id	sid	tid	token	lemma	pos
cana-536	1	1	communications	communication	NOUN
cana-536	1	2	on	on	ADP
cana-536	1	3	applied	apply	VERB
cana-536	1	4	nonlinear	nonlinear	ADJ
cana-536	1	5	analysis	analysis	NOUN
cana-536	1	6	issn	issn	NOUN
cana-536	1	7	:	:	PUNCT
cana-536	1	8	1074	1074	NUM
cana-536	1	9	-	-	PUNCT
cana-536	1	10	133x	133x	NUM
cana-536	1	11	vol	vol	NOUN
cana-536	1	12	31	31	NUM
cana-536	1	13	no	no	NOUN
cana-536	1	14	.	.	NOUN
cana-536	1	15	2	2	NUM
cana-536	1	16	(	(	PUNCT
cana-536	1	17	2024	2024	NUM
cana-536	1	18	)	)	PUNCT
cana-536	1	19	214	214	NUM
cana-536	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-536	1	21	edge	edge	NOUN
cana-536	1	22	irregularity	irregularity	NOUN
cana-536	1	23	strength	strength	NOUN
cana-536	1	24	of	of	ADP
cana-536	1	25	binomial	binomial	ADJ
cana-536	1	26	trees	tree	NOUN
cana-536	1	27	s.	s.	PROPN
cana-536	1	28	muthukkumar1	muthukkumar1	PROPN
cana-536	1	29	,	,	PUNCT
cana-536	1	30	k.	k.	PROPN
cana-536	1	31	rajendran	rajendran	PROPN
cana-536	1	32	vels	vels	PROPN
cana-536	1	33	institute	institute	PROPN
cana-536	1	34	of	of	ADP
cana-536	1	35	science	science	PROPN
cana-536	1	36	technology	technology	NOUN
cana-536	1	37	and	and	CCONJ
cana-536	1	38	advanced	advanced	ADJ
cana-536	1	39	studies	study	NOUN
cana-536	1	40	chennai	chennai	PROPN
cana-536	1	41	,	,	PUNCT
cana-536	1	42	india	india	PROPN
cana-536	1	43	muthumed77@gmail.com	muthumed77@gmail.com	PROPN
cana-536	1	44	,	,	PUNCT
cana-536	1	45	gkrajendra59@gmail.com	gkrajendra59@gmail.com	X
cana-536	1	46	article	article	NOUN
cana-536	1	47	history	history	NOUN
cana-536	1	48	:	:	PUNCT
cana-536	1	49	received	receive	VERB
cana-536	1	50	:	:	PUNCT
cana-536	1	51	28	28	NUM
cana-536	1	52	-	-	SYM
cana-536	1	53	01	01	NUM
cana-536	1	54	-	-	PUNCT
cana-536	1	55	2024	2024	NUM
cana-536	1	56	revised	revise	VERB
cana-536	1	57	:	:	PUNCT
cana-536	1	58	16	16	NUM
cana-536	1	59	-	-	PUNCT
cana-536	1	60	04	04	NUM
cana-536	1	61	-	-	PUNCT
cana-536	1	62	2024	2024	NUM
cana-536	1	63	accepted	accept	VERB
cana-536	1	64	:	:	PUNCT
cana-536	1	65	30	30	NUM
cana-536	1	66	-	-	PUNCT
cana-536	1	67	04	04	NUM
cana-536	1	68	-	-	PUNCT
cana-536	1	69	2024	2024	NUM
cana-536	1	70	abstract	abstract	NOUN
cana-536	1	71	:	:	PUNCT
cana-536	1	72	for	for	ADP
cana-536	1	73	a	a	DET
cana-536	1	74	simple	simple	ADJ
cana-536	1	75	graph	graph	NOUN
cana-536	1	76	g	g	NOUN
cana-536	1	77	,	,	PUNCT
cana-536	1	78	a	a	DET
cana-536	1	79	vertex	vertex	NOUN
cana-536	1	80	labeling	labeling	NOUN
cana-536	2	1	φ	φ	NOUN
cana-536	2	2	:	:	PUNCT
cana-536	2	3	v	v	NOUN
cana-536	2	4	(	(	PUNCT
cana-536	2	5	g	g	NOUN
cana-536	2	6	)	)	PUNCT
cana-536	2	7	→	→	SYM
cana-536	2	8	{	{	PUNCT
cana-536	2	9	1	1	NUM
cana-536	2	10	,	,	PUNCT
cana-536	2	11	2	2	NUM
cana-536	2	12	,	,	PUNCT
cana-536	2	13	·	·	PUNCT
cana-536	2	14	·	·	PUNCT
cana-536	2	15	·	·	PUNCT
cana-536	2	16	,	,	PUNCT
cana-536	2	17	k	k	NOUN
cana-536	2	18	}	}	PUNCT
cana-536	2	19	is	be	AUX
cana-536	2	20	called	call	VERB
cana-536	2	21	klabeling	klabeling	NOUN
cana-536	2	22	.	.	PUNCT
cana-536	3	1	the	the	DET
cana-536	3	2	weight	weight	NOUN
cana-536	3	3	of	of	ADP
cana-536	3	4	an	an	DET
cana-536	3	5	edge	edge	NOUN
cana-536	3	6	uv	uv	NOUN
cana-536	3	7	in	in	ADP
cana-536	3	8	g	g	NOUN
cana-536	3	9	,	,	PUNCT
cana-536	3	10	denoted	denote	VERB
cana-536	3	11	by	by	ADP
cana-536	3	12	wφ(uv	wφ(uv	NOUN
cana-536	3	13	)	)	PUNCT
cana-536	3	14	,	,	PUNCT
cana-536	3	15	is	be	AUX
cana-536	3	16	the	the	DET
cana-536	3	17	sum	sum	NOUN
cana-536	3	18	of	of	ADP
cana-536	3	19	the	the	DET
cana-536	3	20	labels	label	NOUN
cana-536	3	21	of	of	ADP
cana-536	3	22	end	end	NOUN
cana-536	3	23	vertices	vertice	VERB
cana-536	3	24	u	u	NOUN
cana-536	3	25	and	and	CCONJ
cana-536	3	26	v.	v.	ADP
cana-536	3	27	a	a	DET
cana-536	3	28	vertex	vertex	NOUN
cana-536	3	29	k	k	NOUN
cana-536	3	30	-	-	NOUN
cana-536	3	31	labeling	labeling	NOUN
cana-536	3	32	is	be	AUX
cana-536	3	33	defined	define	VERB
cana-536	3	34	to	to	PART
cana-536	3	35	be	be	AUX
cana-536	3	36	an	an	DET
cana-536	3	37	edge	edge	NOUN
cana-536	3	38	irregular	irregular	ADJ
cana-536	3	39	klabeling	klabeling	NOUN
cana-536	3	40	of	of	ADP
cana-536	3	41	the	the	DET
cana-536	3	42	graph	graph	NOUN
cana-536	3	43	g	g	NOUN
cana-536	3	44	if	if	SCONJ
cana-536	3	45	for	for	ADP
cana-536	3	46	every	every	DET
cana-536	3	47	two	two	NUM
cana-536	3	48	different	different	ADJ
cana-536	3	49	edges	edge	NOUN
cana-536	3	50	e	e	NOUN
cana-536	3	51	and	and	CCONJ
cana-536	3	52	f	f	PROPN
cana-536	3	53	,	,	PUNCT
cana-536	3	54	wφ(e	wφ(e	PUNCT
cana-536	3	55	)	)	PUNCT
cana-536	3	56	≠	≠	PROPN
cana-536	3	57	wφ(f	wφ(f	NUM
cana-536	3	58	)	)	PUNCT
cana-536	3	59	.	.	PUNCT
cana-536	4	1	the	the	DET
cana-536	4	2	minimum	minimum	PROPN
cana-536	4	3	k	k	PROPN
cana-536	4	4	for	for	ADP
cana-536	4	5	which	which	PRON
cana-536	4	6	the	the	DET
cana-536	4	7	graph	graph	NOUN
cana-536	4	8	g	g	PROPN
cana-536	4	9	has	have	VERB
cana-536	4	10	an	an	DET
cana-536	4	11	edge	edge	NOUN
cana-536	4	12	irregular	irregular	ADJ
cana-536	4	13	k	k	NOUN
cana-536	4	14	-	-	ADJ
cana-536	4	15	labeling	labeling	NOUN
cana-536	4	16	is	be	AUX
cana-536	4	17	called	call	VERB
cana-536	4	18	the	the	DET
cana-536	4	19	edge	edge	NOUN
cana-536	4	20	irregularity	irregularity	NOUN
cana-536	4	21	strength	strength	NOUN
cana-536	4	22	of	of	ADP
cana-536	4	23	g	g	NOUN
cana-536	4	24	,	,	PUNCT
cana-536	4	25	denoted	denote	VERB
cana-536	4	26	by	by	ADP
cana-536	4	27	es(g	es(g	NOUN
cana-536	4	28	)	)	PUNCT
cana-536	4	29	.	.	PUNCT
cana-536	5	1	in	in	ADP
cana-536	5	2	this	this	DET
cana-536	5	3	paper	paper	NOUN
cana-536	5	4	,	,	PUNCT
cana-536	5	5	we	we	PRON
cana-536	5	6	prove	prove	VERB
cana-536	5	7	that	that	SCONJ
cana-536	5	8	the	the	DET
cana-536	5	9	edge	edge	NOUN
cana-536	5	10	irregularity	irregularity	NOUN
cana-536	5	11	strength	strength	NOUN
cana-536	5	12	of	of	ADP
cana-536	5	13	corona	corona	NOUN
cana-536	5	14	product	product	NOUN
cana-536	5	15	of	of	ADP
cana-536	5	16	a	a	DET
cana-536	5	17	tree	tree	NOUN
cana-536	5	18	t	t	NOUN
cana-536	5	19	with	with	ADP
cana-536	5	20	k1	k1	PROPN
cana-536	5	21	is	be	AUX
cana-536	5	22	es(t	es(t	X
cana-536	5	23	∘	∘	PROPN
cana-536	5	24	k1	k1	NOUN
cana-536	5	25	)	)	PUNCT
cana-536	5	26	=	=	SYM
cana-536	5	27	2es(t	2es(t	NUM
cana-536	5	28	)	)	PUNCT
cana-536	5	29	.	.	PUNCT
cana-536	6	1	further	far	ADV
cana-536	6	2	,	,	PUNCT
cana-536	6	3	we	we	PRON
cana-536	6	4	prove	prove	VERB
cana-536	6	5	that	that	SCONJ
cana-536	6	6	the	the	DET
cana-536	6	7	edge	edge	NOUN
cana-536	6	8	irregularity	irregularity	NOUN
cana-536	6	9	strength	strength	NOUN
cana-536	6	10	of	of	ADP
cana-536	6	11	binomial	binomial	ADJ
cana-536	6	12	trees	tree	NOUN
cana-536	6	13	bk	bk	PRON
cana-536	6	14	is	be	AUX
cana-536	6	15	2k−1	2k−1	NUM
cana-536	6	16	,	,	PUNCT
cana-536	6	17	for	for	ADP
cana-536	6	18	k	k	PROPN
cana-536	6	19	≥	≥	PROPN
cana-536	6	20	1	1	NUM
cana-536	6	21	.	.	PUNCT
cana-536	7	1	keywords	keyword	NOUN
cana-536	7	2	:	:	PUNCT
cana-536	7	3	edge	edge	VERB
cana-536	7	4	irregularity	irregularity	NOUN
cana-536	7	5	strength	strength	NOUN
cana-536	7	6	;	;	PUNCT
cana-536	7	7	corona	corona	NOUN
cana-536	7	8	product	product	NOUN
cana-536	7	9	of	of	ADP
cana-536	7	10	graphs	graph	NOUN
cana-536	7	11	;	;	PUNCT
cana-536	7	12	binomial	binomial	ADJ
cana-536	7	13	trees	tree	NOUN
cana-536	7	14	;	;	PUNCT
cana-536	7	15	1	1	X
cana-536	7	16	.	.	X
cana-536	7	17	introduction	introduction	NOUN
cana-536	7	18	all	all	DET
cana-536	7	19	the	the	DET
cana-536	7	20	graphs	graph	NOUN
cana-536	7	21	considered	consider	VERB
cana-536	7	22	in	in	ADP
cana-536	7	23	this	this	DET
cana-536	7	24	paper	paper	NOUN
cana-536	7	25	are	be	AUX
cana-536	7	26	finite	finite	ADJ
cana-536	7	27	simple	simple	ADJ
cana-536	7	28	graphs	graph	NOUN
cana-536	7	29	.	.	PUNCT
cana-536	8	1	terms	term	NOUN
cana-536	8	2	that	that	PRON
cana-536	8	3	are	be	AUX
cana-536	8	4	not	not	PART
cana-536	8	5	defined	define	VERB
cana-536	8	6	here	here	ADV
cana-536	8	7	can	can	AUX
cana-536	8	8	be	be	AUX
cana-536	8	9	referred	refer	VERB
cana-536	8	10	from	from	ADP
cana-536	8	11	the	the	DET
cana-536	8	12	book	book	NOUN
cana-536	8	13	[	[	X
cana-536	8	14	10	10	NUM
cana-536	8	15	]	]	PUNCT
cana-536	8	16	.	.	PUNCT
cana-536	9	1	for	for	ADP
cana-536	9	2	a	a	DET
cana-536	9	3	simple	simple	ADJ
cana-536	9	4	graph	graph	NOUN
