id	sid	tid	token	lemma	pos
cana-5714	1	1	communications	communication	NOUN
cana-5714	1	2	on	on	ADP
cana-5714	1	3	applied	apply	VERB
cana-5714	1	4	nonlinear	nonlinear	ADJ
cana-5714	1	5	analysis	analysis	NOUN
cana-5714	1	6	issn	issn	NOUN
cana-5714	1	7	:	:	PUNCT
cana-5714	1	8	1074	1074	NUM
cana-5714	1	9	-	-	PUNCT
cana-5714	1	10	133x	133x	NUM
cana-5714	1	11	vol	vol	VERB
cana-5714	1	12	32	32	NUM
cana-5714	1	13	no	no	NOUN
cana-5714	1	14	.	.	PUNCT
cana-5714	2	1	10s	10	NOUN
cana-5714	2	2	(	(	PUNCT
cana-5714	2	3	2025	2025	NUM
cana-5714	2	4	)	)	PUNCT
cana-5714	2	5	2740	2740	NUM
cana-5714	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	2	7	bounds	bound	VERB
cana-5714	2	8	for	for	ADP
cana-5714	2	9	product	product	NOUN
cana-5714	2	10	eccentricity	eccentricity	NOUN
cana-5714	2	11	energy	energy	NOUN
cana-5714	2	12	of	of	ADP
cana-5714	2	13	graphs	graph	NOUN
cana-5714	2	14	priya	priya	PROPN
cana-5714	2	15	karen	karen	PROPN
cana-5714	2	16	s1,a	s1,a	PROPN
cana-5714	2	17	)	)	PUNCT
cana-5714	2	18	,	,	PUNCT
cana-5714	2	19	arokia	arokia	NOUN
cana-5714	2	20	lancy	lancy	PROPN
cana-5714	2	21	a2,b	a2,b	PROPN
cana-5714	2	22	)	)	PUNCT
cana-5714	2	23	1,2	1,2	NUM
cana-5714	2	24	pg	pg	NOUN
cana-5714	2	25	and	and	CCONJ
cana-5714	2	26	research	research	PROPN
cana-5714	2	27	department	department	PROPN
cana-5714	2	28	of	of	ADP
cana-5714	2	29	mathematics	mathematics	PROPN
cana-5714	2	30	,	,	PUNCT
cana-5714	2	31	nirmala	nirmala	PROPN
cana-5714	2	32	college	college	PROPN
cana-5714	2	33	for	for	ADP
cana-5714	2	34	women	woman	NOUN
cana-5714	2	35	,	,	PUNCT
cana-5714	2	36	coimbatore	coimbatore	PROPN
cana-5714	2	37	,	,	PUNCT
cana-5714	2	38	tamilnadu	tamilnadu	NOUN
cana-5714	2	39	,	,	PUNCT
cana-5714	2	40	india	india	PROPN
cana-5714	2	41	a	a	PRON
cana-5714	2	42	)	)	PUNCT
cana-5714	2	43	corresponding	correspond	VERB
cana-5714	2	44	author	author	NOUN
cana-5714	2	45	:	:	PUNCT
cana-5714	2	46	priyakaren2018@gmail.com	priyakaren2018@gmail.com	X
cana-5714	3	1	b)aarokia.lancy@gmail.com	b)aarokia.lancy@gmail.com	PROPN
cana-5714	3	2	article	article	NOUN
cana-5714	3	3	history	history	NOUN
cana-5714	3	4	:	:	PUNCT
cana-5714	3	5	received	receive	VERB
cana-5714	3	6	:	:	PUNCT
cana-5714	3	7	12	12	NUM
cana-5714	3	8	-	-	SYM
cana-5714	3	9	01	01	NUM
cana-5714	3	10	-	-	PUNCT
cana-5714	3	11	2025	2025	NUM
cana-5714	3	12	revised	revise	VERB
cana-5714	3	13	:	:	PUNCT
cana-5714	3	14	15	15	NUM
cana-5714	3	15	-	-	NUM
cana-5714	3	16	02	02	NUM
cana-5714	3	17	-	-	PUNCT
cana-5714	3	18	2025	2025	NUM
cana-5714	3	19	accepted	accept	VERB
cana-5714	3	20	:	:	PUNCT
cana-5714	3	21	01	01	NUM
cana-5714	3	22	-	-	SYM
cana-5714	3	23	03	03	NUM
cana-5714	3	24	-	-	PUNCT
cana-5714	3	25	2025	2025	NUM
cana-5714	3	26	abstract	abstract	NOUN
cana-5714	3	27	:	:	PUNCT
cana-5714	3	28	in	in	ADP
cana-5714	3	29	this	this	DET
cana-5714	3	30	article	article	NOUN
cana-5714	3	31	some	some	PRON
cana-5714	3	32	lower	low	ADJ
cana-5714	3	33	bounds	bound	NOUN
cana-5714	3	34	for	for	ADP
cana-5714	3	35	the	the	DET
cana-5714	3	36	product	product	NOUN
cana-5714	3	37	eccentricity	eccentricity	NOUN
cana-5714	3	38	energy	energy	NOUN
cana-5714	4	1	[	[	X
cana-5714	4	2	𝐸𝑃𝐸	𝐸𝑃𝐸	PROPN
cana-5714	4	3	]	]	PUNCT
cana-5714	4	4	of	of	ADP
cana-5714	4	5	𝐺	𝐺	PROPN
cana-5714	4	6	as	as	ADV
cana-5714	4	7	well	well	ADV
cana-5714	4	8	as	as	ADP
cana-5714	4	9	a	a	DET
cana-5714	4	10	bound	bind	VERB
cana-5714	4	11	for	for	ADP
cana-5714	4	12	the	the	DET
cana-5714	4	13	eigenvalues	eigenvalue	NOUN
cana-5714	4	14	of	of	ADP
cana-5714	4	15	the	the	DET
cana-5714	4	16	product	product	NOUN
cana-5714	4	17	eccentricity	eccentricity	NOUN
cana-5714	4	18	matrix	matrix	NOUN
cana-5714	4	19	[	[	X
cana-5714	4	20	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	4	21	)	)	PUNCT
cana-5714	4	22	]	]	PUNCT
cana-5714	4	23	are	be	AUX
cana-5714	4	24	obtained	obtain	VERB
cana-5714	4	25	.	.	PUNCT
cana-5714	5	1	mathematical	mathematical	ADJ
cana-5714	5	2	subject	subject	ADJ
cana-5714	5	3	classification	classification	NOUN
cana-5714	5	4	:	:	PUNCT
cana-5714	5	5	05c07	05c07	NOUN
cana-5714	5	6	,	,	PUNCT
cana-5714	5	7	050c50	050c50	X
cana-5714	5	8	.	.	PUNCT
cana-5714	5	9	conclusion	conclusion	NOUN
cana-5714	5	10	:	:	PUNCT
cana-5714	6	1	this	this	DET
cana-5714	6	2	article	article	NOUN
cana-5714	6	3	gives	give	VERB
cana-5714	6	4	an	an	DET
cana-5714	6	5	idea	idea	NOUN
cana-5714	6	6	on	on	ADP
cana-5714	6	7	few	few	ADJ
cana-5714	6	8	lower	low	ADJ
cana-5714	6	9	bounds	bound	NOUN
cana-5714	6	10	for	for	ADP
cana-5714	6	11	the	the	DET
cana-5714	6	12	product	product	NOUN
cana-5714	6	13	eccentricity	eccentricity	NOUN
cana-5714	6	14	energy	energy	NOUN
cana-5714	6	15	based	base	VERB
cana-5714	6	16	on	on	ADP
cana-5714	6	17	its	its	PRON
cana-5714	6	18	eigen	eigen	PROPN
cana-5714	6	19	values	value	NOUN
cana-5714	6	20	and	and	CCONJ
cana-5714	6	21	various	various	ADJ
cana-5714	6	22	inequalities	inequality	NOUN
cana-5714	6	23	are	be	AUX
cana-5714	6	24	also	also	ADV
cana-5714	6	25	proved	prove	VERB
cana-5714	6	26	.	.	PUNCT
cana-5714	7	1	keywords	keyword	NOUN
cana-5714	7	2	:	:	PUNCT
cana-5714	7	3	energy	energy	NOUN
cana-5714	7	4	,	,	PUNCT
cana-5714	7	5	bounds	bound	NOUN
cana-5714	7	6	,	,	PUNCT
cana-5714	7	7	product	product	NOUN
cana-5714	7	8	eccentricity	eccentricity	NOUN
cana-5714	7	9	energy	energy	NOUN
cana-5714	7	10	.	.	PUNCT
cana-5714	8	1	1	1	X
cana-5714	8	2	.	.	X
cana-5714	8	3	introduction	introduction	NOUN
cana-5714	8	4	graph	graph	NOUN
cana-5714	8	5	theory	theory	NOUN
cana-5714	8	6	is	be	AUX
cana-5714	8	7	a	a	DET
cana-5714	8	8	branch	branch	NOUN
cana-5714	8	9	of	of	ADP
cana-5714	8	10	discrete	discrete	ADJ
cana-5714	8	11	arithmetic	arithmetic	NOUN
cana-5714	8	12	that	that	PRON
cana-5714	8	13	involves	involve	VERB
cana-5714	8	14	considering	consider	VERB
cana-5714	8	15	structures	structure	NOUN
cana-5714	8	16	together	together	ADV
cana-5714	8	17	with	with	ADP
cana-5714	8	18	their	their	PRON
cana-5714	8	19	attributes	attribute	NOUN
cana-5714	8	20	,	,	PUNCT
cana-5714	8	21	goals	goal	NOUN
cana-5714	8	22	,	,	PUNCT
cana-5714	8	23	and	and	CCONJ
cana-5714	8	24	relationships	relationship	NOUN
cana-5714	8	25	.	.	PUNCT
cana-5714	9	1	originally	originally	ADV
cana-5714	9	2	useful	useful	ADJ
cana-5714	9	3	for	for	ADP
cana-5714	9	4	resolving	resolve	VERB
cana-5714	9	5	a	a	DET
cana-5714	9	6	wide	wide	ADJ
cana-5714	9	7	range	range	NOUN
cana-5714	9	8	of	of	ADP
cana-5714	9	9	mathematical	mathematical	ADJ
cana-5714	9	10	problems	problem	NOUN
cana-5714	9	11	,	,	PUNCT
cana-5714	9	12	it	it	PRON
cana-5714	9	13	periodically	periodically	ADV
cana-5714	9	14	branched	branch	VERB
cana-5714	9	15	out	out	ADP
cana-5714	9	16	into	into	ADP
cana-5714	9	17	new	new	ADJ
cana-5714	9	18	areas	area	NOUN
cana-5714	9	19	of	of	ADP
cana-5714	9	20	mathematical	mathematical	ADJ
cana-5714	9	21	analysis	analysis	NOUN
cana-5714	9	22	when	when	SCONJ
cana-5714	9	23	applied	apply	VERB
cana-5714	9	24	in	in	ADP
cana-5714	9	25	complicated	complicated	ADJ
cana-5714	9	26	science	science	NOUN
cana-5714	9	27	,	,	PUNCT
cana-5714	9	28	computer	computer	NOUN
cana-5714	9	29	science	science	NOUN
cana-5714	9	30	,	,	PUNCT
cana-5714	9	31	chemistry	chemistry	NOUN
cana-5714	9	32	,	,	PUNCT
cana-5714	9	33	and	and	CCONJ
cana-5714	9	34	other	other	ADJ
cana-5714	9	35	disciplines	discipline	NOUN
cana-5714	9	36	.	.	PUNCT
cana-5714	10	1	simple	simple	ADJ
cana-5714	10	2	,	,	PUNCT
cana-5714	10	3	loop	loop	NOUN
cana-5714	10	4	-	-	PUNCT
cana-5714	10	5	less	less	ADJ
cana-5714	10	6	,	,	PUNCT
cana-5714	10	7	and	and	CCONJ
cana-5714	10	8	connected	connected	ADJ
cana-5714	10	9	graphs	graph	NOUN
cana-5714	10	10	are	be	AUX
cana-5714	10	11	the	the	DET
cana-5714	10	12	types	type	NOUN
cana-5714	10	13	of	of	ADP
cana-5714	10	14	graphs	graph	NOUN
cana-5714	10	15	examined	examine	VERB
cana-5714	10	16	in	in	ADP
cana-5714	10	17	this	this	DET
cana-5714	10	18	article	article	NOUN
cana-5714	10	19	.	.	PUNCT
cana-5714	11	1	an	an	DET
cana-5714	11	2	essential	essential	ADJ
cana-5714	11	3	concept	concept	NOUN
cana-5714	11	4	in	in	ADP
cana-5714	11	5	this	this	DET
cana-5714	11	6	theory	theory	NOUN
cana-5714	11	7	is	be	AUX
cana-5714	11	8	a	a	DET
cana-5714	11	9	vertex	vertex	NOUN
cana-5714	11	10	's	's	PART
cana-5714	11	11	eccentricity	eccentricity	NOUN
cana-5714	11	12	,	,	PUNCT
cana-5714	11	13	which	which	PRON
cana-5714	11	14	assesses	assess	VERB
cana-5714	11	15	the	the	DET
cana-5714	11	16	greatest	great	ADJ
cana-5714	11	17	distance	distance	NOUN
cana-5714	11	18	between	between	ADP
cana-5714	11	19	two	two	NUM
cana-5714	11	20	vertices	vertex	NOUN
cana-5714	11	21	.	.	PUNCT
cana-5714	12	1	the	the	DET
cana-5714	12	2	distance	distance	NOUN
cana-5714	12	3	between	between	ADP
cana-5714	12	4	two	two	NUM
cana-5714	12	5	vertices	vertex	NOUN
cana-5714	12	6	𝑎	𝑎	NOUN
cana-5714	12	7	and	and	CCONJ
cana-5714	12	8	𝑏	𝑏	NOUN
cana-5714	12	9	in	in	ADP
cana-5714	12	10	𝑉(𝐺	𝑉(𝐺	NOUN
cana-5714	12	11	)	)	PUNCT
cana-5714	12	12	is	be	AUX
cana-5714	12	13	the	the	DET
cana-5714	12	14	shortest	short	ADJ
cana-5714	12	15	a	a	DET
cana-5714	12	16	-	-	PUNCT
cana-5714	12	17	b	b	NOUN
cana-5714	12	18	path	path	NOUN
cana-5714	12	19	length	length	NOUN
cana-5714	12	20	in	in	ADP
cana-5714	12	21	𝐺.	𝐺.	NOUN
cana-5714	12	22	the	the	DET
cana-5714	12	23	maximum	maximum	ADJ
cana-5714	12	24	distance	distance	NOUN
cana-5714	12	25	between	between	ADP
cana-5714	12	26	a	a	DET
cana-5714	12	27	particular	particular	ADJ
cana-5714	12	28	vertex	vertex	NOUN
cana-5714	12	29	and	and	CCONJ
cana-5714	12	30	any	any	DET
cana-5714	12	31	other	other	ADJ
cana-5714	12	32	vertex	vertex	NOUN
cana-5714	12	33	in	in	ADP
cana-5714	12	34	the	the	DET
cana-5714	12	35	graph	graph	NOUN
cana-5714	12	36	is	be	AUX
cana-5714	12	37	determined	determine	VERB
cana-5714	12	38	by	by	ADP
cana-5714	12	39	the	the	DET
cana-5714	12	40	vertex	vertex	NOUN
cana-5714	12	41	's	's	PART
cana-5714	12	42	eccentricity	eccentricity	NOUN
cana-5714	12	43	.	.	PUNCT
cana-5714	13	1	formally	formally	ADV
cana-5714	13	2	,	,	PUNCT
cana-5714	13	3	it	it	PRON
cana-5714	13	4	can	can	AUX
cana-5714	13	5	be	be	AUX
cana-5714	13	6	expressed	express	VERB
cana-5714	13	7	as	as	ADP
cana-5714	13	8	:	:	PUNCT
cana-5714	13	9	𝜉(𝑏	𝜉(𝑏	NUM
cana-5714	13	10	)	)	PUNCT
cana-5714	13	11	=	=	SYM
cana-5714	14	1	max{𝑑(𝑏	max{𝑑(𝑏	PROPN
cana-5714	14	2	,	,	PUNCT
cana-5714	14	3	𝑎)|∀	𝑎)|∀	ADJ
cana-5714	14	4	𝑎	𝑎	DET
cana-5714	14	5	∈	∈	PROPN
cana-5714	14	6	𝑉(𝐺	𝑉(𝐺	NOUN
cana-5714	14	7	)	)	PUNCT
cana-5714	14	8	}	}	PUNCT
cana-5714	14	9	a	a	DET
cana-5714	14	10	graph	graph	NOUN
cana-5714	14	11	g	g	NOUN
cana-5714	14	12	's	's	PART
cana-5714	14	13	eccentricity	eccentricity	NOUN
cana-5714	14	14	matrix	matrix	NOUN
cana-5714	14	15	𝜉(𝐺	𝜉(𝐺	NOUN
cana-5714	14	16	)	)	PUNCT
cana-5714	14	17	is	be	AUX
cana-5714	14	18	derived	derive	VERB
cana-5714	14	19	from	from	ADP
cana-5714	14	20	its	its	PRON
cana-5714	14	21	distance	distance	NOUN
cana-5714	14	22	matrix	matrix	NOUN
cana-5714	14	23	by	by	ADP
cana-5714	14	24	keeping	keep	VERB
cana-5714	14	25	the	the	DET
cana-5714	14	26	largest	large	ADJ
cana-5714	14	27	distances	distance	NOUN
cana-5714	14	28	in	in	ADP
cana-5714	14	29	each	each	DET
cana-5714	14	30	row	row	NOUN
cana-5714	14	31	and	and	CCONJ
cana-5714	14	32	column	column	NOUN
cana-5714	14	33	and	and	CCONJ
cana-5714	14	34	leaving	leave	VERB
cana-5714	14	35	zeros	zero	NOUN
cana-5714	14	36	in	in	ADP
