id	sid	tid	token	lemma	pos
cana-4846	1	1	communications	communication	NOUN
cana-4846	1	2	on	on	ADP
cana-4846	1	3	applied	apply	VERB
cana-4846	1	4	nonlinear	nonlinear	ADJ
cana-4846	1	5	analysis	analysis	NOUN
cana-4846	1	6	issn	issn	NOUN
cana-4846	1	7	:	:	PUNCT
cana-4846	1	8	1074	1074	NUM
cana-4846	1	9	-	-	PUNCT
cana-4846	1	10	133x	133x	NUM
cana-4846	1	11	vol	vol	VERB
cana-4846	1	12	32	32	NUM
cana-4846	1	13	no	no	NOUN
cana-4846	1	14	.	.	PUNCT
cana-4846	2	1	10s	10	NOUN
cana-4846	2	2	(	(	PUNCT
cana-4846	2	3	2025	2025	NUM
cana-4846	2	4	)	)	PUNCT
cana-4846	2	5	561	561	NUM
cana-4846	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4846	2	7	∪	∪	ADP
cana-4846	2	8	a	a	DET
cana-4846	2	9	study	study	NOUN
cana-4846	2	10	on	on	ADP
cana-4846	2	11	the	the	DET
cana-4846	2	12	differential	differential	ADJ
cana-4846	2	13	value	value	NOUN
cana-4846	2	14	of	of	ADP
cana-4846	2	15	total	total	ADJ
cana-4846	2	16	graph	graph	NOUN
cana-4846	2	17	d.	d.	PROPN
cana-4846	2	18	muralidharan,1	muralidharan,1	PROPN
cana-4846	2	19	m.s	m.s	PROPN
cana-4846	2	20	.	.	PROPN
cana-4846	2	21	paulraj2	paulraj2	PROPN
cana-4846	2	22	and	and	CCONJ
cana-4846	2	23	d.	d.	PROPN
cana-4846	2	24	yokesh3	yokesh3	PROPN
cana-4846	3	1	1department	1department	NUM
cana-4846	3	2	of	of	ADP
cana-4846	3	3	mathematics	mathematic	NOUN
cana-4846	3	4	,	,	PUNCT
cana-4846	3	5	sri	sri	PROPN
cana-4846	3	6	sairam	sairam	PROPN
cana-4846	3	7	institute	institute	PROPN
cana-4846	3	8	of	of	ADP
cana-4846	3	9	technology	technology	PROPN
cana-4846	3	10	,	,	PUNCT
cana-4846	3	11	chennai	chennai	PROPN
cana-4846	3	12	,	,	PUNCT
cana-4846	3	13	tamil	tamil	PROPN
cana-4846	3	14	nadu	nadu	PROPN
cana-4846	3	15	,	,	PUNCT
cana-4846	3	16	india	india	PROPN
cana-4846	3	17	.	.	PUNCT
cana-4846	4	1	2department	2department	NUM
cana-4846	4	2	of	of	ADP
cana-4846	4	3	mathematics	mathematic	NOUN
cana-4846	4	4	,	,	PUNCT
cana-4846	4	5	a.m.	a.m.	PROPN
cana-4846	4	6	jain	jain	PROPN
cana-4846	4	7	college	college	PROPN
cana-4846	4	8	,	,	PUNCT
cana-4846	4	9	chennai	chennai	PROPN
cana-4846	4	10	,	,	PUNCT
cana-4846	4	11	tamil	tamil	PROPN
cana-4846	4	12	nadu	nadu	PROPN
cana-4846	4	13	,	,	PUNCT
cana-4846	4	14	india	india	PROPN
cana-4846	4	15	.	.	PUNCT
cana-4846	5	1	3department	3department	NUM
cana-4846	5	2	of	of	ADP
cana-4846	5	3	mathematics	mathematics	PROPN
cana-4846	5	4	,	,	PUNCT
cana-4846	5	5	anand	anand	PROPN
cana-4846	5	6	institute	institute	PROPN
cana-4846	5	7	of	of	ADP
cana-4846	5	8	technology	technology	PROPN
cana-4846	5	9	,	,	PUNCT
cana-4846	5	10	chennai	chennai	PROPN
cana-4846	5	11	,	,	PUNCT
cana-4846	5	12	tamil	tamil	PROPN
cana-4846	5	13	nadu	nadu	PROPN
cana-4846	5	14	,	,	PUNCT
cana-4846	5	15	india	india	PROPN
cana-4846	5	16	.	.	PUNCT
cana-4846	6	1	e	e	X
cana-4846	6	2	-	-	NOUN
cana-4846	6	3	mail	mail	NOUN
cana-4846	6	4	:	:	PUNCT
cana-4846	6	5	murali.maths@sairamit.edu.in	murali.maths@sairamit.edu.in	PROPN
cana-4846	6	6	article	article	NOUN
cana-4846	6	7	history	history	NOUN
cana-4846	6	8	:	:	PUNCT
cana-4846	6	9	received	receive	VERB
cana-4846	6	10	:	:	PUNCT
cana-4846	6	11	12	12	NUM
cana-4846	6	12	-	-	SYM
cana-4846	6	13	01	01	NUM
cana-4846	6	14	-	-	PUNCT
cana-4846	6	15	2025	2025	NUM
cana-4846	6	16	revised	revise	VERB
cana-4846	6	17	:	:	PUNCT
cana-4846	6	18	15	15	NUM
cana-4846	6	19	-	-	NUM
cana-4846	6	20	02	02	NUM
cana-4846	6	21	-	-	PUNCT
cana-4846	6	22	2025	2025	NUM
cana-4846	6	23	accepted	accept	VERB
cana-4846	6	24	:	:	PUNCT
cana-4846	6	25	01	01	NUM
cana-4846	6	26	-	-	SYM
cana-4846	6	27	03	03	NUM
cana-4846	6	28	-	-	PUNCT
cana-4846	6	29	2025	2025	NUM
cana-4846	6	30	abstract	abstract	NOUN
cana-4846	6	31	:	:	PUNCT
cana-4846	6	32	let	let	VERB
cana-4846	6	33	𝐺	𝐺	PROPN
cana-4846	6	34	=	=	SYM
cana-4846	6	35	(	(	PUNCT
cana-4846	6	36	𝑉	𝑉	PROPN
cana-4846	6	37	,	,	PUNCT
cana-4846	6	38	𝐸	𝐸	PROPN
cana-4846	6	39	)	)	PUNCT
cana-4846	6	40	be	be	VERB
cana-4846	6	41	a	a	DET
cana-4846	6	42	graph	graph	NOUN
cana-4846	6	43	and	and	CCONJ
cana-4846	6	44	x	x	ADJ
cana-4846	6	45	be	be	AUX
cana-4846	6	46	a	a	DET
cana-4846	6	47	subset	subset	NOUN
cana-4846	6	48	of	of	ADP
cana-4846	6	49	v.	v.	INTJ
cana-4846	6	50	let	let	VERB
cana-4846	6	51	𝐵(𝑋	𝐵(𝑋	PROPN
cana-4846	6	52	)	)	PUNCT
cana-4846	6	53	be	be	VERB
cana-4846	6	54	the	the	DET
cana-4846	6	55	set	set	NOUN
cana-4846	6	56	of	of	ADP
cana-4846	6	57	vertices	vertex	NOUN
cana-4846	6	58	in	in	ADP
cana-4846	6	59	v	v	NUM
cana-4846	6	60	−	−	NOUN
cana-4846	6	61	x	x	PUNCT
cana-4846	6	62	that	that	PRON
cana-4846	6	63	has	have	VERB
cana-4846	6	64	a	a	DET
cana-4846	6	65	neighbour	neighbour	NOUN
cana-4846	6	66	in	in	ADP
cana-4846	6	67	a	a	DET
cana-4846	6	68	set	set	NOUN
cana-4846	6	69	x.	x.	NOUN
cana-4846	7	1	the	the	DET
cana-4846	7	2	differential	differential	NOUN
cana-4846	7	3	of	of	ADP
cana-4846	7	4	a	a	DET
cana-4846	7	5	set	set	NOUN
cana-4846	7	6	x	x	NOUN
cana-4846	7	7	,	,	PUNCT
cana-4846	7	8	is	be	AUX
cana-4846	7	9	defined	define	VERB
cana-4846	7	10	as	as	ADP
cana-4846	7	11	∂(x	∂(x	NOUN
cana-4846	7	12	)	)	PUNCT
cana-4846	7	13	which	which	PRON
cana-4846	7	14	is	be	AUX
cana-4846	7	15	|b(x)|	|b(x)|	PROPN
cana-4846	7	16	−	−	PROPN
cana-4846	7	17	|x|	|x|	PROPN
cana-4846	7	18	and	and	CCONJ
cana-4846	7	19	the	the	DET
cana-4846	7	20	differential	differential	NOUN
cana-4846	7	21	of	of	ADP
cana-4846	7	22	a	a	DET
cana-4846	7	23	graph	graph	NOUN
cana-4846	7	24	is	be	AUX
cana-4846	7	25	∂(g	∂(g	ADJ
cana-4846	7	26	)	)	PUNCT
cana-4846	8	1	=	=	SYM
cana-4846	8	2	max	max	X
cana-4846	8	3	{	{	PUNCT
cana-4846	8	4	∂(x)/x	∂(x)/x	X
cana-4846	8	5	⊂	⊂	PROPN
cana-4846	8	6	v	v	ADP
cana-4846	8	7	}	}	PUNCT
cana-4846	8	8	.	.	PUNCT
cana-4846	9	1	the	the	DET
cana-4846	9	2	total	total	ADJ
cana-4846	9	3	graph	graph	NOUN
cana-4846	9	4	t	t	PROPN
cana-4846	9	5	(	(	PUNCT
cana-4846	9	6	g	g	NOUN
cana-4846	9	7	)	)	PUNCT
cana-4846	9	8	of	of	ADP
cana-4846	9	9	a	a	DET
cana-4846	9	10	g	g	PROPN
cana-4846	9	11	raph	raph	PROPN
cana-4846	9	12	g	g	PROPN
cana-4846	9	13	is	be	AUX
cana-4846	9	14	the	the	DET
cana-4846	9	15	graph	graph	NOUN
cana-4846	9	16	whose	whose	DET
cana-4846	9	17	vertex	vertex	NOUN
cana-4846	9	18	set	set	NOUN
cana-4846	9	19	is	be	AUX
cana-4846	9	20	v	v	NOUN
cana-4846	9	21	(	(	PUNCT
cana-4846	9	22	g	g	NOUN
cana-4846	9	23	)	)	PUNCT
cana-4846	9	24	∪	∪	ADP
cana-4846	9	25	e(g	e(g	PROPN
cana-4846	9	26	)	)	PUNCT
cana-4846	9	27	with	with	ADP
cana-4846	9	28	two	two	NUM
cana-4846	9	29	vertices	vertex	NOUN
cana-4846	9	30	of	of	ADP
cana-4846	9	31	t	t	PROPN
cana-4846	9	32	(	(	PUNCT
cana-4846	9	33	g	g	NOUN
cana-4846	9	34	)	)	PUNCT
cana-4846	9	35	being	be	AUX
cana-4846	9	36	adjacent	adjacent	ADJ
cana-4846	9	37	if	if	SCONJ
cana-4846	9	38	and	and	CCONJ
cana-4846	9	39	only	only	ADV
cana-4846	9	40	if	if	SCONJ
cana-4846	9	41	the	the	DET
cana-4846	9	42	corresponding	correspond	VERB
cana-4846	9	43	elements	element	NOUN
cana-4846	9	44	of	of	ADP
cana-4846	9	45	g	g	NOUN
cana-4846	9	46	are	be	AUX
cana-4846	9	47	either	either	CCONJ
cana-4846	9	48	adjacent	adjacent	ADJ
cana-4846	9	49	or	or	CCONJ
cana-4846	9	50	incident	incident	NOUN
cana-4846	9	51	.	.	PUNCT
cana-4846	10	1	in	in	ADP
cana-4846	10	2	this	this	DET
cana-4846	10	3	paper	paper	NOUN
cana-4846	10	4	,	,	PUNCT
cana-4846	10	5	we	we	PRON
cana-4846	10	6	study	study	VERB
cana-4846	10	7	the	the	DET
cana-4846	10	8	differential	differential	ADJ
cana-4846	10	9	value	value	NOUN
cana-4846	10	10	of	of	ADP
cana-4846	10	11	total	total	ADJ
cana-4846	10	12	graph	graph	NOUN
cana-4846	10	13	for	for	ADP
cana-4846	10	14	some	some	DET
cana-4846	10	15	standard	standard	ADJ
cana-4846	10	16	graphs	graph	NOUN
cana-4846	10	17	and	and	CCONJ
cana-4846	10	18	its	its	PRON
cana-4846	10	19	bounds	bound	NOUN
cana-4846	10	20	.	.	PUNCT
cana-4846	11	1	keywords	keyword	NOUN
cana-4846	11	2	:	:	PUNCT
cana-4846	11	3	total	total	ADJ
cana-4846	11	4	graph	graph	NOUN
cana-4846	11	5	,	,	PUNCT
cana-4846	11	6	domination	domination	NOUN
cana-4846	11	7	number	number	NOUN
cana-4846	11	8	mathematics	mathematic	NOUN
cana-4846	11	9	subject	subject	NOUN
cana-4846	11	10	classification	classification	NOUN
cana-4846	11	11	05c38	05c38	NOUN
cana-4846	11	12	,	,	PUNCT
cana-4846	11	13	05c69	05c69	NOUN
cana-4846	11	14	1	1	NUM
cana-4846	11	15	.	.	X
cana-4846	11	16	introduction	introduction	NOUN
cana-4846	11	17	throughout	throughout	ADP
cana-4846	11	18	this	this	DET
cana-4846	11	19	paper	paper	NOUN
cana-4846	11	20	,	,	PUNCT
cana-4846	11	21	𝐺	𝐺	PROPN
cana-4846	11	22	=	=	SYM
cana-4846	11	23	(	(	PUNCT
cana-4846	11	24	𝑉	𝑉	PROPN
cana-4846	11	25	,	,	PUNCT
cana-4846	11	26	𝐸	𝐸	PROPN
cana-4846	11	27	)	)	PUNCT
cana-4846	11	28	is	be	AUX
cana-4846	11	29	a	a	DET
cana-4846	11	30	simple	simple	ADJ
cana-4846	11	31	finite	finite	NOUN
cana-4846	11	32	graph	graph	NOUN
cana-4846	11	33	of	of	ADP
cana-4846	11	34	𝑛	𝑛	DET
cana-4846	11	35	vertices	vertex	NOUN
cana-4846	11	36	.	.	PUNCT
cana-4846	12	1	for	for	ADP
cana-4846	12	2	theoretical	theoretical	ADJ
cana-4846	12	3	terminology	terminology	NOUN
cana-4846	12	4	about	about	ADP
cana-4846	12	5	graph	graph	NOUN
cana-4846	12	6	which	which	PRON
cana-4846	12	7	is	be	AUX
cana-4846	12	8	not	not	PART
cana-4846	12	9	given	give	VERB
cana-4846	12	10	here	here	ADV
cana-4846	12	11	,	,	PUNCT
cana-4846	12	12	we	we	PRON
cana-4846	12	13	refer	refer	VERB
cana-4846	12	14	to	to	ADP
cana-4846	12	15	harary	harary	NOUN
cana-4846	12	16	[	[	X
cana-4846	12	17	7	7	NUM
cana-4846	12	18	]	]	PUNCT
cana-4846	12	19	.	.	PUNCT
cana-4846	13	1	for	for	ADP
cana-4846	13	2	a	a	DET
cana-4846	13	3	vertex	vertex	NOUN
cana-4846	13	4	v	v	ADP
cana-4846	13	5	∈	∈	PROPN
cana-4846	13	6	v	v	NOUN
cana-4846	13	7	,	,	PUNCT
cana-4846	13	8	the	the	DET
cana-4846	13	9	open	open	ADJ
cana-4846	13	10	neighbourhood	neighbourhood	NOUN
cana-4846	13	11	of	of	ADP
cana-4846	13	12	𝑣	𝑣	PRON
cana-4846	13	13	is	be	AUX
cana-4846	13	14	𝑁	𝑁	PROPN
cana-4846	13	15	(	(	PUNCT
cana-4846	13	16	𝑣	𝑣	NOUN
cana-4846	13	17	)	)	PUNCT
cana-4846	13	18	=	=	SYM
cana-4846	13	19	{	{	PUNCT
cana-4846	13	20	𝑢	𝑢	PART
cana-4846	13	21	∈	∈	PROPN
cana-4846	13	22	𝑉/𝑢𝑣	𝑉/𝑢𝑣	NOUN
cana-4846	13	23	∈	∈	PROPN
cana-4846	13	24	𝐸	𝐸	PROPN
cana-4846	13	25	}	}	PUNCT
cana-4846	13	26	and	and	CCONJ
cana-4846	13	27	the	the	DET
cana-4846	13	28	closed	closed	ADJ
cana-4846	13	29	neighbourhood	neighbourhood	NOUN
cana-4846	13	30	of	of	ADP
cana-4846	13	31	the	the	DET
cana-4846	13	32	set	set	NOUN
cana-4846	13	33	𝑁	𝑁	PROPN
cana-4846	13	34	[	[	X
cana-4846	13	35	𝑣	𝑣	X
cana-4846	13	36	]	]	X
cana-4846	13	37	=	=	SYM
cana-4846	13	38	𝑁	𝑁	PROPN
cana-4846	13	39	(	(	PUNCT
cana-4846	13	40	𝑣	𝑣	NOUN
cana-4846	13	41	)	)	PUNCT
cana-4846	13	42	∪	∪	NOUN
cana-4846	13	43	{	{	PUNCT
cana-4846	13	44	𝑣	𝑣	NOUN
cana-4846	13	45	}	}	PUNCT
cana-4846	13	46	.	.	PUNCT
cana-4846	14	1	for	for	ADP
cana-4846	14	2	a	a	DET
cana-4846	14	3	set	set	NOUN
cana-4846	14	4	x	x	X
cana-4846	14	5	⊂	⊂	PROPN
cana-4846	14	6	v	v	PROPN
cana-4846	14	7	,	,	PUNCT
cana-4846	14	8	its	its	PRON
cana-4846	14	9	open	open	ADJ
cana-4846	14	10	neighbourhood	neighbourhood	NOUN
cana-4846	14	11	𝑁(𝑋	𝑁(𝑋	NOUN
cana-4846	14	12	)	)	PUNCT
cana-4846	14	13	=	=	PUNCT
cana-4846	14	14	⋃	⋃	NOUN
cana-4846	14	15	𝑁(𝑣)𝑣𝜖𝑉	𝑁(𝑣)𝑣𝜖𝑉	NOUN
cana-4846	14	16	and	and	CCONJ
cana-4846	14	17	𝑁[𝑋	𝑁[𝑋	NOUN
cana-4846	14	18	]	]	X
cana-4846	14	19	=	=	SYM
cana-4846	14	20	𝑁(𝑋	𝑁(𝑋	NUM
cana-4846	14	21	)	)	PUNCT
cana-4846	14	22	∪	∪	ADP
cana-4846	14	23	𝑋	𝑋	PROPN
cana-4846	14	24	is	be	AUX
cana-4846	14	25	the	the	DET
cana-4846	14	26	closed	closed	ADJ
cana-4846	14	27	neighbourhood	neighbourhood	NOUN
cana-4846	14	28	.	.	PUNCT
cana-4846	15	1	a	a	DET
cana-4846	15	2	set	set	VERB
cana-4846	15	3	𝐷	𝐷	NOUN
cana-4846	15	4	⊆	⊆	NUM
cana-4846	15	5	𝑉	𝑉	PROPN
cana-4846	15	6	is	be	AUX
cana-4846	15	7	a	a	DET
cana-4846	15	8	dominating	dominating	NOUN
cana-4846	15	9	set	set	NOUN
cana-4846	15	10	[	[	X
cana-4846	15	11	6,10	6,10	X
cana-4846	15	12	]	]	PUNCT
cana-4846	15	13	of	of	ADP
cana-4846	15	14	𝐺	𝐺	PROPN
cana-4846	15	15	if	if	SCONJ
cana-4846	15	16	every	every	DET
cana-4846	15	17	vertex	vertex	NOUN
cana-4846	15	18	in	in	ADP
cana-4846	15	19	𝑉	𝑉	PROPN
cana-4846	15	20	−	−	PROPN
cana-4846	15	21	𝐷	𝐷	PROPN
cana-4846	15	22	is	be	AUX
cana-4846	15	23	adjacent	adjacent	ADJ
cana-4846	15	24	to	to	ADP
cana-4846	15	25	some	some	DET
cana-4846	15	26	vertex	vertex	NOUN
cana-4846	15	27	in	in	ADP
cana-4846	15	28	𝐷.	𝐷.	PROPN
cana-4846	15	29	the	the	DET
cana-4846	15	30	boundary	boundary	ADJ
cana-4846	15	31	𝐵(𝑋	𝐵(𝑋	PROPN
cana-4846	15	32	)	)	PUNCT
cana-4846	15	33	of	of	ADP
cana-4846	15	34	a	a	DET
cana-4846	15	35	set	set	NOUN
cana-4846	15	36	𝑋	𝑋	NOUN
cana-4846	15	37	is	be	AUX
cana-4846	15	38	defined	define	VERB
cana-4846	15	39	to	to	PART
cana-4846	15	40	be	be	AUX
cana-4846	15	41	the	the	DET
cana-4846	15	42	set	set	NOUN
cana-4846	15	43	of	of	ADP
cana-4846	15	44	vertices	vertex	NOUN
cana-4846	15	45	in	in	ADP
cana-4846	15	46	𝑉	𝑉	PROPN
cana-4846	15	47	−	−	PROPN
cana-4846	15	48	𝑋	𝑋	PROPN
cana-4846	15	49	dominated	dominate	VERB
cana-4846	15	50	by	by	ADP
cana-4846	15	51	vertices	vertex	NOUN
cana-4846	15	52	in	in	ADP
cana-4846	15	53	𝑋	𝑋	PROPN
cana-4846	15	54	,	,	PUNCT
cana-4846	15	55	that	that	PRON
cana-4846	15	56	is	be	AUX
cana-4846	15	57	𝐵(𝑋	𝐵(𝑋	PROPN
cana-4846	15	58	)	)	PUNCT
cana-4846	16	1	=	=	PUNCT
cana-4846	16	2	(	(	PUNCT
cana-4846	16	3	𝑉	𝑉	PROPN
cana-4846	16	4	−	−	PROPN
cana-4846	16	5	𝑋	𝑋	PROPN
cana-4846	16	6	)	)	PUNCT
cana-4846	16	7	∩	∩	ADJ
cana-4846	16	8	𝑁(𝑋	𝑁(𝑋	NOUN
cana-4846	16	9	)	)	PUNCT
cana-4846	16	10	.	.	PUNCT
cana-4846	17	1	the	the	DET
cana-4846	17	2	differential	differential	ADJ
cana-4846	17	3	𝜕(𝑋	𝜕(𝑋	NUM
cana-4846	17	4	)	)	PUNCT
cana-4846	17	5	of	of	ADP
cana-4846	17	6	𝑋	𝑋	PROPN
cana-4846	17	7	is	be	AUX
cana-4846	17	8	defined	define	VERB
cana-4846	17	9	as	as	ADP
cana-4846	17	10	|𝐵(𝑋)|	|𝐵(𝑋)|	PROPN
cana-4846	17	11	−	−	PROPN
cana-4846	17	12	|𝑋|	|𝑋|	NOUN
cana-4846	17	13	.	.	PUNCT
cana-4846	18	1	the	the	DET
cana-4846	18	2	differential	differential	NOUN
cana-4846	18	3	of	of	ADP
cana-4846	18	4	a	a	DET
cana-4846	18	5	graph	graph	NOUN
cana-4846	18	6	g	g	NOUN
cana-4846	18	7	is	be	AUX
cana-4846	18	8	𝜕(𝐺	𝜕(𝐺	NUM
cana-4846	18	9	)	)	PUNCT
cana-4846	19	1	=	=	SYM
cana-4846	19	2	𝑚𝑎𝑥{𝜕(𝑋)/𝑋	𝑚𝑎𝑥{𝜕(𝑋)/𝑋	NOUN
