id	sid	tid	token	lemma	pos
ejpam-4806	1	1	european	european	PROPN
ejpam-4806	1	2	journal	journal	PROPN
ejpam-4806	1	3	of	of	ADP
ejpam-4806	1	4	pure	pure	ADJ
ejpam-4806	1	5	and	and	CCONJ
ejpam-4806	1	6	applied	apply	VERB
ejpam-4806	1	7	mathematics	mathematic	NOUN
ejpam-4806	1	8	vol	vol	NOUN
ejpam-4806	1	9	.	.	PUNCT
ejpam-4806	2	1	16	16	NUM
ejpam-4806	2	2	,	,	PUNCT
ejpam-4806	2	3	no	no	INTJ
ejpam-4806	2	4	.	.	NOUN
ejpam-4806	2	5	3	3	NUM
ejpam-4806	2	6	,	,	PUNCT
ejpam-4806	2	7	2023	2023	NUM
ejpam-4806	2	8	,	,	PUNCT
ejpam-4806	2	9	1731	1731	NUM
ejpam-4806	2	10	-	-	SYM
ejpam-4806	2	11	1746	1746	NUM
ejpam-4806	2	12	issn	issn	VERB
ejpam-4806	2	13	1307	1307	NUM
ejpam-4806	2	14	-	-	SYM
ejpam-4806	2	15	5543	5543	NUM
ejpam-4806	2	16	–	–	PUNCT
ejpam-4806	3	1	ejpam.com	ejpam.com	X
ejpam-4806	3	2	published	publish	VERB
ejpam-4806	3	3	by	by	ADP
ejpam-4806	3	4	new	new	PROPN
ejpam-4806	3	5	york	york	PROPN
ejpam-4806	3	6	business	business	PROPN
ejpam-4806	3	7	global	global	ADJ
ejpam-4806	3	8	new	new	ADJ
ejpam-4806	3	9	improved	improve	VERB
ejpam-4806	3	10	bounds	bound	NOUN
ejpam-4806	3	11	for	for	ADP
ejpam-4806	3	12	signless	signless	ADJ
ejpam-4806	3	13	laplacian	laplacian	ADJ
ejpam-4806	3	14	spectral	spectral	ADJ
ejpam-4806	3	15	radius	radius	NOUN
ejpam-4806	3	16	and	and	CCONJ
ejpam-4806	3	17	nordhaus	nordhaus	NOUN
ejpam-4806	3	18	-	-	PUNCT
ejpam-4806	3	19	gaddum	gaddum	NOUN
ejpam-4806	3	20	type	type	NOUN
ejpam-4806	3	21	inequalities	inequality	NOUN
ejpam-4806	3	22	for	for	ADP
ejpam-4806	3	23	agave	agave	ADJ
ejpam-4806	3	24	class	class	NOUN
ejpam-4806	3	25	of	of	ADP
ejpam-4806	3	26	graphs	graph	NOUN
ejpam-4806	3	27	malathy	malathy	ADV
ejpam-4806	3	28	v1	v1	NOUN
ejpam-4806	3	29	,	,	PUNCT
ejpam-4806	3	30	kalyani	kalyani	ADJ
ejpam-4806	3	31	desikan1,∗	desikan1,∗	NOUN
ejpam-4806	3	32	1	1	NUM
ejpam-4806	3	33	division	division	NOUN
ejpam-4806	3	34	of	of	ADP
ejpam-4806	3	35	mathematics	mathematic	NOUN
ejpam-4806	3	36	,	,	PUNCT
ejpam-4806	3	37	school	school	NOUN
ejpam-4806	3	38	of	of	ADP
ejpam-4806	3	39	advanced	advanced	ADJ
ejpam-4806	3	40	sciences	science	NOUN
ejpam-4806	3	41	,	,	PUNCT
ejpam-4806	3	42	vellore	vellore	PROPN
ejpam-4806	3	43	institute	institute	PROPN
ejpam-4806	3	44	of	of	ADP
ejpam-4806	3	45	technology	technology	PROPN
ejpam-4806	3	46	,	,	PUNCT
ejpam-4806	3	47	chennai	chennai	PROPN
ejpam-4806	3	48	,	,	PUNCT
ejpam-4806	3	49	india	india	PROPN
ejpam-4806	3	50	abstract	abstract	NOUN
ejpam-4806	3	51	.	.	PUNCT
ejpam-4806	4	1	core	core	NOUN
ejpam-4806	4	2	-	-	PUNCT
ejpam-4806	4	3	satellite	satellite	NOUN
ejpam-4806	4	4	graphs	graph	NOUN
ejpam-4806	4	5	θ(c	θ(c	VERB
ejpam-4806	4	6	,	,	PUNCT
ejpam-4806	4	7	s	s	X
ejpam-4806	4	8	,	,	PUNCT
ejpam-4806	4	9	η	η	NOUN
ejpam-4806	4	10	)	)	PUNCT
ejpam-4806	4	11	∼=	∼=	PROPN
ejpam-4806	4	12	kc	kc	NOUN
ejpam-4806	4	13	▽	▽	X
ejpam-4806	4	14	(	(	PUNCT
ejpam-4806	4	15	ηks	ηk	NOUN
ejpam-4806	4	16	)	)	PUNCT
ejpam-4806	4	17	are	be	AUX
ejpam-4806	4	18	graphs	graph	NOUN
ejpam-4806	4	19	consisting	consist	VERB
ejpam-4806	4	20	of	of	ADP
ejpam-4806	4	21	a	a	DET
ejpam-4806	4	22	central	central	ADJ
ejpam-4806	4	23	clique	clique	NOUN
ejpam-4806	4	24	kc	kc	PROPN
ejpam-4806	4	25	(	(	PUNCT
ejpam-4806	4	26	the	the	DET
ejpam-4806	4	27	core	core	NOUN
ejpam-4806	4	28	)	)	PUNCT
ejpam-4806	4	29	and	and	CCONJ
ejpam-4806	4	30	η	η	PROPN
ejpam-4806	4	31	copies	copy	NOUN
ejpam-4806	4	32	of	of	ADP
ejpam-4806	4	33	ks	ks	PROPN
ejpam-4806	4	34	(	(	PUNCT
ejpam-4806	4	35	the	the	DET
ejpam-4806	4	36	satellites	satellite	NOUN
ejpam-4806	4	37	)	)	PUNCT
ejpam-4806	4	38	meeting	meeting	NOUN
ejpam-4806	4	39	in	in	ADP
ejpam-4806	4	40	a	a	DET
ejpam-4806	4	41	common	common	ADJ
ejpam-4806	4	42	clique	clique	NOUN
ejpam-4806	4	43	.	.	PUNCT
ejpam-4806	5	1	they	they	PRON
ejpam-4806	5	2	belong	belong	VERB
ejpam-4806	5	3	to	to	ADP
ejpam-4806	5	4	the	the	DET
ejpam-4806	5	5	class	class	NOUN
ejpam-4806	5	6	of	of	ADP
ejpam-4806	5	7	graphs	graph	NOUN
ejpam-4806	5	8	of	of	ADP
ejpam-4806	5	9	diameter	diameter	NOUN
ejpam-4806	5	10	two	two	NUM
ejpam-4806	5	11	.	.	PUNCT
ejpam-4806	6	1	agave	agave	VERB
ejpam-4806	6	2	graphs	graph	NOUN
ejpam-4806	6	3	θ(2	θ(2	PROPN
ejpam-4806	6	4	,	,	PUNCT
ejpam-4806	6	5	1	1	NUM
ejpam-4806	6	6	,	,	PUNCT
ejpam-4806	6	7	η	η	NOUN
ejpam-4806	6	8	)	)	PUNCT
ejpam-4806	6	9	∼=	∼=	PROPN
ejpam-4806	6	10	k2	k2	ADJ
ejpam-4806	6	11	▽	▽	X
ejpam-4806	6	12	(	(	PUNCT
ejpam-4806	6	13	ηk1	ηk1	PROPN
ejpam-4806	6	14	)	)	PUNCT
ejpam-4806	6	15	belong	belong	VERB
ejpam-4806	6	16	to	to	ADP
ejpam-4806	6	17	the	the	DET
ejpam-4806	6	18	general	general	ADJ
ejpam-4806	6	19	class	class	NOUN
ejpam-4806	6	20	of	of	ADP
ejpam-4806	6	21	complete	complete	ADJ
ejpam-4806	6	22	split	split	NOUN
ejpam-4806	6	23	graphs	graph	NOUN
ejpam-4806	6	24	,	,	PUNCT
ejpam-4806	6	25	where	where	SCONJ
ejpam-4806	6	26	the	the	DET
ejpam-4806	6	27	graphs	graph	NOUN
ejpam-4806	6	28	consist	consist	VERB
ejpam-4806	6	29	of	of	ADP
ejpam-4806	6	30	a	a	DET
ejpam-4806	6	31	central	central	ADJ
ejpam-4806	6	32	clique	clique	NOUN
ejpam-4806	6	33	k2	k2	PROPN
ejpam-4806	6	34	and	and	CCONJ
ejpam-4806	6	35	η	η	PROPN
ejpam-4806	6	36	copies	copy	NOUN
ejpam-4806	6	37	of	of	ADP
ejpam-4806	6	38	k1	k1	NOUN
ejpam-4806	6	39	which	which	PRON
ejpam-4806	6	40	are	be	AUX
ejpam-4806	6	41	connected	connect	VERB
ejpam-4806	6	42	to	to	ADP
ejpam-4806	6	43	all	all	DET
ejpam-4806	6	44	the	the	DET
ejpam-4806	6	45	nodes	node	NOUN
ejpam-4806	6	46	of	of	ADP
ejpam-4806	6	47	the	the	DET
ejpam-4806	6	48	clique	clique	NOUN
ejpam-4806	6	49	.	.	PUNCT
ejpam-4806	7	1	they	they	PRON
ejpam-4806	7	2	are	be	AUX
ejpam-4806	7	3	the	the	DET
ejpam-4806	7	4	subclass	subclass	NOUN
ejpam-4806	7	5	of	of	ADP
ejpam-4806	7	6	core	core	NOUN
ejpam-4806	7	7	-	-	PUNCT
ejpam-4806	7	8	satellite	satellite	NOUN
ejpam-4806	7	9	graphs	graph	NOUN
ejpam-4806	7	10	.	.	PUNCT
ejpam-4806	8	1	let	let	VERB
ejpam-4806	8	2	µ(g	µ(g	PROPN
ejpam-4806	8	3	)	)	PUNCT
ejpam-4806	8	4	be	be	AUX
ejpam-4806	8	5	the	the	DET
ejpam-4806	8	6	spectral	spectral	ADJ
ejpam-4806	8	7	radius	radius	NOUN
ejpam-4806	8	8	of	of	ADP
ejpam-4806	8	9	the	the	DET
ejpam-4806	8	10	signless	signless	ADJ
ejpam-4806	8	11	laplacian	laplacian	ADJ
ejpam-4806	8	12	matrix	matrix	NOUN
ejpam-4806	8	13	q(g	q(g	PROPN
ejpam-4806	8	14	)	)	PUNCT
ejpam-4806	8	15	.	.	PUNCT
ejpam-4806	9	1	in	in	ADP
ejpam-4806	9	2	this	this	DET
ejpam-4806	9	3	paper	paper	NOUN
ejpam-4806	9	4	,	,	PUNCT
ejpam-4806	9	5	we	we	PRON
ejpam-4806	9	6	have	have	AUX
ejpam-4806	9	7	obtained	obtain	VERB
ejpam-4806	9	8	the	the	DET
ejpam-4806	9	9	greatest	greatest	ADV
ejpam-4806	9	10	lower	lower	ADV
ejpam-4806	9	11	bound	bind	VERB
ejpam-4806	9	12	and	and	CCONJ
ejpam-4806	9	13	the	the	DET
ejpam-4806	9	14	least	least	ADJ
ejpam-4806	9	15	upper	upper	ADJ
ejpam-4806	9	16	bound	bind	VERB
ejpam-4806	9	17	of	of	ADP
ejpam-4806	9	18	signless	signless	PROPN
ejpam-4806	9	19	laplacian	laplacian	ADJ
ejpam-4806	9	20	spectral	spectral	ADJ
ejpam-4806	9	21	radius	radius	NOUN
ejpam-4806	9	22	of	of	ADP
ejpam-4806	9	23	agave	agave	ADJ
ejpam-4806	9	24	graphs	graph	NOUN
ejpam-4806	9	25	.	.	PUNCT
ejpam-4806	10	1	these	these	DET
ejpam-4806	10	2	bounds	bound	NOUN
ejpam-4806	10	3	have	have	AUX
ejpam-4806	10	4	been	be	AUX
ejpam-4806	10	5	expressed	express	VERB
ejpam-4806	10	6	in	in	ADP
ejpam-4806	10	7	terms	term	NOUN
ejpam-4806	10	8	of	of	ADP
ejpam-4806	10	9	graph	graph	NOUN
ejpam-4806	10	10	invariants	invariant	NOUN
ejpam-4806	10	11	like	like	ADP
ejpam-4806	10	12	m	m	VERB
ejpam-4806	10	13	the	the	DET
ejpam-4806	10	14	number	number	NOUN
ejpam-4806	10	15	of	of	ADP
ejpam-4806	10	16	edges	edge	NOUN
ejpam-4806	10	17	,	,	PUNCT
ejpam-4806	10	18	n	n	CCONJ
ejpam-4806	10	19	the	the	DET
ejpam-4806	10	20	number	number	NOUN
ejpam-4806	10	21	of	of	ADP
ejpam-4806	10	22	vertices	vertex	NOUN
ejpam-4806	10	23	,	,	PUNCT
ejpam-4806	10	24	δ	δ	PROPN
ejpam-4806	10	25	the	the	DET
ejpam-4806	10	26	minimum	minimum	NOUN
ejpam-4806	10	27	degree	degree	NOUN
ejpam-4806	10	28	,	,	PUNCT
ejpam-4806	10	29	∆	∆	PROPN
ejpam-4806	10	30	the	the	DET
ejpam-4806	10	31	maximum	maximum	ADJ
ejpam-4806	10	32	degree	degree	NOUN
ejpam-4806	10	33	,	,	PUNCT
ejpam-4806	10	34	and	and	CCONJ
ejpam-4806	10	35	η	η	PROPN
ejpam-4806	10	36	copies	copy	NOUN
ejpam-4806	10	37	of	of	ADP
ejpam-4806	10	38	the	the	DET
ejpam-4806	10	39	satellite	satellite	NOUN
ejpam-4806	10	40	.	.	PUNCT
ejpam-4806	11	1	we	we	PRON
ejpam-4806	11	2	have	have	AUX
ejpam-4806	11	3	made	make	VERB
ejpam-4806	11	4	use	use	NOUN
ejpam-4806	11	5	of	of	ADP
ejpam-4806	11	6	the	the	DET
ejpam-4806	11	7	approximation	approximation	NOUN
ejpam-4806	11	8	technique	technique	NOUN
ejpam-4806	11	9	to	to	PART
ejpam-4806	11	10	derive	derive	VERB
ejpam-4806	11	11	these	these	DET
ejpam-4806	11	12	bounds	bound	NOUN
ejpam-4806	11	13	.	.	PUNCT
ejpam-4806	12	1	this	this	DET
ejpam-4806	12	2	unique	unique	ADJ
ejpam-4806	12	3	approach	approach	NOUN
ejpam-4806	12	4	can	can	AUX
ejpam-4806	12	5	be	be	AUX
ejpam-4806	12	6	utilized	utilize	VERB
ejpam-4806	12	7	to	to	PART
ejpam-4806	12	8	determine	determine	VERB
ejpam-4806	12	9	the	the	DET
ejpam-4806	12	10	bounds	bound	NOUN
ejpam-4806	12	11	for	for	ADP
ejpam-4806	12	12	the	the	DET
ejpam-4806	12	13	signless	signless	PROPN
ejpam-4806	12	14	laplacian	laplacian	ADJ
ejpam-4806	12	15	spectral	spectral	ADJ
ejpam-4806	12	16	radius	radius	NOUN
ejpam-4806	12	17	of	of	ADP
ejpam-4806	12	18	any	any	DET
ejpam-4806	12	19	general	general	ADJ
ejpam-4806	12	20	family	family	NOUN
ejpam-4806	12	21	of	of	ADP
ejpam-4806	12	22	graphs	graph	NOUN
ejpam-4806	12	23	.	.	PUNCT
ejpam-4806	13	1	we	we	PRON
ejpam-4806	13	2	have	have	AUX
ejpam-4806	13	3	also	also	ADV
ejpam-4806	13	4	obtained	obtain	VERB
ejpam-4806	13	5	nordhaus	nordhaus	NOUN
ejpam-4806	13	6	-	-	PUNCT
ejpam-4806	13	7	gaddum	gaddum	NOUN
ejpam-4806	13	8	type	type	NOUN
ejpam-4806	13	9	inequality	inequality	NOUN
ejpam-4806	13	10	using	use	VERB
ejpam-4806	13	11	the	the	DET
ejpam-4806	13	12	derived	derive	VERB
ejpam-4806	13	13	bounds	bound	NOUN
ejpam-4806	13	14	.	.	PUNCT
ejpam-4806	14	1	2020	2020	NUM
ejpam-4806	14	2	mathematics	mathematic	NOUN
ejpam-4806	14	3	subject	subject	NOUN
ejpam-4806	14	4	classifications	classification	NOUN
ejpam-4806	14	5	:	:	PUNCT
ejpam-4806	14	6	05c50	05c50	NUM
ejpam-4806	14	7	key	key	ADJ
ejpam-4806	14	8	words	word	NOUN
ejpam-4806	14	9	and	and	CCONJ
ejpam-4806	14	10	phrases	phrase	NOUN
ejpam-4806	14	11	:	:	PUNCT
ejpam-4806	14	12	spectral	spectral	ADJ
ejpam-4806	14	13	radius	radius	NOUN
ejpam-4806	14	14	,	,	PUNCT
ejpam-4806	14	15	signless	signless	PROPN
ejpam-4806	14	16	laplacian	laplacian	ADJ
ejpam-4806	14	17	spectral	spectral	ADJ
ejpam-4806	14	18	radius	radius	NOUN
ejpam-4806	14	19	,	,	PUNCT
ejpam-4806	14	20	core	core	NOUN
ejpam-4806	14	21	-	-	PUNCT
ejpam-4806	14	22	satellite	satellite	NOUN
ejpam-4806	14	23	graphs	graph	NOUN
ejpam-4806	14	24	,	,	PUNCT
ejpam-4806	14	25	agave	agave	NOUN
ejpam-4806	14	26	graphs	graph	NOUN
ejpam-4806	14	27	,	,	PUNCT
ejpam-4806	14	28	complete	complete	ADJ
ejpam-4806	14	29	split	split	NOUN
ejpam-4806	14	30	graph	graph	NOUN
ejpam-4806	14	31	1	1	NUM
ejpam-4806	14	32	.	.	PUNCT
ejpam-4806	14	33	introduction	introduction	NOUN
ejpam-4806	14	34	hierarchical	hierarchical	ADJ
ejpam-4806	14	35	products	product	NOUN
ejpam-4806	14	36	of	of	ADP
ejpam-4806	14	37	graphs	graph	NOUN
ejpam-4806	14	38	,	,	PUNCT
ejpam-4806	14	39	developed	develop	VERB
ejpam-4806	14	40	through	through	ADP
ejpam-4806	14	41	iterative	iterative	ADJ
ejpam-4806	14	42	hierarchical	hierarchical	ADJ
ejpam-4806	14	43	products	product	NOUN
ejpam-4806	14	44	of	of	ADP
ejpam-4806	14	45	complete	complete	ADJ
ejpam-4806	14	46	graphs	graph	NOUN
ejpam-4806	14	47	,	,	PUNCT
ejpam-4806	14	48	are	be	AUX
ejpam-4806	14	49	used	use	VERB
ejpam-4806	14	50	as	as	ADP
ejpam-4806	14	51	models	model	NOUN
ejpam-4806	14	52	for	for	ADP
ejpam-4806	14	53	real	real	ADJ
ejpam-4806	14	54	-	-	PUNCT
ejpam-4806	14	55	world	world	NOUN
ejpam-4806	14	56	networks	network	NOUN
ejpam-4806	14	57	.	.	PUNCT
ejpam-4806	15	1	core	core	NOUN
ejpam-4806	15	2	-	-	PUNCT
ejpam-4806	15	3	satellite	satellite	NOUN
ejpam-4806	15	4	graphs	graph	NOUN
ejpam-4806	15	5	can	can	AUX
ejpam-4806	15	6	be	be	AUX
ejpam-4806	15	7	redesigned	redesign	VERB
ejpam-4806	15	8	to	to	PART
ejpam-4806	15	9	resemble	resemble	VERB
ejpam-4806	15	10	some	some	DET
ejpam-4806	15	11	important	important	ADJ
ejpam-4806	15	12	characteristics	characteristic	NOUN
ejpam-4806	15	13	of	of	ADP
ejpam-4806	15	14	a	a	DET
ejpam-4806	15	15	complex	complex	ADJ
ejpam-4806	15	16	network	network	NOUN
ejpam-4806	15	17	to	to	PART
ejpam-4806	15	18	exhibit	exhibit	VERB
ejpam-4806	15	19	a	a	DET
ejpam-4806	15	20	hierarchical	hierarchical	ADJ
ejpam-4806	15	21	structure	structure	NOUN
ejpam-4806	15	22	[	[	X
ejpam-4806	15	23	14	14	NUM
ejpam-4806	15	24	,	,	PUNCT
ejpam-4806	15	25	18	18	NUM
ejpam-4806	15	26	]	]	PUNCT
ejpam-4806	15	27	.	.	PUNCT
ejpam-4806	16	1	estrada	estrada	PROPN
ejpam-4806	17	1	[	[	X
ejpam-4806	17	2	6	6	NUM
ejpam-4806	17	3	]	]	PUNCT
ejpam-4806	17	4	demonstrated	demonstrate	VERB
ejpam-4806	17	5	how	how	SCONJ
ejpam-4806	17	6	certain	certain	ADJ
ejpam-4806	17	7	classes	class	NOUN
ejpam-4806	17	8	of	of	ADP
ejpam-4806	17	9	real	real	ADJ
ejpam-4806	17	10	-	-	PUNCT
ejpam-4806	17	11	world	world	NOUN
ejpam-4806	17	12	networks	network	NOUN
ejpam-4806	17	13	can	can	AUX
ejpam-4806	17	14	be	be	AUX
ejpam-4806	17	15	modeled	model	VERB
ejpam-4806	17	16	using	use	VERB
ejpam-4806	17	17	core	core	NOUN
ejpam-4806	17	18	-	-	PUNCT
ejpam-4806	17	19	satellite	satellite	NOUN
ejpam-4806	17	20	graphs	graph	NOUN
ejpam-4806	17	21	.	.	PUNCT
ejpam-4806	18	1	this	this	DET
ejpam-4806	18	2	family	family	NOUN
ejpam-4806	18	3	of	of	ADP
ejpam-4806	18	4	graphs	graph	NOUN
ejpam-4806	18	5	has	have	VERB
ejpam-4806	18	6	topological	topological	ADJ
ejpam-4806	18	7	features	feature	NOUN
ejpam-4806	18	8	of	of	ADP
ejpam-4806	18	9	both	both	CCONJ
ejpam-4806	18	10	small	small	ADJ
ejpam-4806	18	11	world	world	NOUN
ejpam-4806	18	12	and	and	CCONJ
ejpam-4806	18	13	scale	scale	NOUN
ejpam-4806	18	14	-	-	PUNCT
ejpam-4806	18	15	free	free	ADJ
ejpam-4806	18	16	networks	network	NOUN
ejpam-4806	18	17	.	.	PUNCT
ejpam-4806	19	1	using	use	VERB
ejpam-4806	19	2	heuristic	heuristic	ADJ
ejpam-4806	19	3	analysis	analysis	NOUN
ejpam-4806	19	4	,	,	PUNCT
ejpam-4806	19	5	xu	xu	PROPN
ejpam-4806	19	6	and	and	CCONJ
ejpam-4806	19	7	zhang	zhang	PROPN
ejpam-4806	20	1	[	[	X
ejpam-4806	20	2	20	20	NUM
ejpam-4806	20	3	]	]	PUNCT
ejpam-4806	20	4	investigated	investigate	VERB
ejpam-4806	20	5	the	the	DET
ejpam-4806	20	6	effects	effect	NOUN
ejpam-4806	20	7	of	of	ADP
ejpam-4806	20	8	cover	cover	NOUN
ejpam-4806	20	9	time	time	NOUN
ejpam-4806	20	10	.	.	PUNCT
ejpam-4806	21	1	they	they	PRON
ejpam-4806	21	2	showed	show	VERB
ejpam-4806	21	3	that	that	SCONJ
ejpam-4806	21	4	the	the	DET
ejpam-4806	21	5	network	network	NOUN
ejpam-4806	21	6	topology	topology	NOUN
ejpam-4806	21	7	∗corresponding	∗corresponde	VERB
ejpam-4806	21	8	author	author	NOUN
ejpam-4806	21	9	.	.	PUNCT
ejpam-4806	22	1	doi	doi	NOUN
ejpam-4806	22	2	:	:	PUNCT
ejpam-4806	22	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4806	https://doi.org/10.29020/nybg.ejpam.v16i3.4806	ADP
ejpam-4806	22	4	email	email	NOUN
ejpam-4806	22	5	addresses	address	NOUN
ejpam-4806	22	6	:	:	PUNCT
ejpam-4806	22	7	malathy.viswanathan2015@vit.ac.in	malathy.viswanathan2015@vit.ac.in	NOUN
ejpam-4806	22	8	(	(	PUNCT
ejpam-4806	22	9	malathy	malathy	ADV
ejpam-4806	22	10	v	v	NOUN
ejpam-4806	22	11	)	)	PUNCT
ejpam-4806	22	12	,	,	PUNCT
ejpam-4806	22	13	kalyanidesikan@vit.ac.in	kalyanidesikan@vit.ac.in	NOUN
ejpam-4806	22	14	(	(	PUNCT
ejpam-4806	22	15	k.	k.	PROPN
ejpam-4806	22	16	desikan	desikan	PROPN
ejpam-4806	22	17	)	)	PUNCT
ejpam-4806	22	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4806	22	19	1731	1731	NUM
ejpam-4806	22	20	©	©	ADP
ejpam-4806	22	21	2023	2023	NUM
ejpam-4806	22	22	ejpam	ejpam	NOUN
ejpam-4806	22	23	all	all	DET
ejpam-4806	22	24	rights	right	NOUN
ejpam-4806	22	25	reserved	reserve	VERB
ejpam-4806	22	26	.	.	PUNCT
ejpam-4806	23	1	malathy	malathy	PROPN
ejpam-4806	23	2	v	v	NOUN
ejpam-4806	23	3	,	,	PUNCT
ejpam-4806	23	4	kalyani	kalyani	PROPN
ejpam-4806	23	5	desikan	desikan	PROPN
ejpam-4806	23	6	/	/	SYM
ejpam-4806	23	7	eur	eur	PROPN
ejpam-4806	23	8	.	.	PUNCT
ejpam-4806	24	1	j.	j.	PROPN
ejpam-4806	24	2	pure	pure	PROPN
ejpam-4806	24	3	appl	appl	PROPN
ejpam-4806	24	4	.	.	PROPN
ejpam-4806	24	5	math	math	PROPN
ejpam-4806	24	6	,	,	PUNCT
ejpam-4806	24	7	16	16	NUM
ejpam-4806	24	8	(	(	PUNCT
ejpam-4806	24	9	3	3	NUM
ejpam-4806	24	10	)	)	PUNCT
ejpam-4806	24	11	(	(	PUNCT
ejpam-4806	24	12	2023	2023	NUM
ejpam-4806	24	13	)	)	PUNCT
ejpam-4806	24	14	,	,	PUNCT
ejpam-4806	24	15	1731	1731	NUM
ejpam-4806	24	16	-	-	SYM
ejpam-4806	24	17	1746	1746	NUM
ejpam-4806	24	18	1732	1732	NUM
ejpam-4806	24	19	discussed	discuss	VERB
ejpam-4806	24	20	by	by	ADP
ejpam-4806	24	21	them	they	PRON
ejpam-4806	24	22	in	in	ADP
ejpam-4806	24	23	their	their	PRON
ejpam-4806	24	24	work	work	NOUN
ejpam-4806	24	25	,	,	PUNCT
ejpam-4806	24	26	having	have	VERB
ejpam-4806	24	27	both	both	CCONJ
ejpam-4806	24	28	small	small	ADJ
ejpam-4806	24	29	world	world	NOUN
ejpam-4806	24	30	and	and	CCONJ
ejpam-4806	24	31	scale	scale	NOUN
ejpam-4806	24	32	-	-	PUNCT
ejpam-4806	24	33	free	free	ADJ
ejpam-4806	24	34	features	feature	NOUN
ejpam-4806	24	35	,	,	PUNCT
ejpam-4806	24	36	exhibits	exhibit	VERB
ejpam-4806	24	37	the	the	DET
ejpam-4806	24	38	most	most	ADV
ejpam-4806	24	39	minimal	minimal	ADJ
ejpam-4806	24	40	cover	cover	NOUN
ejpam-4806	24	41	time	time	NOUN
ejpam-4806	24	42	.	.	PUNCT
ejpam-4806	25	1	agave	agave	NOUN
ejpam-4806	25	2	graphs	graph	NOUN
ejpam-4806	25	3	are	be	AUX
ejpam-4806	25	4	formed	form	VERB
ejpam-4806	25	5	by	by	ADP
ejpam-4806	25	6	connecting	connect	VERB
ejpam-4806	25	7	η	η	PROPN
ejpam-4806	25	8	disjoint	disjoint	NOUN
ejpam-4806	25	9	vertices	vertex	NOUN
ejpam-4806	25	10	to	to	ADP
ejpam-4806	25	11	both	both	DET
ejpam-4806	25	12	vertices	vertex	NOUN
ejpam-4806	25	13	of	of	ADP
ejpam-4806	25	14	a	a	DET
ejpam-4806	25	15	complete	complete	ADJ
ejpam-4806	25	16	graph	graph	NOUN
ejpam-4806	25	17	k2	k2	NOUN
ejpam-4806	25	18	.	.	PUNCT
ejpam-4806	26	1	they	they	PRON
ejpam-4806	26	2	belong	belong	VERB
ejpam-4806	26	3	to	to	ADP
ejpam-4806	26	4	the	the	DET
ejpam-4806	26	5	general	general	ADJ
ejpam-4806	26	6	class	class	NOUN
ejpam-4806	26	7	of	of	ADP
ejpam-4806	26	8	complete	complete	ADJ
ejpam-4806	26	9	split	split	NOUN
ejpam-4806	26	10	graphs	graph	NOUN
ejpam-4806	26	11	which	which	PRON
ejpam-4806	26	12	are	be	AUX
ejpam-4806	26	13	formed	form	VERB
ejpam-4806	26	14	by	by	ADP
ejpam-4806	26	15	the	the	DET
ejpam-4806	26	16	join	join	NOUN
ejpam-4806	26	17	of	of	ADP
ejpam-4806	26	18	central	central	ADJ
ejpam-4806	26	19	clique	clique	NOUN
ejpam-4806	26	20	kr	kr	PROPN
ejpam-4806	26	21	and	and	CCONJ
ejpam-4806	26	22	a	a	DET
ejpam-4806	26	23	set	set	NOUN
ejpam-4806	26	24	of	of	ADP
ejpam-4806	26	25	(	(	PUNCT
ejpam-4806	26	26	n−	n−	PROPN
ejpam-4806	26	27	r	r	NOUN
ejpam-4806	26	28	)	)	PUNCT
ejpam-4806	26	29	independent	independent	ADJ
ejpam-4806	26	30	vertices	vertex	NOUN
ejpam-4806	26	31	[	[	X
ejpam-4806	26	32	6	6	NUM
ejpam-4806	26	33	,	,	PUNCT
ejpam-4806	26	34	7	7	NUM
ejpam-4806	26	35	]	]	PUNCT
ejpam-4806	26	36	.	.	PUNCT
ejpam-4806	27	1	nair	nair	PROPN
ejpam-4806	27	2	et	et	PROPN
ejpam-4806	27	3	al	al	PROPN
ejpam-4806	27	4	.	.	PUNCT
ejpam-4806	28	1	[	[	X
ejpam-4806	28	2	1	1	X
ejpam-4806	28	3	]	]	PUNCT
ejpam-4806	28	4	distinctly	distinctly	ADV
ejpam-4806	28	5	picturised	picturise	VERB
ejpam-4806	28	6	the	the	DET
ejpam-4806	28	7	complete	complete	ADJ
ejpam-4806	28	8	split	split	NOUN
ejpam-4806	28	9	graphs	graph	NOUN
ejpam-4806	28	10	which	which	PRON
ejpam-4806	28	11	belong	belong	VERB
ejpam-4806	28	12	to	to	ADP
ejpam-4806	28	13	the	the	DET
ejpam-4806	28	14	family	family	NOUN
ejpam-4806	28	15	of	of	ADP
ejpam-4806	28	16	generalized	generalized	ADJ
ejpam-4806	28	17	core	core	NOUN
ejpam-4806	28	18	-	-	PUNCT
ejpam-4806	28	19	satellite	satellite	NOUN
ejpam-4806	28	20	graphs	graph	NOUN
ejpam-4806	28	21	.	.	PUNCT
ejpam-4806	29	1	the	the	DET
ejpam-4806	29	2	generalized	generalize	VERB
ejpam-4806	29	3	core	core	NOUN
ejpam-4806	29	4	-	-	PUNCT
ejpam-4806	29	5	satellite	satellite	NOUN
ejpam-4806	29	6	graphs	graph	NOUN
ejpam-4806	29	7	belong	belong	VERB
ejpam-4806	29	8	to	to	ADP
ejpam-4806	29	9	the	the	DET
ejpam-4806	29	10	larger	large	ADJ
ejpam-4806	29	11	family	family	NOUN
ejpam-4806	29	12	of	of	ADP
ejpam-4806	29	13	graphs	graph	NOUN
ejpam-4806	29	14	of	of	ADP
ejpam-4806	29	15	diameter	diameter	NOUN
ejpam-4806	29	16	two	two	NUM
ejpam-4806	29	17	.	.	PUNCT
ejpam-4806	30	1	the	the	DET
ejpam-4806	30	2	significant	significant	ADJ
ejpam-4806	30	3	features	feature	NOUN
ejpam-4806	30	4	of	of	ADP
ejpam-4806	30	5	agave	agave	ADJ
ejpam-4806	30	6	graphs	graph	NOUN
ejpam-4806	30	7	are	be	AUX
ejpam-4806	30	8	identified	identify	VERB
ejpam-4806	30	9	from	from	ADP
ejpam-4806	30	10	the	the	DET
ejpam-4806	30	11	article	article	NOUN
ejpam-4806	30	12	by	by	ADP
ejpam-4806	30	13	ernesto	ernesto	PROPN
ejpam-4806	30	14	and	and	CCONJ
ejpam-4806	30	15	eusebio	eusebio	PROPN
ejpam-4806	31	1	[	[	X
ejpam-4806	31	2	6	6	NUM
ejpam-4806	31	3	]	]	PUNCT
ejpam-4806	31	4	.	.	PUNCT
ejpam-4806	32	1	the	the	DET
ejpam-4806	32	2	distance	distance	NOUN
ejpam-4806	32	3	-	-	PUNCT
ejpam-4806	32	4	sum	sum	NOUN
ejpam-4806	32	5	heterogeneity	heterogeneity	NOUN
ejpam-4806	32	6	index	index	NOUN
ejpam-4806	32	7	φ(g	φ(g	PROPN
ejpam-4806	32	8	)	)	PUNCT
ejpam-4806	32	9	,	,	PUNCT
ejpam-4806	32	10	plays	play	VERB
ejpam-4806	32	11	a	a	DET
ejpam-4806	32	12	key	key	ADJ
ejpam-4806	32	13	role	role	NOUN
ejpam-4806	32	14	in	in	ADP
ejpam-4806	32	15	the	the	DET
ejpam-4806	32	16	structural	structural	ADJ
ejpam-4806	32	17	analysis	analysis	NOUN
ejpam-4806	32	18	of	of	ADP
ejpam-4806	32	19	complex	complex	ADJ
ejpam-4806	32	20	networks	network	NOUN
ejpam-4806	32	21	.	.	PUNCT
ejpam-4806	33	1	this	this	PRON
ejpam-4806	33	2	allows	allow	VERB
ejpam-4806	33	3	an	an	DET
ejpam-4806	33	4	interpretation	interpretation	NOUN
ejpam-4806	33	5	of	of	ADP
ejpam-4806	33	6	the	the	DET
ejpam-4806	33	7	wiener	wiener	NOUN
ejpam-4806	33	8	and	and	CCONJ
ejpam-4806	33	9	balaban	balaban	PROPN
ejpam-4806	33	10	index	index	NOUN
ejpam-4806	33	11	.	.	PUNCT
ejpam-4806	34	1	it	it	PRON
ejpam-4806	34	2	is	be	AUX
ejpam-4806	34	3	found	find	VERB
ejpam-4806	34	4	to	to	PART
ejpam-4806	34	5	be	be	AUX
ejpam-4806	34	6	useful	useful	ADJ
ejpam-4806	34	7	in	in	ADP
ejpam-4806	34	8	the	the	DET
ejpam-4806	34	9	analysis	analysis	NOUN
ejpam-4806	34	10	of	of	ADP
ejpam-4806	34	11	molecular	molecular	ADJ
ejpam-4806	34	12	graphs	graph	NOUN
ejpam-4806	34	13	.	.	PUNCT
ejpam-4806	35	1	furthermore	furthermore	ADV
ejpam-4806	35	2	,	,	PUNCT
ejpam-4806	35	3	it	it	PRON
ejpam-4806	35	4	is	be	AUX
ejpam-4806	35	5	found	find	VERB
ejpam-4806	35	6	that	that	SCONJ
ejpam-4806	35	7	there	there	PRON
ejpam-4806	35	8	is	be	VERB
ejpam-4806	35	9	proximity	proximity	NOUN
ejpam-4806	35	10	with	with	ADP
ejpam-4806	35	11	a	a	DET
ejpam-4806	35	12	node	node	NOUN
ejpam-4806	35	13	’s	’s	PART
ejpam-4806	35	14	closeness	closeness	NOUN
ejpam-4806	35	15	centrality	centrality	NOUN
ejpam-4806	35	16	and	and	CCONJ
ejpam-4806	35	17	average	average	ADJ
ejpam-4806	35	18	path	path	NOUN
ejpam-4806	35	19	length	length	NOUN
ejpam-4806	35	20	in	in	ADP
ejpam-4806	35	21	the	the	DET
ejpam-4806	35	22	analysis	analysis	NOUN
ejpam-4806	35	23	of	of	ADP
ejpam-4806	35	24	complex	complex	ADJ
ejpam-4806	35	25	networks	network	NOUN
ejpam-4806	35	26	.	.	PUNCT
ejpam-4806	36	1	this	this	DET
ejpam-4806	36	2	index	index	NOUN
ejpam-4806	36	3	highlights	highlight	VERB
ejpam-4806	36	4	more	more	ADJ
ejpam-4806	36	5	of	of	ADP
ejpam-4806	36	6	the	the	DET
ejpam-4806	36	7	structural	structural	ADJ
ejpam-4806	36	8	properties	property	NOUN
ejpam-4806	36	9	of	of	ADP
ejpam-4806	36	10	a	a	DET
ejpam-4806	36	11	network	network	NOUN
ejpam-4806	36	12	which	which	PRON
ejpam-4806	36	13	helps	help	VERB
ejpam-4806	36	14	us	we	PRON
ejpam-4806	36	15	to	to	PART
ejpam-4806	36	16	understand	understand	VERB
ejpam-4806	36	17	the	the	DET
ejpam-4806	36	18	functional	functional	ADJ
ejpam-4806	36	19	and	and	CCONJ
ejpam-4806	36	20	dynamic	dynamic	ADJ
ejpam-4806	36	21	processes	process	NOUN
ejpam-4806	36	22	in	in	ADP
ejpam-4806	36	23	complex	complex	ADJ
ejpam-4806	36	24	systems	system	NOUN
ejpam-4806	36	25	.	.	PUNCT
ejpam-4806	37	1	the	the	DET
ejpam-4806	37	2	authors	author	NOUN
ejpam-4806	37	3	have	have	AUX
ejpam-4806	37	4	conjectured	conjecture	VERB
ejpam-4806	37	5	that	that	SCONJ
ejpam-4806	37	6	among	among	ADP
ejpam-4806	37	7	graphs	graph	NOUN
ejpam-4806	37	8	with	with	ADP
ejpam-4806	37	9	a	a	DET
ejpam-4806	37	10	specific	specific	ADJ
ejpam-4806	37	11	number	number	NOUN
ejpam-4806	37	12	of	of	ADP
ejpam-4806	37	13	nodes	node	NOUN
ejpam-4806	37	14	φ(g	φ(g	ADJ
ejpam-4806	37	15	)	)	PUNCT
ejpam-4806	37	16	is	be	AUX
ejpam-4806	37	17	the	the	DET
ejpam-4806	37	18	maximum	maximum	NOUN
ejpam-4806	37	19	for	for	ADP
ejpam-4806	37	20	graphs	graph	NOUN
ejpam-4806	37	21	with	with	ADP
ejpam-4806	37	22	structures	structure	NOUN
ejpam-4806	37	23	resembling	resemble	VERB
ejpam-4806	37	24	the	the	DET
ejpam-4806	37	25	agave	agave	NOUN
ejpam-4806	37	26	graph	graph	NOUN
ejpam-4806	37	27	.	.	PUNCT
ejpam-4806	38	1	due	due	ADP
ejpam-4806	38	2	to	to	ADP
ejpam-4806	38	3	its	its	PRON
ejpam-4806	38	4	enhanced	enhanced	ADJ
ejpam-4806	38	5	effectiveness	effectiveness	NOUN
ejpam-4806	38	6	,	,	PUNCT
ejpam-4806	38	7	simplicity	simplicity	NOUN
ejpam-4806	38	8	,	,	PUNCT
ejpam-4806	38	9	and	and	CCONJ
ejpam-4806	38	10	better	well	ADJ
ejpam-4806	38	11	performance	performance	NOUN
ejpam-4806	38	12	,	,	PUNCT
ejpam-4806	38	13	the	the	DET
ejpam-4806	38	14	signless	signless	PROPN
ejpam-4806	38	15	laplacian	laplacian	PROPN
ejpam-4806	38	16	spectral	spectral	ADJ
