id	sid	tid	token	lemma	pos
cana-586	1	1	communications	communication	NOUN
cana-586	1	2	on	on	ADP
cana-586	1	3	applied	apply	VERB
cana-586	1	4	nonlinear	nonlinear	ADJ
cana-586	1	5	analysis	analysis	NOUN
cana-586	1	6	issn	issn	NOUN
cana-586	1	7	:	:	PUNCT
cana-586	1	8	1074	1074	NUM
cana-586	1	9	-	-	PUNCT
cana-586	1	10	133x	133x	NUM
cana-586	1	11	vol	vol	NOUN
cana-586	1	12	31	31	NUM
cana-586	1	13	no	no	NOUN
cana-586	1	14	.	.	NOUN
cana-586	1	15	2	2	NUM
cana-586	1	16	(	(	PUNCT
cana-586	1	17	2024	2024	NUM
cana-586	1	18	)	)	PUNCT
cana-586	1	19	404	404	NUM
cana-586	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-586	2	1	every	every	DET
cana-586	2	2	tree	tree	NOUN
cana-586	2	3	is	be	AUX
cana-586	2	4	an	an	DET
cana-586	2	5	integral	integral	ADJ
cana-586	2	6	sum	sum	NOUN
cana-586	2	7	graph	graph	NOUN
cana-586	2	8	s.	s.	PROPN
cana-586	2	9	muthukkumar	muthukkumar	PROPN
cana-586	2	10	,	,	PUNCT
cana-586	2	11	k.	k.	PROPN
cana-586	2	12	rajendran	rajendran	PROPN
cana-586	2	13	vels	vels	PROPN
cana-586	2	14	institute	institute	PROPN
cana-586	2	15	of	of	ADP
cana-586	2	16	science	science	PROPN
cana-586	2	17	technology	technology	NOUN
cana-586	2	18	and	and	CCONJ
cana-586	2	19	advanced	advanced	ADJ
cana-586	2	20	studies	study	NOUN
cana-586	2	21	chennai	chennai	PROPN
cana-586	2	22	,	,	PUNCT
cana-586	2	23	india	india	PROPN
cana-586	2	24	muthumed77@gmail.com	muthumed77@gmail.com	PROPN
cana-586	2	25	,	,	PUNCT
cana-586	2	26	gkrajendra59@gmail.com	gkrajendra59@gmail.com	X
cana-586	2	27	article	article	NOUN
cana-586	2	28	history	history	NOUN
cana-586	2	29	:	:	PUNCT
cana-586	2	30	received	receive	VERB
cana-586	2	31	:	:	PUNCT
cana-586	2	32	26	26	NUM
cana-586	2	33	-	-	PUNCT
cana-586	2	34	02	02	NUM
cana-586	2	35	-	-	PUNCT
cana-586	2	36	2024	2024	NUM
cana-586	2	37	revised	revise	VERB
cana-586	2	38	:	:	PUNCT
cana-586	2	39	25	25	NUM
cana-586	2	40	-	-	PUNCT
cana-586	2	41	04	04	NUM
cana-586	2	42	-	-	PUNCT
cana-586	2	43	2024	2024	NUM
cana-586	2	44	accepted	accept	VERB
cana-586	2	45	:	:	PUNCT
cana-586	2	46	10	10	NUM
cana-586	2	47	-	-	SYM
cana-586	2	48	05	05	NUM
cana-586	2	49	-	-	PUNCT
cana-586	2	50	2024	2024	NUM
cana-586	2	51	abstract	abstract	NOUN
cana-586	2	52	:	:	PUNCT
cana-586	2	53	a	a	DET
cana-586	2	54	finite	finite	ADJ
cana-586	2	55	simple	simple	ADJ
cana-586	2	56	graph	graph	NOUN
cana-586	2	57	g	g	PROPN
cana-586	2	58	is	be	AUX
cana-586	2	59	called	call	VERB
cana-586	2	60	an	an	DET
cana-586	2	61	integral	integral	ADJ
cana-586	2	62	sum	sum	NOUN
cana-586	2	63	graph	graph	NOUN
cana-586	2	64	(	(	PUNCT
cana-586	2	65	respectively	respectively	ADV
cana-586	2	66	,	,	PUNCT
cana-586	2	67	sum	sum	NOUN
cana-586	2	68	graph	graph	NOUN
cana-586	2	69	)	)	PUNCT
cana-586	2	70	if	if	SCONJ
cana-586	2	71	there	there	PRON
cana-586	2	72	is	be	VERB
cana-586	2	73	a	a	DET
cana-586	2	74	bijection	bijection	ADJ
cana-586	2	75	f	f	NOUN
cana-586	2	76	from	from	ADP
cana-586	2	77	the	the	DET
cana-586	2	78	vertices	vertex	NOUN
cana-586	2	79	of	of	ADP
cana-586	2	80	g	g	NOUN
cana-586	2	81	to	to	ADP
cana-586	2	82	a	a	DET
cana-586	2	83	set	set	NOUN
cana-586	2	84	of	of	ADP
cana-586	2	85	integers	integer	NOUN
cana-586	2	86	s	s	PART
cana-586	2	87	(	(	PUNCT
cana-586	2	88	respectively	respectively	ADV
cana-586	2	89	,	,	PUNCT
cana-586	2	90	a	a	DET
cana-586	2	91	set	set	NOUN
cana-586	2	92	of	of	ADP
cana-586	2	93	positive	positive	ADJ
cana-586	2	94	integers	integer	NOUN
cana-586	2	95	s	s	PART
cana-586	2	96	)	)	PUNCT
cana-586	2	97	such	such	ADJ
cana-586	2	98	that	that	SCONJ
cana-586	2	99	uv	uv	NOUN
cana-586	2	100	is	be	AUX
cana-586	2	101	an	an	DET
cana-586	2	102	edge	edge	NOUN
cana-586	2	103	of	of	ADP
cana-586	2	104	g	g	NOUN
cana-586	2	105	if	if	SCONJ
cana-586	2	106	and	and	CCONJ
cana-586	2	107	only	only	ADV
cana-586	2	108	if	if	SCONJ
cana-586	2	109	f	f	PROPN
cana-586	2	110	(	(	PUNCT
cana-586	2	111	u)+f	u)+f	PROPN
cana-586	2	112	(	(	PUNCT
cana-586	2	113	v	v	NOUN
cana-586	2	114	)	)	PUNCT
cana-586	2	115	∈	∈	PROPN
cana-586	2	116	s.	s.	PROPN
cana-586	2	117	in	in	ADP
cana-586	2	118	1999	1999	NUM
cana-586	2	119	,	,	PUNCT
cana-586	2	120	liaw	liaw	PROPN
cana-586	2	121	et	et	PROPN
cana-586	2	122	al	al	PROPN
cana-586	2	123	(	(	PUNCT
cana-586	2	124	ars	ar	NOUN
cana-586	2	125	comb	comb	VERB
cana-586	2	126	.	.	PUNCT
cana-586	3	1	,	,	PUNCT
cana-586	3	2	vol.54	vol.54	PROPN
cana-586	3	3	,	,	PUNCT
cana-586	3	4	259	259	NUM
cana-586	3	5	-	-	SYM
cana-586	3	6	268	268	NUM
cana-586	3	7	)	)	PUNCT
cana-586	3	8	posed	pose	VERB
cana-586	3	9	the	the	DET
cana-586	3	10	conjecture	conjecture	NOUN
cana-586	3	11	that	that	SCONJ
cana-586	3	12	every	every	DET
cana-586	3	13	tree	tree	NOUN
cana-586	3	14	is	be	AUX
cana-586	3	15	an	an	DET
cana-586	3	16	integral	integral	ADJ
cana-586	3	17	sum	sum	NOUN
cana-586	3	18	graph	graph	NOUN
cana-586	3	19	.	.	PUNCT
cana-586	4	1	in	in	ADP
cana-586	4	2	this	this	DET
cana-586	4	3	note	note	NOUN
cana-586	4	4	,	,	PUNCT
cana-586	4	5	we	we	PRON
cana-586	4	6	prove	prove	VERB
cana-586	4	7	that	that	SCONJ
cana-586	4	8	all	all	DET
cana-586	4	9	trees	tree	NOUN
cana-586	4	10	are	be	AUX
cana-586	4	11	integral	integral	ADJ
cana-586	4	12	sum	sum	NOUN
cana-586	4	13	graphs	graph	NOUN
cana-586	4	14	.	.	PUNCT
cana-586	5	1	further	far	ADV
cana-586	5	2	,	,	PUNCT
cana-586	5	3	we	we	PRON
cana-586	5	4	prove	prove	VERB
cana-586	5	5	that	that	SCONJ
cana-586	5	6	every	every	DET
cana-586	5	7	bipartite	bipartite	NOUN
cana-586	5	8	graph	graph	NOUN
cana-586	5	9	is	be	AUX
cana-586	5	10	an	an	DET
cana-586	5	11	induced	induced	ADJ
cana-586	5	12	subgraph	subgraph	NOUN
cana-586	5	13	of	of	ADP
cana-586	5	14	a	a	DET
cana-586	5	15	sum	sum	NOUN
cana-586	5	16	graph	graph	NOUN
cana-586	5	17	g	g	NOUN
cana-586	5	18	with	with	ADP
cana-586	5	19	sum	sum	NOUN
cana-586	5	20	number	number	NOUN
cana-586	5	21	σ(g	σ(g	NOUN
cana-586	5	22	)	)	PUNCT
cana-586	5	23	=	=	SYM
cana-586	6	1	1	1	X
cana-586	6	2	.	.	PUNCT
cana-586	7	1	keywords	keyword	NOUN
cana-586	7	2	:	:	PUNCT
cana-586	7	3	integral	integral	ADJ
cana-586	7	4	sum	sum	NOUN
cana-586	7	5	graphs	graph	NOUN
cana-586	7	6	;	;	PUNCT
cana-586	7	7	sum	sum	NOUN
cana-586	7	8	graphs	graph	NOUN
cana-586	7	9	;	;	PUNCT
cana-586	7	10	integral	integral	ADJ
cana-586	7	11	sum	sum	NOUN
cana-586	7	12	number	number	NOUN
cana-586	7	13	;	;	PUNCT
cana-586	7	14	1	1	X
cana-586	7	15	.	.	X
cana-586	7	16	introduction	introduction	NOUN
cana-586	7	17	all	all	DET
cana-586	7	18	the	the	DET
cana-586	7	19	graphs	graph	NOUN
cana-586	7	20	considered	consider	VERB
cana-586	7	21	in	in	ADP
cana-586	7	22	this	this	DET
cana-586	7	23	paper	paper	NOUN
cana-586	7	24	are	be	AUX
cana-586	7	25	finite	finite	ADJ
cana-586	7	26	simple	simple	ADJ
cana-586	7	27	graphs	graph	NOUN
cana-586	7	28	.	.	PUNCT
cana-586	8	1	terms	term	NOUN
cana-586	8	2	that	that	PRON
cana-586	8	3	are	be	AUX
cana-586	8	4	not	not	PART
cana-586	8	5	defined	define	VERB
cana-586	8	6	here	here	ADV
cana-586	8	7	can	can	AUX
cana-586	8	8	be	be	AUX
cana-586	8	9	referred	refer	VERB
cana-586	8	10	from	from	ADP
cana-586	8	11	the	the	DET
cana-586	8	12	book	book	NOUN
cana-586	8	13	[	[	X
cana-586	8	14	10	10	NUM
cana-586	8	15	]	]	PUNCT
cana-586	8	16	.	.	PUNCT
cana-586	9	1	sum	sum	NOUN
cana-586	9	2	graphs	graph	NOUN
cana-586	9	3	and	and	CCONJ
cana-586	9	4	integral	integral	ADJ
cana-586	9	5	sum	sum	NOUN
cana-586	9	6	graphs	graph	NOUN
cana-586	9	7	were	be	AUX
cana-586	9	8	introduced	introduce	VERB
cana-586	9	9	by	by	ADP
cana-586	9	10	harary	harary	NOUN
cana-586	9	11	[	[	X
cana-586	9	12	5	5	NUM
cana-586	9	13	]	]	PUNCT
cana-586	9	14	.	.	PUNCT
cana-586	10	1	a	a	DET
cana-586	10	2	graph	graph	NOUN
cana-586	10	3	g	g	NOUN
cana-586	10	4	is	be	AUX
cana-586	10	5	called	call	VERB
cana-586	10	6	a	a	DET
cana-586	10	7	sum	sum	NOUN
cana-586	10	8	graph	graph	NOUN
cana-586	10	9	if	if	SCONJ
cana-586	10	10	the	the	DET
cana-586	10	11	vertices	vertex	NOUN
cana-586	10	12	of	of	ADP
cana-586	10	13	g	g	NOUN
cana-586	10	14	can	can	AUX
cana-586	10	15	be	be	AUX
cana-586	10	16	labeled	label	VERB
cana-586	10	17	with	with	ADP
cana-586	10	18	distinct	distinct	ADJ
cana-586	10	19	positive	positive	ADJ
cana-586	10	20	integers	integer	NOUN
cana-586	10	21	so	so	SCONJ
cana-586	10	22	that	that	SCONJ
cana-586	10	23	e	e	NOUN
cana-586	10	24	=	=	NOUN
cana-586	10	25	uv	uv	NOUN
cana-586	10	26	is	be	AUX
cana-586	10	27	an	an	DET
cana-586	10	28	edge	edge	NOUN
cana-586	10	29	of	of	ADP
cana-586	10	30	g	g	NOUN
cana-586	10	31	if	if	SCONJ
cana-586	11	1	and	and	CCONJ
cana-586	11	2	only	only	ADV
cana-586	11	3	if	if	SCONJ
cana-586	11	4	the	the	DET
cana-586	11	5	sum	sum	NOUN
cana-586	11	6	of	of	ADP
cana-586	11	7	the	the	DET
cana-586	11	8	labels	label	NOUN
cana-586	11	9	of	of	ADP
cana-586	11	10	the	the	DET
cana-586	11	11	vertex	vertex	NOUN
cana-586	11	12	u	u	NOUN
cana-586	11	13	and	and	CCONJ
cana-586	11	14	vertex	vertex	NOUN
cana-586	11	15	v	v	NOUN
cana-586	11	16	is	be	AUX
cana-586	11	17	also	also	ADV
cana-586	11	18	a	a	DET
cana-586	11	19	label	label	NOUN
cana-586	11	20	in	in	ADP
cana-586	11	21	g.	g.	PROPN
cana-586	12	1	it	it	PRON
cana-586	12	2	is	be	AUX
cana-586	12	3	clear	clear	ADJ
cana-586	12	4	that	that	SCONJ
cana-586	12	5	if	if	SCONJ
cana-586	12	6	g	g	PROPN
cana-586	12	7	is	be	AUX
cana-586	12	8	a	a	DET
cana-586	12	9	properly	properly	ADV
cana-586	12	10	labeled	label	VERB
