id	sid	tid	token	lemma	pos
cana-1740	1	1	communications	communication	NOUN
cana-1740	1	2	on	on	ADP
cana-1740	1	3	applied	apply	VERB
cana-1740	1	4	nonlinear	nonlinear	ADJ
cana-1740	1	5	analysis	analysis	NOUN
cana-1740	1	6	issn	issn	NOUN
cana-1740	1	7	:	:	PUNCT
cana-1740	1	8	1074	1074	NUM
cana-1740	1	9	-	-	PUNCT
cana-1740	1	10	133x	133x	NUM
cana-1740	1	11	vol	vol	NOUN
cana-1740	1	12	32	32	NUM
cana-1740	1	13	no	no	NOUN
cana-1740	1	14	.	.	NOUN
cana-1740	1	15	2	2	NUM
cana-1740	1	16	(	(	PUNCT
cana-1740	1	17	2025	2025	NUM
cana-1740	1	18	)	)	PUNCT
cana-1740	1	19	254	254	NUM
cana-1740	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	1	21	topological	topological	ADJ
cana-1740	1	22	indices	index	NOUN
cana-1740	1	23	of	of	ADP
cana-1740	1	24	arms	arm	NOUN
cana-1740	1	25	-	-	PUNCT
cana-1740	1	26	product	product	NOUN
cana-1740	1	27	of	of	ADP
cana-1740	1	28	certain	certain	ADJ
cana-1740	1	29	graphs	graph	NOUN
cana-1740	1	30	roy	roy	PROPN
cana-1740	1	31	john.a	john.a	NOUN
cana-1740	1	32	1	1	NUM
cana-1740	1	33	,	,	PUNCT
cana-1740	1	34	akhil	akhil	PROPN
cana-1740	1	35	b.b	b.b	PROPN
cana-1740	1	36	2	2	NUM
cana-1740	1	37	and	and	CCONJ
cana-1740	1	38	manju	manju	PROPN
cana-1740	1	39	v.	v.	ADP
cana-1740	1	40	n.	n.	PROPN
cana-1740	2	1	b	b	PROPN
cana-1740	2	2	3	3	NUM
cana-1740	2	3	a	a	DET
cana-1740	2	4	department	department	NOUN
cana-1740	2	5	of	of	ADP
cana-1740	2	6	mathematics	mathematics	PROPN
cana-1740	2	7	,	,	PUNCT
cana-1740	2	8	st	st	PROPN
cana-1740	2	9	.	.	PROPN
cana-1740	2	10	stephen	stephen	PROPN
cana-1740	2	11	’s	’s	PROPN
cana-1740	2	12	college	college	PROPN
cana-1740	2	13	,	,	PUNCT
cana-1740	2	14	pathanapuram	pathanapuram	PROPN
cana-1740	2	15	,	,	PUNCT
cana-1740	2	16	kollam	kollam	PROPN
cana-1740	2	17	,	,	PUNCT
cana-1740	2	18	kerala	kerala	PROPN
cana-1740	2	19	,	,	PUNCT
cana-1740	2	20	india	india	PROPN
cana-1740	2	21	.	.	PUNCT
cana-1740	3	1	pin:689695	pin:689695	PROPN
cana-1740	3	2	.	.	PUNCT
cana-1740	4	1	b	b	X
cana-1740	4	2	department	department	NOUN
cana-1740	4	3	of	of	ADP
cana-1740	4	4	mathematics	mathematics	PROPN
cana-1740	4	5	,	,	PUNCT
cana-1740	4	6	university	university	NOUN
cana-1740	4	7	of	of	ADP
cana-1740	4	8	kerala	kerala	PROPN
cana-1740	4	9	,	,	PUNCT
cana-1740	4	10	karyavattom	karyavattom	PROPN
cana-1740	4	11	,	,	PUNCT
cana-1740	4	12	thiruvananthapuram	thiruvananthapuram	PROPN
cana-1740	4	13	,	,	PUNCT
cana-1740	4	14	kerala	kerala	PROPN
cana-1740	4	15	,	,	PUNCT
cana-1740	4	16	india	india	PROPN
cana-1740	4	17	.	.	PUNCT
cana-1740	4	18	pin:695581	pin:695581	PROPN
cana-1740	4	19	.	.	PUNCT
cana-1740	5	1	a	a	DET
cana-1740	5	2	1corresponding	1corresponding	PROPN
cana-1740	5	3	author	author	NOUN
cana-1740	5	4	:	:	PUNCT
cana-1740	5	5	roymaruthoor@gmail.com	roymaruthoor@gmail.com	X
cana-1740	6	1	b	b	X
cana-1740	6	2	2	2	NUM
cana-1740	6	3	akhilb@keralauniversity.ac.in	akhilb@keralauniversity.ac.in	PROPN
cana-1740	6	4	b	b	NOUN
cana-1740	6	5	3	3	NUM
cana-1740	6	6	manjushaijulal@gmail.com	manjushaijulal@gmail.com	NOUN
cana-1740	6	7	article	article	NOUN
cana-1740	6	8	history	history	NOUN
cana-1740	6	9	:	:	PUNCT
cana-1740	6	10	received	receive	VERB
cana-1740	6	11	:	:	PUNCT
cana-1740	6	12	01	01	NUM
cana-1740	6	13	-	-	SYM
cana-1740	6	14	08	08	NUM
cana-1740	6	15	-	-	PUNCT
cana-1740	6	16	2024	2024	NUM
cana-1740	6	17	revised	revise	VERB
cana-1740	6	18	:	:	PUNCT
cana-1740	6	19	09	09	NUM
cana-1740	6	20	-	-	SYM
cana-1740	6	21	09	09	NUM
cana-1740	6	22	-	-	PUNCT
cana-1740	6	23	2024	2024	NUM
cana-1740	6	24	accepted	accept	VERB
cana-1740	6	25	:	:	PUNCT
cana-1740	6	26	18	18	NUM
cana-1740	6	27	-	-	SYM
cana-1740	6	28	09	09	NUM
cana-1740	6	29	-	-	PUNCT
cana-1740	6	30	2024	2024	NUM
cana-1740	6	31	abstract	abstract	NOUN
cana-1740	6	32	:	:	PUNCT
cana-1740	6	33	a	a	DET
cana-1740	6	34	mathematical	mathematical	ADJ
cana-1740	6	35	parameter	parameter	NOUN
cana-1740	6	36	known	know	VERB
cana-1740	6	37	as	as	ADP
cana-1740	6	38	the	the	DET
cana-1740	6	39	topological	topological	ADJ
cana-1740	6	40	graph	graph	NOUN
cana-1740	6	41	index	index	NOUN
cana-1740	6	42	,	,	PUNCT
cana-1740	6	43	or	or	CCONJ
cana-1740	6	44	molecular	molecular	ADJ
cana-1740	6	45	descriptor	descriptor	NOUN
cana-1740	6	46	,	,	PUNCT
cana-1740	6	47	can	can	AUX
cana-1740	6	48	be	be	AUX
cana-1740	6	49	applied	apply	VERB
cana-1740	6	50	to	to	ADP
cana-1740	6	51	any	any	DET
cana-1740	6	52	graph	graph	NOUN
cana-1740	6	53	that	that	PRON
cana-1740	6	54	represents	represent	VERB
cana-1740	6	55	a	a	DET
cana-1740	6	56	molecule	molecule	NOUN
cana-1740	6	57	structure	structure	NOUN
cana-1740	6	58	.	.	PUNCT
cana-1740	7	1	this	this	DET
cana-1740	7	2	index	index	NOUN
cana-1740	7	3	can	can	AUX
cana-1740	7	4	be	be	AUX
cana-1740	7	5	used	use	VERB
cana-1740	7	6	to	to	PART
cana-1740	7	7	analyze	analyze	VERB
cana-1740	7	8	mathematical	mathematical	ADJ
cana-1740	7	9	numbers	number	NOUN
cana-1740	7	10	and	and	CCONJ
cana-1740	7	11	look	look	VERB
cana-1740	7	12	into	into	ADP
cana-1740	7	13	specific	specific	ADJ
cana-1740	7	14	physical	physical	ADJ
cana-1740	7	15	and	and	CCONJ
cana-1740	7	16	chemical	chemical	NOUN
cana-1740	7	17	characteristics	characteristic	NOUN
cana-1740	7	18	of	of	ADP
cana-1740	7	19	molecules	molecule	NOUN
cana-1740	7	20	in	in	ADP
cana-1740	7	21	more	more	ADJ
cana-1740	7	22	detail	detail	NOUN
cana-1740	7	23	.	.	PUNCT
cana-1740	8	1	it	it	PRON
cana-1740	8	2	is	be	AUX
cana-1740	8	3	therefore	therefore	ADV
cana-1740	8	4	a	a	DET
cana-1740	8	5	useful	useful	ADJ
cana-1740	8	6	strategy	strategy	NOUN
cana-1740	8	7	for	for	ADP
cana-1740	8	8	avoiding	avoid	VERB
cana-1740	8	9	costly	costly	ADJ
cana-1740	8	10	and	and	CCONJ
cana-1740	8	11	timeconsuming	timeconsuming	ADJ
cana-1740	8	12	laboratory	laboratory	NOUN
cana-1740	8	13	experiments	experiment	NOUN
cana-1740	8	14	.	.	PUNCT
cana-1740	9	1	in	in	ADP
cana-1740	9	2	this	this	DET
cana-1740	9	3	paper	paper	NOUN
cana-1740	9	4	topological	topological	ADJ
cana-1740	9	5	indices	index	NOUN
cana-1740	9	6	of	of	ADP
cana-1740	9	7	a	a	DET
cana-1740	9	8	noval	noval	NOUN
cana-1740	9	9	graph	graph	NOUN
cana-1740	9	10	product	product	NOUN
cana-1740	9	11	called	call	VERB
cana-1740	9	12	arms	arm	NOUN
cana-1740	9	13	-	-	PUNCT
cana-1740	9	14	product	product	NOUN
cana-1740	9	15	of	of	ADP
cana-1740	9	16	certain	certain	ADJ
cana-1740	9	17	graphs	graph	NOUN
cana-1740	9	18	are	be	AUX
cana-1740	9	19	determined	determine	VERB
cana-1740	9	20	.	.	PUNCT
cana-1740	10	1	keywords	keyword	NOUN
cana-1740	10	2	:	:	PUNCT
cana-1740	10	3	arms	arm	NOUN
cana-1740	10	4	-	-	PUNCT
cana-1740	10	5	product	product	NOUN
cana-1740	10	6	,	,	PUNCT
cana-1740	10	7	first	first	PROPN
cana-1740	10	8	zagreb	zagreb	PROPN
cana-1740	10	9	index	index	PROPN
cana-1740	10	10	,	,	PUNCT
cana-1740	10	11	second	second	PROPN
cana-1740	10	12	zagreb	zagreb	PROPN
cana-1740	10	13	index	index	PROPN
cana-1740	10	14	,	,	PUNCT
cana-1740	10	15	harmonic	harmonic	ADJ
cana-1740	10	16	index	index	NOUN
cana-1740	10	17	,	,	PUNCT
cana-1740	10	18	geometric	geometric	ADJ
cana-1740	10	19	-	-	PUNCT
cana-1740	10	20	arithmetic	arithmetic	ADJ
cana-1740	10	21	index	index	NOUN
cana-1740	10	22	,	,	PUNCT
cana-1740	10	23	bull	bull	NOUN
cana-1740	10	24	graph	graph	NOUN
cana-1740	10	25	,	,	PUNCT
cana-1740	10	26	path	path	NOUN
cana-1740	10	27	.	.	PUNCT
cana-1740	11	1	2020	2020	NUM
cana-1740	11	2	ams	am	NOUN
cana-1740	11	3	subject	subject	NOUN
cana-1740	11	4	classifications	classification	NOUN
cana-1740	11	5	:	:	PUNCT
cana-1740	11	6	05c07	05c07	NOUN
cana-1740	11	7	,	,	PUNCT
cana-1740	11	8	05c76	05c76	NUM
cana-1740	11	9	,	,	PUNCT
cana-1740	11	10	05c92	05c92	NOUN
cana-1740	11	11	.	.	PUNCT
cana-1740	12	1	1	1	X
cana-1740	12	2	.	.	X
cana-1740	13	1	introduction	introduction	NOUN
cana-1740	13	2	topological	topological	ADJ
cana-1740	13	3	indices	index	NOUN
cana-1740	13	4	are	be	AUX
cana-1740	13	5	graph	graph	NOUN
cana-1740	13	6	invariant	invariant	ADJ
cana-1740	13	7	.	.	PUNCT
cana-1740	14	1	a	a	DET
cana-1740	14	2	graph	graph	NOUN
cana-1740	14	3	invariant	invariant	ADJ
cana-1740	14	4	is	be	AUX
cana-1740	14	5	any	any	DET
cana-1740	14	6	function	function	NOUN
cana-1740	14	7	on	on	ADP
cana-1740	14	8	a	a	DET
cana-1740	14	9	graph	graph	NOUN
cana-1740	14	10	that	that	PRON
cana-1740	14	11	does	do	AUX
cana-1740	14	12	not	not	PART
cana-1740	14	13	depend	depend	VERB
cana-1740	14	14	on	on	ADP
cana-1740	14	15	labelling	labelling	NOUN
cana-1740	14	16	of	of	ADP
cana-1740	14	17	its	its	PRON
cana-1740	14	18	vertices	vertex	NOUN
cana-1740	14	19	.	.	PUNCT
cana-1740	15	1	in	in	ADP
cana-1740	15	2	chemistry	chemistry	NOUN
cana-1740	15	3	,	,	PUNCT
cana-1740	15	4	graphs	graph	NOUN
cana-1740	15	5	can	can	AUX
cana-1740	15	6	represent	represent	VERB
cana-1740	15	7	different	different	ADJ
cana-1740	15	8	chemical	chemical	NOUN
cana-1740	15	9	objects	object	NOUN
cana-1740	15	10	such	such	ADJ
cana-1740	15	11	as	as	ADP
cana-1740	15	12	molecules	molecule	NOUN
cana-1740	15	13	,	,	PUNCT
cana-1740	15	14	crystals	crystal	NOUN
cana-1740	15	15	,	,	PUNCT
cana-1740	15	16	polymers	polymer	NOUN
cana-1740	15	17	,	,	PUNCT
cana-1740	15	18	etc	etc	X
cana-1740	15	19	.	.	X
cana-1740	15	20	molecular	molecular	ADJ
cana-1740	15	21	graphs	graph	NOUN
cana-1740	15	22	are	be	AUX
cana-1740	15	23	chemical	chemical	NOUN
cana-1740	15	24	graphs	graph	NOUN
cana-1740	15	25	which	which	PRON
cana-1740	15	26	represent	represent	VERB
cana-1740	15	27	the	the	DET
cana-1740	15	28	constitution	constitution	NOUN
cana-1740	15	29	of	of	ADP
cana-1740	15	30	molecules	molecule	NOUN
cana-1740	15	31	.	.	PUNCT
cana-1740	16	1	in	in	ADP
cana-1740	16	2	these	these	DET
cana-1740	16	3	graphs	graph	NOUN
cana-1740	16	4	,	,	PUNCT
cana-1740	16	5	individual	individual	ADJ
cana-1740	16	6	atoms	atom	NOUN
cana-1740	16	7	represent	represent	VERB
cana-1740	16	8	vertices	vertex	NOUN
cana-1740	16	9	and	and	CCONJ
cana-1740	16	10	edges	edge	NOUN
cana-1740	16	11	are	be	AUX
cana-1740	16	12	chemical	chemical	ADJ
cana-1740	16	13	bonds	bond	NOUN
cana-1740	16	14	between	between	ADP
cana-1740	16	15	them	they	PRON
cana-1740	16	16	.	.	PUNCT
cana-1740	17	1	a	a	DET
cana-1740	17	2	single	single	ADJ
cana-1740	17	3	number	number	NOUN
cana-1740	17	4	that	that	PRON
cana-1740	17	5	can	can	AUX
cana-1740	17	6	associate	associate	VERB
cana-1740	17	7	with	with	ADP
cana-1740	17	8	chemical	chemical	NOUN
cana-1740	17	9	graphs	graph	NOUN
cana-1740	17	10	is	be	AUX
cana-1740	17	11	called	call	VERB
cana-1740	17	12	topological	topological	ADJ
cana-1740	17	13	index	index	NOUN
cana-1740	17	14	.	.	PUNCT
cana-1740	18	1	all	all	DET
cana-1740	18	2	graphs	graph	NOUN
cana-1740	18	3	under	under	ADP
cana-1740	18	4	our	our	PRON
cana-1740	18	5	consideration	consideration	NOUN
cana-1740	18	6	are	be	AUX
cana-1740	18	7	simple	simple	ADJ
cana-1740	18	8	,	,	PUNCT
cana-1740	18	9	connected	connected	ADJ
cana-1740	18	10	and	and	CCONJ
cana-1740	18	11	undirected	undirected	ADJ
cana-1740	18	12	.	.	PUNCT
cana-1740	19	1	let	let	VERB
cana-1740	19	2	g	g	PROPN
cana-1740	19	3	=	=	SYM
cana-1740	19	4	(	(	PUNCT
cana-1740	19	5	v	v	NOUN
cana-1740	19	6	(	(	PUNCT
cana-1740	19	7	g	g	NOUN
cana-1740	19	8	)	)	PUNCT
cana-1740	19	9	,	,	PUNCT
cana-1740	19	10	e(g	e(g	PROPN
cana-1740	19	11	)	)	PUNCT
cana-1740	19	12	)	)	PUNCT
cana-1740	20	1	be	be	AUX
cana-1740	20	2	a	a	DET
cana-1740	20	3	graph	graph	NOUN
cana-1740	20	4	with	with	ADP
cana-1740	20	5	order	order	NOUN
cana-1740	20	6	n	n	NOUN
cana-1740	20	7	=	=	SYM
cana-1740	20	8	|v	|v	X
cana-1740	20	9	(	(	PUNCT
cana-1740	20	10	g)|	g)|	NOUN
cana-1740	20	11	and	and	CCONJ
cana-1740	20	12	size	size	NOUN
cana-1740	20	13	m	m	PROPN
cana-1740	20	14	=	=	SYM
cana-1740	20	15	|e(g)|	|e(g)|	NOUN
cana-1740	20	16	.	.	PUNCT
cana-1740	21	1	the	the	DET
cana-1740	21	2	degree	degree	NOUN
cana-1740	21	3	of	of	ADP
cana-1740	21	4	a	a	DET
cana-1740	21	5	vertex	vertex	NOUN
cana-1740	21	6	vi	vi	NOUN
cana-1740	21	7	in	in	ADP
cana-1740	21	8	g	g	PROPN
cana-1740	21	9	is	be	AUX
cana-1740	21	10	the	the	DET
cana-1740	21	11	number	number	NOUN
cana-1740	21	12	of	of	ADP
cana-1740	21	13	edges	edge	NOUN
cana-1740	21	14	incident	incident	NOUN
cana-1740	21	15	on	on	ADP
cana-1740	21	16	vi	vi	PROPN
cana-1740	21	17	and	and	CCONJ
cana-1740	21	18	is	be	AUX
cana-1740	21	19	denoted	denote	VERB
cana-1740	21	20	by	by	ADP
cana-1740	21	21	d(vi	d(vi	PROPN
cana-1740	21	22	)	)	PUNCT
cana-1740	21	23	or	or	CCONJ
cana-1740	21	24	simply	simply	ADV
cana-1740	21	25	di	di	PROPN
cana-1740	21	26	.	.	PUNCT
cana-1740	22	1	the	the	DET
cana-1740	22	2	corona	corona	PROPN
cana-1740	22	3	𝐺1⊙𝐺2	𝐺1⊙𝐺2	NOUN
cana-1740	22	4	of	of	ADP
cana-1740	22	5	two	two	NUM
cana-1740	22	6	graphs	graph	NOUN
cana-1740	22	7	g1	g1	NOUN
cana-1740	22	8	and	and	CCONJ
cana-1740	22	9	g2	g2	PROPN
cana-1740	22	10	is	be	AUX
cana-1740	22	11	defined	define	VERB
cana-1740	22	12	as	as	ADP
cana-1740	22	13	the	the	DET
cana-1740	22	14	graph	graph	NOUN
cana-1740	22	15	g	g	PROPN
cana-1740	22	16	obtained	obtain	VERB
cana-1740	22	17	by	by	ADP
cana-1740	22	18	taking	take	VERB
cana-1740	22	19	one	one	NUM
cana-1740	22	20	copy	copy	NOUN
cana-1740	22	21	of	of	ADP
cana-1740	22	22	g1(which	g1(which	PROPN
cana-1740	22	23	has	have	VERB
cana-1740	22	24	p1	p1	NOUN
cana-1740	22	25	points	point	NOUN
