id	sid	tid	token	lemma	pos
ejpam-4448	1	1	european	european	PROPN
ejpam-4448	1	2	journal	journal	PROPN
ejpam-4448	1	3	of	of	ADP
ejpam-4448	1	4	pure	pure	ADJ
ejpam-4448	1	5	and	and	CCONJ
ejpam-4448	1	6	applied	apply	VERB
ejpam-4448	1	7	mathematics	mathematic	NOUN
ejpam-4448	1	8	vol	vol	NOUN
ejpam-4448	1	9	.	.	PROPN
ejpam-4448	2	1	15	15	NUM
ejpam-4448	2	2	,	,	PUNCT
ejpam-4448	2	3	no	no	INTJ
ejpam-4448	2	4	.	.	NOUN
ejpam-4448	2	5	3	3	NUM
ejpam-4448	2	6	,	,	PUNCT
ejpam-4448	2	7	2022	2022	NUM
ejpam-4448	2	8	,	,	PUNCT
ejpam-4448	2	9	841	841	NUM
ejpam-4448	2	10	-	-	SYM
ejpam-4448	2	11	855	855	NUM
ejpam-4448	2	12	issn	issn	PROPN
ejpam-4448	2	13	1307	1307	NUM
ejpam-4448	2	14	-	-	SYM
ejpam-4448	2	15	5543	5543	NUM
ejpam-4448	2	16	–	–	PUNCT
ejpam-4448	3	1	ejpam.com	ejpam.com	X
ejpam-4448	3	2	published	publish	VERB
ejpam-4448	3	3	by	by	ADP
ejpam-4448	3	4	new	new	PROPN
ejpam-4448	3	5	york	york	PROPN
ejpam-4448	3	6	business	business	PROPN
ejpam-4448	3	7	global	global	PROPN
ejpam-4448	3	8	hermitian	hermitian	ADJ
ejpam-4448	3	9	adjacency	adjacency	NOUN
ejpam-4448	3	10	matrices	matrix	NOUN
ejpam-4448	3	11	of	of	ADP
ejpam-4448	3	12	mixed	mixed	ADJ
ejpam-4448	3	13	graphs	graph	NOUN
ejpam-4448	3	14	mohammad	mohammad	PROPN
ejpam-4448	3	15	abudayah1,∗	abudayah1,∗	PROPN
ejpam-4448	3	16	,	,	PUNCT
ejpam-4448	3	17	omar	omar	PROPN
ejpam-4448	3	18	alomari1	alomari1	PROPN
ejpam-4448	3	19	,	,	PUNCT
ejpam-4448	3	20	torsten	torsten	ADJ
ejpam-4448	3	21	sander2	sander2	ADJ
ejpam-4448	3	22	1	1	NUM
ejpam-4448	3	23	school	school	NOUN
ejpam-4448	3	24	of	of	ADP
ejpam-4448	3	25	basic	basic	ADJ
ejpam-4448	3	26	sciences	science	NOUN
ejpam-4448	3	27	and	and	CCONJ
ejpam-4448	3	28	humanities	humanity	NOUN
ejpam-4448	3	29	,	,	PUNCT
ejpam-4448	3	30	german	german	ADJ
ejpam-4448	3	31	jordanian	jordanian	ADJ
ejpam-4448	3	32	university	university	NOUN
ejpam-4448	3	33	,	,	PUNCT
ejpam-4448	3	34	amman	amman	PROPN
ejpam-4448	3	35	,	,	PUNCT
ejpam-4448	3	36	jordan	jordan	PROPN
ejpam-4448	3	37	2	2	NUM
ejpam-4448	3	38	fakultät	fakultät	PRON
ejpam-4448	3	39	für	für	PROPN
ejpam-4448	3	40	informatik	informatik	NOUN
ejpam-4448	3	41	,	,	PUNCT
ejpam-4448	3	42	ostfalia	ostfalia	PROPN
ejpam-4448	3	43	hochschule	hochschule	NOUN
ejpam-4448	3	44	für	für	PROPN
ejpam-4448	3	45	angewandte	angewandte	NOUN
ejpam-4448	3	46	wissenschaften	wissenschaften	VERB
ejpam-4448	3	47	,	,	PUNCT
ejpam-4448	3	48	wolfenbüttel	wolfenbüttel	PROPN
ejpam-4448	3	49	,	,	PUNCT
ejpam-4448	3	50	germany	germany	PROPN
ejpam-4448	3	51	abstract	abstract	NOUN
ejpam-4448	3	52	.	.	PUNCT
ejpam-4448	4	1	the	the	DET
ejpam-4448	4	2	traditional	traditional	ADJ
ejpam-4448	4	3	adjacency	adjacency	NOUN
ejpam-4448	4	4	matrix	matrix	NOUN
ejpam-4448	4	5	of	of	ADP
ejpam-4448	4	6	a	a	DET
ejpam-4448	4	7	mixed	mixed	ADJ
ejpam-4448	4	8	graph	graph	NOUN
ejpam-4448	4	9	is	be	AUX
ejpam-4448	4	10	not	not	PART
ejpam-4448	4	11	symmetric	symmetric	ADJ
ejpam-4448	4	12	in	in	ADP
ejpam-4448	4	13	general	general	ADJ
ejpam-4448	4	14	,	,	PUNCT
ejpam-4448	4	15	hence	hence	ADV
ejpam-4448	4	16	its	its	PRON
ejpam-4448	4	17	eigenvalues	eigenvalue	NOUN
ejpam-4448	4	18	may	may	AUX
ejpam-4448	4	19	be	be	AUX
ejpam-4448	4	20	not	not	PART
ejpam-4448	4	21	real	real	ADJ
ejpam-4448	4	22	.	.	PUNCT
ejpam-4448	5	1	to	to	PART
ejpam-4448	5	2	overcome	overcome	VERB
ejpam-4448	5	3	this	this	DET
ejpam-4448	5	4	obstacle	obstacle	NOUN
ejpam-4448	5	5	,	,	PUNCT
ejpam-4448	5	6	several	several	ADJ
ejpam-4448	5	7	authors	author	NOUN
ejpam-4448	5	8	have	have	AUX
ejpam-4448	5	9	recently	recently	ADV
ejpam-4448	5	10	defined	define	VERB
ejpam-4448	5	11	and	and	CCONJ
ejpam-4448	5	12	studied	study	VERB
ejpam-4448	5	13	various	various	ADJ
ejpam-4448	5	14	hermitian	hermitian	ADJ
ejpam-4448	5	15	adjacency	adjacency	NOUN
ejpam-4448	5	16	matrices	matrix	NOUN
ejpam-4448	5	17	of	of	ADP
ejpam-4448	5	18	digraphs	digraph	NOUN
ejpam-4448	5	19	or	or	CCONJ
ejpam-4448	5	20	mixed	mixed	ADJ
ejpam-4448	5	21	graphs	graph	NOUN
ejpam-4448	5	22	.	.	PUNCT
ejpam-4448	6	1	in	in	ADP
ejpam-4448	6	2	this	this	DET
ejpam-4448	6	3	work	work	NOUN
ejpam-4448	6	4	we	we	PRON
ejpam-4448	6	5	unify	unify	VERB
ejpam-4448	6	6	previous	previous	ADJ
ejpam-4448	6	7	work	work	NOUN
ejpam-4448	6	8	and	and	CCONJ
ejpam-4448	6	9	offer	offer	VERB
ejpam-4448	6	10	a	a	DET
ejpam-4448	6	11	new	new	ADJ
ejpam-4448	6	12	perspective	perspective	NOUN
ejpam-4448	6	13	on	on	ADP
ejpam-4448	6	14	the	the	DET
ejpam-4448	6	15	subject	subject	NOUN
ejpam-4448	6	16	by	by	ADP
ejpam-4448	6	17	introducing	introduce	VERB
ejpam-4448	6	18	the	the	DET
ejpam-4448	6	19	concept	concept	NOUN
ejpam-4448	6	20	of	of	ADP
ejpam-4448	6	21	monographs	monograph	NOUN
ejpam-4448	6	22	.	.	PUNCT
ejpam-4448	7	1	moreover	moreover	ADV
ejpam-4448	7	2	,	,	PUNCT
ejpam-4448	7	3	we	we	PRON
ejpam-4448	7	4	consider	consider	VERB
ejpam-4448	7	5	questions	question	NOUN
ejpam-4448	7	6	of	of	ADP
ejpam-4448	7	7	cospectrality	cospectrality	NOUN
ejpam-4448	7	8	.	.	PUNCT
ejpam-4448	8	1	2020	2020	NUM
ejpam-4448	8	2	mathematics	mathematic	NOUN
ejpam-4448	8	3	subject	subject	NOUN
ejpam-4448	8	4	classifications	classification	NOUN
ejpam-4448	8	5	:	:	PUNCT
ejpam-4448	8	6	05c50	05c50	NUM
ejpam-4448	8	7	,	,	PUNCT
ejpam-4448	8	8	15a18	15a18	NUM
ejpam-4448	8	9	key	key	ADJ
ejpam-4448	8	10	words	word	NOUN
ejpam-4448	8	11	and	and	CCONJ
ejpam-4448	8	12	phrases	phrase	NOUN
ejpam-4448	8	13	:	:	PUNCT
ejpam-4448	8	14	mixed	mixed	ADJ
ejpam-4448	8	15	graphs	graph	NOUN
ejpam-4448	8	16	,	,	PUNCT
ejpam-4448	8	17	oriented	orient	VERB
ejpam-4448	8	18	graphs	graph	NOUN
ejpam-4448	8	19	,	,	PUNCT
ejpam-4448	8	20	graph	graph	NOUN
ejpam-4448	8	21	spectra	spectra	NOUN
ejpam-4448	8	22	,	,	PUNCT
ejpam-4448	8	23	eigenvalues	eigenvalue	NOUN
ejpam-4448	8	24	,	,	PUNCT
ejpam-4448	8	25	cospectrality	cospectrality	NOUN
ejpam-4448	8	26	1	1	NUM
ejpam-4448	8	27	.	.	PUNCT
ejpam-4448	9	1	introduction	introduction	NOUN
ejpam-4448	9	2	algebraic	algebraic	ADJ
ejpam-4448	9	3	graph	graph	NOUN
ejpam-4448	9	4	theory	theory	NOUN
ejpam-4448	9	5	strives	strive	VERB
ejpam-4448	9	6	to	to	PART
ejpam-4448	9	7	relate	relate	VERB
ejpam-4448	9	8	the	the	DET
ejpam-4448	9	9	structural	structural	ADJ
ejpam-4448	9	10	properties	property	NOUN
ejpam-4448	9	11	of	of	ADP
ejpam-4448	9	12	graphs	graph	NOUN
ejpam-4448	9	13	to	to	ADP
ejpam-4448	9	14	the	the	DET
ejpam-4448	9	15	algebraic	algebraic	ADJ
ejpam-4448	9	16	properties	property	NOUN
ejpam-4448	9	17	of	of	ADP
ejpam-4448	9	18	objects	object	NOUN
ejpam-4448	9	19	associated	associate	VERB
ejpam-4448	9	20	with	with	ADP
ejpam-4448	9	21	them	they	PRON
ejpam-4448	9	22	.	.	PUNCT
ejpam-4448	10	1	specifically	specifically	ADV
ejpam-4448	10	2	,	,	PUNCT
ejpam-4448	10	3	in	in	ADP
ejpam-4448	10	4	spectral	spectral	ADJ
ejpam-4448	10	5	graph	graph	NOUN
ejpam-4448	10	6	theory	theory	NOUN
ejpam-4448	10	7	the	the	DET
ejpam-4448	10	8	eigenvalues	eigenvalue	NOUN
ejpam-4448	10	9	and	and	CCONJ
ejpam-4448	10	10	eigenvectors	eigenvector	NOUN
ejpam-4448	10	11	of	of	ADP
ejpam-4448	10	12	matrices	matrix	NOUN
ejpam-4448	10	13	associated	associate	VERB
ejpam-4448	10	14	with	with	ADP
ejpam-4448	10	15	graphs	graph	NOUN
ejpam-4448	10	16	are	be	AUX
ejpam-4448	10	17	studied	study	VERB
ejpam-4448	10	18	.	.	PUNCT
ejpam-4448	11	1	most	most	ADV
ejpam-4448	11	2	traditionally	traditionally	ADV
ejpam-4448	11	3	,	,	PUNCT
ejpam-4448	11	4	the	the	DET
ejpam-4448	11	5	object	object	NOUN
ejpam-4448	11	6	of	of	ADP
ejpam-4448	11	7	interest	interest	NOUN
ejpam-4448	11	8	would	would	AUX
ejpam-4448	11	9	be	be	AUX
ejpam-4448	11	10	the	the	DET
ejpam-4448	11	11	adjacency	adjacency	NOUN
ejpam-4448	11	12	matrix	matrix	NOUN
ejpam-4448	11	13	of	of	ADP
ejpam-4448	11	14	some	some	DET
ejpam-4448	11	15	undirected	undirected	ADJ
ejpam-4448	11	16	graph	graph	NOUN
ejpam-4448	11	17	,	,	PUNCT
ejpam-4448	11	18	i.e.	i.e.	X
ejpam-4448	11	19	,	,	PUNCT
ejpam-4448	11	20	the	the	DET
ejpam-4448	11	21	square	square	ADJ
ejpam-4448	11	22	matrix	matrix	NOUN
ejpam-4448	12	1	[	[	X
ejpam-4448	12	2	auv	auv	X
ejpam-4448	12	3	]	]	PUNCT
ejpam-4448	12	4	such	such	ADJ
ejpam-4448	12	5	that	that	SCONJ
ejpam-4448	12	6	auv	auv	PROPN
ejpam-4448	12	7	=	=	NOUN
ejpam-4448	12	8	1	1	NUM
ejpam-4448	12	9	if	if	SCONJ
ejpam-4448	12	10	there	there	PRON
ejpam-4448	12	11	is	be	VERB
ejpam-4448	12	12	an	an	DET
ejpam-4448	12	13	edge	edge	NOUN
ejpam-4448	12	14	between	between	ADP
ejpam-4448	12	15	vertices	vertex	NOUN
ejpam-4448	12	16	u	u	NOUN
ejpam-4448	12	17	and	and	CCONJ
ejpam-4448	12	18	v	v	NOUN
ejpam-4448	12	19	,	,	PUNCT
ejpam-4448	12	20	otherwise	otherwise	ADV
ejpam-4448	12	21	auv	auv	PROPN
ejpam-4448	12	22	=	=	NOUN
ejpam-4448	12	23	0	0	PROPN
ejpam-4448	12	24	.	.	PUNCT
ejpam-4448	13	1	by	by	ADP
ejpam-4448	13	2	construction	construction	NOUN
ejpam-4448	13	3	,	,	PUNCT
ejpam-4448	13	4	the	the	DET
ejpam-4448	13	5	adjacency	adjacency	NOUN
ejpam-4448	13	6	matrix	matrix	NOUN
ejpam-4448	13	7	of	of	ADP
ejpam-4448	13	8	an	an	DET
ejpam-4448	13	9	undirected	undirected	ADJ
ejpam-4448	13	10	graph	graph	NOUN
ejpam-4448	13	11	is	be	AUX
ejpam-4448	13	12	symmetric	symmetric	ADJ
ejpam-4448	13	13	.	.	PUNCT
ejpam-4448	14	1	hence	hence	ADV
ejpam-4448	14	2	theorems	theorem	NOUN
ejpam-4448	14	3	from	from	ADP
ejpam-4448	14	4	linear	linear	ADJ
ejpam-4448	14	5	algebra	algebra	NOUN
ejpam-4448	14	6	dealing	deal	VERB
ejpam-4448	14	7	with	with	ADP
ejpam-4448	14	8	non	non	ADJ
ejpam-4448	14	9	-	-	ADJ
ejpam-4448	14	10	negative	negative	ADJ
ejpam-4448	14	11	symmetric	symmetric	ADJ
ejpam-4448	14	12	matrices	matrix	NOUN
ejpam-4448	14	13	can	can	AUX
ejpam-4448	14	14	be	be	AUX
ejpam-4448	14	15	readily	readily	ADV
ejpam-4448	14	16	applied	apply	VERB
ejpam-4448	14	17	to	to	PART
ejpam-4448	14	18	obtain	obtain	VERB
ejpam-4448	14	19	a	a	DET
ejpam-4448	14	20	number	number	NOUN
ejpam-4448	14	21	of	of	ADP
ejpam-4448	14	22	desirable	desirable	ADJ
ejpam-4448	14	23	spectral	spectral	ADJ
ejpam-4448	14	24	properties	property	NOUN
ejpam-4448	14	25	.	.	PUNCT
ejpam-4448	15	1	most	most	ADV
ejpam-4448	15	2	notably	notably	ADV
ejpam-4448	15	3	,	,	PUNCT
ejpam-4448	15	4	the	the	DET
ejpam-4448	15	5	spectrum	spectrum	NOUN
ejpam-4448	15	6	of	of	ADP
ejpam-4448	15	7	the	the	DET
ejpam-4448	15	8	adjacency	adjacency	NOUN
ejpam-4448	15	9	matrix	matrix	NOUN
