id	sid	tid	token	lemma	pos
cana-509	1	1	communications	communication	NOUN
cana-509	1	2	on	on	ADP
cana-509	1	3	applied	apply	VERB
cana-509	1	4	nonlinear	nonlinear	ADJ
cana-509	1	5	analysis	analysis	NOUN
cana-509	1	6	issn	issn	NOUN
cana-509	1	7	:	:	PUNCT
cana-509	1	8	1074	1074	NUM
cana-509	1	9	-	-	PUNCT
cana-509	1	10	133x	133x	NUM
cana-509	1	11	vol	vol	NOUN
cana-509	1	12	31	31	NUM
cana-509	1	13	no	no	NOUN
cana-509	1	14	.	.	NOUN
cana-509	1	15	2	2	NUM
cana-509	1	16	(	(	PUNCT
cana-509	1	17	2024	2024	NUM
cana-509	1	18	)	)	PUNCT
cana-509	1	19	11	11	NUM
cana-509	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	1	21	an	an	DET
cana-509	1	22	application	application	NOUN
cana-509	1	23	on	on	ADP
cana-509	1	24	harmonic	harmonic	ADJ
cana-509	1	25	mean	mean	NOUN
cana-509	1	26	labeling	labeling	NOUN
cana-509	1	27	of	of	ADP
cana-509	1	28	variations	variation	NOUN
cana-509	1	29	in	in	ADP
cana-509	1	30	triangular	triangular	NOUN
cana-509	1	31	snake	snake	NOUN
cana-509	1	32	graphs	graph	NOUN
cana-509	1	33	t.	t.	PROPN
cana-509	1	34	christy1	christy1	PROPN
cana-509	1	35	,	,	PUNCT
cana-509	1	36	g.	g.	PROPN
cana-509	1	37	palani2	palani2	PROPN
cana-509	1	38	,	,	PUNCT
cana-509	1	39	*	*	PROPN
cana-509	1	40	,	,	PUNCT
cana-509	1	41	e.	e.	PROPN
cana-509	1	42	chandrasekaran3	chandrasekaran3	PROPN
cana-509	1	43	1assistant	1assistant	NUM
cana-509	1	44	professor	professor	NOUN
cana-509	1	45	,	,	PUNCT
cana-509	1	46	department	department	NOUN
cana-509	1	47	of	of	ADP
cana-509	1	48	mathematics	mathematic	NOUN
cana-509	1	49	,	,	PUNCT
cana-509	1	50	patrician	patrician	ADJ
cana-509	1	51	college	college	PROPN
cana-509	1	52	of	of	ADP
cana-509	1	53	arts	art	NOUN
cana-509	1	54	and	and	CCONJ
cana-509	1	55	science	science	NOUN
cana-509	1	56	,	,	PUNCT
cana-509	1	57	affiliated	affiliate	VERB
cana-509	1	58	to	to	ADP
cana-509	1	59	university	university	PROPN
cana-509	1	60	of	of	ADP
cana-509	1	61	madras	madras	PROPN
cana-509	1	62	,	,	PUNCT
cana-509	1	63	tamil	tamil	PROPN
cana-509	1	64	nadu	nadu	PROPN
cana-509	1	65	,	,	PUNCT
cana-509	1	66	india	india	PROPN
cana-509	1	67	.	.	PUNCT
cana-509	2	1	2assistant	2assistant	NUM
cana-509	2	2	professor	professor	NOUN
cana-509	2	3	,	,	PUNCT
cana-509	2	4	department	department	NOUN
cana-509	2	5	of	of	ADP
cana-509	2	6	mathematics	mathematics	PROPN
cana-509	2	7	,	,	PUNCT
cana-509	2	8	dr	dr	PROPN
cana-509	2	9	.	.	PROPN
cana-509	2	10	ambedkar	ambedkar	PROPN
cana-509	2	11	government	government	PROPN
cana-509	2	12	arts	arts	PROPN
cana-509	2	13	college	college	PROPN
cana-509	2	14	,	,	PUNCT
cana-509	2	15	tamil	tamil	PROPN
cana-509	2	16	nadu	nadu	PROPN
cana-509	2	17	,	,	PUNCT
cana-509	2	18	india	india	PROPN
cana-509	2	19	.	.	PUNCT
cana-509	3	1	3professor	3professor	NUM
cana-509	3	2	of	of	ADP
cana-509	3	3	mathematics	mathematic	NOUN
cana-509	3	4	,	,	PUNCT
cana-509	3	5	veltech	veltech	PROPN
cana-509	3	6	rangarajan	rangarajan	PROPN
cana-509	3	7	dr	dr	PROPN
cana-509	3	8	sagunthala	sagunthala	PROPN
cana-509	3	9	r&d	r&d	PROPN
cana-509	3	10	institute	institute	PROPN
cana-509	3	11	of	of	ADP
cana-509	3	12	science	science	NOUN
cana-509	3	13	and	and	CCONJ
cana-509	3	14	technology	technology	NOUN
cana-509	3	15	,	,	PUNCT
cana-509	3	16	tamil	tamil	PROPN
cana-509	3	17	nadu	nadu	PROPN
cana-509	3	18	,	,	PUNCT
cana-509	3	19	india	india	PROPN
cana-509	3	20	.	.	PUNCT
cana-509	4	1	*	*	PUNCT
cana-509	4	2	corresponding	correspond	VERB
cana-509	4	3	author	author	NOUN
cana-509	4	4	:	:	PUNCT
cana-509	4	5	e	e	NOUN
cana-509	4	6	-	-	NOUN
cana-509	4	7	mail	mail	NOUN
cana-509	4	8	:	:	PUNCT
cana-509	4	9	gpalani32@yahoo.co.in	gpalani32@yahoo.co.in	NOUN
cana-509	4	10	article	article	NOUN
cana-509	4	11	history	history	NOUN
cana-509	4	12	:	:	PUNCT
cana-509	4	13	received	receive	VERB
cana-509	4	14	:	:	PUNCT
cana-509	4	15	15	15	NUM
cana-509	4	16	-	-	SYM
cana-509	4	17	01	01	NUM
cana-509	4	18	-	-	PUNCT
cana-509	4	19	2024	2024	NUM
cana-509	4	20	revised	revise	VERB
cana-509	4	21	:	:	PUNCT
cana-509	4	22	26	26	NUM
cana-509	4	23	-	-	SYM
cana-509	4	24	03	03	NUM
cana-509	4	25	-	-	PUNCT
cana-509	4	26	2024	2024	NUM
cana-509	4	27	accepted	accept	VERB
cana-509	4	28	:	:	PUNCT
cana-509	4	29	18	18	NUM
cana-509	4	30	-	-	PUNCT
cana-509	4	31	04	04	NUM
cana-509	4	32	-	-	PUNCT
cana-509	4	33	2024	2024	NUM
cana-509	4	34	abstract	abstract	NOUN
cana-509	4	35	:	:	PUNCT
cana-509	4	36	a	a	DET
cana-509	4	37	graph	graph	NOUN
cana-509	4	38	𝐺	𝐺	NOUN
cana-509	4	39	with	with	ADP
cana-509	4	40	𝑝	𝑝	PROPN
cana-509	4	41	vertices	vertex	NOUN
cana-509	4	42	and	and	CCONJ
cana-509	4	43	𝑞	𝑞	PRON
cana-509	4	44	edges	edge	NOUN
cana-509	4	45	is	be	AUX
cana-509	4	46	called	call	VERB
cana-509	4	47	a	a	DET
cana-509	4	48	harmonic	harmonic	ADJ
cana-509	4	49	mean(hm	mean(hm	NOUN
cana-509	4	50	)	)	PUNCT
cana-509	4	51	labeling	labeling	NOUN
cana-509	4	52	if	if	SCONJ
cana-509	4	53	it	it	PRON
cana-509	4	54	is	be	AUX
cana-509	4	55	possible	possible	ADJ
cana-509	4	56	to	to	PART
cana-509	4	57	label	label	VERB
cana-509	4	58	the	the	DET
cana-509	4	59	vertices	vertex	NOUN
cana-509	4	60	𝑥	𝑥	X
cana-509	4	61	∈	∈	NOUN
cana-509	4	62	𝑣	𝑣	X
cana-509	4	63	with	with	ADP
cana-509	4	64	distinct	distinct	ADJ
cana-509	4	65	labels	label	NOUN
cana-509	4	66	𝜌(𝑥	𝜌(𝑥	NOUN
cana-509	4	67	)	)	PUNCT
cana-509	4	68	from	from	ADP
cana-509	4	69	{	{	PUNCT
cana-509	4	70	1,2	1,2	NUM
cana-509	4	71	,	,	PUNCT
cana-509	4	72	⋯	⋯	PROPN
cana-509	4	73	,	,	PUNCT
cana-509	4	74	𝑞	𝑞	X
cana-509	4	75	+	+	NOUN
cana-509	4	76	1	1	NUM
cana-509	4	77	}	}	PUNCT
cana-509	4	78	in	in	ADP
cana-509	4	79	such	such	DET
cana-509	4	80	a	a	DET
cana-509	4	81	way	way	NOUN
cana-509	4	82	that	that	PRON
cana-509	4	83	each	each	DET
cana-509	4	84	edge	edge	VERB
cana-509	4	85	𝑒	𝑒	PROPN
cana-509	4	86	=	=	VERB
cana-509	4	87	𝑎𝑏	𝑎𝑏	PROPN
cana-509	4	88	is	be	AUX
cana-509	4	89	labeled	label	VERB
cana-509	4	90	with	with	ADP
cana-509	4	91	𝜌(𝑎𝑏	𝜌(𝑎𝑏	ADJ
cana-509	4	92	)	)	PUNCT
cana-509	4	93	=	=	SYM
cana-509	4	94	⌈	⌈	NOUN
cana-509	4	95	2𝜌(𝑎)𝜌(𝑏	2𝜌(𝑎)𝜌(𝑏	NUM
cana-509	4	96	)	)	PUNCT
cana-509	4	97	𝜌(𝑎)+𝜌(𝑏	𝜌(𝑎)+𝜌(𝑏	PROPN
cana-509	4	98	)	)	PUNCT
cana-509	4	99	⌉	⌉	PUNCT
cana-509	4	100	or	or	CCONJ
cana-509	4	101	⌊	⌊	PROPN
cana-509	4	102	2𝜌(𝑎)𝜌(𝑏	2𝜌(𝑎)𝜌(𝑏	NUM
cana-509	4	103	)	)	PUNCT
cana-509	4	104	𝜌(𝑎)+𝜌(𝑏	𝜌(𝑎)+𝜌(𝑏	PROPN
cana-509	4	105	)	)	PUNCT
cana-509	5	1	⌋	⌋	NOUN
cana-509	6	1	then	then	ADV
cana-509	6	2	the	the	DET
cana-509	6	3	edge	edge	NOUN
cana-509	6	4	labels	label	NOUN
cana-509	6	5	are	be	AUX
cana-509	6	6	distinct	distinct	ADJ
cana-509	6	7	.	.	PUNCT
cana-509	7	1	in	in	ADP
cana-509	7	2	this	this	DET
cana-509	7	3	case	case	NOUN
cana-509	7	4	𝜌	𝜌	X
cana-509	7	5	is	be	AUX
cana-509	7	6	called	call	VERB
cana-509	7	7	harmonic	harmonic	ADJ
cana-509	7	8	mean(hm	mean(hm	NOUN
cana-509	7	9	)	)	PUNCT
cana-509	7	10	labeling	labeling	NOUN
cana-509	7	11	of	of	ADP
cana-509	7	12	𝐺.	𝐺.	NOUN
cana-509	7	13	in	in	ADP
cana-509	7	14	this	this	DET
cana-509	7	15	paper	paper	NOUN
cana-509	7	16	we	we	PRON
cana-509	7	17	introduce	introduce	VERB
cana-509	7	18	new	new	ADJ
cana-509	7	19	graphs	graph	NOUN
cana-509	7	20	obtained	obtain	VERB
cana-509	7	21	from	from	ADP
cana-509	7	22	triangular	triangular	NOUN
cana-509	7	23	snake	snake	NOUN
cana-509	7	24	graph	graph	NOUN
cana-509	7	25	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	7	26	such	such	ADJ
cana-509	7	27	as	as	ADP
cana-509	7	28	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	7	29	∘	∘	PROPN
cana-509	7	30	𝐾1	𝐾1	PROPN
cana-509	7	31	,	,	PUNCT
cana-509	7	32	and	and	CCONJ
cana-509	7	33	prove	prove	VERB
cana-509	7	34	that	that	SCONJ
cana-509	7	35	they	they	PRON
cana-509	7	36	are	be	AUX
cana-509	7	37	harmonic	harmonic	ADJ
cana-509	7	38	mean	mean	ADJ
cana-509	7	39	labeling	labeling	NOUN
cana-509	7	40	graphs	graph	NOUN
cana-509	7	41	.	.	PUNCT
cana-509	8	1	keywords	keyword	NOUN
cana-509	8	2	:	:	PUNCT
cana-509	8	3	hm	hm	INTJ
cana-509	8	4	labeling	labeling	NOUN
cana-509	8	5	,	,	PUNCT
cana-509	8	6	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	8	7	∘	∘	VERB
cana-509	8	8	𝐾1	𝐾1	PROPN
cana-509	8	9	graphs	graph	NOUN
cana-509	8	10	.	.	PUNCT
cana-509	9	1	1	1	X
cana-509	9	2	.	.	X
cana-509	9	3	introduction	introduction	NOUN
cana-509	9	4	let	let	VERB
cana-509	9	5	𝐺	𝐺	PROPN
cana-509	9	6	=	=	SYM
cana-509	9	7	(	(	PUNCT
cana-509	9	8	𝑉	𝑉	PROPN
cana-509	9	9	,	,	PUNCT
cana-509	9	10	𝐸	𝐸	PROPN
cana-509	9	11	)	)	PUNCT
cana-509	9	12	be	be	VERB
cana-509	9	13	a	a	DET
cana-509	9	14	(	(	PUNCT
cana-509	9	15	𝑝	𝑝	PROPN
cana-509	9	16	,	,	PUNCT
cana-509	9	17	𝑞	𝑞	NOUN
cana-509	9	18	)	)	PUNCT
cana-509	9	19	graph	graph	NOUN
cana-509	9	20	with	with	ADP
cana-509	9	21	𝑝	𝑝	PROPN
cana-509	9	22	=	=	SYM
cana-509	9	23	|𝑉(𝐺)|	|𝑉(𝐺)|	NOUN
cana-509	9	24	vertices	vertex	NOUN
cana-509	9	25	and	and	CCONJ
cana-509	10	1	𝑞	𝑞	X
cana-509	10	2	=	=	ADJ
cana-509	10	3	|𝐸(𝐺)|	|𝐸(𝐺)|	NUM
cana-509	10	4	edges	edge	NOUN
cana-509	10	5	,	,	PUNCT
cana-509	10	6	where	where	SCONJ
cana-509	10	7	𝑉(𝐺	𝑉(𝐺	NOUN
cana-509	10	8	)	)	PUNCT
cana-509	10	9	and	and	CCONJ
cana-509	10	10	𝐸(𝐺	𝐸(𝐺	PROPN
cana-509	10	11	)	)	PUNCT
cana-509	10	12	respectively	respectively	ADV
cana-509	10	13	denote	denote	VERB
cana-509	10	14	the	the	DET
cana-509	10	15	vertex	vertex	NOUN
cana-509	10	16	set	set	NOUN
cana-509	10	17	and	and	CCONJ
cana-509	10	18	edge	edge	NOUN
cana-509	10	19	set	set	NOUN
cana-509	10	20	of	of	ADP
cana-509	10	21	the	the	DET
cana-509	10	22	graph	graph	NOUN
cana-509	10	23	g.	g.	NOUN
cana-509	10	24	in	in	ADP
cana-509	10	25	this	this	DET
cana-509	10	26	paper	paper	NOUN
cana-509	10	27	,	,	PUNCT
cana-509	10	28	we	we	PRON
cana-509	10	29	consider	consider	VERB
cana-509	10	30	the	the	DET
cana-509	10	31	graphs	graph	NOUN
cana-509	10	32	which	which	PRON
cana-509	10	33	are	be	AUX
cana-509	10	34	simple	simple	ADJ
cana-509	10	35	,	,	PUNCT
cana-509	10	36	finite	finite	ADJ
cana-509	10	37	and	and	CCONJ
cana-509	10	38	undirected	undirected	ADJ
cana-509	10	39	for	for	ADP
cana-509	10	40	graph	graph	NOUN
cana-509	10	41	theoretic	theoretic	ADJ
cana-509	10	42	terminology	terminology	NOUN
cana-509	10	43	and	and	CCONJ
cana-509	10	44	notations	notation	NOUN
cana-509	10	45	we	we	PRON
cana-509	10	46	refer	refer	VERB
cana-509	10	47	to	to	PART
cana-509	10	48	haray	haray	VERB
cana-509	10	49	[	[	X
cana-509	10	50	2	2	NUM
cana-509	10	51	]	]	PUNCT
cana-509	10	52	.	.	PUNCT
cana-509	11	1	the	the	DET
cana-509	11	2	concept	concept	NOUN
cana-509	11	3	of	of	ADP
cana-509	11	4	graph	graph	NOUN
cana-509	11	5	labeling	labeling	NOUN
cana-509	11	6	was	be	AUX
cana-509	11	7	introduced	introduce	VERB
cana-509	11	8	by	by	ADP
cana-509	11	9	rosa	rosa	PROPN
cana-509	11	10	in	in	ADP
cana-509	11	11	1967.a	1967.a	NUM
cana-509	11	12	detailed	detailed	ADJ
cana-509	11	13	survey	survey	NOUN
cana-509	11	14	of	of	ADP
cana-509	11	15	graph	graph	NOUN
cana-509	11	16	labeling	labeling	NOUN
cana-509	11	17	is	be	AUX
cana-509	11	18	available	available	ADJ
cana-509	11	19	in	in	ADP
cana-509	11	20	gallian	gallian	ADJ
cana-509	12	1	[	[	X
cana-509	12	2	1	1	NUM
cana-509	12	3	]	]	PUNCT
cana-509	12	4	.	.	PUNCT
cana-509	13	1	the	the	DET
cana-509	13	2	concept	concept	NOUN
cana-509	13	3	of	of	ADP
cana-509	13	4	mean	mean	ADJ
cana-509	13	5	labeling	labeling	NOUN
cana-509	13	6	of	of	ADP
cana-509	13	7	graph	graph	NOUN
cana-509	13	8	was	be	AUX
cana-509	13	9	introduced	introduce	VERB
cana-509	13	10	by	by	ADP
cana-509	13	11	s.	s.	PROPN
cana-509	13	12	somasundaram	somasundaram	PROPN
cana-509	13	13	,	,	PUNCT
cana-509	13	14	r.	r.	PROPN
cana-509	13	15	ponraj	ponraj	PROPN
cana-509	13	16	and	and	CCONJ
cana-509	13	17	s.s	s.s	PROPN
cana-509	13	18	.	.	PROPN
cana-509	13	19	sandhya	sandhya	PROPN
cana-509	14	1	[	[	X
cana-509	14	2	3	3	NUM
cana-509	14	3	]	]	PUNCT
cana-509	14	4	.	.	PUNCT
cana-509	15	1	some	some	PRON
cana-509	15	2	of	of	ADP
cana-509	15	3	the	the	DET
cana-509	15	4	harmonic	harmonic	ADJ
cana-509	15	5	mean	mean	NOUN
cana-509	15	6	graphs	graph	NOUN
cana-509	15	7	are	be	AUX
cana-509	15	8	investigated	investigate	VERB
cana-509	15	9	by	by	ADP
cana-509	15	10	s.	s.	PROPN
cana-509	15	11	meena	meena	PROPN
cana-509	15	12	and	and	CCONJ
cana-509	15	13	m.	m.	NOUN
cana-509	15	14	sivasakthi	sivasakthi	NOUN
cana-509	15	15	in	in	ADP
cana-509	15	16	[	[	X
cana-509	15	17	4	4	NUM
cana-509	15	18	]	]	PUNCT
cana-509	15	19	.	.	PUNCT
cana-509	16	1	the	the	DET
cana-509	16	2	concept	concept	NOUN
cana-509	16	3	of	of	ADP
cana-509	16	4	harmonic	harmonic	ADJ
cana-509	16	5	mean	mean	NOUN
cana-509	16	6	labeling	labeling	NOUN
cana-509	16	7	of	of	ADP
cana-509	16	8	graph	graph	NOUN
cana-509	16	9	was	be	AUX
cana-509	16	10	introduced	introduce	VERB
cana-509	16	11	by	by	ADP
cana-509	16	12	s.	s.	PROPN
cana-509	16	13	somasundaram	somasundaram	PROPN
cana-509	16	14	,	,	PUNCT
cana-509	16	15	r.	r.	PROPN
cana-509	16	16	ponraj	ponraj	PROPN
cana-509	16	17	and	and	CCONJ
cana-509	16	18	s.s	s.s	PROPN
cana-509	16	19	.	.	PROPN
cana-509	16	20	sandhya	sandhya	PROPN
cana-509	17	1	[	[	X
cana-509	17	2	5,6	5,6	NUM
cana-509	17	3	]	]	PUNCT
cana-509	17	4	and	and	CCONJ
cana-509	17	5	they	they	PRON
cana-509	17	6	investigated	investigate	VERB
cana-509	17	7	the	the	DET
cana-509	17	8	existence	existence	NOUN
cana-509	17	9	of	of	ADP
cana-509	17	10	harmonic	harmonic	ADJ
cana-509	17	11	mean	mean	NOUN
cana-509	17	12	labeling	labeling	NOUN
cana-509	17	13	of	of	ADP
cana-509	17	14	several	several	ADJ
cana-509	17	15	family	family	NOUN
cana-509	17	16	of	of	ADP
cana-509	17	17	graphs	graph	NOUN
cana-509	17	18	such	such	ADJ
cana-509	17	19	as	as	ADP
cana-509	17	20	this	this	DET
cana-509	17	21	concept	concept	NOUN
cana-509	17	22	was	be	AUX
cana-509	17	23	then	then	ADV
cana-509	17	24	studied	study	VERB
cana-509	17	25	by	by	ADP
cana-509	17	26	several	several	ADJ
cana-509	17	27	authors	author	NOUN
cana-509	17	28	and	and	CCONJ
cana-509	17	29	studied	study	VERB
cana-509	17	30	their	their	PRON
cana-509	17	31	behavior	behavior	NOUN
cana-509	17	32	in	in	ADP
cana-509	17	33	[	[	X
cana-509	17	34	7	7	NUM
cana-509	17	35	]	]	PUNCT