cana-586	12	11	sum	sum	NOUN
cana-586	12	12	graph	graph	NOUN
cana-586	12	13	,	,	PUNCT
cana-586	12	14	then	then	ADV
cana-586	12	15	the	the	DET
cana-586	12	16	vertex	vertex	NOUN
cana-586	12	17	receiving	receive	VERB
cana-586	12	18	the	the	DET
cana-586	12	19	highest	high	ADJ
cana-586	12	20	label	label	NOUN
cana-586	12	21	can	can	AUX
cana-586	12	22	not	not	PART
cana-586	12	23	be	be	AUX
cana-586	12	24	adjacent	adjacent	ADJ
cana-586	12	25	to	to	ADP
cana-586	12	26	any	any	DET
cana-586	12	27	other	other	ADJ
cana-586	12	28	vertex	vertex	NOUN
cana-586	12	29	.	.	PUNCT
cana-586	13	1	thus	thus	ADV
cana-586	13	2	,	,	PUNCT
cana-586	13	3	every	every	DET
cana-586	13	4	sum	sum	NOUN
cana-586	13	5	graph	graph	NOUN
cana-586	13	6	must	must	AUX
cana-586	13	7	contain	contain	VERB
cana-586	13	8	isolated	isolated	ADJ
cana-586	13	9	vertices	vertex	NOUN
cana-586	13	10	.	.	PUNCT
cana-586	14	1	in	in	ADP
cana-586	14	2	other	other	ADJ
cana-586	14	3	words	word	NOUN
cana-586	14	4	,	,	PUNCT
cana-586	14	5	a	a	DET
cana-586	14	6	connected	connected	ADJ
cana-586	14	7	graph	graph	NOUN
cana-586	14	8	is	be	AUX
cana-586	14	9	not	not	PART
cana-586	14	10	a	a	DET
cana-586	14	11	sum	sum	NOUN
cana-586	14	12	graph	graph	NOUN
cana-586	14	13	.	.	PUNCT
cana-586	15	1	if	if	SCONJ
cana-586	15	2	g	g	PROPN
cana-586	15	3	is	be	AUX
cana-586	15	4	not	not	PART
cana-586	15	5	a	a	DET
cana-586	15	6	sum	sum	NOUN
cana-586	15	7	graph	graph	NOUN
cana-586	15	8	,	,	PUNCT
cana-586	15	9	adding	add	VERB
cana-586	15	10	a	a	DET
cana-586	15	11	finite	finite	ADJ
cana-586	15	12	number	number	NOUN
cana-586	15	13	of	of	ADP
cana-586	15	14	isolated	isolated	ADJ
cana-586	15	15	vertices	vertex	NOUN
cana-586	15	16	to	to	ADP
cana-586	15	17	it	it	PRON
cana-586	15	18	always	always	ADV
cana-586	15	19	yields	yield	VERB
cana-586	15	20	a	a	DET
cana-586	15	21	sum	sum	NOUN
cana-586	15	22	graph	graph	NOUN
cana-586	15	23	.	.	PUNCT
cana-586	16	1	given	give	VERB
cana-586	16	2	any	any	DET
cana-586	16	3	graph	graph	NOUN
cana-586	16	4	g	g	NOUN
cana-586	16	5	with	with	ADP
cana-586	16	6	p	p	NOUN
cana-586	16	7	vertices	vertex	NOUN
cana-586	16	8	and	and	CCONJ
cana-586	16	9	q	q	NOUN
cana-586	16	10	edges	edge	NOUN
cana-586	16	11	,	,	PUNCT
cana-586	16	12	it	it	PRON
cana-586	16	13	is	be	AUX
cana-586	16	14	trivial	trivial	ADJ
cana-586	16	15	that	that	SCONJ
cana-586	16	16	the	the	DET
cana-586	16	17	union	union	NOUN
cana-586	16	18	g	g	PROPN
cana-586	16	19	∪	∪	VERB
cana-586	16	20	qk1	qk1	NOUN
cana-586	16	21	of	of	ADP
cana-586	16	22	g	g	NOUN
cana-586	16	23	with	with	ADP
cana-586	16	24	q	q	ADJ
cana-586	16	25	isolated	isolate	VERB
cana-586	16	26	vertices	vertex	NOUN
cana-586	16	27	is	be	AUX
cana-586	16	28	a	a	DET
cana-586	16	29	sum	sum	NOUN
cana-586	16	30	graph	graph	NOUN
cana-586	16	31	.	.	PUNCT
cana-586	17	1	we	we	PRON
cana-586	17	2	can	can	AUX
cana-586	17	3	define	define	VERB
cana-586	17	4	the	the	DET
cana-586	17	5	sum	sum	NOUN
cana-586	17	6	number	number	NOUN
cana-586	17	7	σ(g	σ(g	NOUN
cana-586	17	8	)	)	PUNCT
cana-586	17	9	of	of	ADP
cana-586	17	10	g	g	PROPN
cana-586	17	11	as	as	ADP
cana-586	17	12	the	the	DET
cana-586	17	13	smallest	small	ADJ
cana-586	17	14	number	number	NOUN
cana-586	17	15	(	(	PUNCT
cana-586	17	16	say	say	VERB
cana-586	17	17	s	s	NOUN
cana-586	17	18	)	)	PUNCT
cana-586	17	19	of	of	ADP
cana-586	17	20	isolated	isolate	VERB
cana-586	17	21	vertices	vertex	NOUN
cana-586	17	22	added	add	VERB
cana-586	17	23	to	to	ADP
cana-586	17	24	g	g	NOUN
cana-586	17	25	such	such	ADJ
cana-586	17	26	that	that	SCONJ
cana-586	17	27	g	g	PROPN
cana-586	17	28	∪	∪	PROPN
cana-586	17	29	sk1	sk1	PROPN
cana-586	17	30	is	be	AUX
cana-586	17	31	a	a	DET
cana-586	17	32	sum	sum	NOUN
cana-586	17	33	graph	graph	NOUN
cana-586	17	34	.	.	PUNCT
cana-586	18	1	an	an	DET
cana-586	18	2	integral	integral	ADJ
cana-586	18	3	sum	sum	NOUN
cana-586	18	4	graph	graph	NOUN
cana-586	18	5	is	be	AUX
cana-586	18	6	also	also	ADV
cana-586	18	7	defined	define	VERB
cana-586	18	8	just	just	ADV
cana-586	18	9	as	as	SCONJ
cana-586	18	10	the	the	DET
cana-586	18	11	sum	sum	NOUN
cana-586	18	12	graph	graph	NOUN
cana-586	18	13	,	,	PUNCT
cana-586	18	14	difference	difference	NOUN
cana-586	18	15	being	be	AUX
cana-586	18	16	that	that	SCONJ
cana-586	18	17	the	the	DET
cana-586	18	18	label	label	NOUN
cana-586	18	19	set	set	NOUN
cana-586	18	20	s	s	VERB
cana-586	18	21	is	be	AUX
cana-586	18	22	a	a	DET
cana-586	18	23	subset	subset	NOUN
cana-586	18	24	of	of	ADP
cana-586	18	25	z	z	PROPN
cana-586	18	26	,	,	PUNCT
cana-586	18	27	the	the	DET
cana-586	18	28	set	set	NOUN
cana-586	18	29	of	of	ADP
cana-586	18	30	integers	integer	NOUN
cana-586	18	31	.	.	PUNCT
cana-586	19	1	the	the	DET
cana-586	19	2	integral	integral	ADJ
cana-586	19	3	sum	sum	NOUN
cana-586	19	4	number	number	NOUN
cana-586	19	5	ζ(g	ζ(g	NOUN
cana-586	19	6	)	)	PUNCT
cana-586	19	7	is	be	AUX
cana-586	19	8	the	the	DET
cana-586	19	9	smallest	small	ADJ
cana-586	19	10	non	non	ADJ
cana-586	19	11	-	-	ADJ
cana-586	19	12	negative	negative	ADJ
cana-586	19	13	integer	integer	NOUN
cana-586	19	14	s	s	PRON
cana-586	19	15	such	such	ADJ
cana-586	19	16	that	that	SCONJ
cana-586	19	17	g	g	PROPN
cana-586	19	18	∪	∪	PROPN
cana-586	19	19	sk1	sk1	PROPN
cana-586	19	20	is	be	AUX
cana-586	19	21	an	an	DET
cana-586	19	22	integral	integral	ADJ
cana-586	19	23	sum	sum	NOUN
cana-586	19	24	graph	graph	NOUN
cana-586	19	25	.	.	PUNCT
cana-586	20	1	clearly	clearly	ADV
cana-586	20	2	for	for	ADP
cana-586	20	3	any	any	DET
cana-586	20	4	graph	graph	NOUN
cana-586	20	5	g	g	NOUN
cana-586	20	6	,	,	PUNCT
cana-586	20	7	ζ(g	ζ(g	NOUN
cana-586	20	8	)	)	PUNCT
cana-586	20	9	≤	≤	NUM
cana-586	20	10	σ(g	σ(g	NOUN
cana-586	20	11	)	)	PUNCT
cana-586	20	12	.	.	PUNCT
cana-586	21	1	for	for	ADP
cana-586	21	2	a	a	DET
cana-586	21	3	survey	survey	NOUN
cana-586	21	4	on	on	ADP
cana-586	21	5	sum	sum	NOUN
cana-586	21	6	graphs	graph	NOUN
cana-586	21	7	and	and	CCONJ
cana-586	21	8	integral	integral	ADJ
cana-586	21	9	sum	sum	NOUN
cana-586	21	10	graphs	graph	NOUN
cana-586	21	11	,	,	PUNCT
cana-586	21	12	we	we	PRON
cana-586	21	13	refer	refer	VERB
cana-586	21	14	to	to	ADP
cana-586	21	15	the	the	DET
cana-586	21	16	dynamic	dynamic	ADJ
cana-586	21	17	survey	survey	NOUN
cana-586	21	18	on	on	ADP
cana-586	21	19	graph	graph	NOUN
cana-586	21	20	labeling	labeling	NOUN
cana-586	21	21	by	by	ADP
cana-586	21	22	gallian	gallian	ADJ
cana-586	21	23	[	[	X
cana-586	21	24	4	4	NUM
cana-586	21	25	]	]	PUNCT
cana-586	21	26	.	.	PUNCT
cana-586	22	1	liaw	liaw	PROPN
cana-586	22	2	et	et	PROPN
cana-586	22	3	al	al	PROPN
cana-586	23	1	[	[	X
cana-586	23	2	7	7	X
cana-586	23	3	]	]	PUNCT
cana-586	23	4	posed	pose	VERB
cana-586	23	5	the	the	DET
cana-586	23	6	conjecture	conjecture	NOUN
cana-586	23	7	that	that	SCONJ
cana-586	23	8	every	every	DET
cana-586	23	9	tree	tree	NOUN
cana-586	23	10	is	be	AUX
cana-586	23	11	an	an	DET
cana-586	23	12	integral	integral	ADJ
cana-586	23	13	sum	sum	NOUN
cana-586	23	14	graph	graph	NOUN
cana-586	23	15	.	.	PUNCT
cana-586	24	1	this	this	DET
cana-586	24	2	conjecture	conjecture	NOUN
cana-586	24	3	was	be	AUX
cana-586	24	4	proved	prove	VERB
cana-586	24	5	only	only	ADV
cana-586	24	6	for	for	ADP
cana-586	24	7	some	some	DET
cana-586	24	8	classes	class	NOUN
cana-586	24	9	of	of	ADP
cana-586	24	10	trees	tree	NOUN
cana-586	24	11	:	:	PUNCT
cana-586	24	12	caterpillars	caterpillar	NOUN
cana-586	24	13	,	,	PUNCT
cana-586	24	14	banana	banana	NOUN
cana-586	24	15	trees	tree	NOUN
cana-586	24	16	,	,	PUNCT
cana-586	24	17	generalized	generalized	ADJ
cana-586	24	18	stars	star	NOUN
cana-586	24	19	and	and	CCONJ
cana-586	24	20	trees	tree	NOUN
cana-586	24	21	whose	whose	DET
cana-586	24	22	forks	fork	NOUN
cana-586	24	23	(	(	PUNCT
cana-586	24	24	by	by	ADP
cana-586	24	25	fork	fork	NOUN
cana-586	24	26	we	we	PRON
cana-586	24	27	mean	mean	VERB
cana-586	24	28	a	a	DET
cana-586	24	29	vertex	vertex	NOUN
cana-586	24	30	of	of	ADP
cana-586	24	31	degree	degree	NOUN
cana-586	24	32	not	not	PART
cana-586	24	33	2	2	NUM
cana-586	24	34	)	)	PUNCT
cana-586	24	35	are	be	AUX
cana-586	24	36	distance	distance	NOUN
cana-586	24	37	at	at	ADV
cana-586	24	38	least	least	ADV
cana-586	24	39	4	4	NUM
cana-586	24	40	from	from	ADP
cana-586	24	41	each	each	DET
cana-586	24	42	other	other	ADJ
cana-586	25	1	[	[	X
cana-586	25	2	1	1	NUM
cana-586	25	3	,	,	PUNCT
cana-586	25	4	2	2	NUM
cana-586	25	5	]	]	PUNCT
cana-586	25	6	.	.	PUNCT
cana-586	26	1	he	he	PRON
cana-586	26	2	et	et	PROPN
cana-586	26	3	al	al	PROPN
cana-586	27	1	[	[	X
cana-586	27	2	6	6	NUM
cana-586	27	3	]	]	PUNCT
cana-586	27	4	reduced	reduce	VERB
cana-586	27	5	this	this	DET
cana-586	27	6	distance	distance	NOUN
cana-586	27	7	to	to	ADP
cana-586	27	8	3	3	NUM
cana-586	27	9	.	.	PUNCT
cana-586	28	1	pyatkin	pyatkin	PROPN
cana-586	28	2	[	[	X
cana-586	28	3	8	8	NUM
cana-586	28	4	]	]	PUNCT
cana-586	28	5	proved	prove	VERB
cana-586	28	6	that	that	SCONJ
cana-586	28	7	every	every	DET
cana-586	28	8	tree	tree	NOUN
cana-586	28	9	whose	whose	DET
cana-586	28	10	forks	fork	NOUN
cana-586	28	11	are	be	AUX
cana-586	28	12	at	at	ADP
cana-586	28	13	least	least	ADJ
cana-586	28	14	distance	distance	NOUN
cana-586	28	15	2	2	NUM
cana-586	28	16	apart	apart	ADV
cana-586	28	17	is	be	AUX
cana-586	28	18	an	an	DET
cana-586	28	19	integral	integral	ADJ
cana-586	28	20	sum	sum	NOUN
cana-586	28	21	graph	graph	NOUN