ejpam-4448	15	10	is	be	AUX
ejpam-4448	15	11	real	real	ADJ
ejpam-4448	15	12	.	.	PUNCT
ejpam-4448	16	1	moreover	moreover	ADV
ejpam-4448	16	2	,	,	PUNCT
ejpam-4448	16	3	there	there	PRON
ejpam-4448	16	4	exists	exist	VERB
ejpam-4448	16	5	a	a	DET
ejpam-4448	16	6	basis	basis	NOUN
ejpam-4448	16	7	of	of	ADP
ejpam-4448	16	8	pairwise	pairwise	NOUN
ejpam-4448	16	9	orthogonal	orthogonal	ADJ
ejpam-4448	16	10	eigenvectors	eigenvector	NOUN
ejpam-4448	16	11	.	.	PUNCT
ejpam-4448	17	1	however	however	ADV
ejpam-4448	17	2	,	,	PUNCT
ejpam-4448	17	3	when	when	SCONJ
ejpam-4448	17	4	dealing	deal	VERB
ejpam-4448	17	5	with	with	ADP
ejpam-4448	17	6	directed	direct	VERB
ejpam-4448	17	7	or	or	CCONJ
ejpam-4448	17	8	mixed	mixed	ADJ
ejpam-4448	17	9	graphs	graph	NOUN
ejpam-4448	17	10	the	the	DET
ejpam-4448	17	11	definition	definition	NOUN
ejpam-4448	17	12	of	of	ADP
ejpam-4448	17	13	the	the	DET
ejpam-4448	17	14	adjacency	adjacency	NOUN
ejpam-4448	17	15	matrix	matrix	NOUN
ejpam-4448	17	16	needs	need	VERB
ejpam-4448	17	17	to	to	PART
ejpam-4448	17	18	be	be	AUX
ejpam-4448	17	19	changed	change	VERB
ejpam-4448	17	20	to	to	PART
ejpam-4448	17	21	accommodate	accommodate	VERB
ejpam-4448	17	22	the	the	DET
ejpam-4448	17	23	fact	fact	NOUN
ejpam-4448	17	24	that	that	SCONJ
ejpam-4448	17	25	the	the	DET
ejpam-4448	17	26	adjacency	adjacency	PROPN
ejpam-4448	17	27	relation	relation	NOUN
ejpam-4448	17	28	of	of	ADP
ejpam-4448	17	29	vertices	vertex	NOUN
ejpam-4448	17	30	is	be	AUX
ejpam-4448	17	31	no	no	PRON
ejpam-4448	17	32	longer	long	ADV
ejpam-4448	17	33	symmetric	symmetric	ADJ
ejpam-4448	17	34	.	.	PUNCT
ejpam-4448	18	1	for	for	ADP
ejpam-4448	18	2	a	a	DET
ejpam-4448	18	3	digraph	digraph	NOUN
ejpam-4448	18	4	we	we	PRON
ejpam-4448	18	5	set	set	VERB
ejpam-4448	18	6	auv	auv	PROPN
ejpam-4448	18	7	=	=	NOUN
ejpam-4448	18	8	1	1	NUM
ejpam-4448	18	9	if	if	SCONJ
ejpam-4448	18	10	there	there	PRON
ejpam-4448	18	11	is	be	VERB
ejpam-4448	18	12	an	an	DET
ejpam-4448	18	13	arc	arc	NOUN
ejpam-4448	18	14	from	from	ADP
ejpam-4448	18	15	vertex	vertex	NOUN
ejpam-4448	18	16	number	number	NOUN
ejpam-4448	18	17	u	u	NOUN
ejpam-4448	18	18	to	to	ADP
ejpam-4448	18	19	v	v	NOUN
ejpam-4448	18	20	and	and	CCONJ
ejpam-4448	18	21	auv	auv	NUM
ejpam-4448	18	22	=	=	PROPN
ejpam-4448	18	23	0	0	PUNCT
ejpam-4448	18	24	otherwise	otherwise	ADV
ejpam-4448	18	25	.	.	PUNCT
ejpam-4448	19	1	the	the	DET
ejpam-4448	19	2	loss	loss	NOUN
ejpam-4448	19	3	of	of	ADP
ejpam-4448	19	4	symmetry	symmetry	NOUN
ejpam-4448	19	5	proves	prove	VERB
ejpam-4448	19	6	a	a	DET
ejpam-4448	19	7	serious	serious	ADJ
ejpam-4448	19	8	impediment	impediment	NOUN
ejpam-4448	19	9	to	to	ADP
ejpam-4448	19	10	relating	relate	VERB
ejpam-4448	19	11	algebraic	algebraic	ADJ
ejpam-4448	19	12	and	and	CCONJ
ejpam-4448	19	13	structural	structural	ADJ
ejpam-4448	19	14	properties	property	NOUN
ejpam-4448	19	15	to	to	ADP
ejpam-4448	19	16	one	one	NUM
ejpam-4448	19	17	another	another	DET
ejpam-4448	19	18	,	,	PUNCT
ejpam-4448	19	19	cf	cf	NOUN
ejpam-4448	19	20	.	.	PUNCT
ejpam-4448	20	1	the	the	DET
ejpam-4448	20	2	survey	survey	NOUN
ejpam-4448	20	3	[	[	X
ejpam-4448	20	4	4	4	NUM
ejpam-4448	20	5	]	]	PUNCT
ejpam-4448	20	6	.	.	PUNCT
ejpam-4448	21	1	∗corresponding	∗corresponde	VERB
ejpam-4448	21	2	author	author	NOUN
ejpam-4448	21	3	.	.	PUNCT
ejpam-4448	22	1	doi	doi	NOUN
ejpam-4448	22	2	:	:	PUNCT
ejpam-4448	22	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4448	https://doi.org/10.29020/nybg.ejpam.v15i3.4448	PRON
ejpam-4448	22	4	email	email	NOUN
ejpam-4448	22	5	addresses	address	NOUN
ejpam-4448	22	6	:	:	PUNCT
ejpam-4448	22	7	mohammad.abudayah@gju.edu.jo	mohammad.abudayah@gju.edu.jo	ADJ
ejpam-4448	22	8	(	(	PUNCT
ejpam-4448	22	9	m.	m.	NOUN
ejpam-4448	22	10	abudayah	abudayah	PROPN
ejpam-4448	22	11	)	)	PUNCT
ejpam-4448	22	12	,	,	PUNCT
ejpam-4448	22	13	omar.alomari@gju.edu.jo	omar.alomari@gju.edu.jo	NUM
ejpam-4448	22	14	(	(	PUNCT
ejpam-4448	22	15	o.	o.	NOUN
ejpam-4448	22	16	alomari	alomari	PROPN
ejpam-4448	22	17	)	)	PUNCT
ejpam-4448	22	18	,	,	PUNCT
ejpam-4448	22	19	t.sander@ostfalia.de	t.sander@ostfalia.de	PUNCT
ejpam-4448	22	20	(	(	PUNCT
ejpam-4448	22	21	t.	t.	NOUN
ejpam-4448	22	22	sander	sander	NOUN
ejpam-4448	22	23	)	)	PUNCT
ejpam-4448	22	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4448	22	25	841	841	NUM
ejpam-4448	23	1	©	©	PROPN
ejpam-4448	23	2	2022	2022	NUM
ejpam-4448	23	3	ejpam	ejpam	VERB
ejpam-4448	23	4	all	all	DET
ejpam-4448	23	5	rights	right	NOUN
ejpam-4448	23	6	reserved	reserve	VERB
ejpam-4448	23	7	.	.	PUNCT
ejpam-4448	24	1	m.	m.	NOUN
ejpam-4448	24	2	abudayah	abudayah	PROPN
ejpam-4448	24	3	,	,	PUNCT
ejpam-4448	24	4	o.	o.	PROPN
ejpam-4448	24	5	alomari	alomari	PROPN
ejpam-4448	24	6	,	,	PUNCT
ejpam-4448	24	7	t.	t.	PROPN
ejpam-4448	24	8	sander	sander	PROPN
ejpam-4448	24	9	/	/	SYM
ejpam-4448	24	10	eur	eur	PROPN
ejpam-4448	24	11	.	.	PUNCT
ejpam-4448	25	1	j.	j.	PROPN
ejpam-4448	25	2	pure	pure	PROPN
ejpam-4448	25	3	appl	appl	PROPN
ejpam-4448	25	4	.	.	PROPN
ejpam-4448	25	5	math	math	PROPN
ejpam-4448	25	6	,	,	PUNCT
ejpam-4448	25	7	15	15	NUM
ejpam-4448	25	8	(	(	PUNCT
ejpam-4448	25	9	3	3	NUM
ejpam-4448	25	10	)	)	PUNCT
ejpam-4448	25	11	(	(	PUNCT
ejpam-4448	25	12	2022	2022	NUM
ejpam-4448	25	13	)	)	PUNCT
ejpam-4448	25	14	,	,	PUNCT
ejpam-4448	25	15	841	841	NUM
ejpam-4448	25	16	-	-	SYM
ejpam-4448	25	17	855	855	NUM
ejpam-4448	25	18	842	842	NUM
ejpam-4448	25	19	quite	quite	ADV
ejpam-4448	25	20	recently	recently	ADV
ejpam-4448	25	21	,	,	PUNCT
ejpam-4448	25	22	the	the	DET
ejpam-4448	25	23	idea	idea	NOUN
ejpam-4448	25	24	has	have	AUX
ejpam-4448	25	25	been	be	AUX
ejpam-4448	25	26	presented	present	VERB
ejpam-4448	25	27	to	to	PART
ejpam-4448	25	28	modify	modify	VERB
ejpam-4448	25	29	the	the	DET
ejpam-4448	25	30	definition	definition	NOUN
ejpam-4448	25	31	of	of	ADP
ejpam-4448	25	32	the	the	DET
ejpam-4448	25	33	adjacency	adjacency	NOUN
ejpam-4448	25	34	matrix	matrix	NOUN
ejpam-4448	25	35	of	of	ADP
ejpam-4448	25	36	a	a	DET
ejpam-4448	25	37	directed	direct	VERB
ejpam-4448	25	38	graph	graph	NOUN
ejpam-4448	25	39	,	,	PUNCT
ejpam-4448	25	40	some	some	DET
ejpam-4448	25	41	authors	author	NOUN
ejpam-4448	25	42	studied	study	VERB
ejpam-4448	25	43	the	the	DET
ejpam-4448	25	44	matrix	matrix	NOUN
ejpam-4448	25	45	aa∗	aa∗	NOUN
ejpam-4448	25	46	,	,	PUNCT
ejpam-4448	25	47	see	see	VERB
ejpam-4448	25	48	[	[	X
ejpam-4448	25	49	1	1	NUM
ejpam-4448	25	50	,	,	PUNCT
ejpam-4448	25	51	2	2	NUM
ejpam-4448	25	52	]	]	PUNCT
ejpam-4448	25	53	,	,	PUNCT
ejpam-4448	25	54	and	and	CCONJ
ejpam-4448	25	55	others	other	NOUN
ejpam-4448	25	56	used	use	VERB
ejpam-4448	25	57	complex	complex	ADJ
ejpam-4448	25	58	numbers	number	NOUN
ejpam-4448	25	59	,	,	PUNCT
ejpam-4448	25	60	in	in	ADP
ejpam-4448	25	61	such	such	DET
ejpam-4448	25	62	a	a	DET
ejpam-4448	25	63	way	way	NOUN
ejpam-4448	25	64	that	that	PRON
ejpam-4448	25	65	it	it	PRON
ejpam-4448	25	66	still	still	ADV
ejpam-4448	25	67	properly	properly	ADV
ejpam-4448	25	68	reflects	reflect	VERB
ejpam-4448	25	69	the	the	DET
ejpam-4448	25	70	adjacency	adjacency	NOUN
ejpam-4448	25	71	relation	relation	NOUN
ejpam-4448	25	72	but	but	CCONJ
ejpam-4448	25	73	at	at	ADP
ejpam-4448	25	74	the	the	DET
ejpam-4448	25	75	same	same	ADJ
ejpam-4448	25	76	time	time	NOUN
ejpam-4448	25	77	constitutes	constitute	VERB
ejpam-4448	25	78	a	a	DET
ejpam-4448	25	79	hermitian	hermitian	ADJ
ejpam-4448	25	80	matrix	matrix	NOUN
ejpam-4448	25	81	.	.	PUNCT
ejpam-4448	26	1	let	let	VERB
ejpam-4448	26	2	us	we	PRON
ejpam-4448	26	3	give	give	VERB
ejpam-4448	26	4	an	an	DET
ejpam-4448	26	5	overview	overview	NOUN
ejpam-4448	26	6	of	of	ADP
ejpam-4448	26	7	some	some	DET
ejpam-4448	26	8	efforts	effort	NOUN
ejpam-4448	26	9	and	and	CCONJ
ejpam-4448	26	10	results	result	NOUN
ejpam-4448	26	11	in	in	ADP
ejpam-4448	26	12	this	this	DET
ejpam-4448	26	13	direction	direction	NOUN
ejpam-4448	26	14	.	.	PUNCT
ejpam-4448	27	1	in	in	ADP
ejpam-4448	27	2	[	[	X
ejpam-4448	27	3	5	5	NUM
ejpam-4448	27	4	]	]	PUNCT
ejpam-4448	27	5	the	the	DET
ejpam-4448	27	6	authors	author	NOUN
ejpam-4448	27	7	use	use	VERB
ejpam-4448	27	8	the	the	DET
ejpam-4448	27	9	imaginary	imaginary	ADJ
ejpam-4448	27	10	number	number	NOUN
ejpam-4448	27	11	i	i	PRON
ejpam-4448	27	12	to	to	PART
ejpam-4448	27	13	specify	specify	VERB
ejpam-4448	27	14	auv	auv	PROPN
ejpam-4448	28	1	=	=	PUNCT
ejpam-4448	29	1	i	i	PROPN
ejpam-4448	29	2	and	and	CCONJ
ejpam-4448	29	3	avu	avu	PRON
ejpam-4448	30	1	=	=	SYM
ejpam-4448	30	2	−i	−i	PROPN
ejpam-4448	30	3	whenever	whenever	SCONJ
ejpam-4448	30	4	there	there	PRON
ejpam-4448	30	5	is	be	VERB
ejpam-4448	30	6	an	an	DET
ejpam-4448	30	7	arc	arc	NOUN
ejpam-4448	30	8	from	from	ADP
ejpam-4448	30	9	u	u	NOUN
ejpam-4448	30	10	to	to	ADP
ejpam-4448	30	11	v	v	NOUN
ejpam-4448	30	12	,	,	PUNCT
ejpam-4448	30	13	but	but	CCONJ
ejpam-4448	30	14	not	not	PART
ejpam-4448	30	15	vice	vice	ADV
ejpam-4448	30	16	versa	versa	ADV
ejpam-4448	30	17	,	,	PUNCT
ejpam-4448	30	18	further	further	PROPN
ejpam-4448	30	19	auv	auv	PROPN
ejpam-4448	30	20	=	=	SYM
ejpam-4448	30	21	1	1	NUM
ejpam-4448	30	22	whenever	whenever	SCONJ
ejpam-4448	30	23	u	u	NOUN
ejpam-4448	30	24	and	and	CCONJ
ejpam-4448	30	25	v	v	NOUN
ejpam-4448	30	26	are	be	AUX
ejpam-4448	30	27	mutually	mutually	ADV
ejpam-4448	30	28	adjacent	adjacent	ADJ
ejpam-4448	30	29	.	.	PUNCT
ejpam-4448	31	1	using	use	VERB
ejpam-4448	31	2	this	this	DET
ejpam-4448	31	3	definition	definition	NOUN
ejpam-4448	31	4	of	of	ADP
ejpam-4448	31	5	a	a	DET
ejpam-4448	31	6	hermitian	hermitian	ADJ
ejpam-4448	31	7	adjacency	adjacency	NOUN
ejpam-4448	31	8	matrix	matrix	NOUN
ejpam-4448	31	9	,	,	PUNCT
ejpam-4448	31	10	it	it	PRON
ejpam-4448	31	11	turns	turn	VERB
ejpam-4448	31	12	out	out	ADP
ejpam-4448	31	13	that	that	SCONJ
ejpam-4448	31	14	many	many	ADJ
ejpam-4448	31	15	results	result	NOUN
ejpam-4448	31	16	from	from	ADP
ejpam-4448	31	17	algebraic	algebraic	ADJ
ejpam-4448	31	18	graph	graph	NOUN
ejpam-4448	31	19	theory	theory	NOUN
ejpam-4448	31	20	known	know	VERB
ejpam-4448	31	21	for	for	ADP
ejpam-4448	31	22	undirected	undirected	ADJ
ejpam-4448	31	23	graphs	graph	NOUN
ejpam-4448	31	24	also	also	ADV
ejpam-4448	31	25	hold	hold	VERB
ejpam-4448	31	26	for	for	ADP
ejpam-4448	31	27	directed	direct	VERB
ejpam-4448	31	28	graphs	graph	NOUN
ejpam-4448	31	29	or	or	CCONJ
