	|	INTJ
cana-369	102	2	h	h	NOUN
cana-369	102	3	(	(	PUNCT
cana-369	102	4	s	s	NOUN
cana-369	102	5	)	)	PUNCT
cana-369	102	6	−	−	PROPN
cana-369	102	7	h	h	NOUN
cana-369	102	8	(	(	PUNCT
cana-369	102	9	f)|	f)|	VERB
cana-369	102	10	apparently	apparently	ADV
cana-369	102	11	,	,	PUNCT
cana-369	102	12	the	the	DET
cana-369	102	13	vertex	vertex	NOUN
cana-369	102	14	label	label	NOUN
cana-369	102	15	satisfies	satisfy	VERB
cana-369	102	16	modulo	modulo	NOUN
cana-369	102	17	division	division	NOUN
cana-369	102	18	by	by	ADP
cana-369	102	19	its	its	PRON
cana-369	102	20	corresponding	corresponding	ADJ
cana-369	102	21	edge	edge	NOUN
cana-369	102	22	label	label	NOUN
cana-369	102	23	.	.	PUNCT
cana-369	103	1	hence	hence	ADV
cana-369	103	2	d2	d2	PROPN
cana-369	103	3	(	(	PUNCT
cana-369	103	4	pt	pt	NOUN
cana-369	103	5	)	)	PUNCT
cana-369	103	6	is	be	AUX
cana-369	103	7	odd	odd	ADJ
cana-369	103	8	-	-	PUNCT
cana-369	103	9	even	even	ADV
cana-369	103	10	congruence	congruence	NOUN
cana-369	103	11	graph	graph	NOUN
cana-369	103	12	.	.	PUNCT
cana-369	103	13	example	example	NOUN
cana-369	103	14	3.12	3.12	NUM
cana-369	103	15	suppose	suppose	VERB
cana-369	103	16	g	g	PROPN
cana-369	103	17	=	=	PROPN
cana-369	103	18	d2	d2	PROPN
cana-369	103	19	(	(	PUNCT
cana-369	103	20	p6	p6	PROPN
cana-369	103	21	)	)	PUNCT
cana-369	103	22	with	with	ADP
cana-369	103	23	t	t	NOUN
cana-369	103	24	=	=	SYM
cana-369	103	25	6	6	NUM
cana-369	103	26	figure	figure	NOUN
cana-369	103	27	7	7	NUM
cana-369	103	28	d2	d2	PROPN
cana-369	103	29	(	(	PUNCT
cana-369	103	30	p6	p6	PROPN
cana-369	103	31	)	)	PUNCT
cana-369	103	32	odd	odd	ADV
cana-369	103	33	-	-	PUNCT
cana-369	103	34	even	even	ADV
cana-369	103	35	congruence	congruence	NOUN
cana-369	103	36	labeling	labeling	NOUN
cana-369	103	37	of	of	ADP
cana-369	103	38	the	the	DET
cana-369	103	39	shadow	shadow	NOUN
cana-369	103	40	graph	graph	NOUN
cana-369	103	41	d2	d2	PROPN
cana-369	103	42	(	(	PUNCT
cana-369	103	43	p6	p6	PROPN
cana-369	103	44	)	)	PUNCT
cana-369	103	45	is	be	AUX
cana-369	103	46	exposed	expose	VERB
cana-369	103	47	in	in	ADP
cana-369	103	48	figure	figure	NOUN
cana-369	103	49	–	–	PUNCT
cana-369	103	50	7	7	NUM
cana-369	103	51	4	4	NUM
cana-369	103	52	.	.	PUNCT
cana-369	104	1	conclusion	conclusion	NOUN
cana-369	104	2	labeling	labeling	NOUN
cana-369	104	3	in	in	ADP
cana-369	104	4	graph	graph	NOUN
cana-369	104	5	theory	theory	NOUN
cana-369	104	6	has	have	AUX
cana-369	104	7	paid	pay	VERB
cana-369	104	8	more	more	ADJ
cana-369	104	9	attention	attention	NOUN
cana-369	104	10	for	for	ADP
cana-369	104	11	many	many	ADJ
cana-369	104	12	researchers	researcher	NOUN
cana-369	104	13	.	.	PUNCT