cana-586	28	22	.	.	PUNCT
cana-586	29	1	also	also	ADV
cana-586	29	2	,	,	PUNCT
cana-586	29	3	pyatkin	pyatkin	PROPN
cana-586	29	4	proved	prove	VERB
cana-586	29	5	that	that	SCONJ
cana-586	29	6	subdivided	subdivide	VERB
cana-586	29	7	trees	tree	NOUN
cana-586	29	8	are	be	AUX
cana-586	29	9	integral	integral	ADJ
cana-586	29	10	sum	sum	NOUN
cana-586	29	11	graphs	graph	NOUN
cana-586	29	12	.	.	PUNCT
cana-586	30	1	ellingham	ellingham	NOUN
cana-586	31	1	[	[	X
cana-586	31	2	3	3	X
cana-586	31	3	]	]	PUNCT
cana-586	31	4	proved	prove	VERB
cana-586	31	5	that	that	SCONJ
cana-586	31	6	σ(t	σ(t	PROPN
cana-586	31	7	)	)	PUNCT
cana-586	32	1	=	=	SYM
cana-586	32	2	1	1	NUM
cana-586	32	3	for	for	ADP
cana-586	32	4	every	every	DET
cana-586	32	5	t≠k1	t≠k1	PROPN
cana-586	32	6	.	.	PUNCT
cana-586	32	7	tiwari	tiwari	PROPN
cana-586	32	8	and	and	CCONJ
cana-586	32	9	tripathi	tripathi	PROPN
cana-586	33	1	[	[	X
cana-586	33	2	9	9	NUM
cana-586	33	3	]	]	PUNCT
cana-586	33	4	gave	give	VERB
cana-586	33	5	some	some	DET
cana-586	33	6	bounds	bound	NOUN
cana-586	33	7	on	on	ADP
cana-586	33	8	the	the	DET
cana-586	33	9	number	number	NOUN
cana-586	33	10	of	of	ADP
cana-586	33	11	edges	edge	NOUN
cana-586	33	12	for	for	ADP
cana-586	33	13	a	a	DET
cana-586	33	14	graph	graph	NOUN
cana-586	33	15	to	to	PART
cana-586	33	16	be	be	AUX
cana-586	33	17	sum	sum	NOUN
cana-586	33	18	graphs	graph	NOUN
cana-586	33	19	and	and	CCONJ
cana-586	33	20	integral	integral	ADJ
cana-586	33	21	sum	sum	NOUN
cana-586	33	22	graphs	graph	NOUN
cana-586	33	23	.	.	PUNCT
cana-586	34	1	in	in	ADP
cana-586	34	2	this	this	DET
cana-586	34	3	paper	paper	NOUN
cana-586	34	4	,	,	PUNCT
cana-586	34	5	we	we	PRON
cana-586	34	6	prove	prove	VERB
cana-586	34	7	that	that	SCONJ
cana-586	34	8	all	all	DET
cana-586	34	9	trees	tree	NOUN
cana-586	34	10	are	be	AUX
cana-586	34	11	integral	integral	ADJ
cana-586	34	12	sum	sum	NOUN
cana-586	34	13	graphs	graph	NOUN
cana-586	34	14	.	.	PUNCT
cana-586	35	1	that	that	PRON
cana-586	35	2	is	is	ADV
cana-586	35	3	,	,	PUNCT
cana-586	35	4	we	we	PRON
cana-586	35	5	prove	prove	VERB
cana-586	35	6	that	that	DET
cana-586	35	7	conjecture	conjecture	NOUN
cana-586	35	8	posed	pose	VERB
cana-586	35	9	by	by	ADP
cana-586	35	10	liaw	liaw	PROPN
cana-586	35	11	et	et	PROPN
cana-586	35	12	al	al	PROPN
cana-586	36	1	[	[	X
cana-586	36	2	7	7	NUM
cana-586	36	3	]	]	X
cana-586	36	4	is	be	AUX
cana-586	36	5	true	true	ADJ
cana-586	36	6	.	.	PUNCT
cana-586	37	1	further	far	ADV
cana-586	37	2	,	,	PUNCT
cana-586	37	3	we	we	PRON
cana-586	37	4	prove	prove	VERB
cana-586	37	5	a	a	DET
cana-586	37	6	characterization	characterization	NOUN
cana-586	37	7	result	result	NOUN
cana-586	37	8	that	that	SCONJ
cana-586	37	9	every	every	DET
cana-586	37	10	bipartite	bipartite	NOUN
cana-586	37	11	graph	graph	NOUN
cana-586	37	12	is	be	AUX
cana-586	37	13	an	an	DET
cana-586	37	14	induced	induced	ADJ
cana-586	37	15	subgraph	subgraph	NOUN
cana-586	37	16	of	of	ADP
cana-586	37	17	a	a	DET
cana-586	37	18	sum	sum	NOUN
cana-586	37	19	graph	graph	NOUN
cana-586	37	20	g	g	NOUN
cana-586	37	21	with	with	ADP
cana-586	37	22	sum	sum	NOUN
cana-586	37	23	number	number	NOUN
cana-586	37	24	σ(g	σ(g	NOUN
cana-586	37	25	)	)	PUNCT
cana-586	37	26	=	=	SYM
cana-586	37	27	1	1	X
cana-586	37	28	.	.	X
cana-586	37	29	communications	communication	NOUN
cana-586	37	30	on	on	ADP
cana-586	37	31	applied	apply	VERB
cana-586	37	32	nonlinear	nonlinear	ADJ
cana-586	37	33	analysis	analysis	NOUN
cana-586	37	34	issn	issn	NOUN
cana-586	37	35	:	:	PUNCT
cana-586	37	36	1074	1074	NUM
cana-586	37	37	-	-	PUNCT
cana-586	37	38	133x	133x	NUM
cana-586	37	39	vol	vol	NOUN
cana-586	37	40	31	31	NUM
cana-586	37	41	no	no	NOUN
cana-586	37	42	.	.	NOUN
cana-586	37	43	2	2	NUM
cana-586	37	44	(	(	PUNCT
cana-586	37	45	2024	2024	NUM
cana-586	37	46	)	)	PUNCT
cana-586	37	47	405	405	NUM
cana-586	37	48	https://internationalpubls.com	https://internationalpubls.com	X
cana-586	37	49	2	2	X
cana-586	37	50	.	.	X
cana-586	38	1	trees	tree	NOUN
cana-586	38	2	are	be	AUX
cana-586	38	3	integral	integral	ADJ
cana-586	38	4	sum	sum	NOUN
cana-586	38	5	graphs	graph	NOUN
cana-586	38	6	in	in	ADP
cana-586	38	7	this	this	DET
cana-586	38	8	section	section	NOUN
cana-586	38	9	,	,	PUNCT
cana-586	38	10	we	we	PRON
cana-586	38	11	prove	prove	VERB
cana-586	38	12	our	our	PRON
cana-586	38	13	main	main	ADJ
cana-586	38	14	result	result	NOUN
cana-586	38	15	that	that	SCONJ
cana-586	38	16	all	all	DET
cana-586	38	17	trees	tree	NOUN
cana-586	38	18	are	be	AUX
cana-586	38	19	integral	integral	ADJ
cana-586	38	20	sum	sum	NOUN
cana-586	38	21	graphs	graph	NOUN
cana-586	38	22	.	.	PUNCT
cana-586	39	1	theorem	theorem	NOUN
cana-586	39	2	1	1	NUM
cana-586	39	3	.	.	PUNCT
cana-586	40	1	every	every	DET
cana-586	40	2	tree	tree	NOUN
cana-586	40	3	t	t	NOUN
cana-586	40	4	with	with	ADP
cana-586	40	5	n	n	PRON
cana-586	40	6	vertices	vertex	NOUN
cana-586	40	7	is	be	AUX
cana-586	40	8	an	an	DET
cana-586	40	9	integral	integral	ADJ
cana-586	40	10	sum	sum	NOUN
cana-586	40	11	graph	graph	NOUN
cana-586	40	12	.	.	PUNCT
cana-586	41	1	proof	proof	NOUN
cana-586	41	2	.	.	PUNCT
cana-586	42	1	let	let	VERB
cana-586	42	2	t	t	NOUN
cana-586	42	3	be	be	AUX
cana-586	42	4	an	an	DET
cana-586	42	5	arbitrary	arbitrary	ADJ
cana-586	42	6	tree	tree	NOUN
cana-586	42	7	with	with	ADP
cana-586	42	8	n	n	ADP
cana-586	42	9	vertices	vertex	NOUN
cana-586	42	10	.	.	PUNCT
cana-586	43	1	since	since	SCONJ
cana-586	43	2	trees	tree	NOUN
cana-586	43	3	are	be	AUX
cana-586	43	4	bipartite	bipartite	ADJ
cana-586	43	5	,	,	PUNCT
cana-586	43	6	consider	consider	VERB
cana-586	43	7	the	the	DET
cana-586	43	8	bipartition	bipartition	NOUN
cana-586	43	9	of	of	ADP
cana-586	43	10	vertex	vertex	NOUN
cana-586	43	11	set	set	NOUN
cana-586	43	12	of	of	ADP
cana-586	43	13	t	t	PROPN
cana-586	43	14	as	as	ADP
cana-586	43	15	v(t	v(t	NOUN
cana-586	43	16	)	)	PUNCT
cana-586	43	17	=	=	SYM
cana-586	43	18	v1	v1	VERB
cana-586	43	19	∪	∪	NOUN
cana-586	43	20	v2	v2	NOUN
cana-586	43	21	.	.	PUNCT
cana-586	44	1	without	without	ADP
cana-586	44	2	loss	loss	NOUN
cana-586	44	3	of	of	ADP
cana-586	44	4	generality	generality	NOUN
cana-586	44	5	,	,	PUNCT
cana-586	44	6	let	let	VERB
cana-586	44	7	|v1|	|v1|	NOUN
cana-586	44	8	≥	≥	NOUN
cana-586	44	9	|v2|	|v2|	ADV
cana-586	44	10	.	.	PUNCT
cana-586	45	1	let	let	VERB
cana-586	45	2	|v1|	|v1|	NOUN
cana-586	45	3	=	=	SYM
cana-586	45	4	k	k	PROPN
cana-586	45	5	and	and	CCONJ
cana-586	45	6	|v2|	|v2|	ADV
cana-586	45	7	=	=	PUNCT
cana-586	45	8	r.	r.	NOUN
cana-586	45	9	consider	consider	VERB
cana-586	45	10	the	the	DET
cana-586	45	11	vertices	vertex	NOUN
cana-586	45	12	in	in	ADP
cana-586	45	13	v1	v1	NOUN
cana-586	45	14	as	as	ADP
cana-586	45	15	v1	v1	NOUN
cana-586	45	16	=	=	SYM
cana-586	45	17	{	{	PUNCT
cana-586	45	18	u1	u1	NOUN
cana-586	45	19	,	,	PUNCT
cana-586	45	20	u2	u2	PROPN
cana-586	45	21	,	,	PUNCT
cana-586	45	22	·	·	PUNCT
cana-586	45	23	·	·	PUNCT
cana-586	45	24	·	·	PUNCT
cana-586	45	25	,	,	PUNCT
cana-586	45	26	uk}and	uk}and	PUNCT
cana-586	45	27	v2	v2	NOUN
cana-586	45	28	as	as	ADP
cana-586	45	29	v2	v2	PROPN
cana-586	45	30	=	=	SYM
cana-586	45	31	{	{	PUNCT
cana-586	45	32	v1	v1	PROPN
cana-586	45	33	,	,	PUNCT
cana-586	45	34	v2	v2	PROPN
cana-586	45	35	,	,	PUNCT
cana-586	45	36	·	·	PUNCT
cana-586	45	37	·	·	PUNCT
cana-586	45	38	·	·	PUNCT
cana-586	45	39	,	,	PUNCT
cana-586	45	40	vr	vr	NOUN
cana-586	45	41	}	}	PUNCT
cana-586	45	42	.	.	PUNCT
cana-586	46	1	label	label	VERB
cana-586	46	2	the	the	DET
cana-586	46	3	vertices	vertex	NOUN
cana-586	46	4	of	of	ADP
cana-586	46	5	tree	tree	NOUN
cana-586	46	6	t	t	PROPN
cana-586	46	7	as	as	SCONJ
cana-586	46	8	follows	follow	VERB
cana-586	46	9	:	:	PUNCT
cana-586	46	10	f	f	PROPN
cana-586	46	11	(	(	PUNCT
cana-586	46	12	ui	ui	PROPN
cana-586	46	13	)	)	PUNCT
cana-586	47	1	=	=	VERB
cana-586	48	1	i	i	PRON
cana-586	48	2	−	−	VERB
cana-586	48	3	1	1	NUM
cana-586	48	4	for	for	ADP
cana-586	48	5	1	1	NUM
cana-586	48	6	≤	≤	NUM
cana-586	48	7	i	i	NOUN
cana-586	48	8	≤	≤	NOUN
cana-586	49	1	k	k	PROPN
cana-586	49	2	and	and	CCONJ
cana-586	49	3	f	f	PROPN
cana-586	49	4	(	(	PUNCT
cana-586	49	5	vi	vi	NOUN
cana-586	49	6	)	)	PUNCT
cana-586	49	7	=	=	SYM
cana-586	49	8	−i	−i	ADJ
cana-586	49	9	for	for	ADP
cana-586	49	10	1	1	NUM
cana-586	49	11	≤	≤	NUM
cana-586	49	12	i	i	PRON
cana-586	49	13	≤	≤	PROPN
cana-586	49	14	r.	r.	X
cana-586	49	15	by	by	ADP
cana-586	49	16	the	the	DET
cana-586	49	17	definition	definition	NOUN
cana-586	49	18	of	of	ADP
cana-586	49	19	function	function	NOUN
cana-586	49	20	f	f	NOUN
cana-586	49	21	,	,	PUNCT
cana-586	49	22	the	the	DET
cana-586	49	23	vertex	vertex	NOUN
cana-586	49	24	labels	label	NOUN
cana-586	49	25	of	of	ADP
cana-586	49	26	vertices	vertex	NOUN
cana-586	49	27	in	in	ADP
cana-586	49	28	v1	v1	NOUN
cana-586	49	29	are	be	AUX
cana-586	49	30	from	from	ADP
cana-586	49	31	the	the	DET
cana-586	49	32	set	set	NOUN
cana-586	49	33	{	{	PUNCT
cana-586	49	34	0	0	NUM
cana-586	49	35	,	,	PUNCT
cana-586	49	36	1	1	NUM
cana-586	49	37	,	,	PUNCT
cana-586	49	38	2	2	NUM
cana-586	49	39	,	,	PUNCT
cana-586	49	40	·	·	PUNCT
cana-586	49	41	·	·	PUNCT
cana-586	50	1	·	·	PUNCT
cana-586	50	2	,	,	PUNCT
cana-586	50	3	k	k	PROPN
cana-586	50	4	−	−	PROPN
cana-586	50	5	1	1	NUM
cana-586	50	6	}	}	PUNCT
cana-586	50	7	and	and	CCONJ
cana-586	50	8	the	the	DET
cana-586	50	9	vertex	vertex	NOUN