cana-536	9	5	g	g	NOUN
cana-536	9	6	,	,	PUNCT
cana-536	9	7	a	a	DET
cana-536	9	8	vertex	vertex	NOUN
cana-536	9	9	labeling	labeling	NOUN
cana-536	9	10	φ	φ	NOUN
cana-536	9	11	:	:	PUNCT
cana-536	9	12	v	v	NOUN
cana-536	9	13	(	(	PUNCT
cana-536	9	14	g	g	NOUN
cana-536	9	15	)	)	PUNCT
cana-536	9	16	→	→	SYM
cana-536	9	17	{	{	PUNCT
cana-536	9	18	1	1	NUM
cana-536	9	19	,	,	PUNCT
cana-536	9	20	2	2	NUM
cana-536	9	21	,	,	PUNCT
cana-536	9	22	·	·	PUNCT
cana-536	9	23	·	·	PUNCT
cana-536	9	24	·	·	PUNCT
cana-536	9	25	,	,	PUNCT
cana-536	9	26	k	k	NOUN
cana-536	9	27	}	}	PUNCT
cana-536	9	28	is	be	AUX
cana-536	9	29	called	call	VERB
cana-536	9	30	k	k	NOUN
cana-536	9	31	-	-	NOUN
cana-536	9	32	labeling	labeling	NOUN
cana-536	9	33	.	.	PUNCT
cana-536	10	1	the	the	DET
cana-536	10	2	weight	weight	NOUN
cana-536	10	3	of	of	ADP
cana-536	10	4	an	an	DET
cana-536	10	5	edge	edge	NOUN
cana-536	10	6	uv	uv	NOUN
cana-536	10	7	in	in	ADP
cana-536	10	8	g	g	NOUN
cana-536	10	9	,	,	PUNCT
cana-536	10	10	denoted	denote	VERB
cana-536	10	11	by	by	ADP
cana-536	10	12	wφ(uv	wφ(uv	NOUN
cana-536	10	13	)	)	PUNCT
cana-536	10	14	,	,	PUNCT
cana-536	10	15	is	be	AUX
cana-536	10	16	the	the	DET
cana-536	10	17	sum	sum	NOUN
cana-536	10	18	of	of	ADP
cana-536	10	19	the	the	DET
cana-536	10	20	labels	label	NOUN
cana-536	10	21	of	of	ADP
cana-536	10	22	end	end	NOUN
cana-536	10	23	vertices	vertice	VERB
cana-536	10	24	u	u	NOUN
cana-536	10	25	and	and	CCONJ
cana-536	10	26	v.	v.	ADP
cana-536	10	27	a	a	DET
cana-536	10	28	vertex	vertex	NOUN
cana-536	10	29	k	k	NOUN
cana-536	10	30	-	-	NOUN
cana-536	10	31	labeling	labeling	NOUN
cana-536	10	32	is	be	AUX
cana-536	10	33	defined	define	VERB
cana-536	10	34	to	to	PART
cana-536	10	35	be	be	AUX
cana-536	10	36	an	an	DET
cana-536	10	37	edge	edge	NOUN
cana-536	10	38	irregular	irregular	ADJ
cana-536	10	39	klabeling	klabeling	NOUN
cana-536	10	40	of	of	ADP
cana-536	10	41	the	the	DET
cana-536	10	42	graph	graph	NOUN
cana-536	10	43	g	g	NOUN
cana-536	10	44	if	if	SCONJ
cana-536	10	45	for	for	ADP
cana-536	10	46	every	every	DET
cana-536	10	47	two	two	NUM
cana-536	10	48	different	different	ADJ
cana-536	10	49	edges	edge	NOUN
cana-536	10	50	e	e	NOUN
cana-536	10	51	and	and	CCONJ
cana-536	10	52	f	f	PROPN
cana-536	10	53	,	,	PUNCT
cana-536	10	54	wφ(e	wφ(e	PUNCT
cana-536	10	55	)	)	PUNCT
cana-536	10	56	≠	≠	PROPN
cana-536	10	57	wφ	wφ	X
cana-536	10	58	(	(	PUNCT
cana-536	10	59	f	f	NOUN
cana-536	10	60	)	)	PUNCT
cana-536	10	61	.	.	PUNCT
cana-536	11	1	the	the	DET
cana-536	11	2	minimum	minimum	PROPN
cana-536	11	3	k	k	PROPN
cana-536	11	4	for	for	ADP
cana-536	11	5	which	which	PRON
cana-536	11	6	the	the	DET
cana-536	11	7	graph	graph	NOUN
cana-536	11	8	g	g	PROPN
cana-536	11	9	has	have	VERB
cana-536	11	10	an	an	DET
cana-536	11	11	edge	edge	NOUN
cana-536	11	12	irregular	irregular	ADJ
cana-536	11	13	k	k	NOUN
cana-536	11	14	-	-	ADJ
cana-536	11	15	labeling	labeling	NOUN
cana-536	11	16	is	be	AUX
cana-536	11	17	called	call	VERB
cana-536	11	18	the	the	DET
cana-536	11	19	edge	edge	NOUN
cana-536	11	20	irregularity	irregularity	NOUN
cana-536	11	21	strength	strength	NOUN
cana-536	11	22	of	of	ADP
cana-536	11	23	g	g	NOUN
cana-536	11	24	,	,	PUNCT
cana-536	11	25	denoted	denote	VERB
cana-536	11	26	by	by	ADP
cana-536	11	27	es(g	es(g	NOUN
cana-536	11	28	)	)	PUNCT
cana-536	11	29	.	.	PUNCT
cana-536	12	1	in	in	ADP
cana-536	12	2	1988	1988	NUM
cana-536	12	3	,	,	PUNCT
cana-536	12	4	chartrand	chartrand	NOUN
cana-536	12	5	et	et	PROPN
cana-536	12	6	al	al	PROPN
cana-536	12	7	.	.	PUNCT
cana-536	13	1	[	[	X
cana-536	13	2	4	4	NUM
cana-536	13	3	]	]	PUNCT
cana-536	13	4	introduced	introduce	VERB
cana-536	13	5	edge	edge	NOUN
cana-536	13	6	k	k	NOUN
cana-536	13	7	-	-	ADJ
cana-536	13	8	labeling	labeling	ADJ
cana-536	13	9	δ	δ	NOUN
cana-536	13	10	of	of	ADP
cana-536	13	11	a	a	DET
cana-536	13	12	graph	graph	NOUN
cana-536	13	13	g	g	ADP
cana-536	13	14	such	such	ADJ
cana-536	13	15	that	that	PRON
cana-536	13	16	wδ(x	wδ(x	X
cana-536	13	17	)	)	PUNCT
cana-536	13	18	≠	≠	NOUN
cana-536	13	19	wδ(y	wδ(y	ADV
cana-536	13	20	)	)	PUNCT
cana-536	13	21	for	for	ADP
cana-536	13	22	all	all	DET
cana-536	13	23	vertices	vertex	NOUN
cana-536	13	24	x	x	X
cana-536	13	25	,	,	PUNCT
cana-536	13	26	y	y	PROPN
cana-536	13	27	∈	∈	PROPN
cana-536	13	28	v	v	ADP
cana-536	13	29	(	(	PUNCT
cana-536	13	30	g	g	NOUN
cana-536	13	31	)	)	PUNCT
cana-536	13	32	with	with	ADP
cana-536	13	33	x≠y	x≠y	NOUN
cana-536	13	34	.	.	PUNCT
cana-536	14	1	such	such	ADJ
cana-536	14	2	labelings	labeling	NOUN
cana-536	14	3	were	be	AUX
cana-536	14	4	called	call	VERB
cana-536	14	5	irregular	irregular	ADJ
cana-536	14	6	assignments	assignment	NOUN
cana-536	14	7	and	and	CCONJ
cana-536	14	8	the	the	DET
cana-536	14	9	irregularity	irregularity	NOUN
cana-536	14	10	strength	strength	NOUN
cana-536	14	11	s(g	s(g	PROPN
cana-536	14	12	)	)	PUNCT
cana-536	14	13	of	of	ADP
cana-536	14	14	a	a	DET
cana-536	14	15	graph	graph	NOUN
cana-536	14	16	g	g	NOUN
cana-536	14	17	is	be	AUX
cana-536	14	18	known	know	VERB
cana-536	14	19	as	as	ADP
cana-536	14	20	the	the	DET
cana-536	14	21	minimum	minimum	NOUN
cana-536	14	22	k	k	NOUN
cana-536	14	23	for	for	ADP
cana-536	14	24	which	which	PRON
cana-536	14	25	g	g	NOUN
cana-536	14	26	has	have	VERB
cana-536	14	27	an	an	DET
cana-536	14	28	irregular	irregular	ADJ
cana-536	14	29	assignment	assignment	NOUN
cana-536	14	30	using	use	VERB
cana-536	14	31	labels	label	NOUN
cana-536	14	32	at	at	ADP
cana-536	14	33	most	most	ADJ
cana-536	14	34	k.	k.	PROPN
cana-536	15	1	this	this	DET
cana-536	15	2	parameter	parameter	NOUN
cana-536	15	3	has	have	AUX
cana-536	15	4	attracted	attract	VERB
cana-536	15	5	many	many	ADJ
cana-536	15	6	researchers	researcher	NOUN
cana-536	15	7	and	and	CCONJ
cana-536	15	8	several	several	ADJ
cana-536	15	9	articles	article	NOUN
cana-536	15	10	[	[	X
cana-536	15	11	2	2	NUM
cana-536	15	12	,	,	PUNCT
cana-536	15	13	3	3	NUM
cana-536	15	14	,	,	PUNCT
cana-536	15	15	6	6	NUM
cana-536	15	16	,	,	PUNCT
cana-536	15	17	9	9	NUM
cana-536	15	18	]	]	PUNCT
cana-536	15	19	were	be	AUX
cana-536	15	20	published	publish	VERB
cana-536	15	21	based	base	VERB
cana-536	15	22	on	on	ADP
cana-536	15	23	irregularity	irregularity	NOUN
cana-536	15	24	strength	strength	NOUN
cana-536	15	25	of	of	ADP
cana-536	15	26	graphs	graph	NOUN
cana-536	15	27	.	.	PUNCT
cana-536	16	1	in	in	ADP
cana-536	16	2	2014	2014	NUM
cana-536	16	3	,	,	PUNCT
cana-536	16	4	ali	ali	PROPN
cana-536	16	5	ahmad	ahmad	PROPN
cana-536	16	6	et	et	PROPN
cana-536	16	7	al	al	PROPN
cana-536	16	8	.	.	PUNCT
cana-536	17	1	[	[	X
cana-536	17	2	1	1	X
cana-536	17	3	]	]	PUNCT
cana-536	17	4	introduced	introduce	VERB
cana-536	17	5	a	a	DET
cana-536	17	6	new	new	ADJ
cana-536	17	7	parameter	parameter	NOUN
cana-536	17	8	called	call	VERB
cana-536	17	9	edge	edge	NOUN
cana-536	17	10	irregularity	irregularity	NOUN
cana-536	17	11	strength	strength	NOUN
cana-536	17	12	of	of	ADP
cana-536	17	13	graphs	graph	NOUN
cana-536	17	14	.	.	PUNCT
cana-536	18	1	a	a	DET
cana-536	18	2	vertex	vertex	NOUN
cana-536	18	3	k	k	NOUN
cana-536	18	4	-	-	NOUN
cana-536	18	5	labeling	labeling	ADJ
cana-536	18	6	φ	φ	NOUN
cana-536	18	7	:	:	PUNCT
cana-536	18	8	v	v	NOUN
cana-536	18	9	(	(	PUNCT
cana-536	18	10	g	g	NOUN
cana-536	18	11	)	)	PUNCT
cana-536	18	12	→	→	SYM
cana-536	18	13	{	{	PUNCT
cana-536	18	14	1	1	NUM
cana-536	18	15	,	,	PUNCT
cana-536	18	16	2	2	NUM
cana-536	18	17	,	,	PUNCT
cana-536	18	18	·	·	PUNCT
cana-536	18	19	·	·	PUNCT
cana-536	18	20	·	·	PUNCT
cana-536	18	21	,	,	PUNCT
cana-536	18	22	k	k	NOUN
cana-536	18	23	}	}	PUNCT
cana-536	18	24	is	be	AUX
cana-536	18	25	called	call	VERB
cana-536	18	26	an	an	DET
cana-536	18	27	edge	edge	NOUN
cana-536	18	28	irregular	irregular	ADJ
cana-536	18	29	klabeling	klabeling	NOUN
cana-536	18	30	of	of	ADP
cana-536	18	31	the	the	DET
cana-536	18	32	graph	graph	NOUN
cana-536	18	33	if	if	SCONJ
cana-536	18	34	for	for	ADP
cana-536	18	35	every	every	DET
cana-536	18	36	two	two	NUM
cana-536	18	37	different	different	ADJ
cana-536	18	38	edges	edge	NOUN
cana-536	18	39	e	e	NOUN
cana-536	18	40	and	and	CCONJ
cana-536	18	41	f	f	NOUN
cana-536	18	42	there	there	PRON
cana-536	18	43	is	be	VERB
cana-536	18	44	wφ(e	wφ(e	PUNCT
cana-536	18	45	)	)	PUNCT
cana-536	18	46	≠	≠	PROPN
cana-536	18	47	wφ	wφ	X
cana-536	18	48	(	(	PUNCT
cana-536	18	49	f	f	X
cana-536	18	50	)	)	PUNCT
