id	sid	tid	token	lemma	pos
cana-3833	1	1	communications	communication	NOUN
cana-3833	1	2	on	on	ADP
cana-3833	1	3	applied	apply	VERB
cana-3833	1	4	nonlinear	nonlinear	ADJ
cana-3833	1	5	analysis	analysis	NOUN
cana-3833	1	6	issn	issn	NOUN
cana-3833	1	7	:	:	PUNCT
cana-3833	1	8	1074	1074	NUM
cana-3833	1	9	-	-	PUNCT
cana-3833	1	10	133x	133x	NUM
cana-3833	1	11	vol	vol	NOUN
cana-3833	1	12	32	32	NUM
cana-3833	1	13	no	no	NOUN
cana-3833	1	14	.	.	PUNCT
cana-3833	2	1	9s	9s	NUM
cana-3833	2	2	(	(	PUNCT
cana-3833	2	3	2025	2025	NUM
cana-3833	2	4	)	)	PUNCT
cana-3833	2	5	1	1	NUM
cana-3833	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	2	7	on	on	ADP
cana-3833	2	8	(	(	PUNCT
cana-3833	2	9	α	α	NOUN
cana-3833	2	10	,	,	PUNCT
cana-3833	2	11	β)-total	β)-total	PUNCT
cana-3833	2	12	edge	edge	NOUN
cana-3833	2	13	irregularity	irregularity	NOUN
cana-3833	2	14	strength	strength	NOUN
cana-3833	2	15	of	of	ADP
cana-3833	2	16	some	some	DET
cana-3833	2	17	graphs	graph	NOUN
cana-3833	2	18	p.	p.	NOUN
cana-3833	2	19	pandiaraj1	pandiaraj1	NOUN
cana-3833	2	20	,	,	PUNCT
cana-3833	2	21	a	a	PRON
cana-3833	2	22	,	,	PUNCT
cana-3833	2	23	k.	k.	PROPN
cana-3833	3	1	muthugurupackiam2	muthugurupackiam2	PROPN
cana-3833	3	2	*	*	PROPN
cana-3833	3	3	,	,	PUNCT
cana-3833	3	4	s.	s.	PROPN
cana-3833	3	5	anbalagan3	anbalagan3	PROPN
cana-3833	3	6	,	,	PUNCT
cana-3833	3	7	r.	r.	PROPN
cana-3833	3	8	gurusamy4	gurusamy4	PROPN
cana-3833	3	9	,	,	PUNCT
cana-3833	3	10	i.	i.	PROPN
cana-3833	3	11	muthuselvam5	muthuselvam5	PROPN
cana-3833	3	12	1	1	NUM
cana-3833	3	13	part	part	NOUN
cana-3833	3	14	time	time	NOUN
cana-3833	3	15	research	research	NOUN
cana-3833	3	16	scholar	scholar	NOUN
cana-3833	3	17	,	,	PUNCT
cana-3833	3	18	rajah	rajah	NOUN
cana-3833	3	19	serfoji	serfoji	PROPN
cana-3833	3	20	government	government	NOUN
cana-3833	3	21	college	college	NOUN
cana-3833	3	22	(	(	PUNCT
cana-3833	3	23	autonomous	autonomous	ADJ
cana-3833	3	24	)	)	PUNCT
cana-3833	3	25	,	,	PUNCT
cana-3833	3	26	affiliated	affiliate	VERB
cana-3833	3	27	to	to	PART
cana-3833	3	28	bharathidasan	bharathidasan	VERB
cana-3833	3	29	university	university	PROPN
cana-3833	3	30	,	,	PUNCT
cana-3833	3	31	tiruchirappalli	tiruchirappalli	PROPN
cana-3833	3	32	,	,	PUNCT
cana-3833	3	33	tamil	tamil	PROPN
cana-3833	3	34	nadu	nadu	PROPN
cana-3833	3	35	,	,	PUNCT
cana-3833	3	36	india	india	PROPN
cana-3833	3	37	.	.	PROPN
cana-3833	3	38	2	2	NUM
cana-3833	3	39	department	department	NOUN
cana-3833	3	40	of	of	ADP
cana-3833	3	41	mathematics	mathematic	NOUN
cana-3833	3	42	,	,	PUNCT
cana-3833	3	43	government	government	NOUN
cana-3833	3	44	arts	art	NOUN
cana-3833	3	45	and	and	CCONJ
cana-3833	3	46	science	science	PROPN
cana-3833	3	47	college	college	NOUN
cana-3833	3	48	,	,	PUNCT
cana-3833	3	49	srivilliputtur	srivilliputtur	NOUN
cana-3833	3	50	–	–	PUNCT
cana-3833	3	51	626	626	NUM
cana-3833	3	52	125	125	NUM
cana-3833	3	53	,	,	PUNCT
cana-3833	3	54	tamil	tamil	PROPN
cana-3833	3	55	nadu	nadu	PROPN
cana-3833	3	56	,	,	PUNCT
cana-3833	3	57	india	india	PROPN
cana-3833	3	58	.	.	PROPN
cana-3833	3	59	3	3	NUM
cana-3833	3	60	department	department	NOUN
cana-3833	3	61	of	of	ADP
cana-3833	3	62	mathematics	mathematics	PROPN
cana-3833	3	63	,	,	PUNCT
cana-3833	3	64	raja	raja	PROPN
cana-3833	3	65	serfoji	serfoji	PROPN
cana-3833	3	66	government	government	PROPN
cana-3833	3	67	college	college	PROPN
cana-3833	3	68	(	(	PUNCT
cana-3833	3	69	autonomous	autonomous	ADJ
cana-3833	3	70	)	)	PUNCT
cana-3833	3	71	,	,	PUNCT
cana-3833	3	72	thanjavur	thanjavur	NOUN
cana-3833	3	73	–	–	PUNCT
cana-3833	3	74	613	613	NUM
cana-3833	3	75	005	005	NUM
cana-3833	3	76	,	,	PUNCT
cana-3833	3	77	tamil	tamil	PROPN
cana-3833	3	78	nadu	nadu	PROPN
cana-3833	3	79	,	,	PUNCT
cana-3833	3	80	india	india	PROPN
cana-3833	3	81	.	.	PROPN
cana-3833	3	82	4	4	NUM
cana-3833	3	83	department	department	NOUN
cana-3833	3	84	of	of	ADP
cana-3833	3	85	mathematics	mathematic	NOUN
cana-3833	3	86	,	,	PUNCT
cana-3833	3	87	mepco	mepco	NOUN
cana-3833	3	88	schlenk	schlenk	PROPN
cana-3833	3	89	engineering	engineering	PROPN
cana-3833	3	90	college	college	PROPN
cana-3833	3	91	,	,	PUNCT
cana-3833	3	92	sivakasi	sivakasi	VERB
cana-3833	3	93	626	626	NUM
cana-3833	3	94	005	005	NUM
cana-3833	3	95	,	,	PUNCT
cana-3833	3	96	tamil	tamil	PROPN
cana-3833	3	97	nadu	nadu	PROPN
cana-3833	3	98	,	,	PUNCT
cana-3833	3	99	india	india	PROPN
cana-3833	3	100	.	.	PROPN
cana-3833	3	101	5	5	NUM
cana-3833	3	102	department	department	NOUN
cana-3833	3	103	of	of	ADP
cana-3833	3	104	mathematics	mathematic	NOUN
cana-3833	3	105	,	,	PUNCT
cana-3833	3	106	kalasalingam	kalasalingam	PROPN
cana-3833	3	107	academy	academy	PROPN
cana-3833	3	108	of	of	ADP
cana-3833	3	109	research	research	NOUN
cana-3833	3	110	and	and	CCONJ
cana-3833	3	111	education	education	NOUN
cana-3833	3	112	,	,	PUNCT
cana-3833	3	113	krishnankoil	krishnankoil	PROPN
cana-3833	3	114	–	–	PUNCT
cana-3833	3	115	626	626	NUM
cana-3833	3	116	126	126	NUM
cana-3833	3	117	,	,	PUNCT
cana-3833	3	118	tamil	tamil	PROPN
cana-3833	3	119	nadu	nadu	PROPN
cana-3833	3	120	,	,	PUNCT
cana-3833	3	121	india	india	PROPN
cana-3833	3	122	.	.	PUNCT
cana-3833	4	1	a	a	DET
cana-3833	4	2	department	department	NOUN
cana-3833	4	3	of	of	ADP
cana-3833	4	4	mathematics	mathematic	NOUN
cana-3833	4	5	,	,	PUNCT
cana-3833	4	6	kamaraj	kamaraj	ADJ
cana-3833	4	7	college	college	NOUN
cana-3833	4	8	of	of	ADP
cana-3833	4	9	engineering	engineering	NOUN
cana-3833	4	10	and	and	CCONJ
cana-3833	4	11	technology	technology	NOUN
cana-3833	4	12	,	,	PUNCT
cana-3833	4	13	virudhunagar	virudhunagar	VERB
cana-3833	4	14	625	625	NUM
cana-3833	4	15	701	701	NUM
cana-3833	4	16	,	,	PUNCT
cana-3833	4	17	tamil	tamil	PROPN
cana-3833	4	18	nadu	nadu	NOUN
cana-3833	4	19	,	,	PUNCT
cana-3833	4	20	india	india	PROPN
cana-3833	4	21	email	email	NOUN
cana-3833	4	22	:	:	PUNCT
cana-3833	4	23	1pandiaraj0@gmail.com	1pandiaraj0@gmail.com	NUM
cana-3833	4	24	,	,	PUNCT
cana-3833	4	25	2gurupackiam@yahoo.com	2gurupackiam@yahoo.com	NUM
cana-3833	4	26	,	,	PUNCT
cana-3833	4	27	3sms.anbu18@gmail.com	3sms.anbu18@gmail.com	NUM
cana-3833	4	28	,	,	PUNCT
cana-3833	4	29	4sahama2010@gmail.com	4sahama2010@gmail.com	NUM
cana-3833	4	30	,	,	PUNCT
cana-3833	4	31	5selvam08mm11@gmail.com	5selvam08mm11@gmail.com	NUM
cana-3833	4	32	article	article	NOUN
cana-3833	4	33	history	history	NOUN
cana-3833	4	34	:	:	PUNCT
cana-3833	4	35	received	receive	VERB
cana-3833	4	36	:	:	PUNCT
cana-3833	4	37	11	11	NUM
cana-3833	4	38	-	-	SYM
cana-3833	4	39	11	11	NUM
cana-3833	4	40	-	-	PUNCT
cana-3833	4	41	2024	2024	NUM
cana-3833	4	42	revised:24	revised:24	X
cana-3833	4	43	-	-	PUNCT
cana-3833	4	44	12	12	NUM
cana-3833	4	45	-	-	PUNCT
cana-3833	4	46	2024	2024	NUM
cana-3833	4	47	accepted:09	accepted:09	NOUN
cana-3833	4	48	-	-	PUNCT
cana-3833	4	49	01	01	NUM
cana-3833	4	50	-	-	PUNCT
cana-3833	4	51	2025	2025	NUM
cana-3833	4	52	abstract	abstract	NOUN
cana-3833	4	53	:	:	PUNCT
cana-3833	4	54	let	let	VERB
cana-3833	4	55	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3833	4	56	)	)	PUNCT
cana-3833	4	57	,	,	PUNCT
cana-3833	4	58	and	and	CCONJ
cana-3833	4	59	𝐸(𝐺	𝐸(𝐺	PROPN
cana-3833	4	60	)	)	PUNCT
cana-3833	4	61	having	have	VERB
cana-3833	4	62	cardinality	cardinality	NOUN
cana-3833	4	63	𝑙	𝑙	X
cana-3833	4	64	and	and	CCONJ
cana-3833	4	65	𝑚	𝑚	X
cana-3833	4	66	respectively	respectively	ADV
cana-3833	4	67	,	,	PUNCT
cana-3833	4	68	symbolize	symbolize	VERB
cana-3833	4	69	the	the	DET
cana-3833	4	70	vertex	vertex	NOUN
cana-3833	4	71	set	set	NOUN
cana-3833	4	72	,	,	PUNCT
cana-3833	4	73	and	and	CCONJ
cana-3833	4	74	edge	edge	NOUN
cana-3833	4	75	set	set	NOUN
cana-3833	4	76	of	of	ADP
cana-3833	4	77	a	a	DET
cana-3833	4	78	simple	simple	ADJ
cana-3833	4	79	graph	graph	NOUN
cana-3833	4	80	𝐺	𝐺	PROPN
cana-3833	4	81	together	together	ADV
cana-3833	4	82	with	with	ADP
cana-3833	4	83	a	a	DET
cana-3833	4	84	total	total	ADJ
cana-3833	4	85	ℏ	ℏ	NOUN
cana-3833	4	86	–	–	PUNCT
cana-3833	4	87	labeling	labeling	NOUN
cana-3833	4	88	ℓ	ℓ	NOUN
cana-3833	4	89	:	:	PUNCT
cana-3833	4	90	ℳ	ℳ	NOUN
cana-3833	4	91	→	→	SYM
cana-3833	4	92	{	{	PUNCT
cana-3833	4	93	1	1	NUM
cana-3833	4	94	,	,	PUNCT
cana-3833	4	95	2	2	NUM
cana-3833	4	96	,	,	PUNCT
cana-3833	4	97	3	3	NUM
cana-3833	4	98	,	,	PUNCT
cana-3833	4	99	…	…	PUNCT
cana-3833	4	100	,	,	PUNCT
cana-3833	4	101	ℏ	ℏ	NOUN
cana-3833	4	102	}	}	PUNCT
cana-3833	4	103	where	where	SCONJ
cana-3833	4	104	ℳ	ℳ	NOUN
cana-3833	4	105	=	=	SYM
cana-3833	4	106	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3833	4	107	)	)	PUNCT
cana-3833	4	108	∪	∪	ADP
cana-3833	4	109	𝐸(𝐺	𝐸(𝐺	PROPN
cana-3833	4	110	)	)	PUNCT
cana-3833	4	111	.	.	PUNCT
cana-3833	5	1	let	let	VERB
cana-3833	5	2	ℂ	ℂ	PROPN
cana-3833	5	3	=	=	SYM
cana-3833	5	4	{	{	PUNCT
cana-3833	5	5	𝛼	𝛼	PROPN
cana-3833	5	6	,	,	PUNCT
cana-3833	5	7	𝛼	𝛼	PROPN
cana-3833	5	8	+	+	X
cana-3833	5	9	𝛽	𝛽	NOUN
cana-3833	5	10	,	,	PUNCT
cana-3833	5	11	𝛼	𝛼	X
cana-3833	5	12	+	+	CCONJ
cana-3833	5	13	2𝛽	2𝛽	NOUN
cana-3833	5	14	,	,	PUNCT
cana-3833	5	15	…	…	PUNCT
cana-3833	5	16	,	,	PUNCT
cana-3833	5	17	𝛼	𝛼	X
cana-3833	6	1	+	+	X
cana-3833	6	2	(	(	PUNCT
cana-3833	6	3	𝑚	𝑚	PROPN
cana-3833	6	4	−	−	PROPN
cana-3833	6	5	1)𝛽	1)𝛽	NOUN
cana-3833	6	6	}	}	PUNCT
cana-3833	6	7	.	.	PUNCT
cana-3833	7	1	if	if	SCONJ
cana-3833	7	2	an	an	DET
cana-3833	7	3	invertible	invertible	ADJ
cana-3833	7	4	function	function	NOUN
cana-3833	7	5	exists	exist	VERB
cana-3833	7	6	,	,	PUNCT
cana-3833	7	7	say	say	VERB
cana-3833	7	8	ℱ	ℱ	PROPN
cana-3833	7	9	:	:	PUNCT
cana-3833	7	10	𝐸(𝐺	𝐸(𝐺	PROPN
cana-3833	7	11	)	)	PUNCT
cana-3833	7	12	→	→	SYM
cana-3833	7	13	ℂ	ℂ	PROPN
cana-3833	7	14	specified	specify	VERB
cana-3833	7	15	by	by	ADP
cana-3833	7	16	ℱ(𝒾𝒿	ℱ(𝒾𝒿	ADJ
cana-3833	7	17	)	)	PUNCT
cana-3833	7	18	=	=	SYM
cana-3833	7	19	ℓ(𝒾	ℓ(𝒾	PROPN
cana-3833	7	20	)	)	PUNCT
cana-3833	7	21	+	+	CCONJ
cana-3833	7	22	ℓ(𝒿	ℓ(𝒿	PROPN
cana-3833	7	23	)	)	PUNCT
cana-3833	8	1	+	+	CCONJ
cana-3833	9	1	ℓ(𝒾𝒿	ℓ(𝒾𝒿	NUM
cana-3833	9	2	)	)	PUNCT
cana-3833	9	3	for	for	ADP
cana-3833	9	4	every	every	DET
cana-3833	9	5	edge	edge	NOUN
cana-3833	9	6	𝒾𝒿	𝒾𝒿	NOUN
cana-3833	9	7	belongs	belong	VERB
cana-3833	9	8	to	to	ADP
cana-3833	9	9	𝐸(𝐺	𝐸(𝐺	PROPN
cana-3833	9	10	)	)	PUNCT
cana-3833	9	11	with	with	ADP
cana-3833	9	12	𝛼	𝛼	PRON
cana-3833	9	13	≥	≥	NUM
cana-3833	9	14	3	3	NUM
cana-3833	9	15	,	,	PUNCT
cana-3833	9	16	𝛽	𝛽	NOUN
cana-3833	9	17	≥	≥	NOUN
cana-3833	9	18	2	2	NUM
cana-3833	9	19	,	,	PUNCT
cana-3833	9	20	then	then	ADV
cana-3833	9	21	ℓ	ℓ	PROPN
cana-3833	9	22	is	be	AUX
cana-3833	9	23	termed	term	VERB
cana-3833	9	24	as	as	ADP
cana-3833	9	25	(	(	PUNCT
cana-3833	9	26	𝛼	𝛼	X
cana-3833	9	27	,	,	PUNCT
cana-3833	9	28	𝛽	𝛽	NOUN
cana-3833	9	29	)	)	PUNCT
cana-3833	9	30	–	–	PUNCT
cana-3833	9	31	total	total	ADJ
cana-3833	9	32	edge	edge	VERB
cana-3833	9	33	irregular	irregular	ADJ
cana-3833	9	34	labeling	labeling	NOUN
cana-3833	9	35	.	.	PUNCT
cana-3833	10	1	also	also	ADV
cana-3833	10	2	,	,	PUNCT
cana-3833	10	3	the	the	DET
cana-3833	10	4	value	value	NOUN
cana-3833	10	5	ℱ(𝒾𝒿	ℱ(𝒾𝒿	NOUN
cana-3833	10	6	)	)	PUNCT
cana-3833	10	7	is	be	AUX
cana-3833	10	8	said	say	VERB
cana-3833	10	9	to	to	PART
cana-3833	10	10	be	be	AUX
cana-3833	10	11	the	the	DET
cana-3833	10	12	edge	edge	NOUN
cana-3833	10	13	weight	weight	NOUN
cana-3833	10	14	of	of	ADP
cana-3833	10	15	𝒾𝒿.	𝒾𝒿.	NOUN
cana-3833	10	16	the	the	DET
cana-3833	10	17	least	least	ADJ
cana-3833	10	18	ℏ	ℏ	NOUN
cana-3833	10	19	for	for	ADP
cana-3833	10	20	which	which	PRON
cana-3833	10	21	𝐺	𝐺	PROPN
cana-3833	10	22	admits	admit	VERB
cana-3833	10	23	(	(	PUNCT
cana-3833	10	24	𝛼	𝛼	PROPN
cana-3833	10	25	,	,	PUNCT
cana-3833	10	26	𝛽	𝛽	NOUN
cana-3833	10	27	)	)	PUNCT
cana-3833	10	28	–	–	PUNCT
cana-3833	10	29	edge	edge	VERB
cana-3833	10	30	irregular	irregular	ADJ
cana-3833	10	31	ℏ	ℏ	PROPN
cana-3833	10	32	is	be	AUX
cana-3833	10	33	indicated	indicate	VERB
cana-3833	10	34	by	by	ADP
cana-3833	10	35	(	(	PUNCT
cana-3833	10	36	𝛼	𝛼	PROPN
cana-3833	10	37	,	,	PUNCT
cana-3833	10	38	𝛽	𝛽	NOUN
cana-3833	10	39	)	)	PUNCT
cana-3833	10	40	−	−	PRON
cana-3833	10	41	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	10	42	)	)	PUNCT
cana-3833	10	43	,	,	PUNCT
cana-3833	10	44	known	know	VERB
cana-3833	10	45	as	as	ADP
cana-3833	10	46	(	(	PUNCT
cana-3833	10	47	𝛼	𝛼	NOUN
cana-3833	10	48	,	,	PUNCT
cana-3833	10	49	𝛽	𝛽	NOUN
cana-3833	10	50	)	)	PUNCT
cana-3833	10	51	−total	−total	ADJ
cana-3833	10	52	edge	edge	NOUN
cana-3833	10	53	irregularity	irregularity	NOUN
cana-3833	10	54	strength	strength	NOUN
cana-3833	10	55	of	of	ADP
cana-3833	10	56	the	the	DET
cana-3833	10	57	graph	graph	NOUN
cana-3833	10	58	𝐺.	𝐺.	NOUN
cana-3833	10	59	the	the	DET
cana-3833	10	60	precise	precise	ADJ
cana-3833	10	61	value	value	NOUN
cana-3833	10	62	of	of	ADP
cana-3833	10	63	(	(	PUNCT
cana-3833	10	64	𝛼	𝛼	PROPN
cana-3833	10	65	,	,	PUNCT
cana-3833	10	66	𝛽	𝛽	NOUN
cana-3833	10	67	)	)	PUNCT
cana-3833	10	68	−	−	PRON
cana-3833	10	69	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	10	70	)	)	PUNCT
cana-3833	10	71	for	for	SCONJ
cana-3833	10	72	standard	standard	ADJ
cana-3833	10	73	graph	graph	NOUN
cana-3833	10	74	families	family	NOUN
cana-3833	10	75	and	and	CCONJ
cana-3833	10	76	mycielskian	mycielskian	ADJ
cana-3833	10	77	path	path	NOUN
cana-3833	10	78	graphs	graph	NOUN
cana-3833	10	79	is	be	AUX
cana-3833	10	80	explored	explore	VERB
cana-3833	10	81	in	in	ADP
cana-3833	10	82	this	this	DET
cana-3833	10	83	work	work	NOUN
cana-3833	10	84	.	.	PUNCT
cana-3833	11	1	in	in	ADP
cana-3833	11	2	addition	addition	NOUN
cana-3833	11	3	,	,	PUNCT
cana-3833	11	4	an	an	DET
cana-3833	11	5	open	open	ADJ
cana-3833	11	6	problem	problem	NOUN
cana-3833	11	7	of	of	ADP
cana-3833	11	8	(	(	PUNCT
cana-3833	11	9	3	3	NUM
cana-3833	11	10	,	,	PUNCT
cana-3833	11	11	2	2	NUM
cana-3833	11	12	)	)	PUNCT
cana-3833	11	13	–	–	PUNCT
cana-3833	11	14	𝑡𝑒𝑠(𝑃𝑛	𝑡𝑒𝑠(𝑃𝑛	NUM
cana-3833	11	15	𝑘	𝑘	NOUN
cana-3833	11	16	)	)	PUNCT
cana-3833	11	17	and	and	CCONJ
cana-3833	11	18	(	(	PUNCT
cana-3833	11	19	3	3	NUM
cana-3833	11	20	,	,	PUNCT
cana-3833	11	21	2	2	NUM
cana-3833	11	22	)	)	PUNCT
cana-3833	11	23	–	–	PUNCT
cana-3833	11	24	𝑡𝑒𝑠(𝑇	𝑡𝑒𝑠(𝑇	NUM
cana-3833	11	25	)	)	PUNCT
cana-3833	11	26	are	be	AUX
cana-3833	11	27	solved	solve	VERB
cana-3833	11	28	partially	partially	ADV
cana-3833	11	29	.	.	PUNCT
cana-3833	12	1	keywords	keyword	NOUN
cana-3833	12	2	:	:	PUNCT
cana-3833	12	3	(	(	PUNCT
cana-3833	12	4	𝛼	𝛼	NOUN
cana-3833	12	5	,	,	PUNCT
cana-3833	12	6	𝛽)irregular	𝛽)irregular	ADJ
cana-3833	12	7	labeling	labeling	NOUN
cana-3833	12	8	,	,	PUNCT
cana-3833	12	9	edge	edge	VERB
cana-3833	12	10	irregular	irregular	ADJ
cana-3833	12	11	labeling	labeling	NOUN
cana-3833	12	12	,	,	PUNCT
cana-3833	12	13	irregular	irregular	ADJ
cana-3833	12	14	labeling	labeling	NOUN
cana-3833	12	15	,	,	PUNCT
cana-3833	12	16	irregularity	irregularity	NOUN
cana-3833	12	17	strength	strength	NOUN
cana-3833	12	18	,	,	PUNCT
cana-3833	12	19	total	total	ADJ
cana-3833	12	20	edge	edge	VERB
cana-3833	12	21	irregular	irregular	ADJ
cana-3833	12	22	labeling	labeling	NOUN
cana-3833	12	23	.	.	PUNCT
cana-3833	13	1	1	1	X
cana-3833	13	2	.	.	X
cana-3833	13	3	introduction	introduction	NOUN
cana-3833	13	4	this	this	DET
cana-3833	13	5	paper	paper	NOUN
cana-3833	13	6	considers	consider	VERB
cana-3833	13	7	finite	finite	ADJ
cana-3833	13	8	simple	simple	ADJ
cana-3833	13	9	undirected	undirected	ADJ
cana-3833	13	10	graphs	graph	NOUN
cana-3833	13	11	.	.	PUNCT
cana-3833	14	1	under	under	ADP
cana-3833	14	2	certain	certain	ADJ
cana-3833	14	3	conditions	condition	NOUN
cana-3833	14	4	,	,	PUNCT
cana-3833	14	5	graph	graph	NOUN
cana-3833	14	6	labeling	labeling	NOUN
cana-3833	14	7	refers	refer	VERB
cana-3833	14	8	to	to	ADP
cana-3833	14	9	assigning	assign	VERB
cana-3833	14	10	numerical	numerical	ADJ
cana-3833	14	11	values	value	NOUN
cana-3833	14	12	to	to	PART
cana-3833	14	13	graph	graph	VERB
cana-3833	14	14	components	component	NOUN
cana-3833	14	15	like	like	ADP
cana-3833	14	16	vertices	vertex	NOUN
cana-3833	14	17	,	,	PUNCT
cana-3833	14	18	edges	edge	NOUN
cana-3833	14	19	,	,	PUNCT
cana-3833	14	20	or	or	CCONJ
cana-3833	14	21	both	both	PRON
cana-3833	14	22	.	.	PUNCT
cana-3833	15	1	these	these	DET
cana-3833	15	2	specifications	specification	NOUN
cana-3833	15	3	described	describe	VERB
cana-3833	15	4	by	by	ADP
cana-3833	15	5	means	mean	NOUN
cana-3833	15	6	of	of	ADP
cana-3833	15	7	some	some	DET
cana-3833	15	8	evaluating	evaluate	VERB
cana-3833	15	9	function	function	NOUN
cana-3833	15	10	values	value	NOUN
cana-3833	15	11	(	(	PUNCT
cana-3833	15	12	weights	weight	NOUN
cana-3833	15	13	)	)	PUNCT
cana-3833	15	14	.	.	PUNCT
cana-3833	16	1	chartrand	chartrand	NOUN
cana-3833	16	2	et	et	PROPN
cana-3833	16	3	al	al	PROPN
cana-3833	16	4	.	.	PUNCT
cana-3833	17	1	[	[	X
cana-3833	17	2	1	1	X
cana-3833	17	3	]	]	PUNCT
cana-3833	17	4	initially	initially	ADV
cana-3833	17	5	proposed	propose	VERB
cana-3833	17	6	irregular	irregular	ADJ
cana-3833	17	7	graph	graph	NOUN
cana-3833	17	8	labeling	labeling	NOUN
cana-3833	17	9	as	as	SCONJ
cana-3833	17	10	follows	follow	VERB
cana-3833	17	11	:	:	PUNCT
cana-3833	17	12	let	let	VERB
cana-3833	17	13	𝐺	𝐺	PROPN
cana-3833	17	14	=	=	SYM
cana-3833	17	15	(	(	PUNCT
cana-3833	17	16	𝑉	𝑉	PROPN
cana-3833	17	17	,	,	PUNCT
cana-3833	17	18	𝐸	𝐸	PROPN
cana-3833	17	19	)	)	PUNCT
cana-3833	17	20	be	be	VERB
cana-3833	17	21	a	a	DET
cana-3833	17	22	connected	connected	ADJ
cana-3833	17	23	graph	graph	NOUN
cana-3833	17	24	of	of	ADP
cana-3833	17	25	order	order	NOUN
cana-3833	17	26	at	at	ADV
cana-3833	17	27	least	least	ADJ
cana-3833	17	28	3	3	NUM
cana-3833	17	29	having	have	VERB
cana-3833	17	30	edge	edge	NOUN
cana-3833	17	31	ℏ	ℏ	NOUN
cana-3833	17	32	–	–	PUNCT
cana-3833	17	33	labeling	labeling	NOUN
cana-3833	17	34	ℊ	ℊ	PROPN
cana-3833	18	1	∶	∶	NOUN
cana-3833	18	2	𝐸	𝐸	X
cana-3833	18	3	→	→	SYM
cana-3833	18	4	{	{	PUNCT
cana-3833	18	5	1	1	NUM
cana-3833	18	6	,	,	PUNCT
cana-3833	18	7	2	2	NUM
cana-3833	18	8	,	,	PUNCT
cana-3833	18	9	.	.	PUNCT
cana-3833	18	10	.	.	PUNCT
cana-3833	19	1	.	.	PUNCT
cana-3833	20	1	,	,	PUNCT
cana-3833	20	2	ℏ	ℏ	NOUN
cana-3833	20	3	}	}	PUNCT
cana-3833	20	4	and	and	CCONJ
cana-3833	20	5	𝓌(𝑣	𝓌(𝑣	PROPN
cana-3833	20	6	)	)	PUNCT
cana-3833	20	7	=	=	SYM
cana-3833	20	8	∑	∑	PUNCT
cana-3833	20	9	ℊ(𝔢)𝓋∈𝔢	ℊ(𝔢)𝓋∈𝔢	NOUN
cana-3833	20	10	,	,	PUNCT
cana-3833	20	11	where	where	SCONJ
cana-3833	20	12	𝓌(𝑣	𝓌(𝑣	NOUN
cana-3833	20	13	)	)	PUNCT
cana-3833	20	14	denotes	denote	NOUN
cana-3833	20	15	vertex	vertex	NOUN
cana-3833	20	16	𝓋	𝓋	PROPN
cana-3833	20	17	weight	weight	NOUN
cana-3833	20	18	.	.	PUNCT
cana-3833	21	1	when	when	SCONJ
cana-3833	21	2	the	the	DET
cana-3833	21	3	vertex	vertex	NOUN
cana-3833	21	4	weights	weight	VERB
cana-3833	21	5	for	for	ADP
cana-3833	21	6	all	all	DET
cana-3833	21	7	vertices	vertex	NOUN
cana-3833	21	8	differ	differ	VERB
cana-3833	21	9	,	,	PUNCT
cana-3833	21	10	this	this	PRON
cana-3833	21	11	is	be	AUX
cana-3833	21	12	termed	term	VERB
cana-3833	21	13	as	as	ADP
cana-3833	21	14	irregular	irregular	ADJ
cana-3833	21	15	labeling	labeling	NOUN
cana-3833	21	16	.	.	PUNCT
cana-3833	22	1	the	the	DET
cana-3833	22	2	irregularity	irregularity	NOUN
cana-3833	22	3	strength	strength	NOUN
cana-3833	22	4	𝑠(𝐺	𝑠(𝐺	PROPN
cana-3833	22	5	)	)	PUNCT
cana-3833	22	6	is	be	AUX
cana-3833	22	7	the	the	DET
cana-3833	22	8	least	least	ADJ
cana-3833	22	9	natural	natural	ADJ
cana-3833	22	10	number	number	NOUN
cana-3833	22	11	ℏ	ℏ	NOUN
cana-3833	22	12	for	for	ADP
cana-3833	22	13	which	which	PRON
cana-3833	22	14	𝐺	𝐺	PROPN
cana-3833	22	15	reveals	reveal	VERB
cana-3833	22	16	irregular	irregular	ADJ
cana-3833	22	17	graph	graph	NOUN
cana-3833	22	18	labeling	labeling	NOUN
cana-3833	22	19	.	.	PUNCT
cana-3833	23	1	baca	baca	PROPN
cana-3833	23	2	et	et	PROPN
cana-3833	23	3	al	al	PROPN
cana-3833	23	4	.	.	PUNCT
cana-3833	24	1	[	[	X
cana-3833	24	2	2	2	X
cana-3833	24	3	]	]	PUNCT
cana-3833	24	4	reshaped	reshape	VERB
cana-3833	24	5	irregular	irregular	ADJ
cana-3833	24	6	labeling	labeling	NOUN
cana-3833	24	7	as	as	ADP
cana-3833	24	8	total	total	ADJ
cana-3833	24	9	vertex	vertex	NOUN
cana-3833	24	10	irregular	irregular	ADJ
cana-3833	24	11	labeling	labeling	NOUN
cana-3833	24	12	and	and	CCONJ
cana-3833	24	13	total	total	ADJ
cana-3833	24	14	edge	edge	NOUN
cana-3833	24	15	irregular	irregular	ADJ
cana-3833	24	16	labeling	labeling	NOUN
cana-3833	24	17	.	.	PUNCT
cana-3833	25	1	the	the	DET
cana-3833	25	2	weight	weight	NOUN
cana-3833	25	3	of	of	ADP
cana-3833	25	4	an	an	DET
cana-3833	25	5	edge	edge	NOUN
cana-3833	25	6	𝑒	𝑒	ADP
cana-3833	25	7	=	=	PUNCT
cana-3833	25	8	𝓈𝓉	𝓈𝓉	NOUN
cana-3833	25	9	∈	∈	PROPN
cana-3833	25	10	𝐸	𝐸	PROPN
cana-3833	25	11	is	be	AUX
cana-3833	25	12	defined	define	VERB
cana-3833	25	13	as	as	ADP
cana-3833	25	14	𝓌(𝑒	𝓌(𝑒	NOUN
cana-3833	25	15	)	)	PUNCT
cana-3833	25	16	=	=	SYM
cana-3833	25	17	𝒻(𝓈	𝒻(𝓈	NOUN
cana-3833	25	18	)	)	PUNCT
cana-3833	25	19	+	+	CCONJ
cana-3833	25	20	𝒻(𝓉	𝒻(𝓉	PROPN
cana-3833	25	21	)	)	PUNCT
cana-3833	26	1	+	+	CCONJ
cana-3833	26	2	𝒻(𝑒	𝒻(𝑒	PROPN
cana-3833	26	3	)	)	PUNCT
cana-3833	26	4	where	where	SCONJ
cana-3833	26	5	𝒻	𝒻	DET
cana-3833	26	6	∶	∶	NOUN
cana-3833	26	7	𝔒	𝔒	PROPN
cana-3833	26	8	→	→	SYM
cana-3833	26	9	𝔔	𝔔	PROPN
cana-3833	26	10	is	be	AUX
cana-3833	26	11	a	a	DET
cana-3833	26	12	function	function	NOUN
cana-3833	26	13	with	with	ADP
cana-3833	26	14	𝔒	𝔒	PROPN
cana-3833	26	15	=	=	SYM
cana-3833	26	16	𝑉	𝑉	PROPN
cana-3833	26	17	∪	∪	NOUN
cana-3833	26	18	𝐸	𝐸	PROPN
cana-3833	26	19	and	and	CCONJ
cana-3833	26	20	𝔔	𝔔	PROPN
cana-3833	26	21	=	=	PUNCT
cana-3833	26	22	{	{	PUNCT
cana-3833	26	23	𝜏	𝜏	NOUN
cana-3833	26	24	∶	∶	NOUN
cana-3833	26	25	1	1	NUM
cana-3833	26	26	≤	≤	NOUN
cana-3833	26	27	𝜏	𝜏	PRON
cana-3833	26	28	≤	≤	NUM
cana-3833	26	29	𝓂	𝓂	PROPN
cana-3833	26	30	and	and	CCONJ
cana-3833	26	31	𝜏𝜖ℕ	𝜏𝜖ℕ	NOUN
cana-3833	26	32	}	}	PUNCT
cana-3833	26	33	.	.	PUNCT
cana-3833	27	1	if	if	SCONJ
cana-3833	27	2	every	every	DET
cana-3833	27	3	one	one	NUM
cana-3833	27	4	of	of	ADP
cana-3833	27	5	the	the	DET
cana-3833	27	6	edge	edge	NOUN
cana-3833	27	7	weights	weight	NOUN
cana-3833	27	8	are	be	AUX
cana-3833	27	9	distinct	distinct	ADJ
cana-3833	27	10	,	,	PUNCT
cana-3833	27	11	𝒻	𝒻	PROPN
cana-3833	27	12	is	be	AUX
cana-3833	27	13	referred	refer	VERB
cana-3833	27	14	to	to	ADP
cana-3833	27	15	as	as	ADP
cana-3833	27	16	a	a	DET
cana-3833	27	17	total	total	ADJ
cana-3833	27	18	edge	edge	NOUN
cana-3833	27	19	irregular	irregular	ADJ
cana-3833	27	20	labeling	labeling	NOUN
cana-3833	27	21	.	.	PUNCT
cana-3833	28	1	the	the	DET
cana-3833	28	2	total	total	ADJ
cana-3833	28	3	edge	edge	NOUN
cana-3833	28	4	irregularity	irregularity	NOUN
cana-3833	28	5	strength	strength	NOUN
cana-3833	28	6	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	PROPN
cana-3833	28	7	)	)	PUNCT
cana-3833	28	8	is	be	AUX
cana-3833	28	9	the	the	DET
cana-3833	28	10	least	least	ADV
cana-3833	28	11	positive	positive	ADJ
cana-3833	28	12	integer	integer	NOUN
cana-3833	28	13	𝓂	𝓂	PROPN
cana-3833	28	14	for	for	ADP
cana-3833	28	15	which	which	PRON
cana-3833	28	16	total	total	ADJ
cana-3833	28	17	edge	edge	VERB
cana-3833	28	18	irregular	irregular	ADJ
cana-3833	28	19	labeling	labeling	NOUN
cana-3833	28	20	𝒻	𝒻	ADP
cana-3833	28	21	∶	∶	NOUN
cana-3833	28	22	𝔒	𝔒	PROPN
cana-3833	28	23	→	→	SYM
cana-3833	28	24	𝔔	𝔔	PROPN
cana-3833	28	25	exists	exist	VERB
cana-3833	28	26	.	.	PUNCT
cana-3833	29	1	for	for	ADP
cana-3833	29	2	a	a	DET
cana-3833	29	3	centralized	centralized	ADJ
cana-3833	29	4	uniform	uniform	ADJ
cana-3833	29	5	theta	theta	NOUN
cana-3833	29	6	graphs	graph	NOUN
cana-3833	29	7	,	,	PUNCT
cana-3833	29	8	riyan	riyan	ADJ
cana-3833	29	9	[	[	X
cana-3833	29	10	3	3	NUM
cana-3833	29	11	]	]	PUNCT
cana-3833	29	12	computed	compute	VERB
cana-3833	29	13	the	the	DET
cana-3833	29	14	total	total	ADJ
cana-3833	29	15	edge	edge	NOUN
cana-3833	29	16	irregularity	irregularity	NOUN
cana-3833	29	17	communications	communication	NOUN
cana-3833	29	18	on	on	ADP
cana-3833	29	19	applied	apply	VERB
cana-3833	29	20	nonlinear	nonlinear	ADJ
cana-3833	29	21	analysis	analysis	NOUN
cana-3833	29	22	issn	issn	NOUN
cana-3833	29	23	:	:	PUNCT
cana-3833	29	24	1074	1074	NUM
cana-3833	29	25	-	-	PUNCT
cana-3833	29	26	133x	133x	NUM
cana-3833	29	27	vol	vol	NOUN
cana-3833	29	28	32	32	NUM
cana-3833	29	29	no	no	NOUN
cana-3833	29	30	.	.	PUNCT
cana-3833	30	1	9s	9s	NUM
cana-3833	30	2	(	(	PUNCT
cana-3833	30	3	2025	2025	NUM
cana-3833	30	4	)	)	PUNCT
cana-3833	30	5	2	2	NUM
cana-3833	30	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	30	7	strength	strength	NOUN
cana-3833	30	8	(	(	PUNCT
cana-3833	30	9	tes	tes	NOUN
cana-3833	30	10	)	)	PUNCT
cana-3833	30	11	.	.	PUNCT
cana-3833	31	1	susanti	susanti	PROPN
cana-3833	31	2	y	y	PROPN
cana-3833	31	3	,	,	PUNCT
cana-3833	31	4	et	et	PROPN
cana-3833	31	5	al	al	PROPN
cana-3833	31	6	.	.	PUNCT
cana-3833	32	1	[	[	X
cana-3833	32	2	4	4	X
cana-3833	32	3	]	]	PUNCT
cana-3833	32	4	predicted	predict	VERB
cana-3833	32	5	asymmetric	asymmetric	ADJ
cana-3833	32	6	graphs	graph	NOUN
cana-3833	32	7	and	and	CCONJ
cana-3833	32	8	symmetric	symmetric	ADJ
cana-3833	32	9	graphs	graph	NOUN
cana-3833	32	10	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	32	11	)	)	PUNCT
cana-3833	32	12	.	.	PUNCT
cana-3833	33	1	rosyida	rosyida	PROPN
cana-3833	33	2	i	i	PRON
cana-3833	33	3	,	,	PUNCT
cana-3833	33	4	et	et	PROPN
cana-3833	33	5	al	al	PROPN
cana-3833	33	6	.	.	PUNCT
cana-3833	34	1	[	[	X
cana-3833	34	2	5	5	NUM
cana-3833	34	3	]	]	PUNCT
cana-3833	34	4	determined	determine	VERB
cana-3833	34	5	the	the	DET
cana-3833	34	6	exact	exact	ADJ
cana-3833	34	7	value	value	NOUN
cana-3833	34	8	of	of	ADP
cana-3833	34	9	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	PROPN
cana-3833	34	10	)	)	PUNCT
cana-3833	34	11	for	for	ADP
cana-3833	34	12	n	n	CCONJ
cana-3833	34	13	-	-	PUNCT
cana-3833	34	14	uniform	uniform	ADJ
cana-3833	34	15	cactus	cactus	NOUN
cana-3833	34	16	chain	chain	NOUN
cana-3833	34	17	graphs	graph	NOUN
cana-3833	34	18	.	.	PUNCT
cana-3833	35	1	muthu	muthu	NOUN
cana-3833	35	2	guru	guru	NOUN
cana-3833	35	3	packiam	packiam	NOUN
cana-3833	35	4	[	[	X
cana-3833	35	5	6	6	NUM
cana-3833	35	6	]	]	PUNCT
cana-3833	35	7	introduced	introduce	VERB
cana-3833	35	8	the	the	DET
cana-3833	35	9	concept	concept	NOUN
cana-3833	35	10	of	of	ADP
cana-3833	35	11	(	(	PUNCT
cana-3833	35	12	𝛼	𝛼	PROPN
cana-3833	35	13	,	,	PUNCT
cana-3833	35	14	𝛽	𝛽	NOUN
cana-3833	35	15	)	)	PUNCT
cana-3833	35	16	−	−	PRON
cana-3833	35	17	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	35	18	)	)	PUNCT
cana-3833	35	19	as	as	SCONJ
cana-3833	35	20	follows	follow	VERB
cana-3833	35	21	:	:	PUNCT
cana-3833	35	22	consider	consider	VERB
cana-3833	35	23	a	a	DET
cana-3833	35	24	simple	simple	ADJ
cana-3833	35	25	graph	graph	NOUN
cana-3833	35	26	𝐺	𝐺	NOUN
cana-3833	35	27	=	=	SYM
cana-3833	35	28	(	(	PUNCT
cana-3833	35	29	𝑉	𝑉	PROPN
cana-3833	35	30	,	,	PUNCT
cana-3833	35	31	𝐸	𝐸	PROPN
cana-3833	35	32	)	)	PUNCT
cana-3833	35	33	on	on	ADP
cana-3833	35	34	𝑙	𝑙	DET
