id	sid	tid	token	lemma	pos
cana-1178	1	1	communications	communication	NOUN
cana-1178	1	2	on	on	ADP
cana-1178	1	3	applied	apply	VERB
cana-1178	1	4	nonlinear	nonlinear	ADJ
cana-1178	1	5	analysis	analysis	NOUN
cana-1178	1	6	issn	issn	NOUN
cana-1178	1	7	:	:	PUNCT
cana-1178	1	8	1074	1074	NUM
cana-1178	1	9	-	-	PUNCT
cana-1178	1	10	133x	133x	NUM
cana-1178	1	11	vol	vol	NOUN
cana-1178	1	12	31	31	NUM
cana-1178	1	13	no	no	NOUN
cana-1178	1	14	.	.	PUNCT
cana-1178	2	1	6s	6s	NUM
cana-1178	2	2	(	(	PUNCT
cana-1178	2	3	2024	2024	NUM
cana-1178	2	4	)	)	PUNCT
cana-1178	2	5	193	193	NUM
cana-1178	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	2	7	the	the	DET
cana-1178	2	8	bounds	bound	NOUN
cana-1178	2	9	of	of	ADP
cana-1178	2	10	energies	energy	NOUN
cana-1178	2	11	of	of	ADP
cana-1178	2	12	rough	rough	ADJ
cana-1178	2	13	complemented	complemented	ADJ
cana-1178	2	14	graph	graph	NOUN
cana-1178	2	15	b.praba1	b.praba1	NOUN
cana-1178	2	16	,	,	PUNCT
cana-1178	2	17	b.sudha1	b.sudha1	NOUN
cana-1178	2	18	,	,	PUNCT
cana-1178	2	19	aathish	aathish	ADJ
cana-1178	2	20	sivasubrahmanian2	sivasubrahmanian2	NOUN
cana-1178	2	21	1,2	1,2	NUM
cana-1178	2	22	department	department	NOUN
cana-1178	2	23	of	of	ADP
cana-1178	2	24	mathematics	mathematic	NOUN
cana-1178	2	25	,	,	PUNCT
cana-1178	2	26	sri	sri	PROPN
cana-1178	2	27	sivasubramaniya	sivasubramaniya	PROPN
cana-1178	2	28	nadar	nadar	PROPN
cana-1178	2	29	college	college	PROPN
cana-1178	2	30	of	of	ADP
cana-1178	2	31	engineering	engineering	PROPN
cana-1178	2	32	,	,	PUNCT
cana-1178	2	33	chennai	chennai	PROPN
cana-1178	2	34	603110	603110	NUM
cana-1178	2	35	,	,	PUNCT
cana-1178	2	36	india	india	PROPN
cana-1178	2	37	.	.	PUNCT
cana-1178	3	1	prabab@ssn.edu.in	prabab@ssn.edu.in	PROPN
cana-1178	3	2	,	,	PUNCT
cana-1178	3	3	sudhab@ssn.edu.in	sudhab@ssn.edu.in	ADV
cana-1178	3	4	,	,	PUNCT
cana-1178	3	5	aathish04@gmail.com	aathish04@gmail.com	PROPN
cana-1178	3	6	article	article	NOUN
cana-1178	3	7	history	history	NOUN
cana-1178	3	8	:	:	PUNCT
cana-1178	3	9	received	receive	VERB
cana-1178	3	10	:	:	PUNCT
cana-1178	3	11	27	27	NUM
cana-1178	3	12	-	-	SYM
cana-1178	3	13	05	05	NUM
cana-1178	3	14	-	-	PUNCT
cana-1178	3	15	2024	2024	NUM
cana-1178	3	16	revised	revise	VERB
cana-1178	3	17	:	:	PUNCT
cana-1178	3	18	20	20	NUM
cana-1178	3	19	-	-	SYM
cana-1178	3	20	07	07	NUM
cana-1178	3	21	-	-	PUNCT
cana-1178	3	22	2024	2024	NUM
cana-1178	3	23	accepted	accept	VERB
cana-1178	3	24	:	:	PUNCT
cana-1178	3	25	30	30	NUM
cana-1178	3	26	-	-	SYM
cana-1178	3	27	07	07	NUM
cana-1178	3	28	-	-	PUNCT
cana-1178	3	29	2024	2024	NUM
cana-1178	3	30	abstract	abstract	NOUN
cana-1178	3	31	:	:	PUNCT
cana-1178	3	32	the	the	DET
cana-1178	3	33	main	main	ADJ
cana-1178	3	34	objective	objective	NOUN
cana-1178	3	35	of	of	ADP
cana-1178	3	36	this	this	DET
cana-1178	3	37	paper	paper	NOUN
cana-1178	3	38	is	be	AUX
cana-1178	3	39	to	to	PART
cana-1178	3	40	study	study	VERB
cana-1178	3	41	the	the	DET
cana-1178	3	42	various	various	ADJ
cana-1178	3	43	energies	energy	NOUN
cana-1178	3	44	and	and	CCONJ
cana-1178	3	45	their	their	PRON
cana-1178	3	46	bounds	bound	NOUN
cana-1178	3	47	of	of	ADP
cana-1178	3	48	the	the	DET
cana-1178	3	49	rough	rough	ADJ
cana-1178	3	50	complemented	complemented	ADJ
cana-1178	3	51	graph	graph	NOUN
cana-1178	3	52	corresponding	correspond	VERB
cana-1178	3	53	to	to	ADP
cana-1178	3	54	the	the	DET
cana-1178	3	55	given	give	VERB
cana-1178	3	56	rough	rough	ADJ
cana-1178	3	57	semiring	semiring	NOUN
cana-1178	3	58	.	.	PUNCT
cana-1178	4	1	in	in	ADP
cana-1178	4	2	this	this	DET
cana-1178	4	3	paper	paper	NOUN
cana-1178	4	4	,	,	PUNCT
cana-1178	4	5	for	for	ADP
cana-1178	4	6	a	a	DET
cana-1178	4	7	given	give	VERB
cana-1178	4	8	approximation	approximation	NOUN
cana-1178	4	9	space	space	NOUN
cana-1178	4	10	i=(u	i=(u	NOUN
cana-1178	4	11	,	,	PUNCT
cana-1178	4	12	r	r	NOUN
cana-1178	4	13	)	)	PUNCT
cana-1178	4	14	where	where	SCONJ
cana-1178	4	15	u	u	NOUN
cana-1178	4	16	is	be	AUX
cana-1178	4	17	the	the	DET
cana-1178	4	18	nonempty	nonempty	ADJ
cana-1178	4	19	finite	finite	ADJ
cana-1178	4	20	set	set	NOUN
cana-1178	4	21	of	of	ADP
cana-1178	4	22	objects	object	NOUN
cana-1178	4	23	and	and	CCONJ
cana-1178	4	24	r	r	NOUN
cana-1178	4	25	is	be	AUX
cana-1178	4	26	an	an	DET
cana-1178	4	27	equivalence	equivalence	NOUN
cana-1178	4	28	relation	relation	NOUN
cana-1178	4	29	on	on	ADP
cana-1178	4	30	u	u	NOUN
cana-1178	4	31	,	,	PUNCT
cana-1178	4	32	the	the	DET
cana-1178	4	33	rough	rough	ADJ
cana-1178	4	34	semiring	semiring	NOUN
cana-1178	4	35	(	(	PUNCT
cana-1178	4	36	t,∆,∇	t,∆,∇	NOUN
cana-1178	4	37	)	)	PUNCT
cana-1178	4	38	is	be	AUX
cana-1178	4	39	taken	take	VERB
cana-1178	4	40	for	for	ADP
cana-1178	4	41	study	study	NOUN
cana-1178	4	42	.	.	PUNCT
cana-1178	5	1	the	the	DET
cana-1178	5	2	rough	rough	ADJ
cana-1178	5	3	complemented	complemented	ADJ
cana-1178	5	4	graph	graph	NOUN
cana-1178	5	5	of	of	ADP
cana-1178	5	6	t	t	PROPN
cana-1178	5	7	denoted	denote	VERB
cana-1178	5	8	by	by	ADP
cana-1178	5	9	grc(t	grc(t	PROPN
cana-1178	5	10	)	)	PUNCT
cana-1178	5	11	is	be	AUX
cana-1178	5	12	a	a	DET
cana-1178	5	13	graph	graph	NOUN
cana-1178	5	14	whose	whose	DET
cana-1178	5	15	vertices	vertex	NOUN
cana-1178	5	16	are	be	AUX
cana-1178	5	17	v(grc(t))={rs(y)|y∈〖℘(e)〗^1	v(grc(t))={rs(y)|y∈〖℘(e)〗^1	VERB
cana-1178	5	18	}	}	PUNCT
cana-1178	5	19	be	be	AUX
cana-1178	5	20	the	the	DET
cana-1178	5	21	set	set	NOUN
cana-1178	5	22	of	of	ADP
cana-1178	5	23	equivalence	equivalence	NOUN
cana-1178	5	24	classes	class	NOUN
cana-1178	5	25	induced	induce	VERB
cana-1178	5	26	by	by	ADP
cana-1178	5	27	i	i	PRON
cana-1178	5	28	and	and	CCONJ
cana-1178	5	29	two	two	NUM
cana-1178	5	30	distinct	distinct	ADJ
cana-1178	5	31	vertices	vertex	NOUN
cana-1178	5	32	rs(x	rs(x	X
cana-1178	5	33	)	)	PUNCT
cana-1178	5	34	and	and	CCONJ
cana-1178	5	35	rs(y	rs(y	NUM
cana-1178	5	36	)	)	PUNCT
cana-1178	5	37	are	be	AUX
cana-1178	5	38	adjacent	adjacent	ADJ
cana-1178	5	39	iff	iff	PROPN
cana-1178	5	40	rs(x)∇rs(y)=rs(∅	rs(x)∇rs(y)=rs(∅	PROPN
cana-1178	5	41	)	)	PUNCT
cana-1178	5	42	.	.	PUNCT
cana-1178	6	1	note	note	VERB
cana-1178	6	2	that	that	SCONJ
cana-1178	6	3	there	there	PRON
cana-1178	6	4	will	will	AUX
cana-1178	6	5	be	be	AUX
cana-1178	6	6	2^n-2	2^n-2	NUM
cana-1178	6	7	vertices	vertex	NOUN
cana-1178	6	8	in	in	ADP
cana-1178	6	9	grc(t	grc(t	PROPN
cana-1178	6	10	)	)	PUNCT
cana-1178	6	11	.	.	PUNCT
cana-1178	7	1	also	also	ADV
cana-1178	7	2	randic	randic	ADJ
cana-1178	7	3	,	,	PUNCT
cana-1178	7	4	seidel	seidel	PROPN
cana-1178	7	5	,	,	PUNCT
cana-1178	7	6	minimum	minimum	ADJ
cana-1178	7	7	dominating	dominating	NOUN
cana-1178	7	8	,	,	PUNCT
cana-1178	7	9	maximal	maximal	ADJ
cana-1178	7	10	independent	independent	ADJ
cana-1178	7	11	and	and	CCONJ
cana-1178	7	12	dominating	dominate	VERB
cana-1178	7	13	energies	energy	NOUN
cana-1178	7	14	of	of	ADP
cana-1178	7	15	grc(t	grc(t	PROPN
cana-1178	7	16	)	)	PUNCT
cana-1178	7	17	are	be	AUX
cana-1178	7	18	obtained	obtain	VERB
cana-1178	7	19	,	,	PUNCT
cana-1178	7	20	the	the	DET
cana-1178	7	21	lower	low	ADJ
cana-1178	7	22	and	and	CCONJ
cana-1178	7	23	upper	upper	ADJ
cana-1178	7	24	bounds	bound	NOUN
cana-1178	7	25	of	of	ADP
cana-1178	7	26	these	these	DET
cana-1178	7	27	energies	energy	NOUN
cana-1178	7	28	are	be	AUX
cana-1178	7	29	also	also	ADV
cana-1178	7	30	established	establish	VERB
cana-1178	7	31	.	.	PUNCT
cana-1178	8	1	these	these	DET
cana-1178	8	2	energies	energy	NOUN
cana-1178	8	3	are	be	AUX
cana-1178	8	4	obtained	obtain	VERB
cana-1178	8	5	through	through	ADP
cana-1178	8	6	python	python	NOUN
cana-1178	8	7	programming	programming	NOUN
cana-1178	8	8	,	,	PUNCT
cana-1178	8	9	and	and	CCONJ
cana-1178	8	10	a	a	DET
cana-1178	8	11	bar	bar	NOUN
cana-1178	8	12	diagram	diagram	NOUN
cana-1178	8	13	is	be	AUX
cana-1178	8	14	used	use	VERB
cana-1178	8	15	to	to	PART
cana-1178	8	16	conduct	conduct	VERB
cana-1178	8	17	a	a	DET
cana-1178	8	18	comparative	comparative	ADJ
cana-1178	8	19	study	study	NOUN
cana-1178	8	20	for	for	ADP
cana-1178	8	21	various	various	ADJ
cana-1178	8	22	values	value	NOUN
cana-1178	8	23	of	of	ADP
cana-1178	8	24	n.	n.	NOUN
cana-1178	8	25	all	all	DET
cana-1178	8	26	the	the	DET
cana-1178	8	27	illustrated	illustrated	ADJ
cana-1178	8	28	concepts	concept	NOUN
cana-1178	8	29	are	be	AUX
cana-1178	8	30	explained	explain	VERB
cana-1178	8	31	with	with	ADP
cana-1178	8	32	suitable	suitable	ADJ
cana-1178	8	33	examples	example	NOUN
cana-1178	8	34	.	.	PUNCT
cana-1178	9	1	keywords	keyword	NOUN
cana-1178	9	2	:	:	PUNCT
cana-1178	9	3	independent	independent	ADJ
cana-1178	9	4	dominating	dominating	NOUN
cana-1178	9	5	set	set	NOUN
cana-1178	9	6	,	,	PUNCT
cana-1178	9	7	minimum	minimum	ADJ
cana-1178	9	8	dominating	dominating	NOUN
cana-1178	9	9	energy	energy	NOUN
cana-1178	9	10	,	,	PUNCT
cana-1178	9	11	randic	randic	ADJ
cana-1178	9	12	energy	energy	NOUN
cana-1178	9	13	,	,	PUNCT
cana-1178	9	14	siedel	siedel	NOUN
cana-1178	9	15	energy	energy	NOUN
cana-1178	9	16	,	,	PUNCT
cana-1178	9	17	python	python	PROPN
cana-1178	9	18	code	code	NOUN
cana-1178	9	19	.	.	PUNCT
cana-1178	10	1	1	1	X
cana-1178	10	2	.	.	X
cana-1178	10	3	introduction	introduction	NOUN
cana-1178	10	4	the	the	DET
cana-1178	10	5	concept	concept	NOUN
cana-1178	10	6	of	of	ADP
cana-1178	10	7	energy	energy	NOUN
cana-1178	10	8	of	of	ADP
cana-1178	10	9	a	a	DET
cana-1178	10	10	graph	graph	NOUN
cana-1178	10	11	was	be	AUX
cana-1178	10	12	introduced	introduce	VERB
cana-1178	10	13	by	by	ADP
cana-1178	10	14	i.	i.	PROPN
cana-1178	10	15	gutman	gutman	PROPN
cana-1178	11	1	[	[	X
cana-1178	11	2	1	1	NUM
cana-1178	11	3	]	]	PUNCT
cana-1178	11	4	in	in	ADP
cana-1178	11	5	the	the	DET
cana-1178	11	6	year	year	NOUN
cana-1178	11	7	1978	1978	NUM
cana-1178	11	8	.	.	PUNCT
cana-1178	12	1	in	in	ADP
cana-1178	12	2	[	[	X
cana-1178	12	3	4	4	X
cana-1178	12	4	]	]	PUNCT
cana-1178	12	5	rajesh	rajesh	PROPN
cana-1178	12	6	kanna	kanna	PROPN
cana-1178	12	7	et	et	PROPN
cana-1178	12	8	al	al	PROPN
cana-1178	12	9	.	.	PROPN
cana-1178	12	10	compute	compute	PROPN
cana-1178	12	11	milovanovic	milovanovic	ADJ
cana-1178	12	12	bounds	bound	NOUN
cana-1178	12	13	of	of	ADP
cana-1178	12	14	the	the	DET
cana-1178	12	15	cocktail	cocktail	NOUN
cana-1178	12	16	party	party	NOUN
cana-1178	12	17	graph	graph	NOUN
cana-1178	12	18	and	and	CCONJ
cana-1178	12	19	crown	crown	NOUN
cana-1178	12	20	graph	graph	NOUN
cana-1178	12	21	.	.	PUNCT
cana-1178	13	1	different	different	ADJ
cana-1178	13	2	results	result	NOUN
cana-1178	13	3	on	on	ADP
cana-1178	13	4	independent	independent	ADJ
cana-1178	13	5	dominance	dominance	NOUN
cana-1178	13	6	in	in	ADP
cana-1178	13	7	graphs	graph	NOUN
cana-1178	13	8	are	be	AUX
cana-1178	13	9	being	be	AUX
cana-1178	13	10	examined	examine	VERB
cana-1178	13	11	by	by	ADP
cana-1178	13	12	the	the	DET
cana-1178	13	13	authors	author	NOUN
cana-1178	13	14	[	[	X
cana-1178	13	15	2	2	NUM
cana-1178	13	16	]	]	PUNCT
cana-1178	13	17	.	.	PUNCT
cana-1178	14	1	a	a	DET
cana-1178	14	2	molecular	molecular	ADJ
cana-1178	14	3	structure	structure	NOUN
cana-1178	14	4	descriptor	descriptor	NOUN
cana-1178	14	5	called	call	VERB
cana-1178	14	6	randic	randic	ADJ
cana-1178	14	7	index	index	NOUN
cana-1178	14	8	was	be	AUX
cana-1178	14	9	created	create	VERB
cana-1178	14	10	by	by	ADP
cana-1178	14	11	milan	milan	PROPN
cana-1178	14	12	randi	randi	PROPN
cana-1178	14	13	in	in	ADP
cana-1178	14	14	1975[6	1975[6	NUM
cana-1178	14	15	]	]	PUNCT
cana-1178	14	16	.	.	PUNCT
cana-1178	15	1	later	later	PROPN
cana-1178	15	2	s.b	s.b	PROPN
cana-1178	15	3	.	.	PROPN
cana-1178	15	4	bozkurt	bozkurt	PROPN
cana-1178	15	5	et	et	PROPN
cana-1178	15	6	al	al	PROPN
cana-1178	16	1	[	[	X
cana-1178	16	2	7	7	NUM
cana-1178	16	3	]	]	X
cana-1178	16	4	defined	define	VERB
cana-1178	16	5	randic	randic	ADJ
cana-1178	16	6	matrix	matrix	NOUN
cana-1178	16	7	and	and	CCONJ
cana-1178	16	8	randic	randic	ADJ
cana-1178	16	9	energy	energy	NOUN
cana-1178	16	10	.	.	PUNCT
cana-1178	17	1	further	further	ADJ
cana-1178	17	2	discussion	discussion	NOUN
cana-1178	17	3	on	on	ADP
cana-1178	17	4	randic	randic	ADJ
cana-1178	17	5	energy	energy	NOUN
cana-1178	17	6	can	can	AUX
cana-1178	17	7	be	be	AUX
cana-1178	17	8	found	find	VERB
cana-1178	17	9	in	in	ADP
cana-1178	17	10	[	[	X
cana-1178	17	11	3	3	NUM
cana-1178	17	12	]	]	PUNCT
cana-1178	17	13	,	,	PUNCT
cana-1178	17	14	[	[	X
cana-1178	17	15	5	5	NUM
cana-1178	17	16	]	]	PUNCT
cana-1178	17	17	.	.	PUNCT
cana-1178	18	1	in	in	ADP
cana-1178	18	2	[	[	X
cana-1178	18	3	8	8	NUM
cana-1178	18	4	]	]	PUNCT
cana-1178	18	5	the	the	DET
cana-1178	18	6	authors	author	NOUN
cana-1178	18	7	find	find	VERB
cana-1178	18	8	the	the	DET
cana-1178	18	9	minimum	minimum	ADJ
cana-1178	18	10	dominating	dominating	NOUN
cana-1178	18	11	energy	energy	NOUN
cana-1178	18	12	for	for	ADP
cana-1178	18	13	various	various	ADJ
cana-1178	18	14	graphs	graph	NOUN
cana-1178	18	15	like	like	ADP
cana-1178	18	16	complete	complete	ADJ
cana-1178	18	17	graph	graph	NOUN
cana-1178	18	18	,	,	PUNCT
cana-1178	18	19	star	star	NOUN
cana-1178	18	20	graph	graph	NOUN
cana-1178	18	21	etc	etc	X
cana-1178	18	22	.	.	X
cana-1178	19	1	this	this	DET
cana-1178	19	2	paper	paper	NOUN
cana-1178	19	3	is	be	AUX
cana-1178	19	4	organized	organize	VERB
cana-1178	19	5	as	as	SCONJ
cana-1178	19	6	follows	follow	VERB
cana-1178	19	7	.	.	PUNCT
cana-1178	20	1	section	section	NOUN
cana-1178	20	2	2	2	NUM
cana-1178	20	3	is	be	AUX
cana-1178	20	4	about	about	ADP
cana-1178	20	5	preliminaries	preliminary	NOUN
cana-1178	20	6	.	.	PUNCT
cana-1178	21	1	in	in	ADP
cana-1178	21	2	section	section	NOUN
cana-1178	21	3	3	3	NUM
cana-1178	21	4	,	,	PUNCT
cana-1178	21	5	we	we	PRON
cana-1178	21	6	introduced	introduce	VERB
cana-1178	21	7	the	the	DET
cana-1178	21	8	rough	rough	ADJ
cana-1178	21	9	complemented	complemented	ADJ
cana-1178	21	10	graph	graph	NOUN
cana-1178	21	11	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	21	12	)	)	PUNCT
cana-1178	21	13	and	and	CCONJ
cana-1178	21	14	explore	explore	VERB
cana-1178	21	15	the	the	DET
cana-1178	21	16	properties	property	NOUN
cana-1178	21	17	of	of	ADP
cana-1178	21	18	the	the	DET
cana-1178	21	19	graph	graph	NOUN
cana-1178	21	20	.	.	PUNCT
cana-1178	22	1	also	also	ADV
cana-1178	22	2	defined	define	VERB
cana-1178	22	3	various	various	ADJ
cana-1178	22	4	energies	energy	NOUN
cana-1178	22	5	like	like	ADP
cana-1178	22	6	minimal	minimal	ADJ
cana-1178	22	7	dominating	dominating	NOUN
cana-1178	22	8	,	,	PUNCT
cana-1178	22	9	seidel	seidel	PROPN
cana-1178	22	10	,	,	PUNCT
cana-1178	22	11	randic	randic	ADJ
cana-1178	22	12	etc	etc	X
cana-1178	22	13	and	and	CCONJ
cana-1178	22	14	look	look	VERB
cana-1178	22	15	at	at	ADP
cana-1178	22	16	further	further	ADJ
cana-1178	22	17	bounds	bound	NOUN
cana-1178	22	18	.	.	PUNCT
cana-1178	23	1	in	in	ADP
cana-1178	23	2	section	section	NOUN
cana-1178	23	3	4	4	NUM
cana-1178	23	4	,	,	PUNCT
cana-1178	23	5	we	we	PRON
cana-1178	23	6	provided	provide	VERB
cana-1178	23	7	the	the	DET
cana-1178	23	8	python	python	NOUN
cana-1178	23	9	coding	coding	NOUN
cana-1178	23	10	for	for	ADP
cana-1178	23	11	the	the	DET
cana-1178	23	12	corresponding	corresponding	ADJ
cana-1178	23	13	graph	graph	NOUN
cana-1178	23	14	energies	energy	NOUN
cana-1178	23	15	and	and	CCONJ
cana-1178	23	16	conclude	conclude	VERB
cana-1178	23	17	in	in	ADP
cana-1178	23	18	section	section	NOUN
cana-1178	23	19	5	5	NUM
cana-1178	23	20	.	.	SYM
cana-1178	23	21	2	2	NUM
cana-1178	23	22	.	.	NUM
cana-1178	23	23	preliminaries	preliminary	NOUN
cana-1178	23	24	in	in	ADP
cana-1178	23	25	this	this	DET
cana-1178	23	26	section	section	NOUN
cana-1178	23	27	,	,	PUNCT
cana-1178	23	28	the	the	DET
cana-1178	23	29	basic	basic	ADJ
cana-1178	23	30	definitions	definition	NOUN
cana-1178	23	31	required	require	VERB
cana-1178	23	32	to	to	PART
cana-1178	23	33	study	study	VERB
cana-1178	23	34	the	the	DET
cana-1178	23	35	article	article	NOUN
cana-1178	23	36	are	be	AUX
cana-1178	23	37	listed	list	VERB
cana-1178	23	38	.	.	PUNCT
cana-1178	24	1	definition	definition	NOUN
cana-1178	24	2	2.1	2.1	NUM
cana-1178	24	3	.	.	PUNCT
cana-1178	25	1	let	let	VERB
cana-1178	25	2	𝐺	𝐺	PROPN
cana-1178	25	3	=	=	SYM
cana-1178	25	4	(	(	PUNCT
cana-1178	25	5	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1178	25	6	)	)	PUNCT
cana-1178	25	7	,	,	PUNCT
cana-1178	25	8	𝐸(𝐺	𝐸(𝐺	NOUN
cana-1178	25	9	)	)	PUNCT
cana-1178	25	10	)	)	PUNCT
cana-1178	25	11	be	be	AUX
cana-1178	25	12	a	a	DET
cana-1178	25	13	simple	simple	ADJ
cana-1178	25	14	graph	graph	NOUN
cana-1178	25	15	.	.	PUNCT
cana-1178	26	1	a	a	DET
cana-1178	26	2	set	set	VERB
cana-1178	26	3	𝐷	𝐷	NOUN
cana-1178	26	4	⊆	⊆	NUM
cana-1178	26	5	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1178	26	6	)	)	PUNCT
cana-1178	26	7	is	be	AUX
cana-1178	26	8	s	s	AUX
cana-1178	26	9	said	say	VERB
cana-1178	26	10	to	to	PART
cana-1178	26	11	be	be	AUX
cana-1178	26	12	a	a	DET
cana-1178	26	13	dominating	dominating	NOUN
cana-1178	26	14	set	set	NOUN
cana-1178	26	15	if	if	SCONJ
cana-1178	26	16	every	every	DET
cana-1178	26	17	vertex	vertex	NOUN
cana-1178	26	18	in	in	ADP
cana-1178	26	19	𝑉(𝐺	𝑉(𝐺	NOUN
cana-1178	26	20	)	)	PUNCT
cana-1178	26	21	−	−	PROPN
cana-1178	26	22	𝐷	𝐷	NOUN
cana-1178	26	23	is	be	AUX
cana-1178	26	24	adjacent	adjacent	ADJ
cana-1178	26	25	to	to	PART
cana-1178	26	26	atleast	atleast	VERB
cana-1178	26	27	one	one	NUM
cana-1178	26	28	vertex	vertex	NOUN
cana-1178	26	29	in	in	ADP
cana-1178	26	30	𝐷.	𝐷.	PROPN
cana-1178	26	31	the	the	DET
cana-1178	26	32	domination	domination	NOUN
cana-1178	26	33	number	number	NOUN
cana-1178	26	34	of	of	ADP
cana-1178	26	35	𝐺	𝐺	PROPN
cana-1178	26	36	,	,	PUNCT
cana-1178	26	37	denoted	denote	VERB
cana-1178	26	38	by	by	ADP
cana-1178	26	39	𝛾(𝐺	𝛾(𝐺	PROPN
cana-1178	26	40	)	)	PUNCT
cana-1178	26	41	,	,	PUNCT
cana-1178	26	42	is	be	AUX
cana-1178	26	43	the	the	DET
cana-1178	26	44	minimum	minimum	ADJ
cana-1178	26	45	cardinality	cardinality	NOUN
cana-1178	26	46	among	among	ADP
cana-1178	26	47	all	all	DET
cana-1178	26	48	dominating	dominating	NOUN
cana-1178	26	49	sets	set	NOUN
cana-1178	26	50	of	of	ADP
cana-1178	26	51	𝐺.	𝐺.	NOUN
cana-1178	26	52	communications	communication	NOUN
cana-1178	26	53	on	on	ADP
cana-1178	26	54	applied	apply	VERB
cana-1178	26	55	nonlinear	nonlinear	ADJ
cana-1178	26	56	analysis	analysis	NOUN
cana-1178	26	57	issn	issn	NOUN
cana-1178	26	58	:	:	PUNCT
cana-1178	26	59	1074	1074	NUM
cana-1178	26	60	-	-	PUNCT
cana-1178	26	61	133x	133x	NUM
cana-1178	26	62	vol	vol	NOUN
cana-1178	26	63	31	31	NUM
cana-1178	26	64	no	no	NOUN
cana-1178	26	65	.	.	PUNCT
cana-1178	27	1	6s	6s	NUM
cana-1178	27	2	(	(	PUNCT
cana-1178	27	3	2024	2024	NUM
cana-1178	27	4	)	)	PUNCT
cana-1178	27	5	194	194	NUM
cana-1178	27	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	27	7	definition	definition	NOUN
cana-1178	27	8	2.2	2.2	NUM
cana-1178	27	9	.	.	PUNCT
cana-1178	28	1	a	a	DET
cana-1178	28	2	set	set	NOUN
cana-1178	28	3	is	be	AUX
cana-1178	28	4	independent	independent	ADJ
cana-1178	28	5	if	if	SCONJ
cana-1178	28	6	no	no	DET
cana-1178	28	7	two	two	NUM
cana-1178	28	8	vertices	vertex	NOUN
cana-1178	28	9	in	in	ADP
cana-1178	28	10	it	it	PRON
cana-1178	28	11	are	be	AUX
cana-1178	28	12	adjacent	adjacent	ADJ
cana-1178	28	13	.	.	PUNCT
cana-1178	29	1	an	an	DET
cana-1178	29	2	independent	independent	ADJ
cana-1178	29	3	dominating	dominating	NOUN
cana-1178	29	4	set	set	NOUN
cana-1178	29	5	of	of	ADP
cana-1178	29	6	𝐺	𝐺	PROPN
cana-1178	29	7	is	be	AUX
cana-1178	29	8	a	a	DET
cana-1178	29	9	set	set	NOUN
cana-1178	29	10	that	that	PRON
cana-1178	29	11	is	be	AUX
cana-1178	29	12	both	both	PRON
cana-1178	29	13	dominating	dominate	VERB
cana-1178	29	14	and	and	CCONJ
cana-1178	29	15	independent	independent	ADJ
cana-1178	29	16	in	in	ADP
cana-1178	29	17	𝐺	𝐺	PROPN
cana-1178	29	18	.the	.the	DET
cana-1178	29	19	independent	independent	ADJ
cana-1178	29	20	domination	domination	NOUN
cana-1178	29	21	number	number	NOUN
cana-1178	29	22	of	of	ADP
cana-1178	29	23	𝐺	𝐺	PROPN
cana-1178	29	24	denoted	denote	VERB
cana-1178	29	25	by	by	ADP
cana-1178	29	26	𝛼	𝛼	NOUN
cana-1178	29	27	,	,	PUNCT
cana-1178	29	28	is	be	AUX
cana-1178	29	29	the	the	DET
cana-1178	29	30	minimum	minimum	ADJ
cana-1178	29	31	size	size	NOUN
cana-1178	29	32	of	of	ADP
cana-1178	29	33	an	an	DET
cana-1178	29	34	independent	independent	ADJ
cana-1178	29	35	dominating	dominating	NOUN
cana-1178	29	36	set	set	NOUN
cana-1178	29	37	.	.	PUNCT
cana-1178	30	1	lemma	lemma	PROPN
cana-1178	30	2	2.1	2.1	NUM
cana-1178	30	3	.	.	PUNCT
cana-1178	31	1	[	[	X
cana-1178	31	2	𝟓	𝟓	X
cana-1178	31	3	]	]	X
cana-1178	31	4	the	the	DET
cana-1178	31	5	randic	randic	ADJ
cana-1178	31	6	spectral	spectral	ADJ
cana-1178	31	7	radius	radius	NOUN
cana-1178	31	8	𝜌1(𝐺	𝜌1(𝐺	PROPN
cana-1178	31	9	)	)	PUNCT