cana-1740	22	26	)	)	PUNCT
cana-1740	22	27	and	and	CCONJ
cana-1740	22	28	p1	p1	PROPN
cana-1740	22	29	copies	copy	NOUN
cana-1740	22	30	of	of	ADP
cana-1740	22	31	g2	g2	PROPN
cana-1740	22	32	,	,	PUNCT
cana-1740	22	33	and	and	CCONJ
cana-1740	22	34	then	then	ADV
cana-1740	22	35	joining	join	VERB
cana-1740	22	36	the	the	PRON
cana-1740	23	1	i	i	NOUN
cana-1740	23	2	th	th	NOUN
cana-1740	23	3	point	point	NOUN
cana-1740	23	4	of	of	ADP
cana-1740	23	5	g1	g1	PROPN
cana-1740	23	6	to	to	ADP
cana-1740	23	7	every	every	DET
cana-1740	23	8	point	point	NOUN
cana-1740	23	9	in	in	ADP
cana-1740	23	10	the	the	DET
cana-1740	23	11	i	i	PROPN
cana-1740	23	12	th	th	PROPN
cana-1740	23	13	copy	copy	NOUN
cana-1740	23	14	of	of	ADP
cana-1740	23	15	g2	g2	PROPN
cana-1740	23	16	.	.	PUNCT
cana-1740	24	1	we	we	PRON
cana-1740	24	2	follow	follow	VERB
cana-1740	24	3	[	[	X
cana-1740	24	4	2	2	X
cana-1740	24	5	]	]	PUNCT
cana-1740	24	6	for	for	ADP
cana-1740	24	7	more	more	ADJ
cana-1740	24	8	terminologies	terminology	NOUN
cana-1740	24	9	and	and	CCONJ
cana-1740	24	10	notations	notation	NOUN
cana-1740	24	11	not	not	PART
cana-1740	24	12	mentioned	mention	VERB
cana-1740	24	13	here	here	ADV
cana-1740	24	14	.	.	PUNCT
cana-1740	25	1	definition	definition	NOUN
cana-1740	25	2	1	1	NUM
cana-1740	25	3	.	.	PUNCT
cana-1740	26	1	a	a	DET
cana-1740	26	2	molecular	molecular	ADJ
cana-1740	26	3	graph	graph	NOUN
cana-1740	26	4	g	g	PROPN
cana-1740	26	5	=	=	SYM
cana-1740	26	6	(	(	PUNCT
cana-1740	26	7	v	v	NOUN
cana-1740	26	8	,	,	PUNCT
cana-1740	26	9	e	e	NOUN
cana-1740	26	10	)	)	PUNCT
cana-1740	26	11	is	be	AUX
cana-1740	26	12	a	a	DET
cana-1740	26	13	simple	simple	ADJ
cana-1740	26	14	graph	graph	NOUN
cana-1740	26	15	having	have	VERB
cana-1740	26	16	p	p	NOUN
cana-1740	26	17	=	=	X
cana-1740	26	18	|v	|v	X
cana-1740	26	19	(	(	PUNCT
cana-1740	26	20	g)|	g)|	NOUN
cana-1740	26	21	vertices	vertex	NOUN
cana-1740	26	22	and	and	CCONJ
cana-1740	26	23	q	q	NOUN
cana-1740	26	24	=	=	PUNCT
cana-1740	26	25	|e(g)|	|e(g)|	ADJ
cana-1740	26	26	edges	edge	NOUN
cana-1740	26	27	.	.	PUNCT
cana-1740	27	1	the	the	DET
cana-1740	27	2	vertices	vertex	NOUN
cana-1740	27	3	vi	vi	PROPN
cana-1740	27	4	∈	∈	PROPN
cana-1740	27	5	v	v	NOUN
cana-1740	27	6	represent	represent	VERB
cana-1740	27	7	non	non	ADJ
cana-1740	27	8	-	-	ADJ
cana-1740	27	9	hydrogen	hydrogen	ADJ
cana-1740	27	10	atoms	atom	NOUN
cana-1740	27	11	and	and	CCONJ
cana-1740	27	12	and	and	CCONJ
cana-1740	27	13	the	the	DET
cana-1740	27	14	edges	edge	NOUN
cana-1740	27	15	(	(	PUNCT
cana-1740	27	16	vi	vi	NOUN
cana-1740	27	17	,	,	PUNCT
cana-1740	27	18	vj	vj	ADJ
cana-1740	27	19	)	)	PUNCT
cana-1740	27	20	∈	∈	PROPN
cana-1740	27	21	e	e	NOUN
cana-1740	27	22	represent	represent	VERB
cana-1740	27	23	covalent	covalent	ADJ
cana-1740	27	24	bonds	bond	NOUN
cana-1740	27	25	between	between	ADP
cana-1740	27	26	the	the	DET
cana-1740	27	27	corresponding	corresponding	ADJ
cana-1740	27	28	atoms	atom	NOUN
cana-1740	27	29	.	.	PUNCT
cana-1740	28	1	in	in	ADP
cana-1740	28	2	particular	particular	ADJ
cana-1740	28	3	,	,	PUNCT
cana-1740	28	4	hydrocarbons	hydrocarbon	NOUN
cana-1740	28	5	are	be	AUX
cana-1740	28	6	formed	form	VERB
cana-1740	28	7	only	only	ADV
cana-1740	28	8	by	by	ADP
cana-1740	28	9	carbon	carbon	NOUN
cana-1740	28	10	and	and	CCONJ
cana-1740	28	11	hydrogen	hydrogen	NOUN
cana-1740	28	12	atoms	atom	NOUN
cana-1740	28	13	and	and	CCONJ
cana-1740	28	14	their	their	PRON
cana-1740	28	15	molecular	molecular	ADJ
cana-1740	28	16	graphs	graph	NOUN
cana-1740	28	17	represent	represent	VERB
cana-1740	28	18	the	the	DET
cana-1740	28	19	carbon	carbon	NOUN
cana-1740	28	20	skeleton	skeleton	NOUN
cana-1740	28	21	of	of	ADP
cana-1740	28	22	the	the	DET
cana-1740	28	23	molecule	molecule	NOUN
cana-1740	28	24	.	.	PUNCT
cana-1740	29	1	definition	definition	NOUN
cana-1740	29	2	2	2	NUM
cana-1740	29	3	.	.	PUNCT
cana-1740	30	1	[	[	X
cana-1740	30	2	3	3	X
cana-1740	30	3	]	]	PUNCT
cana-1740	30	4	let	let	VERB
cana-1740	30	5	g	g	PRON
cana-1740	30	6	be	be	AUX
cana-1740	30	7	a	a	DET
cana-1740	30	8	graph	graph	NOUN
cana-1740	30	9	,	,	PUNCT
cana-1740	30	10	then	then	ADV
cana-1740	30	11	the	the	DET
cana-1740	30	12	first	first	ADJ
cana-1740	30	13	and	and	CCONJ
cana-1740	30	14	second	second	ADJ
cana-1740	30	15	zagreb	zagreb	PROPN
cana-1740	30	16	indices	index	NOUN
cana-1740	30	17	of	of	ADP
cana-1740	30	18	g	g	NOUN
cana-1740	30	19	are	be	AUX
cana-1740	30	20	defined	define	VERB
cana-1740	30	21	as	as	ADP
cana-1740	30	22	𝑀1(𝐺	𝑀1(𝐺	NOUN
cana-1740	30	23	)	)	PUNCT
cana-1740	30	24	=	=	PUNCT
cana-1740	30	25	∑	∑	PUNCT
cana-1740	30	26	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	30	27	)	)	PUNCT
cana-1740	30	28	𝑢∈𝑉(𝐺	𝑢∈𝑉(𝐺	NOUN
cana-1740	30	29	)	)	PUNCT
cana-1740	30	30	+	+	NUM
cana-1740	30	31	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	30	32	)	)	PUNCT
cana-1740	30	33	communications	communication	NOUN
cana-1740	30	34	on	on	ADP
cana-1740	30	35	applied	apply	VERB
cana-1740	30	36	nonlinear	nonlinear	ADJ
cana-1740	30	37	analysis	analysis	NOUN
cana-1740	30	38	issn	issn	NOUN
cana-1740	30	39	:	:	PUNCT
cana-1740	30	40	1074	1074	NUM
cana-1740	30	41	-	-	PUNCT
cana-1740	30	42	133x	133x	NUM
cana-1740	30	43	vol	vol	NOUN
cana-1740	30	44	32	32	NUM
cana-1740	31	1	no	no	NOUN
cana-1740	31	2	.	.	NOUN
cana-1740	31	3	2	2	NUM
cana-1740	31	4	(	(	PUNCT
cana-1740	31	5	2025	2025	NUM
cana-1740	31	6	)	)	PUNCT
cana-1740	31	7	255	255	NUM
cana-1740	32	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	32	2	𝑀2(𝐺	𝑀2(𝐺	NOUN
cana-1740	32	3	)	)	PUNCT
cana-1740	32	4	=	=	SYM
cana-1740	32	5	∑	∑	PUNCT
cana-1740	32	6	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	32	7	)	)	PUNCT
cana-1740	32	8	𝑢𝑣∈𝐸(𝐺	𝑢𝑣∈𝐸(𝐺	ADJ
cana-1740	32	9	)	)	PUNCT
cana-1740	32	10	.	.	PUNCT
cana-1740	33	1	definition	definition	NOUN
cana-1740	33	2	3	3	NUM
cana-1740	33	3	.	.	PUNCT
cana-1740	34	1	[	[	X
cana-1740	34	2	4	4	X
cana-1740	34	3	]	]	PUNCT
cana-1740	34	4	the	the	DET
cana-1740	34	5	harmonic	harmonic	ADJ
cana-1740	34	6	index	index	NOUN
cana-1740	34	7	of	of	ADP
cana-1740	34	8	a	a	DET
cana-1740	34	9	graph	graph	NOUN
cana-1740	34	10	g	g	NOUN
cana-1740	34	11	is	be	AUX
cana-1740	34	12	defined	define	VERB
cana-1740	34	13	as	as	ADP
cana-1740	34	14	𝐻(𝐺	𝐻(𝐺	NOUN
cana-1740	34	15	)	)	PUNCT
cana-1740	34	16	=	=	SYM
cana-1740	35	1	∑	∑	PROPN
cana-1740	35	2	2	2	NUM
cana-1740	35	3	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	35	4	)	)	PUNCT
cana-1740	35	5	+	+	CCONJ
cana-1740	35	6	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	35	7	)	)	PUNCT
cana-1740	35	8	𝑢𝑣∈𝐸(𝐺	𝑢𝑣∈𝐸(𝐺	NOUN
cana-1740	35	9	)	)	PUNCT
cana-1740	35	10	.	.	PUNCT
cana-1740	36	1	definition	definition	NOUN
cana-1740	36	2	4	4	NUM
cana-1740	36	3	.	.	PUNCT
cana-1740	37	1	[	[	X
cana-1740	37	2	5	5	X
cana-1740	37	3	]	]	PUNCT
cana-1740	37	4	the	the	DET
cana-1740	37	5	geometric	geometric	ADJ
cana-1740	37	6	-	-	PUNCT
cana-1740	37	7	arithmetic	arithmetic	ADJ
cana-1740	37	8	index	index	NOUN
cana-1740	37	9	of	of	ADP
cana-1740	37	10	a	a	DET
cana-1740	37	11	graph	graph	NOUN
cana-1740	37	12	g	g	NOUN
cana-1740	37	13	is	be	AUX
cana-1740	37	14	defined	define	VERB
cana-1740	37	15	as	as	ADP
cana-1740	37	16	𝐺𝐴(𝐺	𝐺𝐴(𝐺	NOUN
cana-1740	37	17	)	)	PUNCT
cana-1740	37	18	=	=	PUNCT
cana-1740	37	19	∑	∑	PUNCT
cana-1740	37	20	2√𝑑(𝑢)𝑑(𝑣	2√𝑑(𝑢)𝑑(𝑣	NUM
cana-1740	37	21	)	)	PUNCT
cana-1740	37	22	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	37	23	)	)	PUNCT
cana-1740	38	1	+	+	CCONJ
cana-1740	38	2	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	38	3	)	)	PUNCT
cana-1740	38	4	𝑢𝑣∈𝐸(𝐺	𝑢𝑣∈𝐸(𝐺	NOUN
cana-1740	38	5	)	)	PUNCT
cana-1740	38	6	a	a	DET
cana-1740	38	7	significant	significant	ADJ
cana-1740	38	8	field	field	NOUN
cana-1740	38	9	of	of	ADP
cana-1740	38	10	graph	graph	NOUN
cana-1740	38	11	theory	theory	NOUN
cana-1740	38	12	is	be	AUX
cana-1740	38	13	graph	graph	NOUN
cana-1740	38	14	operations	operation	NOUN
cana-1740	38	15	,	,	PUNCT
cana-1740	38	16	which	which	PRON
cana-1740	38	17	involves	involve	VERB
cana-1740	38	18	creating	create	VERB
cana-1740	38	19	new	new	ADJ
cana-1740	38	20	graphs	graph	NOUN
cana-1740	38	21	from	from	ADP
cana-1740	38	22	existing	exist	VERB
cana-1740	38	23	ones	one	NOUN
cana-1740	38	24	by	by	ADP
cana-1740	38	25	carrying	carry	VERB
cana-1740	38	26	out	out	ADP
cana-1740	38	27	a	a	DET
cana-1740	38	28	series	series	NOUN
cana-1740	38	29	of	of	ADP
cana-1740	38	30	actions	action	NOUN
cana-1740	38	31	on	on	ADP
cana-1740	38	32	the	the	DET
cana-1740	38	33	basis	basis	NOUN
cana-1740	38	34	of	of	ADP
cana-1740	38	35	some	some	DET
cana-1740	38	36	graph	graph	NOUN
cana-1740	38	37	theoretical	theoretical	ADJ
cana-1740	38	38	parameters	parameter	NOUN
cana-1740	38	39	.	.	PUNCT
cana-1740	39	1	they	they	PRON
cana-1740	39	2	comprise	comprise	VERB
cana-1740	39	3	unary	unary	ADJ
cana-1740	39	4	and	and	CCONJ
cana-1740	39	5	binary	binary	ADJ
cana-1740	39	6	operations	operation	NOUN
cana-1740	39	7	.	.	PUNCT
cana-1740	40	1	there	there	PRON
cana-1740	40	2	are	be	VERB
cana-1740	40	3	several	several	ADJ
cana-1740	40	4	operations	operation	NOUN
cana-1740	40	5	on	on	ADP
cana-1740	40	6	two	two	NUM
cana-1740	40	7	graphs	graph	NOUN
cana-1740	40	8	g1	g1	NOUN
cana-1740	40	9	and	and	CCONJ
cana-1740	40	10	g2	g2	NOUN
cana-1740	40	11	which	which	PRON
cana-1740	40	12	result	result	VERB
cana-1740	40	13	in	in	ADP
cana-1740	40	14	a	a	DET
cana-1740	40	15	graph	graph	NOUN
cana-1740	40	16	g	g	ADP
cana-1740	40	17	whose	whose	DET
cana-1740	40	18	set	set	NOUN
cana-1740	40	19	of	of	ADP
cana-1740	40	20	vertices	vertex	NOUN
cana-1740	40	21	is	be	AUX
cana-1740	40	22	the	the	DET
cana-1740	40	23	cartesian	cartesian	ADJ
cana-1740	40	24	product	product	NOUN
cana-1740	40	25	v1	v1	NOUN
cana-1740	40	26	×	×	NOUN
cana-1740	40	27	v2	v2	NOUN
cana-1740	40	28	,	,	PUNCT
cana-1740	40	29	where	where	SCONJ
cana-1740	40	30	vk	vk	PROPN
cana-1740	40	31	is	be	AUX
cana-1740	40	32	the	the	DET
cana-1740	40	33	vertex	vertex	NOUN
cana-1740	40	34	set	set	NOUN
cana-1740	40	35	of	of	ADP
cana-1740	40	36	gk	gk	PROPN
cana-1740	40	37	.	.	PUNCT
cana-1740	41	1	these	these	PRON
cana-1740	41	2	include	include	VERB
cana-1740	41	3	the	the	DET
cana-1740	41	4	cartesian	cartesian	ADJ
cana-1740	41	5	product	product	NOUN
cana-1740	41	6	,	,	PUNCT
cana-1740	41	7	the	the	DET
cana-1740	41	8	composition	composition	NOUN
cana-1740	41	9	,	,	PUNCT
cana-1740	41	10	tensor	tensor	NOUN
cana-1740	41	11	product	product	NOUN
cana-1740	41	12	etc	etc	X
cana-1740	41	13	.	.	X
cana-1740	41	14	in	in	ADP
cana-1740	41	15	2024	2024	NUM
cana-1740	41	16	,	,	PUNCT
cana-1740	41	17	[	[	X
cana-1740	41	18	1	1	NUM
cana-1740	41	19	]	]	X
cana-1740	41	20	akhil	akhil	PROPN
cana-1740	41	21	b.	b.	PROPN
cana-1740	41	22	,	,	PUNCT
cana-1740	41	23	roy	roy	PROPN
cana-1740	41	24	john	john	PROPN
cana-1740	41	25	,	,	PUNCT
cana-1740	41	26	manju	manju	PROPN
cana-1740	41	27	v.n	v.n	PROPN
cana-1740	41	28	.	.	PROPN
cana-1740	41	29	and	and	CCONJ
cana-1740	41	30	g.	g.	PROPN
cana-1740	41	31	suresh	suresh	PROPN
cana-1740	41	32	singh	singh	PROPN
cana-1740	41	33	introduced	introduce	VERB
cana-1740	41	34	a	a	DET
cana-1740	41	35	new	new	ADJ
cana-1740	41	36	graph	graph	NOUN
cana-1740	41	37	product	product	NOUN
cana-1740	41	38	called	call	VERB
cana-1740	41	39	arms	arm	NOUN
cana-1740	41	40	-	-	PUNCT
cana-1740	41	41	product	product	NOUN
cana-1740	41	42	of	of	ADP
cana-1740	41	43	two	two	NUM
cana-1740	41	44	graphs	graph	NOUN
cana-1740	41	45	g1	g1	NOUN
cana-1740	41	46	and	and	CCONJ
cana-1740	41	47	g2	g2	PROPN
cana-1740	41	48	as	as	SCONJ
cana-1740	41	49	follows	follow	VERB
cana-1740	41	50	:	:	PUNCT
cana-1740	41	51	definition	definition	NOUN
cana-1740	41	52	5	5	NUM
cana-1740	41	53	.	.	PUNCT
cana-1740	41	54	let	let	VERB
cana-1740	41	55	g1	g1	PROPN
cana-1740	41	56	=	=	SYM
cana-1740	41	57	(	(	PUNCT
cana-1740	41	58	v1	v1	PROPN
cana-1740	41	59	,	,	PUNCT
cana-1740	41	60	e1	e1	NOUN
cana-1740	41	61	)	)	PUNCT
cana-1740	41	62	and	and	CCONJ
cana-1740	41	63	g2	g2	PROPN
cana-1740	41	64	=	=	PUNCT
cana-1740	41	65	(	(	PUNCT
cana-1740	41	66	v2	v2	PROPN
cana-1740	41	67	,	,	PUNCT
cana-1740	41	68	e2	e2	PROPN
cana-1740	41	69	)	)	PUNCT
cana-1740	41	70	be	be	VERB
cana-1740	41	71	two	two	NUM
cana-1740	41	72	graphs	graph	NOUN
cana-1740	41	73	with	with	ADP
cana-1740	41	74	order	order	NOUN
cana-1740	41	75	m	m	VERB
cana-1740	41	76	and	and	CCONJ
cana-1740	41	77	n	n	PRON
cana-1740	41	78	respectively	respectively	ADV
cana-1740	41	79	.	.	PUNCT
cana-1740	42	1	the	the	DET
cana-1740	42	2	arms	arm	NOUN
cana-1740	42	3	-	-	PUNCT
cana-1740	42	4	product	product	NOUN
cana-1740	42	5	of	of	ADP
cana-1740	42	6	g1	g1	PROPN
cana-1740	42	7	and	and	CCONJ
cana-1740	42	8	g2	g2	PROPN
cana-1740	42	9	denoted	denote	VERB
cana-1740	42	10	by	by	ADP
cana-1740	42	11	g1	g1	PROPN
cana-1740	42	12	⊠	⊠	PROPN
cana-1740	42	13	g2	g2	PROPN
cana-1740	42	14	is	be	AUX
cana-1740	42	15	a	a	DET
cana-1740	42	16	graph	graph	NOUN
cana-1740	42	17	g	g	NOUN
cana-1740	42	18	with	with	ADP
cana-1740	42	19	vertex	vertex	NOUN
cana-1740	42	20	set	set	VERB
cana-1740	42	21	v	v	NOUN
cana-1740	42	22	=	=	SYM
cana-1740	42	23	v1	v1	NOUN
cana-1740	42	24	×v2	×v2	NOUN
cana-1740	42	25	and	and	CCONJ
cana-1740	42	26	two	two	NUM
cana-1740	42	27	vertices	vertex	NOUN
