id	sid	tid	token	lemma	pos
cana-3462	1	1	communications	communication	NOUN
cana-3462	1	2	on	on	ADP
cana-3462	1	3	applied	apply	VERB
cana-3462	1	4	nonlinear	nonlinear	ADJ
cana-3462	1	5	analysis	analysis	NOUN
cana-3462	1	6	issn	issn	NOUN
cana-3462	1	7	:	:	PUNCT
cana-3462	1	8	1074	1074	NUM
cana-3462	1	9	-	-	PUNCT
cana-3462	1	10	133x	133x	NUM
cana-3462	1	11	vol	vol	NOUN
cana-3462	1	12	32	32	NUM
cana-3462	1	13	no	no	NOUN
cana-3462	1	14	.	.	PUNCT
cana-3462	2	1	7s	7	NOUN
cana-3462	2	2	(	(	PUNCT
cana-3462	2	3	2025	2025	NUM
cana-3462	2	4	)	)	PUNCT
cana-3462	2	5	530	530	NUM
cana-3462	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	2	7	quasilaplacian	quasilaplacian	ADJ
cana-3462	2	8	energy	energy	NOUN
cana-3462	2	9	of	of	ADP
cana-3462	2	10	some	some	DET
cana-3462	2	11	novel	novel	ADJ
cana-3462	2	12	classes	class	NOUN
cana-3462	2	13	of	of	ADP
cana-3462	2	14	graphs	graph	NOUN
cana-3462	2	15	manash	manash	PROPN
cana-3462	2	16	protim	protim	PROPN
cana-3462	2	17	borah	borah	PROPN
cana-3462	2	18	a	a	PRON
cana-3462	2	19	,	,	PUNCT
cana-3462	2	20	*	*	PROPN
cana-3462	2	21	,	,	PUNCT
cana-3462	2	22	karam	karam	PROPN
cana-3462	2	23	ratan	ratan	PROPN
cana-3462	3	1	singh	singh	PROPN
cana-3462	3	2	b	b	PROPN
cana-3462	3	3	a	a	PRON
cana-3462	3	4	,	,	PUNCT
cana-3462	3	5	*	*	PUNCT
cana-3462	3	6	department	department	NOUN
cana-3462	3	7	of	of	ADP
cana-3462	3	8	mathematics	mathematic	NOUN
cana-3462	3	9	,	,	PUNCT
cana-3462	3	10	l.t.k	l.t.k	NOUN
cana-3462	3	11	.	.	PUNCT
cana-3462	4	1	college	college	PROPN
cana-3462	4	2	,	,	PUNCT
cana-3462	4	3	azad	azad	PROPN
cana-3462	4	4	,	,	PUNCT
cana-3462	4	5	north	north	NOUN
cana-3462	4	6	lakhimpur	lakhimpur	NOUN
cana-3462	4	7	,	,	PUNCT
cana-3462	4	8	787031	787031	NUM
cana-3462	4	9	,	,	PUNCT
cana-3462	4	10	assam	assam	PROPN
cana-3462	4	11	,	,	PUNCT
cana-3462	4	12	india	india	PROPN
cana-3462	4	13	,	,	PUNCT
cana-3462	4	14	b	b	PROPN
cana-3462	4	15	department	department	NOUN
cana-3462	4	16	of	of	ADP
cana-3462	4	17	basic	basic	ADJ
cana-3462	4	18	and	and	CCONJ
cana-3462	4	19	applied	apply	VERB
cana-3462	4	20	science	science	NOUN
cana-3462	4	21	,	,	PUNCT
cana-3462	4	22	national	national	PROPN
cana-3462	4	23	institute	institute	PROPN
cana-3462	4	24	of	of	ADP
cana-3462	4	25	technology	technology	PROPN
cana-3462	4	26	arunachal	arunachal	PROPN
cana-3462	4	27	pradesh	pradesh	PROPN
cana-3462	4	28	,	,	PUNCT
cana-3462	4	29	papum	papum	PROPN
cana-3462	4	30	pare	pare	PROPN
cana-3462	4	31	,	,	PUNCT
cana-3462	4	32	791113	791113	NUM
cana-3462	4	33	,	,	PUNCT
cana-3462	4	34	arunachal	arunachal	PROPN
cana-3462	4	35	pradesh	pradesh	PROPN
cana-3462	4	36	,	,	PUNCT
cana-3462	4	37	india	india	PROPN
cana-3462	4	38	.	.	PUNCT
cana-3462	5	1	*	*	PUNCT
cana-3462	5	2	corresponding	correspond	VERB
cana-3462	5	3	author	author	NOUN
cana-3462	5	4	email	email	NOUN
cana-3462	5	5	address	address	NOUN
cana-3462	5	6	:	:	PUNCT
cana-3462	5	7	mpborah36@gmail.com	mpborah36@gmail.com	X
cana-3462	5	8	(	(	PUNCT
cana-3462	5	9	manash	manash	PROPN
cana-3462	5	10	protim	protim	PROPN
cana-3462	5	11	borah	borah	PROPN
cana-3462	5	12	)	)	PUNCT
cana-3462	5	13	article	article	NOUN
cana-3462	5	14	history	history	NOUN
cana-3462	5	15	:	:	PUNCT
cana-3462	5	16	received	receive	VERB
cana-3462	5	17	:	:	PUNCT
cana-3462	5	18	26	26	NUM
cana-3462	5	19	-	-	SYM
cana-3462	5	20	10	10	NUM
cana-3462	5	21	-	-	PUNCT
cana-3462	5	22	2024	2024	NUM
cana-3462	5	23	revised:10	revised:10	NOUN
cana-3462	5	24	-	-	PUNCT
cana-3462	5	25	11	11	NUM
cana-3462	5	26	-	-	PUNCT
cana-3462	5	27	2024	2024	NUM
cana-3462	5	28	accepted:18	accepted:18	PROPN
cana-3462	5	29	-	-	PUNCT
cana-3462	5	30	12	12	NUM
cana-3462	5	31	-	-	PUNCT
cana-3462	5	32	2024	2024	NUM
cana-3462	5	33	abstract	abstract	NOUN
cana-3462	5	34	:	:	PUNCT
cana-3462	5	35	we	we	PRON
cana-3462	5	36	formulate	formulate	VERB
cana-3462	5	37	the	the	DET
cana-3462	5	38	relationship	relationship	NOUN
cana-3462	5	39	of	of	ADP
cana-3462	5	40	quasi	quasi	ADJ
cana-3462	5	41	-	-	ADJ
cana-3462	5	42	laplacian	laplacian	ADJ
cana-3462	5	43	energy	energy	NOUN
cana-3462	5	44	of	of	ADP
cana-3462	5	45	some	some	DET
cana-3462	5	46	novel	novel	ADJ
cana-3462	5	47	classes	class	NOUN
cana-3462	5	48	of	of	ADP
cana-3462	5	49	graphs	graph	NOUN
cana-3462	5	50	with	with	ADP
cana-3462	5	51	their	their	PRON
cana-3462	5	52	corresponding	correspond	VERB
cana-3462	5	53	original	original	ADJ
cana-3462	5	54	graphs	graph	NOUN
cana-3462	5	55	.	.	PUNCT
cana-3462	6	1	the	the	DET
cana-3462	6	2	novel	novel	ADJ
cana-3462	6	3	graphs	graph	NOUN
cana-3462	6	4	in	in	ADP
cana-3462	6	5	our	our	PRON
cana-3462	6	6	discussion	discussion	NOUN
cana-3462	6	7	are	be	AUX
cana-3462	6	8	the	the	DET
cana-3462	6	9	𝒮-graph	𝒮-graph	PROPN
cana-3462	6	10	,	,	PUNCT
cana-3462	6	11	ℛ-graph	ℛ-graph	PROPN
cana-3462	6	12	,	,	PUNCT
cana-3462	6	13	𝒬-graph	𝒬-graph	PROPN
cana-3462	6	14	,	,	PUNCT
cana-3462	6	15	total	total	ADJ
cana-3462	6	16	graph	graph	NOUN
cana-3462	6	17	,	,	PUNCT
cana-3462	6	18	and	and	CCONJ
cana-3462	6	19	their	their	PRON
cana-3462	6	20	join	join	NOUN
cana-3462	6	21	and	and	CCONJ
cana-3462	6	22	corona	corona	NOUN
cana-3462	6	23	operations	operation	NOUN
cana-3462	6	24	graphs	graph	NOUN
cana-3462	6	25	.	.	PUNCT
cana-3462	7	1	the	the	DET
cana-3462	7	2	whole	whole	ADJ
cana-3462	7	3	formulation	formulation	NOUN
cana-3462	7	4	is	be	AUX
cana-3462	7	5	based	base	VERB
cana-3462	7	6	on	on	ADP
cana-3462	7	7	the	the	DET
cana-3462	7	8	relationship	relationship	NOUN
cana-3462	7	9	between	between	ADP
cana-3462	7	10	quasi	quasi	ADJ
cana-3462	7	11	-	-	ADJ
cana-3462	7	12	laplacian	laplacian	ADJ
cana-3462	7	13	energy	energy	NOUN
cana-3462	7	14	and	and	CCONJ
cana-3462	7	15	the	the	DET
cana-3462	7	16	vertex	vertex	NOUN
cana-3462	7	17	degrees	degree	NOUN
cana-3462	7	18	of	of	ADP
cana-3462	7	19	the	the	DET
cana-3462	7	20	novel	novel	NOUN
cana-3462	7	21	graph	graph	NOUN
cana-3462	7	22	.	.	PUNCT
cana-3462	8	1	it	it	PRON
cana-3462	8	2	is	be	AUX
cana-3462	8	3	also	also	ADV
cana-3462	8	4	noted	note	VERB
cana-3462	8	5	that	that	SCONJ
cana-3462	8	6	quasilaplacian	quasilaplacian	PROPN
cana-3462	8	7	energy	energy	NOUN
cana-3462	8	8	is	be	AUX
cana-3462	8	9	closely	closely	ADV
cana-3462	8	10	related	relate	VERB
cana-3462	8	11	with	with	ADP
cana-3462	8	12	the	the	DET
cana-3462	8	13	first	first	PROPN
cana-3462	8	14	zagreb	zagreb	PROPN
cana-3462	8	15	index	index	NOUN
cana-3462	8	16	,	,	PUNCT
cana-3462	8	17	number	number	NOUN
cana-3462	8	18	of	of	ADP
cana-3462	8	19	vertices	vertex	NOUN
cana-3462	8	20	and	and	CCONJ
cana-3462	8	21	edges	edge	NOUN
cana-3462	8	22	of	of	ADP
cana-3462	8	23	the	the	DET
cana-3462	8	24	graph	graph	NOUN
cana-3462	8	25	.	.	PUNCT
cana-3462	9	1	the	the	DET
cana-3462	9	2	exact	exact	ADJ
cana-3462	9	3	formulas	formula	NOUN
cana-3462	9	4	of	of	ADP
cana-3462	9	5	quasi	quasi	ADJ
cana-3462	9	6	-	-	ADJ
cana-3462	9	7	laplacian	laplacian	ADJ
cana-3462	9	8	energy	energy	NOUN
cana-3462	9	9	of	of	ADP
cana-3462	9	10	novel	novel	ADJ
cana-3462	9	11	graphs	graph	NOUN
cana-3462	9	12	are	be	AUX
cana-3462	9	13	obtained	obtain	VERB
cana-3462	9	14	in	in	ADP
cana-3462	9	15	terms	term	NOUN
cana-3462	9	16	of	of	ADP
cana-3462	9	17	the	the	DET
cana-3462	9	18	corresponding	corresponding	ADJ
cana-3462	9	19	quasi	quasi	ADJ
cana-3462	9	20	-	-	ADJ
cana-3462	9	21	laplacian	laplacian	ADJ
cana-3462	9	22	energies	energy	NOUN
cana-3462	9	23	,	,	PUNCT
cana-3462	9	24	the	the	DET
cana-3462	9	25	first	first	ADJ
cana-3462	9	26	zagreb	zagreb	PROPN
cana-3462	9	27	indices	index	NOUN
cana-3462	9	28	,	,	PUNCT
cana-3462	9	29	and	and	CCONJ
cana-3462	9	30	the	the	DET
cana-3462	9	31	number	number	NOUN
cana-3462	9	32	of	of	ADP
cana-3462	9	33	vertices	vertex	NOUN
cana-3462	9	34	and	and	CCONJ
cana-3462	9	35	edges	edge	NOUN
cana-3462	9	36	of	of	ADP
cana-3462	9	37	the	the	DET
cana-3462	9	38	original	original	ADJ
cana-3462	9	39	graphs	graph	NOUN
cana-3462	9	40	.	.	PUNCT
cana-3462	10	1	keywords	keyword	NOUN
cana-3462	10	2	:	:	PUNCT
cana-3462	10	3	quasi	quasi	ADJ
cana-3462	10	4	-	-	ADJ
cana-3462	10	5	laplacian	laplacian	ADJ
cana-3462	10	6	energy	energy	NOUN
cana-3462	10	7	,	,	PUNCT
cana-3462	10	8	degree	degree	NOUN
cana-3462	10	9	of	of	ADP
cana-3462	10	10	vertex	vertex	NOUN
cana-3462	10	11	,	,	PUNCT
cana-3462	10	12	zagreb	zagreb	PROPN
cana-3462	10	13	index	index	PROPN
cana-3462	10	14	,	,	PUNCT
cana-3462	10	15	join	join	NOUN
cana-3462	10	16	,	,	PUNCT
cana-3462	10	17	corona	corona	PROPN
cana-3462	10	18	.	.	PUNCT
cana-3462	11	1	mathematics	mathematic	NOUN
cana-3462	11	2	subject	subject	ADJ
cana-3462	11	3	classification	classification	NOUN
cana-3462	11	4	:	:	PUNCT
cana-3462	11	5	05c07	05c07	NOUN
cana-3462	11	6	,	,	PUNCT
cana-3462	11	7	05c09	05c09	NUM
cana-3462	11	8	,	,	PUNCT
cana-3462	11	9	05c50	05c50	NUM
cana-3462	11	10	,	,	PUNCT
cana-3462	11	11	05c76	05c76	NUM
cana-3462	11	12	1	1	X
cana-3462	11	13	.	.	X
cana-3462	12	1	introduction	introduction	NOUN
cana-3462	12	2	all	all	DET
cana-3462	12	3	graphs	graph	NOUN
cana-3462	12	4	discussed	discuss	VERB
cana-3462	12	5	in	in	ADP
cana-3462	12	6	this	this	DET
cana-3462	12	7	paper	paper	NOUN
cana-3462	12	8	are	be	AUX
cana-3462	12	9	simple	simple	ADJ
cana-3462	12	10	and	and	CCONJ
cana-3462	12	11	undirected	undirected	ADJ
cana-3462	12	12	.	.	PUNCT
cana-3462	13	1	let	let	VERB
cana-3462	13	2	𝐺	𝐺	PRON
cana-3462	13	3	be	be	AUX
cana-3462	13	4	a	a	DET
cana-3462	13	5	graph	graph	NOUN
cana-3462	13	6	with	with	ADP
cana-3462	13	7	vertices	vertex	NOUN
cana-3462	13	8	denoted	denote	VERB
cana-3462	13	9	by	by	ADP
cana-3462	13	10	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3462	13	11	)	)	PUNCT
cana-3462	13	12	and	and	CCONJ
cana-3462	13	13	edges	edge	NOUN
cana-3462	13	14	denoted	denote	VERB
cana-3462	13	15	by	by	ADP
cana-3462	13	16	𝐸(𝐺	𝐸(𝐺	PROPN
cana-3462	13	17	)	)	PUNCT
cana-3462	13	18	.	.	PUNCT
cana-3462	14	1	let	let	VERB
cana-3462	14	2	|𝑉(𝐺)|	|𝑉(𝐺)|	VERB
cana-3462	14	3	=	=	SYM
cana-3462	14	4	𝑝	𝑝	PROPN
cana-3462	14	5	and	and	CCONJ
cana-3462	14	6	|𝐸(𝐺)|	|𝐸(𝐺)|	NOUN
cana-3462	14	7	=	=	NOUN
cana-3462	14	8	𝑞.	𝑞.	NOUN
cana-3462	14	9	let	let	VERB
cana-3462	14	10	𝑑𝐺(𝑣	𝑑𝐺(𝑣	PROPN
cana-3462	14	11	)	)	PUNCT
cana-3462	14	12	represent	represent	VERB
cana-3462	14	13	the	the	DET
cana-3462	14	14	degree	degree	NOUN
cana-3462	14	15	of	of	ADP
cana-3462	14	16	vertices	vertex	NOUN
cana-3462	14	17	in	in	ADP
cana-3462	14	18	𝐺	𝐺	PROPN
cana-3462	14	19	,	,	PUNCT
cana-3462	14	20	where	where	SCONJ
cana-3462	14	21	𝑢	𝑢	X
cana-3462	14	22	∈	∈	PROPN
cana-3462	14	23	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3462	14	24	)	)	PUNCT
cana-3462	14	25	.	.	PUNCT
cana-3462	15	1	let	let	AUX
cana-3462	15	2	𝐷(𝐺	𝐷(𝐺	PROPN
cana-3462	15	3	)	)	PUNCT
cana-3462	15	4	denote	denote	VERB
cana-3462	15	5	the	the	DET
cana-3462	15	6	diagonal	diagonal	ADJ
cana-3462	15	7	matrix	matrix	NOUN
cana-3462	15	8	and	and	CCONJ
cana-3462	15	9	𝐴(𝐺	𝐴(𝐺	NOUN
cana-3462	15	10	)	)	PUNCT
cana-3462	15	11	the	the	DET
cana-3462	15	12	adjacency	adjacency	NOUN
cana-3462	15	13	matrix	matrix	NOUN
cana-3462	15	14	.	.	PUNCT
cana-3462	16	1	the	the	DET
cana-3462	16	2	quasi	quasi	PROPN
cana-3462	16	3	laplacian	laplacian	ADJ
cana-3462	16	4	matrix	matrix	NOUN
cana-3462	16	5	,	,	PUNCT
cana-3462	16	6	represented	represent	VERB
cana-3462	16	7	by	by	ADP
cana-3462	16	8	𝑄(𝐺	𝑄(𝐺	PROPN
cana-3462	16	9	)	)	PUNCT
cana-3462	16	10	,	,	PUNCT
cana-3462	16	11	is	be	AUX
cana-3462	16	12	defined	define	VERB
cana-3462	16	13	as	as	ADP
cana-3462	16	14	𝑄(𝐺	𝑄(𝐺	PROPN
cana-3462	16	15	)	)	PUNCT
cana-3462	16	16	=	=	PUNCT
cana-3462	17	1	𝐷(𝐺	𝐷(𝐺	NOUN
cana-3462	17	2	)	)	PUNCT
cana-3462	18	1	+	+	NUM
cana-3462	18	2	𝐴(𝐺	𝐴(𝐺	NOUN
cana-3462	18	3	)	)	PUNCT
cana-3462	18	4	.	.	PUNCT
cana-3462	19	1	let	let	VERB
cana-3462	19	2	𝜇1(𝐺	𝜇1(𝐺	NOUN
cana-3462	19	3	)	)	PUNCT
cana-3462	19	4	≥	≥	NOUN
cana-3462	19	5	𝜇2(𝐺	𝜇2(𝐺	NOUN
cana-3462	19	6	)	)	PUNCT
cana-3462	19	7	≥	≥	NOUN
cana-3462	19	8	⋯	⋯	VERB
cana-3462	19	9	≥	≥	NOUN
cana-3462	19	10	𝜇𝑛(𝐺	𝜇𝑛(𝐺	NOUN
cana-3462	19	11	)	)	PUNCT
cana-3462	19	12	denote	denote	VERB
cana-3462	19	13	the	the	DET
cana-3462	19	14	real	real	ADJ
cana-3462	19	15	,	,	PUNCT
cana-3462	19	16	symmetric	symmetric	ADJ
cana-3462	19	17	,	,	PUNCT
cana-3462	19	18	and	and	CCONJ
cana-3462	19	19	positive	positive	ADJ
cana-3462	19	20	semi	semi	ADJ
cana-3462	19	21	definite	definite	ADJ
cana-3462	19	22	eigenvalues	eigenvalue	NOUN
cana-3462	19	23	of	of	ADP
cana-3462	19	24	𝑄(𝐺	𝑄(𝐺	PROPN
cana-3462	19	25	)	)	PUNCT
cana-3462	19	26	.	.	PUNCT
cana-3462	20	1	it	it	PRON
cana-3462	20	2	is	be	AUX
cana-3462	20	3	known	know	VERB
cana-3462	20	4	that	that	SCONJ
cana-3462	20	5	various	various	ADJ
cana-3462	20	6	graph	graph	NOUN
cana-3462	20	7	operations	operation	NOUN
cana-3462	20	8	can	can	AUX
cana-3462	20	9	create	create	VERB
cana-3462	20	10	novel	novel	ADJ
cana-3462	20	11	classes	class	NOUN
cana-3462	20	12	of	of	ADP
cana-3462	20	13	graphs	graph	NOUN
cana-3462	20	14	from	from	ADP
cana-3462	20	15	the	the	DET
cana-3462	20	16	original	original	ADJ
cana-3462	20	17	graphs	graph	NOUN
cana-3462	20	18	.	.	PUNCT
cana-3462	21	1	therefore	therefore	ADV
cana-3462	21	2	,	,	PUNCT
cana-3462	21	3	understanding	understand	VERB
cana-3462	21	4	the	the	DET
cana-3462	21	5	relationships	relationship	NOUN
cana-3462	21	6	between	between	ADP
cana-3462	21	7	some	some	DET
cana-3462	21	8	invariants	invariant	NOUN
cana-3462	21	9	of	of	ADP
cana-3462	21	10	such	such	ADJ
cana-3462	21	11	novel	novel	ADJ
cana-3462	21	12	graphs	graph	NOUN
cana-3462	21	13	and	and	CCONJ
cana-3462	21	14	the	the	DET
cana-3462	21	15	equivalent	equivalent	ADJ
cana-3462	21	16	invariants	invariant	NOUN
cana-3462	21	17	of	of	ADP
cana-3462	21	18	the	the	DET
cana-3462	21	19	original	original	ADJ
cana-3462	21	20	graphs	graph	NOUN
cana-3462	21	21	is	be	AUX
cana-3462	21	22	relevant	relevant	ADJ
cana-3462	21	23	.	.	PUNCT
cana-3462	22	1	graph	graph	NOUN
cana-3462	22	2	energy	energy	NOUN
cana-3462	22	3	is	be	AUX
cana-3462	22	4	one	one	NUM
cana-3462	22	5	such	such	ADJ
cana-3462	22	6	invariant	invariant	NOUN
cana-3462	22	7	based	base	VERB
cana-3462	22	8	on	on	ADP
cana-3462	22	9	the	the	DET
cana-3462	22	10	graph	graph	NOUN
cana-3462	22	11	spectrum	spectrum	NOUN
cana-3462	22	12	introduced	introduce	VERB
cana-3462	22	13	by	by	ADP
cana-3462	22	14	gutman	gutman	NOUN
cana-3462	22	15	[	[	X
cana-3462	22	16	9	9	NUM
cana-3462	22	17	]	]	PUNCT
cana-3462	22	18	.	.	PUNCT
cana-3462	23	1	we	we	PRON
cana-3462	23	2	discuss	discuss	VERB
cana-3462	23	3	here	here	ADV
cana-3462	23	4	the	the	DET
cana-3462	23	5	quasi	quasi	ADJ
cana-3462	23	6	-	-	ADJ
cana-3462	23	7	laplacian	laplacian	ADJ
cana-3462	23	8	energy	energy	NOUN
cana-3462	23	9	[	[	X
cana-3462	23	10	6	6	NUM
cana-3462	23	11	]	]	PUNCT
cana-3462	23	12	of	of	ADP
cana-3462	23	13	graph	graph	NOUN
cana-3462	23	14	𝐺	𝐺	PROPN
cana-3462	23	15	,	,	PUNCT
cana-3462	23	16	represented	represent	VERB
cana-3462	23	17	as	as	ADP
cana-3462	23	18	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	23	19	)	)	PUNCT
cana-3462	23	20	and	and	CCONJ
cana-3462	23	21	determined	determine	VERB
cana-3462	23	22	by	by	ADP
cana-3462	23	23	the	the	DET
cana-3462	23	24	equation	equation	NOUN
cana-3462	23	25	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	23	26	)	)	PUNCT
cana-3462	24	1	=	=	X
cana-3462	24	2	∑	∑	PART
cana-3462	24	3	  	  	SPACE
cana-3462	24	4	𝑝	𝑝	ADP
cana-3462	24	5	𝑖=1	𝑖=1	PROPN
cana-3462	24	6	𝜇𝑖	𝜇𝑖	ADP
cana-3462	24	7	2	2	NUM
cana-3462	24	8	communications	communication	NOUN
cana-3462	24	9	on	on	ADP
cana-3462	24	10	applied	apply	VERB
cana-3462	24	11	nonlinear	nonlinear	ADJ
cana-3462	24	12	analysis	analysis	NOUN
cana-3462	24	13	issn	issn	NOUN
cana-3462	24	14	:	:	PUNCT
cana-3462	24	15	1074	1074	NUM
cana-3462	24	16	-	-	PUNCT
cana-3462	24	17	133x	133x	NUM
cana-3462	24	18	vol	vol	NOUN
cana-3462	24	19	32	32	NUM
cana-3462	24	20	no	no	NOUN
cana-3462	24	21	.	.	PUNCT
cana-3462	25	1	7s	7	NOUN
cana-3462	25	2	(	(	PUNCT
cana-3462	25	3	2025	2025	NUM
cana-3462	25	4	)	)	PUNCT
cana-3462	25	5	531	531	NUM
cana-3462	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	25	7	yue	yue	PROPN
cana-3462	25	8	,	,	PUNCT
cana-3462	25	9	cao	cao	PROPN
cana-3462	25	10	and	and	CCONJ
cana-3462	25	11	qi	qi	PROPN
cana-3462	25	12	,	,	PUNCT
cana-3462	25	13	in	in	ADP
cana-3462	25	14	the	the	DET
cana-3462	25	15	paper	paper	NOUN
cana-3462	26	1	[	[	X
cana-3462	26	2	6	6	NUM
cana-3462	26	3	]	]	PUNCT
cana-3462	26	4	,	,	PUNCT
cana-3462	26	5	defined	define	VERB
cana-3462	26	6	quasi	quasi	ADJ
cana-3462	26	7	-	-	ADJ
cana-3462	26	8	laplacian	laplacian	ADJ
cana-3462	26	9	energy	energy	NOUN
cana-3462	26	10	of	of	ADP
cana-3462	26	11	𝐺	𝐺	PROPN
cana-3462	26	12	is	be	AUX
cana-3462	26	13	expressed	express	VERB
cana-3462	26	14	as	as	SCONJ
cana-3462	26	15	follows	follow	VERB
cana-3462	26	16	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	26	17	)	)	PUNCT
cana-3462	27	1	=	=	X
cana-3462	27	2	∑	∑	PART
cana-3462	27	3	  	  	SPACE
cana-3462	27	4	𝑝	𝑝	ADP
cana-3462	27	5	𝑖=1	𝑖=1	PROPN
cana-3462	27	6	 	 	SPACE
cana-3462	27	7	𝑑𝐺	𝑑𝐺	PROPN
cana-3462	27	8	2(𝑢𝑖	2(𝑢𝑖	NUM
cana-3462	27	9	)	)	PUNCT
cana-3462	28	1	+	+	ADJ
cana-3462	28	2	∑	∑	PROPN
cana-3462	28	3	  	  	SPACE
cana-3462	28	4	𝑝	𝑝	ADP
cana-3462	28	5	𝑖=1	𝑖=1	PROPN
cana-3462	28	6	 	 	SPACE
cana-3462	28	7	𝑑𝐺(𝑢𝑖	𝑑𝐺(𝑢𝑖	PROPN
cana-3462	28	8	)	)	PUNCT
cana-3462	28	9	,	,	PUNCT
cana-3462	28	10	𝑢	𝑢	PROPN
cana-3462	28	11	∈	∈	PROPN
cana-3462	28	12	𝑉(𝐺)	𝑉(𝐺)	NOUN
cana-3462	28	13	…	…	PUNCT
cana-3462	28	14	…	…	PUNCT
cana-3462	28	15	…	…	X
cana-3462	28	16	(1.1	(1.1	X
cana-3462	28	17	)	)	PUNCT
cana-3462	28	18	also	also	ADV
cana-3462	28	19	,	,	PUNCT
cana-3462	28	20	equation	equation	NOUN
cana-3462	28	21	(	(	PUNCT
cana-3462	28	22	1.1	1.1	NUM
cana-3462	28	23	)	)	PUNCT
cana-3462	28	24	can	can	AUX
cana-3462	28	25	be	be	AUX
cana-3462	28	26	represented	represent	VERB
cana-3462	28	27	as	as	SCONJ
cana-3462	28	28	follows	follow	VERB
cana-3462	28	29	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	28	30	)	)	PUNCT
cana-3462	29	1	=	=	X
cana-3462	29	2	∑	∑	PART
cana-3462	29	3	  	  	SPACE
cana-3462	29	4	𝑝	𝑝	PROPN
cana-3462	29	5	𝑖=1	𝑖=1	PROPN
cana-3462	29	6	 	 	SPACE
cana-3462	29	7	𝑑𝐺	𝑑𝐺	NOUN
cana-3462	29	8	2(𝑣𝑖	2(𝑣𝑖	NUM
cana-3462	29	9	)	)	PUNCT
cana-3462	30	1	+	+	ADJ
cana-3462	30	2	∑	∑	PROPN
cana-3462	30	3	  	  	SPACE
cana-3462	30	4	𝑝	𝑝	PROPN
cana-3462	30	5	𝑖=1	𝑖=1	PROPN
cana-3462	30	6	 	 	SPACE
cana-3462	30	7	𝑑𝐺(𝑣𝑖	𝑑𝐺(𝑣𝑖	PROPN
cana-3462	30	8	)	)	PUNCT
cana-3462	30	9	=	=	SYM
cana-3462	30	10	𝑀1(𝐺	𝑀1(𝐺	NOUN
cana-3462	30	11	)	)	PUNCT
cana-3462	30	12	+	+	NUM
cana-3462	30	13	2𝑞	2𝑞	NUM
cana-3462	30	14	…	…	PUNCT
cana-3462	30	15	…	…	PUNCT
cana-3462	30	16	…	…	PUNCT
cana-3462	30	17	(	(	PUNCT
cana-3462	30	18	1.2	1.2	NUM
cana-3462	30	19	)	)	PUNCT
cana-3462	30	20	where	where	SCONJ
cana-3462	30	21	,	,	PUNCT
cana-3462	30	22	𝑀1	𝑀1	PROPN
cana-3462	30	23	is	be	AUX
cana-3462	30	24	called	call	VERB
cana-3462	30	25	first	first	ADJ
cana-3462	30	26	zagreb	zagreb	PROPN
cana-3462	30	27	index	index	NOUN
cana-3462	30	28	[	[	X
cana-3462	30	29	10	10	NUM
cana-3462	30	30	]	]	PUNCT
cana-3462	30	31	and	and	CCONJ
cana-3462	30	32	second	second	ADJ
cana-3462	30	33	part	part	NOUN
cana-3462	30	34	of	of	ADP
cana-3462	30	35	equation	equation	NOUN
cana-3462	30	36	(	(	PUNCT
cana-3462	30	37	1.2	1.2	NUM
cana-3462	30	38	)	)	PUNCT
cana-3462	30	39	is	be	AUX
cana-3462	30	40	the	the	PRON
cana-3462	30	41	handshaking	handshake	VERB
cana-3462	30	42	lemma	lemma	PROPN
cana-3462	30	43	∑𝑖=1	∑𝑖=1	PROPN
cana-3462	30	44	𝑝	𝑝	PROPN
cana-3462	30	45	 	 	SPACE
cana-3462	30	46	𝑑𝐺(𝑣𝑖	𝑑𝐺(𝑣𝑖	PROPN
cana-3462	30	47	)	)	PUNCT
cana-3462	30	48	=	=	VERB
cana-3462	31	1	2𝑞.	2𝑞.	NUM
cana-3462	32	1	it	it	PRON
cana-3462	32	2	is	be	AUX
cana-3462	32	3	noticed	notice	VERB
cana-3462	32	4	that	that	SCONJ
cana-3462	32	5	the	the	DET
cana-3462	32	6	quasi	quasi	ADJ
cana-3462	32	7	-	-	ADJ
cana-3462	32	8	laplacian	laplacian	ADJ
cana-3462	32	9	energy	energy	NOUN
cana-3462	32	10	of	of	ADP
cana-3462	32	11	any	any	DET
cana-3462	32	12	graph	graph	NOUN
cana-3462	32	13	depends	depend	VERB
cana-3462	32	14	upon	upon	SCONJ
cana-3462	32	15	the	the	DET
cana-3462	32	16	degrees	degree	NOUN
cana-3462	32	17	of	of	ADP
cana-3462	32	18	vertices	vertex	NOUN
cana-3462	32	19	of	of	ADP
cana-3462	32	20	that	that	DET
cana-3462	32	21	graph	graph	NOUN
cana-3462	32	22	.	.	PUNCT
cana-3462	33	1	based	base	VERB
cana-3462	33	2	on	on	ADP
cana-3462	33	3	the	the	DET
cana-3462	33	4	above	above	ADJ
cana-3462	33	5	result	result	NOUN
cana-3462	33	6	discussed	discuss	VERB
cana-3462	33	7	in	in	ADP
cana-3462	33	8	the	the	DET
cana-3462	33	9	paper	paper	NOUN
cana-3462	34	1	[	[	X
cana-3462	34	2	6	6	NUM
cana-3462	34	3	]	]	PUNCT
cana-3462	34	4	.	.	PUNCT
cana-3462	35	1	we	we	PRON
cana-3462	35	2	formulate	formulate	VERB
cana-3462	35	3	the	the	DET
cana-3462	35	4	relations	relation	NOUN
cana-3462	35	5	of	of	ADP
cana-3462	35	6	the	the	DET
cana-3462	35	7	quasilaplacian	quasilaplacian	ADJ
cana-3462	35	8	energy	energy	NOUN
cana-3462	35	9	of	of	ADP
