id	sid	tid	token	lemma	pos
ejpam-5364	1	1	european	european	PROPN
ejpam-5364	1	2	journal	journal	PROPN
ejpam-5364	1	3	of	of	ADP
ejpam-5364	1	4	pure	pure	ADJ
ejpam-5364	1	5	and	and	CCONJ
ejpam-5364	1	6	applied	apply	VERB
ejpam-5364	1	7	mathematics	mathematic	NOUN
ejpam-5364	1	8	vol	vol	NOUN
ejpam-5364	1	9	.	.	PROPN
ejpam-5364	2	1	17	17	NUM
ejpam-5364	2	2	,	,	PUNCT
ejpam-5364	2	3	no	no	INTJ
ejpam-5364	2	4	.	.	NOUN
ejpam-5364	2	5	4	4	NUM
ejpam-5364	2	6	,	,	PUNCT
ejpam-5364	2	7	2024	2024	NUM
ejpam-5364	2	8	,	,	PUNCT
ejpam-5364	2	9	3004	3004	NUM
ejpam-5364	2	10	-	-	SYM
ejpam-5364	2	11	3021	3021	NUM
ejpam-5364	2	12	issn	issn	PROPN
ejpam-5364	2	13	1307	1307	NUM
ejpam-5364	2	14	-	-	SYM
ejpam-5364	2	15	5543	5543	NUM
ejpam-5364	2	16	–	–	PUNCT
ejpam-5364	2	17	ejpam.com	ejpam.com	X
ejpam-5364	2	18	published	publish	VERB
ejpam-5364	2	19	by	by	ADP
ejpam-5364	2	20	new	new	PROPN
ejpam-5364	2	21	york	york	PROPN
ejpam-5364	2	22	business	business	PROPN
ejpam-5364	2	23	global	global	ADJ
ejpam-5364	2	24	relationship	relationship	NOUN
ejpam-5364	2	25	between	between	ADP
ejpam-5364	2	26	the	the	DET
ejpam-5364	2	27	second	second	ADV
ejpam-5364	2	28	largest	large	ADJ
ejpam-5364	2	29	adjacency	adjacency	NOUN
ejpam-5364	2	30	and	and	CCONJ
ejpam-5364	2	31	signless	signless	ADJ
ejpam-5364	2	32	laplacian	laplacian	ADJ
ejpam-5364	2	33	eigenvalues	eigenvalue	NOUN
ejpam-5364	2	34	of	of	ADP
ejpam-5364	2	35	graphs	graph	NOUN
ejpam-5364	2	36	and	and	CCONJ
ejpam-5364	2	37	properties	property	NOUN
ejpam-5364	2	38	of	of	ADP
ejpam-5364	2	39	planar	planar	ADJ
ejpam-5364	2	40	graphs	graph	NOUN
ejpam-5364	2	41	machasri	machasri	PROPN
ejpam-5364	2	42	manickam1	manickam1	PROPN
ejpam-5364	2	43	,	,	PUNCT
ejpam-5364	2	44	kalyani	kalyani	PROPN
ejpam-5364	2	45	desikan1,∗	desikan1,∗	NOUN
ejpam-5364	2	46	1	1	NUM
ejpam-5364	2	47	department	department	NOUN
ejpam-5364	2	48	of	of	ADP
ejpam-5364	2	49	mathematics	mathematic	NOUN
ejpam-5364	2	50	,	,	PUNCT
ejpam-5364	2	51	school	school	NOUN
ejpam-5364	2	52	of	of	ADP
ejpam-5364	2	53	advanced	advanced	ADJ
ejpam-5364	2	54	sciences	science	NOUN
ejpam-5364	2	55	,	,	PUNCT
ejpam-5364	2	56	vellore	vellore	PROPN
ejpam-5364	2	57	institute	institute	PROPN
ejpam-5364	2	58	of	of	ADP
ejpam-5364	2	59	technology	technology	PROPN
ejpam-5364	2	60	,	,	PUNCT
ejpam-5364	2	61	chennai	chennai	PROPN
ejpam-5364	2	62	,	,	PUNCT
ejpam-5364	2	63	tamilnadu	tamilnadu	NOUN
ejpam-5364	2	64	,	,	PUNCT
ejpam-5364	2	65	india	india	PROPN
ejpam-5364	2	66	abstract	abstract	PROPN
ejpam-5364	2	67	.	.	PUNCT
ejpam-5364	3	1	a	a	DET
ejpam-5364	3	2	graph	graph	NOUN
ejpam-5364	3	3	’s	’s	PART
ejpam-5364	3	4	second	second	ADV
ejpam-5364	3	5	largest	large	ADJ
ejpam-5364	3	6	eigenvalue	eigenvalue	NOUN
ejpam-5364	3	7	is	be	AUX
ejpam-5364	3	8	a	a	DET
ejpam-5364	3	9	significant	significant	ADJ
ejpam-5364	3	10	algebraic	algebraic	ADJ
ejpam-5364	3	11	characteristic	characteristic	NOUN
ejpam-5364	3	12	that	that	PRON
ejpam-5364	3	13	provides	provide	VERB
ejpam-5364	3	14	details	detail	NOUN
ejpam-5364	3	15	on	on	ADP
ejpam-5364	3	16	the	the	DET
ejpam-5364	3	17	graph	graph	NOUN
ejpam-5364	3	18	’s	’s	PART
ejpam-5364	3	19	expansion	expansion	NOUN
ejpam-5364	3	20	,	,	PUNCT
ejpam-5364	3	21	connectivity	connectivity	NOUN
ejpam-5364	3	22	,	,	PUNCT
ejpam-5364	3	23	and	and	CCONJ
ejpam-5364	3	24	randomness	randomness	NOUN
ejpam-5364	3	25	.	.	PUNCT
ejpam-5364	4	1	bounds	bound	NOUN
ejpam-5364	4	2	for	for	ADP
ejpam-5364	4	3	the	the	DET
ejpam-5364	4	4	second	second	ADV
ejpam-5364	4	5	largest	large	ADJ
ejpam-5364	4	6	eigenvalue	eigenvalue	NOUN
ejpam-5364	4	7	of	of	ADP
ejpam-5364	4	8	a	a	DET
ejpam-5364	4	9	graph	graph	NOUN
ejpam-5364	4	10	,	,	PUNCT
ejpam-5364	4	11	denoted	denote	VERB
ejpam-5364	4	12	as	as	SCONJ
ejpam-5364	4	13	λ2	λ2	NOUN
ejpam-5364	4	14	were	be	AUX
ejpam-5364	4	15	previously	previously	ADV
ejpam-5364	4	16	established	establish	VERB
ejpam-5364	4	17	in	in	ADP
ejpam-5364	4	18	the	the	DET
ejpam-5364	4	19	literature	literature	NOUN
ejpam-5364	4	20	in	in	ADP
ejpam-5364	4	21	relation	relation	NOUN
ejpam-5364	4	22	to	to	PART
ejpam-5364	4	23	graph	graph	VERB
ejpam-5364	4	24	parameters	parameter	NOUN
ejpam-5364	4	25	like	like	ADP
ejpam-5364	4	26	edge	edge	NOUN
ejpam-5364	4	27	connectivity	connectivity	NOUN
ejpam-5364	4	28	and	and	CCONJ
ejpam-5364	4	29	vertex	vertex	NOUN
ejpam-5364	4	30	connectivity	connectivity	NOUN
ejpam-5364	4	31	,	,	PUNCT
ejpam-5364	4	32	matching	match	VERB
ejpam-5364	4	33	number	number	NOUN
ejpam-5364	4	34	,	,	PUNCT
ejpam-5364	4	35	independence	independence	NOUN
ejpam-5364	4	36	number	number	NOUN
ejpam-5364	4	37	,	,	PUNCT
ejpam-5364	4	38	and	and	CCONJ
ejpam-5364	4	39	edge	edge	NOUN
ejpam-5364	4	40	expansion	expansion	NOUN
ejpam-5364	4	41	constant	constant	ADJ
ejpam-5364	4	42	,	,	PUNCT
ejpam-5364	4	43	among	among	ADP
ejpam-5364	4	44	others	other	NOUN
ejpam-5364	4	45	.	.	PUNCT
ejpam-5364	5	1	a	a	DET
ejpam-5364	5	2	graph	graph	NOUN
ejpam-5364	5	3	is	be	AUX
ejpam-5364	5	4	planar	planar	ADJ
ejpam-5364	5	5	if	if	SCONJ
ejpam-5364	5	6	it	it	PRON
ejpam-5364	5	7	can	can	AUX
ejpam-5364	5	8	be	be	AUX
ejpam-5364	5	9	drawn	draw	VERB
ejpam-5364	5	10	in	in	ADP
ejpam-5364	5	11	a	a	DET
ejpam-5364	5	12	plane	plane	NOUN
ejpam-5364	5	13	without	without	ADP
ejpam-5364	5	14	graph	graph	NOUN
ejpam-5364	5	15	edges	edge	NOUN
ejpam-5364	5	16	crossing	crossing	NOUN
ejpam-5364	5	17	.	.	PUNCT
ejpam-5364	6	1	determining	determine	VERB
ejpam-5364	6	2	the	the	DET
ejpam-5364	6	3	planarity	planarity	NOUN
ejpam-5364	6	4	of	of	ADP
ejpam-5364	6	5	a	a	DET
ejpam-5364	6	6	graph	graph	NOUN
ejpam-5364	6	7	helps	help	VERB
ejpam-5364	6	8	in	in	ADP
ejpam-5364	6	9	optimizing	optimize	VERB
ejpam-5364	6	10	,	,	PUNCT
ejpam-5364	6	11	simplifying	simplifying	NOUN
ejpam-5364	6	12	,	,	PUNCT
ejpam-5364	6	13	and	and	CCONJ
ejpam-5364	6	14	understanding	understand	VERB
ejpam-5364	6	15	complex	complex	ADJ
ejpam-5364	6	16	systems	system	NOUN
ejpam-5364	6	17	across	across	ADP
ejpam-5364	6	18	various	various	ADJ
ejpam-5364	6	19	fields	field	NOUN
ejpam-5364	6	20	.	.	PUNCT
ejpam-5364	7	1	graph	graph	NOUN
ejpam-5364	7	2	skewness	skewness	NOUN
ejpam-5364	7	3	,	,	PUNCT
ejpam-5364	7	4	graph	graph	NOUN
ejpam-5364	7	5	thickness	thickness	NOUN
ejpam-5364	7	6	,	,	PUNCT
ejpam-5364	7	7	and	and	CCONJ
ejpam-5364	7	8	graph	graph	NOUN
ejpam-5364	7	9	crossing	crossing	NOUN
ejpam-5364	7	10	number	number	NOUN
ejpam-5364	7	11	are	be	AUX
ejpam-5364	7	12	a	a	DET
ejpam-5364	7	13	few	few	ADJ
ejpam-5364	7	14	metrics	metric	NOUN
ejpam-5364	7	15	that	that	PRON
ejpam-5364	7	16	describe	describe	VERB
ejpam-5364	7	17	how	how	SCONJ
ejpam-5364	7	18	much	much	ADJ
ejpam-5364	7	19	a	a	DET
ejpam-5364	7	20	graph	graph	NOUN
ejpam-5364	7	21	deviates	deviate	VERB
ejpam-5364	7	22	from	from	ADP
ejpam-5364	7	23	planarity	planarity	NOUN
ejpam-5364	7	24	.	.	PUNCT
ejpam-5364	8	1	in	in	ADP
ejpam-5364	8	2	this	this	DET
ejpam-5364	8	3	work	work	NOUN
ejpam-5364	8	4	,	,	PUNCT
ejpam-5364	8	5	we	we	PRON
ejpam-5364	8	6	ascertain	ascertain	VERB
ejpam-5364	8	7	the	the	DET
ejpam-5364	8	8	relationship	relationship	NOUN
ejpam-5364	8	9	between	between	ADP
ejpam-5364	8	10	the	the	DET
ejpam-5364	8	11	graph	graph	NOUN
ejpam-5364	8	12	’s	’s	PART
ejpam-5364	8	13	properties	property	NOUN
ejpam-5364	8	14	,	,	PUNCT
ejpam-5364	8	15	including	include	VERB
ejpam-5364	8	16	graph	graph	NOUN
ejpam-5364	8	17	skewness	skewness	NOUN
ejpam-5364	8	18	,	,	PUNCT
ejpam-5364	8	19	thickness	thickness	NOUN
ejpam-5364	8	20	,	,	PUNCT
ejpam-5364	8	21	and	and	CCONJ
ejpam-5364	8	22	crossing	crossing	NOUN
ejpam-5364	8	23	number	number	NOUN
ejpam-5364	8	24	,	,	PUNCT
ejpam-5364	8	25	and	and	CCONJ
ejpam-5364	8	26	the	the	DET
ejpam-5364	8	27	graph	graph	NOUN
ejpam-5364	8	28	’s	’s	PART
ejpam-5364	8	29	second	second	ADV
ejpam-5364	8	30	largest	large	ADJ
ejpam-5364	8	31	eigenvalues	eigenvalue	NOUN
ejpam-5364	8	32	of	of	ADP
ejpam-5364	8	33	the	the	DET
ejpam-5364	8	34	adjacency	adjacency	NOUN
ejpam-5364	8	35	matrix	matrix	NOUN
ejpam-5364	8	36	a(g	a(g	PROPN
ejpam-5364	8	37	)	)	PUNCT
ejpam-5364	8	38	and	and	CCONJ
ejpam-5364	8	39	the	the	DET
ejpam-5364	8	40	signless	signless	ADJ
ejpam-5364	8	41	laplacian	laplacian	ADJ
ejpam-5364	8	42	matrix	matrix	NOUN
ejpam-5364	8	43	q(g	q(g	PROPN
ejpam-5364	8	44	)	)	PUNCT
ejpam-5364	8	45	.	.	PUNCT
ejpam-5364	9	1	based	base	VERB
ejpam-5364	9	2	on	on	ADP
ejpam-5364	9	3	the	the	DET
ejpam-5364	9	4	skewness	skewness	NOUN
ejpam-5364	9	5	,	,	PUNCT
ejpam-5364	9	6	thickness	thickness	NOUN
ejpam-5364	9	7	,	,	PUNCT
ejpam-5364	9	8	and	and	CCONJ
ejpam-5364	9	9	crossing	crossing	NOUN
ejpam-5364	9	10	number	number	NOUN
ejpam-5364	9	11	,	,	PUNCT
ejpam-5364	9	12	we	we	PRON
ejpam-5364	9	13	establish	establish	VERB
ejpam-5364	9	14	a	a	DET
ejpam-5364	9	15	lower	lower	ADV
ejpam-5364	9	16	bound	bind	VERB
ejpam-5364	9	17	for	for	ADP
ejpam-5364	9	18	the	the	DET
ejpam-5364	9	19	graph	graph	NOUN
ejpam-5364	9	20	’s	’s	PART
ejpam-5364	9	21	second	second	ADV
ejpam-5364	9	22	largest	large	ADJ
ejpam-5364	9	23	adjacency	adjacency	NOUN
ejpam-5364	9	24	and	and	CCONJ
ejpam-5364	9	25	signless	signless	ADJ
ejpam-5364	9	26	laplacian	laplacian	ADJ
ejpam-5364	9	27	eigenvalues	eigenvalue	VERB
ejpam-5364	9	28	.	.	PUNCT
ejpam-5364	10	1	we	we	PRON
ejpam-5364	10	2	also	also	ADV
ejpam-5364	10	3	determine	determine	VERB
ejpam-5364	10	4	a	a	DET
ejpam-5364	10	5	lower	lower	ADV
ejpam-5364	10	6	bound	bind	VERB
ejpam-5364	10	7	for	for	ADP
ejpam-5364	10	8	these	these	DET
ejpam-5364	10	9	graph	graph	NOUN
ejpam-5364	10	10	properties	property	NOUN
ejpam-5364	10	11	in	in	ADP
ejpam-5364	10	12	terms	term	NOUN
ejpam-5364	10	13	of	of	ADP
ejpam-5364	10	14	the	the	DET
ejpam-5364	10	15	second	second	ADV
ejpam-5364	10	16	largest	large	ADJ
ejpam-5364	10	17	adjacency	adjacency	NOUN
ejpam-5364	10	18	and	and	CCONJ
ejpam-5364	10	19	signless	signless	ADJ
ejpam-5364	10	20	laplacian	laplacian	ADJ
ejpam-5364	10	21	eigenvalues	eigenvalue	NOUN
ejpam-5364	10	22	of	of	ADP
ejpam-5364	10	23	regular	regular	ADJ
ejpam-5364	10	24	graphs	graph	NOUN
ejpam-5364	10	25	.	.	PUNCT
ejpam-5364	11	1	2020	2020	NUM
ejpam-5364	11	2	mathematics	mathematic	NOUN
ejpam-5364	11	3	subject	subject	NOUN
ejpam-5364	11	4	classifications	classification	NOUN
ejpam-5364	11	5	:	:	PUNCT
ejpam-5364	11	6	05c07	05c07	NOUN
ejpam-5364	11	7	,	,	PUNCT
ejpam-5364	11	8	05c10	05c10	ADJ
ejpam-5364	11	9	,	,	PUNCT
ejpam-5364	11	10	05c50	05c50	ADP
ejpam-5364	11	11	key	key	ADJ
ejpam-5364	11	12	words	word	NOUN
ejpam-5364	11	13	and	and	CCONJ
ejpam-5364	11	14	phrases	phrase	NOUN
ejpam-5364	11	15	:	:	PUNCT
ejpam-5364	11	16	second	second	ADV
ejpam-5364	11	17	largest	large	ADJ
ejpam-5364	11	18	eigenvalue	eigenvalue	NOUN
ejpam-5364	11	19	,	,	PUNCT
ejpam-5364	11	20	planar	planar	ADJ
ejpam-5364	11	21	graph	graph	NOUN
ejpam-5364	11	22	,	,	PUNCT
ejpam-5364	11	23	crossing	crossing	NOUN
ejpam-5364	11	24	number	number	NOUN
ejpam-5364	11	25	,	,	PUNCT
ejpam-5364	11	26	graph	graph	NOUN
ejpam-5364	11	27	thickness	thickness	NOUN
ejpam-5364	11	28	,	,	PUNCT
ejpam-5364	11	29	graph	graph	NOUN
ejpam-5364	11	30	skewness	skewness	NOUN
ejpam-5364	11	31	1	1	NUM
ejpam-5364	11	32	.	.	PUNCT
ejpam-5364	12	1	introduction	introduction	NOUN
ejpam-5364	12	2	the	the	DET
ejpam-5364	12	3	goal	goal	NOUN
ejpam-5364	12	4	of	of	ADP
ejpam-5364	12	5	spectral	spectral	ADJ
ejpam-5364	12	6	graph	graph	NOUN
ejpam-5364	12	7	theory	theory	NOUN
ejpam-5364	12	8	,	,	PUNCT
ejpam-5364	12	9	a	a	DET
ejpam-5364	12	10	branch	branch	NOUN
ejpam-5364	12	11	of	of	ADP
ejpam-5364	12	12	algebraic	algebraic	ADJ
ejpam-5364	12	13	graph	graph	NOUN
ejpam-5364	12	14	theory	theory	NOUN
ejpam-5364	12	15	,	,	PUNCT
ejpam-5364	12	16	is	be	AUX
ejpam-5364	12	17	to	to	PART
ejpam-5364	12	18	apply	apply	VERB
ejpam-5364	12	19	ideas	idea	NOUN
ejpam-5364	12	20	from	from	ADP
ejpam-5364	12	21	linear	linear	ADJ
ejpam-5364	12	22	algebra	algebra	PROPN
ejpam-5364	12	23	and	and	CCONJ
ejpam-5364	12	24	spectral	spectral	ADJ
ejpam-5364	12	25	theory	theory	NOUN
ejpam-5364	12	26	to	to	ADP
ejpam-5364	12	27	the	the	DET
ejpam-5364	12	28	study	study	NOUN
ejpam-5364	12	29	of	of	ADP
ejpam-5364	12	30	graph	graph	NOUN
ejpam-5364	12	31	properties	property	NOUN
ejpam-5364	12	32	.	.	PUNCT
ejpam-5364	13	1	graphs	graph	NOUN
ejpam-5364	13	2	are	be	AUX
ejpam-5364	13	3	represented	represent	VERB
ejpam-5364	13	4	as	as	ADP
ejpam-5364	13	5	matrices	matrix	NOUN
ejpam-5364	13	6	in	in	ADP
ejpam-5364	13	7	spectral	spectral	ADJ
ejpam-5364	13	8	graph	graph	NOUN
ejpam-5364	13	9	theory	theory	NOUN
ejpam-5364	13	10	,	,	PUNCT
ejpam-5364	13	11	including	include	VERB
ejpam-5364	13	12	adjacency	adjacency	NOUN
ejpam-5364	13	13	,	,	PUNCT
ejpam-5364	13	14	laplacian	laplacian	ADJ
ejpam-5364	13	15	and	and	CCONJ
ejpam-5364	13	16	signless	signless	ADJ
ejpam-5364	13	17	laplacian	laplacian	ADJ
ejpam-5364	13	18	matrices	matrix	NOUN
ejpam-5364	13	19	.	.	PUNCT
ejpam-5364	14	1	several	several	ADJ
ejpam-5364	14	2	structural	structural	ADJ
ejpam-5364	14	3	characteristics	characteristic	NOUN
ejpam-5364	14	4	and	and	CCONJ
ejpam-5364	14	5	graph	graph	NOUN
ejpam-5364	14	6	properties	property	NOUN
ejpam-5364	14	7	are	be	AUX
ejpam-5364	14	8	analysed	analyse	VERB
ejpam-5364	14	9	using	use	VERB
ejpam-5364	14	10	the	the	DET
ejpam-5364	14	11	eigenvalues	eigenvalue	NOUN
ejpam-5364	14	12	and	and	CCONJ
ejpam-5364	14	13	eigenvectors	eigenvector	NOUN
ejpam-5364	14	14	of	of	ADP
ejpam-5364	14	15	these	these	DET
ejpam-5364	14	16	matrices	matrix	NOUN
ejpam-5364	14	17	.	.	PUNCT
ejpam-5364	15	1	while	while	SCONJ
ejpam-5364	15	2	both	both	CCONJ
ejpam-5364	15	3	the	the	DET
ejpam-5364	15	4	spectral	spectral	ADJ
ejpam-5364	15	5	radius	radius	NOUN
ejpam-5364	15	6	∗corresponding	∗corresponde	VERB
ejpam-5364	15	7	author	author	NOUN
ejpam-5364	15	8	.	.	PUNCT
ejpam-5364	16	1	doi	doi	NOUN
ejpam-5364	16	2	:	:	PUNCT
ejpam-5364	16	3	https://doi.org/10.29020/nybg.ejpam.v17i4	https://doi.org/10.29020/nybg.ejpam.v17i4	PROPN
ejpam-5364	16	4	.	.	PUNCT
ejpam-5364	17	1	email	email	NOUN
ejpam-5364	17	2	addresses	address	NOUN
ejpam-5364	17	3	:	:	PUNCT
ejpam-5364	17	4	machasri.m2019@vitstudent.ac.in	machasri.m2019@vitstudent.ac.in	PUNCT
ejpam-5364	17	5	(	(	PUNCT
ejpam-5364	17	6	m.	m.	NOUN
ejpam-5364	17	7	machasri	machasri	PROPN
ejpam-5364	17	8	)	)	PUNCT
ejpam-5364	17	9	,	,	PUNCT
ejpam-5364	17	10	kalyanidesikan@vit.ac.in	kalyanidesikan@vit.ac.in	NOUN
ejpam-5364	17	11	(	(	PUNCT
ejpam-5364	17	12	k.	k.	PROPN
ejpam-5364	17	13	desikan	desikan	PROPN
ejpam-5364	17	14	)	)	PUNCT
ejpam-5364	17	15	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5364	17	16	3004	3004	NUM
ejpam-5364	18	1	copyright	copyright	NOUN
ejpam-5364	18	2	:	:	PUNCT
ejpam-5364	18	3	©	©	PROPN
ejpam-5364	18	4	2024	2024	NUM
ejpam-5364	18	5	the	the	DET
ejpam-5364	18	6	author(s	author(s	NOUN
ejpam-5364	18	7	)	)	PUNCT
ejpam-5364	18	8	.	.	PUNCT
ejpam-5364	19	1	(	(	PUNCT
ejpam-5364	19	2	cc	cc	NOUN
ejpam-5364	19	3	by	by	ADP
ejpam-5364	19	4	-	-	PUNCT
ejpam-5364	19	5	nc	nc	PROPN
ejpam-5364	19	6	4.0	4.0	NUM
ejpam-5364	19	7	)	)	PUNCT
ejpam-5364	19	8	m.	m.	NOUN
ejpam-5364	19	9	machasri	machasri	PROPN
ejpam-5364	19	10	,	,	PUNCT
ejpam-5364	19	11	d.	d.	PROPN
ejpam-5364	19	12	kalyani	kalyani	PROPN
ejpam-5364	19	13	/	/	SYM
ejpam-5364	19	14	eur	eur	PROPN
ejpam-5364	19	15	.	.	PUNCT
ejpam-5364	20	1	j.	j.	PROPN
ejpam-5364	20	2	pure	pure	PROPN
ejpam-5364	20	3	appl	appl	PROPN
ejpam-5364	20	4	.	.	PROPN
ejpam-5364	20	5	math	math	PROPN
ejpam-5364	20	6	,	,	PUNCT
ejpam-5364	20	7	17	17	NUM
ejpam-5364	20	8	(	(	PUNCT
ejpam-5364	20	9	4	4	NUM
ejpam-5364	20	10	)	)	PUNCT
ejpam-5364	20	11	(	(	PUNCT
ejpam-5364	20	12	2024	2024	NUM
ejpam-5364	20	13	)	)	PUNCT
ejpam-5364	20	14	,	,	PUNCT
ejpam-5364	20	15	3004	3004	NUM
ejpam-5364	20	16	-	-	SYM
ejpam-5364	20	17	3021	3021	NUM
ejpam-5364	20	18	3005	3005	NUM
ejpam-5364	20	19	and	and	CCONJ
ejpam-5364	20	20	the	the	DET
ejpam-5364	20	21	second	second	ADV
ejpam-5364	20	22	largest	large	ADJ
ejpam-5364	20	23	eigenvalue	eigenvalue	NOUN
ejpam-5364	20	24	of	of	ADP
ejpam-5364	20	25	a	a	DET
ejpam-5364	20	26	graph	graph	NOUN
ejpam-5364	20	27	are	be	AUX
ejpam-5364	20	28	crucial	crucial	ADJ
ejpam-5364	20	29	ideas	idea	NOUN
ejpam-5364	20	30	in	in	ADP
ejpam-5364	20	31	spectral	spectral	ADJ
ejpam-5364	20	32	graph	graph	NOUN
ejpam-5364	20	33	theory	theory	NOUN
ejpam-5364	20	34	,	,	PUNCT
ejpam-5364	20	35	they	they	PRON
ejpam-5364	20	36	signify	signify	VERB
ejpam-5364	20	37	distinct	distinct	ADJ
ejpam-5364	20	38	spectral	spectral	ADJ
ejpam-5364	20	39	features	feature	NOUN
ejpam-5364	20	40	of	of	ADP
ejpam-5364	20	41	the	the	DET
ejpam-5364	20	42	graph	graph	NOUN
ejpam-5364	20	43	.	.	PUNCT
ejpam-5364	21	1	in	in	ADP
ejpam-5364	21	2	network	network	NOUN
ejpam-5364	21	3	analysis	analysis	NOUN
ejpam-5364	21	4	,	,	PUNCT
ejpam-5364	21	5	machine	machine	NOUN
ejpam-5364	21	6	learning	learning	NOUN
ejpam-5364	21	7	,	,	PUNCT
ejpam-5364	21	8	optimization	optimization	NOUN
ejpam-5364	21	9	,	,	PUNCT
ejpam-5364	21	10	and	and	CCONJ
ejpam-5364	21	11	computational	computational	ADJ
ejpam-5364	21	12	science	science	NOUN
ejpam-5364	21	13	,	,	PUNCT
ejpam-5364	21	14	the	the	DET
ejpam-5364	21	15	graph	graph	NOUN
ejpam-5364	21	16	’s	’s	PART
ejpam-5364	21	17	second	second	ADV
ejpam-5364	21	18	largest	large	ADJ
ejpam-5364	21	19	eigenvalue	eigenvalue	NOUN
ejpam-5364	21	20	is	be	AUX
ejpam-5364	21	21	widely	widely	ADV
ejpam-5364	21	22	used	use	VERB
ejpam-5364	21	23	and	and	CCONJ
ejpam-5364	21	24	advances	advance	VERB
ejpam-5364	21	25	our	our	PRON
ejpam-5364	21	26	knowledge	knowledge	NOUN
ejpam-5364	21	27	of	of	ADP
ejpam-5364	21	28	graph	graph	NOUN
ejpam-5364	21	29	structures	structure	NOUN
ejpam-5364	21	30	and	and	CCONJ
ejpam-5364	21	31	their	their	PRON
ejpam-5364	21	32	functional	functional	ADJ
ejpam-5364	21	33	characteristics	characteristic	NOUN
ejpam-5364	21	34	.	.	PUNCT
ejpam-5364	22	1	a	a	DET
ejpam-5364	22	2	graph	graph	NOUN
ejpam-5364	22	3	g	g	NOUN
ejpam-5364	22	4	is	be	AUX
ejpam-5364	22	5	made	make	VERB
ejpam-5364	22	6	up	up	ADP
ejpam-5364	22	7	of	of	ADP
ejpam-5364	22	8	two	two	NUM
ejpam-5364	22	9	sets	set	NOUN
ejpam-5364	22	10	(	(	PUNCT
ejpam-5364	22	11	v	v	NOUN
ejpam-5364	22	12	,	,	PUNCT
ejpam-5364	22	13	e	e	NOUN
ejpam-5364	22	14	)	)	PUNCT
ejpam-5364	22	15	,	,	PUNCT
ejpam-5364	22	16	where	where	SCONJ
ejpam-5364	22	17	e	e	NOUN
ejpam-5364	22	18	is	be	AUX
ejpam-5364	22	19	the	the	DET
ejpam-5364	22	20	set	set	NOUN
ejpam-5364	22	21	of	of	ADP
ejpam-5364	22	22	unordered	unordered	ADJ
ejpam-5364	22	23	pairs	pair	NOUN
ejpam-5364	22	24	of	of	ADP
ejpam-5364	22	25	distinct	distinct	ADJ
ejpam-5364	22	26	vertices	vertex	NOUN
ejpam-5364	22	27	,	,	PUNCT
ejpam-5364	22	28	known	know	VERB
ejpam-5364	22	29	as	as	ADP
ejpam-5364	22	30	edges	edge	NOUN
ejpam-5364	22	31	,	,	PUNCT
ejpam-5364	22	32	and	and	CCONJ
ejpam-5364	22	33	v	v	NOUN
ejpam-5364	22	34	is	be	AUX
ejpam-5364	22	35	a	a	DET
ejpam-5364	22	36	finite	finite	ADJ
ejpam-5364	22	37	non	non	ADJ
ejpam-5364	22	38	-	-	ADJ
ejpam-5364	22	39	empty	empty	ADJ
ejpam-5364	22	40	set	set	NOUN
ejpam-5364	22	41	of	of	ADP
ejpam-5364	22	42	elements	element	NOUN
ejpam-5364	22	43	.	.	PUNCT
ejpam-5364	23	1	the	the	DET
ejpam-5364	23	2	vertex	vertex	NOUN
ejpam-5364	23	3	set	set	NOUN
ejpam-5364	23	4	is	be	AUX
ejpam-5364	23	5	denoted	denote	VERB
ejpam-5364	23	6	as	as	ADP
ejpam-5364	23	7	v	v	NOUN
ejpam-5364	23	8	=	=	SYM
ejpam-5364	23	9	{	{	PUNCT
ejpam-5364	23	10	v1	v1	PROPN
ejpam-5364	23	11	,	,	PUNCT
ejpam-5364	23	12	v2	v2	PROPN
ejpam-5364	23	13	,	,	PUNCT
ejpam-5364	23	14	.	.	PUNCT
ejpam-5364	23	15	.	.	PUNCT
ejpam-5364	24	1	.	.	PUNCT
ejpam-5364	25	1	,	,	PUNCT
ejpam-5364	25	2	vn	vn	PROPN
ejpam-5364	25	3	}	}	PUNCT
ejpam-5364	25	4	and	and	CCONJ
ejpam-5364	25	5	the	the	DET
ejpam-5364	25	6	edge	edge	NOUN
ejpam-5364	25	7	set	set	NOUN
ejpam-5364	25	8	is	be	AUX
ejpam-5364	25	9	denoted	denote	VERB
ejpam-5364	25	10	by	by	ADP
ejpam-5364	25	11	e	e	X
ejpam-5364	25	12	=	=	SYM
ejpam-5364	25	13	{	{	PUNCT
ejpam-5364	25	14	e1	e1	PROPN
ejpam-5364	25	15	,	,	PUNCT
ejpam-5364	25	16	e2	e2	PROPN
ejpam-5364	25	17	,	,	PUNCT
ejpam-5364	25	18	.	.	PUNCT
ejpam-5364	25	19	.	.	PUNCT
ejpam-5364	26	1	.	.	PUNCT
ejpam-5364	27	1	,	,	PUNCT
ejpam-5364	27	2	em	em	PRON
ejpam-5364	27	3	}	}	PUNCT
ejpam-5364	27	4	.	.	PUNCT
ejpam-5364	28	1	basic	basic	ADJ
ejpam-5364	28	2	notations	notation	NOUN
ejpam-5364	28	3	and	and	CCONJ
ejpam-5364	28	4	terminology	terminology	NOUN
ejpam-5364	28	5	are	be	AUX
ejpam-5364	28	6	based	base	VERB
ejpam-5364	28	7	on	on	ADP
ejpam-5364	28	8	db	db	PROPN
ejpam-5364	28	9	west	west	PROPN
ejpam-5364	28	10	’s	’s	PART
ejpam-5364	28	11	book	book	NOUN
ejpam-5364	28	12	,	,	PUNCT
ejpam-5364	28	13	introduction	introduction	NOUN
ejpam-5364	28	14	to	to	AUX
ejpam-5364	28	15	graph	graph	NOUN
ejpam-5364	28	16	theory	theory	NOUN
ejpam-5364	28	17	[	[	X
ejpam-5364	28	18	26	26	NUM
ejpam-5364	28	19	]	]	PUNCT
ejpam-5364	28	20	.	.	PUNCT
ejpam-5364	29	1	the	the	DET
ejpam-5364	29	2	adjacency	adjacency	PROPN
ejpam-5364	29	3	matrix	matrix	NOUN
ejpam-5364	29	4	a(g	a(g	PROPN
ejpam-5364	29	5	)	)	PUNCT
ejpam-5364	29	6	of	of	ADP
ejpam-5364	29	7	g	g	PROPN
ejpam-5364	29	8	is	be	AUX
ejpam-5364	29	9	an	an	DET
ejpam-5364	29	10	n×n	n×n	PROPN
ejpam-5364	29	11	matrix	matrix	NOUN
ejpam-5364	29	12	a	a	PRON
ejpam-5364	29	13	=	=	X
ejpam-5364	30	1	[	[	X
ejpam-5364	30	2	aij	aij	X
ejpam-5364	30	3	]	]	X
ejpam-5364	30	4	,	,	PUNCT
ejpam-5364	30	5	where	where	SCONJ
ejpam-5364	30	6	aij	aij	PROPN
ejpam-5364	30	7	=	=	SYM
ejpam-5364	30	8	1	1	PROPN
ejpam-5364	30	9	if	if	SCONJ
ejpam-5364	30	10	vi	vi	PROPN
ejpam-5364	30	11	and	and	CCONJ
ejpam-5364	30	12	vj	vj	NOUN
ejpam-5364	30	13	are	be	AUX
ejpam-5364	30	14	adjacent	adjacent	ADJ
ejpam-5364	30	15	,	,	PUNCT
ejpam-5364	30	16	otherwise	otherwise	ADV
ejpam-5364	30	17	it	it	PRON
ejpam-5364	30	18	is	be	AUX
ejpam-5364	30	19	0	0	NUM
ejpam-5364	30	20	.	.	PUNCT
ejpam-5364	31	1	let	let	VERB
ejpam-5364	31	2	λ1	λ1	ADJ
ejpam-5364	31	3	≥	≥	NOUN
ejpam-5364	31	4	λ2	λ2	NOUN
ejpam-5364	31	5	≥	≥	NOUN
ejpam-5364	31	6	·	·	PUNCT
ejpam-5364	31	7	·	·	PUNCT
ejpam-5364	31	8	·	·	PUNCT
ejpam-5364	32	1	≥	≥	PRON
ejpam-5364	32	2	λn	λn	AUX
ejpam-5364	32	3	be	be	AUX
ejpam-5364	32	4	the	the	DET
ejpam-5364	32	5	eigenvalues	eigenvalue	NOUN
ejpam-5364	32	6	of	of	ADP
ejpam-5364	32	7	a	a	DET
ejpam-5364	32	8	known	know	VERB
ejpam-5364	32	9	as	as	ADP
ejpam-5364	32	10	the	the	DET
ejpam-5364	32	11	spectrum	spectrum	NOUN
ejpam-5364	32	12	of	of	ADP
ejpam-5364	32	13	g.	g.	PROPN
ejpam-5364	32	14	the	the	DET
ejpam-5364	32	15	laplacian	laplacian	ADJ
ejpam-5364	32	16	matrix	matrix	NOUN
ejpam-5364	32	17	of	of	ADP
ejpam-5364	32	18	g	g	PROPN
ejpam-5364	32	19	is	be	AUX
ejpam-5364	32	20	l(g	l(g	NOUN
ejpam-5364	32	21	)	)	PUNCT
ejpam-5364	33	1	=	=	SYM
ejpam-5364	33	2	d(g)−a(g	d(g)−a(g	NOUN
ejpam-5364	33	3	)	)	PUNCT
ejpam-5364	33	4	where	where	SCONJ
ejpam-5364	33	5	d(g	d(g	NOUN
ejpam-5364	33	6	)	)	PUNCT
ejpam-5364	33	7	is	be	AUX
ejpam-5364	33	8	the	the	DET
ejpam-5364	33	9	diagonal	diagonal	ADJ
ejpam-5364	33	10	degree	degree	NOUN
ejpam-5364	33	11	matrix	matrix	NOUN
ejpam-5364	33	12	.	.	PUNCT
ejpam-5364	34	1	let	let	VERB
ejpam-5364	34	2	µ1	µ1	NOUN
ejpam-5364	34	3	≥	≥	NOUN
ejpam-5364	34	4	µ2	µ2	PROPN
ejpam-5364	34	5	≥	≥	NOUN
ejpam-5364	34	6	µ3	µ3	PROPN
ejpam-5364	34	7	≥	≥	X
ejpam-5364	34	8	·	·	PUNCT
ejpam-5364	34	9	·	·	PUNCT
ejpam-5364	34	10	·	·	PUNCT
ejpam-5364	35	1	≥	≥	X
ejpam-5364	35	2	µ(n−1	µ(n−1	NOUN
ejpam-5364	35	3	)	)	PUNCT
ejpam-5364	35	4	≥	≥	NOUN
ejpam-5364	35	5	µn	µn	NOUN
ejpam-5364	35	6	be	be	AUX
ejpam-5364	35	7	the	the	DET
ejpam-5364	35	8	eigenvalues	eigenvalue	NOUN
ejpam-5364	35	9	of	of	ADP
ejpam-5364	35	10	the	the	DET
ejpam-5364	35	11	laplacian	laplacian	ADJ
ejpam-5364	35	12	matrix	matrix	NOUN
ejpam-5364	35	13	.	.	PUNCT
ejpam-5364	36	1	the	the	DET
ejpam-5364	36	2	signless	signless	ADJ
ejpam-5364	36	3	laplacian	laplacian	ADJ
ejpam-5364	36	4	matrix	matrix	NOUN
ejpam-5364	36	5	of	of	ADP
ejpam-5364	36	6	g	g	PROPN
ejpam-5364	36	7	is	be	AUX
ejpam-5364	36	8	q(g	q(g	PROPN
ejpam-5364	36	9	)	)	PUNCT
ejpam-5364	37	1	=	=	SYM
ejpam-5364	37	2	d(g	d(g	PROPN
ejpam-5364	37	3	)	)	PUNCT
ejpam-5364	38	1	+	+	NUM
ejpam-5364	38	2	a(g	a(g	PROPN
ejpam-5364	38	3	)	)	PUNCT
ejpam-5364	38	4	.	.	PUNCT
ejpam-5364	39	1	its	its	PRON
ejpam-5364	39	2	eigenvalues	eigenvalue	NOUN
ejpam-5364	39	3	are	be	AUX
ejpam-5364	39	4	denoted	denote	VERB
ejpam-5364	39	5	as	as	ADP
ejpam-5364	39	6	q1	q1	PROPN
ejpam-5364	39	7	≥	≥	PROPN
ejpam-5364	39	8	q2	q2	PROPN
ejpam-5364	39	9	≥	≥	NUM
ejpam-5364	39	10	·	·	PUNCT
ejpam-5364	39	11	·	·	PUNCT
ejpam-5364	39	12	·	·	PUNCT
ejpam-5364	39	13	≥	≥	NUM
ejpam-5364	40	1	qn	qn	NOUN
ejpam-5364	40	2	.	.	PUNCT
ejpam-5364	41	1	the	the	DET
ejpam-5364	41	2	graph	graph	NOUN
ejpam-5364	41	3	planarization	planarization	NOUN
ejpam-5364	41	4	problem	problem	NOUN
ejpam-5364	41	5	is	be	AUX
ejpam-5364	41	6	to	to	PART
ejpam-5364	41	7	determine	determine	VERB
ejpam-5364	41	8	a	a	DET
ejpam-5364	41	9	minimum	minimum	NOUN
ejpam-5364	41	10	subset	subset	NOUN
ejpam-5364	41	11	of	of	ADP
ejpam-5364	41	12	edges	edge	NOUN
ejpam-5364	41	13	to	to	PART
ejpam-5364	41	14	remove	remove	VERB
ejpam-5364	41	15	from	from	ADP
ejpam-5364	41	16	a	a	DET
ejpam-5364	41	17	non	non	ADJ
ejpam-5364	41	18	-	-	ADJ
ejpam-5364	41	19	planar	planar	ADJ
ejpam-5364	41	20	graph	graph	NOUN
ejpam-5364	41	21	to	to	PART
ejpam-5364	41	22	make	make	VERB
ejpam-5364	41	23	it	it	PRON
ejpam-5364	41	24	planar	planar	VERB
ejpam-5364	41	25	.	.	PUNCT
ejpam-5364	42	1	the	the	DET
ejpam-5364	42	2	problem	problem	NOUN
ejpam-5364	42	3	has	have	VERB
ejpam-5364	42	4	applications	application	NOUN
ejpam-5364	42	5	in	in	ADP
ejpam-5364	42	6	computer	computer	NOUN
ejpam-5364	42	7	science	science	NOUN
ejpam-5364	42	8	with	with	ADP
ejpam-5364	42	9	regard	regard	NOUN
ejpam-5364	42	10	to	to	ADP
ejpam-5364	42	11	printed	print	VERB
ejpam-5364	42	12	circuit	circuit	NOUN
ejpam-5364	42	13	board	board	NOUN
ejpam-5364	42	14	layout	layout	NOUN
ejpam-5364	42	15	and	and	CCONJ
ejpam-5364	42	16	very	very	ADV
ejpam-5364	42	17	-	-	PUNCT
ejpam-5364	42	18	large	large	ADJ
ejpam-5364	42	19	scale	scale	NOUN
ejpam-5364	42	20	integration	integration	NOUN
ejpam-5364	42	21	(	(	PUNCT
ejpam-5364	42	22	vlsi	vlsi	PROPN
ejpam-5364	42	23	)	)	PUNCT
ejpam-5364	42	24	circuit	circuit	NOUN
ejpam-5364	42	25	routing	routing	NOUN
ejpam-5364	42	26	.	.	PUNCT
ejpam-5364	43	1	beyond	beyond	ADP
ejpam-5364	43	2	the	the	DET
ejpam-5364	43	3	standard	standard	ADJ
ejpam-5364	43	4	graph	graph	NOUN
ejpam-5364	43	5	parameters	parameter	NOUN
ejpam-5364	43	6	,	,	PUNCT
ejpam-5364	43	7	we	we	PRON
ejpam-5364	43	8	intend	intend	VERB
ejpam-5364	43	9	to	to	PART
ejpam-5364	43	10	work	work	VERB
ejpam-5364	43	11	on	on	ADP
ejpam-5364	43	12	the	the	DET
ejpam-5364	43	13	parameters	parameter	NOUN
ejpam-5364	43	14	that	that	PRON
ejpam-5364	43	15	are	be	AUX
ejpam-5364	43	16	most	most	ADV
ejpam-5364	43	17	challenging	challenging	ADJ
ejpam-5364	43	18	to	to	PART
ejpam-5364	43	19	compute	compute	VERB
ejpam-5364	43	20	.	.	PUNCT
ejpam-5364	44	1	so	so	ADV
ejpam-5364	44	2	far	far	ADV
ejpam-5364	44	3	,	,	PUNCT
ejpam-5364	44	4	no	no	DET
ejpam-5364	44	5	results	result	NOUN
ejpam-5364	44	6	have	have	AUX
ejpam-5364	44	7	been	be	AUX
ejpam-5364	44	8	found	find	VERB
ejpam-5364	44	9	connecting	connect	VERB
ejpam-5364	44	10	the	the	DET
ejpam-5364	44	11	second	second	ADV
ejpam-5364	44	12	largest	large	ADJ
ejpam-5364	44	13	eigenvalue	eigenvalue	NOUN
ejpam-5364	44	14	to	to	ADP
ejpam-5364	44	15	planarity	planarity	NOUN
ejpam-5364	44	16	-	-	PUNCT
ejpam-5364	44	17	related	relate	VERB
ejpam-5364	44	18	characteristics	characteristic	NOUN
ejpam-5364	44	19	.	.	PUNCT
ejpam-5364	45	1	parameters	parameter	NOUN
ejpam-5364	45	2	such	such	ADJ
ejpam-5364	45	3	as	as	ADP
ejpam-5364	45	4	graph	graph	NOUN
ejpam-5364	45	5	skewness	skewness	NOUN
ejpam-5364	45	6	,	,	PUNCT
ejpam-5364	45	7	thickness	thickness	NOUN
ejpam-5364	45	8	,	,	PUNCT
ejpam-5364	45	9	and	and	CCONJ
ejpam-5364	45	10	crossing	crossing	NOUN
ejpam-5364	45	11	number	number	NOUN
ejpam-5364	45	12	are	be	AUX
ejpam-5364	45	13	used	use	VERB
ejpam-5364	45	14	to	to	PART
ejpam-5364	45	15	determine	determine	VERB
ejpam-5364	45	16	how	how	SCONJ
ejpam-5364	45	17	much	much	ADJ
ejpam-5364	45	18	a	a	DET
ejpam-5364	45	19	graph	graph	NOUN
ejpam-5364	45	20	deviates	deviate	VERB
ejpam-5364	45	21	from	from	ADP
ejpam-5364	45	22	planarity	planarity	NOUN
ejpam-5364	45	23	.	.	PUNCT
ejpam-5364	46	1	computation	computation	NOUN
ejpam-5364	46	2	of	of	ADP
ejpam-5364	46	3	these	these	DET
ejpam-5364	46	4	properties	property	NOUN
ejpam-5364	46	5	are	be	AUX
ejpam-5364	46	6	often	often	ADV
ejpam-5364	46	7	challenging	challenging	ADJ
ejpam-5364	46	8	.	.	PUNCT
ejpam-5364	47	1	as	as	ADP
ejpam-5364	47	2	a	a	DET
ejpam-5364	47	3	result	result	NOUN
ejpam-5364	47	4	,	,	PUNCT
ejpam-5364	47	5	it	it	PRON
ejpam-5364	47	6	is	be	AUX
ejpam-5364	47	7	helpful	helpful	ADJ
ejpam-5364	47	8	to	to	PART
ejpam-5364	47	9	find	find	VERB
ejpam-5364	47	10	bounds	bound	NOUN
ejpam-5364	47	11	for	for	ADP
ejpam-5364	47	12	them	they	PRON
ejpam-5364	47	13	.	.	PUNCT
ejpam-5364	48	1	in	in	ADP
ejpam-5364	48	2	this	this	DET
ejpam-5364	48	3	work	work	NOUN
ejpam-5364	48	4	,	,	PUNCT
ejpam-5364	48	5	we	we	PRON
ejpam-5364	48	6	prove	prove	VERB
ejpam-5364	48	7	the	the	DET
ejpam-5364	48	8	relation	relation	NOUN
ejpam-5364	48	9	of	of	ADP
ejpam-5364	48	10	these	these	DET
ejpam-5364	48	11	graph	graph	NOUN