cana-4846	19	3	⊂	⊂	X
cana-4846	19	4	𝑉	𝑉	PROPN
cana-4846	19	5	}	}	PUNCT
cana-4846	19	6	.	.	PUNCT
cana-4846	20	1	if	if	SCONJ
cana-4846	20	2	𝑆	𝑆	PROPN
cana-4846	20	3	⊂	⊂	PROPN
cana-4846	20	4	𝑉	𝑉	PROPN
cana-4846	20	5	and	and	CCONJ
cana-4846	20	6	𝜕(𝐺	𝜕(𝐺	NUM
cana-4846	20	7	)	)	PUNCT
cana-4846	20	8	=	=	SYM
cana-4846	20	9	𝜕(𝑆	𝜕(𝑆	NUM
cana-4846	20	10	)	)	PUNCT
cana-4846	20	11	,	,	PUNCT
cana-4846	20	12	then	then	ADV
cana-4846	20	13	s	s	VERB
cana-4846	20	14	is	be	AUX
cana-4846	20	15	a	a	DET
cana-4846	20	16	∂-set	∂-set	NOUN
cana-4846	20	17	.	.	PUNCT
cana-4846	21	1	the	the	DET
cana-4846	21	2	differential	differential	NOUN
cana-4846	21	3	of	of	ADP
cana-4846	21	4	a	a	DET
cana-4846	21	5	set	set	NOUN
cana-4846	21	6	was	be	AUX
cana-4846	21	7	first	first	ADV
cana-4846	21	8	defined	define	VERB
cana-4846	21	9	by	by	ADP
cana-4846	21	10	hedetniemi	hedetniemi	ADV
cana-4846	21	11	and	and	CCONJ
cana-4846	21	12	later	later	ADV
cana-4846	21	13	studied	study	VERB
cana-4846	21	14	by	by	ADP
cana-4846	21	15	mashburn	mashburn	NOUN
cana-4846	21	16	et	et	PROPN
cana-4846	21	17	al	al	PROPN
cana-4846	21	18	.	.	PROPN
cana-4846	21	19	and	and	CCONJ
cana-4846	21	20	goddard	goddard	PROPN
cana-4846	21	21	and	and	CCONJ
cana-4846	21	22	henning	henning	NOUN
cana-4846	22	1	[	[	X
cana-4846	22	2	3,4,9,13	3,4,9,13	NUM
cana-4846	22	3	]	]	PUNCT
cana-4846	22	4	.	.	PUNCT
cana-4846	23	1	a	a	DET
cana-4846	23	2	graph	graph	NOUN
cana-4846	23	3	𝐺	𝐺	NOUN
cana-4846	23	4	is	be	AUX
cana-4846	23	5	complete	complete	ADJ
cana-4846	23	6	graph	graph	NOUN
cana-4846	23	7	if	if	SCONJ
cana-4846	23	8	every	every	DET
cana-4846	23	9	distinct	distinct	ADJ
cana-4846	23	10	pair	pair	NOUN
cana-4846	23	11	of	of	ADP
cana-4846	23	12	vertices	vertex	NOUN
cana-4846	23	13	are	be	AUX
cana-4846	23	14	adjacent	adjacent	ADJ
cana-4846	23	15	.	.	PUNCT
cana-4846	24	1	a	a	DET
cana-4846	24	2	complete	complete	ADJ
cana-4846	24	3	bipartite	bipartite	NOUN
cana-4846	24	4	graph	graph	NOUN
cana-4846	24	5	is	be	AUX
cana-4846	24	6	a	a	DET
cana-4846	24	7	special	special	ADJ
cana-4846	24	8	type	type	NOUN
cana-4846	24	9	of	of	ADP
cana-4846	24	10	bipartite	bipartite	NOUN
cana-4846	24	11	graph	graph	NOUN
cana-4846	24	12	where	where	SCONJ
cana-4846	24	13	every	every	DET
cana-4846	24	14	vertex	vertex	NOUN
cana-4846	24	15	of	of	ADP
cana-4846	24	16	one	one	NUM
cana-4846	24	17	set	set	NOUN
cana-4846	24	18	is	be	AUX
cana-4846	24	19	connected	connect	VERB
cana-4846	24	20	to	to	ADP
cana-4846	24	21	every	every	DET
cana-4846	24	22	other	other	ADJ
cana-4846	24	23	vertex	vertex	NOUN
cana-4846	24	24	of	of	ADP
cana-4846	24	25	other	other	ADJ
cana-4846	24	26	set	set	NOUN
cana-4846	24	27	.	.	PUNCT
cana-4846	25	1	a	a	DET
cana-4846	25	2	complete	complete	ADJ
cana-4846	25	3	binary	binary	ADJ
cana-4846	25	4	tree	tree	NOUN
cana-4846	25	5	is	be	AUX
cana-4846	25	6	a	a	DET
cana-4846	25	7	special	special	ADJ
cana-4846	25	8	type	type	NOUN
cana-4846	25	9	of	of	ADP
cana-4846	25	10	binary	binary	ADJ
cana-4846	25	11	tree	tree	NOUN
cana-4846	25	12	where	where	SCONJ
cana-4846	25	13	all	all	DET
cana-4846	25	14	the	the	DET
cana-4846	25	15	levels	level	NOUN
cana-4846	25	16	of	of	ADP
cana-4846	25	17	the	the	DET
cana-4846	25	18	tree	tree	NOUN
cana-4846	25	19	are	be	AUX
cana-4846	25	20	filled	fill	VERB
cana-4846	25	21	completely	completely	ADV
cana-4846	25	22	except	except	SCONJ
cana-4846	25	23	the	the	DET
cana-4846	25	24	lowest	low	ADJ
cana-4846	25	25	level	level	NOUN
cana-4846	25	26	nodes	node	NOUN
cana-4846	25	27	which	which	PRON
cana-4846	25	28	are	be	AUX
cana-4846	25	29	filled	fill	VERB
cana-4846	25	30	from	from	ADP
cana-4846	25	31	as	as	ADV
cana-4846	25	32	left	leave	VERB
cana-4846	25	33	as	as	ADP
cana-4846	25	34	possible	possible	ADJ
cana-4846	25	35	.	.	PUNCT
cana-4846	26	1	a	a	DET
cana-4846	26	2	graph	graph	NOUN
cana-4846	26	3	𝐺	𝐺	NOUN
cana-4846	26	4	is	be	AUX
cana-4846	26	5	said	say	VERB
cana-4846	26	6	to	to	PART
cana-4846	26	7	be	be	AUX
cana-4846	26	8	dominant	dominant	ADJ
cana-4846	26	9	differential	differential	NOUN
cana-4846	26	10	if	if	SCONJ
cana-4846	26	11	it	it	PRON
cana-4846	26	12	contains	contain	VERB
cana-4846	26	13	a	a	DET
cana-4846	26	14	𝜕	𝜕	NOUN
cana-4846	26	15	−	−	NOUN
cana-4846	26	16	set	set	NOUN
cana-4846	26	17	which	which	PRON
cana-4846	26	18	is	be	AUX
cana-4846	26	19	also	also	ADV
cana-4846	26	20	a	a	DET
cana-4846	26	21	dominating	dominating	NOUN
cana-4846	26	22	set	set	NOUN
cana-4846	26	23	.	.	PUNCT
cana-4846	27	1	some	some	DET
cana-4846	27	2	examples	example	NOUN
cana-4846	27	3	are	be	AUX
cana-4846	27	4	complete	complete	ADJ
cana-4846	27	5	graph	graph	NOUN
cana-4846	27	6	and	and	CCONJ
cana-4846	27	7	wheel	wheel	NOUN
cana-4846	27	8	graph	graph	NOUN
cana-4846	27	9	.	.	PUNCT
cana-4846	28	1	mailto:murali.maths@sairamit.edu.in	mailto:murali.maths@sairamit.edu.in	PROPN
cana-4846	28	2	communications	communication	NOUN
cana-4846	28	3	on	on	ADP
cana-4846	28	4	applied	apply	VERB
cana-4846	28	5	nonlinear	nonlinear	ADJ
cana-4846	28	6	analysis	analysis	NOUN
cana-4846	28	7	issn	issn	NOUN
cana-4846	28	8	:	:	PUNCT
cana-4846	28	9	1074	1074	NUM
cana-4846	28	10	-	-	PUNCT
cana-4846	28	11	133x	133x	NUM
cana-4846	28	12	vol	vol	VERB
cana-4846	28	13	32	32	NUM
cana-4846	28	14	no	no	NOUN
cana-4846	28	15	.	.	PUNCT
cana-4846	29	1	10s	10	NOUN
cana-4846	29	2	(	(	PUNCT
cana-4846	29	3	2025	2025	NUM
cana-4846	29	4	)	)	PUNCT
cana-4846	29	5	562	562	NUM
cana-4846	29	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4846	29	7	2	2	NUM
cana-4846	29	8	the	the	DET
cana-4846	29	9	total	total	ADJ
cana-4846	29	10	graph	graph	NOUN
cana-4846	29	11	𝑇	𝑇	PROPN
cana-4846	29	12	(	(	PUNCT
cana-4846	29	13	𝐺	𝐺	NOUN
cana-4846	29	14	)	)	PUNCT
cana-4846	30	1	[	[	X
cana-4846	30	2	1,2,5,8,11,12,14	1,2,5,8,11,12,14	X
cana-4846	30	3	]	]	PUNCT
cana-4846	30	4	of	of	ADP
cana-4846	30	5	𝐺	𝐺	PROPN
cana-4846	30	6	is	be	AUX
cana-4846	30	7	the	the	DET
cana-4846	30	8	graph	graph	NOUN
cana-4846	30	9	whose	whose	DET
cana-4846	30	10	vertex	vertex	NOUN
cana-4846	30	11	set	set	NOUN
cana-4846	30	12	is	be	AUX
cana-4846	30	13	𝑉	𝑉	PROPN
cana-4846	30	14	(	(	PUNCT
cana-4846	30	15	𝐺	𝐺	NOUN
cana-4846	30	16	)	)	PUNCT
cana-4846	30	17	∪	∪	ADP
cana-4846	30	18	𝐸(𝐺	𝐸(𝐺	PROPN
cana-4846	30	19	)	)	PUNCT
cana-4846	30	20	with	with	ADP
cana-4846	30	21	two	two	NUM
cana-4846	30	22	vertices	vertex	NOUN
cana-4846	30	23	of	of	ADP
cana-4846	30	24	𝑇	𝑇	PROPN
cana-4846	30	25	(	(	PUNCT
cana-4846	30	26	𝐺	𝐺	NOUN
cana-4846	30	27	)	)	PUNCT
cana-4846	30	28	being	be	AUX
cana-4846	30	29	adjacent	adjacent	ADJ
cana-4846	30	30	if	if	SCONJ
cana-4846	30	31	and	and	CCONJ
cana-4846	30	32	only	only	ADV
cana-4846	30	33	if	if	SCONJ
cana-4846	30	34	the	the	DET
cana-4846	30	35	corresponding	correspond	VERB
cana-4846	30	36	elements	element	NOUN
cana-4846	30	37	of	of	ADP
cana-4846	30	38	𝐺	𝐺	PROPN
cana-4846	30	39	are	be	AUX
cana-4846	30	40	either	either	CCONJ
cana-4846	30	41	adjacent	adjacent	ADJ
cana-4846	30	42	or	or	CCONJ
cana-4846	30	43	incident	incident	NOUN
cana-4846	30	44	and	and	CCONJ
cana-4846	30	45	|𝑉	|𝑉	NUM
cana-4846	30	46	(	(	PUNCT
cana-4846	30	47	𝑇	𝑇	PROPN
cana-4846	30	48	(	(	PUNCT
cana-4846	30	49	𝐺))|	𝐺))|	PROPN
cana-4846	30	50	=	=	PUNCT
cana-4846	31	1	𝑙.	𝑙.	ADV
cana-4846	31	2	in	in	ADP
cana-4846	31	3	this	this	DET
cana-4846	31	4	paper	paper	NOUN
cana-4846	31	5	,	,	PUNCT
cana-4846	31	6	we	we	PRON
cana-4846	31	7	study	study	VERB
cana-4846	31	8	the	the	DET
cana-4846	31	9	differential	differential	ADJ
cana-4846	31	10	value	value	NOUN
cana-4846	31	11	of	of	ADP
cana-4846	31	12	total	total	ADJ
cana-4846	31	13	graph	graph	NOUN
cana-4846	31	14	of	of	ADP
cana-4846	31	15	some	some	DET
cana-4846	31	16	standard	standard	ADJ
cana-4846	31	17	graphs	graph	NOUN
cana-4846	31	18	and	and	CCONJ
cana-4846	31	19	its	its	PRON
cana-4846	31	20	bounds	bound	NOUN
cana-4846	31	21	.	.	PUNCT
cana-4846	32	1	2	2	X
cana-4846	32	2	.	.	X
cana-4846	32	3	results	result	NOUN
cana-4846	32	4	theorem	theorem	VERB
cana-4846	32	5	1	1	NUM
cana-4846	32	6	.	.	X
cana-4846	33	1	for	for	ADP
cana-4846	33	2	any	any	DET
cana-4846	33	3	graph	graph	NOUN
cana-4846	33	4	g	g	NOUN
cana-4846	33	5	of	of	ADP
cana-4846	33	6	n	n	PRON
cana-4846	33	7	vertices	vertex	NOUN
cana-4846	33	8	,	,	PUNCT
cana-4846	33	9	1	1	NUM
cana-4846	33	10	≤	≤	NUM
cana-4846	33	11	∂(t(g	∂(t(g	PROPN
cana-4846	33	12	)	)	PUNCT
cana-4846	33	13	)	)	PUNCT
cana-4846	33	14	≤	≤	NUM
cana-4846	33	15	n(n−1	n(n−1	X
cana-4846	33	16	)	)	PUNCT
cana-4846	33	17	2	2	NUM
cana-4846	33	18	theorem	theorem	NOUN
cana-4846	33	19	2	2	NUM
cana-4846	33	20	.	.	PUNCT
cana-4846	34	1	if	if	SCONJ
cana-4846	34	2	t	t	PROPN
cana-4846	34	3	(	(	PUNCT
cana-4846	34	4	g	g	NOUN
cana-4846	34	5	)	)	PUNCT
cana-4846	34	6	is	be	AUX
cana-4846	34	7	the	the	DET
cana-4846	34	8	total	total	ADJ
cana-4846	34	9	graph	graph	NOUN
cana-4846	34	10	of	of	ADP
cana-4846	34	11	a	a	DET
cana-4846	34	12	graph	graph	NOUN
cana-4846	34	13	g	g	NOUN
cana-4846	34	14	,	,	PUNCT
cana-4846	34	15	then	then	ADV
cana-4846	34	16	∂(t(g	∂(t(g	PROPN
cana-4846	34	17	)	)	PUNCT
cana-4846	34	18	)	)	PUNCT
cana-4846	34	19	≤	≤	ADV
cana-4846	34	20	n(n−1	n(n−1	NUM
cana-4846	34	21	)	)	PUNCT
cana-4846	34	22	2	2	NUM
cana-4846	34	23	for	for	ADP
cana-4846	34	24	n	n	X
cana-4846	34	25	≥	≥	NUM
cana-4846	34	26	2	2	NUM
cana-4846	34	27	.	.	PUNCT
cana-4846	34	28	proof	proof	NOUN
cana-4846	34	29	.	.	PUNCT
cana-4846	35	1	we	we	PRON
cana-4846	35	2	have	have	VERB
cana-4846	35	3	to	to	PART
cana-4846	35	4	prove	prove	VERB
cana-4846	35	5	by	by	ADP
cana-4846	35	6	induction	induction	NOUN
cana-4846	35	7	method	method	NOUN
cana-4846	35	8	.	.	PUNCT
cana-4846	36	1	we	we	PRON
cana-4846	36	2	have	have	VERB
cana-4846	36	3	to	to	PART
cana-4846	36	4	prove	prove	VERB
cana-4846	36	5	that	that	SCONJ
cana-4846	36	6	the	the	DET
cana-4846	36	7	result	result	NOUN
cana-4846	36	8	is	be	AUX
cana-4846	36	9	true	true	ADJ
cana-4846	36	10	for	for	ADP
cana-4846	36	11	n	n	NOUN
cana-4846	36	12	=	=	SYM
cana-4846	36	13	2	2	NUM
cana-4846	36	14	.	.	PUNCT
cana-4846	36	15	when	when	SCONJ
cana-4846	36	16	n	n	X
cana-4846	36	17	=	=	SYM
cana-4846	36	18	2	2	NUM
cana-4846	36	19	,	,	PUNCT
cana-4846	36	20	then	then	ADV
cana-4846	36	21	𝜕(𝑇	𝜕(𝑇	NUM
cana-4846	36	22	(	(	PUNCT
cana-4846	36	23	𝐺	𝐺	NOUN
cana-4846	36	24	)	)	PUNCT
cana-4846	36	25	)	)	PUNCT
cana-4846	36	26	≤	≤	NUM
cana-4846	36	27	2(2−1	2(2−1	NOUN
cana-4846	36	28	)	)	PUNCT
cana-4846	36	29	2	2	NUM
cana-4846	36	30	=	=	SYM
cana-4846	36	31	1	1	X
cana-4846	36	32	.	.	PUNCT
cana-4846	37	1	it	it	PRON
cana-4846	37	2	is	be	AUX
cana-4846	37	3	always	always	ADV
cana-4846	37	4	true	true	ADJ
cana-4846	37	5	.	.	PUNCT
cana-4846	38	1	we	we	PRON
cana-4846	38	2	assume	assume	VERB
cana-4846	38	3	that	that	SCONJ
cana-4846	38	4	the	the	DET
cana-4846	38	5	result	result	NOUN
cana-4846	38	6	is	be	AUX
cana-4846	38	7	true	true	ADJ
cana-4846	38	8	for	for	ADP
cana-4846	38	9	n	n	PROPN
cana-4846	38	10	=	=	SYM
cana-4846	38	11	k.	k.	PROPN
cana-4846	38	12	then	then	ADV
cana-4846	38	13	,	,	PUNCT
cana-4846	38	14	𝜕(𝑇	𝜕(𝑇	NUM
cana-4846	38	15	(	(	PUNCT
cana-4846	38	16	𝐺	𝐺	NOUN
cana-4846	38	17	)	)	PUNCT
cana-4846	38	18	)	)	PUNCT
cana-4846	38	19	≤	≤	NUM
cana-4846	39	1	𝑘(𝑘−1	𝑘(𝑘−1	PROPN
cana-4846	39	2	)	)	PUNCT
cana-4846	39	3	2	2	NUM
cana-4846	39	4	.	.	PUNCT
cana-4846	40	1	we	we	PRON
cana-4846	40	2	have	have	VERB
cana-4846	40	3	to	to	PART
cana-4846	40	4	prove	prove	VERB
cana-4846	40	5	that	that	SCONJ
cana-4846	40	6	the	the	DET
cana-4846	40	7	result	result	NOUN
cana-4846	40	8	is	be	AUX
cana-4846	40	9	true	true	ADJ
cana-4846	40	10	for	for	ADP
cana-4846	40	11	𝑛	𝑛	PROPN
cana-4846	40	12	=	=	SYM
cana-4846	40	13	𝑘	𝑘	PROPN
cana-4846	41	1	+	+	NOUN
cana-4846	41	2	1	1	X
cana-4846	41	3	.	.	X
cana-4846	41	4	consider	consider	VERB
cana-4846	41	5	𝐺	𝐺	PROPN
cana-4846	41	6	with	with	ADP
cana-4846	41	7	𝑛	𝑛	PROPN
cana-4846	41	8	=	=	SYM
cana-4846	41	9	𝑘	𝑘	PROPN
cana-4846	41	10	+	+	ADJ
cana-4846	41	11	1	1	NUM
cana-4846	41	12	vertices	vertex	NOUN
cana-4846	41	13	.	.	PUNCT
cana-4846	42	1	removing	remove	VERB
cana-4846	42	2	a	a	DET
cana-4846	42	3	vertex	vertex	NOUN
cana-4846	42	4	v	v	ADP
cana-4846	42	5	∈	∈	PROPN
cana-4846	42	6	t	t	NOUN
cana-4846	42	7	(	(	PUNCT
cana-4846	42	8	g	g	NOUN
cana-4846	42	9	)	)	PUNCT
cana-4846	42	10	and	and	CCONJ
cana-4846	42	11	|v	|v	PROPN
cana-4846	42	12	(	(	PUNCT
cana-4846	42	13	t	t	PROPN
cana-4846	42	14	(	(	PUNCT
cana-4846	42	15	g	g	NOUN
cana-4846	42	16	)	)	PUNCT
cana-4846	42	17	)	)	PUNCT
cana-4846	42	18	−	−	NOUN
cana-4846	42	19	{	{	PUNCT
cana-4846	42	20	v}|	v}|	X
cana-4846	42	21	=	=	SYM
cana-4846	42	22	k	k	NOUN
cana-4846	42	23	,	,	PUNCT
cana-4846	42	24	hence	hence	ADV
cana-4846	42	25	by	by	ADP
cana-4846	42	26	induction	induction	NOUN
cana-4846	42	27	hypothesis	hypothesis	NOUN
cana-4846	42	28	,	,	PUNCT
cana-4846	42	29	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	42	30	)	)	PUNCT
cana-4846	42	31	−	−	NOUN
cana-4846	42	32	{	{	PUNCT
cana-4846	42	33	𝑣	𝑣	NOUN
cana-4846	42	34	}	}	PUNCT
cana-4846	42	35	)	)	PUNCT
cana-4846	43	1	+	+	CCONJ
cana-4846	43	2	k	k	X
cana-4846	43	3	≤	≤	NUM
cana-4846	43	4	k(k−1	k(k−1	NOUN
cana-4846	43	5	)	)	PUNCT
cana-4846	43	6	2	2	NUM
cana-4846	43	7	+	+	CCONJ
cana-4846	43	8	k	k	NOUN
cana-4846	43	9	=	=	PUNCT
cana-4846	43	10	k2−k+2k	k2−k+2k	NOUN
cana-4846	43	11	2	2	X
cana-4846	43	12	=	=	SYM
cana-4846	43	13	k2+k	k2+k	X
cana-4846	43	14	2	2	NUM
cana-4846	43	15	=	=	SYM
cana-4846	43	16	k(k+1	k(k+1	X
cana-4846	43	17	)	)	PUNCT
cana-4846	43	18	2	2	NUM
cana-4846	43	19	.	.	PUNCT
cana-4846	44	1	so	so	ADV
cana-4846	44	2	,	,	PUNCT
cana-4846	44	3	𝜕(𝑇	𝜕(𝑇	NUM
cana-4846	44	4	(	(	PUNCT
cana-4846	44	5	𝐺	𝐺	NOUN
cana-4846	44	6	)	)	PUNCT
cana-4846	44	7	)	)	PUNCT
cana-4846	45	1	≤	≤	NUM
cana-4846	46	1	𝑘(𝑘+1	𝑘(𝑘+1	NOUN
cana-4846	46	2	)	)	PUNCT
cana-4846	46	3	2	2	NUM
cana-4846	46	4	.	.	PUNCT
cana-4846	47	1	therefore	therefore	ADV
cana-4846	47	2	,	,	PUNCT
cana-4846	47	3	the	the	DET
cana-4846	47	4	result	result	NOUN
cana-4846	47	5	is	be	AUX
cana-4846	47	6	true	true	ADJ
cana-4846	47	7	for	for	ADP
cana-4846	47	8	𝑛	𝑛	PROPN
cana-4846	47	9	=	=	SYM
cana-4846	47	10	𝑘	𝑘	PROPN
cana-4846	47	11	+	+	NOUN
cana-4846	47	12	1	1	NUM
cana-4846	47	13	and	and	CCONJ