cana-5714	14	37	the	the	DET
cana-5714	14	38	others	other	NOUN
cana-5714	14	39	.	.	PUNCT
cana-5714	15	1	summing	sum	VERB
cana-5714	15	2	the	the	DET
cana-5714	15	3	absolute	absolute	ADJ
cana-5714	15	4	values	value	NOUN
cana-5714	15	5	of	of	ADP
cana-5714	15	6	the	the	DET
cana-5714	15	7	eigenvalues	eigenvalue	NOUN
cana-5714	15	8	of	of	ADP
cana-5714	15	9	𝜉(𝐺	𝜉(𝐺	NOUN
cana-5714	15	10	)	)	PUNCT
cana-5714	15	11	yields	yield	VERB
cana-5714	15	12	the	the	DET
cana-5714	15	13	eccentricity	eccentricity	NOUN
cana-5714	15	14	energy	energy	NOUN
cana-5714	15	15	of	of	ADP
cana-5714	15	16	𝐺.	𝐺.	NOUN
cana-5714	15	17	let	let	VERB
cana-5714	15	18	𝐺	𝐺	PROPN
cana-5714	15	19	be	be	AUX
cana-5714	15	20	a	a	DET
cana-5714	15	21	graph	graph	NOUN
cana-5714	15	22	with	with	ADP
cana-5714	15	23	𝑛	𝑛	DET
cana-5714	15	24	vertices	vertex	NOUN
cana-5714	15	25	and	and	CCONJ
cana-5714	15	26	𝑚	𝑚	ADP
cana-5714	15	27	edges	edge	NOUN
cana-5714	15	28	.	.	PUNCT
cana-5714	16	1	denote	denote	VERB
cana-5714	16	2	the	the	DET
cana-5714	16	3	absolute	absolute	ADJ
cana-5714	16	4	eigen	eigen	PROPN
cana-5714	16	5	values	value	NOUN
cana-5714	16	6	of	of	ADP
cana-5714	16	7	𝐺	𝐺	PROPN
cana-5714	16	8	as	as	ADP
cana-5714	16	9	𝜆𝑖	𝜆𝑖	PROPN
cana-5714	16	10	,	,	PUNCT
cana-5714	16	11	𝑖	𝑖	NOUN
cana-5714	16	12	=	=	SYM
cana-5714	16	13	1,2	1,2	NUM
cana-5714	16	14	,	,	PUNCT
cana-5714	16	15	⋯	⋯	ADP
cana-5714	16	16	𝑛	𝑛	PROPN
cana-5714	16	17	arranged	arrange	VERB
cana-5714	16	18	in	in	ADP
cana-5714	16	19	order	order	NOUN
cana-5714	16	20	that	that	PRON
cana-5714	16	21	is	be	AUX
cana-5714	16	22	not	not	PART
cana-5714	16	23	increasing	increase	VERB
cana-5714	16	24	as	as	ADP
cana-5714	16	25	|𝜆1|	|𝜆1|	NOUN
cana-5714	16	26	≥	≥	NOUN
cana-5714	16	27	|𝜆2|	|𝜆2|	NOUN
cana-5714	16	28	≥	≥	PROPN
cana-5714	16	29	⋯	⋯	NOUN
cana-5714	16	30	≥	≥	NOUN
cana-5714	16	31	|𝜆𝑛|	|𝜆𝑛|	PROPN
cana-5714	16	32	.	.	PUNCT
cana-5714	17	1	in	in	ADP
cana-5714	17	2	1978	1978	NUM
cana-5714	17	3	ivan	ivan	PROPN
cana-5714	17	4	gutman	gutman	NOUN
cana-5714	17	5	[	[	X
cana-5714	17	6	5	5	NUM
cana-5714	17	7	]	]	PUNCT
cana-5714	17	8	computed	compute	VERB
cana-5714	17	9	the	the	DET
cana-5714	17	10	energy	energy	NOUN
cana-5714	17	11	of	of	ADP
cana-5714	17	12	a	a	DET
cana-5714	17	13	graph	graph	NOUN
cana-5714	17	14	𝐺	𝐺	NOUN
cana-5714	17	15	as	as	ADP
cana-5714	17	16	𝐸(𝐺	𝐸(𝐺	NOUN
cana-5714	17	17	)	)	PUNCT
cana-5714	17	18	=	=	PUNCT
cana-5714	18	1	∑	∑	PUNCT
cana-5714	18	2	|𝜆𝑖|𝑛	|𝜆𝑖|𝑛	PROPN
cana-5714	18	3	𝑖=1	𝑖=1	PROPN
cana-5714	18	4	.	.	PUNCT
cana-5714	19	1	li.x	li.x	PROPN
cana-5714	19	2	,	,	PUNCT
cana-5714	19	3	y.	y.	PROPN
cana-5714	19	4	shi	shi	PROPN
cana-5714	19	5	and	and	CCONJ
cana-5714	19	6	i.	i.	PROPN
cana-5714	19	7	gutman	gutman	PROPN
cana-5714	20	1	[	[	X
cana-5714	20	2	6	6	NUM
cana-5714	20	3	]	]	PUNCT
cana-5714	20	4	introduced	introduce	VERB
cana-5714	20	5	the	the	DET
cana-5714	20	6	energy	energy	NOUN
cana-5714	20	7	of	of	ADP
cana-5714	20	8	graph	graph	NOUN
cana-5714	20	9	in	in	ADP
cana-5714	20	10	2012	2012	NUM
cana-5714	20	11	in	in	ADP
cana-5714	20	12	which	which	PRON
cana-5714	20	13	the	the	DET
cana-5714	20	14	adjacency	adjacency	NOUN
cana-5714	20	15	matrix	matrix	NOUN
cana-5714	20	16	of	of	ADP
cana-5714	20	17	a	a	DET
cana-5714	20	18	graph	graph	NOUN
cana-5714	20	19	𝐺is	𝐺is	PROPN
cana-5714	20	20	defined	define	VERB
cana-5714	20	21	as	as	ADP
cana-5714	20	22	𝑎𝑖𝑗	𝑎𝑖𝑗	NOUN
cana-5714	20	23	=	=	SYM
cana-5714	20	24	{	{	PUNCT
cana-5714	20	25	1	1	NUM
cana-5714	20	26	𝑖𝑓	𝑖𝑓	NOUN
cana-5714	20	27	𝑣𝑖𝑣𝑗	𝑣𝑖𝑣𝑗	NOUN
cana-5714	20	28	∈	∈	PROPN
cana-5714	20	29	𝐸	𝐸	PROPN
cana-5714	20	30	0	0	NUM
cana-5714	20	31	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-5714	20	32	spectrum	spectrum	NOUN
cana-5714	20	33	of	of	ADP
cana-5714	20	34	the	the	DET
cana-5714	20	35	graph	graph	NOUN
cana-5714	20	36	is	be	AUX
cana-5714	20	37	denoted	denote	VERB
cana-5714	20	38	by	by	ADP
cana-5714	20	39	𝑆𝑝(𝐺	𝑆𝑝(𝐺	NOUN
cana-5714	20	40	)	)	PUNCT
cana-5714	20	41	=	=	PUNCT
cana-5714	21	1	[	[	PUNCT
cana-5714	21	2	𝜆1	𝜆1	NUM
cana-5714	21	3	𝜆2	𝜆2	PROPN
cana-5714	21	4	⋯	⋯	PROPN
cana-5714	21	5	𝜆𝑛	𝜆𝑛	PROPN
cana-5714	21	6	𝑚1	𝑚1	PROPN
cana-5714	21	7	𝑚2	𝑚2	PROPN
cana-5714	21	8	⋯	⋯	PROPN
cana-5714	21	9	𝑚𝑛	𝑚𝑛	PROPN
cana-5714	21	10	]	]	PUNCT
cana-5714	21	11	mailto:priyakaren2018@gmail.com	mailto:priyakaren2018@gmail.com	PROPN
cana-5714	21	12	mailto:aarokia.lancy@gmail.com	mailto:aarokia.lancy@gmail.com	X
cana-5714	21	13	communications	communication	NOUN
cana-5714	21	14	on	on	ADP
cana-5714	21	15	applied	apply	VERB
cana-5714	21	16	nonlinear	nonlinear	ADJ
cana-5714	21	17	analysis	analysis	NOUN
cana-5714	21	18	issn	issn	NOUN
cana-5714	21	19	:	:	PUNCT
cana-5714	21	20	1074	1074	NUM
cana-5714	21	21	-	-	PUNCT
cana-5714	21	22	133x	133x	NUM
cana-5714	21	23	vol	vol	VERB
cana-5714	21	24	32	32	NUM
cana-5714	21	25	no	no	NOUN
cana-5714	21	26	.	.	PUNCT
cana-5714	22	1	10s	10	NOUN
cana-5714	22	2	(	(	PUNCT
cana-5714	22	3	2025	2025	NUM
cana-5714	22	4	)	)	PUNCT
cana-5714	22	5	2741	2741	NUM
cana-5714	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	22	7	where	where	SCONJ
cana-5714	22	8	𝑚𝑖	𝑚𝑖	NOUN
cana-5714	22	9	′𝑠	′𝑠	PROPN
cana-5714	22	10	denote	denote	VERB
cana-5714	22	11	the	the	DET
cana-5714	22	12	multiplicities	multiplicity	NOUN
cana-5714	22	13	of	of	ADP
cana-5714	22	14	the	the	DET
cana-5714	22	15	corresponding	correspond	VERB
cana-5714	22	16	eigen	eigen	PROPN
cana-5714	22	17	value	value	NOUN
cana-5714	22	18	.	.	PUNCT
cana-5714	23	1	the	the	DET
cana-5714	23	2	total	total	NOUN
cana-5714	23	3	of	of	ADP
cana-5714	23	4	the	the	DET
cana-5714	23	5	absolute	absolute	ADJ
cana-5714	23	6	values	value	NOUN
cana-5714	23	7	of	of	ADP
cana-5714	23	8	the	the	DET
cana-5714	23	9	adjacency	adjacency	NOUN
cana-5714	23	10	matrix	matrix	NOUN
cana-5714	23	11	's	's	PART
cana-5714	23	12	eigenvalues	eigenvalue	NOUN
cana-5714	23	13	equals	equal	VERB
cana-5714	23	14	the	the	DET
cana-5714	23	15	graph	graph	NOUN
cana-5714	23	16	's	's	PART
cana-5714	23	17	energy	energy	NOUN
cana-5714	23	18	.	.	PUNCT
cana-5714	24	1	later	later	ADV
cana-5714	24	2	,	,	PUNCT
cana-5714	24	3	in	in	ADP
cana-5714	24	4	2009	2009	NUM
cana-5714	24	5	,	,	PUNCT
cana-5714	24	6	c.	c.	PROPN
cana-5714	24	7	adiga	adiga	PROPN
cana-5714	24	8	et	et	PROPN
cana-5714	24	9	al	al	PROPN
cana-5714	24	10	.	.	PUNCT
cana-5714	25	1	[	[	X
cana-5714	25	2	1	1	X
cana-5714	25	3	]	]	PUNCT
cana-5714	25	4	defined	define	VERB
cana-5714	25	5	the	the	DET
cana-5714	25	6	graph	graph	NOUN
cana-5714	25	7	's	's	PART
cana-5714	25	8	maximum	maximum	ADJ
cana-5714	25	9	degree	degree	NOUN
cana-5714	25	10	energy	energy	NOUN
cana-5714	25	11	,	,	PUNCT
cana-5714	25	12	which	which	PRON
cana-5714	25	13	is	be	AUX
cana-5714	25	14	dependent	dependent	ADJ
cana-5714	25	15	on	on	ADP
cana-5714	25	16	the	the	DET
cana-5714	25	17	related	relate	VERB
cana-5714	25	18	graph	graph	NOUN
cana-5714	25	19	's	's	PART
cana-5714	25	20	maximum	maximum	ADJ
cana-5714	25	21	degree	degree	NOUN
cana-5714	25	22	matrix	matrix	NOUN
cana-5714	25	23	.	.	PUNCT
cana-5714	26	1	the	the	DET
cana-5714	26	2	maximum	maximum	ADJ
cana-5714	26	3	degree	degree	NOUN
cana-5714	26	4	matrix	matrix	NOUN
cana-5714	26	5	is	be	AUX
cana-5714	26	6	defined	define	VERB
cana-5714	26	7	as	as	ADP
cana-5714	26	8	𝑑𝑖𝑗	𝑑𝑖𝑗	ADJ
cana-5714	26	9	=	=	SYM
cana-5714	26	10	{	{	PUNCT
cana-5714	26	11	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-5714	26	12	{	{	PUNCT
cana-5714	26	13	𝑑(𝑣𝑖	𝑑(𝑣𝑖	NOUN
cana-5714	26	14	)	)	PUNCT
cana-5714	26	15	,	,	PUNCT
cana-5714	26	16	𝑑(𝑣𝑗	𝑑(𝑣𝑗	PROPN
cana-5714	26	17	)	)	PUNCT
cana-5714	26	18	}	}	PUNCT
cana-5714	26	19	,	,	PUNCT
cana-5714	26	20	𝑖𝑓	𝑖𝑓	CCONJ
cana-5714	26	21	𝑣𝑖𝑣𝑗	𝑣𝑖𝑣𝑗	NOUN
cana-5714	26	22	∈	∈	PROPN
cana-5714	26	23	𝐸	𝐸	PROPN
cana-5714	26	24	0	0	NUM
cana-5714	26	25	,	,	PUNCT
cana-5714	26	26	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-5714	26	27	in	in	ADP
cana-5714	26	28	2016	2016	NUM
cana-5714	26	29	,	,	PUNCT
cana-5714	27	1	ahmed	ahmed	PROPN
cana-5714	27	2	m.	m.	PROPN
cana-5714	27	3	naji	naji	PROPN
cana-5714	27	4	et.al	et.al	PROPN
cana-5714	28	1	[	[	X
cana-5714	28	2	2	2	NUM
cana-5714	28	3	]	]	PUNCT
cana-5714	28	4	defined	define	VERB
cana-5714	28	5	the	the	DET
cana-5714	28	6	concept	concept	NOUN
cana-5714	28	7	of	of	ADP
cana-5714	28	8	maximum	maximum	ADJ
cana-5714	28	9	eccentricity	eccentricity	NOUN
cana-5714	28	10	matrix	matrix	NOUN
cana-5714	28	11	.	.	PUNCT
cana-5714	29	1	later	later	ADV
cana-5714	29	2	,	,	PUNCT
cana-5714	29	3	mohammad	mohammad	PROPN
cana-5714	29	4	issa	issa	PROPN
cana-5714	29	5	sowaity	sowaity	PROPN
cana-5714	29	6	and	and	CCONJ
cana-5714	29	7	b.sharada	b.sharada	NOUN
cana-5714	29	8	[	[	X
cana-5714	29	9	7	7	X
cana-5714	29	10	]	]	PUNCT
cana-5714	29	11	in	in	ADP
cana-5714	29	12	2017	2017	NUM
cana-5714	29	13	introduced	introduce	VERB
cana-5714	29	14	the	the	DET
cana-5714	29	15	concept	concept	NOUN
cana-5714	29	16	of	of	ADP
cana-5714	29	17	sum	sum	NOUN
cana-5714	29	18	-	-	PUNCT
cana-5714	29	19	eccentricity	eccentricity	NOUN
cana-5714	29	20	energy	energy	NOUN
cana-5714	29	21	of	of	ADP
cana-5714	29	22	a	a	DET
cana-5714	29	23	graph	graph	NOUN
cana-5714	29	24	in	in	ADP
cana-5714	29	25	2017	2017	NUM
cana-5714	29	26	.	.	PUNCT
cana-5714	30	1	motivated	motivate	VERB
cana-5714	30	2	by	by	ADP
cana-5714	30	3	this	this	PRON
cana-5714	30	4	we	we	PRON
cana-5714	30	5	have	have	AUX
cana-5714	30	6	introduced	introduce	VERB
cana-5714	30	7	the	the	DET
cana-5714	30	8	product	product	NOUN
cana-5714	30	9	eccentricity	eccentricity	NOUN
cana-5714	30	10	energy	energy	NOUN
cana-5714	30	11	of	of	ADP
cana-5714	30	12	a	a	DET
cana-5714	30	13	graph	graph	NOUN
cana-5714	30	14	𝐺.	𝐺.	NOUN
cana-5714	30	15	in	in	ADP
cana-5714	30	16	2025	2025	NUM
cana-5714	30	17	priya	priya	PROPN
cana-5714	30	18	karen	karen	PROPN
cana-5714	30	19	s	s	PROPN
cana-5714	30	20	and	and	CCONJ
cana-5714	30	21	arokia	arokia	PROPN
cana-5714	30	22	lancy	lancy	PROPN
cana-5714	31	1	a	a	PRON
cana-5714	32	1	[	[	X
cana-5714	32	2	10	10	NUM
cana-5714	32	3	]	]	PUNCT
cana-5714	32	4	defined	define	VERB
cana-5714	32	5	the	the	DET
cana-5714	32	6	idea	idea	NOUN
cana-5714	32	7	of	of	ADP
cana-5714	32	8	product	product	NOUN
cana-5714	32	9	eccentricity	eccentricity	NOUN
cana-5714	32	10	energy	energy	NOUN
cana-5714	32	11	as	as	ADP
cana-5714	32	12	𝑃𝑖𝑗	𝑃𝑖𝑗	PROPN
cana-5714	32	13	=	=	PUNCT
cana-5714	32	14	{	{	PUNCT
cana-5714	32	15	𝑒(𝑣𝑖	𝑒(𝑣𝑖	NUM
cana-5714	32	16	)	)	PUNCT
cana-5714	32	17	.	.	PUNCT
cana-5714	33	1	𝑒(𝑣𝑗	𝑒(𝑣𝑗	NOUN
cana-5714	33	2	)	)	PUNCT
cana-5714	34	1	𝑖𝑓	𝑖𝑓	ADP
cana-5714	34	2	𝑣𝑖~	𝑣𝑖~	PROPN
cana-5714	34	3	𝑣𝑗	𝑣𝑗	ADP
cana-5714	34	4	0	0	NUM
cana-5714	34	5	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-5714	34	6	𝑃𝑒(𝐺	𝑃𝑒(𝐺	ADJ
cana-5714	34	7	)	)	PUNCT
cana-5714	34	8	denotes	denote	VERB
cana-5714	34	9	the	the	DET
cana-5714	34	10	product	product	NOUN
cana-5714	34	11	eccentricity	eccentricity	NOUN
cana-5714	34	12	energy	energy	NOUN
cana-5714	34	13	of	of	ADP
cana-5714	34	14	the	the	DET
cana-5714	34	15	graph	graph	NOUN
cana-5714	34	16	.	.	PUNCT
cana-5714	35	1	the	the	DET
cana-5714	35	2	characteristic	characteristic	ADJ
cana-5714	35	3	polynomial	polynomial	NOUN
cana-5714	35	4	of	of	ADP
cana-5714	35	5	the	the	DET
cana-5714	35	6	product	product	NOUN
cana-5714	35	7	eccentricity	eccentricity	NOUN
cana-5714	35	8	matrix	matrix	NOUN
cana-5714	35	9	is	be	AUX
cana-5714	35	10	defined	define	VERB
cana-5714	35	11	by	by	ADP
cana-5714	35	12	|𝜂	|𝜂	PRON