ejpam-5364	48	12	parameters	parameter	NOUN
ejpam-5364	48	13	with	with	ADP
ejpam-5364	48	14	the	the	DET
ejpam-5364	48	15	second	second	ADV
ejpam-5364	48	16	largest	large	ADJ
ejpam-5364	48	17	eigenvalue	eigenvalue	NOUN
ejpam-5364	48	18	of	of	ADP
ejpam-5364	48	19	the	the	DET
ejpam-5364	48	20	graph	graph	NOUN
ejpam-5364	48	21	.	.	PUNCT
ejpam-5364	49	1	we	we	PRON
ejpam-5364	49	2	employ	employ	VERB
ejpam-5364	49	3	quotient	quotient	NOUN
ejpam-5364	49	4	matrix	matrix	NOUN
ejpam-5364	49	5	and	and	CCONJ
ejpam-5364	49	6	eigenvalue	eigenvalue	NOUN
ejpam-5364	49	7	interlacing	interlace	VERB
ejpam-5364	49	8	technique	technique	NOUN
ejpam-5364	49	9	to	to	PART
ejpam-5364	49	10	arrive	arrive	VERB
ejpam-5364	49	11	at	at	ADP
ejpam-5364	49	12	these	these	DET
ejpam-5364	49	13	bounds	bound	NOUN
ejpam-5364	49	14	.	.	PUNCT
ejpam-5364	50	1	determining	determine	VERB
ejpam-5364	50	2	the	the	DET
ejpam-5364	50	3	skewness	skewness	NOUN
ejpam-5364	50	4	of	of	ADP
ejpam-5364	50	5	a	a	DET
ejpam-5364	50	6	non	non	X
ejpam-5364	50	7	planar	planar	ADJ
ejpam-5364	50	8	graph	graph	NOUN
ejpam-5364	50	9	is	be	AUX
ejpam-5364	50	10	the	the	DET
ejpam-5364	50	11	graph	graph	NOUN
ejpam-5364	50	12	-	-	PUNCT
ejpam-5364	50	13	theoretic	theoretic	ADJ
ejpam-5364	50	14	variant	variant	NOUN
ejpam-5364	50	15	of	of	ADP
ejpam-5364	50	16	the	the	DET
ejpam-5364	50	17	problem	problem	NOUN
ejpam-5364	50	18	.	.	PUNCT
ejpam-5364	51	1	the	the	DET
ejpam-5364	51	2	problem	problem	NOUN
ejpam-5364	51	3	is	be	AUX
ejpam-5364	51	4	known	know	VERB
ejpam-5364	51	5	to	to	PART
ejpam-5364	51	6	be	be	AUX
ejpam-5364	51	7	np−complete	np−complete	ADJ
ejpam-5364	51	8	[	[	X
ejpam-5364	51	9	18	18	NUM
ejpam-5364	51	10	]	]	PUNCT
ejpam-5364	51	11	.	.	PUNCT
ejpam-5364	52	1	the	the	DET
ejpam-5364	52	2	concept	concept	NOUN
ejpam-5364	52	3	of	of	ADP
ejpam-5364	52	4	skewness	skewness	NOUN
ejpam-5364	52	5	is	be	AUX
ejpam-5364	52	6	crucial	crucial	ADJ
ejpam-5364	52	7	in	in	ADP
ejpam-5364	52	8	optimizing	optimize	VERB
ejpam-5364	52	9	graph	graph	NOUN
ejpam-5364	52	10	layouts	layout	NOUN
ejpam-5364	52	11	,	,	PUNCT
ejpam-5364	52	12	designing	design	VERB
ejpam-5364	52	13	efficient	efficient	ADJ
ejpam-5364	52	14	algorithms	algorithm	NOUN
ejpam-5364	52	15	,	,	PUNCT
ejpam-5364	52	16	ensuring	ensure	VERB
ejpam-5364	52	17	network	network	NOUN
ejpam-5364	52	18	reliability	reliability	NOUN
ejpam-5364	52	19	,	,	PUNCT
ejpam-5364	52	20	and	and	CCONJ
ejpam-5364	52	21	advancing	advance	VERB
ejpam-5364	52	22	theoretical	theoretical	ADJ
ejpam-5364	52	23	research	research	NOUN
ejpam-5364	52	24	in	in	ADP
ejpam-5364	52	25	graph	graph	NOUN
ejpam-5364	52	26	theory	theory	NOUN
ejpam-5364	52	27	.	.	PUNCT
ejpam-5364	53	1	in	in	ADP
ejpam-5364	53	2	very	very	ADV
ejpam-5364	53	3	large	large	ADJ
ejpam-5364	53	4	scale	scale	NOUN
ejpam-5364	53	5	integration	integration	NOUN
ejpam-5364	53	6	(	(	PUNCT
ejpam-5364	53	7	vlsi	vlsi	PROPN
ejpam-5364	53	8	)	)	PUNCT
ejpam-5364	53	9	circuit	circuit	NOUN
ejpam-5364	53	10	design	design	NOUN
ejpam-5364	53	11	,	,	PUNCT
ejpam-5364	53	12	skewness	skewness	NOUN
ejpam-5364	53	13	helps	help	VERB
ejpam-5364	53	14	in	in	ADP
ejpam-5364	53	15	minimizing	minimize	VERB
ejpam-5364	53	16	the	the	DET
ejpam-5364	53	17	complexity	complexity	NOUN
ejpam-5364	53	18	of	of	ADP
ejpam-5364	53	19	circuit	circuit	NOUN
ejpam-5364	53	20	layouts	layout	NOUN
ejpam-5364	53	21	.	.	PUNCT
ejpam-5364	54	1	by	by	ADP
ejpam-5364	54	2	analyzing	analyze	VERB
ejpam-5364	54	3	the	the	DET
ejpam-5364	54	4	skewness	skewness	NOUN
ejpam-5364	54	5	of	of	ADP
ejpam-5364	54	6	interconnections	interconnection	NOUN
ejpam-5364	54	7	in	in	ADP
ejpam-5364	54	8	a	a	DET
ejpam-5364	54	9	circuit	circuit	NOUN
ejpam-5364	54	10	graph	graph	NOUN
ejpam-5364	54	11	,	,	PUNCT
ejpam-5364	54	12	designers	designer	NOUN
ejpam-5364	54	13	can	can	AUX
ejpam-5364	54	14	strategically	strategically	ADV
ejpam-5364	54	15	remove	remove	VERB
ejpam-5364	54	16	certain	certain	ADJ
ejpam-5364	54	17	connections	connection	NOUN
ejpam-5364	54	18	to	to	PART
ejpam-5364	54	19	create	create	VERB
ejpam-5364	54	20	a	a	DET
ejpam-5364	54	21	more	more	ADV
ejpam-5364	54	22	planar	planar	ADJ
ejpam-5364	54	23	layout	layout	NOUN
ejpam-5364	54	24	,	,	PUNCT
ejpam-5364	54	25	which	which	PRON
ejpam-5364	54	26	is	be	AUX
ejpam-5364	54	27	crucial	crucial	ADJ
ejpam-5364	54	28	for	for	ADP
ejpam-5364	54	29	reducing	reduce	VERB
ejpam-5364	54	30	interference	interference	NOUN
ejpam-5364	54	31	and	and	CCONJ
ejpam-5364	54	32	improving	improve	VERB
ejpam-5364	54	33	signal	signal	ADJ
ejpam-5364	54	34	integrity	integrity	NOUN
ejpam-5364	54	35	.	.	PUNCT
ejpam-5364	55	1	this	this	DET
ejpam-5364	55	2	application	application	NOUN
ejpam-5364	55	3	is	be	AUX
ejpam-5364	55	4	vital	vital	ADJ
ejpam-5364	55	5	in	in	ADP
ejpam-5364	55	6	ensuring	ensure	VERB
ejpam-5364	55	7	that	that	SCONJ
ejpam-5364	55	8	circuits	circuit	NOUN
ejpam-5364	55	9	function	function	VERB
ejpam-5364	55	10	efficiently	efficiently	ADV
ejpam-5364	55	11	without	without	ADP
ejpam-5364	55	12	excessive	excessive	ADJ
ejpam-5364	55	13	heat	heat	NOUN
ejpam-5364	55	14	generation	generation	NOUN
ejpam-5364	55	15	or	or	CCONJ
ejpam-5364	55	16	signal	signal	ADJ
ejpam-5364	55	17	degradation	degradation	NOUN
ejpam-5364	55	18	.	.	PUNCT
ejpam-5364	56	1	the	the	DET
ejpam-5364	56	2	thickness	thickness	NOUN
ejpam-5364	56	3	of	of	ADP
ejpam-5364	56	4	a	a	DET
ejpam-5364	56	5	graph	graph	NOUN
ejpam-5364	56	6	is	be	AUX
ejpam-5364	56	7	a	a	DET
ejpam-5364	56	8	valuable	valuable	ADJ
ejpam-5364	56	9	measure	measure	NOUN
ejpam-5364	56	10	for	for	ADP
ejpam-5364	56	11	simplifying	simplify	VERB
ejpam-5364	56	12	and	and	CCONJ
ejpam-5364	56	13	understanding	understand	VERB
ejpam-5364	56	14	complex	complex	ADJ
ejpam-5364	56	15	graphs	graph	NOUN
ejpam-5364	56	16	by	by	ADP
ejpam-5364	56	17	breaking	break	VERB
ejpam-5364	56	18	them	they	PRON
ejpam-5364	56	19	into	into	ADP
ejpam-5364	56	20	planar	planar	ADJ
ejpam-5364	56	21	components	component	NOUN
ejpam-5364	56	22	.	.	PUNCT
ejpam-5364	57	1	this	this	DET
ejpam-5364	57	2	concept	concept	NOUN
ejpam-5364	57	3	finds	find	VERB
ejpam-5364	57	4	applications	application	NOUN
ejpam-5364	57	5	in	in	ADP
ejpam-5364	57	6	various	various	ADJ
ejpam-5364	57	7	fields	field	NOUN
ejpam-5364	57	8	,	,	PUNCT
ejpam-5364	57	9	enhancing	enhance	VERB
ejpam-5364	57	10	visualization	visualization	NOUN
ejpam-5364	57	11	,	,	PUNCT
ejpam-5364	57	12	improving	improve	VERB
ejpam-5364	57	13	design	design	NOUN
ejpam-5364	57	14	efficiency	efficiency	NOUN
ejpam-5364	57	15	,	,	PUNCT
ejpam-5364	57	16	and	and	CCONJ
ejpam-5364	57	17	aiding	aid	VERB
ejpam-5364	57	18	in	in	ADP
ejpam-5364	57	19	the	the	DET
ejpam-5364	57	20	m.	m.	NOUN
ejpam-5364	57	21	machasri	machasri	PROPN
ejpam-5364	57	22	,	,	PUNCT
ejpam-5364	57	23	d.	d.	PROPN
ejpam-5364	57	24	kalyani	kalyani	PROPN
ejpam-5364	57	25	/	/	SYM
ejpam-5364	57	26	eur	eur	PROPN
ejpam-5364	57	27	.	.	PUNCT
ejpam-5364	58	1	j.	j.	PROPN
ejpam-5364	58	2	pure	pure	PROPN
ejpam-5364	58	3	appl	appl	PROPN
ejpam-5364	58	4	.	.	PROPN
ejpam-5364	58	5	math	math	PROPN
ejpam-5364	58	6	,	,	PUNCT
ejpam-5364	58	7	17	17	NUM
ejpam-5364	58	8	(	(	PUNCT
ejpam-5364	58	9	4	4	NUM
ejpam-5364	58	10	)	)	PUNCT
ejpam-5364	58	11	(	(	PUNCT
ejpam-5364	58	12	2024	2024	NUM
ejpam-5364	58	13	)	)	PUNCT
ejpam-5364	58	14	,	,	PUNCT
ejpam-5364	58	15	3004	3004	NUM
ejpam-5364	58	16	-	-	SYM
ejpam-5364	58	17	3021	3021	NUM
ejpam-5364	58	18	3006	3006	NUM
ejpam-5364	58	19	development	development	NOUN
ejpam-5364	58	20	of	of	ADP
ejpam-5364	58	21	more	more	ADV
ejpam-5364	58	22	effective	effective	ADJ
ejpam-5364	58	23	algorithms	algorithm	NOUN
ejpam-5364	58	24	and	and	CCONJ
ejpam-5364	58	25	systems	system	NOUN
ejpam-5364	58	26	.	.	PUNCT
ejpam-5364	59	1	whether	whether	SCONJ
ejpam-5364	59	2	in	in	ADP
ejpam-5364	59	3	circuit	circuit	NOUN
ejpam-5364	59	4	design	design	NOUN
ejpam-5364	59	5	,	,	PUNCT
ejpam-5364	59	6	network	network	NOUN
ejpam-5364	59	7	optimization	optimization	NOUN
ejpam-5364	59	8	,	,	PUNCT
ejpam-5364	59	9	biological	biological	ADJ
ejpam-5364	59	10	research	research	NOUN
ejpam-5364	59	11	,	,	PUNCT
ejpam-5364	59	12	or	or	CCONJ
ejpam-5364	59	13	software	software	NOUN
ejpam-5364	59	14	engineering	engineering	NOUN
ejpam-5364	59	15	,	,	PUNCT
ejpam-5364	59	16	the	the	DET
ejpam-5364	59	17	ability	ability	NOUN
ejpam-5364	59	18	to	to	PART
ejpam-5364	59	19	work	work	VERB
ejpam-5364	59	20	with	with	ADP
ejpam-5364	59	21	planar	planar	ADJ
ejpam-5364	59	22	subgraphs	subgraphs	NOUN
ejpam-5364	59	23	is	be	AUX
ejpam-5364	59	24	a	a	DET
ejpam-5364	59	25	powerful	powerful	ADJ
ejpam-5364	59	26	tool	tool	NOUN
ejpam-5364	59	27	for	for	ADP
ejpam-5364	59	28	tackling	tackle	VERB
ejpam-5364	59	29	complexity	complexity	NOUN
ejpam-5364	59	30	.	.	PUNCT
ejpam-5364	60	1	particularly	particularly	ADV
ejpam-5364	60	2	in	in	ADP
ejpam-5364	60	3	computational	computational	ADJ
ejpam-5364	60	4	biology	biology	NOUN
ejpam-5364	60	5	,	,	PUNCT
ejpam-5364	60	6	graph	graph	NOUN
ejpam-5364	60	7	thickness	thickness	NOUN
ejpam-5364	60	8	can	can	AUX
ejpam-5364	60	9	aid	aid	VERB
ejpam-5364	60	10	in	in	ADP
ejpam-5364	60	11	the	the	DET
ejpam-5364	60	12	analysis	analysis	NOUN
ejpam-5364	60	13	of	of	ADP
ejpam-5364	60	14	biological	biological	ADJ
ejpam-5364	60	15	networks	network	NOUN
ejpam-5364	60	16	,	,	PUNCT
ejpam-5364	60	17	such	such	ADJ
ejpam-5364	60	18	as	as	ADP
ejpam-5364	60	19	protein	protein	NOUN
ejpam-5364	60	20	-	-	PUNCT
ejpam-5364	60	21	protein	protein	NOUN
ejpam-5364	60	22	interaction	interaction	NOUN
ejpam-5364	60	23	networks	network	NOUN
ejpam-5364	60	24	.	.	PUNCT
ejpam-5364	61	1	understanding	understand	VERB
ejpam-5364	61	2	the	the	DET
ejpam-5364	61	3	thickness	thickness	NOUN
ejpam-5364	61	4	of	of	ADP
ejpam-5364	61	5	these	these	DET
ejpam-5364	61	6	graphs	graph	NOUN
ejpam-5364	61	7	can	can	AUX
ejpam-5364	61	8	help	help	VERB
ejpam-5364	61	9	researchers	researcher	NOUN
ejpam-5364	61	10	identify	identify	VERB
ejpam-5364	61	11	modular	modular	ADJ
ejpam-5364	61	12	structures	structure	NOUN
ejpam-5364	61	13	within	within	ADP
ejpam-5364	61	14	biological	biological	ADJ
ejpam-5364	61	15	systems	system	NOUN
ejpam-5364	61	16	,	,	PUNCT
ejpam-5364	61	17	facilitating	facilitate	VERB
ejpam-5364	61	18	insights	insight	NOUN
ejpam-5364	61	19	into	into	ADP
ejpam-5364	61	20	cellular	cellular	ADJ
ejpam-5364	61	21	functions	function	NOUN
ejpam-5364	61	22	and	and	CCONJ
ejpam-5364	61	23	interactions	interaction	NOUN
ejpam-5364	61	24	.	.	PUNCT
ejpam-5364	62	1	this	this	DET
ejpam-5364	62	2	application	application	NOUN
ejpam-5364	62	3	is	be	AUX
ejpam-5364	62	4	essential	essential	ADJ
ejpam-5364	62	5	for	for	ADP
ejpam-5364	62	6	advancements	advancement	NOUN
ejpam-5364	62	7	in	in	ADP
ejpam-5364	62	8	fields	field	NOUN
ejpam-5364	62	9	like	like	ADP
ejpam-5364	62	10	genomics	genomic	NOUN
ejpam-5364	62	11	and	and	CCONJ
ejpam-5364	62	12	systems	system	NOUN
ejpam-5364	62	13	biology	biology	NOUN
ejpam-5364	62	14	,	,	PUNCT
ejpam-5364	62	15	where	where	SCONJ
ejpam-5364	62	16	complex	complex	ADJ
ejpam-5364	62	17	interactions	interaction	NOUN
ejpam-5364	62	18	need	need	VERB
ejpam-5364	62	19	to	to	PART
ejpam-5364	62	20	be	be	AUX
ejpam-5364	62	21	understood	understand	VERB
ejpam-5364	62	22	and	and	CCONJ
ejpam-5364	62	23	modeled	model	VERB
ejpam-5364	62	24	accurately	accurately	ADV
ejpam-5364	62	25	.	.	PUNCT
ejpam-5364	63	1	crossing	crossing	NOUN
ejpam-5364	63	2	numbers	number	NOUN
ejpam-5364	63	3	play	play	VERB
ejpam-5364	63	4	a	a	DET
ejpam-5364	63	5	significant	significant	ADJ
ejpam-5364	63	6	role	role	NOUN
ejpam-5364	63	7	in	in	ADP
ejpam-5364	63	8	various	various	ADJ
ejpam-5364	63	9	practical	practical	ADJ
ejpam-5364	63	10	and	and	CCONJ
ejpam-5364	63	11	theoretical	theoretical	ADJ
ejpam-5364	63	12	aspects	aspect	NOUN
ejpam-5364	63	13	of	of	ADP
ejpam-5364	63	14	graph	graph	NOUN
ejpam-5364	63	15	theory	theory	NOUN
ejpam-5364	63	16	and	and	CCONJ
ejpam-5364	63	17	related	related	ADJ
ejpam-5364	63	18	disciplines	discipline	NOUN
ejpam-5364	63	19	.	.	PUNCT
ejpam-5364	64	1	in	in	ADP
ejpam-5364	64	2	network	network	NOUN
ejpam-5364	64	3	design	design	NOUN
ejpam-5364	64	4	,	,	PUNCT
ejpam-5364	64	5	such	such	ADJ
ejpam-5364	64	6	as	as	ADP
ejpam-5364	64	7	designing	design	VERB
ejpam-5364	64	8	circuit	circuit	NOUN
ejpam-5364	64	9	layouts	layout	NOUN
ejpam-5364	64	10	or	or	CCONJ
ejpam-5364	64	11	communication	communication	NOUN
ejpam-5364	64	12	networks	network	NOUN
ejpam-5364	64	13	,	,	PUNCT
ejpam-5364	64	14	minimizing	minimize	VERB
ejpam-5364	64	15	crossing	crossing	NOUN
ejpam-5364	64	16	numbers	number	NOUN
ejpam-5364	64	17	can	can	AUX
ejpam-5364	64	18	lead	lead	VERB
ejpam-5364	64	19	to	to	ADP
ejpam-5364	64	20	more	more	ADV
ejpam-5364	64	21	efficient	efficient	ADJ
ejpam-5364	64	22	and	and	CCONJ
ejpam-5364	64	23	less	less	ADV
ejpam-5364	64	24	congested	congested	ADJ
ejpam-5364	64	25	layouts	layout	NOUN
ejpam-5364	64	26	.	.	PUNCT
ejpam-5364	65	1	this	this	DET
ejpam-5364	65	2	optimization	optimization	NOUN
ejpam-5364	65	3	can	can	AUX
ejpam-5364	65	4	improve	improve	VERB
ejpam-5364	65	5	the	the	DET
ejpam-5364	65	6	performance	performance	NOUN
ejpam-5364	65	7	and	and	CCONJ
ejpam-5364	65	8	reliability	reliability	NOUN
ejpam-5364	65	9	of	of	ADP
ejpam-5364	65	10	the	the	DET
ejpam-5364	65	11	network	network	NOUN
ejpam-5364	65	12	.	.	PUNCT
ejpam-5364	66	1	in	in	ADP
ejpam-5364	66	2	map	map	NOUN
ejpam-5364	66	3	labeling	labeling	NOUN
ejpam-5364	66	4	,	,	PUNCT
ejpam-5364	66	5	graphs	graph	NOUN
ejpam-5364	66	6	are	be	AUX
ejpam-5364	66	7	used	use	VERB
ejpam-5364	66	8	to	to	PART
ejpam-5364	66	9	represent	represent	VERB
ejpam-5364	66	10	geographical	geographical	ADJ
ejpam-5364	66	11	features	feature	NOUN
ejpam-5364	66	12	and	and	CCONJ
ejpam-5364	66	13	their	their	PRON
ejpam-5364	66	14	relationships	relationship	NOUN
ejpam-5364	66	15	.	.	PUNCT
ejpam-5364	67	1	minimizing	minimize	VERB
ejpam-5364	67	2	crossing	crossing	NOUN
ejpam-5364	67	3	numbers	number	NOUN
ejpam-5364	67	4	in	in	ADP
ejpam-5364	67	5	these	these	DET
ejpam-5364	67	6	graphs	graph	NOUN
ejpam-5364	67	7	can	can	AUX
ejpam-5364	67	8	lead	lead	VERB
ejpam-5364	67	9	to	to	ADP
ejpam-5364	67	10	clearer	clear	ADJ
ejpam-5364	67	11	and	and	CCONJ
ejpam-5364	67	12	more	more	ADV
ejpam-5364	67	13	informative	informative	ADJ
ejpam-5364	67	14	maps	map	NOUN
ejpam-5364	67	15	.	.	PUNCT
ejpam-5364	68	1	in	in	ADP
ejpam-5364	68	2	transportation	transportation	NOUN
ejpam-5364	68	3	networks	network	NOUN
ejpam-5364	68	4	,	,	PUNCT
ejpam-5364	68	5	such	such	ADJ
ejpam-5364	68	6	as	as	ADP
ejpam-5364	68	7	railway	railway	NOUN
ejpam-5364	68	8	or	or	CCONJ
ejpam-5364	68	9	road	road	NOUN
ejpam-5364	68	10	systems	system	NOUN
ejpam-5364	68	11	,	,	PUNCT
ejpam-5364	68	12	the	the	DET
ejpam-5364	68	13	crossing	crossing	NOUN
ejpam-5364	68	14	number	number	NOUN
ejpam-5364	68	15	can	can	AUX
ejpam-5364	68	16	aid	aid	VERB
ejpam-5364	68	17	the	the	DET
ejpam-5364	68	18	design	design	NOUN
ejpam-5364	68	19	of	of	ADP
ejpam-5364	68	20	routes	route	NOUN
ejpam-5364	68	21	to	to	PART
ejpam-5364	68	22	minimize	minimize	VERB
ejpam-5364	68	23	intersections	intersection	NOUN
ejpam-5364	68	24	,	,	PUNCT
ejpam-5364	68	25	which	which	PRON
ejpam-5364	68	26	can	can	AUX
ejpam-5364	68	27	improve	improve	VERB
ejpam-5364	68	28	traffic	traffic	NOUN
ejpam-5364	68	29	flow	flow	NOUN
ejpam-5364	68	30	and	and	CCONJ
ejpam-5364	68	31	safety	safety	NOUN
ejpam-5364	68	32	.	.	PUNCT
ejpam-5364	69	1	by	by	ADP
ejpam-5364	69	2	analyzing	analyze	VERB
ejpam-5364	69	3	the	the	DET
ejpam-5364	69	4	crossing	crossing	NOUN
ejpam-5364	69	5	number	number	NOUN
ejpam-5364	69	6	of	of	ADP
ejpam-5364	69	7	a	a	DET
ejpam-5364	69	8	graph	graph	NOUN
ejpam-5364	69	9	representing	represent	VERB
ejpam-5364	69	10	a	a	DET
ejpam-5364	69	11	transportation	transportation	NOUN
ejpam-5364	69	12	network	network	NOUN
ejpam-5364	69	13	,	,	PUNCT
ejpam-5364	69	14	planners	planner	NOUN
ejpam-5364	69	15	can	can	AUX
ejpam-5364	69	16	make	make	VERB
ejpam-5364	69	17	decisions	decision	NOUN
ejpam-5364	69	18	that	that	PRON
ejpam-5364	69	19	reduce	reduce	VERB
ejpam-5364	69	20	congestion	congestion	NOUN
ejpam-5364	69	21	and	and	CCONJ
ejpam-5364	69	22	enhance	enhance	VERB
ejpam-5364	69	23	overall	overall	ADJ
ejpam-5364	69	24	efficiency	efficiency	NOUN
ejpam-5364	69	25	.	.	PUNCT
ejpam-5364	70	1	bounds	bound	NOUN
ejpam-5364	70	2	in	in	ADP
ejpam-5364	70	3	graph	graph	NOUN
ejpam-5364	70	4	theory	theory	NOUN
ejpam-5364	70	5	serve	serve	VERB
ejpam-5364	70	6	as	as	ADP
ejpam-5364	70	7	fundamental	fundamental	ADJ
ejpam-5364	70	8	tools	tool	NOUN
ejpam-5364	70	9	for	for	ADP
ejpam-5364	70	10	estimation	estimation	NOUN
ejpam-5364	70	11	,	,	PUNCT
ejpam-5364	70	12	algorithm	algorithm	NOUN
ejpam-5364	70	13	optimization	optimization	NOUN
ejpam-5364	70	14	,	,	PUNCT
ejpam-5364	70	15	structural	structural	ADJ
ejpam-5364	70	16	analysis	analysis	NOUN
ejpam-5364	70	17	,	,	PUNCT
ejpam-5364	70	18	and	and	CCONJ
ejpam-5364	70	19	practical	practical	ADJ
ejpam-5364	70	20	applications	application	NOUN
ejpam-5364	70	21	across	across	ADP
ejpam-5364	70	22	various	various	ADJ
ejpam-5364	70	23	domains	domain	NOUN
ejpam-5364	70	24	.	.	PUNCT
ejpam-5364	71	1	they	they	PRON
ejpam-5364	71	2	not	not	PART
ejpam-5364	71	3	only	only	ADV
ejpam-5364	71	4	facilitate	facilitate	VERB
ejpam-5364	71	5	a	a	DET
ejpam-5364	71	6	deeper	deep	ADJ
ejpam-5364	71	7	understanding	understanding	NOUN
ejpam-5364	71	8	of	of	ADP
ejpam-5364	71	9	graph	graph	NOUN
ejpam-5364	71	10	properties	property	NOUN
ejpam-5364	71	11	but	but	CCONJ
ejpam-5364	71	12	also	also	ADV
ejpam-5364	71	13	enhance	enhance	VERB
ejpam-5364	71	14	computational	computational	ADJ
ejpam-5364	71	15	efficiency	efficiency	NOUN
ejpam-5364	71	16	in	in	ADP
ejpam-5364	71	17	solving	solve	VERB
ejpam-5364	71	18	complex	complex	ADJ
ejpam-5364	71	19	problems	problem	NOUN
ejpam-5364	71	20	.	.	PUNCT
ejpam-5364	72	1	in	in	ADP
ejpam-5364	72	2	the	the	DET
ejpam-5364	72	3	literature	literature	NOUN
ejpam-5364	72	4	,	,	PUNCT
ejpam-5364	72	5	bounds	bound	VERB
ejpam-5364	72	6	for	for	ADP
ejpam-5364	72	7	certain	certain	ADJ
ejpam-5364	72	8	graph	graph	NOUN
ejpam-5364	72	9	parameters	parameter	NOUN
ejpam-5364	72	10	have	have	AUX
ejpam-5364	72	11	been	be	AUX
ejpam-5364	72	12	determined	determine	VERB
ejpam-5364	72	13	in	in	ADP
ejpam-5364	72	14	relation	relation	NOUN
ejpam-5364	72	15	to	to	ADP
ejpam-5364	72	16	other	other	ADJ
ejpam-5364	72	17	graph	graph	NOUN
ejpam-5364	72	18	parameters	parameter	NOUN
ejpam-5364	72	19	.	.	PUNCT
ejpam-5364	73	1	peter	peter	PROPN
ejpam-5364	73	2	firby	firby	PROPN
ejpam-5364	73	3	and	and	CCONJ
ejpam-5364	73	4	julie	julie	PROPN
ejpam-5364	73	5	haviland	haviland	PROPN
ejpam-5364	73	6	in	in	ADP
ejpam-5364	73	7	[	[	X
ejpam-5364	73	8	9	9	NUM
ejpam-5364	73	9	]	]	PUNCT
ejpam-5364	73	10	established	establish	VERB
ejpam-5364	73	11	lower	low	ADJ
ejpam-5364	73	12	bounds	bound	NOUN
ejpam-5364	73	13	for	for	ADP
ejpam-5364	73	14	the	the	DET
ejpam-5364	73	15	average	average	ADJ
ejpam-5364	73	16	distance	distance	NOUN
ejpam-5364	73	17	in	in	ADP
ejpam-5364	73	18	terms	term	NOUN
ejpam-5364	73	19	of	of	ADP
ejpam-5364	73	20	the	the	DET
ejpam-5364	73	21	independence	independence	NOUN
ejpam-5364	73	22	number	number	NOUN
ejpam-5364	73	23	of	of	ADP
ejpam-5364	73	24	the	the	DET
ejpam-5364	73	25	graph	graph	NOUN
ejpam-5364	73	26	.	.	PUNCT
ejpam-5364	74	1	in	in	ADP
ejpam-5364	74	2	[	[	X
ejpam-5364	74	3	4	4	X
ejpam-5364	74	4	]	]	PUNCT
ejpam-5364	74	5	m.aouchiche	m.aouchiche	NOUN
ejpam-5364	74	6	et	et	PROPN
ejpam-5364	74	7	al	al	PROPN
ejpam-5364	74	8	.	.	PROPN
ejpam-5364	74	9	established	establish	VERB
ejpam-5364	74	10	a	a	DET
ejpam-5364	74	11	sharp	sharp	ADJ
ejpam-5364	74	12	upper	upper	ADJ
ejpam-5364	74	13	bound	bind	VERB
ejpam-5364	74	14	on	on	ADP
ejpam-5364	74	15	the	the	DET
ejpam-5364	74	16	algebraic	algebraic	ADJ
ejpam-5364	74	17	connectivity	connectivity	NOUN
ejpam-5364	74	18	of	of	ADP
ejpam-5364	74	19	a	a	DET
ejpam-5364	74	20	connected	connected	ADJ
ejpam-5364	74	21	graph	graph	NOUN
ejpam-5364	74	22	in	in	ADP
ejpam-5364	74	23	terms	term	NOUN
ejpam-5364	74	24	of	of	ADP
ejpam-5364	74	25	the	the	DET
ejpam-5364	74	26	domination	domination	NOUN
ejpam-5364	74	27	number	number	NOUN
ejpam-5364	74	28	.	.	PUNCT
ejpam-5364	75	1	xiaofeng	xiaofeng	PROPN
ejpam-5364	75	2	gu	gu	PROPN
ejpam-5364	75	3	and	and	CCONJ
ejpam-5364	75	4	muhuo	muhuo	PROPN
ejpam-5364	75	5	liu	liu	PROPN
ejpam-5364	75	6	in	in	ADP
ejpam-5364	75	7	[	[	X
ejpam-5364	75	8	11	11	NUM
ejpam-5364	75	9	]	]	PUNCT
ejpam-5364	75	10	,	,	PUNCT
ejpam-5364	75	11	proved	prove	VERB
ejpam-5364	75	12	sharp	sharp	ADJ
ejpam-5364	75	13	lower	low	ADJ
ejpam-5364	75	14	bounds	bound	NOUN
ejpam-5364	75	15	on	on	ADP
ejpam-5364	75	16	the	the	DET
ejpam-5364	75	17	matching	match	VERB
ejpam-5364	75	18	number	number	NOUN
ejpam-5364	75	19	of	of	ADP
ejpam-5364	75	20	graphs	graph	NOUN
ejpam-5364	75	21	in	in	ADP
ejpam-5364	75	22	terms	term	NOUN
ejpam-5364	75	23	of	of	ADP
ejpam-5364	75	24	the	the	DET
ejpam-5364	75	25	laplacian	laplacian	ADJ
ejpam-5364	75	26	eigenvalues	eigenvalue	NOUN
ejpam-5364	75	27	.	.	PUNCT
ejpam-5364	76	1	in	in	ADP
ejpam-5364	76	2	[	[	X
ejpam-5364	76	3	2	2	X
ejpam-5364	76	4	]	]	PUNCT
ejpam-5364	76	5	nasir	nasir	PROPN
ejpam-5364	76	6	ali	ali	PROPN
ejpam-5364	76	7	et	et	PROPN
ejpam-5364	76	8	al	al	PROPN
ejpam-5364	76	9	.	.	PROPN
ejpam-5364	76	10	explored	explore	VERB
ejpam-5364	76	11	commutative	commutative	ADJ
ejpam-5364	76	12	rings	ring	NOUN
ejpam-5364	76	13	such	such	ADJ
ejpam-5364	76	14	as	as	ADP
ejpam-5364	76	15	the	the	DET
ejpam-5364	76	16	ring	ring	NOUN
ejpam-5364	76	17	of	of	ADP
ejpam-5364	76	18	gaussian	gaussian	ADJ
ejpam-5364	76	19	integers	integer	NOUN
ejpam-5364	76	20	,	,	PUNCT
ejpam-5364	76	21	the	the	DET
ejpam-5364	76	22	ring	ring	NOUN
ejpam-5364	76	23	zn	zn	PROPN
ejpam-5364	76	24	of	of	ADP
ejpam-5364	76	25	integers	integer	NOUN
ejpam-5364	76	26	modulo	modulo	PROPN
ejpam-5364	76	27	n	n	CCONJ
ejpam-5364	76	28	,	,	PUNCT
ejpam-5364	76	29	and	and	CCONJ
ejpam-5364	76	30	quotient	quotient	VERB
ejpam-5364	76	31	polynomial	polynomial	ADJ
ejpam-5364	76	32	rings	ring	NOUN
ejpam-5364	76	33	in	in	ADP
ejpam-5364	76	34	order	order	NOUN
ejpam-5364	76	35	to	to	PART
ejpam-5364	76	36	establish	establish	VERB
ejpam-5364	76	37	general	general	ADJ
ejpam-5364	76	38	bounds	bound	NOUN
ejpam-5364	76	39	for	for	ADP
ejpam-5364	76	40	the	the	DET
ejpam-5364	76	41	multiset	multiset	ADJ
ejpam-5364	76	42	dimension	dimension	NOUN
ejpam-5364	76	43	in	in	ADP
ejpam-5364	76	44	zero	zero	NUM
ejpam-5364	76	45	divisor	divisor	NOUN
ejpam-5364	76	46	graphs	graph	NOUN
ejpam-5364	76	47	(	(	PUNCT
ejpam-5364	76	48	zd	zd	NOUN
ejpam-5364	76	49	-	-	PUNCT
ejpam-5364	76	50	graphs	graph	NOUN
ejpam-5364	76	51	)	)	PUNCT
ejpam-5364	76	52	.	.	PUNCT
ejpam-5364	77	1	also	also	ADV
ejpam-5364	77	2	,	,	PUNCT
ejpam-5364	77	3	they	they	PRON
ejpam-5364	77	4	analyzed	analyze	VERB
ejpam-5364	77	5	the	the	DET
ejpam-5364	77	6	behavior	behavior	NOUN
ejpam-5364	77	7	of	of	ADP
ejpam-5364	77	8	mdim	mdim	PROPN
ejpam-5364	77	9	under	under	ADP
ejpam-5364	77	10	algebraic	algebraic	ADJ
ejpam-5364	77	11	operations	operation	NOUN
ejpam-5364	77	12	and	and	CCONJ
ejpam-5364	77	13	discussed	discuss	VERB
ejpam-5364	77	14	bounds	bound	NOUN
ejpam-5364	77	15	in	in	ADP
ejpam-5364	77	16	terms	term	NOUN
ejpam-5364	77	17	of	of	ADP
ejpam-5364	77	18	diameter	diameter	NOUN
ejpam-5364	77	19	and	and	CCONJ
ejpam-5364	77	20	maximum	maximum	ADJ
ejpam-5364	77	21	degree	degree	NOUN
ejpam-5364	77	22	.	.	PUNCT
ejpam-5364	78	1	some	some	DET
ejpam-5364	78	2	general	general	ADJ
ejpam-5364	78	3	bounds	bound	NOUN
ejpam-5364	78	4	on	on	ADP
ejpam-5364	78	5	the	the	DET
ejpam-5364	78	6	dominating	dominating	NOUN
ejpam-5364	78	7	metric	metric	ADJ
ejpam-5364	78	8	dimension	dimension	NOUN
ejpam-5364	78	9	(	(	PUNCT
ejpam-5364	78	10	ddim	ddim	PROPN
ejpam-5364	78	11	)	)	PUNCT
ejpam-5364	78	12	of	of	ADP
ejpam-5364	78	13	the	the	DET
ejpam-5364	78	14	zd	zd	NOUN
ejpam-5364	78	15	-	-	PUNCT
ejpam-5364	78	16	graph	graph	NOUN
ejpam-5364	78	17	of	of	ADP
ejpam-5364	78	18	r	r	NOUN
ejpam-5364	78	19	in	in	ADP
ejpam-5364	78	20	terms	term	NOUN
ejpam-5364	78	21	of	of	ADP
ejpam-5364	78	22	the	the	DET
ejpam-5364	78	23	maximum	maximum	ADJ
ejpam-5364	78	24	degree	degree	NOUN
ejpam-5364	78	25	,	,	PUNCT
ejpam-5364	78	26	girth	girth	NOUN
ejpam-5364	78	27	,	,	PUNCT
ejpam-5364	78	28	clique	clique	ADJ
ejpam-5364	78	29	number	number	NOUN
ejpam-5364	78	30	,	,	PUNCT
ejpam-5364	78	31	and	and	CCONJ
ejpam-5364	78	32	diameter	diameter	NOUN
ejpam-5364	78	33	were	be	AUX
ejpam-5364	78	34	determined	determine	VERB
ejpam-5364	78	35	by	by	ADP
ejpam-5364	78	36	nasir	nasir	PROPN
ejpam-5364	78	37	ali	ali	PROPN
ejpam-5364	78	38	et	et	PROPN
ejpam-5364	78	39	al	al	PROPN
ejpam-5364	78	40	.	.	PUNCT
ejpam-5364	79	1	in	in	ADP
ejpam-5364	79	2	[	[	X
ejpam-5364	79	3	3	3	NUM
ejpam-5364	79	4	]	]	PUNCT
ejpam-5364	79	5	.	.	PUNCT
ejpam-5364	80	1	the	the	DET
ejpam-5364	80	2	second	second	ADV
ejpam-5364	80	3	largest	large	ADJ
ejpam-5364	80	4	eigenvalue	eigenvalue	NOUN
ejpam-5364	80	5	of	of	ADP
ejpam-5364	80	6	a	a	DET
ejpam-5364	80	7	regular	regular	ADJ
ejpam-5364	80	8	graph	graph	NOUN
ejpam-5364	80	9	g	g	NOUN
ejpam-5364	80	10	has	have	VERB
ejpam-5364	80	11	impact	impact	NOUN
ejpam-5364	80	12	on	on	ADP
ejpam-5364	80	13	graph	graph	NOUN
ejpam-5364	80	14	’s	’s	PART
ejpam-5364	80	15	diameter	diameter	NOUN
ejpam-5364	81	1	[	[	X
ejpam-5364	81	2	6	6	NUM
ejpam-5364	81	3	]	]	PUNCT
ejpam-5364	81	4	,	,	PUNCT
ejpam-5364	81	5	covering	cover	VERB
ejpam-5364	81	6	number	number	NOUN
ejpam-5364	81	7	[	[	X
ejpam-5364	81	8	8	8	NUM
ejpam-5364	81	9	]	]	PUNCT
ejpam-5364	81	10	and	and	CCONJ
ejpam-5364	81	11	the	the	DET
ejpam-5364	81	12	convergence	convergence	NOUN
ejpam-5364	81	13	properties	property	NOUN
ejpam-5364	81	14	of	of	ADP
ejpam-5364	81	15	random	random	ADJ
ejpam-5364	81	16	walks	walk	NOUN
ejpam-5364	81	17	[	[	X
ejpam-5364	81	18	10	10	NUM
ejpam-5364	81	19	]	]	PUNCT
ejpam-5364	81	20	.	.	PUNCT
ejpam-5364	82	1	in	in	ADP
ejpam-5364	82	2	2006	2006	NUM
ejpam-5364	82	3	,	,	PUNCT
ejpam-5364	82	4	stanic	stanic	ADJ
ejpam-5364	82	5	[	[	X
ejpam-5364	82	6	23	23	NUM
ejpam-5364	82	7	]	]	PUNCT
ejpam-5364	82	8	discovered	discover	VERB
ejpam-5364	82	9	for	for	ADP
ejpam-5364	82	10	the	the	DET
ejpam-5364	82	11	first	first	ADJ
ejpam-5364	82	12	time	time	NOUN
ejpam-5364	82	13	the	the	DET
ejpam-5364	82	14	star	star	NOUN
ejpam-5364	82	15	complements	complement	VERB
ejpam-5364	82	16	for	for	ADP
ejpam-5364	82	17	the	the	DET
ejpam-5364	82	18	graphs	graph	NOUN
ejpam-5364	82	19	such	such	ADJ
ejpam-5364	82	20	as	as	ADP
ejpam-5364	82	21	complete	complete	ADJ
ejpam-5364	82	22	graphs	graph	NOUN
ejpam-5364	82	23	and	and	CCONJ
ejpam-5364	82	24	trees	tree	NOUN
ejpam-5364	82	25	with	with	ADP
ejpam-5364	82	26	second	second	ADV
ejpam-5364	82	27	largest	large	ADJ
ejpam-5364	82	28	eigenvalue	eigenvalue	NOUN
ejpam-5364	82	29	1	1	NUM
ejpam-5364	82	30	.	.	PUNCT
ejpam-5364	82	31	ramezani	ramezani	PROPN
ejpam-5364	82	32	and	and	CCONJ
ejpam-5364	82	33	tayfeh	tayfeh	NOUN
ejpam-5364	82	34	-	-	PUNCT
ejpam-5364	82	35	rezaiea	rezaiea	NOUN
ejpam-5364	82	36	[	[	X
ejpam-5364	82	37	20	20	NUM
ejpam-5364	82	38	]	]	PUNCT
ejpam-5364	82	39	m.	m.	NOUN
ejpam-5364	82	40	machasri	machasri	PROPN
ejpam-5364	82	41	,	,	PUNCT
ejpam-5364	82	42	d.	d.	PROPN
ejpam-5364	82	43	kalyani	kalyani	PROPN
ejpam-5364	82	44	/	/	SYM
ejpam-5364	82	45	eur	eur	PROPN
ejpam-5364	82	46	.	.	PUNCT
ejpam-5364	83	1	j.	j.	PROPN
ejpam-5364	83	2	pure	pure	PROPN
ejpam-5364	83	3	appl	appl	PROPN
ejpam-5364	83	4	.	.	PROPN
ejpam-5364	83	5	math	math	PROPN
ejpam-5364	83	6	,	,	PUNCT
ejpam-5364	83	7	17	17	NUM
ejpam-5364	83	8	(	(	PUNCT
ejpam-5364	83	9	4	4	NUM
ejpam-5364	83	10	)	)	PUNCT
ejpam-5364	83	11	(	(	PUNCT
ejpam-5364	83	12	2024	2024	NUM
ejpam-5364	83	13	)	)	PUNCT
ejpam-5364	83	14	,	,	PUNCT
ejpam-5364	83	15	3004	3004	NUM
ejpam-5364	83	16	-	-	SYM
ejpam-5364	83	17	3021	3021	NUM
ejpam-5364	83	18	3007	3007	NUM
ejpam-5364	83	19	found	find	VERB
ejpam-5364	83	20	the	the	DET
ejpam-5364	83	21	maximal	maximal	ADJ
ejpam-5364	83	22	graphs	graph	NOUN
ejpam-5364	83	23	and	and	CCONJ
ejpam-5364	83	24	regular	regular	ADJ
ejpam-5364	83	25	graphs	graph	NOUN
ejpam-5364	83	26	which	which	PRON
ejpam-5364	83	27	have	have	VERB
ejpam-5364	83	28	kr	kr	PROPN
ejpam-5364	83	29	,	,	PUNCT
ejpam-5364	83	30	s+	s+	NUM
ejpam-5364	83	31	tk1	tk1	PROPN
ejpam-5364	83	32	as	as	ADP
ejpam-5364	83	33	a	a	DET
ejpam-5364	83	34	star	star	NOUN
ejpam-5364	83	35	complement	complement	NOUN
ejpam-5364	83	36	with	with	ADP
ejpam-5364	83	37	second	second	ADV
ejpam-5364	83	38	largest	large	ADJ
ejpam-5364	83	39	eigenvalue	eigenvalue	NOUN
ejpam-5364	83	40	1	1	NUM
ejpam-5364	83	41	.	.	PUNCT
ejpam-5364	84	1	the	the	DET
ejpam-5364	84	2	relationship	relationship	NOUN
ejpam-5364	84	3	between	between	ADP
ejpam-5364	84	4	the	the	DET
ejpam-5364	84	5	matching	match	VERB
ejpam-5364	84	6	number	number	NOUN
ejpam-5364	84	7	and	and	CCONJ
ejpam-5364	84	8	∂2(g	∂2(g	NOUN
ejpam-5364	84	9	)	)	PUNCT
ejpam-5364	84	10	,	,	PUNCT
ejpam-5364	84	11	the	the	DET
ejpam-5364	84	12	second	second	ADV
ejpam-5364	84	13	largest	large	ADJ
ejpam-5364	84	14	distance	distance	NOUN
ejpam-5364	84	15	laplacian	laplacian	ADJ
ejpam-5364	84	16	eigenvalue	eigenvalue	NOUN
ejpam-5364	84	17	of	of	ADP
ejpam-5364	84	18	g	g	PROPN
ejpam-5364	84	19	,	,	PUNCT