cana-1740	42	28	x	x	X
cana-1740	42	29	=	=	SYM
cana-1740	42	30	(	(	PUNCT
cana-1740	42	31	ui	ui	PROPN
cana-1740	42	32	,	,	PUNCT
cana-1740	42	33	vj	vj	PROPN
cana-1740	42	34	)	)	PUNCT
cana-1740	42	35	and	and	CCONJ
cana-1740	42	36	y	y	PROPN
cana-1740	42	37	=	=	SYM
cana-1740	42	38	(	(	PUNCT
cana-1740	42	39	uk	uk	PROPN
cana-1740	42	40	,	,	PUNCT
cana-1740	42	41	vl	vl	NOUN
cana-1740	42	42	)	)	PUNCT
cana-1740	42	43	are	be	AUX
cana-1740	42	44	adjacent	adjacent	ADJ
cana-1740	42	45	in	in	ADP
cana-1740	42	46	g	g	PROPN
cana-1740	42	47	if	if	SCONJ
cana-1740	42	48	:	:	PUNCT
cana-1740	42	49	1	1	X
cana-1740	42	50	.	.	X
cana-1740	42	51	either	either	CCONJ
cana-1740	42	52	𝑢𝑖	𝑢𝑖	ADP
cana-1740	42	53	=	=	X
cana-1740	42	54	𝑢𝑘	𝑢𝑘	PROPN
cana-1740	42	55	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1740	42	56	gcd	gcd	NOUN
cana-1740	42	57	(	(	PUNCT
cana-1740	42	58	𝑑𝐺2(𝑣𝑗	𝑑𝐺2(𝑣𝑗	NOUN
cana-1740	42	59	)	)	PUNCT
cana-1740	42	60	,	,	PUNCT
cana-1740	42	61	𝑑𝐺2(𝑣𝑙	𝑑𝐺2(𝑣𝑙	NOUN
cana-1740	42	62	)	)	PUNCT
cana-1740	42	63	)	)	PUNCT
cana-1740	43	1	=	=	SYM
cana-1740	43	2	1	1	NUM
cana-1740	43	3	or	or	CCONJ
cana-1740	43	4	2	2	NUM
cana-1740	43	5	.	.	X
cana-1740	43	6	gcd	gcd	NOUN
cana-1740	43	7	(	(	PUNCT
cana-1740	43	8	𝑑𝐺1(𝑢𝑖	𝑑𝐺1(𝑢𝑖	NOUN
cana-1740	43	9	)	)	PUNCT
cana-1740	43	10	,	,	PUNCT
cana-1740	43	11	𝑑𝐺1(𝑢𝑘	𝑑𝐺1(𝑢𝑘	PROPN
cana-1740	43	12	)	)	PUNCT
cana-1740	43	13	)	)	PUNCT
cana-1740	44	1	=	=	SYM
cana-1740	44	2	1	1	NUM
cana-1740	44	3	and	and	CCONJ
cana-1740	44	4	𝑣𝑗	𝑣𝑗	ADP
cana-1740	44	5	=	=	SYM
cana-1740	44	6	𝑣𝑙	𝑣𝑙	NOUN
cana-1740	44	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	44	8	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1740	44	9	𝑖	𝑖	SYM
cana-1740	44	10	,	,	PUNCT
cana-1740	44	11	𝑘	𝑘	X
cana-1740	44	12	=	=	SYM
cana-1740	44	13	1,2	1,2	NUM
cana-1740	44	14	,	,	PUNCT
cana-1740	44	15	…	…	PUNCT
cana-1740	44	16	,	,	PUNCT
cana-1740	44	17	𝑚	𝑚	ADP
cana-1740	44	18	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-1740	44	19	𝑖	𝑖	SYM
cana-1740	44	20	≠	≠	PROPN
cana-1740	44	21	𝑘	𝑘	PRON
cana-1740	44	22	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1740	44	23	𝑗	𝑗	NOUN
cana-1740	44	24	,	,	PUNCT
cana-1740	44	25	𝑙	𝑙	NOUN
cana-1740	44	26	=	=	SYM
cana-1740	44	27	1,2	1,2	NUM
cana-1740	44	28	,	,	PUNCT
cana-1740	44	29	…	…	PUNCT
cana-1740	44	30	,	,	PUNCT
cana-1740	44	31	𝑛	𝑛	DET
cana-1740	44	32	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-1740	44	33	𝑗	𝑗	PRON
cana-1740	44	34	≠	≠	PROPN
cana-1740	44	35	𝑙.	𝑙.	NOUN
cana-1740	44	36	keeping	keep	VERB
cana-1740	44	37	the	the	DET
cana-1740	44	38	importance	importance	NOUN
cana-1740	44	39	of	of	ADP
cana-1740	44	40	this	this	DET
cana-1740	44	41	newly	newly	ADV
cana-1740	44	42	introduced	introduce	VERB
cana-1740	44	43	graph	graph	NOUN
cana-1740	44	44	product	product	NOUN
cana-1740	44	45	,	,	PUNCT
cana-1740	44	46	in	in	ADP
cana-1740	44	47	this	this	DET
cana-1740	44	48	paper	paper	NOUN
cana-1740	44	49	various	various	ADJ
cana-1740	44	50	topological	topological	ADJ
cana-1740	44	51	indices	index	NOUN
cana-1740	44	52	of	of	ADP
cana-1740	44	53	graphs	graph	NOUN
cana-1740	44	54	obtained	obtain	VERB
cana-1740	44	55	through	through	ADP
cana-1740	44	56	arms	arm	NOUN
cana-1740	44	57	-	-	PUNCT
cana-1740	44	58	product	product	NOUN
cana-1740	44	59	is	be	AUX
cana-1740	44	60	evaluated	evaluate	VERB
cana-1740	44	61	.	.	PUNCT
cana-1740	45	1	in	in	ADP
cana-1740	45	2	[	[	X
cana-1740	45	3	1	1	X
cana-1740	45	4	]	]	PUNCT
cana-1740	45	5	it	it	PRON
cana-1740	45	6	is	be	AUX
cana-1740	45	7	shown	show	VERB
cana-1740	45	8	that	that	SCONJ
cana-1740	45	9	the	the	DET
cana-1740	45	10	arms	arm	NOUN
cana-1740	45	11	-	-	PUNCT
cana-1740	45	12	product	product	NOUN
cana-1740	45	13	of	of	ADP
cana-1740	45	14	two	two	NUM
cana-1740	45	15	connected	connected	ADJ
cana-1740	45	16	graph	graph	NOUN
cana-1740	45	17	need	need	AUX
cana-1740	45	18	not	not	PART
cana-1740	45	19	be	be	AUX
cana-1740	45	20	connected	connect	VERB
cana-1740	45	21	.	.	PUNCT
cana-1740	46	1	therefore	therefore	ADV
cana-1740	46	2	,	,	PUNCT
cana-1740	46	3	we	we	PRON
cana-1740	46	4	are	be	AUX
cana-1740	46	5	considering	consider	VERB
cana-1740	46	6	only	only	ADV
cana-1740	46	7	those	those	DET
cana-1740	46	8	graphs	graph	NOUN
cana-1740	46	9	which	which	PRON
cana-1740	46	10	produce	produce	VERB
cana-1740	46	11	connected	connected	ADJ
cana-1740	46	12	graphs	graph	NOUN
cana-1740	46	13	as	as	ADP
cana-1740	46	14	an	an	DET
cana-1740	46	15	outcome	outcome	NOUN
cana-1740	46	16	and	and	CCONJ
cana-1740	46	17	hence	hence	ADV
cana-1740	46	18	compute	compute	VERB
cana-1740	46	19	their	their	PRON
cana-1740	46	20	topological	topological	ADJ
cana-1740	46	21	indices	index	NOUN
cana-1740	46	22	in	in	ADP
cana-1740	46	23	detail	detail	NOUN
cana-1740	46	24	.	.	PUNCT
cana-1740	47	1	2	2	X
cana-1740	47	2	.	.	X
cana-1740	47	3	topological	topological	ADJ
cana-1740	47	4	indices	index	NOUN
cana-1740	47	5	of	of	ADP
cana-1740	47	6	arms	arm	NOUN
cana-1740	47	7	-	-	PUNCT
cana-1740	47	8	product	product	NOUN
cana-1740	47	9	of	of	ADP
cana-1740	47	10	graph	graph	NOUN
cana-1740	47	11	in	in	ADP
cana-1740	47	12	this	this	DET
cana-1740	47	13	section	section	NOUN
cana-1740	47	14	topological	topological	ADJ
cana-1740	47	15	indices	index	NOUN
cana-1740	47	16	namely	namely	ADV
cana-1740	47	17	first	first	ADJ
cana-1740	47	18	,	,	PUNCT
cana-1740	47	19	second	second	ADJ
cana-1740	47	20	zagreb	zagreb	PROPN
cana-1740	47	21	indices	index	NOUN
cana-1740	47	22	,	,	PUNCT
cana-1740	47	23	harmonic	harmonic	ADJ
cana-1740	47	24	index	index	NOUN
cana-1740	47	25	and	and	CCONJ
cana-1740	47	26	geometricarithmetic	geometricarithmetic	ADJ
cana-1740	47	27	index	index	NOUN
cana-1740	47	28	of	of	ADP
cana-1740	47	29	certain	certain	ADJ
cana-1740	47	30	graphs	graph	NOUN
cana-1740	47	31	obtained	obtain	VERB
cana-1740	47	32	through	through	ADP
cana-1740	47	33	the	the	DET
cana-1740	47	34	arms	arm	NOUN
cana-1740	47	35	-	-	PUNCT
cana-1740	47	36	product	product	NOUN
cana-1740	47	37	of	of	ADP
cana-1740	47	38	certain	certain	ADJ
cana-1740	47	39	graphs	graph	NOUN
cana-1740	47	40	are	be	AUX
cana-1740	47	41	determined	determine	VERB
cana-1740	47	42	.	.	PUNCT
cana-1740	48	1	theorem	theorem	VERB
cana-1740	48	2	2.1	2.1	NUM
cana-1740	48	3	.	.	PUNCT
cana-1740	49	1	let	let	VERB
cana-1740	49	2	b	b	X
cana-1740	49	3	be	be	AUX
cana-1740	49	4	the	the	DET
cana-1740	49	5	bull	bull	NOUN
cana-1740	49	6	graph	graph	NOUN
cana-1740	49	7	and	and	CCONJ
cana-1740	49	8	p2	p2	PROPN
cana-1740	49	9	be	be	AUX
cana-1740	49	10	a	a	DET
cana-1740	49	11	path	path	NOUN
cana-1740	49	12	on	on	ADP
cana-1740	49	13	two	two	NUM
cana-1740	49	14	vertices	vertex	NOUN
cana-1740	49	15	.	.	PUNCT
cana-1740	50	1	then	then	ADV
cana-1740	50	2	for	for	ADP
cana-1740	50	3	the	the	DET
cana-1740	50	4	graph	graph	NOUN
cana-1740	50	5	b	b	PROPN
cana-1740	50	6	⊠	⊠	PROPN
cana-1740	50	7	p2	p2	NOUN
cana-1740	50	8	,	,	PUNCT
cana-1740	50	9	first	first	ADJ
cana-1740	50	10	and	and	CCONJ
cana-1740	50	11	second	second	ADJ
cana-1740	50	12	zagreb	zagreb	PROPN
cana-1740	50	13	indices	index	NOUN
cana-1740	50	14	,	,	PUNCT
cana-1740	50	15	harmonic	harmonic	ADJ
cana-1740	50	16	index	index	NOUN
cana-1740	50	17	and	and	CCONJ
cana-1740	50	18	geometric	geometric	ADJ
cana-1740	50	19	-	-	PUNCT
cana-1740	50	20	arithmetic	arithmetic	ADJ
cana-1740	50	21	index	index	NOUN
cana-1740	50	22	are	be	AUX
cana-1740	50	23	respectively	respectively	ADV
cana-1740	50	24	given	give	VERB
cana-1740	50	25	by	by	ADP
cana-1740	50	26	:	:	PUNCT
cana-1740	50	27	1	1	X
cana-1740	50	28	.	.	X
cana-1740	50	29	𝑀1(𝐵	𝑀1(𝐵	NOUN
cana-1740	50	30	⊠	⊠	PROPN
cana-1740	50	31	𝑃2	𝑃2	NOUN
cana-1740	50	32	)	)	PUNCT
cana-1740	51	1	=	=	SYM
cana-1740	51	2	214	214	NUM
cana-1740	51	3	.	.	PUNCT
cana-1740	52	1	communications	communication	NOUN
cana-1740	52	2	on	on	ADP
cana-1740	52	3	applied	apply	VERB
cana-1740	52	4	nonlinear	nonlinear	ADJ
cana-1740	52	5	analysis	analysis	NOUN
cana-1740	52	6	issn	issn	NOUN
cana-1740	52	7	:	:	PUNCT
cana-1740	52	8	1074	1074	NUM
cana-1740	52	9	-	-	PUNCT
cana-1740	52	10	133x	133x	NUM
cana-1740	52	11	vol	vol	NOUN
cana-1740	52	12	32	32	NUM
cana-1740	52	13	no	no	NOUN
cana-1740	52	14	.	.	NOUN
cana-1740	52	15	2	2	NUM
cana-1740	52	16	(	(	PUNCT
cana-1740	52	17	2025	2025	NUM
cana-1740	52	18	)	)	PUNCT
cana-1740	53	1	256	256	NUM
cana-1740	53	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	53	3	2	2	NUM
cana-1740	53	4	.	.	PUNCT
cana-1740	53	5	𝑀2(𝐵	𝑀2(𝐵	PROPN
cana-1740	53	6	⊠	⊠	PROPN
cana-1740	53	7	𝑃2	𝑃2	PROPN
cana-1740	53	8	)	)	PUNCT
cana-1740	53	9	=	=	NOUN
cana-1740	54	1	497	497	NUM
cana-1740	54	2	.	.	NOUN
cana-1740	55	1	3	3	X
cana-1740	55	2	.	.	X
cana-1740	55	3	𝐻	𝐻	PROPN
cana-1740	55	4	(	(	PUNCT
cana-1740	55	5	𝐵	𝐵	NOUN
cana-1740	55	6	⊠	⊠	PROPN
cana-1740	55	7	𝑃	𝑃	NOUN
cana-1740	55	8	2	2	NUM
cana-1740	55	9	)	)	PUNCT
cana-1740	55	10	=	=	NOUN
cana-1740	55	11	149	149	NUM
cana-1740	55	12	30	30	NUM
cana-1740	55	13	.	.	PUNCT
cana-1740	56	1	4	4	X
cana-1740	56	2	.	.	X
cana-1740	56	3	𝐺𝐴(𝐵	𝐺𝐴(𝐵	PRON
cana-1740	56	4	⊠	⊠	PROPN
cana-1740	56	5	𝑃	𝑃	PROPN
cana-1740	56	6	2	2	NUM
cana-1740	56	7	)	)	PUNCT
cana-1740	56	8	=	=	SYM
cana-1740	56	9	11	11	NUM
cana-1740	56	10	+	+	NUM
cana-1740	56	11	16√5	16√5	NUM
cana-1740	56	12	3	3	NUM
cana-1740	56	13	.	.	PUNCT
cana-1740	57	1	proof	proof	NOUN
cana-1740	57	2	:	:	PUNCT
cana-1740	57	3	the	the	DET
cana-1740	57	4	arms	arm	NOUN
cana-1740	57	5	-	-	PUNCT
cana-1740	57	6	product	product	NOUN
cana-1740	57	7	of	of	ADP
cana-1740	57	8	the	the	DET
cana-1740	57	9	bull	bull	NOUN
cana-1740	57	10	graph	graph	NOUN
cana-1740	57	11	,	,	PUNCT
cana-1740	57	12	b	b	NOUN
cana-1740	57	13	and	and	CCONJ
cana-1740	57	14	the	the	DET
cana-1740	57	15	path	path	NOUN
cana-1740	57	16	graph	graph	NOUN
cana-1740	57	17	,	,	PUNCT
cana-1740	57	18	p2	p2	PROPN
cana-1740	57	19	is	be	AUX
cana-1740	57	20	a	a	DET
cana-1740	57	21	graph	graph	NOUN
cana-1740	57	22	with	with	ADP
cana-1740	57	23	10	10	NUM
cana-1740	57	24	vertices	vertex	NOUN
cana-1740	57	25	.	.	PUNCT
cana-1740	58	1	b	b	X
cana-1740	58	2	⊠	⊠	PROPN
cana-1740	58	3	p2	p2	VERB
cana-1740	58	4	the	the	DET
cana-1740	58	5	edge	edge	NOUN
cana-1740	58	6	set	set	NOUN
cana-1740	58	7	of	of	ADP
cana-1740	58	8	b	b	PROPN
cana-1740	58	9	⊠	⊠	PROPN
cana-1740	58	10	p2	p2	NOUN
cana-1740	58	11	can	can	AUX
cana-1740	58	12	be	be	AUX
cana-1740	58	13	partitioned	partition	VERB
cana-1740	58	14	as	as	SCONJ
cana-1740	58	15	follows	follow	VERB
cana-1740	58	16	:	:	PUNCT
cana-1740	58	17	𝐸1	𝐸1	NOUN
cana-1740	58	18	=	=	SYM
cana-1740	58	19	{	{	PUNCT
cana-1740	58	20	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	58	21	|	|	ADV
cana-1740	58	22	𝑑𝑢	𝑑𝑢	NOUN
cana-1740	58	23	=	=	NOUN
cana-1740	58	24	5	5	NUM
cana-1740	58	25	,	,	PUNCT
cana-1740	58	26	𝑑𝑣	𝑑𝑣	X
cana-1740	58	27	=	=	NOUN
cana-1740	58	28	5	5	NUM
cana-1740	58	29	}	}	PUNCT
cana-1740	58	30	,	,	PUNCT
cana-1740	58	31	|𝐸1|	|𝐸1|	NOUN
cana-1740	58	32	=	=	PUNCT
cana-1740	59	1	9	9	X
cana-1740	59	2	.	.	PUNCT
cana-1740	59	3	𝐸2	𝐸2	PROPN
cana-1740	59	4	=	=	X
cana-1740	59	5	{	{	PUNCT
cana-1740	59	6	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	59	7	|	|	ADV
cana-1740	59	8	𝑑𝑢	𝑑𝑢	NOUN
cana-1740	59	9	=	=	NOUN
cana-1740	59	10	5	5	NUM
cana-1740	59	11	,	,	PUNCT
cana-1740	59	12	𝑑𝑣	𝑑𝑣	X
cana-1740	59	13	=	=	NOUN
cana-1740	59	14	4	4	NUM
cana-1740	59	15	}	}	PUNCT
cana-1740	59	16	,	,	PUNCT
cana-1740	59	17	|𝐸2|	|𝐸2|	PROPN
cana-1740	59	18	=	=	NOUN
cana-1740	60	1	12	12	NUM
cana-1740	60	2	.	.	PUNCT
cana-1740	61	1	𝐸3	𝐸3	NOUN
cana-1740	61	2	=	=	PRON
cana-1740	61	3	{	{	PUNCT
cana-1740	61	4	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	61	5	|	|	ADV
cana-1740	61	6	𝑑𝑢	𝑑𝑢	NOUN
cana-1740	61	7	=	=	NOUN
cana-1740	61	8	4	4	NUM
cana-1740	61	9	,	,	PUNCT
cana-1740	61	10	𝑑𝑣	𝑑𝑣	X
cana-1740	61	11	=	=	NOUN
cana-1740	61	12	4	4	NUM
cana-1740	61	13	}	}	PUNCT
cana-1740	61	14	,	,	PUNCT
cana-1740	61	15	|𝐸3|	|𝐸3|	NUM
cana-1740	61	16	=	=	SYM
cana-1740	61	17	2	2	X
cana-1740	61	18	.	.	PUNCT
cana-1740	62	1	𝑀1(𝐵	𝑀1(𝐵	NOUN
cana-1740	62	2	⊠	⊠	PROPN
cana-1740	62	3	𝑃2	𝑃2	NOUN
cana-1740	62	4	)	)	PUNCT
cana-1740	62	5	=	=	PUNCT
cana-1740	62	6	∑	∑	PUNCT
cana-1740	62	7	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	62	8	)	)	PUNCT
cana-1740	62	9	𝑢𝑣∈𝐸(𝐵⊠𝑃2	𝑢𝑣∈𝐸(𝐵⊠𝑃2	PROPN
cana-1740	62	10	)	)	PUNCT
cana-1740	62	11	+	+	NUM
cana-1740	62	12	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	62	13	)	)	PUNCT
cana-1740	62	14	=	=	PUNCT
cana-1740	62	15	∑	∑	PUNCT
cana-1740	62	16	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	62	17	)	)	PUNCT
cana-1740	62	18	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	63	1	+	+	SYM
cana-1740	63	2	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	63	3	)	)	PUNCT
cana-1740	64	1	+	+	CCONJ
cana-1740	64	2	∑	∑	ADP
cana-1740	64	3	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	64	4	)	)	PUNCT
cana-1740	64	5	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	PUNCT
cana-1740	64	6	+	+	CCONJ
cana-1740	64	7	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	64	8	)	)	PUNCT
cana-1740	64	9	+	+	CCONJ
cana-1740	64	10	∑	∑	ADP
cana-1740	64	11	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	64	12	)	)	PUNCT
cana-1740	64	13	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	VERB
cana-1740	64	14	+	+	X
