id	sid	tid	token	lemma	pos
ejpam-5360	1	1	european	european	PROPN
ejpam-5360	1	2	journal	journal	PROPN
ejpam-5360	1	3	of	of	ADP
ejpam-5360	1	4	pure	pure	ADJ
ejpam-5360	1	5	and	and	CCONJ
ejpam-5360	1	6	applied	applied	ADJ
ejpam-5360	1	7	mathematics	mathematic	NOUN
ejpam-5360	1	8	2025	2025	NUM
ejpam-5360	1	9	,	,	PUNCT
ejpam-5360	1	10	vol	vol	NOUN
ejpam-5360	1	11	.	.	PROPN
ejpam-5360	1	12	18	18	NUM
ejpam-5360	1	13	,	,	PUNCT
ejpam-5360	1	14	issue	issue	NOUN
ejpam-5360	1	15	4	4	NUM
ejpam-5360	1	16	,	,	PUNCT
ejpam-5360	1	17	article	article	NOUN
ejpam-5360	1	18	number	number	NOUN
ejpam-5360	1	19	5360	5360	NUM
ejpam-5360	1	20	issn	issn	PROPN
ejpam-5360	1	21	1307	1307	NUM
ejpam-5360	1	22	-	-	SYM
ejpam-5360	1	23	5543	5543	NUM
ejpam-5360	1	24	–	–	PUNCT
ejpam-5360	1	25	ejpam.com	ejpam.com	X
ejpam-5360	1	26	published	publish	VERB
ejpam-5360	1	27	by	by	ADP
ejpam-5360	1	28	new	new	PROPN
ejpam-5360	1	29	york	york	PROPN
ejpam-5360	1	30	business	business	PROPN
ejpam-5360	1	31	global	global	ADJ
ejpam-5360	1	32	bounds	bound	NOUN
ejpam-5360	1	33	on	on	ADP
ejpam-5360	1	34	spectral	spectral	ADJ
ejpam-5360	1	35	radius	radius	NOUN
ejpam-5360	1	36	and	and	CCONJ
ejpam-5360	1	37	signless	signless	PROPN
ejpam-5360	1	38	laplacian	laplacian	ADJ
ejpam-5360	1	39	spectral	spectral	ADJ
ejpam-5360	1	40	radius	radius	NOUN
ejpam-5360	1	41	for	for	ADP
ejpam-5360	1	42	generalized	generalized	ADJ
ejpam-5360	1	43	core	core	NOUN
ejpam-5360	1	44	-	-	PUNCT
ejpam-5360	1	45	satellite	satellite	NOUN
ejpam-5360	1	46	graphs	graph	NOUN
ejpam-5360	1	47	malathy	malathy	ADV
ejpam-5360	1	48	v.1	v.1	NUM
ejpam-5360	1	49	,	,	PUNCT
ejpam-5360	1	50	kalyani	kalyani	ADJ
ejpam-5360	1	51	desikan1,∗	desikan1,∗	NOUN
ejpam-5360	1	52	1	1	NUM
ejpam-5360	1	53	department	department	NOUN
ejpam-5360	1	54	of	of	ADP
ejpam-5360	1	55	mathematics	mathematics	PROPN
ejpam-5360	1	56	,	,	PUNCT
ejpam-5360	1	57	sas	sas	PROPN
ejpam-5360	1	58	,	,	PUNCT
ejpam-5360	1	59	vellore	vellore	PROPN
ejpam-5360	1	60	institute	institute	PROPN
ejpam-5360	1	61	of	of	ADP
ejpam-5360	1	62	technology	technology	PROPN
ejpam-5360	1	63	,	,	PUNCT
ejpam-5360	1	64	chennai	chennai	PROPN
ejpam-5360	1	65	,	,	PUNCT
ejpam-5360	1	66	india	india	PROPN
ejpam-5360	1	67	abstract	abstract	PROPN
ejpam-5360	1	68	.	.	PUNCT
ejpam-5360	2	1	a	a	DET
ejpam-5360	2	2	generalized	generalized	ADJ
ejpam-5360	2	3	core	core	NOUN
ejpam-5360	2	4	-	-	PUNCT
ejpam-5360	2	5	satellite	satellite	NOUN
ejpam-5360	2	6	graph	graph	NOUN
ejpam-5360	2	7	θ(c	θ(c	VERB
ejpam-5360	2	8	,	,	PUNCT
ejpam-5360	2	9	s	s	NOUN
ejpam-5360	2	10	,	,	PUNCT
ejpam-5360	2	11	η∗	η∗	NOUN
ejpam-5360	2	12	)	)	PUNCT
ejpam-5360	2	13	belongs	belong	VERB
ejpam-5360	2	14	to	to	ADP
ejpam-5360	2	15	the	the	DET
ejpam-5360	2	16	family	family	NOUN
ejpam-5360	2	17	of	of	ADP
ejpam-5360	2	18	graphs	graph	NOUN
ejpam-5360	2	19	of	of	ADP
ejpam-5360	2	20	diameter	diameter	NOUN
ejpam-5360	2	21	two	two	NUM
ejpam-5360	2	22	.	.	PUNCT
ejpam-5360	3	1	it	it	PRON
ejpam-5360	3	2	has	have	VERB
ejpam-5360	3	3	a	a	DET
ejpam-5360	3	4	central	central	ADJ
ejpam-5360	3	5	core	core	NOUN
ejpam-5360	3	6	of	of	ADP
ejpam-5360	3	7	nodes	node	NOUN
ejpam-5360	3	8	connected	connect	VERB
ejpam-5360	3	9	to	to	ADP
ejpam-5360	3	10	a	a	DET
ejpam-5360	3	11	few	few	ADJ
ejpam-5360	3	12	satellites	satellite	NOUN
ejpam-5360	3	13	,	,	PUNCT
ejpam-5360	3	14	where	where	SCONJ
ejpam-5360	3	15	all	all	DET
ejpam-5360	3	16	satellite	satellite	NOUN
ejpam-5360	3	17	cliques	clique	NOUN
ejpam-5360	3	18	are	be	AUX
ejpam-5360	3	19	not	not	PART
ejpam-5360	3	20	identical	identical	ADJ
ejpam-5360	3	21	and	and	CCONJ
ejpam-5360	3	22	might	might	AUX
ejpam-5360	3	23	be	be	AUX
ejpam-5360	3	24	of	of	ADP
ejpam-5360	3	25	different	different	ADJ
ejpam-5360	3	26	sizes	size	NOUN
ejpam-5360	3	27	.	.	PUNCT
ejpam-5360	4	1	these	these	DET
ejpam-5360	4	2	graphs	graph	NOUN
ejpam-5360	4	3	can	can	AUX
ejpam-5360	4	4	be	be	AUX
ejpam-5360	4	5	used	use	VERB
ejpam-5360	4	6	to	to	PART
ejpam-5360	4	7	model	model	VERB
ejpam-5360	4	8	any	any	DET
ejpam-5360	4	9	real	real	ADJ
ejpam-5360	4	10	-	-	PUNCT
ejpam-5360	4	11	world	world	NOUN
ejpam-5360	4	12	complex	complex	ADJ
ejpam-5360	4	13	network	network	NOUN
ejpam-5360	4	14	.	.	PUNCT
ejpam-5360	5	1	using	use	VERB
ejpam-5360	5	2	core	core	NOUN
ejpam-5360	5	3	-	-	PUNCT
ejpam-5360	5	4	satellite	satellite	NOUN
ejpam-5360	5	5	graphs	graph	NOUN
ejpam-5360	5	6	,	,	PUNCT
ejpam-5360	5	7	properties	property	NOUN
ejpam-5360	5	8	like	like	ADP
ejpam-5360	5	9	hierarchical	hierarchical	ADJ
ejpam-5360	5	10	structure	structure	NOUN
ejpam-5360	5	11	can	can	AUX
ejpam-5360	5	12	be	be	AUX
ejpam-5360	5	13	conveniently	conveniently	ADV
ejpam-5360	5	14	modeled	model	VERB
ejpam-5360	5	15	for	for	ADP
ejpam-5360	5	16	large	large	ADJ
ejpam-5360	5	17	complex	complex	ADJ
ejpam-5360	5	18	networks	network	NOUN
ejpam-5360	5	19	.	.	PUNCT
ejpam-5360	6	1	in	in	ADP
ejpam-5360	6	2	this	this	DET
ejpam-5360	6	3	paper	paper	NOUN
ejpam-5360	6	4	,	,	PUNCT
ejpam-5360	6	5	we	we	PRON
ejpam-5360	6	6	obtain	obtain	VERB
ejpam-5360	6	7	the	the	DET
ejpam-5360	6	8	lower	low	ADJ
ejpam-5360	6	9	and	and	CCONJ
ejpam-5360	6	10	upper	upper	ADJ
ejpam-5360	6	11	bounds	bound	NOUN
ejpam-5360	6	12	for	for	ADP
ejpam-5360	6	13	the	the	DET
ejpam-5360	6	14	spectral	spectral	ADJ
ejpam-5360	6	15	radius	radius	NOUN
ejpam-5360	6	16	and	and	CCONJ
ejpam-5360	6	17	signless	signless	PROPN
ejpam-5360	6	18	laplacian	laplacian	ADJ
ejpam-5360	6	19	spectral	spectral	ADJ
ejpam-5360	6	20	radius	radius	NOUN
ejpam-5360	6	21	of	of	ADP
ejpam-5360	6	22	the	the	DET
ejpam-5360	6	23	generalized	generalize	VERB
ejpam-5360	6	24	core	core	NOUN
ejpam-5360	6	25	-	-	PUNCT
ejpam-5360	6	26	satellite	satellite	NOUN
ejpam-5360	6	27	graph	graph	NOUN
ejpam-5360	6	28	,	,	PUNCT
ejpam-5360	6	29	in	in	ADP
ejpam-5360	6	30	terms	term	NOUN
ejpam-5360	6	31	of	of	ADP
ejpam-5360	6	32	number	number	NOUN
ejpam-5360	6	33	of	of	ADP
ejpam-5360	6	34	vertices	vertex	NOUN
ejpam-5360	6	35	,	,	PUNCT
ejpam-5360	6	36	number	number	NOUN
ejpam-5360	6	37	of	of	ADP
ejpam-5360	6	38	edges	edge	NOUN
ejpam-5360	6	39	,	,	PUNCT
ejpam-5360	6	40	and	and	CCONJ
ejpam-5360	6	41	the	the	DET
ejpam-5360	6	42	graph	graph	NOUN
ejpam-5360	6	43	parameters	parameter	NOUN
ejpam-5360	6	44	associated	associate	VERB
ejpam-5360	6	45	with	with	ADP
ejpam-5360	6	46	the	the	DET
ejpam-5360	6	47	structure	structure	NOUN
ejpam-5360	6	48	of	of	ADP
ejpam-5360	6	49	the	the	DET
ejpam-5360	6	50	graph	graph	NOUN
ejpam-5360	6	51	in	in	ADP
ejpam-5360	6	52	both	both	DET
ejpam-5360	6	53	satellites	satellite	NOUN
ejpam-5360	6	54	and	and	CCONJ
ejpam-5360	6	55	the	the	DET
ejpam-5360	6	56	core	core	NOUN
ejpam-5360	6	57	.	.	PUNCT
ejpam-5360	7	1	2020	2020	NUM
ejpam-5360	7	2	mathematics	mathematics	PROPN
ejpam-5360	7	3	subject	subject	NOUN
ejpam-5360	7	4	classifications	classification	NOUN
ejpam-5360	7	5	:	:	PUNCT
ejpam-5360	7	6	05c50	05c50	NUM
ejpam-5360	7	7	key	key	ADJ
ejpam-5360	7	8	words	word	NOUN
ejpam-5360	7	9	and	and	CCONJ
ejpam-5360	7	10	phrases	phrase	NOUN
ejpam-5360	7	11	:	:	PUNCT
ejpam-5360	7	12	spectral	spectral	ADJ
ejpam-5360	7	13	radius	radius	NOUN
ejpam-5360	7	14	,	,	PUNCT
ejpam-5360	7	15	signless	signless	PROPN
ejpam-5360	7	16	laplacian	laplacian	ADJ
ejpam-5360	7	17	spectral	spectral	ADJ
ejpam-5360	7	18	radius	radius	NOUN
ejpam-5360	7	19	,	,	PUNCT
ejpam-5360	7	20	generalized	generalized	ADJ
ejpam-5360	7	21	coresatellite	coresatellite	ADJ
ejpam-5360	7	22	graphs	graph	NOUN
ejpam-5360	7	23	1	1	NUM
ejpam-5360	7	24	.	.	PUNCT
ejpam-5360	8	1	introduction	introduction	NOUN
ejpam-5360	8	2	generalized	generalize	VERB
ejpam-5360	8	3	core	core	NOUN
ejpam-5360	8	4	-	-	PUNCT
ejpam-5360	8	5	satellite	satellite	NOUN
ejpam-5360	8	6	graphs	graph	NOUN
ejpam-5360	8	7	can	can	AUX
ejpam-5360	8	8	be	be	AUX
ejpam-5360	8	9	used	use	VERB
ejpam-5360	8	10	to	to	PART
ejpam-5360	8	11	model	model	VERB
ejpam-5360	8	12	any	any	DET
ejpam-5360	8	13	complex	complex	ADJ
ejpam-5360	8	14	real	real	ADJ
ejpam-5360	8	15	-	-	PUNCT
ejpam-5360	8	16	world	world	NOUN
ejpam-5360	8	17	network	network	NOUN
ejpam-5360	8	18	.	.	PUNCT
ejpam-5360	9	1	using	use	VERB
ejpam-5360	9	2	core	core	NOUN
ejpam-5360	9	3	-	-	PUNCT
ejpam-5360	9	4	satellite	satellite	NOUN
ejpam-5360	9	5	graphs	graph	NOUN
ejpam-5360	9	6	,	,	PUNCT
ejpam-5360	9	7	properties	property	NOUN
ejpam-5360	9	8	like	like	ADP
ejpam-5360	9	9	hierarchical	hierarchical	ADJ
ejpam-5360	9	10	structure	structure	NOUN
ejpam-5360	9	11	can	can	AUX
ejpam-5360	9	12	be	be	AUX
ejpam-5360	9	13	conveniently	conveniently	ADV
ejpam-5360	9	14	modeled	model	VERB
ejpam-5360	9	15	for	for	ADP
ejpam-5360	9	16	large	large	ADJ
ejpam-5360	9	17	,	,	PUNCT
ejpam-5360	9	18	complex	complex	ADJ
ejpam-5360	9	19	networks	network	NOUN
ejpam-5360	9	20	.	.	PUNCT
ejpam-5360	10	1	a	a	DET
ejpam-5360	10	2	hierarchical	hierarchical	ADJ
ejpam-5360	10	3	design	design	NOUN
ejpam-5360	10	4	model	model	NOUN
ejpam-5360	10	5	provides	provide	VERB
ejpam-5360	10	6	a	a	DET
ejpam-5360	10	7	reference	reference	NOUN
ejpam-5360	10	8	topology	topology	NOUN
ejpam-5360	10	9	that	that	PRON
ejpam-5360	10	10	separates	separate	VERB
ejpam-5360	10	11	a	a	DET
ejpam-5360	10	12	network	network	NOUN
ejpam-5360	10	13	into	into	ADP
ejpam-5360	10	14	distinct	distinct	ADJ
ejpam-5360	10	15	layers	layer	NOUN
ejpam-5360	10	16	.	.	PUNCT
ejpam-5360	11	1	each	each	PRON
ejpam-5360	11	2	of	of	ADP
ejpam-5360	11	3	these	these	DET
ejpam-5360	11	4	layers	layer	NOUN
ejpam-5360	11	5	has	have	VERB
ejpam-5360	11	6	a	a	DET
ejpam-5360	11	7	series	series	NOUN
ejpam-5360	11	8	of	of	ADP
ejpam-5360	11	9	functions	function	NOUN
ejpam-5360	11	10	that	that	PRON
ejpam-5360	11	11	define	define	VERB
ejpam-5360	11	12	its	its	PRON
ejpam-5360	11	13	role	role	NOUN
ejpam-5360	11	14	in	in	ADP
ejpam-5360	11	15	the	the	DET
ejpam-5360	11	16	network	network	NOUN
ejpam-5360	11	17	.	.	PUNCT
ejpam-5360	12	1	this	this	DET
ejpam-5360	12	2	model	model	NOUN
ejpam-5360	12	3	facilitates	facilitate	VERB
ejpam-5360	12	4	making	make	VERB
ejpam-5360	12	5	the	the	DET
ejpam-5360	12	6	network	network	NOUN
ejpam-5360	12	7	scalable	scalable	ADJ
ejpam-5360	12	8	,	,	PUNCT
ejpam-5360	12	9	stable	stable	ADJ
ejpam-5360	12	10	,	,	PUNCT
ejpam-5360	12	11	deterministic	deterministic	ADJ
ejpam-5360	12	12	,	,	PUNCT
ejpam-5360	12	13	and	and	CCONJ
ejpam-5360	12	14	reliable	reliable	ADJ
ejpam-5360	12	15	,	,	PUNCT
ejpam-5360	12	16	provides	provide	VERB
ejpam-5360	12	17	better	well	ADJ
ejpam-5360	12	18	security	security	NOUN
ejpam-5360	12	19	,	,	PUNCT
ejpam-5360	12	20	is	be	AUX
ejpam-5360	12	21	effortless	effortless	ADJ
ejpam-5360	12	22	to	to	PART
ejpam-5360	12	23	manage	manage	VERB
ejpam-5360	12	24	and	and	CCONJ
ejpam-5360	12	25	design	design	NOUN
ejpam-5360	12	26	,	,	PUNCT
ejpam-5360	12	27	provides	provide	VERB
ejpam-5360	12	28	enhanced	enhanced	ADJ
ejpam-5360	12	29	performance	performance	NOUN
ejpam-5360	12	30	,	,	PUNCT
ejpam-5360	12	31	and	and	CCONJ
ejpam-5360	12	32	is	be	AUX
ejpam-5360	12	33	also	also	ADV
ejpam-5360	12	34	cost	cost	NOUN
ejpam-5360	12	35	-	-	PUNCT
ejpam-5360	12	36	efficient	efficient	ADJ
ejpam-5360	12	37	.	.	PUNCT
ejpam-5360	13	1	factors	factor	NOUN
ejpam-5360	13	2	like	like	ADP
ejpam-5360	13	3	these	these	DET
ejpam-5360	13	4	combine	combine	VERB
ejpam-5360	13	5	to	to	PART
ejpam-5360	13	6	make	make	VERB
ejpam-5360	13	7	core	core	NOUN
ejpam-5360	13	8	-	-	PUNCT
ejpam-5360	13	9	satellite	satellite	NOUN
ejpam-5360	13	10	graphs	graph	NOUN
ejpam-5360	13	11	a	a	DET
ejpam-5360	13	12	dynamic	dynamic	ADJ
ejpam-5360	13	13	model	model	NOUN
ejpam-5360	13	14	design	design	NOUN
ejpam-5360	13	15	for	for	ADP
ejpam-5360	13	16	certain	certain	ADJ
ejpam-5360	13	17	types	type	NOUN
ejpam-5360	13	18	of	of	ADP
ejpam-5360	13	19	real	real	ADJ
ejpam-5360	13	20	-	-	PUNCT
ejpam-5360	13	21	world	world	NOUN
ejpam-5360	13	22	networks	network	NOUN
ejpam-5360	13	23	[	[	X
ejpam-5360	13	24	1	1	NUM
ejpam-5360	13	25	,	,	PUNCT
ejpam-5360	13	26	2	2	NUM
ejpam-5360	13	27	]	]	PUNCT
ejpam-5360	13	28	.	.	PUNCT
ejpam-5360	14	1	generalized	generalize	VERB
ejpam-5360	14	2	core	core	NOUN
ejpam-5360	14	3	-	-	PUNCT
ejpam-5360	14	4	satellite	satellite	NOUN
ejpam-5360	14	5	graphs	graph	NOUN
ejpam-5360	14	6	are	be	AUX
ejpam-5360	14	7	denoted	denote	VERB
ejpam-5360	14	8	as	as	ADP
ejpam-5360	14	9	θ(c	θ(c	VERB
ejpam-5360	14	10	,	,	PUNCT
ejpam-5360	14	11	s	s	NOUN
ejpam-5360	14	12	,	,	PUNCT
ejpam-5360	14	13	η∗	η∗	NOUN
ejpam-5360	14	14	)	)	PUNCT
ejpam-5360	14	15	and	and	CCONJ
ejpam-5360	14	16	belong	belong	VERB
ejpam-5360	14	17	to	to	ADP
ejpam-5360	14	18	the	the	DET
ejpam-5360	14	19	family	family	NOUN
ejpam-5360	14	20	of	of	ADP
ejpam-5360	14	21	graphs	graph	NOUN
ejpam-5360	14	22	of	of	ADP
ejpam-5360	14	23	diameter	diameter	NOUN
ejpam-5360	14	24	two	two	NUM
ejpam-5360	14	25	.	.	PUNCT
ejpam-5360	15	1	it	it	PRON
ejpam-5360	15	2	has	have	VERB
ejpam-5360	15	3	a	a	DET
ejpam-5360	15	4	central	central	ADJ
ejpam-5360	15	5	core	core	NOUN
ejpam-5360	15	6	of	of	ADP
ejpam-5360	15	7	vertices	vertex	NOUN
ejpam-5360	15	8	connected	connect	VERB
ejpam-5360	15	9	to	to	ADP
ejpam-5360	15	10	a	a	DET
ejpam-5360	15	11	few	few	ADJ
ejpam-5360	15	12	satellites	satellite	NOUN
ejpam-5360	15	13	,	,	PUNCT
ejpam-5360	15	14	where	where	SCONJ
ejpam-5360	15	15	all	all	DET
ejpam-5360	15	16	satellite	satellite	NOUN
ejpam-5360	15	17	cliques	clique	NOUN
ejpam-5360	15	18	are	be	AUX
ejpam-5360	15	19	not	not	PART
ejpam-5360	15	20	identical	identical	ADJ
ejpam-5360	15	21	and	and	CCONJ
ejpam-5360	15	22	might	might	AUX
ejpam-5360	15	23	be	be	AUX
ejpam-5360	15	24	of	of	ADP
ejpam-5360	15	25	different	different	ADJ
ejpam-5360	15	26	sizes	size	NOUN
ejpam-5360	15	27	.	.	PUNCT
ejpam-5360	16	1	there	there	PRON
ejpam-5360	16	2	is	be	VERB
ejpam-5360	16	3	no	no	DET
ejpam-5360	16	4	restriction	restriction	NOUN
ejpam-5360	16	5	on	on	ADP
ejpam-5360	16	6	having	have	VERB
ejpam-5360	16	7	the	the	DET
ejpam-5360	16	8	same	same	ADJ
ejpam-5360	16	9	number	number	NOUN
ejpam-5360	16	10	of	of	ADP
ejpam-5360	16	11	vertices	vertex	NOUN
ejpam-5360	16	12	or	or	CCONJ
ejpam-5360	16	13	nodes	node	NOUN
ejpam-5360	16	14	.	.	PUNCT
ejpam-5360	17	1	∗corresponding	∗corresponde	VERB
ejpam-5360	17	2	author	author	NOUN
ejpam-5360	17	3	.	.	PUNCT
ejpam-5360	18	1	doi	doi	NOUN
ejpam-5360	18	2	:	:	PUNCT
ejpam-5360	18	3	https://doi.org/10.29020/nybg.ejpam.v18i4.5360	https://doi.org/10.29020/nybg.ejpam.v18i4.5360	PROPN
ejpam-5360	18	4	email	email	NOUN
ejpam-5360	18	5	addresses	address	NOUN
ejpam-5360	18	6	:	:	PUNCT
ejpam-5360	18	7	malathy.viswanathan2015@vit.ac.in	malathy.viswanathan2015@vit.ac.in	NOUN
ejpam-5360	18	8	(	(	PUNCT
ejpam-5360	18	9	m.	m.	NOUN
ejpam-5360	18	10	v.	v.	PROPN
ejpam-5360	18	11	)	)	PUNCT
ejpam-5360	18	12	,	,	PUNCT
ejpam-5360	18	13	kalyanidesikan@vit.ac.in	kalyanidesikan@vit.ac.in	NOUN
ejpam-5360	18	14	(	(	PUNCT
ejpam-5360	18	15	k.	k.	PROPN
ejpam-5360	18	16	desikan	desikan	PROPN
ejpam-5360	18	17	)	)	PUNCT
ejpam-5360	18	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5360	19	1	1	1	NUM
ejpam-5360	19	2	copyright	copyright	NOUN
ejpam-5360	19	3	:	:	PUNCT
ejpam-5360	19	4	©	©	PROPN
ejpam-5360	19	5	2025	2025	NUM
ejpam-5360	19	6	the	the	DET
ejpam-5360	19	7	author(s	author(s	NOUN
ejpam-5360	19	8	)	)	PUNCT
ejpam-5360	19	9	.	.	PUNCT
ejpam-5360	20	1	(	(	PUNCT
ejpam-5360	20	2	cc	cc	NOUN
ejpam-5360	20	3	by	by	ADP
ejpam-5360	20	4	-	-	PUNCT
ejpam-5360	20	5	nc	nc	PROPN
ejpam-5360	20	6	4.0	4.0	NUM
ejpam-5360	20	7	)	)	PUNCT
ejpam-5360	20	8	m.	m.	NOUN
ejpam-5360	20	9	v.	v.	ADP
ejpam-5360	20	10	,	,	PUNCT
ejpam-5360	20	11	k.	k.	PROPN
ejpam-5360	20	12	desikan	desikan	PROPN
ejpam-5360	20	13	/	/	SYM
ejpam-5360	20	14	eur	eur	PROPN
ejpam-5360	20	15	.	.	PUNCT
ejpam-5360	21	1	j.	j.	PROPN
ejpam-5360	21	2	pure	pure	PROPN
ejpam-5360	21	3	appl	appl	PROPN
ejpam-5360	21	4	.	.	PROPN
ejpam-5360	21	5	math	math	PROPN
ejpam-5360	21	6	,	,	PUNCT
ejpam-5360	21	7	18	18	NUM
ejpam-5360	21	8	(	(	PUNCT
ejpam-5360	21	9	4	4	NUM
ejpam-5360	21	10	)	)	PUNCT
ejpam-5360	21	11	(	(	PUNCT
ejpam-5360	21	12	2025	2025	NUM
ejpam-5360	21	13	)	)	PUNCT
ejpam-5360	21	14	,	,	PUNCT
ejpam-5360	21	15	5360	5360	NUM
ejpam-5360	21	16	2	2	NUM
ejpam-5360	21	17	of	of	ADP
ejpam-5360	21	18	20	20	NUM
ejpam-5360	21	19	estrada	estrada	PROPN
ejpam-5360	21	20	and	and	CCONJ
ejpam-5360	21	21	benzi	benzi	PROPN
ejpam-5360	21	22	[	[	X
ejpam-5360	21	23	1	1	X
ejpam-5360	21	24	]	]	PUNCT
ejpam-5360	21	25	introduced	introduce	VERB
ejpam-5360	21	26	the	the	DET
ejpam-5360	21	27	generalized	generalize	VERB
ejpam-5360	21	28	core	core	NOUN
ejpam-5360	21	29	-	-	PUNCT
ejpam-5360	21	30	satellite	satellite	NOUN
ejpam-5360	21	31	graphs	graph	NOUN
ejpam-5360	21	32	,	,	PUNCT
ejpam-5360	21	33	where	where	SCONJ
ejpam-5360	21	34	they	they	PRON
ejpam-5360	21	35	generalized	generalize	VERB
ejpam-5360	21	36	both	both	CCONJ
ejpam-5360	21	37	the	the	DET
ejpam-5360	21	38	windmill	windmill	NOUN
ejpam-5360	21	39	and	and	CCONJ
ejpam-5360	21	40	complete	complete	ADJ
ejpam-5360	21	41	split	split	NOUN
ejpam-5360	21	42	graphs	graph	NOUN
ejpam-5360	21	43	and	and	CCONJ
ejpam-5360	21	44	analyzed	analyze	VERB
ejpam-5360	21	45	certain	certain	ADJ
ejpam-5360	21	46	features	feature	NOUN
ejpam-5360	21	47	like	like	ADP
ejpam-5360	21	48	clustering	clustering	NOUN
ejpam-5360	21	49	,	,	PUNCT
ejpam-5360	21	50	assortativity	assortativity	NOUN
ejpam-5360	21	51	,	,	PUNCT
ejpam-5360	21	52	and	and	CCONJ
ejpam-5360	21	53	spectral	spectral	ADJ
ejpam-5360	21	54	properties	property	NOUN
ejpam-5360	21	55	.	.	PUNCT
ejpam-5360	22	1	the	the	DET
ejpam-5360	22	2	authors	author	NOUN
ejpam-5360	22	3	characterized	characterize	VERB
ejpam-5360	22	4	the	the	DET
ejpam-5360	22	5	eigenstructure	eigenstructure	NOUN
ejpam-5360	22	6	of	of	ADP
ejpam-5360	22	7	these	these	DET
ejpam-5360	22	8	graphs	graph	NOUN
ejpam-5360	22	9	’	'	PUNCT
ejpam-5360	22	10	adjacency	adjacency	NOUN
ejpam-5360	22	11	and	and	CCONJ
ejpam-5360	22	12	laplacian	laplacian	ADJ
ejpam-5360	22	13	matrices	matrix	NOUN
ejpam-5360	22	14	,	,	PUNCT
ejpam-5360	22	15	observed	observe	VERB
ejpam-5360	22	16	that	that	SCONJ
ejpam-5360	22	17	their	their	PRON
ejpam-5360	22	18	laplacian	laplacian	ADJ
ejpam-5360	22	19	eigenvalues	eigenvalue	NOUN
ejpam-5360	22	20	are	be	AUX
ejpam-5360	22	21	integral	integral	ADJ
ejpam-5360	22	22	,	,	PUNCT
ejpam-5360	22	23	and	and	CCONJ
ejpam-5360	22	24	commented	comment	VERB
ejpam-5360	22	25	on	on	ADP
ejpam-5360	22	26	the	the	DET
ejpam-5360	22	27	asymptotic	asymptotic	ADJ
ejpam-5360	22	28	behavior	behavior	NOUN
ejpam-5360	22	29	of	of	ADP
ejpam-5360	22	30	quantities	quantity	NOUN
ejpam-5360	22	31	such	such	ADJ
ejpam-5360	22	32	as	as	ADP
ejpam-5360	22	33	the	the	DET
ejpam-5360	22	34	synchronizability	synchronizability	NOUN
ejpam-5360	22	35	index	index	NOUN
ejpam-5360	22	36	and	and	CCONJ
ejpam-5360	22	37	infection	infection	NOUN
ejpam-5360	22	38	threshold	threshold	NOUN
ejpam-5360	22	39	.	.	PUNCT
ejpam-5360	23	1	also	also	ADV
ejpam-5360	23	2	,	,	PUNCT
ejpam-5360	23	3	they	they	PRON
ejpam-5360	23	4	determined	determine	VERB
ejpam-5360	23	5	the	the	DET
ejpam-5360	23	6	general	general	ADJ
ejpam-5360	23	7	and	and	CCONJ
ejpam-5360	23	8	spectral	spectral	ADJ
ejpam-5360	23	9	properties	property	NOUN
ejpam-5360	23	10	of	of	ADP
ejpam-5360	23	11	core	core	NOUN
ejpam-5360	23	12	-	-	PUNCT
ejpam-5360	23	13	satellite	satellite	NOUN
ejpam-5360	23	14	graphs	graph	NOUN
ejpam-5360	23	15	and	and	CCONJ
ejpam-5360	23	16	provided	provide	VERB
ejpam-5360	23	17	the	the	DET
ejpam-5360	23	18	bounds	bound	NOUN
ejpam-5360	23	19	of	of	ADP
ejpam-5360	23	20	the	the	DET
ejpam-5360	23	21	spectral	spectral	ADJ
ejpam-5360	23	22	radius	radius	NOUN
ejpam-5360	23	23	of	of	ADP
ejpam-5360	23	24	the	the	DET
ejpam-5360	23	25	generalized	generalize	VERB
ejpam-5360	23	26	core	core	NOUN
ejpam-5360	23	27	-	-	PUNCT
ejpam-5360	23	28	satellite	satellite	NOUN
ejpam-5360	23	29	graph	graph	NOUN
ejpam-5360	23	30	.	.	PUNCT
ejpam-5360	24	1	generalized	generalize	VERB
ejpam-5360	24	2	core	core	NOUN
ejpam-5360	24	3	-	-	PUNCT
ejpam-5360	24	4	satellite	satellite	NOUN
ejpam-5360	24	5	class	class	NOUN
ejpam-5360	24	6	of	of	ADP
ejpam-5360	24	7	graphs	graph	NOUN
ejpam-5360	24	8	is	be	AUX
ejpam-5360	24	9	found	find	VERB
ejpam-5360	24	10	to	to	PART
ejpam-5360	24	11	resolve	resolve	VERB
ejpam-5360	24	12	most	most	ADJ
ejpam-5360	24	13	of	of	ADP
ejpam-5360	24	14	the	the	DET
ejpam-5360	24	15	social	social	ADJ
ejpam-5360	24	16	network	network	NOUN
ejpam-5360	24	17	issues	issue	NOUN
ejpam-5360	24	18	where	where	SCONJ
ejpam-5360	24	19	the	the	DET
ejpam-5360	24	20	network	network	NOUN
ejpam-5360	24	21	’s	’s	PART
ejpam-5360	24	22	graphs	graph	NOUN
ejpam-5360	24	23	are	be	AUX
ejpam-5360	24	24	analyzed	analyze	VERB
ejpam-5360	24	25	.	.	PUNCT
ejpam-5360	25	1	due	due	ADP
ejpam-5360	25	2	to	to	ADP
ejpam-5360	25	3	the	the	DET
ejpam-5360	25	4	expanding	expand	VERB
ejpam-5360	25	5	usage	usage	NOUN
ejpam-5360	25	6	of	of	ADP
ejpam-5360	25	7	social	social	ADJ
ejpam-5360	25	8	networks	network	NOUN
ejpam-5360	25	9	such	such	ADJ
ejpam-5360	25	10	as	as	ADP
ejpam-5360	25	11	linkedin	linkedin	NOUN
ejpam-5360	25	12	,	,	PUNCT
ejpam-5360	25	13	facebook	facebook	PROPN
ejpam-5360	25	14	,	,	PUNCT
ejpam-5360	25	15	instagram	instagram	PROPN
ejpam-5360	25	16	,	,	PUNCT
ejpam-5360	25	17	twitter	twitter	NOUN
ejpam-5360	25	18	,	,	PUNCT
ejpam-5360	25	19	and	and	CCONJ
ejpam-5360	25	20	google+	google+	NOUN
ejpam-5360	25	21	,	,	PUNCT
ejpam-5360	25	22	malicious	malicious	ADJ
ejpam-5360	25	23	users	user	NOUN
ejpam-5360	25	24	seek	seek	VERB
ejpam-5360	25	25	to	to	PART
ejpam-5360	25	26	impinge	impinge	VERB
ejpam-5360	25	27	on	on	ADP
ejpam-5360	25	28	the	the	DET
ejpam-5360	25	29	privacy	privacy	NOUN
ejpam-5360	25	30	of	of	ADP
ejpam-5360	25	31	other	other	ADJ
ejpam-5360	25	32	users	user	NOUN
ejpam-5360	25	33	and	and	CCONJ
ejpam-5360	25	34	exploit	exploit	VERB
ejpam-5360	25	35	their	their	PRON
ejpam-5360	25	36	credentials	credential	NOUN
ejpam-5360	25	37	by	by	ADP
ejpam-5360	25	38	creating	create	VERB
ejpam-5360	25	39	fraudulent	fraudulent	ADJ
ejpam-5360	25	40	accounts	account	NOUN
ejpam-5360	25	41	.	.	PUNCT
ejpam-5360	26	1	this	this	PRON
ejpam-5360	26	2	has	have	AUX
ejpam-5360	26	3	become	become	VERB
ejpam-5360	26	4	a	a	DET
ejpam-5360	26	5	cause	cause	NOUN
ejpam-5360	26	6	of	of	ADP
ejpam-5360	26	7	concern	concern	NOUN
ejpam-5360	26	8	for	for	ADP
ejpam-5360	26	9	users	user	NOUN
ejpam-5360	26	10	.	.	PUNCT
ejpam-5360	27	1	hence	hence	ADV
ejpam-5360	27	2	,	,	PUNCT
ejpam-5360	27	3	social	social	ADJ
ejpam-5360	27	4	network	network	NOUN
ejpam-5360	27	5	providers	provider	NOUN
ejpam-5360	27	6	are	be	AUX
ejpam-5360	27	7	attempting	attempt	VERB
ejpam-5360	27	8	to	to	PART
ejpam-5360	27	9	identify	identify	VERB
ejpam-5360	27	10	these	these	DET
ejpam-5360	27	11	users	user	NOUN
ejpam-5360	27	12	and	and	CCONJ
ejpam-5360	27	13	their	their	PRON
ejpam-5360	27	14	fake	fake	ADJ
ejpam-5360	27	15	accounts	account	NOUN
ejpam-5360	27	16	to	to	PART
ejpam-5360	27	17	remove	remove	VERB
ejpam-5360	27	18	them	they	PRON
ejpam-5360	27	19	from	from	ADP
ejpam-5360	27	20	social	social	ADJ
ejpam-5360	27	21	networking	networking	NOUN
ejpam-5360	27	22	environments	environment	NOUN
ejpam-5360	27	23	using	use	VERB
ejpam-5360	27	24	graph	graph	NOUN
ejpam-5360	27	25	analysis	analysis	NOUN
ejpam-5360	27	26	.	.	PUNCT
ejpam-5360	28	1	graph	graph	NOUN
ejpam-5360	28	2	similarity	similarity	NOUN
ejpam-5360	28	3	measures	measure	NOUN
ejpam-5360	28	4	are	be	AUX
ejpam-5360	28	5	used	use	VERB
ejpam-5360	28	6	in	in	ADP
ejpam-5360	28	7	graph	graph	NOUN
ejpam-5360	28	8	analysis	analysis	NOUN
ejpam-5360	28	9	.	.	PUNCT
ejpam-5360	29	1	some	some	DET
ejpam-5360	29	2	significant	significant	ADJ
ejpam-5360	29	3	similarity	similarity	NOUN
ejpam-5360	29	4	measures	measure	NOUN
ejpam-5360	29	5	namely	namely	ADV
ejpam-5360	29	6	jaccard	jaccard	ADJ
ejpam-5360	29	7	,	,	PUNCT
ejpam-5360	29	8	cosine	cosine	NOUN
ejpam-5360	29	9	,	,	PUNCT
ejpam-5360	29	10	and	and	CCONJ
ejpam-5360	29	11	l1	l1	PROPN
ejpam-5360	29	12	norm	norm	NOUN
ejpam-5360	29	13	are	be	AUX
ejpam-5360	29	14	a	a	DET
ejpam-5360	29	15	few	few	ADJ
ejpam-5360	29	16	measures	measure	NOUN
ejpam-5360	29	17	where	where	SCONJ
ejpam-5360	29	18	friendship	friendship	NOUN
ejpam-5360	29	19	graphs	graph	NOUN
ejpam-5360	29	20	(	(	PUNCT
ejpam-5360	29	21	core	core	NOUN
ejpam-5360	29	22	-	-	PUNCT
ejpam-5360	29	23	satellite	satellite	NOUN
ejpam-5360	29	24	graph	graph	NOUN
ejpam-5360	29	25	)	)	PUNCT
ejpam-5360	29	26	are	be	AUX
ejpam-5360	29	27	utilized	utilize	VERB
ejpam-5360	29	28	to	to	PART
ejpam-5360	29	29	detect	detect	VERB
ejpam-5360	29	30	suspicious	suspicious	ADJ
ejpam-5360	29	31	accounts	account	NOUN
ejpam-5360	29	32	[	[	X
ejpam-5360	29	33	3	3	NUM
ejpam-5360	29	34	]	]	PUNCT
ejpam-5360	29	35	.	.	PUNCT
ejpam-5360	30	1	many	many	ADJ
ejpam-5360	30	2	results	result	NOUN
ejpam-5360	30	3	on	on	ADP
ejpam-5360	30	4	bounds	bound	NOUN
ejpam-5360	30	5	for	for	ADP
ejpam-5360	30	6	the	the	DET
ejpam-5360	30	7	spectral	spectral	ADJ
ejpam-5360	30	8	radius	radius	NOUN
ejpam-5360	30	9	and	and	CCONJ
ejpam-5360	30	10	signless	signless	PROPN
ejpam-5360	30	11	laplacian	laplacian	ADJ
ejpam-5360	30	12	spectral	spectral	ADJ
ejpam-5360	30	13	radius	radius	NOUN
ejpam-5360	30	14	have	have	AUX
ejpam-5360	30	15	been	be	AUX
ejpam-5360	30	16	given	give	VERB
ejpam-5360	30	17	as	as	ADP
ejpam-5360	30	18	functions	function	NOUN
ejpam-5360	30	19	of	of	ADP
ejpam-5360	30	20	graph	graph	NOUN
ejpam-5360	30	21	parameters	parameter	NOUN
ejpam-5360	30	22	such	such	ADJ
ejpam-5360	30	23	as	as	ADP
ejpam-5360	30	24	the	the	DET
ejpam-5360	30	25	number	number	NOUN
ejpam-5360	30	26	of	of	ADP
ejpam-5360	30	27	vertices	vertex	NOUN
ejpam-5360	30	28	,	,	PUNCT
ejpam-5360	30	29	edges	edge	NOUN
ejpam-5360	30	30	,	,	PUNCT
ejpam-5360	30	31	degree	degree	NOUN
ejpam-5360	30	32	sequence	sequence	NOUN
ejpam-5360	30	33	,	,	PUNCT
ejpam-5360	30	34	average	average	ADJ
ejpam-5360	30	35	2	2	NUM
ejpam-5360	30	36	-	-	PUNCT
ejpam-5360	30	37	degree	degree	NOUN
ejpam-5360	30	38	,	,	PUNCT
ejpam-5360	30	39	diameter	diameter	NOUN
ejpam-5360	30	40	,	,	PUNCT
ejpam-5360	30	41	covering	cover	VERB
ejpam-5360	30	42	number	number	NOUN
ejpam-5360	30	43	,	,	PUNCT
ejpam-5360	30	44	domination	domination	NOUN
ejpam-5360	30	45	number	number	NOUN
ejpam-5360	30	46	,	,	PUNCT
ejpam-5360	30	47	independence	independence	NOUN
ejpam-5360	30	48	number	number	NOUN
ejpam-5360	30	49	,	,	PUNCT
ejpam-5360	30	50	and	and	CCONJ
ejpam-5360	30	51	others	other	NOUN
ejpam-5360	31	1	[	[	X
ejpam-5360	31	2	4–7	4–7	X
ejpam-5360	31	3	]	]	X
ejpam-5360	31	4	.	.	PUNCT
ejpam-5360	32	1	in	in	ADP
ejpam-5360	32	2	[	[	X
ejpam-5360	32	3	8	8	NUM
ejpam-5360	32	4	]	]	PUNCT
ejpam-5360	32	5	,	,	PUNCT
ejpam-5360	32	6	nair	nair	PROPN
ejpam-5360	32	7	abreu	abreu	PROPN
ejpam-5360	32	8	et	et	PROPN
ejpam-5360	32	9	al	al	PROPN
ejpam-5360	32	10	.	.	PROPN
ejpam-5360	32	11	mentioned	mention	VERB
ejpam-5360	32	12	that	that	SCONJ
ejpam-5360	32	13	this	this	DET
ejpam-5360	32	14	class	class	NOUN
ejpam-5360	32	15	of	of	ADP
ejpam-5360	32	16	graphs	graph	NOUN
ejpam-5360	32	17	is	be	AUX
ejpam-5360	32	18	equivalent	equivalent	ADJ
ejpam-5360	32	19	to	to	ADP
ejpam-5360	32	20	chordal	chordal	ADJ
ejpam-5360	32	21	graphs	graph	NOUN
ejpam-5360	32	22	having	have	VERB
ejpam-5360	32	23	only	only	ADV
ejpam-5360	32	24	one	one	NUM
ejpam-5360	32	25	minimal	minimal	ADJ
ejpam-5360	32	26	vertex	vertex	NOUN
ejpam-5360	32	27	separator	separator	NOUN
ejpam-5360	32	28	and	and	CCONJ
ejpam-5360	32	29	the	the	DET
ejpam-5360	32	30	subclass	subclass	NOUN
ejpam-5360	32	31	of	of	ADP
ejpam-5360	32	32	a	a	DET
ejpam-5360	32	33	quasi	quasi	ADJ
ejpam-5360	32	34	-	-	ADJ
ejpam-5360	32	35	threshold	threshold	ADJ
ejpam-5360	32	36	graph	graph	NOUN
ejpam-5360	32	37	.	.	PUNCT
ejpam-5360	33	1	they	they	PRON
ejpam-5360	33	2	demonstrated	demonstrate	VERB
ejpam-5360	33	3	that	that	SCONJ
ejpam-5360	33	4	this	this	DET
ejpam-5360	33	5	class	class	NOUN
ejpam-5360	33	6	of	of	ADP
ejpam-5360	33	7	graphs	graph	NOUN
ejpam-5360	33	8	belongs	belong	VERB
ejpam-5360	33	9	to	to	ADP
ejpam-5360	33	10	the	the	DET
ejpam-5360	33	11	hierarchical	hierarchical	ADJ
ejpam-5360	33	12	structure	structure	NOUN
ejpam-5360	33	13	of	of	ADP
ejpam-5360	33	14	chordal	chordal	NOUN
ejpam-5360	33	15	graphs	graph	NOUN
ejpam-5360	33	16	.	.	PUNCT
ejpam-5360	34	1	moreover	moreover	ADV
ejpam-5360	34	2	,	,	PUNCT
ejpam-5360	34	3	in	in	ADP
ejpam-5360	34	4	[	[	PUNCT
ejpam-5360	34	5	9	9	NUM
ejpam-5360	34	6	]	]	PUNCT
ejpam-5360	34	7	,	,	PUNCT
ejpam-5360	34	8	das	das	PROPN
ejpam-5360	34	9	has	have	AUX
ejpam-5360	34	10	proved	prove	VERB
ejpam-5360	34	11	a	a	DET
ejpam-5360	34	12	conjecture	conjecture	NOUN
ejpam-5360	34	13	on	on	ADP
ejpam-5360	34	14	the	the	DET
ejpam-5360	34	15	complete	complete	ADJ
ejpam-5360	34	16	split	split	NOUN
ejpam-5360	34	17	-	-	PUNCT
ejpam-5360	34	18	like	like	ADJ
ejpam-5360	34	19	graph	graph	NOUN
ejpam-5360	34	20	.	.	PUNCT
ejpam-5360	35	1	in	in	ADP
ejpam-5360	35	2	[	[	X
ejpam-5360	35	3	10	10	NUM
ejpam-5360	35	4	]	]	PUNCT
ejpam-5360	35	5	,	,	PUNCT
ejpam-5360	35	6	liu	liu	PROPN
ejpam-5360	35	7	et	et	PROPN
ejpam-5360	35	8	al	al	PROPN
ejpam-5360	35	9	.	.	PROPN
ejpam-5360	35	10	presented	present	VERB
ejpam-5360	35	11	several	several	ADJ
ejpam-5360	35	12	upper	upper	ADJ
ejpam-5360	35	13	and	and	CCONJ
ejpam-5360	35	14	lower	low	ADJ
ejpam-5360	35	15	bounds	bound	NOUN
ejpam-5360	35	16	on	on	ADP
ejpam-5360	35	17	the	the	DET
ejpam-5360	35	18	k	k	NOUN
ejpam-5360	35	19	-	-	PUNCT
ejpam-5360	35	20	th	th	X
ejpam-5360	35	21	largest	large	ADJ
ejpam-5360	35	22	eigenvalue	eigenvalue	NOUN
ejpam-5360	35	23	of	of	ADP
ejpam-5360	35	24	aα	aα	NOUN
ejpam-5360	35	25	matrix	matrix	NOUN
ejpam-5360	35	26	and	and	CCONJ
ejpam-5360	35	27	characterized	characterize	VERB
ejpam-5360	35	28	the	the	DET
ejpam-5360	35	29	extremal	extremal	ADJ
ejpam-5360	35	30	graphs	graph	NOUN
ejpam-5360	35	31	corresponding	correspond	VERB
ejpam-5360	35	32	to	to	ADP
ejpam-5360	35	33	the	the	DET
ejpam-5360	35	34	bounds	bound	NOUN
ejpam-5360	35	35	obtained	obtain	VERB
ejpam-5360	35	36	.	.	PUNCT
ejpam-5360	36	1	we	we	PRON
ejpam-5360	36	2	find	find	VERB
ejpam-5360	36	3	many	many	ADJ
ejpam-5360	36	4	recent	recent	ADJ
ejpam-5360	36	5	articles	article	NOUN
ejpam-5360	36	6	on	on	ADP
ejpam-5360	36	7	spectral	spectral	ADJ
ejpam-5360	36	8	radius	radius	NOUN
ejpam-5360	36	9	and	and	CCONJ
ejpam-5360	36	10	signless	signless	PROPN
ejpam-5360	36	11	laplacian	laplacian	ADJ
ejpam-5360	36	12	spectral	spectral	ADJ
ejpam-5360	36	13	radius	radius	NOUN
ejpam-5360	36	14	.	.	PUNCT
ejpam-5360	37	1	in	in	ADP
ejpam-5360	37	2	[	[	X
ejpam-5360	37	3	11	11	NUM
ejpam-5360	37	4	]	]	PUNCT
ejpam-5360	37	5	,	,	PUNCT
ejpam-5360	37	6	wang	wang	PROPN
ejpam-5360	37	7	and	and	CCONJ
ejpam-5360	37	8	guo	guo	PROPN
ejpam-5360	37	9	investigated	investigate	VERB
ejpam-5360	37	10	the	the	DET
ejpam-5360	37	11	upper	upper	ADJ
ejpam-5360	37	12	bounds	bound	NOUN
ejpam-5360	37	13	of	of	ADP
ejpam-5360	37	14	the	the	DET
ejpam-5360	37	15	spectral	spectral	ADJ
ejpam-5360	37	16	radius	radius	NOUN
ejpam-5360	37	17	of	of	ADP
ejpam-5360	37	18	the	the	DET
ejpam-5360	37	19	coalescense	coalescense	NOUN
ejpam-5360	37	20	of	of	ADP
ejpam-5360	37	21	two	two	NUM
ejpam-5360	37	22	graphs	graph	NOUN
ejpam-5360	37	23	generalizing	generalize	VERB
ejpam-5360	37	24	some	some	DET
ejpam-5360	37	25	results	result	NOUN
ejpam-5360	37	26	by	by	ADP
ejpam-5360	37	27	passbani	passbani	NOUN
ejpam-5360	37	28	and	and	CCONJ
ejpam-5360	37	29	salemi	salemi	NOUN
ejpam-5360	37	30	in	in	ADP
ejpam-5360	37	31	2019	2019	NUM
ejpam-5360	37	32	and	and	CCONJ
ejpam-5360	37	33	as	as	ADP
ejpam-5360	37	34	an	an	DET
ejpam-5360	37	35	application	application	NOUN
ejpam-5360	37	36	provided	provide	VERB
ejpam-5360	37	37	a	a	DET
ejpam-5360	37	38	new	new	ADJ
ejpam-5360	37	39	sharp	sharp	ADJ
ejpam-5360	37	40	upper	upper	ADJ
ejpam-5360	37	41	bound	bind	VERB
ejpam-5360	37	42	on	on	ADP
ejpam-5360	37	43	the	the	DET
ejpam-5360	37	44	spectral	spectral	ADJ
ejpam-5360	37	45	radius	radius	NOUN
ejpam-5360	37	46	of	of	ADP
ejpam-5360	37	47	a	a	DET
ejpam-5360	37	48	tree	tree	NOUN
ejpam-5360	37	49	.	.	PUNCT
ejpam-5360	38	1	ghorbani	ghorbani	NOUN
ejpam-5360	38	2	and	and	CCONJ
ejpam-5360	38	3	amraei	amraei	ADJ
ejpam-5360	38	4	[	[	X
ejpam-5360	38	5	12	12	NUM
ejpam-5360	38	6	]	]	PUNCT
ejpam-5360	38	7	,	,	PUNCT
ejpam-5360	38	8	investigated	investigate	VERB
ejpam-5360	38	9	the	the	DET
ejpam-5360	38	10	spectra	spectra	NOUN
ejpam-5360	38	11	of	of	ADP
ejpam-5360	38	12	certain	certain	ADJ
ejpam-5360	38	13	classes	class	NOUN
ejpam-5360	38	14	of	of	ADP
ejpam-5360	38	15	vertex	vertex	NOUN
ejpam-5360	38	16	or	or	CCONJ
ejpam-5360	38	17	edge	edge	NOUN
ejpam-5360	38	18	-	-	PUNCT
ejpam-5360	38	19	transitive	transitive	ADJ
ejpam-5360	38	20	graphs	graph	NOUN
ejpam-5360	38	21	for	for	ADP
ejpam-5360	38	22	extended	extended	ADJ
ejpam-5360	38	23	adjacency	adjacency	NOUN
ejpam-5360	38	24	matrix	matrix	NOUN
ejpam-5360	38	25	and	and	CCONJ
ejpam-5360	38	26	obtained	obtain	VERB
ejpam-5360	38	27	some	some	DET
ejpam-5360	38	28	new	new	ADJ
ejpam-5360	38	29	bounds	bound	NOUN
ejpam-5360	38	30	for	for	ADP
ejpam-5360	38	31	both	both	CCONJ
ejpam-5360	38	32	the	the	DET
ejpam-5360	38	33	smallest	small	ADJ
ejpam-5360	38	34	and	and	CCONJ
ejpam-5360	38	35	the	the	DET
ejpam-5360	38	36	largest	large	ADJ
ejpam-5360	38	37	eigenvalues	eigenvalue	NOUN
ejpam-5360	38	38	examining	examine	VERB
ejpam-5360	38	39	the	the	DET
ejpam-5360	38	40	behaviour	behaviour	NOUN
ejpam-5360	38	41	of	of	ADP
ejpam-5360	38	42	aex	aex	PROPN
ejpam-5360	38	43	-	-	PUNCT
ejpam-5360	38	44	energy	energy	NOUN
ejpam-5360	38	45	of	of	ADP