cana-536	18	51	,	,	PUNCT
cana-536	18	52	where	where	SCONJ
cana-536	18	53	the	the	DET
cana-536	18	54	weight	weight	NOUN
cana-536	18	55	of	of	ADP
cana-536	18	56	an	an	DET
cana-536	18	57	edge	edge	NOUN
cana-536	18	58	e	e	NOUN
cana-536	18	59	=	=	PUNCT
cana-536	18	60	xy	xy	PROPN
cana-536	18	61	∈	∈	PROPN
cana-536	18	62	e(g	e(g	PROPN
cana-536	18	63	)	)	PUNCT
cana-536	18	64	is	be	AUX
cana-536	18	65	wφ(xy	wφ(xy	VERB
cana-536	18	66	)	)	PUNCT
cana-536	18	67	=	=	SYM
cana-536	18	68	φ(x	φ(x	NOUN
cana-536	18	69	)	)	PUNCT
cana-536	18	70	+	+	NUM
cana-536	18	71	φ(y	φ(y	NOUN
cana-536	18	72	)	)	PUNCT
cana-536	18	73	.	.	PUNCT
cana-536	19	1	the	the	DET
cana-536	19	2	minimum	minimum	PROPN
cana-536	19	3	k	k	PROPN
cana-536	19	4	for	for	ADP
cana-536	19	5	which	which	PRON
cana-536	19	6	the	the	DET
cana-536	19	7	graph	graph	NOUN
cana-536	19	8	g	g	PROPN
cana-536	19	9	has	have	VERB
cana-536	19	10	an	an	DET
cana-536	19	11	edge	edge	NOUN
cana-536	19	12	irregular	irregular	ADJ
cana-536	19	13	k	k	NOUN
cana-536	19	14	-	-	ADJ
cana-536	19	15	labeling	labeling	NOUN
cana-536	19	16	is	be	AUX
cana-536	19	17	called	call	VERB
cana-536	19	18	the	the	DET
cana-536	19	19	edge	edge	NOUN
cana-536	19	20	irregularity	irregularity	NOUN
cana-536	19	21	strength	strength	NOUN
cana-536	19	22	of	of	ADP
cana-536	19	23	g	g	NOUN
cana-536	19	24	,	,	PUNCT
cana-536	19	25	denoted	denote	VERB
cana-536	19	26	by	by	ADP
cana-536	19	27	es(g	es(g	NOUN
cana-536	19	28	)	)	PUNCT
cana-536	19	29	.	.	PUNCT
cana-536	20	1	for	for	ADP
cana-536	20	2	an	an	DET
cana-536	20	3	exhaustive	exhaustive	ADJ
cana-536	20	4	survey	survey	NOUN
cana-536	20	5	on	on	ADP
cana-536	20	6	edge	edge	NOUN
cana-536	20	7	irregularity	irregularity	NOUN
cana-536	20	8	strength	strength	NOUN
cana-536	20	9	of	of	ADP
cana-536	20	10	graphs	graph	NOUN
cana-536	20	11	,	,	PUNCT
cana-536	20	12	we	we	PRON
cana-536	20	13	refer	refer	VERB
cana-536	20	14	to	to	ADP
cana-536	20	15	the	the	DET
cana-536	20	16	dynamic	dynamic	ADJ
cana-536	20	17	survey	survey	NOUN
cana-536	20	18	on	on	ADP
cana-536	20	19	graph	graph	NOUN
cana-536	20	20	labeling	labeling	NOUN
cana-536	20	21	by	by	ADP
cana-536	20	22	gallian	gallian	ADJ
cana-536	21	1	[	[	X
cana-536	21	2	7	7	NUM
cana-536	21	3	]	]	PUNCT
cana-536	21	4	.	.	PUNCT
cana-536	22	1	communications	communication	NOUN
cana-536	22	2	on	on	ADP
cana-536	22	3	applied	apply	VERB
cana-536	22	4	nonlinear	nonlinear	ADJ
cana-536	22	5	analysis	analysis	NOUN
cana-536	22	6	issn	issn	NOUN
cana-536	22	7	:	:	PUNCT
cana-536	22	8	1074	1074	NUM
cana-536	22	9	-	-	PUNCT
cana-536	22	10	133x	133x	NUM
cana-536	22	11	vol	vol	NOUN
cana-536	22	12	31	31	NUM
cana-536	22	13	no	no	NOUN
cana-536	22	14	.	.	NOUN
cana-536	22	15	2	2	NUM
cana-536	22	16	(	(	PUNCT
cana-536	22	17	2024	2024	NUM
cana-536	22	18	)	)	PUNCT
cana-536	22	19	215	215	NUM
cana-536	22	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-536	22	21	in	in	ADP
cana-536	22	22	[	[	X
cana-536	22	23	1	1	NUM
cana-536	22	24	]	]	PUNCT
cana-536	22	25	,	,	PUNCT
cana-536	22	26	ali	ali	PROPN
cana-536	22	27	ahmad	ahmad	PROPN
cana-536	22	28	et	et	PROPN
cana-536	22	29	al	al	PROPN
cana-536	22	30	.	.	PROPN
cana-536	22	31	proved	prove	VERB
cana-536	22	32	that	that	SCONJ
cana-536	22	33	the	the	PRON
cana-536	22	34	lower	lower	ADV
cana-536	22	35	bound	bind	VERB
cana-536	22	36	for	for	ADP
cana-536	22	37	edge	edge	NOUN
cana-536	22	38	irregularity	irregularity	NOUN
cana-536	22	39	strength	strength	NOUN
cana-536	22	40	of	of	ADP
cana-536	22	41	a	a	DET
cana-536	22	42	graph	graph	NOUN
cana-536	22	43	g	g	NOUN
cana-536	22	44	is	be	AUX
cana-536	22	45	given	give	VERB
cana-536	22	46	by	by	ADP
cana-536	22	47	es(g	es(g	NOUN
cana-536	22	48	)	)	PUNCT
cana-536	22	49	≥	≥	PROPN
cana-536	22	50	max	max	PROPN
cana-536	22	51	{	{	PUNCT
cana-536	22	52	⌈	⌈	PROPN
cana-536	22	53	|𝐸(𝐺)|+1	|𝐸(𝐺)|+1	PROPN
cana-536	22	54	2	2	NUM
cana-536	22	55	⌉	⌉	NOUN
cana-536	22	56	,	,	PUNCT
cana-536	22	57	∆(𝐺	∆(𝐺	PROPN
cana-536	22	58	)	)	PUNCT
cana-536	22	59	}	}	PUNCT
cana-536	22	60	.	.	PUNCT
cana-536	23	1	in	in	ADP
cana-536	23	2	this	this	DET
cana-536	23	3	paper	paper	NOUN
cana-536	23	4	,	,	PUNCT
cana-536	23	5	we	we	PRON
cana-536	23	6	prove	prove	VERB
cana-536	23	7	that	that	SCONJ
cana-536	23	8	the	the	DET
cana-536	23	9	edge	edge	NOUN
cana-536	23	10	irregularity	irregularity	NOUN
cana-536	23	11	strength	strength	NOUN
cana-536	23	12	of	of	ADP
cana-536	23	13	corona	corona	NOUN
cana-536	23	14	product	product	NOUN
cana-536	23	15	of	of	ADP
cana-536	23	16	a	a	DET
cana-536	23	17	tree	tree	NOUN
cana-536	23	18	t	t	NOUN
cana-536	23	19	with	with	ADP
cana-536	23	20	k1	k1	PROPN
cana-536	23	21	is	be	AUX
cana-536	23	22	es	es	X
cana-536	23	23	(	(	PUNCT
cana-536	23	24	t	t	NOUN
cana-536	23	25	◦	◦	NOUN
cana-536	23	26	k1	k1	NOUN
cana-536	23	27	)	)	PUNCT
cana-536	24	1	=	=	SYM
cana-536	24	2	2es	2es	NOUN
cana-536	24	3	(	(	PUNCT
cana-536	24	4	t	t	PROPN
cana-536	24	5	)	)	PUNCT
cana-536	24	6	.	.	PUNCT
cana-536	25	1	further	far	ADV
cana-536	25	2	,	,	PUNCT
cana-536	25	3	we	we	PRON
cana-536	25	4	prove	prove	VERB
cana-536	25	5	that	that	SCONJ
cana-536	25	6	the	the	DET
cana-536	25	7	edge	edge	NOUN
cana-536	25	8	irregularity	irregularity	NOUN
cana-536	25	9	strength	strength	NOUN
cana-536	25	10	of	of	ADP
cana-536	25	11	binomial	binomial	ADJ
cana-536	25	12	trees	tree	NOUN
cana-536	25	13	bk	bk	PRON
cana-536	25	14	is	be	AUX
cana-536	25	15	2k−1	2k−1	NUM
cana-536	25	16	,	,	PUNCT
cana-536	25	17	for	for	ADP
cana-536	25	18	k	k	PROPN
cana-536	25	19	≥	≥	PROPN
cana-536	25	20	1	1	NUM
cana-536	25	21	.	.	PUNCT
cana-536	26	1	for	for	SCONJ
cana-536	26	2	all	all	PRON
cana-536	26	3	of	of	ADP
cana-536	26	4	our	our	PRON
cana-536	26	5	results	result	NOUN
cana-536	26	6	proved	prove	VERB
cana-536	26	7	in	in	ADP
cana-536	26	8	this	this	DET
cana-536	26	9	paper	paper	NOUN
cana-536	26	10	,	,	PUNCT
cana-536	26	11	the	the	DET
cana-536	26	12	edge	edge	NOUN
cana-536	26	13	irregularity	irregularity	NOUN
cana-536	26	14	strength	strength	NOUN
cana-536	26	15	of	of	ADP
cana-536	26	16	trees	tree	NOUN
cana-536	26	17	we	we	PRON
cana-536	26	18	proved	prove	VERB
cana-536	26	19	attains	attain	VERB
cana-536	26	20	its	its	PRON
cana-536	26	21	lower	low	ADJ
cana-536	26	22	bound	bind	VERB
cana-536	26	23	.	.	PUNCT
cana-536	27	1	2	2	NUM
cana-536	27	2	corona	corona	NOUN
cana-536	27	3	product	product	NOUN
cana-536	27	4	of	of	ADP
cana-536	27	5	graphs	graph	NOUN
cana-536	27	6	let	let	VERB
cana-536	27	7	g	g	NOUN
cana-536	27	8	and	and	CCONJ
cana-536	27	9	h	h	NOUN
cana-536	27	10	be	be	VERB
cana-536	27	11	two	two	NUM
cana-536	27	12	graphs	graph	NOUN
cana-536	27	13	and	and	CCONJ
cana-536	27	14	let	let	VERB
cana-536	27	15	n	n	PRON
cana-536	27	16	be	be	AUX
cana-536	27	17	the	the	DET
cana-536	27	18	order	order	NOUN
cana-536	27	19	of	of	ADP
cana-536	27	20	g.	g.	PROPN
cana-536	27	21	the	the	DET
cana-536	27	22	corona	corona	NOUN
cana-536	27	23	product	product	NOUN
cana-536	27	24	,	,	PUNCT
cana-536	27	25	or	or	CCONJ
cana-536	27	26	simply	simply	ADV
cana-536	27	27	the	the	DET
cana-536	27	28	corona	corona	NOUN
cana-536	27	29	,	,	PUNCT
cana-536	27	30	of	of	ADP
cana-536	27	31	graphs	graph	NOUN
cana-536	27	32	g	g	NOUN
cana-536	27	33	and	and	CCONJ
cana-536	27	34	h	h	NOUN
cana-536	27	35	is	be	AUX
cana-536	27	36	the	the	DET
cana-536	27	37	graph	graph	NOUN
cana-536	27	38	g	g	PROPN
cana-536	27	39	⊙	⊙	PROPN
cana-536	27	40	h	h	PROPN
cana-536	27	41	obtained	obtain	VERB
cana-536	27	42	by	by	ADP
cana-536	27	43	taking	take	VERB
cana-536	27	44	one	one	NUM
cana-536	27	45	copy	copy	NOUN
cana-536	27	46	of	of	ADP
cana-536	27	47	g	g	PROPN
cana-536	27	48	and	and	CCONJ
cana-536	27	49	n	n	PROPN
cana-536	27	50	copies	copy	NOUN
cana-536	27	51	of	of	ADP
cana-536	27	52	h	h	NOUN
cana-536	27	53	and	and	CCONJ
cana-536	27	54	then	then	ADV
cana-536	27	55	joining	join	VERB
cana-536	27	56	by	by	ADP
cana-536	27	57	an	an	DET
cana-536	27	58	edge	edge	NOUN
cana-536	27	59	the	the	DET
cana-536	27	60	ith	ith	ADJ
cana-536	27	61	vertex	vertex	NOUN
cana-536	27	62	of	of	ADP
cana-536	27	63	g	g	NOUN
cana-536	27	64	to	to	ADP
cana-536	27	65	every	every	DET
cana-536	27	66	vertex	vertex	NOUN
cana-536	27	67	in	in	ADP
cana-536	27	68	the	the	DET
cana-536	27	69	ith	ith	PROPN
cana-536	27	70	copy	copy	NOUN
cana-536	27	71	of	of	ADP
cana-536	27	72	h.	h.	PROPN