cana-3462	35	10	some	some	DET
cana-3462	35	11	classes	class	NOUN
cana-3462	35	12	of	of	ADP
cana-3462	35	13	novel	novel	ADJ
cana-3462	35	14	graphs	graph	NOUN
cana-3462	35	15	in	in	ADP
cana-3462	35	16	terms	term	NOUN
cana-3462	35	17	of	of	ADP
cana-3462	35	18	corresponding	correspond	VERB
cana-3462	35	19	original	original	ADJ
cana-3462	35	20	graphs	graph	NOUN
cana-3462	35	21	.	.	PUNCT
cana-3462	36	1	the	the	DET
cana-3462	36	2	subdivision	subdivision	NOUN
cana-3462	36	3	graph	graph	NOUN
cana-3462	36	4	𝒮(𝐺	𝒮(𝐺	NOUN
cana-3462	36	5	)	)	PUNCT
cana-3462	36	6	,	,	PUNCT
cana-3462	36	7	ℛ-graph	ℛ-graph	PROPN
cana-3462	36	8	ℛ(𝐺	ℛ(𝐺	NUM
cana-3462	36	9	)	)	PUNCT
cana-3462	36	10	,	,	PUNCT
cana-3462	36	11	𝒬	𝒬	PROPN
cana-3462	36	12	graph	graph	NOUN
cana-3462	36	13	𝒬(𝐺	𝒬(𝐺	ADV
cana-3462	36	14	)	)	PUNCT
cana-3462	36	15	,	,	PUNCT
cana-3462	36	16	total	total	ADJ
cana-3462	36	17	graph	graph	NOUN
cana-3462	36	18	𝒯(𝐺	𝒯(𝐺	PROPN
cana-3462	36	19	)	)	PUNCT
cana-3462	37	1	[	[	X
cana-3462	37	2	3	3	X
cana-3462	37	3	]	]	PUNCT
cana-3462	37	4	and	and	CCONJ
cana-3462	37	5	their	their	PRON
cana-3462	37	6	join	join	NOUN
cana-3462	37	7	and	and	CCONJ
cana-3462	37	8	corona	corona	NOUN
cana-3462	37	9	operations	operation	NOUN
cana-3462	37	10	graphs	graph	NOUN
cana-3462	37	11	are	be	AUX
cana-3462	37	12	the	the	DET
cana-3462	37	13	novel	novel	ADJ
cana-3462	37	14	graphs	graph	NOUN
cana-3462	37	15	for	for	ADP
cana-3462	37	16	our	our	PRON
cana-3462	37	17	discussion	discussion	NOUN
cana-3462	37	18	.	.	PUNCT
cana-3462	38	1	indulal	indulal	PROPN
cana-3462	39	1	[	[	X
cana-3462	39	2	8	8	NUM
cana-3462	39	3	]	]	PUNCT
cana-3462	39	4	first	first	ADV
cana-3462	39	5	defined	define	VERB
cana-3462	39	6	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	39	7	and	and	CCONJ
cana-3462	39	8	𝒮	𝒮	PROPN
cana-3462	39	9	edge	edge	NOUN
cana-3462	39	10	join	join	NOUN
cana-3462	39	11	and	and	CCONJ
cana-3462	39	12	liu	liu	PROPN
cana-3462	39	13	and	and	CCONJ
cana-3462	39	14	zhang	zhang	PROPN
cana-3462	40	1	[	[	X
cana-3462	40	2	16	16	NUM
cana-3462	40	3	]	]	PUNCT
cana-3462	40	4	determined	determine	VERB
cana-3462	40	5	their	their	PRON
cana-3462	40	6	spectra	spectra	NOUN
cana-3462	40	7	.	.	PUNCT
cana-3462	41	1	lu	lu	PROPN
cana-3462	41	2	and	and	CCONJ
cana-3462	41	3	miao	miao	NOUN
cana-3462	42	1	[	[	X
cana-3462	42	2	13	13	NUM
cana-3462	42	3	]	]	SYM
cana-3462	42	4	defined	define	VERB
cana-3462	42	5	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	42	6	corona	corona	NOUN
cana-3462	42	7	and	and	CCONJ
cana-3462	42	8	edge	edge	NOUN
cana-3462	42	9	corona	corona	PROPN
cana-3462	42	10	.	.	PUNCT
cana-3462	43	1	liu	liu	PROPN
cana-3462	43	2	and	and	CCONJ
cana-3462	43	3	lu	lu	PROPN
cana-3462	44	1	[	[	X
cana-3462	44	2	14	14	NUM
cana-3462	44	3	]	]	SYM
cana-3462	44	4	defined	define	VERB
cana-3462	44	5	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	44	6	and	and	CCONJ
cana-3462	44	7	edge	edge	VERB
cana-3462	44	8	neighbourhood	neighbourhood	NOUN
cana-3462	44	9	corona	corona	NOUN
cana-3462	44	10	.	.	PUNCT
cana-3462	45	1	sun	sun	PROPN
cana-3462	45	2	,	,	PUNCT
cana-3462	45	3	shang	shang	PROPN
cana-3462	45	4	,	,	PUNCT
cana-3462	45	5	and	and	CCONJ
cana-3462	45	6	bu	bu	VERB
cana-3462	46	1	[	[	X
cana-3462	46	2	16	16	NUM
cana-3462	46	3	]	]	SYM
cana-3462	46	4	defined	define	VERB
cana-3462	46	5	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	46	6	and	and	CCONJ
cana-3462	46	7	edge	edge	VERB
cana-3462	46	8	join	join	NOUN
cana-3462	46	9	graphs	graph	NOUN
cana-3462	46	10	.	.	PUNCT
cana-3462	47	1	the	the	DET
cana-3462	47	2	𝒬vertex	𝒬vertex	PROPN
cana-3462	47	3	and	and	CCONJ
cana-3462	47	4	edge	edge	NOUN
cana-3462	47	5	corona	corona	NOUN
cana-3462	47	6	are	be	AUX
cana-3462	47	7	defined	define	VERB
cana-3462	47	8	by	by	ADP
cana-3462	47	9	najiya	najiya	NOUN
cana-3462	47	10	and	and	CCONJ
cana-3462	47	11	chithra	chithra	NOUN
cana-3462	48	1	[	[	X
cana-3462	48	2	2	2	NUM
cana-3462	48	3	]	]	PUNCT
cana-3462	48	4	.	.	PUNCT
cana-3462	49	1	the	the	DET
cana-3462	49	2	definition	definition	NOUN
cana-3462	49	3	of	of	ADP
cana-3462	49	4	the	the	DET
cana-3462	49	5	corona	corona	NOUN
cana-3462	49	6	of	of	ADP
cana-3462	49	7	the	the	DET
cana-3462	49	8	total	total	ADJ
cana-3462	49	9	graph	graph	NOUN
cana-3462	49	10	of	of	ADP
cana-3462	49	11	one	one	NUM
cana-3462	49	12	regular	regular	ADJ
cana-3462	49	13	and	and	CCONJ
cana-3462	49	14	another	another	DET
cana-3462	49	15	arbitrary	arbitrary	ADJ
cana-3462	49	16	graph	graph	NOUN
cana-3462	49	17	,	,	PUNCT
cana-3462	49	18	and	and	CCONJ
cana-3462	49	19	spectra	spectra	NOUN
cana-3462	49	20	of	of	ADP
cana-3462	49	21	the	the	DET
cana-3462	49	22	corona	corona	NOUN
cana-3462	49	23	of	of	ADP
cana-3462	49	24	the	the	DET
cana-3462	49	25	total	total	ADJ
cana-3462	49	26	graph	graph	NOUN
cana-3462	49	27	is	be	AUX
cana-3462	49	28	determined	determine	VERB
cana-3462	49	29	by	by	ADP
cana-3462	49	30	zhu	zhu	PROPN
cana-3462	49	31	,	,	PUNCT
cana-3462	49	32	tian	tian	ADJ
cana-3462	49	33	,	,	PUNCT
cana-3462	49	34	and	and	CCONJ
cana-3462	49	35	cui	cui	NOUN
cana-3462	49	36	[	[	X
cana-3462	49	37	19	19	NUM
cana-3462	49	38	]	]	PUNCT
cana-3462	49	39	.	.	PUNCT
cana-3462	50	1	also	also	ADV
cana-3462	50	2	,	,	PUNCT
cana-3462	50	3	𝒮	𝒮	PROPN
cana-3462	50	4	vertexℛ	vertexℛ	PROPN
cana-3462	50	5	vertex	vertex	NOUN
cana-3462	50	6	join	join	NOUN
cana-3462	50	7	,	,	PUNCT
cana-3462	50	8	𝒮	𝒮	PROPN
cana-3462	50	9	edgeℛ	edgeℛ	PROPN
cana-3462	50	10	edge	edge	NOUN
cana-3462	50	11	join	join	NOUN
cana-3462	50	12	,	,	PUNCT
cana-3462	50	13	𝒮	𝒮	PROPN
cana-3462	50	14	vertexℛ	vertexℛ	PROPN
cana-3462	50	15	edge	edge	NOUN
cana-3462	50	16	join	join	NOUN
cana-3462	50	17	and	and	CCONJ
cana-3462	50	18	𝒮	𝒮	PROPN
cana-3462	50	19	edgeℛ	edgeℛ	PROPN
cana-3462	50	20	vertex	vertex	NOUN
cana-3462	50	21	join	join	NOUN
cana-3462	50	22	are	be	AUX
cana-3462	50	23	defined	define	VERB
cana-3462	50	24	by	by	ADP
cana-3462	50	25	das	das	PROPN
cana-3462	50	26	and	and	CCONJ
cana-3462	50	27	panigrahi	panigrahi	NOUN
cana-3462	50	28	[	[	X
cana-3462	50	29	5	5	NUM
cana-3462	50	30	,	,	PUNCT
cana-3462	50	31	7	7	NUM
cana-3462	50	32	]	]	PUNCT
cana-3462	50	33	.	.	PUNCT
cana-3462	51	1	the	the	DET
cana-3462	51	2	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	51	3	-	-	PUNCT
cana-3462	51	4	vertex	vertex	NOUN
cana-3462	51	5	-	-	PUNCT
cana-3462	51	6	edge	edge	NOUN
cana-3462	51	7	join	join	NOUN
cana-3462	51	8	of	of	ADP
cana-3462	51	9	three	three	NUM
cana-3462	51	10	𝒮	𝒮	NOUN
cana-3462	51	11	graphs	graph	NOUN
cana-3462	51	12	is	be	AUX
cana-3462	51	13	defined	define	VERB
cana-3462	51	14	by	by	ADP
cana-3462	51	15	wen	wen	PROPN
cana-3462	51	16	,	,	PUNCT
cana-3462	51	17	zhang	zhang	PROPN
cana-3462	51	18	,	,	PUNCT
cana-3462	51	19	and	and	CCONJ
cana-3462	51	20	li	li	X
cana-3462	52	1	[	[	X
cana-3462	52	2	17	17	NUM
cana-3462	52	3	]	]	PUNCT
cana-3462	52	4	.	.	PUNCT
cana-3462	53	1	berberler	berberler	PROPN
cana-3462	54	1	[	[	X
cana-3462	54	2	1	1	X
cana-3462	54	3	]	]	PUNCT
cana-3462	54	4	derived	derive	VERB
cana-3462	54	5	the	the	DET
cana-3462	54	6	quasi	quasi	ADJ
cana-3462	54	7	-	-	ADJ
cana-3462	54	8	laplacian	laplacian	ADJ
cana-3462	54	9	energy	energy	NOUN
cana-3462	54	10	of	of	ADP
cana-3462	54	11	graphs	graph	NOUN
cana-3462	54	12	based	base	VERB
cana-3462	54	13	on	on	ADP
cana-3462	54	14	ℛ	ℛ	PROPN
cana-3462	54	15	graphs	graph	NOUN
cana-3462	54	16	[	[	X
cana-3462	54	17	13	13	NUM
cana-3462	54	18	]	]	PUNCT
cana-3462	54	19	.	.	PUNCT
cana-3462	55	1	we	we	PRON
cana-3462	55	2	derive	derive	VERB
cana-3462	55	3	formulas	formula	NOUN
cana-3462	55	4	of	of	ADP
cana-3462	55	5	the	the	DET
cana-3462	55	6	quasi	quasi	ADJ
cana-3462	55	7	-	-	ADJ
cana-3462	55	8	laplacian	laplacian	ADJ
cana-3462	55	9	energy	energy	NOUN
cana-3462	55	10	of	of	ADP
cana-3462	55	11	𝒮,𝒬	𝒮,𝒬	ADJ
cana-3462	55	12	and	and	CCONJ
cana-3462	55	13	𝒯	𝒯	PROPN
cana-3462	55	14	graphs	graph	NOUN
cana-3462	55	15	and	and	CCONJ
cana-3462	55	16	further	far	ADV
cana-3462	55	17	,	,	PUNCT
cana-3462	55	18	obtain	obtain	VERB
cana-3462	55	19	quasi	quasi	ADJ
cana-3462	55	20	-	-	ADJ
cana-3462	55	21	laplacian	laplacian	ADJ
cana-3462	55	22	energy	energy	NOUN
cana-3462	55	23	of	of	ADP
cana-3462	55	24	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	55	25	and	and	CCONJ
cana-3462	55	26	edge	edge	NOUN
cana-3462	55	27	join	join	NOUN
cana-3462	55	28	,	,	PUNCT
cana-3462	55	29	𝒮-vertex	𝒮-vertex	NOUN
cana-3462	55	30	and	and	CCONJ
cana-3462	55	31	edge	edge	NOUN
cana-3462	55	32	corona	corona	NOUN
cana-3462	55	33	,	,	PUNCT
cana-3462	55	34	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	55	35	and	and	CCONJ
cana-3462	55	36	edge	edge	VERB
cana-3462	55	37	neighbourhood	neighbourhood	NOUN
cana-3462	55	38	corona	corona	NOUN
cana-3462	55	39	in	in	ADP
cana-3462	55	40	terms	term	NOUN
cana-3462	55	41	of	of	ADP
cana-3462	55	42	their	their	PRON
cana-3462	55	43	corresponding	corresponding	ADJ
cana-3462	55	44	original	original	ADJ
cana-3462	55	45	graphs	graph	NOUN
cana-3462	55	46	.	.	PUNCT
cana-3462	56	1	similarly	similarly	ADV
cana-3462	56	2	,	,	PUNCT
cana-3462	56	3	we	we	PRON
cana-3462	56	4	derive	derive	VERB
cana-3462	56	5	quasi	quasi	ADJ
cana-3462	56	6	-	-	ADJ
cana-3462	56	7	laplacian	laplacian	ADJ
cana-3462	56	8	energy	energy	NOUN
cana-3462	56	9	of	of	ADP
cana-3462	56	10	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	56	11	and	and	CCONJ
cana-3462	56	12	edge	edge	NOUN
cana-3462	56	13	join	join	NOUN
cana-3462	56	14	,	,	PUNCT
cana-3462	56	15	𝒬-vertex	𝒬-vertex	NOUN
cana-3462	56	16	and	and	CCONJ
cana-3462	56	17	edge	edge	NOUN
cana-3462	56	18	corona	corona	NOUN
cana-3462	56	19	and	and	CCONJ
cana-3462	56	20	𝒯graph	𝒯graph	PROPN
cana-3462	56	21	corona	corona	NOUN
cana-3462	56	22	.	.	PUNCT
cana-3462	57	1	the	the	DET
cana-3462	57	2	quasi	quasi	ADJ
cana-3462	57	3	-	-	ADJ
cana-3462	57	4	laplacian	laplacian	ADJ
cana-3462	57	5	energy	energy	NOUN
cana-3462	57	6	of	of	ADP
cana-3462	57	7	𝒮	𝒮	PROPN
cana-3462	57	8	vertexℛ	vertexℛ	PROPN
cana-3462	57	9	vertex	vertex	NOUN
cana-3462	57	10	join	join	NOUN
cana-3462	57	11	,	,	PUNCT
cana-3462	57	12	𝒮edgeℛ	𝒮edgeℛ	AUX
cana-3462	57	13	edge	edge	NOUN
cana-3462	57	14	join	join	NOUN
cana-3462	57	15	,	,	PUNCT
cana-3462	57	16	𝒮	𝒮	PROPN
cana-3462	57	17	vertexℛ	vertexℛ	PROPN
cana-3462	57	18	edge	edge	NOUN
cana-3462	57	19	join	join	NOUN
cana-3462	57	20	,	,	PUNCT
cana-3462	57	21	𝒮	𝒮	PROPN
cana-3462	57	22	edgeℛ	edgeℛ	PROPN
cana-3462	57	23	vertex	vertex	NOUN
cana-3462	57	24	join	join	NOUN
cana-3462	57	25	and	and	CCONJ
cana-3462	57	26	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	57	27	-	-	PUNCT
cana-3462	57	28	vertex	vertex	NOUN
cana-3462	57	29	-	-	PUNCT
cana-3462	57	30	edge	edge	NOUN
cana-3462	57	31	of	of	ADP
cana-3462	57	32	three	three	NUM
cana-3462	57	33	graphs	graph	NOUN
cana-3462	57	34	join	join	VERB
cana-3462	57	35	are	be	AUX
cana-3462	57	36	also	also	ADV
cana-3462	57	37	derived	derive	VERB
cana-3462	57	38	in	in	ADP
cana-3462	57	39	terms	term	NOUN
cana-3462	57	40	of	of	ADP
cana-3462	57	41	their	their	PRON
cana-3462	57	42	corresponding	corresponding	ADJ
cana-3462	57	43	original	original	ADJ
cana-3462	57	44	graphs	graph	NOUN
cana-3462	57	45	.	.	PUNCT
cana-3462	58	1	communications	communication	NOUN
cana-3462	58	2	on	on	ADP
cana-3462	58	3	applied	apply	VERB
cana-3462	58	4	nonlinear	nonlinear	ADJ
cana-3462	58	5	analysis	analysis	NOUN
cana-3462	58	6	issn	issn	NOUN
cana-3462	58	7	:	:	PUNCT
cana-3462	58	8	1074	1074	NUM
cana-3462	58	9	-	-	PUNCT
cana-3462	58	10	133x	133x	NUM
cana-3462	58	11	vol	vol	NOUN
cana-3462	58	12	32	32	NUM
cana-3462	58	13	no	no	NOUN
cana-3462	58	14	.	.	PUNCT
cana-3462	59	1	7s	7	NOUN
cana-3462	59	2	(	(	PUNCT
cana-3462	59	3	2025	2025	NUM
cana-3462	59	4	)	)	PUNCT
cana-3462	59	5	532	532	NUM
cana-3462	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	59	7	let	let	VERB
cana-3462	59	8	𝐺	𝐺	PROPN
cana-3462	59	9	be	be	AUX
cana-3462	59	10	the	the	DET
cana-3462	59	11	(	(	PUNCT
cana-3462	59	12	𝑝	𝑝	PROPN
cana-3462	59	13	,	,	PUNCT
cana-3462	59	14	𝑞	𝑞	NOUN
cana-3462	59	15	)	)	PUNCT
cana-3462	59	16	graph	graph	NOUN
cana-3462	59	17	where	where	SCONJ
cana-3462	59	18	sets	set	NOUN
cana-3462	59	19	of	of	ADP
cana-3462	59	20	old	old	ADJ
cana-3462	59	21	vertices	vertex	NOUN
cana-3462	59	22	|𝑉(𝐺)|	|𝑉(𝐺)|	X
cana-3462	59	23	=	=	SYM
cana-3462	59	24	𝑝	𝑝	NOUN
cana-3462	59	25	and	and	CCONJ
cana-3462	59	26	sets	set	NOUN
cana-3462	59	27	of	of	ADP
cana-3462	59	28	inserted	insert	VERB
cana-3462	59	29	new	new	ADJ
cana-3462	59	30	vertices	vertex	NOUN
cana-3462	59	31	𝐼(𝐺	𝐼(𝐺	NOUN
cana-3462	59	32	)	)	PUNCT
cana-3462	59	33	∣=	∣=	PROPN
cana-3462	59	34	𝑞.	𝑞.	NOUN
cana-3462	59	35	let	let	VERB
cana-3462	59	36	𝐻1	𝐻1	NOUN
cana-3462	59	37	be	be	AUX
cana-3462	59	38	the	the	DET
cana-3462	59	39	(	(	PUNCT
cana-3462	59	40	𝑝1	𝑝1	NOUN
cana-3462	59	41	,	,	PUNCT
cana-3462	59	42	𝑞1	𝑞1	PROPN
cana-3462	59	43	)	)	PUNCT
cana-3462	59	44	graph	graph	NOUN
cana-3462	59	45	and	and	CCONJ
cana-3462	59	46	𝐻2	𝐻2	VERB
cana-3462	59	47	be	be	AUX
cana-3462	59	48	the	the	DET
cana-3462	59	49	(	(	PUNCT
cana-3462	59	50	𝑝2	𝑝2	NOUN
cana-3462	59	51	,	,	PUNCT
cana-3462	59	52	𝑞2	𝑞2	NOUN
cana-3462	59	53	)	)	PUNCT
cana-3462	59	54	graph	graph	NOUN
cana-3462	59	55	.	.	PUNCT
cana-3462	60	1	consequently	consequently	ADV
cana-3462	60	2	,	,	PUNCT
cana-3462	60	3	|𝑉(𝐻1)|	|𝑉(𝐻1)|	PROPN
cana-3462	60	4	=	=	PROPN
cana-3462	60	5	𝑝1	𝑝1	NOUN
cana-3462	60	6	,	,	PUNCT
cana-3462	60	7	|𝐼(𝐻1)|	|𝐼(𝐻1)|	NOUN
cana-3462	60	8	=	=	SYM
cana-3462	60	9	𝑞1	𝑞1	X
cana-3462	60	10	,	,	PUNCT
cana-3462	60	11	|𝑉(𝐻2)|	|𝑉(𝐻2)|	NOUN
cana-3462	60	12	=	=	SYM
cana-3462	60	13	𝑝2	𝑝2	NOUN
cana-3462	60	14	and	and	CCONJ
cana-3462	60	15	|𝐼(𝐻2)|	|𝐼(𝐻2)|	PROPN
cana-3462	60	16	=	=	PUNCT
cana-3462	60	17	𝑞2	𝑞2	NOUN
cana-3462	60	18	.	.	PUNCT
cana-3462	61	1	2	2	X
cana-3462	61	2	.	.	X
cana-3462	61	3	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	61	4	based	base	VERB
cana-3462	61	5	on	on	ADP
cana-3462	61	6	𝒮	𝒮	PROPN
cana-3462	61	7	graphs	graph	VERB
cana-3462	61	8	the	the	DET
cana-3462	61	9	quasi	quasi	ADJ
cana-3462	61	10	-	-	ADJ
cana-3462	61	11	laplacian	laplacian	ADJ
cana-3462	61	12	energy	energy	NOUN
cana-3462	61	13	of	of	ADP
cana-3462	61	14	the	the	DET
cana-3462	61	15	𝒮	𝒮	PROPN
cana-3462	61	16	graph	graph	NOUN
cana-3462	61	17	,	,	PUNCT
cana-3462	61	18	𝒮	𝒮	NOUN
cana-3462	61	19	join	join	VERB
cana-3462	61	20	and	and	CCONJ
cana-3462	61	21	𝒮	𝒮	PROPN
cana-3462	61	22	corona	corona	NOUN
cana-3462	61	23	graphs	graph	NOUN
cana-3462	61	24	are	be	AUX
cana-3462	61	25	formulated	formulate	VERB
cana-3462	61	26	in	in	ADP
cana-3462	61	27	this	this	DET
cana-3462	61	28	section	section	NOUN
cana-3462	61	29	based	base	VERB
cana-3462	61	30	on	on	ADP
cana-3462	61	31	the	the	DET
cana-3462	61	32	degree	degree	NOUN
cana-3462	61	33	of	of	ADP
cana-3462	61	34	vertices	vertex	NOUN
cana-3462	61	35	of	of	ADP
cana-3462	61	36	the	the	DET
cana-3462	61	37	𝒮-graph	𝒮-graph	PROPN
cana-3462	61	38	,	,	PUNCT
cana-3462	61	39	𝒮-join	𝒮-join	PROPN
cana-3462	61	40	,	,	PUNCT
cana-3462	61	41	and	and	CCONJ
cana-3462	61	42	𝒮-corona	𝒮-corona	PROPN
cana-3462	61	43	graphs	graph	NOUN
cana-3462	61	44	respectively	respectively	ADV
cana-3462	61	45	.	.	PUNCT
cana-3462	62	1	let	let	VERB
cana-3462	62	2	𝑢	𝑢	PRON
cana-3462	62	3	be	be	AUX
cana-3462	62	4	any	any	DET
cana-3462	62	5	vertex	vertex	NOUN
cana-3462	62	6	in	in	ADP
cana-3462	62	7	𝒮(𝐺	𝒮(𝐺	NOUN
cana-3462	62	8	)	)	PUNCT
cana-3462	62	9	,	,	PUNCT
cana-3462	62	10	then	then	ADV
cana-3462	62	11	the	the	DET
cana-3462	62	12	degree	degree	NOUN
cana-3462	62	13	of	of	ADP
cana-3462	62	14	𝒮(𝐺	𝒮(𝐺	NOUN
cana-3462	62	15	)	)	PUNCT
cana-3462	62	16	is	be	AUX
cana-3462	62	17	represented	represent	VERB
cana-3462	62	18	by	by	ADP
cana-3462	62	19	𝑑𝒮(𝐺)(𝑢𝑖	𝑑𝒮(𝐺)(𝑢𝑖	NOUN
cana-3462	62	20	)	)	PUNCT
cana-3462	62	21	=	=	SYM
cana-3462	62	22	{	{	PUNCT
cana-3462	62	23	𝑑𝐺(𝑢𝑖	𝑑𝐺(𝑢𝑖	PROPN
cana-3462	62	24	)	)	PUNCT
cana-3462	62	25	,	,	PUNCT
cana-3462	62	26	if	if	SCONJ
cana-3462	62	27	𝑢	𝑢	PROPN
cana-3462	62	28	∈	∈	PROPN
cana-3462	62	29	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3462	62	30	)	)	PUNCT
cana-3462	62	31	;	;	PUNCT
cana-3462	63	1	2	2	X
cana-3462	63	2	,	,	PUNCT
cana-3462	63	3	if	if	SCONJ
cana-3462	63	4	𝑢	𝑢	PROPN
cana-3462	63	5	∈	∈	PROPN
cana-3462	63	6	𝐼(𝐺	𝐼(𝐺	NOUN
cana-3462	63	7	)	)	PUNCT
cana-3462	63	8	.	.	PUNCT
cana-3462	63	9	theorem	theorem	VERB
cana-3462	63	10	2.1	2.1	NUM
cana-3462	63	11	.	.	PUNCT
cana-3462	63	12	𝐸𝑄(𝒮(𝐺	𝐸𝑄(𝒮(𝐺	NOUN
cana-3462	63	13	)	)	PUNCT
cana-3462	63	14	)	)	PUNCT
cana-3462	63	15	=	=	SYM
cana-3462	63	16	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	63	17	)	)	PUNCT
cana-3462	64	1	+	+	CCONJ
cana-3462	64	2	6𝑞	6𝑞	ADJ
cana-3462	64	3	proof	proof	NOUN
cana-3462	64	4	.	.	PUNCT
cana-3462	65	1	using	use	VERB
cana-3462	65	2	definition	definition	NOUN
cana-3462	65	3	(	(	PUNCT
cana-3462	65	4	1.1	1.1	NUM
cana-3462	65	5	)	)	PUNCT
cana-3462	65	6	on	on	ADP
cana-3462	65	7	the	the	DET
cana-3462	65	8	𝒮(𝐺)-graph	𝒮(𝐺)-graph	NOUN
cana-3462	65	9	,	,	PUNCT
cana-3462	65	10	we	we	PRON
cana-3462	65	11	have	have	VERB
cana-3462	65	12	𝐸𝑄(𝒮(𝐺))=	𝐸𝑄(𝒮(𝐺))=	PROPN
cana-3462	65	13	∑	∑	PUNCT
cana-3462	65	14	  	  	SPACE
cana-3462	65	15	|𝑉(𝒮(𝐺))|	|𝑉(𝒮(𝐺))|	X
cana-3462	66	1	𝑖=1	𝑖=1	PUNCT
cana-3462	66	2	 	 	SPACE
cana-3462	66	3	𝑑𝒮(𝐺	𝑑𝒮(𝐺	NOUN
cana-3462	66	4	)	)	PUNCT
cana-3462	66	5	2	2	NUM
cana-3462	66	6	(	(	PUNCT
cana-3462	66	7	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	66	8	)	)	PUNCT
cana-3462	66	9	+	+	CCONJ
cana-3462	66	10	∑	∑	PUNCT
cana-3462	66	11	  	  	SPACE
cana-3462	66	12	|𝑉(𝒮(𝐺))|	|𝑉(𝒮(𝐺))|	X
cana-3462	66	13	𝑖=1	𝑖=1	PUNCT
cana-3462	66	14	 	 	SPACE
cana-3462	66	15	𝑑𝒮(𝐺)(𝑢𝑖	𝑑𝒮(𝐺)(𝑢𝑖	NOUN
cana-3462	66	16	)	)	PUNCT
cana-3462	66	17	=	=	NOUN
cana-3462	66	18	∑	∑	PART
cana-3462	66	19	  	  	SPACE
cana-3462	66	20	𝑝	𝑝	PROPN
cana-3462	66	21	𝑖=1	𝑖=1	PROPN
cana-3462	66	22	 	 	SPACE
cana-3462	66	23	𝑑(𝐺	𝑑(𝐺	PROPN
cana-3462	66	24	)	)	PUNCT
cana-3462	66	25	2	2	NUM
cana-3462	66	26	(	(	PUNCT
cana-3462	66	27	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	66	28	)	)	PUNCT
cana-3462	67	1	+	+	ADJ
cana-3462	67	2	∑	∑	PROPN
cana-3462	67	3	  	  	SPACE
cana-3462	67	4	𝑞	𝑞	X
cana-3462	67	5	𝑖=1	𝑖=1	PROPN
cana-3462	67	6	 	 	SPACE
cana-3462	67	7	22	22	NUM
cana-3462	68	1	+	+	ADJ
cana-3462	68	2	∑	∑	PROPN
cana-3462	68	3	  	  	SPACE
cana-3462	68	4	𝑝	𝑝	ADP
cana-3462	68	5	𝑖=1	𝑖=1	PROPN
cana-3462	68	6	 	 	SPACE
cana-3462	68	7	𝑑(𝐺)(𝑢𝑖	𝑑(𝐺)(𝑢𝑖	NOUN
cana-3462	68	8	)	)	PUNCT
cana-3462	69	1	+	+	ADJ
cana-3462	69	2	∑	∑	PROPN
cana-3462	69	3	  	  	SPACE
cana-3462	69	4	𝑞	𝑞	PROPN
cana-3462	69	5	𝑖=1	𝑖=1	PROPN
cana-3462	69	6	 	 	SPACE
cana-3462	69	7	2	2	NUM
cana-3462	69	8	=	=	SYM
cana-3462	69	9	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	69	10	)	)	PUNCT
cana-3462	70	1	+	+	CCONJ
cana-3462	70	2	6𝑞	6𝑞	NOUN
cana-3462	70	3	now	now	ADV
cana-3462	70	4	,	,	PUNCT
cana-3462	70	5	let	let	VERB
cana-3462	70	6	𝑢	𝑢	PRON
cana-3462	70	7	be	be	AUX
cana-3462	70	8	any	any	DET
cana-3462	70	9	vertex	vertex	NOUN
cana-3462	70	10	in	in	ADP
cana-3462	70	11	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	70	12	join	join	NOUN
cana-3462	70	13	(	(	PUNCT
cana-3462	70	14	𝐻1	𝐻1	PROPN
cana-3462	70	15	∨̇	∨̇	PROPN
cana-3462	70	16	𝐻2	𝐻2	PROPN
cana-3462	70	17	)	)	PUNCT
cana-3462	70	18	of	of	ADP
cana-3462	70	19	two	two	NUM
cana-3462	70	20	graphs	graph	NOUN
cana-3462	70	21	𝐻1	𝐻1	NOUN
cana-3462	70	22	and	and	CCONJ
cana-3462	70	23	𝐻2	𝐻2	PROPN
cana-3462	70	24	.	.	PUNCT
cana-3462	71	1	then	then	ADV
cana-3462	71	2	,	,	PUNCT
cana-3462	71	3	the	the	DET
cana-3462	71	4	degree	degree	NOUN
cana-3462	71	5	of	of	ADP
cana-3462	71	6	the	the	DET
cana-3462	71	7	vertex	vertex	NOUN
cana-3462	71	8	𝑢	𝑢	PROPN
cana-3462	71	9	in	in	ADP
cana-3462	71	10	(	(	PUNCT
cana-3462	71	11	𝐻1	𝐻1	NOUN
cana-3462	71	12	∨̇	∨̇	PROPN
cana-3462	71	13	𝐻2	𝐻2	PROPN
cana-3462	71	14	)	)	PUNCT
cana-3462	71	15	is	be	AUX
cana-3462	71	16	given	give	VERB
cana-3462	71	17	by	by	ADP
cana-3462	71	18	𝑑𝐻1v̇𝐻2(𝑢𝑖	𝑑𝐻1v̇𝐻2(𝑢𝑖	NOUN
cana-3462	71	19	)	)	PUNCT
cana-3462	71	20	=	=	SYM
cana-3462	71	21	{	{	PUNCT
cana-3462	71	22	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	NOUN
cana-3462	71	23	)	)	PUNCT
cana-3462	72	1	+	+	NUM
cana-3462	72	2	𝑝2	𝑝2	NOUN
cana-3462	72	3	,	,	PUNCT
cana-3462	72	4	if	if	SCONJ
cana-3462	72	5	𝑢	𝑢	PRON
cana-3462	72	6	∈	∈	PROPN
cana-3462	72	7	𝑉(𝐻1	𝑉(𝐻1	NOUN
cana-3462	72	8	)	)	PUNCT
cana-3462	72	9	,	,	PUNCT
cana-3462	72	10	for	for	ADP
cana-3462	72	11	i	i	PROPN
cana-3462	72	12	=	=	SYM
cana-3462	72	13	1,2	1,2	NUM
cana-3462	72	14	,	,	PUNCT
cana-3462	72	15	…	…	PUNCT
cana-3462	72	16	p1	p1	NOUN
cana-3462	72	17	;	;	PUNCT
cana-3462	72	18	2	2	NUM
cana-3462	72	19	,	,	PUNCT
cana-3462	72	20	if	if	SCONJ
cana-3462	72	21	𝑣	𝑣	PRON
cana-3462	72	22	∈	∈	PROPN
cana-3462	72	23	𝐼(𝐺1	𝐼(𝐺1	NOUN
cana-3462	72	24	)	)	PUNCT
cana-3462	72	25	;	;	PUNCT
cana-3462	72	26	𝑑𝐻2(𝑢𝑗	𝑑𝐻2(𝑢𝑗	NOUN
cana-3462	72	27	)	)	PUNCT
cana-3462	73	1	+	+	CCONJ
cana-3462	73	2	𝑝1	𝑝1	NOUN
cana-3462	73	3	,	,	PUNCT
cana-3462	73	4	if	if	SCONJ
cana-3462	73	5	𝑣	𝑣	PRON