cana-1178	31	10	=	=	SYM
cana-1178	31	11	1	1	X
cana-1178	31	12	.	.	X
cana-1178	31	13	lemma	lemma	PROPN
cana-1178	31	14	2.2	2.2	NUM
cana-1178	31	15	.	.	PUNCT
cana-1178	32	1	[	[	X
cana-1178	32	2	𝟑	𝟑	X
cana-1178	32	3	]	]	X
cana-1178	32	4	if	if	SCONJ
cana-1178	32	5	𝐺	𝐺	PROPN
cana-1178	32	6	posseses	possese	VERB
cana-1178	32	7	isolated	isolated	ADJ
cana-1178	32	8	vertices	vertex	NOUN
cana-1178	32	9	,	,	PUNCT
cana-1178	32	10	then	then	ADV
cana-1178	32	11	𝑑𝑒𝑡𝑅	𝑑𝑒𝑡𝑅	VERB
cana-1178	32	12	=	=	SYM
cana-1178	32	13	𝑑𝑒𝑡𝐴	𝑑𝑒𝑡𝐴	NOUN
cana-1178	32	14	=	=	SYM
cana-1178	32	15	0	0	X
cana-1178	32	16	.	.	PUNCT
cana-1178	33	1	if	if	SCONJ
cana-1178	33	2	𝐺	𝐺	PROPN
cana-1178	33	3	does	do	AUX
cana-1178	33	4	not	not	PART
cana-1178	33	5	possess	possess	VERB
cana-1178	33	6	isolated	isolated	ADJ
cana-1178	33	7	vertices	vertex	NOUN
cana-1178	33	8	,	,	PUNCT
cana-1178	33	9	then	then	ADV
cana-1178	33	10	𝑑𝑒𝑡𝑅	𝑑𝑒𝑡𝑅	VERB
cana-1178	33	11	=	=	SYM
cana-1178	33	12	1	1	NUM
cana-1178	33	13	𝑑1𝑑2	𝑑1𝑑2	NOUN
cana-1178	33	14	…	…	PUNCT
cana-1178	33	15	…	…	PUNCT
cana-1178	33	16	…	…	PUNCT
cana-1178	33	17	…	…	PUNCT
cana-1178	33	18	…	…	SYM
cana-1178	33	19	𝑑𝑛	𝑑𝑛	NOUN
cana-1178	33	20	𝑑𝑒𝑡𝐴.	𝑑𝑒𝑡𝐴.	NOUN
cana-1178	33	21	let	let	VERB
cana-1178	33	22	𝐼	𝐼	PROPN
cana-1178	33	23	=	=	SYM
cana-1178	33	24	(	(	PUNCT
cana-1178	33	25	𝑈	𝑈	PROPN
cana-1178	33	26	,	,	PUNCT
cana-1178	33	27	𝑅	𝑅	PROPN
cana-1178	33	28	)	)	PUNCT
cana-1178	33	29	be	be	VERB
cana-1178	33	30	an	an	DET
cana-1178	33	31	approximation	approximation	NOUN
cana-1178	33	32	space	space	NOUN
cana-1178	33	33	where	where	SCONJ
cana-1178	33	34	𝑈	𝑈	PROPN
cana-1178	33	35	is	be	AUX
cana-1178	33	36	the	the	DET
cana-1178	33	37	nonempty	nonempty	ADJ
cana-1178	33	38	finite	finite	ADJ
cana-1178	33	39	set	set	NOUN
cana-1178	33	40	of	of	ADP
cana-1178	33	41	objects	object	NOUN
cana-1178	33	42	and	and	CCONJ
cana-1178	33	43	𝑅	𝑅	PROPN
cana-1178	33	44	is	be	AUX
cana-1178	33	45	an	an	DET
cana-1178	33	46	equivalence	equivalence	NOUN
cana-1178	33	47	relation	relation	NOUN
cana-1178	33	48	on	on	ADP
cana-1178	33	49	𝑈	𝑈	PROPN
cana-1178	33	50	and	and	CCONJ
cana-1178	33	51	for	for	ADP
cana-1178	33	52	any	any	DET
cana-1178	33	53	𝑥	𝑥	PRON
cana-1178	33	54	∈	∈	PROPN
cana-1178	33	55	𝑈	𝑈	PROPN
cana-1178	33	56	,	,	PUNCT
cana-1178	33	57	[	[	X
cana-1178	33	58	𝑥]𝑅	𝑥]𝑅	NOUN
cana-1178	33	59	=	=	PUNCT
cana-1178	33	60	{	{	PUNCT
cana-1178	33	61	𝑦	𝑦	NOUN
cana-1178	33	62	∈	∈	PROPN
cana-1178	33	63	𝑈|(𝑥	𝑈|(𝑥	PROPN
cana-1178	33	64	,	,	PUNCT
cana-1178	33	65	𝑦	𝑦	NOUN
cana-1178	33	66	)	)	PUNCT
cana-1178	33	67	∈	∈	PROPN
cana-1178	33	68	𝑅	𝑅	PROPN
cana-1178	33	69	}	}	PUNCT
cana-1178	33	70	is	be	AUX
cana-1178	33	71	said	say	VERB
cana-1178	33	72	to	to	PART
cana-1178	33	73	be	be	AUX
cana-1178	33	74	an	an	DET
cana-1178	33	75	equivalence	equivalence	NOUN
cana-1178	33	76	class	class	NOUN
cana-1178	33	77	.	.	PUNCT
cana-1178	34	1	for	for	ADP
cana-1178	34	2	𝑋	𝑋	PROPN
cana-1178	34	3	⊆	⊆	PROPN
cana-1178	34	4	𝑈	𝑈	PROPN
cana-1178	34	5	,	,	PUNCT
cana-1178	34	6	𝑅𝑆(𝑋	𝑅𝑆(𝑋	ADJ
cana-1178	34	7	)	)	PUNCT
cana-1178	34	8	=	=	SYM
cana-1178	34	9	(	(	PUNCT
cana-1178	34	10	𝑅−(𝑋	𝑅−(𝑋	ADJ
cana-1178	34	11	)	)	PUNCT
cana-1178	34	12	,	,	PUNCT
cana-1178	34	13	𝑅−(𝑋	𝑅−(𝑋	NOUN
cana-1178	34	14	)	)	PUNCT
cana-1178	34	15	)	)	PUNCT
cana-1178	34	16	be	be	AUX
cana-1178	34	17	the	the	DET
cana-1178	34	18	rough	rough	ADJ
cana-1178	34	19	set	set	NOUN
cana-1178	34	20	where	where	SCONJ
cana-1178	34	21	r−(x	r−(x	PROPN
cana-1178	34	22	)	)	PUNCT
cana-1178	35	1	=	=	PRON
cana-1178	36	1	{	{	PUNCT
cana-1178	36	2	x	x	PUNCT
cana-1178	36	3	∈	∈	PROPN
cana-1178	36	4	u|[x]r	u|[x]r	NOUN
cana-1178	36	5	x	x	NOUN
cana-1178	36	6	}	}	PUNCT
cana-1178	36	7	is	be	AUX
cana-1178	36	8	said	say	VERB
cana-1178	36	9	to	to	PART
cana-1178	36	10	be	be	AUX
cana-1178	36	11	a	a	DET
cana-1178	36	12	lower	low	ADJ
cana-1178	36	13	approximation	approximation	NOUN
cana-1178	36	14	space	space	NOUN
cana-1178	36	15	and	and	CCONJ
cana-1178	36	16	the	the	DET
cana-1178	36	17	upper	upper	ADJ
cana-1178	36	18	approximation	approximation	NOUN
cana-1178	36	19	space	space	NOUN
cana-1178	36	20	is	be	AUX
cana-1178	36	21	defined	define	VERB
cana-1178	36	22	as	as	ADP
cana-1178	36	23	r−(x	r−(x	PROPN
cana-1178	36	24	)	)	PUNCT
cana-1178	37	1	=	=	PRON
cana-1178	37	2	{	{	PUNCT
cana-1178	37	3	x	x	PUNCT
cana-1178	37	4	∈	∈	PROPN
cana-1178	37	5	u|[x]r	u|[x]r	NOUN
cana-1178	37	6	∩	∩	NOUN
cana-1178	37	7	x	x	SYM
cana-1178	37	8	≠	≠	PROPN
cana-1178	37	9	∅	∅	NOUN
cana-1178	37	10	}	}	PUNCT
cana-1178	37	11	.	.	PUNCT
cana-1178	38	1	also	also	ADV
cana-1178	38	2	we	we	PRON
cana-1178	38	3	defined	define	VERB
cana-1178	38	4	the	the	DET
cana-1178	38	5	set	set	NOUN
cana-1178	38	6	of	of	ADP
cana-1178	38	7	rough	rough	ADJ
cana-1178	38	8	sets	set	NOUN
cana-1178	38	9	as	as	ADP
cana-1178	38	10	t	t	PROPN
cana-1178	38	11	=	=	SYM
cana-1178	38	12	{	{	PUNCT
cana-1178	38	13	rs(x)|x	rs(x)|x	PROPN
cana-1178	38	14	u	u	NOUN
cana-1178	38	15	}	}	PUNCT
cana-1178	38	16	.	.	PUNCT
cana-1178	39	1	it	it	PRON
cana-1178	39	2	has	have	AUX
cana-1178	39	3	been	be	AUX
cana-1178	39	4	established	establish	VERB
cana-1178	39	5	that	that	SCONJ
cana-1178	39	6	if	if	SCONJ
cana-1178	39	7	𝐼	𝐼	PROPN
cana-1178	39	8	=	=	SYM
cana-1178	39	9	(	(	PUNCT
cana-1178	39	10	𝑈	𝑈	PROPN
cana-1178	39	11	,	,	PUNCT
cana-1178	39	12	𝑅	𝑅	PROPN
cana-1178	39	13	)	)	PUNCT
cana-1178	39	14	,	,	PUNCT
cana-1178	39	15	(	(	PUNCT
cana-1178	39	16	𝑇	𝑇	PROPN
cana-1178	39	17	,	,	PUNCT
cana-1178	39	18	∆	∆	X
cana-1178	39	19	,	,	PUNCT
cana-1178	39	20	∇	∇	X
cana-1178	39	21	)	)	PUNCT
cana-1178	39	22	is	be	AUX
cana-1178	39	23	a	a	DET
cana-1178	39	24	rough	rough	ADJ
cana-1178	39	25	semiring	semiring	NOUN
cana-1178	39	26	[	[	X
cana-1178	39	27	10	10	NUM
cana-1178	39	28	]	]	PUNCT
cana-1178	39	29	.	.	PUNCT
cana-1178	40	1	the	the	DET
cana-1178	40	2	partition	partition	NOUN
cana-1178	40	3	created	create	VERB
cana-1178	40	4	by	by	ADP
cana-1178	40	5	𝑅	𝑅	PROPN
cana-1178	40	6	on	on	ADP
cana-1178	40	7	𝑈	𝑈	PROPN
cana-1178	40	8	should	should	AUX
cana-1178	40	9	consist	consist	VERB
cana-1178	40	10	of	of	ADP
cana-1178	40	11	{	{	PUNCT
cana-1178	40	12	𝑋1	𝑋1	PROPN
cana-1178	40	13	,	,	PUNCT
cana-1178	40	14	𝑋2	𝑋2	VERB
cana-1178	40	15	,	,	PUNCT
cana-1178	40	16	…	…	PUNCT
cana-1178	40	17	…	…	PUNCT
cana-1178	41	1	𝑋𝑚	𝑋𝑚	ADJ
cana-1178	41	2	,	,	PUNCT
cana-1178	41	3	𝑋𝑚+1	𝑋𝑚+1	VERB
cana-1178	41	4	…	…	PUNCT
cana-1178	41	5	…	…	PUNCT
cana-1178	41	6	.	.	PUNCT
cana-1178	42	1	𝑋𝑛	𝑋𝑛	NOUN
cana-1178	42	2	}	}	PUNCT
cana-1178	42	3	where	where	SCONJ
cana-1178	42	4	|𝑋𝑖|	|𝑋𝑖|	NOUN
cana-1178	42	5	>	>	X
cana-1178	42	6	1	1	NUM
cana-1178	42	7	,	,	PUNCT
cana-1178	42	8	1	1	NUM
cana-1178	42	9	≤	≤	NUM
cana-1178	42	10	𝑖	𝑖	SYM
cana-1178	42	11	≤	≤	PROPN
cana-1178	42	12	𝑚	𝑚	NOUN
cana-1178	42	13	,	,	PUNCT
cana-1178	42	14	|𝑋𝑗|	|𝑋𝑗|	NOUN
cana-1178	42	15	=	=	SYM
cana-1178	42	16	1	1	NUM
cana-1178	42	17	,	,	PUNCT
cana-1178	42	18	𝑚	𝑚	X
cana-1178	42	19	+	+	PROPN
cana-1178	42	20	1	1	NUM
cana-1178	42	21	≤	≤	NUM
cana-1178	42	22	𝑗	𝑗	PRON
cana-1178	42	23	≤	≤	ADJ
cana-1178	42	24	𝑛.	𝑛.	NOUN
cana-1178	42	25	definition	definition	NOUN
cana-1178	42	26	2.3	2.3	NUM
cana-1178	42	27	[	[	X
cana-1178	42	28	𝟏𝟏	𝟏𝟏	X
cana-1178	42	29	]	]	X
cana-1178	42	30	rough	rough	ADJ
cana-1178	42	31	zero	zero	NUM
cana-1178	42	32	divisor	divisor	NOUN
cana-1178	42	33	graph	graph	NOUN
cana-1178	42	34	of	of	ADP
cana-1178	42	35	rough	rough	ADJ
cana-1178	42	36	semiring	semire	VERB
cana-1178	42	37	the	the	DET
cana-1178	42	38	zero	zero	NUM
cana-1178	42	39	divisor	divisor	NOUN
cana-1178	42	40	graph	graph	NOUN
cana-1178	42	41	of	of	ADP
cana-1178	42	42	the	the	DET
cana-1178	42	43	rough	rough	ADJ
cana-1178	42	44	semiring	semiring	NOUN
cana-1178	42	45	(	(	PUNCT
cana-1178	42	46	t	t	PROPN
cana-1178	42	47	,	,	PUNCT
cana-1178	42	48	∆	∆	PROPN
cana-1178	42	49	,	,	PUNCT
cana-1178	42	50	∇	∇	X
cana-1178	42	51	)	)	PUNCT
cana-1178	42	52	is	be	AUX
cana-1178	42	53	t(g	t(g	NOUN
cana-1178	42	54	)	)	PUNCT
cana-1178	43	1	=	=	SYM
cana-1178	43	2	(	(	PUNCT
cana-1178	43	3	v	v	NOUN
cana-1178	43	4	,	,	PUNCT
cana-1178	43	5	e	e	NOUN
cana-1178	43	6	)	)	PUNCT
cana-1178	43	7	where	where	SCONJ
cana-1178	43	8	v	v	NOUN
cana-1178	43	9	is	be	AUX
cana-1178	43	10	the	the	DET
cana-1178	43	11	set	set	NOUN
cana-1178	43	12	of	of	ADP
cana-1178	43	13	vertices	vertex	NOUN
cana-1178	43	14	in	in	ADP
cana-1178	43	15	t(g	t(g	NOUN
cana-1178	43	16	)	)	PUNCT
cana-1178	43	17	consists	consist	VERB
cana-1178	43	18	of	of	ADP
cana-1178	43	19	nonempty	nonempty	ADJ
cana-1178	43	20	zero	zero	NUM
cana-1178	43	21	divisors	divisor	NOUN
cana-1178	43	22	i.e	i.e	X
cana-1178	43	23	,	,	PUNCT
cana-1178	43	24	v	v	NOUN
cana-1178	43	25	=	=	SYM
cana-1178	43	26	{	{	PUNCT
cana-1178	43	27	rs(x	rs(x	PROPN
cana-1178	43	28	)	)	PUNCT
cana-1178	43	29	∈	∈	PROPN
cana-1178	43	30	t|rs(x	t|rs(x	NOUN
cana-1178	43	31	)	)	PUNCT
cana-1178	43	32	≠	≠	PROPN
cana-1178	43	33	rs(∅	rs(∅	NUM
cana-1178	43	34	)	)	PUNCT
cana-1178	43	35	is	be	AUX
cana-1178	43	36	a	a	DET
cana-1178	43	37	zero	zero	NUM
cana-1178	43	38	divisor	divisor	NOUN
cana-1178	43	39	of	of	ADP
cana-1178	43	40	t	t	PROPN
cana-1178	43	41	}	}	PUNCT
cana-1178	43	42	and	and	CCONJ
cana-1178	43	43	e	e	NOUN
cana-1178	43	44	is	be	AUX
cana-1178	43	45	the	the	DET
cana-1178	43	46	set	set	NOUN
cana-1178	43	47	of	of	ADP
cana-1178	43	48	edges	edge	NOUN
cana-1178	43	49	connecting	connect	VERB
cana-1178	43	50	the	the	DET
cana-1178	43	51	elements	element	NOUN
cana-1178	43	52	of	of	ADP
cana-1178	43	53	v	v	PRON
cana-1178	43	54	such	such	ADJ
cana-1178	43	55	that	that	SCONJ
cana-1178	43	56	there	there	PRON
cana-1178	43	57	is	be	VERB
cana-1178	43	58	an	an	DET
cana-1178	43	59	edge	edge	NOUN
cana-1178	43	60	connecting	connect	VERB
cana-1178	43	61	rs(x	rs(x	NOUN
cana-1178	43	62	)	)	PUNCT
cana-1178	43	63	and	and	CCONJ
cana-1178	43	64	rs(y	rs(y	NUM
cana-1178	43	65	)	)	PUNCT
cana-1178	43	66	in	in	ADP
cana-1178	43	67	v	v	NUM
cana-1178	43	68	iff	iff	PROPN
cana-1178	43	69	rs(x)∇rs(y	rs(x)∇rs(y	NOUN
cana-1178	43	70	)	)	PUNCT
cana-1178	43	71	=	=	SYM
cana-1178	43	72	rs(∅	rs(∅	NUM
cana-1178	43	73	)	)	PUNCT
cana-1178	43	74	.	.	PUNCT
cana-1178	44	1	this	this	DET
cana-1178	44	2	graph	graph	NOUN
cana-1178	44	3	t(g	t(g	PROPN
cana-1178	44	4	)	)	PUNCT
cana-1178	44	5	is	be	AUX
cana-1178	44	6	called	call	VERB
cana-1178	44	7	a	a	DET
cana-1178	44	8	rough	rough	ADJ
cana-1178	44	9	zero	zero	NUM
cana-1178	44	10	divisor	divisor	NOUN
cana-1178	44	11	graph	graph	NOUN
cana-1178	44	12	of	of	ADP
cana-1178	44	13	the	the	DET
cana-1178	44	14	rough	rough	ADJ
cana-1178	44	15	semiring	semiring	NOUN
cana-1178	44	16	(	(	PUNCT
cana-1178	44	17	t	t	PROPN
cana-1178	44	18	,	,	PUNCT
cana-1178	44	19	∆	∆	PROPN
cana-1178	44	20	,	,	PUNCT
cana-1178	44	21	∇	∇	PROPN
cana-1178	44	22	)	)	PUNCT
cana-1178	44	23	.	.	PUNCT
cana-1178	45	1	3	3	X
cana-1178	45	2	.	.	NUM
cana-1178	45	3	bounds	bound	NOUN
cana-1178	45	4	of	of	ADP
cana-1178	45	5	various	various	ADJ
cana-1178	45	6	energies	energy	NOUN
cana-1178	45	7	of	of	ADP
cana-1178	45	8	rough	rough	ADJ
cana-1178	45	9	complemented	complemented	ADJ
cana-1178	45	10	graph	graph	NOUN
cana-1178	45	11	throughout	throughout	ADP
cana-1178	45	12	this	this	DET
cana-1178	45	13	section	section	NOUN
cana-1178	45	14	,	,	PUNCT
cana-1178	45	15	we	we	PRON
cana-1178	45	16	consider	consider	VERB
cana-1178	45	17	an	an	DET
cana-1178	45	18	approximation	approximation	NOUN
cana-1178	45	19	space	space	NOUN
cana-1178	45	20	𝐼	𝐼	PROPN
cana-1178	45	21	=	=	SYM
cana-1178	45	22	(	(	PUNCT
cana-1178	45	23	𝑈	𝑈	PROPN
cana-1178	45	24	,	,	PUNCT
cana-1178	45	25	𝑅	𝑅	PROPN
cana-1178	45	26	)	)	PUNCT
cana-1178	45	27	along	along	ADP
cana-1178	45	28	with	with	ADP
cana-1178	45	29	the	the	DET
cana-1178	45	30	rough	rough	ADJ
cana-1178	45	31	semiring	semiring	NOUN
cana-1178	45	32	(	(	PUNCT
cana-1178	45	33	𝑇	𝑇	PROPN
cana-1178	45	34	,	,	PUNCT
cana-1178	45	35	∆	∆	X
cana-1178	45	36	,	,	PUNCT
cana-1178	45	37	∇	∇	PROPN
cana-1178	45	38	)	)	PUNCT
cana-1178	45	39	.	.	PUNCT
cana-1178	46	1	let	let	VERB
cana-1178	46	2	𝐸	𝐸	PRON
cana-1178	46	3	=	=	PRON
cana-1178	46	4	{	{	PUNCT
cana-1178	46	5	𝑋1	𝑋1	PROPN
cana-1178	46	6	,	,	PUNCT
cana-1178	46	7	𝑋2	𝑋2	VERB
cana-1178	46	8	,	,	PUNCT
cana-1178	46	9	…	…	PUNCT
cana-1178	46	10	…	…	PUNCT
cana-1178	46	11	…	…	PUNCT
cana-1178	46	12	…	…	PUNCT
cana-1178	46	13	.	.	PUNCT
cana-1178	47	1	𝑋𝑛	𝑋𝑛	NOUN
cana-1178	47	2	}	}	PUNCT
cana-1178	47	3	be	be	AUX
cana-1178	47	4	the	the	DET
cana-1178	47	5	equivalence	equivalence	NOUN
cana-1178	47	6	classes	class	NOUN
cana-1178	47	7	induced	induce	VERB
cana-1178	47	8	by	by	ADP
cana-1178	47	9	𝑅	𝑅	PROPN
cana-1178	47	10	in	in	ADP
cana-1178	47	11	which	which	PRON
cana-1178	47	12	{	{	PUNCT
cana-1178	47	13	𝑋1	𝑋1	PROPN
cana-1178	47	14	,	,	PUNCT
cana-1178	47	15	𝑋2	𝑋2	VERB
cana-1178	47	16	,	,	PUNCT
cana-1178	47	17	…	…	PUNCT
cana-1178	47	18	…	…	PUNCT
cana-1178	47	19	…	…	PUNCT
cana-1178	47	20	…	…	PUNCT
cana-1178	47	21	.	.	PUNCT
cana-1178	48	1	𝑋𝑚	𝑋𝑚	NOUN
cana-1178	48	2	}	}	PUNCT
cana-1178	48	3	are	be	AUX
cana-1178	48	4	the	the	DET
cana-1178	48	5	equivalence	equivalence	NOUN
cana-1178	48	6	classes	class	NOUN
cana-1178	48	7	with	with	ADP
cana-1178	48	8	cardinality	cardinality	NOUN
cana-1178	48	9	greater	great	ADJ
cana-1178	48	10	than	than	ADP
cana-1178	48	11	1	1	NUM
cana-1178	48	12	.	.	PUNCT
cana-1178	49	1	in	in	ADP
cana-1178	49	2	this	this	DET
cana-1178	49	3	section	section	NOUN
cana-1178	49	4	,	,	PUNCT
cana-1178	49	5	the	the	DET
cana-1178	49	6	rough	rough	ADJ
cana-1178	49	7	complemented	complemented	ADJ
cana-1178	49	8	graph	graph	NOUN
cana-1178	49	9	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	49	10	)	)	PUNCT
cana-1178	49	11	of	of	ADP
cana-1178	49	12	the	the	DET
cana-1178	49	13	rough	rough	ADJ
cana-1178	49	14	semiring	semiring	NOUN
cana-1178	49	15	is	be	AUX
cana-1178	49	16	introduced	introduce	VERB
cana-1178	49	17	.	.	PUNCT
cana-1178	50	1	properties	property	NOUN
cana-1178	50	2	of	of	ADP
cana-1178	50	3	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	50	4	)	)	PUNCT
cana-1178	50	5	is	be	AUX
cana-1178	50	6	studied	study	VERB
cana-1178	50	7	and	and	CCONJ
cana-1178	50	8	the	the	DET
cana-1178	50	9	various	various	ADJ
cana-1178	50	10	energies	energy	NOUN
cana-1178	50	11	along	along	ADP
cana-1178	50	12	with	with	ADP
cana-1178	50	13	their	their	PRON
cana-1178	50	14	bounds	bound	NOUN
cana-1178	50	15	are	be	AUX
cana-1178	50	16	dealt	deal	VERB
cana-1178	50	17	in	in	ADP
cana-1178	50	18	detail	detail	NOUN
cana-1178	50	19	.	.	PUNCT
cana-1178	51	1	3.1	3.1	NUM
cana-1178	51	2	minimum	minimum	ADJ
cana-1178	51	3	dominating	dominating	NOUN
cana-1178	51	4	energy	energy	NOUN
cana-1178	51	5	definition	definition	NOUN
cana-1178	51	6	3.1	3.1	NUM
cana-1178	51	7	.	.	PUNCT
cana-1178	52	1	rough	rough	ADJ
cana-1178	52	2	complemented	complemented	ADJ
cana-1178	52	3	graph	graph	NOUN
cana-1178	52	4	let	let	VERB
cana-1178	52	5	(	(	PUNCT
cana-1178	52	6	𝑇	𝑇	PROPN
cana-1178	52	7	,	,	PUNCT
cana-1178	52	8	∆	∆	X
cana-1178	52	9	,	,	PUNCT
cana-1178	52	10	∇	∇	X
cana-1178	52	11	)	)	PUNCT
cana-1178	52	12	be	be	AUX
cana-1178	52	13	a	a	DET
cana-1178	52	14	rough	rough	ADJ
cana-1178	52	15	semiring	semiring	NOUN
cana-1178	52	16	.	.	PUNCT
cana-1178	53	1	the	the	DET
cana-1178	53	2	rough	rough	ADJ
cana-1178	53	3	complemented	complemented	ADJ
cana-1178	53	4	graph	graph	NOUN
cana-1178	53	5	of	of	ADP
cana-1178	53	6	𝑇	𝑇	PROPN
cana-1178	53	7	denoted	denote	VERB
cana-1178	53	8	by	by	ADP
cana-1178	53	9	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	53	10	)	)	PUNCT
cana-1178	53	11	is	be	AUX
cana-1178	53	12	a	a	DET
cana-1178	53	13	graph	graph	NOUN
cana-1178	53	14	whose	whose	DET
cana-1178	53	15	vertices	vertex	NOUN
cana-1178	53	16	are	be	AUX
cana-1178	53	17	𝑉(𝐺𝑅𝐶(𝑇	𝑉(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	53	18	)	)	PUNCT
cana-1178	53	19	)	)	PUNCT
cana-1178	54	1	=	=	PRON
cana-1178	54	2	{	{	PUNCT
cana-1178	54	3	𝑅𝑆(𝑌)|𝑌	𝑅𝑆(𝑌)|𝑌	PROPN
cana-1178	54	4	∈	∈	PROPN
cana-1178	54	5	(	(	PUNCT
cana-1178	54	6	℘(𝐸))1	℘(𝐸))1	NOUN
cana-1178	54	7	}	}	PUNCT
cana-1178	54	8	where	where	SCONJ
cana-1178	54	9	(	(	PUNCT
cana-1178	54	10	℘(𝐸))1	℘(𝐸))1	X
cana-1178	54	11	=	=	PUNCT
cana-1178	54	12	℘(𝐸	℘(𝐸	NOUN
cana-1178	54	13	)	)	PUNCT
cana-1178	54	14	−	−	PROPN
cana-1178	54	15	{	{	PUNCT
cana-1178	54	16	𝑅𝑆(𝑈	𝑅𝑆(𝑈	NOUN
cana-1178	54	17	)	)	PUNCT
cana-1178	54	18	,	,	PUNCT
cana-1178	54	19	𝑅𝑆(∅	𝑅𝑆(∅	PUNCT
cana-1178	54	20	)	)	PUNCT
cana-1178	54	21	}	}	PUNCT
cana-1178	54	22	and	and	CCONJ
cana-1178	54	23	two	two	NUM
cana-1178	54	24	distinct	distinct	ADJ
cana-1178	54	25	vertices	vertex	NOUN
cana-1178	54	26	𝑅𝑆(𝑋	𝑅𝑆(𝑋	ADV
cana-1178	54	27	)	)	PUNCT
cana-1178	54	28	,	,	PUNCT
cana-1178	54	29	𝑅𝑆(𝑍	𝑅𝑆(𝑍	PUNCT
cana-1178	54	30	)	)	PUNCT
cana-1178	54	31	are	be	AUX
cana-1178	54	32	adjacent	adjacent	ADJ
cana-1178	54	33	iff	iff	VERB
cana-1178	54	34	𝑅𝑆(𝑋)∇𝑅𝑆(𝑍	𝑅𝑆(𝑋)∇𝑅𝑆(𝑍	PROPN
cana-1178	54	35	)	)	PUNCT
cana-1178	54	36	=	=	PUNCT
cana-1178	54	37	𝑅𝑆(∅	𝑅𝑆(∅	X
cana-1178	54	38	)	)	PUNCT
cana-1178	54	39	.	.	PUNCT
cana-1178	55	1	remarks	remark	VERB
cana-1178	55	2	.	.	PUNCT
cana-1178	56	1	it	it	PRON
cana-1178	56	2	is	be	AUX
cana-1178	56	3	to	to	PART
cana-1178	56	4	be	be	AUX
cana-1178	56	5	noted	note	VERB
cana-1178	56	6	that	that	SCONJ
cana-1178	56	7	•	•	NOUN
cana-1178	56	8	the	the	DET
cana-1178	56	9	number	number	NOUN
cana-1178	56	10	of	of	ADP
cana-1178	56	11	vertices	vertex	NOUN
cana-1178	56	12	in	in	ADP
cana-1178	56	13	rough	rough	ADJ
cana-1178	56	14	complemented	complemented	ADJ
cana-1178	56	15	graph	graph	NOUN
cana-1178	56	16	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	56	17	)	)	PUNCT
cana-1178	56	18	is	be	AUX
cana-1178	56	19	2𝑛	2𝑛	PROPN
cana-1178	56	20	−	−	PROPN
cana-1178	56	21	2	2	NUM
cana-1178	56	22	where	where	SCONJ
cana-1178	56	23	𝑛	𝑛	PROPN
cana-1178	56	24	denotes	denote	VERB
cana-1178	56	25	the	the	DET
cana-1178	56	26	number	number	NOUN
cana-1178	56	27	of	of	ADP
cana-1178	56	28	equivalence	equivalence	NOUN
cana-1178	56	29	classes	class	NOUN
cana-1178	56	30	in	in	ADP
cana-1178	56	31	𝐸.	𝐸.	PROPN
cana-1178	56	32	•	•	ADP
cana-1178	56	33	the	the	DET
cana-1178	56	34	number	number	NOUN
cana-1178	56	35	of	of	ADP
cana-1178	56	36	edges	edge	NOUN
cana-1178	56	37	in	in	ADP
cana-1178	56	38	rough	rough	ADJ
cana-1178	56	39	complemented	complemented	ADJ
cana-1178	56	40	graph	graph	NOUN
cana-1178	56	41	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	56	42	)	)	PUNCT
cana-1178	56	43	is	be	AUX
cana-1178	56	44	1	1	NUM
cana-1178	56	45	2	2	NUM
cana-1178	56	46	(	(	PUNCT
cana-1178	56	47	3𝑛	3𝑛	NUM
cana-1178	56	48	−	−	PROPN
cana-1178	56	49	2𝑛+1	2𝑛+1	NOUN
cana-1178	57	1	+	+	CCONJ
cana-1178	57	2	1	1	NUM
cana-1178	57	3	)	)	PUNCT
cana-1178	57	4	.	.	PUNCT
cana-1178	58	1	communications	communication	NOUN
cana-1178	58	2	on	on	ADP
cana-1178	58	3	applied	apply	VERB
cana-1178	58	4	nonlinear	nonlinear	ADJ
cana-1178	58	5	analysis	analysis	NOUN
cana-1178	58	6	issn	issn	NOUN
cana-1178	58	7	:	:	PUNCT
cana-1178	58	8	1074	1074	NUM
cana-1178	58	9	-	-	PUNCT
cana-1178	58	10	133x	133x	NUM
cana-1178	58	11	vol	vol	NOUN
cana-1178	58	12	31	31	NUM
cana-1178	58	13	no	no	NOUN
cana-1178	58	14	.	.	PUNCT
cana-1178	59	1	6s	6s	NUM
cana-1178	59	2	(	(	PUNCT
cana-1178	59	3	2024	2024	NUM
cana-1178	59	4	)	)	PUNCT
cana-1178	59	5	195	195	NUM
cana-1178	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	59	7	•	•	NOUN
cana-1178	59	8	for	for	ADP
cana-1178	59	9	any	any	DET
cana-1178	59	10	𝑅𝑆(𝑋	𝑅𝑆(𝑋	NOUN
cana-1178	59	11	)	)	PUNCT
cana-1178	59	12	∈	∈	PROPN
cana-1178	59	13	𝑉(𝐺𝑅𝐶(𝑇	𝑉(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	59	14	)	)	PUNCT
cana-1178	59	15	)	)	PUNCT
cana-1178	59	16	,	,	PUNCT
cana-1178	59	17	the	the	DET
cana-1178	59	18	degree	degree	NOUN
cana-1178	59	19	of	of	ADP
cana-1178	59	20	𝑅𝑆(𝑋	𝑅𝑆(𝑋	NOUN
cana-1178	59	21	)	)	PUNCT
cana-1178	59	22	is	be	AUX
cana-1178	59	23	2𝑛−𝑟	2𝑛−𝑟	NUM
cana-1178	59	24	−	−	PROPN
cana-1178	59	25	1	1	NUM
cana-1178	59	26	where	where	SCONJ
cana-1178	59	27	1	1	NUM
cana-1178	59	28	≤	≤	NOUN
cana-1178	59	29	𝑟	𝑟	NOUN
cana-1178	59	30	<	<	X
cana-1178	59	31	𝑚.	𝑚.	ADJ
cana-1178	59	32	•	•	NOUN
cana-1178	59	33	for	for	ADP
cana-1178	59	34	any	any	DET
cana-1178	59	35	𝑅𝑆(𝑋	𝑅𝑆(𝑋	NOUN
cana-1178	59	36	)	)	PUNCT
cana-1178	59	37	,	,	PUNCT
cana-1178	59	38	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	59	39	)	)	PUNCT
cana-1178	59	40	∈	∈	PROPN
cana-1178	59	41	𝑉(𝐺𝑅𝐶(𝑇	𝑉(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	59	42	)	)	PUNCT
cana-1178	59	43	)	)	PUNCT
cana-1178	59	44	,	,	PUNCT
cana-1178	59	45	𝑑2(𝑅𝑆(𝑋	𝑑2(𝑅𝑆(𝑋	NOUN
cana-1178	59	46	)	)	PUNCT
cana-1178	59	47	)	)	PUNCT
cana-1178	60	1	=	=	PRON
cana-1178	60	2	{	{	PUNCT
cana-1178	60	3	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	60	4	)	)	PUNCT