cana-536	27	73	given	give	VERB
cana-536	27	74	a	a	DET
cana-536	27	75	vertex	vertex	NOUN
cana-536	27	76	g	g	PROPN
cana-536	27	77	∈	∈	PROPN
cana-536	27	78	g	g	PROPN
cana-536	27	79	,	,	PUNCT
cana-536	27	80	the	the	DET
cana-536	27	81	copy	copy	NOUN
cana-536	27	82	of	of	ADP
cana-536	27	83	h	h	NOUN
cana-536	27	84	connected	connect	VERB
cana-536	27	85	to	to	ADP
cana-536	27	86	g	g	PROPN
cana-536	27	87	is	be	AUX
cana-536	27	88	denoted	denote	VERB
cana-536	27	89	by	by	ADP
cana-536	27	90	hg	hg	NOUN
cana-536	28	1	[	[	X
cana-536	28	2	8	8	NUM
cana-536	28	3	]	]	PUNCT
cana-536	28	4	.	.	PUNCT
cana-536	29	1	complete	complete	ADJ
cana-536	29	2	graphs	graph	NOUN
cana-536	29	3	,	,	PUNCT
cana-536	29	4	stars	star	NOUN
cana-536	29	5	and	and	CCONJ
cana-536	29	6	wheels	wheel	NOUN
cana-536	29	7	are	be	AUX
cana-536	29	8	basic	basic	ADJ
cana-536	29	9	examples	example	NOUN
cana-536	29	10	of	of	ADP
cana-536	29	11	corona	corona	NOUN
cana-536	29	12	product	product	NOUN
cana-536	29	13	families	family	NOUN
cana-536	29	14	.	.	PUNCT
cana-536	30	1	observation	observation	NOUN
cana-536	30	2	1	1	NUM
cana-536	30	3	:	:	PUNCT
cana-536	30	4	when	when	SCONJ
cana-536	30	5	g	g	PROPN
cana-536	30	6	is	be	AUX
cana-536	30	7	a	a	DET
cana-536	30	8	tree	tree	NOUN
cana-536	30	9	t	t	NOUN
cana-536	30	10	with	with	ADP
cana-536	30	11	m	m	PROPN
cana-536	30	12	edges	edge	NOUN
cana-536	30	13	and	and	CCONJ
cana-536	30	14	h	h	NOUN
cana-536	30	15	≅	≅	PROPN
cana-536	30	16	k1	k1	PROPN
cana-536	30	17	,	,	PUNCT
cana-536	30	18	the	the	DET
cana-536	30	19	corona	corona	NOUN
cana-536	30	20	t	t	PROPN
cana-536	30	21	◦	◦	NOUN
cana-536	30	22	k1	k1	PROPN
cana-536	30	23	is	be	AUX
cana-536	30	24	also	also	ADV
cana-536	30	25	a	a	DET
cana-536	30	26	tree	tree	NOUN
cana-536	30	27	with	with	ADP
cana-536	30	28	2	2	NUM
cana-536	30	29	m	m	NOUN
cana-536	30	30	+	+	ADJ
cana-536	30	31	1	1	NUM
cana-536	30	32	edges	edge	NOUN
cana-536	30	33	.	.	PUNCT
cana-536	31	1	thus	thus	ADV
cana-536	31	2	,	,	PUNCT
cana-536	31	3	the	the	DET
cana-536	31	4	number	number	NOUN
cana-536	31	5	of	of	ADP
cana-536	31	6	newly	newly	ADV
cana-536	31	7	added	add	VERB
cana-536	31	8	vertices	vertex	NOUN
cana-536	31	9	in	in	ADP
cana-536	31	10	the	the	DET
cana-536	31	11	corona	corona	NOUN
cana-536	31	12	product	product	NOUN
cana-536	31	13	of	of	ADP
cana-536	31	14	t	t	PROPN
cana-536	31	15	and	and	CCONJ
cana-536	31	16	k1	k1	PROPN
cana-536	31	17	will	will	AUX
cana-536	31	18	be	be	AUX
cana-536	31	19	m	m	PROPN
cana-536	31	20	+	+	ADJ
cana-536	31	21	1	1	NUM
cana-536	31	22	and	and	CCONJ
cana-536	31	23	all	all	PRON
cana-536	31	24	of	of	ADP
cana-536	31	25	those	those	DET
cana-536	31	26	vertices	vertex	NOUN
cana-536	31	27	are	be	AUX
cana-536	31	28	of	of	ADP
cana-536	31	29	degree	degree	NOUN
cana-536	31	30	1	1	NUM
cana-536	31	31	.	.	PUNCT
cana-536	32	1	notation	notation	NOUN
cana-536	32	2	:	:	PUNCT
cana-536	32	3	for	for	ADP
cana-536	32	4	the	the	DET
cana-536	32	5	sake	sake	NOUN
cana-536	32	6	of	of	ADP
cana-536	32	7	convenience	convenience	NOUN
cana-536	32	8	,	,	PUNCT
cana-536	32	9	let	let	VERB
cana-536	32	10	v	v	NOUN
cana-536	32	11	(	(	PUNCT
cana-536	32	12	k1	k1	NOUN
cana-536	32	13	)	)	PUNCT
cana-536	32	14	=	=	SYM
cana-536	32	15	{	{	PUNCT
cana-536	32	16	w	w	NOUN
cana-536	32	17	}	}	PUNCT
cana-536	32	18	and	and	CCONJ
cana-536	32	19	if	if	SCONJ
cana-536	32	20	u	u	PRON
cana-536	32	21	be	be	VERB
cana-536	32	22	any	any	DET
cana-536	32	23	vertex	vertex	NOUN
cana-536	32	24	in	in	ADP
cana-536	32	25	the	the	DET
cana-536	32	26	tree	tree	NOUN
cana-536	32	27	t	t	PROPN
cana-536	32	28	,	,	PUNCT
cana-536	32	29	then	then	ADV
cana-536	32	30	the	the	DET
cana-536	32	31	corresponding	corresponding	ADJ
cana-536	32	32	vertex	vertex	NOUN
cana-536	32	33	added	add	VERB
cana-536	32	34	in	in	ADP
cana-536	32	35	the	the	DET
cana-536	32	36	corona	corona	NOUN
cana-536	32	37	product	product	NOUN
cana-536	32	38	t	t	PROPN
cana-536	32	39	◦	◦	NOUN
cana-536	32	40	k1	k1	NOUN
cana-536	32	41	will	will	AUX
cana-536	32	42	be	be	AUX
cana-536	32	43	denoted	denote	VERB
cana-536	32	44	as	as	ADP
cana-536	32	45	wu	wu	PROPN
cana-536	32	46	.	.	PUNCT
cana-536	33	1	in	in	ADP
cana-536	33	2	view	view	NOUN
cana-536	33	3	of	of	ADP
cana-536	33	4	this	this	DET
cana-536	33	5	notation	notation	NOUN
cana-536	33	6	,	,	PUNCT
cana-536	33	7	let	let	VERB
cana-536	33	8	us	we	PRON
cana-536	33	9	call	call	VERB
cana-536	33	10	the	the	DET
cana-536	33	11	vertex	vertex	NOUN
cana-536	33	12	u	u	NOUN
cana-536	33	13	as	as	ADP
cana-536	33	14	the	the	DET
cana-536	33	15	original	original	ADJ
cana-536	33	16	vertex	vertex	NOUN
cana-536	33	17	and	and	CCONJ
cana-536	33	18	the	the	DET
cana-536	33	19	newly	newly	ADV
cana-536	33	20	added	add	VERB
cana-536	33	21	vertex	vertex	NOUN
cana-536	33	22	wu	wu	PROPN
cana-536	33	23	as	as	ADP
cana-536	33	24	the	the	DET
cana-536	33	25	duplicate	duplicate	ADJ
cana-536	33	26	vertex	vertex	NOUN
cana-536	33	27	for	for	ADP
cana-536	33	28	u.	u.	NOUN
cana-536	33	29	3	3	NUM
cana-536	33	30	main	main	ADJ
cana-536	33	31	result	result	NOUN
cana-536	33	32	in	in	ADP
cana-536	33	33	this	this	DET
cana-536	33	34	section	section	NOUN
cana-536	33	35	,	,	PUNCT
cana-536	33	36	we	we	PRON
cana-536	33	37	prove	prove	VERB
cana-536	33	38	our	our	PRON
cana-536	33	39	main	main	ADJ
cana-536	33	40	result	result	NOUN
cana-536	33	41	.	.	PUNCT
cana-536	34	1	theorem	theorem	NOUN
cana-536	34	2	1	1	X
cana-536	34	3	.	.	PUNCT
cana-536	35	1	let	let	VERB
cana-536	35	2	t	t	NOUN
cana-536	35	3	be	be	AUX
cana-536	35	4	a	a	DET
cana-536	35	5	tree	tree	NOUN
cana-536	35	6	with	with	ADP
cana-536	35	7	edge	edge	NOUN
cana-536	35	8	irregularity	irregularity	NOUN
cana-536	35	9	strength	strength	NOUN
cana-536	35	10	be	be	AUX
cana-536	35	11	es(t	es(t	PUNCT
cana-536	35	12	)	)	PUNCT
cana-536	35	13	.	.	PUNCT
cana-536	36	1	then	then	ADV
cana-536	36	2	es	es	X
cana-536	36	3	(	(	PUNCT
cana-536	36	4	t	t	NOUN
cana-536	36	5	◦	◦	NOUN
cana-536	36	6	k1	k1	NOUN
cana-536	36	7	)	)	PUNCT
cana-536	36	8	=	=	SYM
cana-536	37	1	2es	2es	NOUN
cana-536	37	2	(	(	PUNCT
cana-536	37	3	t	t	PROPN
cana-536	37	4	)	)	PUNCT
cana-536	37	5	.	.	PUNCT
cana-536	38	1	proof	proof	NOUN
cana-536	38	2	.	.	PUNCT
cana-536	39	1	let	let	VERB
cana-536	39	2	k	k	NOUN
cana-536	39	3	=	=	PUNCT
cana-536	39	4	es	es	X
cana-536	39	5	(	(	PUNCT
cana-536	39	6	t	t	PROPN
cana-536	39	7	)	)	PUNCT
cana-536	39	8	and	and	CCONJ
cana-536	39	9	φ	φ	NOUN
cana-536	39	10	:	:	PUNCT
cana-536	39	11	v	v	PROPN
cana-536	39	12	(	(	PUNCT
cana-536	39	13	t	t	NOUN
cana-536	39	14	)	)	PUNCT
cana-536	39	15	→	→	SYM
cana-536	39	16	{	{	PUNCT
cana-536	39	17	1	1	NUM
cana-536	39	18	,	,	PUNCT
cana-536	39	19	2	2	NUM
cana-536	39	20	,	,	PUNCT
cana-536	39	21	·	·	PUNCT
cana-536	39	22	·	·	PUNCT
cana-536	39	23	·	·	PUNCT
cana-536	39	24	,	,	PUNCT
cana-536	39	25	k	k	X
cana-536	39	26	}	}	PUNCT
cana-536	39	27	be	be	AUX
cana-536	39	28	the	the	DET
cana-536	39	29	edge	edge	NOUN
cana-536	39	30	irregular	irregular	ADJ
cana-536	39	31	k	k	NOUN
cana-536	39	32	-	-	NOUN
cana-536	39	33	labeling	labeling	NOUN
cana-536	39	34	of	of	ADP
cana-536	39	35	t	t	NOUN
cana-536	39	36	case	case	NOUN
cana-536	39	37	1	1	NUM
cana-536	39	38	:	:	PUNCT
cana-536	39	39	k	k	X
cana-536	39	40	is	be	AUX
cana-536	39	41	even	even	ADV
cana-536	39	42	let	let	VERB
cana-536	39	43	v1	v1	NOUN
cana-536	39	44	be	be	AUX
cana-536	39	45	the	the	DET
cana-536	39	46	set	set	NOUN
cana-536	39	47	of	of	ADP
cana-536	39	48	vertices	vertex	NOUN
cana-536	39	49	whose	whose	DET
cana-536	39	50	vertex	vertex	NOUN
cana-536	39	51	labels	label	NOUN
cana-536	39	52	are	be	AUX
cana-536	39	53	from	from	ADP
cana-536	39	54	the	the	DET
cana-536	39	55	set	set	NOUN
cana-536	39	56	{	{	PUNCT
cana-536	39	57	1	1	NUM
cana-536	39	58	,	,	PUNCT
cana-536	39	59	2	2	NUM
cana-536	39	60	,	,	PUNCT
cana-536	39	61	·	·	PUNCT
cana-536	39	62	·	·	PUNCT
cana-536	39	63	·	·	PUNCT
cana-536	39	64	,	,	PUNCT
cana-536	39	65	𝑘	𝑘	DET
cana-536	39	66	2	2	NUM
cana-536	39	67	}	}	PUNCT
cana-536	39	68	and	and	CCONJ
cana-536	39	69	v2	v2	PROPN
cana-536	39	70	be	be	AUX
cana-536	39	71	the	the	DET
cana-536	39	72	set	set	NOUN
cana-536	39	73	of	of	ADP
cana-536	39	74	vertices	vertex	NOUN
cana-536	39	75	whose	whose	DET
cana-536	39	76	vertex	vertex	NOUN
cana-536	39	77	labels	label	NOUN
cana-536	39	78	are	be	AUX
cana-536	39	79	from	from	ADP
cana-536	39	80	the	the	DET