ejpam-5360	38	46	a	a	DET
ejpam-5360	38	47	graph	graph	NOUN
ejpam-5360	38	48	.	.	PUNCT
ejpam-5360	39	1	in	in	ADP
ejpam-5360	39	2	[	[	X
ejpam-5360	39	3	13	13	NUM
ejpam-5360	39	4	]	]	PUNCT
ejpam-5360	39	5	,	,	PUNCT
ejpam-5360	39	6	authors	author	NOUN
ejpam-5360	39	7	characterized	characterize	VERB
ejpam-5360	39	8	irregular	irregular	ADJ
ejpam-5360	39	9	bipartite	bipartite	NOUN
ejpam-5360	39	10	graphs	graph	NOUN
ejpam-5360	39	11	with	with	ADP
ejpam-5360	39	12	maximum	maximum	ADJ
ejpam-5360	39	13	spectral	spectral	ADJ
ejpam-5360	39	14	radius	radius	NOUN
ejpam-5360	39	15	and	and	CCONJ
ejpam-5360	39	16	presented	present	VERB
ejpam-5360	39	17	an	an	DET
ejpam-5360	39	18	upper	upper	ADJ
ejpam-5360	39	19	bound	bind	VERB
ejpam-5360	39	20	on	on	ADP
ejpam-5360	39	21	the	the	DET
ejpam-5360	39	22	spectral	spectral	ADJ
ejpam-5360	39	23	radius	radius	NOUN
ejpam-5360	39	24	in	in	ADP
ejpam-5360	39	25	terms	term	NOUN
ejpam-5360	39	26	of	of	ADP
ejpam-5360	39	27	the	the	DET
ejpam-5360	39	28	order	order	NOUN
ejpam-5360	39	29	and	and	CCONJ
ejpam-5360	39	30	maximum	maximum	ADJ
ejpam-5360	39	31	degree	degree	NOUN
ejpam-5360	39	32	.	.	PUNCT
ejpam-5360	40	1	das	das	PROPN
ejpam-5360	40	2	and	and	CCONJ
ejpam-5360	40	3	liu	liu	PROPN
ejpam-5360	40	4	in	in	ADP
ejpam-5360	40	5	[	[	X
ejpam-5360	40	6	14	14	NUM
ejpam-5360	40	7	]	]	PUNCT
ejpam-5360	40	8	proved	prove	VERB
ejpam-5360	40	9	complete	complete	ADJ
ejpam-5360	40	10	split	split	NOUN
ejpam-5360	40	11	graph	graph	NOUN
ejpam-5360	40	12	cs(n	cs(n	PROPN
ejpam-5360	40	13	,	,	PUNCT
ejpam-5360	40	14	α	α	NOUN
ejpam-5360	40	15	)	)	PUNCT
ejpam-5360	40	16	,	,	PUNCT
ejpam-5360	40	17	the	the	DET
ejpam-5360	40	18	graph	graph	NOUN
ejpam-5360	40	19	on	on	ADP
ejpam-5360	40	20	n	n	DET
ejpam-5360	40	21	vertices	vertex	NOUN
ejpam-5360	40	22	consisting	consist	VERB
ejpam-5360	40	23	of	of	ADP
ejpam-5360	40	24	a	a	DET
ejpam-5360	40	25	clique	clique	NOUN
ejpam-5360	40	26	on	on	ADP
ejpam-5360	40	27	(	(	PUNCT
ejpam-5360	40	28	n	n	CCONJ
ejpam-5360	40	29	−	−	PROPN
ejpam-5360	40	30	α	α	NOUN
ejpam-5360	40	31	)	)	PUNCT
ejpam-5360	40	32	vertices	vertex	NOUN
ejpam-5360	40	33	and	and	CCONJ
ejpam-5360	40	34	an	an	DET
ejpam-5360	40	35	independent	independent	ADJ
ejpam-5360	40	36	set	set	NOUN
ejpam-5360	40	37	on	on	ADP
ejpam-5360	40	38	the	the	DET
ejpam-5360	40	39	remaining	remain	VERB
ejpam-5360	40	40	α	α	NOUN
ejpam-5360	40	41	where	where	SCONJ
ejpam-5360	40	42	(	(	PUNCT
ejpam-5360	40	43	1	1	NUM
ejpam-5360	40	44	≤	≤	NUM
ejpam-5360	40	45	α	α	NOUN
ejpam-5360	40	46	≤	≤	NOUN
ejpam-5360	40	47	n	n	CCONJ
ejpam-5360	40	48	−	−	PROPN
ejpam-5360	40	49	1	1	NUM
ejpam-5360	40	50	)	)	PUNCT
ejpam-5360	40	51	vertices	vertex	NOUN
ejpam-5360	40	52	in	in	ADP
ejpam-5360	40	53	which	which	PRON
ejpam-5360	40	54	each	each	DET
ejpam-5360	40	55	vertex	vertex	NOUN
ejpam-5360	40	56	of	of	ADP
ejpam-5360	40	57	the	the	DET
ejpam-5360	40	58	clique	clique	NOUN
ejpam-5360	40	59	is	be	AUX
ejpam-5360	40	60	adjacent	adjacent	ADJ
ejpam-5360	40	61	to	to	ADP
ejpam-5360	40	62	each	each	DET
ejpam-5360	40	63	vertex	vertex	NOUN
ejpam-5360	40	64	of	of	ADP
ejpam-5360	40	65	m.	m.	NOUN
ejpam-5360	40	66	v.	v.	PROPN
ejpam-5360	40	67	,	,	PUNCT
ejpam-5360	40	68	k.	k.	PROPN
ejpam-5360	40	69	desikan	desikan	PROPN
ejpam-5360	40	70	/	/	SYM
ejpam-5360	40	71	eur	eur	PROPN
ejpam-5360	40	72	.	.	PUNCT
ejpam-5360	41	1	j.	j.	PROPN
ejpam-5360	41	2	pure	pure	PROPN
ejpam-5360	41	3	appl	appl	PROPN
ejpam-5360	41	4	.	.	PROPN
ejpam-5360	41	5	math	math	PROPN
ejpam-5360	41	6	,	,	PUNCT
ejpam-5360	41	7	18	18	NUM
ejpam-5360	41	8	(	(	PUNCT
ejpam-5360	41	9	4	4	NUM
ejpam-5360	41	10	)	)	PUNCT
ejpam-5360	41	11	(	(	PUNCT
ejpam-5360	41	12	2025	2025	NUM
ejpam-5360	41	13	)	)	PUNCT
ejpam-5360	41	14	,	,	PUNCT
ejpam-5360	41	15	5360	5360	NUM
ejpam-5360	41	16	3	3	NUM
ejpam-5360	41	17	of	of	ADP
ejpam-5360	41	18	20	20	NUM
ejpam-5360	41	19	the	the	DET
ejpam-5360	41	20	independent	independent	ADJ
ejpam-5360	41	21	set	set	NOUN
ejpam-5360	41	22	.	.	PUNCT
ejpam-5360	42	1	they	they	PRON
ejpam-5360	42	2	determined	determine	VERB
ejpam-5360	42	3	its	its	PRON
ejpam-5360	42	4	laplacian	laplacian	ADJ
ejpam-5360	42	5	spectrum	spectrum	NOUN
ejpam-5360	42	6	when	when	SCONJ
ejpam-5360	42	7	1	1	NUM
ejpam-5360	42	8	≤	≤	NUM
ejpam-5360	42	9	α	α	NOUN
ejpam-5360	42	10	≤	≤	NOUN
ejpam-5360	42	11	n	n	CCONJ
ejpam-5360	42	12	−	−	PROPN
ejpam-5360	42	13	1	1	NUM
ejpam-5360	42	14	and	and	CCONJ
ejpam-5360	42	15	the	the	DET
ejpam-5360	42	16	signless	signless	PROPN
ejpam-5360	42	17	laplacian	laplacian	NOUN
ejpam-5360	42	18	spectrum	spectrum	NOUN
ejpam-5360	42	19	when	when	SCONJ
ejpam-5360	42	20	1	1	NUM
ejpam-5360	42	21	≤	≤	NUM
ejpam-5360	42	22	α	α	NOUN
ejpam-5360	42	23	≤	≤	NUM
ejpam-5360	42	24	n−	n−	NOUN
ejpam-5360	42	25	1	1	NUM
ejpam-5360	42	26	,	,	PUNCT
ejpam-5360	42	27	α	α	PRON
ejpam-5360	42	28	̸=	̸=	PROPN
ejpam-5360	42	29	3	3	NUM
ejpam-5360	42	30	.	.	PUNCT
ejpam-5360	43	1	in	in	ADP
ejpam-5360	43	2	[	[	X
ejpam-5360	43	3	15	15	NUM
ejpam-5360	43	4	]	]	PUNCT
ejpam-5360	43	5	,	,	PUNCT
ejpam-5360	43	6	the	the	DET
ejpam-5360	43	7	authors	author	NOUN
ejpam-5360	43	8	obtained	obtain	VERB
ejpam-5360	43	9	sharp	sharp	ADJ
ejpam-5360	43	10	upper	upper	ADJ
ejpam-5360	43	11	bounds	bound	NOUN
ejpam-5360	43	12	on	on	ADP
ejpam-5360	43	13	the	the	DET
ejpam-5360	43	14	q	q	NOUN
ejpam-5360	43	15	-	-	PUNCT
ejpam-5360	43	16	index	index	NOUN
ejpam-5360	43	17	of	of	ADP
ejpam-5360	43	18	(	(	PUNCT
ejpam-5360	43	19	minimally	minimally	ADV
ejpam-5360	43	20	)	)	PUNCT
ejpam-5360	43	21	2	2	NUM
ejpam-5360	43	22	-	-	PUNCT
ejpam-5360	43	23	connected	connect	VERB
ejpam-5360	43	24	graphs	graph	NOUN
ejpam-5360	43	25	with	with	ADP
ejpam-5360	43	26	given	give	VERB
ejpam-5360	43	27	size	size	NOUN
ejpam-5360	43	28	,	,	PUNCT
ejpam-5360	43	29	and	and	CCONJ
ejpam-5360	43	30	characterize	characterize	VERB
ejpam-5360	43	31	the	the	DET
ejpam-5360	43	32	corresponding	corresponding	ADJ
ejpam-5360	43	33	extremal	extremal	ADJ
ejpam-5360	43	34	graphs	graph	NOUN
ejpam-5360	43	35	.	.	PUNCT
ejpam-5360	44	1	in	in	ADP
ejpam-5360	44	2	this	this	DET
ejpam-5360	44	3	article	article	NOUN
ejpam-5360	44	4	,	,	PUNCT
ejpam-5360	44	5	we	we	PRON
ejpam-5360	44	6	have	have	AUX
ejpam-5360	44	7	considered	consider	VERB
ejpam-5360	44	8	generalized	generalized	ADJ
ejpam-5360	44	9	core	core	NOUN
ejpam-5360	44	10	-	-	PUNCT
ejpam-5360	44	11	satellite	satellite	NOUN
ejpam-5360	44	12	graph	graph	NOUN
ejpam-5360	44	13	θ(c	θ(c	VERB
ejpam-5360	44	14	,	,	PUNCT
ejpam-5360	44	15	s	s	NOUN
ejpam-5360	44	16	,	,	PUNCT
ejpam-5360	44	17	η∗	η∗	NOUN
ejpam-5360	44	18	)	)	PUNCT
ejpam-5360	44	19	and	and	CCONJ
ejpam-5360	44	20	obtained	obtain	VERB
ejpam-5360	44	21	results	result	NOUN
ejpam-5360	44	22	on	on	ADP
ejpam-5360	44	23	the	the	DET
ejpam-5360	44	24	upper	upper	ADJ
ejpam-5360	44	25	and	and	CCONJ
ejpam-5360	44	26	lower	low	ADJ
ejpam-5360	44	27	bounds	bound	NOUN
ejpam-5360	44	28	for	for	ADP
ejpam-5360	44	29	the	the	DET
ejpam-5360	44	30	spectral	spectral	ADJ
ejpam-5360	44	31	radius	radius	NOUN
ejpam-5360	44	32	and	and	CCONJ
ejpam-5360	44	33	signless	signless	PROPN
ejpam-5360	44	34	laplacian	laplacian	ADJ
ejpam-5360	44	35	spectral	spectral	ADJ
ejpam-5360	44	36	radius	radius	NOUN
ejpam-5360	44	37	in	in	ADP
ejpam-5360	44	38	terms	term	NOUN
ejpam-5360	44	39	of	of	ADP
ejpam-5360	44	40	the	the	DET
ejpam-5360	44	41	parameters	parameter	NOUN
ejpam-5360	44	42	related	relate	VERB
ejpam-5360	44	43	to	to	ADP
ejpam-5360	44	44	the	the	DET
ejpam-5360	44	45	structure	structure	NOUN
ejpam-5360	44	46	of	of	ADP
ejpam-5360	44	47	the	the	DET
ejpam-5360	44	48	graph	graph	NOUN
ejpam-5360	44	49	through	through	ADP
ejpam-5360	44	50	a	a	DET
ejpam-5360	44	51	different	different	ADJ
ejpam-5360	44	52	approach	approach	NOUN
ejpam-5360	44	53	.	.	PUNCT
ejpam-5360	45	1	2	2	X
ejpam-5360	45	2	.	.	X
ejpam-5360	45	3	preliminaries	preliminary	NOUN
ejpam-5360	45	4	figure	figure	VERB
ejpam-5360	45	5	1	1	NUM
ejpam-5360	45	6	:	:	PUNCT
ejpam-5360	45	7	generalized	generalized	ADJ
ejpam-5360	45	8	core	core	NOUN
ejpam-5360	45	9	-	-	PUNCT
ejpam-5360	45	10	satellite	satellite	NOUN
ejpam-5360	45	11	graph	graph	NOUN
ejpam-5360	45	12	(	(	PUNCT
ejpam-5360	45	13	η1kα1)	η1kα1)	VERB
ejpam-5360	45	14	▽	▽	ADJ
ejpam-5360	45	15	kϕ	kϕ	NOUN
ejpam-5360	45	16	∪	∪	X
ejpam-5360	45	17	(	(	PUNCT
ejpam-5360	45	18	η2kα2)	η2kα2)	NOUN
ejpam-5360	45	19	▽	▽	NOUN
ejpam-5360	45	20	kϕ	kϕ	PROPN
ejpam-5360	45	21	∪	∪	NOUN
ejpam-5360	45	22	...	...	PUNCT
ejpam-5360	45	23	∪	∪	X
ejpam-5360	45	24	(	(	PUNCT
ejpam-5360	45	25	ηkkαk	ηkkαk	NOUN
ejpam-5360	45	26	)	)	PUNCT
ejpam-5360	45	27	▽	▽	NOUN
ejpam-5360	45	28	kϕ	kϕ	NOUN
ejpam-5360	45	29	in	in	ADP
ejpam-5360	45	30	this	this	DET
ejpam-5360	45	31	section	section	NOUN
ejpam-5360	45	32	,	,	PUNCT
ejpam-5360	45	33	we	we	PRON
ejpam-5360	45	34	discuss	discuss	VERB
ejpam-5360	45	35	some	some	DET
ejpam-5360	45	36	preliminary	preliminary	ADJ
ejpam-5360	45	37	findings	finding	NOUN
ejpam-5360	45	38	that	that	PRON
ejpam-5360	45	39	will	will	AUX
ejpam-5360	45	40	be	be	AUX
ejpam-5360	45	41	required	require	VERB
ejpam-5360	45	42	to	to	PART
ejpam-5360	45	43	support	support	VERB
ejpam-5360	45	44	our	our	PRON
ejpam-5360	45	45	main	main	ADJ
ejpam-5360	45	46	results	result	NOUN
ejpam-5360	45	47	.	.	PUNCT
ejpam-5360	46	1	let	let	VERB
ejpam-5360	46	2	g	g	PROPN
ejpam-5360	46	3	=	=	SYM
ejpam-5360	46	4	(	(	PUNCT
ejpam-5360	46	5	v	v	NOUN
ejpam-5360	46	6	(	(	PUNCT
ejpam-5360	46	7	g	g	NOUN
ejpam-5360	46	8	)	)	PUNCT
ejpam-5360	46	9	,	,	PUNCT
ejpam-5360	46	10	e(g	e(g	PROPN
ejpam-5360	46	11	)	)	PUNCT
ejpam-5360	46	12	)	)	PUNCT
ejpam-5360	47	1	be	be	AUX
ejpam-5360	47	2	a	a	DET
ejpam-5360	47	3	simple	simple	ADJ
ejpam-5360	47	4	,	,	PUNCT
ejpam-5360	47	5	connected	connected	ADJ
ejpam-5360	47	6	,	,	PUNCT
ejpam-5360	47	7	undirected	undirected	ADJ
ejpam-5360	47	8	,	,	PUNCT
ejpam-5360	47	9	and	and	CCONJ
ejpam-5360	47	10	finite	finite	ADJ
ejpam-5360	47	11	graph	graph	NOUN
ejpam-5360	47	12	.	.	PUNCT
ejpam-5360	48	1	let	let	VERB
ejpam-5360	48	2	the	the	DET
ejpam-5360	48	3	order	order	NOUN
ejpam-5360	48	4	,	,	PUNCT
ejpam-5360	48	5	|v	|v	PROPN
ejpam-5360	48	6	(	(	PUNCT
ejpam-5360	48	7	g)|	g)|	PROPN
ejpam-5360	48	8	,	,	PUNCT
ejpam-5360	48	9	be	be	AUX
ejpam-5360	48	10	n	n	PRON
ejpam-5360	48	11	and	and	CCONJ
ejpam-5360	48	12	let	let	VERB
ejpam-5360	48	13	the	the	DET
ejpam-5360	48	14	size	size	NOUN
ejpam-5360	48	15	of	of	ADP
ejpam-5360	48	16	the	the	DET
ejpam-5360	48	17	graph	graph	NOUN
ejpam-5360	48	18	,	,	PUNCT
ejpam-5360	48	19	|e(g)|	|e(g)|	ADJ
ejpam-5360	48	20	,	,	PUNCT
ejpam-5360	48	21	be	be	VERB
ejpam-5360	48	22	m	m	PROPN
ejpam-5360	48	23	,	,	PUNCT
ejpam-5360	48	24	respectively	respectively	ADV
ejpam-5360	48	25	.	.	PUNCT
ejpam-5360	49	1	let	let	AUX
ejpam-5360	49	2	a(g	a(g	PROPN
ejpam-5360	49	3	)	)	PUNCT
ejpam-5360	49	4	denote	denote	VERB
ejpam-5360	49	5	the	the	DET
ejpam-5360	49	6	(	(	PUNCT
ejpam-5360	49	7	0,1	0,1	NUM
ejpam-5360	49	8	)	)	PUNCT
ejpam-5360	49	9	adjacency	adjacency	NOUN
ejpam-5360	49	10	matrix	matrix	NOUN
ejpam-5360	49	11	and	and	CCONJ
ejpam-5360	49	12	d(g	d(g	PROPN
ejpam-5360	49	13	)	)	PUNCT
ejpam-5360	49	14	the	the	DET
ejpam-5360	49	15	diagonal	diagonal	ADJ
ejpam-5360	49	16	matrix	matrix	NOUN
ejpam-5360	49	17	whose	whose	DET
ejpam-5360	49	18	diagonal	diagonal	ADJ
ejpam-5360	49	19	entries	entry	NOUN
ejpam-5360	49	20	are	be	AUX
ejpam-5360	49	21	degree	degree	NOUN
ejpam-5360	49	22	sequence	sequence	NOUN
ejpam-5360	49	23	of	of	ADP
ejpam-5360	49	24	g.	g.	PROPN
ejpam-5360	49	25	let	let	VERB
ejpam-5360	49	26	q(g	q(g	PROPN
ejpam-5360	49	27	)	)	PUNCT
ejpam-5360	50	1	=	=	SYM
ejpam-5360	50	2	d(g	d(g	PROPN
ejpam-5360	50	3	)	)	PUNCT
ejpam-5360	51	1	+	+	NUM
ejpam-5360	51	2	a(g	a(g	PROPN
ejpam-5360	51	3	)	)	PUNCT
ejpam-5360	51	4	be	be	VERB
ejpam-5360	51	5	the	the	DET
ejpam-5360	51	6	signless	signless	ADJ
ejpam-5360	51	7	laplacian	laplacian	ADJ
ejpam-5360	51	8	matrix	matrix	NOUN
ejpam-5360	51	9	of	of	ADP
ejpam-5360	51	10	the	the	DET
ejpam-5360	51	11	graph	graph	NOUN
ejpam-5360	51	12	g.	g.	NOUN
ejpam-5360	51	13	according	accord	VERB
ejpam-5360	51	14	to	to	ADP
ejpam-5360	51	15	geršgorin	geršgorin	PROPN
ejpam-5360	51	16	’s	’s	PART
ejpam-5360	51	17	theorem	theorem	ADJ
ejpam-5360	51	18	,	,	PUNCT
ejpam-5360	51	19	if	if	SCONJ
ejpam-5360	51	20	c	c	PROPN
ejpam-5360	51	21	is	be	AUX
ejpam-5360	51	22	an	an	DET
ejpam-5360	51	23	n×	n×	PROPN
ejpam-5360	51	24	n	n	CCONJ
ejpam-5360	51	25	real	real	ADJ
ejpam-5360	51	26	symmetric	symmetric	ADJ
ejpam-5360	51	27	matrix	matrix	NOUN
ejpam-5360	51	28	then	then	ADV
ejpam-5360	51	29	its	its	PRON
ejpam-5360	51	30	eigenvalues	eigenvalue	NOUN
ejpam-5360	51	31	are	be	AUX
ejpam-5360	51	32	non	non	ADJ
ejpam-5360	51	33	-	-	ADJ
ejpam-5360	51	34	negative	negative	ADJ
ejpam-5360	51	35	real	real	ADJ
ejpam-5360	51	36	numbers	number	NOUN
ejpam-5360	51	37	.	.	PUNCT
ejpam-5360	52	1	since	since	SCONJ
ejpam-5360	52	2	a(g	a(g	PROPN
ejpam-5360	52	3	)	)	PUNCT
ejpam-5360	52	4	and	and	CCONJ
ejpam-5360	52	5	q(g	q(g	PROPN
ejpam-5360	52	6	)	)	PUNCT
ejpam-5360	52	7	are	be	AUX
ejpam-5360	52	8	real	real	ADJ
ejpam-5360	52	9	symmetric	symmetric	ADJ
ejpam-5360	52	10	matrices	matrix	NOUN
ejpam-5360	52	11	,	,	PUNCT
ejpam-5360	52	12	their	their	PRON
ejpam-5360	52	13	eigenvalues	eigenvalue	NOUN
ejpam-5360	52	14	are	be	AUX
ejpam-5360	52	15	non	non	ADJ
ejpam-5360	52	16	-	-	ADJ
ejpam-5360	52	17	negative	negative	ADJ
ejpam-5360	52	18	real	real	ADJ
ejpam-5360	52	19	numbers	number	NOUN
ejpam-5360	52	20	.	.	PUNCT
ejpam-5360	53	1	the	the	DET
ejpam-5360	53	2	eigenvalues	eigenvalue	NOUN
ejpam-5360	53	3	of	of	ADP
ejpam-5360	53	4	a(g	a(g	PROPN
ejpam-5360	53	5	)	)	PUNCT
ejpam-5360	53	6	and	and	CCONJ
ejpam-5360	53	7	q(g	q(g	PROPN
ejpam-5360	53	8	)	)	PUNCT
ejpam-5360	53	9	are	be	AUX
ejpam-5360	53	10	ordered	order	VERB
ejpam-5360	53	11	as	as	ADP
ejpam-5360	53	12	ρ(g	ρ(g	NOUN
ejpam-5360	53	13	)	)	PUNCT
ejpam-5360	53	14	=	=	SYM
ejpam-5360	53	15	ρ1(g	ρ1(g	PROPN
ejpam-5360	53	16	)	)	PUNCT
ejpam-5360	53	17	≥	≥	NOUN
ejpam-5360	53	18	ρ2(g	ρ2(g	NUM
ejpam-5360	53	19	)	)	PUNCT
ejpam-5360	53	20	≥	≥	NOUN
ejpam-5360	53	21	...	...	PUNCT
ejpam-5360	53	22	≥	≥	X
ejpam-5360	53	23	ρn(g	ρn(g	NUM
ejpam-5360	53	24	)	)	PUNCT
ejpam-5360	53	25	and	and	CCONJ
ejpam-5360	53	26	µ(g	µ(g	NUM
ejpam-5360	53	27	)	)	PUNCT
ejpam-5360	53	28	=	=	SYM
ejpam-5360	53	29	µ1(g	µ1(g	PROPN
ejpam-5360	53	30	)	)	PUNCT
ejpam-5360	53	31	≥	≥	NOUN
ejpam-5360	53	32	µ2(g	µ2(g	NUM
ejpam-5360	53	33	)	)	PUNCT
ejpam-5360	53	34	≥	≥	NUM
ejpam-5360	53	35	...	...	PUNCT
ejpam-5360	53	36	≥	≥	X
ejpam-5360	53	37	µn(g	µn(g	NUM
ejpam-5360	53	38	)	)	PUNCT
ejpam-5360	53	39	,	,	PUNCT
ejpam-5360	53	40	respectively	respectively	ADV
ejpam-5360	53	41	.	.	PUNCT
ejpam-5360	54	1	the	the	DET
ejpam-5360	54	2	largest	large	ADJ
ejpam-5360	54	3	eigenvalue	eigenvalue	NOUN
ejpam-5360	54	4	of	of	ADP
ejpam-5360	54	5	the	the	DET
ejpam-5360	54	6	adjacency	adjacency	NOUN
ejpam-5360	54	7	matrix	matrix	NOUN
ejpam-5360	54	8	of	of	ADP
ejpam-5360	54	9	the	the	DET
ejpam-5360	54	10	graph	graph	NOUN
ejpam-5360	54	11	(	(	PUNCT
ejpam-5360	54	12	known	know	VERB
ejpam-5360	54	13	as	as	ADP
ejpam-5360	54	14	spectral	spectral	ADJ
ejpam-5360	54	15	radius	radius	NOUN
ejpam-5360	54	16	)	)	PUNCT
ejpam-5360	54	17	and	and	CCONJ
ejpam-5360	54	18	signless	signless	ADJ
ejpam-5360	54	19	laplacian	laplacian	ADJ
ejpam-5360	54	20	matrix	matrix	NOUN
ejpam-5360	54	21	(	(	PUNCT
ejpam-5360	54	22	known	know	VERB
ejpam-5360	54	23	as	as	ADP
ejpam-5360	54	24	signless	signless	PROPN
ejpam-5360	54	25	laplacian	laplacian	ADJ
ejpam-5360	54	26	spectral	spectral	ADJ
ejpam-5360	54	27	radius	radius	NOUN
ejpam-5360	54	28	)	)	PUNCT
ejpam-5360	54	29	m.	m.	NOUN
ejpam-5360	54	30	v.	v.	ADP
ejpam-5360	54	31	,	,	PUNCT
ejpam-5360	54	32	k.	k.	PROPN
ejpam-5360	54	33	desikan	desikan	PROPN
ejpam-5360	54	34	/	/	SYM
ejpam-5360	54	35	eur	eur	PROPN
ejpam-5360	54	36	.	.	PUNCT
ejpam-5360	55	1	j.	j.	PROPN
ejpam-5360	55	2	pure	pure	PROPN
ejpam-5360	55	3	appl	appl	PROPN
ejpam-5360	55	4	.	.	PROPN
ejpam-5360	55	5	math	math	PROPN
ejpam-5360	55	6	,	,	PUNCT
ejpam-5360	55	7	18	18	NUM
ejpam-5360	55	8	(	(	PUNCT
ejpam-5360	55	9	4	4	NUM
ejpam-5360	55	10	)	)	PUNCT
ejpam-5360	55	11	(	(	PUNCT
ejpam-5360	55	12	2025	2025	NUM
ejpam-5360	55	13	)	)	PUNCT
ejpam-5360	55	14	,	,	PUNCT
ejpam-5360	55	15	5360	5360	NUM
ejpam-5360	55	16	4	4	NUM
ejpam-5360	55	17	of	of	ADP
ejpam-5360	55	18	20	20	NUM
ejpam-5360	55	19	are	be	AUX
ejpam-5360	55	20	denoted	denote	VERB
ejpam-5360	55	21	as	as	ADP
ejpam-5360	55	22	ρ1(g	ρ1(g	PROPN
ejpam-5360	55	23	)	)	PUNCT
ejpam-5360	55	24	and	and	CCONJ
ejpam-5360	55	25	µ1(q(g	µ1(q(g	NUM
ejpam-5360	55	26	)	)	PUNCT
ejpam-5360	55	27	)	)	PUNCT
ejpam-5360	55	28	,	,	PUNCT
ejpam-5360	55	29	respectively	respectively	ADV
ejpam-5360	55	30	.	.	PUNCT
ejpam-5360	56	1	definition	definition	NOUN
ejpam-5360	56	2	1	1	NUM
ejpam-5360	56	3	.	.	PUNCT
ejpam-5360	57	1	[	[	X
ejpam-5360	57	2	1	1	X
ejpam-5360	57	3	]	]	PUNCT
ejpam-5360	57	4	the	the	DET
ejpam-5360	57	5	join	join	NOUN
ejpam-5360	57	6	(	(	PUNCT
ejpam-5360	57	7	or	or	CCONJ
ejpam-5360	57	8	complete	complete	ADJ
ejpam-5360	57	9	product	product	NOUN
ejpam-5360	57	10	)	)	PUNCT
ejpam-5360	57	11	gψ1	gψ1	PROPN
ejpam-5360	57	12	▽	▽	PUNCT
ejpam-5360	57	13	gψ2	gψ2	PROPN
ejpam-5360	57	14	of	of	ADP
ejpam-5360	57	15	graphs	graph	NOUN
ejpam-5360	57	16	gψ1	gψ1	PROPN
ejpam-5360	57	17	and	and	CCONJ
ejpam-5360	57	18	gψ2	gψ2	PROPN
ejpam-5360	57	19	is	be	AUX
ejpam-5360	57	20	obtained	obtain	VERB
ejpam-5360	57	21	from	from	ADP
ejpam-5360	57	22	gψ1	gψ1	PROPN
ejpam-5360	57	23	∪gψ2	∪gψ2	PROPN
ejpam-5360	57	24	by	by	ADP
ejpam-5360	57	25	joining	join	VERB
ejpam-5360	57	26	every	every	DET
ejpam-5360	57	27	vertex	vertex	NOUN
ejpam-5360	57	28	of	of	ADP
ejpam-5360	57	29	gψ1	gψ1	NOUN
ejpam-5360	57	30	with	with	ADP
ejpam-5360	57	31	every	every	DET
ejpam-5360	57	32	vertex	vertex	NOUN
ejpam-5360	57	33	of	of	ADP
ejpam-5360	57	34	gψ2	gψ2	PROPN
ejpam-5360	57	35	.	.	PUNCT
ejpam-5360	58	1	the	the	DET
ejpam-5360	58	2	generalized	generalize	VERB
ejpam-5360	58	3	core	core	NOUN
ejpam-5360	58	4	-	-	PUNCT
ejpam-5360	58	5	satellite	satellite	NOUN
ejpam-5360	58	6	graph	graph	NOUN
ejpam-5360	58	7	consists	consist	VERB
ejpam-5360	58	8	of	of	ADP
ejpam-5360	58	9	core	core	ADJ
ejpam-5360	58	10	clique	clique	NOUN
ejpam-5360	58	11	c	c	PROPN
ejpam-5360	58	12	=	=	SYM
ejpam-5360	58	13	kϕ	kϕ	PROPN
ejpam-5360	58	14	with	with	ADP
ejpam-5360	58	15	ϕ	ϕ	NOUN
ejpam-5360	58	16	nodes	node	NOUN
ejpam-5360	58	17	and	and	CCONJ
ejpam-5360	58	18	satellites	satellite	NOUN
ejpam-5360	58	19	s1	s1	NOUN
ejpam-5360	58	20	,	,	PUNCT
ejpam-5360	58	21	s2	s2	PROPN
ejpam-5360	58	22	,	,	PUNCT
ejpam-5360	58	23	s3	s3	PROPN
ejpam-5360	58	24	,	,	PUNCT
ejpam-5360	58	25	...	...	PUNCT
ejpam-5360	58	26	,	,	PUNCT
ejpam-5360	58	27	sk	sk	VERB
ejpam-5360	58	28	with	with	ADP
ejpam-5360	58	29	kα1	kα1	PROPN
ejpam-5360	58	30	,	,	PUNCT
ejpam-5360	58	31	kα2	kα2	NOUN
ejpam-5360	58	32	,	,	PUNCT
ejpam-5360	58	33	...	...	PUNCT
ejpam-5360	58	34	,	,	PUNCT
ejpam-5360	58	35	kαk	kαk	VERB
ejpam-5360	58	36	cliques	clique	NOUN
ejpam-5360	58	37	.	.	PUNCT
ejpam-5360	59	1	let	let	VERB
ejpam-5360	59	2	η1	η1	NOUN
ejpam-5360	59	3	,	,	PUNCT
ejpam-5360	59	4	η2	η2	NOUN
ejpam-5360	59	5	,	,	PUNCT
ejpam-5360	59	6	η3,	η3,	NOUN
ejpam-5360	59	7	...	...	PUNCT
ejpam-5360	59	8	,ηk	,ηk	PUNCT
ejpam-5360	59	9	be	be	VERB
ejpam-5360	59	10	the	the	DET
ejpam-5360	59	11	number	number	NOUN
ejpam-5360	59	12	of	of	ADP
ejpam-5360	59	13	copies	copy	NOUN
ejpam-5360	59	14	of	of	ADP
ejpam-5360	59	15	the	the	DET
ejpam-5360	59	16	cliques	clique	NOUN
ejpam-5360	59	17	kα1	kα1	PROPN
ejpam-5360	59	18	,	,	PUNCT
ejpam-5360	59	19	kα2	kα2	NOUN
ejpam-5360	59	20	,	,	PUNCT
ejpam-5360	59	21	...	...	PUNCT
ejpam-5360	59	22	,	,	PUNCT
ejpam-5360	59	23	kαk	kαk	VERB
ejpam-5360	59	24	having	have	VERB
ejpam-5360	59	25	degrees	degree	NOUN
ejpam-5360	59	26	dα1	dα1	NOUN
ejpam-5360	59	27	,	,	PUNCT
ejpam-5360	59	28	dα2	dα2	PROPN
ejpam-5360	59	29	,	,	PUNCT
ejpam-5360	59	30	...	...	PUNCT
ejpam-5360	59	31	,	,	PUNCT
ejpam-5360	59	32	dαk	dαk	VERB
ejpam-5360	59	33	,	,	PUNCT
ejpam-5360	59	34	respectively	respectively	ADV
ejpam-5360	59	35	.	.	PUNCT
ejpam-5360	60	1	let	let	VERB
ejpam-5360	60	2	dα1	dα1	VERB
ejpam-5360	60	3	<	<	X
ejpam-5360	60	4	dα2	dα2	X
ejpam-5360	60	5	<	<	X
ejpam-5360	60	6	....	....	PUNCT
ejpam-5360	61	1	<	<	X
ejpam-5360	61	2	dαk	dαk	ADJ
ejpam-5360	61	3	.	.	PUNCT
ejpam-5360	62	1	let	let	VERB
ejpam-5360	62	2	s	s	PRON
ejpam-5360	62	3	=	=	PUNCT
ejpam-5360	62	4	(	(	PUNCT
ejpam-5360	62	5	s1	s1	PROPN
ejpam-5360	62	6	,	,	PUNCT
ejpam-5360	62	7	s2	s2	PROPN
ejpam-5360	62	8	,	,	PUNCT
ejpam-5360	62	9	...	...	PUNCT
ejpam-5360	62	10	,	,	PUNCT
ejpam-5360	62	11	sk	sk	PROPN
ejpam-5360	62	12	)	)	PUNCT
ejpam-5360	62	13	where	where	SCONJ
ejpam-5360	62	14	each	each	DET
ejpam-5360	62	15	si	si	PROPN
ejpam-5360	62	16	is	be	AUX
ejpam-5360	62	17	the	the	DET
ejpam-5360	62	18	ith	ith	PROPN
ejpam-5360	62	19	satellites	satellite	NOUN
ejpam-5360	62	20	in	in	ADP
ejpam-5360	62	21	the	the	DET
ejpam-5360	62	22	graph	graph	NOUN
ejpam-5360	62	23	having	have	VERB
ejpam-5360	62	24	ηi	ηi	VERB
ejpam-5360	62	25	copies	copy	NOUN
ejpam-5360	62	26	of	of	ADP
ejpam-5360	62	27	kαi	kαi	NOUN
ejpam-5360	62	28	cliques	clique	NOUN
ejpam-5360	62	29	and	and	CCONJ
ejpam-5360	62	30	let	let	VERB
ejpam-5360	62	31	η∗	η∗	NOUN
ejpam-5360	62	32	=	=	SYM
ejpam-5360	62	33	(	(	PUNCT
ejpam-5360	62	34	η1	η1	NOUN
ejpam-5360	62	35	,	,	PUNCT
ejpam-5360	62	36	η2	η2	NOUN
ejpam-5360	62	37	,	,	PUNCT
ejpam-5360	62	38	...	...	PUNCT
ejpam-5360	62	39	,	,	PUNCT
ejpam-5360	62	40	ηk	ηk	PROPN
ejpam-5360	62	41	)	)	PUNCT
ejpam-5360	62	42	.	.	PUNCT
ejpam-5360	63	1	hence	hence	ADV
ejpam-5360	63	2	,	,	PUNCT
ejpam-5360	63	3	the	the	DET
ejpam-5360	63	4	generalized	generalize	VERB
ejpam-5360	63	5	core	core	NOUN
ejpam-5360	63	6	-	-	PUNCT
ejpam-5360	63	7	satellite	satellite	NOUN
ejpam-5360	63	8	graph	graph	NOUN
ejpam-5360	63	9	is	be	AUX
ejpam-5360	63	10	denoted	denote	VERB
ejpam-5360	63	11	as	as	ADP
ejpam-5360	63	12	θ(c	θ(c	VERB
ejpam-5360	63	13	,	,	PUNCT
ejpam-5360	63	14	s	s	NOUN
ejpam-5360	63	15	,	,	PUNCT
ejpam-5360	63	16	η∗	η∗	NOUN
ejpam-5360	63	17	)	)	PUNCT
ejpam-5360	63	18	.	.	PUNCT
ejpam-5360	64	1	theorem	theorem	NOUN
ejpam-5360	64	2	1	1	NUM
ejpam-5360	64	3	.	.	PUNCT
ejpam-5360	65	1	[	[	X
ejpam-5360	65	2	1	1	X
ejpam-5360	65	3	]	]	PUNCT
ejpam-5360	65	4	the	the	DET
ejpam-5360	65	5	spectral	spectral	ADJ
ejpam-5360	65	6	radius	radius	NOUN
ejpam-5360	65	7	(	(	PUNCT
ejpam-5360	65	8	perron	perron	PROPN
ejpam-5360	65	9	eigenvalue	eigenvalue	PROPN
ejpam-5360	65	10	)	)	PUNCT
ejpam-5360	65	11	ρ1(g	ρ1(g	PROPN
ejpam-5360	65	12	)	)	PUNCT
ejpam-5360	65	13	is	be	AUX
ejpam-5360	65	14	given	give	VERB
ejpam-5360	65	15	by	by	ADP
ejpam-5360	65	16	the	the	DET
ejpam-5360	65	17	largest	large	ADJ
ejpam-5360	65	18	root	root	NOUN
ejpam-5360	65	19	of	of	ADP
ejpam-5360	65	20	(	(	PUNCT
ejpam-5360	65	21	λ−	λ−	PROPN
ejpam-5360	65	22	ϕ+	ϕ+	PROPN
ejpam-5360	65	23	1	1	X
ejpam-5360	65	24	)	)	PUNCT
ejpam-5360	65	25	k∏	k∏	PROPN
ejpam-5360	65	26	i=1	i=1	PROPN
ejpam-5360	65	27	(	(	PUNCT
ejpam-5360	65	28	λ−	λ−	PROPN
ejpam-5360	65	29	αi	αi	VERB
ejpam-5360	66	1	+	+	CCONJ
ejpam-5360	66	2	1	1	X
ejpam-5360	66	3	)	)	PUNCT
ejpam-5360	66	4	=	=	PUNCT
ejpam-5360	67	1	ϕ	ϕ	PROPN
ejpam-5360	67	2	k∑	k∑	PROPN
ejpam-5360	68	1	i=1	i=1	PROPN
ejpam-5360	68	2	ηiαi	ηiαi	PROPN
ejpam-5360	68	3	∏	∏	PROPN
ejpam-5360	68	4	j	j	NOUN
ejpam-5360	68	5	̸=i	̸=i	PROPN
ejpam-5360	68	6	(	(	PUNCT
ejpam-5360	68	7	λ−	λ−	PROPN
ejpam-5360	68	8	αj	αj	PROPN
ejpam-5360	68	9	+	+	NOUN
ejpam-5360	68	10	1	1	NUM
ejpam-5360	68	11	)	)	PUNCT
ejpam-5360	68	12	and	and	CCONJ
ejpam-5360	68	13	satisfies	satisfy	VERB
ejpam-5360	68	14	the	the	DET
ejpam-5360	68	15	bounds	bound	NOUN
ejpam-5360	68	16	(	(	PUNCT
ejpam-5360	68	17	ϕ−	ϕ−	PROPN
ejpam-5360	68	18	1	1	NUM
ejpam-5360	68	19	)	)	PUNCT
ejpam-5360	68	20	+	+	CCONJ
ejpam-5360	68	21	max	max	PROPN
ejpam-5360	68	22	1≤i≤k	1≤i≤k	NUM
ejpam-5360	68	23	(	(	PUNCT
ejpam-5360	68	24	αi	αi	NOUN
ejpam-5360	68	25	)	)	PUNCT
ejpam-5360	68	26	<	<	X
ejpam-5360	68	27	ρ1(g	ρ1(g	PROPN
ejpam-5360	68	28	)	)	PUNCT
ejpam-5360	68	29	<	<	X
ejpam-5360	69	1	(	(	PUNCT
ejpam-5360	69	2	ϕ−	ϕ−	PROPN
ejpam-5360	69	3	1	1	NUM
ejpam-5360	69	4	)	)	PUNCT
ejpam-5360	69	5	+	+	CCONJ
ejpam-5360	69	6	k∑	k∑	ADJ
ejpam-5360	69	7	i=1	i=1	PROPN
ejpam-5360	69	8	ηiαi	ηiαi	NOUN
ejpam-5360	69	9	.	.	PUNCT
ejpam-5360	70	1	(	(	PUNCT
ejpam-5360	70	2	1	1	X
ejpam-5360	70	3	)	)	PUNCT
ejpam-5360	70	4	theorem	theorem	NOUN
ejpam-5360	70	5	2	2	NUM
ejpam-5360	70	6	.	.	PUNCT
ejpam-5360	71	1	[	[	X
ejpam-5360	71	2	16	16	NUM
ejpam-5360	71	3	]	]	PUNCT
ejpam-5360	71	4	let	let	VERB
ejpam-5360	71	5	g	g	PROPN
ejpam-5360	71	6	=	=	SYM
ejpam-5360	71	7	(	(	PUNCT
ejpam-5360	71	8	v	v	NOUN
ejpam-5360	71	9	,	,	PUNCT
ejpam-5360	71	10	e	e	NOUN
ejpam-5360	71	11	)	)	PUNCT
ejpam-5360	71	12	be	be	AUX
ejpam-5360	71	13	a	a	DET
ejpam-5360	71	14	graph	graph	NOUN
ejpam-5360	71	15	.	.	PUNCT
ejpam-5360	72	1	then	then	ADV
ejpam-5360	72	2	min	min	PROPN
ejpam-5360	72	3	v∈v	v∈v	PROPN
ejpam-5360	72	4	(	(	PUNCT
ejpam-5360	72	5	g	g	NOUN
ejpam-5360	72	6	)	)	PUNCT
ejpam-5360	72	7	(	(	PUNCT
ejpam-5360	72	8	σ	σ	PROPN
ejpam-5360	72	9	uv∈e	uv∈e	PROPN
ejpam-5360	72	10	d(u	d(u	PROPN
ejpam-5360	72	11	)	)	PUNCT
ejpam-5360	72	12	)	)	PUNCT
ejpam-5360	72	13	1/2	1/2	NUM
ejpam-5360	72	14	≤	≤	NUM
ejpam-5360	72	15	ρ1(g	ρ1(g	NOUN
ejpam-5360	72	16	)	)	PUNCT
ejpam-5360	72	17	≤	≤	NOUN
ejpam-5360	72	18	max	max	PROPN
ejpam-5360	72	19	v∈v	v∈v	NOUN
ejpam-5360	72	20	(	(	PUNCT
ejpam-5360	72	21	g	g	NOUN
ejpam-5360	72	22	)	)	PUNCT
ejpam-5360	72	23	(	(	PUNCT
ejpam-5360	72	24	σ	σ	PROPN
ejpam-5360	72	25	uv∈e	uv∈e	PROPN
ejpam-5360	72	26	d(u	d(u	PROPN
ejpam-5360	72	27	)	)	PUNCT
ejpam-5360	72	28	)	)	PUNCT
ejpam-5360	72	29	1/2	1/2	NUM
ejpam-5360	72	30	.	.	PUNCT
ejpam-5360	73	1	(	(	PUNCT
ejpam-5360	73	2	2	2	X
ejpam-5360	73	3	)	)	PUNCT
ejpam-5360	73	4	moreover	moreover	ADV
ejpam-5360	73	5	,	,	PUNCT
ejpam-5360	73	6	if	if	SCONJ
ejpam-5360	73	7	g	g	PROPN
ejpam-5360	73	8	is	be	AUX
ejpam-5360	73	9	connected	connect	VERB
ejpam-5360	73	10	then	then	ADV
ejpam-5360	73	11	either	either	PRON
ejpam-5360	73	12	of	of	ADP
ejpam-5360	73	13	the	the	DET
ejpam-5360	73	14	equalities	equality	NOUN
ejpam-5360	73	15	holds	hold	VERB
ejpam-5360	73	16	iff	iff	PROPN
ejpam-5360	73	17	σ	σ	PROPN
ejpam-5360	73	18	uv∈e	uv∈e	PROPN
ejpam-5360	73	19	d(u	d(u	PROPN
ejpam-5360	73	20	)	)	PUNCT
ejpam-5360	73	21	is	be	AUX
ejpam-5360	73	22	the	the	DET
ejpam-5360	73	23	same	same	ADJ
ejpam-5360	73	24	∀	∀	NOUN
ejpam-5360	73	25	v	v	ADP
ejpam-5360	73	26	∈	∈	PROPN
ejpam-5360	73	27	v	v	NOUN
ejpam-5360	73	28	.	.	PUNCT
ejpam-5360	74	1	remark	remark	PROPN
ejpam-5360	74	2	1	1	NUM
ejpam-5360	74	3	.	.	PUNCT
ejpam-5360	75	1	(	(	PUNCT
ejpam-5360	75	2	a	a	X
ejpam-5360	75	3	)	)	PUNCT
ejpam-5360	75	4	the	the	DET
ejpam-5360	75	5	number	number	NOUN
ejpam-5360	75	6	of	of	ADP
ejpam-5360	75	7	edges	edge	NOUN
ejpam-5360	75	8	in	in	ADP
ejpam-5360	75	9	si	si	PROPN
ejpam-5360	75	10	alone	alone	ADV
ejpam-5360	75	11	is	be	AUX
ejpam-5360	75	12	ηi	ηi	NOUN
ejpam-5360	75	13	(	(	PUNCT
ejpam-5360	75	14	αi	αi	X
ejpam-5360	75	15	(	(	PUNCT
ejpam-5360	75	16	αi	αi	INTJ
ejpam-5360	75	17	−	−	NOUN
ejpam-5360	75	18	1	1	NUM
ejpam-5360	75	19	)	)	PUNCT
ejpam-5360	75	20	2	2	NUM
ejpam-5360	75	21	)	)	PUNCT
ejpam-5360	75	22	for	for	ADP
ejpam-5360	75	23	i	i	PRON
ejpam-5360	75	24	=	=	SYM
ejpam-5360	75	25	1	1	NUM
ejpam-5360	75	26	,	,	PUNCT
ejpam-5360	75	27	2	2	NUM
ejpam-5360	75	28	....	....	SYM
ejpam-5360	75	29	k.	k.	PROPN
ejpam-5360	76	1	(	(	PUNCT
ejpam-5360	76	2	b	b	X
ejpam-5360	76	3	)	)	PUNCT
ejpam-5360	76	4	the	the	DET
ejpam-5360	76	5	number	number	NOUN
ejpam-5360	76	6	of	of	ADP
ejpam-5360	76	7	edges	edge	NOUN
ejpam-5360	76	8	joining	join	VERB
ejpam-5360	76	9	the	the	DET
ejpam-5360	76	10	satellite	satellite	NOUN
ejpam-5360	76	11	graphs	graph	NOUN
ejpam-5360	76	12	with	with	ADP
ejpam-5360	76	13	vertices	vertex	NOUN
ejpam-5360	76	14	of	of	ADP
ejpam-5360	76	15	the	the	DET
ejpam-5360	76	16	core	core	NOUN
ejpam-5360	76	17	graph	graph	NOUN
ejpam-5360	76	18	is	be	AUX
ejpam-5360	76	19	ϕηiαi	ϕηiαi	NOUN
ejpam-5360	76	20	m.	m.	NOUN
ejpam-5360	76	21	v.	v.	ADP
ejpam-5360	76	22	,	,	PUNCT
ejpam-5360	76	23	k.	k.	PROPN
ejpam-5360	76	24	desikan	desikan	PROPN
ejpam-5360	76	25	/	/	SYM
ejpam-5360	76	26	eur	eur	PROPN
ejpam-5360	76	27	.	.	PUNCT
ejpam-5360	77	1	j.	j.	PROPN
ejpam-5360	77	2	pure	pure	PROPN
ejpam-5360	77	3	appl	appl	PROPN
ejpam-5360	77	4	.	.	PROPN
ejpam-5360	77	5	math	math	PROPN
ejpam-5360	77	6	,	,	PUNCT
ejpam-5360	77	7	18	18	NUM
ejpam-5360	77	8	(	(	PUNCT
ejpam-5360	77	9	4	4	NUM
ejpam-5360	77	10	)	)	PUNCT
ejpam-5360	77	11	(	(	PUNCT
ejpam-5360	77	12	2025	2025	NUM
ejpam-5360	77	13	)	)	PUNCT
ejpam-5360	77	14	,	,	PUNCT
ejpam-5360	77	15	5360	5360	NUM
ejpam-5360	77	16	5	5	NUM
ejpam-5360	77	17	of	of	ADP
ejpam-5360	77	18	20	20	NUM
ejpam-5360	77	19	for	for	ADP
ejpam-5360	77	20	i	i	PRON
ejpam-5360	77	21	=	=	NOUN
ejpam-5360	77	22	1	1	NUM
ejpam-5360	77	23	,	,	PUNCT
ejpam-5360	77	24	2	2	NUM
ejpam-5360	77	25	....	....	PUNCT
ejpam-5360	77	26	k.	k.	PROPN
ejpam-5360	77	27	(	(	PUNCT
ejpam-5360	77	28	c	c	X
ejpam-5360	77	29	)	)	PUNCT
ejpam-5360	77	30	the	the	DET
ejpam-5360	77	31	number	number	NOUN
ejpam-5360	77	32	of	of	ADP
ejpam-5360	77	33	edges	edge	NOUN
ejpam-5360	77	34	in	in	ADP
ejpam-5360	77	35	the	the	DET
ejpam-5360	77	36	core	core	NOUN
ejpam-5360	77	37	graph	graph	NOUN
ejpam-5360	77	38	kϕ	kϕ	PROPN
ejpam-5360	77	39	alone	alone	ADV
ejpam-5360	77	40	is	be	AUX
ejpam-5360	77	41	ϕ	ϕ	NOUN
ejpam-5360	77	42	(	(	PUNCT
ejpam-5360	77	43	ϕ−	ϕ−	PROPN
ejpam-5360	77	44	1	1	NUM
ejpam-5360	77	45	)	)	PUNCT
ejpam-5360	77	46	2	2	NUM
ejpam-5360	77	47	.	.	PUNCT
ejpam-5360	78	1	hence	hence	ADV
ejpam-5360	78	2	,	,	PUNCT
ejpam-5360	78	3	the	the	DET
ejpam-5360	78	4	total	total	ADJ
ejpam-5360	78	5	number	number	NOUN
ejpam-5360	78	6	of	of	ADP
ejpam-5360	78	7	edges	edge	NOUN
ejpam-5360	78	8	in	in	ADP
ejpam-5360	78	9	the	the	DET
ejpam-5360	78	10	core	core	NOUN
ejpam-5360	78	11	-	-	PUNCT
ejpam-5360	78	12	satellite	satellite	NOUN
ejpam-5360	78	13	graph	graph	NOUN
ejpam-5360	78	14	g	g	PROPN
ejpam-5360	78	15	is	be	AUX
ejpam-5360	78	16	m	m	VERB
ejpam-5360	78	17	=	=	PUNCT
ejpam-5360	78	18	{	{	PUNCT
ejpam-5360	78	19	k∑	k∑	PROPN
ejpam-5360	78	20	i=1	i=1	PROPN
ejpam-5360	79	1	(	(	PUNCT
ejpam-5360	79	2	ηiαi(αi	ηiαi(αi	VERB
ejpam-5360	79	3	−	−	PROPN
ejpam-5360	79	4	1	1	NUM
ejpam-5360	79	5	)	)	SYM
ejpam-5360	79	6	2	2	NUM
ejpam-5360	79	7	+	+	NUM
ejpam-5360	79	8	ϕηiαi	ϕηiαi	NOUN
ejpam-5360	79	9	)	)	PUNCT
ejpam-5360	80	1	+	+	CCONJ
ejpam-5360	80	2	ϕ(ϕ−	ϕ(ϕ−	PROPN
ejpam-5360	80	3	1	1	NUM
ejpam-5360	80	4	)	)	PUNCT
ejpam-5360	80	5	2	2	NUM
ejpam-5360	80	6	}	}	PUNCT
ejpam-5360	80	7	and	and	CCONJ
ejpam-5360	80	8	the	the	DET
ejpam-5360	80	9	number	number	NOUN
ejpam-5360	80	10	of	of	ADP
ejpam-5360	80	11	vertices	vertex	NOUN
ejpam-5360	80	12	of	of	ADP
ejpam-5360	80	13	the	the	DET
ejpam-5360	80	14	graph	graph	NOUN
ejpam-5360	80	15	g	g	PROPN
ejpam-5360	80	16	is	be	AUX
ejpam-5360	80	17	n	n	NOUN
ejpam-5360	80	18	=	=	PRON
ejpam-5360	80	19	ϕ+	ϕ+	PROPN
ejpam-5360	80	20	k∑	k∑	PROPN
ejpam-5360	80	21	i=1	i=1	PROPN
ejpam-5360	80	22	ηiαi	ηiαi	PROPN
ejpam-5360	80	23	.	.	PUNCT
ejpam-5360	81	1	theorem	theorem	ADJ
ejpam-5360	81	2	3	3	NUM
ejpam-5360	81	3	and	and	CCONJ
ejpam-5360	81	4	theorem	theorem	VERB
ejpam-5360	81	5	4	4	NUM
ejpam-5360	81	6	derived	derive	VERB
ejpam-5360	81	7	by	by	ADP
ejpam-5360	81	8	duan	duan	PROPN
ejpam-5360	81	9	and	and	CCONJ
ejpam-5360	81	10	zhou	zhou	PROPN
ejpam-5360	81	11	,	,	PUNCT
ejpam-5360	81	12	provide	provide	VERB
ejpam-5360	81	13	the	the	DET
ejpam-5360	81	14	lower	lower	ADV
ejpam-5360	81	15	bound	bind	VERB
ejpam-5360	81	16	and	and	CCONJ
ejpam-5360	81	17	upper	upper	ADJ
ejpam-5360	81	18	bound	bind	VERB
ejpam-5360	81	19	for	for	ADP
ejpam-5360	81	20	the	the	DET
ejpam-5360	81	21	largest	large	ADJ
ejpam-5360	81	22	eigenvalue	eigenvalue	NOUN
ejpam-5360	81	23	of	of	ADP
ejpam-5360	81	24	any	any	DET
ejpam-5360	81	25	general	general	ADJ
ejpam-5360	81	26	non	non	ADJ
ejpam-5360	81	27	-	-	ADJ
ejpam-5360	81	28	negative	negative	ADJ
ejpam-5360	81	29	matrix	matrix	NOUN
ejpam-5360	81	30	.	.	PUNCT
ejpam-5360	82	1	theorem	theorem	NOUN
ejpam-5360	82	2	3	3	NUM
ejpam-5360	82	3	.	.	PUNCT
ejpam-5360	83	1	[	[	X
ejpam-5360	83	2	17	17	NUM
ejpam-5360	83	3	]	]	PUNCT
ejpam-5360	83	4	let	let	VERB
ejpam-5360	83	5	a	a	PRON
ejpam-5360	83	6	=	=	X
ejpam-5360	83	7	(	(	PUNCT
ejpam-5360	83	8	aij	aij	PROPN
ejpam-5360	83	9	)	)	PUNCT
ejpam-5360	83	10	be	be	VERB
ejpam-5360	83	11	an	an	DET
ejpam-5360	83	12	n	n	NUM
ejpam-5360	83	13	×	×	NOUN
ejpam-5360	83	14	n	n	CCONJ
ejpam-5360	83	15	non	non	ADJ
ejpam-5360	83	16	-	-	ADJ
ejpam-5360	83	17	negative	negative	ADJ
ejpam-5360	83	18	matrix	matrix	NOUN
ejpam-5360	83	19	with	with	ADP
ejpam-5360	83	20	row	row	NOUN
ejpam-5360	83	21	sums	sum	NOUN
ejpam-5360	83	22	r1	r1	NOUN
ejpam-5360	83	23	,	,	PUNCT
ejpam-5360	83	24	r2	r2	PROPN
ejpam-5360	83	25	,	,	PUNCT
ejpam-5360	83	26	...	...	PUNCT
ejpam-5360	83	27	,	,	PUNCT
ejpam-5360	83	28	rn	rn	PROPN
ejpam-5360	84	1	where	where	SCONJ
ejpam-5360	84	2	r1	r1	PROPN
ejpam-5360	84	3	≥	≥	NUM
ejpam-5360	84	4	r2	r2	PROPN
ejpam-5360	84	5	≥	≥	NUM
ejpam-5360	84	6	....	....	PUNCT
ejpam-5360	84	7	≥	≥	PROPN
ejpam-5360	84	8	rn	rn	PROPN
ejpam-5360	84	9	.	.	PROPN
ejpam-5360	84	10	let	let	VERB
ejpam-5360	84	11	s	s	PRON
ejpam-5360	84	12	and	and	CCONJ
ejpam-5360	84	13	t	t	PROPN
ejpam-5360	84	14	be	be	AUX
ejpam-5360	84	15	the	the	DET
ejpam-5360	84	16	smallest	small	ADJ
ejpam-5360	84	17	diagonal	diagonal	NOUN
ejpam-5360	84	18	and	and	CCONJ
ejpam-5360	84	19	the	the	DET