ejpam-4806	38	17	radius	radius	NOUN
ejpam-4806	38	18	is	be	AUX
ejpam-4806	38	19	utilized	utilize	VERB
ejpam-4806	38	20	to	to	PART
ejpam-4806	38	21	examine	examine	VERB
ejpam-4806	38	22	graph	graph	NOUN
ejpam-4806	38	23	properties	property	NOUN
ejpam-4806	38	24	.	.	PUNCT
ejpam-4806	39	1	when	when	SCONJ
ejpam-4806	39	2	compared	compare	VERB
ejpam-4806	39	3	to	to	ADP
ejpam-4806	39	4	spectra	spectra	NOUN
ejpam-4806	39	5	of	of	ADP
ejpam-4806	39	6	other	other	ADJ
ejpam-4806	39	7	graph	graph	NOUN
ejpam-4806	39	8	matrices	matrix	NOUN
ejpam-4806	39	9	,	,	PUNCT
ejpam-4806	39	10	the	the	DET
ejpam-4806	39	11	spectrum	spectrum	NOUN
ejpam-4806	39	12	of	of	ADP
ejpam-4806	39	13	the	the	DET
ejpam-4806	39	14	signless	signless	ADJ
ejpam-4806	39	15	laplacian	laplacian	ADJ
ejpam-4806	39	16	matrix	matrix	NOUN
ejpam-4806	39	17	is	be	AUX
ejpam-4806	39	18	thoroughly	thoroughly	ADV
ejpam-4806	39	19	investigated	investigate	VERB
ejpam-4806	39	20	and	and	CCONJ
ejpam-4806	39	21	studied	study	VERB
ejpam-4806	39	22	extensively	extensively	ADV
ejpam-4806	39	23	[	[	X
ejpam-4806	39	24	4	4	NUM
ejpam-4806	39	25	,	,	PUNCT
ejpam-4806	39	26	11	11	NUM
ejpam-4806	39	27	,	,	PUNCT
ejpam-4806	39	28	19	19	NUM
ejpam-4806	39	29	]	]	PUNCT
ejpam-4806	39	30	.	.	PUNCT
ejpam-4806	40	1	through	through	ADP
ejpam-4806	40	2	different	different	ADJ
ejpam-4806	40	3	methods	method	NOUN
ejpam-4806	40	4	,	,	PUNCT
ejpam-4806	40	5	the	the	DET
ejpam-4806	40	6	bounds	bound	NOUN
ejpam-4806	40	7	are	be	AUX
ejpam-4806	40	8	obtained	obtain	VERB
ejpam-4806	40	9	for	for	ADP
ejpam-4806	40	10	µ(q(g	µ(q(g	ADJ
ejpam-4806	40	11	)	)	PUNCT
ejpam-4806	40	12	)	)	PUNCT
ejpam-4806	40	13	.	.	PUNCT
ejpam-4806	41	1	y.	y.	PROPN
ejpam-4806	41	2	chen	chen	PROPN
ejpam-4806	41	3	and	and	CCONJ
ejpam-4806	41	4	l.wang	l.wang	X
ejpam-4806	42	1	[	[	X
ejpam-4806	42	2	3	3	NUM
ejpam-4806	42	3	]	]	PUNCT
ejpam-4806	42	4	provided	provide	VERB
ejpam-4806	42	5	two	two	NUM
ejpam-4806	42	6	sharp	sharp	ADJ
ejpam-4806	42	7	upper	upper	ADJ
ejpam-4806	42	8	bounds	bound	NOUN
ejpam-4806	42	9	for	for	ADP
ejpam-4806	42	10	µ(q(g	µ(q(g	ADJ
ejpam-4806	42	11	)	)	PUNCT
ejpam-4806	42	12	)	)	PUNCT
ejpam-4806	42	13	in	in	ADP
ejpam-4806	42	14	terms	term	NOUN
ejpam-4806	42	15	of	of	ADP
ejpam-4806	42	16	the	the	DET
ejpam-4806	42	17	maximum	maximum	ADJ
ejpam-4806	42	18	degree	degree	NOUN
ejpam-4806	42	19	and	and	CCONJ
ejpam-4806	42	20	the	the	DET
ejpam-4806	42	21	minimum	minimum	NOUN
ejpam-4806	42	22	degree	degree	NOUN
ejpam-4806	42	23	of	of	ADP
ejpam-4806	42	24	the	the	DET
ejpam-4806	42	25	graph	graph	NOUN
ejpam-4806	42	26	g	g	NOUN
ejpam-4806	42	27	and	and	CCONJ
ejpam-4806	42	28	employed	employ	VERB
ejpam-4806	42	29	a	a	DET
ejpam-4806	42	30	new	new	ADJ
ejpam-4806	42	31	technique	technique	NOUN
ejpam-4806	42	32	to	to	PART
ejpam-4806	42	33	obtain	obtain	VERB
ejpam-4806	42	34	another	another	DET
ejpam-4806	42	35	sharp	sharp	ADJ
ejpam-4806	42	36	upper	upper	ADJ
ejpam-4806	42	37	bound	bind	VERB
ejpam-4806	42	38	.	.	PUNCT
ejpam-4806	43	1	in	in	ADP
ejpam-4806	43	2	[	[	X
ejpam-4806	43	3	5	5	NUM
ejpam-4806	43	4	]	]	PUNCT
ejpam-4806	43	5	,	,	PUNCT
ejpam-4806	43	6	xing	xing	PROPN
ejpam-4806	43	7	and	and	CCONJ
ejpam-4806	43	8	zhou	zhou	PROPN
ejpam-4806	43	9	provided	provide	VERB
ejpam-4806	43	10	a	a	DET
ejpam-4806	43	11	generalized	generalized	ADJ
ejpam-4806	43	12	theorem	theorem	NOUN
ejpam-4806	43	13	for	for	ADP
ejpam-4806	43	14	the	the	DET
ejpam-4806	43	15	upper	upper	ADJ
ejpam-4806	43	16	bound	bind	VERB
ejpam-4806	43	17	and	and	CCONJ
ejpam-4806	43	18	lower	low	ADJ
ejpam-4806	43	19	bound	bind	VERB
ejpam-4806	43	20	for	for	ADP
ejpam-4806	43	21	the	the	DET
ejpam-4806	43	22	spectral	spectral	ADJ
ejpam-4806	43	23	radius	radius	NOUN
ejpam-4806	43	24	of	of	ADP
ejpam-4806	43	25	a	a	DET
ejpam-4806	43	26	non	non	ADJ
ejpam-4806	43	27	-	-	ADJ
ejpam-4806	43	28	negative	negative	ADJ
ejpam-4806	43	29	matrix	matrix	NOUN
ejpam-4806	43	30	using	use	VERB
ejpam-4806	43	31	its	its	PRON
ejpam-4806	43	32	row	row	NOUN
ejpam-4806	43	33	sums	sum	NOUN
ejpam-4806	43	34	.	.	PUNCT
ejpam-4806	44	1	they	they	PRON
ejpam-4806	44	2	have	have	AUX
ejpam-4806	44	3	applied	apply	VERB
ejpam-4806	44	4	these	these	DET
ejpam-4806	44	5	bounds	bound	NOUN
ejpam-4806	44	6	to	to	ADP
ejpam-4806	44	7	various	various	ADJ
ejpam-4806	44	8	non	non	ADJ
ejpam-4806	44	9	-	-	ADJ
ejpam-4806	44	10	negative	negative	ADJ
ejpam-4806	44	11	matrices	matrix	NOUN
ejpam-4806	44	12	associated	associate	VERB
ejpam-4806	44	13	with	with	ADP
ejpam-4806	44	14	graphs	graph	NOUN
ejpam-4806	44	15	which	which	PRON
ejpam-4806	44	16	includes	include	VERB
ejpam-4806	44	17	the	the	DET
ejpam-4806	44	18	adjacency	adjacency	NOUN
ejpam-4806	44	19	matrix	matrix	NOUN
ejpam-4806	44	20	,	,	PUNCT
ejpam-4806	44	21	the	the	DET
ejpam-4806	44	22	signless	signless	ADJ
ejpam-4806	44	23	laplacian	laplacian	ADJ
ejpam-4806	44	24	matrix	matrix	NOUN
ejpam-4806	44	25	,	,	PUNCT
ejpam-4806	44	26	the	the	DET
ejpam-4806	44	27	distance	distance	NOUN
ejpam-4806	44	28	matrix	matrix	NOUN
ejpam-4806	44	29	,	,	PUNCT
ejpam-4806	44	30	the	the	DET
ejpam-4806	44	31	distance	distance	NOUN
ejpam-4806	44	32	signless	signless	NOUN
ejpam-4806	44	33	laplacian	laplacian	ADJ
ejpam-4806	44	34	matrix	matrix	NOUN
ejpam-4806	44	35	,	,	PUNCT
ejpam-4806	44	36	and	and	CCONJ
ejpam-4806	44	37	the	the	DET
ejpam-4806	44	38	reciprocal	reciprocal	ADJ
ejpam-4806	44	39	distance	distance	NOUN
ejpam-4806	44	40	matrix	matrix	NOUN
ejpam-4806	44	41	.	.	PUNCT
ejpam-4806	45	1	in	in	ADP
ejpam-4806	45	2	[	[	X
ejpam-4806	45	3	21	21	NUM
ejpam-4806	45	4	]	]	PUNCT
ejpam-4806	45	5	,	,	PUNCT
ejpam-4806	45	6	zhang	zhang	PROPN
ejpam-4806	45	7	et	et	PROPN
ejpam-4806	45	8	al	al	PROPN
ejpam-4806	45	9	.	.	PROPN
ejpam-4806	45	10	determined	determine	VERB
ejpam-4806	45	11	the	the	DET
ejpam-4806	45	12	largest	large	ADJ
ejpam-4806	45	13	signless	signless	ADJ
ejpam-4806	45	14	laplacian	laplacian	ADJ
ejpam-4806	45	15	spectral	spectral	ADJ
ejpam-4806	45	16	radius	radius	NOUN
ejpam-4806	45	17	among	among	ADP
ejpam-4806	45	18	the	the	DET
ejpam-4806	45	19	bicyclic	bicyclic	NOUN
ejpam-4806	45	20	graphs	graph	NOUN
ejpam-4806	45	21	with	with	ADP
ejpam-4806	45	22	perfect	perfect	ADJ
ejpam-4806	45	23	matchings	matching	NOUN
ejpam-4806	45	24	.	.	PUNCT
ejpam-4806	46	1	in	in	ADP
ejpam-4806	46	2	[	[	X
ejpam-4806	46	3	10	10	NUM
ejpam-4806	46	4	]	]	PUNCT
ejpam-4806	46	5	,	,	PUNCT
ejpam-4806	46	6	the	the	DET
ejpam-4806	46	7	authors	author	NOUN
ejpam-4806	46	8	determined	determine	VERB
ejpam-4806	46	9	sharp	sharp	ADJ
ejpam-4806	46	10	upper	upper	ADJ
ejpam-4806	46	11	bounds	bound	NOUN
ejpam-4806	46	12	for	for	ADP
ejpam-4806	46	13	the	the	DET
ejpam-4806	46	14	spectral	spectral	ADJ
ejpam-4806	46	15	radius	radius	NOUN
ejpam-4806	46	16	and	and	CCONJ
ejpam-4806	46	17	signless	signless	PROPN
ejpam-4806	46	18	laplacian	laplacian	ADJ
ejpam-4806	46	19	spectral	spectral	ADJ
ejpam-4806	46	20	radius	radius	NOUN
ejpam-4806	46	21	of	of	ADP
ejpam-4806	46	22	a	a	DET
ejpam-4806	46	23	uniform	uniform	ADJ
ejpam-4806	46	24	hypergraph	hypergraph	NOUN
ejpam-4806	46	25	in	in	ADP
ejpam-4806	46	26	terms	term	NOUN
ejpam-4806	46	27	of	of	ADP
ejpam-4806	46	28	the	the	DET
ejpam-4806	46	29	average	average	ADJ
ejpam-4806	46	30	2	2	NUM
ejpam-4806	46	31	degrees	degree	NOUN
ejpam-4806	46	32	or	or	CCONJ
ejpam-4806	46	33	degrees	degree	NOUN
ejpam-4806	46	34	of	of	ADP
ejpam-4806	46	35	vertices	vertex	NOUN
ejpam-4806	46	36	,	,	PUNCT
ejpam-4806	46	37	degree	degree	NOUN
ejpam-4806	46	38	and	and	CCONJ
ejpam-4806	46	39	the	the	DET
ejpam-4806	46	40	minimum	minimum	NOUN
ejpam-4806	46	41	degree	degree	NOUN
ejpam-4806	46	42	of	of	ADP
ejpam-4806	46	43	the	the	DET
ejpam-4806	46	44	vertices	vertex	NOUN
ejpam-4806	46	45	of	of	ADP
ejpam-4806	46	46	g.	g.	PROPN
ejpam-4806	46	47	moreover	moreover	ADV
ejpam-4806	46	48	,	,	PUNCT
ejpam-4806	46	49	they	they	PRON
ejpam-4806	46	50	proposed	propose	VERB
ejpam-4806	46	51	a	a	DET
ejpam-4806	46	52	new	new	ADJ
ejpam-4806	46	53	proving	proving	NOUN
ejpam-4806	46	54	technique	technique	NOUN
ejpam-4806	46	55	to	to	PART
ejpam-4806	46	56	obtain	obtain	VERB
ejpam-4806	46	57	a	a	DET
ejpam-4806	46	58	sharp	sharp	ADJ
ejpam-4806	46	59	upper	upper	ADJ
ejpam-4806	46	60	bound	bind	VERB
ejpam-4806	46	61	for	for	ADP
ejpam-4806	46	62	µ(q(g	µ(q(g	ADJ
ejpam-4806	46	63	)	)	PUNCT
ejpam-4806	46	64	)	)	PUNCT
ejpam-4806	46	65	.	.	PUNCT
ejpam-4806	47	1	in	in	ADP
ejpam-4806	47	2	[	[	X
ejpam-4806	47	3	13	13	NUM
ejpam-4806	47	4	]	]	PUNCT
ejpam-4806	47	5	,	,	PUNCT
ejpam-4806	47	6	feng	feng	PROPN
ejpam-4806	47	7	and	and	CCONJ
ejpam-4806	47	8	yu	yu	PROPN
ejpam-4806	47	9	studied	study	VERB
ejpam-4806	47	10	the	the	DET
ejpam-4806	47	11	family	family	NOUN
ejpam-4806	47	12	of	of	ADP
ejpam-4806	47	13	connected	connected	ADJ
ejpam-4806	47	14	graphs	graph	NOUN
ejpam-4806	47	15	with	with	ADP
ejpam-4806	47	16	prescribed	prescribed	ADJ
ejpam-4806	47	17	order	order	NOUN
ejpam-4806	47	18	and	and	CCONJ
ejpam-4806	47	19	diameter	diameter	NOUN
ejpam-4806	47	20	and	and	CCONJ
ejpam-4806	47	21	then	then	ADV
ejpam-4806	47	22	determined	determine	VERB
ejpam-4806	47	23	the	the	DET
ejpam-4806	47	24	extremal	extremal	ADJ
ejpam-4806	47	25	graph	graph	NOUN
ejpam-4806	47	26	that	that	PRON
ejpam-4806	47	27	has	have	VERB
ejpam-4806	47	28	the	the	DET
ejpam-4806	47	29	maximal	maximal	ADJ
ejpam-4806	47	30	signless	signless	NOUN
ejpam-4806	47	31	laplacian	laplacian	ADJ
ejpam-4806	47	32	spectral	spectral	ADJ
ejpam-4806	47	33	radius	radius	NOUN
ejpam-4806	47	34	.	.	PUNCT
ejpam-4806	48	1	y.	y.	PROPN
ejpam-4806	48	2	hong	hong	PROPN
ejpam-4806	48	3	and	and	CCONJ
ejpam-4806	48	4	shu	shu	NOUN
ejpam-4806	49	1	[	[	X
ejpam-4806	49	2	8	8	NUM
ejpam-4806	49	3	]	]	PUNCT
ejpam-4806	49	4	obtained	obtain	VERB
ejpam-4806	49	5	a	a	DET
ejpam-4806	49	6	sharp	sharp	ADJ
ejpam-4806	49	7	upper	upper	ADJ
ejpam-4806	49	8	bound	bind	VERB
ejpam-4806	49	9	of	of	ADP
ejpam-4806	49	10	the	the	DET
ejpam-4806	49	11	nordhaus	nordhaus	NOUN
ejpam-4806	49	12	-	-	PUNCT
ejpam-4806	49	13	gaddum	gaddum	NOUN
ejpam-4806	49	14	type	type	NOUN
ejpam-4806	49	15	for	for	ADP
ejpam-4806	49	16	the	the	DET
ejpam-4806	49	17	chromatic	chromatic	ADJ
ejpam-4806	49	18	numbers	number	NOUN
ejpam-4806	49	19	of	of	ADP
ejpam-4806	49	20	g	g	PROPN
ejpam-4806	49	21	and	and	CCONJ
ejpam-4806	49	22	gc	gc	PROPN
ejpam-4806	49	23	,	,	PUNCT
ejpam-4806	49	24	respectively	respectively	ADV
ejpam-4806	49	25	.	.	PUNCT
ejpam-4806	50	1	huiqing	huiqe	VERB
ejpam-4806	50	2	liu	liu	PROPN
ejpam-4806	50	3	et	et	PROPN
ejpam-4806	50	4	al	al	PROPN
ejpam-4806	50	5	.	.	PUNCT
ejpam-4806	51	1	[	[	X
ejpam-4806	51	2	11	11	NUM
ejpam-4806	51	3	]	]	PUNCT
ejpam-4806	51	4	obtained	obtain	VERB
ejpam-4806	51	5	a	a	DET
ejpam-4806	51	6	sharp	sharp	ADJ
ejpam-4806	51	7	upper	upper	ADJ
ejpam-4806	51	8	bound	bind	VERB
ejpam-4806	51	9	for	for	ADP
ejpam-4806	51	10	the	the	DET
ejpam-4806	51	11	nordhaus	nordhaus	NOUN
ejpam-4806	51	12	-	-	PUNCT
ejpam-4806	51	13	gaddam	gaddam	NOUN
ejpam-4806	51	14	type	type	NOUN
ejpam-4806	51	15	relation	relation	NOUN
ejpam-4806	51	16	of	of	ADP
ejpam-4806	51	17	the	the	DET
ejpam-4806	51	18	laplacian	laplacian	ADJ
ejpam-4806	51	19	spectral	spectral	ADJ
ejpam-4806	51	20	radius	radius	PROPN
ejpam-4806	51	21	l(g	l(g	PROPN
ejpam-4806	51	22	)	)	PUNCT
ejpam-4806	51	23	by	by	ADP
ejpam-4806	51	24	making	make	VERB
ejpam-4806	51	25	use	use	NOUN
ejpam-4806	51	26	of	of	ADP
ejpam-4806	51	27	the	the	DET
ejpam-4806	51	28	relation	relation	NOUN
ejpam-4806	51	29	µ(a(g	µ(a(g	PROPN
ejpam-4806	51	30	)	)	PUNCT
ejpam-4806	51	31	)	)	PUNCT
ejpam-4806	51	32	≤	≤	NUM
ejpam-4806	51	33	µ(q(g	µ(q(g	NOUN
ejpam-4806	51	34	)	)	PUNCT
ejpam-4806	51	35	)	)	PUNCT
ejpam-4806	51	36	.	.	PUNCT
ejpam-4806	52	1	shuchao	shuchao	ADJ
ejpam-4806	52	2	and	and	CCONJ
ejpam-4806	52	3	tian	tian	ADJ
ejpam-4806	52	4	[	[	X
ejpam-4806	52	5	9	9	NUM
ejpam-4806	52	6	]	]	PUNCT
ejpam-4806	52	7	and	and	CCONJ
ejpam-4806	52	8	shi	shi	PROPN
ejpam-4806	53	1	[	[	X
ejpam-4806	53	2	19	19	NUM
ejpam-4806	53	3	]	]	PUNCT
ejpam-4806	53	4	,	,	PUNCT
ejpam-4806	53	5	have	have	AUX
ejpam-4806	53	6	derived	derive	VERB
ejpam-4806	53	7	sharp	sharp	ADJ
ejpam-4806	53	8	bounds	bound	NOUN
ejpam-4806	53	9	for	for	ADP
ejpam-4806	53	10	nordhaus	nordhaus	NOUN
ejpam-4806	53	11	-	-	PUNCT
ejpam-4806	53	12	gaddam	gaddam	NOUN
ejpam-4806	53	13	type	type	NOUN
ejpam-4806	53	14	of	of	ADP
ejpam-4806	53	15	relation	relation	NOUN
ejpam-4806	53	16	in	in	ADP
ejpam-4806	53	17	terms	term	NOUN
ejpam-4806	53	18	n	n	CCONJ
ejpam-4806	53	19	,	,	PUNCT
ejpam-4806	53	20	m	m	PROPN
ejpam-4806	53	21	,	,	PUNCT
ejpam-4806	53	22	δ(g	δ(g	PROPN
ejpam-4806	53	23	)	)	PUNCT
ejpam-4806	53	24	,	,	PUNCT
ejpam-4806	53	25	and	and	CCONJ
ejpam-4806	53	26	∆(g	∆(g	NOUN
ejpam-4806	53	27	)	)	PUNCT
ejpam-4806	53	28	.	.	PUNCT
ejpam-4806	54	1	nikiforov	nikiforov	NOUN
ejpam-4806	55	1	[	[	X
ejpam-4806	55	2	15–17	15–17	NUM
ejpam-4806	55	3	]	]	PUNCT
ejpam-4806	55	4	,	,	PUNCT
ejpam-4806	55	5	conjectured	conjecture	VERB
ejpam-4806	55	6	and	and	CCONJ
ejpam-4806	55	7	obtained	obtain	VERB
ejpam-4806	55	8	improved	improved	ADJ
ejpam-4806	55	9	bounds	bound	NOUN
ejpam-4806	55	10	on	on	ADP
ejpam-4806	55	11	nordhausmalathy	nordhausmalathy	ADJ
ejpam-4806	55	12	v	v	NOUN
ejpam-4806	55	13	,	,	PUNCT
ejpam-4806	55	14	kalyani	kalyani	PROPN
ejpam-4806	55	15	desikan	desikan	PROPN
ejpam-4806	55	16	/	/	SYM
ejpam-4806	55	17	eur	eur	PROPN
ejpam-4806	55	18	.	.	PUNCT
ejpam-4806	56	1	j.	j.	PROPN
ejpam-4806	56	2	pure	pure	PROPN
ejpam-4806	56	3	appl	appl	PROPN
ejpam-4806	56	4	.	.	PROPN
ejpam-4806	56	5	math	math	PROPN
ejpam-4806	56	6	,	,	PUNCT
ejpam-4806	56	7	16	16	NUM
ejpam-4806	56	8	(	(	PUNCT
ejpam-4806	56	9	3	3	NUM
ejpam-4806	56	10	)	)	PUNCT
ejpam-4806	56	11	(	(	PUNCT
ejpam-4806	56	12	2023	2023	NUM
ejpam-4806	56	13	)	)	PUNCT
ejpam-4806	56	14	,	,	PUNCT
ejpam-4806	56	15	1731	1731	NUM
ejpam-4806	56	16	-	-	SYM
ejpam-4806	56	17	1746	1746	NUM
ejpam-4806	56	18	1733	1733	NUM
ejpam-4806	56	19	figure	figure	NOUN
ejpam-4806	56	20	1	1	NUM
ejpam-4806	56	21	:	:	PUNCT
ejpam-4806	56	22	agave	agave	NOUN
ejpam-4806	56	23	graph	graph	NOUN
ejpam-4806	56	24	k2	k2	ADJ
ejpam-4806	56	25	▽	▽	NOUN
ejpam-4806	56	26	4k1	4k1	NUM
ejpam-4806	56	27	and	and	CCONJ
ejpam-4806	56	28	k2	k2	ADJ
ejpam-4806	56	29	▽	▽	PUNCT
ejpam-4806	56	30	ηk1	ηk1	PROPN
ejpam-4806	56	31	gaddum	gaddum	PROPN
ejpam-4806	56	32	type	type	NOUN
ejpam-4806	56	33	inequality	inequality	NOUN
ejpam-4806	56	34	.	.	PUNCT
ejpam-4806	57	1	in	in	ADP
ejpam-4806	57	2	[	[	X
ejpam-4806	57	3	2	2	NUM
ejpam-4806	57	4	]	]	PUNCT
ejpam-4806	57	5	,	,	PUNCT
ejpam-4806	57	6	mustapha	mustapha	PROPN
ejpam-4806	57	7	aouchiche	aouchiche	PROPN
ejpam-4806	57	8	and	and	CCONJ
ejpam-4806	57	9	pierre	pierre	PROPN
ejpam-4806	57	10	hansen	hansen	PROPN
ejpam-4806	57	11	mentioned	mention	VERB
ejpam-4806	57	12	many	many	ADJ
ejpam-4806	57	13	results	result	NOUN
ejpam-4806	57	14	on	on	ADP
ejpam-4806	57	15	lower	low	ADJ
ejpam-4806	57	16	and	and	CCONJ
ejpam-4806	57	17	upper	upper	ADJ
ejpam-4806	57	18	bounds	bound	NOUN
ejpam-4806	57	19	in	in	ADP
ejpam-4806	57	20	their	their	PRON
ejpam-4806	57	21	survey	survey	NOUN
ejpam-4806	57	22	article	article	NOUN
ejpam-4806	57	23	on	on	ADP
ejpam-4806	57	24	nordhaus	nordhaus	PROPN
ejpam-4806	57	25	-	-	PUNCT
ejpam-4806	57	26	gaddum	gaddum	NOUN
ejpam-4806	57	27	type	type	NOUN
ejpam-4806	57	28	inequality	inequality	NOUN
ejpam-4806	57	29	on	on	ADP
ejpam-4806	57	30	the	the	DET
ejpam-4806	57	31	sum	sum	NOUN
ejpam-4806	57	32	and	and	CCONJ
ejpam-4806	57	33	the	the	DET
ejpam-4806	57	34	product	product	NOUN
ejpam-4806	57	35	of	of	ADP
ejpam-4806	57	36	many	many	ADJ
ejpam-4806	57	37	other	other	ADJ
ejpam-4806	57	38	graph	graph	NOUN
ejpam-4806	57	39	invariants	invariant	NOUN
ejpam-4806	57	40	of	of	ADP
ejpam-4806	57	41	a	a	DET
ejpam-4806	57	42	graph	graph	NOUN
ejpam-4806	57	43	and	and	CCONJ
ejpam-4806	57	44	its	its	PRON
ejpam-4806	57	45	complements	complement	NOUN
ejpam-4806	57	46	.	.	PUNCT
ejpam-4806	58	1	motivated	motivate	VERB
ejpam-4806	58	2	by	by	ADP
ejpam-4806	58	3	the	the	DET
ejpam-4806	58	4	above	above	ADV
ejpam-4806	58	5	-	-	PUNCT
ejpam-4806	58	6	mentioned	mention	VERB
ejpam-4806	58	7	results	result	NOUN
ejpam-4806	58	8	,	,	PUNCT
ejpam-4806	58	9	we	we	PRON
ejpam-4806	58	10	have	have	AUX
ejpam-4806	58	11	implemented	implement	VERB
ejpam-4806	58	12	some	some	DET
ejpam-4806	58	13	unique	unique	ADJ
ejpam-4806	58	14	techniques	technique	NOUN
ejpam-4806	58	15	to	to	PART
ejpam-4806	58	16	obtain	obtain	VERB
ejpam-4806	58	17	tight	tight	ADJ
ejpam-4806	58	18	upper	upper	ADJ
ejpam-4806	58	19	and	and	CCONJ
ejpam-4806	58	20	lower	low	ADJ
ejpam-4806	58	21	bounds	bound	NOUN
ejpam-4806	58	22	for	for	ADP
ejpam-4806	58	23	the	the	DET
ejpam-4806	58	24	signless	signless	PROPN
ejpam-4806	58	25	laplacian	laplacian	ADJ
ejpam-4806	58	26	spectral	spectral	ADJ
ejpam-4806	58	27	radius	radius	NOUN
ejpam-4806	58	28	for	for	ADP
ejpam-4806	58	29	the	the	DET
ejpam-4806	58	30	agave	agave	ADJ
ejpam-4806	58	31	class	class	NOUN
ejpam-4806	58	32	of	of	ADP
ejpam-4806	58	33	graphs	graph	NOUN
ejpam-4806	58	34	.	.	PUNCT
ejpam-4806	59	1	we	we	PRON
ejpam-4806	59	2	have	have	AUX
ejpam-4806	59	3	utilized	utilize	VERB
ejpam-4806	59	4	these	these	DET
ejpam-4806	59	5	bounds	bound	NOUN
ejpam-4806	59	6	to	to	PART
ejpam-4806	59	7	obtain	obtain	VERB
ejpam-4806	59	8	the	the	DET
ejpam-4806	59	9	corresponding	corresponding	ADJ
ejpam-4806	59	10	tight	tight	ADJ
ejpam-4806	59	11	bounds	bound	NOUN
ejpam-4806	59	12	for	for	ADP
ejpam-4806	59	13	nordhaus	nordhaus	NOUN
ejpam-4806	59	14	-	-	PUNCT
ejpam-4806	59	15	gaddam	gaddam	NOUN
ejpam-4806	59	16	type	type	NOUN
ejpam-4806	59	17	inequality	inequality	NOUN
ejpam-4806	59	18	.	.	PUNCT
ejpam-4806	60	1	2	2	X
ejpam-4806	60	2	.	.	X
ejpam-4806	60	3	preliminaries	preliminary	NOUN
ejpam-4806	60	4	here	here	ADV
ejpam-4806	60	5	we	we	PRON
ejpam-4806	60	6	mention	mention	VERB
ejpam-4806	60	7	some	some	DET
ejpam-4806	60	8	preliminaries	preliminary	NOUN
ejpam-4806	60	9	,	,	PUNCT
ejpam-4806	60	10	definitions	definition	NOUN
ejpam-4806	60	11	,	,	PUNCT
ejpam-4806	60	12	and	and	CCONJ
ejpam-4806	60	13	results	result	NOUN
ejpam-4806	60	14	which	which	PRON
ejpam-4806	60	15	will	will	AUX
ejpam-4806	60	16	be	be	AUX
ejpam-4806	60	17	used	use	VERB
ejpam-4806	60	18	to	to	PART
ejpam-4806	60	19	derive	derive	VERB
ejpam-4806	60	20	our	our	PRON
ejpam-4806	60	21	main	main	ADJ
ejpam-4806	60	22	results	result	NOUN
ejpam-4806	60	23	.	.	PUNCT
ejpam-4806	61	1	let	let	VERB
ejpam-4806	61	2	g	g	PROPN
ejpam-4806	61	3	=	=	SYM
ejpam-4806	61	4	(	(	PUNCT
ejpam-4806	61	5	v	v	NOUN
ejpam-4806	61	6	,	,	PUNCT
ejpam-4806	61	7	e	e	NOUN
ejpam-4806	61	8	)	)	PUNCT
ejpam-4806	61	9	be	be	AUX
ejpam-4806	61	10	a	a	DET
ejpam-4806	61	11	simple	simple	ADJ
ejpam-4806	61	12	,	,	PUNCT
ejpam-4806	61	13	connected	connected	ADJ
ejpam-4806	61	14	graph	graph	NOUN
ejpam-4806	61	15	with	with	ADP
ejpam-4806	61	16	n	n	ADP
ejpam-4806	61	17	vertices	vertex	NOUN
ejpam-4806	61	18	and	and	CCONJ
ejpam-4806	61	19	m	m	PRON
ejpam-4806	61	20	edges	edge	NOUN
ejpam-4806	61	21	.	.	PUNCT
ejpam-4806	62	1	let	let	VERB
ejpam-4806	62	2	δ(g	δ(g	X
ejpam-4806	62	3	)	)	PUNCT
ejpam-4806	63	1	=	=	SYM
ejpam-4806	63	2	δ	δ	PROPN
ejpam-4806	63	3	and	and	CCONJ
ejpam-4806	63	4	∆(g	∆(g	PROPN
ejpam-4806	63	5	)	)	PUNCT
ejpam-4806	63	6	=	=	NOUN
ejpam-4806	63	7	∆	∆	PROPN
ejpam-4806	63	8	be	be	VERB
ejpam-4806	63	9	the	the	DET
ejpam-4806	63	10	minimum	minimum	NOUN
ejpam-4806	63	11	and	and	CCONJ
ejpam-4806	63	12	the	the	DET
ejpam-4806	63	13	maximum	maximum	ADJ
ejpam-4806	63	14	degree	degree	NOUN
ejpam-4806	63	15	of	of	ADP
ejpam-4806	63	16	vertices	vertex	NOUN
ejpam-4806	63	17	of	of	ADP
ejpam-4806	63	18	g	g	NOUN
ejpam-4806	63	19	,	,	PUNCT
ejpam-4806	63	20	respectively	respectively	ADV
ejpam-4806	63	21	.	.	PUNCT
ejpam-4806	64	1	let	let	AUX
ejpam-4806	64	2	gc	gc	PROPN
ejpam-4806	64	3	be	be	AUX
ejpam-4806	64	4	the	the	DET
ejpam-4806	64	5	complement	complement	NOUN
ejpam-4806	64	6	of	of	ADP
ejpam-4806	64	7	graph	graph	NOUN
ejpam-4806	64	8	g.	g.	PROPN
ejpam-4806	64	9	let	let	VERB
ejpam-4806	64	10	a(g	a(g	PROPN
ejpam-4806	64	11	)	)	PUNCT
ejpam-4806	64	12	be	be	AUX
ejpam-4806	64	13	the	the	DET
ejpam-4806	64	14	adjacency	adjacency	NOUN
ejpam-4806	64	15	matrix	matrix	NOUN
ejpam-4806	64	16	of	of	ADP
ejpam-4806	64	17	the	the	DET
ejpam-4806	64	18	graph	graph	NOUN
ejpam-4806	64	19	g	g	NOUN
ejpam-4806	64	20	and	and	CCONJ
ejpam-4806	64	21	the	the	DET
ejpam-4806	64	22	degree	degree	NOUN
ejpam-4806	64	23	diagonal	diagonal	ADJ
ejpam-4806	64	24	matrix	matrix	NOUN
ejpam-4806	64	25	d(g	d(g	NUM
ejpam-4806	64	26	)	)	PUNCT
ejpam-4806	65	1	=	=	SYM
ejpam-4806	65	2	diag	diag	NOUN
ejpam-4806	65	3	(	(	PUNCT
ejpam-4806	65	4	d(v1	d(v1	PROPN
ejpam-4806	65	5	)	)	PUNCT
ejpam-4806	65	6	,	,	PUNCT
ejpam-4806	65	7	d(v2	d(v2	NOUN
ejpam-4806	65	8	)	)	PUNCT
ejpam-4806	65	9	,	,	PUNCT
ejpam-4806	65	10	...	...	PUNCT
ejpam-4806	65	11	,	,	PUNCT
ejpam-4806	65	12	d(vn	d(vn	PROPN
ejpam-4806	65	13	)	)	PUNCT
ejpam-4806	65	14	)	)	PUNCT
ejpam-4806	65	15	is	be	AUX
ejpam-4806	65	16	the	the	DET
ejpam-4806	65	17	diagonal	diagonal	ADJ
ejpam-4806	65	18	matrix	matrix	NOUN
ejpam-4806	65	19	of	of	ADP
ejpam-4806	65	20	vertex	vertex	NOUN
ejpam-4806	65	21	degrees	degree	NOUN
ejpam-4806	65	22	.	.	PUNCT
ejpam-4806	66	1	for	for	ADP
ejpam-4806	66	2	any	any	DET
ejpam-4806	66	3	n	n	NUM
ejpam-4806	66	4	×	×	NOUN
ejpam-4806	66	5	n	n	CCONJ
ejpam-4806	66	6	real	real	ADJ
ejpam-4806	66	7	symmetric	symmetric	ADJ
ejpam-4806	66	8	matrix	matrix	NOUN
ejpam-4806	66	9	m	m	VERB
ejpam-4806	66	10	,	,	PUNCT
ejpam-4806	66	11	by	by	ADP
ejpam-4806	66	12	geršgorin	geršgorin	PROPN
ejpam-4806	66	13	’s	’s	PART
ejpam-4806	66	14	theorem	theorem	PROPN
ejpam-4806	66	15	,	,	PUNCT
ejpam-4806	66	16	its	its	PRON
ejpam-4806	66	17	eigenvalues	eigenvalue	NOUN
ejpam-4806	66	18	are	be	AUX
ejpam-4806	66	19	non	non	ADJ
ejpam-4806	66	20	-	-	ADJ
ejpam-4806	66	21	negative	negative	ADJ
ejpam-4806	66	22	real	real	ADJ
ejpam-4806	66	23	numbers	number	NOUN
ejpam-4806	66	24	.	.	PUNCT
ejpam-4806	67	1	let	let	AUX
ejpam-4806	67	2	ρ(g	ρ(g	NOUN
ejpam-4806	67	3	)	)	PUNCT
ejpam-4806	67	4	be	be	AUX
ejpam-4806	67	5	the	the	DET
ejpam-4806	67	6	spectral	spectral	ADJ
ejpam-4806	67	7	radius	radius	NOUN
ejpam-4806	67	8	of	of	ADP
ejpam-4806	67	9	adjacency	adjacency	NOUN
ejpam-4806	67	10	matrix	matrix	NOUN
ejpam-4806	67	11	a(g	a(g	PROPN
ejpam-4806	67	12	)	)	PUNCT
ejpam-4806	67	13	,	,	PUNCT
ejpam-4806	67	14	with	with	ADP
ejpam-4806	67	15	non	non	ADJ
ejpam-4806	67	16	-	-	ADJ
ejpam-4806	67	17	increasing	increasing	ADJ
ejpam-4806	67	18	sequence	sequence	NOUN
ejpam-4806	67	19	of	of	ADP
ejpam-4806	67	20	real	real	ADJ
ejpam-4806	67	21	eigenvalues	eigenvalue	NOUN
ejpam-4806	67	22	ρ(g	ρ(g	ADP
ejpam-4806	67	23	)	)	PUNCT
ejpam-4806	67	24	=	=	SYM
ejpam-4806	67	25	ρ1(g	ρ1(g	PROPN
ejpam-4806	67	26	)	)	PUNCT
ejpam-4806	67	27	≥	≥	NOUN
ejpam-4806	67	28	ρ2(g	ρ2(g	NUM
ejpam-4806	67	29	)	)	PUNCT
ejpam-4806	67	30	≥	≥	NOUN
ejpam-4806	67	31	...	...	PUNCT
ejpam-4806	67	32	≥	≥	X
ejpam-4806	67	33	ρn(g	ρn(g	NUM
ejpam-4806	67	34	)	)	PUNCT
ejpam-4806	67	35	.	.	PUNCT
ejpam-4806	68	1	let	let	VERB
ejpam-4806	68	2	µ(g	µ(g	PROPN
ejpam-4806	68	3	)	)	PUNCT
ejpam-4806	68	4	be	be	AUX
ejpam-4806	68	5	the	the	DET
ejpam-4806	68	6	signless	signless	PROPN
ejpam-4806	68	7	laplacian	laplacian	ADJ
ejpam-4806	68	8	spectral	spectral	ADJ
ejpam-4806	68	9	radius	radius	NOUN
ejpam-4806	68	10	of	of	ADP
ejpam-4806	68	11	the	the	DET
ejpam-4806	68	12	matrix	matrix	NOUN
ejpam-4806	68	13	q(g	q(g	NOUN
ejpam-4806	68	14	)	)	PUNCT
ejpam-4806	69	1	=	=	SYM
ejpam-4806	69	2	d(g	d(g	PROPN
ejpam-4806	69	3	)	)	PUNCT
ejpam-4806	70	1	+	+	NUM
ejpam-4806	70	2	a(g	a(g	PROPN
ejpam-4806	70	3	)	)	PUNCT
ejpam-4806	70	4	,	,	PUNCT
ejpam-4806	70	5	which	which	PRON
ejpam-4806	70	6	is	be	AUX
ejpam-4806	70	7	irreducible	irreducible	ADJ
ejpam-4806	70	8	,	,	PUNCT
ejpam-4806	70	9	symmetric	symmetric	ADJ
ejpam-4806	70	10	,	,	PUNCT
ejpam-4806	70	11	and	and	CCONJ
ejpam-4806	70	12	non	non	ADJ
ejpam-4806	70	13	-	-	ADJ
ejpam-4806	70	14	negative	negative	ADJ
ejpam-4806	70	15	.	.	PUNCT
ejpam-4806	71	1	thus	thus	ADV
ejpam-4806	71	2	the	the	DET
ejpam-4806	71	3	matrix	matrix	NOUN
ejpam-4806	71	4	q(g	q(g	NOUN
ejpam-4806	71	5	)	)	PUNCT
ejpam-4806	71	6	is	be	AUX
ejpam-4806	71	7	positive	positive	ADJ
ejpam-4806	71	8	and	and	CCONJ
ejpam-4806	71	9	semi	semi	ADJ
ejpam-4806	71	10	-	-	ADJ
ejpam-4806	71	11	definite	definite	ADJ
ejpam-4806	71	12	.	.	PUNCT
ejpam-4806	72	1	it	it	PRON
ejpam-4806	72	2	has	have	VERB
ejpam-4806	72	3	non	non	ADJ
ejpam-4806	72	4	-	-	ADJ
ejpam-4806	72	5	increasing	increase	VERB
ejpam-4806	72	6	sequence	sequence	NOUN
ejpam-4806	72	7	of	of	ADP
ejpam-4806	72	8	real	real	ADJ
ejpam-4806	72	9	eigenvalues	eigenvalue	NOUN
ejpam-4806	72	10	µ(q(g	µ(q(g	ADJ
ejpam-4806	72	11	)	)	PUNCT
ejpam-4806	72	12	)	)	PUNCT
ejpam-4806	73	1	=	=	SYM
ejpam-4806	73	2	µ1(q(g	µ1(q(g	NOUN
ejpam-4806	73	3	)	)	PUNCT
ejpam-4806	73	4	)	)	PUNCT
ejpam-4806	73	5	≥	≥	NOUN
ejpam-4806	73	6	µ2(q(g	µ2(q(g	NUM
ejpam-4806	73	7	)	)	PUNCT
ejpam-4806	73	8	)	)	PUNCT
ejpam-4806	74	1	≥	≥	NUM
ejpam-4806	74	2	...	...	PUNCT
ejpam-4806	74	3	≥	≥	X
ejpam-4806	74	4	µn(q(g	µn(q(g	NOUN
ejpam-4806	74	5	)	)	PUNCT
ejpam-4806	74	6	)	)	PUNCT
ejpam-4806	75	1	=	=	PUNCT
ejpam-4806	75	2	0	0	X
ejpam-4806	75	3	.	.	PUNCT
ejpam-4806	76	1	definition	definition	NOUN
ejpam-4806	76	2	1	1	NUM
ejpam-4806	76	3	.	.	PUNCT
ejpam-4806	77	1	[	[	X
ejpam-4806	77	2	6	6	NUM
ejpam-4806	77	3	]	]	PUNCT
ejpam-4806	77	4	the	the	DET
ejpam-4806	77	5	join	join	NOUN
ejpam-4806	77	6	(	(	PUNCT
ejpam-4806	77	7	or	or	CCONJ
ejpam-4806	77	8	complete	complete	ADJ
ejpam-4806	77	9	product	product	NOUN
ejpam-4806	77	10	)	)	PUNCT
ejpam-4806	77	11	g1	g1	PROPN
ejpam-4806	77	12	▽	▽	ADJ
ejpam-4806	77	13	g2	g2	PROPN
ejpam-4806	77	14	of	of	ADP
ejpam-4806	77	15	graphs	graph	NOUN
ejpam-4806	77	16	g1	g1	PROPN
ejpam-4806	77	17	and	and	CCONJ