cana-3462	73	6	∈	∈	PROPN
cana-3462	73	7	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	73	8	)	)	PUNCT
cana-3462	73	9	,	,	PUNCT
cana-3462	73	10	for	for	ADP
cana-3462	73	11	j	j	PROPN
cana-3462	73	12	=	=	SYM
cana-3462	73	13	1,2	1,2	NUM
cana-3462	73	14	,	,	PUNCT
cana-3462	73	15	…	…	PUNCT
cana-3462	73	16	p2	p2	X
cana-3462	73	17	.	.	PUNCT
cana-3462	74	1	and	and	CCONJ
cana-3462	74	2	𝐻1	𝐻1	NOUN
cana-3462	74	3	∨̇	∨̇	PROPN
cana-3462	74	4	𝐻2	𝐻2	PROPN
cana-3462	74	5	has	have	VERB
cana-3462	74	6	𝑝1	𝑝1	NOUN
cana-3462	74	7	+	+	CCONJ
cana-3462	74	8	𝑞1	𝑞1	PROPN
cana-3462	74	9	+	+	CCONJ
cana-3462	74	10	𝑝2	𝑝2	NOUN
cana-3462	74	11	vertices	vertex	NOUN
cana-3462	74	12	.	.	PUNCT
cana-3462	75	1	we	we	PRON
cana-3462	75	2	derive	derive	VERB
cana-3462	75	3	quasi	quasi	ADJ
cana-3462	75	4	-	-	ADJ
cana-3462	75	5	laplacian	laplacian	ADJ
cana-3462	75	6	energy	energy	NOUN
cana-3462	75	7	of	of	ADP
cana-3462	75	8	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	75	9	join	join	NOUN
cana-3462	75	10	in	in	ADP
cana-3462	75	11	terms	term	NOUN
cana-3462	75	12	of	of	ADP
cana-3462	75	13	corresponding	correspond	VERB
cana-3462	75	14	original	original	ADJ
cana-3462	75	15	graphs	graph	NOUN
cana-3462	75	16	𝐻1	𝐻1	NOUN
cana-3462	75	17	and	and	CCONJ
cana-3462	75	18	𝐻2	𝐻2	PROPN
cana-3462	75	19	.	.	PUNCT
cana-3462	76	1	theorem	theorem	VERB
cana-3462	76	2	2.2	2.2	NUM
cana-3462	76	3	.	.	PUNCT
cana-3462	77	1	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	PROPN
cana-3462	77	2	∨̇	∨̇	PROPN
cana-3462	77	3	𝐻2	𝐻2	PROPN
cana-3462	77	4	)	)	PUNCT
cana-3462	77	5	=	=	PUNCT
cana-3462	77	6	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	77	7	)	)	PUNCT
cana-3462	78	1	+	+	CCONJ
cana-3462	78	2	𝐸𝑄(𝐻2	𝐸𝑄(𝐻2	NOUN
cana-3462	78	3	)	)	PUNCT
cana-3462	79	1	+	+	CCONJ
cana-3462	79	2	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	80	1	+	+	CCONJ
cana-3462	80	2	𝑝1𝑝2	𝑝1𝑝2	PROPN
cana-3462	80	3	2	2	NUM
cana-3462	80	4	+	+	NUM
cana-3462	80	5	6𝑞1	6𝑞1	NUM
cana-3462	80	6	+	+	SYM
cana-3462	80	7	4𝑝1𝑞2	4𝑝1𝑞2	NUM
cana-3462	81	1	+	+	CCONJ
cana-3462	81	2	𝑝1	𝑝1	NOUN
cana-3462	81	3	2𝑝2	2𝑝2	NUM
cana-3462	81	4	+	+	CCONJ
cana-3462	81	5	2𝑝1𝑝2	2𝑝1𝑝2	NUM
cana-3462	81	6	proof	proof	NOUN
cana-3462	81	7	.	.	PUNCT
cana-3462	82	1	using	use	VERB
cana-3462	82	2	definition	definition	NOUN
cana-3462	82	3	(	(	PUNCT
cana-3462	82	4	1.1	1.1	NUM
cana-3462	82	5	)	)	PUNCT
cana-3462	82	6	,	,	PUNCT
cana-3462	82	7	quasi	quasi	ADJ
cana-3462	82	8	-	-	ADJ
cana-3462	82	9	laplacian	laplacian	ADJ
cana-3462	82	10	energy	energy	NOUN
cana-3462	82	11	of	of	ADP
cana-3462	82	12	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	82	13	join	join	NOUN
cana-3462	82	14	is	be	AUX
cana-3462	82	15	obtained	obtain	VERB
cana-3462	82	16	by	by	ADP
cana-3462	82	17	communications	communication	NOUN
cana-3462	82	18	on	on	ADP
cana-3462	82	19	applied	apply	VERB
cana-3462	82	20	nonlinear	nonlinear	ADJ
cana-3462	82	21	analysis	analysis	NOUN
cana-3462	82	22	issn	issn	NOUN
cana-3462	82	23	:	:	PUNCT
cana-3462	82	24	1074	1074	NUM
cana-3462	82	25	-	-	PUNCT
cana-3462	82	26	133x	133x	NUM
cana-3462	82	27	vol	vol	NOUN
cana-3462	82	28	32	32	NUM
cana-3462	82	29	no	no	NOUN
cana-3462	82	30	.	.	PUNCT
cana-3462	83	1	7s	7	NOUN
cana-3462	83	2	(	(	PUNCT
cana-3462	83	3	2025	2025	NUM
cana-3462	83	4	)	)	PUNCT
cana-3462	83	5	533	533	NUM
cana-3462	83	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	83	7	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	PROPN
cana-3462	83	8	∨̇	∨̇	PROPN
cana-3462	83	9	𝐻2	𝐻2	PROPN
cana-3462	83	10	)	)	PUNCT
cana-3462	83	11	=	=	PUNCT
cana-3462	83	12	∑	∑	PUNCT
cana-3462	83	13	  	  	SPACE
cana-3462	83	14	𝑝1+𝑞1+𝑝2	𝑝1+𝑞1+𝑝2	NOUN
cana-3462	83	15	𝑖=1	𝑖=1	PUNCT
cana-3462	83	16	 	 	SPACE
cana-3462	83	17	𝑑𝐻1∨̇𝐺2	𝑑𝐻1∨̇𝐺2	PROPN
cana-3462	83	18	2	2	NUM
cana-3462	83	19	(	(	PUNCT
cana-3462	83	20	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	83	21	)	)	PUNCT
cana-3462	83	22	+	+	CCONJ
cana-3462	83	23	∑	∑	PART
cana-3462	83	24	  	  	SPACE
cana-3462	83	25	𝑝1+𝑞1+𝑝2	𝑝1+𝑞1+𝑝2	NOUN
cana-3462	83	26	𝑖=1	𝑖=1	PROPN
cana-3462	83	27	 	 	SPACE
cana-3462	83	28	𝑑𝐻1∨̇𝐻2(𝑢𝑖	𝑑𝐻1∨̇𝐻2(𝑢𝑖	X
cana-3462	83	29	)	)	PUNCT
cana-3462	84	1	=	=	NOUN
cana-3462	84	2	∑	∑	PART
cana-3462	84	3	  	  	SPACE
cana-3462	84	4	𝑝1	𝑝1	PROPN
cana-3462	84	5	𝑖=1	𝑖=1	PROPN
cana-3462	84	6	 	 	SPACE
cana-3462	84	7	𝑑(𝐻1	𝑑(𝐻1	PROPN
cana-3462	84	8	)	)	PUNCT
cana-3462	84	9	2	2	NUM
cana-3462	84	10	(	(	PUNCT
cana-3462	84	11	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	84	12	)	)	PUNCT
cana-3462	84	13	+	+	NUM
cana-3462	84	14	2𝑝2∑	2𝑝2∑	NUM
cana-3462	84	15	  	  	SPACE
cana-3462	84	16	𝑝1	𝑝1	PROPN
cana-3462	84	17	𝑖=1	𝑖=1	PROPN
cana-3462	84	18	 	 	SPACE
cana-3462	84	19	𝑑(𝐻1)(𝑢𝑖	𝑑(𝐻1)(𝑢𝑖	PROPN
cana-3462	84	20	)	)	PUNCT
cana-3462	85	1	+	+	NUM
cana-3462	85	2	𝑝2	𝑝2	NOUN
cana-3462	85	3	2∑	2∑	NUM
cana-3462	85	4	  	  	SPACE
cana-3462	85	5	𝑝1	𝑝1	NOUN
cana-3462	85	6	𝑖=1	𝑖=1	PROPN
cana-3462	85	7	 	 	SPACE
cana-3462	86	1	+	+	NUM
cana-3462	86	2	4𝑞1	4𝑞1	NUM
cana-3462	86	3	+	+	ADJ
cana-3462	86	4	∑	∑	NOUN
cana-3462	86	5	  	  	SPACE
cana-3462	86	6	𝑝2	𝑝2	NOUN
cana-3462	86	7	𝑗=1	𝑗=1	PROPN
cana-3462	86	8	 	 	SPACE
cana-3462	86	9	𝑑(𝐻2	𝑑(𝐻2	PROPN
cana-3462	86	10	)	)	PUNCT
cana-3462	86	11	2	2	NUM
cana-3462	86	12	(	(	PUNCT
cana-3462	86	13	𝑢𝑗	𝑢𝑗	NOUN
cana-3462	86	14	)	)	PUNCT
cana-3462	86	15	+	+	NUM
cana-3462	86	16	2𝑝1∑	2𝑝1∑	NUM
cana-3462	86	17	  	  	SPACE
cana-3462	86	18	𝑝2	𝑝2	NOUN
cana-3462	86	19	𝑗=1	𝑗=1	PROPN
cana-3462	86	20	 	 	SPACE
cana-3462	86	21	𝑑(𝐻2)(𝑢𝑗	𝑑(𝐻2)(𝑢𝑗	NOUN
cana-3462	86	22	)	)	PUNCT
cana-3462	87	1	+	+	CCONJ
cana-3462	87	2	𝑝1	𝑝1	NOUN
cana-3462	87	3	2∑	2∑	PROPN
cana-3462	87	4	  	  	SPACE
cana-3462	87	5	𝑝2	𝑝2	NOUN
cana-3462	87	6	𝑗=1	𝑗=1	PROPN
cana-3462	87	7	  	  	SPACE
cana-3462	87	8	+	+	PROPN
cana-3462	87	9	∑	∑	ADJ
cana-3462	87	10	  	  	SPACE
cana-3462	87	11	𝑝1	𝑝1	PROPN
cana-3462	87	12	𝑖=1	𝑖=1	PROPN
cana-3462	87	13	 	 	SPACE
cana-3462	87	14	𝑑(𝐻1)(𝑢𝑖	𝑑(𝐻1)(𝑢𝑖	PROPN
cana-3462	87	15	)	)	PUNCT
cana-3462	88	1	+	+	CCONJ
cana-3462	88	2	𝑝2∑	𝑝2∑	PROPN
cana-3462	88	3	  	  	SPACE
cana-3462	88	4	𝑝1	𝑝1	NOUN
cana-3462	88	5	𝑖=1	𝑖=1	PROPN
cana-3462	88	6	 	 	SPACE
cana-3462	89	1	+	+	CCONJ
cana-3462	89	2	2𝑞1	2𝑞1	NUM
cana-3462	89	3	+	+	ADJ
cana-3462	89	4	∑	∑	ADJ
cana-3462	89	5	  	  	SPACE
cana-3462	89	6	𝑝2	𝑝2	NOUN
cana-3462	89	7	𝑗=1	𝑗=1	PROPN
cana-3462	89	8	 	 	SPACE
cana-3462	89	9	𝑑(𝐻2)(𝑢𝑗	𝑑(𝐻2)(𝑢𝑗	NOUN
cana-3462	89	10	)	)	PUNCT
cana-3462	90	1	+	+	CCONJ
cana-3462	90	2	𝑝1∑	𝑝1∑	PROPN
cana-3462	90	3	  	  	SPACE
cana-3462	90	4	𝑝2	𝑝2	NOUN
cana-3462	90	5	𝑗=1	𝑗=1	PROPN
cana-3462	90	6	  	  	SPACE
cana-3462	90	7	=	=	PUNCT
cana-3462	90	8	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	PROPN
cana-3462	90	9	)	)	PUNCT
cana-3462	91	1	+	+	CCONJ
cana-3462	91	2	𝐸𝑄(𝐻2	𝐸𝑄(𝐻2	NOUN
cana-3462	91	3	)	)	PUNCT
cana-3462	92	1	+	+	CCONJ
cana-3462	92	2	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	93	1	+	+	CCONJ
cana-3462	93	2	𝑝1𝑝2	𝑝1𝑝2	PROPN
cana-3462	93	3	2	2	NUM
cana-3462	93	4	+	+	NUM
cana-3462	93	5	6𝑞1	6𝑞1	NUM
cana-3462	93	6	+	+	SYM
cana-3462	93	7	4𝑝1𝑞2	4𝑝1𝑞2	NUM
cana-3462	94	1	+	+	CCONJ
cana-3462	94	2	𝑝1	𝑝1	NOUN
cana-3462	94	3	2𝑝2	2𝑝2	NUM
cana-3462	94	4	+	+	CCONJ
cana-3462	94	5	2𝑝1𝑝2	2𝑝1𝑝2	NOUN
cana-3462	94	6	.	.	PUNCT
cana-3462	95	1	next	next	ADV
cana-3462	95	2	,	,	PUNCT
cana-3462	95	3	we	we	PRON
cana-3462	95	4	formulate	formulate	VERB
cana-3462	95	5	a	a	DET
cana-3462	95	6	relation	relation	NOUN
cana-3462	95	7	of	of	ADP
cana-3462	95	8	quasi	quasi	ADJ
cana-3462	95	9	-	-	ADJ
cana-3462	95	10	laplacian	laplacian	ADJ
cana-3462	95	11	energy	energy	NOUN
cana-3462	95	12	of	of	ADP
cana-3462	95	13	𝐻1	𝐻1	PROPN
cana-3462	95	14	∨	∨	NUM
cana-3462	95	15	𝐻2	𝐻2	PROPN
cana-3462	95	16	join	join	NOUN
cana-3462	95	17	.	.	PUNCT
cana-3462	96	1	let	let	VERB
cana-3462	96	2	𝑢	𝑢	PRON
cana-3462	96	3	be	be	AUX
cana-3462	96	4	any	any	DET
cana-3462	96	5	vertex	vertex	NOUN
cana-3462	96	6	in	in	ADP
cana-3462	96	7	𝒮-edge	𝒮-edge	PROPN
cana-3462	96	8	join	join	NOUN
cana-3462	96	9	.	.	PUNCT
cana-3462	97	1	the	the	DET
cana-3462	97	2	degree	degree	NOUN
cana-3462	97	3	of	of	ADP
cana-3462	97	4	vertex	vertex	NOUN
cana-3462	97	5	𝑢	𝑢	NOUN
cana-3462	97	6	in	in	ADP
cana-3462	97	7	𝐻1	𝐻1	PROPN
cana-3462	97	8	∨	∨	NUM
cana-3462	97	9	𝐻2	𝐻2	PROPN
cana-3462	97	10	join	join	NOUN
cana-3462	97	11	is	be	AUX
cana-3462	97	12	given	give	VERB
cana-3462	97	13	by	by	ADP
cana-3462	97	14	𝑑𝐻1∪𝐻2(𝑢𝑖	𝑑𝐻1∪𝐻2(𝑢𝑖	NOUN
cana-3462	97	15	)	)	PUNCT
cana-3462	97	16	=	=	PRON
cana-3462	97	17	{	{	PUNCT
cana-3462	97	18	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	97	19	)	)	PUNCT
cana-3462	97	20	,	,	PUNCT
cana-3462	97	21	if	if	SCONJ
cana-3462	97	22	𝑢	𝑢	PRON
cana-3462	97	23	∈	∈	PROPN
cana-3462	97	24	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	97	25	)	)	PUNCT
cana-3462	97	26	;	;	PUNCT
cana-3462	97	27	2	2	NUM
cana-3462	97	28	+	+	NUM
cana-3462	97	29	𝑝2	𝑝2	NOUN
cana-3462	97	30	,	,	PUNCT
cana-3462	97	31	if	if	SCONJ
cana-3462	97	32	𝑢	𝑢	PRON
cana-3462	97	33	∈	∈	PROPN
cana-3462	97	34	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	97	35	)	)	PUNCT
cana-3462	97	36	𝑑𝐻2(𝑣𝑖	𝑑𝐻2(𝑣𝑖	NUM
cana-3462	97	37	)	)	PUNCT
cana-3462	98	1	+	+	CCONJ
cana-3462	98	2	𝑞1	𝑞1	ADJ
cana-3462	98	3	,	,	PUNCT
cana-3462	98	4	if	if	SCONJ
cana-3462	98	5	𝑢	𝑢	PRON
cana-3462	98	6	∈	∈	PROPN
cana-3462	98	7	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	98	8	)	)	PUNCT
cana-3462	98	9	.	.	PUNCT
cana-3462	99	1	;	;	PUNCT
cana-3462	99	2	and	and	CCONJ
cana-3462	99	3	𝐻1	𝐻1	PROPN
cana-3462	99	4	∨	∨	NUM
cana-3462	99	5	𝐻2	𝐻2	PROPN
cana-3462	99	6	has	have	VERB
cana-3462	99	7	𝑝1	𝑝1	NOUN
cana-3462	99	8	+	+	CCONJ
cana-3462	99	9	𝑞1	𝑞1	PROPN
cana-3462	99	10	+	+	CCONJ
cana-3462	99	11	𝑝2	𝑝2	NOUN
cana-3462	99	12	vertices	vertex	NOUN
cana-3462	99	13	.	.	PUNCT
cana-3462	100	1	theorem	theorem	VERB
cana-3462	100	2	2.3	2.3	NUM
cana-3462	100	3	.	.	PUNCT
cana-3462	101	1	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	PROPN
cana-3462	101	2	∨	∨	NUM
cana-3462	101	3	𝐻2	𝐻2	NOUN
cana-3462	101	4	)	)	PUNCT
cana-3462	101	5	=	=	PUNCT
cana-3462	101	6	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	101	7	)	)	PUNCT
cana-3462	102	1	+	+	CCONJ
cana-3462	102	2	𝐸𝑄(𝐻2	𝐸𝑄(𝐻2	NOUN
cana-3462	102	3	)	)	PUNCT
cana-3462	103	1	+	+	CCONJ
cana-3462	103	2	6𝑞1𝑝2	6𝑞1𝑝2	NUM
cana-3462	103	3	+	+	CCONJ
cana-3462	103	4	𝑞1𝑝2	𝑞1𝑝2	VERB
cana-3462	103	5	2	2	NUM
cana-3462	103	6	+	+	NUM
cana-3462	103	7	6𝑞1	6𝑞1	NUM
cana-3462	103	8	+	+	CCONJ
cana-3462	103	9	4𝑞1𝑞2	4𝑞1𝑞2	NUM
cana-3462	103	10	+	+	CCONJ
cana-3462	103	11	𝑞1	𝑞1	ADJ
cana-3462	103	12	2𝑝2	2𝑝2	NUM
cana-3462	103	13	proof	proof	NOUN
cana-3462	103	14	.	.	PUNCT
cana-3462	104	1	using	use	VERB
cana-3462	104	2	equation	equation	NOUN
cana-3462	104	3	(	(	PUNCT
cana-3462	104	4	1.1	1.1	NUM
cana-3462	104	5	)	)	PUNCT
cana-3462	104	6	,	,	PUNCT
cana-3462	104	7	we	we	PRON
cana-3462	104	8	get	get	VERB
cana-3462	104	9	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	ADJ
cana-3462	104	10	∨	∨	NUM
cana-3462	104	11	𝐻2	𝐻2	NOUN
cana-3462	104	12	)	)	PUNCT
cana-3462	105	1	=	=	PUNCT
cana-3462	105	2	∑	∑	PUNCT
cana-3462	105	3	  	  	SPACE
cana-3462	105	4	𝑝1+𝑞1+𝑝2	𝑝1+𝑞1+𝑝2	NOUN
cana-3462	105	5	𝑖=1	𝑖=1	PUNCT
cana-3462	105	6	 	 	SPACE
cana-3462	105	7	𝑑𝐻1⊵𝐻2	𝑑𝐻1⊵𝐻2	PROPN
cana-3462	105	8	2	2	NUM
cana-3462	105	9	(	(	PUNCT
cana-3462	105	10	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	105	11	)	)	PUNCT
cana-3462	105	12	+	+	CCONJ
cana-3462	105	13	∑	∑	PART
cana-3462	105	14	  	  	SPACE
cana-3462	105	15	𝑝1+𝑞1+𝑝2	𝑝1+𝑞1+𝑝2	PROPN
cana-3462	105	16	𝑖=1	𝑖=1	PROPN
cana-3462	105	17	 	 	SPACE
cana-3462	105	18	𝑑𝐻1∨𝐻2(𝑢𝑖	𝑑𝐻1∨𝐻2(𝑢𝑖	NOUN
cana-3462	105	19	)	)	PUNCT
cana-3462	106	1	=	=	X
cana-3462	106	2	∑	∑	PART
cana-3462	106	3	  	  	SPACE
cana-3462	106	4	𝑝1	𝑝1	PROPN
cana-3462	106	5	𝑖=1	𝑖=1	PROPN
cana-3462	106	6	 	 	SPACE
cana-3462	106	7	𝑑𝐻1	𝑑𝐻1	PROPN
cana-3462	106	8	2	2	NUM
cana-3462	106	9	(	(	PUNCT
cana-3462	106	10	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	106	11	)	)	PUNCT
cana-3462	106	12	+	+	ADJ
cana-3462	106	13	∑	∑	PROPN
cana-3462	106	14	  	  	SPACE
cana-3462	106	15	𝑞1	𝑞1	PROPN
cana-3462	106	16	𝑖=1	𝑖=1	PUNCT
cana-3462	106	17	  	  	SPACE
cana-3462	106	18	(	(	PUNCT
cana-3462	106	19	2	2	NUM
cana-3462	106	20	+	+	SYM
cana-3462	106	21	𝑝2	𝑝2	NOUN
cana-3462	106	22	)	)	PUNCT
cana-3462	106	23	2	2	NUM
cana-3462	106	24	+	+	ADJ
cana-3462	106	25	∑	∑	NOUN
cana-3462	106	26	  	  	SPACE
cana-3462	106	27	𝑝2	𝑝2	NOUN
cana-3462	106	28	𝑖=1	𝑖=1	PROPN
cana-3462	106	29	  	  	SPACE
cana-3462	106	30	(	(	PUNCT
cana-3462	106	31	𝑑(𝐻2)(𝑢𝑖	𝑑(𝐻2)(𝑢𝑖	PROPN
cana-3462	106	32	)	)	PUNCT
cana-3462	106	33	+	+	CCONJ
cana-3462	106	34	𝑞1	𝑞1	NUM
cana-3462	106	35	)	)	PUNCT
cana-3462	106	36	2	2	NUM
cana-3462	107	1	+	+	ADJ
cana-3462	107	2	∑	∑	NOUN
cana-3462	107	3	  	  	SPACE
cana-3462	107	4	𝑝1	𝑝1	NOUN
cana-3462	107	5	𝑖=1	𝑖=1	PROPN
cana-3462	107	6	 	 	SPACE
cana-3462	107	7	𝑑(𝐻1)(𝑢𝑖	𝑑(𝐻1)(𝑢𝑖	PROPN
cana-3462	107	8	)	)	PUNCT
cana-3462	108	1	+	+	ADJ
cana-3462	108	2	∑	∑	PROPN
cana-3462	108	3	  	  	SPACE
cana-3462	108	4	𝑞1	𝑞1	PROPN
cana-3462	108	5	𝑖=1	𝑖=1	PUNCT
cana-3462	108	6	 	 	SPACE
cana-3462	108	7	2	2	NUM
cana-3462	108	8	+	+	ADJ
cana-3462	108	9	∑	∑	ADJ
cana-3462	108	10	  	  	SPACE
cana-3462	108	11	𝑝2	𝑝2	NOUN
cana-3462	108	12	𝑖=1	𝑖=1	PROPN
cana-3462	108	13	 	 	SPACE
cana-3462	108	14	𝑑(𝐻2)(𝑢𝑖	𝑑(𝐻2)(𝑢𝑖	PROPN
cana-3462	108	15	)	)	PUNCT
cana-3462	108	16	+	+	CCONJ
cana-3462	108	17	𝑞1	𝑞1	ADJ
cana-3462	108	18	the	the	DET
cana-3462	108	19	result	result	NOUN
cana-3462	108	20	can	can	AUX
cana-3462	108	21	be	be	AUX
cana-3462	108	22	derived	derive	VERB
cana-3462	108	23	easily	easily	ADV
cana-3462	108	24	.	.	PUNCT
cana-3462	109	1	2.1	2.1	NUM
cana-3462	109	2	.	.	PUNCT
cana-3462	110	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	110	2	of	of	ADP
cana-3462	110	3	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	110	4	and	and	CCONJ
cana-3462	110	5	edge	edge	NOUN
cana-3462	110	6	corona	corona	NOUN
cana-3462	110	7	let	let	VERB
cana-3462	110	8	𝑢	𝑢	PRON
cana-3462	110	9	be	be	AUX
cana-3462	110	10	any	any	DET
cana-3462	110	11	vertex	vertex	NOUN
cana-3462	110	12	in	in	ADP
cana-3462	110	13	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	110	14	corona	corona	NOUN
cana-3462	110	15	.	.	PUNCT
cana-3462	111	1	the	the	DET
cana-3462	111	2	degree	degree	NOUN
cana-3462	111	3	of	of	ADP
cana-3462	111	4	vertex	vertex	NOUN
cana-3462	111	5	of	of	ADP
cana-3462	111	6	𝑢	𝑢	NOUN
cana-3462	111	7	in	in	ADP
cana-3462	111	8	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	111	9	corona	corona	NOUN
cana-3462	111	10	is	be	AUX
cana-3462	111	11	given	give	VERB
cana-3462	111	12	by	by	ADP
cana-3462	111	13	𝑑𝐻1⊙𝐻2(𝑢𝑖	𝑑𝐻1⊙𝐻2(𝑢𝑖	NOUN
cana-3462	111	14	)	)	PUNCT
cana-3462	111	15	=	=	PRON
cana-3462	111	16	{	{	PUNCT
cana-3462	111	17	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	NOUN
cana-3462	111	18	)	)	PUNCT
cana-3462	112	1	+	+	NUM
cana-3462	112	2	𝑝2	𝑝2	NOUN
cana-3462	112	3	,	,	PUNCT
cana-3462	112	4	if	if	SCONJ
cana-3462	112	5	𝑢	𝑢	PRON
cana-3462	112	6	∈	∈	PROPN
cana-3462	112	7	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	112	8	)	)	PUNCT
cana-3462	112	9	;	;	PUNCT
cana-3462	112	10	2	2	X
cana-3462	112	11	,	,	PUNCT
cana-3462	112	12	if	if	SCONJ
cana-3462	112	13	𝑢	𝑢	PRON
cana-3462	112	14	∈	∈	PROPN
cana-3462	112	15	𝐼(𝐻1	𝐼(𝐻1	PROPN
cana-3462	112	16	)	)	PUNCT
cana-3462	112	17	;	;	PUNCT
cana-3462	112	18	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PUNCT
cana-3462	112	19	𝑖	𝑖	X
cana-3462	112	20	)	)	PUNCT
cana-3462	112	21	+	+	CCONJ
cana-3462	112	22	1	1	NUM
cana-3462	112	23	,	,	PUNCT
cana-3462	112	24	if	if	SCONJ
cana-3462	112	25	𝑢	𝑢	NOUN
cana-3462	112	26	=	=	PUNCT
cana-3462	112	27	𝑣𝑗	𝑣𝑗	ADP
cana-3462	112	28	𝑖	𝑖	X
cana-3462	112	29	,	,	PUNCT
cana-3462	112	30	for	for	ADP
cana-3462	112	31	𝑖	𝑖	PRON
cana-3462	112	32	=	=	SYM
cana-3462	112	33	1,2	1,2	NUM
cana-3462	112	34	,	,	PUNCT
cana-3462	112	35	…	…	PUNCT
cana-3462	112	36	𝑝1	𝑝1	NOUN
cana-3462	112	37	,	,	PUNCT
cana-3462	112	38	for	for	ADP
cana-3462	112	39	𝑗	𝑗	NOUN
cana-3462	112	40	=	=	SYM
cana-3462	112	41	1,2	1,2	NUM
cana-3462	112	42	,	,	PUNCT
cana-3462	112	43	…	…	PUNCT
cana-3462	112	44	𝑝2	𝑝2	NOUN
cana-3462	112	45	.	.	PUNCT
cana-3462	113	1	and	and	CCONJ
cana-3462	113	2	𝐻1⊙𝐻2	𝐻1⊙𝐻2	NOUN
cana-3462	113	3	has	have	VERB
cana-3462	113	4	𝑝1	𝑝1	NOUN
cana-3462	113	5	+	+	CCONJ
cana-3462	113	6	𝑞1	𝑞1	PROPN
cana-3462	113	7	+	+	CCONJ
cana-3462	113	8	𝑝1𝑝2	𝑝1𝑝2	PRON
cana-3462	113	9	vertices	vertice	VERB
cana-3462	113	10	.	.	PUNCT
cana-3462	114	1	the	the	DET
cana-3462	114	2	formula	formula	NOUN
cana-3462	114	3	of	of	ADP
cana-3462	114	4	quasi	quasi	ADJ
cana-3462	114	5	-	-	ADJ
cana-3462	114	6	laplacian	laplacian	ADJ
cana-3462	114	7	energy	energy	NOUN
cana-3462	114	8	of	of	ADP
cana-3462	114	9	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	114	10	corona	corona	NOUN
cana-3462	114	11	is	be	AUX
cana-3462	114	12	given	give	VERB
cana-3462	114	13	below	below	ADP
cana-3462	114	14	communications	communication	NOUN
cana-3462	114	15	on	on	ADP
cana-3462	114	16	applied	apply	VERB
cana-3462	114	17	nonlinear	nonlinear	ADJ
cana-3462	114	18	analysis	analysis	NOUN
cana-3462	114	19	issn	issn	NOUN
cana-3462	114	20	:	:	PUNCT
cana-3462	114	21	1074	1074	NUM
cana-3462	114	22	-	-	PUNCT
cana-3462	114	23	133x	133x	NUM
cana-3462	114	24	vol	vol	NOUN
cana-3462	114	25	32	32	NUM
cana-3462	114	26	no	no	NOUN
cana-3462	114	27	.	.	PUNCT
cana-3462	115	1	7s	7	NOUN
cana-3462	115	2	(	(	PUNCT
cana-3462	115	3	2025	2025	NUM
cana-3462	115	4	)	)	PUNCT
cana-3462	115	5	534	534	NUM
cana-3462	115	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	115	7	theorem2.4	theorem2.4	NOUN
cana-3462	115	8	.	.	PUNCT
cana-3462	116	1	𝐸𝑄(𝐻1⊙𝐻2	𝐸𝑄(𝐻1⊙𝐻2	X
cana-3462	116	2	)	)	PUNCT
cana-3462	117	1	=	=	SYM
cana-3462	117	2	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	117	3	)	)	PUNCT
cana-3462	117	4	+	+	CCONJ
cana-3462	117	5	𝑝1𝐸𝑄(𝐻2	𝑝1𝐸𝑄(𝐻2	PROPN
cana-3462	117	6	)	)	PUNCT
cana-3462	118	1	+	+	CCONJ
cana-3462	118	2	𝑝1𝑝2	𝑝1𝑝2	PROPN
cana-3462	118	3	2	2	NUM
cana-3462	118	4	+	+	CCONJ
cana-3462	118	5	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	118	6	+	+	CCONJ
cana-3462	118	7	3𝑞1	3𝑞1	NUM
cana-3462	118	8	+	+	CCONJ
cana-3462	118	9	3𝑝1𝑝2	3𝑝1𝑝2	NUM
cana-3462	118	10	+	+	CCONJ
cana-3462	118	11	4𝑝1𝑞2	4𝑝1𝑞2	NUM
cana-3462	118	12	proof	proof	NOUN
cana-3462	118	13	.	.	PUNCT
cana-3462	119	1	by	by	ADP
cana-3462	119	2	using	use	VERB
cana-3462	119	3	definition	definition	NOUN
cana-3462	119	4	(	(	PUNCT
cana-3462	119	5	1.1	1.1	NUM
cana-3462	119	6	)	)	PUNCT
cana-3462	119	7	,	,	PUNCT
cana-3462	119	8	we	we	PRON
cana-3462	119	9	get	get	VERB
cana-3462	119	10	𝐸𝑄(𝐻1⊙𝐻2	𝐸𝑄(𝐻1⊙𝐻2	NOUN
cana-3462	119	11	)	)	PUNCT
cana-3462	119	12	=	=	PUNCT
cana-3462	120	1	∑	∑	PUNCT
cana-3462	120	2	  	  	SPACE
cana-3462	120	3	𝑝1+𝑞1+𝑝1𝑝2	𝑝1+𝑞1+𝑝1𝑝2	PROPN
cana-3462	120	4	𝑖=1	𝑖=1	PUNCT
cana-3462	120	5	 	 	SPACE
cana-3462	120	6	𝑑𝐻1⊙𝐻2	𝑑𝐻1⊙𝐻2	PROPN
cana-3462	120	7	2	2	NUM
cana-3462	120	8	(	(	PUNCT
cana-3462	120	9	𝑣𝑖	𝑣𝑖	NOUN
cana-3462	120	10	)	)	PUNCT
cana-3462	120	11	+	+	CCONJ
cana-3462	120	12	∑	∑	PUNCT
cana-3462	120	13	  	  	SPACE
cana-3462	120	14	𝑝1+𝑞1+𝑝1𝑝2	𝑝1+𝑞1+𝑝1𝑝2	PROPN
cana-3462	120	15	𝑖=1	𝑖=1	PROPN
cana-3462	120	16	 	 	SPACE
cana-3462	120	17	𝑑𝐻1⊙𝐻2(𝑢𝑖	𝑑𝐻1⊙𝐻2(𝑢𝑖	NOUN
cana-3462	120	18	)	)	PUNCT
cana-3462	120	19	=	=	NOUN
cana-3462	120	20	∑	∑	PART
cana-3462	120	21	  	  	SPACE
cana-3462	120	22	𝑝1	𝑝1	PROPN
cana-3462	120	23	𝑖=1	𝑖=1	PROPN
cana-3462	120	24	 	 	SPACE
cana-3462	120	25	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	120	26	)	)	PUNCT
cana-3462	120	27	+	+	NUM
cana-3462	120	28	𝑝2	𝑝2	NOUN
cana-3462	120	29	)	)	PUNCT
cana-3462	120	30	2	2	NUM
cana-3462	121	1	+	+	ADJ
cana-3462	121	2	∑	∑	NOUN
cana-3462	121	3	  	  	SPACE
cana-3462	121	4	𝑞1	𝑞1	PROPN
cana-3462	121	5	𝑖=1	𝑖=1	PUNCT
cana-3462	121	6	 	 	SPACE
cana-3462	121	7	22	22	NUM
cana-3462	122	1	+	+	ADJ
cana-3462	122	2	∑	∑	PROPN
cana-3462	122	3	  	  	SPACE
cana-3462	122	4	𝑝1	𝑝1	NOUN
cana-3462	122	5	𝑖=1	𝑖=1	PROPN
cana-3462	122	6	 	 	SPACE
cana-3462	122	7	∑	∑	PROPN
cana-3462	122	8	  	  	SPACE
cana-3462	122	9	𝑝2	𝑝2	NOUN
cana-3462	122	10	𝑗=1	𝑗=1	PROPN
cana-3462	122	11	  	  	SPACE
cana-3462	122	12	(	(	PUNCT
cana-3462	122	13	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	NOUN
cana-3462	122	14	𝑖	𝑖	X
cana-3462	122	15	)	)	PUNCT
cana-3462	122	16	+	+	CCONJ
cana-3462	122	17	1	1	X
cana-3462	122	18	)	)	PUNCT
cana-3462	122	19	2	2	NUM
cana-3462	123	1	+	+	ADJ
cana-3462	123	2	∑	∑	NOUN
cana-3462	123	3	  	  	SPACE
cana-3462	123	4	𝑝1	𝑝1	PROPN
cana-3462	123	5	𝑖=1	𝑖=1	PROPN
cana-3462	123	6	 	 	SPACE
cana-3462	123	7	𝑑(𝐻1)(𝑢𝑖	𝑑(𝐻1)(𝑢𝑖	PROPN
cana-3462	123	8	)	)	PUNCT
cana-3462	124	1	+	+	NUM
cana-3462	125	1	𝑝2	𝑝2	NOUN
cana-3462	125	2	+	+	NOUN
cana-3462	125	3	∑	∑	PROPN
cana-3462	125	4	  	  	SPACE
cana-3462	125	5	𝑞1	𝑞1	PROPN
cana-3462	125	6	𝑖=1	𝑖=1	PUNCT
cana-3462	125	7	 	 	SPACE
cana-3462	125	8	2	2	NUM
cana-3462	125	9	+	+	ADJ
cana-3462	125	10	∑	∑	NOUN
cana-3462	125	11	  	  	SPACE
cana-3462	125	12	𝑝1	𝑝1	NOUN
cana-3462	125	13	𝑖=1	𝑖=1	PROPN
cana-3462	125	14	 	 	SPACE