cana-369	105	1	new	new	ADJ
cana-369	105	2	concept	concept	NOUN
cana-369	105	3	of	of	ADP
cana-369	105	4	labeling	labeling	NOUN
cana-369	105	5	such	such	ADJ
cana-369	105	6	as	as	ADP
cana-369	105	7	odd	odd	ADJ
cana-369	105	8	-	-	PUNCT
cana-369	105	9	even	even	ADV
cana-369	105	10	congruence	congruence	NOUN
cana-369	105	11	labeling	labeling	NOUN
cana-369	105	12	based	base	VERB
cana-369	105	13	on	on	ADP
cana-369	105	14	modulo	modulo	NOUN
cana-369	105	15	division	division	NOUN
cana-369	105	16	has	have	AUX
cana-369	105	17	been	be	AUX
cana-369	105	18	defined	define	VERB
cana-369	105	19	.	.	PUNCT
cana-369	106	1	this	this	DET
cana-369	106	2	paper	paper	NOUN
cana-369	106	3	examines	examine	VERB
cana-369	106	4	the	the	DET
cana-369	106	5	existence	existence	NOUN
cana-369	106	6	of	of	ADP
cana-369	106	7	odd	odd	ADV
cana-369	106	8	-	-	PUNCT
cana-369	106	9	even	even	ADV
cana-369	106	10	congruence	congruence	NOUN
cana-369	106	11	labeling	labeling	NOUN
cana-369	106	12	for	for	ADP
cana-369	106	13	complete	complete	ADJ
cana-369	106	14	bipartite	bipartite	NOUN
cana-369	106	15	graph	graph	NOUN
cana-369	106	16	,	,	PUNCT
cana-369	106	17	comb	comb	NOUN
cana-369	106	18	graph	graph	NOUN
cana-369	106	19	,	,	PUNCT
cana-369	106	20	spliting	split	VERB
cana-369	106	21	graph	graph	NOUN
cana-369	106	22	of	of	ADP
cana-369	106	23	a	a	DET
cana-369	106	24	star	star	NOUN
cana-369	106	25	graph	graph	NOUN
cana-369	106	26	and	and	CCONJ
cana-369	106	27	graph	graph	NOUN
cana-369	106	28	acquired	acquire	VERB
cana-369	106	29	by	by	ADP
cana-369	106	30	connecting	connect	VERB
cana-369	106	31	two	two	NUM
cana-369	106	32	copies	copy	NOUN
cana-369	106	33	of	of	ADP
cana-369	106	34	even	even	ADV
cana-369	106	35	cycle	cycle	NOUN
cana-369	106	36	cr	cr	NOUN
cana-369	106	37	by	by	ADP
cana-369	106	38	a	a	DET
cana-369	106	39	path	path	NOUN
cana-369	106	40	pt	pt	NOUN
cana-369	106	41	were	be	AUX
cana-369	106	42	proved	prove	VERB
cana-369	106	43	.	.	PUNCT
cana-369	107	1	also	also	ADV
cana-369	107	2	,	,	PUNCT
cana-369	107	3	it	it	PRON
cana-369	107	4	is	be	AUX
cana-369	107	5	proved	prove	VERB
cana-369	107	6	that	that	SCONJ
cana-369	107	7	k1,t	k1,t	PROPN
cana-369	107	8	⊗	⊗	PROPN
cana-369	107	9	p2	p2	PROPN
cana-369	107	10	and	and	CCONJ
cana-369	107	11	d2	d2	PROPN
cana-369	107	12	(	(	PUNCT
cana-369	107	13	pt	pt	NOUN
cana-369	107	14	)	)	PUNCT
cana-369	107	15	admits	admit	VERB
cana-369	107	16	odd	odd	ADJ
cana-369	107	17	-	-	PUNCT
cana-369	107	18	even	even	ADV
cana-369	107	19	congruence	congruence	ADJ
cana-369	107	20	labeling	labeling	NOUN
cana-369	107	21	.	.	PUNCT
cana-369	108	1	the	the	DET
cana-369	108	2	odd	odd	ADV