cana-4846	47	14	the	the	DET
cana-4846	47	15	result	result	NOUN
cana-4846	47	16	is	be	AUX
cana-4846	47	17	true	true	ADJ
cana-4846	47	18	for	for	ADP
cana-4846	47	19	all	all	DET
cana-4846	47	20	𝑛.	𝑛.	NOUN
cana-4846	47	21	so	so	ADV
cana-4846	47	22	,	,	PUNCT
cana-4846	47	23	∂(t(g	∂(t(g	PROPN
cana-4846	47	24	)	)	PUNCT
cana-4846	47	25	)	)	PUNCT
cana-4846	47	26	≤	≤	ADV
cana-4846	47	27	n(n−1	n(n−1	X
cana-4846	47	28	)	)	PUNCT
cana-4846	47	29	2	2	NUM
cana-4846	47	30	.	.	PUNCT
cana-4846	48	1	theorem	theorem	NOUN
cana-4846	48	2	3	3	NUM
cana-4846	48	3	.	.	PUNCT
cana-4846	49	1	if	if	SCONJ
cana-4846	49	2	g	g	PROPN
cana-4846	49	3	is	be	AUX
cana-4846	49	4	a	a	DET
cana-4846	49	5	complete	complete	ADJ
cana-4846	49	6	graph	graph	NOUN
cana-4846	49	7	,	,	PUNCT
cana-4846	49	8	then	then	ADV
cana-4846	49	9	∂(t(g	∂(t(g	PROPN
cana-4846	49	10	)	)	PUNCT
cana-4846	49	11	)	)	PUNCT
cana-4846	50	1	=	=	SYM
cana-4846	50	2	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-4846	50	3	)	)	PUNCT
cana-4846	50	4	2	2	NUM
cana-4846	50	5	,	,	PUNCT
cana-4846	50	6	for	for	ADP
cana-4846	50	7	n	n	PRON
cana-4846	50	8	≥	≥	NUM
cana-4846	50	9	2	2	NUM
cana-4846	50	10	.	.	PUNCT
cana-4846	50	11	proof	proof	NOUN
cana-4846	50	12	.	.	PUNCT
cana-4846	51	1	given	give	VERB
cana-4846	51	2	that	that	SCONJ
cana-4846	51	3	𝐺	𝐺	PROPN
cana-4846	51	4	is	be	AUX
cana-4846	51	5	a	a	DET
cana-4846	51	6	complete	complete	ADJ
cana-4846	51	7	graph	graph	NOUN
cana-4846	51	8	.	.	PUNCT
cana-4846	52	1	consider	consider	VERB
cana-4846	52	2	𝑆	𝑆	PROPN
cana-4846	52	3	is	be	AUX
cana-4846	52	4	a	a	DET
cana-4846	52	5	𝜕−	𝜕−	NOUN
cana-4846	52	6	set	set	NOUN
cana-4846	52	7	of	of	ADP
cana-4846	52	8	𝑇(𝐺	𝑇(𝐺	NOUN
cana-4846	52	9	)	)	PUNCT
cana-4846	52	10	.	.	PUNCT
cana-4846	53	1	when	when	SCONJ
cana-4846	53	2	𝑛	𝑛	PROPN
cana-4846	53	3	is	be	AUX
cana-4846	53	4	even	even	ADV
cana-4846	53	5	,	,	PUNCT
cana-4846	53	6	choose	choose	VERB
cana-4846	53	7	any	any	DET
cana-4846	53	8	arbitrary	arbitrary	ADJ
cana-4846	53	9	vertex	vertex	NOUN
cana-4846	53	10	𝑢1	𝑢1	NOUN
cana-4846	53	11	in	in	ADP
cana-4846	53	12	𝑆	𝑆	PROPN
cana-4846	53	13	and	and	CCONJ
cana-4846	53	14	choose	choose	VERB
cana-4846	53	15	next	next	ADJ
cana-4846	53	16	vertex	vertex	NOUN
cana-4846	53	17	𝑢2	𝑢2	PROPN
cana-4846	53	18	which	which	PRON
cana-4846	53	19	is	be	AUX
cana-4846	53	20	not	not	PART
cana-4846	53	21	adjacent	adjacent	ADJ
cana-4846	53	22	with	with	ADP
cana-4846	53	23	𝑢1	𝑢1	PROPN
cana-4846	53	24	.	.	PUNCT
cana-4846	54	1	choose	choose	VERB
cana-4846	54	2	next	next	ADJ
cana-4846	54	3	vertex	vertex	NOUN
cana-4846	54	4	𝑢3	𝑢3	NOUN
cana-4846	54	5	which	which	PRON
cana-4846	54	6	is	be	AUX
cana-4846	54	7	not	not	PART
cana-4846	54	8	adjacent	adjacent	ADJ
cana-4846	54	9	with	with	ADP
cana-4846	54	10	𝑢1	𝑢1	PROPN
cana-4846	54	11	and	and	CCONJ
cana-4846	54	12	𝑢2	𝑢2	PROPN
cana-4846	54	13	.	.	PUNCT
cana-4846	55	1	continuing	continue	VERB
cana-4846	55	2	this	this	DET
cana-4846	55	3	process	process	NOUN
cana-4846	55	4	until	until	ADP
cana-4846	55	5	|s|	|s|	PROPN
cana-4846	55	6	=	=	SYM
cana-4846	55	7	n	n	DET
cana-4846	55	8	2	2	NUM
cana-4846	55	9	.	.	PUNCT
cana-4846	56	1	clearly	clearly	ADV
cana-4846	56	2	,	,	PUNCT
cana-4846	56	3	𝑆	𝑆	PROPN
cana-4846	56	4	is	be	AUX
cana-4846	56	5	a	a	DET
cana-4846	56	6	dominant	dominant	ADJ
cana-4846	56	7	differential	differential	NOUN
cana-4846	56	8	of	of	ADP
cana-4846	56	9	𝑇	𝑇	PROPN
cana-4846	56	10	(	(	PUNCT
cana-4846	56	11	𝐺	𝐺	PROPN
cana-4846	56	12	)	)	PUNCT
cana-4846	56	13	.	.	PUNCT
cana-4846	57	1	therefore	therefore	ADV
cana-4846	57	2	,	,	PUNCT
cana-4846	57	3	∂(t(g	∂(t(g	PROPN
cana-4846	57	4	)	)	PUNCT
cana-4846	57	5	)	)	PUNCT
cana-4846	58	1	=	=	PUNCT
cana-4846	58	2	|b(s)|	|b(s)|	PROPN
cana-4846	58	3	−	−	PROPN
cana-4846	58	4	|s|	|s|	PROPN
cana-4846	58	5	=	=	SYM
cana-4846	58	6	[	[	PUNCT
cana-4846	58	7	n(n+1	n(n+1	ADJ
cana-4846	58	8	)	)	PUNCT
cana-4846	58	9	2	2	NUM
cana-4846	58	10	−	−	NOUN
cana-4846	58	11	n	n	PRON
cana-4846	58	12	2	2	NUM
cana-4846	58	13	]	]	PUNCT
cana-4846	58	14	−	−	PROPN
cana-4846	58	15	n	n	NOUN
cana-4846	58	16	2	2	NUM
cana-4846	58	17	=	=	SYM
cana-4846	58	18	n(n−1	n(n−1	NUM
cana-4846	58	19	)	)	PUNCT
cana-4846	58	20	2	2	NUM
cana-4846	58	21	.	.	PUNCT
cana-4846	59	1	if	if	SCONJ
cana-4846	59	2	𝑛	𝑛	PROPN
cana-4846	59	3	is	be	AUX
cana-4846	59	4	odd	odd	ADJ
cana-4846	59	5	,	,	PUNCT
cana-4846	59	6	we	we	PRON
cana-4846	59	7	choose	choose	VERB
cana-4846	59	8	the	the	DET
cana-4846	59	9	vertices	vertex	NOUN
cana-4846	59	10	in	in	ADP
cana-4846	59	11	s	s	PRON
cana-4846	59	12	as	as	SCONJ
cana-4846	59	13	we	we	PRON
cana-4846	59	14	discussed	discuss	VERB
cana-4846	59	15	in	in	ADP
cana-4846	59	16	the	the	DET
cana-4846	59	17	above	above	ADJ
cana-4846	59	18	case	case	NOUN
cana-4846	59	19	and	and	CCONJ
cana-4846	59	20	s	s	AUX
cana-4846	59	21	dominates	dominate	VERB
cana-4846	59	22	all	all	DET
cana-4846	59	23	the	the	DET
cana-4846	59	24	vertices	vertex	NOUN
cana-4846	59	25	except	except	SCONJ
cana-4846	59	26	one	one	NUM
cana-4846	59	27	in	in	ADP
cana-4846	59	28	the	the	DET
cana-4846	59	29	case	case	NOUN
cana-4846	59	30	.	.	PUNCT
cana-4846	60	1	therefore	therefore	ADV
cana-4846	60	2	,	,	PUNCT
cana-4846	60	3	∂(t(g	∂(t(g	PROPN
cana-4846	60	4	)	)	PUNCT
cana-4846	60	5	)	)	PUNCT
cana-4846	61	1	=	=	PUNCT
cana-4846	61	2	[	[	PUNCT
cana-4846	61	3	n(n+1	n(n+1	ADJ
cana-4846	61	4	)	)	PUNCT
cana-4846	61	5	2	2	NUM
cana-4846	62	1	−	−	PROPN
cana-4846	62	2	n−1	n−1	PROPN
cana-4846	62	3	2	2	NUM
cana-4846	62	4	−	−	PROPN
cana-4846	62	5	1	1	NUM
cana-4846	62	6	]	]	PUNCT
cana-4846	62	7	−	−	PROPN
cana-4846	62	8	n−1	n−1	PROPN
cana-4846	62	9	2	2	NUM
cana-4846	62	10	=	=	SYM
cana-4846	62	11	n(n−1	n(n−1	NUM
cana-4846	62	12	)	)	PUNCT
cana-4846	62	13	2	2	NUM
cana-4846	62	14	.	.	PUNCT
cana-4846	63	1	in	in	ADP
cana-4846	63	2	both	both	CCONJ
cana-4846	63	3	the	the	DET
cana-4846	63	4	cases	case	NOUN
cana-4846	63	5	,	,	PUNCT
cana-4846	63	6	∂(t(g	∂(t(g	PROPN
cana-4846	63	7	)	)	PUNCT
cana-4846	63	8	)	)	PUNCT
cana-4846	63	9	=	=	SYM
cana-4846	63	10	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-4846	63	11	)	)	PUNCT
cana-4846	63	12	2	2	NUM
cana-4846	63	13	.	.	PUNCT
cana-4846	63	14	hence	hence	ADV
cana-4846	63	15	the	the	DET
cana-4846	63	16	proof	proof	NOUN
cana-4846	63	17	.	.	PUNCT
cana-4846	64	1	theorem	theorem	ADJ
cana-4846	64	2	4	4	NUM
cana-4846	64	3	.	.	PUNCT
cana-4846	64	4	given	give	VERB
cana-4846	64	5	a	a	DET
cana-4846	64	6	positive	positive	ADJ
cana-4846	64	7	integer	integer	NOUN
cana-4846	64	8	k	k	NOUN
cana-4846	64	9	,	,	PUNCT
cana-4846	64	10	there	there	PRON
cana-4846	64	11	exist	exist	VERB
cana-4846	64	12	a	a	DET
cana-4846	64	13	graph	graph	NOUN
cana-4846	64	14	on	on	ADP
cana-4846	64	15	n	n	CCONJ
cana-4846	64	16	vertices	vertex	NOUN
cana-4846	64	17	whose	whose	DET
cana-4846	64	18	total	total	ADJ
cana-4846	64	19	graph	graph	NOUN
cana-4846	64	20	on	on	ADP
cana-4846	64	21	𝑙	𝑙	DET
cana-4846	64	22	vertices	vertex	NOUN
cana-4846	64	23	with	with	ADP
cana-4846	64	24	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	64	25	)	)	PUNCT
cana-4846	64	26	)	)	PUNCT
cana-4846	65	1	=	=	PUNCT
cana-4846	66	1	𝑘	𝑘	DET
cana-4846	66	2	proof	proof	NOUN
cana-4846	66	3	.	.	PUNCT
cana-4846	67	1	if	if	SCONJ
cana-4846	67	2	𝑛	𝑛	PROPN
cana-4846	67	3	is	be	AUX
cana-4846	67	4	odd	odd	ADJ
cana-4846	67	5	,	,	PUNCT
cana-4846	67	6	we	we	PRON
cana-4846	67	7	consider	consider	VERB
cana-4846	67	8	the	the	DET
cana-4846	67	9	graph	graph	NOUN
cana-4846	67	10	with	with	ADP
cana-4846	67	11	𝑛	𝑛	PROPN
cana-4846	67	12	=	=	SYM
cana-4846	67	13	𝑘+5	𝑘+5	NOUN
cana-4846	67	14	2	2	NUM
cana-4846	67	15	vertices	vertice	VERB
cana-4846	67	16	whose	whose	DET
cana-4846	67	17	total	total	ADJ
cana-4846	67	18	graph	graph	NOUN
cana-4846	67	19	has	have	VERB
cana-4846	67	20	𝑘+3	𝑘+3	PROPN
cana-4846	67	21	2	2	NUM
cana-4846	67	22	copies	copy	NOUN
cana-4846	67	23	of	of	ADP
cana-4846	67	24	𝑘3	𝑘3	PROPN
cana-4846	67	25	with	with	ADP
cana-4846	67	26	exactly	exactly	ADV
cana-4846	67	27	one	one	NUM
cana-4846	67	28	vertex	vertex	NOUN
cana-4846	67	29	as	as	ADP
cana-4846	67	30	common	common	ADJ
cana-4846	67	31	,	,	PUNCT
cana-4846	67	32	say	say	VERB
cana-4846	67	33	𝑣.	𝑣.	PROPN
cana-4846	67	34	clearly	clearly	ADV
cana-4846	67	35	,	,	PUNCT
cana-4846	67	36	𝜕−	𝜕−	NOUN
cana-4846	67	37	set	set	NOUN
cana-4846	67	38	of	of	ADP
cana-4846	67	39	𝑇(𝐺	𝑇(𝐺	NOUN
cana-4846	67	40	)	)	PUNCT
cana-4846	67	41	contains	contain	VERB
cana-4846	67	42	𝑣	𝑣	PRON
cana-4846	67	43	only	only	ADV
cana-4846	67	44	.	.	PUNCT
cana-4846	68	1	hence	hence	ADV
cana-4846	68	2	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	68	3	)	)	PUNCT
cana-4846	68	4	)	)	PUNCT
cana-4846	69	1	=	=	PUNCT
cana-4846	69	2	𝑘.	𝑘.	NOUN
cana-4846	69	3	if	if	SCONJ
cana-4846	69	4	𝑛	𝑛	PRON
cana-4846	69	5	is	be	AUX
cana-4846	69	6	even	even	ADV
cana-4846	69	7	,	,	PUNCT
cana-4846	69	8	when	when	SCONJ
cana-4846	69	9	𝑛	𝑛	PROPN
cana-4846	69	10	=	=	SYM
cana-4846	69	11	4	4	NUM
cana-4846	69	12	,	,	PUNCT
cana-4846	69	13	we	we	PRON
cana-4846	69	14	consider	consider	VERB
cana-4846	69	15	the	the	DET
cana-4846	69	16	circulant	circulant	ADJ
cana-4846	69	17	graph	graph	NOUN
cana-4846	69	18	c4,2	c4,2	NOUN
cana-4846	69	19	.	.	PUNCT
cana-4846	70	1	for	for	ADP
cana-4846	70	2	other	other	ADJ
cana-4846	70	3	cases	case	NOUN
cana-4846	70	4	,	,	PUNCT
cana-4846	70	5	we	we	PRON
cana-4846	70	6	consider	consider	VERB
cana-4846	70	7	the	the	DET
cana-4846	70	8	circulant	circulant	ADJ
cana-4846	70	9	graph	graph	NOUN
cana-4846	70	10	g	g	NOUN
cana-4846	70	11	=	=	NOUN
cana-4846	70	12	c4,2	c4,2	PROPN
cana-4846	70	13	in	in	ADP
cana-4846	70	14	which	which	PRON
cana-4846	70	15	𝑘−6	𝑘−6	NOUN
cana-4846	70	16	2	2	NUM
cana-4846	70	17	copies	copy	NOUN
cana-4846	70	18	of	of	ADP
cana-4846	70	19	p2	p2	PROPN
cana-4846	70	20	which	which	PRON
cana-4846	70	21	is	be	AUX
cana-4846	70	22	attached	attach	VERB
cana-4846	70	23	with	with	ADP
cana-4846	70	24	a	a	DET
cana-4846	70	25	vertex	vertex	NOUN
cana-4846	70	26	,	,	PUNCT
cana-4846	70	27	say	say	VERB
cana-4846	70	28	u	u	NOUN
cana-4846	70	29	of	of	ADP
cana-4846	70	30	c4,2	c4,2	PROPN
cana-4846	70	31	.	.	PUNCT
cana-4846	71	1	let	let	VERB
cana-4846	71	2	s	s	PRON
cana-4846	71	3	be	be	AUX
cana-4846	71	4	the	the	DET
cana-4846	71	5	𝜕−	𝜕−	NOUN
cana-4846	71	6	set	set	NOUN
cana-4846	71	7	of	of	ADP
cana-4846	71	8	above	above	ADP
cana-4846	71	9	graphs	graph	NOUN
cana-4846	71	10	.	.	PUNCT
cana-4846	72	1	consider	consider	VERB
cana-4846	72	2	s	s	PRON
cana-4846	72	3	=	=	PUNCT
cana-4846	72	4	{	{	PUNCT
cana-4846	72	5	u	u	NOUN
cana-4846	72	6	,	,	PUNCT
cana-4846	72	7	v	v	NOUN
cana-4846	72	8	}	}	PUNCT
cana-4846	72	9	where	where	SCONJ
cana-4846	72	10	𝑣	𝑣	PRON
cana-4846	72	11	∉	∉	X
cana-4846	72	12	𝑁(𝑢	𝑁(𝑢	NUM
cana-4846	72	13	)	)	PUNCT
cana-4846	72	14	.	.	PUNCT
cana-4846	73	1	so	so	ADV
cana-4846	73	2	,	,	PUNCT
cana-4846	73	3	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	73	4	)	)	PUNCT
cana-4846	73	5	)	)	PUNCT
cana-4846	74	1	=	=	PUNCT
cana-4846	74	2	𝑘.	𝑘.	NOUN
cana-4846	74	3	communications	communication	NOUN
cana-4846	74	4	on	on	ADP
cana-4846	74	5	applied	apply	VERB
cana-4846	74	6	nonlinear	nonlinear	ADJ
cana-4846	74	7	analysis	analysis	NOUN
cana-4846	74	8	issn	issn	NOUN
cana-4846	74	9	:	:	PUNCT
cana-4846	74	10	1074	1074	NUM
cana-4846	74	11	-	-	PUNCT
cana-4846	74	12	133x	133x	NUM
cana-4846	74	13	vol	vol	VERB
cana-4846	74	14	32	32	NUM
cana-4846	74	15	no	no	NOUN
cana-4846	74	16	.	.	PUNCT
cana-4846	75	1	10s	10	NOUN
cana-4846	75	2	(	(	PUNCT
cana-4846	75	3	2025	2025	NUM
cana-4846	75	4	)	)	PUNCT
cana-4846	75	5	563	563	NUM
cana-4846	75	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4846	75	7	observation	observation	NOUN
cana-4846	75	8	5	5	NUM
cana-4846	75	9	.	.	PUNCT
cana-4846	76	1	for	for	ADP
cana-4846	76	2	any	any	DET
cana-4846	76	3	graph	graph	NOUN
cana-4846	76	4	g	g	NOUN
cana-4846	76	5	with	with	ADP
cana-4846	76	6	n	n	ADP
cana-4846	76	7	vertices	vertex	NOUN
cana-4846	76	8	,	,	PUNCT
cana-4846	76	9	∂(t(g	∂(t(g	PROPN
cana-4846	76	10	)	)	PUNCT
cana-4846	76	11	)	)	PUNCT
cana-4846	76	12	≠	≠	PROPN
cana-4846	76	13	2	2	NUM
cana-4846	76	14	.	.	X
cana-4846	76	15	theorem	theorem	VERB
cana-4846	76	16	6	6	NUM
cana-4846	76	17	.	.	PUNCT
cana-4846	77	1	for	for	ADP
cana-4846	77	2	any	any	DET
cana-4846	77	3	graph	graph	NOUN
cana-4846	77	4	g	g	NOUN
cana-4846	77	5	with	with	ADP
cana-4846	77	6	n	n	ADP
cana-4846	77	7	vertices	vertex	NOUN
cana-4846	77	8	,	,	PUNCT
cana-4846	77	9	1	1	NUM
cana-4846	77	10	≤	≤	NUM
cana-4846	77	11	∂(t(g	∂(t(g	PROPN
cana-4846	77	12	)	)	PUNCT
cana-4846	77	13	)	)	PUNCT
cana-4846	78	1	−	−	PROPN
cana-4846	78	2	∂(g	∂(g	ADJ
cana-4846	78	3	)	)	PUNCT
cana-4846	78	4	≤	≤	NOUN
cana-4846	78	5	n2−3n+4	n2−3n+4	NOUN
cana-4846	78	6	2	2	NUM
cana-4846	78	7	proof	proof	NOUN
cana-4846	78	8	.	.	PUNCT
cana-4846	79	1	given	give	VERB
cana-4846	79	2	that	that	SCONJ
cana-4846	79	3	𝐺	𝐺	PROPN
cana-4846	79	4	is	be	AUX
cana-4846	79	5	a	a	DET
cana-4846	79	6	graph	graph	NOUN
cana-4846	79	7	of	of	ADP
cana-4846	79	8	𝑛	𝑛	DET
cana-4846	79	9	vertices	vertex	NOUN
cana-4846	79	10	.	.	PUNCT
cana-4846	80	1	the	the	DET
cana-4846	80	2	maximum	maximum	ADJ
cana-4846	80	3	value	value	NOUN
cana-4846	80	4	of	of	ADP
cana-4846	80	5	the	the	DET
cana-4846	80	6	differential	differential	NOUN
cana-4846	80	7	of	of	ADP
cana-4846	80	8	any	any	DET
cana-4846	80	9	total	total	ADJ
cana-4846	80	10	graph	graph	NOUN
cana-4846	80	11	is	be	AUX
cana-4846	80	12	less	less	ADJ
cana-4846	80	13	than	than	ADP
cana-4846	80	14	or	or	CCONJ
cana-4846	80	15	equal	equal	ADJ
cana-4846	80	16	to	to	ADP
cana-4846	80	17	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-4846	80	18	)	)	PUNCT
cana-4846	80	19	2	2	NUM
cana-4846	80	20	and	and	CCONJ
cana-4846	80	21	clearly	clearly	ADV
cana-4846	80	22	𝜕(𝐺	𝜕(𝐺	NUM
cana-4846	80	23	)	)	PUNCT
cana-4846	80	24	≤	≤	NOUN
cana-4846	81	1	𝑛	𝑛	DET
cana-4846	81	2	−	−	PROPN
cana-4846	81	3	2	2	NUM
cana-4846	81	4	.	.	PUNCT
cana-4846	81	5	therefore	therefore	ADV
cana-4846	81	6	,	,	PUNCT
cana-4846	81	7	∂(t(g	∂(t(g	PROPN
cana-4846	81	8	)	)	PUNCT
cana-4846	81	9	)	)	PUNCT
cana-4846	82	1	−	−	PROPN
cana-4846	82	2	∂(g	∂(g	ADJ
cana-4846	82	3	)	)	PUNCT