cana-586	50	10	labels	label	NOUN
cana-586	50	11	of	of	ADP
cana-586	50	12	vertices	vertex	NOUN
cana-586	50	13	in	in	ADP
cana-586	50	14	v2	v2	PROPN
cana-586	50	15	are	be	AUX
cana-586	50	16	from	from	ADP
cana-586	50	17	the	the	DET
cana-586	50	18	set	set	NOUN
cana-586	50	19	{	{	PUNCT
cana-586	50	20	−1	−1	NOUN
cana-586	50	21	,	,	PUNCT
cana-586	50	22	−2	−2	NOUN
cana-586	50	23	,	,	PUNCT
cana-586	50	24	·	·	PUNCT
cana-586	50	25	·	·	PUNCT
cana-586	50	26	·	·	PUNCT
cana-586	50	27	,	,	PUNCT
cana-586	50	28	−r	−r	ADJ
cana-586	50	29	}	}	PUNCT
cana-586	50	30	.	.	PUNCT
cana-586	51	1	therefore	therefore	ADV
cana-586	51	2	,	,	PUNCT
cana-586	51	3	it	it	PRON
cana-586	51	4	is	be	AUX
cana-586	51	5	clear	clear	ADJ
cana-586	51	6	that	that	SCONJ
cana-586	51	7	f	f	PROPN
cana-586	51	8	is	be	AUX
cana-586	51	9	a	a	DET
cana-586	51	10	bijection	bijection	NOUN
cana-586	51	11	from	from	ADP
cana-586	51	12	the	the	DET
cana-586	51	13	set	set	NOUN
cana-586	51	14	of	of	ADP
cana-586	51	15	vertices	vertex	NOUN
cana-586	51	16	of	of	ADP
cana-586	51	17	t	t	PROPN
cana-586	51	18	to	to	ADP
cana-586	51	19	a	a	DET
cana-586	51	20	subset	subset	NOUN
cana-586	51	21	of	of	ADP
cana-586	51	22	integers	integer	NOUN
cana-586	51	23	.	.	PUNCT
cana-586	52	1	the	the	DET
cana-586	52	2	edge	edge	NOUN
cana-586	52	3	label	label	NOUN
cana-586	52	4	of	of	ADP
cana-586	52	5	an	an	DET
cana-586	52	6	arbitrary	arbitrary	ADJ
cana-586	52	7	edge	edge	NOUN
cana-586	52	8	e	e	NOUN
cana-586	52	9	=	=	NOUN
cana-586	52	10	uv	uv	NOUN
cana-586	52	11	defined	define	VERB
cana-586	52	12	by	by	ADP
cana-586	52	13	f	f	PROPN
cana-586	52	14	(	(	PUNCT
cana-586	52	15	e	e	NOUN
cana-586	52	16	)	)	PUNCT
cana-586	52	17	=	=	SYM
cana-586	52	18	f	f	X
cana-586	52	19	(	(	PUNCT
cana-586	52	20	u	u	NOUN
cana-586	52	21	)	)	PUNCT
cana-586	53	1	+	+	NUM
cana-586	53	2	f	f	X
cana-586	53	3	(	(	PUNCT
cana-586	53	4	v	v	NOUN
cana-586	53	5	)	)	PUNCT
cana-586	53	6	.	.	PUNCT
cana-586	54	1	it	it	PRON
cana-586	54	2	is	be	AUX
cana-586	54	3	enough	enough	ADJ
cana-586	54	4	that	that	SCONJ
cana-586	54	5	if	if	SCONJ
cana-586	54	6	we	we	PRON
cana-586	54	7	prove	prove	VERB
cana-586	54	8	that	that	SCONJ
cana-586	54	9	there	there	PRON
cana-586	54	10	exists	exist	VERB
cana-586	54	11	a	a	DET
cana-586	54	12	vertex	vertex	NOUN
cana-586	54	13	in	in	ADP
cana-586	54	14	t	t	NOUN
cana-586	54	15	whose	whose	DET
cana-586	54	16	vertex	vertex	NOUN
cana-586	54	17	label	label	NOUN
cana-586	54	18	is	be	AUX
cana-586	54	19	f	f	PROPN
cana-586	54	20	(	(	PUNCT
cana-586	54	21	u	u	NOUN
cana-586	54	22	)	)	PUNCT
cana-586	55	1	+	+	NUM
cana-586	55	2	f	f	X
cana-586	55	3	(	(	PUNCT
cana-586	55	4	v	v	NOUN
cana-586	55	5	)	)	PUNCT
cana-586	55	6	.	.	PUNCT
cana-586	56	1	let	let	VERB
cana-586	56	2	us	we	PRON
cana-586	56	3	assume	assume	VERB
cana-586	56	4	the	the	DET
cana-586	56	5	contrary	contrary	NOUN
cana-586	56	6	that	that	PRON
cana-586	56	7	exists	exist	VERB
cana-586	56	8	an	an	DET
cana-586	56	9	edge	edge	NOUN
cana-586	56	10	label	label	NOUN
cana-586	56	11	f	f	PROPN
cana-586	56	12	(	(	PUNCT
cana-586	56	13	e	e	NOUN
cana-586	56	14	)	)	PUNCT
cana-586	56	15	=	=	SYM
cana-586	56	16	f	f	X
cana-586	56	17	(	(	PUNCT
cana-586	56	18	u	u	NOUN
cana-586	56	19	)	)	PUNCT
cana-586	56	20	+	+	NUM
cana-586	56	21	f	f	X
cana-586	56	22	(	(	PUNCT
cana-586	56	23	v	v	NOUN
cana-586	56	24	)	)	PUNCT
cana-586	56	25	but	but	CCONJ
cana-586	56	26	there	there	PRON
cana-586	56	27	does	do	AUX
cana-586	56	28	not	not	PART
cana-586	56	29	exist	exist	VERB
cana-586	56	30	a	a	DET
cana-586	56	31	vertex	vertex	NOUN
cana-586	56	32	whose	whose	DET
cana-586	56	33	label	label	NOUN
cana-586	56	34	is	be	AUX
cana-586	56	35	f	f	PROPN
cana-586	56	36	(	(	PUNCT
cana-586	56	37	e	e	NOUN
cana-586	56	38	)	)	PUNCT
cana-586	56	39	.	.	PUNCT
cana-586	57	1	therefore	therefore	ADV
cana-586	57	2	,	,	PUNCT
cana-586	57	3	by	by	ADP
cana-586	57	4	our	our	PRON
cana-586	57	5	assumption	assumption	NOUN
cana-586	57	6	f	f	X
cana-586	57	7	(	(	PUNCT
cana-586	57	8	u	u	NOUN
cana-586	57	9	)	)	PUNCT
cana-586	57	10	+	+	NUM
cana-586	57	11	f	f	X
cana-586	57	12	(	(	PUNCT
cana-586	57	13	v	v	NOUN
cana-586	57	14	)	)	PUNCT
cana-586	57	15	≥	≥	NOUN
cana-586	57	16	k	k	NOUN
cana-586	57	17	or	or	CCONJ
cana-586	57	18	f	f	PROPN
cana-586	57	19	(	(	PUNCT
cana-586	57	20	u	u	NOUN
cana-586	57	21	)	)	PUNCT
cana-586	58	1	+	+	NUM
cana-586	58	2	f	f	X
cana-586	58	3	(	(	PUNCT
cana-586	58	4	v	v	NOUN
cana-586	58	5	)	)	PUNCT
cana-586	58	6	<	<	X
cana-586	58	7	−r	−r	PROPN
cana-586	58	8	.	.	PUNCT
cana-586	59	1	since	since	SCONJ
cana-586	59	2	e	e	PROPN
cana-586	59	3	=	=	NOUN
cana-586	59	4	uv	uv	NOUN
cana-586	59	5	and	and	CCONJ
cana-586	59	6	t	t	PROPN
cana-586	59	7	is	be	AUX
cana-586	59	8	bipartite	bipartite	ADJ
cana-586	59	9	,	,	PUNCT
cana-586	59	10	one	one	NUM
cana-586	59	11	end	end	NOUN
cana-586	59	12	vertex	vertex	NOUN
cana-586	59	13	of	of	ADP
cana-586	59	14	the	the	DET
cana-586	59	15	edge	edge	NOUN
cana-586	59	16	e	e	NOUN
cana-586	59	17	=	=	NOUN
cana-586	59	18	uv	uv	NOUN
cana-586	59	19	is	be	AUX
cana-586	59	20	in	in	ADP
cana-586	59	21	v1	v1	NOUN
cana-586	59	22	and	and	CCONJ
cana-586	59	23	the	the	DET
cana-586	59	24	other	other	ADJ
cana-586	59	25	end	end	NOUN
cana-586	59	26	vertex	vertex	NOUN
cana-586	59	27	is	be	AUX
cana-586	59	28	in	in	ADP
cana-586	59	29	v2	v2	PROPN
cana-586	59	30	.	.	PUNCT
cana-586	60	1	without	without	ADP
cana-586	60	2	loss	loss	NOUN
cana-586	60	3	of	of	ADP
cana-586	60	4	generality	generality	NOUN
cana-586	60	5	,	,	PUNCT
cana-586	60	6	let	let	VERB
cana-586	60	7	us	we	PRON
cana-586	60	8	assume	assume	VERB
cana-586	60	9	that	that	SCONJ
cana-586	60	10	u	u	PROPN
cana-586	60	11	∈	∈	PROPN
cana-586	60	12	v1	v1	NOUN
cana-586	60	13	and	and	CCONJ
cana-586	60	14	v	v	ADP
cana-586	60	15	∈	∈	PROPN
cana-586	60	16	v2	v2	NOUN
cana-586	60	17	.	.	PUNCT
cana-586	61	1	therefore	therefore	ADV
cana-586	61	2	,	,	PUNCT
cana-586	61	3	f	f	PROPN
cana-586	61	4	(	(	PUNCT
cana-586	61	5	u	u	NOUN
cana-586	61	6	)	)	PUNCT
cana-586	61	7	∈	∈	PROPN
cana-586	61	8	{	{	PUNCT
cana-586	61	9	0	0	NUM
cana-586	61	10	,	,	PUNCT
cana-586	61	11	1	1	NUM
cana-586	61	12	,	,	PUNCT
cana-586	61	13	2	2	NUM
cana-586	61	14	,	,	PUNCT
cana-586	61	15	·	·	PUNCT
cana-586	61	16	·	·	PUNCT
cana-586	62	1	·	·	PUNCT
cana-586	62	2	,	,	PUNCT
cana-586	62	3	k	k	PROPN
cana-586	62	4	−	−	PROPN
cana-586	62	5	1	1	NUM
cana-586	62	6	}	}	PUNCT
cana-586	62	7	and	and	CCONJ
cana-586	62	8	f	f	PROPN
cana-586	62	9	(	(	PUNCT
cana-586	62	10	v	v	NOUN
cana-586	62	11	)	)	PUNCT
cana-586	62	12	∈	∈	PROPN
cana-586	62	13	{	{	PUNCT
cana-586	62	14	−1	−1	NOUN
cana-586	62	15	,	,	PUNCT
cana-586	62	16	−2	−2	NOUN
cana-586	62	17	,	,	PUNCT
cana-586	62	18	·	·	PUNCT
cana-586	62	19	·	·	PUNCT
cana-586	62	20	·	·	PUNCT
cana-586	62	21	,	,	PUNCT
cana-586	62	22	−r	−r	ADJ
cana-586	62	23	}	}	PUNCT
cana-586	62	24	case	case	NOUN
cana-586	62	25	1	1	NUM
cana-586	62	26	:	:	PUNCT
cana-586	62	27	|f	|f	PROPN
cana-586	62	28	(	(	PUNCT
cana-586	62	29	u)|	u)|	NOUN
cana-586	62	30	≥	≥	NOUN
cana-586	62	31	|f	|f	PROPN
cana-586	62	32	(	(	PUNCT
cana-586	62	33	v)|	v)|	NOUN
cana-586	62	34	since	since	SCONJ
cana-586	62	35	f	f	PROPN
cana-586	62	36	(	(	PUNCT
cana-586	62	37	v	v	NOUN
cana-586	62	38	)	)	PUNCT
cana-586	62	39	∈	∈	PROPN
cana-586	62	40	{	{	PUNCT
cana-586	62	41	−1	−1	NOUN
cana-586	62	42	,	,	PUNCT
cana-586	62	43	−2	−2	NOUN
cana-586	62	44	,	,	PUNCT
cana-586	62	45	·	·	PUNCT
cana-586	62	46	·	·	PUNCT
cana-586	62	47	·	·	PUNCT
cana-586	62	48	,	,	PUNCT
cana-586	62	49	−r	−r	ADJ
cana-586	62	50	}	}	PUNCT
cana-586	62	51	,	,	PUNCT
cana-586	62	52	implies	imply	VERB
cana-586	62	53	that	that	SCONJ
cana-586	62	54	f	f	PROPN
cana-586	62	55	(	(	PUNCT
cana-586	62	56	u	u	NOUN
cana-586	62	57	)	)	PUNCT
cana-586	62	58	+	+	NUM
cana-586	62	59	f	f	X
cana-586	62	60	(	(	PUNCT
cana-586	62	61	v	v	NOUN
cana-586	62	62	)	)	PUNCT
cana-586	62	63	≥	≥	NOUN
cana-586	62	64	0	0	NUM
cana-586	62	65	and	and	CCONJ
cana-586	62	66	f	f	PROPN
cana-586	62	67	(	(	PUNCT
cana-586	62	68	u)+f	u)+f	PROPN
cana-586	62	69	(	(	PUNCT
cana-586	62	70	v	v	NOUN
cana-586	62	71	)	)	PUNCT
cana-586	62	72	<	<	X
cana-586	62	73	f	f	X
cana-586	62	74	(	(	PUNCT
cana-586	62	75	u	u	NOUN
cana-586	62	76	)	)	PUNCT
cana-586	62	77	.	.	PUNCT
cana-586	63	1	this	this	PRON
cana-586	63	2	implies	imply	VERB
cana-586	63	3	that	that	SCONJ
cana-586	63	4	f	f	PROPN
cana-586	63	5	(	(	PUNCT
cana-586	63	6	u)+f	u)+f	PROPN
cana-586	63	7	(	(	PUNCT
cana-586	63	8	v	v	NOUN
cana-586	63	9	)	)	PUNCT
cana-586	63	10	lies	lie	VERB
cana-586	63	11	in	in	ADP
cana-586	63	12	the	the	DET
cana-586	63	13	set	set	NOUN
cana-586	63	14	{	{	PUNCT
cana-586	63	15	0	0	NUM
cana-586	63	16	,	,	PUNCT
cana-586	63	17	1	1	NUM
cana-586	63	18	,	,	PUNCT
cana-586	63	19	·	·	PUNCT
cana-586	63	20	·	·	PUNCT
cana-586	63	21	·	·	PUNCT
cana-586	63	22	,	,	PUNCT
cana-586	63	23	f	f	PROPN
cana-586	63	24	(	(	PUNCT
cana-586	63	25	u)−|f	u)−|f	PROPN
cana-586	63	26	(	(	PUNCT
cana-586	63	27	v)|	v)|	NOUN
cana-586	63	28	}	}	PUNCT
cana-586	63	29	,	,	PUNCT
cana-586	63	30	a	a	DET
cana-586	63	31	contradiction	contradiction	NOUN
cana-586	63	32	to	to	ADP
cana-586	63	33	our	our	PRON
cana-586	63	34	assumption	assumption	NOUN
cana-586	63	35	that	that	SCONJ
cana-586	63	36	f	f	PROPN
cana-586	63	37	(	(	PUNCT
cana-586	63	38	u)+f	u)+f	PROPN
cana-586	63	39	(	(	PUNCT
cana-586	63	40	v	v	NOUN