ejpam-5360	84	20	smallest	small	ADJ
ejpam-5360	84	21	non	non	ADJ
ejpam-5360	84	22	-	-	ADJ
ejpam-5360	84	23	diagonal	diagonal	ADJ
ejpam-5360	84	24	elements	element	NOUN
ejpam-5360	84	25	of	of	ADP
ejpam-5360	84	26	a	a	PRON
ejpam-5360	84	27	,	,	PUNCT
ejpam-5360	84	28	respectively	respectively	ADV
ejpam-5360	84	29	.	.	PUNCT
ejpam-5360	85	1	let	let	VERB
ejpam-5360	85	2	φn	φn	VERB
ejpam-5360	85	3	=	=	PUNCT
ejpam-5360	85	4	(	(	PUNCT
ejpam-5360	85	5	rn	rn	PROPN
ejpam-5360	85	6	+	+	PROPN
ejpam-5360	85	7	s	s	PART
ejpam-5360	85	8	−	−	PROPN
ejpam-5360	85	9	t	t	NOUN
ejpam-5360	85	10	)	)	PUNCT
ejpam-5360	86	1	+	+	CCONJ
ejpam-5360	86	2	√	√	INTJ
ejpam-5360	86	3	(	(	PUNCT
ejpam-5360	86	4	rn	rn	NOUN
ejpam-5360	86	5	−	−	PROPN
ejpam-5360	86	6	s	s	PART
ejpam-5360	86	7	+	+	NUM
ejpam-5360	86	8	t	t	NOUN
ejpam-5360	86	9	)	)	PUNCT
ejpam-5360	86	10	2	2	NUM
ejpam-5360	86	11	+	+	SYM
ejpam-5360	86	12	4	4	NUM
ejpam-5360	86	13	t	t	NUM
ejpam-5360	86	14	∑n−1	∑n−1	ADJ
ejpam-5360	86	15	i=1	i=1	PROPN
ejpam-5360	86	16	(	(	PUNCT
ejpam-5360	86	17	ri	ri	PROPN
ejpam-5360	86	18	−	−	PROPN
ejpam-5360	86	19	rn	rn	PROPN
ejpam-5360	86	20	)	)	PUNCT
ejpam-5360	86	21	2	2	NUM
ejpam-5360	86	22	(	(	PUNCT
ejpam-5360	86	23	3	3	NUM
ejpam-5360	86	24	)	)	PUNCT
ejpam-5360	86	25	then	then	ADV
ejpam-5360	86	26	ρ1(a(g	ρ1(a(g	NUM
ejpam-5360	86	27	)	)	PUNCT
ejpam-5360	86	28	)	)	PUNCT
ejpam-5360	86	29	≥	≥	NOUN
ejpam-5360	87	1	φn	φn	INTJ
ejpam-5360	87	2	.	.	PUNCT
ejpam-5360	88	1	moreover	moreover	ADV
ejpam-5360	88	2	,	,	PUNCT
ejpam-5360	88	3	if	if	SCONJ
ejpam-5360	88	4	a	a	PRON
ejpam-5360	88	5	is	be	AUX
ejpam-5360	88	6	irreducible	irreducible	ADJ
ejpam-5360	88	7	,	,	PUNCT
ejpam-5360	88	8	ρ1(a(g	ρ1(a(g	PRON
ejpam-5360	88	9	)	)	PUNCT
ejpam-5360	88	10	)	)	PUNCT
ejpam-5360	89	1	=	=	PUNCT
ejpam-5360	89	2	φn	φn	ADP
ejpam-5360	89	3	iff	iff	PROPN
ejpam-5360	89	4	r1	r1	PROPN
ejpam-5360	89	5	=	=	PROPN
ejpam-5360	89	6	r2	r2	PROPN
ejpam-5360	89	7	=	=	PUNCT
ejpam-5360	89	8	...	...	PUNCT
ejpam-5360	90	1	=	=	PUNCT
ejpam-5360	90	2	rn	rn	PROPN
ejpam-5360	90	3	or	or	CCONJ
ejpam-5360	90	4	t	t	PROPN
ejpam-5360	90	5	>	>	X
ejpam-5360	90	6	0	0	NUM
ejpam-5360	90	7	,	,	PUNCT
ejpam-5360	90	8	and	and	CCONJ
ejpam-5360	90	9	for	for	ADP
ejpam-5360	90	10	some	some	DET
ejpam-5360	90	11	2	2	NUM
ejpam-5360	90	12	≤	≤	NOUN
ejpam-5360	90	13	t	t	NOUN
ejpam-5360	90	14	≤	≤	NOUN
ejpam-5360	90	15	n	n	CCONJ
ejpam-5360	90	16	,	,	PUNCT
ejpam-5360	90	17	a	a	DET
ejpam-5360	90	18	satisfies	satisfie	NOUN
ejpam-5360	90	19	the	the	DET
ejpam-5360	90	20	following	follow	VERB
ejpam-5360	90	21	conditions	condition	NOUN
ejpam-5360	90	22	:	:	PUNCT
ejpam-5360	90	23	(	(	PUNCT
ejpam-5360	90	24	1	1	X
ejpam-5360	90	25	)	)	PUNCT
ejpam-5360	90	26	aii	aii	NOUN
ejpam-5360	90	27	=	=	SYM
ejpam-5360	90	28	s	s	PROPN
ejpam-5360	90	29	for	for	ADP
ejpam-5360	90	30	1	1	NUM
ejpam-5360	90	31	≤	≤	NUM
ejpam-5360	90	32	i	i	PRON
ejpam-5360	90	33	≤	≤	NOUN
ejpam-5360	91	1	t−	t−	PROPN
ejpam-5360	91	2	1	1	NUM
ejpam-5360	91	3	.	.	PUNCT
ejpam-5360	91	4	(	(	PUNCT
ejpam-5360	91	5	2	2	X
ejpam-5360	91	6	)	)	PUNCT
ejpam-5360	91	7	aik	aik	NOUN
ejpam-5360	91	8	=	=	SYM
ejpam-5360	91	9	t	t	PROPN
ejpam-5360	91	10	for	for	ADP
ejpam-5360	91	11	1	1	NUM
ejpam-5360	91	12	≤	≤	NUM
ejpam-5360	92	1	i	i	PRON
ejpam-5360	92	2	≤	≤	ADJ
ejpam-5360	92	3	n−	n−	PROPN
ejpam-5360	92	4	1	1	NUM
ejpam-5360	92	5	,	,	PUNCT
ejpam-5360	92	6	1	1	NUM
ejpam-5360	92	7	≤	≤	NUM
ejpam-5360	92	8	k	k	X
ejpam-5360	92	9	≤	≤	NUM
ejpam-5360	92	10	t−	t−	PROPN
ejpam-5360	92	11	1	1	NUM
ejpam-5360	92	12	with	with	ADP
ejpam-5360	92	13	k	k	PROPN
ejpam-5360	92	14	̸=	̸=	PROPN
ejpam-5360	92	15	i.	i.	NOUN
ejpam-5360	92	16	(	(	PUNCT
ejpam-5360	92	17	3	3	NUM
ejpam-5360	92	18	)	)	PUNCT
ejpam-5360	92	19	rt	rt	NOUN
ejpam-5360	92	20	=	=	PUNCT
ejpam-5360	92	21	rt+1	rt+1	PROPN
ejpam-5360	92	22	....	....	PUNCT
ejpam-5360	92	23	=	=	SYM
ejpam-5360	92	24	rn	rn	PROPN
ejpam-5360	92	25	.	.	PROPN
ejpam-5360	93	1	(	(	PUNCT
ejpam-5360	93	2	4	4	X
ejpam-5360	93	3	)	)	PUNCT
ejpam-5360	93	4	ank	ank	PROPN
ejpam-5360	93	5	=	=	PROPN
ejpam-5360	93	6	t	t	PROPN
ejpam-5360	93	7	for	for	ADP
ejpam-5360	93	8	1	1	NUM
ejpam-5360	93	9	≤	≤	NOUN
ejpam-5360	93	10	k	k	X
ejpam-5360	93	11	≤	≤	NOUN
ejpam-5360	93	12	t−	t−	PROPN
ejpam-5360	93	13	1	1	NUM
ejpam-5360	93	14	.	.	PUNCT
ejpam-5360	93	15	theorem	theorem	VERB
ejpam-5360	93	16	4	4	NUM
ejpam-5360	93	17	.	.	PUNCT
ejpam-5360	94	1	[	[	X
ejpam-5360	94	2	17	17	NUM
ejpam-5360	94	3	]	]	PUNCT
ejpam-5360	94	4	let	let	VERB
ejpam-5360	94	5	a	a	PRON
ejpam-5360	94	6	=	=	X
ejpam-5360	94	7	(	(	PUNCT
ejpam-5360	94	8	aij	aij	PROPN
ejpam-5360	94	9	)	)	PUNCT
ejpam-5360	94	10	be	be	VERB
ejpam-5360	94	11	an	an	DET
ejpam-5360	94	12	n	n	NUM
ejpam-5360	94	13	×	×	NOUN
ejpam-5360	94	14	n	n	CCONJ
ejpam-5360	94	15	non	non	ADJ
ejpam-5360	94	16	-	-	ADJ
ejpam-5360	94	17	negative	negative	ADJ
ejpam-5360	94	18	matrix	matrix	NOUN
ejpam-5360	94	19	with	with	ADP
ejpam-5360	94	20	row	row	NOUN
ejpam-5360	94	21	sums	sum	NOUN
ejpam-5360	94	22	r1,r2,	r1,r2,	NOUN
ejpam-5360	94	23	...	...	PUNCT
ejpam-5360	94	24	,rn	,rn	PUNCT
ejpam-5360	94	25	where	where	SCONJ
ejpam-5360	94	26	r1	r1	PROPN
ejpam-5360	94	27	≥	≥	NUM
ejpam-5360	94	28	r2	r2	PROPN
ejpam-5360	94	29	≥	≥	NUM
ejpam-5360	94	30	....	....	PUNCT
ejpam-5360	94	31	≥	≥	PROPN
ejpam-5360	94	32	rn	rn	PROPN
ejpam-5360	94	33	.	.	PROPN
ejpam-5360	94	34	let	let	VERB
ejpam-5360	94	35	m	m	PRON
ejpam-5360	94	36	and	and	CCONJ
ejpam-5360	94	37	n	n	CCONJ
ejpam-5360	94	38	be	be	AUX
ejpam-5360	94	39	the	the	DET
ejpam-5360	94	40	largest	large	ADJ
ejpam-5360	94	41	diagonal	diagonal	ADJ
ejpam-5360	94	42	and	and	CCONJ
ejpam-5360	94	43	the	the	DET
ejpam-5360	94	44	largest	large	ADJ
ejpam-5360	94	45	non	non	ADJ
ejpam-5360	94	46	-	-	ADJ
ejpam-5360	94	47	diagonal	diagonal	ADJ
ejpam-5360	94	48	elements	element	NOUN
ejpam-5360	94	49	of	of	ADP
ejpam-5360	94	50	a	a	PRON
ejpam-5360	94	51	,	,	PUNCT
ejpam-5360	94	52	respectively	respectively	ADV
ejpam-5360	94	53	.	.	PUNCT
ejpam-5360	95	1	suppose	suppose	VERB
ejpam-5360	95	2	n	n	PROPN
ejpam-5360	95	3	>	>	X
ejpam-5360	95	4	0	0	X
ejpam-5360	95	5	.	.	PUNCT
ejpam-5360	96	1	for	for	ADP
ejpam-5360	96	2	1	1	NUM
ejpam-5360	96	3	≤	≤	NUM
ejpam-5360	96	4	l	l	NOUN
ejpam-5360	96	5	≤	≤	NOUN
ejpam-5360	96	6	n	n	CCONJ
ejpam-5360	96	7	,	,	PUNCT
ejpam-5360	96	8	let	let	VERB
ejpam-5360	96	9	φl	φl	PRON
ejpam-5360	96	10	=	=	X
ejpam-5360	96	11	(	(	PUNCT
ejpam-5360	96	12	rl	rl	X
ejpam-5360	96	13	+	+	NOUN
ejpam-5360	96	14	m	m	VERB
ejpam-5360	96	15	−n	−n	ADJ
ejpam-5360	96	16	)	)	PUNCT
ejpam-5360	97	1	+	+	CCONJ
ejpam-5360	97	2	√	√	INTJ
ejpam-5360	97	3	(	(	PUNCT
ejpam-5360	97	4	rl	rl	VERB
ejpam-5360	97	5	−m	−m	PROPN
ejpam-5360	97	6	+	+	NOUN
ejpam-5360	97	7	n)2	n)2	NOUN
ejpam-5360	97	8	+	+	CCONJ
ejpam-5360	97	9	4n	4n	ADJ
ejpam-5360	97	10	∑l−1	∑l−1	NOUN
ejpam-5360	97	11	i=1(ri	i=1(ri	X
ejpam-5360	98	1	−	−	PROPN
ejpam-5360	98	2	rl	rl	NOUN
ejpam-5360	98	3	)	)	PUNCT
ejpam-5360	98	4	2	2	NUM
ejpam-5360	98	5	(	(	PUNCT
ejpam-5360	98	6	4	4	NUM
ejpam-5360	98	7	)	)	PUNCT
ejpam-5360	98	8	then	then	ADV
ejpam-5360	98	9	ρ1(a(g	ρ1(a(g	NUM
ejpam-5360	98	10	)	)	PUNCT
ejpam-5360	98	11	)	)	PUNCT
ejpam-5360	98	12	≤	≤	NOUN
ejpam-5360	98	13	φl	φl	VERB
ejpam-5360	98	14	for	for	ADP
ejpam-5360	98	15	1	1	NUM
ejpam-5360	98	16	≤	≤	NUM
ejpam-5360	98	17	l	l	NOUN
ejpam-5360	98	18	≤	≤	PUNCT
ejpam-5360	98	19	n.	n.	NOUN
ejpam-5360	98	20	moreover	moreover	ADV
ejpam-5360	98	21	,	,	PUNCT
ejpam-5360	98	22	if	if	SCONJ
ejpam-5360	98	23	a	a	PRON
ejpam-5360	98	24	is	be	AUX
ejpam-5360	98	25	irreducible	irreducible	ADJ
ejpam-5360	98	26	,	,	PUNCT
ejpam-5360	98	27	ρ1(a(g	ρ1(a(g	PRON
ejpam-5360	98	28	)	)	PUNCT
ejpam-5360	98	29	)	)	PUNCT
ejpam-5360	99	1	=	=	PUNCT
ejpam-5360	99	2	φl	φl	PRON
ejpam-5360	99	3	iff	iff	PROPN
ejpam-5360	99	4	r1	r1	PROPN
ejpam-5360	99	5	=	=	PROPN
ejpam-5360	99	6	r2	r2	PROPN
ejpam-5360	99	7	=	=	PUNCT
ejpam-5360	99	8	...	...	PUNCT
ejpam-5360	100	1	=	=	PUNCT
ejpam-5360	100	2	rn	rn	PROPN
ejpam-5360	100	3	or	or	CCONJ
ejpam-5360	100	4	for	for	ADP
ejpam-5360	100	5	some	some	DET
ejpam-5360	100	6	2	2	NUM
ejpam-5360	100	7	≤	≤	NOUN
ejpam-5360	100	8	t	t	NOUN
ejpam-5360	100	9	≤	≤	NUM
ejpam-5360	100	10	l	l	NOUN
ejpam-5360	100	11	,	,	PUNCT
ejpam-5360	100	12	a	a	DET
ejpam-5360	100	13	satisfies	satisfie	NOUN
ejpam-5360	100	14	the	the	DET
ejpam-5360	100	15	following	follow	VERB
ejpam-5360	100	16	conditions	condition	NOUN
ejpam-5360	100	17	:	:	PUNCT
ejpam-5360	100	18	(	(	PUNCT
ejpam-5360	100	19	1	1	X
ejpam-5360	100	20	)	)	PUNCT
ejpam-5360	100	21	aii	aii	NOUN
ejpam-5360	100	22	=	=	PUNCT
ejpam-5360	100	23	m	m	VERB
ejpam-5360	100	24	for	for	ADP
ejpam-5360	100	25	1	1	NUM
ejpam-5360	100	26	≤	≤	NUM
ejpam-5360	100	27	i	i	PRON
ejpam-5360	100	28	≤	≤	NOUN
ejpam-5360	101	1	t−	t−	PROPN
ejpam-5360	101	2	1	1	X
ejpam-5360	101	3	.	.	PUNCT
ejpam-5360	101	4	m.	m.	NOUN
ejpam-5360	101	5	v.	v.	ADP
ejpam-5360	101	6	,	,	PUNCT
ejpam-5360	101	7	k.	k.	PROPN
ejpam-5360	101	8	desikan	desikan	PROPN
ejpam-5360	101	9	/	/	SYM
ejpam-5360	101	10	eur	eur	PROPN
ejpam-5360	101	11	.	.	PUNCT
ejpam-5360	102	1	j.	j.	PROPN
ejpam-5360	102	2	pure	pure	PROPN
ejpam-5360	102	3	appl	appl	PROPN
ejpam-5360	102	4	.	.	PROPN
ejpam-5360	102	5	math	math	PROPN
ejpam-5360	102	6	,	,	PUNCT
ejpam-5360	102	7	18	18	NUM
ejpam-5360	102	8	(	(	PUNCT
ejpam-5360	102	9	4	4	NUM
ejpam-5360	102	10	)	)	PUNCT
ejpam-5360	102	11	(	(	PUNCT
ejpam-5360	102	12	2025	2025	NUM
ejpam-5360	102	13	)	)	PUNCT
ejpam-5360	102	14	,	,	PUNCT
ejpam-5360	102	15	5360	5360	NUM
ejpam-5360	102	16	6	6	NUM
ejpam-5360	102	17	of	of	ADP
ejpam-5360	102	18	20	20	NUM
ejpam-5360	102	19	(	(	PUNCT
ejpam-5360	102	20	2	2	NUM
ejpam-5360	102	21	)	)	PUNCT
ejpam-5360	102	22	aik	aik	NOUN
ejpam-5360	102	23	=	=	SYM
ejpam-5360	102	24	n	n	PROPN
ejpam-5360	102	25	for	for	ADP
ejpam-5360	102	26	1	1	NUM
ejpam-5360	102	27	≤	≤	NUM
ejpam-5360	102	28	i	i	NOUN
ejpam-5360	102	29	≤	≤	NOUN
ejpam-5360	103	1	l	l	NOUN
ejpam-5360	104	1	−	−	NOUN
ejpam-5360	104	2	1	1	NUM
ejpam-5360	104	3	,	,	PUNCT
ejpam-5360	104	4	1	1	NUM
ejpam-5360	104	5	≤	≤	NUM
ejpam-5360	104	6	k	k	X
ejpam-5360	104	7	≤	≤	NUM
ejpam-5360	104	8	t−	t−	PROPN
ejpam-5360	104	9	1	1	NUM
ejpam-5360	104	10	with	with	ADP
ejpam-5360	104	11	k	k	PROPN
ejpam-5360	104	12	̸=	̸=	PROPN
ejpam-5360	104	13	i.	i.	NOUN
ejpam-5360	104	14	(	(	PUNCT
ejpam-5360	104	15	3	3	NUM
ejpam-5360	104	16	)	)	PUNCT
ejpam-5360	104	17	rt	rt	NOUN
ejpam-5360	104	18	=	=	PUNCT
ejpam-5360	104	19	rt+1	rt+1	PROPN
ejpam-5360	104	20	...	...	PUNCT
ejpam-5360	104	21	=	=	SYM
ejpam-5360	104	22	rn	rn	PROPN
ejpam-5360	104	23	.	.	PROPN
ejpam-5360	105	1	(	(	PUNCT
ejpam-5360	105	2	4	4	X
ejpam-5360	105	3	)	)	PUNCT
ejpam-5360	105	4	aik	aik	NOUN
ejpam-5360	105	5	=	=	SYM
ejpam-5360	105	6	n	n	PROPN
ejpam-5360	105	7	for	for	ADP
ejpam-5360	105	8	1	1	NUM
ejpam-5360	105	9	≤	≤	NUM
ejpam-5360	105	10	i	i	PRON
ejpam-5360	105	11	≤	≤	PROPN
ejpam-5360	105	12	n	n	CCONJ
ejpam-5360	105	13	,	,	PUNCT
ejpam-5360	105	14	1	1	NUM
ejpam-5360	105	15	≤	≤	NUM
ejpam-5360	105	16	k	k	NOUN
ejpam-5360	105	17	≤	≤	NUM
ejpam-5360	106	1	t−	t−	PROPN
ejpam-5360	106	2	1	1	NUM
ejpam-5360	106	3	.	.	PUNCT
ejpam-5360	106	4	theorem	theorem	NOUN
ejpam-5360	106	5	5	5	NUM
ejpam-5360	106	6	.	.	PUNCT
ejpam-5360	107	1	[	[	X
ejpam-5360	107	2	16	16	NUM
ejpam-5360	107	3	]	]	PUNCT
ejpam-5360	107	4	let	let	VERB
ejpam-5360	107	5	m	m	PRON
ejpam-5360	107	6	be	be	AUX
ejpam-5360	107	7	a	a	DET
ejpam-5360	107	8	real	real	ADV
ejpam-5360	107	9	symmetric	symmetric	ADJ
ejpam-5360	107	10	n×	n×	PRON
ejpam-5360	107	11	n	n	NOUN
ejpam-5360	107	12	matrix	matrix	NOUN
ejpam-5360	107	13	,	,	PUNCT
ejpam-5360	107	14	and	and	CCONJ
ejpam-5360	107	15	let	let	VERB
ejpam-5360	107	16	β	β	NOUN
ejpam-5360	107	17	be	be	AUX
ejpam-5360	107	18	an	an	DET
ejpam-5360	107	19	eigenvalue	eigenvalue	NOUN
ejpam-5360	107	20	of	of	ADP
ejpam-5360	107	21	m	m	PROPN
ejpam-5360	107	22	with	with	ADP
ejpam-5360	107	23	eigenvector	eigenvector	NOUN
ejpam-5360	107	24	x	x	NOUN
ejpam-5360	107	25	all	all	PRON
ejpam-5360	107	26	of	of	ADP
ejpam-5360	107	27	whose	whose	DET
ejpam-5360	107	28	entries	entry	NOUN
ejpam-5360	107	29	are	be	AUX
ejpam-5360	107	30	non	non	ADJ
ejpam-5360	107	31	-	-	ADJ
ejpam-5360	107	32	negative	negative	ADJ
ejpam-5360	107	33	.	.	PUNCT
ejpam-5360	108	1	denote	denote	VERB
ejpam-5360	108	2	the	the	DET
ejpam-5360	108	3	ith	ith	PROPN
ejpam-5360	108	4	row	row	NOUN
ejpam-5360	108	5	sum	sum	NOUN
ejpam-5360	108	6	of	of	ADP
ejpam-5360	108	7	m	m	PRON
ejpam-5360	108	8	by	by	ADP
ejpam-5360	108	9	ri(m	ri(m	NOUN
ejpam-5360	108	10	)	)	PUNCT
ejpam-5360	108	11	.	.	PUNCT
ejpam-5360	109	1	then	then	ADV
ejpam-5360	109	2	min	min	PROPN
ejpam-5360	109	3	1≤i≤n	1≤i≤n	NUM
ejpam-5360	109	4	ri(m	ri(m	NOUN
ejpam-5360	109	5	)	)	PUNCT
ejpam-5360	109	6	≤	≤	NOUN
ejpam-5360	109	7	β(q(g	β(q(g	NUM
ejpam-5360	109	8	)	)	PUNCT
ejpam-5360	109	9	)	)	PUNCT
ejpam-5360	109	10	≤	≤	NUM
ejpam-5360	109	11	max	max	PROPN
ejpam-5360	109	12	1≤i≤n	1≤i≤n	NUM
ejpam-5360	109	13	ri(m	ri(m	NOUN
ejpam-5360	109	14	)	)	PUNCT
ejpam-5360	109	15	.	.	PUNCT
ejpam-5360	110	1	(	(	PUNCT
ejpam-5360	110	2	5	5	X
ejpam-5360	110	3	)	)	PUNCT
ejpam-5360	110	4	moreover	moreover	ADV
ejpam-5360	110	5	,	,	PUNCT
ejpam-5360	110	6	if	if	SCONJ
ejpam-5360	110	7	all	all	DET
ejpam-5360	110	8	entries	entry	NOUN
ejpam-5360	110	9	of	of	ADP
ejpam-5360	110	10	x	x	SYM
ejpam-5360	110	11	are	be	AUX
ejpam-5360	110	12	positive	positive	ADJ
ejpam-5360	110	13	then	then	ADV
ejpam-5360	110	14	either	either	PRON
ejpam-5360	110	15	of	of	ADP
ejpam-5360	110	16	the	the	DET
ejpam-5360	110	17	equalities	equality	NOUN
ejpam-5360	110	18	holds	hold	VERB
ejpam-5360	110	19	if	if	SCONJ
ejpam-5360	110	20	and	and	CCONJ
ejpam-5360	110	21	only	only	ADV
ejpam-5360	110	22	if	if	SCONJ
ejpam-5360	110	23	the	the	DET
ejpam-5360	110	24	row	row	NOUN
ejpam-5360	110	25	sums	sum	NOUN
ejpam-5360	110	26	of	of	ADP
ejpam-5360	110	27	m	m	NOUN
ejpam-5360	110	28	are	be	AUX
ejpam-5360	110	29	all	all	ADV
ejpam-5360	110	30	equal	equal	ADJ
ejpam-5360	110	31	.	.	PUNCT
ejpam-5360	111	1	theorem	theorem	VERB
ejpam-5360	111	2	6	6	NUM
ejpam-5360	111	3	.	.	PUNCT
ejpam-5360	112	1	[	[	X
ejpam-5360	112	2	16	16	NUM
ejpam-5360	112	3	]	]	PUNCT
ejpam-5360	112	4	let	let	VERB
ejpam-5360	112	5	m	m	PRON
ejpam-5360	112	6	be	be	AUX
ejpam-5360	112	7	a	a	DET
ejpam-5360	112	8	real	real	ADV
ejpam-5360	112	9	symmetric	symmetric	ADJ
ejpam-5360	112	10	n×	n×	PRON
ejpam-5360	112	11	n	n	NOUN
ejpam-5360	112	12	matrix	matrix	NOUN
ejpam-5360	112	13	,	,	PUNCT
ejpam-5360	112	14	and	and	CCONJ
ejpam-5360	112	15	let	let	VERB
ejpam-5360	112	16	β	β	NOUN
ejpam-5360	112	17	be	be	AUX
ejpam-5360	112	18	an	an	DET
ejpam-5360	112	19	eigenvalue	eigenvalue	NOUN
ejpam-5360	112	20	of	of	ADP
ejpam-5360	112	21	m	m	PROPN
ejpam-5360	112	22	with	with	ADP
ejpam-5360	112	23	eigenvector	eigenvector	NOUN
ejpam-5360	112	24	x	x	NOUN
ejpam-5360	112	25	all	all	PRON
ejpam-5360	112	26	of	of	ADP
ejpam-5360	112	27	whose	whose	DET
ejpam-5360	112	28	entries	entry	NOUN
ejpam-5360	112	29	are	be	AUX
ejpam-5360	112	30	non	non	ADJ
ejpam-5360	112	31	-	-	ADJ
ejpam-5360	112	32	negative	negative	ADJ
ejpam-5360	112	33	.	.	PUNCT
ejpam-5360	113	1	denote	denote	VERB
ejpam-5360	113	2	the	the	DET
ejpam-5360	113	3	ith	ith	PROPN
ejpam-5360	113	4	row	row	NOUN
ejpam-5360	113	5	sum	sum	NOUN
ejpam-5360	113	6	of	of	ADP
ejpam-5360	113	7	m	m	PRON
ejpam-5360	113	8	by	by	ADP
ejpam-5360	113	9	ri(m	ri(m	NOUN
ejpam-5360	113	10	)	)	PUNCT
ejpam-5360	113	11	.	.	PUNCT
ejpam-5360	114	1	let	let	VERB
ejpam-5360	114	2	p	p	PRON
ejpam-5360	114	3	be	be	AUX
ejpam-5360	114	4	a	a	DET
ejpam-5360	114	5	polynomial	polynomial	NOUN
ejpam-5360	114	6	.	.	PUNCT
ejpam-5360	115	1	then	then	ADV
ejpam-5360	115	2	min	min	PROPN
ejpam-5360	115	3	1≤i≤n	1≤i≤n	NUM
ejpam-5360	115	4	ri(p(m	ri(p(m	NUM
ejpam-5360	115	5	)	)	PUNCT
ejpam-5360	115	6	)	)	PUNCT
ejpam-5360	115	7	≤	≤	NUM
ejpam-5360	115	8	p(β(q(g	p(β(q(g	NUM
ejpam-5360	115	9	)	)	PUNCT
ejpam-5360	115	10	)	)	PUNCT
ejpam-5360	115	11	)	)	PUNCT
ejpam-5360	115	12	≤	≤	NUM
ejpam-5360	115	13	max	max	PROPN
ejpam-5360	115	14	1≤i≤n	1≤i≤n	NUM
ejpam-5360	115	15	ri(p(m	ri(p(m	NOUN
ejpam-5360	115	16	)	)	PUNCT
ejpam-5360	115	17	)	)	PUNCT
ejpam-5360	115	18	.	.	PUNCT
ejpam-5360	116	1	(	(	PUNCT
ejpam-5360	116	2	6	6	NUM
ejpam-5360	116	3	)	)	PUNCT
ejpam-5360	116	4	moreover	moreover	ADV
ejpam-5360	116	5	,	,	PUNCT
ejpam-5360	116	6	if	if	SCONJ
ejpam-5360	116	7	all	all	DET
ejpam-5360	116	8	entries	entry	NOUN
ejpam-5360	116	9	of	of	ADP
ejpam-5360	116	10	x	x	SYM
ejpam-5360	116	11	are	be	AUX
ejpam-5360	116	12	positive	positive	ADJ
ejpam-5360	116	13	then	then	ADV
ejpam-5360	116	14	either	either	PRON
ejpam-5360	116	15	of	of	ADP
ejpam-5360	116	16	the	the	DET
ejpam-5360	116	17	equalities	equality	NOUN
ejpam-5360	116	18	holds	hold	VERB
ejpam-5360	116	19	if	if	SCONJ
ejpam-5360	116	20	and	and	CCONJ
ejpam-5360	116	21	only	only	ADV
ejpam-5360	116	22	if	if	SCONJ
ejpam-5360	116	23	the	the	DET
ejpam-5360	116	24	row	row	NOUN
ejpam-5360	116	25	sums	sum	NOUN
ejpam-5360	116	26	of	of	ADP
ejpam-5360	116	27	m	m	NOUN
ejpam-5360	116	28	are	be	AUX
ejpam-5360	116	29	all	all	ADV
ejpam-5360	116	30	equal	equal	ADJ
ejpam-5360	116	31	.	.	PUNCT
ejpam-5360	117	1	lemma	lemma	PROPN
ejpam-5360	117	2	1	1	NUM
ejpam-5360	117	3	.	.	PUNCT
ejpam-5360	118	1	[	[	X
ejpam-5360	118	2	5	5	NUM
ejpam-5360	118	3	,	,	PUNCT
ejpam-5360	118	4	16	16	NUM
ejpam-5360	118	5	]	]	PUNCT
ejpam-5360	118	6	let	let	VERB
ejpam-5360	118	7	g	g	PROPN
ejpam-5360	118	8	=	=	SYM
ejpam-5360	118	9	(	(	PUNCT
ejpam-5360	118	10	v	v	NOUN
ejpam-5360	118	11	,	,	PUNCT
ejpam-5360	118	12	e	e	NOUN
ejpam-5360	118	13	)	)	PUNCT
ejpam-5360	118	14	be	be	AUX
ejpam-5360	118	15	a	a	DET
ejpam-5360	118	16	simple	simple	ADJ
ejpam-5360	118	17	graph	graph	NOUN
ejpam-5360	118	18	.	.	PUNCT
ejpam-5360	119	1	then	then	ADV
ejpam-5360	119	2	√	√	NUM
ejpam-5360	119	3	2	2	NUM
ejpam-5360	119	4	min	min	NOUN
ejpam-5360	119	5	v∈v	v∈v	NOUN
ejpam-5360	119	6	(	(	PUNCT
ejpam-5360	119	7	g	g	NOUN
ejpam-5360	119	8	)	)	PUNCT
ejpam-5360	119	9	√	√	PROPN
ejpam-5360	119	10	d2	d2	PROPN
ejpam-5360	119	11	(	(	PUNCT
ejpam-5360	119	12	v	v	NOUN
ejpam-5360	119	13	)	)	PUNCT
ejpam-5360	119	14	+	+	CCONJ
ejpam-5360	119	15	∑	∑	PUNCT
ejpam-5360	119	16	uv∈e(g	uv∈e(g	NUM
ejpam-5360	119	17	)	)	PUNCT
ejpam-5360	119	18	d	d	NOUN
ejpam-5360	119	19	(	(	PUNCT
ejpam-5360	119	20	u	u	NOUN
ejpam-5360	119	21	)	)	PUNCT
ejpam-5360	119	22	≤	≤	NOUN
ejpam-5360	119	23	µ1(q(g	µ1(q(g	NUM
ejpam-5360	119	24	)	)	PUNCT
ejpam-5360	119	25	)	)	PUNCT
ejpam-5360	119	26	≤	≤	NUM
ejpam-5360	119	27	√	√	NUM
ejpam-5360	119	28	2	2	NUM
ejpam-5360	119	29	max	max	NOUN
ejpam-5360	119	30	v∈v	v∈v	NOUN
ejpam-5360	119	31	(	(	PUNCT
ejpam-5360	119	32	g	g	NOUN
ejpam-5360	119	33	)	)	PUNCT
ejpam-5360	119	34	√	√	NOUN
ejpam-5360	119	35	d2(v	d2(v	NOUN
ejpam-5360	119	36	)	)	PUNCT
ejpam-5360	119	37	+	+	CCONJ
ejpam-5360	119	38	∑	∑	PUNCT
ejpam-5360	119	39	uv∈e(g	uv∈e(g	NOUN
ejpam-5360	119	40	)	)	PUNCT
ejpam-5360	119	41	d(u	d(u	PROPN
ejpam-5360	119	42	)	)	PUNCT
ejpam-5360	119	43	.	.	PUNCT
ejpam-5360	120	1	moreover	moreover	ADV
ejpam-5360	120	2	,	,	PUNCT
ejpam-5360	120	3	if	if	SCONJ
ejpam-5360	120	4	g	g	PROPN
ejpam-5360	120	5	is	be	AUX
ejpam-5360	120	6	connected	connect	VERB
ejpam-5360	120	7	,	,	PUNCT
ejpam-5360	120	8	both	both	CCONJ
ejpam-5360	120	9	the	the	DET
ejpam-5360	120	10	equalities	equality	NOUN
ejpam-5360	120	11	hold	hold	VERB
ejpam-5360	120	12	iff	iff	PROPN
ejpam-5360	120	13	2d2	2d2	NUM
ejpam-5360	120	14	(	(	PUNCT
ejpam-5360	120	15	v)+2	v)+2	PROPN
ejpam-5360	120	16	σ	σ	PROPN
ejpam-5360	120	17	uv∈e(g	uv∈e(g	NOUN
ejpam-5360	120	18	)	)	PUNCT
ejpam-5360	120	19	d(u	d(u	PROPN
ejpam-5360	120	20	)	)	PUNCT
ejpam-5360	120	21	is	be	AUX
ejpam-5360	120	22	the	the	DET
ejpam-5360	120	23	same	same	ADJ
ejpam-5360	120	24	∀v	∀v	PROPN
ejpam-5360	120	25	∈	∈	PROPN
ejpam-5360	120	26	v	v	NOUN
ejpam-5360	120	27	(	(	PUNCT
ejpam-5360	120	28	g	g	NOUN
ejpam-5360	120	29	)	)	PUNCT
ejpam-5360	120	30	.	.	PUNCT
ejpam-5360	121	1	3	3	X
ejpam-5360	121	2	.	.	X
ejpam-5360	121	3	main	main	ADJ
ejpam-5360	121	4	results	result	NOUN
ejpam-5360	121	5	3.1	3.1	NUM
ejpam-5360	121	6	.	.	PUNCT
ejpam-5360	121	7	bounds	bound	NOUN
ejpam-5360	121	8	on	on	ADP
ejpam-5360	121	9	spectral	spectral	ADJ
ejpam-5360	121	10	radius	radius	NOUN
ejpam-5360	121	11	in	in	ADP
ejpam-5360	121	12	this	this	DET
ejpam-5360	121	13	section	section	NOUN
ejpam-5360	121	14	,	,	PUNCT
ejpam-5360	121	15	we	we	PRON
ejpam-5360	121	16	derive	derive	VERB
ejpam-5360	121	17	tight	tight	ADJ
ejpam-5360	121	18	upper	upper	ADJ
ejpam-5360	121	19	bound	bind	VERB
ejpam-5360	121	20	and	and	CCONJ
ejpam-5360	121	21	lower	low	ADJ
ejpam-5360	121	22	bound	bind	VERB
ejpam-5360	121	23	of	of	ADP
ejpam-5360	121	24	ρ1(g	ρ1(g	NOUN
ejpam-5360	121	25	)	)	PUNCT
ejpam-5360	121	26	for	for	ADP
ejpam-5360	121	27	generalized	generalized	ADJ
ejpam-5360	121	28	core	core	NOUN
ejpam-5360	121	29	-	-	PUNCT
ejpam-5360	121	30	satellite	satellite	NOUN
ejpam-5360	121	31	graphs	graph	NOUN
ejpam-5360	121	32	.	.	PUNCT
ejpam-5360	122	1	theorem	theorem	ADJ
ejpam-5360	122	2	7	7	NUM
ejpam-5360	122	3	.	.	PUNCT
ejpam-5360	123	1	let	let	VERB
ejpam-5360	123	2	g	g	NOUN
ejpam-5360	123	3	=	=	PUNCT
ejpam-5360	123	4	θ(c	θ(c	VERB
ejpam-5360	123	5	,	,	PUNCT
ejpam-5360	123	6	s	s	NOUN
ejpam-5360	123	7	,	,	PUNCT
ejpam-5360	123	8	η∗	η∗	NOUN
ejpam-5360	123	9	)	)	PUNCT
ejpam-5360	123	10	be	be	VERB
ejpam-5360	123	11	the	the	DET
ejpam-5360	123	12	generalized	generalize	VERB
ejpam-5360	123	13	core	core	NOUN
ejpam-5360	123	14	-	-	PUNCT
ejpam-5360	123	15	satellite	satellite	NOUN
ejpam-5360	123	16	graph	graph	NOUN
ejpam-5360	123	17	with	with	ADP
ejpam-5360	123	18	n	n	ADP
ejpam-5360	123	19	vertices	vertex	NOUN
ejpam-5360	123	20	and	and	CCONJ
ejpam-5360	123	21	m	m	PRON
ejpam-5360	123	22	edges	edge	NOUN
ejpam-5360	123	23	.	.	PUNCT
ejpam-5360	124	1	then	then	ADV
ejpam-5360	124	2	the	the	DET
ejpam-5360	124	3	lower	lower	ADV
ejpam-5360	124	4	bound	bind	VERB
ejpam-5360	124	5	and	and	CCONJ
ejpam-5360	124	6	upper	upper	ADJ
ejpam-5360	124	7	bound	bind	VERB
ejpam-5360	124	8	for	for	ADP
ejpam-5360	124	9	the	the	DET
ejpam-5360	124	10	spectral	spectral	ADJ
ejpam-5360	124	11	radius	radius	NOUN
ejpam-5360	124	12	of	of	ADP
ejpam-5360	124	13	g	g	PROPN
ejpam-5360	124	14	are	be	AUX
ejpam-5360	124	15	m.	m.	NOUN
ejpam-5360	124	16	v.	v.	ADP
ejpam-5360	124	17	,	,	PUNCT
ejpam-5360	124	18	k.	k.	PROPN
ejpam-5360	124	19	desikan	desikan	PROPN
ejpam-5360	124	20	/	/	SYM
ejpam-5360	124	21	eur	eur	PROPN
ejpam-5360	124	22	.	.	PUNCT
ejpam-5360	125	1	j.	j.	PROPN
ejpam-5360	125	2	pure	pure	PROPN
ejpam-5360	125	3	appl	appl	PROPN
ejpam-5360	125	4	.	.	PROPN
ejpam-5360	125	5	math	math	PROPN
ejpam-5360	125	6	,	,	PUNCT
ejpam-5360	125	7	18	18	NUM
ejpam-5360	125	8	(	(	PUNCT
ejpam-5360	125	9	4	4	NUM
ejpam-5360	125	10	)	)	PUNCT
ejpam-5360	125	11	(	(	PUNCT
ejpam-5360	125	12	2025	2025	NUM
ejpam-5360	125	13	)	)	PUNCT
ejpam-5360	125	14	,	,	PUNCT
ejpam-5360	125	15	5360	5360	NUM
ejpam-5360	125	16	7	7	NUM
ejpam-5360	125	17	of	of	ADP
ejpam-5360	125	18	20	20	NUM
ejpam-5360	125	19	k∑	k∑	NOUN
ejpam-5360	126	1	j=1	j=1	PROPN
ejpam-5360	126	2	{	{	PUNCT
ejpam-5360	126	3	ϕ	ϕ	PROPN
ejpam-5360	126	4	k∑	k∑	PROPN
ejpam-5360	126	5	i=1	i=1	PROPN
ejpam-5360	126	6	(	(	PUNCT
ejpam-5360	126	7	ηiαi	ηiαi	NOUN
ejpam-5360	126	8	+	+	CCONJ
ejpam-5360	126	9	(	(	PUNCT
ejpam-5360	126	10	ϕ−	ϕ−	PROPN
ejpam-5360	126	11	1	1	NUM
ejpam-5360	126	12	)	)	PUNCT
ejpam-5360	126	13	)	)	PUNCT
ejpam-5360	127	1	+	+	CCONJ
ejpam-5360	127	2	(	(	PUNCT
ejpam-5360	127	3	αj	αj	NOUN
ejpam-5360	127	4	−	−	PROPN
ejpam-5360	127	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	127	6	αj	αj	NOUN
ejpam-5360	127	7	−	−	NOUN
ejpam-5360	127	8	1	1	NUM
ejpam-5360	127	9	)	)	PUNCT
ejpam-5360	127	10	}	}	PUNCT
ejpam-5360	127	11	1/2	1/2	NUM
ejpam-5360	127	12	≤	≤	NUM
ejpam-5360	127	13	kρ1(g	kρ1(g	PROPN
ejpam-5360	127	14	)	)	PUNCT
ejpam-5360	127	15	(	(	PUNCT
ejpam-5360	127	16	7	7	X
ejpam-5360	127	17	)	)	PUNCT
ejpam-5360	127	18	ρ1(g	ρ1(g	NOUN
ejpam-5360	127	19	)	)	PUNCT
ejpam-5360	127	20	≤	≤	NOUN
ejpam-5360	127	21	{	{	PUNCT
ejpam-5360	127	22	ϕ	ϕ	NOUN
ejpam-5360	127	23	(	(	PUNCT
ejpam-5360	127	24	k∑	k∑	PROPN
ejpam-5360	127	25	i=1	i=1	PROPN
ejpam-5360	127	26	ηiαi(ϕ+	ηiαi(ϕ+	X
ejpam-5360	127	27	αi	αi	VERB
ejpam-5360	127	28	−	−	NOUN
ejpam-5360	127	29	1	1	NUM
ejpam-5360	127	30	)	)	PUNCT
ejpam-5360	127	31	)	)	PUNCT
ejpam-5360	127	32	}	}	PUNCT
ejpam-5360	127	33	1/2	1/2	NUM
ejpam-5360	127	34	.	.	PUNCT
ejpam-5360	128	1	(	(	PUNCT
ejpam-5360	128	2	8)	8)	NUM
ejpam-5360	128	3	proof	proof	NOUN
ejpam-5360	128	4	.	.	PUNCT
ejpam-5360	129	1	we	we	PRON
ejpam-5360	129	2	derive	derive	VERB
ejpam-5360	129	3	the	the	DET
ejpam-5360	129	4	lower	low	ADJ
ejpam-5360	129	5	bound	bind	VERB
ejpam-5360	129	6	given	give	VERB
ejpam-5360	129	7	in	in	ADP
ejpam-5360	129	8	equation	equation	NOUN
ejpam-5360	129	9	(	(	PUNCT
ejpam-5360	129	10	7	7	X
ejpam-5360	129	11	)	)	PUNCT
ejpam-5360	129	12	using	use	VERB
ejpam-5360	129	13	the	the	DET
ejpam-5360	129	14	lower	low	ADJ
ejpam-5360	129	15	bound	bind	VERB
ejpam-5360	129	16	of	of	ADP
ejpam-5360	129	17	equation	equation	NOUN
ejpam-5360	129	18	(	(	PUNCT
ejpam-5360	129	19	2	2	NUM
ejpam-5360	129	20	)	)	PUNCT
ejpam-5360	129	21	.	.	PUNCT
ejpam-5360	130	1	from	from	ADP
ejpam-5360	130	2	theorem	theorem	NOUN
ejpam-5360	130	3	2	2	NUM
ejpam-5360	130	4	we	we	PRON
ejpam-5360	130	5	observe	observe	VERB
ejpam-5360	130	6	that	that	SCONJ
ejpam-5360	130	7	the	the	DET
ejpam-5360	130	8	lower	low	ADJ
ejpam-5360	130	9	bound	bind	VERB
ejpam-5360	130	10	is	be	AUX
ejpam-5360	130	11	obtained	obtain	VERB
ejpam-5360	130	12	by	by	ADP
ejpam-5360	130	13	considering	consider	VERB
ejpam-5360	130	14	vertices	vertex	NOUN
ejpam-5360	130	15	having	have	VERB
ejpam-5360	130	16	mimimum	mimimum	ADJ
ejpam-5360	130	17	degree	degree	NOUN
ejpam-5360	130	18	.	.	PUNCT
ejpam-5360	131	1	in	in	ADP
ejpam-5360	131	2	si	si	PROPN
ejpam-5360	131	3	▽	▽	PROPN
ejpam-5360	131	4	kϕ	kϕ	INTJ
ejpam-5360	131	5	we	we	PRON
ejpam-5360	131	6	see	see	VERB
ejpam-5360	131	7	that	that	DET
ejpam-5360	131	8	vertices	vertex	NOUN
ejpam-5360	131	9	with	with	ADP
ejpam-5360	131	10	minimum	minimum	NOUN
ejpam-5360	131	11	degree	degree	NOUN
ejpam-5360	131	12	are	be	AUX
ejpam-5360	131	13	in	in	ADP
ejpam-5360	131	14	the	the	DET
ejpam-5360	131	15	satellite	satellite	NOUN
ejpam-5360	131	16	si	si	PROPN
ejpam-5360	131	17	.	.	PROPN
ejpam-5360	131	18	for	for	ADP
ejpam-5360	131	19	the	the	DET
ejpam-5360	131	20	sthl	sthl	NOUN
ejpam-5360	131	21	clique	clique	NOUN
ejpam-5360	131	22	,	,	PUNCT
ejpam-5360	131	23	we	we	PRON
ejpam-5360	131	24	have	have	VERB
ejpam-5360	131	25	min	min	NOUN
ejpam-5360	131	26	v∈v	v∈v	NOUN
ejpam-5360	131	27	(	(	PUNCT
ejpam-5360	131	28	g	g	NOUN
ejpam-5360	131	29	)	)	PUNCT
ejpam-5360	131	30	(	(	PUNCT
ejpam-5360	131	31	σ	σ	X
ejpam-5360	131	32	u∼v∈e(g	u∼v∈e(g	PROPN
ejpam-5360	131	33	)	)	PUNCT
ejpam-5360	131	34	d(u	d(u	PROPN
ejpam-5360	131	35	)	)	PUNCT
ejpam-5360	131	36	)	)	PUNCT
ejpam-5360	131	37	1/2	1/2	NUM
ejpam-5360	131	38	=	=	SYM
ejpam-5360	131	39	(	(	PUNCT
ejpam-5360	131	40	σ	σ	PROPN
ejpam-5360	131	41	u∼v∈sl	u∼v∈sl	NUM
ejpam-5360	131	42	d(u	d(u	PROPN
ejpam-5360	131	43	)	)	PUNCT
ejpam-5360	131	44	)	)	PUNCT
ejpam-5360	131	45	1/2	1/2	NUM
ejpam-5360	131	46	=	=	SYM
ejpam-5360	131	47	{	{	PUNCT
ejpam-5360	131	48	ϕ	ϕ	PROPN
ejpam-5360	131	49	k∑	k∑	PROPN
ejpam-5360	131	50	i=1	i=1	PROPN
ejpam-5360	131	51	(	(	PUNCT
ejpam-5360	131	52	ηiαi	ηiαi	NOUN
ejpam-5360	131	53	+	+	CCONJ
ejpam-5360	131	54	(	(	PUNCT
ejpam-5360	131	55	ϕ−	ϕ−	PROPN
ejpam-5360	131	56	1	1	NUM
ejpam-5360	131	57	)	)	PUNCT
ejpam-5360	131	58	)	)	PUNCT
ejpam-5360	132	1	+	+	CCONJ
ejpam-5360	132	2	(	(	PUNCT
ejpam-5360	132	3	αl	αl	ADP
ejpam-5360	132	4	−	−	PROPN
ejpam-5360	132	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	132	6	αl	αl	ADP
ejpam-5360	132	7	−	−	PROPN
ejpam-5360	132	8	1	1	NUM
ejpam-5360	132	9	)	)	PUNCT
ejpam-5360	132	10	}	}	PUNCT
ejpam-5360	132	11	1/2	1/2	NUM
ejpam-5360	132	12	≤	≤	NUM
ejpam-5360	132	13	ρ1(g	ρ1(g	NOUN
ejpam-5360	132	14	)	)	PUNCT
ejpam-5360	132	15	.	.	PUNCT
ejpam-5360	133	1	(	(	PUNCT
ejpam-5360	133	2	σ	σ	PROPN
ejpam-5360	133	3	u∼v∈s1	u∼v∈s1	PROPN
ejpam-5360	133	4	d(u	d(u	PROPN
ejpam-5360	133	5	)	)	PUNCT
ejpam-5360	133	6	)	)	PUNCT
ejpam-5360	134	1	1/2	1/2	NUM
ejpam-5360	134	2	+	+	CCONJ
ejpam-5360	134	3	(	(	PUNCT
ejpam-5360	134	4	σ	σ	PROPN
ejpam-5360	134	5	u∼v∈s2	u∼v∈s2	PROPN
ejpam-5360	134	6	d(u	d(u	PROPN
ejpam-5360	134	7	)	)	PUNCT
ejpam-5360	134	8	)	)	PUNCT
ejpam-5360	134	9	1/2	1/2	NUM
ejpam-5360	135	1	+	+	CCONJ
ejpam-5360	135	2	...	...	PUNCT
ejpam-5360	136	1	+	+	CCONJ
ejpam-5360	136	2	(	(	PUNCT
ejpam-5360	136	3	σ	σ	PROPN
ejpam-5360	136	4	u∼v∈sk	u∼v∈sk	PRON
ejpam-5360	136	5	d(u	d(u	PROPN
ejpam-5360	136	6	)	)	PUNCT
ejpam-5360	136	7	)	)	PUNCT
ejpam-5360	136	8	1/2	1/2	NUM
ejpam-5360	136	9	≤	≤	NUM
ejpam-5360	136	10	kρ1(g	kρ1(g	PROPN
ejpam-5360	136	11	)	)	PUNCT
ejpam-5360	136	12	.	.	PUNCT
ejpam-5360	137	1	hence	hence	ADV
ejpam-5360	137	2	,	,	PUNCT
ejpam-5360	137	3	we	we	PRON
ejpam-5360	137	4	have	have	VERB
ejpam-5360	137	5	{	{	PUNCT
ejpam-5360	137	6	ϕ	ϕ	PROPN
ejpam-5360	137	7	k∑	k∑	PROPN
ejpam-5360	137	8	i=1	i=1	PROPN
ejpam-5360	138	1	(	(	PUNCT
ejpam-5360	138	2	ηiαi	ηiαi	NOUN
ejpam-5360	138	3	+	+	CCONJ
ejpam-5360	139	1	(	(	PUNCT
ejpam-5360	139	2	ϕ−	ϕ−	PROPN
ejpam-5360	139	3	1	1	NUM
ejpam-5360	139	4	)	)	PUNCT
ejpam-5360	139	5	)	)	PUNCT
ejpam-5360	140	1	+	+	CCONJ
ejpam-5360	140	2	(	(	PUNCT
ejpam-5360	140	3	α1	α1	PROPN
ejpam-5360	140	4	−	−	PROPN
ejpam-5360	140	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	140	6	α1	α1	PROPN
ejpam-5360	140	7	−	−	PROPN
ejpam-5360	140	8	1	1	NUM
ejpam-5360	140	9	)	)	PUNCT
ejpam-5360	140	10	}	}	PUNCT
ejpam-5360	140	11	1/2	1/2	NUM
ejpam-5360	141	1	+	+	CCONJ
ejpam-5360	141	2	ϕ	ϕ	PROPN
ejpam-5360	141	3	k∑	k∑	PROPN
ejpam-5360	141	4	{	{	PUNCT
ejpam-5360	141	5	i=1	i=1	X
ejpam-5360	141	6	(	(	PUNCT
ejpam-5360	141	7	ηiαi	ηiαi	NOUN
ejpam-5360	141	8	+	+	CCONJ
ejpam-5360	141	9	(	(	PUNCT
ejpam-5360	141	10	ϕ−	ϕ−	PROPN
ejpam-5360	141	11	1	1	NUM
ejpam-5360	141	12	)	)	PUNCT
ejpam-5360	141	13	)	)	PUNCT
ejpam-5360	142	1	+	+	CCONJ
ejpam-5360	142	2	(	(	PUNCT
ejpam-5360	142	3	α2	α2	ADJ
ejpam-5360	142	4	−	−	PROPN
ejpam-5360	142	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	142	6	α2	α2	ADJ
ejpam-5360	142	7	−	−	NOUN
ejpam-5360	142	8	1	1	NUM
ejpam-5360	142	9	)	)	PUNCT
ejpam-5360	142	10			NOUN
ejpam-5360	142	11	1/2	1/2	NUM
ejpam-5360	142	12	+	+	CCONJ
ejpam-5360	142	13	...	...	PUNCT
ejpam-5360	143	1	+	+	CCONJ
ejpam-5360	143	2	{	{	PUNCT
ejpam-5360	143	3	ϕ	ϕ	PROPN
ejpam-5360	143	4	k∑	k∑	PROPN
ejpam-5360	143	5	i=1	i=1	PROPN
ejpam-5360	144	1	(	(	PUNCT
ejpam-5360	144	2	ηiαi	ηiαi	NOUN
ejpam-5360	144	3	+	+	CCONJ
ejpam-5360	145	1	(	(	PUNCT
ejpam-5360	145	2	ϕ−	ϕ−	PROPN
ejpam-5360	145	3	1	1	NUM
ejpam-5360	145	4	)	)	PUNCT
ejpam-5360	145	5	)	)	PUNCT
ejpam-5360	146	1	+	+	CCONJ
ejpam-5360	146	2	(	(	PUNCT
ejpam-5360	146	3	αk	αk	ADP
ejpam-5360	146	4	−	−	PROPN
ejpam-5360	146	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	146	6	αk	αk	NOUN
ejpam-5360	146	7	−	−	NOUN
ejpam-5360	146	8	1	1	NUM
ejpam-5360	146	9	)	)	PUNCT
ejpam-5360	146	10	}	}	PUNCT
ejpam-5360	146	11	1/2	1/2	NUM
ejpam-5360	146	12	≤	≤	NUM
ejpam-5360	146	13	kρ1(g	kρ1(g	PROPN
ejpam-5360	146	14	)	)	PUNCT
ejpam-5360	146	15	.	.	PUNCT
ejpam-5360	147	1	hence	hence	ADV
ejpam-5360	147	2	,	,	PUNCT
ejpam-5360	147	3	k∑	k∑	VERB
ejpam-5360	148	1	l=1	l=1	X
ejpam-5360	148	2	{	{	PUNCT
ejpam-5360	148	3	ϕ	ϕ	PROPN
ejpam-5360	148	4	k∑	k∑	PROPN
ejpam-5360	148	5	i=1	i=1	PROPN
ejpam-5360	149	1	(	(	PUNCT
ejpam-5360	149	2	ηiαi	ηiαi	NOUN
ejpam-5360	149	3	+	+	CCONJ
ejpam-5360	150	1	(	(	PUNCT
ejpam-5360	150	2	ϕ−	ϕ−	PROPN
ejpam-5360	150	3	1	1	NUM
ejpam-5360	150	4	)	)	PUNCT
ejpam-5360	150	5	)	)	PUNCT
ejpam-5360	151	1	+	+	CCONJ
ejpam-5360	151	2	(	(	PUNCT
ejpam-5360	151	3	αl	αl	ADP
ejpam-5360	151	4	−	−	PROPN
ejpam-5360	151	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	151	6	αl	αl	ADP
ejpam-5360	151	7	−	−	PROPN
ejpam-5360	151	8	1	1	NUM
ejpam-5360	151	9	)	)	PUNCT
ejpam-5360	151	10	}	}	PUNCT
ejpam-5360	151	11	1/2	1/2	NUM
ejpam-5360	151	12	≤	≤	NUM
ejpam-5360	151	13	kρ1(g	kρ1(g	PROPN
ejpam-5360	151	14	)	)	PUNCT
ejpam-5360	151	15	.	.	PUNCT
ejpam-5360	152	1	(	(	PUNCT
ejpam-5360	152	2	9	9	X
ejpam-5360	152	3	)	)	PUNCT
ejpam-5360	152	4	m.	m.	NOUN
ejpam-5360	152	5	v.	v.	ADP
ejpam-5360	152	6	,	,	PUNCT
ejpam-5360	152	7	k.	k.	PROPN
ejpam-5360	152	8	desikan	desikan	PROPN
ejpam-5360	152	9	/	/	SYM
ejpam-5360	152	10	eur	eur	PROPN
ejpam-5360	152	11	.	.	PUNCT
ejpam-5360	153	1	j.	j.	PROPN
ejpam-5360	153	2	pure	pure	PROPN
ejpam-5360	153	3	appl	appl	PROPN
ejpam-5360	153	4	.	.	PROPN
ejpam-5360	153	5	math	math	PROPN
ejpam-5360	153	6	,	,	PUNCT
ejpam-5360	153	7	18	18	NUM
ejpam-5360	153	8	(	(	PUNCT
ejpam-5360	153	9	4	4	NUM
ejpam-5360	153	10	)	)	PUNCT
ejpam-5360	153	11	(	(	PUNCT
ejpam-5360	153	12	2025	2025	NUM
ejpam-5360	153	13	)	)	PUNCT
ejpam-5360	153	14	,	,	PUNCT
ejpam-5360	153	15	5360	5360	NUM
ejpam-5360	153	16	8	8	NUM
ejpam-5360	153	17	of	of	ADP
ejpam-5360	153	18	20	20	NUM
ejpam-5360	153	19	from	from	ADP
ejpam-5360	153	20	theorem	theorem	NOUN
ejpam-5360	153	21	2	2	NUM
ejpam-5360	153	22	we	we	PRON
ejpam-5360	153	23	observe	observe	VERB
ejpam-5360	153	24	that	that	SCONJ
ejpam-5360	153	25	the	the	DET
ejpam-5360	153	26	upper	upper	ADJ
ejpam-5360	153	27	bound	bound	NOUN
ejpam-5360	153	28	is	be	AUX
ejpam-5360	153	29	obtained	obtain	VERB
ejpam-5360	153	30	by	by	ADP
ejpam-5360	153	31	considering	consider	VERB
ejpam-5360	153	32	vertices	vertex	NOUN
ejpam-5360	153	33	having	have	VERB
ejpam-5360	153	34	maximum	maximum	ADJ
ejpam-5360	153	35	degree	degree	NOUN
ejpam-5360	153	36	.	.	PUNCT