cana-3462	125	15	∑	∑	PROPN
cana-3462	125	16	  	  	SPACE
cana-3462	125	17	𝑝2	𝑝2	NOUN
cana-3462	125	18	𝑗=1	𝑗=1	PROPN
cana-3462	125	19	 	 	SPACE
cana-3462	125	20	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PROPN
cana-3462	125	21	𝑖	𝑖	X
cana-3462	125	22	)	)	PUNCT
cana-3462	125	23	+	+	CCONJ
cana-3462	125	24	1	1	NUM
cana-3462	125	25	=	=	SYM
cana-3462	125	26	𝑝2	𝑝2	NOUN
cana-3462	125	27	2∑	2∑	NOUN
cana-3462	125	28	  	  	SPACE
cana-3462	125	29	𝑝1	𝑝1	NOUN
cana-3462	125	30	𝑖=1	𝑖=1	PROPN
cana-3462	125	31	 	 	SPACE
cana-3462	126	1	+	+	CCONJ
cana-3462	126	2	2𝑝2∑	2𝑝2∑	NUM
cana-3462	126	3	  	  	SPACE
cana-3462	126	4	𝑝1	𝑝1	NOUN
cana-3462	126	5	𝑖=1	𝑖=1	PROPN
cana-3462	126	6	 	 	SPACE
cana-3462	126	7	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	126	8	)	)	PUNCT
cana-3462	127	1	+	+	ADJ
cana-3462	127	2	∑	∑	PROPN
cana-3462	127	3	  	  	SPACE
cana-3462	127	4	𝑝1	𝑝1	PROPN
cana-3462	127	5	𝑖=1	𝑖=1	PROPN
cana-3462	127	6	 	 	SPACE
cana-3462	127	7	𝑑𝐻1	𝑑𝐻1	PROPN
cana-3462	127	8	2	2	NUM
cana-3462	127	9	(	(	PUNCT
cana-3462	127	10	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	127	11	)	)	PUNCT
cana-3462	127	12	+	+	NUM
cana-3462	127	13	4𝑞1	4𝑞1	NUM
cana-3462	128	1	+	+	ADJ
cana-3462	128	2	∑	∑	NOUN
cana-3462	128	3	  	  	SPACE
cana-3462	128	4	𝑝1	𝑝1	PROPN
cana-3462	128	5	𝑖=1	𝑖=1	PROPN
cana-3462	128	6	 	 	SPACE
cana-3462	128	7	∑	∑	PROPN
cana-3462	128	8	  	  	SPACE
cana-3462	128	9	𝑝2	𝑝2	NOUN
cana-3462	128	10	𝑗=1	𝑗=1	PROPN
cana-3462	128	11	  	  	SPACE
cana-3462	129	1	+2∑	+2∑	NUM
cana-3462	129	2	  	  	SPACE
cana-3462	129	3	𝑝1	𝑝1	NOUN
cana-3462	129	4	𝑖=1	𝑖=1	PROPN
cana-3462	129	5	 	 	SPACE
cana-3462	129	6	∑	∑	PROPN
cana-3462	129	7	  	  	SPACE
cana-3462	129	8	𝑝2	𝑝2	NOUN
cana-3462	129	9	𝑗=1	𝑗=1	PROPN
cana-3462	129	10	 	 	SPACE
cana-3462	129	11	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PUNCT
cana-3462	129	12	)	)	PUNCT
cana-3462	130	1	+	+	ADJ
cana-3462	130	2	∑	∑	PROPN
cana-3462	130	3	  	  	SPACE
cana-3462	130	4	𝑝1	𝑝1	PROPN
cana-3462	130	5	𝑖=1	𝑖=1	PROPN
cana-3462	130	6	 	 	SPACE
cana-3462	130	7	∑	∑	PROPN
cana-3462	130	8	  	  	SPACE
cana-3462	130	9	𝑝2	𝑝2	NOUN
cana-3462	130	10	𝑗=1	𝑗=1	PROPN
cana-3462	130	11	 	 	SPACE
cana-3462	130	12	𝑑𝐻2	𝑑𝐻2	PROPN
cana-3462	130	13	2	2	NUM
cana-3462	130	14	(	(	PUNCT
cana-3462	130	15	𝑣𝑗	𝑣𝑗	NOUN
cana-3462	130	16	)	)	PUNCT
cana-3462	130	17	+	+	CCONJ
cana-3462	130	18	𝑝1𝑝2	𝑝1𝑝2	PRON
cana-3462	130	19	+	+	ADJ
cana-3462	130	20	∑	∑	NOUN
cana-3462	130	21	  	  	SPACE
cana-3462	130	22	𝑝1	𝑝1	PROPN
cana-3462	130	23	𝑖=1	𝑖=1	PROPN
cana-3462	130	24	 	 	SPACE
cana-3462	130	25	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	130	26	)	)	PUNCT
cana-3462	131	1	+2𝑞1	+2𝑞1	NUM
cana-3462	131	2	+	+	ADJ
cana-3462	131	3	∑	∑	NOUN
cana-3462	131	4	  	  	SPACE
cana-3462	131	5	𝑝1	𝑝1	NOUN
cana-3462	131	6	𝑖=1	𝑖=1	PROPN
cana-3462	131	7	 	 	SPACE
cana-3462	131	8	∑	∑	PROPN
cana-3462	131	9	  	  	SPACE
cana-3462	131	10	𝑝2	𝑝2	NOUN
cana-3462	131	11	𝑗=1	𝑗=1	PROPN
cana-3462	131	12	  	  	SPACE
cana-3462	131	13	+	+	PROPN
cana-3462	131	14	∑	∑	ADJ
cana-3462	131	15	  	  	SPACE
cana-3462	131	16	𝑝1	𝑝1	NOUN
cana-3462	131	17	𝑖=1	𝑖=1	PROPN
cana-3462	131	18	 	 	SPACE
cana-3462	131	19	∑	∑	PROPN
cana-3462	131	20	  	  	SPACE
cana-3462	131	21	𝑝2	𝑝2	NOUN
cana-3462	131	22	𝑗=1	𝑗=1	PROPN
cana-3462	131	23	 	 	SPACE
cana-3462	131	24	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	NOUN
cana-3462	131	25	)	)	PUNCT
cana-3462	131	26	hence	hence	ADV
cana-3462	131	27	,	,	PUNCT
cana-3462	131	28	the	the	DET
cana-3462	131	29	result	result	NOUN
cana-3462	131	30	follows	follow	VERB
cana-3462	131	31	.	.	PUNCT
cana-3462	132	1	let	let	VERB
cana-3462	132	2	𝑢	𝑢	PRON
cana-3462	132	3	be	be	AUX
cana-3462	132	4	any	any	DET
cana-3462	132	5	vertex	vertex	NOUN
cana-3462	132	6	in	in	ADP
cana-3462	132	7	𝒮-edge	𝒮-edge	PROPN
cana-3462	132	8	corona	corona	NOUN
cana-3462	132	9	,	,	PUNCT
cana-3462	132	10	then	then	ADV
cana-3462	132	11	the	the	DET
cana-3462	132	12	degree	degree	NOUN
cana-3462	132	13	of	of	ADP
cana-3462	132	14	the	the	DET
cana-3462	132	15	vertex	vertex	NOUN
cana-3462	132	16	𝑢	𝑢	X
cana-3462	132	17	is	be	AUX
cana-3462	132	18	given	give	VERB
cana-3462	132	19	by	by	ADP
cana-3462	132	20	𝑑𝐻1⊖𝐻2(𝑢𝑖	𝑑𝐻1⊖𝐻2(𝑢𝑖	PROPN
cana-3462	132	21	)	)	PUNCT
cana-3462	132	22	=	=	PRON
cana-3462	132	23	{	{	PUNCT
cana-3462	132	24	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	132	25	)	)	PUNCT
cana-3462	132	26	,	,	PUNCT
cana-3462	132	27	if	if	SCONJ
cana-3462	132	28	𝑢	𝑢	PRON
cana-3462	132	29	∈	∈	PROPN
cana-3462	132	30	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	132	31	)	)	PUNCT
cana-3462	132	32	;	;	PUNCT
cana-3462	132	33	2	2	NUM
cana-3462	132	34	+	+	NUM
cana-3462	132	35	𝑝2	𝑝2	NOUN
cana-3462	132	36	,	,	PUNCT
cana-3462	132	37	if	if	SCONJ
cana-3462	132	38	𝑢	𝑢	PRON
cana-3462	132	39	∈	∈	PROPN
cana-3462	132	40	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	132	41	)	)	PUNCT
cana-3462	132	42	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PUNCT
cana-3462	132	43	𝑖	𝑖	X
cana-3462	132	44	)	)	PUNCT
cana-3462	132	45	+	+	CCONJ
cana-3462	132	46	1	1	NUM
cana-3462	132	47	,	,	PUNCT
cana-3462	132	48	if	if	SCONJ
cana-3462	132	49	𝑢	𝑢	NOUN
cana-3462	132	50	=	=	PUNCT
cana-3462	132	51	𝑣𝑗	𝑣𝑗	ADP
cana-3462	132	52	𝑖	𝑖	X
cana-3462	132	53	,	,	PUNCT
cana-3462	132	54	for	for	ADP
cana-3462	132	55	𝑖	𝑖	PRON
cana-3462	132	56	=	=	SYM
cana-3462	132	57	1,2	1,2	NUM
cana-3462	132	58	,	,	PUNCT
cana-3462	132	59	…	…	PUNCT
cana-3462	132	60	𝑞1	𝑞1	ADJ
cana-3462	132	61	,	,	PUNCT
cana-3462	132	62	for	for	ADP
cana-3462	132	63	𝑗	𝑗	NOUN
cana-3462	132	64	=	=	SYM
cana-3462	132	65	1,2	1,2	NUM
cana-3462	132	66	,	,	PUNCT
cana-3462	132	67	…	…	PUNCT
cana-3462	132	68	𝑝2	𝑝2	NOUN
cana-3462	132	69	.	.	PUNCT
cana-3462	132	70	;	;	PUNCT
cana-3462	132	71	and	and	CCONJ
cana-3462	132	72	𝐻1θ𝐻2	𝐻1θ𝐻2	NOUN
cana-3462	132	73	has	have	VERB
cana-3462	132	74	𝑝1	𝑝1	NOUN
cana-3462	132	75	+	+	CCONJ
cana-3462	132	76	𝑞1	𝑞1	PROPN
cana-3462	132	77	+	+	CCONJ
cana-3462	132	78	𝑞1𝑝2	𝑞1𝑝2	NOUN
cana-3462	132	79	vertices	vertex	NOUN
cana-3462	132	80	.	.	PUNCT
cana-3462	133	1	the	the	DET
cana-3462	133	2	formula	formula	NOUN
cana-3462	133	3	of	of	ADP
cana-3462	133	4	quasi	quasi	ADJ
cana-3462	133	5	-	-	ADJ
cana-3462	133	6	laplacian	laplacian	ADJ
cana-3462	133	7	energy	energy	NOUN
cana-3462	133	8	of	of	ADP
cana-3462	133	9	𝒮-edge	𝒮-edge	PROPN
cana-3462	133	10	corona	corona	NOUN
cana-3462	133	11	is	be	AUX
cana-3462	133	12	given	give	VERB
cana-3462	133	13	by	by	ADP
cana-3462	133	14	theorem	theorem	ADJ
cana-3462	133	15	2.5	2.5	NUM
cana-3462	133	16	.	.	PUNCT
cana-3462	134	1	𝐸𝑄(𝐻1⊖𝐻2	𝐸𝑄(𝐻1⊖𝐻2	NOUN
cana-3462	134	2	)	)	PUNCT
cana-3462	134	3	=	=	PUNCT
cana-3462	134	4	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	134	5	)	)	PUNCT
cana-3462	134	6	+	+	CCONJ
cana-3462	134	7	𝑞1𝐸𝑄(𝐻2	𝑞1𝐸𝑄(𝐻2	NOUN
cana-3462	134	8	)	)	PUNCT
cana-3462	134	9	+	+	NUM
cana-3462	134	10	6𝑞1	6𝑞1	NUM
cana-3462	134	11	+	+	CCONJ
cana-3462	134	12	7𝑞1𝑝2	7𝑞1𝑝2	NOUN
cana-3462	134	13	+	+	CCONJ
cana-3462	134	14	4𝑞1𝑞2	4𝑞1𝑞2	NUM
cana-3462	134	15	proof	proof	NOUN
cana-3462	134	16	.	.	PUNCT
cana-3462	135	1	using	use	VERB
cana-3462	135	2	definition	definition	NOUN
cana-3462	135	3	(	(	PUNCT
cana-3462	135	4	1.1	1.1	NUM
cana-3462	135	5	)	)	PUNCT
cana-3462	135	6	,	,	PUNCT
cana-3462	135	7	communications	communication	NOUN
cana-3462	135	8	on	on	ADP
cana-3462	135	9	applied	apply	VERB
cana-3462	135	10	nonlinear	nonlinear	ADJ
cana-3462	135	11	analysis	analysis	NOUN
cana-3462	135	12	issn	issn	NOUN
cana-3462	135	13	:	:	PUNCT
cana-3462	135	14	1074	1074	NUM
cana-3462	135	15	-	-	PUNCT
cana-3462	135	16	133x	133x	NUM
cana-3462	135	17	vol	vol	NOUN
cana-3462	135	18	32	32	NUM
cana-3462	135	19	no	no	NOUN
cana-3462	135	20	.	.	PUNCT
cana-3462	136	1	7s	7	NOUN
cana-3462	136	2	(	(	PUNCT
cana-3462	136	3	2025	2025	NUM
cana-3462	136	4	)	)	PUNCT
cana-3462	136	5	535	535	NUM
cana-3462	136	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	136	7	𝐸𝑄(𝐻1⊙𝐻2	𝐸𝑄(𝐻1⊙𝐻2	NOUN
cana-3462	136	8	)	)	PUNCT
cana-3462	136	9	=	=	PUNCT
cana-3462	137	1	∑	∑	PUNCT
cana-3462	137	2	  	  	SPACE
cana-3462	137	3	|𝑉(𝐻1⊖𝐻2)|	|𝑉(𝐻1⊖𝐻2)|	PROPN
cana-3462	137	4	𝑖=1	𝑖=1	PUNCT
cana-3462	137	5	 	 	SPACE
cana-3462	137	6	𝑑𝐻1⊖𝐻2	𝑑𝐻1⊖𝐻2	PROPN
cana-3462	137	7	2	2	NUM
cana-3462	137	8	(	(	PUNCT
cana-3462	137	9	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	137	10	)	)	PUNCT
cana-3462	137	11	+	+	CCONJ
cana-3462	137	12	∑	∑	PUNCT
cana-3462	137	13	  	  	SPACE
cana-3462	137	14	|𝑉(𝐻1⊖𝐻2)|	|𝑉(𝐻1⊖𝐻2)|	PROPN
cana-3462	137	15	𝑖=1	𝑖=1	PUNCT
cana-3462	137	16	 	 	SPACE
cana-3462	137	17	𝑑𝐻1⊖𝐻2(𝑢𝑖	𝑑𝐻1⊖𝐻2(𝑢𝑖	X
cana-3462	137	18	)	)	PUNCT
cana-3462	137	19	=	=	PUNCT
cana-3462	138	1	∑	∑	PUNCT
cana-3462	138	2	  	  	SPACE
cana-3462	138	3	𝑝1+𝑞1+𝑞1𝑝2	𝑝1+𝑞1+𝑞1𝑝2	PROPN
cana-3462	138	4	𝑖=1	𝑖=1	PUNCT
cana-3462	138	5	 	 	SPACE
cana-3462	139	1	𝑑𝐻1⊖𝐻2	𝑑𝐻1⊖𝐻2	PROPN
cana-3462	139	2	2	2	NUM
cana-3462	139	3	(	(	PUNCT
cana-3462	139	4	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	139	5	)	)	PUNCT
cana-3462	139	6	+	+	CCONJ
cana-3462	139	7	∑	∑	PUNCT
cana-3462	139	8	  	  	SPACE
cana-3462	139	9	𝑝1+𝑞1+𝑞1𝑝2	𝑝1+𝑞1+𝑞1𝑝2	PROPN
cana-3462	139	10	𝑖=1	𝑖=1	PUNCT
cana-3462	139	11	 	 	SPACE
cana-3462	140	1	𝑑𝐻1⊖𝐻2	𝑑𝐻1⊖𝐻2	PROPN
cana-3462	140	2	(	(	PUNCT
cana-3462	140	3	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	140	4	)	)	PUNCT
cana-3462	140	5	=	=	NOUN
cana-3462	140	6	∑	∑	NOUN
cana-3462	140	7	  	  	SPACE
cana-3462	140	8	𝑝1	𝑝1	PROPN
cana-3462	140	9	𝑖=1	𝑖=1	PROPN
cana-3462	140	10	 	 	SPACE
cana-3462	140	11	𝑑𝐻1	𝑑𝐻1	PROPN
cana-3462	140	12	2	2	NUM
cana-3462	140	13	(	(	PUNCT
cana-3462	140	14	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	140	15	)	)	PUNCT
cana-3462	140	16	+	+	ADJ
cana-3462	140	17	∑	∑	PROPN
cana-3462	140	18	  	  	SPACE
cana-3462	140	19	𝑞1	𝑞1	PROPN
cana-3462	140	20	𝑖=1	𝑖=1	PUNCT
cana-3462	140	21	  	  	SPACE
cana-3462	140	22	(	(	PUNCT
cana-3462	140	23	2	2	NUM
cana-3462	140	24	+	+	SYM
cana-3462	140	25	𝑝2	𝑝2	NOUN
cana-3462	140	26	)	)	PUNCT
cana-3462	140	27	2	2	NUM
cana-3462	141	1	+	+	ADJ
cana-3462	141	2	∑	∑	PROPN
cana-3462	141	3	  	  	SPACE
cana-3462	141	4	𝑞1	𝑞1	PROPN
cana-3462	141	5	𝑖=1	𝑖=1	PUNCT
cana-3462	141	6	 	 	SPACE
cana-3462	141	7	∑	∑	PROPN
cana-3462	141	8	  	  	SPACE
cana-3462	141	9	𝑝2	𝑝2	NOUN
cana-3462	141	10	𝑗=1	𝑗=1	PROPN
cana-3462	141	11	  	  	SPACE
cana-3462	141	12	(	(	PUNCT
cana-3462	141	13	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	NOUN
cana-3462	141	14	𝑖	𝑖	X
cana-3462	141	15	)	)	PUNCT
cana-3462	141	16	+	+	CCONJ
cana-3462	141	17	1	1	X
cana-3462	141	18	)	)	PUNCT
cana-3462	141	19	2	2	NUM
cana-3462	141	20	+	+	ADJ
cana-3462	141	21	∑	∑	NOUN
cana-3462	141	22	  	  	SPACE
cana-3462	141	23	𝑝1	𝑝1	PROPN
cana-3462	141	24	𝑖=1	𝑖=1	PROPN
cana-3462	141	25	 	 	SPACE
cana-3462	141	26	𝑑(𝐻1)(𝑢𝑖	𝑑(𝐻1)(𝑢𝑖	PROPN
cana-3462	141	27	)	)	PUNCT
cana-3462	142	1	+	+	ADJ
cana-3462	142	2	∑	∑	PROPN
cana-3462	142	3	  	  	SPACE
cana-3462	142	4	𝑞1	𝑞1	PROPN
cana-3462	142	5	𝑖=1	𝑖=1	PUNCT
cana-3462	142	6	 	 	SPACE
cana-3462	142	7	2	2	NUM
cana-3462	142	8	+	+	ADJ
cana-3462	142	9	∑	∑	PROPN
cana-3462	142	10	  	  	SPACE
cana-3462	142	11	𝑞1	𝑞1	PROPN
cana-3462	142	12	𝑖=1	𝑖=1	PUNCT
cana-3462	142	13	 	 	SPACE
cana-3462	142	14	∑	∑	PROPN
cana-3462	142	15	  	  	SPACE
cana-3462	142	16	𝑝2	𝑝2	NOUN
cana-3462	142	17	𝑗=1	𝑗=1	PROPN
cana-3462	142	18	 	 	SPACE
cana-3462	142	19	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PROPN
cana-3462	142	20	𝑖	𝑖	X
cana-3462	142	21	)	)	PUNCT
cana-3462	142	22	+	+	CCONJ
cana-3462	142	23	1	1	NUM
cana-3462	142	24	hence	hence	ADV
cana-3462	142	25	,	,	PUNCT
cana-3462	142	26	the	the	DET
cana-3462	142	27	result	result	NOUN
cana-3462	142	28	follows	follow	VERB
cana-3462	142	29	.	.	PUNCT
cana-3462	143	1	2.2	2.2	NUM
cana-3462	143	2	.	.	PUNCT
cana-3462	144	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	144	2	of	of	ADP
cana-3462	144	3	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	144	4	and	and	CCONJ
cana-3462	144	5	edge	edge	VERB
cana-3462	144	6	neighbourhood	neighbourhood	NOUN
cana-3462	144	7	corona	corona	NOUN
cana-3462	144	8	let	let	VERB
cana-3462	144	9	𝑢	𝑢	PRON
cana-3462	144	10	be	be	AUX
cana-3462	144	11	any	any	DET
cana-3462	144	12	vertex	vertex	NOUN
cana-3462	144	13	in	in	ADP
cana-3462	144	14	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	144	15	neighbourhood	neighbourhood	NOUN
cana-3462	144	16	corona	corona	NOUN
cana-3462	144	17	.	.	PUNCT
cana-3462	145	1	then	then	ADV
cana-3462	145	2	degree	degree	NOUN
cana-3462	145	3	of	of	ADP
cana-3462	145	4	vertices	vertex	NOUN
cana-3462	145	5	is	be	AUX
cana-3462	145	6	given	give	VERB
cana-3462	145	7	by	by	ADP
cana-3462	145	8	𝑑𝐻1⊟𝐻2(𝑣𝑖	𝑑𝐻1⊟𝐻2(𝑣𝑖	VERB
cana-3462	145	9	)	)	PUNCT
cana-3462	145	10	=	=	SYM
cana-3462	145	11	{	{	PUNCT
cana-3462	145	12	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	145	13	)	)	PUNCT
cana-3462	145	14	,	,	PUNCT
cana-3462	145	15	if	if	SCONJ
cana-3462	145	16	𝑢	𝑢	PRON
cana-3462	145	17	∈	∈	PROPN
cana-3462	145	18	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	145	19	)	)	PUNCT
cana-3462	145	20	;	;	PUNCT
cana-3462	145	21	2	2	NUM
cana-3462	145	22	+	+	SYM
cana-3462	145	23	2𝑝2	2𝑝2	NUM
cana-3462	145	24	,	,	PUNCT
cana-3462	145	25	if	if	SCONJ
cana-3462	145	26	𝑢	𝑢	PRON
cana-3462	145	27	∈	∈	PROPN
cana-3462	145	28	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	145	29	)	)	PUNCT
cana-3462	145	30	;	;	PUNCT
cana-3462	145	31	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	145	32	)	)	PUNCT
cana-3462	146	1	+	+	NUM
cana-3462	146	2	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	X
cana-3462	146	3	)	)	PUNCT
cana-3462	146	4	,	,	PUNCT
cana-3462	146	5	if	if	SCONJ
cana-3462	146	6	𝑢	𝑢	NOUN
cana-3462	146	7	=	=	PUNCT
cana-3462	146	8	𝑣𝑗	𝑣𝑗	ADP
cana-3462	146	9	𝑖	𝑖	X
cana-3462	146	10	,	,	PUNCT
cana-3462	146	11	for	for	ADP
cana-3462	146	12	𝑖	𝑖	PRON
cana-3462	146	13	=	=	SYM
cana-3462	146	14	1,2	1,2	NUM
cana-3462	146	15	,	,	PUNCT
cana-3462	146	16	…	…	PUNCT
cana-3462	146	17	𝑝1	𝑝1	NOUN
cana-3462	146	18	,	,	PUNCT
cana-3462	146	19	for	for	ADP
cana-3462	146	20	𝑗	𝑗	NOUN
cana-3462	146	21	=	=	SYM
cana-3462	146	22	1,2	1,2	NUM
cana-3462	146	23	,	,	PUNCT
cana-3462	146	24	…	…	PUNCT
cana-3462	146	25	𝑝2	𝑝2	NOUN
cana-3462	146	26	.	.	PUNCT
cana-3462	147	1	and	and	CCONJ
cana-3462	147	2	𝐻1⊟𝐻2	𝐻1⊟𝐻2	PROPN
cana-3462	147	3	has	have	VERB
cana-3462	147	4	𝑝1	𝑝1	NOUN
cana-3462	147	5	+	+	CCONJ
cana-3462	147	6	𝑞1	𝑞1	PROPN
cana-3462	148	1	+	+	CCONJ
cana-3462	148	2	𝑝1𝑝2	𝑝1𝑝2	PRON
cana-3462	148	3	vertices	vertice	VERB
cana-3462	148	4	.	.	PUNCT
cana-3462	149	1	the	the	DET
cana-3462	149	2	quasi	quasi	ADJ
cana-3462	149	3	-	-	ADJ
cana-3462	149	4	laplacian	laplacian	ADJ
cana-3462	149	5	energy	energy	NOUN
cana-3462	149	6	of	of	ADP
cana-3462	149	7	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	149	8	neighbourhood	neighbourhood	NOUN
cana-3462	149	9	corona	corona	NOUN
cana-3462	149	10	is	be	AUX
cana-3462	149	11	given	give	VERB
cana-3462	149	12	by	by	ADP
cana-3462	149	13	theorem	theorem	ADJ
cana-3462	149	14	2.6	2.6	NUM
cana-3462	149	15	.	.	PUNCT
cana-3462	150	1	𝐸𝑄(𝐻1⊟𝐻2	𝐸𝑄(𝐻1⊟𝐻2	NOUN
cana-3462	150	2	)	)	PUNCT
cana-3462	151	1	=	=	SYM
cana-3462	151	2	(	(	PUNCT
cana-3462	151	3	1	1	NUM
cana-3462	151	4	+	+	CCONJ
cana-3462	151	5	𝑝2)𝐸𝑄(𝐻1	𝑝2)𝐸𝑄(𝐻1	X
cana-3462	151	6	)	)	PUNCT
cana-3462	151	7	+	+	CCONJ
cana-3462	151	8	𝑝1𝐸𝑄(𝐻2	𝑝1𝐸𝑄(𝐻2	PROPN
cana-3462	151	9	)	)	PUNCT
cana-3462	152	1	+	+	CCONJ
cana-3462	152	2	6𝑞1	6𝑞1	NUM
cana-3462	152	3	+	+	CCONJ
cana-3462	152	4	10𝑞1𝑝2	10𝑞1𝑝2	NUM
cana-3462	153	1	+	+	CCONJ
cana-3462	153	2	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	153	3	2	2	NUM
cana-3462	153	4	+	+	NUM
cana-3462	153	5	8𝑞1𝑞2	8𝑞1𝑞2	NUM
cana-3462	153	6	proof	proof	NOUN
cana-3462	153	7	:	:	PUNCT
cana-3462	153	8	using	use	VERB
cana-3462	153	9	definition	definition	NOUN
cana-3462	153	10	(	(	PUNCT
cana-3462	153	11	1.1	1.1	NUM
cana-3462	153	12	)	)	PUNCT
cana-3462	153	13	,	,	PUNCT
cana-3462	153	14	the	the	DET
cana-3462	153	15	above	above	ADJ
cana-3462	153	16	theorem	theorem	NOUN
cana-3462	153	17	can	can	AUX
cana-3462	153	18	be	be	AUX
cana-3462	153	19	easily	easily	ADV
cana-3462	153	20	proved	prove	VERB
cana-3462	153	21	.	.	PUNCT
cana-3462	154	1	also	also	ADV
cana-3462	154	2	,	,	PUNCT
cana-3462	154	3	the	the	DET
cana-3462	154	4	degree	degree	NOUN
cana-3462	154	5	of	of	ADP
cana-3462	154	6	any	any	DET
cana-3462	154	7	vertex	vertex	NOUN
cana-3462	154	8	of	of	ADP
cana-3462	154	9	𝑢	𝑢	NOUN
cana-3462	154	10	in	in	ADP
cana-3462	154	11	𝒮-edge	𝒮-edge	PROPN
cana-3462	154	12	neighbourhood	neighbourhood	NOUN
cana-3462	154	13	corona	corona	NOUN
cana-3462	154	14	is	be	AUX
cana-3462	154	15	given	give	VERB
cana-3462	154	16	by	by	ADP
cana-3462	154	17	𝑑𝐻1⊟𝐻2(𝑢𝑖	𝑑𝐻1⊟𝐻2(𝑢𝑖	PROPN
cana-3462	154	18	)	)	PUNCT
cana-3462	155	1	=	=	PRON
cana-3462	155	2	{	{	PUNCT
cana-3462	155	3	(	(	PUNCT
cana-3462	155	4	1	1	NUM
cana-3462	155	5	+	+	NOUN
cana-3462	155	6	𝑝2)𝑑𝐻1(𝑢𝑖	𝑝2)𝑑𝐻1(𝑢𝑖	X
cana-3462	155	7	)	)	PUNCT
cana-3462	155	8	,	,	PUNCT
cana-3462	155	9	if	if	SCONJ
cana-3462	155	10	𝑢	𝑢	PRON
cana-3462	155	11	∈	∈	PROPN
cana-3462	155	12	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	155	13	)	)	PUNCT
cana-3462	155	14	;	;	PUNCT
cana-3462	155	15	2	2	X
cana-3462	155	16	,	,	PUNCT
cana-3462	155	17	if	if	SCONJ
cana-3462	155	18	𝑢	𝑢	PRON
cana-3462	155	19	∈	∈	PROPN
cana-3462	155	20	𝐼(𝐻1	𝐼(𝐻1	X
cana-3462	155	21	)	)	PUNCT
cana-3462	155	22	2	2	NUM
cana-3462	155	23	+	+	NUM
cana-3462	155	24	𝑑𝐻2(𝑢𝑗	𝑑𝐻2(𝑢𝑗	NOUN
cana-3462	155	25	)	)	PUNCT
cana-3462	155	26	,	,	PUNCT
cana-3462	155	27	if	if	SCONJ
cana-3462	155	28	𝑢	𝑢	NOUN
cana-3462	155	29	=	=	PUNCT
cana-3462	155	30	𝑣𝑗	𝑣𝑗	ADP
cana-3462	155	31	𝑖	𝑖	X
cana-3462	155	32	,	,	PUNCT
cana-3462	155	33	for	for	ADP
cana-3462	155	34	𝑖	𝑖	PRON
cana-3462	155	35	=	=	SYM
cana-3462	155	36	1,2	1,2	NUM
cana-3462	155	37	,	,	PUNCT
cana-3462	155	38	…	…	PUNCT
cana-3462	155	39	𝑞1	𝑞1	ADJ
cana-3462	155	40	,	,	PUNCT
cana-3462	155	41	for	for	ADP
cana-3462	155	42	𝑗	𝑗	NOUN
cana-3462	155	43	=	=	SYM
cana-3462	155	44	1,2	1,2	NUM
cana-3462	155	45	,	,	PUNCT
cana-3462	155	46	…	…	PUNCT
cana-3462	155	47	𝑝2	𝑝2	NOUN
cana-3462	155	48	.	.	PUNCT
cana-3462	155	49	;	;	PUNCT
cana-3462	155	50	and	and	CCONJ
cana-3462	155	51	𝐻1⊟𝐻2	𝐻1⊟𝐻2	NOUN
cana-3462	155	52	has	have	VERB
cana-3462	155	53	𝑝1	𝑝1	NOUN
cana-3462	155	54	+	+	CCONJ
cana-3462	155	55	𝑞1	𝑞1	PROPN
cana-3462	155	56	+	+	CCONJ
cana-3462	155	57	𝑞1𝑝2	𝑞1𝑝2	NOUN
cana-3462	155	58	vertices	vertex	NOUN
cana-3462	155	59	.	.	PUNCT
cana-3462	156	1	the	the	DET
cana-3462	156	2	quasi	quasi	ADJ
cana-3462	156	3	-	-	ADJ
cana-3462	156	4	laplacian	laplacian	ADJ
cana-3462	156	5	energy	energy	NOUN
cana-3462	156	6	of	of	ADP
cana-3462	156	7	𝒮-edge	𝒮-edge	PROPN
cana-3462	156	8	neighbourhood	neighbourhood	NOUN
cana-3462	156	9	corona	corona	NOUN
cana-3462	156	10	is	be	AUX
cana-3462	156	11	given	give	VERB
cana-3462	156	12	by	by	ADP
cana-3462	156	13	theorem	theorem	ADJ
cana-3462	156	14	2.7	2.7	NUM
cana-3462	156	15	.	.	PUNCT
cana-3462	156	16	𝐸𝑄(𝐻1⊟𝐻2	𝐸𝑄(𝐻1⊟𝐻2	NOUN
cana-3462	156	17	)	)	PUNCT
cana-3462	157	1	=	=	SYM
cana-3462	157	2	(	(	PUNCT
cana-3462	157	3	1	1	NUM
cana-3462	157	4	+	+	NUM
cana-3462	157	5	2𝑝2)𝐸𝑄(𝐻1	2𝑝2)𝐸𝑄(𝐻1	NUM
cana-3462	157	6	)	)	PUNCT
cana-3462	157	7	+	+	CCONJ
cana-3462	157	8	2𝑞1𝐸𝑄(𝐻2	2𝑞1𝐸𝑄(𝐻2	NUM
cana-3462	157	9	)	)	PUNCT
cana-3462	157	10	+	+	NUM
cana-3462	157	11	𝑝2	𝑝2	NOUN
cana-3462	157	12	2𝑀1(𝐻1	2𝑀1(𝐻1	NUM
cana-3462	157	13	)	)	PUNCT
cana-3462	157	14	+	+	CCONJ
cana-3462	157	15	4𝑞1	4𝑞1	NUM
cana-3462	158	1	+	+	CCONJ
cana-3462	158	2	2𝑞1𝑝2	2𝑞1𝑝2	NUM
cana-3462	159	1	+	+	CCONJ
cana-3462	159	2	12𝑞1𝑞2	12𝑞1𝑞2	NUM
cana-3462	159	3	,	,	PUNCT
cana-3462	159	4	where	where	SCONJ
cana-3462	159	5	𝑀1	𝑀1	PROPN
cana-3462	159	6	is	be	AUX
cana-3462	159	7	the	the	DET
cana-3462	159	8	first	first	ADJ
cana-3462	159	9	zagreb	zagreb	PROPN
cana-3462	159	10	index	index	NOUN
cana-3462	159	11	.	.	PUNCT
cana-3462	160	1	using	use	VERB
cana-3462	160	2	definition	definition	NOUN
cana-3462	160	3	(	(	PUNCT
cana-3462	160	4	1.1	1.1	NUM
cana-3462	160	5	)	)	PUNCT
cana-3462	160	6	,	,	PUNCT
cana-3462	160	7	the	the	DET
cana-3462	160	8	above	above	ADJ
cana-3462	160	9	theorem	theorem	NOUN
cana-3462	160	10	can	can	AUX
cana-3462	160	11	be	be	AUX
cana-3462	160	12	easily	easily	ADV
cana-3462	160	13	proved	prove	VERB
cana-3462	160	14	.	.	PUNCT
cana-3462	161	1	communications	communication	NOUN
cana-3462	161	2	on	on	ADP
cana-3462	161	3	applied	apply	VERB
cana-3462	161	4	nonlinear	nonlinear	ADJ
cana-3462	161	5	analysis	analysis	NOUN
cana-3462	161	6	issn	issn	NOUN
cana-3462	161	7	:	:	PUNCT
cana-3462	161	8	1074	1074	NUM
cana-3462	161	9	-	-	PUNCT
cana-3462	161	10	133x	133x	NUM
cana-3462	161	11	vol	vol	NOUN
cana-3462	161	12	32	32	NUM
cana-3462	161	13	no	no	NOUN
cana-3462	161	14	.	.	PUNCT
cana-3462	162	1	7s	7	NOUN
cana-3462	162	2	(	(	PUNCT
cana-3462	162	3	2025	2025	NUM
cana-3462	162	4	)	)	PUNCT
cana-3462	162	5	536	536	NUM
cana-3462	162	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	162	7	3	3	X
cana-3462	162	8	.	.	X
cana-3462	163	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	163	2	based	base	VERB