ejpam-4806	77	18	g2	g2	PROPN
ejpam-4806	77	19	is	be	AUX
ejpam-4806	77	20	the	the	DET
ejpam-4806	77	21	graph	graph	NOUN
ejpam-4806	77	22	obtained	obtain	VERB
ejpam-4806	77	23	from	from	ADP
ejpam-4806	77	24	g1	g1	NOUN
ejpam-4806	77	25	∪g2	∪g2	PROPN
ejpam-4806	77	26	by	by	ADP
ejpam-4806	77	27	joining	join	VERB
ejpam-4806	77	28	every	every	DET
ejpam-4806	77	29	vertex	vertex	NOUN
ejpam-4806	77	30	of	of	ADP
ejpam-4806	77	31	g1	g1	NOUN
ejpam-4806	77	32	with	with	ADP
ejpam-4806	77	33	every	every	DET
ejpam-4806	77	34	vertex	vertex	NOUN
ejpam-4806	77	35	of	of	ADP
ejpam-4806	77	36	g2	g2	PROPN
ejpam-4806	77	37	.	.	PUNCT
ejpam-4806	78	1	definition	definition	NOUN
ejpam-4806	78	2	2	2	NUM
ejpam-4806	78	3	.	.	PUNCT
ejpam-4806	79	1	[	[	X
ejpam-4806	79	2	6	6	NUM
ejpam-4806	79	3	]	]	X
ejpam-4806	79	4	core	core	ADJ
ejpam-4806	79	5	-	-	PUNCT
ejpam-4806	79	6	satellite	satellite	NOUN
ejpam-4806	79	7	graphs	graph	NOUN
ejpam-4806	79	8	θ(c	θ(c	VERB
ejpam-4806	79	9	,	,	PUNCT
ejpam-4806	79	10	s	s	X
ejpam-4806	79	11	,	,	PUNCT
ejpam-4806	79	12	η	η	NOUN
ejpam-4806	79	13	)	)	PUNCT
ejpam-4806	79	14	∼=	∼=	PROPN
ejpam-4806	79	15	kc	kc	NOUN
ejpam-4806	79	16	▽	▽	X
ejpam-4806	79	17	(	(	PUNCT
ejpam-4806	79	18	ηks	ηk	NOUN
ejpam-4806	79	19	)	)	PUNCT
ejpam-4806	79	20	are	be	AUX
ejpam-4806	79	21	the	the	DET
ejpam-4806	79	22	graphs	graph	NOUN
ejpam-4806	79	23	consisting	consist	VERB
ejpam-4806	79	24	of	of	ADP
ejpam-4806	79	25	η	η	PROPN
ejpam-4806	79	26	copies	copy	NOUN
ejpam-4806	79	27	of	of	ADP
ejpam-4806	79	28	ks	ks	PROPN
ejpam-4806	79	29	(	(	PUNCT
ejpam-4806	79	30	the	the	DET
ejpam-4806	79	31	satellites	satellite	NOUN
ejpam-4806	79	32	)	)	PUNCT
ejpam-4806	79	33	meeting	meeting	NOUN
ejpam-4806	79	34	in	in	ADP
ejpam-4806	79	35	a	a	DET
ejpam-4806	79	36	common	common	ADJ
ejpam-4806	79	37	clique	clique	NOUN
ejpam-4806	79	38	kc	kc	PROPN
ejpam-4806	79	39	(	(	PUNCT
ejpam-4806	79	40	the	the	DET
ejpam-4806	79	41	core	core	NOUN
ejpam-4806	79	42	)	)	PUNCT
ejpam-4806	79	43	,	,	PUNCT
ejpam-4806	79	44	where	where	SCONJ
ejpam-4806	79	45	c	c	PROPN
ejpam-4806	79	46	≥	≥	NUM
ejpam-4806	79	47	1	1	NUM
ejpam-4806	79	48	,	,	PUNCT
ejpam-4806	79	49	s	s	VERB
ejpam-4806	79	50	≥	≥	NOUN
ejpam-4806	79	51	1	1	NUM
ejpam-4806	79	52	and	and	CCONJ
ejpam-4806	79	53	η	η	PROPN
ejpam-4806	79	54	≥	≥	PROPN
ejpam-4806	79	55	2	2	NUM
ejpam-4806	79	56	.	.	PUNCT
ejpam-4806	79	57	core	core	NOUN
ejpam-4806	79	58	-	-	PUNCT
ejpam-4806	79	59	satellite	satellite	NOUN
ejpam-4806	79	60	graphs	graph	NOUN
ejpam-4806	79	61	have	have	VERB
ejpam-4806	79	62	central	central	ADJ
ejpam-4806	79	63	nodes	node	NOUN
ejpam-4806	79	64	in	in	ADP
ejpam-4806	79	65	a	a	DET
ejpam-4806	79	66	network	network	NOUN
ejpam-4806	79	67	that	that	PRON
ejpam-4806	79	68	are	be	AUX
ejpam-4806	79	69	connected	connect	VERB
ejpam-4806	79	70	among	among	ADP
ejpam-4806	79	71	themselves	themselves	PRON
ejpam-4806	79	72	.	.	PUNCT
ejpam-4806	80	1	also	also	ADV
ejpam-4806	80	2	,	,	PUNCT
ejpam-4806	80	3	there	there	PRON
ejpam-4806	80	4	are	be	VERB
ejpam-4806	80	5	a	a	DET
ejpam-4806	80	6	few	few	ADJ
ejpam-4806	80	7	cliques	clique	NOUN
ejpam-4806	80	8	of	of	ADP
ejpam-4806	80	9	the	the	DET
ejpam-4806	80	10	same	same	ADJ
ejpam-4806	80	11	size	size	NOUN
ejpam-4806	80	12	which	which	PRON
ejpam-4806	80	13	are	be	AUX
ejpam-4806	80	14	connected	connect	VERB
ejpam-4806	80	15	to	to	ADP
ejpam-4806	80	16	the	the	DET
ejpam-4806	80	17	central	central	ADJ
ejpam-4806	80	18	malathy	malathy	ADJ
ejpam-4806	80	19	v	v	NOUN
ejpam-4806	80	20	,	,	PUNCT
ejpam-4806	80	21	kalyani	kalyani	PROPN
ejpam-4806	80	22	desikan	desikan	PROPN
ejpam-4806	80	23	/	/	SYM
ejpam-4806	80	24	eur	eur	PROPN
ejpam-4806	80	25	.	.	PUNCT
ejpam-4806	81	1	j.	j.	PROPN
ejpam-4806	81	2	pure	pure	PROPN
ejpam-4806	81	3	appl	appl	PROPN
ejpam-4806	81	4	.	.	PROPN
ejpam-4806	81	5	math	math	PROPN
ejpam-4806	81	6	,	,	PUNCT
ejpam-4806	81	7	16	16	NUM
ejpam-4806	81	8	(	(	PUNCT
ejpam-4806	81	9	3	3	NUM
ejpam-4806	81	10	)	)	PUNCT
ejpam-4806	81	11	(	(	PUNCT
ejpam-4806	81	12	2023	2023	NUM
ejpam-4806	81	13	)	)	PUNCT
ejpam-4806	81	14	,	,	PUNCT
ejpam-4806	81	15	1731	1731	NUM
ejpam-4806	81	16	-	-	SYM
ejpam-4806	81	17	1746	1746	NUM
ejpam-4806	81	18	1734	1734	NUM
ejpam-4806	81	19	core	core	NOUN
ejpam-4806	81	20	but	but	CCONJ
ejpam-4806	81	21	not	not	PART
ejpam-4806	81	22	connected	connect	VERB
ejpam-4806	81	23	among	among	ADP
ejpam-4806	81	24	themselves	themselves	PRON
ejpam-4806	81	25	.	.	PUNCT
ejpam-4806	82	1	for	for	ADP
ejpam-4806	82	2	v	v	NUM
ejpam-4806	82	3	∈	∈	PROPN
ejpam-4806	82	4	v	v	NOUN
ejpam-4806	82	5	,	,	PUNCT
ejpam-4806	82	6	the	the	DET
ejpam-4806	82	7	degree	degree	NOUN
ejpam-4806	82	8	of	of	ADP
ejpam-4806	82	9	v	v	NOUN
ejpam-4806	82	10	,	,	PUNCT
ejpam-4806	82	11	written	write	VERB
ejpam-4806	82	12	as	as	ADP
ejpam-4806	82	13	d(v	d(v	PROPN
ejpam-4806	82	14	)	)	PUNCT
ejpam-4806	82	15	,	,	PUNCT
ejpam-4806	82	16	is	be	AUX
ejpam-4806	82	17	the	the	DET
ejpam-4806	82	18	number	number	NOUN
ejpam-4806	82	19	of	of	ADP
ejpam-4806	82	20	edges	edge	NOUN
ejpam-4806	82	21	incident	incident	NOUN
ejpam-4806	82	22	on	on	ADP
ejpam-4806	82	23	v.	v.	ADP
ejpam-4806	82	24	the	the	DET
ejpam-4806	82	25	vertices	vertex	NOUN
ejpam-4806	82	26	in	in	ADP
ejpam-4806	82	27	the	the	DET
ejpam-4806	82	28	satellite	satellite	NOUN
ejpam-4806	82	29	group	group	NOUN
ejpam-4806	82	30	have	have	VERB
ejpam-4806	82	31	minimum	minimum	NOUN
ejpam-4806	82	32	degree	degree	NOUN
ejpam-4806	82	33	δ	δ	PROPN
ejpam-4806	82	34	and	and	CCONJ
ejpam-4806	82	35	the	the	DET
ejpam-4806	82	36	vertices	vertex	NOUN
ejpam-4806	82	37	of	of	ADP
ejpam-4806	82	38	the	the	DET
ejpam-4806	82	39	central	central	ADJ
ejpam-4806	82	40	clique	clique	NOUN
ejpam-4806	82	41	(	(	PUNCT
ejpam-4806	82	42	core	core	NOUN
ejpam-4806	82	43	)	)	PUNCT
ejpam-4806	82	44	have	have	VERB
ejpam-4806	82	45	the	the	DET
ejpam-4806	82	46	maximum	maximum	ADJ
ejpam-4806	82	47	degree	degree	NOUN
ejpam-4806	82	48	∆.	∆.	X
ejpam-4806	82	49	agave	agave	NOUN
ejpam-4806	82	50	graphs	graph	NOUN
ejpam-4806	82	51	are	be	AUX
ejpam-4806	82	52	graphs	graph	NOUN
ejpam-4806	82	53	with	with	ADP
ejpam-4806	82	54	kc	kc	PROPN
ejpam-4806	82	55	as	as	ADP
ejpam-4806	82	56	k2	k2	PROPN
ejpam-4806	82	57	,	,	PUNCT
ejpam-4806	82	58	ks	k	NOUN
ejpam-4806	82	59	as	as	ADP
ejpam-4806	82	60	k1	k1	PROPN
ejpam-4806	82	61	and	and	CCONJ
ejpam-4806	82	62	η	η	PROPN
ejpam-4806	82	63	is	be	AUX
ejpam-4806	82	64	the	the	DET
ejpam-4806	82	65	number	number	NOUN
ejpam-4806	82	66	of	of	ADP
ejpam-4806	82	67	copies	copy	NOUN
ejpam-4806	82	68	of	of	ADP
ejpam-4806	82	69	ks	ks	PROPN
ejpam-4806	82	70	.	.	PUNCT
ejpam-4806	83	1	this	this	DET
ejpam-4806	83	2	class	class	NOUN
ejpam-4806	83	3	belongs	belong	VERB
ejpam-4806	83	4	to	to	ADP
ejpam-4806	83	5	the	the	DET
ejpam-4806	83	6	general	general	ADJ
ejpam-4806	83	7	class	class	NOUN
ejpam-4806	83	8	of	of	ADP
ejpam-4806	83	9	complete	complete	ADJ
ejpam-4806	83	10	split	split	NOUN
ejpam-4806	83	11	graphs	graph	NOUN
ejpam-4806	83	12	[	[	X
ejpam-4806	83	13	6	6	NUM
ejpam-4806	83	14	]	]	PUNCT
ejpam-4806	83	15	.	.	PUNCT
ejpam-4806	84	1	the	the	DET
ejpam-4806	84	2	following	follow	VERB
ejpam-4806	84	3	results	result	NOUN
ejpam-4806	84	4	are	be	AUX
ejpam-4806	84	5	useful	useful	ADJ
ejpam-4806	84	6	in	in	ADP
ejpam-4806	84	7	proving	prove	VERB
ejpam-4806	84	8	the	the	DET
ejpam-4806	84	9	main	main	ADJ
ejpam-4806	84	10	results	result	NOUN
ejpam-4806	84	11	.	.	PUNCT
ejpam-4806	85	1	lemma	lemma	PROPN
ejpam-4806	85	2	1	1	NUM
ejpam-4806	85	3	.	.	PUNCT
ejpam-4806	86	1	[	[	X
ejpam-4806	86	2	19	19	NUM
ejpam-4806	86	3	]	]	X
ejpam-4806	86	4	let	let	VERB
ejpam-4806	86	5	b	b	PRON
ejpam-4806	86	6	be	be	AUX
ejpam-4806	86	7	a	a	DET
ejpam-4806	86	8	real	real	ADJ
ejpam-4806	86	9	symmetric	symmetric	ADJ
ejpam-4806	86	10	n	n	CCONJ
ejpam-4806	86	11	×	×	NOUN
ejpam-4806	86	12	n	n	CCONJ
ejpam-4806	86	13	matrix	matrix	NOUN
ejpam-4806	86	14	and	and	CCONJ
ejpam-4806	86	15	µ	µ	NOUN
ejpam-4806	86	16	be	be	AUX
ejpam-4806	86	17	an	an	DET
ejpam-4806	86	18	eigenvalue	eigenvalue	NOUN
ejpam-4806	86	19	of	of	ADP
ejpam-4806	86	20	b	b	NOUN
ejpam-4806	86	21	with	with	ADP
ejpam-4806	86	22	an	an	DET
ejpam-4806	86	23	eigenvector	eigenvector	NOUN
ejpam-4806	86	24	x	x	NOUN
ejpam-4806	86	25	all	all	PRON
ejpam-4806	86	26	of	of	ADP
ejpam-4806	86	27	whose	whose	DET
ejpam-4806	86	28	entries	entry	NOUN
ejpam-4806	86	29	are	be	AUX
ejpam-4806	86	30	non	non	ADJ
ejpam-4806	86	31	-	-	ADJ
ejpam-4806	86	32	negative	negative	ADJ
ejpam-4806	86	33	.	.	PUNCT
ejpam-4806	87	1	denote	denote	VERB
ejpam-4806	87	2	the	the	DET
ejpam-4806	87	3	ith	ith	PROPN
ejpam-4806	87	4	row	row	NOUN
ejpam-4806	87	5	sum	sum	NOUN
ejpam-4806	87	6	of	of	ADP
ejpam-4806	87	7	b	b	NOUN
ejpam-4806	87	8	by	by	ADP
ejpam-4806	87	9	ri(b	ri(b	PROPN
ejpam-4806	87	10	)	)	PUNCT
ejpam-4806	87	11	.	.	PUNCT
ejpam-4806	88	1	then	then	ADV
ejpam-4806	88	2	min	min	PROPN
ejpam-4806	88	3	1≤i≤n	1≤i≤n	NUM
ejpam-4806	88	4	ri(b	ri(b	PROPN
ejpam-4806	88	5	)	)	PUNCT
ejpam-4806	88	6	≤	≤	NUM
ejpam-4806	88	7	µ(b	µ(b	NOUN
ejpam-4806	88	8	)	)	PUNCT
ejpam-4806	88	9	≤	≤	NUM
ejpam-4806	88	10	max	max	PROPN
ejpam-4806	88	11	1≤i≤n	1≤i≤n	NUM
ejpam-4806	88	12	ri(b	ri(b	PROPN
ejpam-4806	88	13	)	)	PUNCT
ejpam-4806	88	14	.	.	PUNCT
ejpam-4806	89	1	(	(	PUNCT
ejpam-4806	89	2	1	1	X
ejpam-4806	89	3	)	)	PUNCT
ejpam-4806	89	4	moreover	moreover	ADV
ejpam-4806	89	5	,	,	PUNCT
ejpam-4806	89	6	if	if	SCONJ
ejpam-4806	89	7	all	all	DET
ejpam-4806	89	8	entries	entry	NOUN
ejpam-4806	89	9	of	of	ADP
ejpam-4806	89	10	x	x	SYM
ejpam-4806	89	11	are	be	AUX
ejpam-4806	89	12	positive	positive	ADJ
ejpam-4806	89	13	then	then	ADV
ejpam-4806	89	14	either	either	PRON
ejpam-4806	89	15	of	of	ADP
ejpam-4806	89	16	equalities	equality	NOUN
ejpam-4806	89	17	holds	hold	VERB
ejpam-4806	89	18	if	if	SCONJ
ejpam-4806	89	19	and	and	CCONJ
ejpam-4806	89	20	only	only	ADV
ejpam-4806	89	21	if	if	SCONJ
ejpam-4806	89	22	row	row	NOUN
ejpam-4806	89	23	sums	sum	NOUN
ejpam-4806	89	24	of	of	ADP
ejpam-4806	89	25	b	b	NOUN
ejpam-4806	89	26	are	be	AUX
ejpam-4806	89	27	all	all	ADV
ejpam-4806	89	28	equal	equal	ADJ
ejpam-4806	89	29	.	.	PUNCT
ejpam-4806	90	1	lemma	lemma	PROPN
ejpam-4806	90	2	2	2	NUM
ejpam-4806	90	3	.	.	PUNCT
ejpam-4806	91	1	[	[	X
ejpam-4806	91	2	19	19	NUM
ejpam-4806	91	3	]	]	X
ejpam-4806	91	4	let	let	VERB
ejpam-4806	91	5	m	m	PRON
ejpam-4806	91	6	be	be	AUX
ejpam-4806	91	7	a	a	DET
ejpam-4806	91	8	real	real	ADJ
ejpam-4806	91	9	symmetric	symmetric	ADJ
ejpam-4806	91	10	n	n	CCONJ
ejpam-4806	91	11	×	×	NOUN
ejpam-4806	91	12	n	n	PRON
ejpam-4806	91	13	matrix	matrix	NOUN
ejpam-4806	91	14	and	and	CCONJ
ejpam-4806	91	15	λ	λ	PROPN
ejpam-4806	91	16	be	be	AUX
ejpam-4806	91	17	an	an	DET
ejpam-4806	91	18	eigenvalue	eigenvalue	NOUN
ejpam-4806	91	19	of	of	ADP
ejpam-4806	91	20	m	m	PROPN
ejpam-4806	91	21	with	with	ADP
ejpam-4806	91	22	an	an	DET
ejpam-4806	91	23	eigenvector	eigenvector	NOUN
ejpam-4806	91	24	x	x	PUNCT
ejpam-4806	91	25	whose	whose	DET
ejpam-4806	91	26	entries	entry	NOUN
ejpam-4806	91	27	are	be	AUX
ejpam-4806	91	28	all	all	PRON
ejpam-4806	91	29	non	non	ADJ
ejpam-4806	91	30	-	-	ADJ
ejpam-4806	91	31	negative	negative	ADJ
ejpam-4806	91	32	.	.	PUNCT
ejpam-4806	92	1	let	let	VERB
ejpam-4806	92	2	p	p	PRON
ejpam-4806	92	3	be	be	AUX
ejpam-4806	92	4	any	any	DET
ejpam-4806	92	5	polynomial	polynomial	ADJ
ejpam-4806	92	6	.	.	PUNCT
ejpam-4806	93	1	then	then	ADV
ejpam-4806	93	2	min	min	PROPN
ejpam-4806	93	3	1≤i≤n	1≤i≤n	NUM
ejpam-4806	93	4	si(p	si(p	NOUN
ejpam-4806	93	5	(	(	PUNCT
ejpam-4806	93	6	m	m	NOUN
ejpam-4806	93	7	)	)	PUNCT
ejpam-4806	93	8	)	)	PUNCT
ejpam-4806	93	9	≤	≤	NOUN
ejpam-4806	94	1	p	p	NOUN
ejpam-4806	94	2	(	(	PUNCT
ejpam-4806	94	3	λ	λ	NOUN
ejpam-4806	94	4	)	)	PUNCT
ejpam-4806	94	5	≤	≤	NUM
ejpam-4806	94	6	max	max	PROPN
ejpam-4806	94	7	1≤i≤n	1≤i≤n	NUM
ejpam-4806	94	8	si(p	si(p	NOUN
ejpam-4806	94	9	(	(	PUNCT
ejpam-4806	94	10	m	m	NOUN
ejpam-4806	94	11	)	)	PUNCT
ejpam-4806	94	12	)	)	PUNCT
ejpam-4806	94	13	where	where	SCONJ
ejpam-4806	94	14	si(p	si(p	X
ejpam-4806	94	15	(	(	PUNCT
ejpam-4806	94	16	m	m	NOUN
ejpam-4806	94	17	)	)	PUNCT
ejpam-4806	94	18	)	)	PUNCT
ejpam-4806	94	19	is	be	AUX
ejpam-4806	94	20	the	the	DET
ejpam-4806	94	21	row	row	NOUN
ejpam-4806	94	22	sum	sum	NOUN
ejpam-4806	94	23	of	of	ADP
ejpam-4806	94	24	the	the	DET
ejpam-4806	94	25	ith	ith	PROPN
ejpam-4806	94	26	row	row	NOUN
ejpam-4806	94	27	of	of	ADP
ejpam-4806	94	28	the	the	DET
ejpam-4806	94	29	matrix	matrix	NOUN
ejpam-4806	94	30	p	p	X
ejpam-4806	94	31	(	(	PUNCT
ejpam-4806	94	32	m	m	NOUN
ejpam-4806	94	33	)	)	PUNCT
ejpam-4806	94	34	.	.	PUNCT
ejpam-4806	95	1	moreover	moreover	ADV
ejpam-4806	95	2	,	,	PUNCT
ejpam-4806	95	3	if	if	SCONJ
ejpam-4806	95	4	all	all	DET
ejpam-4806	95	5	entries	entry	NOUN
ejpam-4806	95	6	of	of	ADP
ejpam-4806	95	7	x	x	SYM
ejpam-4806	95	8	are	be	AUX
ejpam-4806	95	9	positive	positive	ADJ
ejpam-4806	95	10	then	then	ADV
ejpam-4806	95	11	either	either	PRON
ejpam-4806	95	12	of	of	ADP
ejpam-4806	95	13	the	the	DET
ejpam-4806	95	14	equalities	equality	NOUN
ejpam-4806	95	15	holds	hold	VERB
ejpam-4806	95	16	if	if	SCONJ
ejpam-4806	95	17	and	and	CCONJ
ejpam-4806	95	18	only	only	ADV
ejpam-4806	95	19	if	if	SCONJ
ejpam-4806	95	20	the	the	DET
ejpam-4806	95	21	row	row	NOUN
ejpam-4806	95	22	sums	sum	NOUN
ejpam-4806	95	23	of	of	ADP
ejpam-4806	95	24	p	p	NOUN
ejpam-4806	95	25	(	(	PUNCT
ejpam-4806	95	26	m	m	NOUN
ejpam-4806	95	27	)	)	PUNCT
ejpam-4806	95	28	are	be	AUX
ejpam-4806	95	29	all	all	ADV
ejpam-4806	95	30	equal	equal	ADJ
ejpam-4806	95	31	.	.	PUNCT
ejpam-4806	96	1	lemma	lemma	PROPN
ejpam-4806	96	2	3	3	X
ejpam-4806	96	3	.	.	PUNCT
ejpam-4806	97	1	[	[	X
ejpam-4806	97	2	9	9	NUM
ejpam-4806	97	3	]	]	PUNCT
ejpam-4806	97	4	let	let	VERB
ejpam-4806	97	5	g	g	NOUN
ejpam-4806	97	6	=	=	SYM
ejpam-4806	97	7	(	(	PUNCT
ejpam-4806	97	8	v	v	NOUN
ejpam-4806	97	9	,	,	PUNCT
ejpam-4806	97	10	e	e	NOUN
ejpam-4806	97	11	)	)	PUNCT
ejpam-4806	97	12	be	be	AUX
ejpam-4806	97	13	a	a	DET
ejpam-4806	97	14	simple	simple	ADJ
ejpam-4806	97	15	graph	graph	NOUN
ejpam-4806	97	16	.	.	PUNCT
ejpam-4806	98	1	then	then	ADV
ejpam-4806	98	2	√	√	NUM
ejpam-4806	98	3	2	2	NUM
ejpam-4806	98	4	min	min	NOUN
ejpam-4806	98	5	v∈v	v∈v	NOUN
ejpam-4806	98	6	(	(	PUNCT
ejpam-4806	98	7	g	g	NOUN
ejpam-4806	98	8	)	)	PUNCT
ejpam-4806	98	9	√	√	PROPN
ejpam-4806	98	10	d2	d2	PROPN
ejpam-4806	98	11	(	(	PUNCT
ejpam-4806	98	12	v	v	NOUN
ejpam-4806	98	13	)	)	PUNCT
ejpam-4806	98	14	+	+	CCONJ
ejpam-4806	98	15	∑	∑	PUNCT
ejpam-4806	98	16	uv∈e(g	uv∈e(g	NUM
ejpam-4806	98	17	)	)	PUNCT
ejpam-4806	98	18	d	d	NOUN
ejpam-4806	98	19	(	(	PUNCT
ejpam-4806	98	20	u	u	NOUN
ejpam-4806	98	21	)	)	PUNCT
ejpam-4806	98	22	≤	≤	NOUN
ejpam-4806	98	23	µ(g	µ(g	NOUN
ejpam-4806	98	24	)	)	PUNCT
ejpam-4806	98	25	≤	≤	NOUN
ejpam-4806	98	26	√	√	NUM
ejpam-4806	98	27	2	2	NUM
ejpam-4806	98	28	max	max	NOUN
ejpam-4806	98	29	v∈v	v∈v	NOUN
ejpam-4806	98	30	(	(	PUNCT
ejpam-4806	98	31	g	g	NOUN
ejpam-4806	98	32	)	)	PUNCT
ejpam-4806	98	33	√	√	PROPN
ejpam-4806	98	34	d2	d2	PROPN
ejpam-4806	98	35	(	(	PUNCT
ejpam-4806	98	36	v	v	NOUN
ejpam-4806	98	37	)	)	PUNCT
ejpam-4806	98	38	+	+	CCONJ
ejpam-4806	98	39	∑	∑	PUNCT
ejpam-4806	98	40	uv∈e(g	uv∈e(g	NUM
ejpam-4806	98	41	)	)	PUNCT
ejpam-4806	98	42	d	d	NOUN
ejpam-4806	98	43	(	(	PUNCT
ejpam-4806	98	44	u	u	NOUN
ejpam-4806	98	45	)	)	PUNCT
ejpam-4806	98	46	(	(	PUNCT
ejpam-4806	98	47	2	2	X
ejpam-4806	98	48	)	)	PUNCT
ejpam-4806	98	49	moreover	moreover	ADV
ejpam-4806	98	50	,	,	PUNCT
ejpam-4806	98	51	ifg	ifg	PROPN
ejpam-4806	98	52	is	be	AUX
ejpam-4806	98	53	connected	connect	VERB
ejpam-4806	98	54	,	,	PUNCT
ejpam-4806	98	55	both	both	CCONJ
ejpam-4806	98	56	the	the	DET
ejpam-4806	98	57	equalities	equality	NOUN
ejpam-4806	98	58	hold	hold	VERB
ejpam-4806	98	59	if	if	SCONJ
ejpam-4806	98	60	and	and	CCONJ
ejpam-4806	98	61	only	only	ADV
ejpam-4806	98	62	if	if	SCONJ
ejpam-4806	98	63	2	2	NUM
ejpam-4806	98	64	(	(	PUNCT
ejpam-4806	98	65	d2	d2	PROPN
ejpam-4806	98	66	(	(	PUNCT
ejpam-4806	98	67	v	v	NOUN
ejpam-4806	98	68	)	)	PUNCT
ejpam-4806	98	69	+	+	CCONJ
ejpam-4806	98	70	∑	∑	PUNCT
ejpam-4806	98	71	uv∈e(g	uv∈e(g	NUM
ejpam-4806	98	72	)	)	PUNCT
ejpam-4806	98	73	d	d	NOUN
ejpam-4806	98	74	(	(	PUNCT
ejpam-4806	98	75	u	u	NOUN
ejpam-4806	98	76	)	)	PUNCT
ejpam-4806	98	77	)	)	PUNCT
ejpam-4806	98	78	is	be	AUX
ejpam-4806	98	79	the	the	DET
ejpam-4806	98	80	same	same	ADJ
ejpam-4806	98	81	for	for	ADP
ejpam-4806	98	82	all	all	DET
ejpam-4806	98	83	v	v	ADP
ejpam-4806	98	84	∈	∈	NOUN
ejpam-4806	98	85	v	v	NOUN
ejpam-4806	98	86	(	(	PUNCT
ejpam-4806	98	87	g	g	NOUN
ejpam-4806	98	88	)	)	PUNCT
ejpam-4806	98	89	.	.	PUNCT
ejpam-4806	99	1	lemma	lemma	PROPN
ejpam-4806	99	2	4	4	NUM
ejpam-4806	99	3	.	.	PUNCT
ejpam-4806	100	1	[	[	X
ejpam-4806	100	2	12	12	NUM
ejpam-4806	100	3	]	]	PUNCT
ejpam-4806	100	4	the	the	DET
ejpam-4806	100	5	signless	signless	PROPN
ejpam-4806	100	6	laplacian	laplacian	ADJ
ejpam-4806	100	7	spectrum	spectrum	NOUN
ejpam-4806	100	8	of	of	ADP
ejpam-4806	100	9	ka	ka	PROPN
ejpam-4806	100	10	∨kb	∨kb	PROPN
ejpam-4806	100	11	is	be	AUX
ejpam-4806	100	12	{	{	PUNCT
ejpam-4806	100	13	(	(	PUNCT
ejpam-4806	100	14	3b+	3b+	NUM
ejpam-4806	100	15	a−	a−	PROPN
ejpam-4806	100	16	2)±	2)±	NUM
ejpam-4806	100	17	√	√	NUM
ejpam-4806	100	18	(	(	PUNCT
ejpam-4806	100	19	b+	b+	X
ejpam-4806	100	20	a−	a−	PROPN
ejpam-4806	100	21	2)2	2)2	NUM
ejpam-4806	100	22	+	+	CCONJ
ejpam-4806	100	23	4ab	4ab	ADJ
ejpam-4806	100	24	2	2	NUM
ejpam-4806	100	25	,	,	PUNCT
ejpam-4806	100	26	ba−1	ba−1	NOUN
ejpam-4806	100	27	,	,	PUNCT
ejpam-4806	100	28	(	(	PUNCT
ejpam-4806	100	29	a−	a−	PROPN
ejpam-4806	100	30	b−	b−	NOUN
ejpam-4806	100	31	2)b−1	2)b−1	NUM
ejpam-4806	100	32	}	}	PUNCT
ejpam-4806	100	33	where	where	SCONJ
ejpam-4806	100	34	an	an	DET
ejpam-4806	100	35	exponent	exponent	NOUN
ejpam-4806	100	36	indicates	indicate	VERB
ejpam-4806	100	37	the	the	DET
ejpam-4806	100	38	multiplicity	multiplicity	NOUN
ejpam-4806	100	39	of	of	ADP
ejpam-4806	100	40	the	the	DET
ejpam-4806	100	41	corresponding	corresponding	ADJ
ejpam-4806	100	42	signless	signless	PROPN
ejpam-4806	100	43	laplacian	laplacian	PROPN
ejpam-4806	100	44	eigenvalue	eigenvalue	PROPN
ejpam-4806	100	45	.	.	PUNCT
ejpam-4806	100	46	malathy	malathy	PROPN
ejpam-4806	100	47	v	v	NOUN
ejpam-4806	100	48	,	,	PUNCT
ejpam-4806	100	49	kalyani	kalyani	PROPN
ejpam-4806	100	50	desikan	desikan	PROPN
ejpam-4806	100	51	/	/	SYM
ejpam-4806	100	52	eur	eur	PROPN
ejpam-4806	100	53	.	.	PUNCT
ejpam-4806	101	1	j.	j.	PROPN
ejpam-4806	101	2	pure	pure	PROPN
ejpam-4806	101	3	appl	appl	PROPN
ejpam-4806	101	4	.	.	PROPN
ejpam-4806	101	5	math	math	PROPN
ejpam-4806	101	6	,	,	PUNCT
ejpam-4806	101	7	16	16	NUM
ejpam-4806	101	8	(	(	PUNCT
ejpam-4806	101	9	3	3	NUM
ejpam-4806	101	10	)	)	PUNCT
ejpam-4806	101	11	(	(	PUNCT
ejpam-4806	101	12	2023	2023	NUM
ejpam-4806	101	13	)	)	PUNCT
ejpam-4806	101	14	,	,	PUNCT
ejpam-4806	101	15	1731	1731	NUM
ejpam-4806	101	16	-	-	SYM
ejpam-4806	101	17	1746	1746	NUM
ejpam-4806	101	18	1735	1735	NUM
ejpam-4806	101	19	3	3	NUM
ejpam-4806	101	20	.	.	PUNCT
ejpam-4806	101	21	main	main	ADJ
ejpam-4806	101	22	results	result	NOUN
ejpam-4806	101	23	theorem	theorem	VERB
ejpam-4806	101	24	1	1	X
ejpam-4806	101	25	.	.	PUNCT
ejpam-4806	102	1	let	let	VERB
ejpam-4806	102	2	ga	ga	PROPN
ejpam-4806	102	3	be	be	AUX
ejpam-4806	102	4	an	an	DET
ejpam-4806	102	5	agave	agave	NOUN
ejpam-4806	102	6	graph	graph	NOUN
ejpam-4806	102	7	with	with	ADP
ejpam-4806	102	8	n	n	ADP
ejpam-4806	102	9	vertices	vertex	NOUN
ejpam-4806	102	10	,	,	PUNCT
ejpam-4806	102	11	m	m	VERB
ejpam-4806	102	12	edges	edge	NOUN
ejpam-4806	102	13	,	,	PUNCT
ejpam-4806	102	14	and	and	CCONJ
ejpam-4806	102	15	η	η	PROPN
ejpam-4806	102	16	copies	copy	NOUN
ejpam-4806	102	17	of	of	ADP
ejpam-4806	102	18	the	the	DET
ejpam-4806	102	19	satellite	satellite	NOUN
ejpam-4806	102	20	graph	graph	NOUN
ejpam-4806	102	21	.	.	PUNCT
ejpam-4806	103	1	let	let	VERB
ejpam-4806	103	2	∆	∆	PROPN
ejpam-4806	103	3	and	and	CCONJ
ejpam-4806	103	4	δ	δ	PROPN
ejpam-4806	103	5	be	be	VERB
ejpam-4806	103	6	the	the	DET
ejpam-4806	103	7	maximum	maximum	ADJ
ejpam-4806	103	8	degree	degree	NOUN
ejpam-4806	103	9	and	and	CCONJ
ejpam-4806	103	10	minimum	minimum	NOUN
ejpam-4806	103	11	degree	degree	NOUN
ejpam-4806	103	12	of	of	ADP
ejpam-4806	103	13	ga	ga	PROPN
ejpam-4806	103	14	.	.	PROPN
ejpam-4806	104	1	also	also	ADV
ejpam-4806	104	2	(	(	PUNCT
ejpam-4806	104	3	2δ2	2δ2	NUM
ejpam-4806	104	4	+	+	CCONJ
ejpam-4806	104	5	2(2m−	2(2m−	NUM
ejpam-4806	104	6	ηδ	ηδ	NOUN
ejpam-4806	104	7	)	)	PUNCT
ejpam-4806	104	8	)	)	PUNCT
ejpam-4806	104	9	1/2	1/2	NUM
ejpam-4806	104	10	≤	≤	NOUN
ejpam-4806	104	11	µ	µ	X
ejpam-4806	104	12	(	(	PUNCT
ejpam-4806	104	13	ga	ga	NOUN
ejpam-4806	104	14	)	)	PUNCT
ejpam-4806	104	15	≤	≤	NOUN
ejpam-4806	104	16	(	(	PUNCT
ejpam-4806	104	17	2∆2	2∆2	NUM
ejpam-4806	104	18	+	+	NUM
ejpam-4806	104	19	2(2m−∆	2(2m−∆	NUM
ejpam-4806	104	20	)	)	PUNCT
ejpam-4806	104	21	)	)	PUNCT
ejpam-4806	105	1	1/2	1/2	NUM
ejpam-4806	105	2	(	(	PUNCT
ejpam-4806	105	3	3	3	NUM
ejpam-4806	105	4	)	)	PUNCT
ejpam-4806	105	5	moreover	moreover	ADV
ejpam-4806	105	6	,	,	PUNCT
ejpam-4806	105	7	both	both	CCONJ
ejpam-4806	105	8	the	the	DET
ejpam-4806	105	9	equalities	equality	NOUN
ejpam-4806	105	10	hold	hold	VERB
ejpam-4806	105	11	if	if	SCONJ
ejpam-4806	105	12	and	and	CCONJ
ejpam-4806	105	13	only	only	ADV
ejpam-4806	105	14	if	if	SCONJ
ejpam-4806	105	15	n	n	PROPN
ejpam-4806	105	16	=	=	SYM
ejpam-4806	105	17	3	3	NUM
ejpam-4806	105	18	and	and	CCONJ
ejpam-4806	105	19	η	η	PROPN
ejpam-4806	105	20	=	=	PROPN
ejpam-4806	105	21	1	1	NUM
ejpam-4806	105	22	.	.	PUNCT
ejpam-4806	106	1	proof	proof	NOUN
ejpam-4806	106	2	.	.	PUNCT
ejpam-4806	107	1	using	use	VERB
ejpam-4806	107	2	lemma	lemma	PROPN
ejpam-4806	107	3	3	3	NUM
ejpam-4806	107	4	,	,	PUNCT
ejpam-4806	107	5	the	the	DET
ejpam-4806	107	6	lower	low	ADJ
ejpam-4806	107	7	bound	bind	VERB
ejpam-4806	107	8	is	be	AUX
ejpam-4806	107	9	attained	attain	VERB
ejpam-4806	107	10	when	when	SCONJ
ejpam-4806	107	11	the	the	DET
ejpam-4806	107	12	vertex	vertex	NOUN
ejpam-4806	107	13	v	v	NOUN
ejpam-4806	107	14	is	be	AUX
ejpam-4806	107	15	in	in	ADP
ejpam-4806	107	16	the	the	DET
ejpam-4806	107	17	satellite	satellite	NOUN
ejpam-4806	107	18	graph	graph	NOUN
ejpam-4806	107	19	.	.	PUNCT
ejpam-4806	108	1	√	√	NUM
ejpam-4806	108	2	2	2	NUM
ejpam-4806	108	3	min	min	NOUN
ejpam-4806	108	4	v∈v	v∈v	NOUN
ejpam-4806	108	5	(	(	PUNCT
ejpam-4806	108	6	g	g	NOUN
ejpam-4806	108	7	)	)	PUNCT
ejpam-4806	108	8	√	√	NOUN
ejpam-4806	108	9	d2(v	d2(v	NOUN
ejpam-4806	108	10	)	)	PUNCT
ejpam-4806	108	11	+	+	CCONJ
ejpam-4806	108	12	∑	∑	ADP
ejpam-4806	108	13	uv∈e(ga	uv∈e(ga	PROPN
ejpam-4806	108	14	)	)	PUNCT
ejpam-4806	108	15	d(u	d(u	PROPN
ejpam-4806	108	16	)	)	PUNCT
ejpam-4806	108	17	=	=	SYM
ejpam-4806	109	1	√	√	NUM
ejpam-4806	109	2	2	2	NUM
ejpam-4806	109	3	√√√√√δ2	√√√√√δ2	NOUN
ejpam-4806	109	4	+	+	SYM
ejpam-4806	109	5	2m−	2m−	PROPN
ejpam-4806	109	6	d(u)−	d(u)−	PROPN
ejpam-4806	109	7	∑	∑	PROPN
ejpam-4806	109	8	uv/∈e(ga	uv/∈e(ga	PROPN
ejpam-4806	109	9	)	)	PUNCT
ejpam-4806	109	10	d(u	d(u	PROPN
ejpam-4806	109	11	)	)	PUNCT
ejpam-4806	110	1			PROPN
ejpam-4806	110	2	=	=	SYM
ejpam-4806	110	3	√	√	NUM
ejpam-4806	110	4	2δ2	2δ2	NUM
ejpam-4806	111	1	+	+	CCONJ
ejpam-4806	111	2	2	2	NUM
ejpam-4806	111	3	(	(	PUNCT
ejpam-4806	111	4	2m−	2m−	NUM
ejpam-4806	111	5	δ	δ	NOUN
ejpam-4806	111	6	−	−	PROPN
ejpam-4806	111	7	(	(	PUNCT
ejpam-4806	111	8	η	η	PROPN
ejpam-4806	111	9	−	−	NOUN
ejpam-4806	111	10	1)δ	1)δ	NUM
ejpam-4806	111	11	)	)	PUNCT
ejpam-4806	111	12	=	=	SYM
ejpam-4806	111	13	√	√	NUM
ejpam-4806	111	14	2δ2	2δ2	NUM
ejpam-4806	112	1	+	+	CCONJ
ejpam-4806	112	2	2(2m−	2(2m−	NUM
ejpam-4806	112	3	ηδ	ηδ	NOUN
ejpam-4806	112	4	)	)	PUNCT
ejpam-4806	112	5	≤	≤	NOUN
ejpam-4806	112	6	µ(ga	µ(ga	PROPN
ejpam-4806	112	7	)	)	PUNCT
ejpam-4806	112	8	.	.	PUNCT
ejpam-4806	113	1	similarly	similarly	ADV
ejpam-4806	113	2	,	,	PUNCT
ejpam-4806	113	3	the	the	DET
ejpam-4806	113	4	upper	upper	ADJ
ejpam-4806	113	5	bound	bound	NOUN
ejpam-4806	113	6	is	be	AUX
ejpam-4806	113	7	attained	attain	VERB
ejpam-4806	113	8	when	when	SCONJ
ejpam-4806	113	9	v	v	NOUN
ejpam-4806	113	10	is	be	AUX
ejpam-4806	113	11	the	the	DET
ejpam-4806	113	12	vertex	vertex	NOUN
ejpam-4806	113	13	in	in	ADP
ejpam-4806	113	14	the	the	DET
ejpam-4806	113	15	core	core	NOUN
ejpam-4806	113	16	graph	graph	NOUN
ejpam-4806	113	17	.	.	PUNCT
ejpam-4806	114	1	also	also	ADV
ejpam-4806	114	2	for	for	ADP
ejpam-4806	114	3	vertices	vertex	NOUN
ejpam-4806	114	4	in	in	ADP
ejpam-4806	114	5	the	the	DET
ejpam-4806	114	6	core	core	NOUN
ejpam-4806	114	7	graph	graph	NOUN
ejpam-4806	114	8	,	,	PUNCT
ejpam-4806	114	9	that	that	PRON
ejpam-4806	114	10	is	be	AUX
ejpam-4806	114	11	v	v	ADP
ejpam-4806	114	12	∈	∈	NOUN
ejpam-4806	114	13	c	c	NOUN
ejpam-4806	114	14	,	,	PUNCT
ejpam-4806	114	15	we	we	PRON
ejpam-4806	114	16	have	have	VERB
ejpam-4806	114	17	d(v	d(v	PROPN
ejpam-4806	114	18	)	)	PUNCT
ejpam-4806	114	19	=	=	PUNCT
ejpam-4806	115	1	∆	∆	PUNCT
ejpam-4806	115	2	=	=	SYM
ejpam-4806	115	3	(	(	PUNCT
ejpam-4806	115	4	n−	n−	NOUN
ejpam-4806	115	5	1	1	NUM
ejpam-4806	115	6	)	)	PUNCT
ejpam-4806	115	7	and∑	and∑	PROPN
ejpam-4806	115	8	uv/∈e(g	uv/∈e(g	NOUN
ejpam-4806	115	9	)	)	PUNCT
ejpam-4806	116	1	d	d	NOUN
ejpam-4806	116	2	(	(	PUNCT
ejpam-4806	116	3	u	u	NOUN
ejpam-4806	116	4	)	)	PUNCT
ejpam-4806	116	5	=	=	SYM
ejpam-4806	116	6	0	0	PUNCT
ejpam-4806	116	7	since	since	SCONJ
ejpam-4806	116	8	v	v	NUM
ejpam-4806	116	9	∈	∈	NOUN
ejpam-4806	116	10	c	c	NOUN
ejpam-4806	116	11	is	be	AUX
ejpam-4806	116	12	connected	connect	VERB
ejpam-4806	116	13	to	to	ADP
ejpam-4806	116	14	all	all	DET
ejpam-4806	116	15	the	the	DET
ejpam-4806	116	16	vertices	vertex	NOUN
ejpam-4806	116	17	of	of	ADP
ejpam-4806	116	18	the	the	DET
ejpam-4806	116	19	graph	graph	NOUN
ejpam-4806	116	20	ga	ga	PROPN
ejpam-4806	116	21	.	.	PROPN
ejpam-4806	117	1	therefore	therefore	ADV
ejpam-4806	117	2	√	√	NUM
ejpam-4806	117	3	2	2	NUM
ejpam-4806	117	4	max	max	NOUN
ejpam-4806	117	5	u∈v	u∈v	NOUN
ejpam-4806	117	6	(	(	PUNCT