ejpam-5360	154	1	in	in	ADP
ejpam-5360	154	2	si	si	PROPN
ejpam-5360	154	3	▽	▽	PROPN
ejpam-5360	154	4	kϕ	kϕ	PROPN
ejpam-5360	154	5	,	,	PUNCT
ejpam-5360	154	6	we	we	PRON
ejpam-5360	154	7	see	see	VERB
ejpam-5360	154	8	that	that	PRON
ejpam-5360	154	9	vertices	vertice	VERB
ejpam-5360	154	10	with	with	ADP
ejpam-5360	154	11	a	a	DET
ejpam-5360	154	12	maximum	maximum	ADJ
ejpam-5360	154	13	degree	degree	NOUN
ejpam-5360	154	14	are	be	AUX
ejpam-5360	154	15	in	in	ADP
ejpam-5360	154	16	the	the	DET
ejpam-5360	154	17	core	core	NOUN
ejpam-5360	154	18	kϕ.	kϕ.	NOUN
ejpam-5360	154	19	hence	hence	ADV
ejpam-5360	154	20	,	,	PUNCT
ejpam-5360	154	21	from	from	ADP
ejpam-5360	154	22	theorem	theorem	NOUN
ejpam-5360	154	23	2	2	NUM
ejpam-5360	154	24	,	,	PUNCT
ejpam-5360	154	25	we	we	PRON
ejpam-5360	154	26	have	have	VERB
ejpam-5360	154	27	,	,	PUNCT
ejpam-5360	154	28	max	max	PROPN
ejpam-5360	154	29	v∈v	v∈v	PROPN
ejpam-5360	154	30	(	(	PUNCT
ejpam-5360	154	31	g	g	NOUN
ejpam-5360	154	32	)	)	PUNCT
ejpam-5360	154	33	(	(	PUNCT
ejpam-5360	154	34	σ	σ	PROPN
ejpam-5360	154	35	uv∈e(g	uv∈e(g	NOUN
ejpam-5360	154	36	)	)	PUNCT
ejpam-5360	154	37	d(u	d(u	PROPN
ejpam-5360	154	38	)	)	PUNCT
ejpam-5360	154	39	)	)	PUNCT
ejpam-5360	154	40	1/2	1/2	NUM
ejpam-5360	154	41	=	=	SYM
ejpam-5360	154	42	(	(	PUNCT
ejpam-5360	154	43	σ	σ	PROPN
ejpam-5360	154	44	u∼v∈kϕ	u∼v∈kϕ	PROPN
ejpam-5360	154	45	d(u	d(u	PROPN
ejpam-5360	154	46	)	)	PUNCT
ejpam-5360	154	47	)	)	PUNCT
ejpam-5360	154	48	1/2	1/2	NUM
ejpam-5360	154	49	=	=	SYM
ejpam-5360	154	50	{	{	PUNCT
ejpam-5360	154	51	ϕ	ϕ	PROPN
ejpam-5360	154	52	(	(	PUNCT
ejpam-5360	154	53	k∑	k∑	PROPN
ejpam-5360	154	54	i=1	i=1	PROPN
ejpam-5360	154	55	ηiαi(ϕ+	ηiαi(ϕ+	X
ejpam-5360	154	56	αi	αi	VERB
ejpam-5360	154	57	−	−	NOUN
ejpam-5360	154	58	1	1	NUM
ejpam-5360	154	59	)	)	PUNCT
ejpam-5360	154	60	)	)	PUNCT
ejpam-5360	154	61	}	}	PUNCT
ejpam-5360	154	62	1/2	1/2	NUM
ejpam-5360	154	63	≥	≥	NOUN
ejpam-5360	154	64	ρ1(g	ρ1(g	NOUN
ejpam-5360	154	65	)	)	PUNCT
ejpam-5360	154	66	.	.	PUNCT
ejpam-5360	155	1	(	(	PUNCT
ejpam-5360	155	2	10	10	NUM
ejpam-5360	155	3	)	)	PUNCT
ejpam-5360	155	4	hence	hence	ADV
ejpam-5360	155	5	proved	prove	VERB
ejpam-5360	155	6	.	.	PUNCT
ejpam-5360	156	1	remark	remark	PROPN
ejpam-5360	156	2	2	2	NUM
ejpam-5360	156	3	.	.	PUNCT
ejpam-5360	157	1	we	we	PRON
ejpam-5360	157	2	observe	observe	VERB
ejpam-5360	157	3	that	that	SCONJ
ejpam-5360	157	4	{	{	PUNCT
ejpam-5360	157	5	ϕ	ϕ	X
ejpam-5360	157	6	(	(	PUNCT
ejpam-5360	157	7	k∑	k∑	PROPN
ejpam-5360	157	8	i=1	i=1	PROPN
ejpam-5360	157	9	ηiαi(ϕ+	ηiαi(ϕ+	X
ejpam-5360	157	10	αi	αi	VERB
ejpam-5360	157	11	−	−	NOUN
ejpam-5360	157	12	1	1	NUM
ejpam-5360	157	13	)	)	PUNCT
ejpam-5360	157	14	)	)	PUNCT
ejpam-5360	157	15	}	}	PUNCT
ejpam-5360	157	16	1/2	1/2	NUM
ejpam-5360	157	17	≥	≥	NOUN
ejpam-5360	157	18	{	{	PUNCT
ejpam-5360	157	19	ϕ	ϕ	PROPN
ejpam-5360	157	20	(	(	PUNCT
ejpam-5360	157	21	∑k	∑k	PROPN
ejpam-5360	157	22	i=1	i=1	PROPN
ejpam-5360	157	23	ηiαi	ηiαi	NOUN
ejpam-5360	157	24	(	(	PUNCT
ejpam-5360	157	25	ϕ+	ϕ+	INTJ
ejpam-5360	157	26	αi	αi	ADV
ejpam-5360	157	27	−	−	NOUN
ejpam-5360	157	28	1	1	X
ejpam-5360	157	29	)	)	PUNCT
ejpam-5360	157	30	k	k	NOUN
ejpam-5360	157	31	)	)	PUNCT
ejpam-5360	157	32	}	}	PUNCT
ejpam-5360	157	33	1/2	1/2	NUM
ejpam-5360	157	34	≥	≥	NOUN
ejpam-5360	157	35	ρ1(g	ρ1(g	NUM
ejpam-5360	157	36	)	)	PUNCT
ejpam-5360	157	37	(	(	PUNCT
ejpam-5360	157	38	11	11	NUM
ejpam-5360	157	39	)	)	PUNCT
ejpam-5360	157	40	theorem	theorem	NOUN
ejpam-5360	157	41	8	8	NUM
ejpam-5360	157	42	.	.	PUNCT
ejpam-5360	158	1	let	let	VERB
ejpam-5360	158	2	g	g	NOUN
ejpam-5360	158	3	be	be	AUX
ejpam-5360	158	4	the	the	DET
ejpam-5360	158	5	generalized	generalize	VERB
ejpam-5360	158	6	core	core	NOUN
ejpam-5360	158	7	-	-	PUNCT
ejpam-5360	158	8	satellite	satellite	NOUN
ejpam-5360	158	9	graph	graph	NOUN
ejpam-5360	158	10	of	of	ADP
ejpam-5360	158	11	order	order	NOUN
ejpam-5360	158	12	n	n	CCONJ
ejpam-5360	158	13	with	with	ADP
ejpam-5360	158	14	m	m	PROPN
ejpam-5360	158	15	edges	edge	NOUN
ejpam-5360	158	16	and	and	CCONJ
ejpam-5360	158	17	s1	s1	NOUN
ejpam-5360	158	18	,	,	PUNCT
ejpam-5360	158	19	s2	s2	PROPN
ejpam-5360	158	20	,	,	PUNCT
ejpam-5360	158	21	s3	s3	PROPN
ejpam-5360	158	22	,	,	PUNCT
ejpam-5360	158	23	....	....	PUNCT
ejpam-5360	158	24	,	,	PUNCT
ejpam-5360	158	25	sk	sk	NOUN
ejpam-5360	158	26	be	be	AUX
ejpam-5360	158	27	the	the	DET
ejpam-5360	158	28	k	k	PROPN
ejpam-5360	158	29	satellites	satellite	NOUN
ejpam-5360	158	30	connected	connect	VERB
ejpam-5360	158	31	to	to	ADP
ejpam-5360	158	32	the	the	DET
ejpam-5360	158	33	core	core	NOUN
ejpam-5360	158	34	graph	graph	NOUN
ejpam-5360	158	35	kϕ	kϕ	PROPN
ejpam-5360	158	36	,	,	PUNCT
ejpam-5360	158	37	then√	then√	NOUN
ejpam-5360	158	38	2	2	NUM
ejpam-5360	158	39	m	m	NOUN
ejpam-5360	158	40	k	k	NOUN
ejpam-5360	158	41	<	<	X
ejpam-5360	158	42	ρ1(g	ρ1(g	PROPN
ejpam-5360	158	43	)	)	PUNCT
ejpam-5360	158	44	<	<	X
ejpam-5360	158	45	√	√	NUM
ejpam-5360	158	46	2	2	NUM
ejpam-5360	158	47	m.	m.	NOUN
ejpam-5360	158	48	proof	proof	NOUN
ejpam-5360	158	49	.	.	PUNCT
ejpam-5360	159	1	the	the	DET
ejpam-5360	159	2	sum	sum	NOUN
ejpam-5360	159	3	of	of	ADP
ejpam-5360	159	4	the	the	DET
ejpam-5360	159	5	degrees	degree	NOUN
ejpam-5360	159	6	of	of	ADP
ejpam-5360	159	7	the	the	DET
ejpam-5360	159	8	vertices	vertex	NOUN
ejpam-5360	159	9	of	of	ADP
ejpam-5360	159	10	satellites	satellite	NOUN
ejpam-5360	159	11	s1	s1	NOUN
ejpam-5360	159	12	,	,	PUNCT
ejpam-5360	159	13	s2	s2	PROPN
ejpam-5360	159	14	,	,	PUNCT
ejpam-5360	159	15	s3	s3	PROPN
ejpam-5360	159	16	,	,	PUNCT
ejpam-5360	159	17	...	...	PUNCT
ejpam-5360	159	18	,	,	PUNCT
ejpam-5360	159	19	sk	sk	NOUN
ejpam-5360	159	20	and	and	CCONJ
ejpam-5360	159	21	the	the	DET
ejpam-5360	159	22	sum	sum	NOUN
ejpam-5360	159	23	of	of	ADP
ejpam-5360	159	24	the	the	DET
ejpam-5360	159	25	degrees	degree	NOUN
ejpam-5360	159	26	of	of	ADP
ejpam-5360	159	27	the	the	DET
ejpam-5360	159	28	vertices	vertex	NOUN
ejpam-5360	159	29	of	of	ADP
ejpam-5360	159	30	a	a	DET
ejpam-5360	159	31	core	core	NOUN
ejpam-5360	159	32	graph	graph	NOUN
ejpam-5360	159	33	kϕ	kϕ	NOUN
ejpam-5360	159	34	is	be	AUX
ejpam-5360	159	35	2	2	NUM
ejpam-5360	159	36	m	m	NOUN
ejpam-5360	159	37	,	,	PUNCT
ejpam-5360	159	38	that	that	ADV
ejpam-5360	159	39	is	is	ADV
ejpam-5360	159	40	,	,	PUNCT
ejpam-5360	159	41	k∑	k∑	PROPN
ejpam-5360	159	42	i=1	i=1	PROPN
ejpam-5360	160	1	(	(	PUNCT
ejpam-5360	160	2	σ	σ	PROPN
ejpam-5360	160	3	u∼v∈si	u∼v∈si	PROPN
ejpam-5360	160	4	d(u	d(u	PROPN
ejpam-5360	160	5	)	)	PUNCT
ejpam-5360	160	6	)	)	PUNCT
ejpam-5360	161	1	+	+	CCONJ
ejpam-5360	161	2	(	(	PUNCT
ejpam-5360	161	3	σ	σ	PROPN
ejpam-5360	161	4	u∼v∈kϕ	u∼v∈kϕ	PROPN
ejpam-5360	161	5	d(u	d(u	PROPN
ejpam-5360	161	6	)	)	PUNCT
ejpam-5360	161	7	)	)	PUNCT
ejpam-5360	161	8	=	=	SYM
ejpam-5360	162	1	2	2	NUM
ejpam-5360	162	2	m.	m.	NOUN
ejpam-5360	162	3	considering	consider	VERB
ejpam-5360	162	4	the	the	DET
ejpam-5360	162	5	lower	low	ADJ
ejpam-5360	162	6	bound	bind	VERB
ejpam-5360	162	7	of	of	ADP
ejpam-5360	162	8	equation	equation	NOUN
ejpam-5360	162	9	(	(	PUNCT
ejpam-5360	162	10	2	2	NUM
ejpam-5360	162	11	)	)	PUNCT
ejpam-5360	162	12	,	,	PUNCT
ejpam-5360	162	13	we	we	PRON
ejpam-5360	162	14	have	have	VERB
ejpam-5360	162	15	for	for	ADP
ejpam-5360	162	16	sl	sl	PROPN
ejpam-5360	162	17	min	min	PROPN
ejpam-5360	162	18	v∈v	v∈v	PROPN
ejpam-5360	162	19	(	(	PUNCT
ejpam-5360	162	20	g	g	NOUN
ejpam-5360	162	21	)	)	PUNCT
ejpam-5360	162	22	(	(	PUNCT
ejpam-5360	162	23	σ	σ	X
ejpam-5360	162	24	u∼v∈e(g	u∼v∈e(g	PROPN
ejpam-5360	162	25	)	)	PUNCT
ejpam-5360	162	26	d(u	d(u	PROPN
ejpam-5360	162	27	)	)	PUNCT
ejpam-5360	162	28	)	)	PUNCT
ejpam-5360	162	29	1/2	1/2	NUM
ejpam-5360	162	30	=	=	SYM
ejpam-5360	162	31	(	(	PUNCT
ejpam-5360	162	32	σ	σ	PROPN
ejpam-5360	162	33	u∼v∈sl	u∼v∈sl	NUM
ejpam-5360	162	34	d(u	d(u	PROPN
ejpam-5360	162	35	)	)	PUNCT
ejpam-5360	162	36	)	)	PUNCT
ejpam-5360	162	37	1/2	1/2	NUM
ejpam-5360	162	38	=	=	SYM
ejpam-5360	162	39	{	{	PUNCT
ejpam-5360	162	40	ϕ	ϕ	PROPN
ejpam-5360	162	41	k∑	k∑	PROPN
ejpam-5360	162	42	i=1	i=1	PROPN
ejpam-5360	163	1	(	(	PUNCT
ejpam-5360	163	2	ηiαi	ηiαi	NOUN
ejpam-5360	163	3	+	+	CCONJ
ejpam-5360	164	1	(	(	PUNCT
ejpam-5360	164	2	ϕ−	ϕ−	PROPN
ejpam-5360	164	3	1	1	NUM
ejpam-5360	164	4	)	)	PUNCT
ejpam-5360	164	5	)	)	PUNCT
ejpam-5360	165	1	+	+	CCONJ
ejpam-5360	165	2	(	(	PUNCT
ejpam-5360	165	3	αl	αl	ADP
ejpam-5360	165	4	−	−	PROPN
ejpam-5360	165	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	165	6	αl	αl	ADP
ejpam-5360	165	7	−	−	PROPN
ejpam-5360	165	8	1	1	NUM
ejpam-5360	165	9	)	)	PUNCT
ejpam-5360	165	10	}	}	PUNCT
ejpam-5360	165	11	1/2	1/2	NUM
ejpam-5360	165	12	≤	≤	NUM
ejpam-5360	165	13	ρ1(g	ρ1(g	NOUN
ejpam-5360	165	14	)	)	PUNCT
ejpam-5360	165	15	.	.	PUNCT
ejpam-5360	166	1	hence	hence	ADV
ejpam-5360	166	2	for	for	ADP
ejpam-5360	166	3	the	the	DET
ejpam-5360	166	4	sthl	sthl	NOUN
ejpam-5360	166	5	clique	clique	NOUN
ejpam-5360	166	6	,	,	PUNCT
ejpam-5360	166	7	we	we	PRON
ejpam-5360	166	8	have	have	VERB
ejpam-5360	166	9	(	(	PUNCT
ejpam-5360	166	10	σ	σ	PROPN
ejpam-5360	166	11	u∼v∈sl	u∼v∈sl	NUM
ejpam-5360	166	12	d(u	d(u	PROPN
ejpam-5360	166	13	)	)	PUNCT
ejpam-5360	166	14	)	)	PUNCT
ejpam-5360	167	1	=	=	PRON
ejpam-5360	167	2	{	{	PUNCT
ejpam-5360	167	3	ϕ	ϕ	PROPN
ejpam-5360	167	4	k∑	k∑	PROPN
ejpam-5360	167	5	i=1	i=1	PROPN
ejpam-5360	167	6	(	(	PUNCT
ejpam-5360	167	7	ηiαi	ηiαi	NOUN
ejpam-5360	167	8	+	+	CCONJ
ejpam-5360	167	9	(	(	PUNCT
ejpam-5360	167	10	ϕ−	ϕ−	PROPN
ejpam-5360	167	11	1	1	NUM
ejpam-5360	167	12	)	)	PUNCT
ejpam-5360	167	13	)	)	PUNCT
ejpam-5360	168	1	+	+	CCONJ
ejpam-5360	168	2	(	(	PUNCT
ejpam-5360	168	3	αl	αl	ADP
ejpam-5360	168	4	−	−	PROPN
ejpam-5360	168	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	168	6	αl	αl	ADP
ejpam-5360	168	7	−	−	PROPN
ejpam-5360	168	8	1	1	NUM
ejpam-5360	168	9	)	)	PUNCT
ejpam-5360	168	10	}	}	PUNCT
ejpam-5360	168	11	≤	≤	PUNCT
ejpam-5360	168	12	ρ1(g)2	ρ1(g)2	PROPN
ejpam-5360	168	13	.	.	PUNCT
ejpam-5360	169	1	m.	m.	PROPN
ejpam-5360	169	2	v.	v.	ADP
ejpam-5360	169	3	,	,	PUNCT
ejpam-5360	169	4	k.	k.	PROPN
ejpam-5360	169	5	desikan	desikan	PROPN
ejpam-5360	169	6	/	/	SYM
ejpam-5360	169	7	eur	eur	PROPN
ejpam-5360	169	8	.	.	PUNCT
ejpam-5360	170	1	j.	j.	PROPN
ejpam-5360	170	2	pure	pure	PROPN
ejpam-5360	170	3	appl	appl	PROPN
ejpam-5360	170	4	.	.	PROPN
ejpam-5360	170	5	math	math	PROPN
ejpam-5360	170	6	,	,	PUNCT
ejpam-5360	170	7	18	18	NUM
ejpam-5360	170	8	(	(	PUNCT
ejpam-5360	170	9	4	4	NUM
ejpam-5360	170	10	)	)	PUNCT
ejpam-5360	170	11	(	(	PUNCT
ejpam-5360	170	12	2025	2025	NUM
ejpam-5360	170	13	)	)	PUNCT
ejpam-5360	170	14	,	,	PUNCT
ejpam-5360	170	15	5360	5360	NUM
ejpam-5360	170	16	9	9	NUM
ejpam-5360	170	17	of	of	ADP
ejpam-5360	170	18	20	20	NUM
ejpam-5360	170	19	we	we	PRON
ejpam-5360	170	20	have	have	AUX
ejpam-5360	170	21	k∑	k∑	VERB
ejpam-5360	171	1	l=1	l=1	X
ejpam-5360	171	2	(	(	PUNCT
ejpam-5360	171	3	σ	σ	PROPN
ejpam-5360	171	4	u∼v∈sl	u∼v∈sl	NUM
ejpam-5360	171	5	d(u	d(u	PROPN
ejpam-5360	171	6	)	)	PUNCT
ejpam-5360	171	7	)	)	PUNCT
ejpam-5360	172	1	=	=	PUNCT
ejpam-5360	172	2	k∑	k∑	PROPN
ejpam-5360	173	1	l=1	l=1	X
ejpam-5360	173	2	{	{	PUNCT
ejpam-5360	173	3	ϕ	ϕ	PROPN
ejpam-5360	173	4	k∑	k∑	PROPN
ejpam-5360	173	5	i=1	i=1	PROPN
ejpam-5360	174	1	(	(	PUNCT
ejpam-5360	174	2	ηiαi	ηiαi	NOUN
ejpam-5360	174	3	+	+	CCONJ
ejpam-5360	175	1	(	(	PUNCT
ejpam-5360	175	2	ϕ−	ϕ−	PROPN
ejpam-5360	175	3	1	1	NUM
ejpam-5360	175	4	)	)	PUNCT
ejpam-5360	175	5	)	)	PUNCT
ejpam-5360	176	1	+	+	CCONJ
ejpam-5360	176	2	(	(	PUNCT
ejpam-5360	176	3	αl	αl	ADP
ejpam-5360	176	4	−	−	PROPN
ejpam-5360	176	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	176	6	αl	αl	ADP
ejpam-5360	176	7	−	−	PROPN
ejpam-5360	176	8	1	1	NUM
ejpam-5360	176	9	)	)	PUNCT
ejpam-5360	176	10	}	}	PUNCT
ejpam-5360	176	11	≤	≤	NOUN
ejpam-5360	176	12	kρ1(g)2	kρ1(g)2	PROPN
ejpam-5360	176	13	.	.	PUNCT
ejpam-5360	177	1	also	also	ADV
ejpam-5360	177	2	,	,	PUNCT
ejpam-5360	177	3	we	we	PRON
ejpam-5360	177	4	have	have	AUX
ejpam-5360	177	5	k∑	k∑	VERB
ejpam-5360	177	6	l=1	l=1	X
ejpam-5360	177	7	(	(	PUNCT
ejpam-5360	177	8	σ	σ	PROPN
ejpam-5360	177	9	u∼v∈sl	u∼v∈sl	NUM
ejpam-5360	177	10	d(u	d(u	PROPN
ejpam-5360	177	11	)	)	PUNCT
ejpam-5360	177	12	)	)	PUNCT
ejpam-5360	178	1	=	=	PUNCT
ejpam-5360	178	2	k∑	k∑	PROPN
ejpam-5360	179	1	l=1	l=1	X
ejpam-5360	179	2	{	{	PUNCT
ejpam-5360	179	3	ϕ	ϕ	PROPN
ejpam-5360	179	4	k∑	k∑	PROPN
ejpam-5360	179	5	i=1	i=1	PROPN
ejpam-5360	180	1	(	(	PUNCT
ejpam-5360	180	2	ηiαi	ηiαi	NOUN
ejpam-5360	180	3	+	+	CCONJ
ejpam-5360	181	1	(	(	PUNCT
ejpam-5360	181	2	ϕ−	ϕ−	PROPN
ejpam-5360	181	3	1	1	NUM
ejpam-5360	181	4	)	)	PUNCT
ejpam-5360	181	5	)	)	PUNCT
ejpam-5360	182	1	+	+	CCONJ
ejpam-5360	182	2	(	(	PUNCT
ejpam-5360	182	3	αl	αl	ADP
ejpam-5360	182	4	−	−	PROPN
ejpam-5360	182	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	182	6	αl	αl	ADP
ejpam-5360	182	7	−	−	PROPN
ejpam-5360	182	8	1	1	NUM
ejpam-5360	182	9	)	)	PUNCT
ejpam-5360	182	10	}	}	PUNCT
ejpam-5360	182	11	<	<	X
ejpam-5360	182	12	2	2	NUM
ejpam-5360	182	13	m.	m.	NOUN
ejpam-5360	182	14	we	we	PRON
ejpam-5360	182	15	now	now	ADV
ejpam-5360	182	16	show	show	VERB
ejpam-5360	182	17	that	that	SCONJ
ejpam-5360	182	18	k(ρ1(g))2	k(ρ1(g))2	NOUN
ejpam-5360	182	19	>	>	X
ejpam-5360	182	20	2	2	NUM
ejpam-5360	182	21	m	m	NOUN
ejpam-5360	182	22	,	,	PUNCT
ejpam-5360	182	23	for	for	ADP
ejpam-5360	182	24	when	when	SCONJ
ejpam-5360	182	25	ϕ	ϕ	X
ejpam-5360	182	26	≥	≥	NUM
ejpam-5360	182	27	1	1	NUM
ejpam-5360	182	28	and	and	CCONJ
ejpam-5360	182	29	k	k	PROPN
ejpam-5360	182	30	≥	≥	NUM
ejpam-5360	182	31	2	2	NUM
ejpam-5360	182	32	.	.	X
ejpam-5360	183	1	we	we	PRON
ejpam-5360	183	2	have	have	VERB
ejpam-5360	183	3	2	2	NUM
ejpam-5360	183	4	m	m	NOUN
ejpam-5360	183	5	=	=	SYM
ejpam-5360	183	6	2	2	NUM
ejpam-5360	183	7	{	{	PUNCT
ejpam-5360	183	8	k∑	k∑	NOUN
ejpam-5360	183	9	i=1	i=1	PROPN
ejpam-5360	184	1	(	(	PUNCT
ejpam-5360	184	2	ηiαi(αi	ηiαi(αi	VERB
ejpam-5360	184	3	−	−	PROPN
ejpam-5360	184	4	1	1	NUM
ejpam-5360	184	5	)	)	SYM
ejpam-5360	184	6	2	2	NUM
ejpam-5360	184	7	+	+	NUM
ejpam-5360	184	8	ϕηiαi	ϕηiαi	NOUN
ejpam-5360	184	9	)	)	PUNCT
ejpam-5360	185	1	+	+	CCONJ
ejpam-5360	186	1	ϕ(ϕ−	ϕ(ϕ−	PROPN
ejpam-5360	186	2	1	1	NUM
ejpam-5360	186	3	)	)	PUNCT
ejpam-5360	186	4	2	2	NUM
ejpam-5360	186	5	}	}	PUNCT
ejpam-5360	186	6	=	=	VERB
ejpam-5360	186	7	k∑	k∑	PROPN
ejpam-5360	187	1	i=1	i=1	PROPN
ejpam-5360	188	1	ηiα	ηiα	PROPN
ejpam-5360	189	1	2	2	NUM
ejpam-5360	190	1	i	i	PRON
ejpam-5360	190	2	+	+	CCONJ
ejpam-5360	190	3	(	(	PUNCT
ejpam-5360	190	4	2ϕ−	2ϕ−	NUM
ejpam-5360	190	5	1	1	NUM
ejpam-5360	190	6	)	)	PUNCT
ejpam-5360	190	7	k∑	k∑	VERB
ejpam-5360	191	1	i=1	i=1	PROPN
ejpam-5360	192	1	ηiαi	ηiαi	NOUN
ejpam-5360	193	1	+	+	CCONJ
ejpam-5360	193	2	ϕ(ϕ−	ϕ(ϕ−	PROPN
ejpam-5360	193	3	1	1	NUM
ejpam-5360	193	4	)	)	PUNCT
ejpam-5360	193	5	.	.	PUNCT
ejpam-5360	194	1	consider	consider	VERB
ejpam-5360	194	2	(	(	PUNCT
ejpam-5360	194	3	k(ρ1(g))2	k(ρ1(g))2	NOUN
ejpam-5360	194	4	−	−	PROPN
ejpam-5360	194	5	2	2	NUM
ejpam-5360	194	6	m	m	NOUN
ejpam-5360	194	7	)	)	PUNCT
ejpam-5360	194	8	≥	≥	PRON
ejpam-5360	194	9	k∑	k∑	NOUN
ejpam-5360	195	1	l=1	l=1	X
ejpam-5360	195	2	{	{	PUNCT
ejpam-5360	195	3	ϕ	ϕ	PROPN
ejpam-5360	195	4	k∑	k∑	PROPN
ejpam-5360	195	5	i=1	i=1	PROPN
ejpam-5360	196	1	(	(	PUNCT
ejpam-5360	196	2	ηiαi	ηiαi	NOUN
ejpam-5360	196	3	+	+	CCONJ
ejpam-5360	197	1	(	(	PUNCT
ejpam-5360	197	2	ϕ−	ϕ−	PROPN
ejpam-5360	197	3	1	1	NUM
ejpam-5360	197	4	)	)	PUNCT
ejpam-5360	197	5	)	)	PUNCT
ejpam-5360	198	1	+	+	CCONJ
ejpam-5360	198	2	(	(	PUNCT
ejpam-5360	198	3	αl	αl	ADP
ejpam-5360	198	4	−	−	PROPN
ejpam-5360	198	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	198	6	αl	αl	ADP
ejpam-5360	198	7	−	−	PROPN
ejpam-5360	198	8	1	1	NUM
ejpam-5360	198	9	)	)	PUNCT
ejpam-5360	198	10	}	}	PUNCT
ejpam-5360	198	11	−	−	PROPN
ejpam-5360	198	12	{	{	PUNCT
ejpam-5360	198	13	k∑	k∑	PROPN
ejpam-5360	198	14	i=1	i=1	PROPN
ejpam-5360	199	1	ηiα	ηiα	PROPN
ejpam-5360	199	2	2	2	NUM
ejpam-5360	200	1	i	i	PRON
ejpam-5360	200	2	+	+	CCONJ
ejpam-5360	200	3	(	(	PUNCT
ejpam-5360	200	4	2ϕ−	2ϕ−	NUM
ejpam-5360	200	5	1	1	NUM
ejpam-5360	200	6	)	)	PUNCT
ejpam-5360	200	7	k∑	k∑	VERB
ejpam-5360	201	1	i=1	i=1	PROPN
ejpam-5360	202	1	ηiαi	ηiαi	NOUN
ejpam-5360	203	1	+	+	CCONJ
ejpam-5360	203	2	ϕ(ϕ−	ϕ(ϕ−	PROPN
ejpam-5360	203	3	1	1	NUM
ejpam-5360	203	4	)	)	PUNCT
ejpam-5360	203	5	}	}	PUNCT
ejpam-5360	203	6	=	=	SYM
ejpam-5360	203	7	(	(	PUNCT
ejpam-5360	203	8	(	(	PUNCT
ejpam-5360	203	9	k	k	NOUN
ejpam-5360	203	10	−	−	PROPN
ejpam-5360	203	11	2)ϕ+	2)ϕ+	NUM
ejpam-5360	203	12	1	1	NUM
ejpam-5360	203	13	)	)	PUNCT
ejpam-5360	203	14	k∑	k∑	VERB
ejpam-5360	204	1	i=1	i=1	PROPN
ejpam-5360	204	2	ηiαi	ηiαi	NOUN
ejpam-5360	205	1	−	−	PROPN
ejpam-5360	205	2	k∑	k∑	PROPN
ejpam-5360	205	3	i=1	i=1	PROPN
ejpam-5360	206	1	ηiα	ηiα	PROPN
ejpam-5360	206	2	2	2	NUM
ejpam-5360	207	1	i	i	PRON
ejpam-5360	207	2	+	+	CCONJ
ejpam-5360	207	3	(	(	PUNCT
ejpam-5360	207	4	k2	k2	ADJ
ejpam-5360	207	5	−	−	PROPN
ejpam-5360	207	6	1)ϕ(ϕ−	1)ϕ(ϕ−	NUM
ejpam-5360	207	7	1	1	NUM
ejpam-5360	207	8	)	)	PUNCT
ejpam-5360	207	9	+	+	CCONJ
ejpam-5360	207	10	k∑	k∑	ADJ
ejpam-5360	208	1	l=1	l=1	X
ejpam-5360	208	2	(	(	PUNCT
ejpam-5360	208	3	αl	αl	ADP
ejpam-5360	208	4	−	−	PROPN
ejpam-5360	208	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	208	6	αl	αl	ADP
ejpam-5360	208	7	−	−	PROPN
ejpam-5360	208	8	1	1	NUM
ejpam-5360	208	9	)	)	PUNCT
ejpam-5360	208	10	>	>	X
ejpam-5360	208	11	0	0	X
ejpam-5360	208	12	.	.	PUNCT
ejpam-5360	209	1	therefore	therefore	ADV
ejpam-5360	209	2	2	2	NUM
ejpam-5360	209	3	m	m	NOUN
ejpam-5360	209	4	<	<	X
ejpam-5360	209	5	k(ρ1(g))2	k(ρ1(g))2	NOUN
ejpam-5360	209	6	.	.	PUNCT
ejpam-5360	210	1	this	this	PRON
ejpam-5360	210	2	implies	imply	VERB
ejpam-5360	210	3	√	√	NUM
ejpam-5360	210	4	2	2	NUM
ejpam-5360	210	5	m	m	NOUN
ejpam-5360	210	6	k	k	NOUN
ejpam-5360	210	7	<	<	X
ejpam-5360	210	8	ρ1(g	ρ1(g	PROPN
ejpam-5360	210	9	)	)	PUNCT
ejpam-5360	210	10	(	(	PUNCT
ejpam-5360	210	11	12	12	NUM
ejpam-5360	210	12	)	)	PUNCT
ejpam-5360	210	13	considering	consider	VERB
ejpam-5360	210	14	the	the	DET
ejpam-5360	210	15	upper	upper	ADJ
ejpam-5360	210	16	bound	bind	VERB
ejpam-5360	210	17	of	of	ADP
ejpam-5360	210	18	equation	equation	NOUN
ejpam-5360	210	19	(	(	PUNCT
ejpam-5360	210	20	2	2	NUM
ejpam-5360	210	21	)	)	PUNCT
ejpam-5360	210	22	,	,	PUNCT
ejpam-5360	210	23	we	we	PRON
ejpam-5360	210	24	have	have	VERB
ejpam-5360	210	25	(	(	PUNCT
ejpam-5360	210	26	σ	σ	PROPN
ejpam-5360	210	27	u∼v∈kϕ	u∼v∈kϕ	PROPN
ejpam-5360	210	28	d(u	d(u	PROPN
ejpam-5360	210	29	)	)	PUNCT
ejpam-5360	210	30	)	)	PUNCT
ejpam-5360	211	1	1/2	1/2	NUM
ejpam-5360	211	2	=	=	SYM
ejpam-5360	211	3	{	{	PUNCT
ejpam-5360	211	4	ϕ	ϕ	PROPN
ejpam-5360	211	5	k∑	k∑	PROPN
ejpam-5360	211	6	i=1	i=1	PROPN
ejpam-5360	211	7	(	(	PUNCT
ejpam-5360	211	8	ηiαi	ηiαi	NOUN
ejpam-5360	211	9	(	(	PUNCT
ejpam-5360	211	10	ϕ+	ϕ+	NOUN
ejpam-5360	211	11	αi	αi	ADV
ejpam-5360	211	12	−	−	NOUN
ejpam-5360	211	13	1	1	NUM
ejpam-5360	211	14	)	)	PUNCT
ejpam-5360	211	15	)	)	PUNCT
ejpam-5360	211	16	}	}	PUNCT
ejpam-5360	211	17	1/2	1/2	NUM
ejpam-5360	211	18	≥	≥	NOUN
ejpam-5360	211	19	ρ1(g	ρ1(g	NOUN
ejpam-5360	211	20	)	)	PUNCT
ejpam-5360	211	21	.	.	PUNCT
ejpam-5360	212	1	m.	m.	NOUN
ejpam-5360	212	2	v.	v.	ADP
ejpam-5360	212	3	,	,	PUNCT
ejpam-5360	212	4	k.	k.	PROPN
ejpam-5360	212	5	desikan	desikan	PROPN
ejpam-5360	212	6	/	/	SYM
ejpam-5360	212	7	eur	eur	PROPN
ejpam-5360	212	8	.	.	PUNCT
ejpam-5360	213	1	j.	j.	PROPN
ejpam-5360	213	2	pure	pure	PROPN
ejpam-5360	213	3	appl	appl	PROPN
ejpam-5360	213	4	.	.	PROPN
ejpam-5360	213	5	math	math	PROPN
ejpam-5360	213	6	,	,	PUNCT
ejpam-5360	213	7	18	18	NUM
ejpam-5360	213	8	(	(	PUNCT
ejpam-5360	213	9	4	4	NUM
ejpam-5360	213	10	)	)	PUNCT
ejpam-5360	213	11	(	(	PUNCT
ejpam-5360	213	12	2025	2025	NUM
ejpam-5360	213	13	)	)	PUNCT
ejpam-5360	213	14	,	,	PUNCT
ejpam-5360	213	15	5360	5360	NUM
ejpam-5360	213	16	10	10	NUM
ejpam-5360	213	17	of	of	ADP
ejpam-5360	213	18	20	20	NUM
ejpam-5360	213	19	from	from	ADP
ejpam-5360	213	20	the	the	DET
ejpam-5360	213	21	above	above	ADJ
ejpam-5360	213	22	inequality	inequality	NOUN
ejpam-5360	213	23	,	,	PUNCT
ejpam-5360	213	24	we	we	PRON
ejpam-5360	213	25	get	get	VERB
ejpam-5360	213	26	2	2	NUM
ejpam-5360	213	27	m	m	NOUN
ejpam-5360	213	28	>	>	X
ejpam-5360	213	29	{	{	PUNCT
ejpam-5360	213	30	ϕ	ϕ	PROPN
ejpam-5360	213	31	k∑	k∑	PROPN
ejpam-5360	213	32	i=1	i=1	PROPN
ejpam-5360	214	1	(	(	PUNCT
ejpam-5360	214	2	ηiαi	ηiαi	NOUN
ejpam-5360	214	3	(	(	PUNCT
ejpam-5360	214	4	ϕ+	ϕ+	NOUN
ejpam-5360	214	5	αi	αi	ADV
ejpam-5360	214	6	−	−	NOUN
ejpam-5360	214	7	1	1	NUM
ejpam-5360	214	8	)	)	PUNCT
ejpam-5360	214	9	)	)	PUNCT
ejpam-5360	214	10	}	}	PUNCT
ejpam-5360	214	11	≥	≥	X
ejpam-5360	214	12	(	(	PUNCT
ejpam-5360	214	13	ρ1(g))2	ρ1(g))2	NOUN
ejpam-5360	214	14	.	.	PUNCT
ejpam-5360	215	1	therefore	therefore	ADV
ejpam-5360	215	2	√	√	ADV
ejpam-5360	215	3	2	2	NUM
ejpam-5360	215	4	m	m	NOUN
ejpam-5360	215	5	>	>	X
ejpam-5360	215	6	ρ1(g	ρ1(g	PROPN
ejpam-5360	215	7	)	)	PUNCT
ejpam-5360	215	8	.	.	PUNCT
ejpam-5360	216	1	hence	hence	ADV
ejpam-5360	216	2	√	√	NOUN
ejpam-5360	216	3	2	2	NUM
ejpam-5360	216	4	m	m	NOUN
ejpam-5360	216	5	k	k	X
ejpam-5360	216	6	<	<	X
ejpam-5360	216	7	ρ1(g	ρ1(g	PROPN
ejpam-5360	216	8	)	)	PUNCT
ejpam-5360	216	9	<	<	X
ejpam-5360	216	10	√	√	PROPN
ejpam-5360	216	11	2	2	NUM
ejpam-5360	216	12	m.	m.	NOUN
ejpam-5360	216	13	(	(	PUNCT
ejpam-5360	216	14	13	13	NUM
ejpam-5360	216	15	)	)	PUNCT
ejpam-5360	216	16	hence	hence	ADV
ejpam-5360	216	17	proved	prove	VERB
ejpam-5360	216	18	.	.	PUNCT
ejpam-5360	217	1	in	in	ADP
ejpam-5360	217	2	the	the	DET
ejpam-5360	217	3	following	following	NOUN
ejpam-5360	217	4	theorem	theorem	NOUN
ejpam-5360	217	5	,	,	PUNCT
ejpam-5360	217	6	we	we	PRON
ejpam-5360	217	7	make	make	VERB
ejpam-5360	217	8	use	use	NOUN
ejpam-5360	217	9	of	of	ADP
ejpam-5360	217	10	theorem	theorem	NOUN
ejpam-5360	217	11	4	4	NUM
ejpam-5360	217	12	to	to	PART
ejpam-5360	217	13	obtain	obtain	VERB
ejpam-5360	217	14	the	the	DET
ejpam-5360	217	15	upper	upper	ADJ
ejpam-5360	217	16	bound	bind	VERB
ejpam-5360	217	17	for	for	ADP
ejpam-5360	217	18	the	the	DET
ejpam-5360	217	19	generalized	generalize	VERB
ejpam-5360	217	20	core	core	NOUN
ejpam-5360	217	21	-	-	PUNCT
ejpam-5360	217	22	satellite	satellite	NOUN
ejpam-5360	217	23	graph	graph	NOUN
ejpam-5360	217	24	g.	g.	NOUN
ejpam-5360	217	25	in	in	ADP
ejpam-5360	217	26	order	order	NOUN
ejpam-5360	217	27	to	to	PART
ejpam-5360	217	28	make	make	VERB
ejpam-5360	217	29	use	use	NOUN
ejpam-5360	217	30	of	of	ADP
ejpam-5360	217	31	theorem	theorem	NOUN
ejpam-5360	217	32	4	4	NUM
ejpam-5360	217	33	we	we	PRON
ejpam-5360	217	34	require	require	VERB
ejpam-5360	217	35	the	the	DET
ejpam-5360	217	36	row	row	NOUN
ejpam-5360	217	37	sums	sum	NOUN
ejpam-5360	217	38	of	of	ADP
ejpam-5360	217	39	the	the	DET
ejpam-5360	217	40	adjacency	adjacency	NOUN
ejpam-5360	217	41	matrix	matrix	NOUN
ejpam-5360	217	42	.	.	PUNCT
ejpam-5360	218	1	in	in	ADP
ejpam-5360	218	2	the	the	DET
ejpam-5360	218	3	adjacency	adjacency	NOUN
ejpam-5360	218	4	matrix	matrix	NOUN
ejpam-5360	218	5	a(g	a(g	PROPN
ejpam-5360	218	6	)	)	PUNCT
ejpam-5360	218	7	of	of	ADP
ejpam-5360	218	8	graph	graph	NOUN
ejpam-5360	218	9	g	g	PROPN
ejpam-5360	218	10	,	,	PUNCT
ejpam-5360	218	11	the	the	DET
ejpam-5360	218	12	vertices	vertex	NOUN
ejpam-5360	218	13	are	be	AUX
ejpam-5360	218	14	arranged	arrange	VERB
ejpam-5360	218	15	such	such	ADJ
ejpam-5360	218	16	that	that	SCONJ
ejpam-5360	218	17	the	the	DET
ejpam-5360	218	18	top	top	ADJ
ejpam-5360	218	19	ϕ	ϕ	PROPN
ejpam-5360	218	20	rows	row	NOUN
ejpam-5360	218	21	correspond	correspond	VERB
ejpam-5360	218	22	to	to	ADP
ejpam-5360	218	23	the	the	DET
ejpam-5360	218	24	vertices	vertex	NOUN
ejpam-5360	218	25	in	in	ADP
ejpam-5360	218	26	the	the	DET
ejpam-5360	218	27	core	core	NOUN
ejpam-5360	218	28	kϕ	kϕ	PROPN
ejpam-5360	218	29	,	,	PUNCT
ejpam-5360	218	30	followed	follow	VERB
ejpam-5360	218	31	by	by	ADP
ejpam-5360	218	32	the	the	DET
ejpam-5360	218	33	vertices	vertex	NOUN
ejpam-5360	218	34	of	of	ADP
ejpam-5360	218	35	ηk	ηk	ADP
ejpam-5360	218	36	copies	copy	NOUN
ejpam-5360	218	37	of	of	ADP
ejpam-5360	218	38	the	the	DET
ejpam-5360	218	39	cliques	clique	NOUN
ejpam-5360	218	40	kαk	kαk	X
ejpam-5360	218	41	∈	∈	PROPN
ejpam-5360	218	42	sk	sk	NOUN
ejpam-5360	218	43	.	.	PUNCT
ejpam-5360	219	1	the	the	DET
ejpam-5360	219	2	remaining	remain	VERB
ejpam-5360	219	3	rows	row	NOUN
ejpam-5360	219	4	correspond	correspond	VERB
ejpam-5360	219	5	to	to	ADP
ejpam-5360	219	6	vertices	vertex	NOUN
ejpam-5360	219	7	of	of	ADP
ejpam-5360	219	8	sk−1	sk−1	PROPN
ejpam-5360	219	9	,	,	PUNCT
ejpam-5360	219	10	sk−2,	sk−2,	ADV
ejpam-5360	219	11	...	...	PUNCT
ejpam-5360	219	12	,s1	,s1	PUNCT
ejpam-5360	219	13	.	.	PUNCT
ejpam-5360	220	1	let	let	VERB
ejpam-5360	220	2	r∧	r∧	PROPN
ejpam-5360	220	3	1	1	NUM
ejpam-5360	220	4	,	,	PUNCT
ejpam-5360	220	5	r∧	r∧	PROPN
ejpam-5360	220	6	2	2	NUM
ejpam-5360	220	7	,	,	PUNCT
ejpam-5360	220	8	...	...	PUNCT
ejpam-5360	220	9	,	,	PUNCT
ejpam-5360	220	10	r∧	r∧	PROPN
ejpam-5360	220	11	k	k	PROPN
ejpam-5360	220	12	and	and	CCONJ
ejpam-5360	220	13	r∧	r∧	PROPN
ejpam-5360	220	14	ϕ	ϕ	PROPN
ejpam-5360	220	15	be	be	AUX
ejpam-5360	220	16	the	the	DET
ejpam-5360	220	17	row	row	NOUN
ejpam-5360	220	18	sums	sum	NOUN
ejpam-5360	220	19	corresponding	correspond	VERB
ejpam-5360	220	20	to	to	ADP
ejpam-5360	220	21	the	the	DET
ejpam-5360	220	22	vertices	vertex	NOUN
ejpam-5360	220	23	of	of	ADP
ejpam-5360	220	24	the	the	DET
ejpam-5360	220	25	core	core	NOUN
ejpam-5360	220	26	kϕ	kϕ	NOUN
ejpam-5360	220	27	and	and	CCONJ
ejpam-5360	220	28	the	the	DET
ejpam-5360	220	29	satellites	satellite	NOUN
ejpam-5360	220	30	sk	sk	VERB
ejpam-5360	220	31	,	,	PUNCT
ejpam-5360	220	32	sk−1,	sk−1,	NOUN
ejpam-5360	220	33	...	...	PUNCT
ejpam-5360	220	34	,s1	,s1	PUNCT
ejpam-5360	220	35	,	,	PUNCT
ejpam-5360	221	1	i.e.	i.e.	X
ejpam-5360	221	2	r∧	r∧	PROPN
ejpam-5360	221	3	ϕ,1	ϕ,1	PUNCT
ejpam-5360	222	1	=	=	SYM
ejpam-5360	223	1	r∧	r∧	PROPN
ejpam-5360	223	2	ϕ,2	ϕ,2	NOUN
ejpam-5360	223	3	=	=	PUNCT
ejpam-5360	223	4	...	...	PUNCT
ejpam-5360	224	1	=	=	PUNCT
ejpam-5360	224	2	r∧	r∧	ADV
ejpam-5360	224	3	ϕ,ϕ	ϕ,ϕ	INTJ
ejpam-5360	224	4	=	=	SYM
ejpam-5360	224	5	k∑	k∑	PROPN
ejpam-5360	225	1	i=1	i=1	PROPN
ejpam-5360	226	1	ηiαi	ηiαi	NOUN
ejpam-5360	227	1	+	+	CCONJ
ejpam-5360	227	2	(	(	PUNCT
ejpam-5360	227	3	ϕ−	ϕ−	PROPN
ejpam-5360	227	4	1	1	NUM
ejpam-5360	227	5	)	)	PUNCT
ejpam-5360	227	6	=	=	VERB
ejpam-5360	227	7	r∧	r∧	PROPN
ejpam-5360	227	8	1	1	NUM
ejpam-5360	227	9	r∧	r∧	PROPN
ejpam-5360	227	10	k,1	k,1	PROPN
ejpam-5360	227	11	=	=	PROPN
ejpam-5360	227	12	r∧	r∧	PROPN
ejpam-5360	227	13	k,2	k,2	PROPN
ejpam-5360	227	14	=	=	X
ejpam-5360	227	15	...	...	PUNCT
ejpam-5360	228	1	=	=	PUNCT
ejpam-5360	228	2	r∧	r∧	PROPN
ejpam-5360	229	1	k	k	PROPN
ejpam-5360	229	2	,	,	PUNCT
ejpam-5360	229	3	ηkαk	ηkαk	X
ejpam-5360	229	4	=	=	PUNCT
ejpam-5360	229	5	(	(	PUNCT
ejpam-5360	230	1	ϕ+	ϕ+	INTJ
ejpam-5360	230	2	αk	αk	INTJ
ejpam-5360	230	3	−	−	PROPN
ejpam-5360	230	4	1	1	NUM
ejpam-5360	230	5	)	)	PUNCT
ejpam-5360	230	6	=	=	VERB
ejpam-5360	231	1	r∧	r∧	PROPN
ejpam-5360	231	2	2	2	NUM
ejpam-5360	231	3	r∧	r∧	NOUN
ejpam-5360	231	4	k−1,1	k−1,1	NOUN
ejpam-5360	231	5	=	=	X
ejpam-5360	231	6	r∧	r∧	PROPN
ejpam-5360	231	7	k−1,2	k−1,2	PROPN
ejpam-5360	231	8	=	=	PUNCT
ejpam-5360	231	9	...	...	PUNCT
ejpam-5360	232	1	=	=	SYM
ejpam-5360	232	2	r∧	r∧	PROPN
ejpam-5360	232	3	k−1,ηk−1αk−1	k−1,ηk−1αk−1	PROPN
ejpam-5360	233	1	=	=	SYM
ejpam-5360	234	1	(	(	PUNCT
ejpam-5360	234	2	ϕ+	ϕ+	INTJ
ejpam-5360	234	3	αk−1	αk−1	NOUN
ejpam-5360	234	4	−	−	NOUN
ejpam-5360	234	5	1	1	NUM
ejpam-5360	234	6	)	)	PUNCT
ejpam-5360	234	7	=	=	VERB
ejpam-5360	235	1	r∧	r∧	PROPN
ejpam-5360	235	2	3	3	NUM
ejpam-5360	235	3	r∧	r∧	NOUN
ejpam-5360	235	4	k−2,1	k−2,1	PROPN
ejpam-5360	235	5	=	=	SYM
ejpam-5360	236	1	r∧	r∧	PROPN
ejpam-5360	236	2	k−2,2	k−2,2	PROPN
ejpam-5360	236	3	=	=	PUNCT
ejpam-5360	236	4	...	...	PUNCT
ejpam-5360	237	1	=	=	PUNCT
ejpam-5360	237	2	r∧	r∧	PROPN
ejpam-5360	237	3	k−2,ηk−2αk−2	k−2,ηk−2αk−2	PROPN
ejpam-5360	237	4	=	=	PUNCT
ejpam-5360	238	1	(	(	PUNCT
ejpam-5360	238	2	ϕ+	ϕ+	INTJ
ejpam-5360	238	3	αk−2	αk−2	NOUN
ejpam-5360	238	4	−	−	NOUN
ejpam-5360	238	5	1	1	NUM
ejpam-5360	238	6	)	)	PUNCT
ejpam-5360	238	7	=	=	VERB
ejpam-5360	238	8	r∧	r∧	VERB
ejpam-5360	238	9	4	4	NUM
ejpam-5360	238	10	...	...	PUNCT
ejpam-5360	238	11	...	...	PUNCT
ejpam-5360	238	12	in	in	ADP
ejpam-5360	238	13	general	general	ADJ
ejpam-5360	238	14	,	,	PUNCT
ejpam-5360	238	15	we	we	PRON
ejpam-5360	238	16	have	have	VERB
ejpam-5360	238	17	rk−(j−2),1	rk−(j−2),1	NOUN
ejpam-5360	238	18	=	=	SYM
ejpam-5360	238	19	rk−(j−2),2	rk−(j−2),2	PROPN
ejpam-5360	238	20	=	=	PUNCT
ejpam-5360	238	21	...	...	PUNCT
ejpam-5360	239	1	=	=	PUNCT
ejpam-5360	239	2	rk−(j−2),ηk−(j−2)αk−(j−2	rk−(j−2),ηk−(j−2)αk−(j−2	PROPN
ejpam-5360	239	3	)	)	PUNCT
ejpam-5360	239	4	=	=	PUNCT
ejpam-5360	239	5	(	(	PUNCT
ejpam-5360	239	6	ϕ+	ϕ+	X
ejpam-5360	239	7	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	239	8	)	)	PUNCT
ejpam-5360	239	9	−	−	ADP
ejpam-5360	239	10	1	1	NUM
ejpam-5360	239	11	)	)	PUNCT
ejpam-5360	239	12	=	=	PUNCT
ejpam-5360	240	1	r∧	r∧	PROPN
ejpam-5360	240	2	j	j	PROPN
ejpam-5360	240	3	...	...	PUNCT
ejpam-5360	241	1	r∧	r∧	PROPN
ejpam-5360	242	1	1,1	1,1	NUM
ejpam-5360	242	2	=	=	SYM
ejpam-5360	242	3	r∧	r∧	PROPN
ejpam-5360	242	4	1,2	1,2	NUM
ejpam-5360	242	5	=	=	SYM
ejpam-5360	242	6	...	...	PUNCT
ejpam-5360	243	1	=	=	PUNCT
ejpam-5360	243	2	r∧	r∧	ADJ
ejpam-5360	243	3	1,η1α1	1,η1α1	NUM
ejpam-5360	243	4	=	=	SYM
ejpam-5360	243	5	(	(	PUNCT
ejpam-5360	243	6	ϕ+	ϕ+	INTJ
ejpam-5360	243	7	α1	α1	PROPN
ejpam-5360	243	8	−	−	PROPN
ejpam-5360	243	9	1	1	NUM
ejpam-5360	243	10	)	)	PUNCT
ejpam-5360	243	11	=	=	PUNCT
ejpam-5360	243	12	r∧	r∧	PROPN
ejpam-5360	243	13	k+1	k+1	PROPN
ejpam-5360	243	14	.	.	PUNCT
ejpam-5360	243	15	theorem	theorem	PROPN
ejpam-5360	243	16	9	9	NUM
ejpam-5360	243	17	.	.	X
ejpam-5360	243	18	for	for	ADP
ejpam-5360	243	19	the	the	DET
ejpam-5360	243	20	generalized	generalize	VERB
ejpam-5360	243	21	core	core	NOUN
ejpam-5360	243	22	-	-	PUNCT
ejpam-5360	243	23	satellite	satellite	NOUN
ejpam-5360	243	24	graph	graph	NOUN
ejpam-5360	243	25	g	g	NOUN
ejpam-5360	243	26	of	of	ADP
ejpam-5360	243	27	order	order	NOUN
ejpam-5360	243	28	n	n	NOUN
ejpam-5360	243	29	and	and	CCONJ
ejpam-5360	243	30	size	size	NOUN
ejpam-5360	243	31	m	m	PROPN
ejpam-5360	243	32	,	,	PUNCT
ejpam-5360	243	33	let	let	VERB
ejpam-5360	243	34	s1	s1	NOUN
ejpam-5360	243	35	,	,	PUNCT
ejpam-5360	243	36	s2	s2	PROPN
ejpam-5360	243	37	,	,	PUNCT
ejpam-5360	243	38	s3	s3	PROPN
ejpam-5360	243	39	,	,	PUNCT
ejpam-5360	243	40	..	..	PUNCT
ejpam-5360	243	41	,	,	PUNCT
ejpam-5360	243	42	sk	sk	NOUN
ejpam-5360	243	43	be	be	AUX
ejpam-5360	243	44	the	the	DET
ejpam-5360	243	45	k	k	PROPN
ejpam-5360	243	46	satellites	satellite	NOUN
ejpam-5360	243	47	connected	connect	VERB
ejpam-5360	243	48	to	to	ADP
ejpam-5360	243	49	the	the	DET
ejpam-5360	243	50	core	core	NOUN
ejpam-5360	243	51	graph	graph	NOUN
ejpam-5360	243	52	kϕ.	kϕ.	NOUN
ejpam-5360	243	53	let	let	VERB
ejpam-5360	243	54	r∧	r∧	PROPN
ejpam-5360	243	55	1	1	NUM
ejpam-5360	243	56	,	,	PUNCT
ejpam-5360	243	57	r	r	NOUN
ejpam-5360	243	58	∧	∧	PROPN
ejpam-5360	243	59	2	2	NUM
ejpam-5360	243	60	,	,	PUNCT
ejpam-5360	243	61	...	...	PUNCT
ejpam-5360	243	62	,	,	PUNCT
ejpam-5360	243	63	r	r	NOUN
ejpam-5360	243	64	∧	∧	PROPN
ejpam-5360	243	65	k+1	k+1	X
ejpam-5360	243	66	m.	m.	NOUN
ejpam-5360	243	67	v.	v.	ADP
ejpam-5360	243	68	,	,	PUNCT
ejpam-5360	243	69	k.	k.	PROPN
ejpam-5360	243	70	desikan	desikan	PROPN
ejpam-5360	243	71	/	/	SYM
ejpam-5360	243	72	eur	eur	PROPN
ejpam-5360	243	73	.	.	PUNCT
ejpam-5360	244	1	j.	j.	PROPN
ejpam-5360	244	2	pure	pure	PROPN
ejpam-5360	244	3	appl	appl	PROPN
ejpam-5360	244	4	.	.	PROPN
ejpam-5360	244	5	math	math	PROPN
ejpam-5360	244	6	,	,	PUNCT
ejpam-5360	244	7	18	18	NUM
ejpam-5360	244	8	(	(	PUNCT
ejpam-5360	244	9	4	4	NUM
ejpam-5360	244	10	)	)	PUNCT
ejpam-5360	244	11	(	(	PUNCT
ejpam-5360	244	12	2025	2025	NUM
ejpam-5360	244	13	)	)	PUNCT
ejpam-5360	244	14	,	,	PUNCT
ejpam-5360	244	15	5360	5360	NUM
ejpam-5360	244	16	11	11	NUM
ejpam-5360	244	17	of	of	ADP
ejpam-5360	244	18	20	20	NUM
ejpam-5360	244	19	be	be	AUX
ejpam-5360	244	20	the	the	DET
ejpam-5360	244	21	row	row	NOUN
ejpam-5360	244	22	sums	sum	NOUN
ejpam-5360	244	23	corresponding	correspond	VERB
ejpam-5360	244	24	to	to	ADP
ejpam-5360	244	25	the	the	DET
ejpam-5360	244	26	core	core	NOUN
ejpam-5360	244	27	kϕ	kϕ	NOUN
ejpam-5360	244	28	and	and	CCONJ
ejpam-5360	244	29	the	the	DET
ejpam-5360	244	30	satellites	satellite	NOUN
ejpam-5360	244	31	sk	sk	VERB
ejpam-5360	244	32	,	,	PUNCT
ejpam-5360	244	33	sk−1,	sk−1,	NOUN
ejpam-5360	244	34	...	...	PUNCT
ejpam-5360	244	35	,s1	,s1	PUNCT
ejpam-5360	244	36	.	.	PUNCT
ejpam-5360	245	1	the	the	DET
ejpam-5360	245	2	row	row	NOUN
ejpam-5360	245	3	sums	sum	NOUN
ejpam-5360	245	4	are	be	AUX
ejpam-5360	245	5	such	such	ADJ
ejpam-5360	245	6	that	that	SCONJ
ejpam-5360	245	7	r∧	r∧	PROPN
ejpam-5360	245	8	1	1	NUM
ejpam-5360	245	9	≥	≥	NOUN
ejpam-5360	245	10	r∧	r∧	PROPN
ejpam-5360	245	11	2	2	NUM
ejpam-5360	245	12	≥	≥	NOUN
ejpam-5360	245	13	...	...	PUNCT
ejpam-5360	245	14	≥	≥	X
ejpam-5360	245	15	r∧	r∧	PROPN
ejpam-5360	245	16	k+1	k+1	X
ejpam-5360	245	17	,	,	PUNCT
ejpam-5360	245	18	then	then	ADV
ejpam-5360	245	19	ρ1(g	ρ1(g	PROPN
ejpam-5360	245	20	)	)	PUNCT
ejpam-5360	245	21	≤	≤	NOUN
ejpam-5360	245	22	(	(	PUNCT
ejpam-5360	245	23	ϕ+	ϕ+	NOUN
ejpam-5360	245	24	αl	αl	ADP
ejpam-5360	245	25	−	−	PROPN
ejpam-5360	245	26	2	2	NUM
ejpam-5360	245	27	)	)	PUNCT
ejpam-5360	245	28	2	2	NUM
ejpam-5360	245	29	+	+	CCONJ
ejpam-5360	245	30	√	√	PROPN
ejpam-5360	245	31	(	(	PUNCT
ejpam-5360	245	32	ϕ+	ϕ+	NOUN
ejpam-5360	245	33	αl)2	αl)2	PROPN
ejpam-5360	245	34	+	+	CCONJ
ejpam-5360	245	35	4	4	NUM
ejpam-5360	245	36	∑l−1	∑l−1	NOUN