cana-3833	35	35	vertices	vertex	NOUN
cana-3833	35	36	and	and	CCONJ
cana-3833	35	37	𝑚	𝑚	ADP
cana-3833	35	38	edges	edge	NOUN
cana-3833	35	39	together	together	ADV
cana-3833	35	40	with	with	ADP
cana-3833	35	41	a	a	DET
cana-3833	35	42	total	total	ADJ
cana-3833	35	43	ℎ	ℎ	NOUN
cana-3833	35	44	–	–	PUNCT
cana-3833	35	45	labeling	label	VERB
cana-3833	35	46	𝓅	𝓅	NOUN
cana-3833	35	47	or	or	CCONJ
cana-3833	35	48	𝜌	𝜌	ADP
cana-3833	35	49	:	:	PUNCT
cana-3833	35	50	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3833	35	51	)	)	PUNCT
cana-3833	35	52	∪	∪	ADP
cana-3833	35	53	𝐸(𝐺	𝐸(𝐺	PROPN
cana-3833	35	54	)	)	PUNCT
cana-3833	35	55	→	→	SYM
cana-3833	35	56	{	{	PUNCT
cana-3833	35	57	1	1	NUM
cana-3833	35	58	,	,	PUNCT
cana-3833	35	59	2	2	NUM
cana-3833	35	60	,	,	PUNCT
cana-3833	35	61	3	3	NUM
cana-3833	35	62	,	,	PUNCT
cana-3833	35	63	…	…	PUNCT
cana-3833	35	64	,	,	PUNCT
cana-3833	35	65	ℎ	ℎ	PROPN
cana-3833	35	66	}	}	PUNCT
cana-3833	35	67	.	.	PUNCT
cana-3833	36	1	then	then	ADV
cana-3833	36	2	𝜌	𝜌	X
cana-3833	36	3	is	be	AUX
cana-3833	36	4	called	call	VERB
cana-3833	36	5	(	(	PUNCT
cana-3833	36	6	𝛼	𝛼	PROPN
cana-3833	36	7	,	,	PUNCT
cana-3833	36	8	𝛽)–total	𝛽)–total	ADJ
cana-3833	36	9	edge	edge	VERB
cana-3833	36	10	irregular	irregular	ADJ
cana-3833	36	11	labeling	labeling	NOUN
cana-3833	36	12	if	if	SCONJ
cana-3833	36	13	there	there	PRON
cana-3833	36	14	exists	exist	VERB
cana-3833	36	15	a	a	DET
cana-3833	36	16	one	one	NUM
cana-3833	36	17	-	-	PUNCT
cana-3833	36	18	to	to	ADP
cana-3833	36	19	-	-	PUNCT
cana-3833	36	20	one	one	NUM
cana-3833	36	21	correspondence	correspondence	NOUN
cana-3833	36	22	,	,	PUNCT
cana-3833	36	23	say	say	VERB
cana-3833	36	24	𝜓	𝜓	NOUN
cana-3833	36	25	:	:	PUNCT
cana-3833	36	26	𝐸(𝐺	𝐸(𝐺	X
cana-3833	36	27	)	)	PUNCT
cana-3833	36	28	→	→	SYM
cana-3833	36	29	{	{	PUNCT
cana-3833	36	30	𝛼	𝛼	NOUN
cana-3833	36	31	,	,	PUNCT
cana-3833	36	32	𝛼	𝛼	PROPN
cana-3833	36	33	+	+	X
cana-3833	36	34	𝛽	𝛽	NOUN
cana-3833	36	35	,	,	PUNCT
cana-3833	36	36	𝛼	𝛼	X
cana-3833	36	37	+	+	CCONJ
cana-3833	36	38	2𝛽	2𝛽	NOUN
cana-3833	36	39	,	,	PUNCT
cana-3833	36	40	…	…	PUNCT
cana-3833	36	41	+	+	NUM
cana-3833	36	42	𝛼	𝛼	X
cana-3833	36	43	+	+	X
cana-3833	36	44	(	(	PUNCT
cana-3833	36	45	𝑚	𝑚	PROPN
cana-3833	36	46	−	−	PROPN
cana-3833	36	47	1)𝛽	1)𝛽	NOUN
cana-3833	36	48	}	}	PUNCT
cana-3833	36	49	defined	define	VERB
cana-3833	36	50	by	by	ADP
cana-3833	36	51	𝜓(𝑢𝑣	𝜓(𝑢𝑣	X
cana-3833	36	52	)	)	PUNCT
cana-3833	36	53	=	=	SYM
cana-3833	36	54	𝜌(𝑢	𝜌(𝑢	PROPN
cana-3833	36	55	)	)	PUNCT
cana-3833	36	56	+	+	NUM
cana-3833	36	57	𝜌(𝑣	𝜌(𝑣	NUM
cana-3833	36	58	)	)	PUNCT
cana-3833	36	59	+	+	CCONJ
cana-3833	36	60	𝜌(𝑢𝑣	𝜌(𝑢𝑣	NOUN
cana-3833	36	61	)	)	PUNCT
cana-3833	36	62	for	for	ADP
cana-3833	36	63	all	all	DET
cana-3833	36	64	𝑢𝑣	𝑢𝑣	PROPN
cana-3833	36	65	∈	∈	PROPN
cana-3833	36	66	𝐸(𝐺	𝐸(𝐺	NOUN
cana-3833	36	67	)	)	PUNCT
cana-3833	36	68	,	,	PUNCT
cana-3833	36	69	where	where	SCONJ
cana-3833	36	70	𝛼	𝛼	X
cana-3833	36	71	≥	≥	NOUN
cana-3833	36	72	3	3	NUM
cana-3833	36	73	,	,	PUNCT
cana-3833	36	74	𝛽	𝛽	NOUN
cana-3833	36	75	≥	≥	NOUN
cana-3833	36	76	2	2	NUM
cana-3833	36	77	.	.	PUNCT
cana-3833	36	78	also	also	ADV
cana-3833	36	79	,	,	PUNCT
cana-3833	36	80	the	the	DET
cana-3833	36	81	value	value	NOUN
cana-3833	36	82	𝜓(𝑢𝑣	𝜓(𝑢𝑣	VERB
cana-3833	36	83	)	)	PUNCT
cana-3833	36	84	is	be	AUX
cana-3833	36	85	said	say	VERB
cana-3833	36	86	to	to	PART
cana-3833	36	87	be	be	AUX
cana-3833	36	88	the	the	DET
cana-3833	36	89	edge	edge	NOUN
cana-3833	36	90	weight	weight	NOUN
cana-3833	36	91	of	of	ADP
cana-3833	36	92	𝑢𝑣.	𝑢𝑣.	PROPN
cana-3833	36	93	the	the	DET
cana-3833	36	94	least	least	ADJ
cana-3833	36	95	ℎ	ℎ	NOUN
cana-3833	36	96	for	for	ADP
cana-3833	36	97	which	which	PRON
cana-3833	36	98	𝐺	𝐺	PROPN
cana-3833	36	99	admits	admit	VERB
cana-3833	36	100	(	(	PUNCT
cana-3833	36	101	𝛼	𝛼	PROPN
cana-3833	36	102	,	,	PUNCT
cana-3833	36	103	𝛽	𝛽	NOUN
cana-3833	36	104	)	)	PUNCT
cana-3833	36	105	–	–	PUNCT
cana-3833	36	106	edge	edge	VERB
cana-3833	36	107	irregular	irregular	ADJ
cana-3833	36	108	ℎ	ℎ	NOUN
cana-3833	36	109	is	be	AUX
cana-3833	36	110	indicated	indicate	VERB
cana-3833	36	111	by	by	ADP
cana-3833	36	112	(	(	PUNCT
cana-3833	36	113	𝛼	𝛼	PROPN
cana-3833	36	114	,	,	PUNCT
cana-3833	36	115	𝛽	𝛽	NOUN
cana-3833	36	116	)	)	PUNCT
cana-3833	36	117	−	−	PRON
cana-3833	36	118	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	36	119	)	)	PUNCT
cana-3833	36	120	,	,	PUNCT
cana-3833	36	121	known	know	VERB
cana-3833	36	122	as	as	ADP
cana-3833	36	123	(	(	PUNCT
cana-3833	36	124	𝛼	𝛼	NOUN
cana-3833	36	125	,	,	PUNCT
cana-3833	36	126	𝛽	𝛽	NOUN
cana-3833	36	127	)	)	PUNCT
cana-3833	36	128	−total	−total	ADJ
cana-3833	36	129	edge	edge	NOUN
cana-3833	36	130	irregularity	irregularity	NOUN
cana-3833	36	131	strength	strength	NOUN
cana-3833	36	132	of	of	ADP
cana-3833	36	133	the	the	DET
cana-3833	36	134	graph	graph	NOUN
cana-3833	36	135	𝐺.	𝐺.	NOUN
cana-3833	36	136	additionally	additionally	ADV
cana-3833	36	137	,	,	PUNCT
cana-3833	36	138	they	they	PRON
cana-3833	36	139	presented	present	VERB
cana-3833	36	140	upper	upper	ADJ
cana-3833	36	141	and	and	CCONJ
cana-3833	36	142	lower	low	ADJ
cana-3833	36	143	limitations	limitation	NOUN
cana-3833	36	144	for	for	ADP
cana-3833	36	145	the	the	DET
cana-3833	36	146	parameter	parameter	NOUN
cana-3833	36	147	and	and	CCONJ
cana-3833	36	148	established	establish	VERB
cana-3833	36	149	the	the	DET
cana-3833	36	150	evaluations	evaluation	NOUN
cana-3833	36	151	of	of	ADP
cana-3833	36	152	(	(	PUNCT
cana-3833	36	153	𝛼	𝛼	PROPN
cana-3833	36	154	,	,	PUNCT
cana-3833	36	155	𝛽	𝛽	NOUN
cana-3833	36	156	)	)	PUNCT
cana-3833	36	157	−	−	PRON
cana-3833	36	158	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	36	159	)	)	PUNCT
cana-3833	36	160	for	for	ADP
cana-3833	36	161	some	some	DET
cana-3833	36	162	graph	graph	NOUN
cana-3833	36	163	families	family	NOUN
cana-3833	36	164	.	.	PUNCT
cana-3833	37	1	the	the	DET
cana-3833	37	2	purpose	purpose	NOUN
cana-3833	37	3	of	of	ADP
cana-3833	37	4	this	this	DET
cana-3833	37	5	paper	paper	NOUN
cana-3833	37	6	is	be	AUX
cana-3833	37	7	to	to	PART
cana-3833	37	8	focus	focus	VERB
cana-3833	37	9	on	on	ADP
cana-3833	37	10	the	the	DET
cana-3833	37	11	study	study	NOUN
cana-3833	37	12	of	of	ADP
cana-3833	37	13	(	(	PUNCT
cana-3833	37	14	𝛼	𝛼	PROPN
cana-3833	37	15	,	,	PUNCT
cana-3833	37	16	𝛽	𝛽	NOUN
cana-3833	37	17	)	)	PUNCT
cana-3833	37	18	−	−	PRON
cana-3833	37	19	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	37	20	)	)	PUNCT
cana-3833	37	21	.	.	PUNCT
cana-3833	38	1	subdivision	subdivision	NOUN
cana-3833	38	2	of	of	ADP
cana-3833	38	3	a	a	DET
cana-3833	38	4	graph	graph	NOUN
cana-3833	38	5	g	g	NOUN
cana-3833	38	6	that	that	PRON
cana-3833	38	7	results	result	VERB
cana-3833	38	8	from	from	ADP
cana-3833	38	9	dividing	divide	VERB
cana-3833	38	10	each	each	DET
cana-3833	38	11	edge	edge	NOUN
cana-3833	38	12	by	by	ADP
cana-3833	38	13	a	a	DET
cana-3833	38	14	vertex	vertex	NOUN
cana-3833	38	15	[	[	X
cana-3833	38	16	7	7	NUM
cana-3833	38	17	]	]	PUNCT
cana-3833	38	18	.	.	PUNCT
cana-3833	39	1	in	in	ADP
cana-3833	39	2	a	a	DET
cana-3833	39	3	graph	graph	NOUN
cana-3833	39	4	g	g	NOUN
cana-3833	39	5	,	,	PUNCT
cana-3833	39	6	attaching	attach	VERB
cana-3833	39	7	two	two	NUM
cana-3833	39	8	pendant	pendant	ADJ
cana-3833	39	9	vertices	vertex	NOUN
cana-3833	39	10	to	to	ADP
cana-3833	39	11	each	each	DET
cana-3833	39	12	all	all	DET
cana-3833	39	13	vertices	vertex	NOUN
cana-3833	39	14	of	of	ADP
cana-3833	39	15	g	g	PROPN
cana-3833	39	16	is	be	AUX
cana-3833	39	17	denoted	denote	VERB
cana-3833	39	18	by	by	ADP
cana-3833	39	19	g	g	PROPN
cana-3833	39	20	∘	∘	PROPN
cana-3833	39	21	s2	s2	PROPN
cana-3833	39	22	.	.	PUNCT
cana-3833	40	1	middle	middle	ADJ
cana-3833	40	2	graph	graph	NOUN
cana-3833	40	3	of	of	ADP
cana-3833	40	4	a	a	DET
cana-3833	40	5	graph	graph	NOUN
cana-3833	40	6	𝐺	𝐺	NOUN
cana-3833	40	7	=	=	SYM
cana-3833	40	8	(	(	PUNCT
cana-3833	40	9	𝕍	𝕍	NOUN
cana-3833	40	10	,	,	PUNCT
cana-3833	40	11	𝔼	𝔼	PROPN
cana-3833	40	12	)	)	PUNCT
cana-3833	40	13	denoted	denote	VERB
cana-3833	40	14	by	by	ADP
cana-3833	40	15	𝑀(𝐺	𝑀(𝐺	PROPN
cana-3833	40	16	)	)	PUNCT
cana-3833	40	17	has	have	VERB
cana-3833	40	18	vertex	vertex	NOUN
cana-3833	40	19	set	set	VERB
cana-3833	40	20	𝕍	𝕍	NOUN
cana-3833	40	21	∪	∪	ADJ
cana-3833	40	22	𝔼	𝔼	NOUN
cana-3833	40	23	and	and	CCONJ
cana-3833	40	24	a	a	DET
cana-3833	40	25	pair	pair	NOUN
cana-3833	40	26	of	of	ADP
cana-3833	40	27	vertices	vertex	NOUN
cana-3833	40	28	are	be	AUX
cana-3833	40	29	adjacent	adjacent	ADJ
cana-3833	40	30	if	if	SCONJ
cana-3833	41	1	and	and	CCONJ
cana-3833	41	2	only	only	ADV
cana-3833	41	3	if	if	SCONJ
cana-3833	41	4	they	they	PRON
cana-3833	41	5	are	be	AUX
cana-3833	41	6	incident	incident	NOUN
cana-3833	41	7	with	with	ADP
cana-3833	41	8	one	one	NUM
cana-3833	41	9	other	other	ADJ
cana-3833	41	10	or	or	CCONJ
cana-3833	41	11	adjacent	adjacent	ADJ
cana-3833	41	12	edges	edge	NOUN
cana-3833	41	13	.	.	PUNCT
cana-3833	42	1	any	any	DET
cana-3833	42	2	two	two	NUM
cana-3833	42	3	adjacent	adjacent	ADJ
cana-3833	42	4	vertices	vertex	NOUN
cana-3833	42	5	of	of	ADP
cana-3833	42	6	𝐺	𝐺	PROPN
cana-3833	42	7	are	be	AUX
cana-3833	42	8	adjacent	adjacent	ADJ
cana-3833	42	9	in	in	ADP
cana-3833	42	10	the	the	DET
cana-3833	42	11	middle	middle	ADJ
cana-3833	42	12	graph	graph	NOUN
cana-3833	42	13	of	of	ADP
cana-3833	42	14	𝐺	𝐺	PROPN
cana-3833	42	15	is	be	AUX
cana-3833	42	16	named	name	VERB
cana-3833	42	17	as	as	ADP
cana-3833	42	18	total	total	ADJ
cana-3833	42	19	graph	graph	NOUN
cana-3833	42	20	𝑇(𝐺	𝑇(𝐺	NOUN
cana-3833	42	21	)	)	PUNCT
cana-3833	42	22	of	of	ADP
cana-3833	42	23	a	a	DET
cana-3833	42	24	graph	graph	NOUN
cana-3833	42	25	𝐺.	𝐺.	NOUN
cana-3833	42	26	the	the	DET
cana-3833	42	27	vertices	vertex	NOUN
cana-3833	42	28	of	of	ADP
cana-3833	42	29	path	path	NOUN
cana-3833	42	30	graph	graph	NOUN
cana-3833	42	31	𝑃	𝑃	NOUN
cana-3833	42	32	𝑛	𝑛	PROPN
cana-3833	42	33	are	be	AUX
cana-3833	42	34	𝓋1	𝓋1	NOUN
cana-3833	42	35	,	,	PUNCT
cana-3833	42	36	𝓋2	𝓋2	PROPN
cana-3833	42	37	,	,	PUNCT
cana-3833	42	38	…	…	PUNCT
cana-3833	42	39	,	,	PUNCT
cana-3833	42	40	𝓋𝑛.	𝓋𝑛.	PROPN
cana-3833	42	41	graph	graph	NOUN
cana-3833	42	42	𝑃	𝑃	NOUN
cana-3833	42	43	𝑛	𝑛	PRON
cana-3833	42	44	3	3	NUM
cana-3833	42	45	can	can	AUX
cana-3833	42	46	be	be	AUX
cana-3833	42	47	constructed	construct	VERB
cana-3833	42	48	from	from	ADP
cana-3833	42	49	𝑃	𝑃	NOUN
cana-3833	42	50	𝑛	𝑛	PROPN
cana-3833	42	51	by	by	ADP
cana-3833	42	52	connecting	connect	VERB
cana-3833	42	53	every	every	DET
cana-3833	42	54	𝓋𝑖	𝓋𝑖	NOUN
cana-3833	42	55	𝑡ℎ	𝑡ℎ	NOUN
cana-3833	42	56	vertex	vertex	NOUN
cana-3833	42	57	to	to	ADP
cana-3833	42	58	𝓋𝑖+2	𝓋𝑖+2	NOUN
cana-3833	42	59	𝑡ℎ	𝑡ℎ	PROPN
cana-3833	42	60	vertex	vertex	NOUN
cana-3833	42	61	and	and	CCONJ
cana-3833	42	62	every	every	DET
cana-3833	42	63	𝓋𝑖	𝓋𝑖	NOUN
cana-3833	42	64	𝑡ℎvertex	𝑡ℎvertex	ADJ
cana-3833	42	65	to	to	ADP
cana-3833	42	66	the	the	DET
cana-3833	42	67	𝓋𝑖+3	𝓋𝑖+3	PROPN
cana-3833	42	68	𝑡ℎ	𝑡ℎ	NOUN
cana-3833	42	69	vertex	vertex	NOUN
cana-3833	42	70	by	by	ADP
cana-3833	42	71	an	an	DET
cana-3833	42	72	edge	edge	NOUN
cana-3833	42	73	.	.	PUNCT
cana-3833	43	1	if	if	SCONJ
cana-3833	43	2	all	all	DET
cana-3833	43	3	unordered	unordered	ADJ
cana-3833	43	4	pairs	pair	NOUN
cana-3833	43	5	of	of	ADP
cana-3833	43	6	𝑛	𝑛	DET
cana-3833	43	7	vertices	vertex	NOUN
cana-3833	43	8	are	be	AUX
cana-3833	43	9	adjacent	adjacent	ADJ
cana-3833	43	10	in	in	ADP
cana-3833	43	11	𝐺	𝐺	PROPN
cana-3833	43	12	is	be	AUX
cana-3833	43	13	termed	term	VERB
cana-3833	43	14	as	as	ADP
cana-3833	43	15	complete	complete	ADJ
cana-3833	43	16	graph	graph	NOUN
cana-3833	43	17	[	[	X
cana-3833	43	18	8	8	NUM
cana-3833	43	19	]	]	PUNCT
cana-3833	43	20	denoted	denote	VERB
cana-3833	43	21	by	by	ADP
cana-3833	43	22	𝐾𝑛.	𝐾𝑛.	PROPN
cana-3833	43	23	if	if	SCONJ
cana-3833	43	24	a	a	DET
cana-3833	43	25	vertex	vertex	NOUN
cana-3833	43	26	set	set	NOUN
cana-3833	43	27	is	be	AUX
cana-3833	43	28	separated	separate	VERB
cana-3833	43	29	like	like	ADP
cana-3833	43	30	𝕌	𝕌	NOUN
cana-3833	43	31	and	and	CCONJ
cana-3833	43	32	𝕍	𝕍	NOUN
cana-3833	43	33	with	with	ADP
cana-3833	43	34	|𝕌|	|𝕌|	NOUN
cana-3833	43	35	=	=	NUM
cana-3833	43	36	𝑎	𝑎	PROPN
cana-3833	43	37	and	and	CCONJ
cana-3833	43	38	|𝕍|=	|𝕍|=	NOUN
cana-3833	43	39	𝑏	𝑏	PROPN
cana-3833	43	40	such	such	ADJ
cana-3833	43	41	that	that	SCONJ
cana-3833	43	42	every	every	DET
cana-3833	43	43	vertex	vertex	NOUN
cana-3833	43	44	of	of	ADP
cana-3833	43	45	𝕌	𝕌	PROPN
cana-3833	43	46	is	be	AUX
cana-3833	43	47	adjacent	adjacent	ADJ
cana-3833	43	48	to	to	ADP
cana-3833	43	49	every	every	DET
cana-3833	43	50	vertex	vertex	NOUN
cana-3833	43	51	of	of	ADP
cana-3833	43	52	𝕍	𝕍	PROPN
cana-3833	43	53	is	be	AUX
cana-3833	43	54	referred	refer	VERB
cana-3833	43	55	as	as	ADP
cana-3833	43	56	𝐾𝑎,𝑏complete	𝐾𝑎,𝑏complete	PROPN
cana-3833	43	57	bipartite	bipartite	PROPN
cana-3833	43	58	graph	graph	NOUN
cana-3833	43	59	.	.	PUNCT
cana-3833	44	1	in	in	ADP
cana-3833	44	2	case	case	NOUN
cana-3833	44	3	𝑎	𝑎	X
cana-3833	44	4	=	=	SYM
cana-3833	44	5	1	1	NUM
cana-3833	44	6	𝑜𝑟	𝑜𝑟	PRON
cana-3833	44	7	b	b	NOUN
cana-3833	44	8	=	=	SYM
cana-3833	44	9	1	1	NUM
cana-3833	44	10	in	in	ADP
cana-3833	44	11	𝐾𝑎,𝑏	𝐾𝑎,𝑏	PROPN
cana-3833	44	12	is	be	AUX
cana-3833	44	13	named	name	VERB
cana-3833	44	14	as	as	ADP
cana-3833	44	15	star	star	NOUN
cana-3833	44	16	graph	graph	NOUN
cana-3833	44	17	𝐾1,𝑏	𝐾1,𝑏	PROPN
cana-3833	44	18	𝑜𝑟	𝑜𝑟	X
cana-3833	44	19	𝐾𝑎,1	𝐾𝑎,1	PROPN
cana-3833	44	20	.	.	PUNCT
cana-3833	45	1	the	the	DET
cana-3833	45	2	central	central	ADJ
cana-3833	45	3	vertex	vertex	NOUN
cana-3833	45	4	of	of	ADP
cana-3833	45	5	two	two	NUM
cana-3833	45	6	copies	copy	NOUN
cana-3833	45	7	𝐾1,𝓃	𝐾1,𝓃	PROPN
cana-3833	45	8	is	be	AUX
cana-3833	45	9	joined	join	VERB
cana-3833	45	10	by	by	ADP
cana-3833	45	11	an	an	DET
cana-3833	45	12	edge	edge	NOUN
cana-3833	45	13	is	be	AUX
cana-3833	45	14	𝐵(𝓃	𝐵(𝓃	NOUN
cana-3833	45	15	,	,	PUNCT
cana-3833	45	16	𝓃	𝓃	NOUN
cana-3833	45	17	)	)	PUNCT
cana-3833	45	18	bistar	bistar	NOUN
cana-3833	45	19	graph	graph	NOUN
cana-3833	45	20	.	.	PUNCT
cana-3833	46	1	mycielskian	mycielskian	PROPN
cana-3833	47	1	[	[	X
cana-3833	47	2	9	9	NUM
cana-3833	47	3	]	]	PUNCT
cana-3833	47	4	µ(𝐺	µ(𝐺	PROPN
cana-3833	47	5	)	)	PUNCT
cana-3833	47	6	of	of	ADP
cana-3833	47	7	a	a	DET
cana-3833	47	8	graph	graph	NOUN
cana-3833	47	9	𝐺	𝐺	NOUN
cana-3833	47	10	=	=	SYM
cana-3833	47	11	(	(	PUNCT
cana-3833	47	12	𝕍	𝕍	NOUN
cana-3833	47	13	,	,	PUNCT
cana-3833	47	14	𝔼	𝔼	PROPN
cana-3833	47	15	)	)	PUNCT
cana-3833	47	16	has	have	VERB
cana-3833	47	17	vertex	vertex	NOUN
cana-3833	47	18	set	set	VERB
cana-3833	47	19	𝕍	𝕍	NOUN
cana-3833	47	20	∪	∪	VERB
cana-3833	47	21	𝕍	𝕍	NOUN
cana-3833	47	22	′	′	NOUN
cana-3833	47	23	∪	∪	X
cana-3833	47	24	{	{	PUNCT
cana-3833	47	25	𝑢	𝑢	NOUN
cana-3833	47	26	}	}	PUNCT
cana-3833	47	27	,	,	PUNCT
cana-3833	47	28	where	where	SCONJ
cana-3833	47	29	𝕍	𝕍	NOUN
cana-3833	47	30	′	′	NOUN
cana-3833	48	1	=	=	PUNCT
cana-3833	48	2	{	{	PUNCT
cana-3833	48	3	𝓋𝑖′	𝓋𝑖′	X
cana-3833	48	4	∶	∶	NOUN
cana-3833	48	5	𝓋𝑖	𝓋𝑖	NOUN
cana-3833	48	6	∈	∈	PROPN
cana-3833	48	7	𝕍	𝕍	NOUN
cana-3833	48	8	}	}	PUNCT
cana-3833	48	9	and	and	CCONJ
cana-3833	48	10	edge	edge	VERB
cana-3833	48	11	set	set	VERB
cana-3833	48	12	𝔼	𝔼	PROPN
cana-3833	48	13	∪	∪	NOUN
cana-3833	48	14	{	{	PUNCT
cana-3833	48	15	𝓋𝑖𝓋𝑗′	𝓋𝑖𝓋𝑗′	NOUN
cana-3833	48	16	∶	∶	PROPN
cana-3833	48	17	𝓋𝑖𝓋𝑗	𝓋𝑖𝓋𝑗	NOUN
cana-3833	48	18	∈	∈	PROPN
cana-3833	48	19	𝔼	𝔼	PROPN
cana-3833	48	20	}	}	PUNCT
cana-3833	48	21	∪	∪	NOUN
cana-3833	48	22	{	{	PUNCT
cana-3833	48	23	𝓋𝑖′𝑢	𝓋𝑖′𝑢	NOUN
cana-3833	48	24	∶	∶	NOUN
cana-3833	48	25	𝓋𝑖′	𝓋𝑖′	ADP
cana-3833	48	26	∈	∈	PROPN
cana-3833	48	27	𝕍	𝕍	NOUN
cana-3833	48	28	′	′	NOUN
cana-3833	48	29	}	}	PUNCT
cana-3833	48	30	.	.	PUNCT
cana-3833	49	1	proposition	proposition	NOUN
cana-3833	49	2	1	1	NUM
cana-3833	50	1	[	[	X
cana-3833	50	2	6	6	NUM
cana-3833	50	3	]	]	PUNCT
cana-3833	50	4	:	:	PUNCT
cana-3833	50	5	let	let	VERB
cana-3833	50	6	𝐺	𝐺	PRON
cana-3833	50	7	be	be	AUX
cana-3833	50	8	a	a	DET
cana-3833	50	9	graph	graph	NOUN
cana-3833	50	10	with	with	ADP
cana-3833	50	11	𝑞	𝑞	PROPN
cana-3833	50	12	edges	edge	NOUN
cana-3833	50	13	and	and	CCONJ
cana-3833	50	14	𝑝	𝑝	NOUN
cana-3833	50	15	vertices	vertex	NOUN
cana-3833	50	16	.	.	PUNCT
cana-3833	51	1	then	then	ADV
cana-3833	51	2	⌈	⌈	PROPN
cana-3833	51	3	𝛼+(𝑞−1)𝛽	𝛼+(𝑞−1)𝛽	NUM
cana-3833	51	4	3	3	NUM
cana-3833	51	5	⌉	⌉	X
cana-3833	51	6	≤	≤	X
cana-3833	51	7	(	(	PUNCT
cana-3833	51	8	𝛼	𝛼	X
cana-3833	51	9	,	,	PUNCT
cana-3833	51	10	𝛽	𝛽	NOUN
cana-3833	51	11	)	)	PUNCT
cana-3833	51	12	−	−	PRON
cana-3833	51	13	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	51	14	)	)	PUNCT
cana-3833	51	15	≤	≤	NOUN
cana-3833	52	1	𝛼	𝛼	X
cana-3833	52	2	−	−	PROPN
cana-3833	52	3	2	2	NUM
cana-3833	52	4	+	+	CCONJ
cana-3833	52	5	(	(	PUNCT
cana-3833	52	6	𝑞	𝑞	X
cana-3833	52	7	−	−	PROPN
cana-3833	52	8	1)𝛽	1)𝛽	NUM
cana-3833	52	9	for	for	ADP
cana-3833	52	10	any	any	DET
cana-3833	52	11	two	two	NUM
cana-3833	52	12	natural	natural	ADJ
cana-3833	52	13	number	number	NOUN
cana-3833	52	14	𝛼	𝛼	NOUN
cana-3833	52	15	≥	≥	NOUN
cana-3833	52	16	3	3	NUM
cana-3833	52	17	and	and	CCONJ
cana-3833	52	18	𝛽	𝛽	NOUN
cana-3833	52	19	≥	≥	NOUN
cana-3833	52	20	2	2	NUM
cana-3833	52	21	.	.	PUNCT
cana-3833	52	22	proposition	proposition	NOUN
cana-3833	52	23	2	2	NUM
cana-3833	53	1	[	[	X
cana-3833	53	2	10	10	NUM
cana-3833	53	3	]	]	SYM
cana-3833	53	4	:	:	PUNCT
cana-3833	53	5	(	(	PUNCT
cana-3833	53	6	3	3	NUM
cana-3833	53	7	,	,	PUNCT
cana-3833	53	8	2	2	NUM
cana-3833	53	9	)	)	PUNCT
cana-3833	53	10	−	−	PROPN
cana-3833	53	11	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	53	12	𝑛	𝑛	NOUN
cana-3833	53	13	)	)	PUNCT
cana-3833	53	14	=	=	PUNCT
cana-3833	54	1	⌈	⌈	SYM
cana-3833	54	2	2𝑛−1	2𝑛−1	NUM
cana-3833	54	3	3	3	NUM
cana-3833	54	4	⌉	⌉	X
cana-3833	54	5	for	for	ADP
cana-3833	54	6	path	path	NOUN
cana-3833	54	7	𝑃𝑛.	𝑃𝑛.	PROPN
cana-3833	54	8	2	2	NUM
cana-3833	54	9	.	.	NOUN
cana-3833	54	10	results	result	NOUN
cana-3833	54	11	and	and	CCONJ
cana-3833	54	12	discussion	discussion	NOUN
cana-3833	54	13	theorem	theorem	VERB
cana-3833	54	14	1	1	NUM
cana-3833	54	15	:	:	PUNCT
cana-3833	54	16	(	(	PUNCT
cana-3833	54	17	3	3	NUM
cana-3833	54	18	,	,	PUNCT
cana-3833	54	19	2	2	NUM
cana-3833	54	20	)	)	PUNCT
cana-3833	54	21	–	–	PUNCT
cana-3833	54	22	𝑡𝑒𝑠(𝑆(𝑃	𝑡𝑒𝑠(𝑆(𝑃	NUM
cana-3833	54	23	𝑛	𝑛	PROPN
cana-3833	54	24	∘	∘	PROPN
cana-3833	54	25	𝑆2	𝑆2	PROPN
cana-3833	54	26	)	)	PUNCT
cana-3833	54	27	)	)	PUNCT
cana-3833	55	1	=	=	PUNCT
cana-3833	55	2	4𝑛	4𝑛	NOUN
cana-3833	55	3	−	−	NOUN
cana-3833	56	1	1	1	X
cana-3833	56	2	.	.	PUNCT
cana-3833	57	1	proof	proof	NOUN
cana-3833	57	2	.	.	PUNCT
cana-3833	58	1	consider	consider	VERB
cana-3833	58	2	the	the	DET
cana-3833	58	3	index	index	NOUN
cana-3833	58	4	sets	set	VERB
cana-3833	58	5	𝕀	𝕀	PROPN
cana-3833	58	6	=	=	SYM
cana-3833	58	7	{	{	PUNCT
cana-3833	58	8	1	1	NUM
cana-3833	58	9	,	,	PUNCT
cana-3833	58	10	2	2	NUM
cana-3833	58	11	,	,	PUNCT
cana-3833	58	12	…	…	PUNCT
cana-3833	58	13	,	,	PUNCT
cana-3833	58	14	𝑛	𝑛	NOUN
cana-3833	58	15	}	}	PUNCT
cana-3833	58	16	and	and	CCONJ
cana-3833	58	17	𝒥	𝒥	PROPN
cana-3833	58	18	=	=	PUNCT
cana-3833	58	19	{	{	PUNCT
cana-3833	58	20	1	1	NUM
cana-3833	58	21	,	,	PUNCT
cana-3833	58	22	2	2	NUM
cana-3833	58	23	,	,	PUNCT
cana-3833	58	24	…	…	PUNCT
cana-3833	58	25	,	,	PUNCT
cana-3833	58	26	𝑛	𝑛	DET
cana-3833	58	27	−	−	NOUN
cana-3833	58	28	1	1	NUM
cana-3833	58	29	}	}	PUNCT
cana-3833	58	30	.	.	PUNCT
cana-3833	59	1	let	let	VERB
cana-3833	59	2	the	the	DET
cana-3833	59	3	vertex	vertex	NOUN
cana-3833	59	4	set	set	NOUN
cana-3833	59	5	of	of	ADP
cana-3833	59	6	𝑆(𝑃	𝑆(𝑃	PROPN
cana-3833	59	7	𝑛	𝑛	PROPN
cana-3833	59	8	∘	∘	PROPN
cana-3833	59	9	𝑆2	𝑆2	PROPN
cana-3833	59	10	)	)	PUNCT
cana-3833	60	1	=	=	PUNCT
cana-3833	61	1	⋃	⋃	ADP
cana-3833	61	2	𝕍𝜎	𝕍𝜎	PROPN
cana-3833	61	3	6	6	NUM
cana-3833	61	4	𝜎=1	𝜎=1	ADP
cana-3833	61	5	where	where	SCONJ
cana-3833	61	6	𝑉1	𝑉1	NOUN
cana-3833	61	7	=	=	SYM
cana-3833	61	8	{	{	PUNCT
cana-3833	61	9	𝑢𝔦	𝑢𝔦	NOUN
cana-3833	61	10	}	}	PUNCT
cana-3833	61	11	,	,	PUNCT
cana-3833	61	12	𝑉2	𝑉2	NOUN
cana-3833	61	13	=	=	SYM
cana-3833	61	14	{	{	PUNCT
cana-3833	61	15	𝑣𝔦	𝑣𝔦	NOUN
cana-3833	61	16	}	}	PUNCT
cana-3833	61	17	,	,	PUNCT
cana-3833	61	18	𝑉3	𝑉3	NOUN
cana-3833	61	19	=	=	SYM
cana-3833	61	20	{	{	PUNCT
cana-3833	61	21	𝑤𝔦	𝑤𝔦	VERB
cana-3833	61	22	}	}	PUNCT
cana-3833	61	23	,	,	PUNCT
cana-3833	61	24	𝑉4	𝑉4	NOUN
cana-3833	61	25	=	=	SYM
cana-3833	61	26	{	{	PUNCT
cana-3833	61	27	𝑥𝒿	𝑥𝒿	PROPN
cana-3833	61	28	}	}	PUNCT
cana-3833	61	29	,	,	PUNCT
cana-3833	61	30	𝑉5	𝑉5	PROPN
cana-3833	61	31	=	=	SYM
cana-3833	61	32	{	{	PUNCT
cana-3833	61	33	𝑦𝔦	𝑦𝔦	NOUN
cana-3833	61	34	}	}	PUNCT
cana-3833	61	35	and	and	CCONJ
cana-3833	61	36	𝑉6	𝑉6	NOUN
cana-3833	61	37	=	=	SYM
cana-3833	61	38	{	{	PUNCT
cana-3833	61	39	𝑧𝔦	𝑧𝔦	NOUN
cana-3833	61	40	}	}	PUNCT
cana-3833	61	41	and	and	CCONJ
cana-3833	61	42	edge	edge	VERB
cana-3833	61	43	set	set	NOUN
cana-3833	61	44	of	of	ADP
cana-3833	61	45	(	(	PUNCT
cana-3833	61	46	𝑆(𝑃	𝑆(𝑃	NOUN
cana-3833	61	47	𝑛	𝑛	DET
cana-3833	61	48	∘	∘	PROPN
cana-3833	61	49	𝑆2	𝑆2	PROPN
cana-3833	61	50	)	)	PUNCT
cana-3833	61	51	)	)	PUNCT
cana-3833	62	1	=	=	PUNCT
cana-3833	63	1	⋃	⋃	ADP
cana-3833	63	2	𝔼𝜑	𝔼𝜑	PROPN
cana-3833	63	3	6	6	NUM
cana-3833	63	4	𝜑=1	𝜑=1	PUNCT
cana-3833	63	5	where	where	SCONJ
cana-3833	63	6	𝔼1	𝔼1	NOUN
cana-3833	63	7	=	=	SYM
cana-3833	63	8	{	{	PUNCT
cana-3833	63	9	𝑢𝔦𝑥𝔦	𝑢𝔦𝑥𝔦	NOUN
cana-3833	63	10	}	}	PUNCT
cana-3833	63	11	,	,	PUNCT
cana-3833	63	12	𝔼2	𝔼2	NOUN
cana-3833	63	13	=	=	SYM
cana-3833	63	14	{	{	PUNCT
cana-3833	63	15	𝑥𝒿𝑢𝒿+1	𝑥𝒿𝑢𝒿+1	NOUN
cana-3833	63	16	}	}	PUNCT
cana-3833	63	17	,	,	PUNCT
cana-3833	63	18	𝔼3	𝔼3	NOUN
cana-3833	63	19	=	=	SYM
cana-3833	63	20	{	{	PUNCT
cana-3833	63	21	𝑢𝔦𝑦𝔦	𝑢𝔦𝑦𝔦	ADJ
cana-3833	63	22	}	}	PUNCT
cana-3833	63	23	,	,	PUNCT
cana-3833	63	24	𝔼4	𝔼4	NOUN
cana-3833	63	25	=	=	PUNCT
cana-3833	63	26	{	{	PUNCT
cana-3833	63	27	𝑦𝔦𝑣𝔦	𝑦𝔦𝑣𝔦	NOUN
cana-3833	63	28	}	}	PUNCT
cana-3833	63	29	,	,	PUNCT
cana-3833	63	30	𝔼5	𝔼5	NOUN
cana-3833	63	31	=	=	SYM
cana-3833	63	32	{	{	PUNCT
cana-3833	63	33	𝑢𝔦𝑧𝔦	𝑢𝔦𝑧𝔦	PROPN
cana-3833	63	34	}	}	PUNCT
cana-3833	63	35	and	and	CCONJ
cana-3833	63	36	𝔼6	𝔼6	PROPN
cana-3833	63	37	=	=	SYM
cana-3833	63	38	{	{	PUNCT
cana-3833	63	39	𝑧𝔦𝑤𝔦	𝑧𝔦𝑤𝔦	NOUN
cana-3833	63	40	}	}	PUNCT
cana-3833	63	41	for	for	ADP
cana-3833	63	42	every	every	DET
cana-3833	63	43	𝔦	𝔦	X
cana-3833	63	44	𝜖	𝜖	PROPN
cana-3833	63	45	𝕀	𝕀	PROPN
cana-3833	63	46	and	and	CCONJ
cana-3833	63	47	𝒿	𝒿	PROPN
cana-3833	63	48	𝜖	𝜖	PROPN
cana-3833	63	49	𝒥	𝒥	PROPN
cana-3833	63	50	.	.	PUNCT
cana-3833	64	1	total	total	ADJ
cana-3833	64	2	labeling	labeling	NOUN
cana-3833	64	3	𝜌	𝜌	ADP
cana-3833	64	4	:	:	PUNCT
cana-3833	64	5	ℑ	ℑ	NOUN
cana-3833	64	6	→	→	SYM
cana-3833	64	7	ℵ	ℵ	ADP
cana-3833	64	8	where	where	SCONJ
cana-3833	64	9	ℑ	ℑ	PROPN
cana-3833	64	10	=	=	SYM
cana-3833	64	11	𝑉(𝑆(𝑃	𝑉(𝑆(𝑃	PROPN
cana-3833	64	12	𝑛	𝑛	DET
cana-3833	64	13	∘	∘	PROPN
cana-3833	64	14	𝑆2	𝑆2	PROPN
cana-3833	64	15	)	)	PUNCT
cana-3833	64	16	)	)	PUNCT
cana-3833	64	17	∪	∪	ADP
cana-3833	64	18	𝐸(𝑆(𝑃	𝐸(𝑆(𝑃	PROPN
cana-3833	64	19	𝑛	𝑛	DET
cana-3833	64	20	∘	∘	PROPN
cana-3833	64	21	𝑆2	𝑆2	PROPN
cana-3833	64	22	)	)	PUNCT
cana-3833	64	23	)	)	PUNCT
cana-3833	64	24	and	and	CCONJ
cana-3833	64	25	ℵ	ℵ	X
cana-3833	64	26	=	=	SYM
cana-3833	64	27	{	{	PUNCT
cana-3833	64	28	1	1	NUM
cana-3833	64	29	,	,	PUNCT
cana-3833	64	30	2	2	NUM
cana-3833	64	31	,	,	PUNCT
cana-3833	64	32	…	…	PUNCT
cana-3833	64	33	,	,	PUNCT
cana-3833	64	34	4𝑛	4𝑛	NOUN
cana-3833	64	35	−	−	NOUN
cana-3833	65	1	1	1	NUM
cana-3833	65	2	}	}	PUNCT
cana-3833	65	3	is	be	AUX
cana-3833	65	4	given	give	VERB
cana-3833	65	5	below	below	ADP
cana-3833	65	6	:	:	PUNCT
cana-3833	65	7	𝜌(𝑢1	𝜌(𝑢1	NUM
cana-3833	65	8	)	)	PUNCT
cana-3833	65	9	=	=	SYM
cana-3833	65	10	3	3	NUM
cana-3833	65	11	,	,	PUNCT
cana-3833	65	12	𝜌(𝑢𝑖	𝜌(𝑢𝑖	PROPN
cana-3833	65	13	)	)	PUNCT
cana-3833	65	14	=	=	SYM
cana-3833	66	1	4𝑖	4𝑖	NOUN
cana-3833	66	2	−	−	NOUN
cana-3833	66	3	3	3	NUM
cana-3833	66	4	,	,	PUNCT
cana-3833	66	5	𝜌(𝑣𝑖	𝜌(𝑣𝑖	NUM
cana-3833	66	6	)	)	PUNCT
cana-3833	66	7	=	=	SYM
cana-3833	66	8	4𝑖	4𝑖	NOUN
cana-3833	67	1	−	−	NOUN
cana-3833	67	2	1	1	NUM
cana-3833	67	3	,	,	PUNCT
cana-3833	67	4	𝑖	𝑖	PRON
cana-3833	67	5	≠	≠	PROPN
cana-3833	67	6	1	1	NUM
cana-3833	67	7	.	.	PUNCT
cana-3833	67	8	𝜌(𝑣1	𝜌(𝑣1	ADJ
cana-3833	67	9	)	)	PUNCT
cana-3833	67	10	=	=	SYM
cana-3833	67	11	1	1	NUM
cana-3833	67	12	,	,	PUNCT
cana-3833	67	13	𝜌(𝑤𝑖	𝜌(𝑤𝑖	NOUN
cana-3833	67	14	)	)	PUNCT
cana-3833	67	15	=	=	SYM
cana-3833	67	16	4𝑖	4𝑖	NOUN
cana-3833	67	17	−	−	PROPN
cana-3833	67	18	1	1	NUM
cana-3833	67	19	,	,	PUNCT
cana-3833	67	20	𝜌(𝑦𝑖	𝜌(𝑦𝑖	NUM
cana-3833	67	21	)	)	PUNCT
cana-3833	67	22	=	=	SYM
cana-3833	67	23	4𝑖	4𝑖	NOUN
cana-3833	67	24	−	−	NOUN
cana-3833	67	25	3	3	NUM
cana-3833	67	26	,	,	PUNCT
cana-3833	67	27	𝜌(𝑧𝑖	𝜌(𝑧𝑖	NOUN
cana-3833	67	28	)	)	PUNCT
cana-3833	67	29	=	=	SYM
cana-3833	67	30	4𝑖	4𝑖	NOUN
cana-3833	67	31	−	−	NOUN
cana-3833	67	32	1	1	NUM
cana-3833	67	33	,	,	PUNCT
cana-3833	67	34	communications	communication	NOUN
cana-3833	67	35	on	on	ADP
cana-3833	67	36	applied	apply	VERB
cana-3833	67	37	nonlinear	nonlinear	ADJ
cana-3833	67	38	analysis	analysis	NOUN
cana-3833	67	39	issn	issn	NOUN
cana-3833	67	40	:	:	PUNCT
cana-3833	67	41	1074	1074	NUM
cana-3833	67	42	-	-	PUNCT
cana-3833	67	43	133x	133x	NUM
cana-3833	67	44	vol	vol	NOUN
cana-3833	67	45	32	32	NUM
cana-3833	67	46	no	no	NOUN
cana-3833	67	47	.	.	PUNCT
cana-3833	68	1	9s	9s	NUM
cana-3833	68	2	(	(	PUNCT
cana-3833	68	3	2025	2025	NUM
cana-3833	68	4	)	)	PUNCT
cana-3833	68	5	3	3	NUM
cana-3833	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	68	7	𝜌(𝑥𝒿	𝜌(𝑥𝒿	NOUN
cana-3833	68	8	)	)	PUNCT
cana-3833	68	9	=	=	SYM
cana-3833	68	10	4𝒿	4𝒿	NOUN
cana-3833	68	11	+	+	CCONJ
cana-3833	68	12	1	1	NUM
cana-3833	68	13	,	,	PUNCT
cana-3833	68	14	𝜌(𝑢1𝑥1	𝜌(𝑢1𝑥1	NOUN
cana-3833	68	15	)	)	PUNCT
cana-3833	68	16	=	=	SYM
cana-3833	69	1	3	3	NUM
cana-3833	69	2	,	,	PUNCT
cana-3833	69	3	𝜌(𝑢𝔦𝑥𝔦	𝜌(𝑢𝔦𝑥𝔦	NOUN
cana-3833	69	4	)	)	PUNCT
cana-3833	69	5	=	=	SYM
cana-3833	70	1	1	1	NUM
cana-3833	70	2	+	+	NUM
cana-3833	70	3	4𝔦	4𝔦	NOUN
cana-3833	70	4	,	,	PUNCT
cana-3833	70	5	𝔦	𝔦	X
cana-3833	70	6	≠	≠	PROPN
cana-3833	70	7	1	1	NUM
cana-3833	70	8	.	.	PUNCT
cana-3833	70	9	𝜌(𝑥𝔦𝑢𝔦+1	𝜌(𝑥𝔦𝑢𝔦+1	X
cana-3833	70	10	)	)	PUNCT
cana-3833	70	11	=	=	SYM
cana-3833	70	12	−1	−1	NOUN
cana-3833	70	13	+	+	CCONJ
cana-3833	70	14	4𝔦	4𝔦	NOUN
cana-3833	70	15	,	,	PUNCT
cana-3833	70	16	𝜌(𝑢𝔦𝑦𝔦	𝜌(𝑢𝔦𝑦𝔦	ADJ
cana-3833	70	17	)	)	PUNCT
cana-3833	70	18	=	=	SYM
cana-3833	70	19	𝜌(𝑦𝔦𝑣𝔦	𝜌(𝑦𝔦𝑣𝔦	NOUN
cana-3833	70	20	)	)	PUNCT
cana-3833	70	21	=	=	PUNCT
cana-3833	71	1	−3	−3	PROPN
cana-3833	72	1	+	+	SYM
cana-3833	72	2	4𝑖	4𝑖	ADJ
cana-3833	72	3	,	,	PUNCT
cana-3833	72	4	𝜌(𝑢1𝑧1	𝜌(𝑢1𝑧1	NOUN
cana-3833	72	5	)	)	PUNCT
cana-3833	72	6	=	=	SYM
cana-3833	72	7	1	1	NUM
cana-3833	72	8	,	,	PUNCT
cana-3833	72	9	𝜌(𝑢𝑖𝑧𝑖	𝜌(𝑢𝑖𝑧𝑖	NOUN
cana-3833	72	10	)	)	PUNCT
cana-3833	72	11	=	=	SYM
cana-3833	73	1	4𝑖	4𝑖	NOUN
cana-3833	73	2	−	−	NOUN
cana-3833	73	3	1	1	NUM