ejpam-4448	31	30	at	at	ADP
ejpam-4448	31	31	least	least	ADJ
ejpam-4448	31	32	exist	exist	VERB
ejpam-4448	31	33	in	in	ADP
ejpam-4448	31	34	a	a	DET
ejpam-4448	31	35	slightly	slightly	ADV
ejpam-4448	31	36	modified	modify	VERB
ejpam-4448	31	37	or	or	CCONJ
ejpam-4448	31	38	weaker	weak	ADJ
ejpam-4448	31	39	version	version	NOUN
ejpam-4448	31	40	.	.	PUNCT
ejpam-4448	32	1	for	for	ADP
ejpam-4448	32	2	example	example	NOUN
ejpam-4448	32	3	,	,	PUNCT
ejpam-4448	32	4	if	if	SCONJ
ejpam-4448	32	5	the	the	DET
ejpam-4448	32	6	underlying	underlying	ADJ
ejpam-4448	32	7	undirected	undirected	ADJ
ejpam-4448	32	8	graph	graph	NOUN
ejpam-4448	32	9	of	of	ADP
ejpam-4448	32	10	a	a	DET
ejpam-4448	32	11	given	give	VERB
ejpam-4448	32	12	oriented	orient	VERB
ejpam-4448	32	13	graph	graph	NOUN
ejpam-4448	32	14	is	be	AUX
ejpam-4448	32	15	bipartite	bipartite	ADJ
ejpam-4448	32	16	,	,	PUNCT
ejpam-4448	32	17	then	then	ADV
ejpam-4448	32	18	the	the	DET
ejpam-4448	32	19	spectrum	spectrum	NOUN
ejpam-4448	32	20	of	of	ADP
ejpam-4448	32	21	the	the	DET
ejpam-4448	32	22	hermitian	hermitian	ADJ
ejpam-4448	32	23	adjacency	adjacency	NOUN
ejpam-4448	32	24	matrix	matrix	NOUN
ejpam-4448	32	25	is	be	AUX
ejpam-4448	32	26	symmetric	symmetric	ADJ
ejpam-4448	32	27	with	with	ADP
ejpam-4448	32	28	respect	respect	NOUN
ejpam-4448	32	29	to	to	ADP
ejpam-4448	32	30	zero	zero	NUM
ejpam-4448	32	31	,	,	PUNCT
ejpam-4448	32	32	but	but	CCONJ
ejpam-4448	32	33	–	–	PUNCT
ejpam-4448	32	34	in	in	ADP
ejpam-4448	32	35	contrast	contrast	NOUN
ejpam-4448	32	36	to	to	ADP
ejpam-4448	32	37	the	the	DET
ejpam-4448	32	38	undirected	undirected	ADJ
ejpam-4448	32	39	case	case	NOUN
ejpam-4448	32	40	–	–	PUNCT
ejpam-4448	32	41	the	the	DET
ejpam-4448	32	42	reverse	reverse	NOUN
ejpam-4448	32	43	is	be	AUX
ejpam-4448	32	44	not	not	PART
ejpam-4448	32	45	true	true	ADJ
ejpam-4448	32	46	.	.	PUNCT
ejpam-4448	33	1	independently	independently	ADV
ejpam-4448	33	2	,	,	PUNCT
ejpam-4448	33	3	the	the	DET
ejpam-4448	33	4	authors	author	NOUN
ejpam-4448	33	5	of	of	ADP
ejpam-4448	33	6	[	[	X
ejpam-4448	33	7	10	10	NUM
ejpam-4448	33	8	]	]	PUNCT
ejpam-4448	33	9	introduced	introduce	VERB
ejpam-4448	33	10	the	the	DET
ejpam-4448	33	11	same	same	ADJ
ejpam-4448	33	12	notion	notion	NOUN
ejpam-4448	33	13	of	of	ADP
ejpam-4448	33	14	a	a	DET
ejpam-4448	33	15	hermitian	hermitian	ADJ
ejpam-4448	33	16	adjacency	adjacency	NOUN
ejpam-4448	33	17	matrix	matrix	NOUN
ejpam-4448	33	18	and	and	CCONJ
ejpam-4448	33	19	proved	prove	VERB
ejpam-4448	33	20	many	many	ADJ
ejpam-4448	33	21	fundamental	fundamental	ADJ
ejpam-4448	33	22	results	result	NOUN
ejpam-4448	33	23	.	.	PUNCT
ejpam-4448	34	1	moreover	moreover	ADV
ejpam-4448	34	2	,	,	PUNCT
ejpam-4448	34	3	they	they	PRON
ejpam-4448	34	4	considered	consider	VERB
ejpam-4448	34	5	the	the	DET
ejpam-4448	34	6	gutman	gutman	NOUN
ejpam-4448	34	7	energy	energy	NOUN
ejpam-4448	34	8	(	(	PUNCT
ejpam-4448	34	9	which	which	PRON
ejpam-4448	34	10	is	be	AUX
ejpam-4448	34	11	the	the	DET
ejpam-4448	34	12	sum	sum	NOUN
ejpam-4448	34	13	of	of	ADP
ejpam-4448	34	14	the	the	DET
ejpam-4448	34	15	absolute	absolute	ADJ
ejpam-4448	34	16	values	value	NOUN
ejpam-4448	34	17	of	of	ADP
ejpam-4448	34	18	all	all	DET
ejpam-4448	34	19	eigenvalues	eigenvalue	NOUN
ejpam-4448	34	20	)	)	PUNCT
ejpam-4448	34	21	of	of	ADP
ejpam-4448	34	22	the	the	DET
ejpam-4448	34	23	hermitian	hermitian	ADJ
ejpam-4448	34	24	adjacency	adjacency	NOUN
ejpam-4448	34	25	matrix	matrix	NOUN
ejpam-4448	34	26	.	.	PUNCT
ejpam-4448	35	1	refer	refer	VERB
ejpam-4448	35	2	to	to	ADP
ejpam-4448	35	3	[	[	X
ejpam-4448	35	4	8	8	NUM
ejpam-4448	35	5	,	,	PUNCT
ejpam-4448	35	6	9	9	NUM
ejpam-4448	35	7	,	,	PUNCT
ejpam-4448	35	8	12	12	NUM
ejpam-4448	35	9	]	]	PUNCT
ejpam-4448	35	10	for	for	ADP
ejpam-4448	35	11	other	other	ADJ
ejpam-4448	35	12	related	related	ADJ
ejpam-4448	35	13	work	work	NOUN
ejpam-4448	35	14	.	.	PUNCT
ejpam-4448	36	1	the	the	DET
ejpam-4448	36	2	goal	goal	NOUN
ejpam-4448	36	3	of	of	ADP
ejpam-4448	36	4	the	the	DET
ejpam-4448	36	5	present	present	ADJ
ejpam-4448	36	6	paper	paper	NOUN
ejpam-4448	36	7	is	be	AUX
ejpam-4448	36	8	as	as	SCONJ
ejpam-4448	36	9	follows	follow	VERB
ejpam-4448	36	10	.	.	PUNCT
ejpam-4448	37	1	to	to	PART
ejpam-4448	37	2	begin	begin	VERB
ejpam-4448	37	3	with	with	ADP
ejpam-4448	37	4	,	,	PUNCT
ejpam-4448	37	5	we	we	PRON
ejpam-4448	37	6	generalize	generalize	VERB
ejpam-4448	37	7	and	and	CCONJ
ejpam-4448	37	8	unify	unify	VERB
ejpam-4448	37	9	previous	previous	ADJ
ejpam-4448	37	10	results	result	NOUN
ejpam-4448	37	11	.	.	PUNCT
ejpam-4448	38	1	we	we	PRON
ejpam-4448	38	2	will	will	AUX
ejpam-4448	38	3	then	then	ADV
ejpam-4448	38	4	introduce	introduce	VERB
ejpam-4448	38	5	the	the	DET
ejpam-4448	38	6	concept	concept	NOUN
ejpam-4448	38	7	of	of	ADP
ejpam-4448	38	8	monographs	monograph	NOUN
ejpam-4448	38	9	,	,	PUNCT
ejpam-4448	38	10	permitting	permit	VERB
ejpam-4448	38	11	us	we	PRON
ejpam-4448	38	12	to	to	PART
ejpam-4448	38	13	view	view	VERB
ejpam-4448	38	14	some	some	PRON
ejpam-4448	38	15	of	of	ADP
ejpam-4448	38	16	these	these	DET
ejpam-4448	38	17	results	result	NOUN
ejpam-4448	38	18	from	from	ADP
ejpam-4448	38	19	a	a	DET
ejpam-4448	38	20	new	new	ADJ
ejpam-4448	38	21	perspective	perspective	NOUN
ejpam-4448	38	22	.	.	PUNCT
ejpam-4448	39	1	moreover	moreover	ADV
ejpam-4448	39	2	,	,	PUNCT
ejpam-4448	39	3	we	we	PRON
ejpam-4448	39	4	will	will	AUX
ejpam-4448	39	5	analyze	analyze	VERB
ejpam-4448	39	6	under	under	ADP
ejpam-4448	39	7	which	which	PRON
ejpam-4448	39	8	conditions	condition	NOUN
ejpam-4448	39	9	a	a	DET
ejpam-4448	39	10	mixed	mixed	ADJ
ejpam-4448	39	11	graph	graph	NOUN
ejpam-4448	39	12	has	have	VERB
ejpam-4448	39	13	identical	identical	ADJ
ejpam-4448	39	14	spectra	spectra	NOUN
ejpam-4448	39	15	for	for	ADP
ejpam-4448	39	16	different	different	ADJ
ejpam-4448	39	17	values	value	NOUN
ejpam-4448	39	18	of	of	ADP
ejpam-4448	39	19	α	α	NOUN
ejpam-4448	39	20	.	.	PROPN
ejpam-4448	39	21	2	2	X
ejpam-4448	39	22	.	.	X
ejpam-4448	39	23	preliminaries	preliminary	NOUN
ejpam-4448	39	24	all	all	DET
ejpam-4448	39	25	graphs	graph	NOUN
ejpam-4448	39	26	considered	consider	VERB
ejpam-4448	39	27	hereafter	hereafter	ADV
ejpam-4448	39	28	shall	shall	AUX
ejpam-4448	39	29	not	not	PART
ejpam-4448	39	30	contain	contain	VERB
ejpam-4448	39	31	any	any	DET
ejpam-4448	39	32	loops	loop	NOUN
ejpam-4448	39	33	or	or	CCONJ
ejpam-4448	39	34	multiple	multiple	ADJ
ejpam-4448	39	35	edges	edge	NOUN
ejpam-4448	39	36	.	.	PUNCT
ejpam-4448	40	1	a	a	DET
ejpam-4448	40	2	mixed	mixed	ADJ
ejpam-4448	40	3	graph	graph	NOUN
ejpam-4448	40	4	d	d	NOUN
ejpam-4448	40	5	arises	arise	VERB
ejpam-4448	40	6	from	from	ADP
ejpam-4448	40	7	partially	partially	ADV
ejpam-4448	40	8	orienting	orient	VERB
ejpam-4448	40	9	an	an	DET
ejpam-4448	40	10	undirected	undirected	ADJ
ejpam-4448	40	11	graph	graph	NOUN
ejpam-4448	40	12	g	g	NOUN
ejpam-4448	40	13	,	,	PUNCT
ejpam-4448	40	14	i.e.	i.e.	X
ejpam-4448	40	15	by	by	ADP
ejpam-4448	40	16	turning	turn	VERB
ejpam-4448	40	17	some	some	PRON
ejpam-4448	40	18	of	of	ADP
ejpam-4448	40	19	the	the	DET
ejpam-4448	40	20	undirected	undirected	ADJ
ejpam-4448	40	21	edges	edge	NOUN
ejpam-4448	40	22	into	into	ADP
ejpam-4448	40	23	single	single	ADJ
ejpam-4448	40	24	arcs	arc	NOUN
ejpam-4448	40	25	.	.	PUNCT
ejpam-4448	41	1	thus	thus	ADV
ejpam-4448	41	2	,	,	PUNCT
ejpam-4448	41	3	between	between	ADP
ejpam-4448	41	4	any	any	DET
ejpam-4448	41	5	two	two	NUM
ejpam-4448	41	6	adjacent	adjacent	ADJ
ejpam-4448	41	7	vertices	vertex	NOUN
ejpam-4448	41	8	u	u	NOUN
ejpam-4448	41	9	,	,	PUNCT
ejpam-4448	41	10	v	v	NUM
ejpam-4448	41	11	of	of	ADP
ejpam-4448	41	12	the	the	DET
ejpam-4448	41	13	vertex	vertex	NOUN
ejpam-4448	41	14	set	set	VERB
ejpam-4448	41	15	v	v	NOUN
ejpam-4448	41	16	(	(	PUNCT
ejpam-4448	41	17	g	g	NOUN
ejpam-4448	41	18	)	)	PUNCT
ejpam-4448	41	19	there	there	PRON
ejpam-4448	41	20	exists	exist	VERB
ejpam-4448	41	21	either	either	CCONJ
ejpam-4448	41	22	an	an	DET
ejpam-4448	41	23	arc	arc	NOUN
ejpam-4448	41	24	from	from	ADP
ejpam-4448	41	25	u	u	PRON
ejpam-4448	41	26	to	to	ADP
ejpam-4448	41	27	v	v	NOUN
ejpam-4448	41	28	(	(	PUNCT
ejpam-4448	41	29	indicated	indicate	VERB
ejpam-4448	41	30	by	by	ADP
ejpam-4448	41	31	u→v	u→v	NOUN
ejpam-4448	41	32	)	)	PUNCT
ejpam-4448	41	33	,	,	PUNCT
ejpam-4448	41	34	an	an	DET
ejpam-4448	41	35	arc	arc	NOUN
ejpam-4448	41	36	from	from	ADP
ejpam-4448	41	37	v	v	NUM
ejpam-4448	41	38	to	to	ADP
ejpam-4448	41	39	u	u	NOUN
ejpam-4448	41	40	(	(	PUNCT
ejpam-4448	41	41	indicated	indicate	VERB
ejpam-4448	41	42	by	by	ADP
ejpam-4448	41	43	u←v	u←v	NOUN
ejpam-4448	41	44	)	)	PUNCT
ejpam-4448	41	45	,	,	PUNCT
ejpam-4448	41	46	or	or	CCONJ
ejpam-4448	41	47	an	an	DET
ejpam-4448	41	48	undirected	undirected	ADJ
ejpam-4448	41	49	edge	edge	NOUN
ejpam-4448	41	50	(	(	PUNCT
ejpam-4448	41	51	also	also	ADV
ejpam-4448	41	52	called	call	VERB
ejpam-4448	41	53	a	a	DET
ejpam-4448	41	54	digon	digon	NOUN
ejpam-4448	41	55	)	)	PUNCT
ejpam-4448	41	56	between	between	ADP
ejpam-4448	41	57	u	u	PRON
ejpam-4448	41	58	und	und	VERB
ejpam-4448	41	59	v	v	NOUN
ejpam-4448	41	60	(	(	PUNCT
ejpam-4448	41	61	indicated	indicate	VERB
ejpam-4448	41	62	by	by	ADP
ejpam-4448	41	63	u∼v	u∼v	NOUN
ejpam-4448	41	64	)	)	PUNCT
ejpam-4448	41	65	.	.	PUNCT
ejpam-4448	42	1	altogether	altogether	ADV
ejpam-4448	42	2	,	,	PUNCT
ejpam-4448	42	3	these	these	DET
ejpam-4448	42	4	arcs	arc	NOUN
ejpam-4448	42	5	and	and	CCONJ
ejpam-4448	42	6	digons	digon	NOUN
ejpam-4448	42	7	form	form	VERB
ejpam-4448	42	8	the	the	DET
ejpam-4448	42	9	edge	edge	NOUN
ejpam-4448	42	10	set	set	VERB
ejpam-4448	42	11	e(d	e(d	PROPN
ejpam-4448	42	12	)	)	PUNCT
ejpam-4448	42	13	of	of	ADP
ejpam-4448	42	14	d.	d.	PROPN
ejpam-4448	42	15	the	the	DET
ejpam-4448	42	16	graph	graph	NOUN
ejpam-4448	42	17	g	g	PROPN
ejpam-4448	42	18	is	be	AUX
ejpam-4448	42	19	called	call	VERB
ejpam-4448	42	20	the	the	DET
ejpam-4448	42	21	underlying	underlie	VERB
ejpam-4448	42	22	graph	graph	NOUN
ejpam-4448	42	23	γ(d	γ(d	PROPN
ejpam-4448	42	24	)	)	PUNCT
ejpam-4448	42	25	of	of	ADP
ejpam-4448	42	26	the	the	DET
ejpam-4448	42	27	mixed	mixed	ADJ
ejpam-4448	42	28	graphd	graphd	NOUN
ejpam-4448	42	29	.	.	PUNCT
ejpam-4448	43	1	much	much	ADJ
ejpam-4448	43	2	of	of	ADP
ejpam-4448	43	3	the	the	DET
ejpam-4448	43	4	traditional	traditional	ADJ