cana-4846	82	4	≤	≤	PROPN
cana-4846	82	5	𝑛(𝑛−1	𝑛(𝑛−1	NOUN
cana-4846	82	6	)	)	PUNCT
cana-4846	82	7	2	2	NUM
cana-4846	82	8	−	−	NOUN
cana-4846	82	9	(	(	PUNCT
cana-4846	82	10	𝑛	𝑛	PROPN
cana-4846	82	11	−	−	NOUN
cana-4846	82	12	2	2	NUM
cana-4846	82	13	)	)	PUNCT
cana-4846	82	14	=	=	NOUN
cana-4846	82	15	n2−3n+4	n2−3n+4	NOUN
cana-4846	82	16	2	2	NUM
cana-4846	82	17	.	.	PUNCT
cana-4846	83	1	observation	observation	NOUN
cana-4846	83	2	7	7	NUM
cana-4846	83	3	.	.	PUNCT
cana-4846	84	1	for	for	ADP
cana-4846	84	2	any	any	DET
cana-4846	84	3	graph	graph	NOUN
cana-4846	84	4	g	g	NOUN
cana-4846	84	5	with	with	ADP
cana-4846	84	6	n	n	ADP
cana-4846	84	7	vertices	vertex	NOUN
cana-4846	84	8	,	,	PUNCT
cana-4846	84	9	∂(t(g	∂(t(g	PROPN
cana-4846	84	10	)	)	PUNCT
cana-4846	84	11	)	)	PUNCT
cana-4846	85	1	−	−	PROPN
cana-4846	85	2	∂(g	∂(g	ADJ
cana-4846	85	3	)	)	PUNCT
cana-4846	86	1	=	=	SYM
cana-4846	86	2	n2−3n+4	n2−3n+4	NOUN
cana-4846	86	3	2	2	NUM
cana-4846	86	4	if	if	SCONJ
cana-4846	86	5	and	and	CCONJ
cana-4846	86	6	only	only	ADV
cana-4846	86	7	if	if	SCONJ
cana-4846	86	8	g	g	PROPN
cana-4846	86	9	is	be	AUX
cana-4846	86	10	a	a	DET
cana-4846	86	11	complete	complete	ADJ
cana-4846	86	12	graph	graph	NOUN
cana-4846	86	13	.	.	PUNCT
cana-4846	86	14	theorem	theorem	NOUN
cana-4846	86	15	8	8	NUM
cana-4846	86	16	.	.	PUNCT
cana-4846	87	1	if	if	SCONJ
cana-4846	87	2	g	g	PROPN
cana-4846	87	3	=	=	PUNCT
cana-4846	87	4	𝐾1,𝑛−1	𝐾1,𝑛−1	PROPN
cana-4846	87	5	is	be	AUX
cana-4846	87	6	a	a	DET
cana-4846	87	7	star	star	NOUN
cana-4846	87	8	graph	graph	NOUN
cana-4846	87	9	,	,	PUNCT
cana-4846	87	10	then	then	ADV
cana-4846	87	11	∂	∂	NUM
cana-4846	87	12	(	(	PUNCT
cana-4846	87	13	t(𝐾1,𝑛−1	t(𝐾1,𝑛−1	NOUN
cana-4846	87	14	)	)	PUNCT
cana-4846	87	15	)	)	PUNCT
cana-4846	88	1	=	=	SYM
cana-4846	88	2	2𝑛	2𝑛	PROPN
cana-4846	89	1	−	−	NOUN
cana-4846	89	2	3	3	NUM
cana-4846	89	3	proof	proof	NOUN
cana-4846	89	4	.	.	PUNCT
cana-4846	90	1	let	let	VERB
cana-4846	90	2	𝑉(𝐾1,𝑛−1	𝑉(𝐾1,𝑛−1	VERB
cana-4846	90	3	)	)	PUNCT
cana-4846	90	4	=	=	SYM
cana-4846	90	5	{	{	PUNCT
cana-4846	90	6	𝑣	𝑣	NOUN
cana-4846	90	7	,	,	PUNCT
cana-4846	90	8	𝑣1	𝑣1	NOUN
cana-4846	90	9	,	,	PUNCT
cana-4846	90	10	𝑣2	𝑣2	PROPN
cana-4846	90	11	…	…	SYM
cana-4846	90	12	𝑣𝑛−1	𝑣𝑛−1	NOUN
cana-4846	90	13	}	}	PUNCT
cana-4846	90	14	and	and	CCONJ
cana-4846	90	15	𝐸(𝐾1,𝑛−1	𝐸(𝐾1,𝑛−1	NOUN
cana-4846	90	16	)	)	PUNCT
cana-4846	91	1	=	=	SYM
cana-4846	91	2	{	{	PUNCT
cana-4846	91	3	𝑒1	𝑒1	NOUN
cana-4846	91	4	,	,	PUNCT
cana-4846	91	5	𝑒2	𝑒2	PROPN
cana-4846	91	6	…	…	SYM
cana-4846	91	7	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	91	8	}	}	PUNCT
cana-4846	91	9	where	where	SCONJ
cana-4846	91	10	𝑒𝑖	𝑒𝑖	ADP
cana-4846	91	11	=	=	SYM
cana-4846	91	12	𝑣𝑣𝑖	𝑣𝑣𝑖	PROPN
cana-4846	91	13	,	,	PUNCT
cana-4846	91	14	𝑖	𝑖	X
cana-4846	91	15	=	=	SYM
cana-4846	91	16	1,2	1,2	NUM
cana-4846	91	17	,	,	PUNCT
cana-4846	91	18	…	…	PUNCT
cana-4846	91	19	𝑛	𝑛	PRON
cana-4846	91	20	−	−	NUM
cana-4846	91	21	1	1	NUM
cana-4846	91	22	and	and	CCONJ
cana-4846	91	23	𝑣	𝑣	PROPN
cana-4846	91	24	is	be	AUX
cana-4846	91	25	a	a	DET
cana-4846	91	26	head	head	NOUN
cana-4846	91	27	vertex	vertex	NOUN
cana-4846	91	28	of	of	ADP
cana-4846	91	29	the	the	DET
cana-4846	91	30	star	star	NOUN
cana-4846	91	31	graph	graph	NOUN
cana-4846	91	32	.	.	PUNCT
cana-4846	92	1	by	by	ADP
cana-4846	92	2	the	the	DET
cana-4846	92	3	definition	definition	NOUN
cana-4846	92	4	of	of	ADP
cana-4846	92	5	total	total	ADJ
cana-4846	92	6	graph	graph	NOUN
cana-4846	92	7	,	,	PUNCT
cana-4846	92	8	𝑉(𝐾1,𝑛−1	𝑉(𝐾1,𝑛−1	NOUN
cana-4846	92	9	)	)	PUNCT
cana-4846	92	10	=	=	SYM
cana-4846	92	11	{	{	PUNCT
cana-4846	92	12	𝑣1	𝑣1	PROPN
cana-4846	92	13	,	,	PUNCT
cana-4846	92	14	𝑣2	𝑣2	PROPN
cana-4846	92	15	…	…	SYM
cana-4846	92	16	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4846	92	17	,	,	PUNCT
cana-4846	92	18	𝑒1	𝑒1	NOUN
cana-4846	92	19	,	,	PUNCT
cana-4846	92	20	𝑒2	𝑒2	PROPN
cana-4846	92	21	…	…	SYM
cana-4846	92	22	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	92	23	}	}	PUNCT
cana-4846	92	24	and	and	CCONJ
cana-4846	92	25	|𝑉(𝐾1,𝑛−1)|	|𝑉(𝐾1,𝑛−1)|	PRON
cana-4846	92	26	=	=	SYM
cana-4846	92	27	2𝑛	2𝑛	PROPN
cana-4846	92	28	+	+	CCONJ
cana-4846	92	29	1	1	X
cana-4846	92	30	.	.	PUNCT
cana-4846	92	31	since	since	SCONJ
cana-4846	92	32	𝑑𝑒𝑔(𝑣	𝑑𝑒𝑔(𝑣	NUM
cana-4846	92	33	)	)	PUNCT
cana-4846	92	34	=	=	SYM
cana-4846	92	35	2𝑛	2𝑛	PROPN
cana-4846	92	36	and	and	CCONJ
cana-4846	92	37	𝑆	𝑆	PROPN
cana-4846	92	38	=	=	SYM
cana-4846	92	39	{	{	PUNCT
cana-4846	92	40	𝑣	𝑣	NOUN
cana-4846	92	41	}	}	PUNCT
cana-4846	92	42	is	be	AUX
cana-4846	92	43	the	the	DET
cana-4846	92	44	differential	differential	ADJ
cana-4846	92	45	set	set	NOUN
cana-4846	92	46	,	,	PUNCT
cana-4846	92	47	then	then	ADV
cana-4846	92	48	∂	∂	NUM
cana-4846	92	49	(	(	PUNCT
cana-4846	92	50	t(𝐾1,𝑛−1	t(𝐾1,𝑛−1	NOUN
cana-4846	92	51	)	)	PUNCT
cana-4846	92	52	)	)	PUNCT
cana-4846	93	1	=	=	SYM
cana-4846	94	1	2𝑛	2𝑛	PROPN
cana-4846	94	2	−	−	NOUN
cana-4846	94	3	3	3	NUM
cana-4846	94	4	theorem	theorem	NOUN
cana-4846	94	5	9	9	NUM
cana-4846	94	6	.	.	PUNCT
cana-4846	94	7	for	for	ADP
cana-4846	94	8	any	any	DET
cana-4846	94	9	graph	graph	NOUN
cana-4846	94	10	g	g	PROPN
cana-4846	94	11	=	=	SYM
cana-4846	94	12	𝐶𝑛	𝐶𝑛	PROPN
cana-4846	94	13	,	,	PUNCT
cana-4846	94	14	then	then	ADV
cana-4846	94	15	𝜕(𝑇(𝑃𝑛	𝜕(𝑇(𝑃𝑛	PROPN
cana-4846	94	16	)	)	PUNCT
cana-4846	94	17	)	)	PUNCT
cana-4846	95	1	=	=	PRON
cana-4846	95	2	{	{	PUNCT
cana-4846	95	3	3	3	NUM
cana-4846	95	4	⌊	⌊	PROPN
cana-4846	95	5	2𝑛	2𝑛	PROPN
cana-4846	95	6	5	5	NUM
cana-4846	95	7	⌋	⌋	NOUN
cana-4846	95	8	,	,	PUNCT
cana-4846	95	9	𝑛	𝑛	PRON
cana-4846	95	10	≡	≡	PROPN
cana-4846	95	11	0,1,3	0,1,3	PROPN
cana-4846	95	12	(	(	PUNCT
cana-4846	95	13	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	95	14	5	5	NUM
cana-4846	95	15	)	)	PUNCT
cana-4846	95	16	3	3	NUM
cana-4846	96	1	⌊	⌊	ADP
cana-4846	96	2	2𝑛	2𝑛	NUM
cana-4846	96	3	5	5	NUM
cana-4846	96	4	⌋	⌋	NOUN
cana-4846	96	5	+	+	CCONJ
cana-4846	96	6	1	1	NUM
cana-4846	96	7	,	,	PUNCT
cana-4846	96	8	𝑛	𝑛	DET
cana-4846	96	9	≡	≡	PROPN
cana-4846	96	10	4	4	NUM
cana-4846	96	11	(	(	PUNCT
cana-4846	96	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	96	13	5	5	NUM
cana-4846	96	14	)	)	PUNCT
cana-4846	96	15	3	3	NUM
cana-4846	96	16	⌊	⌊	ADP
cana-4846	96	17	2𝑛	2𝑛	NUM
cana-4846	96	18	5	5	NUM
cana-4846	96	19	⌋	⌋	NOUN
cana-4846	96	20	+	+	CCONJ
cana-4846	96	21	2	2	NUM
cana-4846	96	22	,	,	PUNCT
cana-4846	96	23	𝑛	𝑛	PRON
cana-4846	96	24	≡	≡	PROPN
cana-4846	96	25	2	2	NUM
cana-4846	96	26	(	(	PUNCT
cana-4846	96	27	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	96	28	5	5	NUM
cana-4846	96	29	)	)	PUNCT
cana-4846	96	30	proof	proof	NOUN
cana-4846	96	31	.	.	PUNCT
cana-4846	97	1	let	let	VERB
cana-4846	97	2	𝑉(𝐶𝑛	𝑉(𝐶𝑛	NUM
cana-4846	97	3	)	)	PUNCT
cana-4846	98	1	=	=	PRON
cana-4846	98	2	{	{	PUNCT
cana-4846	98	3	𝑣1	𝑣1	PROPN
cana-4846	98	4	,	,	PUNCT
cana-4846	98	5	𝑣2	𝑣2	PROPN
cana-4846	98	6	,	,	PUNCT
cana-4846	98	7	…	…	PUNCT
cana-4846	98	8	.	.	PUNCT
cana-4846	98	9	.	.	PUNCT
cana-4846	99	1	,	,	PUNCT
cana-4846	99	2	𝑣𝑛	𝑣𝑛	X
cana-4846	99	3	}	}	PUNCT
cana-4846	99	4	be	be	VERB
cana-4846	99	5	the	the	DET
cana-4846	99	6	vertices	vertex	NOUN
cana-4846	99	7	of	of	ADP
cana-4846	99	8	cycle	cycle	NOUN
cana-4846	99	9	of	of	ADP
cana-4846	99	10	length	length	NOUN
cana-4846	99	11	𝑛	𝑛	PROPN
cana-4846	99	12	(	(	PUNCT
cana-4846	99	13	𝑛	𝑛	PRON
cana-4846	99	14	≥	≥	NOUN
cana-4846	99	15	3	3	NUM
cana-4846	99	16	)	)	PUNCT
cana-4846	99	17	and	and	CCONJ
cana-4846	99	18	𝐸(𝐶𝑛	𝐸(𝐶𝑛	NUM
cana-4846	99	19	)	)	PUNCT
cana-4846	100	1	=	=	PRON
cana-4846	100	2	{	{	PUNCT
cana-4846	100	3	𝑒1	𝑒1	NOUN
cana-4846	100	4	,	,	PUNCT
cana-4846	100	5	𝑒2	𝑒2	PROPN
cana-4846	100	6	,	,	PUNCT
cana-4846	100	7	…	…	PUNCT
cana-4846	100	8	.	.	PUNCT
cana-4846	100	9	.	.	PUNCT
cana-4846	101	1	,	,	PUNCT
cana-4846	101	2	𝑒𝑛	𝑒𝑛	AUX
cana-4846	101	3	}	}	PUNCT
cana-4846	101	4	be	be	AUX
cana-4846	101	5	the	the	DET
cana-4846	101	6	edges	edge	NOUN
cana-4846	101	7	of	of	ADP
cana-4846	101	8	the	the	DET
cana-4846	101	9	corresponding	corresponding	ADJ
cana-4846	101	10	vertices	vertex	NOUN
cana-4846	101	11	{	{	PUNCT
cana-4846	101	12	𝑣1	𝑣1	PROPN
cana-4846	101	13	,	,	PUNCT
cana-4846	101	14	𝑣2	𝑣2	PROPN
cana-4846	101	15	,	,	PUNCT
cana-4846	101	16	…	…	PUNCT
cana-4846	101	17	.	.	PUNCT
cana-4846	102	1	.	.	PUNCT
cana-4846	102	2	,	,	PUNCT
cana-4846	102	3	𝑣𝑛	𝑣𝑛	NOUN
cana-4846	102	4	}	}	PUNCT
cana-4846	102	5	.	.	PUNCT
cana-4846	103	1	then	then	ADV
cana-4846	103	2	,	,	PUNCT
cana-4846	103	3	𝑉(𝑇(𝐶𝑛	𝑉(𝑇(𝐶𝑛	PROPN
cana-4846	103	4	)	)	PUNCT
cana-4846	103	5	)	)	PUNCT
cana-4846	104	1	=	=	PRON
cana-4846	104	2	{	{	PUNCT
cana-4846	104	3	𝑣1	𝑣1	PROPN
cana-4846	104	4	,	,	PUNCT
cana-4846	104	5	𝑣2	𝑣2	PROPN
cana-4846	104	6	,	,	PUNCT
cana-4846	104	7	…	…	PUNCT
cana-4846	104	8	.	.	PUNCT
cana-4846	104	9	.	.	PUNCT
cana-4846	105	1	,	,	PUNCT
cana-4846	105	2	𝑣𝑛	𝑣𝑛	NOUN
cana-4846	105	3	,	,	PUNCT
cana-4846	105	4	𝑒1	𝑒1	NOUN
cana-4846	105	5	,	,	PUNCT
cana-4846	105	6	𝑒2	𝑒2	PROPN
cana-4846	105	7	,	,	PUNCT
cana-4846	105	8	…	…	PUNCT
cana-4846	105	9	.	.	PUNCT
cana-4846	105	10	.	.	PUNCT
cana-4846	106	1	,	,	PUNCT
cana-4846	106	2	𝑒𝑛	𝑒𝑛	NOUN
cana-4846	106	3	}	}	PUNCT
cana-4846	106	4	and	and	CCONJ
cana-4846	106	5	𝐸(𝑇(𝐺	𝐸(𝑇(𝐺	NOUN
cana-4846	106	6	)	)	PUNCT
cana-4846	106	7	)	)	PUNCT
cana-4846	107	1	=	=	PRON
cana-4846	107	2	{	{	PUNCT
cana-4846	107	3	𝑣𝑖𝑣𝑖+1/1	𝑣𝑖𝑣𝑖+1/1	PROPN
cana-4846	107	4	≤	≤	PROPN
cana-4846	107	5	𝑖	𝑖	SYM
cana-4846	107	6	≤	≤	NUM
cana-4846	107	7	𝑛	𝑛	PRON
cana-4846	107	8	−	−	PROPN
cana-4846	107	9	1	1	NUM
cana-4846	107	10	}	}	PUNCT
cana-4846	107	11	∪	∪	X
cana-4846	107	12	{	{	PUNCT
cana-4846	107	13	𝑒𝑖𝑒𝑖+1/1	𝑒𝑖𝑒𝑖+1/1	NOUN
cana-4846	107	14	≤	≤	NUM
cana-4846	107	15	𝑖	𝑖	SYM
cana-4846	107	16	≤	≤	NUM
cana-4846	107	17	𝑛	𝑛	PRON
cana-4846	107	18	−	−	PROPN
cana-4846	107	19	1	1	NUM
cana-4846	107	20	}	}	PUNCT
cana-4846	107	21	∪	∪	X
cana-4846	107	22	{	{	PUNCT
cana-4846	107	23	𝑣𝑖𝑒𝑖/1	𝑣𝑖𝑒𝑖/1	VERB
cana-4846	107	24	≤	≤	NUM
cana-4846	107	25	𝑖	𝑖	SYM
cana-4846	107	26	≤	≤	NUM
cana-4846	107	27	𝑛	𝑛	PRON
cana-4846	107	28	}	}	PUNCT
cana-4846	107	29	∪	∪	ADJ
cana-4846	107	30	{	{	PUNCT
cana-4846	107	31	𝑒𝑖𝑣𝑖+1/1	𝑒𝑖𝑣𝑖+1/1	NOUN
cana-4846	107	32	≤	≤	NUM
cana-4846	107	33	𝑖	𝑖	SYM
cana-4846	107	34	≤	≤	NUM
cana-4846	107	35	𝑛	𝑛	PRON
cana-4846	107	36	−	−	PROPN
cana-4846	107	37	1	1	NUM
cana-4846	107	38	}	}	PUNCT
cana-4846	107	39	∪	∪	ADJ
cana-4846	107	40	{	{	PUNCT
cana-4846	107	41	𝑒𝑛𝑣1	𝑒𝑛𝑣1	NOUN
cana-4846	107	42	,	,	PUNCT
cana-4846	107	43	𝑒𝑛𝑒1	𝑒𝑛𝑒1	NOUN
cana-4846	107	44	,	,	PUNCT
cana-4846	107	45	𝑣𝑛𝑣1	𝑣𝑛𝑣1	PROPN
cana-4846	107	46	}	}	PUNCT
cana-4846	107	47	and	and	CCONJ
cana-4846	107	48	hence	hence	ADV
cana-4846	107	49	|𝑉(𝑇(𝐶𝑛))|	|𝑉(𝑇(𝐶𝑛))|	X
cana-4846	107	50	=	=	PRON
cana-4846	107	51	2𝑛.	2𝑛.	NOUN
cana-4846	107	52	let	let	VERB
cana-4846	107	53	𝑆	𝑆	PROPN
cana-4846	107	54	be	be	AUX
cana-4846	107	55	the	the	DET
cana-4846	107	56	differential	differential	ADJ
cana-4846	107	57	set	set	NOUN
cana-4846	107	58	of	of	ADP
cana-4846	107	59	𝑇(𝐶𝑛	𝑇(𝐶𝑛	PROPN
cana-4846	107	60	)	)	PUNCT
cana-4846	107	61	.	.	PUNCT
cana-4846	108	1	when	when	SCONJ
cana-4846	108	2	𝑛	𝑛	DET
cana-4846	108	3	≡	≡	PROPN
cana-4846	108	4	0	0	PUNCT
cana-4846	108	5	(	(	PUNCT
cana-4846	108	6	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	108	7	5	5	NUM
cana-4846	108	8	)	)	PUNCT
cana-4846	108	9	.	.	PUNCT
cana-4846	109	1	consider	consider	VERB
cana-4846	109	2	𝑆	𝑆	PROPN
cana-4846	109	3	=	=	SYM
cana-4846	109	4	{	{	PUNCT
cana-4846	109	5	𝑣1	𝑣1	PROPN
cana-4846	109	6	,	,	PUNCT
cana-4846	109	7	𝑒3	𝑒3	PROPN
cana-4846	109	8	,	,	PUNCT
cana-4846	109	9	𝑣6	𝑣6	PROPN
cana-4846	109	10	,	,	PUNCT
cana-4846	109	11	…	…	PUNCT
cana-4846	109	12	.	.	PUNCT
cana-4846	110	1	.	.	PUNCT
cana-4846	111	1	,	,	PUNCT
cana-4846	111	2	𝑣𝑛−4	𝑣𝑛−4	ADJ
cana-4846	111	3	,	,	PUNCT
cana-4846	111	4	𝑒𝑛−2	𝑒𝑛−2	PROPN
cana-4846	111	5	}	}	PUNCT
cana-4846	111	6	be	be	AUX
cana-4846	111	7	a	a	DET
cana-4846	111	8	𝜕	𝜕	NOUN
cana-4846	111	9	−	−	NOUN
cana-4846	111	10	set	set	NOUN
cana-4846	111	11	and	and	CCONJ
cana-4846	111	12	hence	hence	ADV
cana-4846	111	13	.	.	PUNCT
cana-4846	112	1	|𝑆|	|𝑆|	VERB
cana-4846	112	2	=	=	SYM
cana-4846	112	3	2𝑛	2𝑛	PROPN
cana-4846	112	4	5	5	NUM
cana-4846	112	5	and	and	CCONJ
cana-4846	112	6	|𝐵(𝑆)|	|𝐵(𝑆)|	PROPN
cana-4846	112	7	=	=	NOUN
cana-4846	112	8	8𝑛	8𝑛	PROPN
cana-4846	112	9	5	5	NUM
cana-4846	112	10	.	.	PUNCT
cana-4846	113	1	then	then	ADV
cana-4846	113	2	,	,	PUNCT
cana-4846	113	3	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	113	4	)	)	PUNCT
cana-4846	113	5	)	)	PUNCT
cana-4846	114	1	=	=	PUNCT
cana-4846	115	1	|𝐵(𝑆)|	|𝐵(𝑆)|	AUX
cana-4846	115	2	−	−	NOUN
cana-4846	115	3	|𝑆|	|𝑆|	VERB
cana-4846	115	4	=	=	SYM
cana-4846	115	5	3	3	NUM
cana-4846	115	6	⌊	⌊	PROPN
cana-4846	115	7	2𝑛	2𝑛	NUM
cana-4846	115	8	5	5	NUM
cana-4846	115	9	⌋.	⌋.	NOUN
cana-4846	115	10	when	when	SCONJ
cana-4846	115	11	𝑛	𝑛	PRON
cana-4846	115	12	≡	≡	PROPN
cana-4846	115	13	1	1	NUM
cana-4846	115	14	(	(	PUNCT
cana-4846	115	15	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	115	16	5	5	NUM
cana-4846	115	17	)	)	PUNCT
cana-4846	115	18	.	.	PUNCT
cana-4846	116	1	consider	consider	VERB
cana-4846	116	2	the	the	DET
cana-4846	116	3	differential	differential	NOUN
cana-4846	116	4	set	set	VERB
cana-4846	116	5	𝑆	𝑆	PROPN
cana-4846	116	6	=	=	SYM
cana-4846	116	7	{	{	PUNCT
cana-4846	116	8	𝑣1	𝑣1	PROPN
cana-4846	116	9	,	,	PUNCT
cana-4846	116	10	𝑒3	𝑒3	PROPN
cana-4846	116	11	,	,	PUNCT
cana-4846	116	12	𝑣6	𝑣6	PROPN
cana-4846	116	13	,	,	PUNCT
cana-4846	116	14	…	…	PUNCT
cana-4846	116	15	.	.	PUNCT
cana-4846	116	16	.	.	PUNCT
cana-4846	117	1	,	,	PUNCT
cana-4846	117	2	𝑒𝑛−3	𝑒𝑛−3	PROPN
cana-4846	117	3	,	,	PUNCT
cana-4846	117	4	𝑣𝑛−5	𝑣𝑛−5	NOUN
cana-4846	117	5	}	}	PUNCT
cana-4846	117	6	and	and	CCONJ
cana-4846	117	7	𝐶(𝑆	𝐶(𝑆	NUM
cana-4846	117	8	)	)	PUNCT
cana-4846	118	1	=	=	PRON
cana-4846	118	2	{	{	PUNCT
cana-4846	118	3	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	118	4	,	,	PUNCT
cana-4846	118	5	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4846	118	6	}	}	PUNCT
cana-4846	118	7	.	.	PUNCT
cana-4846	119	1	clearly	clearly	ADV
cana-4846	119	2	,	,	PUNCT
cana-4846	119	3	|𝑆|	|𝑆|	VERB
cana-4846	119	4	=	=	PUNCT
cana-4846	120	1	⌊	⌊	X
cana-4846	120	2	2𝑛	2𝑛	NUM
cana-4846	120	3	5	5	NUM
cana-4846	120	4	⌋	⌋	NOUN