ejpam-5360	245	37	i=1	i=1	X
ejpam-5360	246	1	(	(	PUNCT
ejpam-5360	246	2	r	r	NOUN
ejpam-5360	246	3	∧	∧	PROPN
ejpam-5360	246	4	i	i	NOUN
ejpam-5360	246	5	−r∧	−r∧	NUM
ejpam-5360	246	6	l	l	NOUN
ejpam-5360	246	7	)	)	PUNCT
ejpam-5360	246	8	2	2	NUM
ejpam-5360	246	9	where	where	SCONJ
ejpam-5360	246	10	1	1	NUM
ejpam-5360	246	11	≤	≤	NUM
ejpam-5360	246	12	i	i	NOUN
ejpam-5360	246	13	≤	≤	NOUN
ejpam-5360	247	1	k	k	NOUN
ejpam-5360	247	2	and	and	CCONJ
ejpam-5360	247	3	1	1	NUM
ejpam-5360	247	4	≤	≤	NUM
ejpam-5360	247	5	l	l	NOUN
ejpam-5360	247	6	≤	≤	PUNCT
ejpam-5360	248	1	k	k	X
ejpam-5360	248	2	+	+	NOUN
ejpam-5360	248	3	1	1	X
ejpam-5360	248	4	.	.	X
ejpam-5360	248	5	proof	proof	NOUN
ejpam-5360	248	6	.	.	PUNCT
ejpam-5360	249	1	using	use	VERB
ejpam-5360	249	2	theorem	theorem	NOUN
ejpam-5360	249	3	4	4	NUM
ejpam-5360	249	4	we	we	PRON
ejpam-5360	249	5	have	have	VERB
ejpam-5360	249	6	ρ1(g	ρ1(g	NOUN
ejpam-5360	249	7	)	)	PUNCT
ejpam-5360	249	8	≤	≤	NOUN
ejpam-5360	249	9	(	(	PUNCT
ejpam-5360	249	10	r∧	r∧	NOUN
ejpam-5360	249	11	l	l	PROPN
ejpam-5360	250	1	+	+	NOUN
ejpam-5360	250	2	m	m	VERB
ejpam-5360	250	3	−n	−n	ADJ
ejpam-5360	250	4	)	)	PUNCT
ejpam-5360	251	1	+	+	CCONJ
ejpam-5360	251	2	√	√	INTJ
ejpam-5360	251	3	(	(	PUNCT
ejpam-5360	251	4	r∧	r∧	PROPN
ejpam-5360	251	5	l	l	PROPN
ejpam-5360	251	6	−m	−m	PROPN
ejpam-5360	251	7	+	+	PROPN
ejpam-5360	251	8	n)2	n)2	NOUN
ejpam-5360	251	9	+	+	CCONJ
ejpam-5360	251	10	4n	4n	ADJ
ejpam-5360	251	11	∑l−1	∑l−1	X
ejpam-5360	251	12	i=1	i=1	X
ejpam-5360	251	13	(	(	PUNCT
ejpam-5360	251	14	r	r	NOUN
ejpam-5360	251	15	∧	∧	PROPN
ejpam-5360	251	16	i	i	NOUN
ejpam-5360	251	17	−r∧	−r∧	NUM
ejpam-5360	251	18	l	l	NOUN
ejpam-5360	251	19	)	)	PUNCT
ejpam-5360	251	20	2	2	NUM
ejpam-5360	251	21	.	.	PUNCT
ejpam-5360	252	1	from	from	ADP
ejpam-5360	252	2	the	the	DET
ejpam-5360	252	3	adjacency	adjacency	NOUN
ejpam-5360	252	4	matrix	matrix	NOUN
ejpam-5360	252	5	of	of	ADP
ejpam-5360	252	6	g	g	PROPN
ejpam-5360	252	7	,	,	PUNCT
ejpam-5360	252	8	we	we	PRON
ejpam-5360	252	9	have	have	VERB
ejpam-5360	252	10	m	m	NOUN
ejpam-5360	252	11	=	=	NOUN
ejpam-5360	252	12	0	0	NUM
ejpam-5360	252	13	and	and	CCONJ
ejpam-5360	252	14	n	n	CCONJ
ejpam-5360	252	15	=	=	SYM
ejpam-5360	252	16	1	1	X
ejpam-5360	252	17	.	.	PUNCT
ejpam-5360	252	18	substituting	substitute	VERB
ejpam-5360	252	19	the	the	DET
ejpam-5360	252	20	value	value	NOUN
ejpam-5360	252	21	for	for	ADP
ejpam-5360	252	22	n	n	NOUN
ejpam-5360	252	23	=	=	SYM
ejpam-5360	252	24	1	1	NUM
ejpam-5360	252	25	,	,	PUNCT
ejpam-5360	252	26	the	the	DET
ejpam-5360	252	27	smallest	small	ADJ
ejpam-5360	252	28	non	non	ADJ
ejpam-5360	252	29	-	-	ADJ
ejpam-5360	252	30	diagonal	diagonal	ADJ
ejpam-5360	252	31	number	number	NOUN
ejpam-5360	252	32	in	in	ADP
ejpam-5360	252	33	the	the	DET
ejpam-5360	252	34	above	above	ADJ
ejpam-5360	252	35	equation	equation	NOUN
ejpam-5360	252	36	,	,	PUNCT
ejpam-5360	252	37	we	we	PRON
ejpam-5360	252	38	get	get	VERB
ejpam-5360	252	39	ρ1(g	ρ1(g	NOUN
ejpam-5360	252	40	)	)	PUNCT
ejpam-5360	252	41	≤	≤	NOUN
ejpam-5360	252	42	(	(	PUNCT
ejpam-5360	252	43	r∧	r∧	NOUN
ejpam-5360	252	44	l	l	NOUN
ejpam-5360	252	45	−	−	PROPN
ejpam-5360	252	46	1	1	NUM
ejpam-5360	252	47	)	)	PUNCT
ejpam-5360	253	1	+	+	CCONJ
ejpam-5360	253	2	√	√	INTJ
ejpam-5360	253	3	(	(	PUNCT
ejpam-5360	253	4	r∧	r∧	NOUN
ejpam-5360	253	5	l	l	PROPN
ejpam-5360	253	6	+	+	CCONJ
ejpam-5360	253	7	1)2	1)2	NUM
ejpam-5360	253	8	+	+	CCONJ
ejpam-5360	253	9	4	4	NUM
ejpam-5360	253	10	∑l−1	∑l−1	NOUN
ejpam-5360	253	11	i=1	i=1	X
ejpam-5360	254	1	(	(	PUNCT
ejpam-5360	254	2	r	r	NOUN
ejpam-5360	254	3	∧	∧	PROPN
ejpam-5360	254	4	i	i	NOUN
ejpam-5360	254	5	−r∧	−r∧	NUM
ejpam-5360	254	6	l	l	NOUN
ejpam-5360	254	7	)	)	PUNCT
ejpam-5360	254	8	2	2	NUM
ejpam-5360	254	9	.	.	PUNCT
ejpam-5360	255	1	(	(	PUNCT
ejpam-5360	255	2	14	14	NUM
ejpam-5360	255	3	)	)	PUNCT
ejpam-5360	255	4	here	here	ADV
ejpam-5360	255	5	we	we	PRON
ejpam-5360	255	6	discuss	discuss	VERB
ejpam-5360	255	7	the	the	DET
ejpam-5360	255	8	cases	case	NOUN
ejpam-5360	255	9	corresponding	correspond	VERB
ejpam-5360	255	10	to	to	ADP
ejpam-5360	255	11	the	the	DET
ejpam-5360	255	12	row	row	NOUN
ejpam-5360	255	13	sums	sum	NOUN
ejpam-5360	255	14	.	.	PUNCT
ejpam-5360	256	1	we	we	PRON
ejpam-5360	256	2	have	have	VERB
ejpam-5360	256	3	three	three	NUM
ejpam-5360	256	4	cases	case	NOUN
ejpam-5360	256	5	depending	depend	VERB
ejpam-5360	256	6	on	on	ADP
ejpam-5360	256	7	the	the	DET
ejpam-5360	256	8	choice	choice	NOUN
ejpam-5360	256	9	of	of	ADP
ejpam-5360	256	10	r∧	r∧	PROPN
ejpam-5360	256	11	l	l	PROPN
ejpam-5360	256	12	,	,	PUNCT
ejpam-5360	256	13	i.e.	i.e.	X
ejpam-5360	256	14	when	when	SCONJ
ejpam-5360	256	15	r∧	r∧	PROPN
ejpam-5360	256	16	l	l	PROPN
ejpam-5360	257	1	=	=	PUNCT
ejpam-5360	257	2	r∧	r∧	ADP
ejpam-5360	257	3	1	1	NUM
ejpam-5360	257	4	,	,	PUNCT
ejpam-5360	257	5	r∧	r∧	PROPN
ejpam-5360	257	6	l	l	PROPN
ejpam-5360	257	7	=	=	PUNCT
ejpam-5360	258	1	r∧	r∧	PROPN
ejpam-5360	258	2	2	2	NUM
ejpam-5360	258	3	and	and	CCONJ
ejpam-5360	258	4	r∧	r∧	PROPN
ejpam-5360	258	5	l	l	PROPN
ejpam-5360	258	6	=	=	PUNCT
ejpam-5360	259	1	r∧	r∧	PROPN
ejpam-5360	259	2	j	j	PROPN
ejpam-5360	259	3	,	,	PUNCT
ejpam-5360	259	4	where	where	SCONJ
ejpam-5360	259	5	3	3	NUM
ejpam-5360	259	6	≤	≤	NUM
ejpam-5360	259	7	j	j	PROPN
ejpam-5360	259	8	≤	≤	PROPN
ejpam-5360	259	9	l.	l.	PROPN
ejpam-5360	259	10	case	case	NOUN
ejpam-5360	259	11	(	(	PUNCT
ejpam-5360	259	12	i	i	NOUN
ejpam-5360	259	13	)	)	PUNCT
ejpam-5360	259	14	when	when	SCONJ
ejpam-5360	259	15	r∧	r∧	PROPN
ejpam-5360	259	16	l	l	PROPN
ejpam-5360	259	17	=	=	PUNCT
ejpam-5360	260	1	r∧	r∧	ADP
ejpam-5360	260	2	1	1	NUM
ejpam-5360	260	3	,	,	PUNCT
ejpam-5360	260	4	where	where	SCONJ
ejpam-5360	260	5	r∧	r∧	PROPN
ejpam-5360	260	6	1	1	NUM
ejpam-5360	260	7	=	=	SYM
ejpam-5360	260	8	∑k	∑k	PROPN
ejpam-5360	260	9	i=1	i=1	PROPN
ejpam-5360	261	1	ηiαi	ηiαi	NOUN
ejpam-5360	262	1	+	+	CCONJ
ejpam-5360	262	2	(	(	PUNCT
ejpam-5360	262	3	ϕ−	ϕ−	PROPN
ejpam-5360	262	4	1	1	NUM
ejpam-5360	262	5	)	)	PUNCT
ejpam-5360	262	6	is	be	AUX
ejpam-5360	262	7	the	the	DET
ejpam-5360	262	8	row	row	NOUN
ejpam-5360	262	9	sum	sum	NOUN
ejpam-5360	262	10	corresponding	correspond	VERB
ejpam-5360	262	11	to	to	ADP
ejpam-5360	262	12	the	the	DET
ejpam-5360	262	13	vertex	vertex	NOUN
ejpam-5360	262	14	v	v	ADP
ejpam-5360	262	15	∈	∈	PROPN
ejpam-5360	262	16	kϕ	kϕ	NOUN
ejpam-5360	262	17	,	,	PUNCT
ejpam-5360	262	18	we	we	PRON
ejpam-5360	262	19	get	get	VERB
ejpam-5360	262	20	ρ1(g	ρ1(g	NOUN
ejpam-5360	262	21	)	)	PUNCT
ejpam-5360	262	22	≤	≤	NOUN
ejpam-5360	262	23	(	(	PUNCT
ejpam-5360	262	24	r∧	r∧	PROPN
ejpam-5360	262	25	1	1	NUM
ejpam-5360	262	26	−	−	NOUN
ejpam-5360	262	27	1	1	NUM
ejpam-5360	262	28	)	)	PUNCT
ejpam-5360	262	29	+	+	CCONJ
ejpam-5360	263	1	√	√	INTJ
ejpam-5360	263	2	(	(	PUNCT
ejpam-5360	263	3	r∧	r∧	PROPN
ejpam-5360	263	4	1	1	NUM
ejpam-5360	263	5	+	+	CCONJ
ejpam-5360	263	6	1)2	1)2	NUM
ejpam-5360	263	7	2	2	NUM
ejpam-5360	263	8	=	=	SYM
ejpam-5360	263	9	r∧	r∧	PROPN
ejpam-5360	263	10	1	1	NUM
ejpam-5360	263	11	.	.	PUNCT
ejpam-5360	264	1	(	(	PUNCT
ejpam-5360	264	2	15	15	NUM
ejpam-5360	264	3	)	)	PUNCT
ejpam-5360	264	4	case	case	NOUN
ejpam-5360	264	5	(	(	PUNCT
ejpam-5360	264	6	ii	ii	NOUN
ejpam-5360	264	7	)	)	PUNCT
ejpam-5360	264	8	when	when	SCONJ
ejpam-5360	264	9	r∧	r∧	PROPN
ejpam-5360	264	10	l	l	PROPN
ejpam-5360	264	11	=	=	PUNCT
ejpam-5360	265	1	r∧	r∧	PROPN
ejpam-5360	265	2	2	2	NUM
ejpam-5360	265	3	=	=	SYM
ejpam-5360	265	4	(	(	PUNCT
ejpam-5360	265	5	ϕ+αk−1	ϕ+αk−1	PROPN
ejpam-5360	265	6	)	)	PUNCT
ejpam-5360	265	7	,	,	PUNCT
ejpam-5360	265	8	the	the	DET
ejpam-5360	265	9	row	row	NOUN
ejpam-5360	265	10	sum	sum	NOUN
ejpam-5360	265	11	of	of	ADP
ejpam-5360	265	12	a	a	DET
ejpam-5360	265	13	vertex	vertex	NOUN
ejpam-5360	265	14	belonging	belong	VERB
ejpam-5360	265	15	to	to	ADP
ejpam-5360	265	16	the	the	DET
ejpam-5360	265	17	satellite	satellite	NOUN
ejpam-5360	265	18	sk	sk	NOUN
ejpam-5360	265	19	,	,	PUNCT
ejpam-5360	265	20	we	we	PRON
ejpam-5360	265	21	get	get	VERB
ejpam-5360	265	22	(	(	PUNCT
ejpam-5360	265	23	r∧	r∧	PROPN
ejpam-5360	265	24	1	1	NUM
ejpam-5360	265	25	−r∧	−r∧	NUM
ejpam-5360	265	26	2	2	NUM
ejpam-5360	265	27	)	)	PUNCT
ejpam-5360	265	28	=	=	SYM
ejpam-5360	265	29	ϕ	ϕ	X
ejpam-5360	265	30	{	{	PUNCT
ejpam-5360	265	31	k∑	k∑	PROPN
ejpam-5360	265	32	i=1	i=1	PROPN
ejpam-5360	266	1	ηiαi	ηiαi	NOUN
ejpam-5360	267	1	+	+	CCONJ
ejpam-5360	267	2	(	(	PUNCT
ejpam-5360	267	3	ϕ−	ϕ−	PROPN
ejpam-5360	267	4	1)−	1)−	PROPN
ejpam-5360	267	5	(	(	PUNCT
ejpam-5360	267	6	ϕ+	ϕ+	INTJ
ejpam-5360	267	7	αk	αk	INTJ
ejpam-5360	267	8	−	−	PROPN
ejpam-5360	267	9	1	1	NUM
ejpam-5360	267	10	)	)	PUNCT
ejpam-5360	267	11	}	}	PUNCT
ejpam-5360	268	1	=	=	SYM
ejpam-5360	268	2	ϕ	ϕ	X
ejpam-5360	268	3	(	(	PUNCT
ejpam-5360	268	4	k∑	k∑	NOUN
ejpam-5360	268	5	i=1	i=1	PROPN
ejpam-5360	268	6	ηiαi	ηiαi	NOUN
ejpam-5360	268	7	−	−	NOUN
ejpam-5360	268	8	αk	αk	NOUN
ejpam-5360	268	9	)	)	PUNCT
ejpam-5360	268	10	=	=	SYM
ejpam-5360	269	1	ϕ	ϕ	PROPN
ejpam-5360	269	2	(	(	PUNCT
ejpam-5360	269	3	(	(	PUNCT
ejpam-5360	269	4	n−	n−	NOUN
ejpam-5360	269	5	ϕ)−	ϕ)−	PROPN
ejpam-5360	269	6	αk	αk	CCONJ
ejpam-5360	269	7	)	)	PUNCT
ejpam-5360	269	8	.	.	PUNCT
ejpam-5360	270	1	substituting	substitute	VERB
ejpam-5360	270	2	the	the	DET
ejpam-5360	270	3	values	value	NOUN
ejpam-5360	270	4	of	of	ADP
ejpam-5360	270	5	r∧	r∧	PROPN
ejpam-5360	270	6	2	2	NUM
ejpam-5360	270	7	and	and	CCONJ
ejpam-5360	270	8	(	(	PUNCT
ejpam-5360	270	9	r∧	r∧	PROPN
ejpam-5360	270	10	1	1	NUM
ejpam-5360	270	11	−r∧	−r∧	NUM
ejpam-5360	270	12	2	2	NUM
ejpam-5360	270	13	)	)	PUNCT
ejpam-5360	270	14	in	in	ADP
ejpam-5360	270	15	equation	equation	NOUN
ejpam-5360	270	16	(	(	PUNCT
ejpam-5360	270	17	14	14	NUM
ejpam-5360	270	18	)	)	PUNCT
ejpam-5360	270	19	,	,	PUNCT
ejpam-5360	270	20	we	we	PRON
ejpam-5360	270	21	get	get	VERB
ejpam-5360	270	22	ρ1(g	ρ1(g	NOUN
ejpam-5360	270	23	)	)	PUNCT
ejpam-5360	270	24	≤	≤	NOUN
ejpam-5360	271	1	(	(	PUNCT
ejpam-5360	271	2	ϕ+	ϕ+	INTJ
ejpam-5360	271	3	αk	αk	INTJ
ejpam-5360	271	4	−	−	PROPN
ejpam-5360	271	5	1	1	NUM
ejpam-5360	271	6	)	)	PUNCT
ejpam-5360	271	7	+	+	CCONJ
ejpam-5360	271	8	√	√	INTJ
ejpam-5360	271	9	(	(	PUNCT
ejpam-5360	271	10	ϕ+	ϕ+	ADP
ejpam-5360	271	11	αk)2	αk)2	PROPN
ejpam-5360	271	12	+	+	CCONJ
ejpam-5360	271	13	4(ϕ(n−	4(ϕ(n−	NUM
ejpam-5360	271	14	ϕ)−	ϕ)−	PROPN
ejpam-5360	271	15	αk	αk	NOUN
ejpam-5360	271	16	)	)	PUNCT
ejpam-5360	271	17	2	2	NUM
ejpam-5360	271	18	.	.	PUNCT
ejpam-5360	272	1	(	(	PUNCT
ejpam-5360	272	2	16	16	NUM
ejpam-5360	272	3	)	)	PUNCT
ejpam-5360	272	4	case	case	NOUN
ejpam-5360	272	5	(	(	PUNCT
ejpam-5360	272	6	iii	iii	X
ejpam-5360	272	7	)	)	PUNCT
ejpam-5360	272	8	we	we	PRON
ejpam-5360	272	9	now	now	ADV
ejpam-5360	272	10	discuss	discuss	VERB
ejpam-5360	272	11	the	the	DET
ejpam-5360	272	12	case	case	NOUN
ejpam-5360	272	13	when	when	SCONJ
ejpam-5360	272	14	r∧	r∧	PROPN
ejpam-5360	272	15	l	l	PROPN
ejpam-5360	272	16	=	=	PUNCT
ejpam-5360	273	1	r∧	r∧	PROPN
ejpam-5360	273	2	j	j	PROPN
ejpam-5360	273	3	for	for	ADP
ejpam-5360	273	4	some	some	DET
ejpam-5360	273	5	j	j	NOUN
ejpam-5360	273	6	where	where	SCONJ
ejpam-5360	273	7	3	3	NUM
ejpam-5360	273	8	≤	≤	NUM
ejpam-5360	273	9	j	j	PROPN
ejpam-5360	273	10	≤	≤	PROPN
ejpam-5360	273	11	l	l	NOUN
ejpam-5360	273	12	,	,	PUNCT
ejpam-5360	273	13	the	the	DET
ejpam-5360	273	14	row	row	NOUN
ejpam-5360	273	15	sum	sum	NOUN
ejpam-5360	273	16	of	of	ADP
ejpam-5360	273	17	a	a	DET
ejpam-5360	273	18	vertex	vertex	NOUN
ejpam-5360	273	19	belonging	belong	VERB
ejpam-5360	273	20	to	to	ADP
ejpam-5360	273	21	the	the	DET
ejpam-5360	273	22	satellite	satellite	PROPN
ejpam-5360	273	23	sk−(j−2	sk−(j−2	PROPN
ejpam-5360	273	24	)	)	PUNCT
ejpam-5360	273	25	is	be	AUX
ejpam-5360	273	26	(	(	PUNCT
ejpam-5360	273	27	ϕ+	ϕ+	X
ejpam-5360	273	28	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	273	29	)	)	PUNCT
ejpam-5360	274	1	−	−	ADP
ejpam-5360	274	2	1	1	NUM
ejpam-5360	274	3	)	)	PUNCT
ejpam-5360	274	4	where	where	SCONJ
ejpam-5360	274	5	j	j	PROPN
ejpam-5360	274	6	̸=	̸=	PROPN
ejpam-5360	274	7	2	2	NUM
ejpam-5360	274	8	.	.	PUNCT
ejpam-5360	275	1	m.	m.	NOUN
ejpam-5360	275	2	v.	v.	ADP
ejpam-5360	275	3	,	,	PUNCT
ejpam-5360	275	4	k.	k.	PROPN
ejpam-5360	275	5	desikan	desikan	PROPN
ejpam-5360	275	6	/	/	SYM
ejpam-5360	275	7	eur	eur	PROPN
ejpam-5360	275	8	.	.	PUNCT
ejpam-5360	276	1	j.	j.	PROPN
ejpam-5360	276	2	pure	pure	PROPN
ejpam-5360	276	3	appl	appl	PROPN
ejpam-5360	276	4	.	.	PROPN
ejpam-5360	276	5	math	math	PROPN
ejpam-5360	276	6	,	,	PUNCT
ejpam-5360	276	7	18	18	NUM
ejpam-5360	276	8	(	(	PUNCT
ejpam-5360	276	9	4	4	NUM
ejpam-5360	276	10	)	)	PUNCT
ejpam-5360	276	11	(	(	PUNCT
ejpam-5360	276	12	2025	2025	NUM
ejpam-5360	276	13	)	)	PUNCT
ejpam-5360	276	14	,	,	PUNCT
ejpam-5360	276	15	5360	5360	NUM
ejpam-5360	276	16	12	12	NUM
ejpam-5360	276	17	of	of	ADP
ejpam-5360	276	18	20	20	NUM
ejpam-5360	276	19	consider	consider	VERB
ejpam-5360	276	20	j−1∑	j−1∑	PROPN
ejpam-5360	276	21	i=1	i=1	PROPN
ejpam-5360	277	1	(	(	PUNCT
ejpam-5360	277	2	r∧	r∧	PROPN
ejpam-5360	277	3	i	i	PRON
ejpam-5360	277	4	−r∧	−r∧	NUM
ejpam-5360	277	5	j	j	NOUN
ejpam-5360	277	6	)	)	PUNCT
ejpam-5360	278	1	=	=	PUNCT
ejpam-5360	279	1	ϕ(r∧	ϕ(r∧	ADJ
ejpam-5360	279	2	1	1	NUM
ejpam-5360	279	3	−r∧	−r∧	NUM
ejpam-5360	279	4	j	j	NOUN
ejpam-5360	279	5	)	)	PUNCT
ejpam-5360	280	1	+	+	CCONJ
ejpam-5360	280	2	ηkαk(αk	ηkαk(αk	PROPN
ejpam-5360	280	3	−	−	NOUN
ejpam-5360	280	4	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	280	5	)	)	PUNCT
ejpam-5360	280	6	)	)	PUNCT
ejpam-5360	281	1	+	+	CCONJ
ejpam-5360	281	2	ηk−1αk−1(αk−1	ηk−1αk−1(αk−1	VERB
ejpam-5360	281	3	−	−	NOUN
ejpam-5360	281	4	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	281	5	)	)	PUNCT
ejpam-5360	281	6	)	)	PUNCT
ejpam-5360	282	1	+	+	CCONJ
ejpam-5360	282	2	...	...	PUNCT
ejpam-5360	283	1	+	+	PUNCT
ejpam-5360	283	2	ηk−(j−3)αk−(j−3)(αk−(j−3	ηk−(j−3)αk−(j−3)(αk−(j−3	NUM
ejpam-5360	283	3	)	)	PUNCT
ejpam-5360	283	4	−	−	NOUN
ejpam-5360	283	5	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	283	6	)	)	PUNCT
ejpam-5360	283	7	)	)	PUNCT
ejpam-5360	284	1	=	=	PRON
ejpam-5360	284	2	{	{	PUNCT
ejpam-5360	284	3	ϕ	ϕ	X
ejpam-5360	284	4	(	(	PUNCT
ejpam-5360	284	5	k∑	k∑	NOUN
ejpam-5360	284	6	i=1	i=1	PROPN
ejpam-5360	285	1	ηiαi	ηiαi	NOUN
ejpam-5360	286	1	+	+	CCONJ
ejpam-5360	286	2	(	(	PUNCT
ejpam-5360	286	3	ϕ−	ϕ−	PROPN
ejpam-5360	286	4	1)−	1)−	PROPN
ejpam-5360	286	5	(	(	PUNCT
ejpam-5360	286	6	ϕ+	ϕ+	NOUN
ejpam-5360	286	7	αk−(j−2	αk−(j−2	PROPN
ejpam-5360	286	8	)	)	PUNCT
ejpam-5360	286	9	−	−	PROPN
ejpam-5360	286	10	1	1	NUM
ejpam-5360	286	11	)	)	PUNCT
ejpam-5360	286	12	)	)	PUNCT
ejpam-5360	287	1	+	+	CCONJ
ejpam-5360	287	2	j−3∑	j−3∑	PROPN
ejpam-5360	287	3	i=0	i=0	PROPN
ejpam-5360	287	4	ηk−iαk−i(αk−i	ηk−iαk−i(αk−i	NOUN
ejpam-5360	287	5	−	−	NOUN
ejpam-5360	287	6	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	287	7	)	)	PUNCT
ejpam-5360	287	8	)	)	PUNCT
ejpam-5360	287	9	}	}	PUNCT
ejpam-5360	287	10	where	where	SCONJ
ejpam-5360	287	11	3	3	NUM
ejpam-5360	287	12	≤	≤	NUM
ejpam-5360	287	13	j	j	PROPN
ejpam-5360	287	14	≤	≤	PROPN
ejpam-5360	287	15	l.	l.	NOUN
ejpam-5360	287	16	substituting	substitute	VERB
ejpam-5360	287	17	the	the	DET
ejpam-5360	287	18	above	above	ADJ
ejpam-5360	287	19	expression	expression	NOUN
ejpam-5360	287	20	in	in	ADP
ejpam-5360	287	21	equation	equation	NOUN
ejpam-5360	287	22	(	(	PUNCT
ejpam-5360	287	23	14	14	NUM
ejpam-5360	287	24	)	)	PUNCT
ejpam-5360	287	25	,	,	PUNCT
ejpam-5360	287	26	we	we	PRON
ejpam-5360	287	27	obtain	obtain	VERB
ejpam-5360	287	28	ρ1(g	ρ1(g	NOUN
ejpam-5360	287	29	)	)	PUNCT
ejpam-5360	287	30	≤	≤	NOUN
ejpam-5360	287	31	(	(	PUNCT
ejpam-5360	287	32	ϕ+	ϕ+	X
ejpam-5360	287	33	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	287	34	)	)	PUNCT
ejpam-5360	287	35	−	−	ADP
ejpam-5360	287	36	2	2	NUM
ejpam-5360	287	37	)	)	PUNCT
ejpam-5360	287	38	2	2	NUM
ejpam-5360	287	39	+	+	CCONJ
ejpam-5360	287	40	√	√	PROPN
ejpam-5360	287	41	(	(	PUNCT
ejpam-5360	287	42	ϕ+	ϕ+	NOUN
ejpam-5360	287	43	αk−(j−2))2	αk−(j−2))2	NOUN
ejpam-5360	287	44	+	+	CCONJ
ejpam-5360	287	45	4	4	NUM
ejpam-5360	287	46	(	(	PUNCT
ejpam-5360	287	47	ϕ	ϕ	NOUN
ejpam-5360	287	48	(	(	PUNCT
ejpam-5360	287	49	∑k	∑k	PROPN
ejpam-5360	287	50	i=1	i=1	PROPN
ejpam-5360	288	1	ηiαi	ηiαi	VERB
ejpam-5360	289	1	−	−	PROPN
ejpam-5360	289	2	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	289	3	)	)	PUNCT
ejpam-5360	289	4	)	)	PUNCT
ejpam-5360	290	1	+	+	CCONJ
ejpam-5360	290	2	∑j−3	∑j−3	ADP
ejpam-5360	290	3	i=0	i=0	ADJ
ejpam-5360	290	4	ηk−iαk−i(αk−i	ηk−iαk−i(αk−i	NOUN
ejpam-5360	290	5	−	−	NOUN
ejpam-5360	290	6	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	290	7	)	)	PUNCT
ejpam-5360	290	8	)	)	PUNCT
ejpam-5360	290	9	)	)	PUNCT
ejpam-5360	290	10	2	2	X
ejpam-5360	290	11	.	.	PUNCT
ejpam-5360	291	1	(	(	PUNCT
ejpam-5360	291	2	17	17	NUM
ejpam-5360	291	3	)	)	PUNCT
ejpam-5360	291	4	hence	hence	ADV
ejpam-5360	291	5	proved	prove	VERB
ejpam-5360	291	6	.	.	PUNCT
ejpam-5360	292	1	figure	figure	VERB
ejpam-5360	292	2	2	2	NUM
ejpam-5360	292	3	:	:	PUNCT
ejpam-5360	292	4	s1	s1	NOUN
ejpam-5360	292	5	:	:	PUNCT
ejpam-5360	292	6	3k3	3k3	NUM
ejpam-5360	292	7	,	,	PUNCT
ejpam-5360	292	8	s2	s2	PROPN
ejpam-5360	292	9	:	:	PUNCT
ejpam-5360	292	10	k5	k5	PROPN
ejpam-5360	292	11	,	,	PUNCT
ejpam-5360	292	12	s3	s3	PROPN
ejpam-5360	292	13	:	:	PUNCT
ejpam-5360	292	14	k6	k6	PROPN
ejpam-5360	292	15	join	join	VERB
ejpam-5360	292	16	with	with	ADP
ejpam-5360	292	17	one	one	NUM
ejpam-5360	292	18	of	of	ADP
ejpam-5360	292	19	the	the	DET
ejpam-5360	292	20	vertices	vertex	NOUN
ejpam-5360	292	21	of	of	ADP
ejpam-5360	292	22	k4	k4	PROPN
ejpam-5360	292	23	(	(	PUNCT
ejpam-5360	292	24	core	core	NOUN
ejpam-5360	292	25	)	)	PUNCT
ejpam-5360	292	26	as	as	ADP
ejpam-5360	292	27	illustration	illustration	NOUN
ejpam-5360	292	28	.	.	PUNCT
ejpam-5360	293	1	m.	m.	NOUN
ejpam-5360	293	2	v.	v.	ADP
ejpam-5360	293	3	,	,	PUNCT
ejpam-5360	293	4	k.	k.	PROPN
ejpam-5360	293	5	desikan	desikan	PROPN
ejpam-5360	293	6	/	/	SYM
ejpam-5360	293	7	eur	eur	PROPN
ejpam-5360	293	8	.	.	PUNCT
ejpam-5360	294	1	j.	j.	PROPN
ejpam-5360	294	2	pure	pure	PROPN
ejpam-5360	294	3	appl	appl	PROPN
ejpam-5360	294	4	.	.	PROPN
ejpam-5360	294	5	math	math	PROPN
ejpam-5360	294	6	,	,	PUNCT
ejpam-5360	294	7	18	18	NUM
ejpam-5360	294	8	(	(	PUNCT
ejpam-5360	294	9	4	4	NUM
ejpam-5360	294	10	)	)	PUNCT
ejpam-5360	294	11	(	(	PUNCT
ejpam-5360	294	12	2025	2025	NUM
ejpam-5360	294	13	)	)	PUNCT
ejpam-5360	294	14	,	,	PUNCT
ejpam-5360	294	15	5360	5360	NUM
ejpam-5360	294	16	13	13	NUM
ejpam-5360	294	17	of	of	ADP
ejpam-5360	294	18	20	20	NUM
ejpam-5360	294	19	example	example	NOUN
ejpam-5360	294	20	1	1	NUM
ejpam-5360	294	21	.	.	X
ejpam-5360	294	22	consider	consider	VERB
ejpam-5360	294	23	the	the	DET
ejpam-5360	294	24	core	core	NOUN
ejpam-5360	294	25	-	-	PUNCT
ejpam-5360	294	26	satellite	satellite	NOUN
ejpam-5360	294	27	graph	graph	NOUN
ejpam-5360	294	28	given	give	VERB
ejpam-5360	294	29	in	in	ADP
ejpam-5360	294	30	figure	figure	NOUN
ejpam-5360	294	31	2	2	NUM
ejpam-5360	294	32	.	.	PUNCT
ejpam-5360	295	1	this	this	DET
ejpam-5360	295	2	graph	graph	NOUN
ejpam-5360	295	3	has	have	VERB
ejpam-5360	295	4	s1	s1	NOUN
ejpam-5360	295	5	,	,	PUNCT
ejpam-5360	295	6	s2	s2	NOUN
ejpam-5360	295	7	and	and	CCONJ
ejpam-5360	295	8	s3	s3	PROPN
ejpam-5360	295	9	as	as	ADP
ejpam-5360	295	10	the	the	DET
ejpam-5360	295	11	satellite	satellite	NOUN
ejpam-5360	295	12	graphs	graph	NOUN
ejpam-5360	295	13	and	and	CCONJ
ejpam-5360	295	14	kϕ	kϕ	NOUN
ejpam-5360	295	15	=	=	SYM
ejpam-5360	295	16	k4	k4	PROPN
ejpam-5360	295	17	as	as	ADP
ejpam-5360	295	18	the	the	DET
ejpam-5360	295	19	core	core	NOUN
ejpam-5360	295	20	graph	graph	NOUN
ejpam-5360	295	21	.	.	PUNCT
ejpam-5360	296	1	s1	s1	NOUN
ejpam-5360	296	2	comprises	comprise	NOUN
ejpam-5360	296	3	of	of	ADP
ejpam-5360	296	4	three	three	NUM
ejpam-5360	296	5	copies	copy	NOUN
ejpam-5360	296	6	of	of	ADP
ejpam-5360	296	7	k3	k3	PROPN
ejpam-5360	296	8	,	,	PUNCT
ejpam-5360	296	9	s2	s2	NOUN
ejpam-5360	296	10	comprises	comprise	NOUN
ejpam-5360	296	11	of	of	ADP
ejpam-5360	296	12	one	one	NUM
ejpam-5360	296	13	copy	copy	NOUN
ejpam-5360	296	14	of	of	ADP
ejpam-5360	296	15	k5	k5	PROPN
ejpam-5360	296	16	,	,	PUNCT
ejpam-5360	296	17	s3	s3	PROPN
ejpam-5360	296	18	comprises	comprise	NOUN
ejpam-5360	296	19	of	of	ADP
ejpam-5360	296	20	one	one	NUM
ejpam-5360	296	21	copy	copy	NOUN
ejpam-5360	296	22	k6	k6	NOUN
ejpam-5360	296	23	.	.	PUNCT
ejpam-5360	297	1	as	as	ADP
ejpam-5360	297	2	an	an	DET
ejpam-5360	297	3	illustration	illustration	NOUN
ejpam-5360	297	4	,	,	PUNCT
ejpam-5360	297	5	the	the	DET
ejpam-5360	297	6	join	join	NOUN
ejpam-5360	297	7	of	of	ADP
ejpam-5360	297	8	the	the	DET
ejpam-5360	297	9	cliques	clique	NOUN
ejpam-5360	297	10	in	in	ADP
ejpam-5360	297	11	the	the	DET
ejpam-5360	297	12	satellite	satellite	NOUN
ejpam-5360	297	13	with	with	ADP
ejpam-5360	297	14	one	one	NUM
ejpam-5360	297	15	vertex	vertex	NOUN
ejpam-5360	297	16	of	of	ADP
ejpam-5360	297	17	the	the	DET
ejpam-5360	297	18	core	core	NOUN
ejpam-5360	297	19	graph	graph	NOUN
ejpam-5360	297	20	is	be	AUX
ejpam-5360	297	21	shown	show	VERB
ejpam-5360	297	22	.	.	PUNCT
ejpam-5360	298	1	in	in	ADP
ejpam-5360	298	2	fact	fact	NOUN
ejpam-5360	298	3	,	,	PUNCT
ejpam-5360	298	4	the	the	DET
ejpam-5360	298	5	remaining	remain	VERB
ejpam-5360	298	6	vertices	vertex	NOUN
ejpam-5360	298	7	of	of	ADP
ejpam-5360	298	8	the	the	DET
ejpam-5360	298	9	core	core	NOUN
ejpam-5360	298	10	are	be	AUX
ejpam-5360	298	11	similarly	similarly	ADV
ejpam-5360	298	12	joined	join	VERB
ejpam-5360	298	13	with	with	ADP
ejpam-5360	298	14	cliques	clique	NOUN
ejpam-5360	298	15	of	of	ADP
ejpam-5360	298	16	the	the	DET
ejpam-5360	298	17	satellites	satellite	NOUN
ejpam-5360	298	18	.	.	PUNCT
ejpam-5360	299	1	in	in	ADP
ejpam-5360	299	2	this	this	DET
ejpam-5360	299	3	example	example	NOUN
ejpam-5360	299	4	,	,	PUNCT
ejpam-5360	299	5	the	the	DET
ejpam-5360	299	6	number	number	NOUN
ejpam-5360	299	7	of	of	ADP
ejpam-5360	299	8	vertices	vertex	NOUN
ejpam-5360	299	9	is	be	AUX
ejpam-5360	299	10	24	24	NUM
ejpam-5360	299	11	and	and	CCONJ
ejpam-5360	299	12	the	the	DET
ejpam-5360	299	13	number	number	NOUN
ejpam-5360	299	14	of	of	ADP
ejpam-5360	299	15	edges	edge	NOUN
ejpam-5360	299	16	is	be	AUX
ejpam-5360	299	17	120	120	NUM
ejpam-5360	299	18	.	.	PUNCT
ejpam-5360	300	1	the	the	DET
ejpam-5360	300	2	calculated	calculate	VERB
ejpam-5360	300	3	value	value	NOUN
ejpam-5360	300	4	of	of	ADP
ejpam-5360	300	5	the	the	DET
ejpam-5360	300	6	spectral	spectral	ADJ
ejpam-5360	300	7	radius	radius	NOUN
ejpam-5360	300	8	is	be	AUX
ejpam-5360	300	9	ρ1(g	ρ1(g	PROPN
ejpam-5360	300	10	)	)	PUNCT
ejpam-5360	300	11	=	=	NOUN
ejpam-5360	300	12	12.2485	12.2485	NUM
ejpam-5360	300	13	.	.	PUNCT
ejpam-5360	301	1	we	we	PRON
ejpam-5360	301	2	have	have	VERB
ejpam-5360	301	3	the	the	DET
ejpam-5360	301	4	following	follow	VERB
ejpam-5360	301	5	observations	observation	NOUN
ejpam-5360	301	6	from	from	ADP
ejpam-5360	301	7	the	the	DET
ejpam-5360	301	8	bounds	bound	NOUN
ejpam-5360	301	9	obtained	obtain	VERB
ejpam-5360	301	10	in	in	ADP
ejpam-5360	301	11	theorems	theorem	NOUN
ejpam-5360	301	12	7	7	NUM
ejpam-5360	301	13	,	,	PUNCT
ejpam-5360	301	14	8	8	NUM
ejpam-5360	301	15	and	and	CCONJ
ejpam-5360	301	16	9	9	NUM
ejpam-5360	301	17	.	.	PUNCT
ejpam-5360	302	1	(	(	PUNCT
ejpam-5360	302	2	i	i	NOUN
ejpam-5360	302	3	)	)	PUNCT
ejpam-5360	302	4	from	from	ADP
ejpam-5360	302	5	equation	equation	NOUN
ejpam-5360	302	6	(	(	PUNCT
ejpam-5360	302	7	7	7	NUM
ejpam-5360	302	8	)	)	PUNCT
ejpam-5360	302	9	of	of	ADP
ejpam-5360	302	10	theorem	theorem	NOUN
ejpam-5360	302	11	(	(	PUNCT
ejpam-5360	302	12	7	7	X
ejpam-5360	302	13	)	)	PUNCT
ejpam-5360	302	14	we	we	PRON
ejpam-5360	302	15	get	get	VERB
ejpam-5360	302	16	the	the	DET
ejpam-5360	302	17	lower	lower	ADV
ejpam-5360	302	18	bound	bind	VERB
ejpam-5360	302	19	as	as	ADP
ejpam-5360	302	20	11.013	11.013	NUM
ejpam-5360	302	21	and	and	CCONJ
ejpam-5360	302	22	from	from	ADP
ejpam-5360	302	23	equation	equation	NOUN
ejpam-5360	302	24	(	(	PUNCT
ejpam-5360	302	25	8)	8)	NUM
ejpam-5360	302	26	of	of	ADP
ejpam-5360	302	27	theorem	theorem	NOUN
ejpam-5360	302	28	(	(	PUNCT
ejpam-5360	302	29	7	7	X
ejpam-5360	302	30	)	)	PUNCT
ejpam-5360	302	31	we	we	PRON
ejpam-5360	302	32	get	get	VERB
ejpam-5360	302	33	the	the	DET
ejpam-5360	302	34	upper	upper	ADJ
ejpam-5360	302	35	bound	bind	VERB
ejpam-5360	302	36	as	as	ADP
ejpam-5360	302	37	24.33	24.33	NUM
ejpam-5360	302	38	.	.	PUNCT
ejpam-5360	303	1	(	(	PUNCT
ejpam-5360	303	2	ii	ii	NOUN
ejpam-5360	303	3	)	)	PUNCT
ejpam-5360	303	4	using	use	VERB
ejpam-5360	303	5	equation	equation	NOUN
ejpam-5360	303	6	(	(	PUNCT
ejpam-5360	303	7	11	11	NUM
ejpam-5360	303	8	)	)	PUNCT
ejpam-5360	303	9	we	we	PRON
ejpam-5360	303	10	obtain	obtain	VERB
ejpam-5360	303	11	the	the	DET
ejpam-5360	303	12	improved	improve	VERB
ejpam-5360	303	13	upper	upper	ADJ
ejpam-5360	303	14	bound	bind	VERB
ejpam-5360	303	15	as	as	ADP
ejpam-5360	303	16	14.043	14.043	NUM
ejpam-5360	303	17	.	.	PUNCT
ejpam-5360	304	1	(	(	PUNCT
ejpam-5360	304	2	iii	iii	NOUN
ejpam-5360	304	3	)	)	PUNCT
ejpam-5360	304	4	from	from	ADP
ejpam-5360	304	5	equation	equation	NOUN
ejpam-5360	304	6	(	(	PUNCT
ejpam-5360	304	7	13	13	NUM
ejpam-5360	304	8	)	)	PUNCT
ejpam-5360	304	9	of	of	ADP
ejpam-5360	304	10	theorem	theorem	NOUN
ejpam-5360	304	11	8	8	NUM
ejpam-5360	304	12	,	,	PUNCT
ejpam-5360	304	13	we	we	PRON
ejpam-5360	304	14	get	get	VERB
ejpam-5360	304	15	8.944	8.944	NUM
ejpam-5360	304	16	<	<	X
ejpam-5360	304	17	ρ1(g	ρ1(g	NOUN
ejpam-5360	304	18	)	)	PUNCT
ejpam-5360	304	19	<	<	X
ejpam-5360	305	1	15.49	15.49	NUM
ejpam-5360	305	2	.	.	PUNCT
ejpam-5360	306	1	(	(	PUNCT
ejpam-5360	306	2	iv	iv	X
ejpam-5360	306	3	)	)	PUNCT
ejpam-5360	306	4	using	use	VERB
ejpam-5360	306	5	theorem	theorem	NOUN
ejpam-5360	306	6	9	9	NUM
ejpam-5360	306	7	,	,	PUNCT
ejpam-5360	306	8	three	three	NUM
ejpam-5360	306	9	cases	case	NOUN
ejpam-5360	306	10	are	be	AUX
ejpam-5360	306	11	discussed	discuss	VERB
ejpam-5360	306	12	.	.	PUNCT
ejpam-5360	307	1	(	(	PUNCT
ejpam-5360	307	2	a	a	X
ejpam-5360	307	3	)	)	PUNCT
ejpam-5360	307	4	from	from	ADP
ejpam-5360	307	5	equation	equation	NOUN
ejpam-5360	307	6	(	(	PUNCT
ejpam-5360	307	7	15	15	NUM
ejpam-5360	307	8	)	)	PUNCT
ejpam-5360	307	9	of	of	ADP
ejpam-5360	307	10	case	case	NOUN
ejpam-5360	307	11	(	(	PUNCT
ejpam-5360	307	12	i	i	NOUN
ejpam-5360	307	13	)	)	PUNCT
ejpam-5360	307	14	we	we	PRON
ejpam-5360	307	15	obtain	obtain	VERB
ejpam-5360	307	16	the	the	DET
ejpam-5360	307	17	upper	upper	ADJ
ejpam-5360	307	18	bound	bind	VERB
ejpam-5360	307	19	for	for	ADP
ejpam-5360	307	20	ρ1(g	ρ1(g	PROPN
ejpam-5360	307	21	)	)	PUNCT
ejpam-5360	307	22	as	as	ADP
ejpam-5360	307	23	23	23	NUM
ejpam-5360	307	24	.	.	PUNCT
ejpam-5360	308	1	(	(	PUNCT
ejpam-5360	308	2	b	b	X
ejpam-5360	308	3	)	)	PUNCT
ejpam-5360	308	4	from	from	ADP
ejpam-5360	308	5	equation	equation	NOUN
ejpam-5360	308	6	(	(	PUNCT
ejpam-5360	308	7	16	16	NUM
ejpam-5360	308	8	)	)	PUNCT
ejpam-5360	308	9	of	of	ADP
ejpam-5360	308	10	case	case	NOUN
ejpam-5360	308	11	(	(	PUNCT
ejpam-5360	308	12	ii	ii	NOUN
ejpam-5360	308	13	)	)	PUNCT
ejpam-5360	308	14	we	we	PRON
ejpam-5360	308	15	obtain	obtain	VERB
ejpam-5360	308	16	the	the	DET
ejpam-5360	308	17	tight	tight	ADJ
ejpam-5360	308	18	upper	upper	ADJ
ejpam-5360	308	19	bounds	bound	NOUN
ejpam-5360	308	20	as	as	ADP
ejpam-5360	308	21	13	13	NUM
ejpam-5360	308	22	(	(	PUNCT
ejpam-5360	308	23	c	c	NOUN
ejpam-5360	308	24	)	)	PUNCT
ejpam-5360	308	25	from	from	ADP
ejpam-5360	308	26	equation	equation	NOUN
ejpam-5360	308	27	(	(	PUNCT
ejpam-5360	308	28	17	17	NUM
ejpam-5360	308	29	)	)	PUNCT
ejpam-5360	308	30	of	of	ADP
ejpam-5360	308	31	the	the	DET
ejpam-5360	308	32	case	case	NOUN
ejpam-5360	308	33	(	(	PUNCT
ejpam-5360	308	34	iii	iii	X
ejpam-5360	308	35	)	)	PUNCT
ejpam-5360	308	36	we	we	PRON
ejpam-5360	308	37	obtain	obtain	VERB
ejpam-5360	308	38	the	the	DET
ejpam-5360	308	39	upper	upper	ADJ
ejpam-5360	308	40	bound	bind	VERB
ejpam-5360	308	41	as	as	ADP
ejpam-5360	308	42	15.587	15.587	NUM
ejpam-5360	308	43	.	.	PUNCT
ejpam-5360	309	1	from	from	ADP
ejpam-5360	309	2	the	the	DET
ejpam-5360	309	3	above	above	ADJ
ejpam-5360	309	4	discussions	discussion	NOUN
ejpam-5360	309	5	,	,	PUNCT
ejpam-5360	309	6	we	we	PRON
ejpam-5360	309	7	observe	observe	VERB
ejpam-5360	309	8	that	that	SCONJ
ejpam-5360	309	9	the	the	DET
ejpam-5360	309	10	tight	tight	ADJ
ejpam-5360	309	11	lower	low	ADJ
ejpam-5360	309	12	and	and	CCONJ
ejpam-5360	309	13	upper	upper	ADJ
ejpam-5360	309	14	bounds	bound	NOUN
ejpam-5360	309	15	are	be	AUX
ejpam-5360	309	16	11.013	11.013	NUM
ejpam-5360	309	17	and	and	CCONJ
ejpam-5360	309	18	13	13	NUM
ejpam-5360	309	19	,	,	PUNCT
ejpam-5360	309	20	respectively	respectively	ADV
ejpam-5360	309	21	.	.	PUNCT
ejpam-5360	310	1	3.2	3.2	NUM
ejpam-5360	310	2	.	.	PUNCT
ejpam-5360	310	3	bounds	bound	NOUN
ejpam-5360	310	4	on	on	ADP
ejpam-5360	310	5	signless	signless	PROPN
ejpam-5360	310	6	laplacian	laplacian	ADJ
ejpam-5360	310	7	spectral	spectral	ADJ
ejpam-5360	310	8	radius	radius	NOUN
ejpam-5360	310	9	in	in	ADP
ejpam-5360	310	10	this	this	DET
ejpam-5360	310	11	section	section	NOUN
ejpam-5360	310	12	,	,	PUNCT
ejpam-5360	310	13	we	we	PRON
ejpam-5360	310	14	derive	derive	VERB
ejpam-5360	310	15	the	the	DET
ejpam-5360	310	16	upper	upper	ADJ
ejpam-5360	310	17	and	and	CCONJ
ejpam-5360	310	18	lower	low	ADJ
ejpam-5360	310	19	bounds	bound	NOUN
ejpam-5360	310	20	for	for	ADP
ejpam-5360	310	21	the	the	DET
ejpam-5360	310	22	signless	signless	ADJ
ejpam-5360	310	23	laplacian	laplacian	ADJ
ejpam-5360	310	24	matrix	matrix	NOUN
ejpam-5360	310	25	q(g	q(g	NOUN
ejpam-5360	310	26	)	)	PUNCT
ejpam-5360	311	1	=	=	SYM
ejpam-5360	311	2	d(g	d(g	PROPN
ejpam-5360	311	3	)	)	PUNCT
ejpam-5360	312	1	+	+	NUM
ejpam-5360	312	2	a(g	a(g	PROPN
ejpam-5360	312	3	)	)	PUNCT
ejpam-5360	312	4	for	for	ADP
ejpam-5360	312	5	the	the	DET
ejpam-5360	312	6	graph	graph	NOUN
ejpam-5360	312	7	g.	g.	PROPN
ejpam-5360	312	8	let	let	VERB
ejpam-5360	312	9	µ1(q(g	µ1(q(g	NUM
ejpam-5360	312	10	)	)	PUNCT
ejpam-5360	312	11	)	)	PUNCT
ejpam-5360	313	1	be	be	AUX
ejpam-5360	313	2	the	the	DET
ejpam-5360	313	3	signless	signless	PROPN
ejpam-5360	313	4	laplacian	laplacian	ADJ
ejpam-5360	313	5	spectral	spectral	ADJ
ejpam-5360	313	6	radius	radius	NOUN
ejpam-5360	313	7	.	.	PUNCT
ejpam-5360	314	1	theorem	theorem	VERB
ejpam-5360	314	2	10	10	NUM
ejpam-5360	314	3	.	.	PUNCT
ejpam-5360	315	1	let	let	VERB
ejpam-5360	315	2	g	g	PRON
ejpam-5360	315	3	be	be	AUX
ejpam-5360	315	4	a	a	DET
ejpam-5360	315	5	generalized	generalized	ADJ
ejpam-5360	315	6	core	core	NOUN
ejpam-5360	315	7	-	-	PUNCT
ejpam-5360	315	8	satellite	satellite	NOUN
ejpam-5360	315	9	graph	graph	NOUN
ejpam-5360	315	10	with	with	ADP
ejpam-5360	315	11	v	v	NOUN
ejpam-5360	315	12	(	(	PUNCT
ejpam-5360	315	13	g	g	NOUN
ejpam-5360	315	14	)	)	PUNCT
ejpam-5360	315	15	and	and	CCONJ
ejpam-5360	315	16	e(g	e(g	PROPN
ejpam-5360	315	17	)	)	PUNCT
ejpam-5360	315	18	as	as	ADP
ejpam-5360	315	19	the	the	DET
ejpam-5360	315	20	set	set	NOUN
ejpam-5360	315	21	of	of	ADP
ejpam-5360	315	22	vertices	vertex	NOUN
ejpam-5360	315	23	and	and	CCONJ
ejpam-5360	315	24	edges	edge	NOUN
ejpam-5360	315	25	of	of	ADP
ejpam-5360	315	26	the	the	DET
ejpam-5360	315	27	graph	graph	NOUN
ejpam-5360	316	1	g.	g.	PROPN
ejpam-5360	316	2	then	then	ADV
ejpam-5360	316	3	the	the	DET
ejpam-5360	316	4	upper	upper	ADJ
ejpam-5360	316	5	and	and	CCONJ
ejpam-5360	316	6	lower	low	ADJ
ejpam-5360	316	7	bounds	bound	NOUN
ejpam-5360	316	8	for	for	ADP
ejpam-5360	316	9	signless	signless	ADJ
ejpam-5360	316	10	laplacian	laplacian	ADJ
ejpam-5360	316	11	spectral	spectral	ADJ
ejpam-5360	316	12	radius	radius	NOUN
ejpam-5360	316	13	of	of	ADP
ejpam-5360	316	14	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	316	15	)	)	PUNCT
ejpam-5360	316	16	)	)	PUNCT
ejpam-5360	316	17	are	be	AUX
ejpam-5360	316	18	√	√	ADV
ejpam-5360	316	19	2	2	NUM
ejpam-5360	316	20	[	[	PUNCT
ejpam-5360	316	21	(	(	PUNCT
ejpam-5360	316	22	dαk	dαk	ADJ
ejpam-5360	316	23	)	)	PUNCT
ejpam-5360	316	24	2	2	NUM
ejpam-5360	316	25	+	+	CCONJ
ejpam-5360	316	26	ϕ	ϕ	X
ejpam-5360	316	27	(	(	PUNCT
ejpam-5360	316	28	k∑	k∑	NOUN
ejpam-5360	316	29	i=1	i=1	PROPN
ejpam-5360	317	1	ηiαi	ηiαi	NOUN
ejpam-5360	318	1	+	+	CCONJ
ejpam-5360	318	2	(	(	PUNCT
ejpam-5360	318	3	ϕ−	ϕ−	PROPN
ejpam-5360	318	4	1	1	NUM
ejpam-5360	318	5	)	)	PUNCT
ejpam-5360	318	6	)	)	PUNCT
ejpam-5360	319	1	+	+	CCONJ
ejpam-5360	319	2	(	(	PUNCT
ejpam-5360	319	3	αk	αk	ADP
ejpam-5360	319	4	−	−	PROPN
ejpam-5360	319	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	319	6	αk	αk	NOUN
ejpam-5360	319	7	−	−	NOUN
ejpam-5360	319	8	1	1	NUM
ejpam-5360	319	9	)	)	PUNCT
ejpam-5360	319	10	]	]	PUNCT
ejpam-5360	319	11	1/2	1/2	NUM
ejpam-5360	319	12	≤	≤	NOUN
ejpam-5360	319	13	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	319	14	)	)	PUNCT
ejpam-5360	319	15	)	)	PUNCT
ejpam-5360	319	16	≤	≤	NUM
ejpam-5360	319	17	√	√	ADV
ejpam-5360	319	18	2	2	NUM
ejpam-5360	319	19	[	[	PUNCT
ejpam-5360	319	20	k∑	k∑	NOUN
ejpam-5360	319	21	i=1	i=1	PROPN
ejpam-5360	320	1	ηiαi	ηiαi	NOUN
ejpam-5360	321	1	+	+	CCONJ
ejpam-5360	321	2	(	(	PUNCT
ejpam-5360	321	3	ϕ−	ϕ−	PROPN
ejpam-5360	321	4	1)2	1)2	NUM
ejpam-5360	321	5	+	+	CCONJ
ejpam-5360	321	6	2	2	NUM
ejpam-5360	321	7	(	(	PUNCT
ejpam-5360	321	8	k∑	k∑	NOUN
ejpam-5360	321	9	i=1	i=1	PROPN
ejpam-5360	321	10	(	(	PUNCT
ejpam-5360	321	11	ηiαic2	ηiαic2	PROPN
ejpam-5360	321	12	+	+	CCONJ
ejpam-5360	321	13	ϕηiαi	ϕηiαi	NOUN
ejpam-5360	321	14	)	)	PUNCT
ejpam-5360	322	1	+	+	NUM
ejpam-5360	322	2	ϕc2	ϕc2	NOUN
ejpam-5360	322	3	)	)	PUNCT
ejpam-5360	322	4	−	−	PROPN
ejpam-5360	323	1	(	(	PUNCT
ejpam-5360	323	2	k∑	k∑	NOUN
ejpam-5360	323	3	i=1	i=1	PROPN
ejpam-5360	324	1	ηiαi	ηiαi	NOUN
ejpam-5360	325	1	+	+	CCONJ
ejpam-5360	325	2	(	(	PUNCT
ejpam-5360	325	3	ϕ−	ϕ−	PROPN
