id	sid	tid	token	lemma	pos
ejpam-5057	1	1	european	european	PROPN
ejpam-5057	1	2	journal	journal	PROPN
ejpam-5057	1	3	of	of	ADP
ejpam-5057	1	4	pure	pure	ADJ
ejpam-5057	1	5	and	and	CCONJ
ejpam-5057	1	6	applied	apply	VERB
ejpam-5057	1	7	mathematics	mathematic	NOUN
ejpam-5057	1	8	vol	vol	NOUN
ejpam-5057	1	9	.	.	PROPN
ejpam-5057	2	1	17	17	NUM
ejpam-5057	2	2	,	,	PUNCT
ejpam-5057	2	3	no	no	INTJ
ejpam-5057	2	4	.	.	NOUN
ejpam-5057	2	5	2	2	NUM
ejpam-5057	2	6	,	,	PUNCT
ejpam-5057	2	7	2024	2024	NUM
ejpam-5057	2	8	,	,	PUNCT
ejpam-5057	2	9	772	772	NUM
ejpam-5057	2	10	-	-	SYM
ejpam-5057	2	11	789	789	NUM
ejpam-5057	2	12	issn	issn	PROPN
ejpam-5057	2	13	1307	1307	NUM
ejpam-5057	2	14	-	-	SYM
ejpam-5057	2	15	5543	5543	NUM
ejpam-5057	2	16	–	–	PUNCT
ejpam-5057	3	1	ejpam.com	ejpam.com	X
ejpam-5057	3	2	published	publish	VERB
ejpam-5057	3	3	by	by	ADP
ejpam-5057	3	4	new	new	PROPN
ejpam-5057	3	5	york	york	PROPN
ejpam-5057	3	6	business	business	PROPN
ejpam-5057	3	7	global	global	PROPN
ejpam-5057	3	8	eigenvalue	eigenvalue	PROPN
ejpam-5057	3	9	interlacing	interlacing	NOUN
ejpam-5057	3	10	of	of	ADP
ejpam-5057	3	11	bipartite	bipartite	NOUN
ejpam-5057	3	12	graphs	graph	NOUN
ejpam-5057	3	13	and	and	CCONJ
ejpam-5057	3	14	construction	construction	NOUN
ejpam-5057	3	15	of	of	ADP
ejpam-5057	3	16	expander	expander	NOUN
ejpam-5057	3	17	code	code	NOUN
ejpam-5057	3	18	using	use	VERB
ejpam-5057	3	19	vertex	vertex	NOUN
ejpam-5057	3	20	-	-	PUNCT
ejpam-5057	3	21	split	split	NOUN
ejpam-5057	3	22	of	of	ADP
ejpam-5057	3	23	a	a	DET
ejpam-5057	3	24	bipartite	bipartite	NOUN
ejpam-5057	3	25	graph	graph	NOUN
ejpam-5057	3	26	machasri	machasri	PROPN
ejpam-5057	3	27	manickam1	manickam1	PROPN
ejpam-5057	3	28	,	,	PUNCT
ejpam-5057	3	29	kalyani	kalyani	PROPN
ejpam-5057	3	30	desikan1,∗	desikan1,∗	NOUN
ejpam-5057	3	31	1	1	NUM
ejpam-5057	3	32	department	department	NOUN
ejpam-5057	3	33	of	of	ADP
ejpam-5057	3	34	mathematics	mathematic	NOUN
ejpam-5057	3	35	,	,	PUNCT
ejpam-5057	3	36	school	school	NOUN
ejpam-5057	3	37	of	of	ADP
ejpam-5057	3	38	advanced	advanced	ADJ
ejpam-5057	3	39	sciences	science	NOUN
ejpam-5057	3	40	,	,	PUNCT
ejpam-5057	3	41	vellore	vellore	PROPN
ejpam-5057	3	42	institute	institute	PROPN
ejpam-5057	3	43	of	of	ADP
ejpam-5057	3	44	technology	technology	PROPN
ejpam-5057	3	45	,	,	PUNCT
ejpam-5057	3	46	chennai	chennai	PROPN
ejpam-5057	3	47	,	,	PUNCT
ejpam-5057	3	48	tamilnadu	tamilnadu	NOUN
ejpam-5057	3	49	,	,	PUNCT
ejpam-5057	3	50	india	india	PROPN
ejpam-5057	3	51	.	.	PUNCT
ejpam-5057	3	52	abstract	abstract	PROPN
ejpam-5057	3	53	.	.	PUNCT
ejpam-5057	4	1	the	the	DET
ejpam-5057	4	2	second	second	ADV
ejpam-5057	4	3	largest	large	ADJ
ejpam-5057	4	4	eigenvalue	eigenvalue	NOUN
ejpam-5057	4	5	of	of	ADP
ejpam-5057	4	6	a	a	DET
ejpam-5057	4	7	graph	graph	NOUN
ejpam-5057	4	8	is	be	AUX
ejpam-5057	4	9	an	an	DET
ejpam-5057	4	10	important	important	ADJ
ejpam-5057	4	11	algebraic	algebraic	ADJ
ejpam-5057	4	12	parameter	parameter	NOUN
ejpam-5057	4	13	which	which	PRON
ejpam-5057	4	14	is	be	AUX
ejpam-5057	4	15	related	relate	VERB
ejpam-5057	4	16	with	with	ADP
ejpam-5057	4	17	the	the	DET
ejpam-5057	4	18	expansion	expansion	NOUN
ejpam-5057	4	19	,	,	PUNCT
ejpam-5057	4	20	connectivity	connectivity	NOUN
ejpam-5057	4	21	and	and	CCONJ
ejpam-5057	4	22	randomness	randomness	NOUN
ejpam-5057	4	23	properties	property	NOUN
ejpam-5057	4	24	of	of	ADP
ejpam-5057	4	25	a	a	DET
ejpam-5057	4	26	graph	graph	NOUN
ejpam-5057	4	27	.	.	PUNCT
ejpam-5057	5	1	expanders	expander	NOUN
ejpam-5057	5	2	are	be	AUX
ejpam-5057	5	3	highly	highly	ADV
ejpam-5057	5	4	connected	connect	VERB
ejpam-5057	5	5	sparse	sparse	ADJ
ejpam-5057	5	6	graphs	graph	NOUN
ejpam-5057	5	7	.	.	PUNCT
ejpam-5057	6	1	in	in	ADP
ejpam-5057	6	2	coding	code	VERB
ejpam-5057	6	3	theory	theory	NOUN
ejpam-5057	6	4	,	,	PUNCT
ejpam-5057	6	5	expander	expander	NOUN
ejpam-5057	6	6	codes	code	NOUN
ejpam-5057	6	7	are	be	AUX
ejpam-5057	6	8	error	error	NOUN
ejpam-5057	6	9	correcting	correct	VERB
ejpam-5057	6	10	codes	code	NOUN
ejpam-5057	6	11	made	make	VERB
ejpam-5057	6	12	up	up	ADP
ejpam-5057	6	13	of	of	ADP
ejpam-5057	6	14	bipartite	bipartite	ADJ
ejpam-5057	6	15	expander	expander	NOUN
ejpam-5057	6	16	graphs	graph	NOUN
ejpam-5057	6	17	.	.	PUNCT
ejpam-5057	7	1	in	in	ADP
ejpam-5057	7	2	this	this	DET
ejpam-5057	7	3	paper	paper	NOUN
ejpam-5057	7	4	,	,	PUNCT
ejpam-5057	7	5	first	first	ADV
ejpam-5057	7	6	we	we	PRON
ejpam-5057	7	7	prove	prove	VERB
ejpam-5057	7	8	the	the	DET
ejpam-5057	7	9	interlacing	interlacing	NOUN
ejpam-5057	7	10	of	of	ADP
ejpam-5057	7	11	the	the	DET
ejpam-5057	7	12	eigenvalues	eigenvalue	NOUN
ejpam-5057	7	13	of	of	ADP
ejpam-5057	7	14	the	the	DET
ejpam-5057	7	15	adjacency	adjacency	NOUN
ejpam-5057	7	16	matrix	matrix	NOUN
ejpam-5057	7	17	of	of	ADP
ejpam-5057	7	18	the	the	DET
ejpam-5057	7	19	bipartite	bipartite	PROPN
ejpam-5057	7	20	graph	graph	NOUN
ejpam-5057	7	21	with	with	ADP
ejpam-5057	7	22	the	the	DET
ejpam-5057	7	23	eigenvalues	eigenvalue	NOUN
ejpam-5057	7	24	of	of	ADP
ejpam-5057	7	25	the	the	DET
ejpam-5057	7	26	bipartite	bipartite	PROPN
ejpam-5057	7	27	quotient	quotient	NOUN
ejpam-5057	7	28	matrices	matrix	NOUN
ejpam-5057	7	29	of	of	ADP
ejpam-5057	7	30	the	the	DET
ejpam-5057	7	31	corresponding	corresponding	ADJ
ejpam-5057	7	32	graph	graph	NOUN
ejpam-5057	7	33	matrices	matrix	NOUN
ejpam-5057	7	34	.	.	PUNCT
ejpam-5057	8	1	then	then	ADV
ejpam-5057	8	2	we	we	PRON
ejpam-5057	8	3	obtain	obtain	VERB
ejpam-5057	8	4	bounds	bound	NOUN
ejpam-5057	8	5	for	for	ADP
ejpam-5057	8	6	the	the	DET
ejpam-5057	8	7	second	second	ADV
ejpam-5057	8	8	largest	large	ADJ
ejpam-5057	8	9	and	and	CCONJ
ejpam-5057	8	10	second	second	ADJ
ejpam-5057	8	11	smallest	small	ADJ
ejpam-5057	8	12	eigenvalues	eigenvalue	NOUN
ejpam-5057	8	13	.	.	PUNCT
ejpam-5057	9	1	since	since	SCONJ
ejpam-5057	9	2	the	the	DET
ejpam-5057	9	3	graph	graph	NOUN
ejpam-5057	9	4	is	be	AUX
ejpam-5057	9	5	bipartite	bipartite	ADJ
ejpam-5057	9	6	,	,	PUNCT
ejpam-5057	9	7	the	the	DET
ejpam-5057	9	8	results	result	NOUN
ejpam-5057	9	9	for	for	ADP
ejpam-5057	9	10	laplacian	laplacian	NOUN
ejpam-5057	9	11	will	will	AUX
ejpam-5057	9	12	also	also	ADV
ejpam-5057	9	13	hold	hold	VERB
ejpam-5057	9	14	for	for	ADP
ejpam-5057	9	15	signless	signless	ADJ
ejpam-5057	9	16	laplacian	laplacian	ADJ
ejpam-5057	9	17	matrix	matrix	NOUN
ejpam-5057	9	18	.	.	PUNCT
ejpam-5057	10	1	we	we	PRON
ejpam-5057	10	2	then	then	ADV
ejpam-5057	10	3	introduce	introduce	VERB
ejpam-5057	10	4	a	a	DET
ejpam-5057	10	5	new	new	ADJ
ejpam-5057	10	6	method	method	NOUN
ejpam-5057	10	7	called	call	VERB
ejpam-5057	10	8	vertex	vertex	NOUN
ejpam-5057	10	9	-	-	PUNCT
ejpam-5057	10	10	split	split	NOUN
ejpam-5057	10	11	of	of	ADP
ejpam-5057	10	12	a	a	DET
ejpam-5057	10	13	bipartite	bipartite	ADJ
ejpam-5057	10	14	graph	graph	NOUN
ejpam-5057	10	15	to	to	PART
ejpam-5057	10	16	construct	construct	VERB
ejpam-5057	10	17	asymptotically	asymptotically	ADV
ejpam-5057	10	18	good	good	ADJ
ejpam-5057	10	19	expander	expander	NOUN
ejpam-5057	10	20	codes	code	NOUN
ejpam-5057	10	21	with	with	ADP
ejpam-5057	10	22	expansion	expansion	NOUN
ejpam-5057	10	23	factor	factor	NOUN
ejpam-5057	10	24	d	d	NOUN
ejpam-5057	10	25	2	2	NUM
ejpam-5057	10	26	<	<	X
ejpam-5057	10	27	α	α	X
ejpam-5057	10	28	<	<	X
ejpam-5057	10	29	d	d	PROPN
ejpam-5057	10	30	and	and	CCONJ
ejpam-5057	10	31	ϵ	ϵ	X
ejpam-5057	10	32	<	<	X
ejpam-5057	10	33	1	1	NUM
ejpam-5057	10	34	2	2	NUM
ejpam-5057	10	35	and	and	CCONJ
ejpam-5057	10	36	prove	prove	VERB
ejpam-5057	10	37	a	a	DET
ejpam-5057	10	38	condition	condition	NOUN
ejpam-5057	10	39	for	for	ADP
ejpam-5057	10	40	the	the	DET
ejpam-5057	10	41	vertex	vertex	NOUN
ejpam-5057	10	42	-	-	PUNCT
ejpam-5057	10	43	split	split	NOUN
ejpam-5057	10	44	of	of	ADP
ejpam-5057	10	45	a	a	DET
ejpam-5057	10	46	bipartite	bipartite	NOUN
ejpam-5057	10	47	graph	graph	NOUN
ejpam-5057	10	48	to	to	PART
ejpam-5057	10	49	be	be	AUX
ejpam-5057	10	50	k−connected	k−connecte	VERB
ejpam-5057	10	51	with	with	ADP
ejpam-5057	10	52	respect	respect	NOUN
ejpam-5057	10	53	to	to	ADP
ejpam-5057	10	54	λ2	λ2	NOUN
ejpam-5057	10	55	.	.	PUNCT
ejpam-5057	11	1	further	far	ADV
ejpam-5057	11	2	,	,	PUNCT
ejpam-5057	11	3	we	we	PRON
ejpam-5057	11	4	prove	prove	VERB
ejpam-5057	11	5	that	that	SCONJ
ejpam-5057	11	6	the	the	DET
ejpam-5057	11	7	vertex	vertex	NOUN
ejpam-5057	11	8	-	-	PUNCT
ejpam-5057	11	9	split	split	NOUN
ejpam-5057	11	10	of	of	ADP
ejpam-5057	11	11	g	g	PROPN
ejpam-5057	11	12	is	be	AUX
ejpam-5057	11	13	a	a	DET
ejpam-5057	11	14	bipartite	bipartite	ADJ
ejpam-5057	11	15	expander	expander	NOUN
ejpam-5057	11	16	.	.	PUNCT
ejpam-5057	12	1	finally	finally	ADV
ejpam-5057	12	2	,	,	PUNCT
ejpam-5057	12	3	we	we	PRON
ejpam-5057	12	4	construct	construct	VERB
ejpam-5057	12	5	an	an	DET
ejpam-5057	12	6	asymptotically	asymptotically	ADV
ejpam-5057	12	7	good	good	ADJ
ejpam-5057	12	8	expander	expander	NOUN
ejpam-5057	12	9	code	code	NOUN
ejpam-5057	12	10	whose	whose	DET
ejpam-5057	12	11	factor	factor	NOUN
ejpam-5057	12	12	graph	graph	NOUN
ejpam-5057	12	13	is	be	AUX
ejpam-5057	12	14	a	a	DET
ejpam-5057	12	15	graph	graph	NOUN
ejpam-5057	12	16	obtained	obtain	VERB
ejpam-5057	12	17	by	by	ADP
ejpam-5057	12	18	the	the	DET
ejpam-5057	12	19	vertex	vertex	NOUN
ejpam-5057	12	20	-	-	PUNCT
ejpam-5057	12	21	split	split	NOUN
ejpam-5057	12	22	of	of	ADP
ejpam-5057	12	23	a	a	DET
ejpam-5057	12	24	bipartite	bipartite	NOUN
ejpam-5057	12	25	graph	graph	NOUN
ejpam-5057	12	26	.	.	PUNCT
ejpam-5057	13	1	2020	2020	NUM
ejpam-5057	13	2	mathematics	mathematic	NOUN
ejpam-5057	13	3	subject	subject	NOUN
ejpam-5057	13	4	classifications	classification	NOUN
ejpam-5057	13	5	:	:	PUNCT
ejpam-5057	13	6	05c40	05c40	NUM
ejpam-5057	13	7	,	,	PUNCT
ejpam-5057	13	8	05c48	05c48	NUM
ejpam-5057	13	9	,	,	PUNCT
ejpam-5057	13	10	05c50	05c50	PUNCT
ejpam-5057	13	11	key	key	ADJ
ejpam-5057	13	12	words	word	NOUN
ejpam-5057	13	13	and	and	CCONJ
ejpam-5057	13	14	phrases	phrase	NOUN
ejpam-5057	13	15	:	:	PUNCT
ejpam-5057	13	16	expander	expander	NOUN
ejpam-5057	13	17	code	code	NOUN
ejpam-5057	13	18	,	,	PUNCT
ejpam-5057	13	19	vertex	vertex	NOUN
ejpam-5057	13	20	-	-	PUNCT
ejpam-5057	13	21	split	split	NOUN
ejpam-5057	13	22	,	,	PUNCT
ejpam-5057	13	23	second	second	ADV
ejpam-5057	13	24	largest	large	ADJ
ejpam-5057	13	25	eigenvalue	eigenvalue	NOUN
ejpam-5057	13	26	,	,	PUNCT
ejpam-5057	13	27	bipartite	bipartite	NOUN
ejpam-5057	13	28	graph	graph	NOUN
ejpam-5057	13	29	,	,	PUNCT
ejpam-5057	13	30	quotient	quotient	NOUN
ejpam-5057	13	31	matrix	matrix	NOUN
ejpam-5057	13	32	1	1	NUM
ejpam-5057	13	33	.	.	PUNCT
ejpam-5057	14	1	introduction	introduction	NOUN
ejpam-5057	14	2	let	let	VERB
ejpam-5057	14	3	g	g	PRON
ejpam-5057	14	4	be	be	AUX
ejpam-5057	14	5	a	a	DET
ejpam-5057	14	6	finite	finite	ADJ
ejpam-5057	14	7	graph	graph	NOUN
ejpam-5057	14	8	.	.	PUNCT
ejpam-5057	15	1	the	the	DET
ejpam-5057	15	2	adjacency	adjacency	PROPN
ejpam-5057	15	3	matrix	matrix	NOUN
ejpam-5057	15	4	a(g	a(g	PROPN
ejpam-5057	15	5	)	)	PUNCT
ejpam-5057	15	6	of	of	ADP
ejpam-5057	15	7	g	g	PROPN
ejpam-5057	15	8	is	be	AUX
ejpam-5057	15	9	an	an	DET
ejpam-5057	15	10	n×n	n×n	PROPN
ejpam-5057	15	11	matrix	matrix	NOUN
ejpam-5057	15	12	a	a	PRON
ejpam-5057	15	13	=	=	X
ejpam-5057	16	1	[	[	X
ejpam-5057	16	2	aij	aij	X
ejpam-5057	16	3	]	]	X
ejpam-5057	16	4	,	,	PUNCT
ejpam-5057	16	5	where	where	SCONJ
ejpam-5057	16	6	aij	aij	PROPN
ejpam-5057	16	7	=	=	SYM
ejpam-5057	16	8	1	1	PROPN
ejpam-5057	16	9	if	if	SCONJ
ejpam-5057	16	10	vi	vi	PROPN
ejpam-5057	16	11	and	and	CCONJ
ejpam-5057	16	12	vj	vj	NOUN
ejpam-5057	16	13	are	be	AUX
ejpam-5057	16	14	adjacent	adjacent	ADJ
ejpam-5057	16	15	,	,	PUNCT
ejpam-5057	16	16	otherwise	otherwise	ADV
ejpam-5057	16	17	it	it	PRON
ejpam-5057	16	18	is	be	AUX
ejpam-5057	16	19	0	0	NUM
ejpam-5057	16	20	.	.	PUNCT
ejpam-5057	17	1	let	let	VERB
ejpam-5057	17	2	λ1	λ1	ADJ
ejpam-5057	17	3	≥	≥	NOUN
ejpam-5057	17	4	λ2	λ2	NOUN
ejpam-5057	17	5	≥	≥	NOUN
ejpam-5057	17	6	·	·	PUNCT
ejpam-5057	17	7	·	·	PUNCT
ejpam-5057	17	8	·	·	PUNCT
ejpam-5057	18	1	≥	≥	PRON
ejpam-5057	18	2	λn	λn	AUX
ejpam-5057	18	3	be	be	AUX
ejpam-5057	18	4	the	the	DET
ejpam-5057	18	5	eigenvalues	eigenvalue	NOUN
ejpam-5057	18	6	of	of	ADP
ejpam-5057	18	7	a	a	DET
ejpam-5057	18	8	known	know	VERB
ejpam-5057	18	9	as	as	ADP
ejpam-5057	18	10	the	the	DET
ejpam-5057	18	11	spectrum	spectrum	NOUN
ejpam-5057	18	12	of	of	ADP
ejpam-5057	18	13	g.	g.	PROPN
ejpam-5057	18	14	the	the	DET
ejpam-5057	18	15	laplacian	laplacian	ADJ
ejpam-5057	18	16	matrix	matrix	NOUN
ejpam-5057	18	17	of	of	ADP
ejpam-5057	18	18	g	g	PROPN
ejpam-5057	18	19	is	be	AUX
ejpam-5057	18	20	l(g	l(g	NOUN
ejpam-5057	18	21	)	)	PUNCT
ejpam-5057	18	22	=	=	SYM
ejpam-5057	18	23	d(g	d(g	PROPN
ejpam-5057	18	24	)	)	PUNCT
ejpam-5057	18	25	−	−	PROPN
ejpam-5057	18	26	a(g	a(g	PROPN
ejpam-5057	18	27	)	)	PUNCT
ejpam-5057	18	28	where	where	SCONJ
ejpam-5057	18	29	d(g	d(g	NOUN
ejpam-5057	18	30	)	)	PUNCT
ejpam-5057	18	31	is	be	AUX
ejpam-5057	18	32	the	the	DET
ejpam-5057	18	33	diagonal	diagonal	ADJ
ejpam-5057	18	34	degree	degree	NOUN
ejpam-5057	18	35	matrix	matrix	NOUN
ejpam-5057	18	36	.	.	PUNCT
ejpam-5057	19	1	let	let	VERB
ejpam-5057	19	2	µ1	µ1	NOUN
ejpam-5057	19	3	≥	≥	NOUN
ejpam-5057	19	4	µ2	µ2	PROPN
ejpam-5057	19	5	≥	≥	NOUN
ejpam-5057	19	6	µ3	µ3	PROPN
ejpam-5057	19	7	≥	≥	X
ejpam-5057	19	8	·	·	PUNCT
ejpam-5057	19	9	·	·	PUNCT
ejpam-5057	19	10	·	·	PUNCT
ejpam-5057	19	11	≥	≥	NUM
ejpam-5057	19	12	µn−1	µn−1	ADP
ejpam-5057	19	13	≥	≥	NUM
ejpam-5057	19	14	µn	µn	NOUN
ejpam-5057	19	15	be	be	AUX
ejpam-5057	19	16	the	the	DET
ejpam-5057	19	17	eigenvalues	eigenvalue	NOUN
ejpam-5057	19	18	of	of	ADP
ejpam-5057	19	19	the	the	DET
ejpam-5057	19	20	laplacian	laplacian	ADJ
ejpam-5057	19	21	matrix	matrix	NOUN
ejpam-5057	19	22	.	.	PUNCT
ejpam-5057	20	1	the	the	DET
ejpam-5057	20	2	signless	signless	ADJ
ejpam-5057	20	3	laplacian	laplacian	ADJ
ejpam-5057	20	4	matrix	matrix	NOUN
ejpam-5057	20	5	of	of	ADP
ejpam-5057	20	6	g	g	PROPN
ejpam-5057	20	7	is	be	AUX
ejpam-5057	20	8	q(g	q(g	PROPN
ejpam-5057	20	9	)	)	PUNCT
ejpam-5057	21	1	=	=	SYM
ejpam-5057	21	2	d(g	d(g	PROPN
ejpam-5057	21	3	)	)	PUNCT
ejpam-5057	22	1	+	+	NUM
ejpam-5057	22	2	a(g	a(g	PROPN
ejpam-5057	22	3	)	)	PUNCT
ejpam-5057	22	4	.	.	PUNCT
ejpam-5057	23	1	for	for	ADP
ejpam-5057	23	2	a	a	DET
ejpam-5057	23	3	bipartite	bipartite	ADJ
ejpam-5057	23	4	graph	graph	NOUN
ejpam-5057	23	5	,	,	PUNCT
ejpam-5057	23	6	laplacian	laplacian	ADJ
ejpam-5057	23	7	and	and	CCONJ
ejpam-5057	23	8	signless	signless	ADJ
ejpam-5057	23	9	laplacian	laplacian	ADJ
ejpam-5057	23	10	∗corresponding	∗corresponde	VERB
ejpam-5057	23	11	author	author	NOUN
ejpam-5057	23	12	.	.	PUNCT
ejpam-5057	24	1	doi	doi	NOUN
ejpam-5057	24	2	:	:	PUNCT
ejpam-5057	24	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5102	https://doi.org/10.29020/nybg.ejpam.v17i2.5102	PUNCT
ejpam-5057	24	4	email	email	NOUN
ejpam-5057	24	5	addresses	address	NOUN
ejpam-5057	24	6	:	:	PUNCT
ejpam-5057	24	7	machasri.m2019@vitstudent.ac.in	machasri.m2019@vitstudent.ac.in	PUNCT
ejpam-5057	24	8	(	(	PUNCT
ejpam-5057	24	9	m.machasri	m.machasri	NUM
ejpam-5057	24	10	)	)	PUNCT
ejpam-5057	24	11	,	,	PUNCT
ejpam-5057	24	12	kalyanidesikan@vit.ac.in	kalyanidesikan@vit.ac.in	NOUN
ejpam-5057	24	13	(	(	PUNCT
ejpam-5057	24	14	d.kalyani	d.kalyani	PROPN
ejpam-5057	24	15	)	)	PUNCT
ejpam-5057	24	16	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5057	25	1	772	772	NUM
ejpam-5057	25	2	©	©	ADP
ejpam-5057	25	3	2024	2024	NUM
ejpam-5057	25	4	ejpam	ejpam	NOUN
ejpam-5057	25	5	all	all	DET
ejpam-5057	25	6	rights	right	NOUN
ejpam-5057	25	7	reserved	reserve	VERB
ejpam-5057	25	8	.	.	PUNCT
ejpam-5057	26	1	m.machasri	m.machasri	NUM
ejpam-5057	26	2	,	,	PUNCT
ejpam-5057	26	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	26	4	/	/	SYM
ejpam-5057	26	5	eur	eur	PROPN
ejpam-5057	26	6	.	.	PUNCT
ejpam-5057	27	1	j.	j.	PROPN
ejpam-5057	27	2	pure	pure	PROPN
ejpam-5057	27	3	appl	appl	PROPN
ejpam-5057	27	4	.	.	PROPN
ejpam-5057	27	5	math	math	PROPN
ejpam-5057	27	6	,	,	PUNCT
ejpam-5057	27	7	17	17	NUM
ejpam-5057	27	8	(	(	PUNCT
ejpam-5057	27	9	2	2	NUM
ejpam-5057	27	10	)	)	PUNCT
ejpam-5057	27	11	(	(	PUNCT
ejpam-5057	27	12	2024	2024	NUM
ejpam-5057	27	13	)	)	PUNCT
ejpam-5057	27	14	,	,	PUNCT
ejpam-5057	27	15	772	772	NUM
ejpam-5057	27	16	-	-	SYM
ejpam-5057	27	17	789	789	NUM
ejpam-5057	27	18	773	773	NUM
ejpam-5057	27	19	eigenvalues	eigenvalue	NOUN
ejpam-5057	27	20	are	be	AUX
ejpam-5057	27	21	the	the	DET
ejpam-5057	27	22	same	same	ADJ
ejpam-5057	27	23	.	.	PUNCT
ejpam-5057	28	1	expanders	expander	NOUN
ejpam-5057	28	2	are	be	AUX
ejpam-5057	28	3	graphs	graph	NOUN
ejpam-5057	28	4	which	which	PRON
ejpam-5057	28	5	are	be	AUX
ejpam-5057	28	6	sparse	sparse	ADJ
ejpam-5057	28	7	but	but	CCONJ
ejpam-5057	28	8	highly	highly	ADV
ejpam-5057	28	9	connected	connect	VERB
ejpam-5057	28	10	.	.	PUNCT
ejpam-5057	29	1	the	the	DET
ejpam-5057	29	2	word	word	NOUN
ejpam-5057	29	3	sparse	sparse	ADJ
ejpam-5057	29	4	means	mean	VERB
ejpam-5057	29	5	that	that	SCONJ
ejpam-5057	29	6	the	the	DET
ejpam-5057	29	7	number	number	NOUN
ejpam-5057	29	8	of	of	ADP
ejpam-5057	29	9	edges	edge	NOUN
ejpam-5057	29	10	of	of	ADP
ejpam-5057	29	11	g	g	NOUN
ejpam-5057	29	12	is	be	AUX
ejpam-5057	29	13	much	much	ADV
ejpam-5057	29	14	less	less	ADJ
ejpam-5057	29	15	than	than	ADP
ejpam-5057	29	16	the	the	DET
ejpam-5057	29	17	possible	possible	ADJ
ejpam-5057	29	18	number	number	NOUN
ejpam-5057	29	19	of	of	ADP
ejpam-5057	29	20	edges	edge	NOUN
ejpam-5057	29	21	of	of	ADP
ejpam-5057	29	22	g.	g.	NOUN
ejpam-5057	29	23	expander	expander	NOUN
ejpam-5057	29	24	has	have	VERB
ejpam-5057	29	25	wide	wide	ADJ
ejpam-5057	29	26	application	application	NOUN
ejpam-5057	29	27	in	in	ADP
ejpam-5057	29	28	various	various	ADJ
ejpam-5057	29	29	areas	area	NOUN
ejpam-5057	29	30	of	of	ADP
ejpam-5057	29	31	computer	computer	NOUN
ejpam-5057	29	32	science	science	NOUN
ejpam-5057	29	33	including	include	VERB
ejpam-5057	29	34	pseudorandomness	pseudorandomness	NOUN
ejpam-5057	29	35	,	,	PUNCT
ejpam-5057	29	36	complexity	complexity	NOUN
ejpam-5057	29	37	theory	theory	NOUN
ejpam-5057	29	38	,	,	PUNCT
ejpam-5057	29	39	coding	code	VERB
ejpam-5057	29	40	theory	theory	NOUN
ejpam-5057	29	41	,	,	PUNCT
ejpam-5057	29	42	algorithm	algorithm	PROPN
ejpam-5057	29	43	design	design	NOUN
ejpam-5057	29	44	,	,	PUNCT
ejpam-5057	29	45	cryptography	cryptography	NOUN
ejpam-5057	29	46	,	,	PUNCT
ejpam-5057	29	47	etc	etc	X
ejpam-5057	29	48	.	.	X
ejpam-5057	30	1	depending	depend	VERB
ejpam-5057	30	2	on	on	ADP
ejpam-5057	30	3	the	the	DET
ejpam-5057	30	4	use	use	NOUN
ejpam-5057	30	5	,	,	PUNCT
ejpam-5057	30	6	the	the	DET
ejpam-5057	30	7	term	term	NOUN
ejpam-5057	30	8	expander	expander	NOUN
ejpam-5057	30	9	has	have	VERB
ejpam-5057	30	10	many	many	ADJ
ejpam-5057	30	11	meanings	meaning	NOUN
ejpam-5057	30	12	.	.	PUNCT
ejpam-5057	31	1	an	an	DET
ejpam-5057	31	2	(	(	PUNCT
ejpam-5057	31	3	n	n	X
ejpam-5057	31	4	,	,	PUNCT
ejpam-5057	31	5	m	m	PROPN
ejpam-5057	31	6	,	,	PUNCT
ejpam-5057	31	7	d	d	PROPN
ejpam-5057	31	8	,	,	PUNCT
ejpam-5057	31	9	γ	γ	X
ejpam-5057	31	10	,	,	PUNCT
ejpam-5057	31	11	α)−	α)−	PROPN
ejpam-5057	31	12	expander	expander	NOUN
ejpam-5057	31	13	is	be	AUX
ejpam-5057	31	14	a	a	DET
ejpam-5057	31	15	bipartite	bipartite	ADJ
ejpam-5057	31	16	graph	graph	NOUN
ejpam-5057	31	17	h	h	NOUN
ejpam-5057	31	18	=	=	SYM
ejpam-5057	31	19	(	(	PUNCT
ejpam-5057	31	20	x	x	X
ejpam-5057	31	21	,	,	PUNCT
ejpam-5057	31	22	y	y	PROPN
ejpam-5057	31	23	,	,	PUNCT
ejpam-5057	31	24	e	e	NOUN
ejpam-5057	31	25	)	)	PUNCT
ejpam-5057	31	26	where	where	SCONJ
ejpam-5057	31	27	|x|	|x|	PROPN
ejpam-5057	31	28	=	=	SYM
ejpam-5057	31	29	n	n	CCONJ
ejpam-5057	31	30	,	,	PUNCT
ejpam-5057	31	31	|y	|y	NOUN
ejpam-5057	31	32	|	|	NOUN
ejpam-5057	31	33	=	=	SYM
ejpam-5057	31	34	m	m	NOUN
ejpam-5057	31	35	,	,	PUNCT
ejpam-5057	31	36	d(x	d(x	PROPN
ejpam-5057	31	37	)	)	PUNCT
ejpam-5057	32	1	=	=	SYM
ejpam-5057	33	1	d	d	NOUN
ejpam-5057	33	2	for	for	ADP
ejpam-5057	33	3	all	all	DET
ejpam-5057	33	4	x	x	SYM
ejpam-5057	33	5	∈	∈	ADJ
ejpam-5057	33	6	x	x	X
ejpam-5057	33	7	and	and	CCONJ
ejpam-5057	33	8	for	for	ADP
ejpam-5057	33	9	every	every	PRON
ejpam-5057	33	10	s	s	PROPN
ejpam-5057	33	11	⊆	⊆	NUM
ejpam-5057	33	12	x	x	SYM
ejpam-5057	33	13	,	,	PUNCT
ejpam-5057	33	14	|s|	|s|	NOUN
ejpam-5057	33	15	≤	≤	NUM
ejpam-5057	33	16	γn	γn	NUM
ejpam-5057	33	17	,	,	PUNCT
ejpam-5057	33	18	we	we	PRON
ejpam-5057	33	19	have	have	VERB
ejpam-5057	33	20	the	the	DET
ejpam-5057	33	21	set	set	NOUN
ejpam-5057	33	22	of	of	ADP
ejpam-5057	33	23	vertices	vertex	NOUN
ejpam-5057	33	24	n(s	n(	NOUN
ejpam-5057	33	25	)	)	PUNCT
ejpam-5057	33	26	⊆	⊆	NUM
ejpam-5057	33	27	y	y	NUM
ejpam-5057	33	28	such	such	ADJ
ejpam-5057	33	29	that	that	SCONJ
ejpam-5057	33	30	|n(s)|	|n(s)|	PROPN
ejpam-5057	33	31	≥	≥	NUM
ejpam-5057	33	32	α|s|	α|s|	NOUN
ejpam-5057	33	33	,	,	PUNCT
ejpam-5057	33	34	where	where	SCONJ
ejpam-5057	33	35	γ	γ	PROPN
ejpam-5057	33	36	and	and	CCONJ
ejpam-5057	33	37	α	α	PROPN
ejpam-5057	33	38	are	be	AUX
ejpam-5057	33	39	positive	positive	ADJ
ejpam-5057	33	40	constants	constant	NOUN
ejpam-5057	33	41	.	.	PUNCT
ejpam-5057	34	1	to	to	PART
ejpam-5057	34	2	obtain	obtain	VERB
ejpam-5057	34	3	good	good	ADJ
ejpam-5057	34	4	expansion	expansion	NOUN
ejpam-5057	34	5	,	,	PUNCT
ejpam-5057	34	6	α	α	PRON
ejpam-5057	34	7	should	should	AUX
ejpam-5057	34	8	be	be	AUX
ejpam-5057	34	9	high	high	ADJ
ejpam-5057	34	10	.	.	PUNCT
ejpam-5057	35	1	an	an	DET
ejpam-5057	35	2	expander	expander	NOUN
ejpam-5057	35	3	is	be	AUX
ejpam-5057	35	4	said	say	VERB
ejpam-5057	35	5	to	to	PART
ejpam-5057	35	6	be	be	AUX
ejpam-5057	35	7	a	a	DET
ejpam-5057	35	8	lossless	lossless	NOUN
ejpam-5057	35	9	expander	expander	NOUN
ejpam-5057	35	10	when	when	SCONJ
ejpam-5057	35	11	α	α	NOUN
ejpam-5057	35	12	is	be	AUX
ejpam-5057	35	13	closer	close	ADJ
ejpam-5057	35	14	to	to	ADP
ejpam-5057	35	15	d.	d.	PROPN
ejpam-5057	35	16	expander	expander	PROPN
ejpam-5057	35	17	code	code	NOUN
ejpam-5057	35	18	has	have	AUX
ejpam-5057	35	19	rate	rate	NOUN
ejpam-5057	35	20	at	at	ADP
ejpam-5057	35	21	least	least	ADJ
ejpam-5057	35	22	1−m	1−m	NUM
ejpam-5057	35	23	n	n	NOUN
ejpam-5057	35	24	.	.	PUNCT
ejpam-5057	36	1	therefore	therefore	ADV
ejpam-5057	36	2	,	,	PUNCT
ejpam-5057	36	3	smaller	small	ADJ
ejpam-5057	36	4	m	m	VERB
ejpam-5057	36	5	implies	imply	VERB
ejpam-5057	36	6	codes	code	NOUN
ejpam-5057	36	7	with	with	ADP
ejpam-5057	36	8	higher	high	ADJ
ejpam-5057	36	9	rate	rate	NOUN
ejpam-5057	36	10	.	.	PUNCT
ejpam-5057	37	1	an	an	DET
ejpam-5057	37	2	error	error	NOUN
ejpam-5057	37	3	-	-	PUNCT
ejpam-5057	37	4	correcting	correct	VERB
ejpam-5057	37	5	code	code	NOUN
ejpam-5057	37	6	(	(	PUNCT
ejpam-5057	37	7	ecc	ecc	PROPN
ejpam-5057	37	8	)	)	PUNCT
ejpam-5057	37	9	is	be	AUX
ejpam-5057	37	10	a	a	DET
ejpam-5057	37	11	type	type	NOUN
ejpam-5057	37	12	of	of	ADP
ejpam-5057	37	13	encoding	encoding	NOUN
ejpam-5057	37	14	used	use	VERB
ejpam-5057	37	15	in	in	ADP
ejpam-5057	37	16	the	the	DET
ejpam-5057	37	17	theory	theory	NOUN
ejpam-5057	37	18	of	of	ADP
ejpam-5057	37	19	coding	code	VERB
ejpam-5057	37	20	to	to	PART
ejpam-5057	37	21	transfer	transfer	VERB
ejpam-5057	37	22	messages	message	NOUN
ejpam-5057	37	23	as	as	ADP
ejpam-5057	37	24	binary	binary	ADJ
ejpam-5057	37	25	numbers	number	NOUN
ejpam-5057	37	26	in	in	ADP
ejpam-5057	37	27	a	a	DET
ejpam-5057	37	28	way	way	NOUN
ejpam-5057	37	29	that	that	PRON
ejpam-5057	37	30	allows	allow	VERB
ejpam-5057	37	31	the	the	DET
ejpam-5057	37	32	message	message	NOUN
ejpam-5057	37	33	to	to	PART
ejpam-5057	37	34	be	be	AUX
ejpam-5057	37	35	decoded	decode	VERB
ejpam-5057	37	36	even	even	ADV
ejpam-5057	37	37	if	if	SCONJ
ejpam-5057	37	38	some	some	DET
ejpam-5057	37	39	bits	bit	NOUN
ejpam-5057	37	40	are	be	AUX
ejpam-5057	37	41	reversed	reverse	VERB
ejpam-5057	37	42	.	.	PUNCT
ejpam-5057	38	1	the	the	DET
ejpam-5057	38	2	error	error	NOUN
ejpam-5057	38	3	-	-	PUNCT
ejpam-5057	38	4	correcting	correct	VERB
ejpam-5057	38	5	codes	code	NOUN
ejpam-5057	38	6	known	know	VERB
ejpam-5057	38	7	as	as	ADP
ejpam-5057	38	8	expander	expander	NOUN
ejpam-5057	38	9	codes	code	NOUN
ejpam-5057	38	10	are	be	AUX
ejpam-5057	38	11	constructed	construct	VERB
ejpam-5057	38	12	from	from	ADP
ejpam-5057	38	13	bipartite	bipartite	PROPN
ejpam-5057	38	14	expander	expander	NOUN
ejpam-5057	38	15	graphs	graph	NOUN
ejpam-5057	38	16	.	.	PUNCT
ejpam-5057	39	1	a	a	DET
ejpam-5057	39	2	collection	collection	NOUN
ejpam-5057	39	3	of	of	ADP
ejpam-5057	39	4	strings	string	NOUN
ejpam-5057	39	5	known	know	VERB
ejpam-5057	39	6	as	as	ADP
ejpam-5057	39	7	codewords	codeword	NOUN
ejpam-5057	39	8	form	form	VERB
ejpam-5057	39	9	an	an	DET
ejpam-5057	39	10	error	error	NOUN
ejpam-5057	39	11	correcting	correct	VERB
ejpam-5057	39	12	code	code	NOUN
ejpam-5057	39	13	denoted	denote	VERB
ejpam-5057	39	14	by	by	ADP
ejpam-5057	39	15	c.	c.	PROPN
ejpam-5057	39	16	block	block	PROPN
ejpam-5057	39	17	length	length	NOUN
ejpam-5057	39	18	of	of	ADP
ejpam-5057	39	19	an	an	DET
ejpam-5057	39	20	ecc	ecc	NOUN
ejpam-5057	39	21	is	be	AUX
ejpam-5057	39	22	the	the	DET
ejpam-5057	39	23	number	number	NOUN
ejpam-5057	39	24	of	of	ADP
ejpam-5057	39	25	elements	element	NOUN
ejpam-5057	39	26	in	in	ADP
ejpam-5057	39	27	the	the	DET
ejpam-5057	39	28	code	code	NOUN
ejpam-5057	39	29	word	word	NOUN
ejpam-5057	39	30	and	and	CCONJ
ejpam-5057	39	31	it	it	PRON
ejpam-5057	39	32	is	be	AUX
ejpam-5057	39	33	denoted	denote	VERB
ejpam-5057	39	34	by	by	ADP
ejpam-5057	39	35	n.	n.	NOUN
ejpam-5057	39	36	the	the	DET
ejpam-5057	39	37	codewords	codeword	NOUN
ejpam-5057	39	38	consist	consist	VERB
ejpam-5057	39	39	of	of	ADP
ejpam-5057	39	40	n	n	DET
ejpam-5057	39	41	symbols	symbol	NOUN
ejpam-5057	39	42	from	from	ADP
ejpam-5057	39	43	σ	σ	PROPN
ejpam-5057	39	44	which	which	PRON
ejpam-5057	39	45	is	be	AUX
ejpam-5057	39	46	the	the	DET
ejpam-5057	39	47	alphabet	alphabet	NOUN
ejpam-5057	39	48	set	set	NOUN
ejpam-5057	39	49	.	.	PUNCT
ejpam-5057	40	1	a	a	DET
ejpam-5057	40	2	code	code	NOUN
ejpam-5057	40	3	is	be	AUX
ejpam-5057	40	4	referred	refer	VERB
ejpam-5057	40	5	to	to	ADP
ejpam-5057	40	6	as	as	ADP
ejpam-5057	40	7	(	(	PUNCT
ejpam-5057	40	8	n	n	CCONJ
ejpam-5057	40	9	,	,	PUNCT
ejpam-5057	40	10	k)q	k)q	X
ejpam-5057	40	11	code	code	NOUN
ejpam-5057	40	12	where	where	SCONJ
ejpam-5057	40	13	|σ|	|σ|	PROPN
ejpam-5057	40	14	=	=	SYM
ejpam-5057	40	15	q	q	PROPN
ejpam-5057	40	16	and	and	CCONJ
ejpam-5057	40	17	|c|	|c|	PROPN
ejpam-5057	40	18	=	=	SYM
ejpam-5057	40	19	qk	qk	PROPN
ejpam-5057	40	20	.	.	PUNCT
ejpam-5057	41	1	k	k	PROPN
ejpam-5057	41	2	indicates	indicate	VERB
ejpam-5057	41	3	how	how	SCONJ
ejpam-5057	41	4	many	many	ADJ
ejpam-5057	41	5	informational	informational	ADJ
ejpam-5057	41	6	symbols	symbol	NOUN
ejpam-5057	41	7	are	be	AUX
ejpam-5057	41	8	contained	contain	VERB
ejpam-5057	41	9	in	in	ADP
ejpam-5057	41	10	each	each	DET
ejpam-5057	41	11	codeword	codeword	NOUN
ejpam-5057	41	12	and	and	CCONJ
ejpam-5057	41	13	r(c	r(c	ADJ
ejpam-5057	41	14	)	)	PUNCT
ejpam-5057	41	15	=	=	SYM
ejpam-5057	42	1	k	k	PROPN
ejpam-5057	43	1	n	n	X
ejpam-5057	43	2	is	be	AUX
ejpam-5057	43	3	the	the	DET
ejpam-5057	43	4	rate	rate	NOUN
ejpam-5057	43	5	of	of	ADP
ejpam-5057	43	6	the	the	DET
ejpam-5057	43	7	code	code	NOUN
ejpam-5057	43	8	.	.	PUNCT
ejpam-5057	44	1	the	the	DET
ejpam-5057	44	2	smallest	small	ADJ
ejpam-5057	44	3	hamming	hamming	NOUN
ejpam-5057	44	4	distance	distance	NOUN
ejpam-5057	44	5	between	between	ADP
ejpam-5057	44	6	two	two	NUM
ejpam-5057	44	7	different	different	ADJ
ejpam-5057	44	8	codewords	codeword	NOUN
ejpam-5057	44	9	of	of	ADP
ejpam-5057	44	10	c	c	PROPN
ejpam-5057	44	11	is	be	AUX
ejpam-5057	44	12	the	the	DET
ejpam-5057	44	13	distance	distance	NOUN
ejpam-5057	44	14	d(c	d(c	PROPN
ejpam-5057	44	15	)	)	PUNCT
ejpam-5057	44	16	of	of	ADP
ejpam-5057	44	17	the	the	DET
ejpam-5057	44	18	code	code	NOUN
ejpam-5057	44	19	.	.	PUNCT
ejpam-5057	45	1	the	the	DET
ejpam-5057	45	2	parity	parity	NOUN
ejpam-5057	45	3	-	-	PUNCT
ejpam-5057	45	4	check	check	NOUN
ejpam-5057	45	5	and	and	CCONJ
ejpam-5057	45	6	generator	generator	NOUN
ejpam-5057	45	7	matrix	matrix	NOUN
ejpam-5057	45	8	views	view	NOUN
ejpam-5057	45	9	of	of	ADP
ejpam-5057	45	10	the	the	DET
ejpam-5057	45	11	code	code	NOUN
ejpam-5057	45	12	give	give	VERB
ejpam-5057	45	13	reasons	reason	NOUN
ejpam-5057	45	14	for	for	ADP
ejpam-5057	45	15	seeing	see	VERB
ejpam-5057	45	16	linear	linear	NOUN
ejpam-5057	45	17	codes	code	NOUN
ejpam-5057	45	18	as	as	ADP
ejpam-5057	45	19	graphs	graph	NOUN
ejpam-5057	45	20	and	and	CCONJ
ejpam-5057	45	21	building	build	VERB
ejpam-5057	45	22	them	they	PRON
ejpam-5057	45	23	using	use	VERB
ejpam-5057	45	24	graph	graph	NOUN
ejpam-5057	45	25	-	-	PUNCT
ejpam-5057	45	26	theoretic	theoretic	ADJ
ejpam-5057	45	27	methods	method	NOUN
ejpam-5057	45	28	.	.	PUNCT
ejpam-5057	46	1	to	to	PART
ejpam-5057	46	2	express	express	VERB
ejpam-5057	46	3	the	the	DET
ejpam-5057	46	4	properties	property	NOUN
ejpam-5057	46	5	of	of	ADP
ejpam-5057	46	6	the	the	DET
ejpam-5057	46	7	code	code	NOUN
ejpam-5057	46	8	from	from	ADP
ejpam-5057	46	9	the	the	DET
ejpam-5057	46	10	characteristics	characteristic	NOUN
ejpam-5057	46	11	of	of	ADP
ejpam-5057	46	12	the	the	DET
ejpam-5057	46	13	graph	graph	NOUN
ejpam-5057	46	14	,	,	PUNCT
ejpam-5057	46	15	we	we	PRON
ejpam-5057	46	16	can	can	AUX
ejpam-5057	46	17	interpret	interpret	VERB
ejpam-5057	46	18	the	the	DET
ejpam-5057	46	19	parity	parity	NOUN
ejpam-5057	46	20	check	check	NOUN
ejpam-5057	46	21	matrix	matrix	NOUN
ejpam-5057	46	22	for	for	ADP
ejpam-5057	46	23	a	a	DET
ejpam-5057	46	24	[	[	NOUN
ejpam-5057	46	25	n	n	CCONJ
ejpam-5057	46	26	,	,	PUNCT
ejpam-5057	46	27	n	n	CCONJ
ejpam-5057	46	28	−m]2	−m]2	ADV
ejpam-5057	46	29	code	code	NOUN
ejpam-5057	46	30	as	as	ADP
ejpam-5057	46	31	representing	represent	VERB
ejpam-5057	46	32	an	an	DET
ejpam-5057	46	33	n	n	NUM
ejpam-5057	46	34	×m	×m	NOUN
ejpam-5057	46	35	bipartite	bipartite	NOUN
ejpam-5057	46	36	graph	graph	NOUN
ejpam-5057	46	37	.	.	PUNCT
ejpam-5057	47	1	low	low	ADJ
ejpam-5057	47	2	density	density	NOUN
ejpam-5057	47	3	parity	parity	NOUN
ejpam-5057	47	4	check	check	NOUN
ejpam-5057	47	5	(	(	PUNCT
ejpam-5057	47	6	ldpc	ldpc	NOUN
ejpam-5057	47	7	)	)	PUNCT
ejpam-5057	47	8	codes	code	NOUN
ejpam-5057	47	9	are	be	AUX
ejpam-5057	47	10	an	an	DET
ejpam-5057	47	11	interesting	interesting	ADJ
ejpam-5057	47	12	family	family	NOUN
ejpam-5057	47	13	of	of	ADP
ejpam-5057	47	14	codes	code	NOUN
ejpam-5057	47	15	since	since	SCONJ
ejpam-5057	47	16	they	they	PRON
ejpam-5057	47	17	appear	appear	VERB
ejpam-5057	47	18	as	as	ADP
ejpam-5057	47	19	sparse	sparse	ADJ
ejpam-5057	47	20	graphs	graph	NOUN
ejpam-5057	47	21	in	in	ADP
ejpam-5057	47	22	the	the	DET
ejpam-5057	47	23	graph	graph	NOUN
ejpam-5057	47	24	view	view	NOUN
ejpam-5057	47	25	because	because	SCONJ
ejpam-5057	47	26	there	there	PRON
ejpam-5057	47	27	are	be	VERB
ejpam-5057	47	28	few	few	ADJ
ejpam-5057	47	29	1′s	1′s	NUM
ejpam-5057	47	30	in	in	ADP
ejpam-5057	47	31	each	each	DET