cana-369	108	3	-	-	PUNCT
cana-369	108	4	even	even	ADV
cana-369	108	5	congruence	congruence	NOUN
cana-369	108	6	labeling	labeling	NOUN
cana-369	108	7	is	be	AUX
cana-369	108	8	open	open	ADJ
cana-369	108	9	to	to	PART
cana-369	108	10	investigate	investigate	VERB
cana-369	108	11	for	for	ADP
cana-369	108	12	some	some	DET
cana-369	108	13	other	other	ADJ
cana-369	108	14	family	family	NOUN
cana-369	108	15	of	of	ADP
cana-369	108	16	graphs	graph	NOUN
cana-369	108	17	and	and	CCONJ
cana-369	108	18	can	can	AUX
cana-369	108	19	be	be	AUX
cana-369	108	20	applied	apply	VERB
cana-369	108	21	in	in	ADP
cana-369	108	22	communication	communication	NOUN
cana-369	108	23	networks	network	NOUN
cana-369	108	24	to	to	PART
cana-369	108	25	gaurd	gaurd	VERB
cana-369	108	26	the	the	DET
cana-369	108	27	informations	information	NOUN
cana-369	108	28	.	.	PUNCT
cana-369	109	1	references	reference	NOUN
cana-369	109	2	[	[	X
cana-369	109	3	1	1	X
cana-369	109	4	]	]	PUNCT
cana-369	109	5	bondy	bondy	PROPN
cana-369	109	6	j	j	PROPN
cana-369	109	7	a	a	PROPN
cana-369	109	8	and	and	CCONJ
cana-369	109	9	murty	murty	NOUN
cana-369	109	10	u	u	NOUN
cana-369	109	11	s	s	PROPN
cana-369	109	12	r	r	NOUN
cana-369	109	13	,	,	PUNCT
cana-369	109	14	springer	springer	NOUN
cana-369	109	15	international	international	PROPN
cana-369	109	16	edition	edition	PROPN
cana-369	109	17	(	(	PUNCT
cana-369	109	18	2008	2008	NUM
cana-369	109	19	)	)	PUNCT
cana-369	109	20	.	.	PUNCT
cana-369	110	1	[	[	X
cana-369	110	2	2	2	NUM
cana-369	110	3	]	]	PUNCT
cana-369	110	4	gallian	gallian	PROPN
cana-369	110	5	ja	ja	PROPN
cana-369	110	6	,	,	PUNCT
cana-369	110	7	the	the	DET
cana-369	110	8	eletronic	eletronic	ADJ
cana-369	110	9	journal	journal	NOUN
cana-369	110	10	of	of	ADP
cana-369	110	11	combinatorices	combinatorice	NOUN
cana-369	110	12	,	,	PUNCT
cana-369	110	13	18	18	NUM
cana-369	110	14	ds6	ds6	NOUN
cana-369	110	15	(	(	PUNCT
cana-369	110	16	2011	2011	NUM
cana-369	110	17	)	)	PUNCT
cana-369	110	18	.	.	PUNCT
cana-369	111	1	[	[	X
cana-369	111	2	3	3	X
cana-369	111	3	]	]	X
cana-369	111	4	harary	harary	NOUN
cana-369	111	5	f	f	PROPN
cana-369	111	6	,	,	PUNCT
cana-369	111	7	addison	addison	PROPN
cana-369	111	8	-	-	PUNCT
cana-369	111	9	wesley	wesley	PROPN
cana-369	111	10	,	,	PUNCT
cana-369	111	11	reading	reading	NOUN
cana-369	111	12	,	,	PUNCT
cana-369	111	13	mass	mass	NOUN
cana-369	111	14	(	(	PUNCT
cana-369	111	15	1969	1969	NUM
cana-369	111	16	)	)	PUNCT
cana-369	111	17	.	.	PUNCT
cana-369	112	1	[	[	X
cana-369	112	2	4	4	X
cana-369	112	3	]	]	X
cana-369	112	4	ebin	ebin	PROPN
cana-369	112	5	raja	raja	PROPN
cana-369	112	6	merly	merly	PROPN
cana-369	112	7	e	e	PROPN