cana-3462	163	3	on	on	ADP
cana-3462	163	4	𝒬	𝒬	PROPN
cana-3462	163	5	graph	graph	VERB
cana-3462	163	6	the	the	DET
cana-3462	163	7	energy	energy	NOUN
cana-3462	163	8	of	of	ADP
cana-3462	163	9	the	the	DET
cana-3462	163	10	𝒬	𝒬	PROPN
cana-3462	163	11	graph	graph	NOUN
cana-3462	163	12	,	,	PUNCT
cana-3462	163	13	𝒬	𝒬	PROPN
cana-3462	163	14	join	join	VERB
cana-3462	163	15	and	and	CCONJ
cana-3462	163	16	𝒬	𝒬	PROPN
cana-3462	163	17	corona	corona	NOUN
cana-3462	163	18	graphs	graph	NOUN
cana-3462	163	19	are	be	AUX
cana-3462	163	20	formulated	formulate	VERB
cana-3462	163	21	in	in	ADP
cana-3462	163	22	this	this	DET
cana-3462	163	23	section	section	NOUN
cana-3462	163	24	based	base	VERB
cana-3462	163	25	on	on	ADP
cana-3462	163	26	the	the	DET
cana-3462	163	27	degree	degree	NOUN
cana-3462	163	28	of	of	ADP
cana-3462	163	29	vertices	vertex	NOUN
cana-3462	163	30	of	of	ADP
cana-3462	163	31	the	the	DET
cana-3462	163	32	𝒬-graph	𝒬-graph	PROPN
cana-3462	163	33	,	,	PUNCT
cana-3462	163	34	𝒬-join	𝒬-join	PROPN
cana-3462	163	35	,	,	PUNCT
cana-3462	163	36	and	and	CCONJ
cana-3462	163	37	𝒬-corona	𝒬-corona	PROPN
cana-3462	163	38	graphs	graph	NOUN
cana-3462	163	39	respectively	respectively	ADV
cana-3462	163	40	.	.	PUNCT
cana-3462	164	1	let	let	VERB
cana-3462	164	2	𝑢	𝑢	PRON
cana-3462	164	3	be	be	AUX
cana-3462	164	4	any	any	DET
cana-3462	164	5	vertex	vertex	NOUN
cana-3462	164	6	in	in	ADP
cana-3462	164	7	𝒬(𝐺	𝒬(𝐺	ADV
cana-3462	164	8	)	)	PUNCT
cana-3462	164	9	.	.	PUNCT
cana-3462	165	1	then	then	ADV
cana-3462	165	2	,	,	PUNCT
cana-3462	165	3	the	the	DET
cana-3462	165	4	degree	degree	NOUN
cana-3462	165	5	of	of	ADP
cana-3462	165	6	vertex	vertex	NOUN
cana-3462	165	7	is	be	AUX
cana-3462	165	8	given	give	VERB
cana-3462	165	9	by	by	ADP
cana-3462	165	10	𝑑𝒮(𝐺)(𝑢𝑖	𝑑𝒮(𝐺)(𝑢𝑖	NOUN
cana-3462	165	11	)	)	PUNCT
cana-3462	165	12	=	=	SYM
cana-3462	165	13	{	{	PUNCT
cana-3462	165	14	𝑑𝐺(𝑢𝑖	𝑑𝐺(𝑢𝑖	PROPN
cana-3462	165	15	)	)	PUNCT
cana-3462	165	16	,	,	PUNCT
cana-3462	165	17	if	if	SCONJ
cana-3462	165	18	𝑢	𝑢	PROPN
cana-3462	165	19	∈	∈	PROPN
cana-3462	165	20	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3462	165	21	)	)	PUNCT
cana-3462	165	22	;	;	PUNCT
cana-3462	165	23	4	4	NUM
cana-3462	165	24	,	,	PUNCT
cana-3462	165	25	if	if	SCONJ
cana-3462	165	26	𝑢	𝑢	PROPN
cana-3462	165	27	∈	∈	PROPN
cana-3462	165	28	𝐼(𝐺	𝐼(𝐺	NOUN
cana-3462	165	29	)	)	PUNCT
cana-3462	165	30	.	.	PUNCT
cana-3462	166	1	we	we	PRON
cana-3462	166	2	derive	derive	VERB
cana-3462	166	3	a	a	DET
cana-3462	166	4	formula	formula	NOUN
cana-3462	166	5	of	of	ADP
cana-3462	166	6	energy	energy	NOUN
cana-3462	166	7	of	of	ADP
cana-3462	166	8	𝒬(𝐺	𝒬(𝐺	NOUN
cana-3462	166	9	)	)	PUNCT
cana-3462	166	10	.	.	PUNCT
cana-3462	167	1	theorem	theorem	VERB
cana-3462	167	2	3.1	3.1	NUM
cana-3462	167	3	.	.	PUNCT
cana-3462	167	4	𝐸𝑄(𝒬(𝐺	𝐸𝑄(𝒬(𝐺	PROPN
cana-3462	167	5	)	)	PUNCT
cana-3462	167	6	)	)	PUNCT
cana-3462	168	1	=	=	SYM
cana-3462	168	2	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	168	3	)	)	PUNCT
cana-3462	169	1	+	+	CCONJ
cana-3462	169	2	20𝑞	20𝑞	NOUN
cana-3462	169	3	proof	proof	NOUN
cana-3462	169	4	.	.	PUNCT
cana-3462	170	1	we	we	PRON
cana-3462	170	2	follow	follow	VERB
cana-3462	170	3	from	from	ADP
cana-3462	170	4	definition	definition	NOUN
cana-3462	170	5	(	(	PUNCT
cana-3462	170	6	1.1	1.1	NUM
cana-3462	170	7	)	)	PUNCT
cana-3462	170	8	,	,	PUNCT
cana-3462	170	9	𝐸𝑄(𝒬(𝐺	𝐸𝑄(𝒬(𝐺	NOUN
cana-3462	170	10	)	)	PUNCT
cana-3462	170	11	)	)	PUNCT
cana-3462	171	1	=	=	PUNCT
cana-3462	171	2	=	=	SYM
cana-3462	171	3	∑	∑	PUNCT
cana-3462	171	4	  	  	SPACE
cana-3462	171	5	𝑝+𝑞	𝑝+𝑞	NOUN
cana-3462	171	6	𝑖=1	𝑖=1	PUNCT
cana-3462	171	7	 	 	SPACE
cana-3462	171	8	𝑑𝒬(𝐺	𝑑𝒬(𝐺	NOUN
cana-3462	171	9	)	)	PUNCT
cana-3462	171	10	2	2	NUM
cana-3462	171	11	(	(	PUNCT
cana-3462	171	12	𝑣𝑖	𝑣𝑖	NOUN
cana-3462	171	13	)	)	PUNCT
cana-3462	171	14	+	+	ADJ
cana-3462	171	15	∑	∑	PROPN
cana-3462	171	16	  	  	SPACE
cana-3462	171	17	𝑝+𝑞	𝑝+𝑞	NOUN
cana-3462	171	18	𝑖=1	𝑖=1	PUNCT
cana-3462	171	19	 	 	SPACE
cana-3462	171	20	𝑑𝒬(𝐺)(𝑢𝑖	𝑑𝒬(𝐺)(𝑢𝑖	NOUN
cana-3462	171	21	)	)	PUNCT
cana-3462	171	22	=	=	PUNCT
cana-3462	171	23	∑	∑	PUNCT
cana-3462	171	24	  	  	SPACE
cana-3462	171	25	𝑝	𝑝	PROPN
cana-3462	171	26	𝑖=1	𝑖=1	PROPN
cana-3462	171	27	 	 	SPACE
cana-3462	171	28	𝑑(𝐺	𝑑(𝐺	PROPN
cana-3462	171	29	)	)	PUNCT
cana-3462	171	30	2	2	NUM
cana-3462	171	31	(	(	PUNCT
cana-3462	171	32	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	171	33	)	)	PUNCT
cana-3462	171	34	+	+	ADJ
cana-3462	171	35	∑	∑	PROPN
cana-3462	171	36	  	  	SPACE
cana-3462	171	37	𝑞	𝑞	X
cana-3462	171	38	𝑖=1	𝑖=1	PROPN
cana-3462	171	39	 	 	SPACE
cana-3462	171	40	42	42	NUM
cana-3462	171	41	+	+	ADJ
cana-3462	171	42	∑	∑	PROPN
cana-3462	171	43	  	  	SPACE
cana-3462	171	44	𝑝	𝑝	PROPN
cana-3462	171	45	𝑖=1	𝑖=1	PROPN
cana-3462	171	46	 	 	SPACE
cana-3462	171	47	𝑑(𝐺)(𝑢𝑖	𝑑(𝐺)(𝑢𝑖	NOUN
cana-3462	171	48	)	)	PUNCT
cana-3462	172	1	+	+	ADJ
cana-3462	172	2	∑	∑	PROPN
cana-3462	172	3	  	  	SPACE
cana-3462	172	4	𝑞	𝑞	X
cana-3462	172	5	𝑖=1	𝑖=1	PROPN
cana-3462	172	6	 	 	SPACE
cana-3462	172	7	4	4	NUM
cana-3462	172	8	=	=	SYM
cana-3462	172	9	𝐸𝑄(𝐺	𝐸𝑄(𝐺	NOUN
cana-3462	172	10	)	)	PUNCT
cana-3462	173	1	+	+	CCONJ
cana-3462	173	2	20𝑞	20𝑞	NOUN
cana-3462	173	3	3.1	3.1	NUM
cana-3462	173	4	.	.	PUNCT
cana-3462	174	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	174	2	of	of	ADP
cana-3462	174	3	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	174	4	and	and	CCONJ
cana-3462	174	5	edge	edge	NOUN
cana-3462	174	6	join	join	NOUN
cana-3462	174	7	let	let	VERB
cana-3462	174	8	𝑢	𝑢	PRON
cana-3462	174	9	be	be	AUX
cana-3462	174	10	any	any	DET
cana-3462	174	11	vertex	vertex	NOUN
cana-3462	174	12	in	in	ADP
cana-3462	174	13	𝐻1⟨𝑣⟩𝐻2	𝐻1⟨𝑣⟩𝐻2	NOUN
cana-3462	174	14	.	.	PUNCT
cana-3462	175	1	then	then	ADV
cana-3462	175	2	,	,	PUNCT
cana-3462	175	3	the	the	DET
cana-3462	175	4	degree	degree	NOUN
cana-3462	175	5	of	of	ADP
cana-3462	175	6	vertex	vertex	NOUN
cana-3462	175	7	of	of	ADP
cana-3462	175	8	𝐻1⟨𝑣⟩𝐻2	𝐻1⟨𝑣⟩𝐻2	NOUN
cana-3462	175	9	is	be	AUX
cana-3462	175	10	given	give	VERB
cana-3462	175	11	by	by	ADP
cana-3462	175	12	𝑑𝐻1⟨𝑣⟩𝐻2(𝑢𝑖	𝑑𝐻1⟨𝑣⟩𝐻2(𝑢𝑖	NOUN
cana-3462	175	13	)	)	PUNCT
cana-3462	175	14	=	=	SYM
cana-3462	175	15	{	{	PUNCT
cana-3462	175	16	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	NOUN
cana-3462	175	17	)	)	PUNCT
cana-3462	176	1	+	+	NUM
cana-3462	176	2	𝑝2	𝑝2	NOUN
cana-3462	176	3	,	,	PUNCT
cana-3462	176	4	if	if	SCONJ
cana-3462	176	5	𝑢	𝑢	PRON
cana-3462	176	6	∈	∈	PROPN
cana-3462	176	7	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	176	8	)	)	PUNCT
cana-3462	176	9	;	;	PUNCT
cana-3462	176	10	4	4	NUM
cana-3462	176	11	,	,	PUNCT
cana-3462	176	12	if	if	SCONJ
cana-3462	176	13	𝑢	𝑢	PRON
cana-3462	176	14	∈	∈	PROPN
cana-3462	176	15	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	176	16	)	)	PUNCT
cana-3462	176	17	;	;	PUNCT
cana-3462	176	18	𝑑𝐻2(𝑢𝑖	𝑑𝐻2(𝑢𝑖	X
cana-3462	176	19	)	)	PUNCT
cana-3462	177	1	+	+	CCONJ
cana-3462	177	2	𝑝1	𝑝1	NOUN
cana-3462	177	3	,	,	PUNCT
cana-3462	177	4	if	if	SCONJ
cana-3462	177	5	𝑢	𝑢	PRON
cana-3462	177	6	∈	∈	PROPN
cana-3462	177	7	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	177	8	)	)	PUNCT
cana-3462	177	9	.	.	PUNCT
cana-3462	178	1	𝐻1⟨𝑣⟩𝐻2	𝐻1⟨𝑣⟩𝐻2	PROPN
cana-3462	178	2	has	have	VERB
cana-3462	178	3	𝑝1	𝑝1	NOUN
cana-3462	178	4	+	+	CCONJ
cana-3462	178	5	𝑞1	𝑞1	PROPN
cana-3462	178	6	+	+	CCONJ
cana-3462	178	7	𝑝2	𝑝2	NOUN
cana-3462	178	8	vertices	vertex	NOUN
cana-3462	178	9	.	.	PUNCT
cana-3462	179	1	the	the	DET
cana-3462	179	2	relation	relation	NOUN
cana-3462	179	3	of	of	ADP
cana-3462	179	4	quasi	quasi	ADJ
cana-3462	179	5	-	-	ADJ
cana-3462	179	6	laplacian	laplacian	ADJ
cana-3462	179	7	energy	energy	NOUN
cana-3462	179	8	in	in	ADP
cana-3462	179	9	between	between	ADP
cana-3462	179	10	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	179	11	join	join	NOUN
cana-3462	179	12	with	with	ADP
cana-3462	179	13	corresponding	correspond	VERB
cana-3462	179	14	two	two	NUM
cana-3462	179	15	original	original	ADJ
cana-3462	179	16	graphs	graph	NOUN
cana-3462	179	17	is	be	AUX
cana-3462	179	18	obtained	obtain	VERB
cana-3462	179	19	easily	easily	ADV
cana-3462	179	20	by	by	ADP
cana-3462	179	21	using	use	VERB
cana-3462	179	22	definition	definition	NOUN
cana-3462	179	23	(	(	PUNCT
cana-3462	179	24	1.1	1.1	NUM
cana-3462	179	25	)	)	PUNCT
cana-3462	179	26	as	as	SCONJ
cana-3462	179	27	follows	follow	VERB
cana-3462	179	28	theorem	theorem	VERB
cana-3462	179	29	3.2	3.2	NUM
cana-3462	179	30	.	.	PUNCT
cana-3462	180	1	𝐸𝑄(𝐻1⟨𝑣⟩𝐻2	𝐸𝑄(𝐻1⟨𝑣⟩𝐻2	NUM
cana-3462	180	2	)	)	PUNCT
cana-3462	180	3	=	=	PUNCT
cana-3462	180	4	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	180	5	)	)	PUNCT
cana-3462	180	6	+	+	CCONJ
cana-3462	180	7	𝐸𝑄(𝐻2	𝐸𝑄(𝐻2	NOUN
cana-3462	180	8	)	)	PUNCT
cana-3462	181	1	+	+	CCONJ
cana-3462	181	2	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	182	1	+	+	CCONJ
cana-3462	182	2	𝑝1𝑝2	𝑝1𝑝2	PROPN
cana-3462	182	3	2	2	NUM
cana-3462	182	4	+	+	CCONJ
cana-3462	182	5	20𝑞1	20𝑞1	NUM
cana-3462	183	1	+	+	CCONJ
cana-3462	183	2	4𝑝1𝑞2	4𝑝1𝑞2	NUM
cana-3462	184	1	+	+	CCONJ
cana-3462	184	2	𝑝1	𝑝1	NOUN
cana-3462	184	3	2𝑝2	2𝑝2	NUM
cana-3462	184	4	+	+	CCONJ
cana-3462	184	5	2𝑝1𝑞2	2𝑝1𝑞2	NUM
cana-3462	184	6	also	also	ADV
cana-3462	184	7	,	,	PUNCT
cana-3462	184	8	let	let	VERB
cana-3462	184	9	𝑢	𝑢	PRON
cana-3462	184	10	be	be	AUX
cana-3462	184	11	any	any	DET
cana-3462	184	12	vertex	vertex	NOUN
cana-3462	184	13	in	in	ADP
cana-3462	184	14	𝒬-edge	𝒬-edge	PROPN
cana-3462	184	15	join	join	NOUN
cana-3462	184	16	.	.	PUNCT
cana-3462	185	1	the	the	DET
cana-3462	185	2	degree	degree	NOUN
cana-3462	185	3	of	of	ADP
cana-3462	185	4	vertex	vertex	NOUN
cana-3462	185	5	𝑢	𝑢	NOUN
cana-3462	185	6	of	of	ADP
cana-3462	185	7	𝒬-edge	𝒬-edge	PROPN
cana-3462	185	8	join	join	NOUN
cana-3462	185	9	is	be	AUX
cana-3462	185	10	given	give	VERB
cana-3462	185	11	by	by	ADP
cana-3462	185	12	𝑑𝐻1⟨𝑒⟩𝐻2(𝑢𝑖	𝑑𝐻1⟨𝑒⟩𝐻2(𝑢𝑖	NOUN
cana-3462	185	13	)	)	PUNCT
cana-3462	185	14	=	=	SYM
cana-3462	185	15	{	{	PUNCT
cana-3462	185	16	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	185	17	)	)	PUNCT
cana-3462	185	18	,	,	PUNCT
cana-3462	185	19	if	if	SCONJ
cana-3462	185	20	𝑢	𝑢	PRON
cana-3462	185	21	∈	∈	PROPN
cana-3462	185	22	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	185	23	)	)	PUNCT
cana-3462	185	24	;	;	PUNCT
cana-3462	185	25	2	2	NUM
cana-3462	185	26	+	+	NUM
cana-3462	185	27	𝑝2	𝑝2	NOUN
cana-3462	185	28	,	,	PUNCT
cana-3462	185	29	if	if	SCONJ
cana-3462	185	30	𝑢	𝑢	PRON
cana-3462	185	31	∈	∈	PROPN
cana-3462	185	32	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	185	33	)	)	PUNCT
cana-3462	185	34	𝑑𝐻2(𝑢𝑖	𝑑𝐻2(𝑢𝑖	NOUN
cana-3462	185	35	)	)	PUNCT
cana-3462	186	1	+	+	CCONJ
cana-3462	186	2	𝑞1	𝑞1	ADJ
cana-3462	186	3	,	,	PUNCT
cana-3462	186	4	if	if	SCONJ
cana-3462	186	5	𝑢	𝑢	PRON
cana-3462	186	6	∈	∈	PROPN
cana-3462	186	7	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	186	8	)	)	PUNCT
cana-3462	186	9	.	.	PUNCT
cana-3462	187	1	;	;	PUNCT
cana-3462	187	2	communications	communication	NOUN
cana-3462	187	3	on	on	ADP
cana-3462	187	4	applied	apply	VERB
cana-3462	187	5	nonlinear	nonlinear	ADJ
cana-3462	187	6	analysis	analysis	NOUN
cana-3462	187	7	issn	issn	NOUN
cana-3462	187	8	:	:	PUNCT
cana-3462	187	9	1074	1074	NUM
cana-3462	187	10	-	-	PUNCT
cana-3462	187	11	133x	133x	NUM
cana-3462	187	12	vol	vol	NOUN
cana-3462	187	13	32	32	NUM
cana-3462	187	14	no	no	NOUN
cana-3462	187	15	.	.	PUNCT
cana-3462	188	1	7s	7	NOUN
cana-3462	188	2	(	(	PUNCT
cana-3462	188	3	2025	2025	NUM
cana-3462	188	4	)	)	PUNCT
cana-3462	188	5	537	537	NUM
cana-3462	188	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	188	7	𝐻1⟨𝑒⟩𝐻2	𝐻1⟨𝑒⟩𝐻2	PROPN
cana-3462	188	8	has	have	VERB
cana-3462	188	9	𝑝1	𝑝1	NOUN
cana-3462	188	10	+	+	CCONJ
cana-3462	188	11	𝑞1	𝑞1	PROPN
cana-3462	188	12	+	+	CCONJ
cana-3462	188	13	𝑝2	𝑝2	NOUN
cana-3462	188	14	vertices	vertex	NOUN
cana-3462	188	15	.	.	PUNCT
cana-3462	189	1	the	the	DET
cana-3462	189	2	relation	relation	NOUN
cana-3462	189	3	of	of	ADP
cana-3462	189	4	quasi	quasi	ADJ
cana-3462	189	5	-	-	ADJ
cana-3462	189	6	laplacian	laplacian	ADJ
cana-3462	189	7	energy	energy	NOUN
cana-3462	189	8	in	in	ADP
cana-3462	189	9	between	between	ADP
cana-3462	189	10	𝒬-edge	𝒬-edge	PROPN
cana-3462	189	11	join	join	VERB
cana-3462	189	12	with	with	ADP
cana-3462	189	13	corresponding	correspond	VERB
cana-3462	189	14	two	two	NUM
cana-3462	189	15	original	original	ADJ
cana-3462	189	16	graphs	graph	NOUN
cana-3462	189	17	is	be	AUX
cana-3462	189	18	obtained	obtain	VERB
cana-3462	189	19	easily	easily	ADV
cana-3462	189	20	by	by	ADP
cana-3462	189	21	using	use	VERB
cana-3462	189	22	definition	definition	NOUN
cana-3462	189	23	(	(	PUNCT
cana-3462	189	24	1.1	1.1	NUM
cana-3462	189	25	)	)	PUNCT
cana-3462	189	26	as	as	SCONJ
cana-3462	189	27	follows	follow	VERB
cana-3462	189	28	theorem	theorem	VERB
cana-3462	189	29	3.3	3.3	NUM
cana-3462	189	30	.	.	PUNCT
cana-3462	190	1	𝐸𝑄(𝐻1⟨𝑒⟩𝐻2	𝐸𝑄(𝐻1⟨𝑒⟩𝐻2	PROPN
cana-3462	190	2	)	)	PUNCT
cana-3462	190	3	=	=	PUNCT
cana-3462	190	4	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	190	5	)	)	PUNCT
cana-3462	191	1	+	+	CCONJ
cana-3462	191	2	𝐸𝑄(𝐻2	𝐸𝑄(𝐻2	NOUN
cana-3462	191	3	)	)	PUNCT
cana-3462	192	1	+	+	SYM
cana-3462	192	2	12𝑞1𝑝2	12𝑞1𝑝2	ADJ
cana-3462	193	1	+	+	CCONJ
cana-3462	193	2	𝑞1𝑝2	𝑞1𝑝2	X
cana-3462	193	3	+	+	SYM
cana-3462	193	4	20𝑞1	20𝑞1	NUM
cana-3462	193	5	+	+	CCONJ
cana-3462	193	6	𝑞1	𝑞1	PROPN
cana-3462	193	7	2𝑝2	2𝑝2	NUM
cana-3462	193	8	+	+	CCONJ
cana-3462	193	9	𝑞1𝑝2	𝑞1𝑝2	VERB
cana-3462	193	10	2	2	NUM
cana-3462	193	11	3.2	3.2	NUM
cana-3462	193	12	.	.	PUNCT
cana-3462	194	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	194	2	of	of	ADP
cana-3462	194	3	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	194	4	and	and	CCONJ
cana-3462	194	5	edge	edge	NOUN
cana-3462	194	6	corona	corona	NOUN
cana-3462	194	7	let	let	VERB
cana-3462	194	8	𝑢	𝑢	PRON
cana-3462	194	9	be	be	AUX
cana-3462	194	10	any	any	DET
cana-3462	194	11	vertex	vertex	NOUN
cana-3462	194	12	in	in	ADP
cana-3462	194	13	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	194	14	corona	corona	NOUN
cana-3462	194	15	.	.	PUNCT
cana-3462	195	1	then	then	ADV
cana-3462	195	2	,	,	PUNCT
cana-3462	195	3	the	the	DET
cana-3462	195	4	degree	degree	NOUN
cana-3462	195	5	of	of	ADP
cana-3462	195	6	any	any	DET
cana-3462	195	7	vertex	vertex	NOUN
cana-3462	195	8	𝑢	𝑢	NOUN
cana-3462	195	9	in	in	ADP
cana-3462	195	10	𝒬-vertex	𝒬-vertex	PROPN
cana-3462	195	11	corona	corona	NOUN
cana-3462	195	12	is	be	AUX
cana-3462	195	13	given	give	VERB
cana-3462	195	14	by	by	ADP
cana-3462	195	15	𝑑𝐻1o𝐻2(𝑣𝑖	𝑑𝐻1o𝐻2(𝑣𝑖	SYM
cana-3462	195	16	)	)	PUNCT
cana-3462	195	17	=	=	PRON
cana-3462	195	18	{	{	PUNCT
cana-3462	195	19	𝑑𝐻1(𝑢𝑖)+2	𝑑𝐻1(𝑢𝑖)+2	NOUN
cana-3462	195	20	,	,	PUNCT
cana-3462	195	21	if	if	SCONJ
cana-3462	195	22	𝑢	𝑢	PRON
cana-3462	195	23	∈	∈	PROPN
cana-3462	195	24	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	195	25	)	)	PUNCT
cana-3462	195	26	;	;	PUNCT
cana-3462	195	27	4	4	NUM
cana-3462	195	28	,	,	PUNCT
cana-3462	195	29	if	if	SCONJ
cana-3462	195	30	𝑢	𝑢	PRON
cana-3462	195	31	∈	∈	PROPN
cana-3462	195	32	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	195	33	)	)	PUNCT
cana-3462	195	34	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PUNCT
cana-3462	195	35	𝑖	𝑖	X
cana-3462	195	36	)	)	PUNCT
cana-3462	195	37	+	+	CCONJ
cana-3462	195	38	1	1	NUM
cana-3462	195	39	,	,	PUNCT
cana-3462	195	40	if	if	SCONJ
cana-3462	195	41	𝑢	𝑢	NOUN
cana-3462	195	42	=	=	PUNCT
cana-3462	195	43	𝑣𝑗	𝑣𝑗	ADP
cana-3462	195	44	𝑖	𝑖	X
cana-3462	195	45	,	,	PUNCT
cana-3462	195	46	for	for	ADP
cana-3462	195	47	𝑖	𝑖	PRON
cana-3462	195	48	=	=	NOUN
cana-3462	195	49	1,2,3	1,2,3	NUM
cana-3462	195	50	,	,	PUNCT
cana-3462	195	51	…	…	PUNCT
cana-3462	195	52	𝑝1	𝑝1	NOUN
cana-3462	195	53	,	,	PUNCT
cana-3462	195	54	for	for	ADP
cana-3462	195	55	𝑗	𝑗	NOUN
cana-3462	195	56	=	=	SYM
cana-3462	195	57	1,2,3	1,2,3	NUM
cana-3462	195	58	,	,	PUNCT
cana-3462	195	59	.	.	PUNCT
cana-3462	195	60	.	.	PUNCT
cana-3462	196	1	𝑝2	𝑝2	PROPN
cana-3462	196	2	.	.	PUNCT
cana-3462	197	1	;	;	PUNCT
cana-3462	197	2	𝐻1o𝐻2	𝐻1o𝐻2	PROPN
cana-3462	197	3	has	have	VERB
cana-3462	197	4	𝑝1	𝑝1	NOUN
cana-3462	197	5	+	+	CCONJ
cana-3462	197	6	𝑞1	𝑞1	PROPN
cana-3462	197	7	+	+	CCONJ
cana-3462	197	8	𝑝1𝑝2	𝑝1𝑝2	PRON
cana-3462	197	9	vertices	vertice	VERB
cana-3462	197	10	.	.	PUNCT
cana-3462	198	1	then	then	ADV
cana-3462	198	2	,	,	PUNCT
cana-3462	198	3	by	by	ADP
cana-3462	198	4	using	use	VERB
cana-3462	198	5	equation	equation	NOUN
cana-3462	198	6	(	(	PUNCT
cana-3462	198	7	1.1	1.1	NUM
cana-3462	198	8	)	)	PUNCT
cana-3462	198	9	we	we	PRON
cana-3462	198	10	get	get	AUX
cana-3462	198	11	theorem	theorem	VERB
cana-3462	198	12	3.4	3.4	NUM
cana-3462	198	13	.	.	PUNCT
cana-3462	199	1	𝐸𝑄(𝐻1𝑜𝐻2	𝐸𝑄(𝐻1𝑜𝐻2	NUM
cana-3462	199	2	)	)	PUNCT
cana-3462	200	1	=	=	SYM
cana-3462	200	2	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	200	3	)	)	PUNCT
cana-3462	201	1	+	+	CCONJ
cana-3462	201	2	𝑝1𝐸𝑄(𝐻2	𝑝1𝐸𝑄(𝐻2	PROPN
cana-3462	201	3	)	)	PUNCT
cana-3462	202	1	+	+	CCONJ
cana-3462	202	2	𝑝1𝑝2	𝑝1𝑝2	PROPN
cana-3462	202	3	2	2	NUM
cana-3462	202	4	+	+	CCONJ
cana-3462	202	5	20𝑞1	20𝑞1	NUM
cana-3462	202	6	+	+	CCONJ
cana-3462	202	7	4𝑝1𝑞2	4𝑝1𝑞2	NUM
cana-3462	203	1	+	+	CCONJ
cana-3462	203	2	3𝑝1𝑝2	3𝑝1𝑝2	NUM
cana-3462	203	3	+	+	CCONJ
cana-3462	203	4	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	203	5	next	next	ADJ
cana-3462	203	6	,	,	PUNCT
cana-3462	203	7	the	the	DET
cana-3462	203	8	degree	degree	NOUN
cana-3462	203	9	of	of	ADP
cana-3462	203	10	any	any	DET
cana-3462	203	11	vertex	vertex	NOUN
cana-3462	203	12	of	of	ADP
cana-3462	203	13	𝑢	𝑢	NOUN
cana-3462	203	14	in	in	ADP
cana-3462	203	15	𝒬-edge	𝒬-edge	PROPN
cana-3462	203	16	corona	corona	NOUN
cana-3462	203	17	is	be	AUX
cana-3462	203	18	given	give	VERB
cana-3462	203	19	by	by	ADP
cana-3462	203	20	𝑑𝐻1⊛𝐻2(𝑢𝑖	𝑑𝐻1⊛𝐻2(𝑢𝑖	NOUN
cana-3462	203	21	)	)	PUNCT
cana-3462	203	22	=	=	PRON
cana-3462	203	23	{	{	PUNCT
cana-3462	203	24	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	203	25	)	)	PUNCT
cana-3462	203	26	,	,	PUNCT
cana-3462	203	27	if	if	SCONJ
cana-3462	203	28	𝑢	𝑢	PRON
cana-3462	203	29	∈	∈	PROPN
cana-3462	203	30	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	203	31	)	)	PUNCT
cana-3462	203	32	;	;	PUNCT
cana-3462	203	33	𝑝2	𝑝2	NOUN
cana-3462	203	34	+	+	CCONJ
cana-3462	203	35	4	4	NUM
cana-3462	203	36	,	,	PUNCT
cana-3462	203	37	if	if	SCONJ
cana-3462	203	38	𝑢	𝑢	PRON
cana-3462	203	39	∈	∈	PROPN
cana-3462	203	40	𝐼(𝐻1	𝐼(𝐻1	PROPN
cana-3462	203	41	)	)	PUNCT
cana-3462	203	42	;	;	PUNCT
cana-3462	203	43	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PUNCT
cana-3462	203	44	𝑖	𝑖	X
cana-3462	203	45	)	)	PUNCT
cana-3462	203	46	+	+	CCONJ
cana-3462	203	47	1	1	NUM
cana-3462	203	48	,	,	PUNCT
cana-3462	203	49	if	if	SCONJ
cana-3462	203	50	𝑢	𝑢	NOUN
cana-3462	203	51	=	=	PUNCT
cana-3462	203	52	𝑣𝑗	𝑣𝑗	ADP
cana-3462	203	53	𝑖	𝑖	X
cana-3462	203	54	,	,	PUNCT
cana-3462	203	55	for	for	ADP
cana-3462	203	56	𝑖	𝑖	PRON
cana-3462	203	57	=	=	SYM
cana-3462	203	58	1,2	1,2	NUM
cana-3462	203	59	,	,	PUNCT
cana-3462	203	60	…	…	PUNCT
cana-3462	203	61	𝑝1	𝑝1	NOUN
cana-3462	203	62	,	,	PUNCT
cana-3462	203	63	for	for	ADP
cana-3462	203	64	𝑗	𝑗	NOUN
cana-3462	203	65	=	=	SYM
cana-3462	203	66	1,2	1,2	NUM
cana-3462	203	67	,	,	PUNCT
cana-3462	203	68	…	…	PUNCT
cana-3462	203	69	𝑝2	𝑝2	NOUN
cana-3462	203	70	.	.	PROPN
cana-3462	203	71	𝐺1⊛𝐺2	𝐺1⊛𝐺2	PROPN
cana-3462	203	72	has	have	VERB
cana-3462	203	73	𝑝1	𝑝1	NOUN
cana-3462	203	74	+	+	CCONJ
cana-3462	203	75	𝑞1	𝑞1	PROPN
cana-3462	203	76	+	+	CCONJ
cana-3462	203	77	𝑞1𝑝2	𝑞1𝑝2	NOUN
cana-3462	203	78	vertices	vertex	NOUN
cana-3462	203	79	.	.	PUNCT
cana-3462	204	1	then	then	ADV
cana-3462	204	2	,	,	PUNCT
cana-3462	204	3	the	the	DET
cana-3462	204	4	quasi	quasi	ADJ
cana-3462	204	5	-	-	ADJ
cana-3462	204	6	laplacian	laplacian	ADJ
cana-3462	204	7	energy	energy	NOUN
cana-3462	204	8	of	of	ADP
cana-3462	204	9	𝒬-edge	𝒬-edge	PROPN
cana-3462	204	10	corona	corona	NOUN
cana-3462	204	11	is	be	AUX
cana-3462	204	12	easily	easily	ADV
cana-3462	204	13	obtained	obtain	VERB
cana-3462	204	14	as	as	SCONJ
cana-3462	204	15	follows	follow	NOUN
cana-3462	204	16	theorem	theorem	VERB
cana-3462	204	17	3.5	3.5	NUM
cana-3462	204	18	.	.	PUNCT
cana-3462	205	1	𝐸𝑄(𝐻1⊛𝐻2	𝐸𝑄(𝐻1⊛𝐻2	X
cana-3462	205	2	)	)	PUNCT
cana-3462	206	1	=	=	SYM
cana-3462	206	2	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	206	3	)	)	PUNCT
cana-3462	207	1	+	+	CCONJ
cana-3462	207	2	𝑝1𝐸𝑄(𝐻2	𝑝1𝐸𝑄(𝐻2	PROPN
cana-3462	207	3	)	)	PUNCT
cana-3462	208	1	+	+	CCONJ
cana-3462	208	2	𝑞1𝑝2	𝑞1𝑝2	VERB
cana-3462	208	3	2	2	NUM
cana-3462	208	4	+	+	NUM
cana-3462	208	5	9𝑞1𝑝2	9𝑞1𝑝2	NUM
cana-3462	209	1	+	+	CCONJ
cana-3462	209	2	20𝑞1	20𝑞1	NUM
cana-3462	209	3	+	+	NUM
cana-3462	209	4	4𝑞2𝑝1	4𝑞2𝑝1	NUM
cana-3462	209	5	+	+	CCONJ
cana-3462	209	6	2𝑝1𝑞2	2𝑝1𝑞2	NUM
cana-3462	209	7	4	4	NUM
cana-3462	209	8	.	.	PUNCT
cana-3462	210	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	210	2	based	base	VERB
cana-3462	210	3	on	on	ADP
cana-3462	210	4	𝒯-graph	𝒯-graph	PROPN
cana-3462	210	5	we	we	PRON