ejpam-4806	117	7	g	g	NOUN
ejpam-4806	117	8	)	)	PUNCT
ejpam-4806	117	9	√	√	NOUN
ejpam-4806	117	10	d2(v	d2(v	NOUN
ejpam-4806	117	11	)	)	PUNCT
ejpam-4806	117	12	+	+	CCONJ
ejpam-4806	117	13	∑	∑	ADP
ejpam-4806	117	14	uv∈e(ga	uv∈e(ga	PROPN
ejpam-4806	117	15	)	)	PUNCT
ejpam-4806	117	16	d(u	d(u	PROPN
ejpam-4806	117	17	)	)	PUNCT
ejpam-4806	117	18	=	=	PUNCT
ejpam-4806	118	1	√	√	ADP
ejpam-4806	118	2	2	2	NUM
ejpam-4806	118	3	√	√	PROPN
ejpam-4806	118	4	∆2	∆2	PROPN
ejpam-4806	118	5	+	+	CCONJ
ejpam-4806	118	6	(	(	PUNCT
ejpam-4806	118	7	2m−	2m−	NUM
ejpam-4806	118	8	d	d	PROPN
ejpam-4806	118	9	(	(	PUNCT
ejpam-4806	118	10	v)−	v)−	NOUN
ejpam-4806	118	11	∑	∑	INTJ
ejpam-4806	118	12	uv/∈e(g	uv/∈e(g	PROPN
ejpam-4806	118	13	)	)	PUNCT
ejpam-4806	119	1	d	d	NOUN
ejpam-4806	119	2	(	(	PUNCT
ejpam-4806	119	3	u	u	NOUN
ejpam-4806	119	4	)	)	PUNCT
ejpam-4806	119	5	)	)	PUNCT
ejpam-4806	120	1	=	=	PUNCT
ejpam-4806	121	1	√	√	PROPN
ejpam-4806	121	2	2∆2	2∆2	NUM
ejpam-4806	121	3	+	+	CCONJ
ejpam-4806	121	4	2(2m−∆	2(2m−∆	NUM
ejpam-4806	121	5	)	)	PUNCT
ejpam-4806	121	6	≥	≥	PROPN
ejpam-4806	121	7	µ	µ	X
ejpam-4806	121	8	(	(	PUNCT
ejpam-4806	121	9	ga	ga	NOUN
ejpam-4806	121	10	)	)	PUNCT
ejpam-4806	121	11	hence	hence	ADV
ejpam-4806	121	12	,	,	PUNCT
ejpam-4806	121	13	√	√	PROPN
ejpam-4806	121	14	2δ2	2δ2	NUM
ejpam-4806	122	1	+	+	CCONJ
ejpam-4806	122	2	2(2m−	2(2m−	NUM
ejpam-4806	122	3	ηδ	ηδ	NOUN
ejpam-4806	122	4	)	)	PUNCT
ejpam-4806	122	5	≤	≤	NOUN
ejpam-4806	122	6	µ(ga	µ(ga	ADJ
ejpam-4806	122	7	)	)	PUNCT
ejpam-4806	122	8	≤	≤	NOUN
ejpam-4806	122	9	√	√	ADP
ejpam-4806	122	10	2∆2	2∆2	NUM
ejpam-4806	122	11	+	+	CCONJ
ejpam-4806	122	12	2(2m−∆	2(2m−∆	NUM
ejpam-4806	122	13	)	)	PUNCT
ejpam-4806	122	14	moreover	moreover	ADV
ejpam-4806	122	15	,	,	PUNCT
ejpam-4806	122	16	both	both	CCONJ
ejpam-4806	122	17	the	the	DET
ejpam-4806	122	18	equalities	equality	NOUN
ejpam-4806	122	19	hold	hold	VERB
ejpam-4806	122	20	for	for	ADP
ejpam-4806	122	21	n	n	NOUN
ejpam-4806	122	22	=	=	SYM
ejpam-4806	122	23	3	3	NUM
ejpam-4806	122	24	and	and	CCONJ
ejpam-4806	122	25	η	η	PROPN
ejpam-4806	122	26	=	=	PROPN
ejpam-4806	122	27	1	1	NUM
ejpam-4806	122	28	,	,	PUNCT
ejpam-4806	122	29	that	that	PRON
ejpam-4806	122	30	is	be	AUX
ejpam-4806	122	31	when	when	SCONJ
ejpam-4806	122	32	δ	δ	PROPN
ejpam-4806	122	33	=	=	SYM
ejpam-4806	122	34	∆.	∆.	PROPN
ejpam-4806	122	35	malathy	malathy	ADV
ejpam-4806	122	36	v	v	NOUN
ejpam-4806	122	37	,	,	PUNCT
ejpam-4806	122	38	kalyani	kalyani	PROPN
ejpam-4806	122	39	desikan	desikan	PROPN
ejpam-4806	122	40	/	/	SYM
ejpam-4806	122	41	eur	eur	PROPN
ejpam-4806	122	42	.	.	PUNCT
ejpam-4806	123	1	j.	j.	PROPN
ejpam-4806	123	2	pure	pure	PROPN
ejpam-4806	123	3	appl	appl	PROPN
ejpam-4806	123	4	.	.	PROPN
ejpam-4806	123	5	math	math	PROPN
ejpam-4806	123	6	,	,	PUNCT
ejpam-4806	123	7	16	16	NUM
ejpam-4806	123	8	(	(	PUNCT
ejpam-4806	123	9	3	3	NUM
ejpam-4806	123	10	)	)	PUNCT
ejpam-4806	123	11	(	(	PUNCT
ejpam-4806	123	12	2023	2023	NUM
ejpam-4806	123	13	)	)	PUNCT
ejpam-4806	123	14	,	,	PUNCT
ejpam-4806	123	15	1731	1731	NUM
ejpam-4806	123	16	-	-	SYM
ejpam-4806	123	17	1746	1746	NUM
ejpam-4806	123	18	1736	1736	NUM
ejpam-4806	123	19	example	example	NOUN
ejpam-4806	123	20	1	1	NUM
ejpam-4806	123	21	.	.	PUNCT
ejpam-4806	124	1	according	accord	VERB
ejpam-4806	124	2	to	to	ADP
ejpam-4806	124	3	the	the	DET
ejpam-4806	124	4	result	result	NOUN
ejpam-4806	124	5	derived	derive	VERB
ejpam-4806	124	6	in	in	ADP
ejpam-4806	124	7	theorem	theorem	NOUN
ejpam-4806	124	8	1	1	NUM
ejpam-4806	124	9	,	,	PUNCT
ejpam-4806	124	10	for	for	ADP
ejpam-4806	124	11	n	n	NOUN
ejpam-4806	124	12	=	=	SYM
ejpam-4806	124	13	7	7	NUM
ejpam-4806	124	14	,	,	PUNCT
ejpam-4806	124	15	m	m	VERB
ejpam-4806	124	16	=	=	NOUN
ejpam-4806	124	17	11	11	NUM
ejpam-4806	124	18	,	,	PUNCT
ejpam-4806	124	19	η	η	NOUN
ejpam-4806	124	20	=	=	SYM
ejpam-4806	124	21	5	5	NUM
ejpam-4806	124	22	,	,	PUNCT
ejpam-4806	124	23	∆	∆	X
ejpam-4806	124	24	=	=	SYM
ejpam-4806	124	25	6	6	NUM
ejpam-4806	124	26	,	,	PUNCT
ejpam-4806	124	27	δ	δ	PROPN
ejpam-4806	124	28	=	=	SYM
ejpam-4806	124	29	2	2	NUM
ejpam-4806	124	30	and	and	CCONJ
ejpam-4806	124	31	µ(ga	µ(ga	PROPN
ejpam-4806	124	32	)	)	PUNCT
ejpam-4806	124	33	=	=	SYM
ejpam-4806	124	34	8.5311	8.5311	NUM
ejpam-4806	124	35	,	,	PUNCT
ejpam-4806	124	36	we	we	PRON
ejpam-4806	124	37	have	have	VERB
ejpam-4806	124	38	5.6568	5.6568	NUM
ejpam-4806	124	39	≤	≤	NOUN
ejpam-4806	124	40	µ(ga	µ(ga	ADJ
ejpam-4806	124	41	)	)	PUNCT
ejpam-4806	124	42	≤	≤	NUM
ejpam-4806	124	43	10.198	10.198	NUM
ejpam-4806	124	44	(	(	PUNCT
ejpam-4806	124	45	4	4	NUM
ejpam-4806	124	46	)	)	PUNCT
ejpam-4806	124	47	remark	remark	NOUN
ejpam-4806	124	48	1	1	NUM
ejpam-4806	124	49	.	.	PUNCT
ejpam-4806	125	1	the	the	DET
ejpam-4806	125	2	following	follow	VERB
ejpam-4806	125	3	relationships	relationship	NOUN
ejpam-4806	125	4	between	between	ADP
ejpam-4806	125	5	the	the	DET
ejpam-4806	125	6	parameters	parameter	NOUN
ejpam-4806	125	7	in	in	ADP
ejpam-4806	125	8	the	the	DET
ejpam-4806	125	9	graph	graph	NOUN
ejpam-4806	125	10	ga	ga	PROPN
ejpam-4806	125	11	hold	hold	NOUN
ejpam-4806	125	12	:	:	PUNCT
ejpam-4806	125	13	(	(	PUNCT
ejpam-4806	125	14	i	i	NOUN
ejpam-4806	125	15	)	)	PUNCT
ejpam-4806	125	16	m	m	VERB
ejpam-4806	125	17	=	=	SYM
ejpam-4806	125	18	(	(	PUNCT
ejpam-4806	125	19	n+	n+	X
ejpam-4806	125	20	η	η	PROPN
ejpam-4806	125	21	−	−	PROPN
ejpam-4806	125	22	1	1	NUM
ejpam-4806	125	23	)	)	PUNCT
ejpam-4806	125	24	=	=	NOUN
ejpam-4806	125	25	(	(	PUNCT
ejpam-4806	125	26	2n−	2n−	PROPN
ejpam-4806	125	27	3	3	NUM
ejpam-4806	125	28	)	)	PUNCT
ejpam-4806	125	29	(	(	PUNCT
ejpam-4806	125	30	ii	ii	NOUN
ejpam-4806	125	31	)	)	PUNCT
ejpam-4806	125	32	∆	∆	PROPN
ejpam-4806	125	33	=	=	PUNCT
ejpam-4806	125	34	(	(	PUNCT
ejpam-4806	125	35	n−	n−	NOUN
ejpam-4806	125	36	1	1	NUM
ejpam-4806	125	37	)	)	PUNCT
ejpam-4806	125	38	(	(	PUNCT
ejpam-4806	125	39	iii	iii	NOUN
ejpam-4806	125	40	)	)	PUNCT
ejpam-4806	125	41	(	(	PUNCT
ejpam-4806	125	42	δ	δ	PROPN
ejpam-4806	125	43	+	+	NOUN
ejpam-4806	125	44	∆	∆	X
ejpam-4806	125	45	)	)	PUNCT
ejpam-4806	126	1	=	=	SYM
ejpam-4806	126	2	(	(	PUNCT
ejpam-4806	126	3	n+	n+	NOUN
ejpam-4806	126	4	1	1	NUM
ejpam-4806	126	5	)	)	PUNCT
ejpam-4806	126	6	(	(	PUNCT
ejpam-4806	126	7	iv	iv	X
ejpam-4806	126	8	)	)	PUNCT
ejpam-4806	126	9	(	(	PUNCT
ejpam-4806	126	10	∆−	∆−	NOUN
ejpam-4806	126	11	δ	δ	PROPN
ejpam-4806	126	12	)	)	PUNCT
ejpam-4806	126	13	=	=	SYM
ejpam-4806	126	14	(	(	PUNCT
ejpam-4806	126	15	η	η	PROPN
ejpam-4806	126	16	−	−	PROPN
ejpam-4806	126	17	1	1	NUM
ejpam-4806	126	18	)	)	PUNCT
ejpam-4806	126	19	=	=	SYM
ejpam-4806	126	20	(	(	PUNCT
ejpam-4806	126	21	n−	n−	NOUN
ejpam-4806	126	22	3	3	NUM
ejpam-4806	126	23	)	)	PUNCT
ejpam-4806	126	24	(	(	PUNCT
ejpam-4806	126	25	v	v	NOUN
ejpam-4806	126	26	)	)	PUNCT
ejpam-4806	126	27	(	(	PUNCT
ejpam-4806	126	28	n−	n−	NOUN
ejpam-4806	126	29	η	η	PROPN
ejpam-4806	126	30	)	)	PUNCT
ejpam-4806	126	31	=	=	SYM
ejpam-4806	126	32	δ	δ	NOUN
ejpam-4806	126	33	=	=	SYM
ejpam-4806	127	1	2	2	X
ejpam-4806	127	2	.	.	X
ejpam-4806	127	3	we	we	PRON
ejpam-4806	127	4	prove	prove	VERB
ejpam-4806	127	5	the	the	DET
ejpam-4806	127	6	following	follow	VERB
ejpam-4806	127	7	lemmas	lemmas	PROPN
ejpam-4806	127	8	,	,	PUNCT
ejpam-4806	127	9	which	which	PRON
ejpam-4806	127	10	are	be	AUX
ejpam-4806	127	11	used	use	VERB
ejpam-4806	127	12	to	to	PART
ejpam-4806	127	13	prove	prove	VERB
ejpam-4806	127	14	the	the	DET
ejpam-4806	127	15	theorems	theorem	NOUN
ejpam-4806	127	16	.	.	PUNCT
ejpam-4806	128	1	lemma	lemma	PROPN
ejpam-4806	128	2	5	5	NUM
ejpam-4806	128	3	.	.	PUNCT
ejpam-4806	128	4	for	for	ADP
ejpam-4806	128	5	the	the	DET
ejpam-4806	128	6	graph	graph	NOUN
ejpam-4806	128	7	ga	ga	PROPN
ejpam-4806	128	8	,	,	PUNCT
ejpam-4806	128	9	µ(ga	µ(ga	PROPN
ejpam-4806	128	10	)	)	PUNCT
ejpam-4806	128	11	>	>	X
ejpam-4806	128	12	(	(	PUNCT
ejpam-4806	128	13	2δ2	2δ2	NUM
ejpam-4806	128	14	+	+	CCONJ
ejpam-4806	128	15	2(2m−	2(2m−	NUM
ejpam-4806	128	16	ηδ	ηδ	NOUN
ejpam-4806	128	17	)	)	PUNCT
ejpam-4806	128	18	+	+	CCONJ
ejpam-4806	128	19	(	(	PUNCT
ejpam-4806	128	20	n+	n+	NUM
ejpam-4806	128	21	1)η	1)η	NUM
ejpam-4806	128	22	)	)	PUNCT
ejpam-4806	128	23	1/2	1/2	NUM
ejpam-4806	128	24	>	>	PUNCT
ejpam-4806	128	25	(	(	PUNCT
ejpam-4806	128	26	2δ2	2δ2	NUM
ejpam-4806	128	27	+	+	CCONJ
ejpam-4806	128	28	2(2m−	2(2m−	NUM
ejpam-4806	128	29	ηδ	ηδ	NOUN
ejpam-4806	128	30	)	)	PUNCT
ejpam-4806	128	31	+	+	NUM
ejpam-4806	128	32	nη	nη	X
ejpam-4806	128	33	)	)	PUNCT
ejpam-4806	128	34	1/2	1/2	NUM
ejpam-4806	128	35	for	for	ADP
ejpam-4806	128	36	all	all	PRON
ejpam-4806	128	37	n	n	CCONJ
ejpam-4806	128	38	>	>	X
ejpam-4806	128	39	6	6	NUM
ejpam-4806	128	40	.	.	PUNCT
ejpam-4806	128	41	proof	proof	NOUN
ejpam-4806	128	42	.	.	PUNCT
ejpam-4806	129	1	to	to	PART
ejpam-4806	129	2	prove	prove	VERB
ejpam-4806	129	3	this	this	PRON
ejpam-4806	129	4	we	we	PRON
ejpam-4806	129	5	make	make	VERB
ejpam-4806	129	6	use	use	NOUN
ejpam-4806	129	7	of	of	ADP
ejpam-4806	129	8	lemma	lemma	PROPN
ejpam-4806	129	9	4	4	NUM
ejpam-4806	129	10	.	.	PUNCT
ejpam-4806	130	1	we	we	PRON
ejpam-4806	130	2	consider	consider	VERB
ejpam-4806	130	3	the	the	DET
ejpam-4806	130	4	signless	signless	NOUN
ejpam-4806	130	5	laplacian	laplacian	ADJ
ejpam-4806	130	6	spectral	spectral	ADJ
ejpam-4806	130	7	radius	radius	NOUN
ejpam-4806	130	8	for	for	ADP
ejpam-4806	130	9	ka	ka	PROPN
ejpam-4806	130	10	∨kb	∨kb	PROPN
ejpam-4806	130	11	µ	µ	X
ejpam-4806	130	12	(	(	PUNCT
ejpam-4806	130	13	ga	ga	NOUN
ejpam-4806	130	14	)	)	PUNCT
ejpam-4806	130	15	=	=	PUNCT
ejpam-4806	131	1	(	(	PUNCT
ejpam-4806	131	2	3b+	3b+	NUM
ejpam-4806	131	3	a−	a−	PROPN
ejpam-4806	131	4	2	2	NUM
ejpam-4806	131	5	)	)	PUNCT
ejpam-4806	131	6	+	+	CCONJ
ejpam-4806	131	7	√	√	NUM
ejpam-4806	131	8	(	(	PUNCT
ejpam-4806	131	9	b+	b+	X
ejpam-4806	131	10	a−	a−	PROPN
ejpam-4806	131	11	2)2	2)2	NUM
ejpam-4806	131	12	+	+	CCONJ
ejpam-4806	131	13	4ab	4ab	ADJ
ejpam-4806	131	14	2	2	NUM
ejpam-4806	131	15	(	(	PUNCT
ejpam-4806	131	16	5	5	NUM
ejpam-4806	131	17	)	)	PUNCT
ejpam-4806	131	18	substituting	substitute	VERB
ejpam-4806	131	19	b	b	NOUN
ejpam-4806	131	20	=	=	SYM
ejpam-4806	131	21	2	2	NUM
ejpam-4806	131	22	and	and	CCONJ
ejpam-4806	131	23	a	a	DET
ejpam-4806	131	24	=	=	SYM
ejpam-4806	131	25	η	η	PROPN
ejpam-4806	131	26	,	,	PUNCT
ejpam-4806	131	27	where	where	SCONJ
ejpam-4806	131	28	η	η	PROPN
ejpam-4806	131	29	=	=	PROPN
ejpam-4806	131	30	n	n	CCONJ
ejpam-4806	131	31	−	−	PROPN
ejpam-4806	131	32	2	2	NUM
ejpam-4806	131	33	and	and	CCONJ
ejpam-4806	131	34	making	make	VERB
ejpam-4806	131	35	use	use	NOUN
ejpam-4806	131	36	of	of	ADP
ejpam-4806	131	37	remark	remark	NOUN
ejpam-4806	131	38	1	1	NUM
ejpam-4806	131	39	,	,	PUNCT
ejpam-4806	131	40	we	we	PRON
ejpam-4806	131	41	reduce	reduce	VERB
ejpam-4806	131	42	equation	equation	NOUN
ejpam-4806	131	43	(	(	PUNCT
ejpam-4806	131	44	5	5	NUM
ejpam-4806	131	45	)	)	PUNCT
ejpam-4806	131	46	in	in	ADP
ejpam-4806	131	47	terms	term	NOUN
ejpam-4806	131	48	of	of	ADP
ejpam-4806	131	49	n	n	PRON
ejpam-4806	131	50	as	as	ADP
ejpam-4806	131	51	µ	µ	X
ejpam-4806	131	52	(	(	PUNCT
ejpam-4806	131	53	ga	ga	NOUN
ejpam-4806	131	54	)	)	PUNCT
ejpam-4806	131	55	=	=	PRON
ejpam-4806	131	56	{	{	PUNCT
ejpam-4806	131	57	(	(	PUNCT
ejpam-4806	131	58	n+	n+	NOUN
ejpam-4806	131	59	2	2	NUM
ejpam-4806	131	60	)	)	PUNCT
ejpam-4806	131	61	+	+	CCONJ
ejpam-4806	131	62	√	√	ADJ
ejpam-4806	131	63	(	(	PUNCT
ejpam-4806	131	64	n2	n2	NOUN
ejpam-4806	131	65	+	+	X
ejpam-4806	131	66	4n−	4n−	NUM
ejpam-4806	131	67	12	12	NUM
ejpam-4806	131	68	)	)	SYM
ejpam-4806	131	69	2	2	NUM
ejpam-4806	131	70	}	}	PUNCT
ejpam-4806	131	71	squaring	square	VERB
ejpam-4806	131	72	the	the	DET
ejpam-4806	131	73	above	above	ADJ
ejpam-4806	131	74	expression	expression	NOUN
ejpam-4806	131	75	we	we	PRON
ejpam-4806	131	76	have	have	VERB
ejpam-4806	131	77	(	(	PUNCT
ejpam-4806	131	78	µ	µ	X
ejpam-4806	131	79	(	(	PUNCT
ejpam-4806	131	80	ga	ga	NOUN
ejpam-4806	131	81	)	)	PUNCT
ejpam-4806	131	82	)	)	PUNCT
ejpam-4806	131	83	2	2	NUM
ejpam-4806	131	84	=	=	SYM
ejpam-4806	131	85	{	{	PUNCT
ejpam-4806	131	86	(	(	PUNCT
ejpam-4806	131	87	n+	n+	NUM
ejpam-4806	131	88	2)2	2)2	NUM
ejpam-4806	131	89	+	+	CCONJ
ejpam-4806	131	90	(	(	PUNCT
ejpam-4806	131	91	n2	n2	ADJ
ejpam-4806	131	92	+	+	X
ejpam-4806	131	93	4n−	4n−	NUM
ejpam-4806	131	94	12	12	NUM
ejpam-4806	131	95	)	)	PUNCT
ejpam-4806	131	96	+	+	CCONJ
ejpam-4806	131	97	2(n+	2(n+	NUM
ejpam-4806	131	98	2	2	NUM
ejpam-4806	131	99	)	)	PUNCT
ejpam-4806	131	100	√	√	PROPN
ejpam-4806	131	101	(	(	PUNCT
ejpam-4806	131	102	n2	n2	NOUN
ejpam-4806	131	103	+	+	X
ejpam-4806	131	104	4n−	4n−	NUM
ejpam-4806	131	105	12	12	NUM
ejpam-4806	131	106	)	)	PUNCT
ejpam-4806	131	107	4	4	NUM
ejpam-4806	131	108	}	}	PUNCT
ejpam-4806	131	109	in	in	ADP
ejpam-4806	131	110	the	the	DET
ejpam-4806	131	111	above	above	ADJ
ejpam-4806	131	112	expression	expression	NOUN
ejpam-4806	131	113	,	,	PUNCT
ejpam-4806	131	114	using	use	VERB
ejpam-4806	131	115	the	the	DET
ejpam-4806	131	116	following	follow	VERB
ejpam-4806	131	117	binomial	binomial	ADJ
ejpam-4806	131	118	approximation√	approximation√	NOUN
ejpam-4806	131	119	(	(	PUNCT
ejpam-4806	131	120	n2	n2	NOUN
ejpam-4806	131	121	+	+	X
ejpam-4806	131	122	4n−	4n−	NUM
ejpam-4806	131	123	12	12	NUM
ejpam-4806	131	124	)	)	PUNCT
ejpam-4806	131	125	≃	≃	NOUN
ejpam-4806	131	126	(	(	PUNCT
ejpam-4806	131	127	1	1	NUM
ejpam-4806	131	128	+	+	CCONJ
ejpam-4806	131	129	n	n	PRON
ejpam-4806	131	130	2	2	NUM
ejpam-4806	131	131	(	(	PUNCT
ejpam-4806	131	132	4	4	NUM
ejpam-4806	131	133	n	n	NOUN
ejpam-4806	131	134	−	−	NUM
ejpam-4806	131	135	12	12	NUM
ejpam-4806	131	136	n2	n2	NOUN
ejpam-4806	131	137	)	)	PUNCT
ejpam-4806	131	138	)	)	PUNCT
ejpam-4806	131	139	when	when	SCONJ
ejpam-4806	131	140	1	1	NUM
ejpam-4806	131	141	2	2	NUM
ejpam-4806	131	142	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4806	131	143	4n	4n	NOUN
ejpam-4806	131	144	−	−	NOUN
ejpam-4806	131	145	12	12	NUM
ejpam-4806	131	146	n2	n2	NOUN
ejpam-4806	131	147	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4806	131	148	<	<	X
ejpam-4806	131	149	1	1	NUM
ejpam-4806	131	150	for	for	ADP
ejpam-4806	131	151	n	n	X
ejpam-4806	131	152	≥	≥	NOUN
ejpam-4806	131	153	3	3	NUM
ejpam-4806	131	154	,	,	PUNCT
ejpam-4806	131	155	we	we	PRON
ejpam-4806	131	156	get	get	VERB
ejpam-4806	131	157	(	(	PUNCT
ejpam-4806	131	158	µ	µ	X
ejpam-4806	131	159	(	(	PUNCT
ejpam-4806	131	160	ga	ga	NOUN
ejpam-4806	131	161	)	)	PUNCT
ejpam-4806	131	162	)	)	PUNCT
ejpam-4806	131	163	2	2	NUM
ejpam-4806	131	164	≃	≃	NOUN
ejpam-4806	131	165	{	{	PUNCT
ejpam-4806	131	166	(	(	PUNCT
ejpam-4806	131	167	n+	n+	NUM
ejpam-4806	131	168	2)2	2)2	NUM
ejpam-4806	131	169	+	+	CCONJ
ejpam-4806	131	170	(	(	PUNCT
ejpam-4806	131	171	n2	n2	ADJ
ejpam-4806	131	172	+	+	X
ejpam-4806	132	1	4n−	4n−	NUM
ejpam-4806	132	2	12	12	NUM
ejpam-4806	132	3	)	)	PUNCT
ejpam-4806	133	1	+	+	CCONJ
ejpam-4806	133	2	2(n+	2(n+	NUM
ejpam-4806	133	3	2	2	NUM
ejpam-4806	133	4	)	)	PUNCT
ejpam-4806	133	5	(	(	PUNCT
ejpam-4806	133	6	1	1	NUM
ejpam-4806	133	7	+	+	CCONJ
ejpam-4806	133	8	n	n	PRON
ejpam-4806	133	9	2	2	NUM
ejpam-4806	133	10	(	(	PUNCT
ejpam-4806	133	11	4	4	NUM
ejpam-4806	133	12	n	n	NOUN
ejpam-4806	133	13	−	−	NUM
ejpam-4806	133	14	12	12	NUM
ejpam-4806	133	15	n2	n2	NOUN
ejpam-4806	133	16	)	)	PUNCT
ejpam-4806	133	17	)	)	PUNCT
ejpam-4806	133	18	4	4	NUM
ejpam-4806	133	19	}	}	PUNCT
ejpam-4806	133	20	malathy	malathy	ADV
ejpam-4806	133	21	v	v	NOUN
ejpam-4806	133	22	,	,	PUNCT
ejpam-4806	133	23	kalyani	kalyani	PROPN
ejpam-4806	133	24	desikan	desikan	PROPN
ejpam-4806	133	25	/	/	SYM
ejpam-4806	133	26	eur	eur	PROPN
ejpam-4806	133	27	.	.	PUNCT
ejpam-4806	134	1	j.	j.	PROPN
ejpam-4806	134	2	pure	pure	PROPN
ejpam-4806	134	3	appl	appl	PROPN
ejpam-4806	134	4	.	.	PROPN
ejpam-4806	134	5	math	math	PROPN
ejpam-4806	134	6	,	,	PUNCT
ejpam-4806	134	7	16	16	NUM
ejpam-4806	134	8	(	(	PUNCT
ejpam-4806	134	9	3	3	NUM
ejpam-4806	134	10	)	)	PUNCT
ejpam-4806	134	11	(	(	PUNCT
ejpam-4806	134	12	2023	2023	NUM
ejpam-4806	134	13	)	)	PUNCT
ejpam-4806	134	14	,	,	PUNCT
ejpam-4806	134	15	1731	1731	NUM
ejpam-4806	134	16	-	-	SYM
ejpam-4806	134	17	1746	1746	NUM
ejpam-4806	134	18	1737	1737	NUM
ejpam-4806	134	19	=	=	SYM
ejpam-4806	134	20	(	(	PUNCT
ejpam-4806	134	21	n2	n2	NOUN
ejpam-4806	134	22	+	+	CCONJ
ejpam-4806	135	1	4n−	4n−	NUM
ejpam-4806	135	2	6	6	NUM
ejpam-4806	135	3	n	n	NOUN
ejpam-4806	135	4	−	−	PROPN
ejpam-4806	135	5	3	3	NUM
ejpam-4806	135	6	)	)	PUNCT
ejpam-4806	135	7	.	.	PUNCT
ejpam-4806	136	1	to	to	PART
ejpam-4806	136	2	prove	prove	VERB
ejpam-4806	136	3	µ(ga	µ(ga	NOUN
ejpam-4806	136	4	)	)	PUNCT
ejpam-4806	136	5	>	>	X
ejpam-4806	137	1	(	(	PUNCT
ejpam-4806	137	2	2δ2	2δ2	NUM
ejpam-4806	137	3	+	+	CCONJ
ejpam-4806	137	4	2(2m−	2(2m−	NUM
ejpam-4806	137	5	ηδ	ηδ	NOUN
ejpam-4806	137	6	)	)	PUNCT
ejpam-4806	138	1	+	+	CCONJ
ejpam-4806	138	2	(	(	PUNCT
ejpam-4806	138	3	n+	n+	NUM
ejpam-4806	138	4	1)η	1)η	NUM
ejpam-4806	138	5	)	)	PUNCT
ejpam-4806	138	6	1/2	1/2	NUM
ejpam-4806	138	7	consider	consider	VERB
ejpam-4806	138	8	(	(	PUNCT
ejpam-4806	138	9	µ(ga	µ(ga	NOUN
ejpam-4806	138	10	)	)	PUNCT
ejpam-4806	138	11	)	)	PUNCT
ejpam-4806	138	12	2	2	NUM
ejpam-4806	138	13	−	−	NOUN
ejpam-4806	138	14	(	(	PUNCT
ejpam-4806	138	15	(	(	PUNCT
ejpam-4806	138	16	2δ2	2δ2	NUM
ejpam-4806	138	17	+	+	CCONJ
ejpam-4806	138	18	2(2m−	2(2m−	NUM
ejpam-4806	138	19	ηδ	ηδ	NOUN
ejpam-4806	138	20	)	)	PUNCT
ejpam-4806	138	21	+	+	CCONJ
ejpam-4806	138	22	(	(	PUNCT
ejpam-4806	138	23	n+	n+	NUM
ejpam-4806	138	24	1)η	1)η	NUM
ejpam-4806	138	25	)	)	PUNCT
ejpam-4806	138	26	1/2)2	1/2)2	NUM
ejpam-4806	138	27	expressing	express	VERB
ejpam-4806	138	28	the	the	DET
ejpam-4806	138	29	second	second	ADJ
ejpam-4806	138	30	term	term	NOUN
ejpam-4806	138	31	in	in	ADP
ejpam-4806	138	32	the	the	DET
ejpam-4806	138	33	above	above	ADJ
ejpam-4806	138	34	expression	expression	NOUN
ejpam-4806	138	35	in	in	ADP
ejpam-4806	138	36	terms	term	NOUN
ejpam-4806	138	37	of	of	ADP
ejpam-4806	138	38	n	n	PRON
ejpam-4806	138	39	and	and	CCONJ
ejpam-4806	138	40	using	use	VERB
ejpam-4806	138	41	the	the	DET
ejpam-4806	138	42	approximation	approximation	NOUN
ejpam-4806	138	43	(	(	PUNCT
ejpam-4806	138	44	µ(ga	µ(ga	PROPN
ejpam-4806	138	45	)	)	PUNCT
ejpam-4806	138	46	)	)	PUNCT
ejpam-4806	138	47	2	2	NUM
ejpam-4806	138	48	≃	≃	NOUN
ejpam-4806	138	49	(	(	PUNCT
ejpam-4806	138	50	n2	n2	NOUN
ejpam-4806	138	51	+	+	CCONJ
ejpam-4806	139	1	4n−	4n−	NUM
ejpam-4806	139	2	6	6	NUM
ejpam-4806	139	3	n	n	NOUN
ejpam-4806	139	4	−	−	PROPN
ejpam-4806	139	5	3	3	NUM
ejpam-4806	139	6	)	)	PUNCT
ejpam-4806	139	7	.	.	PUNCT
ejpam-4806	140	1	we	we	PRON
ejpam-4806	140	2	obtain	obtain	VERB
ejpam-4806	140	3	(	(	PUNCT
ejpam-4806	140	4	µ	µ	X
ejpam-4806	140	5	(	(	PUNCT
ejpam-4806	140	6	ga	ga	NOUN
ejpam-4806	140	7	)	)	PUNCT
ejpam-4806	140	8	)	)	PUNCT
ejpam-4806	140	9	2	2	NUM
ejpam-4806	140	10	−	−	PROPN
ejpam-4806	140	11	(	(	PUNCT
ejpam-4806	140	12	n2	n2	NOUN
ejpam-4806	140	13	+	+	CCONJ
ejpam-4806	140	14	3n+	3n+	NUM
ejpam-4806	140	15	2	2	NUM
ejpam-4806	140	16	)	)	PUNCT
ejpam-4806	140	17	≃	≃	NOUN
ejpam-4806	140	18	(	(	PUNCT
ejpam-4806	140	19	n−	n−	NOUN
ejpam-4806	140	20	6	6	NUM
ejpam-4806	140	21	n	n	NOUN
ejpam-4806	140	22	−	−	NUM
ejpam-4806	140	23	5	5	NUM
ejpam-4806	140	24	)	)	PUNCT
ejpam-4806	140	25	>	>	X
ejpam-4806	140	26	0	0	PUNCT
ejpam-4806	140	27	for	for	ADP
ejpam-4806	140	28	n	n	X
ejpam-4806	140	29	>	>	X
ejpam-4806	140	30	6	6	NUM
ejpam-4806	140	31	.	.	PUNCT
ejpam-4806	140	32	hence	hence	ADV
ejpam-4806	140	33	proved	prove	VERB
ejpam-4806	140	34	.	.	PUNCT
ejpam-4806	141	1	lemma	lemma	PROPN
ejpam-4806	141	2	6	6	NUM
ejpam-4806	141	3	.	.	PUNCT
ejpam-4806	142	1	for	for	ADP
ejpam-4806	142	2	the	the	DET
ejpam-4806	142	3	graph	graph	NOUN
ejpam-4806	142	4	ga	ga	PROPN
ejpam-4806	142	5	,	,	PUNCT
ejpam-4806	142	6	µ	µ	X
ejpam-4806	142	7	(	(	PUNCT
ejpam-4806	142	8	ga	ga	PROPN
ejpam-4806	142	9	)	)	PUNCT
ejpam-4806	142	10	>	>	X
ejpam-4806	143	1	(	(	PUNCT
ejpam-4806	143	2	2∆2	2∆2	NUM
ejpam-4806	143	3	+	+	CCONJ
ejpam-4806	143	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	143	5	nη	nη	X
ejpam-4806	143	6	)	)	PUNCT
ejpam-4806	143	7	1/2	1/2	NUM
ejpam-4806	143	8	for	for	ADP
ejpam-4806	143	9	n	n	PRON
ejpam-4806	143	10	≥	≥	NOUN
ejpam-4806	143	11	3	3	NUM
ejpam-4806	143	12	.	.	PUNCT
ejpam-4806	144	1	proof	proof	NOUN
ejpam-4806	144	2	.	.	PUNCT
ejpam-4806	145	1	we	we	PRON
ejpam-4806	145	2	follow	follow	VERB
ejpam-4806	145	3	the	the	DET
ejpam-4806	145	4	procedure	procedure	NOUN
ejpam-4806	145	5	adopted	adopt	VERB
ejpam-4806	145	6	to	to	PART
ejpam-4806	145	7	prove	prove	VERB
ejpam-4806	145	8	lemma	lemma	PROPN
ejpam-4806	145	9	5	5	NUM
ejpam-4806	145	10	.	.	PUNCT
ejpam-4806	146	1	the	the	DET
ejpam-4806	146	2	above	above	ADJ
ejpam-4806	146	3	inequality	inequality	NOUN
ejpam-4806	146	4	is	be	AUX
ejpam-4806	146	5	proved	prove	VERB
ejpam-4806	146	6	by	by	ADP
ejpam-4806	146	7	reducing	reduce	VERB
ejpam-4806	146	8	the	the	DET
ejpam-4806	146	9	rhs	rhs	PROPN
ejpam-4806	146	10	expression	expression	NOUN
ejpam-4806	146	11	in	in	ADP
ejpam-4806	146	12	terms	term	NOUN
ejpam-4806	146	13	of	of	ADP
ejpam-4806	146	14	n	n	PRON
ejpam-4806	146	15	and	and	CCONJ
ejpam-4806	146	16	on	on	ADP
ejpam-4806	146	17	using	use	VERB
ejpam-4806	146	18	the	the	DET
ejpam-4806	146	19	approximation	approximation	NOUN
ejpam-4806	146	20	for	for	ADP
ejpam-4806	146	21	(	(	PUNCT
ejpam-4806	146	22	µ	µ	X
ejpam-4806	146	23	(	(	PUNCT
ejpam-4806	146	24	ga	ga	NOUN
ejpam-4806	146	25	)	)	PUNCT
ejpam-4806	146	26	)	)	PUNCT
ejpam-4806	147	1	2	2	NUM
ejpam-4806	147	2	,	,	PUNCT
ejpam-4806	147	3	we	we	PRON
ejpam-4806	147	4	obtain	obtain	VERB
ejpam-4806	147	5	(	(	PUNCT
ejpam-4806	147	6	µ	µ	X
ejpam-4806	147	7	(	(	PUNCT
ejpam-4806	147	8	ga	ga	NOUN
ejpam-4806	147	9	)	)	PUNCT
ejpam-4806	147	10	)	)	PUNCT
ejpam-4806	147	11	2	2	NUM
ejpam-4806	147	12	−	−	PROPN
ejpam-4806	147	13	(	(	PUNCT
ejpam-4806	147	14	n2	n2	NOUN
ejpam-4806	147	15	+	+	CCONJ
ejpam-4806	148	1	4n−	4n−	NUM
ejpam-4806	148	2	8	8	NUM
ejpam-4806	148	3	)	)	PUNCT
ejpam-4806	148	4	≃	≃	NOUN
ejpam-4806	148	5	(	(	PUNCT
ejpam-4806	148	6	5−	5−	NUM
ejpam-4806	148	7	6	6	NUM
ejpam-4806	148	8	n	n	NOUN
ejpam-4806	148	9	)	)	PUNCT
ejpam-4806	148	10	>	>	X
ejpam-4806	148	11	0	0	PUNCT
ejpam-4806	148	12	for	for	ADP
ejpam-4806	148	13	n	n	X
ejpam-4806	148	14	≥	≥	NOUN
ejpam-4806	148	15	3	3	NUM
ejpam-4806	148	16	.	.	PUNCT
ejpam-4806	149	1	lemma	lemma	PROPN
ejpam-4806	149	2	7	7	NUM
ejpam-4806	149	3	.	.	X
ejpam-4806	149	4	for	for	ADP
ejpam-4806	149	5	the	the	DET
ejpam-4806	149	6	graph	graph	NOUN
ejpam-4806	149	7	ga	ga	PROPN
ejpam-4806	149	8	,	,	PUNCT
ejpam-4806	149	9	µ	µ	X
ejpam-4806	149	10	(	(	PUNCT
ejpam-4806	149	11	ga	ga	PROPN
ejpam-4806	149	12	)	)	PUNCT
ejpam-4806	149	13	<	<	X
ejpam-4806	149	14	(	(	PUNCT
ejpam-4806	149	15	2δ2	2δ2	NUM
ejpam-4806	150	1	+	+	CCONJ
ejpam-4806	150	2	2(2m−	2(2m−	NUM
ejpam-4806	150	3	ηδ	ηδ	NOUN
ejpam-4806	150	4	)	)	PUNCT
ejpam-4806	151	1	+	+	CCONJ
ejpam-4806	151	2	(	(	PUNCT
ejpam-4806	151	3	n+	n+	NUM
ejpam-4806	151	4	2)η	2)η	ADJ
ejpam-4806	151	5	)	)	PUNCT
ejpam-4806	151	6	1/2	1/2	NUM
ejpam-4806	151	7	for	for	ADP
ejpam-4806	151	8	n	n	PRON
ejpam-4806	151	9	≥	≥	NOUN
ejpam-4806	151	10	3	3	NUM
ejpam-4806	151	11	.	.	PUNCT
ejpam-4806	152	1	proof	proof	NOUN
ejpam-4806	152	2	.	.	PUNCT
ejpam-4806	153	1	the	the	DET
ejpam-4806	153	2	proof	proof	NOUN
ejpam-4806	153	3	is	be	AUX
ejpam-4806	153	4	on	on	ADP
ejpam-4806	153	5	the	the	DET
ejpam-4806	153	6	same	same	ADJ
ejpam-4806	153	7	lines	line	NOUN
ejpam-4806	153	8	as	as	ADP
ejpam-4806	153	9	that	that	PRON
ejpam-4806	153	10	of	of	ADP
ejpam-4806	153	11	lemma	lemma	PROPN
ejpam-4806	153	12	5	5	NUM
ejpam-4806	153	13	and	and	CCONJ
ejpam-4806	153	14	we	we	PRON
ejpam-4806	153	15	have	have	VERB
ejpam-4806	153	16	(	(	PUNCT
ejpam-4806	153	17	µ	µ	X
ejpam-4806	153	18	(	(	PUNCT
ejpam-4806	153	19	ga	ga	NOUN
ejpam-4806	153	20	)	)	PUNCT
ejpam-4806	153	21	)	)	PUNCT
ejpam-4806	153	22	2	2	NUM
ejpam-4806	153	23	−	−	PROPN
ejpam-4806	153	24	(	(	PUNCT
ejpam-4806	153	25	n2	n2	NOUN
ejpam-4806	153	26	+	+	CCONJ
ejpam-4806	153	27	4n	4n	NOUN
ejpam-4806	153	28	)	)	PUNCT
ejpam-4806	153	29	≃	≃	NOUN
ejpam-4806	153	30	(	(	PUNCT
ejpam-4806	153	31	−	−	PROPN
ejpam-4806	153	32	6	6	NUM
ejpam-4806	153	33	n	n	NOUN
ejpam-4806	153	34	−	−	PROPN
ejpam-4806	153	35	3	3	NUM
ejpam-4806	153	36	)	)	PUNCT
ejpam-4806	153	37	<	<	X
ejpam-4806	153	38	0	0	NUM
ejpam-4806	153	39	for	for	ADP
ejpam-4806	153	40	n	n	X
ejpam-4806	153	41	≥	≥	NUM
ejpam-4806	153	42	3	3	NUM
ejpam-4806	153	43	.	.	PUNCT
ejpam-4806	153	44	malathy	malathy	NUM
ejpam-4806	154	1	v	v	NOUN
ejpam-4806	154	2	,	,	PUNCT
ejpam-4806	154	3	kalyani	kalyani	PROPN
ejpam-4806	154	4	desikan	desikan	PROPN
ejpam-4806	154	5	/	/	SYM
ejpam-4806	154	6	eur	eur	PROPN
ejpam-4806	154	7	.	.	PUNCT
ejpam-4806	155	1	j.	j.	PROPN
ejpam-4806	155	2	pure	pure	PROPN
ejpam-4806	155	3	appl	appl	PROPN
ejpam-4806	155	4	.	.	PROPN
ejpam-4806	155	5	math	math	PROPN
ejpam-4806	155	6	,	,	PUNCT
ejpam-4806	155	7	16	16	NUM
ejpam-4806	155	8	(	(	PUNCT
ejpam-4806	155	9	3	3	NUM
ejpam-4806	155	10	)	)	PUNCT
ejpam-4806	155	11	(	(	PUNCT
ejpam-4806	155	12	2023	2023	NUM
ejpam-4806	155	13	)	)	PUNCT
ejpam-4806	155	14	,	,	PUNCT
ejpam-4806	155	15	1731	1731	NUM
ejpam-4806	155	16	-	-	SYM
ejpam-4806	155	17	1746	1746	NUM
ejpam-4806	155	18	1738	1738	NUM
ejpam-4806	155	19	corollary	corollary	NOUN
ejpam-4806	155	20	1	1	NUM
ejpam-4806	155	21	.	.	PUNCT
ejpam-4806	155	22	using	use	VERB
ejpam-4806	155	23	lemma	lemma	PROPN
ejpam-4806	155	24	5	5	NUM
ejpam-4806	156	1	,	,	PUNCT
ejpam-4806	156	2	we	we	PRON
ejpam-4806	156	3	obtain	obtain	VERB
ejpam-4806	156	4	the	the	DET
ejpam-4806	156	5	approximation	approximation	NOUN
ejpam-4806	156	6	for	for	ADP
ejpam-4806	156	7	µ	µ	PROPN
ejpam-4806	156	8	(	(	PUNCT
ejpam-4806	156	9	ga	ga	NOUN
ejpam-4806	156	10	)	)	PUNCT
ejpam-4806	156	11	as	as	ADP
ejpam-4806	156	12	µ	µ	X
ejpam-4806	156	13	(	(	PUNCT
ejpam-4806	156	14	ga	ga	NOUN
ejpam-4806	156	15	)	)	PUNCT
ejpam-4806	156	16	≃	≃	NOUN
ejpam-4806	156	17	(	(	PUNCT
ejpam-4806	156	18	n2	n2	NOUN
ejpam-4806	156	19	+	+	X