cana-4846	120	5	and	and	CCONJ
cana-4846	120	6	|𝐵(𝑆)|	|𝐵(𝑆)|	X
cana-4846	120	7	=	=	SYM
cana-4846	120	8	⌊	⌊	VERB
cana-4846	120	9	8𝑛	8𝑛	NOUN
cana-4846	120	10	5	5	NUM
cana-4846	120	11	⌋.	⌋.	ADV
cana-4846	120	12	so	so	ADV
cana-4846	120	13	,	,	PUNCT
cana-4846	120	14	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	120	15	)	)	PUNCT
cana-4846	120	16	)	)	PUNCT
cana-4846	121	1	=	=	SYM
cana-4846	121	2	3	3	NUM
cana-4846	121	3	⌊	⌊	ADP
cana-4846	121	4	2𝑛	2𝑛	NUM
cana-4846	121	5	5	5	NUM
cana-4846	121	6	⌋.	⌋.	NOUN
cana-4846	121	7	when	when	SCONJ
cana-4846	121	8	𝑛	𝑛	DET
cana-4846	121	9	≡	≡	PROPN
cana-4846	121	10	3	3	NUM
cana-4846	121	11	(	(	PUNCT
cana-4846	121	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	121	13	5	5	NUM
cana-4846	121	14	)	)	PUNCT
cana-4846	121	15	.	.	PUNCT
cana-4846	122	1	consider	consider	VERB
cana-4846	122	2	the	the	DET
cana-4846	122	3	differential	differential	NOUN
cana-4846	122	4	set	set	VERB
cana-4846	122	5	𝑆	𝑆	PROPN
cana-4846	122	6	=	=	SYM
cana-4846	122	7	{	{	PUNCT
cana-4846	122	8	𝑣1	𝑣1	PROPN
cana-4846	122	9	,	,	PUNCT
cana-4846	122	10	𝑒3	𝑒3	PROPN
cana-4846	122	11	,	,	PUNCT
cana-4846	122	12	𝑣6	𝑣6	PROPN
cana-4846	122	13	,	,	PUNCT
cana-4846	122	14	…	…	PUNCT
cana-4846	122	15	.	.	PUNCT
cana-4846	122	16	.	.	PUNCT
cana-4846	123	1	,	,	PUNCT
cana-4846	123	2	𝑒𝑛−5	𝑒𝑛−5	NOUN
cana-4846	123	3	,	,	PUNCT
cana-4846	123	4	𝑣𝑛−2	𝑣𝑛−2	PROPN
cana-4846	123	5	}	}	PUNCT
cana-4846	123	6	and	and	CCONJ
cana-4846	123	7	𝐶(𝑆	𝐶(𝑆	NUM
cana-4846	123	8	)	)	PUNCT
cana-4846	124	1	=	=	PRON
cana-4846	124	2	{	{	PUNCT
cana-4846	124	3	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	124	4	}	}	PUNCT
cana-4846	124	5	.	.	PUNCT
cana-4846	125	1	clearly	clearly	ADV
cana-4846	125	2	,	,	PUNCT
cana-4846	125	3	|𝑆|	|𝑆|	VERB
cana-4846	125	4	=	=	PUNCT
cana-4846	126	1	⌊	⌊	X
cana-4846	126	2	2𝑛	2𝑛	NUM
cana-4846	126	3	5	5	NUM
cana-4846	126	4	⌋	⌋	NOUN
cana-4846	126	5	and	and	CCONJ
cana-4846	126	6	|𝐵(𝑆)|	|𝐵(𝑆)|	X
cana-4846	126	7	=	=	SYM
cana-4846	126	8	⌊	⌊	VERB
cana-4846	126	9	8𝑛	8𝑛	NOUN
cana-4846	126	10	5	5	NUM
cana-4846	126	11	⌋.	⌋.	ADV
cana-4846	126	12	so	so	ADV
cana-4846	126	13	,	,	PUNCT
cana-4846	126	14	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	126	15	)	)	PUNCT
cana-4846	126	16	)	)	PUNCT
cana-4846	127	1	=	=	PUNCT
cana-4846	128	1	|𝐵(𝑆)|	|𝐵(𝑆)|	AUX
cana-4846	128	2	−	−	NOUN
cana-4846	128	3	|𝑆|	|𝑆|	VERB
cana-4846	128	4	=	=	SYM
cana-4846	128	5	3	3	NUM
cana-4846	128	6	⌊	⌊	PROPN
cana-4846	128	7	2𝑛	2𝑛	NUM
cana-4846	128	8	5	5	NUM
cana-4846	128	9	⌋.	⌋.	NOUN
cana-4846	128	10	when	when	SCONJ
cana-4846	128	11	𝑛	𝑛	DET
cana-4846	128	12	≡	≡	PROPN
cana-4846	128	13	4	4	NUM
cana-4846	128	14	(	(	PUNCT
cana-4846	128	15	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	128	16	5	5	NUM
cana-4846	128	17	)	)	PUNCT
cana-4846	128	18	consider	consider	VERB
cana-4846	128	19	the	the	DET
cana-4846	128	20	set	set	NOUN
cana-4846	128	21	𝑆	𝑆	PROPN
cana-4846	128	22	=	=	PUNCT
cana-4846	128	23	𝑆1	𝑆1	NOUN
cana-4846	128	24	∪	∪	ADJ
cana-4846	128	25	𝑆2	𝑆2	NOUN
cana-4846	128	26	where	where	SCONJ
cana-4846	128	27	𝑆1	𝑆1	NOUN
cana-4846	128	28	=	=	SYM
cana-4846	128	29	{	{	PUNCT
cana-4846	128	30	𝑣1	𝑣1	PROPN
cana-4846	128	31	,	,	PUNCT
cana-4846	128	32	𝑒3	𝑒3	PROPN
cana-4846	128	33	,	,	PUNCT
cana-4846	128	34	𝑣6	𝑣6	PROPN
cana-4846	128	35	,	,	PUNCT
cana-4846	128	36	…	…	PUNCT
cana-4846	128	37	.	.	PUNCT
cana-4846	128	38	.	.	PUNCT
cana-4846	129	1	,	,	PUNCT
cana-4846	129	2	𝑒𝑛−6	𝑒𝑛−6	PROPN
cana-4846	129	3	,	,	PUNCT
cana-4846	129	4	𝑣𝑛−3	𝑣𝑛−3	NOUN
cana-4846	129	5	}	}	PUNCT
cana-4846	129	6	and	and	CCONJ
cana-4846	129	7	𝑆2	𝑆2	PROPN
cana-4846	129	8	=	=	PUNCT
cana-4846	129	9	{	{	PUNCT
cana-4846	129	10	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4846	129	11	}	}	PUNCT
cana-4846	129	12	.	.	PUNCT
cana-4846	130	1	then	then	ADV
cana-4846	130	2	𝜕(𝑆2	𝜕(𝑆2	PROPN
cana-4846	130	3	)	)	PUNCT
cana-4846	130	4	=	=	SYM
cana-4846	131	1	1	1	X
cana-4846	131	2	.	.	PUNCT
cana-4846	132	1	so	so	ADV
cana-4846	132	2	,	,	PUNCT
cana-4846	132	3	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	132	4	)	)	PUNCT
cana-4846	132	5	)	)	PUNCT
cana-4846	133	1	=	=	SYM
cana-4846	133	2	3	3	NUM
cana-4846	133	3	⌊	⌊	ADP
cana-4846	133	4	2𝑛	2𝑛	PROPN
cana-4846	133	5	5	5	NUM
cana-4846	133	6	⌋	⌋	NOUN
cana-4846	133	7	+	+	CCONJ
cana-4846	133	8	1	1	X
cana-4846	133	9	.	.	X
cana-4846	133	10	when	when	SCONJ
cana-4846	133	11	𝑛	𝑛	DET
cana-4846	133	12	≡	≡	PROPN
cana-4846	133	13	2	2	NUM
cana-4846	133	14	(	(	PUNCT
cana-4846	133	15	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	133	16	5	5	NUM
cana-4846	133	17	)	)	PUNCT
cana-4846	133	18	.	.	PUNCT
cana-4846	134	1	consider	consider	VERB
cana-4846	134	2	the	the	DET
cana-4846	134	3	set	set	NOUN
cana-4846	134	4	𝑆	𝑆	PROPN
cana-4846	134	5	=	=	PUNCT
cana-4846	134	6	𝑆1	𝑆1	NOUN
cana-4846	134	7	∪	∪	ADJ
cana-4846	134	8	𝑆2	𝑆2	NOUN
cana-4846	134	9	where	where	SCONJ
cana-4846	134	10	𝑆1	𝑆1	NOUN
cana-4846	134	11	=	=	SYM
cana-4846	134	12	{	{	PUNCT
cana-4846	134	13	𝑣1	𝑣1	PROPN
cana-4846	134	14	,	,	PUNCT
cana-4846	134	15	𝑒3	𝑒3	PROPN
cana-4846	134	16	,	,	PUNCT
cana-4846	134	17	𝑣6	𝑣6	PROPN
cana-4846	134	18	,	,	PUNCT
cana-4846	134	19	…	…	PUNCT
cana-4846	134	20	.	.	PUNCT
cana-4846	134	21	.	.	PUNCT
cana-4846	135	1	,	,	PUNCT
cana-4846	135	2	𝑒𝑛−4	𝑒𝑛−4	PROPN
cana-4846	135	3	,	,	PUNCT
cana-4846	135	4	𝑣𝑛−6	𝑣𝑛−6	PROPN
cana-4846	135	5	}	}	PUNCT
cana-4846	135	6	and	and	CCONJ
cana-4846	135	7	𝑆2	𝑆2	PROPN
cana-4846	135	8	=	=	PUNCT
cana-4846	135	9	{	{	PUNCT
cana-4846	135	10	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4846	135	11	}	}	PUNCT
cana-4846	135	12	.	.	PUNCT
cana-4846	136	1	then	then	ADV
cana-4846	136	2	𝜕(𝑆2	𝜕(𝑆2	PROPN
cana-4846	136	3	)	)	PUNCT
cana-4846	136	4	=	=	SYM
cana-4846	136	5	2	2	X
cana-4846	136	6	.	.	PUNCT
cana-4846	137	1	so	so	ADV
cana-4846	137	2	,	,	PUNCT
cana-4846	137	3	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	137	4	)	)	PUNCT
cana-4846	137	5	)	)	PUNCT
cana-4846	138	1	=	=	SYM
cana-4846	138	2	3	3	NUM
cana-4846	138	3	⌊	⌊	ADP
cana-4846	138	4	2𝑛	2𝑛	PROPN
cana-4846	138	5	5	5	NUM
cana-4846	138	6	⌋	⌋	NOUN
cana-4846	138	7	+	+	CCONJ
cana-4846	138	8	2	2	X
cana-4846	138	9	.	.	X
cana-4846	138	10	theorem	theorem	VERB
cana-4846	138	11	10	10	NUM
cana-4846	138	12	.	.	PUNCT
cana-4846	139	1	for	for	ADP
cana-4846	139	2	any	any	DET
cana-4846	139	3	path	path	NOUN
cana-4846	139	4	𝐺	𝐺	PROPN
cana-4846	139	5	=	=	PROPN
cana-4846	139	6	𝑃𝑛(𝑛	𝑃𝑛(𝑛	PROPN
cana-4846	139	7	≥	≥	NUM
cana-4846	139	8	5	5	NUM
cana-4846	139	9	)	)	PUNCT
cana-4846	139	10	,	,	PUNCT
cana-4846	139	11	then	then	ADV
cana-4846	139	12	communications	communication	NOUN
cana-4846	139	13	on	on	ADP
cana-4846	139	14	applied	apply	VERB
cana-4846	139	15	nonlinear	nonlinear	ADJ
cana-4846	139	16	analysis	analysis	NOUN
cana-4846	139	17	issn	issn	NOUN
cana-4846	139	18	:	:	PUNCT
cana-4846	139	19	1074	1074	NUM
cana-4846	139	20	-	-	PUNCT
cana-4846	139	21	133x	133x	NUM
cana-4846	139	22	vol	vol	VERB
cana-4846	139	23	32	32	NUM
cana-4846	139	24	no	no	NOUN
cana-4846	139	25	.	.	PUNCT
cana-4846	140	1	10s	10	NOUN
cana-4846	140	2	(	(	PUNCT
cana-4846	140	3	2025	2025	NUM
cana-4846	140	4	)	)	PUNCT
cana-4846	140	5	564	564	NUM
cana-4846	140	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4846	140	7	𝜕(𝑇(𝑃𝑛	𝜕(𝑇(𝑃𝑛	NOUN
cana-4846	140	8	)	)	PUNCT
cana-4846	140	9	)	)	PUNCT
cana-4846	141	1	=	=	PRON
cana-4846	141	2	{	{	PUNCT
cana-4846	141	3	3	3	NUM
cana-4846	141	4	⌊	⌊	PROPN
cana-4846	141	5	2𝑛	2𝑛	NOUN
cana-4846	141	6	−	−	NOUN
cana-4846	141	7	1	1	NUM
cana-4846	141	8	5	5	NUM
cana-4846	141	9	⌋	⌋	NOUN
cana-4846	141	10	,	,	PUNCT
cana-4846	141	11	𝑛	𝑛	PRON
cana-4846	141	12	≡	≡	PROPN
cana-4846	141	13	1,3,4	1,3,4	NUM
cana-4846	141	14	(	(	PUNCT
cana-4846	141	15	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	141	16	5	5	NUM
cana-4846	141	17	)	)	PUNCT
cana-4846	141	18	3	3	NUM
cana-4846	141	19	⌊	⌊	ADP
cana-4846	141	20	2𝑛	2𝑛	NOUN
cana-4846	141	21	−	−	NOUN
cana-4846	141	22	1	1	NUM
cana-4846	141	23	5	5	NUM
cana-4846	141	24	⌋	⌋	NOUN
cana-4846	141	25	+	+	CCONJ
cana-4846	141	26	1	1	NUM
cana-4846	141	27	,	,	PUNCT
cana-4846	141	28	𝑛	𝑛	PRON
cana-4846	141	29	≡	≡	PROPN
cana-4846	141	30	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-4846	141	31	5	5	NUM
cana-4846	141	32	)	)	PUNCT
cana-4846	141	33	3	3	NUM
cana-4846	141	34	⌊	⌊	ADP
cana-4846	141	35	2𝑛	2𝑛	NOUN
cana-4846	141	36	−	−	NOUN
cana-4846	141	37	1	1	NUM
cana-4846	141	38	5	5	NUM
cana-4846	141	39	⌋	⌋	NOUN
cana-4846	141	40	+	+	CCONJ
cana-4846	141	41	2	2	NUM
cana-4846	141	42	,	,	PUNCT
cana-4846	141	43	𝑛	𝑛	DET
cana-4846	141	44	≡	≡	PROPN
cana-4846	141	45	0	0	PUNCT
cana-4846	141	46	(	(	PUNCT
cana-4846	141	47	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	141	48	5	5	NUM
cana-4846	141	49	)	)	PUNCT
cana-4846	141	50	proof	proof	NOUN
cana-4846	141	51	.	.	PUNCT
cana-4846	142	1	let	let	VERB
cana-4846	142	2	𝑉(𝑃𝑛	𝑉(𝑃𝑛	NOUN
cana-4846	142	3	)	)	PUNCT
cana-4846	143	1	=	=	PRON
cana-4846	143	2	{	{	PUNCT
cana-4846	143	3	𝑣1	𝑣1	PROPN
cana-4846	143	4	,	,	PUNCT
cana-4846	143	5	𝑣2	𝑣2	PROPN
cana-4846	143	6	,	,	PUNCT
cana-4846	143	7	…	…	PUNCT
cana-4846	143	8	.	.	PUNCT
cana-4846	144	1	𝑣𝑛	𝑣𝑛	X
cana-4846	144	2	}	}	PUNCT
cana-4846	144	3	be	be	VERB
cana-4846	144	4	the	the	DET
cana-4846	144	5	vertices	vertex	NOUN
cana-4846	144	6	of	of	ADP
cana-4846	144	7	path	path	NOUN
cana-4846	144	8	of	of	ADP
cana-4846	144	9	length	length	NOUN
cana-4846	144	10	𝑛	𝑛	PROPN
cana-4846	144	11	(	(	PUNCT
cana-4846	144	12	𝑛	𝑛	PRON
cana-4846	144	13	≥	≥	NOUN
cana-4846	144	14	3	3	NUM
cana-4846	144	15	)	)	PUNCT
cana-4846	144	16	and	and	CCONJ
cana-4846	144	17	𝐸(𝑃𝑛	𝐸(𝑃𝑛	NOUN
cana-4846	144	18	)	)	PUNCT
cana-4846	145	1	=	=	PRON
cana-4846	145	2	{	{	PUNCT
cana-4846	145	3	𝑒1	𝑒1	NOUN
cana-4846	145	4	,	,	PUNCT
cana-4846	145	5	𝑒2	𝑒2	PROPN
cana-4846	145	6	,	,	PUNCT
cana-4846	145	7	…	…	PUNCT
cana-4846	145	8	.	.	PUNCT
cana-4846	145	9	.	.	PUNCT
cana-4846	146	1	,	,	PUNCT
cana-4846	146	2	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	146	3	}	}	PUNCT
cana-4846	146	4	.	.	PUNCT
cana-4846	147	1	then	then	ADV
cana-4846	147	2	,	,	PUNCT
cana-4846	147	3	𝑉(𝑇(𝑃𝑛	𝑉(𝑇(𝑃𝑛	NOUN
cana-4846	147	4	)	)	PUNCT
cana-4846	147	5	)	)	PUNCT
cana-4846	148	1	=	=	PRON
cana-4846	148	2	{	{	PUNCT
cana-4846	148	3	𝑣1	𝑣1	PROPN
cana-4846	148	4	,	,	PUNCT
cana-4846	148	5	𝑣2	𝑣2	PROPN
cana-4846	148	6	,	,	PUNCT
cana-4846	148	7	…	…	PUNCT
cana-4846	148	8	.	.	PUNCT
cana-4846	148	9	.	.	PUNCT
cana-4846	149	1	,	,	PUNCT
cana-4846	149	2	𝑣𝑛	𝑣𝑛	NOUN
cana-4846	149	3	,	,	PUNCT
cana-4846	149	4	𝑒1	𝑒1	NOUN
cana-4846	149	5	,	,	PUNCT
cana-4846	149	6	𝑒2	𝑒2	PROPN
cana-4846	149	7	,	,	PUNCT
cana-4846	149	8	…	…	PUNCT
cana-4846	149	9	.	.	PUNCT
cana-4846	149	10	.	.	PUNCT
cana-4846	150	1	,	,	PUNCT
cana-4846	150	2	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	150	3	}	}	PUNCT
cana-4846	150	4	and	and	CCONJ
cana-4846	150	5	|𝑉(𝑇(𝑃𝑛))|	|𝑉(𝑇(𝑃𝑛))|	PROPN
cana-4846	150	6	=	=	SYM
cana-4846	150	7	2𝑛	2𝑛	PROPN
cana-4846	151	1	−	−	PROPN
cana-4846	151	2	1	1	X
cana-4846	151	3	.	.	PUNCT
cana-4846	151	4	when	when	SCONJ
cana-4846	151	5	𝑛	𝑛	PRON
cana-4846	151	6	≡	≡	PROPN
cana-4846	151	7	1	1	NUM
cana-4846	151	8	(	(	PUNCT
cana-4846	151	9	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	151	10	5	5	NUM
cana-4846	151	11	)	)	PUNCT
cana-4846	151	12	.	.	PUNCT
cana-4846	152	1	consider	consider	VERB
cana-4846	152	2	the	the	DET
cana-4846	152	3	differential	differential	ADJ
cana-4846	152	4	set	set	NOUN
cana-4846	152	5	of	of	ADP
cana-4846	152	6	a	a	DET
cana-4846	152	7	total	total	ADJ
cana-4846	152	8	graph	graph	NOUN
cana-4846	152	9	𝑆	𝑆	PROPN
cana-4846	152	10	=	=	SYM
cana-4846	152	11	{	{	PUNCT
cana-4846	152	12	𝑣2	𝑣2	PROPN
cana-4846	152	13	,	,	PUNCT
cana-4846	152	14	𝑒4	𝑒4	PROPN
cana-4846	152	15	,	,	PUNCT
cana-4846	152	16	…	…	PUNCT
cana-4846	152	17	.	.	PUNCT
cana-4846	152	18	.	.	PUNCT
cana-4846	153	1	,	,	PUNCT
cana-4846	153	2	𝑒𝑛−2	𝑒𝑛−2	PROPN
cana-4846	153	3	,	,	PUNCT
cana-4846	153	4	𝑣𝑛−4	𝑣𝑛−4	ADJ
cana-4846	153	5	}	}	PUNCT
cana-4846	153	6	and	and	CCONJ
cana-4846	153	7	𝐶(𝑆	𝐶(𝑆	NUM
cana-4846	153	8	)	)	PUNCT
cana-4846	154	1	=	=	PRON
cana-4846	154	2	{	{	PUNCT
cana-4846	154	3	𝑣𝑛	𝑣𝑛	NOUN
cana-4846	154	4	}	}	PUNCT
cana-4846	154	5	.	.	PUNCT
cana-4846	155	1	then	then	ADV
cana-4846	155	2	,	,	PUNCT
cana-4846	155	3	|𝐵(𝑆)|	|𝐵(𝑆)|	PROPN
cana-4846	155	4	=	=	SYM
cana-4846	155	5	⌊	⌊	VERB
cana-4846	155	6	8𝑛−4	8𝑛−4	NUM
cana-4846	155	7	5	5	NUM
cana-4846	155	8	⌋	⌋	NOUN
cana-4846	155	9	and	and	CCONJ
cana-4846	155	10	|𝑆|	|𝑆|	VERB
cana-4846	155	11	=	=	PUNCT
cana-4846	155	12	⌊	⌊	VERB
cana-4846	155	13	2𝑛−1	2𝑛−1	NUM
cana-4846	155	14	5	5	NUM
cana-4846	155	15	⌋.	⌋.	ADV
cana-4846	155	16	so	so	ADV
cana-4846	155	17	,	,	PUNCT
cana-4846	155	18	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	155	19	)	)	PUNCT
cana-4846	155	20	)	)	PUNCT
cana-4846	156	1	=	=	SYM
cana-4846	156	2	3	3	X
cana-4846	156	3	⌊	⌊	PART
cana-4846	156	4	2𝑛−1	2𝑛−1	NUM
cana-4846	156	5	5	5	NUM
cana-4846	156	6	⌋.	⌋.	NOUN
cana-4846	156	7	when	when	SCONJ
cana-4846	156	8	𝑛	𝑛	DET
cana-4846	156	9	≡	≡	PROPN
cana-4846	156	10	3	3	NUM
cana-4846	156	11	(	(	PUNCT
cana-4846	156	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	156	13	5	5	NUM
cana-4846	156	14	)	)	PUNCT
cana-4846	156	15	,	,	PUNCT
cana-4846	156	16	consider	consider	VERB
cana-4846	156	17	the	the	DET
cana-4846	156	18	differential	differential	ADJ
cana-4846	156	19	set	set	NOUN
cana-4846	156	20	of	of	ADP
cana-4846	156	21	a	a	DET
cana-4846	156	22	total	total	ADJ
cana-4846	156	23	graph	graph	NOUN
cana-4846	156	24	𝑆	𝑆	PROPN
cana-4846	156	25	=	=	SYM
cana-4846	156	26	{	{	PUNCT
cana-4846	156	27	𝑣2	𝑣2	PROPN
cana-4846	156	28	,	,	PUNCT
cana-4846	156	29	𝑒4	𝑒4	PROPN
cana-4846	156	30	,	,	PUNCT
cana-4846	156	31	…	…	PUNCT
cana-4846	156	32	.	.	PUNCT
cana-4846	156	33	.	.	PUNCT
cana-4846	157	1	,	,	PUNCT
cana-4846	157	2	𝑒𝑛−4	𝑒𝑛−4	PROPN
cana-4846	157	3	,	,	PUNCT
cana-4846	157	4	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-4846	157	5	}	}	PUNCT
cana-4846	157	6	.	.	PUNCT
cana-4846	158	1	then	then	ADV
cana-4846	158	2	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	158	3	)	)	PUNCT
cana-4846	158	4	)	)	PUNCT
cana-4846	159	1	=	=	SYM
cana-4846	159	2	3	3	X
cana-4846	159	3	⌊	⌊	PART
cana-4846	159	4	2𝑛−1	2𝑛−1	NUM