cana-369	112	8	and	and	CCONJ
cana-369	112	9	anto	anto	PROPN
cana-369	112	10	a	a	DET
cana-369	112	11	m	m	PROPN
cana-369	112	12	,	,	PUNCT
cana-369	112	13	international	international	ADJ
cana-369	112	14	journal	journal	NOUN
cana-369	112	15	of	of	ADP
cana-369	112	16	emerging	emerge	VERB
cana-369	112	17	technologies	technology	NOUN
cana-369	112	18	in	in	ADP
cana-369	112	19	engineering	engineering	NOUN
cana-369	112	20	research	research	NOUN
cana-369	112	21	,	,	PUNCT
cana-369	112	22	4(10	4(10	NUM
cana-369	112	23	)	)	PUNCT
cana-369	112	24	(	(	PUNCT
cana-369	112	25	2016	2016	NUM
cana-369	112	26	)	)	PUNCT
cana-369	112	27	.	.	PUNCT
cana-369	113	1	[	[	X
cana-369	113	2	5	5	X
cana-369	113	3	]	]	X
cana-369	113	4	ganesan	ganesan	PROPN
cana-369	113	5	v	v	PROPN
cana-369	113	6	and	and	CCONJ
cana-369	113	7	lavanya	lavanya	NOUN
cana-369	113	8	s	s	PROPN
cana-369	113	9	,	,	PUNCT
cana-369	113	10	iosr	iosr	ADJ
cana-369	113	11	journal	journal	NOUN
cana-369	113	12	of	of	ADP
cana-369	113	13	mathematics	mathematic	NOUN
cana-369	113	14	,	,	PUNCT
cana-369	113	15	15(6	15(6	NUM
cana-369	113	16	)	)	PUNCT
cana-369	113	17	,	,	PUNCT
cana-369	113	18	04–06	04–06	NUM
cana-369	113	19	(	(	PUNCT
cana-369	113	20	2019	2019	NUM
cana-369	113	21	)	)	PUNCT
cana-369	113	22	.	.	PUNCT
cana-369	114	1	[	[	X
cana-369	114	2	6	6	NUM
cana-369	114	3	]	]	PUNCT
cana-369	114	4	weisstein	weisstein	PROPN
cana-369	114	5	eric	eric	PROPN
cana-369	114	6	w	w	PROPN
cana-369	114	7	,	,	PUNCT
cana-369	114	8	mathworld	mathworld	NOUN
cana-369	114	9	.	.	PUNCT
cana-369	115	1	[	[	X
cana-369	115	2	7	7	X
cana-369	115	3	]	]	X
cana-369	115	4	satyanarayana	satyanarayana	PROPN
cana-369	115	5	bhavanari	bhavanari	PROPN
cana-369	115	6	,	,	PUNCT
cana-369	115	7	srinivasula	srinivasula	PROPN
cana-369	115	8	devanaboina	devanaboina	PROPN
cana-369	115	9	and	and	CCONJ
cana-369	115	10	mallikarjun	mallikarjun	NOUN
cana-369	115	11	bhavanari	bhavanari	NOUN
cana-369	115	12	,	,	PUNCT
cana-369	115	13	research	research	NOUN
cana-369	115	14	journal	journal	NOUN
cana-369	115	15	of	of	ADP
cana-369	115	16	science	science	PROPN
cana-369	115	17	&	&	CCONJ
cana-369	115	18	it	it	PRON
cana-369	115	19	management	management	NOUN
cana-369	115	20	rjsitm	rjsitm	NOUN
cana-369	115	21	,	,	PUNCT
cana-369	115	22	5	5	NUM
cana-369	115	23	(	(	PUNCT
cana-369	115	24	2016	2016	NUM
cana-369	115	25	)	)	PUNCT
cana-369	115	26	.	.	PUNCT
cana-369	116	1	communications	communication	NOUN
cana-369	116	2	on	on	ADP
cana-369	116	3	applied	apply	VERB
cana-369	116	4	nonlinear	nonlinear	ADJ
cana-369	116	5	analysis	analysis	NOUN
cana-369	116	6	issn	issn	NOUN
cana-369	116	7	:	:	PUNCT