ejpam-5057	47	32	row	row	NOUN
ejpam-5057	47	33	and	and	CCONJ
ejpam-5057	47	34	column	column	NOUN
ejpam-5057	47	35	of	of	ADP
ejpam-5057	47	36	the	the	DET
ejpam-5057	47	37	parity	parity	NOUN
ejpam-5057	47	38	-	-	PUNCT
ejpam-5057	47	39	check	check	NOUN
ejpam-5057	47	40	matrix	matrix	NOUN
ejpam-5057	47	41	.	.	PUNCT
ejpam-5057	48	1	in	in	ADP
ejpam-5057	48	2	section	section	NOUN
ejpam-5057	48	3	4	4	NUM
ejpam-5057	48	4	we	we	PRON
ejpam-5057	48	5	obtain	obtain	VERB
ejpam-5057	48	6	the	the	DET
ejpam-5057	48	7	upper	upper	ADJ
ejpam-5057	48	8	bound	bind	VERB
ejpam-5057	48	9	for	for	ADP
ejpam-5057	48	10	the	the	DET
ejpam-5057	48	11	second	second	ADV
ejpam-5057	48	12	largest	large	ADJ
ejpam-5057	48	13	adjacency	adjacency	NOUN
ejpam-5057	48	14	eigenvalue	eigenvalue	NOUN
ejpam-5057	48	15	,	,	PUNCT
ejpam-5057	48	16	lower	low	ADJ
ejpam-5057	48	17	bound	bind	VERB
ejpam-5057	48	18	for	for	ADP
ejpam-5057	48	19	the	the	DET
ejpam-5057	48	20	second	second	ADJ
ejpam-5057	48	21	smallest	small	ADJ
ejpam-5057	48	22	adjacency	adjacency	NOUN
ejpam-5057	48	23	eigenvalue	eigenvalue	NOUN
ejpam-5057	48	24	and	and	CCONJ
ejpam-5057	48	25	upper	upper	ADJ
ejpam-5057	48	26	bound	bind	VERB
ejpam-5057	48	27	for	for	ADP
ejpam-5057	48	28	the	the	DET
ejpam-5057	48	29	second	second	ADV
ejpam-5057	48	30	largest	large	ADJ
ejpam-5057	48	31	laplacian	laplacian	ADJ
ejpam-5057	48	32	eigenvalue	eigenvalue	NOUN
ejpam-5057	48	33	of	of	ADP
ejpam-5057	48	34	a	a	DET
ejpam-5057	48	35	connected	connected	ADJ
ejpam-5057	48	36	bipartite	bipartite	NOUN
ejpam-5057	48	37	graph	graph	NOUN
ejpam-5057	48	38	g.	g.	NOUN
ejpam-5057	48	39	to	to	PART
ejpam-5057	48	40	obtain	obtain	VERB
ejpam-5057	48	41	sharper	sharp	ADJ
ejpam-5057	48	42	bounds	bound	NOUN
ejpam-5057	48	43	,	,	PUNCT
ejpam-5057	48	44	we	we	PRON
ejpam-5057	48	45	discuss	discuss	VERB
ejpam-5057	48	46	some	some	DET
ejpam-5057	48	47	possible	possible	ADJ
ejpam-5057	48	48	cases	case	NOUN
ejpam-5057	48	49	with	with	ADP
ejpam-5057	48	50	respect	respect	NOUN
ejpam-5057	48	51	to	to	ADP
ejpam-5057	48	52	the	the	DET
ejpam-5057	48	53	bipartitions	bipartition	NOUN
ejpam-5057	48	54	of	of	ADP
ejpam-5057	48	55	g	g	NOUN
ejpam-5057	48	56	in	in	ADP
ejpam-5057	48	57	a	a	DET
ejpam-5057	48	58	minimally	minimally	ADV
ejpam-5057	48	59	connected	connect	VERB
ejpam-5057	48	60	bipartite	bipartite	NOUN
ejpam-5057	48	61	graph	graph	NOUN
ejpam-5057	48	62	and	and	CCONJ
ejpam-5057	48	63	derive	derive	ADJ
ejpam-5057	48	64	bounds	bound	NOUN
ejpam-5057	48	65	for	for	ADP
ejpam-5057	48	66	the	the	DET
ejpam-5057	48	67	second	second	ADV
ejpam-5057	48	68	largest	large	ADJ
ejpam-5057	48	69	adjacency	adjacency	NOUN
ejpam-5057	48	70	eigenvalue	eigenvalue	NOUN
ejpam-5057	48	71	and	and	CCONJ
ejpam-5057	48	72	second	second	ADJ
ejpam-5057	48	73	smallest	small	ADJ
ejpam-5057	48	74	adjacency	adjacency	NOUN
ejpam-5057	48	75	eigenvalue	eigenvalue	PROPN
ejpam-5057	48	76	in	in	ADP
ejpam-5057	48	77	terms	term	NOUN
ejpam-5057	48	78	of	of	ADP
ejpam-5057	48	79	n	n	CCONJ
ejpam-5057	48	80	,	,	PUNCT
ejpam-5057	48	81	the	the	DET
ejpam-5057	48	82	number	number	NOUN
ejpam-5057	48	83	of	of	ADP
ejpam-5057	48	84	vertices	vertex	NOUN
ejpam-5057	48	85	.	.	PUNCT
ejpam-5057	49	1	in	in	ADP
ejpam-5057	49	2	section	section	NOUN
ejpam-5057	49	3	5	5	NUM
ejpam-5057	49	4	,	,	PUNCT
ejpam-5057	49	5	we	we	PRON
ejpam-5057	49	6	introduce	introduce	VERB
ejpam-5057	49	7	a	a	DET
ejpam-5057	49	8	new	new	ADJ
ejpam-5057	49	9	concept	concept	NOUN
ejpam-5057	49	10	called	call	VERB
ejpam-5057	49	11	vertex	vertex	NOUN
ejpam-5057	49	12	-	-	PUNCT
ejpam-5057	49	13	split	split	NOUN
ejpam-5057	49	14	of	of	ADP
ejpam-5057	49	15	bipartite	bipartite	NOUN
ejpam-5057	49	16	graph	graph	NOUN
ejpam-5057	49	17	and	and	CCONJ
ejpam-5057	49	18	we	we	PRON
ejpam-5057	49	19	prove	prove	VERB
ejpam-5057	49	20	a	a	DET
ejpam-5057	49	21	condition	condition	NOUN
ejpam-5057	49	22	for	for	ADP
ejpam-5057	49	23	the	the	DET
ejpam-5057	49	24	vertex	vertex	NOUN
ejpam-5057	49	25	-	-	PUNCT
ejpam-5057	49	26	split	split	NOUN
ejpam-5057	49	27	of	of	ADP
ejpam-5057	49	28	a	a	DET
ejpam-5057	49	29	bipartite	bipartite	NOUN
ejpam-5057	49	30	graph	graph	NOUN
ejpam-5057	49	31	to	to	PART
ejpam-5057	49	32	be	be	AUX
ejpam-5057	49	33	k−connected	k−connecte	VERB
ejpam-5057	49	34	with	with	ADP
ejpam-5057	49	35	respect	respect	NOUN
ejpam-5057	49	36	to	to	ADP
ejpam-5057	49	37	λ2	λ2	NOUN
ejpam-5057	49	38	.	.	PUNCT
ejpam-5057	50	1	we	we	PRON
ejpam-5057	50	2	prove	prove	VERB
ejpam-5057	50	3	theorems	theorem	NOUN
ejpam-5057	50	4	related	relate	VERB
ejpam-5057	50	5	to	to	ADP
ejpam-5057	50	6	connectivity	connectivity	NOUN
ejpam-5057	50	7	and	and	CCONJ
ejpam-5057	50	8	prove	prove	VERB
ejpam-5057	50	9	the	the	DET
ejpam-5057	50	10	expansion	expansion	NOUN
ejpam-5057	50	11	of	of	ADP
ejpam-5057	50	12	vertex	vertex	NOUN
ejpam-5057	50	13	-	-	PUNCT
ejpam-5057	50	14	split	split	NOUN
ejpam-5057	50	15	m.machasri	m.machasri	NOUN
ejpam-5057	50	16	,	,	PUNCT
ejpam-5057	50	17	d.kalyani	d.kalyani	NOUN
ejpam-5057	50	18	/	/	SYM
ejpam-5057	50	19	eur	eur	PROPN
ejpam-5057	50	20	.	.	PUNCT
ejpam-5057	51	1	j.	j.	PROPN
ejpam-5057	51	2	pure	pure	PROPN
ejpam-5057	51	3	appl	appl	PROPN
ejpam-5057	51	4	.	.	PROPN
ejpam-5057	51	5	math	math	PROPN
ejpam-5057	51	6	,	,	PUNCT
ejpam-5057	51	7	17	17	NUM
ejpam-5057	51	8	(	(	PUNCT
ejpam-5057	51	9	2	2	NUM
ejpam-5057	51	10	)	)	PUNCT
ejpam-5057	51	11	(	(	PUNCT
ejpam-5057	51	12	2024	2024	NUM
ejpam-5057	51	13	)	)	PUNCT
ejpam-5057	51	14	,	,	PUNCT
ejpam-5057	51	15	772	772	NUM
ejpam-5057	51	16	-	-	SYM
ejpam-5057	51	17	789	789	NUM
ejpam-5057	51	18	774	774	NUM
ejpam-5057	51	19	of	of	ADP
ejpam-5057	51	20	a	a	DET
ejpam-5057	51	21	bipartite	bipartite	NOUN
ejpam-5057	51	22	graph	graph	NOUN
ejpam-5057	51	23	.	.	PUNCT
ejpam-5057	52	1	finally	finally	ADV
ejpam-5057	52	2	,	,	PUNCT
ejpam-5057	52	3	in	in	ADP
ejpam-5057	52	4	section	section	NOUN
ejpam-5057	52	5	6	6	NUM
ejpam-5057	52	6	we	we	PRON
ejpam-5057	52	7	show	show	VERB
ejpam-5057	52	8	that	that	SCONJ
ejpam-5057	52	9	the	the	DET
ejpam-5057	52	10	vertex	vertex	NOUN
ejpam-5057	52	11	-	-	PUNCT
ejpam-5057	52	12	split	split	NOUN
ejpam-5057	52	13	of	of	ADP
ejpam-5057	52	14	a	a	DET
ejpam-5057	52	15	biregular	biregular	ADJ
ejpam-5057	52	16	bipartite	bipartite	NOUN
ejpam-5057	52	17	graph	graph	NOUN
ejpam-5057	52	18	forms	form	VERB
ejpam-5057	52	19	an	an	DET
ejpam-5057	52	20	expander	expander	NOUN
ejpam-5057	52	21	code	code	NOUN
ejpam-5057	52	22	.	.	PUNCT
ejpam-5057	53	1	2	2	X
ejpam-5057	53	2	.	.	X
ejpam-5057	53	3	related	relate	VERB
ejpam-5057	53	4	work	work	NOUN
ejpam-5057	53	5	eigenvalues	eigenvalue	NOUN
ejpam-5057	53	6	are	be	AUX
ejpam-5057	53	7	often	often	ADV
ejpam-5057	53	8	difficult	difficult	ADJ
ejpam-5057	53	9	to	to	PART
ejpam-5057	53	10	compute	compute	VERB
ejpam-5057	53	11	.	.	PUNCT
ejpam-5057	54	1	therefore	therefore	ADV
ejpam-5057	54	2	,	,	PUNCT
ejpam-5057	54	3	obtaining	obtain	VERB
ejpam-5057	54	4	bounds	bound	NOUN
ejpam-5057	54	5	for	for	ADP
ejpam-5057	54	6	eigenvalues	eigenvalue	NOUN
ejpam-5057	54	7	is	be	AUX
ejpam-5057	54	8	useful	useful	ADJ
ejpam-5057	54	9	.	.	PUNCT
ejpam-5057	55	1	in	in	ADP
ejpam-5057	55	2	literature	literature	NOUN
ejpam-5057	55	3	,	,	PUNCT
ejpam-5057	55	4	bounds	bound	NOUN
ejpam-5057	55	5	have	have	AUX
ejpam-5057	55	6	been	be	AUX
ejpam-5057	55	7	found	find	VERB
ejpam-5057	55	8	for	for	ADP
ejpam-5057	55	9	the	the	DET
ejpam-5057	55	10	second	second	ADV
ejpam-5057	55	11	largest	large	ADJ
ejpam-5057	55	12	eigenvalue	eigenvalue	NOUN
ejpam-5057	55	13	of	of	ADP
ejpam-5057	55	14	some	some	DET
ejpam-5057	55	15	family	family	NOUN
ejpam-5057	55	16	of	of	ADP
ejpam-5057	55	17	graphs	graph	NOUN
ejpam-5057	55	18	.	.	PUNCT
ejpam-5057	56	1	chang	chang	PROPN
ejpam-5057	56	2	an	an	PRON
ejpam-5057	57	1	[	[	X
ejpam-5057	57	2	2	2	NUM
ejpam-5057	57	3	]	]	PUNCT
ejpam-5057	57	4	obtained	obtain	VERB
ejpam-5057	57	5	upper	upper	ADJ
ejpam-5057	57	6	and	and	CCONJ
ejpam-5057	57	7	lower	low	ADJ
ejpam-5057	57	8	bounds	bound	NOUN
ejpam-5057	57	9	for	for	ADP
ejpam-5057	57	10	the	the	DET
ejpam-5057	57	11	second	second	ADV
ejpam-5057	57	12	largest	large	ADJ
ejpam-5057	57	13	eigenvalue	eigenvalue	NOUN
ejpam-5057	57	14	of	of	ADP
ejpam-5057	57	15	a	a	DET
ejpam-5057	57	16	tree	tree	NOUN
ejpam-5057	57	17	.	.	PUNCT
ejpam-5057	58	1	the	the	DET
ejpam-5057	58	2	lower	low	ADJ
ejpam-5057	58	3	bounds	bound	NOUN
ejpam-5057	58	4	on	on	ADP
ejpam-5057	58	5	the	the	DET
ejpam-5057	58	6	second	second	ADV
ejpam-5057	58	7	largest	large	ADJ
ejpam-5057	58	8	eigenvalue	eigenvalue	NOUN
ejpam-5057	58	9	of	of	ADP
ejpam-5057	58	10	a	a	DET
ejpam-5057	58	11	regular	regular	ADJ
ejpam-5057	58	12	graph	graph	NOUN
ejpam-5057	58	13	with	with	ADP
ejpam-5057	58	14	given	give	VERB
ejpam-5057	58	15	girth	girth	NOUN
ejpam-5057	58	16	were	be	AUX
ejpam-5057	58	17	obtained	obtain	VERB
ejpam-5057	58	18	by	by	ADP
ejpam-5057	58	19	patrik	patrik	NOUN
ejpam-5057	58	20	solv	solv	NOUN
ejpam-5057	59	1	[	[	X
ejpam-5057	59	2	14	14	NUM
ejpam-5057	59	3	]	]	PUNCT
ejpam-5057	59	4	.	.	PUNCT
ejpam-5057	60	1	mehatari	mehatari	PROPN
ejpam-5057	60	2	and	and	CCONJ
ejpam-5057	60	3	kannan	kannan	PROPN
ejpam-5057	60	4	[	[	X
ejpam-5057	60	5	11	11	NUM
ejpam-5057	60	6	]	]	PUNCT
ejpam-5057	60	7	derived	derive	VERB
ejpam-5057	60	8	bounds	bound	NOUN
ejpam-5057	60	9	for	for	ADP
ejpam-5057	60	10	the	the	DET
ejpam-5057	60	11	second	second	ADV
ejpam-5057	60	12	largest	large	ADJ
ejpam-5057	60	13	and	and	CCONJ
ejpam-5057	60	14	second	second	ADJ
ejpam-5057	60	15	smallest	small	ADJ
ejpam-5057	60	16	eigenvalues	eigenvalue	NOUN
ejpam-5057	60	17	of	of	ADP
ejpam-5057	60	18	adjacency	adjacency	NOUN
ejpam-5057	60	19	matrices	matrix	NOUN
ejpam-5057	60	20	,	,	PUNCT
ejpam-5057	60	21	normalized	normalize	VERB
ejpam-5057	60	22	adjacency	adjacency	NOUN
ejpam-5057	60	23	matrices	matrix	NOUN
ejpam-5057	60	24	and	and	CCONJ
ejpam-5057	60	25	laplacian	laplacian	ADJ
ejpam-5057	60	26	matrices	matrix	NOUN
ejpam-5057	60	27	of	of	ADP
ejpam-5057	60	28	regular	regular	ADJ
ejpam-5057	60	29	graphs	graph	NOUN
ejpam-5057	60	30	.	.	PUNCT
ejpam-5057	61	1	in	in	ADP
ejpam-5057	61	2	1988	1988	NUM
ejpam-5057	61	3	,	,	PUNCT
ejpam-5057	61	4	powers	power	NOUN
ejpam-5057	61	5	[	[	X
ejpam-5057	61	6	12	12	NUM
ejpam-5057	61	7	]	]	PUNCT
ejpam-5057	61	8	gave	give	VERB
ejpam-5057	61	9	some	some	DET
ejpam-5057	61	10	upper	upper	ADJ
ejpam-5057	61	11	bounds	bound	NOUN
ejpam-5057	61	12	of	of	ADP
ejpam-5057	61	13	second	second	ADV
ejpam-5057	61	14	largest	large	ADJ
ejpam-5057	61	15	eigenvalue	eigenvalue	NOUN
ejpam-5057	61	16	for	for	ADP
ejpam-5057	61	17	general	general	ADJ
ejpam-5057	61	18	graphs	graph	NOUN
ejpam-5057	61	19	and	and	CCONJ
ejpam-5057	61	20	bipartite	bipartite	NOUN
ejpam-5057	61	21	graphs	graph	NOUN
ejpam-5057	61	22	.	.	PUNCT
ejpam-5057	62	1	in	in	ADP
ejpam-5057	62	2	2012	2012	NUM
ejpam-5057	62	3	,	,	PUNCT
ejpam-5057	62	4	mingqing	mingqe	VERB
ejpam-5057	62	5	zhaim	zhaim	NOUN
ejpam-5057	62	6	et	et	NOUN
ejpam-5057	62	7	al	al	PROPN
ejpam-5057	62	8	.	.	PUNCT
ejpam-5057	63	1	[	[	X
ejpam-5057	63	2	18	18	NUM
ejpam-5057	63	3	]	]	PUNCT
ejpam-5057	63	4	presented	present	VERB
ejpam-5057	63	5	upper	upper	ADJ
ejpam-5057	63	6	bounds	bound	NOUN
ejpam-5057	63	7	for	for	ADP
ejpam-5057	63	8	the	the	DET
ejpam-5057	63	9	second	second	ADV
ejpam-5057	63	10	largest	large	ADJ
ejpam-5057	63	11	eigenvalue	eigenvalue	NOUN
ejpam-5057	63	12	of	of	ADP
ejpam-5057	63	13	connected	connected	ADJ
ejpam-5057	63	14	graphs	graph	NOUN
ejpam-5057	63	15	and	and	CCONJ
ejpam-5057	63	16	particularly	particularly	ADV
ejpam-5057	63	17	,	,	PUNCT
ejpam-5057	63	18	for	for	ADP
ejpam-5057	63	19	bipartite	bipartite	NOUN
ejpam-5057	63	20	graphs	graph	NOUN
ejpam-5057	63	21	.	.	PUNCT
ejpam-5057	64	1	expander	expander	NOUN
ejpam-5057	64	2	codes	code	NOUN
ejpam-5057	64	3	which	which	PRON
ejpam-5057	64	4	are	be	AUX
ejpam-5057	64	5	constructed	construct	VERB
ejpam-5057	64	6	from	from	ADP
ejpam-5057	64	7	unbalanced	unbalanced	ADJ
ejpam-5057	64	8	bipartite	bipartite	PROPN
ejpam-5057	64	9	expander	expander	NOUN
ejpam-5057	64	10	graphs	graph	VERB
ejpam-5057	64	11	with	with	ADP
ejpam-5057	64	12	expansion	expansion	NOUN
ejpam-5057	64	13	factor	factor	NOUN
ejpam-5057	64	14	(	(	PUNCT
ejpam-5057	64	15	1	1	NUM
ejpam-5057	64	16	−	−	NOUN
ejpam-5057	64	17	ϵ)d	ϵ)d	NOUN
ejpam-5057	64	18	for	for	ADP
ejpam-5057	64	19	ϵ	ϵ	X
ejpam-5057	64	20	<	<	X
ejpam-5057	64	21	1	1	NUM
ejpam-5057	64	22	2	2	NUM
ejpam-5057	64	23	are	be	AUX
ejpam-5057	64	24	said	say	VERB
ejpam-5057	64	25	to	to	PART
ejpam-5057	64	26	be	be	AUX
ejpam-5057	64	27	asymptotically	asymptotically	ADV
ejpam-5057	64	28	good	good	ADJ
ejpam-5057	64	29	codes	code	NOUN
ejpam-5057	64	30	.	.	PUNCT
ejpam-5057	65	1	initially	initially	ADV
ejpam-5057	65	2	,	,	PUNCT
ejpam-5057	65	3	constructions	construction	NOUN
ejpam-5057	65	4	with	with	ADP
ejpam-5057	65	5	α	α	PROPN
ejpam-5057	65	6	≈	≈	PROPN
ejpam-5057	65	7	d	d	NOUN
ejpam-5057	65	8	2	2	NUM
ejpam-5057	65	9	were	be	AUX
ejpam-5057	65	10	known	know	VERB
ejpam-5057	65	11	explicitly	explicitly	ADV
ejpam-5057	65	12	.	.	PUNCT
ejpam-5057	66	1	so	so	ADV
ejpam-5057	66	2	the	the	DET
ejpam-5057	66	3	requirement	requirement	NOUN
ejpam-5057	66	4	is	be	AUX
ejpam-5057	66	5	to	to	PART
ejpam-5057	66	6	give	give	VERB
ejpam-5057	66	7	explicit	explicit	ADJ
ejpam-5057	66	8	construction	construction	NOUN
ejpam-5057	66	9	of	of	ADP
ejpam-5057	66	10	codes	code	NOUN
ejpam-5057	66	11	with	with	ADP
ejpam-5057	66	12	expansion	expansion	NOUN
ejpam-5057	66	13	higher	high	ADJ
ejpam-5057	66	14	than	than	ADP
ejpam-5057	66	15	d	d	PROPN
ejpam-5057	66	16	2	2	NUM
ejpam-5057	66	17	.	.	PUNCT
ejpam-5057	67	1	in	in	ADP
ejpam-5057	67	2	2002	2002	NUM
ejpam-5057	67	3	,	,	PUNCT
ejpam-5057	67	4	capalbo	capalbo	NOUN
ejpam-5057	67	5	et	et	PROPN
ejpam-5057	67	6	al	al	PROPN
ejpam-5057	67	7	.	.	PUNCT
ejpam-5057	68	1	[	[	X
ejpam-5057	68	2	4	4	X
ejpam-5057	68	3	]	]	PUNCT
ejpam-5057	68	4	presented	present	VERB
ejpam-5057	68	5	an	an	DET
ejpam-5057	68	6	explicit	explicit	ADJ
ejpam-5057	68	7	construction	construction	NOUN
ejpam-5057	68	8	with	with	ADP
ejpam-5057	68	9	expansion	expansion	NOUN
ejpam-5057	68	10	(	(	PUNCT
ejpam-5057	68	11	1	1	NUM
ejpam-5057	68	12	−	−	NOUN
ejpam-5057	68	13	ϵ)d	ϵ)d	NOUN
ejpam-5057	68	14	where	where	SCONJ
ejpam-5057	68	15	d	d	NOUN
ejpam-5057	68	16	is	be	AUX
ejpam-5057	68	17	the	the	DET
ejpam-5057	68	18	degree	degree	NOUN
ejpam-5057	68	19	of	of	ADP
ejpam-5057	68	20	the	the	DET
ejpam-5057	68	21	left	left	ADJ
ejpam-5057	68	22	side	side	NOUN
ejpam-5057	68	23	partition	partition	NOUN
ejpam-5057	68	24	of	of	ADP
ejpam-5057	68	25	g	g	NOUN
ejpam-5057	68	26	for	for	ADP
ejpam-5057	68	27	any	any	DET
ejpam-5057	68	28	desired	desire	VERB
ejpam-5057	68	29	ϵ	ϵ	X
ejpam-5057	68	30	>	>	X
ejpam-5057	68	31	0	0	NUM
ejpam-5057	68	32	,	,	PUNCT
ejpam-5057	68	33	and	and	CCONJ
ejpam-5057	68	34	imbalance	imbalance	NOUN
ejpam-5057	68	35	ratio	ratio	NOUN
ejpam-5057	68	36	m	m	VERB
ejpam-5057	68	37	n	n	X
ejpam-5057	68	38	.	.	PUNCT
ejpam-5057	69	1	using	use	VERB
ejpam-5057	69	2	the	the	DET
ejpam-5057	69	3	edge	edge	NOUN
ejpam-5057	69	4	vertex	vertex	NOUN
ejpam-5057	69	5	matrix	matrix	NOUN
ejpam-5057	69	6	is	be	AUX
ejpam-5057	69	7	one	one	NUM
ejpam-5057	69	8	method	method	NOUN
ejpam-5057	69	9	of	of	ADP
ejpam-5057	69	10	building	build	VERB
ejpam-5057	69	11	an	an	DET
ejpam-5057	69	12	unbalanced	unbalanced	ADJ
ejpam-5057	69	13	expander	expander	NOUN
ejpam-5057	69	14	,	,	PUNCT
ejpam-5057	69	15	which	which	DET
ejpam-5057	69	16	tanner	tanner	NOUN
ejpam-5057	69	17	[	[	X
ejpam-5057	69	18	16	16	NUM
ejpam-5057	69	19	]	]	PUNCT
ejpam-5057	69	20	introduced	introduce	VERB
ejpam-5057	69	21	and	and	CCONJ
ejpam-5057	69	22	sipser	sipser	NOUN
ejpam-5057	69	23	and	and	CCONJ
ejpam-5057	69	24	spielman	spielman	NOUN
ejpam-5057	70	1	[	[	X
ejpam-5057	70	2	13]-[15	13]-[15	X
ejpam-5057	70	3	]	]	X
ejpam-5057	70	4	employed	employ	VERB
ejpam-5057	70	5	.	.	PUNCT
ejpam-5057	71	1	also	also	ADV
ejpam-5057	71	2	,	,	PUNCT
ejpam-5057	71	3	it	it	PRON
ejpam-5057	71	4	is	be	AUX
ejpam-5057	71	5	observed	observe	VERB
ejpam-5057	71	6	by	by	ADP
ejpam-5057	71	7	zemor	zemor	PROPN
ejpam-5057	72	1	[	[	X
ejpam-5057	72	2	17	17	NUM
ejpam-5057	72	3	]	]	PUNCT
ejpam-5057	72	4	that	that	SCONJ
ejpam-5057	72	5	if	if	SCONJ
ejpam-5057	72	6	the	the	DET
ejpam-5057	72	7	edge	edge	NOUN
ejpam-5057	72	8	vertex	vertex	NOUN
ejpam-5057	72	9	incidence	incidence	NOUN
ejpam-5057	72	10	graph	graph	NOUN
ejpam-5057	72	11	’s	’s	PART
ejpam-5057	72	12	underlying	underlie	VERB
ejpam-5057	72	13	graph	graph	NOUN
ejpam-5057	72	14	is	be	AUX
ejpam-5057	72	15	a	a	DET
ejpam-5057	72	16	bipartite	bipartite	ADJ
ejpam-5057	72	17	graph	graph	NOUN
ejpam-5057	72	18	,	,	PUNCT
ejpam-5057	72	19	the	the	DET
ejpam-5057	72	20	decoding	decode	VERB
ejpam-5057	72	21	technique	technique	NOUN
ejpam-5057	72	22	is	be	AUX
ejpam-5057	72	23	straightforward	straightforward	ADJ
ejpam-5057	72	24	.	.	PUNCT
ejpam-5057	73	1	in	in	ADP
ejpam-5057	73	2	this	this	DET
ejpam-5057	73	3	work	work	NOUN
ejpam-5057	73	4	,	,	PUNCT
ejpam-5057	73	5	we	we	PRON
ejpam-5057	73	6	use	use	VERB
ejpam-5057	73	7	vertex	vertex	NOUN
ejpam-5057	73	8	-	-	PUNCT
ejpam-5057	73	9	split	split	NOUN
ejpam-5057	73	10	of	of	ADP
ejpam-5057	73	11	a	a	DET
ejpam-5057	73	12	bipartite	bipartite	ADJ
ejpam-5057	73	13	graph	graph	NOUN
ejpam-5057	73	14	to	to	PART
ejpam-5057	73	15	construct	construct	VERB
ejpam-5057	73	16	expanders	expander	NOUN
ejpam-5057	73	17	codes	code	NOUN
ejpam-5057	73	18	with	with	ADP
ejpam-5057	73	19	expansion	expansion	NOUN
ejpam-5057	73	20	factor	factor	NOUN
ejpam-5057	73	21	d	d	NOUN
ejpam-5057	73	22	2	2	NUM
ejpam-5057	73	23	<	<	X
ejpam-5057	73	24	α	α	X
ejpam-5057	73	25	<	<	X
ejpam-5057	73	26	d	d	PROPN
ejpam-5057	73	27	and	and	CCONJ
ejpam-5057	73	28	ϵ	ϵ	X
ejpam-5057	73	29	<	<	X
ejpam-5057	73	30	1	1	NUM
ejpam-5057	73	31	2	2	NUM
ejpam-5057	73	32	.	.	PUNCT
ejpam-5057	74	1	3	3	X
ejpam-5057	74	2	.	.	X
ejpam-5057	74	3	preliminaries	preliminary	NOUN
ejpam-5057	74	4	definition	definition	NOUN
ejpam-5057	74	5	1	1	NUM
ejpam-5057	74	6	.	.	PUNCT
ejpam-5057	75	1	[	[	X
ejpam-5057	75	2	9](quotient	9](quotient	NUM
ejpam-5057	75	3	matrix	matrix	NOUN
ejpam-5057	75	4	)	)	PUNCT
ejpam-5057	75	5	let	let	VERB
ejpam-5057	75	6	m	m	VERB
ejpam-5057	75	7	=	=	VERB
ejpam-5057	75	8	m11	m11	X
ejpam-5057	75	9	·	·	PUNCT
ejpam-5057	75	10	·	·	PUNCT
ejpam-5057	75	11	·	·	PUNCT
ejpam-5057	76	1	m1	m1	NOUN
ejpam-5057	76	2	t	t	NOUN
ejpam-5057	76	3	...	...	PUNCT
ejpam-5057	76	4	.	.	PUNCT
ejpam-5057	76	5	.	.	PUNCT
ejpam-5057	76	6	.	.	PUNCT
ejpam-5057	76	7	...	...	PUNCT
ejpam-5057	77	1	mt1	mt1	PROPN
ejpam-5057	77	2	·	·	PUNCT
ejpam-5057	77	3	·	·	PUNCT
ejpam-5057	77	4	·	·	PUNCT
ejpam-5057	77	5	mtt	mtt	NOUN
ejpam-5057	77	6			PRON
ejpam-5057	77	7	be	be	AUX
ejpam-5057	77	8	a	a	DET
ejpam-5057	77	9	real	real	ADJ
ejpam-5057	77	10	matrix	matrix	NOUN
ejpam-5057	77	11	of	of	ADP
ejpam-5057	77	12	order	order	NOUN
ejpam-5057	77	13	n	n	NOUN
ejpam-5057	77	14	and	and	CCONJ
ejpam-5057	77	15	mij	mij	NOUN
ejpam-5057	77	16	be	be	AUX
ejpam-5057	77	17	the	the	DET
ejpam-5057	77	18	blocks	block	NOUN
ejpam-5057	77	19	of	of	ADP
ejpam-5057	77	20	m	m	PROPN
ejpam-5057	77	21	,	,	PUNCT
ejpam-5057	77	22	where	where	SCONJ
ejpam-5057	77	23	i	i	PRON
ejpam-5057	77	24	,	,	PUNCT
ejpam-5057	77	25	j	j	PROPN
ejpam-5057	77	26	=	=	SYM
ejpam-5057	77	27	1	1	NUM
ejpam-5057	77	28	,	,	PUNCT
ejpam-5057	77	29	2	2	NUM
ejpam-5057	77	30	,	,	PUNCT
ejpam-5057	77	31	.	.	PUNCT
ejpam-5057	77	32	.	.	PUNCT
ejpam-5057	78	1	.	.	PUNCT
ejpam-5057	79	1	,	,	PUNCT
ejpam-5057	79	2	t.	t.	PROPN
ejpam-5057	79	3	then	then	ADV
ejpam-5057	79	4	b(m)=(bij	b(m)=(bij	VERB
ejpam-5057	79	5	)	)	PUNCT
ejpam-5057	79	6	is	be	AUX
ejpam-5057	79	7	called	call	VERB
ejpam-5057	79	8	the	the	DET
ejpam-5057	79	9	quotient	quotient	NOUN
ejpam-5057	79	10	matrix	matrix	NOUN
ejpam-5057	79	11	of	of	ADP
ejpam-5057	79	12	m	m	PRON
ejpam-5057	79	13	where	where	SCONJ
ejpam-5057	79	14	bij	bij	NOUN
ejpam-5057	79	15	is	be	AUX
ejpam-5057	79	16	the	the	DET
ejpam-5057	79	17	sum	sum	NOUN
ejpam-5057	79	18	of	of	ADP
ejpam-5057	79	19	all	all	DET
ejpam-5057	79	20	entries	entry	NOUN
ejpam-5057	79	21	in	in	ADP
ejpam-5057	79	22	mij	mij	NOUN
ejpam-5057	79	23	divided	divide	VERB
ejpam-5057	79	24	by	by	ADP
ejpam-5057	79	25	the	the	DET
ejpam-5057	79	26	number	number	NOUN
ejpam-5057	79	27	of	of	ADP
ejpam-5057	79	28	rows	row	NOUN
ejpam-5057	79	29	of	of	ADP
ejpam-5057	79	30	mij	mij	NOUN
ejpam-5057	79	31	.	.	PUNCT
ejpam-5057	79	32	definition	definition	NOUN
ejpam-5057	79	33	2	2	NUM
ejpam-5057	79	34	.	.	PUNCT
ejpam-5057	80	1	[	[	X
ejpam-5057	80	2	3](interlacing	3](interlace	VERB
ejpam-5057	80	3	)	)	PUNCT
ejpam-5057	80	4	consider	consider	VERB
ejpam-5057	80	5	two	two	NUM
ejpam-5057	80	6	sequences	sequence	NOUN
ejpam-5057	80	7	of	of	ADP
ejpam-5057	80	8	real	real	ADJ
ejpam-5057	80	9	numbers	number	NOUN
ejpam-5057	80	10	:	:	PUNCT
ejpam-5057	80	11	ξ1	ξ1	NOUN
ejpam-5057	80	12	,	,	PUNCT
ejpam-5057	80	13	ξ2	ξ2	NOUN
ejpam-5057	80	14	,	,	PUNCT
ejpam-5057	80	15	.	.	PUNCT
ejpam-5057	80	16	.	.	PUNCT
ejpam-5057	81	1	.	.	PUNCT
ejpam-5057	82	1	,	,	PUNCT
ejpam-5057	82	2	ξn	ξn	NOUN
ejpam-5057	82	3	and	and	CCONJ
ejpam-5057	82	4	η1	η1	NOUN
ejpam-5057	82	5	,	,	PUNCT
ejpam-5057	82	6	η2	η2	NOUN
ejpam-5057	82	7	,	,	PUNCT
ejpam-5057	82	8	.	.	PUNCT
ejpam-5057	82	9	.	.	PUNCT
ejpam-5057	83	1	.	.	PUNCT
ejpam-5057	84	1	,	,	PUNCT
ejpam-5057	84	2	ηm	ηm	PROPN
ejpam-5057	84	3	with	with	ADP
ejpam-5057	84	4	m	m	PROPN
ejpam-5057	84	5	≤	≤	NOUN
ejpam-5057	84	6	n.	n.	NOUN
ejpam-5057	84	7	the	the	DET
ejpam-5057	84	8	second	second	ADJ
ejpam-5057	84	9	sequence	sequence	NOUN
ejpam-5057	84	10	is	be	AUX
ejpam-5057	84	11	said	say	VERB
ejpam-5057	84	12	to	to	PART
ejpam-5057	84	13	interlace	interlace	VERB
ejpam-5057	84	14	the	the	DET
ejpam-5057	84	15	first	first	ADJ
ejpam-5057	84	16	one	one	NUM
ejpam-5057	84	17	whenever	whenever	SCONJ
ejpam-5057	84	18	ξi	ξi	NOUN
ejpam-5057	84	19	≤	≤	NUM
ejpam-5057	84	20	ηi	ηi	VERB
ejpam-5057	84	21	≤	≤	NUM
ejpam-5057	84	22	ξn−m+i	ξn−m+i	PROPN
ejpam-5057	84	23	for	for	ADP
ejpam-5057	84	24	i	i	PRON
ejpam-5057	84	25	=	=	NOUN
ejpam-5057	84	26	1	1	NUM
ejpam-5057	84	27	,	,	PUNCT
ejpam-5057	84	28	2	2	NUM
ejpam-5057	84	29	,	,	PUNCT
ejpam-5057	84	30	.	.	PUNCT
ejpam-5057	84	31	.	.	PUNCT
ejpam-5057	84	32	.	.	PUNCT
ejpam-5057	85	1	,	,	PUNCT
ejpam-5057	85	2	m.	m.	NOUN
ejpam-5057	85	3	the	the	DET
ejpam-5057	85	4	interlacing	interlacing	NOUN
ejpam-5057	85	5	is	be	AUX
ejpam-5057	85	6	called	call	VERB
ejpam-5057	85	7	tight	tight	ADJ
ejpam-5057	85	8	if	if	SCONJ
ejpam-5057	85	9	m.machasri	m.machasri	NUM
ejpam-5057	85	10	,	,	PUNCT
ejpam-5057	85	11	d.kalyani	d.kalyani	NOUN
ejpam-5057	85	12	/	/	SYM
ejpam-5057	85	13	eur	eur	PROPN
ejpam-5057	85	14	.	.	PUNCT
ejpam-5057	86	1	j.	j.	PROPN
ejpam-5057	86	2	pure	pure	PROPN
ejpam-5057	86	3	appl	appl	PROPN
ejpam-5057	86	4	.	.	PROPN
ejpam-5057	86	5	math	math	PROPN
ejpam-5057	86	6	,	,	PUNCT
ejpam-5057	86	7	17	17	NUM
ejpam-5057	86	8	(	(	PUNCT
ejpam-5057	86	9	2	2	NUM
ejpam-5057	86	10	)	)	PUNCT
ejpam-5057	86	11	(	(	PUNCT
ejpam-5057	86	12	2024	2024	NUM
ejpam-5057	86	13	)	)	PUNCT
ejpam-5057	86	14	,	,	PUNCT
ejpam-5057	86	15	772	772	NUM
ejpam-5057	86	16	-	-	SYM
ejpam-5057	86	17	789	789	NUM
ejpam-5057	86	18	775	775	NUM
ejpam-5057	86	19	there	there	PRON
ejpam-5057	86	20	exists	exist	VERB
ejpam-5057	86	21	an	an	DET
ejpam-5057	86	22	integer	integer	NOUN
ejpam-5057	86	23	k	k	PROPN
ejpam-5057	86	24	∈	∈	PROPN
ejpam-5057	87	1	[	[	X
ejpam-5057	87	2	0,m	0,m	X
ejpam-5057	87	3	]	]	X
ejpam-5057	87	4	such	such	ADJ
ejpam-5057	87	5	that	that	DET
ejpam-5057	87	6	ξi	ξi	NOUN
ejpam-5057	87	7	=	=	NUM
ejpam-5057	87	8	ηi	ηi	PROPN
ejpam-5057	87	9	for	for	ADP
ejpam-5057	87	10	1	1	NUM
ejpam-5057	87	11	≤	≤	NUM
ejpam-5057	87	12	i	i	PRON
ejpam-5057	87	13	≤	≤	ADJ
ejpam-5057	87	14	k	k	PROPN
ejpam-5057	87	15	and	and	CCONJ
ejpam-5057	87	16	ξn−m+i	ξn−m+i	PROPN
ejpam-5057	87	17	=	=	SYM
ejpam-5057	87	18	ηi	ηi	PROPN
ejpam-5057	87	19	for	for	ADP
ejpam-5057	87	20	k	k	PROPN
ejpam-5057	88	1	+	+	PROPN
ejpam-5057	88	2	1	1	X
ejpam-5057	88	3	≤	≤	NUM
ejpam-5057	88	4	i	i	PRON
ejpam-5057	88	5	≤	≤	NUM
ejpam-5057	88	6	m.	m.	NOUN
ejpam-5057	88	7	definition	definition	NOUN
ejpam-5057	88	8	3	3	NUM
ejpam-5057	88	9	.	.	PUNCT
ejpam-5057	89	1	a	a	DET
ejpam-5057	89	2	graph	graph	NOUN
ejpam-5057	89	3	is	be	AUX
ejpam-5057	89	4	said	say	VERB
ejpam-5057	89	5	to	to	PART
ejpam-5057	89	6	be	be	AUX
ejpam-5057	89	7	minimally	minimally	ADV
ejpam-5057	89	8	connected	connect	VERB
ejpam-5057	89	9	if	if	SCONJ
ejpam-5057	89	10	removal	removal	NOUN
ejpam-5057	89	11	of	of	ADP
ejpam-5057	89	12	any	any	DET
ejpam-5057	89	13	one	one	NUM
ejpam-5057	89	14	edge	edge	NOUN
ejpam-5057	89	15	disconnects	disconnect	VERB
ejpam-5057	89	16	the	the	DET
ejpam-5057	89	17	graph	graph	NOUN
ejpam-5057	89	18	.	.	PUNCT
ejpam-5057	90	1	lemma	lemma	PROPN
ejpam-5057	90	2	1	1	NUM
ejpam-5057	90	3	.	.	PUNCT
ejpam-5057	91	1	(	(	PUNCT
ejpam-5057	91	2	[	[	X
ejpam-5057	91	3	6]-[10	6]-[10	NUM
ejpam-5057	91	4	]	]	PUNCT
ejpam-5057	91	5	)	)	PUNCT
ejpam-5057	91	6	let	let	VERB
ejpam-5057	91	7	aq	aq	PART
ejpam-5057	91	8	be	be	AUX
ejpam-5057	91	9	the	the	DET
ejpam-5057	91	10	quotient	quotient	NOUN
ejpam-5057	91	11	matrix	matrix	NOUN
ejpam-5057	91	12	of	of	ADP
ejpam-5057	91	13	a	a	DET
ejpam-5057	91	14	symmetric	symmetric	ADJ
ejpam-5057	91	15	matrix	matrix	NOUN
ejpam-5057	91	16	a	a	PRON
ejpam-5057	91	17	whose	whose	DET
ejpam-5057	91	18	rows	row	NOUN
ejpam-5057	91	19	and	and	CCONJ
ejpam-5057	91	20	columns	column	NOUN
ejpam-5057	91	21	are	be	AUX
ejpam-5057	91	22	partitioned	partition	VERB
ejpam-5057	91	23	according	accord	VERB
ejpam-5057	91	24	to	to	ADP
ejpam-5057	91	25	a	a	DET
ejpam-5057	91	26	partitioning	partition	VERB
ejpam-5057	91	27	(	(	PUNCT
ejpam-5057	91	28	x1	x1	PROPN
ejpam-5057	91	29	,	,	PUNCT
ejpam-5057	91	30	x2	x2	PROPN
ejpam-5057	91	31	,	,	PUNCT
ejpam-5057	91	32	.	.	PUNCT
ejpam-5057	91	33	.	.	PUNCT
ejpam-5057	92	1	.	.	PUNCT
ejpam-5057	93	1	,	,	PUNCT
ejpam-5057	93	2	xm	xm	PROPN
ejpam-5057	93	3	)	)	PUNCT
ejpam-5057	93	4	.	.	PUNCT
ejpam-5057	94	1	then	then	ADV
ejpam-5057	94	2	(	(	PUNCT
ejpam-5057	94	3	i	i	NOUN
ejpam-5057	94	4	)	)	PUNCT
ejpam-5057	94	5	the	the	DET
ejpam-5057	94	6	eigenvalues	eigenvalue	NOUN
ejpam-5057	94	7	of	of	ADP
ejpam-5057	94	8	aq	aq	NOUN
ejpam-5057	94	9	interlace	interlace	VERB
ejpam-5057	94	10	the	the	DET
ejpam-5057	94	11	eigenvalues	eigenvalue	NOUN
ejpam-5057	94	12	of	of	ADP
ejpam-5057	94	13	a.	a.	NOUN
ejpam-5057	94	14	(	(	PUNCT
ejpam-5057	94	15	ii	ii	PROPN
ejpam-5057	94	16	)	)	PUNCT
ejpam-5057	94	17	if	if	SCONJ
ejpam-5057	94	18	the	the	DET
ejpam-5057	94	19	interlacing	interlacing	NOUN
ejpam-5057	94	20	is	be	AUX
ejpam-5057	94	21	tight	tight	ADJ
ejpam-5057	94	22	then	then	ADV
ejpam-5057	94	23	the	the	DET
ejpam-5057	94	24	partition	partition	NOUN
ejpam-5057	94	25	is	be	AUX
ejpam-5057	94	26	equitable	equitable	ADJ
ejpam-5057	94	27	.	.	PUNCT
ejpam-5057	95	1	definition	definition	NOUN
ejpam-5057	95	2	4	4	NUM
ejpam-5057	95	3	.	.	PUNCT
ejpam-5057	96	1	[	[	X
ejpam-5057	96	2	8	8	NUM
ejpam-5057	96	3	]	]	PUNCT
ejpam-5057	96	4	(	(	PUNCT
ejpam-5057	96	5	vertex	vertex	NOUN
ejpam-5057	96	6	expander	expander	NOUN
ejpam-5057	96	7	)	)	PUNCT
ejpam-5057	96	8	a	a	DET
ejpam-5057	96	9	graph	graph	NOUN
ejpam-5057	96	10	g	g	NOUN
ejpam-5057	96	11	with	with	ADP
ejpam-5057	96	12	n	n	PRON
ejpam-5057	96	13	vertices	vertex	NOUN
ejpam-5057	96	14	is	be	AUX
ejpam-5057	96	15	said	say	VERB
ejpam-5057	96	16	to	to	PART
ejpam-5057	96	17	be	be	AUX
ejpam-5057	96	18	a	a	DET
ejpam-5057	96	19	vertex	vertex	NOUN
ejpam-5057	96	20	expander	expander	NOUN
ejpam-5057	96	21	if	if	SCONJ
ejpam-5057	96	22	|n(s)|	|n(s)|	PROPN
ejpam-5057	96	23	≥	≥	NOUN
ejpam-5057	96	24	a|s|	a|s|	VERB
ejpam-5057	96	25	for	for	SCONJ
ejpam-5057	96	26	all	all	PRON
ejpam-5057	96	27	s	s	PART
ejpam-5057	96	28	⊆	⊆	NUM
ejpam-5057	96	29	v	v	NOUN
ejpam-5057	96	30	:	:	PUNCT
ejpam-5057	96	31	|s|	|s|	NOUN
ejpam-5057	96	32	≤	≤	PROPN
ejpam-5057	96	33	n	n	PRON
ejpam-5057	96	34	2	2	NUM
ejpam-5057	96	35	,	,	PUNCT
ejpam-5057	96	36	where	where	SCONJ
ejpam-5057	96	37	a	a	PRON
ejpam-5057	96	38	is	be	AUX
ejpam-5057	96	39	a	a	DET
ejpam-5057	96	40	constant	constant	ADJ
ejpam-5057	96	41	and	and	CCONJ
ejpam-5057	96	42	n(s	n(s	NUM
ejpam-5057	96	43	)	)	PUNCT
ejpam-5057	96	44	is	be	AUX
ejpam-5057	96	45	the	the	DET
ejpam-5057	96	46	neighbourhood	neighbourhood	NOUN
ejpam-5057	96	47	of	of	ADP
ejpam-5057	96	48	s	s	NOUN
ejpam-5057	96	49	in	in	ADP
ejpam-5057	96	50	g	g	PROPN
ejpam-5057	96	51	not	not	PART
ejpam-5057	96	52	in	in	ADP
ejpam-5057	96	53	s.	s.	PROPN
ejpam-5057	96	54	definition	definition	NOUN
ejpam-5057	96	55	5	5	NUM
ejpam-5057	96	56	.	.	PUNCT
ejpam-5057	97	1	[	[	X
ejpam-5057	97	2	8	8	NUM
ejpam-5057	97	3	]	]	X
ejpam-5057	97	4	(	(	PUNCT
ejpam-5057	97	5	spectral	spectral	ADJ
ejpam-5057	97	6	expansion	expansion	NOUN
ejpam-5057	97	7	)	)	PUNCT
ejpam-5057	97	8	the	the	DET
ejpam-5057	97	9	spectral	spectral	ADJ
ejpam-5057	97	10	expansion	expansion	NOUN
ejpam-5057	97	11	of	of	ADP
ejpam-5057	97	12	graph	graph	NOUN
ejpam-5057	97	13	g	g	PROPN
ejpam-5057	97	14	is	be	AUX
ejpam-5057	97	15	defined	define	VERB
ejpam-5057	97	16	by	by	ADP
ejpam-5057	97	17	λ	λ	NOUN
ejpam-5057	97	18	=	=	SYM
ejpam-5057	97	19	max{|λ2|	max{|λ2|	PROPN
ejpam-5057	97	20	,	,	PUNCT
ejpam-5057	97	21	|λn|	|λn|	PROPN
ejpam-5057	97	22	}	}	PUNCT
ejpam-5057	97	23	.	.	PUNCT
ejpam-5057	98	1	definition	definition	NOUN
ejpam-5057	98	2	6	6	NUM
ejpam-5057	98	3	.	.	PUNCT
ejpam-5057	99	1	[	[	X
ejpam-5057	99	2	1	1	X
ejpam-5057	99	3	]	]	PUNCT
ejpam-5057	99	4	let	let	AUX
ejpam-5057	99	5	g	g	NOUN
ejpam-5057	99	6	=	=	SYM
ejpam-5057	99	7	(	(	PUNCT
ejpam-5057	99	8	v	v	NOUN
ejpam-5057	99	9	,	,	PUNCT
ejpam-5057	99	10	e	e	NOUN
ejpam-5057	99	11	)	)	PUNCT
ejpam-5057	99	12	be	be	AUX
ejpam-5057	99	13	a	a	DET
ejpam-5057	99	14	graph	graph	NOUN
ejpam-5057	99	15	.	.	PUNCT
ejpam-5057	100	1	for	for	ADP
ejpam-5057	100	2	a	a	DET
ejpam-5057	100	3	subset	subset	NOUN
ejpam-5057	100	4	s	s	NOUN
ejpam-5057	100	5	of	of	ADP
ejpam-5057	100	6	v	v	NOUN
ejpam-5057	100	7	let	let	VERB
ejpam-5057	100	8	n(s	n(s	PRON
ejpam-5057	100	9	)	)	PUNCT
ejpam-5057	100	10	=	=	PRON
ejpam-5057	100	11	{	{	PUNCT
ejpam-5057	100	12	v	v	NUM
ejpam-5057	100	13	∈	∈	NOUN
ejpam-5057	100	14	v	v	NOUN
ejpam-5057	100	15	:	:	PUNCT
ejpam-5057	100	16	vs	vs	PROPN
ejpam-5057	100	17	∈	∈	PROPN
ejpam-5057	100	18	e	e	NOUN
ejpam-5057	100	19	for	for	ADP
ejpam-5057	100	20	some	some	DET
ejpam-5057	100	21	s	s	PART
ejpam-5057	100	22	∈	∈	NOUN
ejpam-5057	100	23	s	s	PART
ejpam-5057	100	24	}	}	PUNCT
ejpam-5057	100	25	.	.	PUNCT
ejpam-5057	101	1	an	an	DET
ejpam-5057	101	2	(	(	PUNCT
ejpam-5057	101	3	n	n	X
ejpam-5057	101	4	,	,	PUNCT
ejpam-5057	101	5	d	d	X
ejpam-5057	101	6	,	,	PUNCT
ejpam-5057	101	7	c)−expander	c)−expander	NOUN
ejpam-5057	101	8	is	be	AUX
ejpam-5057	101	9	a	a	DET
ejpam-5057	101	10	bipartite	bipartite	ADJ