cana-3833	73	4	,	,	PUNCT
cana-3833	73	5	𝜌(𝑧𝑖𝑤𝑖	𝜌(𝑧𝑖𝑤𝑖	NOUN
cana-3833	73	6	)	)	PUNCT
cana-3833	73	7	=	=	SYM
cana-3833	74	1	−1	−1	NOUN
cana-3833	74	2	+	+	CCONJ
cana-3833	74	3	4𝑖	4𝑖	ADJ
cana-3833	74	4	,	,	PUNCT
cana-3833	74	5	𝑖	𝑖	SYM
cana-3833	74	6	≠	≠	PROPN
cana-3833	74	7	1	1	NUM
cana-3833	74	8	.	.	PUNCT
cana-3833	75	1	induced	induce	VERB
cana-3833	75	2	edge	edge	NOUN
cana-3833	75	3	weight	weight	NOUN
cana-3833	75	4	function	function	NOUN
cana-3833	75	5	𝜓	𝜓	PROPN
cana-3833	75	6	:	:	PUNCT
cana-3833	75	7	𝐸(𝑆(𝑃	𝐸(𝑆(𝑃	PROPN
cana-3833	75	8	𝑛	𝑛	DET
cana-3833	75	9	∘	∘	PROPN
cana-3833	75	10	𝑆2	𝑆2	PROPN
cana-3833	75	11	)	)	PUNCT
cana-3833	75	12	)	)	PUNCT
cana-3833	76	1	→	→	PUNCT
cana-3833	76	2	{	{	PUNCT
cana-3833	76	3	3	3	NUM
cana-3833	76	4	,	,	PUNCT
cana-3833	76	5	5	5	NUM
cana-3833	76	6	,	,	PUNCT
cana-3833	76	7	7	7	NUM
cana-3833	76	8	,	,	PUNCT
cana-3833	76	9	…	…	PUNCT
cana-3833	76	10	,	,	PUNCT
cana-3833	76	11	12𝑛	12𝑛	X
cana-3833	76	12	−	−	NOUN
cana-3833	76	13	3	3	NUM
cana-3833	76	14	}	}	PUNCT
cana-3833	76	15	is	be	AUX
cana-3833	76	16	given	give	VERB
cana-3833	76	17	by	by	ADP
cana-3833	76	18	𝜓(𝑢𝔦𝑥𝔦	𝜓(𝑢𝔦𝑥𝔦	NOUN
cana-3833	76	19	)	)	PUNCT
cana-3833	76	20	=	=	SYM
cana-3833	76	21	−1	−1	NOUN
cana-3833	76	22	+	+	NUM
cana-3833	76	23	12𝔦	12𝔦	NOUN
cana-3833	76	24	,	,	PUNCT
cana-3833	76	25	𝜓(𝑥𝔦𝑢𝔦+1	𝜓(𝑥𝔦𝑢𝔦+1	NUM
cana-3833	76	26	)	)	PUNCT
cana-3833	76	27	=	=	SYM
cana-3833	76	28	1	1	NUM
cana-3833	76	29	+	+	NUM
cana-3833	76	30	12𝔦	12𝔦	NOUN
cana-3833	76	31	,	,	PUNCT
cana-3833	76	32	𝜓(𝑢𝑖𝑦𝑖	𝜓(𝑢𝑖𝑦𝑖	NOUN
cana-3833	76	33	)	)	PUNCT
cana-3833	76	34	=	=	NOUN
cana-3833	76	35	12𝑖	12𝑖	NOUN
cana-3833	76	36	−	−	PROPN
cana-3833	76	37	9	9	NUM
cana-3833	76	38	,	,	PUNCT
cana-3833	76	39	𝜓(𝑦𝑖𝑣𝑖	𝜓(𝑦𝑖𝑣𝑖	NOUN
cana-3833	76	40	)	)	PUNCT
cana-3833	76	41	=	=	NOUN
cana-3833	76	42	12𝑖	12𝑖	NOUN
cana-3833	76	43	−	−	PROPN
cana-3833	76	44	7	7	NUM
cana-3833	76	45	,	,	PUNCT
cana-3833	76	46	𝑖	𝑖	PRON
cana-3833	76	47	≠	≠	ADJ
cana-3833	76	48	1	1	NUM
cana-3833	76	49	𝜓(𝑢1𝑦1	𝜓(𝑢1𝑦1	NOUN
cana-3833	76	50	)	)	PUNCT
cana-3833	76	51	=	=	SYM
cana-3833	76	52	5	5	NUM
cana-3833	76	53	,	,	PUNCT
cana-3833	76	54	𝜓(𝑦1𝑣1	𝜓(𝑦1𝑣1	NOUN
cana-3833	76	55	)	)	PUNCT
cana-3833	76	56	=	=	SYM
cana-3833	76	57	3	3	NUM
cana-3833	76	58	,	,	PUNCT
cana-3833	76	59	𝜓(𝑢𝑖𝑧𝑖	𝜓(𝑢𝑖𝑧𝑖	ADJ
cana-3833	76	60	)	)	PUNCT
cana-3833	76	61	=	=	NOUN
cana-3833	76	62	12𝑖	12𝑖	NOUN
cana-3833	76	63	−	−	PROPN
cana-3833	76	64	5	5	NUM
cana-3833	76	65	,	,	PUNCT
cana-3833	76	66	𝜓(𝑧𝑖𝑤𝑖	𝜓(𝑧𝑖𝑤𝑖	NOUN
cana-3833	76	67	)	)	PUNCT
cana-3833	76	68	=	=	NOUN
cana-3833	76	69	12𝑖	12𝑖	NOUN
cana-3833	76	70	−	−	PROPN
cana-3833	76	71	3	3	NUM
cana-3833	76	72	,	,	PUNCT
cana-3833	76	73	therefore	therefore	ADV
cana-3833	76	74	,	,	PUNCT
cana-3833	76	75	the	the	DET
cana-3833	76	76	induced	induced	ADJ
cana-3833	76	77	edge	edge	NOUN
cana-3833	76	78	weights	weight	NOUN
cana-3833	76	79	of	of	ADP
cana-3833	76	80	𝑆(𝑃	𝑆(𝑃	NOUN
cana-3833	76	81	𝑛	𝑛	PRON
cana-3833	76	82	∘	∘	PROPN
cana-3833	76	83	𝑆2	𝑆2	PROPN
cana-3833	76	84	)	)	PUNCT
cana-3833	76	85	differs	differ	VERB
cana-3833	76	86	by	by	ADP
cana-3833	76	87	2	2	NUM
cana-3833	76	88	in	in	ADP
cana-3833	76	89	the	the	DET
cana-3833	76	90	arithmetic	arithmetic	ADJ
cana-3833	76	91	progression	progression	NOUN
cana-3833	76	92	.	.	PUNCT
cana-3833	77	1	thus	thus	ADV
cana-3833	77	2	,	,	PUNCT
cana-3833	77	3	(	(	PUNCT
cana-3833	77	4	3	3	NUM
cana-3833	77	5	,	,	PUNCT
cana-3833	77	6	2	2	NUM
cana-3833	77	7	)	)	PUNCT
cana-3833	77	8	–	–	PUNCT
cana-3833	77	9	𝑡𝑒𝑠(𝑆(𝑃	𝑡𝑒𝑠(𝑆(𝑃	NUM
cana-3833	77	10	𝑛	𝑛	PROPN
cana-3833	77	11	∘	∘	PROPN
cana-3833	77	12	𝑆2	𝑆2	PROPN
cana-3833	77	13	)	)	PUNCT
cana-3833	77	14	)	)	PUNCT
cana-3833	77	15	≤	≤	NUM
cana-3833	77	16	−1	−1	NOUN
cana-3833	77	17	+	+	CCONJ
cana-3833	77	18	4𝑛.	4𝑛.	NUM
cana-3833	77	19	proposition	proposition	NOUN
cana-3833	77	20	1	1	NUM
cana-3833	77	21	confirms	confirm	VERB
cana-3833	77	22	that(3	that(3	NOUN
cana-3833	77	23	,	,	PUNCT
cana-3833	77	24	2	2	NUM
cana-3833	77	25	)	)	PUNCT
cana-3833	77	26	–	–	PUNCT
cana-3833	77	27	𝑡𝑒𝑠(𝑆(𝑃	𝑡𝑒𝑠(𝑆(𝑃	NUM
cana-3833	77	28	𝑛	𝑛	PROPN
cana-3833	77	29	∘	∘	PROPN
cana-3833	77	30	𝑆2	𝑆2	PROPN
cana-3833	77	31	)	)	PUNCT
cana-3833	77	32	)	)	PUNCT
cana-3833	78	1	≥	≥	X
cana-3833	79	1	−1	−1	NOUN
cana-3833	79	2	+	+	CCONJ
cana-3833	79	3	4𝑛	4𝑛	NOUN
cana-3833	79	4	,	,	PUNCT
cana-3833	79	5	this	this	PRON
cana-3833	79	6	concludes	conclude	VERB
cana-3833	79	7	the	the	DET
cana-3833	79	8	proof	proof	NOUN
cana-3833	79	9	.	.	PUNCT
cana-3833	80	1	example	example	NOUN
cana-3833	80	2	1	1	NUM
cana-3833	80	3	:	:	PUNCT
cana-3833	80	4	(	(	PUNCT
cana-3833	80	5	3	3	NUM
cana-3833	80	6	,	,	PUNCT
cana-3833	80	7	2	2	NUM
cana-3833	80	8	)	)	PUNCT
cana-3833	80	9	total	total	ADJ
cana-3833	80	10	edge	edge	VERB
cana-3833	80	11	irregular	irregular	ADJ
cana-3833	80	12	labeling	labeling	NOUN
cana-3833	80	13	for	for	ADP
cana-3833	80	14	𝑆(𝑃	𝑆(𝑃	PROPN
cana-3833	80	15	5	5	NUM
cana-3833	80	16	∘	∘	PROPN
cana-3833	80	17	𝑆2	𝑆2	PROPN
cana-3833	80	18	)	)	PUNCT
cana-3833	80	19	is	be	AUX
cana-3833	80	20	shown	show	VERB
cana-3833	80	21	in	in	ADP
cana-3833	80	22	fig	fig	NOUN
cana-3833	80	23	1	1	NUM
cana-3833	80	24	.	.	PUNCT
cana-3833	81	1	figure	figure	NOUN
cana-3833	81	2	1	1	NUM
cana-3833	81	3	.	.	PUNCT
cana-3833	82	1	(	(	PUNCT
cana-3833	82	2	3	3	NUM
cana-3833	82	3	,	,	PUNCT
cana-3833	82	4	2	2	NUM
cana-3833	82	5	)	)	PUNCT
cana-3833	82	6	–	–	PUNCT
cana-3833	82	7	𝑡𝑒𝑠(𝑆(𝑃	𝑡𝑒𝑠(𝑆(𝑃	NUM
cana-3833	82	8	5	5	NUM
cana-3833	82	9	∘	∘	PROPN
cana-3833	82	10	𝑆2	𝑆2	NOUN
cana-3833	82	11	)	)	PUNCT
cana-3833	82	12	)	)	PUNCT
cana-3833	83	1	=	=	SYM
cana-3833	83	2	19	19	NUM
cana-3833	83	3	.	.	NOUN
cana-3833	83	4	remark	remark	NOUN
cana-3833	83	5	1	1	NUM
cana-3833	83	6	:	:	PUNCT
cana-3833	83	7	when	when	SCONJ
cana-3833	83	8	𝑛	𝑛	PROPN
cana-3833	83	9	=	=	SYM
cana-3833	83	10	1	1	NUM
cana-3833	83	11	,	,	PUNCT
cana-3833	83	12	𝑃	𝑃	PROPN
cana-3833	83	13	1	1	NUM
cana-3833	83	14	∘	∘	NOUN
cana-3833	83	15	𝑆2	𝑆2	PROPN
cana-3833	83	16	≅	≅	PROPN
cana-3833	84	1	𝑃	𝑃	PROPN
cana-3833	84	2	5	5	NUM
cana-3833	84	3	and	and	CCONJ
cana-3833	84	4	hence	hence	ADV
cana-3833	84	5	it	it	PRON
cana-3833	84	6	follows	follow	VERB
cana-3833	84	7	from	from	ADP
cana-3833	84	8	the	the	DET
cana-3833	84	9	proposition	proposition	NOUN
cana-3833	84	10	2	2	NUM
cana-3833	84	11	.	.	PUNCT
cana-3833	85	1	requisition	requisition	NOUN
cana-3833	85	2	theorem	theorem	VERB
cana-3833	85	3	2	2	NUM
cana-3833	85	4	:	:	PUNCT
cana-3833	85	5	(	(	PUNCT
cana-3833	85	6	3	3	NUM
cana-3833	85	7	,	,	PUNCT
cana-3833	85	8	2	2	NUM
cana-3833	85	9	)	)	PUNCT
cana-3833	85	10	–	–	PUNCT
cana-3833	85	11	𝑡𝑒𝑠(𝑀(𝑃	𝑡𝑒𝑠(𝑀(𝑃	X
cana-3833	85	12	𝑛	𝑛	NOUN
cana-3833	85	13	)	)	PUNCT
cana-3833	85	14	)	)	PUNCT
cana-3833	86	1	=	=	SYM
cana-3833	86	2	2𝑛	2𝑛	PROPN
cana-3833	87	1	−	−	NOUN
cana-3833	87	2	2	2	X
cana-3833	87	3	.	.	PUNCT
cana-3833	87	4	proof	proof	NOUN
cana-3833	87	5	.	.	PUNCT
cana-3833	88	1	consider	consider	VERB
cana-3833	88	2	the	the	DET
cana-3833	88	3	index	index	NOUN
cana-3833	88	4	sets	set	VERB
cana-3833	88	5	𝕀	𝕀	PROPN
cana-3833	88	6	=	=	SYM
cana-3833	88	7	{	{	PUNCT
cana-3833	88	8	1	1	NUM
cana-3833	88	9	,	,	PUNCT
cana-3833	88	10	2	2	NUM
cana-3833	88	11	,	,	PUNCT
cana-3833	88	12	…	…	PUNCT
cana-3833	88	13	,	,	PUNCT
cana-3833	88	14	𝑛	𝑛	NOUN
cana-3833	88	15	}	}	PUNCT
cana-3833	88	16	and	and	CCONJ
cana-3833	88	17	𝒥	𝒥	PROPN
cana-3833	88	18	=	=	PUNCT
cana-3833	88	19	{	{	PUNCT
cana-3833	88	20	1	1	NUM
cana-3833	88	21	,	,	PUNCT
cana-3833	88	22	2	2	NUM
cana-3833	88	23	,	,	PUNCT
cana-3833	88	24	…	…	PUNCT
cana-3833	88	25	,	,	PUNCT
cana-3833	88	26	𝑛	𝑛	DET
cana-3833	88	27	−	−	NOUN
cana-3833	88	28	1	1	NUM
cana-3833	88	29	}	}	PUNCT
cana-3833	88	30	.	.	PUNCT
cana-3833	89	1	let	let	VERB
cana-3833	89	2	𝑉(𝑀(𝑃	𝑉(𝑀(𝑃	PROPN
cana-3833	89	3	𝑛	𝑛	NOUN
cana-3833	89	4	)	)	PUNCT
cana-3833	89	5	)	)	PUNCT
cana-3833	90	1	=	=	SYM
cana-3833	90	2	𝑉1	𝑉1	NOUN
cana-3833	90	3	∪	∪	X
cana-3833	90	4	𝑉2	𝑉2	NOUN
cana-3833	90	5	where	where	SCONJ
cana-3833	90	6	𝑉1	𝑉1	NOUN
cana-3833	90	7	=	=	SYM
cana-3833	90	8	{	{	PUNCT
cana-3833	90	9	𝑢𝒾	𝑢𝒾	X
cana-3833	90	10	}	}	PUNCT
cana-3833	90	11	and	and	CCONJ
cana-3833	90	12	𝑉2	𝑉2	NOUN
cana-3833	90	13	=	=	SYM
cana-3833	90	14	{	{	PUNCT
cana-3833	90	15	𝑣𝒿	𝑣𝒿	NOUN
cana-3833	90	16	}	}	PUNCT
cana-3833	90	17	and	and	CCONJ
cana-3833	90	18	let	let	VERB
cana-3833	90	19	𝐸(𝑀(𝑃	𝐸(𝑀(𝑃	PROPN
cana-3833	90	20	𝑛	𝑛	PROPN
cana-3833	90	21	)	)	PUNCT
cana-3833	90	22	)	)	PUNCT
cana-3833	91	1	=	=	PRON
cana-3833	91	2	{	{	PUNCT
cana-3833	91	3	𝑣ℓ𝑣ℓ+1	𝑣ℓ𝑣ℓ+1	VERB
cana-3833	91	4	}	}	PUNCT
cana-3833	91	5	∪	∪	ADJ
cana-3833	91	6	{	{	PUNCT
cana-3833	91	7	𝑢𝓀𝑣𝓀	𝑢𝓀𝑣𝓀	NOUN
cana-3833	91	8	}	}	PUNCT
cana-3833	91	9	∪	∪	ADJ
cana-3833	91	10	{	{	PUNCT
cana-3833	91	11	𝑣𝓀𝑢𝓀+1	𝑣𝓀𝑢𝓀+1	NOUN
cana-3833	91	12	}	}	PUNCT
cana-3833	91	13	denotes	denote	VERB
cana-3833	91	14	the	the	DET
cana-3833	91	15	vertex	vertex	NOUN
cana-3833	91	16	set	set	NOUN
cana-3833	91	17	and	and	CCONJ
cana-3833	91	18	edge	edge	NOUN
cana-3833	91	19	set	set	NOUN
cana-3833	91	20	of	of	ADP
cana-3833	91	21	𝑀(𝑃	𝑀(𝑃	PROPN
cana-3833	91	22	𝑛	𝑛	NOUN
cana-3833	91	23	)	)	PUNCT
cana-3833	91	24	for	for	ADP
cana-3833	91	25	all	all	DET
cana-3833	91	26	𝑖	𝑖	SYM
cana-3833	91	27	𝜖	𝜖	PROPN
cana-3833	91	28	𝕀	𝕀	PROPN
cana-3833	91	29	,	,	PUNCT
cana-3833	91	30	𝒿	𝒿	NOUN
cana-3833	91	31	,	,	PUNCT
cana-3833	91	32	𝓀	𝓀	PROPN
cana-3833	91	33	𝜖	𝜖	PROPN
cana-3833	91	34	𝒥	𝒥	PROPN
cana-3833	91	35	and	and	CCONJ
cana-3833	91	36	ℓ𝜖	ℓ𝜖	PROPN
cana-3833	91	37	𝕀	𝕀	PROPN
cana-3833	91	38	−	−	PROPN
cana-3833	91	39	{	{	PUNCT
cana-3833	91	40	𝑛	𝑛	NOUN
cana-3833	91	41	−	−	PROPN
cana-3833	91	42	1	1	NUM
cana-3833	91	43	,	,	PUNCT
cana-3833	91	44	𝑛	𝑛	ADJ
cana-3833	91	45	}	}	PUNCT
cana-3833	91	46	.	.	PUNCT
cana-3833	92	1	total	total	ADJ
cana-3833	92	2	labeling	labeling	NOUN
cana-3833	92	3	𝜌	𝜌	ADP
cana-3833	92	4	:	:	PUNCT
cana-3833	92	5	𝒜	𝒜	NOUN
cana-3833	92	6	→	→	SYM
cana-3833	92	7	ℬ	ℬ	NOUN
cana-3833	92	8	where	where	SCONJ
cana-3833	92	9	𝒜	𝒜	NOUN
cana-3833	92	10	=	=	SYM
cana-3833	92	11	𝑉(𝑀(𝑃	𝑉(𝑀(𝑃	PROPN
cana-3833	92	12	𝑛	𝑛	NOUN
cana-3833	92	13	)	)	PUNCT
cana-3833	92	14	)	)	PUNCT
cana-3833	92	15	∪	∪	ADP
cana-3833	92	16	𝐸(𝑀(𝑃	𝐸(𝑀(𝑃	PROPN
cana-3833	92	17	𝑛	𝑛	NOUN
cana-3833	92	18	)	)	PUNCT
cana-3833	92	19	)	)	PUNCT
cana-3833	92	20	and	and	CCONJ
cana-3833	92	21	ℬ	ℬ	NOUN
cana-3833	92	22	=	=	SYM
cana-3833	92	23	𝕀	𝕀	PROPN
cana-3833	92	24	⋃{𝑛	⋃{𝑛	NUM
cana-3833	92	25	+	+	NUM
cana-3833	92	26	1	1	NUM
cana-3833	92	27	,	,	PUNCT
cana-3833	92	28	𝑛	𝑛	PRON
cana-3833	92	29	+	+	NOUN
cana-3833	92	30	2	2	NUM
cana-3833	92	31	,	,	PUNCT
cana-3833	92	32	…	…	PUNCT
cana-3833	92	33	,	,	PUNCT
cana-3833	92	34	2𝑛	2𝑛	PROPN
cana-3833	92	35	−	−	PROPN
cana-3833	92	36	2	2	NUM
cana-3833	92	37	}	}	PUNCT
cana-3833	92	38	is	be	AUX
cana-3833	92	39	prescribed	prescribe	VERB
cana-3833	92	40	by	by	ADP
cana-3833	92	41	:	:	PUNCT
cana-3833	92	42	𝜌(𝑣1	𝜌(𝑣1	ADJ
cana-3833	92	43	)	)	PUNCT
cana-3833	92	44	=	=	SYM
cana-3833	92	45	1	1	NUM
cana-3833	92	46	,	,	PUNCT
cana-3833	92	47	𝜌(𝑣2	𝜌(𝑣2	ADJ
cana-3833	92	48	)	)	PUNCT
cana-3833	92	49	=	=	SYM
cana-3833	92	50	3	3	NUM
cana-3833	92	51	,	,	PUNCT
cana-3833	92	52	𝜌(𝑣𝑖	𝜌(𝑣𝑖	NUM
cana-3833	92	53	)	)	PUNCT
cana-3833	92	54	=	=	SYM
cana-3833	92	55	2𝑖	2𝑖	NOUN
cana-3833	92	56	,	,	PUNCT
cana-3833	92	57	𝑖	𝑖	X
cana-3833	92	58	=	=	PUNCT
cana-3833	92	59	3,4,5	3,4,5	NUM
cana-3833	92	60	,	,	PUNCT
cana-3833	92	61	…	…	PUNCT
cana-3833	92	62	,	,	PUNCT
cana-3833	92	63	𝑛	𝑛	PROPN
cana-3833	92	64	𝜌(𝑢1	𝜌(𝑢1	NOUN
cana-3833	92	65	)	)	PUNCT
cana-3833	92	66	=	=	SYM
cana-3833	92	67	1	1	NUM
cana-3833	92	68	,	,	PUNCT
cana-3833	92	69	𝜌(𝑢2	𝜌(𝑢2	ADJ
cana-3833	92	70	)	)	PUNCT
cana-3833	92	71	=	=	SYM
cana-3833	92	72	3,𝜌(𝑢𝔦	3,𝜌(𝑢𝔦	NUM
cana-3833	92	73	)	)	PUNCT
cana-3833	92	74	=	=	SYM
cana-3833	93	1	−2	−2	NOUN
cana-3833	94	1	+	+	CCONJ
cana-3833	94	2	2𝔦	2𝔦	NUM
cana-3833	94	3	,	,	PUNCT
cana-3833	94	4	3	3	NUM
cana-3833	94	5	≤	≤	NUM
cana-3833	95	1	𝔦	𝔦	PRON
cana-3833	95	2	≤	≤	NUM
cana-3833	95	3	𝑛	𝑛	DET
cana-3833	95	4	𝜌(𝑣1𝑣2	𝜌(𝑣1𝑣2	PROPN
cana-3833	95	5	)	)	PUNCT
cana-3833	95	6	=	=	SYM
cana-3833	95	7	3	3	NUM
cana-3833	95	8	,	,	PUNCT
cana-3833	95	9	𝜌(𝑣2𝑣3	𝜌(𝑣2𝑣3	NOUN
cana-3833	95	10	)	)	PUNCT
cana-3833	95	11	=	=	SYM
cana-3833	95	12	4	4	NUM
cana-3833	95	13	,	,	PUNCT
cana-3833	95	14	𝜌(𝑣𝑖𝑣𝑖+1	𝜌(𝑣𝑖𝑣𝑖+1	PROPN
cana-3833	95	15	)	)	PUNCT
cana-3833	95	16	=	=	SYM
cana-3833	95	17	−1	−1	NOUN
cana-3833	95	18	+	+	NUM
cana-3833	95	19	2𝑖	2𝑖	NOUN
cana-3833	95	20	,	,	PUNCT
cana-3833	95	21	𝑖	𝑖	X
cana-3833	95	22	=	=	PUNCT
cana-3833	95	23	3,4,5	3,4,5	NUM
cana-3833	95	24	,	,	PUNCT
cana-3833	95	25	…	…	PUNCT
cana-3833	95	26	,	,	PUNCT
cana-3833	95	27	𝑛	𝑛	DET
cana-3833	95	28	−	−	PROPN
cana-3833	95	29	2	2	NUM
cana-3833	95	30	𝜌(𝑢𝒿𝑣𝒿	𝜌(𝑢𝒿𝑣𝒿	NOUN
cana-3833	95	31	)	)	PUNCT
cana-3833	95	32	=	=	SYM
cana-3833	95	33	−1	−1	NOUN
cana-3833	96	1	+	+	NUM
cana-3833	96	2	2𝒿	2𝒿	NOUN
cana-3833	96	3	,	,	PUNCT
cana-3833	96	4	𝜌(𝑣1𝑢2	𝜌(𝑣1𝑢2	PROPN
cana-3833	96	5	)	)	PUNCT
cana-3833	96	6	=	=	SYM
cana-3833	96	7	1	1	NUM
cana-3833	96	8	,	,	PUNCT
cana-3833	96	9	𝜌(𝑣2𝑢3	𝜌(𝑣2𝑢3	NOUN
cana-3833	96	10	)	)	PUNCT
cana-3833	96	11	=	=	SYM
cana-3833	96	12	4	4	NUM
cana-3833	96	13	,	,	PUNCT
cana-3833	96	14	𝜌(𝑣𝔦𝑢𝔦+1	𝜌(𝑣𝔦𝑢𝔦+1	NOUN
cana-3833	96	15	)	)	PUNCT
cana-3833	96	16	=	=	SYM
cana-3833	97	1	−1	−1	NOUN
cana-3833	98	1	+	+	CCONJ
cana-3833	98	2	2𝔦	2𝔦	NUM
cana-3833	98	3	,	,	PUNCT
cana-3833	98	4	𝔦	𝔦	X
cana-3833	98	5	=	=	SYM
cana-3833	98	6	3,4,5	3,4,5	NUM
cana-3833	98	7	,	,	PUNCT
cana-3833	98	8	…	…	PUNCT
cana-3833	98	9	,	,	PUNCT
cana-3833	98	10	𝑛	𝑛	DET
cana-3833	98	11	−	−	PROPN
cana-3833	98	12	1	1	NUM
cana-3833	98	13	induced	induce	VERB
cana-3833	98	14	edge	edge	NOUN
cana-3833	98	15	weight	weight	NOUN
cana-3833	98	16	function	function	NOUN
cana-3833	98	17	𝜓	𝜓	PROPN
cana-3833	98	18	:	:	PUNCT
cana-3833	98	19	𝐸(𝑀(𝑃	𝐸(𝑀(𝑃	PROPN
cana-3833	98	20	𝑛	𝑛	PROPN
cana-3833	98	21	)	)	PUNCT
cana-3833	98	22	)	)	PUNCT
cana-3833	98	23	→	→	PUNCT
cana-3833	98	24	{	{	PUNCT
cana-3833	98	25	3	3	NUM
cana-3833	98	26	,	,	PUNCT
cana-3833	98	27	5	5	NUM
cana-3833	98	28	,	,	PUNCT
cana-3833	98	29	7	7	NUM
cana-3833	98	30	,	,	PUNCT
cana-3833	98	31	…	…	PUNCT
cana-3833	98	32	,	,	PUNCT
cana-3833	98	33	6𝑛	6𝑛	NOUN
cana-3833	98	34	−	−	NOUN
cana-3833	98	35	7	7	NUM
cana-3833	98	36	}	}	PUNCT
cana-3833	98	37	is	be	AUX
cana-3833	98	38	given	give	VERB
cana-3833	98	39	by	by	ADP
cana-3833	98	40	communications	communication	NOUN
cana-3833	98	41	on	on	ADP
cana-3833	98	42	applied	apply	VERB
cana-3833	98	43	nonlinear	nonlinear	ADJ
cana-3833	98	44	analysis	analysis	NOUN
cana-3833	98	45	issn	issn	NOUN
cana-3833	98	46	:	:	PUNCT
cana-3833	98	47	1074	1074	NUM
cana-3833	98	48	-	-	PUNCT
cana-3833	98	49	133x	133x	NUM
cana-3833	98	50	vol	vol	NOUN
cana-3833	98	51	32	32	NUM
cana-3833	98	52	no	no	NOUN
cana-3833	98	53	.	.	PUNCT
cana-3833	99	1	9s	9s	NUM
cana-3833	99	2	(	(	PUNCT
cana-3833	99	3	2025	2025	NUM
cana-3833	99	4	)	)	PUNCT
cana-3833	99	5	4	4	NUM
cana-3833	99	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	99	7	𝜓(𝑣ℓ𝑣ℓ+1	𝜓(𝑣ℓ𝑣ℓ+1	NOUN
cana-3833	99	8	)	)	PUNCT
cana-3833	99	9	=	=	SYM
cana-3833	99	10	1	1	NUM
cana-3833	99	11	+	+	NUM
cana-3833	99	12	6ℓ	6ℓ	NOUN
cana-3833	99	13	,	,	PUNCT
cana-3833	99	14	ℓ𝜖	ℓ𝜖	ADJ
cana-3833	99	15	𝕀	𝕀	PROPN
cana-3833	99	16	−	−	PROPN
cana-3833	99	17	{	{	PUNCT
cana-3833	99	18	𝑛	𝑛	NOUN
cana-3833	99	19	−	−	PROPN
cana-3833	99	20	1	1	NUM
cana-3833	99	21	,	,	PUNCT
cana-3833	99	22	𝑛	𝑛	ADJ
cana-3833	99	23	}	}	PUNCT
cana-3833	99	24	𝜓(𝑢𝒿𝑣𝒿	𝜓(𝑢𝒿𝑣𝒿	NOUN
cana-3833	99	25	)	)	PUNCT
cana-3833	99	26	=	=	PUNCT
cana-3833	100	1	−3	−3	PROPN
cana-3833	101	1	+	+	NUM
cana-3833	101	2	6𝒿	6𝒿	NUM
cana-3833	101	3	,	,	PUNCT
cana-3833	101	4	𝜓(𝑣𝒿𝑢𝒿+1	𝜓(𝑣𝒿𝑢𝒿+1	ADJ
cana-3833	101	5	)	)	PUNCT
cana-3833	101	6	=	=	SYM
cana-3833	101	7	−1	−1	NOUN
cana-3833	101	8	+	+	NUM
cana-3833	101	9	6𝒿	6𝒿	NOUN
cana-3833	101	10	therefore	therefore	ADV
cana-3833	101	11	,	,	PUNCT
cana-3833	101	12	the	the	DET
cana-3833	101	13	induced	induced	ADJ
cana-3833	101	14	edge	edge	NOUN
cana-3833	101	15	weights	weight	NOUN
cana-3833	101	16	of	of	ADP
cana-3833	101	17	𝑆(𝑀(𝑃	𝑆(𝑀(𝑃	PROPN
cana-3833	101	18	𝑛	𝑛	NOUN
cana-3833	101	19	)	)	PUNCT
cana-3833	101	20	)	)	PUNCT
cana-3833	101	21	differs	differ	VERB
cana-3833	101	22	by	by	ADP
cana-3833	101	23	2	2	NUM
cana-3833	101	24	in	in	ADP
cana-3833	101	25	the	the	DET
cana-3833	101	26	arithmetic	arithmetic	ADJ
cana-3833	101	27	progression	progression	NOUN
cana-3833	101	28	.	.	PUNCT
cana-3833	102	1	hence	hence	ADV
cana-3833	102	2	,	,	PUNCT
cana-3833	102	3	(	(	PUNCT
cana-3833	102	4	3	3	NUM
cana-3833	102	5	,	,	PUNCT
cana-3833	102	6	2	2	NUM
cana-3833	102	7	)	)	PUNCT
cana-3833	102	8	–	–	PUNCT
cana-3833	102	9	𝑡𝑒𝑠(𝑀(𝑃	𝑡𝑒𝑠(𝑀(𝑃	X
cana-3833	102	10	𝑛	𝑛	PROPN
cana-3833	102	11	)	)	PUNCT
cana-3833	102	12	)	)	PUNCT
cana-3833	102	13	≤	≤	NUM
cana-3833	102	14	−2	−2	NOUN
cana-3833	102	15	+	+	CCONJ
cana-3833	102	16	2𝑛.	2𝑛.	NUM
cana-3833	102	17	proposition	proposition	NOUN
cana-3833	102	18	1	1	NUM
cana-3833	102	19	reveals	reveal	VERB
cana-3833	102	20	that(3	that(3	NOUN
cana-3833	102	21	,	,	PUNCT
cana-3833	102	22	2	2	NUM
cana-3833	102	23	)	)	PUNCT
cana-3833	102	24	–	–	PUNCT
cana-3833	102	25	𝑡𝑒𝑠(𝑀(𝑃	𝑡𝑒𝑠(𝑀(𝑃	X
cana-3833	102	26	𝑛	𝑛	PROPN
cana-3833	102	27	)	)	PUNCT
cana-3833	102	28	)	)	PUNCT
cana-3833	102	29	≥	≥	NOUN
cana-3833	102	30	−2	−2	NOUN
cana-3833	103	1	+	+	CCONJ
cana-3833	103	2	2𝑛.	2𝑛.	NOUN
cana-3833	103	3	hence	hence	ADV
cana-3833	103	4	the	the	DET
cana-3833	103	5	result	result	NOUN
cana-3833	103	6	follows	follow	VERB
cana-3833	103	7	.	.	PUNCT
cana-3833	104	1	example	example	NOUN
cana-3833	104	2	2	2	NUM
cana-3833	104	3	:	:	PUNCT
cana-3833	104	4	(	(	PUNCT
cana-3833	104	5	3	3	NUM
cana-3833	104	6	,	,	PUNCT
cana-3833	104	7	2	2	NUM
cana-3833	104	8	)	)	PUNCT
cana-3833	104	9	−	−	NOUN
cana-3833	104	10	𝑡𝑒𝑠(𝑀(𝑃8)is	𝑡𝑒𝑠(𝑀(𝑃8)is	ADV
cana-3833	104	11	shown	show	VERB
cana-3833	104	12	in	in	ADP
cana-3833	104	13	fig	fig	NOUN
cana-3833	104	14	2	2	NUM
cana-3833	104	15	.	.	PUNCT
cana-3833	104	16	figure	figure	NOUN
cana-3833	104	17	2	2	NUM
cana-3833	104	18	.	.	PUNCT
cana-3833	105	1	(	(	PUNCT
cana-3833	105	2	3	3	NUM
cana-3833	105	3	,	,	PUNCT
cana-3833	105	4	2	2	NUM
cana-3833	105	5	)	)	PUNCT
cana-3833	105	6	–	–	PUNCT
cana-3833	105	7	𝑡𝑒𝑠(𝑀(𝑃8	𝑡𝑒𝑠(𝑀(𝑃8	ADJ
cana-3833	105	8	)	)	PUNCT
cana-3833	105	9	)	)	PUNCT
cana-3833	106	1	=	=	SYM
cana-3833	106	2	14	14	NUM
cana-3833	106	3	.	.	PUNCT
cana-3833	107	1	theorem	theorem	VERB
cana-3833	107	2	3	3	NUM
cana-3833	107	3	:	:	PUNCT
cana-3833	107	4	(	(	PUNCT
cana-3833	107	5	3	3	NUM
cana-3833	107	6	,	,	PUNCT
cana-3833	107	7	2	2	NUM
cana-3833	107	8	)	)	PUNCT
cana-3833	107	9	–	–	PUNCT
cana-3833	107	10	𝑡𝑒𝑠(𝑇(𝑃	𝑡𝑒𝑠(𝑇(𝑃	NUM
cana-3833	107	11	𝑛	𝑛	PROPN
cana-3833	107	12	)	)	PUNCT
cana-3833	107	13	)	)	PUNCT
cana-3833	108	1	=	=	PUNCT
cana-3833	108	2	⌈	⌈	NUM
cana-3833	108	3	8𝑛	8𝑛	NOUN
cana-3833	108	4	3	3	NUM
cana-3833	108	5	⌉	⌉	SCONJ
cana-3833	108	6	−	−	PROPN
cana-3833	108	7	3	3	NUM
cana-3833	108	8	for	for	ADP
cana-3833	108	9	every	every	DET
cana-3833	108	10	positive	positive	ADJ
cana-3833	108	11	integer	integer	NOUN
cana-3833	108	12	𝑛	𝑛	ADP
cana-3833	108	13	at	at	ADV
cana-3833	108	14	least	least	ADJ
cana-3833	108	15	3	3	NUM
cana-3833	108	16	.	.	PUNCT
cana-3833	108	17	proof	proof	NOUN
cana-3833	108	18	.	.	PUNCT
cana-3833	109	1	consider	consider	VERB
cana-3833	109	2	the	the	DET
cana-3833	109	3	index	index	NOUN
cana-3833	109	4	sets	set	VERB
cana-3833	109	5	𝕀	𝕀	PROPN
cana-3833	109	6	=	=	SYM
cana-3833	109	7	{	{	PUNCT
cana-3833	109	8	1	1	NUM
cana-3833	109	9	,	,	PUNCT
cana-3833	109	10	2	2	NUM
cana-3833	109	11	,	,	PUNCT
cana-3833	109	12	…	…	PUNCT
cana-3833	109	13	,	,	PUNCT
cana-3833	109	14	𝑛	𝑛	NOUN
cana-3833	109	15	}	}	PUNCT
cana-3833	109	16	and	and	CCONJ
cana-3833	109	17	𝒥	𝒥	PROPN
cana-3833	109	18	=	=	PUNCT
cana-3833	109	19	{	{	PUNCT
cana-3833	109	20	1	1	NUM
cana-3833	109	21	,	,	PUNCT
cana-3833	109	22	2	2	NUM
cana-3833	109	23	,	,	PUNCT
cana-3833	109	24	…	…	PUNCT
cana-3833	109	25	,	,	PUNCT
cana-3833	109	26	𝑛	𝑛	DET
cana-3833	109	27	−	−	NOUN
cana-3833	109	28	1	1	NUM
cana-3833	109	29	}	}	PUNCT
cana-3833	109	30	.	.	PUNCT
cana-3833	110	1	let	let	VERB
cana-3833	110	2	𝑉(𝑇(𝑃	𝑉(𝑇(𝑃	PROPN
cana-3833	110	3	𝑛	𝑛	VERB
cana-3833	110	4	)	)	PUNCT
cana-3833	110	5	)	)	PUNCT
cana-3833	111	1	=	=	SYM
cana-3833	111	2	𝔸1	𝔸1	PROPN
cana-3833	111	3	∪	∪	VERB
cana-3833	111	4	𝕍2	𝕍2	PROPN
cana-3833	111	5	where	where	SCONJ
cana-3833	111	6	𝔸1	𝔸1	NOUN
cana-3833	111	7	=	=	PRON
cana-3833	111	8	{	{	PUNCT
cana-3833	111	9	𝔞𝒾	𝔞𝒾	NOUN
cana-3833	111	10	}	}	PUNCT
cana-3833	111	11	and	and	CCONJ
cana-3833	111	12	𝕍2	𝕍2	PROPN
cana-3833	111	13	=	=	PRON
cana-3833	111	14	{	{	PUNCT
cana-3833	111	15	𝑣𝒿	𝑣𝒿	NOUN
cana-3833	111	16	}	}	PUNCT
cana-3833	111	17	and	and	CCONJ
cana-3833	111	18	𝐸(𝑇(𝑃	𝐸(𝑇(𝑃	PROPN
cana-3833	111	19	𝑛	𝑛	PROPN
cana-3833	111	20	)	)	PUNCT
cana-3833	111	21	)	)	PUNCT
cana-3833	112	1	=	=	PRON
cana-3833	112	2	{	{	PUNCT
cana-3833	112	3	𝔞𝒿𝔞𝒿+1	𝔞𝒿𝔞𝒿+1	ADV
cana-3833	112	4	}	}	PUNCT
cana-3833	112	5	∪	∪	NOUN
cana-3833	112	6	{	{	PUNCT
cana-3833	112	7	𝑣𝓉𝑣𝓉+1	𝑣𝓉𝑣𝓉+1	NOUN
cana-3833	112	8	}	}	PUNCT
cana-3833	112	9	∪	∪	NOUN
cana-3833	112	10	{	{	PUNCT
cana-3833	112	11	𝔞𝒿𝑣𝒿	𝔞𝒿𝑣𝒿	NOUN
cana-3833	112	12	}	}	PUNCT
cana-3833	112	13	∪	∪	NOUN
cana-3833	112	14	{	{	PUNCT
cana-3833	112	15	𝑣𝒿𝔞𝒿+1	𝑣𝒿𝔞𝒿+1	PROPN
cana-3833	112	16	}	}	PUNCT
cana-3833	112	17	denotes	denote	VERB
cana-3833	112	18	the	the	DET
cana-3833	112	19	vertex	vertex	NOUN
cana-3833	112	20	set	set	NOUN
cana-3833	112	21	and	and	CCONJ
cana-3833	112	22	edge	edge	NOUN
cana-3833	112	23	set	set	NOUN
cana-3833	112	24	of	of	ADP
cana-3833	112	25	𝑇(𝑃	𝑇(𝑃	PROPN
cana-3833	112	26	𝑛	𝑛	VERB
cana-3833	112	27	)	)	PUNCT
cana-3833	112	28	for	for	ADP
cana-3833	112	29	all	all	DET
cana-3833	112	30	𝑖	𝑖	SYM
cana-3833	112	31	𝜖	𝜖	PROPN
cana-3833	112	32	𝕀	𝕀	PROPN
cana-3833	112	33	,	,	PUNCT
cana-3833	112	34	𝒿	𝒿	X
cana-3833	112	35	𝜖	𝜖	X
cana-3833	112	36	𝒥	𝒥	PROPN
cana-3833	112	37	and	and	CCONJ
cana-3833	112	38	𝓉	𝓉	PROPN
cana-3833	112	39	𝜖	𝜖	PROPN
cana-3833	112	40	𝒥	𝒥	PROPN
cana-3833	112	41	−	−	PROPN
cana-3833	112	42	{	{	PUNCT
cana-3833	112	43	𝑛	𝑛	PROPN
cana-3833	112	44	−	−	PROPN
cana-3833	112	45	1	1	NUM
cana-3833	112	46	}	}	PUNCT
cana-3833	112	47	.	.	PUNCT
cana-3833	113	1	total	total	ADJ
cana-3833	113	2	labeling	labeling	NOUN
cana-3833	113	3	𝓅	𝓅	NOUN
cana-3833	113	4	:	:	PUNCT
cana-3833	113	5	𝒞	𝒞	PROPN
cana-3833	113	6	→	→	SYM
cana-3833	113	7	𝒟	𝒟	NOUN
cana-3833	113	8	where	where	SCONJ
cana-3833	113	9	𝒞	𝒞	PROPN
cana-3833	113	10	=	=	PUNCT
cana-3833	113	11	𝑉(𝑇(𝑃	𝑉(𝑇(𝑃	PROPN
cana-3833	113	12	𝑛	𝑛	PROPN
cana-3833	113	13	)	)	PUNCT
cana-3833	113	14	)	)	PUNCT
cana-3833	113	15	∪	∪	VERB
cana-3833	113	16	𝐸(𝑇(𝑃	𝐸(𝑇(𝑃	PROPN
cana-3833	113	17	𝑛	𝑛	PROPN
cana-3833	113	18	)	)	PUNCT
cana-3833	113	19	)	)	PUNCT
cana-3833	113	20	and	and	CCONJ
cana-3833	113	21	𝒟	𝒟	NOUN
cana-3833	113	22	=	=	PUNCT
cana-3833	113	23	{	{	PUNCT
cana-3833	113	24	1	1	NUM
cana-3833	113	25	,	,	PUNCT
cana-3833	113	26	2	2	NUM
cana-3833	113	27	,	,	PUNCT
cana-3833	113	28	…	…	PUNCT
cana-3833	113	29	,	,	PUNCT
cana-3833	114	1	⌈	⌈	NOUN
cana-3833	114	2	8𝑛	8𝑛	NOUN
cana-3833	114	3	3	3	NUM
cana-3833	114	4	⌉	⌉	PRON
cana-3833	114	5	−	−	PROPN
cana-3833	114	6	3	3	NUM
cana-3833	114	7	}	}	PUNCT
cana-3833	114	8	is	be	AUX
cana-3833	114	9	represented	represent	VERB
cana-3833	114	10	by	by	ADP
cana-3833	114	11	:	:	PUNCT
cana-3833	114	12	case	case	NOUN
cana-3833	114	13	i	i	PRON
cana-3833	114	14	suppose	suppose	VERB
cana-3833	114	15	𝑛	𝑛	PRON
cana-3833	114	16	≡	≡	PROPN
cana-3833	114	17	0	0	PUNCT
cana-3833	115	1	(	(	PUNCT
cana-3833	115	2	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	115	3	3	3	NUM
cana-3833	115	4	)	)	PUNCT
cana-3833	115	5	𝓅(𝑣1	𝓅(𝑣1	ADJ
cana-3833	115	6	)	)	PUNCT
cana-3833	115	7	=	=	SYM
cana-3833	115	8	1	1	NUM
cana-3833	115	9	,	,	PUNCT
cana-3833	115	10	𝜌(𝑣2	𝜌(𝑣2	ADJ
cana-3833	115	11	)	)	PUNCT
cana-3833	115	12	=	=	SYM
cana-3833	115	13	5	5	NUM
cana-3833	115	14	,	,	PUNCT
cana-3833	115	15	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	115	16	)	)	PUNCT
cana-3833	115	17	=	=	SYM
cana-3833	115	18	8𝔦	8𝔦	NOUN
cana-3833	115	19	−	−	NOUN
cana-3833	115	20	1	1	NUM
cana-3833	115	21	,	,	PUNCT
cana-3833	115	22	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	115	23	)	)	PUNCT
cana-3833	115	24	=	=	SYM
cana-3833	115	25	3	3	NUM
cana-3833	115	26	+	+	NUM
cana-3833	115	27	8𝔦	8𝔦	NOUN
cana-3833	115	28	,	,	PUNCT
cana-3833	115	29	𝓅(𝑣3𝔦+2	𝓅(𝑣3𝔦+2	NOUN
cana-3833	115	30	)	)	PUNCT
cana-3833	115	31	=	=	SYM
cana-3833	115	32	5	5	NUM
cana-3833	115	33	+	+	NUM
cana-3833	115	34	8𝔦	8𝔦	NOUN
cana-3833	115	35	,	,	PUNCT
cana-3833	115	36	𝓅	𝓅	NOUN
cana-3833	115	37	(	(	PUNCT
cana-3833	115	38	𝔞1	𝔞1	NOUN
cana-3833	115	39	)	)	PUNCT
cana-3833	115	40	=	=	SYM
cana-3833	115	41	1	1	NUM
cana-3833	115	42	,	,	PUNCT
cana-3833	115	43	𝓅(𝔞2	𝓅(𝔞2	NOUN
cana-3833	115	44	)	)	PUNCT
cana-3833	115	45	=	=	SYM
cana-3833	116	1	3	3	NUM
cana-3833	116	2	,	,	PUNCT
cana-3833	116	3	𝓅	𝓅	PROPN
cana-3833	116	4	(	(	PUNCT
cana-3833	116	5	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	116	6	)	)	PUNCT
cana-3833	116	7	=	=	PUNCT
cana-3833	117	1	−3	−3	PROPN
cana-3833	118	1	+	+	NUM
cana-3833	118	2	8𝔦	8𝔦	NOUN
cana-3833	118	3	,	,	PUNCT
cana-3833	118	4	𝓅	𝓅	NOUN
cana-3833	118	5	(	(	PUNCT
cana-3833	118	6	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	118	7	)	)	PUNCT
cana-3833	118	8	=	=	SYM
cana-3833	118	9	8𝔦	8𝔦	NOUN
cana-3833	118	10	+	+	CCONJ
cana-3833	118	11	1	1	NUM
cana-3833	118	12	,	,	PUNCT
cana-3833	118	13	𝓅	𝓅	PROPN
cana-3833	118	14	(	(	PUNCT
cana-3833	118	15	𝔞3𝔦+2	𝔞3𝔦+2	PROPN
cana-3833	118	16	)	)	PUNCT
cana-3833	118	17	=	=	SYM
cana-3833	118	18	3	3	NUM
cana-3833	118	19	+	+	NUM
cana-3833	118	20	8𝔦	8𝔦	NOUN
cana-3833	118	21	,	,	PUNCT
cana-3833	118	22	𝓅(𝑣1𝑣2	𝓅(𝑣1𝑣2	NOUN
cana-3833	118	23	)	)	PUNCT
cana-3833	118	24	=	=	SYM
cana-3833	118	25	3	3	NUM
cana-3833	118	26	,	,	PUNCT
cana-3833	118	27	𝓅(𝑣3𝔦−1𝑣3𝔦	𝓅(𝑣3𝔦−1𝑣3𝔦	PROPN
cana-3833	118	28	)	)	PUNCT
cana-3833	119	1	=	=	NOUN
cana-3833	119	2	8𝔦	8𝔦	NOUN
cana-3833	119	3	−	−	PROPN
cana-3833	119	4	3	3	NUM
cana-3833	119	5	,	,	PUNCT
cana-3833	119	6	𝓅(𝑣3𝔦𝑣3𝔦+1	𝓅(𝑣3𝔦𝑣3𝔦+1	NOUN
cana-3833	119	7	)	)	PUNCT
cana-3833	119	8	=	=	SYM
cana-3833	119	9	8𝔦	8𝔦	NOUN
cana-3833	119	10	−	−	NOUN
cana-3833	119	11	1	1	NUM
cana-3833	119	12	,	,	PUNCT
cana-3833	119	13	𝓅(𝑣3𝔦+1𝑣3𝔦+2	𝓅(𝑣3𝔦+1𝑣3𝔦+2	NOUN
cana-3833	119	14	)	)	PUNCT
cana-3833	119	15	=	=	SYM
cana-3833	119	16	8𝔦	8𝔦	NOUN
cana-3833	119	17	+	+	CCONJ
cana-3833	119	18	1	1	NUM
cana-3833	119	19	,	,	PUNCT
cana-3833	119	20	𝓅	𝓅	PROPN
cana-3833	119	21	(	(	PUNCT
cana-3833	119	22	𝔞1	𝔞1	NOUN
cana-3833	119	23	𝔞2	𝔞2	NOUN
cana-3833	119	24	)	)	PUNCT
cana-3833	119	25	=	=	SYM
cana-3833	119	26	1	1	NUM
cana-3833	119	27	,	,	PUNCT
cana-3833	119	28	communications	communication	NOUN
cana-3833	119	29	on	on	ADP
cana-3833	119	30	applied	apply	VERB
cana-3833	119	31	nonlinear	nonlinear	ADJ
cana-3833	119	32	analysis	analysis	NOUN
cana-3833	119	33	issn	issn	NOUN
cana-3833	119	34	:	:	PUNCT
cana-3833	119	35	1074	1074	NUM
cana-3833	119	36	-	-	PUNCT
cana-3833	119	37	133x	133x	NUM