cana-509	17	36	,	,	PUNCT
cana-509	17	37	[	[	X
cana-509	17	38	8	8	NUM
cana-509	17	39	]	]	PUNCT
cana-509	17	40	,	,	PUNCT
cana-509	18	1	[	[	X
cana-509	18	2	9	9	NUM
cana-509	18	3	]	]	PUNCT
cana-509	18	4	,	,	PUNCT
cana-509	18	5	and	and	CCONJ
cana-509	18	6	[	[	X
cana-509	18	7	10	10	NUM
cana-509	18	8	]	]	PUNCT
cana-509	18	9	.	.	PUNCT
cana-509	19	1	2	2	X
cana-509	19	2	.	.	X
cana-509	19	3	preliminaries	preliminary	NOUN
cana-509	19	4	definition	definition	NOUN
cana-509	19	5	2.1	2.1	NUM
cana-509	19	6	.	.	PUNCT
cana-509	20	1	mean	mean	VERB
cana-509	20	2	labeling	label	VERB
cana-509	20	3	a	a	DET
cana-509	20	4	function	function	NOUN
cana-509	20	5	𝜌	𝜌	X
cana-509	20	6	is	be	AUX
cana-509	20	7	called	call	VERB
cana-509	20	8	mean	mean	ADJ
cana-509	20	9	labeling	labeling	NOUN
cana-509	20	10	for	for	ADP
cana-509	20	11	a	a	DET
cana-509	20	12	graph	graph	NOUN
cana-509	20	13	𝐺	𝐺	NOUN
cana-509	20	14	=	=	SYM
cana-509	20	15	(	(	PUNCT
cana-509	20	16	𝑉	𝑉	PROPN
cana-509	20	17	,	,	PUNCT
cana-509	20	18	𝐸	𝐸	PROPN
cana-509	20	19	)	)	PUNCT
cana-509	20	20	if	if	SCONJ
cana-509	20	21	𝜌	𝜌	ADP
cana-509	20	22	:	:	PUNCT
cana-509	20	23	𝑉	𝑉	PROPN
cana-509	20	24	→	→	SYM
cana-509	20	25	{	{	PUNCT
cana-509	20	26	0,1,2,3	0,1,2,3	NUM
cana-509	20	27	,	,	PUNCT
cana-509	20	28	⋯	⋯	PROPN
cana-509	20	29	,	,	PUNCT
cana-509	20	30	𝑞	𝑞	X
cana-509	20	31	}	}	PUNCT
cana-509	20	32	is	be	AUX
cana-509	20	33	injective	injective	ADJ
cana-509	20	34	and	and	CCONJ
cana-509	20	35	the	the	DET
cana-509	20	36	induce	induce	ADJ
cana-509	20	37	function	function	NOUN
cana-509	20	38	𝜌∗	𝜌∗	NOUN
cana-509	20	39	:	:	PUNCT
cana-509	20	40	𝐸	𝐸	PROPN
cana-509	20	41	→	→	SYM
cana-509	20	42	{	{	PUNCT
cana-509	20	43	1,2,3	1,2,3	NUM
cana-509	20	44	,	,	PUNCT
cana-509	20	45	⋯	⋯	PROPN
cana-509	20	46	,	,	PUNCT
cana-509	20	47	𝑞	𝑞	X
cana-509	20	48	}	}	PUNCT
cana-509	20	49	defined	define	VERB
cana-509	20	50	as	as	ADP
cana-509	20	51	𝜌	𝜌	ADP
cana-509	20	52	∗=	∗=	NOUN
cana-509	20	53	⌈	⌈	PROPN
cana-509	20	54	𝜌(𝑎)+𝜌(𝑏	𝜌(𝑎)+𝜌(𝑏	PROPN
cana-509	20	55	)	)	PUNCT
cana-509	20	56	2	2	NUM
cana-509	20	57	⌉	⌉	PUNCT
cana-509	20	58	or	or	CCONJ
cana-509	20	59	⌊	⌊	ADP
cana-509	20	60	𝜌(𝑎)+𝜌(𝑏	𝜌(𝑎)+𝜌(𝑏	PROPN
cana-509	20	61	)	)	PUNCT
cana-509	20	62	2	2	NUM
cana-509	20	63	⌋	⌋	NOUN
cana-509	20	64	is	be	AUX
cana-509	20	65	bijective	bijective	ADJ
cana-509	20	66	for	for	ADP
cana-509	20	67	every	every	DET
cana-509	20	68	edge	edge	NOUN
cana-509	20	69	.	.	PUNCT
cana-509	21	1	a	a	DET
cana-509	21	2	graph	graph	NOUN
cana-509	21	3	𝐺	𝐺	NOUN
cana-509	21	4	is	be	AUX
cana-509	21	5	called	call	VERB
cana-509	21	6	mean	mean	ADJ
cana-509	21	7	labeling	labeling	NOUN
cana-509	21	8	.	.	PUNCT
cana-509	22	1	communications	communication	NOUN
cana-509	22	2	on	on	ADP
cana-509	22	3	applied	apply	VERB
cana-509	22	4	nonlinear	nonlinear	ADJ
cana-509	22	5	analysis	analysis	NOUN
cana-509	22	6	issn	issn	NOUN
cana-509	22	7	:	:	PUNCT
cana-509	22	8	1074	1074	NUM
cana-509	22	9	-	-	PUNCT
cana-509	22	10	133x	133x	NUM
cana-509	22	11	vol	vol	NOUN
cana-509	22	12	31	31	NUM
cana-509	22	13	no	no	NOUN
cana-509	22	14	.	.	NOUN
cana-509	22	15	2	2	NUM
cana-509	22	16	(	(	PUNCT
cana-509	22	17	2024	2024	NUM
cana-509	22	18	)	)	PUNCT
cana-509	22	19	12	12	NUM
cana-509	22	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	22	21	definition	definition	NOUN
cana-509	22	22	2.2	2.2	NUM
cana-509	22	23	.	.	PUNCT
cana-509	23	1	harmonic	harmonic	ADJ
cana-509	23	2	mean	mean	NOUN
cana-509	23	3	(	(	PUNCT
cana-509	23	4	hm	hm	INTJ
cana-509	23	5	)	)	PUNCT
cana-509	23	6	labeling	label	VERB
cana-509	23	7	a	a	DET
cana-509	23	8	graph	graph	NOUN
cana-509	23	9	𝐺	𝐺	NOUN
cana-509	23	10	=	=	SYM
cana-509	23	11	(	(	PUNCT
cana-509	23	12	𝑉	𝑉	PROPN
cana-509	23	13	,	,	PUNCT
cana-509	23	14	𝐸	𝐸	PROPN
cana-509	23	15	)	)	PUNCT
cana-509	23	16	with	with	ADP
cana-509	23	17	𝑝	𝑝	NOUN
cana-509	23	18	vertices	vertex	NOUN
cana-509	23	19	and	and	CCONJ
cana-509	23	20	𝑞	𝑞	DET
cana-509	23	21	edges	edge	NOUN
cana-509	23	22	is	be	AUX
cana-509	23	23	called	call	VERB
cana-509	23	24	a	a	DET
cana-509	23	25	harmonic	harmonic	ADJ
cana-509	23	26	mean	mean	NOUN
cana-509	23	27	(	(	PUNCT
cana-509	23	28	𝐻𝑀	𝐻𝑀	PROPN
cana-509	23	29	)	)	PUNCT
cana-509	23	30	graph	graph	NOUN
cana-509	23	31	if	if	SCONJ
cana-509	23	32	it	it	PRON
cana-509	23	33	is	be	AUX
cana-509	23	34	possible	possible	ADJ
cana-509	23	35	to	to	PART
cana-509	23	36	label	label	VERB
cana-509	23	37	the	the	DET
cana-509	23	38	vertices	vertex	NOUN
cana-509	23	39	𝑥	𝑥	X
cana-509	23	40	∈	∈	NOUN
cana-509	23	41	𝑣	𝑣	X
cana-509	23	42	with	with	ADP
cana-509	23	43	distinct	distinct	ADJ
cana-509	23	44	labels	label	NOUN
cana-509	23	45	𝜌(𝑥	𝜌(𝑥	NOUN
cana-509	23	46	)	)	PUNCT
cana-509	23	47	from	from	ADP
cana-509	23	48	{	{	PUNCT
cana-509	23	49	1,2	1,2	NUM
cana-509	23	50	,	,	PUNCT
cana-509	23	51	⋯	⋯	PROPN
cana-509	23	52	,	,	PUNCT
cana-509	23	53	𝑞	𝑞	X
cana-509	23	54	+	+	NOUN
cana-509	23	55	1	1	NUM
cana-509	23	56	}	}	PUNCT
cana-509	23	57	in	in	ADP
cana-509	23	58	such	such	ADJ
cana-509	23	59	𝑎	𝑎	DET
cana-509	23	60	way	way	NOUN
cana-509	23	61	that	that	SCONJ
cana-509	23	62	when	when	SCONJ
cana-509	23	63	each	each	DET
cana-509	23	64	edge	edge	VERB
cana-509	23	65	𝑒	𝑒	PROPN
cana-509	23	66	=	=	VERB
cana-509	23	67	𝑎𝑏	𝑎𝑏	PROPN
cana-509	23	68	is	be	AUX
cana-509	23	69	labeled	label	VERB
cana-509	23	70	with	with	ADP
cana-509	23	71	𝜌(𝑎𝑏	𝜌(𝑎𝑏	ADJ
cana-509	23	72	)	)	PUNCT
cana-509	23	73	=	=	SYM
cana-509	23	74	⌈	⌈	NOUN
cana-509	23	75	2𝜌(𝑎)𝜌(𝑏	2𝜌(𝑎)𝜌(𝑏	NUM
cana-509	23	76	)	)	PUNCT
cana-509	23	77	𝜌(𝑎)+𝜌(𝑏	𝜌(𝑎)+𝜌(𝑏	PROPN
cana-509	23	78	)	)	PUNCT
cana-509	23	79	⌉	⌉	PUNCT
cana-509	23	80	or	or	CCONJ
cana-509	23	81	⌊	⌊	PROPN
cana-509	23	82	2𝜌(𝑎)𝜌(𝑏	2𝜌(𝑎)𝜌(𝑏	NUM
cana-509	23	83	)	)	PUNCT
cana-509	23	84	𝜌(𝑎)+𝜌(𝑏	𝜌(𝑎)+𝜌(𝑏	PROPN
cana-509	23	85	)	)	PUNCT
cana-509	24	1	⌋	⌋	NOUN
cana-509	24	2	then	then	ADV
cana-509	24	3	the	the	DET
cana-509	24	4	edge	edge	NOUN
cana-509	24	5	labels	label	NOUN
cana-509	24	6	are	be	AUX
cana-509	24	7	distinct	distinct	ADJ
cana-509	24	8	.	.	PUNCT
cana-509	25	1	definition	definition	NOUN
cana-509	25	2	2.3	2.3	NUM
cana-509	25	3	.	.	PUNCT
cana-509	26	1	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	26	2	∘	∘	PROPN
cana-509	26	3	𝐾1	𝐾1	PROPN
cana-509	26	4	graph	graph	NOUN
cana-509	26	5	the	the	DET
cana-509	26	6	triangular	triangular	NOUN
cana-509	26	7	snake	snake	NOUN
cana-509	26	8	is	be	AUX
cana-509	26	9	obtained	obtain	VERB
cana-509	26	10	from	from	ADP
cana-509	26	11	the	the	DET
cana-509	26	12	path	path	NOUN
cana-509	26	13	𝑃𝑛	𝑃𝑛	PROPN
cana-509	26	14	by	by	ADP
cana-509	26	15	replacing	replace	VERB
cana-509	26	16	each	each	DET
cana-509	26	17	edge	edge	NOUN
cana-509	26	18	of	of	ADP
cana-509	26	19	the	the	DET
cana-509	26	20	path	path	NOUN
cana-509	26	21	by	by	ADP
cana-509	26	22	a	a	DET
cana-509	26	23	cycle	cycle	NOUN
cana-509	26	24	𝐶3	𝐶3	NOUN
cana-509	26	25	.	.	PUNCT
cana-509	27	1	each	each	DET
cana-509	27	2	vertex	vertex	NOUN
cana-509	27	3	of	of	ADP
cana-509	27	4	triangular	triangular	NOUN
cana-509	27	5	snake	snake	NOUN
cana-509	27	6	graph	graph	NOUN
cana-509	27	7	𝑇𝑛	𝑇𝑛	PROPN
cana-509	27	8	is	be	AUX
cana-509	27	9	attached	attach	VERB
cana-509	27	10	with	with	ADP
cana-509	27	11	an	an	DET
cana-509	27	12	edge	edge	NOUN
cana-509	27	13	and	and	CCONJ
cana-509	27	14	the	the	DET
cana-509	27	15	graph	graph	NOUN
cana-509	27	16	obtained	obtain	VERB
cana-509	27	17	is	be	AUX
cana-509	27	18	called	call	VERB
cana-509	27	19	as	as	ADP
cana-509	27	20	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	27	21	∘	∘	PROPN
cana-509	27	22	𝐾1	𝐾1	PROPN
cana-509	27	23	3	3	X
cana-509	27	24	.	.	PUNCT
cana-509	27	25	main	main	ADJ
cana-509	27	26	results	result	NOUN
cana-509	27	27	theorem	theorem	VERB
cana-509	27	28	3.1	3.1	NUM
cana-509	27	29	.	.	PUNCT
cana-509	28	1	for	for	ADP
cana-509	28	2	every	every	DET
cana-509	28	3	𝑛	𝑛	DET
cana-509	28	4	≥	≥	NOUN
cana-509	28	5	1	1	NUM
cana-509	28	6	,	,	PUNCT
cana-509	28	7	there	there	PRON
cana-509	28	8	exists	exist	VERB
cana-509	28	9	a	a	DET
cana-509	28	10	triangular	triangular	NOUN
cana-509	28	11	snake	snake	NOUN
cana-509	28	12	graph	graph	NOUN
cana-509	28	13	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	28	14	∘	∘	PROPN
cana-509	28	15	𝐾1	𝐾1	PROPN
cana-509	28	16	,	,	PUNCT
cana-509	28	17	which	which	PRON
cana-509	28	18	admits	admit	VERB
cana-509	28	19	hm	hm	DET
cana-509	28	20	labeling	labeling	NOUN
cana-509	28	21	.	.	PUNCT
cana-509	29	1	proof	proof	NOUN
cana-509	29	2	.	.	PUNCT
cana-509	30	1	consider	consider	VERB
cana-509	30	2	the	the	DET
cana-509	30	3	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	30	4	∘	∘	PROPN
cana-509	30	5	𝐾1	𝐾1	PROPN
cana-509	30	6	graph	graph	NOUN
cana-509	30	7	with	with	ADP
cana-509	30	8	vertex	vertex	NOUN
cana-509	30	9	set	set	VERB
cana-509	30	10	𝑉	𝑉	PROPN
cana-509	30	11	=	=	PROPN
cana-509	30	12	{	{	PUNCT
cana-509	30	13	𝑎𝑖	𝑎𝑖	PROPN
cana-509	30	14	,	,	PUNCT
cana-509	30	15	𝑏𝑖	𝑏𝑖	NOUN
cana-509	30	16	,	,	PUNCT
cana-509	30	17	𝑐𝑖	𝑐𝑖	NOUN
cana-509	30	18	,	,	PUNCT
cana-509	30	19	𝑑𝑖	𝑑𝑖	PART
cana-509	30	20	;	;	PUNCT
cana-509	30	21	1	1	NUM
cana-509	30	22	≤	≤	NUM
cana-509	30	23	𝑖	𝑖	SYM
cana-509	30	24	≤	≤	NUM
cana-509	30	25	𝑛	𝑛	PRON
cana-509	30	26	}	}	PUNCT
cana-509	30	27	and	and	CCONJ
cana-509	30	28	edge	edge	NOUN
cana-509	30	29	set	set	VERB
cana-509	30	30	𝐸	𝐸	NOUN
cana-509	30	31	=	=	SYM
cana-509	30	32	{	{	PUNCT
cana-509	30	33	(	(	PUNCT
cana-509	30	34	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	ADJ
cana-509	30	35	)	)	PUNCT
cana-509	30	36	∪	∪	NOUN
cana-509	30	37	(	(	PUNCT
cana-509	30	38	𝑐𝑖𝑏𝑖	𝑐𝑖𝑏𝑖	ADJ
cana-509	30	39	)	)	PUNCT
cana-509	30	40	∪	∪	NOUN
cana-509	30	41	(	(	PUNCT
cana-509	30	42	𝑐𝑖𝑐𝑖+1	𝑐𝑖𝑐𝑖+1	PROPN
cana-509	30	43	)	)	PUNCT
cana-509	30	44	∪	∪	NOUN
cana-509	30	45	(	(	PUNCT
cana-509	30	46	𝑐𝑖𝑑𝑖	𝑐𝑖𝑑𝑖	NOUN
cana-509	30	47	)	)	PUNCT
cana-509	30	48	;	;	PUNCT
cana-509	30	49	1	1	NUM
cana-509	30	50	≤	≤	NUM
cana-509	30	51	𝑖	𝑖	SYM
cana-509	30	52	≤	≤	NUM
cana-509	30	53	𝑛	𝑛	X
cana-509	30	54	}	}	PUNCT
cana-509	30	55	now	now	ADV
cana-509	30	56	let	let	VERB
cana-509	30	57	us	we	PRON
cana-509	30	58	define	define	VERB
cana-509	30	59	a	a	DET
cana-509	30	60	function	function	NOUN
cana-509	30	61	𝜌	𝜌	ADP
cana-509	30	62	:	:	PUNCT
cana-509	30	63	𝑣	𝑣	X
cana-509	30	64	→	→	PUNCT
cana-509	30	65	{	{	PUNCT
cana-509	30	66	1,2,3	1,2,3	NUM
cana-509	30	67	,	,	PUNCT
cana-509	30	68	⋯	⋯	NOUN
cana-509	30	69	𝑞	𝑞	X
cana-509	30	70	+	+	PROPN
cana-509	30	71	1	1	NUM
cana-509	30	72	}	}	PUNCT
cana-509	30	73	let	let	VERB
cana-509	30	74	us	we	PRON
cana-509	30	75	label	label	VERB
cana-509	30	76	the	the	DET
cana-509	30	77	vertices	vertex	NOUN
cana-509	30	78	as	as	SCONJ
cana-509	30	79	follows	follow	VERB
cana-509	30	80	.	.	PUNCT
cana-509	31	1	𝜌(𝑎1	𝜌(𝑎1	X
cana-509	31	2	)	)	PUNCT
cana-509	31	3	=	=	PUNCT
cana-509	31	4	6	6	NUM
cana-509	31	5	;	;	PUNCT
cana-509	31	6	𝜌(𝑎𝑖+1	𝜌(𝑎𝑖+1	X
cana-509	31	7	)	)	PUNCT
cana-509	31	8	=	=	SYM
cana-509	32	1	5𝑖	5𝑖	NOUN
cana-509	32	2	+	+	CCONJ
cana-509	32	3	3	3	NUM
cana-509	32	4	;	;	PUNCT
cana-509	32	5	1	1	NUM
cana-509	32	6	≤	≤	NUM
cana-509	32	7	𝑖	𝑖	SYM
cana-509	32	8	≤	≤	NUM
cana-509	32	9	𝑛	𝑛	PRON
cana-509	32	10	−	−	NUM
cana-509	32	11	1	1	NUM
cana-509	32	12	𝜌(𝑏𝑖	𝜌(𝑏𝑖	NOUN
cana-509	32	13	)	)	PUNCT
cana-509	33	1	=	=	PUNCT
cana-509	34	1	5𝑖	5𝑖	NOUN
cana-509	34	2	−	−	NOUN
cana-509	34	3	1	1	NUM
cana-509	34	4	;	;	PUNCT
cana-509	34	5	𝜌(𝑐𝑖	𝜌(𝑐𝑖	NOUN
cana-509	34	6	)	)	PUNCT
cana-509	35	1	=	=	NOUN
cana-509	35	2	5𝑖	5𝑖	NOUN
cana-509	35	3	−	−	NOUN
cana-509	35	4	3	3	NUM
cana-509	35	5	;	;	PUNCT
cana-509	35	6	1	1	NUM
cana-509	35	7	≤	≤	NUM
cana-509	35	8	𝑖	𝑖	SYM
cana-509	35	9	≤	≤	NUM
cana-509	35	10	𝑛	𝑛	PRON
cana-509	35	11	𝜌(𝑑1	𝜌(𝑑1	NOUN
cana-509	35	12	)	)	PUNCT
cana-509	35	13	=	=	SYM
cana-509	35	14	1	1	NUM
cana-509	35	15	;	;	PUNCT
cana-509	35	16	𝜌(𝑑𝑖+1	𝜌(𝑑𝑖+1	NOUN
cana-509	35	17	)	)	PUNCT
cana-509	35	18	=	=	SYM
cana-509	35	19	5𝑖	5𝑖	NOUN
cana-509	35	20	;	;	PUNCT
cana-509	35	21	1	1	NUM
cana-509	35	22	≤	≤	NUM
cana-509	35	23	𝑖	𝑖	SYM
cana-509	35	24	≤	≤	NUM
cana-509	35	25	𝑛	𝑛	PRON
cana-509	35	26	−	−	PROPN
cana-509	35	27	1	1	NUM
cana-509	35	28	the	the	DET
cana-509	35	29	induced	induced	ADJ
cana-509	35	30	edge	edge	NOUN
cana-509	35	31	labeling	labeling	NOUN
cana-509	35	32	is	be	AUX
cana-509	35	33	as	as	SCONJ
cana-509	35	34	follows	follow	VERB
cana-509	35	35	𝜌∗(𝑎1𝑏1	𝜌∗(𝑎1𝑏1	PRON
cana-509	35	36	)	)	PUNCT
cana-509	35	37	=	=	SYM
cana-509	35	38	4	4	NUM
cana-509	35	39	;	;	PUNCT
cana-509	35	40	𝜌∗(𝑎𝑖+1𝑏𝑖+1	𝜌∗(𝑎𝑖+1𝑏𝑖+1	PROPN
cana-509	35	41	)	)	PUNCT
cana-509	36	1	=	=	NOUN
cana-509	37	1	5𝑖	5𝑖	NOUN
cana-509	37	2	+	+	CCONJ
cana-509	37	3	3	3	NUM
cana-509	37	4	;	;	PUNCT
cana-509	37	5	1	1	NUM
cana-509	37	6	≤	≤	NUM
cana-509	37	7	𝑖	𝑖	SYM
cana-509	37	8	≤	≤	NUM
cana-509	37	9	𝑛	𝑛	PRON
cana-509	37	10	−	−	PROPN
cana-509	37	11	1	1	NUM
cana-509	37	12	𝜌∗(𝑏𝑖𝑐𝑖	𝜌∗(𝑏𝑖𝑐𝑖	PROPN
cana-509	37	13	)	)	PUNCT
cana-509	37	14	=	=	NOUN
cana-509	38	1	5𝑖	5𝑖	NOUN
cana-509	38	2	−	−	NOUN
cana-509	38	3	3	3	NUM
cana-509	38	4	;	;	PUNCT
cana-509	38	5	𝜌∗(𝑐𝑖+1𝑏𝑖	𝜌∗(𝑐𝑖+1𝑏𝑖	PROPN
cana-509	38	6	)	)	PUNCT
cana-509	39	1	=	=	SYM
cana-509	39	2	5𝑖	5𝑖	NOUN
cana-509	39	3	;	;	PUNCT
cana-509	39	4	1	1	NUM
cana-509	39	5	≤	≤	NUM
cana-509	39	6	𝑖	𝑖	SYM
cana-509	39	7	≤	≤	NOUN
cana-509	39	8	𝑛	𝑛	PRON
cana-509	39	9	𝜌∗(𝑐1𝑐2	𝜌∗(𝑐1𝑐2	NUM
cana-509	39	10	)	)	PUNCT
cana-509	39	11	=	=	SYM
cana-509	39	12	3	3	NUM
cana-509	39	13	;	;	PUNCT
cana-509	39	14	𝜌∗(𝑐𝑖+1𝑐𝑖+2	𝜌∗(𝑐𝑖+1𝑐𝑖+2	PROPN
cana-509	39	15	)	)	PUNCT
cana-509	39	16	=	=	SYM
cana-509	40	1	5𝑖	5𝑖	NOUN
cana-509	40	2	+	+	CCONJ
cana-509	40	3	4	4	NUM
cana-509	40	4	;	;	PUNCT
cana-509	40	5	1	1	NUM
cana-509	40	6	≤	≤	NUM
cana-509	40	7	𝑖	𝑖	SYM
cana-509	40	8	≤	≤	NUM
cana-509	40	9	𝑛	𝑛	PRON
cana-509	40	10	𝜌∗(𝑑𝑖𝑐𝑖	𝜌∗(𝑑𝑖𝑐𝑖	PROPN
cana-509	40	11	)	)	PUNCT
cana-509	40	12	=	=	NOUN
cana-509	40	13	5𝑖	5𝑖	NOUN
cana-509	40	14	−	−	NOUN
cana-509	40	15	4	4	NUM
cana-509	40	16	;	;	PUNCT
cana-509	40	17	1	1	NUM
cana-509	40	18	≤	≤	NUM
cana-509	40	19	𝑖	𝑖	SYM
cana-509	40	20	≤	≤	NUM
cana-509	40	21	𝑛	𝑛	DET
cana-509	40	22	communications	communication	NOUN