cana-1178	60	5	∈	∈	PROPN
cana-1178	60	6	𝑉(𝐺𝑅𝐶(𝑇))|𝑑(𝑅𝑆(𝑋	𝑉(𝐺𝑅𝐶(𝑇))|𝑑(𝑅𝑆(𝑋	PROPN
cana-1178	60	7	)	)	PUNCT
cana-1178	60	8	,	,	PUNCT
cana-1178	60	9	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	60	10	)	)	PUNCT
cana-1178	60	11	=	=	SYM
cana-1178	60	12	2	2	NUM
cana-1178	60	13	)	)	PUNCT
cana-1178	60	14	}	}	PUNCT
cana-1178	60	15	|𝑑2(𝑅𝑆(𝑋))|	|𝑑2(𝑅𝑆(𝑋))|	NOUN
cana-1178	60	16	=	=	PUNCT
cana-1178	60	17	2𝑟	2𝑟	NUM
cana-1178	60	18	−	−	NOUN
cana-1178	60	19	2	2	NUM
cana-1178	60	20	+	+	CCONJ
cana-1178	60	21	(	(	PUNCT
cana-1178	60	22	2𝑟	2𝑟	NUM
cana-1178	60	23	−	−	PROPN
cana-1178	60	24	1)(2𝑛−𝑟	1)(2𝑛−𝑟	NUM
cana-1178	60	25	−	−	NOUN
cana-1178	60	26	2	2	NUM
cana-1178	60	27	)	)	PUNCT
cana-1178	60	28	.	.	PUNCT
cana-1178	61	1	•	•	NOUN
cana-1178	61	2	for	for	ADP
cana-1178	61	3	any	any	DET
cana-1178	61	4	𝑅𝑆(𝑋	𝑅𝑆(𝑋	NOUN
cana-1178	61	5	)	)	PUNCT
cana-1178	61	6	,	,	PUNCT
cana-1178	61	7	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	61	8	)	)	PUNCT
cana-1178	61	9	∈	∈	PROPN
cana-1178	61	10	𝑉(𝐺𝑅𝐶(𝑇	𝑉(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	61	11	)	)	PUNCT
cana-1178	61	12	)	)	PUNCT
cana-1178	61	13	,	,	PUNCT
cana-1178	61	14	𝑑3(𝑅𝑆(𝑋	𝑑3(𝑅𝑆(𝑋	NOUN
cana-1178	61	15	)	)	PUNCT
cana-1178	61	16	)	)	PUNCT
cana-1178	62	1	=	=	PRON
cana-1178	62	2	{	{	PUNCT
cana-1178	62	3	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	62	4	)	)	PUNCT
cana-1178	62	5	∈	∈	PROPN
cana-1178	62	6	𝑉(𝐺𝑅𝐶(𝑇))|𝑑(𝑅𝑆(𝑋	𝑉(𝐺𝑅𝐶(𝑇))|𝑑(𝑅𝑆(𝑋	PROPN
cana-1178	62	7	)	)	PUNCT
cana-1178	62	8	,	,	PUNCT
cana-1178	62	9	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	62	10	)	)	PUNCT
cana-1178	62	11	=	=	SYM
cana-1178	62	12	3	3	X
cana-1178	62	13	)	)	PUNCT
cana-1178	62	14	}	}	PUNCT
cana-1178	62	15	|𝑑3(𝑅𝑆(𝑋))|	|𝑑3(𝑅𝑆(𝑋))|	VERB
cana-1178	62	16	=	=	PUNCT
cana-1178	62	17	2𝑟	2𝑟	NUM
cana-1178	62	18	−	−	NOUN
cana-1178	62	19	2	2	X
cana-1178	62	20	.	.	PUNCT
cana-1178	62	21	here	here	ADV
cana-1178	62	22	𝑑	𝑑	NOUN
cana-1178	62	23	represents	represent	VERB
cana-1178	62	24	the	the	DET
cana-1178	62	25	distance	distance	NOUN
cana-1178	62	26	.	.	PUNCT
cana-1178	63	1	•	•	NUM
cana-1178	63	2	the	the	DET
cana-1178	63	3	diameter	diameter	NOUN
cana-1178	63	4	of	of	ADP
cana-1178	63	5	𝐺𝑅𝐶(𝑇)is	𝐺𝑅𝐶(𝑇)is	ADJ
cana-1178	63	6	3	3	NUM
cana-1178	63	7	.	.	PUNCT
cana-1178	63	8	definition	definition	NOUN
cana-1178	63	9	3.2	3.2	NUM
cana-1178	63	10	.	.	PUNCT
cana-1178	64	1	minimum	minimum	ADJ
cana-1178	64	2	dominating	dominating	NOUN
cana-1178	64	3	set	set	NOUN
cana-1178	64	4	consider	consider	VERB
cana-1178	64	5	the	the	DET
cana-1178	64	6	rough	rough	ADJ
cana-1178	64	7	complemented	complemented	ADJ
cana-1178	64	8	graph	graph	NOUN
cana-1178	64	9	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	64	10	)	)	PUNCT
cana-1178	64	11	=	=	SYM
cana-1178	64	12	(	(	PUNCT
cana-1178	64	13	𝑉(𝐶(𝑇	𝑉(𝐶(𝑇	NOUN
cana-1178	64	14	)	)	PUNCT
cana-1178	64	15	)	)	PUNCT
cana-1178	64	16	,	,	PUNCT
cana-1178	64	17	𝐸(𝐶(𝑇	𝐸(𝐶(𝑇	NOUN
cana-1178	64	18	)	)	PUNCT
cana-1178	64	19	)	)	PUNCT
cana-1178	64	20	)	)	PUNCT
cana-1178	64	21	.	.	PUNCT
cana-1178	65	1	a	a	DET
cana-1178	65	2	subset	subset	NOUN
cana-1178	65	3	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	65	4	)	)	PUNCT
cana-1178	65	5	)	)	PUNCT
cana-1178	65	6	is	be	AUX
cana-1178	65	7	called	call	VERB
cana-1178	65	8	the	the	DET
cana-1178	65	9	dominating	dominating	NOUN
cana-1178	65	10	set	set	NOUN
cana-1178	65	11	of	of	ADP
cana-1178	65	12	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	65	13	)	)	PUNCT
cana-1178	65	14	if	if	SCONJ
cana-1178	65	15	every	every	DET
cana-1178	65	16	vertex	vertex	NOUN
cana-1178	65	17	in	in	ADP
cana-1178	65	18	𝑉(𝐶(𝑇	𝑉(𝐶(𝑇	NOUN
cana-1178	65	19	)	)	PUNCT
cana-1178	65	20	)	)	PUNCT
cana-1178	66	1	−	−	PROPN
cana-1178	66	2	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	66	3	)	)	PUNCT
cana-1178	66	4	)	)	PUNCT
cana-1178	66	5	is	be	AUX
cana-1178	66	6	adjacent	adjacent	ADJ
cana-1178	66	7	to	to	ADP
cana-1178	66	8	some	some	DET
cana-1178	66	9	vertex	vertex	NOUN
cana-1178	66	10	in	in	ADP
cana-1178	66	11	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	66	12	)	)	PUNCT
cana-1178	66	13	)	)	PUNCT
cana-1178	66	14	.	.	PUNCT
cana-1178	67	1	the	the	DET
cana-1178	67	2	minimum	minimum	ADJ
cana-1178	67	3	cardinality	cardinality	NOUN
cana-1178	67	4	of	of	ADP
cana-1178	67	5	a	a	DET
cana-1178	67	6	dominating	dominating	NOUN
cana-1178	67	7	set	set	NOUN
cana-1178	67	8	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	67	9	)	)	PUNCT
cana-1178	67	10	)	)	PUNCT
cana-1178	67	11	is	be	AUX
cana-1178	67	12	called	call	VERB
cana-1178	67	13	the	the	DET
cana-1178	67	14	domination	domination	NOUN
cana-1178	67	15	number	number	NOUN
cana-1178	67	16	of	of	ADP
cana-1178	67	17	the	the	DET
cana-1178	67	18	graph	graph	NOUN
cana-1178	67	19	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	67	20	)	)	PUNCT
cana-1178	67	21	,	,	PUNCT
cana-1178	67	22	denoted	denote	VERB
cana-1178	67	23	by	by	ADP
cana-1178	67	24	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	NOUN
cana-1178	67	25	)	)	PUNCT
cana-1178	67	26	)	)	PUNCT
cana-1178	67	27	.	.	PUNCT
cana-1178	68	1	definition	definition	NOUN
cana-1178	68	2	3.3	3.3	NUM
cana-1178	68	3	.	.	PUNCT
cana-1178	69	1	for	for	ADP
cana-1178	69	2	every	every	DET
cana-1178	69	3	𝑅𝑆(𝑋	𝑅𝑆(𝑋	NOUN
cana-1178	69	4	)	)	PUNCT
cana-1178	69	5	,	,	PUNCT
cana-1178	69	6	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	69	7	)	)	PUNCT
cana-1178	69	8	∈	∈	PROPN
cana-1178	69	9	𝑉(𝐶(𝑇	𝑉(𝐶(𝑇	NOUN
cana-1178	69	10	)	)	PUNCT
cana-1178	69	11	)	)	PUNCT
cana-1178	69	12	,	,	PUNCT
cana-1178	69	13	the	the	DET
cana-1178	69	14	minimum	minimum	ADJ
cana-1178	69	15	dominating	dominating	NOUN
cana-1178	69	16	matrix	matrix	NOUN
cana-1178	69	17	is	be	AUX
cana-1178	69	18	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	69	19	)	)	PUNCT
cana-1178	69	20	)	)	PUNCT
cana-1178	70	1	=	=	PRON
cana-1178	70	2	{	{	PUNCT
cana-1178	70	3	1	1	NUM
cana-1178	70	4	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	NOUN
cana-1178	70	5	)	)	PUNCT
cana-1178	70	6	=	=	PUNCT
cana-1178	70	7	𝑅𝑆(∅	𝑅𝑆(∅	X
cana-1178	70	8	)	)	PUNCT
cana-1178	70	9	1	1	NUM
cana-1178	70	10	𝑖𝑓	𝑖𝑓	ADP
cana-1178	70	11	𝑅𝑆(𝑋	𝑅𝑆(𝑋	PROPN
cana-1178	70	12	)	)	PUNCT
cana-1178	70	13	=	=	SYM
cana-1178	70	14	𝑅𝑆(𝑌)𝑎𝑛𝑑𝑅𝑆(𝑋	𝑅𝑆(𝑌)𝑎𝑛𝑑𝑅𝑆(𝑋	NOUN
cana-1178	70	15	)	)	PUNCT
cana-1178	70	16	∈	∈	PROPN
cana-1178	70	17	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	70	18	)	)	PUNCT
cana-1178	70	19	)	)	PUNCT
cana-1178	70	20	0	0	NUM
cana-1178	71	1	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
cana-1178	71	2	definition	definition	NOUN
cana-1178	71	3	3.4	3.4	NUM
cana-1178	71	4	.	.	PUNCT
cana-1178	72	1	minimum	minimum	ADJ
cana-1178	72	2	dominating	dominating	NOUN
cana-1178	72	3	energy	energy	NOUN
cana-1178	72	4	the	the	DET
cana-1178	72	5	minimum	minimum	ADJ
cana-1178	72	6	dominating	dominating	NOUN
cana-1178	72	7	energy	energy	NOUN
cana-1178	72	8	of	of	ADP
cana-1178	72	9	𝐺𝑅𝐶(𝑇)is	𝐺𝑅𝐶(𝑇)is	ADV
cana-1178	72	10	defined	define	VERB
cana-1178	72	11	as	as	ADP
cana-1178	72	12	ԑ(𝐷(𝐶(𝑇	ԑ(𝐷(𝐶(𝑇	NOUN
cana-1178	72	13	)	)	PUNCT
cana-1178	72	14	)	)	PUNCT
cana-1178	72	15	)	)	PUNCT
cana-1178	73	1	=	=	PUNCT
cana-1178	73	2	∑	∑	PUNCT
cana-1178	73	3	|𝜇𝑖|	|𝜇𝑖|	NOUN
cana-1178	73	4	2𝑛−2	2𝑛−2	NUM
cana-1178	73	5	𝑖=1	𝑖=1	PUNCT
cana-1178	73	6	where	where	SCONJ
cana-1178	73	7	𝜇1	𝜇1	ADJ
cana-1178	73	8	,	,	PUNCT
cana-1178	73	9	𝜇2	𝜇2	PROPN
cana-1178	73	10	…	…	PUNCT
cana-1178	73	11	…	…	PUNCT
cana-1178	73	12	…	…	PUNCT
cana-1178	73	13	.	.	PUNCT
cana-1178	73	14	.	.	PUNCT
cana-1178	74	1	𝜇2𝑛	𝜇2𝑛	PROPN
cana-1178	74	2	−2	−2	PROPN
cana-1178	74	3	are	be	AUX
cana-1178	74	4	the	the	DET
cana-1178	74	5	spectrum	spectrum	NOUN
cana-1178	74	6	of	of	ADP
cana-1178	74	7	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	74	8	)	)	PUNCT
cana-1178	74	9	)	)	PUNCT
cana-1178	74	10	.	.	PUNCT
cana-1178	75	1	the	the	DET
cana-1178	75	2	spectral	spectral	ADJ
cana-1178	75	3	radii	radius	NOUN
cana-1178	75	4	of	of	ADP
cana-1178	75	5	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	75	6	)	)	PUNCT
cana-1178	75	7	)	)	PUNCT
cana-1178	75	8	are	be	AUX
cana-1178	75	9	in	in	ADP
cana-1178	75	10	nonincreasing	nonincrease	VERB
cana-1178	75	11	order	order	NOUN
cana-1178	75	12	i.e.	i.e.	X
cana-1178	75	13	,	,	PUNCT
cana-1178	75	14	𝜇1	𝜇1	PROPN
cana-1178	75	15			PROPN
cana-1178	75	16	𝜇2	𝜇2	PROPN
cana-1178	75	17			NUM
cana-1178	75	18	…	…	PUNCT
cana-1178	75	19	…	…	PUNCT
cana-1178	75	20	…	…	PUNCT
cana-1178	75	21	…	…	PUNCT
cana-1178	75	22			NUM
cana-1178	75	23	𝜇2𝑛	𝜇2𝑛	PROPN
cana-1178	75	24	−2	−2	PROPN
cana-1178	75	25	.	.	PUNCT
cana-1178	76	1	theorem	theorem	VERB
cana-1178	76	2	3.1	3.1	NUM
cana-1178	76	3	.	.	PUNCT
cana-1178	77	1	in	in	ADP
cana-1178	77	2	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	77	3	)	)	PUNCT
cana-1178	77	4	,	,	PUNCT
cana-1178	77	5	the	the	DET
cana-1178	77	6	minimum	minimum	ADJ
cana-1178	77	7	dominating	dominating	NOUN
cana-1178	77	8	set	set	NOUN
cana-1178	77	9	is	be	AUX
cana-1178	77	10	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	77	11	)	)	PUNCT
cana-1178	77	12	)	)	PUNCT
cana-1178	78	1	=	=	PRON
cana-1178	78	2	{	{	PUNCT
cana-1178	78	3	𝑅𝑆(𝑋𝑖)|𝑖	𝑅𝑆(𝑋𝑖)|𝑖	NOUN
cana-1178	78	4	=	=	SYM
cana-1178	78	5	1,2	1,2	NUM
cana-1178	78	6	,	,	PUNCT
cana-1178	78	7	…	…	PUNCT
cana-1178	78	8	…	…	PUNCT
cana-1178	78	9	…	…	PUNCT
cana-1178	78	10	.	.	PUNCT
cana-1178	78	11	.	.	PUNCT
cana-1178	79	1	𝑛	𝑛	X
cana-1178	79	2	}	}	PUNCT
cana-1178	79	3	proof	proof	NOUN
cana-1178	79	4	.	.	PUNCT
cana-1178	80	1	let	let	VERB
cana-1178	80	2	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	80	3	)	)	PUNCT
cana-1178	80	4	∈	∈	PROPN
cana-1178	80	5	𝑉(𝐶(𝑇	𝑉(𝐶(𝑇	NOUN
cana-1178	80	6	)	)	PUNCT
cana-1178	80	7	)	)	PUNCT
cana-1178	81	1	−	−	PROPN
cana-1178	81	2	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	81	3	)	)	PUNCT
cana-1178	81	4	)	)	PUNCT
cana-1178	81	5	.	.	PUNCT
cana-1178	82	1	then	then	ADV
cana-1178	82	2	the	the	DET
cana-1178	82	3	edge	edge	NOUN
cana-1178	82	4	set	set	VERB
cana-1178	82	5	𝜉	𝜉	X
cana-1178	82	6	=	=	SYM
cana-1178	82	7	{	{	PUNCT
cana-1178	82	8	(	(	PUNCT
cana-1178	82	9	𝑅𝑆(𝑌	𝑅𝑆(𝑌	NOUN
cana-1178	82	10	)	)	PUNCT
cana-1178	82	11	,	,	PUNCT
cana-1178	82	12	𝑅𝑆(𝑍))|𝑍	𝑅𝑆(𝑍))|𝑍	PUNCT
cana-1178	82	13	∈	∈	PROPN
cana-1178	82	14	𝐸	𝐸	PROPN
cana-1178	82	15	−	−	PROPN
cana-1178	82	16	𝑌	𝑌	PROPN
cana-1178	82	17	}	}	PUNCT
cana-1178	82	18	next	next	ADV
cana-1178	82	19	to	to	PART
cana-1178	82	20	prove	prove	VERB
cana-1178	82	21	that	that	SCONJ
cana-1178	82	22	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	82	23	)	)	PUNCT
cana-1178	82	24	)	)	PUNCT
cana-1178	82	25	is	be	AUX
cana-1178	82	26	the	the	DET
cana-1178	82	27	minimum	minimum	ADJ
cana-1178	82	28	dominating	dominating	NOUN
cana-1178	82	29	set	set	NOUN
cana-1178	82	30	.	.	PUNCT
cana-1178	83	1	note	note	VERB
cana-1178	83	2	that	that	SCONJ
cana-1178	83	3	if	if	SCONJ
cana-1178	83	4	we	we	PRON
cana-1178	83	5	remove	remove	VERB
cana-1178	83	6	any	any	DET
cana-1178	83	7	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	NOUN
cana-1178	83	8	)	)	PUNCT
cana-1178	83	9	,	,	PUNCT
cana-1178	83	10	𝑖	𝑖	X
cana-1178	83	11	=	=	SYM
cana-1178	83	12	1,2	1,2	NUM
cana-1178	83	13	,	,	PUNCT
cana-1178	83	14	…	…	PUNCT
cana-1178	83	15	…	…	PUNCT
cana-1178	83	16	…	…	PUNCT
cana-1178	83	17	.	.	PUNCT
cana-1178	83	18	.	.	PUNCT
cana-1178	84	1	𝑛	𝑛	PROPN
cana-1178	84	2	from	from	ADP
cana-1178	84	3	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	84	4	)	)	PUNCT
cana-1178	84	5	)	)	PUNCT
cana-1178	85	1	then	then	ADV
cana-1178	85	2	𝑅𝑆(𝑋1𝑋2	𝑅𝑆(𝑋1𝑋2	NUM
cana-1178	85	3	…	…	PUNCT
cana-1178	85	4	…	…	PUNCT
cana-1178	85	5	…	…	PUNCT
cana-1178	85	6	.	.	PUNCT
cana-1178	86	1	𝑋𝑖−1𝑋𝑖+1	𝑋𝑖−1𝑋𝑖+1	NOUN
cana-1178	86	2	…	…	PUNCT
cana-1178	86	3	…	…	PUNCT
cana-1178	86	4	…	…	PUNCT
cana-1178	86	5	.	.	PUNCT
cana-1178	86	6	.	.	PUNCT
cana-1178	87	1	𝑋𝑛	𝑋𝑛	X
cana-1178	87	2	)	)	PUNCT
cana-1178	87	3	will	will	AUX
cana-1178	87	4	not	not	PART
cana-1178	87	5	be	be	AUX
cana-1178	87	6	adjacent	adjacent	ADJ
cana-1178	87	7	to	to	ADP
cana-1178	87	8	any	any	PRON
cana-1178	87	9	of	of	ADP
cana-1178	87	10	the	the	DET
cana-1178	87	11	elements	element	NOUN
cana-1178	87	12	in	in	ADP
cana-1178	87	13	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	87	14	)	)	PUNCT
cana-1178	87	15	)	)	PUNCT
cana-1178	88	1	−	−	PROPN
cana-1178	88	2	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	NOUN
cana-1178	88	3	)	)	PUNCT
cana-1178	88	4	which	which	PRON
cana-1178	88	5	will	will	AUX
cana-1178	88	6	affect	affect	VERB
cana-1178	88	7	the	the	DET
cana-1178	88	8	dominating	dominating	NOUN
cana-1178	88	9	property	property	NOUN
cana-1178	88	10	.	.	PUNCT
cana-1178	89	1	hence	hence	ADV
cana-1178	89	2	removal	removal	NOUN
cana-1178	89	3	of	of	ADP
cana-1178	89	4	any	any	DET
cana-1178	89	5	element	element	NOUN
cana-1178	89	6	from	from	ADP
cana-1178	89	7	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	89	8	)	)	PUNCT
cana-1178	89	9	)	)	PUNCT
cana-1178	89	10	will	will	AUX
cana-1178	89	11	affect	affect	VERB
cana-1178	89	12	the	the	DET
cana-1178	89	13	dominating	dominating	NOUN
cana-1178	89	14	property	property	NOUN
cana-1178	89	15	.	.	PUNCT
cana-1178	90	1	therefore	therefore	ADV
cana-1178	90	2	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	90	3	)	)	PUNCT
cana-1178	90	4	)	)	PUNCT
cana-1178	91	1	=	=	PRON
cana-1178	91	2	{	{	PUNCT
cana-1178	91	3	𝑅𝑆(𝑋𝑖)|𝑖	𝑅𝑆(𝑋𝑖)|𝑖	NOUN
cana-1178	91	4	=	=	SYM
cana-1178	91	5	1,2	1,2	NUM
cana-1178	91	6	,	,	PUNCT
cana-1178	91	7	…	…	PUNCT
cana-1178	91	8	…	…	PUNCT
cana-1178	91	9	…	…	PUNCT
cana-1178	91	10	.	.	PUNCT
cana-1178	91	11	.	.	PUNCT
cana-1178	92	1	𝑛	𝑛	X
cana-1178	92	2	}	}	PUNCT
cana-1178	92	3	is	be	AUX
cana-1178	92	4	the	the	DET
cana-1178	92	5	minimum	minimum	ADJ
cana-1178	92	6	dominating	dominating	NOUN
cana-1178	92	7	set	set	NOUN
cana-1178	92	8	.	.	PUNCT
cana-1178	93	1	theorem	theorem	VERB
cana-1178	93	2	3.2	3.2	NUM
cana-1178	93	3	.	.	PUNCT
cana-1178	94	1	let	let	VERB
cana-1178	94	2	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	94	3	)	)	PUNCT
cana-1178	94	4	be	be	AUX
cana-1178	94	5	the	the	DET
cana-1178	94	6	rough	rough	ADJ
cana-1178	94	7	complemented	complemented	ADJ
cana-1178	94	8	graph	graph	NOUN
cana-1178	94	9	and	and	CCONJ
cana-1178	94	10	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	94	11	)	)	PUNCT
cana-1178	94	12	)	)	PUNCT
cana-1178	95	1	be	be	AUX
cana-1178	95	2	the	the	DET
cana-1178	95	3	minimum	minimum	ADJ
cana-1178	95	4	dominating	dominating	NOUN
cana-1178	95	5	matrix	matrix	NOUN
cana-1178	95	6	of	of	ADP
cana-1178	95	7	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	95	8	)	)	PUNCT
cana-1178	95	9	and	and	CCONJ
cana-1178	95	10	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	PROPN
cana-1178	95	11	)	)	PUNCT
cana-1178	95	12	)	)	PUNCT
cana-1178	96	1	denotes	denote	VERB
cana-1178	96	2	the	the	DET
cana-1178	96	3	minimum	minimum	ADJ
cana-1178	96	4	domination	domination	NOUN
cana-1178	96	5	number	number	NOUN
cana-1178	96	6	then	then	ADV
cana-1178	96	7	communications	communication	NOUN
cana-1178	96	8	on	on	ADP
cana-1178	96	9	applied	apply	VERB
cana-1178	96	10	nonlinear	nonlinear	ADJ
cana-1178	96	11	analysis	analysis	NOUN
cana-1178	96	12	issn	issn	NOUN
cana-1178	96	13	:	:	PUNCT
cana-1178	96	14	1074	1074	NUM
cana-1178	96	15	-	-	PUNCT
cana-1178	96	16	133x	133x	NUM
cana-1178	96	17	vol	vol	NOUN
cana-1178	96	18	31	31	NUM
cana-1178	96	19	no	no	NOUN
cana-1178	96	20	.	.	PUNCT
cana-1178	97	1	6s	6s	NUM
cana-1178	97	2	(	(	PUNCT
cana-1178	97	3	2024	2024	NUM
cana-1178	97	4	)	)	PUNCT
cana-1178	97	5	196	196	NUM
cana-1178	97	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	97	7	•	•	NOUN
cana-1178	97	8	∑	∑	PUNCT
cana-1178	97	9	𝜇𝑖	𝜇𝑖	ADP
cana-1178	97	10	=	=	SYM
cana-1178	97	11	|𝛾(𝐶(𝑇))|	|𝛾(𝐶(𝑇))|	PART
cana-1178	97	12	2𝑛−2	2𝑛−2	NUM
cana-1178	97	13	𝑖=1	𝑖=1	NOUN
cana-1178	97	14	•	•	NUM
cana-1178	97	15	∑	∑	PUNCT
cana-1178	97	16	𝜇𝑖	𝜇𝑖	ADP
cana-1178	97	17	2𝑛−2	2𝑛−2	NUM
cana-1178	97	18	𝑖=1	𝑖=1	SYM
cana-1178	97	19	2	2	NUM
cana-1178	97	20	=	=	SYM
cana-1178	97	21	|3𝑛	|3𝑛	ADP
cana-1178	97	22	−	−	PROPN
cana-1178	97	23	2𝑛+1	2𝑛+1	PROPN
cana-1178	98	1	+	+	CCONJ
cana-1178	98	2	1|	1|	NUM
cana-1178	98	3	+	+	SYM
cana-1178	98	4	|𝛾(𝐶(𝑇))|	|𝛾(𝐶(𝑇))|	NOUN
cana-1178	98	5	proof	proof	NOUN
cana-1178	98	6	.	.	PUNCT
cana-1178	99	1	it	it	PRON
cana-1178	99	2	is	be	AUX
cana-1178	99	3	known	know	VERB
cana-1178	99	4	that	that	SCONJ
cana-1178	99	5	the	the	DET
cana-1178	99	6	sum	sum	NOUN
cana-1178	99	7	of	of	ADP
cana-1178	99	8	eigen	eigen	PROPN
cana-1178	99	9	values	value	NOUN
cana-1178	99	10	of	of	ADP
cana-1178	99	11	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	99	12	)	)	PUNCT
cana-1178	99	13	)	)	PUNCT
cana-1178	99	14	is	be	AUX
cana-1178	99	15	the	the	DET
cana-1178	99	16	trace	trace	NOUN
cana-1178	99	17	of	of	ADP
cana-1178	99	18	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	99	19	)	)	PUNCT
cana-1178	99	20	)	)	PUNCT
cana-1178	99	21	.	.	PUNCT
cana-1178	100	1	∑	∑	PUNCT
cana-1178	100	2	𝜇𝑖	𝜇𝑖	ADP
cana-1178	100	3	2𝑛−2	2𝑛−2	NUM
cana-1178	100	4	𝑖=1	𝑖=1	PUNCT
cana-1178	100	5	=	=	PUNCT
cana-1178	100	6	∑	∑	PUNCT
cana-1178	100	7	𝑑𝑖𝑖	𝑑𝑖𝑖	PROPN
cana-1178	100	8	2𝑛−2	2𝑛−2	NUM
cana-1178	100	9	𝑖=1	𝑖=1	PUNCT
cana-1178	100	10	=	=	SYM
cana-1178	100	11	|𝛾(𝐶(𝑇))|	|𝛾(𝐶(𝑇))|	NOUN
cana-1178	100	12	it	it	PRON
cana-1178	100	13	is	be	AUX
cana-1178	100	14	find	find	VERB
cana-1178	100	15	that	that	SCONJ
cana-1178	100	16	∑	∑	ADP
cana-1178	100	17	𝜇𝑖	𝜇𝑖	ADP
cana-1178	100	18	22𝑛−2	22𝑛−2	NUM
cana-1178	100	19	𝑖=1	𝑖=1	PROPN
cana-1178	100	20	of	of	ADP
cana-1178	100	21	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	100	22	)	)	PUNCT
cana-1178	100	23	)	)	PUNCT
cana-1178	100	24	is	be	AUX
cana-1178	100	25	the	the	DET
cana-1178	100	26	trace	trace	NOUN
cana-1178	100	27	of	of	ADP
cana-1178	100	28	[	[	X
cana-1178	100	29	𝐴𝐷(𝐶(𝑇))]2	𝐴𝐷(𝐶(𝑇))]2	X
cana-1178	100	30	∑	∑	PROPN
cana-1178	100	31	𝜇𝑖	𝜇𝑖	ADP
cana-1178	100	32	2	2	NUM
cana-1178	100	33	2𝑛−2	2𝑛−2	NOUN
cana-1178	100	34	𝑖=1	𝑖=1	PUNCT
cana-1178	101	1	=	=	PUNCT
cana-1178	101	2	∑	∑	PUNCT
cana-1178	101	3	𝑑𝑖𝑗	𝑑𝑖𝑗	PROPN
cana-1178	101	4	2𝑛−2	2𝑛−2	NUM
cana-1178	101	5	𝑖=1	𝑖=1	PUNCT
cana-1178	101	6	∑	∑	SYM
cana-1178	101	7	𝑑𝑗𝑖	𝑑𝑗𝑖	NUM
cana-1178	101	8	2𝑛−2	2𝑛−2	NUM
cana-1178	101	9	𝑗=1	𝑗=1	PUNCT
cana-1178	101	10	=	=	PUNCT
cana-1178	101	11	∑	∑	PUNCT
cana-1178	101	12	𝑑𝑖𝑖	𝑑𝑖𝑖	PROPN
cana-1178	101	13	2	2	NUM
cana-1178	101	14	+2𝑛−2	+2𝑛−2	NOUN
cana-1178	101	15	𝑖=1	𝑖=1	PUNCT
cana-1178	101	16	∑	∑	ADV
cana-1178	101	17	𝑑𝑖𝑗𝑑𝑗𝑖𝑖≠𝑗	𝑑𝑖𝑗𝑑𝑗𝑖𝑖≠𝑗	ADJ
cana-1178	101	18	=	=	PUNCT
cana-1178	101	19	∑	∑	PUNCT
cana-1178	101	20	𝑑𝑖𝑖	𝑑𝑖𝑖	PROPN
cana-1178	101	21	2	2	NUM
cana-1178	101	22	+	+	CCONJ
cana-1178	101	23	2	2	NUM
cana-1178	101	24	∑	∑	ADV
cana-1178	101	25	𝑑𝑖𝑗	𝑑𝑖𝑗	NOUN
cana-1178	101	26	2	2	NUM
cana-1178	101	27	𝑖<𝑗	𝑖<𝑗	NOUN
cana-1178	101	28	2𝑛−2	2𝑛−2	NUM
cana-1178	101	29	𝑖=1	𝑖=1	PUNCT
cana-1178	101	30	=	=	SYM
cana-1178	101	31	|𝛾(𝐶(𝑇))|	|𝛾(𝐶(𝑇))|	PROPN
cana-1178	101	32	+	+	NUM
cana-1178	101	33	|(3𝑛	|(3𝑛	NOUN
cana-1178	101	34	−	−	PROPN
cana-1178	101	35	2𝑛+1	2𝑛+1	PROPN
cana-1178	101	36	+	+	CCONJ
cana-1178	101	37	1)|	1)|	NUM
cana-1178	101	38	theorem	theorem	VERB
cana-1178	101	39	3.3	3.3	NUM
cana-1178	101	40	.	.	PUNCT
cana-1178	102	1	in	in	ADP
cana-1178	102	2	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	102	3	)	)	PUNCT
cana-1178	102	4	,	,	PUNCT
cana-1178	102	5	𝐷(𝐶(𝑇	𝐷(𝐶(𝑇	NOUN
cana-1178	102	6	)	)	PUNCT
cana-1178	102	7	)	)	PUNCT
cana-1178	102	8	be	be	AUX
cana-1178	102	9	the	the	DET
cana-1178	102	10	minimum	minimum	ADJ
cana-1178	102	11	dominant	dominant	NOUN
cana-1178	102	12	set	set	NOUN
cana-1178	102	13	and	and	CCONJ
cana-1178	102	14	∆𝐷	∆𝐷	PROPN
cana-1178	102	15	denote	denote	VERB
cana-1178	102	16	the	the	DET
cana-1178	102	17	determinant	determinant	NOUN
cana-1178	102	18	of	of	ADP
cana-1178	102	19	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	102	20	)	)	PUNCT
cana-1178	102	21	)	)	PUNCT
cana-1178	103	1	then	then	ADV
cana-1178	103	2	√(3𝑛	√(3𝑛	NOUN
cana-1178	103	3	−	−	PROPN
cana-1178	104	1	2𝑛+1	2𝑛+1	PROPN
cana-1178	104	2	+	+	CCONJ
cana-1178	104	3	1	1	NUM
cana-1178	104	4	+	+	NUM
cana-1178	104	5	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	NOUN
cana-1178	104	6	)	)	PUNCT
cana-1178	104	7	)	)	PUNCT
cana-1178	104	8	)	)	PUNCT
cana-1178	105	1	+	+	CCONJ
cana-1178	105	2	(	(	PUNCT
cana-1178	105	3	2𝑛	2𝑛	NUM
cana-1178	105	4	−	−	PROPN
cana-1178	105	5	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	105	6	−	−	PROPN
cana-1178	106	1	3)∆𝐷	3)∆𝐷	NUM
cana-1178	106	2	2	2	NUM
cana-1178	106	3	(	(	PUNCT
cana-1178	106	4	2𝑛−2)(2𝑛−3	2𝑛−2)(2𝑛−3	NUM
cana-1178	106	5	)	)	PUNCT
cana-1178	106	6			NOUN
cana-1178	106	7	ԑ𝐷(𝐺𝑅𝐶(𝑇	ԑ𝐷(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	106	8	)	)	PUNCT
cana-1178	106	9	)	)	PUNCT