cana-1740	64	15	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	64	16	)	)	PUNCT
cana-1740	64	17	=	=	PUNCT
cana-1740	64	18	∑	∑	PUNCT
cana-1740	64	19	5	5	NUM
cana-1740	64	20	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	64	21	+	+	CCONJ
cana-1740	64	22	5	5	NUM
cana-1740	64	23	+	+	CCONJ
cana-1740	64	24	∑	∑	PROPN
cana-1740	64	25	5	5	NUM
cana-1740	64	26	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	64	27	+	+	NUM
cana-1740	64	28	4	4	NUM
cana-1740	64	29	+	+	ADJ
cana-1740	64	30	∑	∑	PROPN
cana-1740	64	31	4	4	NUM
cana-1740	64	32	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	64	33	+	+	CCONJ
cana-1740	64	34	4	4	NUM
cana-1740	64	35	communications	communication	NOUN
cana-1740	64	36	on	on	ADP
cana-1740	64	37	applied	apply	VERB
cana-1740	64	38	nonlinear	nonlinear	ADJ
cana-1740	64	39	analysis	analysis	NOUN
cana-1740	64	40	issn	issn	NOUN
cana-1740	64	41	:	:	PUNCT
cana-1740	64	42	1074	1074	NUM
cana-1740	64	43	-	-	PUNCT
cana-1740	64	44	133x	133x	NUM
cana-1740	64	45	vol	vol	NOUN
cana-1740	64	46	32	32	NUM
cana-1740	64	47	no	no	NOUN
cana-1740	64	48	.	.	NOUN
cana-1740	64	49	2	2	NUM
cana-1740	64	50	(	(	PUNCT
cana-1740	64	51	2025	2025	NUM
cana-1740	64	52	)	)	PUNCT
cana-1740	64	53	257	257	NUM
cana-1740	64	54	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	64	55	=	=	PUNCT
cana-1740	65	1	10	10	NUM
cana-1740	65	2	∑	∑	SYM
cana-1740	65	3	1	1	NUM
cana-1740	65	4	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	65	5	+	+	CCONJ
cana-1740	65	6	9	9	NUM
cana-1740	65	7	∑	∑	SYM
cana-1740	65	8	1	1	NUM
cana-1740	65	9	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	65	10	+	+	CCONJ
cana-1740	65	11	8	8	NUM
cana-1740	65	12	∑	∑	SYM
cana-1740	65	13	1	1	NUM
cana-1740	65	14	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	65	15	=	=	SYM
cana-1740	65	16	10	10	NUM
cana-1740	65	17	×	×	NOUN
cana-1740	65	18	9	9	NUM
cana-1740	65	19	+	+	CCONJ
cana-1740	65	20	9	9	NUM
cana-1740	65	21	×	×	NOUN
cana-1740	65	22	12	12	NUM
cana-1740	65	23	+	+	CCONJ
cana-1740	65	24	8	8	NUM
cana-1740	65	25	×	×	NOUN
cana-1740	65	26	2	2	NUM
cana-1740	65	27	=	=	SYM
cana-1740	65	28	214	214	NUM
cana-1740	65	29	.	.	PUNCT
cana-1740	66	1	𝑀2(𝐵	𝑀2(𝐵	PROPN
cana-1740	66	2	⊠	⊠	PROPN
cana-1740	66	3	𝑃2	𝑃2	PROPN
cana-1740	66	4	)	)	PUNCT
cana-1740	66	5	=	=	PUNCT
cana-1740	66	6	∑	∑	ADP
cana-1740	66	7	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	66	8	)	)	PUNCT
cana-1740	66	9	𝑢𝑣∈𝐸(𝐵⊠𝑃2	𝑢𝑣∈𝐸(𝐵⊠𝑃2	PROPN
cana-1740	66	10	)	)	PUNCT
cana-1740	66	11	=	=	PUNCT
cana-1740	66	12	∑	∑	PUNCT
cana-1740	66	13	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	66	14	)	)	PUNCT
cana-1740	66	15	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	67	1	+	+	CCONJ
cana-1740	67	2	∑	∑	NOUN
cana-1740	67	3	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	67	4	)	)	PUNCT
cana-1740	67	5	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	PUNCT
cana-1740	68	1	+	+	NUM
cana-1740	68	2	∑	∑	NOUN
cana-1740	68	3	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	68	4	)	)	PUNCT
cana-1740	68	5	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	68	6	=	=	SYM
cana-1740	68	7	∑	∑	PROPN
cana-1740	68	8	25	25	NUM
cana-1740	68	9	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	X
cana-1740	68	10	+	+	CCONJ
cana-1740	68	11	∑	∑	PROPN
cana-1740	68	12	20	20	NUM
cana-1740	68	13	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	68	14	+	+	NUM
cana-1740	68	15	∑	∑	PROPN
cana-1740	68	16	16	16	NUM
cana-1740	68	17	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	68	18	=	=	SYM
cana-1740	68	19	25	25	NUM
cana-1740	68	20	∑	∑	SYM
cana-1740	68	21	1	1	NUM
cana-1740	68	22	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	68	23	+	+	CCONJ
cana-1740	68	24	20	20	NUM
cana-1740	68	25	∑	∑	SYM
cana-1740	68	26	1	1	NUM
cana-1740	68	27	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	68	28	+	+	CCONJ
cana-1740	68	29	16	16	NUM
cana-1740	68	30	∑	∑	SYM
cana-1740	68	31	1	1	NUM
cana-1740	68	32	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	68	33	=	=	NOUN
cana-1740	68	34	25	25	NUM
cana-1740	68	35	×	×	NOUN
cana-1740	68	36	9	9	NUM
cana-1740	68	37	+	+	CCONJ
cana-1740	68	38	20	20	NUM
cana-1740	68	39	×	×	NOUN
cana-1740	68	40	12	12	NUM
cana-1740	68	41	+	+	CCONJ
cana-1740	68	42	16	16	NUM
cana-1740	68	43	×	×	NOUN
cana-1740	68	44	2	2	NUM
cana-1740	68	45	=	=	SYM
cana-1740	68	46	497	497	NUM
cana-1740	68	47	.	.	PUNCT
cana-1740	68	48	𝐻(𝐵	𝐻(𝐵	PROPN
cana-1740	68	49	⊠	⊠	PROPN
cana-1740	68	50	𝑃2	𝑃2	PROPN
cana-1740	68	51	)	)	PUNCT
cana-1740	68	52	=	=	PUNCT
cana-1740	68	53	∑	∑	PROPN
cana-1740	68	54	2	2	NUM
cana-1740	68	55	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	68	56	)	)	PUNCT
cana-1740	68	57	+	+	CCONJ
cana-1740	68	58	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	68	59	)	)	PUNCT
cana-1740	68	60	𝑢𝑣∈𝐸(𝐵⊠𝑃2	𝑢𝑣∈𝐸(𝐵⊠𝑃2	PROPN
cana-1740	68	61	)	)	PUNCT
cana-1740	68	62	=	=	PUNCT
cana-1740	69	1	∑	∑	PROPN
cana-1740	69	2	2	2	NUM
cana-1740	69	3	10	10	NUM
cana-1740	69	4	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	70	1	+	+	CCONJ
cana-1740	70	2	∑	∑	SYM
cana-1740	70	3	2	2	NUM
cana-1740	70	4	9	9	NUM
cana-1740	70	5	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	70	6	+	+	NUM
cana-1740	70	7	∑	∑	SYM
cana-1740	70	8	2	2	NUM
cana-1740	70	9	8	8	NUM
cana-1740	70	10	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	70	11	=	=	SYM
cana-1740	70	12	2	2	NUM
cana-1740	70	13	10	10	NUM
cana-1740	70	14	∑	∑	SYM
cana-1740	70	15	1	1	NUM
cana-1740	70	16	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	70	17	+	+	ADJ
cana-1740	70	18	2	2	NUM
cana-1740	70	19	9	9	NUM
cana-1740	70	20	∑	∑	SYM
cana-1740	70	21	1	1	NUM
cana-1740	70	22	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	70	23	+	+	CCONJ
cana-1740	70	24	2	2	NUM
cana-1740	70	25	8	8	NUM
cana-1740	70	26	∑	∑	SYM
cana-1740	70	27	1	1	NUM
cana-1740	70	28	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	70	29	=	=	NOUN
cana-1740	70	30	149	149	NUM
cana-1740	70	31	30	30	NUM
cana-1740	70	32	.	.	PUNCT
cana-1740	71	1	𝐺𝐴	𝐺𝐴	NOUN
cana-1740	71	2	(	(	PUNCT
cana-1740	71	3	𝐵	𝐵	NOUN
cana-1740	71	4	⊠	⊠	PROPN
cana-1740	71	5	𝑃2	𝑃2	PROPN
cana-1740	71	6	)	)	PUNCT
cana-1740	71	7	=	=	PUNCT
cana-1740	71	8	∑	∑	PUNCT
cana-1740	71	9	2√𝑑(𝑢)𝑑(𝑣	2√𝑑(𝑢)𝑑(𝑣	NUM
cana-1740	71	10	)	)	PUNCT
cana-1740	71	11	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	71	12	)	)	PUNCT
cana-1740	71	13	+	+	CCONJ
cana-1740	71	14	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	71	15	)	)	PUNCT
cana-1740	71	16	𝑢𝑣∈𝐸(𝐵⊠𝑃2	𝑢𝑣∈𝐸(𝐵⊠𝑃2	PROPN
cana-1740	71	17	)	)	PUNCT
cana-1740	71	18	=	=	PUNCT
cana-1740	72	1	∑	∑	PUNCT
cana-1740	72	2	2√25	2√25	NOUN
cana-1740	72	3	10	10	NUM
cana-1740	72	4	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	73	1	+	+	CCONJ
cana-1740	73	2	∑	∑	PUNCT
cana-1740	73	3	2√20	2√20	NUM
cana-1740	73	4	9	9	NUM
cana-1740	73	5	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	73	6	+	+	NUM
cana-1740	73	7	∑	∑	PUNCT
cana-1740	73	8	2√16	2√16	NUM
cana-1740	73	9	8	8	NUM
cana-1740	73	10	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	73	11	=	=	SYM
cana-1740	73	12	∑	∑	PROPN
cana-1740	73	13	1	1	NUM
cana-1740	73	14	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	73	15	+	+	CCONJ
cana-1740	73	16	4√5	4√5	PROPN
cana-1740	73	17	9	9	NUM
cana-1740	73	18	∑	∑	SYM
cana-1740	73	19	1	1	NUM
cana-1740	73	20	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	73	21	+	+	NUM
cana-1740	73	22	∑	∑	SYM
cana-1740	73	23	1	1	NUM
cana-1740	73	24	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	73	25	=	=	SYM
cana-1740	73	26	9	9	NUM
cana-1740	73	27	+	+	NUM
cana-1740	73	28	12	12	NUM
cana-1740	74	1	4√5	4√5	NUM
cana-1740	74	2	9	9	NUM
cana-1740	74	3	+	+	NUM
cana-1740	74	4	2	2	NUM
cana-1740	74	5	=	=	SYM
cana-1740	74	6	11	11	NUM
cana-1740	74	7	+	+	NUM
cana-1740	74	8	16√5	16√5	NUM
cana-1740	74	9	3	3	NUM
cana-1740	74	10	.	.	PUNCT
cana-1740	75	1	communications	communication	NOUN
cana-1740	75	2	on	on	ADP
cana-1740	75	3	applied	apply	VERB
cana-1740	75	4	nonlinear	nonlinear	ADJ
cana-1740	75	5	analysis	analysis	NOUN
cana-1740	75	6	issn	issn	NOUN
cana-1740	75	7	:	:	PUNCT
cana-1740	75	8	1074	1074	NUM
cana-1740	75	9	-	-	PUNCT
cana-1740	75	10	133x	133x	NUM
cana-1740	75	11	vol	vol	NOUN
cana-1740	75	12	32	32	NUM
cana-1740	75	13	no	no	NOUN
cana-1740	75	14	.	.	NOUN
cana-1740	75	15	2	2	NUM
cana-1740	75	16	(	(	PUNCT
cana-1740	75	17	2025	2025	NUM
cana-1740	75	18	)	)	PUNCT
cana-1740	75	19	258	258	NUM
cana-1740	75	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	75	21	this	this	PRON
cana-1740	75	22	completes	complete	VERB
cana-1740	75	23	the	the	DET
cana-1740	75	24	proof	proof	NOUN
cana-1740	75	25	.	.	PUNCT
cana-1740	76	1	theorem	theorem	VERB
cana-1740	76	2	2.2	2.2	NUM
cana-1740	76	3	.	.	PUNCT
cana-1740	77	1	let	let	VERB
cana-1740	77	2	k1,n	k1,n	PROPN
cana-1740	77	3	and	and	CCONJ
cana-1740	77	4	k1,m	k1,m	PROPN
cana-1740	77	5	be	be	AUX
cana-1740	77	6	two	two	NUM
cana-1740	77	7	bipartite	bipartite	ADJ
cana-1740	77	8	graphs	graph	NOUN
cana-1740	77	9	.	.	PUNCT
cana-1740	78	1	then	then	ADV
cana-1740	78	2	for	for	ADP
cana-1740	78	3	the	the	DET
cana-1740	78	4	graph	graph	NOUN
cana-1740	78	5	k1,n	k1,n	PROPN
cana-1740	78	6	⊠	⊠	PROPN
cana-1740	78	7	k1,m	k1,m	PROPN
cana-1740	78	8	first	first	ADJ
cana-1740	78	9	and	and	CCONJ
cana-1740	78	10	second	second	ADJ
cana-1740	78	11	zagreb	zagreb	PROPN
cana-1740	78	12	indices	index	NOUN
cana-1740	78	13	,	,	PUNCT
cana-1740	78	14	harmonic	harmonic	ADJ
cana-1740	78	15	index	index	NOUN
cana-1740	78	16	and	and	CCONJ
cana-1740	78	17	geometric	geometric	ADJ
cana-1740	78	18	-	-	PUNCT
cana-1740	78	19	arithmetic	arithmetic	ADJ
cana-1740	78	20	index	index	NOUN
cana-1740	78	21	are	be	AUX
cana-1740	78	22	respectively	respectively	ADV
cana-1740	78	23	given	give	VERB
cana-1740	78	24	by	by	ADP
cana-1740	78	25	:	:	PUNCT
cana-1740	78	26	1	1	NUM
cana-1740	78	27	.	.	X
cana-1740	78	28	𝑀1(𝐾1,𝑛⊠𝐾1,𝑚	𝑀1(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	78	29	)	)	PUNCT
cana-1740	78	30	=	=	PUNCT
cana-1740	79	1	(	(	PUNCT
cana-1740	79	2	𝑚	𝑚	X
cana-1740	79	3	+	+	PUNCT
cana-1740	79	4	𝑛)2(𝑚	𝑛)2(𝑚	PRON
cana-1740	79	5	+	+	CCONJ
cana-1740	79	6	1)(𝑛	1)(𝑛	NUM
cana-1740	79	7	+	+	NUM
cana-1740	79	8	1	1	NUM
cana-1740	79	9	)	)	PUNCT
cana-1740	79	10	.	.	PUNCT
cana-1740	80	1	2.𝑀2(𝐾1,𝑛⊠𝐾1,𝑚	2.𝑀2(𝐾1,𝑛⊠𝐾1,𝑚	X
cana-1740	80	2	)	)	PUNCT
cana-1740	80	3	=	=	SYM
cana-1740	80	4	(	(	PUNCT
cana-1740	80	5	𝑚	𝑚	X
cana-1740	80	6	+	+	PUNCT
cana-1740	80	7	𝑛)2(𝑚	𝑛)2(𝑚	PRON
cana-1740	80	8	+	+	CCONJ
cana-1740	80	9	1)(𝑛	1)(𝑛	NUM
cana-1740	80	10	+	+	NUM
cana-1740	80	11	1	1	NUM
cana-1740	80	12	)	)	PUNCT
cana-1740	80	13	2	2	NUM
cana-1740	80	14	.	.	PUNCT
cana-1740	81	1	3	3	X
cana-1740	81	2	.	.	X
cana-1740	81	3	𝐻(𝐾1,𝑛⊠𝐾1,𝑚	𝐻(𝐾1,𝑛⊠𝐾1,𝑚	PROPN
cana-1740	81	4	)	)	PUNCT
cana-1740	81	5	=	=	PUNCT
cana-1740	81	6	(	(	PUNCT
cana-1740	81	7	𝑚	𝑚	PROPN
cana-1740	81	8	+	+	NOUN
cana-1740	81	9	1)(𝑛	1)(𝑛	NUM
cana-1740	81	10	+	+	NUM
cana-1740	81	11	1	1	NUM
cana-1740	81	12	)	)	PUNCT
cana-1740	81	13	2	2	NUM
cana-1740	81	14	.	.	X
cana-1740	82	1	4	4	X
cana-1740	82	2	.	.	X
cana-1740	82	3	𝐺𝐴(𝐾1,𝑛⊠𝐾1,𝑚	𝐺𝐴(𝐾1,𝑛⊠𝐾1,𝑚	NUM
cana-1740	82	4	)	)	PUNCT
cana-1740	83	1	=	=	NOUN
cana-1740	83	2	(	(	PUNCT
cana-1740	83	3	𝑚	𝑚	X
cana-1740	83	4	+	+	NUM
cana-1740	83	5	𝑛)(𝑚	𝑛)(𝑚	PROPN
cana-1740	84	1	+	+	CCONJ
cana-1740	85	1	1)(𝑛	1)(𝑛	NUM
cana-1740	85	2	+	+	NUM
cana-1740	85	3	1	1	NUM
cana-1740	85	4	)	)	PUNCT
cana-1740	85	5	2	2	NUM
cana-1740	85	6	proof	proof	NOUN
cana-1740	85	7	:	:	PUNCT
cana-1740	85	8	the	the	DET
cana-1740	85	9	armsproduct	armsproduct	NOUN
cana-1740	85	10	of	of	ADP
cana-1740	85	11	k1,n	k1,n	PROPN
cana-1740	85	12	and	and	CCONJ
cana-1740	85	13	k1,m	k1,m	PROPN
cana-1740	85	14	is	be	AUX
cana-1740	85	15	a	a	DET
cana-1740	85	16	(	(	PUNCT
cana-1740	85	17	m	m	PROPN
cana-1740	85	18	+	+	NOUN
cana-1740	85	19	n	n	CCONJ
cana-1740	85	20	)	)	PUNCT
cana-1740	85	21	-regular	-regular	ADJ
cana-1740	85	22	graph	graph	NOUN
cana-1740	85	23	on	on	ADP
cana-1740	85	24	(	(	PUNCT
cana-1740	85	25	m	m	VERB
cana-1740	85	26	+	+	NOUN
cana-1740	85	27	1)(n	1)(n	NUM
cana-1740	85	28	+	+	CCONJ
cana-1740	85	29	1	1	NUM
cana-1740	85	30	)	)	PUNCT
cana-1740	85	31	vertices	vertex	NOUN
cana-1740	85	32	.	.	PUNCT
cana-1740	86	1	the	the	DET
cana-1740	86	2	number	number	NOUN
cana-1740	86	3	of	of	ADP
cana-1740	86	4	edges	edge	NOUN
cana-1740	86	5	of	of	ADP
cana-1740	86	6	k1,n	k1,n	PROPN
cana-1740	86	7	⊠	⊠	PROPN
cana-1740	86	8	k1,m	k1,m	PROPN
cana-1740	86	9	is	be	AUX
cana-1740	86	10	(	(	PUNCT
cana-1740	86	11	𝑚+𝑛)(𝑚+1)(𝑛+1	𝑚+𝑛)(𝑚+1)(𝑛+1	NOUN
cana-1740	86	12	)	)	PUNCT
cana-1740	86	13	2	2	NUM
cana-1740	86	14	.	.	PUNCT
cana-1740	87	1	𝑀1(𝐾1,𝑛⊠𝐾1,𝑚	𝑀1(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	87	2	)	)	PUNCT
cana-1740	88	1	=	=	SYM
cana-1740	88	2	∑	∑	PUNCT
cana-1740	88	3	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	88	4	)	)	PUNCT
cana-1740	88	5	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	88	6	)	)	PUNCT
cana-1740	88	7	+	+	CCONJ
cana-1740	88	8	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	88	9	)	)	PUNCT
cana-1740	88	10	=	=	PUNCT
cana-1740	88	11	∑	∑	PUNCT
cana-1740	88	12	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	88	13	)	)	PUNCT