ejpam-4806	156	20	4n−	4n−	NUM
ejpam-4806	156	21	4.2	4.2	NUM
ejpam-4806	156	22	)	)	PUNCT
ejpam-4806	156	23	1/2	1/2	NUM
ejpam-4806	156	24	(	(	PUNCT
ejpam-4806	156	25	6	6	NUM
ejpam-4806	156	26	)	)	PUNCT
ejpam-4806	156	27	for	for	ADP
ejpam-4806	156	28	all	all	PRON
ejpam-4806	156	29	n	n	CCONJ
ejpam-4806	156	30	>	>	X
ejpam-4806	156	31	6	6	NUM
ejpam-4806	156	32	.	.	PUNCT
ejpam-4806	157	1	we	we	PRON
ejpam-4806	157	2	now	now	ADV
ejpam-4806	157	3	consider	consider	VERB
ejpam-4806	157	4	µ(ga	µ(ga	NOUN
ejpam-4806	157	5	)	)	PUNCT
ejpam-4806	157	6	≃	≃	NOUN
ejpam-4806	157	7	(	(	PUNCT
ejpam-4806	157	8	2δ2	2δ2	NUM
ejpam-4806	157	9	+	+	CCONJ
ejpam-4806	157	10	2(2m−	2(2m−	NUM
ejpam-4806	157	11	ηδ	ηδ	NOUN
ejpam-4806	157	12	)	)	PUNCT
ejpam-4806	157	13	+	+	CCONJ
ejpam-4806	157	14	(	(	PUNCT
ejpam-4806	157	15	n+	n+	NUM
ejpam-4806	157	16	1)η	1)η	NUM
ejpam-4806	157	17	+	+	CCONJ
ejpam-4806	157	18	γ	γ	X
ejpam-4806	157	19	)	)	PUNCT
ejpam-4806	157	20	1/2	1/2	NUM
ejpam-4806	157	21	substituting	substitute	VERB
ejpam-4806	157	22	δ	δ	PROPN
ejpam-4806	157	23	=	=	SYM
ejpam-4806	157	24	2	2	NUM
ejpam-4806	157	25	,	,	PUNCT
ejpam-4806	157	26	m	m	VERB
ejpam-4806	157	27	=	=	PUNCT
ejpam-4806	157	28	(	(	PUNCT
ejpam-4806	157	29	2n−	2n−	PROPN
ejpam-4806	157	30	3	3	NUM
ejpam-4806	157	31	)	)	PUNCT
ejpam-4806	157	32	and	and	CCONJ
ejpam-4806	157	33	η	η	PROPN
ejpam-4806	157	34	=	=	SYM
ejpam-4806	157	35	(	(	PUNCT
ejpam-4806	157	36	n−	n−	NOUN
ejpam-4806	157	37	2	2	NUM
ejpam-4806	157	38	)	)	PUNCT
ejpam-4806	157	39	in	in	ADP
ejpam-4806	157	40	the	the	DET
ejpam-4806	157	41	above	above	ADJ
ejpam-4806	157	42	equation	equation	NOUN
ejpam-4806	157	43	,	,	PUNCT
ejpam-4806	157	44	we	we	PRON
ejpam-4806	157	45	get	get	VERB
ejpam-4806	157	46	µ(ga	µ(ga	NOUN
ejpam-4806	157	47	)	)	PUNCT
ejpam-4806	157	48	≃	≃	NOUN
ejpam-4806	157	49	(	(	PUNCT
ejpam-4806	157	50	8	8	NUM
ejpam-4806	157	51	+	+	SYM
ejpam-4806	157	52	2(2(2n−	2(2(2n−	NUM
ejpam-4806	157	53	3)−	3)−	NUM
ejpam-4806	157	54	2(n−	2(n−	NUM
ejpam-4806	157	55	2	2	NUM
ejpam-4806	157	56	)	)	PUNCT
ejpam-4806	157	57	)	)	PUNCT
ejpam-4806	158	1	+	+	CCONJ
ejpam-4806	158	2	(	(	PUNCT
ejpam-4806	158	3	n+	n+	NUM
ejpam-4806	158	4	1)(n−	1)(n−	NUM
ejpam-4806	158	5	2	2	NUM
ejpam-4806	158	6	)	)	PUNCT
ejpam-4806	158	7	+	+	CCONJ
ejpam-4806	158	8	γ)1/2	γ)1/2	PROPN
ejpam-4806	158	9	µ(ga	µ(ga	X
ejpam-4806	158	10	)	)	PUNCT
ejpam-4806	158	11	≃	≃	NOUN
ejpam-4806	158	12	(	(	PUNCT
ejpam-4806	158	13	n2	n2	NOUN
ejpam-4806	158	14	+	+	CCONJ
ejpam-4806	158	15	3n+	3n+	NUM
ejpam-4806	158	16	2	2	NUM
ejpam-4806	158	17	+	+	CCONJ
ejpam-4806	158	18	γ	γ	NOUN
ejpam-4806	158	19	)	)	PUNCT
ejpam-4806	158	20	1/2	1/2	NUM
ejpam-4806	158	21	for	for	ADP
ejpam-4806	158	22	n	n	NOUN
ejpam-4806	158	23	=	=	SYM
ejpam-4806	158	24	7	7	NUM
ejpam-4806	158	25	,	,	PUNCT
ejpam-4806	158	26	we	we	PRON
ejpam-4806	158	27	have	have	VERB
ejpam-4806	158	28	µ(ga	µ(ga	NOUN
ejpam-4806	158	29	)	)	PUNCT
ejpam-4806	158	30	=	=	PUNCT
ejpam-4806	159	1	8.5311	8.5311	NUM
ejpam-4806	159	2	=	=	SYM
ejpam-4806	159	3	(	(	PUNCT
ejpam-4806	159	4	72	72	NUM
ejpam-4806	159	5	+	+	CCONJ
ejpam-4806	159	6	γ)1/2	γ)1/2	PROPN
ejpam-4806	159	7	for	for	ADP
ejpam-4806	159	8	n	n	NOUN
ejpam-4806	159	9	=	=	SYM
ejpam-4806	159	10	8	8	NUM
ejpam-4806	159	11	,	,	PUNCT
ejpam-4806	159	12	µ(ga	µ(ga	NUM
ejpam-4806	159	13	)	)	PUNCT
ejpam-4806	159	14	=	=	SYM
ejpam-4806	159	15	9.5826	9.5826	NUM
ejpam-4806	159	16	=	=	SYM
ejpam-4806	159	17	(	(	PUNCT
ejpam-4806	159	18	90	90	NUM
ejpam-4806	159	19	+	+	CCONJ
ejpam-4806	159	20	γ)1/2	γ)1/2	PROPN
ejpam-4806	159	21	for	for	ADP
ejpam-4806	159	22	n	n	NOUN
ejpam-4806	159	23	=	=	SYM
ejpam-4806	159	24	9	9	NUM
ejpam-4806	159	25	,	,	PUNCT
ejpam-4806	159	26	µ(ga	µ(ga	NUM
ejpam-4806	159	27	)	)	PUNCT
ejpam-4806	159	28	=	=	SYM
ejpam-4806	160	1	10.6235	10.6235	NUM
ejpam-4806	160	2	=	=	SYM
ejpam-4806	160	3	(	(	PUNCT
ejpam-4806	160	4	110	110	NUM
ejpam-4806	160	5	+	+	CCONJ
ejpam-4806	160	6	γ)1/2	γ)1/2	PROPN
ejpam-4806	160	7	for	for	ADP
ejpam-4806	160	8	n	n	NOUN
ejpam-4806	160	9	=	=	SYM
ejpam-4806	160	10	10	10	NUM
ejpam-4806	160	11	,	,	PUNCT
ejpam-4806	160	12	µ(ga	µ(ga	NUM
ejpam-4806	160	13	)	)	PUNCT
ejpam-4806	160	14	=	=	SYM
ejpam-4806	160	15	11.6569	11.6569	NUM
ejpam-4806	160	16	=	=	SYM
ejpam-4806	160	17	(	(	PUNCT
ejpam-4806	160	18	132	132	NUM
ejpam-4806	160	19	+	+	CCONJ
ejpam-4806	160	20	γ)1/2	γ)1/2	PROPN
ejpam-4806	160	21	for	for	ADP
ejpam-4806	160	22	n	n	NOUN
ejpam-4806	160	23	=	=	SYM
ejpam-4806	160	24	11	11	NUM
ejpam-4806	160	25	,	,	PUNCT
ejpam-4806	160	26	µ(ga	µ(ga	NUM
ejpam-4806	160	27	)	)	PUNCT
ejpam-4806	160	28	=	=	SYM
ejpam-4806	161	1	12.6847	12.6847	NUM
ejpam-4806	161	2	=	=	SYM
ejpam-4806	161	3	(	(	PUNCT
ejpam-4806	161	4	156	156	NUM
ejpam-4806	161	5	+	+	CCONJ
ejpam-4806	161	6	γ)1/2	γ)1/2	PUNCT
ejpam-4806	161	7	by	by	ADP
ejpam-4806	161	8	observing	observe	VERB
ejpam-4806	161	9	the	the	DET
ejpam-4806	161	10	pattern	pattern	NOUN
ejpam-4806	162	1	,	,	PUNCT
ejpam-4806	162	2	we	we	PRON
ejpam-4806	162	3	find	find	VERB
ejpam-4806	162	4	that	that	SCONJ
ejpam-4806	162	5	γ	γ	PROPN
ejpam-4806	162	6	≃	≃	NOUN
ejpam-4806	162	7	(	(	PUNCT
ejpam-4806	162	8	n−	n−	NOUN
ejpam-4806	162	9	6.2	6.2	NUM
ejpam-4806	162	10	)	)	PUNCT
ejpam-4806	162	11	for	for	ADP
ejpam-4806	162	12	all	all	PRON
ejpam-4806	162	13	n	n	CCONJ
ejpam-4806	162	14	>	>	X
ejpam-4806	162	15	6	6	NUM
ejpam-4806	162	16	.	.	X
ejpam-4806	162	17	µ(ga	µ(ga	ADJ
ejpam-4806	162	18	)	)	PUNCT
ejpam-4806	162	19	≃	≃	NOUN
ejpam-4806	162	20	(	(	PUNCT
ejpam-4806	162	21	n2	n2	NOUN
ejpam-4806	162	22	+	+	CCONJ
ejpam-4806	162	23	3n+	3n+	NUM
ejpam-4806	162	24	2	2	NUM
ejpam-4806	162	25	+	+	CCONJ
ejpam-4806	162	26	(	(	PUNCT
ejpam-4806	162	27	n−	n−	NOUN
ejpam-4806	162	28	6.2	6.2	NUM
ejpam-4806	162	29	)	)	PUNCT
ejpam-4806	162	30	)	)	PUNCT
ejpam-4806	162	31	1/2	1/2	NUM
ejpam-4806	162	32	=	=	SYM
ejpam-4806	162	33	(	(	PUNCT
ejpam-4806	162	34	n2	n2	NOUN
ejpam-4806	162	35	+	+	CCONJ
ejpam-4806	163	1	4n−	4n−	NUM
ejpam-4806	163	2	4.2	4.2	NUM
ejpam-4806	163	3	)	)	PUNCT
ejpam-4806	163	4	1/2	1/2	NUM
ejpam-4806	163	5	for	for	ADP
ejpam-4806	163	6	n	n	PROPN
ejpam-4806	163	7	>	>	X
ejpam-4806	163	8	6	6	NUM
ejpam-4806	163	9	.	.	PUNCT
ejpam-4806	164	1	in	in	ADP
ejpam-4806	164	2	the	the	DET
ejpam-4806	164	3	following	following	NOUN
ejpam-4806	164	4	theorem	theorem	NOUN
ejpam-4806	164	5	,	,	PUNCT
ejpam-4806	164	6	we	we	PRON
ejpam-4806	164	7	determine	determine	VERB
ejpam-4806	164	8	the	the	DET
ejpam-4806	164	9	values	value	NOUN
ejpam-4806	164	10	of	of	ADP
ejpam-4806	164	11	κ	κ	NOUN
ejpam-4806	164	12	and	and	CCONJ
ejpam-4806	164	13	κ′	κ′	NOUN
ejpam-4806	164	14	in	in	ADP
ejpam-4806	164	15	order	order	NOUN
ejpam-4806	164	16	to	to	PART
ejpam-4806	164	17	obtain	obtain	VERB
ejpam-4806	164	18	the	the	DET
ejpam-4806	164	19	greatest	greatest	ADV
ejpam-4806	164	20	lower	lower	ADV
ejpam-4806	164	21	bound	bind	VERB
ejpam-4806	164	22	and	and	CCONJ
ejpam-4806	164	23	the	the	DET
ejpam-4806	164	24	least	least	ADJ
ejpam-4806	164	25	upper	upper	ADJ
ejpam-4806	164	26	bound	bind	VERB
ejpam-4806	164	27	for	for	ADP
ejpam-4806	164	28	µ	µ	X
ejpam-4806	164	29	(	(	PUNCT
ejpam-4806	164	30	ga	ga	PROPN
ejpam-4806	164	31	)	)	PUNCT
ejpam-4806	164	32	.	.	PUNCT
ejpam-4806	165	1	theorem	theorem	NOUN
ejpam-4806	165	2	2	2	NUM
ejpam-4806	165	3	.	.	X
ejpam-4806	165	4	for	for	ADP
ejpam-4806	165	5	the	the	DET
ejpam-4806	165	6	agave	agave	NOUN
ejpam-4806	165	7	graph	graph	PROPN
ejpam-4806	165	8	ga	ga	PROPN
ejpam-4806	165	9	,	,	PUNCT
ejpam-4806	165	10	µ	µ	X
ejpam-4806	165	11	(	(	PUNCT
ejpam-4806	165	12	ga	ga	PROPN
ejpam-4806	165	13	)	)	PUNCT
ejpam-4806	165	14	>	>	X
ejpam-4806	166	1	(	(	PUNCT
ejpam-4806	166	2	2δ2	2δ2	NUM
ejpam-4806	166	3	+	+	CCONJ
ejpam-4806	166	4	2(2m−	2(2m−	NUM
ejpam-4806	166	5	ηδ	ηδ	NOUN
ejpam-4806	166	6	)	)	PUNCT
ejpam-4806	166	7	+	+	CCONJ
ejpam-4806	166	8	(	(	PUNCT
ejpam-4806	166	9	n+	n+	NUM
ejpam-4806	166	10	2)η	2)η	NUM
ejpam-4806	166	11	−	−	PROPN
ejpam-4806	166	12	κ	κ	NOUN
ejpam-4806	166	13	)	)	PUNCT
ejpam-4806	166	14	1/2	1/2	NUM
ejpam-4806	166	15	(	(	PUNCT
ejpam-4806	166	16	7	7	NUM
ejpam-4806	166	17	)	)	PUNCT
ejpam-4806	166	18	where	where	SCONJ
ejpam-4806	166	19	5	5	NUM
ejpam-4806	166	20	≤	≤	NOUN
ejpam-4806	166	21	κ	κ	NOUN
ejpam-4806	166	22	≤	≤	NOUN
ejpam-4806	166	23	(	(	PUNCT
ejpam-4806	166	24	n2	n2	ADJ
ejpam-4806	166	25	−	−	PROPN
ejpam-4806	166	26	4	4	NUM
ejpam-4806	166	27	)	)	PUNCT
ejpam-4806	166	28	and	and	CCONJ
ejpam-4806	166	29	the	the	DET
ejpam-4806	166	30	greatest	greatest	ADV
ejpam-4806	166	31	lower	low	ADJ
ejpam-4806	166	32	bound	bind	VERB
ejpam-4806	166	33	is	be	AUX
ejpam-4806	166	34	attained	attain	VERB
ejpam-4806	166	35	when	when	SCONJ
ejpam-4806	166	36	κ	κ	X
ejpam-4806	166	37	=	=	SYM
ejpam-4806	166	38	5	5	NUM
ejpam-4806	166	39	also	also	ADV
ejpam-4806	166	40	µ	µ	X
ejpam-4806	166	41	(	(	PUNCT
ejpam-4806	166	42	ga	ga	PROPN
ejpam-4806	166	43	)	)	PUNCT
ejpam-4806	166	44	<	<	X
ejpam-4806	167	1	(	(	PUNCT
ejpam-4806	167	2	2∆2	2∆2	NUM
ejpam-4806	167	3	+	+	NUM
ejpam-4806	167	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	167	5	nη	nη	NOUN
ejpam-4806	167	6	+	+	NUM
ejpam-4806	167	7	κ′	κ′	NOUN
ejpam-4806	167	8	)	)	PUNCT
ejpam-4806	167	9	1/2	1/2	NUM
ejpam-4806	167	10	(	(	PUNCT
ejpam-4806	167	11	8)	8)	NUM
ejpam-4806	167	12	where	where	SCONJ
ejpam-4806	167	13	4	4	NUM
ejpam-4806	167	14	≤	≤	NUM
ejpam-4806	167	15	κ′	κ′	NOUN
ejpam-4806	167	16	≤	≤	X
ejpam-4806	167	17	n(n−	n(n−	PROPN
ejpam-4806	167	18	2	2	NUM
ejpam-4806	167	19	)	)	PUNCT
ejpam-4806	167	20	and	and	CCONJ
ejpam-4806	167	21	the	the	DET
ejpam-4806	167	22	least	least	ADJ
ejpam-4806	167	23	upper	upper	ADJ
ejpam-4806	167	24	bound	bind	VERB
ejpam-4806	167	25	is	be	AUX
ejpam-4806	167	26	attained	attain	VERB
ejpam-4806	167	27	when	when	SCONJ
ejpam-4806	167	28	κ′	κ′	NOUN
ejpam-4806	167	29	=	=	SYM
ejpam-4806	167	30	4	4	X
ejpam-4806	167	31	.	.	NUM
ejpam-4806	167	32	malathy	malathy	NUM
ejpam-4806	167	33	v	v	NOUN
ejpam-4806	167	34	,	,	PUNCT
ejpam-4806	167	35	kalyani	kalyani	PROPN
ejpam-4806	167	36	desikan	desikan	PROPN
ejpam-4806	167	37	/	/	SYM
ejpam-4806	167	38	eur	eur	PROPN
ejpam-4806	167	39	.	.	PUNCT
ejpam-4806	168	1	j.	j.	PROPN
ejpam-4806	168	2	pure	pure	PROPN
ejpam-4806	168	3	appl	appl	PROPN
ejpam-4806	168	4	.	.	PROPN
ejpam-4806	168	5	math	math	PROPN
ejpam-4806	168	6	,	,	PUNCT
ejpam-4806	168	7	16	16	NUM
ejpam-4806	168	8	(	(	PUNCT
ejpam-4806	168	9	3	3	NUM
ejpam-4806	168	10	)	)	PUNCT
ejpam-4806	168	11	(	(	PUNCT
ejpam-4806	168	12	2023	2023	NUM
ejpam-4806	168	13	)	)	PUNCT
ejpam-4806	168	14	,	,	PUNCT
ejpam-4806	168	15	1731	1731	NUM
ejpam-4806	168	16	-	-	SYM
ejpam-4806	168	17	1746	1746	NUM
ejpam-4806	168	18	1739	1739	NUM
ejpam-4806	168	19	proof	proof	NOUN
ejpam-4806	168	20	.	.	PUNCT
ejpam-4806	169	1	consider	consider	VERB
ejpam-4806	169	2	lemma	lemma	PROPN
ejpam-4806	169	3	7	7	NUM
ejpam-4806	169	4	,	,	PUNCT
ejpam-4806	169	5	µ(ga	µ(ga	NUM
ejpam-4806	169	6	)	)	PUNCT
ejpam-4806	169	7	<	<	X
ejpam-4806	169	8	(	(	PUNCT
ejpam-4806	169	9	2δ2	2δ2	NUM
ejpam-4806	169	10	+	+	CCONJ
ejpam-4806	169	11	2(2m−	2(2m−	NUM
ejpam-4806	169	12	ηδ	ηδ	NOUN
ejpam-4806	169	13	)	)	PUNCT
ejpam-4806	170	1	+	+	CCONJ
ejpam-4806	170	2	(	(	PUNCT
ejpam-4806	170	3	n+	n+	NUM
ejpam-4806	170	4	2)η	2)η	ADJ
ejpam-4806	170	5	)	)	PUNCT
ejpam-4806	170	6	1/2	1/2	NUM
ejpam-4806	170	7	reducing	reduce	VERB
ejpam-4806	170	8	this	this	DET
ejpam-4806	170	9	expression	expression	NOUN
ejpam-4806	170	10	in	in	ADP
ejpam-4806	170	11	terms	term	NOUN
ejpam-4806	170	12	of	of	ADP
ejpam-4806	170	13	n	n	CCONJ
ejpam-4806	170	14	,	,	PUNCT
ejpam-4806	170	15	we	we	PRON
ejpam-4806	170	16	have	have	AUX
ejpam-4806	170	17	µ(ga	µ(ga	NOUN
ejpam-4806	170	18	)	)	PUNCT
ejpam-4806	170	19	≃	≃	NOUN
ejpam-4806	170	20	(	(	PUNCT
ejpam-4806	170	21	n2	n2	NOUN
ejpam-4806	170	22	+	+	X
ejpam-4806	171	1	4n−	4n−	NUM
ejpam-4806	171	2	4.2	4.2	NUM
ejpam-4806	171	3	)	)	PUNCT
ejpam-4806	171	4	1/2	1/2	NUM
ejpam-4806	171	5	<	<	X
ejpam-4806	171	6	(	(	PUNCT
ejpam-4806	171	7	n2	n2	NOUN
ejpam-4806	171	8	+	+	CCONJ
ejpam-4806	171	9	4n	4n	NOUN
ejpam-4806	171	10	)	)	PUNCT
ejpam-4806	171	11	1/2	1/2	NUM
ejpam-4806	171	12	where	where	SCONJ
ejpam-4806	171	13	this	this	PRON
ejpam-4806	171	14	is	be	AUX
ejpam-4806	171	15	true	true	ADJ
ejpam-4806	171	16	for	for	ADP
ejpam-4806	171	17	all	all	PRON
ejpam-4806	171	18	n	n	CCONJ
ejpam-4806	171	19	>	>	X
ejpam-4806	171	20	6	6	NUM
ejpam-4806	171	21	.	.	PUNCT
ejpam-4806	172	1	we	we	PRON
ejpam-4806	172	2	determine	determine	VERB
ejpam-4806	172	3	a	a	DET
ejpam-4806	172	4	positive	positive	ADJ
ejpam-4806	172	5	number	number	NOUN
ejpam-4806	172	6	κ	κ	ADP
ejpam-4806	172	7	such	such	ADJ
ejpam-4806	172	8	that	that	SCONJ
ejpam-4806	172	9	µ(ga	µ(ga	ADJ
ejpam-4806	172	10	)	)	PUNCT
ejpam-4806	172	11	≃	≃	NOUN
ejpam-4806	172	12	(	(	PUNCT
ejpam-4806	172	13	n2	n2	NOUN
ejpam-4806	172	14	+	+	X
ejpam-4806	173	1	4n−	4n−	NUM
ejpam-4806	173	2	4.2	4.2	NUM
ejpam-4806	173	3	)	)	PUNCT
ejpam-4806	173	4	1/2	1/2	NUM
ejpam-4806	173	5	>	>	PUNCT
ejpam-4806	174	1	(	(	PUNCT
ejpam-4806	174	2	2δ2	2δ2	NUM
ejpam-4806	174	3	+	+	CCONJ
ejpam-4806	174	4	2(2m−	2(2m−	NUM
ejpam-4806	174	5	ηδ	ηδ	NOUN
ejpam-4806	174	6	)	)	PUNCT
ejpam-4806	175	1	+	+	CCONJ
ejpam-4806	175	2	(	(	PUNCT
ejpam-4806	175	3	n+	n+	NUM
ejpam-4806	175	4	2)η	2)η	NUM
ejpam-4806	175	5	−	−	PROPN
ejpam-4806	175	6	κ	κ	NOUN
ejpam-4806	175	7	)	)	PUNCT
ejpam-4806	175	8	1/2	1/2	NUM
ejpam-4806	175	9	from	from	ADP
ejpam-4806	175	10	the	the	DET
ejpam-4806	175	11	above	above	ADJ
ejpam-4806	175	12	inequality	inequality	NOUN
ejpam-4806	175	13	,	,	PUNCT
ejpam-4806	175	14	we	we	PRON
ejpam-4806	175	15	have	have	AUX
ejpam-4806	175	16	(	(	PUNCT
ejpam-4806	175	17	n2	n2	ADJ
ejpam-4806	175	18	+	+	CCONJ
ejpam-4806	176	1	4n−	4n−	NUM
ejpam-4806	176	2	4.2	4.2	NUM
ejpam-4806	176	3	)	)	PUNCT
ejpam-4806	176	4	1/2	1/2	NUM
ejpam-4806	176	5	>	>	PUNCT
ejpam-4806	176	6	(	(	PUNCT
ejpam-4806	176	7	n2	n2	PROPN
ejpam-4806	176	8	+	+	CCONJ
ejpam-4806	176	9	4n−	4n−	PROPN
ejpam-4806	176	10	κ	κ	NOUN
ejpam-4806	176	11	)	)	PUNCT
ejpam-4806	176	12	1/2	1/2	NUM
ejpam-4806	176	13	this	this	PRON
ejpam-4806	176	14	implies	imply	VERB
ejpam-4806	176	15	that	that	SCONJ
ejpam-4806	176	16	κ	κ	VERB
ejpam-4806	176	17	>	>	X
ejpam-4806	176	18	4.2	4.2	NUM
ejpam-4806	176	19	⇒	⇒	NOUN
ejpam-4806	176	20	κ	κ	X
ejpam-4806	177	1	=	=	SYM
ejpam-4806	177	2	5	5	NUM
ejpam-4806	177	3	we	we	PRON
ejpam-4806	177	4	can	can	AUX
ejpam-4806	177	5	easily	easily	ADV
ejpam-4806	177	6	verify	verify	VERB
ejpam-4806	177	7	that	that	SCONJ
ejpam-4806	177	8	the	the	DET
ejpam-4806	177	9	expression	expression	NOUN
ejpam-4806	177	10	(	(	PUNCT
ejpam-4806	177	11	2δ2	2δ2	NUM
ejpam-4806	177	12	+	+	CCONJ
ejpam-4806	177	13	2(2m−	2(2m−	NUM
ejpam-4806	177	14	ηδ	ηδ	NOUN
ejpam-4806	177	15	)	)	PUNCT
ejpam-4806	177	16	+	+	CCONJ
ejpam-4806	177	17	(	(	PUNCT
ejpam-4806	177	18	n+	n+	NUM
ejpam-4806	177	19	2)η	2)η	ADJ
ejpam-4806	177	20	−	−	NOUN
ejpam-4806	177	21	5	5	NUM
ejpam-4806	177	22	)	)	PUNCT
ejpam-4806	177	23	1/2	1/2	NUM
ejpam-4806	177	24	is	be	AUX
ejpam-4806	177	25	the	the	DET
ejpam-4806	177	26	greatest	greatest	ADV
ejpam-4806	177	27	lower	lower	ADV
ejpam-4806	177	28	bound	bind	VERB
ejpam-4806	177	29	.	.	PUNCT
ejpam-4806	178	1	consider	consider	VERB
ejpam-4806	178	2	µ(ga	µ(ga	NOUN
ejpam-4806	178	3	)	)	PUNCT
ejpam-4806	178	4	≃	≃	NOUN
ejpam-4806	178	5	(	(	PUNCT
ejpam-4806	178	6	2δ2	2δ2	NUM
ejpam-4806	178	7	+	+	CCONJ
ejpam-4806	178	8	2(2m−	2(2m−	NUM
ejpam-4806	178	9	ηδ	ηδ	NOUN
ejpam-4806	178	10	)	)	PUNCT
ejpam-4806	178	11	+	+	CCONJ
ejpam-4806	178	12	(	(	PUNCT
ejpam-4806	178	13	n+	n+	NUM
ejpam-4806	178	14	2)η	2)η	ADJ
ejpam-4806	178	15	−	−	NUM
ejpam-4806	178	16	5	5	NUM
ejpam-4806	178	17	+	+	CCONJ
ejpam-4806	178	18	1	1	NUM
ejpam-4806	178	19	)	)	PUNCT
ejpam-4806	178	20	1/2	1/2	NUM
ejpam-4806	178	21	to	to	PART
ejpam-4806	178	22	be	be	AUX
ejpam-4806	178	23	the	the	DET
ejpam-4806	178	24	greatest	greatest	ADV
ejpam-4806	178	25	lower	low	ADJ
ejpam-4806	178	26	bound	bind	VERB
ejpam-4806	178	27	of	of	ADP
ejpam-4806	178	28	µ(ga	µ(ga	NOUN
ejpam-4806	178	29	)	)	PUNCT
ejpam-4806	178	30	.	.	PUNCT
ejpam-4806	179	1	that	that	PRON
ejpam-4806	179	2	is	be	AUX
ejpam-4806	179	3	µ(ga	µ(ga	NOUN
ejpam-4806	179	4	)	)	PUNCT
ejpam-4806	179	5	≃	≃	NOUN
ejpam-4806	179	6	(	(	PUNCT
ejpam-4806	179	7	n2	n2	NOUN
ejpam-4806	179	8	+	+	X
ejpam-4806	180	1	4n−	4n−	NUM
ejpam-4806	180	2	4.2	4.2	NUM
ejpam-4806	180	3	)	)	PUNCT
ejpam-4806	180	4	1/2	1/2	NUM
ejpam-4806	180	5	<	<	X
ejpam-4806	180	6	(	(	PUNCT
ejpam-4806	180	7	n2	n2	NOUN
ejpam-4806	180	8	+	+	X
ejpam-4806	181	1	4n−	4n−	NUM
ejpam-4806	181	2	4	4	NUM
ejpam-4806	181	3	)	)	PUNCT
ejpam-4806	181	4	1/2	1/2	NUM
ejpam-4806	181	5	.	.	PUNCT
ejpam-4806	182	1	this	this	PRON
ejpam-4806	182	2	is	be	AUX
ejpam-4806	182	3	a	a	DET
ejpam-4806	182	4	contradiction	contradiction	NOUN
ejpam-4806	182	5	for	for	ADP
ejpam-4806	182	6	all	all	PRON
ejpam-4806	182	7	n	n	CCONJ
ejpam-4806	182	8	>	>	X
ejpam-4806	182	9	6	6	NUM
ejpam-4806	182	10	.	.	PUNCT
ejpam-4806	183	1	hence	hence	ADV
ejpam-4806	183	2	(	(	PUNCT
ejpam-4806	183	3	2δ2	2δ2	NUM
ejpam-4806	183	4	+	+	CCONJ
ejpam-4806	183	5	2(2m−	2(2m−	NUM
ejpam-4806	183	6	ηδ	ηδ	NOUN
ejpam-4806	183	7	)	)	PUNCT
ejpam-4806	183	8	+	+	CCONJ
ejpam-4806	183	9	(	(	PUNCT
ejpam-4806	183	10	n+	n+	NUM
ejpam-4806	183	11	2)η	2)η	ADJ
ejpam-4806	183	12	−	−	NOUN
ejpam-4806	183	13	5	5	NUM
ejpam-4806	183	14	)	)	PUNCT
ejpam-4806	183	15	1/2	1/2	NUM
ejpam-4806	183	16	is	be	AUX
ejpam-4806	183	17	the	the	DET
ejpam-4806	183	18	greatest	greatest	ADV
ejpam-4806	183	19	lower	lower	ADV
ejpam-4806	183	20	bound	bind	VERB
ejpam-4806	183	21	.	.	PUNCT
ejpam-4806	184	1	we	we	PRON
ejpam-4806	184	2	observe	observe	VERB
ejpam-4806	184	3	that	that	SCONJ
ejpam-4806	184	4	the	the	DET
ejpam-4806	184	5	greatest	greatest	ADV
ejpam-4806	184	6	lower	lower	ADV
ejpam-4806	184	7	bound	bind	VERB
ejpam-4806	184	8	occurs	occur	VERB
ejpam-4806	184	9	when	when	SCONJ
ejpam-4806	184	10	κ	κ	NOUN
ejpam-4806	184	11	=	=	SYM
ejpam-4806	184	12	5	5	NUM
ejpam-4806	184	13	and	and	CCONJ
ejpam-4806	184	14	the	the	DET
ejpam-4806	184	15	loose	loose	ADJ
ejpam-4806	184	16	lower	lower	ADV
ejpam-4806	184	17	bound	bind	VERB
ejpam-4806	184	18	is	be	AUX
ejpam-4806	184	19	attained	attain	VERB
ejpam-4806	184	20	when	when	SCONJ
ejpam-4806	184	21	κ	κ	PROPN
ejpam-4806	184	22	=	=	SYM
ejpam-4806	184	23	(	(	PUNCT
ejpam-4806	184	24	n2	n2	ADJ
ejpam-4806	184	25	−	−	PROPN
ejpam-4806	184	26	4	4	NUM
ejpam-4806	184	27	)	)	PUNCT
ejpam-4806	184	28	.	.	PUNCT
ejpam-4806	185	1	similarly	similarly	ADV
ejpam-4806	185	2	,	,	PUNCT
ejpam-4806	185	3	in	in	ADP
ejpam-4806	185	4	order	order	NOUN
ejpam-4806	185	5	to	to	PART
ejpam-4806	185	6	determine	determine	VERB
ejpam-4806	185	7	the	the	DET
ejpam-4806	185	8	least	least	ADJ
ejpam-4806	185	9	upper	upper	ADJ
ejpam-4806	185	10	bound	bind	VERB
ejpam-4806	185	11	for	for	ADP
ejpam-4806	185	12	µ(ga	µ(ga	NOUN
ejpam-4806	185	13	)	)	PUNCT
ejpam-4806	185	14	,	,	PUNCT
ejpam-4806	185	15	we	we	PRON
ejpam-4806	185	16	consider	consider	VERB
ejpam-4806	185	17	the	the	DET
ejpam-4806	185	18	upper	upper	ADJ
ejpam-4806	185	19	bound	bind	VERB
ejpam-4806	185	20	given	give	VERB
ejpam-4806	185	21	in	in	ADP
ejpam-4806	185	22	equation	equation	NOUN
ejpam-4806	185	23	(	(	PUNCT
ejpam-4806	185	24	3	3	NUM
ejpam-4806	185	25	)	)	PUNCT
ejpam-4806	185	26	of	of	ADP
ejpam-4806	185	27	theorem	theorem	NOUN
ejpam-4806	185	28	1	1	X
ejpam-4806	185	29	.	.	X
ejpam-4806	185	30	consider	consider	VERB
ejpam-4806	185	31	lemma	lemma	PROPN
ejpam-4806	185	32	6	6	NUM
ejpam-4806	185	33	,	,	PUNCT
ejpam-4806	185	34	µ(ga	µ(ga	NUM
ejpam-4806	185	35	)	)	PUNCT
ejpam-4806	185	36	>	>	X
ejpam-4806	186	1	(	(	PUNCT
ejpam-4806	186	2	2∆2	2∆2	NUM
ejpam-4806	186	3	+	+	CCONJ
ejpam-4806	186	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	186	5	nη	nη	X
ejpam-4806	186	6	)	)	PUNCT
ejpam-4806	186	7	1/2	1/2	NUM
ejpam-4806	186	8	reducing	reduce	VERB
ejpam-4806	186	9	the	the	DET
ejpam-4806	186	10	above	above	ADJ
ejpam-4806	186	11	expression	expression	NOUN
ejpam-4806	186	12	in	in	ADP
ejpam-4806	186	13	terms	term	NOUN
ejpam-4806	186	14	of	of	ADP
ejpam-4806	186	15	n	n	CCONJ
ejpam-4806	186	16	,	,	PUNCT
ejpam-4806	186	17	we	we	PRON
ejpam-4806	186	18	have	have	VERB
ejpam-4806	186	19	µ(ga	µ(ga	NOUN
ejpam-4806	186	20	)	)	PUNCT
ejpam-4806	186	21	≃	≃	NOUN
ejpam-4806	186	22	(	(	PUNCT
ejpam-4806	186	23	n2	n2	NOUN
ejpam-4806	186	24	+	+	X
ejpam-4806	187	1	4n−	4n−	NUM
ejpam-4806	187	2	4.2	4.2	NUM
ejpam-4806	187	3	)	)	PUNCT
ejpam-4806	187	4	1/2	1/2	NUM
ejpam-4806	187	5	>	>	PUNCT
ejpam-4806	187	6	(	(	PUNCT
ejpam-4806	187	7	n2	n2	NOUN
ejpam-4806	187	8	+	+	CCONJ
ejpam-4806	188	1	4n−	4n−	NUM
ejpam-4806	188	2	8	8	NUM
ejpam-4806	188	3	)	)	PUNCT
ejpam-4806	188	4	1/2	1/2	NUM
ejpam-4806	188	5	in	in	ADP
ejpam-4806	188	6	order	order	NOUN
ejpam-4806	188	7	to	to	PART
ejpam-4806	188	8	determine	determine	VERB
ejpam-4806	188	9	the	the	DET
ejpam-4806	188	10	least	least	ADJ
ejpam-4806	188	11	upper	upper	ADJ
ejpam-4806	188	12	bound	bind	VERB
ejpam-4806	188	13	,	,	PUNCT
ejpam-4806	188	14	we	we	PRON
ejpam-4806	188	15	need	need	VERB
ejpam-4806	188	16	to	to	PART
ejpam-4806	188	17	find	find	VERB
ejpam-4806	188	18	the	the	DET
ejpam-4806	188	19	value	value	NOUN
ejpam-4806	188	20	κ′	κ′	NOUN
ejpam-4806	188	21	such	such	ADJ
ejpam-4806	188	22	that	that	SCONJ
ejpam-4806	188	23	µ(ga	µ(ga	PROPN
ejpam-4806	188	24	)	)	PUNCT
ejpam-4806	188	25	<	<	X
ejpam-4806	189	1	(	(	PUNCT
ejpam-4806	189	2	2∆2	2∆2	NUM
ejpam-4806	189	3	+	+	NUM
ejpam-4806	189	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	189	5	nη	nη	NOUN
ejpam-4806	189	6	+	+	NUM
ejpam-4806	189	7	κ′	κ′	NOUN
ejpam-4806	189	8	)	)	PUNCT
ejpam-4806	189	9	1/2	1/2	NUM
ejpam-4806	189	10	=	=	SYM
ejpam-4806	189	11	(	(	PUNCT
ejpam-4806	189	12	n2	n2	NOUN
ejpam-4806	189	13	+	+	CCONJ
ejpam-4806	190	1	4n−	4n−	NUM
ejpam-4806	190	2	8	8	NUM
ejpam-4806	190	3	+	+	NUM
ejpam-4806	190	4	κ′	κ′	NOUN
ejpam-4806	190	5	)	)	PUNCT
ejpam-4806	190	6	1/2	1/2	NUM
ejpam-4806	190	7	malathy	malathy	NOUN
ejpam-4806	190	8	v	v	NOUN
ejpam-4806	190	9	,	,	PUNCT
ejpam-4806	190	10	kalyani	kalyani	PROPN
ejpam-4806	190	11	desikan	desikan	PROPN
ejpam-4806	190	12	/	/	SYM
ejpam-4806	190	13	eur	eur	PROPN
ejpam-4806	190	14	.	.	PUNCT
ejpam-4806	191	1	j.	j.	PROPN
ejpam-4806	191	2	pure	pure	PROPN
ejpam-4806	191	3	appl	appl	PROPN
ejpam-4806	191	4	.	.	PROPN
ejpam-4806	191	5	math	math	PROPN
ejpam-4806	191	6	,	,	PUNCT
ejpam-4806	191	7	16	16	NUM
ejpam-4806	191	8	(	(	PUNCT
ejpam-4806	191	9	3	3	NUM
ejpam-4806	191	10	)	)	PUNCT
ejpam-4806	191	11	(	(	PUNCT
ejpam-4806	191	12	2023	2023	NUM
ejpam-4806	191	13	)	)	PUNCT
ejpam-4806	191	14	,	,	PUNCT
ejpam-4806	191	15	1731	1731	NUM
ejpam-4806	191	16	-	-	SYM
ejpam-4806	191	17	1746	1746	NUM
ejpam-4806	191	18	1740	1740	NUM
ejpam-4806	191	19	µ(ga	µ(ga	ADJ
ejpam-4806	191	20	)	)	PUNCT
ejpam-4806	191	21	≃	≃	NOUN
ejpam-4806	191	22	(	(	PUNCT
ejpam-4806	191	23	n2	n2	NOUN
ejpam-4806	191	24	+	+	X
ejpam-4806	192	1	4n−	4n−	NUM
ejpam-4806	192	2	4.2	4.2	NUM
ejpam-4806	192	3	)	)	PUNCT
ejpam-4806	192	4	1/2	1/2	NUM
ejpam-4806	192	5	<	<	X
ejpam-4806	192	6	(	(	PUNCT
ejpam-4806	192	7	n2	n2	NOUN
ejpam-4806	192	8	+	+	CCONJ
ejpam-4806	193	1	4n−	4n−	NUM
ejpam-4806	193	2	8	8	NUM
ejpam-4806	193	3	+	+	NUM
ejpam-4806	193	4	κ′	κ′	NOUN
ejpam-4806	193	5	)	)	PUNCT
ejpam-4806	193	6	1/2	1/2	NUM
ejpam-4806	193	7	this	this	PRON
ejpam-4806	193	8	implies	imply	VERB
ejpam-4806	193	9	that	that	SCONJ
ejpam-4806	193	10	κ′	κ′	NOUN
ejpam-4806	193	11	>	>	X
ejpam-4806	193	12	3.8	3.8	NUM
ejpam-4806	193	13	⇒	⇒	X
ejpam-4806	193	14	κ′	κ′	NOUN
ejpam-4806	193	15	=	=	SYM
ejpam-4806	193	16	4	4	NUM
ejpam-4806	193	17	therefore	therefore	ADV
ejpam-4806	193	18	µ(ga	µ(ga	NOUN
ejpam-4806	193	19	)	)	PUNCT
ejpam-4806	193	20	<	<	X
ejpam-4806	193	21	(	(	PUNCT
ejpam-4806	193	22	2∆2	2∆2	NUM
ejpam-4806	193	23	+	+	NUM
ejpam-4806	193	24	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	193	25	nη	nη	NOUN
ejpam-4806	193	26	+	+	CCONJ
ejpam-4806	193	27	4	4	NUM
ejpam-4806	193	28	)	)	SYM
ejpam-4806	193	29	1/2	1/2	NUM
ejpam-4806	193	30	it	it	PRON
ejpam-4806	193	31	can	can	AUX
ejpam-4806	193	32	easily	easily	ADV
ejpam-4806	193	33	be	be	AUX
ejpam-4806	193	34	shown	show	VERB
ejpam-4806	193	35	that	that	SCONJ
ejpam-4806	193	36	the	the	DET
ejpam-4806	193	37	above	above	ADJ
ejpam-4806	193	38	expression	expression	NOUN
ejpam-4806	193	39	is	be	AUX
ejpam-4806	193	40	the	the	DET
ejpam-4806	193	41	least	least	ADJ
ejpam-4806	193	42	upper	upper	ADJ
ejpam-4806	193	43	bound	bind	VERB
ejpam-4806	193	44	.	.	PUNCT
ejpam-4806	194	1	consider	consider	VERB
ejpam-4806	194	2	µ(ga	µ(ga	NOUN
ejpam-4806	194	3	)	)	PUNCT
ejpam-4806	194	4	<	<	X
ejpam-4806	195	1	(	(	PUNCT
ejpam-4806	195	2	2∆2	2∆2	NUM
ejpam-4806	195	3	+	+	NUM
ejpam-4806	195	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	195	5	nη	nη	NOUN
ejpam-4806	195	6	+	+	CCONJ
ejpam-4806	195	7	3	3	NUM
ejpam-4806	195	8	)	)	SYM
ejpam-4806	195	9	1/2	1/2	NUM
ejpam-4806	195	10	=	=	SYM
ejpam-4806	195	11	(	(	PUNCT
ejpam-4806	195	12	n2	n2	NOUN
ejpam-4806	195	13	+	+	CCONJ
ejpam-4806	196	1	4n−	4n−	NUM
ejpam-4806	196	2	5	5	NUM
ejpam-4806	196	3	)	)	PUNCT
ejpam-4806	196	4	1/2	1/2	NUM
ejpam-4806	196	5	to	to	PART
ejpam-4806	196	6	be	be	AUX
ejpam-4806	196	7	the	the	DET
ejpam-4806	196	8	least	least	ADJ
ejpam-4806	196	9	upper	upper	ADJ
ejpam-4806	196	10	bound	bind	VERB
ejpam-4806	196	11	.	.	PUNCT
ejpam-4806	197	1	from	from	ADP
ejpam-4806	197	2	this	this	PRON
ejpam-4806	197	3	,	,	PUNCT
ejpam-4806	197	4	we	we	PRON
ejpam-4806	197	5	notice	notice	VERB
ejpam-4806	197	6	that	that	SCONJ
ejpam-4806	197	7	µ(ga	µ(ga	ADJ
ejpam-4806	197	8	)	)	PUNCT
ejpam-4806	197	9	≃	≃	NOUN
ejpam-4806	197	10	(	(	PUNCT
ejpam-4806	197	11	n2	n2	NOUN
ejpam-4806	197	12	+	+	X
ejpam-4806	198	1	4n−	4n−	NUM
ejpam-4806	198	2	4.2	4.2	NUM
ejpam-4806	198	3	)	)	PUNCT
ejpam-4806	198	4	1/2	1/2	NUM
ejpam-4806	198	5	<	<	X
ejpam-4806	198	6	(	(	PUNCT
ejpam-4806	198	7	n2	n2	NOUN
ejpam-4806	198	8	+	+	X