cana-586	63	41	)	)	PUNCT
cana-586	63	42	≥	≥	NOUN
cana-586	63	43	k	k	NOUN
cana-586	63	44	since	since	SCONJ
cana-586	63	45	|f	|f	PROPN
cana-586	63	46	(	(	PUNCT
cana-586	63	47	v)|	v)|	NOUN
cana-586	63	48	≥	≥	NUM
cana-586	63	49	1	1	NUM
cana-586	63	50	and	and	CCONJ
cana-586	63	51	f	f	PROPN
cana-586	63	52	(	(	PUNCT
cana-586	63	53	u)−|f	u)−|f	ADP
cana-586	63	54	(	(	PUNCT
cana-586	63	55	v)|	v)|	NOUN
cana-586	63	56	<	<	X
cana-586	63	57	k.	k.	PROPN
cana-586	63	58	therefore	therefore	ADV
cana-586	63	59	,	,	PUNCT
cana-586	63	60	in	in	ADP
cana-586	63	61	this	this	DET
cana-586	63	62	case	case	NOUN
cana-586	63	63	,	,	PUNCT
cana-586	63	64	our	our	PRON
cana-586	63	65	assumption	assumption	NOUN
cana-586	63	66	that	that	SCONJ
cana-586	63	67	there	there	PRON
cana-586	63	68	does	do	AUX
cana-586	63	69	not	not	PART
cana-586	63	70	exist	exist	VERB
cana-586	63	71	a	a	DET
cana-586	63	72	vertex	vertex	NOUN
cana-586	63	73	whose	whose	DET
cana-586	63	74	vertex	vertex	NOUN
cana-586	63	75	label	label	NOUN
cana-586	63	76	is	be	AUX
cana-586	63	77	f	f	PROPN
cana-586	63	78	(	(	PUNCT
cana-586	63	79	e	e	NOUN
cana-586	63	80	)	)	PUNCT
cana-586	63	81	=	=	SYM
cana-586	63	82	f	f	X
cana-586	63	83	(	(	PUNCT
cana-586	63	84	u	u	NOUN
cana-586	63	85	)	)	PUNCT
cana-586	63	86	+	+	NUM
cana-586	63	87	f	f	X
cana-586	63	88	(	(	PUNCT
cana-586	63	89	v	v	NOUN
cana-586	63	90	)	)	PUNCT
cana-586	63	91	is	be	AUX
cana-586	63	92	wrong	wrong	ADJ
cana-586	63	93	.	.	PUNCT
cana-586	64	1	case	case	NOUN
cana-586	64	2	2	2	NUM
cana-586	64	3	:	:	PUNCT
cana-586	64	4	|f	|f	PROPN
cana-586	64	5	(	(	PUNCT
cana-586	64	6	u)|	u)|	NOUN
cana-586	64	7	≤	≤	PROPN
cana-586	64	8	|f	|f	PROPN
cana-586	64	9	(	(	PUNCT
cana-586	64	10	v)|	v)|	NOUN
cana-586	64	11	since	since	SCONJ
cana-586	64	12	f	f	PROPN
cana-586	64	13	(	(	PUNCT
cana-586	64	14	v	v	NOUN
cana-586	64	15	)	)	PUNCT
cana-586	64	16	∈	∈	PROPN
cana-586	64	17	{	{	PUNCT
cana-586	64	18	−1	−1	NOUN
cana-586	64	19	,	,	PUNCT
cana-586	64	20	−2	−2	NOUN
cana-586	64	21	,	,	PUNCT
cana-586	64	22	·	·	PUNCT
cana-586	64	23	·	·	PUNCT
cana-586	64	24	·	·	PUNCT
cana-586	64	25	,	,	PUNCT
cana-586	64	26	−r	−r	ADJ
cana-586	64	27	}	}	PUNCT
cana-586	64	28	,	,	PUNCT
cana-586	64	29	implies	imply	VERB
cana-586	64	30	that	that	SCONJ
cana-586	65	1	f	f	PROPN
cana-586	65	2	(	(	PUNCT
cana-586	65	3	u	u	NOUN
cana-586	65	4	)	)	PUNCT
cana-586	66	1	+	+	NUM
cana-586	66	2	f	f	X
cana-586	66	3	(	(	PUNCT
cana-586	66	4	v	v	NOUN
cana-586	66	5	)	)	PUNCT
cana-586	66	6	≤	≤	NOUN
cana-586	66	7	0	0	NUM
cana-586	66	8	and	and	CCONJ
cana-586	66	9	f	f	PROPN
cana-586	66	10	(	(	PUNCT
cana-586	66	11	u	u	NOUN
cana-586	66	12	)	)	PUNCT
cana-586	66	13	+	+	NUM
cana-586	66	14	f	f	X
cana-586	66	15	(	(	PUNCT
cana-586	66	16	v	v	NOUN
cana-586	66	17	)	)	PUNCT
cana-586	66	18	>	>	X
cana-586	66	19	f	f	PROPN
cana-586	66	20	(	(	PUNCT
cana-586	66	21	v	v	NOUN
cana-586	66	22	)	)	PUNCT
cana-586	66	23	.	.	PUNCT
cana-586	67	1	this	this	PRON
cana-586	67	2	implies	imply	VERB
cana-586	67	3	that	that	SCONJ
cana-586	67	4	f	f	PROPN
cana-586	67	5	(	(	PUNCT
cana-586	67	6	u	u	NOUN
cana-586	67	7	)	)	PUNCT
cana-586	68	1	+	+	NUM
cana-586	68	2	f	f	X
cana-586	68	3	(	(	PUNCT
cana-586	68	4	v	v	NOUN
cana-586	68	5	)	)	PUNCT
cana-586	68	6	lies	lie	VERB
cana-586	68	7	in	in	ADP
cana-586	68	8	the	the	DET
cana-586	68	9	set	set	NOUN
cana-586	68	10	{	{	PUNCT
cana-586	68	11	0	0	NUM
cana-586	68	12	,	,	PUNCT
cana-586	68	13	−1	−1	NOUN
cana-586	68	14	,	,	PUNCT
cana-586	68	15	·	·	PUNCT
cana-586	68	16	·	·	PUNCT
cana-586	69	1	·	·	PUNCT
cana-586	69	2	,	,	PUNCT
cana-586	69	3	f	f	PROPN
cana-586	69	4	(	(	PUNCT
cana-586	69	5	u	u	NOUN
cana-586	69	6	)	)	PUNCT
cana-586	69	7	+	+	NUM
cana-586	69	8	f	f	X
cana-586	69	9	(	(	PUNCT
cana-586	69	10	v	v	NOUN
cana-586	69	11	)	)	PUNCT
cana-586	69	12	}	}	PUNCT
cana-586	69	13	,	,	PUNCT
cana-586	69	14	a	a	DET
cana-586	69	15	contradiction	contradiction	NOUN
cana-586	69	16	to	to	ADP
cana-586	69	17	our	our	PRON
cana-586	69	18	assumption	assumption	NOUN
cana-586	69	19	that	that	SCONJ
cana-586	69	20	f	f	PROPN
cana-586	69	21	(	(	PUNCT
cana-586	69	22	u	u	NOUN
cana-586	69	23	)	)	PUNCT
cana-586	69	24	+	+	NUM
cana-586	69	25	f	f	X
cana-586	69	26	(	(	PUNCT
cana-586	69	27	v	v	NOUN
cana-586	69	28	)	)	PUNCT
cana-586	69	29	<	<	X
cana-586	69	30	−r	−r	PROPN
cana-586	69	31	since	since	SCONJ
cana-586	69	32	|f	|f	PROPN
cana-586	69	33	(	(	PUNCT
cana-586	69	34	u)|	u)|	NOUN
cana-586	69	35	≥	≥	NOUN
cana-586	69	36	0	0	NUM
cana-586	69	37	.	.	PUNCT
cana-586	70	1	therefore	therefore	ADV
cana-586	70	2	,	,	PUNCT
cana-586	70	3	in	in	ADP
cana-586	70	4	this	this	DET
cana-586	70	5	case	case	NOUN
cana-586	70	6	,	,	PUNCT
cana-586	70	7	our	our	PRON
cana-586	70	8	assumption	assumption	NOUN
cana-586	70	9	that	that	SCONJ
cana-586	70	10	there	there	PRON
cana-586	70	11	does	do	AUX
cana-586	70	12	not	not	PART
cana-586	70	13	exist	exist	VERB
cana-586	70	14	a	a	DET
cana-586	70	15	vertex	vertex	NOUN
cana-586	70	16	whose	whose	DET
cana-586	70	17	vertex	vertex	NOUN
cana-586	70	18	label	label	NOUN
cana-586	70	19	is	be	AUX
cana-586	70	20	f	f	PROPN
cana-586	70	21	(	(	PUNCT
cana-586	70	22	e	e	NOUN
cana-586	70	23	)	)	PUNCT
cana-586	70	24	=	=	SYM
cana-586	70	25	f	f	X
cana-586	70	26	(	(	PUNCT
cana-586	70	27	u	u	NOUN
cana-586	70	28	)	)	PUNCT
cana-586	70	29	+	+	NUM
cana-586	70	30	f	f	X
cana-586	70	31	(	(	PUNCT
cana-586	70	32	v	v	NOUN
cana-586	70	33	)	)	PUNCT
cana-586	70	34	is	be	AUX
cana-586	70	35	wrong	wrong	ADJ
cana-586	70	36	.	.	PUNCT
cana-586	71	1	therefore	therefore	ADV
cana-586	71	2	,	,	PUNCT
cana-586	71	3	there	there	PRON
cana-586	71	4	exists	exist	VERB
cana-586	71	5	a	a	DET
cana-586	71	6	vertex	vertex	NOUN
cana-586	71	7	in	in	ADP
cana-586	71	8	t	t	NOUN
cana-586	71	9	whose	whose	DET
cana-586	71	10	vertex	vertex	NOUN
cana-586	71	11	label	label	NOUN
cana-586	71	12	is	be	AUX
cana-586	71	13	f	f	PROPN
cana-586	71	14	(	(	PUNCT
cana-586	71	15	u	u	NOUN
cana-586	71	16	)	)	PUNCT
cana-586	71	17	+	+	NUM
cana-586	71	18	f	f	X
cana-586	71	19	(	(	PUNCT
cana-586	71	20	v	v	NOUN
cana-586	71	21	)	)	PUNCT
cana-586	71	22	.	.	PUNCT
cana-586	72	1	this	this	PRON
cana-586	72	2	proves	prove	VERB
cana-586	72	3	that	that	SCONJ
cana-586	72	4	the	the	DET
cana-586	72	5	labeling	labeling	NOUN
cana-586	72	6	function	function	NOUN
cana-586	72	7	f	f	PROPN
cana-586	72	8	satisfies	satisfy	VERB
cana-586	72	9	the	the	DET
cana-586	72	10	conditions	condition	NOUN
cana-586	72	11	of	of	ADP
cana-586	72	12	integral	integral	ADJ
cana-586	72	13	sum	sum	NOUN
cana-586	72	14	graphs	graph	NOUN
cana-586	72	15	.	.	PUNCT
cana-586	73	1	therefore	therefore	ADV
cana-586	73	2	,	,	PUNCT
cana-586	73	3	tree	tree	NOUN
cana-586	73	4	t	t	PROPN
cana-586	73	5	is	be	AUX
cana-586	73	6	an	an	DET
cana-586	73	7	integral	integral	ADJ
cana-586	73	8	sum	sum	NOUN
cana-586	73	9	graph	graph	NOUN
cana-586	73	10	.	.	PUNCT
cana-586	74	1	2.1	2.1	NUM
cana-586	74	2	illustrative	illustrative	ADJ
cana-586	74	3	example	example	NOUN
cana-586	74	4	for	for	ADP
cana-586	74	5	the	the	DET
cana-586	74	6	arbitrary	arbitrary	ADJ
cana-586	74	7	tree	tree	NOUN
cana-586	74	8	t	t	PROPN
cana-586	74	9	in	in	ADP
cana-586	74	10	figure	figure	NOUN
cana-586	74	11	1	1	NUM
cana-586	74	12	,	,	PUNCT
cana-586	74	13	the	the	DET
cana-586	74	14	bipartition	bipartition	NOUN
cana-586	74	15	of	of	ADP
cana-586	74	16	the	the	DET
cana-586	74	17	vertex	vertex	NOUN
cana-586	74	18	set	set	NOUN
cana-586	74	19	of	of	ADP
cana-586	74	20	t	t	PROPN
cana-586	74	21	is	be	AUX
cana-586	74	22	shown	show	VERB
cana-586	74	23	in	in	ADP
cana-586	74	24	figure	figure	NOUN
cana-586	74	25	2	2	NUM
cana-586	74	26	and	and	CCONJ
cana-586	74	27	its	its	PRON
cana-586	74	28	integral	integral	ADJ
cana-586	74	29	sum	sum	NOUN
cana-586	74	30	labeling	labeling	NOUN
cana-586	74	31	is	be	AUX
cana-586	74	32	shown	show	VERB
cana-586	74	33	in	in	ADP
cana-586	74	34	figure	figure	NOUN
cana-586	74	35	3	3	NUM
cana-586	74	36	.	.	PUNCT
cana-586	74	37	figure	figure	NOUN
cana-586	74	38	1	1	NUM
cana-586	74	39	:	:	PUNCT
cana-586	74	40	tree	tree	NOUN
cana-586	74	41	t	t	NOUN
cana-586	74	42	with	with	ADP
cana-586	74	43	23	23	NUM
cana-586	74	44	edges	edge	NOUN
cana-586	74	45	communications	communication	NOUN
cana-586	74	46	on	on	ADP
cana-586	74	47	applied	apply	VERB
cana-586	74	48	nonlinear	nonlinear	ADJ
cana-586	74	49	analysis	analysis	NOUN
cana-586	74	50	issn	issn	NOUN
cana-586	74	51	:	:	PUNCT
cana-586	74	52	1074	1074	NUM
cana-586	74	53	-	-	PUNCT
cana-586	74	54	133x	133x	NUM
cana-586	74	55	vol	vol	NOUN
cana-586	74	56	31	31	NUM
cana-586	74	57	no	no	NOUN
cana-586	74	58	.	.	NOUN
cana-586	74	59	2	2	NUM
cana-586	74	60	(	(	PUNCT
cana-586	74	61	2024	2024	NUM
cana-586	74	62	)	)	PUNCT
cana-586	74	63	406	406	NUM
cana-586	74	64	https://internationalpubls.com	https://internationalpubls.com	X
cana-586	74	65	figure	figure	NOUN
cana-586	74	66	2	2	NUM
cana-586	74	67	:	:	PUNCT
cana-586	74	68	bipartition	bipartition	NOUN
cana-586	74	69	of	of	ADP
cana-586	74	70	the	the	DET
cana-586	74	71	vertex	vertex	NOUN
cana-586	74	72	set	set	NOUN
cana-586	74	73	of	of	ADP
cana-586	74	74	tree	tree	NOUN
cana-586	74	75	t	t	NOUN
cana-586	74	76	0	0	NUM
cana-586	74	77	1	1	NUM
cana-586	74	78	2	2	NUM
cana-586	74	79	3	3	NUM
cana-586	74	80	−1	−1	NOUN
cana-586	74	81	4	4	NUM
cana-586	74	82	−2	−2	NOUN
cana-586	74	83	5	5	NUM