cana-536	39	81	set	set	NOUN
cana-536	39	82	{	{	PUNCT
cana-536	39	83	𝑘	𝑘	PROPN
cana-536	39	84	2	2	NUM
cana-536	39	85	+1	+1	PROPN
cana-536	39	86	,	,	PUNCT
cana-536	39	87	·	·	PUNCT
cana-536	39	88	·	·	PUNCT
cana-536	40	1	·	·	PUNCT
cana-536	40	2	,	,	PUNCT
cana-536	40	3	k	k	NOUN
cana-536	40	4	}	}	PUNCT
cana-536	40	5	.	.	PUNCT
cana-536	41	1	let	let	VERB
cana-536	41	2	us	we	PRON
cana-536	41	3	define	define	VERB
cana-536	41	4	the	the	DET
cana-536	41	5	function	function	NOUN
cana-536	41	6	ψ	ψ	NOUN
cana-536	41	7	:	:	PUNCT
cana-536	41	8	v	v	NOUN
cana-536	41	9	(	(	PUNCT
cana-536	41	10	t	t	NOUN
cana-536	41	11	◦	◦	NOUN
cana-536	41	12	k1	k1	NOUN
cana-536	41	13	)	)	PUNCT
cana-536	41	14	→	→	SYM
cana-536	41	15	{	{	PUNCT
cana-536	41	16	1	1	NUM
cana-536	41	17	,	,	PUNCT
cana-536	41	18	2	2	NUM
cana-536	41	19	,	,	PUNCT
cana-536	41	20	·	·	PUNCT
cana-536	41	21	·	·	PUNCT
cana-536	41	22	·	·	PUNCT
cana-536	41	23	,	,	PUNCT
cana-536	41	24	2k	2k	NUM
cana-536	41	25	}	}	PUNCT
cana-536	41	26	as	as	SCONJ
cana-536	41	27	follows	follow	VERB
cana-536	41	28	:	:	PUNCT
cana-536	41	29	for	for	ADP
cana-536	41	30	any	any	DET
cana-536	41	31	vertex	vertex	NOUN
cana-536	41	32	u	u	NOUN
cana-536	41	33	in	in	ADP
cana-536	41	34	v1	v1	PROPN
cana-536	41	35	⊂	⊂	PROPN
cana-536	41	36	v	v	X
cana-536	41	37	(	(	PUNCT
cana-536	41	38	t	t	PROPN
cana-536	41	39	)	)	PUNCT
cana-536	41	40	,	,	PUNCT
cana-536	41	41	retain	retain	VERB
cana-536	41	42	the	the	DET
cana-536	41	43	vertex	vertex	NOUN
cana-536	41	44	label	label	NOUN
cana-536	41	45	for	for	ADP
cana-536	41	46	the	the	DET
cana-536	41	47	corresponding	corresponding	ADJ
cana-536	41	48	vertex	vertex	NOUN
cana-536	41	49	in	in	ADP
cana-536	41	50	t	t	PROPN
cana-536	41	51	◦	◦	NOUN
cana-536	41	52	k1	k1	PROPN
cana-536	41	53	.	.	PUNCT
cana-536	42	1	that	that	PRON
cana-536	42	2	is	be	AUX
cana-536	42	3	,	,	PUNCT
cana-536	42	4	ψ(u	ψ(u	PROPN
cana-536	42	5	)	)	PUNCT
cana-536	42	6	=	=	SYM
cana-536	43	1	φ(u	φ(u	NOUN
cana-536	43	2	)	)	PUNCT
cana-536	43	3	,	,	PUNCT
cana-536	43	4	for	for	ADP
cana-536	43	5	any	any	DET
cana-536	43	6	vertex	vertex	NOUN
cana-536	43	7	u	u	NOUN
cana-536	43	8	∈	∈	PROPN
cana-536	43	9	v1	v1	NOUN
cana-536	43	10	.	.	PUNCT
cana-536	44	1	for	for	ADP
cana-536	44	2	any	any	DET
cana-536	44	3	vertex	vertex	NOUN
cana-536	44	4	v	v	NOUN
cana-536	44	5	in	in	ADP
cana-536	44	6	v2	v2	PROPN
cana-536	44	7	⊂	⊂	PROPN
cana-536	44	8	v	v	X
cana-536	44	9	(	(	PUNCT
cana-536	44	10	t	t	PROPN
cana-536	44	11	)	)	PUNCT
cana-536	44	12	,	,	PUNCT
cana-536	44	13	ψ(v	ψ(v	PROPN
cana-536	44	14	)	)	PUNCT
cana-536	44	15	=	=	SYM
cana-536	44	16	φ(v	φ(v	PROPN
cana-536	44	17	)	)	PUNCT
cana-536	45	1	+	+	CCONJ
cana-536	45	2	k.	k.	X
cana-536	45	3	at	at	ADP
cana-536	45	4	this	this	DET
cana-536	45	5	stage	stage	NOUN
cana-536	45	6	,	,	PUNCT
cana-536	45	7	the	the	DET
cana-536	45	8	original	original	ADJ
cana-536	45	9	vertices	vertex	NOUN
cana-536	45	10	of	of	ADP
cana-536	45	11	t	t	PROPN
cana-536	45	12	◦	◦	NOUN
cana-536	45	13	k1	k1	NOUN
cana-536	45	14	are	be	AUX
cana-536	45	15	all	all	PRON
cana-536	45	16	labeled	label	VERB
cana-536	45	17	by	by	ADP
cana-536	45	18	ψ	ψ	X
cana-536	45	19	.	.	PUNCT
cana-536	46	1	let	let	VERB
cana-536	46	2	x	x	PRON
cana-536	46	3	be	be	AUX
cana-536	46	4	the	the	DET
cana-536	46	5	set	set	NOUN
cana-536	46	6	of	of	ADP
cana-536	46	7	all	all	DET
cana-536	46	8	weights	weight	NOUN
cana-536	46	9	of	of	ADP
cana-536	46	10	original	original	ADJ
cana-536	46	11	edges	edge	NOUN
cana-536	46	12	of	of	ADP
cana-536	46	13	t	t	NOUN
cana-536	46	14	◦	◦	NOUN
cana-536	46	15	k1	k1	NOUN
cana-536	46	16	as	as	SCONJ
cana-536	46	17	induced	induce	VERB
cana-536	46	18	by	by	ADP
cana-536	46	19	ψ	ψ	X
cana-536	46	20	.	.	PUNCT
cana-536	47	1	let	let	VERB
cana-536	47	2	w	w	VERB
cana-536	47	3	=	=	PUNCT
cana-536	47	4	{	{	PUNCT
cana-536	47	5	2	2	NUM
cana-536	47	6	,	,	PUNCT
cana-536	47	7	3	3	NUM
cana-536	47	8	,	,	PUNCT
cana-536	47	9	·	·	PUNCT
cana-536	47	10	·	·	PUNCT
cana-536	47	11	·	·	PUNCT
cana-536	47	12	,	,	PUNCT
cana-536	47	13	2k	2k	NUM
cana-536	47	14	}	}	PUNCT
cana-536	47	15	be	be	AUX
cana-536	47	16	the	the	DET
cana-536	47	17	set	set	NOUN
cana-536	47	18	of	of	ADP
cana-536	47	19	all	all	DET
cana-536	47	20	weights	weight	NOUN
cana-536	47	21	of	of	ADP
cana-536	47	22	the	the	DET
cana-536	47	23	edges	edge	NOUN
cana-536	47	24	.	.	PUNCT
cana-536	48	1	let	let	VERB
cana-536	48	2	r	r	NOUN
cana-536	48	3	=	=	PUNCT
cana-536	48	4	w	w	NOUN
cana-536	48	5	−	−	NOUN
cana-536	48	6	x	x	PUNCT
cana-536	48	7	be	be	AUX
cana-536	48	8	the	the	DET
cana-536	48	9	required	require	VERB
cana-536	48	10	set	set	NOUN
cana-536	48	11	of	of	ADP
cana-536	48	12	weights	weight	NOUN
cana-536	48	13	of	of	ADP
cana-536	48	14	edges	edge	NOUN
cana-536	48	15	.	.	PUNCT
cana-536	49	1	let	let	VERB
cana-536	49	2	s	s	PRON
cana-536	49	3	be	be	AUX
cana-536	49	4	the	the	DET
cana-536	49	5	sequence	sequence	NOUN
cana-536	49	6	of	of	ADP
cana-536	49	7	arrangement	arrangement	NOUN
cana-536	49	8	of	of	ADP
cana-536	49	9	vertex	vertex	NOUN
cana-536	49	10	labels	label	NOUN
cana-536	49	11	in	in	ADP
cana-536	49	12	the	the	DET
cana-536	49	13	increasing	increase	VERB
cana-536	49	14	order	order	NOUN
cana-536	49	15	as	as	SCONJ
cana-536	49	16	defined	define	VERB
cana-536	49	17	by	by	ADP
cana-536	49	18	ψ	ψ	X
cana-536	49	19	.	.	PUNCT
cana-536	50	1	since	since	SCONJ
cana-536	50	2	t	t	PROPN
cana-536	50	3	◦	◦	PROPN
cana-536	50	4	k1	k1	PROPN
cana-536	50	5	is	be	AUX
cana-536	50	6	also	also	ADV
cana-536	50	7	a	a	DET
cana-536	50	8	tree	tree	NOUN
cana-536	50	9	,	,	PUNCT
cana-536	50	10	we	we	PRON
cana-536	50	11	have	have	VERB
cana-536	50	12	the	the	DET
cana-536	50	13	cardinality	cardinality	NOUN
cana-536	50	14	of	of	ADP
cana-536	50	15	the	the	DET
cana-536	50	16	required	require	VERB
cana-536	50	17	set	set	NOUN
cana-536	50	18	of	of	ADP
cana-536	50	19	weights	weight	NOUN
cana-536	50	20	of	of	ADP
cana-536	50	21	edges	edge	NOUN
cana-536	50	22	is	be	AUX
cana-536	50	23	equal	equal	ADJ
cana-536	50	24	to	to	ADP
cana-536	50	25	the	the	DET
cana-536	50	26	length	length	NOUN
cana-536	50	27	of	of	ADP
cana-536	50	28	the	the	DET
cana-536	50	29	sequence	sequence	NOUN
cana-536	50	30	s.	s.	PROPN
cana-536	50	31	until	until	ADP
cana-536	50	32	r≠ϕ	r≠ϕ	PROPN
cana-536	50	33	,	,	PUNCT
cana-536	50	34	choose	choose	VERB
cana-536	50	35	a	a	DET
cana-536	50	36	minimum	minimum	ADJ
cana-536	50	37	weight	weight	NOUN
cana-536	50	38	(	(	PUNCT
cana-536	50	39	say	say	VERB
cana-536	50	40	r	r	NOUN
cana-536	50	41	)	)	PUNCT
cana-536	50	42	in	in	ADP
cana-536	50	43	r	r	NOUN
cana-536	50	44	and	and	CCONJ
cana-536	50	45	correspondingly	correspondingly	ADV
cana-536	50	46	choose	choose	VERB
cana-536	50	47	the	the	DET
cana-536	50	48	first	first	ADJ
cana-536	50	49	term	term	NOUN
cana-536	50	50	communications	communication	NOUN
cana-536	50	51	on	on	ADP
cana-536	50	52	applied	apply	VERB
cana-536	50	53	nonlinear	nonlinear	ADJ
cana-536	50	54	analysis	analysis	NOUN
cana-536	50	55	issn	issn	NOUN
cana-536	50	56	:	:	PUNCT
cana-536	50	57	1074	1074	NUM
cana-536	50	58	-	-	PUNCT
cana-536	50	59	133x	133x	NUM
cana-536	50	60	vol	vol	NOUN
cana-536	50	61	31	31	NUM
cana-536	50	62	no	no	NOUN
cana-536	50	63	.	.	NOUN
cana-536	50	64	2	2	NUM
cana-536	50	65	(	(	PUNCT
cana-536	50	66	2024	2024	NUM
cana-536	50	67	)	)	PUNCT
cana-536	50	68	216	216	NUM
cana-536	50	69	https://internationalpubls.com	https://internationalpubls.com	X
cana-536	50	70	(	(	PUNCT
cana-536	50	71	say	say	VERB
cana-536	50	72	q	q	INTJ
cana-536	50	73	)	)	PUNCT
cana-536	50	74	in	in	ADP
cana-536	50	75	the	the	DET
cana-536	50	76	sequence	sequence	NOUN
cana-536	50	77	s.	s.	PROPN
cana-536	50	78	now	now	ADV
cana-536	50	79	choose	choose	VERB
cana-536	50	80	the	the	DET
cana-536	50	81	original	original	ADJ
cana-536	50	82	vertex	vertex	NOUN
cana-536	50	83	with	with	ADP
cana-536	50	84	label	label	NOUN
cana-536	50	85	q	q	PUNCT
cana-536	50	86	as	as	SCONJ
cana-536	50	87	defined	define	VERB
cana-536	50	88	by	by	ADP
cana-536	50	89	ψ	ψ	X
cana-536	50	90	in	in	ADP
cana-536	50	91	t	t	PROPN
cana-536	50	92	◦	◦	NOUN
cana-536	50	93	k1	k1	NOUN
cana-536	50	94	and	and	CCONJ
cana-536	50	95	label	label	VERB
cana-536	50	96	its	its	PRON
cana-536	50	97	duplicate	duplicate	ADJ
cana-536	50	98	vertex	vertex	NOUN
cana-536	50	99	as	as	ADP
cana-536	50	100	r	r	NOUN