cana-1740	88	14	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	88	15	)	)	PUNCT
cana-1740	88	16	+	+	CCONJ
cana-1740	88	17	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	88	18	)	)	PUNCT
cana-1740	88	19	=	=	PUNCT
cana-1740	89	1	∑	∑	PUNCT
cana-1740	89	2	2(𝑚	2(𝑚	NUM
cana-1740	89	3	+	+	SYM
cana-1740	89	4	𝑛	𝑛	ADJ
cana-1740	89	5	)	)	PUNCT
cana-1740	89	6	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	89	7	)	)	PUNCT
cana-1740	90	1	=	=	SYM
cana-1740	90	2	2(𝑚	2(𝑚	NUM
cana-1740	90	3	+	+	SYM
cana-1740	90	4	𝑛	𝑛	X
cana-1740	90	5	)	)	PUNCT
cana-1740	90	6	(	(	PUNCT
cana-1740	90	7	𝑚	𝑚	PROPN
cana-1740	90	8	+	+	X
cana-1740	90	9	𝑛)(𝑚	𝑛)(𝑚	PROPN
cana-1740	90	10	+	+	CCONJ
cana-1740	91	1	1)(𝑛	1)(𝑛	NUM
cana-1740	91	2	+	+	NUM
cana-1740	91	3	1	1	NUM
cana-1740	91	4	)	)	PUNCT
cana-1740	91	5	2	2	NUM
cana-1740	91	6	=	=	SYM
cana-1740	91	7	(	(	PUNCT
cana-1740	91	8	𝑚	𝑚	X
cana-1740	91	9	+	+	PUNCT
cana-1740	91	10	𝑛)2(𝑚	𝑛)2(𝑚	PRON
cana-1740	91	11	+	+	CCONJ
cana-1740	91	12	1)(𝑛	1)(𝑛	NUM
cana-1740	91	13	+	+	NUM
cana-1740	91	14	1	1	NUM
cana-1740	91	15	)	)	PUNCT
cana-1740	91	16	.	.	PUNCT
cana-1740	92	1	𝑀2(𝐾1,𝑛⊠𝐾1,𝑚	𝑀2(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	92	2	)	)	PUNCT
cana-1740	92	3	=	=	SYM
cana-1740	92	4	∑	∑	ADP
cana-1740	92	5	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	92	6	)	)	PUNCT
cana-1740	92	7	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	92	8	)	)	PUNCT
cana-1740	92	9	=	=	SYM
cana-1740	92	10	∑	∑	ADP
cana-1740	92	11	𝑑(𝑢)𝑑(𝑣	𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	92	12	)	)	PUNCT
cana-1740	92	13	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	92	14	)	)	PUNCT
cana-1740	92	15	=	=	PUNCT
cana-1740	92	16	∑	∑	PUNCT
cana-1740	92	17	(	(	PUNCT
cana-1740	92	18	𝑚	𝑚	PROPN
cana-1740	92	19	+	+	PROPN
cana-1740	92	20	𝑛)2	𝑛)2	PROPN
cana-1740	92	21	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	92	22	)	)	PUNCT
cana-1740	92	23	=	=	SYM
cana-1740	92	24	(	(	PUNCT
cana-1740	92	25	𝑚	𝑚	PROPN
cana-1740	92	26	+	+	NUM
cana-1740	92	27	𝑛)2	𝑛)2	PROPN
cana-1740	92	28	×	×	PROPN
cana-1740	92	29	(	(	PUNCT
cana-1740	92	30	𝑚	𝑚	PROPN
cana-1740	92	31	+	+	NUM
cana-1740	92	32	𝑛)(𝑚	𝑛)(𝑚	PROPN
cana-1740	92	33	+	+	CCONJ
cana-1740	93	1	1)(𝑛	1)(𝑛	NUM
cana-1740	93	2	+	+	NUM
cana-1740	93	3	1	1	NUM
cana-1740	93	4	)	)	PUNCT
cana-1740	93	5	2	2	NUM
cana-1740	93	6	=	=	SYM
cana-1740	93	7	(	(	PUNCT
cana-1740	93	8	𝑚	𝑚	X
cana-1740	93	9	+	+	PUNCT
cana-1740	93	10	𝑛)2(𝑚	𝑛)2(𝑚	PRON
cana-1740	93	11	+	+	CCONJ
cana-1740	93	12	1)(𝑛	1)(𝑛	NUM
cana-1740	93	13	+	+	NUM
cana-1740	93	14	1	1	NUM
cana-1740	93	15	)	)	PUNCT
cana-1740	93	16	2	2	NUM
cana-1740	93	17	.	.	PUNCT
cana-1740	94	1	communications	communication	NOUN
cana-1740	94	2	on	on	ADP
cana-1740	94	3	applied	apply	VERB
cana-1740	94	4	nonlinear	nonlinear	ADJ
cana-1740	94	5	analysis	analysis	NOUN
cana-1740	94	6	issn	issn	NOUN
cana-1740	94	7	:	:	PUNCT
cana-1740	94	8	1074	1074	NUM
cana-1740	94	9	-	-	PUNCT
cana-1740	94	10	133x	133x	NUM
cana-1740	94	11	vol	vol	NOUN
cana-1740	94	12	32	32	NUM
cana-1740	94	13	no	no	NOUN
cana-1740	94	14	.	.	NOUN
cana-1740	94	15	2	2	NUM
cana-1740	94	16	(	(	PUNCT
cana-1740	94	17	2025	2025	NUM
cana-1740	94	18	)	)	PUNCT
cana-1740	94	19	259	259	NUM
cana-1740	94	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	94	21	𝐻(𝐾1,𝑛⊠𝐾1,𝑚	𝐻(𝐾1,𝑛⊠𝐾1,𝑚	PROPN
cana-1740	94	22	)	)	PUNCT
cana-1740	94	23	=	=	SYM
cana-1740	94	24	∑	∑	PROPN
cana-1740	94	25	2	2	NUM
cana-1740	94	26	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	94	27	)	)	PUNCT
cana-1740	94	28	+	+	CCONJ
cana-1740	94	29	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	94	30	)	)	PUNCT
cana-1740	94	31	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	94	32	)	)	PUNCT
cana-1740	94	33	=	=	SYM
cana-1740	94	34	∑	∑	ADP
cana-1740	94	35	2	2	NUM
cana-1740	94	36	2(𝑚	2(𝑚	NUM
cana-1740	94	37	+	+	CCONJ
cana-1740	94	38	𝑛	𝑛	NOUN
cana-1740	94	39	)	)	PUNCT
cana-1740	94	40	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	94	41	)	)	PUNCT
cana-1740	94	42	=	=	SYM
cana-1740	95	1	1	1	NUM
cana-1740	95	2	𝑚	𝑚	NOUN
cana-1740	95	3	+	+	NOUN
cana-1740	95	4	𝑛	𝑛	DET
cana-1740	95	5	∑	∑	PROPN
cana-1740	95	6	1	1	NUM
cana-1740	95	7	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	95	8	)	)	PUNCT
cana-1740	95	9	=	=	PUNCT
cana-1740	95	10	(	(	PUNCT
cana-1740	95	11	𝑚	𝑚	PROPN
cana-1740	95	12	+	+	NOUN
cana-1740	95	13	1)(𝑛	1)(𝑛	NUM
cana-1740	95	14	+	+	NUM
cana-1740	95	15	1	1	NUM
cana-1740	95	16	)	)	SYM
cana-1740	95	17	2	2	NUM
cana-1740	95	18	.	.	PUNCT
cana-1740	95	19	𝐺𝐴(𝐾1,𝑛⊠𝐾1,𝑚	𝐺𝐴(𝐾1,𝑛⊠𝐾1,𝑚	NUM
cana-1740	95	20	)	)	PUNCT
cana-1740	95	21	=	=	PUNCT
cana-1740	95	22	∑	∑	PUNCT
cana-1740	95	23	2√𝑑(𝑢)𝑑(𝑣	2√𝑑(𝑢)𝑑(𝑣	NUM
cana-1740	95	24	)	)	PUNCT
cana-1740	95	25	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	95	26	)	)	PUNCT
cana-1740	95	27	+	+	CCONJ
cana-1740	95	28	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	95	29	)	)	PUNCT
cana-1740	95	30	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	95	31	)	)	PUNCT
cana-1740	95	32	=	=	PUNCT
cana-1740	96	1	∑	∑	PUNCT
cana-1740	96	2	2√(𝑚	2√(𝑚	NUM
cana-1740	96	3	+	+	NUM
cana-1740	96	4	𝑛)(𝑚	𝑛)(𝑚	PROPN
cana-1740	96	5	+	+	CCONJ
cana-1740	96	6	𝑛	𝑛	X
cana-1740	96	7	)	)	PUNCT
cana-1740	96	8	2(𝑚	2(𝑚	NUM
cana-1740	96	9	+	+	CCONJ
cana-1740	96	10	𝑛	𝑛	NOUN
cana-1740	96	11	)	)	PUNCT
cana-1740	96	12	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚	NOUN
cana-1740	96	13	)	)	PUNCT
cana-1740	96	14	=	=	SYM
cana-1740	96	15	(	(	PUNCT
cana-1740	96	16	𝑚	𝑚	X
cana-1740	96	17	+	+	NUM
cana-1740	96	18	𝑛)(𝑚	𝑛)(𝑚	PROPN
cana-1740	96	19	+	+	CCONJ
cana-1740	96	20	1)(𝑛	1)(𝑛	NUM
cana-1740	97	1	+	+	NUM
cana-1740	97	2	1	1	NUM
cana-1740	97	3	)	)	PUNCT
cana-1740	97	4	2	2	NUM
cana-1740	97	5	.	.	PUNCT
cana-1740	98	1	this	this	PRON
cana-1740	98	2	completes	complete	VERB
cana-1740	98	3	the	the	DET
cana-1740	98	4	proof	proof	NOUN
cana-1740	98	5	.	.	PUNCT
cana-1740	99	1	note	note	VERB
cana-1740	99	2	:	:	PUNCT
cana-1740	99	3	for	for	ADP
cana-1740	99	4	𝑛	𝑛	NOUN
cana-1740	99	5	=	=	SYM
cana-1740	99	6	3	3	NUM
cana-1740	99	7	and	and	CCONJ
cana-1740	99	8	𝑛	𝑛	ADJ
cana-1740	99	9	=	=	SYM
cana-1740	99	10	4	4	NUM
cana-1740	99	11	,	,	PUNCT
cana-1740	99	12	the	the	DET
cana-1740	99	13	armsproduct	armsproduct	NOUN
cana-1740	99	14	of	of	ADP
cana-1740	99	15	k1,3	k1,3	PROPN
cana-1740	99	16	and	and	CCONJ
cana-1740	99	17	k1,4	k1,4	PROPN
cana-1740	99	18	is	be	AUX
cana-1740	99	19	the	the	DET
cana-1740	99	20	rook	rook	NOUN
cana-1740	99	21	graph	graph	NOUN
cana-1740	99	22	and	and	CCONJ
cana-1740	99	23	it	it	PRON
cana-1740	99	24	is	be	AUX
cana-1740	99	25	given	give	VERB
cana-1740	99	26	below	below	ADV
cana-1740	99	27	:	:	PUNCT
cana-1740	99	28	k1,3	k1,3	X
cana-1740	99	29	⊠	⊠	PROPN
cana-1740	99	30	k1,4	k1,4	PROPN
cana-1740	99	31	theorem	theorem	VERB
cana-1740	99	32	2.3	2.3	NUM
cana-1740	99	33	.	.	PUNCT
cana-1740	100	1	let	let	VERB
cana-1740	100	2	pn	pn	PROPN
cana-1740	100	3	⊙	⊙	PROPN
cana-1740	100	4	k1	k1	PROPN
cana-1740	100	5	be	be	AUX
cana-1740	100	6	a	a	DET
cana-1740	100	7	given	give	VERB
cana-1740	100	8	graph	graph	NOUN
cana-1740	100	9	and	and	CCONJ
cana-1740	100	10	p2	p2	PROPN
cana-1740	100	11	be	be	AUX
cana-1740	100	12	a	a	DET
cana-1740	100	13	path	path	NOUN
cana-1740	100	14	graph	graph	NOUN
cana-1740	100	15	.	.	PUNCT
cana-1740	101	1	then	then	ADV
cana-1740	101	2	for	for	ADP
cana-1740	101	3	the	the	DET
cana-1740	101	4	graph	graph	NOUN
cana-1740	101	5	(	(	PUNCT
cana-1740	101	6	pn	pn	PROPN
cana-1740	101	7	⊙	⊙	PROPN
cana-1740	101	8	k1	k1	PROPN
cana-1740	101	9	)	)	PUNCT
cana-1740	101	10	⊠	⊠	PROPN
cana-1740	101	11	p2	p2	NOUN
cana-1740	101	12	,	,	PUNCT
cana-1740	101	13	first	first	ADJ
cana-1740	101	14	and	and	CCONJ
cana-1740	101	15	second	second	ADJ
cana-1740	101	16	zagreb	zagreb	PROPN
cana-1740	101	17	indices	index	NOUN
cana-1740	101	18	,	,	PUNCT
cana-1740	101	19	harmonic	harmonic	ADJ
cana-1740	101	20	index	index	NOUN
cana-1740	101	21	and	and	CCONJ
cana-1740	101	22	geometric	geometric	ADJ
cana-1740	101	23	-	-	PUNCT
cana-1740	101	24	arithmetic	arithmetic	ADJ
cana-1740	101	25	index	index	NOUN
cana-1740	101	26	are	be	AUX
cana-1740	101	27	respectively	respectively	ADV
cana-1740	101	28	given	give	VERB
cana-1740	101	29	by	by	ADP
cana-1740	101	30	:	:	PUNCT
cana-1740	101	31	1	1	NUM
cana-1740	101	32	.	.	PUNCT
cana-1740	102	1	𝑀1((𝑃𝑛⊙𝐾1	𝑀1((𝑃𝑛⊙𝐾1	NUM
cana-1740	102	2	)	)	PUNCT
cana-1740	102	3	⊠	⊠	PROPN
cana-1740	102	4	𝑃2	𝑃2	NOUN
cana-1740	102	5	)	)	PUNCT
cana-1740	102	6	=	=	PRON
cana-1740	102	7	{	{	PUNCT
cana-1740	102	8	100	100	NUM
cana-1740	102	9	,	,	PUNCT
cana-1740	102	10	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	102	11	𝑛	𝑛	X
cana-1740	102	12	=	=	SYM
cana-1740	102	13	2	2	NUM
cana-1740	102	14	𝑛(4𝑛	𝑛(4𝑛	NUM
cana-1740	102	15	−	−	PROPN
cana-1740	102	16	1)2	1)2	NUM
cana-1740	102	17	,	,	PUNCT
cana-1740	102	18	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	102	19	𝑛	𝑛	PROPN
cana-1740	102	20	>	>	SYM
cana-1740	102	21	3	3	NUM
cana-1740	102	22	.	.	PUNCT
cana-1740	103	1	communications	communication	NOUN
cana-1740	103	2	on	on	ADP
cana-1740	103	3	applied	apply	VERB
cana-1740	103	4	nonlinear	nonlinear	ADJ
cana-1740	103	5	analysis	analysis	NOUN
cana-1740	103	6	issn	issn	NOUN
cana-1740	103	7	:	:	PUNCT
cana-1740	103	8	1074	1074	NUM
cana-1740	103	9	-	-	PUNCT
cana-1740	103	10	133x	133x	NUM
cana-1740	103	11	vol	vol	NOUN
cana-1740	103	12	32	32	NUM
cana-1740	103	13	no	no	NOUN
cana-1740	103	14	.	.	NOUN
cana-1740	103	15	2	2	NUM
cana-1740	103	16	(	(	PUNCT
cana-1740	103	17	2025	2025	NUM
cana-1740	103	18	)	)	PUNCT
cana-1740	103	19	260	260	NUM
cana-1740	103	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	103	21	2	2	NUM
cana-1740	103	22	.	.	PUNCT
cana-1740	104	1	𝑀2((𝑃𝑛⊙𝐾1	𝑀2((𝑃𝑛⊙𝐾1	PROPN
cana-1740	104	2	)	)	PUNCT
cana-1740	104	3	⊠	⊠	PROPN
cana-1740	104	4	𝑃2	𝑃2	NOUN
cana-1740	104	5	)	)	PUNCT
cana-1740	105	1	=	=	NOUN
cana-1740	105	2	{	{	PUNCT
cana-1740	105	3	178	178	NUM
cana-1740	105	4	,	,	PUNCT
cana-1740	105	5	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-1740	105	6	𝑛	𝑛	X
cana-1740	105	7	=	=	SYM
cana-1740	105	8	2	2	NUM
cana-1740	105	9	𝑛2(3(2𝑛	𝑛2(3(2𝑛	NOUN
cana-1740	105	10	−	−	PROPN
cana-1740	105	11	1)2	1)2	NUM
cana-1740	105	12	+	+	CCONJ
cana-1740	105	13	4𝑛2	4𝑛2	NUM
cana-1740	105	14	)	)	PUNCT
cana-1740	105	15	,	,	PUNCT
cana-1740	105	16	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	105	17	𝑛	𝑛	X
cana-1740	105	18	>	>	X
cana-1740	105	19	3	3	NUM
cana-1740	105	20	.	.	PUNCT
cana-1740	106	1	3	3	NUM
cana-1740	106	2	.	.	X
cana-1740	106	3	𝐻((𝑃𝑛⊙𝐾1	𝐻((𝑃𝑛⊙𝐾1	NOUN
cana-1740	106	4	)	)	PUNCT
cana-1740	106	5	⊠	⊠	PROPN
cana-1740	106	6	𝑃2	𝑃2	NOUN
cana-1740	106	7	)	)	PUNCT
cana-1740	107	1	=	=	PRON
cana-1740	107	2	{	{	PUNCT
cana-1740	107	3	83	83	NUM
cana-1740	107	4	21	21	NUM
cana-1740	107	5	,	,	PUNCT
cana-1740	107	6	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	107	7	𝑛	𝑛	X
cana-1740	107	8	=	=	SYM
cana-1740	107	9	2	2	NUM
cana-1740	107	10	2𝑛2	2𝑛2	NUM
cana-1740	107	11	4𝑛	4𝑛	NOUN
cana-1740	107	12	−	−	NOUN
cana-1740	107	13	2	2	NUM
cana-1740	107	14	+	+	CCONJ
cana-1740	107	15	2𝑛(2𝑛	2𝑛(2𝑛	NUM
cana-1740	107	16	−	−	NOUN
cana-1740	107	17	1	1	NUM
cana-1740	107	18	)	)	PUNCT
cana-1740	107	19	4𝑛	4𝑛	NOUN
cana-1740	107	20	−	−	NOUN
cana-1740	108	1	1	1	NUM
cana-1740	108	2	+	+	CCONJ
cana-1740	108	3	𝑛	𝑛	DET
cana-1740	108	4	2	2	NUM
cana-1740	108	5	,	,	PUNCT
cana-1740	108	6	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	108	7	𝑛	𝑛	X
cana-1740	108	8	>	>	X
cana-1740	108	9	3	3	NUM
cana-1740	108	10	.	.	NOUN
cana-1740	108	11	4	4	NUM
cana-1740	108	12	.	.	X
cana-1740	108	13	𝐺𝐴((𝑃𝑛⊙𝐾1	𝐺𝐴((𝑃𝑛⊙𝐾1	NUM
cana-1740	108	14	)	)	PUNCT
cana-1740	109	1	⊠	⊠	PROPN
cana-1740	109	2	𝑃2	𝑃2	NOUN
cana-1740	109	3	)	)	PUNCT
cana-1740	110	1	=	=	PRON
cana-1740	110	2	{	{	PUNCT
cana-1740	110	3	42	42	NUM
cana-1740	110	4	+	+	CCONJ
cana-1740	110	5	(	(	PUNCT
cana-1740	110	6	32√3	32√3	NUM
cana-1740	110	7	)	)	PUNCT
cana-1740	110	8	7	7	NUM
cana-1740	110	9	,	,	PUNCT
cana-1740	110	10	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-1740	110	11	𝑛	𝑛	X
cana-1740	110	12	=	=	SYM
cana-1740	110	13	2	2	NUM
cana-1740	110	14	2𝑛2	2𝑛2	NUM
cana-1740	110	15	+	+	CCONJ
cana-1740	110	16	2𝑛(2𝑛	2𝑛(2𝑛	NUM
cana-1740	110	17	−	−	PROPN
cana-1740	110	18	1)√(4𝑛2	1)√(4𝑛2	PROPN
cana-1740	110	19	−	−	PROPN
cana-1740	110	20	2𝑛	2𝑛	NUM
cana-1740	110	21	)	)	PUNCT
cana-1740	110	22	(	(	PUNCT
cana-1740	110	23	4𝑛	4𝑛	NOUN
cana-1740	110	24	−	−	NOUN
cana-1740	110	25	1	1	NUM
cana-1740	110	26	)	)	PUNCT
cana-1740	110	27	,	,	PUNCT
cana-1740	110	28	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1740	110	29	𝑛	𝑛	X
cana-1740	110	30	>	>	X
cana-1740	110	31	3	3	X
cana-1740	110	32	.	.	X
cana-1740	111	1	proof	proof	NOUN
cana-1740	111	2	:	:	PUNCT
cana-1740	111	3	case	case	NOUN
cana-1740	111	4	1	1	NUM
cana-1740	111	5	.	.	PUNCT
cana-1740	112	1	𝑛	𝑛	NOUN
cana-1740	112	2	=	=	SYM
cana-1740	112	3	2	2	NUM
cana-1740	112	4	if	if	SCONJ
cana-1740	112	5	𝑛	𝑛	ADJ
cana-1740	112	6	=	=	SYM
cana-1740	112	7	2	2	NUM
cana-1740	112	8	,	,	PUNCT
cana-1740	112	9	consider	consider	VERB
cana-1740	112	10	(	(	PUNCT
cana-1740	112	11	p2	p2	PROPN
cana-1740	112	12	⊙	⊙	PROPN
cana-1740	112	13	k1	k1	PROPN
cana-1740	112	14	)	)	PUNCT
cana-1740	112	15	⊠	⊠	PROPN
cana-1740	112	16	p2	p2	NOUN
cana-1740	112	17	.	.	PUNCT