ejpam-5364	84	20	was	be	AUX
ejpam-5364	84	21	examined	examine	VERB
ejpam-5364	84	22	by	by	ADP
ejpam-5364	84	23	tian	tian	PROPN
ejpam-5364	84	24	and	and	CCONJ
ejpam-5364	84	25	wong	wong	PROPN
ejpam-5364	85	1	[	[	X
ejpam-5364	85	2	24	24	NUM
ejpam-5364	85	3	]	]	PUNCT
ejpam-5364	85	4	,	,	PUNCT
ejpam-5364	85	5	who	who	PRON
ejpam-5364	85	6	also	also	ADV
ejpam-5364	85	7	provided	provide	VERB
ejpam-5364	85	8	lower	low	ADJ
ejpam-5364	85	9	bounds	bound	NOUN
ejpam-5364	85	10	for	for	ADP
ejpam-5364	85	11	∂2(g	∂2(g	NOUN
ejpam-5364	85	12	)	)	PUNCT
ejpam-5364	85	13	in	in	ADP
ejpam-5364	85	14	terms	term	NOUN
ejpam-5364	85	15	of	of	ADP
ejpam-5364	85	16	m(g	m(g	NOUN
ejpam-5364	85	17	)	)	PUNCT
ejpam-5364	85	18	.	.	PUNCT
ejpam-5364	86	1	additionally	additionally	ADV
ejpam-5364	86	2	,	,	PUNCT
ejpam-5364	86	3	all	all	DET
ejpam-5364	86	4	extremal	extremal	ADJ
ejpam-5364	86	5	graphs	graph	NOUN
ejpam-5364	86	6	that	that	PRON
ejpam-5364	86	7	achieved	achieve	VERB
ejpam-5364	86	8	lower	low	ADJ
ejpam-5364	86	9	bounds	bound	NOUN
ejpam-5364	86	10	were	be	AUX
ejpam-5364	86	11	described	describe	VERB
ejpam-5364	86	12	.	.	PUNCT
ejpam-5364	87	1	in	in	ADP
ejpam-5364	87	2	2019	2019	NUM
ejpam-5364	87	3	,	,	PUNCT
ejpam-5364	87	4	vladislav	vladislav	PROPN
ejpam-5364	87	5	kabanova	kabanova	PROPN
ejpam-5364	87	6	et	et	PROPN
ejpam-5364	87	7	al	al	PROPN
ejpam-5364	87	8	.	.	PUNCT
ejpam-5364	88	1	[	[	X
ejpam-5364	88	2	14	14	NUM
ejpam-5364	88	3	]	]	PUNCT
ejpam-5364	88	4	studied	study	VERB
ejpam-5364	88	5	the	the	DET
ejpam-5364	88	6	eigenfunctions	eigenfunction	NOUN
ejpam-5364	88	7	of	of	ADP
ejpam-5364	88	8	the	the	DET
ejpam-5364	88	9	star	star	NOUN
ejpam-5364	88	10	graph	graph	NOUN
ejpam-5364	88	11	sn	sn	PROPN
ejpam-5364	88	12	with	with	ADP
ejpam-5364	88	13	n	n	PRON
ejpam-5364	88	14	≥	≥	NUM
ejpam-5364	88	15	3	3	NUM
ejpam-5364	88	16	,	,	PUNCT
ejpam-5364	88	17	where	where	SCONJ
ejpam-5364	88	18	sn	sn	PROPN
ejpam-5364	88	19	is	be	AUX
ejpam-5364	88	20	the	the	DET
ejpam-5364	88	21	cayley	cayley	ADJ
ejpam-5364	88	22	graph	graph	NOUN
ejpam-5364	88	23	on	on	ADP
ejpam-5364	88	24	the	the	DET
ejpam-5364	88	25	symmetric	symmetric	ADJ
ejpam-5364	88	26	group	group	NOUN
ejpam-5364	88	27	symn	symn	NOUN
ejpam-5364	88	28	generated	generate	VERB
ejpam-5364	88	29	by	by	ADP
ejpam-5364	88	30	the	the	DET
ejpam-5364	88	31	set	set	NOUN
ejpam-5364	88	32	of	of	ADP
ejpam-5364	88	33	transpositions	transposition	NOUN
ejpam-5364	88	34	{	{	PUNCT
ejpam-5364	88	35	(	(	PUNCT
ejpam-5364	88	36	12	12	NUM
ejpam-5364	88	37	)	)	PUNCT
ejpam-5364	88	38	,	,	PUNCT
ejpam-5364	88	39	(	(	PUNCT
ejpam-5364	88	40	13	13	NUM
ejpam-5364	88	41	)	)	PUNCT
ejpam-5364	88	42	,	,	PUNCT
ejpam-5364	88	43	.	.	PUNCT
ejpam-5364	88	44	.	.	PUNCT
ejpam-5364	89	1	.	.	PUNCT
ejpam-5364	90	1	,	,	PUNCT
ejpam-5364	90	2	(	(	PUNCT
ejpam-5364	90	3	1n	1n	NUM
ejpam-5364	90	4	)	)	PUNCT
ejpam-5364	90	5	}	}	PUNCT
ejpam-5364	90	6	corresponding	correspond	VERB
ejpam-5364	90	7	to	to	ADP
ejpam-5364	90	8	λ2	λ2	NOUN
ejpam-5364	90	9	=	=	SYM
ejpam-5364	90	10	n	n	CCONJ
ejpam-5364	90	11	−	−	PROPN
ejpam-5364	90	12	2	2	NUM
ejpam-5364	90	13	.	.	PUNCT
ejpam-5364	91	1	a	a	DET
ejpam-5364	91	2	characterisation	characterisation	NOUN
ejpam-5364	91	3	of	of	ADP
ejpam-5364	91	4	eigenfunctions	eigenfunction	NOUN
ejpam-5364	91	5	with	with	ADP
ejpam-5364	91	6	the	the	DET
ejpam-5364	91	7	smallest	small	ADJ
ejpam-5364	91	8	cardinality	cardinality	NOUN
ejpam-5364	91	9	of	of	ADP
ejpam-5364	91	10	the	the	DET
ejpam-5364	91	11	support	support	NOUN
ejpam-5364	91	12	was	be	AUX
ejpam-5364	91	13	found	find	VERB
ejpam-5364	91	14	for	for	ADP
ejpam-5364	91	15	n	n	X
ejpam-5364	91	16	≥	≥	NOUN
ejpam-5364	91	17	8	8	NUM
ejpam-5364	91	18	and	and	CCONJ
ejpam-5364	91	19	n	n	NOUN
ejpam-5364	91	20	=	=	NOUN
ejpam-5364	91	21	3	3	X
ejpam-5364	91	22	.	.	X
ejpam-5364	91	23	they	they	PRON
ejpam-5364	91	24	also	also	ADV
ejpam-5364	91	25	got	get	VERB
ejpam-5364	91	26	the	the	DET
ejpam-5364	91	27	minimal	minimal	ADJ
ejpam-5364	91	28	cardinality	cardinality	NOUN
ejpam-5364	91	29	of	of	ADP
ejpam-5364	91	30	the	the	DET
ejpam-5364	91	31	support	support	NOUN
ejpam-5364	91	32	of	of	ADP
ejpam-5364	91	33	an	an	DET
ejpam-5364	91	34	eigenfunction	eigenfunction	NOUN
ejpam-5364	91	35	of	of	ADP
ejpam-5364	91	36	sn	sn	PROPN
ejpam-5364	91	37	corresponding	correspond	VERB
ejpam-5364	91	38	to	to	ADP
ejpam-5364	91	39	the	the	DET
ejpam-5364	91	40	second	second	ADV
ejpam-5364	91	41	largest	large	ADJ
ejpam-5364	91	42	eigenvalue	eigenvalue	NOUN
ejpam-5364	91	43	.	.	PUNCT
ejpam-5364	92	1	in	in	ADP
ejpam-5364	92	2	terms	term	NOUN
ejpam-5364	92	3	of	of	ADP
ejpam-5364	92	4	the	the	DET
ejpam-5364	92	5	order	order	NOUN
ejpam-5364	92	6	and	and	CCONJ
ejpam-5364	92	7	matching	matching	NOUN
ejpam-5364	92	8	number	number	NOUN
ejpam-5364	92	9	of	of	ADP
ejpam-5364	92	10	g	g	NOUN
ejpam-5364	92	11	,	,	PUNCT
ejpam-5364	92	12	shuchao	shuchao	ADJ
ejpam-5364	92	13	li	li	NOUN
ejpam-5364	92	14	and	and	CCONJ
ejpam-5364	92	15	wanting	want	VERB
ejpam-5364	92	16	sun	sun	NOUN
ejpam-5364	92	17	[	[	X
ejpam-5364	92	18	15	15	NUM
ejpam-5364	92	19	]	]	X
ejpam-5364	92	20	set	set	VERB
ejpam-5364	92	21	sharp	sharp	ADJ
ejpam-5364	92	22	lower	low	ADJ
ejpam-5364	92	23	bounds	bound	NOUN
ejpam-5364	92	24	on	on	ADP
ejpam-5364	92	25	q2(g	q2(g	NOUN
ejpam-5364	92	26	)	)	PUNCT
ejpam-5364	92	27	.	.	PUNCT
ejpam-5364	93	1	among	among	ADP
ejpam-5364	93	2	the	the	DET
ejpam-5364	93	3	n−vertex	n−vertex	NOUN
ejpam-5364	93	4	connected	connect	VERB
ejpam-5364	93	5	graphs	graph	NOUN
ejpam-5364	93	6	with	with	ADP
ejpam-5364	93	7	fixed	fix	VERB
ejpam-5364	93	8	connectivity	connectivity	NOUN
ejpam-5364	93	9	,	,	PUNCT
ejpam-5364	93	10	they	they	PRON
ejpam-5364	93	11	found	find	VERB
ejpam-5364	93	12	the	the	DET
ejpam-5364	93	13	one	one	NOUN
ejpam-5364	93	14	and	and	CCONJ
ejpam-5364	93	15	only	only	ADV
ejpam-5364	93	16	graph	graph	NOUN
ejpam-5364	93	17	with	with	ADP
ejpam-5364	93	18	the	the	DET
ejpam-5364	93	19	least	least	ADJ
ejpam-5364	93	20	q2(g	q2(g	NOUN
ejpam-5364	93	21	)	)	PUNCT
ejpam-5364	93	22	.	.	PUNCT
ejpam-5364	94	1	the	the	DET
ejpam-5364	94	2	maximum	maximum	ADJ
ejpam-5364	94	3	number	number	NOUN
ejpam-5364	94	4	of	of	ADP
ejpam-5364	94	5	vertices	vertex	NOUN
ejpam-5364	94	6	of	of	ADP
ejpam-5364	94	7	a	a	DET
ejpam-5364	94	8	connected	connected	ADJ
ejpam-5364	94	9	k−regular	k−regular	NOUN
ejpam-5364	94	10	graph	graph	NOUN
ejpam-5364	94	11	with	with	ADP
ejpam-5364	94	12	the	the	DET
ejpam-5364	94	13	second	second	ADV
ejpam-5364	94	14	largest	large	ADJ
ejpam-5364	94	15	eigenvalue	eigenvalue	NOUN
ejpam-5364	94	16	at	at	ADP
ejpam-5364	94	17	most	most	ADJ
ejpam-5364	94	18	λ	λ	NOUN
ejpam-5364	94	19	is	be	AUX
ejpam-5364	94	20	denoted	denote	VERB
ejpam-5364	94	21	as	as	ADP
ejpam-5364	94	22	v(k	v(k	PROPN
ejpam-5364	94	23	,	,	PUNCT
ejpam-5364	94	24	λ	λ	NOUN
ejpam-5364	94	25	)	)	PUNCT
ejpam-5364	94	26	.	.	PUNCT
ejpam-5364	95	1	the	the	DET
ejpam-5364	95	2	alon	alon	PROPN
ejpam-5364	95	3	-	-	PUNCT
ejpam-5364	95	4	boppana	boppana	PROPN
ejpam-5364	95	5	theorem	theorem	NOUN
ejpam-5364	95	6	implies	imply	VERB
ejpam-5364	95	7	that	that	SCONJ
ejpam-5364	95	8	v(k	v(k	PROPN
ejpam-5364	95	9	,	,	PUNCT
ejpam-5364	95	10	λ	λ	X
ejpam-5364	95	11	)	)	PUNCT
ejpam-5364	95	12	is	be	AUX
ejpam-5364	95	13	finite	finite	ADJ
ejpam-5364	95	14	when	when	SCONJ
ejpam-5364	95	15	k	k	PROPN
ejpam-5364	95	16	>	>	X
ejpam-5364	95	17	λ2	λ2	PROPN
ejpam-5364	95	18	+	+	PROPN
ejpam-5364	95	19	4	4	NUM
ejpam-5364	95	20	4	4	NUM
ejpam-5364	95	21	.	.	PUNCT
ejpam-5364	96	1	jae	jae	PROPN
ejpam-5364	96	2	young	young	PROPN
ejpam-5364	96	3	yang	yang	PROPN
ejpam-5364	96	4	and	and	CCONJ
ejpam-5364	96	5	jack	jack	PROPN
ejpam-5364	96	6	h.	h.	PROPN
ejpam-5364	96	7	koolen	koolen	PROPN
ejpam-5364	97	1	[	[	X
ejpam-5364	97	2	27	27	NUM
ejpam-5364	97	3	]	]	PUNCT
ejpam-5364	97	4	proved	prove	VERB
ejpam-5364	97	5	that	that	SCONJ
ejpam-5364	97	6	for	for	ADP
ejpam-5364	97	7	fixed	fix	VERB
ejpam-5364	97	8	λ	λ	PROPN
ejpam-5364	97	9	≥	≥	NOUN
ejpam-5364	97	10	1	1	NUM
ejpam-5364	97	11	,	,	PUNCT
ejpam-5364	97	12	there	there	PRON
ejpam-5364	97	13	exists	exist	VERB
ejpam-5364	97	14	a	a	DET
ejpam-5364	97	15	constant	constant	ADJ
ejpam-5364	97	16	c(λ	c(λ	PROPN
ejpam-5364	97	17	)	)	PUNCT
ejpam-5364	97	18	such	such	ADJ
ejpam-5364	97	19	that	that	DET
ejpam-5364	97	20	2k	2k	NOUN
ejpam-5364	97	21	+	+	CCONJ
ejpam-5364	97	22	2	2	NUM
ejpam-5364	97	23	≤	≤	NOUN
ejpam-5364	97	24	v(k	v(k	NOUN
ejpam-5364	97	25	,	,	PUNCT
ejpam-5364	97	26	λ	λ	NOUN
ejpam-5364	97	27	)	)	PUNCT
ejpam-5364	97	28	≤	≤	NOUN
ejpam-5364	97	29	2k	2k	NOUN
ejpam-5364	97	30	+	+	CCONJ
ejpam-5364	97	31	c(λ	c(λ	PROPN
ejpam-5364	97	32	)	)	PUNCT
ejpam-5364	97	33	when	when	SCONJ
ejpam-5364	97	34	k	k	PROPN
ejpam-5364	97	35	>	>	X
ejpam-5364	97	36	λ2	λ2	PROPN
ejpam-5364	97	37	+	+	PROPN
ejpam-5364	97	38	4	4	NUM
ejpam-5364	97	39	4	4	NUM
ejpam-5364	97	40	.	.	PUNCT
ejpam-5364	98	1	the	the	DET
ejpam-5364	98	2	relationship	relationship	NOUN
ejpam-5364	98	3	between	between	ADP
ejpam-5364	98	4	the	the	DET
ejpam-5364	98	5	local	local	ADJ
ejpam-5364	98	6	valency	valency	NOUN
ejpam-5364	98	7	of	of	ADP
ejpam-5364	98	8	an	an	DET
ejpam-5364	98	9	edge	edge	NOUN
ejpam-5364	98	10	-	-	PUNCT
ejpam-5364	98	11	regular	regular	ADJ
ejpam-5364	98	12	graph	graph	NOUN
ejpam-5364	98	13	and	and	CCONJ
ejpam-5364	98	14	its	its	PRON
ejpam-5364	98	15	λ2	λ2	NOUN
ejpam-5364	98	16	was	be	AUX
ejpam-5364	98	17	discovered	discover	VERB
ejpam-5364	98	18	by	by	ADP
ejpam-5364	98	19	jongyook	jongyook	NOUN
ejpam-5364	98	20	park	park	NOUN
ejpam-5364	98	21	[	[	X
ejpam-5364	98	22	19	19	NUM
ejpam-5364	98	23	]	]	PUNCT
ejpam-5364	98	24	.	.	PUNCT
ejpam-5364	99	1	for	for	ADP
ejpam-5364	99	2	some	some	DET
ejpam-5364	99	3	connected	connect	VERB
ejpam-5364	99	4	sets	set	NOUN
ejpam-5364	99	5	h	h	NOUN
ejpam-5364	99	6	,	,	PUNCT
ejpam-5364	99	7	siemons	siemon	NOUN
ejpam-5364	99	8	et	et	PROPN
ejpam-5364	99	9	al	al	PROPN
ejpam-5364	99	10	.	.	PUNCT
ejpam-5364	100	1	[	[	X
ejpam-5364	100	2	21	21	NUM
ejpam-5364	100	3	]	]	PUNCT
ejpam-5364	100	4	calculated	calculate	VERB
ejpam-5364	100	5	the	the	DET
ejpam-5364	100	6	value	value	NOUN
ejpam-5364	100	7	of	of	ADP
ejpam-5364	100	8	λ2(γ	λ2(γ	NUM
ejpam-5364	100	9	)	)	PUNCT
ejpam-5364	100	10	by	by	ADP
ejpam-5364	100	11	looking	look	VERB
ejpam-5364	100	12	at	at	ADP
ejpam-5364	100	13	the	the	DET
ejpam-5364	100	14	second	second	ADV
ejpam-5364	100	15	largest	large	ADJ
ejpam-5364	100	16	eigenvalue	eigenvalue	NOUN
ejpam-5364	100	17	of	of	ADP
ejpam-5364	100	18	the	the	DET
ejpam-5364	100	19	cayley	cayley	ADJ
ejpam-5364	100	20	graph	graph	NOUN
ejpam-5364	100	21	γ	γ	X
ejpam-5364	100	22	=	=	SYM
ejpam-5364	100	23	cay(g	cay(g	PROPN
ejpam-5364	100	24	,	,	PUNCT
ejpam-5364	100	25	h	h	NOUN
ejpam-5364	100	26	)	)	PUNCT
ejpam-5364	100	27	over	over	ADP
ejpam-5364	100	28	g	g	PROPN
ejpam-5364	100	29	=	=	PUNCT
ejpam-5364	100	30	sn	sn	PROPN
ejpam-5364	100	31	or	or	CCONJ
ejpam-5364	100	32	an	an	PRON
ejpam-5364	100	33	.	.	PUNCT
ejpam-5364	101	1	this	this	DET
ejpam-5364	101	2	paper	paper	NOUN
ejpam-5364	101	3	is	be	AUX
ejpam-5364	101	4	organised	organise	VERB
ejpam-5364	101	5	as	as	SCONJ
ejpam-5364	101	6	follows	follow	VERB
ejpam-5364	101	7	.	.	PUNCT
ejpam-5364	102	1	we	we	PRON
ejpam-5364	102	2	present	present	VERB
ejpam-5364	102	3	some	some	DET
ejpam-5364	102	4	fundamental	fundamental	ADJ
ejpam-5364	102	5	concepts	concept	NOUN
ejpam-5364	102	6	related	relate	VERB
ejpam-5364	102	7	to	to	ADP
ejpam-5364	102	8	the	the	DET
ejpam-5364	102	9	parameters	parameter	NOUN
ejpam-5364	102	10	of	of	ADP
ejpam-5364	102	11	planar	planar	ADJ
ejpam-5364	102	12	graphs	graph	NOUN
ejpam-5364	102	13	in	in	ADP
ejpam-5364	102	14	section	section	NOUN
ejpam-5364	102	15	2	2	NUM
ejpam-5364	102	16	.	.	PUNCT
ejpam-5364	103	1	in	in	ADP
ejpam-5364	103	2	section	section	NOUN
ejpam-5364	103	3	3	3	NUM
ejpam-5364	103	4	,	,	PUNCT
ejpam-5364	103	5	we	we	PRON
ejpam-5364	103	6	investigate	investigate	VERB
ejpam-5364	103	7	the	the	DET
ejpam-5364	103	8	relation	relation	NOUN
ejpam-5364	103	9	between	between	ADP
ejpam-5364	103	10	the	the	DET
ejpam-5364	103	11	second	second	ADV
ejpam-5364	103	12	largest	large	ADJ
ejpam-5364	103	13	eigenvalue	eigenvalue	NOUN
ejpam-5364	103	14	of	of	ADP
ejpam-5364	103	15	a	a	DET
ejpam-5364	103	16	graph	graph	NOUN
ejpam-5364	103	17	with	with	ADP
ejpam-5364	103	18	the	the	DET
ejpam-5364	103	19	parameters	parameter	NOUN
ejpam-5364	103	20	of	of	ADP
ejpam-5364	103	21	planar	planar	ADJ
ejpam-5364	103	22	graph	graph	NOUN
ejpam-5364	103	23	such	such	ADJ
ejpam-5364	103	24	as	as	ADP
ejpam-5364	103	25	skewness	skewness	NOUN
ejpam-5364	103	26	,	,	PUNCT
ejpam-5364	103	27	thickness	thickness	NOUN
ejpam-5364	103	28	and	and	CCONJ
ejpam-5364	103	29	crossing	crossing	NOUN
ejpam-5364	103	30	number	number	NOUN
ejpam-5364	103	31	.	.	PUNCT
ejpam-5364	104	1	section	section	NOUN
ejpam-5364	104	2	3.1	3.1	NUM
ejpam-5364	104	3	provides	provide	VERB
ejpam-5364	104	4	the	the	DET
ejpam-5364	104	5	relation	relation	NOUN
ejpam-5364	104	6	between	between	ADP
ejpam-5364	104	7	the	the	DET
ejpam-5364	104	8	second	second	ADV
ejpam-5364	104	9	largest	large	ADJ
ejpam-5364	104	10	eigenvalue	eigenvalue	NOUN
ejpam-5364	104	11	of	of	ADP
ejpam-5364	104	12	adjacency	adjacency	NOUN
ejpam-5364	104	13	and	and	CCONJ
ejpam-5364	104	14	signless	signless	ADJ
ejpam-5364	104	15	laplacian	laplacian	ADJ
ejpam-5364	104	16	matrices	matrix	NOUN
ejpam-5364	104	17	with	with	ADP
ejpam-5364	104	18	the	the	DET
ejpam-5364	104	19	skewness	skewness	NOUN
ejpam-5364	104	20	of	of	ADP
ejpam-5364	104	21	the	the	DET
ejpam-5364	104	22	graph	graph	NOUN
ejpam-5364	104	23	sk(g	sk(g	NOUN
ejpam-5364	104	24	)	)	PUNCT
ejpam-5364	104	25	;	;	PUNCT
ejpam-5364	104	26	that	that	PRON
ejpam-5364	104	27	is	be	AUX
ejpam-5364	104	28	,	,	PUNCT
ejpam-5364	104	29	lower	low	ADJ
ejpam-5364	104	30	bounds	bound	NOUN
ejpam-5364	104	31	for	for	ADP
ejpam-5364	104	32	λ2	λ2	NOUN
ejpam-5364	104	33	and	and	CCONJ
ejpam-5364	104	34	q2	q2	NOUN
ejpam-5364	104	35	with	with	ADP
ejpam-5364	104	36	respect	respect	NOUN
ejpam-5364	104	37	to	to	ADP
ejpam-5364	104	38	the	the	DET
ejpam-5364	104	39	skewness	skewness	NOUN
ejpam-5364	104	40	of	of	ADP
ejpam-5364	104	41	the	the	DET
ejpam-5364	104	42	graph	graph	NOUN
ejpam-5364	104	43	.	.	PUNCT
ejpam-5364	105	1	section	section	NOUN
ejpam-5364	105	2	3.2	3.2	NUM
ejpam-5364	105	3	gives	give	VERB
ejpam-5364	105	4	the	the	DET
ejpam-5364	105	5	lower	low	ADJ
ejpam-5364	105	6	bounds	bound	NOUN
ejpam-5364	105	7	for	for	ADP
ejpam-5364	105	8	λ2	λ2	NOUN
ejpam-5364	105	9	and	and	CCONJ
ejpam-5364	105	10	q2	q2	NOUN
ejpam-5364	105	11	with	with	ADP
ejpam-5364	105	12	respect	respect	NOUN
ejpam-5364	105	13	to	to	ADP
ejpam-5364	105	14	the	the	DET
ejpam-5364	105	15	thickness	thickness	NOUN
ejpam-5364	105	16	of	of	ADP
ejpam-5364	105	17	the	the	DET
ejpam-5364	105	18	graph	graph	NOUN
ejpam-5364	105	19	τ(g	τ(g	PROPN
ejpam-5364	105	20	)	)	PUNCT
ejpam-5364	105	21	.	.	PUNCT
ejpam-5364	106	1	section	section	NOUN
ejpam-5364	106	2	3.3	3.3	NUM
ejpam-5364	106	3	presents	present	VERB
ejpam-5364	106	4	the	the	DET
ejpam-5364	106	5	lower	low	ADJ
ejpam-5364	106	6	bounds	bound	NOUN
ejpam-5364	106	7	for	for	ADP
ejpam-5364	106	8	λ2	λ2	NOUN
ejpam-5364	106	9	and	and	CCONJ
ejpam-5364	106	10	q2	q2	NOUN
ejpam-5364	106	11	with	with	ADP
ejpam-5364	106	12	respect	respect	NOUN
ejpam-5364	106	13	to	to	ADP
ejpam-5364	106	14	the	the	DET
ejpam-5364	106	15	crossing	crossing	NOUN
ejpam-5364	106	16	number	number	NOUN
ejpam-5364	106	17	of	of	ADP
ejpam-5364	106	18	the	the	DET
ejpam-5364	106	19	graph	graph	NOUN
ejpam-5364	106	20	cr(g	cr(g	PUNCT
ejpam-5364	106	21	)	)	PUNCT
ejpam-5364	106	22	.	.	PUNCT
ejpam-5364	107	1	in	in	ADP
ejpam-5364	107	2	all	all	DET
ejpam-5364	107	3	the	the	DET
ejpam-5364	107	4	sections	section	NOUN
ejpam-5364	107	5	lower	low	ADJ
ejpam-5364	107	6	bounds	bound	NOUN
ejpam-5364	107	7	for	for	ADP
ejpam-5364	107	8	the	the	DET
ejpam-5364	107	9	graph	graph	NOUN
ejpam-5364	107	10	parameters	parameter	NOUN
ejpam-5364	107	11	sk(g	sk(g	NOUN
ejpam-5364	107	12	)	)	PUNCT
ejpam-5364	107	13	,	,	PUNCT
ejpam-5364	107	14	τ(g	τ(g	PROPN
ejpam-5364	107	15	)	)	PUNCT
ejpam-5364	107	16	,	,	PUNCT
ejpam-5364	107	17	and	and	CCONJ
ejpam-5364	107	18	cr(g	cr(g	PUNCT
ejpam-5364	107	19	)	)	PUNCT
ejpam-5364	107	20	in	in	ADP
ejpam-5364	107	21	terms	term	NOUN
ejpam-5364	107	22	of	of	ADP
ejpam-5364	107	23	the	the	DET
ejpam-5364	107	24	second	second	ADV
ejpam-5364	107	25	largest	large	ADJ
ejpam-5364	107	26	adjacency	adjacency	NOUN
ejpam-5364	107	27	and	and	CCONJ
ejpam-5364	107	28	signless	signless	NOUN
ejpam-5364	107	29	laplacian	laplacian	NOUN
ejpam-5364	107	30	eigenvalues	eigenvalue	VERB
ejpam-5364	107	31	for	for	ADP
ejpam-5364	107	32	regular	regular	ADJ
ejpam-5364	107	33	graphs	graph	NOUN
ejpam-5364	107	34	are	be	AUX
ejpam-5364	107	35	also	also	ADV
ejpam-5364	107	36	presented	present	VERB
ejpam-5364	107	37	.	.	PUNCT
ejpam-5364	108	1	2	2	X
ejpam-5364	108	2	.	.	X
ejpam-5364	108	3	preliminaries	preliminary	NOUN
ejpam-5364	108	4	planar	planar	ADJ
ejpam-5364	108	5	graph	graph	NOUN
ejpam-5364	108	6	is	be	AUX
ejpam-5364	108	7	a	a	DET
ejpam-5364	108	8	graph	graph	NOUN
ejpam-5364	108	9	that	that	PRON
ejpam-5364	108	10	can	can	AUX
ejpam-5364	108	11	be	be	AUX
ejpam-5364	108	12	drawn	draw	VERB
ejpam-5364	108	13	in	in	ADP
ejpam-5364	108	14	such	such	DET
ejpam-5364	108	15	a	a	DET
ejpam-5364	108	16	way	way	NOUN
ejpam-5364	108	17	that	that	PRON
ejpam-5364	108	18	no	no	DET
ejpam-5364	108	19	edges	edge	NOUN
ejpam-5364	108	20	cross	cross	VERB
ejpam-5364	108	21	each	each	DET
ejpam-5364	108	22	other	other	ADJ
ejpam-5364	109	1	[	[	X
ejpam-5364	109	2	25	25	NUM
ejpam-5364	109	3	]	]	PUNCT
ejpam-5364	109	4	.	.	PUNCT
ejpam-5364	110	1	such	such	DET
ejpam-5364	110	2	a	a	DET
ejpam-5364	110	3	drawing	drawing	NOUN
ejpam-5364	110	4	is	be	AUX
ejpam-5364	110	5	called	call	VERB
ejpam-5364	110	6	a	a	DET
ejpam-5364	110	7	plane	plane	NOUN
ejpam-5364	110	8	graph	graph	NOUN
ejpam-5364	110	9	or	or	CCONJ
ejpam-5364	110	10	planar	planar	ADJ
ejpam-5364	110	11	embedding	embed	VERB
ejpam-5364	110	12	of	of	ADP
ejpam-5364	110	13	the	the	DET
ejpam-5364	110	14	graph	graph	NOUN
ejpam-5364	110	15	.	.	PUNCT
ejpam-5364	111	1	the	the	DET
ejpam-5364	111	2	skewness	skewness	NOUN
ejpam-5364	111	3	of	of	ADP
ejpam-5364	111	4	a	a	DET
ejpam-5364	111	5	graph	graph	NOUN
ejpam-5364	111	6	g	g	NOUN
ejpam-5364	111	7	is	be	AUX
ejpam-5364	111	8	the	the	DET
ejpam-5364	111	9	minimum	minimum	ADJ
ejpam-5364	111	10	number	number	NOUN
ejpam-5364	111	11	of	of	ADP
ejpam-5364	111	12	edges	edge	NOUN
ejpam-5364	111	13	whose	whose	DET
ejpam-5364	111	14	removal	removal	NOUN
ejpam-5364	111	15	results	result	VERB
ejpam-5364	111	16	in	in	ADP
ejpam-5364	111	17	a	a	DET
ejpam-5364	111	18	planar	planar	ADJ
ejpam-5364	111	19	graph	graph	NOUN
ejpam-5364	111	20	.	.	PUNCT
ejpam-5364	112	1	it	it	PRON
ejpam-5364	112	2	is	be	AUX
ejpam-5364	112	3	denoted	denote	VERB
ejpam-5364	112	4	by	by	ADP
ejpam-5364	112	5	sk(g	sk(g	NOUN
ejpam-5364	112	6	)	)	PUNCT
ejpam-5364	112	7	.	.	PUNCT
ejpam-5364	113	1	m.	m.	PROPN
ejpam-5364	113	2	machasri	machasri	PROPN
ejpam-5364	113	3	,	,	PUNCT
ejpam-5364	113	4	d.	d.	PROPN
ejpam-5364	113	5	kalyani	kalyani	PROPN
ejpam-5364	113	6	/	/	SYM
ejpam-5364	113	7	eur	eur	PROPN
ejpam-5364	113	8	.	.	PUNCT
ejpam-5364	114	1	j.	j.	PROPN
ejpam-5364	114	2	pure	pure	PROPN
ejpam-5364	114	3	appl	appl	PROPN
ejpam-5364	114	4	.	.	PROPN
ejpam-5364	114	5	math	math	PROPN
ejpam-5364	114	6	,	,	PUNCT
ejpam-5364	114	7	17	17	NUM
ejpam-5364	114	8	(	(	PUNCT
ejpam-5364	114	9	4	4	NUM
ejpam-5364	114	10	)	)	PUNCT
ejpam-5364	114	11	(	(	PUNCT
ejpam-5364	114	12	2024	2024	NUM
ejpam-5364	114	13	)	)	PUNCT
ejpam-5364	114	14	,	,	PUNCT
ejpam-5364	114	15	3004	3004	NUM
ejpam-5364	114	16	-	-	SYM
ejpam-5364	114	17	3021	3021	NUM
ejpam-5364	114	18	3008	3008	NUM
ejpam-5364	114	19	from	from	ADP
ejpam-5364	114	20	[	[	X
ejpam-5364	114	21	7	7	NUM
ejpam-5364	114	22	]	]	PUNCT
ejpam-5364	114	23	,	,	PUNCT
ejpam-5364	114	24	we	we	PRON
ejpam-5364	114	25	have	have	AUX
ejpam-5364	114	26	sk(g	sk(g	NOUN
ejpam-5364	114	27	)	)	PUNCT
ejpam-5364	114	28	≥	≥	NOUN
ejpam-5364	114	29	m−	m−	PROPN
ejpam-5364	114	30	(	(	PUNCT
ejpam-5364	114	31	3n−	3n−	PROPN
ejpam-5364	114	32	6	6	NUM
ejpam-5364	114	33	)	)	PUNCT
ejpam-5364	114	34	(	(	PUNCT
ejpam-5364	114	35	1	1	X
ejpam-5364	114	36	)	)	PUNCT
ejpam-5364	114	37	where	where	SCONJ
ejpam-5364	114	38	m	m	VERB
ejpam-5364	114	39	and	and	CCONJ
ejpam-5364	114	40	n	n	PRON
ejpam-5364	114	41	are	be	AUX
ejpam-5364	114	42	the	the	DET
ejpam-5364	114	43	size	size	NOUN
ejpam-5364	114	44	and	and	CCONJ
ejpam-5364	114	45	the	the	DET
ejpam-5364	114	46	order	order	NOUN
ejpam-5364	114	47	of	of	ADP
ejpam-5364	114	48	g	g	NOUN
ejpam-5364	114	49	,	,	PUNCT
ejpam-5364	114	50	respectively	respectively	ADV
ejpam-5364	114	51	.	.	PUNCT
ejpam-5364	115	1	figure	figure	NOUN
ejpam-5364	115	2	1	1	NUM
ejpam-5364	115	3	:	:	PUNCT
ejpam-5364	115	4	complete	complete	ADJ
ejpam-5364	115	5	graph	graph	NOUN
ejpam-5364	115	6	k6	k6	NOUN
ejpam-5364	115	7	in	in	ADP
ejpam-5364	115	8	figure	figure	NOUN
ejpam-5364	115	9	1	1	NUM
ejpam-5364	115	10	,	,	PUNCT
ejpam-5364	115	11	if	if	SCONJ
ejpam-5364	115	12	we	we	PRON
ejpam-5364	115	13	remove	remove	VERB
ejpam-5364	115	14	one	one	NUM
ejpam-5364	115	15	edge	edge	NOUN
ejpam-5364	115	16	from	from	ADP
ejpam-5364	115	17	each	each	DET
ejpam-5364	115	18	red	red	ADJ
ejpam-5364	115	19	crossing	crossing	NOUN
ejpam-5364	115	20	,	,	PUNCT
ejpam-5364	115	21	it	it	PRON
ejpam-5364	115	22	will	will	AUX
ejpam-5364	115	23	become	become	VERB
ejpam-5364	115	24	planar	planar	ADJ
ejpam-5364	115	25	.	.	PUNCT
ejpam-5364	116	1	skewness	skewness	NOUN
ejpam-5364	116	2	of	of	ADP
ejpam-5364	116	3	the	the	DET
ejpam-5364	116	4	complete	complete	ADJ
ejpam-5364	116	5	graph	graph	NOUN
ejpam-5364	116	6	k6	k6	NOUN
ejpam-5364	116	7	is	be	AUX
ejpam-5364	116	8	given	give	VERB
ejpam-5364	116	9	by	by	ADP
ejpam-5364	116	10	sk(k6	sk(k6	NOUN
ejpam-5364	116	11	)	)	PUNCT
ejpam-5364	116	12	=	=	SYM
ejpam-5364	117	1	3	3	X
ejpam-5364	117	2	.	.	PUNCT
ejpam-5364	117	3	the	the	DET
ejpam-5364	117	4	thickness	thickness	PROPN
ejpam-5364	117	5	τ(g	τ(g	PROPN
ejpam-5364	117	6	)	)	PUNCT
ejpam-5364	117	7	of	of	ADP
ejpam-5364	117	8	a	a	DET
ejpam-5364	117	9	graph	graph	NOUN
ejpam-5364	117	10	g	g	NOUN
ejpam-5364	117	11	is	be	AUX
ejpam-5364	117	12	the	the	DET
ejpam-5364	117	13	minimum	minimum	ADJ
ejpam-5364	117	14	number	number	NOUN
ejpam-5364	117	15	of	of	ADP
ejpam-5364	117	16	planar	planar	ADJ
ejpam-5364	117	17	edge	edge	NOUN
ejpam-5364	117	18	-	-	PUNCT
ejpam-5364	117	19	induced	induce	VERB
ejpam-5364	117	20	subgraphs	subgraph	NOUN
ejpam-5364	117	21	pi	pi	NOUN
ejpam-5364	117	22	of	of	ADP
ejpam-5364	117	23	g	g	PROPN
ejpam-5364	117	24	needed	need	VERB
ejpam-5364	117	25	such	such	ADJ
ejpam-5364	117	26	that	that	SCONJ
ejpam-5364	117	27	the	the	DET
ejpam-5364	117	28	graph	graph	NOUN
ejpam-5364	117	29	union	union	NOUN
ejpam-5364	117	30	⋃	⋃	NOUN
ejpam-5364	117	31	pi	pi	NOUN
ejpam-5364	117	32	=	=	SYM
ejpam-5364	117	33	g	g	PROPN
ejpam-5364	118	1	[	[	X
ejpam-5364	118	2	22	22	NUM
ejpam-5364	118	3	]	]	PUNCT
ejpam-5364	118	4	.	.	PUNCT
ejpam-5364	119	1	from	from	ADP
ejpam-5364	119	2	the	the	DET
ejpam-5364	119	3	definition	definition	NOUN
ejpam-5364	119	4	of	of	ADP
ejpam-5364	119	5	thickness	thickness	NOUN
ejpam-5364	119	6	of	of	ADP
ejpam-5364	119	7	a	a	DET
ejpam-5364	119	8	graph	graph	NOUN
ejpam-5364	119	9	,	,	PUNCT
ejpam-5364	119	10	the	the	DET
ejpam-5364	119	11	graph	graph	NOUN
ejpam-5364	119	12	k6	k6	NOUN
ejpam-5364	119	13	has	have	VERB
ejpam-5364	119	14	2	2	NUM
ejpam-5364	119	15	planar	planar	ADJ
ejpam-5364	119	16	edge	edge	NOUN
ejpam-5364	119	17	-	-	PUNCT
ejpam-5364	119	18	induced	induce	VERB
ejpam-5364	119	19	subgraphs	subgraph	NOUN
ejpam-5364	119	20	.	.	PUNCT
ejpam-5364	120	1	therefore	therefore	ADV
ejpam-5364	120	2	,	,	PUNCT
ejpam-5364	120	3	the	the	DET
ejpam-5364	120	4	thickness	thickness	NOUN
ejpam-5364	120	5	of	of	ADP
ejpam-5364	120	6	k6	k6	PROPN
ejpam-5364	120	7	is	be	AUX
ejpam-5364	120	8	τ(k6	τ(k6	ADJ
ejpam-5364	120	9	)	)	PUNCT
ejpam-5364	120	10	=	=	SYM
ejpam-5364	121	1	2	2	X
ejpam-5364	121	2	.	.	X
ejpam-5364	121	3	a	a	DET
ejpam-5364	121	4	lower	lower	ADV
ejpam-5364	121	5	bound	bind	VERB
ejpam-5364	121	6	for	for	ADP
ejpam-5364	121	7	the	the	DET
ejpam-5364	121	8	thickness	thickness	NOUN
ejpam-5364	121	9	of	of	ADP
ejpam-5364	121	10	a	a	DET
ejpam-5364	121	11	graph	graph	NOUN
ejpam-5364	121	12	is	be	AUX
ejpam-5364	121	13	given	give	VERB
ejpam-5364	121	14	by	by	ADP
ejpam-5364	121	15	[	[	PUNCT
ejpam-5364	121	16	22	22	NUM
ejpam-5364	121	17	]	]	PUNCT
ejpam-5364	121	18	τ(g	τ(g	PROPN
ejpam-5364	121	19	)	)	PUNCT
ejpam-5364	121	20	≥	≥	NOUN
ejpam-5364	121	21	⌈	⌈	NOUN
ejpam-5364	121	22	m	m	PROPN
ejpam-5364	121	23	3n−	3n−	PROPN
ejpam-5364	121	24	6	6	NUM
ejpam-5364	121	25	⌉	⌉	NOUN
ejpam-5364	121	26	(	(	PUNCT
ejpam-5364	121	27	2	2	NUM
ejpam-5364	121	28	)	)	PUNCT
ejpam-5364	121	29	where	where	SCONJ
ejpam-5364	121	30	m	m	NOUN
ejpam-5364	121	31	is	be	AUX
ejpam-5364	121	32	the	the	DET
ejpam-5364	121	33	number	number	NOUN
ejpam-5364	121	34	of	of	ADP
ejpam-5364	121	35	edges	edge	NOUN
ejpam-5364	121	36	,	,	PUNCT
ejpam-5364	121	37	n	n	PRON
ejpam-5364	121	38	≥	≥	NOUN
ejpam-5364	121	39	3	3	NUM
ejpam-5364	121	40	is	be	AUX
ejpam-5364	121	41	the	the	DET
ejpam-5364	121	42	number	number	NOUN
ejpam-5364	121	43	of	of	ADP
ejpam-5364	121	44	vertices	vertex	NOUN
ejpam-5364	121	45	,	,	PUNCT
ejpam-5364	121	46	and	and	CCONJ
ejpam-5364	121	47	⌈x⌉	⌈x⌉	NOUN
ejpam-5364	121	48	is	be	AUX
ejpam-5364	121	49	the	the	DET
ejpam-5364	121	50	ceiling	ceiling	NOUN
ejpam-5364	121	51	function	function	NOUN
ejpam-5364	121	52	.	.	PUNCT
ejpam-5364	122	1	the	the	DET
ejpam-5364	122	2	thickness	thickness	NOUN
ejpam-5364	122	3	of	of	ADP
ejpam-5364	122	4	the	the	DET
ejpam-5364	122	5	hypercube	hypercube	NOUN
ejpam-5364	122	6	graph	graph	NOUN
ejpam-5364	122	7	qn	qn	NOUN
ejpam-5364	122	8	[	[	X
ejpam-5364	122	9	13	13	NUM
ejpam-5364	122	10	]	]	PUNCT
ejpam-5364	122	11	is	be	AUX
ejpam-5364	122	12	given	give	VERB
ejpam-5364	122	13	by	by	ADP
ejpam-5364	122	14	τ(qn	τ(qn	NOUN
ejpam-5364	122	15	)	)	PUNCT
ejpam-5364	122	16	=	=	PUNCT
ejpam-5364	122	17	⌈	⌈	NUM
ejpam-5364	122	18	n+	n+	ADP
ejpam-5364	122	19	1	1	NUM
ejpam-5364	122	20	4	4	NUM
ejpam-5364	122	21	⌉	⌉	X
ejpam-5364	122	22	.	.	PUNCT
ejpam-5364	123	1	figure	figure	VERB
ejpam-5364	123	2	2	2	NUM
ejpam-5364	123	3	:	:	PUNCT
ejpam-5364	123	4	thickness	thickness	NOUN
ejpam-5364	123	5	of	of	ADP
ejpam-5364	123	6	k6	k6	PROPN
ejpam-5364	123	7	m.	m.	PROPN
ejpam-5364	123	8	machasri	machasri	PROPN
ejpam-5364	123	9	,	,	PUNCT
ejpam-5364	123	10	d.	d.	PROPN
ejpam-5364	123	11	kalyani	kalyani	PROPN
ejpam-5364	123	12	/	/	SYM
ejpam-5364	123	13	eur	eur	PROPN
ejpam-5364	123	14	.	.	PUNCT
ejpam-5364	124	1	j.	j.	PROPN
ejpam-5364	124	2	pure	pure	PROPN
ejpam-5364	124	3	appl	appl	PROPN
ejpam-5364	124	4	.	.	PROPN
ejpam-5364	124	5	math	math	PROPN
ejpam-5364	124	6	,	,	PUNCT
ejpam-5364	124	7	17	17	NUM
ejpam-5364	124	8	(	(	PUNCT
ejpam-5364	124	9	4	4	NUM
ejpam-5364	124	10	)	)	PUNCT
ejpam-5364	124	11	(	(	PUNCT
ejpam-5364	124	12	2024	2024	NUM
ejpam-5364	124	13	)	)	PUNCT
ejpam-5364	124	14	,	,	PUNCT
ejpam-5364	124	15	3004	3004	NUM
ejpam-5364	124	16	-	-	SYM
ejpam-5364	124	17	3021	3021	NUM
ejpam-5364	124	18	3009	3009	NUM
ejpam-5364	124	19	figure	figure	NOUN
ejpam-5364	124	20	3	3	NUM
ejpam-5364	124	21	:	:	PUNCT
ejpam-5364	124	22	planar	planar	ADJ
ejpam-5364	124	23	edge	edge	NOUN
ejpam-5364	124	24	-	-	PUNCT
ejpam-5364	124	25	induced	induce	VERB
ejpam-5364	124	26	subgraph	subgraph	NOUN
ejpam-5364	124	27	1	1	NUM
ejpam-5364	124	28	of	of	ADP
ejpam-5364	124	29	k6	k6	PROPN
ejpam-5364	124	30	figure	figure	NOUN
ejpam-5364	124	31	4	4	NUM
ejpam-5364	124	32	:	:	PUNCT
ejpam-5364	124	33	planar	planar	ADJ
ejpam-5364	124	34	edge	edge	NOUN
ejpam-5364	124	35	-	-	PUNCT
ejpam-5364	124	36	induced	induce	VERB
ejpam-5364	124	37	subgraph	subgraph	NOUN
ejpam-5364	124	38	2	2	NUM
ejpam-5364	124	39	of	of	ADP
ejpam-5364	124	40	k6	k6	PROPN
ejpam-5364	124	41	the	the	DET
ejpam-5364	124	42	crossing	crossing	NOUN
ejpam-5364	124	43	number	number	NOUN
ejpam-5364	124	44	cr(g	cr(g	PUNCT
ejpam-5364	124	45	)	)	PUNCT
ejpam-5364	124	46	of	of	ADP
ejpam-5364	124	47	a	a	DET
ejpam-5364	124	48	graph	graph	NOUN
ejpam-5364	124	49	g	g	NOUN
ejpam-5364	124	50	is	be	AUX
ejpam-5364	124	51	the	the	DET
ejpam-5364	124	52	lowest	low	ADJ
ejpam-5364	124	53	number	number	NOUN
ejpam-5364	124	54	of	of	ADP
ejpam-5364	124	55	edge	edge	NOUN
ejpam-5364	124	56	crossings	crossing	NOUN
ejpam-5364	124	57	of	of	ADP
ejpam-5364	124	58	a	a	DET
ejpam-5364	124	59	plane	plane	NOUN
ejpam-5364	124	60	drawing	drawing	NOUN
ejpam-5364	124	61	of	of	ADP
ejpam-5364	124	62	the	the	DET
ejpam-5364	124	63	graph	graph	NOUN
ejpam-5364	124	64	g.	g.	VERB
ejpam-5364	124	65	a	a	DET
ejpam-5364	124	66	graph	graph	NOUN
ejpam-5364	124	67	with	with	ADP
ejpam-5364	124	68	crossing	crossing	NOUN
ejpam-5364	124	69	number	number	NOUN
ejpam-5364	124	70	0	0	NUM
ejpam-5364	124	71	is	be	AUX
ejpam-5364	124	72	known	know	VERB
ejpam-5364	124	73	as	as	ADP
ejpam-5364	124	74	a	a	DET
ejpam-5364	124	75	planar	planar	ADJ
ejpam-5364	124	76	graph	graph	NOUN
ejpam-5364	124	77	.	.	PUNCT
ejpam-5364	125	1	for	for	ADP
ejpam-5364	125	2	example	example	NOUN
ejpam-5364	125	3	,	,	PUNCT
ejpam-5364	125	4	the	the	DET