ejpam-4448	43	5	terminology	terminology	NOUN
ejpam-4448	43	6	(	(	PUNCT
ejpam-4448	43	7	e.g.	e.g.	ADV
ejpam-4448	43	8	being	be	AUX
ejpam-4448	43	9	regular	regular	ADJ
ejpam-4448	43	10	,	,	PUNCT
ejpam-4448	43	11	being	be	AUX
ejpam-4448	43	12	connected	connect	VERB
ejpam-4448	43	13	,	,	PUNCT
ejpam-4448	43	14	vertex	vertex	NOUN
ejpam-4448	43	15	degree	degree	NOUN
ejpam-4448	43	16	deg	deg	PROPN
ejpam-4448	43	17	(	(	PUNCT
ejpam-4448	43	18	·	·	PUNCT
ejpam-4448	43	19	)	)	PUNCT
ejpam-4448	43	20	,	,	PUNCT
ejpam-4448	43	21	maximum	maximum	ADJ
ejpam-4448	43	22	degree	degree	NOUN
ejpam-4448	43	23	∆	∆	PROPN
ejpam-4448	43	24	)	)	PUNCT
ejpam-4448	43	25	that	that	PRON
ejpam-4448	43	26	is	be	AUX
ejpam-4448	43	27	used	use	VERB
ejpam-4448	43	28	for	for	ADP
ejpam-4448	43	29	undirected	undirected	ADJ
ejpam-4448	43	30	graphs	graph	NOUN
ejpam-4448	43	31	simply	simply	ADV
ejpam-4448	43	32	carries	carry	VERB
ejpam-4448	43	33	over	over	ADP
ejpam-4448	43	34	to	to	ADP
ejpam-4448	43	35	mixed	mixed	ADJ
ejpam-4448	43	36	graphs	graph	NOUN
ejpam-4448	43	37	,	,	PUNCT
ejpam-4448	43	38	in	in	ADP
ejpam-4448	43	39	the	the	DET
ejpam-4448	43	40	sense	sense	NOUN
ejpam-4448	43	41	that	that	SCONJ
ejpam-4448	43	42	,	,	PUNCT
ejpam-4448	43	43	d	d	NOUN
ejpam-4448	43	44	is	be	AUX
ejpam-4448	43	45	said	say	VERB
ejpam-4448	43	46	to	to	PART
ejpam-4448	43	47	have	have	VERB
ejpam-4448	43	48	a	a	DET
ejpam-4448	43	49	property	property	NOUN
ejpam-4448	43	50	whenever	whenever	SCONJ
ejpam-4448	43	51	γ(d	γ(d	NOUN
ejpam-4448	43	52	)	)	PUNCT
ejpam-4448	43	53	has	have	VERB
ejpam-4448	43	54	this	this	DET
ejpam-4448	43	55	property	property	NOUN
ejpam-4448	43	56	.	.	PUNCT
ejpam-4448	44	1	in	in	ADP
ejpam-4448	44	2	particular	particular	ADJ
ejpam-4448	44	3	,	,	PUNCT
ejpam-4448	44	4	we	we	PRON
ejpam-4448	44	5	say	say	VERB
ejpam-4448	44	6	that	that	SCONJ
ejpam-4448	44	7	a	a	DET
ejpam-4448	44	8	mixed	mixed	ADJ
ejpam-4448	44	9	graph	graph	NOUN
ejpam-4448	44	10	contains	contain	VERB
ejpam-4448	44	11	a	a	DET
ejpam-4448	44	12	certain	certain	ADJ
ejpam-4448	44	13	undirected	undirected	ADJ
ejpam-4448	44	14	subgraph	subgraph	NOUN
ejpam-4448	44	15	(	(	PUNCT
ejpam-4448	44	16	e.g.	e.g.	ADV
ejpam-4448	44	17	the	the	DET
ejpam-4448	44	18	path	path	NOUN
ejpam-4448	44	19	pk	pk	NOUN
ejpam-4448	44	20	or	or	CCONJ
ejpam-4448	44	21	the	the	DET
ejpam-4448	44	22	cycle	cycle	NOUN
ejpam-4448	45	1	ck	ck	INTJ
ejpam-4448	45	2	on	on	ADP
ejpam-4448	45	3	k	k	PROPN
ejpam-4448	45	4	vertices	vertex	NOUN
ejpam-4448	45	5	)	)	PUNCT
ejpam-4448	45	6	if	if	SCONJ
ejpam-4448	45	7	γ(d	γ(d	NOUN
ejpam-4448	45	8	)	)	PUNCT
ejpam-4448	45	9	contains	contain	VERB
ejpam-4448	45	10	this	this	DET
ejpam-4448	45	11	subgraph	subgraph	NOUN
ejpam-4448	45	12	.	.	PUNCT
ejpam-4448	46	1	a	a	DET
ejpam-4448	46	2	mixed	mixed	ADJ
ejpam-4448	46	3	walk	walk	NOUN
ejpam-4448	46	4	in	in	ADP
ejpam-4448	46	5	d	d	PROPN
ejpam-4448	46	6	is	be	AUX
ejpam-4448	46	7	a	a	DET
ejpam-4448	46	8	sequence	sequence	NOUN
ejpam-4448	46	9	of	of	ADP
ejpam-4448	46	10	vertices	vertex	NOUN
ejpam-4448	46	11	v1	v1	NOUN
ejpam-4448	46	12	,	,	PUNCT
ejpam-4448	46	13	.	.	PUNCT
ejpam-4448	46	14	.	.	PUNCT
ejpam-4448	47	1	.	.	PUNCT
ejpam-4448	48	1	,	,	PUNCT
ejpam-4448	48	2	vk	vk	INTJ
ejpam-4448	48	3	of	of	ADP
ejpam-4448	48	4	d	d	PRON
ejpam-4448	48	5	such	such	ADJ
ejpam-4448	48	6	that	that	SCONJ
ejpam-4448	48	7	there	there	PRON
ejpam-4448	48	8	is	be	VERB
ejpam-4448	48	9	an	an	DET
ejpam-4448	48	10	edge	edge	NOUN
ejpam-4448	48	11	between	between	ADP
ejpam-4448	48	12	any	any	DET
ejpam-4448	48	13	two	two	NUM
ejpam-4448	48	14	subsequent	subsequent	ADJ
ejpam-4448	48	15	vertices	vertex	NOUN
ejpam-4448	48	16	vivi+1	vivi+1	ADV
ejpam-4448	48	17	in	in	ADP
ejpam-4448	48	18	d.	d.	PROPN
ejpam-4448	48	19	the	the	DET
ejpam-4448	48	20	set	set	NOUN
ejpam-4448	48	21	of	of	ADP
ejpam-4448	48	22	all	all	DET
ejpam-4448	48	23	arcs	arc	NOUN
ejpam-4448	48	24	from	from	ADP
ejpam-4448	48	25	some	some	DET
ejpam-4448	48	26	vertex	vertex	NOUN
ejpam-4448	48	27	u	u	NOUN
ejpam-4448	48	28	to	to	ADP
ejpam-4448	48	29	other	other	ADJ
ejpam-4448	48	30	vertices	vertex	NOUN
ejpam-4448	48	31	v	v	NOUN
ejpam-4448	48	32	(	(	PUNCT
ejpam-4448	48	33	resp	resp	NOUN
ejpam-4448	48	34	.	.	PUNCT
ejpam-4448	49	1	from	from	ADP
ejpam-4448	49	2	other	other	ADJ
ejpam-4448	49	3	vertices	vertex	NOUN
ejpam-4448	49	4	to	to	ADP
ejpam-4448	49	5	u	u	NOUN
ejpam-4448	49	6	)	)	PUNCT
ejpam-4448	49	7	is	be	AUX
ejpam-4448	49	8	denoted	denote	VERB
ejpam-4448	49	9	by	by	ADP
ejpam-4448	49	10	n+	n+	ADP
ejpam-4448	49	11	d	d	X
ejpam-4448	49	12	(	(	PUNCT
ejpam-4448	49	13	u	u	NOUN
ejpam-4448	49	14	)	)	PUNCT
ejpam-4448	49	15	(	(	PUNCT
ejpam-4448	49	16	resp	resp	NOUN
ejpam-4448	49	17	.	.	PUNCT
ejpam-4448	50	1	n−	n−	NOUN
ejpam-4448	50	2	d	d	X
ejpam-4448	50	3	(	(	PUNCT
ejpam-4448	50	4	u	u	NOUN
ejpam-4448	50	5	)	)	PUNCT
ejpam-4448	50	6	)	)	PUNCT
ejpam-4448	50	7	.	.	PUNCT
ejpam-4448	51	1	the	the	DET
ejpam-4448	51	2	set	set	NOUN
ejpam-4448	51	3	of	of	ADP
ejpam-4448	51	4	all	all	DET
ejpam-4448	51	5	digons	digon	NOUN
ejpam-4448	51	6	incident	incident	NOUN
ejpam-4448	51	7	with	with	ADP
ejpam-4448	51	8	vertex	vertex	NOUN
ejpam-4448	51	9	u	u	NOUN
ejpam-4448	51	10	is	be	AUX
ejpam-4448	51	11	denoted	denote	VERB
ejpam-4448	51	12	by	by	ADP
ejpam-4448	51	13	nd(u	nd(u	NOUN
ejpam-4448	51	14	)	)	PUNCT
ejpam-4448	51	15	.	.	PUNCT
ejpam-4448	52	1	the	the	DET
ejpam-4448	52	2	(	(	PUNCT
ejpam-4448	52	3	traditional	traditional	ADJ
ejpam-4448	52	4	)	)	PUNCT
ejpam-4448	52	5	adjacency	adjacency	NOUN
ejpam-4448	52	6	matrix	matrix	NOUN
ejpam-4448	52	7	a(g	a(g	PROPN
ejpam-4448	52	8	)	)	PUNCT
ejpam-4448	52	9	=	=	PUNCT
ejpam-4448	53	1	[	[	X
ejpam-4448	53	2	aij	aij	X
ejpam-4448	53	3	]	]	PUNCT
ejpam-4448	53	4	of	of	ADP
ejpam-4448	53	5	a	a	DET
ejpam-4448	53	6	given	give	VERB
ejpam-4448	53	7	,	,	PUNCT
ejpam-4448	53	8	either	either	CCONJ
ejpam-4448	53	9	undirected	undirected	ADJ
ejpam-4448	53	10	or	or	CCONJ
ejpam-4448	53	11	directed	direct	VERB
ejpam-4448	53	12	,	,	PUNCT
ejpam-4448	53	13	graph	graph	NOUN
ejpam-4448	53	14	g	g	NOUN
ejpam-4448	53	15	on	on	ADP
ejpam-4448	53	16	n	n	PRON
ejpam-4448	53	17	vertices	vertex	NOUN
ejpam-4448	53	18	is	be	AUX
ejpam-4448	53	19	the	the	DET
ejpam-4448	53	20	real	real	ADJ
ejpam-4448	53	21	matrix	matrix	NOUN
ejpam-4448	53	22	of	of	ADP
ejpam-4448	53	23	order	order	NOUN
ejpam-4448	53	24	n×n	n×n	PROPN
ejpam-4448	53	25	such	such	ADJ
ejpam-4448	53	26	that	that	SCONJ
ejpam-4448	53	27	aij	aij	PROPN
ejpam-4448	53	28	=	=	SYM
ejpam-4448	53	29	1	1	NUM
ejpam-4448	53	30	if	if	SCONJ
ejpam-4448	53	31	there	there	PRON
ejpam-4448	53	32	is	be	VERB
ejpam-4448	53	33	an	an	DET
ejpam-4448	53	34	edge	edge	NOUN
ejpam-4448	53	35	from	from	ADP
ejpam-4448	53	36	vi	vi	PROPN
ejpam-4448	53	37	to	to	ADP
ejpam-4448	53	38	vj	vj	PROPN
ejpam-4448	53	39	and	and	CCONJ
ejpam-4448	53	40	aij	aij	PROPN
ejpam-4448	53	41	=	=	SYM
ejpam-4448	53	42	0	0	NUM
ejpam-4448	53	43	otherwise	otherwise	ADV
ejpam-4448	53	44	.	.	PUNCT
ejpam-4448	54	1	for	for	ADP
ejpam-4448	54	2	directed	direct	VERB
ejpam-4448	54	3	graphs	graph	NOUN
ejpam-4448	54	4	,	,	PUNCT
ejpam-4448	54	5	the	the	DET
ejpam-4448	54	6	resulting	result	VERB
ejpam-4448	54	7	matrix	matrix	NOUN
ejpam-4448	54	8	a	a	PRON
ejpam-4448	54	9	is	be	AUX
ejpam-4448	54	10	usually	usually	ADV
ejpam-4448	54	11	non	non	ADJ
ejpam-4448	54	12	-	-	ADJ
ejpam-4448	54	13	symmetric	symmetric	ADJ
ejpam-4448	54	14	,	,	PUNCT
ejpam-4448	54	15	thus	thus	ADV
ejpam-4448	54	16	losing	lose	VERB
ejpam-4448	54	17	many	many	ADJ
ejpam-4448	54	18	desirable	desirable	ADJ
ejpam-4448	54	19	algebraic	algebraic	ADJ
ejpam-4448	54	20	properties	property	NOUN
ejpam-4448	54	21	.	.	PUNCT
ejpam-4448	55	1	we	we	PRON
ejpam-4448	55	2	therefore	therefore	ADV
ejpam-4448	55	3	m.	m.	PROPN
ejpam-4448	55	4	abudayah	abudayah	PROPN
ejpam-4448	55	5	,	,	PUNCT
ejpam-4448	55	6	o.	o.	PROPN
ejpam-4448	55	7	alomari	alomari	PROPN
ejpam-4448	55	8	,	,	PUNCT
ejpam-4448	55	9	t.	t.	PROPN
ejpam-4448	55	10	sander	sander	PROPN
ejpam-4448	55	11	/	/	SYM
ejpam-4448	55	12	eur	eur	PROPN
ejpam-4448	55	13	.	.	PUNCT
ejpam-4448	56	1	j.	j.	PROPN
ejpam-4448	56	2	pure	pure	PROPN
ejpam-4448	56	3	appl	appl	PROPN
ejpam-4448	56	4	.	.	PROPN
ejpam-4448	56	5	math	math	PROPN
ejpam-4448	56	6	,	,	PUNCT
ejpam-4448	56	7	15	15	NUM
ejpam-4448	56	8	(	(	PUNCT
ejpam-4448	56	9	3	3	NUM
ejpam-4448	56	10	)	)	PUNCT
ejpam-4448	56	11	(	(	PUNCT
ejpam-4448	56	12	2022	2022	NUM
ejpam-4448	56	13	)	)	PUNCT
ejpam-4448	56	14	,	,	PUNCT
ejpam-4448	56	15	841	841	NUM
ejpam-4448	56	16	-	-	SYM
ejpam-4448	56	17	855	855	NUM
ejpam-4448	56	18	843	843	NUM
ejpam-4448	56	19	define	define	VERB
ejpam-4448	56	20	the	the	DET
ejpam-4448	56	21	following	following	ADJ
ejpam-4448	56	22	alternative	alternative	NOUN
ejpam-4448	56	23	:	:	PUNCT
ejpam-4448	56	24	definition	definition	NOUN
ejpam-4448	56	25	1	1	NUM
ejpam-4448	56	26	.	.	PUNCT
ejpam-4448	56	27	given	give	VERB
ejpam-4448	56	28	a	a	DET
ejpam-4448	56	29	mixed	mixed	ADJ
ejpam-4448	56	30	graph	graph	NOUN
ejpam-4448	56	31	d	d	NOUN
ejpam-4448	56	32	and	and	CCONJ
ejpam-4448	56	33	a	a	DET
ejpam-4448	56	34	unit	unit	NOUN
ejpam-4448	56	35	complex	complex	ADJ
ejpam-4448	56	36	number	number	NOUN
ejpam-4448	56	37	α	α	NOUN
ejpam-4448	56	38	,	,	PUNCT
ejpam-4448	56	39	i.e.	i.e.	X
ejpam-4448	56	40	|α|	|α|	X
ejpam-4448	56	41	=	=	SYM
ejpam-4448	56	42	1	1	NUM
ejpam-4448	56	43	,	,	PUNCT
ejpam-4448	56	44	we	we	PRON
ejpam-4448	56	45	define	define	VERB
ejpam-4448	56	46	the	the	DET
ejpam-4448	56	47	α	α	NOUN
ejpam-4448	56	48	-	-	ADJ
ejpam-4448	56	49	hermitian	hermitian	ADJ
ejpam-4448	56	50	adjacency	adjacency	NOUN
ejpam-4448	56	51	matrix	matrix	NOUN
ejpam-4448	56	52	hα(d	hα(d	PUNCT
ejpam-4448	56	53	)	)	PUNCT
ejpam-4448	57	1	=	=	PUNCT
ejpam-4448	58	1	[	[	X
ejpam-4448	58	2	huv	huv	X
ejpam-4448	58	3	]	]	X
ejpam-4448	58	4	of	of	ADP
ejpam-4448	58	5	d	d	PROPN
ejpam-4448	58	6	by	by	ADP
ejpam-4448	58	7	huv	huv	PROPN
ejpam-4448	58	8	=	=	PUNCT
ejpam-4448	58	9			NOUN
ejpam-4448	58	10	1	1	NUM
ejpam-4448	58	11	if	if	SCONJ
ejpam-4448	58	12	u∼v	u∼v	ADJ
ejpam-4448	58	13	,	,	PUNCT
ejpam-4448	58	14	α	α	NOUN
ejpam-4448	58	15	if	if	SCONJ
ejpam-4448	58	16	u→v	u→v	NOUN
ejpam-4448	58	17	,	,	PUNCT
ejpam-4448	58	18	ᾱ	ᾱ	NOUN
ejpam-4448	58	19	if	if	SCONJ