cana-1178	106	10			NOUN
cana-1178	106	11	√(2𝑛	√(2𝑛	NOUN
cana-1178	106	12	−	−	NOUN
cana-1178	106	13	2	2	NUM
cana-1178	106	14	)	)	PUNCT
cana-1178	106	15	(	(	PUNCT
cana-1178	106	16	3𝑛	3𝑛	NUM
cana-1178	106	17	−	−	PROPN
cana-1178	107	1	2𝑛+1	2𝑛+1	NOUN
cana-1178	107	2	+	+	CCONJ
cana-1178	107	3	1	1	NUM
cana-1178	107	4	+	+	NUM
cana-1178	107	5	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	NOUN
cana-1178	107	6	)	)	PUNCT
cana-1178	107	7	)	)	PUNCT
cana-1178	107	8	)	)	PUNCT
cana-1178	108	1	proof	proof	NOUN
cana-1178	108	2	.	.	PUNCT
cana-1178	109	1	by	by	ADP
cana-1178	109	2	cauchy	cauchy	PROPN
cana-1178	109	3	schwarz	schwarz	PROPN
cana-1178	109	4	inequality	inequality	PROPN
cana-1178	109	5	(	(	PUNCT
cana-1178	109	6	∑	∑	PROPN
cana-1178	109	7	|𝜇𝑖	|𝜇𝑖	X
cana-1178	109	8	2𝑛−2	2𝑛−2	NUM
cana-1178	109	9	𝑖=1	𝑖=1	PUNCT
cana-1178	109	10	|	|	ADV
cana-1178	109	11	)	)	PUNCT
cana-1178	109	12	2	2	NUM
cana-1178	109	13	≤	≤	NOUN
cana-1178	109	14	(	(	PUNCT
cana-1178	109	15	∑	∑	ADV
cana-1178	109	16	1	1	NUM
cana-1178	109	17	2𝑛−2	2𝑛−2	NUM
cana-1178	109	18	𝑖=1	𝑖=1	PUNCT
cana-1178	109	19	)	)	PUNCT
cana-1178	110	1	(	(	PUNCT
cana-1178	110	2	∑	∑	PUNCT
cana-1178	110	3	𝜇𝑖	𝜇𝑖	ADP
cana-1178	110	4	2	2	NUM
cana-1178	110	5	2𝑛−2	2𝑛−2	NUM
cana-1178	110	6	𝑖=1	𝑖=1	PUNCT
cana-1178	110	7	)	)	PUNCT
cana-1178	111	1	[	[	X
cana-1178	111	2	ԑ𝐷(𝐺𝑅𝐶(𝑇	ԑ𝐷(𝐺𝑅𝐶(𝑇	PROPN
cana-1178	111	3	)	)	PUNCT
cana-1178	111	4	)	)	PUNCT
cana-1178	111	5	]	]	PUNCT
cana-1178	112	1	2	2	X
cana-1178	112	2	≤	≤	NOUN
cana-1178	112	3	(	(	PUNCT
cana-1178	112	4	2𝑛	2𝑛	NOUN
cana-1178	112	5	−	−	PROPN
cana-1178	112	6	2	2	NUM
cana-1178	112	7	)	)	PUNCT
cana-1178	112	8	(	(	PUNCT
cana-1178	112	9	3𝑛	3𝑛	NUM
cana-1178	112	10	−	−	PROPN
cana-1178	112	11	2𝑛+1	2𝑛+1	NOUN
cana-1178	112	12	+	+	CCONJ
cana-1178	112	13	1	1	NUM
cana-1178	112	14	+	+	NUM
cana-1178	112	15	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	NOUN
cana-1178	112	16	)	)	PUNCT
cana-1178	112	17	)	)	PUNCT
cana-1178	112	18	)	)	PUNCT
cana-1178	112	19	and	and	CCONJ
cana-1178	112	20	from	from	ADP
cana-1178	112	21	arithmetic	arithmetic	ADJ
cana-1178	112	22	–	–	PUNCT
cana-1178	112	23	geometric	geometric	ADJ
cana-1178	112	24	mean	mean	NOUN
cana-1178	112	25	inequality	inequality	NOUN
cana-1178	112	26	∑	∑	PUNCT
cana-1178	112	27	|𝜇𝑖||𝜇𝑗|	|𝜇𝑖||𝜇𝑗|	VERB
cana-1178	112	28	≥	≥	NUM
cana-1178	112	29	(	(	PUNCT
cana-1178	112	30	2𝑛	2𝑛	PROPN
cana-1178	112	31	−	−	PROPN
cana-1178	112	32	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	112	33	−	−	PROPN
cana-1178	113	1	3)∆𝐷	3)∆𝐷	NUM
cana-1178	113	2	2	2	NUM
cana-1178	113	3	(	(	PUNCT
cana-1178	113	4	2𝑛−2)(2𝑛−3	2𝑛−2)(2𝑛−3	NUM
cana-1178	113	5	)	)	PUNCT
cana-1178	113	6	𝑖≠𝑗	𝑖≠𝑗	PUNCT
cana-1178	114	1	[	[	X
cana-1178	114	2	ԑ𝐷(𝐺𝑅𝐶(𝑇	ԑ𝐷(𝐺𝑅𝐶(𝑇	PROPN
cana-1178	114	3	)	)	PUNCT
cana-1178	114	4	)	)	PUNCT
cana-1178	114	5	]	]	PUNCT
cana-1178	114	6	2	2	X
cana-1178	114	7	=	=	SYM
cana-1178	114	8	(	(	PUNCT
cana-1178	114	9	∑	∑	PUNCT
cana-1178	114	10	|𝜇𝑖|	|𝜇𝑖|	NOUN
cana-1178	114	11	2𝑛−2	2𝑛−2	NUM
cana-1178	114	12	𝑖=1	𝑖=1	PUNCT
cana-1178	114	13	)	)	PUNCT
cana-1178	114	14	2	2	NUM
cana-1178	114	15	=	=	SYM
cana-1178	114	16	∑	∑	PUNCT
cana-1178	114	17	|𝜇𝑖|	|𝜇𝑖|	NOUN
cana-1178	114	18	2𝑛−2	2𝑛−2	NUM
cana-1178	114	19	𝑖=1	𝑖=1	SYM
cana-1178	114	20	2	2	NUM
cana-1178	114	21	+	+	CCONJ
cana-1178	114	22	∑	∑	PROPN
cana-1178	114	23	|𝜇𝑖||𝜇𝑗𝑖≠𝑗	|𝜇𝑖||𝜇𝑗𝑖≠𝑗	NOUN
cana-1178	114	24	|	|	ADV
cana-1178	114	25	[	[	X
cana-1178	114	26	ԑ𝐷(𝐺𝑅𝐶(𝑇	ԑ𝐷(𝐺𝑅𝐶(𝑇	PROPN
cana-1178	114	27	)	)	PUNCT
cana-1178	114	28	)	)	PUNCT
cana-1178	114	29	]	]	PUNCT
cana-1178	114	30	2	2	NUM
cana-1178	114	31	≥	≥	NOUN
cana-1178	114	32	(	(	PUNCT
cana-1178	114	33	3𝑛	3𝑛	NUM
cana-1178	114	34	−	−	PROPN
cana-1178	114	35	2𝑛+1	2𝑛+1	NOUN
cana-1178	114	36	+	+	CCONJ
cana-1178	114	37	1	1	NUM
cana-1178	114	38	+	+	NUM
cana-1178	114	39	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	NOUN
cana-1178	114	40	)	)	PUNCT
cana-1178	114	41	)	)	PUNCT
cana-1178	114	42	)	)	PUNCT
cana-1178	115	1	+	+	CCONJ
cana-1178	115	2	(	(	PUNCT
cana-1178	115	3	2𝑛	2𝑛	NUM
cana-1178	115	4	−	−	PROPN
cana-1178	115	5	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	115	6	−	−	PROPN
cana-1178	116	1	3)∆𝐷	3)∆𝐷	NUM
cana-1178	116	2	2	2	NUM
cana-1178	116	3	(	(	PUNCT
cana-1178	116	4	2𝑛−2)(2𝑛−3	2𝑛−2)(2𝑛−3	NUM
cana-1178	116	5	)	)	PUNCT
cana-1178	116	6	ԑ𝐷(𝐺𝑅𝐶(𝑇	ԑ𝐷(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	116	7	)	)	PUNCT
cana-1178	116	8	)	)	PUNCT
cana-1178	116	9	≥	≥	PROPN
cana-1178	116	10	√(3𝑛	√(3𝑛	NOUN
cana-1178	117	1	−	−	PROPN
cana-1178	117	2	2𝑛+1	2𝑛+1	PROPN
cana-1178	117	3	+	+	CCONJ
cana-1178	117	4	1	1	NUM
cana-1178	117	5	+	+	NUM
cana-1178	117	6	𝛾(𝐶(𝑇	𝛾(𝐶(𝑇	NOUN
cana-1178	117	7	)	)	PUNCT
cana-1178	117	8	)	)	PUNCT
cana-1178	117	9	)	)	PUNCT
cana-1178	118	1	+	+	CCONJ
cana-1178	118	2	(	(	PUNCT
cana-1178	118	3	2𝑛	2𝑛	NUM
cana-1178	118	4	−	−	PROPN
cana-1178	118	5	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	118	6	−	−	PROPN
cana-1178	119	1	3)∆𝐷	3)∆𝐷	NUM
cana-1178	119	2	2	2	NUM
cana-1178	119	3	(	(	PUNCT
cana-1178	119	4	2𝑛−2)(2𝑛−3	2𝑛−2)(2𝑛−3	NUM
cana-1178	119	5	)	)	PUNCT
cana-1178	119	6	communications	communication	NOUN
cana-1178	119	7	on	on	ADP
cana-1178	119	8	applied	apply	VERB
cana-1178	119	9	nonlinear	nonlinear	ADJ
cana-1178	119	10	analysis	analysis	NOUN
cana-1178	119	11	issn	issn	NOUN
cana-1178	119	12	:	:	PUNCT
cana-1178	119	13	1074	1074	NUM
cana-1178	119	14	-	-	PUNCT
cana-1178	119	15	133x	133x	NUM
cana-1178	119	16	vol	vol	NOUN
cana-1178	119	17	31	31	NUM
cana-1178	119	18	no	no	NOUN
cana-1178	119	19	.	.	PUNCT
cana-1178	120	1	6s	6s	NUM
cana-1178	120	2	(	(	PUNCT
cana-1178	120	3	2024	2024	NUM
cana-1178	120	4	)	)	PUNCT
cana-1178	120	5	197	197	NUM
cana-1178	120	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	120	7	3.2	3.2	NUM
cana-1178	120	8	.	.	PUNCT
cana-1178	121	1	maximal	maximal	ADJ
cana-1178	121	2	independent	independent	ADJ
cana-1178	121	3	and	and	CCONJ
cana-1178	121	4	dominating	dominate	VERB
cana-1178	121	5	energy	energy	NOUN
cana-1178	121	6	definition	definition	NOUN
cana-1178	121	7	3.5	3.5	NUM
cana-1178	121	8	.	.	PUNCT
cana-1178	122	1	the	the	DET
cana-1178	122	2	maximal	maximal	ADJ
cana-1178	122	3	independent	independent	ADJ
cana-1178	122	4	and	and	CCONJ
cana-1178	122	5	dominating	dominating	NOUN
cana-1178	122	6	set	set	NOUN
cana-1178	122	7	of	of	ADP
cana-1178	122	8	the	the	DET
cana-1178	122	9	rough	rough	ADJ
cana-1178	122	10	complemented	complemented	ADJ
cana-1178	122	11	graph	graph	NOUN
cana-1178	122	12	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	122	13	)	)	PUNCT
cana-1178	122	14	is	be	AUX
cana-1178	122	15	denoted	denote	VERB
cana-1178	122	16	by	by	ADP
cana-1178	122	17	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	122	18	)	)	PUNCT
cana-1178	122	19	)	)	PUNCT
cana-1178	122	20	.	.	PUNCT
cana-1178	123	1	note	note	VERB
cana-1178	123	2	that	that	SCONJ
cana-1178	123	3	the	the	DET
cana-1178	123	4	elements	element	NOUN
cana-1178	123	5	of	of	ADP
cana-1178	123	6	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	123	7	)	)	PUNCT
cana-1178	123	8	)	)	PUNCT
cana-1178	123	9	will	will	AUX
cana-1178	123	10	be	be	AUX
cana-1178	123	11	a	a	DET
cana-1178	123	12	maximum	maximum	ADJ
cana-1178	123	13	independent	independent	ADJ
cana-1178	123	14	set	set	NOUN
cana-1178	123	15	as	as	ADV
cana-1178	123	16	well	well	ADV
cana-1178	123	17	as	as	ADP
cana-1178	123	18	a	a	DET
cana-1178	123	19	dominating	dominating	NOUN
cana-1178	123	20	set	set	NOUN
cana-1178	123	21	.	.	PUNCT
cana-1178	124	1	definition	definition	NOUN
cana-1178	124	2	3.6	3.6	NUM
cana-1178	124	3	.	.	PUNCT
cana-1178	125	1	let	let	VERB
cana-1178	125	2	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	125	3	)	)	PUNCT
cana-1178	125	4	)	)	PUNCT
cana-1178	126	1	be	be	AUX
cana-1178	126	2	a	a	DET
cana-1178	126	3	maximal	maximal	ADJ
cana-1178	126	4	independent	independent	ADJ
cana-1178	126	5	and	and	CCONJ
cana-1178	126	6	dominating	dominating	NOUN
cana-1178	126	7	set	set	NOUN
cana-1178	126	8	of	of	ADP
cana-1178	126	9	a	a	DET
cana-1178	126	10	graph	graph	NOUN
cana-1178	126	11	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	126	12	)	)	PUNCT
cana-1178	126	13	,	,	PUNCT
cana-1178	126	14	then	then	ADV
cana-1178	126	15	the	the	DET
cana-1178	126	16	corresponding	corresponding	ADJ
cana-1178	126	17	adjacency	adjacency	NOUN
cana-1178	126	18	matrix	matrix	NOUN
cana-1178	126	19	is	be	AUX
cana-1178	126	20	denoted	denote	VERB
cana-1178	126	21	by	by	ADP
cana-1178	126	22	𝐴𝐼𝐷(𝐶(𝑇	𝐴𝐼𝐷(𝐶(𝑇	PROPN
cana-1178	126	23	)	)	PUNCT
cana-1178	126	24	)	)	PUNCT
cana-1178	126	25	is	be	AUX
cana-1178	126	26	defined	define	VERB
cana-1178	126	27	as	as	ADP
cana-1178	126	28	𝐴𝐼𝐷(𝐶(𝑇	𝐴𝐼𝐷(𝐶(𝑇	PROPN
cana-1178	126	29	)	)	PUNCT
cana-1178	126	30	)	)	PUNCT
cana-1178	127	1	=	=	PRON
cana-1178	127	2	{	{	PUNCT
cana-1178	127	3	1	1	NUM
cana-1178	127	4	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	NOUN
cana-1178	127	5	)	)	PUNCT
cana-1178	127	6	=	=	PUNCT
cana-1178	127	7	𝑅𝑆(∅	𝑅𝑆(∅	X
cana-1178	127	8	)	)	PUNCT
cana-1178	127	9	1	1	NUM
cana-1178	127	10	𝑖𝑓	𝑖𝑓	ADP
cana-1178	127	11	𝑅𝑆(𝑋	𝑅𝑆(𝑋	PROPN
cana-1178	127	12	)	)	PUNCT
cana-1178	127	13	=	=	SYM
cana-1178	127	14	𝑅𝑆(𝑌)𝑎𝑛𝑑𝑅𝑆(𝑋	𝑅𝑆(𝑌)𝑎𝑛𝑑𝑅𝑆(𝑋	NOUN
cana-1178	127	15	)	)	PUNCT
cana-1178	127	16	∈	∈	PROPN
cana-1178	127	17	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	127	18	)	)	PUNCT
cana-1178	127	19	)	)	PUNCT
cana-1178	127	20	0	0	NUM
cana-1178	127	21	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
cana-1178	127	22	definition	definition	NOUN
cana-1178	127	23	3.7	3.7	NUM
cana-1178	127	24	.	.	PUNCT
cana-1178	128	1	the	the	DET
cana-1178	128	2	maximal	maximal	ADJ
cana-1178	128	3	independent	independent	ADJ
cana-1178	128	4	and	and	CCONJ
cana-1178	128	5	dominating	dominate	VERB
cana-1178	128	6	energy	energy	NOUN
cana-1178	128	7	of	of	ADP
cana-1178	128	8	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	128	9	)	)	PUNCT
cana-1178	128	10	is	be	AUX
cana-1178	128	11	defined	define	VERB
cana-1178	128	12	by	by	ADP
cana-1178	128	13	ԑ𝐼𝐷(𝐶(𝑇	ԑ𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	128	14	)	)	PUNCT
cana-1178	128	15	)	)	PUNCT
cana-1178	129	1	=	=	PUNCT
cana-1178	129	2	∑	∑	PUNCT
cana-1178	129	3	|𝜔𝑖|	|𝜔𝑖|	PROPN
cana-1178	129	4	2𝑛−2	2𝑛−2	NUM
cana-1178	129	5	𝑖=1	𝑖=1	PUNCT
cana-1178	129	6	where	where	SCONJ
cana-1178	129	7	𝜔𝑖	𝜔𝑖	PART
cana-1178	129	8	are	be	AUX
cana-1178	129	9	the	the	DET
cana-1178	129	10	eigen	eigen	PROPN
cana-1178	129	11	values	value	NOUN
cana-1178	129	12	of	of	ADP
cana-1178	129	13	𝐴𝐼𝐷(𝐶(𝑇	𝐴𝐼𝐷(𝐶(𝑇	PROPN
cana-1178	129	14	)	)	PUNCT
cana-1178	129	15	)	)	PUNCT
cana-1178	130	1	and	and	CCONJ
cana-1178	130	2	are	be	AUX
cana-1178	130	3	in	in	ADP
cana-1178	130	4	non	non	ADJ
cana-1178	130	5	increasing	increase	VERB
cana-1178	130	6	order	order	NOUN
cana-1178	130	7	.	.	PUNCT
cana-1178	131	1	theorem	theorem	VERB
cana-1178	131	2	3.4	3.4	NUM
cana-1178	131	3	.	.	PUNCT
cana-1178	132	1	for	for	ADP
cana-1178	132	2	the	the	DET
cana-1178	132	3	rough	rough	ADJ
cana-1178	132	4	complemented	complemented	ADJ
cana-1178	132	5	graph	graph	NOUN
cana-1178	132	6	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	132	7	)	)	PUNCT
cana-1178	132	8	,	,	PUNCT
cana-1178	132	9	|𝐼𝐷(𝐶(𝑇))|	|𝐼𝐷(𝐶(𝑇))|	X
cana-1178	132	10	is	be	AUX
cana-1178	132	11	2𝑛−1	2𝑛−1	NUM
cana-1178	132	12	−	−	NOUN
cana-1178	132	13	1	1	NUM
cana-1178	132	14	.	.	PUNCT
cana-1178	132	15	proof	proof	NOUN
cana-1178	132	16	.	.	PUNCT
cana-1178	133	1	consider	consider	VERB
cana-1178	133	2	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	133	3	)	)	PUNCT
cana-1178	133	4	)	)	PUNCT
cana-1178	134	1	=	=	PRON
cana-1178	134	2	{	{	PUNCT
cana-1178	134	3	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	SCONJ
cana-1178	134	4	∪	∪	ADJ
cana-1178	134	5	𝑌)|𝑌	𝑌)|𝑌	NOUN
cana-1178	134	6	∈	∈	NOUN
cana-1178	134	7	℘(𝐸	℘(𝐸	NOUN
cana-1178	134	8	−	−	PROPN
cana-1178	135	1	𝑋𝑖	𝑋𝑖	NOUN
cana-1178	135	2	)	)	PUNCT
cana-1178	135	3	}	}	PUNCT
cana-1178	135	4	to	to	PART
cana-1178	135	5	prove	prove	VERB
cana-1178	135	6	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	135	7	)	)	PUNCT
cana-1178	135	8	)	)	PUNCT
cana-1178	135	9	is	be	AUX
cana-1178	135	10	an	an	DET
cana-1178	135	11	independent	independent	ADJ
cana-1178	135	12	set	set	NOUN
cana-1178	135	13	.	.	PUNCT
cana-1178	136	1	let	let	VERB
cana-1178	136	2	𝑅𝑆(𝑋	𝑅𝑆(𝑋	ADV
cana-1178	136	3	)	)	PUNCT
cana-1178	136	4	,	,	PUNCT
cana-1178	136	5	𝑅𝑆(𝑍	𝑅𝑆(𝑍	PUNCT
cana-1178	136	6	)	)	PUNCT
cana-1178	136	7	∈	∈	PROPN
cana-1178	136	8	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	136	9	)	)	PUNCT
cana-1178	136	10	)	)	PUNCT
cana-1178	136	11	,	,	PUNCT
cana-1178	136	12	where	where	SCONJ
cana-1178	136	13	𝑅𝑆(𝑋	𝑅𝑆(𝑋	ADV
cana-1178	136	14	)	)	PUNCT
cana-1178	136	15	=	=	PUNCT
cana-1178	136	16	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	ADP
cana-1178	136	17	∪	∪	ADP
cana-1178	136	18	𝑌1	𝑌1	NOUN
cana-1178	136	19	)	)	PUNCT
cana-1178	137	1	and𝑅𝑆(𝑍	and𝑅𝑆(𝑍	PROPN
cana-1178	137	2	)	)	PUNCT
cana-1178	137	3	=	=	PUNCT
cana-1178	137	4	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	ADP
cana-1178	137	5	∪	∪	ADJ
cana-1178	137	6	𝑌2	𝑌2	NOUN
cana-1178	137	7	)	)	PUNCT
cana-1178	137	8	where	where	SCONJ
cana-1178	137	9	𝑌1	𝑌1	NOUN
cana-1178	137	10	,	,	PUNCT
cana-1178	137	11	𝑌2	𝑌2	PROPN
cana-1178	137	12	∈	∈	PROPN
cana-1178	137	13	℘(𝐸	℘(𝐸	NOUN
cana-1178	137	14	−	−	PROPN
cana-1178	137	15	𝑋𝑖	𝑋𝑖	NOUN
cana-1178	137	16	)	)	PUNCT
cana-1178	137	17	this	this	PRON
cana-1178	137	18	implies	implies	AUX
cana-1178	137	19	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	NOUN
cana-1178	137	20	)	)	PUNCT
cana-1178	137	21	∈	∈	PROPN
cana-1178	137	22	𝑅𝑆(𝑋)∇𝑅𝑆(𝑍	𝑅𝑆(𝑋)∇𝑅𝑆(𝑍	PROPN
cana-1178	137	23	)	)	PUNCT
cana-1178	137	24	≠	≠	PROPN
cana-1178	137	25	𝑅𝑆(∅	𝑅𝑆(∅	PUNCT
cana-1178	137	26	)	)	PUNCT
cana-1178	137	27	there	there	PRON
cana-1178	137	28	is	be	VERB
cana-1178	137	29	no	no	DET
cana-1178	137	30	edge	edge	NOUN
cana-1178	137	31	between	between	ADP
cana-1178	137	32	𝑅𝑆(𝑋)𝑎𝑛𝑑	𝑅𝑆(𝑋)𝑎𝑛𝑑	PROPN
cana-1178	137	33	𝑅𝑆(𝑍	𝑅𝑆(𝑍	PRON
cana-1178	137	34	)	)	PUNCT
cana-1178	137	35	∴	∴	PROPN
cana-1178	137	36	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	137	37	)	)	PUNCT
cana-1178	137	38	)	)	PUNCT
cana-1178	137	39	is	be	AUX
cana-1178	137	40	an	an	DET
cana-1178	137	41	independent	independent	ADJ
cana-1178	137	42	set	set	NOUN
cana-1178	137	43	.	.	PUNCT
cana-1178	138	1	it	it	PRON
cana-1178	138	2	is	be	AUX
cana-1178	138	3	clear	clear	ADJ
cana-1178	138	4	to	to	PART
cana-1178	138	5	verify	verify	VERB
cana-1178	138	6	that	that	SCONJ
cana-1178	138	7	addition	addition	NOUN
cana-1178	138	8	of	of	ADP
cana-1178	138	9	any	any	DET
cana-1178	138	10	vertex	vertex	NOUN
cana-1178	138	11	to	to	ADP
cana-1178	138	12	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	138	13	)	)	PUNCT
cana-1178	138	14	)	)	PUNCT
cana-1178	138	15	will	will	AUX
cana-1178	138	16	affect	affect	VERB
cana-1178	138	17	the	the	DET
cana-1178	138	18	independence	independence	NOUN
cana-1178	138	19	property	property	NOUN
cana-1178	138	20	.	.	PUNCT
cana-1178	139	1	hence	hence	ADV
cana-1178	139	2	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	139	3	)	)	PUNCT
cana-1178	139	4	)	)	PUNCT
cana-1178	139	5	is	be	AUX
cana-1178	139	6	the	the	DET
cana-1178	139	7	maximal	maximal	ADJ
cana-1178	139	8	independent	independent	ADJ
cana-1178	139	9	set	set	NOUN
cana-1178	139	10	.	.	PUNCT
cana-1178	140	1	next	next	ADJ
cana-1178	140	2	to	to	PART
cana-1178	140	3	prove	prove	VERB
cana-1178	140	4	that	that	PRON
cana-1178	140	5	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	140	6	)	)	PUNCT
cana-1178	140	7	)	)	PUNCT
cana-1178	140	8	is	be	AUX
cana-1178	140	9	a	a	DET
cana-1178	140	10	dominating	dominating	NOUN
cana-1178	140	11	set	set	NOUN
cana-1178	140	12	.	.	PUNCT
cana-1178	141	1	let	let	VERB
cana-1178	141	2	𝑅𝑆(𝑍	𝑅𝑆(𝑍	PRON
cana-1178	141	3	)	)	PUNCT
cana-1178	141	4	∈	∈	PROPN
cana-1178	141	5	𝑉(𝐺𝑅𝐶(𝑇	𝑉(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	141	6	)	)	PUNCT
cana-1178	141	7	)	)	PUNCT
cana-1178	142	1	−	−	PROPN
cana-1178	142	2	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	142	3	)	)	PUNCT
cana-1178	142	4	)	)	PUNCT
cana-1178	142	5	since	since	SCONJ
cana-1178	142	6	the	the	DET
cana-1178	142	7	elements	element	NOUN
cana-1178	142	8	of	of	ADP
cana-1178	142	9	𝑉(𝐺𝑅𝐶(𝑇	𝑉(𝐺𝑅𝐶(𝑇	NOUN
cana-1178	142	10	)	)	PUNCT
cana-1178	142	11	)	)	PUNCT
cana-1178	143	1	−	−	PROPN
cana-1178	143	2	𝐼𝐷(𝐶(𝑇	𝐼𝐷(𝐶(𝑇	NOUN
cana-1178	143	3	)	)	PUNCT
cana-1178	143	4	)	)	PUNCT
cana-1178	143	5	are	be	AUX
cana-1178	143	6	from	from	ADP
cana-1178	143	7	℘(𝐸	℘(𝐸	ADP
cana-1178	143	8	−	−	PROPN
cana-1178	143	9	𝑋𝑖	𝑋𝑖	NOUN
cana-1178	143	10	)	)	PUNCT
cana-1178	143	11	and	and	CCONJ
cana-1178	143	12	so	so	ADV
cana-1178	143	13	𝑅𝑆(𝑍	𝑅𝑆(𝑍	PRON
cana-1178	143	14	)	)	PUNCT
cana-1178	144	1	is	be	AUX
cana-1178	144	2	adjacent	adjacent	ADJ
cana-1178	144	3	to	to	PART
cana-1178	144	4	𝑅𝑆(𝑋𝑖	𝑅𝑆(𝑋𝑖	VERB
cana-1178	144	5	)	)	PUNCT
cana-1178	144	6	and	and	CCONJ
cana-1178	144	7	hence	hence	ADV
cana-1178	144	8	it	it	PRON
cana-1178	144	9	is	be	AUX
cana-1178	144	10	a	a	DET
cana-1178	144	11	dominating	dominating	NOUN
cana-1178	144	12	set	set	NOUN
cana-1178	144	13	.	.	PUNCT
cana-1178	145	1	hence	hence	ADV
cana-1178	145	2	it	it	PRON
cana-1178	145	3	is	be	AUX
cana-1178	145	4	clear	clear	ADJ
cana-1178	145	5	that	that	SCONJ
cana-1178	145	6	|𝐼𝐷(𝐶(𝑇))|	|𝐼𝐷(𝐶(𝑇))|	NOUN
cana-1178	145	7	=	=	SYM
cana-1178	145	8	2𝑛−1	2𝑛−1	NUM
cana-1178	145	9	−	−	NOUN
cana-1178	145	10	1	1	NUM
cana-1178	145	11	.	.	PUNCT
cana-1178	145	12	remarks	remark	VERB
cana-1178	145	13	3.1	3.1	NUM
cana-1178	145	14	.	.	PUNCT
cana-1178	146	1	for	for	ADP
cana-1178	146	2	the	the	DET
cana-1178	146	3	rough	rough	ADJ
cana-1178	146	4	complemented	complemented	ADJ
cana-1178	146	5	graph	graph	NOUN
cana-1178	146	6	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	146	7	)	)	PUNCT
cana-1178	146	8	,	,	PUNCT
cana-1178	146	9	|𝐼𝐷(𝐶(𝑇))|	|𝐼𝐷(𝐶(𝑇))|	VERB
cana-1178	146	10	≤	≤	PROPN
cana-1178	146	11	2𝑛	2𝑛	PROPN
cana-1178	146	12	−	−	PROPN
cana-1178	146	13	3	3	X
cana-1178	146	14	.	.	PUNCT
cana-1178	146	15	also	also	ADV
cana-1178	146	16	|𝐼𝐷(𝐶(𝑇))|	|𝐼𝐷(𝐶(𝑇))|	VERB
cana-1178	146	17	≥	≥	NUM
cana-1178	146	18	2𝑛−2	2𝑛−2	NUM
cana-1178	146	19	𝑛	𝑛	PROPN
cana-1178	146	20	.	.	PUNCT
cana-1178	147	1	communications	communication	NOUN
cana-1178	147	2	on	on	ADP
cana-1178	147	3	applied	apply	VERB
cana-1178	147	4	nonlinear	nonlinear	ADJ
cana-1178	147	5	analysis	analysis	NOUN
cana-1178	147	6	issn	issn	NOUN
cana-1178	147	7	:	:	PUNCT
cana-1178	147	8	1074	1074	NUM
cana-1178	147	9	-	-	PUNCT
cana-1178	147	10	133x	133x	NUM
cana-1178	147	11	vol	vol	NOUN
cana-1178	147	12	31	31	NUM
cana-1178	147	13	no	no	NOUN
cana-1178	147	14	.	.	PUNCT
cana-1178	148	1	6s	6s	NUM
cana-1178	148	2	(	(	PUNCT
cana-1178	148	3	2024	2024	NUM
cana-1178	148	4	)	)	PUNCT
cana-1178	148	5	198	198	NUM
cana-1178	148	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	148	7	3.3	3.3	NUM
cana-1178	148	8	.	.	PUNCT
cana-1178	149	1	siedel	siedel	PROPN
cana-1178	149	2	energy	energy	NOUN
cana-1178	149	3	definition	definition	NOUN
cana-1178	149	4	3.8	3.8	NUM
cana-1178	149	5	.	.	PUNCT
cana-1178	150	1	the	the	DET
cana-1178	150	2	siedel	siedel	NOUN
cana-1178	150	3	matrix	matrix	NOUN
cana-1178	150	4	of	of	ADP
cana-1178	150	5	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	150	6	)	)	PUNCT
cana-1178	150	7	is	be	AUX
cana-1178	150	8	defined	define	VERB
cana-1178	150	9	by	by	ADP
cana-1178	150	10	𝑆(𝐶(𝑇	𝑆(𝐶(𝑇	NOUN
cana-1178	150	11	)	)	PUNCT
cana-1178	150	12	)	)	PUNCT
cana-1178	151	1	=	=	PRON
cana-1178	151	2	{	{	PUNCT
cana-1178	151	3	−1	−1	NOUN
cana-1178	151	4	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	NOUN
cana-1178	151	5	)	)	PUNCT
cana-1178	151	6	=	=	PUNCT
cana-1178	151	7	𝑅𝑆(∅	𝑅𝑆(∅	X
cana-1178	151	8	)	)	PUNCT
cana-1178	151	9	1	1	NUM
cana-1178	151	10	𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	PROPN
cana-1178	151	11	)	)	PUNCT
cana-1178	151	12	≠	≠	PROPN
cana-1178	151	13	𝑅𝑆(∅	𝑅𝑆(∅	PUNCT
cana-1178	151	14	)	)	PUNCT
cana-1178	151	15	0	0	NUM
cana-1178	152	1	𝑖𝑓	𝑖𝑓	ADP
cana-1178	152	2	𝑅𝑆(𝑋	𝑅𝑆(𝑋	PROPN
cana-1178	152	3	)	)	PUNCT
cana-1178	152	4	=	=	SYM
cana-1178	152	5	𝑅𝑆(𝑌	𝑅𝑆(𝑌	PROPN
cana-1178	152	6	)	)	PUNCT
cana-1178	152	7	the	the	DET
cana-1178	152	8	siedel	siedel	NOUN
cana-1178	152	9	energy	energy	NOUN
cana-1178	152	10	of	of	ADP
cana-1178	152	11	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	152	12	)	)	PUNCT
cana-1178	152	13	is	be	AUX
cana-1178	152	14	defined	define	VERB
cana-1178	152	15	as	as	ADP
cana-1178	152	16	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	NOUN
cana-1178	152	17	)	)	PUNCT
cana-1178	152	18	)	)	PUNCT
cana-1178	153	1	=	=	PUNCT
cana-1178	153	2	∑	∑	PUNCT
cana-1178	154	1	|	|	INTJ
cana-1178	154	2	i	i	PRON