cana-586	74	84	−3	−3	NOUN
cana-586	74	85	6	6	NUM
cana-586	74	86	−4	−4	SYM
cana-586	74	87	7	7	NUM
cana-586	74	88	−5	−5	NOUN
cana-586	74	89	8	8	NUM
cana-586	74	90	−6	−6	NOUN
cana-586	74	91	9	9	NUM
cana-586	74	92	−7	−7	NOUN
cana-586	74	93	10	10	NUM
cana-586	74	94	−8	−8	SYM
cana-586	74	95	11	11	NUM
cana-586	74	96	−9	−9	NOUN
cana-586	74	97	12	12	NUM
cana-586	74	98	13	13	NUM
cana-586	74	99	14	14	NUM
cana-586	74	100	figure	figure	NOUN
cana-586	74	101	3	3	NUM
cana-586	74	102	:	:	PUNCT
cana-586	74	103	integral	integral	ADJ
cana-586	74	104	sum	sum	NOUN
cana-586	74	105	labeling	labeling	NOUN
cana-586	74	106	for	for	ADP
cana-586	74	107	tree	tree	NOUN
cana-586	74	108	t	t	NOUN
cana-586	74	109	3	3	NUM
cana-586	74	110	characterization	characterization	NOUN
cana-586	74	111	of	of	ADP
cana-586	74	112	sum	sum	NOUN
cana-586	74	113	graphs	graph	NOUN
cana-586	74	114	in	in	ADP
cana-586	74	115	this	this	DET
cana-586	74	116	section	section	NOUN
cana-586	74	117	,	,	PUNCT
cana-586	74	118	we	we	PRON
cana-586	74	119	prove	prove	VERB
cana-586	74	120	that	that	SCONJ
cana-586	74	121	any	any	DET
cana-586	74	122	bipartite	bipartite	NOUN
cana-586	74	123	graph	graph	NOUN
cana-586	74	124	is	be	AUX
cana-586	74	125	an	an	DET
cana-586	74	126	induced	induced	ADJ
cana-586	74	127	subgraph	subgraph	NOUN
cana-586	74	128	of	of	ADP
cana-586	74	129	a	a	DET
cana-586	74	130	sum	sum	NOUN
cana-586	74	131	graph	graph	NOUN
cana-586	74	132	communications	communication	NOUN
cana-586	74	133	on	on	ADP
cana-586	74	134	applied	apply	VERB
cana-586	74	135	nonlinear	nonlinear	ADJ
cana-586	74	136	analysis	analysis	NOUN
cana-586	74	137	issn	issn	NOUN
cana-586	74	138	:	:	PUNCT
cana-586	74	139	1074	1074	NUM
cana-586	74	140	-	-	PUNCT
cana-586	74	141	133x	133x	NUM
cana-586	74	142	vol	vol	NOUN
cana-586	74	143	31	31	NUM
cana-586	74	144	no	no	NOUN
cana-586	74	145	.	.	NOUN
cana-586	74	146	2	2	NUM
cana-586	74	147	(	(	PUNCT
cana-586	74	148	2024	2024	NUM
cana-586	74	149	)	)	PUNCT
cana-586	74	150	407	407	NUM
cana-586	74	151	https://internationalpubls.com	https://internationalpubls.com	X
cana-586	74	152	g	g	NOUN
cana-586	74	153	with	with	ADP
cana-586	74	154	sum	sum	NOUN
cana-586	74	155	number	number	NOUN
cana-586	74	156	σ(g	σ(g	NOUN
cana-586	74	157	)	)	PUNCT
cana-586	74	158	=	=	SYM
cana-586	74	159	1	1	X
cana-586	74	160	.	.	X
cana-586	74	161	theorem	theorem	NOUN
cana-586	74	162	2	2	NUM
cana-586	74	163	.	.	PUNCT
cana-586	75	1	let	let	VERB
cana-586	75	2	b	b	NOUN
cana-586	75	3	=	=	SYM
cana-586	75	4	(	(	PUNCT
cana-586	75	5	v1	v1	PROPN
cana-586	75	6	,	,	PUNCT
cana-586	75	7	v2	v2	PROPN
cana-586	75	8	)	)	PUNCT
cana-586	75	9	be	be	AUX
cana-586	75	10	any	any	DET
cana-586	75	11	bipartite	bipartite	NOUN
cana-586	75	12	graph	graph	NOUN
cana-586	75	13	with	with	ADP
cana-586	75	14	|v1|	|v1|	NOUN
cana-586	75	15	≥	≥	NOUN
cana-586	75	16	|v2|	|v2|	NOUN
cana-586	75	17	.	.	PUNCT
cana-586	76	1	then	then	ADV
cana-586	76	2	there	there	PRON
cana-586	76	3	exists	exist	VERB
cana-586	76	4	a	a	DET
cana-586	76	5	sum	sum	NOUN
cana-586	76	6	graph	graph	NOUN
cana-586	76	7	g	g	NOUN
cana-586	76	8	with	with	ADP
cana-586	76	9	σ(g	σ(g	NOUN
cana-586	76	10	)	)	PUNCT
cana-586	76	11	=	=	PUNCT
cana-586	77	1	1	1	NUM
cana-586	77	2	such	such	ADJ
cana-586	77	3	that	that	SCONJ
cana-586	77	4	b	b	PROPN
cana-586	77	5	is	be	AUX
cana-586	77	6	an	an	DET
cana-586	77	7	induced	induced	ADJ
cana-586	77	8	subgraph	subgraph	NOUN
cana-586	77	9	of	of	ADP
cana-586	77	10	g.	g.	PROPN
cana-586	77	11	proof	proof	PROPN
cana-586	77	12	.	.	PUNCT
cana-586	78	1	given	give	VERB
cana-586	78	2	that	that	DET
cana-586	78	3	b	b	NOUN
cana-586	78	4	=	=	SYM
cana-586	78	5	(	(	PUNCT
cana-586	78	6	v1	v1	PROPN
cana-586	78	7	,	,	PUNCT
cana-586	78	8	v2	v2	PROPN
cana-586	78	9	)	)	PUNCT
cana-586	78	10	is	be	AUX
cana-586	78	11	a	a	DET
cana-586	78	12	bipartite	bipartite	ADJ
cana-586	78	13	graph	graph	NOUN
cana-586	78	14	with	with	ADP
cana-586	78	15	|v1|	|v1|	PROPN
cana-586	78	16	≥	≥	NOUN
cana-586	78	17	|v2|	|v2|	ADV
cana-586	78	18	.	.	PUNCT
cana-586	79	1	let	let	VERB
cana-586	79	2	|v1|	|v1|	NOUN
cana-586	79	3	=	=	NOUN
cana-586	79	4	r	r	NOUN
cana-586	79	5	and	and	CCONJ
cana-586	79	6	|v2|	|v2|	ADV
cana-586	79	7	=	=	PUNCT
cana-586	79	8	s.	s.	PROPN
cana-586	79	9	consider	consider	VERB
cana-586	79	10	the	the	DET
cana-586	79	11	vertices	vertex	NOUN
cana-586	79	12	in	in	ADP
cana-586	79	13	v1	v1	NOUN
cana-586	79	14	as	as	ADP
cana-586	79	15	{	{	PUNCT
cana-586	79	16	u1	u1	NOUN
cana-586	79	17	,	,	PUNCT
cana-586	79	18	u2	u2	PROPN
cana-586	79	19	,	,	PUNCT
cana-586	79	20	·	·	PUNCT
cana-586	79	21	·	·	PUNCT
cana-586	79	22	·	·	PUNCT
cana-586	79	23	,	,	PUNCT
cana-586	79	24	ur	ur	INTJ
cana-586	79	25	}	}	PUNCT
cana-586	79	26	and	and	CCONJ
cana-586	79	27	the	the	DET
cana-586	79	28	vertices	vertex	NOUN
cana-586	79	29	in	in	ADP
cana-586	79	30	v2	v2	PROPN
cana-586	79	31	as	as	ADP
cana-586	79	32	{	{	PUNCT
cana-586	79	33	v1	v1	NOUN
cana-586	79	34	,	,	PUNCT
cana-586	79	35	v2	v2	PROPN
cana-586	79	36	,	,	PUNCT
cana-586	79	37	·	·	PUNCT
cana-586	79	38	·	·	PUNCT
cana-586	79	39	·	·	PUNCT
cana-586	79	40	,	,	PUNCT
cana-586	79	41	vs	vs	ADP
cana-586	79	42	}	}	PUNCT
cana-586	79	43	.	.	PUNCT
cana-586	80	1	define	define	VERB
cana-586	80	2	the	the	DET
cana-586	80	3	vertex	vertex	NOUN
cana-586	80	4	labeling	labeling	NOUN
cana-586	80	5	function	function	NOUN
cana-586	80	6	f	f	PROPN
cana-586	80	7	as	as	ADP
cana-586	80	8	f	f	PROPN
cana-586	80	9	(	(	PUNCT
cana-586	80	10	ui	ui	PROPN
cana-586	80	11	)	)	PUNCT
cana-586	80	12	=	=	SYM
cana-586	80	13	2i	2i	NOUN
cana-586	80	14	−	−	NOUN
cana-586	80	15	1	1	NUM
cana-586	80	16	,	,	PUNCT
cana-586	80	17	for	for	ADP
cana-586	80	18	1	1	NUM
cana-586	80	19	≤	≤	NUM
cana-586	80	20	i	i	PRON
cana-586	80	21	≤	≤	ADJ
cana-586	80	22	r	r	NOUN
cana-586	80	23	and	and	CCONJ
cana-586	80	24	f	f	PROPN
cana-586	80	25	(	(	PUNCT
cana-586	80	26	vi	vi	NOUN
cana-586	80	27	)	)	PUNCT
cana-586	80	28	=	=	SYM
cana-586	80	29	2i	2i	NOUN
cana-586	80	30	,	,	PUNCT
cana-586	80	31	for	for	ADP
cana-586	80	32	1	1	NUM
cana-586	80	33	≤	≤	NUM
cana-586	80	34	i	i	PRON
cana-586	80	35	≤	≤	ADJ
cana-586	80	36	s.	s.	PROPN
cana-586	80	37	by	by	ADP
cana-586	80	38	the	the	DET
cana-586	80	39	definition	definition	NOUN
cana-586	80	40	of	of	ADP
cana-586	80	41	f	f	PROPN
cana-586	80	42	,	,	PUNCT
cana-586	80	43	it	it	PRON
cana-586	80	44	is	be	AUX
cana-586	80	45	clear	clear	ADJ
cana-586	80	46	that	that	SCONJ
cana-586	80	47	f	f	PROPN
cana-586	80	48	is	be	AUX
cana-586	80	49	one	one	NUM
cana-586	80	50	-	-	PUNCT
cana-586	80	51	toone	toone	NOUN
cana-586	80	52	.	.	PUNCT
cana-586	81	1	now	now	ADV
cana-586	81	2	,	,	PUNCT
cana-586	81	3	define	define	VERB
cana-586	81	4	the	the	DET
cana-586	81	5	edge	edge	NOUN
cana-586	81	6	label	label	NOUN
cana-586	81	7	for	for	ADP
cana-586	81	8	an	an	DET
cana-586	81	9	edge	edge	NOUN
cana-586	81	10	e	e	NOUN
cana-586	81	11	=	=	NOUN
cana-586	81	12	uv	uv	NOUN
cana-586	81	13	as	as	ADP
cana-586	81	14	f	f	PROPN
cana-586	81	15	(	(	PUNCT
cana-586	81	16	e	e	NOUN
cana-586	81	17	)	)	PUNCT
cana-586	82	1	=	=	SYM
cana-586	82	2	f	f	X
cana-586	82	3	(	(	PUNCT
cana-586	82	4	u	u	NOUN
cana-586	82	5	)	)	PUNCT
cana-586	83	1	+	+	NUM
cana-586	83	2	f	f	X
cana-586	83	3	(	(	PUNCT
cana-586	83	4	v	v	NOUN
cana-586	83	5	)	)	PUNCT
cana-586	83	6	.	.	PUNCT
cana-586	84	1	since	since	SCONJ
cana-586	84	2	b	b	PROPN
cana-586	84	3	is	be	AUX
cana-586	84	4	a	a	DET
cana-586	84	5	bipartite	bipartite	ADJ
cana-586	84	6	graph	graph	NOUN
cana-586	84	7	,	,	PUNCT
cana-586	84	8	every	every	DET
cana-586	84	9	edge	edge	NOUN
cana-586	84	10	in	in	ADP
cana-586	84	11	b	b	PROPN
cana-586	84	12	has	have	VERB
cana-586	84	13	one	one	NUM
cana-586	84	14	end	end	NOUN
cana-586	84	15	in	in	ADP
cana-586	84	16	v1	v1	NOUN
cana-586	84	17	and	and	CCONJ
cana-586	84	18	the	the	DET
cana-586	84	19	other	other	ADJ
cana-586	84	20	end	end	NOUN
cana-586	84	21	in	in	ADP
cana-586	84	22	v2	v2	NOUN
cana-586	84	23	,	,	PUNCT
cana-586	84	24	and	and	CCONJ
cana-586	84	25	being	be	AUX
cana-586	84	26	all	all	DET
cana-586	84	27	the	the	DET
cana-586	84	28	vertex	vertex	NOUN
cana-586	84	29	labels	label	NOUN
cana-586	84	30	of	of	ADP
cana-586	84	31	vertices	vertex	NOUN
cana-586	84	32	in	in	ADP
cana-586	84	33	v1	v1	NOUN
cana-586	84	34	are	be	AUX
cana-586	84	35	odd	odd	ADJ
cana-586	84	36	whereas	whereas	SCONJ
cana-586	84	37	vertex	vertex	NOUN
cana-586	84	38	labels	label	NOUN
cana-586	84	39	of	of	ADP
cana-586	84	40	vertices	vertex	NOUN
cana-586	84	41	in	in	ADP
cana-586	84	42	v2	v2	PROPN
cana-586	84	43	are	be	AUX
cana-586	84	44	even	even	ADV
cana-586	84	45	,	,	PUNCT
cana-586	84	46	by	by	ADP
cana-586	84	47	the	the	DET
cana-586	84	48	definition	definition	NOUN
cana-586	84	49	of	of	ADP
cana-586	84	50	the	the	DET
cana-586	84	51	edge	edge	NOUN
cana-586	84	52	labels	label	NOUN
cana-586	84	53	,	,	PUNCT
cana-586	84	54	the	the	DET
cana-586	84	55	edge	edge	NOUN
cana-586	84	56	label	label	NOUN
cana-586	84	57	of	of	ADP
cana-586	84	58	any	any	DET
cana-586	84	59	edge	edge	NOUN
cana-586	84	60	in	in	ADP
cana-586	84	61	b	b	PROPN
cana-586	84	62	is	be	AUX
cana-586	84	63	an	an	DET
cana-586	84	64	odd	odd	ADJ
cana-586	84	65	number	number	NOUN
cana-586	84	66	.	.	PUNCT
cana-586	85	1	define	define	VERB
cana-586	85	2	vf	vf	X
cana-586	85	3	=	=	PUNCT
cana-586	85	4	{	{	PUNCT