cana-3833	119	38	vol	vol	NOUN
cana-3833	119	39	32	32	NUM
cana-3833	120	1	no	no	NOUN
cana-3833	120	2	.	.	PUNCT
cana-3833	121	1	9s	9s	NUM
cana-3833	121	2	(	(	PUNCT
cana-3833	121	3	2025	2025	NUM
cana-3833	121	4	)	)	PUNCT
cana-3833	121	5	5	5	NUM
cana-3833	121	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	121	7	𝓅	𝓅	PROPN
cana-3833	121	8	(	(	PUNCT
cana-3833	121	9	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	121	10	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	121	11	)	)	PUNCT
cana-3833	121	12	=	=	PUNCT
cana-3833	122	1	−3	−3	PROPN
cana-3833	123	1	+	+	NUM
cana-3833	123	2	8𝔦	8𝔦	NOUN
cana-3833	123	3	,	,	PUNCT
cana-3833	123	4	𝓅	𝓅	PROPN
cana-3833	123	5	(	(	PUNCT
cana-3833	123	6	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	123	7	𝔞3𝔦+1	𝔞3𝔦+1	X
cana-3833	123	8	)	)	PUNCT
cana-3833	123	9	=	=	SYM
cana-3833	124	1	−1	−1	NOUN
cana-3833	124	2	+	+	NUM
cana-3833	124	3	8𝔦	8𝔦	NOUN
cana-3833	124	4	,	,	PUNCT
cana-3833	124	5	𝓅	𝓅	PROPN
cana-3833	124	6	(	(	PUNCT
cana-3833	124	7	𝔞3𝔦+1	𝔞3𝔦+1	ADJ
cana-3833	124	8	𝔞3𝔦+2	𝔞3𝔦+2	ADJ
cana-3833	124	9	)	)	PUNCT
cana-3833	124	10	=	=	NOUN
cana-3833	124	11	8𝔦	8𝔦	NOUN
cana-3833	124	12	+	+	CCONJ
cana-3833	124	13	1	1	NUM
cana-3833	124	14	,	,	PUNCT
cana-3833	124	15	𝓅	𝓅	PROPN
cana-3833	124	16	(	(	PUNCT
cana-3833	124	17	𝔞1𝑣1	𝔞1𝑣1	NOUN
cana-3833	124	18	)	)	PUNCT
cana-3833	124	19	=	=	SYM
cana-3833	124	20	1	1	NUM
cana-3833	124	21	,	,	PUNCT
cana-3833	124	22	𝓅	𝓅	PROPN
cana-3833	124	23	(	(	PUNCT
cana-3833	124	24	𝔞3𝔦𝑣3𝔦	𝔞3𝔦𝑣3𝔦	NOUN
cana-3833	124	25	)	)	PUNCT
cana-3833	124	26	=	=	SYM
cana-3833	124	27	𝓅	𝓅	PROPN
cana-3833	124	28	(	(	PUNCT
cana-3833	124	29	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	124	30	𝑣3𝔦+1	𝑣3𝔦+1	PROPN
cana-3833	124	31	)	)	PUNCT
cana-3833	124	32	=	=	SYM
cana-3833	124	33	−1	−1	NOUN
cana-3833	124	34	+	+	NUM
cana-3833	124	35	8𝔦	8𝔦	NOUN
cana-3833	124	36	,	,	PUNCT
cana-3833	124	37	𝓅	𝓅	PROPN
cana-3833	124	38	(	(	PUNCT
cana-3833	124	39	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	124	40	𝑣3𝔦−1	𝑣3𝔦−1	PROPN
cana-3833	124	41	)	)	PUNCT
cana-3833	124	42	=	=	PUNCT
cana-3833	125	1	−5	−5	NOUN
cana-3833	126	1	+	+	X
cana-3833	126	2	8𝔦	8𝔦	NOUN
cana-3833	126	3	,	,	PUNCT
cana-3833	126	4	𝓅(𝑣1	𝓅(𝑣1	ADJ
cana-3833	126	5	𝔞2	𝔞2	PROPN
cana-3833	126	6	)	)	PUNCT
cana-3833	127	1	=	=	SYM
cana-3833	127	2	3	3	NUM
cana-3833	127	3	,	,	PUNCT
cana-3833	127	4	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	NOUN
cana-3833	127	5	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	127	6	)	)	PUNCT
cana-3833	127	7	=	=	PUNCT
cana-3833	128	1	−3	−3	PROPN
cana-3833	129	1	+	+	NUM
cana-3833	129	2	8𝔦	8𝔦	NOUN
cana-3833	129	3	,	,	PUNCT
cana-3833	129	4	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	129	5	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	129	6	)	)	PUNCT
cana-3833	129	7	=	=	NOUN
cana-3833	129	8	8𝔦	8𝔦	NOUN
cana-3833	129	9	−	−	NOUN
cana-3833	129	10	1	1	NUM
cana-3833	129	11	,	,	PUNCT
cana-3833	129	12	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	129	13	𝔞3𝔦+2	𝔞3𝔦+2	NOUN
cana-3833	129	14	)	)	PUNCT
cana-3833	129	15	=	=	NOUN
cana-3833	129	16	8𝔦	8𝔦	NOUN
cana-3833	129	17	+	+	CCONJ
cana-3833	129	18	1	1	NUM
cana-3833	129	19	,	,	PUNCT
cana-3833	129	20	case	case	NOUN
cana-3833	129	21	ii	ii	NOUN
cana-3833	129	22	let	let	VERB
cana-3833	129	23	𝑛	𝑛	PRON
cana-3833	129	24	≡	≡	PROPN
cana-3833	129	25	1	1	NUM
cana-3833	129	26	(	(	PUNCT
cana-3833	129	27	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	129	28	3	3	NUM
cana-3833	129	29	)	)	PUNCT
cana-3833	129	30	𝓅(𝑣1	𝓅(𝑣1	ADJ
cana-3833	129	31	)	)	PUNCT
cana-3833	129	32	=	=	SYM
cana-3833	129	33	1	1	NUM
cana-3833	129	34	,	,	PUNCT
cana-3833	129	35	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	NOUN
cana-3833	129	36	)	)	PUNCT
cana-3833	129	37	=	=	SYM
cana-3833	129	38	8𝔦	8𝔦	NOUN
cana-3833	129	39	−	−	NOUN
cana-3833	129	40	3	3	NUM
cana-3833	129	41	,	,	PUNCT
cana-3833	129	42	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	129	43	)	)	PUNCT
cana-3833	129	44	=	=	SYM
cana-3833	129	45	8𝔦	8𝔦	NOUN
cana-3833	129	46	−	−	NOUN
cana-3833	129	47	1	1	NUM
cana-3833	129	48	,	,	PUNCT
cana-3833	129	49	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	129	50	)	)	PUNCT
cana-3833	129	51	=	=	SYM
cana-3833	129	52	3	3	NUM
cana-3833	129	53	+	+	NUM
cana-3833	129	54	8𝔦	8𝔦	NOUN
cana-3833	129	55	,	,	PUNCT
cana-3833	129	56	𝓅	𝓅	NOUN
cana-3833	129	57	(	(	PUNCT
cana-3833	129	58	𝔞1	𝔞1	NOUN
cana-3833	129	59	)	)	PUNCT
cana-3833	129	60	=	=	SYM
cana-3833	129	61	1	1	NUM
cana-3833	129	62	,	,	PUNCT
cana-3833	129	63	𝓅	𝓅	PROPN
cana-3833	129	64	(	(	PUNCT
cana-3833	129	65	𝔞2	𝔞2	PROPN
cana-3833	129	66	)	)	PUNCT
cana-3833	129	67	=	=	SYM
cana-3833	130	1	3	3	NUM
cana-3833	130	2	,	,	PUNCT
cana-3833	130	3	𝓅	𝓅	PROPN
cana-3833	130	4	(	(	PUNCT
cana-3833	130	5	𝔞𝑛	𝔞𝑛	NOUN
cana-3833	130	6	)	)	PUNCT
cana-3833	130	7	=	=	SYM
cana-3833	130	8	8	8	NUM
cana-3833	130	9	3	3	NUM
cana-3833	130	10	(	(	PUNCT
cana-3833	130	11	𝑛	𝑛	PROPN
cana-3833	130	12	−	−	PROPN
cana-3833	130	13	1	1	NUM
cana-3833	130	14	)	)	PUNCT
cana-3833	130	15	𝓅	𝓅	PROPN
cana-3833	130	16	(	(	PUNCT
cana-3833	130	17	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	130	18	)	)	PUNCT
cana-3833	130	19	=	=	PUNCT
cana-3833	130	20	−3	−3	PROPN
cana-3833	131	1	+	+	NUM
cana-3833	131	2	8𝔦	8𝔦	NOUN
cana-3833	131	3	,	,	PUNCT
cana-3833	131	4	𝓅	𝓅	NOUN
cana-3833	131	5	(	(	PUNCT
cana-3833	131	6	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	131	7	)	)	PUNCT
cana-3833	131	8	=	=	SYM
cana-3833	131	9	8𝔦	8𝔦	NOUN
cana-3833	131	10	+	+	CCONJ
cana-3833	131	11	1	1	NUM
cana-3833	131	12	,	,	PUNCT
cana-3833	131	13	𝓅	𝓅	PROPN
cana-3833	131	14	(	(	PUNCT
cana-3833	131	15	𝔞3𝔦+2	𝔞3𝔦+2	PROPN
cana-3833	131	16	)	)	PUNCT
cana-3833	131	17	=	=	SYM
cana-3833	131	18	3	3	NUM
cana-3833	131	19	+	+	NUM
cana-3833	131	20	8𝔦	8𝔦	NOUN
cana-3833	131	21	,	,	PUNCT
cana-3833	131	22	𝓅(𝑣1𝑣2	𝓅(𝑣1𝑣2	NOUN
cana-3833	131	23	)	)	PUNCT
cana-3833	131	24	=	=	SYM
cana-3833	131	25	3	3	NUM
cana-3833	131	26	,	,	PUNCT
cana-3833	131	27	𝓅(𝑣3𝔦−1𝑣3𝔦	𝓅(𝑣3𝔦−1𝑣3𝔦	PROPN
cana-3833	131	28	)	)	PUNCT
cana-3833	131	29	=	=	PUNCT
cana-3833	132	1	−3	−3	PROPN
cana-3833	133	1	+	+	NUM
cana-3833	133	2	8𝔦	8𝔦	NOUN
cana-3833	133	3	,	,	PUNCT
cana-3833	133	4	𝓅(𝑣3𝔦𝑣3𝔦+1	𝓅(𝑣3𝔦𝑣3𝔦+1	NOUN
cana-3833	133	5	)	)	PUNCT
cana-3833	133	6	=	=	SYM
cana-3833	133	7	−1	−1	NOUN
cana-3833	133	8	+	+	NUM
cana-3833	133	9	8𝔦	8𝔦	NOUN
cana-3833	133	10	,	,	PUNCT
cana-3833	133	11	𝓅(𝑣3𝔦+1𝑣3𝔦+2	𝓅(𝑣3𝔦+1𝑣3𝔦+2	PROPN
cana-3833	133	12	)	)	PUNCT
cana-3833	133	13	=	=	SYM
cana-3833	133	14	1	1	NUM
cana-3833	133	15	+	+	NUM
cana-3833	133	16	8𝔦	8𝔦	NOUN
cana-3833	133	17	,	,	PUNCT
cana-3833	133	18	𝓅	𝓅	NOUN
cana-3833	133	19	(	(	PUNCT
cana-3833	133	20	𝔞1	𝔞1	NOUN
cana-3833	133	21	𝔞2	𝔞2	NOUN
cana-3833	133	22	)	)	PUNCT
cana-3833	133	23	=	=	SYM
cana-3833	134	1	1	1	NUM
cana-3833	134	2	,	,	PUNCT
cana-3833	134	3	𝓅	𝓅	PROPN
cana-3833	134	4	(	(	PUNCT
cana-3833	134	5	𝔞𝑛−1	𝔞𝑛−1	PROPN
cana-3833	134	6	𝔞𝑛	𝔞𝑛	NOUN
cana-3833	134	7	)	)	PUNCT
cana-3833	134	8	=	=	SYM
cana-3833	135	1	8	8	NUM
cana-3833	135	2	3	3	NUM
cana-3833	135	3	(	(	PUNCT
cana-3833	135	4	𝑛	𝑛	PROPN
cana-3833	135	5	−	−	PROPN
cana-3833	135	6	1	1	NUM
cana-3833	135	7	)	)	PUNCT
cana-3833	135	8	𝓅	𝓅	PROPN
cana-3833	135	9	(	(	PUNCT
cana-3833	135	10	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	135	11	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	135	12	)	)	PUNCT
cana-3833	135	13	=	=	PUNCT
cana-3833	135	14	−3	−3	PROPN
cana-3833	136	1	+	+	NUM
cana-3833	136	2	8𝔦	8𝔦	NOUN
cana-3833	136	3	,	,	PUNCT
cana-3833	136	4	𝓅	𝓅	PROPN
cana-3833	136	5	(	(	PUNCT
cana-3833	136	6	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	136	7	𝔞3𝔦+1	𝔞3𝔦+1	X
cana-3833	136	8	)	)	PUNCT
cana-3833	136	9	=	=	NOUN
cana-3833	136	10	8𝔦	8𝔦	NOUN
cana-3833	136	11	−	−	NOUN
cana-3833	136	12	1	1	NUM
cana-3833	136	13	,	,	PUNCT
cana-3833	136	14	𝓅	𝓅	PROPN
cana-3833	136	15	(	(	PUNCT
cana-3833	136	16	𝔞3𝔦+1	𝔞3𝔦+1	ADJ
cana-3833	136	17	𝔞3𝔦+2	𝔞3𝔦+2	NOUN
cana-3833	136	18	)	)	PUNCT
cana-3833	136	19	=	=	SYM
cana-3833	136	20	1	1	NUM
cana-3833	136	21	+	+	NUM
cana-3833	136	22	8𝔦	8𝔦	NOUN
cana-3833	136	23	,	,	PUNCT
cana-3833	136	24	𝓅(𝑣1	𝓅(𝑣1	ADJ
cana-3833	136	25	𝔞2	𝔞2	PROPN
cana-3833	136	26	)	)	PUNCT
cana-3833	136	27	=	=	SYM
cana-3833	137	1	3	3	NUM
cana-3833	137	2	,	,	PUNCT
cana-3833	137	3	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	NOUN
cana-3833	137	4	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	137	5	)	)	PUNCT
cana-3833	137	6	=	=	PUNCT
cana-3833	138	1	−3	−3	PROPN
cana-3833	139	1	+	+	NUM
cana-3833	139	2	8𝔦	8𝔦	NOUN
cana-3833	139	3	,	,	PUNCT
cana-3833	139	4	𝓅(𝑣𝑛−1	𝓅(𝑣𝑛−1	NOUN
cana-3833	139	5	𝔞𝑛	𝔞𝑛	NOUN
cana-3833	139	6	)	)	PUNCT
cana-3833	139	7	=	=	SYM
cana-3833	139	8	8	8	NUM
cana-3833	139	9	3	3	NUM
cana-3833	139	10	(	(	PUNCT
cana-3833	139	11	𝑛	𝑛	PROPN
cana-3833	139	12	−	−	PROPN
cana-3833	139	13	1	1	NUM
cana-3833	139	14	)	)	PUNCT
cana-3833	139	15	,	,	PUNCT
cana-3833	139	16	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	139	17	𝔞3𝔦+1	𝔞3𝔦+1	X
cana-3833	139	18	)	)	PUNCT
cana-3833	139	19	=	=	NOUN
cana-3833	139	20	8𝔦	8𝔦	NOUN
cana-3833	139	21	−	−	NOUN
cana-3833	139	22	1	1	NUM
cana-3833	139	23	,	,	PUNCT
cana-3833	139	24	communications	communication	NOUN
cana-3833	139	25	on	on	ADP
cana-3833	139	26	applied	apply	VERB
cana-3833	139	27	nonlinear	nonlinear	ADJ
cana-3833	139	28	analysis	analysis	NOUN
cana-3833	139	29	issn	issn	NOUN
cana-3833	139	30	:	:	PUNCT
cana-3833	139	31	1074	1074	NUM
cana-3833	139	32	-	-	PUNCT
cana-3833	139	33	133x	133x	NUM
cana-3833	139	34	vol	vol	NOUN
cana-3833	139	35	32	32	NUM
cana-3833	139	36	no	no	NOUN
cana-3833	139	37	.	.	PUNCT
cana-3833	140	1	9s	9s	NUM
cana-3833	140	2	(	(	PUNCT
cana-3833	140	3	2025	2025	NUM
cana-3833	140	4	)	)	PUNCT
cana-3833	140	5	6	6	NUM
cana-3833	140	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	140	7	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	140	8	𝔞3𝔦+2	𝔞3𝔦+2	PROPN
cana-3833	140	9	)	)	PUNCT
cana-3833	140	10	=	=	SYM
cana-3833	140	11	1	1	NUM
cana-3833	140	12	+	+	NUM
cana-3833	140	13	8𝔦	8𝔦	NOUN
cana-3833	140	14	,	,	PUNCT
cana-3833	140	15	𝓅	𝓅	PROPN
cana-3833	140	16	(	(	PUNCT
cana-3833	140	17	𝔞1𝑣1	𝔞1𝑣1	NOUN
cana-3833	140	18	)	)	PUNCT
cana-3833	140	19	=	=	SYM
cana-3833	140	20	1	1	NUM
cana-3833	140	21	𝓅	𝓅	PROPN
cana-3833	140	22	(	(	PUNCT
cana-3833	140	23	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	140	24	𝑣3𝔦−1	𝑣3𝔦−1	PROPN
cana-3833	140	25	)	)	PUNCT
cana-3833	140	26	=	=	PUNCT
cana-3833	141	1	−5	−5	NOUN
cana-3833	142	1	+	+	X
cana-3833	142	2	8𝔦	8𝔦	NOUN
cana-3833	142	3	,	,	PUNCT
cana-3833	142	4	𝓅	𝓅	NOUN
cana-3833	142	5	(	(	PUNCT
cana-3833	142	6	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	142	7	𝑣3𝔦	𝑣3𝔦	NOUN
cana-3833	142	8	)	)	PUNCT
cana-3833	142	9	=	=	SYM
cana-3833	143	1	−1	−1	NOUN
cana-3833	143	2	+	+	NUM
cana-3833	143	3	8𝔦	8𝔦	NOUN
cana-3833	143	4	,	,	PUNCT
cana-3833	143	5	𝓅	𝓅	PROPN
cana-3833	143	6	(	(	PUNCT
cana-3833	143	7	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	143	8	𝑣3𝔦+1	𝑣3𝔦+1	PROPN
cana-3833	143	9	)	)	PUNCT
cana-3833	143	10	=	=	SYM
cana-3833	143	11	−1	−1	NOUN
cana-3833	143	12	+	+	NUM
cana-3833	143	13	8𝔦	8𝔦	NOUN
cana-3833	143	14	,	,	PUNCT
cana-3833	143	15	case	case	NOUN
cana-3833	143	16	iii	iii	NOUN
cana-3833	143	17	consider	consider	VERB
cana-3833	143	18	𝑛	𝑛	PRON
cana-3833	143	19	≡	≡	PROPN
cana-3833	143	20	2	2	NUM
cana-3833	143	21	(	(	PUNCT
cana-3833	143	22	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	143	23	3	3	NUM
cana-3833	143	24	)	)	PUNCT
cana-3833	143	25	𝓅(𝑣1	𝓅(𝑣1	ADJ
cana-3833	143	26	)	)	PUNCT
cana-3833	143	27	=	=	SYM
cana-3833	143	28	1	1	NUM
cana-3833	143	29	,	,	PUNCT
cana-3833	143	30	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	NOUN
cana-3833	143	31	)	)	PUNCT
cana-3833	143	32	=	=	PUNCT
cana-3833	143	33	−3	−3	PROPN
cana-3833	144	1	+	+	NUM
cana-3833	144	2	8𝔦	8𝔦	NOUN
cana-3833	144	3	,	,	PUNCT
cana-3833	144	4	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	144	5	)	)	PUNCT
cana-3833	144	6	=	=	SYM
cana-3833	144	7	8𝔦	8𝔦	NOUN
cana-3833	144	8	−	−	NOUN
cana-3833	144	9	1	1	NUM
cana-3833	144	10	,	,	PUNCT
cana-3833	144	11	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	144	12	)	)	PUNCT
cana-3833	144	13	=	=	SYM
cana-3833	144	14	3	3	NUM
cana-3833	144	15	+	+	NUM
cana-3833	144	16	8𝔦	8𝔦	NOUN
cana-3833	144	17	,	,	PUNCT
cana-3833	144	18	𝓅	𝓅	NOUN
cana-3833	144	19	(	(	PUNCT
cana-3833	144	20	𝔞1	𝔞1	NOUN
cana-3833	144	21	)	)	PUNCT
cana-3833	144	22	=	=	SYM
cana-3833	144	23	1	1	NUM
cana-3833	144	24	𝓅	𝓅	PROPN
cana-3833	144	25	(	(	PUNCT
cana-3833	144	26	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	144	27	)	)	PUNCT
cana-3833	144	28	=	=	PUNCT
cana-3833	145	1	−5	−5	NOUN
cana-3833	146	1	+	+	X
cana-3833	146	2	8𝔦	8𝔦	NOUN
cana-3833	146	3	,	,	PUNCT
cana-3833	146	4	𝓅	𝓅	X
cana-3833	146	5	(	(	PUNCT
cana-3833	146	6	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	146	7	)	)	PUNCT
cana-3833	146	8	=	=	PUNCT
cana-3833	147	1	−3	−3	PROPN
cana-3833	148	1	+	+	NUM
cana-3833	148	2	8𝔦	8𝔦	NOUN
cana-3833	148	3	,	,	PUNCT
cana-3833	148	4	𝓅	𝓅	NOUN
cana-3833	148	5	(	(	PUNCT
cana-3833	148	6	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	148	7	)	)	PUNCT
cana-3833	148	8	=	=	SYM
cana-3833	148	9	8𝔦	8𝔦	NOUN
cana-3833	148	10	+	+	CCONJ
cana-3833	148	11	1	1	NUM
cana-3833	148	12	,	,	PUNCT
cana-3833	148	13	𝓅(𝑣1𝑣2	𝓅(𝑣1𝑣2	NOUN
cana-3833	148	14	)	)	PUNCT
cana-3833	148	15	=	=	SYM
cana-3833	148	16	3	3	NUM
cana-3833	148	17	,	,	PUNCT
cana-3833	148	18	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	ADJ
cana-3833	148	19	𝑣3𝔦	𝑣3𝔦	NOUN
cana-3833	148	20	)	)	PUNCT
cana-3833	148	21	=	=	PUNCT
cana-3833	149	1	−3	−3	PROPN
cana-3833	150	1	+	+	NUM
cana-3833	150	2	8𝔦	8𝔦	NOUN
cana-3833	150	3	,	,	PUNCT
cana-3833	150	4	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	PRON
cana-3833	150	5	𝑣3𝔦+1	𝑣3𝔦+1	ADJ
cana-3833	150	6	)	)	PUNCT
cana-3833	150	7	=	=	NOUN
cana-3833	150	8	8𝔦	8𝔦	NOUN
cana-3833	150	9	−	−	NOUN
cana-3833	150	10	1	1	NUM
cana-3833	150	11	,	,	PUNCT
cana-3833	150	12	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	150	13	𝑣3𝔦+2	𝑣3𝔦+2	PROPN
cana-3833	150	14	)	)	PUNCT
cana-3833	150	15	=	=	SYM
cana-3833	150	16	8𝔦	8𝔦	NOUN
cana-3833	150	17	+	+	CCONJ
cana-3833	150	18	1	1	NUM
cana-3833	150	19	,	,	PUNCT
cana-3833	150	20	𝓅	𝓅	PROPN
cana-3833	150	21	(	(	PUNCT
cana-3833	150	22	𝔞1	𝔞1	NOUN
cana-3833	150	23	𝔞2	𝔞2	NOUN
cana-3833	150	24	)	)	PUNCT
cana-3833	150	25	=	=	SYM
cana-3833	150	26	1	1	NUM
cana-3833	150	27	𝓅	𝓅	PROPN
cana-3833	150	28	(	(	PUNCT
cana-3833	150	29	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	150	30	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	150	31	)	)	PUNCT
cana-3833	150	32	=	=	PUNCT
cana-3833	151	1	−3	−3	PROPN
cana-3833	152	1	+	+	NUM
cana-3833	152	2	8𝔦	8𝔦	NOUN
cana-3833	152	3	,	,	PUNCT
cana-3833	152	4	𝓅	𝓅	PROPN
cana-3833	152	5	(	(	PUNCT
cana-3833	152	6	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	152	7	𝔞3𝔦+1	𝔞3𝔦+1	X
cana-3833	152	8	)	)	PUNCT
cana-3833	152	9	=	=	SYM
cana-3833	153	1	−1	−1	NOUN
cana-3833	153	2	+	+	NUM
cana-3833	153	3	8𝔦	8𝔦	NOUN
cana-3833	153	4	,	,	PUNCT
cana-3833	153	5	𝓅	𝓅	PROPN
cana-3833	153	6	(	(	PUNCT
cana-3833	153	7	𝔞3𝔦+1	𝔞3𝔦+1	ADJ
cana-3833	153	8	𝔞3𝔦+2	𝔞3𝔦+2	ADJ
cana-3833	153	9	)	)	PUNCT
cana-3833	153	10	=	=	NOUN
cana-3833	153	11	8𝔦	8𝔦	NOUN
cana-3833	153	12	+	+	CCONJ
cana-3833	153	13	1	1	NUM
cana-3833	153	14	,	,	PUNCT
cana-3833	153	15	𝓅	𝓅	PROPN
cana-3833	153	16	(	(	PUNCT
cana-3833	153	17	𝔞1	𝔞1	NOUN
cana-3833	153	18	𝑣1	𝑣1	NOUN
cana-3833	153	19	)	)	PUNCT
cana-3833	153	20	=	=	SYM
cana-3833	153	21	1	1	NUM
cana-3833	153	22	𝓅	𝓅	PROPN
cana-3833	153	23	(	(	PUNCT
cana-3833	153	24	𝔞3𝔦−1	𝔞3𝔦−1	PROPN
cana-3833	153	25	𝑣3𝔦−1	𝑣3𝔦−1	PROPN
cana-3833	153	26	)	)	PUNCT
cana-3833	153	27	=	=	PUNCT
cana-3833	153	28	−5	−5	NOUN
cana-3833	154	1	+	+	X
cana-3833	154	2	8𝔦	8𝔦	NOUN
cana-3833	154	3	,	,	PUNCT
cana-3833	154	4	𝓅	𝓅	NOUN
cana-3833	154	5	(	(	PUNCT
cana-3833	154	6	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	154	7	𝑣3𝔦	𝑣3𝔦	NOUN
cana-3833	154	8	)	)	PUNCT
cana-3833	154	9	=	=	SYM
cana-3833	154	10	𝓅	𝓅	PROPN
cana-3833	154	11	(	(	PUNCT
cana-3833	154	12	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	154	13	𝑣3𝔦+1	𝑣3𝔦+1	PROPN
cana-3833	154	14	)	)	PUNCT
cana-3833	154	15	=	=	NOUN
cana-3833	154	16	8𝔦	8𝔦	NOUN
cana-3833	154	17	−	−	NOUN
cana-3833	154	18	1	1	NUM
cana-3833	154	19	,	,	PUNCT
cana-3833	154	20	𝓅(𝑣1	𝓅(𝑣1	ADJ
cana-3833	154	21	𝔞2	𝔞2	PROPN
cana-3833	154	22	)	)	PUNCT
cana-3833	154	23	=	=	SYM
cana-3833	154	24	3	3	NUM
cana-3833	154	25	,	,	PUNCT
cana-3833	154	26	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	NOUN
cana-3833	154	27	𝔞3𝔦	𝔞3𝔦	NUM
cana-3833	154	28	)	)	PUNCT
cana-3833	154	29	=	=	PUNCT
cana-3833	154	30	−3	−3	PROPN
cana-3833	155	1	+	+	NUM
cana-3833	155	2	8𝔦	8𝔦	NOUN
cana-3833	155	3	,	,	PUNCT
cana-3833	155	4	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	155	5	𝔞3𝔦+1	𝔞3𝔦+1	NOUN
cana-3833	155	6	)	)	PUNCT
cana-3833	155	7	=	=	NOUN
cana-3833	155	8	8𝔦	8𝔦	NOUN
cana-3833	155	9	−	−	NOUN
cana-3833	155	10	1	1	NUM
cana-3833	155	11	,	,	PUNCT
cana-3833	155	12	𝓅(𝑣3𝔦+1	𝓅(𝑣3𝔦+1	NOUN
cana-3833	155	13	𝔞3𝔦+2	𝔞3𝔦+2	NOUN
cana-3833	155	14	)	)	PUNCT
cana-3833	155	15	=	=	NOUN
cana-3833	155	16	8𝔦	8𝔦	NOUN
cana-3833	155	17	+	+	CCONJ
cana-3833	155	18	1	1	X
cana-3833	155	19	.	.	PUNCT
cana-3833	156	1	induced	induce	VERB
cana-3833	156	2	edge	edge	NOUN
cana-3833	156	3	weight	weight	NOUN
cana-3833	156	4	function	function	NOUN
cana-3833	156	5	𝜓	𝜓	PROPN
cana-3833	156	6	:	:	PUNCT
cana-3833	156	7	𝐸(𝑇(𝑃	𝐸(𝑇(𝑃	PROPN
cana-3833	156	8	𝑛	𝑛	PROPN
cana-3833	156	9	)	)	PUNCT
cana-3833	156	10	)	)	PUNCT
cana-3833	156	11	→	→	PUNCT
cana-3833	156	12	{	{	PUNCT
cana-3833	156	13	3	3	NUM
cana-3833	156	14	,	,	PUNCT
cana-3833	156	15	5	5	NUM
cana-3833	156	16	,	,	PUNCT
cana-3833	156	17	7	7	NUM
cana-3833	156	18	,	,	PUNCT
cana-3833	156	19	…	…	PUNCT
cana-3833	156	20	,	,	PUNCT
cana-3833	156	21	8𝑛	8𝑛	NOUN
cana-3833	156	22	−	−	NOUN
cana-3833	156	23	9	9	NUM
cana-3833	156	24	}	}	PUNCT
cana-3833	156	25	is	be	AUX
cana-3833	156	26	given	give	VERB
cana-3833	156	27	by	by	ADP
cana-3833	156	28	𝜓(𝔞𝒿𝔞𝒿+1	𝜓(𝔞𝒿𝔞𝒿+1	NOUN
cana-3833	156	29	)	)	PUNCT
cana-3833	156	30	=	=	PUNCT
cana-3833	157	1	−3	−3	PROPN
cana-3833	158	1	+	+	SYM
cana-3833	158	2	8𝒿	8𝒿	NUM
cana-3833	158	3	,	,	PUNCT
cana-3833	158	4	𝜓(𝑣𝓉𝑣𝓉+1	𝜓(𝑣𝓉𝑣𝓉+1	NOUN
cana-3833	158	5	)	)	PUNCT
cana-3833	158	6	=	=	SYM
cana-3833	158	7	1	1	NUM
cana-3833	158	8	+	+	NUM
cana-3833	158	9	8𝓉	8𝓉	NUM
cana-3833	158	10	,	,	PUNCT
cana-3833	158	11	𝜓(𝔞𝒿𝑣𝒿	𝜓(𝔞𝒿𝑣𝒿	NOUN
cana-3833	158	12	)	)	PUNCT
cana-3833	158	13	=	=	SYM
cana-3833	159	1	−5	−5	PROPN
cana-3833	160	1	+	+	CCONJ
cana-3833	160	2	8𝒿	8𝒿	NUM
cana-3833	160	3	,	,	PUNCT
cana-3833	160	4	communications	communication	NOUN
cana-3833	160	5	on	on	ADP
cana-3833	160	6	applied	apply	VERB
cana-3833	160	7	nonlinear	nonlinear	ADJ
cana-3833	160	8	analysis	analysis	NOUN
cana-3833	160	9	issn	issn	NOUN
cana-3833	160	10	:	:	PUNCT
cana-3833	160	11	1074	1074	NUM
cana-3833	160	12	-	-	PUNCT
cana-3833	160	13	133x	133x	NUM
cana-3833	160	14	vol	vol	NOUN
cana-3833	160	15	32	32	NUM
cana-3833	160	16	no	no	NOUN
cana-3833	160	17	.	.	PUNCT
cana-3833	161	1	9s	9s	NUM
cana-3833	161	2	(	(	PUNCT
cana-3833	161	3	2025	2025	NUM
cana-3833	161	4	)	)	PUNCT
cana-3833	161	5	7	7	NUM
cana-3833	161	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	161	7	𝜓(𝑣𝒿𝔞𝒿+1	𝜓(𝑣𝒿𝔞𝒿+1	PROPN
cana-3833	161	8	)	)	PUNCT
cana-3833	161	9	=	=	SYM
cana-3833	161	10	8𝒿	8𝒿	NOUN
cana-3833	162	1	−	−	NOUN
cana-3833	162	2	1	1	NUM
cana-3833	162	3	,	,	PUNCT
cana-3833	162	4	therefore	therefore	ADV
cana-3833	162	5	,	,	PUNCT
cana-3833	162	6	the	the	DET
cana-3833	162	7	induced	induced	ADJ
cana-3833	162	8	edge	edge	NOUN
cana-3833	162	9	weights	weight	NOUN
cana-3833	162	10	of	of	ADP
cana-3833	162	11	𝑇(𝑃	𝑇(𝑃	PROPN
cana-3833	162	12	𝑛)generates	𝑛)generate	VERB
cana-3833	162	13	an	an	DET
cana-3833	162	14	arithmetic	arithmetic	ADJ
cana-3833	162	15	progression	progression	NOUN
cana-3833	163	1	and	and	CCONJ
cana-3833	163	2	it	it	PRON
cana-3833	163	3	differs	differ	VERB
cana-3833	163	4	by	by	ADP
cana-3833	163	5	2	2	NUM
cana-3833	163	6	and	and	CCONJ
cana-3833	163	7	hence	hence	ADV
cana-3833	163	8	(	(	PUNCT
cana-3833	163	9	3	3	NUM
cana-3833	163	10	,	,	PUNCT
cana-3833	163	11	2	2	NUM
cana-3833	163	12	)	)	PUNCT
cana-3833	163	13	–	–	PUNCT
cana-3833	163	14	𝑡𝑒𝑠(𝑇(𝑃	𝑡𝑒𝑠(𝑇(𝑃	NUM
cana-3833	163	15	𝑛	𝑛	PROPN
cana-3833	163	16	)	)	PUNCT
cana-3833	163	17	)	)	PUNCT
cana-3833	164	1	≤	≤	NUM
cana-3833	165	1	⌈	⌈	X
cana-3833	165	2	8𝑛	8𝑛	NOUN
cana-3833	165	3	3	3	NUM
cana-3833	165	4	⌉	⌉	SCONJ
cana-3833	165	5	−	−	PROPN
cana-3833	165	6	3	3	NUM
cana-3833	165	7	.	.	PUNCT
cana-3833	165	8	from	from	ADP
cana-3833	165	9	proposition	proposition	NOUN
cana-3833	165	10	1	1	NUM
cana-3833	165	11	,	,	PUNCT
cana-3833	165	12	we	we	PRON
cana-3833	165	13	get	get	VERB
cana-3833	165	14	(	(	PUNCT
cana-3833	165	15	3	3	NUM
cana-3833	165	16	,	,	PUNCT
cana-3833	165	17	2	2	NUM
cana-3833	165	18	)	)	PUNCT
cana-3833	165	19	–	–	PUNCT
cana-3833	165	20	𝑡𝑒𝑠(𝑇(𝑃	𝑡𝑒𝑠(𝑇(𝑃	NUM
cana-3833	165	21	𝑛	𝑛	PROPN
cana-3833	165	22	)	)	PUNCT
cana-3833	165	23	)	)	PUNCT
cana-3833	166	1	≥	≥	PROPN
cana-3833	167	1	⌈	⌈	X
cana-3833	167	2	8𝑛	8𝑛	NOUN
cana-3833	167	3	3	3	NUM
cana-3833	167	4	⌉	⌉	SCONJ
cana-3833	167	5	−	−	PROPN
cana-3833	167	6	3	3	NUM
cana-3833	167	7	and	and	CCONJ
cana-3833	167	8	hence	hence	ADV
cana-3833	167	9	the	the	DET
cana-3833	167	10	result	result	NOUN
cana-3833	167	11	follows	follow	VERB
cana-3833	167	12	.	.	PUNCT
cana-3833	168	1	example	example	NOUN
cana-3833	169	1	3	3	NUM
cana-3833	169	2	:	:	PUNCT
cana-3833	169	3	(	(	PUNCT
cana-3833	169	4	3	3	NUM
cana-3833	169	5	,	,	PUNCT
cana-3833	169	6	2	2	NUM
cana-3833	169	7	)	)	PUNCT
cana-3833	169	8	−	−	PRON
cana-3833	169	9	𝑡𝑒𝑠(𝑇(𝑃9	𝑡𝑒𝑠(𝑇(𝑃9	X
cana-3833	169	10	)	)	PUNCT
cana-3833	169	11	)	)	PUNCT
cana-3833	169	12	is	be	AUX
cana-3833	169	13	given	give	VERB
cana-3833	169	14	in	in	ADP
cana-3833	169	15	fig	fig	NOUN
cana-3833	169	16	3	3	NUM
cana-3833	169	17	.	.	PUNCT
cana-3833	169	18	figure	figure	NOUN
cana-3833	169	19	3	3	NUM
cana-3833	169	20	.	.	PUNCT
cana-3833	169	21	(	(	PUNCT
cana-3833	169	22	3	3	NUM
cana-3833	169	23	,	,	PUNCT
cana-3833	169	24	2	2	NUM
cana-3833	169	25	)	)	PUNCT
cana-3833	169	26	–	–	PUNCT
cana-3833	169	27	𝑡𝑒𝑠(𝑇(𝑃9	𝑡𝑒𝑠(𝑇(𝑃9	X
cana-3833	169	28	)	)	PUNCT
cana-3833	169	29	)	)	PUNCT
cana-3833	170	1	=	=	SYM
cana-3833	170	2	21	21	X
cana-3833	170	3	.	.	PUNCT
cana-3833	170	4	remark	remark	NOUN
cana-3833	170	5	2	2	NUM
cana-3833	170	6	:	:	PUNCT
cana-3833	170	7	when	when	SCONJ
cana-3833	170	8	𝑛	𝑛	PROPN
cana-3833	170	9	=	=	SYM
cana-3833	170	10	2	2	NUM
cana-3833	170	11	,	,	PUNCT
cana-3833	170	12	𝑇(𝑃	𝑇(𝑃	PROPN
cana-3833	170	13	2	2	NUM
cana-3833	170	14	)	)	PUNCT
cana-3833	170	15	≅	≅	PROPN
cana-3833	170	16	𝐶3	𝐶3	NOUN
cana-3833	170	17	and	and	CCONJ
cana-3833	170	18	hence	hence	ADV
cana-3833	170	19	(	(	PUNCT
cana-3833	170	20	3	3	NUM
cana-3833	170	21	,	,	PUNCT
cana-3833	170	22	2	2	NUM
cana-3833	170	23	)	)	PUNCT
cana-3833	170	24	–	–	PUNCT
cana-3833	170	25	𝑡𝑒𝑠(𝑇(𝑃2	𝑡𝑒𝑠(𝑇(𝑃2	NOUN
cana-3833	170	26	)	)	PUNCT
cana-3833	170	27	)	)	PUNCT
cana-3833	171	1	=	=	SYM
cana-3833	171	2	3	3	X
cana-3833	171	3	.	.	PUNCT
cana-3833	171	4	(	(	PUNCT
cana-3833	171	5	3	3	NUM
cana-3833	171	6	,	,	PUNCT
cana-3833	171	7	2	2	NUM
cana-3833	171	8	)	)	PUNCT
cana-3833	171	9	total	total	ADJ
cana-3833	171	10	edge	edge	VERB
cana-3833	171	11	irregular	irregular	ADJ
cana-3833	171	12	labeling	labeling	NOUN
cana-3833	171	13	for	for	ADP
cana-3833	171	14	𝑇(𝑃9	𝑇(𝑃9	PROPN
cana-3833	171	15	)	)	PUNCT
cana-3833	171	16	is	be	AUX
cana-3833	171	17	shown	show	VERB
cana-3833	171	18	in	in	ADP
cana-3833	171	19	fig	fig	NOUN
cana-3833	171	20	4	4	NUM
cana-3833	171	21	.	.	PUNCT
cana-3833	171	22	figure	figure	VERB
cana-3833	171	23	4	4	NUM
cana-3833	171	24	.	.	PUNCT
cana-3833	172	1	(	(	PUNCT
cana-3833	172	2	3	3	NUM
cana-3833	172	3	,	,	PUNCT
cana-3833	172	4	2	2	NUM
cana-3833	172	5	)	)	PUNCT
cana-3833	172	6	–	–	PUNCT
cana-3833	172	7	𝑡𝑒𝑠(𝑇(𝑃2	𝑡𝑒𝑠(𝑇(𝑃2	NOUN
cana-3833	172	8	)	)	PUNCT
cana-3833	172	9	)	)	PUNCT
cana-3833	173	1	=	=	SYM
cana-3833	173	2	3	3	X
cana-3833	173	3	.	.	X
cana-3833	173	4	theorem	theorem	VERB
cana-3833	173	5	4	4	NUM
cana-3833	173	6	:	:	PUNCT
cana-3833	173	7	(	(	PUNCT
cana-3833	173	8	3	3	NUM
cana-3833	173	9	,	,	PUNCT
cana-3833	173	10	2	2	NUM
cana-3833	173	11	)	)	PUNCT
cana-3833	173	12	–	–	PUNCT
cana-3833	173	13	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	X
cana-3833	173	14	𝑛	𝑛	DET
cana-3833	173	15	∘	∘	PROPN
cana-3833	173	16	𝐾2	𝐾2	NOUN
cana-3833	173	17	)	)	PUNCT
cana-3833	173	18	=	=	PUNCT
cana-3833	173	19	⌈	⌈	NOUN
cana-3833	173	20	8(𝑛+1	8(𝑛+1	NUM
cana-3833	173	21	)	)	PUNCT
cana-3833	173	22	3	3	NUM
cana-3833	173	23	⌉	⌉	PRON
cana-3833	173	24	−	−	PROPN
cana-3833	173	25	3	3	NUM
cana-3833	173	26	.	.	PUNCT
cana-3833	173	27	proof	proof	NOUN
cana-3833	173	28	.	.	PUNCT
cana-3833	174	1	consider	consider	VERB
cana-3833	174	2	the	the	DET
cana-3833	174	3	index	index	NOUN
cana-3833	174	4	sets	set	VERB
cana-3833	174	5	𝕀	𝕀	PROPN
cana-3833	174	6	=	=	SYM
cana-3833	174	7	{	{	PUNCT
cana-3833	174	8	1	1	NUM
cana-3833	174	9	,	,	PUNCT
cana-3833	174	10	2	2	NUM
cana-3833	174	11	,	,	PUNCT
cana-3833	174	12	…	…	PUNCT
cana-3833	174	13	,	,	PUNCT
cana-3833	174	14	𝑛	𝑛	NOUN
cana-3833	174	15	}	}	PUNCT
cana-3833	174	16	and	and	CCONJ
cana-3833	174	17	𝒥	𝒥	PROPN
cana-3833	174	18	=	=	PUNCT
cana-3833	174	19	{	{	PUNCT
cana-3833	174	20	1	1	NUM
cana-3833	174	21	,	,	PUNCT
cana-3833	174	22	2	2	NUM
cana-3833	174	23	,	,	PUNCT
cana-3833	174	24	…	…	PUNCT
cana-3833	174	25	,	,	PUNCT
cana-3833	174	26	𝑛	𝑛	DET
cana-3833	174	27	−	−	NOUN
cana-3833	174	28	1	1	NUM
cana-3833	174	29	}	}	PUNCT
cana-3833	174	30	.	.	PUNCT
cana-3833	175	1	let	let	VERB
cana-3833	175	2	𝑉(𝑃	𝑉(𝑃	NOUN
cana-3833	175	3	𝑛	𝑛	DET
cana-3833	175	4	∘	∘	NOUN
cana-3833	175	5	𝐾2	𝐾2	NOUN
cana-3833	175	6	)	)	PUNCT
cana-3833	176	1	=	=	PUNCT
cana-3833	177	1	⋃	⋃	PUNCT
cana-3833	177	2	𝑉𝑛	𝑉𝑛	PROPN
cana-3833	177	3	3	3	NUM
cana-3833	177	4	𝑛=1	𝑛=1	NOUN
cana-3833	177	5	where	where	SCONJ
cana-3833	177	6	𝑉1	𝑉1	NOUN
cana-3833	177	7	=	=	SYM
cana-3833	177	8	{	{	PUNCT
cana-3833	177	9	𝔰𝑖	𝔰𝑖	NOUN
cana-3833	177	10	}	}	PUNCT
cana-3833	177	11	,	,	PUNCT
cana-3833	177	12	𝑉2	𝑉2	NOUN
cana-3833	177	13	=	=	SYM
cana-3833	177	14	{	{	PUNCT
cana-3833	177	15	𝑣𝑖	𝑣𝑖	NOUN
cana-3833	177	16	}	}	PUNCT
cana-3833	177	17	and	and	CCONJ
cana-3833	177	18	𝑉3	𝑉3	NOUN
cana-3833	177	19	=	=	SYM
cana-3833	177	20	{	{	PUNCT
cana-3833	177	21	𝔱𝑖	𝔱𝑖	NOUN
cana-3833	177	22	}	}	PUNCT
cana-3833	177	23	and	and	CCONJ
cana-3833	177	24	let	let	VERB
cana-3833	177	25	𝐸(𝑃	𝐸(𝑃	NOUN
cana-3833	177	26	𝑛	𝑛	DET
cana-3833	177	27	∘	∘	NOUN
cana-3833	177	28	𝐾2	𝐾2	NOUN
cana-3833	177	29	)	)	PUNCT
cana-3833	177	30	=	=	SYM
cana-3833	177	31	{	{	PUNCT
cana-3833	177	32	𝔰𝒿𝔰𝒿+1	𝔰𝒿𝔰𝒿+1	PROPN
cana-3833	177	33	}	}	PUNCT
cana-3833	177	34	∪	∪	ADJ
cana-3833	177	35	{	{	PUNCT
cana-3833	177	36	𝔰𝑖𝑣𝑖	𝔰𝑖𝑣𝑖	NOUN
cana-3833	177	37	}	}	PUNCT