cana-509	40	23	on	on	ADP
cana-509	40	24	applied	apply	VERB
cana-509	40	25	nonlinear	nonlinear	ADJ
cana-509	40	26	analysis	analysis	NOUN
cana-509	40	27	issn	issn	NOUN
cana-509	40	28	:	:	PUNCT
cana-509	40	29	1074	1074	NUM
cana-509	40	30	-	-	PUNCT
cana-509	40	31	133x	133x	NUM
cana-509	40	32	vol	vol	NOUN
cana-509	40	33	31	31	NUM
cana-509	40	34	no	no	NOUN
cana-509	40	35	.	.	NOUN
cana-509	40	36	2	2	NUM
cana-509	40	37	(	(	PUNCT
cana-509	40	38	2024	2024	NUM
cana-509	40	39	)	)	PUNCT
cana-509	40	40	13	13	NUM
cana-509	40	41	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	40	42	thus	thus	ADV
cana-509	40	43	,	,	PUNCT
cana-509	40	44	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	40	45	∘	∘	PROPN
cana-509	40	46	𝐾1	𝐾1	PROPN
cana-509	40	47	is	be	AUX
cana-509	40	48	a	a	DET
cana-509	40	49	hm	hm	INTJ
cana-509	40	50	labeling	labeling	NOUN
cana-509	40	51	.	.	PUNCT
cana-509	41	1	fig	fig	NOUN
cana-509	41	2	1	1	NUM
cana-509	41	3	.	.	PUNCT
cana-509	42	1	𝑇𝑆4	𝑇𝑆4	PROPN
cana-509	42	2	∘	∘	PROPN
cana-509	42	3	𝐾1	𝐾1	PROPN
cana-509	42	4	theorem	theorem	ADJ
cana-509	42	5	3.2	3.2	NUM
cana-509	42	6	.	.	PUNCT
cana-509	43	1	for	for	ADP
cana-509	43	2	every	every	DET
cana-509	43	3	𝑛	𝑛	DET
cana-509	43	4	≥	≥	NOUN
cana-509	43	5	1	1	NUM
cana-509	43	6	,	,	PUNCT
cana-509	43	7	there	there	PRON
cana-509	43	8	exists	exist	VERB
cana-509	43	9	a	a	DET
cana-509	43	10	alternate	alternate	ADJ
cana-509	43	11	triangular	triangular	NOUN
cana-509	43	12	snake	snake	NOUN
cana-509	43	13	graph	graph	NOUN
cana-509	43	14	𝐴(𝑇𝑆)𝑛	𝐴(𝑇𝑆)𝑛	PROPN
cana-509	43	15	∘	∘	PROPN
cana-509	43	16	𝐾1	𝐾1	PROPN
cana-509	43	17	,	,	PUNCT
cana-509	43	18	which	which	PRON
cana-509	43	19	admits	admit	VERB
cana-509	43	20	𝐻𝑀	𝐻𝑀	PROPN
cana-509	43	21	labeling	labeling	NOUN
cana-509	43	22	.	.	PUNCT
cana-509	44	1	proof	proof	NOUN
cana-509	44	2	.	.	PUNCT
cana-509	45	1	consider	consider	VERB
cana-509	45	2	the	the	DET
cana-509	45	3	𝐴(𝑇𝑆)𝑛	𝐴(𝑇𝑆)𝑛	NOUN
cana-509	45	4	∘	∘	VERB
cana-509	45	5	𝐾1	𝐾1	PROPN
cana-509	45	6	graph	graph	NOUN
cana-509	45	7	with	with	ADP
cana-509	45	8	vertex	vertex	NOUN
cana-509	45	9	set	set	VERB
cana-509	45	10	𝑉	𝑉	PROPN
cana-509	45	11	=	=	PROPN
cana-509	45	12	{	{	PUNCT
cana-509	45	13	𝑎𝑖	𝑎𝑖	PROPN
cana-509	45	14	,	,	PUNCT
cana-509	45	15	𝑏𝑖	𝑏𝑖	NOUN
cana-509	45	16	,	,	PUNCT
cana-509	45	17	𝑐𝑖	𝑐𝑖	NOUN
cana-509	45	18	,	,	PUNCT
cana-509	45	19	𝑑𝑖	𝑑𝑖	PART
cana-509	45	20	;	;	PUNCT
cana-509	45	21	1	1	NUM
cana-509	45	22	≤	≤	NUM
cana-509	45	23	𝑖	𝑖	SYM
cana-509	45	24	≤	≤	NUM
cana-509	45	25	𝑛	𝑛	PRON
cana-509	45	26	}	}	PUNCT
cana-509	45	27	and	and	CCONJ
cana-509	45	28	edge	edge	NOUN
cana-509	45	29	set	set	VERB
cana-509	45	30	𝐸	𝐸	NOUN
cana-509	45	31	=	=	SYM
cana-509	45	32	{	{	PUNCT
cana-509	45	33	(	(	PUNCT
cana-509	45	34	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	ADJ
cana-509	45	35	)	)	PUNCT
cana-509	45	36	∪	∪	NOUN
cana-509	45	37	(	(	PUNCT
cana-509	45	38	𝑐𝑖𝑏𝑖	𝑐𝑖𝑏𝑖	ADJ
cana-509	45	39	)	)	PUNCT
cana-509	45	40	∪	∪	NOUN
cana-509	45	41	(	(	PUNCT
cana-509	45	42	𝑐𝑖𝑐𝑖+1	𝑐𝑖𝑐𝑖+1	PROPN
cana-509	45	43	)	)	PUNCT
cana-509	45	44	∪	∪	NOUN
cana-509	45	45	(	(	PUNCT
cana-509	45	46	𝑐𝑖𝑑𝑖	𝑐𝑖𝑑𝑖	NOUN
cana-509	45	47	)	)	PUNCT
cana-509	45	48	;	;	PUNCT
cana-509	45	49	1	1	NUM
cana-509	45	50	≤	≤	NUM
cana-509	45	51	𝑖	𝑖	SYM
cana-509	45	52	≤	≤	NUM
cana-509	45	53	𝑛	𝑛	X
cana-509	45	54	}	}	PUNCT
cana-509	45	55	now	now	ADV
cana-509	45	56	let	let	VERB
cana-509	45	57	us	we	PRON
cana-509	45	58	define	define	VERB
cana-509	45	59	a	a	DET
cana-509	45	60	function	function	NOUN
cana-509	45	61	𝜌	𝜌	ADP
cana-509	45	62	:	:	PUNCT
cana-509	45	63	𝑣	𝑣	X
cana-509	45	64	→	→	PUNCT
cana-509	45	65	{	{	PUNCT
cana-509	45	66	1,2,3	1,2,3	NUM
cana-509	45	67	,	,	PUNCT
cana-509	45	68	⋯	⋯	NOUN
cana-509	45	69	𝑞	𝑞	X
cana-509	45	70	+	+	PROPN
cana-509	45	71	1	1	NUM
cana-509	45	72	}	}	PUNCT
cana-509	45	73	let	let	VERB
cana-509	45	74	us	we	PRON
cana-509	45	75	label	label	VERB
cana-509	45	76	the	the	DET
cana-509	45	77	vertices	vertex	NOUN
cana-509	45	78	as	as	SCONJ
cana-509	45	79	follows	follow	VERB
cana-509	45	80	𝜌(𝑎𝑖	𝜌(𝑎𝑖	NOUN
cana-509	45	81	)	)	PUNCT
cana-509	45	82	=	=	SYM
cana-509	46	1	7𝑖	7𝑖	ADJ
cana-509	46	2	−	−	NOUN
cana-509	46	3	6	6	NUM
cana-509	46	4	;	;	PUNCT
cana-509	46	5	𝜌(𝑏𝑖	𝜌(𝑏𝑖	NOUN
cana-509	46	6	)	)	PUNCT
cana-509	46	7	=	=	SYM
cana-509	47	1	7𝑖	7𝑖	ADJ
cana-509	47	2	−	−	NOUN
cana-509	47	3	5	5	NUM
cana-509	47	4	;	;	PUNCT
cana-509	47	5	1	1	NUM
cana-509	47	6	≤	≤	NUM
cana-509	47	7	𝑖	𝑖	PRON
cana-509	47	8	≤	≤	NOUN
cana-509	47	9	𝑛.	𝑛.	NOUN
cana-509	47	10	𝜌(𝑐1	𝜌(𝑐1	NOUN
cana-509	47	11	)	)	PUNCT
cana-509	47	12	=	=	SYM
cana-509	47	13	4	4	NUM
cana-509	47	14	;	;	PUNCT
cana-509	47	15	𝜌(𝑐2𝑖	𝜌(𝑐2𝑖	NUM
cana-509	47	16	)	)	PUNCT
cana-509	47	17	=	=	SYM
cana-509	48	1	7𝑖	7𝑖	ADJ
cana-509	48	2	−	−	NOUN
cana-509	48	3	2	2	NUM
cana-509	48	4	;	;	PUNCT
cana-509	48	5	𝑖	𝑖	NUM
cana-509	48	6	≡	≡	PROPN
cana-509	48	7	0(mod2	0(mod2	PROPN
cana-509	48	8	)	)	PUNCT
cana-509	48	9	𝜌(𝑐2𝑖+1	𝜌(𝑐2𝑖+1	PUNCT
cana-509	48	10	)	)	PUNCT
cana-509	49	1	=	=	SYM
cana-509	50	1	7𝑖	7𝑖	ADJ
cana-509	50	2	+	+	NOUN
cana-509	50	3	3	3	NUM
cana-509	50	4	;	;	PUNCT
cana-509	50	5	𝑖	𝑖	NUM
cana-509	50	6	≡	≡	PROPN
cana-509	50	7	1(mod2	1(mod2	NUM
cana-509	50	8	)	)	PUNCT
cana-509	50	9	𝜌(𝑑2𝑖	𝜌(𝑑2𝑖	NOUN
cana-509	50	10	)	)	PUNCT
cana-509	50	11	=	=	SYM
cana-509	50	12	7𝑖	7𝑖	ADJ
cana-509	50	13	;	;	PUNCT
cana-509	50	14	𝑖	𝑖	NUM
cana-509	50	15	≡	≡	PROPN
cana-509	50	16	0(mod2	0(mod2	PROPN
cana-509	50	17	)	)	PUNCT
cana-509	50	18	𝜌(𝑑2𝑖−1	𝜌(𝑑2𝑖−1	PROPN
cana-509	50	19	)	)	PUNCT
cana-509	50	20	=	=	SYM
cana-509	51	1	7𝑖	7𝑖	ADJ
cana-509	51	2	−	−	NOUN
cana-509	51	3	1	1	NUM
cana-509	51	4	;	;	PUNCT
cana-509	51	5	𝑖	𝑖	NUM
cana-509	51	6	≡	≡	PROPN
cana-509	51	7	1(mod2	1(mod2	NUM
cana-509	51	8	)	)	PUNCT
cana-509	51	9	the	the	DET
cana-509	51	10	induced	induced	ADJ
cana-509	51	11	edge	edge	NOUN
cana-509	51	12	labeling	labeling	NOUN
cana-509	51	13	is	be	AUX
cana-509	51	14	as	as	SCONJ
cana-509	51	15	follows	follow	VERB
cana-509	51	16	𝜌∗(𝑎𝑖𝑏𝑖	𝜌∗(𝑎𝑖𝑏𝑖	ADV
cana-509	51	17	)	)	PUNCT
cana-509	51	18	=	=	SYM
cana-509	52	1	7𝑖	7𝑖	ADJ
cana-509	52	2	−	−	NOUN
cana-509	52	3	6	6	NUM
cana-509	52	4	;	;	PUNCT
cana-509	52	5	1	1	NUM
cana-509	52	6	≤	≤	NUM
cana-509	52	7	𝑖	𝑖	SYM
cana-509	52	8	≤	≤	NOUN
cana-509	52	9	𝑛	𝑛	DET
cana-509	52	10	𝜌∗(𝑏2𝑖−1𝑐2𝑖−1	𝜌∗(𝑏2𝑖−1𝑐2𝑖−1	NOUN
cana-509	52	11	)	)	PUNCT
cana-509	52	12	=	=	SYM
cana-509	53	1	7𝑖	7𝑖	ADJ
cana-509	53	2	−	−	NOUN
cana-509	53	3	5	5	NUM
cana-509	53	4	;	;	PUNCT
cana-509	53	5	𝑖	𝑖	NUM
cana-509	53	6	≡	≡	PROPN
cana-509	53	7	1(mod2	1(mod2	NUM
cana-509	53	8	)	)	PUNCT
cana-509	53	9	𝜌∗(𝑏2𝑖𝑐2𝑖	𝜌∗(𝑏2𝑖𝑐2𝑖	NOUN
cana-509	53	10	)	)	PUNCT
cana-509	53	11	=	=	SYM
cana-509	54	1	7𝑖	7𝑖	ADJ
cana-509	54	2	−	−	NOUN
cana-509	54	3	4	4	NUM
cana-509	54	4	;	;	PUNCT
cana-509	54	5	𝑖	𝑖	NUM
cana-509	54	6	≡	≡	PROPN
cana-509	54	7	0(mod2	0(mod2	PROPN
cana-509	54	8	)	)	PUNCT
cana-509	54	9	𝜌∗(𝑐2𝑖−1𝑐2𝑖	𝜌∗(𝑐2𝑖−1𝑐2𝑖	NUM
cana-509	54	10	)	)	PUNCT
cana-509	55	1	=	=	SYM
cana-509	56	1	7𝑖	7𝑖	ADJ
cana-509	56	2	−	−	NOUN
cana-509	56	3	3	3	NUM
cana-509	56	4	;	;	PUNCT
cana-509	56	5	𝜌∗(𝑐2𝑖+1𝑐2𝑖	𝜌∗(𝑐2𝑖+1𝑐2𝑖	NUM
cana-509	56	6	)	)	PUNCT
cana-509	56	7	=	=	SYM
cana-509	57	1	7𝑖	7𝑖	ADJ
cana-509	57	2	;	;	PUNCT
cana-509	57	3	1	1	NUM
cana-509	57	4	≤	≤	NUM
cana-509	57	5	𝑖	𝑖	SYM
cana-509	57	6	≤	≤	NUM
cana-509	57	7	𝑛	𝑛	DET
cana-509	57	8	𝜌∗(𝑑2𝑖−1𝑐2𝑖−1	𝜌∗(𝑑2𝑖−1𝑐2𝑖−1	NOUN
cana-509	57	9	)	)	PUNCT
cana-509	57	10	=	=	SYM
cana-509	57	11	7𝑖	7𝑖	ADJ
cana-509	57	12	−	−	NOUN
cana-509	57	13	2	2	NUM
cana-509	57	14	;	;	PUNCT
cana-509	57	15	𝑖	𝑖	NUM
cana-509	57	16	≡	≡	PROPN
cana-509	57	17	1(mod2	1(mod2	NUM
cana-509	57	18	)	)	PUNCT
cana-509	57	19	𝜌∗(𝑑2𝑖𝑐2𝑖	𝜌∗(𝑑2𝑖𝑐2𝑖	NOUN
cana-509	57	20	)	)	PUNCT
cana-509	57	21	=	=	SYM
cana-509	58	1	7𝑖	7𝑖	ADJ
cana-509	58	2	−	−	NOUN
cana-509	58	3	1	1	NUM
cana-509	58	4	;	;	PUNCT
cana-509	58	5	𝑖	𝑖	NUM
cana-509	58	6	≡	≡	PROPN
cana-509	58	7	0(mod2	0(mod2	PROPN
cana-509	58	8	)	)	PUNCT
cana-509	58	9	thus	thus	ADV
cana-509	58	10	,	,	PUNCT
cana-509	58	11	𝐴(𝑇𝑆)𝑛	𝐴(𝑇𝑆)𝑛	PROPN
cana-509	58	12	∘	∘	PROPN
cana-509	58	13	𝐾1	𝐾1	PROPN
cana-509	58	14	is	be	AUX
cana-509	58	15	a	a	DET
cana-509	58	16	hm	hm	INTJ
cana-509	58	17	labeling	labeling	NOUN
cana-509	58	18	.	.	PUNCT
cana-509	59	1	communications	communication	NOUN
cana-509	59	2	on	on	ADP
cana-509	59	3	applied	apply	VERB
cana-509	59	4	nonlinear	nonlinear	ADJ
cana-509	59	5	analysis	analysis	NOUN
cana-509	59	6	issn	issn	NOUN
cana-509	59	7	:	:	PUNCT
cana-509	59	8	1074	1074	NUM
cana-509	59	9	-	-	PUNCT
cana-509	59	10	133x	133x	NUM
cana-509	59	11	vol	vol	NOUN
cana-509	59	12	31	31	NUM
cana-509	59	13	no	no	NOUN
cana-509	59	14	.	.	NOUN
cana-509	59	15	2	2	NUM
cana-509	59	16	(	(	PUNCT
cana-509	59	17	2024	2024	NUM
cana-509	59	18	)	)	PUNCT
cana-509	59	19	14	14	NUM
cana-509	59	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	59	21	fig	fig	NOUN
cana-509	59	22	.	.	PUNCT
cana-509	60	1	2	2	X
cana-509	60	2	.	.	X
cana-509	60	3	𝐴(𝑇𝑆)3	𝐴(𝑇𝑆)3	PROPN
cana-509	60	4	∘	∘	PROPN
cana-509	60	5	𝐾1	𝐾1	PROPN
cana-509	60	6	theorem	theorem	VERB
cana-509	60	7	3.3	3.3	NUM
cana-509	60	8	.	.	PUNCT
cana-509	61	1	for	for	ADP
cana-509	61	2	every	every	DET
cana-509	61	3	𝑛	𝑛	DET
cana-509	61	4	≥	≥	NOUN
cana-509	61	5	1	1	NUM
cana-509	61	6	,	,	PUNCT
cana-509	61	7	there	there	PRON
cana-509	61	8	exists	exist	VERB
cana-509	61	9	a	a	DET
cana-509	61	10	double	double	ADJ
cana-509	61	11	triangular	triangular	NOUN
cana-509	61	12	snake	snake	NOUN
cana-509	61	13	graph	graph	NOUN
cana-509	61	14	dts	dts	NOUN
cana-509	61	15	𝑆𝑛	𝑆𝑛	PROPN
cana-509	61	16	。	。	PROPN
cana-509	61	17	𝐾1	𝐾1	PROPN
cana-509	61	18	,	,	PUNCT
cana-509	61	19	which	which	PRON
cana-509	61	20	admits	admit	VERB
cana-509	61	21	hm	hm	DET
cana-509	61	22	labeling	labeling	NOUN
cana-509	61	23	.	.	PUNCT
cana-509	62	1	proof	proof	NOUN
cana-509	62	2	.	.	PUNCT
cana-509	63	1	consider	consider	VERB
cana-509	63	2	the	the	DET
cana-509	63	3	𝐷𝑇𝑆𝑛	𝐷𝑇𝑆𝑛	PROPN
cana-509	63	4	∘	∘	PROPN
cana-509	63	5	𝐾1	𝐾1	PROPN
cana-509	63	6	graph	graph	NOUN
cana-509	63	7	with	with	ADP
cana-509	63	8	vertex	vertex	NOUN
cana-509	63	9	set	set	VERB
cana-509	63	10	𝑉	𝑉	PROPN
cana-509	63	11	=	=	PROPN
cana-509	63	12	{	{	PUNCT
cana-509	63	13	𝑎𝑖	𝑎𝑖	PROPN
cana-509	63	14	,	,	PUNCT
cana-509	63	15	𝑏𝑖	𝑏𝑖	NOUN
cana-509	63	16	,	,	PUNCT
cana-509	63	17	𝑐𝑖	𝑐𝑖	NOUN
cana-509	63	18	,	,	PUNCT
cana-509	63	19	𝑑𝑖	𝑑𝑖	PROPN
cana-509	63	20	,	,	PUNCT
cana-509	63	21	𝑎𝑖	𝑎𝑖	ADP
cana-509	63	22	′	′	NOUN
cana-509	63	23	,	,	PUNCT
cana-509	63	24	𝑏𝑖	𝑏𝑖	ADP
cana-509	63	25	′	′	NUM
cana-509	63	26	;	;	PUNCT
cana-509	63	27	1	1	NUM
cana-509	63	28	≤	≤	NUM
cana-509	63	29	𝑖	𝑖	SYM
cana-509	63	30	≤	≤	NUM
cana-509	63	31	𝑛	𝑛	PRON
cana-509	63	32	}	}	PUNCT
cana-509	63	33	and	and	CCONJ
cana-509	63	34	edge	edge	NOUN
cana-509	63	35	set	set	VERB
cana-509	63	36	𝐸	𝐸	NOUN
cana-509	63	37	=	=	SYM
cana-509	63	38	{	{	PUNCT
cana-509	63	39	(	(	PUNCT
cana-509	63	40	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	ADJ
cana-509	63	41	)	)	PUNCT
cana-509	63	42	∪	∪	NOUN
cana-509	63	43	(	(	PUNCT
cana-509	63	44	𝑑𝑖𝑏𝑖	𝑑𝑖𝑏𝑖	NOUN
cana-509	63	45	)	)	PUNCT
cana-509	63	46	∪	∪	NOUN
cana-509	63	47	(	(	PUNCT
cana-509	63	48	𝑐𝑖𝑑𝑖	𝑐𝑖𝑑𝑖	NOUN
cana-509	63	49	)	)	PUNCT
cana-509	63	50	∪	∪	NOUN
cana-509	63	51	(	(	PUNCT
cana-509	63	52	𝑑𝑖𝑑𝑖+1	𝑑𝑖𝑑𝑖+1	NOUN
cana-509	63	53	)	)	PUNCT
cana-509	63	54	∪	∪	NOUN
cana-509	63	55	(	(	PUNCT
cana-509	63	56	𝑏𝑖	𝑏𝑖	ADP
cana-509	63	57	′𝑑𝑖	′𝑑𝑖	NOUN
cana-509	63	58	′),∪	′),∪	NOUN
cana-509	63	59	(	(	PUNCT
cana-509	63	60	𝑎𝑖	𝑎𝑖	INTJ
cana-509	63	61	′𝑏𝑖	′𝑏𝑖	NOUN
cana-509	63	62	′	′	NUM
cana-509	63	63	)	)	PUNCT
cana-509	63	64	;	;	PUNCT
cana-509	63	65	1	1	NUM
cana-509	63	66	≤	≤	NUM
cana-509	63	67	𝑖	𝑖	PUNCT
cana-509	63	68	≤	≤	NUM
cana-509	63	69	𝑛	𝑛	NOUN
cana-509	63	70	}	}	PUNCT
cana-509	63	71	.	.	PUNCT
cana-509	64	1	now	now	ADV
cana-509	64	2	let	let	VERB
cana-509	64	3	us	we	PRON
cana-509	64	4	define	define	VERB
cana-509	64	5	a	a	DET
cana-509	64	6	function	function	NOUN
cana-509	64	7	𝜌	𝜌	ADP
cana-509	64	8	:	:	PUNCT
cana-509	64	9	𝑣	𝑣	X
cana-509	64	10	→	→	PUNCT
cana-509	64	11	{	{	PUNCT
cana-509	64	12	1,2,3	1,2,3	NUM
cana-509	64	13	,	,	PUNCT
cana-509	64	14	⋯	⋯	NOUN
cana-509	64	15	𝑞	𝑞	X
cana-509	64	16	+	+	PROPN
cana-509	64	17	1	1	NUM
cana-509	64	18	}	}	PUNCT
cana-509	64	19	let	let	VERB
cana-509	64	20	us	we	PRON
cana-509	64	21	label	label	VERB
cana-509	64	22	the	the	DET
cana-509	64	23	vertices	vertex	NOUN
cana-509	64	24	as	as	SCONJ
cana-509	64	25	follows	follow	VERB
cana-509	64	26	.	.	PUNCT
cana-509	65	1	𝜌(𝑎1	𝜌(𝑎1	X
cana-509	65	2	)	)	PUNCT
cana-509	65	3	=	=	SYM
cana-509	65	4	1	1	NUM
cana-509	65	5	;	;	PUNCT
cana-509	65	6	𝜌(𝑎𝑖+1	𝜌(𝑎𝑖+1	X
cana-509	65	7	)	)	PUNCT
cana-509	65	8	=	=	SYM
cana-509	65	9	8𝑖	8𝑖	ADJ
cana-509	65	10	+	+	CCONJ
cana-509	65	11	3	3	NUM
cana-509	65	12	;	;	PUNCT
cana-509	65	13	1	1	NUM
cana-509	65	14	≤	≤	NUM
cana-509	65	15	𝑖	𝑖	SYM
cana-509	65	16	≤	≤	NUM
cana-509	65	17	𝑛	𝑛	PRON
cana-509	65	18	−	−	PROPN
cana-509	65	19	1	1	NUM
cana-509	65	20	𝜌(𝑏1	𝜌(𝑏1	NOUN
cana-509	65	21	)	)	PUNCT
cana-509	65	22	=	=	SYM
cana-509	65	23	2	2	NUM
cana-509	65	24	;	;	PUNCT
cana-509	65	25	𝜌(𝑏𝑖+1	𝜌(𝑏𝑖+1	NUM
cana-509	65	26	)	)	PUNCT
cana-509	65	27	=	=	SYM
cana-509	65	28	8𝑖	8𝑖	ADJ
cana-509	66	1	+	+	CCONJ
cana-509	66	2	4	4	NUM
cana-509	66	3	;	;	PUNCT
cana-509	66	4	1	1	NUM
cana-509	66	5	≤	≤	NUM
cana-509	66	6	𝑖	𝑖	SYM
cana-509	66	7	≤	≤	NUM
cana-509	66	8	𝑛	𝑛	PRON
cana-509	66	9	−	−	NUM
cana-509	66	10	1	1	NUM
cana-509	66	11	𝜌(𝑐1	𝜌(𝑐1	NOUN
cana-509	66	12	)	)	PUNCT
cana-509	66	13	=	=	SYM
cana-509	66	14	4	4	NUM
cana-509	66	15	;	;	PUNCT
cana-509	66	16	𝜌(𝑐𝑖+1	𝜌(𝑐𝑖+1	NUM
cana-509	66	17	)	)	PUNCT
cana-509	66	18	=	=	SYM
cana-509	66	19	8𝑖	8𝑖	ADJ
cana-509	66	20	−	−	NOUN
cana-509	66	21	2	2	NUM
cana-509	66	22	;	;	PUNCT
cana-509	66	23	1	1	NUM
cana-509	66	24	≤	≤	NUM
cana-509	66	25	𝑖	𝑖	SYM
cana-509	66	26	≤	≤	NUM
cana-509	66	27	𝑛	𝑛	PRON
cana-509	66	28	−	−	NUM
cana-509	66	29	1	1	NUM
cana-509	66	30	𝜌(𝑑1	𝜌(𝑑1	NOUN
cana-509	66	31	)	)	PUNCT
cana-509	66	32	=	=	SYM
cana-509	66	33	5	5	NUM
cana-509	66	34	;	;	PUNCT
cana-509	66	35	𝜌(𝑑𝑖+1	𝜌(𝑑𝑖+1	NOUN
cana-509	66	36	)	)	PUNCT