cana-1740	113	1	(	(	PUNCT
cana-1740	113	2	p2	p2	PROPN
cana-1740	113	3	⊙	⊙	PROPN
cana-1740	113	4	k1	k1	PROPN
cana-1740	113	5	)	)	PUNCT
cana-1740	113	6	⊠	⊠	PROPN
cana-1740	113	7	p2	p2	VERB
cana-1740	113	8	the	the	DET
cana-1740	113	9	edge	edge	NOUN
cana-1740	113	10	set	set	NOUN
cana-1740	113	11	of	of	ADP
cana-1740	113	12	(	(	PUNCT
cana-1740	113	13	p2	p2	PROPN
cana-1740	113	14	⊙	⊙	PROPN
cana-1740	113	15	k1	k1	PROPN
cana-1740	113	16	)	)	PUNCT
cana-1740	113	17	⊠	⊠	PROPN
cana-1740	113	18	p2	p2	NOUN
cana-1740	113	19	can	can	AUX
cana-1740	113	20	be	be	AUX
cana-1740	113	21	partitioned	partition	VERB
cana-1740	113	22	as	as	SCONJ
cana-1740	113	23	follows	follow	VERB
cana-1740	113	24	:	:	PUNCT
cana-1740	113	25	𝐸1	𝐸1	NOUN
cana-1740	113	26	=	=	SYM
cana-1740	113	27	{	{	PUNCT
cana-1740	113	28	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	113	29	|	|	ADV
cana-1740	113	30	𝑑𝑢	𝑑𝑢	PROPN
cana-1740	113	31	=	=	NOUN
cana-1740	113	32	3	3	NUM
cana-1740	113	33	,	,	PUNCT
cana-1740	113	34	𝑑𝑣	𝑑𝑣	X
cana-1740	113	35	=	=	NOUN
cana-1740	113	36	3	3	NUM
cana-1740	113	37	}	}	PUNCT
cana-1740	113	38	,	,	PUNCT
cana-1740	113	39	|𝐸1|	|𝐸1|	NOUN
cana-1740	113	40	=	=	PUNCT
cana-1740	114	1	2	2	X
cana-1740	114	2	.	.	PUNCT
cana-1740	114	3	𝐸2	𝐸2	PROPN
cana-1740	114	4	=	=	X
cana-1740	114	5	{	{	PUNCT
cana-1740	114	6	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	114	7	|	|	ADV
cana-1740	114	8	𝑑𝑢	𝑑𝑢	PROPN
cana-1740	114	9	=	=	NOUN
cana-1740	114	10	3	3	NUM
cana-1740	114	11	,	,	PUNCT
cana-1740	114	12	𝑑𝑣	𝑑𝑣	X
cana-1740	114	13	=	=	NOUN
cana-1740	114	14	4	4	NUM
cana-1740	114	15	}	}	PUNCT
cana-1740	114	16	,	,	PUNCT
cana-1740	114	17	|𝐸2|	|𝐸2|	X
cana-1740	114	18	=	=	NOUN
cana-1740	114	19	8	8	X
cana-1740	114	20	.	.	PUNCT
cana-1740	115	1	𝐸3	𝐸3	NOUN
cana-1740	115	2	=	=	PRON
cana-1740	115	3	{	{	PUNCT
cana-1740	115	4	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	115	5	|	|	ADV
cana-1740	115	6	𝑑𝑢	𝑑𝑢	NOUN
cana-1740	115	7	=	=	NOUN
cana-1740	115	8	4	4	NUM
cana-1740	115	9	,	,	PUNCT
cana-1740	115	10	𝑑𝑣	𝑑𝑣	X
cana-1740	115	11	=	=	NOUN
cana-1740	115	12	4	4	NUM
cana-1740	115	13	}	}	PUNCT
cana-1740	115	14	,	,	PUNCT
cana-1740	115	15	|𝐸3|	|𝐸3|	NUM
cana-1740	115	16	=	=	SYM
cana-1740	115	17	4	4	X
cana-1740	115	18	.	.	PUNCT
cana-1740	115	19	𝑀1((𝑃2⊙𝐾1	𝑀1((𝑃2⊙𝐾1	NUM
cana-1740	115	20	)	)	PUNCT
cana-1740	115	21	⊠	⊠	PROPN
cana-1740	115	22	𝑃2	𝑃2	NOUN
cana-1740	115	23	)	)	PUNCT
cana-1740	115	24	=	=	PUNCT
cana-1740	115	25	∑	∑	PUNCT
cana-1740	115	26	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	115	27	)	)	PUNCT
cana-1740	115	28	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	NOUN
cana-1740	115	29	)	)	PUNCT
cana-1740	115	30	+	+	NUM
cana-1740	115	31	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	115	32	)	)	PUNCT
cana-1740	115	33	=	=	PUNCT
cana-1740	115	34	∑	∑	PROPN
cana-1740	115	35	3	3	NUM
cana-1740	115	36	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	115	37	+	+	CCONJ
cana-1740	115	38	3	3	NUM
cana-1740	115	39	+	+	CCONJ
cana-1740	115	40	∑	∑	PROPN
cana-1740	115	41	3	3	NUM
cana-1740	115	42	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	115	43	+	+	NUM
cana-1740	115	44	4	4	NUM
cana-1740	115	45	+	+	ADJ
cana-1740	115	46	∑	∑	PROPN
cana-1740	115	47	4	4	NUM
cana-1740	115	48	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	115	49	+	+	CCONJ
cana-1740	115	50	4	4	NUM
cana-1740	115	51	=	=	SYM
cana-1740	115	52	6	6	NUM
cana-1740	115	53	∑	∑	SYM
cana-1740	115	54	1	1	NUM
cana-1740	115	55	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	115	56	+	+	CCONJ
cana-1740	115	57	7	7	NUM
cana-1740	115	58	∑	∑	SYM
cana-1740	115	59	1	1	NUM
cana-1740	115	60	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	115	61	+	+	CCONJ
cana-1740	115	62	8	8	NUM
cana-1740	115	63	∑	∑	SYM
cana-1740	115	64	1	1	NUM
cana-1740	115	65	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	115	66	communications	communication	NOUN
cana-1740	115	67	on	on	ADP
cana-1740	115	68	applied	apply	VERB
cana-1740	115	69	nonlinear	nonlinear	ADJ
cana-1740	115	70	analysis	analysis	NOUN
cana-1740	115	71	issn	issn	NOUN
cana-1740	115	72	:	:	PUNCT
cana-1740	115	73	1074	1074	NUM
cana-1740	115	74	-	-	PUNCT
cana-1740	115	75	133x	133x	NUM
cana-1740	115	76	vol	vol	NOUN
cana-1740	115	77	32	32	NUM
cana-1740	115	78	no	no	NOUN
cana-1740	115	79	.	.	NOUN
cana-1740	115	80	2	2	NUM
cana-1740	115	81	(	(	PUNCT
cana-1740	115	82	2025	2025	NUM
cana-1740	115	83	)	)	PUNCT
cana-1740	115	84	261	261	NUM
cana-1740	115	85	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	115	86	=	=	SYM
cana-1740	115	87	6	6	NUM
cana-1740	115	88	×	×	NOUN
cana-1740	115	89	2	2	NUM
cana-1740	115	90	+	+	CCONJ
cana-1740	115	91	7	7	NUM
cana-1740	115	92	×	×	NOUN
cana-1740	115	93	8	8	NUM
cana-1740	115	94	+	+	CCONJ
cana-1740	115	95	8	8	NUM
cana-1740	115	96	×	×	NOUN
cana-1740	115	97	4	4	NUM
cana-1740	115	98	=	=	SYM
cana-1740	115	99	100	100	NUM
cana-1740	115	100	.	.	PUNCT
cana-1740	116	1	𝑀2((𝑃2⊙𝐾1	𝑀2((𝑃2⊙𝐾1	PROPN
cana-1740	116	2	)	)	PUNCT
cana-1740	116	3	⊠	⊠	PROPN
cana-1740	116	4	𝑃2	𝑃2	PROPN
cana-1740	116	5	)	)	PUNCT
cana-1740	116	6	=	=	PUNCT
cana-1740	116	7	∑	∑	PUNCT
cana-1740	116	8	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	116	9	)	)	PUNCT
cana-1740	116	10	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	NOUN
cana-1740	116	11	)	)	PUNCT
cana-1740	116	12	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	116	13	)	)	PUNCT
cana-1740	116	14	=	=	PUNCT
cana-1740	117	1	∑	∑	PROPN
cana-1740	117	2	9	9	NUM
cana-1740	117	3	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	X
cana-1740	117	4	+	+	CCONJ
cana-1740	117	5	∑	∑	PROPN
cana-1740	117	6	12	12	NUM
cana-1740	117	7	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	117	8	+	+	NUM
cana-1740	117	9	∑	∑	PROPN
cana-1740	117	10	16	16	NUM
cana-1740	117	11	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	117	12	=	=	SYM
cana-1740	117	13	9	9	NUM
cana-1740	117	14	×	×	NOUN
cana-1740	117	15	2	2	NUM
cana-1740	117	16	+	+	CCONJ
cana-1740	117	17	12	12	NUM
cana-1740	117	18	×	×	NOUN
cana-1740	117	19	8	8	NUM
cana-1740	117	20	+	+	CCONJ
cana-1740	117	21	16	16	NUM
cana-1740	117	22	×	×	NOUN
cana-1740	117	23	4	4	NUM
cana-1740	117	24	=	=	SYM
cana-1740	117	25	178	178	NUM
cana-1740	117	26	.	.	PUNCT
cana-1740	118	1	𝐻((𝑃2⊙𝐾1	𝐻((𝑃2⊙𝐾1	NOUN
cana-1740	118	2	)	)	PUNCT
cana-1740	118	3	⊠	⊠	PROPN
cana-1740	118	4	𝑃2	𝑃2	NOUN
cana-1740	118	5	)	)	PUNCT
cana-1740	118	6	=	=	PUNCT
cana-1740	119	1	∑	∑	PROPN
cana-1740	119	2	2	2	NUM
cana-1740	119	3	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	119	4	)	)	PUNCT
cana-1740	119	5	+	+	CCONJ
cana-1740	119	6	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	119	7	)	)	PUNCT
cana-1740	119	8	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	NOUN
cana-1740	119	9	)	)	PUNCT
cana-1740	119	10	=	=	PUNCT
cana-1740	120	1	∑	∑	PROPN
cana-1740	120	2	2	2	NUM
cana-1740	120	3	6	6	NUM
cana-1740	120	4	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	120	5	+	+	CCONJ
cana-1740	120	6	∑	∑	SYM
cana-1740	120	7	2	2	NUM
cana-1740	120	8	7	7	NUM
cana-1740	120	9	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	120	10	+	+	NUM
cana-1740	120	11	∑	∑	SYM
cana-1740	120	12	2	2	NUM
cana-1740	120	13	8	8	NUM
cana-1740	120	14	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	120	15	=	=	SYM
cana-1740	120	16	2	2	NUM
cana-1740	120	17	6	6	NUM
cana-1740	120	18	×	×	NOUN
cana-1740	120	19	2	2	NUM
cana-1740	120	20	+	+	CCONJ
cana-1740	120	21	2	2	NUM
cana-1740	120	22	7	7	NUM
cana-1740	120	23	×	×	NOUN
cana-1740	120	24	8	8	NUM
cana-1740	120	25	+	+	CCONJ
cana-1740	120	26	2	2	NUM
cana-1740	120	27	8	8	NUM
cana-1740	120	28	×	×	NOUN
cana-1740	120	29	4	4	NUM
cana-1740	120	30	=	=	SYM
cana-1740	120	31	83	83	NUM
cana-1740	120	32	21	21	NUM
cana-1740	120	33	.	.	PUNCT
cana-1740	120	34	𝐺𝐴((𝑃2⊙𝐾1	𝐺𝐴((𝑃2⊙𝐾1	NOUN
cana-1740	120	35	)	)	PUNCT
cana-1740	120	36	⊠	⊠	PROPN
cana-1740	120	37	𝑃2	𝑃2	NOUN
cana-1740	120	38	)	)	PUNCT
cana-1740	120	39	=	=	PUNCT
cana-1740	120	40	∑	∑	PROPN
cana-1740	120	41	2	2	NUM
cana-1740	120	42	√𝑑(𝑢)𝑑(𝑣	√𝑑(𝑢)𝑑(𝑣	NOUN
cana-1740	120	43	)	)	PUNCT
cana-1740	120	44	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	120	45	)	)	PUNCT
cana-1740	120	46	+	+	CCONJ
cana-1740	120	47	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	120	48	)	)	PUNCT
cana-1740	120	49	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2	NOUN
cana-1740	120	50	)	)	PUNCT
cana-1740	120	51	=	=	PUNCT
cana-1740	121	1	∑	∑	PUNCT
cana-1740	121	2	2√3	2√3	NUM
cana-1740	121	3	×	×	NOUN
cana-1740	121	4	3	3	NUM
cana-1740	121	5	6	6	NUM
cana-1740	121	6	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	121	7	+	+	CCONJ
cana-1740	121	8	∑	∑	PUNCT
cana-1740	121	9	2√3	2√3	NUM
cana-1740	121	10	×	×	NOUN
cana-1740	121	11	4	4	NUM
cana-1740	121	12	7	7	NUM
cana-1740	121	13	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	121	14	+	+	NUM
cana-1740	121	15	∑	∑	PROPN
cana-1740	121	16	2√4	2√4	NUM
cana-1740	121	17	×	×	NOUN
cana-1740	121	18	4	4	NUM
cana-1740	121	19	8	8	NUM
cana-1740	121	20	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	121	21	=	=	SYM
cana-1740	121	22	∑	∑	PROPN
cana-1740	121	23	1	1	NUM
cana-1740	121	24	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	121	25	+	+	CCONJ
cana-1740	121	26	4√3	4√3	NUM
cana-1740	121	27	7	7	NUM
cana-1740	121	28	∑	∑	SYM
cana-1740	121	29	1	1	NUM
cana-1740	121	30	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	121	31	+	+	NUM
cana-1740	121	32	∑	∑	SYM
cana-1740	121	33	1	1	NUM
cana-1740	121	34	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	121	35	=	=	SYM
cana-1740	121	36	2	2	NUM
cana-1740	121	37	+	+	NUM
cana-1740	121	38	4√3	4√3	NUM
cana-1740	121	39	7	7	NUM
cana-1740	121	40	×	×	NOUN
cana-1740	121	41	8	8	NUM
cana-1740	121	42	+	+	CCONJ
cana-1740	121	43	4	4	NUM
cana-1740	121	44	=	=	SYM
cana-1740	121	45	42	42	NUM
cana-1740	121	46	+	+	NUM
cana-1740	121	47	32√3	32√3	NUM
cana-1740	121	48	7	7	NUM
cana-1740	121	49	.	.	PUNCT
cana-1740	122	1	case	case	NOUN
cana-1740	122	2	2	2	NUM
cana-1740	122	3	.	.	X
cana-1740	123	1	𝑛	𝑛	NOUN
cana-1740	123	2	>	>	SYM
cana-1740	123	3	3	3	NUM
cana-1740	123	4	if	if	SCONJ
cana-1740	123	5	𝑛	𝑛	PROPN
cana-1740	123	6	>	>	X
cana-1740	123	7	3	3	NUM
cana-1740	123	8	,	,	PUNCT
cana-1740	123	9	the	the	DET
cana-1740	123	10	edge	edge	NOUN
cana-1740	123	11	set	set	NOUN
cana-1740	123	12	of	of	ADP
cana-1740	123	13	(	(	PUNCT
cana-1740	123	14	pn	pn	PROPN
cana-1740	123	15	⊙	⊙	PROPN
cana-1740	123	16	k1	k1	PROPN
cana-1740	123	17	)	)	PUNCT
cana-1740	123	18	⊠	⊠	PROPN
cana-1740	123	19	p2	p2	NOUN
cana-1740	123	20	can	can	AUX
cana-1740	123	21	be	be	AUX
cana-1740	123	22	partitioned	partition	VERB
cana-1740	123	23	as	as	SCONJ
cana-1740	123	24	follows	follow	VERB
cana-1740	123	25	:	:	PUNCT
cana-1740	123	26	𝐸1	𝐸1	NOUN
cana-1740	123	27	=	=	SYM
cana-1740	123	28	{	{	PUNCT
cana-1740	123	29	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	123	30	|	|	ADV
cana-1740	123	31	𝑑𝑢	𝑑𝑢	PROPN
cana-1740	123	32	=	=	PROPN
cana-1740	123	33	2𝑛	2𝑛	PROPN
cana-1740	124	1	−	−	PROPN
cana-1740	124	2	1	1	NUM
cana-1740	124	3	,	,	PUNCT
cana-1740	124	4	𝑑𝑣	𝑑𝑣	PROPN
cana-1740	124	5	=	=	PROPN
cana-1740	124	6	2𝑛	2𝑛	PROPN
cana-1740	124	7	−	−	PROPN
cana-1740	124	8	1	1	NUM
cana-1740	124	9	}	}	PUNCT
cana-1740	124	10	,	,	PUNCT
cana-1740	124	11	|𝐸1|	|𝐸1|	NOUN
cana-1740	124	12	=	=	SYM
cana-1740	124	13	𝑛2	𝑛2	PROPN
cana-1740	124	14	.	.	PUNCT
cana-1740	125	1	𝐸2	𝐸2	PROPN
cana-1740	125	2	=	=	X
cana-1740	125	3	{	{	PUNCT
cana-1740	125	4	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	125	5	|	|	ADV
cana-1740	125	6	𝑑𝑢	𝑑𝑢	PROPN
cana-1740	125	7	=	=	PROPN
cana-1740	125	8	2𝑛	2𝑛	PROPN
cana-1740	126	1	−	−	PROPN
cana-1740	126	2	1	1	NUM
cana-1740	126	3	,	,	PUNCT
cana-1740	126	4	𝑑𝑣	𝑑𝑣	PROPN
cana-1740	126	5	=	=	SYM
cana-1740	126	6	2𝑛	2𝑛	NUM
cana-1740	126	7	}	}	PUNCT
cana-1740	126	8	,	,	PUNCT
cana-1740	126	9	|𝐸2|	|𝐸2|	X
cana-1740	126	10	=	=	NOUN
cana-1740	126	11	2𝑛2	2𝑛2	NUM
cana-1740	126	12	−	−	PROPN
cana-1740	126	13	𝑛.	𝑛.	NOUN
cana-1740	126	14	𝐸3	𝐸3	NOUN
cana-1740	126	15	=	=	PRON
cana-1740	126	16	{	{	PUNCT
cana-1740	126	17	𝑢𝑣	𝑢𝑣	NOUN
cana-1740	126	18	|	|	ADV
cana-1740	126	19	𝑑𝑢	𝑑𝑢	PROPN
cana-1740	126	20	=	=	PROPN
cana-1740	126	21	2𝑛	2𝑛	PROPN
cana-1740	126	22	,	,	PUNCT
cana-1740	126	23	𝑑𝑣	𝑑𝑣	PROPN
cana-1740	126	24	=	=	SYM
cana-1740	126	25	2𝑛	2𝑛	NUM
cana-1740	126	26	}	}	PUNCT
cana-1740	126	27	,	,	PUNCT
cana-1740	126	28	|𝐸3|	|𝐸3|	NUM
cana-1740	126	29	=	=	SYM
cana-1740	126	30	𝑛	𝑛	PRON
cana-1740	126	31	2	2	NUM
cana-1740	126	32	.	.	PUNCT
cana-1740	126	33	communications	communication	NOUN
cana-1740	126	34	on	on	ADP
cana-1740	126	35	applied	apply	VERB
cana-1740	126	36	nonlinear	nonlinear	ADJ
cana-1740	126	37	analysis	analysis	NOUN
cana-1740	126	38	issn	issn	NOUN
cana-1740	126	39	:	:	PUNCT
cana-1740	126	40	1074	1074	NUM
cana-1740	126	41	-	-	PUNCT
cana-1740	126	42	133x	133x	NUM
cana-1740	126	43	vol	vol	NOUN
cana-1740	126	44	32	32	NUM
cana-1740	126	45	no	no	NOUN
cana-1740	126	46	.	.	NOUN
cana-1740	126	47	2	2	NUM
cana-1740	126	48	(	(	PUNCT
cana-1740	126	49	2025	2025	NUM
cana-1740	126	50	)	)	PUNCT
cana-1740	126	51	262	262	NUM
cana-1740	126	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	126	53	𝑀1((𝑃𝑛⊙𝐾1	𝑀1((𝑃𝑛⊙𝐾1	PROPN
cana-1740	126	54	)	)	PUNCT
cana-1740	126	55	⊠	⊠	PROPN