cana-5714	35	13	𝐼	𝐼	PROPN
cana-5714	35	14	−	−	PROPN
cana-5714	35	15	𝑃𝑒(𝐺)|	𝑃𝑒(𝐺)|	NOUN
cana-5714	35	16	and	and	CCONJ
cana-5714	35	17	the	the	DET
cana-5714	35	18	corresponding	corresponding	ADJ
cana-5714	35	19	characteristic	characteristic	ADJ
cana-5714	35	20	equation	equation	NOUN
cana-5714	35	21	is	be	AUX
cana-5714	35	22	𝜂𝐼	𝜂𝐼	NOUN
cana-5714	35	23	−	−	NOUN
cana-5714	35	24	𝑃𝑒(𝐺	𝑃𝑒(𝐺	ADJ
cana-5714	35	25	)	)	PUNCT
cana-5714	36	1	=	=	SYM
cana-5714	36	2	0	0	X
cana-5714	36	3	.	.	PUNCT
cana-5714	37	1	here	here	ADV
cana-5714	37	2	,	,	PUNCT
cana-5714	37	3	𝐼	𝐼	PROPN
cana-5714	37	4	denotes	denote	VERB
cana-5714	37	5	the	the	DET
cana-5714	37	6	identity	identity	NOUN
cana-5714	37	7	matrix	matrix	NOUN
cana-5714	37	8	of	of	ADP
cana-5714	37	9	order	order	NOUN
cana-5714	37	10	𝑛.	𝑛.	NOUN
cana-5714	37	11	𝑃𝑒(𝐺	𝑃𝑒(𝐺	ADJ
cana-5714	37	12	)	)	PUNCT
cana-5714	37	13	is	be	AUX
cana-5714	37	14	a	a	DET
cana-5714	37	15	real	real	ADJ
cana-5714	37	16	symmetric	symmetric	ADJ
cana-5714	37	17	matrix	matrix	NOUN
cana-5714	37	18	with	with	ADP
cana-5714	37	19	its	its	PRON
cana-5714	37	20	trace	trace	NOUN
cana-5714	37	21	zero	zero	NUM
cana-5714	37	22	.	.	PUNCT
cana-5714	38	1	since	since	SCONJ
cana-5714	38	2	𝐺	𝐺	PROPN
cana-5714	38	3	is	be	AUX
cana-5714	38	4	a	a	DET
cana-5714	38	5	simple	simple	ADJ
cana-5714	38	6	loopless	loopless	NOUN
cana-5714	38	7	graph	graph	NOUN
cana-5714	38	8	all	all	PRON
cana-5714	38	9	𝑎𝑖𝑖	𝑎𝑖𝑖	NOUN
cana-5714	38	10	=	=	SYM
cana-5714	38	11	0	0	NUM
cana-5714	38	12	and	and	CCONJ
cana-5714	38	13	its	its	PRON
cana-5714	38	14	eigen	eigen	PROPN
cana-5714	38	15	values	value	NOUN
cana-5714	38	16	with	with	ADP
cana-5714	38	17	real	real	ADJ
cana-5714	38	18	sum	sum	NOUN
cana-5714	38	19	equals	equal	VERB
cana-5714	38	20	zero	zero	NUM
cana-5714	38	21	(	(	PUNCT
cana-5714	38	22	𝑡𝑟(𝑃𝑒(𝐺	𝑡𝑟(𝑃𝑒(𝐺	PROPN
cana-5714	38	23	)	)	PUNCT
cana-5714	38	24	=	=	NOUN
cana-5714	38	25	0	0	NUM
cana-5714	38	26	)	)	PUNCT
cana-5714	38	27	.	.	PUNCT
cana-5714	39	1	eigen	eigen	PROPN
cana-5714	39	2	values	value	NOUN
cana-5714	39	3	of	of	ADP
cana-5714	39	4	the	the	DET
cana-5714	39	5	product	product	NOUN
cana-5714	39	6	eccentricity	eccentricity	NOUN
cana-5714	39	7	matrix	matrix	NOUN
cana-5714	39	8	are	be	AUX
cana-5714	39	9	the	the	DET
cana-5714	39	10	roots	root	NOUN
cana-5714	39	11	of	of	ADP
cana-5714	39	12	the	the	DET
cana-5714	39	13	corresponding	corresponding	ADJ
cana-5714	39	14	characteristic	characteristic	ADJ
cana-5714	39	15	polynomial	polynomial	NOUN
cana-5714	39	16	.	.	PUNCT
cana-5714	40	1	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	40	2	)	)	PUNCT
cana-5714	40	3	is	be	AUX
cana-5714	40	4	defined	define	VERB
cana-5714	40	5	as	as	ADP
cana-5714	40	6	the	the	DET
cana-5714	40	7	sum	sum	NOUN
cana-5714	40	8	of	of	ADP
cana-5714	40	9	the	the	DET
cana-5714	40	10	absolute	absolute	ADJ
cana-5714	40	11	eigen	eigen	PROPN
cana-5714	40	12	values	value	NOUN
cana-5714	40	13	,	,	PUNCT
cana-5714	40	14	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	40	15	)	)	PUNCT
cana-5714	40	16	=	=	PUNCT
cana-5714	41	1	∑	∑	PUNCT
cana-5714	41	2	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	41	3	𝑛	𝑛	PRON
cana-5714	41	4	𝑖=1	𝑖=1	PROPN
cana-5714	41	5	𝜂1	𝜂1	PROPN
cana-5714	41	6	,	,	PUNCT
cana-5714	41	7	𝜂2	𝜂2	PROPN
cana-5714	41	8	,	,	PUNCT
cana-5714	41	9	⋯	⋯	VERB
cana-5714	41	10	𝜂𝑛	𝜂𝑛	NOUN
cana-5714	41	11	are	be	AUX
cana-5714	41	12	the	the	DET
cana-5714	41	13	eigen	eigen	PROPN
cana-5714	41	14	values	value	NOUN
cana-5714	41	15	of	of	ADP
cana-5714	41	16	the	the	DET
cana-5714	41	17	given	give	VERB
cana-5714	41	18	product	product	NOUN
cana-5714	41	19	eccentricity	eccentricity	NOUN
cana-5714	41	20	matrix	matrix	NOUN
cana-5714	41	21	.	.	PUNCT
cana-5714	42	1	2	2	X
cana-5714	42	2	.	.	NUM
cana-5714	42	3	preliminaries	preliminary	NOUN
cana-5714	42	4	a	a	DET
cana-5714	42	5	few	few	ADJ
cana-5714	42	6	important	important	ADJ
cana-5714	42	7	theorems	theorem	NOUN
cana-5714	42	8	that	that	PRON
cana-5714	42	9	are	be	AUX
cana-5714	42	10	utilized	utilize	VERB
cana-5714	42	11	throughout	throughout	ADP
cana-5714	42	12	the	the	DET
cana-5714	42	13	work	work	NOUN
cana-5714	42	14	are	be	AUX
cana-5714	42	15	listed	list	VERB
cana-5714	42	16	below	below	ADV
cana-5714	42	17	in	in	ADP
cana-5714	42	18	order	order	NOUN
cana-5714	42	19	to	to	PART
cana-5714	42	20	show	show	VERB
cana-5714	42	21	the	the	DET
cana-5714	42	22	complete	complete	ADJ
cana-5714	42	23	results	result	NOUN
cana-5714	42	24	.	.	PUNCT
cana-5714	43	1	theorem	theorem	VERB
cana-5714	43	2	2.1:[9	2.1:[9	NUM
cana-5714	43	3	]	]	PUNCT
cana-5714	43	4	suppose	suppose	VERB
cana-5714	43	5	𝑎𝑖	𝑎𝑖	ADP
cana-5714	43	6	and	and	CCONJ
cana-5714	43	7	𝑏𝑖	𝑏𝑖	ADP
cana-5714	43	8	,	,	PUNCT
cana-5714	43	9	1	1	NUM
cana-5714	43	10	≤	≤	NUM
cana-5714	43	11	𝑖	𝑖	SYM
cana-5714	43	12	≤	≤	NOUN
cana-5714	44	1	𝑛	𝑛	PROPN
cana-5714	44	2	are	be	AUX
cana-5714	44	3	non	non	ADJ
cana-5714	44	4	-	-	ADJ
cana-5714	44	5	negative	negative	ADJ
cana-5714	44	6	real	real	ADJ
cana-5714	44	7	numbers	number	NOUN
cana-5714	44	8	,	,	PUNCT
cana-5714	44	9	then	then	ADV
cana-5714	44	10	∑	∑	PUNCT
cana-5714	44	11	𝑎𝑖	𝑎𝑖	ADP
cana-5714	44	12	2	2	NUM
cana-5714	44	13	𝑛	𝑛	PRON
cana-5714	44	14	𝑖=1	𝑖=1	PUNCT
cana-5714	44	15	∑	∑	PROPN
cana-5714	44	16	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	44	17	2	2	NUM
cana-5714	44	18	𝑛	𝑛	DET
cana-5714	44	19	𝑖=1	𝑖=1	PUNCT
cana-5714	44	20	≤	≤	NUM
cana-5714	44	21	1	1	NUM
cana-5714	44	22	4	4	NUM
cana-5714	44	23	(	(	PUNCT
cana-5714	44	24	√	√	NUM
cana-5714	44	25	𝑀1𝑀2	𝑀1𝑀2	NOUN
cana-5714	45	1	𝑚1𝑚2	𝑚1𝑚2	X
cana-5714	45	2	+	+	CCONJ
cana-5714	45	3	√	√	ADJ
cana-5714	45	4	𝑚1𝑚2	𝑚1𝑚2	NUM
cana-5714	45	5	𝑀1𝑀2	𝑀1𝑀2	PROPN
cana-5714	45	6	)	)	PUNCT
cana-5714	45	7	2	2	NUM
cana-5714	45	8	(	(	PUNCT
cana-5714	45	9	∑	∑	ADV
cana-5714	45	10	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	45	11	𝑛	𝑛	PRON
cana-5714	45	12	𝑖=1	𝑖=1	PROPN
cana-5714	45	13	)	)	PUNCT
cana-5714	45	14	2	2	NUM
cana-5714	45	15	where	where	SCONJ
cana-5714	45	16	𝑀1	𝑀1	PROPN
cana-5714	45	17	=	=	SYM
cana-5714	45	18	max	max	PROPN
cana-5714	45	19	1≤i≤n	1≤i≤n	NUM
cana-5714	45	20	(	(	PUNCT
cana-5714	45	21	𝑎𝑖	𝑎𝑖	PROPN
cana-5714	45	22	)	)	PUNCT
cana-5714	45	23	;	;	PUNCT
cana-5714	45	24	𝑀2	𝑀2	PROPN
cana-5714	45	25	=	=	SYM
cana-5714	45	26	max	max	PROPN
cana-5714	45	27	1≤i≤n	1≤i≤n	NUM
cana-5714	45	28	(	(	PUNCT
cana-5714	45	29	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	45	30	)	)	PUNCT
cana-5714	45	31	;	;	PUNCT
cana-5714	45	32	𝑚1	𝑚1	NOUN
cana-5714	45	33	=	=	SYM
cana-5714	45	34	max	max	PROPN
cana-5714	45	35	1≤i≤n	1≤i≤n	NUM
cana-5714	45	36	(	(	PUNCT
cana-5714	45	37	𝑎𝑖	𝑎𝑖	PROPN
cana-5714	45	38	)	)	PUNCT
cana-5714	45	39	;	;	PUNCT
cana-5714	45	40	𝑚2	𝑚2	NOUN
cana-5714	45	41	=	=	SYM
cana-5714	45	42	max	max	PROPN
cana-5714	45	43	1≤i≤n	1≤i≤n	NUM
cana-5714	45	44	(	(	PUNCT
cana-5714	45	45	𝑏𝑖	𝑏𝑖	NOUN
cana-5714	45	46	)	)	PUNCT
cana-5714	45	47	theorem	theorem	NOUN
cana-5714	45	48	2.2:[8	2.2:[8	NUM
cana-5714	45	49	]	]	PUNCT
cana-5714	45	50	let	let	VERB
cana-5714	45	51	𝑎𝑖	𝑎𝑖	INTJ
cana-5714	45	52	and	and	CCONJ
cana-5714	45	53	𝑏𝑖	𝑏𝑖	NOUN
cana-5714	45	54	,	,	PUNCT
cana-5714	45	55	1	1	NUM
cana-5714	45	56	≤	≤	NUM
cana-5714	45	57	𝑖	𝑖	SYM
cana-5714	45	58	≤	≤	NOUN
cana-5714	45	59	𝑛	𝑛	PROPN
cana-5714	45	60	are	be	AUX
cana-5714	45	61	non	non	ADJ
cana-5714	45	62	-	-	ADJ
cana-5714	45	63	negative	negative	ADJ
cana-5714	45	64	real	real	ADJ
cana-5714	45	65	numbers	number	NOUN
cana-5714	45	66	,	,	PUNCT
cana-5714	45	67	then	then	ADV
cana-5714	45	68	communications	communication	NOUN
cana-5714	45	69	on	on	ADP
cana-5714	45	70	applied	apply	VERB
cana-5714	45	71	nonlinear	nonlinear	ADJ
cana-5714	45	72	analysis	analysis	NOUN
cana-5714	45	73	issn	issn	NOUN
cana-5714	45	74	:	:	PUNCT
cana-5714	45	75	1074	1074	NUM
cana-5714	45	76	-	-	PUNCT
cana-5714	45	77	133x	133x	NUM
cana-5714	45	78	vol	vol	VERB
cana-5714	45	79	32	32	NUM
cana-5714	45	80	no	no	NOUN
cana-5714	45	81	.	.	PUNCT
cana-5714	46	1	10s	10	NOUN
cana-5714	46	2	(	(	PUNCT
cana-5714	46	3	2025	2025	NUM
cana-5714	46	4	)	)	PUNCT
cana-5714	46	5	2742	2742	NUM
cana-5714	46	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	46	7	∑	∑	PUNCT
cana-5714	46	8	𝑎𝑖	𝑎𝑖	ADP
cana-5714	46	9	2	2	NUM
cana-5714	46	10	𝑛	𝑛	PRON
cana-5714	46	11	𝑖=1	𝑖=1	PUNCT
cana-5714	46	12	∑	∑	PROPN
cana-5714	46	13	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	46	14	2	2	NUM
cana-5714	46	15	𝑛	𝑛	PRON
cana-5714	46	16	𝑖=1	𝑖=1	PROPN
cana-5714	46	17	−	−	PROPN
cana-5714	46	18	(	(	PUNCT
cana-5714	46	19	∑	∑	PUNCT
cana-5714	46	20	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	46	21	𝑛	𝑛	PRON
cana-5714	46	22	𝑖=1	𝑖=1	PROPN
cana-5714	46	23	)	)	PUNCT
cana-5714	46	24	2	2	NUM
cana-5714	46	25	≤	≤	NOUN
cana-5714	46	26	𝑛2	𝑛2	NOUN
cana-5714	46	27	4	4	NUM
cana-5714	46	28	(	(	PUNCT
cana-5714	46	29	𝑀1𝑀2	𝑀1𝑀2	NOUN
cana-5714	46	30	−	−	NOUN
cana-5714	46	31	𝑚1𝑚2)2	𝑚1𝑚2)2	NOUN
cana-5714	46	32	where	where	SCONJ
cana-5714	46	33	𝑀1𝑀2	𝑀1𝑀2	PROPN
cana-5714	46	34	and	and	CCONJ
cana-5714	46	35	𝑚1𝑚2	𝑚1𝑚2	PRON
cana-5714	46	36	are	be	AUX
cana-5714	46	37	defined	define	VERB
cana-5714	46	38	similarly	similarly	ADV
cana-5714	46	39	to	to	PART
cana-5714	46	40	theorem	theorem	VERB
cana-5714	46	41	2.1	2.1	NUM
cana-5714	46	42	theorem	theorem	NOUN
cana-5714	46	43	2.3:[3	2.3:[3	NUM
cana-5714	46	44	]	]	PUNCT
cana-5714	46	45	suppose	suppose	VERB
cana-5714	46	46	𝑎𝑖	𝑎𝑖	ADP
cana-5714	46	47	and	and	CCONJ
cana-5714	46	48	𝑏𝑖	𝑏𝑖	ADP
cana-5714	46	49	,	,	PUNCT
cana-5714	46	50	1	1	NUM
cana-5714	46	51	≤	≤	NUM
cana-5714	46	52	𝑖	𝑖	SYM
cana-5714	46	53	≤	≤	NOUN
cana-5714	46	54	𝑛	𝑛	PRON
cana-5714	46	55	are	be	AUX
cana-5714	46	56	positive	positive	ADJ
cana-5714	46	57	real	real	ADJ
cana-5714	46	58	numbers	number	NOUN
cana-5714	46	59	,	,	PUNCT
cana-5714	46	60	then	then	ADV
cana-5714	46	61	|𝑛	|𝑛	VERB
cana-5714	46	62	∑	∑	PUNCT
cana-5714	46	63	𝑎𝑖	𝑎𝑖	ADV
cana-5714	46	64	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	46	65	𝑛	𝑛	PRON
cana-5714	46	66	𝑖=1	𝑖=1	PROPN
cana-5714	47	1	−	−	PROPN
cana-5714	47	2	∑	∑	PROPN
cana-5714	47	3	𝑎𝑖	𝑎𝑖	ADP
cana-5714	47	4	𝑛	𝑛	PROPN
cana-5714	47	5	𝑖=1	𝑖=1	PUNCT
cana-5714	47	6	∑	∑	PROPN
cana-5714	47	7	𝑏𝑖	𝑏𝑖	ADP
cana-5714	47	8	𝑛	𝑛	PRON
cana-5714	47	9	𝑖=1	𝑖=1	PROPN
cana-5714	48	1	|	|	ADV
cana-5714	48	2	≤	≤	NOUN
cana-5714	48	3	𝜇(𝑛)(𝐴	𝜇(𝑛)(𝐴	ADP
cana-5714	48	4	−	−	PROPN
cana-5714	48	5	𝑎)(𝐵	𝑎)(𝐵	PROPN
cana-5714	48	6	−	−	ADP
cana-5714	48	7	𝑏	𝑏	NOUN
cana-5714	48	8	)	)	PUNCT
cana-5714	48	9	here	here	ADV
cana-5714	48	10	𝑎	𝑎	PROPN
cana-5714	48	11	,	,	PUNCT
cana-5714	48	12	𝑏	𝑏	NOUN
cana-5714	48	13	,	,	PUNCT
cana-5714	48	14	𝐴	𝐴	PROPN
cana-5714	48	15	and	and	CCONJ
cana-5714	48	16	𝐵	𝐵	PROPN
cana-5714	48	17	are	be	AUX
cana-5714	48	18	the	the	DET
cana-5714	48	19	real	real	ADJ
cana-5714	48	20	constants	constant	NOUN
cana-5714	48	21	,	,	PUNCT
cana-5714	48	22	1	1	NUM
cana-5714	48	23	≤	≤	NUM
cana-5714	48	24	𝑖	𝑖	SYM
cana-5714	48	25	≤	≤	NUM
cana-5714	48	26	𝑛	𝑛	NOUN
cana-5714	48	27	,	,	PUNCT
cana-5714	48	28	𝑎	𝑎	ADJ
cana-5714	48	29	≤	≤	NUM
cana-5714	48	30	𝑎𝑖	𝑎𝑖	ADP
cana-5714	48	31	≤	≤	ADJ
cana-5714	48	32	𝐴	𝐴	PROPN
cana-5714	48	33	and	and	CCONJ
cana-5714	48	34	𝑏	𝑏	DET
cana-5714	48	35	≤	≤	NOUN
cana-5714	48	36	𝑏𝑖	𝑏𝑖	ADP
cana-5714	48	37	≤	≤	NUM
cana-5714	48	38	𝐵.	𝐵.	NOUN
cana-5714	48	39	further	far	ADV
cana-5714	48	40	we	we	PRON
cana-5714	48	41	have	have	VERB
cana-5714	48	42	,	,	PUNCT
cana-5714	48	43	𝜇(𝑛	𝜇(𝑛	NOUN
cana-5714	48	44	)	)	PUNCT