cana-369	116	8	1074	1074	NUM
cana-369	116	9	-	-	PUNCT
cana-369	116	10	133x	133x	NUM
cana-369	116	11	vol	vol	NOUN
cana-369	116	12	31	31	NUM
cana-369	116	13	no	no	NOUN
cana-369	116	14	.	.	NOUN
cana-369	116	15	1	1	NUM
cana-369	116	16	(	(	PUNCT
cana-369	116	17	2024	2024	NUM
cana-369	116	18	)	)	PUNCT
cana-369	116	19	149	149	NUM
cana-369	116	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-369	117	1	[	[	X
cana-369	117	2	8	8	X
cana-369	117	3	]	]	X
cana-369	117	4	rosa	rosa	PROPN
cana-369	117	5	a	a	PROPN
cana-369	117	6	,	,	PUNCT
cana-369	117	7	graph	graph	NOUN
cana-369	117	8	theory	theory	NOUN
cana-369	117	9	:	:	PUNCT
cana-369	117	10	int	int	NOUN
cana-369	117	11	symp	symp	NOUN
cana-369	117	12	.	.	PUNCT
cana-369	117	13	,	,	PUNCT
cana-369	117	14	349–355(1966	349–355(1966	NUM
cana-369	117	15	)	)	PUNCT
cana-369	117	16	.	.	PUNCT
cana-369	118	1	[	[	X
cana-369	118	2	9	9	NUM
cana-369	118	3	]	]	PUNCT
cana-369	118	4	lakshmi	lakshmi	PROPN
cana-369	118	5	prasanna	prasanna	PROPN
cana-369	118	6	n	n	CCONJ
cana-369	118	7	,	,	PUNCT
cana-369	118	8	sravanthi	sravanthi	PROPN
cana-369	118	9	k	k	PROPN
cana-369	118	10	and	and	CCONJ
cana-369	118	11	nagalla	nagalla	PROPN
cana-369	118	12	sundhakar	sundhakar	PROPN
cana-369	118	13	,	,	PUNCT
cana-369	118	14	oriental	oriental	ADJ
cana-369	118	15	journal	journal	NOUN
cana-369	118	16	of	of	ADP
cana-369	118	17	computer	computer	NOUN
cana-369	118	18	science	science	PROPN
cana-369	118	19	&	&	CCONJ
cana-369	118	20	technology	technology	PROPN
cana-369	118	21	,	,	PUNCT
cana-369	118	22	7(1	7(1	NUM
cana-369	118	23	)	)	PUNCT
cana-369	118	24	,	,	PUNCT
cana-369	118	25	139	139	NUM
cana-369	118	26	-	-	SYM
cana-369	118	27	145	145	NUM
cana-369	118	28	(	(	PUNCT
cana-369	118	29	2014	2014	NUM
cana-369	118	30	)	)	PUNCT
cana-369	118	31	.	.	PUNCT
cana-369	119	1	[	[	X
cana-369	119	2	10	10	NUM
cana-369	119	3	]	]	X
cana-369	119	4	sridevi	sridevi	PROPN
cana-369	119	5	s	s	PROPN
cana-369	119	6	,	,	PUNCT
cana-369	119	7	navaneethakrishnan	navaneethakrishnan	PROPN
cana-369	119	8	s	s	PROPN
cana-369	119	9	,	,	PUNCT
cana-369	119	10	nagarajan	nagarajan	NOUN
cana-369	119	11	a	a	DET
cana-369	119	12	and	and	CCONJ
cana-369	119	13	nagarajan	nagarajan	PROPN
cana-369	119	14	k	k	PROPN
cana-369	119	15	,	,	PUNCT
cana-369	119	16	journal	journal	NOUN
cana-369	119	17	of	of	ADP
cana-369	119	18	applied	apply	VERB
cana-369	119	19	mathematics	mathematics	PROPN
cana-369	119	20	&	&	CCONJ
cana-369	119	21	informatics	informatic	NOUN
cana-369	119	22	,	,	PUNCT
cana-369	119	23	30	30	NUM
cana-369	119	24	,	,	PUNCT
cana-369	119	25	913–923	913–923	NUM
cana-369	119	26	(	(	PUNCT
cana-369	119	27	2012	2012	NUM