cana-3833	177	38	∪	∪	ADJ
cana-3833	177	39	{	{	PUNCT
cana-3833	177	40	𝑣𝑖𝔱𝑖	𝑣𝑖𝔱𝑖	ADJ
cana-3833	177	41	}	}	PUNCT
cana-3833	177	42	∪	∪	NOUN
cana-3833	177	43	{	{	PUNCT
cana-3833	177	44	𝔰𝑖𝔱𝑖	𝔰𝑖𝔱𝑖	NOUN
cana-3833	177	45	}	}	PUNCT
cana-3833	177	46	denotes	denote	VERB
cana-3833	177	47	the	the	DET
cana-3833	177	48	vertex	vertex	NOUN
cana-3833	177	49	set	set	NOUN
cana-3833	177	50	and	and	CCONJ
cana-3833	177	51	edge	edge	NOUN
cana-3833	177	52	of	of	ADP
cana-3833	177	53	𝑃	𝑃	NOUN
cana-3833	177	54	𝑛	𝑛	PRON
cana-3833	177	55	∘	∘	NOUN
cana-3833	177	56	𝐾2	𝐾2	NOUN
cana-3833	177	57	respectively	respectively	ADV
cana-3833	177	58	for	for	ADP
cana-3833	177	59	all	all	DET
cana-3833	177	60	𝑖	𝑖	SYM
cana-3833	177	61	𝜖	𝜖	PROPN
cana-3833	177	62	𝕀	𝕀	PROPN
cana-3833	177	63	,	,	PUNCT
cana-3833	177	64	𝒿	𝒿	X
cana-3833	177	65	𝜖	𝜖	X
cana-3833	177	66	𝒥.	𝒥.	NOUN
cana-3833	177	67	total	total	NOUN
cana-3833	177	68	labeling	labeling	NOUN
cana-3833	177	69	𝓅	𝓅	NOUN
cana-3833	177	70	:	:	PUNCT
cana-3833	177	71	ℒ	ℒ	PROPN
cana-3833	177	72	→	→	SYM
cana-3833	177	73	ℳwhere	ℳwhere	PROPN
cana-3833	177	74	ℒ	ℒ	PROPN
cana-3833	177	75	=	=	SYM
cana-3833	177	76	𝑉(𝑃	𝑉(𝑃	NUM
cana-3833	177	77	𝑛	𝑛	ADJ
cana-3833	177	78	∘	∘	PROPN
cana-3833	177	79	𝐾2	𝐾2	NOUN
cana-3833	177	80	)	)	PUNCT
cana-3833	177	81	∪	∪	VERB
cana-3833	177	82	𝐸(𝑃	𝐸(𝑃	NOUN
cana-3833	177	83	𝑛	𝑛	DET
cana-3833	177	84	∘	∘	PROPN
cana-3833	177	85	𝐾2	𝐾2	NOUN
cana-3833	177	86	)	)	PUNCT
cana-3833	177	87	and	and	CCONJ
cana-3833	177	88	ℳ	ℳ	PROPN
cana-3833	177	89	=	=	SYM
cana-3833	177	90	{	{	PUNCT
cana-3833	177	91	1	1	NUM
cana-3833	177	92	,	,	PUNCT
cana-3833	177	93	2	2	NUM
cana-3833	177	94	,	,	PUNCT
cana-3833	177	95	…	…	PUNCT
cana-3833	177	96	,	,	PUNCT
cana-3833	177	97	⌈	⌈	NOUN
cana-3833	177	98	8(𝑛+1	8(𝑛+1	NUM
cana-3833	177	99	)	)	PUNCT
cana-3833	177	100	3	3	NUM
cana-3833	177	101	⌉	⌉	PRON
cana-3833	177	102	−	−	PROPN
cana-3833	177	103	3	3	NUM
cana-3833	177	104	}	}	PUNCT
cana-3833	177	105	is	be	AUX
cana-3833	177	106	described	describe	VERB
cana-3833	177	107	by	by	ADP
cana-3833	177	108	:	:	PUNCT
cana-3833	177	109	case	case	NOUN
cana-3833	177	110	i	i	PRON
cana-3833	177	111	consider	consider	VERB
cana-3833	177	112	𝑛	𝑛	DET
cana-3833	177	113	≡	≡	PROPN
cana-3833	177	114	0	0	PUNCT
cana-3833	178	1	(	(	PUNCT
cana-3833	178	2	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	178	3	3	3	NUM
cana-3833	178	4	)	)	PUNCT
cana-3833	178	5	communications	communication	NOUN
cana-3833	178	6	on	on	ADP
cana-3833	178	7	applied	apply	VERB
cana-3833	178	8	nonlinear	nonlinear	ADJ
cana-3833	178	9	analysis	analysis	NOUN
cana-3833	178	10	issn	issn	NOUN
cana-3833	178	11	:	:	PUNCT
cana-3833	178	12	1074	1074	NUM
cana-3833	178	13	-	-	PUNCT
cana-3833	178	14	133x	133x	NUM
cana-3833	178	15	vol	vol	NOUN
cana-3833	178	16	32	32	NUM
cana-3833	178	17	no	no	NOUN
cana-3833	178	18	.	.	PUNCT
cana-3833	179	1	9s	9s	NUM
cana-3833	179	2	(	(	PUNCT
cana-3833	179	3	2025	2025	NUM
cana-3833	179	4	)	)	PUNCT
cana-3833	179	5	8	8	NUM
cana-3833	179	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	179	7	𝓅	𝓅	PROPN
cana-3833	179	8	(	(	PUNCT
cana-3833	179	9	𝔰𝑛	𝔰𝑛	PROPN
cana-3833	179	10	)	)	PUNCT
cana-3833	179	11	=	=	SYM
cana-3833	180	1	8𝑛	8𝑛	NOUN
cana-3833	180	2	3	3	NUM
cana-3833	180	3	,	,	PUNCT
cana-3833	180	4	𝓅	𝓅	PROPN
cana-3833	180	5	(	(	PUNCT
cana-3833	180	6	𝑣𝑛	𝑣𝑛	NOUN
cana-3833	180	7	)	)	PUNCT
cana-3833	180	8	=	=	SYM
cana-3833	180	9	8𝑛	8𝑛	NOUN
cana-3833	180	10	3	3	NUM
cana-3833	180	11	−	−	NOUN
cana-3833	180	12	3	3	NUM
cana-3833	180	13	,	,	PUNCT
cana-3833	180	14	𝓅	𝓅	PROPN
cana-3833	180	15	(	(	PUNCT
cana-3833	180	16	𝔱𝑛	𝔱𝑛	X
cana-3833	180	17	)	)	PUNCT
cana-3833	180	18	=	=	SYM
cana-3833	180	19	8𝑛	8𝑛	NOUN
cana-3833	180	20	3	3	NUM
cana-3833	180	21	𝓅	𝓅	PROPN
cana-3833	180	22	(	(	PUNCT
cana-3833	180	23	𝔰3𝔦	𝔰3𝔦	NOUN
cana-3833	180	24	)	)	PUNCT
cana-3833	180	25	=	=	SYM
cana-3833	180	26	1	1	NUM
cana-3833	180	27	+	+	NUM
cana-3833	180	28	8𝔦	8𝔦	NOUN
cana-3833	180	29	,	,	PUNCT
cana-3833	180	30	𝓅	𝓅	PROPN
cana-3833	180	31	(	(	PUNCT
cana-3833	180	32	𝔰3𝔦−1	𝔰3𝔦−1	NOUN
cana-3833	180	33	)	)	PUNCT
cana-3833	180	34	=	=	PUNCT
cana-3833	181	1	−3	−3	PROPN
cana-3833	182	1	+	+	NUM
cana-3833	182	2	8𝔦	8𝔦	NOUN
cana-3833	182	3	,	,	PUNCT
cana-3833	182	4	𝓅	𝓅	PROPN
cana-3833	182	5	(	(	PUNCT
cana-3833	182	6	𝔰3𝑖−2	𝔰3𝑖−2	PROPN
cana-3833	182	7	)	)	PUNCT
cana-3833	182	8	=	=	SYM
cana-3833	182	9	8𝑖	8𝑖	ADJ
cana-3833	183	1	−	−	NOUN
cana-3833	183	2	5	5	NUM
cana-3833	183	3	,	,	PUNCT
cana-3833	183	4	𝓅	𝓅	PROPN
cana-3833	183	5	(	(	PUNCT
cana-3833	183	6	𝑣3𝔦	𝑣3𝔦	NOUN
cana-3833	183	7	)	)	PUNCT
cana-3833	183	8	=	=	PUNCT
cana-3833	184	1	−3	−3	PROPN
cana-3833	185	1	+	+	NUM
cana-3833	185	2	8𝔦	8𝔦	NOUN
cana-3833	185	3	,	,	PUNCT
cana-3833	185	4	𝓅	𝓅	PROPN
cana-3833	185	5	(	(	PUNCT
cana-3833	185	6	𝑣3𝑖−1	𝑣3𝑖−1	PROPN
cana-3833	185	7	)	)	PUNCT
cana-3833	185	8	=	=	SYM
cana-3833	185	9	8𝑖	8𝑖	ADJ
cana-3833	185	10	−	−	NOUN
cana-3833	185	11	5	5	NUM
cana-3833	185	12	,	,	PUNCT
cana-3833	185	13	𝓅	𝓅	PROPN
cana-3833	185	14	(	(	PUNCT
cana-3833	185	15	𝑣3𝑖−2	𝑣3𝑖−2	PROPN
cana-3833	185	16	)	)	PUNCT
cana-3833	185	17	=	=	SYM
cana-3833	185	18	8𝑖	8𝑖	ADJ
cana-3833	186	1	−	−	NOUN
cana-3833	186	2	7	7	NUM
cana-3833	186	3	,	,	PUNCT
cana-3833	186	4	𝓅	𝓅	PROPN
cana-3833	186	5	(	(	PUNCT
cana-3833	186	6	𝔱1	𝔱1	PROPN
cana-3833	186	7	)	)	PUNCT
cana-3833	186	8	=	=	SYM
cana-3833	186	9	1	1	NUM
cana-3833	186	10	𝓅	𝓅	PROPN
cana-3833	186	11	(	(	PUNCT
cana-3833	186	12	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	186	13	)	)	PUNCT
cana-3833	186	14	=	=	SYM
cana-3833	186	15	8𝑖	8𝑖	ADJ
cana-3833	187	1	+	+	CCONJ
cana-3833	187	2	1	1	NUM
cana-3833	187	3	,	,	PUNCT
cana-3833	187	4	𝓅	𝓅	PROPN
cana-3833	187	5	(	(	PUNCT
cana-3833	187	6	𝔱3𝔦−1	𝔱3𝔦−1	NOUN
cana-3833	187	7	)	)	PUNCT
cana-3833	187	8	=	=	PUNCT
cana-3833	188	1	−3	−3	PROPN
cana-3833	189	1	+	+	NUM
cana-3833	189	2	8𝔦	8𝔦	NOUN
cana-3833	189	3	,	,	PUNCT
cana-3833	189	4	𝓅	𝓅	NOUN
cana-3833	189	5	(	(	PUNCT
cana-3833	189	6	𝔱3𝑖−2	𝔱3𝑖−2	PROPN
cana-3833	189	7	)	)	PUNCT
cana-3833	189	8	=	=	SYM
cana-3833	189	9	8𝑖	8𝑖	ADJ
cana-3833	189	10	−	−	NOUN
cana-3833	189	11	5	5	NUM
cana-3833	189	12	,	,	PUNCT
cana-3833	189	13	𝑖	𝑖	SYM
cana-3833	189	14	≠	≠	PROPN
cana-3833	189	15	1	1	NUM
cana-3833	189	16	𝓅	𝓅	PROPN
cana-3833	189	17	(	(	PUNCT
cana-3833	189	18	𝔰3𝔦−2	𝔰3𝔦−2	NOUN
cana-3833	189	19	𝔰3𝔦−1	𝔰3𝔦−1	NOUN
cana-3833	189	20	)	)	PUNCT
cana-3833	190	1	=	=	PRON
cana-3833	190	2	−7	−7	NOUN
cana-3833	190	3	+	+	NOUN
cana-3833	190	4	8𝔦	8𝔦	NOUN
cana-3833	190	5	,	,	PUNCT
cana-3833	190	6	𝓅	𝓅	PROPN
cana-3833	190	7	(	(	PUNCT
cana-3833	190	8	𝔰3𝑖−1	𝔰3𝑖−1	PROPN
cana-3833	190	9	𝔰3𝑖	𝔰3𝑖	PROPN
cana-3833	190	10	)	)	PUNCT
cana-3833	190	11	=	=	SYM
cana-3833	190	12	8𝑖	8𝑖	ADJ
cana-3833	190	13	−	−	NOUN
cana-3833	190	14	5	5	NUM
cana-3833	190	15	,	,	PUNCT
cana-3833	190	16	𝓅	𝓅	PROPN
cana-3833	190	17	(	(	PUNCT
cana-3833	190	18	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	190	19	𝔰3𝑖+1	𝔰3𝑖+1	PROPN
cana-3833	190	20	)	)	PUNCT
cana-3833	190	21	=	=	SYM
cana-3833	190	22	8𝑖	8𝑖	ADJ
cana-3833	190	23	−	−	NOUN
cana-3833	190	24	3	3	NUM
cana-3833	190	25	,	,	PUNCT
cana-3833	190	26	𝓅	𝓅	PROPN
cana-3833	190	27	(	(	PUNCT
cana-3833	190	28	𝔰1𝑣1	𝔰1𝑣1	NOUN
cana-3833	190	29	)	)	PUNCT
cana-3833	190	30	=	=	SYM
cana-3833	190	31	1	1	NUM
cana-3833	190	32	𝓅	𝓅	PROPN
cana-3833	190	33	(	(	PUNCT
cana-3833	190	34	𝔰3𝑖−1𝑣3𝑖−1	𝔰3𝑖−1𝑣3𝑖−1	NOUN
cana-3833	190	35	)	)	PUNCT
cana-3833	190	36	=	=	SYM
cana-3833	190	37	8𝑖	8𝑖	ADJ
cana-3833	190	38	−	−	NOUN
cana-3833	190	39	5	5	NUM
cana-3833	190	40	,	,	PUNCT
cana-3833	190	41	𝓅	𝓅	PROPN
cana-3833	190	42	(	(	PUNCT
cana-3833	190	43	𝔰3𝑖𝑣3𝑖	𝔰3𝑖𝑣3𝑖	NUM
cana-3833	190	44	)	)	PUNCT
cana-3833	190	45	=	=	SYM
cana-3833	190	46	8𝑖	8𝑖	ADJ
cana-3833	190	47	−	−	NOUN
cana-3833	190	48	3	3	NUM
cana-3833	190	49	,	,	PUNCT
cana-3833	190	50	𝓅	𝓅	PROPN
cana-3833	190	51	(	(	PUNCT
cana-3833	190	52	𝔰3𝑖−2𝑣3𝑖−2	𝔰3𝑖−2𝑣3𝑖−2	PROPN
cana-3833	190	53	)	)	PUNCT
cana-3833	190	54	=	=	SYM
cana-3833	190	55	8𝑖	8𝑖	ADJ
cana-3833	190	56	−	−	NOUN
cana-3833	190	57	9	9	NUM
cana-3833	190	58	,	,	PUNCT
cana-3833	190	59	𝑖	𝑖	PRON
cana-3833	190	60	≠	≠	PROPN
cana-3833	190	61	1	1	NUM
cana-3833	190	62	𝓅(𝑣3𝔦−2𝔱3𝔦−2	𝓅(𝑣3𝔦−2𝔱3𝔦−2	NOUN
cana-3833	190	63	)	)	PUNCT
cana-3833	190	64	=	=	PRON
cana-3833	190	65	−7	−7	NOUN
cana-3833	190	66	+	+	NOUN
cana-3833	190	67	8𝔦	8𝔦	NOUN
cana-3833	190	68	,	,	PUNCT
cana-3833	190	69	𝓅(𝑣3𝔦−1𝔱3𝔦−1	𝓅(𝑣3𝔦−1𝔱3𝔦−1	NOUN
cana-3833	190	70	)	)	PUNCT
cana-3833	190	71	=	=	PUNCT
cana-3833	191	1	−3	−3	PROPN
cana-3833	192	1	+	+	NUM
cana-3833	192	2	8𝔦	8𝔦	NOUN
cana-3833	192	3	,	,	PUNCT
cana-3833	192	4	𝓅(𝑣3𝑖𝔱3𝑖	𝓅(𝑣3𝑖𝔱3𝑖	NOUN
cana-3833	192	5	)	)	PUNCT
cana-3833	192	6	=	=	SYM
cana-3833	192	7	8𝑖	8𝑖	ADJ
cana-3833	193	1	−	−	NOUN
cana-3833	193	2	1	1	NUM
cana-3833	193	3	,	,	PUNCT
cana-3833	193	4	𝓅	𝓅	PROPN
cana-3833	193	5	(	(	PUNCT
cana-3833	193	6	𝔰1𝔱1	𝔰1𝔱1	NOUN
cana-3833	193	7	)	)	PUNCT
cana-3833	193	8	=	=	SYM
cana-3833	193	9	3	3	NUM
cana-3833	193	10	𝓅	𝓅	PROPN
cana-3833	193	11	(	(	PUNCT
cana-3833	193	12	𝔰3𝑖−1𝔱3𝑖−1	𝔰3𝑖−1𝔱3𝑖−1	PROPN
cana-3833	193	13	)	)	PUNCT
cana-3833	193	14	=	=	SYM
cana-3833	193	15	8𝑖	8𝑖	ADJ
cana-3833	193	16	−	−	NOUN
cana-3833	193	17	3	3	NUM
cana-3833	193	18	,	,	PUNCT
cana-3833	193	19	𝓅	𝓅	PROPN
cana-3833	193	20	(	(	PUNCT
cana-3833	193	21	𝔰3𝑖𝔱3𝑖	𝔰3𝑖𝔱3𝑖	NUM
cana-3833	193	22	)	)	PUNCT
cana-3833	193	23	=	=	SYM
cana-3833	193	24	8𝑖	8𝑖	ADJ
cana-3833	193	25	−	−	NOUN
cana-3833	193	26	3	3	NUM
cana-3833	193	27	,	,	PUNCT
cana-3833	193	28	𝓅	𝓅	PROPN
cana-3833	193	29	(	(	PUNCT
cana-3833	193	30	𝔰3𝑖−2𝔱3𝑖−2	𝔰3𝑖−2𝔱3𝑖−2	PROPN
cana-3833	193	31	)	)	PUNCT
cana-3833	193	32	=	=	SYM
cana-3833	193	33	8𝑖	8𝑖	ADJ
cana-3833	193	34	−	−	NOUN
cana-3833	193	35	7	7	NUM
cana-3833	193	36	,	,	PUNCT
cana-3833	193	37	𝑖	𝑖	PRON
cana-3833	193	38	≠	≠	PROPN
cana-3833	193	39	1	1	NUM
cana-3833	193	40	𝓅	𝓅	PROPN
cana-3833	193	41	(	(	PUNCT
cana-3833	193	42	𝔰𝑛−1	𝔰𝑛−1	NOUN
cana-3833	193	43	𝔰𝑛	𝔰𝑛	PROPN
cana-3833	193	44	)	)	PUNCT
cana-3833	193	45	=	=	NOUN
cana-3833	193	46	8𝑛	8𝑛	NOUN
cana-3833	193	47	3	3	NUM
cana-3833	193	48	−	−	NOUN
cana-3833	193	49	4	4	NUM
cana-3833	193	50	,	,	PUNCT
cana-3833	193	51	𝓅	𝓅	PROPN
cana-3833	193	52	(	(	PUNCT
cana-3833	193	53	𝔰𝑛	𝔰𝑛	NOUN
cana-3833	193	54	𝑣𝑛	𝑣𝑛	NOUN
cana-3833	193	55	)	)	PUNCT
cana-3833	193	56	=	=	SYM
cana-3833	193	57	8𝑛	8𝑛	NOUN
cana-3833	193	58	3	3	NUM
cana-3833	193	59	−	−	NOUN
cana-3833	193	60	2	2	NUM
cana-3833	193	61	,	,	PUNCT
cana-3833	193	62	𝓅	𝓅	PROPN
cana-3833	193	63	(	(	PUNCT
cana-3833	193	64	𝔰𝑛	𝔰𝑛	NUM
cana-3833	193	65	𝔱𝑛	𝔱𝑛	NOUN
cana-3833	193	66	)	)	PUNCT
cana-3833	193	67	=	=	SYM
cana-3833	193	68	8𝑛	8𝑛	NOUN
cana-3833	193	69	3	3	NUM
cana-3833	193	70	−	−	NOUN
cana-3833	193	71	1	1	NUM
cana-3833	193	72	,	,	PUNCT
cana-3833	193	73	𝓅	𝓅	PROPN
cana-3833	193	74	(	(	PUNCT
cana-3833	193	75	𝑣𝑛	𝑣𝑛	ADJ
cana-3833	193	76	𝔱𝑛	𝔱𝑛	NOUN
cana-3833	193	77	)	)	PUNCT
cana-3833	193	78	=	=	SYM
cana-3833	193	79	8𝑛	8𝑛	NOUN
cana-3833	193	80	3	3	NUM
cana-3833	193	81	case	case	NOUN
cana-3833	193	82	ii	ii	NOUN
cana-3833	193	83	if	if	SCONJ
cana-3833	193	84	𝑛	𝑛	PRON
cana-3833	193	85	−	−	PROPN
cana-3833	193	86	1	1	NUM
cana-3833	193	87	≡	≡	PROPN
cana-3833	193	88	0	0	NUM
cana-3833	193	89	(	(	PUNCT
cana-3833	193	90	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	193	91	3	3	NUM
cana-3833	193	92	)	)	PUNCT
cana-3833	193	93	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	193	94	)	)	PUNCT
cana-3833	193	95	=	=	PUNCT
cana-3833	194	1	−3	−3	PROPN
cana-3833	195	1	+	+	NUM
cana-3833	195	2	8𝔦	8𝔦	NOUN
cana-3833	195	3	,	,	PUNCT
cana-3833	195	4	𝓅(𝑣3𝑖−1	𝓅(𝑣3𝑖−1	NUM
cana-3833	195	5	)	)	PUNCT
cana-3833	195	6	=	=	SYM
cana-3833	195	7	8𝑖	8𝑖	ADJ
cana-3833	196	1	−	−	NOUN
cana-3833	196	2	5	5	NUM
cana-3833	196	3	,	,	PUNCT
cana-3833	196	4	𝓅(𝑣3𝑖−2	𝓅(𝑣3𝑖−2	PROPN
cana-3833	196	5	)	)	PUNCT
cana-3833	196	6	=	=	SYM
cana-3833	196	7	8𝑖	8𝑖	ADJ
cana-3833	196	8	−	−	NOUN
cana-3833	196	9	7	7	NUM
cana-3833	196	10	,	,	PUNCT
cana-3833	196	11	communications	communication	NOUN
cana-3833	196	12	on	on	ADP
cana-3833	196	13	applied	apply	VERB
cana-3833	196	14	nonlinear	nonlinear	ADJ
cana-3833	196	15	analysis	analysis	NOUN
cana-3833	196	16	issn	issn	NOUN
cana-3833	196	17	:	:	PUNCT
cana-3833	196	18	1074	1074	NUM
cana-3833	196	19	-	-	PUNCT
cana-3833	196	20	133x	133x	NUM
cana-3833	196	21	vol	vol	NOUN
cana-3833	196	22	32	32	NUM
cana-3833	196	23	no	no	NOUN
cana-3833	196	24	.	.	PUNCT
cana-3833	197	1	9s	9s	NUM
cana-3833	197	2	(	(	PUNCT
cana-3833	197	3	2025	2025	NUM
cana-3833	197	4	)	)	PUNCT
cana-3833	198	1	9	9	NUM
cana-3833	198	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	198	3	𝓅	𝓅	PROPN
cana-3833	198	4	(	(	PUNCT
cana-3833	198	5	𝔰3𝔦	𝔰3𝔦	NOUN
cana-3833	198	6	)	)	PUNCT
cana-3833	198	7	=	=	SYM
cana-3833	198	8	1	1	NUM
cana-3833	198	9	+	+	NUM
cana-3833	198	10	8𝔦	8𝔦	NOUN
cana-3833	198	11	,	,	PUNCT
cana-3833	198	12	𝓅	𝓅	PROPN
cana-3833	198	13	(	(	PUNCT
cana-3833	198	14	𝔰3𝔦−1	𝔰3𝔦−1	NOUN
cana-3833	198	15	)	)	PUNCT
cana-3833	198	16	=	=	PUNCT
cana-3833	199	1	−3	−3	PROPN
cana-3833	200	1	+	+	NUM
cana-3833	200	2	8𝔦	8𝔦	NOUN
cana-3833	200	3	,	,	PUNCT
cana-3833	200	4	𝓅	𝓅	PROPN
cana-3833	200	5	(	(	PUNCT
cana-3833	200	6	𝔰3𝔦−2	𝔰3𝔦−2	NOUN
cana-3833	200	7	)	)	PUNCT
cana-3833	200	8	=	=	SYM
cana-3833	200	9	8𝑖	8𝑖	ADJ
cana-3833	201	1	−	−	NOUN
cana-3833	201	2	5	5	NUM
cana-3833	201	3	,	,	PUNCT
cana-3833	201	4	𝓅	𝓅	PROPN
cana-3833	201	5	(	(	PUNCT
cana-3833	201	6	𝔱1	𝔱1	PROPN
cana-3833	201	7	)	)	PUNCT
cana-3833	201	8	=	=	SYM
cana-3833	201	9	1	1	NUM
cana-3833	201	10	𝓅	𝓅	PROPN
cana-3833	201	11	(	(	PUNCT
cana-3833	201	12	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	201	13	)	)	PUNCT
cana-3833	201	14	=	=	SYM
cana-3833	201	15	8𝑖	8𝑖	ADJ
cana-3833	202	1	+	+	CCONJ
cana-3833	202	2	1	1	NUM
cana-3833	202	3	,	,	PUNCT
cana-3833	202	4	𝓅	𝓅	PROPN
cana-3833	202	5	(	(	PUNCT
cana-3833	202	6	𝔱3𝔦−1	𝔱3𝔦−1	NOUN
cana-3833	202	7	)	)	PUNCT
cana-3833	202	8	=	=	PUNCT
cana-3833	203	1	−3	−3	PROPN
cana-3833	204	1	+	+	NUM
cana-3833	204	2	8𝔦	8𝔦	NOUN
cana-3833	204	3	,	,	PUNCT
cana-3833	204	4	𝓅	𝓅	PROPN
cana-3833	204	5	(	(	PUNCT
cana-3833	204	6	𝔱3𝔦−2	𝔱3𝔦−2	PROPN
cana-3833	204	7	)	)	PUNCT
cana-3833	204	8	=	=	SYM
cana-3833	205	1	−5	−5	NOUN
cana-3833	206	1	+	+	NUM
cana-3833	206	2	8𝔦	8𝔦	NOUN
cana-3833	206	3	,	,	PUNCT
cana-3833	206	4	𝑖	𝑖	SYM
cana-3833	206	5	≠	≠	PROPN
cana-3833	206	6	1	1	NUM
cana-3833	206	7	𝓅	𝓅	PROPN
cana-3833	206	8	(	(	PUNCT
cana-3833	206	9	𝔰3𝑖−1	𝔰3𝑖−1	PROPN
cana-3833	206	10	𝔰3𝑖	𝔰3𝑖	PROPN
cana-3833	206	11	)	)	PUNCT
cana-3833	206	12	=	=	SYM
cana-3833	206	13	8𝑖	8𝑖	ADJ
cana-3833	207	1	−	−	NOUN
cana-3833	207	2	5	5	NUM
cana-3833	207	3	,	,	PUNCT
cana-3833	207	4	𝓅	𝓅	PROPN
cana-3833	207	5	(	(	PUNCT
cana-3833	207	6	𝔰3𝔦−2	𝔰3𝔦−2	NOUN
cana-3833	207	7	𝔰3𝔦−1	𝔰3𝔦−1	NOUN
cana-3833	207	8	)	)	PUNCT
cana-3833	208	1	=	=	PRON
cana-3833	208	2	−7	−7	NOUN
cana-3833	208	3	+	+	NOUN
cana-3833	208	4	8𝔦	8𝔦	NOUN
cana-3833	208	5	,	,	PUNCT
cana-3833	208	6	𝓅	𝓅	X
cana-3833	208	7	(	(	PUNCT
cana-3833	208	8	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	208	9	𝔰3𝑖+1	𝔰3𝑖+1	PROPN
cana-3833	208	10	)	)	PUNCT
cana-3833	208	11	=	=	SYM
cana-3833	208	12	8𝑖	8𝑖	ADJ
cana-3833	208	13	−	−	NOUN
cana-3833	208	14	3	3	NUM
cana-3833	208	15	,	,	PUNCT
cana-3833	208	16	𝓅	𝓅	PROPN
cana-3833	208	17	(	(	PUNCT
cana-3833	208	18	𝔰1𝑣1	𝔰1𝑣1	NOUN
cana-3833	208	19	)	)	PUNCT
cana-3833	208	20	=	=	SYM
cana-3833	208	21	1	1	NUM
cana-3833	208	22	𝓅	𝓅	PROPN
cana-3833	208	23	(	(	PUNCT
cana-3833	208	24	𝔰3𝑖−1	𝔰3𝑖−1	PROPN
cana-3833	208	25	𝑣3𝑖−1	𝑣3𝑖−1	PROPN
cana-3833	208	26	)	)	PUNCT
cana-3833	208	27	=	=	SYM
cana-3833	208	28	8𝑖	8𝑖	ADJ
cana-3833	208	29	−	−	NOUN
cana-3833	208	30	5	5	NUM
cana-3833	208	31	,	,	PUNCT
cana-3833	208	32	𝓅	𝓅	PROPN
cana-3833	208	33	(	(	PUNCT
cana-3833	208	34	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	208	35	𝑣3𝑖	𝑣3𝑖	NOUN
cana-3833	208	36	)	)	PUNCT
cana-3833	208	37	=	=	SYM
cana-3833	208	38	8𝑖	8𝑖	ADJ
cana-3833	208	39	−	−	NOUN
cana-3833	208	40	3	3	NUM
cana-3833	208	41	,	,	PUNCT
cana-3833	208	42	𝓅	𝓅	PROPN
cana-3833	208	43	(	(	PUNCT
cana-3833	208	44	𝔰3𝑖−2	𝔰3𝑖−2	PROPN
cana-3833	208	45	𝑣3𝑖−2	𝑣3𝑖−2	PROPN
cana-3833	208	46	)	)	PUNCT
cana-3833	208	47	=	=	SYM
cana-3833	208	48	8𝑖	8𝑖	ADJ
cana-3833	208	49	−	−	NOUN
cana-3833	208	50	9	9	NUM
cana-3833	208	51	,	,	PUNCT
cana-3833	208	52	𝓅(𝑣3𝔦−2	𝓅(𝑣3𝔦−2	PROPN
cana-3833	208	53	𝔱3𝔦−2	𝔱3𝔦−2	PROPN
cana-3833	208	54	)	)	PUNCT
cana-3833	208	55	=	=	PRON
cana-3833	208	56	−7	−7	NOUN
cana-3833	208	57	+	+	NOUN
cana-3833	208	58	8𝔦	8𝔦	NOUN
cana-3833	208	59	,	,	PUNCT
cana-3833	208	60	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	ADJ
cana-3833	208	61	𝔱3𝔦−1	𝔱3𝔦−1	NOUN
cana-3833	208	62	)	)	PUNCT
cana-3833	208	63	=	=	PUNCT
cana-3833	209	1	−3	−3	PROPN
cana-3833	210	1	+	+	NUM
cana-3833	210	2	8𝔦	8𝔦	NOUN
cana-3833	210	3	,	,	PUNCT
cana-3833	210	4	𝓅(𝑣3𝑖	𝓅(𝑣3𝑖	PROPN
cana-3833	210	5	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	210	6	)	)	PUNCT
cana-3833	210	7	=	=	SYM
cana-3833	210	8	8𝑖	8𝑖	ADJ
cana-3833	210	9	−	−	NOUN
cana-3833	210	10	1	1	NUM
cana-3833	210	11	,	,	PUNCT
cana-3833	210	12	𝓅(𝔰1𝔱1	𝓅(𝔰1𝔱1	ADJ
cana-3833	210	13	)	)	PUNCT
cana-3833	210	14	=	=	SYM
cana-3833	210	15	3	3	NUM
cana-3833	210	16	𝓅(𝔰3𝑖−1	𝓅(𝔰3𝑖−1	NOUN
cana-3833	210	17	𝔱3𝑖−1	𝔱3𝑖−1	PROPN
cana-3833	210	18	)	)	PUNCT
cana-3833	210	19	=	=	SYM
cana-3833	210	20	8𝑖	8𝑖	ADJ
cana-3833	210	21	−	−	NOUN
cana-3833	210	22	3	3	NUM
cana-3833	210	23	,	,	PUNCT
cana-3833	210	24	𝓅	𝓅	PROPN
cana-3833	210	25	(	(	PUNCT
cana-3833	210	26	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	210	27	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	210	28	)	)	PUNCT
cana-3833	210	29	=	=	SYM
cana-3833	210	30	8𝑖	8𝑖	ADJ
cana-3833	210	31	−	−	NOUN
cana-3833	210	32	3	3	NUM
cana-3833	210	33	,	,	PUNCT
cana-3833	210	34	𝓅	𝓅	PROPN
cana-3833	210	35	(	(	PUNCT
cana-3833	210	36	𝔰3𝑖−2	𝔰3𝑖−2	PROPN
cana-3833	210	37	𝔱3𝑖−2	𝔱3𝑖−2	PROPN
cana-3833	210	38	)	)	PUNCT
cana-3833	210	39	=	=	SYM
cana-3833	210	40	8𝑖	8𝑖	ADJ
cana-3833	211	1	−	−	NOUN
cana-3833	211	2	7	7	NUM
cana-3833	211	3	,	,	PUNCT
cana-3833	211	4	𝑖	𝑖	PRON
cana-3833	211	5	≠	≠	ADJ
cana-3833	211	6	1	1	NUM
cana-3833	211	7	case	case	NOUN
cana-3833	211	8	iii	iii	NOUN
cana-3833	211	9	if	if	SCONJ
cana-3833	211	10	𝑛	𝑛	PRON
cana-3833	211	11	≡	≡	PROPN
cana-3833	211	12	2	2	NUM
cana-3833	211	13	(	(	PUNCT
cana-3833	211	14	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	211	15	3	3	NUM
cana-3833	211	16	)	)	PUNCT
cana-3833	211	17	𝓅	𝓅	PROPN
cana-3833	211	18	(	(	PUNCT
cana-3833	211	19	𝔰3𝑖−2	𝔰3𝑖−2	PROPN
cana-3833	211	20	)	)	PUNCT
cana-3833	211	21	=	=	SYM
cana-3833	211	22	8𝑖	8𝑖	ADJ
cana-3833	211	23	−	−	NOUN
cana-3833	211	24	5	5	NUM
cana-3833	211	25	,	,	PUNCT
cana-3833	211	26	𝓅	𝓅	PROPN
cana-3833	211	27	(	(	PUNCT
cana-3833	211	28	𝔰3𝔦−1	𝔰3𝔦−1	NOUN
cana-3833	211	29	)	)	PUNCT
cana-3833	211	30	=	=	PUNCT
cana-3833	212	1	−3	−3	PROPN
cana-3833	213	1	+	+	NUM
cana-3833	213	2	8𝔦	8𝔦	NOUN
cana-3833	213	3	,	,	PUNCT
cana-3833	213	4	𝓅	𝓅	NOUN
cana-3833	213	5	(	(	PUNCT
cana-3833	213	6	𝔰3𝔦	𝔰3𝔦	NOUN
cana-3833	213	7	)	)	PUNCT
cana-3833	213	8	=	=	SYM
cana-3833	214	1	1	1	NUM
cana-3833	214	2	+	+	NUM
cana-3833	214	3	8𝔦	8𝔦	NOUN
cana-3833	214	4	,	,	PUNCT
cana-3833	214	5	𝓅(𝑣3𝔦−2	𝓅(𝑣3𝔦−2	NOUN
cana-3833	214	6	)	)	PUNCT
cana-3833	214	7	=	=	PRON
cana-3833	215	1	−7	−7	NOUN
cana-3833	215	2	+	+	NOUN
cana-3833	215	3	8𝔦	8𝔦	NOUN
cana-3833	215	4	,	,	PUNCT
cana-3833	215	5	𝓅(𝑣3𝑖−1	𝓅(𝑣3𝑖−1	NUM
cana-3833	215	6	)	)	PUNCT
cana-3833	215	7	=	=	SYM
cana-3833	215	8	8𝑖	8𝑖	ADJ
cana-3833	216	1	−	−	NOUN
cana-3833	216	2	5	5	NUM
cana-3833	216	3	,	,	PUNCT
cana-3833	216	4	𝓅(𝑣3𝔦	𝓅(𝑣3𝔦	NOUN
cana-3833	216	5	)	)	PUNCT
cana-3833	216	6	=	=	PUNCT
cana-3833	216	7	−3	−3	PROPN
cana-3833	217	1	+	+	NUM
cana-3833	217	2	8𝔦	8𝔦	NOUN
cana-3833	217	3	,	,	PUNCT
cana-3833	217	4	𝓅	𝓅	PROPN
cana-3833	217	5	(	(	PUNCT
cana-3833	217	6	𝔱1	𝔱1	PROPN
cana-3833	217	7	)	)	PUNCT
cana-3833	217	8	=	=	SYM
cana-3833	217	9	1	1	NUM
cana-3833	217	10	𝓅	𝓅	PROPN
cana-3833	217	11	(	(	PUNCT
cana-3833	217	12	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	217	13	)	)	PUNCT
cana-3833	217	14	=	=	SYM
cana-3833	217	15	8𝑖	8𝑖	ADJ
cana-3833	218	1	+	+	CCONJ
cana-3833	218	2	1	1	NUM
cana-3833	218	3	,	,	PUNCT
cana-3833	218	4	𝓅	𝓅	PROPN
cana-3833	218	5	(	(	PUNCT
cana-3833	218	6	𝔱3𝔦−1	𝔱3𝔦−1	NOUN
cana-3833	218	7	)	)	PUNCT
cana-3833	218	8	=	=	PUNCT
cana-3833	219	1	−3	−3	PROPN
cana-3833	220	1	+	+	NUM
cana-3833	220	2	8𝔦	8𝔦	NOUN
cana-3833	220	3	,	,	PUNCT
cana-3833	220	4	communications	communication	NOUN
cana-3833	220	5	on	on	ADP
cana-3833	220	6	applied	apply	VERB
cana-3833	220	7	nonlinear	nonlinear	ADJ
cana-3833	220	8	analysis	analysis	NOUN
cana-3833	220	9	issn	issn	NOUN
cana-3833	220	10	:	:	PUNCT
cana-3833	220	11	1074	1074	NUM
cana-3833	220	12	-	-	PUNCT
cana-3833	220	13	133x	133x	NUM
cana-3833	220	14	vol	vol	NOUN
cana-3833	220	15	32	32	NUM
cana-3833	220	16	no	no	NOUN
cana-3833	220	17	.	.	PUNCT
cana-3833	221	1	9s	9s	NUM
cana-3833	221	2	(	(	PUNCT
cana-3833	221	3	2025	2025	NUM
cana-3833	221	4	)	)	PUNCT
cana-3833	221	5	10	10	NUM
cana-3833	222	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	222	2	𝓅	𝓅	PROPN
cana-3833	222	3	(	(	PUNCT
cana-3833	222	4	𝔱3𝔦−2	𝔱3𝔦−2	PROPN
cana-3833	222	5	)	)	PUNCT
cana-3833	222	6	=	=	SYM
cana-3833	222	7	−5	−5	NOUN
cana-3833	223	1	+	+	NUM
cana-3833	223	2	8𝔦	8𝔦	NOUN
cana-3833	223	3	,	,	PUNCT
cana-3833	223	4	𝑖	𝑖	SYM
cana-3833	223	5	≠	≠	PROPN
cana-3833	223	6	1	1	NUM
cana-3833	223	7	𝓅	𝓅	PROPN
cana-3833	223	8	(	(	PUNCT
cana-3833	223	9	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	223	10	𝔰3𝑖+1	𝔰3𝑖+1	PROPN
cana-3833	223	11	)	)	PUNCT
cana-3833	223	12	=	=	SYM
cana-3833	223	13	8𝑖	8𝑖	ADJ
cana-3833	223	14	−	−	NOUN
cana-3833	223	15	3	3	NUM
cana-3833	223	16	,	,	PUNCT
cana-3833	223	17	𝓅	𝓅	PROPN
cana-3833	223	18	(	(	PUNCT
cana-3833	223	19	𝔰3𝑖−1	𝔰3𝑖−1	PROPN
cana-3833	223	20	𝔰3𝑖	𝔰3𝑖	PROPN
cana-3833	223	21	)	)	PUNCT
cana-3833	223	22	=	=	SYM
cana-3833	223	23	8𝑖	8𝑖	ADJ
cana-3833	224	1	−	−	NOUN
cana-3833	224	2	5	5	NUM
cana-3833	224	3	,	,	PUNCT
cana-3833	224	4	𝓅	𝓅	PROPN
cana-3833	224	5	(	(	PUNCT
cana-3833	224	6	𝔰3𝔦−2	𝔰3𝔦−2	NOUN
cana-3833	224	7	𝔰3𝔦−1	𝔰3𝔦−1	NOUN
cana-3833	224	8	)	)	PUNCT
cana-3833	225	1	=	=	PRON
cana-3833	225	2	−7	−7	NOUN
cana-3833	225	3	+	+	NOUN
cana-3833	225	4	8𝔦	8𝔦	NOUN
cana-3833	225	5	,	,	PUNCT
cana-3833	225	6	𝓅	𝓅	PROPN
cana-3833	225	7	(	(	PUNCT
cana-3833	225	8	𝔰1𝑣1	𝔰1𝑣1	NOUN
cana-3833	225	9	)	)	PUNCT
cana-3833	225	10	=	=	SYM
cana-3833	225	11	1	1	NUM
cana-3833	225	12	𝓅	𝓅	PROPN
cana-3833	225	13	(	(	PUNCT
cana-3833	225	14	𝔰3𝑖−1𝑣3𝑖−1	𝔰3𝑖−1𝑣3𝑖−1	NOUN
cana-3833	225	15	)	)	PUNCT
cana-3833	225	16	=	=	SYM
cana-3833	225	17	8𝑖	8𝑖	ADJ
cana-3833	225	18	−	−	NOUN
cana-3833	225	19	5	5	NUM
cana-3833	225	20	,	,	PUNCT
cana-3833	225	21	𝓅	𝓅	PROPN
cana-3833	225	22	(	(	PUNCT
cana-3833	225	23	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	225	24	𝑣3𝑖	𝑣3𝑖	NOUN
cana-3833	225	25	)	)	PUNCT
cana-3833	225	26	=	=	SYM
cana-3833	225	27	8𝑖	8𝑖	ADJ
cana-3833	225	28	−	−	NOUN
cana-3833	225	29	3	3	NUM
cana-3833	225	30	,	,	PUNCT
cana-3833	225	31	𝓅	𝓅	PROPN
cana-3833	225	32	(	(	PUNCT
cana-3833	225	33	𝔰3𝑖−2	𝔰3𝑖−2	PROPN
cana-3833	225	34	𝑣3𝑖−2	𝑣3𝑖−2	PROPN
cana-3833	225	35	)	)	PUNCT
cana-3833	225	36	=	=	SYM
cana-3833	225	37	8𝑖	8𝑖	ADJ
cana-3833	225	38	−	−	NOUN
cana-3833	225	39	9	9	NUM
cana-3833	225	40	,	,	PUNCT
cana-3833	225	41	𝑖	𝑖	PRON
cana-3833	225	42	≠	≠	PROPN
cana-3833	225	43	1	1	NUM
cana-3833	225	44	𝓅(𝑣3𝔦−2	𝓅(𝑣3𝔦−2	PROPN
cana-3833	225	45	𝔱3𝔦−2	𝔱3𝔦−2	PROPN
cana-3833	225	46	)	)	PUNCT
cana-3833	225	47	=	=	PRON
cana-3833	225	48	−7	−7	NOUN
cana-3833	225	49	+	+	NOUN
cana-3833	225	50	8𝔦	8𝔦	NOUN
cana-3833	225	51	,	,	PUNCT
cana-3833	225	52	𝓅(𝑣3𝔦−1	𝓅(𝑣3𝔦−1	ADJ
cana-3833	225	53	𝔱3𝔦−1	𝔱3𝔦−1	NOUN
cana-3833	225	54	)	)	PUNCT
cana-3833	225	55	=	=	PUNCT
cana-3833	226	1	−3	−3	PROPN
cana-3833	227	1	+	+	NUM
cana-3833	227	2	8𝔦	8𝔦	NOUN
cana-3833	227	3	,	,	PUNCT
cana-3833	227	4	𝓅(𝑣3𝑖	𝓅(𝑣3𝑖	PROPN
cana-3833	227	5	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	227	6	)	)	PUNCT
cana-3833	227	7	=	=	SYM
cana-3833	227	8	8𝑖	8𝑖	ADJ
cana-3833	228	1	−	−	NOUN
cana-3833	228	2	1	1	NUM
cana-3833	228	3	,	,	PUNCT
cana-3833	228	4	𝓅	𝓅	PROPN
cana-3833	228	5	(	(	PUNCT
cana-3833	228	6	𝔰1	𝔰1	NOUN
cana-3833	228	7	𝔱1	𝔱1	PROPN
cana-3833	228	8	)	)	PUNCT
cana-3833	228	9	=	=	SYM
cana-3833	228	10	3	3	NUM
cana-3833	228	11	𝓅	𝓅	PROPN
cana-3833	228	12	(	(	PUNCT
cana-3833	228	13	𝔰3𝑖−1	𝔰3𝑖−1	PROPN
cana-3833	228	14	𝔱3𝑖−1	𝔱3𝑖−1	NUM
cana-3833	228	15	)	)	PUNCT
cana-3833	228	16	=	=	SYM
cana-3833	228	17	8𝑖	8𝑖	ADJ
cana-3833	228	18	−	−	NOUN
cana-3833	228	19	3	3	NUM
cana-3833	228	20	,	,	PUNCT
cana-3833	228	21	𝓅	𝓅	PROPN
cana-3833	228	22	(	(	PUNCT
cana-3833	228	23	𝔰3𝑖	𝔰3𝑖	NOUN
cana-3833	228	24	𝔱3𝑖	𝔱3𝑖	NOUN
cana-3833	228	25	)	)	PUNCT
cana-3833	228	26	=	=	SYM
cana-3833	228	27	8𝑖	8𝑖	ADJ
cana-3833	228	28	−	−	NOUN
cana-3833	228	29	3	3	NUM
cana-3833	228	30	,	,	PUNCT
cana-3833	228	31	𝓅	𝓅	PROPN
cana-3833	228	32	(	(	PUNCT
cana-3833	228	33	𝔰3𝑖−2	𝔰3𝑖−2	PROPN
cana-3833	228	34	𝔱3𝑖−2	𝔱3𝑖−2	PROPN
cana-3833	228	35	)	)	PUNCT
cana-3833	228	36	=	=	SYM
cana-3833	228	37	8𝑖	8𝑖	ADJ
cana-3833	228	38	−	−	NOUN
cana-3833	228	39	7	7	NUM
cana-3833	228	40	,	,	PUNCT
cana-3833	228	41	𝑖	𝑖	PRON
cana-3833	228	42	≠	≠	SYM
cana-3833	228	43	1	1	NUM
cana-3833	228	44	induced	induce	VERB
cana-3833	228	45	weight	weight	NOUN
cana-3833	228	46	function	function	NOUN
cana-3833	228	47	for	for	ADP
cana-3833	228	48	edge	edge	NOUN
cana-3833	228	49	𝜓	𝜓	PROPN
cana-3833	228	50	:	:	PUNCT
cana-3833	228	51	𝐸(𝑃𝑛	𝐸(𝑃𝑛	PROPN
cana-3833	228	52	∘	∘	PROPN
cana-3833	228	53	𝐾2	𝐾2	NOUN
cana-3833	228	54	)	)	PUNCT
cana-3833	228	55	→	→	SYM
cana-3833	228	56	{	{	PUNCT
cana-3833	228	57	3	3	NUM
cana-3833	228	58	,	,	PUNCT
cana-3833	228	59	5	5	NUM
cana-3833	228	60	,	,	PUNCT
cana-3833	228	61	7	7	NUM
cana-3833	228	62	,	,	PUNCT
cana-3833	228	63	…	…	PUNCT
cana-3833	228	64	,	,	PUNCT
cana-3833	228	65	8𝑛	8𝑛	NOUN
cana-3833	228	66	−	−	NOUN
cana-3833	228	67	1	1	NUM
cana-3833	228	68	}	}	PUNCT
cana-3833	228	69	is	be	AUX
cana-3833	228	70	given	give	VERB
cana-3833	228	71	by	by	ADP
cana-3833	228	72	𝜓(𝔰𝒿𝔰𝒿+1	𝜓(𝔰𝒿𝔰𝒿+1	NOUN
cana-3833	228	73	)	)	PUNCT
cana-3833	228	74	=	=	SYM
cana-3833	229	1	1	1	NUM
cana-3833	229	2	+	+	NUM
cana-3833	229	3	8𝒿	8𝒿	NUM
cana-3833	229	4	,	,	PUNCT
cana-3833	229	5	𝜓(𝔰1𝑣1	𝜓(𝔰1𝑣1	NOUN
cana-3833	229	6	)	)	PUNCT
cana-3833	229	7	=	=	SYM
cana-3833	229	8	5	5	NUM
cana-3833	229	9	𝜓(𝔰𝔦𝑣𝔦	𝜓(𝔰𝔦𝑣𝔦	NOUN
cana-3833	229	10	)	)	PUNCT
cana-3833	229	11	=	=	SYM
cana-3833	230	1	−5	−5	NOUN
cana-3833	231	1	+	+	NUM
cana-3833	231	2	8𝔦	8𝔦	NOUN
cana-3833	231	3	,	,	PUNCT
cana-3833	231	4	𝑖	𝑖	SYM
cana-3833	231	5	≠	≠	PROPN
cana-3833	231	6	1	1	NUM
cana-3833	231	7	𝜓(𝔰𝔦𝔱𝔦	𝜓(𝔰𝔦𝔱𝔦	NOUN
cana-3833	231	8	)	)	PUNCT
cana-3833	231	9	=	=	SYM
cana-3833	232	1	−1	−1	NOUN
cana-3833	232	2	+	+	NUM
cana-3833	232	3	8𝔦	8𝔦	NOUN
cana-3833	232	4	,	,	PUNCT
cana-3833	232	5	𝜓(𝑣1𝔱1	𝜓(𝑣1𝔱1	ADJ
cana-3833	232	6	)	)	PUNCT
cana-3833	232	7	=	=	SYM
cana-3833	232	8	3	3	NUM
cana-3833	232	9	𝜓(𝑣𝔦𝔱𝔦	𝜓(𝑣𝔦𝔱𝔦	NOUN
cana-3833	232	10	)	)	PUNCT
cana-3833	232	11	=	=	PUNCT
cana-3833	233	1	−3	−3	PROPN
cana-3833	234	1	+	+	NUM