cana-5714	48	45	=	=	SYM
cana-5714	49	1	𝑛	𝑛	PART
cana-5714	49	2	⌊	⌊	NOUN
cana-5714	49	3	𝑛	𝑛	DET
cana-5714	49	4	2	2	NUM
cana-5714	49	5	⌋	⌋	NOUN
cana-5714	49	6	(	(	PUNCT
cana-5714	49	7	1	1	NUM
cana-5714	49	8	−	−	NUM
cana-5714	49	9	1	1	NUM
cana-5714	49	10	𝑛	𝑛	VERB
cana-5714	49	11	⌊	⌊	VERB
cana-5714	49	12	𝑛	𝑛	DET
cana-5714	49	13	2	2	NUM
cana-5714	49	14	⌋	⌋	NOUN
cana-5714	49	15	)	)	PUNCT
cana-5714	49	16	theorem	theorem	VERB
cana-5714	49	17	2.4:[4	2.4:[4	NUM
cana-5714	49	18	]	]	X
cana-5714	49	19	let	let	VERB
cana-5714	49	20	𝑎𝑖	𝑎𝑖	PRON
cana-5714	49	21	and	and	CCONJ
cana-5714	49	22	𝑏𝑖	𝑏𝑖	NOUN
cana-5714	49	23	,	,	PUNCT
cana-5714	49	24	1	1	NUM
cana-5714	49	25	≤	≤	NUM
cana-5714	49	26	𝑖	𝑖	SYM
cana-5714	49	27	≤	≤	NOUN
cana-5714	49	28	𝑛	𝑛	PROPN
cana-5714	49	29	are	be	AUX
cana-5714	49	30	non	non	ADJ
cana-5714	49	31	-	-	ADJ
cana-5714	49	32	negative	negative	ADJ
cana-5714	49	33	real	real	ADJ
cana-5714	49	34	numbers	number	NOUN
cana-5714	49	35	,	,	PUNCT
cana-5714	49	36	then	then	ADV
cana-5714	49	37	∑	∑	ADP
cana-5714	49	38	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	49	39	2	2	NUM
cana-5714	49	40	𝑛	𝑛	PRON
cana-5714	49	41	𝑖=1	𝑖=1	PROPN
cana-5714	50	1	+	+	CCONJ
cana-5714	50	2	𝑟𝑅	𝑟𝑅	NOUN
cana-5714	50	3	∑	∑	PUNCT
cana-5714	50	4	𝑎𝑖	𝑎𝑖	PRON
cana-5714	50	5	2	2	NUM
cana-5714	50	6	𝑛	𝑛	PRON
cana-5714	50	7	𝑖=1	𝑖=1	PROPN
cana-5714	50	8	≤	≤	NUM
cana-5714	50	9	(	(	PUNCT
cana-5714	50	10	𝑟	𝑟	X
cana-5714	50	11	+	+	CCONJ
cana-5714	50	12	𝑅	𝑅	NOUN
cana-5714	50	13	)	)	PUNCT
cana-5714	50	14	(	(	PUNCT
cana-5714	50	15	∑	∑	ADV
cana-5714	50	16	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	50	17	𝑛	𝑛	PRON
cana-5714	50	18	𝑖=1	𝑖=1	PROPN
cana-5714	50	19	)	)	PUNCT
cana-5714	50	20	here	here	ADV
cana-5714	50	21	𝑟	𝑟	X
cana-5714	50	22	and	and	CCONJ
cana-5714	50	23	𝑅	𝑅	PROPN
cana-5714	50	24	are	be	AUX
cana-5714	50	25	real	real	ADJ
cana-5714	50	26	constants	constant	NOUN
cana-5714	50	27	,	,	PUNCT
cana-5714	50	28	1	1	NUM
cana-5714	50	29	≤	≤	NUM
cana-5714	50	30	𝑖	𝑖	SYM
cana-5714	50	31	≤	≤	NOUN
cana-5714	50	32	𝑛	𝑛	PROPN
cana-5714	50	33	holds	hold	VERB
cana-5714	50	34	𝑟𝑎𝑖	𝑟𝑎𝑖	NOUN
cana-5714	50	35	≤	≤	X
cana-5714	50	36	𝑏𝑖	𝑏𝑖	ADP
cana-5714	50	37	≤	≤	NUM
cana-5714	50	38	𝑅𝑎𝑖	𝑅𝑎𝑖	PROPN
cana-5714	50	39	3.bounds	3.bounds	NUM
cana-5714	50	40	for	for	ADP
cana-5714	50	41	the	the	DET
cana-5714	50	42	eigen	eigen	PROPN
cana-5714	50	43	values	value	NOUN
cana-5714	50	44	of	of	ADP
cana-5714	50	45	product	product	NOUN
cana-5714	50	46	eccentricity	eccentricity	NOUN
cana-5714	50	47	matrix	matrix	NOUN
cana-5714	50	48	of	of	ADP
cana-5714	50	49	a	a	DET
cana-5714	50	50	graph	graph	NOUN
cana-5714	50	51	the	the	DET
cana-5714	50	52	following	follow	VERB
cana-5714	50	53	lemma	lemma	PROPN
cana-5714	50	54	is	be	AUX
cana-5714	50	55	required	require	VERB
cana-5714	50	56	to	to	PART
cana-5714	50	57	support	support	VERB
cana-5714	50	58	the	the	DET
cana-5714	50	59	subsequent	subsequent	ADJ
cana-5714	50	60	findings	finding	NOUN
cana-5714	50	61	.	.	PUNCT
cana-5714	51	1	lemma	lemma	PROPN
cana-5714	51	2	3.1	3.1	NUM
cana-5714	51	3	.	.	PUNCT
cana-5714	52	1	if	if	SCONJ
cana-5714	52	2	the	the	DET
cana-5714	52	3	trace	trace	NOUN
cana-5714	52	4	of	of	ADP
cana-5714	52	5	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	52	6	)	)	PUNCT
cana-5714	52	7	=	=	SYM
cana-5714	52	8	0	0	NUM
cana-5714	52	9	,	,	PUNCT
cana-5714	52	10	then	then	ADV
cana-5714	52	11	the	the	DET
cana-5714	52	12	eigen	eigen	PROPN
cana-5714	52	13	values	value	NOUN
cana-5714	52	14	obtained	obtain	VERB
cana-5714	52	15	from	from	ADP
cana-5714	52	16	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	52	17	)	)	PUNCT
cana-5714	52	18	matrix	matrix	NOUN
cana-5714	52	19	satisfies	satisfy	VERB
cana-5714	52	20	the	the	DET
cana-5714	52	21	following	follow	VERB
cana-5714	52	22	3	3	NUM
cana-5714	52	23	.	.	PUNCT
cana-5714	52	24	∑	∑	PROPN
cana-5714	52	25	𝜂𝑖	𝜂𝑖	X
cana-5714	52	26	=	=	SYM
cana-5714	52	27	0𝑛	0𝑛	VERB
cana-5714	52	28	𝑖=1	𝑖=1	PROPN
cana-5714	52	29	4	4	X
cana-5714	52	30	.	.	X
cana-5714	53	1	∑	∑	PROPN
cana-5714	53	2	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	53	3	2𝑛	2𝑛	PROPN
cana-5714	53	4	𝑖=1	𝑖=1	PUNCT
cana-5714	53	5	=	=	SYM
cana-5714	53	6	𝑡𝑟𝑎𝑐𝑒	𝑡𝑟𝑎𝑐𝑒	ADV
cana-5714	53	7	(	(	PUNCT
cana-5714	53	8	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	53	9	)	)	PUNCT
cana-5714	53	10	)	)	PUNCT
cana-5714	53	11	2	2	NUM
cana-5714	53	12	∑	∑	NOUN
cana-5714	53	13	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	53	14	2	2	NUM
cana-5714	53	15	𝑛	𝑛	PRON
cana-5714	53	16	𝑖=1	𝑖=1	PUNCT
cana-5714	53	17	=	=	SYM
cana-5714	53	18	∑(𝜂𝑖)2	∑(𝜂𝑖)2	PROPN
cana-5714	53	19	𝑛	𝑛	PRON
cana-5714	54	1	𝑖=1	𝑖=1	PUNCT
cana-5714	55	1	=	=	SYM
cana-5714	56	1	(	(	PUNCT
cana-5714	56	2	𝑡𝑟𝑎𝑐𝑒(𝑃𝑒(𝐺	𝑡𝑟𝑎𝑐𝑒(𝑃𝑒(𝐺	PROPN
cana-5714	56	3	)	)	PUNCT
cana-5714	56	4	)	)	PUNCT
cana-5714	56	5	)	)	PUNCT
cana-5714	57	1	2	2	NUM
cana-5714	58	1	=	=	SYM
cana-5714	58	2	∑	∑	PUNCT
cana-5714	58	3	∑	∑	ADV
cana-5714	58	4	𝑝𝑖𝑘𝑝𝑘𝑖	𝑝𝑖𝑘𝑝𝑘𝑖	ADV
cana-5714	58	5	𝑛	𝑛	ADP
cana-5714	58	6	𝑘=1	𝑘=1	NOUN
cana-5714	58	7	𝑛	𝑛	PRON
cana-5714	58	8	𝑖=1	𝑖=1	PUNCT
cana-5714	58	9	=	=	SYM
cana-5714	58	10	2	2	NUM
cana-5714	58	11	∑	∑	PUNCT
cana-5714	58	12	∑(𝑝𝑖𝑘)2	∑(𝑝𝑖𝑘)2	PROPN
cana-5714	58	13	𝑛	𝑛	PROPN
cana-5714	58	14	𝑖<𝑘	𝑖<𝑘	NOUN
cana-5714	58	15	𝑛	𝑛	PROPN
cana-5714	58	16	𝑖=1	𝑖=1	PUNCT
cana-5714	58	17	=	=	SYM
cana-5714	58	18	2	2	NUM
cana-5714	58	19	∑	∑	PUNCT
cana-5714	58	20	(	(	PUNCT
cana-5714	58	21	𝑒(𝑣𝑖	𝑒(𝑣𝑖	NUM
cana-5714	58	22	)	)	PUNCT
cana-5714	58	23	.	.	PUNCT
cana-5714	59	1	𝑒(𝑣𝑗	𝑒(𝑣𝑗	NOUN
cana-5714	59	2	)	)	PUNCT
cana-5714	59	3	)	)	PUNCT
cana-5714	59	4	2	2	NUM
cana-5714	59	5	𝑛	𝑛	NOUN
cana-5714	59	6	𝑖=1,𝑖<𝑘	𝑖=1,𝑖<𝑘	NOUN
cana-5714	59	7	∑(𝜂𝑖)2	∑(𝜂𝑖)2	PROPN
cana-5714	59	8	=	=	SYM
cana-5714	60	1	2	2	NUM
cana-5714	60	2	𝐻	𝐻	NOUN
cana-5714	60	3	𝑛	𝑛	PRON
cana-5714	60	4	𝑖=1	𝑖=1	PROPN
cana-5714	60	5	communications	communication	NOUN
cana-5714	60	6	on	on	ADP
cana-5714	60	7	applied	apply	VERB
cana-5714	60	8	nonlinear	nonlinear	ADJ
cana-5714	60	9	analysis	analysis	NOUN
cana-5714	60	10	issn	issn	NOUN
cana-5714	60	11	:	:	PUNCT
cana-5714	60	12	1074	1074	NUM
cana-5714	60	13	-	-	PUNCT
cana-5714	60	14	133x	133x	NUM
cana-5714	60	15	vol	vol	VERB
cana-5714	60	16	32	32	NUM
cana-5714	60	17	no	no	NOUN
cana-5714	60	18	.	.	PUNCT
cana-5714	61	1	10s	10	NOUN
cana-5714	61	2	(	(	PUNCT
cana-5714	61	3	2025	2025	NUM
cana-5714	61	4	)	)	PUNCT
cana-5714	61	5	2743	2743	NUM
cana-5714	61	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	61	7	where	where	SCONJ
cana-5714	61	8	𝐻	𝐻	PROPN
cana-5714	61	9	=	=	SYM
cana-5714	61	10	∑	∑	PUNCT
cana-5714	61	11	(	(	PUNCT
cana-5714	61	12	𝑒(𝑣𝑖	𝑒(𝑣𝑖	NUM
cana-5714	61	13	)	)	PUNCT
cana-5714	61	14	.	.	PUNCT
cana-5714	62	1	𝑒(𝑣𝑗	𝑒(𝑣𝑗	NOUN
cana-5714	62	2	)	)	PUNCT
cana-5714	62	3	)	)	PUNCT
cana-5714	62	4	2	2	NUM
cana-5714	62	5	𝑛	𝑛	NOUN
cana-5714	62	6	𝑖=1,𝑖<𝑘	𝑖=1,𝑖<𝑘	X
cana-5714	62	7	theorem	theorem	VERB
cana-5714	62	8	3.2	3.2	NUM
cana-5714	62	9	let	let	VERB
cana-5714	62	10	𝐺	𝐺	PROPN
cana-5714	62	11	be	be	AUX
cana-5714	62	12	a	a	DET
cana-5714	62	13	graph	graph	NOUN
cana-5714	62	14	with	with	ADP
cana-5714	62	15	𝑛	𝑛	DET
cana-5714	62	16	−	−	PROPN
cana-5714	62	17	vertices	vertex	NOUN
cana-5714	62	18	,	,	PUNCT
cana-5714	62	19	then	then	ADV
cana-5714	62	20	𝜂𝑖	𝜂𝑖	X
cana-5714	62	21	≤	≤	NUM
cana-5714	63	1	√	√	NUM
cana-5714	63	2	2𝐻(𝑛	2𝐻(𝑛	NUM
cana-5714	63	3	−	−	NUM
cana-5714	63	4	1	1	NUM
cana-5714	63	5	)	)	PUNCT
cana-5714	63	6	𝑛	𝑛	DET
cana-5714	63	7	proof	proof	NOUN
cana-5714	63	8	.	.	PUNCT
cana-5714	64	1	consider	consider	VERB
cana-5714	64	2	a	a	DET
cana-5714	64	3	graph	graph	NOUN
cana-5714	64	4	𝐺	𝐺	NOUN
cana-5714	64	5	with	with	ADP
cana-5714	64	6	𝑛	𝑛	DET
cana-5714	64	7	−vertices	−vertices	PROPN
cana-5714	64	8	.	.	PUNCT
cana-5714	65	1	let	let	VERB
cana-5714	65	2	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	65	3	)	)	PUNCT
cana-5714	65	4	be	be	AUX
cana-5714	65	5	the	the	DET
cana-5714	65	6	product	product	NOUN
cana-5714	65	7	eccentricity	eccentricity	NOUN
cana-5714	65	8	matrix	matrix	NOUN
cana-5714	65	9	of	of	ADP
cana-5714	65	10	graph	graph	NOUN
cana-5714	65	11	𝐺	𝐺	PROPN
cana-5714	65	12	and	and	CCONJ
cana-5714	65	13	𝜂1	𝜂1	PROPN
cana-5714	65	14	,	,	PUNCT
cana-5714	65	15	𝜂2	𝜂2	PROPN
cana-5714	65	16	,	,	PUNCT
cana-5714	65	17	⋯	⋯	VERB
cana-5714	65	18	𝜂𝑛	𝜂𝑛	NOUN
cana-5714	65	19	are	be	AUX
cana-5714	65	20	the	the	DET
cana-5714	65	21	eigen	eigen	PROPN
cana-5714	65	22	values	value	NOUN
cana-5714	65	23	obtained	obtain	VERB
cana-5714	65	24	from	from	ADP
cana-5714	65	25	the	the	DET
cana-5714	65	26	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	65	27	)	)	PUNCT
cana-5714	65	28	matrix	matrix	NOUN
cana-5714	65	29	,	,	PUNCT
cana-5714	65	30	where	where	SCONJ
cana-5714	65	31	𝜂1	𝜂1	PROPN
cana-5714	65	32	is	be	AUX
cana-5714	65	33	the	the	DET
cana-5714	65	34	largest	large	ADJ
cana-5714	65	35	eigen	eigen	NOUN
cana-5714	65	36	value	value	NOUN
cana-5714	65	37	among	among	ADP
cana-5714	65	38	all	all	DET
cana-5714	65	39	the	the	DET
cana-5714	65	40	eigen	eigen	PROPN
cana-5714	65	41	values	value	NOUN
cana-5714	65	42	computed	compute	VERB
cana-5714	65	43	and	and	CCONJ
cana-5714	65	44	using	use	VERB
cana-5714	65	45	the	the	DET
cana-5714	65	46	cauchy	cauchy	NOUN
cana-5714	65	47	-	-	PUNCT
cana-5714	65	48	schwarz	schwarz	PROPN
cana-5714	65	49	inequality	inequality	NOUN
cana-5714	65	50	is	be	AUX
cana-5714	65	51	used	use	VERB
cana-5714	65	52	to	to	PART
cana-5714	65	53	obtain	obtain	VERB
cana-5714	65	54	the	the	DET
cana-5714	65	55	bound	bind	VERB
cana-5714	65	56	for	for	ADP
cana-5714	65	57	𝜂1	𝜂1	PROPN
cana-5714	65	58	.	.	PUNCT
cana-5714	66	1	(	(	PUNCT
cana-5714	66	2	∑	∑	ADV
cana-5714	66	3	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	66	4	𝑛	𝑛	PRON
cana-5714	66	5	𝑖=1	𝑖=1	PROPN
cana-5714	66	6	)	)	PUNCT
cana-5714	66	7	2	2	NUM
cana-5714	66	8	≤	≤	NOUN
cana-5714	66	9	(	(	PUNCT
cana-5714	66	10	∑	∑	PROPN
cana-5714	66	11	𝑎𝑖	𝑎𝑖	PRON
cana-5714	66	12	2	2	NUM
cana-5714	66	13	𝑛	𝑛	PRON
cana-5714	66	14	𝑖=1	𝑖=1	PROPN
cana-5714	66	15	)	)	PUNCT
cana-5714	66	16	(	(	PUNCT
cana-5714	66	17	∑	∑	PROPN
cana-5714	66	18	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	66	19	2	2	NUM
cana-5714	66	20	𝑛	𝑛	PRON
cana-5714	66	21	𝑖=1	𝑖=1	PUNCT
cana-5714	66	22	)	)	PUNCT
cana-5714	66	23	let	let	VERB
cana-5714	66	24	𝑎𝑖	𝑎𝑖	PRON
cana-5714	66	25	=	=	SYM
cana-5714	66	26	1	1	NUM
cana-5714	66	27	and	and	CCONJ
cana-5714	66	28	𝑏𝑖	𝑏𝑖	ADP
cana-5714	66	29	=	=	PUNCT
cana-5714	66	30	𝜂𝑖	𝜂𝑖	X
cana-5714	66	31	∀	∀	NOUN
cana-5714	66	32	𝑖	𝑖	NOUN
cana-5714	67	1	=	=	NOUN
cana-5714	67	2	1,2,3	1,2,3	NUM
cana-5714	67	3	,	,	PUNCT
cana-5714	67	4	⋯	⋯	NOUN
cana-5714	67	5	𝑛	𝑛	PROPN
cana-5714	67	6	then	then	ADV
cana-5714	67	7	the	the	DET
cana-5714	67	8	inequality	inequality	NOUN
cana-5714	67	9	becomes	become	VERB
cana-5714	67	10	,	,	PUNCT
cana-5714	67	11	(	(	PUNCT
cana-5714	67	12	∑	∑	PROPN
cana-5714	67	13	1	1	X
cana-5714	67	14	.	.	PUNCT
cana-5714	67	15	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	67	16	𝑛	𝑛	PRON
cana-5714	67	17	𝑖=1	𝑖=1	PROPN
cana-5714	67	18	)	)	PUNCT
cana-5714	67	19	2	2	NUM
cana-5714	67	20	≤	≤	NOUN
cana-5714	67	21	(	(	PUNCT
cana-5714	67	22	∑	∑	PROPN
cana-5714	67	23	12	12	NUM
cana-5714	67	24	𝑛	𝑛	PRON
cana-5714	67	25	𝑖=1	𝑖=1	PROPN
cana-5714	67	26	)	)	PUNCT
cana-5714	68	1	(	(	PUNCT
cana-5714	68	2	∑	∑	PUNCT
cana-5714	68	3	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	68	4	2	2	NUM
cana-5714	68	5	𝑛	𝑛	PRON
cana-5714	68	6	𝑖=1	𝑖=1	PROPN