cana-509	66	37	=	=	SYM
cana-509	66	38	8𝑖	8𝑖	ADJ
cana-509	67	1	+	+	CCONJ
cana-509	67	2	2	2	NUM
cana-509	67	3	;	;	PUNCT
cana-509	67	4	1	1	NUM
cana-509	67	5	≤	≤	NUM
cana-509	67	6	𝑖	𝑖	SYM
cana-509	67	7	≤	≤	NUM
cana-509	67	8	𝑛	𝑛	PRON
cana-509	67	9	−	−	PROPN
cana-509	67	10	1	1	NUM
cana-509	67	11	𝜌(𝑏𝑖	𝜌(𝑏𝑖	NOUN
cana-509	67	12	′	′	NUM
cana-509	67	13	)	)	PUNCT
cana-509	67	14	=	=	SYM
cana-509	67	15	8𝑖	8𝑖	ADJ
cana-509	67	16	−	−	NOUN
cana-509	67	17	1	1	NUM
cana-509	67	18	;	;	PUNCT
cana-509	67	19	𝜌(𝑎𝑖	𝜌(𝑎𝑖	NUM
cana-509	67	20	′	′	NUM
cana-509	67	21	)	)	PUNCT
cana-509	67	22	=	=	PRON
cana-509	67	23	8𝑖	8𝑖	ADJ
cana-509	67	24	+	+	CCONJ
cana-509	67	25	1	1	NUM
cana-509	67	26	;	;	PUNCT
cana-509	67	27	1	1	NUM
cana-509	67	28	≤	≤	NUM
cana-509	67	29	𝑖	𝑖	SYM
cana-509	67	30	≤	≤	NUM
cana-509	67	31	𝑛	𝑛	ADP
cana-509	67	32	the	the	DET
cana-509	67	33	induced	induced	ADJ
cana-509	67	34	edge	edge	NOUN
cana-509	67	35	labeling	labeling	NOUN
cana-509	67	36	is	be	AUX
cana-509	67	37	as	as	SCONJ
cana-509	67	38	follows	follow	VERB
cana-509	67	39	𝜌∗(𝑎1𝑏1	𝜌∗(𝑎1𝑏1	PRON
cana-509	67	40	)	)	PUNCT
cana-509	67	41	=	=	SYM
cana-509	67	42	1	1	NUM
cana-509	67	43	;	;	PUNCT
cana-509	67	44	𝜌∗(𝑎𝑖+1𝑏𝑖+1	𝜌∗(𝑎𝑖+1𝑏𝑖+1	PROPN
cana-509	67	45	)	)	PUNCT
cana-509	67	46	=	=	PRON
cana-509	67	47	8𝑖	8𝑖	ADJ
cana-509	68	1	+	+	CCONJ
cana-509	68	2	3	3	NUM
cana-509	68	3	;	;	PUNCT
cana-509	68	4	1	1	NUM
cana-509	68	5	≤	≤	NUM
cana-509	68	6	𝑖	𝑖	SYM
cana-509	68	7	≤	≤	NUM
cana-509	68	8	𝑛	𝑛	PRON
cana-509	68	9	−	−	PROPN
cana-509	68	10	1	1	NUM
cana-509	68	11	𝜌∗(𝑑1𝑐1	𝜌∗(𝑑1𝑐1	PROPN
cana-509	68	12	)	)	PUNCT
cana-509	68	13	=	=	SYM
cana-509	68	14	4	4	NUM
cana-509	68	15	;	;	PUNCT
cana-509	68	16	𝜌∗(𝑑𝑖+1𝑐𝑖+1	𝜌∗(𝑑𝑖+1𝑐𝑖+1	NUM
cana-509	68	17	)	)	PUNCT
cana-509	68	18	=	=	SYM
cana-509	68	19	8𝑖	8𝑖	ADJ
cana-509	68	20	−	−	NOUN
cana-509	68	21	1	1	NUM
cana-509	68	22	;	;	PUNCT
cana-509	68	23	1	1	NUM
cana-509	68	24	≤	≤	NUM
cana-509	68	25	𝑖	𝑖	SYM
cana-509	68	26	≤	≤	NUM
cana-509	68	27	𝑛	𝑛	DET
cana-509	68	28	−	−	NUM
cana-509	68	29	1	1	NUM
cana-509	68	30	𝜌∗(𝑎𝑖	𝜌∗(𝑎𝑖	PROPN
cana-509	68	31	′𝑏𝑖	′𝑏𝑖	NOUN
cana-509	68	32	′	′	NUM
cana-509	68	33	)	)	PUNCT
cana-509	69	1	=	=	SYM
cana-509	69	2	8𝑖	8𝑖	ADJ
cana-509	69	3	;	;	PUNCT
cana-509	69	4	𝜌∗(𝑑𝑖𝑏𝑖	𝜌∗(𝑑𝑖𝑏𝑖	PROPN
cana-509	69	5	)	)	PUNCT
cana-509	69	6	=	=	SYM
cana-509	70	1	8𝑖	8𝑖	ADJ
cana-509	70	2	−	−	NOUN
cana-509	70	3	6	6	NUM
cana-509	70	4	;	;	PUNCT
cana-509	70	5	1	1	NUM
cana-509	70	6	≤	≤	NUM
cana-509	70	7	𝑖	𝑖	SYM
cana-509	70	8	≤	≤	NUM
cana-509	70	9	𝑛	𝑛	PRON
cana-509	70	10	𝜌∗(𝑑2𝑏1	𝜌∗(𝑑2𝑏1	PROPN
cana-509	70	11	)	)	PUNCT
cana-509	70	12	=	=	SYM
cana-509	71	1	3	3	NUM
cana-509	71	2	;	;	PUNCT
cana-509	71	3	𝜌∗(𝑑𝑖+1𝑏𝑖	𝜌∗(𝑑𝑖+1𝑏𝑖	NUM
cana-509	71	4	)	)	PUNCT
cana-509	71	5	=	=	SYM
cana-509	71	6	8𝑖	8𝑖	ADJ
cana-509	71	7	+	+	CCONJ
cana-509	71	8	6	6	NUM
cana-509	71	9	;	;	PUNCT
cana-509	71	10	2	2	NUM
cana-509	71	11	≤	≤	NUM
cana-509	71	12	𝑖	𝑖	SYM
cana-509	71	13	≤	≤	NUM
cana-509	71	14	𝑛	𝑛	DET
cana-509	71	15	communications	communication	NOUN
cana-509	71	16	on	on	ADP
cana-509	71	17	applied	apply	VERB
cana-509	71	18	nonlinear	nonlinear	ADJ
cana-509	71	19	analysis	analysis	NOUN
cana-509	71	20	issn	issn	NOUN
cana-509	71	21	:	:	PUNCT
cana-509	71	22	1074	1074	NUM
cana-509	71	23	-	-	PUNCT
cana-509	71	24	133x	133x	NUM
cana-509	71	25	vol	vol	NOUN
cana-509	71	26	31	31	NUM
cana-509	71	27	no	no	NOUN
cana-509	71	28	.	.	NOUN
cana-509	71	29	2	2	NUM
cana-509	71	30	(	(	PUNCT
cana-509	71	31	2024	2024	NUM
cana-509	71	32	)	)	PUNCT
cana-509	71	33	15	15	NUM
cana-509	71	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	71	35	𝜌∗(𝑏1	𝜌∗(𝑏1	PROPN
cana-509	71	36	′	′	NUM
cana-509	71	37	𝑑1	𝑑1	NOUN
cana-509	71	38	)	)	PUNCT
cana-509	71	39	=	=	SYM
cana-509	71	40	5	5	NUM
cana-509	71	41	;	;	PUNCT
cana-509	71	42	𝜌∗(𝑏𝑖+1	𝜌∗(𝑏𝑖+1	PROPN
cana-509	71	43	′	′	NUM
cana-509	71	44	𝑑𝑖+1	𝑑𝑖+1	NUM
cana-509	71	45	)	)	PUNCT
cana-509	71	46	=	=	VERB
cana-509	71	47	8𝑖	8𝑖	ADJ
cana-509	72	1	+	+	CCONJ
cana-509	72	2	4	4	NUM
cana-509	72	3	;	;	PUNCT
cana-509	72	4	𝜌∗(𝑏𝑖	𝜌∗(𝑏𝑖	NUM
cana-509	72	5	′𝑑𝑖+1	′𝑑𝑖+1	NOUN
cana-509	72	6	)	)	PUNCT
cana-509	72	7	=	=	PRON
cana-509	72	8	8𝑖	8𝑖	ADJ
cana-509	73	1	+	+	CCONJ
cana-509	73	2	1	1	NUM
cana-509	73	3	;	;	PUNCT
cana-509	73	4	1	1	NUM
cana-509	73	5	≤	≤	NUM
cana-509	73	6	𝑖	𝑖	SYM
cana-509	73	7	≤	≤	NUM
cana-509	73	8	𝑛	𝑛	PRON
cana-509	73	9	𝜌∗(𝑑1𝑑2	𝜌∗(𝑑1𝑑2	PROPN
cana-509	73	10	)	)	PUNCT
cana-509	73	11	=	=	SYM
cana-509	73	12	6	6	NUM
cana-509	73	13	;	;	PUNCT
cana-509	73	14	𝜌∗(𝑑𝑖+1𝑑𝑖+2	𝜌∗(𝑑𝑖+1𝑑𝑖+2	PROPN
cana-509	73	15	)	)	PUNCT
cana-509	73	16	=	=	PRON
cana-509	73	17	8𝑖	8𝑖	ADJ
cana-509	73	18	+	+	CCONJ
cana-509	73	19	5	5	NUM
cana-509	73	20	;	;	PUNCT
cana-509	73	21	1	1	NUM
cana-509	73	22	≤	≤	NUM
cana-509	73	23	𝑖	𝑖	SYM
cana-509	73	24	≤	≤	NUM
cana-509	73	25	𝑛	𝑛	PRON
cana-509	73	26	−	−	PROPN
cana-509	73	27	1	1	NUM
cana-509	73	28	thus	thus	ADV
cana-509	73	29	,	,	PUNCT
cana-509	73	30	𝐷𝑇𝑆𝑛	𝐷𝑇𝑆𝑛	PROPN
cana-509	73	31	∘	∘	PROPN
cana-509	73	32	𝐾1	𝐾1	PROPN
cana-509	73	33	is	be	AUX
cana-509	73	34	a	a	DET
cana-509	73	35	hm	hm	INTJ
cana-509	73	36	labeling	labeling	NOUN
cana-509	73	37	.	.	PUNCT
cana-509	74	1	fig	fig	NOUN
cana-509	74	2	.	.	PUNCT
cana-509	75	1	3	3	X
cana-509	75	2	.	.	X
cana-509	75	3	𝐷𝑇𝑆4	𝐷𝑇𝑆4	VERB
cana-509	75	4	∘	∘	PROPN
cana-509	75	5	𝐾1	𝐾1	PROPN
cana-509	75	6	theorem	theorem	ADJ
cana-509	75	7	3.4	3.4	NUM
cana-509	75	8	.	.	PUNCT
cana-509	76	1	for	for	ADP
cana-509	76	2	every	every	DET
cana-509	76	3	𝑛	𝑛	DET
cana-509	76	4	≥	≥	NOUN
cana-509	76	5	1	1	NUM
cana-509	76	6	,	,	PUNCT
cana-509	76	7	there	there	PRON
cana-509	76	8	exists	exist	VERB
cana-509	76	9	a	a	DET
cana-509	76	10	alternate	alternate	ADJ
cana-509	76	11	double	double	ADJ
cana-509	76	12	triangular	triangular	NOUN
cana-509	76	13	snake	snake	NOUN
cana-509	76	14	graph	graph	NOUN
cana-509	76	15	𝐴(𝐷𝑇𝑆)𝑛	𝐴(𝐷𝑇𝑆)𝑛	PROPN
cana-509	76	16	∘	∘	PROPN
cana-509	76	17	𝐾1	𝐾1	PROPN
cana-509	76	18	,	,	PUNCT
cana-509	76	19	which	which	PRON
cana-509	76	20	admits	admit	VERB
cana-509	76	21	𝐻𝑀	𝐻𝑀	PROPN
cana-509	76	22	labeling	labeling	NOUN
cana-509	76	23	.	.	PUNCT
cana-509	77	1	proof	proof	NOUN
cana-509	77	2	.	.	PUNCT
cana-509	78	1	consider	consider	VERB
cana-509	78	2	the	the	DET
cana-509	78	3	𝐴(𝐷𝑇𝑆)𝑛	𝐴(𝐷𝑇𝑆)𝑛	PROPN
cana-509	78	4	∘	∘	PROPN
cana-509	78	5	𝐾1	𝐾1	PROPN
cana-509	78	6	graph	graph	NOUN
cana-509	78	7	with	with	ADP
cana-509	78	8	vertex	vertex	NOUN
cana-509	78	9	set	set	VERB
cana-509	78	10	𝑉	𝑉	PROPN
cana-509	78	11	=	=	PROPN
cana-509	78	12	{	{	PUNCT
cana-509	78	13	𝑎𝑖	𝑎𝑖	PROPN
cana-509	78	14	,	,	PUNCT
cana-509	78	15	𝑏𝑖	𝑏𝑖	ADP
cana-509	78	16	,	,	PUNCT
cana-509	78	17	𝑐𝑖	𝑐𝑖	PROPN
cana-509	78	18	,	,	PUNCT
cana-509	78	19	𝑑𝑖	𝑑𝑖	PROPN
cana-509	78	20	,	,	PUNCT
cana-509	78	21	𝑎𝑖	𝑎𝑖	ADP
cana-509	78	22	′	′	NOUN
cana-509	78	23	,	,	PUNCT
cana-509	78	24	𝑏𝑖	𝑏𝑖	ADP
cana-509	78	25	′	′	NUM
cana-509	78	26	;	;	PUNCT
cana-509	78	27	1	1	NUM
cana-509	78	28	≤	≤	NUM
cana-509	78	29	𝑖	𝑖	SYM
cana-509	78	30	≤	≤	NUM
cana-509	78	31	𝑛	𝑛	PRON
cana-509	78	32	}	}	PUNCT
cana-509	78	33	and	and	CCONJ
cana-509	78	34	edge	edge	NOUN
cana-509	78	35	set	set	VERB
cana-509	78	36	𝐸	𝐸	NOUN
cana-509	78	37	=	=	SYM
cana-509	78	38	{	{	PUNCT
cana-509	78	39	(	(	PUNCT
cana-509	78	40	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	ADJ
cana-509	78	41	)	)	PUNCT
cana-509	78	42	∪	∪	NOUN
cana-509	78	43	(	(	PUNCT
cana-509	78	44	𝑑𝑖𝑏𝑖	𝑑𝑖𝑏𝑖	NOUN
cana-509	78	45	)	)	PUNCT
cana-509	78	46	∪	∪	NOUN
cana-509	78	47	(	(	PUNCT
cana-509	78	48	𝑐𝑖𝑑𝑖	𝑐𝑖𝑑𝑖	NOUN
cana-509	78	49	)	)	PUNCT
cana-509	78	50	∪	∪	NOUN
cana-509	78	51	(	(	PUNCT
cana-509	78	52	𝑑𝑖𝑑𝑖+1	𝑑𝑖𝑑𝑖+1	NOUN
cana-509	78	53	)	)	PUNCT
cana-509	78	54	∪	∪	NOUN
cana-509	78	55	(	(	PUNCT
cana-509	78	56	𝑏𝑖	𝑏𝑖	ADP
cana-509	78	57	′𝑑𝑖	′𝑑𝑖	NOUN
cana-509	78	58	)	)	PUNCT
cana-509	78	59	∪	∪	NOUN
cana-509	78	60	(	(	PUNCT
cana-509	78	61	𝑎𝑖	𝑎𝑖	PRON
cana-509	78	62	′𝑏𝑖	′𝑏𝑖	NOUN
cana-509	78	63	′	′	NUM
cana-509	78	64	)	)	PUNCT
cana-509	78	65	;	;	PUNCT
cana-509	78	66	1	1	NUM
cana-509	78	67	≤	≤	NUM
cana-509	78	68	𝑖	𝑖	SYM
cana-509	78	69	≤	≤	NUM
cana-509	78	70	𝑛	𝑛	X
cana-509	78	71	}	}	PUNCT
cana-509	78	72	now	now	ADV
cana-509	78	73	let	let	VERB
cana-509	78	74	us	we	PRON
cana-509	78	75	define	define	VERB
cana-509	78	76	a	a	DET
cana-509	78	77	function	function	NOUN
cana-509	78	78	𝜌	𝜌	ADP
cana-509	78	79	:	:	PUNCT
cana-509	78	80	𝑣	𝑣	X
cana-509	78	81	→	→	PUNCT
cana-509	78	82	{	{	PUNCT
cana-509	78	83	1,2,3	1,2,3	NUM
cana-509	78	84	,	,	PUNCT
cana-509	78	85	⋯	⋯	NOUN
cana-509	78	86	𝑞	𝑞	X
cana-509	78	87	+	+	PROPN
cana-509	78	88	1	1	NUM
cana-509	78	89	}	}	PUNCT
cana-509	78	90	let	let	VERB
cana-509	78	91	us	we	PRON
cana-509	78	92	label	label	VERB
cana-509	78	93	the	the	DET
cana-509	78	94	vertices	vertex	NOUN
cana-509	78	95	as	as	SCONJ
cana-509	78	96	follows	follow	VERB
cana-509	78	97	.	.	PUNCT
cana-509	79	1	𝜌(𝑎1	𝜌(𝑎1	X
cana-509	79	2	)	)	PUNCT
cana-509	79	3	=	=	SYM
cana-509	79	4	1	1	NUM
cana-509	79	5	;	;	PUNCT
cana-509	79	6	𝜌(𝑎𝑖+1	𝜌(𝑎𝑖+1	X
cana-509	79	7	)	)	PUNCT
cana-509	79	8	=	=	SYM
cana-509	79	9	10𝑖	10𝑖	NOUN
cana-509	80	1	+	+	CCONJ
cana-509	80	2	3	3	NUM
cana-509	80	3	;	;	PUNCT
cana-509	80	4	1	1	NUM
cana-509	80	5	≤	≤	NUM
cana-509	80	6	𝑖	𝑖	SYM
cana-509	80	7	≤	≤	NUM
cana-509	80	8	𝑛	𝑛	PRON
cana-509	80	9	−	−	NUM
cana-509	80	10	1	1	NUM
cana-509	80	11	𝜌(𝑏𝑖	𝜌(𝑏𝑖	NOUN
cana-509	80	12	)	)	PUNCT
cana-509	80	13	=	=	SYM
cana-509	80	14	10𝑖	10𝑖	PROPN
cana-509	81	1	−	−	PROPN
cana-509	81	2	8	8	NUM
cana-509	81	3	;	;	PUNCT
cana-509	81	4	1	1	NUM
cana-509	81	5	≤	≤	NUM
cana-509	81	6	𝑖	𝑖	SYM
cana-509	81	7	≤	≤	NUM
cana-509	81	8	𝑛	𝑛	PRON
cana-509	81	9	𝜌(𝑑1	𝜌(𝑑1	NOUN
cana-509	81	10	)	)	PUNCT
cana-509	81	11	=	=	SYM
cana-509	81	12	5	5	NUM
cana-509	81	13	;	;	PUNCT
cana-509	81	14	𝜌(𝑑2𝑖	𝜌(𝑑2𝑖	X
cana-509	81	15	)	)	PUNCT
cana-509	81	16	=	=	SYM
cana-509	81	17	10𝑖	10𝑖	NUM
cana-509	81	18	;	;	PUNCT
cana-509	81	19	𝑖	𝑖	NUM
cana-509	81	20	≡	≡	PROPN
cana-509	81	21	0(mod2	0(mod2	PROPN
cana-509	81	22	)	)	PUNCT
cana-509	81	23	𝜌(𝑑2𝑖−1	𝜌(𝑑2𝑖−1	PROPN
cana-509	81	24	)	)	PUNCT
cana-509	81	25	=	=	SYM
cana-509	81	26	10𝑖	10𝑖	NOUN
cana-509	82	1	+	+	ADJ
cana-509	82	2	1	1	NUM
cana-509	82	3	;	;	PUNCT
cana-509	82	4	𝑖	𝑖	NUM
cana-509	82	5	≡	≡	PROPN
cana-509	82	6	1(mod2	1(mod2	NUM
cana-509	82	7	)	)	PUNCT
cana-509	82	8	𝜌(𝑐2𝑖	𝜌(𝑐2𝑖	PUNCT
cana-509	82	9	)	)	PUNCT
cana-509	83	1	=	=	PRON
cana-509	83	2	10𝑖	10𝑖	PROPN
cana-509	84	1	−	−	PROPN
cana-509	84	2	4	4	NUM
cana-509	84	3	;	;	PUNCT
cana-509	84	4	𝑖	𝑖	NUM
cana-509	84	5	≡	≡	PROPN
cana-509	84	6	0(mod2	0(mod2	PROPN
cana-509	84	7	)	)	PUNCT
cana-509	84	8	𝜌(𝑐2𝑖−1	𝜌(𝑐2𝑖−1	ADV
cana-509	84	9	)	)	PUNCT
cana-509	85	1	=	=	SYM
cana-509	85	2	10𝑖	10𝑖	PROPN
cana-509	86	1	−	−	PROPN
cana-509	86	2	6	6	NUM
cana-509	86	3	;	;	PUNCT
cana-509	86	4	𝑖	𝑖	NUM
cana-509	86	5	≡	≡	PROPN
cana-509	86	6	1(mod2	1(mod2	NUM
cana-509	86	7	)	)	PUNCT
cana-509	86	8	𝜌(𝑏𝑖	𝜌(𝑏𝑖	NOUN
cana-509	86	9	′	′	NUM
cana-509	86	10	)	)	PUNCT
cana-509	86	11	=	=	PRON
cana-509	86	12	10𝑖	10𝑖	PROPN
cana-509	86	13	−	−	PROPN
cana-509	86	14	3	3	NUM
cana-509	86	15	;	;	PUNCT
cana-509	86	16	𝜌(𝑎𝑖	𝜌(𝑎𝑖	NUM
cana-509	86	17	′	′	NUM
cana-509	86	18	)	)	PUNCT
cana-509	86	19	=	=	PRON
cana-509	86	20	10𝑖	10𝑖	PROPN
cana-509	87	1	−	−	PROPN
cana-509	87	2	1	1	NUM
cana-509	87	3	;	;	PUNCT
cana-509	87	4	1	1	NUM
cana-509	87	5	≤	≤	NUM
cana-509	87	6	𝑖	𝑖	SYM
cana-509	87	7	≤	≤	NUM
cana-509	87	8	𝑛	𝑛	DET
cana-509	87	9	communications	communication	NOUN
cana-509	87	10	on	on	ADP
cana-509	87	11	applied	apply	VERB
cana-509	87	12	nonlinear	nonlinear	ADJ
cana-509	87	13	analysis	analysis	NOUN
cana-509	87	14	issn	issn	NOUN
cana-509	87	15	:	:	PUNCT
cana-509	87	16	1074	1074	NUM
cana-509	87	17	-	-	PUNCT
cana-509	87	18	133x	133x	NUM
cana-509	87	19	vol	vol	NOUN
cana-509	87	20	31	31	NUM
cana-509	87	21	no	no	NOUN
cana-509	87	22	.	.	NOUN
cana-509	87	23	2	2	NUM
cana-509	87	24	(	(	PUNCT
cana-509	87	25	2024	2024	NUM
cana-509	87	26	)	)	PUNCT
cana-509	87	27	16	16	NUM
cana-509	87	28	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	87	29	the	the	DET
cana-509	87	30	induced	induce	VERB
cana-509	87	31	edge	edge	NOUN
cana-509	87	32	labeling	labeling	NOUN
cana-509	87	33	is	be	AUX
cana-509	87	34	as	as	SCONJ
cana-509	87	35	follows	follow	VERB
cana-509	87	36	𝜌∗(𝑎1𝑏1	𝜌∗(𝑎1𝑏1	PRON
cana-509	87	37	)	)	PUNCT
cana-509	87	38	=	=	SYM
cana-509	87	39	1	1	NUM
cana-509	87	40	;	;	PUNCT
cana-509	87	41	𝜌∗(𝑎𝑖+1𝑏𝑖+1	𝜌∗(𝑎𝑖+1𝑏𝑖+1	PROPN
cana-509	87	42	)	)	PUNCT
cana-509	87	43	=	=	SYM
cana-509	87	44	10𝑖	10𝑖	NOUN
cana-509	88	1	+	+	ADJ
cana-509	88	2	2	2	NUM
cana-509	88	3	;	;	PUNCT
cana-509	88	4	1	1	NUM
cana-509	88	5	≤	≤	NUM
cana-509	88	6	𝑖	𝑖	SYM
cana-509	88	7	≤	≤	NUM
cana-509	88	8	𝑛	𝑛	PRON
cana-509	88	9	−	−	PROPN
cana-509	88	10	1	1	NUM
cana-509	88	11	𝜌∗(𝑑1𝑐1	𝜌∗(𝑑1𝑐1	PROPN
cana-509	88	12	)	)	PUNCT
cana-509	88	13	=	=	SYM
cana-509	89	1	4	4	NUM
cana-509	89	2	;	;	PUNCT
cana-509	89	3	𝜌∗(𝑑2𝑖𝑐2𝑖	𝜌∗(𝑑2𝑖𝑐2𝑖	NOUN
cana-509	89	4	)	)	PUNCT
cana-509	89	5	=	=	SYM
cana-509	89	6	10𝑖	10𝑖	PROPN
cana-509	90	1	−	−	PROPN
cana-509	90	2	3	3	NUM
cana-509	90	3	;	;	PUNCT
cana-509	90	4	𝑖	𝑖	NUM
cana-509	90	5	≡	≡	PROPN
cana-509	90	6	0(mod2	0(mod2	PROPN
cana-509	90	7	)	)	PUNCT
cana-509	90	8	𝜌(𝑑2𝑖+1𝑐2𝑖+1	𝜌(𝑑2𝑖+1𝑐2𝑖+1	PROPN
cana-509	90	9	)	)	PUNCT
cana-509	90	10	=	=	SYM
cana-509	90	11	10𝑖	10𝑖	NOUN
cana-509	91	1	+	+	CCONJ
cana-509	91	2	3	3	NUM
cana-509	91	3	;	;	PUNCT
cana-509	91	4	𝑖	𝑖	NUM
cana-509	91	5	≡	≡	PROPN
cana-509	91	6	1(mod2	1(mod2	NUM
cana-509	91	7	)	)	PUNCT
cana-509	91	8	𝜌∗(𝑑1𝑏1	𝜌∗(𝑑1𝑏1	PROPN
cana-509	91	9	)	)	PUNCT
cana-509	91	10	=	=	SYM
cana-509	91	11	2	2	NUM
cana-509	91	12	;	;	PUNCT
cana-509	91	13	𝜌∗(𝑑2𝑖+1𝑏𝑖+1	𝜌∗(𝑑2𝑖+1𝑏𝑖+1	NOUN
cana-509	91	14	)	)	PUNCT
cana-509	91	15	=	=	SYM
cana-509	91	16	10𝑖	10𝑖	NOUN
cana-509	92	1	+	+	ADJ
cana-509	92	2	1	1	NUM
cana-509	92	3	;	;	PUNCT
cana-509	92	4	1	1	NUM
cana-509	92	5	≤	≤	NUM
cana-509	92	6	𝑖	𝑖	SYM
cana-509	92	7	≤	≤	NUM
cana-509	92	8	𝑛	𝑛	PRON
cana-509	92	9	𝜌∗(𝑑2𝑏1	𝜌∗(𝑑2𝑏1	PROPN
cana-509	92	10	)	)	PUNCT
cana-509	92	11	=	=	SYM
cana-509	92	12	3	3	NUM
cana-509	92	13	;	;	PUNCT
cana-509	92	14	𝜌∗(𝑑2𝑖+2𝑏𝑖+1	𝜌∗(𝑑2𝑖+2𝑏𝑖+1	X
cana-509	92	15	)	)	PUNCT
cana-509	92	16	=	=	SYM