cana-586	85	5	the	the	DET
cana-586	85	6	set	set	NOUN
cana-586	85	7	of	of	ADP
cana-586	85	8	all	all	DET
cana-586	85	9	vertex	vertex	NOUN
cana-586	85	10	labels	label	NOUN
cana-586	85	11	of	of	ADP
cana-586	85	12	vertices	vertex	NOUN
cana-586	85	13	of	of	ADP
cana-586	85	14	b	b	NOUN
cana-586	85	15	}	}	PUNCT
cana-586	85	16	and	and	CCONJ
cana-586	85	17	ef	ef	X
cana-586	86	1	=	=	PUNCT
cana-586	86	2	{	{	PUNCT
cana-586	86	3	the	the	DET
cana-586	86	4	set	set	NOUN
cana-586	86	5	of	of	ADP
cana-586	86	6	all	all	DET
cana-586	86	7	edge	edge	NOUN
cana-586	86	8	labels	label	NOUN
cana-586	86	9	of	of	ADP
cana-586	86	10	edges	edge	NOUN
cana-586	86	11	of	of	ADP
cana-586	86	12	b	b	NOUN
cana-586	86	13	}	}	PUNCT
cana-586	86	14	.	.	PUNCT
cana-586	87	1	let	let	VERB
cana-586	87	2	l	l	NOUN
cana-586	87	3	=	=	PUNCT
cana-586	87	4	e	e	X
cana-586	87	5	−	−	PROPN
cana-586	87	6	(	(	PUNCT
cana-586	87	7	vf	vf	X
cana-586	87	8	∩	∩	X
cana-586	87	9	ef	ef	X
cana-586	87	10	)	)	PUNCT
cana-586	87	11	=	=	PRON
cana-586	87	12	{	{	PUNCT
cana-586	87	13	x1	x1	PROPN
cana-586	87	14	,	,	PUNCT
cana-586	87	15	x2	x2	PROPN
cana-586	87	16	,	,	PUNCT
cana-586	87	17	·	·	PUNCT
cana-586	87	18	·	·	PUNCT
cana-586	87	19	·	·	PUNCT
cana-586	87	20	,	,	PUNCT
cana-586	87	21	xt	xt	ADP
cana-586	87	22	}	}	PUNCT
cana-586	87	23	and	and	CCONJ
cana-586	87	24	let	let	VERB
cana-586	87	25	z	z	NOUN
cana-586	87	26	be	be	AUX
cana-586	87	27	the	the	DET
cana-586	87	28	maximum	maximum	ADJ
cana-586	87	29	label	label	NOUN
cana-586	87	30	among	among	ADP
cana-586	87	31	the	the	DET
cana-586	87	32	labels	label	NOUN
cana-586	87	33	in	in	ADP
cana-586	87	34	l.	l.	PROPN
cana-586	87	35	now	now	ADV
cana-586	87	36	,	,	PUNCT
cana-586	87	37	let	let	VERB
cana-586	87	38	us	we	PRON
cana-586	87	39	construct	construct	VERB
cana-586	87	40	the	the	DET
cana-586	87	41	bipartite	bipartite	PROPN
cana-586	87	42	graph	graph	NOUN
cana-586	87	43	g	g	NOUN
cana-586	87	44	as	as	SCONJ
cana-586	87	45	follows	follow	VERB
cana-586	87	46	:	:	PUNCT
cana-586	87	47	start	start	VERB
cana-586	87	48	with	with	ADP
cana-586	87	49	the	the	DET
cana-586	87	50	bipartite	bipartite	PROPN
cana-586	87	51	graph	graph	NOUN
cana-586	87	52	b	b	PROPN
cana-586	87	53	along	along	ADP
cana-586	87	54	with	with	ADP
cana-586	87	55	their	their	PRON
cana-586	87	56	labels	label	NOUN
cana-586	87	57	as	as	SCONJ
cana-586	87	58	defined	define	VERB
cana-586	87	59	by	by	ADP
cana-586	87	60	the	the	DET
cana-586	87	61	function	function	NOUN
cana-586	87	62	f.	f.	PROPN
cana-586	87	63	add	add	VERB
cana-586	87	64	an	an	DET
cana-586	87	65	isolated	isolated	ADJ
cana-586	87	66	vertex	vertex	NOUN
cana-586	87	67	with	with	ADP
cana-586	87	68	label	label	NOUN
cana-586	87	69	z.	z.	PROPN
cana-586	87	70	define	define	VERB
cana-586	87	71	l′	l′	PROPN
cana-586	88	1	=	=	SYM
cana-586	88	2	l	l	NOUN
cana-586	88	3	−	−	PROPN
cana-586	88	4	{	{	PUNCT
cana-586	88	5	z	z	NOUN
cana-586	88	6	}	}	PUNCT
cana-586	88	7	.	.	PUNCT
cana-586	89	1	until	until	ADP
cana-586	89	2	l′	l′	NOUN
cana-586	89	3	=	=	SYM
cana-586	89	4	ϕ	ϕ	NOUN
cana-586	89	5	,	,	PUNCT
cana-586	89	6	choose	choose	VERB
cana-586	89	7	a	a	DET
cana-586	89	8	label	label	NOUN
cana-586	89	9	(	(	PUNCT
cana-586	89	10	say	say	VERB
cana-586	89	11	y	y	PROPN
cana-586	89	12	)	)	PUNCT
cana-586	89	13	from	from	ADP
cana-586	89	14	l′	l′	NUM
cana-586	89	15	,	,	PUNCT
cana-586	89	16	add	add	VERB
cana-586	89	17	a	a	DET
cana-586	89	18	vertex	vertex	NOUN
cana-586	89	19	with	with	ADP
cana-586	89	20	vertex	vertex	NOUN
cana-586	89	21	label	label	NOUN
cana-586	89	22	y	y	PROPN
cana-586	89	23	to	to	ADP
cana-586	89	24	the	the	DET
cana-586	89	25	vertex	vertex	NOUN
cana-586	89	26	with	with	ADP
cana-586	89	27	vertex	vertex	NOUN
cana-586	89	28	label	label	NOUN
cana-586	90	1	z	z	NOUN
cana-586	90	2	−	−	NOUN
cana-586	90	3	y	y	PROPN
cana-586	91	1	and	and	CCONJ
cana-586	91	2	remove	remove	VERB
cana-586	91	3	the	the	DET
cana-586	91	4	label	label	NOUN
cana-586	91	5	y	y	NOUN
cana-586	91	6	from	from	ADP
cana-586	91	7	the	the	DET
cana-586	91	8	set	set	NOUN
cana-586	91	9	l′.	l′.	NOUN
cana-586	91	10	observe	observe	VERB
cana-586	91	11	that	that	SCONJ
cana-586	91	12	by	by	ADP
cana-586	91	13	the	the	DET
cana-586	91	14	construction	construction	NOUN
cana-586	91	15	of	of	ADP
cana-586	91	16	graph	graph	NOUN
cana-586	91	17	g	g	PROPN
cana-586	91	18	,	,	PUNCT
cana-586	91	19	g	g	PROPN
cana-586	91	20	is	be	AUX
cana-586	91	21	a	a	DET
cana-586	91	22	sum	sum	NOUN
cana-586	91	23	graph	graph	NOUN
cana-586	91	24	with	with	ADP
cana-586	91	25	one	one	NUM
cana-586	91	26	isolated	isolated	ADJ
cana-586	91	27	vertex	vertex	NOUN
cana-586	91	28	.	.	PUNCT
cana-586	92	1	therefore	therefore	ADV
cana-586	92	2	,	,	PUNCT
cana-586	92	3	σ(g	σ(g	NOUN
cana-586	92	4	)	)	PUNCT
cana-586	92	5	=	=	SYM
cana-586	92	6	1	1	X
cana-586	92	7	.	.	PUNCT
cana-586	92	8	thus	thus	ADV
cana-586	92	9	,	,	PUNCT
cana-586	92	10	we	we	PRON
cana-586	92	11	have	have	AUX
cana-586	92	12	constructed	construct	VERB
cana-586	92	13	a	a	DET
cana-586	92	14	sum	sum	NOUN
cana-586	92	15	graph	graph	NOUN
cana-586	92	16	g	g	NOUN
cana-586	92	17	with	with	ADP
cana-586	92	18	σ(g	σ(g	NOUN
cana-586	92	19	)	)	PUNCT
cana-586	92	20	=	=	PUNCT
cana-586	93	1	1	1	NUM
cana-586	93	2	in	in	ADP
cana-586	93	3	such	such	DET
cana-586	93	4	a	a	DET
cana-586	93	5	way	way	NOUN
cana-586	93	6	that	that	PRON
cana-586	93	7	given	give	VERB
cana-586	93	8	bipartite	bipartite	PROPN
cana-586	93	9	graph	graph	NOUN
cana-586	93	10	b	b	PROPN
cana-586	93	11	is	be	AUX
cana-586	93	12	an	an	DET
cana-586	93	13	induced	induced	ADJ
cana-586	93	14	subgraph	subgraph	NOUN
cana-586	93	15	of	of	ADP
cana-586	93	16	g.	g.	PROPN
cana-586	93	17	hence	hence	ADV
cana-586	93	18	the	the	DET
cana-586	93	19	proof	proof	NOUN
cana-586	93	20	.	.	PUNCT
cana-586	94	1	3.1	3.1	NUM
cana-586	94	2	illustrative	illustrative	ADJ
cana-586	94	3	example	example	NOUN
cana-586	94	4	for	for	ADP
cana-586	94	5	the	the	DET
cana-586	94	6	bipartite	bipartite	PROPN
cana-586	94	7	graph	graph	NOUN
cana-586	94	8	in	in	ADP
cana-586	94	9	figure	figure	NOUN
cana-586	94	10	4	4	NUM
cana-586	94	11	,	,	PUNCT
cana-586	94	12	the	the	DET
cana-586	94	13	corresponding	corresponding	ADJ
cana-586	94	14	sum	sum	NOUN
cana-586	94	15	graph	graph	NOUN
cana-586	94	16	g	g	NOUN
cana-586	94	17	with	with	ADP
cana-586	94	18	σ(g	σ(g	NOUN
cana-586	94	19	)	)	PUNCT
cana-586	94	20	=	=	SYM
cana-586	94	21	1	1	NUM
cana-586	94	22	is	be	AUX
cana-586	94	23	shown	show	VERB
cana-586	94	24	in	in	ADP
cana-586	94	25	figure	figure	NOUN
cana-586	94	26	5	5	NUM
cana-586	94	27	.	.	PUNCT
cana-586	94	28	figure	figure	VERB
cana-586	94	29	4	4	NUM
cana-586	94	30	:	:	PUNCT
cana-586	94	31	bipartite	bipartite	ADJ
cana-586	94	32	graph	graph	NOUN
cana-586	94	33	b(v1	b(v1	NOUN
cana-586	94	34	,	,	PUNCT
cana-586	94	35	v2	v2	PROPN
cana-586	94	36	)	)	PUNCT
cana-586	94	37	1	1	NUM
cana-586	94	38	15	15	NUM
cana-586	94	39	2	2	NUM
cana-586	94	40	3	3	NUM
cana-586	94	41	13	13	NUM
cana-586	94	42	4	4	NUM
cana-586	94	43	5	5	NUM
cana-586	94	44	11	11	NUM
cana-586	94	45	6	6	NUM
cana-586	94	46	7	7	NUM
cana-586	94	47	8	8	NUM
cana-586	94	48	9	9	NUM
cana-586	94	49	17	17	NUM
cana-586	94	50	figure	figure	NOUN
cana-586	94	51	5	5	NUM
cana-586	94	52	:	:	PUNCT
cana-586	94	53	sum	sum	NOUN
cana-586	94	54	graph	graph	NOUN
cana-586	94	55	g	g	PROPN
cana-586	94	56	with	with	ADP
cana-586	94	57	bipartite	bipartite	PROPN
cana-586	94	58	graph	graph	NOUN
cana-586	94	59	b	b	PROPN
cana-586	94	60	as	as	ADP
cana-586	94	61	an	an	DET
cana-586	94	62	induced	induced	ADJ
cana-586	94	63	subgraph	subgraph	NOUN
cana-586	94	64	communications	communication	NOUN
cana-586	94	65	on	on	ADP
cana-586	94	66	applied	apply	VERB
cana-586	94	67	nonlinear	nonlinear	ADJ
cana-586	94	68	analysis	analysis	NOUN
cana-586	94	69	issn	issn	NOUN
cana-586	94	70	:	:	PUNCT
cana-586	94	71	1074	1074	NUM
cana-586	94	72	-	-	PUNCT
cana-586	94	73	133x	133x	NUM
cana-586	94	74	vol	vol	NOUN
cana-586	94	75	31	31	NUM
cana-586	94	76	no	no	NOUN
cana-586	94	77	.	.	NOUN
cana-586	94	78	2	2	NUM
cana-586	94	79	(	(	PUNCT
cana-586	94	80	2024	2024	NUM
cana-586	94	81	)	)	PUNCT
cana-586	94	82	408	408	NUM
cana-586	94	83	https://internationalpubls.com	https://internationalpubls.com	X
cana-586	94	84	4	4	NUM
cana-586	94	85	conclusion	conclusion	NOUN
cana-586	94	86	in	in	ADP
cana-586	94	87	this	this	DET
cana-586	94	88	note	note	NOUN
cana-586	94	89	,	,	PUNCT
cana-586	94	90	we	we	PRON
cana-586	94	91	proved	prove	VERB
cana-586	94	92	that	that	SCONJ
cana-586	94	93	all	all	DET
cana-586	94	94	trees	tree	NOUN
cana-586	94	95	are	be	AUX
cana-586	94	96	integral	integral	ADJ
cana-586	94	97	sum	sum	NOUN
cana-586	94	98	graphs	graph	NOUN
cana-586	94	99	.	.	PUNCT
cana-586	95	1	further	far	ADV
cana-586	95	2	,	,	PUNCT
cana-586	95	3	we	we	PRON
cana-586	95	4	proved	prove	VERB
cana-586	95	5	a	a	DET
cana-586	95	6	characterization	characterization	NOUN
cana-586	95	7	result	result	NOUN
cana-586	95	8	that	that	SCONJ
cana-586	95	9	any	any	DET
cana-586	95	10	bipartite	bipartite	NOUN
cana-586	95	11	graph	graph	NOUN
cana-586	95	12	is	be	AUX
cana-586	95	13	an	an	DET
cana-586	95	14	induced	induced	ADJ
cana-586	95	15	subgraph	subgraph	NOUN
cana-586	95	16	of	of	ADP
cana-586	95	17	a	a	DET
cana-586	95	18	sum	sum	NOUN
cana-586	95	19	graph	graph	NOUN
cana-586	95	20	g	g	NOUN