cana-3833	234	2	8𝔦	8𝔦	NOUN
cana-3833	234	3	,	,	PUNCT
cana-3833	234	4	𝑖	𝑖	SYM
cana-3833	234	5	≠	≠	PROPN
cana-3833	234	6	1	1	NUM
cana-3833	234	7	therefore	therefore	ADV
cana-3833	234	8	,	,	PUNCT
cana-3833	234	9	the	the	DET
cana-3833	234	10	induced	induced	ADJ
cana-3833	234	11	edge	edge	NOUN
cana-3833	234	12	weights	weight	NOUN
cana-3833	234	13	of	of	ADP
cana-3833	234	14	𝑃	𝑃	NOUN
cana-3833	234	15	𝑛	𝑛	PRON
cana-3833	234	16	∘	∘	NOUN
cana-3833	234	17	𝐾2differs	𝐾2differ	NOUN
cana-3833	234	18	by	by	ADP
cana-3833	234	19	2	2	NUM
cana-3833	234	20	in	in	ADP
cana-3833	234	21	the	the	DET
cana-3833	234	22	arithmetic	arithmetic	ADJ
cana-3833	234	23	progression	progression	NOUN
cana-3833	234	24	.	.	PUNCT
cana-3833	235	1	thus	thus	ADV
cana-3833	235	2	,	,	PUNCT
cana-3833	235	3	(	(	PUNCT
cana-3833	235	4	3	3	NUM
cana-3833	235	5	,	,	PUNCT
cana-3833	235	6	2	2	NUM
cana-3833	235	7	)	)	PUNCT
cana-3833	235	8	–	–	PUNCT
cana-3833	235	9	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	X
cana-3833	235	10	𝑛	𝑛	DET
cana-3833	235	11	∘	∘	PROPN
cana-3833	235	12	𝐾2	𝐾2	NOUN
cana-3833	235	13	)	)	PUNCT
cana-3833	235	14	≤	≤	NUM
cana-3833	236	1	⌈	⌈	SYM
cana-3833	236	2	8(𝑛+1	8(𝑛+1	NUM
cana-3833	236	3	)	)	PUNCT
cana-3833	236	4	3	3	NUM
cana-3833	236	5	⌉	⌉	PRON
cana-3833	236	6	−	−	PROPN
cana-3833	236	7	3	3	NUM
cana-3833	236	8	.	.	PUNCT
cana-3833	236	9	proposition	proposition	NOUN
cana-3833	236	10	1	1	NUM
cana-3833	236	11	exhibits	exhibit	VERB
cana-3833	236	12	that(3	that(3	NOUN
cana-3833	236	13	,	,	PUNCT
cana-3833	236	14	2	2	NUM
cana-3833	236	15	)	)	PUNCT
cana-3833	236	16	–	–	PUNCT
cana-3833	236	17	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	X
cana-3833	236	18	𝑛	𝑛	PRON
cana-3833	236	19	∘	∘	PROPN
cana-3833	236	20	𝐾2	𝐾2	NOUN
cana-3833	236	21	)	)	PUNCT
cana-3833	236	22	≥	≥	NOUN
cana-3833	236	23	⌈	⌈	NOUN
cana-3833	236	24	8(𝑛+1	8(𝑛+1	NUM
cana-3833	236	25	)	)	PUNCT
cana-3833	236	26	3	3	NUM
cana-3833	236	27	⌉	⌉	PRON
cana-3833	236	28	−	−	PROPN
cana-3833	236	29	3	3	NUM
cana-3833	236	30	.	.	NOUN
cana-3833	236	31	example	example	NOUN
cana-3833	237	1	4	4	NUM
cana-3833	237	2	:	:	PUNCT
cana-3833	237	3	(	(	PUNCT
cana-3833	237	4	3	3	NUM
cana-3833	237	5	,	,	PUNCT
cana-3833	237	6	2	2	NUM
cana-3833	237	7	)	)	PUNCT
cana-3833	237	8	−	−	PROPN
cana-3833	237	9	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	237	10	8	8	NUM
cana-3833	237	11	∘	∘	NOUN
cana-3833	237	12	𝐾2)is	𝐾2)is	NOUN
cana-3833	237	13	shown	show	VERB
cana-3833	237	14	in	in	ADP
cana-3833	237	15	fig	fig	NOUN
cana-3833	237	16	5	5	NUM
cana-3833	237	17	.	.	PUNCT
cana-3833	237	18	communications	communication	NOUN
cana-3833	237	19	on	on	ADP
cana-3833	237	20	applied	apply	VERB
cana-3833	237	21	nonlinear	nonlinear	ADJ
cana-3833	237	22	analysis	analysis	NOUN
cana-3833	237	23	issn	issn	NOUN
cana-3833	237	24	:	:	PUNCT
cana-3833	237	25	1074	1074	NUM
cana-3833	237	26	-	-	PUNCT
cana-3833	237	27	133x	133x	NUM
cana-3833	237	28	vol	vol	NOUN
cana-3833	237	29	32	32	NUM
cana-3833	237	30	no	no	NOUN
cana-3833	237	31	.	.	PUNCT
cana-3833	238	1	9s	9s	NUM
cana-3833	238	2	(	(	PUNCT
cana-3833	238	3	2025	2025	NUM
cana-3833	238	4	)	)	PUNCT
cana-3833	238	5	11	11	NUM
cana-3833	239	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	239	2	figure	figure	NOUN
cana-3833	239	3	5	5	NUM
cana-3833	239	4	.	.	PUNCT
cana-3833	239	5	(	(	PUNCT
cana-3833	239	6	3	3	NUM
cana-3833	239	7	,	,	PUNCT
cana-3833	239	8	2	2	NUM
cana-3833	239	9	)	)	PUNCT
cana-3833	239	10	–	–	PUNCT
cana-3833	239	11	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	239	12	8	8	NUM
cana-3833	239	13	∘	∘	NUM
cana-3833	239	14	𝐾2	𝐾2	NOUN
cana-3833	239	15	)	)	PUNCT
cana-3833	239	16	=	=	SYM
cana-3833	240	1	21	21	NUM
cana-3833	240	2	.	.	PUNCT
cana-3833	240	3	theorem	theorem	VERB
cana-3833	240	4	5	5	NUM
cana-3833	240	5	:	:	PUNCT
cana-3833	240	6	(	(	PUNCT
cana-3833	240	7	3	3	NUM
cana-3833	240	8	,	,	PUNCT
cana-3833	240	9	2	2	NUM
cana-3833	240	10	)	)	PUNCT
cana-3833	240	11	–	–	PUNCT
cana-3833	240	12	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	240	13	𝑛	𝑛	PRON
cana-3833	240	14	3	3	NUM
cana-3833	240	15	)	)	PUNCT
cana-3833	240	16	=	=	SYM
cana-3833	240	17	2𝑛	2𝑛	PROPN
cana-3833	241	1	−	−	NOUN
cana-3833	241	2	3	3	NUM
cana-3833	241	3	for	for	ADP
cana-3833	241	4	every	every	DET
cana-3833	241	5	positive	positive	ADJ
cana-3833	241	6	integer	integer	NOUN
cana-3833	241	7	𝑛	𝑛	ADP
cana-3833	241	8	more	more	ADJ
cana-3833	241	9	than	than	ADP
cana-3833	241	10	3	3	NUM
cana-3833	241	11	.	.	PUNCT
cana-3833	241	12	proof	proof	NOUN
cana-3833	241	13	.	.	PUNCT
cana-3833	242	1	consider	consider	VERB
cana-3833	242	2	the	the	DET
cana-3833	242	3	index	index	NOUN
cana-3833	242	4	set	set	VERB
cana-3833	242	5	𝕀	𝕀	PROPN
cana-3833	242	6	=	=	SYM
cana-3833	242	7	{	{	PUNCT
cana-3833	242	8	1	1	NUM
cana-3833	242	9	,	,	PUNCT
cana-3833	242	10	2	2	NUM
cana-3833	242	11	,	,	PUNCT
cana-3833	242	12	…	…	PUNCT
cana-3833	242	13	,	,	PUNCT
cana-3833	242	14	𝑛	𝑛	NOUN
cana-3833	242	15	}	}	PUNCT
cana-3833	242	16	.	.	PUNCT
cana-3833	243	1	let	let	VERB
cana-3833	243	2	𝑉(𝑃	𝑉(𝑃	PROPN
cana-3833	243	3	𝑛	𝑛	PRON
cana-3833	243	4	3	3	NUM
cana-3833	243	5	)	)	PUNCT
cana-3833	243	6	=	=	PRON
cana-3833	243	7	{	{	PUNCT
cana-3833	243	8	𝑣𝑖	𝑣𝑖	NOUN
cana-3833	243	9	}	}	PUNCT
cana-3833	243	10	and	and	CCONJ
cana-3833	243	11	let	let	VERB
cana-3833	243	12	𝐸(𝑃	𝐸(𝑃	NOUN
cana-3833	243	13	𝑛	𝑛	DET
cana-3833	243	14	3	3	NUM
cana-3833	243	15	)	)	PUNCT
cana-3833	243	16	=	=	PUNCT
cana-3833	244	1	⋃	⋃	NOUN
cana-3833	244	2	𝐸𝑛	𝐸𝑛	ADP
cana-3833	244	3	3	3	NUM
cana-3833	244	4	𝑛=1	𝑛=1	NOUN
cana-3833	244	5	where	where	SCONJ
cana-3833	244	6	𝐸1	𝐸1	NOUN
cana-3833	244	7	=	=	SYM
cana-3833	244	8	{	{	PUNCT
cana-3833	244	9	𝑣𝑖𝑣𝑖+1	𝑣𝑖𝑣𝑖+1	PROPN
cana-3833	244	10	}	}	PUNCT
cana-3833	244	11	,	,	PUNCT
cana-3833	244	12	𝐸2	𝐸2	PROPN
cana-3833	244	13	=	=	SYM
cana-3833	244	14	{	{	PUNCT
cana-3833	244	15	𝑣𝑖𝑣𝑖+2	𝑣𝑖𝑣𝑖+2	NOUN
cana-3833	244	16	}	}	PUNCT
cana-3833	244	17	and	and	CCONJ
cana-3833	244	18	𝐸3	𝐸3	NOUN
cana-3833	244	19	=	=	SYM
cana-3833	244	20	{	{	PUNCT
cana-3833	244	21	𝑣𝑖𝑣𝑖+3	𝑣𝑖𝑣𝑖+3	NOUN
cana-3833	244	22	}	}	PUNCT
cana-3833	244	23	denotes	denote	VERB
cana-3833	244	24	the	the	DET
cana-3833	244	25	vertex	vertex	NOUN
cana-3833	244	26	set	set	NOUN
cana-3833	244	27	and	and	CCONJ
cana-3833	244	28	edge	edge	NOUN
cana-3833	244	29	set	set	NOUN
cana-3833	244	30	of	of	ADP
cana-3833	244	31	𝑃	𝑃	NOUN
cana-3833	244	32	𝑛	𝑛	ADP
cana-3833	244	33	3	3	NUM
cana-3833	244	34	for	for	ADP
cana-3833	244	35	all	all	PRON
cana-3833	244	36	𝑖	𝑖	X
cana-3833	244	37	𝜖	𝜖	PROPN
cana-3833	244	38	𝕀.	𝕀.	ADJ
cana-3833	244	39	total	total	NOUN
cana-3833	244	40	labeling	labeling	NOUN
cana-3833	244	41	𝜌	𝜌	ADP
cana-3833	244	42	:	:	PUNCT
cana-3833	244	43	𝒫	𝒫	PROPN
cana-3833	244	44	→	→	SYM
cana-3833	244	45	𝒬	𝒬	PROPN
cana-3833	244	46	where	where	SCONJ
cana-3833	244	47	𝒫	𝒫	NOUN
cana-3833	244	48	=	=	SYM
cana-3833	244	49	𝑉(𝑃	𝑉(𝑃	NUM
cana-3833	244	50	𝑛	𝑛	PRON
cana-3833	244	51	3	3	NUM
cana-3833	244	52	)	)	PUNCT
cana-3833	244	53	∪	∪	ADP
cana-3833	244	54	𝐸(𝑃	𝐸(𝑃	X
cana-3833	244	55	𝑛	𝑛	DET
cana-3833	244	56	3	3	NUM
cana-3833	244	57	)	)	PUNCT
cana-3833	244	58	and	and	CCONJ
cana-3833	244	59	𝒬	𝒬	PROPN
cana-3833	244	60	=	=	PUNCT
cana-3833	244	61	{	{	PUNCT
cana-3833	244	62	1	1	NUM
cana-3833	244	63	,	,	PUNCT
cana-3833	244	64	2	2	NUM
cana-3833	244	65	,	,	PUNCT
cana-3833	244	66	…	…	PUNCT
cana-3833	244	67	,	,	PUNCT
cana-3833	244	68	2𝑛	2𝑛	PROPN
cana-3833	244	69	−	−	PROPN
cana-3833	244	70	3	3	NUM
cana-3833	244	71	}	}	PUNCT
cana-3833	244	72	is	be	AUX
cana-3833	244	73	predicted	predict	VERB
cana-3833	244	74	by	by	ADP
cana-3833	244	75	:	:	PUNCT
cana-3833	244	76	𝜌(𝑣1	𝜌(𝑣1	ADJ
cana-3833	244	77	)	)	PUNCT
cana-3833	244	78	=	=	SYM
cana-3833	244	79	1	1	NUM
cana-3833	244	80	,	,	PUNCT
cana-3833	244	81	𝑓(𝑣2	𝑓(𝑣2	NOUN
cana-3833	244	82	)	)	PUNCT
cana-3833	244	83	=	=	SYM
cana-3833	244	84	1	1	NUM
cana-3833	244	85	,	,	PUNCT
cana-3833	244	86	𝜌(𝑣𝔦	𝜌(𝑣𝔦	X
cana-3833	244	87	)	)	PUNCT
cana-3833	244	88	=	=	SYM
cana-3833	244	89	2𝔦	2𝔦	NUM
cana-3833	245	1	−	−	PROPN
cana-3833	245	2	3	3	NUM
cana-3833	245	3	,	,	PUNCT
cana-3833	245	4	𝑖	𝑖	PRON
cana-3833	245	5	≠	≠	PROPN
cana-3833	245	6	1	1	NUM
cana-3833	245	7	,	,	PUNCT
cana-3833	245	8	2	2	NUM
cana-3833	245	9	.	.	PUNCT
cana-3833	245	10	𝜌(𝑣𝔦𝑣𝔦+1	𝜌(𝑣𝔦𝑣𝔦+1	PROPN
cana-3833	245	11	)	)	PUNCT
cana-3833	245	12	=	=	PRON
cana-3833	245	13	{	{	PUNCT
cana-3833	245	14	1	1	NUM
cana-3833	245	15	𝔦	𝔦	NOUN
cana-3833	245	16	=	=	SYM
cana-3833	245	17	1	1	NUM
cana-3833	245	18	−1	−1	NOUN
cana-3833	246	1	+	+	X
cana-3833	246	2	2𝔦	2𝔦	NUM
cana-3833	246	3	𝔦	𝔦	PROPN
cana-3833	246	4	≠	≠	PROPN
cana-3833	246	5	1	1	NUM
cana-3833	246	6	𝜌(𝑣𝔦𝑣𝔦+2	𝜌(𝑣𝔦𝑣𝔦+2	PROPN
cana-3833	246	7	)	)	PUNCT
cana-3833	246	8	=	=	PRON
cana-3833	246	9	{	{	PUNCT
cana-3833	247	1	1	1	NUM
cana-3833	247	2	𝑖	𝑖	NOUN
cana-3833	247	3	=	=	NOUN
cana-3833	247	4	1	1	NUM
cana-3833	247	5	1	1	NUM
cana-3833	247	6	+	+	NUM
cana-3833	247	7	2𝔦	2𝔦	NUM
cana-3833	247	8	𝔦	𝔦	PROPN
cana-3833	247	9	≠	≠	PROPN
cana-3833	247	10	1	1	NUM
cana-3833	247	11	𝜌(𝑣𝔦𝑣𝔦+3	𝜌(𝑣𝔦𝑣𝔦+3	NOUN
cana-3833	247	12	)	)	PUNCT
cana-3833	247	13	=	=	NOUN
cana-3833	247	14	{	{	PUNCT
cana-3833	247	15	3	3	NUM
cana-3833	247	16	𝔦	𝔦	NOUN
cana-3833	247	17	=	=	SYM
cana-3833	247	18	1	1	NUM
cana-3833	247	19	2𝑖	2𝑖	NOUN
cana-3833	247	20	+	+	CCONJ
cana-3833	247	21	3	3	NUM
cana-3833	247	22	𝔦	𝔦	X
cana-3833	247	23	≠	≠	PROPN
cana-3833	247	24	1	1	NUM
cana-3833	247	25	induced	induce	VERB
cana-3833	247	26	edge	edge	NOUN
cana-3833	247	27	weight	weight	NOUN
cana-3833	247	28	function	function	NOUN
cana-3833	247	29	𝜓	𝜓	PROPN
cana-3833	247	30	:	:	PUNCT
cana-3833	247	31	𝐸(𝑃	𝐸(𝑃	X
cana-3833	247	32	𝑛	𝑛	DET
cana-3833	247	33	3	3	NUM
cana-3833	247	34	)	)	PUNCT
cana-3833	247	35	→	→	SYM
cana-3833	247	36	{	{	PUNCT
cana-3833	247	37	3	3	NUM
cana-3833	247	38	,	,	PUNCT
cana-3833	247	39	5	5	NUM
cana-3833	247	40	,	,	PUNCT
cana-3833	247	41	7	7	NUM
cana-3833	247	42	,	,	PUNCT
cana-3833	247	43	…	…	PUNCT
cana-3833	247	44	,	,	PUNCT
cana-3833	247	45	6𝑛	6𝑛	NOUN
cana-3833	247	46	−	−	NOUN
cana-3833	247	47	11	11	NUM
cana-3833	247	48	}	}	PUNCT
cana-3833	247	49	is	be	AUX
cana-3833	247	50	given	give	VERB
cana-3833	247	51	by	by	ADP
cana-3833	247	52	𝜓(𝑣1𝑣2	𝜓(𝑣1𝑣2	NOUN
cana-3833	247	53	)	)	PUNCT
cana-3833	247	54	=	=	SYM
cana-3833	247	55	3	3	NUM
cana-3833	247	56	𝜓(𝑣𝔦𝑣𝔦+1	𝜓(𝑣𝔦𝑣𝔦+1	NOUN
cana-3833	247	57	)	)	PUNCT
cana-3833	248	1	=	=	SYM
cana-3833	248	2	−5	−5	PROPN
cana-3833	249	1	+	+	X
cana-3833	249	2	6𝔦	6𝔦	NOUN
cana-3833	249	3	,	,	PUNCT
cana-3833	249	4	𝔦	𝔦	X
cana-3833	249	5	≠	≠	PROPN
cana-3833	249	6	1	1	NUM
cana-3833	249	7	𝜓(𝑣𝔦𝑣𝔦+2	𝜓(𝑣𝔦𝑣𝔦+2	NOUN
cana-3833	249	8	)	)	PUNCT
cana-3833	249	9	=	=	SYM
cana-3833	249	10	−1	−1	NOUN
cana-3833	249	11	+	+	CCONJ
cana-3833	249	12	6𝔦	6𝔦	NOUN
cana-3833	249	13	,	,	PUNCT
cana-3833	249	14	𝜓(𝑣𝔦𝑣𝔦+3	𝜓(𝑣𝔦𝑣𝔦+3	PROPN
cana-3833	249	15	)	)	PUNCT
cana-3833	249	16	=	=	SYM
cana-3833	249	17	3	3	NUM
cana-3833	249	18	+	+	CCONJ
cana-3833	249	19	6𝔦	6𝔦	NOUN
cana-3833	249	20	,	,	PUNCT
cana-3833	249	21	therefore	therefore	ADV
cana-3833	249	22	,	,	PUNCT
cana-3833	249	23	induced	induce	VERB
cana-3833	249	24	edge	edge	NOUN
cana-3833	249	25	weights	weight	NOUN
cana-3833	249	26	of	of	ADP
cana-3833	249	27	𝑃	𝑃	NOUN
cana-3833	249	28	𝑛	𝑛	ADP
cana-3833	249	29	3differs	3differs	NUM
cana-3833	249	30	by	by	ADP
cana-3833	249	31	2	2	NUM
cana-3833	249	32	in	in	ADP
cana-3833	249	33	arithmetic	arithmetic	ADJ
cana-3833	249	34	progression	progression	NOUN
cana-3833	249	35	.	.	PUNCT
cana-3833	250	1	thus	thus	ADV
cana-3833	250	2	,	,	PUNCT
cana-3833	250	3	(	(	PUNCT
cana-3833	250	4	3	3	NUM
cana-3833	250	5	,	,	PUNCT
cana-3833	250	6	2	2	NUM
cana-3833	250	7	)	)	PUNCT
cana-3833	250	8	–	–	PUNCT
cana-3833	250	9	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	250	10	𝑛	𝑛	DET
cana-3833	250	11	3	3	NUM
cana-3833	250	12	)	)	PUNCT
cana-3833	250	13	≤	≤	NOUN
cana-3833	250	14	−3	−3	ADV
cana-3833	251	1	+	+	CCONJ
cana-3833	251	2	2𝑛.	2𝑛.	NUM
cana-3833	251	3	by	by	ADP
cana-3833	251	4	proposition	proposition	NOUN
cana-3833	251	5	1	1	NUM
cana-3833	251	6	,	,	PUNCT
cana-3833	251	7	we	we	PRON
cana-3833	251	8	get	get	VERB
cana-3833	251	9	(	(	PUNCT
cana-3833	251	10	3	3	NUM
cana-3833	251	11	,	,	PUNCT
cana-3833	251	12	2	2	NUM
cana-3833	251	13	)	)	PUNCT
cana-3833	251	14	–	–	PUNCT
cana-3833	251	15	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	251	16	𝑛	𝑛	PRON
cana-3833	251	17	3	3	NUM
cana-3833	251	18	)	)	PUNCT
cana-3833	251	19	≥	≥	NOUN
cana-3833	251	20	−3	−3	NOUN
cana-3833	252	1	+	+	CCONJ
cana-3833	252	2	2𝑛.	2𝑛.	NOUN
cana-3833	252	3	this	this	PRON
cana-3833	252	4	concludes	conclude	VERB
cana-3833	252	5	the	the	DET
cana-3833	252	6	proof	proof	NOUN
cana-3833	252	7	.	.	PUNCT
cana-3833	253	1	example	example	NOUN
cana-3833	253	2	5	5	NUM
cana-3833	253	3	:	:	PUNCT
cana-3833	253	4	(	(	PUNCT
cana-3833	253	5	3	3	NUM
cana-3833	253	6	,	,	PUNCT
cana-3833	253	7	2	2	NUM
cana-3833	253	8	)	)	PUNCT
cana-3833	253	9	−	−	PROPN
cana-3833	253	10	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	253	11	7	7	NUM
cana-3833	253	12	3	3	NUM
cana-3833	253	13	)	)	PUNCT
cana-3833	253	14	is	be	AUX
cana-3833	253	15	shown	show	VERB
cana-3833	253	16	in	in	ADP
cana-3833	253	17	fig	fig	NOUN
cana-3833	253	18	6	6	NUM
cana-3833	253	19	.	.	PUNCT
cana-3833	254	1	communications	communication	NOUN
cana-3833	254	2	on	on	ADP
cana-3833	254	3	applied	apply	VERB
cana-3833	254	4	nonlinear	nonlinear	ADJ
cana-3833	254	5	analysis	analysis	NOUN
cana-3833	254	6	issn	issn	NOUN
cana-3833	254	7	:	:	PUNCT
cana-3833	254	8	1074	1074	NUM
cana-3833	254	9	-	-	PUNCT
cana-3833	254	10	133x	133x	NUM
cana-3833	254	11	vol	vol	NOUN
cana-3833	254	12	32	32	NUM
cana-3833	254	13	no	no	NOUN
cana-3833	254	14	.	.	PUNCT
cana-3833	255	1	9s	9s	NUM
cana-3833	255	2	(	(	PUNCT
cana-3833	255	3	2025	2025	NUM
cana-3833	255	4	)	)	PUNCT
cana-3833	255	5	12	12	NUM
cana-3833	255	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	255	7	figure	figure	NOUN
cana-3833	255	8	6	6	NUM
cana-3833	255	9	.	.	PUNCT
cana-3833	256	1	(	(	PUNCT
cana-3833	256	2	3	3	NUM
cana-3833	256	3	,	,	PUNCT
cana-3833	256	4	2	2	NUM
cana-3833	256	5	)	)	PUNCT
cana-3833	256	6	–	–	PUNCT
cana-3833	256	7	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	256	8	7	7	NUM
cana-3833	256	9	3	3	NUM
cana-3833	256	10	)	)	PUNCT
cana-3833	256	11	=	=	SYM
cana-3833	257	1	11	11	NUM
cana-3833	257	2	.	.	PUNCT
cana-3833	257	3	theorem	theorem	VERB
cana-3833	257	4	6	6	NUM
cana-3833	257	5	:	:	PUNCT
cana-3833	257	6	if	if	SCONJ
cana-3833	257	7	𝑇	𝑇	PROPN
cana-3833	257	8	is	be	AUX
cana-3833	257	9	a	a	DET
cana-3833	257	10	tree	tree	NOUN
cana-3833	257	11	other	other	ADJ
cana-3833	257	12	than	than	ADP
cana-3833	257	13	star	star	NOUN
cana-3833	257	14	on	on	ADP
cana-3833	257	15	𝑝	𝑝	PROPN
cana-3833	257	16	vertices	vertex	NOUN
cana-3833	257	17	,	,	PUNCT
cana-3833	257	18	then	then	ADV
cana-3833	257	19	(	(	PUNCT
cana-3833	257	20	3	3	NUM
cana-3833	257	21	,	,	PUNCT
cana-3833	257	22	2	2	NUM
cana-3833	257	23	)	)	PUNCT
cana-3833	257	24	–	–	PUNCT
cana-3833	257	25	𝑡𝑒𝑠(𝑇	𝑡𝑒𝑠(𝑇	NUM
cana-3833	257	26	)	)	PUNCT
cana-3833	257	27	=	=	PUNCT
cana-3833	258	1	⌈	⌈	NUM
cana-3833	258	2	2𝑝−1	2𝑝−1	NUM
cana-3833	258	3	3	3	NUM
cana-3833	258	4	⌉.	⌉.	ADV
cana-3833	258	5	proof	proof	NOUN
cana-3833	258	6	.	.	PUNCT
cana-3833	259	1	step	step	NOUN
cana-3833	260	1	i	i	PRON
cana-3833	260	2	:	:	PUNCT
cana-3833	260	3	arrange	arrange	VERB
cana-3833	260	4	the	the	DET
cana-3833	260	5	given	give	VERB
cana-3833	260	6	tree	tree	NOUN
cana-3833	260	7	from	from	ADP
cana-3833	260	8	left	left	ADJ
cana-3833	260	9	to	to	ADP
cana-3833	260	10	right	right	NOUN
cana-3833	260	11	by	by	ADP
cana-3833	260	12	considering	consider	VERB
cana-3833	260	13	one	one	NUM
cana-3833	260	14	pendant	pendant	ADJ
cana-3833	260	15	as	as	ADP
cana-3833	260	16	starting	start	VERB
cana-3833	260	17	node	node	NOUN
cana-3833	260	18	.	.	PUNCT
cana-3833	261	1	step	step	PROPN
cana-3833	261	2	ii	ii	PROPN
cana-3833	261	3	:	:	PUNCT
cana-3833	261	4	label	label	VERB
cana-3833	261	5	the	the	DET
cana-3833	261	6	edges	edge	NOUN
cana-3833	261	7	of	of	ADP
cana-3833	261	8	tree	tree	NOUN
cana-3833	261	9	from	from	ADP
cana-3833	261	10	left	left	ADJ
cana-3833	261	11	to	to	ADP
cana-3833	261	12	right	right	NOUN
cana-3833	261	13	by	by	ADP
cana-3833	261	14	completing	complete	VERB
cana-3833	261	15	lower	low	ADJ
cana-3833	261	16	branch	branch	NOUN
cana-3833	261	17	and	and	CCONJ
cana-3833	261	18	then	then	ADV
cana-3833	261	19	upper	upper	ADJ
cana-3833	261	20	branch	branch	NOUN
cana-3833	261	21	using	use	VERB
cana-3833	261	22	the	the	DET
cana-3833	261	23	induced	induced	ADJ
cana-3833	261	24	edge	edge	NOUN
cana-3833	261	25	weights	weight	NOUN
cana-3833	261	26	{	{	PUNCT
cana-3833	261	27	3	3	NUM
cana-3833	261	28	,	,	PUNCT
cana-3833	261	29	5	5	NUM
cana-3833	261	30	,	,	PUNCT
cana-3833	261	31	7	7	NUM
cana-3833	261	32	,	,	PUNCT
cana-3833	261	33	…	…	PUNCT
cana-3833	261	34	,	,	PUNCT
cana-3833	261	35	2𝑝	2𝑝	NOUN
cana-3833	261	36	−	−	NOUN
cana-3833	261	37	1	1	NUM
cana-3833	261	38	}	}	PUNCT
cana-3833	261	39	.	.	PUNCT
cana-3833	262	1	step	step	NOUN
cana-3833	262	2	iii	iii	PROPN
cana-3833	262	3	:	:	PUNCT
cana-3833	262	4	each	each	DET
cana-3833	262	5	pair	pair	NOUN
cana-3833	262	6	of	of	ADP
cana-3833	262	7	vertices	vertex	NOUN
cana-3833	262	8	and	and	CCONJ
cana-3833	262	9	its	its	PRON
cana-3833	262	10	edge	edge	NOUN
cana-3833	262	11	should	should	AUX
cana-3833	262	12	be	be	AUX
cana-3833	262	13	labeled	label	VERB
cana-3833	262	14	by	by	ADP
cana-3833	262	15	at	at	ADP
cana-3833	262	16	most	most	ADV
cana-3833	262	17	odd	odd	ADJ
cana-3833	262	18	numbers	number	NOUN
cana-3833	262	19	which	which	PRON
cana-3833	262	20	are	be	AUX
cana-3833	262	21	less	less	ADJ
cana-3833	262	22	than	than	ADP
cana-3833	262	23	or	or	CCONJ
cana-3833	262	24	equal	equal	ADJ
cana-3833	262	25	to	to	ADP
cana-3833	262	26	either	either	CCONJ
cana-3833	262	27	the	the	DET
cana-3833	262	28	corresponding	corresponding	ADJ
cana-3833	262	29	induced	induce	VERB
cana-3833	262	30	edge	edge	NOUN
cana-3833	262	31	weights	weight	NOUN
cana-3833	262	32	or	or	CCONJ
cana-3833	262	33	⌈	⌈	NOUN
cana-3833	262	34	2𝑝−1	2𝑝−1	PROPN
cana-3833	262	35	3	3	NUM
cana-3833	262	36	⌉	⌉	X
cana-3833	262	37	with	with	ADP
cana-3833	262	38	non	non	ADJ
cana-3833	262	39	-	-	ADJ
cana-3833	262	40	decreasing	decrease	VERB
cana-3833	262	41	order	order	NOUN
cana-3833	262	42	.	.	PUNCT
cana-3833	263	1	step	step	NOUN
cana-3833	263	2	iv	iv	NUM
cana-3833	263	3	:	:	PUNCT
cana-3833	263	4	from	from	ADP
cana-3833	263	5	the	the	DET
cana-3833	263	6	above	above	ADJ
cana-3833	263	7	labeling	labeling	NOUN
cana-3833	263	8	,	,	PUNCT
cana-3833	263	9	(	(	PUNCT
cana-3833	263	10	3	3	NUM
cana-3833	263	11	,	,	PUNCT
cana-3833	263	12	2	2	NUM
cana-3833	263	13	)	)	PUNCT
cana-3833	263	14	–	–	PUNCT
cana-3833	263	15	𝑡𝑒𝑠(𝑇	𝑡𝑒𝑠(𝑇	NUM
cana-3833	263	16	)	)	PUNCT
cana-3833	263	17	≤	≤	NOUN
cana-3833	264	1	⌈	⌈	ADP
cana-3833	264	2	2𝑝−1	2𝑝−1	NOUN
cana-3833	264	3	3	3	NUM
cana-3833	264	4	⌉.	⌉.	ADV
cana-3833	264	5	proposition	proposition	NOUN
cana-3833	264	6	1exhibits	1exhibits	NUM
cana-3833	264	7	that(3	that(3	NOUN
cana-3833	264	8	,	,	PUNCT
cana-3833	264	9	2	2	NUM
cana-3833	264	10	)	)	PUNCT
cana-3833	264	11	–	–	PUNCT
cana-3833	264	12	𝑡𝑒𝑠(𝑇	𝑡𝑒𝑠(𝑇	NUM
cana-3833	264	13	)	)	PUNCT
cana-3833	264	14	≥	≥	NOUN
cana-3833	264	15	⌈	⌈	NOUN
cana-3833	264	16	2𝑝−1	2𝑝−1	PROPN
cana-3833	264	17	3	3	NUM
cana-3833	264	18	⌉	⌉	PUNCT
cana-3833	264	19	and	and	CCONJ
cana-3833	264	20	hence	hence	ADV
cana-3833	264	21	the	the	DET
cana-3833	264	22	result	result	NOUN
cana-3833	264	23	follows	follow	VERB
cana-3833	264	24	.	.	PUNCT
cana-3833	265	1	example	example	NOUN
cana-3833	265	2	6	6	NUM
cana-3833	265	3	:	:	PUNCT
cana-3833	265	4	(	(	PUNCT
cana-3833	265	5	3	3	NUM
cana-3833	265	6	,	,	PUNCT
cana-3833	265	7	2	2	NUM
cana-3833	265	8	)	)	PUNCT
cana-3833	265	9	total	total	ADJ
cana-3833	265	10	edge	edge	VERB
cana-3833	265	11	irregular	irregular	ADJ
cana-3833	265	12	labeling	labeling	NOUN
cana-3833	265	13	for	for	ADP
cana-3833	265	14	tree	tree	NOUN
cana-3833	265	15	with	with	ADP
cana-3833	265	16	10	10	NUM
cana-3833	265	17	vertices	vertex	NOUN
cana-3833	265	18	is	be	AUX
cana-3833	265	19	shown	show	VERB
cana-3833	265	20	in	in	ADP
cana-3833	265	21	fig	fig	NOUN
cana-3833	265	22	7	7	NUM
cana-3833	265	23	.	.	PUNCT
cana-3833	265	24	figure	figure	NOUN
cana-3833	265	25	7	7	NUM
cana-3833	265	26	.	.	PUNCT
cana-3833	266	1	(	(	PUNCT
cana-3833	266	2	3	3	NUM
cana-3833	266	3	,	,	PUNCT
cana-3833	266	4	2	2	NUM
cana-3833	266	5	)	)	PUNCT
cana-3833	266	6	–	–	PUNCT
cana-3833	266	7	𝑡𝑒𝑠(𝑇	𝑡𝑒𝑠(𝑇	NUM
cana-3833	266	8	)	)	PUNCT
cana-3833	266	9	=	=	SYM
cana-3833	266	10	7	7	NUM
cana-3833	266	11	communications	communication	NOUN
cana-3833	266	12	on	on	ADP
cana-3833	266	13	applied	apply	VERB
cana-3833	266	14	nonlinear	nonlinear	ADJ
cana-3833	266	15	analysis	analysis	NOUN
cana-3833	266	16	issn	issn	NOUN
cana-3833	266	17	:	:	PUNCT
cana-3833	266	18	1074	1074	NUM
cana-3833	266	19	-	-	PUNCT
cana-3833	266	20	133x	133x	NUM
cana-3833	266	21	vol	vol	NOUN
cana-3833	266	22	32	32	NUM
cana-3833	266	23	no	no	NOUN
cana-3833	266	24	.	.	PUNCT
cana-3833	267	1	9s	9s	NUM
cana-3833	267	2	(	(	PUNCT
cana-3833	267	3	2025	2025	NUM
cana-3833	267	4	)	)	PUNCT
cana-3833	267	5	13	13	NUM
cana-3833	267	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	267	7	theorem	theorem	VERB
cana-3833	267	8	7	7	NUM
cana-3833	267	9	:	:	PUNCT
cana-3833	267	10	(	(	PUNCT
cana-3833	267	11	3	3	NUM
cana-3833	267	12	,	,	PUNCT
cana-3833	267	13	2	2	NUM
cana-3833	267	14	)	)	PUNCT
cana-3833	267	15	–	–	PUNCT
cana-3833	267	16	𝑡𝑒𝑠(𝐾𝑛	𝑡𝑒𝑠(𝐾𝑛	NUM
cana-3833	267	17	)	)	PUNCT
cana-3833	267	18	=	=	SYM
cana-3833	267	19	𝑛2	𝑛2	NOUN
cana-3833	267	20	−	−	PROPN
cana-3833	267	21	5𝑛	5𝑛	NUM
cana-3833	268	1	+	+	CCONJ
cana-3833	268	2	9	9	NUM
cana-3833	268	3	for	for	ADP
cana-3833	268	4	𝑛	𝑛	PRON
cana-3833	268	5	≥	≥	NUM
cana-3833	268	6	4	4	NUM
cana-3833	268	7	.	.	PUNCT
cana-3833	269	1	proof	proof	NOUN
cana-3833	269	2	.	.	PUNCT
cana-3833	270	1	consider	consider	VERB
cana-3833	270	2	the	the	DET
cana-3833	270	3	index	index	NOUN
cana-3833	270	4	set	set	VERB
cana-3833	270	5	𝕀	𝕀	PROPN
cana-3833	270	6	=	=	SYM
cana-3833	270	7	{	{	PUNCT
cana-3833	270	8	1	1	NUM
cana-3833	270	9	,	,	PUNCT
cana-3833	270	10	2	2	NUM
cana-3833	270	11	,	,	PUNCT
cana-3833	270	12	…	…	PUNCT
cana-3833	270	13	,	,	PUNCT
cana-3833	270	14	𝑛	𝑛	NOUN
cana-3833	270	15	}	}	PUNCT
cana-3833	270	16	.	.	PUNCT
cana-3833	271	1	let	let	VERB
cana-3833	271	2	𝑉(𝐾𝑛	𝑉(𝐾𝑛	NOUN
cana-3833	271	3	)	)	PUNCT
cana-3833	271	4	=	=	PRON
cana-3833	271	5	{	{	PUNCT
cana-3833	271	6	𝑢𝑖	𝑢𝑖	INTJ
cana-3833	271	7	}	}	PUNCT
cana-3833	271	8	and	and	CCONJ
cana-3833	271	9	let	let	VERB
cana-3833	271	10	𝐸(𝐾𝑛	𝐸(𝐾𝑛	NOUN
cana-3833	271	11	)	)	PUNCT
cana-3833	271	12	=	=	SYM
cana-3833	271	13	{	{	PUNCT
cana-3833	271	14	𝑢𝔦𝑢𝔦+1	𝑢𝔦𝑢𝔦+1	PROPN
cana-3833	271	15	}	}	PUNCT
cana-3833	271	16	∪	∪	VERB
cana-3833	271	17	{	{	PUNCT
cana-3833	271	18	𝑢𝔦𝑢𝑗/	𝑢𝔦𝑢𝑗/	NUM
cana-3833	271	19	1	1	NUM
cana-3833	271	20	≤	≤	NOUN
cana-3833	271	21	𝔦	𝔦	X
cana-3833	271	22	<	<	X
cana-3833	271	23	𝑗	𝑗	PROPN
cana-3833	271	24	≤	≤	NUM
cana-3833	271	25	𝑛	𝑛	PROPN
cana-3833	271	26	}	}	PUNCT
cana-3833	271	27	denotes	denote	VERB
cana-3833	271	28	the	the	DET
cana-3833	271	29	vertex	vertex	NOUN
cana-3833	271	30	set	set	NOUN
cana-3833	271	31	and	and	CCONJ
cana-3833	271	32	edge	edge	NOUN
cana-3833	271	33	set	set	NOUN
cana-3833	271	34	of	of	ADP
cana-3833	271	35	𝐾𝑛	𝐾𝑛	PROPN
cana-3833	271	36	for	for	ADP
cana-3833	271	37	all	all	PRON
cana-3833	271	38	𝑖	𝑖	X
cana-3833	271	39	𝜖	𝜖	PROPN
cana-3833	271	40	𝕀.	𝕀.	ADJ
cana-3833	271	41	total	total	NOUN
cana-3833	271	42	labeling	labeling	NOUN
cana-3833	271	43	𝜌	𝜌	ADP
cana-3833	271	44	:	:	PUNCT
cana-3833	271	45	𝒫	𝒫	PROPN
cana-3833	271	46	→	→	SYM
cana-3833	271	47	𝒬	𝒬	PROPN
cana-3833	271	48	where	where	SCONJ
cana-3833	271	49	𝒫	𝒫	NOUN
cana-3833	271	50	=	=	PUNCT
cana-3833	271	51	𝑉(𝐾𝑛	𝑉(𝐾𝑛	PROPN
cana-3833	271	52	)	)	PUNCT
cana-3833	271	53	∪	∪	ADP
cana-3833	271	54	𝐸(𝐾𝑛	𝐸(𝐾𝑛	NOUN
cana-3833	271	55	)	)	PUNCT
cana-3833	271	56	and	and	CCONJ
cana-3833	271	57	𝒬	𝒬	NOUN
cana-3833	271	58	=	=	PUNCT
cana-3833	271	59	{	{	PUNCT
cana-3833	271	60	1	1	NUM
cana-3833	271	61	,	,	PUNCT
cana-3833	271	62	2	2	NUM
cana-3833	271	63	,	,	PUNCT
cana-3833	271	64	…	…	PUNCT
cana-3833	271	65	,	,	PUNCT
cana-3833	271	66	𝑛2	𝑛2	NOUN
cana-3833	271	67	−	−	PROPN
cana-3833	271	68	5𝑛	5𝑛	NUM
cana-3833	271	69	+	+	CCONJ
cana-3833	271	70	9	9	NUM
cana-3833	271	71	}	}	PUNCT
cana-3833	271	72	is	be	AUX
cana-3833	271	73	given	give	VERB
cana-3833	271	74	below	below	ADP
cana-3833	271	75	:	:	PUNCT
cana-3833	271	76	𝜌(𝑢1	𝜌(𝑢1	NUM
cana-3833	271	77	)	)	PUNCT
cana-3833	272	1	=	=	PUNCT
cana-3833	272	2	1	1	NUM
cana-3833	272	3	𝜌(𝑢𝔦	𝜌(𝑢𝔦	NOUN
cana-3833	272	4	)	)	PUNCT
cana-3833	272	5	=	=	PUNCT
cana-3833	273	1	−3	−3	PROPN
cana-3833	274	1	+	+	PUNCT
cana-3833	274	2	2𝔦	2𝔦	NUM
cana-3833	274	3	,	,	PUNCT
cana-3833	274	4	𝔦	𝔦	X
cana-3833	274	5	≠	≠	PROPN
cana-3833	274	6	1	1	NUM
cana-3833	274	7	𝜌(𝑢1𝑢𝔦	𝜌(𝑢1𝑢𝔦	NUM
cana-3833	274	8	)	)	PUNCT
cana-3833	274	9	=	=	SYM
cana-3833	274	10	1	1	NUM
cana-3833	274	11	,	,	PUNCT
cana-3833	274	12	𝜌(𝑢𝔦𝑢𝑗	𝜌(𝑢𝔦𝑢𝑗	ADJ
cana-3833	274	13	)	)	PUNCT
cana-3833	274	14	=	=	VERB
cana-3833	275	1	2𝑛𝔦	2𝑛𝔦	NOUN
cana-3833	275	2	−	−	PROPN
cana-3833	275	3	2𝑛	2𝑛	PROPN
cana-3833	276	1	−	−	PROPN
cana-3833	276	2	(	(	PUNCT
cana-3833	276	3	𝔦2	𝔦2	NOUN
cana-3833	276	4	+	+	CCONJ
cana-3833	276	5	3𝔦	3𝔦	NUM
cana-3833	276	6	−	−	PROPN
cana-3833	276	7	7	7	NUM
cana-3833	276	8	)	)	PUNCT
cana-3833	276	9	,	,	PUNCT
cana-3833	276	10	𝔦	𝔦	X
cana-3833	276	11	≠	≠	PROPN
cana-3833	276	12	1	1	NUM
cana-3833	276	13	,	,	PUNCT
cana-3833	276	14	𝑛	𝑛	DET
cana-3833	276	15	−	−	PROPN
cana-3833	276	16	1	1	NUM
cana-3833	276	17	,	,	PUNCT
cana-3833	276	18	𝔦	𝔦	X
cana-3833	276	19	+	+	CCONJ
cana-3833	276	20	1	1	NUM
cana-3833	276	21	≤	≤	NUM
cana-3833	276	22	𝑗	𝑗	PRON
cana-3833	276	23	≤	≤	NUM
cana-3833	276	24	𝑛	𝑛	NOUN
cana-3833	276	25	,	,	PUNCT
cana-3833	276	26	𝜌(𝑢𝑛−1𝑢𝑛	𝜌(𝑢𝑛−1𝑢𝑛	NOUN
cana-3833	276	27	)	)	PUNCT
cana-3833	276	28	=	=	SYM
cana-3833	276	29	𝑛2	𝑛2	NOUN
cana-3833	276	30	−	−	PROPN
cana-3833	276	31	5𝑛	5𝑛	NUM
cana-3833	276	32	+	+	SYM
cana-3833	276	33	9	9	NUM
cana-3833	276	34	induced	induced	ADJ
cana-3833	276	35	weight	weight	NOUN
cana-3833	276	36	function	function	NOUN
cana-3833	276	37	for	for	ADP
cana-3833	276	38	edge	edge	NOUN
cana-3833	276	39	𝜓	𝜓	PROPN
cana-3833	276	40	:	:	PUNCT
cana-3833	276	41	𝐸(𝐾𝑛	𝐸(𝐾𝑛	NUM
cana-3833	276	42	)	)	PUNCT
cana-3833	276	43	→	→	SYM
cana-3833	276	44	{	{	PUNCT
cana-3833	276	45	3	3	NUM
cana-3833	276	46	,	,	PUNCT
cana-3833	276	47	5	5	NUM
cana-3833	276	48	,	,	PUNCT
cana-3833	276	49	7	7	NUM
cana-3833	276	50	,	,	PUNCT
cana-3833	276	51	…	…	PUNCT
cana-3833	276	52	,	,	PUNCT
cana-3833	277	1	1	1	NUM
cana-3833	277	2	−	−	PROPN
cana-3833	277	3	𝑛	𝑛	PRON
cana-3833	277	4	+	+	NUM
cana-3833	277	5	𝑛2	𝑛2	NOUN
cana-3833	277	6	}	}	PUNCT
cana-3833	277	7	is	be	AUX