ejpam-5364	125	5	complete	complete	ADJ
ejpam-5364	125	6	graph	graph	NOUN
ejpam-5364	125	7	k6	k6	PROPN
ejpam-5364	125	8	illustrated	illustrate	VERB
ejpam-5364	125	9	in	in	ADP
ejpam-5364	125	10	figure	figure	NOUN
ejpam-5364	125	11	1	1	NUM
ejpam-5364	125	12	has	have	AUX
ejpam-5364	125	13	crossing	cross	VERB
ejpam-5364	125	14	number	number	NOUN
ejpam-5364	125	15	3	3	NUM
ejpam-5364	125	16	.	.	PUNCT
ejpam-5364	126	1	ajtai	ajtai	PROPN
ejpam-5364	126	2	et	et	PROPN
ejpam-5364	126	3	al	al	PROPN
ejpam-5364	126	4	.	.	PROPN
ejpam-5364	127	1	(	(	PUNCT
ejpam-5364	127	2	1982	1982	NUM
ejpam-5364	127	3	)	)	PUNCT
ejpam-5364	127	4	showed	show	VERB
ejpam-5364	127	5	that	that	SCONJ
ejpam-5364	127	6	there	there	PRON
ejpam-5364	127	7	is	be	VERB
ejpam-5364	127	8	an	an	DET
ejpam-5364	127	9	absolute	absolute	ADJ
ejpam-5364	127	10	constant	constant	ADJ
ejpam-5364	127	11	c	c	NOUN
ejpam-5364	127	12	>	>	X
ejpam-5364	127	13	0	0	NUM
ejpam-5364	127	14	such	such	ADJ
ejpam-5364	127	15	that	that	PRON
ejpam-5364	127	16	cr(g	cr(g	NUM
ejpam-5364	127	17	)	)	PUNCT
ejpam-5364	127	18	≥	≥	PROPN
ejpam-5364	127	19	cm3	cm3	NOUN
ejpam-5364	127	20	n2	n2	NOUN
ejpam-5364	127	21	.	.	PUNCT
ejpam-5364	128	1	this	this	DET
ejpam-5364	128	2	inequality	inequality	NOUN
ejpam-5364	128	3	is	be	AUX
ejpam-5364	128	4	known	know	VERB
ejpam-5364	128	5	as	as	ADP
ejpam-5364	128	6	crossing	crossing	NOUN
ejpam-5364	128	7	number	number	NOUN
ejpam-5364	128	8	inequality	inequality	NOUN
ejpam-5364	128	9	or	or	CCONJ
ejpam-5364	128	10	crossing	cross	VERB
ejpam-5364	128	11	lemma	lemma	PROPN
ejpam-5364	128	12	.	.	PUNCT
ejpam-5364	129	1	due	due	ADP
ejpam-5364	129	2	to	to	ADP
ejpam-5364	129	3	ackerman	ackerman	PROPN
ejpam-5364	129	4	[	[	X
ejpam-5364	129	5	1	1	NUM
ejpam-5364	129	6	]	]	PUNCT
ejpam-5364	129	7	,	,	PUNCT
ejpam-5364	129	8	the	the	DET
ejpam-5364	129	9	constant	constant	ADJ
ejpam-5364	129	10	c	c	NOUN
ejpam-5364	129	11	=	=	SYM
ejpam-5364	129	12	1	1	NUM
ejpam-5364	129	13	29	29	NUM
ejpam-5364	129	14	is	be	AUX
ejpam-5364	129	15	the	the	DET
ejpam-5364	129	16	best	well	ADV
ejpam-5364	129	17	known	know	VERB
ejpam-5364	129	18	to	to	ADP
ejpam-5364	129	19	date	date	NOUN
ejpam-5364	129	20	.	.	PUNCT
ejpam-5364	130	1	therefore	therefore	ADV
ejpam-5364	130	2	,	,	PUNCT
ejpam-5364	130	3	cr(g	cr(g	X
ejpam-5364	130	4	)	)	PUNCT
ejpam-5364	130	5	≥	≥	PROPN
ejpam-5364	130	6	m3	m3	PROPN
ejpam-5364	130	7	29n2	29n2	NUM
ejpam-5364	130	8	.	.	PUNCT
ejpam-5364	131	1	(	(	PUNCT
ejpam-5364	131	2	3	3	X
ejpam-5364	131	3	)	)	PUNCT
ejpam-5364	131	4	definition	definition	NOUN
ejpam-5364	131	5	1	1	NUM
ejpam-5364	131	6	.	.	PUNCT
ejpam-5364	132	1	[	[	X
ejpam-5364	132	2	5	5	NUM
ejpam-5364	132	3	]	]	PUNCT
ejpam-5364	132	4	consider	consider	VERB
ejpam-5364	132	5	two	two	NUM
ejpam-5364	132	6	sequences	sequence	NOUN
ejpam-5364	132	7	of	of	ADP
ejpam-5364	132	8	real	real	ADJ
ejpam-5364	132	9	numbers	number	NOUN
ejpam-5364	132	10	:	:	PUNCT
ejpam-5364	132	11	ξ1	ξ1	NOUN
ejpam-5364	132	12	,	,	PUNCT
ejpam-5364	132	13	ξ2	ξ2	NOUN
ejpam-5364	132	14	,	,	PUNCT
ejpam-5364	132	15	.	.	PUNCT
ejpam-5364	132	16	.	.	PUNCT
ejpam-5364	133	1	.	.	PUNCT
ejpam-5364	134	1	,	,	PUNCT
ejpam-5364	134	2	ξn	ξn	NOUN
ejpam-5364	134	3	and	and	CCONJ
ejpam-5364	134	4	η1	η1	NOUN
ejpam-5364	134	5	,	,	PUNCT
ejpam-5364	134	6	η2	η2	NOUN
ejpam-5364	134	7	,	,	PUNCT
ejpam-5364	134	8	.	.	PUNCT
ejpam-5364	134	9	.	.	PUNCT
ejpam-5364	135	1	.	.	PUNCT
ejpam-5364	136	1	,	,	PUNCT
ejpam-5364	136	2	ηm	ηm	PROPN
ejpam-5364	136	3	with	with	ADP
ejpam-5364	136	4	m	m	PROPN
ejpam-5364	136	5	≤	≤	NOUN
ejpam-5364	136	6	n.	n.	NOUN
ejpam-5364	136	7	the	the	DET
ejpam-5364	136	8	second	second	ADJ
ejpam-5364	136	9	sequence	sequence	NOUN
ejpam-5364	136	10	is	be	AUX
ejpam-5364	136	11	said	say	VERB
ejpam-5364	136	12	to	to	PART
ejpam-5364	136	13	interlace	interlace	VERB
ejpam-5364	136	14	the	the	DET
ejpam-5364	136	15	first	first	ADJ
ejpam-5364	136	16	one	one	NUM
ejpam-5364	136	17	whenever	whenever	SCONJ
ejpam-5364	136	18	ξi	ξi	NOUN
ejpam-5364	136	19	≤	≤	NUM
ejpam-5364	136	20	ηi	ηi	VERB
ejpam-5364	136	21	≤	≤	NUM
ejpam-5364	136	22	ξn−m+i	ξn−m+i	PROPN
ejpam-5364	136	23	for	for	ADP
ejpam-5364	136	24	i	i	PRON
ejpam-5364	136	25	=	=	NOUN
ejpam-5364	136	26	1	1	NUM
ejpam-5364	136	27	,	,	PUNCT
ejpam-5364	136	28	2	2	NUM
ejpam-5364	136	29	,	,	PUNCT
ejpam-5364	136	30	.	.	PUNCT
ejpam-5364	136	31	.	.	PUNCT
ejpam-5364	136	32	.	.	PUNCT
ejpam-5364	137	1	,	,	PUNCT
ejpam-5364	137	2	m.	m.	NOUN
ejpam-5364	137	3	the	the	DET
ejpam-5364	137	4	interlacing	interlacing	NOUN
ejpam-5364	137	5	is	be	AUX
ejpam-5364	137	6	called	call	VERB
ejpam-5364	137	7	tight	tight	ADJ
ejpam-5364	137	8	if	if	SCONJ
ejpam-5364	137	9	there	there	PRON
ejpam-5364	137	10	exists	exist	VERB
ejpam-5364	137	11	an	an	DET
ejpam-5364	137	12	integer	integer	NOUN
ejpam-5364	137	13	k	k	PROPN
ejpam-5364	137	14	∈	∈	PROPN
ejpam-5364	138	1	[	[	X
ejpam-5364	138	2	0,m	0,m	X
ejpam-5364	138	3	]	]	X
ejpam-5364	138	4	such	such	ADJ
ejpam-5364	138	5	that	that	DET
ejpam-5364	138	6	ξi	ξi	NOUN
ejpam-5364	138	7	=	=	NUM
ejpam-5364	138	8	ηi	ηi	PROPN
ejpam-5364	138	9	for	for	ADP
ejpam-5364	138	10	1	1	NUM
ejpam-5364	138	11	≤	≤	NUM
ejpam-5364	138	12	i	i	PRON
ejpam-5364	138	13	≤	≤	ADJ
ejpam-5364	138	14	k	k	PROPN
ejpam-5364	138	15	and	and	CCONJ
ejpam-5364	138	16	ξn−m+i	ξn−m+i	PROPN
ejpam-5364	138	17	=	=	SYM
ejpam-5364	138	18	ηi	ηi	PROPN
ejpam-5364	138	19	for	for	ADP
ejpam-5364	138	20	k	k	PROPN
ejpam-5364	139	1	+	+	PROPN
ejpam-5364	139	2	1	1	X
ejpam-5364	139	3	≤	≤	NUM
ejpam-5364	139	4	i	i	PRON
ejpam-5364	139	5	≤	≤	NUM
ejpam-5364	139	6	m.	m.	NOUN
ejpam-5364	139	7	definition	definition	NOUN
ejpam-5364	139	8	2	2	NUM
ejpam-5364	139	9	.	.	PUNCT
ejpam-5364	140	1	[	[	X
ejpam-5364	140	2	16	16	NUM
ejpam-5364	140	3	]	]	X
ejpam-5364	140	4	let	let	VERB
ejpam-5364	140	5	m	m	VERB
ejpam-5364	140	6	=	=	SYM
ejpam-5364	140	7	(	(	PUNCT
ejpam-5364	140	8	mi	mi	PROPN
ejpam-5364	140	9	,	,	PUNCT
ejpam-5364	140	10	j)t×t	j)t×t	PROPN
ejpam-5364	140	11	be	be	VERB
ejpam-5364	140	12	a	a	DET
ejpam-5364	140	13	real	real	ADJ
ejpam-5364	140	14	matrix	matrix	NOUN
ejpam-5364	140	15	of	of	ADP
ejpam-5364	140	16	order	order	NOUN
ejpam-5364	140	17	n	n	PRON
ejpam-5364	140	18	where	where	SCONJ
ejpam-5364	140	19	mij	mij	NOUN
ejpam-5364	140	20	are	be	AUX
ejpam-5364	140	21	the	the	DET
ejpam-5364	140	22	blocks	block	NOUN
ejpam-5364	140	23	of	of	ADP
ejpam-5364	140	24	m	m	NOUN
ejpam-5364	141	1	and	and	CCONJ
ejpam-5364	141	2	i	i	PRON
ejpam-5364	141	3	,	,	PUNCT
ejpam-5364	141	4	j	j	PROPN
ejpam-5364	141	5	=	=	SYM
ejpam-5364	141	6	1	1	NUM
ejpam-5364	141	7	,	,	PUNCT
ejpam-5364	141	8	2	2	NUM
ejpam-5364	141	9	,	,	PUNCT
ejpam-5364	141	10	.	.	PUNCT
ejpam-5364	141	11	.	.	PUNCT
ejpam-5364	142	1	.	.	PUNCT
ejpam-5364	143	1	,	,	PUNCT
ejpam-5364	143	2	t.	t.	PROPN
ejpam-5364	143	3	then	then	ADV
ejpam-5364	143	4	the	the	DET
ejpam-5364	143	5	quotient	quotient	NOUN
ejpam-5364	143	6	matrix	matrix	NOUN
ejpam-5364	143	7	of	of	ADP
ejpam-5364	143	8	m	m	PROPN
ejpam-5364	143	9	is	be	AUX
ejpam-5364	143	10	a	a	DET
ejpam-5364	143	11	matrix	matrix	NOUN
ejpam-5364	143	12	b(m	b(m	NOUN
ejpam-5364	143	13	)	)	PUNCT
ejpam-5364	143	14	=	=	PUNCT
ejpam-5364	143	15	(	(	PUNCT
ejpam-5364	143	16	bij	bij	NOUN
ejpam-5364	143	17	)	)	PUNCT
ejpam-5364	143	18	where	where	SCONJ
ejpam-5364	143	19	bij	bij	NOUN
ejpam-5364	143	20	is	be	AUX
ejpam-5364	143	21	the	the	DET
ejpam-5364	143	22	sum	sum	NOUN
ejpam-5364	143	23	of	of	ADP
ejpam-5364	143	24	all	all	DET
ejpam-5364	143	25	entries	entry	NOUN
ejpam-5364	143	26	in	in	ADP
ejpam-5364	143	27	mij	mij	NOUN
ejpam-5364	143	28	divided	divide	VERB
ejpam-5364	143	29	by	by	ADP
ejpam-5364	143	30	the	the	DET
ejpam-5364	143	31	number	number	NOUN
ejpam-5364	143	32	of	of	ADP
ejpam-5364	143	33	rows	row	NOUN
ejpam-5364	143	34	of	of	ADP
ejpam-5364	143	35	mij	mij	NOUN
ejpam-5364	143	36	.	.	PUNCT
ejpam-5364	144	1	lemma	lemma	PROPN
ejpam-5364	144	2	1	1	NUM
ejpam-5364	144	3	.	.	PUNCT
ejpam-5364	145	1	[	[	X
ejpam-5364	145	2	12	12	NUM
ejpam-5364	145	3	,	,	PUNCT
ejpam-5364	145	4	17	17	NUM
ejpam-5364	145	5	]	]	PUNCT
ejpam-5364	145	6	let	let	VERB
ejpam-5364	145	7	aq	aq	PART
ejpam-5364	145	8	be	be	AUX
ejpam-5364	145	9	the	the	DET
ejpam-5364	145	10	quotient	quotient	NOUN
ejpam-5364	145	11	matrix	matrix	NOUN
ejpam-5364	145	12	of	of	ADP
ejpam-5364	145	13	a	a	DET
ejpam-5364	145	14	symmetric	symmetric	ADJ
ejpam-5364	145	15	matrix	matrix	NOUN
ejpam-5364	145	16	a	a	PRON
ejpam-5364	145	17	whose	whose	DET
ejpam-5364	145	18	rows	row	NOUN
ejpam-5364	145	19	and	and	CCONJ
ejpam-5364	145	20	columns	column	NOUN
ejpam-5364	145	21	are	be	AUX
ejpam-5364	145	22	partitioned	partition	VERB
ejpam-5364	145	23	according	accord	VERB
ejpam-5364	145	24	to	to	ADP
ejpam-5364	145	25	a	a	DET
ejpam-5364	145	26	partitioning	partition	VERB
ejpam-5364	145	27	(	(	PUNCT
ejpam-5364	145	28	x1	x1	PROPN
ejpam-5364	145	29	,	,	PUNCT
ejpam-5364	145	30	x2	x2	PROPN
ejpam-5364	145	31	,	,	PUNCT
ejpam-5364	145	32	.	.	PUNCT
ejpam-5364	145	33	.	.	PUNCT
ejpam-5364	146	1	.	.	PUNCT
ejpam-5364	147	1	,	,	PUNCT
ejpam-5364	147	2	xm	xm	PROPN
ejpam-5364	147	3	)	)	PUNCT
ejpam-5364	147	4	.	.	PUNCT
ejpam-5364	148	1	then	then	ADV
ejpam-5364	148	2	i.	i.	VERB
ejpam-5364	148	3	the	the	DET
ejpam-5364	148	4	eigenvalues	eigenvalues	PROPN
ejpam-5364	148	5	of	of	ADP
ejpam-5364	148	6	aq	aq	NOUN
ejpam-5364	148	7	interlace	interlace	VERB
ejpam-5364	148	8	the	the	DET
ejpam-5364	148	9	eigenvalues	eigenvalue	NOUN
ejpam-5364	148	10	of	of	ADP
ejpam-5364	148	11	a.	a.	PROPN
ejpam-5364	148	12	ii	ii	PROPN
ejpam-5364	148	13	.	.	PUNCT
ejpam-5364	149	1	if	if	SCONJ
ejpam-5364	149	2	the	the	DET
ejpam-5364	149	3	interlacing	interlacing	NOUN
ejpam-5364	149	4	is	be	AUX
ejpam-5364	149	5	tight	tight	ADJ
ejpam-5364	149	6	then	then	ADV
ejpam-5364	149	7	the	the	DET
ejpam-5364	149	8	partition	partition	NOUN
ejpam-5364	149	9	is	be	AUX
ejpam-5364	149	10	equitable	equitable	ADJ
ejpam-5364	149	11	.	.	PUNCT
ejpam-5364	150	1	3	3	X
ejpam-5364	150	2	.	.	X
ejpam-5364	150	3	main	main	ADJ
ejpam-5364	150	4	results	result	NOUN
ejpam-5364	150	5	in	in	ADP
ejpam-5364	150	6	this	this	DET
ejpam-5364	150	7	section	section	NOUN
ejpam-5364	150	8	,	,	PUNCT
ejpam-5364	150	9	the	the	DET
ejpam-5364	150	10	relation	relation	NOUN
ejpam-5364	150	11	between	between	ADP
ejpam-5364	150	12	the	the	DET
ejpam-5364	150	13	second	second	ADV
ejpam-5364	150	14	largest	large	ADJ
ejpam-5364	150	15	eigenvalue	eigenvalue	NOUN
ejpam-5364	150	16	of	of	ADP
ejpam-5364	150	17	graph	graph	NOUN
ejpam-5364	150	18	with	with	ADP
ejpam-5364	150	19	the	the	DET
ejpam-5364	150	20	parameters	parameter	NOUN
ejpam-5364	150	21	of	of	ADP
ejpam-5364	150	22	planar	planar	ADJ
ejpam-5364	150	23	graphs	graph	NOUN
ejpam-5364	150	24	such	such	ADJ
ejpam-5364	150	25	as	as	ADP
ejpam-5364	150	26	skewness	skewness	NOUN
ejpam-5364	150	27	,	,	PUNCT
ejpam-5364	150	28	thickness	thickness	NOUN
ejpam-5364	150	29	and	and	CCONJ
ejpam-5364	150	30	crossing	crossing	NOUN
ejpam-5364	150	31	number	number	NOUN
ejpam-5364	150	32	are	be	AUX
ejpam-5364	150	33	discussed	discuss	VERB
ejpam-5364	150	34	.	.	PUNCT
ejpam-5364	151	1	m.	m.	PROPN
ejpam-5364	151	2	machasri	machasri	PROPN
ejpam-5364	151	3	,	,	PUNCT
ejpam-5364	151	4	d.	d.	PROPN
ejpam-5364	151	5	kalyani	kalyani	PROPN
ejpam-5364	151	6	/	/	SYM
ejpam-5364	151	7	eur	eur	PROPN
ejpam-5364	151	8	.	.	PUNCT
ejpam-5364	152	1	j.	j.	PROPN
ejpam-5364	152	2	pure	pure	PROPN
ejpam-5364	152	3	appl	appl	PROPN
ejpam-5364	152	4	.	.	PROPN
ejpam-5364	152	5	math	math	PROPN
ejpam-5364	152	6	,	,	PUNCT
ejpam-5364	152	7	17	17	NUM
ejpam-5364	152	8	(	(	PUNCT
ejpam-5364	152	9	4	4	NUM
ejpam-5364	152	10	)	)	PUNCT
ejpam-5364	152	11	(	(	PUNCT
ejpam-5364	152	12	2024	2024	NUM
ejpam-5364	152	13	)	)	PUNCT
ejpam-5364	152	14	,	,	PUNCT
ejpam-5364	152	15	3004	3004	NUM
ejpam-5364	152	16	-	-	SYM
ejpam-5364	152	17	3021	3021	NUM
ejpam-5364	152	18	3010	3010	NUM
ejpam-5364	152	19	3.1	3.1	NUM
ejpam-5364	152	20	.	.	PUNCT
ejpam-5364	153	1	skewness	skewness	NOUN
ejpam-5364	153	2	the	the	DET
ejpam-5364	153	3	theorems	theorem	NOUN
ejpam-5364	153	4	relating	relate	VERB
ejpam-5364	153	5	the	the	DET
ejpam-5364	153	6	second	second	ADV
ejpam-5364	153	7	largest	large	ADJ
ejpam-5364	153	8	adjacency	adjacency	NOUN
ejpam-5364	153	9	and	and	CCONJ
ejpam-5364	153	10	signless	signless	ADJ
ejpam-5364	153	11	laplacian	laplacian	ADJ
ejpam-5364	153	12	eigenvalues	eigenvalue	NOUN
ejpam-5364	153	13	of	of	ADP
ejpam-5364	153	14	a	a	DET
ejpam-5364	153	15	graph	graph	NOUN
ejpam-5364	153	16	and	and	CCONJ
ejpam-5364	153	17	skewness	skewness	NOUN
ejpam-5364	153	18	of	of	ADP
ejpam-5364	153	19	the	the	DET
ejpam-5364	153	20	graph	graph	NOUN
ejpam-5364	153	21	are	be	AUX
ejpam-5364	153	22	presented	present	VERB
ejpam-5364	153	23	in	in	ADP
ejpam-5364	153	24	this	this	DET
ejpam-5364	153	25	section	section	NOUN
ejpam-5364	153	26	.	.	PUNCT
ejpam-5364	154	1	we	we	PRON
ejpam-5364	154	2	establish	establish	VERB
ejpam-5364	154	3	lower	low	ADJ
ejpam-5364	154	4	bounds	bound	NOUN
ejpam-5364	154	5	for	for	ADP
ejpam-5364	154	6	λ2	λ2	NOUN
ejpam-5364	154	7	and	and	CCONJ
ejpam-5364	154	8	q2	q2	NOUN
ejpam-5364	154	9	in	in	ADP
ejpam-5364	154	10	terms	term	NOUN
ejpam-5364	154	11	of	of	ADP
ejpam-5364	154	12	sk(g	sk(g	NOUN
ejpam-5364	154	13	)	)	PUNCT
ejpam-5364	154	14	.	.	PUNCT
ejpam-5364	155	1	also	also	ADV
ejpam-5364	155	2	,	,	PUNCT
ejpam-5364	155	3	lower	low	ADJ
ejpam-5364	155	4	bounds	bound	NOUN
ejpam-5364	155	5	for	for	ADP
ejpam-5364	155	6	the	the	DET
ejpam-5364	155	7	skewness	skewness	NOUN
ejpam-5364	155	8	of	of	ADP
ejpam-5364	155	9	regular	regular	ADJ
ejpam-5364	155	10	graphs	graph	NOUN
ejpam-5364	155	11	in	in	ADP
ejpam-5364	155	12	terms	term	NOUN
ejpam-5364	155	13	of	of	ADP
ejpam-5364	155	14	λ2	λ2	NOUN
ejpam-5364	155	15	and	and	CCONJ
ejpam-5364	155	16	q2	q2	NOUN
ejpam-5364	155	17	are	be	AUX
ejpam-5364	155	18	obtained	obtain	VERB
ejpam-5364	155	19	.	.	PUNCT
ejpam-5364	156	1	theorem	theorem	NOUN
ejpam-5364	156	2	1	1	NUM
ejpam-5364	156	3	.	.	PUNCT
ejpam-5364	157	1	let	let	VERB
ejpam-5364	157	2	g	g	PRON
ejpam-5364	157	3	be	be	AUX
ejpam-5364	157	4	a	a	DET
ejpam-5364	157	5	connected	connected	ADJ
ejpam-5364	157	6	graph	graph	NOUN
ejpam-5364	157	7	with	with	ADP
ejpam-5364	157	8	skewness	skewness	NOUN
ejpam-5364	157	9	sk(g	sk(g	NOUN
ejpam-5364	157	10	)	)	PUNCT
ejpam-5364	157	11	.	.	PUNCT
ejpam-5364	158	1	then	then	ADV
ejpam-5364	158	2	λ2	λ2	PRON
ejpam-5364	158	3	≥	≥	NUM
ejpam-5364	158	4	1	1	NUM
ejpam-5364	158	5	2	2	NUM
ejpam-5364	158	6	{	{	PUNCT
ejpam-5364	158	7	[	[	PUNCT
ejpam-5364	158	8	δ	δ	NOUN
ejpam-5364	158	9	−	−	PROPN
ejpam-5364	159	1	3∆	3∆	NUM
ejpam-5364	159	2	m−	m−	PROPN
ejpam-5364	159	3	sk(g	sk(g	NOUN
ejpam-5364	159	4	)	)	PUNCT
ejpam-5364	160	1	+	+	CCONJ
ejpam-5364	160	2	3	3	X
ejpam-5364	160	3	]	]	SYM
ejpam-5364	160	4	−	−	NUM
ejpam-5364	160	5	√	√	PROPN
ejpam-5364	160	6	[	[	PUNCT
ejpam-5364	160	7	δ	δ	NOUN
ejpam-5364	160	8	+	+	CCONJ
ejpam-5364	160	9	3∆	3∆	NUM
ejpam-5364	160	10	m−	m−	PROPN
ejpam-5364	160	11	sk(g	sk(g	NOUN
ejpam-5364	160	12	)	)	PUNCT
ejpam-5364	160	13	+	+	CCONJ
ejpam-5364	160	14	3	3	NUM
ejpam-5364	160	15	]	]	SYM
ejpam-5364	160	16	2	2	NUM
ejpam-5364	160	17	+	+	SYM
ejpam-5364	160	18	12∆(∆−	12∆(∆−	NUM
ejpam-5364	160	19	δ	δ	NOUN
ejpam-5364	160	20	)	)	PUNCT
ejpam-5364	160	21	m−	m−	PROPN
ejpam-5364	160	22	sk(g	sk(g	NOUN
ejpam-5364	160	23	)	)	PUNCT
ejpam-5364	161	1	+	+	CCONJ
ejpam-5364	161	2	3	3	X
ejpam-5364	161	3	}	}	PUNCT
ejpam-5364	161	4	.	.	PUNCT
ejpam-5364	162	1	proof	proof	NOUN
ejpam-5364	162	2	.	.	PUNCT
ejpam-5364	163	1	let	let	VERB
ejpam-5364	163	2	aq	aq	PART
ejpam-5364	163	3	be	be	AUX
ejpam-5364	163	4	the	the	DET
ejpam-5364	163	5	quotient	quotient	NOUN
ejpam-5364	163	6	matrix	matrix	NOUN
ejpam-5364	163	7	of	of	ADP
ejpam-5364	163	8	the	the	DET
ejpam-5364	163	9	adjacency	adjacency	NOUN
ejpam-5364	163	10	matrix	matrix	NOUN
ejpam-5364	163	11	a	a	PRON
ejpam-5364	163	12	of	of	ADP
ejpam-5364	163	13	g	g	NOUN
ejpam-5364	163	14	with	with	ADP
ejpam-5364	163	15	respect	respect	NOUN
ejpam-5364	163	16	to	to	ADP
ejpam-5364	163	17	the	the	DET
ejpam-5364	163	18	partitions	partition	NOUN
ejpam-5364	163	19	p1	p1	PROPN
ejpam-5364	163	20	and	and	CCONJ
ejpam-5364	163	21	p2	p2	PROPN
ejpam-5364	163	22	where	where	SCONJ
ejpam-5364	163	23	p1	p1	NOUN
ejpam-5364	163	24	consists	consist	VERB
ejpam-5364	163	25	of	of	ADP
ejpam-5364	163	26	one	one	NUM
ejpam-5364	163	27	vertex	vertex	NOUN
ejpam-5364	163	28	of	of	ADP
ejpam-5364	163	29	g	g	NOUN
ejpam-5364	163	30	with	with	ADP
ejpam-5364	163	31	maximum	maximum	ADJ
ejpam-5364	163	32	degree	degree	NOUN
ejpam-5364	163	33	∆.	∆.	X
ejpam-5364	163	34	then	then	ADV
ejpam-5364	163	35	,	,	PUNCT
ejpam-5364	163	36	we	we	PRON
ejpam-5364	163	37	have	have	VERB
ejpam-5364	163	38	|p1|	|p1|	ADJ
ejpam-5364	163	39	=	=	SYM
ejpam-5364	163	40	n1	n1	PROPN
ejpam-5364	163	41	=	=	SYM
ejpam-5364	163	42	1	1	NUM
ejpam-5364	163	43	and	and	CCONJ
ejpam-5364	163	44	|p2|	|p2|	NOUN
ejpam-5364	163	45	=	=	SYM
ejpam-5364	163	46	n2	n2	PROPN
ejpam-5364	163	47	=	=	SYM
ejpam-5364	163	48	(	(	PUNCT
ejpam-5364	163	49	n−	n−	NOUN
ejpam-5364	163	50	1	1	NUM
ejpam-5364	163	51	)	)	PUNCT
ejpam-5364	163	52	.	.	PUNCT
ejpam-5364	164	1	then	then	ADV
ejpam-5364	164	2	aq	aq	VERB
ejpam-5364	164	3	=	=	PUNCT
ejpam-5364	165	1	[	[	PUNCT
ejpam-5364	165	2	d′1	d′1	CCONJ
ejpam-5364	165	3	−	−	PROPN
ejpam-5364	165	4	t	t	PROPN
ejpam-5364	165	5	t	t	PROPN
ejpam-5364	165	6	t	t	PROPN
ejpam-5364	165	7	(	(	PUNCT
ejpam-5364	165	8	n−1	n−1	PROPN
ejpam-5364	165	9	)	)	PUNCT
ejpam-5364	165	10	d′2	d′2	NOUN
ejpam-5364	165	11	−	−	PROPN
ejpam-5364	165	12	t	t	PROPN
ejpam-5364	165	13	(	(	PUNCT
ejpam-5364	165	14	n−1	n−1	PROPN
ejpam-5364	165	15	)	)	PUNCT
ejpam-5364	165	16	]	]	PUNCT
ejpam-5364	165	17	where	where	SCONJ
ejpam-5364	165	18	d′1	d′1	NOUN
ejpam-5364	165	19	,	,	PUNCT
ejpam-5364	165	20	d	d	NOUN
ejpam-5364	165	21	′	′	NOUN
ejpam-5364	165	22	2	2	NUM
ejpam-5364	165	23	are	be	AUX
ejpam-5364	165	24	the	the	DET
ejpam-5364	165	25	average	average	ADJ
ejpam-5364	165	26	degrees	degree	NOUN
ejpam-5364	165	27	of	of	ADP
ejpam-5364	165	28	the	the	DET
ejpam-5364	165	29	two	two	NUM
ejpam-5364	165	30	partitions	partition	NOUN
ejpam-5364	165	31	p1	p1	NOUN
ejpam-5364	165	32	,	,	PUNCT
ejpam-5364	165	33	p2	p2	PROPN
ejpam-5364	165	34	and	and	CCONJ
ejpam-5364	165	35	t	t	NOUN
ejpam-5364	165	36	is	be	AUX
ejpam-5364	165	37	the	the	DET
ejpam-5364	165	38	number	number	NOUN
ejpam-5364	165	39	of	of	ADP
ejpam-5364	165	40	edges	edge	NOUN
ejpam-5364	165	41	between	between	ADP
ejpam-5364	165	42	p1	p1	NOUN
ejpam-5364	165	43	and	and	CCONJ
ejpam-5364	165	44	p2	p2	NOUN
ejpam-5364	165	45	.	.	PUNCT
ejpam-5364	166	1	the	the	DET
ejpam-5364	166	2	characteristic	characteristic	ADJ
ejpam-5364	166	3	equation	equation	NOUN
ejpam-5364	166	4	of	of	ADP
ejpam-5364	166	5	the	the	DET
ejpam-5364	166	6	above	above	ADJ
ejpam-5364	166	7	matrix	matrix	NOUN
ejpam-5364	166	8	is	be	AUX
ejpam-5364	166	9	given	give	VERB
ejpam-5364	166	10	by	by	ADP
ejpam-5364	166	11	η2	η2	PUNCT
ejpam-5364	166	12	−	−	PROPN
ejpam-5364	167	1	[	[	PUNCT
ejpam-5364	167	2	d′1	d′1	NOUN
ejpam-5364	167	3	+	+	CCONJ
ejpam-5364	167	4	d′2	d′2	ADJ
ejpam-5364	167	5	−	−	PROPN
ejpam-5364	167	6	t−	t−	PROPN
ejpam-5364	167	7	t	t	PROPN
ejpam-5364	167	8	(	(	PUNCT
ejpam-5364	167	9	n−	n−	NOUN
ejpam-5364	167	10	1	1	NUM
ejpam-5364	167	11	)	)	PUNCT
ejpam-5364	167	12	]	]	PUNCT
ejpam-5364	168	1	η	η	PROPN
ejpam-5364	168	2	+	+	PUNCT
ejpam-5364	168	3	(	(	PUNCT
ejpam-5364	168	4	d′1	d′1	ADV
ejpam-5364	168	5	−	−	PROPN
ejpam-5364	168	6	t	t	NOUN
ejpam-5364	168	7	)	)	PUNCT
ejpam-5364	168	8	(	(	PUNCT
ejpam-5364	168	9	d′2	d′2	ADP
ejpam-5364	168	10	−	−	PROPN
ejpam-5364	168	11	t	t	PROPN
ejpam-5364	168	12	(	(	PUNCT
ejpam-5364	168	13	n−	n−	NOUN
ejpam-5364	168	14	1	1	NUM
ejpam-5364	168	15	)	)	PUNCT
ejpam-5364	168	16	)	)	PUNCT
ejpam-5364	169	1	−	−	PROPN
ejpam-5364	169	2	t2	t2	NOUN
ejpam-5364	169	3	(	(	PUNCT
ejpam-5364	169	4	n−	n−	NOUN
ejpam-5364	169	5	1	1	NUM
ejpam-5364	169	6	)	)	PUNCT
ejpam-5364	169	7	=	=	SYM
ejpam-5364	169	8	0	0	X
ejpam-5364	169	9	.	.	PUNCT
ejpam-5364	170	1	the	the	DET
ejpam-5364	170	2	roots	root	NOUN
ejpam-5364	170	3	of	of	ADP
ejpam-5364	170	4	the	the	DET
ejpam-5364	170	5	above	above	ADJ
ejpam-5364	170	6	characteristic	characteristic	ADJ
ejpam-5364	170	7	equation	equation	NOUN
ejpam-5364	170	8	are	be	AUX
ejpam-5364	170	9	η1	η1	NOUN
ejpam-5364	170	10	=	=	SYM
ejpam-5364	170	11	1	1	NUM
ejpam-5364	170	12	2	2	NUM
ejpam-5364	170	13	{	{	PUNCT
ejpam-5364	170	14	(	(	PUNCT
ejpam-5364	170	15	d′1	d′1	NOUN
ejpam-5364	171	1	+	+	CCONJ
ejpam-5364	171	2	d′2	d′2	ADJ
ejpam-5364	171	3	−	−	PROPN
ejpam-5364	171	4	tn	tn	PROPN
ejpam-5364	171	5	(	(	PUNCT
ejpam-5364	171	6	n−1	n−1	PROPN
ejpam-5364	171	7	)	)	PUNCT
ejpam-5364	171	8	)	)	PUNCT
ejpam-5364	172	1	+	+	CCONJ
ejpam-5364	172	2	√	√	VERB
ejpam-5364	172	3	r	r	NOUN
ejpam-5364	172	4	}	}	PUNCT
ejpam-5364	172	5	and	and	CCONJ
ejpam-5364	172	6	η2	η2	ADJ
ejpam-5364	172	7	=	=	SYM
ejpam-5364	172	8	1	1	NUM
ejpam-5364	172	9	2	2	NUM
ejpam-5364	172	10	{	{	PUNCT
ejpam-5364	172	11	(	(	PUNCT
ejpam-5364	172	12	d′1	d′1	NOUN
ejpam-5364	172	13	+	+	CCONJ
ejpam-5364	172	14	d′2	d′2	ADJ
ejpam-5364	172	15	−	−	PROPN
ejpam-5364	172	16	tn	tn	PROPN
ejpam-5364	172	17	(	(	PUNCT
ejpam-5364	172	18	n−1	n−1	PROPN
ejpam-5364	172	19	)	)	PUNCT
ejpam-5364	172	20	)	)	PUNCT
ejpam-5364	173	1	−	−	ADP
ejpam-5364	174	1	√	√	NUM
ejpam-5364	174	2	r	r	NOUN
ejpam-5364	174	3	}	}	PUNCT
ejpam-5364	174	4	where	where	SCONJ
ejpam-5364	174	5	r	r	NOUN
ejpam-5364	174	6	=	=	PUNCT
ejpam-5364	175	1	[	[	X
ejpam-5364	175	2	(	(	PUNCT
ejpam-5364	175	3	d′1	d′1	NOUN
ejpam-5364	175	4	−	−	PROPN
ejpam-5364	175	5	t	t	PROPN
ejpam-5364	175	6	)	)	PUNCT
ejpam-5364	175	7	−	−	PROPN
ejpam-5364	176	1	(	(	PUNCT
ejpam-5364	176	2	d′2	d′2	ADJ
ejpam-5364	176	3	−	−	PROPN
ejpam-5364	176	4	t	t	PROPN
ejpam-5364	176	5	(	(	PUNCT
ejpam-5364	176	6	n−	n−	NOUN
ejpam-5364	176	7	1	1	NUM
ejpam-5364	176	8	)	)	PUNCT
ejpam-5364	176	9	)	)	PUNCT
ejpam-5364	177	1	]	]	PUNCT
ejpam-5364	177	2	2	2	NUM
ejpam-5364	177	3	+	+	NUM
ejpam-5364	177	4	4t2	4t2	NUM
ejpam-5364	177	5	(	(	PUNCT
ejpam-5364	177	6	n−	n−	NOUN
ejpam-5364	177	7	1	1	NUM
ejpam-5364	177	8	)	)	PUNCT
ejpam-5364	177	9	=	=	NOUN
ejpam-5364	177	10	(	(	PUNCT
ejpam-5364	177	11	d′1	d′1	ADV
ejpam-5364	177	12	−	−	PROPN
ejpam-5364	177	13	d′2	d′2	ADJ
ejpam-5364	177	14	−	−	PROPN
ejpam-5364	177	15	t−	t−	PROPN
ejpam-5364	177	16	t	t	PROPN
ejpam-5364	177	17	(	(	PUNCT
ejpam-5364	177	18	n−	n−	NOUN
ejpam-5364	177	19	1	1	NUM
ejpam-5364	177	20	)	)	PUNCT
ejpam-5364	177	21	)	)	PUNCT
ejpam-5364	177	22	2	2	NUM
ejpam-5364	178	1	+	+	NUM
ejpam-5364	178	2	4t(d′1	4t(d′1	NUM
ejpam-5364	178	3	−	−	NUM
ejpam-5364	178	4	d′2	d′2	NOUN
ejpam-5364	178	5	)	)	PUNCT
ejpam-5364	178	6	(	(	PUNCT
ejpam-5364	178	7	n−	n−	NOUN
ejpam-5364	178	8	1	1	NUM
ejpam-5364	178	9	)	)	PUNCT
ejpam-5364	178	10	.	.	PUNCT
ejpam-5364	179	1	therefore	therefore	ADV
ejpam-5364	179	2	,	,	PUNCT
ejpam-5364	179	3	we	we	PRON
ejpam-5364	179	4	have	have	VERB
ejpam-5364	179	5	η2	η2	ADJ
ejpam-5364	179	6	=	=	SYM
ejpam-5364	179	7	1	1	NUM
ejpam-5364	179	8	2	2	NUM
ejpam-5364	179	9	{	{	PUNCT
ejpam-5364	179	10	(	(	PUNCT
ejpam-5364	179	11	d′1	d′1	NOUN
ejpam-5364	179	12	+	+	CCONJ
ejpam-5364	179	13	d′2	d′2	ADJ
ejpam-5364	179	14	−	−	PROPN
ejpam-5364	179	15	tn	tn	PROPN
ejpam-5364	180	1	(	(	PUNCT
ejpam-5364	180	2	n−	n−	NOUN
ejpam-5364	180	3	1	1	NUM
ejpam-5364	180	4	)	)	PUNCT
ejpam-5364	180	5	)	)	PUNCT
ejpam-5364	181	1	−	−	PROPN
ejpam-5364	181	2	√√√√(d′1	√√√√(d′1	PROPN
ejpam-5364	181	3	−	−	PROPN
ejpam-5364	181	4	d′2	d′2	ADJ
ejpam-5364	181	5	−	−	PROPN
ejpam-5364	181	6	t−	t−	PROPN
ejpam-5364	181	7	t	t	PROPN
ejpam-5364	181	8	(	(	PUNCT
ejpam-5364	181	9	n−	n−	NOUN
ejpam-5364	181	10	1	1	NUM
ejpam-5364	181	11	)	)	PUNCT
ejpam-5364	181	12	)	)	PUNCT
ejpam-5364	181	13	2	2	NUM
ejpam-5364	181	14	+	+	NUM
ejpam-5364	181	15	4t(d′1	4t(d′1	NUM
ejpam-5364	181	16	−	−	NUM
ejpam-5364	181	17	d′2	d′2	ADJ
ejpam-5364	181	18	)	)	PUNCT
ejpam-5364	181	19	n2	n2	NOUN
ejpam-5364	181	20	}	}	PUNCT
ejpam-5364	181	21	.	.	PUNCT
ejpam-5364	182	1	since	since	SCONJ
ejpam-5364	182	2	d′1	d′1	NOUN
ejpam-5364	182	3	=	=	SYM
ejpam-5364	182	4	∆	∆	PROPN
ejpam-5364	182	5	and	and	CCONJ
ejpam-5364	182	6	d′2	d′2	ADJ
ejpam-5364	182	7	≥	≥	NOUN
ejpam-5364	182	8	δ	δ	NOUN
ejpam-5364	182	9	we	we	PRON
ejpam-5364	182	10	get	get	VERB
ejpam-5364	182	11	,	,	PUNCT
ejpam-5364	182	12	η2	η2	X
ejpam-5364	182	13	≥	≥	NOUN
ejpam-5364	182	14	1	1	NUM
ejpam-5364	182	15	2	2	NUM
ejpam-5364	182	16	{	{	PUNCT
ejpam-5364	182	17	(	(	PUNCT
ejpam-5364	182	18	∆+	∆+	NUM
ejpam-5364	182	19	δ	δ	PROPN
ejpam-5364	182	20	−	−	PROPN
ejpam-5364	182	21	tn	tn	PROPN
ejpam-5364	182	22	(	(	PUNCT
ejpam-5364	182	23	n−	n−	NOUN
ejpam-5364	182	24	1	1	NUM
ejpam-5364	182	25	)	)	PUNCT
ejpam-5364	182	26	)	)	PUNCT
ejpam-5364	183	1	−	−	PROPN
ejpam-5364	184	1	√√√√(∆−	√√√√(∆−	PROPN
ejpam-5364	184	2	δ	δ	X
ejpam-5364	184	3	−	−	PROPN
ejpam-5364	184	4	t−	t−	PROPN
ejpam-5364	184	5	t	t	PROPN
ejpam-5364	184	6	(	(	PUNCT
ejpam-5364	184	7	n−	n−	NOUN
ejpam-5364	184	8	1	1	NUM
ejpam-5364	184	9	)	)	PUNCT
ejpam-5364	184	10	)	)	PUNCT
ejpam-5364	184	11	2	2	NUM
ejpam-5364	184	12	+	+	SYM
ejpam-5364	184	13	4t(∆−	4t(∆−	NUM
ejpam-5364	184	14	δ	δ	NOUN
ejpam-5364	184	15	)	)	PUNCT
ejpam-5364	184	16	(	(	PUNCT
ejpam-5364	184	17	n−	n−	NOUN
ejpam-5364	184	18	1	1	NUM
ejpam-5364	184	19	)	)	PUNCT
ejpam-5364	184	20	}	}	PUNCT
ejpam-5364	184	21	.	.	PUNCT
ejpam-5364	185	1	m.	m.	PROPN
ejpam-5364	185	2	machasri	machasri	PROPN
ejpam-5364	185	3	,	,	PUNCT
ejpam-5364	185	4	d.	d.	PROPN
ejpam-5364	185	5	kalyani	kalyani	PROPN
ejpam-5364	185	6	/	/	SYM
ejpam-5364	185	7	eur	eur	PROPN
ejpam-5364	185	8	.	.	PUNCT
ejpam-5364	186	1	j.	j.	PROPN
ejpam-5364	186	2	pure	pure	PROPN
ejpam-5364	186	3	appl	appl	PROPN
ejpam-5364	186	4	.	.	PROPN
ejpam-5364	186	5	math	math	PROPN
ejpam-5364	186	6	,	,	PUNCT
ejpam-5364	186	7	17	17	NUM
ejpam-5364	186	8	(	(	PUNCT
ejpam-5364	186	9	4	4	NUM
ejpam-5364	186	10	)	)	PUNCT
ejpam-5364	186	11	(	(	PUNCT
ejpam-5364	186	12	2024	2024	NUM
ejpam-5364	186	13	)	)	PUNCT
ejpam-5364	186	14	,	,	PUNCT
ejpam-5364	186	15	3004	3004	NUM
ejpam-5364	186	16	-	-	SYM
ejpam-5364	186	17	3021	3021	NUM
ejpam-5364	186	18	3011	3011	NUM
ejpam-5364	186	19	by	by	ADP
ejpam-5364	186	20	eigenvalue	eigenvalue	PROPN
ejpam-5364	186	21	interlacing	interlacing	NOUN
ejpam-5364	187	1	,	,	PUNCT
ejpam-5364	187	2	we	we	PRON
ejpam-5364	187	3	have	have	VERB
ejpam-5364	187	4	λ2	λ2	NOUN
ejpam-5364	187	5	≥	≥	NUM
ejpam-5364	187	6	1	1	NUM
ejpam-5364	187	7	2	2	NUM
ejpam-5364	187	8	{	{	PUNCT
ejpam-5364	187	9	(	(	PUNCT
ejpam-5364	187	10	∆+	∆+	NUM
ejpam-5364	187	11	δ	δ	PROPN
ejpam-5364	187	12	−	−	PROPN
ejpam-5364	187	13	tn	tn	PROPN
ejpam-5364	187	14	(	(	PUNCT
ejpam-5364	187	15	n−	n−	NOUN
ejpam-5364	187	16	1	1	NUM
ejpam-5364	187	17	)	)	PUNCT
ejpam-5364	187	18	)	)	PUNCT
ejpam-5364	188	1	−	−	ADP
ejpam-5364	188	2	√	√	NUM
ejpam-5364	188	3	(	(	PUNCT
ejpam-5364	188	4	∆−	∆−	NOUN
ejpam-5364	188	5	δ	δ	PROPN
ejpam-5364	188	6	−	−	PROPN
ejpam-5364	188	7	tn	tn	PROPN
ejpam-5364	188	8	(	(	PUNCT
ejpam-5364	188	9	n−	n−	NOUN
ejpam-5364	188	10	1	1	NUM
ejpam-5364	188	11	)	)	PUNCT
ejpam-5364	188	12	)	)	PUNCT
ejpam-5364	189	1	2	2	NUM
ejpam-5364	189	2	+	+	SYM
ejpam-5364	189	3	4t(∆−	4t(∆−	NUM
ejpam-5364	189	4	δ	δ	NOUN
ejpam-5364	189	5	)	)	PUNCT
ejpam-5364	189	6	(	(	PUNCT
ejpam-5364	189	7	n−	n−	NOUN
ejpam-5364	189	8	1	1	NUM
ejpam-5364	189	9	)	)	PUNCT
ejpam-5364	189	10	}	}	PUNCT
ejpam-5364	189	11	.	.	PUNCT
ejpam-5364	190	1	the	the	DET
ejpam-5364	190	2	above	above	ADJ
ejpam-5364	190	3	inequality	inequality	NOUN
ejpam-5364	190	4	can	can	AUX
ejpam-5364	190	5	be	be	AUX
ejpam-5364	190	6	rewritten	rewrite	VERB
ejpam-5364	190	7	as	as	ADP
ejpam-5364	190	8	λ2	λ2	NOUN
ejpam-5364	190	9	≥	≥	NUM
ejpam-5364	190	10	1	1	NUM
ejpam-5364	190	11	2	2	NUM
ejpam-5364	190	12	{	{	PUNCT
ejpam-5364	190	13	[	[	PUNCT
ejpam-5364	190	14	∆+	∆+	NUM
ejpam-5364	190	15	δ	δ	PROPN
ejpam-5364	190	16	−	−	PROPN
ejpam-5364	190	17	t	t	PROPN
ejpam-5364	190	18	(	(	PUNCT
ejpam-5364	190	19	1	1	NUM
ejpam-5364	190	20	+	+	NUM
ejpam-5364	190	21	1	1	NUM
ejpam-5364	190	22	(	(	PUNCT
ejpam-5364	190	23	n−	n−	NOUN
ejpam-5364	190	24	1	1	NUM
ejpam-5364	190	25	)	)	PUNCT
ejpam-5364	190	26	)	)	PUNCT
ejpam-5364	190	27	]	]	PUNCT
ejpam-5364	191	1	−	−	PUNCT
ejpam-5364	191	2	√	√	NUM
ejpam-5364	191	3	[	[	PUNCT
ejpam-5364	191	4	∆−	∆−	NUM
ejpam-5364	191	5	δ	δ	PROPN
ejpam-5364	191	6	−	−	PROPN
ejpam-5364	191	7	t	t	PROPN
ejpam-5364	191	8	(	(	PUNCT
ejpam-5364	191	9	1	1	NUM
ejpam-5364	191	10	+	+	NUM
ejpam-5364	191	11	1	1	NUM
ejpam-5364	191	12	(	(	PUNCT
ejpam-5364	191	13	n−	n−	NOUN
ejpam-5364	191	14	1	1	NUM
ejpam-5364	191	15	)	)	PUNCT
ejpam-5364	191	16	)	)	PUNCT
ejpam-5364	192	1	]	]	X
ejpam-5364	192	2	2	2	NUM
ejpam-5364	192	3	+	+	SYM
ejpam-5364	192	4	4t(∆−	4t(∆−	NUM
ejpam-5364	192	5	δ	δ	NOUN
ejpam-5364	192	6	)	)	PUNCT
ejpam-5364	192	7	(	(	PUNCT
ejpam-5364	192	8	n−	n−	NOUN
ejpam-5364	192	9	1	1	NUM
ejpam-5364	192	10	)	)	PUNCT
ejpam-5364	192	11	}	}	PUNCT
ejpam-5364	192	12	.	.	PUNCT
ejpam-5364	193	1	since	since	SCONJ
ejpam-5364	193	2	t	t	NOUN
ejpam-5364	193	3	=	=	SYM
ejpam-5364	193	4	∆	∆	PROPN
ejpam-5364	193	5	,	,	PUNCT
ejpam-5364	193	6	the	the	DET
ejpam-5364	193	7	above	above	ADJ
ejpam-5364	193	8	inequality	inequality	NOUN
ejpam-5364	193	9	becomes	become	VERB
ejpam-5364	193	10	λ2	λ2	PROPN
ejpam-5364	193	11	≥	≥	NUM
ejpam-5364	193	12	1	1	NUM
ejpam-5364	193	13	2	2	NUM
ejpam-5364	193	14	{	{	PUNCT