cana-1178	154	3	|2n−2	|2n−2	VERB
cana-1178	154	4	i=1	i=1	X
cana-1178	154	5	,	,	PUNCT
cana-1178	154	6	𝑖	𝑖	PUNCT
cana-1178	154	7	=	=	SYM
cana-1178	154	8	1,2	1,2	NUM
cana-1178	154	9	…	…	PUNCT
cana-1178	154	10	…	…	PUNCT
cana-1178	154	11	…	…	PUNCT
cana-1178	154	12	2n	2n	NUM
cana-1178	154	13	−	−	NOUN
cana-1178	154	14	2	2	NUM
cana-1178	154	15	,	,	PUNCT
cana-1178	154	16	where	where	SCONJ
cana-1178	154	17	i	i	PROPN
cana-1178	154	18	s	s	VERB
cana-1178	154	19	are	be	AUX
cana-1178	154	20	the	the	DET
cana-1178	154	21	eigen	eigen	PROPN
cana-1178	154	22	values	value	NOUN
cana-1178	154	23	of	of	ADP
cana-1178	154	24	𝑆(𝐶(𝑇	𝑆(𝐶(𝑇	NOUN
cana-1178	154	25	)	)	PUNCT
cana-1178	154	26	)	)	PUNCT
cana-1178	154	27	.	.	PUNCT
cana-1178	155	1	lemma	lemma	PROPN
cana-1178	155	2	3.1	3.1	NUM
cana-1178	155	3	.	.	PUNCT
cana-1178	156	1	let	let	VERB
cana-1178	156	2	𝜆1	𝜆1	NOUN
cana-1178	156	3	,	,	PUNCT
cana-1178	156	4	𝜆2	𝜆2	PROPN
cana-1178	156	5	…	…	PUNCT
cana-1178	156	6	…	…	PUNCT
cana-1178	156	7	…	…	PUNCT
cana-1178	156	8	…	…	PUNCT
cana-1178	156	9	.	.	PUNCT
cana-1178	156	10	.	.	PUNCT
cana-1178	157	1	𝜆2𝑛−2	𝜆2𝑛−2	PROPN
cana-1178	157	2	denote	denote	VERB
cana-1178	157	3	the	the	DET
cana-1178	157	4	siedel	siedel	NOUN
cana-1178	157	5	eigenvalues	eigenvalue	VERB
cana-1178	157	6	of	of	ADP
cana-1178	157	7	𝑆(𝐶(𝑇	𝑆(𝐶(𝑇	NOUN
cana-1178	157	8	)	)	PUNCT
cana-1178	157	9	)	)	PUNCT
cana-1178	158	1	then	then	ADV
cana-1178	158	2	•	•	VERB
cana-1178	158	3	∑	∑	ADP
cana-1178	158	4	𝑖	𝑖	PROPN
cana-1178	158	5	2𝑛−2	2𝑛−2	NUM
cana-1178	158	6	𝑖=1	𝑖=1	PUNCT
cana-1178	158	7	=	=	SYM
cana-1178	158	8	0	0	NUM
cana-1178	158	9	•	•	NUM
cana-1178	158	10	∑	∑	ADP
cana-1178	158	11	𝑖	𝑖	PROPN
cana-1178	158	12	2	2	NUM
cana-1178	158	13	=	=	SYM
cana-1178	158	14	(	(	PUNCT
cana-1178	158	15	2𝑛	2𝑛	NOUN
cana-1178	158	16	−	−	PROPN
cana-1178	158	17	2	2	NUM
cana-1178	158	18	)	)	PUNCT
cana-1178	158	19	(	(	PUNCT
cana-1178	158	20	2𝑛	2𝑛	X
cana-1178	158	21	−	−	PROPN
cana-1178	158	22	3)2𝑛−2	3)2𝑛−2	NUM
cana-1178	158	23	𝑖=1	𝑖=1	PROPN
cana-1178	158	24	proof	proof	NOUN
cana-1178	158	25	.	.	PUNCT
cana-1178	159	1	it	it	PRON
cana-1178	159	2	is	be	AUX
cana-1178	159	3	known	know	VERB
cana-1178	159	4	that	that	SCONJ
cana-1178	159	5	∑	∑	ADP
cana-1178	159	6	𝑖	𝑖	PROPN
cana-1178	159	7	2𝑛−2	2𝑛−2	NUM
cana-1178	159	8	𝑖=1	𝑖=1	PUNCT
cana-1178	159	9	=	=	PUNCT
cana-1178	159	10	∑	∑	SCONJ
cana-1178	159	11	𝑠𝑖𝑖	𝑠𝑖𝑖	ADJ
cana-1178	159	12	2𝑛−2	2𝑛−2	NUM
cana-1178	159	13	𝑖=1	𝑖=1	PUNCT
cana-1178	159	14	=	=	SYM
cana-1178	159	15	0	0	NUM
cana-1178	159	16	sum	sum	NOUN
cana-1178	159	17	of	of	ADP
cana-1178	159	18	squares	square	NOUN
cana-1178	159	19	of	of	ADP
cana-1178	159	20	eigen	eigen	PROPN
cana-1178	159	21	values	value	NOUN
cana-1178	159	22	of	of	ADP
cana-1178	159	23	𝑆(𝐶(𝑇	𝑆(𝐶(𝑇	NOUN
cana-1178	159	24	)	)	PUNCT
cana-1178	159	25	)	)	PUNCT
cana-1178	159	26	is	be	AUX
cana-1178	159	27	the	the	DET
cana-1178	159	28	trace	trace	NOUN
cana-1178	159	29	of	of	ADP
cana-1178	159	30	(	(	PUNCT
cana-1178	159	31	𝑆(𝐶(𝑇	𝑆(𝐶(𝑇	NOUN
cana-1178	159	32	)	)	PUNCT
cana-1178	159	33	)	)	PUNCT
cana-1178	159	34	)	)	PUNCT
cana-1178	160	1	2	2	X
cana-1178	160	2	.	.	PUNCT
cana-1178	161	1	∑	∑	PUNCT
cana-1178	161	2	𝑖	𝑖	PROPN
cana-1178	161	3	2	2	NUM
cana-1178	161	4	=	=	SYM
cana-1178	161	5	2𝑛−2	2𝑛−2	NUM
cana-1178	161	6	𝑖=1	𝑖=1	PUNCT
cana-1178	161	7	∑	∑	PROPN
cana-1178	161	8	𝑠𝑖𝑗	𝑠𝑖𝑗	PROPN
cana-1178	161	9	2𝑛−2	2𝑛−2	NUM
cana-1178	161	10	𝑖=1	𝑖=1	PUNCT
cana-1178	161	11	∑	∑	VERB
cana-1178	161	12	𝑠𝑗𝑖	𝑠𝑗𝑖	VERB
cana-1178	161	13	2𝑛−2	2𝑛−2	NUM
cana-1178	161	14	𝑗=1	𝑗=1	PUNCT
cana-1178	161	15	=	=	PUNCT
cana-1178	161	16	∑	∑	PUNCT
cana-1178	161	17	(	(	PUNCT
cana-1178	161	18	𝑠𝑖𝑖	𝑠𝑖𝑖	ADJ
cana-1178	161	19	)	)	PUNCT
cana-1178	161	20	2	2	NUM
cana-1178	161	21	2𝑛−2	2𝑛−2	NUM
cana-1178	161	22	𝑖=1	𝑖=1	PUNCT
cana-1178	162	1	+	+	CCONJ
cana-1178	162	2	∑	∑	ADV
cana-1178	162	3	𝑠𝑖𝑗𝑠𝑗𝑖	𝑠𝑖𝑗𝑠𝑗𝑖	NOUN
cana-1178	162	4	𝑖≠𝑗	𝑖≠𝑗	NOUN
cana-1178	162	5	=	=	PUNCT
cana-1178	162	6	∑	∑	PUNCT
cana-1178	162	7	(	(	PUNCT
cana-1178	162	8	𝑠𝑖𝑖	𝑠𝑖𝑖	ADJ
cana-1178	162	9	)	)	PUNCT
cana-1178	162	10	22𝑛−2	22𝑛−2	X
cana-1178	162	11	𝑖=1	𝑖=1	PUNCT
cana-1178	163	1	+	+	CCONJ
cana-1178	163	2	2	2	NUM
cana-1178	163	3	∑	∑	NOUN
cana-1178	163	4	𝑠𝑖𝑗	𝑠𝑖𝑗	PROPN
cana-1178	163	5	2	2	NUM
cana-1178	163	6	𝑖<𝑗	𝑖<𝑗	NOUN
cana-1178	163	7	=	=	SYM
cana-1178	163	8	2	2	NUM
cana-1178	163	9	{	{	PUNCT
cana-1178	163	10	1	1	NUM
cana-1178	163	11	2	2	NUM
cana-1178	163	12	(	(	PUNCT
cana-1178	163	13	3𝑛	3𝑛	NUM
cana-1178	163	14	−	−	PROPN
cana-1178	164	1	2𝑛+1	2𝑛+1	NOUN
cana-1178	164	2	+	+	CCONJ
cana-1178	164	3	1)(−1)2	1)(−1)2	NUM
cana-1178	165	1	+	+	CCONJ
cana-1178	165	2	(	(	PUNCT
cana-1178	165	3	(	(	PUNCT
cana-1178	165	4	2𝑛−2)2	2𝑛−2)2	NUM
cana-1178	165	5	–	–	PUNCT
cana-1178	165	6	(	(	PUNCT
cana-1178	165	7	2𝑛−2	2𝑛−2	NOUN
cana-1178	165	8	)	)	PUNCT
cana-1178	165	9	2	2	NUM
cana-1178	165	10	−	−	NOUN
cana-1178	165	11	1	1	NUM
cana-1178	165	12	2	2	NUM
cana-1178	165	13	(	(	PUNCT
cana-1178	165	14	3𝑛	3𝑛	NUM
cana-1178	165	15	−	−	PROPN
cana-1178	165	16	2𝑛+1	2𝑛+1	NOUN
cana-1178	165	17	+	+	CCONJ
cana-1178	165	18	1	1	NUM
cana-1178	165	19	)	)	PUNCT
cana-1178	165	20	)	)	PUNCT
cana-1178	166	1	(	(	PUNCT
cana-1178	166	2	1)2	1)2	NUM
cana-1178	166	3	}	}	PUNCT
cana-1178	166	4	=	=	SYM
cana-1178	166	5	(	(	PUNCT
cana-1178	166	6	2𝑛	2𝑛	PROPN
cana-1178	166	7	−	−	PROPN
cana-1178	166	8	2)2	2)2	NUM
cana-1178	166	9	−	−	PROPN
cana-1178	166	10	(	(	PUNCT
cana-1178	166	11	2𝑛	2𝑛	NOUN
cana-1178	166	12	−	−	PROPN
cana-1178	166	13	2	2	NUM
cana-1178	166	14	)	)	PUNCT
cana-1178	166	15	∑	∑	ADP
cana-1178	166	16	𝑖	𝑖	PROPN
cana-1178	166	17	2	2	NUM
cana-1178	166	18	2𝑛−2	2𝑛−2	NUM
cana-1178	166	19	𝑖=1	𝑖=1	PUNCT
cana-1178	166	20	=	=	SYM
cana-1178	166	21	2𝑛(2𝑛	2𝑛(2𝑛	NUM
cana-1178	166	22	−	−	NUM
cana-1178	166	23	5	5	NUM
cana-1178	166	24	)	)	PUNCT
cana-1178	166	25	+	+	CCONJ
cana-1178	166	26	6	6	NUM
cana-1178	166	27	theorem	theorem	VERB
cana-1178	166	28	3.5	3.5	NUM
cana-1178	166	29	.	.	PUNCT
cana-1178	167	1	in	in	ADP
cana-1178	167	2	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	167	3	)	)	PUNCT
cana-1178	167	4	,	,	PUNCT
cana-1178	167	5	∆𝑆=	∆𝑆=	NOUN
cana-1178	167	6	|𝑑𝑒𝑡𝑆(𝐶(𝑇))|	|𝑑𝑒𝑡𝑆(𝐶(𝑇))|	NOUN
cana-1178	167	7	then	then	ADV
cana-1178	167	8	√2𝑛(2𝑛	√2𝑛(2𝑛	ADP
cana-1178	167	9	−	−	PROPN
cana-1178	167	10	5	5	NUM
cana-1178	167	11	)	)	PUNCT
cana-1178	167	12	+	+	CCONJ
cana-1178	167	13	6	6	NUM
cana-1178	167	14	+	+	CCONJ
cana-1178	167	15	(	(	PUNCT
cana-1178	167	16	2𝑛	2𝑛	PROPN
cana-1178	167	17	−	−	PROPN
cana-1178	167	18	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	168	1	−	−	PROPN
cana-1178	168	2	3)∆s	3)∆s	NUM
cana-1178	168	3	2	2	NUM
cana-1178	168	4	2𝑛−2≤	2𝑛−2≤	NUM
cana-1178	168	5	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	NOUN
cana-1178	168	6	)	)	PUNCT
cana-1178	168	7	)	)	PUNCT
cana-1178	168	8	≤	≤	NUM
cana-1178	168	9	√(2𝑛	√(2𝑛	NOUN
cana-1178	168	10	−	−	PROPN
cana-1178	168	11	2){2𝑛(2𝑛	2){2𝑛(2𝑛	NOUN
cana-1178	168	12	−	−	PROPN
cana-1178	168	13	5	5	NUM
cana-1178	168	14	)	)	PUNCT
cana-1178	168	15	+	+	CCONJ
cana-1178	168	16	6	6	NUM
cana-1178	168	17	}	}	PUNCT
cana-1178	168	18	where	where	SCONJ
cana-1178	168	19	|𝑑𝑒𝑡𝑆(𝐶(𝑇))|	|𝑑𝑒𝑡𝑆(𝐶(𝑇))|	NOUN
cana-1178	168	20	means	mean	VERB
cana-1178	168	21	the	the	DET
cana-1178	168	22	absolute	absolute	ADJ
cana-1178	168	23	value	value	NOUN
cana-1178	168	24	.	.	PUNCT
cana-1178	169	1	proof	proof	NOUN
cana-1178	169	2	.	.	PUNCT
cana-1178	170	1	taking	take	VERB
cana-1178	170	2	𝑎𝑖	𝑎𝑖	X
cana-1178	170	3	=	=	NOUN
cana-1178	170	4	1	1	NUM
cana-1178	170	5	,	,	PUNCT
cana-1178	170	6	𝑏𝑖	𝑏𝑖	ADP
cana-1178	170	7	=	=	PUNCT
cana-1178	170	8	|𝑖|	|𝑖|	NOUN
cana-1178	170	9	cauchy	cauchy	PROPN
cana-1178	170	10	schwarz	schwarz	PROPN
cana-1178	170	11	inequality	inequality	NOUN
cana-1178	170	12	becomes	become	VERB
cana-1178	170	13	(	(	PUNCT
cana-1178	170	14	∑	∑	PUNCT
cana-1178	170	15	|𝑖|	|𝑖|	NOUN
cana-1178	170	16	2𝑛−2	2𝑛−2	NUM
cana-1178	170	17	𝑖=1	𝑖=1	SYM
cana-1178	170	18	)	)	PUNCT
cana-1178	170	19	2	2	NUM
cana-1178	170	20	≤	≤	NOUN
cana-1178	170	21	(	(	PUNCT
cana-1178	170	22	∑	∑	PUNCT
cana-1178	170	23	12𝑛−2	12𝑛−2	NUM
cana-1178	170	24	𝑖=1	𝑖=1	PUNCT
cana-1178	170	25	)	)	PUNCT
cana-1178	170	26	(	(	PUNCT
cana-1178	170	27	∑	∑	PUNCT
cana-1178	170	28	𝑖	𝑖	PROPN
cana-1178	170	29	22𝑛−2	22𝑛−2	X
cana-1178	170	30	𝑖=1	𝑖=1	PROPN
cana-1178	170	31	)	)	PUNCT
cana-1178	170	32	(	(	PUNCT
cana-1178	170	33	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	NOUN
cana-1178	170	34	)	)	PUNCT
cana-1178	170	35	)	)	PUNCT
cana-1178	170	36	)	)	PUNCT
cana-1178	170	37	2	2	NUM
cana-1178	170	38	≤	≤	NOUN
cana-1178	170	39	(	(	PUNCT
cana-1178	170	40	2𝑛	2𝑛	PROPN
cana-1178	170	41	−	−	PROPN
cana-1178	170	42	2)(2𝑛(2𝑛	2)(2𝑛(2𝑛	NUM
cana-1178	170	43	−	−	NOUN
cana-1178	170	44	5	5	NUM
cana-1178	170	45	)	)	PUNCT
cana-1178	170	46	+	+	CCONJ
cana-1178	170	47	6	6	NUM
cana-1178	170	48	)	)	PUNCT
cana-1178	170	49	(	(	PUNCT
cana-1178	170	50	𝑏𝑦	𝑏𝑦	NOUN
cana-1178	170	51	𝑙𝑒𝑚𝑚𝑎	𝑙𝑒𝑚𝑚𝑎	VERB
cana-1178	170	52	3.1	3.1	NUM
cana-1178	170	53	)	)	PUNCT
cana-1178	170	54	communications	communication	NOUN
cana-1178	170	55	on	on	ADP
cana-1178	170	56	applied	apply	VERB
cana-1178	170	57	nonlinear	nonlinear	ADJ
cana-1178	170	58	analysis	analysis	NOUN
cana-1178	170	59	issn	issn	NOUN
cana-1178	170	60	:	:	PUNCT
cana-1178	170	61	1074	1074	NUM
cana-1178	170	62	-	-	PUNCT
cana-1178	170	63	133x	133x	NUM
cana-1178	170	64	vol	vol	NOUN
cana-1178	170	65	31	31	NUM
cana-1178	170	66	no	no	NOUN
cana-1178	170	67	.	.	PUNCT
cana-1178	171	1	6s	6s	NUM
cana-1178	171	2	(	(	PUNCT
cana-1178	171	3	2024	2024	NUM
cana-1178	171	4	)	)	PUNCT
cana-1178	171	5	199	199	NUM
cana-1178	171	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	171	7	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	PROPN
cana-1178	171	8	)	)	PUNCT
cana-1178	171	9	)	)	PUNCT
cana-1178	171	10	≤	≤	NUM
cana-1178	171	11	√(2n	√(2n	NOUN
cana-1178	171	12	−	−	ADP
cana-1178	171	13	2)(2n(2n	2)(2n(2n	NUM
cana-1178	171	14	−	−	NOUN
cana-1178	171	15	5	5	NUM
cana-1178	171	16	)	)	PUNCT
cana-1178	171	17	+	+	CCONJ
cana-1178	171	18	6	6	NUM
cana-1178	171	19	)	)	PUNCT
cana-1178	171	20	by	by	ADP
cana-1178	171	21	arithmetic	arithmetic	ADJ
cana-1178	171	22	geometric	geometric	ADJ
cana-1178	171	23	mean	mean	NOUN
cana-1178	171	24	inequality	inequality	NOUN
cana-1178	171	25	∑	∑	PUNCT
cana-1178	171	26	|	|	PROPN
cana-1178	171	27	i	i	PRON
cana-1178	171	28	||	||	VERB
cana-1178	171	29	j	j	PROPN
cana-1178	171	30	|	|	ADJ
cana-1178	171	31	≥	≥	PROPN
cana-1178	171	32	(	(	PUNCT
cana-1178	171	33	2𝑛	2𝑛	PROPN
cana-1178	171	34	−	−	PROPN
cana-1178	171	35	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	171	36	−	−	PROPN
cana-1178	171	37	3)∆s	3)∆s	NUM
cana-1178	171	38	2	2	NUM
cana-1178	171	39	2𝑛−2i≠j	2𝑛−2i≠j	NUM
cana-1178	171	40	(	(	PUNCT
cana-1178	171	41	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	PROPN
cana-1178	171	42	)	)	PUNCT
cana-1178	171	43	)	)	PUNCT
cana-1178	171	44	)	)	PUNCT
cana-1178	172	1	2	2	NUM
cana-1178	172	2	=	=	SYM
cana-1178	172	3	(	(	PUNCT
cana-1178	172	4	∑	∑	ADV
cana-1178	172	5	|i|	|i|	ADJ
cana-1178	172	6	2n−2	2n−2	NUM
cana-1178	172	7	i=1	i=1	NUM
cana-1178	172	8	)	)	PUNCT
cana-1178	172	9	2	2	NUM
cana-1178	172	10	=	=	SYM
cana-1178	172	11	∑	∑	PUNCT
cana-1178	172	12	|i|	|i|	ADJ
cana-1178	172	13	2n−2	2n−2	NUM
cana-1178	172	14	i=1	i=1	PROPN
cana-1178	172	15	2	2	NUM
cana-1178	172	16	+	+	CCONJ
cana-1178	172	17	∑	∑	PUNCT
cana-1178	172	18	|	|	PRON
cana-1178	172	19	i	i	PRON
cana-1178	172	20	||	||	VERB
cana-1178	172	21	j	j	PROPN
cana-1178	172	22	|i≠j	|i≠j	NOUN
cana-1178	172	23	(	(	PUNCT
cana-1178	172	24	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	PROPN
cana-1178	172	25	)	)	PUNCT
cana-1178	172	26	)	)	PUNCT
cana-1178	172	27	)	)	PUNCT
cana-1178	172	28	2	2	NUM
cana-1178	172	29	≥	≥	NOUN
cana-1178	172	30	2𝑛(2𝑛	2𝑛(2𝑛	NUM
cana-1178	172	31	−	−	NOUN
cana-1178	172	32	5	5	NUM
cana-1178	172	33	)	)	PUNCT
cana-1178	172	34	+	+	CCONJ
cana-1178	172	35	6	6	NUM
cana-1178	172	36	+	+	CCONJ
cana-1178	172	37	(	(	PUNCT
cana-1178	172	38	2𝑛	2𝑛	PROPN
cana-1178	172	39	−	−	PROPN
cana-1178	172	40	2)(2𝑛	2)(2𝑛	PROPN
cana-1178	172	41	−	−	PROPN
cana-1178	172	42	3)∆s	3)∆s	NUM
cana-1178	172	43	2	2	NUM
cana-1178	172	44	2𝑛−2	2𝑛−2	NUM
cana-1178	172	45	𝑆𝐸(𝐶(𝑇	𝑆𝐸(𝐶(𝑇	NOUN
cana-1178	172	46	)	)	PUNCT
cana-1178	172	47	)	)	PUNCT
cana-1178	172	48	≥	≥	PROPN
cana-1178	172	49	√(2𝑛	√(2𝑛	NOUN
cana-1178	172	50	−	−	PROPN
cana-1178	172	51	2)(2𝑛	2)(2𝑛	NOUN
cana-1178	172	52	−	−	NOUN
cana-1178	172	53	3	3	NUM
cana-1178	172	54	)	)	PUNCT
cana-1178	172	55	(	(	PUNCT
cana-1178	172	56	1	1	NUM
cana-1178	172	57	+	+	CCONJ
cana-1178	172	58	∆s	∆s	NOUN
cana-1178	172	59	2	2	NUM
cana-1178	172	60	2𝑛−2	2𝑛−2	NUM
cana-1178	172	61	)	)	PUNCT
cana-1178	172	62	3.4	3.4	NUM
cana-1178	172	63	.	.	PUNCT
cana-1178	173	1	randic	randic	ADJ
cana-1178	173	2	energy	energy	NOUN
cana-1178	173	3	definition	definition	NOUN
cana-1178	173	4	3.9	3.9	NUM
cana-1178	173	5	.	.	PUNCT
cana-1178	174	1	the	the	DET
cana-1178	174	2	randic	randic	ADJ
cana-1178	174	3	matrix	matrix	NOUN
cana-1178	174	4	𝑅(𝐶(𝑇	𝑅(𝐶(𝑇	NOUN
cana-1178	174	5	)	)	PUNCT
cana-1178	174	6	)	)	PUNCT
cana-1178	175	1	=	=	PUNCT
cana-1178	175	2	𝑅𝑥𝑦	𝑅𝑥𝑦	ADP
cana-1178	175	3	of	of	ADP
cana-1178	175	4	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	175	5	)	)	PUNCT
cana-1178	175	6	is	be	AUX
cana-1178	175	7	a	a	DET
cana-1178	175	8	square	square	ADJ
cana-1178	175	9	matrix	matrix	NOUN
cana-1178	175	10	of	of	ADP
cana-1178	175	11	order	order	NOUN
cana-1178	175	12	2𝑛	2𝑛	PROPN
cana-1178	175	13	−	−	PROPN
cana-1178	175	14	2	2	NUM
cana-1178	175	15	whose	whose	DET
cana-1178	175	16	(	(	PUNCT
cana-1178	175	17	𝑥	𝑥	NOUN
cana-1178	175	18	,	,	PUNCT
cana-1178	175	19	𝑦	𝑦	NOUN
cana-1178	175	20	)	)	PUNCT
cana-1178	175	21	entry	entry	NOUN
cana-1178	175	22	is	be	AUX
cana-1178	175	23	𝑅𝑥𝑦	𝑅𝑥𝑦	ADV
cana-1178	175	24	=	=	PUNCT
cana-1178	175	25	{	{	PUNCT
cana-1178	175	26	1	1	NUM
cana-1178	175	27	√𝑑𝑥𝑑𝑦	√𝑑𝑥𝑑𝑦	NOUN
cana-1178	175	28	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌	NOUN
cana-1178	175	29	)	)	PUNCT
cana-1178	175	30	=	=	PUNCT
cana-1178	176	1	𝑅𝑆(∅	𝑅𝑆(∅	X
cana-1178	176	2	)	)	PUNCT
cana-1178	176	3	0	0	NUM
cana-1178	176	4	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-1178	176	5	where	where	SCONJ
cana-1178	176	6	𝑑𝑥	𝑑𝑥	AUX
cana-1178	176	7	denote	denote	VERB
cana-1178	176	8	the	the	DET
cana-1178	176	9	degree	degree	NOUN
cana-1178	176	10	of	of	ADP
cana-1178	176	11	the	the	DET
cana-1178	176	12	vertex	vertex	NOUN
cana-1178	176	13	𝑅𝑆(𝑋	𝑅𝑆(𝑋	PROPN
cana-1178	176	14	)	)	PUNCT
cana-1178	176	15	.	.	PUNCT
cana-1178	177	1	the	the	DET
cana-1178	177	2	eigenvalues	eigenvalue	NOUN
cana-1178	177	3	of	of	ADP
cana-1178	177	4	𝑅(𝐶(𝑇	𝑅(𝐶(𝑇	NOUN
cana-1178	177	5	)	)	PUNCT
cana-1178	177	6	)	)	PUNCT
cana-1178	177	7	are	be	AUX
cana-1178	177	8	called	call	VERB
cana-1178	177	9	randic	randic	ADJ
cana-1178	177	10	eigenvalues	eigenvalue	NOUN
cana-1178	177	11	and	and	CCONJ
cana-1178	177	12	are	be	AUX
cana-1178	177	13	denoted	denote	VERB
cana-1178	177	14	by	by	ADP
cana-1178	177	15	𝜌1	𝜌1	PROPN
cana-1178	177	16	,	,	PUNCT
cana-1178	177	17	𝜌2	𝜌2	ADJ
cana-1178	177	18	…	…	PUNCT
cana-1178	177	19	…	…	PUNCT
cana-1178	177	20	…	…	PUNCT
cana-1178	177	21	.	.	PUNCT
cana-1178	177	22	.	.	PUNCT
cana-1178	178	1	𝜌2𝑛−2	𝜌2𝑛−2	NOUN
cana-1178	178	2	.	.	PUNCT
cana-1178	179	1	if	if	SCONJ
cana-1178	179	2	all	all	DET
cana-1178	179	3	the	the	DET
cana-1178	179	4	𝜌𝑖	𝜌𝑖	NOUN
cana-1178	179	5	’s	’s	NOUN
cana-1178	179	6	,	,	PUNCT
cana-1178	179	7	1	1	NUM
cana-1178	179	8	≤	≤	NUM
cana-1178	179	9	𝑖	𝑖	PRON
cana-1178	179	10	≤	≤	NOUN
cana-1178	179	11	2𝑛	2𝑛	NOUN
cana-1178	179	12	−	−	PROPN
cana-1178	179	13	2	2	NUM
cana-1178	179	14	are	be	AUX
cana-1178	179	15	distinct	distinct	ADJ
cana-1178	179	16	then	then	ADV
cana-1178	179	17	the	the	DET
cana-1178	179	18	randic	randic	ADJ
cana-1178	179	19	spectrum	spectrum	NOUN
cana-1178	179	20	of	of	ADP
cana-1178	179	21	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	179	22	)	)	PUNCT
cana-1178	179	23	can	can	AUX
cana-1178	179	24	be	be	AUX
cana-1178	179	25	denoted	denote	VERB
cana-1178	179	26	as	as	ADP
cana-1178	179	27	𝑆𝑝𝑒𝑐	𝑆𝑝𝑒𝑐	PROPN
cana-1178	179	28	(	(	PUNCT
cana-1178	179	29	𝑅(𝐶(𝑇	𝑅(𝐶(𝑇	PROPN
cana-1178	179	30	)	)	PUNCT
cana-1178	179	31	)	)	PUNCT
cana-1178	179	32	)	)	PUNCT
cana-1178	180	1	=	=	PRON
cana-1178	180	2	(	(	PUNCT
cana-1178	180	3	𝜌1	𝜌1	NOUN
cana-1178	180	4	𝜌2	𝜌2	ADJ
cana-1178	180	5	…	…	PUNCT
cana-1178	180	6	…	…	PUNCT
cana-1178	180	7	…	…	PUNCT
cana-1178	180	8	.	.	PUNCT
cana-1178	180	9	.	.	PUNCT
cana-1178	181	1	𝜌2𝑛−2	𝜌2𝑛−2	NOUN
cana-1178	181	2	𝑚1	𝑚1	NOUN
cana-1178	181	3	𝑚2	𝑚2	PROPN
cana-1178	181	4	…	…	PUNCT
cana-1178	181	5	…	…	PUNCT
cana-1178	181	6	…	…	PUNCT
cana-1178	181	7	𝑚2𝑛−2	𝑚2𝑛−2	NOUN
cana-1178	181	8	)	)	PUNCT
cana-1178	181	9	where	where	SCONJ
cana-1178	181	10	𝑚𝑗	𝑚𝑗	PROPN
cana-1178	181	11	indicates	indicate	VERB
cana-1178	181	12	the	the	DET
cana-1178	181	13	algebraic	algebraic	ADJ
cana-1178	181	14	multiplicity	multiplicity	NOUN
cana-1178	181	15	of	of	ADP
cana-1178	181	16	the	the	DET
cana-1178	181	17	eigenvalue	eigenvalue	PROPN
cana-1178	181	18			ADP
cana-1178	181	19	𝑗	𝑗	PROPN
cana-1178	181	20	,	,	PUNCT
cana-1178	181	21	1	1	NUM
cana-1178	181	22	≤	≤	NUM
cana-1178	181	23	𝑗	𝑗	PRON
cana-1178	181	24	≤	≤	X
cana-1178	181	25	2𝑛	2𝑛	PROPN
cana-1178	181	26	−	−	PROPN
cana-1178	181	27	2	2	NUM
cana-1178	181	28	of	of	ADP
cana-1178	181	29	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	181	30	)	)	PUNCT
cana-1178	181	31	.	.	PUNCT
cana-1178	182	1	the	the	DET
cana-1178	182	2	randic	randic	ADJ
cana-1178	182	3	energy	energy	NOUN
cana-1178	182	4	of	of	ADP
cana-1178	182	5	𝐺𝑅𝐶(𝑇)is	𝐺𝑅𝐶(𝑇)is	ADV
cana-1178	182	6	defined	define	VERB
cana-1178	182	7	as	as	ADP
cana-1178	182	8	𝑅𝐸(𝐶(𝑇	𝑅𝐸(𝐶(𝑇	NOUN
cana-1178	182	9	)	)	PUNCT
cana-1178	182	10	)	)	PUNCT
cana-1178	183	1	=	=	PUNCT
cana-1178	183	2	∑	∑	PUNCT
cana-1178	183	3	|	|	PROPN
cana-1178	183	4	𝑖	𝑖	VERB
cana-1178	183	5	|2𝑛−2	|2𝑛−2	NOUN
cana-1178	183	6	𝑖=1	𝑖=1	PUNCT
cana-1178	183	7	,	,	PUNCT
cana-1178	183	8	𝑖	𝑖	SYM
cana-1178	183	9	=	=	SYM
cana-1178	183	10	1,2	1,2	NUM
cana-1178	183	11	,	,	PUNCT
cana-1178	183	12	…	…	PUNCT
cana-1178	183	13	…	…	PUNCT
cana-1178	183	14	…	…	PUNCT
cana-1178	183	15	.	.	PUNCT
cana-1178	184	1	2𝑛	2𝑛	NOUN
cana-1178	185	1	−	−	NOUN
cana-1178	185	2	2	2	X
cana-1178	185	3	.	.	PUNCT
cana-1178	185	4	theorem	theorem	VERB
cana-1178	185	5	3.6	3.6	NUM
cana-1178	185	6	.	.	PUNCT
cana-1178	186	1	for	for	ADP
cana-1178	186	2	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	186	3	)	)	PUNCT
cana-1178	186	4	,	,	PUNCT
cana-1178	186	5	𝑅𝐸(𝐶(𝑇	𝑅𝐸(𝐶(𝑇	NOUN
cana-1178	186	6	)	)	PUNCT
cana-1178	186	7	)	)	PUNCT
cana-1178	186	8	≤	≤	ADV
cana-1178	186	9	1	1	NUM
cana-1178	186	10	+	+	NUM
cana-1178	186	11	√	√	PROPN
cana-1178	186	12	(	(	PUNCT
cana-1178	186	13	2𝑛−3)(2𝑛−2−(𝐶(𝑇	2𝑛−3)(2𝑛−2−(𝐶(𝑇	NOUN
cana-1178	186	14	)	)	PUNCT
cana-1178	186	15	)	)	PUNCT
cana-1178	186	16	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	186	17	)	)	PUNCT
cana-1178	186	18	)	)	PUNCT
cana-1178	186	19	where	where	SCONJ
cana-1178	186	20	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	186	21	)	)	PUNCT
cana-1178	186	22	)	)	PUNCT
cana-1178	186	23	is	be	AUX
cana-1178	186	24	the	the	DET
cana-1178	186	25	minimum	minimum	ADJ
cana-1178	186	26	degree	degree	NOUN
cana-1178	186	27	.	.	PUNCT
cana-1178	187	1	proof	proof	NOUN
cana-1178	187	2	.	.	PUNCT
cana-1178	188	1	it	it	PRON
cana-1178	188	2	is	be	AUX
cana-1178	188	3	known	know	VERB
cana-1178	188	4	that	that	SCONJ
cana-1178	188	5	∑	∑	PUNCT
cana-1178	188	6	𝜌𝑖	𝜌𝑖	ADP
cana-1178	188	7	22𝑛−2	22𝑛−2	X
cana-1178	188	8	𝑖=1	𝑖=1	PUNCT
cana-1178	188	9	=	=	SYM
cana-1178	188	10	2	2	NUM
cana-1178	188	11	∑	∑	SYM
cana-1178	188	12	1	1	NUM
cana-1178	188	13	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	PROPN
cana-1178	188	14	𝑥,𝑦∈𝐸(𝐶(𝑇	𝑥,𝑦∈𝐸(𝐶(𝑇	PROPN
cana-1178	188	15	)	)	PUNCT
cana-1178	188	16	)	)	PUNCT
cana-1178	189	1	=	=	PUNCT
cana-1178	189	2	∑	∑	PUNCT
cana-1178	189	3	1	1	NUM
cana-1178	189	4	𝑑𝑥	𝑑𝑥	NOUN
cana-1178	189	5	2𝑛−2	2𝑛−2	NUM
cana-1178	189	6	𝑥=1	𝑥=1	X
cana-1178	189	7	∑	∑	ADP
cana-1178	189	8	1	1	NUM
cana-1178	189	9	𝑑𝑦	𝑑𝑦	ADJ
cana-1178	189	10	𝑥,𝑦∈𝐸(𝐶(𝑇	𝑥,𝑦∈𝐸(𝐶(𝑇	NOUN
cana-1178	189	11	)	)	PUNCT
cana-1178	189	12	)	)	PUNCT
cana-1178	189	13	≤	≤	ADV
cana-1178	189	14	∑	∑	ADV
cana-1178	189	15	1	1	NUM
cana-1178	189	16	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	189	17	)	)	PUNCT
cana-1178	189	18	)	)	PUNCT
cana-1178	189	19	2𝑛−2	2𝑛−2	X
cana-1178	189	20	𝑥=1	𝑥=1	SYM
cana-1178	189	21	∑	∑	SYM
cana-1178	189	22	1	1	NUM
cana-1178	189	23	𝑑𝑦	𝑑𝑦	ADJ
cana-1178	189	24	𝑥,𝑦∈𝐸(𝐶(𝑇	𝑥,𝑦∈𝐸(𝐶(𝑇	NOUN
cana-1178	189	25	)	)	PUNCT
cana-1178	189	26	)	)	PUNCT
cana-1178	190	1	𝑆𝑖𝑛𝑐𝑒	𝑆𝑖𝑛𝑐𝑒	PROPN
cana-1178	190	2	𝑑𝑥	𝑑𝑥	VERB
cana-1178	190	3	≥	≥	NOUN
cana-1178	190	4	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	190	5	)	)	PUNCT