cana-3833	277	8	given	give	VERB
cana-3833	277	9	by	by	ADP
cana-3833	277	10	𝜌(𝑢1𝑢𝔦	𝜌(𝑢1𝑢𝔦	NOUN
cana-3833	277	11	)	)	PUNCT
cana-3833	277	12	=	=	SYM
cana-3833	278	1	−1	−1	NOUN
cana-3833	278	2	+	+	CCONJ
cana-3833	278	3	2𝔦	2𝔦	NOUN
cana-3833	278	4	,	,	PUNCT
cana-3833	278	5	𝜌(𝑢𝑛−1𝑢𝑛	𝜌(𝑢𝑛−1𝑢𝑛	NOUN
cana-3833	278	6	)	)	PUNCT
cana-3833	278	7	=	=	SYM
cana-3833	279	1	1	1	NUM
cana-3833	279	2	−	−	NUM
cana-3833	279	3	𝑛	𝑛	PROPN
cana-3833	279	4	+	+	NUM
cana-3833	279	5	𝑛2	𝑛2	NOUN
cana-3833	279	6	𝜌(𝑢𝔦𝑢𝑗	𝜌(𝑢𝔦𝑢𝑗	ADJ
cana-3833	279	7	)	)	PUNCT
cana-3833	280	1	=	=	PUNCT
cana-3833	281	1	2𝑛𝔦	2𝑛𝔦	NOUN
cana-3833	281	2	−	−	PROPN
cana-3833	281	3	2𝑛	2𝑛	PROPN
cana-3833	282	1	−	−	PROPN
cana-3833	282	2	(	(	PUNCT
cana-3833	282	3	3	3	NUM
cana-3833	282	4	+	+	CCONJ
cana-3833	282	5	𝔦	𝔦	CCONJ
cana-3833	282	6	−	−	NOUN
cana-3833	282	7	𝔦2	𝔦2	NOUN
cana-3833	282	8	)	)	PUNCT
cana-3833	282	9	−	−	PROPN
cana-3833	282	10	2	2	NUM
cana-3833	282	11	−	−	PROPN
cana-3833	282	12	2𝔦	2𝔦	NOUN
cana-3833	282	13	+	+	CCONJ
cana-3833	282	14	2𝑗	2𝑗	NOUN
cana-3833	282	15	,	,	PUNCT
cana-3833	282	16	𝔦	𝔦	X
cana-3833	282	17	≠	≠	PROPN
cana-3833	282	18	1	1	NUM
cana-3833	282	19	,	,	PUNCT
cana-3833	282	20	𝑛	𝑛	PRON
cana-3833	282	21	−	−	PROPN
cana-3833	282	22	1	1	NUM
cana-3833	282	23	,	,	PUNCT
cana-3833	282	24	𝔦	𝔦	X
cana-3833	282	25	+	+	CCONJ
cana-3833	282	26	1	1	NUM
cana-3833	282	27	≤	≤	NUM
cana-3833	282	28	𝑗	𝑗	PRON
cana-3833	282	29	≤	≤	NUM
cana-3833	282	30	𝑛	𝑛	NOUN
cana-3833	282	31	,	,	PUNCT
cana-3833	282	32	the	the	DET
cana-3833	282	33	induced	induced	ADJ
cana-3833	282	34	edge	edge	NOUN
cana-3833	282	35	weights	weight	NOUN
cana-3833	282	36	of	of	ADP
cana-3833	282	37	𝐾𝑛	𝐾𝑛	PROPN
cana-3833	282	38	differs	differ	VERB
cana-3833	282	39	by	by	ADP
cana-3833	282	40	2	2	NUM
cana-3833	282	41	in	in	ADP
cana-3833	282	42	arithmetic	arithmetic	ADJ
cana-3833	282	43	progression	progression	NOUN
cana-3833	282	44	.	.	PUNCT
cana-3833	283	1	hence	hence	ADV
cana-3833	283	2	,	,	PUNCT
cana-3833	283	3	(	(	PUNCT
cana-3833	283	4	3	3	NUM
cana-3833	283	5	,	,	PUNCT
cana-3833	283	6	2	2	NUM
cana-3833	283	7	)	)	PUNCT
cana-3833	283	8	–	–	PUNCT
cana-3833	283	9	𝑡𝑒𝑠(𝐾𝑛	𝑡𝑒𝑠(𝐾𝑛	NUM
cana-3833	283	10	)	)	PUNCT
cana-3833	283	11	≤	≤	NOUN
cana-3833	283	12	9	9	NUM
cana-3833	283	13	−	−	NOUN
cana-3833	283	14	5𝑛+𝑛2	5𝑛+𝑛2	NOUN
cana-3833	283	15	.	.	PUNCT
cana-3833	284	1	further	far	ADV
cana-3833	284	2	it	it	PRON
cana-3833	284	3	is	be	AUX
cana-3833	284	4	not	not	PART
cana-3833	284	5	possible	possible	ADJ
cana-3833	284	6	to	to	PART
cana-3833	284	7	obtain	obtain	VERB
cana-3833	284	8	(	(	PUNCT
cana-3833	284	9	3	3	NUM
cana-3833	284	10	,	,	PUNCT
cana-3833	284	11	2	2	NUM
cana-3833	284	12	)	)	PUNCT
cana-3833	284	13	–	–	PUNCT
cana-3833	284	14	total	total	ADJ
cana-3833	284	15	edge	edge	VERB
cana-3833	284	16	irregular	irregular	ADJ
cana-3833	284	17	labelling	labelling	NOUN
cana-3833	284	18	by	by	ADP
cana-3833	284	19	assigning	assign	VERB
cana-3833	284	20	label	label	NOUN
cana-3833	284	21	fewer	few	ADJ
cana-3833	284	22	than	than	ADP
cana-3833	284	23	𝑛2	𝑛2	NOUN
cana-3833	284	24	−	−	PROPN
cana-3833	284	25	5𝑛	5𝑛	NUM
cana-3833	284	26	+	+	X
cana-3833	284	27	9	9	NUM
cana-3833	284	28	,	,	PUNCT
cana-3833	284	29	hence	hence	ADV
cana-3833	284	30	(	(	PUNCT
cana-3833	284	31	3	3	NUM
cana-3833	284	32	,	,	PUNCT
cana-3833	284	33	2	2	NUM
cana-3833	284	34	)	)	PUNCT
cana-3833	284	35	–	–	PUNCT
cana-3833	284	36	𝑡𝑒𝑠(𝐾𝑛	𝑡𝑒𝑠(𝐾𝑛	NUM
cana-3833	284	37	)	)	PUNCT
cana-3833	284	38	=	=	SYM
cana-3833	284	39	𝑛2	𝑛2	NOUN
cana-3833	284	40	−	−	PROPN
cana-3833	284	41	5𝑛	5𝑛	NUM
cana-3833	284	42	+	+	X
cana-3833	284	43	9	9	NUM
cana-3833	284	44	.	.	NOUN
cana-3833	284	45	example	example	NOUN
cana-3833	284	46	7	7	NUM
cana-3833	284	47	:	:	PUNCT
cana-3833	284	48	(	(	PUNCT
cana-3833	284	49	3	3	NUM
cana-3833	284	50	,	,	PUNCT
cana-3833	284	51	2	2	NUM
cana-3833	284	52	)	)	PUNCT
cana-3833	284	53	−	−	PROPN
cana-3833	284	54	𝑡𝑒𝑠(𝐾6	𝑡𝑒𝑠(𝐾6	NOUN
cana-3833	284	55	)	)	PUNCT
cana-3833	284	56	is	be	AUX
cana-3833	284	57	given	give	VERB
cana-3833	284	58	in	in	ADP
cana-3833	284	59	fig	fig	NOUN
cana-3833	284	60	8	8	NUM
cana-3833	284	61	.	.	PUNCT
cana-3833	285	1	figure	figure	NOUN
cana-3833	285	2	8	8	NUM
cana-3833	285	3	.	.	PUNCT
cana-3833	286	1	(	(	PUNCT
cana-3833	286	2	3	3	NUM
cana-3833	286	3	,	,	PUNCT
cana-3833	286	4	2	2	NUM
cana-3833	286	5	)	)	PUNCT
cana-3833	286	6	–	–	PUNCT
cana-3833	286	7	𝑡𝑒𝑠(𝐾6	𝑡𝑒𝑠(𝐾6	NOUN
cana-3833	286	8	)	)	PUNCT
cana-3833	286	9	=	=	SYM
cana-3833	287	1	15	15	X
cana-3833	287	2	.	.	PUNCT
cana-3833	287	3	remark	remark	NOUN
cana-3833	287	4	2	2	NUM
cana-3833	287	5	:	:	PUNCT
cana-3833	287	6	when	when	SCONJ
cana-3833	287	7	𝑛	𝑛	PROPN
cana-3833	287	8	=	=	SYM
cana-3833	287	9	3	3	NUM
cana-3833	287	10	,	,	PUNCT
cana-3833	287	11	(	(	PUNCT
cana-3833	287	12	3	3	NUM
cana-3833	287	13	,	,	PUNCT
cana-3833	287	14	2	2	NUM
cana-3833	287	15	)	)	PUNCT
cana-3833	287	16	–	–	PUNCT
cana-3833	287	17	𝑡𝑒𝑠(𝐾3	𝑡𝑒𝑠(𝐾3	NOUN
cana-3833	287	18	)	)	PUNCT
cana-3833	287	19	=	=	SYM
cana-3833	287	20	3	3	NUM
cana-3833	287	21	,	,	PUNCT
cana-3833	287	22	total	total	ADJ
cana-3833	287	23	edge	edge	VERB
cana-3833	287	24	irregular	irregular	ADJ
cana-3833	287	25	labelling	labelling	NOUN
cana-3833	287	26	for	for	ADP
cana-3833	287	27	𝐾3	𝐾3	NOUN
cana-3833	287	28	is	be	AUX
cana-3833	287	29	shown	show	VERB
cana-3833	287	30	in	in	ADP
cana-3833	287	31	fig	fig	NOUN
cana-3833	287	32	9	9	NUM
cana-3833	287	33	.	.	PUNCT
cana-3833	288	1	communications	communication	NOUN
cana-3833	288	2	on	on	ADP
cana-3833	288	3	applied	apply	VERB
cana-3833	288	4	nonlinear	nonlinear	ADJ
cana-3833	288	5	analysis	analysis	NOUN
cana-3833	288	6	issn	issn	NOUN
cana-3833	288	7	:	:	PUNCT
cana-3833	288	8	1074	1074	NUM
cana-3833	288	9	-	-	PUNCT
cana-3833	288	10	133x	133x	NUM
cana-3833	288	11	vol	vol	NOUN
cana-3833	288	12	32	32	NUM
cana-3833	288	13	no	no	NOUN
cana-3833	288	14	.	.	PUNCT
cana-3833	289	1	9s	9s	NUM
cana-3833	289	2	(	(	PUNCT
cana-3833	289	3	2025	2025	NUM
cana-3833	289	4	)	)	PUNCT
cana-3833	289	5	14	14	NUM
cana-3833	290	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	290	2	figure	figure	NOUN
cana-3833	290	3	9	9	NUM
cana-3833	290	4	.	.	PUNCT
cana-3833	291	1	(	(	PUNCT
cana-3833	291	2	3	3	NUM
cana-3833	291	3	,	,	PUNCT
cana-3833	291	4	2	2	NUM
cana-3833	291	5	)	)	PUNCT
cana-3833	291	6	–	–	PUNCT
cana-3833	291	7	𝑡𝑒𝑠(𝐾3	𝑡𝑒𝑠(𝐾3	NOUN
cana-3833	291	8	)	)	PUNCT
cana-3833	291	9	=	=	SYM
cana-3833	292	1	3	3	X
cana-3833	292	2	.	.	PUNCT
cana-3833	292	3	corollary	corollary	ADJ
cana-3833	292	4	1	1	NUM
cana-3833	292	5	.	.	PUNCT
cana-3833	293	1	if	if	SCONJ
cana-3833	293	2	a	a	DET
cana-3833	293	3	simple	simple	ADJ
cana-3833	293	4	connected	connected	ADJ
cana-3833	293	5	graph	graph	NOUN
cana-3833	293	6	𝐺	𝐺	PROPN
cana-3833	293	7	has	have	VERB
cana-3833	293	8	𝑝	𝑝	PROPN
cana-3833	293	9	vertices	vertex	NOUN
cana-3833	293	10	and	and	CCONJ
cana-3833	293	11	𝑞	𝑞	NOUN
cana-3833	293	12	edges	edge	NOUN
cana-3833	293	13	,	,	PUNCT
cana-3833	293	14	then	then	ADV
cana-3833	293	15	⌈	⌈	NOUN
cana-3833	293	16	2𝑞+1	2𝑞+1	NUM
cana-3833	293	17	3	3	NUM
cana-3833	293	18	⌉	⌉	X
cana-3833	293	19	≤	≤	X
cana-3833	293	20	(	(	PUNCT
cana-3833	293	21	3	3	NUM
cana-3833	293	22	,	,	PUNCT
cana-3833	293	23	2	2	NUM
cana-3833	293	24	)	)	PUNCT
cana-3833	293	25	−	−	PRON
cana-3833	293	26	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	293	27	)	)	PUNCT
cana-3833	293	28	≤	≤	NOUN
cana-3833	293	29	9	9	NUM
cana-3833	293	30	−	−	PROPN
cana-3833	293	31	5𝑝	5𝑝	PROPN
cana-3833	293	32	+	+	CCONJ
cana-3833	293	33	𝑝2	𝑝2	NOUN
cana-3833	293	34	.	.	PUNCT
cana-3833	294	1	theorem	theorem	VERB
cana-3833	294	2	8	8	NUM
cana-3833	294	3	:	:	PUNCT
cana-3833	294	4	(	(	PUNCT
cana-3833	294	5	3	3	NUM
cana-3833	294	6	,	,	PUNCT
cana-3833	294	7	2	2	NUM
cana-3833	294	8	)	)	PUNCT
cana-3833	294	9	–	–	PUNCT
cana-3833	294	10	𝑡𝑒𝑠(𝜇(𝑃𝑛	𝑡𝑒𝑠(𝜇(𝑃𝑛	NOUN
cana-3833	294	11	)	)	PUNCT
cana-3833	294	12	)	)	PUNCT
cana-3833	295	1	=	=	PUNCT
cana-3833	295	2	⌊	⌊	VERB
cana-3833	295	3	8𝑛	8𝑛	NOUN
cana-3833	295	4	3	3	NUM
cana-3833	295	5	⌋	⌋	NOUN
cana-3833	295	6	−	−	PROPN
cana-3833	295	7	1	1	NUM
cana-3833	295	8	for	for	ADP
cana-3833	295	9	𝑛	𝑛	PRON
cana-3833	295	10	≥	≥	NUM
cana-3833	295	11	3	3	NUM
cana-3833	295	12	.	.	PUNCT
cana-3833	296	1	proof	proof	NOUN
cana-3833	296	2	.	.	PUNCT
cana-3833	297	1	consider	consider	VERB
cana-3833	297	2	the	the	DET
cana-3833	297	3	index	index	NOUN
cana-3833	297	4	set	set	VERB
cana-3833	297	5	𝕀	𝕀	PROPN
cana-3833	297	6	=	=	SYM
cana-3833	297	7	{	{	PUNCT
cana-3833	297	8	1	1	NUM
cana-3833	297	9	,	,	PUNCT
cana-3833	297	10	2	2	NUM
cana-3833	297	11	,	,	PUNCT
cana-3833	297	12	…	…	PUNCT
cana-3833	297	13	,	,	PUNCT
cana-3833	297	14	𝑛	𝑛	PROPN
cana-3833	297	15	}	}	PUNCT
cana-3833	297	16	.	.	PUNCT
cana-3833	298	1	the	the	DET
cana-3833	298	2	vertex	vertex	NOUN
cana-3833	298	3	set	set	NOUN
cana-3833	298	4	and	and	CCONJ
cana-3833	298	5	edge	edge	NOUN
cana-3833	298	6	set	set	NOUN
cana-3833	298	7	of	of	ADP
cana-3833	298	8	𝑃𝑛	𝑃𝑛	PROPN
cana-3833	298	9	are	be	AUX
cana-3833	298	10	𝒱1and	𝒱1and	PROPN
cana-3833	298	11	ℰ1	ℰ1	NOUN
cana-3833	298	12	respectively	respectively	ADV
cana-3833	298	13	.	.	PUNCT
cana-3833	299	1	let	let	VERB
cana-3833	299	2	𝑉(𝜇(𝑃𝑛	𝑉(𝜇(𝑃𝑛	PROPN
cana-3833	299	3	)	)	PUNCT
cana-3833	299	4	)	)	PUNCT
cana-3833	300	1	=	=	PUNCT
cana-3833	301	1	⋃	⋃	ADP
cana-3833	301	2	𝒱𝜑	𝒱𝜑	PROPN
cana-3833	301	3	3	3	NUM
cana-3833	301	4	𝜑=1	𝜑=1	NOUN
cana-3833	301	5	and	and	CCONJ
cana-3833	301	6	𝐸(𝜇(𝑃𝑛	𝐸(𝜇(𝑃𝑛	ADJ
cana-3833	301	7	)	)	PUNCT
cana-3833	301	8	)	)	PUNCT
cana-3833	302	1	=	=	PUNCT
cana-3833	302	2	⋃	⋃	VERB
cana-3833	302	3	ℰℴ	ℰℴ	PROPN
cana-3833	302	4	4	4	NUM
cana-3833	302	5	ℴ=1	ℴ=1	PUNCT
cana-3833	302	6	denotes	denote	VERB
cana-3833	302	7	the	the	DET
cana-3833	302	8	vertex	vertex	NOUN
cana-3833	302	9	set	set	NOUN
cana-3833	302	10	and	and	CCONJ
cana-3833	302	11	edge	edge	NOUN
cana-3833	302	12	set	set	NOUN
cana-3833	302	13	of	of	ADP
cana-3833	302	14	𝜇(𝑃𝑛	𝜇(𝑃𝑛	NOUN
cana-3833	302	15	)	)	PUNCT
cana-3833	302	16	where	where	SCONJ
cana-3833	302	17	𝒱1	𝒱1	NOUN
cana-3833	302	18	=	=	SYM
cana-3833	302	19	{	{	PUNCT
cana-3833	302	20	𝔵𝑖	𝔵𝑖	X
cana-3833	302	21	}	}	PUNCT
cana-3833	302	22	,	,	PUNCT
cana-3833	302	23	𝒱2	𝒱2	NOUN
cana-3833	302	24	=	=	SYM
cana-3833	302	25	{	{	PUNCT
cana-3833	302	26	𝔵𝑖′	𝔵𝑖′	PROPN
cana-3833	302	27	}	}	PUNCT
cana-3833	302	28	,	,	PUNCT
cana-3833	302	29	𝒱3	𝒱3	NOUN
cana-3833	302	30	=	=	SYM
cana-3833	302	31	{	{	PUNCT
cana-3833	302	32	𝑤	𝑤	PART
cana-3833	302	33	}	}	PUNCT
cana-3833	302	34	,	,	PUNCT
cana-3833	302	35	ℰ1	ℰ1	NOUN
cana-3833	302	36	=	=	SYM
cana-3833	302	37	{	{	PUNCT
cana-3833	302	38	𝔵𝔦𝔵𝔦+1	𝔵𝔦𝔵𝔦+1	PROPN
cana-3833	302	39	}	}	PUNCT
cana-3833	302	40	,	,	PUNCT
cana-3833	302	41	ℰ2	ℰ2	PROPN
cana-3833	302	42	=	=	SYM
cana-3833	302	43	{	{	PUNCT
cana-3833	302	44	𝔵𝔦′𝔵𝔦+1	𝔵𝔦′𝔵𝔦+1	X
cana-3833	302	45	}	}	PUNCT
cana-3833	302	46	,	,	PUNCT
cana-3833	302	47	ℰ3	ℰ3	NOUN
cana-3833	302	48	=	=	SYM
cana-3833	302	49	{	{	PUNCT
cana-3833	302	50	𝔵𝔦𝔵𝔦+1′	𝔵𝔦𝔵𝔦+1′	NOUN
cana-3833	302	51	}	}	PUNCT
cana-3833	302	52	and	and	CCONJ
cana-3833	302	53	ℰ4	ℰ4	VERB
cana-3833	302	54	=	=	SYM
cana-3833	302	55	{	{	PUNCT
cana-3833	302	56	𝔵𝔦′𝑤	𝔵𝔦′𝑤	NOUN
cana-3833	302	57	}	}	PUNCT
cana-3833	302	58	for	for	ADP
cana-3833	302	59	all	all	PRON
cana-3833	302	60	𝑖	𝑖	X
cana-3833	302	61	𝜖	𝜖	PROPN
cana-3833	302	62	𝕀.	𝕀.	ADJ
cana-3833	302	63	total	total	NOUN
cana-3833	302	64	labeling	labeling	NOUN
cana-3833	302	65	𝓅	𝓅	NOUN
cana-3833	302	66	:	:	PUNCT
cana-3833	302	67	𝒳	𝒳	PROPN
cana-3833	302	68	→	→	SYM
cana-3833	302	69	𝒴	𝒴	PROPN
cana-3833	302	70	where	where	SCONJ
cana-3833	302	71	𝒳	𝒳	PROPN
cana-3833	302	72	=	=	SYM
cana-3833	302	73	𝑉(𝜇(𝑃𝑛	𝑉(𝜇(𝑃𝑛	PROPN
cana-3833	302	74	)	)	PUNCT
cana-3833	302	75	)	)	PUNCT
cana-3833	302	76	∪	∪	ADP
cana-3833	302	77	𝐸(𝜇(𝑃𝑛	𝐸(𝜇(𝑃𝑛	NOUN
cana-3833	302	78	)	)	PUNCT
cana-3833	302	79	)	)	PUNCT
cana-3833	302	80	and	and	CCONJ
cana-3833	302	81	𝒴	𝒴	PROPN
cana-3833	302	82	=	=	SYM
cana-3833	302	83	{	{	PUNCT
cana-3833	302	84	1	1	NUM
cana-3833	302	85	,	,	PUNCT
cana-3833	302	86	2	2	NUM
cana-3833	302	87	,	,	PUNCT
cana-3833	302	88	…	…	PUNCT
cana-3833	302	89	,	,	PUNCT
cana-3833	302	90	⌊	⌊	VERB
cana-3833	302	91	8𝑛	8𝑛	NUM
cana-3833	302	92	3	3	NUM
cana-3833	302	93	⌋	⌋	NOUN
cana-3833	302	94	−	−	NOUN
cana-3833	302	95	1	1	NUM
cana-3833	302	96	}	}	PUNCT
cana-3833	302	97	is	be	AUX
cana-3833	302	98	given	give	VERB
cana-3833	302	99	below	below	ADP
cana-3833	302	100	:	:	PUNCT
cana-3833	302	101	case	case	NOUN
cana-3833	302	102	i	i	PRON
cana-3833	302	103	suppose	suppose	VERB
cana-3833	302	104	𝑛	𝑛	PRON
cana-3833	302	105	≡	≡	PROPN
cana-3833	302	106	0	0	PUNCT
cana-3833	303	1	(	(	PUNCT
cana-3833	303	2	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	303	3	3	3	NUM
cana-3833	303	4	)	)	PUNCT
cana-3833	303	5	𝓅	𝓅	PROPN
cana-3833	303	6	(	(	PUNCT
cana-3833	303	7	𝔵1	𝔵1	NOUN
cana-3833	303	8	)	)	PUNCT
cana-3833	303	9	=	=	SYM
cana-3833	303	10	8𝑛	8𝑛	NOUN
cana-3833	303	11	3	3	NUM
cana-3833	303	12	−	−	NOUN
cana-3833	303	13	1	1	NUM
cana-3833	303	14	,	,	PUNCT
cana-3833	303	15	𝓅	𝓅	PROPN
cana-3833	303	16	(	(	PUNCT
cana-3833	303	17	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	303	18	)	)	PUNCT
cana-3833	303	19	=	=	SYM
cana-3833	304	1	1	1	NUM
cana-3833	304	2	+	+	NUM
cana-3833	304	3	2𝔦	2𝔦	NUM
cana-3833	304	4	,	,	PUNCT
cana-3833	304	5	𝓅	𝓅	PROPN
cana-3833	304	6	(	(	PUNCT
cana-3833	304	7	𝔵3𝔦−1	𝔵3𝔦−1	PROPN
cana-3833	304	8	)	)	PUNCT
cana-3833	304	9	=	=	SYM
cana-3833	304	10	2𝔦	2𝔦	NUM
cana-3833	305	1	−	−	NOUN
cana-3833	305	2	1	1	NUM
cana-3833	305	3	,	,	PUNCT
cana-3833	305	4	𝓅	𝓅	PROPN
cana-3833	305	5	(	(	PUNCT
cana-3833	305	6	𝔵3𝔦−2	𝔵3𝔦−2	NOUN
cana-3833	305	7	)	)	PUNCT
cana-3833	305	8	=	=	SYM
cana-3833	305	9	2𝔦	2𝔦	NUM
cana-3833	306	1	−	−	NOUN
cana-3833	306	2	1	1	NUM
cana-3833	306	3	,	,	PUNCT
cana-3833	306	4	𝓅(𝔵′3𝔦	𝓅(𝔵′3𝔦	NOUN
cana-3833	306	5	)	)	PUNCT
cana-3833	306	6	=	=	PUNCT
cana-3833	306	7	2𝑛	2𝑛	PROPN
cana-3833	307	1	+	+	CCONJ
cana-3833	307	2	2𝑖	2𝑖	NUM
cana-3833	307	3	−	−	NOUN
cana-3833	307	4	1	1	NUM
cana-3833	307	5	,	,	PUNCT
cana-3833	307	6	𝓅	𝓅	PROPN
cana-3833	307	7	(	(	PUNCT
cana-3833	307	8	𝔵′3𝔦−1	𝔵′3𝔦−1	NUM
cana-3833	307	9	)	)	PUNCT
cana-3833	307	10	=	=	SYM
cana-3833	308	1	2𝑛	2𝑛	PROPN
cana-3833	309	1	+	+	CCONJ
cana-3833	309	2	2𝔦	2𝔦	NUM
cana-3833	309	3	−	−	PROPN
cana-3833	309	4	3	3	NUM
cana-3833	309	5	,	,	PUNCT
cana-3833	309	6	𝓅	𝓅	PROPN
cana-3833	309	7	(	(	PUNCT
cana-3833	309	8	𝔵′3𝔦−2	𝔵′3𝔦−2	NUM
cana-3833	309	9	)	)	PUNCT
cana-3833	310	1	=	=	SYM
cana-3833	311	1	2𝑛	2𝑛	PROPN
cana-3833	312	1	+	+	CCONJ
cana-3833	312	2	2𝔦	2𝔦	NUM
cana-3833	312	3	−	−	PROPN
cana-3833	312	4	3	3	NUM
cana-3833	312	5	,	,	PUNCT
cana-3833	312	6	𝓅	𝓅	PROPN
cana-3833	312	7	(	(	PUNCT
cana-3833	312	8	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	312	9	𝔵3𝔦	𝔵3𝔦	NOUN
cana-3833	312	10	)	)	PUNCT
cana-3833	312	11	=	=	SYM
cana-3833	312	12	𝓅	𝓅	PROPN
cana-3833	312	13	(	(	PUNCT
cana-3833	312	14	𝔵3𝔦−2	𝔵3𝔦−2	PROPN
cana-3833	312	15	𝔵3𝔦−1	𝔵3𝔦−1	PROPN
cana-3833	312	16	)	)	PUNCT
cana-3833	312	17	=	=	SYM
cana-3833	312	18	2𝔦	2𝔦	NUM
cana-3833	313	1	−	−	NOUN
cana-3833	313	2	1	1	NUM
cana-3833	313	3	,	,	PUNCT
cana-3833	313	4	𝓅	𝓅	PROPN
cana-3833	313	5	(	(	PUNCT
cana-3833	313	6	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	313	7	𝔵3𝔦+1	𝔵3𝔦+1	PROPN
cana-3833	313	8	)	)	PUNCT
cana-3833	313	9	=	=	SYM
cana-3833	313	10	2𝔦	2𝔦	NUM
cana-3833	314	1	−	−	PROPN
cana-3833	314	2	1	1	NUM
cana-3833	314	3	,	,	PUNCT
cana-3833	314	4	𝓅	𝓅	PROPN
cana-3833	314	5	(	(	PUNCT
cana-3833	314	6	𝔵3𝔦−2	𝔵3𝔦−2	NOUN
cana-3833	314	7	𝔵′3𝔦−1	𝔵′3𝔦−1	NOUN
cana-3833	314	8	)	)	PUNCT
cana-3833	315	1	=	=	PRON
cana-3833	315	2	−7	−7	NOUN
cana-3833	315	3	+	+	NOUN
cana-3833	315	4	8𝔦	8𝔦	NOUN
cana-3833	315	5	,	,	PUNCT
cana-3833	315	6	𝓅	𝓅	PROPN
cana-3833	315	7	(	(	PUNCT
cana-3833	315	8	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	315	9	𝔵′3𝔦−2	𝔵′3𝔦−2	NUM
cana-3833	315	10	)	)	PUNCT
cana-3833	316	1	=	=	SYM
cana-3833	316	2	𝓅	𝓅	PROPN
cana-3833	316	3	(	(	PUNCT
cana-3833	316	4	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	316	5	𝔵′3𝔦	𝔵′3𝔦	NUM
cana-3833	316	6	)	)	PUNCT
cana-3833	316	7	=	=	NOUN
cana-3833	316	8	8𝔦	8𝔦	NOUN
cana-3833	316	9	−	−	PROPN
cana-3833	316	10	5	5	NUM
cana-3833	316	11	,	,	PUNCT
cana-3833	316	12	𝓅	𝓅	PROPN
cana-3833	316	13	(	(	PUNCT
cana-3833	316	14	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	316	15	𝔵′3𝔦+1	𝔵′3𝔦+1	NUM
cana-3833	316	16	)	)	PUNCT
cana-3833	316	17	=	=	PUNCT
cana-3833	316	18	−3	−3	PROPN
cana-3833	317	1	+	+	NUM
cana-3833	317	2	8𝔦	8𝔦	NOUN
cana-3833	317	3	,	,	PUNCT
cana-3833	317	4	𝓅	𝓅	PROPN
cana-3833	317	5	(	(	PUNCT
cana-3833	317	6	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	317	7	𝔵′3𝔦−1	𝔵′3𝔦−1	NUM
cana-3833	317	8	)	)	PUNCT
cana-3833	318	1	=	=	NOUN
cana-3833	318	2	8𝔦	8𝔦	NOUN
cana-3833	318	3	−	−	NOUN
cana-3833	318	4	3	3	NUM
cana-3833	318	5	,	,	PUNCT
cana-3833	318	6	communications	communication	NOUN
cana-3833	318	7	on	on	ADP
cana-3833	318	8	applied	apply	VERB
cana-3833	318	9	nonlinear	nonlinear	ADJ
cana-3833	318	10	analysis	analysis	NOUN
cana-3833	318	11	issn	issn	NOUN
cana-3833	318	12	:	:	PUNCT
cana-3833	318	13	1074	1074	NUM
cana-3833	318	14	-	-	PUNCT
cana-3833	318	15	133x	133x	NUM
cana-3833	318	16	vol	vol	NOUN
cana-3833	318	17	32	32	NUM
cana-3833	319	1	no	no	NOUN
cana-3833	319	2	.	.	PUNCT
cana-3833	320	1	9s	9s	NUM
cana-3833	320	2	(	(	PUNCT
cana-3833	320	3	2025	2025	NUM
cana-3833	320	4	)	)	PUNCT
cana-3833	320	5	15	15	NUM
cana-3833	321	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	321	2	𝓅	𝓅	PROPN
cana-3833	321	3	(	(	PUNCT
cana-3833	321	4	𝔵3𝔦+1	𝔵3𝔦+1	PROPN
cana-3833	321	5	𝔵′3𝔦	𝔵′3𝔦	NOUN
cana-3833	321	6	)	)	PUNCT
cana-3833	321	7	=	=	SYM
cana-3833	321	8	−1	−1	NOUN
cana-3833	321	9	+	+	NUM
cana-3833	321	10	8𝔦	8𝔦	NOUN
cana-3833	321	11	,	,	PUNCT
cana-3833	321	12	𝓅	𝓅	NOUN
cana-3833	321	13	(	(	PUNCT
cana-3833	321	14	𝔵′3𝔦−1𝑤	𝔵′3𝔦−1𝑤	NOUN
cana-3833	321	15	)	)	PUNCT
cana-3833	322	1	=	=	SYM
cana-3833	322	2	𝓅	𝓅	PROPN
cana-3833	322	3	(	(	PUNCT
cana-3833	322	4	𝔵′3𝔦𝑤	𝔵′3𝔦𝑤	PROPN
cana-3833	322	5	)	)	PUNCT
cana-3833	322	6	=	=	PUNCT
cana-3833	322	7	4𝑛	4𝑛	NOUN
cana-3833	322	8	3	3	NUM
cana-3833	323	1	+	+	CCONJ
cana-3833	323	2	1	1	NUM
cana-3833	323	3	+	+	NUM
cana-3833	323	4	4(𝔦	4(𝔦	NUM
cana-3833	323	5	−	−	NUM
cana-3833	323	6	1	1	NUM
cana-3833	323	7	)	)	PUNCT
cana-3833	323	8	.	.	PUNCT
cana-3833	324	1	𝓅	𝓅	NOUN
cana-3833	324	2	(	(	PUNCT
cana-3833	324	3	𝔵′3𝔦−2𝑤	𝔵′3𝔦−2𝑤	NUM
cana-3833	324	4	)	)	PUNCT
cana-3833	324	5	=	=	PUNCT
cana-3833	325	1	4𝑛	4𝑛	NOUN
cana-3833	325	2	3	3	NUM
cana-3833	325	3	−	−	NOUN
cana-3833	325	4	1	1	NUM
cana-3833	326	1	+	+	NUM
cana-3833	326	2	4(𝔦	4(𝔦	NUM
cana-3833	326	3	−	−	NUM
cana-3833	326	4	1	1	NUM
cana-3833	326	5	)	)	PUNCT
cana-3833	326	6	,	,	PUNCT
cana-3833	326	7	case	case	NOUN
cana-3833	326	8	ii	ii	NOUN
cana-3833	326	9	consider	consider	VERB
cana-3833	326	10	𝑛	𝑛	PRON
cana-3833	326	11	≡	≡	PROPN
cana-3833	326	12	1	1	NUM
cana-3833	326	13	(	(	PUNCT
cana-3833	326	14	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	326	15	3	3	NUM
cana-3833	326	16	)	)	PUNCT
cana-3833	326	17	𝓅	𝓅	PROPN
cana-3833	326	18	(	(	PUNCT
cana-3833	326	19	𝔵1	𝔵1	NOUN
cana-3833	326	20	)	)	PUNCT
cana-3833	326	21	=	=	SYM
cana-3833	326	22	8	8	NUM
cana-3833	326	23	3	3	NUM
cana-3833	326	24	(	(	PUNCT
cana-3833	326	25	𝑛	𝑛	PROPN
cana-3833	326	26	−	−	PROPN
cana-3833	326	27	1	1	NUM
cana-3833	326	28	)	)	PUNCT
cana-3833	326	29	+	+	NUM
cana-3833	326	30	1	1	NUM
cana-3833	326	31	,	,	PUNCT
cana-3833	326	32	𝓅(𝔵3𝔦	𝓅(𝔵3𝔦	ADV
cana-3833	326	33	)	)	PUNCT
cana-3833	326	34	=	=	SYM
cana-3833	326	35	2𝔦	2𝔦	NUM
cana-3833	327	1	+	+	CCONJ
cana-3833	327	2	1	1	NUM
cana-3833	327	3	,	,	PUNCT
cana-3833	327	4	𝓅	𝓅	PROPN
cana-3833	327	5	(	(	PUNCT
cana-3833	327	6	𝔵3𝔦−1	𝔵3𝔦−1	NOUN
cana-3833	327	7	)	)	PUNCT
cana-3833	327	8	=	=	SYM
cana-3833	328	1	−1	−1	NOUN
cana-3833	329	1	+	+	CCONJ
cana-3833	329	2	2𝔦	2𝔦	NUM
cana-3833	329	3	,	,	PUNCT
cana-3833	329	4	𝓅	𝓅	PROPN
cana-3833	329	5	(	(	PUNCT
cana-3833	329	6	𝔵3𝔦−2	𝔵3𝔦−2	NOUN
cana-3833	329	7	)	)	PUNCT
cana-3833	329	8	=	=	SYM
cana-3833	329	9	2𝔦	2𝔦	NUM
cana-3833	330	1	−	−	NOUN
cana-3833	330	2	1	1	NUM
cana-3833	330	3	,	,	PUNCT
cana-3833	330	4	𝓅	𝓅	PROPN
cana-3833	330	5	(	(	PUNCT
cana-3833	330	6	𝔵′	𝔵′	NOUN
cana-3833	330	7	3𝔦	3𝔦	NUM
cana-3833	330	8	)	)	PUNCT
cana-3833	330	9	=	=	SYM
cana-3833	330	10	2(𝑛	2(𝑛	NUM
cana-3833	330	11	+	+	CCONJ
cana-3833	330	12	𝔦	𝔦	X
cana-3833	330	13	)	)	PUNCT
cana-3833	330	14	−	−	PROPN
cana-3833	330	15	1	1	NUM
cana-3833	330	16	,	,	PUNCT
cana-3833	330	17	𝓅	𝓅	PROPN
cana-3833	330	18	(	(	PUNCT
cana-3833	330	19	𝔵′3𝔦−1	𝔵′3𝔦−1	NUM
cana-3833	330	20	)	)	PUNCT
cana-3833	330	21	=	=	SYM
cana-3833	330	22	2(𝑛	2(𝑛	NUM
cana-3833	330	23	+	+	CCONJ
cana-3833	330	24	𝔦	𝔦	X
cana-3833	330	25	)	)	PUNCT
cana-3833	330	26	−	−	ADP
cana-3833	330	27	3	3	NUM
cana-3833	330	28	,	,	PUNCT
cana-3833	330	29	𝓅	𝓅	PROPN
cana-3833	330	30	(	(	PUNCT
cana-3833	330	31	𝔵′3𝔦−2	𝔵′3𝔦−2	NUM
cana-3833	330	32	)	)	PUNCT
cana-3833	331	1	=	=	SYM
cana-3833	332	1	2𝑛	2𝑛	PROPN
cana-3833	333	1	+	+	CCONJ
cana-3833	333	2	2𝔦	2𝔦	NUM
cana-3833	333	3	−	−	PROPN
cana-3833	333	4	3	3	NUM
cana-3833	333	5	,	,	PUNCT
cana-3833	333	6	𝓅	𝓅	PROPN
cana-3833	333	7	(	(	PUNCT
cana-3833	333	8	𝔵3𝔦−2	𝔵3𝔦−2	PROPN
cana-3833	333	9	𝔵3𝔦−1	𝔵3𝔦−1	PROPN
cana-3833	333	10	)	)	PUNCT
cana-3833	333	11	=	=	SYM
cana-3833	333	12	𝓅	𝓅	PROPN
cana-3833	333	13	(	(	PUNCT
cana-3833	333	14	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	333	15	𝔵3𝔦	𝔵3𝔦	NOUN
cana-3833	333	16	)	)	PUNCT
cana-3833	333	17	=	=	SYM
cana-3833	333	18	𝓅	𝓅	PROPN
cana-3833	333	19	(	(	PUNCT
cana-3833	333	20	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	333	21	𝔵3𝔦+1	𝔵3𝔦+1	PROPN
cana-3833	333	22	)	)	PUNCT
cana-3833	333	23	=	=	SYM
cana-3833	333	24	2𝔦	2𝔦	NUM
cana-3833	333	25	−	−	PROPN
cana-3833	333	26	1	1	NUM
cana-3833	333	27	,	,	PUNCT
cana-3833	333	28	𝓅	𝓅	PROPN
cana-3833	333	29	(	(	PUNCT
cana-3833	333	30	𝔵3𝔦−2	𝔵3𝔦−2	NOUN
cana-3833	333	31	𝔵′3𝔦−1	𝔵′3𝔦−1	NOUN
cana-3833	333	32	)	)	PUNCT
cana-3833	333	33	=	=	NOUN
cana-3833	333	34	8𝔦	8𝔦	NOUN
cana-3833	333	35	−	−	PROPN
cana-3833	333	36	7	7	NUM
cana-3833	333	37	,	,	PUNCT
cana-3833	333	38	𝓅	𝓅	PROPN
cana-3833	333	39	(	(	PUNCT
cana-3833	333	40	𝔵3𝑖−1	𝔵3𝑖−1	PROPN
cana-3833	333	41	𝔵′3𝑖	𝔵′3𝑖	PROPN
cana-3833	333	42	)	)	PUNCT
cana-3833	333	43	=	=	SYM
cana-3833	333	44	8𝑖	8𝑖	ADJ
cana-3833	333	45	−	−	NOUN
cana-3833	333	46	5	5	NUM
cana-3833	333	47	,	,	PUNCT
cana-3833	333	48	𝓅	𝓅	PROPN
cana-3833	333	49	(	(	PUNCT
cana-3833	333	50	𝔵3𝑖	𝔵3𝑖	NOUN
cana-3833	333	51	𝔵′3𝑖+1	𝔵′3𝑖+1	PART
cana-3833	333	52	)	)	PUNCT
cana-3833	333	53	=	=	SYM
cana-3833	333	54	8𝑖	8𝑖	ADJ
cana-3833	333	55	−	−	NOUN
cana-3833	333	56	3	3	NUM
cana-3833	333	57	,	,	PUNCT
cana-3833	333	58	𝓅(𝔵3𝑖−1	𝓅(𝔵3𝑖−1	ADJ
cana-3833	333	59	𝔵′3𝑖−2	𝔵′3𝑖−2	NOUN
cana-3833	333	60	)	)	PUNCT
cana-3833	333	61	=	=	SYM
cana-3833	333	62	8𝑖	8𝑖	ADJ
cana-3833	333	63	−	−	NOUN
cana-3833	333	64	5	5	NUM
cana-3833	333	65	,	,	PUNCT
cana-3833	333	66	𝓅	𝓅	PROPN
cana-3833	333	67	(	(	PUNCT
cana-3833	333	68	𝔵3𝑖	𝔵3𝑖	PROPN
cana-3833	333	69	𝔵′3𝑖−1	𝔵′3𝑖−1	PRON
cana-3833	333	70	)	)	PUNCT
cana-3833	333	71	=	=	SYM
cana-3833	333	72	8𝑖	8𝑖	ADJ
cana-3833	333	73	−	−	NOUN
cana-3833	333	74	3	3	NUM
cana-3833	333	75	,	,	PUNCT
cana-3833	333	76	𝓅	𝓅	PROPN
cana-3833	333	77	(	(	PUNCT
cana-3833	333	78	𝔵3𝑖+1	𝔵3𝑖+1	ADV
cana-3833	333	79	𝔵′3𝑖	𝔵′3𝑖	NUM
cana-3833	333	80	)	)	PUNCT
cana-3833	333	81	=	=	SYM
cana-3833	333	82	8𝑖	8𝑖	ADJ
cana-3833	333	83	−	−	NOUN
cana-3833	333	84	1	1	NUM
cana-3833	333	85	,	,	PUNCT
cana-3833	333	86	𝓅	𝓅	PROPN
cana-3833	333	87	(	(	PUNCT
cana-3833	333	88	𝔵′3𝔦−1𝑤	𝔵′3𝔦−1𝑤	NOUN
cana-3833	333	89	)	)	PUNCT
cana-3833	333	90	=	=	SYM
cana-3833	333	91	𝓅	𝓅	PROPN
cana-3833	333	92	(	(	PUNCT
cana-3833	333	93	𝔵′3𝔦𝑤	𝔵′3𝔦𝑤	PROPN
cana-3833	333	94	)	)	PUNCT
cana-3833	333	95	=	=	SYM
cana-3833	333	96	4(𝑛	4(𝑛	NUM
cana-3833	333	97	−	−	NOUN
cana-3833	333	98	1	1	NUM
cana-3833	333	99	)	)	PUNCT
cana-3833	333	100	3	3	NUM
cana-3833	333	101	+	+	CCONJ
cana-3833	333	102	4𝔦	4𝔦	NOUN
cana-3833	333	103	−	−	NOUN
cana-3833	333	104	1	1	NUM
cana-3833	333	105	.	.	X
cana-3833	334	1	𝓅	𝓅	NOUN
cana-3833	334	2	(	(	PUNCT
cana-3833	334	3	𝔵′3𝔦−2𝑤	𝔵′3𝔦−2𝑤	NUM
cana-3833	334	4	)	)	PUNCT
cana-3833	334	5	=	=	PUNCT
cana-3833	335	1	4𝑛+12𝔦−7	4𝑛+12𝔦−7	NUM
cana-3833	335	2	3	3	NUM
cana-3833	335	3	,	,	PUNCT
cana-3833	335	4	case	case	NOUN
cana-3833	335	5	iii	iii	NOUN
cana-3833	335	6	suppose	suppose	VERB
cana-3833	335	7	𝑛	𝑛	PRON
cana-3833	335	8	≡	≡	PROPN
cana-3833	335	9	2	2	NUM
cana-3833	335	10	(	(	PUNCT
cana-3833	335	11	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3833	335	12	3	3	NUM
cana-3833	335	13	)	)	PUNCT
cana-3833	335	14	𝓅	𝓅	PROPN
cana-3833	335	15	(	(	PUNCT
cana-3833	335	16	𝔵1	𝔵1	NOUN
cana-3833	335	17	)	)	PUNCT
cana-3833	335	18	=	=	SYM
cana-3833	336	1	8𝑛−4	8𝑛−4	NUM
cana-3833	336	2	3	3	NUM
cana-3833	336	3	,	,	PUNCT
cana-3833	336	4	𝓅	𝓅	PROPN
cana-3833	336	5	(	(	PUNCT
cana-3833	336	6	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	336	7	)	)	PUNCT
cana-3833	337	1	=	=	SYM
cana-3833	337	2	2𝔦	2𝔦	NUM
cana-3833	338	1	+	+	CCONJ
cana-3833	338	2	1	1	NUM
cana-3833	338	3	,	,	PUNCT
cana-3833	338	4	𝓅	𝓅	PROPN
cana-3833	338	5	(	(	PUNCT
cana-3833	338	6	𝔵3𝔦−1	𝔵3𝔦−1	PROPN
cana-3833	338	7	)	)	PUNCT
cana-3833	338	8	=	=	PUNCT
cana-3833	339	1	𝜌	𝜌	X
cana-3833	339	2	(	(	PUNCT
cana-3833	339	3	𝔵3𝔦−2	𝔵3𝔦−2	NOUN
cana-3833	339	4	)	)	PUNCT
cana-3833	339	5	=	=	SYM
cana-3833	339	6	2𝔦	2𝔦	NUM
cana-3833	340	1	−	−	NOUN
cana-3833	340	2	1	1	NUM
cana-3833	340	3	,	,	PUNCT
cana-3833	340	4	𝓅	𝓅	PROPN
cana-3833	340	5	(	(	PUNCT
cana-3833	340	6	𝔵′3𝔦	𝔵′3𝔦	NOUN
cana-3833	340	7	)	)	PUNCT
cana-3833	340	8	=	=	SYM
cana-3833	340	9	2𝔦	2𝔦	NUM
cana-3833	341	1	−	−	NOUN
cana-3833	341	2	1	1	NUM
cana-3833	341	3	+	+	CCONJ
cana-3833	341	4	2𝑛	2𝑛	NUM
cana-3833	341	5	,	,	PUNCT
cana-3833	341	6	𝓅	𝓅	X
cana-3833	341	7	(	(	PUNCT
cana-3833	341	8	𝔵′3𝔦−1	𝔵′3𝔦−1	NUM
cana-3833	341	9	)	)	PUNCT
cana-3833	341	10	=	=	SYM
cana-3833	341	11	𝓅	𝓅	PROPN
cana-3833	341	12	(	(	PUNCT
cana-3833	341	13	𝔵′3𝔦−2	𝔵′3𝔦−2	NUM
cana-3833	341	14	)	)	PUNCT
cana-3833	341	15	=	=	PUNCT
cana-3833	342	1	−3	−3	PROPN
cana-3833	343	1	+	+	NUM
cana-3833	343	2	2𝔦	2𝔦	NOUN
cana-3833	343	3	+	+	CCONJ
cana-3833	343	4	2𝑛	2𝑛	NUM
cana-3833	343	5	,	,	PUNCT
cana-3833	343	6	𝓅	𝓅	PROPN
cana-3833	343	7	(	(	PUNCT
cana-3833	343	8	𝔵3𝔦−2	𝔵3𝔦−2	PROPN
cana-3833	343	9	𝔵3𝔦−1	𝔵3𝔦−1	PROPN
cana-3833	343	10	)	)	PUNCT
cana-3833	344	1	=	=	SYM
cana-3833	345	1	−1	−1	NOUN
cana-3833	346	1	+	+	CCONJ
cana-3833	346	2	2𝔦	2𝔦	NUM
cana-3833	346	3	,	,	PUNCT
cana-3833	346	4	𝓅	𝓅	PROPN
cana-3833	346	5	(	(	PUNCT
cana-3833	346	6	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	346	7	𝔵3𝔦	𝔵3𝔦	NOUN
cana-3833	346	8	)	)	PUNCT
cana-3833	346	9	=	=	SYM
cana-3833	346	10	𝓅	𝓅	PROPN
cana-3833	346	11	(	(	PUNCT
cana-3833	346	12	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	346	13	𝔵3𝔦+1	𝔵3𝔦+1	PROPN
cana-3833	346	14	)	)	PUNCT
cana-3833	346	15	=	=	SYM
cana-3833	346	16	2𝔦	2𝔦	NUM
cana-3833	347	1	−	−	NOUN
cana-3833	347	2	1	1	NUM
cana-3833	347	3	,	,	PUNCT