ejpam-5057	101	11	graph	graph	NOUN
ejpam-5057	101	12	on	on	ADP
ejpam-5057	101	13	the	the	DET
ejpam-5057	101	14	sets	set	NOUN
ejpam-5057	101	15	of	of	ADP
ejpam-5057	101	16	vertices	vertex	NOUN
ejpam-5057	101	17	x	x	PUNCT
ejpam-5057	101	18	and	and	CCONJ
ejpam-5057	101	19	y	y	PROPN
ejpam-5057	101	20	,	,	PUNCT
ejpam-5057	101	21	where	where	SCONJ
ejpam-5057	101	22	|x|	|x|	PROPN
ejpam-5057	101	23	=	=	NOUN
ejpam-5057	101	24	|y	|y	NOUN
ejpam-5057	101	25	|	|	NOUN
ejpam-5057	101	26	=	=	SYM
ejpam-5057	101	27	n	n	NOUN
ejpam-5057	101	28	,	,	PUNCT
ejpam-5057	101	29	the	the	DET
ejpam-5057	101	30	maximal	maximal	ADJ
ejpam-5057	101	31	degree	degree	NOUN
ejpam-5057	101	32	of	of	ADP
ejpam-5057	101	33	a	a	DET
ejpam-5057	101	34	vertex	vertex	NOUN
ejpam-5057	101	35	is	be	AUX
ejpam-5057	101	36	d	d	NOUN
ejpam-5057	101	37	,	,	PUNCT
ejpam-5057	101	38	and	and	CCONJ
ejpam-5057	101	39	for	for	ADP
ejpam-5057	101	40	every	every	DET
ejpam-5057	101	41	set	set	NOUN
ejpam-5057	101	42	s	s	NOUN
ejpam-5057	101	43	⊆	⊆	NUM
ejpam-5057	101	44	x	x	X
ejpam-5057	101	45	of	of	ADP
ejpam-5057	101	46	cardinality	cardinality	NOUN
ejpam-5057	101	47	|s|	|s|	PROPN
ejpam-5057	101	48	=	=	SYM
ejpam-5057	101	49	α	α	NOUN
ejpam-5057	101	50	≤	≤	NOUN
ejpam-5057	102	1	n	n	PRON
ejpam-5057	102	2	2	2	NUM
ejpam-5057	102	3	,	,	PUNCT
ejpam-5057	102	4	|n(s)|	|n(s)|	PROPN
ejpam-5057	102	5	≥	≥	X
ejpam-5057	102	6	(	(	PUNCT
ejpam-5057	102	7	1	1	NUM
ejpam-5057	102	8	+	+	NUM
ejpam-5057	102	9	c(1−	c(1−	NOUN
ejpam-5057	102	10	α	α	NOUN
ejpam-5057	102	11	n	n	NOUN
ejpam-5057	102	12	)	)	PUNCT
ejpam-5057	102	13	)	)	PUNCT
ejpam-5057	103	1	α	α	X
ejpam-5057	103	2	.	.	PUNCT
ejpam-5057	103	3	definition	definition	NOUN
ejpam-5057	103	4	7	7	NUM
ejpam-5057	103	5	.	.	PUNCT
ejpam-5057	104	1	[	[	X
ejpam-5057	104	2	4	4	X
ejpam-5057	104	3	]	]	PUNCT
ejpam-5057	104	4	a	a	DET
ejpam-5057	104	5	d−left	d−left	ADV
ejpam-5057	104	6	regular	regular	ADJ
ejpam-5057	104	7	bipartite	bipartite	PROPN
ejpam-5057	104	8	graph	graph	NOUN
ejpam-5057	104	9	g	g	PROPN
ejpam-5057	104	10	=	=	PUNCT
ejpam-5057	104	11	(	(	PUNCT
ejpam-5057	104	12	x	x	SYM
ejpam-5057	104	13	∪	∪	PROPN
ejpam-5057	104	14	y	y	PROPN
ejpam-5057	104	15	,	,	PUNCT
ejpam-5057	104	16	e	e	NOUN
ejpam-5057	104	17	)	)	PUNCT
ejpam-5057	104	18	where	where	SCONJ
ejpam-5057	104	19	|x|	|x|	PROPN
ejpam-5057	104	20	=	=	SYM
ejpam-5057	104	21	n	n	PROPN
ejpam-5057	104	22	and	and	CCONJ
ejpam-5057	104	23	|y	|y	NOUN
ejpam-5057	104	24	|	|	ADV
ejpam-5057	104	25	=	=	PUNCT
ejpam-5057	104	26	m	m	VERB
ejpam-5057	104	27	such	such	ADJ
ejpam-5057	104	28	that	that	SCONJ
ejpam-5057	104	29	for	for	ADP
ejpam-5057	104	30	all	all	PRON
ejpam-5057	104	31	s	s	PART
ejpam-5057	104	32	⊆	⊆	NUM
ejpam-5057	104	33	x	x	X
ejpam-5057	104	34	with	with	ADP
ejpam-5057	104	35	|s|	|s|	NOUN
ejpam-5057	104	36	≤	≤	NUM
ejpam-5057	104	37	γn	γn	NUM
ejpam-5057	104	38	,	,	PUNCT
ejpam-5057	104	39	|n(s)|	|n(s)|	PROPN
ejpam-5057	104	40	≥	≥	NOUN
ejpam-5057	104	41	α|s|	α|s|	NOUN
ejpam-5057	104	42	,	,	PUNCT
ejpam-5057	104	43	where	where	SCONJ
ejpam-5057	104	44	γ	γ	PROPN
ejpam-5057	104	45	and	and	CCONJ
ejpam-5057	104	46	α	α	PROPN
ejpam-5057	104	47	are	be	AUX
ejpam-5057	104	48	positive	positive	ADJ
ejpam-5057	104	49	constants	constant	NOUN
ejpam-5057	104	50	is	be	AUX
ejpam-5057	104	51	known	know	VERB
ejpam-5057	104	52	as	as	ADP
ejpam-5057	104	53	an	an	DET
ejpam-5057	104	54	(	(	PUNCT
ejpam-5057	104	55	n	n	X
ejpam-5057	104	56	,	,	PUNCT
ejpam-5057	104	57	m	m	PROPN
ejpam-5057	104	58	,	,	PUNCT
ejpam-5057	104	59	d	d	PROPN
ejpam-5057	104	60	,	,	PUNCT
ejpam-5057	104	61	γ	γ	X
ejpam-5057	104	62	,	,	PUNCT
ejpam-5057	104	63	d(1−	d(1−	PROPN
ejpam-5057	104	64	ϵ	ϵ	NUM
ejpam-5057	104	65	)	)	PUNCT
ejpam-5057	104	66	)	)	PUNCT
ejpam-5057	104	67	expander	expander	NOUN
ejpam-5057	104	68	.	.	PUNCT
ejpam-5057	105	1	note	note	NOUN
ejpam-5057	105	2	:	:	PUNCT
ejpam-5057	105	3	let	let	VERB
ejpam-5057	105	4	λ1	λ1	PROPN
ejpam-5057	105	5	≥	≥	NOUN
ejpam-5057	105	6	λ2	λ2	NOUN
ejpam-5057	105	7	≥	≥	NOUN
ejpam-5057	105	8	·	·	PUNCT
ejpam-5057	105	9	·	·	PUNCT
ejpam-5057	105	10	·	·	PUNCT
ejpam-5057	106	1	≥	≥	PRON
ejpam-5057	106	2	λn	λn	AUX
ejpam-5057	106	3	be	be	AUX
ejpam-5057	106	4	the	the	DET
ejpam-5057	106	5	adjacency	adjacency	NOUN
ejpam-5057	106	6	eigenvalues	eigenvalue	VERB
ejpam-5057	106	7	of	of	ADP
ejpam-5057	106	8	g	g	NOUN
ejpam-5057	106	9	and	and	CCONJ
ejpam-5057	106	10	let	let	VERB
ejpam-5057	106	11	λ′	λ′	PROPN
ejpam-5057	106	12	1	1	NUM
ejpam-5057	106	13	≥	≥	NOUN
ejpam-5057	106	14	λ′	λ′	X
ejpam-5057	106	15	2	2	NUM
ejpam-5057	106	16	≥	≥	NOUN
ejpam-5057	106	17	·	·	PUNCT
ejpam-5057	106	18	·	·	PUNCT
ejpam-5057	106	19	·	·	PUNCT
ejpam-5057	106	20	≥	≥	NUM
ejpam-5057	106	21	λ′	λ′	X
ejpam-5057	106	22	n	n	CCONJ
ejpam-5057	106	23	be	be	AUX
ejpam-5057	106	24	the	the	DET
ejpam-5057	106	25	adjacency	adjacency	NOUN
ejpam-5057	106	26	eigenvalues	eigenvalue	NOUN
ejpam-5057	106	27	of	of	ADP
ejpam-5057	106	28	g′.	g′.	NOUN
ejpam-5057	106	29	assume	assume	VERB
ejpam-5057	106	30	that	that	SCONJ
ejpam-5057	106	31	γi	γi	NOUN
ejpam-5057	106	32	=	=	SYM
ejpam-5057	106	33	λ′	λ′	X
ejpam-5057	107	1	i	i	PRON
ejpam-5057	107	2	d	d	AUX
ejpam-5057	107	3	be	be	VERB
ejpam-5057	107	4	the	the	DET
ejpam-5057	107	5	normalized	normalize	VERB
ejpam-5057	107	6	eigenvalues	eigenvalue	NOUN
ejpam-5057	107	7	of	of	ADP
ejpam-5057	107	8	g′.	g′.	NOUN
ejpam-5057	107	9	let	let	VERB
ejpam-5057	107	10	λ′	λ′	PRON
ejpam-5057	107	11	be	be	AUX
ejpam-5057	107	12	the	the	DET
ejpam-5057	107	13	spectral	spectral	ADJ
ejpam-5057	107	14	expansion	expansion	NOUN
ejpam-5057	107	15	of	of	ADP
ejpam-5057	107	16	g′.	g′.	PROPN
ejpam-5057	107	17	then	then	ADV
ejpam-5057	107	18	γ	γ	X
ejpam-5057	107	19	=	=	SYM
ejpam-5057	107	20	λ′	λ′	X
ejpam-5057	108	1	d	d	X
ejpam-5057	108	2	is	be	AUX
ejpam-5057	108	3	the	the	DET
ejpam-5057	108	4	spectral	spectral	ADJ
ejpam-5057	108	5	expansion	expansion	NOUN
ejpam-5057	108	6	of	of	ADP
ejpam-5057	108	7	g′	g′	NOUN
ejpam-5057	108	8	with	with	ADP
ejpam-5057	108	9	respect	respect	NOUN
ejpam-5057	108	10	to	to	ADP
ejpam-5057	108	11	the	the	DET
ejpam-5057	108	12	normalized	normalize	VERB
ejpam-5057	108	13	adjacency	adjacency	NOUN
ejpam-5057	108	14	eigenvalues	eigenvalue	VERB
ejpam-5057	108	15	.	.	PUNCT
ejpam-5057	109	1	lemma	lemma	PROPN
ejpam-5057	109	2	2	2	NUM
ejpam-5057	109	3	.	.	PUNCT
ejpam-5057	110	1	[	[	X
ejpam-5057	110	2	8	8	NUM
ejpam-5057	110	3	]	]	PUNCT
ejpam-5057	110	4	(	(	PUNCT
ejpam-5057	110	5	vertex	vertex	NOUN
ejpam-5057	110	6	expansion	expansion	NOUN
ejpam-5057	110	7	to	to	ADP
ejpam-5057	110	8	spectral	spectral	ADJ
ejpam-5057	110	9	expansion	expansion	NOUN
ejpam-5057	110	10	)	)	PUNCT
ejpam-5057	110	11	.	.	PUNCT
ejpam-5057	111	1	let	let	VERB
ejpam-5057	111	2	g	g	PRON
ejpam-5057	111	3	be	be	AUX
ejpam-5057	111	4	a	a	DET
ejpam-5057	111	5	d−regular	d−regular	NUM
ejpam-5057	111	6	graph	graph	NOUN
ejpam-5057	111	7	.	.	PUNCT
ejpam-5057	112	1	for	for	ADP
ejpam-5057	112	2	every	every	DET
ejpam-5057	112	3	ϵ	ϵ	PROPN
ejpam-5057	112	4	>	>	X
ejpam-5057	112	5	0	0	PUNCT
ejpam-5057	113	1	and	and	CCONJ
ejpam-5057	113	2	d	d	X
ejpam-5057	113	3	>	>	X
ejpam-5057	113	4	0	0	PROPN
ejpam-5057	113	5	,	,	PUNCT
ejpam-5057	113	6	there	there	PRON
ejpam-5057	113	7	exists	exist	VERB
ejpam-5057	113	8	γ	γ	PROPN
ejpam-5057	113	9	>	>	X
ejpam-5057	113	10	0	0	NUM
ejpam-5057	113	11	such	such	ADJ
ejpam-5057	113	12	that	that	SCONJ
ejpam-5057	113	13	if	if	SCONJ
ejpam-5057	113	14	g	g	PROPN
ejpam-5057	113	15	is	be	AUX
ejpam-5057	113	16	a	a	DET
ejpam-5057	113	17	d−regular	d−regular	NUM
ejpam-5057	113	18	(	(	PUNCT
ejpam-5057	113	19	1	1	NUM
ejpam-5057	114	1	+	+	X
ejpam-5057	114	2	ϵ)−expander	ϵ)−expander	NOUN
ejpam-5057	114	3	then	then	ADV
ejpam-5057	114	4	g	g	PROPN
ejpam-5057	114	5	has	have	VERB
ejpam-5057	114	6	spectral	spectral	ADJ
ejpam-5057	114	7	expansion	expansion	NOUN
ejpam-5057	114	8	(	(	PUNCT
ejpam-5057	114	9	1−	1−	NUM
ejpam-5057	114	10	γ	γ	NOUN
ejpam-5057	114	11	)	)	PUNCT
ejpam-5057	114	12	.	.	PUNCT
ejpam-5057	115	1	specifically	specifically	ADV
ejpam-5057	115	2	,	,	PUNCT
ejpam-5057	115	3	we	we	PRON
ejpam-5057	115	4	can	can	AUX
ejpam-5057	115	5	take	take	VERB
ejpam-5057	115	6	γ	γ	X
ejpam-5057	115	7	=	=	PUNCT
ejpam-5057	115	8	ω(ϵ2	ω(ϵ2	X
ejpam-5057	115	9	/	/	SYM
ejpam-5057	115	10	d	d	NOUN
ejpam-5057	115	11	)	)	PUNCT
ejpam-5057	115	12	.	.	PUNCT
ejpam-5057	116	1	lemma	lemma	PROPN
ejpam-5057	116	2	3	3	X
ejpam-5057	116	3	.	.	PUNCT
ejpam-5057	117	1	[	[	X
ejpam-5057	117	2	5	5	NUM
ejpam-5057	117	3	]	]	PUNCT
ejpam-5057	117	4	let	let	VERB
ejpam-5057	117	5	g	g	PRON
ejpam-5057	117	6	be	be	AUX
ejpam-5057	117	7	an	an	DET
ejpam-5057	117	8	(	(	PUNCT
ejpam-5057	117	9	n	n	X
ejpam-5057	117	10	,	,	PUNCT
ejpam-5057	117	11	m	m	PROPN
ejpam-5057	117	12	,	,	PUNCT
ejpam-5057	117	13	d	d	PROPN
ejpam-5057	117	14	,	,	PUNCT
ejpam-5057	117	15	γ	γ	X
ejpam-5057	117	16	,	,	PUNCT
ejpam-5057	117	17	d(1	d(1	VERB
ejpam-5057	117	18	−	−	PROPN
ejpam-5057	117	19	ϵ	ϵ	NOUN
ejpam-5057	117	20	)	)	PUNCT
ejpam-5057	117	21	)	)	PUNCT
ejpam-5057	117	22	expander	expander	NOUN
ejpam-5057	117	23	.	.	PUNCT
ejpam-5057	118	1	then	then	ADV
ejpam-5057	118	2	the	the	DET
ejpam-5057	118	3	distance	distance	NOUN
ejpam-5057	118	4	of	of	ADP
ejpam-5057	118	5	the	the	DET
ejpam-5057	118	6	code	code	NOUN
ejpam-5057	118	7	corresponding	correspond	VERB
ejpam-5057	118	8	to	to	ADP
ejpam-5057	118	9	the	the	DET
ejpam-5057	118	10	graph	graph	NOUN
ejpam-5057	118	11	g	g	NOUN
ejpam-5057	118	12	is	be	AUX
ejpam-5057	118	13	∆(c(g	∆(c(g	NOUN
ejpam-5057	118	14	)	)	PUNCT
ejpam-5057	118	15	)	)	PUNCT
ejpam-5057	118	16	≥	≥	NUM
ejpam-5057	118	17	2γ(1−	2γ(1−	NUM
ejpam-5057	118	18	ϵ)n	ϵ)n	PROPN
ejpam-5057	118	19	where	where	SCONJ
ejpam-5057	118	20	c(g	c(g	PROPN
ejpam-5057	118	21	)	)	PUNCT
ejpam-5057	118	22	denotes	denote	VERB
ejpam-5057	118	23	the	the	DET
ejpam-5057	118	24	code	code	NOUN
ejpam-5057	118	25	corresponding	correspond	VERB
ejpam-5057	118	26	to	to	ADP
ejpam-5057	118	27	an	an	DET
ejpam-5057	118	28	expander	expander	NOUN
ejpam-5057	118	29	graph	graph	NOUN
ejpam-5057	118	30	g.	g.	PROPN
ejpam-5057	118	31	m.machasri	m.machasri	PROPN
ejpam-5057	118	32	,	,	PUNCT
ejpam-5057	118	33	d.kalyani	d.kalyani	NOUN
ejpam-5057	118	34	/	/	SYM
ejpam-5057	118	35	eur	eur	PROPN
ejpam-5057	118	36	.	.	PUNCT
ejpam-5057	119	1	j.	j.	PROPN
ejpam-5057	119	2	pure	pure	PROPN
ejpam-5057	119	3	appl	appl	PROPN
ejpam-5057	119	4	.	.	PROPN
ejpam-5057	119	5	math	math	PROPN
ejpam-5057	119	6	,	,	PUNCT
ejpam-5057	119	7	17	17	NUM
ejpam-5057	119	8	(	(	PUNCT
ejpam-5057	119	9	2	2	NUM
ejpam-5057	119	10	)	)	PUNCT
ejpam-5057	119	11	(	(	PUNCT
ejpam-5057	119	12	2024	2024	NUM
ejpam-5057	119	13	)	)	PUNCT
ejpam-5057	119	14	,	,	PUNCT
ejpam-5057	119	15	772	772	NUM
ejpam-5057	119	16	-	-	SYM
ejpam-5057	119	17	789	789	NUM
ejpam-5057	119	18	776	776	NUM
ejpam-5057	119	19	4	4	NUM
ejpam-5057	119	20	.	.	PUNCT
ejpam-5057	120	1	eigenvalue	eigenvalue	ADJ
ejpam-5057	120	2	interlacing	interlacing	NOUN
ejpam-5057	120	3	of	of	ADP
ejpam-5057	120	4	bipartite	bipartite	PROPN
ejpam-5057	120	5	quotient	quotient	NOUN
ejpam-5057	120	6	matrix	matrix	NOUN
ejpam-5057	120	7	and	and	CCONJ
ejpam-5057	120	8	spectral	spectral	ADJ
ejpam-5057	120	9	bounds	bound	NOUN
ejpam-5057	120	10	in	in	ADP
ejpam-5057	120	11	this	this	DET
ejpam-5057	120	12	section	section	NOUN
ejpam-5057	120	13	we	we	PRON
ejpam-5057	120	14	derive	derive	VERB
ejpam-5057	120	15	the	the	DET
ejpam-5057	120	16	upper	upper	ADJ
ejpam-5057	120	17	bounds	bound	NOUN
ejpam-5057	120	18	for	for	ADP
ejpam-5057	120	19	the	the	DET
ejpam-5057	120	20	second	second	ADJ
ejpam-5057	120	21	largest	large	ADJ
ejpam-5057	120	22	eigenvalues	eigenvalue	NOUN
ejpam-5057	120	23	of	of	ADP
ejpam-5057	120	24	both	both	CCONJ
ejpam-5057	120	25	the	the	DET
ejpam-5057	120	26	adjacency	adjacency	NOUN
ejpam-5057	120	27	matrix	matrix	NOUN
ejpam-5057	120	28	and	and	CCONJ
ejpam-5057	120	29	laplacian	laplacian	ADJ
ejpam-5057	120	30	matrix	matrix	NOUN
ejpam-5057	120	31	of	of	ADP
ejpam-5057	120	32	a	a	DET
ejpam-5057	120	33	connected	connected	ADJ
ejpam-5057	120	34	bipartite	bipartite	NOUN
ejpam-5057	120	35	graph	graph	NOUN
ejpam-5057	120	36	.	.	PUNCT
ejpam-5057	121	1	also	also	ADV
ejpam-5057	121	2	,	,	PUNCT
ejpam-5057	121	3	we	we	PRON
ejpam-5057	121	4	derive	derive	VERB
ejpam-5057	121	5	the	the	DET
ejpam-5057	121	6	lower	low	ADJ
ejpam-5057	121	7	bounds	bound	NOUN
ejpam-5057	121	8	for	for	ADP
ejpam-5057	121	9	the	the	DET
ejpam-5057	121	10	second	second	ADJ
ejpam-5057	121	11	smallest	small	ADJ
ejpam-5057	121	12	eigenvalue	eigenvalue	NOUN
ejpam-5057	121	13	of	of	ADP
ejpam-5057	121	14	the	the	DET
ejpam-5057	121	15	adjacency	adjacency	NOUN
ejpam-5057	121	16	matrix	matrix	NOUN
ejpam-5057	121	17	of	of	ADP
ejpam-5057	121	18	a	a	DET
ejpam-5057	121	19	connected	connected	ADJ
ejpam-5057	121	20	bipartite	bipartite	NOUN
ejpam-5057	121	21	graph	graph	NOUN
ejpam-5057	121	22	.	.	PUNCT
ejpam-5057	122	1	to	to	PART
ejpam-5057	122	2	arrive	arrive	VERB
ejpam-5057	122	3	at	at	ADP
ejpam-5057	122	4	these	these	DET
ejpam-5057	122	5	bounds	bound	NOUN
ejpam-5057	122	6	,	,	PUNCT
ejpam-5057	122	7	we	we	PRON
ejpam-5057	122	8	define	define	VERB
ejpam-5057	122	9	the	the	DET
ejpam-5057	122	10	bipartite	bipartite	PROPN
ejpam-5057	122	11	quotient	quotient	NOUN
ejpam-5057	122	12	matrix	matrix	NOUN
ejpam-5057	122	13	as	as	SCONJ
ejpam-5057	122	14	follows	follow	VERB
ejpam-5057	122	15	:	:	PUNCT
ejpam-5057	122	16	definition	definition	NOUN
ejpam-5057	122	17	8	8	NUM
ejpam-5057	122	18	.	.	PUNCT
ejpam-5057	123	1	(	(	PUNCT
ejpam-5057	123	2	bipartite	bipartite	VERB
ejpam-5057	123	3	quotient	quotient	NOUN
ejpam-5057	123	4	matrix	matrix	NOUN
ejpam-5057	123	5	)	)	PUNCT
ejpam-5057	123	6	let	let	VERB
ejpam-5057	123	7	g	g	PRON
ejpam-5057	123	8	be	be	AUX
ejpam-5057	123	9	a	a	DET
ejpam-5057	123	10	bipartite	bipartite	ADJ
ejpam-5057	123	11	graph	graph	NOUN
ejpam-5057	123	12	.	.	PUNCT
ejpam-5057	124	1	the	the	DET
ejpam-5057	124	2	bipartite	bipartite	PROPN
ejpam-5057	124	3	quotient	quotient	NOUN
ejpam-5057	124	4	matrix	matrix	NOUN
ejpam-5057	124	5	of	of	ADP
ejpam-5057	124	6	the	the	DET
ejpam-5057	124	7	adjacency	adjacency	NOUN
ejpam-5057	124	8	matrix	matrix	NOUN
ejpam-5057	124	9	a	a	DET
ejpam-5057	124	10	(	(	PUNCT
ejpam-5057	124	11	laplacian	laplacian	ADJ
ejpam-5057	124	12	matrix	matrix	NOUN
ejpam-5057	124	13	l	l	NOUN
ejpam-5057	124	14	)	)	PUNCT
ejpam-5057	124	15	of	of	ADP
ejpam-5057	124	16	g	g	PROPN
ejpam-5057	124	17	is	be	AUX
ejpam-5057	124	18	the	the	DET
ejpam-5057	124	19	quotient	quotient	NOUN
ejpam-5057	124	20	matrix	matrix	NOUN
ejpam-5057	124	21	of	of	ADP
ejpam-5057	124	22	a	a	DET
ejpam-5057	124	23	/	/	SYM
ejpam-5057	124	24	l	l	NOUN
ejpam-5057	124	25	whose	whose	DET
ejpam-5057	124	26	rows	row	NOUN
ejpam-5057	124	27	and	and	CCONJ
ejpam-5057	124	28	columns	column	NOUN
ejpam-5057	124	29	are	be	AUX
ejpam-5057	124	30	partitioned	partition	VERB
ejpam-5057	124	31	according	accord	VERB
ejpam-5057	124	32	to	to	ADP
ejpam-5057	124	33	the	the	DET
ejpam-5057	124	34	bipartition	bipartition	NOUN
ejpam-5057	124	35	of	of	ADP
ejpam-5057	124	36	g.	g.	PROPN
ejpam-5057	124	37	4.1	4.1	NUM
ejpam-5057	124	38	.	.	PUNCT
ejpam-5057	125	1	bounds	bound	NOUN
ejpam-5057	125	2	for	for	ADP
ejpam-5057	125	3	the	the	DET
ejpam-5057	125	4	second	second	ADV
ejpam-5057	125	5	largest	large	ADJ
ejpam-5057	125	6	eigenvalue	eigenvalue	NOUN
ejpam-5057	125	7	and	and	CCONJ
ejpam-5057	125	8	second	second	ADJ
ejpam-5057	125	9	smallest	small	ADJ
ejpam-5057	125	10	eigenvalue	eigenvalue	NOUN
ejpam-5057	125	11	of	of	ADP
ejpam-5057	125	12	adjacency	adjacency	NOUN
ejpam-5057	125	13	matrix	matrix	NOUN
ejpam-5057	125	14	in	in	ADP
ejpam-5057	125	15	this	this	DET
ejpam-5057	125	16	section	section	NOUN
ejpam-5057	125	17	we	we	PRON
ejpam-5057	125	18	derive	derive	VERB
ejpam-5057	125	19	the	the	DET
ejpam-5057	125	20	upper	upper	ADJ
ejpam-5057	125	21	bounds	bound	NOUN
ejpam-5057	125	22	for	for	ADP
ejpam-5057	125	23	the	the	DET
ejpam-5057	125	24	second	second	ADV
ejpam-5057	125	25	largest	large	ADJ
ejpam-5057	125	26	eigenvalue	eigenvalue	NOUN
ejpam-5057	125	27	and	and	CCONJ
ejpam-5057	125	28	lower	low	ADJ
ejpam-5057	125	29	bounds	bound	NOUN
ejpam-5057	125	30	for	for	ADP
ejpam-5057	125	31	the	the	DET
ejpam-5057	125	32	second	second	ADJ
ejpam-5057	125	33	smallest	small	ADJ
ejpam-5057	125	34	eigenvalue	eigenvalue	NOUN
ejpam-5057	125	35	of	of	ADP
ejpam-5057	125	36	the	the	DET
ejpam-5057	125	37	adjacency	adjacency	NOUN
ejpam-5057	125	38	matrix	matrix	NOUN
ejpam-5057	125	39	of	of	ADP
ejpam-5057	125	40	a	a	DET
ejpam-5057	125	41	connected	connected	ADJ
ejpam-5057	125	42	bipartite	bipartite	NOUN
ejpam-5057	125	43	graph	graph	NOUN
ejpam-5057	125	44	g.	g.	PROPN
ejpam-5057	125	45	theorem	theorem	NOUN
ejpam-5057	125	46	1	1	X
ejpam-5057	125	47	.	.	X
ejpam-5057	126	1	consider	consider	VERB
ejpam-5057	126	2	a	a	DET
ejpam-5057	126	3	bipartite	bipartite	NOUN
ejpam-5057	126	4	graph	graph	NOUN
ejpam-5057	126	5	g	g	NOUN
ejpam-5057	126	6	with	with	ADP
ejpam-5057	126	7	bipartition	bipartition	NOUN
ejpam-5057	126	8	v	v	NOUN
ejpam-5057	126	9	=	=	SYM
ejpam-5057	126	10	(	(	PUNCT
ejpam-5057	126	11	x	x	X
ejpam-5057	126	12	,	,	PUNCT
ejpam-5057	126	13	y	y	PROPN
ejpam-5057	126	14	)	)	PUNCT
ejpam-5057	126	15	where	where	SCONJ
ejpam-5057	126	16	|x|	|x|	PROPN
ejpam-5057	126	17	=	=	SYM
ejpam-5057	126	18	n1	n1	PROPN
ejpam-5057	126	19	,	,	PUNCT
ejpam-5057	126	20	|y	|y	NOUN
ejpam-5057	126	21	|	|	NOUN
ejpam-5057	126	22	=	=	SYM
ejpam-5057	126	23	n2	n2	NOUN
ejpam-5057	126	24	and	and	CCONJ
ejpam-5057	126	25	|v	|v	ADJ
ejpam-5057	126	26	|	|	NOUN
ejpam-5057	126	27	=	=	SYM
ejpam-5057	126	28	n1	n1	PROPN
ejpam-5057	126	29	+	+	CCONJ
ejpam-5057	126	30	n2	n2	NOUN
ejpam-5057	126	31	=	=	SYM
ejpam-5057	126	32	n.	n.	NOUN
ejpam-5057	126	33	let	let	VERB
ejpam-5057	126	34	λ1	λ1	PROPN
ejpam-5057	126	35	≥	≥	NOUN
ejpam-5057	126	36	λ2	λ2	NOUN
ejpam-5057	126	37	≥	≥	NOUN
ejpam-5057	126	38	·	·	PUNCT
ejpam-5057	126	39	·	·	PUNCT
ejpam-5057	127	1	·	·	PUNCT
ejpam-5057	127	2	≥	≥	PRON
ejpam-5057	127	3	λn−1	λn−1	X
ejpam-5057	127	4	≥	≥	PUNCT
ejpam-5057	127	5	λn	λn	AUX
ejpam-5057	127	6	be	be	AUX
ejpam-5057	127	7	the	the	DET
ejpam-5057	127	8	eigenvalues	eigenvalue	NOUN
ejpam-5057	127	9	of	of	ADP
ejpam-5057	127	10	the	the	DET
ejpam-5057	127	11	adjacency	adjacency	NOUN
ejpam-5057	127	12	matrix	matrix	NOUN
ejpam-5057	127	13	a	a	PRON
ejpam-5057	127	14	of	of	ADP
ejpam-5057	127	15	g	g	NOUN
ejpam-5057	127	16	,	,	PUNCT
ejpam-5057	127	17	aq	aq	X
ejpam-5057	127	18	be	be	AUX
ejpam-5057	127	19	the	the	DET
ejpam-5057	127	20	bipartite	bipartite	PROPN
ejpam-5057	127	21	quotient	quotient	NOUN
ejpam-5057	127	22	matrix	matrix	NOUN
ejpam-5057	127	23	of	of	ADP
ejpam-5057	127	24	a	a	PRON
ejpam-5057	127	25	and	and	CCONJ
ejpam-5057	127	26	η1	η1	NOUN
ejpam-5057	127	27	and	and	CCONJ
ejpam-5057	127	28	η2	η2	VERB
ejpam-5057	127	29	the	the	DET
ejpam-5057	127	30	eigenvalues	eigenvalue	NOUN
ejpam-5057	127	31	of	of	ADP
ejpam-5057	127	32	aq	aq	NOUN
ejpam-5057	127	33	.	.	PUNCT
ejpam-5057	128	1	then	then	ADV
ejpam-5057	128	2	(	(	PUNCT
ejpam-5057	128	3	i	i	NOUN
ejpam-5057	128	4	)	)	PUNCT
ejpam-5057	128	5	λ1	λ1	PROPN
ejpam-5057	128	6	≥	≥	NOUN
ejpam-5057	128	7	η1	η1	NOUN
ejpam-5057	128	8	≥	≥	NOUN
ejpam-5057	128	9	λ2	λ2	PROPN
ejpam-5057	128	10	(	(	PUNCT
ejpam-5057	128	11	ii	ii	NOUN
ejpam-5057	128	12	)	)	PUNCT
ejpam-5057	128	13	λn−1	λn−1	PROPN
ejpam-5057	128	14	≥	≥	NUM
ejpam-5057	128	15	η2	η2	VERB
ejpam-5057	128	16	≥	≥	PROPN
ejpam-5057	128	17	λn	λn	PROPN
ejpam-5057	128	18	(	(	PUNCT
ejpam-5057	128	19	iii	iii	NOUN
ejpam-5057	128	20	)	)	PUNCT
ejpam-5057	128	21	λ2	λ2	NOUN
ejpam-5057	128	22	≤	≤	NUM
ejpam-5057	128	23	m√	m√	PROPN
ejpam-5057	128	24	n1n2	n1n2	NUM
ejpam-5057	128	25	(	(	PUNCT
ejpam-5057	128	26	iv	iv	X
ejpam-5057	128	27	)	)	PUNCT
ejpam-5057	128	28	λn−1	λn−1	PROPN
ejpam-5057	128	29	≥	≥	NUM
ejpam-5057	128	30	−m√	−m√	NOUN
ejpam-5057	128	31	n1n2	n1n2	ADV
ejpam-5057	128	32	proof	proof	ADJ
ejpam-5057	128	33	.	.	PUNCT
ejpam-5057	129	1	let	let	VERB
ejpam-5057	129	2	a	a	PRON
ejpam-5057	129	3	be	be	AUX
ejpam-5057	129	4	the	the	DET
ejpam-5057	129	5	adjacency	adjacency	NOUN
ejpam-5057	129	6	matrix	matrix	NOUN
ejpam-5057	129	7	of	of	ADP
ejpam-5057	129	8	g	g	PROPN
ejpam-5057	129	9	represented	represent	VERB
ejpam-5057	129	10	in	in	ADP
ejpam-5057	129	11	the	the	DET
ejpam-5057	129	12	following	follow	VERB
ejpam-5057	129	13	block	block	NOUN
ejpam-5057	129	14	matrix	matrix	NOUN
ejpam-5057	129	15	form	form	NOUN
ejpam-5057	129	16	with	with	ADP
ejpam-5057	129	17	respect	respect	NOUN
ejpam-5057	129	18	to	to	ADP
ejpam-5057	129	19	the	the	DET
ejpam-5057	129	20	bipartition	bipartition	NOUN
ejpam-5057	129	21	v	v	ADP
ejpam-5057	129	22	=	=	SYM
ejpam-5057	129	23	(	(	PUNCT
ejpam-5057	129	24	x	x	X
ejpam-5057	129	25	,	,	PUNCT
ejpam-5057	129	26	y	y	PROPN
ejpam-5057	129	27	)	)	PUNCT
ejpam-5057	129	28	.	.	PUNCT
ejpam-5057	130	1	a	a	DET
ejpam-5057	130	2	=	=	X
ejpam-5057	130	3	[	[	PUNCT
ejpam-5057	130	4	a11	a11	PROPN
ejpam-5057	130	5	a12	a12	PROPN
ejpam-5057	130	6	a21	a21	PROPN
ejpam-5057	130	7	a22	a22	PROPN
ejpam-5057	130	8	]	]	PUNCT
ejpam-5057	130	9	.	.	PUNCT
ejpam-5057	131	1	let	let	VERB
ejpam-5057	131	2	aq	aq	PART
ejpam-5057	131	3	be	be	AUX
ejpam-5057	131	4	the	the	DET
ejpam-5057	131	5	bipartite	bipartite	PROPN
ejpam-5057	131	6	quotient	quotient	NOUN
ejpam-5057	131	7	matrix	matrix	NOUN
ejpam-5057	131	8	of	of	ADP
ejpam-5057	131	9	a.	a.	NOUN
ejpam-5057	131	10	then	then	ADV
ejpam-5057	131	11	aq	aq	VERB
ejpam-5057	132	1	=	=	PUNCT
ejpam-5057	133	1	[	[	PUNCT
ejpam-5057	133	2	0	0	NUM
ejpam-5057	133	3	m	m	NOUN
ejpam-5057	133	4	n1	n1	PROPN
ejpam-5057	133	5	m	m	NOUN
ejpam-5057	133	6	n2	n2	NOUN
ejpam-5057	133	7	0	0	NUM
ejpam-5057	133	8	]	]	PUNCT
ejpam-5057	133	9	.	.	PUNCT
ejpam-5057	134	1	the	the	DET
ejpam-5057	134	2	characteristic	characteristic	ADJ
ejpam-5057	134	3	equation	equation	NOUN
ejpam-5057	134	4	of	of	ADP
ejpam-5057	134	5	aq	aq	PROPN
ejpam-5057	134	6	is	be	AUX
ejpam-5057	134	7	λ2	λ2	NOUN
ejpam-5057	134	8	−	−	PROPN
ejpam-5057	134	9	m2	m2	PROPN
ejpam-5057	134	10	n1n2	n1n2	PROPN
ejpam-5057	134	11	=	=	PROPN
ejpam-5057	134	12	0	0	NUM
ejpam-5057	134	13	and	and	CCONJ
ejpam-5057	134	14	the	the	DET
ejpam-5057	134	15	corresponding	corresponding	ADJ
ejpam-5057	134	16	eigenvalues	eigenvalue	NOUN
ejpam-5057	134	17	of	of	ADP
ejpam-5057	134	18	aq	aq	NOUN
ejpam-5057	134	19	are	be	AUX
ejpam-5057	134	20	η1	η1	NOUN
ejpam-5057	134	21	=	=	SYM
ejpam-5057	134	22	m√	m√	PROPN
ejpam-5057	134	23	n1n2	n1n2	NUM
ejpam-5057	134	24	and	and	CCONJ
ejpam-5057	134	25	η2	η2	ADJ
ejpam-5057	134	26	=	=	PUNCT
ejpam-5057	134	27	−m√	−m√	NOUN
ejpam-5057	134	28	n1n2	n1n2	VERB
ejpam-5057	134	29	.	.	PUNCT
ejpam-5057	135	1	to	to	PART
ejpam-5057	135	2	get	get	VERB
ejpam-5057	135	3	the	the	DET
ejpam-5057	135	4	generalized	generalized	ADJ
ejpam-5057	135	5	interlacing	interlacing	NOUN
ejpam-5057	135	6	,	,	PUNCT
ejpam-5057	135	7	we	we	PRON
ejpam-5057	135	8	need	need	VERB
ejpam-5057	135	9	n	n	ADV
ejpam-5057	135	10	m.machasri	m.machasri	NUM
ejpam-5057	135	11	,	,	PUNCT
ejpam-5057	135	12	d.kalyani	d.kalyani	NOUN
ejpam-5057	135	13	/	/	SYM
ejpam-5057	135	14	eur	eur	PROPN
ejpam-5057	135	15	.	.	PUNCT
ejpam-5057	136	1	j.	j.	PROPN
ejpam-5057	136	2	pure	pure	PROPN
ejpam-5057	136	3	appl	appl	PROPN
ejpam-5057	136	4	.	.	PROPN
ejpam-5057	136	5	math	math	PROPN
ejpam-5057	136	6	,	,	PUNCT
ejpam-5057	136	7	17	17	NUM
ejpam-5057	136	8	(	(	PUNCT
ejpam-5057	136	9	2	2	NUM
ejpam-5057	136	10	)	)	PUNCT
ejpam-5057	136	11	(	(	PUNCT
ejpam-5057	136	12	2024	2024	NUM
ejpam-5057	136	13	)	)	PUNCT
ejpam-5057	136	14	,	,	PUNCT
ejpam-5057	136	15	772	772	NUM
ejpam-5057	136	16	-	-	SYM
ejpam-5057	136	17	789	789	NUM
ejpam-5057	136	18	777	777	NUM
ejpam-5057	136	19	eigenvalues	eigenvalue	NOUN
ejpam-5057	136	20	.	.	PUNCT
ejpam-5057	137	1	to	to	PART
ejpam-5057	137	2	obtain	obtain	VERB
ejpam-5057	137	3	this	this	PRON
ejpam-5057	137	4	,	,	PUNCT
ejpam-5057	137	5	consider	consider	VERB
ejpam-5057	137	6	the	the	DET
ejpam-5057	137	7	following	following	NOUN
ejpam-5057	137	8	.	.	PUNCT
ejpam-5057	138	1	the	the	DET
ejpam-5057	138	2	characteristic	characteristic	ADJ
ejpam-5057	138	3	matrix	matrix	NOUN
ejpam-5057	138	4	of	of	ADP
ejpam-5057	138	5	g	g	PROPN
ejpam-5057	138	6	is	be	AUX
ejpam-5057	138	7	given	give	VERB
ejpam-5057	138	8	by	by	ADP
ejpam-5057	138	9	s̃	s̃	PROPN
ejpam-5057	138	10	=	=	PUNCT
ejpam-5057	138	11	[	[	PUNCT
ejpam-5057	138	12	jn1×1	jn1×1	NOUN
ejpam-5057	138	13	0n1×1	0n1×1	VERB
ejpam-5057	138	14	0n2×1	0n2×1	NUM
ejpam-5057	138	15	jn2×1	jn2×1	PROPN
ejpam-5057	138	16	.	.	PUNCT
ejpam-5057	138	17	]	]	PUNCT
ejpam-5057	139	1	n×2	n×2	NOUN
ejpam-5057	139	2	where	where	SCONJ
ejpam-5057	139	3	j	j	PROPN
ejpam-5057	139	4	is	be	AUX
ejpam-5057	139	5	the	the	DET
ejpam-5057	139	6	all	all	DET
ejpam-5057	139	7	-	-	PUNCT
ejpam-5057	139	8	ones	one	NOUN
ejpam-5057	139	9	matrix	matrix	NOUN
ejpam-5057	139	10	.	.	PUNCT
ejpam-5057	140	1	let	let	VERB
ejpam-5057	140	2	s	s	PRON
ejpam-5057	140	3	=	=	VERB
ejpam-5057	140	4	s̃k	s̃k	ADP
ejpam-5057	140	5	−1	−1	NOUN
ejpam-5057	140	6	2	2	NUM
ejpam-5057	140	7	where	where	SCONJ
ejpam-5057	140	8	k	k	PROPN
ejpam-5057	140	9	=	=	SYM
ejpam-5057	140	10	diag(|x|	diag(|x|	PROPN
ejpam-5057	140	11	,	,	PUNCT
ejpam-5057	140	12	|y	|y	NOUN
ejpam-5057	140	13	|	|	NOUN
ejpam-5057	140	14	)	)	PUNCT
ejpam-5057	140	15	.	.	PUNCT
ejpam-5057	141	1	i.e.	i.e.	X
ejpam-5057	141	2	,	,	PUNCT
ejpam-5057	141	3	k	k	PROPN
ejpam-5057	141	4	=[	=[	NOUN
ejpam-5057	141	5	n1	n1	PROPN
ejpam-5057	141	6	0	0	NUM
ejpam-5057	141	7	0	0	NUM
ejpam-5057	141	8	n2	n2	NOUN
ejpam-5057	141	9	]	]	PUNCT
ejpam-5057	141	10	.	.	PUNCT
ejpam-5057	142	1	then	then	ADV
ejpam-5057	142	2	s	s	VERB
ejpam-5057	142	3	=	=	PUNCT
ejpam-5057	142	4	[	[	PUNCT
ejpam-5057	142	5	pn1×1	pn1×1	NOUN
ejpam-5057	142	6	0n1×1	0n1×1	NOUN
ejpam-5057	142	7	0n2×1	0n2×1	NUM
ejpam-5057	142	8	rn2×1	rn2×1	NOUN
ejpam-5057	142	9	]	]	PUNCT
ejpam-5057	142	10	n×2	n×2	NOUN
ejpam-5057	142	11	.	.	PUNCT
ejpam-5057	143	1	where	where	SCONJ
ejpam-5057	143	2	pn1×1	pn1×1	NOUN
ejpam-5057	143	3	and	and	CCONJ
ejpam-5057	143	4	rn2×1	rn2×1	NOUN
ejpam-5057	143	5	are	be	AUX
ejpam-5057	143	6	column	column	NOUN
ejpam-5057	143	7	matrices	matrix	NOUN
ejpam-5057	143	8	with	with	ADP
ejpam-5057	143	9	entries	entry	NOUN
ejpam-5057	143	10	(	(	PUNCT
ejpam-5057	143	11	pi1	pi1	NOUN
ejpam-5057	143	12	)	)	PUNCT
ejpam-5057	143	13	=	=	SYM
ejpam-5057	143	14	1√	1√	ADJ
ejpam-5057	143	15	n1	n1	NOUN
ejpam-5057	143	16	for	for	ADP
ejpam-5057	143	17	all	all	DET
ejpam-5057	143	18	vi	vi	NOUN
ejpam-5057	143	19	∈	∈	PROPN
ejpam-5057	143	20	x	x	X
ejpam-5057	143	21	and	and	CCONJ
ejpam-5057	143	22	(	(	PUNCT
ejpam-5057	143	23	ri1	ri1	NOUN
ejpam-5057	143	24	)	)	PUNCT
ejpam-5057	143	25	=	=	SYM
ejpam-5057	143	26	1√	1√	ADJ
ejpam-5057	143	27	n2	n2	NOUN
ejpam-5057	143	28	for	for	ADP
ejpam-5057	143	29	all	all	DET
ejpam-5057	143	30	vi	vi	NOUN
ejpam-5057	143	31	∈	∈	PROPN
ejpam-5057	143	32	y	y	PROPN
ejpam-5057	143	33	,	,	PUNCT
ejpam-5057	143	34	respectively	respectively	ADV
ejpam-5057	143	35	.	.	PUNCT
ejpam-5057	144	1	from	from	ADP
ejpam-5057	144	2	the	the	DET
ejpam-5057	144	3	proof	proof	NOUN
ejpam-5057	144	4	of	of	ADP
ejpam-5057	144	5	lemma	lemma	PROPN
ejpam-5057	144	6	1	1	NUM
ejpam-5057	144	7	[	[	X
ejpam-5057	144	8	7	7	X
ejpam-5057	144	9	]	]	PUNCT
ejpam-5057	144	10	we	we	PRON
ejpam-5057	144	11	have	have	VERB
ejpam-5057	144	12	,	,	PUNCT
ejpam-5057	144	13	aq	aq	VERB
ejpam-5057	144	14	=	=	NOUN
ejpam-5057	144	15	stas	sta	NOUN
ejpam-5057	144	16	.	.	PUNCT
ejpam-5057	145	1	(	(	PUNCT
ejpam-5057	145	2	1	1	X
ejpam-5057	145	3	)	)	PUNCT
ejpam-5057	145	4	left	leave	VERB
ejpam-5057	145	5	and	and	CCONJ
ejpam-5057	145	6	right	right	ADJ
ejpam-5057	145	7	multiplying	multiply	VERB
ejpam-5057	145	8	equation	equation	NOUN
ejpam-5057	145	9	(	(	PUNCT
ejpam-5057	145	10	1	1	NUM
ejpam-5057	145	11	)	)	PUNCT
ejpam-5057	145	12	by	by	ADP
ejpam-5057	145	13	s	s	PRON
ejpam-5057	145	14	and	and	CCONJ
ejpam-5057	145	15	st	st	PROPN
ejpam-5057	145	16	respectively	respectively	ADV
ejpam-5057	145	17	,	,	PUNCT
ejpam-5057	145	18	we	we	PRON
ejpam-5057	145	19	get	get	VERB
ejpam-5057	145	20	,	,	PUNCT
ejpam-5057	145	21	saqs	saq	NOUN
ejpam-5057	145	22	t	t	NOUN
ejpam-5057	145	23	=	=	SYM
ejpam-5057	145	24	a.	a.	NOUN
ejpam-5057	145	25	now	now	ADV
ejpam-5057	145	26	consider	consider	VERB
ejpam-5057	145	27	saqs	saq	NOUN
ejpam-5057	145	28	t	t	PROPN
ejpam-5057	145	29	and	and	CCONJ
ejpam-5057	145	30	denote	denote	VERB
ejpam-5057	145	31	it	it	PRON
ejpam-5057	145	32	by	by	ADP
ejpam-5057	145	33	c.then	c.then	NOUN
ejpam-5057	145	34	c	c	NOUN
ejpam-5057	145	35	=	=	SYM
ejpam-5057	145	36	(	(	PUNCT
ejpam-5057	145	37	cij	cij	PROPN
ejpam-5057	145	38	)	)	PUNCT
ejpam-5057	145	39	=	=	PUNCT
ejpam-5057	146	1			NUM
ejpam-5057	146	2	m	m	VERB
ejpam-5057	146	3	n1	n1	ADJ
ejpam-5057	146	4	√	√	PROPN
ejpam-5057	146	5	n1	n1	PROPN
ejpam-5057	146	6	vi	vi	PROPN
ejpam-5057	146	7	∈	∈	PROPN
ejpam-5057	146	8	x	x	X
ejpam-5057	146	9	and	and	CCONJ
ejpam-5057	146	10	vj	vj	INTJ
ejpam-5057	146	11	∈	∈	PROPN
ejpam-5057	146	12	y	y	PROPN
ejpam-5057	146	13	m	m	PROPN
ejpam-5057	146	14	n2	n2	ADJ
ejpam-5057	146	15	√	√	PROPN
ejpam-5057	146	16	n2	n2	PROPN
ejpam-5057	146	17	vi	vi	PROPN
ejpam-5057	146	18	∈	∈	PROPN
ejpam-5057	146	19	y	y	PROPN
ejpam-5057	146	20	and	and	CCONJ
ejpam-5057	146	21	vj	vj	PRON
ejpam-5057	146	22	∈	∈	PROPN
ejpam-5057	146	23	x	x	SYM
ejpam-5057	146	24	0	0	NUM
ejpam-5057	146	25	otherwise	otherwise	ADV
ejpam-5057	146	26	.	.	PUNCT
ejpam-5057	147	1	the	the	DET
ejpam-5057	147	2	above	above	ADJ
ejpam-5057	147	3	matrix	matrix	NOUN
ejpam-5057	147	4	can	can	AUX
ejpam-5057	147	5	be	be	AUX
ejpam-5057	147	6	represented	represent	VERB
ejpam-5057	147	7	as	as	ADP
ejpam-5057	147	8	a	a	DET
ejpam-5057	147	9	block	block	NOUN
ejpam-5057	147	10	matrix	matrix	NOUN
ejpam-5057	147	11	as	as	SCONJ
ejpam-5057	147	12	follows	follow	VERB
ejpam-5057	147	13	:	:	PUNCT
ejpam-5057	147	14	c	c	X
ejpam-5057	147	15	=	=	PUNCT
ejpam-5057	148	1	[	[	PUNCT
ejpam-5057	148	2	0	0	NUM
ejpam-5057	148	3	m	m	VERB
ejpam-5057	148	4	n	n	ADJ
ejpam-5057	148	5	0	0	NUM
ejpam-5057	148	6	]	]	PUNCT
ejpam-5057	149	1	n×n	n×n	PROPN
ejpam-5057	149	2	.	.	PUNCT
ejpam-5057	150	1	denote	denote	VERB
ejpam-5057	150	2	the	the	DET
ejpam-5057	150	3	eigenvalues	eigenvalue	NOUN
ejpam-5057	150	4	of	of	ADP
ejpam-5057	150	5	c	c	NOUN
ejpam-5057	150	6	by	by	ADP
ejpam-5057	150	7	γi	γi	ADP
ejpam-5057	150	8	,	,	PUNCT
ejpam-5057	150	9	i	i	NOUN
ejpam-5057	150	10	=	=	NOUN
ejpam-5057	150	11	1	1	NUM
ejpam-5057	150	12	,	,	PUNCT
ejpam-5057	150	13	2	2	NUM
ejpam-5057	150	14	,	,	PUNCT
ejpam-5057	150	15	.	.	PUNCT
ejpam-5057	150	16	.	.	PUNCT
ejpam-5057	151	1	.	.	PUNCT
ejpam-5057	152	1	,	,	PUNCT
ejpam-5057	152	2	n.	n.	VERB
ejpam-5057	152	3	the	the	DET
ejpam-5057	152	4	eigenvalues	eigenvalue	NOUN
ejpam-5057	152	5	of	of	ADP
ejpam-5057	152	6	c	c	NOUN
ejpam-5057	152	7	are	be	AUX
ejpam-5057	152	8	the	the	DET