ejpam-4448	58	20	u←v	u←v	PROPN
ejpam-4448	58	21	,	,	PUNCT
ejpam-4448	58	22	0	0	NUM
ejpam-4448	58	23	otherwise	otherwise	ADV
ejpam-4448	58	24	.	.	PUNCT
ejpam-4448	59	1	(	(	PUNCT
ejpam-4448	59	2	1	1	X
ejpam-4448	59	3	)	)	PUNCT
ejpam-4448	59	4	when	when	SCONJ
ejpam-4448	59	5	there	there	PRON
ejpam-4448	59	6	is	be	VERB
ejpam-4448	59	7	no	no	DET
ejpam-4448	59	8	ambiguity	ambiguity	NOUN
ejpam-4448	59	9	regarding	regard	VERB
ejpam-4448	59	10	the	the	DET
ejpam-4448	59	11	reference	reference	NOUN
ejpam-4448	59	12	graph	graph	NOUN
ejpam-4448	59	13	d	d	SCONJ
ejpam-4448	59	14	we	we	PRON
ejpam-4448	59	15	will	will	AUX
ejpam-4448	59	16	often	often	ADV
ejpam-4448	59	17	omit	omit	VERB
ejpam-4448	59	18	any	any	DET
ejpam-4448	59	19	symbolic	symbolic	ADJ
ejpam-4448	59	20	reference	reference	NOUN
ejpam-4448	59	21	to	to	ADP
ejpam-4448	59	22	d	d	PROPN
ejpam-4448	59	23	,	,	PUNCT
ejpam-4448	59	24	e.g.	e.g.	ADV
ejpam-4448	59	25	write	write	VERB
ejpam-4448	59	26	hα	hα	NOUN
ejpam-4448	59	27	instead	instead	ADV
ejpam-4448	59	28	of	of	ADP
ejpam-4448	59	29	hα(d	hα(d	PRON
ejpam-4448	59	30	)	)	PUNCT
ejpam-4448	59	31	.	.	PUNCT
ejpam-4448	60	1	clearly	clearly	ADV
ejpam-4448	60	2	,	,	PUNCT
ejpam-4448	60	3	the	the	DET
ejpam-4448	60	4	matrix	matrix	NOUN
ejpam-4448	60	5	hα	hα	ADP
ejpam-4448	60	6	from	from	ADP
ejpam-4448	60	7	definition	definition	NOUN
ejpam-4448	60	8	1	1	NUM
ejpam-4448	60	9	is	be	AUX
ejpam-4448	60	10	hermitian	hermitian	ADJ
ejpam-4448	60	11	,	,	PUNCT
ejpam-4448	60	12	i.e.	i.e.	X
ejpam-4448	60	13	(	(	PUNCT
ejpam-4448	60	14	hα)∗	hα)∗	X
ejpam-4448	60	15	=	=	SYM
ejpam-4448	60	16	(	(	PUNCT
ejpam-4448	60	17	hα	hα	NOUN
ejpam-4448	60	18	)	)	PUNCT
ejpam-4448	60	19	where	where	SCONJ
ejpam-4448	60	20	m∗	m∗	PROPN
ejpam-4448	60	21	denotes	denote	VERB
ejpam-4448	60	22	the	the	DET
ejpam-4448	60	23	conjugate	conjugate	ADJ
ejpam-4448	60	24	transpose	transpose	NOUN
ejpam-4448	60	25	of	of	ADP
ejpam-4448	60	26	matrix	matrix	NOUN
ejpam-4448	60	27	m	m	NOUN
ejpam-4448	60	28	.	.	PUNCT
ejpam-4448	61	1	by	by	ADP
ejpam-4448	61	2	χα(d	χα(d	X
ejpam-4448	61	3	,	,	PUNCT
ejpam-4448	61	4	x	x	X
ejpam-4448	61	5	)	)	PUNCT
ejpam-4448	61	6	=	=	SYM
ejpam-4448	61	7	det(xi−hα(d	det(xi−hα(d	PROPN
ejpam-4448	61	8	)	)	PUNCT
ejpam-4448	61	9	)	)	PUNCT
ejpam-4448	61	10	,	,	PUNCT
ejpam-4448	61	11	where	where	SCONJ
ejpam-4448	61	12	i	i	PRON
ejpam-4448	61	13	is	be	AUX
ejpam-4448	61	14	the	the	DET
ejpam-4448	61	15	identity	identity	NOUN
ejpam-4448	61	16	matrix	matrix	NOUN
ejpam-4448	61	17	,	,	PUNCT
ejpam-4448	61	18	we	we	PRON
ejpam-4448	61	19	denote	denote	VERB
ejpam-4448	61	20	the	the	DET
ejpam-4448	61	21	characteristic	characteristic	ADJ
ejpam-4448	61	22	polynomial	polynomial	NOUN
ejpam-4448	61	23	of	of	ADP
ejpam-4448	61	24	the	the	DET
ejpam-4448	61	25	matrix	matrix	NOUN
ejpam-4448	61	26	hα(d	hα(d	PRON
ejpam-4448	61	27	)	)	PUNCT
ejpam-4448	61	28	,	,	PUNCT
ejpam-4448	61	29	calling	call	VERB
ejpam-4448	61	30	this	this	PRON
ejpam-4448	61	31	the	the	DET
ejpam-4448	61	32	α	α	NOUN
ejpam-4448	61	33	-	-	ADJ
ejpam-4448	61	34	characteristic	characteristic	ADJ
ejpam-4448	61	35	polynomial	polynomial	NOUN
ejpam-4448	61	36	of	of	ADP
ejpam-4448	61	37	d.	d.	PROPN
ejpam-4448	61	38	the	the	DET
ejpam-4448	61	39	multiset	multiset	PROPN
ejpam-4448	61	40	σα(d	σα(d	NOUN
ejpam-4448	61	41	)	)	PUNCT
ejpam-4448	61	42	of	of	ADP
ejpam-4448	61	43	all	all	DET
ejpam-4448	61	44	roots	root	NOUN
ejpam-4448	61	45	of	of	ADP
ejpam-4448	61	46	χα(d	χα(d	X
ejpam-4448	61	47	,	,	PUNCT
ejpam-4448	61	48	x	x	X
ejpam-4448	61	49	)	)	PUNCT
ejpam-4448	61	50	is	be	AUX
ejpam-4448	61	51	called	call	VERB
ejpam-4448	61	52	the	the	DET
ejpam-4448	61	53	α	α	NOUN
ejpam-4448	61	54	-	-	NOUN
ejpam-4448	61	55	spectrum	spectrum	NOUN
ejpam-4448	61	56	of	of	ADP
ejpam-4448	61	57	d	d	PROPN
ejpam-4448	61	58	,	,	PUNCT
ejpam-4448	61	59	as	as	SCONJ
ejpam-4448	61	60	opposed	oppose	VERB
ejpam-4448	61	61	to	to	ADP
ejpam-4448	61	62	the	the	DET
ejpam-4448	61	63	(	(	PUNCT
ejpam-4448	61	64	traditional	traditional	ADJ
ejpam-4448	61	65	)	)	PUNCT
ejpam-4448	61	66	spectrum	spectrum	NOUN
ejpam-4448	61	67	σ(γ(d	σ(γ(d	NOUN
ejpam-4448	61	68	)	)	PUNCT
ejpam-4448	61	69	)	)	PUNCT
ejpam-4448	61	70	of	of	ADP
ejpam-4448	61	71	the	the	DET
ejpam-4448	61	72	underlying	underlying	ADJ
ejpam-4448	61	73	undirected	undirected	ADJ
ejpam-4448	61	74	graph	graph	NOUN
ejpam-4448	61	75	γ(d	γ(d	NOUN
ejpam-4448	61	76	)	)	PUNCT
ejpam-4448	61	77	.	.	PUNCT
ejpam-4448	62	1	consequently	consequently	ADV
ejpam-4448	62	2	,	,	PUNCT
ejpam-4448	62	3	we	we	PRON
ejpam-4448	62	4	shall	shall	AUX
ejpam-4448	62	5	refer	refer	VERB
ejpam-4448	62	6	to	to	ADP
ejpam-4448	62	7	the	the	DET
ejpam-4448	62	8	elements	element	NOUN
ejpam-4448	62	9	of	of	ADP
ejpam-4448	62	10	σα(d	σα(d	NOUN
ejpam-4448	62	11	)	)	PUNCT
ejpam-4448	62	12	as	as	ADP
ejpam-4448	62	13	the	the	DET
ejpam-4448	62	14	α	α	NOUN
ejpam-4448	62	15	-	-	PUNCT
ejpam-4448	62	16	eigenvalues	eigenvalue	NOUN
ejpam-4448	62	17	of	of	ADP
ejpam-4448	62	18	d.	d.	PROPN
ejpam-4448	62	19	likewise	likewise	ADV
ejpam-4448	62	20	,	,	PUNCT
ejpam-4448	62	21	we	we	PRON
ejpam-4448	62	22	speak	speak	VERB
ejpam-4448	62	23	of	of	ADP
ejpam-4448	62	24	α	α	NOUN
ejpam-4448	62	25	-	-	PUNCT
ejpam-4448	62	26	eigenvectors	eigenvector	NOUN
ejpam-4448	62	27	.	.	PUNCT
ejpam-4448	63	1	note	note	VERB
ejpam-4448	63	2	that	that	SCONJ
ejpam-4448	63	3	α	α	PRON
ejpam-4448	63	4	-	-	PUNCT
ejpam-4448	63	5	eigenvalues	eigenvalue	NOUN
ejpam-4448	63	6	are	be	AUX
ejpam-4448	63	7	always	always	ADV
ejpam-4448	63	8	real	real	ADJ
ejpam-4448	63	9	.	.	PUNCT
ejpam-4448	64	1	a	a	DET
ejpam-4448	64	2	direct	direct	ADJ
ejpam-4448	64	3	consequence	consequence	NOUN
ejpam-4448	64	4	of	of	ADP
ejpam-4448	64	5	definition	definition	NOUN
ejpam-4448	64	6	1	1	NUM
ejpam-4448	64	7	is	be	AUX
ejpam-4448	64	8	the	the	DET
ejpam-4448	64	9	following	follow	VERB
ejpam-4448	64	10	summation	summation	NOUN
ejpam-4448	64	11	rule	rule	NOUN
ejpam-4448	64	12	characterizing	characterize	VERB
ejpam-4448	64	13	α	α	NOUN
ejpam-4448	64	14	-	-	NOUN
ejpam-4448	64	15	eigenvectors	eigenvector	NOUN
ejpam-4448	64	16	:	:	PUNCT
ejpam-4448	64	17	proposition	proposition	NOUN
ejpam-4448	64	18	1	1	NUM
ejpam-4448	64	19	.	.	PUNCT
ejpam-4448	65	1	let	let	VERB
ejpam-4448	65	2	d	d	PRON
ejpam-4448	65	3	be	be	AUX
ejpam-4448	65	4	a	a	DET
ejpam-4448	65	5	mixed	mixed	ADJ
ejpam-4448	65	6	graph	graph	NOUN
ejpam-4448	65	7	.	.	PUNCT
ejpam-4448	66	1	then	then	ADV
ejpam-4448	66	2	x	x	X
ejpam-4448	66	3	is	be	AUX
ejpam-4448	66	4	an	an	DET
ejpam-4448	66	5	α	α	NOUN
ejpam-4448	66	6	-	-	NOUN
ejpam-4448	66	7	eigenvector	eigenvector	NOUN
ejpam-4448	66	8	of	of	ADP
ejpam-4448	66	9	d	d	PROPN
ejpam-4448	66	10	corresponding	correspond	VERB
ejpam-4448	66	11	to	to	ADP
ejpam-4448	66	12	α	α	NOUN
ejpam-4448	66	13	-	-	PUNCT
ejpam-4448	66	14	eigenvalue	eigenvalue	ADJ
ejpam-4448	66	15	λ	λ	NOUN
ejpam-4448	66	16	if	if	SCONJ
ejpam-4448	67	1	and	and	CCONJ
ejpam-4448	67	2	only	only	ADV
ejpam-4448	67	3	if	if	SCONJ
ejpam-4448	67	4	,	,	PUNCT
ejpam-4448	67	5	for	for	ADP
ejpam-4448	67	6	each	each	DET
ejpam-4448	67	7	u	u	PROPN
ejpam-4448	67	8	∈	∈	PROPN
ejpam-4448	67	9	v	v	NOUN
ejpam-4448	67	10	(	(	PUNCT
ejpam-4448	67	11	d	d	NOUN
ejpam-4448	67	12	)	)	PUNCT
ejpam-4448	67	13	,	,	PUNCT
ejpam-4448	67	14	λx(u	λx(u	X
ejpam-4448	67	15	)	)	PUNCT
ejpam-4448	67	16	=	=	SYM
ejpam-4448	67	17	∑	∑	PUNCT
ejpam-4448	67	18	u∼v	u∼v	ADJ
ejpam-4448	67	19	x(v	x(v	NUM
ejpam-4448	67	20	)	)	PUNCT
ejpam-4448	68	1	+	+	CCONJ
ejpam-4448	68	2	(	(	PUNCT
ejpam-4448	68	3	α	α	NOUN
ejpam-4448	68	4	∑	∑	PROPN
ejpam-4448	68	5	u→v	u→v	PROPN
ejpam-4448	68	6	x(v	x(v	NUM
ejpam-4448	68	7	)	)	PUNCT
ejpam-4448	68	8	)	)	PUNCT
ejpam-4448	69	1	+	+	CCONJ
ejpam-4448	69	2	(	(	PUNCT
ejpam-4448	69	3	ᾱ	ᾱ	NOUN
ejpam-4448	69	4	∑	∑	PART
ejpam-4448	69	5	u←v	u←v	PUNCT
ejpam-4448	69	6	x(v	x(v	PROPN
ejpam-4448	69	7	)	)	PUNCT
ejpam-4448	69	8	)	)	PUNCT
ejpam-4448	69	9	.	.	PUNCT
ejpam-4448	70	1	(	(	PUNCT
ejpam-4448	70	2	2	2	X
ejpam-4448	70	3	)	)	PUNCT
ejpam-4448	70	4	throughout	throughout	ADP
ejpam-4448	70	5	this	this	DET
ejpam-4448	70	6	paper	paper	NOUN
ejpam-4448	70	7	we	we	PRON
ejpam-4448	70	8	shall	shall	AUX
ejpam-4448	70	9	assume	assume	VERB
ejpam-4448	70	10	|α|	|α|	PROPN
ejpam-4448	70	11	=	=	SYM
ejpam-4448	70	12	1	1	NUM
ejpam-4448	70	13	,	,	PUNCT
ejpam-4448	70	14	i.e.	i.e.	X
ejpam-4448	70	15	α	α	X
ejpam-4448	70	16	=	=	SYM
ejpam-4448	70	17	eiθ	eiθ	PROPN
ejpam-4448	70	18	for	for	ADP
ejpam-4448	70	19	some	some	DET
ejpam-4448	70	20	θ	θ	PROPN
ejpam-4448	70	21	∈	∈	PROPN
ejpam-4448	70	22	r.	r.	PROPN
ejpam-4448	70	23	moreover	moreover	ADV
ejpam-4448	70	24	,	,	PUNCT
ejpam-4448	70	25	we	we	PRON
ejpam-4448	70	26	make	make	VERB
ejpam-4448	70	27	use	use	NOUN
ejpam-4448	70	28	of	of	ADP
ejpam-4448	70	29	the	the	DET
ejpam-4448	70	30	constants	constant	NOUN
ejpam-4448	70	31	ω	ω	NOUN
ejpam-4448	70	32	:	:	PUNCT
ejpam-4448	70	33	=	=	SYM
ejpam-4448	70	34	e	e	SYM
ejpam-4448	70	35	π	π	PROPN
ejpam-4448	70	36	3	3	X
ejpam-4448	70	37	i	i	NOUN
ejpam-4448	70	38	(	(	PUNCT
ejpam-4448	70	39	a	a	DET
ejpam-4448	70	40	sixth	sixth	ADJ
ejpam-4448	70	41	root	root	NOUN
ejpam-4448	70	42	of	of	ADP
ejpam-4448	70	43	unity	unity	NOUN
ejpam-4448	70	44	)	)	PUNCT
ejpam-4448	70	45	and	and	CCONJ
ejpam-4448	70	46	γ	γ	X
ejpam-4448	70	47	:	:	PUNCT
ejpam-4448	70	48	=	=	SYM
ejpam-4448	70	49	e	e	X
ejpam-4448	70	50	2π	2π	NUM
ejpam-4448	70	51	3	3	NUM
ejpam-4448	70	52	i	i	NOUN
ejpam-4448	70	53	(	(	PUNCT
ejpam-4448	70	54	a	a	DET
ejpam-4448	70	55	third	third	ADJ
ejpam-4448	70	56	root	root	NOUN
ejpam-4448	70	57	of	of	ADP
ejpam-4448	70	58	unity	unity	NOUN
ejpam-4448	70	59	)	)	PUNCT
ejpam-4448	70	60	.	.	PUNCT
ejpam-4448	71	1	in	in	ADP
ejpam-4448	71	2	the	the	DET
ejpam-4448	71	3	context	context	NOUN
ejpam-4448	71	4	of	of	ADP
ejpam-4448	71	5	hermitian	hermitian	ADJ
ejpam-4448	71	6	adjacency	adjacency	NOUN
ejpam-4448	71	7	matrices	matrix	NOUN
ejpam-4448	71	8	,	,	PUNCT
ejpam-4448	71	9	the	the	DET
ejpam-4448	71	10	former	former	ADJ
ejpam-4448	71	11	constant	constant	NOUN
ejpam-4448	71	12	has	have	AUX
ejpam-4448	71	13	been	be	AUX
ejpam-4448	71	14	endorsed	endorse	VERB
ejpam-4448	71	15	in	in	ADP
ejpam-4448	71	16	[	[	PUNCT
ejpam-4448	71	17	11	11	NUM
ejpam-4448	71	18	]	]	PUNCT
ejpam-4448	71	19	,	,	PUNCT