cana-509	92	17	10𝑖	10𝑖	NOUN
cana-509	93	1	+	+	ADJ
cana-509	93	2	6	6	NUM
cana-509	93	3	;	;	PUNCT
cana-509	93	4	1	1	NUM
cana-509	93	5	≤	≤	NUM
cana-509	93	6	𝑖	𝑖	SYM
cana-509	93	7	≤	≤	NUM
cana-509	93	8	𝑛	𝑛	PRON
cana-509	93	9	𝜌∗(𝑑2𝑑1	𝜌∗(𝑑2𝑑1	PROPN
cana-509	93	10	)	)	PUNCT
cana-509	93	11	=	=	SYM
cana-509	93	12	3	3	NUM
cana-509	93	13	;	;	PUNCT
cana-509	93	14	𝜌∗(𝑑2𝑖𝑑2𝑖+1	𝜌∗(𝑑2𝑖𝑑2𝑖+1	PROPN
cana-509	93	15	)	)	PUNCT
cana-509	93	16	=	=	SYM
cana-509	93	17	10𝑖	10𝑖	NOUN
cana-509	93	18	;	;	PUNCT
cana-509	93	19	𝜌∗(𝑑2𝑖+1𝑑2𝑖+2	𝜌∗(𝑑2𝑖+1𝑑2𝑖+2	NUM
cana-509	93	20	)	)	PUNCT
cana-509	93	21	=	=	SYM
cana-509	93	22	10𝑖	10𝑖	NOUN
cana-509	94	1	+	+	X
cana-509	94	2	5	5	NUM
cana-509	94	3	;	;	PUNCT
cana-509	94	4	1	1	NUM
cana-509	94	5	≤	≤	NUM
cana-509	94	6	𝑖	𝑖	SYM
cana-509	94	7	≤	≤	NUM
cana-509	94	8	𝑛	𝑛	PRON
cana-509	94	9	𝜌∗(𝑏𝑖+1	𝜌∗(𝑏𝑖+1	PROPN
cana-509	94	10	′	′	NUM
cana-509	94	11	𝑑2𝑖+1	𝑑2𝑖+1	PROPN
cana-509	94	12	)	)	PUNCT
cana-509	94	13	=	=	SYM
cana-509	94	14	10𝑖	10𝑖	NOUN
cana-509	94	15	+	+	CCONJ
cana-509	94	16	4	4	NUM
cana-509	94	17	;	;	PUNCT
cana-509	94	18	𝜌∗(𝑏𝑖	𝜌∗(𝑏𝑖	NUM
cana-509	94	19	′𝑑𝑖+1	′𝑑𝑖+1	NOUN
cana-509	94	20	)	)	PUNCT
cana-509	94	21	=	=	PRON
cana-509	94	22	10𝑖	10𝑖	PROPN
cana-509	94	23	−	−	PROPN
cana-509	94	24	1	1	NUM
cana-509	94	25	;	;	PUNCT
cana-509	94	26	1	1	NUM
cana-509	94	27	≤	≤	NUM
cana-509	94	28	𝑖	𝑖	SYM
cana-509	94	29	≤	≤	NOUN
cana-509	94	30	𝑛	𝑛	PRON
cana-509	94	31	𝜌∗(𝑎1	𝜌∗(𝑎1	ADJ
cana-509	94	32	′	′	NOUN
cana-509	94	33	𝑏𝑖	𝑏𝑖	ADP
cana-509	94	34	′	′	NUM
cana-509	94	35	)	)	PUNCT
cana-509	95	1	=	=	PRON
cana-509	95	2	10𝑖	10𝑖	PROPN
cana-509	96	1	−	−	PROPN
cana-509	96	2	2	2	NUM
cana-509	96	3	;	;	PUNCT
cana-509	96	4	1	1	NUM
cana-509	96	5	≤	≤	NUM
cana-509	96	6	𝑖	𝑖	SYM
cana-509	96	7	≤	≤	NOUN
cana-509	96	8	𝑛	𝑛	PRON
cana-509	96	9	thus	thus	ADV
cana-509	96	10	,	,	PUNCT
cana-509	96	11	𝐴(𝐷𝑇𝑆)𝑛	𝐴(𝐷𝑇𝑆)𝑛	PROPN
cana-509	96	12	∘	∘	PROPN
cana-509	96	13	𝐾1	𝐾1	PROPN
cana-509	96	14	is	be	AUX
cana-509	96	15	a	a	DET
cana-509	96	16	hm	hm	INTJ
cana-509	96	17	labeling	labeling	NOUN
cana-509	96	18	.	.	PUNCT
cana-509	97	1	fig	fig	NOUN
cana-509	97	2	.	.	PUNCT
cana-509	98	1	4	4	X
cana-509	98	2	.	.	X
cana-509	98	3	𝐴(𝐷𝑇𝑆)3	𝐴(𝐷𝑇𝑆)3	PROPN
cana-509	98	4	∘	∘	PROPN
cana-509	98	5	𝐾1	𝐾1	PROPN
cana-509	98	6	theorem	theorem	ADJ
cana-509	98	7	3.5	3.5	NUM
cana-509	98	8	.	.	PUNCT
cana-509	99	1	for	for	ADP
cana-509	99	2	every	every	DET
cana-509	99	3	𝑛	𝑛	PRON
cana-509	99	4	≥	≥	NOUN
cana-509	99	5	1	1	NUM
cana-509	99	6	there	there	PRON
cana-509	99	7	exists	exist	VERB
cana-509	99	8	a	a	DET
cana-509	99	9	triple	triple	ADJ
cana-509	99	10	triangular	triangular	NOUN
cana-509	99	11	snake	snake	NOUN
cana-509	99	12	graph	graph	NOUN
cana-509	99	13	𝑇3𝑆𝑛	𝑇3𝑆𝑛	PROPN
cana-509	99	14	∘	∘	PROPN
cana-509	99	15	𝐾1	𝐾1	PROPN
cana-509	99	16	,	,	PUNCT
cana-509	99	17	which	which	PRON
cana-509	99	18	admits	admit	VERB
cana-509	99	19	hm	hm	DET
cana-509	99	20	labeling	labeling	NOUN
cana-509	99	21	.	.	PUNCT
cana-509	100	1	proof	proof	NOUN
cana-509	100	2	.	.	PUNCT
cana-509	101	1	consider	consider	VERB
cana-509	101	2	the	the	DET
cana-509	101	3	𝑇3𝑆𝑛	𝑇3𝑆𝑛	PROPN
cana-509	101	4	∘	∘	PROPN
cana-509	101	5	𝐾1	𝐾1	PROPN
cana-509	101	6	graph	graph	NOUN
cana-509	101	7	with	with	ADP
cana-509	101	8	vertex	vertex	NOUN
cana-509	101	9	set	set	VERB
cana-509	101	10	𝑉	𝑉	PROPN
cana-509	101	11	=	=	PROPN
cana-509	101	12	{	{	PUNCT
cana-509	101	13	𝑎𝑖	𝑎𝑖	PROPN
cana-509	101	14	,	,	PUNCT
cana-509	101	15	𝑏𝑖	𝑏𝑖	NOUN
cana-509	101	16	,	,	PUNCT
cana-509	101	17	𝑐𝑖	𝑐𝑖	NOUN
cana-509	101	18	,	,	PUNCT
cana-509	101	19	𝑑𝑖	𝑑𝑖	PROPN
cana-509	101	20	,	,	PUNCT
cana-509	101	21	𝑎𝑖	𝑎𝑖	ADP
cana-509	101	22	′	′	NOUN
cana-509	101	23	,	,	PUNCT
cana-509	101	24	𝑏𝑖	𝑏𝑖	ADP
cana-509	101	25	′	′	NUM
cana-509	101	26	,	,	PUNCT
cana-509	101	27	𝑑𝑖	𝑑𝑖	PROPN
cana-509	101	28	′	′	NUM
cana-509	101	29	;	;	PUNCT
cana-509	101	30	1	1	NUM
cana-509	101	31	≤	≤	NUM
cana-509	101	32	𝑖	𝑖	SYM
cana-509	101	33	≤	≤	NUM
cana-509	101	34	𝑛	𝑛	PRON
cana-509	101	35	}	}	PUNCT
cana-509	101	36	and	and	CCONJ
cana-509	101	37	edge	edge	NOUN
cana-509	101	38	set	set	VERB
cana-509	101	39	𝐸	𝐸	NOUN
cana-509	101	40	=	=	SYM
cana-509	101	41	{	{	PUNCT
cana-509	101	42	(	(	PUNCT
cana-509	101	43	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	ADJ
cana-509	101	44	)	)	PUNCT
cana-509	101	45	∪	∪	NOUN
cana-509	101	46	(	(	PUNCT
cana-509	101	47	𝑑𝑖𝑏𝑖	𝑑𝑖𝑏𝑖	NOUN
cana-509	101	48	)	)	PUNCT
cana-509	101	49	∪	∪	NOUN
cana-509	101	50	(	(	PUNCT
cana-509	101	51	𝑐𝑖𝑑𝑖	𝑐𝑖𝑑𝑖	NOUN
cana-509	101	52	)	)	PUNCT
cana-509	101	53	∪	∪	NOUN
cana-509	101	54	(	(	PUNCT
cana-509	101	55	𝑑𝑖𝑑𝑖+1	𝑑𝑖𝑑𝑖+1	NOUN
cana-509	101	56	)	)	PUNCT
cana-509	101	57	∪	∪	NOUN
cana-509	101	58	(	(	PUNCT
cana-509	101	59	𝑏𝑖	𝑏𝑖	ADP
cana-509	101	60	′𝑑𝑖),∪	′𝑑𝑖),∪	NOUN
cana-509	101	61	(	(	PUNCT
cana-509	101	62	𝑎𝑖	𝑎𝑖	DET
cana-509	101	63	′𝑏𝑖	′𝑏𝑖	NOUN
cana-509	101	64	′	′	NUM
cana-509	101	65	)	)	PUNCT
cana-509	101	66	∪	∪	NOUN
cana-509	101	67	(	(	PUNCT
cana-509	101	68	𝑑𝑖	𝑑𝑖	NOUN
cana-509	101	69	′𝑑𝑖	′𝑑𝑖	NOUN
cana-509	101	70	)	)	PUNCT
cana-509	101	71	;	;	PUNCT
cana-509	101	72	1	1	NUM
cana-509	101	73	≤	≤	NUM
cana-509	101	74	𝑖	𝑖	SYM
cana-509	101	75	≤	≤	NUM
cana-509	101	76	𝑛	𝑛	X
cana-509	101	77	}	}	PUNCT
cana-509	101	78	now	now	ADV
cana-509	101	79	let	let	VERB
cana-509	101	80	us	we	PRON
cana-509	101	81	define	define	VERB
cana-509	101	82	a	a	DET
cana-509	101	83	function	function	NOUN
cana-509	101	84	𝜌	𝜌	ADP
cana-509	101	85	:	:	PUNCT
cana-509	101	86	𝑣	𝑣	X
cana-509	101	87	→	→	PUNCT
cana-509	101	88	{	{	PUNCT
cana-509	101	89	1,2,3	1,2,3	NUM
cana-509	101	90	,	,	PUNCT
cana-509	101	91	⋯	⋯	NOUN
cana-509	101	92	𝑞	𝑞	NOUN
cana-509	101	93	+	+	NOUN
cana-509	101	94	1	1	NUM
cana-509	101	95	}	}	PUNCT
cana-509	101	96	communications	communication	NOUN
cana-509	101	97	on	on	ADP
cana-509	101	98	applied	apply	VERB
cana-509	101	99	nonlinear	nonlinear	ADJ
cana-509	101	100	analysis	analysis	NOUN
cana-509	101	101	issn	issn	NOUN
cana-509	101	102	:	:	PUNCT
cana-509	101	103	1074	1074	NUM
cana-509	101	104	-	-	PUNCT
cana-509	101	105	133x	133x	NUM
cana-509	101	106	vol	vol	NOUN
cana-509	101	107	31	31	NUM
cana-509	101	108	no	no	NOUN
cana-509	101	109	.	.	NOUN
cana-509	101	110	2	2	NUM
cana-509	101	111	(	(	PUNCT
cana-509	101	112	2024	2024	NUM
cana-509	101	113	)	)	PUNCT
cana-509	101	114	17	17	NUM
cana-509	101	115	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	101	116	let	let	VERB
cana-509	101	117	us	we	PRON
cana-509	101	118	label	label	VERB
cana-509	101	119	the	the	DET
cana-509	101	120	vertices	vertex	NOUN
cana-509	101	121	as	as	SCONJ
cana-509	101	122	follows	follow	VERB
cana-509	101	123	.	.	PUNCT
cana-509	102	1	𝜌(𝑎1	𝜌(𝑎1	X
cana-509	102	2	)	)	PUNCT
cana-509	102	3	=	=	SYM
cana-509	102	4	1	1	NUM
cana-509	102	5	;	;	PUNCT
cana-509	102	6	𝜌(𝑎𝑖+1	𝜌(𝑎𝑖+1	X
cana-509	102	7	)	)	PUNCT
cana-509	102	8	=	=	SYM
cana-509	102	9	10𝑖	10𝑖	NOUN
cana-509	103	1	+	+	ADJ
cana-509	103	2	4	4	NUM
cana-509	103	3	;	;	PUNCT
cana-509	103	4	1	1	NUM
cana-509	103	5	≤	≤	NUM
cana-509	103	6	𝑖	𝑖	SYM
cana-509	103	7	≤	≤	NUM
cana-509	103	8	𝑛	𝑛	PRON
cana-509	103	9	−	−	PROPN
cana-509	103	10	1	1	NUM
cana-509	103	11	𝜌(𝑏1	𝜌(𝑏1	NOUN
cana-509	103	12	)	)	PUNCT
cana-509	103	13	=	=	SYM
cana-509	103	14	2	2	NUM
cana-509	103	15	;	;	PUNCT
cana-509	103	16	𝜌(𝑏𝑖+1	𝜌(𝑏𝑖+1	NUM
cana-509	103	17	)	)	PUNCT
cana-509	103	18	=	=	SYM
cana-509	103	19	10𝑖	10𝑖	NOUN
cana-509	103	20	+	+	X
cana-509	103	21	5	5	NUM
cana-509	103	22	;	;	PUNCT
cana-509	103	23	1	1	NUM
cana-509	103	24	≤	≤	NUM
cana-509	103	25	𝑖	𝑖	SYM
cana-509	103	26	≤	≤	NUM
cana-509	103	27	𝑛	𝑛	PRON
cana-509	103	28	−	−	NUM
cana-509	103	29	1	1	NUM
cana-509	103	30	𝜌(𝑑1	𝜌(𝑑1	NOUN
cana-509	103	31	)	)	PUNCT
cana-509	103	32	=	=	SYM
cana-509	103	33	5	5	NUM
cana-509	103	34	;	;	PUNCT
cana-509	103	35	𝜌(𝑑𝑖+1	𝜌(𝑑𝑖+1	NUM
cana-509	103	36	′	′	NUM
cana-509	103	37	)	)	PUNCT
cana-509	104	1	=	=	SYM
cana-509	104	2	10𝑖	10𝑖	NOUN
cana-509	105	1	+	+	ADJ
cana-509	105	2	2	2	NUM
cana-509	105	3	;	;	PUNCT
cana-509	105	4	1	1	NUM
cana-509	105	5	≤	≤	NUM
cana-509	105	6	𝑖	𝑖	SYM
cana-509	105	7	≤	≤	NUM
cana-509	105	8	𝑛	𝑛	PRON
cana-509	105	9	−	−	NUM
cana-509	105	10	1	1	NUM
cana-509	105	11	𝜌(𝑑1	𝜌(𝑑1	NOUN
cana-509	105	12	)	)	PUNCT
cana-509	105	13	=	=	SYM
cana-509	105	14	6	6	NUM
cana-509	105	15	;	;	PUNCT
cana-509	105	16	𝜌(𝑑𝑖+1	𝜌(𝑑𝑖+1	NOUN
cana-509	105	17	)	)	PUNCT
cana-509	105	18	=	=	SYM
cana-509	105	19	10𝑖	10𝑖	NOUN
cana-509	105	20	+	+	ADJ
cana-509	105	21	1	1	NUM
cana-509	105	22	;	;	PUNCT
cana-509	105	23	1	1	NUM
cana-509	105	24	≤	≤	NUM
cana-509	105	25	𝑖	𝑖	SYM
cana-509	105	26	≤	≤	NUM
cana-509	105	27	𝑛	𝑛	PRON
cana-509	105	28	−	−	NUM
cana-509	105	29	1	1	NUM
cana-509	105	30	𝜌(𝑐𝑖	𝜌(𝑐𝑖	NOUN
cana-509	105	31	)	)	PUNCT
cana-509	105	32	=	=	SYM
cana-509	105	33	10𝑖	10𝑖	NOUN
cana-509	105	34	+	+	CCONJ
cana-509	105	35	3	3	NUM
cana-509	105	36	;	;	PUNCT
cana-509	105	37	𝜌(𝑏𝑖	𝜌(𝑏𝑖	PROPN
cana-509	105	38	′	′	NUM
cana-509	105	39	)	)	PUNCT
cana-509	106	1	=	=	PRON
cana-509	106	2	10𝑖	10𝑖	PROPN
cana-509	107	1	−	−	PROPN
cana-509	107	2	2	2	NUM
cana-509	107	3	;	;	PUNCT
cana-509	107	4	𝜌(𝑐𝑖	𝜌(𝑐𝑖	NOUN
cana-509	107	5	′	′	NUM
cana-509	107	6	)	)	PUNCT
cana-509	107	7	=	=	PRON
cana-509	107	8	10𝑖	10𝑖	PROPN
cana-509	107	9	−	−	PROPN
cana-509	107	10	1	1	NUM
cana-509	107	11	;	;	PUNCT
cana-509	107	12	1	1	NUM
cana-509	107	13	≤	≤	NUM
cana-509	107	14	𝑖	𝑖	SYM
cana-509	107	15	≤	≤	NUM
cana-509	107	16	𝑛	𝑛	ADP
cana-509	107	17	the	the	DET
cana-509	107	18	induced	induced	ADJ
cana-509	107	19	edge	edge	NOUN
cana-509	107	20	labeling	labeling	NOUN
cana-509	107	21	is	be	AUX
cana-509	107	22	as	as	SCONJ
cana-509	107	23	follows	follow	VERB
cana-509	107	24	𝜌∗(𝑎1𝑏1	𝜌∗(𝑎1𝑏1	PRON
cana-509	107	25	)	)	PUNCT
cana-509	107	26	=	=	SYM
cana-509	107	27	1	1	NUM
cana-509	107	28	;	;	PUNCT
cana-509	107	29	𝜌∗(𝑎𝑖+1𝑏𝑖+1	𝜌∗(𝑎𝑖+1𝑏𝑖+1	PROPN
cana-509	107	30	)	)	PUNCT
cana-509	107	31	=	=	SYM
cana-509	107	32	10𝑖	10𝑖	NOUN
cana-509	108	1	+	+	ADJ
cana-509	108	2	4	4	NUM
cana-509	108	3	;	;	PUNCT
cana-509	108	4	1	1	NUM
cana-509	108	5	≤	≤	NUM
cana-509	108	6	𝑖	𝑖	SYM
cana-509	108	7	≤	≤	NUM
cana-509	108	8	𝑛	𝑛	PRON
cana-509	108	9	−	−	NUM
cana-509	108	10	1	1	NUM
cana-509	108	11	𝜌∗(𝑑1	𝜌∗(𝑑1	PROPN
cana-509	108	12	′	′	NUM
cana-509	108	13	𝑑1	𝑑1	NOUN
cana-509	108	14	)	)	PUNCT
cana-509	109	1	=	=	SYM
cana-509	109	2	6	6	NUM
cana-509	109	3	;	;	PUNCT
cana-509	109	4	𝜌∗(𝑑𝑖+1	𝜌∗(𝑑𝑖+1	PROPN
cana-509	109	5	′	′	NOUN
cana-509	109	6	𝑑𝑖+1	𝑑𝑖+1	NUM
cana-509	109	7	)	)	PUNCT
cana-509	109	8	=	=	SYM
cana-509	109	9	10𝑖	10𝑖	NOUN
cana-509	110	1	+	+	ADJ
cana-509	110	2	1	1	NUM
cana-509	110	3	;	;	PUNCT
cana-509	110	4	1	1	NUM
cana-509	110	5	≤	≤	NUM
cana-509	110	6	𝑖	𝑖	SYM
cana-509	110	7	≤	≤	NUM
cana-509	110	8	𝑛	𝑛	PRON
cana-509	110	9	−	−	NUM
cana-509	110	10	1	1	NUM
cana-509	110	11	𝜌∗(𝑑1	𝜌∗(𝑑1	PROPN
cana-509	110	12	′	′	NUM
cana-509	110	13	𝑏1	𝑏1	NOUN
cana-509	110	14	)	)	PUNCT
cana-509	110	15	=	=	SYM
cana-509	111	1	2	2	NUM
cana-509	111	2	;	;	PUNCT
cana-509	111	3	𝜌∗(𝑑𝑖+2	𝜌∗(𝑑𝑖+2	ADJ
cana-509	111	4	′	′	NUM
cana-509	111	5	𝑏𝑖+1	𝑏𝑖+1	NOUN
cana-509	111	6	)	)	PUNCT
cana-509	111	7	=	=	SYM
cana-509	111	8	10𝑖	10𝑖	NOUN
cana-509	112	1	+	+	CCONJ
cana-509	112	2	3	3	NUM
cana-509	112	3	;	;	PUNCT
cana-509	112	4	1	1	NUM
cana-509	112	5	≤	≤	NUM
cana-509	112	6	𝑖	𝑖	SYM
cana-509	112	7	≤	≤	NUM
cana-509	112	8	𝑛	𝑛	PRON
cana-509	112	9	−	−	PROPN
cana-509	112	10	1	1	NUM
cana-509	112	11	𝜌∗(𝑑2	𝜌∗(𝑑2	PROPN
cana-509	112	12	′	′	NOUN
cana-509	112	13	𝑏1	𝑏1	NOUN
cana-509	112	14	)	)	PUNCT
cana-509	112	15	=	=	SYM
cana-509	112	16	3	3	NUM
cana-509	112	17	;	;	PUNCT
cana-509	112	18	𝜌∗(𝑑2𝑖	𝜌∗(𝑑2𝑖	NUM
cana-509	112	19	′	′	NUM
cana-509	112	20	𝑏𝑖+1	𝑏𝑖+1	NOUN
cana-509	112	21	)	)	PUNCT
cana-509	112	22	=	=	SYM
cana-509	112	23	10𝑖	10𝑖	NOUN
cana-509	113	1	+	+	CCONJ
cana-509	113	2	8	8	NUM
cana-509	113	3	;	;	PUNCT
cana-509	113	4	1	1	NUM
cana-509	113	5	≤	≤	NUM
cana-509	113	6	𝑖	𝑖	SYM
cana-509	113	7	≤	≤	NUM
cana-509	113	8	𝑛	𝑛	DET
cana-509	113	9	−	−	NUM
cana-509	113	10	1	1	NUM
cana-509	113	11	𝜌∗(𝑑1	𝜌∗(𝑑1	PROPN
cana-509	113	12	′	′	NUM
cana-509	113	13	𝑐1	𝑐1	NOUN
cana-509	113	14	)	)	PUNCT
cana-509	113	15	=	=	SYM
cana-509	114	1	4	4	NUM
cana-509	114	2	;	;	PUNCT
cana-509	114	3	𝜌∗(𝑑2𝑖+1	𝜌∗(𝑑2𝑖+1	ADJ
cana-509	114	4	′	′	NUM
cana-509	114	5	𝑐𝑖+1	𝑐𝑖+1	NOUN
cana-509	114	6	)	)	PUNCT
cana-509	114	7	=	=	SYM
cana-509	114	8	10𝑖	10𝑖	NOUN
cana-509	114	9	+	+	ADJ
cana-509	114	10	2	2	NUM
cana-509	114	11	;	;	PUNCT
cana-509	114	12	1	1	NUM
cana-509	114	13	≤	≤	NUM
cana-509	114	14	𝑖	𝑖	SYM
cana-509	114	15	≤	≤	NUM
cana-509	114	16	𝑛	𝑛	DET
cana-509	114	17	−	−	PROPN
cana-509	114	18	1	1	NUM
cana-509	114	19	𝜌∗(𝑑2	𝜌∗(𝑑2	PROPN
cana-509	114	20	′	′	NUM
cana-509	114	21	𝑐1	𝑐1	NOUN
cana-509	114	22	)	)	PUNCT
cana-509	114	23	=	=	SYM
cana-509	114	24	5	5	NUM
cana-509	114	25	;	;	PUNCT
cana-509	114	26	𝜌∗(𝑑2𝑖	𝜌∗(𝑑2𝑖	NUM
cana-509	114	27	′	′	NUM
cana-509	114	28	𝑐𝑖+1	𝑐𝑖+1	NOUN
cana-509	114	29	)	)	PUNCT
cana-509	114	30	=	=	SYM
cana-509	114	31	10𝑖	10𝑖	NOUN
cana-509	114	32	+	+	ADJ
cana-509	114	33	7	7	NUM
cana-509	114	34	;	;	PUNCT
cana-509	114	35	1	1	NUM
cana-509	114	36	≤	≤	NUM
cana-509	114	37	𝑖	𝑖	SYM
cana-509	114	38	≤	≤	NUM
cana-509	114	39	𝑛	𝑛	DET
cana-509	114	40	−	−	PROPN
cana-509	114	41	1	1	NUM
cana-509	115	1	𝜌∗(𝑑2	𝜌∗(𝑑2	PROPN
cana-509	115	2	′	′	NUM
cana-509	115	3	𝑑2	𝑑2	NOUN
cana-509	115	4	)	)	PUNCT
cana-509	115	5	=	=	SYM
cana-509	115	6	8	8	NUM
cana-509	115	7	;	;	PUNCT
cana-509	115	8	𝜌∗(𝑑2𝑖+1	𝜌∗(𝑑2𝑖+1	ADJ
cana-509	115	9	′	′	NUM
cana-509	115	10	𝑐2𝑖+2	𝑐2𝑖+2	NOUN
cana-509	115	11	)	)	PUNCT
cana-509	115	12	=	=	SYM
cana-509	115	13	10𝑖	10𝑖	NOUN
cana-509	116	1	+	+	ADJ
cana-509	116	2	6	6	NUM
cana-509	116	3	;	;	PUNCT
cana-509	116	4	1	1	NUM
cana-509	116	5	≤	≤	NUM
cana-509	116	6	𝑖	𝑖	SYM
cana-509	116	7	≤	≤	NUM
cana-509	116	8	𝑛	𝑛	DET
cana-509	116	9	−	−	PROPN
cana-509	116	10	1	1	NUM
cana-509	116	11	𝜌∗(𝑏1	𝜌∗(𝑏1	PROPN
cana-509	116	12	′	′	NUM
cana-509	116	13	𝑑1	𝑑1	NOUN
cana-509	116	14	′	′	NUM
cana-509	116	15	)	)	PUNCT
cana-509	117	1	=	=	PUNCT
cana-509	117	2	7	7	NUM
cana-509	117	3	;	;	PUNCT
cana-509	117	4	𝜌∗(𝑑𝑖+2	𝜌∗(𝑑𝑖+2	ADJ
cana-509	117	5	′	′	NUM
cana-509	117	6	𝑏𝑖+1	𝑏𝑖+1	ADP
cana-509	117	7	′	′	NUM
cana-509	117	8	)	)	PUNCT