cana-1740	126	56	𝑃2	𝑃2	NOUN
cana-1740	126	57	)	)	PUNCT
cana-1740	126	58	=	=	PUNCT
cana-1740	126	59	∑	∑	PUNCT
cana-1740	126	60	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	126	61	)	)	PUNCT
cana-1740	126	62	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	NUM
cana-1740	126	63	)	)	PUNCT
cana-1740	127	1	+	+	CCONJ
cana-1740	127	2	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	127	3	)	)	PUNCT
cana-1740	127	4	=	=	SYM
cana-1740	127	5	∑	∑	PUNCT
cana-1740	127	6	(	(	PUNCT
cana-1740	127	7	2𝑛	2𝑛	NOUN
cana-1740	127	8	−	−	PROPN
cana-1740	127	9	1	1	NUM
cana-1740	127	10	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	127	11	)	)	PUNCT
cana-1740	128	1	+	+	CCONJ
cana-1740	128	2	(	(	PUNCT
cana-1740	128	3	2𝑛	2𝑛	NUM
cana-1740	128	4	−	−	PROPN
cana-1740	128	5	1	1	NUM
cana-1740	128	6	)	)	PUNCT
cana-1740	128	7	+	+	CCONJ
cana-1740	128	8	∑	∑	PROPN
cana-1740	128	9	(	(	PUNCT
cana-1740	128	10	2𝑛	2𝑛	NOUN
cana-1740	128	11	−	−	PROPN
cana-1740	128	12	1	1	NUM
cana-1740	128	13	)	)	PUNCT
cana-1740	128	14	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	PUNCT
cana-1740	129	1	+	+	CCONJ
cana-1740	129	2	(	(	PUNCT
cana-1740	129	3	2𝑛	2𝑛	NUM
cana-1740	129	4	)	)	PUNCT
cana-1740	130	1	+	+	CCONJ
cana-1740	130	2	∑	∑	PROPN
cana-1740	130	3	(	(	PUNCT
cana-1740	130	4	2𝑛	2𝑛	NUM
cana-1740	130	5	)	)	PUNCT
cana-1740	130	6	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	VERB
cana-1740	130	7	+	+	CCONJ
cana-1740	130	8	(	(	PUNCT
cana-1740	130	9	2𝑛	2𝑛	NUM
cana-1740	130	10	)	)	PUNCT
cana-1740	131	1	=	=	SYM
cana-1740	131	2	(	(	PUNCT
cana-1740	131	3	4𝑛	4𝑛	NOUN
cana-1740	131	4	−	−	NOUN
cana-1740	131	5	2	2	NUM
cana-1740	131	6	)	)	PUNCT
cana-1740	131	7	∑	∑	ADV
cana-1740	131	8	1	1	NUM
cana-1740	131	9	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	131	10	+	+	CCONJ
cana-1740	131	11	(	(	PUNCT
cana-1740	131	12	4𝑛	4𝑛	NOUN
cana-1740	131	13	−	−	NOUN
cana-1740	131	14	1	1	NUM
cana-1740	131	15	)	)	PUNCT
cana-1740	131	16	∑	∑	ADV
cana-1740	131	17	1	1	NUM
cana-1740	131	18	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	131	19	+	+	NUM
cana-1740	131	20	(	(	PUNCT
cana-1740	131	21	4𝑛	4𝑛	NUM
cana-1740	131	22	)	)	PUNCT
cana-1740	131	23	∑	∑	ADP
cana-1740	131	24	1	1	NUM
cana-1740	131	25	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	131	26	=	=	SYM
cana-1740	131	27	(	(	PUNCT
cana-1740	131	28	4𝑛	4𝑛	NOUN
cana-1740	131	29	−	−	NOUN
cana-1740	131	30	2	2	X
cana-1740	131	31	)	)	PUNCT
cana-1740	131	32	×	×	NOUN
cana-1740	131	33	𝑛2	𝑛2	NOUN
cana-1740	131	34	+	+	CCONJ
cana-1740	131	35	(	(	PUNCT
cana-1740	131	36	4𝑛	4𝑛	NOUN
cana-1740	131	37	−	−	NOUN
cana-1740	131	38	1	1	X
cana-1740	131	39	)	)	PUNCT
cana-1740	131	40	×	×	NOUN
cana-1740	131	41	(	(	PUNCT
cana-1740	131	42	2𝑛2	2𝑛2	NUM
cana-1740	131	43	−	−	NOUN
cana-1740	131	44	𝑛	𝑛	NOUN
cana-1740	131	45	)	)	PUNCT
cana-1740	131	46	+	+	CCONJ
cana-1740	131	47	(	(	PUNCT
cana-1740	131	48	4𝑛	4𝑛	NUM
cana-1740	131	49	)	)	PUNCT
cana-1740	131	50	×	×	NOUN
cana-1740	131	51	𝑛2	𝑛2	NOUN
cana-1740	131	52	=	=	SYM
cana-1740	131	53	𝑛(4𝑛	𝑛(4𝑛	PROPN
cana-1740	131	54	−	−	PROPN
cana-1740	131	55	1)2	1)2	NUM
cana-1740	131	56	.	.	PUNCT
cana-1740	132	1	𝑀2((𝑃𝑛⊙𝐾1	𝑀2((𝑃𝑛⊙𝐾1	PROPN
cana-1740	132	2	)	)	PUNCT
cana-1740	132	3	⊠	⊠	PROPN
cana-1740	132	4	𝑃2	𝑃2	NOUN
cana-1740	132	5	)	)	PUNCT
cana-1740	132	6	=	=	PUNCT
cana-1740	132	7	∑	∑	PUNCT
cana-1740	132	8	𝑑(𝑢	𝑑(𝑢	ADJ
cana-1740	132	9	)	)	PUNCT
cana-1740	132	10	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	NOUN
cana-1740	132	11	)	)	PUNCT
cana-1740	132	12	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	132	13	)	)	PUNCT
cana-1740	132	14	=	=	SYM
cana-1740	132	15	∑	∑	PUNCT
cana-1740	132	16	(	(	PUNCT
cana-1740	132	17	2𝑛	2𝑛	NOUN
cana-1740	132	18	−	−	PROPN
cana-1740	132	19	1	1	NUM
cana-1740	132	20	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	NOUN
cana-1740	132	21	)	)	PUNCT
cana-1740	133	1	(	(	PUNCT
cana-1740	133	2	2𝑛	2𝑛	NOUN
cana-1740	133	3	−	−	PROPN
cana-1740	133	4	1	1	NUM
cana-1740	133	5	)	)	PUNCT
cana-1740	133	6	+	+	CCONJ
cana-1740	133	7	∑	∑	PROPN
cana-1740	133	8	(	(	PUNCT
cana-1740	133	9	2𝑛	2𝑛	NOUN
cana-1740	133	10	−	−	PROPN
cana-1740	133	11	1	1	NUM
cana-1740	133	12	)	)	PUNCT
cana-1740	133	13	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	PUNCT
cana-1740	133	14	(	(	PUNCT
cana-1740	133	15	2𝑛	2𝑛	NUM
cana-1740	133	16	)	)	PUNCT
cana-1740	134	1	+	+	CCONJ
cana-1740	134	2	∑	∑	PROPN
cana-1740	134	3	(	(	PUNCT
cana-1740	134	4	2𝑛	2𝑛	NUM
cana-1740	134	5	)	)	PUNCT
cana-1740	134	6	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	134	7	(	(	PUNCT
cana-1740	134	8	2𝑛	2𝑛	NUM
cana-1740	134	9	)	)	PUNCT
cana-1740	134	10	=	=	SYM
cana-1740	135	1	(	(	PUNCT
cana-1740	135	2	2𝑛	2𝑛	X
cana-1740	135	3	−	−	PROPN
cana-1740	135	4	1)2	1)2	NUM
cana-1740	135	5	∑	∑	SYM
cana-1740	135	6	1	1	NUM
cana-1740	135	7	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	X
cana-1740	135	8	+	+	CCONJ
cana-1740	135	9	(	(	PUNCT
cana-1740	135	10	4𝑛2	4𝑛2	NUM
cana-1740	135	11	−	−	PROPN
cana-1740	135	12	2𝑛	2𝑛	NUM
cana-1740	135	13	)	)	PUNCT
cana-1740	135	14	∑	∑	ADV
cana-1740	135	15	1	1	NUM
cana-1740	135	16	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	135	17	+	+	NUM
cana-1740	135	18	(	(	PUNCT
cana-1740	135	19	4𝑛2	4𝑛2	NUM
cana-1740	135	20	)	)	PUNCT
cana-1740	135	21	∑	∑	ADP
cana-1740	135	22	1	1	NUM
cana-1740	135	23	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	135	24	=	=	SYM
cana-1740	135	25	(	(	PUNCT
cana-1740	135	26	2𝑛	2𝑛	NOUN
cana-1740	135	27	−	−	PROPN
cana-1740	135	28	1)2	1)2	NUM
cana-1740	135	29	×	×	NOUN
cana-1740	135	30	𝑛2	𝑛2	NOUN
cana-1740	135	31	+	+	CCONJ
cana-1740	135	32	(	(	PUNCT
cana-1740	135	33	4𝑛2	4𝑛2	NUM
cana-1740	135	34	−	−	PROPN
cana-1740	135	35	2𝑛	2𝑛	NUM
cana-1740	135	36	)	)	PUNCT
cana-1740	135	37	×	×	NOUN
cana-1740	135	38	(	(	PUNCT
cana-1740	135	39	2𝑛2	2𝑛2	NUM
cana-1740	135	40	−	−	NOUN
cana-1740	135	41	𝑛	𝑛	NOUN
cana-1740	135	42	)	)	PUNCT
cana-1740	135	43	+	+	CCONJ
cana-1740	135	44	(	(	PUNCT
cana-1740	135	45	4𝑛2	4𝑛2	NUM
cana-1740	135	46	)	)	PUNCT
cana-1740	135	47	×	×	NOUN
cana-1740	135	48	𝑛2	𝑛2	NOUN
cana-1740	135	49	=	=	PROPN
cana-1740	135	50	𝑛2(3(2𝑛	𝑛2(3(2𝑛	PROPN
cana-1740	135	51	−	−	PROPN
cana-1740	135	52	1)2	1)2	NUM
cana-1740	135	53	+	+	CCONJ
cana-1740	135	54	4𝑛2	4𝑛2	NUM
cana-1740	135	55	)	)	PUNCT
cana-1740	135	56	.	.	PUNCT
cana-1740	136	1	𝐻((𝑃𝑛⊙𝐾1	𝐻((𝑃𝑛⊙𝐾1	PROPN
cana-1740	136	2	)	)	PUNCT
cana-1740	136	3	⊠	⊠	PROPN
cana-1740	136	4	𝑃2	𝑃2	PROPN
cana-1740	136	5	)	)	PUNCT
cana-1740	136	6	=	=	PUNCT
cana-1740	137	1	∑	∑	PROPN
cana-1740	137	2	2	2	NUM
cana-1740	137	3	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	137	4	)	)	PUNCT
cana-1740	137	5	+	+	CCONJ
cana-1740	137	6	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	137	7	)	)	PUNCT
cana-1740	137	8	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	NUM
cana-1740	137	9	)	)	PUNCT
cana-1740	137	10	=	=	SYM
cana-1740	138	1	∑	∑	PUNCT
cana-1740	138	2	2	2	NUM
cana-1740	138	3	(	(	PUNCT
cana-1740	138	4	4𝑛	4𝑛	NOUN
cana-1740	138	5	−	−	NOUN
cana-1740	138	6	2	2	NUM
cana-1740	138	7	)	)	PUNCT
cana-1740	138	8	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	139	1	+	+	CCONJ
cana-1740	139	2	∑	∑	SYM
cana-1740	139	3	2	2	NUM
cana-1740	139	4	(	(	PUNCT
cana-1740	139	5	4𝑛	4𝑛	NOUN
cana-1740	139	6	−	−	NOUN
cana-1740	139	7	1	1	NUM
cana-1740	139	8	)	)	PUNCT
cana-1740	139	9	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	PUNCT
cana-1740	140	1	+	+	CCONJ
cana-1740	140	2	∑	∑	SYM
cana-1740	140	3	2	2	NUM
cana-1740	140	4	(	(	PUNCT
cana-1740	140	5	4𝑛	4𝑛	NOUN
cana-1740	140	6	)	)	PUNCT
cana-1740	140	7	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	NOUN
cana-1740	141	1	=	=	SYM
cana-1740	141	2	2	2	NUM
cana-1740	141	3	(	(	PUNCT
cana-1740	141	4	4𝑛	4𝑛	NOUN
cana-1740	141	5	−	−	NOUN
cana-1740	141	6	2	2	X
cana-1740	141	7	)	)	PUNCT
cana-1740	141	8	×	×	NOUN
cana-1740	141	9	𝑛2	𝑛2	NOUN
cana-1740	141	10	+	+	CCONJ
cana-1740	141	11	2	2	NUM
cana-1740	141	12	(	(	PUNCT
cana-1740	141	13	4𝑛	4𝑛	NOUN
cana-1740	141	14	−	−	NOUN
cana-1740	141	15	1	1	NUM
cana-1740	141	16	)	)	PUNCT
cana-1740	141	17	×	×	NOUN
cana-1740	141	18	(	(	PUNCT
cana-1740	141	19	2𝑛2	2𝑛2	NUM
cana-1740	141	20	−	−	NOUN
cana-1740	141	21	𝑛	𝑛	NOUN
cana-1740	141	22	)	)	PUNCT
cana-1740	141	23	+	+	CCONJ
cana-1740	141	24	2	2	NUM
cana-1740	141	25	(	(	PUNCT
cana-1740	141	26	4𝑛	4𝑛	NOUN
cana-1740	141	27	)	)	PUNCT
cana-1740	141	28	×	×	NOUN
cana-1740	141	29	𝑛2	𝑛2	NOUN
cana-1740	141	30	=	=	PROPN
cana-1740	141	31	𝑛2	𝑛2	PROPN
cana-1740	141	32	(	(	PUNCT
cana-1740	141	33	2𝑛	2𝑛	PROPN
cana-1740	141	34	−	−	PROPN
cana-1740	141	35	1	1	NUM
cana-1740	141	36	)	)	PUNCT
cana-1740	141	37	+	+	CCONJ
cana-1740	141	38	2𝑛(2𝑛	2𝑛(2𝑛	NUM
cana-1740	141	39	−	−	NOUN
cana-1740	141	40	1	1	NUM
cana-1740	141	41	)	)	PUNCT
cana-1740	141	42	(	(	PUNCT
cana-1740	141	43	4𝑛	4𝑛	NOUN
cana-1740	141	44	−	−	NOUN
cana-1740	141	45	1	1	X
cana-1740	141	46	)	)	PUNCT
cana-1740	141	47	+	+	CCONJ
cana-1740	141	48	𝑛	𝑛	DET
cana-1740	141	49	2	2	NUM
cana-1740	141	50	.	.	PUNCT
cana-1740	142	1	𝐺𝐴((𝑃𝑛⊙𝐾1	𝐺𝐴((𝑃𝑛⊙𝐾1	NOUN
cana-1740	142	2	)	)	PUNCT
cana-1740	143	1	⊠	⊠	PROPN
cana-1740	143	2	𝑃2	𝑃2	NOUN
cana-1740	143	3	)	)	PUNCT
cana-1740	143	4	=	=	PUNCT
cana-1740	144	1	∑	∑	PROPN
cana-1740	144	2	2	2	NUM
cana-1740	144	3	√𝑑(𝑢	√𝑑(𝑢	NUM
cana-1740	144	4	)	)	PUNCT
cana-1740	144	5	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	144	6	)	)	PUNCT
cana-1740	144	7	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1740	144	8	)	)	PUNCT
cana-1740	144	9	+	+	CCONJ
cana-1740	144	10	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1740	144	11	)	)	PUNCT
cana-1740	144	12	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2	PRON
cana-1740	144	13	)	)	PUNCT
cana-1740	144	14	=	=	PUNCT
cana-1740	144	15	∑	∑	PUNCT
cana-1740	144	16	2√2𝑛	2√2𝑛	NUM
cana-1740	144	17	−	−	NUM
cana-1740	144	18	1	1	NUM
cana-1740	144	19	×	×	NOUN
cana-1740	144	20	2𝑛	2𝑛	PROPN
cana-1740	144	21	−	−	PROPN
cana-1740	144	22	1	1	NUM
cana-1740	144	23	4𝑛	4𝑛	NOUN
cana-1740	144	24	−	−	NOUN
cana-1740	144	25	2	2	NUM
cana-1740	144	26	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	X
cana-1740	145	1	+	+	CCONJ
cana-1740	145	2	∑	∑	PROPN
cana-1740	145	3	2√2𝑛	2√2𝑛	NUM
cana-1740	145	4	×	×	NOUN
cana-1740	145	5	2𝑛	2𝑛	PROPN
cana-1740	145	6	−	−	PROPN
cana-1740	145	7	1	1	NUM
cana-1740	145	8	4𝑛	4𝑛	NOUN
cana-1740	145	9	−	−	NOUN
cana-1740	145	10	1	1	NUM
cana-1740	145	11	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	ADV
cana-1740	146	1	+	+	NUM
cana-1740	146	2	∑	∑	PROPN
cana-1740	146	3	2√2𝑛	2√2𝑛	NUM
cana-1740	146	4	×	×	PROPN
cana-1740	146	5	2𝑛	2𝑛	PROPN
cana-1740	146	6	4𝑛	4𝑛	PROPN
cana-1740	146	7	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	PROPN
cana-1740	147	1	=	=	PUNCT
cana-1740	147	2	∑	∑	PUNCT
cana-1740	147	3	2(2𝑛	2(2𝑛	NUM
cana-1740	147	4	−	−	NOUN
cana-1740	147	5	1	1	NUM
cana-1740	147	6	)	)	PUNCT
cana-1740	147	7	(	(	PUNCT
cana-1740	147	8	4𝑛	4𝑛	NOUN
cana-1740	147	9	−	−	NOUN
cana-1740	147	10	2	2	NUM
cana-1740	147	11	)	)	PUNCT
cana-1740	147	12	𝑢𝑣∈𝐸1	𝑢𝑣∈𝐸1	PUNCT
cana-1740	148	1	+	+	CCONJ
cana-1740	148	2	∑	∑	PROPN
cana-1740	148	3	2√(4𝑛2	2√(4𝑛2	NUM
cana-1740	148	4	−	−	PROPN
cana-1740	148	5	2𝑛	2𝑛	NUM
cana-1740	148	6	)	)	PUNCT
cana-1740	149	1	(	(	PUNCT
cana-1740	149	2	4𝑛	4𝑛	NOUN
cana-1740	149	3	−	−	NOUN
cana-1740	149	4	1	1	NUM
cana-1740	149	5	)	)	PUNCT
cana-1740	149	6	𝑢𝑣∈𝐸2	𝑢𝑣∈𝐸2	PUNCT
cana-1740	150	1	+	+	CCONJ
cana-1740	150	2	∑	∑	PROPN
cana-1740	150	3	2×	2×	NUM
cana-1740	150	4	2𝑛	2𝑛	PROPN
cana-1740	150	5	4𝑛	4𝑛	PROPN
cana-1740	150	6	𝑢𝑣∈𝐸3	𝑢𝑣∈𝐸3	VERB
cana-1740	150	7	communications	communication	NOUN
cana-1740	150	8	on	on	ADP
cana-1740	150	9	applied	apply	VERB
cana-1740	150	10	nonlinear	nonlinear	ADJ
cana-1740	150	11	analysis	analysis	NOUN
cana-1740	150	12	issn	issn	NOUN
cana-1740	150	13	:	:	PUNCT
cana-1740	150	14	1074	1074	NUM
cana-1740	150	15	-	-	PUNCT
cana-1740	150	16	133x	133x	NUM
cana-1740	150	17	vol	vol	NOUN
cana-1740	150	18	32	32	NUM
cana-1740	150	19	no	no	NOUN
cana-1740	150	20	.	.	NOUN
cana-1740	150	21	2	2	NUM
cana-1740	150	22	(	(	PUNCT
cana-1740	150	23	2025	2025	NUM
cana-1740	150	24	)	)	PUNCT
cana-1740	150	25	263	263	NUM
cana-1740	150	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-1740	150	27	=	=	SYM
cana-1740	150	28	2𝑛2	2𝑛2	NUM
cana-1740	150	29	+	+	CCONJ
cana-1740	150	30	2𝑛(2𝑛	2𝑛(2𝑛	NUM
cana-1740	150	31	−	−	PROPN
cana-1740	150	32	1)√(4𝑛2	1)√(4𝑛2	PROPN
cana-1740	150	33	−	−	PROPN
cana-1740	150	34	2𝑛	2𝑛	NUM
cana-1740	150	35	)	)	PUNCT
cana-1740	151	1	(	(	PUNCT
cana-1740	151	2	4𝑛	4𝑛	NOUN
cana-1740	151	3	−	−	NOUN
cana-1740	151	4	1	1	NUM
cana-1740	151	5	)	)	PUNCT
cana-1740	151	6	.	.	PUNCT
cana-1740	152	1	this	this	PRON
cana-1740	152	2	completes	complete	VERB
cana-1740	152	3	the	the	DET
cana-1740	152	4	proof	proof	NOUN
cana-1740	152	5	.	.	PUNCT
cana-1740	153	1	note	note	VERB
cana-1740	153	2	:	:	PUNCT
cana-1740	153	3	for	for	ADP
cana-1740	153	4	𝑛	𝑛	PROPN
cana-1740	153	5	=	=	SYM
cana-1740	153	6	3	3	NUM
cana-1740	153	7	,	,	PUNCT
cana-1740	153	8	first	first	ADJ
cana-1740	153	9	and	and	CCONJ
cana-1740	153	10	second	second	ADJ
cana-1740	153	11	zagreb	zagreb	PROPN
cana-1740	153	12	indices	index	NOUN
cana-1740	153	13	,	,	PUNCT
cana-1740	153	14	harmonic	harmonic	ADJ
cana-1740	153	15	index	index	NOUN
cana-1740	153	16	and	and	CCONJ