cana-4846	159	5	5	5	NUM
cana-4846	159	6	⌋.	⌋.	NOUN
cana-4846	159	7	when	when	SCONJ
cana-4846	159	8	𝑛	𝑛	DET
cana-4846	159	9	≡	≡	PROPN
cana-4846	159	10	4	4	NUM
cana-4846	159	11	(	(	PUNCT
cana-4846	159	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	159	13	5	5	NUM
cana-4846	159	14	)	)	PUNCT
cana-4846	159	15	.	.	PUNCT
cana-4846	160	1	consider	consider	VERB
cana-4846	160	2	the	the	DET
cana-4846	160	3	differential	differential	NOUN
cana-4846	160	4	set	set	VERB
cana-4846	160	5	𝑆	𝑆	PROPN
cana-4846	160	6	=	=	SYM
cana-4846	160	7	{	{	PUNCT
cana-4846	160	8	𝑣2	𝑣2	PROPN
cana-4846	160	9	,	,	PUNCT
cana-4846	160	10	𝑒4	𝑒4	PROPN
cana-4846	160	11	,	,	PUNCT
cana-4846	160	12	…	…	PUNCT
cana-4846	160	13	.	.	PUNCT
cana-4846	161	1	.	.	PUNCT
cana-4846	162	1	,	,	PUNCT
cana-4846	162	2	𝑒𝑛−5	𝑒𝑛−5	NOUN
cana-4846	162	3	,	,	PUNCT
cana-4846	162	4	𝑣𝑛−2	𝑣𝑛−2	PROPN
cana-4846	162	5	}	}	PUNCT
cana-4846	162	6	and	and	CCONJ
cana-4846	162	7	𝐶(𝑆	𝐶(𝑆	NUM
cana-4846	162	8	)	)	PUNCT
cana-4846	163	1	=	=	PRON
cana-4846	163	2	{	{	PUNCT
cana-4846	163	3	𝑣𝑛	𝑣𝑛	PROPN
cana-4846	163	4	,	,	PUNCT
cana-4846	163	5	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	163	6	}	}	PUNCT
cana-4846	163	7	.	.	PUNCT
cana-4846	164	1	then	then	ADV
cana-4846	164	2	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	164	3	)	)	PUNCT
cana-4846	164	4	)	)	PUNCT
cana-4846	165	1	=	=	SYM
cana-4846	165	2	3	3	X
cana-4846	165	3	⌊	⌊	PART
cana-4846	165	4	2𝑛−1	2𝑛−1	NUM
cana-4846	165	5	5	5	NUM
cana-4846	165	6	⌋.	⌋.	NOUN
cana-4846	165	7	when	when	SCONJ
cana-4846	165	8	𝑛	𝑛	DET
cana-4846	165	9	≡	≡	PROPN
cana-4846	165	10	2	2	NUM
cana-4846	165	11	(	(	PUNCT
cana-4846	165	12	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	165	13	5	5	NUM
cana-4846	165	14	)	)	PUNCT
cana-4846	165	15	,	,	PUNCT
cana-4846	165	16	there	there	PRON
cana-4846	165	17	are	be	VERB
cana-4846	165	18	three	three	NUM
cana-4846	165	19	possible	possible	ADJ
cana-4846	165	20	differential	differential	NOUN
cana-4846	165	21	sets	set	NOUN
cana-4846	166	1	𝑆	𝑆	PROPN
cana-4846	166	2	=	=	PUNCT
cana-4846	166	3	𝑆1	𝑆1	NOUN
cana-4846	166	4	∪	∪	NOUN
cana-4846	166	5	{	{	PUNCT
cana-4846	166	6	𝑣𝑛−1	𝑣𝑛−1	NOUN
cana-4846	166	7	}	}	PUNCT
cana-4846	166	8	or	or	CCONJ
cana-4846	166	9	𝑆	𝑆	PROPN
cana-4846	166	10	=	=	PUNCT
cana-4846	166	11	𝑆1	𝑆1	NOUN
cana-4846	166	12	∪	∪	NOUN
cana-4846	166	13	{	{	PUNCT
cana-4846	166	14	𝑣𝑛	𝑣𝑛	NOUN
cana-4846	166	15	}	}	PUNCT
cana-4846	166	16	or	or	CCONJ
cana-4846	166	17	𝑆	𝑆	PROPN
cana-4846	166	18	=	=	PUNCT
cana-4846	166	19	𝑆1	𝑆1	NOUN
cana-4846	166	20	∪	∪	NOUN
cana-4846	166	21	{	{	PUNCT
cana-4846	166	22	𝑒𝑛−1	𝑒𝑛−1	PROPN
cana-4846	166	23	}	}	PUNCT
cana-4846	166	24	where	where	SCONJ
cana-4846	166	25	𝑆1	𝑆1	NOUN
cana-4846	166	26	=	=	SYM
cana-4846	166	27	{	{	PUNCT
cana-4846	166	28	𝑣2	𝑣2	PROPN
cana-4846	166	29	,	,	PUNCT
cana-4846	166	30	𝑒4	𝑒4	PROPN
cana-4846	166	31	,	,	PUNCT
cana-4846	166	32	…	…	PUNCT
cana-4846	166	33	.	.	PUNCT
cana-4846	167	1	,	,	PUNCT
cana-4846	167	2	𝑒𝑛−3	𝑒𝑛−3	PROPN
cana-4846	167	3	,	,	PUNCT
cana-4846	167	4	𝑣𝑛−5	𝑣𝑛−5	NOUN
cana-4846	167	5	}	}	PUNCT
cana-4846	167	6	.	.	PUNCT
cana-4846	168	1	then	then	ADV
cana-4846	168	2	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	168	3	)	)	PUNCT
cana-4846	168	4	)	)	PUNCT
cana-4846	169	1	=	=	SYM
cana-4846	169	2	3	3	X
cana-4846	169	3	⌊	⌊	PART
cana-4846	169	4	2𝑛−1	2𝑛−1	NUM
cana-4846	169	5	5	5	NUM
cana-4846	169	6	⌋	⌋	NOUN
cana-4846	169	7	+	+	CCONJ
cana-4846	169	8	1	1	X
cana-4846	169	9	.	.	X
cana-4846	169	10	when	when	SCONJ
cana-4846	169	11	𝑛	𝑛	DET
cana-4846	169	12	≡	≡	PROPN
cana-4846	169	13	0	0	PUNCT
cana-4846	169	14	(	(	PUNCT
cana-4846	169	15	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-4846	169	16	5	5	NUM
cana-4846	169	17	)	)	PUNCT
cana-4846	169	18	,	,	PUNCT
cana-4846	169	19	the	the	DET
cana-4846	169	20	possible	possible	ADJ
cana-4846	169	21	differential	differential	NOUN
cana-4846	169	22	set	set	VERB
cana-4846	169	23	𝑆	𝑆	PROPN
cana-4846	169	24	=	=	PUNCT
cana-4846	169	25	𝑆1	𝑆1	NOUN
cana-4846	169	26	∪	∪	ADJ
cana-4846	169	27	𝑆2	𝑆2	NOUN
cana-4846	169	28	where	where	SCONJ
cana-4846	169	29	𝑆1	𝑆1	NOUN
cana-4846	169	30	=	=	SYM
cana-4846	169	31	{	{	PUNCT
cana-4846	169	32	𝑣2	𝑣2	PROPN
cana-4846	169	33	,	,	PUNCT
cana-4846	169	34	𝑒4	𝑒4	PROPN
cana-4846	169	35	,	,	PUNCT
cana-4846	169	36	…	…	PUNCT
cana-4846	169	37	.	.	PUNCT
cana-4846	170	1	,	,	PUNCT
cana-4846	171	1	𝑒𝑛−6	𝑒𝑛−6	PROPN
cana-4846	171	2	,	,	PUNCT
cana-4846	171	3	𝑣𝑛−3	𝑣𝑛−3	NOUN
cana-4846	171	4	}	}	PUNCT
cana-4846	171	5	and	and	CCONJ
cana-4846	171	6	𝑆2	𝑆2	PROPN
cana-4846	171	7	=	=	PUNCT
cana-4846	171	8	{	{	PUNCT
cana-4846	171	9	𝑒𝑛−2	𝑒𝑛−2	PROPN
cana-4846	171	10	}	}	PUNCT
cana-4846	171	11	.	.	PUNCT
cana-4846	172	1	then	then	ADV
cana-4846	172	2	,	,	PUNCT
cana-4846	172	3	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	172	4	)	)	PUNCT
cana-4846	172	5	)	)	PUNCT
cana-4846	173	1	=	=	SYM
cana-4846	173	2	3	3	X
cana-4846	173	3	⌊	⌊	PART
cana-4846	173	4	2𝑛−1	2𝑛−1	NUM
cana-4846	173	5	5	5	NUM
cana-4846	173	6	⌋	⌋	NOUN
cana-4846	173	7	+	+	CCONJ
cana-4846	173	8	2	2	X
cana-4846	173	9	.	.	X
cana-4846	173	10	theorem	theorem	NOUN
cana-4846	173	11	11	11	NUM
cana-4846	173	12	.	.	PUNCT
cana-4846	174	1	if	if	SCONJ
cana-4846	174	2	𝐺	𝐺	PROPN
cana-4846	174	3	=	=	NOUN
cana-4846	174	4	𝐾𝑚1×𝑛1	𝐾𝑚1×𝑛1	PRON
cana-4846	174	5	is	be	AUX
cana-4846	174	6	a	a	DET
cana-4846	174	7	complete	complete	ADJ
cana-4846	174	8	bipartite	bipartite	NOUN
cana-4846	174	9	graph	graph	NOUN
cana-4846	174	10	,	,	PUNCT
cana-4846	174	11	then	then	ADV
cana-4846	174	12	𝜕	𝜕	NOUN
cana-4846	174	13	(	(	PUNCT
cana-4846	174	14	𝑇(𝐾𝑚1×𝑛1	𝑇(𝐾𝑚1×𝑛1	NUM
cana-4846	174	15	)	)	PUNCT
cana-4846	174	16	)	)	PUNCT
cana-4846	175	1	=	=	PRON
cana-4846	175	2	(	(	PUNCT
cana-4846	175	3	𝑚1	𝑚1	NOUN
cana-4846	175	4	+	+	CCONJ
cana-4846	176	1	1)𝑛	1)𝑛	NUM
cana-4846	176	2	−	−	PROPN
cana-4846	176	3	𝑚1	𝑚1	NOUN
cana-4846	176	4	.	.	PUNCT
cana-4846	177	1	proof	proof	NOUN
cana-4846	177	2	.	.	PUNCT
cana-4846	178	1	since	since	SCONJ
cana-4846	178	2	𝐺	𝐺	PROPN
cana-4846	178	3	=	=	NOUN
cana-4846	178	4	𝐾𝑚1×𝑛1	𝐾𝑚1×𝑛1	PRON
cana-4846	178	5	is	be	AUX
cana-4846	178	6	a	a	DET
cana-4846	178	7	complete	complete	ADJ
cana-4846	178	8	bipartite	bipartite	NOUN
cana-4846	178	9	graph	graph	NOUN
cana-4846	178	10	,	,	PUNCT
cana-4846	178	11	the	the	DET
cana-4846	178	12	vertex	vertex	NOUN
cana-4846	178	13	set	set	NOUN
cana-4846	178	14	can	can	AUX
cana-4846	178	15	be	be	AUX
cana-4846	178	16	partitioned	partition	VERB
cana-4846	178	17	into	into	ADP
cana-4846	178	18	two	two	NUM
cana-4846	178	19	disjoint	disjoint	NOUN
cana-4846	178	20	non	non	ADJ
cana-4846	178	21	empty	empty	ADJ
cana-4846	178	22	sets	set	VERB
cana-4846	178	23	𝑉1(𝐺	𝑉1(𝐺	NOUN
cana-4846	178	24	)	)	PUNCT
cana-4846	178	25	=	=	PRON
cana-4846	178	26	{	{	PUNCT
cana-4846	178	27	𝑢1	𝑢1	PROPN
cana-4846	178	28	,	,	PUNCT
cana-4846	178	29	𝑢2	𝑢2	PROPN
cana-4846	178	30	…	…	SYM
cana-4846	178	31	𝑢𝑚1	𝑢𝑚1	NOUN
cana-4846	178	32	}	}	PUNCT
cana-4846	178	33	and	and	CCONJ
cana-4846	178	34	𝑉2(𝐺	𝑉2(𝐺	NOUN
cana-4846	178	35	)	)	PUNCT
cana-4846	178	36	=	=	PRON
cana-4846	178	37	{	{	PUNCT
cana-4846	178	38	𝑣1	𝑣1	PROPN
cana-4846	178	39	,	,	PUNCT
cana-4846	178	40	𝑣2	𝑣2	PROPN
cana-4846	178	41	,	,	PUNCT
cana-4846	178	42	…	…	PUNCT
cana-4846	178	43	𝑣𝑛1	𝑣𝑛1	X
cana-4846	178	44	}	}	PUNCT
cana-4846	178	45	.	.	PUNCT
cana-4846	179	1	here	here	ADV
cana-4846	179	2	,	,	PUNCT
cana-4846	179	3	𝑉	𝑉	PROPN
cana-4846	179	4	(	(	PUNCT
cana-4846	179	5	𝑇(𝐾𝑚1×𝑛1	𝑇(𝐾𝑚1×𝑛1	NUM
cana-4846	179	6	)	)	PUNCT
cana-4846	179	7	)	)	PUNCT
cana-4846	180	1	=	=	PRON
cana-4846	180	2	{	{	PUNCT
cana-4846	180	3	𝑢𝑖/1	𝑢𝑖/1	NOUN
cana-4846	180	4	≤	≤	NOUN
cana-4846	180	5	𝑖	𝑖	SYM
cana-4846	180	6	≤	≤	NUM
cana-4846	180	7	𝑚1	𝑚1	NOUN
cana-4846	180	8	}	}	PUNCT
cana-4846	180	9	∪	∪	NOUN
cana-4846	180	10	{	{	PUNCT
cana-4846	180	11	𝑣𝑗/1	𝑣𝑗/1	NOUN
cana-4846	180	12	≤	≤	X
cana-4846	180	13	𝑗	𝑗	PRON
cana-4846	180	14	≤	≤	NOUN
cana-4846	180	15	𝑛1	𝑛1	NOUN
cana-4846	180	16	}	}	PUNCT
cana-4846	180	17	∪	∪	NOUN
cana-4846	180	18	{	{	PUNCT
cana-4846	180	19	𝑒𝑖𝑗/1	𝑒𝑖𝑗/1	PROPN
cana-4846	180	20	≤	≤	PROPN
cana-4846	180	21	𝑖	𝑖	PUNCT
cana-4846	180	22	≤	≤	NUM
cana-4846	180	23	𝑚1	𝑚1	NOUN
cana-4846	180	24	,	,	PUNCT
cana-4846	180	25	1	1	NUM
cana-4846	180	26	≤	≤	NUM
cana-4846	180	27	𝑗	𝑗	PRON
cana-4846	180	28	≤	≤	NOUN
cana-4846	180	29	𝑛1	𝑛1	NOUN
cana-4846	180	30	}	}	PUNCT
cana-4846	180	31	and	and	CCONJ
cana-4846	180	32	|𝑉	|𝑉	NUM
cana-4846	180	33	(	(	PUNCT
cana-4846	180	34	𝑇(𝐾𝑚1×𝑛1	𝑇(𝐾𝑚1×𝑛1	NUM
cana-4846	180	35	)	)	PUNCT
cana-4846	180	36	)	)	PUNCT
cana-4846	180	37	|	|	ADV
cana-4846	180	38	=	=	SYM
cana-4846	180	39	𝑚1	𝑚1	NOUN
cana-4846	180	40	+	+	CCONJ
cana-4846	180	41	𝑛1	𝑛1	NOUN
cana-4846	180	42	+	+	ADJ
cana-4846	180	43	𝑚1𝑛1	𝑚1𝑛1	X
cana-4846	180	44	.	.	PUNCT
cana-4846	181	1	clearly	clearly	ADV
cana-4846	181	2	,	,	PUNCT
cana-4846	181	3	the	the	DET
cana-4846	181	4	differential	differential	NOUN
cana-4846	181	5	set	set	VERB
cana-4846	181	6	𝑆	𝑆	PROPN
cana-4846	181	7	=	=	SYM
cana-4846	181	8	{	{	PUNCT
cana-4846	181	9	𝑢1	𝑢1	PROPN
cana-4846	181	10	,	,	PUNCT
cana-4846	181	11	𝑢2	𝑢2	PROPN
cana-4846	181	12	…	…	SYM
cana-4846	181	13	𝑢𝑚1	𝑢𝑚1	NOUN
cana-4846	181	14	}	}	PUNCT
cana-4846	181	15	.	.	PUNCT
cana-4846	182	1	since	since	SCONJ
cana-4846	182	2	𝑢1	𝑢1	PROPN
cana-4846	182	3	is	be	AUX
cana-4846	182	4	adjacent	adjacent	ADJ
cana-4846	182	5	with	with	ADP
cana-4846	182	6	2𝑛1	2𝑛1	NUM
cana-4846	182	7	vertices	vertex	NOUN
cana-4846	182	8	and	and	CCONJ
cana-4846	182	9	𝑢2	𝑢2	PROPN
cana-4846	182	10	is	be	AUX
cana-4846	182	11	adjacent	adjacent	ADJ
cana-4846	182	12	with	with	ADP
cana-4846	182	13	𝑛1	𝑛1	ADJ
cana-4846	182	14	vertices	vertex	NOUN
cana-4846	182	15	and	and	CCONJ
cana-4846	182	16	so	so	ADV
cana-4846	182	17	on	on	ADV
cana-4846	182	18	.	.	PUNCT
cana-4846	183	1	therefore	therefore	ADV
cana-4846	183	2	𝜕	𝜕	NOUN
cana-4846	183	3	(	(	PUNCT
cana-4846	183	4	𝑇(𝐾𝑚1×𝑛1	𝑇(𝐾𝑚1×𝑛1	NUM
cana-4846	183	5	)	)	PUNCT
cana-4846	183	6	)	)	PUNCT
cana-4846	184	1	=	=	PUNCT
cana-4846	184	2	(	(	PUNCT
cana-4846	184	3	2𝑛1	2𝑛1	NUM
cana-4846	184	4	−	−	NOUN
cana-4846	184	5	1	1	NUM
cana-4846	184	6	)	)	PUNCT
cana-4846	184	7	+	+	CCONJ
cana-4846	184	8	(	(	PUNCT
cana-4846	184	9	𝑛1	𝑛1	ADJ
cana-4846	184	10	−	−	PROPN
cana-4846	184	11	1	1	NUM
cana-4846	184	12	)	)	PUNCT
cana-4846	184	13	+	+	NUM
cana-4846	184	14	⋯+	⋯+	NOUN
cana-4846	184	15	(	(	PUNCT
cana-4846	184	16	𝑛1	𝑛1	NOUN
cana-4846	184	17	−	−	PROPN
cana-4846	184	18	1	1	NUM
cana-4846	184	19	)	)	PUNCT
cana-4846	184	20	=	=	SYM
cana-4846	184	21	(	(	PUNCT
cana-4846	184	22	𝑚1	𝑚1	X
cana-4846	184	23	+	+	CCONJ
cana-4846	185	1	1)𝑛1	1)𝑛1	NUM
cana-4846	185	2	−	−	PROPN
cana-4846	185	3	𝑚1	𝑚1	NOUN
cana-4846	185	4	.	.	PUNCT
cana-4846	186	1	theorem	theorem	PROPN
cana-4846	186	2	12	12	NUM
cana-4846	186	3	.	.	PUNCT
cana-4846	187	1	if	if	SCONJ
cana-4846	187	2	𝐺	𝐺	PROPN
cana-4846	187	3	is	be	AUX
cana-4846	187	4	a	a	DET
cana-4846	187	5	complete	complete	ADJ
cana-4846	187	6	binary	binary	ADJ
cana-4846	187	7	tree	tree	NOUN
cana-4846	187	8	,	,	PUNCT
cana-4846	187	9	then	then	ADV
cana-4846	187	10	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	187	11	)	)	PUNCT
cana-4846	187	12	)	)	PUNCT
cana-4846	188	1	=	=	PRON
cana-4846	188	2	{	{	PUNCT
cana-4846	188	3	2𝑘+3−7	2𝑘+3−7	NUM
cana-4846	188	4	3	3	NUM
cana-4846	188	5	,	,	PUNCT
cana-4846	188	6	𝑘	𝑘	PRON
cana-4846	188	7	𝑖𝑠	𝑖𝑠	ADV
cana-4846	188	8	𝑎𝑛	𝑎𝑛	ADP
cana-4846	188	9	𝑜𝑑𝑑	𝑜𝑑𝑑	ADV
cana-4846	188	10	𝑖𝑛𝑡𝑒𝑔𝑒𝑟	𝑖𝑛𝑡𝑒𝑔𝑒𝑟	NOUN
cana-4846	188	11	2𝑘+3−5	2𝑘+3−5	NUM
cana-4846	188	12	3	3	NUM
cana-4846	188	13	,	,	PUNCT
cana-4846	188	14	𝑘	𝑘	INTJ
cana-4846	188	15	𝑖𝑠	𝑖𝑠	NOUN
cana-4846	188	16	𝑎𝑛	𝑎𝑛	PRON
cana-4846	188	17	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-4846	188	18	𝑖𝑛𝑡𝑒𝑔𝑒𝑟	𝑖𝑛𝑡𝑒𝑔𝑒𝑟	ADJ
cana-4846	188	19	proof	proof	NOUN
cana-4846	188	20	.	.	PUNCT
cana-4846	189	1	let	let	VERB
cana-4846	189	2	sk	sk	NOUN
cana-4846	189	3	be	be	AUX
cana-4846	189	4	the	the	DET
cana-4846	189	5	set	set	NOUN
cana-4846	189	6	of	of	ADP
cana-4846	189	7	all	all	DET
cana-4846	189	8	vertices	vertex	NOUN
cana-4846	189	9	in	in	ADP
cana-4846	189	10	level	level	NOUN
cana-4846	189	11	𝑘	𝑘	NOUN
cana-4846	189	12	and	and	CCONJ
cana-4846	189	13	|𝑆𝑘|	|𝑆𝑘|	NOUN
cana-4846	189	14	=	=	SYM
cana-4846	190	1	2	2	NUM
cana-4846	190	2	𝑘	𝑘	X
cana-4846	190	3	where	where	SCONJ
cana-4846	190	4	0	0	NUM
cana-4846	190	5	≤	≤	NOUN
cana-4846	190	6	𝑘	𝑘	DET
cana-4846	190	7	≤	≤	NOUN
cana-4846	190	8	𝑚	𝑚	X
cana-4846	190	9	and	and	CCONJ
cana-4846	190	10	𝑚	𝑚	PROPN
cana-4846	190	11	is	be	AUX
cana-4846	190	12	a	a	DET
cana-4846	190	13	positive	positive	ADJ
cana-4846	190	14	integer	integer	NOUN
cana-4846	190	15	.	.	PUNCT
cana-4846	191	1	case(i	case(i	PROPN
cana-4846	191	2	)	)	PUNCT
cana-4846	191	3	𝑘	𝑘	PROPN
cana-4846	191	4	is	be	AUX
cana-4846	191	5	an	an	DET
cana-4846	191	6	odd	odd	ADJ
cana-4846	191	7	integer	integer	NOUN
cana-4846	191	8	.	.	PUNCT
cana-4846	192	1	clearly	clearly	ADV
cana-4846	192	2	,	,	PUNCT
cana-4846	192	3	𝑆𝑘−1	𝑆𝑘−1	PROPN
cana-4846	192	4	∪	∪	VERB
cana-4846	192	5	𝑆𝑘−3	𝑆𝑘−3	NOUN
cana-4846	192	6	∪	∪	ADP
cana-4846	192	7	…	…	X
cana-4846	192	8	∪	∪	ADJ
cana-4846	192	9	𝑆2	𝑆2	PROPN
cana-4846	192	10	∪	∪	ADJ
cana-4846	192	11	𝑆0	𝑆0	PROPN
cana-4846	192	12	is	be	AUX
cana-4846	192	13	a	a	DET
cana-4846	192	14	𝜕	𝜕	NOUN
cana-4846	192	15	−set	−set	NOUN
cana-4846	192	16	.	.	PUNCT
cana-4846	193	1	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	193	2	)	)	PUNCT
cana-4846	193	3	)	)	PUNCT
cana-4846	194	1	=	=	SYM
cana-4846	194	2	2𝑘+1	2𝑘+1	NUM
cana-4846	194	3	+	+	NUM
cana-4846	194	4	2𝑘−1	2𝑘−1	NUM
cana-4846	194	5	+	+	CCONJ
cana-4846	194	6	2𝑘−3	2𝑘−3	NUM
cana-4846	194	7	+	+	ADJ
cana-4846	194	8	⋯+	⋯+	NOUN
cana-4846	194	9	24	24	NUM
cana-4846	194	10	+	+	CCONJ
cana-4846	194	11	22	22	NUM
cana-4846	194	12	−	−	NOUN
cana-4846	194	13	1	1	NUM
cana-4846	194	14	=	=	SYM
cana-4846	194	15	2𝑘+1	2𝑘+1	NUM
cana-4846	195	1	[	[	X
cana-4846	195	2	1	1	NUM
cana-4846	195	3	+	+	NUM
cana-4846	195	4	2𝑘−1	2𝑘−1	NUM
cana-4846	195	5	2𝑘+1	2𝑘+1	NUM
cana-4846	195	6	+	+	CCONJ
cana-4846	195	7	2𝑘−3	2𝑘−3	NUM
cana-4846	195	8	2𝑘+1	2𝑘+1	NUM
cana-4846	195	9	+	+	NOUN
cana-4846	195	10	⋯+	⋯+	NOUN
cana-4846	195	11	24	24	NUM
cana-4846	195	12	2𝑘+1	2𝑘+1	NUM
cana-4846	195	13	+	+	CCONJ