cana-509	118	1	=	=	PRON
cana-509	118	2	10𝑖	10𝑖	NOUN
cana-509	119	1	+	+	X
cana-509	119	2	5	5	NUM
cana-509	119	3	;	;	PUNCT
cana-509	119	4	1	1	NUM
cana-509	119	5	≤	≤	NUM
cana-509	119	6	𝑖	𝑖	SYM
cana-509	119	7	≤	≤	NUM
cana-509	119	8	𝑛	𝑛	PRON
cana-509	119	9	−	−	NUM
cana-509	119	10	1	1	NUM
cana-509	119	11	𝜌∗(𝑏𝑖	𝜌∗(𝑏𝑖	NOUN
cana-509	119	12	′𝑑2𝑖	′𝑑2𝑖	ADJ
cana-509	119	13	′	′	NUM
cana-509	119	14	)	)	PUNCT
cana-509	120	1	=	=	PUNCT
cana-509	120	2	10𝑖	10𝑖	NOUN
cana-509	120	3	;	;	PUNCT
cana-509	120	4	𝜌∗(𝑎𝑖	𝜌∗(𝑎𝑖	NUM
cana-509	120	5	′𝑏𝑖	′𝑏𝑖	NOUN
cana-509	120	6	′	′	NUM
cana-509	120	7	)	)	PUNCT
cana-509	121	1	=	=	PRON
cana-509	121	2	10𝑖	10𝑖	PROPN
cana-509	122	1	−	−	PROPN
cana-509	122	2	1	1	NUM
cana-509	122	3	;	;	PUNCT
cana-509	122	4	1	1	NUM
cana-509	122	5	≤	≤	NUM
cana-509	122	6	𝑖	𝑖	SYM
cana-509	122	7	≤	≤	NOUN
cana-509	122	8	𝑛	𝑛	PRON
cana-509	122	9	thus	thus	ADV
cana-509	122	10	,	,	PUNCT
cana-509	122	11	𝑇3𝑆𝑛	𝑇3𝑆𝑛	PROPN
cana-509	122	12	∘	∘	PROPN
cana-509	122	13	𝐾1	𝐾1	PROPN
cana-509	122	14	is	be	AUX
cana-509	122	15	a	a	DET
cana-509	122	16	hm	hm	INTJ
cana-509	122	17	labeling	labeling	NOUN
cana-509	122	18	.	.	PUNCT
cana-509	123	1	fig	fig	NOUN
cana-509	123	2	.	.	PUNCT
cana-509	124	1	5	5	X
cana-509	124	2	.	.	PUNCT
cana-509	124	3	𝑇3𝑆3	𝑇3𝑆3	PROPN
cana-509	124	4	∘	∘	PROPN
cana-509	124	5	𝐾1	𝐾1	PROPN
cana-509	124	6	communications	communication	NOUN
cana-509	124	7	on	on	ADP
cana-509	124	8	applied	apply	VERB
cana-509	124	9	nonlinear	nonlinear	ADJ
cana-509	124	10	analysis	analysis	NOUN
cana-509	124	11	issn	issn	NOUN
cana-509	124	12	:	:	PUNCT
cana-509	124	13	1074	1074	NUM
cana-509	124	14	-	-	PUNCT
cana-509	124	15	133x	133x	NUM
cana-509	124	16	vol	vol	NOUN
cana-509	124	17	31	31	NUM
cana-509	124	18	no	no	NOUN
cana-509	124	19	.	.	NOUN
cana-509	124	20	2	2	NUM
cana-509	124	21	(	(	PUNCT
cana-509	124	22	2024	2024	NUM
cana-509	124	23	)	)	PUNCT
cana-509	124	24	18	18	NUM
cana-509	124	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	124	26	4	4	NUM
cana-509	124	27	.	.	PUNCT
cana-509	124	28	application	application	NOUN
cana-509	124	29	definition	definition	NOUN
cana-509	124	30	4.1	4.1	NUM
cana-509	124	31	.	.	PUNCT
cana-509	125	1	plain	plain	ADJ
cana-509	125	2	text	text	NOUN
cana-509	125	3	an	an	DET
cana-509	125	4	original	original	ADJ
cana-509	125	5	intelligible	intelligible	ADJ
cana-509	125	6	message	message	NOUN
cana-509	125	7	is	be	AUX
cana-509	125	8	called	call	VERB
cana-509	125	9	as	as	ADV
cana-509	125	10	plain	plain	ADJ
cana-509	125	11	text	text	NOUN
cana-509	125	12	.	.	PUNCT
cana-509	126	1	definition	definition	NOUN
cana-509	126	2	4.2	4.2	NUM
cana-509	126	3	.	.	PUNCT
cana-509	127	1	cipher	cipher	ADJ
cana-509	127	2	text	text	NOUN
cana-509	127	3	the	the	DET
cana-509	127	4	transformed	transform	VERB
cana-509	127	5	message	message	NOUN
cana-509	127	6	after	after	SCONJ
cana-509	127	7	coding	code	VERB
cana-509	127	8	is	be	AUX
cana-509	127	9	called	call	VERB
cana-509	127	10	as	as	ADP
cana-509	127	11	cipher	cipher	ADJ
cana-509	127	12	text	text	NOUN
cana-509	127	13	.	.	PUNCT
cana-509	128	1	in	in	ADP
cana-509	128	2	this	this	DET
cana-509	128	3	section	section	NOUN
cana-509	128	4	,	,	PUNCT
cana-509	128	5	we	we	PRON
cana-509	128	6	used	use	VERB
cana-509	128	7	hm	hm	INTJ
cana-509	128	8	labeling	labeling	NOUN
cana-509	128	9	of	of	ADP
cana-509	128	10	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	128	11	∘	∘	PROPN
cana-509	128	12	𝐾1	𝐾1	PROPN
cana-509	128	13	graph	graph	NOUN
cana-509	128	14	to	to	PART
cana-509	128	15	encode	encode	VERB
cana-509	128	16	a	a	DET
cana-509	128	17	message	message	NOUN
cana-509	128	18	and	and	CCONJ
cana-509	128	19	created	create	VERB
cana-509	128	20	novel	novel	NOUN
cana-509	128	21	encoding	encoding	NOUN
cana-509	128	22	and	and	CCONJ
cana-509	128	23	decoding	decode	VERB
cana-509	128	24	techniques	technique	NOUN
cana-509	128	25	that	that	PRON
cana-509	128	26	increase	increase	VERB
cana-509	128	27	the	the	DET
cana-509	128	28	secrecy	secrecy	NOUN
cana-509	128	29	of	of	ADP
cana-509	128	30	the	the	DET
cana-509	128	31	coded	code	VERB
cana-509	128	32	message	message	NOUN
cana-509	128	33	.	.	PUNCT
cana-509	129	1	4.3	4.3	NUM
cana-509	129	2	.	.	PUNCT
cana-509	129	3	algorithm	algorithm	NOUN
cana-509	129	4	for	for	ADP
cana-509	129	5	encoding	encode	VERB
cana-509	129	6	•	•	NOUN
cana-509	129	7	first	first	ADJ
cana-509	129	8	assign	assign	VERB
cana-509	129	9	the	the	DET
cana-509	129	10	values	value	NOUN
cana-509	129	11	of	of	ADP
cana-509	129	12	the	the	DET
cana-509	129	13	alphabet	alphabet	NOUN
cana-509	129	14	ranges	range	VERB
cana-509	129	15	over	over	ADP
cana-509	129	16	1	1	NUM
cana-509	129	17	-	-	SYM
cana-509	129	18	26	26	NUM
cana-509	129	19	.	.	PUNCT
cana-509	130	1	•	•	INTJ
cana-509	130	2	it	it	PRON
cana-509	130	3	is	be	AUX
cana-509	130	4	the	the	DET
cana-509	130	5	original	original	ADJ
cana-509	130	6	position	position	NOUN
cana-509	130	7	of	of	ADP
cana-509	130	8	the	the	DET
cana-509	130	9	alphabet	alphabet	NOUN
cana-509	130	10	.	.	PUNCT
cana-509	131	1	then	then	ADV
cana-509	131	2	shift	shift	VERB
cana-509	131	3	the	the	DET
cana-509	131	4	alphabet	alphabet	NOUN
cana-509	131	5	using	use	VERB
cana-509	131	6	the	the	DET
cana-509	131	7	formula	formula	NOUN
cana-509	131	8	𝑦	𝑦	NOUN
cana-509	131	9	=	=	PUNCT
cana-509	131	10	(	(	PUNCT
cana-509	131	11	𝑥	𝑥	X
cana-509	132	1	+	+	CCONJ
cana-509	132	2	𝑛)(mod26),where	𝑛)(mod26),where	PRON
cana-509	132	3	x	x	PART
cana-509	132	4	denotes	denote	VERB
cana-509	132	5	the	the	DET
cana-509	132	6	original	original	ADJ
cana-509	132	7	position	position	NOUN
cana-509	132	8	of	of	ADP
cana-509	132	9	the	the	DET
cana-509	132	10	alphabet	alphabet	NOUN
cana-509	132	11	and	and	CCONJ
cana-509	132	12	n	n	NOUN
cana-509	132	13	denotes	denote	VERB
cana-509	132	14	the	the	DET
cana-509	132	15	length	length	NOUN
cana-509	132	16	of	of	ADP
cana-509	132	17	the	the	DET
cana-509	132	18	word	word	NOUN
cana-509	132	19	.	.	PUNCT
cana-509	133	1	in	in	ADP
cana-509	133	2	this	this	DET
cana-509	133	3	way	way	NOUN
cana-509	133	4	,	,	PUNCT
cana-509	133	5	encoding	encode	VERB
cana-509	133	6	table	table	NOUN
cana-509	133	7	was	be	AUX
cana-509	133	8	created	create	VERB
cana-509	133	9	.	.	PUNCT
cana-509	134	1	•	•	NUM
cana-509	134	2	convert	convert	VERB
cana-509	134	3	the	the	DET
cana-509	134	4	plaintext	plaintext	NOUN
cana-509	134	5	message	message	NOUN
cana-509	134	6	into	into	ADP
cana-509	134	7	a	a	DET
cana-509	134	8	sequence	sequence	NOUN
cana-509	134	9	of	of	ADP
cana-509	134	10	integers	integer	NOUN
cana-509	134	11	using	use	VERB
cana-509	134	12	the	the	DET
cana-509	134	13	encoding	encoding	NOUN
cana-509	134	14	table	table	NOUN
cana-509	134	15	.	.	PUNCT
cana-509	135	1	•	•	NUM
cana-509	135	2	construct	construct	VERB
cana-509	135	3	the	the	DET
cana-509	135	4	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	135	5	∘	∘	PROPN
cana-509	135	6	𝐾1	𝐾1	PROPN
cana-509	135	7	graph	graph	NOUN
cana-509	135	8	corresponding	correspond	VERB
cana-509	135	9	to	to	ADP
cana-509	135	10	the	the	DET
cana-509	135	11	length	length	NOUN
cana-509	135	12	'	'	PUNCT
cana-509	135	13	𝑛	𝑛	NOUN
cana-509	135	14	'	'	PUNCT
cana-509	135	15	of	of	ADP
cana-509	135	16	the	the	DET
cana-509	135	17	plaintext	plaintext	NOUN
cana-509	135	18	message	message	NOUN
cana-509	135	19	𝑇𝑆𝑛	𝑇𝑆𝑛	PROPN
cana-509	135	20	∘	∘	VERB
cana-509	135	21	𝐾1	𝐾1	PROPN
cana-509	135	22	graph	graph	NOUN
cana-509	135	23	,	,	PUNCT
cana-509	135	24	let	let	VERB
cana-509	135	25	𝑏𝑖	𝑏𝑖	ADP
cana-509	135	26	,	,	PUNCT
cana-509	135	27	𝑐𝑖	𝑐𝑖	NOUN
cana-509	135	28	be	be	AUX
cana-509	135	29	the	the	DET
cana-509	135	30	vertices	vertex	NOUN
cana-509	135	31	of	of	ADP
cana-509	135	32	a	a	DET
cana-509	135	33	triangular	triangular	NOUN
cana-509	135	34	snake	snake	NOUN
cana-509	135	35	.	.	PUNCT
cana-509	136	1	let	let	VERB
cana-509	136	2	𝑎𝑖	𝑎𝑖	NUM
cana-509	136	3	,	,	PUNCT
cana-509	136	4	𝑑𝑖	𝑑𝑖	PUNCT
cana-509	136	5	be	be	VERB
cana-509	136	6	the	the	DET
cana-509	136	7	pendent	pendent	NOUN
cana-509	136	8	vertices	vertex	NOUN
cana-509	136	9	join	join	VERB
cana-509	136	10	𝑎𝑖	𝑎𝑖	ADP
cana-509	136	11	,	,	PUNCT
cana-509	136	12	𝑏𝑖	𝑏𝑖	ADP
cana-509	137	1	and	and	CCONJ
cana-509	137	2	𝑐𝑖	𝑐𝑖	NOUN
cana-509	137	3	,	,	PUNCT
cana-509	137	4	𝑑𝑖	𝑑𝑖	VERB
cana-509	137	5	•	•	NUM
cana-509	138	1	𝑣𝑖	𝑣𝑖	ADP
cana-509	138	2	denotes	denote	VERB
cana-509	138	3	the	the	DET
cana-509	138	4	vertices	vertex	NOUN
cana-509	138	5	in	in	ADP
cana-509	138	6	the	the	DET
cana-509	138	7	graph	graph	NOUN
cana-509	138	8	.	.	PUNCT
cana-509	139	1	𝑒𝑖	𝑒𝑖	ADP
cana-509	139	2	denotes	denote	VERB
cana-509	139	3	the	the	DET
cana-509	139	4	edges	edge	NOUN
cana-509	139	5	in	in	ADP
cana-509	139	6	the	the	DET
cana-509	139	7	graph	graph	NOUN
cana-509	139	8	.	.	PUNCT
cana-509	140	1	𝑤𝑖(𝑒𝑖	𝑤𝑖(𝑒𝑖	PROPN
cana-509	140	2	)	)	PUNCT
cana-509	140	3	denotes	denote	VERB
cana-509	140	4	the	the	DET
cana-509	140	5	weights	weight	NOUN
cana-509	140	6	of	of	ADP
cana-509	140	7	each	each	DET
cana-509	140	8	edges	edge	NOUN
cana-509	140	9	.	.	PUNCT
cana-509	141	1	•	•	NUM
cana-509	141	2	next	next	ADV
cana-509	141	3	,	,	PUNCT
cana-509	141	4	the	the	DET
cana-509	141	5	weights	weight	NOUN
cana-509	141	6	𝑤1	𝑤1	VERB
cana-509	141	7	,	,	PUNCT
cana-509	141	8	𝑤2	𝑤2	NOUN
cana-509	141	9	,	,	PUNCT
cana-509	141	10	…	…	PUNCT
cana-509	141	11	𝑤𝑛	𝑤𝑛	NOUN
cana-509	141	12	to	to	ADP
cana-509	141	13	each	each	PRON
cana-509	141	14	to	to	ADP
cana-509	141	15	the	the	DET
cana-509	141	16	edges	edge	NOUN
cana-509	141	17	𝑒1	𝑒1	NOUN
cana-509	141	18	,	,	PUNCT
cana-509	141	19	𝑒2	𝑒2	NOUN
cana-509	141	20	,	,	PUNCT
cana-509	141	21	…	…	PUNCT
cana-509	141	22	𝑒𝑛	𝑒𝑛	NOUN
cana-509	141	23	in	in	ADP
cana-509	141	24	such	such	DET
cana-509	141	25	a	a	DET
cana-509	141	26	way	way	NOUN
cana-509	141	27	that	that	PRON
cana-509	141	28	𝑊1(𝑒1	𝑊1(𝑒1	NOUN
cana-509	141	29	)	)	PUNCT
cana-509	141	30	<	<	X
cana-509	141	31	𝑊2(𝑒2	𝑊2(𝑒2	NOUN
cana-509	141	32	)	)	PUNCT
cana-509	141	33	…	…	PUNCT
cana-509	141	34	𝑊𝑛(𝑒𝑛	𝑊𝑛(𝑒𝑛	NOUN
cana-509	141	35	)	)	PUNCT
cana-509	141	36	.	.	PUNCT
cana-509	142	1	4.3.1	4.3.1	X
cana-509	142	2	.	.	PUNCT
cana-509	142	3	method	method	NOUN
cana-509	142	4	for	for	ADP
cana-509	142	5	calculating	calculate	VERB
cana-509	142	6	weight	weight	NOUN
cana-509	142	7	of	of	ADP
cana-509	142	8	edges	edge	NOUN
cana-509	142	9	•	•	ADV
cana-509	142	10	find	find	VERB
cana-509	142	11	the	the	DET
cana-509	142	12	weight	weight	NOUN
cana-509	142	13	of	of	ADP
cana-509	142	14	each	each	DET
cana-509	142	15	edge	edge	NOUN
cana-509	142	16	𝑤𝑖(𝑒𝑖	𝑤𝑖(𝑒𝑖	NOUN
cana-509	142	17	)	)	PUNCT
cana-509	142	18	by	by	ADP
cana-509	142	19	the	the	DET
cana-509	142	20	formulation	formulation	NOUN
cana-509	142	21	.	.	PUNCT
cana-509	143	1	𝑤𝑖(𝑒𝑖	𝑤𝑖(𝑒𝑖	PROPN
cana-509	143	2	)	)	PUNCT
cana-509	144	1	=	=	PRON
cana-509	144	2	(	(	PUNCT
cana-509	144	3	numerical	numerical	ADJ
cana-509	144	4	value	value	NOUN
cana-509	144	5	of	of	ADP
cana-509	144	6	vertex	vertex	NOUN
cana-509	144	7	𝑣𝑖	𝑣𝑖	ADP
cana-509	144	8	corresponding	correspond	VERB
cana-509	144	9	to	to	ADP
cana-509	144	10	the	the	DET
cana-509	144	11	edge	edge	NOUN
cana-509	144	12	𝑒𝑖	𝑒𝑖	NOUN
cana-509	144	13	)	)	PUNCT
cana-509	144	14	10𝑖	10𝑖	NOUN
cana-509	144	15	,	,	PUNCT
cana-509	144	16	where	where	SCONJ
cana-509	144	17	𝑖	𝑖	ADP
cana-509	144	18	=	=	SYM
cana-509	144	19	1,2,3	1,2,3	NUM
cana-509	144	20	…	…	PUNCT
cana-509	144	21	𝑛.	𝑛.	NOUN
cana-509	144	22	•	•	ADP
cana-509	144	23	the	the	DET
cana-509	144	24	resulting	result	VERB
cana-509	144	25	values	value	NOUN
cana-509	144	26	are	be	AUX
cana-509	144	27	the	the	DET
cana-509	144	28	weight	weight	NOUN
cana-509	144	29	of	of	ADP
cana-509	144	30	each	each	DET
cana-509	144	31	edge	edge	NOUN
cana-509	144	32	.	.	PUNCT
cana-509	145	1	now	now	ADV
cana-509	145	2	assign	assign	VERB
cana-509	145	3	the	the	DET
cana-509	145	4	weight	weight	NOUN
cana-509	145	5	of	of	ADP
cana-509	145	6	each	each	DET
cana-509	145	7	edge	edge	NOUN
cana-509	145	8	in	in	ADP
cana-509	145	9	the	the	DET
cana-509	145	10	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	145	11	∘	∘	VERB
cana-509	145	12	𝐾1	𝐾1	PROPN
cana-509	145	13	graph	graph	NOUN
cana-509	145	14	.	.	PUNCT
cana-509	146	1	•	•	NOUN
cana-509	146	2	finally	finally	ADV
cana-509	146	3	send	send	VERB
cana-509	146	4	this	this	DET
cana-509	146	5	graph	graph	NOUN
cana-509	146	6	to	to	ADP
cana-509	146	7	the	the	DET
cana-509	146	8	receiver	receiver	NOUN
cana-509	146	9	by	by	ADP
cana-509	146	10	hiding	hide	VERB
cana-509	146	11	the	the	DET
cana-509	146	12	values	value	NOUN
cana-509	146	13	of	of	ADP
cana-509	146	14	the	the	DET
cana-509	146	15	vertex	vertex	NOUN
cana-509	146	16	which	which	PRON
cana-509	146	17	is	be	AUX
cana-509	146	18	known	know	VERB
cana-509	146	19	as	as	ADP
cana-509	146	20	encrypted	encrypt	VERB
cana-509	146	21	message	message	NOUN
cana-509	146	22	along	along	ADP
cana-509	146	23	with	with	ADP
cana-509	146	24	the	the	DET
cana-509	146	25	public	public	ADJ
cana-509	146	26	key	key	NOUN
cana-509	146	27	10	10	NUM
cana-509	146	28	.	.	PUNCT
cana-509	147	1	4.4	4.4	NUM
cana-509	147	2	.	.	PUNCT
cana-509	147	3	algorithm	algorithm	NOUN
cana-509	147	4	for	for	ADP
cana-509	147	5	decoding	decode	VERB
cana-509	147	6	•	•	NUM
cana-509	147	7	arrange	arrange	NOUN
cana-509	147	8	the	the	DET
cana-509	147	9	weights	weight	NOUN
cana-509	147	10	of	of	ADP
cana-509	147	11	edge	edge	NOUN
cana-509	147	12	in	in	ADP
cana-509	147	13	ascending	ascend	VERB
cana-509	147	14	order	order	NOUN
cana-509	147	15	of	of	ADP
cana-509	147	16	mod	mod	ADJ
cana-509	147	17	values	value	NOUN
cana-509	147	18	and	and	CCONJ
cana-509	147	19	add	add	VERB
cana-509	147	20	the	the	DET
cana-509	147	21	multiples	multiple	NOUN
cana-509	147	22	of	of	ADP
cana-509	147	23	10	10	NUM
cana-509	147	24	respectively	respectively	ADV
cana-509	147	25	.	.	PUNCT
cana-509	148	1	communications	communication	NOUN
cana-509	148	2	on	on	ADP
cana-509	148	3	applied	apply	VERB
cana-509	148	4	nonlinear	nonlinear	ADJ
cana-509	148	5	analysis	analysis	NOUN