ejpam-4448	71	20	whereas	whereas	SCONJ
ejpam-4448	71	21	the	the	DET
ejpam-4448	71	22	suitability	suitability	NOUN
ejpam-4448	71	23	of	of	ADP
ejpam-4448	71	24	the	the	DET
ejpam-4448	71	25	latter	latter	ADJ
ejpam-4448	71	26	constant	constant	NOUN
ejpam-4448	71	27	will	will	AUX
ejpam-4448	71	28	become	become	VERB
ejpam-4448	71	29	evident	evident	ADJ
ejpam-4448	71	30	later	later	ADV
ejpam-4448	71	31	on	on	ADV
ejpam-4448	71	32	.	.	PUNCT
ejpam-4448	72	1	3	3	X
ejpam-4448	72	2	.	.	X
ejpam-4448	72	3	characteristic	characteristic	ADJ
ejpam-4448	72	4	polynomial	polynomial	NOUN
ejpam-4448	72	5	of	of	ADP
ejpam-4448	72	6	hα(d	hα(d	NOUN
ejpam-4448	72	7	)	)	PUNCT
ejpam-4448	72	8	in	in	ADP
ejpam-4448	72	9	this	this	DET
ejpam-4448	72	10	section	section	NOUN
ejpam-4448	72	11	we	we	PRON
ejpam-4448	72	12	will	will	AUX
ejpam-4448	72	13	expand	expand	VERB
ejpam-4448	72	14	the	the	DET
ejpam-4448	72	15	determinant	determinant	NOUN
ejpam-4448	72	16	of	of	ADP
ejpam-4448	72	17	hα	hα	NOUN
ejpam-4448	72	18	and	and	CCONJ
ejpam-4448	72	19	study	study	VERB
ejpam-4448	72	20	the	the	DET
ejpam-4448	72	21	α	α	NOUN
ejpam-4448	72	22	-	-	ADJ
ejpam-4448	72	23	characteristic	characteristic	ADJ
ejpam-4448	72	24	polynomial	polynomial	NOUN
ejpam-4448	72	25	,	,	PUNCT
ejpam-4448	72	26	in	in	ADP
ejpam-4448	72	27	particular	particular	ADJ
ejpam-4448	72	28	with	with	ADP
ejpam-4448	72	29	respect	respect	NOUN
ejpam-4448	72	30	to	to	ADP
ejpam-4448	72	31	the	the	DET
ejpam-4448	72	32	three	three	NUM
ejpam-4448	72	33	instances	instance	NOUN
ejpam-4448	73	1	h	h	NOUN
ejpam-4448	74	1	i	i	PRON
ejpam-4448	74	2	,	,	PUNCT
ejpam-4448	74	3	hw	hw	PRON
ejpam-4448	74	4	and	and	CCONJ
ejpam-4448	74	5	hγ	hγ	INTJ
ejpam-4448	74	6	.	.	PUNCT
ejpam-4448	75	1	a	a	DET
ejpam-4448	75	2	classic	classic	ADJ
ejpam-4448	75	3	result	result	NOUN
ejpam-4448	75	4	from	from	ADP
ejpam-4448	75	5	linear	linear	PROPN
ejpam-4448	75	6	algebra	algebra	PROPN
ejpam-4448	75	7	,	,	PUNCT
ejpam-4448	75	8	concerning	concern	VERB
ejpam-4448	75	9	determinant	determinant	ADJ
ejpam-4448	75	10	expansion	expansion	NOUN
ejpam-4448	75	11	,	,	PUNCT
ejpam-4448	75	12	is	be	AUX
ejpam-4448	75	13	the	the	DET
ejpam-4448	75	14	following	following	NOUN
ejpam-4448	75	15	:	:	PUNCT
ejpam-4448	75	16	theorem	theorem	NOUN
ejpam-4448	75	17	1	1	NUM
ejpam-4448	75	18	.	.	PUNCT
ejpam-4448	76	1	if	if	SCONJ
ejpam-4448	76	2	a	a	PRON
ejpam-4448	76	3	=	=	PRON
ejpam-4448	77	1	[	[	X
ejpam-4448	77	2	ai	ai	NOUN
ejpam-4448	77	3	,	,	PUNCT
ejpam-4448	77	4	j	j	PROPN
ejpam-4448	77	5	]	]	PUNCT
ejpam-4448	77	6	is	be	AUX
ejpam-4448	77	7	a	a	DET
ejpam-4448	77	8	square	square	ADJ
ejpam-4448	77	9	matrix	matrix	NOUN
ejpam-4448	77	10	of	of	ADP
ejpam-4448	77	11	order	order	NOUN
ejpam-4448	77	12	n	n	NOUN
ejpam-4448	77	13	then	then	ADV
ejpam-4448	77	14	det(a	det(a	PROPN
ejpam-4448	77	15	)	)	PUNCT
ejpam-4448	77	16	=	=	SYM
ejpam-4448	77	17	∑	∑	PUNCT
ejpam-4448	77	18	η∈sn	η∈sn	ADJ
ejpam-4448	77	19	sgn(η)a1,η(1)a2,η(2)a3,η(3	sgn(η)a1,η(1)a2,η(2)a3,η(3	NOUN
ejpam-4448	77	20	)	)	PUNCT
ejpam-4448	77	21	.	.	PUNCT
ejpam-4448	77	22	.	.	PUNCT
ejpam-4448	77	23	.	.	PUNCT
ejpam-4448	78	1	an	an	DET
ejpam-4448	78	2	,	,	PUNCT
ejpam-4448	78	3	η(n	η(n	NOUN
ejpam-4448	78	4	)	)	PUNCT
ejpam-4448	78	5	.	.	PUNCT
ejpam-4448	79	1	(	(	PUNCT
ejpam-4448	79	2	3	3	X
ejpam-4448	79	3	)	)	PUNCT
ejpam-4448	79	4	m.	m.	NOUN
ejpam-4448	79	5	abudayah	abudayah	NOUN
ejpam-4448	79	6	,	,	PUNCT
ejpam-4448	79	7	o.	o.	PROPN
ejpam-4448	79	8	alomari	alomari	PROPN
ejpam-4448	79	9	,	,	PUNCT
ejpam-4448	79	10	t.	t.	PROPN
ejpam-4448	79	11	sander	sander	PROPN
ejpam-4448	79	12	/	/	SYM
ejpam-4448	79	13	eur	eur	PROPN
ejpam-4448	79	14	.	.	PUNCT
ejpam-4448	80	1	j.	j.	PROPN
ejpam-4448	80	2	pure	pure	PROPN
ejpam-4448	80	3	appl	appl	PROPN
ejpam-4448	80	4	.	.	PROPN
ejpam-4448	80	5	math	math	PROPN
ejpam-4448	80	6	,	,	PUNCT
ejpam-4448	80	7	15	15	NUM
ejpam-4448	80	8	(	(	PUNCT
ejpam-4448	80	9	3	3	NUM
ejpam-4448	80	10	)	)	PUNCT
ejpam-4448	80	11	(	(	PUNCT
ejpam-4448	80	12	2022	2022	NUM
ejpam-4448	80	13	)	)	PUNCT
ejpam-4448	80	14	,	,	PUNCT
ejpam-4448	80	15	841	841	NUM
ejpam-4448	80	16	-	-	SYM
ejpam-4448	80	17	855	855	NUM
ejpam-4448	80	18	844	844	NUM
ejpam-4448	80	19	decades	decade	NOUN
ejpam-4448	80	20	ago	ago	ADV
ejpam-4448	80	21	,	,	PUNCT
ejpam-4448	80	22	the	the	DET
ejpam-4448	80	23	above	above	ADJ
ejpam-4448	80	24	theorem	theorem	NOUN
ejpam-4448	80	25	has	have	AUX
ejpam-4448	80	26	been	be	AUX
ejpam-4448	80	27	applied	apply	VERB
ejpam-4448	80	28	to	to	ADP
ejpam-4448	80	29	adjacency	adjacency	NOUN
ejpam-4448	80	30	matrices	matrix	NOUN
ejpam-4448	80	31	of	of	ADP
ejpam-4448	80	32	graphs	graph	NOUN
ejpam-4448	80	33	.	.	PUNCT
ejpam-4448	81	1	the	the	DET
ejpam-4448	81	2	permutations	permutation	NOUN
ejpam-4448	81	3	over	over	ADP
ejpam-4448	81	4	which	which	PRON
ejpam-4448	81	5	the	the	DET
ejpam-4448	81	6	sum	sum	NOUN
ejpam-4448	81	7	ranges	range	VERB
ejpam-4448	81	8	can	can	AUX
ejpam-4448	81	9	be	be	AUX
ejpam-4448	81	10	put	put	VERB
ejpam-4448	81	11	into	into	ADP
ejpam-4448	81	12	correspondence	correspondence	NOUN
ejpam-4448	81	13	with	with	ADP
ejpam-4448	81	14	certain	certain	ADJ
ejpam-4448	81	15	subgraphs	subgraph	NOUN
ejpam-4448	81	16	of	of	ADP
ejpam-4448	81	17	the	the	DET
ejpam-4448	81	18	given	give	VERB
ejpam-4448	81	19	graph	graph	NOUN
ejpam-4448	81	20	.	.	PUNCT
ejpam-4448	82	1	to	to	ADP
ejpam-4448	82	2	this	this	DET
ejpam-4448	82	3	end	end	NOUN
ejpam-4448	82	4	,	,	PUNCT
ejpam-4448	82	5	we	we	PRON
ejpam-4448	82	6	define	define	VERB
ejpam-4448	82	7	the	the	DET
ejpam-4448	82	8	following	follow	VERB
ejpam-4448	82	9	terms	term	NOUN
ejpam-4448	82	10	and	and	CCONJ
ejpam-4448	82	11	notation	notation	NOUN
ejpam-4448	82	12	:	:	PUNCT
ejpam-4448	82	13	definition	definition	NOUN
ejpam-4448	82	14	2	2	NUM
ejpam-4448	82	15	.	.	PUNCT
ejpam-4448	83	1	let	let	VERB
ejpam-4448	83	2	d	d	PRON
ejpam-4448	83	3	be	be	AUX
ejpam-4448	83	4	a	a	DET
ejpam-4448	83	5	mixed	mixed	ADJ
ejpam-4448	83	6	graph	graph	NOUN
ejpam-4448	83	7	.	.	PUNCT
ejpam-4448	84	1	(	(	PUNCT
ejpam-4448	84	2	i	i	NOUN
ejpam-4448	84	3	)	)	PUNCT
ejpam-4448	85	1	d	d	NOUN
ejpam-4448	85	2	is	be	AUX
ejpam-4448	85	3	called	call	VERB
ejpam-4448	85	4	elementary	elementary	ADJ
ejpam-4448	85	5	if	if	SCONJ
ejpam-4448	85	6	,	,	PUNCT
ejpam-4448	85	7	for	for	ADP
ejpam-4448	85	8	every	every	DET
ejpam-4448	85	9	component	component	NOUN
ejpam-4448	85	10	c	c	NOUN
ejpam-4448	85	11	of	of	ADP
ejpam-4448	85	12	d	d	PROPN
ejpam-4448	85	13	,	,	PUNCT
ejpam-4448	85	14	γ(c	γ(c	PROPN
ejpam-4448	85	15	)	)	PUNCT
ejpam-4448	85	16	is	be	AUX
ejpam-4448	85	17	either	either	CCONJ
ejpam-4448	85	18	isomorphic	isomorphic	ADJ
ejpam-4448	85	19	to	to	ADP
ejpam-4448	85	20	p2	p2	PROPN
ejpam-4448	85	21	or	or	CCONJ
ejpam-4448	85	22	ck	ck	INTJ
ejpam-4448	85	23	(	(	PUNCT
ejpam-4448	85	24	for	for	ADP
ejpam-4448	85	25	some	some	DET
ejpam-4448	85	26	k	k	PROPN
ejpam-4448	85	27	≥	≥	NUM
ejpam-4448	85	28	3	3	NUM
ejpam-4448	85	29	)	)	PUNCT
ejpam-4448	85	30	.	.	PUNCT
ejpam-4448	86	1	(	(	PUNCT
ejpam-4448	86	2	ii	ii	X
ejpam-4448	86	3	)	)	PUNCT
ejpam-4448	86	4	let	let	VERB
ejpam-4448	86	5	d	d	PRON
ejpam-4448	86	6	be	be	AUX
ejpam-4448	86	7	elementary	elementary	ADJ
ejpam-4448	86	8	.	.	PUNCT
ejpam-4448	87	1	the	the	DET
ejpam-4448	87	2	rank	rank	NOUN
ejpam-4448	87	3	of	of	ADP
ejpam-4448	87	4	d	d	PROPN
ejpam-4448	87	5	is	be	AUX
ejpam-4448	87	6	defined	define	VERB
ejpam-4448	87	7	as	as	ADP
ejpam-4448	87	8	r(d	r(d	NOUN
ejpam-4448	87	9	)	)	PUNCT
ejpam-4448	88	1	=	=	SYM
ejpam-4448	88	2	n−c	n−c	NOUN
ejpam-4448	88	3	,	,	PUNCT
ejpam-4448	88	4	where	where	SCONJ
ejpam-4448	88	5	n	n	NOUN
ejpam-4448	88	6	=	=	SYM
ejpam-4448	88	7	|v	|v	X
ejpam-4448	88	8	(	(	PUNCT
ejpam-4448	88	9	d)|	d)|	PROPN
ejpam-4448	88	10	and	and	CCONJ
ejpam-4448	88	11	c	c	PROPN
ejpam-4448	88	12	is	be	AUX
ejpam-4448	88	13	the	the	DET
ejpam-4448	88	14	number	number	NOUN
ejpam-4448	88	15	of	of	ADP
ejpam-4448	88	16	its	its	PRON
ejpam-4448	88	17	components	component	NOUN
ejpam-4448	88	18	.	.	PUNCT
ejpam-4448	89	1	the	the	DET
ejpam-4448	89	2	co	co	NOUN
ejpam-4448	89	3	-	-	NOUN
ejpam-4448	89	4	rank	rank	NOUN
ejpam-4448	89	5	of	of	ADP
ejpam-4448	89	6	d	d	PROPN
ejpam-4448	89	7	is	be	AUX
ejpam-4448	89	8	defined	define	VERB
ejpam-4448	89	9	as	as	ADP
ejpam-4448	89	10	s(d	s(d	PROPN
ejpam-4448	89	11	)	)	PUNCT
ejpam-4448	90	1	=	=	SYM
ejpam-4448	90	2	m−	m−	PROPN
ejpam-4448	90	3	r(d	r(d	NOUN
ejpam-4448	90	4	)	)	PUNCT
ejpam-4448	90	5	,	,	PUNCT
ejpam-4448	90	6	where	where	SCONJ
ejpam-4448	90	7	m	m	VERB
ejpam-4448	90	8	=	=	SYM
ejpam-4448	90	9	|e(d)|	|e(d)|	PROPN
ejpam-4448	90	10	.	.	PROPN
ejpam-4448	90	11	note	note	VERB
ejpam-4448	90	12	that	that	SCONJ
ejpam-4448	90	13	the	the	DET
ejpam-4448	90	14	co	co	NOUN
ejpam-4448	90	15	-	-	NOUN
ejpam-4448	90	16	rank	rank	NOUN
ejpam-4448	90	17	s(d	s(d	NOUN
ejpam-4448	90	18	)	)	PUNCT
ejpam-4448	90	19	is	be	AUX
ejpam-4448	90	20	equal	equal	ADJ
ejpam-4448	90	21	to	to	ADP
ejpam-4448	90	22	the	the	DET
ejpam-4448	90	23	number	number	NOUN
ejpam-4448	90	24	of	of	ADP
ejpam-4448	90	25	ck	ck	PROPN
ejpam-4448	90	26	components	component	NOUN
ejpam-4448	90	27	of	of	ADP
ejpam-4448	90	28	d.	d.	PROPN
ejpam-4448	90	29	now	now	ADV
ejpam-4448	90	30	we	we	PRON
ejpam-4448	90	31	are	be	AUX
ejpam-4448	90	32	ready	ready	ADJ
ejpam-4448	90	33	to	to	PART
ejpam-4448	90	34	state	state	VERB
ejpam-4448	90	35	the	the	DET
ejpam-4448	90	36	following	follow	VERB
ejpam-4448	90	37	classic	classic	ADJ
ejpam-4448	90	38	theorem	theorem	NOUN
ejpam-4448	90	39	by	by	ADP
ejpam-4448	90	40	harary	harary	NOUN
ejpam-4448	90	41	(	(	PUNCT
ejpam-4448	90	42	cf	cf	NOUN
ejpam-4448	90	43	.	.	PUNCT
ejpam-4448	91	1	[	[	X
ejpam-4448	91	2	6	6	NUM
ejpam-4448	91	3	]	]	PUNCT
ejpam-4448	91	4	):	):	PUNCT
ejpam-4448	91	5	theorem	theorem	ADJ
ejpam-4448	91	6	2	2	NUM
ejpam-4448	91	7	(	(	PUNCT
ejpam-4448	91	8	determinant	determinant	ADJ
ejpam-4448	91	9	expansion	expansion	NOUN
ejpam-4448	91	10	(	(	PUNCT
ejpam-4448	91	11	harary	harary	NOUN
ejpam-4448	91	12	,	,	PUNCT
ejpam-4448	91	13	1962	1962	NUM
ejpam-4448	91	14	)	)	PUNCT
ejpam-4448	91	15	)	)	PUNCT
ejpam-4448	91	16	.	.	PUNCT
ejpam-4448	92	1	let	let	VERB
ejpam-4448	92	2	d	d	PRON
ejpam-4448	92	3	be	be	AUX
ejpam-4448	92	4	a	a	DET
ejpam-4448	92	5	graph	graph	NOUN
ejpam-4448	92	6	with	with	ADP