cana-369	119	28	)	)	PUNCT
cana-369	119	29	.	.	PUNCT
cana-369	120	1	[	[	X
cana-369	120	2	11	11	NUM
cana-369	120	3	]	]	X
cana-369	120	4	sirous	sirous	ADJ
cana-369	120	5	moradi	moradi	NOUN
cana-369	120	6	,	,	PUNCT
cana-369	120	7	iranian	iranian	ADJ
cana-369	120	8	journal	journal	PROPN
cana-369	120	9	of	of	ADP
cana-369	120	10	mathematical	mathematical	ADJ
cana-369	120	11	sciences	sciences	PROPN
cana-369	120	12	and	and	CCONJ
cana-369	120	13	informatics	informatic	NOUN
cana-369	120	14	,	,	PUNCT
cana-369	120	15	7(1	7(1	NUM
cana-369	120	16	)	)	PUNCT
cana-369	120	17	,	,	PUNCT
cana-369	120	18	73–81	73–81	PROPN
cana-369	120	19	(	(	PUNCT
cana-369	120	20	2012	2012	NUM
cana-369	120	21	)	)	PUNCT
cana-369	120	22	.	.	PUNCT
cana-369	121	1	[	[	X
cana-369	121	2	12	12	NUM
cana-369	121	3	]	]	PUNCT
cana-369	121	4	jayasekaran	jayasekaran	NOUN
cana-369	121	5	c	c	PROPN
cana-369	121	6	and	and	CCONJ
cana-369	121	7	little	little	ADJ
cana-369	121	8	flower	flower	NOUN
cana-369	121	9	j	j	PROPN
cana-369	121	10	,	,	PUNCT
cana-369	121	11	international	international	ADJ
cana-369	121	12	journal	journal	NOUN
cana-369	121	13	of	of	ADP
cana-369	121	14	pure	pure	ADJ
cana-369	121	15	and	and	CCONJ
cana-369	121	16	applied	applied	ADJ
cana-369	121	17	mathematics	mathematic	NOUN
cana-369	121	18	,	,	PUNCT
cana-369	121	19	120(3	120(3	NUM
cana-369	121	20	)	)	PUNCT
cana-369	121	21	,	,	PUNCT
cana-369	121	22	303–313	303–313	NUM
cana-369	121	23	(	(	PUNCT
cana-369	121	24	2018	2018	NUM
cana-369	121	25	)	)	PUNCT
cana-369	121	26	.	.	PUNCT
cana-369	122	1	[	[	X
cana-369	122	2	13	13	NUM
cana-369	122	3	]	]	X
cana-369	122	4	kanakambika	kanakambika	PROPN
cana-369	122	5	k	k	PROPN
cana-369	122	6	and	and	CCONJ
cana-369	122	7	thamizhendhi	thamizhendhi	VERB
cana-369	122	8	g	g	PROPN
cana-369	122	9	,	,	PUNCT
cana-369	122	10	journal	journal	NOUN
cana-369	122	11	of	of	ADP
cana-369	122	12	xidian	xidian	PROPN
cana-369	122	13	university	university	PROPN
cana-369	122	14	,	,	PUNCT
cana-369	122	15	14(4	14(4	NUM
cana-369	122	16	)	)	PUNCT
cana-369	122	17	,	,	PUNCT
cana-369	122	18	3551–3565	3551–3565	NUM
cana-369	122	19	(	(	PUNCT
cana-369	122	20	2020	2020	NUM
cana-369	122	21	)	)	PUNCT
cana-369	122	22	.	.	PUNCT
cana-369	123	1	[	[	X
cana-369	123	2	14	14	NUM
cana-369	123	3	]	]	X
cana-369	123	4	kanakambika	kanakambika	PROPN
cana-369	123	5	k	k	PROPN
cana-369	123	6	and	and	CCONJ
cana-369	123	7	thamizhendhi	thamizhendhi	VERB
cana-369	123	8	g	g	PROPN
cana-369	123	9	,	,	PUNCT
cana-369	123	10	igi	igi	PROPN
cana-369	123	11	global	global	ADJ
cana-369	123	12	(	(	PUNCT
cana-369	123	13	2022	2022	NUM
cana-369	123	14	)	)	PUNCT
cana-369	123	15	.	.	PUNCT