cana-536	50	101	−	−	NOUN
cana-536	50	102	q	q	NOUN
cana-536	51	1	so	so	SCONJ
cana-536	51	2	that	that	SCONJ
cana-536	51	3	the	the	DET
cana-536	51	4	induced	induced	ADJ
cana-536	51	5	weight	weight	NOUN
cana-536	51	6	of	of	ADP
cana-536	51	7	the	the	DET
cana-536	51	8	newly	newly	ADV
cana-536	51	9	added	add	VERB
cana-536	51	10	edge	edge	NOUN
cana-536	51	11	between	between	ADP
cana-536	51	12	the	the	DET
cana-536	51	13	original	original	ADJ
cana-536	51	14	vertex	vertex	NOUN
cana-536	51	15	and	and	CCONJ
cana-536	51	16	its	its	PRON
cana-536	51	17	duplicate	duplicate	ADJ
cana-536	51	18	vertex	vertex	NOUN
cana-536	51	19	will	will	AUX
cana-536	51	20	be	be	AUX
cana-536	51	21	r.	r.	PROPN
cana-536	51	22	then	then	ADV
cana-536	51	23	,	,	PUNCT
cana-536	51	24	delete	delete	ADJ
cana-536	51	25	r	r	NOUN
cana-536	51	26	from	from	ADP
cana-536	51	27	r	r	NOUN
cana-536	51	28	and	and	CCONJ
cana-536	51	29	construct	construct	VERB
cana-536	51	30	a	a	DET
cana-536	51	31	new	new	ADJ
cana-536	51	32	sequence	sequence	NOUN
cana-536	51	33	s	s	VERB
cana-536	51	34	by	by	ADP
cana-536	51	35	deleting	delete	VERB
cana-536	51	36	its	its	PRON
cana-536	51	37	first	first	ADJ
cana-536	51	38	term	term	NOUN
cana-536	51	39	q.	q.	NOUN
cana-536	51	40	this	this	DET
cana-536	51	41	procedure	procedure	NOUN
cana-536	51	42	can	can	AUX
cana-536	51	43	be	be	AUX
cana-536	51	44	done	do	VERB
cana-536	51	45	until	until	SCONJ
cana-536	51	46	r	r	NOUN
cana-536	51	47	becomes	become	VERB
cana-536	51	48	an	an	DET
cana-536	51	49	empty	empty	ADJ
cana-536	51	50	set	set	NOUN
cana-536	51	51	.	.	PUNCT
cana-536	52	1	when	when	SCONJ
cana-536	52	2	r	r	NOUN
cana-536	52	3	=	=	SYM
cana-536	52	4	ϕ	ϕ	PROPN
cana-536	52	5	,	,	PUNCT
cana-536	52	6	all	all	DET
cana-536	52	7	the	the	DET
cana-536	52	8	vertices	vertex	NOUN
cana-536	52	9	have	have	AUX
cana-536	52	10	been	be	AUX
cana-536	52	11	labeled	label	VERB
cana-536	52	12	by	by	ADP
cana-536	52	13	the	the	DET
cana-536	52	14	function	function	NOUN
cana-536	52	15	ψ	ψ	NOUN
cana-536	52	16	in	in	ADP
cana-536	52	17	t	t	PROPN
cana-536	52	18	◦	◦	NOUN
cana-536	52	19	k1	k1	NOUN
cana-536	52	20	.	.	PUNCT
cana-536	53	1	therefore	therefore	ADV
cana-536	53	2	,	,	PUNCT
cana-536	53	3	by	by	ADP
cana-536	53	4	the	the	DET
cana-536	53	5	construction	construction	NOUN
cana-536	53	6	of	of	ADP
cana-536	53	7	t	t	PROPN
cana-536	53	8	◦	◦	NOUN
cana-536	53	9	k1	k1	NOUN
cana-536	53	10	and	and	CCONJ
cana-536	53	11	the	the	DET
cana-536	53	12	definition	definition	NOUN
cana-536	53	13	of	of	ADP
cana-536	53	14	ψ	ψ	PROPN
cana-536	53	15	,	,	PUNCT
cana-536	53	16	es	es	X
cana-536	53	17	(	(	PUNCT
cana-536	53	18	t	t	NOUN
cana-536	53	19	◦	◦	NOUN
cana-536	53	20	k1	k1	NOUN
cana-536	53	21	)	)	PUNCT
cana-536	53	22	=	=	SYM
cana-536	53	23	2k	2k	NUM
cana-536	53	24	.	.	PUNCT
cana-536	54	1	case	case	NOUN
cana-536	54	2	2	2	NUM
cana-536	54	3	:	:	PUNCT
cana-536	54	4	k	k	X
cana-536	54	5	is	be	AUX
cana-536	54	6	odd	odd	ADJ
cana-536	54	7	:	:	PUNCT
cana-536	54	8	let	let	VERB
cana-536	54	9	v1	v1	NOUN
cana-536	54	10	be	be	AUX
cana-536	54	11	the	the	DET
cana-536	54	12	set	set	NOUN
cana-536	54	13	of	of	ADP
cana-536	54	14	vertices	vertex	NOUN
cana-536	54	15	whose	whose	DET
cana-536	54	16	vertex	vertex	NOUN
cana-536	54	17	labels	label	NOUN
cana-536	54	18	are	be	AUX
cana-536	54	19	from	from	ADP
cana-536	54	20	the	the	DET
cana-536	54	21	set	set	NOUN
cana-536	54	22	{	{	PUNCT
cana-536	54	23	1	1	NUM
cana-536	54	24	,	,	PUNCT
cana-536	54	25	2	2	NUM
cana-536	54	26	,	,	PUNCT
cana-536	54	27	·	·	PUNCT
cana-536	54	28	·	·	PUNCT
cana-536	54	29	·	·	PUNCT
cana-536	54	30	,	,	PUNCT
cana-536	54	31	⌈	⌈	PROPN
cana-536	54	32	𝑘	𝑘	ADP
cana-536	54	33	2	2	NUM
cana-536	54	34	⌉+	⌉+	SYM
cana-536	54	35	1	1	NUM
cana-536	54	36	}	}	PUNCT
cana-536	54	37	and	and	CCONJ
cana-536	54	38	v2	v2	PROPN
cana-536	54	39	be	be	AUX
cana-536	54	40	the	the	DET
cana-536	54	41	set	set	NOUN
cana-536	54	42	of	of	ADP
cana-536	54	43	vertices	vertex	NOUN
cana-536	54	44	whose	whose	DET
cana-536	54	45	vertex	vertex	NOUN
cana-536	54	46	labels	label	NOUN
cana-536	54	47	are	be	AUX
cana-536	54	48	from	from	ADP
cana-536	54	49	the	the	DET
cana-536	54	50	set	set	NOUN
cana-536	54	51	{	{	PUNCT
cana-536	54	52	⌈	⌈	NOUN
cana-536	54	53	𝑘	𝑘	DET
cana-536	54	54	2	2	NUM
cana-536	54	55	⌉	⌉	NOUN
cana-536	54	56	+2	+2	PROPN
cana-536	54	57	,	,	PUNCT
cana-536	54	58	·	·	PUNCT
cana-536	54	59	·	·	PUNCT
cana-536	55	1	·	·	PUNCT
cana-536	55	2	,	,	PUNCT
cana-536	55	3	k	k	NOUN
cana-536	55	4	}	}	PUNCT
cana-536	55	5	.	.	PUNCT
cana-536	56	1	as	as	SCONJ
cana-536	56	2	defined	define	VERB
cana-536	56	3	for	for	ADP
cana-536	56	4	the	the	DET
cana-536	56	5	even	even	ADJ
cana-536	56	6	case	case	NOUN
cana-536	56	7	,	,	PUNCT
cana-536	56	8	similarly	similarly	ADV
cana-536	56	9	we	we	PRON
cana-536	56	10	define	define	VERB
cana-536	56	11	ψ	ψ	NOUN
cana-536	56	12	:	:	PUNCT
cana-536	56	13	v	v	NOUN
cana-536	56	14	(	(	PUNCT
cana-536	56	15	t	t	NOUN
cana-536	56	16	◦	◦	NOUN
cana-536	56	17	k1	k1	NOUN
cana-536	56	18	)	)	PUNCT
cana-536	56	19	→	→	SYM
cana-536	56	20	{	{	PUNCT
cana-536	56	21	1	1	NUM
cana-536	56	22	,	,	PUNCT
cana-536	56	23	2	2	NUM
cana-536	56	24	,	,	PUNCT
cana-536	56	25	·	·	PUNCT
cana-536	56	26	·	·	PUNCT
cana-536	56	27	·	·	PUNCT
cana-536	56	28	,	,	PUNCT
cana-536	56	29	2k	2k	NUM
cana-536	56	30	}	}	PUNCT
cana-536	56	31	.	.	PUNCT
cana-536	57	1	therefore	therefore	ADV
cana-536	57	2	,	,	PUNCT
cana-536	57	3	either	either	CCONJ
cana-536	57	4	es	es	X
cana-536	57	5	(	(	PUNCT
cana-536	57	6	t	t	PROPN
cana-536	57	7	)	)	PUNCT
cana-536	57	8	is	be	AUX
cana-536	57	9	odd	odd	ADJ
cana-536	57	10	or	or	CCONJ
cana-536	57	11	even	even	ADV
cana-536	57	12	,	,	PUNCT
cana-536	57	13	es	es	X
cana-536	57	14	(	(	PUNCT
cana-536	57	15	t	t	NOUN
cana-536	57	16	◦	◦	NOUN
cana-536	57	17	k1	k1	NOUN
cana-536	57	18	)	)	PUNCT
cana-536	58	1	=	=	SYM
cana-536	58	2	2es	2es	NOUN
cana-536	58	3	(	(	PUNCT
cana-536	58	4	t	t	PROPN
cana-536	58	5	)	)	PUNCT
cana-536	58	6	.	.	PUNCT
cana-536	59	1	4	4	NUM
cana-536	59	2	binomial	binomial	ADJ
cana-536	59	3	trees	tree	NOUN
cana-536	59	4	the	the	DET
cana-536	59	5	binomial	binomial	ADJ
cana-536	59	6	tree	tree	NOUN
cana-536	59	7	b0	b0	NOUN
cana-536	59	8	consists	consist	VERB
cana-536	59	9	of	of	ADP
cana-536	59	10	a	a	DET
cana-536	59	11	single	single	ADJ
cana-536	59	12	vertex	vertex	NOUN
cana-536	59	13	.	.	PUNCT
cana-536	60	1	the	the	DET
cana-536	60	2	binomial	binomial	ADJ
cana-536	60	3	tree	tree	NOUN
cana-536	60	4	bk	bk	PROPN
cana-536	60	5	is	be	AUX
cana-536	60	6	an	an	DET
cana-536	60	7	ordered	order	VERB
cana-536	60	8	tree	tree	NOUN
cana-536	60	9	defined	define	VERB
cana-536	60	10	recursively	recursively	ADV
cana-536	60	11	.	.	PUNCT
cana-536	61	1	the	the	DET
cana-536	61	2	binomial	binomial	ADJ
cana-536	61	3	tree	tree	NOUN
cana-536	61	4	bk	bk	PROPN
cana-536	61	5	consists	consist	VERB
cana-536	61	6	of	of	ADP
cana-536	61	7	two	two	NUM
cana-536	61	8	binomial	binomial	ADJ
cana-536	61	9	trees	tree	NOUN
cana-536	61	10	bk−1	bk−1	ADJ
cana-536	61	11	that	that	PRON
cana-536	61	12	are	be	AUX
cana-536	61	13	linked	link	VERB
cana-536	61	14	together	together	ADV
cana-536	61	15	:	:	PUNCT
cana-536	61	16	the	the	DET
cana-536	61	17	root	root	NOUN
cana-536	61	18	of	of	ADP
cana-536	61	19	one	one	NUM
cana-536	61	20	is	be	AUX
cana-536	61	21	the	the	DET
cana-536	61	22	leftmost	leftmost	ADJ
cana-536	61	23	child	child	NOUN
cana-536	61	24	of	of	ADP
cana-536	61	25	the	the	DET
cana-536	61	26	root	root	NOUN
cana-536	61	27	of	of	ADP
cana-536	61	28	the	the	DET
cana-536	61	29	other	other	ADJ
cana-536	61	30	.	.	PUNCT
cana-536	62	1	note	note	VERB
cana-536	62	2	that	that	SCONJ
cana-536	62	3	there	there	PRON
cana-536	62	4	are	be	VERB
cana-536	62	5	2k	2k	NUM
cana-536	62	6	vertices	vertex	NOUN
cana-536	62	7	in	in	ADP
cana-536	62	8	the	the	DET
cana-536	62	9	binomial	binomial	ADJ
cana-536	62	10	tree	tree	NOUN
cana-536	62	11	bk	bk	PROPN
cana-536	62	12	.	.	PUNCT
cana-536	63	1	for	for	ADP
cana-536	63	2	more	more	ADJ
cana-536	63	3	details	detail	NOUN
cana-536	63	4	about	about	ADP
cana-536	63	5	binomial	binomial	ADJ
cana-536	63	6	trees	tree	NOUN
cana-536	63	7	refer	refer	VERB
cana-536	63	8	[	[	X
cana-536	63	9	5	5	NUM
cana-536	63	10	]	]	PUNCT
cana-536	63	11	.	.	PUNCT
cana-536	64	1	4.1	4.1	NUM
cana-536	64	2	edge	edge	NOUN
cana-536	64	3	irregularity	irregularity	NOUN