cana-586	95	21	with	with	ADP
cana-586	95	22	σ(g	σ(g	NOUN
cana-586	95	23	)	)	PUNCT
cana-586	95	24	=	=	SYM
cana-586	96	1	1	1	X
cana-586	96	2	.	.	X
cana-586	97	1	we	we	PRON
cana-586	97	2	have	have	AUX
cana-586	97	3	given	give	VERB
cana-586	97	4	a	a	DET
cana-586	97	5	constructive	constructive	ADJ
cana-586	97	6	procedure	procedure	NOUN
cana-586	97	7	on	on	ADP
cana-586	97	8	to	to	PART
cana-586	97	9	generate	generate	VERB
cana-586	97	10	such	such	ADJ
cana-586	97	11	sum	sum	NOUN
cana-586	97	12	graphs	graph	NOUN
cana-586	97	13	g	g	NOUN
cana-586	97	14	for	for	ADP
cana-586	97	15	a	a	DET
cana-586	97	16	given	give	VERB
cana-586	97	17	bipartite	bipartite	NOUN
cana-586	97	18	graph	graph	NOUN
cana-586	97	19	.	.	PUNCT
cana-586	98	1	references	reference	NOUN
cana-586	98	2	[	[	X
cana-586	98	3	1	1	NUM
cana-586	98	4	]	]	PUNCT
cana-586	98	5	z.	z.	PROPN
cana-586	98	6	chen	chen	PROPN
cana-586	98	7	,	,	PUNCT
cana-586	98	8	integral	integral	ADJ
cana-586	98	9	sum	sum	NOUN
cana-586	98	10	graphs	graph	NOUN
cana-586	98	11	from	from	ADP
cana-586	98	12	identification	identification	NOUN
cana-586	98	13	,	,	PUNCT
cana-586	98	14	disc	disc	NOUN
cana-586	98	15	.	.	PUNCT
cana-586	98	16	math	math	NOUN
cana-586	98	17	.	.	PUNCT
cana-586	98	18	,	,	PUNCT
cana-586	98	19	181	181	NUM
cana-586	98	20	,	,	PUNCT
cana-586	98	21	(	(	PUNCT
cana-586	98	22	1998	1998	NUM
cana-586	98	23	)	)	PUNCT
cana-586	98	24	,	,	PUNCT
cana-586	98	25	77	77	NUM
cana-586	98	26	-	-	SYM
cana-586	98	27	90	90	NUM
cana-586	98	28	.	.	PUNCT
cana-586	99	1	[	[	X
cana-586	99	2	2	2	NUM
cana-586	99	3	]	]	PUNCT
cana-586	99	4	z.	z.	PROPN
cana-586	99	5	chen	chen	PROPN
cana-586	99	6	,	,	PUNCT
cana-586	99	7	on	on	ADP
cana-586	99	8	integral	integral	ADJ
cana-586	99	9	sum	sum	NOUN
cana-586	99	10	graphs	graph	NOUN
cana-586	99	11	,	,	PUNCT
cana-586	99	12	disc	disc	NOUN
cana-586	99	13	.	.	PUNCT
cana-586	99	14	math	math	NOUN
cana-586	99	15	.	.	PUNCT
cana-586	99	16	,	,	PUNCT
cana-586	99	17	306	306	NUM
cana-586	99	18	,	,	PUNCT
cana-586	99	19	(	(	PUNCT
cana-586	99	20	2006	2006	NUM
cana-586	99	21	)	)	PUNCT
cana-586	99	22	,	,	PUNCT
cana-586	99	23	19	19	NUM
cana-586	99	24	-	-	SYM
cana-586	99	25	25	25	NUM
cana-586	99	26	.	.	PUNCT
cana-586	100	1	[	[	X
cana-586	100	2	3	3	X
cana-586	100	3	]	]	X
cana-586	100	4	m.n	m.n	PROPN
cana-586	100	5	.	.	PROPN
cana-586	100	6	ellingham	ellingham	PROPN
cana-586	100	7	,	,	PUNCT
cana-586	100	8	sum	sum	NOUN
cana-586	100	9	graphs	graph	NOUN
cana-586	100	10	from	from	ADP
cana-586	100	11	trees	tree	NOUN
cana-586	100	12	,	,	PUNCT
cana-586	100	13	ars	ar	NOUN
cana-586	100	14	comb	comb	VERB
cana-586	100	15	.	.	PUNCT
cana-586	100	16	,	,	PUNCT
cana-586	100	17	35	35	NUM
cana-586	100	18	,	,	PUNCT
cana-586	100	19	(	(	PUNCT
cana-586	100	20	1993	1993	NUM
cana-586	100	21	)	)	PUNCT
cana-586	100	22	,	,	PUNCT
cana-586	100	23	335	335	NUM
cana-586	100	24	-	-	SYM
cana-586	100	25	349	349	NUM
cana-586	100	26	.	.	PUNCT
cana-586	101	1	[	[	X
cana-586	101	2	4	4	NUM
cana-586	101	3	]	]	PUNCT
cana-586	101	4	gallian	gallian	ADJ
cana-586	101	5	j.a	j.a	PROPN
cana-586	101	6	.	.	PROPN
cana-586	101	7	,	,	PUNCT
cana-586	101	8	a	a	DET
cana-586	101	9	dynamic	dynamic	ADJ
cana-586	101	10	survey	survey	NOUN
cana-586	101	11	of	of	ADP
cana-586	101	12	graph	graph	NOUN
cana-586	101	13	labeling	labeling	NOUN
cana-586	101	14	,	,	PUNCT
cana-586	101	15	the	the	DET
cana-586	101	16	electronic	electronic	ADJ
cana-586	101	17	journal	journal	NOUN
cana-586	101	18	of	of	ADP
cana-586	101	19	combinatorics	combinatorics	PROPN
cana-586	101	20	,	,	PUNCT
cana-586	101	21	22nd	22nd	NOUN
cana-586	101	22	edition	edition	NOUN
cana-586	101	23	,	,	PUNCT
cana-586	101	24	(	(	PUNCT
cana-586	101	25	2019	2019	NUM
cana-586	101	26	)	)	PUNCT
cana-586	101	27	,	,	PUNCT
cana-586	101	28	#	#	NOUN
cana-586	101	29	ds6	ds6	NOUN
cana-586	101	30	.	.	PUNCT
cana-586	102	1	[	[	X
cana-586	102	2	5	5	NUM
cana-586	102	3	]	]	PUNCT
cana-586	102	4	f.	f.	PROPN
cana-586	102	5	harary	harary	PROPN
cana-586	102	6	,	,	PUNCT
cana-586	102	7	sum	sum	NOUN
cana-586	102	8	graphs	graph	NOUN
cana-586	102	9	over	over	ADP
cana-586	102	10	all	all	DET
cana-586	102	11	integers	integer	NOUN
cana-586	102	12	,	,	PUNCT
cana-586	102	13	disc	disc	NOUN
cana-586	102	14	.	.	PUNCT
cana-586	102	15	math	math	NOUN
cana-586	102	16	.	.	PUNCT
cana-586	102	17	,	,	PUNCT
cana-586	102	18	124	124	NUM
cana-586	102	19	,	,	PUNCT
cana-586	102	20	(	(	PUNCT
cana-586	102	21	1994	1994	NUM
cana-586	102	22	)	)	PUNCT
cana-586	102	23	,	,	PUNCT
cana-586	102	24	99	99	NUM
cana-586	102	25	-	-	SYM
cana-586	102	26	105	105	NUM
cana-586	102	27	.	.	PUNCT
cana-586	103	1	[	[	X
cana-586	103	2	6	6	NUM
cana-586	103	3	]	]	PUNCT
cana-586	103	4	w.	w.	PROPN
cana-586	103	5	he	he	PROPN
cana-586	103	6	,	,	PUNCT
cana-586	103	7	l.	l.	PROPN
cana-586	103	8	wang	wang	PROPN
cana-586	103	9	,	,	PUNCT
cana-586	103	10	h.	h.	PROPN
cana-586	103	11	mi	mi	PROPN
cana-586	103	12	,	,	PUNCT
cana-586	103	13	y.	y.	PROPN
cana-586	103	14	shen	shen	PROPN
cana-586	103	15	,	,	PUNCT
cana-586	103	16	x.	x.	PROPN
cana-586	103	17	yu	yu	PROPN
cana-586	103	18	,	,	PUNCT
cana-586	103	19	integral	integral	ADJ
cana-586	103	20	sum	sum	NOUN
cana-586	103	21	graphs	graph	NOUN
cana-586	103	22	from	from	ADP
cana-586	103	23	a	a	DET
cana-586	103	24	class	class	NOUN
cana-586	103	25	of	of	ADP
cana-586	103	26	tree	tree	NOUN
cana-586	103	27	,	,	PUNCT
cana-586	103	28	ars	ar	VERB
cana-586	103	29	combin	combin	NOUN
cana-586	103	30	.	.	PROPN
cana-586	103	31	,	,	PUNCT
cana-586	103	32	lxx	lxx	PROPN
cana-586	103	33	,	,	PUNCT
cana-586	103	34	(	(	PUNCT
cana-586	103	35	2004	2004	NUM
cana-586	103	36	)	)	PUNCT
cana-586	103	37	.	.	PUNCT
cana-586	104	1	[	[	X
cana-586	104	2	7	7	X
cana-586	104	3	]	]	X
cana-586	104	4	s.c	s.c	PROPN
cana-586	104	5	.	.	PROPN
cana-586	104	6	liaw	liaw	PROPN
cana-586	104	7	,	,	PUNCT
cana-586	104	8	d.	d.	PROPN
cana-586	104	9	kuo	kuo	PROPN
cana-586	104	10	,	,	PUNCT
cana-586	104	11	g.	g.	PROPN
cana-586	104	12	chang	chang	PROPN
cana-586	104	13	,	,	PUNCT
cana-586	104	14	integral	integral	ADJ
cana-586	104	15	sum	sum	NOUN
cana-586	104	16	numbers	number	NOUN
cana-586	104	17	of	of	ADP
cana-586	104	18	graphs	graph	NOUN
cana-586	104	19	,	,	PUNCT
cana-586	104	20	ars	ar	VERB
cana-586	104	21	combin	combin	NOUN
cana-586	104	22	.	.	PROPN
cana-586	104	23	,	,	PUNCT
cana-586	104	24	54	54	NUM
cana-586	104	25	,	,	PUNCT
cana-586	104	26	(	(	PUNCT
cana-586	104	27	1999	1999	NUM
cana-586	104	28	)	)	PUNCT
cana-586	104	29	,	,	PUNCT
cana-586	104	30	259	259	NUM
cana-586	104	31	-	-	SYM
cana-586	104	32	268	268	NUM
cana-586	104	33	.	.	PUNCT
cana-586	105	1	[	[	X
cana-586	105	2	8	8	NUM
cana-586	105	3	]	]	X
cana-586	105	4	a.v	a.v	PROPN
cana-586	105	5	.	.	PROPN
cana-586	105	6	pyatkin	pyatkin	PROPN
cana-586	105	7	,	,	PUNCT
cana-586	105	8	subdivided	subdivide	VERB
cana-586	105	9	trees	tree	NOUN
cana-586	105	10	are	be	AUX
cana-586	105	11	integral	integral	ADJ
cana-586	105	12	sum	sum	NOUN
cana-586	105	13	graphs	graph	NOUN
cana-586	105	14	,	,	PUNCT
cana-586	105	15	disc	disc	NOUN
cana-586	105	16	.	.	PUNCT
cana-586	105	17	math	math	NOUN
cana-586	105	18	.	.	PUNCT
cana-586	106	1	,	,	PUNCT
cana-586	106	2	308	308	NUM
cana-586	106	3	,	,	PUNCT
cana-586	106	4	(	(	PUNCT
cana-586	106	5	2008	2008	NUM
cana-586	106	6	)	)	PUNCT
cana-586	106	7	,	,	PUNCT
cana-586	106	8	1749	1749	NUM
cana-586	106	9	-	-	SYM
cana-586	106	10	1750	1750	NUM
cana-586	106	11	.	.	PUNCT
cana-586	107	1	[	[	X
cana-586	107	2	9	9	NUM
cana-586	107	3	]	]	SYM
cana-586	107	4	a.	a.	NOUN
cana-586	107	5	tiwari	tiwari	NOUN
cana-586	107	6	and	and	CCONJ
cana-586	107	7	a.	a.	NOUN
cana-586	107	8	tripathi	tripathi	PROPN
cana-586	107	9	,	,	PUNCT
cana-586	107	10	on	on	ADP
cana-586	107	11	the	the	DET
cana-586	107	12	range	range	NOUN
cana-586	107	13	of	of	ADP
cana-586	107	14	size	size	NOUN
cana-586	107	15	of	of	ADP
cana-586	107	16	sum	sum	NOUN
cana-586	107	17	graphs	graph	NOUN
cana-586	107	18	and	and	CCONJ
cana-586	107	19	integral	integral	ADJ
cana-586	107	20	sum	sum	NOUN
cana-586	107	21	graphs	graph	NOUN
cana-586	107	22	of	of	ADP
cana-586	107	23	a	a	DET
cana-586	107	24	given	give	VERB
cana-586	107	25	order	order	NOUN
cana-586	107	26	,	,	PUNCT
cana-586	107	27	disc	disc	NOUN
cana-586	108	1	.	.	PUNCT
cana-586	108	2	appl	appl	PROPN
cana-586	108	3	.	.	PROPN
cana-586	108	4	math	math	PROPN
cana-586	108	5	.	.	PUNCT
cana-586	108	6	,	,	PUNCT
cana-586	108	7	161	161	NUM
cana-586	108	8	,	,	PUNCT
cana-586	108	9	(	(	PUNCT
cana-586	108	10	2013	2013	NUM
cana-586	108	11	)	)	PUNCT
cana-586	108	12	,	,	PUNCT
cana-586	108	13	2653	2653	NUM
cana-586	108	14	-	-	SYM
cana-586	108	15	2661	2661	NUM
cana-586	108	16	.	.	PUNCT
cana-586	109	1	[	[	X
cana-586	109	2	10	10	NUM
cana-586	109	3	]	]	X
cana-586	109	4	west	west	NOUN
cana-586	109	5	d.b	d.b	PROPN
cana-586	109	6	.	.	PROPN
cana-586	109	7	,	,	PUNCT
cana-586	109	8	introduction	introduction	NOUN
cana-586	109	9	to	to	AUX
cana-586	109	10	graph	graph	NOUN
cana-586	109	11	theory	theory	NOUN
cana-586	109	12	,	,	PUNCT
cana-586	109	13	prentice	prentice	NOUN
cana-586	109	14	hall	hall	NOUN
cana-586	109	15	of	of	ADP
cana-586	109	16	india	india	PROPN
cana-586	109	17	,	,	PUNCT
cana-586	109	18	2nd	2nd	PROPN
cana-586	109	19	edition	edition	NOUN
cana-586	109	20	,	,	PUNCT
cana-586	109	21	2001	2001	NUM
cana-586	109	22	.	.	PUNCT