cana-3462	210	6	derive	derive	VERB
cana-3462	210	7	energy	energy	NOUN
cana-3462	210	8	of	of	ADP
cana-3462	210	9	𝒯	𝒯	PROPN
cana-3462	210	10	graph	graph	NOUN
cana-3462	210	11	and	and	CCONJ
cana-3462	210	12	𝒯	𝒯	PROPN
cana-3462	210	13	graph	graph	NOUN
cana-3462	210	14	corona	corona	NOUN
cana-3462	210	15	based	base	VERB
cana-3462	210	16	on	on	ADP
cana-3462	210	17	the	the	DET
cana-3462	210	18	degree	degree	NOUN
cana-3462	210	19	of	of	ADP
cana-3462	210	20	vertices	vertex	NOUN
cana-3462	210	21	of	of	ADP
cana-3462	210	22	the	the	DET
cana-3462	210	23	𝒯graph	𝒯graph	PROPN
cana-3462	210	24	and	and	CCONJ
cana-3462	210	25	𝒯-corona	𝒯-corona	PROPN
cana-3462	210	26	graphs	graph	NOUN
cana-3462	210	27	are	be	AUX
cana-3462	210	28	determined	determine	VERB
cana-3462	210	29	here	here	ADV
cana-3462	210	30	.	.	PUNCT
cana-3462	211	1	let	let	VERB
cana-3462	211	2	𝑢	𝑢	PRON
cana-3462	211	3	be	be	AUX
cana-3462	211	4	any	any	DET
cana-3462	211	5	vertex	vertex	NOUN
cana-3462	211	6	in	in	ADP
cana-3462	211	7	𝒯(𝐺	𝒯(𝐺	PROPN
cana-3462	211	8	)	)	PUNCT
cana-3462	211	9	,	,	PUNCT
cana-3462	211	10	then	then	ADV
cana-3462	211	11	the	the	DET
cana-3462	211	12	degree	degree	NOUN
cana-3462	211	13	of	of	ADP
cana-3462	211	14	vertex	vertex	NOUN
cana-3462	211	15	𝑢	𝑢	NOUN
cana-3462	211	16	in	in	ADP
cana-3462	211	17	𝒯(𝐺	𝒯(𝐺	PROPN
cana-3462	211	18	)	)	PUNCT
cana-3462	211	19	is	be	AUX
cana-3462	211	20	given	give	VERB
cana-3462	211	21	by	by	ADP
cana-3462	211	22	𝑑𝒯(𝐺)(𝑢𝑖	𝑑𝒯(𝐺)(𝑢𝑖	NOUN
cana-3462	211	23	)	)	PUNCT
cana-3462	211	24	=	=	SYM
cana-3462	211	25	{	{	PUNCT
cana-3462	211	26	2𝑑𝐺(𝑣𝑖	2𝑑𝐺(𝑣𝑖	NUM
cana-3462	211	27	)	)	PUNCT
cana-3462	211	28	,	,	PUNCT
cana-3462	211	29	if	if	SCONJ
cana-3462	211	30	𝑢	𝑢	PROPN
cana-3462	211	31	∈	∈	PROPN
cana-3462	211	32	𝑉(𝐺	𝑉(𝐺	NOUN
cana-3462	211	33	)	)	PUNCT
cana-3462	211	34	;	;	PUNCT
cana-3462	211	35	4	4	NUM
cana-3462	211	36	,	,	PUNCT
cana-3462	211	37	if	if	SCONJ
cana-3462	211	38	𝑢	𝑢	PROPN
cana-3462	211	39	∈	∈	PROPN
cana-3462	211	40	𝐼(𝐺	𝐼(𝐺	NOUN
cana-3462	211	41	)	)	PUNCT
cana-3462	211	42	.	.	PUNCT
cana-3462	212	1	the	the	DET
cana-3462	212	2	vertices	vertex	NOUN
cana-3462	212	3	of	of	ADP
cana-3462	212	4	𝒯(𝐺	𝒯(𝐺	PROPN
cana-3462	212	5	)	)	PUNCT
cana-3462	212	6	is	be	AUX
cana-3462	212	7	𝑝	𝑝	NOUN
cana-3462	212	8	+	+	PRON
cana-3462	212	9	𝑞.	𝑞.	VERB
cana-3462	212	10	the	the	DET
cana-3462	212	11	𝑄	𝑄	PROPN
cana-3462	212	12	energy	energy	NOUN
cana-3462	212	13	of	of	ADP
cana-3462	212	14	𝒯(𝐺	𝒯(𝐺	PROPN
cana-3462	212	15	)	)	PUNCT
cana-3462	212	16	is	be	AUX
cana-3462	212	17	given	give	VERB
cana-3462	212	18	by	by	ADP
cana-3462	212	19	theorem	theorem	NOUN
cana-3462	212	20	4.1	4.1	NUM
cana-3462	212	21	.	.	PUNCT
cana-3462	212	22	𝐸𝑄(𝒯(𝐺	𝐸𝑄(𝒯(𝐺	NOUN
cana-3462	212	23	)	)	PUNCT
cana-3462	212	24	)	)	PUNCT
cana-3462	213	1	=	=	PUNCT
cana-3462	213	2	2𝐸𝑄(𝐺	2𝐸𝑄(𝐺	NUM
cana-3462	213	3	)	)	PUNCT
cana-3462	214	1	+	+	NUM
cana-3462	214	2	20𝑞	20𝑞	NOUN
cana-3462	214	3	+	+	CCONJ
cana-3462	214	4	2𝑀1(𝐺	2𝑀1(𝐺	NOUN
cana-3462	214	5	)	)	PUNCT
cana-3462	214	6	where	where	SCONJ
cana-3462	214	7	,	,	PUNCT
cana-3462	214	8	𝑀1	𝑀1	PROPN
cana-3462	214	9	is	be	AUX
cana-3462	214	10	the	the	DET
cana-3462	214	11	first	first	ADJ
cana-3462	214	12	zagreb	zagreb	PROPN
cana-3462	214	13	index	index	NOUN
cana-3462	214	14	.	.	PUNCT
cana-3462	215	1	communications	communication	NOUN
cana-3462	215	2	on	on	ADP
cana-3462	215	3	applied	apply	VERB
cana-3462	215	4	nonlinear	nonlinear	ADJ
cana-3462	215	5	analysis	analysis	NOUN
cana-3462	215	6	issn	issn	NOUN
cana-3462	215	7	:	:	PUNCT
cana-3462	215	8	1074	1074	NUM
cana-3462	215	9	-	-	PUNCT
cana-3462	215	10	133x	133x	NUM
cana-3462	215	11	vol	vol	NOUN
cana-3462	215	12	32	32	NUM
cana-3462	215	13	no	no	NOUN
cana-3462	215	14	.	.	PUNCT
cana-3462	216	1	7s	7	NOUN
cana-3462	216	2	(	(	PUNCT
cana-3462	216	3	2025	2025	NUM
cana-3462	216	4	)	)	PUNCT
cana-3462	216	5	538	538	NUM
cana-3462	216	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	216	7	4.1	4.1	NUM
cana-3462	216	8	.	.	PUNCT
cana-3462	217	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	217	2	of	of	ADP
cana-3462	217	3	𝒯-graph	𝒯-graph	PROPN
cana-3462	217	4	corona	corona	NOUN
cana-3462	217	5	let	let	VERB
cana-3462	217	6	𝐻1	𝐻1	NOUN
cana-3462	217	7	be	be	AUX
cana-3462	217	8	𝑟1	𝑟1	NOUN
cana-3462	217	9	regular	regular	ADV
cana-3462	217	10	and	and	CCONJ
cana-3462	217	11	𝐻2	𝐻2	ADJ
cana-3462	217	12	be	be	AUX
cana-3462	217	13	any	any	DET
cana-3462	217	14	graph	graph	NOUN
cana-3462	217	15	.	.	PUNCT
cana-3462	218	1	also	also	ADV
cana-3462	218	2	,	,	PUNCT
cana-3462	218	3	let	let	VERB
cana-3462	218	4	𝑢	𝑢	PRON
cana-3462	218	5	be	be	AUX
cana-3462	218	6	any	any	DET
cana-3462	218	7	vertex	vertex	NOUN
cana-3462	218	8	in	in	ADP
cana-3462	218	9	𝒯(𝐺	𝒯(𝐺	PROPN
cana-3462	218	10	)	)	PUNCT
cana-3462	218	11	.	.	PUNCT
cana-3462	219	1	then	then	ADV
cana-3462	219	2	,	,	PUNCT
cana-3462	219	3	the	the	DET
cana-3462	219	4	degree	degree	NOUN
cana-3462	219	5	of	of	ADP
cana-3462	219	6	vertex	vertex	NOUN
cana-3462	219	7	of	of	ADP
cana-3462	219	8	𝒯	𝒯	PROPN
cana-3462	219	9	corona	corona	NOUN
cana-3462	219	10	is	be	AUX
cana-3462	219	11	given	give	VERB
cana-3462	219	12	by	by	ADP
cana-3462	219	13	𝑑𝐻1⋆𝐻2(𝑢𝑖	𝑑𝐻1⋆𝐻2(𝑢𝑖	NOUN
cana-3462	219	14	)	)	PUNCT
cana-3462	219	15	=	=	SYM
cana-3462	219	16	{	{	PUNCT
cana-3462	219	17	2𝑑𝐻1(𝑢𝑖	2𝑑𝐻1(𝑢𝑖	NUM
cana-3462	219	18	)	)	PUNCT
cana-3462	220	1	+	+	NUM
cana-3462	220	2	𝑝2	𝑝2	NOUN
cana-3462	220	3	,	,	PUNCT
cana-3462	220	4	if	if	SCONJ
cana-3462	220	5	𝑢	𝑢	PRON
cana-3462	220	6	∈	∈	PROPN
cana-3462	220	7	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	220	8	)	)	PUNCT
cana-3462	220	9	;	;	PUNCT
cana-3462	220	10	2𝑟1	2𝑟1	NOUN
cana-3462	220	11	,	,	PUNCT
cana-3462	220	12	if	if	SCONJ
cana-3462	220	13	𝑢	𝑢	PRON
cana-3462	220	14	∈	∈	PROPN
cana-3462	220	15	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	220	16	)	)	PUNCT
cana-3462	220	17	𝑑𝐻2(𝑣𝑗	𝑑𝐻2(𝑣𝑗	PUNCT
cana-3462	220	18	𝑖	𝑖	X
cana-3462	220	19	)	)	PUNCT
cana-3462	220	20	+	+	CCONJ
cana-3462	220	21	1	1	NUM
cana-3462	220	22	,	,	PUNCT
cana-3462	220	23	if	if	SCONJ
cana-3462	220	24	𝑢	𝑢	NOUN
cana-3462	220	25	=	=	PUNCT
cana-3462	220	26	𝑣𝑗	𝑣𝑗	ADP
cana-3462	220	27	𝑖	𝑖	X
cana-3462	220	28	,	,	PUNCT
cana-3462	220	29	for	for	ADP
cana-3462	220	30	𝑖	𝑖	PRON
cana-3462	220	31	=	=	SYM
cana-3462	220	32	1,2	1,2	NUM
cana-3462	220	33	,	,	PUNCT
cana-3462	220	34	…	…	PUNCT
cana-3462	220	35	𝑝1	𝑝1	NOUN
cana-3462	220	36	,	,	PUNCT
cana-3462	220	37	for	for	ADP
cana-3462	220	38	𝑗	𝑗	NOUN
cana-3462	220	39	=	=	SYM
cana-3462	220	40	1,2	1,2	NUM
cana-3462	220	41	,	,	PUNCT
cana-3462	220	42	…	…	PUNCT
cana-3462	220	43	𝑝2	𝑝2	NOUN
cana-3462	220	44	.	.	PUNCT
cana-3462	220	45	;	;	PUNCT
cana-3462	220	46	𝐻1	𝐻1	PROPN
cana-3462	220	47	⋆	⋆	VERB
cana-3462	220	48	𝐻2	𝐻2	PROPN
cana-3462	220	49	has	have	VERB
cana-3462	220	50	𝑝1	𝑝1	NOUN
cana-3462	220	51	+	+	CCONJ
cana-3462	220	52	𝑞1	𝑞1	PROPN
cana-3462	220	53	+	+	CCONJ
cana-3462	220	54	𝑝1𝑝2	𝑝1𝑝2	PRON
cana-3462	220	55	vertices	vertice	VERB
cana-3462	220	56	.	.	PUNCT
cana-3462	221	1	the	the	DET
cana-3462	221	2	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	221	3	of	of	ADP
cana-3462	221	4	𝒯-graph	𝒯-graph	PROPN
cana-3462	221	5	corona	corona	NOUN
cana-3462	221	6	is	be	AUX
cana-3462	221	7	easily	easily	ADV
cana-3462	221	8	obtained	obtain	VERB
cana-3462	221	9	by	by	ADP
cana-3462	221	10	using	use	VERB
cana-3462	221	11	equation	equation	NOUN
cana-3462	221	12	(	(	PUNCT
cana-3462	221	13	1.1	1.1	NUM
cana-3462	221	14	)	)	PUNCT
cana-3462	221	15	as	as	SCONJ
cana-3462	221	16	follows	follow	VERB
cana-3462	221	17	theorem	theorem	VERB
cana-3462	221	18	4.2	4.2	NUM
cana-3462	221	19	.	.	PUNCT
cana-3462	222	1	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	VERB
cana-3462	222	2	⋆	⋆	VERB
cana-3462	222	3	𝐻2	𝐻2	NOUN
cana-3462	222	4	)	)	PUNCT
cana-3462	222	5	=	=	SYM
cana-3462	223	1	2𝐸𝑄(𝐻1	2𝐸𝑄(𝐻1	NUM
cana-3462	223	2	)	)	PUNCT
cana-3462	224	1	+	+	CCONJ
cana-3462	224	2	𝑝1𝐸𝑄(𝐻2	𝑝1𝐸𝑄(𝐻2	PROPN
cana-3462	224	3	)	)	PUNCT
cana-3462	225	1	+	+	CCONJ
cana-3462	225	2	2𝑀1(𝐻1	2𝑀1(𝐻1	X
cana-3462	225	3	)	)	PUNCT
cana-3462	226	1	+	+	NUM
cana-3462	226	2	5𝑝1𝑝2	5𝑝1𝑝2	NUM
cana-3462	226	3	+	+	CCONJ
cana-3462	226	4	𝑝2	𝑝2	NOUN
cana-3462	226	5	2𝑝1	2𝑝1	NUM
cana-3462	226	6	+	+	CCONJ
cana-3462	226	7	4𝑟1	4𝑟1	NUM
cana-3462	226	8	2𝑞1	2𝑞1	NUM
cana-3462	227	1	+	+	CCONJ
cana-3462	227	2	2𝑟1𝑞1	2𝑟1𝑞1	NUM
cana-3462	228	1	+	+	CCONJ
cana-3462	228	2	4𝑝1𝑞2	4𝑝1𝑞2	NUM
cana-3462	229	1	+	+	CCONJ
cana-3462	229	2	2𝑝1𝑝2	2𝑝1𝑝2	NUM
cana-3462	229	3	where	where	SCONJ
cana-3462	229	4	,	,	PUNCT
cana-3462	229	5	𝑀1	𝑀1	PROPN
cana-3462	229	6	is	be	AUX
cana-3462	229	7	the	the	DET
cana-3462	229	8	first	first	ADJ
cana-3462	229	9	zagreb	zagreb	PROPN
cana-3462	229	10	index	index	NOUN
cana-3462	229	11	.	.	PUNCT
cana-3462	230	1	5	5	X
cana-3462	230	2	.	.	X
cana-3462	230	3	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	230	4	based	base	VERB
cana-3462	230	5	on	on	ADP
cana-3462	230	6	𝒮-graph	𝒮-graph	PROPN
cana-3462	230	7	and	and	CCONJ
cana-3462	230	8	ℛ-graph	ℛ-graph	PROPN
cana-3462	230	9	join	join	VERB
cana-3462	230	10	the	the	DET
cana-3462	230	11	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	230	12	energy	energy	NOUN
cana-3462	230	13	of	of	ADP
cana-3462	230	14	four	four	NUM
cana-3462	230	15	𝒮-graph	𝒮-graph	PROPN
cana-3462	230	16	and	and	CCONJ
cana-3462	230	17	ℛ-graph	ℛ-graph	PROPN
cana-3462	230	18	joins	join	VERB
cana-3462	230	19	with	with	ADP
cana-3462	230	20	their	their	PRON
cana-3462	230	21	corresponding	correspond	VERB
cana-3462	230	22	original	original	ADJ
cana-3462	230	23	graphs	graph	NOUN
cana-3462	230	24	𝐻1	𝐻1	NOUN
cana-3462	230	25	and	and	CCONJ
cana-3462	230	26	𝐻2	𝐻2	NOUN
cana-3462	230	27	are	be	AUX
cana-3462	230	28	formulated	formulate	VERB
cana-3462	230	29	in	in	ADP
cana-3462	230	30	this	this	DET
cana-3462	230	31	part	part	NOUN
cana-3462	230	32	.	.	PUNCT
cana-3462	231	1	5.1	5.1	NUM
cana-3462	231	2	.	.	PUNCT
cana-3462	232	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	232	2	of	of	ADP
cana-3462	232	3	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	232	4	and	and	CCONJ
cana-3462	232	5	ℛ-vertex	ℛ-vertex	PROPN
cana-3462	232	6	join	join	VERB
cana-3462	232	7	first	first	ADV
cana-3462	232	8	,	,	PUNCT
cana-3462	232	9	we	we	PRON
cana-3462	232	10	formulate	formulate	VERB
cana-3462	232	11	the	the	DET
cana-3462	232	12	quasi	quasi	ADJ
cana-3462	232	13	-	-	ADJ
cana-3462	232	14	laplacian	laplacian	ADJ
cana-3462	232	15	energy	energy	NOUN
cana-3462	232	16	of	of	ADP
cana-3462	232	17	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	232	18	and	and	CCONJ
cana-3462	232	19	ℛ-vertex	ℛ-vertex	PROPN
cana-3462	232	20	join	join	NOUN
cana-3462	232	21	.	.	PUNCT
cana-3462	233	1	let	let	VERB
cana-3462	233	2	𝑢	𝑢	PRON
cana-3462	233	3	be	be	AUX
cana-3462	233	4	any	any	DET
cana-3462	233	5	vertex	vertex	NOUN
cana-3462	233	6	in	in	ADP
cana-3462	233	7	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	233	8	and	and	CCONJ
cana-3462	233	9	ℛ-vertex	ℛ-vertex	PROPN
cana-3462	233	10	join	join	NOUN
cana-3462	233	11	.	.	PUNCT
cana-3462	234	1	then	then	ADV
cana-3462	234	2	,	,	PUNCT
cana-3462	234	3	the	the	DET
cana-3462	234	4	degrre	degrre	NOUN
cana-3462	234	5	of	of	ADP
cana-3462	234	6	any	any	DET
cana-3462	234	7	vertex	vertex	NOUN
cana-3462	234	8	in	in	ADP
cana-3462	234	9	𝒮(𝐻1	𝒮(𝐻1	NOUN
cana-3462	234	10	)	)	PUNCT
cana-3462	234	11	∨̈	∨̈	PROPN
cana-3462	234	12	ℛ(𝐻2	ℛ(𝐻2	NUM
cana-3462	234	13	)	)	PUNCT
cana-3462	234	14	is	be	AUX
cana-3462	234	15	given	give	VERB
cana-3462	234	16	by	by	ADP
cana-3462	234	17	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	NOUN
cana-3462	234	18	)	)	PUNCT
cana-3462	234	19	=	=	PRON
cana-3462	234	20	{	{	PUNCT
cana-3462	234	21	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	NOUN
cana-3462	234	22	)	)	PUNCT
cana-3462	235	1	+	+	NUM
cana-3462	235	2	𝑝2	𝑝2	NOUN
cana-3462	235	3	,	,	PUNCT
cana-3462	235	4	if	if	SCONJ
cana-3462	235	5	𝑢	𝑢	PRON
cana-3462	235	6	∈	∈	PROPN
cana-3462	235	7	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	235	8	)	)	PUNCT
cana-3462	235	9	;	;	PUNCT
cana-3462	235	10	2	2	X
cana-3462	235	11	,	,	PUNCT
cana-3462	235	12	if	if	SCONJ
cana-3462	235	13	𝑢	𝑢	PRON
cana-3462	235	14	∈	∈	PROPN
cana-3462	235	15	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	235	16	)	)	PUNCT
cana-3462	235	17	∪	∪	ADP
cana-3462	235	18	𝐼(𝐻2	𝐼(𝐻2	PROPN
cana-3462	235	19	)	)	PUNCT
cana-3462	235	20	;	;	PUNCT
cana-3462	235	21	2𝑑𝐻2(𝑢𝑖	2𝑑𝐻2(𝑢𝑖	X
cana-3462	235	22	)	)	PUNCT
cana-3462	235	23	+	+	CCONJ
cana-3462	235	24	𝑝1	𝑝1	NOUN
cana-3462	235	25	,	,	PUNCT
cana-3462	235	26	if	if	SCONJ
cana-3462	235	27	𝑢	𝑢	PRON
cana-3462	235	28	∈	∈	PROPN
cana-3462	235	29	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	235	30	)	)	PUNCT
cana-3462	235	31	.	.	PUNCT
cana-3462	236	1	𝒮(𝐻1	𝒮(𝐻1	PUNCT
cana-3462	236	2	)	)	PUNCT
cana-3462	236	3	∨̈	∨̈	PROPN
cana-3462	237	1	ℛ(𝐻2	ℛ(𝐻2	NUM
cana-3462	237	2	)	)	PUNCT
cana-3462	237	3	has	have	VERB
cana-3462	237	4	𝑝1	𝑝1	NOUN
cana-3462	237	5	+	+	CCONJ
cana-3462	237	6	𝑞1	𝑞1	PROPN
cana-3462	237	7	+	+	CCONJ
cana-3462	237	8	𝑝2	𝑝2	NOUN
cana-3462	237	9	+	+	CCONJ
cana-3462	237	10	𝑞2	𝑞2	PROPN
cana-3462	237	11	vertices	vertex	NOUN
cana-3462	237	12	.	.	PUNCT
cana-3462	238	1	theorem	theorem	VERB
cana-3462	238	2	5.1	5.1	NUM
cana-3462	238	3	.	.	PUNCT
cana-3462	239	1	the	the	DET
cana-3462	239	2	quasi	quasi	ADJ
cana-3462	239	3	-	-	ADJ
cana-3462	239	4	laplacian	laplacian	ADJ
cana-3462	239	5	energy	energy	NOUN
cana-3462	239	6	of	of	ADP
cana-3462	239	7	𝐸𝑄(𝒮(𝐻1	𝐸𝑄(𝒮(𝐻1	NOUN
cana-3462	239	8	)	)	PUNCT
cana-3462	239	9	∨̈	∨̈	PROPN
cana-3462	239	10	ℛ(𝐻2	ℛ(𝐻2	NUM
cana-3462	239	11	)	)	PUNCT
cana-3462	239	12	)	)	PUNCT
cana-3462	240	1	=	=	PUNCT
cana-3462	240	2	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	240	3	)	)	PUNCT
cana-3462	241	1	+	+	CCONJ
cana-3462	242	1	4𝐸𝑄(𝐻2	4𝐸𝑄(𝐻2	X
cana-3462	242	2	)	)	PUNCT
cana-3462	242	3	+	+	NUM
cana-3462	242	4	𝑝2	𝑝2	NOUN
cana-3462	242	5	2𝑝1	2𝑝1	NUM
cana-3462	242	6	+	+	CCONJ
cana-3462	242	7	𝑝1	𝑝1	NOUN
cana-3462	242	8	2𝑝2	2𝑝2	NUM
cana-3462	242	9	+	+	CCONJ
cana-3462	242	10	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	243	1	+	+	PUNCT
cana-3462	243	2	8𝑞2𝑝1	8𝑞2𝑝1	NUM
cana-3462	243	3	+	+	NUM
cana-3462	243	4	2𝑞2	2𝑞2	NUM
cana-3462	243	5	+	+	CCONJ
cana-3462	243	6	6𝑞1	6𝑞1	NUM
cana-3462	243	7	+	+	CCONJ
cana-3462	243	8	2𝑝1𝑝2	2𝑝1𝑝2	NUM
cana-3462	243	9	proof	proof	NOUN
cana-3462	243	10	.	.	PUNCT
cana-3462	244	1	by	by	ADP
cana-3462	244	2	using	use	VERB
cana-3462	244	3	equation	equation	NOUN
cana-3462	244	4	(	(	PUNCT
cana-3462	244	5	1.1	1.1	NUM
cana-3462	244	6	)	)	PUNCT
cana-3462	244	7	,	,	PUNCT
cana-3462	244	8	we	we	PRON
cana-3462	244	9	get	get	VERB
cana-3462	244	10	communications	communication	NOUN
cana-3462	244	11	on	on	ADP
cana-3462	244	12	applied	apply	VERB
cana-3462	244	13	nonlinear	nonlinear	ADJ
cana-3462	244	14	analysis	analysis	NOUN
cana-3462	244	15	issn	issn	NOUN
cana-3462	244	16	:	:	PUNCT
cana-3462	244	17	1074	1074	NUM
cana-3462	244	18	-	-	PUNCT
cana-3462	244	19	133x	133x	NUM
cana-3462	244	20	vol	vol	NOUN
cana-3462	244	21	32	32	NUM
cana-3462	244	22	no	no	NOUN
cana-3462	244	23	.	.	PUNCT
cana-3462	245	1	7s	7	NOUN
cana-3462	245	2	(	(	PUNCT
cana-3462	245	3	2025	2025	NUM
cana-3462	245	4	)	)	PUNCT
cana-3462	245	5	539	539	NUM
cana-3462	245	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	245	7	𝐸𝑄(𝒮(𝐻1	𝐸𝑄(𝒮(𝐻1	NOUN
cana-3462	245	8	)	)	PUNCT
cana-3462	245	9	∨̈	∨̈	PROPN
cana-3462	246	1	ℛ(𝐻2	ℛ(𝐻2	NUM
cana-3462	246	2	)	)	PUNCT
cana-3462	246	3	)	)	PUNCT
cana-3462	247	1	=	=	PUNCT
cana-3462	247	2	∑	∑	PUNCT
cana-3462	247	3	  	  	SPACE
cana-3462	247	4	𝑝1+𝑞1+𝑝2+𝑞2	𝑝1+𝑞1+𝑝2+𝑞2	PROPN
cana-3462	247	5	𝑖=1	𝑖=1	PROPN
cana-3462	247	6	 	 	SPACE
cana-3462	247	7	𝑑𝒮(𝐻1	𝑑𝒮(𝐻1	NOUN
cana-3462	247	8	)	)	PUNCT
cana-3462	247	9	2	2	NUM
cana-3462	247	10	∨̈	∨̈	PROPN
cana-3462	247	11	ℛ(𝐻2)(𝑢𝑖	ℛ(𝐻2)(𝑢𝑖	X
cana-3462	247	12	)	)	PUNCT
cana-3462	248	1	+	+	CCONJ
cana-3462	248	2	∑	∑	PART
cana-3462	248	3	  	  	SPACE
cana-3462	248	4	𝑝1+𝑞1+𝑝2+𝑞2	𝑝1+𝑞1+𝑝2+𝑞2	PROPN
cana-3462	248	5	𝑖=1	𝑖=1	PUNCT
cana-3462	248	6	 	 	SPACE
cana-3462	248	7	𝑑𝒮(𝐺1)∨̈ℛ(𝐺2)(𝑢𝑖	𝑑𝒮(𝐺1)∨̈ℛ(𝐺2)(𝑢𝑖	NOUN
cana-3462	248	8	)	)	PUNCT
cana-3462	249	1	=	=	X
cana-3462	249	2	∑	∑	PART
cana-3462	249	3	  	  	SPACE
cana-3462	249	4	𝑝1	𝑝1	PROPN
cana-3462	249	5	𝑖=1	𝑖=1	PROPN
cana-3462	249	6	 	 	SPACE
cana-3462	249	7	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	PROPN
cana-3462	249	8	)	)	PUNCT
cana-3462	249	9	2	2	NUM
cana-3462	249	10	(	(	PUNCT
cana-3462	249	11	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	249	12	)	)	PUNCT
cana-3462	249	13	+	+	ADJ
cana-3462	249	14	∑	∑	PROPN
cana-3462	249	15	  	  	SPACE
cana-3462	249	16	𝑞1	𝑞1	PROPN
cana-3462	249	17	𝑖=1	𝑖=1	PROPN
cana-3462	249	18	 	 	SPACE
cana-3462	249	19	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	PROPN
cana-3462	249	20	)	)	PUNCT
cana-3462	249	21	2	2	NUM
cana-3462	249	22	(	(	PUNCT
cana-3462	249	23	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	249	24	)	)	PUNCT
cana-3462	249	25	+	+	ADJ
cana-3462	249	26	∑	∑	ADJ
cana-3462	249	27	  	  	SPACE
cana-3462	249	28	𝑝2	𝑝2	NOUN
cana-3462	249	29	𝑗=1	𝑗=1	PROPN
cana-3462	249	30	 	 	SPACE
cana-3462	249	31	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	PROPN
cana-3462	249	32	)	)	PUNCT
cana-3462	249	33	2	2	NUM
cana-3462	249	34	(	(	PUNCT
cana-3462	249	35	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	249	36	)	)	PUNCT
cana-3462	249	37	+	+	ADJ
cana-3462	249	38	∑	∑	ADJ
cana-3462	249	39	  	  	SPACE
cana-3462	249	40	𝑞2	𝑞2	NOUN
cana-3462	249	41	𝑖=1	𝑖=1	PROPN
cana-3462	249	42	 	 	SPACE
cana-3462	249	43	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2	PROPN
cana-3462	249	44	)	)	PUNCT
cana-3462	249	45	2	2	NUM
cana-3462	249	46	(	(	PUNCT
cana-3462	249	47	𝑢𝑖	𝑢𝑖	NOUN
cana-3462	249	48	)	)	PUNCT
cana-3462	249	49	+	+	ADJ
cana-3462	249	50	∑	∑	PROPN
cana-3462	249	51	  	  	SPACE
cana-3462	249	52	𝑝1	𝑝1	PROPN
cana-3462	249	53	𝑖=1	𝑖=1	PROPN
cana-3462	249	54	 	 	SPACE
cana-3462	249	55	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	NOUN
cana-3462	249	56	)	)	PUNCT
cana-3462	250	1	+	+	ADJ
cana-3462	250	2	∑	∑	PROPN
cana-3462	250	3	  	  	SPACE
cana-3462	250	4	𝑞1	𝑞1	PROPN
cana-3462	250	5	𝑖=1	𝑖=1	PROPN
cana-3462	250	6	 	 	SPACE
cana-3462	250	7	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	NOUN
cana-3462	250	8	)	)	PUNCT
cana-3462	251	1	+	+	ADJ
cana-3462	251	2	∑	∑	ADJ
cana-3462	251	3	  	  	SPACE
cana-3462	251	4	𝑝2	𝑝2	NOUN
cana-3462	251	5	𝑖=1	𝑖=1	PROPN
cana-3462	251	6	 	 	SPACE
cana-3462	251	7	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	NOUN
cana-3462	251	8	)	)	PUNCT
cana-3462	252	1	+	+	ADJ
cana-3462	252	2	∑	∑	ADJ
cana-3462	252	3	  	  	SPACE
cana-3462	252	4	𝑞2	𝑞2	NOUN
cana-3462	252	5	𝑖=1	𝑖=1	PROPN
cana-3462	252	6	 	 	SPACE
cana-3462	252	7	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖	NOUN
cana-3462	252	8	)	)	PUNCT
cana-3462	252	9	hence	hence	ADV
cana-3462	252	10	,	,	PUNCT
cana-3462	252	11	the	the	DET
cana-3462	252	12	result	result	NOUN
cana-3462	252	13	follows	follow	VERB
cana-3462	252	14	.	.	PUNCT
cana-3462	253	1	5.2	5.2	NUM
cana-3462	253	2	.	.	PUNCT
cana-3462	254	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	254	2	of	of	ADP
cana-3462	254	3	𝒮-edge	𝒮-edge	PROPN
cana-3462	254	4	and	and	CCONJ
cana-3462	254	5	ℛ-edge	ℛ-edge	PROPN
cana-3462	254	6	join	join	VERB
cana-3462	254	7	let	let	VERB
cana-3462	254	8	𝑢	𝑢	PRON
cana-3462	254	9	be	be	AUX
cana-3462	254	10	any	any	DET
cana-3462	254	11	vertex	vertex	NOUN
cana-3462	254	12	in	in	ADP
cana-3462	254	13	𝒮-edge	𝒮-edge	PROPN
cana-3462	254	14	and	and	CCONJ
cana-3462	254	15	ℛ-edge	ℛ-edge	PROPN
cana-3462	254	16	join	join	NOUN
cana-3462	254	17	.	.	PUNCT
cana-3462	255	1	then	then	ADV
cana-3462	255	2	,	,	PUNCT
cana-3462	255	3	the	the	DET
cana-3462	255	4	degree	degree	NOUN
cana-3462	255	5	of	of	ADP
cana-3462	255	6	any	any	DET
cana-3462	255	7	vertex	vertex	NOUN
cana-3462	255	8	of	of	ADP
cana-3462	255	9	𝑢	𝑢	NOUN
cana-3462	255	10	in	in	ADP
cana-3462	255	11	𝒮-edge	𝒮-edge	PROPN
cana-3462	255	12	and	and	CCONJ
cana-3462	255	13	ℛ-edge	ℛ-edge	PROPN
cana-3462	255	14	join	join	VERB
cana-3462	255	15	𝒮(𝐻1)𝑉‾ℛ(𝐻2	𝒮(𝐻1)𝑉‾ℛ(𝐻2	NOUN
cana-3462	255	16	)	)	PUNCT
cana-3462	255	17	is	be	AUX
cana-3462	255	18	given	give	VERB
cana-3462	255	19	by	by	ADP
cana-3462	255	20	𝑑𝒮(𝐻1)𝑉‾ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)𝑉‾ℛ(𝐻2)(𝑢𝑖	PROPN
cana-3462	255	21	)	)	PUNCT
cana-3462	256	1	=	=	PRON
cana-3462	256	2	{	{	PUNCT
cana-3462	256	3	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	256	4	)	)	PUNCT
cana-3462	256	5	,	,	PUNCT
cana-3462	256	6	if	if	SCONJ
cana-3462	256	7	𝑢	𝑢	PRON
cana-3462	256	8	∈	∈	PROPN
cana-3462	256	9	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	256	10	)	)	PUNCT
cana-3462	256	11	;	;	PUNCT
cana-3462	256	12	2	2	NUM
cana-3462	256	13	+	+	NUM
cana-3462	256	14	𝑞2	𝑞2	NOUN
cana-3462	256	15	,	,	PUNCT
cana-3462	256	16	if	if	SCONJ
cana-3462	256	17	𝑣	𝑣	PRON
cana-3462	256	18	∈	∈	PROPN
cana-3462	256	19	𝐼(𝐻1	𝐼(𝐻1	PROPN
cana-3462	256	20	)	)	PUNCT
cana-3462	256	21	;	;	PUNCT