cana-5714	68	7	)	)	PUNCT
cana-5714	68	8	using	use	VERB
cana-5714	68	9	the	the	DET
cana-5714	68	10	idea	idea	NOUN
cana-5714	68	11	of	of	ADP
cana-5714	68	12	lemma	lemma	PROPN
cana-5714	68	13	3.1	3.1	NUM
cana-5714	68	14	(	(	PUNCT
cana-5714	68	15	i	i	NOUN
cana-5714	68	16	)	)	PUNCT
cana-5714	68	17	∑	∑	ADV
cana-5714	68	18	𝜂𝑖	𝜂𝑖	X
cana-5714	68	19	𝑛	𝑛	PRON
cana-5714	68	20	𝑖=1	𝑖=1	PUNCT
cana-5714	68	21	=	=	SYM
cana-5714	68	22	0	0	NUM
cana-5714	68	23	𝜂1	𝜂1	NOUN
cana-5714	68	24	+	+	CCONJ
cana-5714	68	25	∑	∑	PROPN
cana-5714	68	26	𝜂𝑖	𝜂𝑖	X
cana-5714	68	27	𝑛	𝑛	PROPN
cana-5714	68	28	𝑖=2	𝑖=2	PUNCT
cana-5714	68	29	=	=	SYM
cana-5714	68	30	0	0	PUNCT
cana-5714	68	31	∑	∑	ADV
cana-5714	68	32	𝜂𝑖	𝜂𝑖	X
cana-5714	68	33	𝑛	𝑛	PROPN
cana-5714	68	34	𝑖=2	𝑖=2	PUNCT
cana-5714	68	35	=	=	SYM
cana-5714	68	36	−𝜂1	−𝜂1	PROPN
cana-5714	68	37	on	on	ADP
cana-5714	68	38	squaring	square	VERB
cana-5714	68	39	we	we	PRON
cana-5714	68	40	obtain	obtain	VERB
cana-5714	68	41	(	(	PUNCT
cana-5714	68	42	∑	∑	PART
cana-5714	68	43	𝜂𝑖	𝜂𝑖	X
cana-5714	68	44	𝑛	𝑛	PROPN
cana-5714	68	45	𝑖=2	𝑖=2	PUNCT
cana-5714	68	46	)	)	PUNCT
cana-5714	68	47	2	2	NUM
cana-5714	68	48	=	=	SYM
cana-5714	68	49	(	(	PUNCT
cana-5714	68	50	−𝜂1)2	−𝜂1)2	PROPN
cana-5714	68	51	=	=	SYM
cana-5714	68	52	𝜂1	𝜂1	NOUN
cana-5714	68	53	2	2	NUM
cana-5714	68	54	by	by	ADP
cana-5714	68	55	(	(	PUNCT
cana-5714	68	56	ii	ii	NOUN
cana-5714	68	57	)	)	PUNCT
cana-5714	68	58	of	of	ADP
cana-5714	68	59	lemma	lemma	PROPN
cana-5714	68	60	3.1	3.1	NUM
cana-5714	68	61	(	(	PUNCT
cana-5714	68	62	∑	∑	PART
cana-5714	68	63	𝜂𝑖	𝜂𝑖	X
cana-5714	68	64	𝑛	𝑛	PROPN
cana-5714	68	65	𝑖=2	𝑖=2	PUNCT
cana-5714	68	66	)	)	PUNCT
cana-5714	68	67	2	2	NUM
cana-5714	68	68	=	=	SYM
cana-5714	68	69	2𝐻	2𝐻	PROPN
cana-5714	68	70	(	(	PUNCT
cana-5714	68	71	𝜂1)2	𝜂1)2	X
cana-5714	68	72	+	+	CCONJ
cana-5714	68	73	(	(	PUNCT
cana-5714	68	74	∑	∑	INTJ
cana-5714	68	75	𝜂𝑖	𝜂𝑖	X
cana-5714	68	76	𝑛	𝑛	PROPN
cana-5714	68	77	𝑖=2	𝑖=2	PUNCT
cana-5714	68	78	)	)	PUNCT
cana-5714	68	79	2	2	NUM
cana-5714	68	80	=	=	SYM
cana-5714	68	81	2𝐻	2𝐻	PROPN
cana-5714	68	82	communications	communication	NOUN
cana-5714	68	83	on	on	ADP
cana-5714	68	84	applied	apply	VERB
cana-5714	68	85	nonlinear	nonlinear	ADJ
cana-5714	68	86	analysis	analysis	NOUN
cana-5714	68	87	issn	issn	NOUN
cana-5714	68	88	:	:	PUNCT
cana-5714	68	89	1074	1074	NUM
cana-5714	68	90	-	-	PUNCT
cana-5714	68	91	133x	133x	NUM
cana-5714	68	92	vol	vol	VERB
cana-5714	68	93	32	32	NUM
cana-5714	68	94	no	no	NOUN
cana-5714	68	95	.	.	PUNCT
cana-5714	69	1	10s	10	NOUN
cana-5714	69	2	(	(	PUNCT
cana-5714	69	3	2025	2025	NUM
cana-5714	69	4	)	)	PUNCT
cana-5714	69	5	2744	2744	NUM
cana-5714	69	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	69	7	(	(	PUNCT
cana-5714	69	8	∑	∑	PUNCT
cana-5714	69	9	𝜂𝑖	𝜂𝑖	X
cana-5714	69	10	𝑛	𝑛	PROPN
cana-5714	69	11	𝑖=2	𝑖=2	PUNCT
cana-5714	69	12	)	)	PUNCT
cana-5714	69	13	2	2	NUM
cana-5714	69	14	=	=	SYM
cana-5714	69	15	2𝐻	2𝐻	PROPN
cana-5714	69	16	−	−	PROPN
cana-5714	69	17	(	(	PUNCT
cana-5714	69	18	𝜂1)2	𝜂1)2	NUM
cana-5714	69	19	equation	equation	NOUN
cana-5714	69	20	one	one	NOUN
cana-5714	69	21	becomes	become	VERB
cana-5714	69	22	(	(	PUNCT
cana-5714	69	23	−𝜂1)2	−𝜂1)2	PROPN
cana-5714	69	24	≤	≤	NOUN
cana-5714	69	25	(	(	PUNCT
cana-5714	69	26	𝑛	𝑛	PRON
cana-5714	69	27	−	−	PROPN
cana-5714	69	28	1)(2𝐻	1)(2𝐻	NUM
cana-5714	69	29	−	−	NOUN
cana-5714	69	30	𝜂1	𝜂1	NOUN
cana-5714	69	31	2	2	NUM
cana-5714	69	32	)	)	PUNCT
cana-5714	69	33	𝜂1	𝜂1	NOUN
cana-5714	69	34	2	2	NUM
cana-5714	69	35	≤	≤	NUM
cana-5714	69	36	2𝐻	2𝐻	PROPN
cana-5714	69	37	(	(	PUNCT
cana-5714	69	38	𝑛	𝑛	PROPN
cana-5714	69	39	−	−	PROPN
cana-5714	69	40	1	1	NUM
cana-5714	69	41	)	)	PUNCT
cana-5714	69	42	−	−	PROPN
cana-5714	70	1	𝜂1	𝜂1	PROPN
cana-5714	70	2	2(𝑛	2(𝑛	NUM
cana-5714	70	3	−	−	NUM
cana-5714	70	4	1	1	NUM
cana-5714	70	5	)	)	PUNCT
cana-5714	70	6	𝜂1	𝜂1	NOUN
cana-5714	70	7	2	2	NUM
cana-5714	70	8	+	+	CCONJ
cana-5714	70	9	𝜂1	𝜂1	PROPN
cana-5714	70	10	2(𝑛	2(𝑛	NUM
cana-5714	70	11	−	−	PROPN
cana-5714	70	12	1	1	NUM
cana-5714	70	13	)	)	PUNCT
cana-5714	70	14	≤	≤	NOUN
cana-5714	70	15	2𝐻(𝑛	2𝐻(𝑛	NUM
cana-5714	71	1	−	−	NOUN
cana-5714	71	2	1	1	NUM
cana-5714	71	3	)	)	PUNCT
cana-5714	71	4	𝜂1	𝜂1	NOUN
cana-5714	71	5	2(𝑛	2(𝑛	NUM
cana-5714	71	6	)	)	PUNCT
cana-5714	71	7	≤	≤	NOUN
cana-5714	71	8	2𝐻(𝑛	2𝐻(𝑛	NUM
cana-5714	71	9	−	−	NOUN
cana-5714	71	10	1	1	NUM
cana-5714	71	11	)	)	PUNCT
cana-5714	71	12	𝜂1	𝜂1	NOUN
cana-5714	71	13	≤	≤	PUNCT
cana-5714	71	14	√	√	NUM
cana-5714	71	15	2𝐻(𝑛	2𝐻(𝑛	NUM
cana-5714	71	16	−	−	NUM
cana-5714	71	17	1	1	NUM
cana-5714	71	18	)	)	PUNCT
cana-5714	71	19	𝑛	𝑛	PROPN
cana-5714	71	20	theorem	theorem	VERB
cana-5714	71	21	3.3	3.3	NUM
cana-5714	71	22	if	if	SCONJ
cana-5714	71	23	𝐺be	𝐺be	PROPN
cana-5714	71	24	a	a	DET
cana-5714	71	25	graph	graph	NOUN
cana-5714	71	26	with	with	ADP
cana-5714	71	27	𝑛	𝑛	DET
cana-5714	71	28	−vertices	−vertice	NOUN
cana-5714	71	29	then	then	ADV
cana-5714	71	30	√2𝐻	√2𝐻	X
cana-5714	71	31	≤	≤	NUM
cana-5714	71	32	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	71	33	)	)	PUNCT
cana-5714	71	34	≤	≤	NUM
cana-5714	71	35	√2𝑛𝐻	√2𝑛𝐻	ADJ
cana-5714	71	36	proof	proof	NOUN
cana-5714	71	37	.	.	PUNCT
cana-5714	72	1	consider	consider	VERB
cana-5714	72	2	a	a	DET
cana-5714	72	3	graph	graph	NOUN
cana-5714	72	4	𝐺	𝐺	NOUN
cana-5714	72	5	with	with	ADP
cana-5714	72	6	𝑛	𝑛	DET
cana-5714	72	7	−vertices	−vertices	PROPN
cana-5714	72	8	.	.	PUNCT
cana-5714	73	1	let	let	VERB
cana-5714	73	2	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	73	3	)	)	PUNCT
cana-5714	73	4	be	be	AUX
cana-5714	73	5	the	the	DET
cana-5714	73	6	product	product	NOUN
cana-5714	73	7	eccentricity	eccentricity	NOUN
cana-5714	73	8	matrix	matrix	NOUN
cana-5714	73	9	of	of	ADP
cana-5714	73	10	a	a	DET
cana-5714	73	11	graph	graph	NOUN
cana-5714	73	12	𝐺	𝐺	NOUN
cana-5714	73	13	and	and	CCONJ
cana-5714	73	14	𝜂1	𝜂1	PROPN
cana-5714	73	15	,	,	PUNCT
cana-5714	73	16	𝜂2	𝜂2	PROPN
cana-5714	73	17	,	,	PUNCT
cana-5714	73	18	⋯	⋯	VERB
cana-5714	73	19	𝜂𝑛	𝜂𝑛	NOUN
cana-5714	73	20	are	be	AUX
cana-5714	73	21	the	the	DET
cana-5714	73	22	eigen	eigen	PROPN
cana-5714	73	23	values	value	NOUN
cana-5714	73	24	obtained	obtain	VERB
cana-5714	73	25	from	from	ADP
cana-5714	73	26	the	the	DET
cana-5714	73	27	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	73	28	)	)	PUNCT
cana-5714	73	29	matrix	matrix	NOUN
cana-5714	73	30	.	.	PUNCT
cana-5714	74	1	by	by	ADP
cana-5714	74	2	the	the	DET
cana-5714	74	3	idea	idea	NOUN
cana-5714	74	4	of	of	ADP
cana-5714	74	5	cauchy	cauchy	PROPN
cana-5714	74	6	schwartz	schwartz	PROPN
cana-5714	74	7	inequality	inequality	PROPN
cana-5714	74	8	the	the	DET
cana-5714	74	9	theorem	theorem	NOUN
cana-5714	74	10	is	be	AUX
cana-5714	74	11	proved	prove	VERB
cana-5714	74	12	(	(	PUNCT
cana-5714	74	13	∑	∑	ADV
cana-5714	74	14	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	74	15	𝑛	𝑛	PRON
cana-5714	74	16	𝑖=1	𝑖=1	PROPN
cana-5714	74	17	)	)	PUNCT
cana-5714	74	18	2	2	NUM
cana-5714	74	19	≤	≤	NOUN
cana-5714	74	20	(	(	PUNCT
cana-5714	74	21	∑	∑	PROPN
cana-5714	74	22	𝑎𝑖	𝑎𝑖	PRON
cana-5714	74	23	2	2	NUM
cana-5714	74	24	𝑛	𝑛	PRON
cana-5714	74	25	𝑖=1	𝑖=1	PROPN
cana-5714	74	26	)	)	PUNCT
cana-5714	75	1	(	(	PUNCT
cana-5714	75	2	∑	∑	PROPN
cana-5714	75	3	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	75	4	2	2	NUM
cana-5714	75	5	𝑛	𝑛	PRON
cana-5714	75	6	𝑖=1	𝑖=1	PUNCT
cana-5714	75	7	)	)	PUNCT
cana-5714	75	8	let	let	VERB
cana-5714	75	9	us	we	PRON
cana-5714	75	10	assume	assume	VERB
cana-5714	75	11	that	that	SCONJ
cana-5714	75	12	𝑎𝑖	𝑎𝑖	ADV
cana-5714	75	13	=	=	SYM
cana-5714	75	14	1	1	NUM
cana-5714	75	15	and	and	CCONJ
cana-5714	75	16	𝑏𝑖	𝑏𝑖	ADP
cana-5714	75	17	=	=	PUNCT
cana-5714	75	18	𝜂𝑖	𝜂𝑖	X
cana-5714	75	19	∀	∀	NOUN
cana-5714	75	20	𝑖	𝑖	NOUN
cana-5714	76	1	=	=	NOUN
cana-5714	76	2	1,2,3	1,2,3	NUM
cana-5714	76	3	,	,	PUNCT
cana-5714	76	4	⋯	⋯	NOUN
cana-5714	76	5	𝑛.	𝑛.	NOUN
cana-5714	76	6	(	(	PUNCT
cana-5714	76	7	∑	∑	PROPN
cana-5714	76	8	1	1	X
cana-5714	76	9	.	.	PUNCT
cana-5714	76	10	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	76	11	𝑛	𝑛	PRON
cana-5714	76	12	𝑖=1	𝑖=1	PROPN
cana-5714	76	13	)	)	PUNCT
cana-5714	76	14	2	2	NUM
cana-5714	76	15	≤	≤	NOUN
cana-5714	76	16	(	(	PUNCT
cana-5714	76	17	∑	∑	PROPN
cana-5714	76	18	12	12	NUM
cana-5714	76	19	𝑛	𝑛	PRON
cana-5714	76	20	𝑖=1	𝑖=1	PROPN
cana-5714	76	21	)	)	PUNCT
cana-5714	77	1	(	(	PUNCT
cana-5714	77	2	∑	∑	PUNCT
cana-5714	77	3	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	77	4	2	2	NUM
cana-5714	77	5	𝑛	𝑛	PRON
cana-5714	77	6	𝑖=1	𝑖=1	PROPN
cana-5714	77	7	)	)	PUNCT
cana-5714	77	8	(	(	PUNCT
cana-5714	77	9	∑	∑	PUNCT
cana-5714	77	10	𝜂𝑖	𝜂𝑖	X
cana-5714	77	11	𝑛	𝑛	PRON
cana-5714	77	12	𝑖=1	𝑖=1	PROPN
cana-5714	77	13	)	)	PUNCT
cana-5714	77	14	2	2	NUM
cana-5714	77	15	≤	≤	NOUN
cana-5714	77	16	𝑛	𝑛	PRON
cana-5714	77	17	(	(	PUNCT
cana-5714	77	18	∑	∑	PROPN
cana-5714	77	19	𝜂𝑖	𝜂𝑖	PROPN
cana-5714	77	20	2	2	NUM
cana-5714	77	21	𝑛	𝑛	PRON
cana-5714	77	22	𝑖=1	𝑖=1	PROPN
cana-5714	77	23	)	)	PUNCT
cana-5714	77	24	(	(	PUNCT
cana-5714	77	25	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	77	26	)	)	PUNCT
cana-5714	77	27	)	)	PUNCT
cana-5714	77	28	2	2	NUM
cana-5714	77	29	≤	≤	NOUN
cana-5714	77	30	2𝑛𝐻	2𝑛𝐻	NUM
cana-5714	77	31	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	77	32	)	)	PUNCT
cana-5714	77	33	≤	≤	NUM
cana-5714	77	34	√2𝑛𝐻	√2𝑛𝐻	NOUN
cana-5714	77	35	this	this	PRON
cana-5714	77	36	is	be	AUX
cana-5714	77	37	an	an	DET
cana-5714	77	38	upper	upper	ADJ
cana-5714	77	39	bound	bind	VERB
cana-5714	77	40	,	,	PUNCT
cana-5714	77	41	we	we	PRON
cana-5714	77	42	have	have	VERB
cana-5714	77	43	,	,	PUNCT
cana-5714	77	44	(	(	PUNCT
cana-5714	77	45	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	77	46	)	)	PUNCT
cana-5714	77	47	)	)	PUNCT
cana-5714	77	48	2	2	NUM
cana-5714	78	1	=	=	SYM
cana-5714	78	2	(	(	PUNCT
cana-5714	78	3	∑	∑	PROPN
cana-5714	78	4	|𝜂𝑖|𝑛	|𝜂𝑖|𝑛	PROPN
cana-5714	78	5	𝑖=1	𝑖=1	PROPN
cana-5714	78	6	)	)	PUNCT
cana-5714	78	7	2	2	NUM
cana-5714	78	8	≥	≥	NOUN
cana-5714	78	9	∑	∑	PUNCT
cana-5714	78	10	|𝜂𝑖|2	|𝜂𝑖|2	PUNCT
cana-5714	78	11	=	=	SYM
cana-5714	78	12	2	2	NUM
cana-5714	78	13	𝐻𝑛	𝐻𝑛	PROPN
cana-5714	78	14	𝑖=1	𝑖=1	PROPN
cana-5714	78	15	.	.	PUNCT
cana-5714	79	1	thus	thus	ADV
cana-5714	79	2	,	,	PUNCT
cana-5714	79	3	we	we	PRON
cana-5714	79	4	obtain	obtain	VERB
cana-5714	79	5	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	79	6	)	)	PUNCT
cana-5714	79	7	≥	≥	NOUN
cana-5714	79	8	√2𝐻	√2𝐻	NUM
cana-5714	79	9	which	which	PRON
cana-5714	79	10	is	be	AUX
cana-5714	79	11	the	the	DET
cana-5714	79	12	lower	lower	ADV
cana-5714	79	13	bound	bind	VERB
cana-5714	79	14	.	.	PUNCT
cana-5714	80	1	hence	hence	ADV
cana-5714	80	2	the	the	DET
cana-5714	80	3	inequality	inequality	NOUN
cana-5714	80	4	holds	hold	VERB
cana-5714	80	5	.	.	PUNCT
cana-5714	81	1	communications	communication	NOUN
cana-5714	81	2	on	on	ADP
cana-5714	81	3	applied	apply	VERB
cana-5714	81	4	nonlinear	nonlinear	ADJ
cana-5714	81	5	analysis	analysis	NOUN
cana-5714	81	6	issn	issn	NOUN
cana-5714	81	7	:	:	PUNCT
cana-5714	81	8	1074	1074	NUM
cana-5714	81	9	-	-	PUNCT
cana-5714	81	10	133x	133x	NUM
cana-5714	81	11	vol	vol	VERB
cana-5714	81	12	32	32	NUM
cana-5714	81	13	no	no	NOUN
cana-5714	81	14	.	.	PUNCT
cana-5714	82	1	10s	10	NOUN
cana-5714	82	2	(	(	PUNCT
cana-5714	82	3	2025	2025	NUM
cana-5714	82	4	)	)	PUNCT
cana-5714	82	5	2745	2745	NUM
cana-5714	82	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	82	7	thus	thus	ADV
cana-5714	82	8	we	we	PRON