ejpam-5364	193	15	(	(	PUNCT
ejpam-5364	193	16	δ	δ	PROPN
ejpam-5364	193	17	−	−	PROPN
ejpam-5364	193	18	∆	∆	PROPN
ejpam-5364	193	19	(	(	PUNCT
ejpam-5364	193	20	n−	n−	NOUN
ejpam-5364	193	21	1	1	NUM
ejpam-5364	193	22	)	)	PUNCT
ejpam-5364	193	23	)	)	PUNCT
ejpam-5364	194	1	−	−	ADP
ejpam-5364	194	2	√	√	PROPN
ejpam-5364	194	3	(	(	PUNCT
ejpam-5364	194	4	δ	δ	PROPN
ejpam-5364	194	5	+	+	X
ejpam-5364	194	6	∆	∆	PROPN
ejpam-5364	194	7	(	(	PUNCT
ejpam-5364	194	8	n−	n−	NOUN
ejpam-5364	194	9	1	1	NUM
ejpam-5364	194	10	)	)	PUNCT
ejpam-5364	194	11	)	)	PUNCT
ejpam-5364	194	12	2	2	NUM
ejpam-5364	194	13	+	+	CCONJ
ejpam-5364	194	14	4∆(∆−	4∆(∆−	PROPN
ejpam-5364	194	15	δ	δ	NOUN
ejpam-5364	194	16	)	)	PUNCT
ejpam-5364	194	17	(	(	PUNCT
ejpam-5364	194	18	n−	n−	NOUN
ejpam-5364	194	19	1	1	NUM
ejpam-5364	194	20	)	)	PUNCT
ejpam-5364	194	21	}	}	PUNCT
ejpam-5364	194	22	.	.	PUNCT
ejpam-5364	195	1	by	by	ADP
ejpam-5364	195	2	making	make	VERB
ejpam-5364	195	3	use	use	NOUN
ejpam-5364	195	4	of	of	ADP
ejpam-5364	195	5	inequality	inequality	NOUN
ejpam-5364	195	6	(	(	PUNCT
ejpam-5364	195	7	1	1	NUM
ejpam-5364	195	8	)	)	PUNCT
ejpam-5364	195	9	,	,	PUNCT
ejpam-5364	195	10	the	the	DET
ejpam-5364	195	11	result	result	NOUN
ejpam-5364	195	12	is	be	AUX
ejpam-5364	195	13	obtained	obtain	VERB
ejpam-5364	195	14	.	.	PUNCT
ejpam-5364	196	1	theorem	theorem	NOUN
ejpam-5364	196	2	2	2	NUM
ejpam-5364	196	3	.	.	PUNCT
ejpam-5364	197	1	let	let	VERB
ejpam-5364	197	2	g	g	PRON
ejpam-5364	197	3	be	be	AUX
ejpam-5364	197	4	a	a	DET
ejpam-5364	197	5	d−regular	d−regular	NUM
ejpam-5364	197	6	graph	graph	NOUN
ejpam-5364	197	7	with	with	ADP
ejpam-5364	197	8	n	n	PRON
ejpam-5364	197	9	≥	≥	NUM
ejpam-5364	197	10	3	3	NUM
ejpam-5364	197	11	.	.	PUNCT
ejpam-5364	198	1	then	then	ADV
ejpam-5364	198	2	λ2	λ2	PROPN
ejpam-5364	198	3	≥	≥	NUM
ejpam-5364	198	4	3d	3d	NUM
ejpam-5364	198	5	sk(g)−m−	sk(g)−m−	NUM
ejpam-5364	198	6	3	3	NUM
ejpam-5364	198	7	.	.	PUNCT
ejpam-5364	199	1	proof	proof	NOUN
ejpam-5364	199	2	.	.	PUNCT
ejpam-5364	200	1	let	let	VERB
ejpam-5364	200	2	a	a	PRON
ejpam-5364	200	3	be	be	AUX
ejpam-5364	200	4	the	the	DET
ejpam-5364	200	5	adjacency	adjacency	NOUN
ejpam-5364	200	6	matrix	matrix	NOUN
ejpam-5364	200	7	of	of	ADP
ejpam-5364	200	8	g	g	PROPN
ejpam-5364	200	9	represented	represent	VERB
ejpam-5364	200	10	in	in	ADP
ejpam-5364	200	11	the	the	DET
ejpam-5364	200	12	following	follow	VERB
ejpam-5364	200	13	block	block	NOUN
ejpam-5364	200	14	matrix	matrix	NOUN
ejpam-5364	200	15	form	form	NOUN
ejpam-5364	200	16	a	a	DET
ejpam-5364	200	17	=	=	X
ejpam-5364	200	18	[	[	PUNCT
ejpam-5364	200	19	a11	a11	PROPN
ejpam-5364	200	20	a12	a12	PROPN
ejpam-5364	200	21	a21	a21	PROPN
ejpam-5364	200	22	a22	a22	PROPN
ejpam-5364	200	23	]	]	PUNCT
ejpam-5364	200	24	.	.	PUNCT
ejpam-5364	201	1	let	let	VERB
ejpam-5364	201	2	aq	aq	PART
ejpam-5364	201	3	be	be	AUX
ejpam-5364	201	4	the	the	DET
ejpam-5364	201	5	quotient	quotient	NOUN
ejpam-5364	201	6	matrix	matrix	NOUN
ejpam-5364	201	7	of	of	ADP
ejpam-5364	201	8	a	a	PRON
ejpam-5364	201	9	of	of	ADP
ejpam-5364	201	10	g	g	NOUN
ejpam-5364	201	11	with	with	ADP
ejpam-5364	201	12	respect	respect	NOUN
ejpam-5364	201	13	to	to	ADP
ejpam-5364	201	14	the	the	DET
ejpam-5364	201	15	partitions	partition	NOUN
ejpam-5364	201	16	p1	p1	PROPN
ejpam-5364	201	17	and	and	CCONJ
ejpam-5364	201	18	p2	p2	NOUN
ejpam-5364	201	19	.	.	PUNCT
ejpam-5364	202	1	let	let	VERB
ejpam-5364	202	2	|p1|	|p1|	ADV
ejpam-5364	202	3	=	=	SYM
ejpam-5364	202	4	n1	n1	PROPN
ejpam-5364	202	5	=	=	SYM
ejpam-5364	202	6	1	1	NUM
ejpam-5364	202	7	and	and	CCONJ
ejpam-5364	202	8	|p2|	|p2|	NOUN
ejpam-5364	202	9	=	=	SYM
ejpam-5364	202	10	n2	n2	PROPN
ejpam-5364	202	11	=	=	SYM
ejpam-5364	202	12	(	(	PUNCT
ejpam-5364	202	13	n−	n−	NOUN
ejpam-5364	202	14	1	1	NUM
ejpam-5364	202	15	)	)	PUNCT
ejpam-5364	202	16	.	.	PUNCT
ejpam-5364	203	1	then	then	ADV
ejpam-5364	203	2	aq	aq	VERB
ejpam-5364	203	3	=	=	PUNCT
ejpam-5364	204	1	[	[	PUNCT
ejpam-5364	204	2	d−	d−	PROPN
ejpam-5364	204	3	t	t	PROPN
ejpam-5364	204	4	t	t	PROPN
ejpam-5364	204	5	t	t	PROPN
ejpam-5364	204	6	(	(	PUNCT
ejpam-5364	204	7	n−1	n−1	PROPN
ejpam-5364	204	8	)	)	PUNCT
ejpam-5364	204	9	d−	d−	PROPN
ejpam-5364	204	10	t	t	PROPN
ejpam-5364	204	11	(	(	PUNCT
ejpam-5364	204	12	n−1	n−1	PROPN
ejpam-5364	204	13	)	)	PUNCT
ejpam-5364	204	14	]	]	PUNCT
ejpam-5364	204	15	where	where	SCONJ
ejpam-5364	204	16	t	t	PROPN
ejpam-5364	204	17	is	be	AUX
ejpam-5364	204	18	the	the	DET
ejpam-5364	204	19	number	number	NOUN
ejpam-5364	204	20	of	of	ADP
ejpam-5364	204	21	edges	edge	NOUN
ejpam-5364	204	22	between	between	ADP
ejpam-5364	204	23	p1	p1	NOUN
ejpam-5364	204	24	and	and	CCONJ
ejpam-5364	204	25	p2	p2	NOUN
ejpam-5364	204	26	.	.	PUNCT
ejpam-5364	205	1	the	the	DET
ejpam-5364	205	2	characteristic	characteristic	ADJ
ejpam-5364	205	3	equation	equation	NOUN
ejpam-5364	205	4	of	of	ADP
ejpam-5364	205	5	the	the	DET
ejpam-5364	205	6	above	above	ADJ
ejpam-5364	205	7	matrix	matrix	NOUN
ejpam-5364	205	8	is	be	AUX
ejpam-5364	205	9	given	give	VERB
ejpam-5364	205	10	by	by	ADP
ejpam-5364	205	11	[	[	PUNCT
ejpam-5364	205	12	η	η	PROPN
ejpam-5364	205	13	−	−	PROPN
ejpam-5364	205	14	(	(	PUNCT
ejpam-5364	205	15	d−	d−	PROPN
ejpam-5364	205	16	t	t	PROPN
ejpam-5364	205	17	)	)	PUNCT
ejpam-5364	205	18	]	]	X
ejpam-5364	205	19	[	[	PUNCT
ejpam-5364	205	20	η	η	X
ejpam-5364	205	21	−	−	PROPN
ejpam-5364	205	22	(	(	PUNCT
ejpam-5364	205	23	d−	d−	PROPN
ejpam-5364	205	24	t	t	PROPN
ejpam-5364	205	25	(	(	PUNCT
ejpam-5364	205	26	n−	n−	NOUN
ejpam-5364	205	27	1	1	NUM
ejpam-5364	205	28	)	)	PUNCT
ejpam-5364	205	29	)	)	PUNCT
ejpam-5364	205	30	]	]	PUNCT
ejpam-5364	206	1	−	−	PROPN
ejpam-5364	206	2	t2	t2	NOUN
ejpam-5364	206	3	(	(	PUNCT
ejpam-5364	206	4	n−	n−	NOUN
ejpam-5364	206	5	1	1	NUM
ejpam-5364	206	6	)	)	PUNCT
ejpam-5364	206	7	=	=	SYM
ejpam-5364	206	8	0	0	X
ejpam-5364	206	9	.	.	PUNCT
ejpam-5364	207	1	the	the	DET
ejpam-5364	207	2	roots	root	NOUN
ejpam-5364	207	3	of	of	ADP
ejpam-5364	207	4	the	the	DET
ejpam-5364	207	5	above	above	ADJ
ejpam-5364	207	6	equation	equation	NOUN
ejpam-5364	207	7	are	be	AUX
ejpam-5364	207	8	η1	η1	NOUN
ejpam-5364	207	9	=	=	SYM
ejpam-5364	207	10	d	d	PROPN
ejpam-5364	207	11	and	and	CCONJ
ejpam-5364	207	12	η2	η2	ADJ
ejpam-5364	207	13	=	=	SYM
ejpam-5364	207	14	d−	d−	PROPN
ejpam-5364	207	15	tn	tn	PROPN
ejpam-5364	207	16	(	(	PUNCT
ejpam-5364	207	17	n−1	n−1	PROPN
ejpam-5364	207	18	)	)	PUNCT
ejpam-5364	207	19	.	.	PUNCT
ejpam-5364	208	1	from	from	ADP
ejpam-5364	208	2	eigenvalue	eigenvalue	PROPN
ejpam-5364	208	3	interlacing	interlace	VERB
ejpam-5364	208	4	we	we	PRON
ejpam-5364	208	5	have	have	VERB
ejpam-5364	208	6	,	,	PUNCT
ejpam-5364	208	7	λ2	λ2	PROPN
ejpam-5364	208	8	≥	≥	NUM
ejpam-5364	208	9	d−	d−	PROPN
ejpam-5364	208	10	tn	tn	PROPN
ejpam-5364	208	11	(	(	PUNCT
ejpam-5364	208	12	n−	n−	NOUN
ejpam-5364	208	13	1	1	NUM
ejpam-5364	208	14	)	)	PUNCT
ejpam-5364	208	15	.	.	PUNCT
ejpam-5364	209	1	since	since	SCONJ
ejpam-5364	209	2	t	t	PROPN
ejpam-5364	209	3	=	=	SYM
ejpam-5364	209	4	d	d	PROPN
ejpam-5364	209	5	,	,	PUNCT
ejpam-5364	209	6	the	the	DET
ejpam-5364	209	7	above	above	ADJ
ejpam-5364	209	8	inequality	inequality	NOUN
ejpam-5364	209	9	can	can	AUX
ejpam-5364	209	10	be	be	AUX
ejpam-5364	209	11	rewritten	rewrite	VERB
ejpam-5364	209	12	as	as	ADP
ejpam-5364	209	13	λ2	λ2	NOUN
ejpam-5364	209	14	≥	≥	PROPN
ejpam-5364	209	15	d	d	NOUN
ejpam-5364	209	16	(	(	PUNCT
ejpam-5364	209	17	1−	1−	NUM
ejpam-5364	209	18	n	n	CCONJ
ejpam-5364	209	19	)	)	PUNCT
ejpam-5364	209	20	.	.	PUNCT
ejpam-5364	210	1	now	now	ADV
ejpam-5364	210	2	,	,	PUNCT
ejpam-5364	210	3	using	use	VERB
ejpam-5364	210	4	inequality	inequality	NOUN
ejpam-5364	210	5	(	(	PUNCT
ejpam-5364	210	6	1	1	NUM
ejpam-5364	210	7	)	)	PUNCT
ejpam-5364	210	8	,	,	PUNCT
ejpam-5364	210	9	we	we	PRON
ejpam-5364	210	10	get	get	VERB
ejpam-5364	210	11	the	the	DET
ejpam-5364	210	12	final	final	ADJ
ejpam-5364	210	13	result	result	NOUN
ejpam-5364	210	14	.	.	PUNCT
ejpam-5364	211	1	m.	m.	NOUN
ejpam-5364	211	2	machasri	machasri	PROPN
ejpam-5364	211	3	,	,	PUNCT
ejpam-5364	211	4	d.	d.	PROPN
ejpam-5364	211	5	kalyani	kalyani	PROPN
ejpam-5364	211	6	/	/	SYM
ejpam-5364	211	7	eur	eur	PROPN
ejpam-5364	211	8	.	.	PUNCT
ejpam-5364	212	1	j.	j.	PROPN
ejpam-5364	212	2	pure	pure	PROPN
ejpam-5364	212	3	appl	appl	PROPN
ejpam-5364	212	4	.	.	PROPN
ejpam-5364	212	5	math	math	PROPN
ejpam-5364	212	6	,	,	PUNCT
ejpam-5364	212	7	17	17	NUM
ejpam-5364	212	8	(	(	PUNCT
ejpam-5364	212	9	4	4	NUM
ejpam-5364	212	10	)	)	PUNCT
ejpam-5364	212	11	(	(	PUNCT
ejpam-5364	212	12	2024	2024	NUM
ejpam-5364	212	13	)	)	PUNCT
ejpam-5364	212	14	,	,	PUNCT
ejpam-5364	212	15	3004	3004	NUM
ejpam-5364	212	16	-	-	SYM
ejpam-5364	212	17	3021	3021	NUM
ejpam-5364	212	18	3012	3012	NUM
ejpam-5364	212	19	note	note	NOUN
ejpam-5364	212	20	1	1	NUM
ejpam-5364	212	21	.	.	PUNCT
ejpam-5364	213	1	if	if	SCONJ
ejpam-5364	213	2	g	g	PROPN
ejpam-5364	213	3	is	be	AUX
ejpam-5364	213	4	a	a	DET
ejpam-5364	213	5	d−regular	d−regular	NUM
ejpam-5364	213	6	graph	graph	NOUN
ejpam-5364	213	7	except	except	SCONJ
ejpam-5364	213	8	complete	complete	ADJ
ejpam-5364	213	9	split	split	NOUN
ejpam-5364	213	10	graph	graph	NOUN
ejpam-5364	213	11	and	and	CCONJ
ejpam-5364	213	12	complete	complete	ADJ
ejpam-5364	213	13	multipartite	multipartite	ADJ
ejpam-5364	213	14	graph	graph	NOUN
ejpam-5364	213	15	with	with	ADP
ejpam-5364	213	16	n	n	NUM
ejpam-5364	213	17	≥	≥	NOUN
ejpam-5364	213	18	3	3	NUM
ejpam-5364	213	19	then	then	ADV
ejpam-5364	213	20	sk(g	sk(g	NOUN
ejpam-5364	213	21	)	)	PUNCT
ejpam-5364	213	22	≤	≤	NOUN
ejpam-5364	214	1	3d	3d	NUM
ejpam-5364	214	2	λ2	λ2	NOUN
ejpam-5364	214	3	+	+	CCONJ
ejpam-5364	214	4	(	(	PUNCT
ejpam-5364	214	5	m+	m+	NOUN
ejpam-5364	214	6	3	3	NUM
ejpam-5364	214	7	)	)	PUNCT
ejpam-5364	214	8	.	.	PUNCT
ejpam-5364	215	1	theorem	theorem	NOUN
ejpam-5364	215	2	3	3	X
ejpam-5364	215	3	.	.	PUNCT
ejpam-5364	216	1	let	let	VERB
ejpam-5364	216	2	g	g	PRON
ejpam-5364	216	3	be	be	AUX
ejpam-5364	216	4	a	a	DET
ejpam-5364	216	5	connected	connected	ADJ
ejpam-5364	216	6	graph	graph	NOUN
ejpam-5364	216	7	with	with	ADP
ejpam-5364	216	8	skewness	skewness	NOUN
ejpam-5364	216	9	sk(g	sk(g	NOUN
ejpam-5364	216	10	)	)	PUNCT
ejpam-5364	216	11	.	.	PUNCT
ejpam-5364	217	1	then	then	ADV
ejpam-5364	217	2	q2	q2	PROPN
ejpam-5364	217	3	≥	≥	NUM
ejpam-5364	217	4	1	1	NUM
ejpam-5364	217	5	2	2	NUM
ejpam-5364	217	6	{	{	PUNCT
ejpam-5364	217	7	[	[	PUNCT
ejpam-5364	217	8	2δ	2δ	NUM
ejpam-5364	217	9	+	+	SYM
ejpam-5364	217	10	∆(1−	∆(1−	NOUN
ejpam-5364	217	11	3u	3u	NOUN
ejpam-5364	217	12	)	)	PUNCT
ejpam-5364	217	13	]	]	PUNCT
ejpam-5364	218	1	−	−	ADP
ejpam-5364	218	2	√	√	PROPN
ejpam-5364	218	3	[	[	PUNCT
ejpam-5364	218	4	∆(1−	∆(1−	PROPN
ejpam-5364	218	5	3u)−	3u)−	NUM
ejpam-5364	218	6	2δ	2δ	NOUN
ejpam-5364	218	7	]	]	SYM
ejpam-5364	218	8	2	2	NUM
ejpam-5364	218	9	+	+	CCONJ
ejpam-5364	218	10	24∆(∆−	24∆(∆−	NUM
ejpam-5364	218	11	δ)u	δ)u	NOUN
ejpam-5364	218	12	}	}	PUNCT
ejpam-5364	218	13	where	where	SCONJ
ejpam-5364	218	14	u	u	NOUN
ejpam-5364	218	15	=	=	NOUN
ejpam-5364	218	16	1	1	NUM
ejpam-5364	218	17	(	(	PUNCT
ejpam-5364	218	18	m−sk(g)+3	m−sk(g)+3	NOUN
ejpam-5364	218	19	)	)	PUNCT
ejpam-5364	218	20	.	.	PUNCT
ejpam-5364	219	1	proof	proof	NOUN
ejpam-5364	219	2	.	.	PUNCT
ejpam-5364	220	1	let	let	VERB
ejpam-5364	220	2	bq	bq	INTJ
ejpam-5364	220	3	be	be	AUX
ejpam-5364	220	4	the	the	DET
ejpam-5364	220	5	quotient	quotient	NOUN
ejpam-5364	220	6	matrix	matrix	NOUN
ejpam-5364	220	7	of	of	ADP
ejpam-5364	220	8	the	the	DET
ejpam-5364	220	9	signless	signless	ADJ
ejpam-5364	220	10	laplacian	laplacian	ADJ
ejpam-5364	220	11	matrix	matrix	NOUN
ejpam-5364	220	12	q	q	NOUN
ejpam-5364	220	13	of	of	ADP
ejpam-5364	220	14	g	g	NOUN
ejpam-5364	220	15	with	with	ADP
ejpam-5364	220	16	respect	respect	NOUN
ejpam-5364	220	17	to	to	ADP
ejpam-5364	220	18	the	the	DET
ejpam-5364	220	19	partitions	partition	NOUN
ejpam-5364	220	20	p1	p1	PROPN
ejpam-5364	220	21	and	and	CCONJ
ejpam-5364	220	22	p2	p2	PROPN
ejpam-5364	220	23	where	where	SCONJ
ejpam-5364	220	24	p1	p1	NOUN
ejpam-5364	220	25	consists	consist	VERB
ejpam-5364	220	26	of	of	ADP
ejpam-5364	220	27	one	one	NUM
ejpam-5364	220	28	vertex	vertex	NOUN
ejpam-5364	220	29	of	of	ADP
ejpam-5364	220	30	g	g	NOUN
ejpam-5364	220	31	with	with	ADP
ejpam-5364	220	32	maximum	maximum	ADJ
ejpam-5364	220	33	degree	degree	NOUN
ejpam-5364	220	34	∆.	∆.	NOUN
ejpam-5364	220	35	we	we	PRON
ejpam-5364	220	36	have	have	VERB
ejpam-5364	220	37	|p1|	|p1|	ADJ
ejpam-5364	220	38	=	=	SYM
ejpam-5364	220	39	n1	n1	PROPN
ejpam-5364	220	40	=	=	SYM
ejpam-5364	220	41	1	1	NUM
ejpam-5364	220	42	and	and	CCONJ
ejpam-5364	220	43	|p2|	|p2|	NOUN
ejpam-5364	220	44	=	=	SYM
ejpam-5364	220	45	n2	n2	PROPN
ejpam-5364	220	46	=	=	SYM
ejpam-5364	220	47	(	(	PUNCT
ejpam-5364	220	48	n−	n−	NOUN
ejpam-5364	220	49	1	1	NUM
ejpam-5364	220	50	)	)	PUNCT
ejpam-5364	220	51	.	.	PUNCT
ejpam-5364	221	1	then	then	ADV
ejpam-5364	221	2	bq	bq	INTJ
ejpam-5364	221	3	=	=	PUNCT
ejpam-5364	222	1	[	[	PUNCT
ejpam-5364	222	2	2d′1	2d′1	NUM
ejpam-5364	222	3	−	−	PROPN
ejpam-5364	222	4	t	t	PROPN
ejpam-5364	222	5	t	t	PROPN
ejpam-5364	222	6	t	t	PROPN
ejpam-5364	222	7	(	(	PUNCT
ejpam-5364	222	8	n−1	n−1	PROPN
ejpam-5364	222	9	)	)	PUNCT
ejpam-5364	222	10	2d′2	2d′2	PROPN
ejpam-5364	222	11	−	−	PROPN
ejpam-5364	222	12	t	t	PROPN
ejpam-5364	222	13	(	(	PUNCT
ejpam-5364	222	14	n−1	n−1	PROPN
ejpam-5364	222	15	)	)	PUNCT
ejpam-5364	222	16	]	]	PUNCT
ejpam-5364	222	17	where	where	SCONJ
ejpam-5364	222	18	d′1	d′1	NOUN
ejpam-5364	222	19	,	,	PUNCT
ejpam-5364	222	20	d	d	NOUN
ejpam-5364	222	21	′	′	NOUN
ejpam-5364	222	22	2	2	NUM
ejpam-5364	222	23	are	be	AUX
ejpam-5364	222	24	the	the	DET
ejpam-5364	222	25	average	average	ADJ
ejpam-5364	222	26	degrees	degree	NOUN
ejpam-5364	222	27	of	of	ADP
ejpam-5364	222	28	the	the	DET
ejpam-5364	222	29	two	two	NUM
ejpam-5364	222	30	partitions	partition	NOUN
ejpam-5364	222	31	p1	p1	NOUN
ejpam-5364	222	32	,	,	PUNCT
ejpam-5364	222	33	p2	p2	PROPN
ejpam-5364	222	34	and	and	CCONJ
ejpam-5364	222	35	t	t	NOUN
ejpam-5364	222	36	is	be	AUX
ejpam-5364	222	37	the	the	DET
ejpam-5364	222	38	number	number	NOUN
ejpam-5364	222	39	of	of	ADP
ejpam-5364	222	40	edges	edge	NOUN
ejpam-5364	222	41	between	between	ADP
ejpam-5364	222	42	p1	p1	NOUN
ejpam-5364	222	43	and	and	CCONJ
ejpam-5364	222	44	p2	p2	NOUN
ejpam-5364	222	45	.	.	PUNCT
ejpam-5364	223	1	the	the	DET
ejpam-5364	223	2	characteristic	characteristic	ADJ
ejpam-5364	223	3	equation	equation	NOUN
ejpam-5364	223	4	of	of	ADP
ejpam-5364	223	5	the	the	DET
ejpam-5364	223	6	above	above	ADJ
ejpam-5364	223	7	matrix	matrix	NOUN
ejpam-5364	223	8	is	be	AUX
ejpam-5364	223	9	given	give	VERB
ejpam-5364	223	10	by	by	ADP
ejpam-5364	223	11	ζ2	ζ2	NOUN
ejpam-5364	223	12	−	−	PROPN
ejpam-5364	223	13	(	(	PUNCT
ejpam-5364	223	14	2d	2d	NUM
ejpam-5364	223	15	′	′	NOUN
ejpam-5364	223	16	1	1	NUM
ejpam-5364	224	1	+	+	NUM
ejpam-5364	224	2	2d	2d	NUM
ejpam-5364	224	3	′	′	NOUN
ejpam-5364	224	4	2	2	NUM
ejpam-5364	224	5	−	−	PROPN
ejpam-5364	224	6	t−	t−	PROPN
ejpam-5364	224	7	t	t	PROPN
ejpam-5364	224	8	(	(	PUNCT
ejpam-5364	224	9	n−	n−	NOUN
ejpam-5364	224	10	1	1	NUM
ejpam-5364	224	11	)	)	PUNCT
ejpam-5364	224	12	)	)	PUNCT
ejpam-5364	225	1	ζ	ζ	NOUN
ejpam-5364	225	2	+	+	CCONJ
ejpam-5364	225	3	(	(	PUNCT
ejpam-5364	225	4	2d	2d	NUM
ejpam-5364	225	5	′	′	NOUN
ejpam-5364	225	6	1	1	NUM
ejpam-5364	225	7	−	−	PROPN
ejpam-5364	225	8	t	t	NOUN
ejpam-5364	225	9	)	)	PUNCT
ejpam-5364	225	10	(	(	PUNCT
ejpam-5364	225	11	2d	2d	NUM
ejpam-5364	225	12	′	′	NOUN
ejpam-5364	225	13	2	2	NUM
ejpam-5364	225	14	−	−	NOUN
ejpam-5364	225	15	t	t	NOUN
ejpam-5364	225	16	(	(	PUNCT
ejpam-5364	225	17	n−	n−	NOUN
ejpam-5364	225	18	1	1	NUM
ejpam-5364	225	19	)	)	PUNCT
ejpam-5364	225	20	)	)	PUNCT
ejpam-5364	226	1	−	−	PROPN
ejpam-5364	226	2	t2	t2	NOUN
ejpam-5364	226	3	(	(	PUNCT
ejpam-5364	226	4	n−	n−	NOUN
ejpam-5364	226	5	1	1	NUM
ejpam-5364	226	6	)	)	PUNCT
ejpam-5364	226	7	=	=	SYM
ejpam-5364	226	8	0	0	X
ejpam-5364	226	9	.	.	PUNCT
ejpam-5364	227	1	the	the	DET
ejpam-5364	227	2	roots	root	NOUN
ejpam-5364	227	3	of	of	ADP
ejpam-5364	227	4	the	the	DET
ejpam-5364	227	5	above	above	ADJ
ejpam-5364	227	6	characteristic	characteristic	ADJ
ejpam-5364	227	7	equation	equation	NOUN
ejpam-5364	227	8	are	be	AUX
ejpam-5364	227	9	ζ1	ζ1	NOUN
ejpam-5364	227	10	=	=	SYM
ejpam-5364	227	11	1	1	NUM
ejpam-5364	227	12	2	2	NUM
ejpam-5364	227	13	{	{	PUNCT
ejpam-5364	227	14	(	(	PUNCT
ejpam-5364	227	15	2d′1	2d′1	NUM
ejpam-5364	228	1	+	+	CCONJ
ejpam-5364	228	2	2d′2	2d′2	PROPN
ejpam-5364	228	3	−	−	PROPN
ejpam-5364	228	4	tn	tn	PROPN
ejpam-5364	228	5	(	(	PUNCT
ejpam-5364	228	6	n−1	n−1	PROPN
ejpam-5364	228	7	)	)	PUNCT
ejpam-5364	228	8	)	)	PUNCT
ejpam-5364	229	1	+	+	CCONJ
ejpam-5364	229	2	√	√	VERB
ejpam-5364	229	3	r	r	NOUN
ejpam-5364	229	4	}	}	PUNCT
ejpam-5364	229	5	and	and	CCONJ
ejpam-5364	229	6	ζ2	ζ2	NOUN
ejpam-5364	229	7	=	=	SYM
ejpam-5364	229	8	1	1	NUM
ejpam-5364	229	9	2	2	NUM
ejpam-5364	229	10	{	{	PUNCT
ejpam-5364	229	11	(	(	PUNCT
ejpam-5364	229	12	2d′1	2d′1	NUM
ejpam-5364	229	13	+	+	CCONJ
ejpam-5364	229	14	2d′2	2d′2	PROPN
ejpam-5364	229	15	−	−	PROPN
ejpam-5364	229	16	tn	tn	PROPN
ejpam-5364	229	17	(	(	PUNCT
ejpam-5364	229	18	n−1	n−1	PROPN
ejpam-5364	229	19	)	)	PUNCT
ejpam-5364	229	20	)	)	PUNCT
ejpam-5364	230	1	−	−	ADP
ejpam-5364	231	1	√	√	NUM
ejpam-5364	231	2	r	r	NOUN
ejpam-5364	231	3	}	}	PUNCT
ejpam-5364	231	4	where	where	SCONJ
ejpam-5364	231	5	r	r	NOUN
ejpam-5364	231	6	=	=	PUNCT
ejpam-5364	231	7	(	(	PUNCT
ejpam-5364	231	8	2d	2d	NUM
ejpam-5364	231	9	′	′	NOUN
ejpam-5364	231	10	1	1	NUM
ejpam-5364	231	11	−	−	PROPN
ejpam-5364	231	12	2d	2d	NOUN
ejpam-5364	231	13	′	′	NOUN
ejpam-5364	231	14	2	2	NUM
ejpam-5364	231	15	−	−	NOUN
ejpam-5364	231	16	t+	t+	PUNCT
ejpam-5364	231	17	t	t	PROPN
ejpam-5364	231	18	(	(	PUNCT
ejpam-5364	231	19	n−	n−	NOUN
ejpam-5364	231	20	1	1	NUM
ejpam-5364	231	21	)	)	PUNCT
ejpam-5364	231	22	)	)	PUNCT
ejpam-5364	231	23	2	2	NUM
ejpam-5364	232	1	+	+	NUM
ejpam-5364	232	2	4t2	4t2	NUM
ejpam-5364	232	3	(	(	PUNCT
ejpam-5364	232	4	n−	n−	NOUN
ejpam-5364	232	5	1	1	NUM
ejpam-5364	232	6	)	)	PUNCT
ejpam-5364	232	7	=	=	SYM
ejpam-5364	232	8	(	(	PUNCT
ejpam-5364	232	9	2d	2d	NUM
ejpam-5364	232	10	′	′	NOUN
ejpam-5364	232	11	1	1	NUM
ejpam-5364	232	12	−	−	PROPN
ejpam-5364	232	13	2d	2d	NOUN
ejpam-5364	232	14	′	′	NOUN
ejpam-5364	232	15	2	2	NUM
ejpam-5364	232	16	−	−	PROPN
ejpam-5364	232	17	t−	t−	PROPN
ejpam-5364	232	18	t	t	PROPN
ejpam-5364	232	19	(	(	PUNCT
ejpam-5364	232	20	n−	n−	NOUN
ejpam-5364	232	21	1	1	NUM
ejpam-5364	232	22	)	)	PUNCT
ejpam-5364	232	23	)	)	PUNCT
ejpam-5364	232	24	2	2	NUM
ejpam-5364	233	1	+	+	NUM
ejpam-5364	233	2	8t(d	8t(d	NOUN
ejpam-5364	233	3	′	′	NOUN
ejpam-5364	233	4	1	1	NUM
ejpam-5364	233	5	−	−	NOUN
ejpam-5364	234	1	d	d	NOUN
ejpam-5364	234	2	′	′	NOUN
ejpam-5364	234	3	2	2	NUM
ejpam-5364	234	4	)	)	PUNCT
ejpam-5364	234	5	(	(	PUNCT
ejpam-5364	234	6	n−	n−	NOUN
ejpam-5364	234	7	1	1	NUM
ejpam-5364	234	8	)	)	PUNCT
ejpam-5364	234	9	.	.	PUNCT
ejpam-5364	235	1	therefore	therefore	ADV
ejpam-5364	235	2	,	,	PUNCT
ejpam-5364	235	3	we	we	PRON
ejpam-5364	235	4	have	have	VERB
ejpam-5364	235	5	ζ2	ζ2	NOUN
ejpam-5364	235	6	=	=	SYM
ejpam-5364	235	7	1	1	NUM
ejpam-5364	235	8	2	2	NUM
ejpam-5364	235	9	{	{	PUNCT
ejpam-5364	235	10	(	(	PUNCT
ejpam-5364	235	11	2d	2d	NUM
ejpam-5364	235	12	′	′	NOUN
ejpam-5364	235	13	1	1	NUM
ejpam-5364	235	14	+	+	NUM
ejpam-5364	235	15	2d	2d	NUM
ejpam-5364	235	16	′	′	NOUN
ejpam-5364	235	17	2	2	NUM
ejpam-5364	235	18	−	−	PROPN
ejpam-5364	235	19	tn	tn	NOUN
ejpam-5364	235	20	(	(	PUNCT
ejpam-5364	235	21	n−	n−	NOUN
ejpam-5364	235	22	1	1	NUM
ejpam-5364	235	23	)	)	PUNCT
ejpam-5364	235	24	)	)	PUNCT
ejpam-5364	236	1	−	−	ADP
ejpam-5364	236	2	√	√	PROPN
ejpam-5364	236	3	(	(	PUNCT
ejpam-5364	236	4	2d	2d	NUM
ejpam-5364	236	5	′	′	NOUN
ejpam-5364	236	6	1	1	NUM
ejpam-5364	236	7	−	−	PROPN
ejpam-5364	236	8	2d	2d	NOUN
ejpam-5364	236	9	′	′	NOUN
ejpam-5364	236	10	2	2	NUM
ejpam-5364	236	11	−	−	PROPN
ejpam-5364	236	12	tn	tn	NOUN
ejpam-5364	236	13	(	(	PUNCT
ejpam-5364	236	14	n−	n−	NOUN
ejpam-5364	236	15	1	1	NUM
ejpam-5364	236	16	)	)	PUNCT
ejpam-5364	236	17	)	)	PUNCT
ejpam-5364	236	18	2	2	NUM
ejpam-5364	236	19	+	+	NUM
ejpam-5364	236	20	8t(d	8t(d	NOUN
ejpam-5364	236	21	′	′	NOUN
ejpam-5364	236	22	1	1	NUM
ejpam-5364	236	23	−	−	NOUN
ejpam-5364	236	24	d	d	NOUN
ejpam-5364	236	25	′	′	NOUN
ejpam-5364	236	26	2	2	NUM
ejpam-5364	236	27	)	)	PUNCT
ejpam-5364	236	28	(	(	PUNCT
ejpam-5364	236	29	n−	n−	NOUN
ejpam-5364	236	30	1	1	NUM
ejpam-5364	236	31	)	)	PUNCT
ejpam-5364	236	32	}	}	PUNCT
ejpam-5364	236	33	.	.	PUNCT
ejpam-5364	237	1	by	by	ADP
ejpam-5364	237	2	eigenvalue	eigenvalue	PROPN
ejpam-5364	237	3	interlacing	interlacing	NOUN
ejpam-5364	237	4	,	,	PUNCT
ejpam-5364	237	5	we	we	PRON
ejpam-5364	237	6	have	have	VERB
ejpam-5364	237	7	q2	q2	NOUN
ejpam-5364	237	8	≥	≥	NUM
ejpam-5364	237	9	1	1	NUM
ejpam-5364	237	10	2	2	NUM
ejpam-5364	237	11	{	{	PUNCT
ejpam-5364	237	12	(	(	PUNCT
ejpam-5364	237	13	2d	2d	NUM
ejpam-5364	237	14	′	′	NOUN
ejpam-5364	237	15	1	1	NUM
ejpam-5364	238	1	+	+	NUM
ejpam-5364	238	2	2d	2d	NUM
ejpam-5364	238	3	′	′	NOUN
ejpam-5364	238	4	2	2	NUM
ejpam-5364	238	5	−	−	PROPN
ejpam-5364	238	6	tn	tn	NOUN
ejpam-5364	238	7	(	(	PUNCT
ejpam-5364	238	8	n−	n−	NOUN
ejpam-5364	238	9	1	1	NUM
ejpam-5364	238	10	)	)	PUNCT
ejpam-5364	238	11	)	)	PUNCT
ejpam-5364	239	1	−	−	ADP
ejpam-5364	239	2	√	√	PROPN
ejpam-5364	239	3	(	(	PUNCT
ejpam-5364	239	4	2d	2d	NUM
ejpam-5364	239	5	′	′	NOUN
ejpam-5364	239	6	1	1	NUM
ejpam-5364	239	7	−	−	PROPN
ejpam-5364	239	8	2d	2d	NOUN
ejpam-5364	239	9	′	′	NOUN
ejpam-5364	239	10	2	2	NUM
ejpam-5364	239	11	−	−	PROPN
ejpam-5364	239	12	tn	tn	NOUN
ejpam-5364	239	13	(	(	PUNCT
ejpam-5364	239	14	n−	n−	NOUN
ejpam-5364	239	15	1	1	NUM
ejpam-5364	239	16	)	)	PUNCT
ejpam-5364	239	17	)	)	PUNCT
ejpam-5364	239	18	2	2	NUM
ejpam-5364	239	19	+	+	NUM
ejpam-5364	239	20	8t(d	8t(d	NOUN
ejpam-5364	239	21	′	′	NOUN
ejpam-5364	239	22	1	1	NUM
ejpam-5364	239	23	−	−	NOUN
ejpam-5364	239	24	d	d	NOUN
ejpam-5364	239	25	′	′	NOUN
ejpam-5364	239	26	2	2	NUM
ejpam-5364	239	27	)	)	PUNCT
ejpam-5364	239	28	(	(	PUNCT
ejpam-5364	239	29	n−	n−	NOUN
ejpam-5364	239	30	1	1	NUM
ejpam-5364	239	31	)	)	PUNCT
ejpam-5364	239	32	}	}	PUNCT
ejpam-5364	239	33	.	.	PUNCT
ejpam-5364	240	1	m.	m.	PROPN
ejpam-5364	240	2	machasri	machasri	PROPN
ejpam-5364	240	3	,	,	PUNCT
ejpam-5364	240	4	d.	d.	PROPN
ejpam-5364	240	5	kalyani	kalyani	PROPN
ejpam-5364	240	6	/	/	SYM
ejpam-5364	240	7	eur	eur	PROPN
ejpam-5364	240	8	.	.	PUNCT
ejpam-5364	241	1	j.	j.	PROPN
ejpam-5364	241	2	pure	pure	PROPN
ejpam-5364	241	3	appl	appl	PROPN
ejpam-5364	241	4	.	.	PROPN
ejpam-5364	241	5	math	math	PROPN
ejpam-5364	241	6	,	,	PUNCT
ejpam-5364	241	7	17	17	NUM
ejpam-5364	241	8	(	(	PUNCT
ejpam-5364	241	9	4	4	NUM
ejpam-5364	241	10	)	)	PUNCT
ejpam-5364	241	11	(	(	PUNCT
ejpam-5364	241	12	2024	2024	NUM
ejpam-5364	241	13	)	)	PUNCT
ejpam-5364	241	14	,	,	PUNCT
ejpam-5364	241	15	3004	3004	NUM
ejpam-5364	241	16	-	-	SYM
ejpam-5364	241	17	3021	3021	NUM
ejpam-5364	241	18	3013	3013	NUM
ejpam-5364	241	19	since	since	SCONJ
ejpam-5364	241	20	d′1	d′1	NOUN
ejpam-5364	241	21	=	=	SYM
ejpam-5364	241	22	∆	∆	PROPN
ejpam-5364	241	23	and	and	CCONJ
ejpam-5364	242	1	d′2	d′2	ADJ
ejpam-5364	242	2	≥	≥	NOUN
ejpam-5364	242	3	δ	δ	NOUN
ejpam-5364	242	4	we	we	PRON
ejpam-5364	242	5	get	get	VERB
ejpam-5364	242	6	,	,	PUNCT
ejpam-5364	242	7	q2	q2	X
ejpam-5364	242	8	≥	≥	NUM
ejpam-5364	242	9	1	1	NUM
ejpam-5364	242	10	2	2	NUM
ejpam-5364	242	11	{	{	PUNCT
ejpam-5364	242	12	(	(	PUNCT
ejpam-5364	242	13	2∆	2∆	NUM
ejpam-5364	242	14	+	+	NUM
ejpam-5364	242	15	2δ	2δ	NUM
ejpam-5364	242	16	−	−	PROPN
ejpam-5364	242	17	tn	tn	NOUN
ejpam-5364	242	18	(	(	PUNCT
ejpam-5364	242	19	n−	n−	NOUN
ejpam-5364	242	20	1	1	NUM
ejpam-5364	242	21	)	)	PUNCT
ejpam-5364	242	22	)	)	PUNCT
ejpam-5364	243	1	−	−	ADP
ejpam-5364	243	2	√	√	NUM
ejpam-5364	243	3	(	(	PUNCT
ejpam-5364	243	4	2∆−	2∆−	NUM
ejpam-5364	243	5	2δ	2δ	NUM
ejpam-5364	243	6	−	−	PROPN
ejpam-5364	243	7	tn	tn	PROPN
ejpam-5364	243	8	(	(	PUNCT
ejpam-5364	243	9	n−	n−	NOUN
ejpam-5364	243	10	1	1	NUM
ejpam-5364	243	11	)	)	PUNCT
ejpam-5364	243	12	)	)	PUNCT
ejpam-5364	243	13	2	2	NUM
ejpam-5364	243	14	+	+	SYM
ejpam-5364	243	15	8t(∆−	8t(∆−	NUM
ejpam-5364	243	16	δ	δ	NOUN
ejpam-5364	243	17	)	)	PUNCT
ejpam-5364	243	18	(	(	PUNCT
ejpam-5364	243	19	n−	n−	NOUN
ejpam-5364	243	20	1	1	NUM
ejpam-5364	243	21	)	)	PUNCT
ejpam-5364	243	22	}	}	PUNCT
ejpam-5364	243	23	.	.	PUNCT
ejpam-5364	244	1	the	the	DET
ejpam-5364	244	2	above	above	ADJ
ejpam-5364	244	3	inequality	inequality	NOUN
ejpam-5364	244	4	can	can	AUX
ejpam-5364	244	5	be	be	AUX
ejpam-5364	244	6	rewritten	rewrite	VERB
ejpam-5364	244	7	as	as	ADP
ejpam-5364	244	8	q2	q2	NOUN
ejpam-5364	244	9	≥	≥	NUM
ejpam-5364	244	10	1	1	NUM
ejpam-5364	244	11	2	2	NUM
ejpam-5364	244	12	{	{	PUNCT
ejpam-5364	244	13	[	[	PUNCT
ejpam-5364	244	14	2∆	2∆	NUM
ejpam-5364	244	15	+	+	NUM
ejpam-5364	244	16	2δ	2δ	NUM
ejpam-5364	244	17	−	−	PROPN
ejpam-5364	244	18	t	t	PROPN
ejpam-5364	244	19	(	(	PUNCT
ejpam-5364	244	20	1	1	NUM
ejpam-5364	244	21	+	+	NUM
ejpam-5364	244	22	1	1	NUM
ejpam-5364	244	23	(	(	PUNCT
ejpam-5364	244	24	n−	n−	NOUN
ejpam-5364	244	25	1	1	NUM
ejpam-5364	244	26	)	)	PUNCT
ejpam-5364	244	27	)	)	PUNCT
ejpam-5364	244	28	]	]	PUNCT
ejpam-5364	245	1	−	−	PUNCT
ejpam-5364	245	2	√	√	NUM
ejpam-5364	245	3	[	[	PUNCT
ejpam-5364	245	4	2∆−	2∆−	NUM
ejpam-5364	245	5	2δ	2δ	NUM
ejpam-5364	245	6	−	−	PROPN
ejpam-5364	245	7	t	t	PROPN
ejpam-5364	245	8	(	(	PUNCT
ejpam-5364	245	9	1	1	NUM
ejpam-5364	245	10	+	+	NUM
ejpam-5364	245	11	1	1	NUM
ejpam-5364	245	12	(	(	PUNCT
ejpam-5364	245	13	n−	n−	NOUN
ejpam-5364	245	14	1	1	NUM
ejpam-5364	245	15	)	)	PUNCT
ejpam-5364	245	16	)	)	PUNCT
ejpam-5364	246	1	]	]	PUNCT
ejpam-5364	246	2	2	2	NUM
ejpam-5364	246	3	+	+	SYM
ejpam-5364	246	4	8t(∆−	8t(∆−	NUM
ejpam-5364	246	5	δ	δ	NOUN
ejpam-5364	246	6	)	)	PUNCT
ejpam-5364	246	7	(	(	PUNCT
ejpam-5364	246	8	n−	n−	NOUN
ejpam-5364	246	9	1	1	NUM
ejpam-5364	246	10	)	)	PUNCT
ejpam-5364	246	11	}	}	PUNCT
ejpam-5364	246	12	.	.	PUNCT
ejpam-5364	247	1	since	since	SCONJ
ejpam-5364	247	2	t	t	NOUN
ejpam-5364	247	3	=	=	SYM
ejpam-5364	247	4	∆	∆	PROPN
ejpam-5364	247	5	,	,	PUNCT
ejpam-5364	247	6	the	the	DET
ejpam-5364	247	7	above	above	ADJ
ejpam-5364	247	8	inequality	inequality	NOUN
ejpam-5364	247	9	becomes	become	VERB
ejpam-5364	247	10	q2	q2	NOUN
ejpam-5364	247	11	≥	≥	NUM
ejpam-5364	247	12	1	1	NUM
ejpam-5364	247	13	2	2	NUM
ejpam-5364	247	14	{	{	PUNCT
ejpam-5364	247	15	(	(	PUNCT
ejpam-5364	247	16	∆+	∆+	NUM
ejpam-5364	247	17	2δ	2δ	NOUN
ejpam-5364	247	18	−	−	PROPN
ejpam-5364	247	19	∆	∆	PROPN
ejpam-5364	247	20	(	(	PUNCT
ejpam-5364	247	21	n−	n−	NOUN
ejpam-5364	247	22	1	1	NUM
ejpam-5364	247	23	)	)	PUNCT
ejpam-5364	247	24	)	)	PUNCT
ejpam-5364	248	1	−	−	ADP
ejpam-5364	248	2	√	√	NUM
ejpam-5364	248	3	(	(	PUNCT
ejpam-5364	248	4	∆−	∆−	VERB
ejpam-5364	248	5	2δ	2δ	NOUN
ejpam-5364	248	6	−	−	PROPN
ejpam-5364	248	7	∆	∆	PROPN
ejpam-5364	248	8	(	(	PUNCT
ejpam-5364	248	9	n−	n−	NOUN
ejpam-5364	248	10	1	1	NUM
ejpam-5364	248	11	)	)	PUNCT
ejpam-5364	248	12	)	)	PUNCT
ejpam-5364	248	13	2	2	NUM
ejpam-5364	249	1	+	+	NUM
ejpam-5364	249	2	8∆(∆−	8∆(∆−	NUM
ejpam-5364	249	3	δ	δ	NOUN
ejpam-5364	249	4	)	)	PUNCT
ejpam-5364	249	5	(	(	PUNCT
ejpam-5364	249	6	n−	n−	NOUN
ejpam-5364	249	7	1	1	NUM
ejpam-5364	249	8	)	)	PUNCT
ejpam-5364	249	9	}	}	PUNCT
ejpam-5364	249	10	.	.	PUNCT
ejpam-5364	250	1	we	we	PRON
ejpam-5364	250	2	obtain	obtain	VERB
ejpam-5364	250	3	the	the	DET
ejpam-5364	250	4	result	result	NOUN
ejpam-5364	250	5	by	by	ADP
ejpam-5364	250	6	making	make	VERB
ejpam-5364	250	7	use	use	NOUN
ejpam-5364	250	8	of	of	ADP
ejpam-5364	250	9	inequality	inequality	NOUN
ejpam-5364	250	10	(	(	PUNCT
ejpam-5364	250	11	1	1	NUM
ejpam-5364	250	12	)	)	PUNCT
ejpam-5364	250	13	.	.	PUNCT
ejpam-5364	251	1	theorem	theorem	ADJ
ejpam-5364	251	2	4	4	NUM
ejpam-5364	251	3	.	.	PUNCT
ejpam-5364	252	1	let	let	VERB
ejpam-5364	252	2	g	g	PRON
ejpam-5364	252	3	be	be	AUX
ejpam-5364	252	4	a	a	DET
ejpam-5364	252	5	d−regular	d−regular	NUM
ejpam-5364	252	6	graph	graph	NOUN
ejpam-5364	252	7	with	with	ADP
ejpam-5364	252	8	n	n	PRON
ejpam-5364	252	9	≥	≥	NUM
ejpam-5364	252	10	3	3	NUM
ejpam-5364	252	11	.	.	PUNCT
ejpam-5364	253	1	then	then	ADV
ejpam-5364	253	2	q2	q2	PROPN
ejpam-5364	253	3	≥	≥	PROPN
ejpam-5364	253	4	d	d	PROPN
ejpam-5364	253	5	(	(	PUNCT
ejpam-5364	253	6	1−	1−	NUM
ejpam-5364	253	7	3	3	NUM
ejpam-5364	253	8	m−	m−	PROPN
ejpam-5364	253	9	sk(g	sk(g	NOUN
ejpam-5364	253	10	)	)	PUNCT
ejpam-5364	253	11	+	+	CCONJ
ejpam-5364	253	12	3	3	X
ejpam-5364	253	13	)	)	PUNCT
ejpam-5364	253	14	.	.	PUNCT
ejpam-5364	254	1	proof	proof	NOUN
ejpam-5364	254	2	.	.	PUNCT
ejpam-5364	255	1	let	let	VERB
ejpam-5364	255	2	bq	bq	INTJ
ejpam-5364	255	3	be	be	AUX
ejpam-5364	255	4	the	the	DET
ejpam-5364	255	5	quotient	quotient	NOUN
ejpam-5364	255	6	matrix	matrix	NOUN
ejpam-5364	255	7	of	of	ADP
ejpam-5364	255	8	the	the	DET
ejpam-5364	255	9	signless	signless	ADJ
ejpam-5364	255	10	laplacian	laplacian	ADJ
ejpam-5364	255	11	matrix	matrix	NOUN
ejpam-5364	255	12	q	q	NOUN
ejpam-5364	255	13	of	of	ADP
ejpam-5364	255	14	g	g	NOUN
ejpam-5364	255	15	with	with	ADP
ejpam-5364	255	16	respect	respect	NOUN
ejpam-5364	255	17	to	to	ADP
ejpam-5364	255	18	the	the	DET
ejpam-5364	255	19	partitions	partition	NOUN
ejpam-5364	255	20	p1	p1	PROPN
ejpam-5364	255	21	and	and	CCONJ
ejpam-5364	255	22	p2	p2	NOUN
ejpam-5364	255	23	.	.	PUNCT
ejpam-5364	256	1	let	let	VERB
ejpam-5364	256	2	|p1|	|p1|	ADV
ejpam-5364	256	3	=	=	SYM