cana-536	64	4	strength	strength	NOUN
cana-536	64	5	of	of	ADP
cana-536	64	6	binomial	binomial	ADJ
cana-536	64	7	trees	tree	NOUN
cana-536	64	8	in	in	ADP
cana-536	64	9	this	this	DET
cana-536	64	10	section	section	NOUN
cana-536	65	1	,	,	PUNCT
cana-536	65	2	we	we	PRON
cana-536	65	3	will	will	AUX
cana-536	65	4	calculate	calculate	VERB
cana-536	65	5	the	the	DET
cana-536	65	6	edge	edge	NOUN
cana-536	65	7	irregularity	irregularity	NOUN
cana-536	65	8	strength	strength	NOUN
cana-536	65	9	of	of	ADP
cana-536	65	10	binomial	binomial	ADJ
cana-536	65	11	trees	tree	NOUN
cana-536	65	12	.	.	PUNCT
cana-536	66	1	observation	observation	NOUN
cana-536	66	2	2	2	NUM
cana-536	66	3	:	:	PUNCT
cana-536	66	4	one	one	PRON
cana-536	66	5	can	can	AUX
cana-536	66	6	easily	easily	ADV
cana-536	66	7	observe	observe	VERB
cana-536	66	8	that	that	SCONJ
cana-536	66	9	corona	corona	NOUN
cana-536	66	10	product	product	NOUN
cana-536	66	11	of	of	ADP
cana-536	66	12	binomial	binomial	ADJ
cana-536	66	13	tree	tree	NOUN
cana-536	66	14	bk−1	bk−1	NOUN
cana-536	66	15	with	with	ADP
cana-536	66	16	k1	k1	NOUN
cana-536	66	17	leads	lead	NOUN
cana-536	66	18	to	to	ADP
cana-536	66	19	the	the	DET
cana-536	66	20	binomial	binomial	ADJ
cana-536	66	21	tree	tree	NOUN
cana-536	66	22	bk	bk	PROPN
cana-536	66	23	,	,	PUNCT
cana-536	66	24	for	for	ADP
cana-536	66	25	k	k	PROPN
cana-536	66	26	≥	≥	PROPN
cana-536	66	27	1	1	NUM
cana-536	66	28	.	.	PUNCT
cana-536	67	1	thus	thus	ADV
cana-536	67	2	,	,	PUNCT
cana-536	67	3	we	we	PRON
cana-536	67	4	have	have	VERB
cana-536	67	5	bk−1	bk−1	ADJ
cana-536	67	6	◦	◦	NOUN
cana-536	67	7	k1	k1	NOUN
cana-536	67	8	=	=	SYM
cana-536	67	9	bk	bk	PROPN
cana-536	67	10	.	.	PUNCT
cana-536	68	1	theorem	theorem	NOUN
cana-536	68	2	2	2	NUM
cana-536	68	3	.	.	PUNCT
cana-536	69	1	let	let	VERB
cana-536	69	2	bk	bk	PRON
cana-536	69	3	be	be	AUX
cana-536	69	4	the	the	DET
cana-536	69	5	binomial	binomial	ADJ
cana-536	69	6	trees	tree	NOUN
cana-536	69	7	for	for	ADP
cana-536	69	8	k	k	PROPN
cana-536	69	9	≥	≥	PROPN
cana-536	69	10	1	1	NUM
cana-536	69	11	.	.	PUNCT
cana-536	70	1	then	then	ADV
cana-536	70	2	es	es	X
cana-536	70	3	(	(	PUNCT
cana-536	70	4	bk	bk	NOUN
cana-536	70	5	)	)	PUNCT
cana-536	70	6	=	=	SYM
cana-536	70	7	2k−1	2k−1	NUM
cana-536	70	8	.	.	PUNCT
cana-536	71	1	proof	proof	NOUN
cana-536	71	2	.	.	PUNCT
cana-536	72	1	we	we	PRON
cana-536	72	2	prove	prove	VERB
cana-536	72	3	this	this	PRON
cana-536	72	4	by	by	ADP
cana-536	72	5	the	the	DET
cana-536	72	6	method	method	NOUN
cana-536	72	7	of	of	ADP
cana-536	72	8	induction	induction	NOUN
cana-536	72	9	on	on	ADP
cana-536	72	10	k	k	PROPN
cana-536	72	11	≥	≥	NUM
cana-536	72	12	1	1	NUM
cana-536	72	13	.	.	PUNCT
cana-536	73	1	it	it	PRON
cana-536	73	2	is	be	AUX
cana-536	73	3	clear	clear	ADJ
cana-536	73	4	that	that	SCONJ
cana-536	73	5	es(b1	es(b1	NOUN
cana-536	73	6	)	)	PUNCT
cana-536	73	7	=	=	SYM
cana-536	73	8	1	1	NUM
cana-536	73	9	and	and	CCONJ
cana-536	73	10	es(b2	es(b2	NOUN
cana-536	73	11	)	)	PUNCT
cana-536	73	12	=	=	SYM
cana-536	74	1	2	2	X
cana-536	74	2	.	.	PUNCT
cana-536	74	3	let	let	VERB
cana-536	74	4	us	we	PRON
cana-536	74	5	assume	assume	VERB
cana-536	74	6	that	that	SCONJ
cana-536	74	7	es	es	INTJ
cana-536	74	8	(	(	PUNCT
cana-536	74	9	bk−1	bk−1	NOUN
cana-536	74	10	)	)	PUNCT
cana-536	74	11	=	=	SYM
cana-536	74	12	2k−2	2k−2	PROPN
cana-536	74	13	.	.	PUNCT
cana-536	75	1	since	since	SCONJ
cana-536	75	2	binomial	binomial	ADJ
cana-536	75	3	tree	tree	NOUN
cana-536	75	4	bk	bk	ADP
cana-536	75	5	=	=	PUNCT
cana-536	75	6	bk−1	bk−1	VERB
cana-536	75	7	◦	◦	NOUN
cana-536	75	8	k1	k1	NOUN
cana-536	75	9	and	and	CCONJ
cana-536	75	10	using	use	VERB
cana-536	75	11	theorem	theorem	NOUN
cana-536	75	12	1	1	NUM
cana-536	75	13	,	,	PUNCT
cana-536	75	14	we	we	PRON
cana-536	75	15	have	have	VERB
cana-536	75	16	es	es	X
cana-536	75	17	(	(	PUNCT
cana-536	75	18	bk	bk	NOUN
cana-536	75	19	)	)	PUNCT
cana-536	76	1	=	=	SYM
cana-536	76	2	2es	2es	NOUN
cana-536	76	3	(	(	PUNCT
cana-536	76	4	bk−1	bk−1	NOUN
cana-536	76	5	)	)	PUNCT
cana-536	76	6	=	=	SYM
cana-536	77	1	2.2k−2	2.2k−2	NUM
cana-536	77	2	=	=	SYM
cana-536	77	3	2k−1	2k−1	NUM
cana-536	77	4	.	.	PUNCT
cana-536	78	1	5	5	NUM
cana-536	78	2	conclusion	conclusion	NOUN
cana-536	78	3	in	in	ADP
cana-536	78	4	this	this	DET
cana-536	78	5	paper	paper	NOUN
cana-536	78	6	,	,	PUNCT
cana-536	78	7	we	we	PRON
cana-536	78	8	proved	prove	VERB
cana-536	78	9	that	that	SCONJ
cana-536	78	10	the	the	DET
cana-536	78	11	edge	edge	NOUN
cana-536	78	12	irregularity	irregularity	NOUN
cana-536	78	13	strength	strength	NOUN
cana-536	78	14	of	of	ADP
cana-536	78	15	corona	corona	NOUN
cana-536	78	16	product	product	NOUN
cana-536	78	17	of	of	ADP
cana-536	78	18	a	a	DET
cana-536	78	19	tree	tree	NOUN
cana-536	78	20	t	t	NOUN
cana-536	78	21	with	with	ADP
cana-536	78	22	k1	k1	PROPN
cana-536	78	23	is	be	AUX
cana-536	78	24	es	es	X
cana-536	78	25	(	(	PUNCT
cana-536	78	26	t	t	NOUN
cana-536	78	27	◦	◦	NOUN
cana-536	78	28	k1	k1	NOUN
cana-536	78	29	)	)	PUNCT
cana-536	79	1	=	=	SYM
cana-536	79	2	2es	2es	NOUN
cana-536	79	3	(	(	PUNCT
cana-536	79	4	t	t	PROPN
cana-536	79	5	)	)	PUNCT
cana-536	79	6	.	.	PUNCT
cana-536	80	1	also	also	ADV
cana-536	80	2	,	,	PUNCT
cana-536	80	3	we	we	PRON
cana-536	80	4	proved	prove	VERB
cana-536	80	5	that	that	SCONJ
cana-536	80	6	the	the	DET
cana-536	80	7	edge	edge	NOUN
cana-536	80	8	irregularity	irregularity	NOUN
cana-536	80	9	strength	strength	NOUN
cana-536	80	10	of	of	ADP
cana-536	80	11	binomial	binomial	ADJ
cana-536	80	12	trees	tree	NOUN
cana-536	80	13	bk	bk	PRON
cana-536	80	14	is	be	AUX
cana-536	80	15	2k−1	2k−1	NUM
cana-536	80	16	,	,	PUNCT
cana-536	80	17	for	for	ADP
cana-536	80	18	k	k	PROPN
cana-536	80	19	≥	≥	PROPN
cana-536	80	20	1	1	NUM
cana-536	80	21	.	.	PUNCT
cana-536	81	1	the	the	DET
cana-536	81	2	edge	edge	NOUN
cana-536	81	3	irregularity	irregularity	NOUN
cana-536	81	4	strength	strength	NOUN
cana-536	81	5	of	of	ADP
cana-536	81	6	corona	corona	NOUN
cana-536	81	7	product	product	NOUN
cana-536	81	8	of	of	ADP
cana-536	81	9	a	a	DET
cana-536	81	10	tree	tree	NOUN
cana-536	81	11	and	and	CCONJ
cana-536	81	12	k1	k1	NOUN
cana-536	81	13	attains	attain	NOUN
cana-536	81	14	its	its	PRON
cana-536	81	15	lower	low	ADJ
cana-536	81	16	bound	bind	VERB
cana-536	81	17	provided	provide	VERB
cana-536	81	18	es	es	X
cana-536	81	19	(	(	PUNCT
cana-536	81	20	t	t	NOUN
cana-536	81	21	)	)	PUNCT
cana-536	81	22	attains	attain	VERB
cana-536	81	23	its	its	PRON
cana-536	81	24	lower	low	ADJ
cana-536	81	25	bound	bind	VERB
cana-536	81	26	.	.	PUNCT
cana-536	82	1	further	far	ADV
cana-536	82	2	,	,	PUNCT
cana-536	82	3	the	the	DET
cana-536	82	4	edge	edge	NOUN
cana-536	82	5	irregularity	irregularity	NOUN
cana-536	82	6	strength	strength	NOUN
cana-536	82	7	of	of	ADP
cana-536	82	8	binomial	binomial	ADJ
cana-536	82	9	trees	tree	NOUN
cana-536	82	10	bk	bk	ADP
cana-536	82	11	attains	attain	VERB
cana-536	82	12	its	its	PRON
cana-536	82	13	lower	low	ADJ
cana-536	82	14	bound	bind	VERB
cana-536	82	15	2k−1	2k−1	NUM
cana-536	82	16	.	.	PUNCT
cana-536	83	1	references	reference	NOUN
cana-536	83	2	[	[	X
cana-536	83	3	1	1	NUM
cana-536	83	4	]	]	X
cana-536	83	5	ali	ali	PROPN
cana-536	83	6	ahmad	ahmad	PROPN
cana-536	83	7	,	,	PUNCT
cana-536	83	8	omar	omar	PROPN
cana-536	83	9	bin	bin	PROPN
cana-536	83	10	saeed	saeed	PROPN
cana-536	83	11	al	al	PROPN
cana-536	83	12	-	-	PUNCT
cana-536	83	13	mushayt	mushayt	PROPN
cana-536	83	14	,	,	PUNCT
cana-536	83	15	martin	martin	PROPN
cana-536	83	16	baca	baca	PROPN
cana-536	83	17	,	,	PUNCT
cana-536	83	18	on	on	ADP
cana-536	83	19	edge	edge	NOUN
cana-536	83	20	irregularity	irregularity	NOUN
cana-536	83	21	strength	strength	NOUN
cana-536	83	22	of	of	ADP
cana-536	83	23	graphs	graph	NOUN
cana-536	83	24	,	,	PUNCT
cana-536	83	25	appl	appl	PROPN
cana-536	83	26	.	.	PROPN
cana-536	83	27	,	,	PUNCT
cana-536	83	28	math	math	NOUN
cana-536	83	29	.	.	PUNCT
cana-536	83	30	,	,	PUNCT
cana-536	83	31	and	and	CCONJ
cana-536	83	32	comput	comput	NOUN
cana-536	83	33	.	.	PUNCT
cana-536	83	34	,	,	PUNCT
cana-536	83	35	243	243	NUM
cana-536	83	36	,	,	PUNCT
cana-536	83	37	(	(	PUNCT
cana-536	83	38	2014	2014	NUM
cana-536	83	39	)	)	PUNCT
cana-536	83	40	,	,	PUNCT
cana-536	83	41	607	607	NUM
cana-536	83	42	-	-	SYM
cana-536	83	43	610	610	NUM
cana-536	83	44	.	.	PUNCT