cana-3462	256	22	2𝑑𝐻2(𝑢𝑖	2𝑑𝐻2(𝑢𝑖	NUM
cana-3462	256	23	)	)	PUNCT
cana-3462	256	24	,	,	PUNCT
cana-3462	256	25	if	if	SCONJ
cana-3462	256	26	𝑢	𝑢	PRON
cana-3462	256	27	∈	∈	PROPN
cana-3462	256	28	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	256	29	)	)	PUNCT
cana-3462	256	30	;	;	PUNCT
cana-3462	256	31	2	2	NUM
cana-3462	256	32	+	+	CCONJ
cana-3462	256	33	𝑞1	𝑞1	ADJ
cana-3462	256	34	,	,	PUNCT
cana-3462	256	35	if	if	SCONJ
cana-3462	256	36	𝑢	𝑢	PROPN
cana-3462	256	37	∈	∈	PROPN
cana-3462	256	38	𝐼(𝐻2	𝐼(𝐻2	PROPN
cana-3462	256	39	)	)	PUNCT
cana-3462	256	40	.	.	PUNCT
cana-3462	257	1	𝒮(𝐻1)𝑉‾ℛ(𝐻2	𝒮(𝐻1)𝑉‾ℛ(𝐻2	NOUN
cana-3462	257	2	)	)	PUNCT
cana-3462	257	3	has	have	VERB
cana-3462	257	4	𝑝1	𝑝1	NOUN
cana-3462	257	5	+	+	CCONJ
cana-3462	257	6	𝑞1	𝑞1	PROPN
cana-3462	257	7	+	+	CCONJ
cana-3462	257	8	𝑝2	𝑝2	NOUN
cana-3462	257	9	+	+	CCONJ
cana-3462	257	10	𝑞2	𝑞2	NOUN
cana-3462	257	11	vertices	vertex	NOUN
cana-3462	257	12	.	.	PUNCT
cana-3462	258	1	the	the	DET
cana-3462	258	2	𝑄-energy	𝑄-energy	PROPN
cana-3462	258	3	is	be	AUX
cana-3462	258	4	obtained	obtain	VERB
cana-3462	258	5	easily	easily	ADV
cana-3462	258	6	by	by	ADP
cana-3462	258	7	using	use	VERB
cana-3462	258	8	equation	equation	NOUN
cana-3462	258	9	(	(	PUNCT
cana-3462	258	10	1.1	1.1	NUM
cana-3462	258	11	)	)	PUNCT
cana-3462	258	12	.	.	PUNCT
cana-3462	259	1	theorem	theorem	VERB
cana-3462	259	2	5.2	5.2	NUM
cana-3462	259	3	.	.	PUNCT
cana-3462	260	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	260	2	(	(	PUNCT
cana-3462	260	3	𝒮(𝐻1)𝑉‾ℛ(𝐻2	𝒮(𝐻1)𝑉‾ℛ(𝐻2	NOUN
cana-3462	260	4	)	)	PUNCT
cana-3462	260	5	)	)	PUNCT
cana-3462	261	1	=	=	PUNCT
cana-3462	261	2	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	261	3	)	)	PUNCT
cana-3462	262	1	+	+	CCONJ
cana-3462	263	1	4𝐸𝑄(𝐻2	4𝐸𝑄(𝐻2	X
cana-3462	263	2	)	)	PUNCT
cana-3462	263	3	+	+	NUM
cana-3462	263	4	6𝑞1	6𝑞1	NUM
cana-3462	263	5	+	+	CCONJ
cana-3462	263	6	10𝑞1𝑞2	10𝑞1𝑞2	X
cana-3462	263	7	+	+	CCONJ
cana-3462	263	8	𝑞1𝑞2(𝑞1	𝑞1𝑞2(𝑞1	ADJ
cana-3462	263	9	+	+	SYM
cana-3462	263	10	𝑞2	𝑞2	NOUN
cana-3462	263	11	)	)	PUNCT
cana-3462	263	12	+	+	CCONJ
cana-3462	263	13	6𝑞2	6𝑞2	NUM
cana-3462	263	14	5.3	5.3	NUM
cana-3462	263	15	.	.	PUNCT
cana-3462	264	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	264	2	of	of	ADP
cana-3462	264	3	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	264	4	and	and	CCONJ
cana-3462	264	5	ℛ-edge	ℛ-edge	PROPN
cana-3462	264	6	join	join	VERB
cana-3462	264	7	let	let	VERB
cana-3462	264	8	𝑢	𝑢	PRON
cana-3462	264	9	be	be	AUX
cana-3462	264	10	any	any	DET
cana-3462	264	11	vertex	vertex	NOUN
cana-3462	264	12	in	in	ADP
cana-3462	264	13	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	264	14	and	and	CCONJ
cana-3462	264	15	ℛ-edge	ℛ-edge	PROPN
cana-3462	264	16	join	join	NOUN
cana-3462	264	17	.	.	PUNCT
cana-3462	265	1	then	then	ADV
cana-3462	265	2	,	,	PUNCT
cana-3462	265	3	the	the	DET
cana-3462	265	4	degree	degree	NOUN
cana-3462	265	5	of	of	ADP
cana-3462	265	6	any	any	DET
cana-3462	265	7	vertex	vertex	NOUN
cana-3462	265	8	of	of	ADP
cana-3462	265	9	𝑢	𝑢	NOUN
cana-3462	265	10	in	in	ADP
cana-3462	265	11	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	265	12	and	and	CCONJ
cana-3462	265	13	ℛ-edge	ℛ-edge	PROPN
cana-3462	265	14	join	join	VERB
cana-3462	265	15	𝒮(𝐻1)𝑉‾̇ℛ(𝐻2	𝒮(𝐻1)𝑉‾̇ℛ(𝐻2	NOUN
cana-3462	265	16	)	)	PUNCT
cana-3462	265	17	is	be	AUX
cana-3462	265	18	given	give	VERB
cana-3462	265	19	by	by	ADP
cana-3462	265	20	𝑑𝒮(𝐻1)	𝑑𝒮(𝐻1)	PROPN
cana-3462	265	21	�	�	PROPN
cana-3462	265	22	̇	̇	PROPN
cana-3462	265	23	�	�	NOUN
cana-3462	265	24	ℛ(𝐻2)(𝑢𝑖	ℛ(𝐻2)(𝑢𝑖	PROPN
cana-3462	265	25	)	)	PUNCT
cana-3462	266	1	=	=	PRON
cana-3462	266	2	{	{	PUNCT
cana-3462	266	3	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	NOUN
cana-3462	266	4	)	)	PUNCT
cana-3462	267	1	+	+	NUM
cana-3462	267	2	𝑞2	𝑞2	NOUN
cana-3462	267	3	,	,	PUNCT
cana-3462	267	4	if	if	SCONJ
cana-3462	267	5	𝑢	𝑢	PRON
cana-3462	267	6	∈	∈	PROPN
cana-3462	267	7	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	267	8	)	)	PUNCT
cana-3462	267	9	;	;	PUNCT
cana-3462	267	10	2	2	X
cana-3462	267	11	,	,	PUNCT
cana-3462	267	12	if	if	SCONJ
cana-3462	267	13	𝑢	𝑢	PRON
cana-3462	267	14	∈	∈	PROPN
cana-3462	267	15	𝐼(𝐻1	𝐼(𝐻1	PROPN
cana-3462	267	16	)	)	PUNCT
cana-3462	267	17	;	;	PUNCT
cana-3462	267	18	2𝑑𝐻2(𝑢𝑖	2𝑑𝐻2(𝑢𝑖	NUM
cana-3462	267	19	)	)	PUNCT
cana-3462	267	20	,	,	PUNCT
cana-3462	267	21	if	if	SCONJ
cana-3462	267	22	𝑢	𝑢	PRON
cana-3462	267	23	∈	∈	PROPN
cana-3462	267	24	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	267	25	)	)	PUNCT
cana-3462	267	26	;	;	PUNCT
cana-3462	267	27	2	2	NUM
cana-3462	267	28	+	+	NUM
cana-3462	267	29	𝑝1	𝑝1	NOUN
cana-3462	267	30	,	,	PUNCT
cana-3462	267	31	if	if	SCONJ
cana-3462	267	32	𝑢	𝑢	PROPN
cana-3462	267	33	∈	∈	PROPN
cana-3462	267	34	𝐼(𝐻2	𝐼(𝐻2	PROPN
cana-3462	267	35	)	)	PUNCT
cana-3462	267	36	.	.	PUNCT
cana-3462	268	1	𝒮(𝐺1)𝑉‾̇ℛ(𝐺2	𝒮(𝐺1)𝑉‾̇ℛ(𝐺2	PROPN
cana-3462	268	2	)	)	PUNCT
cana-3462	268	3	has	have	VERB
cana-3462	268	4	𝑝1	𝑝1	NOUN
cana-3462	268	5	+	+	CCONJ
cana-3462	268	6	𝑞1	𝑞1	PROPN
cana-3462	268	7	+	+	CCONJ
cana-3462	268	8	𝑝2	𝑝2	NOUN
cana-3462	268	9	+	+	CCONJ
cana-3462	268	10	𝑞2	𝑞2	NOUN
cana-3462	268	11	vertices	vertex	NOUN
cana-3462	268	12	.	.	PUNCT
cana-3462	269	1	we	we	PRON
cana-3462	269	2	get	get	VERB
cana-3462	269	3	the	the	DET
cana-3462	269	4	following	follow	VERB
cana-3462	269	5	result	result	NOUN
cana-3462	269	6	by	by	ADP
cana-3462	269	7	using	use	VERB
cana-3462	269	8	equation	equation	NOUN
cana-3462	269	9	(	(	PUNCT
cana-3462	269	10	1.1	1.1	NUM
cana-3462	269	11	)	)	PUNCT
cana-3462	269	12	.	.	PUNCT
cana-3462	270	1	communications	communication	NOUN
cana-3462	270	2	on	on	ADP
cana-3462	270	3	applied	apply	VERB
cana-3462	270	4	nonlinear	nonlinear	ADJ
cana-3462	270	5	analysis	analysis	NOUN
cana-3462	270	6	issn	issn	NOUN
cana-3462	270	7	:	:	PUNCT
cana-3462	270	8	1074	1074	NUM
cana-3462	270	9	-	-	PUNCT
cana-3462	270	10	133x	133x	NUM
cana-3462	270	11	vol	vol	NOUN
cana-3462	270	12	32	32	NUM
cana-3462	270	13	no	no	NOUN
cana-3462	270	14	.	.	PUNCT
cana-3462	271	1	7s	7	NOUN
cana-3462	271	2	(	(	PUNCT
cana-3462	271	3	2025	2025	NUM
cana-3462	271	4	)	)	PUNCT
cana-3462	271	5	540	540	NUM
cana-3462	271	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	271	7	theorem	theorem	VERB
cana-3462	271	8	5.3	5.3	NUM
cana-3462	271	9	.	.	PUNCT
cana-3462	272	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	272	2	(	(	PUNCT
cana-3462	272	3	𝒮(𝐻1)𝑉‾̇ℛ(𝐻2	𝒮(𝐻1)𝑉‾̇ℛ(𝐻2	PROPN
cana-3462	272	4	)	)	PUNCT
cana-3462	272	5	)	)	PUNCT
cana-3462	272	6	=	=	PUNCT
cana-3462	272	7	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	272	8	)	)	PUNCT
cana-3462	272	9	+	+	CCONJ
cana-3462	272	10	4𝐸𝑄(𝐻2	4𝐸𝑄(𝐻2	X
cana-3462	272	11	)	)	PUNCT
cana-3462	273	1	+	+	NUM
cana-3462	273	2	𝑞2	𝑞2	NOUN
cana-3462	273	3	2𝑝1	2𝑝1	NUM
cana-3462	273	4	+	+	CCONJ
cana-3462	273	5	4𝑞1𝑞2	4𝑞1𝑞2	NUM
cana-3462	273	6	+	+	NUM
cana-3462	273	7	6𝑞1	6𝑞1	NUM
cana-3462	273	8	+	+	SYM
cana-3462	273	9	6𝑞2𝑝1	6𝑞2𝑝1	NUM
cana-3462	273	10	+	+	CCONJ
cana-3462	273	11	𝑝2	𝑝2	NOUN
cana-3462	273	12	2𝑞2	2𝑞2	NUM
cana-3462	273	13	+	+	CCONJ
cana-3462	273	14	4𝑞2	4𝑞2	NUM
cana-3462	273	15	5.4	5.4	NUM
cana-3462	273	16	.	.	PUNCT
cana-3462	274	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	274	2	of	of	ADP
cana-3462	274	3	𝒮-edge	𝒮-edge	PROPN
cana-3462	274	4	and	and	CCONJ
cana-3462	274	5	ℛ-vertex	ℛ-vertex	PROPN
cana-3462	274	6	join	join	VERB
cana-3462	274	7	let	let	VERB
cana-3462	274	8	𝑢	𝑢	PRON
cana-3462	274	9	be	be	AUX
cana-3462	274	10	any	any	DET
cana-3462	274	11	vertex	vertex	NOUN
cana-3462	274	12	in	in	ADP
cana-3462	274	13	𝒮-edge	𝒮-edge	PROPN
cana-3462	274	14	and	and	CCONJ
cana-3462	274	15	ℛ-vertex	ℛ-vertex	PROPN
cana-3462	274	16	join	join	NOUN
cana-3462	274	17	.	.	PUNCT
cana-3462	275	1	then	then	ADV
cana-3462	275	2	,	,	PUNCT
cana-3462	275	3	the	the	DET
cana-3462	275	4	degree	degree	NOUN
cana-3462	275	5	of	of	ADP
cana-3462	275	6	any	any	DET
cana-3462	275	7	vertex	vertex	NOUN
cana-3462	275	8	of	of	ADP
cana-3462	275	9	𝑢	𝑢	NOUN
cana-3462	275	10	in	in	ADP
cana-3462	275	11	𝒮-edge	𝒮-edge	PROPN
cana-3462	275	12	and	and	CCONJ
cana-3462	275	13	ℛ-vertex	ℛ-vertex	PROPN
cana-3462	275	14	join	join	VERB
cana-3462	275	15	𝒮(𝐻1)𝑉‾̇ℛ(𝐻2	𝒮(𝐻1)𝑉‾̇ℛ(𝐻2	NOUN
cana-3462	275	16	)	)	PUNCT
cana-3462	275	17	is	be	AUX
cana-3462	275	18	given	give	VERB
cana-3462	275	19	by	by	ADP
cana-3462	275	20	𝑑𝒮(𝐻1)𝑉‾ℛ(𝐻2)(𝑢𝑖	𝑑𝒮(𝐻1)𝑉‾ℛ(𝐻2)(𝑢𝑖	PROPN
cana-3462	275	21	)	)	PUNCT
cana-3462	275	22	=	=	PRON
cana-3462	275	23	{	{	PUNCT
cana-3462	275	24	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	PROPN
cana-3462	275	25	)	)	PUNCT
cana-3462	275	26	,	,	PUNCT
cana-3462	275	27	if	if	SCONJ
cana-3462	275	28	𝑢	𝑢	PRON
cana-3462	275	29	∈	∈	PROPN
cana-3462	275	30	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	275	31	)	)	PUNCT
cana-3462	275	32	;	;	PUNCT
cana-3462	275	33	2	2	NUM
cana-3462	275	34	+	+	NUM
cana-3462	275	35	𝑝2	𝑝2	NOUN
cana-3462	275	36	,	,	PUNCT
cana-3462	275	37	if	if	SCONJ
cana-3462	275	38	𝑢	𝑢	PRON
cana-3462	275	39	∈	∈	PROPN
cana-3462	275	40	𝐼(𝐻1	𝐼(𝐻1	PROPN
cana-3462	275	41	)	)	PUNCT
cana-3462	275	42	;	;	PUNCT
cana-3462	275	43	2𝑑𝐻2(𝑢𝑖	2𝑑𝐻2(𝑢𝑖	X
cana-3462	275	44	)	)	PUNCT
cana-3462	276	1	+	+	CCONJ
cana-3462	277	1	𝑞1	𝑞1	ADJ
cana-3462	277	2	,	,	PUNCT
cana-3462	277	3	if	if	SCONJ
cana-3462	277	4	𝑢	𝑢	PRON
cana-3462	277	5	∈	∈	PROPN
cana-3462	277	6	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	277	7	)	)	PUNCT
cana-3462	277	8	;	;	PUNCT
cana-3462	277	9	2	2	X
cana-3462	277	10	,	,	PUNCT
cana-3462	277	11	if	if	SCONJ
cana-3462	277	12	𝑢	𝑢	PROPN
cana-3462	277	13	∈	∈	PROPN
cana-3462	277	14	𝐼(𝐻2	𝐼(𝐻2	PROPN
cana-3462	277	15	)	)	PUNCT
cana-3462	277	16	.	.	PUNCT
cana-3462	278	1	𝒮(𝐻1)	𝒮(𝐻1)	PROPN
cana-3462	278	2	�	�	PROPN
cana-3462	278	3	̇	̇	PROPN
cana-3462	278	4	�	�	PROPN
cana-3462	278	5	ℛ(𝐻2	ℛ(𝐻2	NUM
cana-3462	278	6	)	)	PUNCT
cana-3462	278	7	has	have	VERB
cana-3462	278	8	𝑝1	𝑝1	NOUN
cana-3462	278	9	+	+	CCONJ
cana-3462	278	10	𝑞1	𝑞1	PROPN
cana-3462	278	11	+	+	CCONJ
cana-3462	278	12	𝑝2	𝑝2	NOUN
cana-3462	278	13	+	+	CCONJ
cana-3462	278	14	𝑞2	𝑞2	NOUN
cana-3462	278	15	vertices	vertex	NOUN
cana-3462	278	16	.	.	PUNCT
cana-3462	279	1	we	we	PRON
cana-3462	279	2	easily	easily	ADV
cana-3462	279	3	get	get	VERB
cana-3462	279	4	the	the	DET
cana-3462	279	5	following	following	NOUN
cana-3462	279	6	theorem	theorem	VERB
cana-3462	279	7	by	by	ADP
cana-3462	279	8	using	use	VERB
cana-3462	279	9	equation	equation	NOUN
cana-3462	279	10	(	(	PUNCT
cana-3462	279	11	1.1	1.1	NUM
cana-3462	279	12	)	)	PUNCT
cana-3462	279	13	theorem	theorem	VERB
cana-3462	279	14	5.4	5.4	NUM
cana-3462	279	15	.	.	PUNCT
cana-3462	280	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	280	2	(	(	PUNCT
cana-3462	280	3	𝒮(𝐻1)	𝒮(𝐻1)	PROPN
cana-3462	280	4	�	�	PROPN
cana-3462	280	5	̇	̇	NOUN
cana-3462	280	6	�	�	PROPN
cana-3462	280	7	ℛ(𝐻2	ℛ(𝐻2	NUM
cana-3462	280	8	)	)	PUNCT
cana-3462	280	9	)	)	PUNCT
cana-3462	280	10	=	=	PUNCT
cana-3462	280	11	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	280	12	)	)	PUNCT
cana-3462	281	1	+	+	CCONJ
cana-3462	282	1	4𝐸𝑄(𝐻2	4𝐸𝑄(𝐻2	X
cana-3462	282	2	)	)	PUNCT
cana-3462	282	3	+	+	NUM
cana-3462	282	4	6𝑞1	6𝑞1	NUM
cana-3462	282	5	+	+	CCONJ
cana-3462	282	6	6𝑞1𝑝2	6𝑞1𝑝2	NUM
cana-3462	282	7	+	+	CCONJ
cana-3462	282	8	𝑞1	𝑞1	PROPN
cana-3462	282	9	2𝑝2	2𝑝2	NUM
cana-3462	283	1	+	+	CCONJ
cana-3462	283	2	𝑞1𝑝2	𝑞1𝑝2	VERB
cana-3462	283	3	2	2	NUM
cana-3462	283	4	+	+	NUM
cana-3462	283	5	10𝑞2	10𝑞2	NUM
cana-3462	283	6	6	6	NUM
cana-3462	283	7	.	.	PUNCT
cana-3462	284	1	𝐸𝑄	𝐸𝑄	PROPN
cana-3462	284	2	of	of	ADP
cana-3462	284	3	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	284	4	-	-	PUNCT
cana-3462	284	5	vertex	vertex	NOUN
cana-3462	284	6	-	-	PUNCT
cana-3462	284	7	edge	edge	NOUN
cana-3462	284	8	join	join	NOUN
cana-3462	284	9	of	of	ADP
cana-3462	284	10	triple	triple	ADJ
cana-3462	284	11	graphs	graph	NOUN
cana-3462	284	12	let	let	VERB
cana-3462	284	13	𝐻3	𝐻3	NOUN
cana-3462	284	14	be	be	AUX
cana-3462	284	15	(	(	PUNCT
cana-3462	284	16	𝑝3	𝑝3	ADV
cana-3462	284	17	,	,	PUNCT
cana-3462	284	18	𝑞3	𝑞3	NOUN
cana-3462	284	19	)	)	PUNCT
cana-3462	284	20	graph	graph	NOUN
cana-3462	284	21	.	.	PUNCT
cana-3462	285	1	then	then	ADV
cana-3462	285	2	𝒮-vertex	𝒮-vertex	PROPN
cana-3462	285	3	-	-	PUNCT
cana-3462	285	4	vertex	vertex	NOUN
cana-3462	285	5	-	-	PUNCT
cana-3462	285	6	edge	edge	NOUN
cana-3462	285	7	join	join	NOUN
cana-3462	285	8	of	of	ADP
cana-3462	285	9	three	three	NUM
cana-3462	285	10	𝒮	𝒮	NOUN
cana-3462	285	11	graphs	graph	NOUN
cana-3462	285	12	is	be	AUX
cana-3462	285	13	denoted	denote	VERB
cana-3462	285	14	by	by	ADP
cana-3462	285	15	𝐻1	𝐻1	NOUN
cana-3462	285	16	𝑆	𝑆	PROPN
cana-3462	285	17	▹	▹	PROPN
cana-3462	285	18	(	(	PUNCT
cana-3462	285	19	𝐻2	𝐻2	PROPN
cana-3462	285	20	𝑉	𝑉	PROPN
cana-3462	285	21	∪	∪	ADJ
cana-3462	285	22	𝐻3	𝐻3	PROPN
cana-3462	285	23	𝐸	𝐸	PROPN
cana-3462	285	24	)	)	PUNCT
cana-3462	285	25	.	.	PUNCT
cana-3462	286	1	the	the	DET
cana-3462	286	2	degree	degree	NOUN
cana-3462	286	3	of	of	ADP
cana-3462	286	4	vertex	vertex	NOUN
cana-3462	286	5	𝑢	𝑢	PROPN
cana-3462	286	6	∈	∈	PROPN
cana-3462	286	7	𝐻1	𝐻1	NOUN
cana-3462	286	8	𝑆	𝑆	PROPN
cana-3462	286	9	▹	▹	PROPN
cana-3462	286	10	(	(	PUNCT
cana-3462	286	11	𝐻2	𝐻2	PROPN
cana-3462	286	12	𝑉	𝑉	PROPN
cana-3462	286	13	∪	∪	ADJ
cana-3462	286	14	𝐻3	𝐻3	PROPN
cana-3462	286	15	𝐸	𝐸	PROPN
cana-3462	286	16	)	)	PUNCT
cana-3462	286	17	is	be	AUX
cana-3462	286	18	given	give	VERB
cana-3462	286	19	by	by	ADP
cana-3462	286	20	𝑑𝐻1𝑆	𝑑𝐻1𝑆	NOUN
cana-3462	286	21	▹	▹	NOUN
cana-3462	286	22	(𝐻2𝑉∪𝐻3𝐸	(𝐻2𝑉∪𝐻3𝐸	PROPN
cana-3462	286	23	)	)	PUNCT
cana-3462	286	24	(	(	PUNCT
cana-3462	286	25	𝑢𝑖	𝑢𝑖	X
cana-3462	286	26	)	)	PUNCT
cana-3462	286	27	=	=	PRON
cana-3462	286	28	{	{	PUNCT
cana-3462	286	29	𝑑𝐻1(𝑢𝑖	𝑑𝐻1(𝑢𝑖	NOUN
cana-3462	286	30	)	)	PUNCT
cana-3462	286	31	+	+	NUM
cana-3462	287	1	𝑝2	𝑝2	NOUN
cana-3462	287	2	,	,	PUNCT
cana-3462	287	3	if	if	SCONJ
cana-3462	287	4	𝑢	𝑢	PRON
cana-3462	287	5	∈	∈	PROPN
cana-3462	287	6	𝑉(𝐻1	𝑉(𝐻1	PROPN
cana-3462	287	7	)	)	PUNCT
cana-3462	287	8	;	;	PUNCT
cana-3462	287	9	2	2	NUM
cana-3462	287	10	+	+	NUM
cana-3462	287	11	𝑝1	𝑝1	NOUN
cana-3462	287	12	,	,	PUNCT
cana-3462	287	13	if	if	SCONJ
cana-3462	287	14	𝑢	𝑢	PRON
cana-3462	287	15	∈	∈	PROPN
cana-3462	287	16	𝐼(𝐻1	𝐼(𝐻1	NOUN
cana-3462	287	17	)	)	PUNCT
cana-3462	287	18	;	;	PUNCT
cana-3462	287	19	𝑑𝐻2(𝑢𝑗	𝑑𝐻2(𝑢𝑗	NOUN
cana-3462	287	20	)	)	PUNCT
cana-3462	288	1	+	+	CCONJ
cana-3462	288	2	𝑝1	𝑝1	NOUN
cana-3462	288	3	,	,	PUNCT
cana-3462	288	4	if	if	SCONJ
cana-3462	288	5	𝑢	𝑢	PRON
cana-3462	288	6	∈	∈	PROPN
cana-3462	288	7	𝑉(𝐻2	𝑉(𝐻2	VERB
cana-3462	288	8	)	)	PUNCT
cana-3462	288	9	,	,	PUNCT
cana-3462	288	10	𝑗	𝑗	NOUN
cana-3462	288	11	=	=	SYM
cana-3462	288	12	1,2	1,2	NUM
cana-3462	288	13	,	,	PUNCT
cana-3462	288	14	…	…	PUNCT
cana-3462	288	15	𝑝2	𝑝2	NOUN
cana-3462	288	16	;	;	PUNCT
cana-3462	288	17	𝑑𝐻3(𝑤𝑘	𝑑𝐻3(𝑤𝑘	NOUN
cana-3462	288	18	)	)	PUNCT
cana-3462	288	19	,	,	PUNCT
cana-3462	288	20	if	if	SCONJ
cana-3462	288	21	𝑢	𝑢	X
cana-3462	288	22	=	=	VERB
cana-3462	288	23	𝑤𝑘	𝑤𝑘	PRON
cana-3462	288	24	∈	∈	PROPN
cana-3462	288	25	𝑉(𝐻3	𝑉(𝐻3	PROPN
cana-3462	288	26	)	)	PUNCT
cana-3462	288	27	,	,	PUNCT
cana-3462	288	28	𝑘	𝑘	X
cana-3462	288	29	=	=	SYM
cana-3462	288	30	1,2	1,2	NUM
cana-3462	288	31	…	…	PUNCT
cana-3462	288	32	𝑝3	𝑝3	NOUN
cana-3462	288	33	.	.	PUNCT
cana-3462	289	1	𝐻1	𝐻1	PROPN
cana-3462	290	1	𝑆	𝑆	PROPN
cana-3462	290	2	▹	▹	PROPN
cana-3462	290	3	(	(	PUNCT
cana-3462	290	4	𝐻2	𝐻2	PROPN
cana-3462	290	5	𝑉	𝑉	PROPN
cana-3462	290	6	∪	∪	ADJ
cana-3462	290	7	𝐻3	𝐻3	PROPN
cana-3462	290	8	𝐸	𝐸	PROPN
cana-3462	290	9	)	)	PUNCT
cana-3462	290	10	has	have	VERB
cana-3462	290	11	𝑝1	𝑝1	NOUN
cana-3462	290	12	+	+	CCONJ
cana-3462	290	13	𝑞1	𝑞1	PROPN
cana-3462	290	14	+	+	CCONJ
cana-3462	290	15	𝑝2	𝑝2	NOUN
cana-3462	290	16	+	+	CCONJ
cana-3462	290	17	𝑝3	𝑝3	ADJ
cana-3462	290	18	vertices	vertex	NOUN
cana-3462	290	19	.	.	PUNCT
cana-3462	291	1	the	the	DET
cana-3462	291	2	following	follow	VERB
cana-3462	291	3	theorem	theorem	NOUN
cana-3462	291	4	is	be	AUX
cana-3462	291	5	easily	easily	ADV
cana-3462	291	6	obtained	obtain	VERB
cana-3462	291	7	by	by	ADP
cana-3462	291	8	using	use	VERB
cana-3462	291	9	equation	equation	NOUN
cana-3462	291	10	(	(	PUNCT
cana-3462	291	11	1.1	1.1	NUM
cana-3462	291	12	)	)	PUNCT
cana-3462	291	13	.	.	PUNCT
cana-3462	292	1	theorem	theorem	VERB
cana-3462	292	2	6.1	6.1	NUM
cana-3462	292	3	.	.	PUNCT
cana-3462	293	1	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	PROPN
cana-3462	293	2	𝑆	𝑆	PROPN
cana-3462	293	3	▹	▹	NOUN
cana-3462	293	4	(	(	PUNCT
cana-3462	293	5	𝐻2	𝐻2	PROPN
cana-3462	293	6	𝑉	𝑉	PROPN
cana-3462	293	7	∪	∪	ADJ
cana-3462	293	8	𝐻3	𝐻3	PROPN
cana-3462	293	9	𝐸	𝐸	PROPN
cana-3462	293	10	)	)	PUNCT
cana-3462	293	11	=	=	PUNCT
cana-3462	293	12	𝐸𝑄(𝐻1	𝐸𝑄(𝐻1	X
cana-3462	293	13	)	)	PUNCT
cana-3462	294	1	+	+	CCONJ
cana-3462	294	2	𝐸𝑄(𝐻2	𝐸𝑄(𝐻2	PROPN
cana-3462	294	3	)	)	PUNCT
cana-3462	295	1	+	+	PUNCT
cana-3462	295	2	𝐸𝑄(𝐻3	𝐸𝑄(𝐻3	NOUN
cana-3462	295	3	)	)	PUNCT
cana-3462	296	1	+	+	CCONJ
cana-3462	296	2	4𝑞1𝑝2	4𝑞1𝑝2	NUM
cana-3462	297	1	+	+	ADJ
cana-3462	297	2	𝑞1𝑝2	𝑞1𝑝2	PROPN
cana-3462	297	3	2	2	NUM
cana-3462	297	4	+	+	NUM
cana-3462	297	5	6𝑞1	6𝑞1	NUM
cana-3462	297	6	+	+	CCONJ
cana-3462	297	7	5𝑞1𝑝1	5𝑞1𝑝1	NOUN
cana-3462	297	8	+	+	CCONJ
cana-3462	297	9	𝑝1	𝑝1	NOUN
cana-3462	297	10	2𝑞1	2𝑞1	NUM
cana-3462	297	11	+	+	CCONJ
cana-3462	297	12	4𝑞2𝑝1	4𝑞2𝑝1	NUM
cana-3462	297	13	+	+	SYM
cana-3462	297	14	𝑝1	𝑝1	NOUN
cana-3462	297	15	2𝑝2	2𝑝2	NUM
cana-3462	297	16	+	+	CCONJ
cana-3462	297	17	4𝑞1𝑞3	4𝑞1𝑞3	NUM
cana-3462	298	1	+	+	CCONJ
cana-3462	298	2	𝑞1	𝑞1	ADJ
cana-3462	298	3	2𝑝3	2𝑝3	NUM
cana-3462	299	1	+	+	CCONJ
cana-3462	299	2	2𝑝1𝑝2	2𝑝1𝑝2	NUM
cana-3462	299	3	+	+	CCONJ
cana-3462	299	4	𝑞1𝑝3	𝑞1𝑝3	NOUN
cana-3462	299	5	.	.	NOUN
cana-3462	299	6	coclusion	coclusion	NOUN
cana-3462	299	7	:	:	PUNCT
cana-3462	299	8	we	we	PRON
cana-3462	299	9	establish	establish	VERB
cana-3462	299	10	a	a	DET
cana-3462	299	11	relation	relation	NOUN
cana-3462	299	12	between	between	ADP
cana-3462	299	13	the	the	DET
cana-3462	299	14	quasi	quasi	ADJ
cana-3462	299	15	-	-	ADJ
cana-3462	299	16	laplacian	laplacian	ADJ
cana-3462	299	17	energy	energy	NOUN
cana-3462	299	18	of	of	ADP
cana-3462	299	19	few	few	ADJ
cana-3462	299	20	novel	novel	ADJ
cana-3462	299	21	graphs	graph	NOUN
cana-3462	299	22	and	and	CCONJ
cana-3462	299	23	their	their	PRON
cana-3462	299	24	respective	respective	ADJ
cana-3462	299	25	original	original	ADJ
cana-3462	299	26	graphs	graph	NOUN
cana-3462	299	27	.	.	PUNCT
cana-3462	300	1	the	the	DET
cana-3462	300	2	formulation	formulation	NOUN
cana-3462	300	3	is	be	AUX
cana-3462	300	4	predicated	predicate	VERB
cana-3462	300	5	on	on	ADP
cana-3462	300	6	the	the	DET
cana-3462	300	7	relationships	relationship	NOUN
cana-3462	300	8	between	between	ADP
cana-3462	300	9	quasi	quasi	PROPN
cana-3462	300	10	laplacian	laplacian	ADJ
cana-3462	300	11	energy	energy	NOUN
cana-3462	300	12	and	and	CCONJ
cana-3462	300	13	vertex	vertex	NOUN
cana-3462	300	14	degree	degree	NOUN
cana-3462	300	15	in	in	ADP
cana-3462	300	16	the	the	DET
cana-3462	300	17	novel	novel	NOUN
cana-3462	300	18	graphs	graph	NOUN
cana-3462	300	19	.	.	PUNCT
cana-3462	301	1	moreover	moreover	ADV
cana-3462	301	2	,	,	PUNCT
cana-3462	301	3	it	it	PRON
cana-3462	301	4	is	be	AUX
cana-3462	301	5	directly	directly	ADV
cana-3462	301	6	associated	associate	VERB
cana-3462	301	7	with	with	ADP
cana-3462	301	8	the	the	DET
cana-3462	301	9	first	first	PROPN
cana-3462	301	10	zagreb	zagreb	PROPN
cana-3462	301	11	index	index	NOUN
cana-3462	301	12	,	,	PUNCT
cana-3462	301	13	quantity	quantity	NOUN
cana-3462	301	14	of	of	ADP
cana-3462	301	15	vertices	vertex	NOUN
cana-3462	301	16	and	and	CCONJ
cana-3462	301	17	edges	edge	NOUN
cana-3462	301	18	of	of	ADP
cana-3462	301	19	the	the	DET
cana-3462	301	20	graph	graph	NOUN
cana-3462	301	21	.	.	PUNCT
cana-3462	302	1	communications	communication	NOUN
cana-3462	302	2	on	on	ADP
cana-3462	302	3	applied	apply	VERB
cana-3462	302	4	nonlinear	nonlinear	ADJ
cana-3462	302	5	analysis	analysis	NOUN
cana-3462	302	6	issn	issn	NOUN
cana-3462	302	7	:	:	PUNCT
cana-3462	302	8	1074	1074	NUM
cana-3462	302	9	-	-	PUNCT
cana-3462	302	10	133x	133x	NUM
cana-3462	302	11	vol	vol	NOUN
cana-3462	302	12	32	32	NUM
cana-3462	302	13	no	no	NOUN
cana-3462	302	14	.	.	PUNCT
cana-3462	303	1	7s	7	NOUN
cana-3462	303	2	(	(	PUNCT
cana-3462	303	3	2025	2025	NUM
cana-3462	303	4	)	)	PUNCT
cana-3462	303	5	541	541	NUM
cana-3462	303	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3462	303	7	refrences	refrence	VERB
cana-3462	303	8	[	[	X
cana-3462	303	9	1	1	X
cana-3462	303	10	]	]	PUNCT
cana-3462	303	11	m.	m.	PROPN
cana-3462	303	12	e.	e.	PROPN
cana-3462	303	13	berberler	berberler	PROPN
cana-3462	303	14	,	,	PUNCT
cana-3462	303	15	quasi	quasi	ADJ
cana-3462	303	16	-	-	ADJ
cana-3462	303	17	laplacian	laplacian	ADJ
cana-3462	303	18	energy	energy	NOUN
cana-3462	303	19	of	of	ADP
cana-3462	303	20	fractal	fractal	ADJ
cana-3462	303	21	graphs	graph	NOUN
cana-3462	303	22	,	,	PUNCT
cana-3462	303	23	acta	acta	PROPN
cana-3462	303	24	et	et	PROPN