ejpam-4806	199	1	4n−	4n−	NUM
ejpam-4806	199	2	5	5	NUM
ejpam-4806	199	3	)	)	PUNCT
ejpam-4806	199	4	1/2	1/2	NUM
ejpam-4806	199	5	this	this	PRON
ejpam-4806	199	6	is	be	AUX
ejpam-4806	199	7	a	a	DET
ejpam-4806	199	8	contradiction	contradiction	NOUN
ejpam-4806	199	9	.	.	PUNCT
ejpam-4806	200	1	therefore	therefore	ADV
ejpam-4806	200	2	µ(ga	µ(ga	NOUN
ejpam-4806	200	3	)	)	PUNCT
ejpam-4806	200	4	<	<	X
ejpam-4806	201	1	(	(	PUNCT
ejpam-4806	201	2	2∆2	2∆2	NUM
ejpam-4806	201	3	+	+	NUM
ejpam-4806	201	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	201	5	nη	nη	NOUN
ejpam-4806	201	6	+	+	CCONJ
ejpam-4806	201	7	4	4	NUM
ejpam-4806	201	8	)	)	PUNCT
ejpam-4806	201	9	1/2	1/2	NUM
ejpam-4806	201	10	.	.	PUNCT
ejpam-4806	202	1	(	(	PUNCT
ejpam-4806	202	2	9	9	X
ejpam-4806	202	3	)	)	PUNCT
ejpam-4806	202	4	is	be	AUX
ejpam-4806	202	5	the	the	DET
ejpam-4806	202	6	least	least	ADJ
ejpam-4806	202	7	upper	upper	ADJ
ejpam-4806	202	8	bound	bind	VERB
ejpam-4806	202	9	.	.	PUNCT
ejpam-4806	203	1	clearly	clearly	ADV
ejpam-4806	203	2	,	,	PUNCT
ejpam-4806	203	3	we	we	PRON
ejpam-4806	203	4	observe	observe	VERB
ejpam-4806	203	5	that	that	SCONJ
ejpam-4806	203	6	the	the	DET
ejpam-4806	203	7	least	least	ADV
ejpam-4806	203	8	upper	upper	ADJ
ejpam-4806	203	9	bound	bind	VERB
ejpam-4806	203	10	occurs	occur	VERB
ejpam-4806	203	11	when	when	SCONJ
ejpam-4806	203	12	κ′	κ′	NOUN
ejpam-4806	203	13	=	=	SYM
ejpam-4806	203	14	4	4	NUM
ejpam-4806	203	15	and	and	CCONJ
ejpam-4806	203	16	the	the	DET
ejpam-4806	203	17	loose	loose	ADJ
ejpam-4806	203	18	upper	upper	ADJ
ejpam-4806	203	19	bound	bind	VERB
ejpam-4806	203	20	is	be	AUX
ejpam-4806	203	21	attained	attain	VERB
ejpam-4806	203	22	when	when	SCONJ
ejpam-4806	203	23	κ′	κ′	PROPN
ejpam-4806	203	24	=	=	SYM
ejpam-4806	203	25	n(n−	n(n−	VERB
ejpam-4806	203	26	2	2	NUM
ejpam-4806	203	27	)	)	PUNCT
ejpam-4806	203	28	.	.	PUNCT
ejpam-4806	204	1	we	we	PRON
ejpam-4806	204	2	prove	prove	VERB
ejpam-4806	204	3	that	that	SCONJ
ejpam-4806	204	4	the	the	DET
ejpam-4806	204	5	lower	lower	ADV
ejpam-4806	204	6	bound	bind	VERB
ejpam-4806	204	7	and	and	CCONJ
ejpam-4806	204	8	the	the	DET
ejpam-4806	204	9	upper	upper	ADJ
ejpam-4806	204	10	bound	bind	VERB
ejpam-4806	204	11	are	be	AUX
ejpam-4806	204	12	the	the	DET
ejpam-4806	204	13	greatest	greatest	ADV
ejpam-4806	204	14	lower	lower	ADV
ejpam-4806	204	15	bound	bind	VERB
ejpam-4806	204	16	and	and	CCONJ
ejpam-4806	204	17	the	the	DET
ejpam-4806	204	18	least	least	ADJ
ejpam-4806	204	19	upper	upper	ADJ
ejpam-4806	204	20	bound	bind	VERB
ejpam-4806	204	21	for	for	ADP
ejpam-4806	204	22	µ(ga	µ(ga	NOUN
ejpam-4806	204	23	)	)	PUNCT
ejpam-4806	204	24	by	by	ADP
ejpam-4806	204	25	making	make	VERB
ejpam-4806	204	26	use	use	NOUN
ejpam-4806	204	27	of	of	ADP
ejpam-4806	204	28	the	the	DET
ejpam-4806	204	29	relation	relation	NOUN
ejpam-4806	204	30	between	between	ADP
ejpam-4806	204	31	the	the	DET
ejpam-4806	204	32	parameters	parameter	NOUN
ejpam-4806	204	33	given	give	VERB
ejpam-4806	204	34	in	in	ADP
ejpam-4806	204	35	remark	remark	NOUN
ejpam-4806	204	36	1	1	NUM
ejpam-4806	204	37	.	.	PUNCT
ejpam-4806	205	1	lemma	lemma	PROPN
ejpam-4806	205	2	8	8	NUM
ejpam-4806	205	3	.	.	PUNCT
ejpam-4806	206	1	for	for	ADP
ejpam-4806	206	2	the	the	DET
ejpam-4806	206	3	graph	graph	NOUN
ejpam-4806	206	4	ga	ga	PROPN
ejpam-4806	206	5	,	,	PUNCT
ejpam-4806	206	6	the	the	DET
ejpam-4806	206	7	lower	lower	ADV
ejpam-4806	206	8	bound	bind	VERB
ejpam-4806	206	9	and	and	CCONJ
ejpam-4806	206	10	upper	upper	ADJ
ejpam-4806	206	11	bound	bind	VERB
ejpam-4806	206	12	are	be	AUX
ejpam-4806	206	13	improved	improve	VERB
ejpam-4806	206	14	as	as	ADP
ejpam-4806	206	15	(	(	PUNCT
ejpam-4806	206	16	2δ2	2δ2	NUM
ejpam-4806	206	17	+	+	SYM
ejpam-4806	206	18	2(2m−	2(2m−	NUM
ejpam-4806	206	19	ηδ	ηδ	NOUN
ejpam-4806	206	20	)	)	PUNCT
ejpam-4806	206	21	+	+	CCONJ
ejpam-4806	206	22	(	(	PUNCT
ejpam-4806	206	23	n+	n+	NUM
ejpam-4806	206	24	2)η	2)η	ADJ
ejpam-4806	206	25	−	−	PROPN
ejpam-4806	207	1	5)1/2	5)1/2	NUM
ejpam-4806	207	2	<	<	X
ejpam-4806	207	3	µ	µ	X
ejpam-4806	207	4	(	(	PUNCT
ejpam-4806	207	5	ga	ga	PROPN
ejpam-4806	207	6	)	)	PUNCT
ejpam-4806	207	7	(	(	PUNCT
ejpam-4806	207	8	10	10	NUM
ejpam-4806	207	9	)	)	PUNCT
ejpam-4806	207	10	and	and	CCONJ
ejpam-4806	207	11	(	(	PUNCT
ejpam-4806	207	12	2∆2	2∆2	NUM
ejpam-4806	207	13	+	+	NUM
ejpam-4806	207	14	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	207	15	(	(	PUNCT
ejpam-4806	207	16	nη	nη	ADP
ejpam-4806	207	17	−	−	PROPN
ejpam-4806	207	18	4	4	NUM
ejpam-4806	207	19	)	)	PUNCT
ejpam-4806	207	20	)	)	PUNCT
ejpam-4806	208	1	1/2	1/2	NUM
ejpam-4806	208	2	>	>	SYM
ejpam-4806	208	3	µ	µ	X
ejpam-4806	208	4	(	(	PUNCT
ejpam-4806	208	5	ga	ga	PROPN
ejpam-4806	208	6	)	)	PUNCT
ejpam-4806	208	7	(	(	PUNCT
ejpam-4806	208	8	11	11	X
ejpam-4806	208	9	)	)	PUNCT
ejpam-4806	208	10	we	we	PRON
ejpam-4806	208	11	verify	verify	VERB
ejpam-4806	208	12	that	that	SCONJ
ejpam-4806	208	13	when	when	SCONJ
ejpam-4806	208	14	(	(	PUNCT
ejpam-4806	208	15	(	(	PUNCT
ejpam-4806	208	16	n	n	X
ejpam-4806	208	17	+	+	CCONJ
ejpam-4806	208	18	2)η	2)η	NUM
ejpam-4806	208	19	−	−	NOUN
ejpam-4806	208	20	5	5	NUM
ejpam-4806	208	21	)	)	PUNCT
ejpam-4806	208	22	is	be	AUX
ejpam-4806	208	23	added	add	VERB
ejpam-4806	208	24	to	to	ADP
ejpam-4806	208	25	the	the	DET
ejpam-4806	208	26	expression	expression	NOUN
ejpam-4806	208	27	in	in	ADP
ejpam-4806	208	28	the	the	DET
ejpam-4806	208	29	lower	lower	ADV
ejpam-4806	208	30	bound	bind	VERB
ejpam-4806	208	31	in	in	ADP
ejpam-4806	208	32	equation	equation	NOUN
ejpam-4806	208	33	(	(	PUNCT
ejpam-4806	208	34	3	3	NUM
ejpam-4806	208	35	)	)	PUNCT
ejpam-4806	208	36	of	of	ADP
ejpam-4806	208	37	theorem	theorem	NOUN
ejpam-4806	208	38	1	1	NUM
ejpam-4806	208	39	,	,	PUNCT
ejpam-4806	208	40	we	we	PRON
ejpam-4806	208	41	get	get	VERB
ejpam-4806	208	42	the	the	DET
ejpam-4806	208	43	greatest	greatest	ADV
ejpam-4806	208	44	lower	lower	ADV
ejpam-4806	208	45	bound	bind	VERB
ejpam-4806	208	46	and	and	CCONJ
ejpam-4806	208	47	when	when	SCONJ
ejpam-4806	208	48	the	the	DET
ejpam-4806	208	49	term	term	NOUN
ejpam-4806	208	50	(	(	PUNCT
ejpam-4806	208	51	nη	nη	ADP
ejpam-4806	208	52	−	−	PROPN
ejpam-4806	208	53	4	4	NUM
ejpam-4806	208	54	)	)	PUNCT
ejpam-4806	208	55	is	be	AUX
ejpam-4806	208	56	subtracted	subtract	VERB
ejpam-4806	208	57	from	from	ADP
ejpam-4806	208	58	the	the	DET
ejpam-4806	208	59	expression	expression	NOUN
ejpam-4806	208	60	in	in	ADP
ejpam-4806	208	61	the	the	DET
ejpam-4806	208	62	upper	upper	ADJ
ejpam-4806	208	63	bound	bind	VERB
ejpam-4806	208	64	in	in	ADP
ejpam-4806	208	65	equation	equation	NOUN
ejpam-4806	208	66	(	(	PUNCT
ejpam-4806	208	67	3	3	NUM
ejpam-4806	208	68	)	)	PUNCT
ejpam-4806	208	69	of	of	ADP
ejpam-4806	208	70	theorem	theorem	NOUN
ejpam-4806	208	71	1	1	NUM
ejpam-4806	208	72	,	,	PUNCT
ejpam-4806	208	73	we	we	PRON
ejpam-4806	208	74	obtain	obtain	VERB
ejpam-4806	208	75	the	the	DET
ejpam-4806	208	76	least	least	ADJ
ejpam-4806	208	77	upper	upper	ADJ
ejpam-4806	208	78	bound	bind	VERB
ejpam-4806	208	79	for	for	ADP
ejpam-4806	208	80	the	the	DET
ejpam-4806	208	81	signless	signless	PROPN
ejpam-4806	208	82	laplacian	laplacian	PROPN
ejpam-4806	208	83	spectral	spectral	PROPN
ejpam-4806	208	84	radius	radius	PROPN
ejpam-4806	208	85	µ	µ	X
ejpam-4806	208	86	(	(	PUNCT
ejpam-4806	208	87	ga	ga	PROPN
ejpam-4806	208	88	)	)	PUNCT
ejpam-4806	208	89	.	.	PUNCT
ejpam-4806	209	1	proof	proof	NOUN
ejpam-4806	209	2	.	.	PUNCT
ejpam-4806	210	1	let	let	VERB
ejpam-4806	210	2	us	we	PRON
ejpam-4806	210	3	assume	assume	VERB
ejpam-4806	210	4	l.b∗	l.b∗	VERB
ejpam-4806	210	5	=	=	PUNCT
ejpam-4806	210	6	(	(	PUNCT
ejpam-4806	210	7	2δ2	2δ2	NUM
ejpam-4806	210	8	+	+	CCONJ
ejpam-4806	210	9	2(2m−	2(2m−	NUM
ejpam-4806	210	10	ηδ	ηδ	NOUN
ejpam-4806	210	11	)	)	PUNCT
ejpam-4806	211	1	+	+	CCONJ
ejpam-4806	211	2	(	(	PUNCT
ejpam-4806	211	3	ϵ1	ϵ1	ADJ
ejpam-4806	211	4	+	+	NOUN
ejpam-4806	211	5	1	1	NUM
ejpam-4806	211	6	)	)	PUNCT
ejpam-4806	211	7	)	)	PUNCT
ejpam-4806	212	1	1/2	1/2	NUM
ejpam-4806	212	2	=	=	SYM
ejpam-4806	212	3	(	(	PUNCT
ejpam-4806	212	4	2δ2	2δ2	NUM
ejpam-4806	212	5	+	+	CCONJ
ejpam-4806	212	6	2(2m−	2(2m−	NUM
ejpam-4806	212	7	ηδ	ηδ	NOUN
ejpam-4806	212	8	)	)	PUNCT
ejpam-4806	213	1	+	+	CCONJ
ejpam-4806	213	2	(	(	PUNCT
ejpam-4806	213	3	(	(	PUNCT
ejpam-4806	213	4	n+	n+	X
ejpam-4806	213	5	2)η	2)η	ADJ
ejpam-4806	213	6	−	−	NUM
ejpam-4806	213	7	5	5	NUM
ejpam-4806	213	8	+	+	SYM
ejpam-4806	213	9	1))1/2	1))1/2	NUM
ejpam-4806	213	10	malathy	malathy	ADJ
ejpam-4806	213	11	v	v	NOUN
ejpam-4806	213	12	,	,	PUNCT
ejpam-4806	213	13	kalyani	kalyani	PROPN
ejpam-4806	213	14	desikan	desikan	PROPN
ejpam-4806	213	15	/	/	SYM
ejpam-4806	213	16	eur	eur	PROPN
ejpam-4806	213	17	.	.	PUNCT
ejpam-4806	214	1	j.	j.	PROPN
ejpam-4806	214	2	pure	pure	PROPN
ejpam-4806	214	3	appl	appl	PROPN
ejpam-4806	214	4	.	.	PROPN
ejpam-4806	214	5	math	math	PROPN
ejpam-4806	214	6	,	,	PUNCT
ejpam-4806	214	7	16	16	NUM
ejpam-4806	214	8	(	(	PUNCT
ejpam-4806	214	9	3	3	NUM
ejpam-4806	214	10	)	)	PUNCT
ejpam-4806	214	11	(	(	PUNCT
ejpam-4806	214	12	2023	2023	NUM
ejpam-4806	214	13	)	)	PUNCT
ejpam-4806	214	14	,	,	PUNCT
ejpam-4806	214	15	1731	1731	NUM
ejpam-4806	214	16	-	-	SYM
ejpam-4806	214	17	1746	1746	NUM
ejpam-4806	214	18	1741	1741	NUM
ejpam-4806	214	19	to	to	PART
ejpam-4806	214	20	be	be	AUX
ejpam-4806	214	21	the	the	DET
ejpam-4806	214	22	greatest	greatest	ADV
ejpam-4806	214	23	lower	low	ADJ
ejpam-4806	214	24	bound	bind	VERB
ejpam-4806	214	25	of	of	ADP
ejpam-4806	214	26	µ	µ	X
ejpam-4806	214	27	(	(	PUNCT
ejpam-4806	214	28	ga	ga	NOUN
ejpam-4806	214	29	)	)	PUNCT
ejpam-4806	214	30	and	and	CCONJ
ejpam-4806	214	31	let	let	VERB
ejpam-4806	214	32	u.b	u.b	PROPN
ejpam-4806	214	33	=	=	PUNCT
ejpam-4806	214	34	(	(	PUNCT
ejpam-4806	214	35	2∆2	2∆2	NUM
ejpam-4806	214	36	+	+	NUM
ejpam-4806	214	37	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	214	38	(	(	PUNCT
ejpam-4806	214	39	nη	nη	ADP
ejpam-4806	214	40	−	−	PROPN
ejpam-4806	214	41	4))1/2	4))1/2	NOUN
ejpam-4806	214	42	.	.	PUNCT
ejpam-4806	215	1	by	by	ADP
ejpam-4806	215	2	making	make	VERB
ejpam-4806	215	3	use	use	NOUN
ejpam-4806	215	4	of	of	ADP
ejpam-4806	215	5	the	the	DET
ejpam-4806	215	6	relationship	relationship	NOUN
ejpam-4806	215	7	between	between	ADP
ejpam-4806	215	8	the	the	DET
ejpam-4806	215	9	parameters	parameter	NOUN
ejpam-4806	215	10	given	give	VERB
ejpam-4806	215	11	in	in	ADP
ejpam-4806	215	12	remark	remark	NOUN
ejpam-4806	215	13	1	1	NUM
ejpam-4806	215	14	,	,	PUNCT
ejpam-4806	215	15	we	we	PRON
ejpam-4806	215	16	observe	observe	VERB
ejpam-4806	215	17	that	that	SCONJ
ejpam-4806	215	18	(	(	PUNCT
ejpam-4806	215	19	l.b∗)2	l.b∗)2	INTJ
ejpam-4806	215	20	−	−	PROPN
ejpam-4806	215	21	(	(	PUNCT
ejpam-4806	215	22	u.b)2	u.b)2	NOUN
ejpam-4806	215	23	=	=	SYM
ejpam-4806	215	24	(	(	PUNCT
ejpam-4806	215	25	2δ2	2δ2	NUM
ejpam-4806	215	26	+	+	CCONJ
ejpam-4806	215	27	2(2m−	2(2m−	NUM
ejpam-4806	215	28	ηδ	ηδ	NOUN
ejpam-4806	215	29	)	)	PUNCT
ejpam-4806	216	1	+	+	CCONJ
ejpam-4806	216	2	(	(	PUNCT
ejpam-4806	216	3	n+	n+	NUM
ejpam-4806	216	4	2)η	2)η	ADJ
ejpam-4806	216	5	−	−	NOUN
ejpam-4806	217	1	4)−	4)−	INTJ
ejpam-4806	217	2	(	(	PUNCT
ejpam-4806	217	3	2∆2	2∆2	NUM
ejpam-4806	217	4	+	+	NUM
ejpam-4806	217	5	4m−	4m−	PROPN
ejpam-4806	217	6	2∆−	2∆−	NUM
ejpam-4806	217	7	(	(	PUNCT
ejpam-4806	217	8	nη	nη	ADP
ejpam-4806	217	9	−	−	PROPN
ejpam-4806	217	10	4	4	NUM
ejpam-4806	217	11	)	)	PUNCT
ejpam-4806	217	12	)	)	PUNCT
ejpam-4806	218	1	=	=	PUNCT
ejpam-4806	219	1	2(δ	2(δ	NUM
ejpam-4806	219	2	+	+	ADJ
ejpam-4806	219	3	∆)(δ	∆)(δ	PROPN
ejpam-4806	219	4	−∆)−	−∆)−	PROPN
ejpam-4806	219	5	2ηδ	2ηδ	NOUN
ejpam-4806	220	1	+	+	CCONJ
ejpam-4806	220	2	nη	nη	CCONJ
ejpam-4806	220	3	+	+	NUM
ejpam-4806	220	4	2η	2η	NUM
ejpam-4806	221	1	−	−	NOUN
ejpam-4806	221	2	4	4	NUM
ejpam-4806	221	3	+	+	CCONJ
ejpam-4806	221	4	2(η	2(η	NUM
ejpam-4806	221	5	+	+	SYM
ejpam-4806	221	6	1	1	NUM
ejpam-4806	221	7	)	)	PUNCT
ejpam-4806	221	8	+	+	NOUN
ejpam-4806	221	9	nη	nη	ADP
ejpam-4806	221	10	−	−	NUM
ejpam-4806	221	11	4	4	NUM
ejpam-4806	221	12	=	=	SYM
ejpam-4806	221	13	2(n+	2(n+	NOUN
ejpam-4806	221	14	1)(1−	1)(1−	NUM
ejpam-4806	222	1	η)−	η)−	ADP
ejpam-4806	222	2	4η	4η	NOUN
ejpam-4806	222	3	+	+	CCONJ
ejpam-4806	222	4	nη	nη	PROPN
ejpam-4806	222	5	+	+	NUM
ejpam-4806	222	6	2η	2η	PROPN
ejpam-4806	222	7	+	+	CCONJ
ejpam-4806	222	8	2η	2η	NUM
ejpam-4806	222	9	−	−	NOUN
ejpam-4806	222	10	4	4	NUM
ejpam-4806	222	11	+	+	SYM
ejpam-4806	222	12	2	2	NUM
ejpam-4806	222	13	+	+	NUM
ejpam-4806	222	14	nη	nη	ADP
ejpam-4806	222	15	−	−	NUM
ejpam-4806	222	16	4	4	NUM
ejpam-4806	222	17	=	=	SYM
ejpam-4806	222	18	2(n−	2(n−	NUM
ejpam-4806	222	19	η)−	η)−	PROPN
ejpam-4806	222	20	2nη	2nη	NOUN
ejpam-4806	223	1	+	+	CCONJ
ejpam-4806	223	2	4	4	NUM
ejpam-4806	223	3	+	+	CCONJ
ejpam-4806	223	4	2nη	2nη	ADJ
ejpam-4806	223	5	−	−	NOUN
ejpam-4806	223	6	8	8	NUM
ejpam-4806	223	7	=	=	SYM
ejpam-4806	223	8	0	0	NUM
ejpam-4806	224	1	this	this	PRON
ejpam-4806	224	2	implies	imply	VERB
ejpam-4806	224	3	l.b∗	l.b∗	PROPN
ejpam-4806	224	4	=	=	SYM
ejpam-4806	224	5	u.b	u.b	PROPN
ejpam-4806	224	6	which	which	PRON
ejpam-4806	224	7	is	be	AUX
ejpam-4806	224	8	a	a	DET
ejpam-4806	224	9	contradiction	contradiction	NOUN
ejpam-4806	224	10	.	.	PUNCT
ejpam-4806	225	1	therefore	therefore	ADV
ejpam-4806	225	2	the	the	DET
ejpam-4806	225	3	following	follow	VERB
ejpam-4806	225	4	expression	expression	NOUN
ejpam-4806	225	5	is	be	AUX
ejpam-4806	225	6	the	the	DET
ejpam-4806	225	7	greatest	greatest	ADV
ejpam-4806	225	8	lower	lower	ADV
ejpam-4806	225	9	bound	bind	VERB
ejpam-4806	225	10	(	(	PUNCT
ejpam-4806	225	11	2δ2	2δ2	NUM
ejpam-4806	225	12	+	+	CCONJ
ejpam-4806	225	13	2(2m−	2(2m−	NUM
ejpam-4806	225	14	ηδ	ηδ	NOUN
ejpam-4806	225	15	)	)	PUNCT
ejpam-4806	225	16	+	+	CCONJ
ejpam-4806	225	17	(	(	PUNCT
ejpam-4806	225	18	n+	n+	NUM
ejpam-4806	225	19	2)η	2)η	ADJ
ejpam-4806	225	20	−	−	PROPN
ejpam-4806	226	1	5)1/2	5)1/2	NUM
ejpam-4806	226	2	<	<	X
ejpam-4806	226	3	µ	µ	X
ejpam-4806	226	4	(	(	PUNCT
ejpam-4806	226	5	ga	ga	PROPN
ejpam-4806	226	6	)	)	PUNCT
ejpam-4806	226	7	.	.	PUNCT
ejpam-4806	227	1	similarly	similarly	ADV
ejpam-4806	227	2	,	,	PUNCT
ejpam-4806	227	3	we	we	PRON
ejpam-4806	227	4	prove	prove	VERB
ejpam-4806	227	5	that	that	SCONJ
ejpam-4806	227	6	ϵ2	ϵ2	NOUN
ejpam-4806	227	7	=	=	PUNCT
ejpam-4806	228	1	(	(	PUNCT
ejpam-4806	228	2	nη	nη	INTJ
ejpam-4806	228	3	−	−	PROPN
ejpam-4806	228	4	4	4	NUM
ejpam-4806	228	5	)	)	PUNCT
ejpam-4806	228	6	is	be	AUX
ejpam-4806	228	7	the	the	DET
ejpam-4806	228	8	maximum	maximum	ADJ
ejpam-4806	228	9	integral	integral	ADJ
ejpam-4806	228	10	value	value	NOUN
ejpam-4806	228	11	required	require	VERB
ejpam-4806	228	12	to	to	PART
ejpam-4806	228	13	obtain	obtain	VERB
ejpam-4806	228	14	the	the	DET
ejpam-4806	228	15	least	least	ADJ
ejpam-4806	228	16	upper	upper	ADJ
ejpam-4806	228	17	bound	bind	VERB
ejpam-4806	228	18	.	.	PUNCT
ejpam-4806	229	1	let	let	VERB
ejpam-4806	229	2	us	we	PRON
ejpam-4806	229	3	assume	assume	VERB
ejpam-4806	229	4	that	that	SCONJ
ejpam-4806	229	5	u.b∗	u.b∗	ADJ
ejpam-4806	229	6	=	=	PUNCT
ejpam-4806	229	7	(	(	PUNCT
ejpam-4806	229	8	(	(	PUNCT
ejpam-4806	229	9	2∆2	2∆2	NUM
ejpam-4806	229	10	+	+	NUM
ejpam-4806	229	11	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	229	12	(	(	PUNCT
ejpam-4806	229	13	ϵ2	ϵ2	VERB
ejpam-4806	229	14	+	+	CCONJ
ejpam-4806	229	15	1	1	NUM
ejpam-4806	229	16	)	)	PUNCT
ejpam-4806	229	17	)	)	PUNCT
ejpam-4806	229	18	1/2	1/2	NUM
ejpam-4806	229	19	=	=	SYM
ejpam-4806	229	20	(	(	PUNCT
ejpam-4806	229	21	(	(	PUNCT
ejpam-4806	229	22	2∆2	2∆2	NUM
ejpam-4806	229	23	+	+	NUM
ejpam-4806	229	24	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	229	25	(	(	PUNCT
ejpam-4806	229	26	nη	nη	ADP
ejpam-4806	229	27	−	−	PROPN
ejpam-4806	229	28	3	3	NUM
ejpam-4806	229	29	)	)	PUNCT
ejpam-4806	229	30	)	)	PUNCT
ejpam-4806	230	1	1/2	1/2	NUM
ejpam-4806	230	2	is	be	AUX
ejpam-4806	230	3	the	the	DET
ejpam-4806	230	4	least	least	ADJ
ejpam-4806	230	5	upper	upper	ADJ
ejpam-4806	230	6	bound	bind	VERB
ejpam-4806	230	7	and	and	CCONJ
ejpam-4806	230	8	let	let	VERB
ejpam-4806	230	9	l.b	l.b	PROPN
ejpam-4806	230	10	=	=	X
ejpam-4806	230	11	(	(	PUNCT
ejpam-4806	230	12	2δ2	2δ2	NUM
ejpam-4806	230	13	+	+	CCONJ
ejpam-4806	230	14	2(2m−	2(2m−	NUM
ejpam-4806	230	15	ηδ	ηδ	NOUN
ejpam-4806	230	16	)	)	PUNCT
ejpam-4806	230	17	+	+	CCONJ
ejpam-4806	230	18	(	(	PUNCT
ejpam-4806	230	19	n+	n+	NUM
ejpam-4806	230	20	2)η	2)η	ADJ
ejpam-4806	230	21	−	−	PROPN
ejpam-4806	231	1	5)1/2	5)1/2	PRON
ejpam-4806	231	2	.	.	PUNCT
ejpam-4806	232	1	consider	consider	VERB
ejpam-4806	232	2	(	(	PUNCT
ejpam-4806	232	3	u.b∗)2	u.b∗)2	ADJ
ejpam-4806	232	4	−	−	PROPN
ejpam-4806	232	5	(	(	PUNCT
ejpam-4806	232	6	l.b)2	l.b)2	NOUN
ejpam-4806	232	7	=	=	SYM
ejpam-4806	232	8	(	(	PUNCT
ejpam-4806	232	9	2∆2	2∆2	NUM
ejpam-4806	232	10	+	+	NUM
ejpam-4806	232	11	4m−	4m−	PROPN
ejpam-4806	232	12	2∆−	2∆−	NUM
ejpam-4806	232	13	(	(	PUNCT
ejpam-4806	232	14	nη	nη	ADP
ejpam-4806	232	15	−	−	PROPN
ejpam-4806	233	1	3))−	3))−	NUM
ejpam-4806	233	2	(	(	PUNCT
ejpam-4806	233	3	2δ2	2δ2	NUM
ejpam-4806	234	1	+	+	CCONJ
ejpam-4806	234	2	2(2m−	2(2m−	NUM
ejpam-4806	234	3	ηδ	ηδ	NOUN
ejpam-4806	234	4	)	)	PUNCT
ejpam-4806	235	1	+	+	CCONJ
ejpam-4806	235	2	(	(	PUNCT
ejpam-4806	235	3	n+	n+	NUM
ejpam-4806	235	4	2)η	2)η	ADJ
ejpam-4806	235	5	−	−	NUM
ejpam-4806	235	6	5	5	NUM
ejpam-4806	235	7	)	)	PUNCT
ejpam-4806	235	8	=	=	PUNCT
ejpam-4806	236	1	2(δ	2(δ	NUM
ejpam-4806	236	2	+	+	ADJ
ejpam-4806	236	3	∆)(δ	∆)(δ	X
ejpam-4806	236	4	−∆	−∆	NOUN
ejpam-4806	236	5	)	)	PUNCT
ejpam-4806	237	1	+	+	CCONJ
ejpam-4806	237	2	2ηδ	2ηδ	ADJ
ejpam-4806	237	3	−	−	PROPN
ejpam-4806	237	4	2∆−	2∆−	NUM
ejpam-4806	237	5	(	(	PUNCT
ejpam-4806	237	6	nη	nη	ADP
ejpam-4806	237	7	−	−	PROPN
ejpam-4806	237	8	3	3	NUM
ejpam-4806	237	9	)	)	PUNCT
ejpam-4806	237	10	+	+	CCONJ
ejpam-4806	237	11	(	(	PUNCT
ejpam-4806	237	12	n+	n+	NUM
ejpam-4806	237	13	2)η	2)η	ADJ
ejpam-4806	237	14	−	−	NUM
ejpam-4806	237	15	5	5	NUM
ejpam-4806	237	16	=	=	SYM
ejpam-4806	237	17	0	0	NUM
ejpam-4806	237	18	using	use	VERB
ejpam-4806	237	19	the	the	DET
ejpam-4806	237	20	relationship	relationship	NOUN
ejpam-4806	237	21	between	between	ADP
ejpam-4806	237	22	the	the	DET
ejpam-4806	237	23	parameters	parameter	NOUN
ejpam-4806	237	24	given	give	VERB
ejpam-4806	237	25	in	in	ADP
ejpam-4806	237	26	remark	remark	NOUN
ejpam-4806	237	27	1	1	NUM
ejpam-4806	237	28	,	,	PUNCT
ejpam-4806	237	29	we	we	PRON
ejpam-4806	237	30	observe	observe	VERB
ejpam-4806	237	31	that	that	SCONJ
ejpam-4806	237	32	u.b∗	u.b∗	ADJ
ejpam-4806	238	1	−	−	PROPN
ejpam-4806	238	2	l.b	l.b	PROPN
ejpam-4806	238	3	=	=	SYM
ejpam-4806	238	4	0	0	NUM
ejpam-4806	238	5	which	which	PRON
ejpam-4806	238	6	implies	imply	VERB
ejpam-4806	238	7	u.b∗	u.b∗	PROPN
ejpam-4806	238	8	=	=	SYM
ejpam-4806	238	9	l.b	l.b	PROPN
ejpam-4806	238	10	,	,	PUNCT
ejpam-4806	238	11	which	which	PRON
ejpam-4806	238	12	is	be	AUX
ejpam-4806	238	13	a	a	DET
ejpam-4806	238	14	contradiction	contradiction	NOUN
ejpam-4806	238	15	.	.	PUNCT
ejpam-4806	239	1	therefore	therefore	ADV
ejpam-4806	239	2	,	,	PUNCT
ejpam-4806	239	3	we	we	PRON
ejpam-4806	239	4	conclude	conclude	VERB
ejpam-4806	239	5	that	that	SCONJ
ejpam-4806	239	6	u.b	u.b	PROPN
ejpam-4806	239	7	is	be	AUX
ejpam-4806	239	8	the	the	DET
ejpam-4806	239	9	greatest	great	ADJ
ejpam-4806	239	10	upper	upper	ADJ
ejpam-4806	239	11	bound	bind	VERB
ejpam-4806	239	12	.	.	PUNCT
ejpam-4806	240	1	therefore	therefore	ADV
ejpam-4806	240	2	we	we	PRON
ejpam-4806	240	3	have	have	VERB
ejpam-4806	240	4	l.b	l.b	NOUN
ejpam-4806	240	5	=	=	X
ejpam-4806	240	6	(	(	PUNCT
ejpam-4806	240	7	2δ2	2δ2	NUM
ejpam-4806	241	1	+	+	CCONJ
ejpam-4806	241	2	2(2m−	2(2m−	NUM
ejpam-4806	241	3	ηδ	ηδ	NOUN
ejpam-4806	241	4	)	)	PUNCT
ejpam-4806	242	1	+	+	CCONJ
ejpam-4806	242	2	(	(	PUNCT
ejpam-4806	242	3	n+	n+	NUM
ejpam-4806	242	4	2)η	2)η	ADJ
ejpam-4806	242	5	−	−	NOUN
ejpam-4806	242	6	5	5	NUM
ejpam-4806	242	7	)	)	PUNCT
ejpam-4806	242	8	1/2	1/2	NUM
ejpam-4806	242	9	<	<	X
ejpam-4806	242	10	µ	µ	X
ejpam-4806	242	11	(	(	PUNCT
ejpam-4806	242	12	ga	ga	PROPN
ejpam-4806	242	13	)	)	PUNCT
ejpam-4806	243	1	<	<	X
ejpam-4806	243	2	(	(	PUNCT
ejpam-4806	243	3	2∆2	2∆2	NUM
ejpam-4806	243	4	+	+	NUM
ejpam-4806	243	5	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	243	6	(	(	PUNCT
ejpam-4806	243	7	nη	nη	ADP
ejpam-4806	243	8	−	−	PROPN
ejpam-4806	243	9	4	4	NUM
ejpam-4806	243	10	)	)	PUNCT
ejpam-4806	243	11	)	)	PUNCT
ejpam-4806	243	12	1/2	1/2	NUM
ejpam-4806	243	13	=	=	SYM
ejpam-4806	243	14	u.b	u.b	PROPN
ejpam-4806	243	15	.	.	PUNCT
ejpam-4806	244	1	(	(	PUNCT
ejpam-4806	244	2	12	12	NUM
ejpam-4806	244	3	)	)	PUNCT
ejpam-4806	244	4	remark	remark	NOUN
ejpam-4806	244	5	2	2	NUM
ejpam-4806	244	6	.	.	PUNCT
ejpam-4806	244	7	by	by	ADP
ejpam-4806	244	8	using	use	VERB
ejpam-4806	244	9	the	the	DET
ejpam-4806	244	10	improved	improve	VERB
ejpam-4806	244	11	bounds	bound	NOUN
ejpam-4806	244	12	derived	derive	VERB
ejpam-4806	244	13	in	in	ADP
ejpam-4806	244	14	theorem	theorem	NOUN
ejpam-4806	244	15	2	2	NUM
ejpam-4806	244	16	,	,	PUNCT
ejpam-4806	244	17	for	for	ADP
ejpam-4806	244	18	the	the	DET
ejpam-4806	244	19	graph	graph	NOUN
ejpam-4806	244	20	k2	k2	ADJ
ejpam-4806	244	21	▽	▽	NOUN
ejpam-4806	244	22	5k1	5k1	NUM
ejpam-4806	244	23	,	,	PUNCT
ejpam-4806	244	24	with	with	ADP
ejpam-4806	244	25	n	n	NOUN
ejpam-4806	244	26	=	=	SYM
ejpam-4806	244	27	7	7	NUM
ejpam-4806	244	28	,	,	PUNCT
ejpam-4806	244	29	m	m	VERB
ejpam-4806	244	30	=	=	NOUN
ejpam-4806	244	31	11	11	NUM
ejpam-4806	244	32	,	,	PUNCT
ejpam-4806	244	33	η	η	NOUN
ejpam-4806	244	34	=	=	SYM
ejpam-4806	244	35	5	5	NUM
ejpam-4806	244	36	,	,	PUNCT
ejpam-4806	244	37	∆	∆	X
ejpam-4806	244	38	=	=	SYM
ejpam-4806	244	39	6	6	NUM
ejpam-4806	244	40	,	,	PUNCT
ejpam-4806	244	41	δ	δ	PROPN
ejpam-4806	244	42	=	=	SYM
ejpam-4806	244	43	2	2	NUM
ejpam-4806	244	44	and	and	CCONJ
ejpam-4806	244	45	µ(ga	µ(ga	PROPN
ejpam-4806	244	46	)	)	PUNCT
ejpam-4806	244	47	=	=	SYM
ejpam-4806	244	48	8.5311	8.5311	NUM
ejpam-4806	244	49	,	,	PUNCT
ejpam-4806	244	50	we	we	PRON
ejpam-4806	244	51	have	have	VERB
ejpam-4806	244	52	8.4852	8.4852	NUM
ejpam-4806	244	53	<	<	X
ejpam-4806	244	54	µ(ga	µ(ga	NOUN
ejpam-4806	244	55	)	)	PUNCT
ejpam-4806	244	56	<	<	X
ejpam-4806	244	57	8.544	8.544	NUM
ejpam-4806	244	58	malathy	malathy	NOUN
ejpam-4806	244	59	v	v	NOUN
ejpam-4806	244	60	,	,	PUNCT
ejpam-4806	244	61	kalyani	kalyani	PROPN
ejpam-4806	244	62	desikan	desikan	PROPN
ejpam-4806	244	63	/	/	SYM
ejpam-4806	244	64	eur	eur	PROPN
ejpam-4806	244	65	.	.	PUNCT
ejpam-4806	245	1	j.	j.	PROPN
ejpam-4806	245	2	pure	pure	PROPN
ejpam-4806	245	3	appl	appl	PROPN
ejpam-4806	245	4	.	.	PROPN
ejpam-4806	245	5	math	math	PROPN
ejpam-4806	245	6	,	,	PUNCT
ejpam-4806	245	7	16	16	NUM
ejpam-4806	245	8	(	(	PUNCT
ejpam-4806	245	9	3	3	NUM
ejpam-4806	245	10	)	)	PUNCT
ejpam-4806	245	11	(	(	PUNCT
ejpam-4806	245	12	2023	2023	NUM
ejpam-4806	245	13	)	)	PUNCT
ejpam-4806	245	14	,	,	PUNCT
ejpam-4806	245	15	1731	1731	NUM
ejpam-4806	245	16	-	-	SYM
ejpam-4806	245	17	1746	1746	NUM
ejpam-4806	245	18	1742	1742	NUM
ejpam-4806	245	19	4	4	NUM
ejpam-4806	245	20	.	.	PUNCT
ejpam-4806	245	21	bounds	bound	NOUN
ejpam-4806	245	22	of	of	ADP
ejpam-4806	245	23	the	the	DET
ejpam-4806	245	24	nordhaus	nordhaus	NOUN
ejpam-4806	245	25	-	-	PUNCT
ejpam-4806	245	26	gaddam	gaddam	NOUN
ejpam-4806	245	27	type	type	NOUN
ejpam-4806	245	28	in	in	ADP
ejpam-4806	245	29	this	this	DET
ejpam-4806	245	30	section	section	NOUN
ejpam-4806	245	31	,	,	PUNCT
ejpam-4806	245	32	we	we	PRON
ejpam-4806	245	33	obtain	obtain	VERB
ejpam-4806	245	34	the	the	DET
ejpam-4806	245	35	upper	upper	NOUN
ejpam-4806	245	36	and	and	CCONJ
ejpam-4806	245	37	the	the	DET
ejpam-4806	245	38	lower	low	ADJ
ejpam-4806	245	39	bounds	bound	NOUN
ejpam-4806	245	40	on	on	ADP
ejpam-4806	245	41	µ	µ	PROPN
ejpam-4806	245	42	(	(	PUNCT
ejpam-4806	245	43	ga)+µ	ga)+µ	PROPN
ejpam-4806	245	44	(	(	PUNCT
ejpam-4806	245	45	gc	gc	PROPN
ejpam-4806	245	46	a	a	NOUN
ejpam-4806	245	47	)	)	PUNCT
ejpam-4806	245	48	(	(	PUNCT
ejpam-4806	245	49	nordhausgaddam	nordhausgaddam	ADJ
ejpam-4806	245	50	type	type	NOUN
ejpam-4806	245	51	inequality	inequality	NOUN
ejpam-4806	245	52	)	)	PUNCT
ejpam-4806	245	53	for	for	ADP
ejpam-4806	245	54	the	the	DET
ejpam-4806	245	55	signless	signless	PROPN
ejpam-4806	245	56	laplacian	laplacian	ADJ
ejpam-4806	245	57	spectral	spectral	ADJ
ejpam-4806	245	58	radius	radius	NOUN
ejpam-4806	245	59	of	of	ADP
ejpam-4806	245	60	the	the	DET
ejpam-4806	245	61	agave	agave	ADJ
ejpam-4806	245	62	graphs	graph	NOUN
ejpam-4806	245	63	in	in	ADP
ejpam-4806	245	64	terms	term	NOUN
ejpam-4806	245	65	of	of	ADP
ejpam-4806	245	66	the	the	DET
ejpam-4806	245	67	maximum	maximum	ADJ
ejpam-4806	245	68	degree	degree	NOUN
ejpam-4806	245	69	∆	∆	PROPN
ejpam-4806	245	70	,	,	PUNCT
ejpam-4806	245	71	the	the	DET
ejpam-4806	245	72	minimum	minimum	NOUN
ejpam-4806	245	73	degree	degree	NOUN
ejpam-4806	245	74	δ	δ	PROPN
ejpam-4806	245	75	,	,	PUNCT
ejpam-4806	245	76	the	the	DET
ejpam-4806	245	77	order	order	NOUN
ejpam-4806	245	78	n	n	CCONJ
ejpam-4806	245	79	,	,	PUNCT
ejpam-4806	245	80	and	and	CCONJ
ejpam-4806	245	81	the	the	DET
ejpam-4806	245	82	size	size	NOUN
ejpam-4806	245	83	m	m	PROPN
ejpam-4806	245	84	of	of	ADP
ejpam-4806	245	85	the	the	DET
ejpam-4806	245	86	graph	graph	NOUN
ejpam-4806	245	87	g.	g.	NOUN
ejpam-4806	245	88	theorem	theorem	NOUN
ejpam-4806	245	89	3	3	X
ejpam-4806	245	90	.	.	PUNCT
ejpam-4806	246	1	let	let	VERB
ejpam-4806	246	2	ga	ga	PROPN
ejpam-4806	246	3	be	be	AUX
ejpam-4806	246	4	an	an	DET
ejpam-4806	246	5	agave	agave	NOUN
ejpam-4806	246	6	graph	graph	NOUN
ejpam-4806	246	7	with	with	ADP
ejpam-4806	246	8	n	n	ADP
ejpam-4806	246	9	vertices	vertex	NOUN
ejpam-4806	246	10	,	,	PUNCT
ejpam-4806	246	11	m	m	VERB
ejpam-4806	246	12	edges	edge	NOUN
ejpam-4806	246	13	and	and	CCONJ
ejpam-4806	246	14	η	η	PROPN
ejpam-4806	246	15	copies	copy	NOUN
ejpam-4806	246	16	of	of	ADP
ejpam-4806	246	17	the	the	DET
ejpam-4806	246	18	satellite	satellite	NOUN
ejpam-4806	246	19	graph	graph	NOUN
ejpam-4806	246	20	k1	k1	NOUN
ejpam-4806	246	21	.	.	PUNCT