ejpam-5360	325	4	1	1	NUM
ejpam-5360	325	5	)	)	PUNCT
ejpam-5360	325	6	)	)	PUNCT
ejpam-5360	325	7	]	]	X
ejpam-5360	325	8	1/2	1/2	NUM
ejpam-5360	325	9	(	(	PUNCT
ejpam-5360	325	10	18	18	NUM
ejpam-5360	325	11	)	)	PUNCT
ejpam-5360	325	12	m.	m.	NOUN
ejpam-5360	326	1	v.	v.	ADP
ejpam-5360	326	2	,	,	PUNCT
ejpam-5360	326	3	k.	k.	PROPN
ejpam-5360	326	4	desikan	desikan	PROPN
ejpam-5360	326	5	/	/	SYM
ejpam-5360	326	6	eur	eur	PROPN
ejpam-5360	326	7	.	.	PUNCT
ejpam-5360	327	1	j.	j.	PROPN
ejpam-5360	327	2	pure	pure	PROPN
ejpam-5360	327	3	appl	appl	PROPN
ejpam-5360	327	4	.	.	PROPN
ejpam-5360	327	5	math	math	PROPN
ejpam-5360	327	6	,	,	PUNCT
ejpam-5360	327	7	18	18	NUM
ejpam-5360	327	8	(	(	PUNCT
ejpam-5360	327	9	4	4	NUM
ejpam-5360	327	10	)	)	PUNCT
ejpam-5360	327	11	(	(	PUNCT
ejpam-5360	327	12	2025	2025	NUM
ejpam-5360	327	13	)	)	PUNCT
ejpam-5360	327	14	,	,	PUNCT
ejpam-5360	327	15	5360	5360	NUM
ejpam-5360	327	16	14	14	NUM
ejpam-5360	327	17	of	of	ADP
ejpam-5360	327	18	20	20	NUM
ejpam-5360	327	19	proof	proof	NOUN
ejpam-5360	327	20	.	.	PUNCT
ejpam-5360	328	1	since	since	SCONJ
ejpam-5360	328	2	q(g	q(g	PROPN
ejpam-5360	328	3	)	)	PUNCT
ejpam-5360	328	4	=	=	SYM
ejpam-5360	328	5	d(g)+a(g	d(g)+a(g	NOUN
ejpam-5360	328	6	)	)	PUNCT
ejpam-5360	328	7	,	,	PUNCT
ejpam-5360	328	8	for	for	ADP
ejpam-5360	328	9	vertex	vertex	NOUN
ejpam-5360	328	10	v	v	ADP
ejpam-5360	328	11	its	its	PRON
ejpam-5360	328	12	row	row	NOUN
ejpam-5360	328	13	sum	sum	NOUN
ejpam-5360	328	14	is	be	AUX
ejpam-5360	328	15	denoted	denote	VERB
ejpam-5360	328	16	as	as	ADP
ejpam-5360	328	17	rv(q(g	rv(q(g	VERB
ejpam-5360	328	18	)	)	PUNCT
ejpam-5360	328	19	)	)	PUNCT
ejpam-5360	328	20	and	and	CCONJ
ejpam-5360	328	21	we	we	PRON
ejpam-5360	328	22	have	have	VERB
ejpam-5360	328	23	rv(q(g	rv(q(g	VERB
ejpam-5360	328	24	)	)	PUNCT
ejpam-5360	328	25	)	)	PUNCT
ejpam-5360	329	1	=	=	SYM
ejpam-5360	329	2	2d(v	2d(v	NUM
ejpam-5360	329	3	)	)	PUNCT
ejpam-5360	329	4	.	.	PUNCT
ejpam-5360	330	1	also	also	ADV
ejpam-5360	330	2	we	we	PRON
ejpam-5360	330	3	have	have	VERB
ejpam-5360	330	4	rv(ad	rv(ad	NOUN
ejpam-5360	330	5	)	)	PUNCT
ejpam-5360	330	6	=	=	SYM
ejpam-5360	330	7	rv(a	rv(a	NOUN
ejpam-5360	331	1	2	2	NUM
ejpam-5360	331	2	)	)	PUNCT
ejpam-5360	331	3	=	=	SYM
ejpam-5360	331	4	σ	σ	PROPN
ejpam-5360	331	5	u∼v	u∼v	ADJ
ejpam-5360	331	6	d(v	d(v	PROPN
ejpam-5360	331	7	)	)	PUNCT
ejpam-5360	331	8	is	be	AUX
ejpam-5360	331	9	true	true	ADJ
ejpam-5360	331	10	for	for	ADP
ejpam-5360	331	11	the	the	DET
ejpam-5360	331	12	graph	graph	NOUN
ejpam-5360	331	13	g.	g.	NOUN
ejpam-5360	331	14	then	then	ADV
ejpam-5360	331	15	rv(q(g))2	rv(q(g))2	PROPN
ejpam-5360	331	16	=	=	PUNCT
ejpam-5360	331	17	rv(d(d	rv(d(d	PROPN
ejpam-5360	331	18	+	+	NOUN
ejpam-5360	331	19	a	a	X
ejpam-5360	331	20	)	)	PUNCT
ejpam-5360	332	1	+	+	ADP
ejpam-5360	332	2	ad	ad	NOUN
ejpam-5360	332	3	+	+	NOUN
ejpam-5360	332	4	a2	a2	NOUN
ejpam-5360	332	5	)	)	PUNCT
ejpam-5360	332	6	=	=	SYM
ejpam-5360	332	7	dvrv(q	dvrv(q	PROPN
ejpam-5360	332	8	)	)	PUNCT
ejpam-5360	332	9	+	+	CCONJ
ejpam-5360	332	10	2	2	NUM
ejpam-5360	332	11	(	(	PUNCT
ejpam-5360	332	12	σ	σ	PROPN
ejpam-5360	332	13	u∼v	u∼v	ADJ
ejpam-5360	332	14	d(v	d(v	PROPN
ejpam-5360	332	15	)	)	PUNCT
ejpam-5360	332	16	)	)	PUNCT
ejpam-5360	333	1	=	=	PUNCT
ejpam-5360	333	2	2(d2v	2(d2v	NOUN
ejpam-5360	333	3	+	+	CCONJ
ejpam-5360	333	4	σ	σ	NOUN
ejpam-5360	333	5	u∼v	u∼v	ADJ
ejpam-5360	333	6	d(v	d(v	PROPN
ejpam-5360	333	7	)	)	PUNCT
ejpam-5360	333	8	)	)	PUNCT
ejpam-5360	334	1	=	=	PUNCT
ejpam-5360	334	2	2(d2v	2(d2v	NOUN
ejpam-5360	334	3	+	+	CCONJ
ejpam-5360	334	4	(	(	PUNCT
ejpam-5360	334	5	2m−	2m−	PROPN
ejpam-5360	334	6	d(v)−	d(v)−	PROPN
ejpam-5360	334	7	σ	σ	PROPN
ejpam-5360	334	8	uv/∈e(g	uv/∈e(g	PROPN
ejpam-5360	334	9	)	)	PUNCT
ejpam-5360	334	10	d(u	d(u	PROPN
ejpam-5360	334	11	)	)	PUNCT
ejpam-5360	334	12	)	)	PUNCT
ejpam-5360	334	13	(	(	PUNCT
ejpam-5360	334	14	19	19	NUM
ejpam-5360	334	15	)	)	PUNCT
ejpam-5360	334	16	from	from	ADP
ejpam-5360	334	17	lemma	lemma	PROPN
ejpam-5360	334	18	1	1	NUM
ejpam-5360	334	19	,	,	PUNCT
ejpam-5360	334	20	we	we	PRON
ejpam-5360	334	21	have	have	VERB
ejpam-5360	334	22	√	√	NUM
ejpam-5360	334	23	2	2	NUM
ejpam-5360	334	24	min	min	NOUN
ejpam-5360	334	25	v∈v	v∈v	NOUN
ejpam-5360	334	26	(	(	PUNCT
ejpam-5360	334	27	g	g	NOUN
ejpam-5360	334	28	)	)	PUNCT
ejpam-5360	334	29	√	√	NOUN
ejpam-5360	334	30	d2(v	d2(v	NOUN
ejpam-5360	334	31	)	)	PUNCT
ejpam-5360	334	32	+	+	CCONJ
ejpam-5360	334	33	∑	∑	PUNCT
ejpam-5360	334	34	uv∈e(g	uv∈e(g	NOUN
ejpam-5360	334	35	)	)	PUNCT
ejpam-5360	334	36	d(u	d(u	PROPN
ejpam-5360	334	37	)	)	PUNCT
ejpam-5360	334	38	≤	≤	NUM
ejpam-5360	334	39	µ1(q(g	µ1(q(g	NUM
ejpam-5360	334	40	)	)	PUNCT
ejpam-5360	334	41	)	)	PUNCT
ejpam-5360	334	42	≤	≤	NUM
ejpam-5360	334	43	√	√	NUM
ejpam-5360	334	44	2	2	NUM
ejpam-5360	334	45	max	max	NOUN
ejpam-5360	334	46	v∈v	v∈v	NOUN
ejpam-5360	334	47	(	(	PUNCT
ejpam-5360	334	48	g	g	NOUN
ejpam-5360	334	49	)	)	PUNCT
ejpam-5360	334	50	√√√√d2(v	√√√√d2(v	PROPN
ejpam-5360	334	51	)	)	PUNCT
ejpam-5360	335	1	+	+	CCONJ
ejpam-5360	335	2	(	(	PUNCT
ejpam-5360	335	3	2m−	2m−	PROPN
ejpam-5360	335	4	d(v)−	d(v)−	PROPN
ejpam-5360	335	5	∑	∑	PROPN
ejpam-5360	335	6	uv/∈e(g	uv/∈e(g	PROPN
ejpam-5360	335	7	)	)	PUNCT
ejpam-5360	335	8	d(u	d(u	PROPN
ejpam-5360	335	9	)	)	PUNCT
ejpam-5360	335	10	)	)	PUNCT
ejpam-5360	336	1	(	(	PUNCT
ejpam-5360	336	2	20	20	NUM
ejpam-5360	336	3	)	)	PUNCT
ejpam-5360	336	4	in	in	ADP
ejpam-5360	336	5	this	this	DET
ejpam-5360	336	6	graph	graph	NOUN
ejpam-5360	336	7	g	g	NOUN
ejpam-5360	336	8	,	,	PUNCT
ejpam-5360	336	9	consider	consider	VERB
ejpam-5360	336	10	the	the	DET
ejpam-5360	336	11	vertex	vertex	NOUN
ejpam-5360	336	12	v	v	ADP
ejpam-5360	336	13	taken	take	VERB
ejpam-5360	336	14	from	from	ADP
ejpam-5360	336	15	satellite	satellite	NOUN
ejpam-5360	336	16	graph	graph	NOUN
ejpam-5360	336	17	sk	sk	NOUN
ejpam-5360	336	18	,	,	PUNCT
ejpam-5360	336	19	since	since	SCONJ
ejpam-5360	336	20	dαk	dαk	VERB
ejpam-5360	336	21	=	=	SYM
ejpam-5360	336	22	max(dαi	max(dαi	NOUN
ejpam-5360	336	23	)	)	PUNCT
ejpam-5360	336	24	,	,	PUNCT
ejpam-5360	336	25	for	for	ADP
ejpam-5360	336	26	i	i	PROPN
ejpam-5360	336	27	=	=	SYM
ejpam-5360	336	28	1	1	NUM
ejpam-5360	336	29	,	,	PUNCT
ejpam-5360	336	30	2	2	NUM
ejpam-5360	336	31	,	,	PUNCT
ejpam-5360	336	32	...	...	PUNCT
ejpam-5360	336	33	,	,	PUNCT
ejpam-5360	336	34	k.	k.	PROPN
ejpam-5360	336	35	the	the	DET
ejpam-5360	336	36	sum	sum	NOUN
ejpam-5360	336	37	of	of	ADP
ejpam-5360	336	38	the	the	DET
ejpam-5360	336	39	degrees	degree	NOUN
ejpam-5360	336	40	of	of	ADP
ejpam-5360	336	41	vertices	vertex	NOUN
ejpam-5360	336	42	adjacent	adjacent	ADJ
ejpam-5360	336	43	to	to	ADP
ejpam-5360	336	44	v	v	NUM
ejpam-5360	336	45	is	be	AUX
ejpam-5360	336	46	the	the	DET
ejpam-5360	336	47	sum	sum	NOUN
ejpam-5360	336	48	of	of	ADP
ejpam-5360	336	49	the	the	DET
ejpam-5360	336	50	degrees	degree	NOUN
ejpam-5360	336	51	of	of	ADP
ejpam-5360	336	52	the	the	DET
ejpam-5360	336	53	vertices	vertex	NOUN
ejpam-5360	336	54	of	of	ADP
ejpam-5360	336	55	the	the	DET
ejpam-5360	336	56	core	core	NOUN
ejpam-5360	336	57	graph	graph	NOUN
ejpam-5360	336	58	and	and	CCONJ
ejpam-5360	336	59	the	the	DET
ejpam-5360	336	60	remaining	remain	VERB
ejpam-5360	336	61	(	(	PUNCT
ejpam-5360	336	62	αk	αk	INTJ
ejpam-5360	336	63	−	−	ADP
ejpam-5360	336	64	1	1	NUM
ejpam-5360	336	65	)	)	PUNCT
ejpam-5360	336	66	vertices	vertex	NOUN
ejpam-5360	336	67	of	of	ADP
ejpam-5360	336	68	the	the	DET
ejpam-5360	336	69	clique	clique	NOUN
ejpam-5360	336	70	αk	αk	INTJ
ejpam-5360	336	71	∈	∈	PROPN
ejpam-5360	336	72	sk	sk	PROPN
ejpam-5360	336	73	σ	σ	PROPN
ejpam-5360	336	74	u∼v	u∼v	PROPN
ejpam-5360	336	75	d(u	d(u	PROPN
ejpam-5360	336	76	)	)	PUNCT
ejpam-5360	336	77	)	)	PUNCT
ejpam-5360	337	1	=	=	PUNCT
ejpam-5360	337	2	[	[	PUNCT
ejpam-5360	337	3	ϕ	ϕ	X
ejpam-5360	337	4	(	(	PUNCT
ejpam-5360	337	5	k∑	k∑	NOUN
ejpam-5360	337	6	i=1	i=1	PROPN
ejpam-5360	338	1	ηiαi	ηiαi	NOUN
ejpam-5360	339	1	+	+	CCONJ
ejpam-5360	339	2	(	(	PUNCT
ejpam-5360	339	3	ϕ−	ϕ−	PROPN
ejpam-5360	339	4	1	1	NUM
ejpam-5360	339	5	)	)	PUNCT
ejpam-5360	339	6	)	)	PUNCT
ejpam-5360	340	1	+	+	CCONJ
ejpam-5360	340	2	(	(	PUNCT
ejpam-5360	340	3	αk	αk	ADP
ejpam-5360	340	4	−	−	PROPN
ejpam-5360	340	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	340	6	αk	αk	NOUN
ejpam-5360	340	7	−	−	NOUN
ejpam-5360	340	8	1	1	NUM
ejpam-5360	340	9	)	)	PUNCT
ejpam-5360	340	10	]	]	PUNCT
ejpam-5360	340	11	.	.	PUNCT
ejpam-5360	341	1	we	we	PRON
ejpam-5360	341	2	have	have	VERB
ejpam-5360	341	3	µ1(q(g	µ1(q(g	NUM
ejpam-5360	341	4	)	)	PUNCT
ejpam-5360	341	5	)	)	PUNCT
ejpam-5360	341	6	≥	≥	NOUN
ejpam-5360	341	7	√	√	NUM
ejpam-5360	341	8	2	2	NUM
ejpam-5360	341	9	√√√√[(dαk	√√√√[(dαk	ADJ
ejpam-5360	341	10	)	)	PUNCT
ejpam-5360	341	11	2	2	NUM
ejpam-5360	342	1	+	+	CCONJ
ejpam-5360	342	2	ϕ	ϕ	X
ejpam-5360	342	3	(	(	PUNCT
ejpam-5360	342	4	k∑	k∑	NOUN
ejpam-5360	342	5	i=1	i=1	PROPN
ejpam-5360	343	1	ηiαi	ηiαi	NOUN
ejpam-5360	344	1	+	+	CCONJ
ejpam-5360	344	2	(	(	PUNCT
ejpam-5360	344	3	ϕ−	ϕ−	PROPN
ejpam-5360	344	4	1	1	NUM
ejpam-5360	344	5	)	)	PUNCT
ejpam-5360	344	6	)	)	PUNCT
ejpam-5360	345	1	+	+	CCONJ
ejpam-5360	345	2	(	(	PUNCT
ejpam-5360	345	3	αk	αk	ADP
ejpam-5360	345	4	−	−	PROPN
ejpam-5360	345	5	1)(ϕ+	1)(ϕ+	NUM
ejpam-5360	345	6	αk	αk	NOUN
ejpam-5360	345	7	−	−	NOUN
ejpam-5360	345	8	1	1	NUM
ejpam-5360	345	9	)	)	PUNCT
ejpam-5360	345	10	]	]	PUNCT
ejpam-5360	345	11	(	(	PUNCT
ejpam-5360	345	12	21	21	NUM
ejpam-5360	345	13	)	)	PUNCT
ejpam-5360	345	14	similarly	similarly	ADV
ejpam-5360	345	15	consider	consider	VERB
ejpam-5360	345	16	the	the	DET
ejpam-5360	345	17	vertex	vertex	NOUN
ejpam-5360	345	18	v	v	NOUN
ejpam-5360	345	19	from	from	ADP
ejpam-5360	345	20	core	core	PROPN
ejpam-5360	345	21	kϕ	kϕ	PROPN
ejpam-5360	345	22	,	,	PUNCT
ejpam-5360	345	23	the	the	DET
ejpam-5360	345	24	sum	sum	NOUN
ejpam-5360	345	25	of	of	ADP
ejpam-5360	345	26	the	the	DET
ejpam-5360	345	27	degrees	degree	NOUN
ejpam-5360	345	28	of	of	ADP
ejpam-5360	345	29	the	the	DET
ejpam-5360	345	30	vertices	vertex	NOUN
ejpam-5360	345	31	adjacent	adjacent	ADJ
ejpam-5360	345	32	to	to	ADP
ejpam-5360	345	33	v	v	NUM
ejpam-5360	345	34	is	be	AUX
ejpam-5360	345	35	the	the	DET
ejpam-5360	345	36	sum	sum	NOUN
ejpam-5360	345	37	of	of	ADP
ejpam-5360	345	38	the	the	DET
ejpam-5360	345	39	degrees	degree	NOUN
ejpam-5360	345	40	of	of	ADP
ejpam-5360	345	41	the	the	DET
ejpam-5360	345	42	vertices	vertex	NOUN
ejpam-5360	345	43	of	of	ADP
ejpam-5360	345	44	the	the	DET
ejpam-5360	345	45	satellites	satellite	NOUN
ejpam-5360	345	46	,	,	PUNCT
ejpam-5360	345	47	and	and	CCONJ
ejpam-5360	345	48	the	the	DET
ejpam-5360	345	49	remaining	remain	VERB
ejpam-5360	345	50	(	(	PUNCT
ejpam-5360	345	51	ϕ	ϕ	NOUN
ejpam-5360	345	52	−	−	PROPN
ejpam-5360	345	53	1	1	NUM
ejpam-5360	345	54	)	)	PUNCT
ejpam-5360	345	55	vertices	vertex	NOUN
ejpam-5360	345	56	of	of	ADP
ejpam-5360	345	57	kϕ.	kϕ.	NOUN
ejpam-5360	345	58	we	we	PRON
ejpam-5360	345	59	have	have	VERB
ejpam-5360	345	60	,	,	PUNCT
ejpam-5360	345	61	for	for	ADP
ejpam-5360	345	62	v	v	ADP
ejpam-5360	345	63	∈	∈	PROPN
ejpam-5360	345	64	kϕ	kϕ	X
ejpam-5360	345	65	d(v	d(v	PROPN
ejpam-5360	345	66	)	)	PUNCT
ejpam-5360	346	1	=	=	PRON
ejpam-5360	346	2	k∑	k∑	VERB
ejpam-5360	347	1	i=1	i=1	PROPN
ejpam-5360	348	1	ηiαi	ηiαi	NOUN
ejpam-5360	349	1	+	+	CCONJ
ejpam-5360	349	2	(	(	PUNCT
ejpam-5360	349	3	ϕ−	ϕ−	PROPN
ejpam-5360	349	4	1	1	NUM
ejpam-5360	349	5	)	)	PUNCT
ejpam-5360	349	6	.	.	PUNCT
ejpam-5360	350	1	also	also	ADV
ejpam-5360	350	2	,	,	PUNCT
ejpam-5360	350	3	∑	∑	PROPN
ejpam-5360	350	4	u∼v∈kϕ	u∼v∈kϕ	ADJ
ejpam-5360	350	5	d(u	d(u	PROPN
ejpam-5360	350	6	)	)	PUNCT
ejpam-5360	350	7	=	=	PRON
ejpam-5360	351	1	(	(	PUNCT
ejpam-5360	351	2	2m−	2m−	NUM
ejpam-5360	351	3	d(v)−	d(v)−	PROPN
ejpam-5360	351	4	∑	∑	PUNCT
ejpam-5360	351	5	u∼v∈kϕ	u∼v∈kϕ	PROPN
ejpam-5360	351	6	d(u	d(u	PROPN
ejpam-5360	351	7	)	)	PUNCT
ejpam-5360	351	8	)	)	PUNCT
ejpam-5360	351	9	therefore	therefore	ADV
ejpam-5360	351	10	,	,	PUNCT
ejpam-5360	351	11	µ1(q(g	µ1(q(g	NUM
ejpam-5360	351	12	)	)	PUNCT
ejpam-5360	351	13	)	)	PUNCT
ejpam-5360	351	14	≤	≤	NOUN
ejpam-5360	352	1	√	√	ADP
ejpam-5360	352	2	2	2	NUM
ejpam-5360	352	3	√√√√	√√√√	NOUN
ejpam-5360	352	4	(	(	PUNCT
ejpam-5360	352	5	k∑	k∑	NOUN
ejpam-5360	352	6	i=1	i=1	PROPN
ejpam-5360	353	1	ηiαi	ηiαi	NOUN
ejpam-5360	354	1	+	+	CCONJ
ejpam-5360	354	2	(	(	PUNCT
ejpam-5360	354	3	ϕ−	ϕ−	PROPN
ejpam-5360	354	4	1	1	NUM
ejpam-5360	354	5	)	)	PUNCT
ejpam-5360	354	6	)	)	PUNCT
ejpam-5360	354	7	2	2	X
ejpam-5360	355	1	+	+	CCONJ
ejpam-5360	355	2	[	[	PUNCT
ejpam-5360	355	3	2	2	NUM
ejpam-5360	355	4	(	(	PUNCT
ejpam-5360	355	5	k∑	k∑	NOUN
ejpam-5360	355	6	i=1	i=1	PROPN
ejpam-5360	355	7	(	(	PUNCT
ejpam-5360	355	8	ηiαic2	ηiαic2	PROPN
ejpam-5360	355	9	+	+	CCONJ
ejpam-5360	355	10	ϕηiαi	ϕηiαi	NOUN
ejpam-5360	355	11	)	)	PUNCT
ejpam-5360	356	1	+	+	NUM
ejpam-5360	356	2	ϕc2	ϕc2	NOUN
ejpam-5360	356	3	)	)	PUNCT
ejpam-5360	356	4	−	−	PROPN
ejpam-5360	357	1	(	(	PUNCT
ejpam-5360	357	2	k∑	k∑	NOUN
ejpam-5360	357	3	i=1	i=1	PROPN
ejpam-5360	358	1	ηiαi	ηiαi	NOUN
ejpam-5360	359	1	+	+	CCONJ
ejpam-5360	359	2	(	(	PUNCT
ejpam-5360	359	3	ϕ−	ϕ−	PROPN
ejpam-5360	359	4	1	1	NUM
ejpam-5360	359	5	)	)	PUNCT
ejpam-5360	359	6	)	)	PUNCT
ejpam-5360	359	7	]	]	PUNCT
ejpam-5360	359	8	.	.	PUNCT
ejpam-5360	360	1	m.	m.	NOUN
ejpam-5360	360	2	v.	v.	ADP
ejpam-5360	360	3	,	,	PUNCT
ejpam-5360	360	4	k.	k.	PROPN
ejpam-5360	360	5	desikan	desikan	PROPN
ejpam-5360	360	6	/	/	SYM
ejpam-5360	360	7	eur	eur	PROPN
ejpam-5360	360	8	.	.	PUNCT
ejpam-5360	361	1	j.	j.	PROPN
ejpam-5360	361	2	pure	pure	PROPN
ejpam-5360	361	3	appl	appl	PROPN
ejpam-5360	361	4	.	.	PROPN
ejpam-5360	361	5	math	math	PROPN
ejpam-5360	361	6	,	,	PUNCT
ejpam-5360	361	7	18	18	NUM
ejpam-5360	361	8	(	(	PUNCT
ejpam-5360	361	9	4	4	NUM
ejpam-5360	361	10	)	)	PUNCT
ejpam-5360	361	11	(	(	PUNCT
ejpam-5360	361	12	2025	2025	NUM
ejpam-5360	361	13	)	)	PUNCT
ejpam-5360	361	14	,	,	PUNCT
ejpam-5360	361	15	5360	5360	NUM
ejpam-5360	361	16	15	15	NUM
ejpam-5360	361	17	of	of	ADP
ejpam-5360	361	18	20	20	NUM
ejpam-5360	361	19	(	(	PUNCT
ejpam-5360	361	20	22	22	NUM
ejpam-5360	361	21	)	)	PUNCT
ejpam-5360	361	22	in	in	ADP
ejpam-5360	361	23	the	the	DET
ejpam-5360	361	24	following	following	NOUN
ejpam-5360	361	25	theorem	theorem	NOUN
ejpam-5360	361	26	,	,	PUNCT
ejpam-5360	361	27	we	we	PRON
ejpam-5360	361	28	make	make	VERB
ejpam-5360	361	29	use	use	NOUN
ejpam-5360	361	30	of	of	ADP
ejpam-5360	361	31	theorem	theorem	NOUN
ejpam-5360	361	32	3	3	NUM
ejpam-5360	361	33	to	to	PART
ejpam-5360	361	34	obtain	obtain	VERB
ejpam-5360	361	35	the	the	DET
ejpam-5360	361	36	lower	low	ADJ
ejpam-5360	361	37	and	and	CCONJ
ejpam-5360	361	38	theorem	theorem	VERB
ejpam-5360	361	39	4	4	NUM
ejpam-5360	361	40	to	to	PART
ejpam-5360	361	41	obtain	obtain	VERB
ejpam-5360	361	42	the	the	DET
ejpam-5360	361	43	upper	upper	ADJ
ejpam-5360	361	44	bounds	bound	NOUN
ejpam-5360	361	45	for	for	ADP
ejpam-5360	361	46	the	the	DET
ejpam-5360	361	47	generalized	generalize	VERB
ejpam-5360	361	48	core	core	NOUN
ejpam-5360	361	49	-	-	PUNCT
ejpam-5360	361	50	satellite	satellite	NOUN
ejpam-5360	361	51	graph	graph	NOUN
ejpam-5360	361	52	g.	g.	NOUN
ejpam-5360	361	53	in	in	ADP
ejpam-5360	361	54	the	the	DET
ejpam-5360	361	55	signless	signless	ADJ
ejpam-5360	361	56	laplacian	laplacian	ADJ
ejpam-5360	361	57	matrix	matrix	NOUN
ejpam-5360	361	58	q(g	q(g	NOUN
ejpam-5360	361	59	)	)	PUNCT
ejpam-5360	361	60	=	=	PUNCT
ejpam-5360	361	61	a(g	a(g	PROPN
ejpam-5360	361	62	)	)	PUNCT
ejpam-5360	362	1	+	+	NUM
ejpam-5360	363	1	d(g	d(g	NOUN
ejpam-5360	363	2	)	)	PUNCT
ejpam-5360	363	3	of	of	ADP
ejpam-5360	363	4	the	the	DET
ejpam-5360	363	5	graph	graph	NOUN
ejpam-5360	363	6	g	g	PROPN
ejpam-5360	363	7	,	,	PUNCT
ejpam-5360	363	8	the	the	DET
ejpam-5360	363	9	vertices	vertex	NOUN
ejpam-5360	363	10	are	be	AUX
ejpam-5360	363	11	arranged	arrange	VERB
ejpam-5360	363	12	such	such	ADJ
ejpam-5360	363	13	that	that	SCONJ
ejpam-5360	363	14	the	the	DET
ejpam-5360	363	15	top	top	ADJ
ejpam-5360	363	16	ϕ	ϕ	PROPN
ejpam-5360	363	17	rows	row	NOUN
ejpam-5360	363	18	correspond	correspond	VERB
ejpam-5360	363	19	to	to	ADP
ejpam-5360	363	20	the	the	DET
ejpam-5360	363	21	vertices	vertex	NOUN
ejpam-5360	363	22	in	in	ADP
ejpam-5360	363	23	the	the	DET
ejpam-5360	363	24	core	core	NOUN
ejpam-5360	363	25	kϕ	kϕ	PROPN
ejpam-5360	363	26	,	,	PUNCT
ejpam-5360	363	27	followed	follow	VERB
ejpam-5360	363	28	by	by	ADP
ejpam-5360	363	29	the	the	DET
ejpam-5360	363	30	vertices	vertex	NOUN
ejpam-5360	363	31	of	of	ADP
ejpam-5360	363	32	ηk	ηk	ADP
ejpam-5360	363	33	copies	copy	NOUN
ejpam-5360	363	34	of	of	ADP
ejpam-5360	363	35	the	the	DET
ejpam-5360	363	36	cliques	clique	NOUN
ejpam-5360	363	37	kαk	kαk	X
ejpam-5360	363	38	∈	∈	PROPN
ejpam-5360	363	39	sk	sk	NOUN
ejpam-5360	363	40	.	.	PUNCT
ejpam-5360	364	1	the	the	DET
ejpam-5360	364	2	remaining	remain	VERB
ejpam-5360	364	3	rows	row	NOUN
ejpam-5360	364	4	correspond	correspond	VERB
ejpam-5360	364	5	to	to	ADP
ejpam-5360	364	6	vertices	vertex	NOUN
ejpam-5360	364	7	of	of	ADP
ejpam-5360	364	8	sk−1	sk−1	PROPN
ejpam-5360	364	9	,	,	PUNCT
ejpam-5360	364	10	sk−2,	sk−2,	ADV
ejpam-5360	364	11	...	...	PUNCT
ejpam-5360	364	12	,s1	,s1	PUNCT
ejpam-5360	364	13	.	.	PUNCT
ejpam-5360	365	1	let	let	VERB
ejpam-5360	365	2	r∧	r∧	PROPN
ejpam-5360	365	3	1	1	NUM
ejpam-5360	365	4	,	,	PUNCT
ejpam-5360	365	5	r∧	r∧	PROPN
ejpam-5360	365	6	2	2	NUM
ejpam-5360	365	7	,	,	PUNCT
ejpam-5360	365	8	...	...	PUNCT
ejpam-5360	365	9	,	,	PUNCT
ejpam-5360	365	10	r∧	r∧	PROPN
ejpam-5360	365	11	k	k	PROPN
ejpam-5360	365	12	and	and	CCONJ
ejpam-5360	365	13	r∧	r∧	PROPN
ejpam-5360	365	14	ϕ	ϕ	PROPN
ejpam-5360	365	15	be	be	AUX
ejpam-5360	365	16	the	the	DET
ejpam-5360	365	17	row	row	NOUN
ejpam-5360	365	18	sums	sum	NOUN
ejpam-5360	365	19	corresponding	correspond	VERB
ejpam-5360	365	20	to	to	ADP
ejpam-5360	365	21	the	the	DET
ejpam-5360	365	22	vertices	vertex	NOUN
ejpam-5360	365	23	of	of	ADP
ejpam-5360	365	24	the	the	DET
ejpam-5360	365	25	core	core	NOUN
ejpam-5360	365	26	kϕ	kϕ	NOUN
ejpam-5360	365	27	and	and	CCONJ
ejpam-5360	365	28	the	the	DET
ejpam-5360	365	29	satellites	satellite	NOUN
ejpam-5360	365	30	sk	sk	VERB
ejpam-5360	365	31	,	,	PUNCT
ejpam-5360	365	32	sk−1,	sk−1,	NOUN
ejpam-5360	365	33	...	...	PUNCT
ejpam-5360	365	34	,s1	,s1	PUNCT
ejpam-5360	365	35	,	,	PUNCT
ejpam-5360	366	1	i.e.	i.e.	X
ejpam-5360	366	2	r∧	r∧	PROPN
ejpam-5360	366	3	ϕ,1	ϕ,1	PUNCT
ejpam-5360	367	1	=	=	SYM
ejpam-5360	368	1	r∧	r∧	PROPN
ejpam-5360	368	2	ϕ,2	ϕ,2	NOUN
ejpam-5360	368	3	=	=	PUNCT
ejpam-5360	368	4	...	...	PUNCT
ejpam-5360	369	1	=	=	PUNCT
ejpam-5360	369	2	r∧	r∧	ADV
ejpam-5360	369	3	ϕ,ϕ	ϕ,ϕ	INTJ
ejpam-5360	369	4	=	=	SYM
ejpam-5360	369	5	2	2	NUM
ejpam-5360	369	6	k∑	k∑	NOUN
ejpam-5360	369	7	i=1	i=1	PROPN
ejpam-5360	370	1	ηiαi	ηiαi	NOUN
ejpam-5360	371	1	+	+	CCONJ
ejpam-5360	371	2	(	(	PUNCT
ejpam-5360	371	3	ϕ−	ϕ−	PROPN
ejpam-5360	371	4	1	1	NUM
ejpam-5360	371	5	)	)	PUNCT
ejpam-5360	371	6	=	=	VERB
ejpam-5360	371	7	r∧	r∧	PROPN
ejpam-5360	371	8	1	1	NUM
ejpam-5360	371	9	r∧	r∧	PROPN
ejpam-5360	371	10	k,1	k,1	PROPN
ejpam-5360	371	11	=	=	PROPN
ejpam-5360	371	12	r∧	r∧	PROPN
ejpam-5360	371	13	k,2	k,2	PROPN
ejpam-5360	371	14	=	=	X
ejpam-5360	371	15	...	...	PUNCT
ejpam-5360	372	1	=	=	PUNCT
ejpam-5360	372	2	r∧	r∧	PROPN
ejpam-5360	373	1	k	k	PROPN
ejpam-5360	373	2	,	,	PUNCT
ejpam-5360	373	3	ηkαk	ηkαk	NOUN
ejpam-5360	373	4	=	=	NOUN
ejpam-5360	373	5	2(ϕ+	2(ϕ+	NUM
ejpam-5360	373	6	αk	αk	INTJ
ejpam-5360	373	7	−	−	NOUN
ejpam-5360	373	8	1	1	NUM
ejpam-5360	373	9	)	)	PUNCT
ejpam-5360	373	10	=	=	VERB
ejpam-5360	374	1	r∧	r∧	PROPN
ejpam-5360	374	2	2	2	NUM
ejpam-5360	374	3	r∧	r∧	NOUN
ejpam-5360	374	4	k−1,1	k−1,1	NOUN
ejpam-5360	374	5	=	=	X
ejpam-5360	374	6	r∧	r∧	PROPN
ejpam-5360	374	7	k−1,2	k−1,2	PROPN
ejpam-5360	374	8	=	=	PUNCT
ejpam-5360	374	9	...	...	PUNCT
ejpam-5360	375	1	=	=	SYM
ejpam-5360	375	2	r∧	r∧	PROPN
ejpam-5360	375	3	k−1,ηk−1αk−1	k−1,ηk−1αk−1	PROPN
ejpam-5360	375	4	=	=	SYM
ejpam-5360	375	5	2(ϕ+	2(ϕ+	NUM
ejpam-5360	375	6	αk−1	αk−1	NOUN
ejpam-5360	375	7	−	−	NOUN
ejpam-5360	375	8	1	1	NUM
ejpam-5360	375	9	)	)	PUNCT
ejpam-5360	375	10	=	=	VERB
ejpam-5360	376	1	r∧	r∧	PROPN
ejpam-5360	376	2	3	3	NUM
ejpam-5360	376	3	r∧	r∧	NOUN
ejpam-5360	376	4	k−2,1	k−2,1	PROPN
ejpam-5360	376	5	=	=	SYM
ejpam-5360	377	1	r∧	r∧	PROPN
ejpam-5360	377	2	k−2,2	k−2,2	PROPN
ejpam-5360	377	3	=	=	PUNCT
ejpam-5360	377	4	...	...	PUNCT
ejpam-5360	378	1	=	=	PUNCT
ejpam-5360	378	2	r∧	r∧	PROPN
ejpam-5360	378	3	k−2,ηk−2αk−2	k−2,ηk−2αk−2	PROPN
ejpam-5360	378	4	=	=	SYM
ejpam-5360	378	5	2(ϕ+	2(ϕ+	NUM
ejpam-5360	379	1	αk−2	αk−2	NOUN
ejpam-5360	379	2	−	−	NOUN
ejpam-5360	379	3	1	1	NUM
ejpam-5360	379	4	)	)	PUNCT
ejpam-5360	379	5	=	=	VERB
ejpam-5360	380	1	r∧	r∧	VERB
ejpam-5360	380	2	4	4	NUM
ejpam-5360	380	3	...	...	PUNCT
ejpam-5360	380	4	...	...	PUNCT
ejpam-5360	381	1	in	in	ADP
ejpam-5360	381	2	general	general	ADJ
ejpam-5360	381	3	for	for	ADP
ejpam-5360	381	4	the	the	DET
ejpam-5360	381	5	jth	jth	PROPN
ejpam-5360	381	6	term	term	NOUN
ejpam-5360	381	7	,	,	PUNCT
ejpam-5360	381	8	we	we	PRON
ejpam-5360	381	9	have	have	VERB
ejpam-5360	381	10	rk−(j−2),1	rk−(j−2),1	NOUN
ejpam-5360	381	11	=	=	SYM
ejpam-5360	381	12	rk−(j−2),2	rk−(j−2),2	PROPN
ejpam-5360	381	13	=	=	PUNCT
ejpam-5360	381	14	...	...	PUNCT
ejpam-5360	382	1	=	=	PUNCT
ejpam-5360	382	2	rk−(j−2),ηk−(j−2)αk−(j−2	rk−(j−2),ηk−(j−2)αk−(j−2	PROPN
ejpam-5360	382	3	)	)	PUNCT
ejpam-5360	382	4	=	=	SYM
ejpam-5360	382	5	2(ϕ+	2(ϕ+	NUM
ejpam-5360	382	6	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	382	7	)	)	PUNCT
ejpam-5360	382	8	−	−	ADP
ejpam-5360	382	9	1	1	NUM
ejpam-5360	382	10	)	)	PUNCT
ejpam-5360	382	11	=	=	PUNCT
ejpam-5360	383	1	r∧	r∧	PROPN
ejpam-5360	383	2	j	j	PROPN
ejpam-5360	383	3	..	..	PUNCT
ejpam-5360	383	4	..	..	PUNCT
ejpam-5360	384	1	r1,1	r1,1	NOUN
ejpam-5360	384	2	=	=	PUNCT
ejpam-5360	384	3	r1,2	r1,2	ADJ
ejpam-5360	384	4	=	=	PUNCT
ejpam-5360	384	5	...	...	PUNCT
ejpam-5360	385	1	=	=	PUNCT
ejpam-5360	385	2	r1,ηkαk	r1,ηkαk	NUM
ejpam-5360	385	3	=	=	SYM
ejpam-5360	385	4	2(ϕ+	2(ϕ+	NUM
ejpam-5360	385	5	α1	α1	NOUN
ejpam-5360	385	6	−	−	PROPN
ejpam-5360	385	7	1	1	NUM
ejpam-5360	385	8	)	)	PUNCT
ejpam-5360	385	9	=	=	VERB
ejpam-5360	386	1	r∧	r∧	PROPN
ejpam-5360	386	2	k+1	k+1	X
ejpam-5360	386	3	theorem	theorem	VERB
ejpam-5360	386	4	11	11	NUM
ejpam-5360	386	5	.	.	PUNCT
ejpam-5360	387	1	for	for	ADP
ejpam-5360	387	2	the	the	DET
ejpam-5360	387	3	generalized	generalize	VERB
ejpam-5360	387	4	core	core	NOUN
ejpam-5360	387	5	-	-	PUNCT
ejpam-5360	387	6	satellite	satellite	NOUN
ejpam-5360	387	7	graph	graph	NOUN
ejpam-5360	387	8	g	g	NOUN
ejpam-5360	387	9	of	of	ADP
ejpam-5360	387	10	order	order	NOUN
ejpam-5360	387	11	n	n	NOUN
ejpam-5360	387	12	and	and	CCONJ
ejpam-5360	387	13	size	size	NOUN
ejpam-5360	387	14	m	m	PROPN
ejpam-5360	387	15	,	,	PUNCT
ejpam-5360	387	16	let	let	VERB
ejpam-5360	387	17	s1	s1	NOUN
ejpam-5360	387	18	,	,	PUNCT
ejpam-5360	387	19	s2	s2	PROPN
ejpam-5360	387	20	,	,	PUNCT
ejpam-5360	387	21	s3	s3	PROPN
ejpam-5360	387	22	,	,	PUNCT
ejpam-5360	387	23	...	...	PUNCT
ejpam-5360	387	24	,	,	PUNCT
ejpam-5360	387	25	sk	sk	X
ejpam-5360	387	26	be	be	AUX
ejpam-5360	387	27	the	the	DET
ejpam-5360	387	28	k	k	PROPN
ejpam-5360	387	29	satellites	satellite	NOUN
ejpam-5360	387	30	connected	connect	VERB
ejpam-5360	387	31	to	to	ADP
ejpam-5360	387	32	the	the	DET
ejpam-5360	387	33	core	core	NOUN
ejpam-5360	387	34	graph	graph	NOUN
ejpam-5360	387	35	kϕ.	kϕ.	NOUN
ejpam-5360	387	36	let	let	VERB
ejpam-5360	387	37	r∧	r∧	PROPN
ejpam-5360	387	38	1	1	NUM
ejpam-5360	387	39	,	,	PUNCT
ejpam-5360	387	40	r	r	NOUN
ejpam-5360	387	41	∧	∧	PROPN
ejpam-5360	387	42	2	2	NUM
ejpam-5360	387	43	,	,	PUNCT
ejpam-5360	387	44	...	...	PUNCT
ejpam-5360	387	45	,	,	PUNCT
ejpam-5360	387	46	r	r	NOUN
ejpam-5360	387	47	∧	∧	PROPN
ejpam-5360	387	48	k+1	k+1	AUX
ejpam-5360	387	49	be	be	AUX
ejpam-5360	387	50	the	the	DET
ejpam-5360	387	51	row	row	NOUN
ejpam-5360	387	52	sums	sum	NOUN
ejpam-5360	387	53	corresponding	correspond	VERB
ejpam-5360	387	54	to	to	ADP
ejpam-5360	387	55	the	the	DET
ejpam-5360	387	56	core	core	NOUN
ejpam-5360	387	57	kϕ	kϕ	NOUN
ejpam-5360	387	58	and	and	CCONJ
ejpam-5360	387	59	the	the	DET
ejpam-5360	387	60	satellites	satellite	NOUN
ejpam-5360	387	61	sk	sk	VERB
ejpam-5360	387	62	,	,	PUNCT
ejpam-5360	387	63	sk−1,	sk−1,	NOUN
ejpam-5360	387	64	...	...	PUNCT
ejpam-5360	387	65	,s1	,s1	PUNCT
ejpam-5360	387	66	.	.	PUNCT
ejpam-5360	388	1	the	the	DET
ejpam-5360	388	2	row	row	NOUN
ejpam-5360	388	3	sums	sum	NOUN
ejpam-5360	388	4	are	be	AUX
ejpam-5360	388	5	such	such	ADJ
ejpam-5360	388	6	that	that	SCONJ
ejpam-5360	388	7	r∧	r∧	PROPN
ejpam-5360	388	8	1	1	NUM
ejpam-5360	388	9	≥	≥	NOUN
ejpam-5360	388	10	r∧	r∧	PROPN
ejpam-5360	388	11	2	2	NUM
ejpam-5360	388	12	≥	≥	NOUN
ejpam-5360	388	13	....	....	PUNCT
ejpam-5360	388	14	≥	≥	PROPN
ejpam-5360	388	15	r∧	r∧	PROPN
ejpam-5360	388	16	k+1	k+1	X
ejpam-5360	388	17	,	,	PUNCT
ejpam-5360	388	18	then	then	ADV
ejpam-5360	388	19	µ1(q(g	µ1(q(g	NUM
ejpam-5360	388	20	)	)	PUNCT
ejpam-5360	388	21	)	)	PUNCT
ejpam-5360	388	22	≥	≥	NOUN
ejpam-5360	388	23	(	(	PUNCT
ejpam-5360	388	24	3ϕ+	3ϕ+	NUM
ejpam-5360	388	25	2αk−(j−2	2αk−(j−2	NUM
ejpam-5360	388	26	)	)	PUNCT
ejpam-5360	388	27	+	+	NUM
ejpam-5360	388	28	α1	α1	PROPN
ejpam-5360	388	29	−	−	PROPN
ejpam-5360	388	30	3	3	NUM
ejpam-5360	388	31	)	)	PUNCT
ejpam-5360	388	32	2	2	NUM
ejpam-5360	389	1	+	+	CCONJ
ejpam-5360	389	2	√	√	PROPN
ejpam-5360	389	3	(	(	PUNCT
ejpam-5360	389	4	ϕ+	ϕ+	NOUN
ejpam-5360	389	5	2αk−(j−2	2αk−(j−2	NUM
ejpam-5360	389	6	)	)	PUNCT
ejpam-5360	390	1	−	−	PROPN
ejpam-5360	390	2	α1)2	α1)2	NUM
ejpam-5360	390	3	2	2	NUM
ejpam-5360	390	4	(	(	PUNCT
ejpam-5360	390	5	23	23	NUM
ejpam-5360	390	6	)	)	PUNCT
ejpam-5360	390	7	where	where	SCONJ
ejpam-5360	390	8	1	1	NUM
ejpam-5360	390	9	≤	≤	NUM
ejpam-5360	390	10	i	i	NOUN
ejpam-5360	390	11	≤	≤	NOUN
ejpam-5360	391	1	k	k	PROPN
ejpam-5360	391	2	and	and	CCONJ
ejpam-5360	391	3	3	3	NUM
ejpam-5360	391	4	≤	≤	NUM
ejpam-5360	391	5	j	j	PROPN
ejpam-5360	391	6	≤	≤	PROPN
ejpam-5360	392	1	k	k	PROPN
ejpam-5360	392	2	+	+	NOUN
ejpam-5360	392	3	1	1	X
ejpam-5360	392	4	.	.	X
ejpam-5360	392	5	proof	proof	NOUN
ejpam-5360	392	6	.	.	PUNCT
ejpam-5360	393	1	to	to	PART
ejpam-5360	393	2	prove	prove	VERB
ejpam-5360	393	3	this	this	DET
ejpam-5360	393	4	theorem	theorem	ADJ
ejpam-5360	393	5	,	,	PUNCT
ejpam-5360	393	6	applying	apply	VERB
ejpam-5360	393	7	theorem	theorem	NOUN
ejpam-5360	393	8	3	3	NUM
ejpam-5360	393	9	for	for	ADP
ejpam-5360	393	10	the	the	DET
ejpam-5360	393	11	signless	signless	ADJ
ejpam-5360	393	12	laplacian	laplacian	ADJ
ejpam-5360	393	13	matrix	matrix	NOUN
ejpam-5360	393	14	q(g	q(g	NOUN
ejpam-5360	393	15	)	)	PUNCT
ejpam-5360	394	1	=	=	PUNCT
ejpam-5360	394	2	a(g	a(g	PROPN
ejpam-5360	394	3	)	)	PUNCT
ejpam-5360	395	1	+	+	NOUN
ejpam-5360	395	2	d(g	d(g	NOUN
ejpam-5360	395	3	)	)	PUNCT
ejpam-5360	395	4	of	of	ADP
ejpam-5360	395	5	the	the	DET
ejpam-5360	395	6	graph	graph	NOUN
ejpam-5360	395	7	g	g	NOUN
ejpam-5360	395	8	,	,	PUNCT
ejpam-5360	395	9	we	we	PRON
ejpam-5360	395	10	have	have	VERB
ejpam-5360	395	11	µ1(q(g	µ1(q(g	NUM
ejpam-5360	395	12	)	)	PUNCT
ejpam-5360	395	13	)	)	PUNCT
ejpam-5360	396	1	≥	≥	NOUN
ejpam-5360	396	2	(	(	PUNCT
ejpam-5360	396	3	r∧	r∧	PROPN
ejpam-5360	396	4	l	l	PROPN
ejpam-5360	396	5	+	+	SYM
ejpam-5360	396	6	s	s	VERB
ejpam-5360	396	7	−	−	PROPN
ejpam-5360	396	8	t	t	NOUN
ejpam-5360	396	9	)	)	PUNCT
ejpam-5360	397	1	+	+	CCONJ
ejpam-5360	397	2	√	√	INTJ
ejpam-5360	397	3	(	(	PUNCT
ejpam-5360	397	4	r∧	r∧	PROPN
ejpam-5360	397	5	l	l	NOUN
ejpam-5360	397	6	−	−	PROPN
ejpam-5360	397	7	s	s	PART
ejpam-5360	398	1	+	+	NUM
ejpam-5360	398	2	t	t	NOUN
ejpam-5360	398	3	)	)	PUNCT
ejpam-5360	398	4	2	2	NUM
ejpam-5360	398	5	+	+	SYM
ejpam-5360	398	6	4	4	NUM
ejpam-5360	398	7	t	t	NOUN
ejpam-5360	398	8	∑l−1	∑l−1	X
ejpam-5360	398	9	i=1	i=1	X
ejpam-5360	399	1	(	(	PUNCT
ejpam-5360	399	2	r	r	NOUN
ejpam-5360	399	3	∧	∧	PROPN
ejpam-5360	399	4	i	i	NOUN
ejpam-5360	399	5	−r∧	−r∧	NUM
ejpam-5360	399	6	l	l	NOUN
ejpam-5360	399	7	)	)	PUNCT
ejpam-5360	399	8	2	2	NUM
ejpam-5360	399	9	.	.	PUNCT
ejpam-5360	400	1	(	(	PUNCT
ejpam-5360	400	2	24	24	NUM
ejpam-5360	400	3	)	)	PUNCT
ejpam-5360	400	4	m.	m.	NOUN
ejpam-5360	400	5	v.	v.	ADP
ejpam-5360	400	6	,	,	PUNCT
ejpam-5360	400	7	k.	k.	PROPN
ejpam-5360	400	8	desikan	desikan	PROPN
ejpam-5360	400	9	/	/	SYM
ejpam-5360	400	10	eur	eur	PROPN
ejpam-5360	400	11	.	.	PUNCT
ejpam-5360	401	1	j.	j.	PROPN
ejpam-5360	401	2	pure	pure	PROPN
ejpam-5360	401	3	appl	appl	PROPN
ejpam-5360	401	4	.	.	PROPN
ejpam-5360	401	5	math	math	PROPN
ejpam-5360	401	6	,	,	PUNCT
ejpam-5360	401	7	18	18	NUM
ejpam-5360	401	8	(	(	PUNCT
ejpam-5360	401	9	4	4	NUM
ejpam-5360	401	10	)	)	PUNCT
ejpam-5360	401	11	(	(	PUNCT
ejpam-5360	401	12	2025	2025	NUM
ejpam-5360	401	13	)	)	PUNCT
ejpam-5360	401	14	,	,	PUNCT
ejpam-5360	401	15	5360	5360	NUM
ejpam-5360	401	16	16	16	NUM
ejpam-5360	401	17	of	of	ADP
ejpam-5360	401	18	20	20	NUM
ejpam-5360	401	19	for	for	ADP
ejpam-5360	401	20	the	the	DET
ejpam-5360	401	21	matrix	matrix	NOUN
ejpam-5360	401	22	q(g	q(g	PROPN
ejpam-5360	401	23	)	)	PUNCT
ejpam-5360	402	1	,	,	PUNCT
ejpam-5360	402	2	we	we	PRON
ejpam-5360	402	3	observe	observe	VERB
ejpam-5360	402	4	that	that	SCONJ
ejpam-5360	402	5	the	the	DET
ejpam-5360	402	6	smallest	small	ADJ
ejpam-5360	402	7	diagonal	diagonal	ADJ
ejpam-5360	402	8	element	element	NOUN
ejpam-5360	402	9	s	s	PART
ejpam-5360	402	10	=	=	PUNCT
ejpam-5360	402	11	(	(	PUNCT
ejpam-5360	402	12	ϕ+α1−1	ϕ+α1−1	PROPN
ejpam-5360	402	13	)	)	PUNCT
ejpam-5360	402	14	,	,	PUNCT
ejpam-5360	402	15	and	and	CCONJ
ejpam-5360	402	16	the	the	DET
ejpam-5360	402	17	smallest	small	ADJ
ejpam-5360	402	18	non	non	ADJ
ejpam-5360	402	19	-	-	ADJ
ejpam-5360	402	20	diagonal	diagonal	ADJ
ejpam-5360	402	21	element	element	NOUN
ejpam-5360	402	22	t	t	NOUN
ejpam-5360	402	23	=	=	SYM
ejpam-5360	402	24	0	0	X
ejpam-5360	402	25	.	.	PUNCT
ejpam-5360	403	1	here	here	ADV
ejpam-5360	403	2	we	we	PRON
ejpam-5360	403	3	discuss	discuss	VERB
ejpam-5360	403	4	the	the	DET
ejpam-5360	403	5	cases	case	NOUN
ejpam-5360	403	6	corresponding	correspond	VERB
ejpam-5360	403	7	to	to	ADP
ejpam-5360	403	8	the	the	DET
ejpam-5360	403	9	row	row	NOUN
ejpam-5360	403	10	sums	sum	NOUN
ejpam-5360	403	11	r∧	r∧	NOUN
ejpam-5360	403	12	l	l	PROPN
ejpam-5360	403	13	=	=	PUNCT
ejpam-5360	404	1	r∧	r∧	PROPN
ejpam-5360	404	2	2	2	NUM
ejpam-5360	404	3	and	and	CCONJ
ejpam-5360	404	4	r∧	r∧	PROPN
ejpam-5360	404	5	j	j	PROPN
ejpam-5360	404	6	.	.	PUNCT
ejpam-5360	405	1	also	also	ADV
ejpam-5360	405	2	r∧	r∧	PROPN
ejpam-5360	405	3	l	l	PROPN
ejpam-5360	405	4	̸=	̸=	PROPN
ejpam-5360	405	5	r∧	r∧	VERB
ejpam-5360	405	6	1	1	NUM
ejpam-5360	405	7	as	as	SCONJ
ejpam-5360	405	8	the	the	DET
ejpam-5360	405	9	lower	low	ADJ
ejpam-5360	405	10	bound	bind	VERB
ejpam-5360	405	11	is	be	AUX
ejpam-5360	405	12	obtained	obtain	VERB
ejpam-5360	405	13	by	by	ADP
ejpam-5360	405	14	considering	consider	VERB
ejpam-5360	405	15	the	the	DET
ejpam-5360	405	16	row	row	NOUN
ejpam-5360	405	17	sum	sum	NOUN
ejpam-5360	405	18	of	of	ADP
ejpam-5360	405	19	any	any	DET
ejpam-5360	405	20	vertex	vertex	NOUN
ejpam-5360	405	21	vi	vi	NOUN
ejpam-5360	405	22	∈	∈	PROPN
ejpam-5360	405	23	si	si	X
ejpam-5360	405	24	.	.	PUNCT
ejpam-5360	406	1	the	the	DET
ejpam-5360	406	2	row	row	NOUN
ejpam-5360	406	3	sums	sum	NOUN
ejpam-5360	406	4	of	of	ADP
ejpam-5360	406	5	the	the	DET
ejpam-5360	406	6	vertices	vertex	NOUN
ejpam-5360	406	7	belonging	belong	VERB
ejpam-5360	406	8	to	to	ADP
ejpam-5360	406	9	the	the	DET
ejpam-5360	406	10	satellite	satellite	NOUN
ejpam-5360	406	11	sk	sk	NOUN
ejpam-5360	406	12	are	be	AUX
ejpam-5360	406	13	greater	great	ADJ
ejpam-5360	406	14	compared	compare	VERB
ejpam-5360	406	15	to	to	ADP
ejpam-5360	406	16	the	the	DET
ejpam-5360	406	17	row	row	NOUN
ejpam-5360	406	18	sums	sum	NOUN
ejpam-5360	406	19	of	of	ADP
ejpam-5360	406	20	the	the	DET
ejpam-5360	406	21	vertices	vertex	NOUN
ejpam-5360	406	22	belonging	belong	VERB
ejpam-5360	406	23	to	to	ADP
ejpam-5360	406	24	the	the	DET
ejpam-5360	406	25	satellites	satellite	NOUN
ejpam-5360	406	26	sk−1	sk−1	ADV
ejpam-5360	406	27	,	,	PUNCT
ejpam-5360	406	28	sk−2,	sk−2,	ADV
ejpam-5360	406	29	...	...	PUNCT
ejpam-5360	406	30	,s1	,s1	PUNCT
ejpam-5360	406	31	.	.	PUNCT
ejpam-5360	407	1	case	case	NOUN
ejpam-5360	407	2	(	(	PUNCT
ejpam-5360	407	3	i	i	NOUN
ejpam-5360	407	4	)	)	PUNCT
ejpam-5360	407	5	we	we	PRON
ejpam-5360	407	6	now	now	ADV
ejpam-5360	407	7	consider	consider	VERB
ejpam-5360	407	8	r∧	r∧	PROPN
ejpam-5360	407	9	l	l	PROPN
ejpam-5360	407	10	=	=	PUNCT
ejpam-5360	407	11	r∧	r∧	PROPN
ejpam-5360	407	12	2	2	NUM
ejpam-5360	407	13	=	=	SYM
ejpam-5360	407	14	2(ϕ	2(ϕ	NUM
ejpam-5360	408	1	+	+	CCONJ
ejpam-5360	408	2	αk	αk	ADP
ejpam-5360	408	3	−	−	NUM
ejpam-5360	408	4	1	1	NUM
ejpam-5360	408	5	)	)	PUNCT
ejpam-5360	408	6	and	and	CCONJ
ejpam-5360	408	7	by	by	ADP
ejpam-5360	408	8	substituting	substitute	VERB
ejpam-5360	408	9	the	the	DET
ejpam-5360	408	10	values	value	NOUN
ejpam-5360	408	11	of	of	ADP
ejpam-5360	408	12	r∧	r∧	PROPN
ejpam-5360	408	13	l	l	PROPN
ejpam-5360	408	14	and	and	CCONJ
ejpam-5360	408	15	t	t	PROPN
ejpam-5360	408	16	in	in	ADP
ejpam-5360	408	17	equation	equation	NOUN
ejpam-5360	408	18	(	(	PUNCT
ejpam-5360	408	19	24	24	NUM
ejpam-5360	408	20	)	)	PUNCT
ejpam-5360	408	21	,	,	PUNCT
ejpam-5360	408	22	we	we	PRON
ejpam-5360	408	23	get	get	VERB
ejpam-5360	408	24	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	408	25	)	)	PUNCT
ejpam-5360	408	26	)	)	PUNCT
ejpam-5360	409	1	≥	≥	NOUN
ejpam-5360	409	2	(	(	PUNCT
ejpam-5360	409	3	r∧	r∧	PROPN
ejpam-5360	409	4	2	2	NUM
ejpam-5360	409	5	+	+	SYM
ejpam-5360	409	6	s	s	X
ejpam-5360	409	7	)	)	PUNCT
ejpam-5360	409	8	+	+	CCONJ
ejpam-5360	409	9	√	√	INTJ
ejpam-5360	409	10	(	(	PUNCT
ejpam-5360	409	11	r∧	r∧	PROPN