cana-5714	82	9	have	have	VERB
cana-5714	82	10	,	,	PUNCT
cana-5714	82	11	√2𝐻	√2𝐻	AUX
cana-5714	82	12	≤	≤	NUM
cana-5714	82	13	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	82	14	)	)	PUNCT
cana-5714	82	15	≤	≤	NUM
cana-5714	82	16	√2𝑛𝐻	√2𝑛𝐻	NOUN
cana-5714	82	17	theorem	theorem	VERB
cana-5714	82	18	4.4	4.4	NUM
cana-5714	82	19	:	:	PUNCT
cana-5714	82	20	let	let	VERB
cana-5714	82	21	𝐺	𝐺	PRON
cana-5714	82	22	be	be	AUX
cana-5714	82	23	a	a	DET
cana-5714	82	24	graph	graph	NOUN
cana-5714	82	25	with	with	ADP
cana-5714	82	26	𝑛	𝑛	DET
cana-5714	82	27	−vertices	−vertice	NOUN
cana-5714	82	28	and	and	CCONJ
cana-5714	82	29	𝑚	𝑚	ADP
cana-5714	82	30	−	−	NOUN
cana-5714	82	31	edges	edge	NOUN
cana-5714	82	32	.	.	PUNCT
cana-5714	83	1	if	if	SCONJ
cana-5714	83	2	|𝜂1|	|𝜂1|	ADJ
cana-5714	83	3	≥	≥	X
cana-5714	83	4	|𝜂2|	|𝜂2|	VERB
cana-5714	83	5	≥	≥	PROPN
cana-5714	83	6	⋯	⋯	PROPN
cana-5714	83	7	≥	≥	PROPN
cana-5714	83	8	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	83	9	are	be	AUX
cana-5714	83	10	the	the	DET
cana-5714	83	11	eigen	eigen	PROPN
cana-5714	83	12	values	value	NOUN
cana-5714	83	13	of	of	ADP
cana-5714	83	14	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	83	15	)	)	PUNCT
cana-5714	83	16	,	,	PUNCT
cana-5714	83	17	then	then	ADV
cana-5714	83	18	the	the	DET
cana-5714	83	19	following	follow	VERB
cana-5714	83	20	inequality	inequality	NOUN
cana-5714	83	21	holds	hold	VERB
cana-5714	83	22	.	.	PUNCT
cana-5714	84	1	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	84	2	)	)	PUNCT
cana-5714	84	3	≥	≥	PROPN
cana-5714	84	4	2	2	NUM
cana-5714	84	5	√2𝑛𝐻|𝜂1||𝜂𝑛|	√2𝑛𝐻|𝜂1||𝜂𝑛|	VERB
cana-5714	84	6	|𝜂1|	|𝜂1|	NOUN
cana-5714	84	7	+	+	X
cana-5714	84	8	𝜂𝑛|	𝜂𝑛|	NOUN
cana-5714	84	9	proof	proof	NOUN
cana-5714	84	10	.	.	PUNCT
cana-5714	85	1	consider	consider	VERB
cana-5714	85	2	a	a	DET
cana-5714	85	3	graph	graph	NOUN
cana-5714	85	4	𝐺	𝐺	NOUN
cana-5714	85	5	with	with	ADP
cana-5714	85	6	𝑛	𝑛	DET
cana-5714	85	7	−vertices	−vertice	NOUN
cana-5714	85	8	and	and	CCONJ
cana-5714	85	9	|𝜂1|	|𝜂1|	PRON
cana-5714	85	10	≥	≥	NOUN
cana-5714	85	11	|𝜂2|	|𝜂2|	PROPN
cana-5714	85	12	≥	≥	PROPN
cana-5714	85	13	⋯	⋯	PROPN
cana-5714	85	14	≥	≥	PROPN
cana-5714	85	15	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	85	16	are	be	AUX
cana-5714	85	17	the	the	DET
cana-5714	85	18	eigen	eigen	PROPN
cana-5714	85	19	values	value	NOUN
cana-5714	85	20	of	of	ADP
cana-5714	85	21	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	85	22	)	)	PUNCT
cana-5714	85	23	,	,	PUNCT
cana-5714	85	24	where	where	SCONJ
cana-5714	85	25	|𝜂1|	|𝜂1|	ADJ
cana-5714	85	26	and	and	CCONJ
cana-5714	85	27	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	85	28	are	be	AUX
cana-5714	85	29	the	the	DET
cana-5714	85	30	maximum	maximum	ADJ
cana-5714	85	31	and	and	CCONJ
cana-5714	85	32	minimum	minimum	ADJ
cana-5714	85	33	eigen	eigen	PROPN
cana-5714	85	34	values	value	NOUN
cana-5714	85	35	of	of	ADP
cana-5714	85	36	|𝜂𝑖|	|𝜂𝑖|	NOUN
cana-5714	85	37	respectively	respectively	ADV
cana-5714	85	38	.	.	PUNCT
cana-5714	86	1	from	from	ADP
cana-5714	86	2	theorem	theorem	ADJ
cana-5714	86	3	2.1	2.1	NUM
cana-5714	86	4	∑	∑	PUNCT
cana-5714	86	5	𝑎𝑖	𝑎𝑖	DET
cana-5714	86	6	2	2	NUM
cana-5714	86	7	𝑛	𝑛	PRON
cana-5714	86	8	𝑖=1	𝑖=1	PUNCT
cana-5714	86	9	∑	∑	PROPN
cana-5714	86	10	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	86	11	2	2	NUM
cana-5714	86	12	𝑛	𝑛	DET
cana-5714	86	13	𝑖=1	𝑖=1	PUNCT
cana-5714	86	14	≤	≤	NUM
cana-5714	86	15	1	1	NUM
cana-5714	86	16	4	4	NUM
cana-5714	86	17	(	(	PUNCT
cana-5714	86	18	√	√	NUM
cana-5714	86	19	𝑀1𝑀2	𝑀1𝑀2	NOUN
cana-5714	86	20	𝑚1𝑚2	𝑚1𝑚2	X
cana-5714	86	21	+	+	CCONJ
cana-5714	86	22	√	√	ADJ
cana-5714	86	23	𝑚1𝑚2	𝑚1𝑚2	NUM
cana-5714	86	24	𝑀1𝑀2	𝑀1𝑀2	PROPN
cana-5714	86	25	)	)	PUNCT
cana-5714	86	26	2	2	NUM
cana-5714	86	27	(	(	PUNCT
cana-5714	86	28	∑	∑	ADV
cana-5714	86	29	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	86	30	𝑛	𝑛	DET
cana-5714	86	31	𝑖=1	𝑖=1	PROPN
cana-5714	86	32	)	)	PUNCT
cana-5714	86	33	2	2	NUM
cana-5714	86	34	assume	assume	VERB
cana-5714	86	35	𝑎𝑖	𝑎𝑖	ADP
cana-5714	86	36	=	=	NOUN
cana-5714	86	37	1	1	NUM
cana-5714	86	38	and	and	CCONJ
cana-5714	86	39	𝑏𝑖	𝑏𝑖	ADP
cana-5714	86	40	=	=	SYM
cana-5714	86	41	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	86	42	,	,	PUNCT
cana-5714	86	43	𝑀1𝑀2	𝑀1𝑀2	X
cana-5714	86	44	=	=	PRON
cana-5714	86	45	|𝜂1|	|𝜂1|	PROPN
cana-5714	86	46	and	and	CCONJ
cana-5714	86	47	𝑚1𝑚2	𝑚1𝑚2	NOUN
cana-5714	87	1	=	=	SYM
cana-5714	87	2	|𝛼𝑛|	|𝛼𝑛|	NOUN
cana-5714	87	3	then	then	ADV
cana-5714	87	4	,	,	PUNCT
cana-5714	87	5	∑	∑	PROPN
cana-5714	87	6	12	12	NUM
cana-5714	87	7	𝑛	𝑛	PRON
cana-5714	87	8	𝑖=1	𝑖=1	PROPN
cana-5714	87	9	∑|𝜂𝑖|2	∑|𝜂𝑖|2	X
cana-5714	87	10	𝑛	𝑛	DET
cana-5714	87	11	𝑖=1	𝑖=1	PUNCT
cana-5714	87	12	≤	≤	NOUN
cana-5714	87	13	1	1	NUM
cana-5714	87	14	4	4	NUM
cana-5714	87	15	(	(	PUNCT
cana-5714	87	16	√	√	NUM
cana-5714	87	17	|𝜂1|	|𝜂1|	VERB
cana-5714	87	18	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	87	19	+	+	CCONJ
cana-5714	87	20	√	√	VERB
cana-5714	87	21	|𝜂𝑛|	|𝜂𝑛|	ADJ
cana-5714	87	22	|𝜂1|	|𝜂1|	ADJ
cana-5714	87	23	)	)	PUNCT
cana-5714	87	24	2	2	NUM
cana-5714	87	25	(	(	PUNCT
cana-5714	87	26	∑	∑	PROPN
cana-5714	87	27	1	1	NUM
cana-5714	87	28	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	87	29	𝑛	𝑛	PRON
cana-5714	87	30	𝑖=1	𝑖=1	PROPN
cana-5714	87	31	)	)	PUNCT
cana-5714	87	32	2	2	NUM
cana-5714	87	33	from	from	ADP
cana-5714	87	34	lemma	lemma	PROPN
cana-5714	87	35	3.1	3.1	NUM
cana-5714	87	36	and	and	CCONJ
cana-5714	87	37	using	use	VERB
cana-5714	87	38	the	the	DET
cana-5714	87	39	idea	idea	NOUN
cana-5714	87	40	of	of	ADP
cana-5714	87	41	arithmetic	arithmetic	ADJ
cana-5714	87	42	-	-	PUNCT
cana-5714	87	43	geometric	geometric	ADJ
cana-5714	87	44	inequality	inequality	NOUN
cana-5714	87	45	we	we	PRON
cana-5714	87	46	obtain	obtain	VERB
cana-5714	87	47	,	,	PUNCT
cana-5714	87	48	2𝑛𝐻	2𝑛𝐻	NUM
cana-5714	87	49	≤	≤	NUM
cana-5714	87	50	1	1	NUM
cana-5714	87	51	4	4	NUM
cana-5714	87	52	[	[	PUNCT
cana-5714	87	53	(	(	PUNCT
cana-5714	87	54	|𝜂1|	|𝜂1|	X
cana-5714	87	55	+	+	X
cana-5714	87	56	|𝜂𝑛|)2	|𝜂𝑛|)2	DET
cana-5714	87	57	|𝜂1||𝜂𝑛|	|𝜂1||𝜂𝑛|	X
cana-5714	87	58	]	]	PUNCT
cana-5714	87	59	(	(	PUNCT
cana-5714	87	60	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	87	61	)	)	PUNCT
cana-5714	87	62	)	)	PUNCT
cana-5714	87	63	2	2	NUM
cana-5714	87	64	(	(	PUNCT
cana-5714	87	65	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	87	66	)	)	PUNCT
cana-5714	87	67	)	)	PUNCT
cana-5714	87	68	2	2	NUM
cana-5714	87	69	≥	≥	NUM
cana-5714	87	70	8𝑛𝐻	8𝑛𝐻	NUM
cana-5714	87	71	|𝜂1||𝜂𝑛|	|𝜂1||𝜂𝑛|	NOUN
cana-5714	87	72	(	(	PUNCT
cana-5714	87	73	|𝜂1|	|𝜂1|	ADV
cana-5714	87	74	+	+	X
cana-5714	87	75	|𝜂𝑛|)2	|𝜂𝑛|)2	NUM
cana-5714	87	76	(	(	PUNCT
cana-5714	87	77	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	87	78	)	)	PUNCT
cana-5714	87	79	)	)	PUNCT
cana-5714	87	80	2	2	NUM
cana-5714	87	81	≥	≥	NOUN
cana-5714	87	82	2√2𝑛𝐻|𝜂1||𝜂𝑛|	2√2𝑛𝐻|𝜂1||𝜂𝑛|	NUM
cana-5714	87	83	|𝜂1|	|𝜂1|	VERB
cana-5714	87	84	+	+	X
cana-5714	87	85	|𝜂𝑛|	|𝜂𝑛|	ADJ
cana-5714	87	86	theorem	theorem	VERB
cana-5714	87	87	3.5	3.5	NUM
cana-5714	87	88	let	let	VERB
cana-5714	87	89	𝐺	𝐺	PROPN
cana-5714	87	90	be	be	AUX
cana-5714	87	91	a	a	DET
cana-5714	87	92	graph	graph	NOUN
cana-5714	87	93	with	with	ADP
cana-5714	87	94	𝑛	𝑛	DET
cana-5714	87	95	−vertices	−vertice	NOUN
cana-5714	87	96	,	,	PUNCT
cana-5714	87	97	then	then	ADV
cana-5714	87	98	the	the	DET
cana-5714	87	99	following	follow	VERB
cana-5714	87	100	inequalities	inequality	NOUN
cana-5714	87	101	holds	hold	VERB
cana-5714	87	102	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	87	103	)	)	PUNCT
cana-5714	87	104	≥	≥	NOUN
cana-5714	87	105	2𝐻_𝑛|𝜂1||𝜂𝑛|	2𝐻_𝑛|𝜂1||𝜂𝑛|	NUM
cana-5714	87	106	|𝜂1|	|𝜂1|	X
cana-5714	87	107	+	+	CCONJ
cana-5714	87	108	|𝜂𝑛|	|𝜂𝑛|	ADJ
cana-5714	87	109	proof	proof	NOUN
cana-5714	87	110	.	.	PUNCT
cana-5714	88	1	consider	consider	VERB
cana-5714	88	2	a	a	DET
cana-5714	88	3	graph	graph	NOUN
cana-5714	88	4	𝐺	𝐺	NOUN
cana-5714	88	5	with	with	ADP
cana-5714	88	6	order	order	NOUN
cana-5714	88	7	𝑛	𝑛	NOUN
cana-5714	88	8	and	and	CCONJ
cana-5714	88	9	size	size	NOUN
cana-5714	88	10	𝑚.	𝑚.	ADV
cana-5714	88	11	let	let	VERB
cana-5714	88	12	|𝜂1|	|𝜂1|	ADJ
cana-5714	88	13	≥	≥	NOUN
cana-5714	88	14	|𝜂2|	|𝜂2|	PROPN
cana-5714	88	15	≥	≥	PROPN
cana-5714	88	16	⋯	⋯	PROPN
cana-5714	88	17	≥	≥	PRON
cana-5714	88	18	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	88	19	be	be	VERB
cana-5714	88	20	the	the	DET
cana-5714	88	21	eigen	eigen	PROPN
cana-5714	88	22	balues	balue	NOUN
cana-5714	88	23	of	of	ADP
cana-5714	88	24	the	the	DET
cana-5714	88	25	product	product	NOUN
cana-5714	88	26	eccentricity	eccentricity	NOUN
cana-5714	88	27	matrix	matrix	NOUN
cana-5714	88	28	,	,	PUNCT
cana-5714	88	29	arranged	arrange	VERB
cana-5714	88	30	in	in	ADP
cana-5714	88	31	nonincreasing	nonincrease	VERB
cana-5714	88	32	order	order	NOUN
cana-5714	88	33	,	,	PUNCT
cana-5714	88	34	where	where	SCONJ
cana-5714	88	35	|𝜂1|	|𝜂1|	ADJ
cana-5714	88	36	and	and	CCONJ
cana-5714	88	37	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	88	38	are	be	AUX
cana-5714	88	39	the	the	DET
cana-5714	88	40	maximum	maximum	ADJ
cana-5714	88	41	and	and	CCONJ
cana-5714	88	42	minimum	minimum	ADJ
cana-5714	88	43	eigen	eigen	PROPN
cana-5714	88	44	values	value	NOUN
cana-5714	88	45	respectively	respectively	ADV
cana-5714	88	46	.	.	PUNCT
cana-5714	89	1	using	use	VERB
cana-5714	89	2	the	the	DET
cana-5714	89	3	inequality	inequality	NOUN
cana-5714	89	4	from	from	ADP
cana-5714	89	5	theorem	theorem	ADJ
cana-5714	89	6	2.4	2.4	NUM
cana-5714	89	7	∑	∑	NOUN
cana-5714	89	8	𝑏𝑖	𝑏𝑖	ADP
cana-5714	89	9	2	2	NUM
cana-5714	89	10	𝑛	𝑛	PRON
cana-5714	89	11	𝑖=1	𝑖=1	PROPN
cana-5714	90	1	+	+	CCONJ
cana-5714	90	2	𝑟𝑅	𝑟𝑅	NOUN
cana-5714	90	3	∑	∑	PUNCT
cana-5714	90	4	𝑎𝑖	𝑎𝑖	PRON
cana-5714	90	5	2	2	NUM
cana-5714	90	6	𝑛	𝑛	PRON
cana-5714	90	7	𝑖=1	𝑖=1	PROPN
cana-5714	90	8	≤	≤	NUM
cana-5714	90	9	(	(	PUNCT
cana-5714	90	10	𝑟	𝑟	X
cana-5714	90	11	+	+	CCONJ
cana-5714	90	12	𝑅	𝑅	NOUN
cana-5714	90	13	)	)	PUNCT
cana-5714	90	14	(	(	PUNCT
cana-5714	90	15	∑	∑	ADV
cana-5714	90	16	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	90	17	𝑛	𝑛	PRON
cana-5714	90	18	𝑖=1	𝑖=1	PUNCT
cana-5714	90	19	)	)	PUNCT
cana-5714	90	20	assume	assume	VERB
cana-5714	90	21	𝑏𝑖	𝑏𝑖	ADP
cana-5714	90	22	=	=	SYM
cana-5714	90	23	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	90	24	,	,	PUNCT
cana-5714	90	25	𝑎𝑖	𝑎𝑖	X
cana-5714	90	26	=	=	NOUN
cana-5714	90	27	1	1	NUM
cana-5714	90	28	,	,	PUNCT
cana-5714	90	29	𝑟	𝑟	NOUN
cana-5714	90	30	=	=	SYM
cana-5714	90	31	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	90	32	and	and	CCONJ
cana-5714	90	33	𝑅	𝑅	PROPN
cana-5714	90	34	=	=	PUNCT
cana-5714	90	35	|𝜂1|	|𝜂1|	NOUN
cana-5714	90	36	,	,	PUNCT
cana-5714	90	37	then	then	ADV
cana-5714	90	38	the	the	DET
cana-5714	90	39	inequality	inequality	NOUN
cana-5714	90	40	implies	imply	VERB
cana-5714	90	41	to	to	ADP
cana-5714	90	42	communications	communication	NOUN
cana-5714	90	43	on	on	ADP
cana-5714	90	44	applied	apply	VERB
cana-5714	90	45	nonlinear	nonlinear	ADJ
cana-5714	90	46	analysis	analysis	NOUN
cana-5714	90	47	issn	issn	NOUN
cana-5714	90	48	:	:	PUNCT
cana-5714	90	49	1074	1074	NUM
cana-5714	90	50	-	-	PUNCT
cana-5714	90	51	133x	133x	NUM
cana-5714	90	52	vol	vol	VERB
cana-5714	90	53	32	32	NUM
cana-5714	90	54	no	no	NOUN
cana-5714	90	55	.	.	PUNCT
cana-5714	91	1	10s	10	NOUN
cana-5714	91	2	(	(	PUNCT