cana-509	148	6	issn	issn	NOUN
cana-509	148	7	:	:	PUNCT
cana-509	148	8	1074	1074	NUM
cana-509	148	9	-	-	PUNCT
cana-509	148	10	133x	133x	NUM
cana-509	148	11	vol	vol	NOUN
cana-509	148	12	31	31	NUM
cana-509	148	13	no	no	NOUN
cana-509	148	14	.	.	NOUN
cana-509	148	15	2	2	NUM
cana-509	148	16	(	(	PUNCT
cana-509	148	17	2024	2024	NUM
cana-509	148	18	)	)	PUNCT
cana-509	148	19	19	19	NUM
cana-509	148	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	148	21	•	•	NOUN
cana-509	148	22	decode	decode	VERB
cana-509	148	23	the	the	DET
cana-509	148	24	cipher	cipher	ADJ
cana-509	148	25	text	text	NOUN
cana-509	148	26	into	into	ADP
cana-509	148	27	plain	plain	ADJ
cana-509	148	28	text	text	NOUN
cana-509	148	29	message	message	NOUN
cana-509	148	30	from	from	ADP
cana-509	148	31	the	the	DET
cana-509	148	32	encoding	encoding	NOUN
cana-509	148	33	table	table	NOUN
cana-509	148	34	to	to	PART
cana-509	148	35	get	get	VERB
cana-509	148	36	the	the	DET
cana-509	148	37	original	original	ADJ
cana-509	148	38	plain	plain	ADJ
cana-509	148	39	text	text	NOUN
cana-509	148	40	message	message	NOUN
cana-509	148	41	.	.	PUNCT
cana-509	149	1	•	•	NUM
cana-509	149	2	plaintext	plaintext	NOUN
cana-509	149	3	:	:	PUNCT
cana-509	149	4	bishop	bishop	PROPN
cana-509	149	5	4.5	4.5	NUM
cana-509	149	6	.	.	PUNCT
cana-509	150	1	encoding	encode	VERB
cana-509	150	2	•	•	NUM
cana-509	150	3	length	length	NOUN
cana-509	150	4	of	of	ADP
cana-509	150	5	the	the	DET
cana-509	150	6	message	message	NOUN
cana-509	150	7	=	=	NOUN
cana-509	150	8	6	6	NUM
cana-509	150	9	,	,	PUNCT
cana-509	150	10	𝑦	𝑦	NOUN
cana-509	150	11	=	=	PUNCT
cana-509	150	12	(	(	PUNCT
cana-509	150	13	𝑥	𝑥	X
cana-509	150	14	+	+	CCONJ
cana-509	150	15	𝑛)(mod26	𝑛)(mod26	ADJ
cana-509	150	16	)	)	PUNCT
cana-509	150	17	•	•	NUM
cana-509	150	18	construct	construct	VERB
cana-509	150	19	the	the	DET
cana-509	150	20	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	150	21	∘	∘	PROPN
cana-509	150	22	𝐾1	𝐾1	PROPN
cana-509	150	23	graph	graph	NOUN
cana-509	150	24	by	by	ADP
cana-509	150	25	assigning	assign	VERB
cana-509	150	26	the	the	DET
cana-509	150	27	above	above	ADJ
cana-509	150	28	sequence	sequence	NOUN
cana-509	150	29	of	of	ADP
cana-509	150	30	integers	integer	NOUN
cana-509	150	31	to	to	ADP
cana-509	150	32	the	the	DET
cana-509	150	33	vertices	vertex	NOUN
cana-509	150	34	of	of	ADP
cana-509	150	35	the	the	DET
cana-509	150	36	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	150	37	∘	∘	VERB
cana-509	150	38	𝐾1	𝐾1	PROPN
cana-509	150	39	graph	graph	NOUN
cana-509	150	40	.	.	PUNCT
cana-509	150	41	table	table	NOUN
cana-509	150	42	1	1	NUM
cana-509	150	43	.	.	X
cana-509	150	44	encoding	encode	VERB
cana-509	150	45	table	table	NOUN
cana-509	150	46	a	a	DET
cana-509	150	47	b	b	NOUN
cana-509	150	48	c	c	NOUN
cana-509	151	1	d	d	X
cana-509	151	2	e	e	X
cana-509	151	3	f	f	PROPN
cana-509	151	4	g	g	PROPN
cana-509	151	5	h	h	NOUN
cana-509	152	1	i	i	PRON
cana-509	152	2	j	j	PROPN
cana-509	153	1	k	k	PROPN
cana-509	153	2	l	l	PROPN
cana-509	153	3	m	m	VERB
cana-509	153	4	7	7	NUM
cana-509	153	5	8	8	NUM
cana-509	153	6	9	9	NUM
cana-509	153	7	10	10	NUM
cana-509	153	8	11	11	NUM
cana-509	153	9	12	12	NUM
cana-509	153	10	13	13	NUM
cana-509	153	11	14	14	NUM
cana-509	153	12	15	15	NUM
cana-509	153	13	16	16	NUM
cana-509	153	14	17	17	NUM
cana-509	153	15	18	18	NUM
cana-509	153	16	19	19	NUM
cana-509	154	1	n	n	NUM
cana-509	154	2	o	o	NOUN
cana-509	154	3	p	p	X
cana-509	154	4	q	q	NOUN
cana-509	154	5	r	r	NOUN
cana-509	154	6	s	s	PROPN
cana-509	154	7	t	t	NOUN
cana-509	154	8	u	u	NOUN
cana-509	154	9	v	v	PROPN
cana-509	154	10	w	w	NOUN
cana-509	154	11	x	x	SYM
cana-509	154	12	y	y	PROPN
cana-509	154	13	z	z	PROPN
cana-509	154	14	20	20	NUM
cana-509	154	15	21	21	NUM
cana-509	154	16	22	22	NUM
cana-509	154	17	22	22	NUM
cana-509	154	18	23	23	NUM
cana-509	154	19	24	24	NUM
cana-509	154	20	25	25	NUM
cana-509	154	21	26	26	NUM
cana-509	154	22	1	1	NUM
cana-509	154	23	2	2	NUM
cana-509	154	24	3	3	NUM
cana-509	154	25	5	5	NUM
cana-509	154	26	6	6	NUM
cana-509	154	27	each	each	DET
cana-509	154	28	letter	letter	NOUN
cana-509	154	29	from	from	ADP
cana-509	154	30	the	the	DET
cana-509	154	31	table	table	NOUN
cana-509	154	32	above	above	ADV
cana-509	154	33	represented	represent	VERB
cana-509	154	34	as	as	SCONJ
cana-509	154	35	follows	follow	VERB
cana-509	154	36	b	b	X
cana-509	155	1	i	i	PRON
cana-509	155	2	s	s	VERB
cana-509	155	3	h	h	NOUN
cana-509	156	1	o	o	NOUN
cana-509	156	2	p	p	NOUN
cana-509	156	3	8	8	NUM
cana-509	156	4	15	15	NUM
cana-509	156	5	25	25	NUM
cana-509	156	6	14	14	NUM
cana-509	156	7	21	21	NUM
cana-509	156	8	22	22	NUM
cana-509	156	9	fig	fig	NOUN
cana-509	156	10	.	.	PUNCT
cana-509	157	1	6	6	X
cana-509	157	2	.	.	X
cana-509	157	3	encoding	encode	VERB
cana-509	157	4	communications	communication	NOUN
cana-509	157	5	on	on	ADP
cana-509	157	6	applied	apply	VERB
cana-509	157	7	nonlinear	nonlinear	ADJ
cana-509	157	8	analysis	analysis	NOUN
cana-509	157	9	issn	issn	NOUN
cana-509	157	10	:	:	PUNCT
cana-509	157	11	1074	1074	NUM
cana-509	157	12	-	-	PUNCT
cana-509	157	13	133x	133x	NUM
cana-509	157	14	vol	vol	NOUN
cana-509	157	15	31	31	NUM
cana-509	157	16	no	no	NOUN
cana-509	157	17	.	.	NOUN
cana-509	157	18	2	2	NUM
cana-509	157	19	(	(	PUNCT
cana-509	157	20	2024	2024	NUM
cana-509	157	21	)	)	PUNCT
cana-509	157	22	20	20	NUM
cana-509	157	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	157	24	•	•	NUM
cana-509	157	25	applying	apply	VERB
cana-509	157	26	weights	weight	NOUN
cana-509	157	27	to	to	ADP
cana-509	157	28	the	the	DET
cana-509	157	29	𝑤𝑖	𝑤𝑖	NOUN
cana-509	157	30	,	,	PUNCT
cana-509	157	31	𝑖	𝑖	NOUN
cana-509	157	32	=	=	NOUN
cana-509	157	33	1,2,3,4,5,6	1,2,3,4,5,6	NUM
cana-509	157	34	to	to	ADP
cana-509	157	35	the	the	DET
cana-509	157	36	corresponding	corresponding	ADJ
cana-509	157	37	edges	edge	NOUN
cana-509	157	38	of	of	ADP
cana-509	157	39	the	the	DET
cana-509	157	40	vertices	vertex	NOUN
cana-509	157	41	𝑤1(8	𝑤1(8	PROPN
cana-509	157	42	)	)	PUNCT
cana-509	157	43	<	<	X
cana-509	157	44	𝑤2(15	𝑤2(15	NOUN
cana-509	157	45	)	)	PUNCT
cana-509	157	46	<	<	X
cana-509	158	1	𝑊3(25	𝑊3(25	PROPN
cana-509	158	2	)	)	PUNCT
cana-509	158	3	<	<	X
cana-509	158	4	𝑤4(14	𝑤4(14	NOUN
cana-509	158	5	)	)	PUNCT
cana-509	158	6	<	<	X
cana-509	158	7	𝑤5(21	𝑤5(21	PROPN
cana-509	158	8	)	)	PUNCT
cana-509	158	9	<	<	X
cana-509	158	10	𝑤6(22	𝑤6(22	PROPN
cana-509	158	11	)	)	PUNCT
cana-509	158	12	.	.	PUNCT
cana-509	159	1	•	•	NUM
cana-509	159	2	weights	weight	NOUN
cana-509	159	3	are	be	AUX
cana-509	159	4	obtained	obtain	VERB
cana-509	159	5	by	by	ADP
cana-509	159	6	subtracting	subtract	VERB
cana-509	159	7	the	the	DET
cana-509	159	8	multiples	multiple	NOUN
cana-509	159	9	of	of	ADP
cana-509	159	10	10	10	NUM
cana-509	159	11	from	from	ADP
cana-509	159	12	each	each	DET
cana-509	159	13	adjacent	adjacent	ADJ
cana-509	159	14	numeric	numeric	ADJ
cana-509	159	15	value	value	NOUN
cana-509	159	16	of	of	ADP
cana-509	159	17	the	the	DET
cana-509	159	18	vertices	vertex	NOUN
cana-509	159	19	.	.	PUNCT
cana-509	160	1	•	•	NUM
cana-509	160	2	weight	weight	NOUN
cana-509	160	3	of	of	ADP
cana-509	160	4	edge	edge	NOUN
cana-509	160	5	𝑒1	𝑒1	NOUN
cana-509	160	6	=	=	PUNCT
cana-509	160	7	𝑤1	𝑤1	NOUN
cana-509	160	8	=	=	SYM
cana-509	160	9	8	8	NUM
cana-509	160	10	−	−	NUM
cana-509	160	11	10	10	NUM
cana-509	160	12	=	=	SYM
cana-509	160	13	−2	−2	NOUN
cana-509	160	14	𝑒2	𝑒2	NOUN
cana-509	160	15	=	=	SYM
cana-509	160	16	𝑤2	𝑤2	NOUN
cana-509	160	17	=	=	SYM
cana-509	160	18	15	15	NUM
cana-509	160	19	−	−	NOUN
cana-509	160	20	20	20	NUM
cana-509	160	21	=	=	SYM
cana-509	160	22	−5	−5	NOUN
cana-509	160	23	𝑒3	𝑒3	NOUN
cana-509	160	24	=	=	SYM
cana-509	160	25	𝑤3	𝑤3	NOUN
cana-509	160	26	=	=	NOUN
cana-509	161	1	25	25	NUM
cana-509	161	2	−	−	NUM
cana-509	161	3	30	30	NUM
cana-509	161	4	=	=	SYM
cana-509	161	5	−5	−5	NOUN
cana-509	161	6	𝑒4	𝑒4	ADJ
cana-509	161	7	=	=	SYM
cana-509	161	8	𝑤4	𝑤4	NOUN
cana-509	161	9	=	=	SYM
cana-509	161	10	14	14	NUM
cana-509	161	11	−	−	NUM
cana-509	161	12	40	40	NUM
cana-509	161	13	=	=	SYM
cana-509	161	14	−26	−26	NUM
cana-509	161	15	𝑒5	𝑒5	NOUN
cana-509	161	16	=	=	NOUN
cana-509	161	17	𝑤5	𝑤5	NOUN
cana-509	161	18	=	=	SYM
cana-509	161	19	21	21	NUM
cana-509	161	20	−	−	NUM
cana-509	161	21	50	50	NUM
cana-509	161	22	=	=	NOUN
cana-509	161	23	−29	−29	NOUN
cana-509	161	24	𝑒6	𝑒6	NOUN
cana-509	161	25	=	=	SYM
cana-509	161	26	𝑤6	𝑤6	NOUN
cana-509	161	27	=	=	SYM
cana-509	161	28	22	22	NUM
cana-509	161	29	−	−	NUM
cana-509	161	30	60	60	NUM
cana-509	161	31	=	=	SYM
cana-509	161	32	−38	−38	NOUN
cana-509	161	33	•	•	NOUN
cana-509	161	34	cipher	cipher	ADJ
cana-509	161	35	graph	graph	NOUN
cana-509	161	36	is	be	AUX
cana-509	161	37	created	create	VERB
cana-509	161	38	by	by	ADP
cana-509	161	39	assigning	assign	VERB
cana-509	161	40	the	the	DET
cana-509	161	41	weight	weight	NOUN
cana-509	161	42	of	of	ADP
cana-509	161	43	each	each	DET
cana-509	161	44	edge	edge	NOUN
cana-509	161	45	and	and	CCONJ
cana-509	161	46	hiding	hide	VERB
cana-509	161	47	the	the	DET
cana-509	161	48	values	value	NOUN
cana-509	161	49	of	of	ADP
cana-509	161	50	the	the	DET
cana-509	161	51	each	each	DET
cana-509	161	52	vertices	vertex	NOUN
cana-509	161	53	.	.	PUNCT
cana-509	162	1	4.6	4.6	NUM
cana-509	162	2	.	.	PUNCT
cana-509	162	3	encrypted	encrypt	VERB
cana-509	162	4	message	message	NOUN
cana-509	162	5	fig	fig	NOUN
cana-509	162	6	.	.	PUNCT
cana-509	163	1	7	7	X
cana-509	163	2	.	.	X
cana-509	163	3	encryption	encryption	NOUN
cana-509	163	4	4.7	4.7	NUM
cana-509	163	5	.	.	PUNCT
cana-509	164	1	decryption	decryption	NOUN
cana-509	164	2	•	•	ADP
cana-509	164	3	the	the	DET
cana-509	164	4	receiver	receiver	NOUN
cana-509	164	5	receives	receive	VERB
cana-509	164	6	the	the	DET
cana-509	164	7	encrypted	encrypt	VERB
cana-509	164	8	message	message	NOUN
cana-509	164	9	.	.	PUNCT
cana-509	165	1	•	•	NUM
cana-509	165	2	arrange	arrange	VERB
cana-509	165	3	the	the	DET
cana-509	165	4	weights	weight	NOUN
cana-509	165	5	of	of	ADP
cana-509	165	6	edges	edge	NOUN
cana-509	165	7	in	in	ADP
cana-509	165	8	ascending	ascend	VERB
cana-509	165	9	order	order	NOUN
cana-509	165	10	of	of	ADP
cana-509	165	11	mod	mod	ADJ
cana-509	165	12	values	value	NOUN
cana-509	166	1	|	|	ADV
cana-509	166	2	−	−	PROPN
cana-509	167	1	2|	2|	INTJ
cana-509	167	2	<	<	X
cana-509	168	1	|	|	ADV
cana-509	168	2	−	−	PROPN
cana-509	168	3	5|	5|	NUM
cana-509	168	4	<	<	X
cana-509	168	5	|	|	ADV
cana-509	168	6	−	−	PROPN
cana-509	168	7	5|	5|	NUM
cana-509	168	8	<	<	X
cana-509	168	9	|	|	ADV
cana-509	168	10	−	−	PROPN
cana-509	168	11	26|	26|	PROPN
cana-509	168	12	<	<	X
cana-509	168	13	|	|	ADV
cana-509	168	14	−	−	PROPN
cana-509	168	15	29|	29|	NUM
cana-509	168	16	<	<	X
cana-509	168	17	|	|	ADV
cana-509	168	18	−	−	PROPN
cana-509	168	19	38|	38|	NUM
cana-509	168	20	•	•	NUM
cana-509	168	21	adding	add	VERB
cana-509	168	22	the	the	DET
cana-509	168	23	multiples	multiple	NOUN
cana-509	168	24	of	of	ADP
cana-509	168	25	10	10	NUM
cana-509	168	26	to	to	ADP
cana-509	168	27	each	each	DET
cana-509	168	28	adjacent	adjacent	ADJ
cana-509	168	29	value	value	NOUN
cana-509	168	30	:	:	PUNCT
cana-509	168	31	|	|	ADV
cana-509	168	32	−	−	PROPN
cana-509	168	33	2	2	NUM
cana-509	168	34	+	+	CCONJ
cana-509	168	35	1(10)|	1(10)|	NUM
cana-509	168	36	<	<	X
cana-509	168	37	|	|	ADV
cana-509	168	38	−	−	PROPN
cana-509	168	39	5	5	NUM
cana-509	168	40	+	+	CCONJ
cana-509	168	41	2(10)|	2(10)|	NUM
cana-509	168	42	<	<	X
cana-509	168	43	|	|	ADV
cana-509	168	44	−	−	PROPN
cana-509	168	45	5	5	NUM
cana-509	169	1	+	+	NUM
cana-509	169	2	3(10)|	3(10)|	NUM
cana-509	169	3	<	<	X
cana-509	169	4	|	|	ADV
cana-509	169	5	−	−	PROPN
cana-509	169	6	26	26	NUM
cana-509	169	7	+	+	CCONJ
cana-509	169	8	4(10)|	4(10)|	NUM
cana-509	169	9	<	<	X
cana-509	169	10	|	|	ADV
cana-509	169	11	−	−	PROPN
cana-509	169	12	29	29	NUM
cana-509	169	13	+	+	CCONJ
cana-509	169	14	5(10)|	5(10)|	NUM
cana-509	169	15	<	<	X
cana-509	169	16	|	|	ADV
cana-509	169	17	−	−	PROPN
cana-509	169	18	38	38	NUM
cana-509	169	19	+	+	SYM
cana-509	169	20	6(10)|	6(10)|	NUM
cana-509	169	21	•	•	NUM
cana-509	169	22	sequence	sequence	NOUN
cana-509	169	23	of	of	ADP
cana-509	169	24	integers	integer	NOUN
cana-509	169	25	is	be	AUX
cana-509	169	26	8,15,25,14,21,22	8,15,25,14,21,22	NUM
cana-509	169	27	•	•	NUM
cana-509	169	28	8,15,25,14,21,22	8,15,25,14,21,22	PROPN
cana-509	169	29	=	=	SYM
cana-509	169	30	b	b	PROPN
cana-509	169	31	,	,	PUNCT
cana-509	169	32	i	i	PRON
cana-509	169	33	,	,	PUNCT
cana-509	169	34	s	s	PROPN
cana-509	169	35	,	,	PUNCT
cana-509	169	36	h	h	NOUN
cana-509	169	37	,	,	PUNCT
cana-509	169	38	o	o	NOUN
cana-509	169	39	,	,	PUNCT
cana-509	169	40	p	p	NOUN
cana-509	169	41	communications	communication	NOUN
cana-509	169	42	on	on	ADP
cana-509	169	43	applied	apply	VERB
cana-509	169	44	nonlinear	nonlinear	ADJ
cana-509	169	45	analysis	analysis	NOUN
cana-509	169	46	issn	issn	NOUN
cana-509	169	47	:	:	PUNCT
cana-509	169	48	1074	1074	NUM
cana-509	169	49	-	-	PUNCT
cana-509	169	50	133x	133x	NUM
cana-509	169	51	vol	vol	NOUN
cana-509	169	52	31	31	NUM
cana-509	169	53	no	no	NOUN
cana-509	169	54	.	.	NOUN
cana-509	169	55	2	2	NUM
cana-509	169	56	(	(	PUNCT
cana-509	169	57	2024	2024	NUM
cana-509	169	58	)	)	PUNCT
cana-509	169	59	21	21	NUM
cana-509	169	60	https://internationalpubls.com	https://internationalpubls.com	X
cana-509	169	61	therefore	therefore	ADV
cana-509	169	62	,	,	PUNCT
cana-509	169	63	we	we	PRON
cana-509	169	64	deduce	deduce	VERB
cana-509	169	65	from	from	ADP
cana-509	169	66	the	the	DET
cana-509	169	67	encoding	encoding	NOUN
cana-509	169	68	table	table	NOUN
cana-509	169	69	that	that	SCONJ
cana-509	169	70	the	the	DET
cana-509	169	71	plain	plain	ADJ
cana-509	169	72	text	text	NOUN
cana-509	169	73	is	be	AUX
cana-509	169	74	bishop	bishop	NOUN
cana-509	169	75	.	.	PUNCT
cana-509	170	1	5	5	X
cana-509	170	2	.	.	X
cana-509	170	3	conclusion	conclusion	NOUN
cana-509	170	4	in	in	ADP
cana-509	170	5	this	this	DET
cana-509	170	6	paper	paper	NOUN
cana-509	170	7	,	,	PUNCT