ejpam-5360	409	12	2	2	NUM
ejpam-5360	409	13	−	−	NOUN
ejpam-5360	409	14	s)2	s)2	NOUN
ejpam-5360	409	15	2	2	NUM
ejpam-5360	409	16	now	now	ADV
ejpam-5360	409	17	substituting	substitute	VERB
ejpam-5360	409	18	for	for	ADP
ejpam-5360	409	19	r∧	r∧	PROPN
ejpam-5360	409	20	2	2	NUM
ejpam-5360	409	21	and	and	CCONJ
ejpam-5360	409	22	s	s	X
ejpam-5360	409	23	in	in	ADP
ejpam-5360	409	24	the	the	DET
ejpam-5360	409	25	above	above	ADJ
ejpam-5360	409	26	equation	equation	NOUN
ejpam-5360	409	27	,	,	PUNCT
ejpam-5360	409	28	we	we	PRON
ejpam-5360	409	29	get	get	VERB
ejpam-5360	409	30	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	409	31	)	)	PUNCT
ejpam-5360	409	32	)	)	PUNCT
ejpam-5360	410	1	≥	≥	NOUN
ejpam-5360	410	2	(	(	PUNCT
ejpam-5360	410	3	3ϕ+	3ϕ+	NUM
ejpam-5360	410	4	2αk	2αk	ADJ
ejpam-5360	410	5	+	+	CCONJ
ejpam-5360	410	6	α1	α1	PROPN
ejpam-5360	410	7	−	−	PROPN
ejpam-5360	410	8	3	3	NUM
ejpam-5360	410	9	)	)	PUNCT
ejpam-5360	410	10	2	2	NUM
ejpam-5360	410	11	+	+	CCONJ
ejpam-5360	410	12	√	√	PROPN
ejpam-5360	410	13	(	(	PUNCT
ejpam-5360	410	14	ϕ+	ϕ+	NOUN
ejpam-5360	410	15	2αk	2αk	ADJ
ejpam-5360	410	16	−	−	PROPN
ejpam-5360	410	17	α1	α1	PROPN
ejpam-5360	410	18	−	−	PROPN
ejpam-5360	410	19	1)2	1)2	NUM
ejpam-5360	410	20	2	2	NUM
ejpam-5360	410	21	.	.	PUNCT
ejpam-5360	411	1	(	(	PUNCT
ejpam-5360	411	2	25	25	NUM
ejpam-5360	411	3	)	)	PUNCT
ejpam-5360	411	4	case	case	NOUN
ejpam-5360	411	5	(	(	PUNCT
ejpam-5360	411	6	ii	ii	NOUN
ejpam-5360	411	7	)	)	PUNCT
ejpam-5360	411	8	we	we	PRON
ejpam-5360	411	9	discuss	discuss	VERB
ejpam-5360	411	10	the	the	DET
ejpam-5360	411	11	case	case	NOUN
ejpam-5360	411	12	for	for	ADP
ejpam-5360	411	13	r∧	r∧	PROPN
ejpam-5360	411	14	l	l	PROPN
ejpam-5360	411	15	=	=	PUNCT
ejpam-5360	412	1	r∧	r∧	PROPN
ejpam-5360	412	2	j	j	PROPN
ejpam-5360	412	3	=	=	PUNCT
ejpam-5360	412	4	2(ϕ	2(ϕ	NUM
ejpam-5360	412	5	+	+	CCONJ
ejpam-5360	412	6	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	412	7	)	)	PUNCT
ejpam-5360	412	8	−	−	ADP
ejpam-5360	412	9	1	1	NUM
ejpam-5360	412	10	)	)	PUNCT
ejpam-5360	412	11	,	,	PUNCT
ejpam-5360	412	12	the	the	DET
ejpam-5360	412	13	row	row	NOUN
ejpam-5360	412	14	sum	sum	NOUN
ejpam-5360	412	15	of	of	ADP
ejpam-5360	412	16	the	the	DET
ejpam-5360	412	17	vertex	vertex	NOUN
ejpam-5360	412	18	belonging	belong	VERB
ejpam-5360	412	19	to	to	ADP
ejpam-5360	412	20	the	the	DET
ejpam-5360	412	21	satellite	satellite	PROPN
ejpam-5360	412	22	sk−(j−2	sk−(j−2	PROPN
ejpam-5360	412	23	)	)	PUNCT
ejpam-5360	412	24	,	,	PUNCT
ejpam-5360	412	25	where	where	SCONJ
ejpam-5360	412	26	j	j	PROPN
ejpam-5360	412	27	̸=	̸=	PROPN
ejpam-5360	412	28	2	2	NUM
ejpam-5360	412	29	and	and	CCONJ
ejpam-5360	412	30	3	3	NUM
ejpam-5360	412	31	≤	≤	NUM
ejpam-5360	412	32	j	j	PROPN
ejpam-5360	412	33	≤	≤	PROPN
ejpam-5360	412	34	l.	l.	NOUN
ejpam-5360	412	35	equation	equation	NOUN
ejpam-5360	412	36	(	(	PUNCT
ejpam-5360	412	37	24	24	NUM
ejpam-5360	412	38	)	)	PUNCT
ejpam-5360	412	39	reduces	reduce	VERB
ejpam-5360	412	40	to	to	ADP
ejpam-5360	412	41	µ1(q(g	µ1(q(g	NUM
ejpam-5360	412	42	)	)	PUNCT
ejpam-5360	412	43	)	)	PUNCT
ejpam-5360	412	44	≥	≥	PROPN
ejpam-5360	413	1	(	(	PUNCT
ejpam-5360	413	2	r∧	r∧	PROPN
ejpam-5360	413	3	j	j	PROPN
ejpam-5360	413	4	+	+	PROPN
ejpam-5360	413	5	s	s	X
ejpam-5360	413	6	)	)	PUNCT
ejpam-5360	413	7	+	+	CCONJ
ejpam-5360	413	8	√	√	PROPN
ejpam-5360	413	9	(	(	PUNCT
ejpam-5360	413	10	r∧	r∧	PROPN
ejpam-5360	413	11	j	j	PROPN
ejpam-5360	413	12	−	−	PROPN
ejpam-5360	413	13	s)2	s)2	NOUN
ejpam-5360	413	14	2	2	NUM
ejpam-5360	413	15	.	.	PUNCT
ejpam-5360	414	1	we	we	PRON
ejpam-5360	414	2	see	see	VERB
ejpam-5360	414	3	that	that	SCONJ
ejpam-5360	414	4	the	the	DET
ejpam-5360	414	5	smallest	small	ADJ
ejpam-5360	414	6	diagonal	diagonal	ADJ
ejpam-5360	414	7	element	element	NOUN
ejpam-5360	414	8	of	of	ADP
ejpam-5360	414	9	q(g	q(g	PROPN
ejpam-5360	414	10	)	)	PUNCT
ejpam-5360	414	11	is	be	AUX
ejpam-5360	414	12	s	s	NOUN
ejpam-5360	414	13	=	=	PUNCT
ejpam-5360	414	14	(	(	PUNCT
ejpam-5360	414	15	ϕ+α1−1	ϕ+α1−1	PROPN
ejpam-5360	414	16	)	)	PUNCT
ejpam-5360	414	17	and	and	CCONJ
ejpam-5360	414	18	the	the	DET
ejpam-5360	414	19	smallest	small	ADJ
ejpam-5360	414	20	non	non	ADJ
ejpam-5360	414	21	-	-	ADJ
ejpam-5360	414	22	diagonal	diagonal	ADJ
ejpam-5360	414	23	element	element	NOUN
ejpam-5360	414	24	t	t	NOUN
ejpam-5360	414	25	=	=	SYM
ejpam-5360	414	26	0	0	X
ejpam-5360	414	27	.	.	X
ejpam-5360	415	1	substituting	substitute	VERB
ejpam-5360	415	2	for	for	ADP
ejpam-5360	415	3	r∧	r∧	PROPN
ejpam-5360	415	4	j	j	PROPN
ejpam-5360	415	5	and	and	CCONJ
ejpam-5360	415	6	s	s	PROPN
ejpam-5360	415	7	in	in	ADP
ejpam-5360	415	8	the	the	DET
ejpam-5360	415	9	above	above	ADJ
ejpam-5360	415	10	equation	equation	NOUN
ejpam-5360	415	11	,	,	PUNCT
ejpam-5360	415	12	we	we	PRON
ejpam-5360	415	13	get	get	VERB
ejpam-5360	415	14	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	415	15	)	)	PUNCT
ejpam-5360	415	16	)	)	PUNCT
ejpam-5360	416	1	≥	≥	NOUN
ejpam-5360	416	2	(	(	PUNCT
ejpam-5360	416	3	3ϕ+	3ϕ+	NUM
ejpam-5360	416	4	2αk−(j−2	2αk−(j−2	NUM
ejpam-5360	416	5	)	)	PUNCT
ejpam-5360	416	6	+	+	NUM
ejpam-5360	416	7	α1	α1	PROPN
ejpam-5360	416	8	−	−	PROPN
ejpam-5360	416	9	3	3	NUM
ejpam-5360	416	10	)	)	PUNCT
ejpam-5360	416	11	2	2	NUM
ejpam-5360	416	12	+	+	CCONJ
ejpam-5360	416	13	√	√	PROPN
ejpam-5360	416	14	(	(	PUNCT
ejpam-5360	416	15	ϕ+	ϕ+	NOUN
ejpam-5360	416	16	2αk−(j−2	2αk−(j−2	NUM
ejpam-5360	416	17	)	)	PUNCT
ejpam-5360	416	18	−	−	PROPN
ejpam-5360	416	19	α1	α1	PROPN
ejpam-5360	416	20	−	−	PROPN
ejpam-5360	416	21	1)2	1)2	NUM
ejpam-5360	416	22	2	2	NUM
ejpam-5360	416	23	(	(	PUNCT
ejpam-5360	416	24	26	26	NUM
ejpam-5360	416	25	)	)	PUNCT
ejpam-5360	416	26	hence	hence	ADV
ejpam-5360	416	27	proved	prove	VERB
ejpam-5360	416	28	.	.	PUNCT
ejpam-5360	417	1	theorem	theorem	NOUN
ejpam-5360	417	2	12	12	NUM
ejpam-5360	417	3	.	.	PUNCT
ejpam-5360	418	1	for	for	ADP
ejpam-5360	418	2	the	the	DET
ejpam-5360	418	3	generalized	generalize	VERB
ejpam-5360	418	4	core	core	NOUN
ejpam-5360	418	5	-	-	PUNCT
ejpam-5360	418	6	satellite	satellite	NOUN
ejpam-5360	418	7	graph	graph	NOUN
ejpam-5360	418	8	g	g	NOUN
ejpam-5360	418	9	of	of	ADP
ejpam-5360	418	10	order	order	NOUN
ejpam-5360	418	11	n	n	NOUN
ejpam-5360	418	12	and	and	CCONJ
ejpam-5360	418	13	size	size	NOUN
ejpam-5360	418	14	m	m	PROPN
ejpam-5360	418	15	,	,	PUNCT
ejpam-5360	418	16	let	let	VERB
ejpam-5360	418	17	s1	s1	NOUN
ejpam-5360	418	18	,	,	PUNCT
ejpam-5360	418	19	s2	s2	PROPN
ejpam-5360	418	20	,	,	PUNCT
ejpam-5360	418	21	s3	s3	PROPN
ejpam-5360	418	22	,	,	PUNCT
ejpam-5360	418	23	...	...	PUNCT
ejpam-5360	418	24	,	,	PUNCT
ejpam-5360	418	25	sk	sk	X
ejpam-5360	418	26	be	be	AUX
ejpam-5360	418	27	the	the	DET
ejpam-5360	418	28	k	k	PROPN
ejpam-5360	418	29	satellites	satellite	NOUN
ejpam-5360	418	30	connected	connect	VERB
ejpam-5360	418	31	to	to	ADP
ejpam-5360	418	32	the	the	DET
ejpam-5360	418	33	core	core	NOUN
ejpam-5360	418	34	graph	graph	NOUN
ejpam-5360	418	35	kϕ.	kϕ.	NOUN
ejpam-5360	418	36	let	let	VERB
ejpam-5360	418	37	r∧	r∧	PROPN
ejpam-5360	418	38	1	1	NUM
ejpam-5360	418	39	,	,	PUNCT
ejpam-5360	418	40	r	r	NOUN
ejpam-5360	418	41	∧	∧	PROPN
ejpam-5360	418	42	2	2	NUM
ejpam-5360	418	43	,	,	PUNCT
ejpam-5360	418	44	..	..	PUNCT
ejpam-5360	418	45	,	,	PUNCT
ejpam-5360	418	46	r	r	NOUN
ejpam-5360	418	47	∧	∧	PROPN
ejpam-5360	418	48	k+1	k+1	AUX
ejpam-5360	418	49	be	be	AUX
ejpam-5360	418	50	the	the	DET
ejpam-5360	418	51	row	row	NOUN
ejpam-5360	418	52	sums	sum	NOUN
ejpam-5360	418	53	corresponding	correspond	VERB
ejpam-5360	418	54	to	to	ADP
ejpam-5360	418	55	the	the	DET
ejpam-5360	418	56	core	core	NOUN
ejpam-5360	418	57	kϕ	kϕ	NOUN
ejpam-5360	418	58	and	and	CCONJ
ejpam-5360	418	59	the	the	DET
ejpam-5360	418	60	satellites	satellite	NOUN
ejpam-5360	418	61	sk	sk	VERB
ejpam-5360	418	62	,	,	PUNCT
ejpam-5360	418	63	sk−1,	sk−1,	NOUN
ejpam-5360	418	64	...	...	PUNCT
ejpam-5360	418	65	,s1	,s1	PUNCT
ejpam-5360	418	66	.	.	PUNCT
ejpam-5360	419	1	the	the	DET
ejpam-5360	419	2	row	row	NOUN
ejpam-5360	419	3	sums	sum	NOUN
ejpam-5360	419	4	are	be	AUX
ejpam-5360	419	5	such	such	ADJ
ejpam-5360	419	6	that	that	SCONJ
ejpam-5360	419	7	r∧	r∧	PROPN
ejpam-5360	419	8	1	1	NUM
ejpam-5360	419	9	≥	≥	NOUN
ejpam-5360	419	10	r∧	r∧	PROPN
ejpam-5360	419	11	2	2	NUM
ejpam-5360	419	12	≥	≥	NOUN
ejpam-5360	419	13	...	...	PUNCT
ejpam-5360	419	14	≥	≥	X
ejpam-5360	419	15	r∧	r∧	PROPN
ejpam-5360	419	16	k+1	k+1	X
ejpam-5360	419	17	,	,	PUNCT
ejpam-5360	419	18	then	then	ADV
ejpam-5360	419	19	µ1(q(g	µ1(q(g	NUM
ejpam-5360	419	20	)	)	PUNCT
ejpam-5360	419	21	)	)	PUNCT
ejpam-5360	419	22	≤	≤	NOUN
ejpam-5360	419	23	(	(	PUNCT
ejpam-5360	419	24	n+	n+	NUM
ejpam-5360	419	25	2(ϕ+	2(ϕ+	NUM
ejpam-5360	419	26	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	419	27	)	)	PUNCT
ejpam-5360	419	28	−	−	ADP
ejpam-5360	419	29	2	2	NUM
ejpam-5360	419	30	)	)	PUNCT
ejpam-5360	419	31	)	)	PUNCT
ejpam-5360	419	32	2	2	NUM
ejpam-5360	420	1	+	+	CCONJ
ejpam-5360	420	2	√	√	ADJ
ejpam-5360	420	3	2(ϕ+	2(ϕ+	NUM
ejpam-5360	420	4	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	420	5	)	)	PUNCT
ejpam-5360	420	6	−	−	PROPN
ejpam-5360	420	7	n)2	n)2	NOUN
ejpam-5360	420	8	+	+	CCONJ
ejpam-5360	420	9	8	8	NUM
ejpam-5360	420	10	{	{	PUNCT
ejpam-5360	420	11	ϕ	ϕ	NOUN
ejpam-5360	420	12	(	(	PUNCT
ejpam-5360	420	13	∑k	∑k	PROPN
ejpam-5360	420	14	i=1	i=1	PROPN
ejpam-5360	420	15	ηiαi	ηiαi	VERB
ejpam-5360	420	16	−	−	PROPN
ejpam-5360	420	17	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	420	18	)	)	PUNCT
ejpam-5360	420	19	)	)	PUNCT
ejpam-5360	421	1	+	+	CCONJ
ejpam-5360	421	2	∑j−3	∑j−3	ADP
ejpam-5360	421	3	i=0	i=0	ADJ
ejpam-5360	421	4	ηk−iαk−i(αk−i	ηk−iαk−i(αk−i	NOUN
ejpam-5360	421	5	−	−	NOUN
ejpam-5360	421	6	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	421	7	)	)	PUNCT
ejpam-5360	421	8	)	)	PUNCT
ejpam-5360	421	9	}	}	PUNCT
ejpam-5360	421	10	2	2	NUM
ejpam-5360	421	11	where	where	SCONJ
ejpam-5360	421	12	3	3	NUM
ejpam-5360	421	13	≤	≤	NUM
ejpam-5360	421	14	j	j	PROPN
ejpam-5360	421	15	≤	≤	PROPN
ejpam-5360	422	1	k	k	PROPN
ejpam-5360	423	1	+	+	PUNCT
ejpam-5360	423	2	1	1	X
ejpam-5360	423	3	.	.	X
ejpam-5360	423	4	m.	m.	NOUN
ejpam-5360	424	1	v.	v.	ADP
ejpam-5360	424	2	,	,	PUNCT
ejpam-5360	424	3	k.	k.	PROPN
ejpam-5360	424	4	desikan	desikan	PROPN
ejpam-5360	424	5	/	/	SYM
ejpam-5360	424	6	eur	eur	PROPN
ejpam-5360	424	7	.	.	PUNCT
ejpam-5360	425	1	j.	j.	PROPN
ejpam-5360	425	2	pure	pure	PROPN
ejpam-5360	425	3	appl	appl	PROPN
ejpam-5360	425	4	.	.	PROPN
ejpam-5360	425	5	math	math	PROPN
ejpam-5360	425	6	,	,	PUNCT
ejpam-5360	425	7	18	18	NUM
ejpam-5360	425	8	(	(	PUNCT
ejpam-5360	425	9	4	4	NUM
ejpam-5360	425	10	)	)	PUNCT
ejpam-5360	425	11	(	(	PUNCT
ejpam-5360	425	12	2025	2025	NUM
ejpam-5360	425	13	)	)	PUNCT
ejpam-5360	425	14	,	,	PUNCT
ejpam-5360	425	15	5360	5360	NUM
ejpam-5360	425	16	17	17	NUM
ejpam-5360	425	17	of	of	ADP
ejpam-5360	425	18	20	20	NUM
ejpam-5360	425	19	proof	proof	NOUN
ejpam-5360	425	20	.	.	PUNCT
ejpam-5360	426	1	we	we	PRON
ejpam-5360	426	2	apply	apply	VERB
ejpam-5360	426	3	theorem	theorem	VERB
ejpam-5360	426	4	4	4	NUM
ejpam-5360	426	5	to	to	PART
ejpam-5360	426	6	prove	prove	VERB
ejpam-5360	426	7	our	our	PRON
ejpam-5360	426	8	result	result	NOUN
ejpam-5360	426	9	.	.	PUNCT
ejpam-5360	427	1	we	we	PRON
ejpam-5360	427	2	apply	apply	VERB
ejpam-5360	427	3	the	the	DET
ejpam-5360	427	4	values	value	NOUN
ejpam-5360	427	5	of	of	ADP
ejpam-5360	427	6	m	m	NOUN
ejpam-5360	427	7	=	=	PUNCT
ejpam-5360	428	1	(	(	PUNCT
ejpam-5360	428	2	∑k	∑k	PROPN
ejpam-5360	428	3	i=1	i=1	PROPN
ejpam-5360	429	1	ηiαi	ηiαi	VERB
ejpam-5360	430	1	+	+	CCONJ
ejpam-5360	430	2	(	(	PUNCT
ejpam-5360	430	3	ϕ	ϕ	NOUN
ejpam-5360	430	4	−	−	PROPN
ejpam-5360	430	5	1	1	NUM
ejpam-5360	430	6	)	)	PUNCT
ejpam-5360	430	7	)	)	PUNCT
ejpam-5360	430	8	,	,	PUNCT
ejpam-5360	430	9	the	the	DET
ejpam-5360	430	10	largest	large	ADJ
ejpam-5360	430	11	diagonal	diagonal	ADJ
ejpam-5360	430	12	element	element	NOUN
ejpam-5360	430	13	and	and	CCONJ
ejpam-5360	430	14	n	n	NOUN
ejpam-5360	430	15	=	=	SYM
ejpam-5360	430	16	1	1	NUM
ejpam-5360	430	17	,	,	PUNCT
ejpam-5360	430	18	the	the	DET
ejpam-5360	430	19	largest	large	ADJ
ejpam-5360	430	20	non	non	ADJ
ejpam-5360	430	21	-	-	ADJ
ejpam-5360	430	22	diagonal	diagonal	ADJ
ejpam-5360	430	23	element	element	NOUN
ejpam-5360	430	24	.	.	PUNCT
ejpam-5360	431	1	consider	consider	VERB
ejpam-5360	431	2	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	431	3	)	)	PUNCT
ejpam-5360	431	4	)	)	PUNCT
ejpam-5360	431	5	≤	≤	NOUN
ejpam-5360	431	6	(	(	PUNCT
ejpam-5360	431	7	r∧	r∧	NOUN
ejpam-5360	431	8	l	l	PROPN
ejpam-5360	432	1	+	+	NOUN
ejpam-5360	432	2	m	m	VERB
ejpam-5360	432	3	−n	−n	ADJ
ejpam-5360	432	4	)	)	PUNCT
ejpam-5360	433	1	+	+	CCONJ
ejpam-5360	433	2	√	√	INTJ
ejpam-5360	433	3	(	(	PUNCT
ejpam-5360	433	4	r∧	r∧	PROPN
ejpam-5360	433	5	l	l	PROPN
ejpam-5360	433	6	−m	−m	PROPN
ejpam-5360	433	7	+	+	PROPN
ejpam-5360	433	8	n)2	n)2	NOUN
ejpam-5360	433	9	+	+	CCONJ
ejpam-5360	433	10	4n	4n	ADJ
ejpam-5360	433	11	∑l−1	∑l−1	X
ejpam-5360	433	12	i=1	i=1	X
ejpam-5360	433	13	(	(	PUNCT
ejpam-5360	433	14	r	r	NOUN
ejpam-5360	433	15	∧	∧	PROPN
ejpam-5360	433	16	i	i	NOUN
ejpam-5360	433	17	−r∧	−r∧	NUM
ejpam-5360	433	18	l	l	NOUN
ejpam-5360	433	19	)	)	PUNCT
ejpam-5360	433	20	2	2	NUM
ejpam-5360	433	21	.	.	PUNCT
ejpam-5360	434	1	here	here	ADV
ejpam-5360	434	2	we	we	PRON
ejpam-5360	434	3	apply	apply	VERB
ejpam-5360	434	4	the	the	DET
ejpam-5360	434	5	value	value	NOUN
ejpam-5360	434	6	of	of	ADP
ejpam-5360	434	7	n	n	NOUN
ejpam-5360	434	8	=	=	SYM
ejpam-5360	434	9	1	1	NUM
ejpam-5360	434	10	,	,	PUNCT
ejpam-5360	434	11	the	the	DET
ejpam-5360	434	12	largest	large	ADJ
ejpam-5360	434	13	non	non	ADJ
ejpam-5360	434	14	-	-	ADJ
ejpam-5360	434	15	diagonal	diagonal	ADJ
ejpam-5360	434	16	element	element	NOUN
ejpam-5360	434	17	in	in	ADP
ejpam-5360	434	18	the	the	DET
ejpam-5360	434	19	above	above	ADJ
ejpam-5360	434	20	equation	equation	NOUN
ejpam-5360	434	21	,	,	PUNCT
ejpam-5360	434	22	we	we	PRON
ejpam-5360	434	23	get	get	VERB
ejpam-5360	434	24	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	434	25	)	)	PUNCT
ejpam-5360	434	26	)	)	PUNCT
ejpam-5360	435	1	≤	≤	NOUN
ejpam-5360	435	2	(	(	PUNCT
ejpam-5360	435	3	r∧	r∧	NOUN
ejpam-5360	435	4	l	l	PROPN
ejpam-5360	436	1	+	+	NOUN
ejpam-5360	436	2	m	m	VERB
ejpam-5360	436	3	−	−	NOUN
ejpam-5360	436	4	1	1	NUM
ejpam-5360	436	5	)	)	PUNCT
ejpam-5360	436	6	+	+	CCONJ
ejpam-5360	436	7	√	√	INTJ
ejpam-5360	436	8	(	(	PUNCT
ejpam-5360	436	9	r∧	r∧	PROPN
ejpam-5360	436	10	l	l	PROPN
ejpam-5360	436	11	−m	−m	NOUN
ejpam-5360	436	12	+	+	CCONJ
ejpam-5360	436	13	1)2	1)2	NUM
ejpam-5360	436	14	+	+	CCONJ
ejpam-5360	436	15	4	4	NUM
ejpam-5360	436	16	∑l−1	∑l−1	NOUN
ejpam-5360	436	17	i=1	i=1	X
ejpam-5360	437	1	(	(	PUNCT
ejpam-5360	437	2	r	r	NOUN
ejpam-5360	437	3	∧	∧	PROPN
ejpam-5360	437	4	i	i	NOUN
ejpam-5360	437	5	−r∧	−r∧	NUM
ejpam-5360	437	6	l	l	NOUN
ejpam-5360	437	7	)	)	PUNCT
ejpam-5360	437	8	2	2	NUM
ejpam-5360	437	9	.	.	PUNCT
ejpam-5360	438	1	(	(	PUNCT
ejpam-5360	438	2	27	27	NUM
ejpam-5360	438	3	)	)	PUNCT
ejpam-5360	438	4	we	we	PRON
ejpam-5360	438	5	discuss	discuss	VERB
ejpam-5360	438	6	the	the	DET
ejpam-5360	438	7	cases	case	NOUN
ejpam-5360	438	8	corresponding	correspond	VERB
ejpam-5360	438	9	to	to	ADP
ejpam-5360	438	10	the	the	DET
ejpam-5360	438	11	row	row	NOUN
ejpam-5360	438	12	sums	sum	NOUN
ejpam-5360	438	13	r∧	r∧	NOUN
ejpam-5360	438	14	l	l	PROPN
ejpam-5360	438	15	=	=	PUNCT
ejpam-5360	439	1	r∧	r∧	ADP
ejpam-5360	439	2	1	1	NUM
ejpam-5360	439	3	,	,	PUNCT
ejpam-5360	439	4	r∧	r∧	PROPN
ejpam-5360	439	5	2	2	NUM
ejpam-5360	439	6	and	and	CCONJ
ejpam-5360	439	7	r∧	r∧	PROPN
ejpam-5360	439	8	j	j	PROPN
ejpam-5360	439	9	.	.	PUNCT
ejpam-5360	440	1	case	case	NOUN
ejpam-5360	440	2	(	(	PUNCT
ejpam-5360	440	3	i	i	NOUN
ejpam-5360	440	4	)	)	PUNCT
ejpam-5360	440	5	consider	consider	VERB
ejpam-5360	440	6	r∧	r∧	PROPN
ejpam-5360	440	7	l	l	PROPN
ejpam-5360	440	8	=	=	PUNCT
ejpam-5360	441	1	r∧	r∧	ADP
ejpam-5360	441	2	1	1	NUM
ejpam-5360	441	3	,	,	PUNCT
ejpam-5360	441	4	where	where	SCONJ
ejpam-5360	441	5	r∧	r∧	PROPN
ejpam-5360	441	6	1	1	NUM
ejpam-5360	441	7	=	=	SYM
ejpam-5360	441	8	2	2	NUM
ejpam-5360	441	9	(	(	PUNCT
ejpam-5360	441	10	∑k	∑k	PROPN
ejpam-5360	441	11	i=1	i=1	PROPN
ejpam-5360	441	12	ηiαi+	ηiαi+	X
ejpam-5360	441	13	(	(	PUNCT
ejpam-5360	441	14	ϕ−	ϕ−	PROPN
ejpam-5360	441	15	1	1	NUM
ejpam-5360	441	16	)	)	PUNCT
ejpam-5360	441	17	)	)	PUNCT
ejpam-5360	441	18	.	.	PUNCT
ejpam-5360	442	1	the	the	DET
ejpam-5360	442	2	vertex	vertex	NOUN
ejpam-5360	442	3	v	v	ADP
ejpam-5360	442	4	∈	∈	NOUN
ejpam-5360	442	5	kϕ	kϕ	NOUN
ejpam-5360	442	6	has	have	VERB
ejpam-5360	442	7	the	the	DET
ejpam-5360	442	8	maximum	maximum	ADJ
ejpam-5360	442	9	degree	degree	NOUN
ejpam-5360	442	10	.	.	PUNCT
ejpam-5360	443	1	also	also	ADV
ejpam-5360	443	2	,	,	PUNCT
ejpam-5360	443	3	the	the	DET
ejpam-5360	443	4	row	row	NOUN
ejpam-5360	443	5	sum	sum	NOUN
ejpam-5360	443	6	of	of	ADP
ejpam-5360	443	7	any	any	DET
ejpam-5360	443	8	vertex	vertex	NOUN
ejpam-5360	443	9	vi	vi	PROPN
ejpam-5360	443	10	∈	∈	PROPN
ejpam-5360	443	11	si	si	X
ejpam-5360	443	12	has	have	VERB
ejpam-5360	443	13	the	the	DET
ejpam-5360	443	14	row	row	NOUN
ejpam-5360	443	15	sum	sum	NOUN
ejpam-5360	444	1	r∧	r∧	PROPN
ejpam-5360	444	2	i	i	PRON
ejpam-5360	444	3	=	=	SYM
ejpam-5360	444	4	2(ϕ+	2(ϕ+	NUM
ejpam-5360	444	5	αi	αi	VERB
ejpam-5360	444	6	−	−	NOUN
ejpam-5360	444	7	1	1	NUM
ejpam-5360	444	8	)	)	PUNCT
ejpam-5360	444	9	.	.	PUNCT
ejpam-5360	445	1	µ1(q(g	µ1(q(g	NUM
ejpam-5360	445	2	)	)	PUNCT
ejpam-5360	445	3	)	)	PUNCT
ejpam-5360	446	1	≤	≤	NOUN
ejpam-5360	446	2	(	(	PUNCT
ejpam-5360	446	3	r∧	r∧	PROPN
ejpam-5360	446	4	1	1	NUM
ejpam-5360	446	5	+	+	NOUN
ejpam-5360	446	6	m	m	VERB
ejpam-5360	446	7	−	−	NOUN
ejpam-5360	446	8	1	1	NUM
ejpam-5360	446	9	)	)	PUNCT
ejpam-5360	446	10	+	+	CCONJ
ejpam-5360	446	11	√	√	INTJ
ejpam-5360	446	12	(	(	PUNCT
ejpam-5360	446	13	r∧	r∧	PROPN
ejpam-5360	446	14	1	1	NUM
ejpam-5360	446	15	−m	−m	NOUN
ejpam-5360	446	16	+	+	CCONJ
ejpam-5360	447	1	1)2	1)2	NUM
ejpam-5360	447	2	+	+	CCONJ
ejpam-5360	447	3	4	4	NUM
ejpam-5360	447	4	∑l−1	∑l−1	NOUN
ejpam-5360	447	5	i=1	i=1	X
ejpam-5360	448	1	(	(	PUNCT
ejpam-5360	448	2	r	r	NOUN
ejpam-5360	448	3	∧	∧	PROPN
ejpam-5360	448	4	i	i	PRON
ejpam-5360	448	5	−r∧	−r∧	PRON
ejpam-5360	448	6	1	1	NUM
ejpam-5360	448	7	)	)	PUNCT
ejpam-5360	448	8	2	2	NUM
ejpam-5360	448	9	.	.	PUNCT
ejpam-5360	449	1	(	(	PUNCT
ejpam-5360	449	2	28	28	NUM
ejpam-5360	449	3	)	)	PUNCT
ejpam-5360	449	4	we	we	PRON
ejpam-5360	449	5	observe	observe	VERB
ejpam-5360	449	6	that	that	SCONJ
ejpam-5360	449	7	the	the	DET
ejpam-5360	449	8	summation	summation	NOUN
ejpam-5360	449	9	l−1∑	l−1∑	X
ejpam-5360	449	10	i=1	i=1	PROPN
ejpam-5360	450	1	(	(	PUNCT
ejpam-5360	450	2	r∧	r∧	PROPN
ejpam-5360	450	3	i	i	PRON
ejpam-5360	450	4	−r∧	−r∧	X
ejpam-5360	450	5	1	1	X
ejpam-5360	450	6	)	)	PUNCT
ejpam-5360	450	7	=	=	SYM
ejpam-5360	450	8	0	0	X
ejpam-5360	450	9	.	.	PUNCT
ejpam-5360	451	1	also	also	ADV
ejpam-5360	451	2	(	(	PUNCT
ejpam-5360	451	3	r∧	r∧	PROPN
ejpam-5360	451	4	1	1	NUM
ejpam-5360	451	5	+	+	NOUN
ejpam-5360	451	6	m	m	VERB
ejpam-5360	451	7	−	−	NOUN
ejpam-5360	451	8	1	1	NUM
ejpam-5360	451	9	)	)	PUNCT
ejpam-5360	451	10	=	=	SYM
ejpam-5360	451	11	(	(	PUNCT
ejpam-5360	451	12	3n−	3n−	PROPN
ejpam-5360	451	13	4	4	NUM
ejpam-5360	451	14	)	)	PUNCT
ejpam-5360	451	15	and	and	CCONJ
ejpam-5360	451	16	(	(	PUNCT
ejpam-5360	451	17	r∧	r∧	PROPN
ejpam-5360	451	18	1	1	NUM
ejpam-5360	451	19	−m	−m	NOUN
ejpam-5360	451	20	+	+	CCONJ
ejpam-5360	451	21	1	1	NUM
ejpam-5360	451	22	)	)	PUNCT
ejpam-5360	451	23	=	=	PUNCT
ejpam-5360	451	24	n	n	NOUN
ejpam-5360	451	25	substituting	substitute	VERB
ejpam-5360	451	26	the	the	DET
ejpam-5360	451	27	above	above	ADJ
ejpam-5360	451	28	expressions	expression	NOUN
ejpam-5360	451	29	in	in	ADP
ejpam-5360	451	30	equation	equation	NOUN
ejpam-5360	451	31	(	(	PUNCT
ejpam-5360	451	32	28	28	NUM
ejpam-5360	451	33	)	)	PUNCT
ejpam-5360	451	34	,	,	PUNCT
ejpam-5360	451	35	we	we	PRON
ejpam-5360	451	36	get	get	VERB
ejpam-5360	451	37	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	451	38	)	)	PUNCT
ejpam-5360	451	39	)	)	PUNCT
ejpam-5360	452	1	≤	≤	NOUN
ejpam-5360	452	2	(	(	PUNCT
ejpam-5360	452	3	2n−	2n−	PROPN
ejpam-5360	452	4	1	1	NUM
ejpam-5360	452	5	)	)	PUNCT
ejpam-5360	452	6	.	.	PUNCT
ejpam-5360	453	1	(	(	PUNCT
ejpam-5360	453	2	29	29	NUM
ejpam-5360	453	3	)	)	PUNCT
ejpam-5360	453	4	case	case	NOUN
ejpam-5360	453	5	(	(	PUNCT
ejpam-5360	453	6	ii	ii	NOUN
ejpam-5360	453	7	)	)	PUNCT
ejpam-5360	453	8	consider	consider	VERB
ejpam-5360	453	9	r∧	r∧	PROPN
ejpam-5360	453	10	l	l	PROPN
ejpam-5360	453	11	=	=	PUNCT
ejpam-5360	453	12	r∧	r∧	PROPN
ejpam-5360	453	13	2	2	NUM
ejpam-5360	453	14	=	=	SYM
ejpam-5360	453	15	2(ϕ+	2(ϕ+	NUM
ejpam-5360	453	16	αk	αk	INTJ
ejpam-5360	453	17	−	−	NOUN
ejpam-5360	453	18	1	1	NUM
ejpam-5360	453	19	)	)	PUNCT
ejpam-5360	453	20	,	,	PUNCT
ejpam-5360	453	21	the	the	DET
ejpam-5360	453	22	row	row	NOUN
ejpam-5360	453	23	sum	sum	NOUN
ejpam-5360	453	24	of	of	ADP
ejpam-5360	453	25	the	the	DET
ejpam-5360	453	26	vertex	vertex	NOUN
ejpam-5360	453	27	belonging	belong	VERB
ejpam-5360	453	28	to	to	ADP
ejpam-5360	453	29	the	the	DET
ejpam-5360	453	30	satellite	satellite	NOUN
ejpam-5360	453	31	sk	sk	NOUN
ejpam-5360	453	32	,	,	PUNCT
ejpam-5360	453	33	as	as	SCONJ
ejpam-5360	453	34	the	the	DET
ejpam-5360	453	35	row	row	NOUN
ejpam-5360	453	36	sum	sum	NOUN
ejpam-5360	453	37	of	of	ADP
ejpam-5360	453	38	the	the	DET
ejpam-5360	453	39	vertex	vertex	NOUN
ejpam-5360	453	40	belonging	belong	VERB
ejpam-5360	453	41	to	to	ADP
ejpam-5360	453	42	the	the	DET
ejpam-5360	453	43	satellite	satellite	NOUN
ejpam-5360	453	44	sk	sk	NOUN
ejpam-5360	453	45	is	be	AUX
ejpam-5360	453	46	the	the	DET
ejpam-5360	453	47	maximum	maximum	NOUN
ejpam-5360	453	48	compared	compare	VERB
ejpam-5360	453	49	to	to	ADP
ejpam-5360	453	50	the	the	DET
ejpam-5360	453	51	row	row	NOUN
ejpam-5360	453	52	sums	sum	NOUN
ejpam-5360	453	53	of	of	ADP
ejpam-5360	453	54	the	the	DET
ejpam-5360	453	55	vertices	vertex	NOUN
ejpam-5360	453	56	belonging	belong	VERB
ejpam-5360	453	57	to	to	ADP
ejpam-5360	453	58	the	the	DET
ejpam-5360	453	59	satellite	satellite	NOUN
ejpam-5360	453	60	s1	s1	NOUN
ejpam-5360	453	61	,	,	PUNCT
ejpam-5360	453	62	s2	s2	PROPN
ejpam-5360	453	63	,	,	PUNCT
ejpam-5360	453	64	...	...	PUNCT
ejpam-5360	453	65	,	,	PUNCT
ejpam-5360	453	66	sk−1	sk−1	ADJ
ejpam-5360	453	67	.	.	PUNCT
ejpam-5360	453	68	µ1(q(g	µ1(q(g	NUM
ejpam-5360	453	69	)	)	PUNCT
ejpam-5360	453	70	)	)	PUNCT
ejpam-5360	454	1	≤	≤	NOUN
ejpam-5360	454	2	(	(	PUNCT
ejpam-5360	454	3	r∧	r∧	PROPN
ejpam-5360	454	4	2	2	NUM
ejpam-5360	454	5	+	+	NOUN
ejpam-5360	454	6	m	m	VERB
ejpam-5360	454	7	−	−	NOUN
ejpam-5360	454	8	1	1	NUM
ejpam-5360	454	9	)	)	PUNCT
ejpam-5360	454	10	+	+	CCONJ
ejpam-5360	454	11	√	√	INTJ
ejpam-5360	454	12	(	(	PUNCT
ejpam-5360	454	13	r∧	r∧	PROPN
ejpam-5360	454	14	2	2	NUM
ejpam-5360	454	15	−m	−m	NOUN
ejpam-5360	454	16	+	+	X
ejpam-5360	455	1	1)2	1)2	NUM
ejpam-5360	455	2	+	+	CCONJ
ejpam-5360	455	3	4	4	NUM
ejpam-5360	455	4	∑l−1	∑l−1	NOUN
ejpam-5360	455	5	i=1	i=1	X
ejpam-5360	456	1	(	(	PUNCT
ejpam-5360	456	2	r	r	NOUN
ejpam-5360	456	3	∧	∧	PROPN
ejpam-5360	456	4	i	i	PRON
ejpam-5360	456	5	−r∧	−r∧	NUM
ejpam-5360	456	6	2	2	NUM
ejpam-5360	456	7	)	)	PUNCT
ejpam-5360	456	8	2	2	NUM
ejpam-5360	456	9	.	.	PUNCT
ejpam-5360	457	1	(	(	PUNCT
ejpam-5360	457	2	30	30	NUM
ejpam-5360	457	3	)	)	PUNCT
ejpam-5360	457	4	consider	consider	VERB
ejpam-5360	457	5	l−1∑	l−1∑	PRON
ejpam-5360	457	6	i=1	i=1	PROPN
ejpam-5360	458	1	(	(	PUNCT
ejpam-5360	458	2	r∧	r∧	PROPN
ejpam-5360	458	3	i	i	PRON
ejpam-5360	458	4	−r∧	−r∧	NUM
ejpam-5360	458	5	2	2	NUM
ejpam-5360	458	6	)	)	PUNCT
ejpam-5360	458	7	=	=	SYM
ejpam-5360	459	1	ϕ(r∧	ϕ(r∧	ADJ
ejpam-5360	459	2	1	1	NUM
ejpam-5360	460	1	−r∧	−r∧	NUM
ejpam-5360	460	2	2	2	NUM
ejpam-5360	460	3	)	)	PUNCT
ejpam-5360	460	4	=	=	NOUN
ejpam-5360	460	5	2ϕ	2ϕ	NUM
ejpam-5360	460	6	(	(	PUNCT
ejpam-5360	460	7	k∑	k∑	NOUN
ejpam-5360	460	8	i=1	i=1	PROPN
ejpam-5360	461	1	ηiαi	ηiαi	NOUN
ejpam-5360	462	1	+	+	CCONJ
ejpam-5360	462	2	(	(	PUNCT
ejpam-5360	462	3	ϕ−	ϕ−	PROPN
ejpam-5360	462	4	1)−	1)−	PROPN
ejpam-5360	462	5	(	(	PUNCT
ejpam-5360	462	6	ϕ+	ϕ+	INTJ
ejpam-5360	462	7	αk	αk	INTJ
ejpam-5360	462	8	−	−	PROPN
ejpam-5360	462	9	1	1	NUM
ejpam-5360	462	10	)	)	PUNCT
ejpam-5360	462	11	)	)	PUNCT
ejpam-5360	463	1	m.	m.	NOUN
ejpam-5360	463	2	v.	v.	ADP
ejpam-5360	463	3	,	,	PUNCT
ejpam-5360	463	4	k.	k.	PROPN
ejpam-5360	463	5	desikan	desikan	PROPN
ejpam-5360	463	6	/	/	SYM
ejpam-5360	463	7	eur	eur	PROPN
ejpam-5360	463	8	.	.	PUNCT
ejpam-5360	464	1	j.	j.	PROPN
ejpam-5360	464	2	pure	pure	PROPN
ejpam-5360	464	3	appl	appl	PROPN
ejpam-5360	464	4	.	.	PROPN
ejpam-5360	464	5	math	math	PROPN
ejpam-5360	464	6	,	,	PUNCT
ejpam-5360	464	7	18	18	NUM
ejpam-5360	464	8	(	(	PUNCT
ejpam-5360	464	9	4	4	NUM
ejpam-5360	464	10	)	)	PUNCT
ejpam-5360	464	11	(	(	PUNCT
ejpam-5360	464	12	2025	2025	NUM
ejpam-5360	464	13	)	)	PUNCT
ejpam-5360	464	14	,	,	PUNCT
ejpam-5360	464	15	5360	5360	NUM
ejpam-5360	464	16	18	18	NUM
ejpam-5360	464	17	of	of	ADP
ejpam-5360	464	18	20	20	NUM
ejpam-5360	464	19	=	=	NOUN
ejpam-5360	464	20	2ϕ	2ϕ	NOUN
ejpam-5360	464	21	(	(	PUNCT
ejpam-5360	464	22	(	(	PUNCT
ejpam-5360	464	23	n−	n−	NOUN
ejpam-5360	464	24	ϕ)−	ϕ)−	PROPN
ejpam-5360	464	25	αk	αk	AUX
ejpam-5360	464	26	)	)	PUNCT
ejpam-5360	464	27	.	.	PUNCT
ejpam-5360	465	1	substituting	substitute	VERB
ejpam-5360	465	2	the	the	DET
ejpam-5360	465	3	values	value	NOUN
ejpam-5360	465	4	of	of	ADP
ejpam-5360	465	5	r∧	r∧	PROPN
ejpam-5360	465	6	2	2	NUM
ejpam-5360	465	7	,	,	PUNCT
ejpam-5360	465	8	m	m	PRON
ejpam-5360	465	9	and	and	CCONJ
ejpam-5360	465	10	∑l−1	∑l−1	NUM
ejpam-5360	466	1	i=1	i=1	X
ejpam-5360	467	1	(	(	PUNCT
ejpam-5360	467	2	r	r	NOUN
ejpam-5360	467	3	∧	∧	PROPN
ejpam-5360	467	4	i	i	PRON
ejpam-5360	467	5	−r∧	−r∧	X
ejpam-5360	467	6	2	2	NUM
ejpam-5360	467	7	)	)	PUNCT
ejpam-5360	467	8	in	in	ADP
ejpam-5360	467	9	equation	equation	NOUN
ejpam-5360	467	10	(	(	PUNCT
ejpam-5360	467	11	30	30	NUM
ejpam-5360	467	12	)	)	PUNCT
ejpam-5360	467	13	,	,	PUNCT
ejpam-5360	467	14	we	we	PRON
ejpam-5360	467	15	get	get	VERB
ejpam-5360	467	16	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	467	17	)	)	PUNCT
ejpam-5360	467	18	)	)	PUNCT
ejpam-5360	468	1	≤	≤	NOUN
ejpam-5360	468	2	(	(	PUNCT
ejpam-5360	468	3	n+	n+	NUM
ejpam-5360	468	4	2(ϕ+	2(ϕ+	NUM
ejpam-5360	468	5	αk	αk	CCONJ
ejpam-5360	468	6	−	−	PROPN
ejpam-5360	468	7	2	2	NUM
ejpam-5360	468	8	)	)	PUNCT
ejpam-5360	468	9	)	)	PUNCT
ejpam-5360	468	10	2	2	NUM
ejpam-5360	469	1	+	+	CCONJ
ejpam-5360	469	2	√	√	NUM
ejpam-5360	469	3	2(ϕ+	2(ϕ+	NUM
ejpam-5360	469	4	2αk	2αk	ADJ
ejpam-5360	469	5	−	−	PROPN
ejpam-5360	469	6	n)2	n)2	NOUN
ejpam-5360	469	7	+	+	CCONJ
ejpam-5360	469	8	8	8	NUM
ejpam-5360	469	9	{	{	PUNCT
ejpam-5360	469	10	ϕ	ϕ	NOUN
ejpam-5360	469	11	(	(	PUNCT
ejpam-5360	469	12	n−	n−	NOUN
ejpam-5360	469	13	ϕ)−	ϕ)−	PROPN
ejpam-5360	469	14	αk	αk	AUX
ejpam-5360	469	15	}	}	SYM
ejpam-5360	469	16	2	2	NUM
ejpam-5360	469	17	.	.	PUNCT
ejpam-5360	470	1	(	(	PUNCT
ejpam-5360	470	2	31	31	NUM
ejpam-5360	470	3	)	)	PUNCT
ejpam-5360	470	4	case	case	NOUN
ejpam-5360	470	5	(	(	PUNCT
ejpam-5360	470	6	iii	iii	X
ejpam-5360	470	7	)	)	PUNCT
ejpam-5360	470	8	consider	consider	VERB
ejpam-5360	470	9	the	the	DET
ejpam-5360	470	10	case	case	NOUN
ejpam-5360	470	11	r∧	r∧	PROPN
ejpam-5360	470	12	l	l	PROPN
ejpam-5360	470	13	=	=	PUNCT
ejpam-5360	471	1	r∧	r∧	PROPN
ejpam-5360	471	2	j	j	PROPN
ejpam-5360	471	3	for	for	ADP
ejpam-5360	471	4	some	some	DET
ejpam-5360	471	5	general	general	PROPN
ejpam-5360	471	6	j	j	PROPN
ejpam-5360	471	7	,	,	PUNCT
ejpam-5360	471	8	where	where	SCONJ
ejpam-5360	471	9	3	3	NUM
ejpam-5360	471	10	≤	≤	NUM
ejpam-5360	471	11	j	j	PROPN
ejpam-5360	471	12	≤	≤	PROPN
ejpam-5360	471	13	l.	l.	PROPN
ejpam-5360	471	14	here	here	ADV
ejpam-5360	471	15	r∧	r∧	PROPN
ejpam-5360	471	16	j	j	PROPN
ejpam-5360	471	17	=	=	PUNCT
ejpam-5360	471	18	2(ϕ	2(ϕ	NUM
ejpam-5360	471	19	+	+	CCONJ
ejpam-5360	471	20	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	471	21	)	)	PUNCT
ejpam-5360	472	1	−	−	ADP
ejpam-5360	472	2	1	1	NUM
ejpam-5360	472	3	)	)	PUNCT
ejpam-5360	472	4	,	,	PUNCT
ejpam-5360	472	5	the	the	DET
ejpam-5360	472	6	row	row	NOUN
ejpam-5360	472	7	sum	sum	NOUN
ejpam-5360	472	8	of	of	ADP
ejpam-5360	472	9	the	the	DET
ejpam-5360	472	10	vertex	vertex	NOUN
ejpam-5360	472	11	belonging	belong	VERB
ejpam-5360	472	12	to	to	ADP
ejpam-5360	472	13	the	the	DET
ejpam-5360	472	14	satellite	satellite	PROPN
ejpam-5360	472	15	sk−(j−2	sk−(j−2	PROPN
ejpam-5360	472	16	)	)	PUNCT
ejpam-5360	472	17	where	where	SCONJ
ejpam-5360	472	18	j	j	PROPN
ejpam-5360	472	19	̸=	̸=	PROPN
ejpam-5360	472	20	2	2	NUM
ejpam-5360	472	21	.	.	NUM
ejpam-5360	472	22	µ1(q(g	µ1(q(g	NUM
ejpam-5360	472	23	)	)	PUNCT
ejpam-5360	472	24	)	)	PUNCT
ejpam-5360	472	25	≤	≤	NOUN
ejpam-5360	472	26	(	(	PUNCT
ejpam-5360	472	27	r∧	r∧	PROPN
ejpam-5360	472	28	j	j	PROPN
ejpam-5360	472	29	+	+	PROPN
ejpam-5360	472	30	m	m	VERB
ejpam-5360	472	31	−	−	NOUN
ejpam-5360	472	32	1	1	NUM
ejpam-5360	472	33	)	)	PUNCT
ejpam-5360	472	34	+	+	CCONJ
ejpam-5360	472	35	√	√	INTJ
ejpam-5360	472	36	(	(	PUNCT
ejpam-5360	472	37	r∧	r∧	PROPN
ejpam-5360	472	38	j	j	PROPN
ejpam-5360	472	39	−m	−m	PROPN
ejpam-5360	472	40	+	+	CCONJ
ejpam-5360	472	41	1)2	1)2	NUM
ejpam-5360	472	42	+	+	CCONJ
ejpam-5360	472	43	4	4	NUM
ejpam-5360	472	44	∑l−1	∑l−1	NOUN
ejpam-5360	472	45	i=1	i=1	X
ejpam-5360	473	1	(	(	PUNCT
ejpam-5360	473	2	r	r	NOUN
ejpam-5360	473	3	∧	∧	PROPN
ejpam-5360	473	4	i	i	PRON
ejpam-5360	473	5	−r∧	−r∧	NUM
ejpam-5360	473	6	j	j	NOUN
ejpam-5360	473	7	)	)	PUNCT
ejpam-5360	473	8	2	2	NUM
ejpam-5360	473	9	.	.	PUNCT
ejpam-5360	474	1	(	(	PUNCT
ejpam-5360	474	2	32	32	NUM
ejpam-5360	474	3	)	)	PUNCT
ejpam-5360	474	4	consider	consider	VERB
ejpam-5360	474	5	j−1∑	j−1∑	PRON
ejpam-5360	474	6	i=1	i=1	PROPN
ejpam-5360	475	1	(	(	PUNCT
ejpam-5360	475	2	r∧	r∧	PROPN
ejpam-5360	475	3	i	i	PRON
ejpam-5360	475	4	−r∧	−r∧	NUM
ejpam-5360	475	5	j	j	NOUN
ejpam-5360	475	6	)	)	PUNCT
ejpam-5360	476	1	=	=	PRON
ejpam-5360	476	2	(	(	PUNCT
ejpam-5360	476	3	2ϕ	2ϕ	NUM
ejpam-5360	476	4	k∑	k∑	NOUN
ejpam-5360	476	5	i=1	i=1	X
ejpam-5360	476	6	(	(	PUNCT
ejpam-5360	476	7	ηiαi−αk−(j−2))+2ηkαk(αk−αk−(j−2))+2ηk−1αk−1(αk−1−αk−(j−2))+	ηiαi−αk−(j−2))+2ηkαk(αk−αk−(j−2))+2ηk−1αk−1(αk−1−αk−(j−2))+	PROPN
ejpam-5360	476	8	....	....	SYM
ejpam-5360	477	1	+2ηk−(j−3)αk−(j−3)(αk−(j−3	+2ηk−(j−3)αk−(j−3)(αk−(j−3	NOUN
ejpam-5360	477	2	)	)	PUNCT
ejpam-5360	477	3	−	−	NOUN
ejpam-5360	477	4	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	477	5	)	)	PUNCT
ejpam-5360	477	6	)	)	PUNCT
ejpam-5360	478	1	=	=	PRON
ejpam-5360	478	2	{	{	PUNCT
ejpam-5360	478	3	2ϕ	2ϕ	NUM
ejpam-5360	478	4	(	(	PUNCT
ejpam-5360	478	5	k∑	k∑	NOUN
ejpam-5360	478	6	i=1	i=1	PROPN
ejpam-5360	479	1	ηiαi	ηiαi	PROPN
ejpam-5360	479	2	−	−	PROPN
ejpam-5360	479	3	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	479	4	)	)	PUNCT
ejpam-5360	479	5	)	)	PUNCT
ejpam-5360	480	1	+	+	CCONJ
ejpam-5360	480	2	2	2	NUM
ejpam-5360	480	3	j−3∑	j−3∑	NOUN
ejpam-5360	480	4	i=0	i=0	PROPN
ejpam-5360	480	5	ηk−iαk−i(αk−i	ηk−iαk−i(αk−i	NOUN
ejpam-5360	480	6	−	−	NOUN
ejpam-5360	480	7	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	480	8	)	)	PUNCT
ejpam-5360	480	9	)	)	PUNCT
ejpam-5360	480	10	}	}	PUNCT
ejpam-5360	480	11	where	where	SCONJ
ejpam-5360	480	12	3	3	NUM
ejpam-5360	480	13	≤	≤	NUM
ejpam-5360	480	14	j	j	PROPN
ejpam-5360	480	15	≤	≤	PROPN
ejpam-5360	480	16	l.	l.	NOUN
ejpam-5360	480	17	substituting	substituting	NOUN
ejpam-5360	480	18	for	for	ADP
ejpam-5360	480	19	r∧	r∧	PROPN
ejpam-5360	480	20	j	j	PROPN
ejpam-5360	480	21	,	,	PUNCT
ejpam-5360	480	22	m	m	PROPN
ejpam-5360	480	23	and	and	CCONJ
ejpam-5360	480	24	∑l−1	∑l−1	NUM
ejpam-5360	480	25	i=1	i=1	X
ejpam-5360	480	26	(	(	PUNCT
ejpam-5360	480	27	r∧	r∧	PROPN
ejpam-5360	480	28	i	i	PRON
ejpam-5360	480	29	−r∧	−r∧	NUM
ejpam-5360	480	30	j	j	NOUN
ejpam-5360	480	31	)	)	PUNCT
ejpam-5360	480	32	in	in	ADP
ejpam-5360	480	33	equation	equation	NOUN
ejpam-5360	480	34	(	(	PUNCT
ejpam-5360	480	35	32	32	NUM
ejpam-5360	480	36	)	)	PUNCT
ejpam-5360	480	37	,	,	PUNCT
ejpam-5360	480	38	we	we	PRON
ejpam-5360	480	39	get	get	VERB
ejpam-5360	480	40	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	480	41	)	)	PUNCT
ejpam-5360	480	42	)	)	PUNCT
ejpam-5360	481	1	≤	≤	NOUN
ejpam-5360	481	2	(	(	PUNCT
ejpam-5360	481	3	n+	n+	NUM
ejpam-5360	481	4	2(ϕ+	2(ϕ+	NUM
ejpam-5360	481	5	2αk−(j−2	2αk−(j−2	NUM
ejpam-5360	481	6	)	)	PUNCT
ejpam-5360	481	7	−	−	ADP
ejpam-5360	481	8	2	2	NUM
ejpam-5360	481	9	)	)	PUNCT
ejpam-5360	481	10	)	)	PUNCT
ejpam-5360	481	11	2	2	NUM
ejpam-5360	481	12	.	.	PUNCT
ejpam-5360	482	1	+	+	CCONJ
ejpam-5360	482	2	√	√	ADJ
ejpam-5360	482	3	2(ϕ+	2(ϕ+	NUM
ejpam-5360	482	4	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	482	5	)	)	PUNCT
ejpam-5360	482	6	−	−	PROPN
ejpam-5360	482	7	n)2	n)2	NOUN
ejpam-5360	482	8	+	+	CCONJ
ejpam-5360	483	1	8{ϕ	8{ϕ	NUM
ejpam-5360	484	1	(	(	PUNCT
ejpam-5360	484	2	(	(	PUNCT
ejpam-5360	484	3	n−	n−	NOUN
ejpam-5360	484	4	ϕ)−	ϕ)−	PROPN
ejpam-5360	484	5	αk−(j−2	αk−(j−2	VERB
ejpam-5360	484	6	)	)	PUNCT
ejpam-5360	484	7	)	)	PUNCT
ejpam-5360	485	1	+	+	CCONJ
ejpam-5360	485	2	∑j−3	∑j−3	ADP
ejpam-5360	485	3	i=0	i=0	ADJ
ejpam-5360	485	4	ηk−iαk−i(αk−i	ηk−iαk−i(αk−i	NOUN
ejpam-5360	485	5	−	−	NOUN
ejpam-5360	485	6	αk−(j−2	αk−(j−2	NOUN
ejpam-5360	485	7	)	)	PUNCT
ejpam-5360	485	8	)	)	PUNCT
ejpam-5360	485	9	}	}	PUNCT
ejpam-5360	485	10	2	2	NUM
ejpam-5360	485	11	(	(	PUNCT
ejpam-5360	485	12	33	33	NUM
ejpam-5360	485	13	)	)	PUNCT
ejpam-5360	485	14	hence	hence	ADV
ejpam-5360	485	15	proved	prove	VERB
ejpam-5360	485	16	.	.	PUNCT
ejpam-5360	486	1	remark	remark	PROPN
ejpam-5360	486	2	3	3	NUM
ejpam-5360	486	3	.	.	PUNCT
ejpam-5360	487	1	for	for	ADP
ejpam-5360	487	2	the	the	DET