ejpam-4806	247	1	let	let	VERB
ejpam-4806	247	2	∆	∆	PROPN
ejpam-4806	247	3	and	and	CCONJ
ejpam-4806	247	4	δ	δ	PROPN
ejpam-4806	247	5	be	be	VERB
ejpam-4806	247	6	the	the	DET
ejpam-4806	247	7	maximum	maximum	ADJ
ejpam-4806	247	8	degree	degree	NOUN
ejpam-4806	247	9	and	and	CCONJ
ejpam-4806	247	10	minimum	minimum	NOUN
ejpam-4806	247	11	degree	degree	NOUN
ejpam-4806	247	12	of	of	ADP
ejpam-4806	247	13	ga	ga	PROPN
ejpam-4806	247	14	,	,	PUNCT
ejpam-4806	247	15	respectively	respectively	ADV
ejpam-4806	247	16	.	.	PUNCT
ejpam-4806	248	1	2	2	NUM
ejpam-4806	249	1	(	(	PUNCT
ejpam-4806	249	2	(	(	PUNCT
ejpam-4806	249	3	4n2	4n2	NUM
ejpam-4806	249	4	−	−	NOUN
ejpam-4806	249	5	20n+	20n+	NUM
ejpam-4806	249	6	40	40	NUM
ejpam-4806	249	7	)	)	PUNCT
ejpam-4806	250	1	+	+	CCONJ
ejpam-4806	250	2	(	(	PUNCT
ejpam-4806	250	3	(	(	PUNCT
ejpam-4806	250	4	n+	n+	X
ejpam-4806	250	5	2)η	2)η	NUM
ejpam-4806	250	6	−	−	PROPN
ejpam-4806	250	7	κ	κ	NOUN
ejpam-4806	250	8	)	)	PUNCT
ejpam-4806	250	9	2	2	NUM
ejpam-4806	250	10	)	)	PUNCT
ejpam-4806	250	11	1/2	1/2	NUM
ejpam-4806	250	12	<	<	X
ejpam-4806	250	13	µ	µ	X
ejpam-4806	250	14	(	(	PUNCT
ejpam-4806	250	15	ga	ga	NOUN
ejpam-4806	250	16	)	)	PUNCT
ejpam-4806	250	17	+	+	X
ejpam-4806	250	18	µ	µ	X
ejpam-4806	250	19	(	(	PUNCT
ejpam-4806	250	20	gc	gc	PROPN
ejpam-4806	250	21	a	a	NOUN
ejpam-4806	250	22	)	)	PUNCT
ejpam-4806	250	23	<	<	X
ejpam-4806	250	24	2	2	NUM
ejpam-4806	250	25	(	(	PUNCT
ejpam-4806	250	26	(	(	PUNCT
ejpam-4806	250	27	6n2	6n2	NUM
ejpam-4806	250	28	−	−	NUM
ejpam-4806	250	29	22n+	22n+	NUM
ejpam-4806	250	30	28)−	28)−	NUM
ejpam-4806	250	31	(	(	PUNCT
ejpam-4806	250	32	nη	nη	NUM
ejpam-4806	250	33	−	−	PROPN
ejpam-4806	250	34	κ′	κ′	NOUN
ejpam-4806	250	35	)	)	PUNCT
ejpam-4806	250	36	2	2	NUM
ejpam-4806	250	37	)	)	PUNCT
ejpam-4806	250	38	1/2	1/2	NUM
ejpam-4806	250	39	(	(	PUNCT
ejpam-4806	250	40	13	13	NUM
ejpam-4806	250	41	)	)	PUNCT
ejpam-4806	250	42	for	for	ADP
ejpam-4806	250	43	5	5	NUM
ejpam-4806	250	44	≤	≤	NOUN
ejpam-4806	250	45	κ	κ	NOUN
ejpam-4806	250	46	≤	≤	NOUN
ejpam-4806	250	47	(	(	PUNCT
ejpam-4806	250	48	n2	n2	ADJ
ejpam-4806	250	49	−	−	PROPN
ejpam-4806	250	50	4	4	NUM
ejpam-4806	250	51	)	)	PUNCT
ejpam-4806	250	52	and	and	CCONJ
ejpam-4806	250	53	4	4	NUM
ejpam-4806	250	54	≤	≤	NUM
ejpam-4806	250	55	κ′	κ′	NOUN
ejpam-4806	250	56	≤	≤	PROPN
ejpam-4806	250	57	(	(	PUNCT
ejpam-4806	250	58	n(n−	n(n−	NOUN
ejpam-4806	250	59	2	2	NUM
ejpam-4806	250	60	)	)	PUNCT
ejpam-4806	250	61	)	)	PUNCT
ejpam-4806	250	62	.	.	PUNCT
ejpam-4806	251	1	proof	proof	NOUN
ejpam-4806	251	2	.	.	PUNCT
ejpam-4806	252	1	consider	consider	VERB
ejpam-4806	252	2	the	the	DET
ejpam-4806	252	3	family	family	NOUN
ejpam-4806	252	4	of	of	ADP
ejpam-4806	252	5	agave	agave	PROPN
ejpam-4806	252	6	graphs	graph	NOUN
ejpam-4806	252	7	ga	ga	PROPN
ejpam-4806	252	8	.	.	PROPN
ejpam-4806	252	9	let	let	VERB
ejpam-4806	252	10	η	η	PROPN
ejpam-4806	252	11	be	be	AUX
ejpam-4806	252	12	the	the	DET
ejpam-4806	252	13	number	number	NOUN
ejpam-4806	252	14	of	of	ADP
ejpam-4806	252	15	satellites	satellite	NOUN
ejpam-4806	252	16	k1	k1	PROPN
ejpam-4806	252	17	join	join	VERB
ejpam-4806	252	18	with	with	ADP
ejpam-4806	252	19	k2	k2	PROPN
ejpam-4806	252	20	and	and	CCONJ
ejpam-4806	252	21	let	let	VERB
ejpam-4806	252	22	gc	gc	PROPN
ejpam-4806	252	23	a	a	DET
ejpam-4806	252	24	be	be	AUX
ejpam-4806	252	25	the	the	DET
ejpam-4806	252	26	complement	complement	NOUN
ejpam-4806	252	27	having	have	VERB
ejpam-4806	252	28	mc	mc	PROPN
ejpam-4806	252	29	edges	edge	NOUN
ejpam-4806	252	30	,	,	PUNCT
ejpam-4806	252	31	where	where	SCONJ
ejpam-4806	252	32	mc	mc	PROPN
ejpam-4806	252	33	=	=	PUNCT
ejpam-4806	252	34	nc2	nc2	DET
ejpam-4806	252	35	−m	−m	NOUN
ejpam-4806	252	36	,	,	PUNCT
ejpam-4806	252	37	∆c	∆c	NOUN
ejpam-4806	252	38	=	=	SYM
ejpam-4806	252	39	δc	δc	NOUN
ejpam-4806	252	40	=	=	SYM
ejpam-4806	252	41	(	(	PUNCT
ejpam-4806	252	42	n−	n−	NOUN
ejpam-4806	252	43	3	3	NUM
ejpam-4806	252	44	)	)	PUNCT
ejpam-4806	252	45	from	from	ADP
ejpam-4806	252	46	equation	equation	NOUN
ejpam-4806	252	47	(	(	PUNCT
ejpam-4806	252	48	7	7	NUM
ejpam-4806	252	49	)	)	PUNCT
ejpam-4806	252	50	of	of	ADP
ejpam-4806	252	51	theorem	theorem	NOUN
ejpam-4806	252	52	2	2	NUM
ejpam-4806	252	53	,	,	PUNCT
ejpam-4806	252	54	we	we	PRON
ejpam-4806	252	55	consider	consider	VERB
ejpam-4806	252	56	the	the	DET
ejpam-4806	252	57	lower	low	ADJ
ejpam-4806	252	58	bound	bind	VERB
ejpam-4806	252	59	µ(ga	µ(ga	NOUN
ejpam-4806	252	60	)	)	PUNCT
ejpam-4806	252	61	>	>	X
ejpam-4806	253	1	(	(	PUNCT
ejpam-4806	253	2	2δ2	2δ2	NUM
ejpam-4806	253	3	+	+	CCONJ
ejpam-4806	253	4	2(2m−	2(2m−	NUM
ejpam-4806	253	5	ηδ	ηδ	NOUN
ejpam-4806	253	6	)	)	PUNCT
ejpam-4806	253	7	+	+	CCONJ
ejpam-4806	253	8	(	(	PUNCT
ejpam-4806	253	9	n+	n+	NUM
ejpam-4806	253	10	2)η	2)η	NUM
ejpam-4806	253	11	−	−	PROPN
ejpam-4806	253	12	κ	κ	NOUN
ejpam-4806	253	13	)	)	PUNCT
ejpam-4806	253	14	1/2	1/2	NUM
ejpam-4806	253	15	and	and	CCONJ
ejpam-4806	253	16	the	the	DET
ejpam-4806	253	17	signless	signless	PROPN
ejpam-4806	253	18	laplacian	laplacian	ADJ
ejpam-4806	253	19	spectral	spectral	ADJ
ejpam-4806	253	20	radius	radius	NOUN
ejpam-4806	253	21	of	of	ADP
ejpam-4806	253	22	complement	complement	NOUN
ejpam-4806	253	23	of	of	ADP
ejpam-4806	253	24	ga	ga	PROPN
ejpam-4806	253	25	µ(gc	µ(gc	PROPN
ejpam-4806	253	26	a	a	X
ejpam-4806	253	27	)	)	PUNCT
ejpam-4806	253	28	=	=	PUNCT
ejpam-4806	253	29	µ(kη−1	µ(kη−1	NOUN
ejpam-4806	253	30	)	)	PUNCT
ejpam-4806	253	31	=	=	SYM
ejpam-4806	253	32	(	(	PUNCT
ejpam-4806	253	33	2(∆−	2(∆−	NUM
ejpam-4806	253	34	δ)2	δ)2	NOUN
ejpam-4806	253	35	+	+	CCONJ
ejpam-4806	253	36	2(n(n−	2(n(n−	NUM
ejpam-4806	253	37	1)−	1)−	PROPN
ejpam-4806	253	38	2m−	2m−	PROPN
ejpam-4806	253	39	(	(	PUNCT
ejpam-4806	253	40	n−	n−	NOUN
ejpam-4806	253	41	3))1/2	3))1/2	NUM
ejpam-4806	253	42	we	we	PRON
ejpam-4806	253	43	have	have	VERB
ejpam-4806	253	44	µ(ga	µ(ga	NOUN
ejpam-4806	253	45	)	)	PUNCT
ejpam-4806	254	1	+	+	CCONJ
ejpam-4806	254	2	µ(gc	µ(gc	PROPN
ejpam-4806	254	3	a	a	X
ejpam-4806	254	4	)	)	PUNCT
ejpam-4806	254	5	>	>	PUNCT
ejpam-4806	255	1	(	(	PUNCT
ejpam-4806	255	2	2δ2	2δ2	NUM
ejpam-4806	255	3	+	+	CCONJ
ejpam-4806	255	4	2(2m−	2(2m−	NUM
ejpam-4806	255	5	ηδ	ηδ	NOUN
ejpam-4806	255	6	)	)	PUNCT
ejpam-4806	256	1	+	+	CCONJ
ejpam-4806	256	2	(	(	PUNCT
ejpam-4806	256	3	n+	n+	NUM
ejpam-4806	256	4	2)η	2)η	NUM
ejpam-4806	256	5	−	−	PROPN
ejpam-4806	256	6	κ	κ	NOUN
ejpam-4806	256	7	)	)	PUNCT
ejpam-4806	256	8	1/2	1/2	NUM
ejpam-4806	257	1	+	+	CCONJ
ejpam-4806	257	2	(	(	PUNCT
ejpam-4806	257	3	2(∆−	2(∆−	NUM
ejpam-4806	257	4	δ)2	δ)2	PROPN
ejpam-4806	257	5	+	+	CCONJ
ejpam-4806	257	6	2(n(n−	2(n(n−	NUM
ejpam-4806	257	7	1)−	1)−	PROPN
ejpam-4806	257	8	2m−	2m−	PROPN
ejpam-4806	257	9	(	(	PUNCT
ejpam-4806	257	10	n−	n−	NOUN
ejpam-4806	257	11	3	3	NUM
ejpam-4806	257	12	)	)	PUNCT
ejpam-4806	257	13	)	)	PUNCT
ejpam-4806	257	14	1/2	1/2	NUM
ejpam-4806	257	15	=	=	VERB
ejpam-4806	257	16	g(m	g(m	PROPN
ejpam-4806	257	17	)	)	PUNCT
ejpam-4806	257	18	(	(	PUNCT
ejpam-4806	257	19	14	14	NUM
ejpam-4806	257	20	)	)	PUNCT
ejpam-4806	257	21	then	then	ADV
ejpam-4806	257	22	dg	dg	VERB
ejpam-4806	257	23	dm	dm	INTJ
ejpam-4806	257	24	<	<	X
ejpam-4806	257	25	0	0	PUNCT
ejpam-4806	258	1	if	if	SCONJ
ejpam-4806	258	2	and	and	CCONJ
ejpam-4806	258	3	only	only	ADV
ejpam-4806	258	4	if	if	SCONJ
ejpam-4806	258	5	{	{	PUNCT
ejpam-4806	258	6	2√	2√	PROPN
ejpam-4806	258	7	(	(	PUNCT
ejpam-4806	258	8	2δ2	2δ2	NUM
ejpam-4806	258	9	+	+	NUM
ejpam-4806	258	10	4m−	4m−	PROPN
ejpam-4806	258	11	2ηδ	2ηδ	NOUN
ejpam-4806	258	12	+	+	CCONJ
ejpam-4806	259	1	(	(	PUNCT
ejpam-4806	259	2	n+	n+	NUM
ejpam-4806	259	3	2)η	2)η	NUM
ejpam-4806	259	4	−	−	PROPN
ejpam-4806	259	5	κ	κ	NOUN
ejpam-4806	259	6	)	)	PUNCT
ejpam-4806	259	7	}	}	PUNCT
ejpam-4806	259	8	>	>	X
ejpam-4806	259	9	{	{	PUNCT
ejpam-4806	259	10	(	(	PUNCT
ejpam-4806	259	11	−2)√	−2)√	PROPN
ejpam-4806	259	12	(	(	PUNCT
ejpam-4806	259	13	2(∆−	2(∆−	PROPN
ejpam-4806	259	14	δ)2	δ)2	PROPN
ejpam-4806	259	15	+	+	CCONJ
ejpam-4806	259	16	2n(n−	2n(n−	NUM
ejpam-4806	259	17	1)−	1)−	NUM
ejpam-4806	259	18	4m−	4m−	PROPN
ejpam-4806	259	19	2(n−	2(n−	NUM
ejpam-4806	259	20	3	3	NUM
ejpam-4806	259	21	)	)	PUNCT
ejpam-4806	259	22	)	)	PUNCT
ejpam-4806	259	23	}	}	PUNCT
ejpam-4806	259	24	malathy	malathy	ADV
ejpam-4806	259	25	v	v	NOUN
ejpam-4806	259	26	,	,	PUNCT
ejpam-4806	259	27	kalyani	kalyani	PROPN
ejpam-4806	259	28	desikan	desikan	PROPN
ejpam-4806	259	29	/	/	SYM
ejpam-4806	259	30	eur	eur	PROPN
ejpam-4806	259	31	.	.	PUNCT
ejpam-4806	260	1	j.	j.	PROPN
ejpam-4806	260	2	pure	pure	PROPN
ejpam-4806	260	3	appl	appl	PROPN
ejpam-4806	260	4	.	.	PROPN
ejpam-4806	260	5	math	math	PROPN
ejpam-4806	260	6	,	,	PUNCT
ejpam-4806	260	7	16	16	NUM
ejpam-4806	260	8	(	(	PUNCT
ejpam-4806	260	9	3	3	NUM
ejpam-4806	260	10	)	)	PUNCT
ejpam-4806	260	11	(	(	PUNCT
ejpam-4806	260	12	2023	2023	NUM
ejpam-4806	260	13	)	)	PUNCT
ejpam-4806	260	14	,	,	PUNCT
ejpam-4806	260	15	1731	1731	NUM
ejpam-4806	260	16	-	-	SYM
ejpam-4806	260	17	1746	1746	NUM
ejpam-4806	260	18	1743	1743	NUM
ejpam-4806	260	19	by	by	ADP
ejpam-4806	260	20	reducing	reduce	VERB
ejpam-4806	260	21	∆	∆	PROPN
ejpam-4806	260	22	and	and	CCONJ
ejpam-4806	260	23	δ	δ	PROPN
ejpam-4806	260	24	in	in	ADP
ejpam-4806	260	25	terms	term	NOUN
ejpam-4806	260	26	of	of	ADP
ejpam-4806	260	27	n	n	PRON
ejpam-4806	260	28	and	and	CCONJ
ejpam-4806	260	29	solving	solve	VERB
ejpam-4806	260	30	for	for	ADP
ejpam-4806	260	31	m	m	PROPN
ejpam-4806	260	32	,	,	PUNCT
ejpam-4806	260	33	we	we	PRON
ejpam-4806	260	34	have	have	AUX
ejpam-4806	260	35	m	m	PRON
ejpam-4806	260	36	>	>	X
ejpam-4806	260	37	(	(	PUNCT
ejpam-4806	260	38	(	(	PUNCT
ejpam-4806	260	39	4n2	4n2	NUM
ejpam-4806	260	40	−	−	NOUN
ejpam-4806	260	41	12n+	12n+	NUM
ejpam-4806	260	42	8)−	8)−	NOUN
ejpam-4806	260	43	(	(	PUNCT
ejpam-4806	260	44	(	(	PUNCT
ejpam-4806	260	45	n+	n+	X
ejpam-4806	260	46	2)η	2)η	NUM
ejpam-4806	260	47	−	−	PROPN
ejpam-4806	260	48	κ	κ	NOUN
ejpam-4806	260	49	)	)	PUNCT
ejpam-4806	260	50	8	8	NUM
ejpam-4806	260	51	)	)	PUNCT
ejpam-4806	260	52	substituting	substitute	VERB
ejpam-4806	260	53	in	in	ADP
ejpam-4806	260	54	(	(	PUNCT
ejpam-4806	260	55	14	14	NUM
ejpam-4806	260	56	)	)	PUNCT
ejpam-4806	260	57	,	,	PUNCT
ejpam-4806	260	58	we	we	PRON
ejpam-4806	260	59	get	get	VERB
ejpam-4806	260	60	µ	µ	X
ejpam-4806	260	61	(	(	PUNCT
ejpam-4806	260	62	ga	ga	NOUN
ejpam-4806	260	63	)	)	PUNCT
ejpam-4806	261	1	+	+	X
ejpam-4806	261	2	µ	µ	X
ejpam-4806	261	3	(	(	PUNCT
ejpam-4806	261	4	gc	gc	PROPN
ejpam-4806	261	5	a	a	NOUN
ejpam-4806	261	6	)	)	PUNCT
ejpam-4806	261	7	>	>	PUNCT
ejpam-4806	261	8	(	(	PUNCT
ejpam-4806	261	9	2δ2	2δ2	NUM
ejpam-4806	261	10	+	+	CCONJ
ejpam-4806	261	11	4	4	NUM
ejpam-4806	261	12	(	(	PUNCT
ejpam-4806	261	13	(	(	PUNCT
ejpam-4806	261	14	4n2	4n2	NUM
ejpam-4806	261	15	−	−	NOUN
ejpam-4806	262	1	12n+	12n+	NUM
ejpam-4806	262	2	8)−	8)−	NOUN
ejpam-4806	262	3	(	(	PUNCT
ejpam-4806	262	4	(	(	PUNCT
ejpam-4806	262	5	n+	n+	X
ejpam-4806	262	6	2)η	2)η	NUM
ejpam-4806	262	7	−	−	PROPN
ejpam-4806	262	8	κ	κ	NOUN
ejpam-4806	262	9	)	)	PUNCT
ejpam-4806	262	10	8	8	NUM
ejpam-4806	262	11	)	)	PUNCT
ejpam-4806	262	12	−	−	PROPN
ejpam-4806	262	13	2ηδ	2ηδ	ADJ
ejpam-4806	262	14	)	)	PUNCT
ejpam-4806	263	1	+	+	CCONJ
ejpam-4806	263	2	(	(	PUNCT
ejpam-4806	263	3	n+	n+	NUM
ejpam-4806	263	4	2)η	2)η	NUM
ejpam-4806	263	5	−	−	PROPN
ejpam-4806	263	6	κ	κ	NOUN
ejpam-4806	263	7	)	)	PUNCT
ejpam-4806	263	8	1/2	1/2	NUM
ejpam-4806	264	1	+	+	CCONJ
ejpam-4806	264	2	(	(	PUNCT
ejpam-4806	264	3	2(∆−	2(∆−	PROPN
ejpam-4806	264	4	δ)2	δ)2	PROPN
ejpam-4806	264	5	+	+	CCONJ
ejpam-4806	264	6	2n(n−	2n(n−	NUM
ejpam-4806	264	7	1)−	1)−	NUM
ejpam-4806	264	8	4	4	NUM
ejpam-4806	264	9	(	(	PUNCT
ejpam-4806	264	10	(	(	PUNCT
ejpam-4806	264	11	4n2	4n2	NUM
ejpam-4806	264	12	−	−	NOUN
ejpam-4806	264	13	12n+	12n+	NUM
ejpam-4806	264	14	8)−	8)−	NOUN
ejpam-4806	264	15	(	(	PUNCT
ejpam-4806	264	16	(	(	PUNCT
ejpam-4806	264	17	n+	n+	X
ejpam-4806	264	18	2)η	2)η	NUM
ejpam-4806	264	19	−	−	PROPN
ejpam-4806	264	20	κ	κ	NOUN
ejpam-4806	264	21	)	)	PUNCT
ejpam-4806	264	22	8	8	NUM
ejpam-4806	264	23	)	)	PUNCT
ejpam-4806	264	24	−	−	PROPN
ejpam-4806	264	25	2(n−	2(n−	NUM
ejpam-4806	264	26	3	3	NUM
ejpam-4806	264	27	)	)	PUNCT
ejpam-4806	264	28	)	)	PUNCT
ejpam-4806	264	29	1/2	1/2	NUM
ejpam-4806	264	30	therefore	therefore	ADV
ejpam-4806	264	31	µ	µ	X
ejpam-4806	264	32	(	(	PUNCT
ejpam-4806	264	33	ga	ga	NOUN
ejpam-4806	264	34	)	)	PUNCT
ejpam-4806	264	35	+	+	X
ejpam-4806	264	36	µ	µ	X
ejpam-4806	264	37	(	(	PUNCT
ejpam-4806	264	38	gc	gc	PROPN
ejpam-4806	264	39	a	a	NOUN
ejpam-4806	264	40	)	)	PUNCT
ejpam-4806	264	41	>	>	SYM
ejpam-4806	264	42	2	2	NUM
ejpam-4806	264	43	(	(	PUNCT
ejpam-4806	264	44	(	(	PUNCT
ejpam-4806	264	45	4n2	4n2	NUM
ejpam-4806	264	46	−	−	NOUN
ejpam-4806	264	47	20n+	20n+	NUM
ejpam-4806	264	48	40	40	NUM
ejpam-4806	264	49	)	)	PUNCT
ejpam-4806	265	1	+	+	CCONJ
ejpam-4806	265	2	(	(	PUNCT
ejpam-4806	265	3	(	(	PUNCT
ejpam-4806	265	4	n+	n+	X
ejpam-4806	265	5	2)η	2)η	NUM
ejpam-4806	265	6	−	−	PROPN
ejpam-4806	265	7	κ	κ	NOUN
ejpam-4806	265	8	)	)	PUNCT
ejpam-4806	265	9	2	2	NUM
ejpam-4806	265	10	)	)	PUNCT
ejpam-4806	265	11	1/2	1/2	NUM
ejpam-4806	265	12	for	for	ADP
ejpam-4806	265	13	5	5	NUM
ejpam-4806	265	14	≤	≤	NOUN
ejpam-4806	265	15	κ	κ	NOUN
ejpam-4806	265	16	≤	≤	NOUN
ejpam-4806	265	17	(	(	PUNCT
ejpam-4806	265	18	n2	n2	ADJ
ejpam-4806	265	19	−	−	PROPN
ejpam-4806	265	20	4	4	NUM
ejpam-4806	265	21	)	)	PUNCT
ejpam-4806	265	22	.	.	PUNCT
ejpam-4806	266	1	(	(	PUNCT
ejpam-4806	266	2	15	15	NUM
ejpam-4806	266	3	)	)	PUNCT
ejpam-4806	266	4	similarly	similarly	ADV
ejpam-4806	266	5	,	,	PUNCT
ejpam-4806	266	6	we	we	PRON
ejpam-4806	266	7	derive	derive	VERB
ejpam-4806	266	8	the	the	DET
ejpam-4806	266	9	condition	condition	NOUN
ejpam-4806	266	10	for	for	ADP
ejpam-4806	266	11	the	the	DET
ejpam-4806	266	12	existence	existence	NOUN
ejpam-4806	266	13	of	of	ADP
ejpam-4806	266	14	the	the	DET
ejpam-4806	266	15	upper	upper	ADJ
ejpam-4806	266	16	bound	bind	VERB
ejpam-4806	266	17	.	.	PUNCT
ejpam-4806	267	1	we	we	PRON
ejpam-4806	267	2	consider	consider	VERB
ejpam-4806	267	3	the	the	DET
ejpam-4806	267	4	upper	upper	ADJ
ejpam-4806	267	5	bound	bind	VERB
ejpam-4806	267	6	given	give	VERB
ejpam-4806	267	7	in	in	ADP
ejpam-4806	267	8	equation	equation	NOUN
ejpam-4806	267	9	(	(	PUNCT
ejpam-4806	267	10	8)	8)	NUM
ejpam-4806	267	11	of	of	ADP
ejpam-4806	267	12	theorem	theorem	ADJ
ejpam-4806	267	13	2	2	NUM
ejpam-4806	267	14	µ	µ	X
ejpam-4806	267	15	(	(	PUNCT
ejpam-4806	267	16	ga	ga	PROPN
ejpam-4806	267	17	)	)	PUNCT
ejpam-4806	267	18	<	<	X
ejpam-4806	268	1	(	(	PUNCT
ejpam-4806	268	2	2∆2	2∆2	NUM
ejpam-4806	268	3	+	+	NUM
ejpam-4806	268	4	2(2m−∆)−	2(2m−∆)−	NUM
ejpam-4806	268	5	(	(	PUNCT
ejpam-4806	268	6	nη	nη	ADP
ejpam-4806	268	7	−	−	PROPN
ejpam-4806	268	8	κ′	κ′	NOUN
ejpam-4806	268	9	)	)	PUNCT
ejpam-4806	268	10	)	)	PUNCT
ejpam-4806	268	11	1/2	1/2	NUM
ejpam-4806	268	12	(	(	PUNCT
ejpam-4806	268	13	16	16	NUM
ejpam-4806	268	14	)	)	PUNCT
ejpam-4806	268	15	µ	µ	X
ejpam-4806	268	16	(	(	PUNCT
ejpam-4806	268	17	gc	gc	PROPN
ejpam-4806	268	18	a	a	PRON
ejpam-4806	268	19	)	)	PUNCT
ejpam-4806	268	20	=	=	SYM
ejpam-4806	268	21	(	(	PUNCT
ejpam-4806	268	22	2(∆−	2(∆−	NUM
ejpam-4806	268	23	δ)2	δ)2	NOUN
ejpam-4806	268	24	+	+	CCONJ
ejpam-4806	268	25	2(2mc	2(2mc	PROPN
ejpam-4806	268	26	−∆c	−∆c	NUM
ejpam-4806	268	27	)	)	PUNCT
ejpam-4806	268	28	)	)	PUNCT
ejpam-4806	269	1	1/2	1/2	NUM
ejpam-4806	269	2	where	where	SCONJ
ejpam-4806	269	3	mc	mc	PROPN
ejpam-4806	269	4	=	=	PUNCT
ejpam-4806	269	5	nc2	nc2	PROPN
ejpam-4806	269	6	−m	−m	PROPN
ejpam-4806	269	7	µ	µ	X
ejpam-4806	269	8	(	(	PUNCT
ejpam-4806	269	9	gc	gc	PROPN
ejpam-4806	269	10	a	a	PRON
ejpam-4806	269	11	)	)	PUNCT
ejpam-4806	269	12	=	=	SYM
ejpam-4806	269	13	(	(	PUNCT
ejpam-4806	269	14	2(∆−	2(∆−	NUM
ejpam-4806	269	15	δ)2	δ)2	NOUN
ejpam-4806	269	16	+	+	CCONJ
ejpam-4806	270	1	2(n(n−	2(n(n−	NUM
ejpam-4806	270	2	1)−	1)−	PROPN
ejpam-4806	270	3	2m−	2m−	PROPN
ejpam-4806	270	4	(	(	PUNCT
ejpam-4806	270	5	n−	n−	NOUN
ejpam-4806	270	6	3	3	NUM
ejpam-4806	270	7	)	)	PUNCT
ejpam-4806	270	8	)	)	PUNCT
ejpam-4806	270	9	1/2	1/2	NUM
ejpam-4806	270	10	(	(	PUNCT
ejpam-4806	270	11	17	17	NUM
ejpam-4806	270	12	)	)	PUNCT
ejpam-4806	270	13	µ	µ	X
ejpam-4806	270	14	(	(	PUNCT
ejpam-4806	270	15	ga	ga	NOUN
ejpam-4806	270	16	)	)	PUNCT
ejpam-4806	270	17	+	+	X
ejpam-4806	270	18	µ	µ	X
ejpam-4806	270	19	(	(	PUNCT
ejpam-4806	270	20	gc	gc	PROPN
ejpam-4806	270	21	a	a	NOUN
ejpam-4806	270	22	)	)	PUNCT
ejpam-4806	270	23	<	<	X
ejpam-4806	270	24	(	(	PUNCT
ejpam-4806	270	25	2∆2	2∆2	NUM
ejpam-4806	270	26	+	+	NUM
ejpam-4806	270	27	4m−	4m−	PROPN
ejpam-4806	270	28	2∆−	2∆−	NUM
ejpam-4806	270	29	(	(	PUNCT
ejpam-4806	270	30	nη	nη	PROPN
ejpam-4806	270	31	−	−	PROPN
ejpam-4806	270	32	κ′	κ′	NOUN
ejpam-4806	270	33	)	)	PUNCT
ejpam-4806	270	34	)	)	PUNCT
ejpam-4806	270	35	1/2	1/2	NUM
ejpam-4806	271	1	+	+	CCONJ
ejpam-4806	271	2	(	(	PUNCT
ejpam-4806	271	3	2(∆−	2(∆−	PROPN
ejpam-4806	271	4	δ)2	δ)2	PROPN
ejpam-4806	271	5	+	+	CCONJ
ejpam-4806	271	6	2n(n−	2n(n−	NUM
ejpam-4806	272	1	1)−	1)−	NUM
ejpam-4806	272	2	4m−	4m−	PROPN
ejpam-4806	272	3	2(n−	2(n−	NUM
ejpam-4806	272	4	3	3	NUM
ejpam-4806	272	5	)	)	PUNCT
ejpam-4806	272	6	)	)	PUNCT
ejpam-4806	272	7	1/2	1/2	NUM
ejpam-4806	272	8	=	=	VERB
ejpam-4806	272	9	g(m	g(m	PROPN
ejpam-4806	272	10	)	)	PUNCT
ejpam-4806	272	11	(	(	PUNCT
ejpam-4806	272	12	18	18	NUM
ejpam-4806	272	13	)	)	PUNCT
ejpam-4806	272	14	then	then	ADV
ejpam-4806	272	15	dg	dg	VERB
ejpam-4806	272	16	dm	dm	INTJ
ejpam-4806	272	17	>	>	X
ejpam-4806	272	18	0	0	PUNCT
ejpam-4806	273	1	if	if	SCONJ
ejpam-4806	273	2	and	and	CCONJ
ejpam-4806	273	3	only	only	ADV
ejpam-4806	273	4	if	if	SCONJ
ejpam-4806	273	5	{	{	PUNCT
ejpam-4806	273	6	2√	2√	PROPN
ejpam-4806	273	7	(	(	PUNCT
ejpam-4806	273	8	2∆2	2∆2	NUM
ejpam-4806	273	9	+	+	NUM
ejpam-4806	273	10	4m−	4m−	PROPN
ejpam-4806	273	11	2∆−	2∆−	NUM
ejpam-4806	273	12	(	(	PUNCT
ejpam-4806	273	13	nη	nη	PROPN
ejpam-4806	273	14	−	−	PROPN
ejpam-4806	273	15	κ′	κ′	NOUN
ejpam-4806	273	16	)	)	PUNCT
ejpam-4806	273	17	)	)	PUNCT
ejpam-4806	273	18	}	}	PUNCT
ejpam-4806	273	19	<	<	X
ejpam-4806	273	20	{	{	PUNCT
ejpam-4806	273	21	(	(	PUNCT
ejpam-4806	273	22	−2)√	−2)√	PROPN
ejpam-4806	273	23	(	(	PUNCT
ejpam-4806	273	24	2(∆−	2(∆−	PROPN
ejpam-4806	273	25	δ)2	δ)2	PROPN
ejpam-4806	273	26	+	+	CCONJ
ejpam-4806	273	27	2n(n−	2n(n−	NUM
ejpam-4806	274	1	1)−	1)−	NUM
ejpam-4806	274	2	4m−	4m−	PROPN
ejpam-4806	274	3	2(n−	2(n−	NUM
ejpam-4806	274	4	3	3	NUM
ejpam-4806	274	5	)	)	PUNCT
ejpam-4806	274	6	)	)	PUNCT
ejpam-4806	274	7	}	}	PUNCT
ejpam-4806	274	8	malathy	malathy	ADV
ejpam-4806	274	9	v	v	NOUN
ejpam-4806	274	10	,	,	PUNCT
ejpam-4806	274	11	kalyani	kalyani	PROPN
ejpam-4806	274	12	desikan	desikan	PROPN
ejpam-4806	274	13	/	/	SYM
ejpam-4806	274	14	eur	eur	PROPN
ejpam-4806	274	15	.	.	PUNCT
ejpam-4806	275	1	j.	j.	PROPN
ejpam-4806	275	2	pure	pure	PROPN
ejpam-4806	275	3	appl	appl	PROPN
ejpam-4806	275	4	.	.	PROPN
ejpam-4806	275	5	math	math	PROPN
ejpam-4806	275	6	,	,	PUNCT
ejpam-4806	275	7	16	16	NUM
ejpam-4806	275	8	(	(	PUNCT
ejpam-4806	275	9	3	3	NUM
ejpam-4806	275	10	)	)	PUNCT
ejpam-4806	275	11	(	(	PUNCT
ejpam-4806	275	12	2023	2023	NUM
ejpam-4806	275	13	)	)	PUNCT
ejpam-4806	275	14	,	,	PUNCT
ejpam-4806	275	15	1731	1731	NUM
ejpam-4806	275	16	-	-	SYM
ejpam-4806	275	17	1746	1746	NUM
ejpam-4806	275	18	1744	1744	NUM
ejpam-4806	275	19	reducing	reduce	VERB
ejpam-4806	275	20	∆	∆	PROPN
ejpam-4806	275	21	and	and	CCONJ
ejpam-4806	275	22	δ	δ	PROPN
ejpam-4806	275	23	in	in	ADP
ejpam-4806	275	24	terms	term	NOUN
ejpam-4806	275	25	of	of	ADP
ejpam-4806	275	26	n	n	PRON
ejpam-4806	275	27	and	and	CCONJ
ejpam-4806	275	28	solving	solve	VERB
ejpam-4806	275	29	for	for	ADP
ejpam-4806	275	30	m	m	PROPN
ejpam-4806	275	31	,	,	PUNCT
ejpam-4806	275	32	we	we	PRON
ejpam-4806	275	33	get	get	VERB
ejpam-4806	275	34	m	m	PRON
ejpam-4806	275	35	<	<	X
ejpam-4806	275	36	(	(	PUNCT
ejpam-4806	275	37	(	(	PUNCT
ejpam-4806	275	38	2n2	2n2	NUM
ejpam-4806	275	39	−	−	PROPN
ejpam-4806	275	40	10n+	10n+	NUM
ejpam-4806	275	41	20	20	NUM
ejpam-4806	275	42	)	)	PUNCT
ejpam-4806	276	1	+	+	CCONJ
ejpam-4806	276	2	(	(	PUNCT
ejpam-4806	276	3	nη	nη	ADP
ejpam-4806	276	4	−	−	PROPN
ejpam-4806	276	5	κ′	κ′	NOUN
ejpam-4806	276	6	)	)	PUNCT
ejpam-4806	276	7	8	8	NUM
ejpam-4806	276	8	)	)	PUNCT
ejpam-4806	276	9	substituting	substitute	VERB
ejpam-4806	276	10	the	the	DET
ejpam-4806	276	11	value	value	NOUN
ejpam-4806	276	12	of	of	ADP
ejpam-4806	276	13	m	m	PRON
ejpam-4806	276	14	in	in	ADP
ejpam-4806	276	15	(	(	PUNCT
ejpam-4806	276	16	18	18	NUM
ejpam-4806	276	17	)	)	PUNCT
ejpam-4806	276	18	,	,	PUNCT
ejpam-4806	276	19	we	we	PRON
ejpam-4806	276	20	get	get	VERB
ejpam-4806	276	21	µ	µ	X
ejpam-4806	276	22	(	(	PUNCT
ejpam-4806	276	23	ga	ga	NOUN
ejpam-4806	276	24	)	)	PUNCT
ejpam-4806	276	25	+	+	X
ejpam-4806	276	26	µ	µ	X
ejpam-4806	276	27	(	(	PUNCT
ejpam-4806	276	28	gc	gc	PROPN
ejpam-4806	276	29	a	a	NOUN
ejpam-4806	276	30	)	)	PUNCT
ejpam-4806	276	31	<	<	X
ejpam-4806	276	32	2	2	NUM
ejpam-4806	276	33	(	(	PUNCT
ejpam-4806	276	34	(	(	PUNCT
ejpam-4806	276	35	6n2	6n2	NUM
ejpam-4806	276	36	−	−	NUM
ejpam-4806	276	37	22n+	22n+	NUM
ejpam-4806	276	38	28)−	28)−	NUM
ejpam-4806	276	39	(	(	PUNCT
ejpam-4806	276	40	nη	nη	NUM
ejpam-4806	276	41	−	−	PROPN
ejpam-4806	276	42	κ′	κ′	NOUN
ejpam-4806	276	43	)	)	PUNCT
ejpam-4806	276	44	2	2	NUM
ejpam-4806	276	45	)	)	PUNCT
ejpam-4806	276	46	1/2	1/2	NUM
ejpam-4806	276	47	for	for	ADP
ejpam-4806	276	48	4	4	NUM
ejpam-4806	276	49	≤	≤	NUM
ejpam-4806	276	50	κ′	κ′	NOUN
ejpam-4806	276	51	≤	≤	PROPN
ejpam-4806	276	52	(	(	PUNCT
ejpam-4806	276	53	n(n−	n(n−	NOUN
ejpam-4806	276	54	2	2	NUM
ejpam-4806	276	55	)	)	PUNCT
ejpam-4806	276	56	)	)	PUNCT
ejpam-4806	276	57	.	.	PUNCT
ejpam-4806	277	1	therefore	therefore	ADV
ejpam-4806	277	2	2	2	NUM
ejpam-4806	277	3	(	(	PUNCT
ejpam-4806	277	4	(	(	PUNCT
ejpam-4806	277	5	4n2	4n2	NUM
ejpam-4806	277	6	−	−	NOUN
ejpam-4806	277	7	20n+	20n+	NUM
ejpam-4806	277	8	40	40	NUM
ejpam-4806	277	9	)	)	PUNCT
ejpam-4806	278	1	+	+	CCONJ
ejpam-4806	278	2	(	(	PUNCT
ejpam-4806	278	3	(	(	PUNCT
ejpam-4806	278	4	n+	n+	X
ejpam-4806	278	5	2)η	2)η	NUM
ejpam-4806	278	6	−	−	PROPN
ejpam-4806	278	7	κ	κ	NOUN
ejpam-4806	278	8	)	)	PUNCT
ejpam-4806	278	9	2	2	NUM
ejpam-4806	278	10	)	)	PUNCT
ejpam-4806	278	11	1/2	1/2	NUM
ejpam-4806	278	12	<	<	X
ejpam-4806	278	13	µ	µ	X
ejpam-4806	278	14	(	(	PUNCT
ejpam-4806	278	15	ga	ga	NOUN
ejpam-4806	278	16	)	)	PUNCT
ejpam-4806	278	17	+	+	X
ejpam-4806	278	18	µ	µ	X
ejpam-4806	278	19	(	(	PUNCT
ejpam-4806	278	20	gc	gc	PROPN
ejpam-4806	278	21	a	a	NOUN
ejpam-4806	278	22	)	)	PUNCT
ejpam-4806	278	23	<	<	X
ejpam-4806	278	24	2	2	NUM
ejpam-4806	278	25	(	(	PUNCT
ejpam-4806	278	26	(	(	PUNCT
ejpam-4806	278	27	6n2	6n2	NUM
ejpam-4806	278	28	−	−	NUM
ejpam-4806	278	29	22n+	22n+	NUM
ejpam-4806	278	30	28)−	28)−	NUM
ejpam-4806	278	31	(	(	PUNCT
ejpam-4806	278	32	nη	nη	NUM
ejpam-4806	278	33	−	−	PROPN
ejpam-4806	278	34	κ′	κ′	NOUN
ejpam-4806	278	35	)	)	PUNCT
ejpam-4806	278	36	2	2	NUM
ejpam-4806	278	37	)	)	PUNCT
ejpam-4806	278	38	1/2	1/2	NUM
ejpam-4806	278	39	for	for	ADP
ejpam-4806	278	40	5	5	NUM
ejpam-4806	278	41	≤	≤	NOUN
ejpam-4806	278	42	κ	κ	NOUN
ejpam-4806	278	43	≤	≤	NOUN
ejpam-4806	278	44	(	(	PUNCT
ejpam-4806	278	45	n2	n2	ADJ
ejpam-4806	278	46	−	−	PROPN
ejpam-4806	278	47	4	4	NUM
ejpam-4806	278	48	)	)	PUNCT
ejpam-4806	278	49	and	and	CCONJ
ejpam-4806	278	50	4	4	NUM
ejpam-4806	278	51	≤	≤	NUM
ejpam-4806	278	52	κ′	κ′	NOUN
ejpam-4806	278	53	≤	≤	X
ejpam-4806	278	54	n(n−	n(n−	PROPN
ejpam-4806	278	55	2	2	NUM
ejpam-4806	278	56	)	)	PUNCT
ejpam-4806	278	57	.	.	PUNCT
ejpam-4806	279	1	hence	hence	ADV
ejpam-4806	279	2	proved	prove	VERB
ejpam-4806	279	3	.	.	PUNCT
ejpam-4806	280	1	remark	remark	PROPN
ejpam-4806	280	2	3	3	NUM
ejpam-4806	280	3	.	.	PUNCT
ejpam-4806	281	1	it	it	PRON
ejpam-4806	281	2	can	can	AUX
ejpam-4806	281	3	be	be	AUX
ejpam-4806	281	4	easily	easily	ADV
ejpam-4806	281	5	verified	verify	VERB
ejpam-4806	281	6	from	from	ADP
ejpam-4806	281	7	equation	equation	NOUN
ejpam-4806	281	8	(	(	PUNCT
ejpam-4806	281	9	13	13	NUM
ejpam-4806	281	10	)	)	PUNCT
ejpam-4806	281	11	of	of	ADP
ejpam-4806	281	12	theorem	theorem	NOUN
ejpam-4806	281	13	3	3	NUM
ejpam-4806	281	14	,	,	PUNCT
ejpam-4806	281	15	for	for	ADP
ejpam-4806	281	16	the	the	DET
ejpam-4806	281	17	graph	graph	NOUN
ejpam-4806	281	18	k2	k2	ADJ
ejpam-4806	281	19	▽	▽	ADJ
ejpam-4806	281	20	5k1	5k1	PROPN
ejpam-4806	281	21	,	,	PUNCT
ejpam-4806	281	22	where	where	SCONJ
ejpam-4806	281	23	n	n	NOUN
ejpam-4806	281	24	=	=	SYM
ejpam-4806	281	25	7	7	NUM
ejpam-4806	281	26	,	,	PUNCT
ejpam-4806	281	27	m	m	VERB
ejpam-4806	281	28	=	=	NOUN
ejpam-4806	281	29	11	11	NUM
ejpam-4806	281	30	,	,	PUNCT
ejpam-4806	281	31	∆	∆	X
ejpam-4806	281	32	=	=	SYM
ejpam-4806	281	33	6	6	NUM
ejpam-4806	281	34	,	,	PUNCT
ejpam-4806	281	35	δ	δ	PROPN
ejpam-4806	281	36	=	=	SYM
ejpam-4806	281	37	2	2	NUM
ejpam-4806	281	38	,	,	PUNCT
ejpam-4806	281	39	η	η	NOUN
ejpam-4806	281	40	=	=	SYM
ejpam-4806	281	41	5	5	NUM
ejpam-4806	281	42	,	,	PUNCT
ejpam-4806	281	43	κ	κ	X
ejpam-4806	281	44	=	=	SYM
ejpam-4806	281	45	5	5	NUM
ejpam-4806	281	46	and	and	CCONJ
ejpam-4806	281	47	κ′	κ′	NOUN
ejpam-4806	282	1	=	=	NOUN
ejpam-4806	282	2	4	4	NUM
ejpam-4806	282	3	,	,	PUNCT
ejpam-4806	282	4	the	the	DET
ejpam-4806	282	5	bounds	bound	NOUN
ejpam-4806	282	6	are	be	AUX
ejpam-4806	282	7	16.4924	16.4924	NUM
ejpam-4806	282	8	<	<	X
ejpam-4806	282	9	µ	µ	X
ejpam-4806	282	10	(	(	PUNCT
ejpam-4806	282	11	ga	ga	NOUN
ejpam-4806	282	12	)	)	PUNCT
ejpam-4806	282	13	+	+	X
ejpam-4806	282	14	µ	µ	X
ejpam-4806	282	15	(	(	PUNCT
ejpam-4806	282	16	gc	gc	PROPN
ejpam-4806	282	17	a	a	NOUN
ejpam-4806	282	18	)	)	PUNCT
ejpam-4806	282	19	<	<	X
ejpam-4806	282	20	16.553	16.553	NUM
ejpam-4806	282	21	5	5	NUM
ejpam-4806	282	22	.	.	PUNCT
ejpam-4806	282	23	discussion	discussion	NOUN
ejpam-4806	282	24	in	in	ADP
ejpam-4806	282	25	this	this	DET
ejpam-4806	282	26	paper	paper	NOUN
ejpam-4806	282	27	,	,	PUNCT
ejpam-4806	282	28	we	we	PRON
ejpam-4806	282	29	have	have	AUX