ejpam-5364	256	4	n1	n1	PROPN
ejpam-5364	256	5	=	=	SYM
ejpam-5364	256	6	1	1	NUM
ejpam-5364	256	7	and	and	CCONJ
ejpam-5364	256	8	|p2|	|p2|	NOUN
ejpam-5364	256	9	=	=	SYM
ejpam-5364	256	10	n2	n2	PROPN
ejpam-5364	256	11	=	=	SYM
ejpam-5364	256	12	(	(	PUNCT
ejpam-5364	256	13	n−	n−	NOUN
ejpam-5364	256	14	1	1	NUM
ejpam-5364	256	15	)	)	PUNCT
ejpam-5364	256	16	.	.	PUNCT
ejpam-5364	257	1	then	then	ADV
ejpam-5364	257	2	bq	bq	INTJ
ejpam-5364	257	3	=	=	PUNCT
ejpam-5364	258	1	[	[	PUNCT
ejpam-5364	258	2	2d−	2d−	PROPN
ejpam-5364	258	3	t	t	PROPN
ejpam-5364	258	4	t	t	PROPN
ejpam-5364	258	5	t	t	PROPN
ejpam-5364	258	6	(	(	PUNCT
ejpam-5364	258	7	n−1	n−1	PROPN
ejpam-5364	258	8	)	)	PUNCT
ejpam-5364	258	9	2d−	2d−	PROPN
ejpam-5364	258	10	t	t	PROPN
ejpam-5364	258	11	(	(	PUNCT
ejpam-5364	258	12	n−1	n−1	PROPN
ejpam-5364	258	13	)	)	PUNCT
ejpam-5364	258	14	]	]	PUNCT
ejpam-5364	258	15	where	where	SCONJ
ejpam-5364	258	16	t	t	PROPN
ejpam-5364	258	17	is	be	AUX
ejpam-5364	258	18	the	the	DET
ejpam-5364	258	19	number	number	NOUN
ejpam-5364	258	20	of	of	ADP
ejpam-5364	258	21	edges	edge	NOUN
ejpam-5364	258	22	between	between	ADP
ejpam-5364	258	23	p1	p1	NOUN
ejpam-5364	258	24	and	and	CCONJ
ejpam-5364	258	25	p2	p2	NOUN
ejpam-5364	258	26	.	.	PUNCT
ejpam-5364	259	1	the	the	DET
ejpam-5364	259	2	characteristic	characteristic	ADJ
ejpam-5364	259	3	equation	equation	NOUN
ejpam-5364	259	4	of	of	ADP
ejpam-5364	259	5	the	the	DET
ejpam-5364	259	6	above	above	ADJ
ejpam-5364	259	7	matrix	matrix	NOUN
ejpam-5364	259	8	is	be	AUX
ejpam-5364	259	9	given	give	VERB
ejpam-5364	259	10	by	by	ADP
ejpam-5364	259	11	[	[	PUNCT
ejpam-5364	259	12	ζ	ζ	NOUN
ejpam-5364	259	13	−	−	PROPN
ejpam-5364	259	14	(	(	PUNCT
ejpam-5364	259	15	2d−	2d−	PROPN
ejpam-5364	259	16	t	t	PROPN
ejpam-5364	259	17	)	)	PUNCT
ejpam-5364	259	18	]	]	X
ejpam-5364	259	19	[	[	PUNCT
ejpam-5364	259	20	ζ	ζ	NOUN
ejpam-5364	259	21	−	−	PROPN
ejpam-5364	259	22	(	(	PUNCT
ejpam-5364	259	23	2d−	2d−	PROPN
ejpam-5364	259	24	t	t	PROPN
ejpam-5364	259	25	(	(	PUNCT
ejpam-5364	259	26	n−	n−	NOUN
ejpam-5364	259	27	1	1	NUM
ejpam-5364	259	28	)	)	PUNCT
ejpam-5364	259	29	)	)	PUNCT
ejpam-5364	259	30	]	]	PUNCT
ejpam-5364	260	1	−	−	PROPN
ejpam-5364	260	2	t2	t2	NOUN
ejpam-5364	260	3	(	(	PUNCT
ejpam-5364	260	4	n−	n−	NOUN
ejpam-5364	260	5	1	1	NUM
ejpam-5364	260	6	)	)	PUNCT
ejpam-5364	260	7	=	=	SYM
ejpam-5364	260	8	0	0	X
ejpam-5364	260	9	.	.	PUNCT
ejpam-5364	261	1	the	the	DET
ejpam-5364	261	2	roots	root	NOUN
ejpam-5364	261	3	of	of	ADP
ejpam-5364	261	4	the	the	DET
ejpam-5364	261	5	above	above	ADJ
ejpam-5364	261	6	equation	equation	NOUN
ejpam-5364	261	7	are	be	AUX
ejpam-5364	261	8	ζ1	ζ1	NOUN
ejpam-5364	261	9	=	=	SYM
ejpam-5364	261	10	2d	2d	NOUN
ejpam-5364	261	11	and	and	CCONJ
ejpam-5364	261	12	ζ2	ζ2	NOUN
ejpam-5364	261	13	=	=	SYM
ejpam-5364	261	14	d−	d−	PROPN
ejpam-5364	261	15	d	d	PROPN
ejpam-5364	261	16	(	(	PUNCT
ejpam-5364	261	17	n−1	n−1	PROPN
ejpam-5364	261	18	)	)	PUNCT
ejpam-5364	261	19	.	.	PUNCT
ejpam-5364	262	1	from	from	ADP
ejpam-5364	262	2	eigenvalue	eigenvalue	PROPN
ejpam-5364	262	3	interlacing	interlace	VERB
ejpam-5364	262	4	we	we	PRON
ejpam-5364	262	5	have	have	VERB
ejpam-5364	262	6	,	,	PUNCT
ejpam-5364	262	7	q2	q2	PROPN
ejpam-5364	262	8	≥	≥	NOUN
ejpam-5364	262	9	d−	d−	PROPN
ejpam-5364	262	10	d	d	PROPN
ejpam-5364	262	11	(	(	PUNCT
ejpam-5364	262	12	n−	n−	NOUN
ejpam-5364	262	13	1	1	NUM
ejpam-5364	262	14	)	)	PUNCT
ejpam-5364	262	15	.	.	PUNCT
ejpam-5364	263	1	substituting	substitute	VERB
ejpam-5364	263	2	t	t	PROPN
ejpam-5364	264	1	=	=	SYM
ejpam-5364	264	2	d	d	NOUN
ejpam-5364	264	3	in	in	ADP
ejpam-5364	264	4	the	the	DET
ejpam-5364	264	5	above	above	ADJ
ejpam-5364	264	6	inequality	inequality	NOUN
ejpam-5364	264	7	we	we	PRON
ejpam-5364	264	8	have	have	VERB
ejpam-5364	264	9	,	,	PUNCT
ejpam-5364	264	10	q2	q2	PROPN
ejpam-5364	264	11	≥	≥	NOUN
ejpam-5364	264	12	d−	d−	PROPN
ejpam-5364	264	13	d	d	PROPN
ejpam-5364	264	14	(	(	PUNCT
ejpam-5364	264	15	n−	n−	NOUN
ejpam-5364	264	16	1	1	NUM
ejpam-5364	264	17	)	)	PUNCT
ejpam-5364	264	18	.	.	PUNCT
ejpam-5364	265	1	the	the	DET
ejpam-5364	265	2	result	result	NOUN
ejpam-5364	265	3	is	be	AUX
ejpam-5364	265	4	obtained	obtain	VERB
ejpam-5364	265	5	by	by	ADP
ejpam-5364	265	6	applying	apply	VERB
ejpam-5364	265	7	inequality	inequality	NOUN
ejpam-5364	265	8	(	(	PUNCT
ejpam-5364	265	9	1	1	NUM
ejpam-5364	265	10	)	)	PUNCT
ejpam-5364	265	11	in	in	ADP
ejpam-5364	265	12	the	the	DET
ejpam-5364	265	13	above	above	ADJ
ejpam-5364	265	14	inequality	inequality	NOUN
ejpam-5364	265	15	.	.	PUNCT
ejpam-5364	266	1	m.	m.	PROPN
ejpam-5364	266	2	machasri	machasri	PROPN
ejpam-5364	266	3	,	,	PUNCT
ejpam-5364	266	4	d.	d.	PROPN
ejpam-5364	266	5	kalyani	kalyani	PROPN
ejpam-5364	266	6	/	/	SYM
ejpam-5364	266	7	eur	eur	PROPN
ejpam-5364	266	8	.	.	PUNCT
ejpam-5364	267	1	j.	j.	PROPN
ejpam-5364	267	2	pure	pure	PROPN
ejpam-5364	267	3	appl	appl	PROPN
ejpam-5364	267	4	.	.	PROPN
ejpam-5364	267	5	math	math	PROPN
ejpam-5364	267	6	,	,	PUNCT
ejpam-5364	267	7	17	17	NUM
ejpam-5364	267	8	(	(	PUNCT
ejpam-5364	267	9	4	4	NUM
ejpam-5364	267	10	)	)	PUNCT
ejpam-5364	267	11	(	(	PUNCT
ejpam-5364	267	12	2024	2024	NUM
ejpam-5364	267	13	)	)	PUNCT
ejpam-5364	267	14	,	,	PUNCT
ejpam-5364	267	15	3004	3004	NUM
ejpam-5364	267	16	-	-	SYM
ejpam-5364	267	17	3021	3021	NUM
ejpam-5364	267	18	3014	3014	NUM
ejpam-5364	267	19	note	note	NOUN
ejpam-5364	267	20	2	2	NUM
ejpam-5364	267	21	.	.	PUNCT
ejpam-5364	268	1	if	if	SCONJ
ejpam-5364	268	2	g	g	PROPN
ejpam-5364	268	3	is	be	AUX
ejpam-5364	268	4	a	a	DET
ejpam-5364	268	5	d−regular	d−regular	NUM
ejpam-5364	268	6	graph	graph	NOUN
ejpam-5364	268	7	except	except	SCONJ
ejpam-5364	268	8	complete	complete	ADJ
ejpam-5364	268	9	split	split	NOUN
ejpam-5364	268	10	graph	graph	NOUN
ejpam-5364	268	11	and	and	CCONJ
ejpam-5364	268	12	complete	complete	ADJ
ejpam-5364	268	13	multipartite	multipartite	ADJ
ejpam-5364	268	14	graph	graph	NOUN
ejpam-5364	268	15	then	then	ADV
ejpam-5364	268	16	sk(g	sk(g	NOUN
ejpam-5364	268	17	)	)	PUNCT
ejpam-5364	268	18	≤	≤	NUM
ejpam-5364	268	19	m−	m−	PROPN
ejpam-5364	268	20	3q2	3q2	NUM
ejpam-5364	268	21	(	(	PUNCT
ejpam-5364	268	22	d−	d−	PROPN
ejpam-5364	268	23	q2	q2	NOUN
ejpam-5364	268	24	)	)	PUNCT
ejpam-5364	268	25	.	.	PUNCT
ejpam-5364	269	1	example	example	NOUN
ejpam-5364	270	1	1	1	NUM
ejpam-5364	270	2	.	.	PUNCT
ejpam-5364	271	1	the	the	DET
ejpam-5364	271	2	graph	graph	NOUN
ejpam-5364	271	3	illustrated	illustrate	VERB
ejpam-5364	271	4	in	in	ADP
ejpam-5364	271	5	figure	figure	NOUN
ejpam-5364	271	6	5	5	NUM
ejpam-5364	271	7	has	have	VERB
ejpam-5364	271	8	8	8	NUM
ejpam-5364	271	9	vertices	vertex	NOUN
ejpam-5364	271	10	,	,	PUNCT
ejpam-5364	271	11	20	20	NUM
ejpam-5364	271	12	edges	edge	NOUN
ejpam-5364	271	13	with	with	ADP
ejpam-5364	271	14	∆	∆	PROPN
ejpam-5364	271	15	=	=	SYM
ejpam-5364	271	16	7	7	NUM
ejpam-5364	271	17	,	,	PUNCT
ejpam-5364	271	18	δ	δ	X
ejpam-5364	271	19	=	=	SYM
ejpam-5364	271	20	3	3	NUM
ejpam-5364	271	21	and	and	CCONJ
ejpam-5364	271	22	λ2	λ2	NOUN
ejpam-5364	271	23	=	=	NOUN
ejpam-5364	271	24	0.828	0.828	NUM
ejpam-5364	271	25	.	.	PUNCT
ejpam-5364	272	1	its	its	PRON
ejpam-5364	272	2	skewness	skewness	NOUN
ejpam-5364	272	3	is	be	AUX
ejpam-5364	272	4	given	give	VERB
ejpam-5364	272	5	by	by	ADP
ejpam-5364	272	6	sk(g	sk(g	NOUN
ejpam-5364	272	7	)	)	PUNCT
ejpam-5364	272	8	=	=	SYM
ejpam-5364	272	9	2	2	X
ejpam-5364	272	10	.	.	X
ejpam-5364	272	11	also	also	ADV
ejpam-5364	272	12	its	its	PRON
ejpam-5364	272	13	crossing	crossing	NOUN
ejpam-5364	272	14	number	number	NOUN
ejpam-5364	272	15	cr(g	cr(g	PUNCT
ejpam-5364	272	16	)	)	PUNCT
ejpam-5364	272	17	=	=	SYM
ejpam-5364	273	1	5	5	X
ejpam-5364	273	2	.	.	X
ejpam-5364	273	3	lower	lower	ADV
ejpam-5364	273	4	bound	bind	VERB
ejpam-5364	273	5	for	for	ADP
ejpam-5364	273	6	λ2	λ2	NOUN
ejpam-5364	273	7	from	from	ADP
ejpam-5364	273	8	theorem	theorem	ADJ
ejpam-5364	273	9	1	1	NUM
ejpam-5364	273	10	is	be	AUX
ejpam-5364	273	11	λ2	λ2	PROPN
ejpam-5364	273	12	≥	≥	NUM
ejpam-5364	273	13	−1.828	−1.828	NOUN
ejpam-5364	273	14	.	.	PUNCT
ejpam-5364	274	1	figure	figure	NOUN
ejpam-5364	274	2	5	5	NUM
ejpam-5364	274	3	:	:	PUNCT
ejpam-5364	274	4	graph	graph	VERB
ejpam-5364	274	5	with	with	ADP
ejpam-5364	274	6	sk(g	sk(g	NOUN
ejpam-5364	274	7	)	)	PUNCT
ejpam-5364	274	8	=	=	SYM
ejpam-5364	274	9	2	2	NUM
ejpam-5364	274	10	and	and	CCONJ
ejpam-5364	274	11	cr(g	cr(g	PUNCT
ejpam-5364	274	12	)	)	PUNCT
ejpam-5364	275	1	=	=	SYM
ejpam-5364	275	2	5	5	X
ejpam-5364	275	3	.	.	NOUN
ejpam-5364	275	4	example	example	NOUN
ejpam-5364	275	5	2	2	NUM
ejpam-5364	275	6	.	.	PUNCT
ejpam-5364	276	1	the	the	DET
ejpam-5364	276	2	graph	graph	NOUN
ejpam-5364	276	3	illustrated	illustrate	VERB
ejpam-5364	276	4	in	in	ADP
ejpam-5364	276	5	figure	figure	NOUN
ejpam-5364	276	6	6	6	NUM
ejpam-5364	276	7	has	have	VERB
ejpam-5364	276	8	6	6	NUM
ejpam-5364	276	9	vertices	vertex	NOUN
ejpam-5364	276	10	,	,	PUNCT
ejpam-5364	276	11	9	9	NUM
ejpam-5364	276	12	edges	edge	NOUN
ejpam-5364	276	13	with	with	ADP
ejpam-5364	276	14	d	d	PROPN
ejpam-5364	276	15	=	=	SYM
ejpam-5364	276	16	3	3	NUM
ejpam-5364	276	17	,	,	PUNCT
ejpam-5364	276	18	λ2	λ2	NOUN
ejpam-5364	276	19	=	=	SYM
ejpam-5364	276	20	0	0	NUM
ejpam-5364	276	21	and	and	CCONJ
ejpam-5364	276	22	q2	q2	NOUN
ejpam-5364	276	23	=	=	SYM
ejpam-5364	277	1	3	3	X
ejpam-5364	277	2	.	.	X
ejpam-5364	277	3	also	also	ADV
ejpam-5364	277	4	it	it	PRON
ejpam-5364	277	5	has	have	VERB
ejpam-5364	277	6	sk(g	sk(g	NOUN
ejpam-5364	277	7	)	)	PUNCT
ejpam-5364	277	8	=	=	SYM
ejpam-5364	278	1	1	1	X
ejpam-5364	278	2	.	.	PUNCT
ejpam-5364	278	3	lower	lower	ADV
ejpam-5364	278	4	bound	bind	VERB
ejpam-5364	278	5	from	from	ADP
ejpam-5364	278	6	theorem	theorem	ADJ
ejpam-5364	278	7	2	2	NUM
ejpam-5364	278	8	is	be	AUX
ejpam-5364	278	9	λ2	λ2	NOUN
ejpam-5364	278	10	≥	≥	NUM
ejpam-5364	278	11	−0.818	−0.818	NOUN
ejpam-5364	278	12	and	and	CCONJ
ejpam-5364	278	13	from	from	ADP
ejpam-5364	278	14	theorem	theorem	NOUN
ejpam-5364	278	15	4	4	NUM
ejpam-5364	278	16	we	we	PRON
ejpam-5364	278	17	have	have	VERB
ejpam-5364	278	18	q2	q2	NOUN
ejpam-5364	278	19	≥	≥	NUM
ejpam-5364	278	20	2.182	2.182	NUM
ejpam-5364	278	21	.	.	PUNCT
ejpam-5364	279	1	figure	figure	VERB
ejpam-5364	279	2	6	6	NUM
ejpam-5364	279	3	:	:	PUNCT
ejpam-5364	279	4	a	a	DET
ejpam-5364	279	5	3−regular	3−regular	NUM
ejpam-5364	279	6	graph	graph	NOUN
ejpam-5364	279	7	with	with	ADP
ejpam-5364	279	8	sk(g	sk(g	NOUN
ejpam-5364	279	9	)	)	PUNCT
ejpam-5364	279	10	=	=	SYM
ejpam-5364	279	11	cr(g	cr(g	X
ejpam-5364	279	12	)	)	PUNCT
ejpam-5364	280	1	=	=	SYM
ejpam-5364	280	2	1	1	NUM
ejpam-5364	280	3	3.2	3.2	NUM
ejpam-5364	280	4	.	.	PUNCT
ejpam-5364	281	1	thickness	thickness	NOUN
ejpam-5364	281	2	this	this	DET
ejpam-5364	281	3	section	section	NOUN
ejpam-5364	281	4	presents	present	VERB
ejpam-5364	281	5	the	the	DET
ejpam-5364	281	6	theorems	theorem	NOUN
ejpam-5364	281	7	connecting	connect	VERB
ejpam-5364	281	8	a	a	DET
ejpam-5364	281	9	graph	graph	NOUN
ejpam-5364	281	10	’s	’s	PART
ejpam-5364	281	11	thickness	thickness	NOUN
ejpam-5364	281	12	τ(g	τ(g	PROPN
ejpam-5364	281	13	)	)	PUNCT
ejpam-5364	281	14	and	and	CCONJ
ejpam-5364	281	15	second	second	ADV
ejpam-5364	281	16	largest	large	ADJ
ejpam-5364	281	17	adjacency	adjacency	NOUN
ejpam-5364	281	18	and	and	CCONJ
ejpam-5364	281	19	signless	signless	ADJ
ejpam-5364	281	20	laplacian	laplacian	ADJ
ejpam-5364	281	21	eigenvalues	eigenvalue	NOUN
ejpam-5364	281	22	.	.	PUNCT
ejpam-5364	282	1	lower	low	ADJ
ejpam-5364	282	2	bounds	bound	NOUN
ejpam-5364	282	3	are	be	AUX
ejpam-5364	282	4	established	establish	VERB
ejpam-5364	282	5	for	for	ADP
ejpam-5364	282	6	λ2	λ2	NOUN
ejpam-5364	282	7	and	and	CCONJ
ejpam-5364	282	8	q2	q2	NOUN
ejpam-5364	282	9	in	in	ADP
ejpam-5364	282	10	terms	term	NOUN
ejpam-5364	282	11	of	of	ADP
ejpam-5364	282	12	τ(g	τ(g	PROPN
ejpam-5364	282	13	)	)	PUNCT
ejpam-5364	282	14	.	.	PUNCT
ejpam-5364	283	1	also	also	ADV
ejpam-5364	283	2	,	,	PUNCT
ejpam-5364	283	3	lower	low	ADJ
ejpam-5364	283	4	bounds	bound	NOUN
ejpam-5364	283	5	for	for	ADP
ejpam-5364	283	6	the	the	DET
ejpam-5364	283	7	thickness	thickness	NOUN
ejpam-5364	283	8	τ(g	τ(g	PROPN
ejpam-5364	283	9	)	)	PUNCT
ejpam-5364	283	10	of	of	ADP
ejpam-5364	283	11	regular	regular	ADJ
ejpam-5364	283	12	graphs	graph	NOUN
ejpam-5364	283	13	g	g	NOUN
ejpam-5364	283	14	are	be	AUX
ejpam-5364	283	15	presented	present	VERB
ejpam-5364	283	16	in	in	ADP
ejpam-5364	283	17	terms	term	NOUN
ejpam-5364	283	18	of	of	ADP
ejpam-5364	283	19	λ2	λ2	NOUN
ejpam-5364	283	20	and	and	CCONJ
ejpam-5364	283	21	q2	q2	NOUN
ejpam-5364	283	22	.	.	PUNCT
ejpam-5364	284	1	theorem	theorem	NOUN
ejpam-5364	284	2	5	5	NUM
ejpam-5364	284	3	.	.	PUNCT
ejpam-5364	285	1	let	let	VERB
ejpam-5364	285	2	g	g	PRON
ejpam-5364	285	3	be	be	AUX
ejpam-5364	285	4	a	a	DET
ejpam-5364	285	5	connected	connected	ADJ
ejpam-5364	285	6	graph	graph	NOUN
ejpam-5364	285	7	with	with	ADP
ejpam-5364	285	8	thickness	thickness	NOUN
ejpam-5364	285	9	τ(g	τ(g	PROPN
ejpam-5364	285	10	)	)	PUNCT
ejpam-5364	285	11	.	.	PUNCT
ejpam-5364	286	1	then	then	ADV
ejpam-5364	286	2	λ2	λ2	PRON
ejpam-5364	286	3	≥	≥	NUM
ejpam-5364	286	4	1	1	NUM
ejpam-5364	286	5	2	2	NUM
ejpam-5364	286	6	{	{	PUNCT
ejpam-5364	286	7	[	[	PUNCT
ejpam-5364	286	8	δ	δ	PROPN
ejpam-5364	286	9	−	−	PROPN
ejpam-5364	287	1	3τ∆	3τ∆	NUM
ejpam-5364	287	2	m+	m+	NUM
ejpam-5364	287	3	3τ	3τ	NOUN
ejpam-5364	287	4	]	]	PUNCT
ejpam-5364	287	5	−	−	NUM
ejpam-5364	287	6	√	√	PROPN
ejpam-5364	287	7	[	[	PUNCT
ejpam-5364	287	8	δ	δ	PROPN
ejpam-5364	287	9	+	+	CCONJ
ejpam-5364	287	10	3τ∆	3τ∆	NUM
ejpam-5364	287	11	m+	m+	NUM
ejpam-5364	287	12	3τ	3τ	PROPN
ejpam-5364	287	13	]	]	X
ejpam-5364	287	14	2	2	NUM
ejpam-5364	287	15	+	+	CCONJ
ejpam-5364	287	16	12τ∆(∆−	12τ∆(∆−	PROPN
ejpam-5364	287	17	δ	δ	PROPN
ejpam-5364	287	18	)	)	PUNCT
ejpam-5364	287	19	m+	m+	NUM
ejpam-5364	287	20	3τ	3τ	PROPN
ejpam-5364	287	21	}	}	PUNCT
ejpam-5364	287	22	.	.	PUNCT
ejpam-5364	288	1	proof	proof	NOUN
ejpam-5364	288	2	.	.	PUNCT
ejpam-5364	289	1	the	the	DET
ejpam-5364	289	2	proof	proof	NOUN
ejpam-5364	289	3	is	be	AUX
ejpam-5364	289	4	similar	similar	ADJ
ejpam-5364	289	5	to	to	ADP
ejpam-5364	289	6	the	the	DET
ejpam-5364	289	7	proof	proof	NOUN
ejpam-5364	289	8	of	of	ADP
ejpam-5364	289	9	theorem	theorem	NOUN
ejpam-5364	289	10	1	1	NUM
ejpam-5364	289	11	.	.	PUNCT
ejpam-5364	289	12	by	by	ADP
ejpam-5364	289	13	making	make	VERB
ejpam-5364	289	14	use	use	NOUN
ejpam-5364	289	15	of	of	ADP
ejpam-5364	289	16	inequality	inequality	NOUN
ejpam-5364	289	17	(	(	PUNCT
ejpam-5364	289	18	2	2	NUM
ejpam-5364	289	19	)	)	PUNCT
ejpam-5364	289	20	,	,	PUNCT
ejpam-5364	289	21	we	we	PRON
ejpam-5364	289	22	get	get	VERB
ejpam-5364	289	23	the	the	DET
ejpam-5364	289	24	result	result	NOUN
ejpam-5364	289	25	.	.	PUNCT
ejpam-5364	290	1	m.	m.	NOUN
ejpam-5364	290	2	machasri	machasri	PROPN
ejpam-5364	290	3	,	,	PUNCT
ejpam-5364	290	4	d.	d.	PROPN
ejpam-5364	290	5	kalyani	kalyani	PROPN
ejpam-5364	290	6	/	/	SYM
ejpam-5364	290	7	eur	eur	PROPN
ejpam-5364	290	8	.	.	PUNCT
ejpam-5364	291	1	j.	j.	PROPN
ejpam-5364	291	2	pure	pure	PROPN
ejpam-5364	291	3	appl	appl	PROPN
ejpam-5364	291	4	.	.	PROPN
ejpam-5364	291	5	math	math	PROPN
ejpam-5364	291	6	,	,	PUNCT
ejpam-5364	291	7	17	17	NUM
ejpam-5364	291	8	(	(	PUNCT
ejpam-5364	291	9	4	4	NUM
ejpam-5364	291	10	)	)	PUNCT
ejpam-5364	291	11	(	(	PUNCT
ejpam-5364	291	12	2024	2024	NUM
ejpam-5364	291	13	)	)	PUNCT
ejpam-5364	291	14	,	,	PUNCT
ejpam-5364	291	15	3004	3004	NUM
ejpam-5364	291	16	-	-	SYM
ejpam-5364	291	17	3021	3021	NUM
ejpam-5364	291	18	3015	3015	NUM
ejpam-5364	291	19	theorem	theorem	VERB
ejpam-5364	291	20	6	6	NUM
ejpam-5364	291	21	.	.	PUNCT
ejpam-5364	292	1	let	let	VERB
ejpam-5364	292	2	g	g	PRON
ejpam-5364	292	3	be	be	AUX
ejpam-5364	292	4	a	a	DET
ejpam-5364	292	5	d−regular	d−regular	NUM
ejpam-5364	292	6	graph	graph	NOUN
ejpam-5364	292	7	with	with	ADP
ejpam-5364	292	8	n	n	PRON
ejpam-5364	292	9	≥	≥	NOUN
ejpam-5364	292	10	3	3	NUM
ejpam-5364	292	11	.	.	PUNCT
ejpam-5364	293	1	λ2	λ2	PROPN
ejpam-5364	293	2	≥	≥	NUM
ejpam-5364	293	3	−3τd	−3τd	PROPN
ejpam-5364	293	4	m+	m+	NUM
ejpam-5364	293	5	3τ	3τ	PROPN
ejpam-5364	293	6	.	.	PUNCT
ejpam-5364	294	1	proof	proof	NOUN
ejpam-5364	294	2	.	.	PUNCT
ejpam-5364	295	1	the	the	DET
ejpam-5364	295	2	proof	proof	NOUN
ejpam-5364	295	3	is	be	AUX
ejpam-5364	295	4	on	on	ADP
ejpam-5364	295	5	the	the	DET
ejpam-5364	295	6	same	same	ADJ
ejpam-5364	295	7	lines	line	NOUN
ejpam-5364	295	8	as	as	ADP
ejpam-5364	295	9	the	the	DET
ejpam-5364	295	10	proof	proof	NOUN
ejpam-5364	295	11	of	of	ADP
ejpam-5364	295	12	theorem	theorem	ADJ
ejpam-5364	295	13	2	2	NUM
ejpam-5364	295	14	.	.	X
ejpam-5364	295	15	inequality	inequality	NOUN
ejpam-5364	295	16	(	(	PUNCT
ejpam-5364	295	17	2	2	NUM
ejpam-5364	295	18	)	)	PUNCT
ejpam-5364	295	19	is	be	AUX
ejpam-5364	295	20	used	use	VERB
ejpam-5364	295	21	to	to	PART
ejpam-5364	295	22	get	get	VERB
ejpam-5364	295	23	the	the	DET
ejpam-5364	295	24	final	final	ADJ
ejpam-5364	295	25	result	result	NOUN
ejpam-5364	295	26	.	.	PUNCT
ejpam-5364	296	1	note	note	VERB
ejpam-5364	296	2	3	3	X
ejpam-5364	296	3	.	.	PUNCT
ejpam-5364	297	1	if	if	SCONJ
ejpam-5364	297	2	g	g	PROPN
ejpam-5364	297	3	is	be	AUX
ejpam-5364	297	4	a	a	DET
ejpam-5364	297	5	d−regular	d−regular	NUM
ejpam-5364	297	6	graph	graph	NOUN
ejpam-5364	297	7	with	with	ADP
ejpam-5364	297	8	n	n	NUM
ejpam-5364	297	9	≥	≥	NOUN
ejpam-5364	297	10	3	3	NUM
ejpam-5364	297	11	then	then	ADV
ejpam-5364	297	12	τ	τ	PROPN
ejpam-5364	297	13	≥	≥	PROPN
ejpam-5364	297	14	−mλ2	−mλ2	NOUN
ejpam-5364	297	15	3(λ2	3(λ2	NUM
ejpam-5364	298	1	+	+	CCONJ
ejpam-5364	298	2	d	d	NOUN
ejpam-5364	298	3	)	)	PUNCT
ejpam-5364	298	4	.	.	PUNCT
ejpam-5364	299	1	theorem	theorem	ADJ
ejpam-5364	299	2	7	7	NUM
ejpam-5364	299	3	.	.	PUNCT
ejpam-5364	300	1	let	let	VERB
ejpam-5364	300	2	g	g	PRON
ejpam-5364	300	3	be	be	AUX
ejpam-5364	300	4	a	a	DET
ejpam-5364	300	5	connected	connected	ADJ
ejpam-5364	300	6	graph	graph	NOUN
ejpam-5364	300	7	.	.	PUNCT
ejpam-5364	301	1	then	then	ADV
ejpam-5364	301	2	q2	q2	PROPN
ejpam-5364	301	3	≥	≥	NUM
ejpam-5364	301	4	1	1	NUM
ejpam-5364	301	5	2	2	NUM
ejpam-5364	301	6	{	{	PUNCT
ejpam-5364	301	7	[	[	X
ejpam-5364	301	8	∆(1−	∆(1−	ADV
ejpam-5364	301	9	3τu	3τu	NOUN
ejpam-5364	301	10	)	)	PUNCT
ejpam-5364	302	1	+	+	CCONJ
ejpam-5364	302	2	2δ]−	2δ]−	NUM
ejpam-5364	302	3	√	√	NOUN
ejpam-5364	303	1	[	[	X
ejpam-5364	303	2	∆(1−	∆(1−	ADV
ejpam-5364	303	3	3τu)−	3τu)−	NUM
ejpam-5364	303	4	2δ]2	2δ]2	NUM
ejpam-5364	303	5	+	+	CCONJ
ejpam-5364	303	6	24τ∆(∆−	24τ∆(∆−	NUM
ejpam-5364	303	7	δ)u	δ)u	NOUN
ejpam-5364	303	8	}	}	PUNCT
ejpam-5364	303	9	where	where	SCONJ
ejpam-5364	303	10	u	u	NOUN
ejpam-5364	303	11	=	=	NOUN
ejpam-5364	303	12	1	1	NUM
ejpam-5364	303	13	m+3τ	m+3τ	NOUN
ejpam-5364	303	14	.	.	PUNCT
ejpam-5364	304	1	proof	proof	NOUN
ejpam-5364	304	2	.	.	PUNCT
ejpam-5364	305	1	the	the	DET
ejpam-5364	305	2	proof	proof	NOUN
ejpam-5364	305	3	is	be	AUX
ejpam-5364	305	4	similar	similar	ADJ
ejpam-5364	305	5	to	to	ADP
ejpam-5364	305	6	the	the	DET
ejpam-5364	305	7	proof	proof	NOUN
ejpam-5364	305	8	of	of	ADP
ejpam-5364	305	9	theorem	theorem	NOUN
ejpam-5364	305	10	3	3	NUM
ejpam-5364	305	11	.	.	PUNCT
ejpam-5364	305	12	by	by	ADP
ejpam-5364	305	13	using	use	VERB
ejpam-5364	305	14	inequality	inequality	NOUN
ejpam-5364	305	15	(	(	PUNCT
ejpam-5364	305	16	2	2	NUM
ejpam-5364	305	17	)	)	PUNCT
ejpam-5364	305	18	,	,	PUNCT
ejpam-5364	305	19	we	we	PRON
ejpam-5364	305	20	get	get	VERB
ejpam-5364	305	21	the	the	DET
ejpam-5364	305	22	result	result	NOUN
ejpam-5364	305	23	.	.	PUNCT
ejpam-5364	306	1	theorem	theorem	ADJ
ejpam-5364	306	2	8	8	NUM
ejpam-5364	306	3	.	.	PUNCT
ejpam-5364	307	1	let	let	VERB
ejpam-5364	307	2	g	g	PRON
ejpam-5364	307	3	be	be	AUX
ejpam-5364	307	4	a	a	DET
ejpam-5364	307	5	d−regular	d−regular	NUM
ejpam-5364	307	6	graph	graph	NOUN
ejpam-5364	307	7	with	with	ADP
ejpam-5364	307	8	n	n	PRON
ejpam-5364	307	9	≥	≥	NUM
ejpam-5364	307	10	3	3	NUM
ejpam-5364	307	11	.	.	PUNCT
ejpam-5364	308	1	then	then	ADV
ejpam-5364	308	2	q2	q2	PROPN
ejpam-5364	308	3	≥	≥	PRON
ejpam-5364	309	1	dm	dm	NUM
ejpam-5364	309	2	m+	m+	NUM
ejpam-5364	309	3	3τ	3τ	PROPN
ejpam-5364	309	4	.	.	PUNCT
ejpam-5364	310	1	proof	proof	NOUN
ejpam-5364	310	2	.	.	PUNCT
ejpam-5364	311	1	the	the	DET
ejpam-5364	311	2	proof	proof	NOUN
ejpam-5364	311	3	is	be	AUX
ejpam-5364	311	4	similar	similar	ADJ
ejpam-5364	311	5	to	to	ADP
ejpam-5364	311	6	the	the	DET
ejpam-5364	311	7	proof	proof	NOUN
ejpam-5364	311	8	of	of	ADP
ejpam-5364	311	9	theorem	theorem	ADJ
ejpam-5364	311	10	4	4	NUM
ejpam-5364	311	11	.	.	PUNCT
ejpam-5364	311	12	by	by	ADP
ejpam-5364	311	13	using	use	VERB
ejpam-5364	311	14	inequality	inequality	NOUN
ejpam-5364	311	15	(	(	PUNCT
ejpam-5364	311	16	2	2	NUM
ejpam-5364	311	17	)	)	PUNCT
ejpam-5364	311	18	,	,	PUNCT
ejpam-5364	311	19	we	we	PRON
ejpam-5364	311	20	get	get	VERB
ejpam-5364	311	21	the	the	DET
ejpam-5364	311	22	result	result	NOUN
ejpam-5364	311	23	.	.	PUNCT
ejpam-5364	312	1	note	note	VERB
ejpam-5364	312	2	4	4	NUM
ejpam-5364	312	3	.	.	PUNCT
ejpam-5364	313	1	if	if	SCONJ
ejpam-5364	313	2	g	g	PROPN
ejpam-5364	313	3	is	be	AUX
ejpam-5364	313	4	a	a	DET
ejpam-5364	313	5	d−regular	d−regular	NUM
ejpam-5364	313	6	graph	graph	NOUN
ejpam-5364	313	7	except	except	SCONJ
ejpam-5364	313	8	complete	complete	ADJ
ejpam-5364	313	9	split	split	NOUN
ejpam-5364	313	10	graph	graph	NOUN
ejpam-5364	313	11	and	and	CCONJ
ejpam-5364	313	12	complete	complete	ADJ
ejpam-5364	313	13	multipartite	multipartite	ADJ
ejpam-5364	313	14	graph	graph	NOUN
ejpam-5364	313	15	with	with	ADP
ejpam-5364	313	16	n	n	NUM
ejpam-5364	313	17	≥	≥	NOUN
ejpam-5364	313	18	3	3	NUM
ejpam-5364	313	19	then	then	ADV
ejpam-5364	313	20	τ	τ	PROPN
ejpam-5364	313	21	≥	≥	PROPN
ejpam-5364	313	22	m	m	VERB
ejpam-5364	313	23	3	3	NUM
ejpam-5364	313	24	(	(	PUNCT
ejpam-5364	313	25	d	d	PROPN
ejpam-5364	313	26	q2	q2	NOUN
ejpam-5364	313	27	−	−	PROPN
ejpam-5364	313	28	1	1	NUM
ejpam-5364	313	29	)	)	PUNCT
ejpam-5364	313	30	.	.	PUNCT
ejpam-5364	314	1	example	example	NOUN
ejpam-5364	315	1	3	3	NUM
ejpam-5364	315	2	.	.	PUNCT
ejpam-5364	316	1	the	the	DET
ejpam-5364	316	2	hypercube	hypercube	NOUN
ejpam-5364	316	3	q4	q4	PROPN
ejpam-5364	316	4	is	be	AUX
ejpam-5364	316	5	a	a	DET
ejpam-5364	316	6	4−regular	4−regular	ADJ
ejpam-5364	316	7	graph	graph	NOUN
ejpam-5364	316	8	on	on	ADP
ejpam-5364	316	9	16	16	NUM
ejpam-5364	316	10	vertices	vertex	NOUN
ejpam-5364	316	11	and	and	CCONJ
ejpam-5364	316	12	32	32	NUM
ejpam-5364	316	13	edges	edge	NOUN
ejpam-5364	316	14	with	with	ADP
ejpam-5364	316	15	thickness	thickness	NOUN
ejpam-5364	316	16	τ(g	τ(g	NOUN
ejpam-5364	316	17	)	)	PUNCT
ejpam-5364	317	1	=	=	SYM
ejpam-5364	317	2	2	2	NUM
ejpam-5364	317	3	,	,	PUNCT
ejpam-5364	317	4	λ2	λ2	NOUN
ejpam-5364	317	5	=	=	SYM
ejpam-5364	317	6	2	2	NUM
ejpam-5364	317	7	and	and	CCONJ
ejpam-5364	317	8	q2	q2	NOUN
ejpam-5364	317	9	=	=	SYM
ejpam-5364	318	1	6	6	NUM
ejpam-5364	318	2	.	.	PUNCT
ejpam-5364	319	1	the	the	DET
ejpam-5364	319	2	lower	low	ADJ
ejpam-5364	319	3	bounds	bound	NOUN
ejpam-5364	319	4	from	from	ADP
ejpam-5364	319	5	theorems	theorem	NOUN
ejpam-5364	319	6	6	6	NUM
ejpam-5364	319	7	and	and	CCONJ
ejpam-5364	319	8	8	8	NUM
ejpam-5364	319	9	are	be	AUX
ejpam-5364	319	10	given	give	VERB
ejpam-5364	319	11	by	by	ADP
ejpam-5364	319	12	λ2	λ2	PROPN
ejpam-5364	319	13	≥	≥	NOUN
ejpam-5364	319	14	−0.631	−0.631	PROPN
ejpam-5364	319	15	and	and	CCONJ
ejpam-5364	319	16	q2	q2	PROPN
ejpam-5364	319	17	≥	≥	NUM
ejpam-5364	319	18	3.368	3.368	NUM
ejpam-5364	319	19	.	.	PUNCT
ejpam-5364	320	1	in	in	ADP
ejpam-5364	320	2	table	table	NOUN
ejpam-5364	320	3	1	1	NUM
ejpam-5364	320	4	,	,	PUNCT
ejpam-5364	320	5	numerical	numerical	ADJ
ejpam-5364	320	6	values	value	NOUN
ejpam-5364	320	7	of	of	ADP
ejpam-5364	320	8	the	the	DET
ejpam-5364	320	9	bounds	bound	NOUN
ejpam-5364	320	10	that	that	PRON
ejpam-5364	320	11	we	we	PRON
ejpam-5364	320	12	have	have	AUX
ejpam-5364	320	13	proved	prove	VERB
ejpam-5364	320	14	in	in	ADP
ejpam-5364	320	15	theorem	theorem	ADJ
ejpam-5364	320	16	1	1	NUM
ejpam-5364	320	17	and	and	CCONJ
ejpam-5364	320	18	theorem	theorem	VERB
ejpam-5364	320	19	5	5	NUM
ejpam-5364	320	20	are	be	AUX
ejpam-5364	320	21	presented	present	VERB
ejpam-5364	320	22	for	for	ADP
ejpam-5364	320	23	complete	complete	ADJ
ejpam-5364	320	24	bipartite	bipartite	NOUN
ejpam-5364	320	25	graph	graph	NOUN
ejpam-5364	320	26	km.n	km.n	VERB
ejpam-5364	320	27	with	with	ADP
ejpam-5364	320	28	m	m	PROPN
ejpam-5364	320	29	̸=	̸=	PROPN
ejpam-5364	320	30	n.	n.	NOUN
ejpam-5364	320	31	the	the	DET
ejpam-5364	320	32	graphical	graphical	ADJ
ejpam-5364	320	33	illustration	illustration	NOUN
ejpam-5364	320	34	of	of	ADP
ejpam-5364	320	35	these	these	DET
ejpam-5364	320	36	bounds	bound	NOUN
ejpam-5364	320	37	are	be	AUX
ejpam-5364	320	38	given	give	VERB
ejpam-5364	320	39	in	in	ADP
ejpam-5364	320	40	figure	figure	NOUN
ejpam-5364	320	41	7	7	NUM
ejpam-5364	320	42	and	and	CCONJ
ejpam-5364	320	43	figure	figure	VERB
ejpam-5364	320	44	8	8	NUM
ejpam-5364	320	45	.	.	PUNCT
ejpam-5364	321	1	m.	m.	PROPN
ejpam-5364	321	2	machasri	machasri	PROPN
ejpam-5364	321	3	,	,	PUNCT
ejpam-5364	321	4	d.	d.	PROPN
ejpam-5364	321	5	kalyani	kalyani	PROPN
ejpam-5364	321	6	/	/	SYM
ejpam-5364	321	7	eur	eur	PROPN
ejpam-5364	321	8	.	.	PUNCT
ejpam-5364	322	1	j.	j.	PROPN
ejpam-5364	322	2	pure	pure	PROPN
ejpam-5364	322	3	appl	appl	PROPN
ejpam-5364	322	4	.	.	PROPN
ejpam-5364	322	5	math	math	PROPN
ejpam-5364	322	6	,	,	PUNCT
ejpam-5364	322	7	17	17	NUM
ejpam-5364	322	8	(	(	PUNCT
ejpam-5364	322	9	4	4	NUM
ejpam-5364	322	10	)	)	PUNCT
ejpam-5364	322	11	(	(	PUNCT
ejpam-5364	322	12	2024	2024	NUM
ejpam-5364	322	13	)	)	PUNCT
ejpam-5364	322	14	,	,	PUNCT
ejpam-5364	322	15	3004	3004	NUM
ejpam-5364	322	16	-	-	SYM
ejpam-5364	322	17	3021	3021	NUM
ejpam-5364	322	18	3016	3016	NUM
ejpam-5364	322	19	s.no	s.no	NOUN
ejpam-5364	322	20	.	.	PUNCT
ejpam-5364	322	21	graph	graph	NOUN
ejpam-5364	322	22	λ2	λ2	NOUN
ejpam-5364	322	23	theorem	theorem	ADJ
ejpam-5364	322	24	1	1	NUM
ejpam-5364	322	25	bound	bind	VERB
ejpam-5364	322	26	theorem	theorem	NOUN
ejpam-5364	322	27	5	5	NUM
ejpam-5364	322	28	bound	bind	VERB
ejpam-5364	322	29	1	1	NUM
ejpam-5364	322	30	k3,5	k3,5	PROPN
ejpam-5364	322	31	0	0	NUM
ejpam-5364	322	32	-1.4494	-1.4494	NOUN
ejpam-5364	322	33	-2.0717	-2.0717	NOUN
ejpam-5364	323	1	2	2	NUM
ejpam-5364	323	2	k4,6	k4,6	NOUN
ejpam-5364	323	3	0	0	PUNCT
ejpam-5364	323	4	-1.3042	-1.3042	NUM
ejpam-5364	323	5	-1.626	-1.626	PUNCT
ejpam-5364	324	1	3	3	NUM
ejpam-5364	324	2	k5,7	k5,7	NOUN
ejpam-5364	324	3	0	0	NUM
ejpam-5364	324	4	-1.2072	-1.2072	NOUN
ejpam-5364	324	5	-1.3472	-1.3472	ADJ
ejpam-5364	324	6	4	4	NUM
ejpam-5364	324	7	k6,8	k6,8	NOUN
ejpam-5364	324	8	0	0	NUM
ejpam-5364	324	9	-1.138	-1.138	NUM
ejpam-5364	324	10	-1.138	-1.138	NUM
ejpam-5364	324	11	5	5	NUM
ejpam-5364	324	12	k7,9	k7,9	PROPN
ejpam-5364	324	13	0	0	NUM
ejpam-5364	324	14	-1.0863	-1.0863	NOUN
ejpam-5364	324	15	-1.3931	-1.3931	NOUN
ejpam-5364	324	16	6	6	NUM
ejpam-5364	324	17	k8,10	k8,10	PROPN
ejpam-5364	324	18	0	0	NUM
ejpam-5364	324	19	-1.0466	-1.0466	PROPN
ejpam-5364	324	20	-1.2303	-1.2303	PROPN
ejpam-5364	324	21	7	7	NUM
ejpam-5364	324	22	k9,11	k9,11	NOUN
ejpam-5364	324	23	0	0	NUM
ejpam-5364	324	24	-1.0151	-1.0151	NOUN
ejpam-5364	324	25	-1.0982	-1.0982	NOUN
ejpam-5364	324	26	8	8	NUM
ejpam-5364	324	27	k10,12	k10,12	PROPN
ejpam-5364	324	28	0	0	NUM
ejpam-5364	324	29	-0.9895	-0.9895	ADJ
ejpam-5364	324	30	-0.9896	-0.9896	NOUN
ejpam-5364	325	1	9	9	NUM
ejpam-5364	325	2	k11,13	k11,13	NOUN
ejpam-5364	325	3	0	0	NUM
ejpam-5364	325	4	-0.9683	-0.9683	NOUN
ejpam-5364	325	5	-1.1714	-1.1714	VERB
ejpam-5364	325	6	10	10	NUM
ejpam-5364	325	7	k12,14	k12,14	ADP
ejpam-5364	325	8	0	0	NUM
ejpam-5364	326	1	-0.9503	-0.9503	NOUN
ejpam-5364	326	2	-1.0702	-1.0702	NOUN
ejpam-5364	327	1	11	11	NUM
ejpam-5364	327	2	k13,15	k13,15	NOUN
ejpam-5364	327	3	0	0	NUM
ejpam-5364	327	4	-0.9355	-0.9355	NOUN
ejpam-5364	327	5	-0.9938	-0.9938	PROPN
ejpam-5364	327	6	table	table	NOUN
ejpam-5364	327	7	1	1	NUM
ejpam-5364	327	8	:	:	PUNCT
ejpam-5364	327	9	lower	low	ADJ
ejpam-5364	327	10	bounds	bound	NOUN
ejpam-5364	327	11	of	of	ADP
ejpam-5364	327	12	λ2	λ2	NOUN
ejpam-5364	327	13	for	for	ADP
ejpam-5364	327	14	km	km	PROPN
ejpam-5364	327	15	,	,	PUNCT
ejpam-5364	327	16	n	n	PRON
ejpam-5364	327	17	figure	figure	VERB
ejpam-5364	327	18	7	7	NUM
ejpam-5364	327	19	:	:	PUNCT
ejpam-5364	327	20	skewness	skewness	NOUN
ejpam-5364	327	21	figure	figure	NOUN
ejpam-5364	327	22	8	8	NUM
ejpam-5364	327	23	:	:	PUNCT
ejpam-5364	327	24	thickness	thickness	PROPN
ejpam-5364	327	25	numerical	numerical	ADJ
ejpam-5364	327	26	values	value	NOUN
ejpam-5364	327	27	for	for	ADP
ejpam-5364	327	28	the	the	DET
ejpam-5364	327	29	lower	low	ADJ
ejpam-5364	327	30	bounds	bound	NOUN
ejpam-5364	327	31	of	of	ADP
ejpam-5364	327	32	λ2	λ2	NOUN
ejpam-5364	327	33	proved	prove	VERB
ejpam-5364	327	34	in	in	ADP
ejpam-5364	327	35	theorems	theorem	NOUN
ejpam-5364	327	36	2	2	NUM
ejpam-5364	327	37	and	and	CCONJ
ejpam-5364	327	38	6	6	NUM
ejpam-5364	327	39	are	be	AUX
ejpam-5364	327	40	presented	present	VERB
ejpam-5364	327	41	for	for	ADP
ejpam-5364	327	42	complete	complete	ADJ
ejpam-5364	327	43	bipartite	bipartite	PROPN
ejpam-5364	327	44	graphs	graph	NOUN
ejpam-5364	327	45	kn	kn	PROPN
ejpam-5364	327	46	,	,	PUNCT
ejpam-5364	327	47	n	n	CCONJ
ejpam-5364	327	48	in	in	ADP
ejpam-5364	327	49	table	table	NOUN
ejpam-5364	327	50	2	2	NUM
ejpam-5364	327	51	.	.	PUNCT
ejpam-5364	328	1	the	the	DET
ejpam-5364	328	2	graphical	graphical	ADJ