cana-3833	347	4	communications	communication	NOUN
cana-3833	347	5	on	on	ADP
cana-3833	347	6	applied	apply	VERB
cana-3833	347	7	nonlinear	nonlinear	ADJ
cana-3833	347	8	analysis	analysis	NOUN
cana-3833	347	9	issn	issn	NOUN
cana-3833	347	10	:	:	PUNCT
cana-3833	347	11	1074	1074	NUM
cana-3833	347	12	-	-	PUNCT
cana-3833	347	13	133x	133x	NUM
cana-3833	347	14	vol	vol	NOUN
cana-3833	347	15	32	32	NUM
cana-3833	347	16	no	no	NOUN
cana-3833	347	17	.	.	PUNCT
cana-3833	348	1	9s	9s	NUM
cana-3833	348	2	(	(	PUNCT
cana-3833	348	3	2025	2025	NUM
cana-3833	348	4	)	)	PUNCT
cana-3833	348	5	16	16	NUM
cana-3833	349	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	349	2	𝓅	𝓅	PROPN
cana-3833	349	3	(	(	PUNCT
cana-3833	349	4	𝔵3𝔦−2	𝔵3𝔦−2	NOUN
cana-3833	349	5	𝔵′3𝔦−1	𝔵′3𝔦−1	NOUN
cana-3833	349	6	)	)	PUNCT
cana-3833	350	1	=	=	NOUN
cana-3833	350	2	8𝔦	8𝔦	NOUN
cana-3833	350	3	−	−	PROPN
cana-3833	350	4	7	7	NUM
cana-3833	350	5	,	,	PUNCT
cana-3833	350	6	𝓅	𝓅	PROPN
cana-3833	350	7	(	(	PUNCT
cana-3833	350	8	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	350	9	𝔵′3𝔦−1	𝔵′3𝔦−1	NUM
cana-3833	350	10	)	)	PUNCT
cana-3833	351	1	=	=	NOUN
cana-3833	351	2	8𝔦	8𝔦	NOUN
cana-3833	351	3	−	−	PROPN
cana-3833	351	4	3	3	NUM
cana-3833	351	5	,	,	PUNCT
cana-3833	351	6	𝓅	𝓅	PROPN
cana-3833	351	7	(	(	PUNCT
cana-3833	351	8	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	351	9	𝔵′3𝔦	𝔵′3𝔦	NUM
cana-3833	351	10	)	)	PUNCT
cana-3833	351	11	=	=	NOUN
cana-3833	351	12	8𝔦	8𝔦	NOUN
cana-3833	351	13	−	−	PROPN
cana-3833	351	14	5	5	NUM
cana-3833	351	15	,	,	PUNCT
cana-3833	351	16	𝓅	𝓅	PROPN
cana-3833	351	17	(	(	PUNCT
cana-3833	351	18	𝔵3𝔦+1	𝔵3𝔦+1	X
cana-3833	351	19	𝔵′3𝔦	𝔵′3𝔦	NOUN
cana-3833	351	20	)	)	PUNCT
cana-3833	351	21	=	=	SYM
cana-3833	351	22	−1	−1	NOUN
cana-3833	351	23	+	+	NUM
cana-3833	351	24	8𝔦	8𝔦	NOUN
cana-3833	351	25	,	,	PUNCT
cana-3833	351	26	𝓅	𝓅	PROPN
cana-3833	351	27	(	(	PUNCT
cana-3833	351	28	𝔵3𝔦	𝔵3𝔦	ADV
cana-3833	351	29	𝔵′3𝔦+1	𝔵′3𝔦+1	NUM
cana-3833	351	30	)	)	PUNCT
cana-3833	352	1	=	=	VERB
cana-3833	352	2	8𝔦	8𝔦	NOUN
cana-3833	352	3	−	−	PROPN
cana-3833	352	4	3	3	NUM
cana-3833	352	5	,	,	PUNCT
cana-3833	352	6	𝓅	𝓅	PROPN
cana-3833	352	7	(	(	PUNCT
cana-3833	352	8	𝔵3𝔦−1	𝔵3𝔦−1	ADJ
cana-3833	352	9	𝔵′3𝔦−2	𝔵′3𝔦−2	NUM
cana-3833	352	10	)	)	PUNCT
cana-3833	353	1	=	=	NOUN
cana-3833	353	2	8𝔦	8𝔦	NOUN
cana-3833	353	3	−	−	PROPN
cana-3833	353	4	5	5	NUM
cana-3833	353	5	,	,	PUNCT
cana-3833	353	6	𝓅	𝓅	PROPN
cana-3833	353	7	(	(	PUNCT
cana-3833	353	8	𝔵′3𝔦	𝔵′3𝔦	PROPN
cana-3833	353	9	𝑤	𝑤	ADP
cana-3833	353	10	)	)	PUNCT
cana-3833	353	11	=	=	PUNCT
cana-3833	354	1	4𝑛	4𝑛	PROPN
cana-3833	355	1	+	+	CCONJ
cana-3833	355	2	12𝔦	12𝔦	NOUN
cana-3833	355	3	−	−	PROPN
cana-3833	355	4	8	8	NUM
cana-3833	355	5	3	3	NUM
cana-3833	355	6	,	,	PUNCT
cana-3833	355	7	𝓅	𝓅	PROPN
cana-3833	355	8	(	(	PUNCT
cana-3833	355	9	𝔵′3𝔦−1	𝔵′3𝔦−1	NOUN
cana-3833	355	10	𝑤	𝑤	X
cana-3833	355	11	)	)	PUNCT
cana-3833	356	1	=	=	PUNCT
cana-3833	356	2	4𝑛	4𝑛	PROPN
cana-3833	357	1	+	+	CCONJ
cana-3833	357	2	12𝔦	12𝔦	NOUN
cana-3833	357	3	−	−	PROPN
cana-3833	357	4	8	8	NUM
cana-3833	357	5	3	3	NUM
cana-3833	357	6	,	,	PUNCT
cana-3833	357	7	𝓅	𝓅	PROPN
cana-3833	357	8	(	(	PUNCT
cana-3833	357	9	𝔵′3𝔦−2	𝔵′3𝔦−2	NUM
cana-3833	357	10	𝑤	𝑤	ADP
cana-3833	357	11	)	)	PUNCT
cana-3833	357	12	=	=	PUNCT
cana-3833	358	1	4𝑛	4𝑛	PROPN
cana-3833	359	1	+	+	SYM
cana-3833	359	2	12𝔦	12𝔦	NUM
cana-3833	359	3	−	−	PROPN
cana-3833	359	4	14	14	NUM
cana-3833	359	5	3	3	NUM
cana-3833	359	6	,	,	PUNCT
cana-3833	359	7	induced	induce	VERB
cana-3833	359	8	weight	weight	NOUN
cana-3833	359	9	function	function	NOUN
cana-3833	359	10	for	for	ADP
cana-3833	359	11	edge𝜓	edge𝜓	NOUN
cana-3833	359	12	:	:	PUNCT
cana-3833	359	13	𝐸(𝜇(𝑃𝑛	𝐸(𝜇(𝑃𝑛	ADJ
cana-3833	359	14	)	)	PUNCT
cana-3833	359	15	)	)	PUNCT
cana-3833	360	1	→	→	PUNCT
cana-3833	360	2	{	{	PUNCT
cana-3833	360	3	3	3	NUM
cana-3833	360	4	,	,	PUNCT
cana-3833	360	5	5	5	NUM
cana-3833	360	6	,	,	PUNCT
cana-3833	360	7	7	7	NUM
cana-3833	360	8	,	,	PUNCT
cana-3833	360	9	…	…	PUNCT
cana-3833	360	10	,	,	PUNCT
cana-3833	360	11	−5	−5	ADP
cana-3833	360	12	+	+	CCONJ
cana-3833	360	13	8𝑛	8𝑛	NOUN
cana-3833	360	14	}	}	PUNCT
cana-3833	360	15	is	be	AUX
cana-3833	360	16	given	give	VERB
cana-3833	360	17	by	by	ADP
cana-3833	360	18	𝜓(𝔵𝔦𝔵𝔦+1	𝜓(𝔵𝔦𝔵𝔦+1	NOUN
cana-3833	360	19	)	)	PUNCT
cana-3833	360	20	=	=	SYM
cana-3833	360	21	2𝔦	2𝔦	NUM
cana-3833	361	1	+	+	CCONJ
cana-3833	361	2	1	1	NUM
cana-3833	361	3	,	,	PUNCT
cana-3833	361	4	𝔦	𝔦	X
cana-3833	361	5	≠	≠	PROPN
cana-3833	361	6	1	1	NUM
cana-3833	361	7	𝜓(𝔵𝔦𝔵′𝔦+1	𝜓(𝔵𝔦𝔵′𝔦+1	NUM
cana-3833	361	8	)	)	PUNCT
cana-3833	361	9	=	=	PUNCT
cana-3833	362	1	−3	−3	PROPN
cana-3833	363	1	+	+	NUM
cana-3833	363	2	2𝑛	2𝑛	PROPN
cana-3833	363	3	+	+	CCONJ
cana-3833	363	4	4𝔦	4𝔦	NOUN
cana-3833	363	5	,	,	PUNCT
cana-3833	363	6	𝔦	𝔦	X
cana-3833	363	7	≠	≠	PROPN
cana-3833	363	8	1	1	NUM
cana-3833	363	9	𝜓(𝔵𝔦+1𝔵′𝔦	𝜓(𝔵𝔦+1𝔵′𝔦	NOUN
cana-3833	363	10	)	)	PUNCT
cana-3833	363	11	=	=	PUNCT
cana-3833	364	1	−1	−1	NOUN
cana-3833	364	2	+	+	CCONJ
cana-3833	364	3	4𝔦	4𝔦	PROPN
cana-3833	364	4	+	+	CCONJ
cana-3833	364	5	2𝑛	2𝑛	NUM
cana-3833	364	6	,	,	PUNCT
cana-3833	364	7	𝔦	𝔦	PROPN
cana-3833	364	8	≠	≠	PROPN
cana-3833	364	9	1	1	NUM
cana-3833	364	10	𝜓(𝔵′𝔦𝑤	𝜓(𝔵′𝔦𝑤	NOUN
cana-3833	364	11	)	)	PUNCT
cana-3833	364	12	=	=	SYM
cana-3833	365	1	−5	−5	NOUN
cana-3833	366	1	+	+	NUM
cana-3833	366	2	6𝑛	6𝑛	NOUN
cana-3833	366	3	+	+	CCONJ
cana-3833	366	4	2𝔦.	2𝔦.	NUM
cana-3833	366	5	therefore	therefore	ADV
cana-3833	366	6	,	,	PUNCT
cana-3833	366	7	induced	induce	VERB
cana-3833	366	8	weights	weight	NOUN
cana-3833	366	9	in	in	ADP
cana-3833	366	10	edges	edge	NOUN
cana-3833	366	11	of	of	ADP
cana-3833	366	12	𝜇(𝑃𝑛)generates	𝜇(𝑃𝑛)generate	NOUN
cana-3833	366	13	an	an	DET
cana-3833	366	14	arithmetic	arithmetic	ADJ
cana-3833	366	15	progression	progression	NOUN
cana-3833	366	16	and	and	CCONJ
cana-3833	366	17	it	it	PRON
cana-3833	366	18	differed	differ	VERB
cana-3833	366	19	by	by	ADP
cana-3833	366	20	2	2	NUM
cana-3833	366	21	.	.	PUNCT
cana-3833	367	1	hence(3	hence(3	PROPN
cana-3833	367	2	,	,	PUNCT
cana-3833	367	3	2	2	NUM
cana-3833	367	4	)	)	PUNCT
cana-3833	367	5	–	–	PUNCT
cana-3833	367	6	𝑡𝑒𝑠(𝜇(𝑃𝑛	𝑡𝑒𝑠(𝜇(𝑃𝑛	NOUN
cana-3833	367	7	)	)	PUNCT
cana-3833	367	8	)	)	PUNCT
cana-3833	367	9	≤	≤	NUM
cana-3833	368	1	⌊	⌊	ADP
cana-3833	368	2	8𝑛	8𝑛	NOUN
cana-3833	368	3	3	3	NUM
cana-3833	368	4	⌋	⌋	NOUN
cana-3833	368	5	−	−	NOUN
cana-3833	368	6	1	1	X
cana-3833	368	7	.	.	PUNCT
cana-3833	369	1	from	from	ADP
cana-3833	369	2	proposition	proposition	NOUN
cana-3833	369	3	1	1	NUM
cana-3833	369	4	,	,	PUNCT
cana-3833	369	5	we	we	PRON
cana-3833	369	6	get	get	VERB
cana-3833	369	7	(	(	PUNCT
cana-3833	369	8	3	3	NUM
cana-3833	369	9	,	,	PUNCT
cana-3833	369	10	2	2	NUM
cana-3833	369	11	)	)	PUNCT
cana-3833	369	12	–	–	PUNCT
cana-3833	369	13	𝑡𝑒𝑠(𝜇(𝑃𝑛	𝑡𝑒𝑠(𝜇(𝑃𝑛	NOUN
cana-3833	369	14	)	)	PUNCT
cana-3833	369	15	)	)	PUNCT
cana-3833	370	1	≥	≥	X
cana-3833	370	2	−1	−1	NOUN
cana-3833	371	1	+	+	CCONJ
cana-3833	371	2	⌊	⌊	PROPN
cana-3833	371	3	8𝑛	8𝑛	NOUN
cana-3833	371	4	3	3	NUM
cana-3833	371	5	⌋	⌋	NOUN
cana-3833	371	6	and	and	CCONJ
cana-3833	371	7	it	it	PRON
cana-3833	371	8	concludes	conclude	VERB
cana-3833	371	9	the	the	DET
cana-3833	371	10	result	result	NOUN
cana-3833	371	11	.	.	PUNCT
cana-3833	372	1	example	example	NOUN
cana-3833	372	2	8	8	NUM
cana-3833	372	3	:	:	PUNCT
cana-3833	372	4	(	(	PUNCT
cana-3833	372	5	3	3	NUM
cana-3833	372	6	,	,	PUNCT
cana-3833	372	7	2	2	NUM
cana-3833	372	8	)	)	PUNCT
cana-3833	372	9	total	total	ADJ
cana-3833	372	10	edge	edge	VERB
cana-3833	372	11	irregular	irregular	ADJ
cana-3833	372	12	labeling	labeling	NOUN
cana-3833	372	13	of	of	ADP
cana-3833	372	14	𝜇(𝑃6)is	𝜇(𝑃6)is	NOUN
cana-3833	372	15	given	give	VERB
cana-3833	372	16	in	in	ADP
cana-3833	372	17	fig	fig	NOUN
cana-3833	372	18	10	10	NUM
cana-3833	372	19	.	.	PUNCT
cana-3833	373	1	figure	figure	NOUN
cana-3833	373	2	10	10	NUM
cana-3833	373	3	.	.	PUNCT
cana-3833	374	1	(	(	PUNCT
cana-3833	374	2	3	3	NUM
cana-3833	374	3	,	,	PUNCT
cana-3833	374	4	2	2	NUM
cana-3833	374	5	)	)	PUNCT
cana-3833	374	6	–	–	PUNCT
cana-3833	374	7	𝑡𝑒𝑠(𝜇(𝑃6	𝑡𝑒𝑠(𝜇(𝑃6	NUM
cana-3833	374	8	)	)	PUNCT
cana-3833	374	9	)	)	PUNCT
cana-3833	375	1	=	=	SYM
cana-3833	375	2	15	15	X
cana-3833	375	3	.	.	PUNCT
cana-3833	376	1	communications	communication	NOUN
cana-3833	376	2	on	on	ADP
cana-3833	376	3	applied	apply	VERB
cana-3833	376	4	nonlinear	nonlinear	ADJ
cana-3833	376	5	analysis	analysis	NOUN
cana-3833	376	6	issn	issn	NOUN
cana-3833	376	7	:	:	PUNCT
cana-3833	376	8	1074	1074	NUM
cana-3833	376	9	-	-	PUNCT
cana-3833	376	10	133x	133x	NUM
cana-3833	376	11	vol	vol	NOUN
cana-3833	376	12	32	32	NUM
cana-3833	376	13	no	no	NOUN
cana-3833	376	14	.	.	PUNCT
cana-3833	377	1	9s	9s	NUM
cana-3833	377	2	(	(	PUNCT
cana-3833	377	3	2025	2025	NUM
cana-3833	377	4	)	)	PUNCT
cana-3833	377	5	17	17	NUM
cana-3833	378	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3833	378	2	remark	remark	NOUN
cana-3833	378	3	3	3	NUM
cana-3833	378	4	:	:	PUNCT
cana-3833	378	5	when	when	SCONJ
cana-3833	378	6	𝑛	𝑛	PROPN
cana-3833	378	7	=	=	SYM
cana-3833	378	8	2	2	NUM
cana-3833	378	9	,	,	PUNCT
cana-3833	378	10	𝜇(𝑃2	𝜇(𝑃2	PROPN
cana-3833	378	11	)	)	PUNCT
cana-3833	378	12	≅	≅	PROPN
cana-3833	378	13	𝐶5	𝐶5	PROPN
cana-3833	378	14	,	,	PUNCT
cana-3833	378	15	and	and	CCONJ
cana-3833	378	16	(	(	PUNCT
cana-3833	378	17	3	3	NUM
cana-3833	378	18	,	,	PUNCT
cana-3833	378	19	2	2	NUM
cana-3833	378	20	)	)	PUNCT
cana-3833	378	21	–	–	PUNCT
cana-3833	378	22	𝑡𝑒𝑠(𝐶	𝑡𝑒𝑠(𝐶	NUM
cana-3833	378	23	5	5	X
cana-3833	378	24	)	)	PUNCT
cana-3833	378	25	=	=	SYM
cana-3833	378	26	4	4	NUM
cana-3833	378	27	,	,	PUNCT
cana-3833	378	28	total	total	ADJ
cana-3833	378	29	edge	edge	VERB
cana-3833	378	30	irregular	irregular	ADJ
cana-3833	378	31	labelling	labelling	NOUN
cana-3833	378	32	for	for	ADP
cana-3833	378	33	𝐶	𝐶	PROPN
cana-3833	378	34	5	5	NUM
cana-3833	378	35	is	be	AUX
cana-3833	378	36	shown	show	VERB
cana-3833	378	37	in	in	ADP
cana-3833	378	38	fig	fig	NOUN
cana-3833	378	39	11	11	NUM
cana-3833	378	40	.	.	PUNCT
cana-3833	379	1	figure	figure	VERB
cana-3833	379	2	11	11	NUM
cana-3833	379	3	.	.	PUNCT
cana-3833	380	1	(	(	PUNCT
cana-3833	380	2	3	3	NUM
cana-3833	380	3	,	,	PUNCT
cana-3833	380	4	2	2	NUM
cana-3833	380	5	)	)	PUNCT
cana-3833	380	6	–	–	PUNCT
cana-3833	380	7	𝑡𝑒𝑠(𝐶5	𝑡𝑒𝑠(𝐶5	X
cana-3833	380	8	)	)	PUNCT
cana-3833	380	9	=	=	PUNCT
cana-3833	381	1	4	4	NUM
cana-3833	381	2	.	.	NOUN
cana-3833	381	3	3	3	NUM
cana-3833	381	4	.	.	X
cana-3833	381	5	conclusion	conclusion	NOUN
cana-3833	381	6	the	the	DET
cana-3833	381	7	exact	exact	ADJ
cana-3833	381	8	value	value	NOUN
cana-3833	381	9	of	of	ADP
cana-3833	381	10	(	(	PUNCT
cana-3833	381	11	3	3	NUM
cana-3833	381	12	,	,	PUNCT
cana-3833	381	13	2	2	NUM
cana-3833	381	14	)	)	PUNCT
cana-3833	381	15	–	–	PUNCT
cana-3833	381	16	𝑡𝑒𝑠(𝐺	𝑡𝑒𝑠(𝐺	NUM
cana-3833	381	17	)	)	PUNCT
cana-3833	381	18	is	be	AUX
cana-3833	381	19	presented	present	VERB
cana-3833	381	20	in	in	ADP
cana-3833	381	21	this	this	DET
cana-3833	381	22	article	article	NOUN
cana-3833	381	23	namely	namely	ADV
cana-3833	381	24	:	:	PUNCT
cana-3833	381	25	(	(	PUNCT
cana-3833	381	26	3	3	NUM
cana-3833	381	27	,	,	PUNCT
cana-3833	381	28	2	2	NUM
cana-3833	381	29	)	)	PUNCT
cana-3833	381	30	–	–	PUNCT
cana-3833	381	31	𝑡𝑒𝑠(𝑆(𝑃	𝑡𝑒𝑠(𝑆(𝑃	NUM
cana-3833	381	32	𝑛	𝑛	PROPN
cana-3833	381	33	∘	∘	PROPN
cana-3833	381	34	𝑆2	𝑆2	PROPN
cana-3833	381	35	)	)	PUNCT
cana-3833	381	36	)	)	PUNCT
cana-3833	382	1	=	=	PUNCT
cana-3833	382	2	4𝑛	4𝑛	NOUN
cana-3833	382	3	−	−	NOUN
cana-3833	383	1	1	1	NUM
cana-3833	383	2	,	,	PUNCT
cana-3833	383	3	(	(	PUNCT
cana-3833	383	4	3	3	NUM
cana-3833	383	5	,	,	PUNCT
cana-3833	383	6	2	2	NUM
cana-3833	383	7	)	)	PUNCT
cana-3833	383	8	–	–	PUNCT
cana-3833	383	9	𝑡𝑒𝑠(𝑀(𝑃	𝑡𝑒𝑠(𝑀(𝑃	X
cana-3833	383	10	𝑛	𝑛	NOUN
cana-3833	383	11	)	)	PUNCT
cana-3833	383	12	)	)	PUNCT
cana-3833	384	1	=	=	SYM
cana-3833	384	2	2𝑛	2𝑛	PROPN
cana-3833	384	3	−	−	PROPN
cana-3833	384	4	2	2	NUM
cana-3833	384	5	,	,	PUNCT
cana-3833	384	6	(	(	PUNCT
cana-3833	384	7	3	3	NUM
cana-3833	384	8	,	,	PUNCT
cana-3833	384	9	2	2	NUM
cana-3833	384	10	)	)	PUNCT
cana-3833	384	11	–	–	PUNCT
cana-3833	384	12	𝑡𝑒𝑠(𝐾𝑛	𝑡𝑒𝑠(𝐾𝑛	NUM
cana-3833	384	13	)	)	PUNCT
cana-3833	384	14	=	=	SYM
cana-3833	384	15	𝑛2	𝑛2	NOUN
cana-3833	384	16	−	−	PROPN
cana-3833	384	17	5𝑛	5𝑛	NUM
cana-3833	384	18	+	+	CCONJ
cana-3833	384	19	9	9	NUM
cana-3833	384	20	for	for	ADP
cana-3833	384	21	𝑛	𝑛	PRON
cana-3833	384	22	≥	≥	NUM
cana-3833	384	23	4	4	NUM
cana-3833	384	24	,	,	PUNCT
cana-3833	384	25	(	(	PUNCT
cana-3833	384	26	3	3	NUM
cana-3833	384	27	,	,	PUNCT
cana-3833	384	28	2	2	NUM
cana-3833	384	29	)	)	PUNCT
cana-3833	384	30	–	–	PUNCT
cana-3833	384	31	𝑡𝑒𝑠(𝑇(𝑃	𝑡𝑒𝑠(𝑇(𝑃	NUM
cana-3833	384	32	𝑛	𝑛	PROPN
cana-3833	384	33	)	)	PUNCT
cana-3833	384	34	)	)	PUNCT
cana-3833	385	1	=	=	PUNCT
cana-3833	385	2	⌈	⌈	NUM
cana-3833	385	3	8𝑛	8𝑛	NOUN
cana-3833	385	4	3	3	NUM
cana-3833	385	5	⌉	⌉	SCONJ
cana-3833	385	6	−	−	PROPN
cana-3833	385	7	3	3	NUM
cana-3833	385	8	for	for	ADP
cana-3833	385	9	𝑛	𝑛	PRON
cana-3833	385	10	≥	≥	NUM
cana-3833	385	11	3	3	NUM
cana-3833	385	12	,	,	PUNCT
cana-3833	385	13	(	(	PUNCT
cana-3833	385	14	3	3	NUM
cana-3833	385	15	,	,	PUNCT
cana-3833	385	16	2	2	NUM
cana-3833	385	17	)	)	PUNCT
cana-3833	385	18	–	–	PUNCT
cana-3833	385	19	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	X
cana-3833	385	20	𝑛	𝑛	DET
cana-3833	385	21	∘	∘	PROPN
cana-3833	385	22	𝐾2	𝐾2	NOUN
cana-3833	385	23	)	)	PUNCT
cana-3833	385	24	=	=	PUNCT
cana-3833	386	1	⌈	⌈	NOUN
cana-3833	386	2	8(𝑛+1	8(𝑛+1	NUM
cana-3833	386	3	)	)	PUNCT
cana-3833	386	4	3	3	NUM
cana-3833	386	5	⌉	⌉	PRON
cana-3833	386	6	−	−	PROPN
cana-3833	386	7	3	3	NUM
cana-3833	386	8	,	,	PUNCT
cana-3833	386	9	and	and	CCONJ
cana-3833	386	10	(	(	PUNCT
cana-3833	386	11	3	3	NUM
cana-3833	386	12	,	,	PUNCT
cana-3833	386	13	2	2	NUM
cana-3833	386	14	)	)	PUNCT
cana-3833	386	15	–	–	PUNCT
cana-3833	386	16	𝑡𝑒𝑠(𝜇(𝑃𝑛	𝑡𝑒𝑠(𝜇(𝑃𝑛	NOUN
cana-3833	386	17	)	)	PUNCT
cana-3833	386	18	)	)	PUNCT
cana-3833	387	1	=	=	PUNCT
cana-3833	387	2	⌊	⌊	VERB
cana-3833	387	3	8𝑛	8𝑛	NOUN
cana-3833	387	4	3	3	NUM
cana-3833	387	5	⌋	⌋	NOUN
cana-3833	387	6	−	−	PROPN
cana-3833	387	7	1	1	NUM
cana-3833	387	8	for	for	ADP
cana-3833	387	9	𝑛	𝑛	PRON
cana-3833	387	10	≥	≥	NUM
cana-3833	387	11	3	3	NUM
cana-3833	387	12	.	.	PUNCT
cana-3833	388	1	in	in	ADP
cana-3833	388	2	addition	addition	NOUN
cana-3833	388	3	,	,	PUNCT
cana-3833	388	4	an	an	DET
cana-3833	388	5	open	open	ADJ
cana-3833	388	6	problem	problem	NOUN
cana-3833	388	7	of	of	ADP
cana-3833	388	8	(	(	PUNCT
cana-3833	388	9	3	3	NUM
cana-3833	388	10	,	,	PUNCT
cana-3833	388	11	2	2	NUM
cana-3833	388	12	)	)	PUNCT
cana-3833	388	13	–	–	PUNCT
cana-3833	388	14	𝑡𝑒𝑠(𝑃	𝑡𝑒𝑠(𝑃	NUM
cana-3833	388	15	𝑛	𝑛	PRON
cana-3833	388	16	3	3	NUM
cana-3833	388	17	)	)	PUNCT
cana-3833	388	18	=	=	SYM
cana-3833	388	19	2𝑛	2𝑛	PROPN
cana-3833	389	1	−	−	NOUN
cana-3833	389	2	3	3	NUM
cana-3833	389	3	for	for	ADP
cana-3833	389	4	𝑛	𝑛	PROPN
cana-3833	389	5	>	>	SYM
cana-3833	389	6	3	3	NUM
cana-3833	389	7	and	and	CCONJ
cana-3833	389	8	(	(	PUNCT
cana-3833	389	9	3	3	NUM
cana-3833	389	10	,	,	PUNCT
cana-3833	389	11	2	2	NUM
cana-3833	389	12	)	)	PUNCT
cana-3833	389	13	–	–	PUNCT
cana-3833	389	14	𝑡𝑒𝑠(𝑇	𝑡𝑒𝑠(𝑇	NUM
cana-3833	389	15	)	)	PUNCT
cana-3833	389	16	=	=	PUNCT
cana-3833	390	1	⌈	⌈	X
cana-3833	390	2	2𝑝−1	2𝑝−1	NUM
cana-3833	390	3	3	3	NUM
cana-3833	390	4	⌉	⌉	X
cana-3833	390	5	where	where	SCONJ
cana-3833	390	6	𝑇	𝑇	PROPN
cana-3833	390	7	is	be	AUX
cana-3833	390	8	a	a	DET
cana-3833	390	9	tree	tree	NOUN
cana-3833	390	10	other	other	ADJ
cana-3833	390	11	than	than	ADP
cana-3833	390	12	star	star	NOUN
cana-3833	390	13	are	be	AUX
cana-3833	390	14	solved	solve	VERB
cana-3833	390	15	positively	positively	ADV
cana-3833	390	16	.	.	PUNCT
cana-3833	391	1	there	there	PRON
cana-3833	391	2	is	be	VERB
cana-3833	391	3	still	still	ADV
cana-3833	391	4	an	an	DET
cana-3833	391	5	opportunity	opportunity	NOUN
cana-3833	391	6	for	for	ADP
cana-3833	391	7	an	an	DET
cana-3833	391	8	investigation	investigation	NOUN
cana-3833	391	9	towards	towards	ADP
cana-3833	391	10	similar	similar	ADJ
cana-3833	391	11	findings	finding	NOUN
cana-3833	391	12	for	for	ADP
cana-3833	391	13	other	other	ADJ
cana-3833	391	14	graph	graph	NOUN
cana-3833	391	15	families	family	NOUN
cana-3833	391	16	.	.	PUNCT
cana-3833	392	1	references	reference	NOUN
cana-3833	392	2	[	[	X
cana-3833	392	3	1	1	NUM
cana-3833	392	4	]	]	PUNCT
cana-3833	392	5	chartrand	chartrand	NOUN
cana-3833	392	6	g	g	PROPN
cana-3833	392	7	,	,	PUNCT
cana-3833	392	8	jacobson	jacobson	PROPN
cana-3833	392	9	ms	ms	PROPN
cana-3833	392	10	,	,	PUNCT
cana-3833	392	11	lehel	lehel	PROPN
cana-3833	392	12	j	j	PROPN
cana-3833	392	13	,	,	PUNCT
cana-3833	392	14	oellermann	oellermann	VERB
cana-3833	392	15	or	or	CCONJ
cana-3833	392	16	,	,	PUNCT
cana-3833	392	17	ruiz	ruiz	NOUN
cana-3833	392	18	s	s	PROPN
cana-3833	392	19	,	,	PUNCT
cana-3833	392	20	saba	saba	PROPN
cana-3833	392	21	f.	f.	PROPN
cana-3833	392	22	irregular	irregular	ADJ
cana-3833	392	23	networks	network	NOUN
cana-3833	392	24	.	.	PUNCT
cana-3833	393	1	congr	congr	PROPN
cana-3833	393	2	numer	numer	PROPN
cana-3833	393	3	.	.	PUNCT
cana-3833	394	1	1988;64	1988;64	NUM
cana-3833	394	2	:	:	PUNCT
cana-3833	394	3	187	187	NUM
cana-3833	394	4	-	-	SYM
cana-3833	394	5	192	192	NUM
cana-3833	394	6	.	.	PUNCT
cana-3833	395	1	[	[	X
cana-3833	395	2	2	2	X
cana-3833	395	3	]	]	PUNCT
cana-3833	395	4	baca	baca	PROPN
cana-3833	395	5	m	m	PROPN
cana-3833	395	6	,	,	PUNCT
cana-3833	395	7	jendrol	jendrol	PROPN
cana-3833	395	8	s	s	PROPN
cana-3833	395	9	,	,	PUNCT
cana-3833	395	10	miller	miller	PROPN
cana-3833	395	11	m	m	PROPN
cana-3833	395	12	,	,	PUNCT
cana-3833	395	13	ryan	ryan	PROPN
cana-3833	395	14	j.	j.	PROPN
cana-3833	395	15	on	on	ADP
cana-3833	395	16	irregular	irregular	ADJ
cana-3833	395	17	total	total	ADJ
cana-3833	395	18	labelings	labeling	NOUN
cana-3833	395	19	.	.	PUNCT
cana-3833	396	1	discrete	discrete	ADJ
cana-3833	396	2	math	math	NOUN
cana-3833	396	3	.	.	PUNCT
cana-3833	397	1	2007	2007	NUM
cana-3833	397	2	;	;	PUNCT
cana-3833	397	3	307	307	NUM
cana-3833	397	4	:	:	SYM
cana-3833	397	5	1378	1378	NUM
cana-3833	397	6	-	-	SYM
cana-3833	397	7	1388	1388	NUM
cana-3833	397	8	.	.	PUNCT
cana-3833	398	1	[	[	X
cana-3833	398	2	3	3	X
cana-3833	398	3	]	]	PUNCT
cana-3833	398	4	riyan	riyan	PROPN
cana-3833	398	5	wicaksana	wicaksana	PROPN
cana-3833	398	6	putra	putra	PROPN
cana-3833	398	7	,	,	PUNCT
cana-3833	398	8	yeni	yeni	PROPN
cana-3833	398	9	susanti	susanti	X
cana-3833	398	10	.	.	PUNCT
cana-3833	399	1	on	on	ADP
cana-3833	399	2	total	total	ADJ
cana-3833	399	3	edge	edge	NOUN
cana-3833	399	4	irregularity	irregularity	NOUN
cana-3833	399	5	strength	strength	NOUN
cana-3833	399	6	of	of	ADP
cana-3833	399	7	centralized	centralized	ADJ
cana-3833	399	8	uniform	uniform	ADJ
cana-3833	399	9	theta	theta	NOUN
cana-3833	399	10	graphs	graph	NOUN
cana-3833	399	11	.	.	PUNCT
cana-3833	400	1	akce	akce	PROPN
cana-3833	400	2	int	int	PROPN
cana-3833	400	3	.	.	PUNCT
cana-3833	401	1	j.	j.	PROPN
cana-3833	401	2	graphs	graphs	PROPN
cana-3833	401	3	comb	comb	NOUN
cana-3833	401	4	.	.	PUNCT
cana-3833	402	1	2018	2018	NUM
cana-3833	402	2	;	;	PUNCT
cana-3833	402	3	15	15	NUM
cana-3833	402	4	:	:	SYM
cana-3833	402	5	7	7	NUM
cana-3833	402	6	-	-	SYM
cana-3833	402	7	13	13	NUM
cana-3833	402	8	.	.	PUNCT
cana-3833	403	1	[	[	X
cana-3833	403	2	4	4	X
cana-3833	403	3	]	]	PUNCT
cana-3833	403	4	susanti	susanti	PROPN
cana-3833	403	5	y	y	PROPN
cana-3833	403	6	,	,	PUNCT
cana-3833	403	7	wahyuni	wahyuni	PROPN
cana-3833	403	8	s	s	PROPN
cana-3833	403	9	,	,	PUNCT
cana-3833	403	10	sutjijana	sutjijana	VERB
cana-3833	403	11	a	a	PRON
cana-3833	403	12	,	,	PUNCT
cana-3833	403	13	sutopo	sutopo	PROPN
cana-3833	403	14	s	s	NOUN
cana-3833	403	15	,	,	PUNCT
cana-3833	403	16	ernanto	ernanto	ADP
cana-3833	403	17	i.	i.	PROPN
cana-3833	403	18	generalized	generalize	VERB
cana-3833	403	19	arithmetic	arithmetic	ADJ
cana-3833	403	20	staircase	staircase	NOUN
cana-3833	403	21	graphs	graph	NOUN
cana-3833	403	22	and	and	CCONJ
cana-3833	403	23	their	their	PRON
cana-3833	403	24	total	total	ADJ
cana-3833	403	25	edge	edge	NOUN
cana-3833	403	26	irregularity	irregularity	NOUN
cana-3833	403	27	strengths	strength	NOUN
cana-3833	403	28	.	.	PUNCT
cana-3833	404	1	symmetry	symmetry	NOUN
cana-3833	404	2	.	.	PUNCT
cana-3833	405	1	2022	2022	NUM
cana-3833	405	2	;	;	PUNCT
cana-3833	406	1	14(19	14(19	NUM
cana-3833	406	2	):	):	PUNCT
cana-3833	406	3	1853	1853	NUM
cana-3833	406	4	.	.	PUNCT
cana-3833	407	1	[	[	X
cana-3833	407	2	5	5	X
cana-3833	407	3	]	]	PUNCT
cana-3833	407	4	rosyida	rosyida	PROPN
cana-3833	407	5	i	i	PROPN
cana-3833	407	6	,	,	PUNCT
cana-3833	407	7	indriati	indriati	PROPN
cana-3833	407	8	d.	d.	PROPN
cana-3833	407	9	computing	compute	VERB
cana-3833	407	10	total	total	ADJ
cana-3833	407	11	edge	edge	NOUN
cana-3833	407	12	irregularity	irregularity	NOUN
cana-3833	407	13	strength	strength	NOUN
cana-3833	407	14	of	of	ADP
cana-3833	407	15	some	some	DET
cana-3833	407	16	n	n	CCONJ
cana-3833	407	17	-	-	PUNCT
cana-3833	407	18	uniform	uniform	ADJ
cana-3833	407	19	cactus	cactus	NOUN
cana-3833	407	20	chain	chain	NOUN
cana-3833	407	21	graphs	graph	NOUN
cana-3833	407	22	and	and	CCONJ
cana-3833	407	23	related	relate	VERB
cana-3833	407	24	chain	chain	NOUN
cana-3833	407	25	graphs	graph	NOUN
cana-3833	407	26	.	.	PUNCT
cana-3833	408	1	indones	indone	NOUN
cana-3833	408	2	.	.	PUNCT
cana-3833	409	1	j.	j.	PROPN
cana-3833	409	2	combin	combin	PROPN
cana-3833	409	3	.	.	PUNCT
cana-3833	410	1	2020	2020	NUM
cana-3833	410	2	;	;	PUNCT
cana-3833	410	3	4(1	4(1	NOUN
cana-3833	410	4	):	):	PUNCT
cana-3833	410	5	53	53	NUM
cana-3833	410	6	-	-	SYM
cana-3833	410	7	75	75	NUM
cana-3833	410	8	.	.	PUNCT
cana-3833	411	1	[	[	X
cana-3833	411	2	6	6	NUM
cana-3833	411	3	]	]	PUNCT
cana-3833	411	4	muthu	muthu	NOUN
cana-3833	411	5	guru	guru	NOUN
cana-3833	411	6	packiam	packiam	PROPN
cana-3833	411	7	k	k	PROPN
cana-3833	411	8	,	,	PUNCT
cana-3833	411	9	padmapriya	padmapriya	PROPN
cana-3833	411	10	r.	r.	PROPN
cana-3833	411	11	(	(	PUNCT
cana-3833	411	12	a	a	DET
cana-3833	411	13	,	,	PUNCT
cana-3833	411	14	d	d	NOUN
cana-3833	411	15	)	)	PUNCT
cana-3833	411	16	–	–	PUNCT
cana-3833	411	17	total	total	ADJ
cana-3833	411	18	edge	edge	NOUN
cana-3833	411	19	irregularity	irregularity	NOUN
cana-3833	411	20	strength	strength	NOUN
cana-3833	411	21	of	of	ADP
cana-3833	411	22	graphs	graph	NOUN
cana-3833	411	23	.	.	PUNCT
cana-3833	412	1	j	j	PROPN
cana-3833	412	2	math	math	PROPN
cana-3833	412	3	comput	comput	PROPN
cana-3833	412	4	sci	sci	PROPN
cana-3833	412	5	.	.	PROPN
cana-3833	412	6	2021	2021	NUM
cana-3833	412	7	;	;	PUNCT
cana-3833	412	8	11(4	11(4	NUM
cana-3833	412	9	):	):	PUNCT
cana-3833	412	10	4436	4436	NUM
cana-3833	412	11	-	-	SYM
cana-3833	412	12	4453	4453	NUM
cana-3833	412	13	.	.	PUNCT
cana-3833	413	1	[	[	X
cana-3833	413	2	7	7	X
cana-3833	413	3	]	]	PUNCT
cana-3833	413	4	s.muthukumar	s.muthukumar	NOUN
cana-3833	413	5	,	,	PUNCT
cana-3833	413	6	k.rajendran	k.rajendran	NOUN
cana-3833	413	7	,	,	PUNCT
cana-3833	413	8	edge	edge	VERB
cana-3833	413	9	irregularity	irregularity	NOUN
cana-3833	413	10	strength	strength	NOUN
cana-3833	413	11	of	of	ADP
cana-3833	413	12	binomial	binomial	ADJ
cana-3833	413	13	trees	tree	NOUN
cana-3833	413	14	.	.	PUNCT
cana-3833	414	1	communications	communication	NOUN
cana-3833	414	2	on	on	ADP
cana-3833	414	3	applied	apply	VERB
cana-3833	414	4	nonlinear	nonlinear	ADJ
cana-3833	414	5	analysis	analysis	NOUN
cana-3833	414	6	.	.	PUNCT
cana-3833	415	1	issn	issn	PROPN
cana-3833	415	2	:	:	PUNCT
cana-3833	415	3	1074	1074	NUM
cana-3833	415	4	-	-	PUNCT
cana-3833	415	5	133x	133x	NUM
cana-3833	415	6	vol	vol	NOUN
cana-3833	415	7	31no	31no	NOUN
cana-3833	415	8	.	.	PUNCT
cana-3833	416	1	2(2024	2(2024	NUM
cana-3833	416	2	)	)	PUNCT
cana-3833	416	3	.	.	PUNCT
cana-3833	417	1	[	[	X
cana-3833	417	2	8	8	NUM
cana-3833	417	3	]	]	PUNCT
cana-3833	417	4	ashwini	ashwini	PROPN
cana-3833	417	5	j	j	PROPN
cana-3833	417	6	,	,	PUNCT
cana-3833	417	7	selvam	selvam	PROPN
cana-3833	417	8	s	s	PROPN
cana-3833	417	9	,	,	PUNCT
cana-3833	417	10	gnanajothi	gnanajothi	PROPN
cana-3833	417	11	r.	r.	PROPN
cana-3833	417	12	some	some	DET
cana-3833	417	13	new	new	ADJ
cana-3833	417	14	results	result	NOUN
cana-3833	417	15	on	on	ADP
cana-3833	417	16	lucky	lucky	ADJ
cana-3833	417	17	labeling	labeling	NOUN
cana-3833	417	18	.	.	PUNCT
cana-3833	418	1	baghdad	baghdad	PROPN
cana-3833	418	2	sci	sci	PROPN
cana-3833	418	3	j.	j.	PROPN
cana-3833	418	4	2023	2023	NUM
cana-3833	418	5	mar	mar	PROPN
cana-3833	418	6	.	.	PROPN
cana-3833	418	7	1	1	NUM
cana-3833	418	8	;	;	PUNCT
cana-3833	418	9	20(1	20(1	NUM
cana-3833	418	10	):	):	PUNCT
cana-3833	418	11	0365	0365	NUM
cana-3833	418	12	.	.	PUNCT
cana-3833	419	1	[	[	X
cana-3833	419	2	9	9	NUM
cana-3833	419	3	]	]	SYM
cana-3833	419	4	gallian	gallian	ADJ
cana-3833	419	5	j	j	PROPN
cana-3833	419	6	a.	a.	NOUN
cana-3833	419	7	a	a	DET
cana-3833	419	8	dynamic	dynamic	ADJ
cana-3833	419	9	survey	survey	NOUN
cana-3833	419	10	of	of	ADP
cana-3833	419	11	graph	graph	NOUN
cana-3833	419	12	labeling	labeling	NOUN
cana-3833	419	13	.	.	PUNCT
cana-3833	420	1	electron	electron	PROPN
cana-3833	420	2	j	j	PROPN
cana-3833	420	3	comb	comb	NOUN
cana-3833	420	4	.	.	PUNCT
cana-3833	421	1	2018	2018	NUM
cana-3833	421	2	;	;	PUNCT
cana-3833	421	3	1	1	NUM
cana-3833	421	4	(	(	PUNCT
cana-3833	421	5	dynamic	dynamic	ADJ
cana-3833	421	6	surveys	survey	NOUN
cana-3833	421	7	)	)	PUNCT
cana-3833	421	8	.	.	PUNCT
cana-3833	422	1	[	[	X
cana-3833	422	2	10	10	NUM
cana-3833	422	3	]	]	X
cana-3833	422	4	muthugurupackiam	muthugurupackiam	PROPN
cana-3833	422	5	,	,	PUNCT
cana-3833	422	6	k.	k.	PROPN
cana-3833	422	7	,	,	PUNCT
cana-3833	422	8	pandiaraj	pandiaraj	ADJ
cana-3833	422	9	,	,	PUNCT
cana-3833	422	10	p.	p.	NOUN
cana-3833	422	11	,	,	PUNCT
cana-3833	422	12	gurusamy	gurusamy	PROPN
cana-3833	422	13	,	,	PUNCT
cana-3833	422	14	r.	r.	PROPN
cana-3833	422	15	,	,	PUNCT
cana-3833	422	16	&	&	CCONJ
cana-3833	422	17	muthuselvam	muthuselvam	PROPN
cana-3833	422	18	,	,	PUNCT
cana-3833	422	19	i.	i.	PROPN
cana-3833	422	20	further	further	ADJ
cana-3833	422	21	results	result	VERB
cana-3833	422	22	on	on	ADP
cana-3833	422	23	(	(	PUNCT
cana-3833	422	24	a	a	DET
cana-3833	422	25	,	,	PUNCT
cana-3833	422	26	d	d	NOUN
cana-3833	422	27	)	)	PUNCT
cana-3833	422	28	-total	-total	ADJ
cana-3833	422	29	edge	edge	NOUN
cana-3833	422	30	irregularity	irregularity	NOUN
cana-3833	422	31	strength	strength	NOUN
cana-3833	422	32	of	of	ADP
cana-3833	422	33	graphs	graph	NOUN
cana-3833	422	34	.	.	PUNCT
cana-3833	423	1	baghdad	baghdad	PROPN
cana-3833	423	2	science	science	PROPN
cana-3833	423	3	journal	journal	PROPN
cana-3833	423	4	.	.	PUNCT
cana-3833	424	1	2023	2023	NUM
cana-3833	424	2	;	;	PUNCT
cana-3833	424	3	20(6	20(6	NUM
cana-3833	424	4	):	):	PUNCT
cana-3833	424	5	2498	2498	NUM
cana-3833	424	6	-	-	SYM
cana-3833	424	7	2507	2507	NUM
cana-3833	424	8	.	.	PUNCT