cana-5714	91	3	2025	2025	NUM
cana-5714	91	4	)	)	PUNCT
cana-5714	91	5	2746	2746	NUM
cana-5714	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	91	7	∑|𝜂𝑖|2	∑|𝜂𝑖|2	X
cana-5714	91	8	𝑛	𝑛	PROPN
cana-5714	91	9	𝑖=1	𝑖=1	PROPN
cana-5714	92	1	+	+	CCONJ
cana-5714	92	2	|𝜂𝑛||𝜂1|	|𝜂𝑛||𝜂1|	PROPN
cana-5714	92	3	∑	∑	PROPN
cana-5714	92	4	12	12	NUM
cana-5714	92	5	𝑛	𝑛	DET
cana-5714	92	6	𝑖=1	𝑖=1	PROPN
cana-5714	92	7	≤	≤	NOUN
cana-5714	92	8	(	(	PUNCT
cana-5714	92	9	|𝜂𝑛|	|𝜂𝑛|	ADJ
cana-5714	92	10	+	+	CCONJ
cana-5714	92	11	|𝜂1	|𝜂1	NOUN
cana-5714	92	12	)	)	PUNCT
cana-5714	92	13	(	(	PUNCT
cana-5714	92	14	∑	∑	PUNCT
cana-5714	92	15	1|𝜂𝑖|	1|𝜂𝑖|	PROPN
cana-5714	92	16	𝑛	𝑛	PRON
cana-5714	92	17	𝑖=1	𝑖=1	PROPN
cana-5714	92	18	)	)	PUNCT
cana-5714	92	19	using	use	VERB
cana-5714	92	20	lemma	lemma	PROPN
cana-5714	92	21	3.1	3.1	NUM
cana-5714	92	22	2𝐻	2𝐻	NOUN
cana-5714	93	1	+	+	CCONJ
cana-5714	93	2	(	(	PUNCT
cana-5714	93	3	|𝜂𝑛||𝜂1|)𝑛	|𝜂𝑛||𝜂1|)𝑛	PROPN
cana-5714	93	4	≤	≤	NUM
cana-5714	93	5	(	(	PUNCT
cana-5714	93	6	|𝜂𝑛|	|𝜂𝑛|	ADJ
cana-5714	93	7	+	+	CCONJ
cana-5714	93	8	|𝜂1|)𝐸𝑃𝐸(𝐺	|𝜂1|)𝐸𝑃𝐸(𝐺	ADJ
cana-5714	93	9	)	)	PUNCT
cana-5714	93	10	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	93	11	)	)	PUNCT
cana-5714	93	12	≥	≥	NOUN
cana-5714	94	1	2𝐻	2𝐻	NOUN
cana-5714	94	2	+	+	CCONJ
cana-5714	94	3	|𝜂𝑛||𝜂1|	|𝜂𝑛||𝜂1|	PROPN
cana-5714	94	4	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	94	5	+	+	CCONJ
cana-5714	94	6	|𝜂1|	|𝜂1|	VERB
cana-5714	94	7	theorem	theorem	ADJ
cana-5714	94	8	3.6	3.6	NUM
cana-5714	94	9	:	:	PUNCT
cana-5714	94	10	let	let	VERB
cana-5714	94	11	𝐺	𝐺	PRON
cana-5714	94	12	be	be	AUX
cana-5714	94	13	a	a	DET
cana-5714	94	14	graph	graph	NOUN
cana-5714	94	15	with	with	ADP
cana-5714	94	16	𝑛	𝑛	DET
cana-5714	94	17	−	−	PROPN
cana-5714	94	18	vertices	vertex	NOUN
cana-5714	94	19	,	,	PUNCT
cana-5714	94	20	then	then	ADV
cana-5714	94	21	the	the	DET
cana-5714	94	22	following	follow	VERB
cana-5714	94	23	inequality	inequality	NOUN
cana-5714	94	24	holds	hold	VERB
cana-5714	94	25	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	94	26	)	)	PUNCT
cana-5714	94	27	≥	≥	NOUN
cana-5714	94	28	√2𝑛𝐻	√2𝑛𝐻	ADJ
cana-5714	94	29	−	−	PROPN
cana-5714	94	30	𝑛2	𝑛2	NOUN
cana-5714	94	31	4	4	NUM
cana-5714	94	32	(	(	PUNCT
cana-5714	94	33	|𝜂1|	|𝜂1|	ADJ
cana-5714	94	34	−	−	PROPN
cana-5714	94	35	|𝜂𝑛|)2	|𝜂𝑛|)2	DET
cana-5714	94	36	proof	proof	NOUN
cana-5714	94	37	.	.	PUNCT
cana-5714	95	1	consider	consider	VERB
cana-5714	95	2	a	a	DET
cana-5714	95	3	graph	graph	NOUN
cana-5714	95	4	𝐺	𝐺	NOUN
cana-5714	95	5	with	with	ADP
cana-5714	95	6	order	order	NOUN
cana-5714	95	7	𝑛	𝑛	NOUN
cana-5714	95	8	and	and	CCONJ
cana-5714	95	9	size	size	NOUN
cana-5714	95	10	𝑚.	𝑚.	ADV
cana-5714	95	11	let	let	VERB
cana-5714	95	12	|𝜂1|	|𝜂1|	ADJ
cana-5714	95	13	≥	≥	NOUN
cana-5714	95	14	|𝜂2|	|𝜂2|	VERB
cana-5714	95	15	≥	≥	NOUN
cana-5714	95	16	⋯	⋯	PROPN
cana-5714	95	17	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	95	18	be	be	AUX
cana-5714	95	19	the	the	DET
cana-5714	95	20	eigen	eigen	PROPN
cana-5714	95	21	values	value	NOUN
cana-5714	95	22	of	of	ADP
cana-5714	95	23	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	95	24	)	)	PUNCT
cana-5714	95	25	matrix	matrix	NOUN
cana-5714	95	26	,	,	PUNCT
cana-5714	95	27	where	where	SCONJ
cana-5714	95	28	|𝜂1|	|𝜂1|	ADJ
cana-5714	95	29	and	and	CCONJ
cana-5714	95	30	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	95	31	are	be	AUX
cana-5714	95	32	the	the	DET
cana-5714	95	33	maximum	maximum	ADJ
cana-5714	95	34	and	and	CCONJ
cana-5714	95	35	minimum	minimum	ADJ
cana-5714	95	36	eigen	eigen	PROPN
cana-5714	95	37	values	value	NOUN
cana-5714	95	38	respectively	respectively	ADV
cana-5714	95	39	.	.	PUNCT
cana-5714	96	1	from	from	ADP
cana-5714	96	2	theorem	theorem	ADJ
cana-5714	96	3	2.2	2.2	NUM
cana-5714	96	4	we	we	PRON
cana-5714	96	5	have	have	VERB
cana-5714	96	6	the	the	DET
cana-5714	96	7	inequality	inequality	NOUN
cana-5714	96	8	∑	∑	PUNCT
cana-5714	96	9	𝑎𝑖	𝑎𝑖	ADP
cana-5714	96	10	2	2	NUM
cana-5714	96	11	𝑛	𝑛	PRON
cana-5714	96	12	𝑖=1	𝑖=1	PUNCT
cana-5714	96	13	∑	∑	PROPN
cana-5714	96	14	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	96	15	2	2	NUM
cana-5714	96	16	𝑛	𝑛	PRON
cana-5714	96	17	𝑖=1	𝑖=1	PROPN
cana-5714	96	18	−	−	PROPN
cana-5714	97	1	(	(	PUNCT
cana-5714	97	2	∑	∑	PUNCT
cana-5714	97	3	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	NOUN
cana-5714	97	4	𝑛	𝑛	PRON
cana-5714	97	5	𝑖=1	𝑖=1	PROPN
cana-5714	97	6	)	)	PUNCT
cana-5714	97	7	2	2	NUM
cana-5714	97	8	≤	≤	NOUN
cana-5714	97	9	𝑛2	𝑛2	NOUN
cana-5714	97	10	4	4	NUM
cana-5714	97	11	(	(	PUNCT
cana-5714	97	12	𝑀1𝑀2	𝑀1𝑀2	X
cana-5714	97	13	−	−	NOUN
cana-5714	97	14	𝑚1𝑚2)2	𝑚1𝑚2)2	NOUN
cana-5714	97	15	assume	assume	VERB
cana-5714	97	16	𝑎𝑖	𝑎𝑖	ADP
cana-5714	97	17	=	=	NOUN
cana-5714	97	18	1	1	NUM
cana-5714	97	19	,	,	PUNCT
cana-5714	97	20	𝑏𝑖	𝑏𝑖	ADP
cana-5714	97	21	=	=	SYM
cana-5714	97	22	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	97	23	,	,	PUNCT
cana-5714	97	24	𝑀1𝑀2	𝑀1𝑀2	X
cana-5714	97	25	=	=	PRON
cana-5714	97	26	|𝜂1|	|𝜂1|	PROPN
cana-5714	97	27	and	and	CCONJ
cana-5714	97	28	𝑚1𝑚2	𝑚1𝑚2	NOUN
cana-5714	98	1	=	=	SYM
cana-5714	98	2	|𝛼𝑛|	|𝛼𝑛|	NOUN
cana-5714	98	3	then	then	ADV
cana-5714	98	4	∑	∑	PROPN
cana-5714	98	5	12	12	NUM
cana-5714	98	6	𝑛	𝑛	PRON
cana-5714	98	7	𝑖=1	𝑖=1	PROPN
cana-5714	98	8	∑|𝜂𝑖|2	∑|𝜂𝑖|2	X
cana-5714	98	9	𝑛	𝑛	PRON
cana-5714	98	10	𝑖=1	𝑖=1	PROPN
cana-5714	99	1	−	−	PROPN
cana-5714	99	2	(	(	PUNCT
cana-5714	99	3	∑	∑	PROPN
cana-5714	99	4	1	1	NUM
cana-5714	99	5	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	99	6	𝑛	𝑛	PRON
cana-5714	99	7	𝑖=1	𝑖=1	PROPN
cana-5714	99	8	)	)	PUNCT
cana-5714	99	9	2	2	NUM
cana-5714	99	10	≤	≤	NOUN
cana-5714	99	11	𝑛2	𝑛2	NOUN
cana-5714	99	12	4	4	NUM
cana-5714	99	13	(	(	PUNCT
cana-5714	99	14	|𝜂1|	|𝜂1|	X
cana-5714	99	15	+	+	X
cana-5714	99	16	|𝜂𝑛|)2	|𝜂𝑛|)2	NUM
cana-5714	99	17	from	from	ADP
cana-5714	99	18	lemma	lemma	PROPN
cana-5714	99	19	3.1	3.1	NUM
cana-5714	99	20	2𝑛𝐻	2𝑛𝐻	NUM
cana-5714	99	21	−	−	PROPN
cana-5714	99	22	(	(	PUNCT
cana-5714	99	23	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	99	24	)	)	PUNCT
cana-5714	99	25	)	)	PUNCT
cana-5714	99	26	2	2	NUM
cana-5714	99	27	≤	≤	NOUN
cana-5714	99	28	𝑛2	𝑛2	NOUN
cana-5714	99	29	4	4	NUM
cana-5714	99	30	(	(	PUNCT
cana-5714	99	31	|𝜂1|	|𝜂1|	ADJ
cana-5714	99	32	−	−	PROPN
cana-5714	99	33	|𝜂𝑛|)2	|𝜂𝑛|)2	PROPN
cana-5714	99	34	(	(	PUNCT
cana-5714	99	35	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	99	36	)	)	PUNCT
cana-5714	99	37	)	)	PUNCT
cana-5714	99	38	2	2	NUM
cana-5714	99	39	≥	≥	NOUN
cana-5714	99	40	√2𝑛𝐻	√2𝑛𝐻	ADJ
cana-5714	99	41	−	−	PROPN
cana-5714	99	42	𝑛2	𝑛2	NOUN
cana-5714	99	43	4	4	NUM
cana-5714	99	44	(	(	PUNCT
cana-5714	99	45	|𝜂1|	|𝜂1|	ADJ
cana-5714	99	46	−	−	PROPN
cana-5714	99	47	|𝜂𝑛|)2	|𝜂𝑛|)2	NUM
cana-5714	99	48	hence	hence	ADV
cana-5714	99	49	the	the	DET
cana-5714	99	50	inequality	inequality	NOUN
cana-5714	99	51	holds	hold	VERB
cana-5714	99	52	true	true	ADJ
cana-5714	99	53	.	.	PUNCT
cana-5714	100	1	theorem	theorem	VERB
cana-5714	100	2	3.7	3.7	NUM
cana-5714	100	3	:	:	PUNCT
cana-5714	100	4	let	let	VERB
cana-5714	100	5	𝐺	𝐺	PRON
cana-5714	100	6	be	be	AUX
cana-5714	100	7	a	a	DET
cana-5714	100	8	graph	graph	NOUN
cana-5714	100	9	with	with	ADP
cana-5714	100	10	𝑛	𝑛	DET
cana-5714	100	11	−	−	PROPN
cana-5714	100	12	vertices	vertex	NOUN
cana-5714	100	13	,	,	PUNCT
cana-5714	100	14	then	then	ADV
cana-5714	100	15	the	the	DET
cana-5714	100	16	following	follow	VERB
cana-5714	100	17	inequality	inequality	NOUN
cana-5714	100	18	holds	hold	VERB
cana-5714	100	19	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	100	20	)	)	PUNCT
cana-5714	100	21	≥	≥	NOUN
cana-5714	100	22	√2𝑛𝐻	√2𝑛𝐻	ADJ
cana-5714	100	23	−	−	PROPN
cana-5714	100	24	𝜇(𝑛)(|𝜂1|−|𝜂𝑛|)2	𝜇(𝑛)(|𝜂1|−|𝜂𝑛|)2	ADJ
cana-5714	100	25	proof	proof	NOUN
cana-5714	100	26	.	.	PUNCT
cana-5714	101	1	consider	consider	VERB
cana-5714	101	2	a	a	DET
cana-5714	101	3	graph	graph	NOUN
cana-5714	101	4	𝐺	𝐺	NOUN
cana-5714	101	5	with	with	ADP
cana-5714	101	6	order	order	NOUN
cana-5714	101	7	𝑛	𝑛	NOUN
cana-5714	101	8	and	and	CCONJ
cana-5714	101	9	size	size	NOUN
cana-5714	101	10	𝑚.	𝑚.	ADV
cana-5714	101	11	let	let	VERB
cana-5714	101	12	|𝜂1|	|𝜂1|	ADJ
cana-5714	101	13	≥	≥	NOUN
cana-5714	101	14	|𝜂2|	|𝜂2|	VERB
cana-5714	101	15	≥	≥	NOUN
cana-5714	101	16	⋯	⋯	PROPN
cana-5714	101	17	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	101	18	be	be	AUX
cana-5714	101	19	the	the	DET
cana-5714	101	20	eigen	eigen	PROPN
cana-5714	101	21	values	value	NOUN
cana-5714	101	22	of	of	ADP
cana-5714	101	23	𝑃𝐸(𝐺	𝑃𝐸(𝐺	NOUN
cana-5714	101	24	)	)	PUNCT
cana-5714	101	25	matrix	matrix	NOUN
cana-5714	101	26	,	,	PUNCT
cana-5714	101	27	where	where	SCONJ
cana-5714	101	28	|𝜂1|	|𝜂1|	ADJ
cana-5714	101	29	and	and	CCONJ
cana-5714	101	30	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	101	31	are	be	AUX
cana-5714	101	32	the	the	DET
cana-5714	101	33	maximum	maximum	ADJ
cana-5714	101	34	and	and	CCONJ
cana-5714	101	35	minimum	minimum	ADJ
cana-5714	101	36	eigen	eigen	PROPN
cana-5714	101	37	values	value	NOUN
cana-5714	101	38	respectively	respectively	ADV
cana-5714	101	39	.	.	PUNCT
cana-5714	102	1	consider	consider	VERB
cana-5714	102	2	the	the	DET
cana-5714	102	3	inequality	inequality	NOUN
cana-5714	102	4	from	from	ADP
cana-5714	102	5	the	the	DET
cana-5714	102	6	theorem	theorem	ADJ
cana-5714	102	7	2.3	2.3	NUM
cana-5714	102	8	communications	communication	NOUN
cana-5714	102	9	on	on	ADP
cana-5714	102	10	applied	apply	VERB
cana-5714	102	11	nonlinear	nonlinear	ADJ
cana-5714	102	12	analysis	analysis	NOUN
cana-5714	102	13	issn	issn	NOUN
cana-5714	102	14	:	:	PUNCT
cana-5714	102	15	1074	1074	NUM
cana-5714	102	16	-	-	PUNCT
cana-5714	102	17	133x	133x	NUM
cana-5714	102	18	vol	vol	VERB
cana-5714	102	19	32	32	NUM
cana-5714	102	20	no	no	NOUN
cana-5714	102	21	.	.	PUNCT
cana-5714	103	1	10s	10	NOUN
cana-5714	103	2	(	(	PUNCT
cana-5714	103	3	2025	2025	NUM
cana-5714	103	4	)	)	PUNCT
cana-5714	103	5	2747	2747	NUM
cana-5714	103	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5714	103	7	|𝑛	|𝑛	VERB
cana-5714	103	8	∑	∑	PUNCT
cana-5714	103	9	𝑎𝑖𝑏𝑖	𝑎𝑖𝑏𝑖	ADV
cana-5714	103	10	−	−	PROPN
cana-5714	103	11	∑	∑	PROPN
cana-5714	103	12	𝑎𝑖	𝑎𝑖	ADP
cana-5714	103	13	𝑛	𝑛	PROPN
cana-5714	103	14	𝑖=1	𝑖=1	PUNCT
cana-5714	103	15	∑	∑	PROPN
cana-5714	103	16	𝑏𝑖	𝑏𝑖	ADP
cana-5714	103	17	𝑛	𝑛	PRON
cana-5714	103	18	𝑖=1	𝑖=1	PUNCT
cana-5714	103	19	𝑛	𝑛	PRON
cana-5714	103	20	𝑖=1	𝑖=1	PROPN
cana-5714	104	1	|	|	ADV
cana-5714	104	2	≤	≤	NOUN
cana-5714	104	3	𝜇(𝑛)(𝐴	𝜇(𝑛)(𝐴	ADP
cana-5714	104	4	−	−	PROPN
cana-5714	104	5	𝑎)(𝐵	𝑎)(𝐵	PROPN
cana-5714	104	6	−	−	ADP
cana-5714	104	7	𝑏	𝑏	NOUN
cana-5714	104	8	)	)	PUNCT
cana-5714	104	9	now	now	ADV
cana-5714	104	10	assume	assume	VERB
cana-5714	104	11	that	that	SCONJ
cana-5714	104	12	𝑎𝑖	𝑎𝑖	ADV
cana-5714	104	13	=	=	NOUN
cana-5714	104	14	𝑏𝑖	𝑏𝑖	PROPN
cana-5714	104	15	=	=	PUNCT
cana-5714	104	16	|𝜂𝑖|	|𝜂𝑖|	PROPN
cana-5714	104	17	,	,	PUNCT
cana-5714	104	18	𝐴	𝐴	NOUN
cana-5714	104	19	=	=	SYM
cana-5714	104	20	𝐵	𝐵	PROPN
cana-5714	104	21	=	=	PUNCT
cana-5714	104	22	|𝜂1|	|𝜂1|	NOUN
cana-5714	104	23	and	and	CCONJ
cana-5714	104	24	𝑎	𝑎	NOUN
cana-5714	104	25	=	=	X
cana-5714	104	26	𝑏	𝑏	NOUN
cana-5714	104	27	=	=	SYM
cana-5714	104	28	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	104	29	,	,	PUNCT
cana-5714	104	30	then	then	ADV
cana-5714	104	31	the	the	DET
cana-5714	104	32	inequality	inequality	NOUN
cana-5714	104	33	reduces	reduce	VERB
cana-5714	104	34	to	to	ADP