ejpam-4448	92	7	adjacency	adjacency	NOUN
ejpam-4448	92	8	matrix	matrix	NOUN
ejpam-4448	92	9	a(g	a(g	PROPN
ejpam-4448	92	10	)	)	PUNCT
ejpam-4448	92	11	.	.	PUNCT
ejpam-4448	93	1	then	then	ADV
ejpam-4448	93	2	,	,	PUNCT
ejpam-4448	93	3	det(a(g	det(a(g	NOUN
ejpam-4448	93	4	)	)	PUNCT
ejpam-4448	93	5	)	)	PUNCT
ejpam-4448	94	1	=	=	PUNCT
ejpam-4448	94	2	∑	∑	PUNCT
ejpam-4448	94	3	s	s	PROPN
ejpam-4448	94	4	(	(	PUNCT
ejpam-4448	94	5	−1)r(s)2s(s	−1)r(s)2s(s	NOUN
ejpam-4448	94	6	)	)	PUNCT
ejpam-4448	94	7	,	,	PUNCT
ejpam-4448	94	8	(	(	PUNCT
ejpam-4448	94	9	4	4	X
ejpam-4448	94	10	)	)	PUNCT
ejpam-4448	94	11	where	where	SCONJ
ejpam-4448	94	12	the	the	DET
ejpam-4448	94	13	sum	sum	NOUN
ejpam-4448	94	14	ranges	range	VERB
ejpam-4448	94	15	over	over	ADP
ejpam-4448	94	16	all	all	DET
ejpam-4448	94	17	spanning	span	VERB
ejpam-4448	94	18	elementary	elementary	ADJ
ejpam-4448	94	19	subgraphs	subgraph	NOUN
ejpam-4448	94	20	s	s	PROPN
ejpam-4448	94	21	of	of	ADP
ejpam-4448	94	22	g.	g.	NOUN
ejpam-4448	94	23	following	follow	VERB
ejpam-4448	94	24	the	the	DET
ejpam-4448	94	25	classic	classic	ADJ
ejpam-4448	94	26	proof	proof	NOUN
ejpam-4448	94	27	strategy	strategy	NOUN
ejpam-4448	94	28	used	use	VERB
ejpam-4448	94	29	in	in	ADP
ejpam-4448	94	30	theorem	theorem	NOUN
ejpam-4448	94	31	2	2	NUM
ejpam-4448	94	32	,	,	PUNCT
ejpam-4448	94	33	the	the	DET
ejpam-4448	94	34	result	result	NOUN
ejpam-4448	94	35	readily	readily	ADV
ejpam-4448	94	36	generalizes	generalize	VERB
ejpam-4448	94	37	to	to	ADP
ejpam-4448	94	38	any	any	DET
ejpam-4448	94	39	α	α	NOUN
ejpam-4448	94	40	-	-	ADJ
ejpam-4448	94	41	hermitian	hermitian	ADJ
ejpam-4448	94	42	adjacency	adjacency	NOUN
ejpam-4448	94	43	matrix	matrix	NOUN
ejpam-4448	94	44	.	.	PUNCT
ejpam-4448	95	1	but	but	CCONJ
ejpam-4448	95	2	first	first	ADV
ejpam-4448	95	3	we	we	PRON
ejpam-4448	95	4	require	require	VERB
ejpam-4448	95	5	the	the	DET
ejpam-4448	95	6	following	follow	VERB
ejpam-4448	95	7	definition	definition	NOUN
ejpam-4448	95	8	:	:	PUNCT
ejpam-4448	95	9	definition	definition	NOUN
ejpam-4448	95	10	3	3	X
ejpam-4448	95	11	.	.	PUNCT
ejpam-4448	96	1	let	let	VERB
ejpam-4448	96	2	d	d	PRON
ejpam-4448	96	3	be	be	AUX
ejpam-4448	96	4	a	a	DET
ejpam-4448	96	5	mixed	mixed	ADJ
ejpam-4448	96	6	graph	graph	NOUN
ejpam-4448	96	7	and	and	CCONJ
ejpam-4448	96	8	hα(d	hα(d	PRON
ejpam-4448	96	9	)	)	PUNCT
ejpam-4448	97	1	=	=	PUNCT
ejpam-4448	98	1	[	[	X
ejpam-4448	98	2	huv	huv	X
ejpam-4448	98	3	]	]	X
ejpam-4448	98	4	.	.	PUNCT
ejpam-4448	99	1	with	with	ADP
ejpam-4448	99	2	respect	respect	NOUN
ejpam-4448	99	3	to	to	ADP
ejpam-4448	99	4	this	this	PRON
ejpam-4448	99	5	,	,	PUNCT
ejpam-4448	99	6	the	the	DET
ejpam-4448	99	7	value	value	NOUN
ejpam-4448	99	8	hα(w	hα(w	NOUN
ejpam-4448	99	9	)	)	PUNCT
ejpam-4448	99	10	of	of	ADP
ejpam-4448	99	11	a	a	DET
ejpam-4448	99	12	mixed	mixed	ADJ
ejpam-4448	99	13	walk	walk	NOUN
ejpam-4448	99	14	w	w	NOUN
ejpam-4448	99	15	with	with	ADP
ejpam-4448	99	16	vertices	vertex	NOUN
ejpam-4448	99	17	v1	v1	NOUN
ejpam-4448	99	18	,	,	PUNCT
ejpam-4448	99	19	v2	v2	NOUN
ejpam-4448	99	20	,	,	PUNCT
ejpam-4448	99	21	.	.	PUNCT
ejpam-4448	99	22	.	.	PUNCT
ejpam-4448	100	1	.	.	PUNCT
ejpam-4448	101	1	,	,	PUNCT
ejpam-4448	101	2	vk	vk	PROPN
ejpam-4448	101	3	is	be	AUX
ejpam-4448	101	4	defined	define	VERB
ejpam-4448	101	5	as	as	ADP
ejpam-4448	101	6	hα(w	hα(w	NOUN
ejpam-4448	101	7	)	)	PUNCT
ejpam-4448	102	1	=	=	SYM
ejpam-4448	102	2	(	(	PUNCT
ejpam-4448	102	3	hv1v2hv2v3hv3v4	hv1v2hv2v3hv3v4	PROPN
ejpam-4448	102	4	·	·	PUNCT
ejpam-4448	102	5	·	·	PUNCT
ejpam-4448	102	6	·	·	PUNCT
ejpam-4448	102	7	hvk−1vk	hvk−1vk	NOUN
ejpam-4448	102	8	)	)	PUNCT
ejpam-4448	102	9	∈	∈	PROPN
ejpam-4448	102	10	{	{	PUNCT
ejpam-4448	102	11	α	α	NOUN
ejpam-4448	102	12	r}r∈z	r}r∈z	NOUN
ejpam-4448	102	13	.	.	PUNCT
ejpam-4448	103	1	(	(	PUNCT
ejpam-4448	103	2	5	5	X
ejpam-4448	103	3	)	)	PUNCT
ejpam-4448	103	4	theorem	theorem	NOUN
ejpam-4448	103	5	3	3	NUM
ejpam-4448	103	6	(	(	PUNCT
ejpam-4448	103	7	determinant	determinant	ADJ
ejpam-4448	103	8	expansion	expansion	NOUN
ejpam-4448	103	9	for	for	ADP
ejpam-4448	103	10	hα	hα	NOUN
ejpam-4448	103	11	)	)	PUNCT
ejpam-4448	103	12	.	.	PUNCT
ejpam-4448	104	1	let	let	VERB
ejpam-4448	104	2	d	d	PRON
ejpam-4448	104	3	be	be	AUX
ejpam-4448	104	4	a	a	DET
ejpam-4448	104	5	mixed	mixed	ADJ
ejpam-4448	104	6	graph	graph	NOUN
ejpam-4448	104	7	.	.	PUNCT
ejpam-4448	105	1	then	then	ADV
ejpam-4448	105	2	det(hα	det(hα	NUM
ejpam-4448	105	3	)	)	PUNCT
ejpam-4448	105	4	=	=	PUNCT
ejpam-4448	106	1	∑	∑	PUNCT
ejpam-4448	106	2	d′	d′	X
ejpam-4448	106	3	(	(	PUNCT
ejpam-4448	106	4	−1)r(d′	−1)r(d′	NOUN
ejpam-4448	106	5	)	)	PUNCT
ejpam-4448	106	6	2s(d	2s(d	NUM
ejpam-4448	106	7	′)re	′)re	NOUN
ejpam-4448	106	8	(	(	PUNCT
ejpam-4448	106	9	∏	∏	PROPN
ejpam-4448	106	10	c	c	PROPN
ejpam-4448	106	11	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	106	12	)	)	PUNCT
ejpam-4448	106	13	)	)	PUNCT
ejpam-4448	106	14	,	,	PUNCT
ejpam-4448	106	15	(	(	PUNCT
ejpam-4448	106	16	6	6	NUM
ejpam-4448	106	17	)	)	PUNCT
ejpam-4448	106	18	where	where	SCONJ
ejpam-4448	106	19	the	the	DET
ejpam-4448	106	20	sum	sum	NOUN
ejpam-4448	106	21	ranges	range	VERB
ejpam-4448	106	22	over	over	ADP
ejpam-4448	106	23	all	all	DET
ejpam-4448	106	24	spanning	span	VERB
ejpam-4448	106	25	elementary	elementary	ADJ
ejpam-4448	106	26	mixed	mixed	ADJ
ejpam-4448	106	27	subgraphs	subgraph	NOUN
ejpam-4448	106	28	d′	d′	NUM
ejpam-4448	106	29	of	of	ADP
ejpam-4448	106	30	d	d	PROPN
ejpam-4448	106	31	,	,	PUNCT
ejpam-4448	106	32	the	the	DET
ejpam-4448	106	33	product	product	NOUN
ejpam-4448	106	34	ranges	range	VERB
ejpam-4448	106	35	over	over	ADP
ejpam-4448	106	36	all	all	DET
ejpam-4448	106	37	mixed	mixed	ADJ
ejpam-4448	106	38	cycles	cycle	NOUN
ejpam-4448	106	39	c	c	NOUN
ejpam-4448	106	40	in	in	ADP
ejpam-4448	106	41	d′	d′	NUM
ejpam-4448	106	42	,	,	PUNCT
ejpam-4448	106	43	and	and	CCONJ
ejpam-4448	106	44	c⃗	c⃗	NOUN
ejpam-4448	106	45	is	be	AUX
ejpam-4448	106	46	any	any	DET
ejpam-4448	106	47	closed	closed	ADJ
ejpam-4448	106	48	walk	walk	NOUN
ejpam-4448	106	49	traversing	traverse	VERB
ejpam-4448	106	50	c.	c.	NOUN
ejpam-4448	106	51	proof	proof	NOUN
ejpam-4448	106	52	.	.	PUNCT
ejpam-4448	107	1	consider	consider	VERB
ejpam-4448	107	2	the	the	DET
ejpam-4448	107	3	matrix	matrix	NOUN
ejpam-4448	107	4	hα	hα	ADP
ejpam-4448	107	5	and	and	CCONJ
ejpam-4448	107	6	apply	apply	VERB
ejpam-4448	107	7	the	the	DET
ejpam-4448	107	8	classic	classic	ADJ
ejpam-4448	107	9	proof	proof	NOUN
ejpam-4448	107	10	strategy	strategy	NOUN
ejpam-4448	107	11	for	for	ADP
ejpam-4448	107	12	determinant	determinant	ADJ
ejpam-4448	107	13	expansion	expansion	NOUN
ejpam-4448	107	14	on	on	ADP
ejpam-4448	107	15	graphs	graph	NOUN
ejpam-4448	107	16	,	,	PUNCT
ejpam-4448	107	17	cf	cf	NOUN
ejpam-4448	107	18	.	.	PUNCT
ejpam-4448	108	1	the	the	DET
ejpam-4448	108	2	proof	proof	NOUN
ejpam-4448	108	3	of	of	ADP
ejpam-4448	108	4	theorem	theorem	NOUN
ejpam-4448	108	5	2	2	NUM
ejpam-4448	108	6	in	in	ADP
ejpam-4448	108	7	[	[	X
ejpam-4448	108	8	3	3	NUM
ejpam-4448	108	9	]	]	PUNCT
ejpam-4448	108	10	.	.	PUNCT
ejpam-4448	109	1	considering	consider	VERB
ejpam-4448	109	2	specific	specific	ADJ
ejpam-4448	109	3	values	value	NOUN
ejpam-4448	109	4	of	of	ADP
ejpam-4448	109	5	α	α	NOUN
ejpam-4448	109	6	,	,	PUNCT
ejpam-4448	109	7	the	the	DET
ejpam-4448	109	8	formula	formula	NOUN
ejpam-4448	109	9	in	in	ADP
ejpam-4448	109	10	theorem	theorem	NOUN
ejpam-4448	109	11	3	3	NUM
ejpam-4448	109	12	becomes	become	VERB
ejpam-4448	109	13	more	more	ADV
ejpam-4448	109	14	specific	specific	ADJ
ejpam-4448	109	15	,	,	PUNCT
ejpam-4448	109	16	too	too	ADV
ejpam-4448	109	17	.	.	PUNCT
ejpam-4448	110	1	for	for	ADP
ejpam-4448	110	2	α	α	NOUN
ejpam-4448	110	3	=	=	PUNCT
ejpam-4448	110	4	i	i	PRON
ejpam-4448	110	5	we	we	PRON
ejpam-4448	110	6	may	may	AUX
ejpam-4448	110	7	rediscover	rediscover	VERB
ejpam-4448	110	8	a	a	DET
ejpam-4448	110	9	result	result	NOUN
ejpam-4448	110	10	given	give	VERB
ejpam-4448	110	11	in	in	ADP
ejpam-4448	110	12	[	[	X
ejpam-4448	110	13	10	10	NUM
ejpam-4448	110	14	]	]	PUNCT
ejpam-4448	110	15	.	.	PUNCT
ejpam-4448	111	1	moreover	moreover	ADV
ejpam-4448	111	2	,	,	PUNCT
ejpam-4448	111	3	theorem	theorem	VERB
ejpam-4448	111	4	3	3	NUM
ejpam-4448	111	5	immediately	immediately	ADV
ejpam-4448	111	6	allows	allow	VERB
ejpam-4448	111	7	us	we	PRON
ejpam-4448	111	8	to	to	PART
ejpam-4448	111	9	compute	compute	VERB
ejpam-4448	111	10	the	the	DET
ejpam-4448	111	11	α	α	NOUN
ejpam-4448	111	12	-	-	ADJ
ejpam-4448	111	13	characteristic	characteristic	ADJ
ejpam-4448	111	14	polynomial	polynomial	NOUN
ejpam-4448	111	15	:	:	PUNCT
ejpam-4448	111	16	m.	m.	NOUN
ejpam-4448	111	17	abudayah	abudayah	PROPN
ejpam-4448	111	18	,	,	PUNCT
ejpam-4448	111	19	o.	o.	PROPN
ejpam-4448	111	20	alomari	alomari	PROPN
ejpam-4448	111	21	,	,	PUNCT
ejpam-4448	111	22	t.	t.	PROPN
ejpam-4448	111	23	sander	sander	PROPN
ejpam-4448	111	24	/	/	SYM
ejpam-4448	111	25	eur	eur	PROPN
ejpam-4448	111	26	.	.	PUNCT
ejpam-4448	112	1	j.	j.	PROPN
ejpam-4448	112	2	pure	pure	PROPN
ejpam-4448	112	3	appl	appl	PROPN
ejpam-4448	112	4	.	.	PROPN
ejpam-4448	112	5	math	math	PROPN
ejpam-4448	112	6	,	,	PUNCT
ejpam-4448	112	7	15	15	NUM
ejpam-4448	112	8	(	(	PUNCT
ejpam-4448	112	9	3	3	NUM
ejpam-4448	112	10	)	)	PUNCT
ejpam-4448	112	11	(	(	PUNCT
ejpam-4448	112	12	2022	2022	NUM
ejpam-4448	112	13	)	)	PUNCT
ejpam-4448	112	14	,	,	PUNCT
ejpam-4448	112	15	841	841	NUM
ejpam-4448	112	16	-	-	SYM
ejpam-4448	112	17	855	855	NUM
ejpam-4448	112	18	845	845	NUM
ejpam-4448	112	19	1	1	NUM
ejpam-4448	112	20	2	2	NUM
ejpam-4448	112	21	3	3	NUM
ejpam-4448	112	22	4	4	NUM
ejpam-4448	112	23	5	5	NUM
ejpam-4448	112	24	figure	figure	NOUN
ejpam-4448	112	25	1	1	NUM
ejpam-4448	112	26	:	:	PUNCT
ejpam-4448	112	27	a	a	DET
ejpam-4448	112	28	mixed	mixed	ADJ
ejpam-4448	112	29	graph	graph	NOUN
ejpam-4448	112	30	where	where	SCONJ
ejpam-4448	112	31	σγ	σγ	PROPN
ejpam-4448	112	32	,	,	PUNCT
ejpam-4448	112	33	σω	σω	AUX
ejpam-4448	112	34	,	,	PUNCT
ejpam-4448	112	35	σi	σi	PRON
ejpam-4448	112	36	are	be	AUX
ejpam-4448	112	37	different	different	ADJ
ejpam-4448	112	38	from	from	ADP
ejpam-4448	112	39	one	one	NUM
ejpam-4448	112	40	another	another	DET
ejpam-4448	112	41	corollary	corollary	ADJ
ejpam-4448	112	42	1	1	NUM
ejpam-4448	112	43	.	.	PUNCT