ejpam-5360	487	3	graph	graph	NOUN
ejpam-5360	487	4	g	g	NOUN
ejpam-5360	487	5	in	in	ADP
ejpam-5360	487	6	example	example	NOUN
ejpam-5360	487	7	1	1	NUM
ejpam-5360	487	8	the	the	DET
ejpam-5360	487	9	calculated	calculate	VERB
ejpam-5360	487	10	value	value	NOUN
ejpam-5360	487	11	of	of	ADP
ejpam-5360	487	12	the	the	DET
ejpam-5360	487	13	signless	signless	PROPN
ejpam-5360	487	14	laplacian	laplacian	PROPN
ejpam-5360	487	15	spectral	spectral	ADJ
ejpam-5360	487	16	radius	radius	NOUN
ejpam-5360	487	17	is	be	AUX
ejpam-5360	487	18	µ1(q(g	µ1(q(g	NOUN
ejpam-5360	487	19	)	)	PUNCT
ejpam-5360	487	20	)	)	PUNCT
ejpam-5360	488	1	=	=	PUNCT
ejpam-5360	488	2	30.2016	30.2016	NUM
ejpam-5360	488	3	.	.	PUNCT
ejpam-5360	489	1	we	we	PRON
ejpam-5360	489	2	have	have	VERB
ejpam-5360	489	3	the	the	DET
ejpam-5360	489	4	following	follow	VERB
ejpam-5360	489	5	observations	observation	NOUN
ejpam-5360	489	6	from	from	ADP
ejpam-5360	489	7	theorem	theorem	ADJ
ejpam-5360	489	8	10	10	NUM
ejpam-5360	489	9	,	,	PUNCT
ejpam-5360	489	10	theorem	theorem	VERB
ejpam-5360	489	11	11	11	NUM
ejpam-5360	489	12	,	,	PUNCT
ejpam-5360	489	13	and	and	CCONJ
ejpam-5360	489	14	theorem	theorem	VERB
ejpam-5360	489	15	12	12	NUM
ejpam-5360	489	16	.	.	PUNCT
ejpam-5360	490	1	m.	m.	NOUN
ejpam-5360	490	2	v.	v.	ADP
ejpam-5360	490	3	,	,	PUNCT
ejpam-5360	490	4	k.	k.	PROPN
ejpam-5360	490	5	desikan	desikan	PROPN
ejpam-5360	490	6	/	/	SYM
ejpam-5360	490	7	eur	eur	PROPN
ejpam-5360	490	8	.	.	PUNCT
ejpam-5360	491	1	j.	j.	PROPN
ejpam-5360	491	2	pure	pure	PROPN
ejpam-5360	491	3	appl	appl	PROPN
ejpam-5360	491	4	.	.	PROPN
ejpam-5360	491	5	math	math	PROPN
ejpam-5360	491	6	,	,	PUNCT
ejpam-5360	491	7	18	18	NUM
ejpam-5360	491	8	(	(	PUNCT
ejpam-5360	491	9	4	4	NUM
ejpam-5360	491	10	)	)	PUNCT
ejpam-5360	491	11	(	(	PUNCT
ejpam-5360	491	12	2025	2025	NUM
ejpam-5360	491	13	)	)	PUNCT
ejpam-5360	491	14	,	,	PUNCT
ejpam-5360	491	15	5360	5360	NUM
ejpam-5360	491	16	19	19	NUM
ejpam-5360	491	17	of	of	ADP
ejpam-5360	491	18	20	20	NUM
ejpam-5360	491	19	(	(	PUNCT
ejpam-5360	491	20	i	i	NOUN
ejpam-5360	491	21	)	)	PUNCT
ejpam-5360	491	22	we	we	PRON
ejpam-5360	491	23	obtain	obtain	VERB
ejpam-5360	491	24	the	the	DET
ejpam-5360	491	25	lower	low	ADJ
ejpam-5360	491	26	and	and	CCONJ
ejpam-5360	491	27	upper	upper	ADJ
ejpam-5360	491	28	bounds	bound	NOUN
ejpam-5360	491	29	from	from	ADP
ejpam-5360	491	30	equation	equation	NOUN
ejpam-5360	491	31	(	(	PUNCT
ejpam-5360	491	32	18	18	NUM
ejpam-5360	491	33	)	)	PUNCT
ejpam-5360	491	34	of	of	ADP
ejpam-5360	491	35	theorem	theorem	ADJ
ejpam-5360	491	36	10	10	NUM
ejpam-5360	491	37	as	as	ADP
ejpam-5360	491	38	18	18	NUM
ejpam-5360	491	39	≤	≤	NUM
ejpam-5360	491	40	µ1(q(g	µ1(q(g	NUM
ejpam-5360	491	41	)	)	PUNCT
ejpam-5360	491	42	)	)	PUNCT
ejpam-5360	491	43	≤	≤	NOUN
ejpam-5360	492	1	38.626	38.626	NUM
ejpam-5360	492	2	.	.	PUNCT
ejpam-5360	493	1	(	(	PUNCT
ejpam-5360	493	2	ii	ii	NOUN
ejpam-5360	493	3	)	)	PUNCT
ejpam-5360	493	4	we	we	PRON
ejpam-5360	493	5	obtain	obtain	VERB
ejpam-5360	493	6	the	the	DET
ejpam-5360	493	7	lower	lower	ADV
ejpam-5360	493	8	bound	bind	VERB
ejpam-5360	493	9	from	from	ADP
ejpam-5360	493	10	theorem	theorem	ADJ
ejpam-5360	493	11	11	11	NUM
ejpam-5360	493	12	and	and	CCONJ
ejpam-5360	493	13	to	to	PART
ejpam-5360	493	14	determine	determine	VERB
ejpam-5360	493	15	the	the	DET
ejpam-5360	493	16	tight	tight	ADV
ejpam-5360	493	17	lower	lower	ADV
ejpam-5360	493	18	bound	bind	VERB
ejpam-5360	493	19	,	,	PUNCT
ejpam-5360	493	20	we	we	PRON
ejpam-5360	493	21	discuss	discuss	VERB
ejpam-5360	493	22	the	the	DET
ejpam-5360	493	23	following	follow	VERB
ejpam-5360	493	24	cases	case	NOUN
ejpam-5360	493	25	.	.	PUNCT
ejpam-5360	494	1	(	(	PUNCT
ejpam-5360	494	2	a	a	X
ejpam-5360	494	3	)	)	PUNCT
ejpam-5360	494	4	when	when	SCONJ
ejpam-5360	494	5	r∧	r∧	PROPN
ejpam-5360	494	6	l	l	PROPN
ejpam-5360	494	7	=	=	PUNCT
ejpam-5360	495	1	r∧	r∧	ADP
ejpam-5360	495	2	2	2	NUM
ejpam-5360	495	3	,	,	PUNCT
ejpam-5360	495	4	we	we	PRON
ejpam-5360	495	5	have	have	VERB
ejpam-5360	495	6	from	from	ADP
ejpam-5360	495	7	equation	equation	NOUN
ejpam-5360	495	8	(	(	PUNCT
ejpam-5360	495	9	25	25	NUM
ejpam-5360	495	10	)	)	PUNCT
ejpam-5360	495	11	the	the	PRON
ejpam-5360	495	12	lower	lower	ADV
ejpam-5360	495	13	bound	bind	VERB
ejpam-5360	495	14	as	as	ADP
ejpam-5360	495	15	µ1(q(g	µ1(q(g	NUM
ejpam-5360	495	16	)	)	PUNCT
ejpam-5360	495	17	)	)	PUNCT
ejpam-5360	495	18	≥	≥	NOUN
ejpam-5360	496	1	18	18	NUM
ejpam-5360	496	2	.	.	PUNCT
ejpam-5360	497	1	(	(	PUNCT
ejpam-5360	497	2	b	b	X
ejpam-5360	497	3	)	)	PUNCT
ejpam-5360	497	4	when	when	SCONJ
ejpam-5360	497	5	r∧	r∧	PROPN
ejpam-5360	497	6	l	l	PROPN
ejpam-5360	497	7	=	=	PUNCT
ejpam-5360	498	1	r∧	r∧	ADP
ejpam-5360	498	2	3	3	NUM
ejpam-5360	498	3	,	,	PUNCT
ejpam-5360	498	4	we	we	PRON
ejpam-5360	498	5	have	have	VERB
ejpam-5360	498	6	from	from	ADP
ejpam-5360	498	7	equation	equation	NOUN
ejpam-5360	498	8	(	(	PUNCT
ejpam-5360	498	9	26	26	NUM
ejpam-5360	498	10	)	)	PUNCT
ejpam-5360	498	11	the	the	PRON
ejpam-5360	498	12	lower	lower	ADV
ejpam-5360	498	13	bound	bind	VERB
ejpam-5360	498	14	as	as	ADP
ejpam-5360	498	15	µ1(q(g	µ1(q(g	NUM
ejpam-5360	498	16	)	)	PUNCT
ejpam-5360	498	17	)	)	PUNCT
ejpam-5360	498	18	≥	≥	NOUN
ejpam-5360	498	19	16	16	NUM
ejpam-5360	498	20	,	,	PUNCT
ejpam-5360	498	21	for	for	ADP
ejpam-5360	498	22	any	any	DET
ejpam-5360	498	23	j	j	NOUN
ejpam-5360	498	24	,	,	PUNCT
ejpam-5360	498	25	where	where	SCONJ
ejpam-5360	498	26	3	3	NUM
ejpam-5360	498	27	≤	≤	NUM
ejpam-5360	498	28	j	j	PROPN
ejpam-5360	498	29	≤	≤	ADJ
ejpam-5360	498	30	l	l	NOUN
ejpam-5360	498	31	we	we	PRON
ejpam-5360	498	32	obtain	obtain	VERB
ejpam-5360	498	33	this	this	DET
ejpam-5360	498	34	value	value	NOUN
ejpam-5360	498	35	by	by	ADP
ejpam-5360	498	36	taking	take	VERB
ejpam-5360	498	37	j	j	PROPN
ejpam-5360	498	38	=	=	SYM
ejpam-5360	498	39	3	3	X
ejpam-5360	498	40	.	.	PUNCT
ejpam-5360	498	41	(	(	PUNCT
ejpam-5360	498	42	iii	iii	X
ejpam-5360	498	43	)	)	PUNCT
ejpam-5360	498	44	we	we	PRON
ejpam-5360	498	45	obtain	obtain	VERB
ejpam-5360	498	46	the	the	DET
ejpam-5360	498	47	upper	upper	ADJ
ejpam-5360	498	48	bound	bind	VERB
ejpam-5360	498	49	from	from	ADP
ejpam-5360	498	50	theorem	theorem	ADJ
ejpam-5360	498	51	12	12	NUM
ejpam-5360	498	52	and	and	CCONJ
ejpam-5360	498	53	to	to	PART
ejpam-5360	498	54	determine	determine	VERB
ejpam-5360	498	55	the	the	DET
ejpam-5360	498	56	tight	tight	ADJ
ejpam-5360	498	57	upper	upper	ADJ
ejpam-5360	498	58	bound	bind	VERB
ejpam-5360	498	59	,	,	PUNCT
ejpam-5360	498	60	we	we	PRON
ejpam-5360	498	61	discuss	discuss	VERB
ejpam-5360	498	62	the	the	DET
ejpam-5360	498	63	following	follow	VERB
ejpam-5360	498	64	cases	case	NOUN
ejpam-5360	498	65	.	.	PUNCT
ejpam-5360	499	1	(	(	PUNCT
ejpam-5360	499	2	a	a	X
ejpam-5360	499	3	)	)	PUNCT
ejpam-5360	499	4	when	when	SCONJ
ejpam-5360	499	5	r∧	r∧	PROPN
ejpam-5360	499	6	l	l	PROPN
ejpam-5360	499	7	=	=	PUNCT
ejpam-5360	500	1	r∧	r∧	ADJ
ejpam-5360	500	2	1	1	NUM
ejpam-5360	500	3	we	we	PRON
ejpam-5360	500	4	get	get	VERB
ejpam-5360	500	5	from	from	ADP
ejpam-5360	500	6	equation	equation	NOUN
ejpam-5360	500	7	(	(	PUNCT
ejpam-5360	500	8	29	29	NUM
ejpam-5360	500	9	)	)	PUNCT
ejpam-5360	500	10	,	,	PUNCT
ejpam-5360	500	11	the	the	DET
ejpam-5360	500	12	upper	upper	ADJ
ejpam-5360	500	13	bound	bind	VERB
ejpam-5360	500	14	as	as	ADP
ejpam-5360	500	15	46	46	NUM
ejpam-5360	500	16	.	.	PUNCT
ejpam-5360	501	1	(	(	PUNCT
ejpam-5360	501	2	b	b	X
ejpam-5360	501	3	)	)	PUNCT
ejpam-5360	501	4	when	when	SCONJ
ejpam-5360	501	5	r∧	r∧	PROPN
ejpam-5360	501	6	l	l	PROPN
ejpam-5360	501	7	=	=	PUNCT
ejpam-5360	502	1	r∧	r∧	ADP
ejpam-5360	502	2	2	2	NUM
ejpam-5360	502	3	we	we	PRON
ejpam-5360	502	4	get	get	VERB
ejpam-5360	502	5	from	from	ADP
ejpam-5360	502	6	equation	equation	NOUN
ejpam-5360	502	7	(	(	PUNCT
ejpam-5360	502	8	31	31	NUM
ejpam-5360	502	9	)	)	PUNCT
ejpam-5360	502	10	,	,	PUNCT
ejpam-5360	502	11	the	the	DET
ejpam-5360	502	12	upper	upper	ADJ
ejpam-5360	502	13	bound	bind	VERB
ejpam-5360	502	14	as	as	ADP
ejpam-5360	502	15	µ1(q(g	µ1(q(g	NUM
ejpam-5360	502	16	)	)	PUNCT
ejpam-5360	502	17	)	)	PUNCT
ejpam-5360	503	1	≤	≤	ADV
ejpam-5360	503	2	30.77	30.77	NUM
ejpam-5360	503	3	.	.	PUNCT
ejpam-5360	504	1	(	(	PUNCT
ejpam-5360	504	2	c	c	X
ejpam-5360	504	3	)	)	PUNCT
ejpam-5360	504	4	when	when	SCONJ
ejpam-5360	504	5	r∧	r∧	PROPN
ejpam-5360	504	6	l	l	PROPN
ejpam-5360	504	7	=	=	PUNCT
ejpam-5360	505	1	r∧	r∧	ADP
ejpam-5360	505	2	3	3	NUM
ejpam-5360	505	3	we	we	PRON
ejpam-5360	505	4	get	get	VERB
ejpam-5360	505	5	from	from	ADP
ejpam-5360	505	6	equation	equation	NOUN
ejpam-5360	505	7	(	(	PUNCT
ejpam-5360	505	8	33	33	NUM
ejpam-5360	505	9	)	)	PUNCT
ejpam-5360	505	10	,	,	PUNCT
ejpam-5360	505	11	the	the	DET
ejpam-5360	505	12	upper	upper	ADJ
ejpam-5360	505	13	bound	bind	VERB
ejpam-5360	505	14	as	as	ADP
ejpam-5360	505	15	µ1(q(g	µ1(q(g	NUM
ejpam-5360	505	16	)	)	PUNCT
ejpam-5360	505	17	)	)	PUNCT
ejpam-5360	506	1	≤	≤	NOUN
ejpam-5360	506	2	30.79	30.79	NUM
ejpam-5360	506	3	.	.	PUNCT
ejpam-5360	507	1	from	from	ADP
ejpam-5360	507	2	the	the	DET
ejpam-5360	507	3	above	above	ADJ
ejpam-5360	507	4	discussions	discussion	NOUN
ejpam-5360	507	5	,	,	PUNCT
ejpam-5360	507	6	we	we	PRON
ejpam-5360	507	7	infer	infer	VERB
ejpam-5360	507	8	that	that	SCONJ
ejpam-5360	507	9	the	the	DET
ejpam-5360	507	10	closest	close	ADV
ejpam-5360	507	11	lower	low	ADJ
ejpam-5360	507	12	and	and	CCONJ
ejpam-5360	507	13	upper	upper	ADJ
ejpam-5360	507	14	bounds	bound	NOUN
ejpam-5360	507	15	are	be	AUX
ejpam-5360	507	16	18	18	NUM
ejpam-5360	507	17	and	and	CCONJ
ejpam-5360	507	18	30.77	30.77	NUM
ejpam-5360	507	19	,	,	PUNCT
ejpam-5360	507	20	respectively	respectively	ADV
ejpam-5360	507	21	.	.	PUNCT
ejpam-5360	508	1	4	4	X
ejpam-5360	508	2	.	.	X
ejpam-5360	508	3	conclusion	conclusion	NOUN
ejpam-5360	508	4	generalized	generalize	VERB
ejpam-5360	508	5	core	core	NOUN
ejpam-5360	508	6	-	-	PUNCT
ejpam-5360	508	7	satellite	satellite	NOUN
ejpam-5360	508	8	graphs	graph	NOUN
ejpam-5360	508	9	are	be	AUX
ejpam-5360	508	10	hierarchical	hierarchical	ADJ
ejpam-5360	508	11	network	network	NOUN
ejpam-5360	508	12	structures	structure	NOUN
ejpam-5360	508	13	that	that	PRON
ejpam-5360	508	14	are	be	AUX
ejpam-5360	508	15	much	much	ADV
ejpam-5360	508	16	more	more	ADV
ejpam-5360	508	17	suitable	suitable	ADJ
ejpam-5360	508	18	to	to	AUX
ejpam-5360	508	19	model	model	VERB
ejpam-5360	508	20	real	real	ADJ
ejpam-5360	508	21	-	-	PUNCT
ejpam-5360	508	22	world	world	NOUN
ejpam-5360	508	23	complex	complex	ADJ
ejpam-5360	508	24	networks	network	NOUN
ejpam-5360	508	25	.	.	PUNCT
ejpam-5360	509	1	this	this	DET
ejpam-5360	509	2	structural	structural	ADJ
ejpam-5360	509	3	design	design	NOUN
ejpam-5360	509	4	has	have	AUX
ejpam-5360	509	5	been	be	AUX
ejpam-5360	509	6	applied	apply	VERB
ejpam-5360	509	7	as	as	SCONJ
ejpam-5360	509	8	this	this	PRON
ejpam-5360	509	9	has	have	VERB
ejpam-5360	509	10	a	a	DET
ejpam-5360	509	11	wide	wide	ADJ
ejpam-5360	509	12	range	range	NOUN
ejpam-5360	509	13	of	of	ADP
ejpam-5360	509	14	benefits	benefit	NOUN
ejpam-5360	509	15	,	,	PUNCT
ejpam-5360	509	16	for	for	ADP
ejpam-5360	509	17	developing	develop	VERB
ejpam-5360	509	18	a	a	DET
ejpam-5360	509	19	network	network	NOUN
ejpam-5360	509	20	that	that	PRON
ejpam-5360	509	21	is	be	AUX
ejpam-5360	509	22	reliable	reliable	ADJ
ejpam-5360	509	23	,	,	PUNCT
ejpam-5360	509	24	resilient	resilient	ADJ
ejpam-5360	509	25	,	,	PUNCT
ejpam-5360	509	26	scalable	scalable	ADJ
ejpam-5360	509	27	,	,	PUNCT
ejpam-5360	509	28	flexible	flexible	ADJ
ejpam-5360	509	29	,	,	PUNCT
ejpam-5360	509	30	cost	cost	NOUN
ejpam-5360	509	31	-	-	PUNCT
ejpam-5360	509	32	effective	effective	ADJ
ejpam-5360	509	33	,	,	PUNCT
ejpam-5360	509	34	has	have	VERB
ejpam-5360	509	35	better	well	ADJ
ejpam-5360	509	36	security	security	NOUN
ejpam-5360	509	37	,	,	PUNCT
ejpam-5360	509	38	easier	easy	ADJ
ejpam-5360	509	39	management	management	NOUN
ejpam-5360	509	40	design	design	NOUN
ejpam-5360	509	41	,	,	PUNCT
ejpam-5360	509	42	enhanced	enhanced	ADJ
ejpam-5360	509	43	performance	performance	NOUN
ejpam-5360	509	44	,	,	PUNCT
ejpam-5360	509	45	and	and	CCONJ
ejpam-5360	509	46	improved	improved	ADJ
ejpam-5360	509	47	cost	cost	NOUN
ejpam-5360	509	48	-	-	PUNCT
ejpam-5360	509	49	efficiency	efficiency	NOUN
ejpam-5360	509	50	.	.	PUNCT
ejpam-5360	510	1	in	in	ADP
ejpam-5360	510	2	this	this	DET
ejpam-5360	510	3	article	article	NOUN
ejpam-5360	510	4	,	,	PUNCT
ejpam-5360	510	5	we	we	PRON
ejpam-5360	510	6	have	have	AUX
ejpam-5360	510	7	provided	provide	VERB
ejpam-5360	510	8	varied	varied	ADJ
ejpam-5360	510	9	results	result	NOUN
ejpam-5360	510	10	on	on	ADP
ejpam-5360	510	11	bounds	bound	NOUN
ejpam-5360	510	12	for	for	ADP
ejpam-5360	510	13	the	the	DET
ejpam-5360	510	14	spectral	spectral	ADJ
ejpam-5360	510	15	radius	radius	NOUN
ejpam-5360	510	16	in	in	ADP
ejpam-5360	510	17	terms	term	NOUN
ejpam-5360	510	18	of	of	ADP
ejpam-5360	510	19	m	m	PROPN
ejpam-5360	510	20	and	and	CCONJ
ejpam-5360	510	21	k	k	PROPN
ejpam-5360	510	22	(	(	PUNCT
ejpam-5360	510	23	number	number	NOUN
ejpam-5360	510	24	of	of	ADP
ejpam-5360	510	25	satellites	satellite	NOUN
ejpam-5360	510	26	)	)	PUNCT
ejpam-5360	510	27	.	.	PUNCT
ejpam-5360	511	1	we	we	PRON
ejpam-5360	511	2	have	have	AUX
ejpam-5360	511	3	arrived	arrive	VERB
ejpam-5360	511	4	at	at	ADP
ejpam-5360	511	5	the	the	DET
ejpam-5360	511	6	tight	tight	ADJ
ejpam-5360	511	7	bounds	bound	NOUN
ejpam-5360	511	8	from	from	ADP
ejpam-5360	511	9	among	among	ADP
ejpam-5360	511	10	the	the	DET
ejpam-5360	511	11	bounds	bound	NOUN
ejpam-5360	511	12	derived	derive	VERB
ejpam-5360	511	13	.	.	PUNCT
ejpam-5360	512	1	we	we	PRON
ejpam-5360	512	2	have	have	AUX
ejpam-5360	512	3	also	also	ADV
ejpam-5360	512	4	derived	derive	VERB
ejpam-5360	512	5	bounds	bound	NOUN
ejpam-5360	512	6	on	on	ADP
ejpam-5360	512	7	the	the	DET
ejpam-5360	512	8	signless	signless	NOUN
ejpam-5360	512	9	laplacian	laplacian	ADJ
ejpam-5360	512	10	spectral	spectral	ADJ
ejpam-5360	512	11	radius	radius	NOUN
ejpam-5360	512	12	by	by	ADP
ejpam-5360	512	13	applying	apply	VERB
ejpam-5360	512	14	a	a	DET
ejpam-5360	512	15	few	few	ADJ
ejpam-5360	512	16	conditions	condition	NOUN
ejpam-5360	512	17	on	on	ADP
ejpam-5360	512	18	the	the	DET
ejpam-5360	512	19	general	general	ADJ
ejpam-5360	512	20	bounds	bound	NOUN
ejpam-5360	512	21	derived	derive	VERB
ejpam-5360	512	22	by	by	ADP
ejpam-5360	512	23	duan	duan	PROPN
ejpam-5360	512	24	and	and	CCONJ
ejpam-5360	512	25	zhou	zhou	PROPN
ejpam-5360	512	26	,	,	PUNCT
ejpam-5360	512	27	using	use	VERB
ejpam-5360	512	28	row	row	NOUN
ejpam-5360	512	29	sums	sum	NOUN
ejpam-5360	512	30	.	.	PUNCT
ejpam-5360	513	1	we	we	PRON
ejpam-5360	513	2	discussed	discuss	VERB
ejpam-5360	513	3	various	various	ADJ
ejpam-5360	513	4	cases	case	NOUN
ejpam-5360	513	5	and	and	CCONJ
ejpam-5360	513	6	identified	identify	VERB
ejpam-5360	513	7	the	the	DET
ejpam-5360	513	8	optimal	optimal	ADJ
ejpam-5360	513	9	conditions	condition	NOUN
ejpam-5360	513	10	for	for	SCONJ
ejpam-5360	513	11	the	the	DET
ejpam-5360	513	12	bounds	bound	NOUN
ejpam-5360	513	13	to	to	PART
ejpam-5360	513	14	be	be	AUX
ejpam-5360	513	15	proximate	proximate	NOUN
ejpam-5360	513	16	to	to	ADP
ejpam-5360	513	17	µ1(q(g	µ1(q(g	NUM
ejpam-5360	513	18	)	)	PUNCT
ejpam-5360	513	19	)	)	PUNCT
ejpam-5360	513	20	.	.	PUNCT
ejpam-5360	514	1	references	reference	NOUN
ejpam-5360	514	2	[	[	X
ejpam-5360	514	3	1	1	X
ejpam-5360	514	4	]	]	PUNCT
ejpam-5360	514	5	ernesto	ernesto	PROPN
ejpam-5360	514	6	estrada	estrada	PROPN
ejpam-5360	514	7	and	and	CCONJ
ejpam-5360	514	8	michele	michele	PROPN
ejpam-5360	514	9	benzi	benzi	PROPN
ejpam-5360	514	10	.	.	PUNCT
ejpam-5360	514	11	core	core	PROPN
ejpam-5360	514	12	–	–	PUNCT
ejpam-5360	514	13	satellite	satellite	NOUN
ejpam-5360	514	14	graphs	graph	NOUN
ejpam-5360	514	15	:	:	PUNCT
ejpam-5360	514	16	clustering	clustering	NOUN
ejpam-5360	514	17	,	,	PUNCT
ejpam-5360	514	18	assortativity	assortativity	NOUN
ejpam-5360	514	19	and	and	CCONJ
ejpam-5360	514	20	spectral	spectral	ADJ
ejpam-5360	514	21	properties	property	NOUN
ejpam-5360	514	22	.	.	PUNCT
ejpam-5360	515	1	linear	linear	ADJ
ejpam-5360	515	2	algebra	algebra	NOUN
ejpam-5360	515	3	and	and	CCONJ
ejpam-5360	515	4	its	its	PRON
ejpam-5360	515	5	applications	application	NOUN
ejpam-5360	515	6	,	,	PUNCT
ejpam-5360	515	7	517:30–52	517:30–52	NUM
ejpam-5360	515	8	,	,	PUNCT
ejpam-5360	515	9	2017	2017	NUM
ejpam-5360	515	10	.	.	PUNCT
ejpam-5360	516	1	[	[	X
ejpam-5360	516	2	2	2	NUM
ejpam-5360	516	3	]	]	X
ejpam-5360	516	4	erzsébet	erzsébet	PROPN
ejpam-5360	516	5	ravasz	ravasz	NOUN
ejpam-5360	516	6	and	and	CCONJ
ejpam-5360	516	7	albert	albert	PROPN
ejpam-5360	516	8	-	-	PUNCT
ejpam-5360	516	9	lászló	lászló	PROPN
ejpam-5360	516	10	barabási	barabási	NOUN
ejpam-5360	516	11	.	.	PUNCT
ejpam-5360	517	1	hierarchical	hierarchical	ADJ
ejpam-5360	517	2	organization	organization	NOUN
ejpam-5360	517	3	in	in	ADP
ejpam-5360	517	4	complex	complex	ADJ
ejpam-5360	517	5	networks	network	NOUN
ejpam-5360	517	6	.	.	PUNCT
ejpam-5360	518	1	physical	physical	ADJ
ejpam-5360	518	2	review	review	PROPN
ejpam-5360	518	3	e	e	PROPN
ejpam-5360	518	4	,	,	PUNCT
ejpam-5360	518	5	67(2):026–112	67(2):026–112	PROPN
ejpam-5360	518	6	,	,	PUNCT
ejpam-5360	518	7	2003	2003	NUM
ejpam-5360	518	8	.	.	PUNCT
ejpam-5360	519	1	m.	m.	NOUN
ejpam-5360	519	2	v.	v.	ADP
ejpam-5360	519	3	,	,	PUNCT
ejpam-5360	519	4	k.	k.	PROPN
ejpam-5360	519	5	desikan	desikan	PROPN
ejpam-5360	519	6	/	/	SYM
ejpam-5360	519	7	eur	eur	PROPN
ejpam-5360	519	8	.	.	PUNCT
ejpam-5360	520	1	j.	j.	PROPN
ejpam-5360	520	2	pure	pure	PROPN
ejpam-5360	520	3	appl	appl	PROPN
ejpam-5360	520	4	.	.	PROPN
ejpam-5360	520	5	math	math	PROPN
ejpam-5360	520	6	,	,	PUNCT
ejpam-5360	520	7	18	18	NUM
ejpam-5360	520	8	(	(	PUNCT
ejpam-5360	520	9	4	4	NUM
ejpam-5360	520	10	)	)	PUNCT
ejpam-5360	520	11	(	(	PUNCT
ejpam-5360	520	12	2025	2025	NUM
ejpam-5360	520	13	)	)	PUNCT
ejpam-5360	520	14	,	,	PUNCT
ejpam-5360	520	15	5360	5360	NUM
ejpam-5360	520	16	20	20	NUM
ejpam-5360	520	17	of	of	ADP
ejpam-5360	520	18	20	20	NUM
ejpam-5360	520	19	[	[	SYM
ejpam-5360	520	20	3	3	NUM
ejpam-5360	520	21	]	]	X
ejpam-5360	520	22	mohammadreza	mohammadreza	NOUN
ejpam-5360	520	23	mohammadrezaei	mohammadrezaei	NOUN
ejpam-5360	520	24	,	,	PUNCT
ejpam-5360	520	25	mohammad	mohammad	PROPN
ejpam-5360	520	26	ebrahim	ebrahim	PROPN
ejpam-5360	520	27	shiri	shiri	PROPN
ejpam-5360	520	28	,	,	PUNCT
ejpam-5360	520	29	amir	amir	PROPN
ejpam-5360	520	30	masoud	masoud	PROPN
ejpam-5360	520	31	rahmani	rahmani	PROPN
ejpam-5360	520	32	,	,	PUNCT
ejpam-5360	520	33	et	et	PROPN
ejpam-5360	520	34	al	al	PROPN
ejpam-5360	520	35	.	.	PUNCT
ejpam-5360	520	36	identifying	identify	VERB
ejpam-5360	520	37	fake	fake	ADJ
ejpam-5360	520	38	accounts	account	NOUN
ejpam-5360	520	39	on	on	ADP
ejpam-5360	520	40	social	social	ADJ
ejpam-5360	520	41	networks	network	NOUN
ejpam-5360	520	42	based	base	VERB
ejpam-5360	520	43	on	on	ADP
ejpam-5360	520	44	graph	graph	NOUN
ejpam-5360	520	45	analysis	analysis	NOUN
ejpam-5360	520	46	and	and	CCONJ
ejpam-5360	520	47	classification	classification	NOUN
ejpam-5360	520	48	algorithms	algorithm	NOUN
ejpam-5360	520	49	.	.	PUNCT
ejpam-5360	521	1	security	security	NOUN
ejpam-5360	521	2	and	and	CCONJ
ejpam-5360	521	3	communication	communication	NOUN
ejpam-5360	521	4	networks	network	NOUN
ejpam-5360	521	5	,	,	PUNCT
ejpam-5360	521	6	2018	2018	NUM
ejpam-5360	521	7	,	,	PUNCT
ejpam-5360	521	8	2018	2018	NUM
ejpam-5360	521	9	.	.	PUNCT
ejpam-5360	522	1	[	[	X
ejpam-5360	522	2	4	4	X
ejpam-5360	522	3	]	]	PUNCT
ejpam-5360	522	4	yanqing	yanqe	VERB
ejpam-5360	522	5	chen	chen	PROPN
ejpam-5360	522	6	and	and	CCONJ
ejpam-5360	522	7	ligong	ligong	PROPN
ejpam-5360	522	8	wang	wang	PROPN
ejpam-5360	522	9	.	.	PUNCT
ejpam-5360	523	1	sharp	sharp	ADJ
ejpam-5360	523	2	bounds	bound	NOUN
ejpam-5360	523	3	for	for	ADP
ejpam-5360	523	4	the	the	DET
ejpam-5360	523	5	largest	large	ADJ
ejpam-5360	523	6	eigenvalue	eigenvalue	NOUN
ejpam-5360	523	7	of	of	ADP
ejpam-5360	523	8	the	the	DET
ejpam-5360	523	9	signless	signless	NOUN
ejpam-5360	523	10	laplacian	laplacian	NOUN
ejpam-5360	523	11	of	of	ADP
ejpam-5360	523	12	a	a	DET
ejpam-5360	523	13	graph	graph	NOUN
ejpam-5360	523	14	.	.	PUNCT
ejpam-5360	524	1	linear	linear	ADJ
ejpam-5360	524	2	algebra	algebra	NOUN
ejpam-5360	524	3	and	and	CCONJ
ejpam-5360	524	4	its	its	PRON
ejpam-5360	524	5	applications	application	NOUN
ejpam-5360	524	6	,	,	PUNCT
ejpam-5360	524	7	433(5):908–913	433(5):908–913	NUM
ejpam-5360	524	8	,	,	PUNCT
ejpam-5360	524	9	2010	2010	NUM
ejpam-5360	524	10	.	.	PUNCT
ejpam-5360	525	1	[	[	X
ejpam-5360	525	2	5	5	NUM
ejpam-5360	525	3	]	]	X
ejpam-5360	525	4	shuchao	shuchao	ADJ
ejpam-5360	525	5	li	li	PROPN
ejpam-5360	525	6	and	and	CCONJ
ejpam-5360	525	7	yi	yi	PROPN
ejpam-5360	525	8	tian	tian	PROPN
ejpam-5360	525	9	.	.	PUNCT
ejpam-5360	526	1	some	some	DET
ejpam-5360	526	2	bounds	bound	NOUN
ejpam-5360	526	3	on	on	ADP
ejpam-5360	526	4	the	the	DET
ejpam-5360	526	5	largest	large	ADJ
ejpam-5360	526	6	eigenvalues	eigenvalue	NOUN
ejpam-5360	526	7	of	of	ADP
ejpam-5360	526	8	graphs	graph	NOUN
ejpam-5360	526	9	.	.	PUNCT
ejpam-5360	527	1	applied	apply	VERB
ejpam-5360	527	2	mathematics	mathematics	NOUN
ejpam-5360	527	3	letters	letter	NOUN
ejpam-5360	527	4	,	,	PUNCT
ejpam-5360	527	5	25(3):326–332	25(3):326–332	PROPN
ejpam-5360	527	6	,	,	PUNCT
ejpam-5360	527	7	2012	2012	NUM
ejpam-5360	527	8	.	.	PUNCT
ejpam-5360	528	1	[	[	X
ejpam-5360	528	2	6	6	NUM
ejpam-5360	528	3	]	]	PUNCT
ejpam-5360	528	4	vladimir	vladimir	NOUN
ejpam-5360	528	5	nikiforov	nikiforov	PROPN
ejpam-5360	528	6	.	.	PUNCT
ejpam-5360	529	1	more	more	ADJ
ejpam-5360	529	2	spectral	spectral	ADJ
ejpam-5360	529	3	bounds	bound	NOUN
ejpam-5360	529	4	on	on	ADP
ejpam-5360	529	5	the	the	DET
ejpam-5360	529	6	clique	clique	NOUN
ejpam-5360	529	7	and	and	CCONJ
ejpam-5360	529	8	independence	independence	NOUN
ejpam-5360	529	9	numbers	number	NOUN
ejpam-5360	529	10	.	.	PUNCT
ejpam-5360	530	1	journal	journal	NOUN
ejpam-5360	530	2	of	of	ADP
ejpam-5360	530	3	combinatorial	combinatorial	ADJ
ejpam-5360	530	4	theory	theory	NOUN
ejpam-5360	530	5	,	,	PUNCT
ejpam-5360	530	6	series	series	NOUN
ejpam-5360	530	7	b	b	PROPN
ejpam-5360	530	8	,	,	PUNCT
ejpam-5360	530	9	99(6):819–826	99(6):819–826	NUM
ejpam-5360	530	10	,	,	PUNCT
ejpam-5360	530	11	2009	2009	NUM
ejpam-5360	530	12	.	.	PUNCT
ejpam-5360	531	1	[	[	X
ejpam-5360	531	2	7	7	X
ejpam-5360	531	3	]	]	X
ejpam-5360	531	4	kamal	kamal	PROPN
ejpam-5360	531	5	lochan	lochan	PROPN
ejpam-5360	531	6	patra	patra	PROPN
ejpam-5360	531	7	and	and	CCONJ
ejpam-5360	531	8	binod	binod	PROPN
ejpam-5360	531	9	kumar	kumar	PROPN
ejpam-5360	531	10	sahoo	sahoo	PROPN
ejpam-5360	531	11	.	.	PROPN
ejpam-5360	531	12	bounds	bound	VERB
ejpam-5360	531	13	for	for	ADP
ejpam-5360	531	14	the	the	DET
ejpam-5360	531	15	laplacian	laplacian	ADJ
ejpam-5360	531	16	spectral	spectral	ADJ
ejpam-5360	531	17	radius	radius	NOUN
ejpam-5360	531	18	of	of	ADP
ejpam-5360	531	19	graphs	graph	NOUN
ejpam-5360	531	20	.	.	PUNCT
ejpam-5360	532	1	electronic	electronic	ADJ
ejpam-5360	532	2	journal	journal	NOUN
ejpam-5360	532	3	of	of	ADP
ejpam-5360	532	4	graph	graph	NOUN
ejpam-5360	532	5	theory	theory	NOUN
ejpam-5360	532	6	&	&	CCONJ
ejpam-5360	532	7	applications	application	NOUN
ejpam-5360	532	8	,	,	PUNCT
ejpam-5360	532	9	5(2	5(2	NUM
ejpam-5360	532	10	)	)	PUNCT
ejpam-5360	532	11	,	,	PUNCT
ejpam-5360	532	12	2017	2017	NUM
ejpam-5360	532	13	.	.	PUNCT
ejpam-5360	533	1	[	[	X
ejpam-5360	533	2	8	8	NUM
ejpam-5360	533	3	]	]	PUNCT
ejpam-5360	533	4	nair	nair	NOUN
ejpam-5360	533	5	abreu	abreu	PROPN
ejpam-5360	533	6	,	,	PUNCT
ejpam-5360	533	7	claudia	claudia	PROPN
ejpam-5360	533	8	marcela	marcela	PROPN
ejpam-5360	533	9	justel	justel	NOUN
ejpam-5360	533	10	,	,	PUNCT
ejpam-5360	533	11	and	and	CCONJ
ejpam-5360	533	12	lilian	lilian	ADJ
ejpam-5360	533	13	markenzon	markenzon	NOUN
ejpam-5360	533	14	.	.	PUNCT
ejpam-5360	534	1	integer	integer	PROPN
ejpam-5360	534	2	laplacian	laplacian	PROPN
ejpam-5360	534	3	eigenvalues	eigenvalue	NOUN
ejpam-5360	534	4	of	of	ADP
ejpam-5360	534	5	chordal	chordal	NOUN
ejpam-5360	534	6	graphs	graph	NOUN
ejpam-5360	534	7	.	.	PUNCT
ejpam-5360	535	1	linear	linear	ADJ
ejpam-5360	535	2	algebra	algebra	NOUN
ejpam-5360	535	3	and	and	CCONJ
ejpam-5360	535	4	its	its	PRON
ejpam-5360	535	5	applications	application	NOUN
ejpam-5360	535	6	,	,	PUNCT
ejpam-5360	535	7	614:68–81	614:68–81	NUM
ejpam-5360	535	8	,	,	PUNCT
ejpam-5360	535	9	2021	2021	NUM
ejpam-5360	535	10	.	.	PUNCT
ejpam-5360	536	1	[	[	X
ejpam-5360	536	2	9	9	NUM
ejpam-5360	536	3	]	]	X
ejpam-5360	536	4	kinkar	kinkar	PROPN
ejpam-5360	536	5	chandra	chandra	PROPN
ejpam-5360	536	6	das	das	PROPN
ejpam-5360	536	7	.	.	PUNCT
ejpam-5360	537	1	proof	proof	NOUN
ejpam-5360	537	2	of	of	ADP
ejpam-5360	537	3	a	a	DET
ejpam-5360	537	4	conjecture	conjecture	NOUN
ejpam-5360	537	5	on	on	ADP
ejpam-5360	537	6	the	the	DET
ejpam-5360	537	7	complete	complete	ADJ
ejpam-5360	537	8	split	split	NOUN
ejpam-5360	537	9	-	-	PUNCT
ejpam-5360	537	10	like	like	ADJ
ejpam-5360	537	11	graphs	graph	NOUN
ejpam-5360	537	12	.	.	PUNCT
ejpam-5360	538	1	utilitas	utilitas	PROPN
ejpam-5360	538	2	mathematica	mathematica	PROPN
ejpam-5360	538	3	,	,	PUNCT
ejpam-5360	538	4	117	117	NUM
ejpam-5360	538	5	,	,	PUNCT
ejpam-5360	538	6	2020	2020	NUM
ejpam-5360	538	7	.	.	PUNCT
ejpam-5360	539	1	[	[	X
ejpam-5360	539	2	10	10	NUM
ejpam-5360	539	3	]	]	X
ejpam-5360	539	4	shuting	shut	VERB
ejpam-5360	539	5	liu	liu	PROPN
ejpam-5360	539	6	,	,	PUNCT
ejpam-5360	539	7	kinkar	kinkar	PROPN
ejpam-5360	539	8	chandra	chandra	PROPN
ejpam-5360	539	9	das	das	PROPN
ejpam-5360	539	10	,	,	PUNCT
ejpam-5360	539	11	and	and	CCONJ
ejpam-5360	539	12	jinlong	jinlong	PROPN
ejpam-5360	539	13	shu	shu	PROPN
ejpam-5360	539	14	.	.	PUNCT
ejpam-5360	540	1	on	on	ADP
ejpam-5360	540	2	the	the	DET
ejpam-5360	540	3	eigenvalues	eigenvalue	NOUN
ejpam-5360	540	4	of	of	ADP
ejpam-5360	540	5	aα	aα	NOUN
ejpam-5360	540	6	-	-	PUNCT
ejpam-5360	540	7	matrix	matrix	NOUN
ejpam-5360	540	8	of	of	ADP
ejpam-5360	540	9	graphs	graph	NOUN
ejpam-5360	540	10	.	.	PUNCT
ejpam-5360	541	1	discrete	discrete	ADJ
ejpam-5360	541	2	mathematics	mathematic	NOUN
ejpam-5360	541	3	,	,	PUNCT
ejpam-5360	541	4	343(8):111917	343(8):111917	NUM
ejpam-5360	541	5	,	,	PUNCT
ejpam-5360	541	6	2020	2020	NUM
ejpam-5360	541	7	.	.	PUNCT
ejpam-5360	542	1	[	[	X
ejpam-5360	542	2	11	11	NUM
ejpam-5360	542	3	]	]	X
ejpam-5360	542	4	zhi	zhi	PROPN
ejpam-5360	542	5	-	-	PUNCT
ejpam-5360	542	6	wen	wen	PROPN
ejpam-5360	542	7	wang	wang	PROPN
ejpam-5360	542	8	and	and	CCONJ
ejpam-5360	542	9	ji	ji	PROPN
ejpam-5360	542	10	-	-	PUNCT
ejpam-5360	542	11	ming	ming	PROPN
ejpam-5360	542	12	guo	guo	PROPN
ejpam-5360	542	13	.	.	PUNCT
ejpam-5360	543	1	some	some	DET
ejpam-5360	543	2	upper	upper	ADJ
ejpam-5360	543	3	bounds	bound	NOUN
ejpam-5360	543	4	on	on	ADP
ejpam-5360	543	5	the	the	DET
ejpam-5360	543	6	spectral	spectral	ADJ
ejpam-5360	543	7	radius	radius	NOUN
ejpam-5360	543	8	of	of	ADP
ejpam-5360	543	9	a	a	DET
ejpam-5360	543	10	graph	graph	NOUN
ejpam-5360	543	11	.	.	PUNCT
ejpam-5360	544	1	linear	linear	ADJ
ejpam-5360	544	2	algebra	algebra	NOUN
ejpam-5360	544	3	and	and	CCONJ
ejpam-5360	544	4	its	its	PRON
ejpam-5360	544	5	applications	application	NOUN
ejpam-5360	544	6	,	,	PUNCT
ejpam-5360	544	7	601:101–112	601:101–112	NUM
ejpam-5360	544	8	,	,	PUNCT
ejpam-5360	544	9	2020	2020	NUM
ejpam-5360	544	10	.	.	PUNCT
ejpam-5360	545	1	[	[	X
ejpam-5360	545	2	12	12	NUM
ejpam-5360	545	3	]	]	PUNCT
ejpam-5360	545	4	modjtaba	modjtaba	NOUN
ejpam-5360	545	5	ghorbani	ghorbani	NOUN
ejpam-5360	545	6	and	and	CCONJ
ejpam-5360	545	7	najaf	najaf	PROPN
ejpam-5360	545	8	amraei	amraei	NOUN
ejpam-5360	545	9	.	.	PUNCT
ejpam-5360	546	1	a	a	DET
ejpam-5360	546	2	note	note	NOUN
ejpam-5360	546	3	on	on	ADP
ejpam-5360	546	4	eigenvalue	eigenvalue	PROPN
ejpam-5360	546	5	,	,	PUNCT
ejpam-5360	546	6	spectral	spectral	ADJ
ejpam-5360	546	7	radius	radius	NOUN
ejpam-5360	546	8	and	and	CCONJ
ejpam-5360	546	9	energy	energy	NOUN
ejpam-5360	546	10	of	of	ADP
ejpam-5360	546	11	extended	extended	ADJ
ejpam-5360	546	12	adjacency	adjacency	NOUN
ejpam-5360	546	13	matrix	matrix	NOUN
ejpam-5360	546	14	.	.	PUNCT
ejpam-5360	547	1	discrete	discrete	ADJ
ejpam-5360	547	2	applied	apply	VERB
ejpam-5360	547	3	mathematics	mathematic	NOUN
ejpam-5360	547	4	,	,	PUNCT
ejpam-5360	547	5	322:102–116	322:102–116	NUM
ejpam-5360	547	6	,	,	PUNCT
ejpam-5360	547	7	2022	2022	NUM
ejpam-5360	547	8	.	.	PUNCT
ejpam-5360	548	1	[	[	X
ejpam-5360	548	2	13	13	NUM
ejpam-5360	548	3	]	]	SYM
ejpam-5360	548	4	jie	jie	PROPN
ejpam-5360	548	5	xue	xue	PROPN
ejpam-5360	548	6	,	,	PUNCT
ejpam-5360	548	7	ruifang	ruifang	PROPN
ejpam-5360	548	8	liu	liu	PROPN
ejpam-5360	548	9	,	,	PUNCT
ejpam-5360	548	10	jiaxin	jiaxin	NOUN
ejpam-5360	548	11	guo	guo	PROPN
ejpam-5360	548	12	,	,	PUNCT
ejpam-5360	548	13	and	and	CCONJ
ejpam-5360	548	14	jinlong	jinlong	PROPN
ejpam-5360	548	15	shu	shu	PROPN
ejpam-5360	548	16	.	.	PUNCT
ejpam-5360	549	1	the	the	DET
ejpam-5360	549	2	maximum	maximum	ADJ
ejpam-5360	549	3	spectral	spectral	ADJ
ejpam-5360	549	4	radius	radius	NOUN
ejpam-5360	549	5	of	of	ADP
ejpam-5360	549	6	irregular	irregular	ADJ
ejpam-5360	549	7	bipartite	bipartite	NOUN
ejpam-5360	549	8	graphs	graph	NOUN
ejpam-5360	549	9	.	.	PUNCT
ejpam-5360	550	1	advances	advance	NOUN
ejpam-5360	550	2	in	in	ADP
ejpam-5360	550	3	applied	apply	VERB
ejpam-5360	550	4	mathematics	mathematic	NOUN
ejpam-5360	550	5	,	,	PUNCT
ejpam-5360	550	6	142:102433	142:102433	NUM
ejpam-5360	550	7	,	,	PUNCT
ejpam-5360	550	8	2023	2023	NUM
ejpam-5360	550	9	.	.	PUNCT
ejpam-5360	551	1	[	[	X
ejpam-5360	551	2	14	14	NUM
ejpam-5360	551	3	]	]	X
ejpam-5360	551	4	kinkar	kinkar	PROPN
ejpam-5360	551	5	ch	ch	PROPN
ejpam-5360	551	6	das	das	PROPN
ejpam-5360	551	7	and	and	CCONJ
ejpam-5360	551	8	muhuo	muhuo	PROPN
ejpam-5360	551	9	liu	liu	PROPN
ejpam-5360	551	10	.	.	PUNCT
ejpam-5360	552	1	complete	complete	ADJ
ejpam-5360	552	2	split	split	NOUN
ejpam-5360	552	3	graph	graph	NOUN
ejpam-5360	552	4	determined	determine	VERB
ejpam-5360	552	5	by	by	ADP
ejpam-5360	552	6	its	its	PRON
ejpam-5360	552	7	(	(	PUNCT
ejpam-5360	552	8	signless	signless	NOUN
ejpam-5360	552	9	)	)	PUNCT
ejpam-5360	552	10	laplacian	laplacian	ADJ
ejpam-5360	552	11	spectrum	spectrum	NOUN
ejpam-5360	552	12	.	.	PUNCT
ejpam-5360	553	1	discrete	discrete	ADJ
ejpam-5360	553	2	applied	apply	VERB
ejpam-5360	553	3	mathematics	mathematic	NOUN
ejpam-5360	553	4	,	,	PUNCT
ejpam-5360	553	5	205:45–51	205:45–51	NUM
ejpam-5360	553	6	,	,	PUNCT
ejpam-5360	553	7	2016	2016	NUM
ejpam-5360	553	8	.	.	PUNCT
ejpam-5360	554	1	[	[	X
ejpam-5360	554	2	15	15	NUM
ejpam-5360	554	3	]	]	X
ejpam-5360	554	4	ji	ji	PROPN
ejpam-5360	554	5	-	-	PUNCT
ejpam-5360	554	6	ming	ming	PROPN
ejpam-5360	554	7	guo	guo	PROPN
ejpam-5360	554	8	and	and	CCONJ
ejpam-5360	554	9	meng	meng	PROPN
ejpam-5360	554	10	-	-	PUNCT
ejpam-5360	554	11	ni	ni	PROPN
ejpam-5360	554	12	yu	yu	PROPN
ejpam-5360	554	13	.	.	PUNCT
ejpam-5360	555	1	a	a	DET
ejpam-5360	555	2	sharp	sharp	ADV
ejpam-5360	555	3	lower	lower	ADV
ejpam-5360	555	4	bound	bind	VERB
ejpam-5360	555	5	of	of	ADP
ejpam-5360	555	6	the	the	DET
ejpam-5360	555	7	spectral	spectral	ADJ
ejpam-5360	555	8	radius	radius	NOUN
ejpam-5360	555	9	with	with	ADP
ejpam-5360	555	10	application	application	NOUN
ejpam-5360	555	11	to	to	ADP
ejpam-5360	555	12	the	the	DET
ejpam-5360	555	13	energy	energy	NOUN
ejpam-5360	555	14	of	of	ADP
ejpam-5360	555	15	a	a	DET
ejpam-5360	555	16	graph	graph	NOUN
ejpam-5360	555	17	.	.	PUNCT
ejpam-5360	556	1	discrete	discrete	ADJ
ejpam-5360	556	2	applied	apply	VERB
ejpam-5360	556	3	mathematics	mathematic	NOUN
ejpam-5360	556	4	,	,	PUNCT
ejpam-5360	556	5	293:59–63	293:59–63	NUM
ejpam-5360	556	6	,	,	PUNCT
ejpam-5360	556	7	2021	2021	NUM
ejpam-5360	556	8	.	.	PUNCT
ejpam-5360	557	1	[	[	X
ejpam-5360	557	2	16	16	NUM
ejpam-5360	557	3	]	]	X
ejpam-5360	557	4	lingsheng	lingsheng	PROPN
ejpam-5360	557	5	shi	shi	PROPN
ejpam-5360	557	6	.	.	PUNCT
ejpam-5360	558	1	bounds	bound	VERB
ejpam-5360	558	2	on	on	ADP
ejpam-5360	558	3	the	the	DET
ejpam-5360	558	4	(	(	PUNCT
ejpam-5360	558	5	laplacian	laplacian	ADJ
ejpam-5360	558	6	)	)	PUNCT
ejpam-5360	558	7	spectral	spectral	ADJ
ejpam-5360	558	8	radius	radius	NOUN
ejpam-5360	558	9	of	of	ADP
ejpam-5360	558	10	graphs	graph	NOUN
ejpam-5360	558	11	.	.	PUNCT
ejpam-5360	559	1	linear	linear	ADJ
ejpam-5360	559	2	algebra	algebra	NOUN
ejpam-5360	559	3	and	and	CCONJ
ejpam-5360	559	4	its	its	PRON
ejpam-5360	559	5	applications	application	NOUN
ejpam-5360	559	6	,	,	PUNCT
ejpam-5360	559	7	422(2	422(2	NUM
ejpam-5360	559	8	-	-	SYM
ejpam-5360	559	9	3):755–770	3):755–770	NUM
ejpam-5360	559	10	,	,	PUNCT
ejpam-5360	559	11	2007	2007	NUM
ejpam-5360	559	12	.	.	PUNCT
ejpam-5360	560	1	[	[	X
ejpam-5360	560	2	17	17	NUM
ejpam-5360	560	3	]	]	PUNCT
ejpam-5360	560	4	xing	xing	PROPN
ejpam-5360	560	5	duan	duan	PROPN
ejpam-5360	560	6	and	and	CCONJ
ejpam-5360	560	7	bo	bo	PROPN
ejpam-5360	560	8	zhou	zhou	PROPN
ejpam-5360	560	9	.	.	PUNCT
ejpam-5360	561	1	sharp	sharp	ADJ
ejpam-5360	561	2	bounds	bound	NOUN
ejpam-5360	561	3	on	on	ADP
ejpam-5360	561	4	the	the	DET
ejpam-5360	561	5	spectral	spectral	ADJ
ejpam-5360	561	6	radius	radius	NOUN
ejpam-5360	561	7	of	of	ADP
ejpam-5360	561	8	a	a	DET
ejpam-5360	561	9	nonnegative	nonnegative	ADJ
ejpam-5360	561	10	matrix	matrix	NOUN
ejpam-5360	561	11	.	.	PUNCT
ejpam-5360	562	1	linear	linear	ADJ
ejpam-5360	562	2	algebra	algebra	NOUN
ejpam-5360	562	3	and	and	CCONJ
ejpam-5360	562	4	its	its	PRON
ejpam-5360	562	5	applications	application	NOUN
ejpam-5360	562	6	,	,	PUNCT
ejpam-5360	562	7	439(10):2961–2970	439(10):2961–2970	PROPN
ejpam-5360	562	8	,	,	PUNCT
ejpam-5360	562	9	2013	2013	NUM
ejpam-5360	562	10	.	.	PUNCT