ejpam-5057	152	9	square	square	ADJ
ejpam-5057	152	10	roots	root	NOUN
ejpam-5057	152	11	of	of	ADP
ejpam-5057	152	12	the	the	DET
ejpam-5057	152	13	non	non	ADJ
ejpam-5057	152	14	-	-	ADJ
ejpam-5057	152	15	zero	zero	NUM
ejpam-5057	152	16	eigenvalues	eigenvalue	NOUN
ejpam-5057	152	17	of	of	ADP
ejpam-5057	152	18	mn	mn	PROPN
ejpam-5057	152	19	.	.	PUNCT
ejpam-5057	153	1	that	that	PRON
ejpam-5057	153	2	is	be	AUX
ejpam-5057	153	3	γ1	γ1	NOUN
ejpam-5057	153	4	=	=	SYM
ejpam-5057	153	5	m√	m√	PROPN
ejpam-5057	153	6	n1n2	n1n2	NUM
ejpam-5057	153	7	,	,	PUNCT
ejpam-5057	153	8	γn	γn	ADP
ejpam-5057	153	9	=	=	PUNCT
ejpam-5057	153	10	−m√	−m√	VERB
ejpam-5057	153	11	n1n2	n1n2	NOUN
ejpam-5057	153	12	and	and	CCONJ
ejpam-5057	153	13	γi	γi	X
ejpam-5057	153	14	=	=	SYM
ejpam-5057	153	15	0	0	PROPN
ejpam-5057	153	16	for	for	ADP
ejpam-5057	153	17	i	i	PRON
ejpam-5057	153	18	=	=	SYM
ejpam-5057	153	19	2	2	NUM
ejpam-5057	153	20	,	,	PUNCT
ejpam-5057	153	21	3	3	NUM
ejpam-5057	153	22	,	,	PUNCT
ejpam-5057	153	23	.	.	PUNCT
ejpam-5057	153	24	.	.	PUNCT
ejpam-5057	154	1	.	.	PUNCT
ejpam-5057	155	1	,	,	PUNCT
ejpam-5057	155	2	n−	n−	NOUN
ejpam-5057	155	3	1	1	NUM
ejpam-5057	155	4	.	.	PUNCT
ejpam-5057	156	1	then	then	ADV
ejpam-5057	156	2	by	by	ADP
ejpam-5057	156	3	interlacing	interlace	VERB
ejpam-5057	156	4	we	we	PRON
ejpam-5057	156	5	get	get	VERB
ejpam-5057	156	6	,	,	PUNCT
ejpam-5057	156	7	λ1	λ1	PROPN
ejpam-5057	156	8	≥	≥	NOUN
ejpam-5057	156	9	γ1	γ1	PROPN
ejpam-5057	156	10	≥	≥	NOUN
ejpam-5057	156	11	λ2	λ2	PROPN
ejpam-5057	156	12	≥	≥	NUM
ejpam-5057	156	13	·	·	PUNCT
ejpam-5057	156	14	·	·	PUNCT
ejpam-5057	156	15	·	·	PUNCT
ejpam-5057	157	1	≥	≥	PRON
ejpam-5057	158	1	λn−1	λn−1	PROPN
ejpam-5057	158	2	≥	≥	NUM
ejpam-5057	158	3	γn−1	γn−1	PROPN
ejpam-5057	158	4	≥	≥	NUM
ejpam-5057	158	5	λn	λn	NOUN
ejpam-5057	158	6	≥	≥	NUM
ejpam-5057	158	7	γn	γn	NOUN
ejpam-5057	158	8	.	.	PUNCT
ejpam-5057	159	1	by	by	ADP
ejpam-5057	159	2	comparing	compare	VERB
ejpam-5057	159	3	the	the	DET
ejpam-5057	159	4	eigenvalues	eigenvalue	NOUN
ejpam-5057	159	5	of	of	ADP
ejpam-5057	159	6	aq	aq	NOUN
ejpam-5057	159	7	and	and	CCONJ
ejpam-5057	159	8	c	c	AUX
ejpam-5057	159	9	we	we	PRON
ejpam-5057	159	10	get	get	VERB
ejpam-5057	159	11	,	,	PUNCT
ejpam-5057	159	12	γ1	γ1	NOUN
ejpam-5057	159	13	=	=	SYM
ejpam-5057	159	14	m√	m√	ADJ
ejpam-5057	159	15	n1n2	n1n2	NOUN
ejpam-5057	159	16	=	=	SYM
ejpam-5057	159	17	η1	η1	NOUN
ejpam-5057	159	18	and	and	CCONJ
ejpam-5057	159	19	γn	γn	NOUN
ejpam-5057	159	20	=	=	PUNCT
ejpam-5057	159	21	−m√	−m√	VERB
ejpam-5057	159	22	n1n2	n1n2	NOUN
ejpam-5057	159	23	=	=	SYM
ejpam-5057	159	24	η2	η2	PROPN
ejpam-5057	159	25	.	.	PUNCT
ejpam-5057	160	1	now	now	ADV
ejpam-5057	160	2	let	let	VERB
ejpam-5057	160	3	us	we	PRON
ejpam-5057	160	4	first	first	ADV
ejpam-5057	160	5	consider	consider	VERB
ejpam-5057	160	6	λ1	λ1	PROPN
ejpam-5057	160	7	≥	≥	NOUN
ejpam-5057	160	8	γ1	γ1	PROPN
ejpam-5057	160	9	≥	≥	NOUN
ejpam-5057	160	10	λ2	λ2	PROPN
ejpam-5057	160	11	.	.	PUNCT
ejpam-5057	161	1	since	since	SCONJ
ejpam-5057	161	2	γ1	γ1	NOUN
ejpam-5057	161	3	=	=	NOUN
ejpam-5057	161	4	η1	η1	NOUN
ejpam-5057	161	5	we	we	PRON
ejpam-5057	161	6	have	have	VERB
ejpam-5057	161	7	λ1	λ1	VERB
ejpam-5057	161	8	≥	≥	NOUN
ejpam-5057	161	9	η1	η1	NOUN
ejpam-5057	161	10	≥	≥	NOUN
ejpam-5057	161	11	λ2	λ2	NOUN
ejpam-5057	161	12	.	.	PUNCT
ejpam-5057	162	1	this	this	PRON
ejpam-5057	162	2	proves	prove	VERB
ejpam-5057	162	3	(	(	PUNCT
ejpam-5057	162	4	i	i	NOUN
ejpam-5057	162	5	)	)	PUNCT
ejpam-5057	162	6	.	.	PUNCT
ejpam-5057	163	1	since	since	SCONJ
ejpam-5057	163	2	g	g	PROPN
ejpam-5057	163	3	is	be	AUX
ejpam-5057	163	4	bipartite	bipartite	ADJ
ejpam-5057	163	5	,	,	PUNCT
ejpam-5057	163	6	its	its	PRON
ejpam-5057	163	7	eigenvalues	eigenvalue	NOUN
ejpam-5057	163	8	are	be	AUX
ejpam-5057	163	9	symmetric	symmetric	ADJ
ejpam-5057	163	10	about	about	ADP
ejpam-5057	163	11	the	the	DET
ejpam-5057	163	12	origin	origin	NOUN
ejpam-5057	163	13	.	.	PUNCT
ejpam-5057	164	1	now	now	ADV
ejpam-5057	164	2	from	from	ADP
ejpam-5057	164	3	(	(	PUNCT
ejpam-5057	164	4	i	i	NOUN
ejpam-5057	164	5	)	)	PUNCT
ejpam-5057	164	6	and	and	CCONJ
ejpam-5057	164	7	since	since	SCONJ
ejpam-5057	164	8	g	g	PROPN
ejpam-5057	164	9	is	be	AUX
ejpam-5057	164	10	bipartite	bipartite	ADJ
ejpam-5057	164	11	,	,	PUNCT
ejpam-5057	164	12	we	we	PRON
ejpam-5057	164	13	get	get	VERB
ejpam-5057	165	1	λn	λn	NOUN
ejpam-5057	165	2	≤	≤	NUM
ejpam-5057	165	3	η2	η2	VERB
ejpam-5057	165	4	≤	≤	PUNCT
ejpam-5057	165	5	λn−1	λn−1	PROPN
ejpam-5057	165	6	.	.	PUNCT
ejpam-5057	166	1	this	this	PRON
ejpam-5057	166	2	proves	prove	VERB
ejpam-5057	166	3	(	(	PUNCT
ejpam-5057	166	4	ii	ii	NOUN
ejpam-5057	166	5	)	)	PUNCT
ejpam-5057	166	6	.	.	PUNCT
ejpam-5057	167	1	to	to	PART
ejpam-5057	167	2	prove	prove	VERB
ejpam-5057	167	3	(	(	PUNCT
ejpam-5057	167	4	iii	iii	NOUN
ejpam-5057	167	5	)	)	PUNCT
ejpam-5057	167	6	,	,	PUNCT
ejpam-5057	167	7	using	use	VERB
ejpam-5057	167	8	(	(	PUNCT
ejpam-5057	167	9	i	i	NOUN
ejpam-5057	167	10	)	)	PUNCT
ejpam-5057	167	11	we	we	PRON
ejpam-5057	167	12	have	have	VERB
ejpam-5057	167	13	,	,	PUNCT
ejpam-5057	167	14	λ2	λ2	NOUN
ejpam-5057	167	15	≤	≤	NUM
ejpam-5057	167	16	η1	η1	NOUN
ejpam-5057	167	17	=	=	SYM
ejpam-5057	167	18	m	m	VERB
ejpam-5057	167	19	√	√	ADJ
ejpam-5057	167	20	n1n2	n1n2	NOUN
ejpam-5057	167	21	.	.	PUNCT
ejpam-5057	168	1	m.machasri	m.machasri	NUM
ejpam-5057	168	2	,	,	PUNCT
ejpam-5057	168	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	168	4	/	/	SYM
ejpam-5057	168	5	eur	eur	PROPN
ejpam-5057	168	6	.	.	PUNCT
ejpam-5057	169	1	j.	j.	PROPN
ejpam-5057	169	2	pure	pure	PROPN
ejpam-5057	169	3	appl	appl	PROPN
ejpam-5057	169	4	.	.	PROPN
ejpam-5057	169	5	math	math	PROPN
ejpam-5057	169	6	,	,	PUNCT
ejpam-5057	169	7	17	17	NUM
ejpam-5057	169	8	(	(	PUNCT
ejpam-5057	169	9	2	2	NUM
ejpam-5057	169	10	)	)	PUNCT
ejpam-5057	169	11	(	(	PUNCT
ejpam-5057	169	12	2024	2024	NUM
ejpam-5057	169	13	)	)	PUNCT
ejpam-5057	169	14	,	,	PUNCT
ejpam-5057	169	15	772	772	NUM
ejpam-5057	169	16	-	-	SYM
ejpam-5057	169	17	789	789	NUM
ejpam-5057	169	18	778	778	NUM
ejpam-5057	169	19	to	to	PART
ejpam-5057	169	20	prove	prove	VERB
ejpam-5057	169	21	(	(	PUNCT
ejpam-5057	169	22	iv	iv	NUM
ejpam-5057	169	23	)	)	PUNCT
ejpam-5057	169	24	,	,	PUNCT
ejpam-5057	169	25	using	use	VERB
ejpam-5057	169	26	(	(	PUNCT
ejpam-5057	169	27	ii	ii	NOUN
ejpam-5057	169	28	)	)	PUNCT
ejpam-5057	169	29	,	,	PUNCT
ejpam-5057	169	30	we	we	PRON
ejpam-5057	169	31	have	have	AUX
ejpam-5057	169	32	,	,	PUNCT
ejpam-5057	169	33	λn−1	λn−1	PROPN
ejpam-5057	169	34	≥	≥	NUM
ejpam-5057	169	35	η2	η2	ADJ
ejpam-5057	169	36	=	=	PUNCT
ejpam-5057	169	37	−m	−m	NOUN
ejpam-5057	169	38	√	√	NUM
ejpam-5057	169	39	n1n2	n1n2	NOUN
ejpam-5057	169	40	.	.	PUNCT
ejpam-5057	170	1	this	this	PRON
ejpam-5057	170	2	completes	complete	VERB
ejpam-5057	170	3	the	the	DET
ejpam-5057	170	4	proof	proof	NOUN
ejpam-5057	170	5	.	.	PUNCT
ejpam-5057	171	1	corollary	corollary	ADJ
ejpam-5057	171	2	1	1	NUM
ejpam-5057	171	3	.	.	PUNCT
ejpam-5057	172	1	if	if	SCONJ
ejpam-5057	172	2	g	g	PROPN
ejpam-5057	172	3	is	be	AUX
ejpam-5057	172	4	a	a	DET
ejpam-5057	172	5	regular	regular	ADJ
ejpam-5057	172	6	bipartite	bipartite	NOUN
ejpam-5057	172	7	graph	graph	NOUN
ejpam-5057	172	8	then	then	ADV
ejpam-5057	172	9	λ2	λ2	PROPN
ejpam-5057	172	10	≤	≤	PUNCT
ejpam-5057	172	11	m	m	VERB
ejpam-5057	172	12	n1	n1	ADJ
ejpam-5057	172	13	proof	proof	NOUN
ejpam-5057	172	14	.	.	PUNCT
ejpam-5057	173	1	for	for	ADP
ejpam-5057	173	2	a	a	DET
ejpam-5057	173	3	regular	regular	ADJ
ejpam-5057	173	4	bipartite	bipartite	NOUN
ejpam-5057	173	5	graph	graph	NOUN
ejpam-5057	173	6	g	g	NOUN
ejpam-5057	173	7	,	,	PUNCT
ejpam-5057	173	8	n1	n1	PROPN
ejpam-5057	173	9	=	=	SYM
ejpam-5057	173	10	n2	n2	NOUN
ejpam-5057	173	11	.	.	PUNCT
ejpam-5057	174	1	substituting	substitute	VERB
ejpam-5057	174	2	this	this	PRON
ejpam-5057	174	3	in	in	ADP
ejpam-5057	174	4	(	(	PUNCT
ejpam-5057	174	5	iii	iii	NOUN
ejpam-5057	174	6	)	)	PUNCT
ejpam-5057	174	7	of	of	ADP
ejpam-5057	174	8	theorem	theorem	NOUN
ejpam-5057	174	9	1	1	NUM
ejpam-5057	174	10	,	,	PUNCT
ejpam-5057	174	11	we	we	PRON
ejpam-5057	174	12	get	get	VERB
ejpam-5057	174	13	the	the	DET
ejpam-5057	174	14	result	result	NOUN
ejpam-5057	174	15	.	.	PUNCT
ejpam-5057	175	1	corollary	corollary	ADJ
ejpam-5057	175	2	2	2	NUM
ejpam-5057	175	3	.	.	PUNCT
ejpam-5057	176	1	if	if	SCONJ
ejpam-5057	176	2	g	g	PROPN
ejpam-5057	176	3	is	be	AUX
ejpam-5057	176	4	a	a	DET
ejpam-5057	176	5	regular	regular	ADJ
ejpam-5057	176	6	bipartite	bipartite	NOUN
ejpam-5057	176	7	graph	graph	NOUN
ejpam-5057	176	8	then	then	ADV
ejpam-5057	176	9	λn−1	λn−1	PROPN
ejpam-5057	176	10	≥	≥	NUM
ejpam-5057	176	11	−m	−m	NOUN
ejpam-5057	176	12	n1	n1	ADJ
ejpam-5057	176	13	proof	proof	NOUN
ejpam-5057	176	14	.	.	PUNCT
ejpam-5057	177	1	for	for	ADP
ejpam-5057	177	2	a	a	DET
ejpam-5057	177	3	regular	regular	ADJ
ejpam-5057	177	4	bipartite	bipartite	NOUN
ejpam-5057	177	5	graph	graph	NOUN
ejpam-5057	177	6	g	g	NOUN
ejpam-5057	177	7	,	,	PUNCT
ejpam-5057	177	8	n1	n1	PROPN
ejpam-5057	177	9	=	=	SYM
ejpam-5057	177	10	n2	n2	NOUN
ejpam-5057	177	11	.	.	PUNCT
ejpam-5057	178	1	substituting	substitute	VERB
ejpam-5057	178	2	this	this	PRON
ejpam-5057	178	3	in	in	ADP
ejpam-5057	178	4	(	(	PUNCT
ejpam-5057	178	5	iv	iv	NOUN
ejpam-5057	178	6	)	)	PUNCT
ejpam-5057	178	7	of	of	ADP
ejpam-5057	178	8	theorem	theorem	NOUN
ejpam-5057	178	9	1	1	NUM
ejpam-5057	178	10	,	,	PUNCT
ejpam-5057	178	11	we	we	PRON
ejpam-5057	178	12	get	get	VERB
ejpam-5057	178	13	the	the	DET
ejpam-5057	178	14	result	result	NOUN
ejpam-5057	178	15	.	.	PUNCT
ejpam-5057	179	1	in	in	ADP
ejpam-5057	179	2	the	the	DET
ejpam-5057	179	3	following	follow	VERB
ejpam-5057	179	4	theorems	theorem	NOUN
ejpam-5057	179	5	,	,	PUNCT
ejpam-5057	179	6	we	we	PRON
ejpam-5057	179	7	present	present	VERB
ejpam-5057	179	8	simpler	simple	ADJ
ejpam-5057	179	9	bounds	bound	NOUN
ejpam-5057	179	10	,	,	PUNCT
ejpam-5057	179	11	involving	involve	VERB
ejpam-5057	179	12	a	a	DET
ejpam-5057	179	13	single	single	ADJ
ejpam-5057	179	14	parameter	parameter	NOUN
ejpam-5057	179	15	n	n	CCONJ
ejpam-5057	179	16	,	,	PUNCT
ejpam-5057	179	17	when	when	SCONJ
ejpam-5057	179	18	compared	compare	VERB
ejpam-5057	179	19	to	to	ADP
ejpam-5057	179	20	the	the	DET
ejpam-5057	179	21	bounds	bound	NOUN
ejpam-5057	179	22	obtained	obtain	VERB
ejpam-5057	179	23	in	in	ADP
ejpam-5057	179	24	theorem	theorem	NOUN
ejpam-5057	179	25	1	1	NUM
ejpam-5057	179	26	.	.	PUNCT
ejpam-5057	179	27	to	to	PART
ejpam-5057	179	28	achieve	achieve	VERB
ejpam-5057	179	29	this	this	PRON
ejpam-5057	179	30	we	we	PRON
ejpam-5057	179	31	require	require	VERB
ejpam-5057	179	32	m	m	VERB
ejpam-5057	179	33	to	to	PART
ejpam-5057	179	34	be	be	AUX
ejpam-5057	179	35	minimum	minimum	ADJ
ejpam-5057	179	36	.	.	PUNCT
ejpam-5057	180	1	m	m	PROPN
ejpam-5057	180	2	would	would	AUX
ejpam-5057	180	3	be	be	AUX
ejpam-5057	180	4	minimum	minimum	ADJ
ejpam-5057	180	5	only	only	ADV
ejpam-5057	180	6	if	if	SCONJ
ejpam-5057	180	7	g	g	PROPN
ejpam-5057	180	8	is	be	AUX
ejpam-5057	180	9	minimally	minimally	ADV
ejpam-5057	180	10	connected	connect	VERB
ejpam-5057	180	11	,	,	PUNCT
ejpam-5057	180	12	that	that	PRON
ejpam-5057	180	13	is	be	AUX
ejpam-5057	180	14	m	m	VERB
ejpam-5057	180	15	=	=	ADJ
ejpam-5057	180	16	n−	n−	NOUN
ejpam-5057	180	17	1	1	NUM
ejpam-5057	180	18	.	.	PUNCT
ejpam-5057	181	1	we	we	PRON
ejpam-5057	181	2	consider	consider	VERB
ejpam-5057	181	3	three	three	NUM
ejpam-5057	181	4	types	type	NOUN
ejpam-5057	181	5	of	of	ADP
ejpam-5057	181	6	bipartitions	bipartition	NOUN
ejpam-5057	181	7	such	such	ADJ
ejpam-5057	181	8	as	as	ADP
ejpam-5057	181	9	balanced	balanced	ADJ
ejpam-5057	181	10	,	,	PUNCT
ejpam-5057	181	11	unbalanced	unbalanced	ADJ
ejpam-5057	181	12	and	and	CCONJ
ejpam-5057	181	13	average	average	ADJ
ejpam-5057	181	14	bipartitions	bipartition	NOUN
ejpam-5057	181	15	to	to	PART
ejpam-5057	181	16	arrive	arrive	VERB
ejpam-5057	181	17	at	at	ADP
ejpam-5057	181	18	our	our	PRON
ejpam-5057	181	19	result	result	NOUN
ejpam-5057	181	20	.	.	PUNCT
ejpam-5057	182	1	for	for	ADP
ejpam-5057	182	2	a	a	DET
ejpam-5057	182	3	balanced	balanced	ADJ
ejpam-5057	182	4	bipartition	bipartition	NOUN
ejpam-5057	182	5	we	we	PRON
ejpam-5057	182	6	have	have	VERB
ejpam-5057	182	7	(	(	PUNCT
ejpam-5057	182	8	n1	n1	NOUN
ejpam-5057	182	9	,	,	PUNCT
ejpam-5057	182	10	n2	n2	NOUN
ejpam-5057	182	11	)	)	PUNCT
ejpam-5057	182	12	=	=	SYM
ejpam-5057	182	13	(n−1	(n−1	PROPN
ejpam-5057	182	14	2	2	NUM
ejpam-5057	182	15	,	,	PUNCT
ejpam-5057	182	16	n+1	n+1	PROPN
ejpam-5057	182	17	2	2	NUM
ejpam-5057	182	18	)	)	PUNCT
ejpam-5057	182	19	n	n	CCONJ
ejpam-5057	182	20	is	be	AUX
ejpam-5057	182	21	odd	odd	ADJ
ejpam-5057	182	22	(	(	PUNCT
ejpam-5057	182	23	n2	n2	ADJ
ejpam-5057	182	24	,	,	PUNCT
ejpam-5057	182	25	n	n	PRON
ejpam-5057	182	26	2	2	NUM
ejpam-5057	182	27	)	)	PUNCT
ejpam-5057	182	28	n	n	CCONJ
ejpam-5057	182	29	is	be	AUX
ejpam-5057	182	30	even	even	ADV
ejpam-5057	182	31	.	.	PUNCT
ejpam-5057	183	1	unbalanced	unbalanced	ADJ
ejpam-5057	183	2	bipartition	bipartition	NOUN
ejpam-5057	183	3	is	be	AUX
ejpam-5057	183	4	given	give	VERB
ejpam-5057	183	5	by	by	ADP
ejpam-5057	183	6	,	,	PUNCT
ejpam-5057	183	7	(	(	PUNCT
ejpam-5057	183	8	n1	n1	NOUN
ejpam-5057	183	9	,	,	PUNCT
ejpam-5057	183	10	n2	n2	ADJ
ejpam-5057	183	11	)	)	PUNCT
ejpam-5057	183	12	=	=	PUNCT
ejpam-5057	183	13	(	(	PUNCT
ejpam-5057	183	14	1	1	NUM
ejpam-5057	183	15	,	,	PUNCT
ejpam-5057	183	16	n−	n−	NOUN
ejpam-5057	183	17	1	1	NUM
ejpam-5057	183	18	)	)	PUNCT
ejpam-5057	183	19	.	.	PUNCT
ejpam-5057	184	1	average	average	ADJ
ejpam-5057	184	2	bipartition	bipartition	NOUN
ejpam-5057	184	3	is	be	AUX
ejpam-5057	184	4	given	give	VERB
ejpam-5057	184	5	by	by	ADP
ejpam-5057	184	6	(	(	PUNCT
ejpam-5057	184	7	n1	n1	NOUN
ejpam-5057	184	8	,	,	PUNCT
ejpam-5057	184	9	n2	n2	ADJ
ejpam-5057	184	10	)	)	PUNCT
ejpam-5057	184	11	=	=	SYM
ejpam-5057	185	1			PROPN
ejpam-5057	185	2	(	(	PUNCT
ejpam-5057	185	3	3n−4	3n−4	NOUN
ejpam-5057	185	4	4	4	NUM
ejpam-5057	185	5	,	,	PUNCT
ejpam-5057	185	6	n+4	n+4	NUM
ejpam-5057	185	7	4	4	X
ejpam-5057	185	8	)	)	PUNCT
ejpam-5057	185	9	if	if	SCONJ
ejpam-5057	185	10	n	n	PRON
ejpam-5057	185	11	is	be	AUX
ejpam-5057	185	12	even	even	ADV
ejpam-5057	185	13	(	(	PUNCT
ejpam-5057	185	14	3n−3	3n−3	NUM
ejpam-5057	185	15	4	4	NUM
ejpam-5057	185	16	,	,	PUNCT
ejpam-5057	185	17	n+3	n+3	PROPN
ejpam-5057	185	18	4	4	NUM
ejpam-5057	185	19	)	)	PUNCT
ejpam-5057	185	20	if	if	SCONJ
ejpam-5057	185	21	n	n	NOUN
ejpam-5057	185	22	=	=	SYM
ejpam-5057	185	23	4k	4k	NUM
ejpam-5057	185	24	+	+	NOUN
ejpam-5057	185	25	3	3	NUM
ejpam-5057	185	26	where	where	SCONJ
ejpam-5057	185	27	k	k	PROPN
ejpam-5057	185	28	=	=	SYM
ejpam-5057	185	29	0	0	NUM
ejpam-5057	185	30	,	,	PUNCT
ejpam-5057	185	31	1	1	NUM
ejpam-5057	185	32	,	,	PUNCT
ejpam-5057	185	33	2	2	NUM
ejpam-5057	185	34	,	,	PUNCT
ejpam-5057	185	35	.	.	PUNCT
ejpam-5057	185	36	.	.	PUNCT
ejpam-5057	185	37	.	.	PUNCT
ejpam-5057	185	38	.	.	PUNCT
ejpam-5057	186	1	(	(	PUNCT
ejpam-5057	186	2	3n−5	3n−5	NUM
ejpam-5057	186	3	4	4	NUM
ejpam-5057	186	4	,	,	PUNCT
ejpam-5057	186	5	n+5	n+5	PRON
ejpam-5057	186	6	4	4	X
ejpam-5057	186	7	)	)	PUNCT
ejpam-5057	186	8	if	if	SCONJ
ejpam-5057	186	9	n	n	NOUN
ejpam-5057	186	10	=	=	SYM
ejpam-5057	186	11	4k	4k	NOUN
ejpam-5057	186	12	+	+	NOUN
ejpam-5057	186	13	1	1	NUM
ejpam-5057	186	14	where	where	SCONJ
ejpam-5057	186	15	k	k	PROPN
ejpam-5057	186	16	=	=	SYM
ejpam-5057	186	17	0	0	NUM
ejpam-5057	186	18	,	,	PUNCT
ejpam-5057	186	19	1	1	NUM
ejpam-5057	186	20	,	,	PUNCT
ejpam-5057	186	21	2	2	NUM
ejpam-5057	186	22	,	,	PUNCT
ejpam-5057	186	23	.	.	PUNCT
ejpam-5057	186	24	.	.	PUNCT
ejpam-5057	186	25	.	.	PUNCT
ejpam-5057	186	26	.	.	PUNCT
ejpam-5057	187	1	let	let	VERB
ejpam-5057	187	2	η11	η11	NOUN
ejpam-5057	187	3	,	,	PUNCT
ejpam-5057	187	4	η12	η12	NOUN
ejpam-5057	187	5	and	and	CCONJ
ejpam-5057	187	6	η13	η13	NOUN
ejpam-5057	187	7	represent	represent	VERB
ejpam-5057	187	8	the	the	DET
ejpam-5057	187	9	largest	large	ADJ
ejpam-5057	187	10	eigenvalues	eigenvalue	NOUN
ejpam-5057	187	11	of	of	ADP
ejpam-5057	187	12	the	the	DET
ejpam-5057	187	13	bipartite	bipartite	PROPN
ejpam-5057	187	14	quotient	quotient	NOUN
ejpam-5057	187	15	matrices	matrix	NOUN
ejpam-5057	187	16	corresponding	correspond	VERB
ejpam-5057	187	17	to	to	ADP
ejpam-5057	187	18	the	the	DET
ejpam-5057	187	19	three	three	NUM
ejpam-5057	187	20	different	different	ADJ
ejpam-5057	187	21	bipartitions	bipartition	NOUN
ejpam-5057	187	22	.	.	PUNCT
ejpam-5057	188	1	let	let	VERB
ejpam-5057	188	2	η1	η1	NOUN
ejpam-5057	188	3	=	=	SYM
ejpam-5057	188	4	min{η11	min{η11	PROPN
ejpam-5057	188	5	,	,	PUNCT
ejpam-5057	188	6	η12	η12	NOUN
ejpam-5057	188	7	,	,	PUNCT
ejpam-5057	188	8	η13	η13	NOUN
ejpam-5057	188	9	}	}	PUNCT
ejpam-5057	188	10	.	.	PUNCT
ejpam-5057	189	1	let	let	VERB
ejpam-5057	189	2	η21	η21	PROPN
ejpam-5057	189	3	,	,	PUNCT
ejpam-5057	189	4	η22	η22	PROPN
ejpam-5057	189	5	and	and	CCONJ
ejpam-5057	189	6	η23	η23	PROPN
ejpam-5057	189	7	represent	represent	VERB
ejpam-5057	189	8	the	the	DET
ejpam-5057	189	9	smallest	small	ADJ
ejpam-5057	189	10	eigenvalues	eigenvalue	NOUN
ejpam-5057	189	11	of	of	ADP
ejpam-5057	189	12	the	the	DET
ejpam-5057	189	13	bipartite	bipartite	PROPN
ejpam-5057	189	14	quotient	quotient	NOUN
ejpam-5057	189	15	matrices	matrix	NOUN
ejpam-5057	189	16	corresponding	correspond	VERB
ejpam-5057	189	17	to	to	ADP
ejpam-5057	189	18	the	the	DET
ejpam-5057	189	19	three	three	NUM
ejpam-5057	189	20	different	different	ADJ
ejpam-5057	189	21	bipartitions	bipartition	NOUN
ejpam-5057	189	22	.	.	PUNCT
ejpam-5057	190	1	let	let	VERB
ejpam-5057	190	2	η2	η2	ADJ
ejpam-5057	190	3	=	=	SYM
ejpam-5057	190	4	max{η21	max{η21	NOUN
ejpam-5057	190	5	,	,	PUNCT
ejpam-5057	190	6	η22	η22	PROPN
ejpam-5057	190	7	,	,	PUNCT
ejpam-5057	190	8	η23	η23	NOUN
ejpam-5057	190	9	}	}	PUNCT
ejpam-5057	190	10	.	.	PUNCT
ejpam-5057	191	1	m.machasri	m.machasri	NUM
ejpam-5057	191	2	,	,	PUNCT
ejpam-5057	191	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	191	4	/	/	SYM
ejpam-5057	191	5	eur	eur	PROPN
ejpam-5057	191	6	.	.	PUNCT
ejpam-5057	192	1	j.	j.	PROPN
ejpam-5057	192	2	pure	pure	PROPN
ejpam-5057	192	3	appl	appl	PROPN
ejpam-5057	192	4	.	.	PROPN
ejpam-5057	192	5	math	math	PROPN
ejpam-5057	192	6	,	,	PUNCT
ejpam-5057	192	7	17	17	NUM
ejpam-5057	192	8	(	(	PUNCT
ejpam-5057	192	9	2	2	NUM
ejpam-5057	192	10	)	)	PUNCT
ejpam-5057	192	11	(	(	PUNCT
ejpam-5057	192	12	2024	2024	NUM
ejpam-5057	192	13	)	)	PUNCT
ejpam-5057	192	14	,	,	PUNCT
ejpam-5057	192	15	772	772	NUM
ejpam-5057	192	16	-	-	SYM
ejpam-5057	192	17	789	789	NUM
ejpam-5057	192	18	779	779	NUM
ejpam-5057	192	19	theorem	theorem	NOUN
ejpam-5057	192	20	2	2	NUM
ejpam-5057	192	21	.	.	PUNCT
ejpam-5057	193	1	let	let	VERB
ejpam-5057	193	2	g	g	PRON
ejpam-5057	193	3	be	be	AUX
ejpam-5057	193	4	a	a	DET
ejpam-5057	193	5	connected	connected	ADJ
ejpam-5057	193	6	bipartite	bipartite	NOUN
ejpam-5057	193	7	graph	graph	NOUN
ejpam-5057	193	8	of	of	ADP
ejpam-5057	193	9	order	order	NOUN
ejpam-5057	193	10	n.	n.	NOUN
ejpam-5057	193	11	then	then	ADV
ejpam-5057	193	12	λ2	λ2	PROPN
ejpam-5057	193	13	≤	≤	NUM
ejpam-5057	193	14			PUNCT
ejpam-5057	194	1	2(n−1)√	2(n−1)√	NUM
ejpam-5057	194	2	n2−1	n2−1	NOUN
ejpam-5057	194	3	n	n	NOUN
ejpam-5057	194	4	is	be	AUX
ejpam-5057	194	5	odd	odd	ADJ
ejpam-5057	194	6	2(n−1	2(n−1	ADJ
ejpam-5057	194	7	)	)	PUNCT
ejpam-5057	194	8	n	n	CCONJ
ejpam-5057	194	9	n	n	ADV
ejpam-5057	194	10	is	be	AUX
ejpam-5057	194	11	even	even	ADV
ejpam-5057	194	12	.	.	PUNCT
ejpam-5057	195	1	proof	proof	NOUN
ejpam-5057	195	2	.	.	PUNCT
ejpam-5057	196	1	from	from	ADP
ejpam-5057	196	2	theorem	theorem	NOUN
ejpam-5057	196	3	1	1	NUM
ejpam-5057	196	4	we	we	PRON
ejpam-5057	196	5	have	have	VERB
ejpam-5057	196	6	,	,	PUNCT
ejpam-5057	196	7	λ2	λ2	PROPN
ejpam-5057	196	8	≤	≤	NOUN
ejpam-5057	196	9	m	m	VERB
ejpam-5057	196	10	√	√	NUM
ejpam-5057	196	11	n1n2	n1n2	NOUN
ejpam-5057	196	12	.	.	PUNCT
ejpam-5057	197	1	(	(	PUNCT
ejpam-5057	197	2	2	2	X
ejpam-5057	197	3	)	)	PUNCT
ejpam-5057	197	4	consider	consider	VERB
ejpam-5057	197	5	the	the	DET
ejpam-5057	197	6	following	follow	VERB
ejpam-5057	197	7	cases	case	NOUN
ejpam-5057	197	8	.	.	PUNCT
ejpam-5057	198	1	case	case	NOUN
ejpam-5057	198	2	1	1	NUM
ejpam-5057	198	3	:	:	PUNCT
ejpam-5057	198	4	let	let	VERB
ejpam-5057	198	5	g	g	PRON
ejpam-5057	198	6	be	be	AUX
ejpam-5057	198	7	minimally	minimally	ADV
ejpam-5057	198	8	connected	connect	VERB
ejpam-5057	198	9	with	with	ADP
ejpam-5057	198	10	balanced	balanced	ADJ
ejpam-5057	198	11	bipartition	bipartition	NOUN
ejpam-5057	198	12	.	.	PUNCT
ejpam-5057	199	1	substituting	substitute	VERB
ejpam-5057	199	2	for	for	ADP
ejpam-5057	199	3	m	m	PROPN
ejpam-5057	199	4	,	,	PUNCT
ejpam-5057	199	5	n1	n1	NOUN
ejpam-5057	199	6	and	and	CCONJ
ejpam-5057	199	7	n2	n2	ADJ
ejpam-5057	199	8	for	for	ADP
ejpam-5057	199	9	a	a	DET
ejpam-5057	199	10	balanced	balanced	ADJ
ejpam-5057	199	11	bipartition	bipartition	NOUN
ejpam-5057	199	12	in	in	ADP
ejpam-5057	199	13	equation	equation	NOUN
ejpam-5057	199	14	(	(	PUNCT
ejpam-5057	199	15	2	2	NUM
ejpam-5057	199	16	)	)	PUNCT
ejpam-5057	199	17	,	,	PUNCT
ejpam-5057	199	18	we	we	PRON
ejpam-5057	199	19	have	have	VERB
ejpam-5057	199	20	η11	η11	NOUN
ejpam-5057	199	21	=	=	SYM
ejpam-5057	199	22			PUNCT
ejpam-5057	199	23	2(n−1)√	2(n−1)√	NUM
ejpam-5057	199	24	n2−1	n2−1	NOUN
ejpam-5057	199	25	n	n	NOUN
ejpam-5057	199	26	is	be	AUX
ejpam-5057	199	27	odd	odd	ADJ
ejpam-5057	199	28	2(n−1	2(n−1	ADJ
ejpam-5057	199	29	)	)	PUNCT
ejpam-5057	199	30	n	n	CCONJ
ejpam-5057	199	31	n	n	ADV
ejpam-5057	199	32	is	be	AUX
ejpam-5057	199	33	even	even	ADV
ejpam-5057	199	34	.	.	PUNCT
ejpam-5057	200	1	case	case	NOUN
ejpam-5057	200	2	2	2	NUM
ejpam-5057	200	3	:	:	PUNCT
ejpam-5057	200	4	let	let	VERB
ejpam-5057	200	5	g	g	PRON
ejpam-5057	200	6	be	be	AUX
ejpam-5057	200	7	minimally	minimally	ADV
ejpam-5057	200	8	connected	connect	VERB
ejpam-5057	200	9	with	with	ADP
ejpam-5057	200	10	unbalanced	unbalanced	ADJ
ejpam-5057	200	11	bipartition	bipartition	NOUN
ejpam-5057	200	12	.	.	PUNCT
ejpam-5057	201	1	substituting	substitute	VERB
ejpam-5057	201	2	for	for	ADP
ejpam-5057	201	3	m	m	PROPN
ejpam-5057	201	4	,	,	PUNCT
ejpam-5057	201	5	n1	n1	NOUN
ejpam-5057	201	6	and	and	CCONJ
ejpam-5057	201	7	n2	n2	NOUN
ejpam-5057	201	8	for	for	ADP
ejpam-5057	201	9	an	an	DET
ejpam-5057	201	10	unbalanced	unbalanced	ADJ
ejpam-5057	201	11	bipartition	bipartition	NOUN
ejpam-5057	201	12	in	in	ADP
ejpam-5057	201	13	equation	equation	NOUN
ejpam-5057	201	14	(	(	PUNCT
ejpam-5057	201	15	2	2	NUM
ejpam-5057	201	16	)	)	PUNCT
ejpam-5057	201	17	,	,	PUNCT
ejpam-5057	201	18	we	we	PRON
ejpam-5057	201	19	have	have	AUX
ejpam-5057	201	20	η12	η12	VERB
ejpam-5057	201	21	=	=	PUNCT
ejpam-5057	201	22	n−	n−	PROPN
ejpam-5057	201	23	1√	1√	PROPN
ejpam-5057	201	24	n−	n−	PROPN
ejpam-5057	201	25	1	1	NUM
ejpam-5057	201	26	=	=	SYM
ejpam-5057	201	27	√	√	PROPN
ejpam-5057	201	28	n−	n−	NOUN
ejpam-5057	201	29	1	1	NUM
ejpam-5057	201	30	.	.	PUNCT
ejpam-5057	201	31	case	case	NOUN
ejpam-5057	201	32	3	3	X
ejpam-5057	201	33	:	:	PUNCT
ejpam-5057	201	34	let	let	VERB
ejpam-5057	201	35	g	g	PRON
ejpam-5057	201	36	be	be	AUX
ejpam-5057	201	37	minimally	minimally	ADV
ejpam-5057	201	38	connected	connect	VERB
ejpam-5057	201	39	with	with	ADP
ejpam-5057	201	40	the	the	DET
ejpam-5057	201	41	average	average	ADJ
ejpam-5057	201	42	bipartition	bipartition	NOUN
ejpam-5057	201	43	.	.	PUNCT
ejpam-5057	202	1	substituting	substitute	VERB
ejpam-5057	202	2	for	for	ADP
ejpam-5057	202	3	m	m	PROPN
ejpam-5057	202	4	,	,	PUNCT
ejpam-5057	202	5	n1	n1	NOUN
ejpam-5057	202	6	and	and	CCONJ
ejpam-5057	202	7	n2	n2	NOUN
ejpam-5057	202	8	for	for	ADP
ejpam-5057	202	9	an	an	DET
ejpam-5057	202	10	average	average	ADJ
ejpam-5057	202	11	bipartition	bipartition	NOUN
ejpam-5057	202	12	in	in	ADP
ejpam-5057	202	13	equation	equation	NOUN
ejpam-5057	202	14	(	(	PUNCT
ejpam-5057	202	15	2	2	NUM
ejpam-5057	202	16	)	)	PUNCT
ejpam-5057	202	17	,	,	PUNCT
ejpam-5057	202	18	we	we	PRON
ejpam-5057	202	19	have	have	VERB
ejpam-5057	202	20	when	when	SCONJ
ejpam-5057	202	21	n	n	X
ejpam-5057	202	22	is	be	AUX
ejpam-5057	202	23	even	even	ADV
ejpam-5057	202	24	,	,	PUNCT
ejpam-5057	202	25	η13	η13	NOUN
ejpam-5057	202	26	=	=	PUNCT
ejpam-5057	202	27			NOUN
ejpam-5057	202	28	4(n−1)√	4(n−1)√	NUM
ejpam-5057	203	1	3n2	3n2	NUM
ejpam-5057	204	1	+	+	NOUN
ejpam-5057	204	2	8n−16	8n−16	NOUN
ejpam-5057	204	3	if	if	SCONJ
ejpam-5057	204	4	n	n	PRON
ejpam-5057	204	5	is	be	AUX
ejpam-5057	204	6	even	even	ADV
ejpam-5057	204	7	4(n−1)√	4(n−1)√	PROPN
ejpam-5057	204	8	3n2	3n2	NUM
ejpam-5057	204	9	+	+	NOUN
ejpam-5057	204	10	6n−9	6n−9	NOUN
ejpam-5057	204	11	if	if	SCONJ
ejpam-5057	204	12	n	n	NOUN
ejpam-5057	204	13	=	=	SYM
ejpam-5057	204	14	4k	4k	NUM
ejpam-5057	204	15	+	+	NOUN
ejpam-5057	204	16	3	3	NUM
ejpam-5057	204	17	where	where	SCONJ
ejpam-5057	204	18	k	k	PROPN
ejpam-5057	204	19	=	=	SYM
ejpam-5057	204	20	0	0	NUM
ejpam-5057	204	21	,	,	PUNCT
ejpam-5057	204	22	1	1	NUM
ejpam-5057	204	23	,	,	PUNCT
ejpam-5057	204	24	2	2	NUM
ejpam-5057	204	25	,	,	PUNCT
ejpam-5057	204	26	.	.	PUNCT
ejpam-5057	204	27	.	.	PUNCT
ejpam-5057	204	28	.	.	PUNCT
ejpam-5057	205	1	.	.	PUNCT
ejpam-5057	206	1	4(n−1)√	4(n−1)√	PROPN
ejpam-5057	206	2	3n2	3n2	NUM
ejpam-5057	206	3	+	+	NOUN
ejpam-5057	206	4	10n−25	10n−25	NOUN
ejpam-5057	206	5	if	if	SCONJ
ejpam-5057	206	6	n	n	NOUN
ejpam-5057	206	7	=	=	SYM
ejpam-5057	206	8	4k	4k	NOUN
ejpam-5057	206	9	+	+	NOUN
ejpam-5057	206	10	1	1	NUM
ejpam-5057	206	11	where	where	SCONJ
ejpam-5057	206	12	k	k	PROPN
ejpam-5057	206	13	=	=	SYM
ejpam-5057	206	14	0	0	NUM
ejpam-5057	206	15	,	,	PUNCT
ejpam-5057	206	16	1	1	NUM
ejpam-5057	206	17	,	,	PUNCT
ejpam-5057	206	18	2	2	NUM
ejpam-5057	206	19	,	,	PUNCT
ejpam-5057	206	20	.	.	PUNCT
ejpam-5057	206	21	.	.	PUNCT
ejpam-5057	206	22	.	.	PUNCT
ejpam-5057	206	23	.	.	PUNCT
ejpam-5057	207	1	when	when	SCONJ
ejpam-5057	207	2	n	n	X
ejpam-5057	207	3	is	be	AUX
ejpam-5057	207	4	even	even	ADV
ejpam-5057	207	5	,	,	PUNCT
ejpam-5057	207	6	comparing	compare	VERB
ejpam-5057	207	7	all	all	DET
ejpam-5057	207	8	the	the	DET
ejpam-5057	207	9	three	three	NUM
ejpam-5057	207	10	cases	case	NOUN
ejpam-5057	207	11	,	,	PUNCT
ejpam-5057	207	12	since	since	SCONJ
ejpam-5057	207	13	the	the	DET
ejpam-5057	207	14	numerator	numerator	NOUN
ejpam-5057	207	15	contains	contain	VERB
ejpam-5057	207	16	(	(	PUNCT
ejpam-5057	207	17	n−	n−	NOUN
ejpam-5057	207	18	1	1	NUM
ejpam-5057	207	19	)	)	PUNCT
ejpam-5057	207	20	as	as	ADP
ejpam-5057	207	21	a	a	DET
ejpam-5057	207	22	common	common	ADJ
ejpam-5057	207	23	term	term	NOUN
ejpam-5057	207	24	,	,	PUNCT
ejpam-5057	207	25	we	we	PRON
ejpam-5057	207	26	have	have	VERB
ejpam-5057	207	27	√	√	NUM
ejpam-5057	207	28	n−	n−	NOUN
ejpam-5057	207	29	1	1	NUM
ejpam-5057	207	30	≤	≤	NOUN
ejpam-5057	207	31	√	√	ADP
ejpam-5057	207	32	3n2	3n2	NUM
ejpam-5057	207	33	+	+	CCONJ
ejpam-5057	207	34	8n−	8n−	NUM
ejpam-5057	207	35	16	16	NUM
ejpam-5057	207	36	4	4	NUM
ejpam-5057	207	37	≤	≤	NOUN
ejpam-5057	207	38	n	n	PRON
ejpam-5057	207	39	2	2	NUM
ejpam-5057	207	40	.	.	PUNCT
ejpam-5057	208	1	which	which	PRON
ejpam-5057	208	2	implies	imply	VERB
ejpam-5057	208	3	that	that	SCONJ
ejpam-5057	208	4	2(n−	2(n−	NUM
ejpam-5057	208	5	1	1	NUM
ejpam-5057	208	6	)	)	PUNCT
ejpam-5057	208	7	n	n	PRON
ejpam-5057	208	8	≤	≤	NOUN
ejpam-5057	208	9	4(n−	4(n−	NUM
ejpam-5057	208	10	1)√	1)√	NUM
ejpam-5057	208	11	3n2	3n2	NUM
ejpam-5057	208	12	+	+	NUM
ejpam-5057	208	13	8n−	8n−	NUM
ejpam-5057	208	14	16	16	NUM
ejpam-5057	208	15	≤	≤	NUM
ejpam-5057	208	16	√	√	NUM
ejpam-5057	208	17	n−	n−	NOUN
ejpam-5057	208	18	1	1	NUM
ejpam-5057	208	19	.	.	PUNCT
ejpam-5057	208	20	when	when	SCONJ
ejpam-5057	208	21	n	n	X
ejpam-5057	208	22	is	be	AUX
ejpam-5057	208	23	odd	odd	ADJ
ejpam-5057	208	24	and	and	CCONJ
ejpam-5057	208	25	n	n	CCONJ
ejpam-5057	209	1	=	=	SYM
ejpam-5057	209	2	4k	4k	NUM
ejpam-5057	209	3	+	+	NOUN
ejpam-5057	209	4	3	3	NUM
ejpam-5057	209	5	where	where	SCONJ
ejpam-5057	209	6	k	k	PROPN
ejpam-5057	209	7	=	=	SYM
ejpam-5057	209	8	0	0	NUM
ejpam-5057	209	9	,	,	PUNCT
ejpam-5057	209	10	1	1	NUM
ejpam-5057	209	11	,	,	PUNCT
ejpam-5057	209	12	2	2	NUM
ejpam-5057	209	13	,	,	PUNCT
ejpam-5057	209	14	.	.	PUNCT
ejpam-5057	209	15	.	.	PUNCT
ejpam-5057	209	16	.	.	PUNCT
ejpam-5057	210	1	,	,	PUNCT
ejpam-5057	210	2	comparing	compare	VERB
ejpam-5057	210	3	all	all	DET
ejpam-5057	210	4	the	the	DET
ejpam-5057	210	5	three	three	NUM
ejpam-5057	210	6	cases	case	NOUN
ejpam-5057	210	7	,	,	PUNCT
ejpam-5057	210	8	since	since	SCONJ
ejpam-5057	210	9	the	the	DET
ejpam-5057	210	10	numerator	numerator	NOUN
ejpam-5057	210	11	contains	contain	VERB
ejpam-5057	210	12	(	(	PUNCT
ejpam-5057	210	13	n−	n−	NOUN
ejpam-5057	210	14	1	1	NUM
ejpam-5057	210	15	)	)	PUNCT
ejpam-5057	210	16	as	as	ADP
ejpam-5057	210	17	a	a	DET
ejpam-5057	210	18	common	common	ADJ
ejpam-5057	210	19	term	term	NOUN
ejpam-5057	210	20	,	,	PUNCT
ejpam-5057	210	21	we	we	PRON
ejpam-5057	210	22	have	have	VERB
ejpam-5057	210	23	√	√	NUM
ejpam-5057	210	24	n−	n−	NOUN
ejpam-5057	210	25	1	1	NUM
ejpam-5057	210	26	≤	≤	NOUN
ejpam-5057	210	27	√	√	ADP
ejpam-5057	210	28	3n2	3n2	NUM
ejpam-5057	211	1	+	+	CCONJ
ejpam-5057	212	1	6n−	6n−	NUM
ejpam-5057	212	2	9	9	NUM
ejpam-5057	212	3	4	4	NUM
ejpam-5057	212	4	≤	≤	NUM
ejpam-5057	212	5	√	√	ADJ
ejpam-5057	212	6	n2	n2	NOUN
ejpam-5057	212	7	−	−	PROPN
ejpam-5057	212	8	1	1	NUM
ejpam-5057	212	9	2	2	NUM
ejpam-5057	212	10	.	.	PUNCT
ejpam-5057	213	1	m.machasri	m.machasri	NUM
ejpam-5057	213	2	,	,	PUNCT
ejpam-5057	213	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	213	4	/	/	SYM
ejpam-5057	213	5	eur	eur	PROPN
ejpam-5057	213	6	.	.	PUNCT
ejpam-5057	214	1	j.	j.	PROPN
ejpam-5057	214	2	pure	pure	PROPN
ejpam-5057	214	3	appl	appl	PROPN
ejpam-5057	214	4	.	.	PROPN
ejpam-5057	214	5	math	math	PROPN
ejpam-5057	214	6	,	,	PUNCT
ejpam-5057	214	7	17	17	NUM
ejpam-5057	214	8	(	(	PUNCT
ejpam-5057	214	9	2	2	NUM
ejpam-5057	214	10	)	)	PUNCT
ejpam-5057	214	11	(	(	PUNCT
ejpam-5057	214	12	2024	2024	NUM
ejpam-5057	214	13	)	)	PUNCT
ejpam-5057	214	14	,	,	PUNCT
ejpam-5057	214	15	772	772	NUM
ejpam-5057	214	16	-	-	SYM
ejpam-5057	214	17	789	789	NUM
ejpam-5057	214	18	780	780	NUM
ejpam-5057	214	19	which	which	PRON
ejpam-5057	214	20	implies	imply	VERB
ejpam-5057	214	21	that	that	SCONJ
ejpam-5057	214	22	2(n−	2(n−	NUM
ejpam-5057	214	23	1)√	1)√	NUM
ejpam-5057	214	24	n2	n2	NOUN
ejpam-5057	214	25	−	−	PROPN
ejpam-5057	214	26	1	1	NUM
ejpam-5057	214	27	≤	≤	NUM
ejpam-5057	214	28	4(n−	4(n−	NUM
ejpam-5057	215	1	1)√	1)√	NUM
ejpam-5057	215	2	3n2	3n2	NUM
ejpam-5057	215	3	+	+	CCONJ
ejpam-5057	215	4	6n−	6n−	NUM
ejpam-5057	215	5	9	9	NUM
ejpam-5057	215	6	≤	≤	NOUN
ejpam-5057	215	7	√	√	NUM
ejpam-5057	215	8	n−	n−	NOUN
ejpam-5057	215	9	1	1	NUM
ejpam-5057	215	10	.	.	PUNCT
ejpam-5057	216	1	when	when	SCONJ
ejpam-5057	216	2	n	n	X
ejpam-5057	216	3	is	be	AUX
ejpam-5057	216	4	odd	odd	ADJ
ejpam-5057	216	5	and	and	CCONJ
ejpam-5057	216	6	n	n	CCONJ
ejpam-5057	216	7	=	=	SYM
ejpam-5057	216	8	4k	4k	NOUN
ejpam-5057	216	9	+	+	NOUN
ejpam-5057	216	10	1	1	NUM