cana-1178	190	6	)	)	PUNCT
cana-1178	190	7	communications	communication	NOUN
cana-1178	190	8	on	on	ADP
cana-1178	190	9	applied	apply	VERB
cana-1178	190	10	nonlinear	nonlinear	ADJ
cana-1178	190	11	analysis	analysis	NOUN
cana-1178	190	12	issn	issn	NOUN
cana-1178	190	13	:	:	PUNCT
cana-1178	190	14	1074	1074	NUM
cana-1178	190	15	-	-	PUNCT
cana-1178	190	16	133x	133x	NUM
cana-1178	190	17	vol	vol	NOUN
cana-1178	190	18	31	31	NUM
cana-1178	190	19	no	no	NOUN
cana-1178	190	20	.	.	PUNCT
cana-1178	191	1	6s	6s	NUM
cana-1178	191	2	(	(	PUNCT
cana-1178	191	3	2024	2024	NUM
cana-1178	191	4	)	)	PUNCT
cana-1178	191	5	200	200	NUM
cana-1178	192	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	192	2	=	=	SYM
cana-1178	192	3	2𝑛	2𝑛	PROPN
cana-1178	192	4	−	−	NOUN
cana-1178	192	5	2	2	NUM
cana-1178	192	6	𝑅𝐸(𝐶(𝑇	𝑅𝐸(𝐶(𝑇	NOUN
cana-1178	192	7	)	)	PUNCT
cana-1178	192	8	)	)	PUNCT
cana-1178	193	1	=	=	PUNCT
cana-1178	193	2	∑	∑	PUNCT
cana-1178	193	3	|	|	PROPN
cana-1178	193	4	𝑖	𝑖	VERB
cana-1178	193	5	|	|	NOUN
cana-1178	193	6	=	=	SYM
cana-1178	193	7	1	1	NUM
cana-1178	193	8	+	+	CCONJ
cana-1178	193	9	∑	∑	SYM
cana-1178	193	10	|	|	PROPN
cana-1178	193	11	𝑖	𝑖	VERB
cana-1178	193	12	|2𝑛−2	|2𝑛−2	NOUN
cana-1178	193	13	𝑖=2	𝑖=2	VERB
cana-1178	193	14	2𝑛−2	2𝑛−2	NUM
cana-1178	193	15	𝑖=1	𝑖=1	PUNCT
cana-1178	194	1	(	(	PUNCT
cana-1178	194	2	𝑏𝑦	𝑏𝑦	NOUN
cana-1178	194	3	𝑙𝑒𝑚𝑚𝑎	𝑙𝑒𝑚𝑚𝑎	VERB
cana-1178	194	4	2.1	2.1	NUM
cana-1178	194	5	)	)	PUNCT
cana-1178	194	6	≤	≤	NUM
cana-1178	194	7	1	1	NUM
cana-1178	194	8	+	+	CCONJ
cana-1178	194	9	√(2𝑛	√(2𝑛	NOUN
cana-1178	194	10	−	−	PROPN
cana-1178	194	11	3)(∑	3)(∑	NUM
cana-1178	194	12	𝜌𝑖	𝜌𝑖	ADP
cana-1178	194	13	2	2	NUM
cana-1178	194	14	−	−	PROPN
cana-1178	194	15	12𝑛−2	12𝑛−2	NUM
cana-1178	194	16	𝑖=1	𝑖=1	PUNCT
cana-1178	194	17	)	)	PUNCT
cana-1178	194	18	(	(	PUNCT
cana-1178	194	19	by	by	ADP
cana-1178	194	20	cauchy	cauchy	PROPN
cana-1178	194	21	schwarz	schwarz	PROPN
cana-1178	194	22	inequality	inequality	PROPN
cana-1178	194	23	)	)	PUNCT
cana-1178	194	24	≤	≤	NOUN
cana-1178	194	25	2𝑛	2𝑛	PROPN
cana-1178	194	26	−	−	PROPN
cana-1178	194	27	2	2	NUM
cana-1178	194	28	theorem	theorem	VERB
cana-1178	194	29	3.7	3.7	NUM
cana-1178	194	30	.	.	PUNCT
cana-1178	195	1	in	in	ADP
cana-1178	195	2	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	195	3	)	)	PUNCT
cana-1178	195	4	,	,	PUNCT
cana-1178	195	5	with	with	ADP
cana-1178	195	6	maximum	maximum	ADJ
cana-1178	195	7	degree	degree	NOUN
cana-1178	195	8			NOUN
cana-1178	195	9	(	(	PUNCT
cana-1178	195	10	c(t	c(t	PROPN
cana-1178	195	11	)	)	PUNCT
cana-1178	195	12	)	)	PUNCT
cana-1178	195	13	𝑅𝐸(𝐶(𝑇	𝑅𝐸(𝐶(𝑇	NOUN
cana-1178	195	14	)	)	PUNCT
cana-1178	195	15	)	)	PUNCT
cana-1178	195	16	≥	≥	NOUN
cana-1178	195	17	1	1	NUM
cana-1178	195	18	+	+	CCONJ
cana-1178	195	19	√	√	PROPN
cana-1178	195	20	2𝑛	2𝑛	PROPN
cana-1178	195	21	−	−	PROPN
cana-1178	195	22	2	2	NUM
cana-1178	195	23			NOUN
cana-1178	195	24	(	(	PUNCT
cana-1178	195	25	𝐶(𝑇	𝐶(𝑇	ADJ
cana-1178	195	26	)	)	PUNCT
cana-1178	195	27	)	)	PUNCT
cana-1178	196	1	−	−	NOUN
cana-1178	196	2	1	1	NUM
cana-1178	197	1	+	+	CCONJ
cana-1178	197	2	(	(	PUNCT
cana-1178	197	3	2𝑛	2𝑛	PROPN
cana-1178	197	4	−	−	PROPN
cana-1178	198	1	3)(2𝑛	3)(2𝑛	NUM
cana-1178	198	2	−	−	NOUN
cana-1178	198	3	4	4	NUM
cana-1178	198	4	)	)	PUNCT
cana-1178	198	5	(	(	PUNCT
cana-1178	198	6	|	|	ADV
cana-1178	198	7	det	det	PROPN
cana-1178	198	8	𝐴(𝐶(𝑇))|	𝐴(𝐶(𝑇))|	PROPN
cana-1178	198	9	∏	∏	PROPN
cana-1178	198	10	𝑑𝑖	𝑑𝑖	PROPN
cana-1178	198	11	2𝑛−2	2𝑛−2	NUM
cana-1178	198	12	𝑖=1	𝑖=1	PUNCT
cana-1178	198	13	)	)	PUNCT
cana-1178	198	14	2	2	NUM
cana-1178	198	15	2𝑛−3	2𝑛−3	NUM
cana-1178	198	16	where	where	SCONJ
cana-1178	198	17	𝑑𝑒𝑡	𝑑𝑒𝑡	NOUN
cana-1178	198	18	𝐴(𝐶(𝑇	𝐴(𝐶(𝑇	PROPN
cana-1178	198	19	)	)	PUNCT
cana-1178	198	20	)	)	PUNCT
cana-1178	198	21	denotes	denote	VERB
cana-1178	198	22	the	the	DET
cana-1178	198	23	determinant	determinant	NOUN
cana-1178	198	24	of	of	ADP
cana-1178	198	25	adjacency	adjacency	NOUN
cana-1178	198	26	matrix	matrix	NOUN
cana-1178	198	27	of	of	ADP
cana-1178	198	28	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	198	29	)	)	PUNCT
cana-1178	198	30	.	.	PUNCT
cana-1178	199	1	proof	proof	NOUN
cana-1178	199	2	.	.	PUNCT
cana-1178	200	1	proceeding	proceed	VERB
cana-1178	200	2	as	as	ADP
cana-1178	200	3	in	in	ADP
cana-1178	200	4	the	the	DET
cana-1178	200	5	above	above	NOUN
cana-1178	200	6	,	,	PUNCT
cana-1178	200	7	we	we	PRON
cana-1178	200	8	have	have	VERB
cana-1178	200	9	∑	∑	ADV
cana-1178	200	10	𝜌𝑖	𝜌𝑖	ADP
cana-1178	200	11	2𝑛−2	2𝑛−2	NUM
cana-1178	200	12	𝑖=1	𝑖=1	SYM
cana-1178	200	13	2	2	NUM
cana-1178	200	14	=	=	SYM
cana-1178	200	15	2𝑛−2	2𝑛−2	NUM
cana-1178	200	16	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	200	17	)	)	PUNCT
cana-1178	200	18	)	)	PUNCT
cana-1178	200	19	using	use	VERB
cana-1178	200	20	arithmetic	arithmetic	ADJ
cana-1178	200	21	geometric	geometric	ADJ
cana-1178	200	22	mean	mean	NOUN
cana-1178	200	23	inequality	inequality	NOUN
cana-1178	200	24	2	2	NUM
cana-1178	200	25	∑	∑	PUNCT
cana-1178	200	26	|𝜌𝑖||𝜌𝑗|	|𝜌𝑖||𝜌𝑗|	X
cana-1178	200	27	≥	≥	PROPN
cana-1178	200	28	2≤𝑖<𝑗≤2𝑛−2	2≤𝑖<𝑗≤2𝑛−2	NUM
cana-1178	200	29	(	(	PUNCT
cana-1178	200	30	2𝑛	2𝑛	PROPN
cana-1178	200	31	−	−	PROPN
cana-1178	201	1	3)(2𝑛	3)(2𝑛	NUM
cana-1178	202	1	−	−	NOUN
cana-1178	202	2	4	4	NUM
cana-1178	202	3	)	)	PUNCT
cana-1178	202	4	(	(	PUNCT
cana-1178	202	5	∏	∏	X
cana-1178	202	6	|𝜌𝑖|	|𝜌𝑖|	NOUN
cana-1178	202	7	2𝑛−2	2𝑛−2	NUM
cana-1178	202	8	𝑖=2	𝑖=2	PUNCT
cana-1178	202	9	)	)	PUNCT
cana-1178	202	10	2	2	NUM
cana-1178	202	11	2𝑛−3	2𝑛−3	NUM
cana-1178	202	12	=	=	SYM
cana-1178	202	13	(	(	PUNCT
cana-1178	202	14	2𝑛	2𝑛	PROPN
cana-1178	202	15	−	−	PROPN
cana-1178	203	1	3)(2𝑛	3)(2𝑛	NUM
cana-1178	203	2	−	−	NOUN
cana-1178	203	3	4)(|det	4)(|det	NOUN
cana-1178	203	4	re(𝐶(𝑇	re(𝐶(𝑇	NOUN
cana-1178	203	5	)	)	PUNCT
cana-1178	203	6	)	)	PUNCT
cana-1178	204	1	|	|	ADV
cana-1178	204	2	)	)	PUNCT
cana-1178	204	3	2	2	NUM
cana-1178	204	4	2𝑛−3	2𝑛−3	NUM
cana-1178	204	5	=	=	SYM
cana-1178	204	6	(	(	PUNCT
cana-1178	204	7	2𝑛	2𝑛	PROPN
cana-1178	204	8	−	−	PROPN
cana-1178	205	1	3)(2𝑛	3)(2𝑛	NUM
cana-1178	206	1	−	−	NOUN
cana-1178	206	2	4	4	NUM
cana-1178	206	3	)	)	PUNCT
cana-1178	206	4	(	(	PUNCT
cana-1178	206	5	|	|	ADV
cana-1178	206	6	det	det	PROPN
cana-1178	206	7	𝐴(𝐶(𝑇))|	𝐴(𝐶(𝑇))|	PROPN
cana-1178	206	8	∏	∏	PROPN
cana-1178	206	9	𝑑𝑖	𝑑𝑖	PROPN
cana-1178	206	10	2𝑛−2	2𝑛−2	NUM
cana-1178	206	11	𝑖=1	𝑖=1	PUNCT
cana-1178	206	12	)	)	PUNCT
cana-1178	206	13	2	2	NUM
cana-1178	206	14	2𝑛−3	2𝑛−3	NUM
cana-1178	206	15	(	(	PUNCT
cana-1178	206	16	𝑏𝑦	𝑏𝑦	NOUN
cana-1178	206	17	𝑙𝑒𝑚𝑚𝑎	𝑙𝑒𝑚𝑚𝑎	VERB
cana-1178	206	18	2.2	2.2	NUM
cana-1178	206	19	)	)	PUNCT
cana-1178	206	20	now	now	ADV
cana-1178	206	21	,	,	PUNCT
cana-1178	206	22	(	(	PUNCT
cana-1178	206	23	∑	∑	ADP
cana-1178	206	24	|𝜌𝑖	|𝜌𝑖	ADP
cana-1178	206	25	2𝑛−2	2𝑛−2	NUM
cana-1178	206	26	𝑖=2	𝑖=2	PUNCT
cana-1178	206	27	|	|	NOUN
cana-1178	206	28	)	)	PUNCT
cana-1178	206	29	2	2	NUM
cana-1178	206	30	=	=	SYM
cana-1178	206	31	∑	∑	PROPN
cana-1178	206	32	𝜌𝑖	𝜌𝑖	ADP
cana-1178	206	33	2𝑛−2	2𝑛−2	NUM
cana-1178	206	34	𝑖=2	𝑖=2	SYM
cana-1178	206	35	2	2	NUM
cana-1178	206	36	+	+	NUM
cana-1178	206	37	2	2	NUM
cana-1178	206	38	∑	∑	NOUN
cana-1178	206	39	|𝜌𝑖||𝜌𝑗|2≤𝑖<𝑗≤2𝑛−2	|𝜌𝑖||𝜌𝑗|2≤𝑖<𝑗≤2𝑛−2	PROPN
cana-1178	206	40	∑	∑	PUNCT
cana-1178	206	41	|	|	PROPN
cana-1178	206	42	𝑖	𝑖	ADP
cana-1178	206	43	|	|	ADV
cana-1178	206	44	2𝑛−2	2𝑛−2	NUM
cana-1178	206	45	𝑖=2	𝑖=2	VERB
cana-1178	206	46	≥	≥	NOUN
cana-1178	206	47	√	√	ADP
cana-1178	206	48	2𝑛	2𝑛	PROPN
cana-1178	206	49	−	−	PROPN
cana-1178	206	50	2	2	NUM
cana-1178	206	51	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	206	52	)	)	PUNCT
cana-1178	206	53	)	)	PUNCT
cana-1178	207	1	−	−	ADP
cana-1178	207	2	1	1	NUM
cana-1178	208	1	+	+	CCONJ
cana-1178	208	2	(	(	PUNCT
cana-1178	208	3	2𝑛	2𝑛	PROPN
cana-1178	208	4	−	−	PROPN
cana-1178	209	1	3)(2𝑛	3)(2𝑛	NUM
cana-1178	209	2	−	−	NOUN
cana-1178	209	3	4	4	NUM
cana-1178	209	4	)	)	PUNCT
cana-1178	209	5	(	(	PUNCT
cana-1178	209	6	|	|	ADV
cana-1178	209	7	det	det	PROPN
cana-1178	209	8	𝐴(𝐶(𝑇))|	𝐴(𝐶(𝑇))|	PROPN
cana-1178	209	9	∏	∏	PROPN
cana-1178	209	10	𝑑𝑖	𝑑𝑖	PROPN
cana-1178	209	11	2𝑛−2	2𝑛−2	NUM
cana-1178	209	12	𝑖=1	𝑖=1	PUNCT
cana-1178	209	13	)	)	PUNCT
cana-1178	209	14	2	2	NUM
cana-1178	209	15	2𝑛−3	2𝑛−3	NUM
cana-1178	209	16	𝑅𝐸(𝐶(𝑇	𝑅𝐸(𝐶(𝑇	NOUN
cana-1178	209	17	)	)	PUNCT
cana-1178	209	18	)	)	PUNCT
cana-1178	210	1	=	=	PUNCT
cana-1178	210	2	∑	∑	PUNCT
cana-1178	210	3	|	|	PROPN
cana-1178	210	4	𝑖	𝑖	VERB
cana-1178	210	5	|2𝑛−2	|2𝑛−2	NOUN
cana-1178	210	6	𝑖=1	𝑖=1	PROPN
cana-1178	210	7	𝑅𝐸(𝐶(𝑇	𝑅𝐸(𝐶(𝑇	NOUN
cana-1178	210	8	)	)	PUNCT
cana-1178	210	9	)	)	PUNCT
cana-1178	211	1	≥	≥	NOUN
cana-1178	211	2	1	1	NUM
cana-1178	212	1	+	+	CCONJ
cana-1178	212	2	√	√	ADP
cana-1178	212	3	2𝑛−2	2𝑛−2	NUM
cana-1178	212	4	(𝐶(𝑇	(𝐶(𝑇	NOUN
cana-1178	212	5	)	)	PUNCT
cana-1178	212	6	)	)	PUNCT
cana-1178	213	1	−	−	ADP
cana-1178	213	2	1	1	NUM
cana-1178	214	1	+	+	CCONJ
cana-1178	214	2	(	(	PUNCT
cana-1178	214	3	2𝑛	2𝑛	PROPN
cana-1178	214	4	−	−	PROPN
cana-1178	215	1	3)(2𝑛	3)(2𝑛	NUM
cana-1178	215	2	−	−	NOUN
cana-1178	215	3	4	4	NUM
cana-1178	215	4	)	)	PUNCT
cana-1178	215	5	(	(	PUNCT
cana-1178	215	6	|	|	ADV
cana-1178	215	7	det	det	PROPN
cana-1178	215	8	𝐴(𝐶(𝑇))|	𝐴(𝐶(𝑇))|	PROPN
cana-1178	215	9	∏	∏	PROPN
cana-1178	215	10	𝑑𝑖	𝑑𝑖	PROPN
cana-1178	215	11	2𝑛−2	2𝑛−2	NUM
cana-1178	215	12	𝑖=1	𝑖=1	PUNCT
cana-1178	215	13	)	)	PUNCT
cana-1178	216	1	2	2	NUM
cana-1178	216	2	2𝑛−3	2𝑛−3	NUM
cana-1178	216	3	example	example	NOUN
cana-1178	216	4	3.1	3.1	NUM
cana-1178	216	5	.	.	PUNCT
cana-1178	217	1	the	the	DET
cana-1178	217	2	following	follow	VERB
cana-1178	217	3	graph	graph	NOUN
cana-1178	217	4	is	be	AUX
cana-1178	217	5	corresponding	correspond	VERB
cana-1178	217	6	to	to	ADP
cana-1178	217	7	the	the	DET
cana-1178	217	8	approximation	approximation	NOUN
cana-1178	217	9	space	space	NOUN
cana-1178	217	10	𝐼	𝐼	PROPN
cana-1178	217	11	=	=	SYM
cana-1178	217	12	(	(	PUNCT
cana-1178	217	13	𝑈	𝑈	PROPN
cana-1178	217	14	,	,	PUNCT
cana-1178	217	15	𝑅	𝑅	PROPN
cana-1178	217	16	)	)	PUNCT
cana-1178	217	17	where	where	SCONJ
cana-1178	217	18	𝑅	𝑅	PROPN
cana-1178	217	19	induces	induce	VERB
cana-1178	217	20	3	3	NUM
cana-1178	217	21	equivalence	equivalence	NOUN
cana-1178	217	22	classes	class	NOUN
cana-1178	217	23	.	.	PUNCT
cana-1178	218	1	let	let	VERB
cana-1178	218	2	𝑈	𝑈	PROPN
cana-1178	218	3	=	=	SYM
cana-1178	218	4	(	(	PUNCT
cana-1178	218	5	𝑥1	𝑥1	NOUN
cana-1178	218	6	,	,	PUNCT
cana-1178	218	7	𝑥2	𝑥2	NOUN
cana-1178	218	8	,	,	PUNCT
cana-1178	218	9	𝑥3	𝑥3	NOUN
cana-1178	218	10	,	,	PUNCT
cana-1178	218	11	𝑥4	𝑥4	NOUN
cana-1178	218	12	,	,	PUNCT
cana-1178	218	13	𝑥5	𝑥5	PROPN
cana-1178	218	14	,	,	PUNCT
cana-1178	218	15	𝑥6	𝑥6	PROPN
cana-1178	218	16	)	)	PUNCT
cana-1178	218	17	here	here	ADV
cana-1178	218	18	𝐸	𝐸	PROPN
cana-1178	218	19	=	=	SYM
cana-1178	218	20	{	{	PUNCT
cana-1178	218	21	𝑋1	𝑋1	PROPN
cana-1178	218	22	,	,	PUNCT
cana-1178	218	23	𝑋2	𝑋2	VERB
cana-1178	218	24	,	,	PUNCT
cana-1178	218	25	𝑋3	𝑋3	NOUN
cana-1178	218	26	}	}	PUNCT
cana-1178	218	27	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1178	218	28	𝑋1	𝑋1	NOUN
cana-1178	218	29	=	=	SYM
cana-1178	218	30	{	{	PUNCT
cana-1178	218	31	𝑥1	𝑥1	NOUN
cana-1178	218	32	,	,	PUNCT
cana-1178	218	33	𝑥3	𝑥3	NOUN
cana-1178	218	34	}	}	PUNCT
cana-1178	218	35	,	,	PUNCT
cana-1178	218	36	𝑋2	𝑋2	VERB
cana-1178	218	37	=	=	PUNCT
cana-1178	218	38	{	{	PUNCT
cana-1178	218	39	𝑥2	𝑥2	NOUN
cana-1178	218	40	,	,	PUNCT
cana-1178	218	41	𝑥4	𝑥4	NOUN
cana-1178	218	42	,	,	PUNCT
cana-1178	218	43	𝑥6	𝑥6	PROPN
cana-1178	218	44	}	}	PUNCT
cana-1178	218	45	,	,	PUNCT
cana-1178	218	46	𝑋3	𝑋3	NOUN
cana-1178	218	47	=	=	PUNCT
cana-1178	218	48	{	{	PUNCT
cana-1178	218	49	𝑥5	𝑥5	PROPN
cana-1178	218	50	}	}	PUNCT
cana-1178	218	51	communications	communication	NOUN
cana-1178	218	52	on	on	ADP
cana-1178	218	53	applied	apply	VERB
cana-1178	218	54	nonlinear	nonlinear	ADJ
cana-1178	218	55	analysis	analysis	NOUN
cana-1178	218	56	issn	issn	NOUN
cana-1178	218	57	:	:	PUNCT
cana-1178	218	58	1074	1074	NUM
cana-1178	218	59	-	-	PUNCT
cana-1178	218	60	133x	133x	NUM
cana-1178	218	61	vol	vol	NOUN
cana-1178	218	62	31	31	NUM
cana-1178	218	63	no	no	NOUN
cana-1178	218	64	.	.	PUNCT
cana-1178	219	1	6s	6s	NUM
cana-1178	219	2	(	(	PUNCT
cana-1178	219	3	2024	2024	NUM
cana-1178	219	4	)	)	PUNCT
cana-1178	219	5	201	201	NUM
cana-1178	220	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	220	2	𝐻𝑒𝑟𝑒	𝐻𝑒𝑟𝑒	PROPN
cana-1178	220	3	𝑉(𝐶(𝑇	𝑉(𝐶(𝑇	NOUN
cana-1178	220	4	)	)	PUNCT
cana-1178	220	5	)	)	PUNCT
cana-1178	221	1	=	=	PRON
cana-1178	221	2	{	{	PUNCT
cana-1178	221	3	𝑅𝑆(𝑋1	𝑅𝑆(𝑋1	NUM
cana-1178	221	4	)	)	PUNCT
cana-1178	221	5	,	,	PUNCT
cana-1178	221	6	𝑅𝑆(𝑋2	𝑅𝑆(𝑋2	PROPN
cana-1178	221	7	)	)	PUNCT
cana-1178	221	8	,	,	PUNCT
cana-1178	221	9	𝑅𝑆(𝑋3	𝑅𝑆(𝑋3	NOUN
cana-1178	221	10	)	)	PUNCT
cana-1178	221	11	,	,	PUNCT
cana-1178	221	12	𝑅𝑆(𝑥1	𝑅𝑆(𝑥1	NUM
cana-1178	221	13	)	)	PUNCT
cana-1178	221	14	,	,	PUNCT
cana-1178	221	15	𝑅𝑆(𝑥2	𝑅𝑆(𝑥2	PROPN
cana-1178	221	16	)	)	PUNCT
cana-1178	221	17	,	,	PUNCT
cana-1178	221	18	𝑅𝑆(𝑥1	𝑅𝑆(𝑥1	ADV
cana-1178	221	19	∪	∪	ADP
cana-1178	221	20	𝑋2	𝑋2	PROPN
cana-1178	221	21	)	)	PUNCT
cana-1178	221	22	,	,	PUNCT
cana-1178	221	23	𝑅𝑆(𝑋1	𝑅𝑆(𝑋1	PUNCT
cana-1178	221	24	∪	∪	ADP
cana-1178	221	25	𝑥2	𝑥2	NOUN
cana-1178	221	26	)	)	PUNCT
cana-1178	221	27	,	,	PUNCT
cana-1178	221	28	𝑅𝑆(𝑥1	𝑅𝑆(𝑥1	ADV
cana-1178	221	29	∪	∪	ADJ
cana-1178	221	30	𝑥2	𝑥2	NOUN
cana-1178	221	31	)	)	PUNCT
cana-1178	221	32	,	,	PUNCT
cana-1178	221	33	𝑅𝑆(𝑋1	𝑅𝑆(𝑋1	PUNCT
cana-1178	221	34	∪	∪	ADP
cana-1178	221	35	𝑋2	𝑋2	VERB
cana-1178	221	36	)	)	PUNCT
cana-1178	221	37	,	,	PUNCT
cana-1178	221	38	𝑅𝑆(𝑥1	𝑅𝑆(𝑥1	ADV
cana-1178	221	39	∪	∪	ADP
cana-1178	221	40	𝑋3	𝑋3	NOUN
cana-1178	221	41	)	)	PUNCT
cana-1178	221	42	,	,	PUNCT
cana-1178	221	43	𝑅𝑆(𝑋1	𝑅𝑆(𝑋1	PUNCT
cana-1178	221	44	∪	∪	ADP
cana-1178	221	45	𝑋3	𝑋3	NOUN
cana-1178	221	46	)	)	PUNCT
cana-1178	221	47	,	,	PUNCT
cana-1178	221	48	𝑅𝑆(𝑥2	𝑅𝑆(𝑥2	PROPN
cana-1178	221	49	∪	∪	ADP
cana-1178	221	50	𝑋3	𝑋3	NOUN
cana-1178	221	51	)	)	PUNCT
cana-1178	221	52	,	,	PUNCT
cana-1178	221	53	𝑅𝑆(𝑋2	𝑅𝑆(𝑋2	ADP
cana-1178	221	54	∪	∪	ADP
cana-1178	221	55	𝑋3	𝑋3	NOUN
cana-1178	221	56	)	)	PUNCT
cana-1178	221	57	,	,	PUNCT
cana-1178	221	58	𝑅𝑆(𝑥1	𝑅𝑆(𝑥1	PROPN
cana-1178	221	59	∪	∪	X
cana-1178	221	60	𝑋2	𝑋2	VERB
cana-1178	221	61	∪	∪	ADJ
cana-1178	221	62	𝑋3	𝑋3	NOUN
cana-1178	221	63	)	)	PUNCT
cana-1178	221	64	,	,	PUNCT
cana-1178	221	65	𝑅𝑆(𝑋1	𝑅𝑆(𝑋1	PUNCT
cana-1178	221	66	∪	∪	ADP
cana-1178	221	67	𝑥2	𝑥2	NOUN
cana-1178	221	68	∪	∪	ADP
cana-1178	221	69	𝑋3	𝑋3	NOUN
cana-1178	221	70	)	)	PUNCT
cana-1178	221	71	,	,	PUNCT
cana-1178	221	72	𝑅𝑆(𝑥1	𝑅𝑆(𝑥1	ADV
cana-1178	221	73	∪	∪	ADP
cana-1178	221	74	𝑥2	𝑥2	NOUN
cana-1178	221	75	∪	∪	ADP
cana-1178	221	76	𝑋3	𝑋3	NOUN
cana-1178	221	77	)	)	PUNCT
cana-1178	221	78	}	}	PUNCT
cana-1178	221	79	the	the	DET
cana-1178	221	80	minimum	minimum	ADJ
cana-1178	221	81	dominating	dominating	NOUN
cana-1178	221	82	matrix	matrix	NOUN
cana-1178	221	83	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	PROPN
cana-1178	221	84	)	)	PUNCT
cana-1178	221	85	)	)	PUNCT
cana-1178	221	86	is	be	AUX
cana-1178	221	87	given	give	VERB
cana-1178	221	88	by	by	ADP
cana-1178	221	89	𝐴𝐷(𝐶(𝑇	𝐴𝐷(𝐶(𝑇	NOUN
cana-1178	221	90	)	)	PUNCT
cana-1178	221	91	)	)	PUNCT
cana-1178	222	1	=	=	PRON
cana-1178	222	2	figure	figure	NOUN
cana-1178	222	3	1	1	NUM
cana-1178	222	4	:	:	PUNCT
cana-1178	222	5	rough	rough	ADJ
cana-1178	222	6	complemented	complemented	ADJ
cana-1178	222	7	graph	graph	NOUN
cana-1178	222	8	the	the	DET
cana-1178	222	9	spectrum	spectrum	NOUN
cana-1178	222	10	of	of	ADP
cana-1178	222	11	minimum	minimum	ADJ
cana-1178	222	12	dominating	dominating	NOUN
cana-1178	222	13	energy	energy	NOUN
cana-1178	222	14	are	be	AUX
cana-1178	222	15	[	[	X
cana-1178	222	16	3.3028	3.3028	NUM
cana-1178	222	17	,	,	PUNCT
cana-1178	222	18	1(2	1(2	NUM
cana-1178	222	19	)	)	PUNCT
cana-1178	222	20	,	,	PUNCT
cana-1178	222	21	−0.3028	−0.3028	PROPN
cana-1178	222	22	,	,	PUNCT
cana-1178	222	23	(	(	PUNCT
cana-1178	222	24	−1)(2	−1)(2	NOUN
cana-1178	222	25	)	)	PUNCT
cana-1178	222	26	]	]	PUNCT
cana-1178	223	1	the	the	DET
cana-1178	223	2	maximal	maximal	ADJ
cana-1178	223	3	independent	independent	ADJ
cana-1178	223	4	and	and	CCONJ
cana-1178	223	5	dominating	dominating	NOUN
cana-1178	223	6	matrix	matrix	NOUN
cana-1178	223	7	of	of	ADP
cana-1178	223	8	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	223	9	)	)	PUNCT
cana-1178	223	10	is	be	AUX
cana-1178	223	11	given	give	VERB
cana-1178	223	12	by	by	ADP
cana-1178	223	13	the	the	DET
cana-1178	223	14	characteristic	characteristic	ADJ
cana-1178	223	15	polynomial	polynomial	ADJ
cana-1178	223	16	,	,	PUNCT
cana-1178	223	17	spectrum	spectrum	NOUN
cana-1178	223	18	and	and	CCONJ
cana-1178	223	19	maximal	maximal	ADJ
cana-1178	223	20	independent	independent	ADJ
cana-1178	223	21	domination	domination	NOUN
cana-1178	223	22	energy	energy	NOUN
cana-1178	223	23	are	be	AUX
cana-1178	223	24	as	as	SCONJ
cana-1178	223	25	follows	follow	VERB
cana-1178	223	26	.	.	PUNCT
cana-1178	224	1	f(𝐺𝑅𝐶(𝑇),	f(𝐺𝑅𝐶(𝑇),	NUM
cana-1178	224	2	)	)	PUNCT
cana-1178	225	1	=	=	VERB
cana-1178	225	2	(2	(2	ADP
cana-1178	225	3	−	−	NUM
cana-1178	225	4	2)(3	2)(3	NUM
cana-1178	226	1	−	−	PROPN
cana-1178	226	2	32	32	NUM
cana-1178	226	3	−	−	PROPN
cana-1178	226	4			ADJ
cana-1178	226	5	+	+	X
cana-1178	226	6	4	4	X
cana-1178	226	7	)	)	PUNCT
cana-1178	226	8	spec	spec	NOUN
cana-1178	226	9	(	(	PUNCT
cana-1178	226	10	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	PROPN
cana-1178	226	11	)	)	PUNCT
cana-1178	226	12	)	)	PUNCT
cana-1178	227	1	=	=	SYM
cana-1178	227	2	(	(	PUNCT
cana-1178	227	3	−1.4142	−1.4142	NOUN
cana-1178	227	4	1	1	NUM
cana-1178	227	5	−1.1149	−1.1149	NOUN
cana-1178	228	1	1	1	NUM
cana-1178	228	2	0	0	NUM
cana-1178	228	3	1	1	NUM
cana-1178	228	4	1.2541	1.2541	NUM
cana-1178	228	5	1	1	NUM
cana-1178	228	6	1.4142	1.4142	NUM
cana-1178	228	7	1	1	NUM
cana-1178	228	8	2.8608	2.8608	NUM
cana-1178	228	9	1	1	NUM
cana-1178	228	10	)	)	PUNCT
cana-1178	228	11	communications	communication	NOUN
cana-1178	228	12	on	on	ADP
cana-1178	228	13	applied	apply	VERB
cana-1178	228	14	nonlinear	nonlinear	ADJ
cana-1178	228	15	analysis	analysis	NOUN
cana-1178	228	16	issn	issn	NOUN
cana-1178	228	17	:	:	PUNCT
cana-1178	228	18	1074	1074	NUM
cana-1178	228	19	-	-	PUNCT
cana-1178	228	20	133x	133x	NUM
cana-1178	228	21	vol	vol	NOUN
cana-1178	228	22	31	31	NUM
cana-1178	228	23	no	no	NOUN
cana-1178	228	24	.	.	PUNCT
cana-1178	229	1	6s	6s	NUM
cana-1178	229	2	(	(	PUNCT
cana-1178	229	3	2024	2024	NUM
cana-1178	229	4	)	)	PUNCT
cana-1178	229	5	202	202	NUM
cana-1178	229	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	229	7	ԑ𝐼𝐷(𝐶(𝑇	ԑ𝐼𝐷(𝐶(𝑇	PROPN
cana-1178	229	8	)	)	PUNCT
cana-1178	229	9	)	)	PUNCT
cana-1178	230	1	=	=	PUNCT
cana-1178	230	2	8.0582	8.0582	NUM
cana-1178	230	3	.	.	PUNCT
cana-1178	231	1	here	here	ADV
cana-1178	231	2	the	the	DET
cana-1178	231	3	maximal	maximal	ADJ
cana-1178	231	4	independent	independent	ADJ
cana-1178	231	5	and	and	CCONJ
cana-1178	231	6	domination	domination	NOUN
cana-1178	231	7	number	number	NOUN
cana-1178	231	8	|𝐼𝐷(𝐶(𝑇))|	|𝐼𝐷(𝐶(𝑇))|	NOUN
cana-1178	231	9	=	=	SYM
cana-1178	231	10	2n−1	2n−1	NUM
cana-1178	231	11	−	−	NOUN
cana-1178	231	12	1	1	NUM
cana-1178	231	13	=	=	SYM
cana-1178	231	14	3	3	NUM
cana-1178	231	15	.	.	NOUN
cana-1178	231	16	4	4	NUM
cana-1178	231	17	.	.	X
cana-1178	231	18	generation	generation	NOUN
cana-1178	231	19	of	of	ADP
cana-1178	231	20	various	various	ADJ
cana-1178	231	21	graph	graph	NOUN
cana-1178	231	22	energies	energy	NOUN
cana-1178	231	23	using	use	VERB
cana-1178	231	24	python	python	NOUN
cana-1178	231	25	for	for	ADP
cana-1178	231	26	the	the	DET
cana-1178	231	27	rough	rough	ADJ
cana-1178	231	28	complemented	complemented	ADJ
cana-1178	231	29	graph	graph	NOUN
cana-1178	231	30	in	in	ADP
cana-1178	231	31	this	this	DET
cana-1178	231	32	section	section	NOUN
cana-1178	231	33	,	,	PUNCT
cana-1178	231	34	python	python	PROPN
cana-1178	231	35	code	code	NOUN
cana-1178	231	36	is	be	AUX
cana-1178	231	37	provided	provide	VERB
cana-1178	231	38	for	for	ADP
cana-1178	231	39	siedel	siedel	NOUN
cana-1178	231	40	,	,	PUNCT
cana-1178	231	41	randic	randic	ADJ
cana-1178	231	42	and	and	CCONJ
cana-1178	231	43	minimum	minimum	ADJ
cana-1178	231	44	dominating	dominating	NOUN
cana-1178	231	45	energies	energy	NOUN
cana-1178	231	46	for	for	ADP
cana-1178	231	47	the	the	DET
cana-1178	231	48	graph	graph	NOUN
cana-1178	231	49	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	231	50	)	)	PUNCT
cana-1178	231	51	.	.	PUNCT
cana-1178	232	1	additionally	additionally	ADV
cana-1178	232	2	,	,	PUNCT
cana-1178	232	3	the	the	DET
cana-1178	232	4	energies	energy	NOUN
cana-1178	232	5	are	be	AUX
cana-1178	232	6	compared	compare	VERB
cana-1178	232	7	using	use	VERB