cana-4846	195	14	22	22	NUM
cana-4846	195	15	2𝑘+1	2𝑘+1	NUM
cana-4846	195	16	]	]	PUNCT
cana-4846	195	17	−	−	PROPN
cana-4846	195	18	1	1	NUM
cana-4846	195	19	communications	communication	NOUN
cana-4846	195	20	on	on	ADP
cana-4846	195	21	applied	apply	VERB
cana-4846	195	22	nonlinear	nonlinear	ADJ
cana-4846	195	23	analysis	analysis	NOUN
cana-4846	195	24	issn	issn	NOUN
cana-4846	195	25	:	:	PUNCT
cana-4846	195	26	1074	1074	NUM
cana-4846	195	27	-	-	PUNCT
cana-4846	195	28	133x	133x	NUM
cana-4846	195	29	vol	vol	VERB
cana-4846	195	30	32	32	NUM
cana-4846	195	31	no	no	NOUN
cana-4846	195	32	.	.	PUNCT
cana-4846	196	1	10s	10	NOUN
cana-4846	196	2	(	(	PUNCT
cana-4846	196	3	2025	2025	NUM
cana-4846	196	4	)	)	PUNCT
cana-4846	196	5	565	565	NUM
cana-4846	196	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4846	196	7	=	=	SYM
cana-4846	196	8	2𝑘+1[1	2𝑘+1[1	NUM
cana-4846	197	1	+	+	NUM
cana-4846	197	2	2−2	2−2	NUM
cana-4846	197	3	+	+	CCONJ
cana-4846	197	4	2−4	2−4	NUM
cana-4846	197	5	+	+	NUM
cana-4846	197	6	⋯+	⋯+	NOUN
cana-4846	197	7	2−𝑘+3	2−𝑘+3	NUM
cana-4846	197	8	+	+	CCONJ
cana-4846	197	9	2−𝑘+1	2−𝑘+1	NUM
cana-4846	197	10	]	]	X
cana-4846	197	11	−	−	PROPN
cana-4846	197	12	1	1	NUM
cana-4846	197	13	=	=	SYM
cana-4846	197	14	2𝑘+1	2𝑘+1	NUM
cana-4846	198	1	[	[	X
cana-4846	198	2	1	1	NUM
cana-4846	198	3	+	+	CCONJ
cana-4846	198	4	(	(	PUNCT
cana-4846	198	5	1	1	NUM
cana-4846	198	6	22	22	NUM
cana-4846	198	7	)	)	PUNCT
cana-4846	198	8	1	1	NUM
cana-4846	199	1	+	+	CCONJ
cana-4846	199	2	(	(	PUNCT
cana-4846	199	3	1	1	NUM
cana-4846	199	4	22	22	NUM
cana-4846	199	5	)	)	PUNCT
cana-4846	199	6	2	2	NUM
cana-4846	200	1	+	+	CCONJ
cana-4846	200	2	…	…	PUNCT
cana-4846	200	3	+	+	CCONJ
cana-4846	200	4	(	(	PUNCT
cana-4846	200	5	1	1	NUM
cana-4846	200	6	22	22	NUM
cana-4846	200	7	)	)	PUNCT
cana-4846	200	8	𝑘−3	𝑘−3	PROPN
cana-4846	200	9	2	2	NUM
cana-4846	201	1	+	+	CCONJ
cana-4846	201	2	(	(	PUNCT
cana-4846	201	3	1	1	NUM
cana-4846	201	4	22	22	NUM
cana-4846	201	5	)	)	PUNCT
cana-4846	201	6	𝑘−1	𝑘−1	PROPN
cana-4846	201	7	2	2	NUM
cana-4846	201	8	]	]	PUNCT
cana-4846	201	9	−	−	PROPN
cana-4846	201	10	1	1	NUM
cana-4846	201	11	=	=	SYM
cana-4846	201	12	2𝑘+1	2𝑘+1	PROPN
cana-4846	201	13	[	[	PUNCT
cana-4846	201	14	1−	1−	NUM
cana-4846	201	15	(	(	PUNCT
cana-4846	201	16	1	1	NUM
cana-4846	201	17	22	22	NUM
cana-4846	201	18	)	)	PUNCT
cana-4846	201	19	𝑘+1	𝑘+1	NOUN
cana-4846	201	20	2	2	NUM
cana-4846	201	21	1−	1−	NUM
cana-4846	201	22	(	(	PUNCT
cana-4846	201	23	1	1	NUM
cana-4846	201	24	22	22	NUM
cana-4846	201	25	)	)	PUNCT
cana-4846	201	26	]	]	PUNCT
cana-4846	202	1	−	−	PROPN
cana-4846	202	2	1	1	NUM
cana-4846	202	3	=	=	SYM
cana-4846	202	4	2𝑘+1	2𝑘+1	PROPN
cana-4846	202	5	[	[	PUNCT
cana-4846	202	6	1−	1−	NUM
cana-4846	202	7	(	(	PUNCT
cana-4846	202	8	1	1	NUM
cana-4846	202	9	22	22	NUM
cana-4846	202	10	)	)	PUNCT
cana-4846	202	11	𝑘+1	𝑘+1	NOUN
cana-4846	202	12	2	2	NUM
cana-4846	202	13	(	(	PUNCT
cana-4846	202	14	3	3	NUM
cana-4846	202	15	22	22	NUM
cana-4846	202	16	)	)	PUNCT
cana-4846	202	17	]	]	PUNCT
cana-4846	203	1	−	−	PROPN
cana-4846	203	2	1	1	NUM
cana-4846	203	3	=	=	SYM
cana-4846	203	4	2𝑘+1	2𝑘+1	NUM
cana-4846	203	5	×	×	NOUN
cana-4846	203	6	22	22	NUM
cana-4846	203	7	3	3	NUM
cana-4846	203	8	×	×	NOUN
cana-4846	203	9	[	[	PUNCT
cana-4846	203	10	1	1	NUM
cana-4846	203	11	−	−	PROPN
cana-4846	203	12	1	1	NUM
cana-4846	203	13	2𝑘+1	2𝑘+1	NUM
cana-4846	203	14	]	]	PUNCT
cana-4846	203	15	−	−	PROPN
cana-4846	204	1	1	1	NUM
cana-4846	204	2	=	=	SYM
cana-4846	205	1	[	[	PUNCT
cana-4846	205	2	2𝑘+3−4	2𝑘+3−4	NUM
cana-4846	205	3	3	3	NUM
cana-4846	205	4	]	]	PUNCT
cana-4846	205	5	−	−	PROPN
cana-4846	205	6	1	1	NUM
cana-4846	205	7	=	=	SYM
cana-4846	205	8	[	[	PUNCT
cana-4846	205	9	2𝑘+3−7	2𝑘+3−7	NUM
cana-4846	205	10	3	3	NUM
cana-4846	205	11	]	]	X
cana-4846	205	12	case(ii	case(ii	PROPN
cana-4846	205	13	)	)	PUNCT
cana-4846	205	14	𝑘	𝑘	NOUN
cana-4846	205	15	is	be	AUX
cana-4846	205	16	an	an	DET
cana-4846	205	17	even	even	ADV
cana-4846	205	18	integer	integer	NOUN
cana-4846	205	19	.	.	PUNCT
cana-4846	206	1	clearly	clearly	ADV
cana-4846	206	2	,	,	PUNCT
cana-4846	206	3	𝑆𝑘−1	𝑆𝑘−1	PROPN
cana-4846	206	4	∪	∪	VERB
cana-4846	206	5	𝑆𝑘−3	𝑆𝑘−3	NOUN
cana-4846	206	6	∪	∪	ADP
cana-4846	206	7	…	…	SYM
cana-4846	206	8	∪	∪	ADJ
cana-4846	206	9	𝑆3	𝑆3	PROPN
cana-4846	206	10	∪	∪	ADJ
cana-4846	206	11	𝑆1	𝑆1	NOUN
cana-4846	206	12	is	be	AUX
cana-4846	206	13	a	a	DET
cana-4846	206	14	𝜕	𝜕	NOUN
cana-4846	206	15	−set	−set	NOUN
cana-4846	206	16	.	.	PUNCT
cana-4846	207	1	𝜕(𝑇(𝐺	𝜕(𝑇(𝐺	NUM
cana-4846	207	2	)	)	PUNCT
cana-4846	207	3	)	)	PUNCT
cana-4846	208	1	=	=	SYM
cana-4846	208	2	2𝑘+1	2𝑘+1	NUM
cana-4846	208	3	+	+	NUM
cana-4846	208	4	2𝑘−1	2𝑘−1	NUM
cana-4846	208	5	+	+	CCONJ
cana-4846	208	6	2𝑘−3	2𝑘−3	NUM
cana-4846	208	7	+	+	NOUN
cana-4846	208	8	⋯+	⋯+	NOUN
cana-4846	208	9	25	25	NUM
cana-4846	208	10	+	+	NUM
cana-4846	208	11	23	23	NUM
cana-4846	208	12	+	+	SYM
cana-4846	208	13	1	1	NUM
cana-4846	208	14	=	=	SYM
cana-4846	208	15	2𝑘+1	2𝑘+1	NUM
cana-4846	209	1	[	[	X
cana-4846	209	2	1	1	NUM
cana-4846	209	3	+	+	NUM
cana-4846	209	4	2𝑘−1	2𝑘−1	NUM
cana-4846	209	5	2𝑘+1	2𝑘+1	NUM
cana-4846	209	6	+	+	CCONJ
cana-4846	209	7	2𝑘−3	2𝑘−3	NUM
cana-4846	209	8	2𝑘+1	2𝑘+1	NUM
cana-4846	209	9	+	+	NOUN
cana-4846	209	10	⋯+	⋯+	NOUN
cana-4846	209	11	25	25	NUM
cana-4846	209	12	2𝑘+1	2𝑘+1	NUM
cana-4846	209	13	+	+	CCONJ
cana-4846	209	14	23	23	NUM
cana-4846	209	15	2𝑘+1	2𝑘+1	NUM
cana-4846	209	16	]	]	PUNCT
cana-4846	210	1	+	+	CCONJ
cana-4846	210	2	1	1	NUM
cana-4846	210	3	=	=	SYM
cana-4846	210	4	2𝑘+1[1	2𝑘+1[1	NUM
cana-4846	210	5	+	+	NUM
cana-4846	210	6	2−2	2−2	NUM
cana-4846	210	7	+	+	CCONJ
cana-4846	210	8	2−4	2−4	NUM
cana-4846	210	9	+	+	ADJ
cana-4846	210	10	⋯+	⋯+	NOUN
cana-4846	210	11	2−𝑘+4	2−𝑘+4	NUM
cana-4846	210	12	+	+	CCONJ
cana-4846	210	13	2−𝑘+2	2−𝑘+2	NUM
cana-4846	210	14	]	]	PUNCT
cana-4846	211	1	+	+	CCONJ
cana-4846	211	2	1	1	NUM
cana-4846	211	3	=	=	SYM
cana-4846	211	4	2𝑘+1	2𝑘+1	NUM
cana-4846	212	1	[	[	X
cana-4846	212	2	1	1	NUM
cana-4846	212	3	+	+	CCONJ
cana-4846	212	4	(	(	PUNCT
cana-4846	212	5	1	1	NUM
cana-4846	212	6	22	22	NUM
cana-4846	212	7	)	)	PUNCT
cana-4846	212	8	1	1	NUM
cana-4846	213	1	+	+	CCONJ
cana-4846	213	2	(	(	PUNCT
cana-4846	213	3	1	1	NUM
cana-4846	213	4	22	22	NUM
cana-4846	213	5	)	)	PUNCT
cana-4846	213	6	2	2	NUM
cana-4846	214	1	+	+	CCONJ
cana-4846	214	2	…	…	PUNCT
cana-4846	214	3	+	+	CCONJ
cana-4846	214	4	(	(	PUNCT
cana-4846	214	5	1	1	NUM
cana-4846	214	6	22	22	NUM
cana-4846	214	7	)	)	PUNCT
cana-4846	215	1	𝑘−4	𝑘−4	NOUN
cana-4846	215	2	2	2	NUM
cana-4846	216	1	+	+	CCONJ
cana-4846	216	2	(	(	PUNCT
cana-4846	216	3	1	1	NUM
cana-4846	216	4	22	22	NUM
cana-4846	216	5	)	)	PUNCT
cana-4846	216	6	𝑘−2	𝑘−2	ADP
cana-4846	216	7	2	2	NUM
cana-4846	216	8	]	]	PUNCT
cana-4846	216	9	+	+	CCONJ
cana-4846	216	10	1	1	NUM
cana-4846	216	11	=	=	SYM
cana-4846	216	12	2𝑘+1	2𝑘+1	PROPN
cana-4846	216	13	[	[	PUNCT
cana-4846	216	14	1−	1−	NUM
cana-4846	216	15	(	(	PUNCT
cana-4846	216	16	1	1	NUM
cana-4846	216	17	22	22	NUM
cana-4846	216	18	)	)	PUNCT
cana-4846	216	19	𝑘	𝑘	PRON
cana-4846	216	20	2	2	NUM
cana-4846	216	21	1−	1−	NUM
cana-4846	216	22	(	(	PUNCT
cana-4846	216	23	1	1	NUM
cana-4846	216	24	22	22	NUM
cana-4846	216	25	)	)	PUNCT
cana-4846	216	26	]	]	PUNCT
cana-4846	217	1	+	+	CCONJ
cana-4846	217	2	1	1	X
cana-4846	217	3	=	=	SYM
cana-4846	217	4	2𝑘+1	2𝑘+1	NUM
cana-4846	217	5	×	×	NOUN
cana-4846	217	6	22	22	NUM
cana-4846	217	7	3	3	NUM
cana-4846	217	8	×	×	NOUN
cana-4846	217	9	[	[	PUNCT
cana-4846	217	10	1	1	NUM
cana-4846	217	11	−	−	NUM
cana-4846	217	12	1	1	NUM
cana-4846	217	13	2𝑘	2𝑘	NUM
cana-4846	217	14	]	]	PUNCT
cana-4846	218	1	+	+	CCONJ
cana-4846	218	2	1	1	X
cana-4846	218	3	=	=	SYM
cana-4846	218	4	[	[	PUNCT
cana-4846	218	5	2𝑘+3−8	2𝑘+3−8	NUM
cana-4846	218	6	3	3	NUM
cana-4846	218	7	]	]	PUNCT
cana-4846	218	8	+	+	CCONJ
cana-4846	218	9	1	1	X
cana-4846	218	10	=	=	SYM
cana-4846	218	11	[	[	PUNCT
cana-4846	218	12	2𝑘+3−5	2𝑘+3−5	NUM
cana-4846	218	13	3	3	NUM
cana-4846	218	14	]	]	PUNCT
cana-4846	218	15	theorem	theorem	VERB
cana-4846	218	16	13	13	NUM
cana-4846	218	17	.	.	PUNCT
cana-4846	219	1	if	if	SCONJ
cana-4846	219	2	𝐺	𝐺	PROPN
cana-4846	219	3	=	=	PUNCT
cana-4846	219	4	𝑃2	𝑃2	PROPN
cana-4846	219	5	×	×	PROPN
cana-4846	219	6	𝑃𝑛1	𝑃𝑛1	NOUN
cana-4846	219	7	is	be	AUX
cana-4846	219	8	a	a	DET
cana-4846	219	9	grid	grid	NOUN
cana-4846	219	10	graph	graph	NOUN
cana-4846	219	11	with	with	ADP
cana-4846	219	12	then	then	ADV
cana-4846	219	13	𝜕(𝑇(𝑃2	𝜕(𝑇(𝑃2	PROPN
cana-4846	219	14	×	×	PROPN
cana-4846	219	15	𝑃𝑛	𝑃𝑛	PROPN
cana-4846	219	16	)	)	PUNCT
cana-4846	219	17	)	)	PUNCT
cana-4846	220	1	=	=	PUNCT
cana-4846	221	1	𝑙	𝑙	NOUN
cana-4846	221	2	−	−	NUM
cana-4846	221	3	2𝑛1when	2𝑛1when	NOUN
cana-4846	221	4	𝑛1	𝑛1	ADJ
cana-4846	221	5	≥	≥	NUM
cana-4846	221	6	2	2	NUM
cana-4846	221	7	.	.	PUNCT
cana-4846	222	1	proof	proof	NOUN
cana-4846	222	2	.	.	PUNCT
cana-4846	223	1	let	let	VERB
cana-4846	223	2	𝑇(𝑃2	𝑇(𝑃2	VERB
cana-4846	223	3	×	×	NOUN
cana-4846	223	4	𝑃𝑛1	𝑃𝑛1	NOUN
cana-4846	223	5	)	)	PUNCT
cana-4846	223	6	be	be	AUX
cana-4846	223	7	the	the	DET
cana-4846	223	8	total	total	ADJ
cana-4846	223	9	graph	graph	NOUN
cana-4846	223	10	of	of	ADP
cana-4846	223	11	the	the	DET
cana-4846	223	12	grid	grid	NOUN
cana-4846	223	13	graph	graph	NOUN
cana-4846	223	14	and	and	CCONJ
cana-4846	223	15	the	the	DET
cana-4846	223	16	vertex	vertex	NOUN
cana-4846	223	17	set	set	NOUN
cana-4846	223	18	of	of	ADP
cana-4846	223	19	the	the	DET
cana-4846	223	20	graph	graph	NOUN
cana-4846	223	21	is	be	AUX
cana-4846	223	22	𝑉	𝑉	PROPN
cana-4846	223	23	(	(	PUNCT
cana-4846	223	24	𝑇(𝑃2	𝑇(𝑃2	ADJ
cana-4846	223	25	×	×	NOUN
cana-4846	223	26	𝑃𝑛1	𝑃𝑛1	NOUN
cana-4846	223	27	)	)	PUNCT
cana-4846	223	28	)	)	PUNCT
cana-4846	224	1	=	=	PRON
cana-4846	224	2	{	{	PUNCT
cana-4846	224	3	𝑣11	𝑣11	PROPN
cana-4846	224	4	,	,	PUNCT
cana-4846	224	5	…	…	PUNCT
cana-4846	224	6	𝑣1𝑛1	𝑣1𝑛1	PROPN
cana-4846	224	7	,	,	PUNCT
cana-4846	224	8	𝑣21	𝑣21	NUM
cana-4846	224	9	,	,	PUNCT
cana-4846	224	10	…	…	PUNCT
cana-4846	224	11	𝑣2𝑛1	𝑣2𝑛1	X
cana-4846	224	12	,	,	PUNCT
cana-4846	224	13	𝑢11	𝑢11	PROPN
cana-4846	224	14	,	,	PUNCT
cana-4846	224	15	.	.	PUNCT
cana-4846	224	16	.	.	PUNCT
cana-4846	225	1	,	,	PUNCT
cana-4846	225	2	𝑢1𝑛1	𝑢1𝑛1	PROPN
cana-4846	225	3	,	,	PUNCT
cana-4846	225	4	𝑒11	𝑒11	PROPN
cana-4846	225	5	,	,	PUNCT
cana-4846	225	6	…	…	PUNCT
cana-4846	225	7	𝑒1(𝑛1−1	𝑒1(𝑛1−1	NOUN
cana-4846	225	8	)	)	PUNCT
cana-4846	225	9	,	,	PUNCT
cana-4846	225	10	𝑒21	𝑒21	ADJ
cana-4846	225	11	,	,	PUNCT
cana-4846	225	12	…	…	PUNCT
cana-4846	225	13	.	.	PUNCT
cana-4846	226	1	𝑒2(𝑛1−1	𝑒2(𝑛1−1	ADJ
cana-4846	226	2	)	)	PUNCT
cana-4846	226	3	}	}	PUNCT
cana-4846	226	4	where	where	SCONJ
cana-4846	226	5	𝑢𝑖𝑗	𝑢𝑖𝑗	NOUN
cana-4846	226	6	=	=	SYM
cana-4846	226	7	𝑣𝑖𝑗𝑣(𝑖+1)𝑗	𝑣𝑖𝑗𝑣(𝑖+1)𝑗	NOUN
cana-4846	226	8	,	,	PUNCT
cana-4846	226	9	𝑖	𝑖	AUX
cana-4846	226	10	,	,	PUNCT
cana-4846	226	11	𝑗	𝑗	NOUN
cana-4846	226	12	=	=	SYM
cana-4846	226	13	1,2	1,2	NUM
cana-4846	226	14	,	,	PUNCT
cana-4846	226	15	…	…	PUNCT
cana-4846	226	16	,	,	PUNCT
cana-4846	226	17	𝑛1	𝑛1	NOUN
cana-4846	226	18	and	and	CCONJ
cana-4846	226	19	|𝑉	|𝑉	NUM
cana-4846	226	20	(	(	PUNCT
cana-4846	226	21	𝑇(𝑃2	𝑇(𝑃2	PUNCT
cana-4846	226	22	×	×	NOUN
cana-4846	226	23	𝑃𝑛1))|	𝑃𝑛1))|	PROPN
cana-4846	226	24	=	=	SYM
cana-4846	226	25	5𝑛1	5𝑛1	NUM
cana-4846	226	26	−	−	NOUN
cana-4846	226	27	2	2	NUM
cana-4846	226	28	=	=	PRON
cana-4846	226	29	𝑙.	𝑙.	NOUN
cana-4846	226	30	consider	consider	VERB
cana-4846	226	31	the	the	DET
cana-4846	226	32	differential	differential	ADJ
cana-4846	226	33	set	set	NOUN
cana-4846	226	34	of	of	ADP
cana-4846	226	35	the	the	DET
cana-4846	226	36	given	give	VERB
cana-4846	226	37	grid	grid	NOUN
cana-4846	226	38	graph	graph	NOUN
cana-4846	226	39	𝐺	𝐺	NOUN
cana-4846	226	40	is	be	AUX
cana-4846	226	41	𝑆	𝑆	PROPN
cana-4846	226	42	=	=	PRON
cana-4846	226	43	{	{	PUNCT
cana-4846	226	44	𝑣11	𝑣11	PROPN
cana-4846	226	45	,	,	PUNCT
cana-4846	226	46	𝑣14	𝑣14	NUM
cana-4846	226	47	,	,	PUNCT
cana-4846	226	48	𝑢12	𝑢12	NUM
cana-4846	226	49	,	,	PUNCT
cana-4846	226	50	𝑢15	𝑢15	NOUN
cana-4846	226	51	,	,	PUNCT
cana-4846	226	52	𝑣23	𝑣23	PROPN
cana-4846	226	53	,	,	PUNCT
cana-4846	226	54	…	…	PUNCT
cana-4846	226	55	.	.	PUNCT
cana-4846	226	56	.	.	PUNCT
cana-4846	227	1	𝑣1(𝑛1−1	𝑣1(𝑛1−1	ADV
cana-4846	227	2	)	)	PUNCT
cana-4846	227	3	,	,	PUNCT
cana-4846	227	4	𝑢1𝑛1	𝑢1𝑛1	PROPN
cana-4846	227	5	,	,	PUNCT
cana-4846	227	6	𝑣2(𝑛1−2	𝑣2(𝑛1−2	PROPN
cana-4846	227	7	)	)	PUNCT
cana-4846	227	8	}	}	PUNCT
cana-4846	227	9	which	which	PRON
cana-4846	227	10	is	be	AUX
cana-4846	227	11	also	also	ADV
cana-4846	227	12	an	an	DET
cana-4846	227	13	𝛾	𝛾	NOUN
cana-4846	227	14	−set	−set	NOUN
cana-4846	227	15	.	.	PUNCT
cana-4846	228	1	therefore	therefore	ADV
cana-4846	228	2	,	,	PUNCT
cana-4846	228	3	𝜕	𝜕	NOUN
cana-4846	228	4	(	(	PUNCT
cana-4846	228	5	𝑇(𝑃2	𝑇(𝑃2	PROPN
cana-4846	228	6	×	×	NOUN
cana-4846	228	7	𝑃𝑛1	𝑃𝑛1	NOUN
cana-4846	228	8	)	)	PUNCT
cana-4846	228	9	)	)	PUNCT
cana-4846	229	1	=	=	PUNCT
cana-4846	230	1	𝑙	𝑙	PRON
cana-4846	230	2	−	−	NUM
cana-4846	230	3	2𝛾(𝐺	2𝛾(𝐺	NUM
cana-4846	230	4	)	)	PUNCT
cana-4846	230	5	=	=	SYM
cana-4846	231	1	1	1	NUM
cana-4846	231	2	−	−	NUM
cana-4846	231	3	2𝑛1	2𝑛1	NUM
cana-4846	231	4	.	.	PUNCT
cana-4846	232	1	3	3	X
cana-4846	232	2	.	.	NUM
cana-4846	232	3	references	reference	NOUN
cana-4846	232	4	[	[	X
cana-4846	232	5	1	1	NUM
cana-4846	232	6	]	]	PUNCT
cana-4846	232	7	athul	athul	PROPN
cana-4846	232	8	,	,	PUNCT
cana-4846	232	9	t.b	t.b	PROPN
cana-4846	232	10	.	.	PROPN
cana-4846	232	11	,	,	PUNCT
cana-4846	232	12	and	and	CCONJ
cana-4846	232	13	suresh	suresh	PROPN
cana-4846	232	14	singh	singh	PROPN
cana-4846	232	15	,	,	PUNCT
cana-4846	232	16	g.	g.	PROPN
cana-4846	232	17	total	total	ADJ
cana-4846	232	18	graph	graph	NOUN
cana-4846	232	19	of	of	ADP
cana-4846	232	20	regular	regular	ADJ
cana-4846	232	21	graphs	graph	NOUN
cana-4846	232	22	.	.	PUNCT
cana-4846	233	1	advances	advance	NOUN
cana-4846	233	2	in	in	ADP
cana-4846	233	3	mathematics	mathematic	NOUN
cana-4846	233	4	:	:	PUNCT
cana-4846	233	5	scientific	scientific	ADJ
cana-4846	233	6	journal	journal	NOUN
cana-4846	233	7	.	.	PUNCT
cana-4846	234	1	2020	2020	NUM
cana-4846	234	2	,	,	PUNCT
cana-4846	234	3	9	9	NUM
cana-4846	234	4	,	,	PUNCT
cana-4846	234	5	4213	4213	NUM
cana-4846	234	6	-	-	SYM