ejpam-5057	216	11	where	where	SCONJ
ejpam-5057	216	12	k	k	PROPN
ejpam-5057	216	13	=	=	SYM
ejpam-5057	216	14	0	0	NUM
ejpam-5057	216	15	,	,	PUNCT
ejpam-5057	216	16	1	1	NUM
ejpam-5057	216	17	,	,	PUNCT
ejpam-5057	216	18	2	2	NUM
ejpam-5057	216	19	,	,	PUNCT
ejpam-5057	216	20	.	.	PUNCT
ejpam-5057	216	21	.	.	PUNCT
ejpam-5057	216	22	.	.	PUNCT
ejpam-5057	217	1	,	,	PUNCT
ejpam-5057	217	2	comparing	compare	VERB
ejpam-5057	217	3	all	all	DET
ejpam-5057	217	4	the	the	DET
ejpam-5057	217	5	three	three	NUM
ejpam-5057	217	6	cases	case	NOUN
ejpam-5057	217	7	,	,	PUNCT
ejpam-5057	217	8	since	since	SCONJ
ejpam-5057	217	9	the	the	DET
ejpam-5057	217	10	numerator	numerator	NOUN
ejpam-5057	217	11	contains	contain	VERB
ejpam-5057	217	12	(	(	PUNCT
ejpam-5057	217	13	n−	n−	NOUN
ejpam-5057	217	14	1	1	NUM
ejpam-5057	217	15	)	)	PUNCT
ejpam-5057	217	16	as	as	ADP
ejpam-5057	217	17	a	a	DET
ejpam-5057	217	18	common	common	ADJ
ejpam-5057	217	19	term	term	NOUN
ejpam-5057	217	20	,	,	PUNCT
ejpam-5057	217	21	we	we	PRON
ejpam-5057	217	22	have	have	VERB
ejpam-5057	217	23	√	√	NUM
ejpam-5057	217	24	n−	n−	NOUN
ejpam-5057	217	25	1	1	NUM
ejpam-5057	217	26	≤	≤	NOUN
ejpam-5057	217	27	√	√	ADP
ejpam-5057	217	28	3n2	3n2	NUM
ejpam-5057	218	1	+	+	CCONJ
ejpam-5057	218	2	10n−	10n−	NUM
ejpam-5057	218	3	25	25	NUM
ejpam-5057	218	4	4	4	NUM
ejpam-5057	218	5	≤	≤	NUM
ejpam-5057	218	6	√	√	ADJ
ejpam-5057	218	7	n2	n2	NOUN
ejpam-5057	218	8	−	−	PROPN
ejpam-5057	218	9	1	1	NUM
ejpam-5057	218	10	2	2	NUM
ejpam-5057	218	11	.	.	PUNCT
ejpam-5057	219	1	which	which	PRON
ejpam-5057	219	2	implies	imply	VERB
ejpam-5057	219	3	that	that	SCONJ
ejpam-5057	219	4	2(n−	2(n−	NUM
ejpam-5057	219	5	1)√	1)√	NUM
ejpam-5057	219	6	n2	n2	NOUN
ejpam-5057	219	7	−	−	PROPN
ejpam-5057	219	8	1	1	NUM
ejpam-5057	219	9	≤	≤	NUM
ejpam-5057	219	10	4(n−	4(n−	NUM
ejpam-5057	219	11	1)√	1)√	NUM
ejpam-5057	219	12	3n2	3n2	NUM
ejpam-5057	219	13	+	+	CCONJ
ejpam-5057	219	14	10n−	10n−	NUM
ejpam-5057	219	15	25	25	NUM
ejpam-5057	219	16	≤	≤	NUM
ejpam-5057	219	17	√	√	NUM
ejpam-5057	219	18	n−	n−	NOUN
ejpam-5057	219	19	1	1	NUM
ejpam-5057	219	20	.	.	PUNCT
ejpam-5057	219	21	comparing	compare	VERB
ejpam-5057	219	22	all	all	DET
ejpam-5057	219	23	the	the	DET
ejpam-5057	219	24	three	three	NUM
ejpam-5057	219	25	cases	case	NOUN
ejpam-5057	219	26	,	,	PUNCT
ejpam-5057	219	27	we	we	PRON
ejpam-5057	219	28	get	get	VERB
ejpam-5057	219	29	η11	η11	ADJ
ejpam-5057	219	30	≤	≤	NOUN
ejpam-5057	219	31	η13	η13	NOUN
ejpam-5057	219	32	≤	≤	NUM
ejpam-5057	219	33	η12	η12	NOUN
ejpam-5057	219	34	.	.	PUNCT
ejpam-5057	220	1	from	from	ADP
ejpam-5057	220	2	this	this	PRON
ejpam-5057	220	3	we	we	PRON
ejpam-5057	220	4	have	have	VERB
ejpam-5057	220	5	,	,	PUNCT
ejpam-5057	220	6	η1	η1	NOUN
ejpam-5057	220	7	=	=	SYM
ejpam-5057	220	8	η11	η11	NOUN
ejpam-5057	220	9	.	.	PUNCT
ejpam-5057	221	1	applying	apply	VERB
ejpam-5057	221	2	this	this	PRON
ejpam-5057	221	3	in	in	ADP
ejpam-5057	221	4	equation	equation	NOUN
ejpam-5057	221	5	(	(	PUNCT
ejpam-5057	221	6	2	2	NUM
ejpam-5057	221	7	)	)	PUNCT
ejpam-5057	221	8	,	,	PUNCT
ejpam-5057	221	9	we	we	PRON
ejpam-5057	221	10	get	get	VERB
ejpam-5057	221	11	a	a	DET
ejpam-5057	221	12	tight	tight	ADJ
ejpam-5057	221	13	upper	upper	ADJ
ejpam-5057	221	14	bound	bind	VERB
ejpam-5057	221	15	in	in	ADP
ejpam-5057	221	16	terms	term	NOUN
ejpam-5057	221	17	of	of	ADP
ejpam-5057	221	18	n	n	PRON
ejpam-5057	221	19	as	as	ADP
ejpam-5057	221	20	λ2	λ2	NOUN
ejpam-5057	221	21	≤	≤	NOUN
ejpam-5057	221	22			PUNCT
ejpam-5057	222	1	2(n−1)√	2(n−1)√	NUM
ejpam-5057	222	2	n2−1	n2−1	NOUN
ejpam-5057	222	3	n	n	NOUN
ejpam-5057	222	4	is	be	AUX
ejpam-5057	222	5	odd	odd	ADJ
ejpam-5057	222	6	2(n−1	2(n−1	ADJ
ejpam-5057	222	7	)	)	PUNCT
ejpam-5057	222	8	n	n	CCONJ
ejpam-5057	222	9	n	n	ADV
ejpam-5057	222	10	is	be	AUX
ejpam-5057	222	11	even	even	ADV
ejpam-5057	222	12	.	.	PUNCT
ejpam-5057	223	1	theorem	theorem	NOUN
ejpam-5057	223	2	3	3	X
ejpam-5057	223	3	.	.	PUNCT
ejpam-5057	224	1	let	let	VERB
ejpam-5057	224	2	g	g	PRON
ejpam-5057	224	3	be	be	AUX
ejpam-5057	224	4	a	a	DET
ejpam-5057	224	5	connected	connected	ADJ
ejpam-5057	224	6	bipartite	bipartite	NOUN
ejpam-5057	224	7	graph	graph	NOUN
ejpam-5057	224	8	of	of	ADP
ejpam-5057	224	9	order	order	NOUN
ejpam-5057	224	10	n.	n.	NOUN
ejpam-5057	224	11	then	then	ADV
ejpam-5057	224	12	λn−1	λn−1	PROPN
ejpam-5057	224	13	≥	≥	NUM
ejpam-5057	224	14			PUNCT
ejpam-5057	224	15	−(2(n−1))√	−(2(n−1))√	NOUN
ejpam-5057	224	16	n2−1	n2−1	NOUN
ejpam-5057	224	17	n	n	PROPN
ejpam-5057	224	18	is	be	AUX
ejpam-5057	224	19	odd	odd	ADJ
ejpam-5057	224	20	−(2(n−1	−(2(n−1	NUM
ejpam-5057	224	21	)	)	PUNCT
ejpam-5057	224	22	)	)	PUNCT
ejpam-5057	225	1	n	n	CCONJ
ejpam-5057	225	2	n	n	ADV
ejpam-5057	225	3	is	be	AUX
ejpam-5057	225	4	even	even	ADV
ejpam-5057	225	5	.	.	PUNCT
ejpam-5057	226	1	proof	proof	NOUN
ejpam-5057	226	2	.	.	PUNCT
ejpam-5057	227	1	in	in	ADP
ejpam-5057	227	2	theorem	theorem	NOUN
ejpam-5057	227	3	1	1	NUM
ejpam-5057	227	4	,	,	PUNCT
ejpam-5057	227	5	we	we	PRON
ejpam-5057	227	6	have	have	AUX
ejpam-5057	227	7	proved	prove	VERB
ejpam-5057	227	8	that	that	SCONJ
ejpam-5057	228	1	λn−1	λn−1	PROPN
ejpam-5057	228	2	≥	≥	PUNCT
ejpam-5057	228	3	−m	−m	NOUN
ejpam-5057	228	4	√	√	NUM
ejpam-5057	228	5	n1n2	n1n2	NOUN
ejpam-5057	228	6	.	.	PUNCT
ejpam-5057	229	1	(	(	PUNCT
ejpam-5057	229	2	3	3	X
ejpam-5057	229	3	)	)	PUNCT
ejpam-5057	229	4	consider	consider	VERB
ejpam-5057	229	5	the	the	DET
ejpam-5057	229	6	following	follow	VERB
ejpam-5057	229	7	cases	case	NOUN
ejpam-5057	229	8	.	.	PUNCT
ejpam-5057	230	1	case	case	NOUN
ejpam-5057	230	2	1	1	NUM
ejpam-5057	230	3	:	:	PUNCT
ejpam-5057	230	4	g	g	PROPN
ejpam-5057	230	5	is	be	AUX
ejpam-5057	230	6	minimally	minimally	ADV
ejpam-5057	230	7	connected	connect	VERB
ejpam-5057	230	8	with	with	ADP
ejpam-5057	230	9	balanced	balanced	ADJ
ejpam-5057	230	10	bipartition	bipartition	NOUN
ejpam-5057	230	11	.	.	PUNCT
ejpam-5057	231	1	substituting	substitute	VERB
ejpam-5057	231	2	for	for	ADP
ejpam-5057	231	3	m	m	PROPN
ejpam-5057	231	4	,	,	PUNCT
ejpam-5057	231	5	n1	n1	NOUN
ejpam-5057	231	6	and	and	CCONJ
ejpam-5057	231	7	n2	n2	ADJ
ejpam-5057	231	8	for	for	ADP
ejpam-5057	231	9	a	a	DET
ejpam-5057	231	10	balanced	balanced	ADJ
ejpam-5057	231	11	bipartition	bipartition	NOUN
ejpam-5057	231	12	in	in	ADP
ejpam-5057	231	13	equation	equation	NOUN
ejpam-5057	231	14	(	(	PUNCT
ejpam-5057	231	15	3	3	NUM
ejpam-5057	231	16	)	)	PUNCT
ejpam-5057	231	17	,	,	PUNCT
ejpam-5057	231	18	we	we	PRON
ejpam-5057	231	19	have	have	VERB
ejpam-5057	231	20	η21	η21	NOUN
ejpam-5057	231	21	=	=	SYM
ejpam-5057	231	22			X
ejpam-5057	231	23	−(4(n−1))√	−(4(n−1))√	X
ejpam-5057	232	1	3n2	3n2	NUM
ejpam-5057	232	2	+	+	NOUN
ejpam-5057	232	3	8n−16	8n−16	NOUN
ejpam-5057	232	4	if	if	SCONJ
ejpam-5057	232	5	n	n	PRON
ejpam-5057	232	6	is	be	AUX
ejpam-5057	232	7	even	even	ADV
ejpam-5057	232	8	−(4(n−1))√	−(4(n−1))√	X
ejpam-5057	233	1	3n2	3n2	NUM
ejpam-5057	233	2	+	+	NOUN
ejpam-5057	233	3	6n−9	6n−9	NOUN
ejpam-5057	233	4	if	if	SCONJ
ejpam-5057	233	5	n	n	NOUN
ejpam-5057	233	6	=	=	SYM
ejpam-5057	233	7	4k	4k	NUM
ejpam-5057	233	8	+	+	NOUN
ejpam-5057	233	9	3	3	NUM
ejpam-5057	233	10	where	where	SCONJ
ejpam-5057	233	11	k	k	PROPN
ejpam-5057	233	12	=	=	SYM
ejpam-5057	233	13	0	0	NUM
ejpam-5057	233	14	,	,	PUNCT
ejpam-5057	233	15	1	1	NUM
ejpam-5057	233	16	,	,	PUNCT
ejpam-5057	233	17	2	2	NUM
ejpam-5057	233	18	,	,	PUNCT
ejpam-5057	233	19	.	.	PUNCT
ejpam-5057	233	20	.	.	PUNCT
ejpam-5057	233	21	.	.	PUNCT
ejpam-5057	234	1	.	.	PUNCT
ejpam-5057	235	1	−(4(n−1))√	−(4(n−1))√	X
ejpam-5057	236	1	3n2	3n2	NUM
ejpam-5057	237	1	+	+	NOUN
ejpam-5057	237	2	10n−25	10n−25	NOUN
ejpam-5057	237	3	if	if	SCONJ
ejpam-5057	237	4	n	n	NOUN
ejpam-5057	237	5	=	=	SYM
ejpam-5057	237	6	4k	4k	NOUN
ejpam-5057	237	7	+	+	NOUN
ejpam-5057	237	8	1	1	NUM
ejpam-5057	237	9	where	where	SCONJ
ejpam-5057	237	10	k	k	PROPN
ejpam-5057	237	11	=	=	SYM
ejpam-5057	237	12	0	0	NUM
ejpam-5057	237	13	,	,	PUNCT
ejpam-5057	237	14	1	1	NUM
ejpam-5057	237	15	,	,	PUNCT
ejpam-5057	237	16	2	2	NUM
ejpam-5057	237	17	,	,	PUNCT
ejpam-5057	237	18	.	.	PUNCT
ejpam-5057	237	19	.	.	PUNCT
ejpam-5057	237	20	.	.	PUNCT
ejpam-5057	237	21	.	.	PUNCT
ejpam-5057	238	1	case	case	NOUN
ejpam-5057	238	2	2	2	NUM
ejpam-5057	238	3	:	:	PUNCT
ejpam-5057	238	4	g	g	PROPN
ejpam-5057	238	5	is	be	AUX
ejpam-5057	238	6	minimally	minimally	ADV
ejpam-5057	238	7	connected	connect	VERB
ejpam-5057	238	8	with	with	ADP
ejpam-5057	238	9	unbalanced	unbalanced	ADJ
ejpam-5057	238	10	bipartition	bipartition	NOUN
ejpam-5057	238	11	.	.	PUNCT
ejpam-5057	239	1	substituting	substitute	VERB
ejpam-5057	239	2	for	for	ADP
ejpam-5057	239	3	m	m	PROPN
ejpam-5057	239	4	,	,	PUNCT
ejpam-5057	239	5	n1	n1	NOUN
ejpam-5057	239	6	and	and	CCONJ
ejpam-5057	239	7	n2	n2	NOUN
ejpam-5057	239	8	for	for	ADP
ejpam-5057	239	9	an	an	DET
ejpam-5057	239	10	unbalanced	unbalanced	ADJ
ejpam-5057	239	11	bipartition	bipartition	NOUN
ejpam-5057	239	12	in	in	ADP
ejpam-5057	239	13	equation	equation	NOUN
ejpam-5057	239	14	(	(	PUNCT
ejpam-5057	239	15	3	3	NUM
ejpam-5057	239	16	)	)	PUNCT
ejpam-5057	239	17	,	,	PUNCT
ejpam-5057	239	18	we	we	PRON
ejpam-5057	239	19	have	have	VERB
ejpam-5057	239	20	η22	η22	NOUN
ejpam-5057	239	21	=	=	NOUN
ejpam-5057	239	22	−(n−	−(n−	ADJ
ejpam-5057	239	23	1)√	1)√	NUM
ejpam-5057	239	24	n−	n−	NOUN
ejpam-5057	239	25	1	1	NUM
ejpam-5057	239	26	=	=	SYM
ejpam-5057	239	27	−	−	NOUN
ejpam-5057	239	28	√	√	NUM
ejpam-5057	239	29	n−	n−	NOUN
ejpam-5057	239	30	1	1	NUM
ejpam-5057	239	31	.	.	PUNCT
ejpam-5057	240	1	m.machasri	m.machasri	NUM
ejpam-5057	240	2	,	,	PUNCT
ejpam-5057	240	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	240	4	/	/	SYM
ejpam-5057	240	5	eur	eur	PROPN
ejpam-5057	240	6	.	.	PUNCT
ejpam-5057	241	1	j.	j.	PROPN
ejpam-5057	241	2	pure	pure	PROPN
ejpam-5057	241	3	appl	appl	PROPN
ejpam-5057	241	4	.	.	PROPN
ejpam-5057	241	5	math	math	PROPN
ejpam-5057	241	6	,	,	PUNCT
ejpam-5057	241	7	17	17	NUM
ejpam-5057	241	8	(	(	PUNCT
ejpam-5057	241	9	2	2	NUM
ejpam-5057	241	10	)	)	PUNCT
ejpam-5057	241	11	(	(	PUNCT
ejpam-5057	241	12	2024	2024	NUM
ejpam-5057	241	13	)	)	PUNCT
ejpam-5057	241	14	,	,	PUNCT
ejpam-5057	241	15	772	772	NUM
ejpam-5057	241	16	-	-	SYM
ejpam-5057	241	17	789	789	NUM
ejpam-5057	241	18	781	781	NUM
ejpam-5057	241	19	case	case	NOUN
ejpam-5057	241	20	3	3	NUM
ejpam-5057	241	21	:	:	PUNCT
ejpam-5057	241	22	g	g	PROPN
ejpam-5057	241	23	is	be	AUX
ejpam-5057	241	24	minimally	minimally	ADV
ejpam-5057	241	25	connected	connect	VERB
ejpam-5057	241	26	with	with	ADP
ejpam-5057	241	27	average	average	ADJ
ejpam-5057	241	28	bipartition	bipartition	NOUN
ejpam-5057	241	29	.	.	PUNCT
ejpam-5057	242	1	substituting	substitute	VERB
ejpam-5057	242	2	for	for	ADP
ejpam-5057	242	3	m	m	PROPN
ejpam-5057	242	4	,	,	PUNCT
ejpam-5057	242	5	n1	n1	NOUN
ejpam-5057	242	6	and	and	CCONJ
ejpam-5057	242	7	n2	n2	NOUN
ejpam-5057	242	8	for	for	ADP
ejpam-5057	242	9	an	an	DET
ejpam-5057	242	10	average	average	ADJ
ejpam-5057	242	11	bipartition	bipartition	NOUN
ejpam-5057	242	12	in	in	ADP
ejpam-5057	242	13	equation	equation	NOUN
ejpam-5057	242	14	(	(	PUNCT
ejpam-5057	242	15	3	3	NUM
ejpam-5057	242	16	)	)	PUNCT
ejpam-5057	242	17	,	,	PUNCT
ejpam-5057	242	18	we	we	PRON
ejpam-5057	242	19	have	have	VERB
ejpam-5057	242	20	η23	η23	VERB
ejpam-5057	242	21	=	=	SYM
ejpam-5057	242	22			X
ejpam-5057	242	23	−(2(n−1	−(2(n−1	NUM
ejpam-5057	242	24	)	)	PUNCT
ejpam-5057	242	25	)	)	PUNCT
ejpam-5057	243	1	n	n	CCONJ
ejpam-5057	243	2	n	n	ADV
ejpam-5057	243	3	is	be	AUX
ejpam-5057	243	4	even	even	ADV
ejpam-5057	243	5	−(2(n−1))√	−(2(n−1))√	X
ejpam-5057	243	6	n2−1	n2−1	NOUN
ejpam-5057	243	7	n	n	PROPN
ejpam-5057	243	8	is	be	AUX
ejpam-5057	243	9	odd	odd	ADJ
ejpam-5057	243	10	.	.	PUNCT
ejpam-5057	244	1	when	when	SCONJ
ejpam-5057	244	2	n	n	X
ejpam-5057	244	3	is	be	AUX
ejpam-5057	244	4	even	even	ADV
ejpam-5057	244	5	,	,	PUNCT
ejpam-5057	244	6	comparing	compare	VERB
ejpam-5057	244	7	all	all	DET
ejpam-5057	244	8	the	the	DET
ejpam-5057	244	9	three	three	NUM
ejpam-5057	244	10	cases	case	NOUN
ejpam-5057	244	11	,	,	PUNCT
ejpam-5057	244	12	since	since	SCONJ
ejpam-5057	244	13	the	the	DET
ejpam-5057	244	14	numerator	numerator	NOUN
ejpam-5057	244	15	contains	contain	VERB
ejpam-5057	244	16	(	(	PUNCT
ejpam-5057	244	17	n−	n−	NOUN
ejpam-5057	244	18	1	1	NUM
ejpam-5057	244	19	)	)	PUNCT
ejpam-5057	244	20	as	as	ADP
ejpam-5057	244	21	a	a	DET
ejpam-5057	244	22	common	common	ADJ
ejpam-5057	244	23	term	term	NOUN
ejpam-5057	244	24	,	,	PUNCT
ejpam-5057	244	25	we	we	PRON
ejpam-5057	244	26	have	have	VERB
ejpam-5057	244	27	√	√	NUM
ejpam-5057	244	28	n−	n−	NOUN
ejpam-5057	244	29	1	1	NUM
ejpam-5057	244	30	≤	≤	NOUN
ejpam-5057	244	31	√	√	ADP
ejpam-5057	244	32	3n2	3n2	NUM
ejpam-5057	244	33	+	+	CCONJ
ejpam-5057	244	34	8n−	8n−	NUM
ejpam-5057	244	35	16	16	NUM
ejpam-5057	244	36	4	4	NUM
ejpam-5057	244	37	≤	≤	NOUN
ejpam-5057	244	38	n	n	PRON
ejpam-5057	244	39	2	2	NUM
ejpam-5057	244	40	.	.	PUNCT
ejpam-5057	245	1	which	which	PRON
ejpam-5057	245	2	implies	imply	VERB
ejpam-5057	245	3	that	that	SCONJ
ejpam-5057	245	4	−(2(n−	−(2(n−	ADJ
ejpam-5057	245	5	1	1	NUM
ejpam-5057	245	6	)	)	PUNCT
ejpam-5057	245	7	)	)	PUNCT
ejpam-5057	246	1	n	n	PRON
ejpam-5057	246	2	≥	≥	NOUN
ejpam-5057	246	3	−(4(n−	−(4(n−	NUM
ejpam-5057	246	4	1))√	1))√	NUM
ejpam-5057	246	5	3n2	3n2	NUM
ejpam-5057	247	1	+	+	CCONJ
ejpam-5057	247	2	8n−	8n−	NUM
ejpam-5057	247	3	16	16	NUM
ejpam-5057	247	4	≥	≥	NOUN
ejpam-5057	247	5	−	−	ADP
ejpam-5057	247	6	√	√	NUM
ejpam-5057	247	7	n−	n−	NOUN
ejpam-5057	247	8	1	1	NUM
ejpam-5057	247	9	.	.	PUNCT
ejpam-5057	248	1	when	when	SCONJ
ejpam-5057	248	2	n	n	X
ejpam-5057	248	3	=	=	SYM
ejpam-5057	248	4	4k+3	4k+3	PROPN
ejpam-5057	248	5	where	where	SCONJ
ejpam-5057	248	6	k	k	PROPN
ejpam-5057	248	7	=	=	SYM
ejpam-5057	248	8	0	0	NUM
ejpam-5057	248	9	,	,	PUNCT
ejpam-5057	248	10	1	1	NUM
ejpam-5057	248	11	,	,	PUNCT
ejpam-5057	248	12	2	2	NUM
ejpam-5057	248	13	,	,	PUNCT
ejpam-5057	248	14	.	.	PUNCT
ejpam-5057	248	15	.	.	PUNCT
ejpam-5057	248	16	.	.	PUNCT
ejpam-5057	249	1	,	,	PUNCT
ejpam-5057	249	2	comparing	compare	VERB
ejpam-5057	249	3	all	all	DET
ejpam-5057	249	4	the	the	DET
ejpam-5057	249	5	three	three	NUM
ejpam-5057	249	6	cases	case	NOUN
ejpam-5057	249	7	,	,	PUNCT
ejpam-5057	249	8	since	since	SCONJ
ejpam-5057	249	9	the	the	DET
ejpam-5057	249	10	numerator	numerator	NOUN
ejpam-5057	249	11	contains	contain	VERB
ejpam-5057	249	12	(	(	PUNCT
ejpam-5057	249	13	n−	n−	NOUN
ejpam-5057	249	14	1	1	NUM
ejpam-5057	249	15	)	)	PUNCT
ejpam-5057	249	16	as	as	ADP
ejpam-5057	249	17	a	a	DET
ejpam-5057	249	18	common	common	ADJ
ejpam-5057	249	19	term	term	NOUN
ejpam-5057	249	20	,	,	PUNCT
ejpam-5057	249	21	we	we	PRON
ejpam-5057	249	22	have	have	VERB
ejpam-5057	249	23	√	√	NUM
ejpam-5057	249	24	n−	n−	NOUN
ejpam-5057	249	25	1	1	NUM
ejpam-5057	249	26	≤	≤	NOUN
ejpam-5057	249	27	√	√	ADP
ejpam-5057	249	28	3n2	3n2	NUM
ejpam-5057	250	1	+	+	CCONJ
ejpam-5057	251	1	6n−	6n−	NUM
ejpam-5057	251	2	9	9	NUM
ejpam-5057	251	3	4	4	NUM
ejpam-5057	251	4	≤	≤	NUM
ejpam-5057	251	5	√	√	ADJ
ejpam-5057	251	6	n2	n2	NOUN
ejpam-5057	251	7	−	−	PROPN
ejpam-5057	251	8	1	1	NUM
ejpam-5057	251	9	2	2	NUM
ejpam-5057	251	10	.	.	PUNCT
ejpam-5057	252	1	which	which	PRON
ejpam-5057	252	2	implies	imply	VERB
ejpam-5057	252	3	that	that	SCONJ
ejpam-5057	252	4	2(n−	2(n−	NUM
ejpam-5057	252	5	1)√	1)√	NUM
ejpam-5057	252	6	n2	n2	NOUN
ejpam-5057	252	7	−	−	PROPN
ejpam-5057	252	8	1	1	NUM
ejpam-5057	252	9	≤	≤	NUM
ejpam-5057	252	10	4(n−	4(n−	NUM
ejpam-5057	252	11	1)√	1)√	NUM
ejpam-5057	252	12	3n2	3n2	NUM
ejpam-5057	252	13	+	+	CCONJ
ejpam-5057	252	14	6n−	6n−	NUM
ejpam-5057	252	15	9	9	NUM
ejpam-5057	252	16	≤	≤	NOUN
ejpam-5057	252	17	√	√	NUM
ejpam-5057	252	18	n−	n−	NOUN
ejpam-5057	252	19	1	1	NUM
ejpam-5057	252	20	.	.	PUNCT
ejpam-5057	252	21	when	when	SCONJ
ejpam-5057	252	22	n	n	PROPN
ejpam-5057	252	23	=	=	SYM
ejpam-5057	252	24	4k+1	4k+1	PROPN
ejpam-5057	252	25	where	where	SCONJ
ejpam-5057	252	26	k	k	PROPN
ejpam-5057	252	27	=	=	SYM
ejpam-5057	252	28	0	0	NUM
ejpam-5057	252	29	,	,	PUNCT
ejpam-5057	252	30	1	1	NUM
ejpam-5057	252	31	,	,	PUNCT
ejpam-5057	252	32	2	2	NUM
ejpam-5057	252	33	,	,	PUNCT
ejpam-5057	252	34	.	.	PUNCT
ejpam-5057	253	1	.	.	PUNCT
ejpam-5057	254	1	.	.	PUNCT
ejpam-5057	255	1	,	,	PUNCT
ejpam-5057	255	2	comparing	compare	VERB
ejpam-5057	255	3	all	all	DET
ejpam-5057	255	4	the	the	DET
ejpam-5057	255	5	three	three	NUM
ejpam-5057	255	6	cases	case	NOUN
ejpam-5057	255	7	,	,	PUNCT
ejpam-5057	255	8	since	since	SCONJ
ejpam-5057	255	9	the	the	DET
ejpam-5057	255	10	numerator	numerator	NOUN
ejpam-5057	255	11	contains	contain	VERB
ejpam-5057	255	12	(	(	PUNCT
ejpam-5057	255	13	n−	n−	NOUN
ejpam-5057	255	14	1	1	NUM
ejpam-5057	255	15	)	)	PUNCT
ejpam-5057	255	16	as	as	ADP
ejpam-5057	255	17	a	a	DET
ejpam-5057	255	18	common	common	ADJ
ejpam-5057	255	19	term	term	NOUN
ejpam-5057	255	20	,	,	PUNCT
ejpam-5057	255	21	we	we	PRON
ejpam-5057	255	22	have	have	VERB
ejpam-5057	255	23	√	√	NUM
ejpam-5057	255	24	n−	n−	NOUN
ejpam-5057	255	25	1	1	NUM
ejpam-5057	255	26	≤	≤	NOUN
ejpam-5057	255	27	√	√	ADP
ejpam-5057	255	28	3n2	3n2	NUM
ejpam-5057	256	1	+	+	CCONJ
ejpam-5057	256	2	10n−	10n−	NUM
ejpam-5057	256	3	25	25	NUM
ejpam-5057	256	4	4	4	NUM
ejpam-5057	256	5	≤	≤	NUM
ejpam-5057	256	6	√	√	ADJ
ejpam-5057	256	7	n2	n2	NOUN
ejpam-5057	256	8	−	−	PROPN
ejpam-5057	256	9	1	1	NUM
ejpam-5057	256	10	2	2	NUM
ejpam-5057	256	11	.	.	PUNCT
ejpam-5057	257	1	which	which	PRON
ejpam-5057	257	2	implies	imply	VERB
ejpam-5057	257	3	that	that	SCONJ
ejpam-5057	257	4	2(n−	2(n−	NUM
ejpam-5057	257	5	1)√	1)√	NUM
ejpam-5057	257	6	n2	n2	NOUN
ejpam-5057	257	7	−	−	PROPN
ejpam-5057	257	8	1	1	NUM
ejpam-5057	257	9	≤	≤	NUM
ejpam-5057	257	10	4(n−	4(n−	NUM
ejpam-5057	257	11	1)√	1)√	NUM
ejpam-5057	257	12	3n2	3n2	NUM
ejpam-5057	257	13	+	+	CCONJ
ejpam-5057	257	14	10n−	10n−	NUM
ejpam-5057	257	15	25	25	NUM
ejpam-5057	257	16	≤	≤	NUM
ejpam-5057	257	17	√	√	NUM
ejpam-5057	257	18	n−	n−	NOUN
ejpam-5057	257	19	1	1	NUM
ejpam-5057	257	20	.	.	PUNCT
ejpam-5057	257	21	comparing	compare	VERB
ejpam-5057	257	22	all	all	DET
ejpam-5057	257	23	the	the	DET
ejpam-5057	257	24	three	three	NUM
ejpam-5057	257	25	cases	case	NOUN
ejpam-5057	257	26	,	,	PUNCT
ejpam-5057	257	27	we	we	PRON
ejpam-5057	257	28	get	get	VERB
ejpam-5057	257	29	η21	η21	PROPN
ejpam-5057	257	30	≥	≥	NOUN
ejpam-5057	257	31	η23	η23	PROPN
ejpam-5057	257	32	≥	≥	NUM
ejpam-5057	257	33	η22	η22	PROPN
ejpam-5057	257	34	.	.	PUNCT
ejpam-5057	258	1	from	from	ADP
ejpam-5057	258	2	this	this	PRON
ejpam-5057	258	3	we	we	PRON
ejpam-5057	258	4	have	have	VERB
ejpam-5057	258	5	,	,	PUNCT
ejpam-5057	258	6	η2	η2	ADJ
ejpam-5057	258	7	=	=	SYM
ejpam-5057	258	8	η21	η21	PROPN
ejpam-5057	258	9	.	.	PUNCT
ejpam-5057	259	1	applying	apply	VERB
ejpam-5057	259	2	this	this	PRON
ejpam-5057	259	3	in	in	ADP
ejpam-5057	259	4	equation	equation	NOUN
ejpam-5057	259	5	(	(	PUNCT
ejpam-5057	259	6	3	3	NUM
ejpam-5057	259	7	)	)	PUNCT
ejpam-5057	259	8	,	,	PUNCT
ejpam-5057	259	9	we	we	PRON
ejpam-5057	259	10	get	get	VERB
ejpam-5057	259	11	a	a	DET
ejpam-5057	259	12	tight	tight	ADV
ejpam-5057	259	13	lower	lower	ADV
ejpam-5057	259	14	bound	bind	VERB
ejpam-5057	259	15	in	in	ADP
ejpam-5057	259	16	terms	term	NOUN
ejpam-5057	259	17	of	of	ADP
ejpam-5057	259	18	n	n	PRON
ejpam-5057	259	19	as	as	SCONJ
ejpam-5057	259	20	λn−1	λn−1	PROPN
ejpam-5057	259	21	≥	≥	NUM
ejpam-5057	259	22			PUNCT
ejpam-5057	259	23	−(2(n−1))√	−(2(n−1))√	NOUN
ejpam-5057	259	24	n2−1	n2−1	NOUN
ejpam-5057	259	25	n	n	PROPN
ejpam-5057	259	26	is	be	AUX
ejpam-5057	259	27	odd	odd	ADJ
ejpam-5057	259	28	−(2(n−1	−(2(n−1	NUM
ejpam-5057	259	29	)	)	PUNCT
ejpam-5057	259	30	)	)	PUNCT
ejpam-5057	260	1	n	n	CCONJ
ejpam-5057	260	2	n	n	ADV
ejpam-5057	260	3	is	be	AUX
ejpam-5057	260	4	even	even	ADV
ejpam-5057	260	5	.	.	PUNCT
ejpam-5057	261	1	m.machasri	m.machasri	NUM
ejpam-5057	261	2	,	,	PUNCT
ejpam-5057	261	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	261	4	/	/	SYM
ejpam-5057	261	5	eur	eur	PROPN
ejpam-5057	261	6	.	.	PUNCT
ejpam-5057	262	1	j.	j.	PROPN
ejpam-5057	262	2	pure	pure	PROPN
ejpam-5057	262	3	appl	appl	PROPN
ejpam-5057	262	4	.	.	PROPN
ejpam-5057	262	5	math	math	PROPN
ejpam-5057	262	6	,	,	PUNCT
ejpam-5057	262	7	17	17	NUM
ejpam-5057	262	8	(	(	PUNCT
ejpam-5057	262	9	2	2	NUM
ejpam-5057	262	10	)	)	PUNCT
ejpam-5057	262	11	(	(	PUNCT
ejpam-5057	262	12	2024	2024	NUM
ejpam-5057	262	13	)	)	PUNCT
ejpam-5057	262	14	,	,	PUNCT
ejpam-5057	262	15	772	772	NUM
ejpam-5057	262	16	-	-	SYM
ejpam-5057	262	17	789	789	NUM
ejpam-5057	262	18	782	782	NUM
ejpam-5057	262	19	4.2	4.2	NUM
ejpam-5057	262	20	.	.	PUNCT
ejpam-5057	262	21	bounds	bound	NOUN
ejpam-5057	262	22	for	for	ADP
ejpam-5057	262	23	the	the	DET
ejpam-5057	262	24	second	second	ADV
ejpam-5057	262	25	largest	large	ADJ
ejpam-5057	262	26	laplacian	laplacian	ADJ
ejpam-5057	262	27	eigenvalue	eigenvalue	NOUN
ejpam-5057	262	28	in	in	ADP
ejpam-5057	262	29	this	this	DET
ejpam-5057	262	30	section	section	NOUN
ejpam-5057	262	31	we	we	PRON
ejpam-5057	262	32	derive	derive	VERB
ejpam-5057	262	33	the	the	DET
ejpam-5057	262	34	upper	upper	ADJ
ejpam-5057	262	35	bound	bind	VERB
ejpam-5057	262	36	for	for	ADP
ejpam-5057	262	37	the	the	DET
ejpam-5057	262	38	second	second	ADV
ejpam-5057	262	39	largest	large	ADJ
ejpam-5057	262	40	laplacian	laplacian	ADJ
ejpam-5057	262	41	eigenvalue	eigenvalue	NOUN
ejpam-5057	262	42	of	of	ADP
ejpam-5057	262	43	a	a	DET
ejpam-5057	262	44	connected	connected	ADJ
ejpam-5057	262	45	bipartite	bipartite	NOUN
ejpam-5057	262	46	graph	graph	NOUN
ejpam-5057	262	47	g.	g.	NOUN
ejpam-5057	262	48	theorem	theorem	NOUN
ejpam-5057	262	49	4	4	NUM
ejpam-5057	262	50	.	.	PUNCT
ejpam-5057	262	51	consider	consider	VERB
ejpam-5057	262	52	a	a	DET
ejpam-5057	262	53	bipartite	bipartite	NOUN
ejpam-5057	262	54	graph	graph	NOUN
ejpam-5057	262	55	g	g	NOUN
ejpam-5057	262	56	with	with	ADP
ejpam-5057	262	57	bipartition	bipartition	NOUN
ejpam-5057	262	58	v	v	NOUN
ejpam-5057	262	59	=	=	SYM
ejpam-5057	262	60	(	(	PUNCT
ejpam-5057	262	61	x	x	X
ejpam-5057	262	62	,	,	PUNCT
ejpam-5057	262	63	y	y	PROPN
ejpam-5057	262	64	)	)	PUNCT
ejpam-5057	262	65	.	.	PUNCT
ejpam-5057	263	1	let	let	VERB
ejpam-5057	263	2	µ1	µ1	NOUN
ejpam-5057	263	3	≥	≥	NOUN
ejpam-5057	263	4	µ2	µ2	PROPN
ejpam-5057	263	5	≥	≥	PROPN
ejpam-5057	263	6	·	·	PUNCT
ejpam-5057	263	7	·	·	PUNCT
ejpam-5057	263	8	·	·	PUNCT
ejpam-5057	263	9	≥	≥	NUM
ejpam-5057	263	10	µn−1	µn−1	ADP
ejpam-5057	263	11	≥	≥	NUM
ejpam-5057	263	12	µn	µn	NOUN
ejpam-5057	263	13	be	be	AUX
ejpam-5057	263	14	the	the	DET
ejpam-5057	263	15	eigenvalues	eigenvalue	NOUN
ejpam-5057	263	16	of	of	ADP
ejpam-5057	263	17	the	the	DET
ejpam-5057	263	18	laplacian	laplacian	ADJ
ejpam-5057	263	19	matrix	matrix	NOUN
ejpam-5057	263	20	l	l	NOUN
ejpam-5057	263	21	of	of	ADP
ejpam-5057	263	22	g	g	NOUN
ejpam-5057	263	23	,	,	PUNCT
ejpam-5057	263	24	lq	lq	X
ejpam-5057	263	25	be	be	AUX
ejpam-5057	263	26	the	the	DET
ejpam-5057	263	27	bipartite	bipartite	PROPN
ejpam-5057	263	28	quotient	quotient	NOUN
ejpam-5057	263	29	matrix	matrix	NOUN
ejpam-5057	263	30	of	of	ADP
ejpam-5057	263	31	l	l	NOUN
ejpam-5057	263	32	and	and	CCONJ
ejpam-5057	263	33	θ1	θ1	NOUN
ejpam-5057	263	34	and	and	CCONJ
ejpam-5057	263	35	θ2	θ2	ADV
ejpam-5057	263	36	the	the	DET
ejpam-5057	263	37	eigenvalues	eigenvalue	NOUN
ejpam-5057	263	38	of	of	ADP
ejpam-5057	263	39	lq	lq	PROPN
ejpam-5057	263	40	.	.	PUNCT
ejpam-5057	264	1	then	then	ADV
ejpam-5057	264	2	(	(	PUNCT
ejpam-5057	264	3	i	i	NOUN
ejpam-5057	264	4	)	)	PUNCT
ejpam-5057	264	5	µ1	µ1	PROPN
ejpam-5057	264	6	≥	≥	PROPN
ejpam-5057	264	7	θ1	θ1	PROPN
ejpam-5057	264	8	≥	≥	PRON
ejpam-5057	264	9	µ2	µ2	PROPN
ejpam-5057	264	10	(	(	PUNCT
ejpam-5057	264	11	ii	ii	NOUN
ejpam-5057	264	12	)	)	PUNCT
ejpam-5057	264	13	µ2	µ2	PROPN
ejpam-5057	264	14	≤	≤	PUNCT
ejpam-5057	264	15	mn	mn	PROPN
ejpam-5057	264	16	n1n2	n1n2	NOUN
ejpam-5057	264	17	proof	proof	NOUN
ejpam-5057	264	18	.	.	PUNCT
ejpam-5057	265	1	the	the	DET
ejpam-5057	265	2	proof	proof	NOUN
ejpam-5057	265	3	is	be	AUX
ejpam-5057	265	4	similar	similar	ADJ
ejpam-5057	265	5	to	to	ADP
ejpam-5057	265	6	that	that	PRON
ejpam-5057	265	7	of	of	ADP
ejpam-5057	265	8	theorem	theorem	NOUN
ejpam-5057	265	9	1	1	X
ejpam-5057	265	10	.	.	PUNCT
ejpam-5057	266	1	let	let	VERB
ejpam-5057	266	2	l	l	NOUN
ejpam-5057	266	3	be	be	AUX
ejpam-5057	266	4	the	the	DET
ejpam-5057	266	5	laplacian	laplacian	ADJ
ejpam-5057	266	6	matrix	matrix	NOUN
ejpam-5057	266	7	of	of	ADP
ejpam-5057	266	8	g	g	PROPN
ejpam-5057	266	9	represented	represent	VERB
ejpam-5057	266	10	in	in	ADP
ejpam-5057	266	11	the	the	DET
ejpam-5057	266	12	following	follow	VERB
ejpam-5057	266	13	block	block	NOUN
ejpam-5057	266	14	matrix	matrix	NOUN
ejpam-5057	266	15	form	form	NOUN
ejpam-5057	266	16	with	with	ADP
ejpam-5057	266	17	respect	respect	NOUN
ejpam-5057	266	18	to	to	ADP
ejpam-5057	266	19	the	the	DET
ejpam-5057	266	20	bipartition	bipartition	NOUN
ejpam-5057	266	21	v	v	ADP
ejpam-5057	266	22	=	=	SYM
ejpam-5057	266	23	(	(	PUNCT
ejpam-5057	266	24	x	x	X
ejpam-5057	266	25	,	,	PUNCT
ejpam-5057	266	26	y	y	PROPN
ejpam-5057	266	27	)	)	PUNCT
ejpam-5057	266	28	l	l	NOUN
ejpam-5057	267	1	=	=	PUNCT
ejpam-5057	267	2	[	[	PUNCT
ejpam-5057	267	3	l11	l11	PROPN
ejpam-5057	267	4	l12	l12	NOUN
ejpam-5057	267	5	l21	l21	NOUN
ejpam-5057	267	6	l22	l22	NOUN
ejpam-5057	267	7	]	]	PUNCT
ejpam-5057	267	8	.	.	PUNCT
ejpam-5057	268	1	let	let	VERB
ejpam-5057	268	2	lq	lq	NOUN
ejpam-5057	268	3	be	be	AUX
ejpam-5057	268	4	the	the	DET
ejpam-5057	268	5	bipartite	bipartite	PROPN
ejpam-5057	268	6	quotient	quotient	NOUN
ejpam-5057	268	7	matrix	matrix	NOUN
ejpam-5057	268	8	of	of	ADP
ejpam-5057	268	9	the	the	DET
ejpam-5057	268	10	laplacian	laplacian	ADJ
ejpam-5057	268	11	matrix	matrix	NOUN
ejpam-5057	268	12	l	l	NOUN
ejpam-5057	268	13	of	of	ADP
ejpam-5057	268	14	g.	g.	PROPN
ejpam-5057	268	15	then	then	ADV
ejpam-5057	268	16	lq	lq	ADV
ejpam-5057	268	17	=	=	PUNCT
ejpam-5057	268	18	[	[	PUNCT
ejpam-5057	268	19	m	m	VERB
ejpam-5057	268	20	n1	n1	ADJ
ejpam-5057	268	21	−m	−m	NOUN
ejpam-5057	268	22	n1−m	n1−m	PROPN
ejpam-5057	268	23	n2	n2	ADJ
ejpam-5057	268	24	m	m	PROPN
ejpam-5057	268	25	n2	n2	NOUN
ejpam-5057	268	26	]	]	PUNCT
ejpam-5057	268	27	.	.	PUNCT
ejpam-5057	269	1	the	the	DET
ejpam-5057	269	2	characteristic	characteristic	ADJ
ejpam-5057	269	3	equation	equation	NOUN
ejpam-5057	269	4	of	of	ADP
ejpam-5057	269	5	lq	lq	NOUN
ejpam-5057	269	6	is	be	AUX
ejpam-5057	269	7	µ2	µ2	PROPN
ejpam-5057	269	8	−	−	PROPN
ejpam-5057	269	9	mn	mn	PROPN
ejpam-5057	269	10	n1n2	n1n2	PROPN
ejpam-5057	269	11	µ	µ	X
ejpam-5057	269	12	=	=	SYM
ejpam-5057	269	13	0	0	PUNCT
ejpam-5057	270	1	then	then	ADV
ejpam-5057	270	2	the	the	DET
ejpam-5057	270	3	eigenvalues	eigenvalue	NOUN
ejpam-5057	270	4	of	of	ADP
ejpam-5057	270	5	lq	lq	NOUN
ejpam-5057	270	6	are	be	AUX
ejpam-5057	270	7	θ1	θ1	NOUN
ejpam-5057	270	8	=	=	SYM
ejpam-5057	270	9	mn	mn	PROPN
ejpam-5057	270	10	n1n2	n1n2	NOUN
ejpam-5057	270	11	and	and	CCONJ
ejpam-5057	270	12	θ2	θ2	ADV
ejpam-5057	270	13	=	=	PROPN
ejpam-5057	270	14	0	0	X
ejpam-5057	270	15	.	.	PUNCT
ejpam-5057	271	1	to	to	PART
ejpam-5057	271	2	get	get	VERB
ejpam-5057	271	3	the	the	DET
ejpam-5057	271	4	generalized	generalized	ADJ
ejpam-5057	271	5	interlacing	interlacing	NOUN
ejpam-5057	271	6	consider	consider	VERB
ejpam-5057	271	7	the	the	DET
ejpam-5057	271	8	following	following	NOUN
ejpam-5057	271	9	.	.	PUNCT
ejpam-5057	272	1	the	the	DET
ejpam-5057	272	2	characteristic	characteristic	ADJ
ejpam-5057	272	3	matrix	matrix	NOUN
ejpam-5057	272	4	of	of	ADP
ejpam-5057	272	5	g	g	PROPN
ejpam-5057	272	6	is	be	AUX
ejpam-5057	272	7	given	give	VERB
ejpam-5057	272	8	by	by	ADP
ejpam-5057	272	9	s̃	s̃	PROPN
ejpam-5057	272	10	=	=	PUNCT
ejpam-5057	272	11	[	[	PUNCT
ejpam-5057	272	12	jn1×1	jn1×1	NOUN
ejpam-5057	272	13	0n1×1	0n1×1	VERB
ejpam-5057	272	14	0n2×1	0n2×1	NOUN
ejpam-5057	272	15	jn2×1	jn2×1	PROPN
ejpam-5057	272	16	]	]	PUNCT
ejpam-5057	272	17	n×2	n×2	PROPN
ejpam-5057	272	18	let	let	VERB
ejpam-5057	272	19	s	s	PRON
ejpam-5057	272	20	=	=	VERB
ejpam-5057	272	21	s̃k	s̃k	ADP
ejpam-5057	272	22	−1	−1	NOUN
ejpam-5057	272	23	2	2	NUM
ejpam-5057	272	24	,	,	PUNCT
ejpam-5057	272	25	where	where	SCONJ
ejpam-5057	272	26	k	k	PROPN
ejpam-5057	272	27	=	=	SYM
ejpam-5057	272	28	diag(|x|	diag(|x|	PROPN
ejpam-5057	272	29	,	,	PUNCT
ejpam-5057	272	30	|y	|y	NOUN
ejpam-5057	272	31	|	|	NOUN
ejpam-5057	272	32	)	)	PUNCT
ejpam-5057	272	33	i.e.	i.e.	X
ejpam-5057	272	34	,	,	PUNCT
ejpam-5057	272	35	k	k	X
ejpam-5057	273	1	=	=	PUNCT
ejpam-5057	274	1	[	[	PUNCT
ejpam-5057	274	2	n1	n1	NOUN
ejpam-5057	274	3	0	0	NUM
ejpam-5057	274	4	0	0	NUM
ejpam-5057	274	5	n2	n2	NOUN
ejpam-5057	274	6	]	]	PUNCT
ejpam-5057	274	7	.	.	PUNCT
ejpam-5057	275	1	then	then	ADV
ejpam-5057	275	2	s	s	VERB
ejpam-5057	275	3	=	=	PUNCT
ejpam-5057	275	4	[	[	PUNCT
ejpam-5057	275	5	pn1×1	pn1×1	NOUN
ejpam-5057	275	6	0n1×1	0n1×1	NOUN
ejpam-5057	275	7	0n2×1	0n2×1	NUM
ejpam-5057	275	8	rn2×1	rn2×1	NOUN
ejpam-5057	275	9	]	]	PUNCT
ejpam-5057	275	10	n×2	n×2	NOUN
ejpam-5057	275	11	.	.	PUNCT
ejpam-5057	276	1	where	where	SCONJ
ejpam-5057	276	2	pn1×1	pn1×1	NOUN
ejpam-5057	276	3	and	and	CCONJ
ejpam-5057	276	4	rn2×1	rn2×1	NOUN
ejpam-5057	276	5	are	be	AUX
ejpam-5057	276	6	column	column	NOUN
ejpam-5057	276	7	matrices	matrix	NOUN
ejpam-5057	276	8	with	with	ADP
ejpam-5057	276	9	entries	entry	NOUN
ejpam-5057	276	10	(	(	PUNCT
ejpam-5057	276	11	pi1	pi1	NOUN
ejpam-5057	276	12	)	)	PUNCT
ejpam-5057	276	13	=	=	SYM
ejpam-5057	276	14	1√	1√	NUM
ejpam-5057	276	15	n1	n1	NOUN
ejpam-5057	276	16	where	where	SCONJ
ejpam-5057	276	17	i	i	PRON
ejpam-5057	276	18	=	=	NOUN
ejpam-5057	276	19	1	1	NUM
ejpam-5057	276	20	,	,	PUNCT
ejpam-5057	276	21	2	2	NUM
ejpam-5057	276	22	,	,	PUNCT
ejpam-5057	276	23	.	.	PUNCT
ejpam-5057	276	24	.	.	PUNCT
ejpam-5057	277	1	.	.	PUNCT
ejpam-5057	278	1	,	,	PUNCT
ejpam-5057	278	2	n1	n1	PROPN
ejpam-5057	278	3	and	and	CCONJ
ejpam-5057	278	4	(	(	PUNCT
ejpam-5057	278	5	rj1	rj1	NOUN
ejpam-5057	278	6	)	)	PUNCT
ejpam-5057	278	7	=	=	SYM
ejpam-5057	278	8	1√	1√	NUM
ejpam-5057	278	9	n2	n2	NOUN
ejpam-5057	278	10	where	where	SCONJ
ejpam-5057	278	11	j	j	PROPN
ejpam-5057	278	12	=	=	SYM
ejpam-5057	278	13	1	1	NUM
ejpam-5057	278	14	,	,	PUNCT
ejpam-5057	278	15	2	2	NUM
ejpam-5057	278	16	,	,	PUNCT
ejpam-5057	278	17	.	.	PUNCT
ejpam-5057	278	18	.	.	PUNCT
ejpam-5057	279	1	.	.	PUNCT
ejpam-5057	280	1	,	,	PUNCT
ejpam-5057	280	2	n2	n2	PROPN
ejpam-5057	280	3	.	.	PROPN
ejpam-5057	281	1	from	from	ADP
ejpam-5057	281	2	the	the	DET
ejpam-5057	281	3	proof	proof	NOUN
ejpam-5057	281	4	of	of	ADP
ejpam-5057	281	5	lemma	lemma	PROPN
ejpam-5057	281	6	1	1	NUM
ejpam-5057	281	7	[	[	X
ejpam-5057	281	8	7	7	X
ejpam-5057	281	9	]	]	PUNCT
ejpam-5057	281	10	we	we	PRON
ejpam-5057	281	11	have	have	AUX
ejpam-5057	281	12	,	,	PUNCT
ejpam-5057	281	13	lq	lq	NOUN
ejpam-5057	281	14	=	=	NOUN
ejpam-5057	281	15	stls	stls	NOUN
ejpam-5057	281	16	.	.	PUNCT
ejpam-5057	282	1	(	(	PUNCT
ejpam-5057	282	2	4	4	X
ejpam-5057	282	3	)	)	PUNCT
ejpam-5057	282	4	left	leave	VERB
ejpam-5057	282	5	and	and	CCONJ