cana-1178	232	8	bar	bar	NOUN
cana-1178	232	9	diagram	diagram	NOUN
cana-1178	232	10	for	for	ADP
cana-1178	232	11	various	various	ADJ
cana-1178	232	12	values	value	NOUN
cana-1178	232	13	of	of	ADP
cana-1178	232	14	𝑛.	𝑛.	NOUN
cana-1178	232	15	certain	certain	ADJ
cana-1178	232	16	auxiliary	auxiliary	ADJ
cana-1178	232	17	functions	function	NOUN
cana-1178	232	18	used	use	VERB
cana-1178	232	19	for	for	ADP
cana-1178	232	20	displaying	display	VERB
cana-1178	232	21	and	and	CCONJ
cana-1178	232	22	increasing	increase	VERB
cana-1178	232	23	the	the	DET
cana-1178	232	24	visual	visual	ADJ
cana-1178	232	25	appeal	appeal	NOUN
cana-1178	232	26	of	of	ADP
cana-1178	232	27	the	the	DET
cana-1178	232	28	graphs	graph	NOUN
cana-1178	232	29	and	and	CCONJ
cana-1178	232	30	their	their	PRON
cana-1178	232	31	energies	energy	NOUN
cana-1178	232	32	are	be	AUX
cana-1178	232	33	not	not	PART
cana-1178	232	34	included	include	VERB
cana-1178	232	35	in	in	ADP
cana-1178	232	36	the	the	DET
cana-1178	232	37	code	code	NOUN
cana-1178	232	38	presented	present	VERB
cana-1178	232	39	.	.	PUNCT
cana-1178	233	1	libraries	library	NOUN
cana-1178	233	2	used	use	VERB
cana-1178	233	3	to	to	PART
cana-1178	233	4	aid	aid	VERB
cana-1178	233	5	code	code	NOUN
cana-1178	233	6	reusability	reusability	NOUN
cana-1178	233	7	have	have	AUX
cana-1178	233	8	been	be	AUX
cana-1178	233	9	imported	import	VERB
cana-1178	233	10	at	at	ADP
cana-1178	233	11	the	the	DET
cana-1178	233	12	top	top	NOUN
cana-1178	233	13	of	of	ADP
cana-1178	233	14	the	the	DET
cana-1178	233	15	first	first	ADJ
cana-1178	233	16	code	code	NOUN
cana-1178	233	17	section	section	NOUN
cana-1178	233	18	.	.	PUNCT
cana-1178	234	1	graph	graph	NOUN
cana-1178	234	2	generation	generation	NOUN
cana-1178	234	3	code	code	NOUN
cana-1178	234	4	import	import	NOUN
cana-1178	234	5	numpy	numpy	NOUN
cana-1178	234	6	as	as	ADP
cana-1178	234	7	np	np	NUM
cana-1178	234	8	import	import	NOUN
cana-1178	234	9	networkx	networkx	NOUN
cana-1178	234	10	as	as	ADP
cana-1178	234	11	nx	nx	NUM
cana-1178	234	12	from	from	ADP
cana-1178	234	13	itertools	itertools	PROPN
cana-1178	234	14	import	import	NOUN
cana-1178	234	15	chain	chain	NOUN
cana-1178	234	16	,	,	PUNCT
cana-1178	234	17	combinations	combination	NOUN
cana-1178	234	18	import	import	NOUN
cana-1178	234	19	matplotlib.pyplot	matplotlib.pyplot	NOUN
cana-1178	234	20	as	as	ADP
cana-1178	234	21	plt	plt	NOUN
cana-1178	234	22	import	import	NOUN
cana-1178	234	23	csv	csv	VERB
cana-1178	234	24	#	#	NOUN
cana-1178	234	25	utility	utility	NOUN
cana-1178	234	26	functions	function	NOUN
cana-1178	234	27	def	def	ADJ
cana-1178	234	28	powerset(iterable	powerset(iterable	PROPN
cana-1178	234	29	):	):	PUNCT
cana-1178	234	30	s	s	X
cana-1178	234	31	=	=	ADJ
cana-1178	234	32	set(iterable	set(iterable	X
cana-1178	234	33	)	)	PUNCT
cana-1178	234	34	return	return	NOUN
cana-1178	234	35	set(chain.from_iterable(combinations(s	set(chain.from_iterable(combinations(s	NOUN
cana-1178	234	36	,	,	PUNCT
cana-1178	234	37	r	r	NOUN
cana-1178	234	38	)	)	PUNCT
cana-1178	234	39	for	for	ADP
cana-1178	234	40	r	r	NOUN
cana-1178	234	41	in	in	ADP
cana-1178	234	42	range(len(s)+1	range(len(s)+1	NOUN
cana-1178	234	43	)	)	PUNCT
cana-1178	234	44	)	)	PUNCT
cana-1178	234	45	)	)	PUNCT
cana-1178	235	1	def	def	VERB
cana-1178	235	2	energyofmatrix(a	energyofmatrix(a	NOUN
cana-1178	235	3	):	):	PUNCT
cana-1178	235	4	eigvals	eigval	NOUN
cana-1178	235	5	=	=	SYM
cana-1178	235	6	np.linalg.eigvals(a	np.linalg.eigvals(a	PROPN
cana-1178	235	7	)	)	PUNCT
cana-1178	235	8	#	#	NOUN
cana-1178	235	9	compute	compute	NOUN
cana-1178	235	10	eigenvalues	eigenvalue	NOUN
cana-1178	235	11	of	of	ADP
cana-1178	235	12	a	a	DET
cana-1178	235	13	sumabseigvals	sumabseigval	NOUN
cana-1178	235	14	=	=	SYM
cana-1178	235	15	sum(abs(eigvals	sum(abs(eigval	NOUN
cana-1178	235	16	)	)	PUNCT
cana-1178	235	17	)	)	PUNCT
cana-1178	235	18	#	#	NOUN
cana-1178	235	19	compute	compute	NOUN
cana-1178	235	20	sum	sum	NOUN
cana-1178	235	21	of	of	ADP
cana-1178	235	22	absolute	absolute	ADJ
cana-1178	235	23	values	value	NOUN
cana-1178	235	24	of	of	ADP
cana-1178	235	25	eigenvalues	eigenvalue	NOUN
cana-1178	235	26	return	return	NOUN
cana-1178	235	27	sumabseigvals	sumabseigval	NOUN
cana-1178	235	28	def	def	VERB
cana-1178	235	29	energyofgraph(g	energyofgraph(g	NOUN
cana-1178	235	30	):	):	PUNCT
cana-1178	235	31	return	return	NOUN
cana-1178	235	32	energyofmatrix(nx.adjacency_matrix(g).todense	energyofmatrix(nx.adjacency_matrix(g).todense	NOUN
cana-1178	235	33	(	(	PUNCT
cana-1178	235	34	)	)	PUNCT
cana-1178	235	35	)	)	PUNCT
cana-1178	236	1	n	n	NOUN
cana-1178	236	2	=	=	PUNCT
cana-1178	236	3	int(input("enter	int(input("enter	VERB
cana-1178	236	4	the	the	DET
cana-1178	236	5	value	value	NOUN
cana-1178	236	6	of	of	ADP
cana-1178	236	7	n	n	CCONJ
cana-1178	236	8	:	:	PUNCT
cana-1178	236	9	"	"	PUNCT
cana-1178	236	10	)	)	PUNCT
cana-1178	236	11	)	)	PUNCT
cana-1178	237	1	n_nat	n_nat	NOUN
cana-1178	238	1	=	=	SYM
cana-1178	238	2	set(range(1	set(range(1	X
cana-1178	238	3	,	,	PUNCT
cana-1178	238	4	n	n	PROPN
cana-1178	238	5	+	+	NOUN
cana-1178	238	6	1	1	NUM
cana-1178	238	7	)	)	PUNCT
cana-1178	238	8	)	)	PUNCT
cana-1178	238	9	powerset_n_nat	powerset_n_nat	NOUN
cana-1178	239	1	=	=	SYM
cana-1178	239	2	powerset(n_nat	powerset(n_nat	PROPN
cana-1178	239	3	)	)	PUNCT
cana-1178	239	4	powerset_n_nat_min_1	powerset_n_nat_min_1	NOUN
cana-1178	239	5	=	=	SYM
cana-1178	239	6	powerset(range(2	powerset(range(2	NOUN
cana-1178	239	7	,	,	PUNCT
cana-1178	239	8	n	n	PROPN
cana-1178	239	9	+	+	NOUN
cana-1178	239	10	1	1	NUM
cana-1178	239	11	)	)	PUNCT
cana-1178	239	12	)	)	PUNCT
cana-1178	240	1	i_d_set	i_d_set	VERB
cana-1178	241	1	=	=	PRON
cana-1178	242	1	set	set	NOUN
cana-1178	242	2	(	(	PUNCT
cana-1178	242	3	#	#	NOUN
cana-1178	242	4	independent	independent	ADJ
cana-1178	242	5	dominating	dominating	NOUN
cana-1178	242	6	set	set	NOUN
cana-1178	242	7	(	(	PUNCT
cana-1178	242	8	1	1	NUM
cana-1178	242	9	,	,	PUNCT
cana-1178	242	10	x	x	NOUN
cana-1178	242	11	)	)	PUNCT
cana-1178	242	12	for	for	ADP
cana-1178	242	13	x	x	SYM
cana-1178	242	14	in	in	ADP
cana-1178	242	15	powerset_n_nat_min_1.difference(set(range(2	powerset_n_nat_min_1.difference(set(range(2	NOUN
cana-1178	242	16	,	,	PUNCT
cana-1178	242	17	n	n	PROPN
cana-1178	242	18	+	+	NOUN
cana-1178	242	19	1	1	NUM
cana-1178	242	20	)	)	PUNCT
cana-1178	242	21	)	)	PUNCT
cana-1178	242	22	)	)	PUNCT
cana-1178	242	23	)	)	PUNCT
cana-1178	242	24	vertices	vertice	VERB
cana-1178	242	25	=	=	X
cana-1178	242	26	set	set	NOUN
cana-1178	242	27	(	(	PUNCT
cana-1178	242	28	elem	elem	NOUN
cana-1178	242	29	for	for	ADP
cana-1178	242	30	elem	elem	NOUN
cana-1178	242	31	in	in	ADP
cana-1178	242	32	powerset(n_nat	powerset(n_nat	NOUN
cana-1178	242	33	)	)	PUNCT
cana-1178	242	34	if	if	SCONJ
cana-1178	242	35	(	(	PUNCT
cana-1178	242	36	elem	elem	NOUN
cana-1178	242	37	not	not	PART
cana-1178	242	38	in	in	ADP
cana-1178	242	39	[	[	X
cana-1178	242	40	tuple	tuple	NOUN
cana-1178	242	41	(	(	PUNCT
cana-1178	242	42	)	)	PUNCT
cana-1178	242	43	,	,	PUNCT
cana-1178	242	44	tuple(n_nat	tuple(n_nat	NOUN
cana-1178	242	45	)	)	PUNCT
cana-1178	242	46	]	]	PUNCT
cana-1178	242	47	)	)	PUNCT
cana-1178	242	48	)	)	PUNCT
cana-1178	242	49	edges	edge	NOUN
cana-1178	242	50	=	=	PUNCT
cana-1178	242	51	set	set	NOUN
cana-1178	242	52	(	(	PUNCT
cana-1178	242	53	communications	communication	NOUN
cana-1178	242	54	on	on	ADP
cana-1178	242	55	applied	apply	VERB
cana-1178	242	56	nonlinear	nonlinear	ADJ
cana-1178	242	57	analysis	analysis	NOUN
cana-1178	242	58	issn	issn	NOUN
cana-1178	242	59	:	:	PUNCT
cana-1178	242	60	1074	1074	NUM
cana-1178	242	61	-	-	PUNCT
cana-1178	242	62	133x	133x	NUM
cana-1178	242	63	vol	vol	NOUN
cana-1178	242	64	31	31	NUM
cana-1178	242	65	no	no	NOUN
cana-1178	242	66	.	.	PUNCT
cana-1178	243	1	6s	6s	NUM
cana-1178	243	2	(	(	PUNCT
cana-1178	243	3	2024	2024	NUM
cana-1178	243	4	)	)	PUNCT
cana-1178	243	5	203	203	NUM
cana-1178	243	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	243	7	(	(	PUNCT
cana-1178	243	8	vert1	vert1	PROPN
cana-1178	243	9	,	,	PUNCT
cana-1178	243	10	vert2	vert2	PROPN
cana-1178	243	11	)	)	PUNCT
cana-1178	243	12	for	for	ADP
cana-1178	243	13	vert1	vert1	NOUN
cana-1178	243	14	in	in	ADP
cana-1178	243	15	vertices	vertex	NOUN
cana-1178	243	16	for	for	ADP
cana-1178	243	17	vert2	vert2	NOUN
cana-1178	243	18	in	in	ADP
cana-1178	243	19	vertices	vertex	NOUN
cana-1178	243	20	if	if	SCONJ
cana-1178	243	21	(	(	PUNCT
cana-1178	243	22	(	(	PUNCT
cana-1178	243	23	len(set(vert1).intersection(set(vert2	len(set(vert1).intersection(set(vert2	NOUN
cana-1178	243	24	)	)	PUNCT
cana-1178	243	25	)	)	PUNCT
cana-1178	243	26	)	)	PUNCT
cana-1178	244	1	=	=	PUNCT
cana-1178	244	2	=	=	SYM
cana-1178	244	3	0	0	NUM
cana-1178	244	4	)	)	PUNCT
cana-1178	244	5	or	or	CCONJ
cana-1178	244	6	(	(	PUNCT
cana-1178	244	7	vert1	vert1	NOUN
cana-1178	244	8	=	=	SYM
cana-1178	244	9	=	=	NOUN
cana-1178	244	10	vert2	vert2	PROPN
cana-1178	244	11	and	and	CCONJ
cana-1178	244	12	(	(	PUNCT
cana-1178	244	13	vert1	vert1	ADV
cana-1178	244	14	in	in	ADP
cana-1178	244	15	i_d_set	i_d_set	NOUN
cana-1178	244	16	)	)	PUNCT
cana-1178	244	17	)	)	PUNCT
cana-1178	244	18	)	)	PUNCT
cana-1178	244	19	)	)	PUNCT
cana-1178	245	1	#	#	NOUN
cana-1178	245	2	%	%	NOUN
cana-1178	245	3	%	%	NOUN
cana-1178	245	4	graph	graph	NOUN
cana-1178	245	5	=	=	SYM
cana-1178	245	6	nx.graph	nx.graph	X
cana-1178	245	7	(	(	PUNCT
cana-1178	245	8	)	)	PUNCT
cana-1178	245	9	graph.add_nodes_from(vertices	graph.add_nodes_from(vertice	NOUN
cana-1178	245	10	)	)	PUNCT
cana-1178	245	11	graph.add_edges_from(edges	graph.add_edges_from(edge	NOUN
cana-1178	245	12	)	)	PUNCT
cana-1178	245	13	graphdegrees	graphdegree	NOUN
cana-1178	245	14	=	=	SYM
cana-1178	245	15	{	{	PUNCT
cana-1178	245	16	node	node	NOUN
cana-1178	245	17	:	:	PUNCT
cana-1178	245	18	val	val	NOUN
cana-1178	245	19	for	for	ADP
cana-1178	245	20	(	(	PUNCT
cana-1178	245	21	node	node	NOUN
cana-1178	245	22	,	,	PUNCT
cana-1178	245	23	val	val	NOUN
cana-1178	245	24	)	)	PUNCT
cana-1178	245	25	in	in	ADP
cana-1178	245	26	graph.degree	graph.degree	PROPN
cana-1178	245	27	(	(	PUNCT
cana-1178	245	28	)	)	PUNCT
cana-1178	245	29	}	}	PUNCT
cana-1178	245	30	siedel_edges	siedel_edge	NOUN
cana-1178	245	31	=	=	PUNCT
cana-1178	246	1	[	[	X
cana-1178	246	2	]	]	X
cana-1178	246	3	#	#	NOUN
cana-1178	246	4	stores	store	NOUN
cana-1178	246	5	edges	edge	NOUN
cana-1178	246	6	as	as	ADP
cana-1178	246	7	randic_edges	randic_edge	NOUN
cana-1178	246	8	=	=	PUNCT
cana-1178	247	1	[	[	X
cana-1178	247	2	]	]	X
cana-1178	247	3	#	#	NOUN
cana-1178	247	4	(	(	PUNCT
cana-1178	247	5	v1,v2,weight	v1,v2,weight	ADJ
cana-1178	247	6	)	)	PUNCT
cana-1178	247	7	triples	triple	NOUN
cana-1178	247	8	for	for	ADP
cana-1178	247	9	vert1	vert1	NOUN
cana-1178	247	10	in	in	ADP
cana-1178	247	11	vertices	vertex	NOUN
cana-1178	247	12	:	:	PUNCT
cana-1178	247	13	for	for	ADP
cana-1178	247	14	vert2	vert2	NOUN
cana-1178	247	15	in	in	ADP
cana-1178	247	16	vertices	vertex	NOUN
cana-1178	247	17	:	:	PUNCT
cana-1178	247	18	if	if	SCONJ
cana-1178	247	19	set(vert1	set(vert1	ADV
cana-1178	247	20	)	)	PUNCT
cana-1178	247	21	=	=	NOUN
cana-1178	247	22	=	=	NOUN
cana-1178	247	23	set(vert2	set(vert2	NOUN
cana-1178	247	24	):	):	PUNCT
cana-1178	247	25	if	if	SCONJ
cana-1178	247	26	len(vert1	len(vert1	NOUN
cana-1178	247	27	)	)	PUNCT
cana-1178	247	28	=	=	SYM
cana-1178	247	29	=	=	SYM
cana-1178	247	30	1	1	NUM
cana-1178	247	31	:	:	PUNCT
cana-1178	247	32	pass	pass	VERB
cana-1178	247	33	elif	elif	PROPN
cana-1178	247	34	len(set(vert1).intersection(set(vert2	len(set(vert1).intersection(set(vert2	PROPN
cana-1178	247	35	)	)	PUNCT
cana-1178	247	36	)	)	PUNCT
cana-1178	247	37	)	)	PUNCT
cana-1178	248	1	=	=	PUNCT
cana-1178	248	2	=	=	SYM
cana-1178	248	3	0	0	NUM
cana-1178	248	4	:	:	PUNCT
cana-1178	248	5	siedel_edges.append((vert1	siedel_edges.append((vert1	NOUN
cana-1178	248	6	,	,	PUNCT
cana-1178	248	7	vert2	vert2	NOUN
cana-1178	248	8	,	,	PUNCT
cana-1178	248	9	-1	-1	NOUN
cana-1178	248	10	)	)	PUNCT
cana-1178	248	11	)	)	PUNCT
cana-1178	249	1	#	#	NOUN
cana-1178	249	2	min_dom.append((vert1,vert2,1	min_dom.append((vert1,vert2,1	NOUN
cana-1178	249	3	)	)	PUNCT
cana-1178	249	4	)	)	PUNCT
cana-1178	250	1	randic_edges.append	randic_edges.append	PROPN
cana-1178	250	2	(	(	PUNCT
cana-1178	250	3	(	(	PUNCT
cana-1178	250	4	vert1	vert1	ADV
cana-1178	250	5	,	,	PUNCT
cana-1178	250	6	vert2	vert2	PROPN
cana-1178	250	7	,	,	PUNCT
cana-1178	250	8	1	1	NUM
cana-1178	250	9	/	/	SYM
cana-1178	250	10	np.sqrt((graphdegrees[vert1	np.sqrt((graphdegrees[vert1	NOUN
cana-1178	250	11	]	]	X
cana-1178	250	12	*	*	PUNCT
cana-1178	250	13	graphdegrees[vert2	graphdegrees[vert2	PROPN
cana-1178	250	14	]	]	PUNCT
cana-1178	250	15	)	)	PUNCT
cana-1178	250	16	)	)	PUNCT
cana-1178	250	17	)	)	PUNCT
cana-1178	250	18	)	)	PUNCT
cana-1178	251	1	elif	elif	PROPN
cana-1178	251	2	len(set(vert1).intersection(set(vert2	len(set(vert1).intersection(set(vert2	PROPN
cana-1178	251	3	)	)	PUNCT
cana-1178	251	4	)	)	PUNCT
cana-1178	251	5	)	)	PUNCT
cana-1178	251	6	!	!	PUNCT
cana-1178	252	1	=	=	PUNCT
cana-1178	253	1	0	0	NUM
cana-1178	253	2	:	:	PUNCT
cana-1178	253	3	#	#	NOUN
cana-1178	253	4	min_dom.append((vert1,vert2,0	min_dom.append((vert1,vert2,0	PROPN
cana-1178	253	5	)	)	PUNCT
cana-1178	253	6	)	)	PUNCT
cana-1178	253	7	siedel_edges.append((vert1	siedel_edges.append((vert1	NOUN
cana-1178	253	8	,	,	PUNCT
cana-1178	253	9	vert2	vert2	NOUN
cana-1178	253	10	,	,	PUNCT
cana-1178	253	11	1	1	NUM
cana-1178	253	12	)	)	PUNCT
cana-1178	253	13	)	)	PUNCT
cana-1178	253	14	randic_edges.append((vert1	randic_edges.append((vert1	PROPN
cana-1178	253	15	,	,	PUNCT
cana-1178	253	16	vert2	vert2	PROPN
cana-1178	253	17	,	,	PUNCT
cana-1178	253	18	0	0	NUM
cana-1178	253	19	)	)	PUNCT
cana-1178	253	20	)	)	PUNCT
cana-1178	254	1	seidel_graph	seidel_graph	PROPN
cana-1178	254	2	=	=	SYM
cana-1178	254	3	nx.graph	nx.graph	X
cana-1178	254	4	(	(	PUNCT
cana-1178	254	5	)	)	PUNCT
cana-1178	254	6	seidel_graph.add_weighted_edges_from(siedel_edges	seidel_graph.add_weighted_edges_from(siedel_edge	NOUN
cana-1178	254	7	)	)	PUNCT
cana-1178	254	8	randic_graph	randic_graph	NOUN
cana-1178	254	9	=	=	SYM
cana-1178	254	10	nx.graph	nx.graph	X
cana-1178	254	11	(	(	PUNCT
cana-1178	254	12	)	)	PUNCT
cana-1178	254	13	randic_graph.add_weighted_edges_from(randic_edges	randic_graph.add_weighted_edges_from(randic_edge	NOUN
cana-1178	254	14	)	)	PUNCT
cana-1178	254	15	min_dom_adj	min_dom_adj	NOUN
cana-1178	254	16	=	=	SYM
cana-1178	254	17	nx.adjacency_matrix(graph).todense	nx.adjacency_matrix(graph).todense	NOUN
cana-1178	254	18	(	(	PUNCT
cana-1178	254	19	)	)	PUNCT
cana-1178	254	20	#	#	NOUN
cana-1178	254	21	grc	grc	NOUN
cana-1178	254	22	t	t	PROPN
cana-1178	254	23	for	for	ADP
cana-1178	254	24	i	i	PRON
cana-1178	254	25	in	in	ADP
cana-1178	254	26	range(len(min_dom_adj	range(len(min_dom_adj	NOUN
cana-1178	254	27	)	)	PUNCT
cana-1178	254	28	):	):	PUNCT
cana-1178	254	29	for	for	SCONJ
cana-1178	254	30	j	j	PROPN
cana-1178	254	31	in	in	ADP
cana-1178	254	32	range(len(min_dom_adj[0	range(len(min_dom_adj[0	PROPN
cana-1178	254	33	]	]	PUNCT
cana-1178	254	34	)	)	PUNCT
cana-1178	254	35	):	):	PUNCT
cana-1178	254	36	if	if	SCONJ
cana-1178	254	37	i==j	i==j	NOUN
cana-1178	254	38	:	:	PUNCT
cana-1178	254	39	if	if	SCONJ
cana-1178	254	40	len(list(vertices)[i	len(list(vertices)[i	PROPN
cana-1178	254	41	]	]	X
cana-1178	254	42	)	)	PUNCT
cana-1178	255	1	=	=	SYM
cana-1178	255	2	=	=	SYM
cana-1178	255	3	1	1	NUM
cana-1178	255	4	:	:	PUNCT
cana-1178	255	5	min_dom_adj[i][i	min_dom_adj[i][i	X
cana-1178	255	6	]	]	X
cana-1178	255	7	=	=	SYM
cana-1178	255	8	1	1	NUM
cana-1178	255	9	communications	communication	NOUN
cana-1178	255	10	on	on	ADP
cana-1178	255	11	applied	apply	VERB
cana-1178	255	12	nonlinear	nonlinear	ADJ
cana-1178	255	13	analysis	analysis	NOUN
cana-1178	255	14	issn	issn	NOUN
cana-1178	255	15	:	:	PUNCT
cana-1178	255	16	1074	1074	NUM
cana-1178	255	17	-	-	PUNCT
cana-1178	255	18	133x	133x	NUM
cana-1178	255	19	vol	vol	NOUN
cana-1178	255	20	31	31	NUM
cana-1178	255	21	no	no	NOUN
cana-1178	255	22	.	.	PUNCT
cana-1178	256	1	6s	6s	NUM
cana-1178	256	2	(	(	PUNCT
cana-1178	256	3	2024	2024	NUM
cana-1178	256	4	)	)	PUNCT
cana-1178	256	5	204	204	NUM
cana-1178	256	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	256	7	min_dom_graph	min_dom_graph	NOUN
cana-1178	256	8	=	=	SYM
cana-1178	256	9	nx.relabel_nodes(nx.from_numpy_array(min_dom_adj),{i	nx.relabel_nodes(nx.from_numpy_array(min_dom_adj),{i	NOUN
cana-1178	256	10	:	:	PUNCT
cana-1178	256	11	vert	vert	NOUN
cana-1178	256	12	for	for	ADP
cana-1178	256	13	i	i	PROPN
cana-1178	256	14	,	,	PUNCT
cana-1178	256	15	vert	vert	X
cana-1178	256	16	in	in	ADP
cana-1178	256	17	enumerate(vertices	enumerate(vertice	NOUN
cana-1178	256	18	)	)	PUNCT
cana-1178	256	19	}	}	PUNCT
cana-1178	256	20	)	)	PUNCT
cana-1178	256	21	#	#	NOUN
cana-1178	256	22	mapping	mapping	NOUN
cana-1178	256	23	=	=	SYM
cana-1178	256	24	{	{	PUNCT
cana-1178	256	25	i	i	PRON
cana-1178	256	26	:	:	PUNCT
cana-1178	256	27	vert	vert	NOUN
cana-1178	256	28	for	for	ADP
cana-1178	256	29	i	i	PROPN
cana-1178	256	30	,	,	PUNCT
cana-1178	256	31	vert	vert	X
cana-1178	256	32	in	in	ADP
cana-1178	256	33	enumerate(vertices	enumerate(vertice	NOUN
cana-1178	256	34	)	)	PUNCT
cana-1178	256	35	}	}	PUNCT
cana-1178	256	36	energy	energy	NOUN
cana-1178	256	37	calculation	calculation	NOUN
cana-1178	256	38	code	code	NOUN
cana-1178	256	39	functions	function	NOUN
cana-1178	256	40	were	be	AUX
cana-1178	256	41	written	write	VERB
cana-1178	256	42	that	that	PRON
cana-1178	256	43	would	would	AUX
cana-1178	256	44	ease	ease	VERB
cana-1178	256	45	the	the	DET
cana-1178	256	46	process	process	NOUN
cana-1178	256	47	of	of	ADP
cana-1178	256	48	calculating	calculate	VERB
cana-1178	256	49	the	the	DET
cana-1178	256	50	graph	graph	NOUN
cana-1178	256	51	energies	energy	NOUN
cana-1178	256	52	for	for	ADP
cana-1178	256	53	any	any	DET
cana-1178	256	54	graph	graph	NOUN
cana-1178	256	55	:	:	PUNCT
cana-1178	256	56	def	def	ADJ
cana-1178	256	57	energyofmatrix(a	energyofmatrix(a	NOUN
cana-1178	256	58	):	):	PUNCT
cana-1178	256	59	eigvals	eigval	NOUN
cana-1178	256	60	=	=	SYM
cana-1178	256	61	np.linalg.eigvals(a	np.linalg.eigvals(a	PROPN
cana-1178	256	62	)	)	PUNCT
cana-1178	256	63	#	#	NOUN
cana-1178	256	64	compute	compute	NOUN
cana-1178	256	65	eigenvalues	eigenvalue	NOUN
cana-1178	256	66	of	of	ADP
cana-1178	256	67	a	a	DET
cana-1178	256	68	sumabseigvals	sumabseigval	NOUN
cana-1178	256	69	=	=	SYM
cana-1178	256	70	sum(abs(eigvals	sum(abs(eigval	NOUN
cana-1178	256	71	)	)	PUNCT
cana-1178	256	72	)	)	PUNCT
cana-1178	256	73	#	#	NOUN
cana-1178	256	74	compute	compute	NOUN
cana-1178	256	75	sum	sum	NOUN
cana-1178	256	76	of	of	ADP
cana-1178	256	77	absolute	absolute	ADJ
cana-1178	256	78	values	value	NOUN
cana-1178	256	79	of	of	ADP
cana-1178	256	80	eigenvalues	eigenvalue	NOUN
cana-1178	256	81	return	return	NOUN
cana-1178	256	82	sumabseigvals	sumabseigval	NOUN
cana-1178	256	83	def	def	VERB
cana-1178	256	84	energyofgraph(g	energyofgraph(g	NOUN
cana-1178	256	85	):	):	PUNCT
cana-1178	256	86	return	return	NOUN
cana-1178	256	87	energyofmatrix(nx.adjacency_matrix(g).todense	energyofmatrix(nx.adjacency_matrix(g).todense	NOUN
cana-1178	256	88	(	(	PUNCT
cana-1178	256	89	)	)	PUNCT
cana-1178	256	90	)	)	PUNCT
cana-1178	256	91	generated	generate	VERB
cana-1178	256	92	graphs	graph	NOUN
cana-1178	256	93	and	and	CCONJ
cana-1178	256	94	charts	chart	NOUN
cana-1178	256	95	fig	fig	NOUN
cana-1178	256	96	2	2	NUM
cana-1178	256	97	:	:	PUNCT
cana-1178	256	98	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	256	99	)	)	PUNCT
cana-1178	256	100	graph	graph	NOUN
cana-1178	256	101	when	when	SCONJ
cana-1178	256	102	𝑛	𝑛	PROPN
cana-1178	256	103	=	=	SYM
cana-1178	256	104	4	4	NUM
cana-1178	256	105	fig	fig	NOUN
cana-1178	256	106	3	3	NUM
cana-1178	256	107	:	:	PUNCT
cana-1178	256	108	the	the	DET
cana-1178	256	109	siedel	siedel	NOUN
cana-1178	256	110	graph	graph	NOUN
cana-1178	256	111	communications	communication	NOUN
cana-1178	256	112	on	on	ADP
cana-1178	256	113	applied	apply	VERB
cana-1178	256	114	nonlinear	nonlinear	ADJ
cana-1178	256	115	analysis	analysis	NOUN
cana-1178	256	116	issn	issn	NOUN
cana-1178	256	117	:	:	PUNCT
cana-1178	256	118	1074	1074	NUM
cana-1178	256	119	-	-	PUNCT
cana-1178	256	120	133x	133x	NUM
cana-1178	256	121	vol	vol	NOUN
cana-1178	256	122	31	31	NUM
cana-1178	256	123	no	no	NOUN
cana-1178	256	124	.	.	PUNCT
cana-1178	257	1	6s	6s	NUM
cana-1178	257	2	(	(	PUNCT
cana-1178	257	3	2024	2024	NUM
cana-1178	257	4	)	)	PUNCT
cana-1178	257	5	205	205	NUM
cana-1178	257	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	257	7	fig	fig	NOUN