ejpam-5364	328	3	representation	representation	NOUN
ejpam-5364	328	4	of	of	ADP
ejpam-5364	328	5	these	these	DET
ejpam-5364	328	6	bounds	bound	NOUN
ejpam-5364	328	7	are	be	AUX
ejpam-5364	328	8	given	give	VERB
ejpam-5364	328	9	in	in	ADP
ejpam-5364	328	10	figure	figure	NOUN
ejpam-5364	328	11	9	9	NUM
ejpam-5364	328	12	and	and	CCONJ
ejpam-5364	328	13	figure	figure	VERB
ejpam-5364	328	14	10	10	NUM
ejpam-5364	328	15	.	.	PUNCT
ejpam-5364	329	1	s.no	s.no	AUX
ejpam-5364	329	2	.	.	PUNCT
ejpam-5364	329	3	graph	graph	NOUN
ejpam-5364	329	4	λ2	λ2	NOUN
ejpam-5364	329	5	theorem	theorem	ADJ
ejpam-5364	329	6	2	2	NUM
ejpam-5364	329	7	bound	bind	VERB
ejpam-5364	329	8	theorem	theorem	NOUN
ejpam-5364	329	9	6	6	NUM
ejpam-5364	329	10	bound	bind	VERB
ejpam-5364	329	11	1	1	NUM
ejpam-5364	329	12	k3,3	k3,3	PROPN
ejpam-5364	329	13	0	0	NUM
ejpam-5364	329	14	-0.8181	-0.8181	X
ejpam-5364	330	1	-1.2	-1.2	NOUN
ejpam-5364	330	2	2	2	NUM
ejpam-5364	330	3	k4,4	k4,4	PROPN
ejpam-5364	330	4	0	0	NUM
ejpam-5364	330	5	-0.8	-0.8	PROPN
ejpam-5364	330	6	-1.0909	-1.0909	NOUN
ejpam-5364	330	7	3	3	NUM
ejpam-5364	330	8	k5,5	k5,5	PROPN
ejpam-5364	330	9	0	0	NUM
ejpam-5364	330	10	-0.7895	-0.7895	PRON
ejpam-5364	330	11	-0.9677	-0.9677	PROPN
ejpam-5364	330	12	4	4	NUM
ejpam-5364	330	13	k6,6	k6,6	PROPN
ejpam-5364	330	14	0	0	NUM
ejpam-5364	330	15	-0.783	-0.783	NOUN
ejpam-5364	330	16	-0.8571	-0.8571	PROPN
ejpam-5364	330	17	5	5	NUM
ejpam-5364	330	18	k7,7	k7,7	NOUN
ejpam-5364	330	19	0	0	NUM
ejpam-5364	330	20	-0.7778	-0.7778	NOUN
ejpam-5364	330	21	-1.0862	-1.0862	X
ejpam-5364	330	22	6	6	NUM
ejpam-5364	330	23	k8,8	k8,8	PROPN
ejpam-5364	330	24	0	0	NUM
ejpam-5364	330	25	-0.7742	-0.7742	NOUN
ejpam-5364	330	26	-0.9863	-0.9863	NOUN
ejpam-5364	331	1	7	7	NUM
ejpam-5364	331	2	k9,9	k9,9	PROPN
ejpam-5364	331	3	0	0	NUM
ejpam-5364	331	4	-0.7714	-0.7714	NOUN
ejpam-5364	331	5	-0.9	-0.9	NOUN
ejpam-5364	331	6	8	8	NUM
ejpam-5364	331	7	k10,10	k10,10	PROPN
ejpam-5364	331	8	0	0	NUM
ejpam-5364	331	9	-0.7692	-0.7692	PROPN
ejpam-5364	331	10	-0.8257	-0.8257	NOUN
ejpam-5364	331	11	9	9	NUM
ejpam-5364	331	12	k11,11	k11,11	PROPN
ejpam-5364	331	13	0	0	NUM
ejpam-5364	331	14	-0.7674	-0.7674	PROPN
ejpam-5364	331	15	-0.9925	-0.9925	VERB
ejpam-5364	332	1	10	10	NUM
ejpam-5364	332	2	k12,12	k12,12	PROPN
ejpam-5364	332	3	0	0	NUM
ejpam-5364	332	4	-0.7659	-0.7659	PROPN
ejpam-5364	332	5	-0.9231	-0.9231	PROPN
ejpam-5364	332	6	table	table	NOUN
ejpam-5364	332	7	2	2	NUM
ejpam-5364	332	8	:	:	PUNCT
ejpam-5364	332	9	lower	low	ADJ
ejpam-5364	332	10	bounds	bound	NOUN
ejpam-5364	332	11	of	of	ADP
ejpam-5364	332	12	λ2	λ2	NOUN
ejpam-5364	332	13	for	for	ADP
ejpam-5364	332	14	kn	kn	PROPN
ejpam-5364	332	15	,	,	PUNCT
ejpam-5364	332	16	n	n	PRON
ejpam-5364	332	17	m.	m.	NOUN
ejpam-5364	332	18	machasri	machasri	PROPN
ejpam-5364	332	19	,	,	PUNCT
ejpam-5364	332	20	d.	d.	PROPN
ejpam-5364	332	21	kalyani	kalyani	PROPN
ejpam-5364	332	22	/	/	SYM
ejpam-5364	332	23	eur	eur	PROPN
ejpam-5364	332	24	.	.	PUNCT
ejpam-5364	333	1	j.	j.	PROPN
ejpam-5364	333	2	pure	pure	PROPN
ejpam-5364	333	3	appl	appl	PROPN
ejpam-5364	333	4	.	.	PROPN
ejpam-5364	333	5	math	math	PROPN
ejpam-5364	333	6	,	,	PUNCT
ejpam-5364	333	7	17	17	NUM
ejpam-5364	333	8	(	(	PUNCT
ejpam-5364	333	9	4	4	NUM
ejpam-5364	333	10	)	)	PUNCT
ejpam-5364	333	11	(	(	PUNCT
ejpam-5364	333	12	2024	2024	NUM
ejpam-5364	333	13	)	)	PUNCT
ejpam-5364	333	14	,	,	PUNCT
ejpam-5364	333	15	3004	3004	NUM
ejpam-5364	333	16	-	-	SYM
ejpam-5364	333	17	3021	3021	NUM
ejpam-5364	333	18	3017	3017	NUM
ejpam-5364	333	19	figure	figure	NOUN
ejpam-5364	333	20	9	9	NUM
ejpam-5364	333	21	:	:	PUNCT
ejpam-5364	333	22	skewness	skewness	NOUN
ejpam-5364	333	23	figure	figure	NOUN
ejpam-5364	333	24	10	10	NUM
ejpam-5364	333	25	:	:	PUNCT
ejpam-5364	333	26	thickness	thickness	PROPN
ejpam-5364	333	27	numerical	numerical	ADJ
ejpam-5364	333	28	values	value	NOUN
ejpam-5364	333	29	of	of	ADP
ejpam-5364	333	30	the	the	DET
ejpam-5364	333	31	lower	low	ADJ
ejpam-5364	333	32	bounds	bound	NOUN
ejpam-5364	333	33	of	of	ADP
ejpam-5364	333	34	q2	q2	NOUN
ejpam-5364	333	35	proved	prove	VERB
ejpam-5364	333	36	in	in	ADP
ejpam-5364	333	37	theorems	theorem	NOUN
ejpam-5364	333	38	3	3	NUM
ejpam-5364	333	39	and	and	CCONJ
ejpam-5364	333	40	7	7	NUM
ejpam-5364	333	41	are	be	AUX
ejpam-5364	333	42	presented	present	VERB
ejpam-5364	333	43	for	for	ADP
ejpam-5364	333	44	complete	complete	ADJ
ejpam-5364	333	45	bipartite	bipartite	PROPN
ejpam-5364	333	46	graphs	graph	NOUN
ejpam-5364	333	47	km	km	PROPN
ejpam-5364	333	48	,	,	PUNCT
ejpam-5364	333	49	n	n	PROPN
ejpam-5364	333	50	with	with	ADP
ejpam-5364	333	51	m	m	PROPN
ejpam-5364	333	52	̸=	̸=	PROPN
ejpam-5364	333	53	n	n	NOUN
ejpam-5364	333	54	in	in	ADP
ejpam-5364	333	55	table	table	NOUN
ejpam-5364	333	56	3	3	NUM
ejpam-5364	333	57	.	.	PUNCT
ejpam-5364	334	1	the	the	DET
ejpam-5364	334	2	graphical	graphical	ADJ
ejpam-5364	334	3	illustration	illustration	NOUN
ejpam-5364	334	4	of	of	ADP
ejpam-5364	334	5	these	these	DET
ejpam-5364	334	6	bounds	bound	NOUN
ejpam-5364	334	7	are	be	AUX
ejpam-5364	334	8	represented	represent	VERB
ejpam-5364	334	9	in	in	ADP
ejpam-5364	334	10	figure	figure	NOUN
ejpam-5364	334	11	11	11	NUM
ejpam-5364	334	12	and	and	CCONJ
ejpam-5364	334	13	figure	figure	VERB
ejpam-5364	334	14	12	12	NUM
ejpam-5364	334	15	.	.	PUNCT
ejpam-5364	335	1	s.no	s.no	NOUN
ejpam-5364	335	2	.	.	PUNCT
ejpam-5364	335	3	graph	graph	NOUN
ejpam-5364	335	4	q2	q2	NOUN
ejpam-5364	335	5	theorem	theorem	VERB
ejpam-5364	335	6	3	3	NUM
ejpam-5364	335	7	bound	bind	VERB
ejpam-5364	335	8	theorem	theorem	VERB
ejpam-5364	335	9	7	7	NUM
ejpam-5364	335	10	bound	bind	VERB
ejpam-5364	335	11	1	1	NUM
ejpam-5364	335	12	k3,5	k3,5	PROPN
ejpam-5364	335	13	5	5	NUM
ejpam-5364	335	14	2.764	2.764	NUM
ejpam-5364	335	15	2.1045	2.1045	NUM
ejpam-5364	335	16	2	2	NUM
ejpam-5364	335	17	k4,6	k4,6	NOUN
ejpam-5364	335	18	6	6	NUM
ejpam-5364	335	19	4.0848	4.0848	NUM
ejpam-5364	335	20	3.6871	3.6871	NUM
ejpam-5364	335	21	3	3	NUM
ejpam-5364	335	22	k5,7	k5,7	PROPN
ejpam-5364	335	23	7	7	NUM
ejpam-5364	335	24	5.3086	5.3086	NUM
ejpam-5364	335	25	5.1336	5.1336	NUM
ejpam-5364	335	26	4	4	NUM
ejpam-5364	335	27	k6,8	k6,8	NOUN
ejpam-5364	335	28	8	8	NUM
ejpam-5364	335	29	6.4683	6.4683	NUM
ejpam-5364	335	30	6.4684	6.4684	NUM
ejpam-5364	335	31	5	5	NUM
ejpam-5364	335	32	k7,9	k7,9	NOUN
ejpam-5364	335	33	9	9	NUM
ejpam-5364	335	34	7.5859	7.5859	NUM
ejpam-5364	335	35	7.2121	7.2121	NUM
ejpam-5364	335	36	6	6	NUM
ejpam-5364	335	37	k8,10	k8,10	NUM
ejpam-5364	335	38	10	10	NUM
ejpam-5364	335	39	8.6749	8.6749	NUM
ejpam-5364	335	40	8.4528	8.4528	NUM
ejpam-5364	335	41	7	7	NUM
ejpam-5364	335	42	k9,11	k9,11	NOUN
ejpam-5364	335	43	11	11	NUM
ejpam-5364	335	44	9.7439	9.7439	NUM
ejpam-5364	335	45	9.6445	9.6445	NUM
ejpam-5364	335	46	8	8	NUM
ejpam-5364	335	47	k10,12	k10,12	PROPN
ejpam-5364	335	48	12	12	NUM
ejpam-5364	335	49	10.799	10.799	NUM
ejpam-5364	335	50	10.7989	10.7989	NUM
ejpam-5364	335	51	9	9	NUM
ejpam-5364	335	52	k11,13	k11,13	PROPN
ejpam-5364	335	53	13	13	NUM
ejpam-5364	335	54	11.8584	11.8584	NUM
ejpam-5364	335	55	11.6062	11.6062	NUM
ejpam-5364	335	56	10	10	NUM
ejpam-5364	335	57	k12,14	k12,14	ADP
ejpam-5364	335	58	14	14	NUM
ejpam-5364	335	59	12.8802	12.8802	NUM
ejpam-5364	335	60	12.7353	12.7353	NUM
ejpam-5364	335	61	table	table	NOUN
ejpam-5364	335	62	3	3	NUM
ejpam-5364	335	63	:	:	PUNCT
ejpam-5364	335	64	lower	low	ADJ
ejpam-5364	335	65	bounds	bound	NOUN
ejpam-5364	335	66	of	of	ADP
ejpam-5364	335	67	q2	q2	NOUN
ejpam-5364	335	68	for	for	ADP
ejpam-5364	335	69	km	km	PROPN
ejpam-5364	335	70	,	,	PUNCT
ejpam-5364	335	71	n	n	PRON
ejpam-5364	335	72	figure	figure	VERB
ejpam-5364	335	73	11	11	NUM
ejpam-5364	335	74	:	:	PUNCT
ejpam-5364	335	75	skewness	skewness	NOUN
ejpam-5364	335	76	figure	figure	NOUN
ejpam-5364	335	77	12	12	NUM
ejpam-5364	335	78	:	:	PUNCT
ejpam-5364	335	79	thickness	thickness	PROPN
ejpam-5364	335	80	numerical	numerical	ADJ
ejpam-5364	335	81	values	value	NOUN
ejpam-5364	335	82	of	of	ADP
ejpam-5364	335	83	the	the	DET
ejpam-5364	335	84	lower	low	ADJ
ejpam-5364	335	85	bounds	bound	NOUN
ejpam-5364	335	86	of	of	ADP
ejpam-5364	335	87	q2	q2	NOUN
ejpam-5364	335	88	proved	prove	VERB
ejpam-5364	335	89	in	in	ADP
ejpam-5364	335	90	theorems	theorem	NOUN
ejpam-5364	335	91	4	4	NUM
ejpam-5364	335	92	and	and	CCONJ
ejpam-5364	335	93	8	8	NUM
ejpam-5364	335	94	are	be	AUX
ejpam-5364	335	95	presented	present	VERB
ejpam-5364	335	96	for	for	ADP
ejpam-5364	335	97	complete	complete	ADJ
ejpam-5364	335	98	bipartite	bipartite	PROPN
ejpam-5364	335	99	graphs	graph	NOUN
ejpam-5364	335	100	kn	kn	PROPN
ejpam-5364	335	101	,	,	PUNCT
ejpam-5364	335	102	n	n	CCONJ
ejpam-5364	335	103	in	in	ADP
ejpam-5364	335	104	table	table	NOUN
ejpam-5364	335	105	4	4	NUM
ejpam-5364	335	106	.	.	PUNCT
ejpam-5364	336	1	its	its	PRON
ejpam-5364	336	2	graphical	graphical	ADJ
ejpam-5364	336	3	representations	representation	NOUN
ejpam-5364	336	4	are	be	AUX
ejpam-5364	336	5	given	give	VERB
ejpam-5364	336	6	in	in	ADP
ejpam-5364	336	7	figure	figure	NOUN
ejpam-5364	336	8	13	13	NUM
ejpam-5364	336	9	and	and	CCONJ
ejpam-5364	336	10	figure	figure	VERB
ejpam-5364	336	11	14	14	NUM
ejpam-5364	336	12	.	.	PUNCT
ejpam-5364	337	1	m.	m.	PROPN
ejpam-5364	337	2	machasri	machasri	PROPN
ejpam-5364	337	3	,	,	PUNCT
ejpam-5364	337	4	d.	d.	PROPN
ejpam-5364	337	5	kalyani	kalyani	PROPN
ejpam-5364	337	6	/	/	SYM
ejpam-5364	337	7	eur	eur	PROPN
ejpam-5364	337	8	.	.	PUNCT
ejpam-5364	338	1	j.	j.	PROPN
ejpam-5364	338	2	pure	pure	PROPN
ejpam-5364	338	3	appl	appl	PROPN
ejpam-5364	338	4	.	.	PROPN
ejpam-5364	338	5	math	math	PROPN
ejpam-5364	338	6	,	,	PUNCT
ejpam-5364	338	7	17	17	NUM
ejpam-5364	338	8	(	(	PUNCT
ejpam-5364	338	9	4	4	NUM
ejpam-5364	338	10	)	)	PUNCT
ejpam-5364	338	11	(	(	PUNCT
ejpam-5364	338	12	2024	2024	NUM
ejpam-5364	338	13	)	)	PUNCT
ejpam-5364	338	14	,	,	PUNCT
ejpam-5364	338	15	3004	3004	NUM
ejpam-5364	338	16	-	-	SYM
ejpam-5364	338	17	3021	3021	NUM
ejpam-5364	338	18	3018	3018	NUM
ejpam-5364	338	19	s.no	s.no	PROPN
ejpam-5364	338	20	.	.	PROPN
ejpam-5364	338	21	graph	graph	NOUN
ejpam-5364	338	22	q2	q2	NOUN
ejpam-5364	338	23	theorem	theorem	VERB
ejpam-5364	338	24	4	4	NUM
ejpam-5364	338	25	bound	bind	VERB
ejpam-5364	338	26	theorem	theorem	NOUN
ejpam-5364	338	27	8	8	NUM
ejpam-5364	338	28	bound	bind	VERB
ejpam-5364	338	29	1	1	NUM
ejpam-5364	338	30	k3,3	k3,3	PROPN
ejpam-5364	338	31	3	3	NUM
ejpam-5364	338	32	2.1818	2.1818	NUM
ejpam-5364	338	33	1.8	1.8	NUM
ejpam-5364	338	34	2	2	NUM
ejpam-5364	338	35	k4,4	k4,4	PROPN
ejpam-5364	338	36	4	4	NUM
ejpam-5364	338	37	3.2	3.2	NUM
ejpam-5364	338	38	2.909	2.909	NUM
ejpam-5364	338	39	3	3	NUM
ejpam-5364	338	40	k5,5	k5,5	PROPN
ejpam-5364	338	41	5	5	NUM
ejpam-5364	338	42	4.2105	4.2105	NUM
ejpam-5364	338	43	4.032	4.032	NUM
ejpam-5364	338	44	4	4	NUM
ejpam-5364	338	45	k6,6	k6,6	PROPN
ejpam-5364	338	46	6	6	NUM
ejpam-5364	338	47	5.2174	5.2174	NUM
ejpam-5364	338	48	5.143	5.143	NUM
ejpam-5364	338	49	5	5	NUM
ejpam-5364	338	50	k7,7	k7,7	NOUN
ejpam-5364	338	51	7	7	NUM
ejpam-5364	338	52	6.222	6.222	NUM
ejpam-5364	338	53	5.914	5.914	NUM
ejpam-5364	338	54	6	6	NUM
ejpam-5364	338	55	k8,8	k8,8	PROPN
ejpam-5364	338	56	8	8	NUM
ejpam-5364	338	57	7.22	7.22	NUM
ejpam-5364	338	58	7.014	7.014	NUM
ejpam-5364	338	59	7	7	NUM
ejpam-5364	338	60	k9,9	k9,9	PROPN
ejpam-5364	338	61	9	9	NUM
ejpam-5364	338	62	8.2286	8.2286	NUM
ejpam-5364	338	63	8.1	8.1	NUM
ejpam-5364	338	64	8	8	NUM
ejpam-5364	338	65	k10,10	k10,10	PROPN
ejpam-5364	338	66	10	10	NUM
ejpam-5364	338	67	9.2307	9.2307	NUM
ejpam-5364	338	68	9.174	9.174	NUM
ejpam-5364	338	69	9	9	NUM
ejpam-5364	338	70	k11,11	k11,11	PROPN
ejpam-5364	338	71	11	11	NUM
ejpam-5364	338	72	10.2326	10.2326	NUM
ejpam-5364	338	73	10.008	10.008	NUM
ejpam-5364	338	74	10	10	NUM
ejpam-5364	338	75	k12,12	k12,12	PROPN
ejpam-5364	338	76	12	12	NUM
ejpam-5364	338	77	11.234	11.234	NUM
ejpam-5364	338	78	11.0769	11.0769	NUM
ejpam-5364	338	79	table	table	NOUN
ejpam-5364	338	80	4	4	NUM
ejpam-5364	338	81	:	:	PUNCT
ejpam-5364	338	82	lower	low	ADJ
ejpam-5364	338	83	bounds	bound	NOUN
ejpam-5364	338	84	of	of	ADP
ejpam-5364	338	85	q2	q2	NOUN
ejpam-5364	338	86	for	for	ADP
ejpam-5364	338	87	kn	kn	PROPN
ejpam-5364	338	88	,	,	PUNCT
ejpam-5364	338	89	n	n	PRON
ejpam-5364	338	90	figure	figure	VERB
ejpam-5364	338	91	13	13	NUM
ejpam-5364	338	92	:	:	PUNCT
ejpam-5364	338	93	skewness	skewness	NOUN
ejpam-5364	338	94	figure	figure	NOUN
ejpam-5364	338	95	14	14	NUM
ejpam-5364	338	96	:	:	PUNCT
ejpam-5364	338	97	thickness	thickness	NOUN
ejpam-5364	338	98	3.3	3.3	NUM
ejpam-5364	338	99	.	.	PUNCT
ejpam-5364	339	1	crossing	crossing	NOUN
ejpam-5364	339	2	number	number	NOUN
ejpam-5364	339	3	this	this	DET
ejpam-5364	339	4	section	section	NOUN
ejpam-5364	339	5	presents	present	VERB
ejpam-5364	339	6	the	the	DET
ejpam-5364	339	7	theorems	theorem	NOUN
ejpam-5364	339	8	connecting	connect	VERB
ejpam-5364	339	9	a	a	DET
ejpam-5364	339	10	graph	graph	NOUN
ejpam-5364	339	11	’s	’s	PART
ejpam-5364	339	12	crossing	crossing	NOUN
ejpam-5364	339	13	number	number	NOUN
ejpam-5364	339	14	cr(g	cr(g	PUNCT
ejpam-5364	339	15	)	)	PUNCT
ejpam-5364	339	16	and	and	CCONJ
ejpam-5364	339	17	second	second	ADV
ejpam-5364	339	18	largest	large	ADJ
ejpam-5364	339	19	adjacency	adjacency	NOUN
ejpam-5364	339	20	and	and	CCONJ
ejpam-5364	339	21	signless	signless	ADJ
ejpam-5364	339	22	laplacian	laplacian	ADJ
ejpam-5364	339	23	eigenvalues	eigenvalue	NOUN
ejpam-5364	339	24	.	.	PUNCT
ejpam-5364	340	1	lower	low	ADJ
ejpam-5364	340	2	bounds	bound	NOUN
ejpam-5364	340	3	are	be	AUX
ejpam-5364	340	4	established	establish	VERB
ejpam-5364	340	5	for	for	ADP
ejpam-5364	340	6	λ2	λ2	NOUN
ejpam-5364	340	7	and	and	CCONJ
ejpam-5364	340	8	q2	q2	NOUN
ejpam-5364	340	9	in	in	ADP
ejpam-5364	340	10	terms	term	NOUN
ejpam-5364	340	11	of	of	ADP
ejpam-5364	340	12	cr(g	cr(g	PROPN
ejpam-5364	340	13	)	)	PUNCT
ejpam-5364	340	14	and	and	CCONJ
ejpam-5364	340	15	lower	low	ADJ
ejpam-5364	340	16	bounds	bound	NOUN
ejpam-5364	340	17	for	for	ADP
ejpam-5364	340	18	the	the	DET
ejpam-5364	340	19	crossing	crossing	NOUN
ejpam-5364	340	20	number	number	NOUN
ejpam-5364	340	21	of	of	ADP
ejpam-5364	340	22	regular	regular	ADJ
ejpam-5364	340	23	graphs	graph	NOUN
ejpam-5364	340	24	are	be	AUX
ejpam-5364	340	25	determined	determine	VERB
ejpam-5364	340	26	in	in	ADP
ejpam-5364	340	27	terms	term	NOUN
ejpam-5364	340	28	of	of	ADP
ejpam-5364	340	29	λ2	λ2	NOUN
ejpam-5364	340	30	and	and	CCONJ
ejpam-5364	340	31	q2	q2	NOUN
ejpam-5364	340	32	.	.	PUNCT
ejpam-5364	341	1	theorem	theorem	VERB
ejpam-5364	341	2	9	9	NUM
ejpam-5364	341	3	.	.	PUNCT
ejpam-5364	342	1	let	let	VERB
ejpam-5364	342	2	g	g	PRON
ejpam-5364	342	3	be	be	AUX
ejpam-5364	342	4	a	a	DET
ejpam-5364	342	5	connected	connected	ADJ
ejpam-5364	342	6	graph	graph	NOUN
ejpam-5364	342	7	with	with	ADP
ejpam-5364	342	8	crossing	crossing	NOUN
ejpam-5364	342	9	number	number	NOUN
ejpam-5364	342	10	cr(g	cr(g	PUNCT
ejpam-5364	342	11	)	)	PUNCT
ejpam-5364	342	12	.	.	PUNCT
ejpam-5364	343	1	then	then	ADV
ejpam-5364	343	2	λ2	λ2	PRON
ejpam-5364	343	3	≥	≥	NUM
ejpam-5364	343	4	1	1	NUM
ejpam-5364	343	5	2	2	NUM
ejpam-5364	343	6	{	{	PUNCT
ejpam-5364	343	7	[	[	X
ejpam-5364	343	8	δ	δ	X
ejpam-5364	343	9	+	+	NOUN
ejpam-5364	343	10	∆(1−	∆(1−	ADV
ejpam-5364	343	11	u)]−	u)]−	NOUN
ejpam-5364	343	12	√	√	PROPN
ejpam-5364	344	1	[	[	X
ejpam-5364	344	2	δ	δ	X
ejpam-5364	344	3	−∆(1−	−∆(1−	VERB
ejpam-5364	345	1	u)]2	u)]2	DET
ejpam-5364	345	2	−	−	PROPN
ejpam-5364	345	3	4∆(∆−	4∆(∆−	NUM
ejpam-5364	345	4	δ)(1−	δ)(1−	PROPN
ejpam-5364	345	5	u	u	NOUN
ejpam-5364	345	6	)	)	PUNCT
ejpam-5364	345	7	}	}	PUNCT
ejpam-5364	345	8	where	where	SCONJ
ejpam-5364	345	9	u	u	NOUN
ejpam-5364	345	10	=	=	NOUN
ejpam-5364	345	11	m	m	VERB
ejpam-5364	345	12	√	√	VERB
ejpam-5364	345	13	m	m	VERB
ejpam-5364	345	14	m	m	VERB
ejpam-5364	345	15	√	√	ADJ
ejpam-5364	345	16	m−	m−	PROPN
ejpam-5364	345	17	√	√	NUM
ejpam-5364	345	18	29cr	29cr	NOUN
ejpam-5364	345	19	.	.	PUNCT
ejpam-5364	346	1	proof	proof	NOUN
ejpam-5364	346	2	.	.	PUNCT
ejpam-5364	347	1	the	the	DET
ejpam-5364	347	2	proof	proof	NOUN
ejpam-5364	347	3	is	be	AUX
ejpam-5364	347	4	similar	similar	ADJ
ejpam-5364	347	5	to	to	ADP
ejpam-5364	347	6	the	the	DET
ejpam-5364	347	7	proof	proof	NOUN
ejpam-5364	347	8	of	of	ADP
ejpam-5364	347	9	theorem	theorem	NOUN
ejpam-5364	347	10	1	1	NUM
ejpam-5364	347	11	.	.	PUNCT
ejpam-5364	348	1	the	the	DET
ejpam-5364	348	2	result	result	NOUN
ejpam-5364	348	3	is	be	AUX
ejpam-5364	348	4	obtained	obtain	VERB
ejpam-5364	348	5	by	by	ADP
ejpam-5364	348	6	using	use	VERB
ejpam-5364	348	7	inequality	inequality	NOUN
ejpam-5364	348	8	(	(	PUNCT
ejpam-5364	348	9	3	3	NUM
ejpam-5364	348	10	)	)	PUNCT
ejpam-5364	348	11	.	.	PUNCT
ejpam-5364	349	1	theorem	theorem	ADJ
ejpam-5364	349	2	10	10	NUM
ejpam-5364	349	3	.	.	PUNCT
ejpam-5364	350	1	let	let	VERB
ejpam-5364	350	2	g	g	PRON
ejpam-5364	350	3	be	be	AUX
ejpam-5364	350	4	a	a	DET
ejpam-5364	350	5	d−regular	d−regular	NUM
ejpam-5364	350	6	graph	graph	NOUN
ejpam-5364	350	7	with	with	ADP
ejpam-5364	350	8	n	n	PRON
ejpam-5364	350	9	≥	≥	NUM
ejpam-5364	350	10	3	3	NUM
ejpam-5364	350	11	.	.	PUNCT
ejpam-5364	351	1	then	then	ADV
ejpam-5364	351	2	λ2	λ2	PROPN
ejpam-5364	351	3	≥	≥	PROPN
ejpam-5364	351	4	d	d	NOUN
ejpam-5364	351	5	(	(	PUNCT
ejpam-5364	351	6	1−	1−	NUM
ejpam-5364	351	7	m	m	NOUN
ejpam-5364	351	8	√	√	NOUN
ejpam-5364	351	9	m	m	VERB
ejpam-5364	351	10	m	m	VERB
ejpam-5364	351	11	√	√	ADJ
ejpam-5364	351	12	m−	m−	PROPN
ejpam-5364	351	13	√	√	NUM
ejpam-5364	351	14	29cr	29cr	NOUN
ejpam-5364	351	15	)	)	PUNCT
ejpam-5364	351	16	.	.	PUNCT
ejpam-5364	352	1	m.	m.	PROPN
ejpam-5364	352	2	machasri	machasri	PROPN
ejpam-5364	352	3	,	,	PUNCT
ejpam-5364	352	4	d.	d.	PROPN
ejpam-5364	352	5	kalyani	kalyani	PROPN
ejpam-5364	352	6	/	/	SYM
ejpam-5364	352	7	eur	eur	PROPN
ejpam-5364	352	8	.	.	PUNCT
ejpam-5364	353	1	j.	j.	PROPN
ejpam-5364	353	2	pure	pure	PROPN
ejpam-5364	353	3	appl	appl	PROPN
ejpam-5364	353	4	.	.	PROPN
ejpam-5364	353	5	math	math	PROPN
ejpam-5364	353	6	,	,	PUNCT
ejpam-5364	353	7	17	17	NUM
ejpam-5364	353	8	(	(	PUNCT
ejpam-5364	353	9	4	4	NUM
ejpam-5364	353	10	)	)	PUNCT
ejpam-5364	353	11	(	(	PUNCT
ejpam-5364	353	12	2024	2024	NUM
ejpam-5364	353	13	)	)	PUNCT
ejpam-5364	353	14	,	,	PUNCT
ejpam-5364	353	15	3004	3004	NUM
ejpam-5364	353	16	-	-	SYM
ejpam-5364	353	17	3021	3021	NUM
ejpam-5364	353	18	3019	3019	NUM
ejpam-5364	353	19	proof	proof	NOUN
ejpam-5364	353	20	.	.	PUNCT
ejpam-5364	354	1	the	the	DET
ejpam-5364	354	2	proof	proof	NOUN
ejpam-5364	354	3	is	be	AUX
ejpam-5364	354	4	on	on	ADP
ejpam-5364	354	5	the	the	DET
ejpam-5364	354	6	same	same	ADJ
ejpam-5364	354	7	lines	line	NOUN
ejpam-5364	354	8	as	as	ADP
ejpam-5364	354	9	the	the	DET
ejpam-5364	354	10	proof	proof	NOUN
ejpam-5364	354	11	of	of	ADP
ejpam-5364	354	12	theorem	theorem	NOUN
ejpam-5364	354	13	2	2	NUM
ejpam-5364	354	14	.	.	PUNCT
ejpam-5364	354	15	by	by	ADP
ejpam-5364	354	16	using	use	VERB
ejpam-5364	354	17	inequality	inequality	NOUN
ejpam-5364	354	18	(	(	PUNCT
ejpam-5364	354	19	3	3	NUM
ejpam-5364	354	20	)	)	PUNCT
ejpam-5364	354	21	,	,	PUNCT
ejpam-5364	354	22	we	we	PRON
ejpam-5364	354	23	get	get	VERB
ejpam-5364	354	24	the	the	DET
ejpam-5364	354	25	result	result	NOUN
ejpam-5364	354	26	.	.	PUNCT
ejpam-5364	355	1	note	note	VERB
ejpam-5364	355	2	5	5	NUM
ejpam-5364	355	3	.	.	PUNCT
ejpam-5364	356	1	if	if	SCONJ
ejpam-5364	356	2	g	g	PROPN
ejpam-5364	356	3	is	be	AUX
ejpam-5364	356	4	a	a	DET
ejpam-5364	356	5	d−regular	d−regular	NUM
ejpam-5364	356	6	graph	graph	NOUN
ejpam-5364	356	7	with	with	ADP
ejpam-5364	356	8	n	n	NUM
ejpam-5364	356	9	≥	≥	NOUN
ejpam-5364	356	10	3	3	NUM
ejpam-5364	356	11	then	then	ADV
ejpam-5364	356	12	cr	cr	NOUN
ejpam-5364	356	13	≤	≤	NUM
ejpam-5364	356	14	m3λ2	m3λ2	PUNCT
ejpam-5364	356	15	2	2	NUM
ejpam-5364	356	16	29(d−	29(d−	PROPN
ejpam-5364	356	17	λ2)2	λ2)2	X
ejpam-5364	356	18	.	.	PUNCT
ejpam-5364	357	1	theorem	theorem	ADJ
ejpam-5364	357	2	11	11	NUM
ejpam-5364	357	3	.	.	PUNCT
ejpam-5364	358	1	let	let	VERB
ejpam-5364	358	2	g	g	PRON
ejpam-5364	358	3	be	be	AUX
ejpam-5364	358	4	a	a	DET
ejpam-5364	358	5	connected	connected	ADJ
ejpam-5364	358	6	graph	graph	NOUN
ejpam-5364	358	7	.	.	PUNCT
ejpam-5364	359	1	then	then	ADV
ejpam-5364	359	2	q2	q2	PROPN
ejpam-5364	359	3	≥	≥	NUM
ejpam-5364	359	4	1	1	NUM
ejpam-5364	359	5	2	2	NUM
ejpam-5364	359	6	{	{	PUNCT
ejpam-5364	359	7	[	[	X
ejpam-5364	359	8	2(∆	2(∆	NUM
ejpam-5364	359	9	+	+	CCONJ
ejpam-5364	359	10	δ)−∆u]−	δ)−∆u]−	NOUN
ejpam-5364	359	11	√	√	PROPN
ejpam-5364	360	1	[	[	X
ejpam-5364	360	2	2(∆−	2(∆−	NUM
ejpam-5364	360	3	δ)−∆u]2	δ)−∆u]2	NUM
ejpam-5364	360	4	−	−	NOUN
ejpam-5364	360	5	8∆(∆−	8∆(∆−	NUM
ejpam-5364	360	6	δ)(1−	δ)(1−	PROPN
ejpam-5364	360	7	u	u	NOUN
ejpam-5364	360	8	)	)	PUNCT
ejpam-5364	360	9	}	}	PUNCT
ejpam-5364	360	10	where	where	SCONJ
ejpam-5364	360	11	u	u	NOUN
ejpam-5364	360	12	=	=	NOUN
ejpam-5364	360	13	m	m	VERB
ejpam-5364	360	14	√	√	VERB
ejpam-5364	360	15	m	m	VERB
ejpam-5364	360	16	m	m	VERB
ejpam-5364	360	17	√	√	ADJ
ejpam-5364	360	18	m−	m−	PROPN
ejpam-5364	360	19	√	√	NUM
ejpam-5364	360	20	29cr	29cr	NOUN
ejpam-5364	360	21	.	.	PUNCT
ejpam-5364	361	1	proof	proof	NOUN
ejpam-5364	361	2	.	.	PUNCT
ejpam-5364	362	1	the	the	DET
ejpam-5364	362	2	proof	proof	NOUN
ejpam-5364	362	3	is	be	AUX
ejpam-5364	362	4	similar	similar	ADJ
ejpam-5364	362	5	to	to	ADP
ejpam-5364	362	6	the	the	DET
ejpam-5364	362	7	proof	proof	NOUN
ejpam-5364	362	8	of	of	ADP
ejpam-5364	362	9	theorem	theorem	ADJ
ejpam-5364	362	10	3	3	NUM
ejpam-5364	362	11	.	.	X
ejpam-5364	362	12	inequality	inequality	NOUN
ejpam-5364	362	13	(	(	PUNCT
ejpam-5364	362	14	3	3	NUM
ejpam-5364	362	15	)	)	PUNCT
ejpam-5364	362	16	is	be	AUX
ejpam-5364	362	17	used	use	VERB
ejpam-5364	362	18	to	to	PART
ejpam-5364	362	19	get	get	VERB
ejpam-5364	362	20	the	the	DET
ejpam-5364	362	21	result	result	NOUN
ejpam-5364	362	22	.	.	PUNCT
ejpam-5364	363	1	theorem	theorem	NOUN
ejpam-5364	363	2	12	12	NUM
ejpam-5364	363	3	.	.	PUNCT
ejpam-5364	364	1	let	let	VERB
ejpam-5364	364	2	g	g	PRON
ejpam-5364	364	3	be	be	AUX
ejpam-5364	364	4	a	a	DET
ejpam-5364	364	5	d−regular	d−regular	NUM
ejpam-5364	364	6	graph	graph	NOUN
ejpam-5364	364	7	with	with	ADP
ejpam-5364	364	8	n	n	PRON
ejpam-5364	364	9	≥	≥	NUM
ejpam-5364	364	10	3	3	NUM
ejpam-5364	364	11	.	.	PUNCT
ejpam-5364	365	1	then	then	ADV
ejpam-5364	365	2	q2	q2	PROPN
ejpam-5364	365	3	≥	≥	PRON
ejpam-5364	366	1	2d−	2d−	PROPN
ejpam-5364	366	2	dm	dm	VERB
ejpam-5364	366	3	√	√	NUM
ejpam-5364	366	4	m	m	VERB
ejpam-5364	366	5	m	m	VERB
ejpam-5364	366	6	√	√	ADJ
ejpam-5364	366	7	m−	m−	PROPN
ejpam-5364	366	8	√	√	NUM
ejpam-5364	366	9	29cr	29cr	NOUN
ejpam-5364	366	10	.	.	PUNCT
ejpam-5364	367	1	proof	proof	NOUN
ejpam-5364	367	2	.	.	PUNCT
ejpam-5364	368	1	the	the	DET
ejpam-5364	368	2	proof	proof	NOUN
ejpam-5364	368	3	is	be	AUX
ejpam-5364	368	4	on	on	ADP
ejpam-5364	368	5	the	the	DET
ejpam-5364	368	6	same	same	ADJ
ejpam-5364	368	7	lines	line	NOUN
ejpam-5364	368	8	as	as	ADP
ejpam-5364	368	9	the	the	DET
ejpam-5364	368	10	proof	proof	NOUN
ejpam-5364	368	11	of	of	ADP
ejpam-5364	368	12	theorem	theorem	ADJ
ejpam-5364	368	13	4	4	NUM
ejpam-5364	368	14	.	.	PUNCT
ejpam-5364	368	15	inequality	inequality	NOUN
ejpam-5364	368	16	(	(	PUNCT
ejpam-5364	368	17	3	3	NUM
ejpam-5364	368	18	)	)	PUNCT
ejpam-5364	368	19	is	be	AUX
ejpam-5364	368	20	used	use	VERB
ejpam-5364	368	21	to	to	PART
ejpam-5364	368	22	obtain	obtain	VERB
ejpam-5364	368	23	the	the	DET
ejpam-5364	368	24	result	result	NOUN
ejpam-5364	368	25	.	.	PUNCT
ejpam-5364	369	1	note	note	VERB
ejpam-5364	369	2	6	6	NUM
ejpam-5364	369	3	.	.	PUNCT
ejpam-5364	370	1	if	if	SCONJ
ejpam-5364	370	2	g	g	PROPN
ejpam-5364	370	3	is	be	AUX
ejpam-5364	370	4	a	a	DET
ejpam-5364	370	5	d−regular	d−regular	NUM
ejpam-5364	370	6	graph	graph	NOUN
ejpam-5364	370	7	with	with	ADP
ejpam-5364	370	8	n	n	NUM
ejpam-5364	370	9	≥	≥	NOUN
ejpam-5364	370	10	3	3	NUM
ejpam-5364	370	11	then	then	ADV
ejpam-5364	370	12	cr	cr	PROPN
ejpam-5364	370	13	≤	≤	PROPN
ejpam-5364	370	14	m3	m3	PROPN
ejpam-5364	370	15	29	29	NUM
ejpam-5364	370	16	(	(	PUNCT
ejpam-5364	370	17	d−	d−	PROPN
ejpam-5364	370	18	q2	q2	NOUN
ejpam-5364	370	19	2d−	2d−	PROPN
ejpam-5364	370	20	q2	q2	NOUN
ejpam-5364	370	21	)	)	PUNCT
ejpam-5364	370	22	2	2	NUM
ejpam-5364	370	23	.	.	PUNCT
ejpam-5364	370	24	example	example	NOUN
ejpam-5364	370	25	4	4	NUM
ejpam-5364	370	26	.	.	X
ejpam-5364	371	1	for	for	ADP
ejpam-5364	371	2	the	the	DET
ejpam-5364	371	3	graph	graph	NOUN
ejpam-5364	371	4	represented	represent	VERB
ejpam-5364	371	5	in	in	ADP
ejpam-5364	371	6	figure	figure	NOUN
ejpam-5364	371	7	5	5	NUM
ejpam-5364	371	8	with	with	ADP
ejpam-5364	371	9	crossing	crossing	NOUN
ejpam-5364	371	10	number	number	NOUN
ejpam-5364	371	11	cr(g	cr(g	PUNCT
ejpam-5364	371	12	)	)	PUNCT
ejpam-5364	371	13	=	=	SYM
ejpam-5364	371	14	5	5	NUM
ejpam-5364	371	15	,	,	PUNCT
ejpam-5364	371	16	the	the	DET
ejpam-5364	371	17	lower	lower	ADV
ejpam-5364	371	18	bound	bind	VERB
ejpam-5364	371	19	from	from	ADP
ejpam-5364	371	20	theorem	theorem	ADJ
ejpam-5364	371	21	9	9	NUM
ejpam-5364	371	22	is	be	AUX
ejpam-5364	371	23	given	give	VERB
ejpam-5364	371	24	by	by	ADP
ejpam-5364	371	25	λ2	λ2	PROPN
ejpam-5364	371	26	≥	≥	NUM
ejpam-5364	371	27	−1.971	−1.971	NOUN
ejpam-5364	371	28	.	.	PUNCT
ejpam-5364	371	29	example	example	NOUN
ejpam-5364	372	1	5	5	NUM
ejpam-5364	372	2	.	.	X
ejpam-5364	372	3	for	for	ADP
ejpam-5364	372	4	the	the	DET
ejpam-5364	372	5	graph	graph	NOUN
ejpam-5364	372	6	illustrated	illustrate	VERB
ejpam-5364	372	7	in	in	ADP
ejpam-5364	372	8	figure	figure	NOUN
ejpam-5364	372	9	6	6	NUM
ejpam-5364	372	10	with	with	ADP
ejpam-5364	372	11	crossing	crossing	NOUN
ejpam-5364	372	12	number	number	NOUN
ejpam-5364	372	13	cr(g	cr(g	PUNCT
ejpam-5364	372	14	)	)	PUNCT
ejpam-5364	373	1	=	=	SYM
ejpam-5364	373	2	1	1	NUM
ejpam-5364	373	3	,	,	PUNCT
ejpam-5364	373	4	the	the	DET
ejpam-5364	373	5	lower	lower	ADV
ejpam-5364	373	6	bound	bind	VERB
ejpam-5364	373	7	obtained	obtain	VERB
ejpam-5364	373	8	using	use	VERB
ejpam-5364	373	9	theorem	theorem	NOUN
ejpam-5364	373	10	10	10	NUM
ejpam-5364	373	11	is	be	AUX
ejpam-5364	373	12	λ2	λ2	NUM
ejpam-5364	373	13	≥	≥	NOUN
ejpam-5364	373	14	−0.747	−0.747	PROPN
ejpam-5364	373	15	and	and	CCONJ
ejpam-5364	373	16	lower	low	ADJ
ejpam-5364	373	17	bound	bind	VERB
ejpam-5364	373	18	obtained	obtain	VERB
ejpam-5364	373	19	using	use	VERB
ejpam-5364	373	20	theorem	theorem	NOUN
ejpam-5364	373	21	12	12	NUM
ejpam-5364	373	22	is	be	AUX
ejpam-5364	373	23	q2	q2	NOUN
ejpam-5364	373	24	≥	≥	NUM
ejpam-5364	373	25	2.253	2.253	NUM
ejpam-5364	373	26	.	.	PUNCT
ejpam-5364	373	27	example	example	NOUN
ejpam-5364	374	1	6	6	NUM
ejpam-5364	374	2	.	.	PUNCT
ejpam-5364	374	3	let	let	VERB
ejpam-5364	374	4	us	we	PRON
ejpam-5364	374	5	consider	consider	VERB
ejpam-5364	374	6	the	the	DET
ejpam-5364	374	7	petersen	petersen	NOUN
ejpam-5364	374	8	graph	graph	NOUN
ejpam-5364	374	9	.	.	PUNCT
ejpam-5364	375	1	it	it	PRON
ejpam-5364	375	2	has	have	VERB
ejpam-5364	375	3	n	n	NOUN
ejpam-5364	375	4	=	=	SYM
ejpam-5364	375	5	10	10	NUM
ejpam-5364	375	6	,	,	PUNCT
ejpam-5364	375	7	m	m	VERB
ejpam-5364	375	8	=	=	NOUN
ejpam-5364	375	9	15	15	NUM
ejpam-5364	375	10	,	,	PUNCT
ejpam-5364	375	11	d	d	NOUN
ejpam-5364	375	12	=	=	SYM
ejpam-5364	375	13	3	3	NUM
ejpam-5364	375	14	,	,	PUNCT
ejpam-5364	375	15	λ2	λ2	NOUN
ejpam-5364	375	16	=	=	SYM
ejpam-5364	375	17	1	1	NUM
ejpam-5364	375	18	and	and	CCONJ
ejpam-5364	375	19	q2	q2	NOUN
ejpam-5364	375	20	=	=	SYM
ejpam-5364	376	1	4	4	X
ejpam-5364	376	2	.	.	PUNCT
ejpam-5364	376	3	then	then	ADV
ejpam-5364	376	4	the	the	DET
ejpam-5364	376	5	lower	low	ADJ
ejpam-5364	376	6	bounds	bound	NOUN
ejpam-5364	376	7	from	from	ADP
ejpam-5364	376	8	theorems	theorem	NOUN
ejpam-5364	376	9	10	10	NUM
ejpam-5364	376	10	and	and	CCONJ
ejpam-5364	376	11	12	12	NUM
ejpam-5364	376	12	are	be	AUX
ejpam-5364	376	13	given	give	VERB
ejpam-5364	376	14	by	by	ADP
ejpam-5364	376	15	λ2	λ2	PROPN
ejpam-5364	376	16	≥	≥	NOUN
ejpam-5364	376	17	−0.7844	−0.7844	NOUN
ejpam-5364	376	18	and	and	CCONJ
ejpam-5364	376	19	q2	q2	PROPN
ejpam-5364	376	20	≥	≥	NUM
ejpam-5364	376	21	2.2156	2.2156	NUM
ejpam-5364	376	22	.	.	PUNCT
ejpam-5364	377	1	references	reference	NOUN
ejpam-5364	377	2	3020	3020	NUM
ejpam-5364	377	3	4	4	NUM
ejpam-5364	377	4	.	.	PUNCT
ejpam-5364	377	5	conclusion	conclusion	NOUN
ejpam-5364	377	6	planarity	planarity	NOUN
ejpam-5364	377	7	is	be	AUX
ejpam-5364	377	8	an	an	DET
ejpam-5364	377	9	important	important	ADJ
ejpam-5364	377	10	field	field	NOUN
ejpam-5364	377	11	of	of	ADP
ejpam-5364	377	12	study	study	NOUN
ejpam-5364	377	13	in	in	ADP
ejpam-5364	377	14	graph	graph	NOUN
ejpam-5364	377	15	theory	theory	NOUN
ejpam-5364	377	16	since	since	SCONJ
ejpam-5364	377	17	it	it	PRON
ejpam-5364	377	18	is	be	AUX
ejpam-5364	377	19	essential	essential	ADJ
ejpam-5364	377	20	for	for	ADP
ejpam-5364	377	21	both	both	CCONJ
ejpam-5364	377	22	theoretical	theoretical	ADJ
ejpam-5364	377	23	understanding	understanding	NOUN
ejpam-5364	377	24	and	and	CCONJ
ejpam-5364	377	25	real	real	ADJ
ejpam-5364	377	26	-	-	PUNCT
ejpam-5364	377	27	world	world	NOUN
ejpam-5364	377	28	applications	application	NOUN
ejpam-5364	377	29	.	.	PUNCT
ejpam-5364	378	1	in	in	ADP
ejpam-5364	378	2	this	this	DET
ejpam-5364	378	3	work	work	NOUN
ejpam-5364	378	4	,	,	PUNCT
ejpam-5364	378	5	the	the	DET
ejpam-5364	378	6	relationship	relationship	NOUN
ejpam-5364	378	7	between	between	ADP
ejpam-5364	378	8	the	the	DET
ejpam-5364	378	9	parameters	parameter	NOUN
ejpam-5364	378	10	associated	associate	VERB