ejpam-5057	282	6	right	right	ADJ
ejpam-5057	282	7	multiplying	multiply	VERB
ejpam-5057	282	8	equation	equation	NOUN
ejpam-5057	282	9	(	(	PUNCT
ejpam-5057	282	10	4	4	NUM
ejpam-5057	282	11	)	)	PUNCT
ejpam-5057	282	12	by	by	ADP
ejpam-5057	282	13	s	s	PRON
ejpam-5057	282	14	and	and	CCONJ
ejpam-5057	282	15	st	st	PROPN
ejpam-5057	283	1	we	we	PRON
ejpam-5057	283	2	have	have	VERB
ejpam-5057	283	3	slqs	slqs	ADJ
ejpam-5057	283	4	t	t	PROPN
ejpam-5057	283	5	=	=	SYM
ejpam-5057	283	6	l.	l.	PROPN
ejpam-5057	283	7	now	now	ADV
ejpam-5057	283	8	consider	consider	VERB
ejpam-5057	283	9	slqs	slqs	ADJ
ejpam-5057	283	10	t	t	PROPN
ejpam-5057	283	11	and	and	CCONJ
ejpam-5057	283	12	denote	denote	VERB
ejpam-5057	283	13	it	it	PRON
ejpam-5057	283	14	by	by	ADP
ejpam-5057	283	15	u	u	PROPN
ejpam-5057	283	16	.	.	PUNCT
ejpam-5057	284	1	let	let	VERB
ejpam-5057	284	2	u	u	PRON
ejpam-5057	284	3	=	=	X
ejpam-5057	284	4	(	(	PUNCT
ejpam-5057	284	5	uij	uij	X
ejpam-5057	284	6	)	)	PUNCT
ejpam-5057	284	7	=	=	SYM
ejpam-5057	285	1			PROPN
ejpam-5057	285	2	m	m	VERB
ejpam-5057	285	3	n2	n2	ADJ
ejpam-5057	285	4	1	1	NUM
ejpam-5057	285	5	vi	vi	NOUN
ejpam-5057	285	6	,	,	PUNCT
ejpam-5057	285	7	vj	vj	X
ejpam-5057	285	8	∈	∈	PROPN
ejpam-5057	285	9	x	x	PUNCT
ejpam-5057	285	10	m	m	NOUN
ejpam-5057	285	11	n2	n2	ADJ
ejpam-5057	285	12	2	2	NUM
ejpam-5057	285	13	vi	vi	NOUN
ejpam-5057	285	14	,	,	PUNCT
ejpam-5057	285	15	vj	vj	PROPN
ejpam-5057	285	16	∈	∈	PROPN
ejpam-5057	285	17	y	y	PROPN
ejpam-5057	285	18	−m	−m	PROPN
ejpam-5057	285	19	n1	n1	PROPN
ejpam-5057	285	20	√	√	PROPN
ejpam-5057	285	21	n1n2	n1n2	SYM
ejpam-5057	285	22	vi	vi	NOUN
ejpam-5057	285	23	∈	∈	PROPN
ejpam-5057	285	24	x	x	X
ejpam-5057	285	25	and	and	CCONJ
ejpam-5057	285	26	vj	vj	INTJ
ejpam-5057	285	27	∈	∈	PROPN
ejpam-5057	285	28	y	y	PROPN
ejpam-5057	285	29	−m	−m	PROPN
ejpam-5057	285	30	n2	n2	PROPN
ejpam-5057	285	31	√	√	NUM
ejpam-5057	285	32	n1n2	n1n2	SYM
ejpam-5057	285	33	vi	vi	PROPN
ejpam-5057	285	34	∈	∈	PROPN
ejpam-5057	285	35	y	y	PROPN
ejpam-5057	285	36	and	and	CCONJ
ejpam-5057	285	37	vj	vj	PRON
ejpam-5057	285	38	∈	∈	PROPN
ejpam-5057	285	39	x	x	SYM
ejpam-5057	285	40	m.machasri	m.machasri	NOUN
ejpam-5057	285	41	,	,	PUNCT
ejpam-5057	285	42	d.kalyani	d.kalyani	NOUN
ejpam-5057	285	43	/	/	SYM
ejpam-5057	285	44	eur	eur	PROPN
ejpam-5057	285	45	.	.	PUNCT
ejpam-5057	286	1	j.	j.	PROPN
ejpam-5057	286	2	pure	pure	PROPN
ejpam-5057	286	3	appl	appl	PROPN
ejpam-5057	286	4	.	.	PROPN
ejpam-5057	286	5	math	math	PROPN
ejpam-5057	286	6	,	,	PUNCT
ejpam-5057	286	7	17	17	NUM
ejpam-5057	286	8	(	(	PUNCT
ejpam-5057	286	9	2	2	NUM
ejpam-5057	286	10	)	)	PUNCT
ejpam-5057	286	11	(	(	PUNCT
ejpam-5057	286	12	2024	2024	NUM
ejpam-5057	286	13	)	)	PUNCT
ejpam-5057	286	14	,	,	PUNCT
ejpam-5057	286	15	772	772	NUM
ejpam-5057	286	16	-	-	SYM
ejpam-5057	286	17	789	789	NUM
ejpam-5057	286	18	783	783	NUM
ejpam-5057	286	19	then	then	ADV
ejpam-5057	286	20	the	the	DET
ejpam-5057	286	21	block	block	NOUN
ejpam-5057	286	22	matrix	matrix	NOUN
ejpam-5057	286	23	representation	representation	NOUN
ejpam-5057	286	24	of	of	ADP
ejpam-5057	286	25	u	u	NOUN
ejpam-5057	286	26	is	be	AUX
ejpam-5057	286	27	given	give	VERB
ejpam-5057	286	28	by	by	ADP
ejpam-5057	286	29	u	u	NOUN
ejpam-5057	286	30	=	=	PUNCT
ejpam-5057	286	31	[	[	PUNCT
ejpam-5057	286	32	e	e	X
ejpam-5057	286	33	f	f	PROPN
ejpam-5057	286	34	g	g	PROPN
ejpam-5057	286	35	h	h	NOUN
ejpam-5057	286	36	]	]	X
ejpam-5057	287	1	n×n	n×n	PROPN
ejpam-5057	287	2	.	.	PUNCT
ejpam-5057	288	1	let	let	VERB
ejpam-5057	288	2	the	the	DET
ejpam-5057	288	3	eigenvalues	eigenvalue	NOUN
ejpam-5057	288	4	of	of	ADP
ejpam-5057	288	5	u	u	NOUN
ejpam-5057	288	6	be	be	AUX
ejpam-5057	288	7	β1	β1	PROPN
ejpam-5057	288	8	≥	≥	NUM
ejpam-5057	288	9	β2	β2	PROPN
ejpam-5057	288	10	≥	≥	PRON
ejpam-5057	288	11	·	·	PUNCT
ejpam-5057	288	12	·	·	PUNCT
ejpam-5057	288	13	·	·	PUNCT
ejpam-5057	288	14	≥	≥	NUM
ejpam-5057	288	15	βn	βn	NOUN
ejpam-5057	288	16	.	.	PUNCT
ejpam-5057	289	1	the	the	DET
ejpam-5057	289	2	eigenvalues	eigenvalue	NOUN
ejpam-5057	289	3	of	of	ADP
ejpam-5057	289	4	u	u	NOUN
ejpam-5057	289	5	are	be	AUX
ejpam-5057	289	6	the	the	DET
ejpam-5057	289	7	trace	trace	NOUN
ejpam-5057	289	8	of	of	ADP
ejpam-5057	289	9	u	u	NOUN
ejpam-5057	289	10	and	and	CCONJ
ejpam-5057	289	11	0	0	NUM
ejpam-5057	289	12	,	,	PUNCT
ejpam-5057	289	13	that	that	PRON
ejpam-5057	289	14	is	is	ADV
ejpam-5057	289	15	β1	β1	PROPN
ejpam-5057	289	16	=	=	PUNCT
ejpam-5057	289	17	mn	mn	PROPN
ejpam-5057	289	18	n1n2	n1n2	PROPN
ejpam-5057	289	19	and	and	CCONJ
ejpam-5057	289	20	βi	βi	PRON
ejpam-5057	290	1	=	=	NOUN
ejpam-5057	290	2	0	0	NUM
ejpam-5057	291	1	for	for	ADP
ejpam-5057	291	2	i	i	PRON
ejpam-5057	291	3	=	=	SYM
ejpam-5057	291	4	2	2	NUM
ejpam-5057	291	5	,	,	PUNCT
ejpam-5057	291	6	3	3	NUM
ejpam-5057	291	7	,	,	PUNCT
ejpam-5057	291	8	.	.	PUNCT
ejpam-5057	291	9	.	.	PUNCT
ejpam-5057	291	10	.	.	PUNCT
ejpam-5057	292	1	,	,	PUNCT
ejpam-5057	292	2	n.	n.	PROPN
ejpam-5057	292	3	then	then	ADV
ejpam-5057	292	4	the	the	DET
ejpam-5057	292	5	interlacing	interlacing	NOUN
ejpam-5057	292	6	becomes	become	VERB
ejpam-5057	292	7	,	,	PUNCT
ejpam-5057	292	8	µ1	µ1	PROPN
ejpam-5057	292	9	≥	≥	NOUN
ejpam-5057	292	10	β1	β1	PROPN
ejpam-5057	292	11	≥	≥	PUNCT
ejpam-5057	292	12	µ2	µ2	PROPN
ejpam-5057	292	13	≥	≥	X
ejpam-5057	292	14	·	·	PUNCT
ejpam-5057	292	15	·	·	PUNCT
ejpam-5057	292	16	·	·	PUNCT
ejpam-5057	293	1	≥	≥	NUM
ejpam-5057	293	2	µn−1	µn−1	ADP
ejpam-5057	293	3	≥	≥	NOUN
ejpam-5057	293	4	βm	βm	VERB
ejpam-5057	293	5	≥	≥	PROPN
ejpam-5057	293	6	µn	µn	PROPN
ejpam-5057	293	7	which	which	PRON
ejpam-5057	293	8	implies	imply	VERB
ejpam-5057	293	9	that	that	SCONJ
ejpam-5057	293	10	µ1	µ1	PROPN
ejpam-5057	293	11	≥	≥	NOUN
ejpam-5057	293	12	β1	β1	PROPN
ejpam-5057	293	13	≥	≥	NUM
ejpam-5057	293	14	µ2	µ2	PROPN
ejpam-5057	293	15	.	.	PUNCT
ejpam-5057	294	1	by	by	ADP
ejpam-5057	294	2	comparing	compare	VERB
ejpam-5057	294	3	the	the	DET
ejpam-5057	294	4	eigenvalues	eigenvalue	NOUN
ejpam-5057	294	5	of	of	ADP
ejpam-5057	294	6	lq	lq	NOUN
ejpam-5057	294	7	and	and	CCONJ
ejpam-5057	294	8	u	u	NOUN
ejpam-5057	294	9	we	we	PRON
ejpam-5057	294	10	have	have	VERB
ejpam-5057	294	11	,	,	PUNCT
ejpam-5057	294	12	β1	β1	PROPN
ejpam-5057	294	13	=	=	SYM
ejpam-5057	294	14	θ1	θ1	PROPN
ejpam-5057	294	15	.	.	PUNCT
ejpam-5057	295	1	let	let	VERB
ejpam-5057	295	2	us	we	PRON
ejpam-5057	295	3	consider	consider	VERB
ejpam-5057	295	4	µ1	µ1	PROPN
ejpam-5057	295	5	≥	≥	NOUN
ejpam-5057	295	6	β1	β1	PROPN
ejpam-5057	295	7	≥	≥	NUM
ejpam-5057	295	8	µ2	µ2	PROPN
ejpam-5057	295	9	.	.	PUNCT
ejpam-5057	296	1	since	since	SCONJ
ejpam-5057	296	2	β1	β1	PROPN
ejpam-5057	296	3	=	=	SYM
ejpam-5057	296	4	θ1	θ1	NOUN
ejpam-5057	296	5	,	,	PUNCT
ejpam-5057	296	6	we	we	PRON
ejpam-5057	296	7	have	have	VERB
ejpam-5057	296	8	µ1	µ1	PROPN
ejpam-5057	296	9	≥	≥	NOUN
ejpam-5057	296	10	θ1	θ1	NOUN
ejpam-5057	296	11	≥	≥	NOUN
ejpam-5057	296	12	µ2	µ2	PROPN
ejpam-5057	296	13	.	.	PUNCT
ejpam-5057	297	1	this	this	PRON
ejpam-5057	297	2	proves	prove	VERB
ejpam-5057	297	3	(	(	PUNCT
ejpam-5057	297	4	i	i	NOUN
ejpam-5057	297	5	)	)	PUNCT
ejpam-5057	297	6	.	.	PUNCT
ejpam-5057	298	1	to	to	PART
ejpam-5057	298	2	prove	prove	VERB
ejpam-5057	298	3	(	(	PUNCT
ejpam-5057	298	4	ii	ii	NOUN
ejpam-5057	298	5	)	)	PUNCT
ejpam-5057	298	6	,	,	PUNCT
ejpam-5057	298	7	using	use	VERB
ejpam-5057	298	8	(	(	PUNCT
ejpam-5057	298	9	i	i	NOUN
ejpam-5057	298	10	)	)	PUNCT
ejpam-5057	298	11	we	we	PRON
ejpam-5057	298	12	have	have	VERB
ejpam-5057	298	13	,	,	PUNCT
ejpam-5057	298	14	µ2	µ2	VERB
ejpam-5057	298	15	≤	≤	NUM
ejpam-5057	298	16	θ1	θ1	NOUN
ejpam-5057	298	17	=	=	SYM
ejpam-5057	298	18	mn	mn	PROPN
ejpam-5057	298	19	n1n2	n1n2	NOUN
ejpam-5057	298	20	.	.	PUNCT
ejpam-5057	299	1	corollary	corollary	ADJ
ejpam-5057	299	2	3	3	X
ejpam-5057	299	3	.	.	PUNCT
ejpam-5057	300	1	if	if	SCONJ
ejpam-5057	300	2	g	g	PROPN
ejpam-5057	300	3	is	be	AUX
ejpam-5057	300	4	a	a	DET
ejpam-5057	300	5	regular	regular	ADJ
ejpam-5057	300	6	bipartite	bipartite	NOUN
ejpam-5057	300	7	graph	graph	NOUN
ejpam-5057	300	8	then	then	ADV
ejpam-5057	300	9	µ2	µ2	VERB
ejpam-5057	300	10	≤	≤	ADJ
ejpam-5057	300	11	2	2	NUM
ejpam-5057	300	12	m	m	NOUN
ejpam-5057	300	13	n1	n1	NOUN
ejpam-5057	300	14	.	.	PUNCT
ejpam-5057	301	1	proof	proof	NOUN
ejpam-5057	301	2	.	.	PUNCT
ejpam-5057	302	1	for	for	ADP
ejpam-5057	302	2	a	a	DET
ejpam-5057	302	3	regular	regular	ADJ
ejpam-5057	302	4	bipartite	bipartite	NOUN
ejpam-5057	302	5	graph	graph	NOUN
ejpam-5057	302	6	g	g	NOUN
ejpam-5057	302	7	,	,	PUNCT
ejpam-5057	302	8	n1	n1	PROPN
ejpam-5057	302	9	=	=	SYM
ejpam-5057	302	10	n2	n2	NOUN
ejpam-5057	302	11	.	.	PUNCT
ejpam-5057	303	1	substituting	substitute	VERB
ejpam-5057	303	2	this	this	PRON
ejpam-5057	303	3	in	in	ADP
ejpam-5057	303	4	theorem	theorem	NOUN
ejpam-5057	303	5	4	4	NUM
ejpam-5057	303	6	,	,	PUNCT
ejpam-5057	303	7	we	we	PRON
ejpam-5057	303	8	get	get	VERB
ejpam-5057	303	9	the	the	DET
ejpam-5057	303	10	result	result	NOUN
ejpam-5057	303	11	.	.	PUNCT
ejpam-5057	304	1	note	note	NOUN
ejpam-5057	304	2	:	:	PUNCT
ejpam-5057	304	3	for	for	ADP
ejpam-5057	304	4	a	a	DET
ejpam-5057	304	5	complete	complete	ADJ
ejpam-5057	304	6	bipartite	bipartite	NOUN
ejpam-5057	304	7	graph	graph	NOUN
ejpam-5057	304	8	g	g	NOUN
ejpam-5057	304	9	,	,	PUNCT
ejpam-5057	304	10	the	the	DET
ejpam-5057	304	11	size	size	NOUN
ejpam-5057	304	12	of	of	ADP
ejpam-5057	304	13	g	g	PROPN
ejpam-5057	304	14	is	be	AUX
ejpam-5057	304	15	equal	equal	ADJ
ejpam-5057	304	16	to	to	ADP
ejpam-5057	304	17	the	the	DET
ejpam-5057	304	18	product	product	NOUN
ejpam-5057	304	19	of	of	ADP
ejpam-5057	304	20	the	the	DET
ejpam-5057	304	21	orders	order	NOUN
ejpam-5057	304	22	of	of	ADP
ejpam-5057	304	23	the	the	DET
ejpam-5057	304	24	bipartitions	bipartition	NOUN
ejpam-5057	304	25	of	of	ADP
ejpam-5057	304	26	g	g	NOUN
ejpam-5057	304	27	,	,	PUNCT
ejpam-5057	304	28	that	that	PRON
ejpam-5057	304	29	is	be	AUX
ejpam-5057	304	30	m	m	NOUN
ejpam-5057	304	31	=	=	SYM
ejpam-5057	304	32	n1n2	n1n2	NOUN
ejpam-5057	304	33	.	.	NOUN
ejpam-5057	304	34	hence	hence	ADV
ejpam-5057	304	35	µ2	µ2	VERB
ejpam-5057	304	36	≤	≤	PROPN
ejpam-5057	304	37	n.	n.	NOUN
ejpam-5057	304	38	5	5	NUM
ejpam-5057	304	39	.	.	PUNCT
ejpam-5057	304	40	vertex	vertex	NOUN
ejpam-5057	304	41	-	-	PUNCT
ejpam-5057	304	42	split	split	NOUN
ejpam-5057	304	43	of	of	ADP
ejpam-5057	304	44	a	a	DET
ejpam-5057	304	45	bipartite	bipartite	ADJ
ejpam-5057	304	46	graph	graph	NOUN
ejpam-5057	304	47	in	in	ADP
ejpam-5057	304	48	this	this	DET
ejpam-5057	304	49	section	section	NOUN
ejpam-5057	304	50	,	,	PUNCT
ejpam-5057	304	51	we	we	PRON
ejpam-5057	304	52	define	define	VERB
ejpam-5057	304	53	vertex	vertex	NOUN
ejpam-5057	304	54	split	split	NOUN
ejpam-5057	304	55	of	of	ADP
ejpam-5057	304	56	a	a	DET
ejpam-5057	304	57	bipartite	bipartite	NOUN
ejpam-5057	304	58	graph	graph	NOUN
ejpam-5057	304	59	.	.	PUNCT
ejpam-5057	305	1	next	next	ADV
ejpam-5057	305	2	,	,	PUNCT
ejpam-5057	305	3	we	we	PRON
ejpam-5057	305	4	prove	prove	VERB
ejpam-5057	305	5	theorems	theorem	NOUN
ejpam-5057	305	6	related	relate	VERB
ejpam-5057	305	7	to	to	ADP
ejpam-5057	305	8	the	the	DET
ejpam-5057	305	9	connectivity	connectivity	NOUN
ejpam-5057	305	10	and	and	CCONJ
ejpam-5057	305	11	expansion	expansion	NOUN
ejpam-5057	305	12	of	of	ADP
ejpam-5057	305	13	vertex	vertex	NOUN
ejpam-5057	305	14	split	split	NOUN
ejpam-5057	305	15	of	of	ADP
ejpam-5057	305	16	a	a	DET
ejpam-5057	305	17	bipartite	bipartite	NOUN
ejpam-5057	305	18	graph	graph	NOUN
ejpam-5057	305	19	.	.	PUNCT
ejpam-5057	306	1	definition	definition	NOUN
ejpam-5057	306	2	9	9	NUM
ejpam-5057	306	3	.	.	PUNCT
ejpam-5057	307	1	(	(	PUNCT
ejpam-5057	307	2	vertex	vertex	NOUN
ejpam-5057	307	3	-	-	PUNCT
ejpam-5057	307	4	split	split	NOUN
ejpam-5057	307	5	of	of	ADP
ejpam-5057	307	6	a	a	DET
ejpam-5057	307	7	bipartite	bipartite	NOUN
ejpam-5057	307	8	graph	graph	NOUN
ejpam-5057	307	9	)	)	PUNCT
ejpam-5057	307	10	let	let	VERB
ejpam-5057	307	11	g	g	NOUN
ejpam-5057	307	12	=	=	SYM
ejpam-5057	307	13	(	(	PUNCT
ejpam-5057	307	14	x	x	SYM
ejpam-5057	307	15	∪	∪	PROPN
ejpam-5057	307	16	y	y	PROPN
ejpam-5057	307	17	,	,	PUNCT
ejpam-5057	307	18	e	e	NOUN
ejpam-5057	307	19	)	)	PUNCT
ejpam-5057	307	20	be	be	AUX
ejpam-5057	307	21	a	a	DET
ejpam-5057	307	22	connected	connected	ADJ
ejpam-5057	307	23	bipartite	bipartite	NOUN
ejpam-5057	307	24	graph	graph	NOUN
ejpam-5057	307	25	with	with	ADP
ejpam-5057	307	26	δ	δ	PROPN
ejpam-5057	307	27	≥	≥	NUM
ejpam-5057	307	28	4	4	NUM
ejpam-5057	307	29	where	where	SCONJ
ejpam-5057	307	30	|x|	|x|	PROPN
ejpam-5057	307	31	=	=	SYM
ejpam-5057	307	32	n1	n1	PROPN
ejpam-5057	307	33	≥	≥	NOUN
ejpam-5057	307	34	4	4	NUM
ejpam-5057	307	35	,	,	PUNCT
ejpam-5057	307	36	|y	|y	NOUN
ejpam-5057	307	37	|	|	NOUN
ejpam-5057	307	38	=	=	SYM
ejpam-5057	307	39	n2	n2	NOUN
ejpam-5057	307	40	≥	≥	NUM
ejpam-5057	307	41	3	3	NUM
ejpam-5057	307	42	with	with	ADP
ejpam-5057	307	43	n1	n1	PROPN
ejpam-5057	307	44	>	>	X
ejpam-5057	307	45	n2	n2	PROPN
ejpam-5057	307	46	.	.	PUNCT
ejpam-5057	308	1	a	a	DET
ejpam-5057	308	2	graph	graph	NOUN
ejpam-5057	308	3	g′	g′	NOUN
ejpam-5057	308	4	=	=	SYM
ejpam-5057	308	5	(	(	PUNCT
ejpam-5057	308	6	x	x	SYM
ejpam-5057	308	7	′	′	NOUN
ejpam-5057	308	8	∪	∪	ADP
ejpam-5057	308	9	y	y	PROPN
ejpam-5057	308	10	′	′	NUM
ejpam-5057	308	11	,	,	PUNCT
ejpam-5057	308	12	e′	e′	ADJ
ejpam-5057	308	13	)	)	PUNCT
ejpam-5057	308	14	is	be	AUX
ejpam-5057	308	15	said	say	VERB
ejpam-5057	308	16	to	to	PART
ejpam-5057	308	17	be	be	AUX
ejpam-5057	308	18	a	a	DET
ejpam-5057	308	19	vertex	vertex	NOUN
ejpam-5057	308	20	-	-	PUNCT
ejpam-5057	308	21	split	split	NOUN
ejpam-5057	308	22	of	of	ADP
ejpam-5057	308	23	g	g	NOUN
ejpam-5057	308	24	if	if	SCONJ
ejpam-5057	308	25	(	(	PUNCT
ejpam-5057	308	26	i	i	NOUN
ejpam-5057	308	27	)	)	PUNCT
ejpam-5057	308	28	|x	|x	VERB
ejpam-5057	308	29	′|	′|	NUM
ejpam-5057	308	30	=	=	SYM
ejpam-5057	308	31	|x|	|x|	PROPN
ejpam-5057	308	32	and	and	CCONJ
ejpam-5057	308	33	|y	|y	VERB
ejpam-5057	308	34	′|	′|	NUM
ejpam-5057	308	35	=	=	SYM
ejpam-5057	308	36	|ya|	|ya|	NOUN
ejpam-5057	308	37	+	+	CCONJ
ejpam-5057	308	38	|yb|	|yb|	NOUN
ejpam-5057	308	39	=	=	SYM
ejpam-5057	309	1	2|y	2|y	NUM
ejpam-5057	310	1	|	|	INTJ
ejpam-5057	310	2	where	where	SCONJ
ejpam-5057	310	3	ya	ya	PRON
ejpam-5057	310	4	=	=	SYM
ejpam-5057	310	5	{	{	PUNCT
ejpam-5057	310	6	y1a	y1a	PROPN
ejpam-5057	310	7	,	,	PUNCT
ejpam-5057	310	8	y2a	y2a	NOUN
ejpam-5057	310	9	,	,	PUNCT
ejpam-5057	310	10	.	.	PUNCT
ejpam-5057	310	11	.	.	PUNCT
ejpam-5057	310	12	.	.	PUNCT
ejpam-5057	311	1	,	,	PUNCT
ejpam-5057	311	2	yn1a	yn1a	PROPN
ejpam-5057	311	3	}	}	PUNCT
ejpam-5057	311	4	,	,	PUNCT
ejpam-5057	311	5	yb	yb	PROPN
ejpam-5057	311	6	=	=	SYM
ejpam-5057	311	7	{	{	PUNCT
ejpam-5057	311	8	y1b	y1b	PROPN
ejpam-5057	311	9	,	,	PUNCT
ejpam-5057	311	10	y2b	y2b	PROPN
ejpam-5057	311	11	,	,	PUNCT
ejpam-5057	311	12	.	.	PUNCT
ejpam-5057	311	13	.	.	PUNCT
ejpam-5057	311	14	.	.	PUNCT
ejpam-5057	312	1	,	,	PUNCT
ejpam-5057	312	2	yn2b	yn2b	PROPN
ejpam-5057	312	3	}	}	PUNCT
ejpam-5057	312	4	.	.	PUNCT
ejpam-5057	313	1	(	(	PUNCT
ejpam-5057	313	2	ii	ii	NOUN
ejpam-5057	313	3	)	)	PUNCT
ejpam-5057	313	4	let	let	VERB
ejpam-5057	313	5	deg(y	deg(y	PROPN
ejpam-5057	313	6	)	)	PUNCT
ejpam-5057	313	7	=	=	PUNCT
ejpam-5057	314	1	dy	dy	NOUN
ejpam-5057	314	2	=	=	SYM
ejpam-5057	314	3	da	da	PROPN
ejpam-5057	315	1	+	+	PUNCT
ejpam-5057	315	2	db	db	PROPN
ejpam-5057	315	3	∀y	∀y	NUM
ejpam-5057	315	4	∈	∈	PROPN
ejpam-5057	315	5	y	y	NOUN
ejpam-5057	315	6	such	such	ADJ
ejpam-5057	315	7	that	that	PRON
ejpam-5057	315	8	da	da	ADJ
ejpam-5057	315	9	=	=	SYM
ejpam-5057	315	10	deg(ya	deg(ya	NOUN
ejpam-5057	315	11	)	)	PUNCT
ejpam-5057	315	12	∀ya	∀ya	PROPN
ejpam-5057	315	13	∈	∈	PROPN
ejpam-5057	315	14	ya	ya	PROPN
ejpam-5057	315	15	and	and	CCONJ
ejpam-5057	315	16	db	db	PROPN
ejpam-5057	315	17	=	=	PUNCT
ejpam-5057	315	18	deg(yb	deg(yb	PROPN
ejpam-5057	315	19	)	)	PUNCT
ejpam-5057	315	20	∀yb	∀yb	PROPN
ejpam-5057	315	21	∈	∈	PROPN
ejpam-5057	315	22	yb	yb	PROPN
ejpam-5057	315	23	with	with	ADP
ejpam-5057	315	24	the	the	DET
ejpam-5057	315	25	condition	condition	NOUN
ejpam-5057	315	26	that	that	PRON
ejpam-5057	315	27	|da	|da	VERB
ejpam-5057	315	28	−	−	PROPN
ejpam-5057	315	29	db|	db|	NOUN
ejpam-5057	315	30	=	=	PUNCT
ejpam-5057	315	31	{	{	PUNCT
ejpam-5057	315	32	1	1	NUM
ejpam-5057	315	33	if	if	SCONJ
ejpam-5057	315	34	dy	dy	NOUN
ejpam-5057	315	35	is	be	AUX
ejpam-5057	315	36	odd	odd	ADJ
ejpam-5057	315	37	0	0	NUM
ejpam-5057	315	38	if	if	SCONJ
ejpam-5057	315	39	dy	dy	NOUN
ejpam-5057	315	40	is	be	AUX
ejpam-5057	315	41	even	even	ADV
ejpam-5057	315	42	.	.	PUNCT
ejpam-5057	316	1	(	(	PUNCT
ejpam-5057	316	2	iii	iii	NOUN
ejpam-5057	316	3	)	)	PUNCT
ejpam-5057	316	4	n(ya	n(ya	NUM
ejpam-5057	316	5	)	)	PUNCT
ejpam-5057	316	6	∩n(yb	∩n(yb	NOUN
ejpam-5057	316	7	)	)	PUNCT
ejpam-5057	317	1	̸=	̸=	PROPN
ejpam-5057	317	2	∅	∅	NOUN
ejpam-5057	317	3	example	example	NOUN
ejpam-5057	317	4	:	:	PUNCT
ejpam-5057	317	5	vertex	vertex	NOUN
ejpam-5057	317	6	-	-	PUNCT
ejpam-5057	317	7	split	split	NOUN
ejpam-5057	317	8	of	of	ADP
ejpam-5057	317	9	a	a	DET
ejpam-5057	317	10	complete	complete	ADJ
ejpam-5057	317	11	graph	graph	NOUN
ejpam-5057	317	12	k8,4	k8,4	PROPN
ejpam-5057	317	13	is	be	AUX
ejpam-5057	317	14	shown	show	VERB
ejpam-5057	317	15	in	in	ADP
ejpam-5057	317	16	figure	figure	NOUN
ejpam-5057	317	17	1	1	NUM
ejpam-5057	317	18	where	where	SCONJ
ejpam-5057	317	19	|x|	|x|	PROPN
ejpam-5057	317	20	=	=	PUNCT
ejpam-5057	317	21	|x	|x	X
ejpam-5057	317	22	′|	′|	NUM
ejpam-5057	317	23	=	=	SYM
ejpam-5057	317	24	8	8	NUM
ejpam-5057	317	25	,	,	PUNCT
ejpam-5057	317	26	|y	|y	NOUN
ejpam-5057	317	27	|	|	NOUN
ejpam-5057	317	28	=	=	SYM
ejpam-5057	317	29	4	4	NUM
ejpam-5057	317	30	,	,	PUNCT
ejpam-5057	317	31	|y	|y	ADJ
ejpam-5057	317	32	′|	′|	NUM
ejpam-5057	317	33	=	=	SYM
ejpam-5057	317	34	8	8	NUM
ejpam-5057	317	35	m.machasri	m.machasri	NUM
ejpam-5057	317	36	,	,	PUNCT
ejpam-5057	317	37	d.kalyani	d.kalyani	NOUN
ejpam-5057	317	38	/	/	SYM
ejpam-5057	317	39	eur	eur	PROPN
ejpam-5057	317	40	.	.	PUNCT
ejpam-5057	318	1	j.	j.	PROPN
ejpam-5057	318	2	pure	pure	PROPN
ejpam-5057	318	3	appl	appl	PROPN
ejpam-5057	318	4	.	.	PROPN
ejpam-5057	318	5	math	math	PROPN
ejpam-5057	318	6	,	,	PUNCT
ejpam-5057	318	7	17	17	NUM
ejpam-5057	318	8	(	(	PUNCT
ejpam-5057	318	9	2	2	NUM
ejpam-5057	318	10	)	)	PUNCT
ejpam-5057	318	11	(	(	PUNCT
ejpam-5057	318	12	2024	2024	NUM
ejpam-5057	318	13	)	)	PUNCT
ejpam-5057	318	14	,	,	PUNCT
ejpam-5057	318	15	772	772	NUM
ejpam-5057	318	16	-	-	SYM
ejpam-5057	318	17	789	789	NUM
ejpam-5057	318	18	784	784	NUM
ejpam-5057	318	19	figure	figure	NOUN
ejpam-5057	318	20	1	1	NUM
ejpam-5057	318	21	:	:	PUNCT
ejpam-5057	318	22	vertex	vertex	NOUN
ejpam-5057	318	23	-	-	PUNCT
ejpam-5057	318	24	split	split	NOUN
ejpam-5057	318	25	of	of	ADP
ejpam-5057	318	26	k8,4	k8,4	PROPN
ejpam-5057	318	27	5.1	5.1	NUM
ejpam-5057	318	28	.	.	PUNCT
ejpam-5057	319	1	connectivity	connectivity	NOUN
ejpam-5057	319	2	of	of	ADP
ejpam-5057	319	3	vertex	vertex	NOUN
ejpam-5057	319	4	-	-	PUNCT
ejpam-5057	319	5	split	split	NOUN
ejpam-5057	319	6	of	of	ADP
ejpam-5057	319	7	a	a	DET
ejpam-5057	319	8	bipartite	bipartite	NOUN
ejpam-5057	319	9	graph	graph	NOUN
ejpam-5057	319	10	let	let	VERB
ejpam-5057	319	11	g′	g′	NOUN
ejpam-5057	319	12	be	be	AUX
ejpam-5057	319	13	a	a	DET
ejpam-5057	319	14	vertex	vertex	NOUN
ejpam-5057	319	15	-	-	PUNCT
ejpam-5057	319	16	split	split	NOUN
ejpam-5057	319	17	of	of	ADP
ejpam-5057	319	18	a	a	DET
ejpam-5057	319	19	bipartite	bipartite	NOUN
ejpam-5057	319	20	graph	graph	NOUN
ejpam-5057	319	21	g	g	NOUN
ejpam-5057	319	22	with	with	ADP
ejpam-5057	319	23	minimum	minimum	NOUN
ejpam-5057	319	24	degree	degree	NOUN
ejpam-5057	319	25	δ′	δ′	PROPN
ejpam-5057	319	26	and	and	CCONJ
ejpam-5057	319	27	edge	edge	VERB
ejpam-5057	319	28	connectivity	connectivity	NOUN
ejpam-5057	319	29	κ′(g′	κ′(g′	NOUN
ejpam-5057	319	30	)	)	PUNCT
ejpam-5057	319	31	.	.	PUNCT
ejpam-5057	320	1	the	the	DET
ejpam-5057	320	2	adjacency	adjacency	PROPN
ejpam-5057	320	3	eigenvalues	eigenvalue	VERB
ejpam-5057	320	4	of	of	ADP
ejpam-5057	320	5	g′	g′	NOUN
ejpam-5057	320	6	are	be	AUX
ejpam-5057	320	7	denoted	denote	VERB
ejpam-5057	320	8	as	as	ADP
ejpam-5057	320	9	λ′	λ′	X
ejpam-5057	320	10	1	1	NUM
ejpam-5057	320	11	≥	≥	NOUN
ejpam-5057	320	12	λ′	λ′	X
ejpam-5057	320	13	2	2	NUM
ejpam-5057	320	14	≥	≥	NOUN
ejpam-5057	320	15	·	·	PUNCT
ejpam-5057	320	16	·	·	PUNCT
ejpam-5057	320	17	·	·	PUNCT
ejpam-5057	320	18	≥	≥	NUM
ejpam-5057	320	19	λ′	λ′	X
ejpam-5057	320	20	n.	n.	NOUN
ejpam-5057	320	21	theorem	theorem	VERB
ejpam-5057	320	22	5	5	NUM
ejpam-5057	320	23	.	.	PUNCT
ejpam-5057	321	1	let	let	VERB
ejpam-5057	321	2	δ′	δ′	PROPN
ejpam-5057	321	3	≥	≥	PRON
ejpam-5057	321	4	k	k	X
ejpam-5057	321	5	≥	≥	NUM
ejpam-5057	321	6	2	2	NUM
ejpam-5057	321	7	be	be	AUX
ejpam-5057	321	8	a	a	DET
ejpam-5057	321	9	constant	constant	ADJ
ejpam-5057	321	10	and	and	CCONJ
ejpam-5057	321	11	let	let	VERB
ejpam-5057	321	12	g(x	g(x	PROPN
ejpam-5057	321	13	∪	∪	VERB
ejpam-5057	321	14	y	y	PROPN
ejpam-5057	321	15	,	,	PUNCT
ejpam-5057	321	16	e	e	NOUN
ejpam-5057	321	17	)	)	PUNCT
ejpam-5057	321	18	be	be	AUX
ejpam-5057	321	19	a	a	DET
ejpam-5057	321	20	d−regular	d−regular	NUM
ejpam-5057	321	21	bipartite	bipartite	NOUN
ejpam-5057	321	22	graph	graph	NOUN
ejpam-5057	321	23	and	and	CCONJ
ejpam-5057	321	24	g′	g′	NOUN
ejpam-5057	321	25	the	the	DET
ejpam-5057	321	26	vertex	vertex	NOUN
ejpam-5057	321	27	-	-	PUNCT
ejpam-5057	321	28	split	split	NOUN
ejpam-5057	321	29	of	of	ADP
ejpam-5057	321	30	g.	g.	PROPN
ejpam-5057	321	31	if	if	SCONJ
ejpam-5057	321	32	λ′	λ′	PROPN
ejpam-5057	321	33	2	2	NUM
ejpam-5057	321	34	≥	≥	NOUN
ejpam-5057	321	35	2k	2k	NOUN
ejpam-5057	322	1	−	−	PROPN
ejpam-5057	322	2	1√	1√	PROPN
ejpam-5057	322	3	2	2	NUM
ejpam-5057	322	4	then	then	ADV
ejpam-5057	322	5	κ′(g′	κ′(g′	PRON
ejpam-5057	322	6	)	)	PUNCT
ejpam-5057	322	7	≥	≥	PROPN
ejpam-5057	322	8	k.	k.	PROPN
ejpam-5057	323	1	proof	proof	PROPN
ejpam-5057	323	2	.	.	PUNCT
ejpam-5057	324	1	our	our	PRON
ejpam-5057	324	2	proof	proof	NOUN
ejpam-5057	324	3	is	be	AUX
ejpam-5057	324	4	by	by	ADP
ejpam-5057	324	5	contradiction	contradiction	NOUN
ejpam-5057	324	6	.	.	PUNCT
ejpam-5057	325	1	we	we	PRON
ejpam-5057	325	2	prove	prove	VERB
ejpam-5057	325	3	that	that	SCONJ
ejpam-5057	325	4	if	if	SCONJ
ejpam-5057	325	5	g′	g′	NOUN
ejpam-5057	325	6	is	be	AUX
ejpam-5057	325	7	a	a	DET
ejpam-5057	325	8	connected	connected	ADJ
ejpam-5057	325	9	bipartite	bipartite	NOUN
ejpam-5057	325	10	graph	graph	NOUN
ejpam-5057	325	11	of	of	ADP
ejpam-5057	325	12	order	order	NOUN
ejpam-5057	325	13	n′	n′	NOUN
ejpam-5057	325	14	and	and	CCONJ
ejpam-5057	325	15	minimum	minimum	NOUN
ejpam-5057	325	16	degree	degree	NOUN
ejpam-5057	325	17	δ′	δ′	NOUN
ejpam-5057	325	18	such	such	ADJ
ejpam-5057	325	19	that	that	SCONJ
ejpam-5057	325	20	κ′(g′	κ′(g′	VERB
ejpam-5057	325	21	)	)	PUNCT
ejpam-5057	325	22	≤	≤	NOUN
ejpam-5057	325	23	2k′	2k′	NUM
ejpam-5057	325	24	−	−	NOUN
ejpam-5057	325	25	1	1	NUM
ejpam-5057	325	26	,	,	PUNCT
ejpam-5057	325	27	then	then	ADV
ejpam-5057	325	28	λ′	λ′	X
ejpam-5057	325	29	2	2	NUM
ejpam-5057	325	30	≤	≤	NOUN
ejpam-5057	325	31	2k	2k	NOUN
ejpam-5057	325	32	−	−	PROPN
ejpam-5057	325	33	1√	1√	PROPN
ejpam-5057	325	34	2	2	NUM
ejpam-5057	325	35	.	.	PUNCT
ejpam-5057	326	1	(	(	PUNCT
ejpam-5057	326	2	5	5	X
ejpam-5057	326	3	)	)	PUNCT
ejpam-5057	326	4	let	let	VERB
ejpam-5057	326	5	n′	n′	NOUN
ejpam-5057	326	6	=	=	PUNCT
ejpam-5057	326	7	n′	n′	NOUN
ejpam-5057	326	8	1	1	NUM
ejpam-5057	326	9	+	+	CCONJ
ejpam-5057	326	10	n′	n′	PROPN
ejpam-5057	326	11	2	2	NUM
ejpam-5057	326	12	where	where	SCONJ
ejpam-5057	326	13	n′	n′	PROPN
ejpam-5057	326	14	1	1	NUM
ejpam-5057	326	15	=	=	SYM
ejpam-5057	326	16	n1	n1	PROPN
ejpam-5057	326	17	and	and	CCONJ
ejpam-5057	326	18	n′	n′	ADV
ejpam-5057	326	19	2	2	NUM
ejpam-5057	326	20	=	=	SYM
ejpam-5057	326	21	2n2	2n2	NUM
ejpam-5057	326	22	and	and	CCONJ
ejpam-5057	326	23	δ′	δ′	NOUN
ejpam-5057	326	24	=	=	SYM
ejpam-5057	326	25	δ	δ	PROPN
ejpam-5057	326	26	2	2	NUM
ejpam-5057	326	27	.	.	PUNCT
ejpam-5057	327	1	let	let	VERB
ejpam-5057	327	2	δ	δ	PRON
ejpam-5057	327	3	′	′	NOUN
ejpam-5057	327	4	≥	≥	PROPN
ejpam-5057	327	5	k	k	X
ejpam-5057	327	6	≥	≥	NUM
ejpam-5057	327	7	2	2	NUM
ejpam-5057	327	8	.	.	PUNCT
ejpam-5057	328	1	from	from	ADP
ejpam-5057	328	2	this	this	PRON
ejpam-5057	328	3	we	we	PRON
ejpam-5057	328	4	have	have	VERB
ejpam-5057	328	5	δ	δ	PROPN
ejpam-5057	328	6	2	2	NUM
ejpam-5057	328	7	≥	≥	NOUN
ejpam-5057	328	8	k	k	X
ejpam-5057	328	9	≥	≥	NUM
ejpam-5057	328	10	2	2	X
ejpam-5057	328	11	.	.	PUNCT
ejpam-5057	328	12	consider	consider	VERB
ejpam-5057	328	13	δ	δ	PROPN
ejpam-5057	328	14	2	2	NUM
ejpam-5057	328	15	≥	≥	NOUN
ejpam-5057	328	16	k.	k.	X
ejpam-5057	329	1	by	by	ADP
ejpam-5057	329	2	(	(	PUNCT
ejpam-5057	329	3	iii	iii	NOUN
ejpam-5057	329	4	)	)	PUNCT
ejpam-5057	329	5	of	of	ADP
ejpam-5057	329	6	theorem	theorem	NOUN
ejpam-5057	329	7	1	1	NUM
ejpam-5057	329	8	we	we	PRON
ejpam-5057	329	9	have	have	VERB
ejpam-5057	329	10	,	,	PUNCT
ejpam-5057	329	11	λ2	λ2	PROPN
ejpam-5057	329	12	≤	≤	NOUN
ejpam-5057	329	13	m	m	VERB
ejpam-5057	329	14	√	√	NUM
ejpam-5057	329	15	n1n2	n1n2	NOUN
ejpam-5057	329	16	.	.	PUNCT
ejpam-5057	330	1	(	(	PUNCT
ejpam-5057	330	2	6	6	NUM
ejpam-5057	330	3	)	)	PUNCT
ejpam-5057	330	4	for	for	ADP
ejpam-5057	330	5	a	a	DET
ejpam-5057	330	6	regular	regular	ADJ
ejpam-5057	330	7	bipartite	bipartite	NOUN
ejpam-5057	330	8	graph	graph	NOUN
ejpam-5057	330	9	n1	n1	PROPN
ejpam-5057	330	10	=	=	SYM
ejpam-5057	330	11	n2	n2	NOUN
ejpam-5057	330	12	=	=	PROPN
ejpam-5057	330	13	n	n	CCONJ
ejpam-5057	330	14	2	2	NUM
ejpam-5057	330	15	and	and	CCONJ
ejpam-5057	330	16	m	m	PROPN
ejpam-5057	330	17	=	=	NOUN
ejpam-5057	330	18	nd	nd	SYM
ejpam-5057	330	19	2	2	NUM
ejpam-5057	330	20	.	.	PUNCT
ejpam-5057	330	21	suppose	suppose	VERB
ejpam-5057	330	22	δ	δ	PROPN
ejpam-5057	330	23	≤	≤	NOUN
ejpam-5057	330	24	2k	2k	NOUN
ejpam-5057	330	25	−	−	NOUN
ejpam-5057	330	26	1	1	NUM
ejpam-5057	330	27	.	.	PUNCT
ejpam-5057	331	1	substituting	substitute	VERB
ejpam-5057	331	2	this	this	PRON
ejpam-5057	331	3	in	in	ADP
ejpam-5057	331	4	equation	equation	NOUN
ejpam-5057	331	5	(	(	PUNCT
ejpam-5057	331	6	6	6	NUM
ejpam-5057	331	7	)	)	PUNCT
ejpam-5057	331	8	we	we	PRON
ejpam-5057	331	9	get	get	VERB
ejpam-5057	331	10	,	,	PUNCT
ejpam-5057	331	11	λ′	λ′	X
ejpam-5057	331	12	2	2	NUM
ejpam-5057	331	13	≤	≤	NUM
ejpam-5057	331	14	d√	d√	PROPN
ejpam-5057	331	15	2	2	NUM
ejpam-5057	331	16	=	=	SYM
ejpam-5057	331	17	δ√	δ√	NUM
ejpam-5057	331	18	2	2	NUM
ejpam-5057	331	19	λ′	λ′	NOUN
ejpam-5057	331	20	2	2	NUM
ejpam-5057	331	21	≤	≤	NOUN
ejpam-5057	331	22	2k	2k	NOUN
ejpam-5057	331	23	−	−	PROPN
ejpam-5057	331	24	1√	1√	PROPN
ejpam-5057	331	25	2	2	NUM
ejpam-5057	331	26	this	this	PRON
ejpam-5057	331	27	completes	complete	VERB
ejpam-5057	331	28	the	the	DET
ejpam-5057	331	29	proof	proof	NOUN
ejpam-5057	331	30	.	.	PUNCT
ejpam-5057	332	1	example	example	NOUN
ejpam-5057	332	2	for	for	ADP
ejpam-5057	332	3	connectivity	connectivity	NOUN
ejpam-5057	332	4	of	of	ADP
ejpam-5057	332	5	vertex	vertex	NOUN
ejpam-5057	332	6	-	-	PUNCT
ejpam-5057	332	7	split	split	NOUN
ejpam-5057	332	8	of	of	ADP
ejpam-5057	332	9	a	a	DET
ejpam-5057	332	10	regular	regular	ADJ
ejpam-5057	332	11	graph	graph	NOUN
ejpam-5057	332	12	:	:	PUNCT
ejpam-5057	332	13	consider	consider	VERB
ejpam-5057	332	14	a	a	DET
ejpam-5057	332	15	complete	complete	ADJ
ejpam-5057	332	16	bipartite	bipartite	NOUN
ejpam-5057	332	17	graph	graph	NOUN
ejpam-5057	332	18	k5,5	k5,5	PROPN
ejpam-5057	332	19	.	.	PUNCT
ejpam-5057	333	1	the	the	DET
ejpam-5057	333	2	vertex	vertex	NOUN
ejpam-5057	333	3	-	-	PUNCT
ejpam-5057	333	4	split	split	NOUN
ejpam-5057	333	5	of	of	ADP
ejpam-5057	333	6	k5,5	k5,5	PROPN
ejpam-5057	333	7	is	be	AUX
ejpam-5057	333	8	a	a	DET
ejpam-5057	333	9	graph	graph	NOUN
ejpam-5057	333	10	with	with	ADP
ejpam-5057	333	11	minimum	minimum	NOUN
ejpam-5057	333	12	degree	degree	NOUN
ejpam-5057	333	13	δ′	δ′	NOUN
ejpam-5057	333	14	=	=	SYM
ejpam-5057	333	15	2	2	NUM
ejpam-5057	333	16	and	and	CCONJ
ejpam-5057	333	17	the	the	DET
ejpam-5057	333	18	second	second	ADV
ejpam-5057	333	19	largest	large	ADJ
ejpam-5057	333	20	eigenvalue	eigenvalue	ADJ
ejpam-5057	333	21	λ′	λ′	X
ejpam-5057	333	22	2	2	NUM
ejpam-5057	333	23	=	=	SYM
ejpam-5057	333	24	2.34	2.34	NUM
ejpam-5057	333	25	.	.	PUNCT
ejpam-5057	334	1	since	since	SCONJ
ejpam-5057	334	2	it	it	PRON
ejpam-5057	334	3	is	be	AUX
ejpam-5057	334	4	regular	regular	ADJ
ejpam-5057	334	5	graph	graph	NOUN
ejpam-5057	334	6	,	,	PUNCT
ejpam-5057	334	7	from	from	ADP
ejpam-5057	334	8	theorem	theorem	NOUN
ejpam-5057	334	9	1	1	NUM
ejpam-5057	334	10	we	we	PRON
ejpam-5057	334	11	have	have	VERB
ejpam-5057	334	12	,	,	PUNCT
ejpam-5057	334	13	for	for	ADP
ejpam-5057	334	14	k	k	PROPN
ejpam-5057	334	15	=	=	SYM
ejpam-5057	334	16	2	2	NUM
ejpam-5057	334	17	,	,	PUNCT
ejpam-5057	334	18	2k−1√	2k−1√	NUM
ejpam-5057	334	19	2	2	NUM
ejpam-5057	334	20	=	=	SYM
ejpam-5057	334	21	2.121	2.121	NUM
ejpam-5057	334	22	m.machasri	m.machasri	NUM
ejpam-5057	334	23	,	,	PUNCT
ejpam-5057	334	24	d.kalyani	d.kalyani	NOUN
ejpam-5057	334	25	/	/	SYM
ejpam-5057	334	26	eur	eur	PROPN
ejpam-5057	334	27	.	.	PUNCT
ejpam-5057	335	1	j.	j.	PROPN
ejpam-5057	335	2	pure	pure	PROPN
ejpam-5057	335	3	appl	appl	PROPN
ejpam-5057	335	4	.	.	PROPN
ejpam-5057	335	5	math	math	PROPN
ejpam-5057	335	6	,	,	PUNCT
ejpam-5057	335	7	17	17	NUM
ejpam-5057	335	8	(	(	PUNCT
ejpam-5057	335	9	2	2	NUM
ejpam-5057	335	10	)	)	PUNCT
ejpam-5057	335	11	(	(	PUNCT
ejpam-5057	335	12	2024	2024	NUM
ejpam-5057	335	13	)	)	PUNCT
ejpam-5057	335	14	,	,	PUNCT
ejpam-5057	335	15	772	772	NUM
ejpam-5057	335	16	-	-	SYM
ejpam-5057	335	17	789	789	NUM
ejpam-5057	335	18	785	785	NUM
ejpam-5057	335	19	figure	figure	NOUN
ejpam-5057	335	20	2	2	NUM
ejpam-5057	335	21	:	:	PUNCT
ejpam-5057	335	22	vertex	vertex	NOUN
ejpam-5057	335	23	-	-	PUNCT
ejpam-5057	335	24	split	split	NOUN
ejpam-5057	335	25	of	of	ADP
ejpam-5057	335	26	k5,5	k5,5	PROPN
ejpam-5057	335	27	theorem	theorem	VERB
ejpam-5057	335	28	6	6	NUM
ejpam-5057	335	29	.	.	PUNCT
ejpam-5057	336	1	let	let	VERB
ejpam-5057	336	2	g	g	PRON
ejpam-5057	336	3	be	be	AUX
ejpam-5057	336	4	a	a	DET
ejpam-5057	336	5	biregular	biregular	ADJ
ejpam-5057	336	6	bipartite	bipartite	NOUN
ejpam-5057	336	7	graph	graph	NOUN
ejpam-5057	336	8	with	with	ADP
ejpam-5057	336	9	v	v	NOUN
ejpam-5057	336	10	=	=	SYM
ejpam-5057	336	11	(	(	PUNCT
ejpam-5057	336	12	x	x	X
ejpam-5057	336	13	,	,	PUNCT
ejpam-5057	336	14	y	y	PROPN
ejpam-5057	336	15	)	)	PUNCT
ejpam-5057	336	16	where	where	SCONJ
ejpam-5057	336	17	|x|	|x|	PROPN
ejpam-5057	336	18	=	=	SYM
ejpam-5057	336	19	n1	n1	PROPN
ejpam-5057	336	20	,	,	PUNCT
ejpam-5057	336	21	|y	|y	NOUN
ejpam-5057	336	22	|	|	NOUN
ejpam-5057	336	23	=	=	SYM
ejpam-5057	336	24	n2	n2	ADJ
ejpam-5057	336	25	and	and	CCONJ
ejpam-5057	336	26	g′	g′	NOUN
ejpam-5057	336	27	the	the	DET
ejpam-5057	336	28	vertex	vertex	NOUN
ejpam-5057	336	29	-	-	PUNCT
ejpam-5057	336	30	split	split	NOUN
ejpam-5057	336	31	of	of	ADP