cana-4846	234	7	4220	4220	NUM
cana-4846	234	8	.	.	PUNCT
cana-4846	235	1	issn	issn	PROPN
cana-4846	235	2	1857	1857	NUM
cana-4846	235	3	-	-	SYM
cana-4846	235	4	8365	8365	NUM
cana-4846	235	5	.	.	PUNCT
cana-4846	236	1	[	[	X
cana-4846	236	2	2	2	NUM
cana-4846	236	3	]	]	X
cana-4846	236	4	behzad	behzad	PROPN
cana-4846	236	5	,	,	PUNCT
cana-4846	236	6	m.	m.	VERB
cana-4846	236	7	the	the	DET
cana-4846	236	8	connectivity	connectivity	NOUN
cana-4846	236	9	of	of	ADP
cana-4846	236	10	total	total	ADJ
cana-4846	236	11	graphs	graph	NOUN
cana-4846	236	12	.	.	PUNCT
cana-4846	237	1	bulletin	bulletin	NOUN
cana-4846	237	2	australian	australian	ADJ
cana-4846	237	3	mathematical	mathematical	ADJ
cana-4846	237	4	society	society	NOUN
cana-4846	237	5	,	,	PUNCT
cana-4846	237	6	1969	1969	NUM
cana-4846	237	7	,	,	PUNCT
cana-4846	237	8	1	1	NUM
cana-4846	237	9	,	,	PUNCT
cana-4846	237	10	175	175	NUM
cana-4846	237	11	-	-	SYM
cana-4846	237	12	181	181	NUM
cana-4846	238	1	[	[	X
cana-4846	238	2	3	3	NUM
cana-4846	238	3	]	]	X
cana-4846	238	4	bermudo	bermudo	PROPN
cana-4846	238	5	,	,	PUNCT
cana-4846	238	6	s.	s.	PROPN
cana-4846	238	7	,	,	PUNCT
cana-4846	238	8	and	and	CCONJ
cana-4846	238	9	fernau	fernau	PROPN
cana-4846	238	10	,	,	PUNCT
cana-4846	238	11	h.	h.	PROPN
cana-4846	238	12	lower	lower	ADV
cana-4846	238	13	bound	bind	VERB
cana-4846	238	14	on	on	ADP
cana-4846	238	15	the	the	DET
cana-4846	238	16	differential	differential	NOUN
cana-4846	238	17	of	of	ADP
cana-4846	238	18	a	a	DET
cana-4846	238	19	graph	graph	NOUN
cana-4846	238	20	.	.	PUNCT
cana-4846	239	1	discret	discret	PROPN
cana-4846	239	2	.	.	PUNCT
cana-4846	239	3	math	math	NOUN
cana-4846	239	4	.	.	PUNCT
cana-4846	240	1	2012	2012	NUM
cana-4846	240	2	,	,	PUNCT
cana-4846	240	3	communications	communication	NOUN
cana-4846	240	4	on	on	ADP
cana-4846	240	5	applied	apply	VERB
cana-4846	240	6	nonlinear	nonlinear	ADJ
cana-4846	240	7	analysis	analysis	NOUN
cana-4846	240	8	issn	issn	NOUN
cana-4846	240	9	:	:	PUNCT
cana-4846	240	10	1074	1074	NUM
cana-4846	240	11	-	-	PUNCT
cana-4846	240	12	133x	133x	NUM
cana-4846	240	13	vol	vol	VERB
cana-4846	240	14	32	32	NUM
cana-4846	240	15	no	no	NOUN
cana-4846	240	16	.	.	PUNCT
cana-4846	241	1	10s	10	NOUN
cana-4846	241	2	(	(	PUNCT
cana-4846	241	3	2025	2025	NUM
cana-4846	241	4	)	)	PUNCT
cana-4846	241	5	566	566	NUM
cana-4846	241	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4846	241	7	312	312	NUM
cana-4846	241	8	,	,	PUNCT
cana-4846	241	9	32363250	32363250	NUM
cana-4846	241	10	,	,	PUNCT
cana-4846	241	11	doi:10.1016	doi:10.1016	PROPN
cana-4846	241	12	/	/	SYM
cana-4846	241	13	j.disc.2012.07.021	j.disc.2012.07.021	PROPN
cana-4846	241	14	.	.	PUNCT
cana-4846	242	1	[	[	X
cana-4846	242	2	4	4	NUM
cana-4846	242	3	]	]	X
cana-4846	242	4	bermudo	bermudo	PROPN
cana-4846	242	5	,	,	PUNCT
cana-4846	242	6	s.	s.	PROPN
cana-4846	242	7	,	,	PUNCT
cana-4846	242	8	rodrguez	rodrguez	ADJ
cana-4846	242	9	,	,	PUNCT
cana-4846	242	10	j.m	j.m	PROPN
cana-4846	242	11	.	.	PROPN
cana-4846	242	12	,	,	PUNCT
cana-4846	242	13	and	and	CCONJ
cana-4846	242	14	sigarreta	sigarreta	NOUN
cana-4846	242	15	,	,	PUNCT
cana-4846	242	16	j.m	j.m	PROPN
cana-4846	242	17	.	.	PROPN
cana-4846	242	18	on	on	ADP
cana-4846	242	19	the	the	DET
cana-4846	242	20	differential	differential	NOUN
cana-4846	242	21	in	in	ADP
cana-4846	242	22	graphs	graph	NOUN
cana-4846	242	23	.	.	PUNCT
cana-4846	243	1	util	util	NOUN
cana-4846	243	2	.	.	PUNCT
cana-4846	244	1	math	math	NOUN
cana-4846	244	2	.	.	PUNCT
cana-4846	245	1	2015	2015	NUM
cana-4846	245	2	,	,	PUNCT
cana-4846	245	3	97	97	NUM
cana-4846	245	4	,	,	PUNCT
cana-4846	245	5	257270	257270	NUM
cana-4846	245	6	.	.	PUNCT
cana-4846	246	1	[	[	X
cana-4846	246	2	5	5	NUM
cana-4846	246	3	]	]	SYM
cana-4846	246	4	ganghua	ganghua	PROPN
cana-4846	246	5	xie	xie	PROPN
cana-4846	246	6	.	.	PROPN
cana-4846	246	7	,	,	PUNCT
cana-4846	246	8	yinkui	yinkui	PROPN
cana-4846	246	9	li	li	PROPN
cana-4846	246	10	.	.	PROPN
cana-4846	246	11	cycle	cycle	NOUN
cana-4846	246	12	multiplicity	multiplicity	NOUN
cana-4846	246	13	of	of	ADP
cana-4846	246	14	total	total	ADJ
cana-4846	246	15	graph	graph	NOUN
cana-4846	246	16	of	of	ADP
cana-4846	246	17	complete	complete	ADJ
cana-4846	246	18	bipartite	bipartite	NOUN
cana-4846	246	19	graph	graph	NOUN
cana-4846	246	20	.	.	PUNCT
cana-4846	247	1	open	open	ADJ
cana-4846	247	2	journal	journal	NOUN
cana-4846	247	3	of	of	ADP
cana-4846	247	4	discrete	discrete	ADJ
cana-4846	247	5	mathematics	mathematic	NOUN
cana-4846	247	6	,	,	PUNCT
cana-4846	247	7	2013	2013	NUM
cana-4846	247	8	,	,	PUNCT
cana-4846	247	9	13	13	NUM
cana-4846	247	10	,	,	PUNCT
cana-4846	247	11	95	95	NUM
cana-4846	247	12	-	-	SYM
cana-4846	247	13	99	99	NUM
cana-4846	247	14	.	.	PUNCT
cana-4846	248	1	[	[	X
cana-4846	248	2	6	6	NUM
cana-4846	248	3	]	]	X
cana-4846	248	4	goddard	goddard	NOUN
cana-4846	248	5	,	,	PUNCT
cana-4846	248	6	w	w	NOUN
cana-4846	248	7	and	and	CCONJ
cana-4846	248	8	henning	henning	PROPN
cana-4846	248	9	,	,	PUNCT
cana-4846	248	10	m.a	m.a	PROPN
cana-4846	248	11	generalized	generalize	VERB
cana-4846	248	12	domination	domination	NOUN
cana-4846	248	13	and	and	CCONJ
cana-4846	248	14	independence	independence	NOUN
cana-4846	248	15	in	in	ADP
cana-4846	248	16	graphs	graph	NOUN
cana-4846	248	17	,	,	PUNCT
cana-4846	248	18	congr	congr	NOUN
cana-4846	248	19	.	.	PUNCT
cana-4846	249	1	(	(	PUNCT
cana-4846	249	2	1997	1997	NUM
cana-4846	249	3	)	)	PUNCT
cana-4846	249	4	161	161	NUM
cana-4846	249	5	-	-	SYM
cana-4846	249	6	171	171	NUM
cana-4846	249	7	.	.	PUNCT
cana-4846	250	1	[	[	X
cana-4846	250	2	7	7	NUM
cana-4846	250	3	]	]	X
cana-4846	250	4	harary	harary	NOUN
cana-4846	250	5	,	,	PUNCT
cana-4846	250	6	f	f	PROPN
cana-4846	250	7	graph	graph	NOUN
cana-4846	250	8	theory	theory	NOUN
cana-4846	250	9	,	,	PUNCT
cana-4846	250	10	addison	addison	PROPN
cana-4846	250	11	wesley	wesley	PROPN
cana-4846	250	12	,	,	PUNCT
cana-4846	250	13	reading	read	VERB
cana-4846	250	14	mass	mass	PROPN
cana-4846	250	15	.	.	PROPN
cana-4846	250	16	,	,	PUNCT
cana-4846	250	17	1972	1972	NUM
cana-4846	250	18	.	.	PUNCT
cana-4846	251	1	[	[	X
cana-4846	251	2	8	8	NUM
cana-4846	251	3	]	]	SYM
cana-4846	251	4	li	li	PROPN
cana-4846	251	5	,	,	PUNCT
cana-4846	251	6	y.k	y.k	PROPN
cana-4846	251	7	.	.	PROPN
cana-4846	251	8	cycle	cycle	NOUN
cana-4846	251	9	multiplicity	multiplicity	NOUN
cana-4846	251	10	of	of	ADP
cana-4846	251	11	some	some	DET
cana-4846	251	12	total	total	ADJ
cana-4846	251	13	graphs	graph	NOUN
cana-4846	251	14	.	.	PUNCT
cana-4846	252	1	applied	apply	VERB
cana-4846	252	2	mathematics	mathematic	NOUN
cana-4846	252	3	and	and	CCONJ
cana-4846	252	4	computation	computation	NOUN
cana-4846	252	5	,	,	PUNCT
cana-4846	252	6	2017	2017	NUM
cana-4846	252	7	,	,	PUNCT
cana-4846	252	8	107	107	NUM
cana-4846	252	9	-	-	SYM
cana-4846	252	10	113	113	NUM
cana-4846	252	11	.	.	PUNCT
cana-4846	253	1	[	[	X
cana-4846	253	2	9	9	NUM
cana-4846	253	3	]	]	SYM
cana-4846	253	4	mashburn.j.l	mashburn.j.l	NOUN
cana-4846	253	5	.	.	PUNCT
cana-4846	253	6	,	,	PUNCT
cana-4846	253	7	haynes	haynes	PROPN
cana-4846	253	8	,	,	PUNCT
cana-4846	253	9	t.w	t.w	PROPN
cana-4846	253	10	.	.	PROPN
cana-4846	253	11	,	,	PUNCT
cana-4846	253	12	hedetniemi	hedetniemi	PROPN
cana-4846	253	13	,	,	PUNCT
cana-4846	253	14	s.m	s.m	PROPN
cana-4846	253	15	.	.	PROPN
cana-4846	253	16	,	,	PUNCT
cana-4846	253	17	hedeteniemi	hedeteniemi	PROPN
cana-4846	253	18	,	,	PUNCT
cana-4846	253	19	s.t	s.t	PROPN
cana-4846	253	20	.	.	PROPN
cana-4846	253	21	,	,	PUNCT
cana-4846	253	22	and	and	CCONJ
cana-4846	253	23	slater	slater	NOUN
cana-4846	253	24	,	,	PUNCT
cana-4846	253	25	p.j	p.j	PROPN
cana-4846	253	26	.	.	PROPN
cana-4846	253	27	,	,	PUNCT
cana-4846	253	28	differential	differential	VERB
cana-4846	253	29	in	in	ADP
cana-4846	253	30	graphs	graph	NOUN
cana-4846	253	31	,	,	PUNCT
cana-4846	253	32	utilitas	utilitas	PROPN
cana-4846	253	33	mathematica	mathematica	PROPN
cana-4846	253	34	69	69	NUM
cana-4846	253	35	,	,	PUNCT
cana-4846	253	36	43	43	NUM
cana-4846	253	37	-	-	SYM
cana-4846	253	38	54	54	NUM
cana-4846	253	39	,	,	PUNCT
cana-4846	253	40	2006	2006	NUM
cana-4846	253	41	.	.	PUNCT
cana-4846	254	1	[	[	X
cana-4846	254	2	10	10	NUM
cana-4846	254	3	]	]	X
cana-4846	254	4	mohammed	mohammed	PROPN
cana-4846	254	5	alatif	alatif	PROPN
cana-4846	254	6	,	,	PUNCT
cana-4846	254	7	puttaswamy	puttaswamy	NOUN
cana-4846	254	8	and	and	CCONJ
cana-4846	254	9	nayaka	nayaka	PROPN
cana-4846	254	10	,	,	PUNCT
cana-4846	254	11	s.r	s.r	PROPN
cana-4846	254	12	,	,	PUNCT
cana-4846	254	13	boundary	boundary	ADJ
cana-4846	254	14	domination	domination	NOUN
cana-4846	254	15	in	in	ADP
cana-4846	254	16	total	total	ADJ
cana-4846	254	17	graphs	graph	NOUN
cana-4846	254	18	,	,	PUNCT
cana-4846	254	19	gen.math.notes	gen.math.notes	X
cana-4846	254	20	,	,	PUNCT
cana-4846	254	21	vol	vol	NOUN
cana-4846	254	22	.	.	PROPN
cana-4846	254	23	32	32	NUM
cana-4846	254	24	,	,	PUNCT
cana-4846	254	25	no	no	INTJ
cana-4846	254	26	.	.	NOUN
cana-4846	254	27	1	1	NUM
cana-4846	254	28	,	,	PUNCT
cana-4846	254	29	january	january	PROPN
cana-4846	254	30	2016	2016	NUM
cana-4846	254	31	,	,	PUNCT
cana-4846	254	32	pp.12	pp.12	NOUN
cana-4846	254	33	-	-	ADJ
cana-4846	254	34	20	20	NUM
cana-4846	254	35	,	,	PUNCT
cana-4846	254	36	issn	issn	PROPN
cana-4846	254	37	2219	2219	NUM
cana-4846	254	38	-	-	SYM
cana-4846	254	39	7184	7184	NUM
cana-4846	254	40	[	[	X
cana-4846	254	41	11	11	NUM
cana-4846	254	42	]	]	SYM
cana-4846	254	43	murugesan.n	murugesan.n	NUM
cana-4846	254	44	,	,	PUNCT
cana-4846	254	45	and	and	CCONJ
cana-4846	254	46	nair	nair	NOUN
cana-4846	254	47	,	,	PUNCT
cana-4846	254	48	d	d	X
cana-4846	254	49	,	,	PUNCT
cana-4846	254	50	s	s	X
cana-4846	254	51	,	,	PUNCT
cana-4846	254	52	(	(	PUNCT
cana-4846	254	53	1,2	1,2	NUM
cana-4846	254	54	)	)	PUNCT
cana-4846	254	55	domination	domination	NOUN
cana-4846	254	56	in	in	ADP
cana-4846	254	57	total	total	ADJ
cana-4846	254	58	graph	graph	NOUN
cana-4846	254	59	of	of	ADP
cana-4846	254	60	k1,n	k1,n	PROPN
cana-4846	254	61	,	,	PUNCT
cana-4846	254	62	cn	cn	PROPN
cana-4846	254	63	and	and	CCONJ
cana-4846	254	64	pn	pn	PROPN
cana-4846	254	65	,	,	PUNCT
cana-4846	254	66	advances	advance	NOUN
cana-4846	254	67	in	in	ADP
cana-4846	254	68	mathematics	mathematic	NOUN
cana-4846	254	69	,	,	PUNCT
cana-4846	254	70	4(3	4(3	NUM
cana-4846	254	71	)	)	PUNCT
cana-4846	254	72	(	(	PUNCT
cana-4846	254	73	2013	2013	NUM
cana-4846	254	74	)	)	PUNCT
cana-4846	254	75	,	,	PUNCT
cana-4846	254	76	516	516	NUM
cana-4846	254	77	-	-	SYM
cana-4846	254	78	526	526	NUM
cana-4846	254	79	[	[	X
cana-4846	254	80	12	12	NUM
cana-4846	254	81	]	]	X
cana-4846	254	82	muthuramakrishnan	muthuramakrishnan	PROPN
cana-4846	254	83	,	,	PUNCT
cana-4846	254	84	d.	d.	PROPN
cana-4846	254	85	,	,	PUNCT
cana-4846	254	86	and	and	CCONJ
cana-4846	254	87	jayaraman	jayaraman	NOUN
cana-4846	254	88	,	,	PUNCT
cana-4846	254	89	g.	g.	PROPN
cana-4846	254	90	,total	,total	PUNCT
cana-4846	254	91	chromatic	chromatic	ADJ
cana-4846	254	92	number	number	NOUN
cana-4846	254	93	of	of	ADP
cana-4846	254	94	middle	middle	ADJ
cana-4846	254	95	and	and	CCONJ
cana-4846	254	96	total	total	ADJ
cana-4846	254	97	graph	graph	NOUN
cana-4846	254	98	of	of	ADP
cana-4846	254	99	path	path	NOUN
cana-4846	254	100	and	and	CCONJ
cana-4846	254	101	sunlet	sunlet	NOUN
cana-4846	254	102	graph	graph	NOUN
cana-4846	254	103	,	,	PUNCT
cana-4846	254	104	international	international	ADJ
cana-4846	254	105	journal	journal	NOUN
cana-4846	254	106	of	of	ADP
cana-4846	254	107	scientific	scientific	ADJ
cana-4846	254	108	and	and	CCONJ
cana-4846	254	109	innovative	innovative	ADJ
cana-4846	254	110	mathematical	mathematical	ADJ
cana-4846	254	111	research	research	NOUN
cana-4846	254	112	,	,	PUNCT
cana-4846	254	113	m2018	m2018	NOUN
cana-4846	254	114	,	,	PUNCT
cana-4846	254	115	4	4	NUM
cana-4846	254	116	,	,	PUNCT
cana-4846	254	117	1	1	NUM
cana-4846	254	118	-	-	SYM
cana-4846	254	119	9	9	NUM
cana-4846	254	120	.	.	PUNCT
cana-4846	255	1	[	[	X
cana-4846	255	2	13	13	NUM
cana-4846	255	3	]	]	SYM
cana-4846	255	4	pushpam	pushpam	NOUN
cana-4846	255	5	,	,	PUNCT
cana-4846	255	6	p.r.l	p.r.l	NOUN
cana-4846	255	7	.	.	PROPN
cana-4846	255	8	,	,	PUNCT
cana-4846	255	9	and	and	CCONJ
cana-4846	255	10	yokesh.d	yokesh.d	NOUN
cana-4846	255	11	,	,	PUNCT
cana-4846	255	12	differential	differential	NOUN
cana-4846	255	13	in	in	ADP
cana-4846	255	14	certain	certain	ADJ
cana-4846	255	15	classes	class	NOUN
cana-4846	255	16	of	of	ADP
cana-4846	255	17	graphs	graph	NOUN
cana-4846	255	18	,	,	PUNCT
cana-4846	255	19	tamkang	tamkang	PROPN
cana-4846	255	20	j.	j.	PROPN
cana-4846	255	21	math	math	PROPN
cana-4846	255	22	.	.	PUNCT
cana-4846	255	23	,	,	PUNCT
cana-4846	255	24	41	41	NUM
cana-4846	255	25	(	(	PUNCT
cana-4846	255	26	2	2	NUM
cana-4846	255	27	)	)	PUNCT
cana-4846	255	28	(	(	PUNCT
cana-4846	255	29	2010	2010	NUM
cana-4846	255	30	)	)	PUNCT
cana-4846	256	1	129–138	129–138	NUM
cana-4846	256	2	.	.	PUNCT
cana-4846	257	1	[	[	X
cana-4846	257	2	14	14	NUM
cana-4846	257	3	]	]	PUNCT
cana-4846	257	4	rajeswari	rajeswari	NOUN
cana-4846	257	5	,	,	PUNCT
cana-4846	257	6	v.	v.	ADV
cana-4846	257	7	,	,	PUNCT
cana-4846	257	8	and	and	CCONJ
cana-4846	257	9	thiagarajan	thiagarajan	NOUN
cana-4846	257	10	,	,	PUNCT
cana-4846	257	11	k.	k.	PROPN
cana-4846	257	12	,	,	PUNCT
cana-4846	257	13	sibede	sibede	NOUN
cana-4846	257	14	approach	approach	NOUN
cana-4846	257	15	for	for	ADP
cana-4846	257	16	total	total	ADJ
cana-4846	257	17	graph	graph	NOUN
cana-4846	257	18	of	of	ADP
cana-4846	257	19	path	path	NOUN
cana-4846	257	20	and	and	CCONJ
cana-4846	257	21	cycle	cycle	NOUN
cana-4846	257	22	graphs	graph	NOUN
cana-4846	257	23	,	,	PUNCT
cana-4846	257	24	middle	middle	ADJ
cana-4846	257	25	-	-	PUNCT
cana-4846	257	26	east	east	NOUN
cana-4846	257	27	journal	journal	NOUN
cana-4846	257	28	of	of	ADP
cana-4846	257	29	scientific	scientific	ADJ
cana-4846	257	30	research	research	NOUN
cana-4846	257	31	,	,	PUNCT
cana-4846	257	32	2017	2017	NUM
cana-4846	257	33	,	,	PUNCT
cana-4846	257	34	25	25	NUM
cana-4846	257	35	,	,	PUNCT
cana-4846	257	36	1553	1553	NUM
cana-4846	257	37	-	-	SYM
cana-4846	257	38	1558	1558	NUM
cana-4846	257	39	.	.	PUNCT