ejpam-5364	378	11	with	with	ADP
ejpam-5364	378	12	planar	planar	ADJ
ejpam-5364	378	13	graphs	graph	NOUN
ejpam-5364	378	14	and	and	CCONJ
ejpam-5364	378	15	a	a	DET
ejpam-5364	378	16	graph	graph	NOUN
ejpam-5364	378	17	’s	’s	PART
ejpam-5364	378	18	second	second	ADV
ejpam-5364	378	19	largest	large	ADJ
ejpam-5364	378	20	adjacency	adjacency	NOUN
ejpam-5364	378	21	,	,	PUNCT
ejpam-5364	378	22	laplacian	laplacian	ADJ
ejpam-5364	378	23	and	and	CCONJ
ejpam-5364	378	24	signless	signless	ADJ
ejpam-5364	378	25	laplacian	laplacian	ADJ
ejpam-5364	378	26	eigenvalues	eigenvalue	NOUN
ejpam-5364	378	27	have	have	AUX
ejpam-5364	378	28	been	be	AUX
ejpam-5364	378	29	determined	determine	VERB
ejpam-5364	378	30	.	.	PUNCT
ejpam-5364	379	1	lower	low	ADJ
ejpam-5364	379	2	bounds	bound	NOUN
ejpam-5364	379	3	on	on	ADP
ejpam-5364	379	4	the	the	DET
ejpam-5364	379	5	second	second	ADV
ejpam-5364	379	6	largest	large	ADJ
ejpam-5364	379	7	adjacency	adjacency	NOUN
ejpam-5364	379	8	and	and	CCONJ
ejpam-5364	379	9	signless	signless	ADJ
ejpam-5364	379	10	laplacian	laplacian	ADJ
ejpam-5364	379	11	eigenvalues	eigenvalue	NOUN
ejpam-5364	379	12	of	of	ADP
ejpam-5364	379	13	a	a	DET
ejpam-5364	379	14	graph	graph	NOUN
ejpam-5364	379	15	are	be	AUX
ejpam-5364	379	16	determined	determine	VERB
ejpam-5364	379	17	in	in	ADP
ejpam-5364	379	18	terms	term	NOUN
ejpam-5364	379	19	of	of	ADP
ejpam-5364	379	20	skewness	skewness	NOUN
ejpam-5364	379	21	sk(g	sk(g	NOUN
ejpam-5364	379	22	)	)	PUNCT
ejpam-5364	379	23	,	,	PUNCT
ejpam-5364	379	24	thickness	thickness	NOUN
ejpam-5364	379	25	τ(g	τ(g	PROPN
ejpam-5364	379	26	)	)	PUNCT
ejpam-5364	379	27	,	,	PUNCT
ejpam-5364	379	28	and	and	CCONJ
ejpam-5364	379	29	crossing	crossing	NOUN
ejpam-5364	379	30	number	number	NOUN
ejpam-5364	379	31	cr(g	cr(g	PUNCT
ejpam-5364	379	32	)	)	PUNCT
ejpam-5364	379	33	of	of	ADP
ejpam-5364	379	34	graphs	graph	NOUN
ejpam-5364	379	35	.	.	PUNCT
ejpam-5364	380	1	in	in	ADP
ejpam-5364	380	2	the	the	DET
ejpam-5364	380	3	future	future	NOUN
ejpam-5364	380	4	,	,	PUNCT
ejpam-5364	380	5	we	we	PRON
ejpam-5364	380	6	will	will	AUX
ejpam-5364	380	7	investigate	investigate	VERB
ejpam-5364	380	8	how	how	SCONJ
ejpam-5364	380	9	the	the	DET
ejpam-5364	380	10	other	other	ADJ
ejpam-5364	380	11	graph	graph	NOUN
ejpam-5364	380	12	parameters	parameter	NOUN
ejpam-5364	380	13	are	be	AUX
ejpam-5364	380	14	related	relate	VERB
ejpam-5364	380	15	with	with	ADP
ejpam-5364	380	16	the	the	DET
ejpam-5364	380	17	second	second	ADV
ejpam-5364	380	18	largest	large	ADJ
ejpam-5364	380	19	adjacency	adjacency	NOUN
ejpam-5364	380	20	and	and	CCONJ
ejpam-5364	380	21	signless	signless	ADJ
ejpam-5364	380	22	laplacian	laplacian	ADJ
ejpam-5364	380	23	eigenvalues	eigenvalue	NOUN
ejpam-5364	380	24	of	of	ADP
ejpam-5364	380	25	graphs	graph	NOUN
ejpam-5364	380	26	.	.	PUNCT
ejpam-5364	381	1	furthermore	furthermore	ADV
ejpam-5364	381	2	,	,	PUNCT
ejpam-5364	381	3	using	use	VERB
ejpam-5364	381	4	λ2	λ2	NOUN
ejpam-5364	381	5	,	,	PUNCT
ejpam-5364	381	6	we	we	PRON
ejpam-5364	381	7	will	will	AUX
ejpam-5364	381	8	focus	focus	VERB
ejpam-5364	381	9	on	on	ADP
ejpam-5364	381	10	characterising	characterise	VERB
ejpam-5364	381	11	graphs	graph	NOUN
ejpam-5364	381	12	.	.	PUNCT
ejpam-5364	382	1	references	reference	NOUN
ejpam-5364	382	2	[	[	X
ejpam-5364	382	3	1	1	NUM
ejpam-5364	382	4	]	]	PUNCT
ejpam-5364	382	5	eyal	eyal	PROPN
ejpam-5364	382	6	ackerman	ackerman	PROPN
ejpam-5364	382	7	.	.	PUNCT
ejpam-5364	383	1	on	on	ADP
ejpam-5364	383	2	topological	topological	ADJ
ejpam-5364	383	3	graphs	graph	NOUN
ejpam-5364	383	4	with	with	ADP
ejpam-5364	383	5	at	at	ADP
ejpam-5364	383	6	most	most	ADJ
ejpam-5364	383	7	four	four	NUM
ejpam-5364	383	8	crossings	crossing	NOUN
ejpam-5364	383	9	per	per	ADP
ejpam-5364	383	10	edge	edge	NOUN
ejpam-5364	383	11	.	.	PUNCT
ejpam-5364	384	1	computational	computational	ADJ
ejpam-5364	384	2	geometry	geometry	NOUN
ejpam-5364	384	3	,	,	PUNCT
ejpam-5364	384	4	85:101574	85:101574	NUM
ejpam-5364	384	5	,	,	PUNCT
ejpam-5364	384	6	2019	2019	NUM
ejpam-5364	384	7	.	.	PUNCT
ejpam-5364	385	1	[	[	X
ejpam-5364	385	2	2	2	X
ejpam-5364	385	3	]	]	X
ejpam-5364	385	4	nasir	nasir	PROPN
ejpam-5364	385	5	ali	ali	PROPN
ejpam-5364	385	6	,	,	PUNCT
ejpam-5364	385	7	hafiz	hafiz	PROPN
ejpam-5364	385	8	muhammad	muhammad	PROPN
ejpam-5364	385	9	afzal	afzal	PROPN
ejpam-5364	385	10	siddiqui	siddiqui	PROPN
ejpam-5364	385	11	,	,	PUNCT
ejpam-5364	385	12	and	and	CCONJ
ejpam-5364	385	13	muhammad	muhammad	PROPN
ejpam-5364	385	14	imran	imran	PROPN
ejpam-5364	385	15	qureshi	qureshi	PROPN
ejpam-5364	385	16	.	.	PUNCT
ejpam-5364	386	1	on	on	ADP
ejpam-5364	386	2	certain	certain	ADJ
ejpam-5364	386	3	bounds	bound	NOUN
ejpam-5364	386	4	for	for	ADP
ejpam-5364	386	5	multiset	multiset	ADJ
ejpam-5364	386	6	dimensions	dimension	NOUN
ejpam-5364	386	7	of	of	ADP
ejpam-5364	386	8	zero	zero	NUM
ejpam-5364	386	9	-	-	PUNCT
ejpam-5364	386	10	divisor	divisor	NOUN
ejpam-5364	386	11	graphs	graph	NOUN
ejpam-5364	386	12	associated	associate	VERB
ejpam-5364	386	13	with	with	ADP
ejpam-5364	386	14	rings	ring	NOUN
ejpam-5364	386	15	.	.	PUNCT
ejpam-5364	387	1	arxiv	arxiv	PROPN
ejpam-5364	387	2	preprint	preprint	NOUN
ejpam-5364	387	3	arxiv:2405.06180	arxiv:2405.06180	NUM
ejpam-5364	387	4	,	,	PUNCT
ejpam-5364	387	5	2024	2024	NUM
ejpam-5364	387	6	.	.	PUNCT
ejpam-5364	388	1	[	[	X
ejpam-5364	388	2	3	3	X
ejpam-5364	388	3	]	]	X
ejpam-5364	388	4	nasir	nasir	PROPN
ejpam-5364	388	5	ali	ali	PROPN
ejpam-5364	388	6	,	,	PUNCT
ejpam-5364	388	7	hafiz	hafiz	PROPN
ejpam-5364	388	8	muhammad	muhammad	PROPN
ejpam-5364	388	9	afzal	afzal	PROPN
ejpam-5364	388	10	siddiqui	siddiqui	PROPN
ejpam-5364	388	11	,	,	PUNCT
ejpam-5364	388	12	muhammad	muhammad	PROPN
ejpam-5364	388	13	bilal	bilal	PROPN
ejpam-5364	388	14	riaz	riaz	PROPN
ejpam-5364	388	15	,	,	PUNCT
ejpam-5364	388	16	muhammad	muhammad	PROPN
ejpam-5364	388	17	imran	imran	PROPN
ejpam-5364	388	18	qureshi	qureshi	PROPN
ejpam-5364	388	19	,	,	PUNCT
ejpam-5364	388	20	and	and	CCONJ
ejpam-5364	388	21	ali	ali	PROPN
ejpam-5364	388	22	akgül	akgül	PROPN
ejpam-5364	388	23	.	.	PUNCT
ejpam-5364	389	1	a	a	DET
ejpam-5364	389	2	graph	graph	NOUN
ejpam-5364	389	3	-	-	PUNCT
ejpam-5364	389	4	theoretic	theoretic	ADJ
ejpam-5364	389	5	approach	approach	NOUN
ejpam-5364	389	6	to	to	PART
ejpam-5364	389	7	ring	ring	VERB
ejpam-5364	389	8	analysis	analysis	NOUN
ejpam-5364	389	9	:	:	PUNCT
ejpam-5364	389	10	dominant	dominant	ADJ
ejpam-5364	389	11	metric	metric	ADJ
ejpam-5364	389	12	dimensions	dimension	NOUN
ejpam-5364	389	13	in	in	ADP
ejpam-5364	389	14	zero	zero	NUM
ejpam-5364	389	15	-	-	PUNCT
ejpam-5364	389	16	divisor	divisor	NOUN
ejpam-5364	389	17	graphs	graph	NOUN
ejpam-5364	389	18	.	.	PUNCT
ejpam-5364	390	1	heliyon	heliyon	NOUN
ejpam-5364	390	2	,	,	PUNCT
ejpam-5364	390	3	10(10	10(10	NUM
ejpam-5364	390	4	)	)	PUNCT
ejpam-5364	390	5	,	,	PUNCT
ejpam-5364	390	6	2024	2024	NUM
ejpam-5364	390	7	.	.	PUNCT
ejpam-5364	391	1	[	[	X
ejpam-5364	391	2	4	4	X
ejpam-5364	391	3	]	]	X
ejpam-5364	391	4	mustapha	mustapha	PROPN
ejpam-5364	391	5	aouchiche	aouchiche	PROPN
ejpam-5364	391	6	,	,	PUNCT
ejpam-5364	391	7	pierre	pierre	PROPN
ejpam-5364	391	8	hansen	hansen	PROPN
ejpam-5364	391	9	,	,	PUNCT
ejpam-5364	391	10	and	and	CCONJ
ejpam-5364	391	11	dragan	dragan	VERB
ejpam-5364	391	12	stevanović.	stevanović.	PROPN
ejpam-5364	391	13	a	a	DET
ejpam-5364	391	14	sharp	sharp	ADJ
ejpam-5364	391	15	upper	upper	ADJ
ejpam-5364	391	16	bound	bind	VERB
ejpam-5364	391	17	on	on	ADP
ejpam-5364	391	18	algebraic	algebraic	ADJ
ejpam-5364	391	19	connectivity	connectivity	NOUN
ejpam-5364	391	20	using	use	VERB
ejpam-5364	391	21	domination	domination	NOUN
ejpam-5364	391	22	number	number	NOUN
ejpam-5364	391	23	.	.	PUNCT
ejpam-5364	392	1	linear	linear	ADJ
ejpam-5364	392	2	algebra	algebra	NOUN
ejpam-5364	392	3	and	and	CCONJ
ejpam-5364	392	4	its	its	PRON
ejpam-5364	392	5	applications	application	NOUN
ejpam-5364	392	6	,	,	PUNCT
ejpam-5364	392	7	432(11):2879–2893	432(11):2879–2893	NUM
ejpam-5364	392	8	,	,	PUNCT
ejpam-5364	392	9	2010	2010	NUM
ejpam-5364	392	10	.	.	PUNCT
ejpam-5364	393	1	[	[	X
ejpam-5364	393	2	5	5	X
ejpam-5364	393	3	]	]	X
ejpam-5364	393	4	andries	andries	PROPN
ejpam-5364	393	5	e	e	PROPN
ejpam-5364	393	6	brouwer	brouwer	PROPN
ejpam-5364	393	7	and	and	CCONJ
ejpam-5364	393	8	willem	willem	PROPN
ejpam-5364	393	9	h	h	PROPN
ejpam-5364	393	10	haemers	haemer	NOUN
ejpam-5364	393	11	.	.	PUNCT
ejpam-5364	394	1	spectra	spectra	NOUN
ejpam-5364	394	2	of	of	ADP
ejpam-5364	394	3	graphs	graph	NOUN
ejpam-5364	394	4	.	.	PUNCT
ejpam-5364	395	1	springer	springer	NOUN
ejpam-5364	395	2	science	science	PROPN
ejpam-5364	395	3	&	&	CCONJ
ejpam-5364	395	4	business	business	NOUN
ejpam-5364	395	5	media	medium	NOUN
ejpam-5364	395	6	,	,	PUNCT
ejpam-5364	395	7	2011	2011	NUM
ejpam-5364	395	8	.	.	PUNCT
ejpam-5364	396	1	[	[	X
ejpam-5364	396	2	6	6	NUM
ejpam-5364	396	3	]	]	PUNCT
ejpam-5364	396	4	fan	fan	NOUN
ejpam-5364	396	5	rk	rk	PROPN
ejpam-5364	396	6	chung	chung	PROPN
ejpam-5364	396	7	.	.	PUNCT
ejpam-5364	397	1	diameters	diameter	NOUN
ejpam-5364	397	2	and	and	CCONJ
ejpam-5364	397	3	eigenvalues	eigenvalue	NOUN
ejpam-5364	397	4	.	.	PUNCT
ejpam-5364	398	1	journal	journal	NOUN
ejpam-5364	398	2	of	of	ADP
ejpam-5364	398	3	the	the	DET
ejpam-5364	398	4	american	american	PROPN
ejpam-5364	398	5	mathematical	mathematical	PROPN
ejpam-5364	398	6	society	society	NOUN
ejpam-5364	398	7	,	,	PUNCT
ejpam-5364	398	8	2(2):187–196	2(2):187–196	NUM
ejpam-5364	398	9	,	,	PUNCT
ejpam-5364	398	10	1989	1989	NUM
ejpam-5364	398	11	.	.	PUNCT
ejpam-5364	399	1	[	[	X
ejpam-5364	399	2	7	7	X
ejpam-5364	399	3	]	]	X
ejpam-5364	399	4	robert	robert	PROPN
ejpam-5364	399	5	j	j	PROPN
ejpam-5364	399	6	cimikowski	cimikowski	PROPN
ejpam-5364	399	7	.	.	PUNCT
ejpam-5364	400	1	graph	graph	NOUN
ejpam-5364	400	2	planarization	planarization	NOUN
ejpam-5364	400	3	and	and	CCONJ
ejpam-5364	400	4	skewness	skewness	NOUN
ejpam-5364	400	5	.	.	PUNCT
ejpam-5364	401	1	congressus	congressus	PROPN
ejpam-5364	401	2	numerantium	numerantium	PROPN
ejpam-5364	401	3	,	,	PUNCT
ejpam-5364	401	4	pages	page	NOUN
ejpam-5364	401	5	21–21	21–21	NUM
ejpam-5364	401	6	,	,	PUNCT
ejpam-5364	401	7	1992	1992	NUM
ejpam-5364	401	8	.	.	PUNCT
ejpam-5364	402	1	[	[	X
ejpam-5364	402	2	8	8	NUM
ejpam-5364	402	3	]	]	X
ejpam-5364	402	4	charles	charles	PROPN
ejpam-5364	402	5	delorme	delorme	PROPN
ejpam-5364	402	6	and	and	CCONJ
ejpam-5364	402	7	patrick	patrick	PROPN
ejpam-5364	402	8	solé.	solé.	PROPN
ejpam-5364	402	9	diameter	diameter	NOUN
ejpam-5364	402	10	,	,	PUNCT
ejpam-5364	402	11	covering	cover	VERB
ejpam-5364	402	12	index	index	NOUN
ejpam-5364	402	13	,	,	PUNCT
ejpam-5364	402	14	covering	cover	VERB
ejpam-5364	402	15	radius	radius	NOUN
ejpam-5364	402	16	and	and	CCONJ
ejpam-5364	402	17	eigenvalues	eigenvalue	NOUN
ejpam-5364	402	18	.	.	PUNCT
ejpam-5364	403	1	european	european	PROPN
ejpam-5364	403	2	journal	journal	PROPN
ejpam-5364	403	3	of	of	ADP
ejpam-5364	403	4	combinatorics	combinatorics	PROPN
ejpam-5364	403	5	,	,	PUNCT
ejpam-5364	403	6	12(2):95–108	12(2):95–108	NUM
ejpam-5364	403	7	,	,	PUNCT
ejpam-5364	403	8	1991	1991	NUM
ejpam-5364	403	9	.	.	PUNCT
ejpam-5364	404	1	[	[	X
ejpam-5364	404	2	9	9	NUM
ejpam-5364	404	3	]	]	X
ejpam-5364	404	4	peter	peter	PROPN
ejpam-5364	404	5	firby	firby	PROPN
ejpam-5364	404	6	and	and	CCONJ
ejpam-5364	404	7	julie	julie	PROPN
ejpam-5364	404	8	haviland	haviland	PROPN
ejpam-5364	404	9	.	.	PUNCT
ejpam-5364	405	1	independence	independence	NOUN
ejpam-5364	405	2	and	and	CCONJ
ejpam-5364	405	3	average	average	ADJ
ejpam-5364	405	4	distance	distance	NOUN
ejpam-5364	405	5	in	in	ADP
ejpam-5364	405	6	graphs	graph	NOUN
ejpam-5364	405	7	.	.	PUNCT
ejpam-5364	406	1	discrete	discrete	ADJ
ejpam-5364	406	2	applied	apply	VERB
ejpam-5364	406	3	mathematics	mathematic	NOUN
ejpam-5364	406	4	,	,	PUNCT
ejpam-5364	406	5	75(1):27–37	75(1):27–37	NUM
ejpam-5364	406	6	,	,	PUNCT
ejpam-5364	406	7	1997	1997	NUM
ejpam-5364	406	8	.	.	PUNCT
ejpam-5364	407	1	[	[	X
ejpam-5364	407	2	10	10	NUM
ejpam-5364	407	3	]	]	X
ejpam-5364	407	4	joel	joel	PROPN
ejpam-5364	407	5	friedman	friedman	PROPN
ejpam-5364	407	6	.	.	PUNCT
ejpam-5364	408	1	on	on	ADP
ejpam-5364	408	2	the	the	DET
ejpam-5364	408	3	second	second	ADJ
ejpam-5364	408	4	eigenvalue	eigenvalue	NOUN
ejpam-5364	408	5	and	and	CCONJ
ejpam-5364	408	6	random	random	ADJ
ejpam-5364	408	7	walks	walk	NOUN
ejpam-5364	408	8	in	in	ADP
ejpam-5364	408	9	random	random	ADJ
ejpam-5364	408	10	d	d	ADJ
ejpam-5364	408	11	-	-	ADJ
ejpam-5364	408	12	regular	regular	ADJ
ejpam-5364	408	13	graphs	graph	NOUN
ejpam-5364	408	14	.	.	PUNCT
ejpam-5364	409	1	combinatorica	combinatorica	PROPN
ejpam-5364	409	2	,	,	PUNCT
ejpam-5364	409	3	11(4):331–362	11(4):331–362	NUM
ejpam-5364	409	4	,	,	PUNCT
ejpam-5364	409	5	1991	1991	NUM
ejpam-5364	409	6	.	.	PUNCT
ejpam-5364	410	1	references	reference	NOUN
ejpam-5364	410	2	3021	3021	NUM
ejpam-5364	410	3	[	[	X
ejpam-5364	410	4	11	11	NUM
ejpam-5364	410	5	]	]	X
ejpam-5364	410	6	xiaofeng	xiaofeng	PROPN
ejpam-5364	410	7	gu	gu	PROPN
ejpam-5364	410	8	and	and	CCONJ
ejpam-5364	410	9	muhuo	muhuo	PROPN
ejpam-5364	410	10	liu	liu	PROPN
ejpam-5364	410	11	.	.	PUNCT
ejpam-5364	411	1	a	a	DET
ejpam-5364	411	2	tight	tight	ADV
ejpam-5364	411	3	lower	lower	ADV
ejpam-5364	411	4	bound	bind	VERB
ejpam-5364	411	5	on	on	ADP
ejpam-5364	411	6	the	the	DET
ejpam-5364	411	7	matching	match	VERB
ejpam-5364	411	8	number	number	NOUN
ejpam-5364	411	9	of	of	ADP
ejpam-5364	411	10	graphs	graph	NOUN
ejpam-5364	411	11	via	via	ADP
ejpam-5364	411	12	laplacian	laplacian	ADJ
ejpam-5364	411	13	eigenvalues	eigenvalue	NOUN
ejpam-5364	411	14	.	.	PUNCT
ejpam-5364	412	1	european	european	PROPN
ejpam-5364	412	2	journal	journal	PROPN
ejpam-5364	412	3	of	of	ADP
ejpam-5364	412	4	combinatorics	combinatorics	PROPN
ejpam-5364	412	5	,	,	PUNCT
ejpam-5364	412	6	101:103468	101:103468	NUM
ejpam-5364	412	7	,	,	PUNCT
ejpam-5364	412	8	2022	2022	NUM
ejpam-5364	412	9	.	.	PUNCT
ejpam-5364	413	1	[	[	X
ejpam-5364	413	2	12	12	NUM
ejpam-5364	413	3	]	]	PUNCT
ejpam-5364	413	4	wilhelmus	wilhelmus	PROPN
ejpam-5364	413	5	hubertus	hubertus	PROPN
ejpam-5364	413	6	haemers	haemer	NOUN
ejpam-5364	413	7	.	.	PUNCT
ejpam-5364	414	1	eigenvalue	eigenvalue	NOUN
ejpam-5364	414	2	techniques	technique	NOUN
ejpam-5364	414	3	in	in	ADP
ejpam-5364	414	4	design	design	NOUN
ejpam-5364	414	5	and	and	CCONJ
ejpam-5364	414	6	graph	graph	NOUN
ejpam-5364	414	7	theory	theory	NOUN
ejpam-5364	414	8	.	.	PUNCT
ejpam-5364	415	1	1979	1979	NUM
ejpam-5364	415	2	.	.	PUNCT
ejpam-5364	416	1	[	[	X
ejpam-5364	416	2	13	13	NUM
ejpam-5364	416	3	]	]	X
ejpam-5364	416	4	frank	frank	PROPN
ejpam-5364	416	5	harary	harary	PROPN
ejpam-5364	416	6	.	.	PUNCT
ejpam-5364	417	1	graph	graph	NOUN
ejpam-5364	417	2	theory	theory	NOUN
ejpam-5364	417	3	(	(	PUNCT
ejpam-5364	417	4	on	on	ADP
ejpam-5364	417	5	demand	demand	NOUN
ejpam-5364	417	6	printing	printing	NOUN
ejpam-5364	417	7	of	of	ADP
ejpam-5364	417	8	02787	02787	NUM
ejpam-5364	417	9	)	)	PUNCT
ejpam-5364	417	10	.	.	PUNCT
ejpam-5364	418	1	crc	crc	PROPN
ejpam-5364	418	2	press	press	PROPN
ejpam-5364	418	3	,	,	PUNCT
ejpam-5364	418	4	2018	2018	NUM
ejpam-5364	418	5	.	.	PUNCT
ejpam-5364	419	1	[	[	X
ejpam-5364	419	2	14	14	NUM
ejpam-5364	419	3	]	]	X
ejpam-5364	419	4	vladislav	vladislav	PROPN
ejpam-5364	419	5	kabanov	kabanov	PROPN
ejpam-5364	419	6	,	,	PUNCT
ejpam-5364	419	7	elena	elena	PROPN
ejpam-5364	419	8	v	v	NUM
ejpam-5364	419	9	konstantinova	konstantinova	PROPN
ejpam-5364	419	10	,	,	PUNCT
ejpam-5364	419	11	leonid	leonid	PROPN
ejpam-5364	419	12	shalaginov	shalaginov	PROPN
ejpam-5364	419	13	,	,	PUNCT
ejpam-5364	419	14	and	and	CCONJ
ejpam-5364	419	15	alexandr	alexandr	PROPN
ejpam-5364	419	16	valyuzhenich	valyuzhenich	PROPN
ejpam-5364	419	17	.	.	PUNCT
ejpam-5364	420	1	minimum	minimum	ADJ
ejpam-5364	420	2	supports	support	NOUN
ejpam-5364	420	3	of	of	ADP
ejpam-5364	420	4	eigenfunctions	eigenfunction	NOUN
ejpam-5364	420	5	with	with	ADP
ejpam-5364	420	6	the	the	DET
ejpam-5364	420	7	second	second	ADV
ejpam-5364	420	8	largest	large	ADJ
ejpam-5364	420	9	eigenvalue	eigenvalue	NOUN
ejpam-5364	420	10	of	of	ADP
ejpam-5364	420	11	the	the	DET
ejpam-5364	420	12	star	star	NOUN
ejpam-5364	420	13	graph	graph	NOUN
ejpam-5364	420	14	.	.	PUNCT
ejpam-5364	421	1	arxiv	arxiv	PROPN
ejpam-5364	421	2	preprint	preprint	NOUN
ejpam-5364	421	3	arxiv:1910.01374	arxiv:1910.01374	NOUN
ejpam-5364	421	4	,	,	PUNCT
ejpam-5364	421	5	2019	2019	NUM
ejpam-5364	421	6	.	.	PUNCT
ejpam-5364	422	1	[	[	X
ejpam-5364	422	2	15	15	NUM
ejpam-5364	422	3	]	]	X
ejpam-5364	422	4	shuchao	shuchao	ADJ
ejpam-5364	422	5	li	li	NOUN
ejpam-5364	422	6	and	and	CCONJ
ejpam-5364	422	7	wanting	want	VERB
ejpam-5364	422	8	sun	sun	NOUN
ejpam-5364	422	9	.	.	PUNCT
ejpam-5364	423	1	matching	match	VERB
ejpam-5364	423	2	number	number	NOUN
ejpam-5364	423	3	,	,	PUNCT
ejpam-5364	423	4	connectivity	connectivity	NOUN
ejpam-5364	423	5	and	and	CCONJ
ejpam-5364	423	6	eigenvalues	eigenvalue	NOUN
ejpam-5364	423	7	of	of	ADP
ejpam-5364	423	8	distance	distance	NOUN
ejpam-5364	423	9	signless	signless	NOUN
ejpam-5364	423	10	laplacians	laplacian	NOUN
ejpam-5364	423	11	.	.	PUNCT
ejpam-5364	424	1	linear	linear	ADJ
ejpam-5364	424	2	and	and	CCONJ
ejpam-5364	424	3	multilinear	multilinear	PROPN
ejpam-5364	424	4	algebra	algebra	PROPN
ejpam-5364	424	5	,	,	PUNCT
ejpam-5364	424	6	69(1):74–92	69(1):74–92	NUM
ejpam-5364	424	7	,	,	PUNCT
ejpam-5364	424	8	2021	2021	NUM
ejpam-5364	424	9	.	.	PUNCT
ejpam-5364	425	1	[	[	X
ejpam-5364	425	2	16	16	NUM
ejpam-5364	425	3	]	]	X
ejpam-5364	425	4	chia	chia	X
ejpam-5364	425	5	-	-	PUNCT
ejpam-5364	425	6	an	an	DET
ejpam-5364	425	7	liu	liu	PROPN
ejpam-5364	425	8	and	and	CCONJ
ejpam-5364	425	9	chih	chih	PROPN
ejpam-5364	425	10	-	-	PUNCT
ejpam-5364	425	11	wen	wen	PROPN
ejpam-5364	425	12	weng	weng	PROPN
ejpam-5364	425	13	.	.	PUNCT
ejpam-5364	426	1	spectral	spectral	ADJ
ejpam-5364	426	2	radius	radius	PROPN
ejpam-5364	426	3	of	of	ADP
ejpam-5364	426	4	bipartite	bipartite	PROPN
ejpam-5364	426	5	graphs	graph	NOUN
ejpam-5364	426	6	.	.	PUNCT
ejpam-5364	427	1	linear	linear	ADJ
ejpam-5364	427	2	algebra	algebra	NOUN
ejpam-5364	427	3	and	and	CCONJ
ejpam-5364	427	4	its	its	PRON
ejpam-5364	427	5	applications	application	NOUN
ejpam-5364	427	6	,	,	PUNCT
ejpam-5364	427	7	474:30–43	474:30–43	NUM
ejpam-5364	427	8	,	,	PUNCT
ejpam-5364	427	9	2015	2015	NUM
ejpam-5364	427	10	.	.	PUNCT
ejpam-5364	428	1	[	[	X
ejpam-5364	428	2	17	17	NUM
ejpam-5364	428	3	]	]	PUNCT
ejpam-5364	428	4	huiqing	huiqe	VERB
ejpam-5364	428	5	liu	liu	PROPN
ejpam-5364	428	6	,	,	PUNCT
ejpam-5364	428	7	mei	mei	PROPN
ejpam-5364	428	8	lu	lu	PROPN
ejpam-5364	428	9	,	,	PUNCT
ejpam-5364	428	10	and	and	CCONJ
ejpam-5364	428	11	feng	feng	PROPN
ejpam-5364	428	12	tian	tian	PROPN
ejpam-5364	428	13	.	.	PUNCT
ejpam-5364	429	1	edge	edge	NOUN
ejpam-5364	429	2	-	-	PUNCT
ejpam-5364	429	3	connectivity	connectivity	NOUN
ejpam-5364	429	4	and	and	CCONJ
ejpam-5364	429	5	(	(	PUNCT
ejpam-5364	429	6	signless	signless	NOUN
ejpam-5364	429	7	)	)	PUNCT
ejpam-5364	429	8	laplacian	laplacian	ADJ
ejpam-5364	429	9	eigenvalue	eigenvalue	NOUN
ejpam-5364	429	10	of	of	ADP
ejpam-5364	429	11	graphs	graph	NOUN
ejpam-5364	429	12	.	.	PUNCT
ejpam-5364	430	1	linear	linear	ADJ
ejpam-5364	430	2	algebra	algebra	NOUN
ejpam-5364	430	3	and	and	CCONJ
ejpam-5364	430	4	its	its	PRON
ejpam-5364	430	5	applications	application	NOUN
ejpam-5364	430	6	,	,	PUNCT
ejpam-5364	430	7	439(12):3777–3784	439(12):3777–3784	NUM
ejpam-5364	430	8	,	,	PUNCT
ejpam-5364	430	9	2013	2013	NUM
ejpam-5364	430	10	.	.	PUNCT
ejpam-5364	431	1	[	[	X
ejpam-5364	431	2	18	18	NUM
ejpam-5364	431	3	]	]	X
ejpam-5364	431	4	peter	peter	PROPN
ejpam-5364	431	5	c	c	PROPN
ejpam-5364	431	6	liu	liu	PROPN
ejpam-5364	431	7	.	.	PUNCT
ejpam-5364	432	1	on	on	ADP
ejpam-5364	432	2	the	the	DET
ejpam-5364	432	3	deletion	deletion	NOUN
ejpam-5364	432	4	of	of	ADP
ejpam-5364	432	5	nonplanar	nonplanar	ADJ
ejpam-5364	432	6	edges	edge	NOUN
ejpam-5364	432	7	of	of	ADP
ejpam-5364	432	8	a	a	DET
ejpam-5364	432	9	graph	graph	NOUN
ejpam-5364	432	10	.	.	PUNCT
ejpam-5364	433	1	in	in	ADP
ejpam-5364	433	2	proc	proc	PROPN
ejpam-5364	433	3	.	.	PUNCT
ejpam-5364	434	1	10th	10th	ADJ
ejpam-5364	434	2	south	south	PROPN
ejpam-5364	434	3	-	-	PUNCT
ejpam-5364	434	4	east	east	NOUN
ejpam-5364	434	5	conference	conference	NOUN
ejpam-5364	434	6	on	on	ADP
ejpam-5364	434	7	combinatorics	combinatoric	NOUN
ejpam-5364	434	8	,	,	PUNCT
ejpam-5364	434	9	graph	graph	NOUN
ejpam-5364	434	10	theory	theory	NOUN
ejpam-5364	434	11	,	,	PUNCT
ejpam-5364	434	12	and	and	CCONJ
ejpam-5364	434	13	computing	computing	NOUN
ejpam-5364	434	14	,	,	PUNCT
ejpam-5364	434	15	pages	page	NOUN
ejpam-5364	434	16	727–738	727–738	NUM
ejpam-5364	434	17	,	,	PUNCT
ejpam-5364	434	18	1977	1977	NUM
ejpam-5364	434	19	.	.	PUNCT
ejpam-5364	435	1	[	[	X
ejpam-5364	435	2	19	19	NUM
ejpam-5364	435	3	]	]	PUNCT
ejpam-5364	435	4	jongyook	jongyook	NOUN
ejpam-5364	435	5	park	park	NOUN
ejpam-5364	435	6	.	.	PUNCT
ejpam-5364	436	1	a	a	DET
ejpam-5364	436	2	relationship	relationship	NOUN
ejpam-5364	436	3	between	between	ADP
ejpam-5364	436	4	the	the	DET
ejpam-5364	436	5	second	second	ADV
ejpam-5364	436	6	largest	large	ADJ
ejpam-5364	436	7	eigenvalue	eigenvalue	ADJ
ejpam-5364	436	8	and	and	CCONJ
ejpam-5364	436	9	local	local	ADJ
ejpam-5364	436	10	valency	valency	NOUN
ejpam-5364	436	11	of	of	ADP
ejpam-5364	436	12	an	an	DET
ejpam-5364	436	13	edge	edge	NOUN
ejpam-5364	436	14	-	-	PUNCT
ejpam-5364	436	15	regular	regular	ADJ
ejpam-5364	436	16	graph	graph	NOUN
ejpam-5364	436	17	.	.	PUNCT
ejpam-5364	437	1	kyungpook	kyungpook	PROPN
ejpam-5364	437	2	mathematical	mathematical	PROPN
ejpam-5364	437	3	journal	journal	PROPN
ejpam-5364	437	4	,	,	PUNCT
ejpam-5364	437	5	61(3):671–677	61(3):671–677	PROPN
ejpam-5364	437	6	,	,	PUNCT
ejpam-5364	437	7	2021	2021	NUM
ejpam-5364	437	8	.	.	PUNCT
ejpam-5364	438	1	[	[	X
ejpam-5364	438	2	20	20	NUM
ejpam-5364	438	3	]	]	PUNCT
ejpam-5364	438	4	farzaneh	farzaneh	NOUN
ejpam-5364	438	5	ramezani	ramezani	NOUN
ejpam-5364	438	6	and	and	CCONJ
ejpam-5364	438	7	behruz	behruz	NOUN
ejpam-5364	438	8	tayfeh	tayfeh	NOUN
ejpam-5364	438	9	-	-	PUNCT
ejpam-5364	438	10	rezaie	rezaie	NOUN
ejpam-5364	438	11	.	.	PUNCT
ejpam-5364	439	1	graphs	graph	NOUN
ejpam-5364	439	2	with	with	ADP
ejpam-5364	439	3	prescribed	prescribed	ADJ
ejpam-5364	439	4	star	star	NOUN
ejpam-5364	439	5	complement	complement	NOUN
ejpam-5364	439	6	for	for	ADP
ejpam-5364	439	7	1	1	NUM
ejpam-5364	439	8	as	as	ADP
ejpam-5364	439	9	the	the	DET
ejpam-5364	439	10	second	second	ADV
ejpam-5364	439	11	largest	large	ADJ
ejpam-5364	439	12	eigenvalue	eigenvalue	NOUN
ejpam-5364	439	13	.	.	PROPN
ejpam-5364	439	14	ars	ars	PROPN
ejpam-5364	439	15	combinatoria	combinatoria	PROPN
ejpam-5364	439	16	,	,	PUNCT
ejpam-5364	439	17	116:129–145	116:129–145	NUM
ejpam-5364	439	18	,	,	PUNCT
ejpam-5364	439	19	2014	2014	NUM
ejpam-5364	439	20	.	.	PUNCT
ejpam-5364	440	1	[	[	X
ejpam-5364	440	2	21	21	NUM
ejpam-5364	440	3	]	]	X
ejpam-5364	440	4	johannes	johanne	NOUN
ejpam-5364	440	5	siemons	siemon	NOUN
ejpam-5364	440	6	and	and	CCONJ
ejpam-5364	440	7	alexandre	alexandre	PROPN
ejpam-5364	440	8	zalesski	zalesski	PROPN
ejpam-5364	440	9	.	.	PUNCT
ejpam-5364	441	1	on	on	ADP
ejpam-5364	441	2	the	the	DET
ejpam-5364	441	3	second	second	ADV
ejpam-5364	441	4	largest	large	ADJ
ejpam-5364	441	5	eigenvalue	eigenvalue	NOUN
ejpam-5364	441	6	of	of	ADP
ejpam-5364	441	7	some	some	DET
ejpam-5364	441	8	cayley	cayley	ADJ
ejpam-5364	441	9	graphs	graph	NOUN
ejpam-5364	441	10	of	of	ADP
ejpam-5364	441	11	the	the	DET
ejpam-5364	441	12	symmetric	symmetric	ADJ
ejpam-5364	441	13	group	group	NOUN
ejpam-5364	441	14	.	.	PUNCT
ejpam-5364	442	1	journal	journal	PROPN
ejpam-5364	442	2	of	of	ADP
ejpam-5364	442	3	algebraic	algebraic	PROPN
ejpam-5364	442	4	combinatorics	combinatoric	NOUN
ejpam-5364	442	5	,	,	PUNCT
ejpam-5364	442	6	pages	page	NOUN
ejpam-5364	442	7	1–17	1–17	PROPN
ejpam-5364	442	8	,	,	PUNCT
ejpam-5364	442	9	2022	2022	NUM
ejpam-5364	442	10	.	.	PUNCT
ejpam-5364	443	1	[	[	X
ejpam-5364	443	2	22	22	NUM
ejpam-5364	443	3	]	]	X
ejpam-5364	443	4	steven	steven	PROPN
ejpam-5364	443	5	skiena	skiena	PROPN
ejpam-5364	443	6	.	.	PUNCT
ejpam-5364	444	1	combinatorics	combinatoric	NOUN
ejpam-5364	444	2	and	and	CCONJ
ejpam-5364	444	3	graph	graph	NOUN
ejpam-5364	444	4	theory	theory	NOUN
ejpam-5364	444	5	with	with	ADP
ejpam-5364	444	6	mathematica	mathematica	PROPN
ejpam-5364	444	7	.	.	PUNCT
ejpam-5364	445	1	perseus	perseus	NOUN
ejpam-5364	445	2	books	book	NOUN
ejpam-5364	445	3	(	(	PUNCT
ejpam-5364	445	4	sd	sd	NOUN
ejpam-5364	445	5	)	)	PUNCT
ejpam-5364	445	6	,	,	PUNCT
ejpam-5364	445	7	1990	1990	NUM
ejpam-5364	445	8	.	.	PUNCT
ejpam-5364	446	1	[	[	X
ejpam-5364	446	2	23	23	NUM
ejpam-5364	446	3	]	]	PUNCT
ejpam-5364	446	4	zoran	zoran	PROPN
ejpam-5364	446	5	stanić.	stanić.	PROPN
ejpam-5364	446	6	on	on	ADP
ejpam-5364	446	7	graphs	graph	NOUN
ejpam-5364	446	8	whose	whose	DET
ejpam-5364	446	9	second	second	ADV
ejpam-5364	446	10	largest	large	ADJ
ejpam-5364	446	11	eigenvalue	eigenvalue	NOUN
ejpam-5364	446	12	equals	equal	VERB
ejpam-5364	446	13	1	1	NUM
ejpam-5364	446	14	–	–	PUNCT
ejpam-5364	446	15	the	the	DET
ejpam-5364	446	16	star	star	NOUN
ejpam-5364	446	17	complement	complement	NOUN
ejpam-5364	446	18	technique	technique	NOUN
ejpam-5364	446	19	.	.	PUNCT
ejpam-5364	447	1	linear	linear	ADJ
ejpam-5364	447	2	algebra	algebra	NOUN
ejpam-5364	447	3	and	and	CCONJ
ejpam-5364	447	4	its	its	PRON
ejpam-5364	447	5	applications	application	NOUN
ejpam-5364	447	6	,	,	PUNCT
ejpam-5364	447	7	420(2	420(2	NUM
ejpam-5364	447	8	-	-	SYM
ejpam-5364	447	9	3):700–710	3):700–710	NUM
ejpam-5364	447	10	,	,	PUNCT
ejpam-5364	447	11	2007	2007	NUM
ejpam-5364	447	12	.	.	PUNCT
ejpam-5364	448	1	[	[	X
ejpam-5364	448	2	24	24	NUM
ejpam-5364	448	3	]	]	X
ejpam-5364	448	4	fenglei	fenglei	PROPN
ejpam-5364	448	5	tian	tian	PROPN
ejpam-5364	448	6	and	and	CCONJ
ejpam-5364	448	7	dein	dein	PROPN
ejpam-5364	448	8	wong	wong	PROPN
ejpam-5364	448	9	.	.	PUNCT
ejpam-5364	449	1	relation	relation	NOUN
ejpam-5364	449	2	between	between	ADP
ejpam-5364	449	3	the	the	DET
ejpam-5364	449	4	matching	matching	NOUN
ejpam-5364	449	5	number	number	NOUN
ejpam-5364	449	6	and	and	CCONJ
ejpam-5364	449	7	the	the	DET
ejpam-5364	449	8	second	second	ADV
ejpam-5364	449	9	largest	large	ADJ
ejpam-5364	449	10	distance	distance	NOUN
ejpam-5364	449	11	laplacian	laplacian	ADJ
ejpam-5364	449	12	eigenvalue	eigenvalue	NOUN
ejpam-5364	449	13	of	of	ADP
ejpam-5364	449	14	a	a	DET
ejpam-5364	449	15	graph	graph	NOUN
ejpam-5364	449	16	.	.	PUNCT
ejpam-5364	450	1	linear	linear	ADJ
ejpam-5364	450	2	algebra	algebra	NOUN
ejpam-5364	450	3	and	and	CCONJ
ejpam-5364	450	4	its	its	PRON
ejpam-5364	450	5	applications	application	NOUN
ejpam-5364	450	6	,	,	PUNCT
ejpam-5364	450	7	558:174–185	558:174–185	NUM
ejpam-5364	450	8	,	,	PUNCT
ejpam-5364	450	9	2018	2018	NUM
ejpam-5364	450	10	.	.	PUNCT
ejpam-5364	451	1	[	[	X
ejpam-5364	451	2	25	25	NUM
ejpam-5364	451	3	]	]	PUNCT
ejpam-5364	451	4	richard	richard	PROPN
ejpam-5364	451	5	j	j	PROPN
ejpam-5364	451	6	trudeau	trudeau	PROPN
ejpam-5364	451	7	.	.	PUNCT
ejpam-5364	452	1	introduction	introduction	NOUN
ejpam-5364	452	2	to	to	AUX
ejpam-5364	452	3	graph	graph	NOUN
ejpam-5364	452	4	theory	theory	NOUN
ejpam-5364	452	5	.	.	PUNCT
ejpam-5364	453	1	courier	courier	NOUN
ejpam-5364	453	2	corporation	corporation	NOUN
ejpam-5364	453	3	,	,	PUNCT
ejpam-5364	453	4	2013	2013	NUM
ejpam-5364	453	5	.	.	PUNCT
ejpam-5364	454	1	[	[	X
ejpam-5364	454	2	26	26	NUM
ejpam-5364	454	3	]	]	X
ejpam-5364	454	4	douglas	douglas	PROPN
ejpam-5364	454	5	brent	brent	PROPN
ejpam-5364	454	6	west	west	PROPN
ejpam-5364	454	7	et	et	PROPN
ejpam-5364	454	8	al	al	PROPN
ejpam-5364	454	9	.	.	PROPN
ejpam-5364	454	10	introduction	introduction	NOUN
ejpam-5364	454	11	to	to	AUX
ejpam-5364	454	12	graph	graph	NOUN
ejpam-5364	454	13	theory	theory	NOUN
ejpam-5364	454	14	,	,	PUNCT
ejpam-5364	454	15	volume	volume	NOUN
ejpam-5364	454	16	2	2	NUM
ejpam-5364	454	17	.	.	PUNCT
ejpam-5364	454	18	prentice	prentice	PROPN
ejpam-5364	454	19	hall	hall	PROPN
ejpam-5364	454	20	upper	upper	PROPN
ejpam-5364	454	21	saddle	saddle	PROPN
ejpam-5364	454	22	river	river	NOUN
ejpam-5364	454	23	,	,	PUNCT
ejpam-5364	454	24	2001	2001	NUM
ejpam-5364	454	25	.	.	PUNCT
ejpam-5364	455	1	[	[	X
ejpam-5364	455	2	27	27	NUM
ejpam-5364	455	3	]	]	X
ejpam-5364	455	4	jae	jae	PROPN
ejpam-5364	455	5	young	young	PROPN
ejpam-5364	455	6	yang	yang	PROPN
ejpam-5364	455	7	and	and	CCONJ
ejpam-5364	455	8	jack	jack	PROPN
ejpam-5364	455	9	h	h	PROPN
ejpam-5364	455	10	koolen	koolen	VERB
ejpam-5364	455	11	.	.	PUNCT
ejpam-5364	456	1	on	on	ADP
ejpam-5364	456	2	the	the	DET
ejpam-5364	456	3	order	order	NOUN
ejpam-5364	456	4	of	of	ADP
ejpam-5364	456	5	regular	regular	ADJ
ejpam-5364	456	6	graphs	graph	NOUN
ejpam-5364	456	7	with	with	ADP
ejpam-5364	456	8	fixed	fix	VERB
ejpam-5364	456	9	second	second	ADV
ejpam-5364	456	10	largest	large	ADJ
ejpam-5364	456	11	eigenvalue	eigenvalue	NOUN
ejpam-5364	456	12	.	.	PUNCT
ejpam-5364	457	1	linear	linear	PROPN
ejpam-5364	457	2	algebra	algebra	PROPN
ejpam-5364	457	3	and	and	CCONJ
ejpam-5364	457	4	its	its	PRON
ejpam-5364	457	5	applications	application	NOUN
ejpam-5364	457	6	,	,	PUNCT
ejpam-5364	457	7	610:29–39	610:29–39	NUM
ejpam-5364	457	8	,	,	PUNCT
ejpam-5364	457	9	2021	2021	NUM
ejpam-5364	457	10	.	.	PUNCT