ejpam-5057	336	32	g.	g.	PROPN
ejpam-5057	336	33	if	if	SCONJ
ejpam-5057	336	34	λ′	λ′	PROPN
ejpam-5057	336	35	2	2	NUM
ejpam-5057	336	36	≥	≥	NOUN
ejpam-5057	336	37	n2(2k	n2(2k	ADJ
ejpam-5057	336	38	−	−	PROPN
ejpam-5057	336	39	1)√	1)√	NUM
ejpam-5057	336	40	2n1n2	2n1n2	NUM
ejpam-5057	336	41	then	then	ADV
ejpam-5057	336	42	κ′(g′	κ′(g′	NOUN
ejpam-5057	336	43	)	)	PUNCT
ejpam-5057	336	44	≥	≥	PROPN
ejpam-5057	336	45	k.	k.	PROPN
ejpam-5057	336	46	proof	proof	PROPN
ejpam-5057	336	47	.	.	PUNCT
ejpam-5057	337	1	the	the	DET
ejpam-5057	337	2	proof	proof	NOUN
ejpam-5057	337	3	is	be	AUX
ejpam-5057	337	4	by	by	ADP
ejpam-5057	337	5	contradiction	contradiction	NOUN
ejpam-5057	337	6	.	.	PUNCT
ejpam-5057	338	1	we	we	PRON
ejpam-5057	338	2	prove	prove	VERB
ejpam-5057	338	3	that	that	SCONJ
ejpam-5057	338	4	if	if	SCONJ
ejpam-5057	338	5	g′	g′	NOUN
ejpam-5057	338	6	is	be	AUX
ejpam-5057	338	7	a	a	DET
ejpam-5057	338	8	connected	connected	ADJ
ejpam-5057	338	9	bipartite	bipartite	NOUN
ejpam-5057	338	10	graph	graph	NOUN
ejpam-5057	338	11	of	of	ADP
ejpam-5057	338	12	order	order	NOUN
ejpam-5057	338	13	n′	n′	NOUN
ejpam-5057	338	14	and	and	CCONJ
ejpam-5057	338	15	minimum	minimum	NOUN
ejpam-5057	338	16	degree	degree	NOUN
ejpam-5057	338	17	δ′	δ′	NOUN
ejpam-5057	338	18	such	such	ADJ
ejpam-5057	338	19	that	that	SCONJ
ejpam-5057	338	20	κ′(g′	κ′(g′	VERB
ejpam-5057	338	21	)	)	PUNCT
ejpam-5057	338	22	≤	≤	NOUN
ejpam-5057	338	23	2k	2k	NOUN
ejpam-5057	338	24	−	−	NOUN
ejpam-5057	338	25	1	1	NUM
ejpam-5057	338	26	,	,	PUNCT
ejpam-5057	338	27	then	then	ADV
ejpam-5057	338	28	λ′	λ′	X
ejpam-5057	338	29	2	2	NUM
ejpam-5057	338	30	≥	≥	NOUN
ejpam-5057	338	31	n2(2k	n2(2k	ADJ
ejpam-5057	338	32	−	−	PROPN
ejpam-5057	338	33	1)√	1)√	NUM
ejpam-5057	338	34	2n1n2	2n1n2	NUM
ejpam-5057	338	35	.	.	PUNCT
ejpam-5057	339	1	let	let	VERB
ejpam-5057	339	2	n′	n′	NOUN
ejpam-5057	339	3	=	=	PUNCT
ejpam-5057	339	4	n′	n′	NOUN
ejpam-5057	339	5	1	1	NUM
ejpam-5057	339	6	+	+	CCONJ
ejpam-5057	339	7	n′	n′	PROPN
ejpam-5057	339	8	2	2	NUM
ejpam-5057	339	9	where	where	SCONJ
ejpam-5057	339	10	n′	n′	PROPN
ejpam-5057	339	11	1	1	NUM
ejpam-5057	339	12	=	=	SYM
ejpam-5057	339	13	n1	n1	PROPN
ejpam-5057	339	14	and	and	CCONJ
ejpam-5057	339	15	n′	n′	ADV
ejpam-5057	339	16	2	2	NUM
ejpam-5057	339	17	=	=	SYM
ejpam-5057	339	18	2n2	2n2	NUM
ejpam-5057	339	19	and	and	CCONJ
ejpam-5057	339	20	δ′	δ′	NOUN
ejpam-5057	339	21	=	=	SYM
ejpam-5057	339	22	δ	δ	PROPN
ejpam-5057	339	23	2	2	NUM
ejpam-5057	339	24	.	.	PUNCT
ejpam-5057	340	1	let	let	VERB
ejpam-5057	340	2	δ	δ	PRON
ejpam-5057	340	3	′	′	NOUN
ejpam-5057	340	4	≥	≥	PROPN
ejpam-5057	340	5	k	k	X
ejpam-5057	340	6	≥	≥	NUM
ejpam-5057	340	7	2	2	NUM
ejpam-5057	340	8	.	.	PUNCT
ejpam-5057	341	1	therefore	therefore	ADV
ejpam-5057	341	2	we	we	PRON
ejpam-5057	341	3	have	have	AUX
ejpam-5057	341	4	,	,	PUNCT
ejpam-5057	341	5	δ	δ	PROPN
ejpam-5057	341	6	2	2	NUM
ejpam-5057	341	7	≥	≥	NOUN
ejpam-5057	341	8	k	k	X
ejpam-5057	341	9	≥	≥	NUM
ejpam-5057	341	10	2	2	X
ejpam-5057	341	11	.	.	PUNCT
ejpam-5057	341	12	consider	consider	VERB
ejpam-5057	341	13	δ	δ	PROPN
ejpam-5057	341	14	2	2	NUM
ejpam-5057	341	15	≥	≥	NOUN
ejpam-5057	341	16	k.	k.	PROPN
ejpam-5057	341	17	suppose	suppose	VERB
ejpam-5057	341	18	δ	δ	PROPN
ejpam-5057	341	19	≤	≤	NOUN
ejpam-5057	341	20	2k	2k	NOUN
ejpam-5057	341	21	−	−	NOUN
ejpam-5057	341	22	1	1	X
ejpam-5057	341	23	.	.	PUNCT
ejpam-5057	342	1	for	for	ADP
ejpam-5057	342	2	a	a	DET
ejpam-5057	342	3	biregular	biregular	ADJ
ejpam-5057	342	4	bipartite	bipartite	NOUN
ejpam-5057	342	5	graph	graph	NOUN
ejpam-5057	342	6	n1d1	n1d1	NOUN
ejpam-5057	342	7	=	=	PUNCT
ejpam-5057	342	8	n2d2	n2d2	X
ejpam-5057	342	9	=	=	VERB
ejpam-5057	342	10	m	m	NOUN
ejpam-5057	342	11	,	,	PUNCT
ejpam-5057	342	12	where	where	SCONJ
ejpam-5057	342	13	d1	d1	PROPN
ejpam-5057	342	14	and	and	CCONJ
ejpam-5057	342	15	d2	d2	PROPN
ejpam-5057	342	16	are	be	AUX
ejpam-5057	342	17	the	the	DET
ejpam-5057	342	18	degrees	degree	NOUN
ejpam-5057	342	19	of	of	ADP
ejpam-5057	342	20	the	the	DET
ejpam-5057	342	21	bipartition	bipartition	NOUN
ejpam-5057	342	22	with	with	ADP
ejpam-5057	342	23	d1	d1	PROPN
ejpam-5057	342	24	>	>	X
ejpam-5057	342	25	d2	d2	PROPN
ejpam-5057	342	26	.	.	PUNCT
ejpam-5057	343	1	similarly	similarly	ADV
ejpam-5057	343	2	n′	n′	PROPN
ejpam-5057	343	3	1d	1d	NUM
ejpam-5057	343	4	′	′	NOUN
ejpam-5057	343	5	1	1	NUM
ejpam-5057	343	6	=	=	SYM
ejpam-5057	343	7	n′	n′	PRON
ejpam-5057	343	8	2d	2d	NOUN
ejpam-5057	343	9	′	′	NOUN
ejpam-5057	344	1	1	1	NUM
ejpam-5057	344	2	=	=	SYM
ejpam-5057	344	3	m′	m′	NOUN
ejpam-5057	344	4	=	=	SYM
ejpam-5057	344	5	m.	m.	NOUN
ejpam-5057	344	6	substituting	substitute	VERB
ejpam-5057	344	7	this	this	PRON
ejpam-5057	344	8	in	in	ADP
ejpam-5057	344	9	(	(	PUNCT
ejpam-5057	344	10	iii	iii	NOUN
ejpam-5057	344	11	)	)	PUNCT
ejpam-5057	344	12	of	of	ADP
ejpam-5057	344	13	theorem	theorem	NOUN
ejpam-5057	344	14	1	1	NUM
ejpam-5057	344	15	we	we	PRON
ejpam-5057	344	16	have	have	VERB
ejpam-5057	344	17	,	,	PUNCT
ejpam-5057	344	18	λ′	λ′	X
ejpam-5057	344	19	2	2	NUM
ejpam-5057	344	20	≤	≤	NUM
ejpam-5057	344	21	n2d2√	n2d2√	NOUN
ejpam-5057	344	22	2n1n2	2n1n2	NUM
ejpam-5057	344	23	=	=	PUNCT
ejpam-5057	344	24	n2δ√	n2δ√	NOUN
ejpam-5057	344	25	2n1n2	2n1n2	NUM
ejpam-5057	344	26	λ′	λ′	NOUN
ejpam-5057	344	27	2	2	NUM
ejpam-5057	344	28	≤	≤	NOUN
ejpam-5057	344	29	n2(2k	n2(2k	ADJ
ejpam-5057	344	30	−	−	PROPN
ejpam-5057	344	31	1)√	1)√	NUM
ejpam-5057	344	32	2n1n2	2n1n2	NUM
ejpam-5057	344	33	this	this	PRON
ejpam-5057	344	34	completes	complete	VERB
ejpam-5057	344	35	the	the	DET
ejpam-5057	344	36	proof	proof	NOUN
ejpam-5057	344	37	.	.	PUNCT
ejpam-5057	345	1	m.machasri	m.machasri	NUM
ejpam-5057	345	2	,	,	PUNCT
ejpam-5057	345	3	d.kalyani	d.kalyani	NOUN
ejpam-5057	345	4	/	/	SYM
ejpam-5057	345	5	eur	eur	PROPN
ejpam-5057	345	6	.	.	PUNCT
ejpam-5057	346	1	j.	j.	PROPN
ejpam-5057	346	2	pure	pure	PROPN
ejpam-5057	346	3	appl	appl	PROPN
ejpam-5057	346	4	.	.	PROPN
ejpam-5057	346	5	math	math	PROPN
ejpam-5057	346	6	,	,	PUNCT
ejpam-5057	346	7	17	17	NUM
ejpam-5057	346	8	(	(	PUNCT
ejpam-5057	346	9	2	2	NUM
ejpam-5057	346	10	)	)	PUNCT
ejpam-5057	346	11	(	(	PUNCT
ejpam-5057	346	12	2024	2024	NUM
ejpam-5057	346	13	)	)	PUNCT
ejpam-5057	346	14	,	,	PUNCT
ejpam-5057	346	15	772	772	NUM
ejpam-5057	346	16	-	-	SYM
ejpam-5057	346	17	789	789	NUM
ejpam-5057	346	18	786	786	NUM
ejpam-5057	346	19	5.2	5.2	NUM
ejpam-5057	346	20	.	.	PUNCT
ejpam-5057	347	1	expansion	expansion	NOUN
ejpam-5057	347	2	of	of	ADP
ejpam-5057	347	3	vertex	vertex	NOUN
ejpam-5057	347	4	-	-	PUNCT
ejpam-5057	347	5	split	split	NOUN
ejpam-5057	347	6	of	of	ADP
ejpam-5057	347	7	a	a	DET
ejpam-5057	347	8	bipartite	bipartite	ADJ
ejpam-5057	347	9	graph	graph	NOUN
ejpam-5057	347	10	in	in	ADP
ejpam-5057	347	11	this	this	DET
ejpam-5057	347	12	section	section	NOUN
ejpam-5057	347	13	,	,	PUNCT
ejpam-5057	347	14	we	we	PRON
ejpam-5057	347	15	prove	prove	VERB
ejpam-5057	347	16	that	that	SCONJ
ejpam-5057	347	17	the	the	DET
ejpam-5057	347	18	vertex	vertex	NOUN
ejpam-5057	347	19	split	split	NOUN
ejpam-5057	347	20	of	of	ADP
ejpam-5057	347	21	a	a	DET
ejpam-5057	347	22	complete	complete	ADJ
ejpam-5057	347	23	bipartite	bipartite	NOUN
ejpam-5057	347	24	graph	graph	NOUN
ejpam-5057	347	25	is	be	AUX
ejpam-5057	347	26	a	a	DET
ejpam-5057	347	27	bipartite	bipartite	ADJ
ejpam-5057	347	28	expander	expander	NOUN
ejpam-5057	347	29	.	.	PUNCT
ejpam-5057	348	1	theorem	theorem	VERB
ejpam-5057	348	2	7	7	NUM
ejpam-5057	348	3	.	.	PUNCT
ejpam-5057	349	1	the	the	DET
ejpam-5057	349	2	vertex	vertex	NOUN
ejpam-5057	349	3	-	-	PUNCT
ejpam-5057	349	4	split	split	NOUN
ejpam-5057	349	5	of	of	ADP
ejpam-5057	349	6	a	a	DET
ejpam-5057	349	7	complete	complete	ADJ
ejpam-5057	349	8	bipartite	bipartite	NOUN
ejpam-5057	349	9	graph	graph	NOUN
ejpam-5057	349	10	km	km	PROPN
ejpam-5057	349	11	,	,	PUNCT
ejpam-5057	349	12	n	n	PRON
ejpam-5057	349	13	is	be	AUX
ejpam-5057	349	14	a	a	DET
ejpam-5057	349	15	bipartite	bipartite	ADJ
ejpam-5057	349	16	expander	expander	NOUN
ejpam-5057	349	17	if	if	SCONJ
ejpam-5057	349	18	(	(	PUNCT
ejpam-5057	349	19	i	i	NOUN
ejpam-5057	349	20	)	)	PUNCT
ejpam-5057	349	21	n	n	CCONJ
ejpam-5057	349	22	is	be	AUX
ejpam-5057	349	23	even	even	ADV
ejpam-5057	349	24	(	(	PUNCT
ejpam-5057	349	25	ii	ii	NOUN
ejpam-5057	349	26	)	)	PUNCT
ejpam-5057	350	1	n	n	PRON
ejpam-5057	351	1	≤	≤	NUM
ejpam-5057	351	2	m	m	VERB
ejpam-5057	351	3	≤	≤	NOUN
ejpam-5057	351	4	2n	2n	NUM
ejpam-5057	351	5	proof	proof	NOUN
ejpam-5057	351	6	.	.	PUNCT
ejpam-5057	352	1	let	let	VERB
ejpam-5057	352	2	g′	g′	NOUN
ejpam-5057	352	3	=	=	PUNCT
ejpam-5057	353	1	(	(	PUNCT
ejpam-5057	353	2	x	x	SYM
ejpam-5057	353	3	′	′	NOUN
ejpam-5057	353	4	∪	∪	ADP
ejpam-5057	353	5	y	y	PROPN
ejpam-5057	353	6	′	′	NUM
ejpam-5057	353	7	,	,	PUNCT
ejpam-5057	353	8	e′	e′	ADJ
ejpam-5057	353	9	)	)	PUNCT
ejpam-5057	353	10	be	be	VERB
ejpam-5057	353	11	the	the	DET
ejpam-5057	353	12	vertex	vertex	NOUN
ejpam-5057	353	13	-	-	PUNCT
ejpam-5057	353	14	split	split	NOUN
ejpam-5057	353	15	of	of	ADP
ejpam-5057	353	16	the	the	DET
ejpam-5057	353	17	complete	complete	ADJ
ejpam-5057	353	18	bipartite	bipartite	PROPN
ejpam-5057	353	19	graph	graph	NOUN
ejpam-5057	353	20	g.	g.	PROPN
ejpam-5057	353	21	here	here	ADV
ejpam-5057	353	22	|x	|x	VERB
ejpam-5057	353	23	′|	′|	NUM
ejpam-5057	353	24	=	=	SYM
ejpam-5057	353	25	|x|	|x|	PROPN
ejpam-5057	353	26	=	=	SYM
ejpam-5057	353	27	m	m	PROPN
ejpam-5057	353	28	,	,	PUNCT
ejpam-5057	353	29	|y	|y	VERB
ejpam-5057	353	30	′|	′|	NUM
ejpam-5057	353	31	=	=	SYM
ejpam-5057	353	32	2|y	2|y	NUM
ejpam-5057	353	33	|	|	NOUN
ejpam-5057	353	34	=	=	SYM
ejpam-5057	353	35	2n	2n	X
ejpam-5057	353	36	.	.	PUNCT
ejpam-5057	354	1	assume	assume	VERB
ejpam-5057	354	2	that	that	SCONJ
ejpam-5057	354	3	s	s	VERB
ejpam-5057	354	4	⊆	⊆	NUM
ejpam-5057	354	5	x	x	SYM
ejpam-5057	354	6	′	′	NOUN
ejpam-5057	354	7	with	with	ADP
ejpam-5057	354	8	|s|	|s|	NOUN
ejpam-5057	354	9	=	=	SYM
ejpam-5057	354	10	n	n	PRON
ejpam-5057	354	11	2	2	NUM
ejpam-5057	354	12	.	.	PUNCT
ejpam-5057	355	1	to	to	PART
ejpam-5057	355	2	prove	prove	VERB
ejpam-5057	355	3	this	this	DET
ejpam-5057	355	4	theorem	theorem	NOUN
ejpam-5057	355	5	,	,	PUNCT
ejpam-5057	355	6	consider	consider	VERB
ejpam-5057	355	7	the	the	DET
ejpam-5057	355	8	following	follow	VERB
ejpam-5057	355	9	cases	case	NOUN
ejpam-5057	355	10	.	.	PUNCT
ejpam-5057	356	1	case	case	NOUN
ejpam-5057	356	2	1	1	NUM
ejpam-5057	356	3	:	:	PUNCT
ejpam-5057	356	4	m	m	VERB
ejpam-5057	356	5	=	=	VERB
ejpam-5057	356	6	n.	n.	ADJ
ejpam-5057	356	7	in	in	ADP
ejpam-5057	356	8	this	this	DET
ejpam-5057	356	9	case	case	NOUN
ejpam-5057	356	10	,	,	PUNCT
ejpam-5057	356	11	when	when	SCONJ
ejpam-5057	356	12	|s|	|s|	PROPN
ejpam-5057	356	13	=	=	SYM
ejpam-5057	356	14	m	m	PROPN
ejpam-5057	356	15	2	2	NUM
ejpam-5057	356	16	the	the	DET
ejpam-5057	356	17	cardinality	cardinality	NOUN
ejpam-5057	356	18	of	of	ADP
ejpam-5057	356	19	the	the	DET
ejpam-5057	356	20	neighbourhood	neighbourhood	NOUN
ejpam-5057	356	21	of	of	ADP
ejpam-5057	356	22	s	s	NOUN
ejpam-5057	356	23	is	be	AUX
ejpam-5057	356	24	|n(s)|	|n(s)|	X
ejpam-5057	356	25	=	=	SYM
ejpam-5057	356	26	m+2	m+2	SYM
ejpam-5057	356	27	2	2	NUM
ejpam-5057	356	28	which	which	PRON
ejpam-5057	356	29	implies	imply	VERB
ejpam-5057	356	30	that	that	SCONJ
ejpam-5057	356	31	|n(s)|	|n(s)|	PROPN
ejpam-5057	356	32	≥	≥	NUM
ejpam-5057	356	33	α|s|	α|s|	NOUN
ejpam-5057	356	34	where	where	SCONJ
ejpam-5057	356	35	α	α	NOUN
ejpam-5057	356	36	=	=	X
ejpam-5057	356	37	(	(	PUNCT
ejpam-5057	356	38	1	1	NUM
ejpam-5057	356	39	+	+	SYM
ejpam-5057	356	40	2	2	NUM
ejpam-5057	356	41	n	n	NUM
ejpam-5057	356	42	)	)	PUNCT
ejpam-5057	356	43	.	.	PUNCT
ejpam-5057	357	1	case	case	NOUN
ejpam-5057	357	2	2	2	NUM
ejpam-5057	357	3	:	:	PUNCT
ejpam-5057	357	4	m	m	PROPN
ejpam-5057	357	5	=	=	ADJ
ejpam-5057	357	6	2n	2n	NUM
ejpam-5057	357	7	.	.	PUNCT
ejpam-5057	358	1	in	in	ADP
ejpam-5057	358	2	this	this	DET
ejpam-5057	358	3	case	case	NOUN
ejpam-5057	358	4	,	,	PUNCT
ejpam-5057	358	5	when	when	SCONJ
ejpam-5057	358	6	|s|	|s|	PROPN
ejpam-5057	358	7	=	=	SYM
ejpam-5057	358	8	n	n	PRON
ejpam-5057	358	9	2	2	NUM
ejpam-5057	358	10	the	the	DET
ejpam-5057	358	11	cardinality	cardinality	NOUN
ejpam-5057	358	12	of	of	ADP
ejpam-5057	358	13	the	the	DET
ejpam-5057	358	14	neighbourhood	neighbourhood	NOUN
ejpam-5057	358	15	of	of	ADP
ejpam-5057	358	16	s	s	NOUN
ejpam-5057	358	17	is	be	AUX
ejpam-5057	358	18	|n(s)|	|n(s)|	X
ejpam-5057	358	19	=	=	SYM
ejpam-5057	358	20	n	n	NUM
ejpam-5057	358	21	which	which	PRON
ejpam-5057	358	22	implies	imply	VERB
ejpam-5057	358	23	that	that	SCONJ
ejpam-5057	358	24	|n(s)|	|n(s)|	PROPN
ejpam-5057	358	25	≥	≥	NUM
ejpam-5057	358	26	α|s|	α|s|	NOUN
ejpam-5057	358	27	where	where	SCONJ
ejpam-5057	358	28	α	α	NOUN
ejpam-5057	358	29	=	=	SYM
ejpam-5057	358	30	1	1	X
ejpam-5057	358	31	.	.	PUNCT
ejpam-5057	358	32	case	case	NOUN
ejpam-5057	358	33	3	3	NUM
ejpam-5057	358	34	:	:	PUNCT
ejpam-5057	358	35	n	n	ADP
ejpam-5057	358	36	<	<	X
ejpam-5057	358	37	m	m	X
ejpam-5057	358	38	<	<	X
ejpam-5057	358	39	2n	2n	NUM
ejpam-5057	358	40	.	.	PUNCT
ejpam-5057	359	1	in	in	ADP
ejpam-5057	359	2	this	this	DET
ejpam-5057	359	3	case	case	NOUN
ejpam-5057	359	4	,	,	PUNCT
ejpam-5057	359	5	when	when	SCONJ
ejpam-5057	359	6	|s|	|s|	PROPN
ejpam-5057	359	7	=	=	SYM
ejpam-5057	359	8	n	n	PRON
ejpam-5057	359	9	2	2	NUM
ejpam-5057	359	10	the	the	DET
ejpam-5057	359	11	cardinality	cardinality	NOUN
ejpam-5057	359	12	of	of	ADP
ejpam-5057	359	13	the	the	DET
ejpam-5057	359	14	neighbourhood	neighbourhood	NOUN
ejpam-5057	359	15	of	of	ADP
ejpam-5057	359	16	s	s	NOUN
ejpam-5057	359	17	is	be	AUX
ejpam-5057	359	18	|n(s)|	|n(s)|	PROPN
ejpam-5057	359	19	=	=	SYM
ejpam-5057	359	20	n+(i−	n+(i−	NOUN
ejpam-5057	359	21	n	n	PRON
ejpam-5057	359	22	2	2	NUM
ejpam-5057	359	23	)	)	PUNCT
ejpam-5057	359	24	,	,	PUNCT
ejpam-5057	359	25	i	i	PRON
ejpam-5057	359	26	=	=	NOUN
ejpam-5057	359	27	2	2	NUM
ejpam-5057	359	28	,	,	PUNCT
ejpam-5057	359	29	3	3	NUM
ejpam-5057	359	30	,	,	PUNCT
ejpam-5057	359	31	.	.	PUNCT
ejpam-5057	359	32	.	.	PUNCT
ejpam-5057	359	33	.	.	PUNCT
ejpam-5057	360	1	,	,	PUNCT
ejpam-5057	360	2	n2	n2	PROPN
ejpam-5057	360	3	which	which	PRON
ejpam-5057	360	4	implies	imply	VERB
ejpam-5057	360	5	that	that	SCONJ
ejpam-5057	360	6	|n(s)|	|n(s)|	PROPN
ejpam-5057	360	7	≥	≥	NUM
ejpam-5057	360	8	α|s|	α|s|	NOUN
ejpam-5057	360	9	where	where	SCONJ
ejpam-5057	360	10	α	α	NOUN
ejpam-5057	360	11	=	=	SYM
ejpam-5057	360	12	1	1	NUM
ejpam-5057	360	13	+	+	CCONJ
ejpam-5057	360	14	(	(	PUNCT
ejpam-5057	360	15	i−n	i−n	PROPN
ejpam-5057	360	16	2	2	NUM
ejpam-5057	360	17	)	)	PUNCT
ejpam-5057	360	18	n	n	CCONJ
ejpam-5057	360	19	,	,	PUNCT
ejpam-5057	360	20	where	where	SCONJ
ejpam-5057	360	21	i	i	PRON
ejpam-5057	360	22	=	=	NOUN
ejpam-5057	360	23	2	2	NUM
ejpam-5057	360	24	,	,	PUNCT
ejpam-5057	360	25	3	3	NUM
ejpam-5057	360	26	,	,	PUNCT
ejpam-5057	360	27	.	.	PUNCT
ejpam-5057	360	28	.	.	PUNCT
ejpam-5057	361	1	.	.	PUNCT
ejpam-5057	362	1	,	,	PUNCT
ejpam-5057	362	2	n2	n2	PROPN
ejpam-5057	362	3	.	.	PUNCT
ejpam-5057	363	1	by	by	ADP
ejpam-5057	363	2	definition	definition	NOUN
ejpam-5057	363	3	(	(	PUNCT
ejpam-5057	363	4	4	4	NUM
ejpam-5057	363	5	)	)	PUNCT
ejpam-5057	363	6	,	,	PUNCT
ejpam-5057	363	7	in	in	ADP
ejpam-5057	363	8	all	all	DET
ejpam-5057	363	9	the	the	DET
ejpam-5057	363	10	cases	case	NOUN
ejpam-5057	363	11	we	we	PRON
ejpam-5057	363	12	conclude	conclude	VERB
ejpam-5057	363	13	that	that	SCONJ
ejpam-5057	363	14	the	the	DET
ejpam-5057	363	15	vertex	vertex	NOUN
ejpam-5057	363	16	-	-	PUNCT
ejpam-5057	363	17	split	split	NOUN
ejpam-5057	363	18	of	of	ADP
ejpam-5057	363	19	km	km	PROPN
ejpam-5057	363	20	,	,	PUNCT
ejpam-5057	363	21	n	n	PRON
ejpam-5057	363	22	is	be	AUX
ejpam-5057	363	23	a	a	DET
ejpam-5057	363	24	bipartite	bipartite	ADJ
ejpam-5057	363	25	expander	expander	NOUN
ejpam-5057	363	26	.	.	PUNCT
ejpam-5057	364	1	note	note	NOUN
ejpam-5057	364	2	:	:	PUNCT
ejpam-5057	364	3	from	from	ADP
ejpam-5057	364	4	the	the	DET
ejpam-5057	364	5	above	above	ADJ
ejpam-5057	364	6	theorem	theorem	NOUN
ejpam-5057	364	7	we	we	PRON
ejpam-5057	364	8	say	say	VERB
ejpam-5057	364	9	that	that	SCONJ
ejpam-5057	364	10	,	,	PUNCT
ejpam-5057	364	11	vertex	vertex	NOUN
ejpam-5057	364	12	-	-	PUNCT
ejpam-5057	364	13	split	split	NOUN
ejpam-5057	364	14	g′	g′	NOUN
ejpam-5057	364	15	of	of	ADP
ejpam-5057	364	16	g	g	PROPN
ejpam-5057	364	17	is	be	AUX
ejpam-5057	364	18	a	a	DET
ejpam-5057	364	19	α−	α−	PROPN
ejpam-5057	364	20	vertex	vertex	NOUN
ejpam-5057	364	21	expander	expander	NOUN
ejpam-5057	364	22	.	.	PUNCT
ejpam-5057	365	1	corollary	corollary	ADJ
ejpam-5057	365	2	4	4	NUM
ejpam-5057	365	3	.	.	PUNCT
ejpam-5057	366	1	the	the	DET
ejpam-5057	366	2	vertex	vertex	NOUN
ejpam-5057	366	3	-	-	PUNCT
ejpam-5057	366	4	split	split	NOUN
ejpam-5057	366	5	g′	g′	NOUN
ejpam-5057	366	6	of	of	ADP
ejpam-5057	366	7	a	a	DET
ejpam-5057	366	8	complete	complete	ADJ
ejpam-5057	366	9	bipartite	bipartite	NOUN
ejpam-5057	366	10	graph	graph	NOUN
ejpam-5057	366	11	g	g	PROPN
ejpam-5057	366	12	=	=	SYM
ejpam-5057	366	13	km	km	PROPN
ejpam-5057	366	14	,	,	PUNCT
ejpam-5057	366	15	n	n	PRON
ejpam-5057	366	16	where	where	SCONJ
ejpam-5057	366	17	n	n	NOUN
ejpam-5057	366	18	=	=	SYM
ejpam-5057	366	19	m	m	PRON
ejpam-5057	366	20	2	2	NUM
ejpam-5057	366	21	is	be	AUX
ejpam-5057	366	22	a	a	DET
ejpam-5057	366	23	(	(	PUNCT
ejpam-5057	366	24	1−	1−	NUM
ejpam-5057	366	25	γ)−spectral	γ)−spectral	ADJ
ejpam-5057	366	26	expander	expander	NOUN
ejpam-5057	366	27	.	.	PUNCT
ejpam-5057	367	1	proof	proof	NOUN
ejpam-5057	367	2	.	.	PUNCT
ejpam-5057	368	1	from	from	ADP
ejpam-5057	368	2	the	the	DET
ejpam-5057	368	3	above	above	ADJ
ejpam-5057	368	4	theorem	theorem	NOUN
ejpam-5057	368	5	we	we	PRON
ejpam-5057	368	6	know	know	VERB
ejpam-5057	368	7	that	that	SCONJ
ejpam-5057	368	8	g′	g′	NOUN
ejpam-5057	368	9	is	be	AUX
ejpam-5057	368	10	a	a	DET
ejpam-5057	368	11	α−	α−	PROPN
ejpam-5057	368	12	vertex	vertex	NOUN
ejpam-5057	368	13	expander	expander	NOUN
ejpam-5057	368	14	where	where	SCONJ
ejpam-5057	368	15	α	α	NOUN
ejpam-5057	368	16	=	=	NOUN
ejpam-5057	368	17	1	1	NUM
ejpam-5057	368	18	+	+	CCONJ
ejpam-5057	368	19	ϵ	ϵ	X
ejpam-5057	368	20	where	where	SCONJ
ejpam-5057	368	21	ϵ	ϵ	NOUN
ejpam-5057	368	22	=	=	SYM
ejpam-5057	368	23	1	1	NUM
ejpam-5057	368	24	n	n	NOUN
ejpam-5057	368	25	.	.	PUNCT
ejpam-5057	369	1	from	from	ADP
ejpam-5057	369	2	lemma	lemma	PROPN
ejpam-5057	369	3	2	2	NUM
ejpam-5057	369	4	,	,	PUNCT
ejpam-5057	369	5	we	we	PRON
ejpam-5057	369	6	can	can	AUX
ejpam-5057	369	7	conclude	conclude	VERB
ejpam-5057	369	8	that	that	PRON
ejpam-5057	369	9	g′	g′	NOUN
ejpam-5057	369	10	is	be	AUX
ejpam-5057	369	11	(	(	PUNCT
ejpam-5057	369	12	1	1	NUM
ejpam-5057	369	13	−	−	PROPN
ejpam-5057	369	14	γ)−spectral	γ)−spectral	ADJ
ejpam-5057	369	15	expander	expander	NOUN
ejpam-5057	369	16	where	where	SCONJ
ejpam-5057	369	17	γ	γ	X
ejpam-5057	369	18	=	=	SYM
ejpam-5057	369	19	1	1	NUM
ejpam-5057	369	20	n2d′	n2d′	NOUN
ejpam-5057	369	21	,	,	PUNCT
ejpam-5057	369	22	where	where	SCONJ
ejpam-5057	369	23	d′	d′	PRON
ejpam-5057	369	24	is	be	AUX
ejpam-5057	369	25	the	the	DET
ejpam-5057	369	26	degree	degree	NOUN
ejpam-5057	369	27	of	of	ADP
ejpam-5057	369	28	g′.	g′.	NOUN
ejpam-5057	369	29	6	6	NUM
ejpam-5057	369	30	.	.	PUNCT
ejpam-5057	369	31	construction	construction	NOUN
ejpam-5057	369	32	of	of	ADP
ejpam-5057	369	33	error	error	NOUN
ejpam-5057	369	34	correcting	correct	VERB
ejpam-5057	369	35	code	code	NOUN
ejpam-5057	369	36	consider	consider	VERB
ejpam-5057	369	37	a	a	DET
ejpam-5057	369	38	biregular	biregular	ADJ
ejpam-5057	369	39	graphg	graphg	NOUN
ejpam-5057	369	40	with	with	ADP
ejpam-5057	369	41	bipartitions	bipartition	NOUN
ejpam-5057	369	42	(	(	PUNCT
ejpam-5057	369	43	x	x	X
ejpam-5057	369	44	,	,	PUNCT
ejpam-5057	369	45	y	y	PROPN
ejpam-5057	369	46	)	)	PUNCT
ejpam-5057	369	47	and	and	CCONJ
ejpam-5057	369	48	the	the	DET
ejpam-5057	369	49	corresponding	corresponding	ADJ
ejpam-5057	369	50	degrees	degree	NOUN
ejpam-5057	369	51	d1	d1	PROPN
ejpam-5057	369	52	and	and	CCONJ
ejpam-5057	369	53	d2	d2	PROPN
ejpam-5057	369	54	.	.	PUNCT
ejpam-5057	370	1	let	let	VERB
ejpam-5057	370	2	d1	d1	PROPN
ejpam-5057	370	3	=	=	PUNCT
ejpam-5057	370	4	|x|	|x|	PROPN
ejpam-5057	370	5	2	2	NUM
ejpam-5057	370	6	.	.	PUNCT
ejpam-5057	371	1	in	in	ADP
ejpam-5057	371	2	this	this	DET
ejpam-5057	371	3	section	section	NOUN
ejpam-5057	371	4	,	,	PUNCT
ejpam-5057	371	5	we	we	PRON
ejpam-5057	371	6	prove	prove	VERB
ejpam-5057	371	7	that	that	SCONJ
ejpam-5057	371	8	the	the	DET
ejpam-5057	371	9	vertex	vertex	NOUN
ejpam-5057	371	10	-	-	PUNCT
ejpam-5057	371	11	split	split	NOUN
ejpam-5057	371	12	of	of	ADP
ejpam-5057	371	13	a	a	DET
ejpam-5057	371	14	biregular	biregular	ADJ
ejpam-5057	371	15	bipartite	bipartite	NOUN
ejpam-5057	371	16	graph	graph	NOUN
ejpam-5057	371	17	is	be	AUX
ejpam-5057	371	18	a	a	DET
ejpam-5057	371	19	bipartite	bipartite	ADJ
ejpam-5057	371	20	expander	expander	NOUN
ejpam-5057	371	21	and	and	CCONJ
ejpam-5057	371	22	then	then	ADV
ejpam-5057	371	23	we	we	PRON
ejpam-5057	371	24	show	show	VERB
ejpam-5057	371	25	that	that	SCONJ
ejpam-5057	371	26	the	the	DET
ejpam-5057	371	27	vertex	vertex	NOUN
ejpam-5057	371	28	-	-	PUNCT
ejpam-5057	371	29	split	split	NOUN
ejpam-5057	371	30	of	of	ADP
ejpam-5057	371	31	a	a	DET
ejpam-5057	371	32	biregular	biregular	ADJ
ejpam-5057	371	33	bipartite	bipartite	NOUN
ejpam-5057	371	34	graph	graph	NOUN
ejpam-5057	371	35	forms	form	VERB
ejpam-5057	371	36	an	an	DET
ejpam-5057	371	37	expander	expander	NOUN
ejpam-5057	371	38	code	code	NOUN
ejpam-5057	371	39	.	.	PUNCT
ejpam-5057	372	1	let	let	AUX
ejpam-5057	372	2	c(g	c(g	PROPN
ejpam-5057	372	3	)	)	PUNCT
ejpam-5057	372	4	be	be	AUX
ejpam-5057	372	5	the	the	DET
ejpam-5057	372	6	error	error	NOUN
ejpam-5057	372	7	correcting	correct	VERB
ejpam-5057	372	8	code	code	NOUN
ejpam-5057	372	9	of	of	ADP
ejpam-5057	372	10	block	block	NOUN
ejpam-5057	372	11	length	length	NOUN
ejpam-5057	372	12	n	n	NUM
ejpam-5057	372	13	obtained	obtain	VERB
ejpam-5057	372	14	from	from	ADP
ejpam-5057	372	15	a	a	DET
ejpam-5057	372	16	bipartite	bipartite	NOUN
ejpam-5057	372	17	graph	graph	NOUN
ejpam-5057	372	18	g	g	NOUN
ejpam-5057	372	19	and	and	CCONJ
ejpam-5057	372	20	let	let	VERB
ejpam-5057	372	21	∆(c(g	∆(c(g	NOUN
ejpam-5057	372	22	)	)	PUNCT
ejpam-5057	372	23	)	)	PUNCT
ejpam-5057	372	24	be	be	AUX
ejpam-5057	372	25	the	the	DET
ejpam-5057	372	26	distance	distance	NOUN
ejpam-5057	372	27	of	of	ADP
ejpam-5057	372	28	the	the	DET
ejpam-5057	372	29	code	code	NOUN
ejpam-5057	372	30	c.	c.	PROPN
ejpam-5057	372	31	m.machasri	m.machasri	PROPN
ejpam-5057	372	32	,	,	PUNCT
ejpam-5057	372	33	d.kalyani	d.kalyani	NOUN
ejpam-5057	372	34	/	/	SYM
ejpam-5057	372	35	eur	eur	PROPN
ejpam-5057	372	36	.	.	PUNCT
ejpam-5057	373	1	j.	j.	PROPN
ejpam-5057	373	2	pure	pure	PROPN
ejpam-5057	373	3	appl	appl	PROPN
ejpam-5057	373	4	.	.	PROPN
ejpam-5057	373	5	math	math	PROPN
ejpam-5057	373	6	,	,	PUNCT
ejpam-5057	373	7	17	17	NUM
ejpam-5057	373	8	(	(	PUNCT
ejpam-5057	373	9	2	2	NUM
ejpam-5057	373	10	)	)	PUNCT
ejpam-5057	373	11	(	(	PUNCT
ejpam-5057	373	12	2024	2024	NUM
ejpam-5057	373	13	)	)	PUNCT
ejpam-5057	373	14	,	,	PUNCT
ejpam-5057	373	15	772	772	NUM
ejpam-5057	373	16	-	-	SYM
ejpam-5057	373	17	789	789	NUM
ejpam-5057	373	18	787	787	NUM
ejpam-5057	373	19	lemma	lemma	PROPN
ejpam-5057	373	20	4	4	NUM
ejpam-5057	373	21	.	.	PUNCT
ejpam-5057	374	1	let	let	AUX
ejpam-5057	374	2	g	g	NOUN
ejpam-5057	374	3	=	=	SYM
ejpam-5057	374	4	(	(	PUNCT
ejpam-5057	374	5	x	x	X
ejpam-5057	374	6	∪y	∪y	NUM
ejpam-5057	374	7	,	,	PUNCT
ejpam-5057	374	8	e	e	NOUN
ejpam-5057	374	9	)	)	PUNCT
ejpam-5057	374	10	be	be	AUX
ejpam-5057	374	11	a	a	DET
ejpam-5057	374	12	biregular	biregular	ADJ
ejpam-5057	374	13	bipartite	bipartite	NOUN
ejpam-5057	374	14	graph	graph	NOUN
ejpam-5057	374	15	with	with	ADP
ejpam-5057	374	16	|x|	|x|	PROPN
ejpam-5057	374	17	=	=	SYM
ejpam-5057	374	18	n1	n1	PROPN
ejpam-5057	374	19	and	and	CCONJ
ejpam-5057	374	20	|y	|y	NOUN
ejpam-5057	374	21	|	|	NOUN
ejpam-5057	374	22	=	=	SYM
ejpam-5057	374	23	n2	n2	PROPN
ejpam-5057	374	24	.	.	PUNCT
ejpam-5057	375	1	let	let	VERB
ejpam-5057	375	2	d1	d1	PROPN
ejpam-5057	375	3	,	,	PUNCT
ejpam-5057	375	4	d2	d2	PROPN
ejpam-5057	375	5	be	be	AUX
ejpam-5057	375	6	the	the	DET
ejpam-5057	375	7	degrees	degree	NOUN
ejpam-5057	375	8	of	of	ADP
ejpam-5057	375	9	the	the	DET
ejpam-5057	375	10	corresponding	correspond	VERB
ejpam-5057	375	11	bipartitions	bipartition	NOUN
ejpam-5057	375	12	x	x	X
ejpam-5057	375	13	,	,	PUNCT
ejpam-5057	375	14	y	y	PROPN
ejpam-5057	375	15	where	where	SCONJ
ejpam-5057	375	16	d1	d1	NOUN
ejpam-5057	375	17	=	=	PUNCT
ejpam-5057	376	1	⌊	⌊	VERB
ejpam-5057	376	2	|x|	|x|	PROPN
ejpam-5057	376	3	2	2	NUM
ejpam-5057	376	4	⌋.	⌋.	ADV
ejpam-5057	376	5	then	then	ADV
ejpam-5057	376	6	vertex	vertex	NOUN
ejpam-5057	376	7	-	-	PUNCT
ejpam-5057	376	8	split	split	NOUN
ejpam-5057	376	9	g′	g′	NOUN
ejpam-5057	376	10	of	of	ADP
ejpam-5057	376	11	g	g	PROPN
ejpam-5057	376	12	is	be	AUX
ejpam-5057	376	13	a	a	DET
ejpam-5057	376	14	bipartite	bipartite	ADJ
ejpam-5057	376	15	expander	expander	NOUN
ejpam-5057	376	16	.	.	PUNCT
ejpam-5057	377	1	proof	proof	NOUN
ejpam-5057	377	2	.	.	PUNCT
ejpam-5057	378	1	let	let	VERB
ejpam-5057	378	2	g′	g′	NOUN
ejpam-5057	378	3	=	=	PUNCT
ejpam-5057	379	1	(	(	PUNCT
ejpam-5057	379	2	x	x	SYM
ejpam-5057	379	3	′	′	NOUN
ejpam-5057	379	4	∪	∪	ADP
ejpam-5057	379	5	y	y	PROPN
ejpam-5057	379	6	′	′	NUM
ejpam-5057	379	7	,	,	PUNCT
ejpam-5057	379	8	e′	e′	ADJ
ejpam-5057	379	9	)	)	PUNCT
ejpam-5057	379	10	be	be	VERB
ejpam-5057	379	11	the	the	DET
ejpam-5057	379	12	vertex	vertex	NOUN
ejpam-5057	379	13	-	-	PUNCT
ejpam-5057	379	14	split	split	NOUN
ejpam-5057	379	15	of	of	ADP
ejpam-5057	379	16	a	a	DET
ejpam-5057	379	17	biregular	biregular	ADJ
ejpam-5057	379	18	bipartite	bipartite	NOUN
ejpam-5057	379	19	graph	graph	NOUN
ejpam-5057	379	20	g	g	NOUN
ejpam-5057	379	21	with	with	ADP
ejpam-5057	379	22	|x	|x	NOUN
ejpam-5057	379	23	′|	′|	NUM
ejpam-5057	379	24	=	=	SYM
ejpam-5057	379	25	n1	n1	NOUN
ejpam-5057	379	26	and	and	CCONJ
ejpam-5057	379	27	|y	|y	VERB
ejpam-5057	379	28	′|	′|	NUM
ejpam-5057	379	29	=	=	SYM
ejpam-5057	379	30	2n2	2n2	NUM
ejpam-5057	379	31	.	.	PUNCT
ejpam-5057	380	1	the	the	DET
ejpam-5057	380	2	degrees	degree	NOUN
ejpam-5057	380	3	of	of	ADP
ejpam-5057	380	4	the	the	DET
ejpam-5057	380	5	bipartitions	bipartition	NOUN
ejpam-5057	380	6	x	x	NOUN
ejpam-5057	380	7	′	′	NUM
ejpam-5057	380	8	,	,	PUNCT
ejpam-5057	380	9	y	y	PROPN
ejpam-5057	380	10	′	′	NOUN
ejpam-5057	380	11	are	be	AUX
ejpam-5057	380	12	denoted	denote	VERB
ejpam-5057	380	13	by	by	ADP
ejpam-5057	380	14	d′1	d′1	NOUN
ejpam-5057	380	15	and	and	CCONJ
ejpam-5057	380	16	d′2	d′2	ADJ
ejpam-5057	380	17	respectively	respectively	ADV
ejpam-5057	380	18	.	.	PUNCT
ejpam-5057	381	1	let	let	VERB
ejpam-5057	381	2	s	s	PRON
ejpam-5057	381	3	be	be	AUX
ejpam-5057	381	4	a	a	DET
ejpam-5057	381	5	subset	subset	NOUN
ejpam-5057	381	6	of	of	ADP
ejpam-5057	381	7	x	x	SYM
ejpam-5057	381	8	′	′	NOUN
ejpam-5057	381	9	with	with	ADP
ejpam-5057	381	10	|s|	|s|	NOUN
ejpam-5057	381	11	=	=	SYM
ejpam-5057	381	12	n1	n1	PROPN
ejpam-5057	381	13	d′1	d′1	INTJ
ejpam-5057	381	14	.	.	PUNCT
ejpam-5057	382	1	since	since	SCONJ
ejpam-5057	382	2	d′1	d′1	NOUN
ejpam-5057	382	3	=	=	PUNCT
ejpam-5057	382	4	|x′|	|x′|	PROPN
ejpam-5057	382	5	2	2	NUM
ejpam-5057	382	6	,	,	PUNCT
ejpam-5057	382	7	we	we	PRON
ejpam-5057	382	8	have	have	VERB
ejpam-5057	382	9	|s|	|s|	NOUN
ejpam-5057	382	10	=	=	SYM
ejpam-5057	382	11	2	2	X
ejpam-5057	382	12	.	.	X
ejpam-5057	383	1	for	for	ADP
ejpam-5057	383	2	any	any	DET
ejpam-5057	383	3	subset	subset	NOUN
ejpam-5057	383	4	s	s	NOUN
ejpam-5057	383	5	of	of	ADP
ejpam-5057	383	6	x	x	SYM
ejpam-5057	383	7	′	′	NUM
ejpam-5057	383	8	the	the	DET
ejpam-5057	383	9	neighbourhood	neighbourhood	NOUN
ejpam-5057	383	10	of	of	ADP
ejpam-5057	383	11	s	s	NOUN
ejpam-5057	383	12	is	be	AUX
ejpam-5057	383	13	|n(s)|	|n(s)|	PROPN
ejpam-5057	383	14	≥	≥	NOUN
ejpam-5057	383	15	n1	n1	NOUN
ejpam-5057	383	16	2	2	NUM
ejpam-5057	383	17	+	+	NUM
ejpam-5057	383	18	1	1	NUM
ejpam-5057	383	19	,	,	PUNCT
ejpam-5057	383	20	which	which	PRON
ejpam-5057	383	21	implies	imply	VERB
ejpam-5057	383	22	that	that	SCONJ
ejpam-5057	383	23	|n(s)|	|n(s)|	PROPN
ejpam-5057	383	24	|s|	|s|	PRON
ejpam-5057	383	25	≥	≥	PROPN
ejpam-5057	383	26	α	α	NUM
ejpam-5057	383	27	where	where	SCONJ
ejpam-5057	383	28	α	α	NOUN
ejpam-5057	383	29	=	=	X
ejpam-5057	383	30	(	(	PUNCT
ejpam-5057	383	31	n1	n1	PROPN
ejpam-5057	383	32	+	+	PROPN
ejpam-5057	383	33	2	2	NUM
ejpam-5057	383	34	4	4	NUM
ejpam-5057	383	35	)	)	PUNCT
ejpam-5057	383	36	is	be	AUX
ejpam-5057	383	37	the	the	DET
ejpam-5057	383	38	vertex	vertex	NOUN
ejpam-5057	383	39	expansion	expansion	NOUN
ejpam-5057	383	40	constant	constant	ADJ
ejpam-5057	383	41	.	.	PUNCT
ejpam-5057	384	1	that	that	PRON
ejpam-5057	384	2	is	be	AUX
ejpam-5057	384	3	|n(s)|	|n(s)|	PROPN
ejpam-5057	384	4	≥	≥	NOUN
ejpam-5057	384	5	α|s|	α|s|	NOUN
ejpam-5057	384	6	.	.	PUNCT
ejpam-5057	385	1	this	this	PRON
ejpam-5057	385	2	proves	prove	VERB
ejpam-5057	385	3	the	the	DET
ejpam-5057	385	4	result	result	NOUN
ejpam-5057	385	5	.	.	PUNCT
ejpam-5057	386	1	theorem	theorem	ADJ
ejpam-5057	386	2	8	8	NUM
ejpam-5057	386	3	.	.	PUNCT
ejpam-5057	387	1	the	the	DET
ejpam-5057	387	2	vertex	vertex	NOUN
ejpam-5057	387	3	-	-	PUNCT
ejpam-5057	387	4	split	split	NOUN
ejpam-5057	387	5	of	of	ADP
ejpam-5057	387	6	a	a	DET
ejpam-5057	387	7	biregular	biregular	ADJ
ejpam-5057	387	8	bipartite	bipartite	NOUN
ejpam-5057	387	9	graph	graph	NOUN
ejpam-5057	387	10	g	g	PROPN
ejpam-5057	387	11	=	=	PUNCT
ejpam-5057	387	12	(	(	PUNCT
ejpam-5057	387	13	x∪y	x∪y	ADJ
ejpam-5057	387	14	,	,	PUNCT
ejpam-5057	387	15	e	e	NOUN
ejpam-5057	387	16	)	)	PUNCT
ejpam-5057	387	17	where	where	SCONJ
ejpam-5057	387	18	|x|	|x|	PROPN
ejpam-5057	387	19	=	=	SYM
ejpam-5057	387	20	n1	n1	PROPN
ejpam-5057	387	21	,	,	PUNCT
ejpam-5057	387	22	|y	|y	NOUN
ejpam-5057	387	23	|	|	NOUN
ejpam-5057	387	24	=	=	SYM
ejpam-5057	387	25	n2	n2	NOUN
ejpam-5057	387	26	with	with	ADP
ejpam-5057	387	27	degree	degree	NOUN
ejpam-5057	387	28	d1	d1	NOUN
ejpam-5057	387	29	=	=	SYM
ejpam-5057	387	30	|x|	|x|	PROPN
ejpam-5057	387	31	2	2	NUM
ejpam-5057	387	32	is	be	AUX
ejpam-5057	387	33	an	an	DET
ejpam-5057	387	34	expander	expander	NOUN
ejpam-5057	387	35	code	code	NOUN
ejpam-5057	387	36	.	.	PUNCT
ejpam-5057	388	1	proof	proof	NOUN
ejpam-5057	388	2	.	.	PUNCT
ejpam-5057	389	1	we	we	PRON
ejpam-5057	389	2	proved	prove	VERB
ejpam-5057	389	3	that	that	SCONJ
ejpam-5057	389	4	the	the	DET
ejpam-5057	389	5	vertex	vertex	NOUN
ejpam-5057	389	6	-	-	PUNCT
ejpam-5057	389	7	split	split	NOUN
ejpam-5057	389	8	of	of	ADP
ejpam-5057	389	9	a	a	DET
ejpam-5057	389	10	biregular	biregular	ADJ
ejpam-5057	389	11	bipartite	bipartite	NOUN
ejpam-5057	389	12	graph	graph	NOUN
ejpam-5057	389	13	g	g	PROPN
ejpam-5057	389	14	is	be	AUX
ejpam-5057	389	15	a	a	DET
ejpam-5057	389	16	bipartite	bipartite	ADJ
ejpam-5057	389	17	expander	expander	NOUN
ejpam-5057	389	18	in	in	ADP