cana-509	170	8	we	we	PRON
cana-509	170	9	investigated	investigate	VERB
cana-509	170	10	some	some	DET
cana-509	170	11	families	family	NOUN
cana-509	170	12	triangular	triangular	NOUN
cana-509	170	13	snake	snake	NOUN
cana-509	170	14	graphs	graph	NOUN
cana-509	170	15	satisfy	satisfy	VERB
cana-509	170	16	the	the	DET
cana-509	170	17	condition	condition	NOUN
cana-509	170	18	of	of	ADP
cana-509	170	19	harmonic	harmonic	ADJ
cana-509	170	20	mean	mean	ADJ
cana-509	170	21	labeling	labeling	NOUN
cana-509	170	22	.	.	PUNCT
cana-509	171	1	we	we	PRON
cana-509	171	2	presented	present	VERB
cana-509	171	3	secret	secret	ADJ
cana-509	171	4	coding	coding	NOUN
cana-509	171	5	by	by	ADP
cana-509	171	6	using	use	VERB
cana-509	171	7	a	a	DET
cana-509	171	8	revised	revise	VERB
cana-509	171	9	gmj	gmj	NOUN
cana-509	171	10	technique	technique	NOUN
cana-509	171	11	with	with	ADP
cana-509	171	12	new	new	ADJ
cana-509	171	13	labeling	labeling	NOUN
cana-509	171	14	and	and	CCONJ
cana-509	171	15	numbering	numbering	NOUN
cana-509	171	16	of	of	ADP
cana-509	171	17	alphabets	alphabet	NOUN
cana-509	171	18	based	base	VERB
cana-509	171	19	on	on	ADP
cana-509	171	20	vowels	vowel	NOUN
cana-509	171	21	.	.	PUNCT
cana-509	172	1	in	in	ADP
cana-509	172	2	the	the	DET
cana-509	172	3	future	future	NOUN
cana-509	172	4	,	,	PUNCT
cana-509	172	5	we	we	PRON
cana-509	172	6	intend	intend	VERB
cana-509	172	7	to	to	PART
cana-509	172	8	introduce	introduce	VERB
cana-509	172	9	new	new	ADJ
cana-509	172	10	labeling	labeling	NOUN
cana-509	172	11	technique	technique	NOUN
cana-509	172	12	and	and	CCONJ
cana-509	172	13	prove	prove	VERB
cana-509	172	14	coding	code	VERB
cana-509	172	15	,	,	PUNCT
cana-509	172	16	utilising	utilise	VERB
cana-509	172	17	various	various	ADJ
cana-509	172	18	graphs	graph	NOUN
cana-509	172	19	in	in	ADP
cana-509	172	20	conjunction	conjunction	NOUN
cana-509	172	21	with	with	ADP
cana-509	172	22	various	various	ADJ
cana-509	172	23	methods	method	NOUN
cana-509	172	24	of	of	ADP
cana-509	172	25	alphabet	alphabet	PROPN
cana-509	172	26	numbering	number	VERB
cana-509	172	27	.	.	PUNCT
cana-509	173	1	it	it	PRON
cana-509	173	2	is	be	AUX
cana-509	173	3	very	very	ADV
cana-509	173	4	interesting	interesting	ADJ
cana-509	173	5	and	and	CCONJ
cana-509	173	6	challenging	challenging	ADJ
cana-509	173	7	as	as	ADV
cana-509	173	8	well	well	ADV
cana-509	173	9	as	as	ADP
cana-509	173	10	to	to	PART
cana-509	173	11	investigate	investigate	VERB
cana-509	173	12	graph	graph	NOUN
cana-509	173	13	families	family	NOUN
cana-509	173	14	which	which	PRON
cana-509	173	15	admit	admit	VERB
cana-509	173	16	harmonic	harmonic	ADJ
cana-509	173	17	mean	mean	NOUN
cana-509	173	18	labeling	labeling	NOUN
cana-509	173	19	graph	graph	NOUN
cana-509	173	20	.	.	PUNCT
cana-509	174	1	we	we	PRON
cana-509	174	2	have	have	AUX
cana-509	174	3	presented	present	VERB
cana-509	174	4	new	new	ADJ
cana-509	174	5	results	result	NOUN
cana-509	174	6	on	on	ADP
cana-509	174	7	the	the	DET
cana-509	174	8	harmonic	harmonic	ADJ
cana-509	174	9	mean	mean	NOUN
cana-509	174	10	labeling	labeling	NOUN
cana-509	174	11	of	of	ADP
cana-509	174	12	certain	certain	ADJ
cana-509	174	13	classes	class	NOUN
cana-509	174	14	of	of	ADP
cana-509	174	15	graphs	graph	NOUN
cana-509	174	16	like	like	ADP
cana-509	174	17	𝑇𝑆𝑛	𝑇𝑆𝑛	NOUN
cana-509	174	18	∘	∘	VERB
cana-509	175	1	𝐾1.analogous	𝐾1.analogous	ADJ
cana-509	175	2	work	work	NOUN
cana-509	175	3	can	can	AUX
cana-509	175	4	be	be	AUX
cana-509	175	5	carried	carry	VERB
cana-509	175	6	out	out	ADP
cana-509	175	7	for	for	ADP
cana-509	175	8	other	other	ADJ
cana-509	175	9	families	family	NOUN
cana-509	175	10	and	and	CCONJ
cana-509	175	11	in	in	ADP
cana-509	175	12	the	the	DET
cana-509	175	13	context	context	NOUN
cana-509	175	14	of	of	ADP
cana-509	175	15	different	different	ADJ
cana-509	175	16	types	type	NOUN
cana-509	175	17	of	of	ADP
cana-509	175	18	graph	graph	NOUN
cana-509	175	19	labeling	labeling	NOUN
cana-509	175	20	techniques	technique	NOUN
cana-509	175	21	.	.	PUNCT
cana-509	176	1	references	reference	NOUN
cana-509	176	2	[	[	X
cana-509	176	3	1	1	NUM
cana-509	176	4	]	]	PUNCT
cana-509	176	5	gallian	gallian	NOUN
cana-509	176	6	,	,	PUNCT
cana-509	176	7	j.	j.	PROPN
cana-509	176	8	a.	a.	PROPN
cana-509	176	9	(	(	PUNCT
cana-509	176	10	2018	2018	NUM
cana-509	176	11	)	)	PUNCT
cana-509	176	12	.	.	PUNCT
cana-509	177	1	a	a	DET
cana-509	177	2	dynamic	dynamic	ADJ
cana-509	177	3	survey	survey	NOUN
cana-509	177	4	of	of	ADP
cana-509	177	5	graph	graph	NOUN
cana-509	177	6	labeling	labeling	NOUN
cana-509	177	7	.	.	PUNCT
cana-509	178	1	electronic	electronic	ADJ
cana-509	178	2	journal	journal	NOUN
cana-509	178	3	of	of	ADP
cana-509	178	4	combinatorics	combinatoric	NOUN
cana-509	178	5	,	,	PUNCT
cana-509	178	6	1(dynamicsurveys	1(dynamicsurveys	NUM
cana-509	178	7	)	)	PUNCT
cana-509	178	8	,	,	PUNCT
cana-509	178	9	ds6	ds6	NOUN
cana-509	178	10	.	.	PUNCT
cana-509	179	1	[	[	X
cana-509	179	2	2	2	NUM
cana-509	179	3	]	]	PUNCT
cana-509	179	4	harary	harary	NOUN
cana-509	179	5	,	,	PUNCT
cana-509	179	6	graph	graph	NOUN
cana-509	179	7	theory	theory	NOUN
cana-509	179	8	,	,	PUNCT
cana-509	179	9	addison	addison	PROPN
cana-509	179	10	wesley	wesley	PROPN
cana-509	179	11	,	,	PUNCT
cana-509	179	12	massachusetts	massachusetts	PROPN
cana-509	179	13	,	,	PUNCT
cana-509	179	14	1969	1969	NUM
cana-509	179	15	.	.	PUNCT
cana-509	180	1	[	[	X
cana-509	180	2	3	3	NUM
cana-509	180	3	]	]	X
cana-509	180	4	somasundaram	somasundaram	PROPN
cana-509	180	5	,	,	PUNCT
cana-509	180	6	s.	s.	PROPN
cana-509	180	7	,	,	PUNCT
cana-509	180	8	&	&	CCONJ
cana-509	180	9	ponraj	ponraj	VERB
cana-509	180	10	,	,	PUNCT
cana-509	180	11	r.	r.	PROPN
cana-509	180	12	(	(	PUNCT
cana-509	180	13	2003	2003	NUM
cana-509	180	14	)	)	PUNCT
cana-509	180	15	.	.	PUNCT
cana-509	181	1	mean	mean	VERB
cana-509	181	2	labelings	labeling	NOUN
cana-509	181	3	of	of	ADP
cana-509	181	4	graphs	graph	NOUN
cana-509	181	5	.	.	PUNCT
cana-509	182	1	national	national	PROPN
cana-509	182	2	academy	academy	PROPN
cana-509	182	3	science	science	PROPN
cana-509	182	4	letters	letter	NOUN
cana-509	182	5	,	,	PUNCT
cana-509	182	6	26(7	26(7	NOUN
cana-509	182	7	-	-	SYM
cana-509	182	8	8)	8)	NUM
cana-509	182	9	,	,	PUNCT
cana-509	182	10	210213	210213	NUM
cana-509	182	11	.	.	PUNCT
cana-509	183	1	[	[	X
cana-509	183	2	4	4	X
cana-509	183	3	]	]	X
cana-509	183	4	david	david	PROPN
cana-509	183	5	raj	raj	PROPN
cana-509	183	6	,	,	PUNCT
cana-509	183	7	c.	c.	PROPN
cana-509	183	8	,	,	PUNCT
cana-509	183	9	jayasekaran	jayasekaran	NOUN
cana-509	183	10	,	,	PUNCT
cana-509	183	11	c.	c.	PROPN
cana-509	183	12	,	,	PUNCT
cana-509	183	13	&	&	CCONJ
cana-509	183	14	sandhya	sandhya	PROPN
cana-509	183	15	,	,	PUNCT
cana-509	183	16	s.	s.	PROPN
cana-509	183	17	s.	s.	PROPN
cana-509	183	18	(	(	PUNCT
cana-509	183	19	2016	2016	NUM
cana-509	183	20	)	)	PUNCT
cana-509	183	21	.	.	PUNCT
cana-509	184	1	few	few	ADJ
cana-509	184	2	families	family	NOUN
cana-509	184	3	of	of	ADP
cana-509	184	4	harmonic	harmonic	ADJ
cana-509	184	5	mean	mean	NOUN
cana-509	184	6	graphs	graph	NOUN
cana-509	184	7	.	.	PUNCT
cana-509	185	1	journal	journal	NOUN
cana-509	185	2	of	of	ADP
cana-509	185	3	combinatorial	combinatorial	ADJ
cana-509	185	4	mathematics	mathematic	NOUN
cana-509	185	5	and	and	CCONJ
cana-509	185	6	combinatorial	combinatorial	ADJ
cana-509	185	7	computing	computing	NOUN
cana-509	185	8	,	,	PUNCT
cana-509	185	9	96	96	NUM
cana-509	185	10	,	,	PUNCT
cana-509	185	11	235	235	NUM
cana-509	185	12	-	-	SYM
cana-509	185	13	243	243	NUM
cana-509	185	14	.	.	PUNCT
cana-509	186	1	[	[	X
cana-509	186	2	5	5	NUM
cana-509	186	3	]	]	X
cana-509	186	4	meena	meena	PROPN
cana-509	186	5	,	,	PUNCT
cana-509	186	6	s.	s.	PROPN
cana-509	186	7	,	,	PUNCT
cana-509	186	8	&	&	CCONJ
cana-509	186	9	sivasakthi	sivasakthi	PROPN
cana-509	186	10	,	,	PUNCT
cana-509	186	11	m.	m.	NOUN
cana-509	186	12	(	(	PUNCT
cana-509	186	13	2019	2019	NUM
cana-509	186	14	)	)	PUNCT
cana-509	186	15	.	.	PUNCT
cana-509	187	1	harmonic	harmonic	ADJ
cana-509	187	2	mean	mean	VERB
cana-509	187	3	labeling	labeling	NOUN
cana-509	187	4	subdivision	subdivision	NOUN
cana-509	187	5	graphs	graph	NOUN
cana-509	187	6	.	.	PUNCT
cana-509	188	1	international	international	ADJ
cana-509	188	2	journal	journal	PROPN
cana-509	188	3	of	of	ADP
cana-509	188	4	research	research	NOUN
cana-509	188	5	and	and	CCONJ
cana-509	188	6	analytical	analytical	ADJ
cana-509	188	7	reviews	review	NOUN
cana-509	188	8	,	,	PUNCT
cana-509	188	9	6(1	6(1	NUM
cana-509	188	10	)	)	PUNCT
cana-509	188	11	,	,	PUNCT
cana-509	188	12	2348	2348	NUM
cana-509	188	13	-	-	SYM
cana-509	188	14	1269	1269	NUM
cana-509	188	15	.	.	PUNCT
cana-509	189	1	[	[	X
cana-509	189	2	6	6	NUM
cana-509	189	3	]	]	X
cana-509	189	4	somasundaram	somasundaram	PROPN
cana-509	189	5	,	,	PUNCT
cana-509	189	6	s.	s.	PROPN
cana-509	189	7	,	,	PUNCT
cana-509	189	8	sandhya	sandhya	PROPN
cana-509	189	9	,	,	PUNCT
cana-509	189	10	s.	s.	PROPN
cana-509	189	11	s.	s.	PROPN
cana-509	189	12	,	,	PUNCT
cana-509	189	13	&	&	CCONJ
cana-509	189	14	ponraj	ponraj	VERB
cana-509	189	15	,	,	PUNCT
cana-509	189	16	r.	r.	PROPN
cana-509	189	17	harmonic	harmonic	PROPN
cana-509	189	18	mean	mean	VERB
cana-509	189	19	labeling	labeling	NOUN
cana-509	189	20	of	of	ADP
cana-509	189	21	graphs	graph	NOUN
cana-509	189	22	,	,	PUNCT
cana-509	189	23	communicated	communicate	VERB
cana-509	189	24	to	to	PART
cana-509	189	25	journal	journal	VERB
cana-509	189	26	of	of	ADP
cana-509	189	27	combinatorial	combinatorial	ADJ
cana-509	189	28	mathematics	mathematic	NOUN
cana-509	189	29	and	and	CCONJ
cana-509	189	30	combinational	combinational	ADJ
cana-509	189	31	computing	computing	NOUN
cana-509	189	32	.	.	PUNCT
cana-509	190	1	[	[	X
cana-509	190	2	7	7	NUM
cana-509	190	3	]	]	X
cana-509	190	4	sandhya	sandhya	PROPN
cana-509	190	5	,	,	PUNCT
cana-509	190	6	s.	s.	PROPN
cana-509	190	7	s.	s.	PROPN
cana-509	190	8	,	,	PUNCT
cana-509	190	9	somasundaram	somasundaram	PROPN
cana-509	190	10	,	,	PUNCT
cana-509	190	11	s.	s.	PROPN
cana-509	190	12	,	,	PUNCT
cana-509	190	13	&	&	CCONJ
cana-509	190	14	ponraj	ponraj	VERB
cana-509	190	15	,	,	PUNCT
cana-509	190	16	r.	r.	PROPN
cana-509	190	17	(	(	PUNCT
cana-509	190	18	2012	2012	NUM
cana-509	190	19	)	)	PUNCT
cana-509	190	20	.	.	PUNCT
cana-509	191	1	some	some	DET
cana-509	191	2	more	more	ADJ
cana-509	191	3	results	result	NOUN
cana-509	191	4	on	on	ADP
cana-509	191	5	harmonic	harmonic	ADJ
cana-509	191	6	mean	mean	NOUN
cana-509	191	7	graphs	graph	NOUN
cana-509	191	8	.	.	PUNCT
cana-509	192	1	journal	journal	NOUN
cana-509	192	2	of	of	ADP
cana-509	192	3	mathematics	mathematics	PROPN
cana-509	192	4	research	research	NOUN
cana-509	192	5	,	,	PUNCT
cana-509	192	6	4(1	4(1	NOUN
cana-509	192	7	)	)	PUNCT
cana-509	192	8	,	,	PUNCT
cana-509	192	9	21	21	NUM
cana-509	192	10	.	.	PUNCT
cana-509	193	1	[	[	X
cana-509	193	2	8	8	NUM
cana-509	193	3	]	]	X
cana-509	193	4	sandhya	sandhya	PROPN
cana-509	193	5	,	,	PUNCT
cana-509	193	6	s.	s.	PROPN
cana-509	193	7	s.	s.	PROPN
cana-509	193	8	,	,	PUNCT
cana-509	193	9	somasundaram	somasundaram	PROPN
cana-509	193	10	,	,	PUNCT
cana-509	193	11	s.	s.	PROPN
cana-509	193	12	,	,	PUNCT
cana-509	193	13	&	&	CCONJ
cana-509	193	14	ponraj	ponraj	VERB
cana-509	193	15	,	,	PUNCT
cana-509	193	16	r.	r.	PROPN
cana-509	193	17	(	(	PUNCT
cana-509	193	18	2012	2012	NUM
cana-509	193	19	)	)	PUNCT
cana-509	193	20	.	.	PUNCT
cana-509	194	1	some	some	DET
cana-509	194	2	results	result	NOUN
cana-509	194	3	on	on	ADP
cana-509	194	4	harmonic	harmonic	ADJ
cana-509	194	5	mean	mean	NOUN
cana-509	194	6	graphs	graph	NOUN
cana-509	194	7	.	.	PUNCT
cana-509	195	1	int	int	NOUN
cana-509	195	2	.	.	PUNCT
cana-509	196	1	j.	j.	PROPN
cana-509	196	2	contemp	contemp	PROPN
cana-509	196	3	.	.	PUNCT
cana-509	197	1	math	math	NOUN
cana-509	197	2	.	.	PUNCT
cana-509	198	1	sci	sci	PROPN
cana-509	198	2	,	,	PUNCT
cana-509	198	3	7	7	NUM
cana-509	198	4	,	,	PUNCT
cana-509	198	5	197	197	NUM
cana-509	198	6	-	-	SYM
cana-509	198	7	208	208	NUM
cana-509	198	8	.	.	PUNCT
cana-509	199	1	[	[	X
cana-509	199	2	9	9	NUM
cana-509	199	3	]	]	SYM
cana-509	199	4	sandhya	sandhya	PROPN
cana-509	199	5	,	,	PUNCT
cana-509	199	6	s.	s.	PROPN
cana-509	199	7	s.	s.	PROPN
cana-509	199	8	,	,	PUNCT
cana-509	199	9	somasundaram	somasundaram	PROPN
cana-509	199	10	,	,	PUNCT
cana-509	199	11	s.	s.	PROPN
cana-509	199	12	,	,	PUNCT
cana-509	199	13	&	&	CCONJ
cana-509	199	14	ponraj	ponraj	VERB
cana-509	199	15	,	,	PUNCT
cana-509	199	16	r.	r.	PROPN
cana-509	199	17	(	(	PUNCT
cana-509	199	18	2012	2012	NUM
cana-509	199	19	)	)	PUNCT
cana-509	199	20	.	.	PUNCT
cana-509	200	1	harmonic	harmonic	ADJ
cana-509	200	2	mean	mean	VERB
cana-509	200	3	labeling	labeling	NOUN
cana-509	200	4	of	of	ADP
cana-509	200	5	some	some	DET
cana-509	200	6	cycle	cycle	NOUN
cana-509	200	7	related	relate	VERB
cana-509	200	8	graphs	graph	NOUN
cana-509	200	9	,	,	PUNCT
cana-509	200	10	international	international	ADJ
cana-509	200	11	journal	journal	NOUN
cana-509	200	12	of	of	ADP
cana-509	200	13	mathematical	mathematical	ADJ
cana-509	200	14	analysis	analysis	NOUN
cana-509	200	15	,	,	PUNCT
cana-509	200	16	6(37	6(37	NUM
cana-509	200	17	)	)	PUNCT
cana-509	200	18	,	,	PUNCT
cana-509	200	19	1997	1997	NUM
cana-509	200	20	-	-	SYM
cana-509	200	21	2005	2005	NUM
cana-509	200	22	.	.	PUNCT
cana-509	201	1	[	[	X
cana-509	201	2	10	10	NUM
cana-509	201	3	]	]	X
cana-509	201	4	somasundaram	somasundaram	NOUN
cana-509	201	5	.	.	PUNCT
cana-509	202	1	s	s	PART
cana-509	202	2	,	,	PUNCT
cana-509	203	1	ponraj	ponraj	VERB
cana-509	203	2	.	.	PUNCT
cana-509	204	1	r	r	NOUN
cana-509	204	2	and	and	CCONJ
cana-509	204	3	sandhya	sandhya	PROPN
cana-509	204	4	.	.	PUNCT
cana-509	205	1	s.s	s.s	PROPN
cana-509	205	2	,	,	PUNCT
cana-509	205	3	harmonic	harmonic	ADJ
cana-509	205	4	mean	mean	ADJ
cana-509	205	5	labeling	labeling	NOUN
cana-509	205	6	of	of	ADP
cana-509	205	7	graphs	graph	NOUN
cana-509	205	8	communicated	communicate	VERB
cana-509	205	9	.	.	PUNCT