ejpam-4806	282	30	utilized	utilize	VERB
ejpam-4806	282	31	a	a	DET
ejpam-4806	282	32	new	new	ADJ
ejpam-4806	282	33	technique	technique	NOUN
ejpam-4806	282	34	to	to	PART
ejpam-4806	282	35	improve	improve	VERB
ejpam-4806	282	36	the	the	DET
ejpam-4806	282	37	bounds	bound	NOUN
ejpam-4806	282	38	and	and	CCONJ
ejpam-4806	282	39	obtain	obtain	VERB
ejpam-4806	282	40	the	the	DET
ejpam-4806	282	41	tight	tight	ADJ
ejpam-4806	282	42	upper	upper	ADJ
ejpam-4806	282	43	and	and	CCONJ
ejpam-4806	282	44	lower	low	ADJ
ejpam-4806	282	45	bounds	bound	NOUN
ejpam-4806	282	46	for	for	ADP
ejpam-4806	282	47	the	the	DET
ejpam-4806	282	48	signless	signless	PROPN
ejpam-4806	282	49	laplacian	laplacian	ADJ
ejpam-4806	282	50	spectral	spectral	ADJ
ejpam-4806	282	51	radius	radius	NOUN
ejpam-4806	282	52	of	of	ADP
ejpam-4806	282	53	agave	agave	ADJ
ejpam-4806	282	54	class	class	NOUN
ejpam-4806	282	55	of	of	ADP
ejpam-4806	282	56	graphs	graph	NOUN
ejpam-4806	282	57	.	.	PUNCT
ejpam-4806	283	1	these	these	DET
ejpam-4806	283	2	bounds	bound	NOUN
ejpam-4806	283	3	are	be	AUX
ejpam-4806	283	4	in	in	ADP
ejpam-4806	283	5	terms	term	NOUN
ejpam-4806	283	6	of	of	ADP
ejpam-4806	283	7	the	the	DET
ejpam-4806	283	8	maximum	maximum	ADJ
ejpam-4806	283	9	degree	degree	NOUN
ejpam-4806	283	10	∆	∆	PROPN
ejpam-4806	283	11	,	,	PUNCT
ejpam-4806	283	12	the	the	DET
ejpam-4806	283	13	minimum	minimum	NOUN
ejpam-4806	283	14	degree	degree	NOUN
ejpam-4806	283	15	δ	δ	PROPN
ejpam-4806	283	16	,	,	PUNCT
ejpam-4806	283	17	number	number	NOUN
ejpam-4806	283	18	of	of	ADP
ejpam-4806	283	19	satellites	satellite	NOUN
ejpam-4806	283	20	η	η	PROPN
ejpam-4806	283	21	,	,	PUNCT
ejpam-4806	283	22	n	n	CCONJ
ejpam-4806	283	23	the	the	DET
ejpam-4806	283	24	number	number	NOUN
ejpam-4806	283	25	of	of	ADP
ejpam-4806	283	26	vertices	vertex	NOUN
ejpam-4806	283	27	and	and	CCONJ
ejpam-4806	283	28	m	m	VERB
ejpam-4806	283	29	the	the	DET
ejpam-4806	283	30	number	number	NOUN
ejpam-4806	283	31	of	of	ADP
ejpam-4806	283	32	edges	edge	NOUN
ejpam-4806	283	33	of	of	ADP
ejpam-4806	283	34	the	the	DET
ejpam-4806	283	35	graph	graph	NOUN
ejpam-4806	283	36	ga	ga	PROPN
ejpam-4806	283	37	.	.	PUNCT
ejpam-4806	284	1	they	they	PRON
ejpam-4806	284	2	are	be	AUX
ejpam-4806	284	3	exceptionally	exceptionally	ADV
ejpam-4806	284	4	proximate	proximate	ADJ
ejpam-4806	284	5	to	to	ADP
ejpam-4806	284	6	µ	µ	PROPN
ejpam-4806	284	7	(	(	PUNCT
ejpam-4806	284	8	ga	ga	PROPN
ejpam-4806	284	9	)	)	PUNCT
ejpam-4806	284	10	.	.	PUNCT
ejpam-4806	285	1	also	also	ADV
ejpam-4806	285	2	,	,	PUNCT
ejpam-4806	285	3	we	we	PRON
ejpam-4806	285	4	have	have	AUX
ejpam-4806	285	5	derived	derive	VERB
ejpam-4806	285	6	new	new	ADJ
ejpam-4806	285	7	improved	improve	VERB
ejpam-4806	285	8	upper	upper	ADJ
ejpam-4806	285	9	and	and	CCONJ
ejpam-4806	285	10	lower	low	ADJ
ejpam-4806	285	11	bounds	bound	NOUN
ejpam-4806	285	12	for	for	ADP
ejpam-4806	285	13	the	the	DET
ejpam-4806	285	14	nordhaus	nordhaus	NOUN
ejpam-4806	285	15	-	-	PUNCT
ejpam-4806	285	16	gaddum	gaddum	NOUN
ejpam-4806	285	17	type	type	NOUN
ejpam-4806	285	18	inequality	inequality	NOUN
ejpam-4806	285	19	.	.	PUNCT
ejpam-4806	286	1	we	we	PRON
ejpam-4806	286	2	anticipate	anticipate	VERB
ejpam-4806	286	3	that	that	SCONJ
ejpam-4806	286	4	the	the	DET
ejpam-4806	286	5	technique	technique	NOUN
ejpam-4806	286	6	used	use	VERB
ejpam-4806	286	7	to	to	PART
ejpam-4806	286	8	obtain	obtain	VERB
ejpam-4806	286	9	these	these	DET
ejpam-4806	286	10	bounds	bound	NOUN
ejpam-4806	286	11	will	will	AUX
ejpam-4806	286	12	be	be	AUX
ejpam-4806	286	13	helpful	helpful	ADJ
ejpam-4806	286	14	in	in	ADP
ejpam-4806	286	15	determining	determine	VERB
ejpam-4806	286	16	the	the	DET
ejpam-4806	286	17	tight	tight	ADV
ejpam-4806	286	18	lower	low	ADJ
ejpam-4806	286	19	and	and	CCONJ
ejpam-4806	286	20	the	the	DET
ejpam-4806	286	21	upper	upper	ADJ
ejpam-4806	286	22	bounds	bound	NOUN
ejpam-4806	286	23	for	for	ADP
ejpam-4806	286	24	the	the	DET
ejpam-4806	286	25	signless	signless	PROPN
ejpam-4806	286	26	laplacian	laplacian	ADJ
ejpam-4806	286	27	spectral	spectral	ADJ
ejpam-4806	286	28	radius	radius	NOUN
ejpam-4806	286	29	of	of	ADP
ejpam-4806	286	30	any	any	DET
ejpam-4806	286	31	general	general	ADJ
ejpam-4806	286	32	class	class	NOUN
ejpam-4806	286	33	of	of	ADP
ejpam-4806	286	34	graphs	graph	NOUN
ejpam-4806	286	35	.	.	PUNCT
ejpam-4806	287	1	acknowledgements	acknowledgement	NOUN
ejpam-4806	287	2	the	the	DET
ejpam-4806	287	3	authors	author	NOUN
ejpam-4806	287	4	are	be	AUX
ejpam-4806	287	5	grateful	grateful	ADJ
ejpam-4806	287	6	to	to	ADP
ejpam-4806	287	7	the	the	DET
ejpam-4806	287	8	reviewers	reviewer	NOUN
ejpam-4806	287	9	for	for	ADP
ejpam-4806	287	10	their	their	PRON
ejpam-4806	287	11	valuable	valuable	ADJ
ejpam-4806	287	12	comments	comment	NOUN
ejpam-4806	287	13	which	which	PRON
ejpam-4806	287	14	have	have	AUX
ejpam-4806	287	15	enhanced	enhance	VERB
ejpam-4806	287	16	the	the	DET
ejpam-4806	287	17	article	article	NOUN
ejpam-4806	287	18	.	.	PUNCT
ejpam-4806	288	1	references	reference	NOUN
ejpam-4806	288	2	1745	1745	NUM
ejpam-4806	288	3	references	reference	NOUN
ejpam-4806	288	4	[	[	X
ejpam-4806	288	5	1	1	NUM
ejpam-4806	288	6	]	]	PUNCT
ejpam-4806	288	7	n	n	PRON
ejpam-4806	288	8	abreu	abreu	NOUN
ejpam-4806	288	9	,	,	PUNCT
ejpam-4806	288	10	justel	justel	NOUN
ejpam-4806	288	11	c.m	c.m	PROPN
ejpam-4806	288	12	,	,	PUNCT
ejpam-4806	288	13	and	and	CCONJ
ejpam-4806	288	14	l.	l.	PROPN
ejpam-4806	288	15	markenzon	markenzon	NOUN
ejpam-4806	288	16	.	.	PUNCT
ejpam-4806	289	1	integer	integer	PROPN
ejpam-4806	289	2	laplacian	laplacian	PROPN
ejpam-4806	289	3	eigenvalues	eigenvalue	NOUN
ejpam-4806	289	4	of	of	ADP
ejpam-4806	289	5	chordal	chordal	NOUN
ejpam-4806	289	6	graphs	graph	NOUN
ejpam-4806	289	7	.	.	PUNCT
ejpam-4806	290	1	linear	linear	ADJ
ejpam-4806	290	2	algebra	algebra	NOUN
ejpam-4806	290	3	and	and	CCONJ
ejpam-4806	290	4	its	its	PRON
ejpam-4806	290	5	applications	application	NOUN
ejpam-4806	290	6	,	,	PUNCT
ejpam-4806	290	7	614:68–81	614:68–81	NUM
ejpam-4806	290	8	,	,	PUNCT
ejpam-4806	290	9	2021	2021	NUM
ejpam-4806	290	10	.	.	PUNCT
ejpam-4806	291	1	[	[	X
ejpam-4806	291	2	2	2	NUM
ejpam-4806	291	3	]	]	PUNCT
ejpam-4806	291	4	m	m	VERB
ejpam-4806	291	5	aouchiche	aouchiche	NOUN
ejpam-4806	291	6	and	and	CCONJ
ejpam-4806	291	7	hansen	hansen	PROPN
ejpam-4806	291	8	.	.	PUNCT
ejpam-4806	292	1	p.	p.	NOUN
ejpam-4806	292	2	a	a	DET
ejpam-4806	292	3	survey	survey	NOUN
ejpam-4806	292	4	of	of	ADP
ejpam-4806	292	5	nordhaus	nordhaus	NOUN
ejpam-4806	292	6	–	–	PUNCT
ejpam-4806	292	7	gaddum	gaddum	NOUN
ejpam-4806	292	8	type	type	NOUN
ejpam-4806	292	9	relations	relation	NOUN
ejpam-4806	292	10	.	.	PUNCT
ejpam-4806	293	1	discrete	discrete	ADJ
ejpam-4806	293	2	applied	applied	ADJ
ejpam-4806	293	3	mathematics	mathematic	NOUN
ejpam-4806	293	4	,	,	PUNCT
ejpam-4806	293	5	161:466	161:466	ADJ
ejpam-4806	293	6	–	–	PUNCT
ejpam-4806	293	7	546	546	NUM
ejpam-4806	293	8	,	,	PUNCT
ejpam-4806	293	9	2013	2013	NUM
ejpam-4806	293	10	.	.	PUNCT
ejpam-4806	294	1	[	[	X
ejpam-4806	294	2	3	3	X
ejpam-4806	294	3	]	]	PUNCT
ejpam-4806	294	4	yanqing	yanqe	VERB
ejpam-4806	294	5	chen	chen	PROPN
ejpam-4806	294	6	and	and	CCONJ
ejpam-4806	294	7	ligong	ligong	PROPN
ejpam-4806	294	8	wang	wang	PROPN
ejpam-4806	294	9	.	.	PUNCT
ejpam-4806	295	1	sharp	sharp	ADJ
ejpam-4806	295	2	bounds	bound	NOUN
ejpam-4806	295	3	for	for	ADP
ejpam-4806	295	4	the	the	DET
ejpam-4806	295	5	largest	large	ADJ
ejpam-4806	295	6	eigenvalue	eigenvalue	NOUN
ejpam-4806	295	7	of	of	ADP
ejpam-4806	295	8	the	the	DET
ejpam-4806	295	9	signless	signless	NOUN
ejpam-4806	295	10	laplacian	laplacian	NOUN
ejpam-4806	295	11	of	of	ADP
ejpam-4806	295	12	a	a	DET
ejpam-4806	295	13	graph	graph	NOUN
ejpam-4806	295	14	.	.	PUNCT
ejpam-4806	296	1	linear	linear	ADJ
ejpam-4806	296	2	algebra	algebra	NOUN
ejpam-4806	296	3	and	and	CCONJ
ejpam-4806	296	4	its	its	PRON
ejpam-4806	296	5	applications	application	NOUN
ejpam-4806	296	6	,	,	PUNCT
ejpam-4806	296	7	433:908–913	433:908–913	NUM
ejpam-4806	296	8	,	,	PUNCT
ejpam-4806	296	9	2010	2010	NUM
ejpam-4806	296	10	.	.	PUNCT
ejpam-4806	297	1	[	[	X
ejpam-4806	297	2	4	4	X
ejpam-4806	297	3	]	]	X
ejpam-4806	297	4	dragoš	dragoš	VERB
ejpam-4806	297	5	cvetkovic	cvetkovic	VERB
ejpam-4806	297	6	,	,	PUNCT
ejpam-4806	297	7	peter	peter	PROPN
ejpam-4806	297	8	rowlinson	rowlinson	PROPN
ejpam-4806	297	9	,	,	PUNCT
ejpam-4806	297	10	and	and	CCONJ
ejpam-4806	297	11	slobodan	slobodan	PROPN
ejpam-4806	297	12	k.	k.	PROPN
ejpam-4806	297	13	simic	simic	PROPN
ejpam-4806	297	14	.	.	PUNCT
ejpam-4806	298	1	signless	signless	ADJ
ejpam-4806	298	2	laplacians	laplacian	NOUN
ejpam-4806	298	3	of	of	ADP
ejpam-4806	298	4	finite	finite	ADJ
ejpam-4806	298	5	graphs	graph	NOUN
ejpam-4806	298	6	.	.	PUNCT
ejpam-4806	299	1	linear	linear	ADJ
ejpam-4806	299	2	algebra	algebra	NOUN
ejpam-4806	299	3	and	and	CCONJ
ejpam-4806	299	4	its	its	PRON
ejpam-4806	299	5	applications	application	NOUN
ejpam-4806	299	6	,	,	PUNCT
ejpam-4806	299	7	423:155–171	423:155–171	NUM
ejpam-4806	299	8	,	,	PUNCT
ejpam-4806	299	9	2007	2007	NUM
ejpam-4806	299	10	.	.	PUNCT
ejpam-4806	300	1	[	[	X
ejpam-4806	300	2	5	5	NUM
ejpam-4806	300	3	]	]	PUNCT
ejpam-4806	300	4	x	x	PROPN
ejpam-4806	300	5	duan	duan	PROPN
ejpam-4806	300	6	and	and	CCONJ
ejpam-4806	300	7	zhou	zhou	PROPN
ejpam-4806	300	8	.	.	PUNCT
ejpam-4806	301	1	sharp	sharp	ADJ
ejpam-4806	301	2	bounds	bound	NOUN
ejpam-4806	301	3	on	on	ADP
ejpam-4806	301	4	the	the	DET
ejpam-4806	301	5	spectral	spectral	ADJ
ejpam-4806	301	6	radius	radius	NOUN
ejpam-4806	301	7	of	of	ADP
ejpam-4806	301	8	a	a	DET
ejpam-4806	301	9	nonnegative	nonnegative	ADJ
ejpam-4806	301	10	matrix	matrix	NOUN
ejpam-4806	301	11	.	.	PUNCT
ejpam-4806	302	1	linear	linear	ADJ
ejpam-4806	302	2	algebra	algebra	NOUN
ejpam-4806	302	3	and	and	CCONJ
ejpam-4806	302	4	its	its	PRON
ejpam-4806	302	5	applications	application	NOUN
ejpam-4806	302	6	,	,	PUNCT
ejpam-4806	302	7	439(10):2961–2970	439(10):2961–2970	PROPN
ejpam-4806	302	8	,	,	PUNCT
ejpam-4806	302	9	2013	2013	NUM
ejpam-4806	302	10	.	.	PUNCT
ejpam-4806	303	1	[	[	X
ejpam-4806	303	2	6	6	NUM
ejpam-4806	303	3	]	]	X
ejpam-4806	303	4	ernesto	ernesto	PROPN
ejpam-4806	303	5	estrada	estrada	PROPN
ejpam-4806	303	6	and	and	CCONJ
ejpam-4806	303	7	michele	michele	PROPN
ejpam-4806	303	8	benzi	benzi	PROPN
ejpam-4806	303	9	.	.	PUNCT
ejpam-4806	303	10	core	core	PROPN
ejpam-4806	303	11	–	–	PUNCT
ejpam-4806	303	12	satellite	satellite	NOUN
ejpam-4806	303	13	graphs	graph	NOUN
ejpam-4806	303	14	:	:	PUNCT
ejpam-4806	303	15	clustering	clustering	NOUN
ejpam-4806	303	16	,	,	PUNCT
ejpam-4806	303	17	assortativity	assortativity	NOUN
ejpam-4806	303	18	,	,	PUNCT
ejpam-4806	303	19	and	and	CCONJ
ejpam-4806	303	20	spectral	spectral	ADJ
ejpam-4806	303	21	properties	property	NOUN
ejpam-4806	303	22	.	.	PUNCT
ejpam-4806	304	1	linear	linear	ADJ
ejpam-4806	304	2	algebra	algebra	NOUN
ejpam-4806	304	3	and	and	CCONJ
ejpam-4806	304	4	its	its	PRON
ejpam-4806	304	5	applications	application	NOUN
ejpam-4806	304	6	,	,	PUNCT
ejpam-4806	304	7	517:30–52	517:30–52	NUM
ejpam-4806	304	8	,	,	PUNCT
ejpam-4806	304	9	2017	2017	NUM
ejpam-4806	304	10	.	.	PUNCT
ejpam-4806	305	1	[	[	X
ejpam-4806	305	2	7	7	X
ejpam-4806	305	3	]	]	X
ejpam-4806	305	4	ernesto	ernesto	PROPN
ejpam-4806	305	5	estrada	estrada	PROPN
ejpam-4806	305	6	and	and	CCONJ
ejpam-4806	305	7	eusebio	eusebio	PROPN
ejpam-4806	305	8	vargas	vargas	PROPN
ejpam-4806	305	9	-	-	PUNCT
ejpam-4806	305	10	estrada	estrada	PROPN
ejpam-4806	305	11	.	.	PUNCT
ejpam-4806	306	1	distance	distance	NOUN
ejpam-4806	306	2	-	-	PUNCT
ejpam-4806	306	3	sum	sum	NOUN
ejpam-4806	306	4	heterogeneity	heterogeneity	NOUN
ejpam-4806	306	5	in	in	ADP
ejpam-4806	306	6	graphs	graph	NOUN
ejpam-4806	306	7	and	and	CCONJ
ejpam-4806	306	8	complex	complex	ADJ
ejpam-4806	306	9	networks	network	NOUN
ejpam-4806	306	10	.	.	PUNCT
ejpam-4806	307	1	applied	apply	VERB
ejpam-4806	307	2	mathematics	mathematic	NOUN
ejpam-4806	307	3	and	and	CCONJ
ejpam-4806	307	4	computation	computation	NOUN
ejpam-4806	307	5	,	,	PUNCT
ejpam-4806	307	6	218:10393–10405	218:10393–10405	NUM
ejpam-4806	307	7	,	,	PUNCT
ejpam-4806	307	8	2012	2012	NUM
ejpam-4806	307	9	.	.	PUNCT
ejpam-4806	308	1	[	[	X
ejpam-4806	308	2	8	8	NUM
ejpam-4806	308	3	]	]	SYM
ejpam-4806	308	4	yuan	yuan	NOUN
ejpam-4806	308	5	hong	hong	PROPN
ejpam-4806	308	6	and	and	CCONJ
ejpam-4806	308	7	jin	jin	NOUN
ejpam-4806	308	8	-	-	PUNCT
ejpam-4806	308	9	long	long	ADJ
ejpam-4806	308	10	shu	shu	NOUN
ejpam-4806	308	11	.	.	PUNCT
ejpam-4806	309	1	a	a	DET
ejpam-4806	309	2	sharp	sharp	ADJ
ejpam-4806	309	3	upper	upper	ADJ
ejpam-4806	309	4	bound	bind	VERB
ejpam-4806	309	5	for	for	ADP
ejpam-4806	309	6	the	the	DET
ejpam-4806	309	7	spectral	spectral	ADJ
ejpam-4806	309	8	radius	radius	NOUN
ejpam-4806	309	9	of	of	ADP
ejpam-4806	309	10	the	the	DET
ejpam-4806	309	11	nordhaus	nordhaus	NOUN
ejpam-4806	309	12	-	-	PUNCT
ejpam-4806	309	13	gaddum	gaddum	NOUN
ejpam-4806	309	14	type	type	NOUN
ejpam-4806	309	15	.	.	PUNCT
ejpam-4806	310	1	discrete	discrete	ADJ
ejpam-4806	310	2	mathematics	mathematic	NOUN
ejpam-4806	310	3	,	,	PUNCT
ejpam-4806	310	4	211:229–232	211:229–232	NUM
ejpam-4806	310	5	,	,	PUNCT
ejpam-4806	310	6	2000	2000	NUM
ejpam-4806	310	7	.	.	PUNCT
ejpam-4806	311	1	[	[	X
ejpam-4806	311	2	9	9	NUM
ejpam-4806	311	3	]	]	X
ejpam-4806	311	4	shuchao	shuchao	ADJ
ejpam-4806	311	5	li	li	PROPN
ejpam-4806	311	6	and	and	CCONJ
ejpam-4806	311	7	yi	yi	PROPN
ejpam-4806	311	8	tian	tian	PROPN
ejpam-4806	311	9	.	.	PUNCT
ejpam-4806	312	1	some	some	DET
ejpam-4806	312	2	bounds	bound	NOUN
ejpam-4806	312	3	on	on	ADP
ejpam-4806	312	4	the	the	DET
ejpam-4806	312	5	largest	large	ADJ
ejpam-4806	312	6	eigenvalues	eigenvalue	NOUN
ejpam-4806	312	7	of	of	ADP
ejpam-4806	312	8	graphs	graph	NOUN
ejpam-4806	312	9	.	.	PUNCT
ejpam-4806	313	1	applied	apply	VERB
ejpam-4806	313	2	mathematics	mathematics	NOUN
ejpam-4806	313	3	letters	letter	NOUN
ejpam-4806	313	4	,	,	PUNCT
ejpam-4806	313	5	25:326–332	25:326–332	NUM
ejpam-4806	313	6	,	,	PUNCT
ejpam-4806	313	7	2012	2012	NUM
ejpam-4806	313	8	.	.	PUNCT
ejpam-4806	314	1	[	[	X
ejpam-4806	314	2	10	10	NUM
ejpam-4806	314	3	]	]	X
ejpam-4806	314	4	hongying	hongye	VERB
ejpam-4806	314	5	lin	lin	PROPN
ejpam-4806	314	6	,	,	PUNCT
ejpam-4806	314	7	biao	biao	PROPN
ejpam-4806	314	8	mo	mo	PROPN
ejpam-4806	314	9	,	,	PUNCT
ejpam-4806	314	10	bo	bo	PROPN
ejpam-4806	314	11	zhou	zhou	PROPN
ejpam-4806	314	12	,	,	PUNCT
ejpam-4806	314	13	and	and	CCONJ
ejpam-4806	314	14	weiming	weime	VERB
ejpam-4806	314	15	weng	weng	PROPN
ejpam-4806	314	16	.	.	PUNCT
ejpam-4806	315	1	sharp	sharp	ADJ
ejpam-4806	315	2	bounds	bound	NOUN
ejpam-4806	315	3	for	for	ADP
ejpam-4806	315	4	ordinary	ordinary	ADJ
ejpam-4806	315	5	and	and	CCONJ
ejpam-4806	315	6	signless	signless	ADJ
ejpam-4806	315	7	laplacian	laplacian	ADJ
ejpam-4806	315	8	spectral	spectral	ADJ
ejpam-4806	315	9	radii	radius	NOUN
ejpam-4806	315	10	of	of	ADP
ejpam-4806	315	11	uniform	uniform	ADJ
ejpam-4806	315	12	hypergraphs	hypergraph	NOUN
ejpam-4806	315	13	.	.	PUNCT
ejpam-4806	316	1	applied	apply	VERB
ejpam-4806	316	2	mathematics	mathematic	NOUN
ejpam-4806	316	3	and	and	CCONJ
ejpam-4806	316	4	computation	computation	NOUN
ejpam-4806	316	5	,	,	PUNCT
ejpam-4806	316	6	285:217–227	285:217–227	NUM
ejpam-4806	316	7	,	,	PUNCT
ejpam-4806	316	8	2016	2016	NUM
ejpam-4806	316	9	.	.	PUNCT
ejpam-4806	317	1	[	[	X
ejpam-4806	317	2	11	11	NUM
ejpam-4806	317	3	]	]	PUNCT
ejpam-4806	317	4	huiqing	huiqe	VERB
ejpam-4806	317	5	liu	liu	PROPN
ejpam-4806	317	6	,	,	PUNCT
ejpam-4806	317	7	mei	mei	PROPN
ejpam-4806	317	8	lu	lu	PROPN
ejpam-4806	317	9	,	,	PUNCT
ejpam-4806	317	10	and	and	CCONJ
ejpam-4806	317	11	feng	feng	PROPN
ejpam-4806	317	12	tian	tian	PROPN
ejpam-4806	317	13	.	.	PUNCT
ejpam-4806	318	1	on	on	ADP
ejpam-4806	318	2	the	the	DET
ejpam-4806	318	3	laplacian	laplacian	ADJ
ejpam-4806	318	4	spectral	spectral	ADJ
ejpam-4806	318	5	radius	radius	NOUN
ejpam-4806	318	6	of	of	ADP
ejpam-4806	318	7	a	a	DET
ejpam-4806	318	8	graph	graph	NOUN
ejpam-4806	318	9	.	.	PUNCT
ejpam-4806	319	1	linear	linear	ADJ
ejpam-4806	319	2	algebra	algebra	NOUN
ejpam-4806	319	3	and	and	CCONJ
ejpam-4806	319	4	its	its	PRON
ejpam-4806	319	5	applications	application	NOUN
ejpam-4806	319	6	,	,	PUNCT
ejpam-4806	319	7	376:135–141	376:135–141	NUM
ejpam-4806	319	8	,	,	PUNCT
ejpam-4806	319	9	2004	2004	NUM
ejpam-4806	319	10	.	.	PUNCT
ejpam-4806	320	1	[	[	X
ejpam-4806	320	2	12	12	NUM
ejpam-4806	320	3	]	]	PUNCT
ejpam-4806	320	4	x.	x.	NOUN
ejpam-4806	320	5	liu	liu	PROPN
ejpam-4806	320	6	and	and	CCONJ
ejpam-4806	320	7	p.	p.	PROPN
ejpam-4806	320	8	lu	lu	PROPN
ejpam-4806	320	9	.	.	PUNCT
ejpam-4806	321	1	signless	signless	PROPN
ejpam-4806	321	2	laplacian	laplacian	ADJ
ejpam-4806	321	3	spectral	spectral	ADJ
ejpam-4806	321	4	characterization	characterization	NOUN
ejpam-4806	321	5	of	of	ADP
ejpam-4806	321	6	some	some	DET
ejpam-4806	321	7	joins	join	NOUN
ejpam-4806	321	8	.	.	PUNCT
ejpam-4806	322	1	the	the	DET
ejpam-4806	322	2	electronic	electronic	ADJ
ejpam-4806	322	3	journal	journal	NOUN
ejpam-4806	322	4	of	of	ADP
ejpam-4806	322	5	linear	linear	PROPN
ejpam-4806	322	6	algebra	algebra	PROPN
ejpam-4806	322	7	,	,	PUNCT
ejpam-4806	322	8	30:443–454	30:443–454	PROPN
ejpam-4806	322	9	,	,	PUNCT
ejpam-4806	322	10	2015	2015	NUM
ejpam-4806	322	11	.	.	PUNCT
ejpam-4806	323	1	[	[	X
ejpam-4806	323	2	13	13	NUM
ejpam-4806	323	3	]	]	X
ejpam-4806	323	4	feng	feng	PROPN
ejpam-4806	323	5	li	li	PROPN
ejpam-4806	323	6	-	-	PROPN
ejpam-4806	323	7	hua	hua	PROPN
ejpam-4806	323	8	and	and	CCONJ
ejpam-4806	323	9	yu	yu	PROPN
ejpam-4806	323	10	gui	gui	PROPN
ejpam-4806	323	11	-	-	PUNCT
ejpam-4806	323	12	hai	hai	NOUN
ejpam-4806	323	13	.	.	PUNCT
ejpam-4806	324	1	the	the	DET
ejpam-4806	324	2	signless	signless	PROPN
ejpam-4806	324	3	laplacian	laplacian	ADJ
ejpam-4806	324	4	spectral	spectral	ADJ
ejpam-4806	324	5	radius	radius	NOUN
ejpam-4806	324	6	of	of	ADP
ejpam-4806	324	7	graphs	graph	NOUN
ejpam-4806	324	8	with	with	ADP
ejpam-4806	324	9	given	give	VERB
ejpam-4806	324	10	diameter	diameter	NOUN
ejpam-4806	324	11	.	.	PUNCT
ejpam-4806	325	1	utilitas	utilitas	PROPN
ejpam-4806	325	2	mathematica	mathematica	PROPN
ejpam-4806	325	3	,	,	PUNCT
ejpam-4806	325	4	83:265	83:265	NUM
ejpam-4806	325	5	-	-	SYM
ejpam-4806	325	6	276	276	NUM
ejpam-4806	325	7	,	,	PUNCT
ejpam-4806	325	8	2010	2010	NUM
ejpam-4806	325	9	.	.	PUNCT
ejpam-4806	326	1	[	[	X
ejpam-4806	326	2	14	14	NUM
ejpam-4806	326	3	]	]	PUNCT
ejpam-4806	326	4	m.	m.	PROPN
ejpam-4806	326	5	e.	e.	PROPN
ejpam-4806	326	6	j.	j.	PROPN
ejpam-4806	326	7	newman	newman	PROPN
ejpam-4806	326	8	.	.	PUNCT
ejpam-4806	327	1	the	the	DET
ejpam-4806	327	2	structure	structure	NOUN
ejpam-4806	327	3	and	and	CCONJ
ejpam-4806	327	4	function	function	NOUN
ejpam-4806	327	5	of	of	ADP
ejpam-4806	327	6	complex	complex	ADJ
ejpam-4806	327	7	networks	network	NOUN
ejpam-4806	327	8	.	.	PUNCT
ejpam-4806	328	1	siam	siam	PROPN
ejpam-4806	328	2	rev	rev	PROPN
ejpam-4806	328	3	,	,	PUNCT
ejpam-4806	328	4	45:167–256	45:167–256	PROPN
ejpam-4806	328	5	,	,	PUNCT
ejpam-4806	328	6	2003	2003	NUM
ejpam-4806	328	7	.	.	PUNCT
ejpam-4806	329	1	[	[	X
ejpam-4806	329	2	15	15	X
ejpam-4806	329	3	]	]	X
ejpam-4806	329	4	v.	v.	ADP
ejpam-4806	329	5	nikiforov	nikiforov	PROPN
ejpam-4806	329	6	and	and	CCONJ
ejpam-4806	329	7	x.	x.	NOUN
ejpam-4806	329	8	yuan	yuan	PROPN
ejpam-4806	329	9	.	.	PUNCT
ejpam-4806	330	1	more	more	ADJ
ejpam-4806	330	2	eigenvalue	eigenvalue	ADJ
ejpam-4806	330	3	problems	problem	NOUN
ejpam-4806	330	4	of	of	ADP
ejpam-4806	330	5	nordhaus	nordhaus	NOUN
ejpam-4806	330	6	–	–	PUNCT
ejpam-4806	330	7	gaddum	gaddum	NOUN
ejpam-4806	330	8	type	type	NOUN
ejpam-4806	330	9	.	.	PUNCT
ejpam-4806	331	1	linear	linear	ADJ
ejpam-4806	331	2	algebra	algebra	NOUN
ejpam-4806	331	3	and	and	CCONJ
ejpam-4806	331	4	its	its	PRON
ejpam-4806	331	5	applications	application	NOUN
ejpam-4806	331	6	,	,	PUNCT
ejpam-4806	331	7	451:231	451:231	NOUN
ejpam-4806	331	8	–	–	PUNCT
ejpam-4806	331	9	245	245	NUM
ejpam-4806	331	10	,	,	PUNCT
ejpam-4806	331	11	2014	2014	NUM
ejpam-4806	331	12	.	.	PUNCT
ejpam-4806	332	1	references	reference	NOUN
ejpam-4806	332	2	1746	1746	NUM
ejpam-4806	332	3	[	[	X
ejpam-4806	332	4	16	16	NUM
ejpam-4806	332	5	]	]	PUNCT
ejpam-4806	332	6	vladimir	vladimir	PROPN
ejpam-4806	332	7	nikiforov	nikiforov	PROPN
ejpam-4806	332	8	.	.	PUNCT
ejpam-4806	333	1	eigenvalue	eigenvalue	VERB
ejpam-4806	333	2	problems	problem	NOUN
ejpam-4806	333	3	of	of	ADP
ejpam-4806	333	4	nordhaus	nordhaus	NOUN
ejpam-4806	333	5	-	-	PUNCT
ejpam-4806	333	6	gaddum	gaddum	NOUN
ejpam-4806	333	7	type	type	NOUN
ejpam-4806	333	8	.	.	PUNCT
ejpam-4806	334	1	discrete	discrete	ADJ
ejpam-4806	334	2	mathematics	mathematic	NOUN
ejpam-4806	334	3	,	,	PUNCT
ejpam-4806	334	4	307:774	307:774	NOUN
ejpam-4806	334	5	–	–	PUNCT
ejpam-4806	334	6	780	780	NUM
ejpam-4806	334	7	,	,	PUNCT
ejpam-4806	334	8	2007	2007	NUM
ejpam-4806	334	9	.	.	PUNCT
ejpam-4806	335	1	[	[	X
ejpam-4806	335	2	17	17	NUM
ejpam-4806	335	3	]	]	PUNCT
ejpam-4806	335	4	csikvári	csikvári	PROPN
ejpam-4806	335	5	.	.	PUNCT
ejpam-4806	336	1	p.	p.	NOUN
ejpam-4806	336	2	on	on	ADP
ejpam-4806	336	3	a	a	DET
ejpam-4806	336	4	conjecture	conjecture	NOUN
ejpam-4806	336	5	of	of	ADP
ejpam-4806	336	6	v.	v.	ADP
ejpam-4806	336	7	nikiforov	nikiforov	PROPN
ejpam-4806	336	8	.	.	PUNCT
ejpam-4806	337	1	discrete	discrete	ADJ
ejpam-4806	337	2	mathematics	mathematic	NOUN
ejpam-4806	337	3	,	,	PUNCT
ejpam-4806	337	4	309(13):4522	309(13):4522	PROPN
ejpam-4806	337	5	–	–	PUNCT
ejpam-4806	337	6	4526	4526	NUM
ejpam-4806	337	7	,	,	PUNCT
ejpam-4806	337	8	2009	2009	NUM
ejpam-4806	337	9	.	.	PUNCT
ejpam-4806	338	1	[	[	X
ejpam-4806	338	2	18	18	NUM
ejpam-4806	338	3	]	]	PUNCT
ejpam-4806	338	4	erzsebet	erzsebet	NOUN
ejpam-4806	338	5	ravasz	ravasz	NOUN
ejpam-4806	338	6	and	and	CCONJ
ejpam-4806	338	7	albert	albert	PROPN
ejpam-4806	338	8	-	-	PUNCT
ejpam-4806	338	9	laszlo	laszlo	ADJ
ejpam-4806	338	10	’	'	PUNCT
ejpam-4806	338	11	barabasi	barabasi	NOUN
ejpam-4806	338	12	.	.	PUNCT
ejpam-4806	339	1	hierarchical	hierarchical	ADJ
ejpam-4806	339	2	organization	organization	NOUN
ejpam-4806	339	3	in	in	ADP
ejpam-4806	339	4	complex	complex	ADJ
ejpam-4806	339	5	networks	network	NOUN
ejpam-4806	339	6	.	.	PUNCT
ejpam-4806	340	1	physical	physical	ADJ
ejpam-4806	340	2	review	review	PROPN
ejpam-4806	340	3	,	,	PUNCT
ejpam-4806	340	4	e67:026–112	e67:026–112	PROPN
ejpam-4806	340	5	,	,	PUNCT
ejpam-4806	340	6	2003	2003	NUM
ejpam-4806	340	7	.	.	PUNCT
ejpam-4806	341	1	[	[	X
ejpam-4806	341	2	19	19	NUM
ejpam-4806	341	3	]	]	X
ejpam-4806	341	4	lingsheng	lingsheng	PROPN
ejpam-4806	341	5	shi	shi	PROPN
ejpam-4806	341	6	.	.	PUNCT
ejpam-4806	342	1	bounds	bound	VERB
ejpam-4806	342	2	on	on	ADP
ejpam-4806	342	3	the	the	DET
ejpam-4806	342	4	(	(	PUNCT
ejpam-4806	342	5	laplacian	laplacian	ADJ
ejpam-4806	342	6	)	)	PUNCT
ejpam-4806	342	7	spectral	spectral	ADJ
ejpam-4806	342	8	radius	radius	NOUN
ejpam-4806	342	9	of	of	ADP
ejpam-4806	342	10	graphs	graph	NOUN
ejpam-4806	342	11	.	.	PUNCT
ejpam-4806	343	1	linear	linear	ADJ
ejpam-4806	343	2	algebra	algebra	NOUN
ejpam-4806	343	3	and	and	CCONJ
ejpam-4806	343	4	its	its	PRON
ejpam-4806	343	5	applications	application	NOUN
ejpam-4806	343	6	,	,	PUNCT
ejpam-4806	343	7	422:755–770	422:755–770	NUM
ejpam-4806	343	8	,	,	PUNCT
ejpam-4806	343	9	2007	2007	NUM
ejpam-4806	343	10	.	.	PUNCT
ejpam-4806	344	1	[	[	X
ejpam-4806	344	2	20	20	NUM
ejpam-4806	344	3	]	]	PUNCT
ejpam-4806	344	4	xu	xu	PROPN
ejpam-4806	344	5	wanyue	wanyue	PROPN
ejpam-4806	344	6	and	and	CCONJ
ejpam-4806	344	7	zhongzhi	zhongzhi	PROPN
ejpam-4806	344	8	zhang	zhang	PROPN
ejpam-4806	344	9	.	.	PUNCT
ejpam-4806	345	1	optimal	optimal	ADJ
ejpam-4806	345	2	scale	scale	NOUN
ejpam-4806	345	3	-	-	PUNCT
ejpam-4806	345	4	free	free	ADJ
ejpam-4806	345	5	small	small	ADJ
ejpam-4806	345	6	-	-	PUNCT
ejpam-4806	345	7	world	world	NOUN
ejpam-4806	345	8	graphs	graph	NOUN
ejpam-4806	345	9	with	with	ADP
ejpam-4806	345	10	minimum	minimum	ADJ
ejpam-4806	345	11	scaling	scaling	NOUN
ejpam-4806	345	12	of	of	ADP
ejpam-4806	345	13	cover	cover	NOUN
ejpam-4806	345	14	time	time	NOUN
ejpam-4806	345	15	.	.	PUNCT
ejpam-4806	346	1	acm	acm	PROPN
ejpam-4806	346	2	transactions	transaction	NOUN
ejpam-4806	346	3	on	on	ADP
ejpam-4806	346	4	knowledge	knowledge	NOUN
ejpam-4806	346	5	discovery	discovery	NOUN
ejpam-4806	346	6	from	from	ADP
ejpam-4806	346	7	data	data	PROPN
ejpam-4806	346	8	,	,	PUNCT
ejpam-4806	346	9	17:1–19	17:1–19	NUM
ejpam-4806	346	10	,	,	PUNCT
ejpam-4806	346	11	2023	2023	NUM
ejpam-4806	346	12	.	.	PUNCT
ejpam-4806	347	1	[	[	X
ejpam-4806	347	2	21	21	NUM
ejpam-4806	347	3	]	]	X
ejpam-4806	347	4	zhang	zhang	PROPN
ejpam-4806	347	5	,	,	PUNCT
ejpam-4806	347	6	jing	jing	PROPN
ejpam-4806	347	7	-	-	PUNCT
ejpam-4806	347	8	ming	ming	PROPN
ejpam-4806	347	9	,	,	PUNCT
ejpam-4806	347	10	huang	huang	PROPN
ejpam-4806	347	11	,	,	PUNCT
ejpam-4806	347	12	ting	ting	PROPN
ejpam-4806	347	13	-	-	PUNCT
ejpam-4806	347	14	zhu	zhu	PROPN
ejpam-4806	347	15	,	,	PUNCT
ejpam-4806	347	16	and	and	CCONJ
ejpam-4806	347	17	ji	ji	PROPN
ejpam-4806	347	18	-	-	PUNCT
ejpam-4806	347	19	ming	ming	PROPN
ejpam-4806	347	20	guo	guo	PROPN
ejpam-4806	347	21	.	.	PUNCT
ejpam-4806	348	1	on	on	ADP
ejpam-4806	348	2	the	the	DET
ejpam-4806	348	3	signless	signless	PROPN
ejpam-4806	348	4	laplacian	laplacian	ADJ
ejpam-4806	348	5	spectral	spectral	ADJ
ejpam-4806	348	6	radius	radius	NOUN
ejpam-4806	348	7	of	of	ADP
ejpam-4806	348	8	bicyclic	bicyclic	NOUN
ejpam-4806	348	9	graphs	graph	NOUN
ejpam-4806	348	10	with	with	ADP
ejpam-4806	348	11	perfect	perfect	ADJ
ejpam-4806	348	12	matchings	matching	NOUN
ejpam-4806	348	13	.	.	PUNCT
ejpam-4806	349	1	the	the	DET
ejpam-4806	349	2	scientific	scientific	ADJ
ejpam-4806	349	3	world	world	NOUN
ejpam-4806	349	4	journal	journal	NOUN
ejpam-4806	349	5	,	,	PUNCT
ejpam-4806	349	6	2014	2014	NUM
ejpam-4806	349	7	,	,	PUNCT
ejpam-4806	349	8	2014	2014	NUM
ejpam-4806	349	9	.	.	PUNCT