cana-536	84	1	[	[	X
cana-536	84	2	2	2	NUM
cana-536	84	3	]	]	X
cana-536	84	4	d.	d.	PROPN
cana-536	84	5	amar	amar	PROPN
cana-536	84	6	,	,	PUNCT
cana-536	84	7	o.	o.	PROPN
cana-536	84	8	togni	togni	PROPN
cana-536	84	9	,	,	PUNCT
cana-536	84	10	irregularity	irregularity	NOUN
cana-536	84	11	strength	strength	NOUN
cana-536	84	12	of	of	ADP
cana-536	84	13	trees	tree	NOUN
cana-536	84	14	,	,	PUNCT
cana-536	84	15	discrete	discrete	ADJ
cana-536	84	16	math	math	NOUN
cana-536	84	17	.	.	PUNCT
cana-536	85	1	190	190	NUM
cana-536	85	2	,	,	PUNCT
cana-536	85	3	(	(	PUNCT
cana-536	85	4	1998	1998	NUM
cana-536	85	5	)	)	PUNCT
cana-536	85	6	,	,	PUNCT
cana-536	86	1	15–38	15–38	NUM
cana-536	86	2	.	.	PUNCT
cana-536	87	1	[	[	X
cana-536	87	2	3	3	X
cana-536	87	3	]	]	X
cana-536	87	4	t.	t.	NOUN
cana-536	87	5	bohman	bohman	NOUN
cana-536	87	6	,	,	PUNCT
cana-536	87	7	d.	d.	PROPN
cana-536	87	8	kravitz	kravitz	PROPN
cana-536	87	9	,	,	PUNCT
cana-536	87	10	on	on	ADP
cana-536	87	11	the	the	DET
cana-536	87	12	irregularity	irregularity	NOUN
cana-536	87	13	strength	strength	NOUN
cana-536	87	14	of	of	ADP
cana-536	87	15	trees	tree	NOUN
cana-536	87	16	,	,	PUNCT
cana-536	87	17	j.	j.	PROPN
cana-536	87	18	graph	graph	PROPN
cana-536	87	19	theory	theory	NOUN
cana-536	87	20	,	,	PUNCT
cana-536	87	21	45	45	NUM
cana-536	87	22	,	,	PUNCT
cana-536	87	23	(	(	PUNCT
cana-536	87	24	2004	2004	NUM
cana-536	87	25	)	)	PUNCT
cana-536	87	26	,	,	PUNCT
cana-536	87	27	241–254	241–254	NUM
cana-536	87	28	.	.	PUNCT
cana-536	88	1	communications	communication	NOUN
cana-536	88	2	on	on	ADP
cana-536	88	3	applied	apply	VERB
cana-536	88	4	nonlinear	nonlinear	ADJ
cana-536	88	5	analysis	analysis	NOUN
cana-536	88	6	issn	issn	NOUN
cana-536	88	7	:	:	PUNCT
cana-536	88	8	1074	1074	NUM
cana-536	88	9	-	-	PUNCT
cana-536	88	10	133x	133x	NUM
cana-536	88	11	vol	vol	NOUN
cana-536	88	12	31	31	NUM
cana-536	88	13	no	no	NOUN
cana-536	88	14	.	.	NOUN
cana-536	88	15	2	2	NUM
cana-536	88	16	(	(	PUNCT
cana-536	88	17	2024	2024	NUM
cana-536	88	18	)	)	PUNCT
cana-536	88	19	217	217	NUM
cana-536	88	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-536	89	1	[	[	X
cana-536	89	2	4	4	X
cana-536	89	3	]	]	X
cana-536	89	4	g.	g.	PROPN
cana-536	89	5	chartrand	chartrand	PROPN
cana-536	89	6	,	,	PUNCT
cana-536	89	7	m.s	m.s	PROPN
cana-536	89	8	.	.	PROPN
cana-536	89	9	jacobson	jacobson	PROPN
cana-536	89	10	,	,	PUNCT
cana-536	89	11	j.	j.	PROPN
cana-536	89	12	lehel	lehel	PROPN
cana-536	89	13	,	,	PUNCT
cana-536	89	14	o.r	o.r	PROPN
cana-536	89	15	.	.	PROPN
cana-536	89	16	oellermann	oellermann	PROPN
cana-536	89	17	,	,	PUNCT
cana-536	89	18	s.	s.	PROPN
cana-536	89	19	ruiz	ruiz	PROPN
cana-536	89	20	,	,	PUNCT
cana-536	89	21	f.	f.	PROPN
cana-536	89	22	saba	saba	PROPN
cana-536	89	23	,	,	PUNCT
cana-536	89	24	irregular	irregular	ADJ
cana-536	89	25	networks	network	NOUN
cana-536	89	26	,	,	PUNCT
cana-536	89	27	congr	congr	NOUN
cana-536	89	28	.	.	PUNCT
cana-536	90	1	numer	numer	PROPN
cana-536	90	2	.	.	PROPN
cana-536	90	3	,	,	PUNCT
cana-536	90	4	64	64	NUM
cana-536	90	5	,	,	PUNCT
cana-536	90	6	(	(	PUNCT
cana-536	90	7	1988	1988	NUM
cana-536	90	8	)	)	PUNCT
cana-536	90	9	,	,	PUNCT
cana-536	90	10	187–192	187–192	NUM
cana-536	90	11	.	.	PUNCT
cana-536	91	1	[	[	X
cana-536	91	2	5	5	X
cana-536	91	3	]	]	PUNCT
cana-536	91	4	thomas	thomas	PROPN
cana-536	91	5	cormen	cormen	PROPN
cana-536	91	6	,	,	PUNCT
cana-536	91	7	charles	charles	PROPN
cana-536	91	8	e.	e.	PROPN
cana-536	91	9	leiserson	leiserson	PROPN
cana-536	91	10	,	,	PUNCT
cana-536	91	11	ronald	ronald	PROPN
cana-536	91	12	l.	l.	PROPN
cana-536	91	13	rivest	rivest	PROPN
cana-536	91	14	,	,	PUNCT
cana-536	91	15	clifford	clifford	PROPN
cana-536	91	16	stein	stein	PROPN
cana-536	91	17	,	,	PUNCT
cana-536	91	18	introduction	introduction	NOUN
cana-536	91	19	to	to	ADP
cana-536	91	20	algorithms	algorithm	NOUN
cana-536	91	21	,	,	PUNCT
cana-536	91	22	second	second	ADJ
cana-536	91	23	edition	edition	NOUN
cana-536	91	24	,	,	PUNCT
cana-536	91	25	mit	mit	PROPN
cana-536	91	26	press	press	NOUN
cana-536	91	27	.	.	PUNCT
cana-536	92	1	[	[	X
cana-536	92	2	6	6	NUM
cana-536	92	3	]	]	PUNCT
cana-536	92	4	a.	a.	NOUN
cana-536	92	5	frieze	frieze	PROPN
cana-536	92	6	,	,	PUNCT
cana-536	92	7	r.j	r.j	PROPN
cana-536	92	8	.	.	PROPN
cana-536	92	9	gould	gould	PROPN
cana-536	92	10	,	,	PUNCT
cana-536	92	11	m.	m.	PROPN
cana-536	92	12	karonski	karonski	PROPN
cana-536	92	13	,	,	PUNCT
cana-536	92	14	f.	f.	PROPN
cana-536	92	15	pfender	pfender	PROPN
cana-536	92	16	,	,	PUNCT
cana-536	92	17	on	on	ADP
cana-536	92	18	graph	graph	NOUN
cana-536	92	19	irregularity	irregularity	NOUN
cana-536	92	20	strength	strength	NOUN
cana-536	92	21	,	,	PUNCT
cana-536	92	22	j.	j.	PROPN
cana-536	92	23	graph	graph	PROPN
cana-536	92	24	theory	theory	NOUN
cana-536	92	25	,	,	PUNCT
cana-536	92	26	41	41	NUM
cana-536	92	27	,	,	PUNCT
cana-536	92	28	(	(	PUNCT
cana-536	92	29	2002	2002	NUM
cana-536	92	30	)	)	PUNCT
cana-536	92	31	,	,	PUNCT
cana-536	92	32	120–137	120–137	NUM
cana-536	92	33	.	.	PUNCT
cana-536	93	1	[	[	X
cana-536	93	2	7	7	NUM
cana-536	93	3	]	]	PUNCT
cana-536	93	4	gallian	gallian	ADJ
cana-536	93	5	j.a	j.a	PROPN
cana-536	93	6	.	.	PROPN
cana-536	93	7	,	,	PUNCT
cana-536	93	8	a	a	DET
cana-536	93	9	dynamic	dynamic	ADJ
cana-536	93	10	survey	survey	NOUN
cana-536	93	11	of	of	ADP
cana-536	93	12	graph	graph	NOUN
cana-536	93	13	labeling	labeling	NOUN
cana-536	93	14	,	,	PUNCT
cana-536	93	15	the	the	DET
cana-536	93	16	electronic	electronic	ADJ
cana-536	93	17	journal	journal	NOUN
cana-536	93	18	of	of	ADP
cana-536	93	19	combinatorics	combinatorics	PROPN
cana-536	93	20	,	,	PUNCT
cana-536	93	21	22nd	22nd	NOUN
cana-536	93	22	edition	edition	NOUN
cana-536	93	23	,	,	PUNCT
cana-536	93	24	(	(	PUNCT
cana-536	93	25	2019	2019	NUM
cana-536	93	26	)	)	PUNCT
cana-536	93	27	,	,	PUNCT
cana-536	93	28	#	#	NOUN
cana-536	93	29	ds6	ds6	NOUN
cana-536	93	30	.	.	PUNCT
cana-536	94	1	[	[	X
cana-536	94	2	8	8	NUM
cana-536	94	3	]	]	X
cana-536	94	4	farrugia	farrugia	PROPN
cana-536	94	5	r.	r.	PROPN
cana-536	94	6	and	and	CCONJ
cana-536	94	7	harary	harary	PROPN
cana-536	94	8	f	f	X
cana-536	94	9	,	,	PUNCT
cana-536	94	10	on	on	ADP
cana-536	94	11	the	the	DET
cana-536	94	12	corona	corona	NOUN
cana-536	94	13	of	of	ADP
cana-536	94	14	two	two	NUM
cana-536	94	15	graphs	graph	NOUN
cana-536	94	16	,	,	PUNCT
cana-536	94	17	aequationes	aequatione	VERB
cana-536	94	18	math	math	PROPN
cana-536	94	19	.	.	PUNCT
cana-536	94	20	,	,	PUNCT
cana-536	94	21	4	4	NUM
cana-536	94	22	,	,	PUNCT
cana-536	94	23	(	(	PUNCT
cana-536	94	24	1970	1970	NUM
cana-536	94	25	)	)	PUNCT
cana-536	94	26	,	,	PUNCT
cana-536	94	27	322	322	NUM
cana-536	94	28	-	-	SYM
cana-536	94	29	325	325	NUM
cana-536	94	30	.	.	PUNCT
cana-536	95	1	[	[	X
cana-536	95	2	9	9	NUM
cana-536	95	3	]	]	PUNCT
cana-536	95	4	p.	p.	NOUN
cana-536	95	5	majerski	majerski	PROPN
cana-536	95	6	,	,	PUNCT
cana-536	95	7	j.	j.	PROPN
cana-536	95	8	przybylo	przybylo	PROPN
cana-536	95	9	,	,	PUNCT
cana-536	95	10	on	on	ADP
cana-536	95	11	the	the	DET
cana-536	95	12	irregularity	irregularity	NOUN
cana-536	95	13	strength	strength	NOUN
cana-536	95	14	of	of	ADP
cana-536	95	15	dense	dense	ADJ
cana-536	95	16	graphs	graph	NOUN
cana-536	95	17	,	,	PUNCT
cana-536	95	18	siam	siam	ADJ
cana-536	95	19	j.	j.	PROPN
cana-536	95	20	discrete	discrete	PROPN
cana-536	95	21	math	math	PROPN
cana-536	95	22	.	.	PUNCT
cana-536	96	1	,	,	PUNCT
cana-536	96	2	28	28	NUM
cana-536	96	3	(	(	PUNCT
cana-536	96	4	1	1	NUM
cana-536	96	5	)	)	PUNCT
cana-536	96	6	,	,	PUNCT
cana-536	96	7	(	(	PUNCT
cana-536	96	8	2014	2014	NUM
cana-536	96	9	)	)	PUNCT
cana-536	96	10	,	,	PUNCT
cana-536	96	11	197–205	197–205	NUM
cana-536	96	12	.	.	PUNCT
cana-536	97	1	[	[	X
cana-536	97	2	10	10	NUM
cana-536	97	3	]	]	X
cana-536	97	4	west	west	NOUN
cana-536	97	5	d.b	d.b	PROPN
cana-536	97	6	.	.	PROPN
cana-536	97	7	,	,	PUNCT
cana-536	97	8	introduction	introduction	NOUN
cana-536	97	9	to	to	AUX
cana-536	97	10	graph	graph	NOUN
cana-536	97	11	theory	theory	NOUN
cana-536	97	12	,	,	PUNCT
cana-536	97	13	prentice	prentice	NOUN
cana-536	97	14	hall	hall	NOUN
cana-536	97	15	of	of	ADP
cana-536	97	16	india	india	PROPN
cana-536	97	17	,	,	PUNCT
cana-536	97	18	2nd	2nd	PROPN
cana-536	97	19	edition	edition	NOUN
cana-536	97	20	,	,	PUNCT
cana-536	97	21	2001	2001	NUM
cana-536	97	22	.	.	PUNCT