ejpam-5057	389	19	lemma	lemma	PROPN
ejpam-5057	389	20	4	4	NUM
ejpam-5057	389	21	.	.	PUNCT
ejpam-5057	389	22	for	for	ADP
ejpam-5057	389	23	constructing	construct	VERB
ejpam-5057	389	24	an	an	DET
ejpam-5057	389	25	efficient	efficient	ADJ
ejpam-5057	389	26	error	error	NOUN
ejpam-5057	389	27	correcting	correct	VERB
ejpam-5057	389	28	code	code	NOUN
ejpam-5057	389	29	,	,	PUNCT
ejpam-5057	389	30	let	let	VERB
ejpam-5057	389	31	us	we	PRON
ejpam-5057	389	32	fix	fix	VERB
ejpam-5057	389	33	|y	|y	NOUN
ejpam-5057	389	34	|	|	ADV
ejpam-5057	389	35	as	as	SCONJ
ejpam-5057	389	36	|y	|y	NOUN
ejpam-5057	389	37	|	|	ADV
ejpam-5057	389	38	=	=	SYM
ejpam-5057	389	39	n2	n2	NOUN
ejpam-5057	389	40	=	=	PUNCT
ejpam-5057	390	1	max{x	max{x	PROPN
ejpam-5057	390	2	:	:	PUNCT
ejpam-5057	390	3	x|n	x|n	PROPN
ejpam-5057	390	4	2	2	NUM
ejpam-5057	390	5	1	1	NUM
ejpam-5057	390	6	2	2	NUM
ejpam-5057	390	7	and	and	CCONJ
ejpam-5057	390	8	n1	n1	NOUN
ejpam-5057	390	9	+	+	PROPN
ejpam-5057	390	10	2	2	NUM
ejpam-5057	390	11	4	4	NUM
ejpam-5057	390	12	<	<	X
ejpam-5057	390	13	x	x	X
ejpam-5057	390	14	<	<	X
ejpam-5057	390	15	n1	n1	PROPN
ejpam-5057	390	16	2	2	NUM
ejpam-5057	390	17	}	}	PUNCT
ejpam-5057	390	18	.	.	PUNCT
ejpam-5057	391	1	usually	usually	ADV
ejpam-5057	391	2	α	α	PRON
ejpam-5057	391	3	is	be	AUX
ejpam-5057	391	4	close	close	ADJ
ejpam-5057	391	5	to	to	ADP
ejpam-5057	391	6	d	d	PROPN
ejpam-5057	391	7	2	2	NUM
ejpam-5057	391	8	where	where	SCONJ
ejpam-5057	391	9	d	d	NOUN
ejpam-5057	391	10	is	be	AUX
ejpam-5057	391	11	the	the	DET
ejpam-5057	391	12	degree	degree	NOUN
ejpam-5057	391	13	of	of	ADP
ejpam-5057	391	14	the	the	DET
ejpam-5057	391	15	larger	large	ADJ
ejpam-5057	391	16	side	side	NOUN
ejpam-5057	391	17	partition	partition	NOUN
ejpam-5057	391	18	.	.	PUNCT
ejpam-5057	392	1	here	here	ADV
ejpam-5057	392	2	d	d	X
ejpam-5057	392	3	=	=	SYM
ejpam-5057	392	4	d1	d1	NOUN
ejpam-5057	392	5	=	=	SYM
ejpam-5057	392	6	n1	n1	PROPN
ejpam-5057	392	7	2	2	NUM
ejpam-5057	392	8	.	.	PUNCT
ejpam-5057	393	1	then	then	ADV
ejpam-5057	393	2	d	d	X
ejpam-5057	393	3	2	2	X
ejpam-5057	393	4	=	=	SYM
ejpam-5057	393	5	n1	n1	PROPN
ejpam-5057	393	6	4	4	NUM
ejpam-5057	393	7	.	.	PUNCT
ejpam-5057	394	1	clearly	clearly	ADV
ejpam-5057	394	2	d	d	ADP
ejpam-5057	394	3	2	2	NUM
ejpam-5057	394	4	<	<	X
ejpam-5057	394	5	α	α	X
ejpam-5057	394	6	<	<	X
ejpam-5057	394	7	d	d	PROPN
ejpam-5057	394	8	,	,	PUNCT
ejpam-5057	394	9	where	where	SCONJ
ejpam-5057	394	10	α	α	NOUN
ejpam-5057	394	11	=	=	SYM
ejpam-5057	394	12	(	(	PUNCT
ejpam-5057	394	13	n1	n1	PROPN
ejpam-5057	394	14	+	+	PROPN
ejpam-5057	394	15	2	2	NUM
ejpam-5057	394	16	4	4	NUM
ejpam-5057	394	17	)	)	PUNCT
ejpam-5057	394	18	.	.	PUNCT
ejpam-5057	395	1	putting	put	VERB
ejpam-5057	395	2	α	α	NOUN
ejpam-5057	395	3	=	=	SYM
ejpam-5057	395	4	d(1−	d(1−	PROPN
ejpam-5057	395	5	ϵ	ϵ	X
ejpam-5057	395	6	)	)	PUNCT
ejpam-5057	395	7	,	,	PUNCT
ejpam-5057	395	8	we	we	PRON
ejpam-5057	395	9	get	get	VERB
ejpam-5057	395	10	ϵ	ϵ	X
ejpam-5057	395	11	=	=	PUNCT
ejpam-5057	395	12	n1−2	n1−2	X
ejpam-5057	395	13	2n1	2n1	PROPN
ejpam-5057	395	14	<	<	X
ejpam-5057	395	15	1	1	NUM
ejpam-5057	395	16	2	2	NUM
ejpam-5057	395	17	.	.	PUNCT
ejpam-5057	396	1	corollary	corollary	ADJ
ejpam-5057	396	2	5	5	NUM
ejpam-5057	396	3	.	.	PUNCT
ejpam-5057	397	1	let	let	VERB
ejpam-5057	397	2	the	the	DET
ejpam-5057	397	3	vertex	vertex	NOUN
ejpam-5057	397	4	-	-	PUNCT
ejpam-5057	397	5	split	split	NOUN
ejpam-5057	397	6	g′	g′	NOUN
ejpam-5057	397	7	of	of	ADP
ejpam-5057	397	8	g	g	PROPN
ejpam-5057	397	9	be	be	AUX
ejpam-5057	397	10	a	a	DET
ejpam-5057	397	11	(	(	PUNCT
ejpam-5057	397	12	n1	n1	NOUN
ejpam-5057	397	13	,	,	PUNCT
ejpam-5057	397	14	n2	n2	ADJ
ejpam-5057	397	15	,	,	PUNCT
ejpam-5057	397	16	d1	d1	PROPN
ejpam-5057	397	17	,	,	PUNCT
ejpam-5057	397	18	γ	γ	X
ejpam-5057	397	19	,	,	PUNCT
ejpam-5057	397	20	d(1−	d(1−	PROPN
ejpam-5057	397	21	ϵ	ϵ	NUM
ejpam-5057	397	22	)	)	PUNCT
ejpam-5057	397	23	)	)	PUNCT
ejpam-5057	397	24	expander	expander	NOUN
ejpam-5057	397	25	.	.	PUNCT
ejpam-5057	398	1	then	then	ADV
ejpam-5057	398	2	the	the	DET
ejpam-5057	398	3	distance	distance	NOUN
ejpam-5057	398	4	of	of	ADP
ejpam-5057	398	5	the	the	DET
ejpam-5057	398	6	ecc	ecc	NOUN
ejpam-5057	398	7	corresponding	corresponding	NOUN
ejpam-5057	398	8	to	to	PART
ejpam-5057	398	9	graph	graph	VERB
ejpam-5057	398	10	g′	g′	NOUN
ejpam-5057	398	11	is	be	AUX
ejpam-5057	398	12	∆(c(g′	∆(c(g′	NOUN
ejpam-5057	398	13	)	)	PUNCT
ejpam-5057	398	14	)	)	PUNCT
ejpam-5057	398	15	≥	≥	PROPN
ejpam-5057	398	16	n1(n1	n1(n1	PROPN
ejpam-5057	398	17	+	+	NOUN
ejpam-5057	398	18	2	2	NUM
ejpam-5057	398	19	)	)	PUNCT
ejpam-5057	398	20	2d21	2d21	NUM
ejpam-5057	398	21	.	.	PUNCT
ejpam-5057	399	1	proof	proof	NOUN
ejpam-5057	399	2	.	.	PUNCT
ejpam-5057	400	1	we	we	PRON
ejpam-5057	400	2	proved	prove	VERB
ejpam-5057	400	3	in	in	ADP
ejpam-5057	400	4	lemma	lemma	PROPN
ejpam-5057	400	5	4	4	NUM
ejpam-5057	400	6	that	that	SCONJ
ejpam-5057	400	7	the	the	DET
ejpam-5057	400	8	vertex	vertex	NOUN
ejpam-5057	400	9	-	-	PUNCT
ejpam-5057	400	10	split	split	NOUN
ejpam-5057	400	11	g′	g′	NOUN
ejpam-5057	400	12	of	of	ADP
ejpam-5057	400	13	g	g	PROPN
ejpam-5057	400	14	is	be	AUX
ejpam-5057	400	15	a	a	DET
ejpam-5057	400	16	(	(	PUNCT
ejpam-5057	400	17	n1	n1	NOUN
ejpam-5057	400	18	,	,	PUNCT
ejpam-5057	400	19	n2	n2	ADJ
ejpam-5057	400	20	,	,	PUNCT
ejpam-5057	400	21	d1	d1	PROPN
ejpam-5057	400	22	,	,	PUNCT
ejpam-5057	400	23	γ	γ	X
ejpam-5057	400	24	,	,	PUNCT
ejpam-5057	400	25	d(1−	d(1−	PROPN
ejpam-5057	400	26	ϵ	ϵ	NUM
ejpam-5057	400	27	)	)	PUNCT
ejpam-5057	400	28	)	)	PUNCT
ejpam-5057	400	29	expander	expander	NOUN
ejpam-5057	400	30	,	,	PUNCT
ejpam-5057	400	31	where	where	SCONJ
ejpam-5057	400	32	γ	γ	X
ejpam-5057	400	33	=	=	SYM
ejpam-5057	400	34	1	1	NUM
ejpam-5057	400	35	d1	d1	NOUN
ejpam-5057	400	36	,	,	PUNCT
ejpam-5057	400	37	α	α	NOUN
ejpam-5057	400	38	=	=	SYM
ejpam-5057	400	39	n1	n1	PROPN
ejpam-5057	400	40	+	+	PROPN
ejpam-5057	400	41	2	2	NUM
ejpam-5057	400	42	4	4	NUM
ejpam-5057	400	43	and	and	CCONJ
ejpam-5057	400	44	ϵ	ϵ	NOUN
ejpam-5057	400	45	=	=	PUNCT
ejpam-5057	400	46	n1−2	n1−2	X
ejpam-5057	400	47	2n1	2n1	NUM
ejpam-5057	400	48	.	.	PUNCT
ejpam-5057	401	1	substituting	substitute	VERB
ejpam-5057	401	2	all	all	DET
ejpam-5057	401	3	these	these	PRON
ejpam-5057	401	4	in	in	ADP
ejpam-5057	401	5	lemma	lemma	PROPN
ejpam-5057	401	6	3	3	NUM
ejpam-5057	401	7	,	,	PUNCT
ejpam-5057	401	8	we	we	PRON
ejpam-5057	401	9	get	get	VERB
ejpam-5057	401	10	the	the	DET
ejpam-5057	401	11	result	result	NOUN
ejpam-5057	401	12	.	.	PUNCT
ejpam-5057	402	1	7	7	X
ejpam-5057	402	2	.	.	X
ejpam-5057	402	3	conclusion	conclusion	NOUN
ejpam-5057	402	4	in	in	ADP
ejpam-5057	402	5	the	the	DET
ejpam-5057	402	6	first	first	ADJ
ejpam-5057	402	7	part	part	NOUN
ejpam-5057	402	8	of	of	ADP
ejpam-5057	402	9	this	this	DET
ejpam-5057	402	10	paper	paper	NOUN
ejpam-5057	402	11	,	,	PUNCT
ejpam-5057	402	12	we	we	PRON
ejpam-5057	402	13	defined	define	VERB
ejpam-5057	402	14	bipartite	bipartite	ADJ
ejpam-5057	402	15	quotient	quotient	NOUN
ejpam-5057	402	16	matrix	matrix	NOUN
ejpam-5057	402	17	of	of	ADP
ejpam-5057	402	18	matrices	matrix	NOUN
ejpam-5057	402	19	associated	associate	VERB
ejpam-5057	402	20	with	with	ADP
ejpam-5057	402	21	bipartite	bipartite	NOUN
ejpam-5057	402	22	graphs	graph	NOUN
ejpam-5057	402	23	and	and	CCONJ
ejpam-5057	402	24	proved	prove	VERB
ejpam-5057	402	25	that	that	SCONJ
ejpam-5057	402	26	the	the	DET
ejpam-5057	402	27	two	two	NUM
ejpam-5057	402	28	eigenvalues	eigenvalue	NOUN
ejpam-5057	402	29	of	of	ADP
ejpam-5057	402	30	the	the	DET
ejpam-5057	402	31	bipartite	bipartite	PROPN
ejpam-5057	402	32	quotient	quotient	NOUN
ejpam-5057	402	33	matrix	matrix	NOUN
ejpam-5057	402	34	interlace	interlace	NOUN
ejpam-5057	402	35	at	at	ADP
ejpam-5057	402	36	the	the	DET
ejpam-5057	402	37	two	two	NUM
ejpam-5057	402	38	ends	end	NOUN
ejpam-5057	402	39	of	of	ADP
ejpam-5057	402	40	the	the	DET
ejpam-5057	402	41	spectrum	spectrum	NOUN
ejpam-5057	402	42	of	of	ADP
ejpam-5057	402	43	the	the	DET
ejpam-5057	402	44	adjacency	adjacency	NOUN
ejpam-5057	402	45	matrix	matrix	NOUN
ejpam-5057	402	46	of	of	ADP
ejpam-5057	402	47	a	a	DET
ejpam-5057	402	48	bipartite	bipartite	NOUN
ejpam-5057	402	49	graph	graph	NOUN
ejpam-5057	402	50	.	.	PUNCT
ejpam-5057	403	1	from	from	ADP
ejpam-5057	403	2	this	this	DET
ejpam-5057	403	3	interlacing	interlacing	NOUN
ejpam-5057	403	4	,	,	PUNCT
ejpam-5057	403	5	we	we	PRON
ejpam-5057	403	6	obtained	obtain	VERB
ejpam-5057	403	7	the	the	DET
ejpam-5057	403	8	upper	upper	ADJ
ejpam-5057	403	9	bound	bind	VERB
ejpam-5057	403	10	for	for	ADP
ejpam-5057	403	11	the	the	DET
ejpam-5057	403	12	second	second	ADV
ejpam-5057	403	13	largest	large	ADJ
ejpam-5057	403	14	eigenvalue	eigenvalue	NOUN
ejpam-5057	403	15	and	and	CCONJ
ejpam-5057	403	16	the	the	DET
ejpam-5057	403	17	lower	lower	ADV
ejpam-5057	403	18	bound	bind	VERB
ejpam-5057	403	19	for	for	ADP
ejpam-5057	403	20	the	the	DET
ejpam-5057	403	21	second	second	ADJ
ejpam-5057	403	22	smallest	small	ADJ
ejpam-5057	403	23	eigenvalue	eigenvalue	NOUN
ejpam-5057	403	24	of	of	ADP
ejpam-5057	403	25	a	a	DET
ejpam-5057	403	26	bipartite	bipartite	NOUN
ejpam-5057	403	27	graph	graph	NOUN
ejpam-5057	403	28	.	.	PUNCT
ejpam-5057	404	1	to	to	PART
ejpam-5057	404	2	achieve	achieve	VERB
ejpam-5057	404	3	a	a	DET
ejpam-5057	404	4	more	more	ADV
ejpam-5057	404	5	sharper	sharp	ADJ
ejpam-5057	404	6	bound	bind	VERB
ejpam-5057	404	7	we	we	PRON
ejpam-5057	404	8	considered	consider	VERB
ejpam-5057	404	9	the	the	DET
ejpam-5057	404	10	minimally	minimally	ADV
ejpam-5057	404	11	connected	connect	VERB
ejpam-5057	404	12	graph	graph	NOUN
ejpam-5057	404	13	for	for	ADP
ejpam-5057	404	14	three	three	NUM
ejpam-5057	404	15	different	different	ADJ
ejpam-5057	404	16	bipartitions	bipartition	NOUN
ejpam-5057	404	17	and	and	CCONJ
ejpam-5057	404	18	obtained	obtain	VERB
ejpam-5057	404	19	two	two	NUM
ejpam-5057	404	20	new	new	ADJ
ejpam-5057	404	21	bounds	bound	NOUN
ejpam-5057	404	22	which	which	PRON
ejpam-5057	404	23	depend	depend	VERB
ejpam-5057	404	24	only	only	ADV
ejpam-5057	404	25	on	on	ADP
ejpam-5057	404	26	the	the	DET
ejpam-5057	404	27	order	order	NOUN
ejpam-5057	404	28	of	of	ADP
ejpam-5057	404	29	the	the	DET
ejpam-5057	404	30	graph	graph	NOUN
ejpam-5057	404	31	.	.	PUNCT
ejpam-5057	405	1	also	also	ADV
ejpam-5057	405	2	,	,	PUNCT
ejpam-5057	405	3	we	we	PRON
ejpam-5057	405	4	proved	prove	VERB
ejpam-5057	405	5	the	the	DET
ejpam-5057	405	6	eigenvalue	eigenvalue	NOUN
ejpam-5057	405	7	interlacing	interlace	VERB
ejpam-5057	405	8	for	for	ADP
ejpam-5057	405	9	the	the	DET
ejpam-5057	405	10	laplacian	laplacian	ADJ
ejpam-5057	405	11	matrix	matrix	NOUN
ejpam-5057	405	12	and	and	CCONJ
ejpam-5057	405	13	obtained	obtain	VERB
ejpam-5057	405	14	an	an	DET
ejpam-5057	405	15	upper	upper	ADJ
ejpam-5057	405	16	bound	bind	VERB
ejpam-5057	405	17	for	for	ADP
ejpam-5057	405	18	the	the	DET
ejpam-5057	405	19	second	second	ADV
ejpam-5057	405	20	largest	large	ADJ
ejpam-5057	405	21	laplacian	laplacian	ADJ
ejpam-5057	405	22	eigenvalue	eigenvalue	NOUN
ejpam-5057	405	23	.	.	PUNCT
ejpam-5057	406	1	in	in	ADP
ejpam-5057	406	2	the	the	DET
ejpam-5057	406	3	second	second	ADJ
ejpam-5057	406	4	part	part	NOUN
ejpam-5057	406	5	of	of	ADP
ejpam-5057	406	6	this	this	DET
ejpam-5057	406	7	paper	paper	NOUN
ejpam-5057	406	8	,	,	PUNCT
ejpam-5057	406	9	we	we	PRON
ejpam-5057	406	10	defined	define	VERB
ejpam-5057	406	11	vertex	vertex	NOUN
ejpam-5057	406	12	-	-	PUNCT
ejpam-5057	406	13	split	split	NOUN
ejpam-5057	406	14	of	of	ADP
ejpam-5057	406	15	a	a	DET
ejpam-5057	406	16	bipartite	bipartite	NOUN
ejpam-5057	406	17	graph	graph	NOUN
ejpam-5057	406	18	and	and	CCONJ
ejpam-5057	406	19	proved	prove	VERB
ejpam-5057	406	20	its	its	PRON
ejpam-5057	406	21	connectivity	connectivity	NOUN
ejpam-5057	406	22	with	with	ADP
ejpam-5057	406	23	respect	respect	NOUN
ejpam-5057	406	24	to	to	ADP
ejpam-5057	406	25	the	the	DET
ejpam-5057	406	26	second	second	ADV
ejpam-5057	406	27	largest	large	ADJ
ejpam-5057	406	28	eigenvalue	eigenvalue	NOUN
ejpam-5057	406	29	.	.	PUNCT
ejpam-5057	407	1	finally	finally	ADV
ejpam-5057	407	2	,	,	PUNCT
ejpam-5057	407	3	we	we	PRON
ejpam-5057	407	4	proved	prove	VERB
ejpam-5057	407	5	that	that	SCONJ
ejpam-5057	407	6	the	the	DET
ejpam-5057	407	7	vertex	vertex	NOUN
ejpam-5057	407	8	-	-	PUNCT
ejpam-5057	407	9	splits	split	NOUN
ejpam-5057	407	10	of	of	ADP
ejpam-5057	407	11	complete	complete	ADJ
ejpam-5057	407	12	bipartite	bipartite	PROPN
ejpam-5057	407	13	graph	graph	NOUN
ejpam-5057	407	14	km	km	PROPN
ejpam-5057	407	15	,	,	PUNCT
ejpam-5057	407	16	n	n	NOUN
ejpam-5057	407	17	and	and	CCONJ
ejpam-5057	407	18	biregular	biregular	ADJ
ejpam-5057	407	19	bipartite	bipartite	NOUN
ejpam-5057	407	20	graph	graph	NOUN
ejpam-5057	407	21	g(x	g(x	PROPN
ejpam-5057	407	22	,	,	PUNCT
ejpam-5057	407	23	y	y	PROPN
ejpam-5057	407	24	)	)	PUNCT
ejpam-5057	407	25	are	be	AUX
ejpam-5057	407	26	bipartite	bipartite	ADJ
ejpam-5057	407	27	expanders	expander	NOUN
ejpam-5057	407	28	and	and	CCONJ
ejpam-5057	407	29	constructed	construct	VERB
ejpam-5057	407	30	an	an	DET
ejpam-5057	407	31	efficient	efficient	ADJ
ejpam-5057	407	32	ecc	ecc	NOUN
ejpam-5057	407	33	with	with	ADP
ejpam-5057	407	34	expansion	expansion	NOUN
ejpam-5057	407	35	factor	factor	NOUN
ejpam-5057	407	36	α	α	NOUN
ejpam-5057	407	37	>	>	X
ejpam-5057	407	38	d	d	PROPN
ejpam-5057	407	39	2	2	NUM
ejpam-5057	407	40	.	.	PUNCT
ejpam-5057	408	1	references	reference	NOUN
ejpam-5057	408	2	788	788	NUM
ejpam-5057	408	3	acknowledgements	acknowledgement	NOUN
ejpam-5057	408	4	we	we	PRON
ejpam-5057	408	5	are	be	AUX
ejpam-5057	408	6	highly	highly	ADV
ejpam-5057	408	7	thankful	thankful	ADJ
ejpam-5057	408	8	to	to	ADP
ejpam-5057	408	9	the	the	DET
ejpam-5057	408	10	anonymous	anonymous	ADJ
ejpam-5057	408	11	referees	referee	NOUN
ejpam-5057	408	12	for	for	ADP
ejpam-5057	408	13	their	their	PRON
ejpam-5057	408	14	comments	comment	NOUN
ejpam-5057	408	15	and	and	CCONJ
ejpam-5057	408	16	suggestions	suggestion	NOUN
ejpam-5057	408	17	to	to	PART
ejpam-5057	408	18	enhance	enhance	VERB
ejpam-5057	408	19	our	our	PRON
ejpam-5057	408	20	paper	paper	NOUN
ejpam-5057	408	21	.	.	PUNCT
ejpam-5057	409	1	references	reference	NOUN
ejpam-5057	409	2	[	[	X
ejpam-5057	409	3	1	1	NUM
ejpam-5057	409	4	]	]	PUNCT
ejpam-5057	409	5	noga	noga	PROPN
ejpam-5057	409	6	alon	alon	PROPN
ejpam-5057	409	7	.	.	PUNCT
ejpam-5057	410	1	eigenvalues	eigenvalues	PROPN
ejpam-5057	410	2	and	and	CCONJ
ejpam-5057	410	3	expanders	expander	NOUN
ejpam-5057	410	4	.	.	PUNCT
ejpam-5057	411	1	combinatorica	combinatorica	PROPN
ejpam-5057	411	2	,	,	PUNCT
ejpam-5057	411	3	6(2):83–96	6(2):83–96	NUM
ejpam-5057	411	4	,	,	PUNCT
ejpam-5057	411	5	1986	1986	NUM
ejpam-5057	411	6	.	.	PUNCT
ejpam-5057	412	1	[	[	X
ejpam-5057	412	2	2	2	X
ejpam-5057	412	3	]	]	X
ejpam-5057	412	4	chang	chang	PROPN
ejpam-5057	412	5	an	an	PROPN
ejpam-5057	412	6	.	.	PROPN
ejpam-5057	412	7	bounds	bound	NOUN
ejpam-5057	412	8	on	on	ADP
ejpam-5057	412	9	the	the	DET
ejpam-5057	412	10	second	second	ADV
ejpam-5057	412	11	largest	large	ADJ
ejpam-5057	412	12	eigenvalue	eigenvalue	NOUN
ejpam-5057	412	13	of	of	ADP
ejpam-5057	412	14	a	a	DET
ejpam-5057	412	15	tree	tree	NOUN
ejpam-5057	412	16	with	with	ADP
ejpam-5057	412	17	perfect	perfect	ADJ
ejpam-5057	412	18	matchings	matching	NOUN
ejpam-5057	412	19	.	.	PUNCT
ejpam-5057	413	1	linear	linear	ADJ
ejpam-5057	413	2	algebra	algebra	NOUN
ejpam-5057	413	3	and	and	CCONJ
ejpam-5057	413	4	its	its	PRON
ejpam-5057	413	5	applications	application	NOUN
ejpam-5057	413	6	,	,	PUNCT
ejpam-5057	413	7	283(1	283(1	NUM
ejpam-5057	413	8	-	-	SYM
ejpam-5057	413	9	3):247–255	3):247–255	NUM
ejpam-5057	413	10	,	,	PUNCT
ejpam-5057	413	11	1998	1998	NUM
ejpam-5057	413	12	.	.	PUNCT
ejpam-5057	414	1	[	[	X
ejpam-5057	414	2	3	3	X
ejpam-5057	414	3	]	]	X
ejpam-5057	414	4	andries	andries	PROPN
ejpam-5057	414	5	e	e	PROPN
ejpam-5057	414	6	brouwer	brouwer	PROPN
ejpam-5057	414	7	and	and	CCONJ
ejpam-5057	414	8	willem	willem	PROPN
ejpam-5057	414	9	h	h	PROPN
ejpam-5057	414	10	haemers	haemer	NOUN
ejpam-5057	414	11	.	.	PUNCT
ejpam-5057	415	1	spectra	spectra	NOUN
ejpam-5057	415	2	of	of	ADP
ejpam-5057	415	3	graphs	graph	NOUN
ejpam-5057	415	4	.	.	PUNCT
ejpam-5057	416	1	springer	springer	NOUN
ejpam-5057	416	2	science	science	PROPN
ejpam-5057	416	3	&	&	CCONJ
ejpam-5057	416	4	business	business	NOUN
ejpam-5057	416	5	media	medium	NOUN
ejpam-5057	416	6	,	,	PUNCT
ejpam-5057	416	7	2011	2011	NUM
ejpam-5057	416	8	.	.	PUNCT
ejpam-5057	417	1	[	[	X
ejpam-5057	417	2	4	4	NUM
ejpam-5057	417	3	]	]	X
ejpam-5057	417	4	michael	michael	PROPN
ejpam-5057	417	5	capalbo	capalbo	PROPN
ejpam-5057	417	6	,	,	PUNCT
ejpam-5057	417	7	omer	omer	PROPN
ejpam-5057	417	8	reingold	reingold	PROPN
ejpam-5057	417	9	,	,	PUNCT
ejpam-5057	417	10	salil	salil	PROPN
ejpam-5057	417	11	vadhan	vadhan	PROPN
ejpam-5057	417	12	,	,	PUNCT
ejpam-5057	417	13	and	and	CCONJ
ejpam-5057	417	14	avi	avi	VERB
ejpam-5057	417	15	wigderson	wigderson	NOUN
ejpam-5057	417	16	.	.	PUNCT
ejpam-5057	418	1	randomness	randomness	NOUN
ejpam-5057	418	2	conductors	conductor	NOUN
ejpam-5057	418	3	and	and	CCONJ
ejpam-5057	418	4	constant	constant	ADJ
ejpam-5057	418	5	-	-	PUNCT
ejpam-5057	418	6	degree	degree	NOUN
ejpam-5057	418	7	lossless	lossless	NOUN
ejpam-5057	418	8	expanders	expander	NOUN
ejpam-5057	418	9	.	.	PUNCT
ejpam-5057	419	1	in	in	ADP
ejpam-5057	419	2	proceedings	proceeding	NOUN
ejpam-5057	419	3	of	of	ADP
ejpam-5057	419	4	the	the	DET
ejpam-5057	419	5	thiry	thiry	NOUN
ejpam-5057	419	6	-	-	PUNCT
ejpam-5057	419	7	fourth	fourth	ADJ
ejpam-5057	419	8	annual	annual	ADJ
ejpam-5057	419	9	acm	acm	NOUN
ejpam-5057	419	10	symposium	symposium	NOUN
ejpam-5057	419	11	on	on	ADP
ejpam-5057	419	12	theory	theory	NOUN
ejpam-5057	419	13	of	of	ADP
ejpam-5057	419	14	computing	computing	NOUN
ejpam-5057	419	15	,	,	PUNCT
ejpam-5057	419	16	pages	page	NOUN
ejpam-5057	419	17	659–668	659–668	NUM
ejpam-5057	419	18	,	,	PUNCT
ejpam-5057	419	19	2002	2002	NUM
ejpam-5057	419	20	.	.	PUNCT
ejpam-5057	420	1	[	[	X
ejpam-5057	420	2	5	5	NUM
ejpam-5057	420	3	]	]	X
ejpam-5057	420	4	venkatesan	venkatesan	ADJ
ejpam-5057	420	5	guruswami	guruswami	NOUN
ejpam-5057	420	6	.	.	PUNCT
ejpam-5057	421	1	expander	expander	NOUN
ejpam-5057	421	2	codes	code	NOUN
ejpam-5057	421	3	and	and	CCONJ
ejpam-5057	421	4	their	their	PRON
ejpam-5057	421	5	decoding	decoding	NOUN
ejpam-5057	421	6	.	.	PUNCT
ejpam-5057	422	1	https://www.cs.cmu.edu/	https://www.cs.cmu.edu/	PROPN
ejpam-5057	422	2	venkatg	venkatg	PROPN
ejpam-5057	422	3	/	/	SYM
ejpam-5057	422	4	teaching	teaching	NOUN
ejpam-5057	422	5	/	/	SYM
ejpam-5057	422	6	codingtheory	codingtheory	NOUN
ejpam-5057	422	7	/	/	SYM
ejpam-5057	422	8	notes	note	NOUN
ejpam-5057	422	9	/	/	SYM
ejpam-5057	422	10	notes8.pdf	notes8.pdf	NOUN
ejpam-5057	422	11	,	,	PUNCT
ejpam-5057	422	12	2010	2010	NUM
ejpam-5057	422	13	.	.	PUNCT
ejpam-5057	423	1	[	[	X
ejpam-5057	423	2	6	6	NUM
ejpam-5057	423	3	]	]	PUNCT
ejpam-5057	423	4	w	w	NOUN
ejpam-5057	423	5	haemers	haemer	NOUN
ejpam-5057	423	6	.	.	PUNCT
ejpam-5057	424	1	eigenvalue	eigenvalue	NOUN
ejpam-5057	424	2	techniques	technique	NOUN
ejpam-5057	424	3	in	in	ADP
ejpam-5057	424	4	design	design	NOUN
ejpam-5057	424	5	and	and	CCONJ
ejpam-5057	424	6	graph	graph	NOUN
ejpam-5057	424	7	theory	theory	NOUN
ejpam-5057	424	8	(	(	PUNCT
ejpam-5057	424	9	thesis	thesis	NOUN
ejpam-5057	424	10	)	)	PUNCT
ejpam-5057	424	11	,	,	PUNCT
ejpam-5057	424	12	tract	tract	VERB
ejpam-5057	424	13	121	121	NUM
ejpam-5057	424	14	,	,	PUNCT
ejpam-5057	424	15	math	math	NOUN
ejpam-5057	424	16	.	.	PUNCT
ejpam-5057	425	1	centrum	centrum	PROPN
ejpam-5057	425	2	amsterdam	amsterdam	PROPN
ejpam-5057	425	3	,	,	PUNCT
ejpam-5057	425	4	1980	1980	NUM
ejpam-5057	425	5	.	.	PUNCT
ejpam-5057	426	1	[	[	X
ejpam-5057	426	2	7	7	NUM
ejpam-5057	426	3	]	]	X
ejpam-5057	426	4	willem	willem	NOUN
ejpam-5057	426	5	h	h	PROPN
ejpam-5057	426	6	haemers	haemer	NOUN
ejpam-5057	426	7	.	.	PUNCT
ejpam-5057	427	1	interlacing	interlace	VERB
ejpam-5057	427	2	eigenvalues	eigenvalue	NOUN
ejpam-5057	427	3	and	and	CCONJ
ejpam-5057	427	4	graphs	graph	NOUN
ejpam-5057	427	5	.	.	PUNCT
ejpam-5057	428	1	linear	linear	ADJ
ejpam-5057	428	2	algebra	algebra	NOUN
ejpam-5057	428	3	and	and	CCONJ
ejpam-5057	428	4	its	its	PRON
ejpam-5057	428	5	applications	application	NOUN
ejpam-5057	428	6	,	,	PUNCT
ejpam-5057	428	7	226:593–616	226:593–616	NUM
ejpam-5057	428	8	,	,	PUNCT
ejpam-5057	428	9	1995	1995	NUM
ejpam-5057	428	10	.	.	PUNCT
ejpam-5057	429	1	[	[	X
ejpam-5057	429	2	8	8	NUM
ejpam-5057	429	3	]	]	PUNCT
ejpam-5057	429	4	shlomo	shlomo	NOUN
ejpam-5057	429	5	hoory	hoory	PROPN
ejpam-5057	429	6	,	,	PUNCT
ejpam-5057	429	7	nathan	nathan	PROPN
ejpam-5057	429	8	linial	linial	PROPN
ejpam-5057	429	9	,	,	PUNCT
ejpam-5057	429	10	and	and	CCONJ
ejpam-5057	429	11	avi	avi	VERB
ejpam-5057	429	12	wigderson	wigderson	NOUN
ejpam-5057	429	13	.	.	PUNCT
ejpam-5057	430	1	expander	expander	NOUN
ejpam-5057	430	2	graphs	graph	NOUN
ejpam-5057	430	3	and	and	CCONJ
ejpam-5057	430	4	their	their	PRON
ejpam-5057	430	5	applications	application	NOUN
ejpam-5057	430	6	.	.	PUNCT
ejpam-5057	431	1	bulletin	bulletin	NOUN
ejpam-5057	431	2	of	of	ADP
ejpam-5057	431	3	the	the	DET
ejpam-5057	431	4	american	american	PROPN
ejpam-5057	431	5	mathematical	mathematical	PROPN
ejpam-5057	431	6	society	society	NOUN
ejpam-5057	431	7	,	,	PUNCT
ejpam-5057	431	8	43(4):439–561	43(4):439–561	PROPN
ejpam-5057	431	9	,	,	PUNCT
ejpam-5057	431	10	2006	2006	NUM
ejpam-5057	431	11	.	.	PUNCT
ejpam-5057	432	1	[	[	X
ejpam-5057	432	2	9	9	NUM
ejpam-5057	432	3	]	]	X
ejpam-5057	432	4	chia	chia	X
ejpam-5057	432	5	-	-	PUNCT
ejpam-5057	432	6	an	an	DET
ejpam-5057	432	7	liu	liu	PROPN
ejpam-5057	432	8	and	and	CCONJ
ejpam-5057	432	9	chih	chih	PROPN
ejpam-5057	432	10	-	-	PUNCT
ejpam-5057	432	11	wen	wen	PROPN
ejpam-5057	432	12	weng	weng	PROPN
ejpam-5057	432	13	.	.	PUNCT
ejpam-5057	433	1	spectral	spectral	ADJ
ejpam-5057	433	2	radius	radius	PROPN
ejpam-5057	433	3	of	of	ADP
ejpam-5057	433	4	bipartite	bipartite	PROPN
ejpam-5057	433	5	graphs	graph	NOUN
ejpam-5057	433	6	.	.	PUNCT
ejpam-5057	434	1	linear	linear	ADJ
ejpam-5057	434	2	algebra	algebra	NOUN
ejpam-5057	434	3	and	and	CCONJ
ejpam-5057	434	4	its	its	PRON
ejpam-5057	434	5	applications	application	NOUN
ejpam-5057	434	6	,	,	PUNCT
ejpam-5057	434	7	474:30–43	474:30–43	NUM
ejpam-5057	434	8	,	,	PUNCT
ejpam-5057	434	9	2015	2015	NUM
ejpam-5057	434	10	.	.	PUNCT
ejpam-5057	435	1	[	[	X
ejpam-5057	435	2	10	10	NUM
ejpam-5057	435	3	]	]	PUNCT
ejpam-5057	435	4	huiqing	huiqe	VERB
ejpam-5057	435	5	liu	liu	PROPN
ejpam-5057	435	6	,	,	PUNCT
ejpam-5057	435	7	mei	mei	PROPN
ejpam-5057	435	8	lu	lu	PROPN
ejpam-5057	435	9	,	,	PUNCT
ejpam-5057	435	10	and	and	CCONJ
ejpam-5057	435	11	feng	feng	PROPN
ejpam-5057	435	12	tian	tian	PROPN
ejpam-5057	435	13	.	.	PUNCT
ejpam-5057	436	1	edge	edge	NOUN
ejpam-5057	436	2	-	-	PUNCT
ejpam-5057	436	3	connectivity	connectivity	NOUN
ejpam-5057	436	4	and	and	CCONJ
ejpam-5057	436	5	(	(	PUNCT
ejpam-5057	436	6	signless	signless	NOUN
ejpam-5057	436	7	)	)	PUNCT
ejpam-5057	436	8	laplacian	laplacian	ADJ
ejpam-5057	436	9	eigenvalue	eigenvalue	NOUN
ejpam-5057	436	10	of	of	ADP
ejpam-5057	436	11	graphs	graph	NOUN
ejpam-5057	436	12	.	.	PUNCT
ejpam-5057	437	1	linear	linear	ADJ
ejpam-5057	437	2	algebra	algebra	NOUN
ejpam-5057	437	3	and	and	CCONJ
ejpam-5057	437	4	its	its	PRON
ejpam-5057	437	5	applications	application	NOUN
ejpam-5057	437	6	,	,	PUNCT
ejpam-5057	437	7	439(12):3777–3784	439(12):3777–3784	NUM
ejpam-5057	437	8	,	,	PUNCT
ejpam-5057	437	9	2013	2013	NUM
ejpam-5057	437	10	.	.	PUNCT
ejpam-5057	438	1	[	[	X
ejpam-5057	438	2	11	11	NUM
ejpam-5057	438	3	]	]	PUNCT
ejpam-5057	438	4	ranjit	ranjit	VERB
ejpam-5057	438	5	mehatari	mehatari	PROPN
ejpam-5057	438	6	and	and	CCONJ
ejpam-5057	438	7	m	m	PROPN
ejpam-5057	438	8	rajesh	rajesh	PROPN
ejpam-5057	438	9	kannan	kannan	PROPN
ejpam-5057	438	10	.	.	PUNCT
ejpam-5057	439	1	eigenvalue	eigenvalue	PROPN
ejpam-5057	439	2	bounds	bound	NOUN
ejpam-5057	439	3	for	for	ADP
ejpam-5057	439	4	some	some	DET
ejpam-5057	439	5	classes	class	NOUN
ejpam-5057	439	6	of	of	ADP
ejpam-5057	439	7	matrices	matrix	NOUN
ejpam-5057	439	8	associated	associate	VERB
ejpam-5057	439	9	with	with	ADP
ejpam-5057	439	10	graphs	graph	NOUN
ejpam-5057	439	11	.	.	PUNCT
ejpam-5057	440	1	czechoslovak	czechoslovak	ADJ
ejpam-5057	440	2	mathematical	mathematical	PROPN
ejpam-5057	440	3	journal	journal	NOUN
ejpam-5057	440	4	,	,	PUNCT
ejpam-5057	440	5	71:231–251	71:231–251	NUM
ejpam-5057	440	6	,	,	PUNCT
ejpam-5057	440	7	2021	2021	NUM
ejpam-5057	440	8	.	.	PUNCT
ejpam-5057	441	1	[	[	X
ejpam-5057	441	2	12	12	NUM
ejpam-5057	441	3	]	]	X
ejpam-5057	441	4	david	david	PROPN
ejpam-5057	441	5	l	l	PROPN
ejpam-5057	441	6	powers	power	NOUN
ejpam-5057	441	7	.	.	PUNCT
ejpam-5057	442	1	graph	graph	NOUN
ejpam-5057	442	2	partitioning	partition	VERB
ejpam-5057	442	3	by	by	ADP
ejpam-5057	442	4	eigenvectors	eigenvector	NOUN
ejpam-5057	442	5	.	.	PUNCT
ejpam-5057	443	1	linear	linear	ADJ
ejpam-5057	443	2	algebra	algebra	NOUN
ejpam-5057	443	3	and	and	CCONJ
ejpam-5057	443	4	its	its	PRON
ejpam-5057	443	5	applications	application	NOUN
ejpam-5057	443	6	,	,	PUNCT
ejpam-5057	443	7	101:121–133	101:121–133	NUM
ejpam-5057	443	8	,	,	PUNCT
ejpam-5057	443	9	1988	1988	NUM
ejpam-5057	443	10	.	.	PUNCT
ejpam-5057	444	1	[	[	X
ejpam-5057	444	2	13	13	NUM
ejpam-5057	444	3	]	]	PUNCT
ejpam-5057	444	4	michael	michael	PROPN
ejpam-5057	444	5	sipser	sipser	PROPN
ejpam-5057	444	6	and	and	CCONJ
ejpam-5057	444	7	daniel	daniel	PROPN
ejpam-5057	444	8	a	a	DET
ejpam-5057	444	9	spielman	spielman	PROPN
ejpam-5057	444	10	.	.	PUNCT
ejpam-5057	445	1	expander	expander	NOUN
ejpam-5057	445	2	codes	code	NOUN
ejpam-5057	445	3	.	.	PUNCT
ejpam-5057	446	1	ieee	ieee	NOUN
ejpam-5057	446	2	transactions	transaction	NOUN
ejpam-5057	446	3	on	on	ADP
ejpam-5057	446	4	information	information	NOUN
ejpam-5057	446	5	theory	theory	NOUN
ejpam-5057	446	6	,	,	PUNCT
ejpam-5057	446	7	42(6):1710–1722	42(6):1710–1722	NUM
ejpam-5057	446	8	,	,	PUNCT
ejpam-5057	446	9	1996	1996	NUM
ejpam-5057	446	10	.	.	PUNCT
ejpam-5057	447	1	[	[	X
ejpam-5057	447	2	14	14	NUM
ejpam-5057	447	3	]	]	X
ejpam-5057	447	4	patrick	patrick	PROPN
ejpam-5057	447	5	solé.	solé.	PROPN
ejpam-5057	447	6	the	the	DET
ejpam-5057	447	7	second	second	ADJ
ejpam-5057	447	8	eigenvalue	eigenvalue	NOUN
ejpam-5057	447	9	of	of	ADP
ejpam-5057	447	10	regular	regular	ADJ
ejpam-5057	447	11	graphs	graph	NOUN
ejpam-5057	447	12	of	of	ADP
ejpam-5057	447	13	given	give	VERB
ejpam-5057	447	14	girth	girth	NOUN
ejpam-5057	447	15	.	.	PUNCT
ejpam-5057	448	1	journal	journal	NOUN
ejpam-5057	448	2	of	of	ADP
ejpam-5057	448	3	combinatorial	combinatorial	ADJ
ejpam-5057	448	4	theory	theory	NOUN
ejpam-5057	448	5	,	,	PUNCT
ejpam-5057	448	6	series	series	PROPN
ejpam-5057	448	7	b	b	PROPN
ejpam-5057	448	8	,	,	PUNCT
ejpam-5057	448	9	56(2):239–249	56(2):239–249	PROPN
ejpam-5057	448	10	,	,	PUNCT
ejpam-5057	448	11	1992	1992	NUM
ejpam-5057	448	12	.	.	PUNCT
ejpam-5057	449	1	references	reference	NOUN
ejpam-5057	449	2	789	789	NUM
ejpam-5057	449	3	[	[	X
ejpam-5057	449	4	15	15	NUM
ejpam-5057	449	5	]	]	X
ejpam-5057	449	6	daniel	daniel	PROPN
ejpam-5057	449	7	a	a	DET
ejpam-5057	449	8	spielman	spielman	PROPN
ejpam-5057	449	9	.	.	PUNCT
ejpam-5057	450	1	linear	linear	ADJ
ejpam-5057	450	2	-	-	PUNCT
ejpam-5057	450	3	time	time	NOUN
ejpam-5057	450	4	encodable	encodable	ADJ
ejpam-5057	450	5	and	and	CCONJ
ejpam-5057	450	6	decodable	decodable	ADJ
ejpam-5057	450	7	error	error	NOUN
ejpam-5057	450	8	-	-	PUNCT
ejpam-5057	450	9	correcting	correct	VERB
ejpam-5057	450	10	codes	code	NOUN
ejpam-5057	450	11	.	.	PUNCT
ejpam-5057	451	1	in	in	ADP
ejpam-5057	451	2	proceedings	proceeding	NOUN
ejpam-5057	451	3	of	of	ADP
ejpam-5057	451	4	the	the	DET
ejpam-5057	451	5	twenty	twenty	NUM
ejpam-5057	451	6	-	-	PUNCT
ejpam-5057	451	7	seventh	seventh	ADJ
ejpam-5057	451	8	annual	annual	ADJ
ejpam-5057	451	9	acm	acm	NOUN
ejpam-5057	451	10	symposium	symposium	NOUN
ejpam-5057	451	11	on	on	ADP
ejpam-5057	451	12	theory	theory	NOUN
ejpam-5057	451	13	of	of	ADP
ejpam-5057	451	14	computing	computing	NOUN
ejpam-5057	451	15	,	,	PUNCT
ejpam-5057	451	16	pages	page	NOUN
ejpam-5057	451	17	388–397	388–397	NUM
ejpam-5057	451	18	,	,	PUNCT
ejpam-5057	451	19	1995	1995	NUM
ejpam-5057	451	20	.	.	PUNCT
ejpam-5057	452	1	[	[	X
ejpam-5057	452	2	16	16	NUM
ejpam-5057	452	3	]	]	X
ejpam-5057	452	4	r	r	NOUN
ejpam-5057	452	5	tanner	tanner	NOUN
ejpam-5057	452	6	.	.	PUNCT
ejpam-5057	453	1	a	a	DET
ejpam-5057	453	2	recursive	recursive	ADJ
ejpam-5057	453	3	approach	approach	NOUN
ejpam-5057	453	4	to	to	ADP
ejpam-5057	453	5	low	low	ADJ
ejpam-5057	453	6	complexity	complexity	NOUN
ejpam-5057	453	7	codes	code	NOUN
ejpam-5057	453	8	.	.	PUNCT
ejpam-5057	454	1	ieee	ieee	NOUN
ejpam-5057	454	2	transactions	transaction	NOUN
ejpam-5057	454	3	on	on	ADP
ejpam-5057	454	4	information	information	NOUN
ejpam-5057	454	5	theory	theory	NOUN
ejpam-5057	454	6	,	,	PUNCT
ejpam-5057	454	7	27(5):533–547	27(5):533–547	PROPN
ejpam-5057	454	8	,	,	PUNCT
ejpam-5057	454	9	1981	1981	NUM
ejpam-5057	454	10	.	.	PUNCT
ejpam-5057	455	1	[	[	X
ejpam-5057	455	2	17	17	NUM
ejpam-5057	455	3	]	]	X
ejpam-5057	455	4	gillés	gillés	PROPN
ejpam-5057	455	5	zémor	zémor	PROPN
ejpam-5057	455	6	.	.	PUNCT
ejpam-5057	456	1	on	on	ADP
ejpam-5057	456	2	expander	expander	NOUN
ejpam-5057	456	3	codes	code	NOUN
ejpam-5057	456	4	.	.	PUNCT
ejpam-5057	457	1	ieee	ieee	NOUN
ejpam-5057	457	2	transactions	transaction	NOUN
ejpam-5057	457	3	on	on	ADP
ejpam-5057	457	4	information	information	NOUN
ejpam-5057	457	5	theory	theory	NOUN
ejpam-5057	457	6	,	,	PUNCT
ejpam-5057	457	7	47(2):835–837	47(2):835–837	PROPN
ejpam-5057	457	8	,	,	PUNCT
ejpam-5057	457	9	2001	2001	NUM
ejpam-5057	457	10	.	.	PUNCT
ejpam-5057	458	1	[	[	X
ejpam-5057	458	2	18	18	NUM
ejpam-5057	458	3	]	]	PUNCT
ejpam-5057	458	4	mingqing	mingqe	VERB
ejpam-5057	458	5	zhai	zhai	PROPN
ejpam-5057	458	6	,	,	PUNCT
ejpam-5057	458	7	huiqiu	huiqiu	VERB
ejpam-5057	458	8	lin	lin	PROPN
ejpam-5057	458	9	,	,	PUNCT
ejpam-5057	458	10	and	and	CCONJ
ejpam-5057	458	11	bingwang	bingwang	PROPN
ejpam-5057	458	12	.	.	PUNCT
ejpam-5057	459	1	sharp	sharp	ADJ
ejpam-5057	459	2	upper	upper	ADJ
ejpam-5057	459	3	bounds	bound	NOUN
ejpam-5057	459	4	on	on	ADP
ejpam-5057	459	5	the	the	DET
ejpam-5057	459	6	second	second	ADJ
ejpam-5057	459	7	largest	large	ADJ
ejpam-5057	459	8	eigenvalues	eigenvalue	NOUN
ejpam-5057	459	9	of	of	ADP
ejpam-5057	459	10	connected	connected	ADJ
ejpam-5057	459	11	graphs	graph	NOUN
ejpam-5057	459	12	.	.	PUNCT
ejpam-5057	460	1	linear	linear	ADJ
ejpam-5057	460	2	algebra	algebra	NOUN
ejpam-5057	460	3	and	and	CCONJ
ejpam-5057	460	4	its	its	PRON
ejpam-5057	460	5	applications	application	NOUN
ejpam-5057	460	6	,	,	PUNCT
ejpam-5057	460	7	437(1):236–241	437(1):236–241	NUM
ejpam-5057	460	8	,	,	PUNCT
ejpam-5057	460	9	2012	2012	NUM
ejpam-5057	460	10	.	.	PUNCT