cana-1740	153	17	geometric	geometric	ADJ
cana-1740	153	18	-	-	PUNCT
cana-1740	153	19	arithmetic	arithmetic	ADJ
cana-1740	153	20	index	index	NOUN
cana-1740	153	21	of	of	ADP
cana-1740	153	22	(	(	PUNCT
cana-1740	153	23	𝑃3⊙𝐾1	𝑃3⊙𝐾1	PROPN
cana-1740	153	24	)	)	PUNCT
cana-1740	153	25	⊠	⊠	PROPN
cana-1740	153	26	𝑃2	𝑃2	NOUN
cana-1740	153	27	are	be	AUX
cana-1740	153	28	388	388	NUM
cana-1740	153	29	,	,	PUNCT
cana-1740	153	30	1106	1106	NUM
cana-1740	153	31	,	,	PUNCT
cana-1740	153	32	986	986	NUM
cana-1740	153	33	165	165	NUM
cana-1740	153	34	and	and	CCONJ
cana-1740	153	35	198	198	NUM
cana-1740	153	36	+32√30	+32√30	NUM
cana-1740	153	37	11	11	NUM
cana-1740	153	38	respectively	respectively	ADV
cana-1740	153	39	.	.	PUNCT
cana-1740	154	1	3	3	X
cana-1740	154	2	.	.	X
cana-1740	154	3	conclusion	conclusion	NOUN
cana-1740	154	4	in	in	ADP
cana-1740	154	5	this	this	DET
cana-1740	154	6	work	work	NOUN
cana-1740	154	7	,	,	PUNCT
cana-1740	154	8	we	we	PRON
cana-1740	154	9	determined	determine	VERB
cana-1740	154	10	various	various	ADJ
cana-1740	154	11	topological	topological	ADJ
cana-1740	154	12	indices	index	NOUN
cana-1740	154	13	of	of	ADP
cana-1740	154	14	arms	arm	NOUN
cana-1740	154	15	-	-	PUNCT
cana-1740	154	16	product	product	NOUN
cana-1740	154	17	of	of	ADP
cana-1740	154	18	certain	certain	ADJ
cana-1740	154	19	classes	class	NOUN
cana-1740	154	20	of	of	ADP
cana-1740	154	21	graphs	graph	NOUN
cana-1740	154	22	.	.	PUNCT
cana-1740	155	1	since	since	SCONJ
cana-1740	155	2	the	the	DET
cana-1740	155	3	arms	arm	NOUN
cana-1740	155	4	-	-	PUNCT
cana-1740	155	5	product	product	NOUN
cana-1740	155	6	is	be	AUX
cana-1740	155	7	a	a	DET
cana-1740	155	8	newly	newly	ADV
cana-1740	155	9	introduced	introduce	VERB
cana-1740	155	10	graph	graph	NOUN
cana-1740	155	11	product	product	NOUN
cana-1740	155	12	,	,	PUNCT
cana-1740	155	13	it	it	PRON
cana-1740	155	14	has	have	VERB
cana-1740	155	15	great	great	ADJ
cana-1740	155	16	significance	significance	NOUN
cana-1740	155	17	in	in	ADP
cana-1740	155	18	the	the	DET
cana-1740	155	19	field	field	NOUN
cana-1740	155	20	of	of	ADP
cana-1740	155	21	topological	topological	ADJ
cana-1740	155	22	indices	index	NOUN
cana-1740	155	23	.	.	PUNCT
cana-1740	156	1	acknowledgement	acknowledgement	NOUN
cana-1740	156	2	the	the	DET
cana-1740	156	3	authors	author	NOUN
cana-1740	156	4	would	would	AUX
cana-1740	156	5	like	like	VERB
cana-1740	156	6	to	to	PART
cana-1740	156	7	thank	thank	VERB
cana-1740	156	8	university	university	PROPN
cana-1740	156	9	of	of	ADP
cana-1740	156	10	kerala	kerala	PROPN
cana-1740	156	11	for	for	ADP
cana-1740	156	12	providing	provide	VERB
cana-1740	156	13	all	all	DET
cana-1740	156	14	facilities	facility	NOUN
cana-1740	156	15	.	.	PUNCT
cana-1740	157	1	the	the	DET
cana-1740	157	2	first	first	ADJ
cana-1740	157	3	author	author	NOUN
cana-1740	157	4	would	would	AUX
cana-1740	157	5	like	like	VERB
cana-1740	157	6	to	to	PART
cana-1740	157	7	thank	thank	VERB
cana-1740	157	8	the	the	DET
cana-1740	157	9	technical	technical	ADJ
cana-1740	157	10	help	help	NOUN
cana-1740	157	11	received	receive	VERB
cana-1740	157	12	from	from	ADP
cana-1740	157	13	dst	dst	NOUN
cana-1740	157	14	-	-	PUNCT
cana-1740	157	15	fist	fist	NOUN
cana-1740	157	16	facility	facility	NOUN
cana-1740	157	17	of	of	ADP
cana-1740	157	18	department	department	PROPN
cana-1740	157	19	of	of	ADP
cana-1740	157	20	mathematics	mathematics	PROPN
cana-1740	157	21	,	,	PUNCT
cana-1740	157	22	st	st	PROPN
cana-1740	157	23	.	.	PROPN
cana-1740	157	24	stephen	stephen	PROPN
cana-1740	157	25	’s	’s	PROPN
cana-1740	157	26	college	college	PROPN
cana-1740	157	27	,	,	PUNCT
cana-1740	157	28	pathanapuram	pathanapuram	PROPN
cana-1740	157	29	,	,	PUNCT
cana-1740	157	30	kollam	kollam	PROPN
cana-1740	157	31	,	,	PUNCT
cana-1740	157	32	kerala	kerala	PROPN
cana-1740	157	33	.	.	PUNCT
cana-1740	158	1	the	the	DET
cana-1740	158	2	second	second	ADJ
cana-1740	158	3	author	author	NOUN
cana-1740	158	4	would	would	AUX
cana-1740	158	5	like	like	VERB
cana-1740	158	6	to	to	PART
cana-1740	158	7	thank	thank	VERB
cana-1740	158	8	the	the	DET
cana-1740	158	9	kerala	kerala	PROPN
cana-1740	158	10	state	state	PROPN
cana-1740	158	11	council	council	PROPN
cana-1740	158	12	for	for	ADP
cana-1740	158	13	science	science	NOUN
cana-1740	158	14	,	,	PUNCT
cana-1740	158	15	technology	technology	NOUN
cana-1740	158	16	and	and	CCONJ
cana-1740	158	17	environment	environment	NOUN
cana-1740	158	18	(	(	PUNCT
cana-1740	158	19	kscste	kscste	NOUN
cana-1740	158	20	)	)	PUNCT
cana-1740	158	21	for	for	ADP
cana-1740	158	22	the	the	DET
cana-1740	158	23	financial	financial	ADJ
cana-1740	158	24	support	support	NOUN
cana-1740	158	25	.	.	PUNCT
cana-1740	159	1	further	far	ADV
cana-1740	159	2	we	we	PRON
cana-1740	159	3	would	would	AUX
cana-1740	159	4	like	like	VERB
cana-1740	159	5	to	to	PART
cana-1740	159	6	inform	inform	VERB
cana-1740	159	7	that	that	SCONJ
cana-1740	159	8	all	all	DET
cana-1740	159	9	the	the	DET
cana-1740	159	10	authors	author	NOUN
cana-1740	159	11	contributed	contribute	VERB
cana-1740	159	12	equally	equally	ADV
cana-1740	159	13	in	in	ADP
cana-1740	159	14	this	this	DET
cana-1740	159	15	work	work	NOUN
cana-1740	159	16	.	.	PUNCT
cana-1740	160	1	the	the	DET
cana-1740	160	2	authors	author	NOUN
cana-1740	160	3	would	would	AUX
cana-1740	160	4	like	like	VERB
cana-1740	160	5	to	to	PART
cana-1740	160	6	thank	thank	VERB
cana-1740	160	7	dr	dr	PROPN
cana-1740	160	8	.	.	PROPN
cana-1740	160	9	g.	g.	PROPN
cana-1740	160	10	suresh	suresh	PROPN
cana-1740	160	11	singh	singh	PROPN
cana-1740	160	12	,	,	PUNCT
cana-1740	160	13	professor	professor	NOUN
cana-1740	160	14	,	,	PUNCT
cana-1740	160	15	department	department	NOUN
cana-1740	160	16	of	of	ADP
cana-1740	160	17	mathematics	mathematics	PROPN
cana-1740	160	18	,	,	PUNCT
cana-1740	160	19	university	university	NOUN
cana-1740	160	20	of	of	ADP
cana-1740	160	21	kerala	kerala	PROPN
cana-1740	160	22	for	for	ADP
cana-1740	160	23	his	his	PRON
cana-1740	160	24	continuous	continuous	ADJ
cana-1740	160	25	support	support	NOUN
cana-1740	160	26	and	and	CCONJ
cana-1740	160	27	valuable	valuable	ADJ
cana-1740	160	28	suggestions	suggestion	NOUN
cana-1740	160	29	.	.	PUNCT
cana-1740	161	1	declarations	declaration	NOUN
cana-1740	161	2	•	•	NUM
cana-1740	161	3	funding	funding	NOUN
cana-1740	161	4	:	:	PUNCT
cana-1740	161	5	this	this	DET
cana-1740	161	6	work	work	NOUN
cana-1740	161	7	was	be	AUX
cana-1740	161	8	supported	support	VERB
cana-1740	161	9	by	by	ADP
cana-1740	161	10	kerala	kerala	PROPN
cana-1740	161	11	state	state	PROPN
cana-1740	161	12	council	council	PROPN
cana-1740	161	13	for	for	ADP
cana-1740	161	14	science	science	NOUN
cana-1740	161	15	,	,	PUNCT
cana-1740	161	16	technology	technology	NOUN
cana-1740	161	17	and	and	CCONJ
cana-1740	161	18	environment	environment	NOUN
cana-1740	161	19	(	(	PUNCT
cana-1740	161	20	kscste	kscste	NOUN
cana-1740	161	21	)	)	PUNCT
cana-1740	161	22	.	.	PUNCT
cana-1740	162	1	•	•	NUM
cana-1740	162	2	authors	author	NOUN
cana-1740	162	3	’	'	PUNCT
cana-1740	162	4	contributions	contribution	NOUN
cana-1740	162	5	:	:	PUNCT
cana-1740	162	6	all	all	DET
cana-1740	162	7	authors	author	NOUN
cana-1740	162	8	contributed	contribute	VERB
cana-1740	162	9	the	the	DET
cana-1740	162	10	study	study	NOUN
cana-1740	162	11	conception	conception	NOUN
cana-1740	162	12	and	and	CCONJ
cana-1740	162	13	design	design	NOUN
cana-1740	162	14	.	.	PUNCT
cana-1740	163	1	all	all	DET
cana-1740	163	2	authors	author	NOUN
cana-1740	163	3	read	read	VERB
cana-1740	163	4	and	and	CCONJ
cana-1740	163	5	approved	approve	VERB
cana-1740	163	6	the	the	DET
cana-1740	163	7	final	final	ADJ
cana-1740	163	8	manuscript	manuscript	NOUN
cana-1740	163	9	.	.	PUNCT
cana-1740	164	1	conflicts	conflict	NOUN
cana-1740	164	2	of	of	ADP
cana-1740	164	3	interest	interest	NOUN
cana-1740	164	4	:	:	PUNCT
cana-1740	164	5	the	the	DET
cana-1740	164	6	authors	author	NOUN
cana-1740	164	7	declare	declare	VERB
cana-1740	164	8	that	that	SCONJ
cana-1740	164	9	they	they	PRON
cana-1740	164	10	have	have	VERB
cana-1740	164	11	no	no	DET
cana-1740	164	12	conflicts	conflict	NOUN
cana-1740	164	13	of	of	ADP
cana-1740	164	14	interest	interest	NOUN
cana-1740	164	15	regarding	regard	VERB
cana-1740	164	16	the	the	DET
cana-1740	164	17	publication	publication	NOUN
cana-1740	164	18	of	of	ADP
cana-1740	164	19	this	this	DET
cana-1740	164	20	article	article	NOUN
cana-1740	164	21	.	.	PUNCT
cana-1740	165	1	references	reference	NOUN
cana-1740	165	2	[	[	X
cana-1740	165	3	1	1	NUM
cana-1740	165	4	]	]	PUNCT
cana-1740	165	5	akhil	akhil	PROPN
cana-1740	165	6	b.	b.	PROPN
cana-1740	165	7	et.al	et.al	PROPN
cana-1740	165	8	.	.	PROPN
cana-1740	165	9	,	,	PUNCT
cana-1740	165	10	arms	arm	NOUN
cana-1740	165	11	-	-	PUNCT
cana-1740	165	12	product	product	NOUN
cana-1740	165	13	and	and	CCONJ
cana-1740	165	14	limit	limit	NOUN
cana-1740	165	15	graph	graph	NOUN
cana-1740	165	16	of	of	ADP
cana-1740	165	17	graphs	graph	NOUN
cana-1740	165	18	.	.	PUNCT
cana-1740	166	1	global	global	ADJ
cana-1740	166	2	and	and	CCONJ
cana-1740	166	3	stochastic	stochastic	ADJ
cana-1740	166	4	analysis	analysis	NOUN
cana-1740	166	5	,	,	PUNCT
cana-1740	166	6	11(2	11(2	NOUN
cana-1740	166	7	)	)	PUNCT
cana-1740	166	8	,	,	PUNCT
cana-1740	166	9	2024	2024	NUM
cana-1740	166	10	.	.	PUNCT
cana-1740	167	1	[	[	X
cana-1740	167	2	2	2	X
cana-1740	167	3	]	]	PUNCT
cana-1740	167	4	g.	g.	PROPN
cana-1740	167	5	suresh	suresh	PROPN
cana-1740	167	6	singh	singh	PROPN
cana-1740	167	7	,	,	PUNCT
cana-1740	167	8	graph	graph	NOUN
cana-1740	167	9	theory	theory	NOUN
cana-1740	167	10	,	,	PUNCT
cana-1740	167	11	phi	phi	NOUN
cana-1740	167	12	,	,	PUNCT
cana-1740	167	13	new	new	ADJ
cana-1740	167	14	delhi	delhi	PROPN
cana-1740	167	15	,	,	PUNCT
cana-1740	167	16	(	(	PUNCT
cana-1740	167	17	2010	2010	NUM
cana-1740	167	18	)	)	PUNCT
cana-1740	167	19	.	.	PUNCT
cana-1740	168	1	[	[	X
cana-1740	168	2	3	3	NUM
cana-1740	168	3	]	]	X
cana-1740	168	4	i.	i.	NOUN
cana-1740	168	5	gutman	gutman	PROPN
cana-1740	168	6	and	and	CCONJ
cana-1740	168	7	n.	n.	PROPN
cana-1740	168	8	trinajstic	trinajstic	PROPN
cana-1740	168	9	´	´	PROPN
cana-1740	168	10	,	,	PUNCT
cana-1740	168	11	graph	graph	NOUN
cana-1740	168	12	theory	theory	NOUN
cana-1740	168	13	and	and	CCONJ
cana-1740	168	14	molecular	molecular	ADJ
cana-1740	168	15	orbitals	orbital	NOUN
cana-1740	168	16	.	.	PUNCT
cana-1740	169	1	total	total	ADJ
cana-1740	169	2	π	π	ADJ
cana-1740	169	3	-	-	PUNCT
cana-1740	169	4	electron	electron	NOUN
cana-1740	169	5	energy	energy	NOUN
cana-1740	169	6	of	of	ADP
cana-1740	169	7	alternant	alternant	ADJ
cana-1740	169	8	hydrocarbons	hydrocarbon	NOUN
cana-1740	169	9	,	,	PUNCT
cana-1740	169	10	chemical	chemical	NOUN
cana-1740	169	11	physics	physics	NOUN
cana-1740	169	12	letters,17(4	letters,17(4	PROPN
cana-1740	169	13	)	)	PUNCT
cana-1740	169	14	,	,	PUNCT
cana-1740	169	15	1972	1972	NUM
cana-1740	169	16	.	.	PUNCT
cana-1740	170	1	[	[	X
cana-1740	170	2	4	4	NUM
cana-1740	170	3	]	]	X
cana-1740	170	4	iranmanesh	iranmanesh	PROPN
cana-1740	170	5	m.a	m.a	PROPN
cana-1740	170	6	.	.	PROPN
cana-1740	170	7	and	and	CCONJ
cana-1740	170	8	mahboubeh	mahboubeh	PROPN
cana-1740	170	9	s.	s.	PROPN
cana-1740	170	10	,	,	PUNCT
cana-1740	170	11	on	on	ADP
cana-1740	170	12	the	the	DET
cana-1740	170	13	harmonic	harmonic	ADJ
cana-1740	170	14	index	index	NOUN
cana-1740	170	15	and	and	CCONJ
cana-1740	170	16	harmonic	harmonic	ADJ
cana-1740	170	17	polynomial	polynomial	NOUN
cana-1740	170	18	of	of	ADP
cana-1740	170	19	caterpillars	caterpillar	NOUN
cana-1740	170	20	with	with	ADP
cana-1740	170	21	diameter	diameter	NOUN
cana-1740	170	22	four	four	NUM
cana-1740	170	23	,	,	PUNCT
cana-1740	170	24	iran	iran	PROPN
cana-1740	170	25	.	.	PUNCT
cana-1740	171	1	j.	j.	PROPN
cana-1740	171	2	math	math	PROPN
cana-1740	171	3	.	.	PUNCT
cana-1740	172	1	chem	chem	PROPN
cana-1740	172	2	.	.	PUNCT
cana-1740	172	3	,	,	PUNCT
cana-1740	172	4	5	5	NUM
cana-1740	172	5	,	,	PUNCT
cana-1740	172	6	pn	pn	NOUN
cana-1740	172	7	:	:	PUNCT
cana-1740	172	8	35	35	NUM
cana-1740	172	9	-	-	SYM
cana-1740	172	10	43	43	NUM
cana-1740	172	11	,	,	PUNCT
cana-1740	172	12	2015	2015	NUM
cana-1740	172	13	.	.	PUNCT
cana-1740	173	1	[	[	X
cana-1740	173	2	5	5	NUM
cana-1740	173	3	]	]	PUNCT
cana-1740	173	4	yuan	yuan	PROPN
cana-1740	173	5	yan	yan	PROPN
cana-1740	173	6	and	and	CCONJ
cana-1740	173	7	zhou	zhou	PROPN
cana-1740	173	8	bo	bo	PROPN
cana-1740	173	9	and	and	CCONJ
cana-1740	173	10	trinajstic	trinajstic	ADJ
cana-1740	173	11	nenad	nenad	PROPN
cana-1740	173	12	´	´	PROPN
cana-1740	173	13	,	,	PUNCT
cana-1740	173	14	on	on	ADP
cana-1740	173	15	geometric	geometric	ADJ
cana-1740	173	16	-	-	PUNCT
cana-1740	173	17	arithmetic	arithmetic	ADJ
cana-1740	173	18	index	index	NOUN
cana-1740	173	19	,	,	PUNCT
cana-1740	173	20	journal	journal	NOUN
cana-1740	173	21	of	of	ADP
cana-1740	173	22	mathematical	mathematical	ADJ
cana-1740	173	23	chemistry	chemistry	NOUN
cana-1740	173	24	,	,	PUNCT
cana-1740	173	25	47(2	47(2	PROPN
cana-1740	173	26	)	)	PUNCT
cana-1740	173	27	,	,	PUNCT
cana-1740	173	28	833	833	NUM
cana-1740	173	29	-	-	SYM
cana-1740	173	30	841	841	NUM
cana-1740	173	31	,	,	PUNCT
cana-1740	173	32	2010	2010	NUM
cana-1740	173	33	.	.	PUNCT