cana-1178	257	8	4	4	NUM
cana-1178	257	9	:	:	PUNCT
cana-1178	257	10	the	the	DET
cana-1178	257	11	randic	randic	ADJ
cana-1178	257	12	graph	graph	NOUN
cana-1178	257	13	fig	fig	NOUN
cana-1178	257	14	5	5	NUM
cana-1178	257	15	:	:	PUNCT
cana-1178	257	16	minimum	minimum	ADJ
cana-1178	257	17	dominating	dominating	NOUN
cana-1178	257	18	graph	graph	NOUN
cana-1178	257	19	fig	fig	NOUN
cana-1178	257	20	6	6	NUM
cana-1178	257	21	:	:	PUNCT
cana-1178	257	22	a	a	DET
cana-1178	257	23	comparison	comparison	NOUN
cana-1178	257	24	of	of	ADP
cana-1178	257	25	the	the	DET
cana-1178	257	26	adjacency	adjacency	NOUN
cana-1178	257	27	energy	energy	NOUN
cana-1178	257	28	,	,	PUNCT
cana-1178	257	29	siedel	siedel	NOUN
cana-1178	257	30	energy	energy	NOUN
cana-1178	257	31	,	,	PUNCT
cana-1178	257	32	randic	randic	ADJ
cana-1178	257	33	energy	energy	NOUN
cana-1178	257	34	and	and	CCONJ
cana-1178	257	35	minimum	minimum	NOUN
cana-1178	257	36	dominating	dominating	NOUN
cana-1178	257	37	energy	energy	NOUN
cana-1178	257	38	for	for	ADP
cana-1178	257	39	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	257	40	)	)	PUNCT
cana-1178	257	41	graph	graph	NOUN
cana-1178	257	42	for	for	ADP
cana-1178	257	43	varying	vary	VERB
cana-1178	257	44	values	value	NOUN
cana-1178	257	45	of	of	ADP
cana-1178	257	46	𝑛.	𝑛.	NOUN
cana-1178	257	47	5	5	NUM
cana-1178	257	48	.	.	PUNCT
cana-1178	257	49	conclusion	conclusion	NOUN
cana-1178	257	50	in	in	ADP
cana-1178	257	51	this	this	DET
cana-1178	257	52	study	study	NOUN
cana-1178	257	53	,	,	PUNCT
cana-1178	257	54	the	the	DET
cana-1178	257	55	rough	rough	ADJ
cana-1178	257	56	complemented	complemented	ADJ
cana-1178	257	57	graph	graph	NOUN
cana-1178	257	58	of	of	ADP
cana-1178	257	59	the	the	DET
cana-1178	257	60	rough	rough	ADJ
cana-1178	257	61	semiring	semiring	NOUN
cana-1178	257	62	is	be	AUX
cana-1178	257	63	defined	define	VERB
cana-1178	257	64	using	use	VERB
cana-1178	257	65	the	the	DET
cana-1178	257	66	equivalence	equivalence	NOUN
cana-1178	257	67	classes	class	NOUN
cana-1178	257	68	.	.	PUNCT
cana-1178	258	1	also	also	ADV
cana-1178	258	2	the	the	DET
cana-1178	258	3	maximal	maximal	ADJ
cana-1178	258	4	independent	independent	ADJ
cana-1178	258	5	and	and	CCONJ
cana-1178	258	6	dominating	dominating	NOUN
cana-1178	258	7	set	set	NOUN
cana-1178	258	8	of	of	ADP
cana-1178	258	9	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	258	10	)	)	PUNCT
cana-1178	258	11	graph	graph	NOUN
cana-1178	258	12	is	be	AUX
cana-1178	258	13	established	establish	VERB
cana-1178	258	14	.	.	PUNCT
cana-1178	259	1	additionally	additionally	ADV
cana-1178	259	2	,	,	PUNCT
cana-1178	259	3	the	the	DET
cana-1178	259	4	minimum	minimum	ADJ
cana-1178	259	5	dominating	dominating	NOUN
cana-1178	259	6	,	,	PUNCT
cana-1178	259	7	siedel	siedel	NOUN
cana-1178	259	8	and	and	CCONJ
cana-1178	259	9	randic	randic	ADJ
cana-1178	259	10	energies	energy	NOUN
cana-1178	259	11	of	of	ADP
cana-1178	259	12	the	the	DET
cana-1178	259	13	𝐺𝑅𝐶(𝑇	𝐺𝑅𝐶(𝑇	NOUN
cana-1178	259	14	)	)	PUNCT
cana-1178	259	15	graph	graph	NOUN
cana-1178	259	16	are	be	AUX
cana-1178	259	17	defined	define	VERB
cana-1178	259	18	and	and	CCONJ
cana-1178	259	19	the	the	DET
cana-1178	259	20	lower	low	ADJ
cana-1178	259	21	and	and	CCONJ
cana-1178	259	22	upper	upper	ADJ
cana-1178	259	23	bounds	bound	NOUN
cana-1178	259	24	are	be	AUX
cana-1178	259	25	derived	derive	VERB
cana-1178	259	26	.	.	PUNCT
cana-1178	260	1	all	all	DET
cana-1178	260	2	the	the	DET
cana-1178	260	3	above	above	ADJ
cana-1178	260	4	mentioned	mention	VERB
cana-1178	260	5	energies	energy	NOUN
cana-1178	260	6	can	can	AUX
cana-1178	260	7	be	be	AUX
cana-1178	260	8	found	find	VERB
cana-1178	260	9	using	use	VERB
cana-1178	260	10	python	python	NOUN
cana-1178	260	11	and	and	CCONJ
cana-1178	260	12	several	several	ADJ
cana-1178	260	13	values	value	NOUN
cana-1178	260	14	of	of	ADP
cana-1178	260	15	𝑛	𝑛	PROPN
cana-1178	260	16	are	be	AUX
cana-1178	260	17	compared	compare	VERB
cana-1178	260	18	.	.	PUNCT
cana-1178	261	1	each	each	DET
cana-1178	261	2	notion	notion	NOUN
cana-1178	261	3	is	be	AUX
cana-1178	261	4	illustrated	illustrate	VERB
cana-1178	261	5	and	and	CCONJ
cana-1178	261	6	supported	support	VERB
cana-1178	261	7	by	by	ADP
cana-1178	261	8	an	an	DET
cana-1178	261	9	example	example	NOUN
cana-1178	261	10	.	.	PUNCT
cana-1178	262	1	communications	communication	NOUN
cana-1178	262	2	on	on	ADP
cana-1178	262	3	applied	apply	VERB
cana-1178	262	4	nonlinear	nonlinear	ADJ
cana-1178	262	5	analysis	analysis	NOUN
cana-1178	262	6	issn	issn	NOUN
cana-1178	262	7	:	:	PUNCT
cana-1178	262	8	1074	1074	NUM
cana-1178	262	9	-	-	PUNCT
cana-1178	262	10	133x	133x	NUM
cana-1178	262	11	vol	vol	NOUN
cana-1178	262	12	31	31	NUM
cana-1178	262	13	no	no	NOUN
cana-1178	262	14	.	.	PUNCT
cana-1178	263	1	6s	6s	NUM
cana-1178	263	2	(	(	PUNCT
cana-1178	263	3	2024	2024	NUM
cana-1178	263	4	)	)	PUNCT
cana-1178	263	5	206	206	NUM
cana-1178	264	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1178	264	2	conflicts	conflict	NOUN
cana-1178	264	3	of	of	ADP
cana-1178	264	4	interest	interest	NOUN
cana-1178	264	5	:	:	PUNCT
cana-1178	264	6	the	the	DET
cana-1178	264	7	authors	author	NOUN
cana-1178	264	8	declares	declare	VERB
cana-1178	264	9	that	that	SCONJ
cana-1178	264	10	there	there	PRON
cana-1178	264	11	is	be	VERB
cana-1178	264	12	no	no	DET
cana-1178	264	13	conflict	conflict	NOUN
cana-1178	264	14	of	of	ADP
cana-1178	264	15	interest	interest	NOUN
cana-1178	264	16	regarding	regard	VERB
cana-1178	264	17	the	the	DET
cana-1178	264	18	publication	publication	NOUN
cana-1178	264	19	of	of	ADP
cana-1178	264	20	this	this	DET
cana-1178	264	21	article	article	NOUN
cana-1178	264	22	.	.	PUNCT
cana-1178	265	1	funding	funding	NOUN
cana-1178	265	2	:	:	PUNCT
cana-1178	265	3	this	this	DET
cana-1178	265	4	research	research	NOUN
cana-1178	265	5	did	do	AUX
cana-1178	265	6	not	not	PART
cana-1178	265	7	receive	receive	VERB
cana-1178	265	8	any	any	DET
cana-1178	265	9	specific	specific	ADJ
cana-1178	265	10	grant	grant	NOUN
cana-1178	265	11	from	from	ADP
cana-1178	265	12	funding	fund	VERB
cana-1178	265	13	agencies	agency	NOUN
cana-1178	265	14	in	in	ADP
cana-1178	265	15	the	the	DET
cana-1178	265	16	public	public	ADJ
cana-1178	265	17	,	,	PUNCT
cana-1178	265	18	commercial	commercial	ADJ
cana-1178	265	19	,	,	PUNCT
cana-1178	265	20	or	or	CCONJ
cana-1178	265	21	not	not	PART
cana-1178	265	22	-	-	PUNCT
cana-1178	265	23	for	for	ADP
cana-1178	265	24	-	-	PUNCT
cana-1178	265	25	profit	profit	NOUN
cana-1178	265	26	sectors	sector	NOUN
cana-1178	265	27	acknowledgement	acknowledgement	NOUN
cana-1178	265	28	:	:	PUNCT
cana-1178	265	29	the	the	DET
cana-1178	265	30	authors	author	NOUN
cana-1178	265	31	would	would	AUX
cana-1178	265	32	express	express	VERB
cana-1178	265	33	their	their	PRON
cana-1178	265	34	sincere	sincere	ADJ
cana-1178	265	35	gratitude	gratitude	NOUN
cana-1178	265	36	to	to	ADP
cana-1178	265	37	the	the	DET
cana-1178	265	38	management	management	NOUN
cana-1178	265	39	,	,	PUNCT
cana-1178	265	40	and	and	CCONJ
cana-1178	265	41	the	the	DET
cana-1178	265	42	principal	principal	NOUN
cana-1178	265	43	,	,	PUNCT
cana-1178	265	44	of	of	ADP
cana-1178	265	45	sri	sri	PROPN
cana-1178	265	46	sivasubramaniya	sivasubramaniya	PROPN
cana-1178	265	47	nadar	nadar	PROPN
cana-1178	265	48	college	college	PROPN
cana-1178	265	49	of	of	ADP
cana-1178	265	50	engineering	engineering	NOUN
cana-1178	265	51	for	for	ADP
cana-1178	265	52	their	their	PRON
cana-1178	265	53	constant	constant	ADJ
cana-1178	265	54	support	support	NOUN
cana-1178	265	55	.	.	PUNCT
cana-1178	266	1	references	reference	NOUN
cana-1178	266	2	:	:	PUNCT
cana-1178	267	1	[	[	X
cana-1178	267	2	1	1	NUM
cana-1178	267	3	]	]	X
cana-1178	267	4	i.	i.	PROPN
cana-1178	267	5	gutman	gutman	PROPN
cana-1178	267	6	,	,	PUNCT
cana-1178	267	7	the	the	DET
cana-1178	267	8	energy	energy	NOUN
cana-1178	267	9	of	of	ADP
cana-1178	267	10	a	a	DET
cana-1178	267	11	graph	graph	NOUN
cana-1178	267	12	,	,	PUNCT
cana-1178	267	13	ber	ber	PROPN
cana-1178	267	14	.	.	PUNCT
cana-1178	267	15	math	math	NOUN
cana-1178	267	16	.	.	PUNCT
cana-1178	268	1	stat	stat	PROPN
cana-1178	268	2	.	.	PUNCT
cana-1178	269	1	sekt	sekt	PROPN
cana-1178	269	2	.	.	PUNCT
cana-1178	270	1	forschungszentrum	forschungszentrum	PROPN
cana-1178	270	2	graz	graz	PROPN
cana-1178	270	3	103	103	NUM
cana-1178	270	4	,	,	PUNCT
cana-1178	270	5	(	(	PUNCT
cana-1178	270	6	1978	1978	NUM
cana-1178	270	7	)	)	PUNCT
cana-1178	270	8	1	1	NUM
cana-1178	270	9	-	-	SYM
cana-1178	270	10	22	22	NUM
cana-1178	270	11	.	.	PUNCT
cana-1178	271	1	[	[	X
cana-1178	271	2	2	2	NUM
cana-1178	271	3	]	]	PUNCT
cana-1178	271	4	wayne	wayne	PROPN
cana-1178	271	5	goddard	goddard	PROPN
cana-1178	271	6	,	,	PUNCT
cana-1178	271	7	michael	michael	PROPN
cana-1178	271	8	a.	a.	PROPN
cana-1178	271	9	henning	henning	PROPN
cana-1178	271	10	,	,	PUNCT
cana-1178	271	11	independent	independent	ADJ
cana-1178	271	12	domination	domination	NOUN
cana-1178	271	13	in	in	ADP
cana-1178	271	14	graphs	graph	NOUN
cana-1178	271	15	:	:	PUNCT
cana-1178	271	16	a	a	DET
cana-1178	271	17	survey	survey	NOUN
cana-1178	271	18	and	and	CCONJ
cana-1178	271	19	recent	recent	ADJ
cana-1178	271	20	results	result	NOUN
cana-1178	271	21	,	,	PUNCT
cana-1178	271	22	discrete	discrete	ADJ
cana-1178	271	23	mathematics	mathematic	NOUN
cana-1178	271	24	313	313	NUM
cana-1178	271	25	(	(	PUNCT
cana-1178	271	26	2013	2013	NUM
cana-1178	271	27	)	)	PUNCT
cana-1178	271	28	839–854	839–854	NUM
cana-1178	271	29	.	.	PUNCT
cana-1178	272	1	[	[	X
cana-1178	272	2	3	3	X
cana-1178	272	3	]	]	X
cana-1178	272	4	ivan	ivan	PROPN
cana-1178	272	5	gutman	gutman	PROPN
cana-1178	272	6	,	,	PUNCT
cana-1178	272	7	boris	boris	PROPN
cana-1178	272	8	furtula	furtula	PROPN
cana-1178	272	9	̧s.burcubozkurt	̧s.burcubozkurt	PROPN
cana-1178	272	10	,	,	PUNCT
cana-1178	272	11	on	on	ADP
cana-1178	272	12	randic	randic	ADJ
cana-1178	272	13	energy	energy	NOUN
cana-1178	272	14	,	,	PUNCT
cana-1178	272	15	linear	linear	ADJ
cana-1178	272	16	algebra	algebra	NOUN
cana-1178	272	17	and	and	CCONJ
cana-1178	272	18	its	its	PRON
cana-1178	272	19	applications	application	NOUN
cana-1178	272	20	442	442	NUM
cana-1178	272	21	(	(	PUNCT
cana-1178	272	22	2014	2014	NUM
cana-1178	272	23	)	)	PUNCT
cana-1178	272	24	,	,	PUNCT
cana-1178	272	25	50	50	NUM
cana-1178	272	26	-	-	SYM
cana-1178	272	27	57	57	NUM
cana-1178	272	28	.	.	PUNCT
cana-1178	273	1	[	[	X
cana-1178	273	2	4	4	X
cana-1178	273	3	]	]	PUNCT
cana-1178	273	4	m.	m.	PROPN
cana-1178	273	5	r.	r.	PROPN
cana-1178	273	6	rajesh	rajesh	PROPN
cana-1178	273	7	kanna	kanna	PROPN
cana-1178	273	8	,	,	PUNCT
cana-1178	273	9	r.	r.	PROPN
cana-1178	273	10	pradeep	pradeep	PROPN
cana-1178	273	11	kumar	kumar	PROPN
cana-1178	273	12	,	,	PUNCT
cana-1178	273	13	mohammad	mohammad	PROPN
cana-1178	273	14	reza	reza	PROPN
cana-1178	273	15	farahani	farahani	PROPN
cana-1178	273	16	,	,	PUNCT
cana-1178	273	17	milovanovic	milovanovic	ADJ
cana-1178	273	18	bounds	bound	NOUN
cana-1178	273	19	for	for	ADP
cana-1178	273	20	seidel	seidel	PROPN
cana-1178	273	21	energy	energy	NOUN
cana-1178	273	22	of	of	ADP
cana-1178	273	23	a	a	DET
cana-1178	273	24	graph	graph	NOUN
cana-1178	273	25	,	,	PUNCT
cana-1178	273	26	advances	advance	NOUN
cana-1178	273	27	in	in	ADP
cana-1178	273	28	theoretical	theoretical	ADJ
cana-1178	273	29	and	and	CCONJ
cana-1178	273	30	applied	applied	ADJ
cana-1178	273	31	mathematics	mathematic	NOUN
cana-1178	273	32	,	,	PUNCT
cana-1178	273	33	volume	volume	NOUN
cana-1178	273	34	10	10	NUM
cana-1178	273	35	,	,	PUNCT
cana-1178	273	36	number	number	NOUN
cana-1178	273	37	1	1	NUM
cana-1178	273	38	(	(	PUNCT
cana-1178	273	39	2016	2016	NUM
cana-1178	273	40	)	)	PUNCT
cana-1178	273	41	,	,	PUNCT
cana-1178	273	42	37–44	37–44	NUM
cana-1178	273	43	.	.	PUNCT
cana-1178	274	1	[	[	X
cana-1178	274	2	5	5	NUM
cana-1178	274	3	]	]	PUNCT
cana-1178	274	4	bolian	bolian	PROPN
cana-1178	274	5	liu	liu	PROPN
cana-1178	274	6	,	,	PUNCT
cana-1178	274	7	yufei	yufei	PROPN
cana-1178	274	8	huang	huang	PROPN
cana-1178	274	9	,	,	PUNCT
cana-1178	274	10	jingfang	jingfang	PROPN
cana-1178	274	11	feng	feng	PROPN
cana-1178	274	12	,	,	PUNCT
cana-1178	274	13	a	a	DET
cana-1178	274	14	note	note	NOUN
cana-1178	274	15	on	on	ADP
cana-1178	274	16	the	the	DET
cana-1178	274	17	randic	randic	ADJ
cana-1178	274	18	spectral	spectral	ADJ
cana-1178	274	19	radius	radius	NOUN
cana-1178	274	20	,	,	PUNCT
cana-1178	274	21	match	match	NOUN
cana-1178	274	22	commun	commun	PROPN
cana-1178	274	23	.	.	PUNCT
cana-1178	274	24	math	math	PROPN
cana-1178	274	25	.	.	PUNCT
cana-1178	275	1	comput	comput	NOUN
cana-1178	275	2	.	.	PUNCT
cana-1178	276	1	chem	chem	NOUN
cana-1178	276	2	.	.	PUNCT
cana-1178	277	1	68	68	NUM
cana-1178	277	2	(	(	PUNCT
cana-1178	277	3	2012	2012	NUM
cana-1178	277	4	)	)	PUNCT
cana-1178	277	5	913	913	NUM
cana-1178	277	6	.	.	PUNCT
cana-1178	278	1	[	[	X
cana-1178	278	2	6	6	NUM
cana-1178	278	3	]	]	PUNCT
cana-1178	278	4	m.	m.	NOUN
cana-1178	278	5	randic	randic	NOUN
cana-1178	278	6	,	,	PUNCT
cana-1178	278	7	on	on	ADP
cana-1178	278	8	characterization	characterization	NOUN
cana-1178	278	9	of	of	ADP
cana-1178	278	10	molecular	molecular	ADJ
cana-1178	278	11	branching	branching	NOUN
cana-1178	278	12	,	,	PUNCT
cana-1178	278	13	journal	journal	NOUN
cana-1178	278	14	of	of	ADP
cana-1178	278	15	the	the	DET
cana-1178	278	16	american	american	PROPN
cana-1178	278	17	chemical	chemical	PROPN
cana-1178	278	18	society	society	PROPN
cana-1178	278	19	,	,	PUNCT
cana-1178	278	20	97(1975	97(1975	NUM
cana-1178	278	21	)	)	PUNCT
cana-1178	278	22	,	,	PUNCT
cana-1178	278	23	6609	6609	NUM
cana-1178	278	24	-	-	SYM
cana-1178	278	25	6615	6615	NUM
cana-1178	278	26	.	.	PUNCT
cana-1178	279	1	[	[	X
cana-1178	279	2	7	7	X
cana-1178	279	3	]	]	PUNCT
cana-1178	279	4	s.	s.	PROPN
cana-1178	279	5	b.	b.	PROPN
cana-1178	279	6	bozkurt	bozkurt	PROPN
cana-1178	279	7	,	,	PUNCT
cana-1178	279	8	a.	a.	PROPN
cana-1178	279	9	d.	d.	PROPN
cana-1178	279	10	gungor	gungor	PROPN
cana-1178	279	11	,	,	PUNCT
cana-1178	279	12	and	and	CCONJ
cana-1178	279	13	i.gutman	i.gutman	ADJ
cana-1178	279	14	,	,	PUNCT
cana-1178	279	15	randic	randic	ADJ
cana-1178	279	16	spectral	spectral	ADJ
cana-1178	279	17	radius	radius	NOUN
cana-1178	279	18	and	and	CCONJ
cana-1178	279	19	randic	randic	ADJ
cana-1178	279	20	energy	energy	NOUN
cana-1178	279	21	,	,	PUNCT
cana-1178	279	22	communications	communication	NOUN
cana-1178	279	23	in	in	ADP
cana-1178	279	24	mathematical	mathematical	ADJ
cana-1178	279	25	and	and	CCONJ
cana-1178	279	26	in	in	ADP
cana-1178	279	27	computer	computer	NOUN
cana-1178	279	28	chemistry	chemistry	NOUN
cana-1178	279	29	,	,	PUNCT
cana-1178	279	30	64(2010	64(2010	NOUN
cana-1178	279	31	)	)	PUNCT
cana-1178	279	32	,	,	PUNCT
cana-1178	279	33	239	239	NUM
cana-1178	279	34	-	-	SYM
cana-1178	279	35	250	250	NUM
cana-1178	279	36	.	.	PUNCT
cana-1178	280	1	[	[	X
cana-1178	280	2	8	8	NUM
cana-1178	280	3	]	]	PUNCT
cana-1178	280	4	m.	m.	PROPN
cana-1178	280	5	r.	r.	PROPN
cana-1178	280	6	rajesh	rajesh	PROPN
cana-1178	280	7	kanna	kanna	PROPN
cana-1178	280	8	,	,	PUNCT
cana-1178	280	9	b.	b.	PROPN
cana-1178	280	10	n.	n.	PROPN
cana-1178	280	11	dharmendra	dharmendra	PROPN
cana-1178	280	12	,	,	PUNCT
cana-1178	280	13	g.	g.	PROPN
cana-1178	280	14	sridhara	sridhara	PROPN
cana-1178	280	15	,	,	PUNCT
cana-1178	280	16	the	the	DET
cana-1178	280	17	minimum	minimum	ADJ
cana-1178	280	18	dominating	dominating	NOUN
cana-1178	280	19	energy	energy	NOUN
cana-1178	280	20	of	of	ADP
cana-1178	280	21	a	a	DET
cana-1178	280	22	graph	graph	NOUN
cana-1178	280	23	,	,	PUNCT
cana-1178	280	24	international	international	ADJ
cana-1178	280	25	journal	journal	NOUN
cana-1178	280	26	of	of	ADP
cana-1178	280	27	pure	pure	ADJ
cana-1178	280	28	and	and	CCONJ
cana-1178	280	29	applied	applied	ADJ
cana-1178	280	30	mathematics	mathematic	NOUN
cana-1178	280	31	,	,	PUNCT
cana-1178	280	32	volume	volume	NOUN
cana-1178	280	33	85	85	NUM
cana-1178	280	34	no	no	NOUN
cana-1178	280	35	.	.	NOUN
cana-1178	280	36	4	4	NUM
cana-1178	280	37	(	(	PUNCT
cana-1178	280	38	2013	2013	NUM
cana-1178	280	39	)	)	PUNCT
cana-1178	280	40	,	,	PUNCT
cana-1178	280	41	707	707	NUM
cana-1178	280	42	-	-	SYM
cana-1178	280	43	718	718	NUM
cana-1178	280	44	.	.	PUNCT
cana-1178	281	1	[	[	X
cana-1178	281	2	9	9	NUM
cana-1178	281	3	]	]	SYM
cana-1178	281	4	madhukar	madhukar	PROPN
cana-1178	281	5	m.	m.	PROPN
cana-1178	281	6	pawar	pawar	PROPN
cana-1178	281	7	,	,	PUNCT
cana-1178	281	8	shilpa	shilpa	PROPN
cana-1178	281	9	t.	t.	PROPN
cana-1178	281	10	bhangale	bhangale	PROPN
cana-1178	281	11	,	,	PUNCT
cana-1178	281	12	minimum	minimum	ADJ
cana-1178	281	13	independent	independent	ADJ
cana-1178	281	14	dominating	dominating	NOUN
cana-1178	281	15	energy	energy	NOUN
cana-1178	281	16	of	of	ADP
cana-1178	281	17	graphs	graph	NOUN
cana-1178	281	18	,	,	PUNCT
cana-1178	281	19	asian	asian	ADJ
cana-1178	281	20	-	-	PUNCT
cana-1178	281	21	european	european	ADJ
cana-1178	281	22	journal	journal	NOUN
cana-1178	281	23	of	of	ADP
cana-1178	281	24	mathematics	mathematic	NOUN
cana-1178	281	25	,	,	PUNCT
cana-1178	281	26	(	(	PUNCT
cana-1178	281	27	2021	2021	NUM
cana-1178	281	28	)	)	PUNCT
cana-1178	281	29	.	.	PUNCT
cana-1178	282	1	[	[	X
cana-1178	282	2	10	10	NUM
cana-1178	282	3	]	]	X
cana-1178	282	4	b.	b.	PROPN
cana-1178	282	5	praba	praba	PROPN
cana-1178	282	6	,	,	PUNCT
cana-1178	282	7	v.	v.	ADP
cana-1178	282	8	m.	m.	NOUN
cana-1178	282	9	chandrasekaran	chandrasekaran	VERB
cana-1178	282	10	,	,	PUNCT
cana-1178	282	11	a.	a.	NOUN
cana-1178	282	12	manimaran	manimaran	NOUN
cana-1178	282	13	,	,	PUNCT
cana-1178	282	14	semiring	semire	VERB
cana-1178	282	15	on	on	ADP
cana-1178	282	16	rough	rough	ADJ
cana-1178	282	17	sets	set	NOUN
cana-1178	282	18	,	,	PUNCT
cana-1178	282	19	ind	ind	PROPN
cana-1178	282	20	.	.	PUNCT
cana-1178	283	1	j.	j.	PROPN
cana-1178	283	2	of	of	ADP
cana-1178	283	3	sci	sci	PROPN
cana-1178	283	4	and	and	CCONJ
cana-1178	283	5	tech	tech	NOUN
cana-1178	283	6	.	.	PUNCT
cana-1178	283	7	3	3	NUM
cana-1178	283	8	(	(	PUNCT
cana-1178	283	9	2015	2015	NUM
cana-1178	283	10	)	)	PUNCT
cana-1178	283	11	280	280	NUM
cana-1178	283	12	-	-	SYM
cana-1178	283	13	286	286	NUM
cana-1178	283	14	.	.	PUNCT
cana-1178	284	1	[	[	X
cana-1178	284	2	11	11	NUM
cana-1178	284	3	]	]	X
cana-1178	284	4	b.	b.	PROPN
cana-1178	284	5	praba	praba	PROPN
cana-1178	284	6	,	,	PUNCT
cana-1178	284	7	a.	a.	NOUN
cana-1178	284	8	manimaran	manimaran	PROPN
cana-1178	284	9	,	,	PUNCT
cana-1178	284	10	v.	v.	ADP
cana-1178	284	11	m.	m.	NOUN
cana-1178	284	12	chandrasekaran	chandrasekaran	VERB
cana-1178	284	13	,	,	PUNCT
cana-1178	284	14	the	the	DET
cana-1178	284	15	zero	zero	NUM
cana-1178	284	16	divisor	divisor	NOUN
cana-1178	284	17	graph	graph	NOUN
cana-1178	284	18	of	of	ADP
cana-1178	284	19	a	a	DET
cana-1178	284	20	rough	rough	ADJ
cana-1178	284	21	semiring	semiring	NOUN
cana-1178	284	22	,	,	PUNCT
cana-1178	284	23	int	int	NOUN
cana-1178	284	24	.	.	PUNCT
cana-1178	285	1	j.	j.	PROPN
cana-1178	285	2	pure	pure	PROPN
cana-1178	285	3	and	and	CCONJ
cana-1178	285	4	appl	appl	PROPN
cana-1178	285	5	.	.	PROPN
cana-1178	285	6	math	math	NOUN
cana-1178	285	7	.	.	PUNCT
cana-1178	286	1	98	98	NUM
cana-1178	286	2	(	(	PUNCT
cana-1178	286	3	2015	2015	NUM
cana-1178	286	4	)	)	PUNCT
cana-1178	286	5	33	33	NUM
cana-1178	286	6	-	-	SYM
cana-1178	286	7	37	37	NUM
cana-1178	286	8	.	.	PUNCT
cana-1178	287	1	[	[	X
cana-1178	287	2	12	12	NUM
cana-1178	287	3	]	]	X
cana-1178	287	4	m.r	m.r	PROPN
cana-1178	287	5	.	.	PROPN
cana-1178	287	6	rajesh	rajesh	PROPN
cana-1178	287	7	kanna	kanna	PROPN
cana-1178	287	8	,	,	PUNCT
cana-1178	287	9	r.	r.	PROPN
cana-1178	287	10	jagadeesh	jagadeesh	PROPN
cana-1178	287	11	,	,	PUNCT
cana-1178	287	12	b.k	b.k	PROPN
cana-1178	287	13	.	.	PROPN
cana-1178	287	14	kempegowda	kempegowda	PROPN
cana-1178	287	15	,	,	PUNCT
cana-1178	287	16	minimum	minimum	ADJ
cana-1178	287	17	dominating	dominating	NOUN
cana-1178	287	18	seidel	seidel	NOUN
cana-1178	287	19	energy	energy	NOUN
cana-1178	287	20	of	of	ADP
cana-1178	287	21	a	a	DET
cana-1178	287	22	graph	graph	NOUN
cana-1178	287	23	,	,	PUNCT
cana-1178	287	24	international	international	ADJ
cana-1178	287	25	journal	journal	NOUN
cana-1178	287	26	of	of	ADP
cana-1178	287	27	scientific	scientific	PROPN
cana-1178	287	28	&	&	CCONJ
cana-1178	287	29	engineering	engineering	NOUN
cana-1178	287	30	research	research	NOUN
cana-1178	287	31	,	,	PUNCT
cana-1178	287	32	volume	volume	NOUN
cana-1178	287	33	7	7	NUM
cana-1178	287	34	,	,	PUNCT
cana-1178	287	35	issue	issue	NOUN
cana-1178	287	36	5	5	NUM
cana-1178	287	37	,	,	PUNCT
cana-1178	287	38	may-2016	may-2016	NOUN
cana-1178	287	39	.	.	PUNCT