cana-5714	104	35	|𝑛	|𝑛	PROPN
cana-5714	105	1	∑|𝜂𝑖|2	∑|𝜂𝑖|2	PROPN
cana-5714	105	2	−	−	PROPN
cana-5714	105	3	(	(	PUNCT
cana-5714	105	4	∑|𝜂𝑖|	∑|𝜂𝑖|	NOUN
cana-5714	105	5	𝑛	𝑛	PRON
cana-5714	105	6	𝑖=1	𝑖=1	PROPN
cana-5714	105	7	)	)	PUNCT
cana-5714	105	8	2𝑛	2𝑛	PROPN
cana-5714	106	1	𝑖=1	𝑖=1	PUNCT
cana-5714	107	1	|	|	ADV
cana-5714	107	2	≤	≤	NUM
cana-5714	107	3	𝜇(𝑛)(|𝜂1|	𝜇(𝑛)(|𝜂1|	NOUN
cana-5714	107	4	−	−	PROPN
cana-5714	107	5	|𝜂𝑛|)(|𝜂1|	|𝜂𝑛|)(|𝜂1|	INTJ
cana-5714	107	6	−	−	PROPN
cana-5714	107	7	|𝜂𝑛|	|𝜂𝑛|	PROPN
cana-5714	107	8	)	)	PUNCT
cana-5714	107	9	from	from	ADP
cana-5714	107	10	lemma	lemma	PROPN
cana-5714	107	11	3.1	3.1	NUM
cana-5714	107	12	|2𝑛𝐻	|2𝑛𝐻	NOUN
cana-5714	107	13	−	−	PROPN
cana-5714	107	14	(	(	PUNCT
cana-5714	107	15	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	107	16	)	)	PUNCT
cana-5714	107	17	)	)	PUNCT
cana-5714	107	18	2	2	NUM
cana-5714	108	1	|	|	ADV
cana-5714	108	2	≤	≤	PUNCT
cana-5714	108	3	𝜇(𝑛)(|𝜂1|	𝜇(𝑛)(|𝜂1|	NOUN
cana-5714	108	4	−	−	PROPN
cana-5714	108	5	|𝜂𝑛|)2	|𝜂𝑛|)2	PRON
cana-5714	108	6	𝐸𝑃𝐸(𝐺	𝐸𝑃𝐸(𝐺	NOUN
cana-5714	108	7	)	)	PUNCT
cana-5714	108	8	≥	≥	NOUN
cana-5714	108	9	√2𝑛𝐻	√2𝑛𝐻	ADJ
cana-5714	108	10	−	−	PROPN
cana-5714	108	11	𝜇(𝑛)(|𝜂1|	𝜇(𝑛)(|𝜂1|	NOUN
cana-5714	108	12	−	−	PROPN
cana-5714	108	13	|𝜂𝑛|)2	|𝜂𝑛|)2	PRON
cana-5714	108	14	hence	hence	ADV
cana-5714	108	15	the	the	DET
cana-5714	108	16	inequality	inequality	NOUN
cana-5714	108	17	holds	hold	VERB
cana-5714	108	18	true	true	ADJ
cana-5714	108	19	.	.	PUNCT
cana-5714	109	1	refrences	refrence	VERB
cana-5714	110	1	[	[	X
cana-5714	110	2	1	1	X
cana-5714	110	3	]	]	PUNCT
cana-5714	110	4	adiga.c	adiga.c	PUNCT
cana-5714	110	5	and	and	CCONJ
cana-5714	110	6	smitha	smitha	PROPN
cana-5714	110	7	,	,	PUNCT
cana-5714	110	8	m	m	PROPN
cana-5714	110	9	,	,	PUNCT
cana-5714	110	10	on	on	ADP
cana-5714	110	11	maximum	maximum	ADJ
cana-5714	110	12	degree	degree	NOUN
cana-5714	110	13	energy	energy	NOUN
cana-5714	110	14	of	of	ADP
cana-5714	110	15	a	a	DET
cana-5714	110	16	graph	graph	NOUN
cana-5714	110	17	,	,	PUNCT
cana-5714	110	18	int	int	NOUN
cana-5714	110	19	.	.	PUNCT
cana-5714	111	1	j.	j.	PROPN
cana-5714	111	2	contempt	contempt	PROPN
cana-5714	111	3	.	.	PUNCT
cana-5714	112	1	math	math	NOUN
cana-5714	112	2	.	.	PUNCT
cana-5714	113	1	sci	sci	PROPN
cana-5714	113	2	.	.	PROPN
cana-5714	113	3	,	,	PUNCT
cana-5714	113	4	4(2009	4(2009	NUM
cana-5714	113	5	)	)	PUNCT
cana-5714	113	6	,	,	PUNCT
cana-5714	113	7	385	385	NUM
cana-5714	113	8	-	-	SYM
cana-5714	113	9	396	396	NUM
cana-5714	113	10	.	.	PUNCT
cana-5714	114	1	[	[	X
cana-5714	114	2	2	2	X
cana-5714	114	3	]	]	PUNCT
cana-5714	114	4	ahmed	ahme	VERB
cana-5714	114	5	naji.m	naji.m	PROPN
cana-5714	114	6	and	and	CCONJ
cana-5714	114	7	soner	soner	NOUN
cana-5714	114	8	n.d	n.d	PROPN
cana-5714	114	9	,	,	PUNCT
cana-5714	114	10	the	the	DET
cana-5714	114	11	maximum	maximum	ADJ
cana-5714	114	12	eccentricity	eccentricity	NOUN
cana-5714	114	13	energy	energy	NOUN
cana-5714	114	14	of	of	ADP
cana-5714	114	15	a	a	DET
cana-5714	114	16	graph	graph	NOUN
cana-5714	114	17	,	,	PUNCT
cana-5714	114	18	int	int	NOUN
cana-5714	114	19	.	.	PUNCT
cana-5714	115	1	j.	j.	PROPN
cana-5714	115	2	sci	sci	PROPN
cana-5714	115	3	.	.	PUNCT
cana-5714	116	1	engin	engin	PROPN
cana-5714	116	2	.	.	PUNCT
cana-5714	117	1	research	research	NOUN
cana-5714	117	2	,	,	PUNCT
cana-5714	117	3	7(2016	7(2016	NUM
cana-5714	117	4	)	)	PUNCT
cana-5714	117	5	,	,	PUNCT
cana-5714	117	6	5	5	NUM
cana-5714	117	7	-	-	SYM
cana-5714	117	8	13	13	NUM
cana-5714	117	9	[	[	X
cana-5714	117	10	3	3	NUM
cana-5714	117	11	]	]	PUNCT
cana-5714	117	12	m.	m.	NOUN
cana-5714	117	13	biernacki	biernacki	PROPN
cana-5714	117	14	,	,	PUNCT
cana-5714	117	15	h.	h.	PROPN
cana-5714	117	16	pidek	pidek	PROPN
cana-5714	117	17	and	and	CCONJ
cana-5714	117	18	c.	c.	PROPN
cana-5714	117	19	ryll	ryll	PROPN
cana-5714	117	20	-	-	PUNCT
cana-5714	117	21	nardzewsk	nardzewsk	NOUN
cana-5714	117	22	,	,	PUNCT
cana-5714	117	23	sur	sur	PROPN
cana-5714	117	24	une	une	PROPN
cana-5714	117	25	iné	iné	PROPN
cana-5714	117	26	galité	galité	PROPN
cana-5714	117	27	entre	entre	PROPN
cana-5714	117	28	des	des	PROPN
cana-5714	117	29	intégrales	intégrales	PROPN
cana-5714	117	30	définies	définies	PROPN
cana-5714	117	31	,	,	PUNCT
cana-5714	117	32	maria	maria	PROPN
cana-5714	117	33	curie	curie	PROPN
cana-5714	117	34	skåćodowska	skåćodowska	PROPN
cana-5714	117	35	university	university	PROPN
cana-5714	117	36	,	,	PUNCT
cana-5714	117	37	a4	a4	PROPN
cana-5714	117	38	,	,	PUNCT
cana-5714	117	39	(	(	PUNCT
cana-5714	117	40	2009	2009	NUM
cana-5714	117	41	)	)	PUNCT
cana-5714	117	42	,	,	PUNCT
cana-5714	117	43	1	1	NUM
cana-5714	117	44	-	-	SYM
cana-5714	117	45	4	4	NUM
cana-5714	117	46	.	.	PUNCT
cana-5714	118	1	[	[	X
cana-5714	118	2	4	4	X
cana-5714	118	3	]	]	PUNCT
cana-5714	118	4	j.	j.	PROPN
cana-5714	118	5	b.	b.	PROPN
cana-5714	118	6	diaz	diaz	PROPN
cana-5714	118	7	and	and	CCONJ
cana-5714	118	8	f.	f.	PROPN
cana-5714	118	9	t.	t.	PROPN
cana-5714	118	10	metcalf	metcalf	PROPN
cana-5714	118	11	,	,	PUNCT
cana-5714	118	12	stronger	strong	ADJ
cana-5714	118	13	forms	form	NOUN
cana-5714	118	14	of	of	ADP
cana-5714	118	15	a	a	DET
cana-5714	118	16	class	class	NOUN
cana-5714	118	17	of	of	ADP
cana-5714	118	18	inequalities	inequality	NOUN
cana-5714	118	19	of	of	ADP
cana-5714	118	20	g.	g.	PROPN
cana-5714	118	21	pólya	pólya	PROPN
cana-5714	118	22	-	-	PUNCT
cana-5714	118	23	g.	g.	PROPN
cana-5714	118	24	szegȍ	szegȍ	PROPN
cana-5714	118	25	and	and	CCONJ
cana-5714	118	26	l.	l.	PROPN
cana-5714	118	27	v.	v.	PROPN
cana-5714	118	28	kantorovich	kantorovich	PROPN
cana-5714	118	29	,	,	PUNCT
cana-5714	118	30	bulletin	bulletin	NOUN
cana-5714	118	31	of	of	ADP
cana-5714	118	32	the	the	DET
cana-5714	118	33	ams	ams	PROPN
cana-5714	118	34	-	-	PUNCT
cana-5714	118	35	american	american	PROPN
cana-5714	118	36	mathematical	mathematical	PROPN
cana-5714	118	37	society	society	NOUN
cana-5714	118	38	,	,	PUNCT
cana-5714	118	39	69(1963	69(1963	NUM
cana-5714	118	40	)	)	PUNCT
cana-5714	118	41	,	,	PUNCT
cana-5714	118	42	415	415	NUM
cana-5714	118	43	-	-	SYM
cana-5714	118	44	418	418	NUM
cana-5714	118	45	.	.	PUNCT
cana-5714	119	1	[	[	X
cana-5714	119	2	5	5	X
cana-5714	119	3	]	]	PUNCT
cana-5714	119	4	i.gutman	i.gutman	NOUN
cana-5714	119	5	,	,	PUNCT
cana-5714	119	6	the	the	DET
cana-5714	119	7	energy	energy	NOUN
cana-5714	119	8	of	of	ADP
cana-5714	119	9	graph	graph	NOUN
cana-5714	119	10	,	,	PUNCT
cana-5714	119	11	ber	ber	PROPN
cana-5714	119	12	.	.	PUNCT
cana-5714	119	13	math.-stat	math.-stat	PROPN
cana-5714	119	14	.	.	PROPN
cana-5714	119	15	sekt	sekt	PROPN
cana-5714	119	16	.	.	PUNCT
cana-5714	120	1	forshungsz	forshungsz	NOUN
cana-5714	120	2	.	.	PUNCT
cana-5714	121	1	graz	graz	PROPN
cana-5714	121	2	103	103	NUM
cana-5714	121	3	(	(	PUNCT
cana-5714	121	4	1978	1978	NUM
cana-5714	121	5	)	)	PUNCT
cana-5714	121	6	1–22	1–22	NOUN
cana-5714	121	7	.	.	PUNCT
cana-5714	122	1	[	[	X
cana-5714	122	2	6	6	NUM
cana-5714	122	3	]	]	SYM
cana-5714	122	4	li.x	li.x	NOUN
cana-5714	122	5	,	,	PUNCT
cana-5714	122	6	y.	y.	PROPN
cana-5714	122	7	shi	shi	PROPN
cana-5714	122	8	and	and	CCONJ
cana-5714	122	9	i.	i.	PROPN
cana-5714	122	10	gutman	gutman	PROPN
cana-5714	122	11	,	,	PUNCT
cana-5714	122	12	graph	graph	NOUN
cana-5714	122	13	energy	energy	NOUN
cana-5714	122	14	,	,	PUNCT
cana-5714	122	15	springer	springer	NOUN
cana-5714	122	16	,	,	PUNCT
cana-5714	122	17	2012	2012	NUM
cana-5714	122	18	.	.	PUNCT
cana-5714	123	1	[	[	X
cana-5714	123	2	7	7	X
cana-5714	123	3	]	]	X
cana-5714	123	4	mohammad	mohammad	PROPN
cana-5714	123	5	issa	issa	PROPN
cana-5714	123	6	sowaity	sowaity	PROPN
cana-5714	123	7	and	and	CCONJ
cana-5714	123	8	b.sharada	b.sharada	PROPN
cana-5714	123	9	,	,	PUNCT
cana-5714	123	10	the	the	DET
cana-5714	123	11	sum	sum	NOUN
cana-5714	123	12	-	-	PUNCT
cana-5714	123	13	eccentricity	eccentricity	NOUN
cana-5714	123	14	energy	energy	NOUN
cana-5714	123	15	of	of	ADP
cana-5714	123	16	a	a	DET
cana-5714	123	17	graph	graph	NOUN
cana-5714	123	18	,	,	PUNCT
cana-5714	123	19	international	international	ADJ
cana-5714	123	20	jorunal	jorunal	NOUN
cana-5714	123	21	on	on	ADP
cana-5714	123	22	recent	recent	ADJ
cana-5714	123	23	and	and	CCONJ
cana-5714	123	24	innovation	innovation	NOUN
cana-5714	123	25	trends	trend	NOUN
cana-5714	123	26	in	in	ADP
cana-5714	123	27	computing	computing	NOUN
cana-5714	123	28	and	and	CCONJ
cana-5714	123	29	communication	communication	NOUN
cana-5714	123	30	,	,	PUNCT
cana-5714	123	31	5(2017	5(2017	NUM
cana-5714	123	32	)	)	PUNCT
cana-5714	123	33	,	,	PUNCT
cana-5714	123	34	293	293	NUM
cana-5714	123	35	-	-	SYM
cana-5714	123	36	304	304	NUM
cana-5714	123	37	.	.	PUNCT
cana-5714	124	1	[	[	X
cana-5714	124	2	8	8	NUM
cana-5714	124	3	]	]	X
cana-5714	124	4	n.	n.	PROPN
cana-5714	124	5	ozeki	ozeki	PROPN
cana-5714	124	6	,	,	PUNCT
cana-5714	124	7	on	on	ADP
cana-5714	124	8	the	the	DET
cana-5714	124	9	estimation	estimation	NOUN
cana-5714	124	10	of	of	ADP
cana-5714	124	11	inequalities	inequality	NOUN
cana-5714	124	12	by	by	ADP
cana-5714	124	13	maximum	maximum	ADJ
cana-5714	124	14	and	and	CCONJ
cana-5714	124	15	minimum	minimum	ADJ
cana-5714	124	16	values	value	NOUN
cana-5714	124	17	,	,	PUNCT
cana-5714	124	18	journal	journal	NOUN
cana-5714	124	19	of	of	ADP
cana-5714	124	20	college	college	NOUN
cana-5714	124	21	arts	art	NOUN
cana-5714	124	22	and	and	CCONJ
cana-5714	124	23	science	science	NOUN
cana-5714	124	24	,	,	PUNCT
cana-5714	124	25	chiba	chiba	PROPN
cana-5714	124	26	university	university	PROPN
cana-5714	124	27	,	,	PUNCT
cana-5714	124	28	5(1968	5(1968	NUM
cana-5714	124	29	)	)	PUNCT
cana-5714	124	30	,	,	PUNCT
cana-5714	124	31	199	199	NUM
cana-5714	124	32	-	-	SYM
cana-5714	124	33	203	203	NUM
cana-5714	124	34	.	.	PUNCT
cana-5714	125	1	[	[	X
cana-5714	125	2	9	9	NUM
cana-5714	125	3	]	]	X
cana-5714	125	4	g.	g.	NOUN
cana-5714	125	5	pólya	pólya	NOUN
cana-5714	125	6	and	and	CCONJ
cana-5714	125	7	g.	g.	PROPN
cana-5714	125	8	szegȍ	szegȍ	PROPN
cana-5714	125	9	,	,	PUNCT
cana-5714	125	10	problems	problem	NOUN
cana-5714	125	11	and	and	CCONJ
cana-5714	125	12	theorems	theorem	NOUN
cana-5714	125	13	in	in	ADP
cana-5714	125	14	analysis	analysis	NOUN
cana-5714	125	15	,	,	PUNCT
cana-5714	125	16	series	series	NOUN
cana-5714	125	17	,	,	PUNCT
cana-5714	125	18	integral	integral	ADJ
cana-5714	125	19	calculus	calculus	NOUN
cana-5714	125	20	,	,	PUNCT
cana-5714	125	21	theory	theory	NOUN
cana-5714	125	22	of	of	ADP
cana-5714	125	23	functions	function	NOUN
cana-5714	125	24	,	,	PUNCT
cana-5714	125	25	springer	springer	NOUN
cana-5714	125	26	,	,	PUNCT
cana-5714	125	27	berlin	berlin	PROPN
cana-5714	125	28	,	,	PUNCT
cana-5714	125	29	(	(	PUNCT
cana-5714	125	30	1972	1972	NUM
cana-5714	125	31	)	)	PUNCT
cana-5714	125	32	.	.	PUNCT
cana-5714	126	1	[	[	X
cana-5714	126	2	10	10	NUM
cana-5714	126	3	]	]	X
cana-5714	126	4	priya	priya	PROPN
cana-5714	126	5	karen	karen	PROPN
cana-5714	126	6	s	s	PROPN
cana-5714	126	7	and	and	CCONJ
cana-5714	126	8	arokia	arokia	PROPN
cana-5714	126	9	lancy	lancy	PROPN
cana-5714	126	10	a	a	X
cana-5714	126	11	,	,	PUNCT
cana-5714	126	12	on	on	ADP
cana-5714	126	13	product	product	NOUN
cana-5714	126	14	eccentricity	eccentricity	NOUN
cana-5714	126	15	energy	energy	NOUN
cana-5714	126	16	of	of	ADP
cana-5714	126	17	graphs	graph	NOUN
cana-5714	126	18	.	.	PUNCT
cana-5714	127	1	(	(	PUNCT
cana-5714	127	2	communicated	communicate	VERB
cana-5714	127	3	)	)	PUNCT