cana-3462	303	25	commentationes	commentatione	VERB
cana-3462	303	26	universitatis	universitatis	PROPN
cana-3462	303	27	tartuensis	tartuensis	PROPN
cana-3462	303	28	de	de	X
cana-3462	303	29	mathematica	mathematica	PROPN
cana-3462	303	30	,	,	PUNCT
cana-3462	303	31	28(1	28(1	NOUN
cana-3462	303	32	)	)	PUNCT
cana-3462	303	33	(	(	PUNCT
cana-3462	303	34	2024	2024	NUM
cana-3462	303	35	)	)	PUNCT
cana-3462	303	36	,	,	PUNCT
cana-3462	303	37	5	5	NUM
cana-3462	303	38	-	-	SYM
cana-3462	303	39	18	18	NUM
cana-3462	303	40	.	.	PUNCT
cana-3462	304	1	[	[	X
cana-3462	304	2	2	2	X
cana-3462	304	3	]	]	PUNCT
cana-3462	304	4	v.	v.	PROPN
cana-3462	304	5	k.	k.	PROPN
cana-3462	304	6	najiya	najiya	PROPN
cana-3462	304	7	and	and	CCONJ
cana-3462	304	8	a.v	a.v	PROPN
cana-3462	304	9	.	.	PROPN
cana-3462	304	10	chithra	chithra	PROPN
cana-3462	304	11	,	,	PUNCT
cana-3462	304	12	constructions	construction	NOUN
cana-3462	304	13	of	of	ADP
cana-3462	304	14	𝐴𝛼-cospectral	𝐴𝛼-cospectral	ADJ
cana-3462	304	15	graphs	graph	NOUN
cana-3462	304	16	using	use	VERB
cana-3462	304	17	some	some	DET
cana-3462	304	18	corona	corona	NOUN
cana-3462	304	19	operations	operation	NOUN
cana-3462	304	20	,	,	PUNCT
cana-3462	304	21	arxiv	arxiv	PROPN
cana-3462	304	22	preprint	preprint	VERB
cana-3462	304	23	arxiv:2406.07183	arxiv:2406.07183	X
cana-3462	304	24	(	(	PUNCT
cana-3462	304	25	2024	2024	NUM
cana-3462	304	26	)	)	PUNCT
cana-3462	304	27	.	.	PUNCT
cana-3462	305	1	[	[	X
cana-3462	305	2	3	3	X
cana-3462	305	3	]	]	X
cana-3462	305	4	d.	d.	PROPN
cana-3462	305	5	m.	m.	PROPN
cana-3462	305	6	cvetkovic	cvetkovic	PROPN
cana-3462	305	7	,	,	PUNCT
cana-3462	305	8	p.	p.	NOUN
cana-3462	305	9	rowlinson	rowlinson	NOUN
cana-3462	305	10	and	and	CCONJ
cana-3462	305	11	s.	s.	PROPN
cana-3462	305	12	simic	simic	PROPN
cana-3462	305	13	,	,	PUNCT
cana-3462	305	14	an	an	DET
cana-3462	305	15	introduction	introduction	NOUN
cana-3462	305	16	to	to	ADP
cana-3462	305	17	the	the	DET
cana-3462	305	18	theory	theory	NOUN
cana-3462	305	19	of	of	ADP
cana-3462	305	20	graph	graph	NOUN
cana-3462	305	21	spectra	spectra	PROPN
cana-3462	305	22	,	,	PUNCT
cana-3462	305	23	cambridge	cambridge	PROPN
cana-3462	305	24	university	university	PROPN
cana-3462	305	25	press	press	NOUN
cana-3462	305	26	,	,	PUNCT
cana-3462	305	27	(	(	PUNCT
cana-3462	305	28	2010	2010	NUM
cana-3462	305	29	)	)	PUNCT
cana-3462	305	30	.	.	PUNCT
cana-3462	306	1	[	[	X
cana-3462	306	2	4	4	X
cana-3462	306	3	]	]	PUNCT
cana-3462	306	4	s.	s.	PROPN
cana-3462	306	5	y.	y.	PROPN
cana-3462	306	6	cui	cui	PROPN
cana-3462	306	7	and	and	CCONJ
cana-3462	306	8	g.	g.	PROPN
cana-3462	306	9	x.	x.	PROPN
cana-3462	306	10	tian	tian	PROPN
cana-3462	306	11	,	,	PUNCT
cana-3462	306	12	the	the	DET
cana-3462	306	13	spectrum	spectrum	NOUN
cana-3462	306	14	and	and	CCONJ
cana-3462	306	15	the	the	DET
cana-3462	306	16	signless	signless	PROPN
cana-3462	306	17	laplacian	laplacian	ADJ
cana-3462	306	18	spectrum	spectrum	NOUN
cana-3462	306	19	of	of	ADP
cana-3462	306	20	coronae	coronae	NOUN
cana-3462	306	21	,	,	PUNCT
cana-3462	306	22	linear	linear	ADJ
cana-3462	306	23	algebra	algebra	NOUN
cana-3462	306	24	and	and	CCONJ
cana-3462	306	25	its	its	PRON
cana-3462	306	26	applications	application	NOUN
cana-3462	306	27	,	,	PUNCT
cana-3462	306	28	437(7	437(7	NUM
cana-3462	306	29	)	)	PUNCT
cana-3462	306	30	(	(	PUNCT
cana-3462	306	31	2012	2012	NUM
cana-3462	306	32	)	)	PUNCT
cana-3462	306	33	,	,	PUNCT
cana-3462	306	34	1692	1692	NUM
cana-3462	306	35	-	-	SYM
cana-3462	306	36	703	703	NUM
cana-3462	306	37	.	.	PUNCT
cana-3462	307	1	[	[	X
cana-3462	307	2	5	5	NUM
cana-3462	307	3	]	]	PUNCT
cana-3462	307	4	a.	a.	NOUN
cana-3462	307	5	das	das	PROPN
cana-3462	307	6	and	and	CCONJ
cana-3462	307	7	p.	p.	NOUN
cana-3462	307	8	panigrahi	panigrahi	PROPN
cana-3462	307	9	,	,	PUNCT
cana-3462	307	10	spectra	spectra	NOUN
cana-3462	307	11	of	of	ADP
cana-3462	307	12	r	r	NOUN
cana-3462	307	13	-	-	PUNCT
cana-3462	307	14	vertex	vertex	NOUN
cana-3462	307	15	join	join	NOUN
cana-3462	307	16	and	and	CCONJ
cana-3462	307	17	r	r	NOUN
cana-3462	307	18	-	-	PUNCT
cana-3462	307	19	edge	edge	NOUN
cana-3462	307	20	join	join	NOUN
cana-3462	307	21	of	of	ADP
cana-3462	307	22	two	two	NUM
cana-3462	307	23	graphs	graph	NOUN
cana-3462	307	24	,	,	PUNCT
cana-3462	307	25	discussiones	discussione	VERB
cana-3462	307	26	mathematicaegeneral	mathematicaegeneral	ADJ
cana-3462	307	27	algebra	algebra	NOUN
cana-3462	307	28	and	and	CCONJ
cana-3462	307	29	applications	application	NOUN
cana-3462	307	30	,	,	PUNCT
cana-3462	307	31	38(1	38(1	NUM
cana-3462	307	32	)	)	PUNCT
cana-3462	307	33	(	(	PUNCT
cana-3462	307	34	2018	2018	NUM
cana-3462	307	35	)	)	PUNCT
cana-3462	307	36	,	,	PUNCT
cana-3462	307	37	19	19	NUM
cana-3462	307	38	-	-	SYM
cana-3462	307	39	32	32	NUM
cana-3462	307	40	.	.	PUNCT
cana-3462	308	1	[	[	X
cana-3462	308	2	6	6	NUM
cana-3462	308	3	]	]	PUNCT
cana-3462	308	4	m.	m.	NOUN
cana-3462	308	5	yue	yue	PROPN
cana-3462	308	6	,	,	PUNCT
cana-3462	308	7	z.	z.	PROPN
cana-3462	308	8	cao	cao	PROPN
cana-3462	308	9	,	,	PUNCT
cana-3462	308	10	and	and	CCONJ
cana-3462	308	11	x.	x.	NOUN
cana-3462	308	12	qi	qi	PROPN
cana-3462	308	13	,	,	PUNCT
cana-3462	308	14	quasi	quasi	ADJ
cana-3462	308	15	-	-	ADJ
cana-3462	308	16	laplacian	laplacian	ADJ
cana-3462	308	17	centrality	centrality	NOUN
cana-3462	308	18	:	:	PUNCT
cana-3462	308	19	a	a	DET
cana-3462	308	20	new	new	ADJ
cana-3462	308	21	vertex	vertex	NOUN
cana-3462	308	22	centrality	centrality	NOUN
cana-3462	308	23	measurement	measurement	NOUN
cana-3462	308	24	based	base	VERB
cana-3462	308	25	on	on	ADP
cana-3462	308	26	quasilaplacian	quasilaplacian	ADJ
cana-3462	308	27	energy	energy	NOUN
cana-3462	308	28	of	of	ADP
cana-3462	308	29	networks	network	NOUN
cana-3462	308	30	,	,	PUNCT
cana-3462	308	31	physica	physica	VERB
cana-3462	308	32	a	a	DET
cana-3462	308	33	:	:	PUNCT
cana-3462	308	34	statistical	statistical	ADJ
cana-3462	308	35	mechanics	mechanic	NOUN
cana-3462	308	36	and	and	CCONJ
cana-3462	308	37	its	its	PRON
cana-3462	308	38	applications	application	NOUN
cana-3462	308	39	527	527	NUM
cana-3462	308	40	(	(	PUNCT
cana-3462	308	41	2019	2019	NUM
cana-3462	308	42	):	):	PUNCT
cana-3462	308	43	121130	121130	NUM
cana-3462	308	44	.	.	PUNCT
cana-3462	309	1	[	[	X
cana-3462	309	2	7	7	NUM
cana-3462	309	3	]	]	PUNCT
cana-3462	309	4	a.	a.	NOUN
cana-3462	309	5	das	das	PROPN
cana-3462	309	6	and	and	CCONJ
cana-3462	309	7	p.	p.	NOUN
cana-3462	309	8	panigrahi	panigrahi	PROPN
cana-3462	309	9	,	,	PUNCT
cana-3462	309	10	new	new	ADJ
cana-3462	309	11	classes	class	NOUN
cana-3462	309	12	of	of	ADP
cana-3462	309	13	simultaneous	simultaneous	ADJ
cana-3462	309	14	cospectral	cospectral	ADJ
cana-3462	309	15	graphs	graph	NOUN
cana-3462	309	16	for	for	ADP
cana-3462	309	17	adjacency	adjacency	NOUN
cana-3462	309	18	,	,	PUNCT
cana-3462	309	19	laplacian	laplacian	ADJ
cana-3462	309	20	and	and	CCONJ
cana-3462	309	21	normalized	normalize	VERB
cana-3462	309	22	laplacian	laplacian	ADJ
cana-3462	309	23	matrices	matrix	NOUN
cana-3462	309	24	,	,	PUNCT
cana-3462	309	25	kragujevac	kragujevac	PROPN
cana-3462	309	26	journal	journal	NOUN
cana-3462	309	27	of	of	ADP
cana-3462	309	28	mathematics	mathematic	NOUN
cana-3462	309	29	,	,	PUNCT
cana-3462	309	30	43(2	43(2	NUM
cana-3462	309	31	)	)	PUNCT
cana-3462	309	32	(	(	PUNCT
cana-3462	309	33	2019	2019	NUM
cana-3462	309	34	)	)	PUNCT
cana-3462	309	35	,	,	PUNCT
cana-3462	309	36	303	303	NUM
cana-3462	309	37	-	-	SYM
cana-3462	309	38	323	323	NUM
cana-3462	309	39	.	.	PUNCT
cana-3462	310	1	[	[	X
cana-3462	310	2	8	8	NUM
cana-3462	310	3	]	]	X
cana-3462	310	4	i.	i.	NOUN
cana-3462	310	5	gopal	gopal	PROPN
cana-3462	310	6	,	,	PUNCT
cana-3462	310	7	spectrum	spectrum	NOUN
cana-3462	310	8	of	of	ADP
cana-3462	310	9	two	two	NUM
cana-3462	310	10	new	new	ADJ
cana-3462	310	11	joins	join	NOUN
cana-3462	310	12	of	of	ADP
cana-3462	310	13	graphs	graph	NOUN
cana-3462	310	14	and	and	CCONJ
cana-3462	310	15	infinite	infinite	ADJ
cana-3462	310	16	families	family	NOUN
cana-3462	310	17	of	of	ADP
cana-3462	310	18	integral	integral	ADJ
cana-3462	310	19	graphs	graph	NOUN
cana-3462	310	20	.	.	PUNCT
cana-3462	311	1	kragujevac	kragujevac	PROPN
cana-3462	311	2	journal	journal	PROPN
cana-3462	311	3	of	of	ADP
cana-3462	311	4	mathematics	mathematic	NOUN
cana-3462	311	5	,	,	PUNCT
cana-3462	311	6	36(1	36(1	NUM
cana-3462	311	7	)	)	PUNCT
cana-3462	311	8	(	(	PUNCT
cana-3462	311	9	2012	2012	NUM
cana-3462	311	10	)	)	PUNCT
cana-3462	311	11	,	,	PUNCT
cana-3462	311	12	133	133	NUM
cana-3462	311	13	-	-	SYM
cana-3462	311	14	139	139	NUM
cana-3462	311	15	.	.	PUNCT
cana-3462	312	1	[	[	X
cana-3462	312	2	9	9	NUM
cana-3462	312	3	]	]	SYM
cana-3462	312	4	i.	i.	NOUN
cana-3462	312	5	gutman	gutman	PROPN
cana-3462	312	6	,	,	PUNCT
cana-3462	312	7	the	the	DET
cana-3462	312	8	energy	energy	NOUN
cana-3462	312	9	of	of	ADP
cana-3462	312	10	a	a	DET
cana-3462	312	11	graph	graph	NOUN
cana-3462	312	12	:	:	PUNCT
cana-3462	312	13	old	old	ADJ
cana-3462	312	14	and	and	CCONJ
cana-3462	312	15	new	new	ADJ
cana-3462	312	16	results	result	NOUN
cana-3462	312	17	,	,	PUNCT
cana-3462	312	18	in	in	ADP
cana-3462	312	19	:	:	PUNCT
cana-3462	312	20	algebraic	algebraic	ADJ
cana-3462	312	21	combinatorics	combinatoric	NOUN
cana-3462	312	22	and	and	CCONJ
cana-3462	312	23	applications	application	NOUN
cana-3462	312	24	:	:	PUNCT
cana-3462	312	25	proceedings	proceeding	NOUN
cana-3462	312	26	of	of	ADP
cana-3462	312	27	the	the	DET
cana-3462	312	28	euroconference	euroconference	NOUN
cana-3462	312	29	,	,	PUNCT
cana-3462	312	30	algebraic	algebraic	ADJ
cana-3462	312	31	combinatorics	combinatoric	NOUN
cana-3462	312	32	and	and	CCONJ
cana-3462	312	33	applications	application	NOUN
cana-3462	312	34	(	(	PUNCT
cana-3462	312	35	alcoma	alcoma	ADJ
cana-3462	312	36	)	)	PUNCT
cana-3462	312	37	,	,	PUNCT
cana-3462	312	38	springer	springer	NOUN
cana-3462	312	39	,	,	PUNCT
cana-3462	312	40	berlin	berlin	PROPN
cana-3462	312	41	,	,	PUNCT
cana-3462	312	42	heidelberg	heidelberg	PROPN
cana-3462	312	43	,	,	PUNCT
cana-3462	312	44	germany	germany	PROPN
cana-3462	312	45	2001	2001	NUM
cana-3462	312	46	(	(	PUNCT
cana-3462	312	47	pp	pp	ADJ
cana-3462	312	48	.	.	PUNCT
cana-3462	312	49	196	196	NUM
cana-3462	312	50	-	-	SYM
cana-3462	312	51	211	211	NUM
cana-3462	312	52	)	)	PUNCT
cana-3462	312	53	.	.	PUNCT
cana-3462	313	1	[	[	X
cana-3462	313	2	10	10	NUM
cana-3462	313	3	]	]	X
cana-3462	313	4	i.	i.	PROPN
cana-3462	313	5	gutman	gutman	PROPN
cana-3462	313	6	and	and	CCONJ
cana-3462	313	7	n.	n.	PROPN
cana-3462	313	8	trinajstić	trinajstić	PROPN
cana-3462	313	9	,	,	PUNCT
cana-3462	313	10	graph	graph	NOUN
cana-3462	313	11	theory	theory	NOUN
cana-3462	313	12	and	and	CCONJ
cana-3462	313	13	molecular	molecular	ADJ
cana-3462	313	14	orbitals	orbital	NOUN
cana-3462	313	15	:	:	PUNCT
cana-3462	313	16	total	total	ADJ
cana-3462	313	17	𝜙-electron	𝜙-electron	NOUN
cana-3462	313	18	energy	energy	NOUN
cana-3462	313	19	of	of	ADP
cana-3462	313	20	alternant	alternant	ADJ
cana-3462	313	21	hydrocarbons	hydrocarbon	NOUN
cana-3462	313	22	,	,	PUNCT
cana-3462	313	23	chemical	chemical	NOUN
cana-3462	313	24	physics	physics	NOUN
cana-3462	313	25	letters	letter	NOUN
cana-3462	313	26	,	,	PUNCT
cana-3462	313	27	17(4	17(4	NUM
cana-3462	313	28	)	)	PUNCT
cana-3462	313	29	(	(	PUNCT
cana-3462	313	30	1972	1972	NUM
cana-3462	313	31	)	)	PUNCT
cana-3462	313	32	,	,	PUNCT
cana-3462	313	33	535	535	NUM
cana-3462	313	34	-	-	SYM
cana-3462	313	35	8	8	NUM
cana-3462	313	36	.	.	PUNCT
cana-3462	314	1	[	[	X
cana-3462	314	2	11	11	NUM
cana-3462	314	3	]	]	X
cana-3462	314	4	y.	y.	PROPN
cana-3462	314	5	hou	hou	PROPN
cana-3462	314	6	and	and	CCONJ
cana-3462	314	7	w.	w.	PROPN
cana-3462	314	8	c.	c.	PROPN
cana-3462	314	9	shiu	shiu	PROPN
cana-3462	314	10	,	,	PUNCT
cana-3462	314	11	the	the	DET
cana-3462	314	12	spectrum	spectrum	NOUN
cana-3462	314	13	of	of	ADP
cana-3462	314	14	the	the	DET
cana-3462	314	15	edge	edge	NOUN
cana-3462	314	16	corona	corona	NOUN
cana-3462	314	17	of	of	ADP
cana-3462	314	18	two	two	NUM
cana-3462	314	19	graphs	graph	NOUN
cana-3462	314	20	,	,	PUNCT
cana-3462	314	21	the	the	DET
cana-3462	314	22	electronic	electronic	ADJ
cana-3462	314	23	journal	journal	NOUN
cana-3462	314	24	of	of	ADP
cana-3462	314	25	linear	linear	PROPN
cana-3462	314	26	algebra	algebra	PROPN
cana-3462	314	27	,	,	PUNCT
cana-3462	314	28	20(2010	20(2010	NUM
cana-3462	314	29	)	)	PUNCT
cana-3462	314	30	,	,	PUNCT
cana-3462	314	31	586	586	NUM
cana-3462	314	32	-	-	SYM
cana-3462	314	33	94	94	NUM
cana-3462	314	34	.	.	PUNCT
cana-3462	315	1	[	[	X
cana-3462	315	2	12	12	NUM
cana-3462	315	3	]	]	X
cana-3462	315	4	lan	lan	PROPN
cana-3462	315	5	and	and	CCONJ
cana-3462	315	6	b.	b.	PROPN
cana-3462	315	7	zhou	zhou	PROPN
cana-3462	315	8	,	,	PUNCT
cana-3462	315	9	spectra	spectra	NOUN
cana-3462	315	10	of	of	ADP
cana-3462	315	11	graph	graph	NOUN
cana-3462	315	12	operations	operation	NOUN
cana-3462	315	13	based	base	VERB
cana-3462	315	14	on	on	ADP
cana-3462	315	15	r	r	NOUN
cana-3462	315	16	-	-	PUNCT
cana-3462	315	17	graph	graph	NOUN
cana-3462	315	18	,	,	PUNCT
cana-3462	315	19	linear	linear	ADJ
cana-3462	315	20	and	and	CCONJ
cana-3462	315	21	multilinear	multilinear	PROPN
cana-3462	315	22	algebra	algebra	PROPN
cana-3462	315	23	,	,	PUNCT
cana-3462	315	24	63(7	63(7	NUM
cana-3462	315	25	)	)	PUNCT
cana-3462	315	26	(	(	PUNCT
cana-3462	315	27	2015	2015	NUM
cana-3462	315	28	)	)	PUNCT
cana-3462	315	29	,	,	PUNCT
cana-3462	315	30	1401	1401	NUM
cana-3462	315	31	-	-	SYM
cana-3462	315	32	22	22	NUM
cana-3462	315	33	.	.	PUNCT
cana-3462	316	1	[	[	X
cana-3462	316	2	13	13	NUM
cana-3462	316	3	]	]	PUNCT
cana-3462	316	4	p.	p.	NOUN
cana-3462	316	5	lu	lu	PROPN
cana-3462	316	6	and	and	CCONJ
cana-3462	316	7	y.	y.	PROPN
cana-3462	316	8	miao	miao	PROPN
cana-3462	316	9	,	,	PUNCT
cana-3462	316	10	spectra	spectra	NOUN
cana-3462	316	11	of	of	ADP
cana-3462	316	12	the	the	DET
cana-3462	316	13	subdivision	subdivision	NOUN
cana-3462	316	14	-vertex	-vertex	PROPN
cana-3462	316	15	and	and	CCONJ
cana-3462	316	16	subdivision	subdivision	NOUN
cana-3462	316	17	-edge	-edge	PROPN
cana-3462	316	18	coronae	coronae	NOUN
cana-3462	316	19	,	,	PUNCT
cana-3462	316	20	https://arxiv.org/abs/1302.0457(2013	https://arxiv.org/abs/1302.0457(2013	VERB
cana-3462	316	21	)	)	PUNCT
cana-3462	316	22	.	.	PUNCT
cana-3462	317	1	[	[	X
cana-3462	317	2	14	14	NUM
cana-3462	317	3	]	]	PUNCT
cana-3462	317	4	x.	x.	NOUN
cana-3462	317	5	liu	liu	PROPN
cana-3462	317	6	and	and	CCONJ
cana-3462	317	7	p.	p.	PROPN
cana-3462	317	8	lu	lu	PROPN
cana-3462	317	9	,	,	PUNCT
cana-3462	317	10	spectra	spectra	NOUN
cana-3462	317	11	of	of	ADP
cana-3462	317	12	subdivision	subdivision	NOUN
cana-3462	317	13	-	-	PUNCT
cana-3462	317	14	vertex	vertex	NOUN
cana-3462	317	15	and	and	CCONJ
cana-3462	317	16	subdivision	subdivision	NOUN
cana-3462	317	17	-	-	PUNCT
cana-3462	317	18	edge	edge	NOUN
cana-3462	317	19	neighbourhood	neighbourhood	NOUN
cana-3462	317	20	coronae	coronae	NOUN
cana-3462	317	21	,	,	PUNCT
cana-3462	317	22	linear	linear	ADJ
cana-3462	317	23	algebra	algebra	NOUN
cana-3462	317	24	and	and	CCONJ
cana-3462	317	25	its	its	PRON
cana-3462	317	26	applications	application	NOUN
cana-3462	317	27	,	,	PUNCT
cana-3462	317	28	438(8	438(8	NUM
cana-3462	317	29	)	)	PUNCT
cana-3462	317	30	(	(	PUNCT
cana-3462	317	31	2013	2013	NUM
cana-3462	317	32	)	)	PUNCT
cana-3462	317	33	,	,	PUNCT
cana-3462	317	34	3547	3547	NUM
cana-3462	317	35	-	-	SYM
cana-3462	317	36	59	59	NUM
cana-3462	317	37	.	.	PUNCT
cana-3462	318	1	[	[	X
cana-3462	318	2	15	15	NUM
cana-3462	318	3	]	]	PUNCT
cana-3462	318	4	x.	x.	NOUN
cana-3462	318	5	liu	liu	PROPN
cana-3462	318	6	and	and	CCONJ
cana-3462	318	7	z.	z.	PROPN
cana-3462	318	8	zhang	zhang	PROPN
cana-3462	318	9	,	,	PUNCT
cana-3462	318	10	spectra	spectra	NOUN
cana-3462	318	11	of	of	ADP
cana-3462	318	12	subdivision	subdivision	NOUN
cana-3462	318	13	-	-	PUNCT
cana-3462	318	14	vertex	vertex	NOUN
cana-3462	318	15	join	join	NOUN
cana-3462	318	16	and	and	CCONJ
cana-3462	318	17	subdivision	subdivision	NOUN
cana-3462	318	18	-	-	PUNCT
cana-3462	318	19	edge	edge	NOUN
cana-3462	318	20	join	join	NOUN
cana-3462	318	21	of	of	ADP
cana-3462	318	22	two	two	NUM
cana-3462	318	23	graphs	graph	NOUN
cana-3462	318	24	,	,	PUNCT
cana-3462	318	25	bulletin	bulletin	NOUN
cana-3462	318	26	of	of	ADP
cana-3462	318	27	the	the	DET
cana-3462	318	28	malaysian	malaysian	PROPN
cana-3462	318	29	mathematical	mathematical	PROPN
cana-3462	318	30	sciences	sciences	PROPN
cana-3462	318	31	society	society	NOUN
cana-3462	318	32	,	,	PUNCT
cana-3462	318	33	42(2019	42(2019	X
cana-3462	318	34	)	)	PUNCT
cana-3462	318	35	.	.	PUNCT
cana-3462	319	1	[	[	X
cana-3462	319	2	16	16	NUM
cana-3462	319	3	]	]	X
cana-3462	319	4	l.	l.	PROPN
cana-3462	319	5	sun	sun	PROPN
cana-3462	319	6	,	,	PUNCT
cana-3462	319	7	z.	z.	PROPN
cana-3462	319	8	shang	shang	PROPN
cana-3462	319	9	and	and	CCONJ
cana-3462	319	10	c.	c.	PROPN
cana-3462	319	11	bu	bu	PROPN
cana-3462	319	12	,	,	PUNCT
cana-3462	319	13	resistance	resistance	NOUN
cana-3462	319	14	distance	distance	NOUN
cana-3462	319	15	and	and	CCONJ
cana-3462	319	16	kirchhoff	kirchhoff	NOUN
cana-3462	319	17	index	index	NOUN
cana-3462	319	18	of	of	ADP
cana-3462	319	19	the	the	DET
cana-3462	319	20	q	q	NOUN
cana-3462	319	21	-	-	PUNCT
cana-3462	319	22	vertex	vertex	NOUN
cana-3462	319	23	(	(	PUNCT
cana-3462	319	24	or	or	CCONJ
cana-3462	319	25	edge	edge	NOUN
cana-3462	319	26	)	)	PUNCT
cana-3462	319	27	join	join	VERB
cana-3462	319	28	graphs	graph	NOUN
cana-3462	319	29	,	,	PUNCT
cana-3462	319	30	discrete	discrete	ADJ
cana-3462	319	31	mathematics	mathematic	NOUN
cana-3462	319	32	,	,	PUNCT
cana-3462	319	33	344(8)(2021	344(8)(2021	NUM
cana-3462	319	34	)	)	PUNCT
cana-3462	319	35	,	,	PUNCT
cana-3462	319	36	112433	112433	NUM
cana-3462	319	37	.	.	PUNCT
cana-3462	320	1	[	[	X
cana-3462	320	2	17	17	NUM
cana-3462	320	3	]	]	X
cana-3462	320	4	f.	f.	PROPN
cana-3462	320	5	wen	wen	PROPN
cana-3462	320	6	,	,	PUNCT
cana-3462	320	7	y.	y.	PROPN
cana-3462	320	8	zhang	zhang	PROPN
cana-3462	320	9	and	and	CCONJ
cana-3462	320	10	m.	m.	PROPN
cana-3462	320	11	li	li	PROPN
cana-3462	320	12	,	,	PUNCT
cana-3462	320	13	spectra	spectra	NOUN
cana-3462	320	14	of	of	ADP
cana-3462	320	15	subdivision	subdivision	NOUN
cana-3462	320	16	vertex	vertex	NOUN
cana-3462	320	17	-	-	PUNCT
cana-3462	320	18	edge	edge	NOUN
cana-3462	320	19	join	join	NOUN
cana-3462	320	20	of	of	ADP
cana-3462	320	21	three	three	NUM
cana-3462	320	22	graphs	graph	NOUN
cana-3462	320	23	.	.	PUNCT
cana-3462	321	1	mathematics	mathematic	NOUN
cana-3462	321	2	,	,	PUNCT
cana-3462	321	3	𝟕(2)(2019),171	𝟕(2)(2019),171	NOUN
cana-3462	321	4	.	.	PUNCT
cana-3462	322	1	[	[	X
cana-3462	322	2	18	18	NUM
cana-3462	322	3	]	]	PUNCT
cana-3462	322	4	l.	l.	PROPN
cana-3462	322	5	xiaogang	xiaogang	PROPN
cana-3462	322	6	and	and	CCONJ
cana-3462	322	7	l.	l.	PROPN
cana-3462	322	8	pengli	pengli	PROPN
cana-3462	322	9	,	,	PUNCT
cana-3462	322	10	spectra	spectra	NOUN
cana-3462	322	11	of	of	ADP
cana-3462	322	12	subdivision	subdivision	NOUN
cana-3462	322	13	-	-	PUNCT
cana-3462	322	14	vertex	vertex	NOUN
cana-3462	322	15	and	and	CCONJ
cana-3462	322	16	subdivision	subdivision	NOUN
cana-3462	322	17	-	-	PUNCT
cana-3462	322	18	edge	edge	NOUN
cana-3462	322	19	neighbourhood	neighbourhood	NOUN
cana-3462	322	20	coronae	coronae	NOUN
cana-3462	322	21	,	,	PUNCT
cana-3462	322	22	linear	linear	ADJ
cana-3462	322	23	algebra	algebra	NOUN
cana-3462	322	24	and	and	CCONJ
cana-3462	322	25	its	its	PRON
cana-3462	322	26	applications	application	NOUN
cana-3462	322	27	,	,	PUNCT
cana-3462	322	28	438(8	438(8	NUM
cana-3462	322	29	)	)	PUNCT
cana-3462	322	30	(	(	PUNCT
cana-3462	322	31	2013	2013	NUM
cana-3462	322	32	)	)	PUNCT
cana-3462	322	33	,	,	PUNCT
cana-3462	322	34	3547	3547	NUM
cana-3462	322	35	-	-	SYM
cana-3462	322	36	3559	3559	NUM
cana-3462	322	37	.	.	PUNCT
cana-3462	323	1	[	[	X
cana-3462	323	2	19	19	NUM
cana-3462	323	3	]	]	PUNCT
cana-3462	323	4	x.	x.	NOUN
cana-3462	323	5	q.	q.	PROPN
cana-3462	323	6	zhu	zhu	PROPN
cana-3462	323	7	,	,	PUNCT
cana-3462	323	8	g.	g.	PROPN
cana-3462	323	9	x.	x.	PROPN
cana-3462	323	10	tian	tian	PROPN
cana-3462	323	11	and	and	CCONJ
cana-3462	323	12	s.y	s.y	PROPN
cana-3462	323	13	cui	cui	PROPN
cana-3462	323	14	,	,	PUNCT
cana-3462	323	15	spectra	spectra	NOUN
cana-3462	323	16	of	of	ADP
cana-3462	323	17	corona	corona	NOUN
cana-3462	323	18	based	base	VERB
cana-3462	323	19	on	on	ADP
cana-3462	323	20	total	total	ADJ
cana-3462	323	21	graph	graph	NOUN
cana-3462	323	22	.	.	PUNCT
cana-3462	324	1	journal	journal	PROPN
cana-3462	324	2	of	of	ADP
cana-3462	324	3	mathematical	mathematical	ADJ
cana-3462	324	4	study	study	NOUN
cana-3462	324	5	,	,	PUNCT
cana-3462	324	6	49(1	49(1	NUM
cana-3462	324	7	)	)	PUNCT
cana-3462	324	8	(	(	PUNCT
cana-3462	324	9	2016	2016	NUM
cana-3462	324	10	)	)	PUNCT
cana-3462	324	11	,	,	PUNCT
cana-3462	324	12	72	72	NUM
cana-3462	324	13	-	-	SYM
cana-3462	324	14	81	81	NUM
cana-3462	324	15	.	.	PUNCT