ejpam-4448	113	1	if	if	SCONJ
ejpam-4448	113	2	χα(d	χα(d	X
ejpam-4448	113	3	,	,	PUNCT
ejpam-4448	113	4	λ	λ	X
ejpam-4448	113	5	)	)	PUNCT
ejpam-4448	113	6	=	=	SYM
ejpam-4448	113	7	λn	λn	PROPN
ejpam-4448	113	8	+	+	CCONJ
ejpam-4448	113	9	c1λ	c1λ	PROPN
ejpam-4448	113	10	n−1	n−1	PROPN
ejpam-4448	113	11	+	+	NUM
ejpam-4448	113	12	c2λ	c2λ	NOUN
ejpam-4448	113	13	n−2	n−2	PROPN
ejpam-4448	113	14	+	+	PROPN
ejpam-4448	113	15	·	·	PUNCT
ejpam-4448	113	16	·	·	PUNCT
ejpam-4448	113	17	·	·	PUNCT
ejpam-4448	114	1	+	+	CCONJ
ejpam-4448	114	2	cn	cn	PROPN
ejpam-4448	114	3	is	be	AUX
ejpam-4448	114	4	the	the	DET
ejpam-4448	114	5	α	α	NOUN
ejpam-4448	114	6	-	-	ADJ
ejpam-4448	114	7	characteristic	characteristic	ADJ
ejpam-4448	114	8	polynomial	polynomial	NOUN
ejpam-4448	114	9	of	of	ADP
ejpam-4448	114	10	a	a	DET
ejpam-4448	114	11	mixed	mixed	ADJ
ejpam-4448	114	12	graph	graph	NOUN
ejpam-4448	114	13	d	d	NOUN
ejpam-4448	114	14	,	,	PUNCT
ejpam-4448	114	15	then	then	ADV
ejpam-4448	114	16	(	(	PUNCT
ejpam-4448	114	17	−1)kck	−1)kck	PROPN
ejpam-4448	114	18	=	=	PUNCT
ejpam-4448	114	19	∑	∑	PROPN
ejpam-4448	114	20	(	(	PUNCT
ejpam-4448	114	21	−1)r(d′	−1)r(d′	NOUN
ejpam-4448	114	22	)	)	PUNCT
ejpam-4448	114	23	2s(d	2s(d	NUM
ejpam-4448	114	24	′)re	′)re	NOUN
ejpam-4448	114	25	(	(	PUNCT
ejpam-4448	114	26	∏	∏	PROPN
ejpam-4448	114	27	c	c	PROPN
ejpam-4448	114	28	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	114	29	)	)	PUNCT
ejpam-4448	114	30	)	)	PUNCT
ejpam-4448	114	31	,	,	PUNCT
ejpam-4448	114	32	(	(	PUNCT
ejpam-4448	114	33	7	7	X
ejpam-4448	114	34	)	)	PUNCT
ejpam-4448	114	35	where	where	SCONJ
ejpam-4448	114	36	the	the	DET
ejpam-4448	114	37	sum	sum	NOUN
ejpam-4448	114	38	ranges	range	VERB
ejpam-4448	114	39	over	over	ADP
ejpam-4448	114	40	all	all	DET
ejpam-4448	114	41	elementary	elementary	ADJ
ejpam-4448	114	42	mixed	mixed	ADJ
ejpam-4448	114	43	subgraphs	subgraph	NOUN
ejpam-4448	114	44	d′	d′	PROPN
ejpam-4448	114	45	with	with	ADP
ejpam-4448	114	46	k	k	PROPN
ejpam-4448	114	47	vertices	vertex	NOUN
ejpam-4448	114	48	,	,	PUNCT
ejpam-4448	114	49	the	the	DET
ejpam-4448	114	50	product	product	NOUN
ejpam-4448	114	51	ranges	range	VERB
ejpam-4448	114	52	over	over	ADP
ejpam-4448	114	53	all	all	DET
ejpam-4448	114	54	mixed	mixed	ADJ
ejpam-4448	114	55	cycles	cycle	NOUN
ejpam-4448	114	56	c	c	NOUN
ejpam-4448	114	57	in	in	ADP
ejpam-4448	114	58	d′	d′	NUM
ejpam-4448	114	59	,	,	PUNCT
ejpam-4448	114	60	and	and	CCONJ
ejpam-4448	114	61	c⃗	c⃗	NOUN
ejpam-4448	114	62	is	be	AUX
ejpam-4448	114	63	any	any	DET
ejpam-4448	114	64	closed	closed	ADJ
ejpam-4448	114	65	walk	walk	NOUN
ejpam-4448	114	66	traversing	traverse	VERB
ejpam-4448	114	67	c.	c.	NOUN
ejpam-4448	114	68	proof	proof	NOUN
ejpam-4448	114	69	.	.	PUNCT
ejpam-4448	115	1	this	this	PRON
ejpam-4448	115	2	follows	follow	VERB
ejpam-4448	115	3	immediately	immediately	ADV
ejpam-4448	115	4	from	from	ADP
ejpam-4448	115	5	the	the	DET
ejpam-4448	115	6	fact	fact	NOUN
ejpam-4448	115	7	that	that	SCONJ
ejpam-4448	115	8	(	(	PUNCT
ejpam-4448	115	9	−1)kck	−1)kck	PROPN
ejpam-4448	115	10	equals	equal	VERB
ejpam-4448	115	11	the	the	DET
ejpam-4448	115	12	sum	sum	NOUN
ejpam-4448	115	13	of	of	ADP
ejpam-4448	115	14	all	all	DET
ejpam-4448	115	15	principal	principal	ADJ
ejpam-4448	115	16	minors	minor	NOUN
ejpam-4448	115	17	of	of	ADP
ejpam-4448	115	18	hα(d	hα(d	PRON
ejpam-4448	115	19	)	)	PUNCT
ejpam-4448	115	20	with	with	ADP
ejpam-4448	115	21	k	k	PROPN
ejpam-4448	115	22	rows	row	NOUN
ejpam-4448	115	23	and	and	CCONJ
ejpam-4448	115	24	columns	column	NOUN
ejpam-4448	115	25	.	.	PUNCT
ejpam-4448	116	1	4	4	X
ejpam-4448	116	2	.	.	X
ejpam-4448	116	3	cospectrality	cospectrality	NOUN
ejpam-4448	116	4	a	a	DET
ejpam-4448	116	5	recurring	recur	VERB
ejpam-4448	116	6	theme	theme	NOUN
ejpam-4448	116	7	in	in	ADP
ejpam-4448	116	8	algebraic	algebraic	ADJ
ejpam-4448	116	9	graph	graph	NOUN
ejpam-4448	116	10	theory	theory	NOUN
ejpam-4448	116	11	is	be	AUX
ejpam-4448	116	12	the	the	DET
ejpam-4448	116	13	hunt	hunt	NOUN
ejpam-4448	116	14	for	for	ADP
ejpam-4448	116	15	pairs	pair	NOUN
ejpam-4448	116	16	of	of	ADP
ejpam-4448	116	17	non	non	ADJ
ejpam-4448	116	18	-	-	ADJ
ejpam-4448	116	19	isomorphic	isomorphic	ADJ
ejpam-4448	116	20	graphs	graph	NOUN
ejpam-4448	116	21	having	have	VERB
ejpam-4448	116	22	the	the	DET
ejpam-4448	116	23	same	same	ADJ
ejpam-4448	116	24	spectrum	spectrum	NOUN
ejpam-4448	116	25	.	.	PUNCT
ejpam-4448	117	1	such	such	ADJ
ejpam-4448	117	2	graphs	graph	NOUN
ejpam-4448	117	3	are	be	AUX
ejpam-4448	117	4	called	call	VERB
ejpam-4448	117	5	cospectral	cospectral	ADJ
ejpam-4448	117	6	.	.	PUNCT
ejpam-4448	118	1	in	in	ADP
ejpam-4448	118	2	contrast	contrast	NOUN
ejpam-4448	118	3	to	to	ADP
ejpam-4448	118	4	this	this	PRON
ejpam-4448	118	5	,	,	PUNCT
ejpam-4448	118	6	we	we	PRON
ejpam-4448	118	7	shall	shall	AUX
ejpam-4448	118	8	look	look	VERB
ejpam-4448	118	9	into	into	ADP
ejpam-4448	118	10	the	the	DET
ejpam-4448	118	11	question	question	NOUN
ejpam-4448	118	12	under	under	ADP
ejpam-4448	118	13	which	which	PRON
ejpam-4448	118	14	conditions	condition	NOUN
ejpam-4448	118	15	the	the	DET
ejpam-4448	118	16	same	same	ADJ
ejpam-4448	118	17	graph	graph	NOUN
ejpam-4448	118	18	has	have	VERB
ejpam-4448	118	19	identical	identical	ADJ
ejpam-4448	118	20	α	α	NOUN
ejpam-4448	118	21	-	-	NOUN
ejpam-4448	118	22	spectrum	spectrum	NOUN
ejpam-4448	118	23	for	for	ADP
ejpam-4448	118	24	different	different	ADJ
ejpam-4448	118	25	values	value	NOUN
ejpam-4448	118	26	of	of	ADP
ejpam-4448	118	27	α	α	NOUN
ejpam-4448	118	28	.	.	PUNCT
ejpam-4448	119	1	it	it	PRON
ejpam-4448	119	2	comes	come	VERB
ejpam-4448	119	3	as	as	ADP
ejpam-4448	119	4	no	no	DET
ejpam-4448	119	5	surprise	surprise	NOUN
ejpam-4448	119	6	that	that	SCONJ
ejpam-4448	119	7	such	such	ADJ
ejpam-4448	119	8	spectra	spectra	NOUN
ejpam-4448	119	9	may	may	AUX
ejpam-4448	119	10	be	be	AUX
ejpam-4448	119	11	completely	completely	ADV
ejpam-4448	119	12	different	different	ADJ
ejpam-4448	119	13	:	:	PUNCT
ejpam-4448	119	14	example	example	NOUN
ejpam-4448	120	1	1	1	NUM
ejpam-4448	120	2	.	.	X
ejpam-4448	121	1	for	for	ADP
ejpam-4448	121	2	the	the	DET
ejpam-4448	121	3	mixed	mixed	ADJ
ejpam-4448	121	4	graph	graph	NOUN
ejpam-4448	121	5	shown	show	VERB
ejpam-4448	121	6	in	in	ADP
ejpam-4448	121	7	figure	figure	NOUN
ejpam-4448	121	8	1	1	NUM
ejpam-4448	121	9	we	we	PRON
ejpam-4448	121	10	have	have	VERB
ejpam-4448	121	11	:	:	PUNCT
ejpam-4448	121	12	σγ	σγ	ADP
ejpam-4448	121	13	=	=	PUNCT
ejpam-4448	121	14	{	{	PUNCT
ejpam-4448	121	15	2.57083,−2.3222	2.57083,−2.3222	NUM
ejpam-4448	121	16	,	,	PUNCT
ejpam-4448	121	17	1.50413,−1.19239,−0.560369	1.50413,−1.19239,−0.560369	NUM
ejpam-4448	121	18	}	}	PUNCT
ejpam-4448	121	19	σω	σω	NOUN
ejpam-4448	121	20	=	=	SYM
ejpam-4448	121	21	{	{	PUNCT
ejpam-4448	121	22	−2.93033	−2.93033	PROPN
ejpam-4448	121	23	,	,	PUNCT
ejpam-4448	121	24	2.30034	2.30034	NUM
ejpam-4448	121	25	,	,	PUNCT
ejpam-4448	121	26	1.15439,−0.832963	1.15439,−0.832963	NUM
ejpam-4448	121	27	,	,	PUNCT
ejpam-4448	121	28	0.308565	0.308565	NUM
ejpam-4448	121	29	}	}	PUNCT
ejpam-4448	121	30	σi	σi	NOUN
ejpam-4448	121	31	=	=	SYM
ejpam-4448	121	32	{	{	PUNCT
ejpam-4448	121	33	−2.71687	−2.71687	NUM
ejpam-4448	121	34	,	,	PUNCT
ejpam-4448	121	35	2.2803	2.2803	NUM
ejpam-4448	121	36	,	,	PUNCT
ejpam-4448	121	37	1.50739,−1.07082	1.50739,−1.07082	NUM
ejpam-4448	121	38	,	,	PUNCT
ejpam-4448	121	39	0.0	0.0	NUM
ejpam-4448	121	40	}	}	PUNCT
ejpam-4448	121	41	however	however	ADV
ejpam-4448	121	42	,	,	PUNCT
ejpam-4448	121	43	there	there	PRON
ejpam-4448	121	44	exist	exist	VERB
ejpam-4448	121	45	mixed	mixed	ADJ
ejpam-4448	121	46	graphs	graph	NOUN
ejpam-4448	121	47	exhibiting	exhibit	VERB
ejpam-4448	121	48	the	the	DET
ejpam-4448	121	49	same	same	ADJ
ejpam-4448	121	50	α	α	NOUN
ejpam-4448	121	51	-	-	NOUN
ejpam-4448	121	52	spectrum	spectrum	NOUN
ejpam-4448	121	53	for	for	ADP
ejpam-4448	121	54	two	two	NUM
ejpam-4448	121	55	different	different	ADJ
ejpam-4448	121	56	values	value	NOUN
ejpam-4448	121	57	of	of	ADP
ejpam-4448	121	58	α	α	NOUN
ejpam-4448	121	59	,	,	PUNCT
ejpam-4448	121	60	say	say	VERB
ejpam-4448	121	61	α1	α1	PROPN
ejpam-4448	121	62	,	,	PUNCT
ejpam-4448	121	63	α2	α2	PROPN
ejpam-4448	121	64	.	.	PUNCT
ejpam-4448	122	1	we	we	PRON
ejpam-4448	122	2	call	call	VERB
ejpam-4448	122	3	such	such	DET
ejpam-4448	122	4	a	a	DET
ejpam-4448	122	5	mixed	mixed	ADJ
ejpam-4448	122	6	graph	graph	NOUN
ejpam-4448	122	7	α1	α1	NOUN
ejpam-4448	122	8	-	-	PUNCT
ejpam-4448	122	9	α2	α2	VERB
ejpam-4448	122	10	-	-	PUNCT
ejpam-4448	122	11	cospectral	cospectral	NOUN
ejpam-4448	122	12	.	.	PUNCT
ejpam-4448	123	1	let	let	VERB
ejpam-4448	123	2	us	we	PRON
ejpam-4448	123	3	give	give	VERB
ejpam-4448	123	4	an	an	DET
ejpam-4448	123	5	example	example	NOUN
ejpam-4448	123	6	for	for	ADP
ejpam-4448	123	7	a	a	DET
ejpam-4448	123	8	γ	γ	PROPN
ejpam-4448	123	9	-	-	PUNCT
ejpam-4448	123	10	ω	ω	VERB
ejpam-4448	123	11	-	-	ADJ
ejpam-4448	123	12	cospectral	cospectral	ADJ
ejpam-4448	123	13	mixed	mixed	ADJ
ejpam-4448	123	14	graph	graph	NOUN
ejpam-4448	123	15	:	:	PUNCT
ejpam-4448	123	16	example	example	NOUN
ejpam-4448	124	1	2	2	NUM
ejpam-4448	124	2	.	.	PUNCT
ejpam-4448	124	3	the	the	DET
ejpam-4448	124	4	mixed	mixed	ADJ
ejpam-4448	124	5	graph	graph	NOUN
ejpam-4448	124	6	d	d	NOUN
ejpam-4448	124	7	shown	show	VERB
ejpam-4448	124	8	in	in	ADP
ejpam-4448	124	9	figure	figure	NOUN
ejpam-4448	124	10	2	2	NUM
ejpam-4448	124	11	is	be	AUX
ejpam-4448	124	12	γ	γ	PROPN
ejpam-4448	124	13	-	-	PUNCT
ejpam-4448	124	14	ω	ω	NOUN
ejpam-4448	124	15	-	-	NOUN
ejpam-4448	124	16	cospectral	cospectral	ADJ
ejpam-4448	124	17	,	,	PUNCT
ejpam-4448	124	18	i.e.	i.e.	X
ejpam-4448	124	19	σγ(d	σγ(d	NOUN
ejpam-4448	124	20	)	)	PUNCT
ejpam-4448	124	21	=	=	SYM
ejpam-4448	124	22	σω(d	σω(d	NOUN
ejpam-4448	124	23	)	)	PUNCT
ejpam-4448	124	24	.	.	PUNCT
ejpam-4448	125	1	this	this	PRON
ejpam-4448	125	2	is	be	AUX
ejpam-4448	125	3	not	not	PART
ejpam-4448	125	4	difficult	difficult	ADJ
ejpam-4448	125	5	to	to	PART
ejpam-4448	125	6	see	see	VERB
ejpam-4448	125	7	:	:	PUNCT
ejpam-4448	125	8	with	with	ADP
ejpam-4448	125	9	respect	respect	NOUN
ejpam-4448	125	10	to	to	ADP
ejpam-4448	125	11	corollary	corollary	ADJ
ejpam-4448	125	12	1	1	NUM
ejpam-4448	125	13	note	note	NOUN
ejpam-4448	125	14	that	that	SCONJ
ejpam-4448	125	15	d	d	NOUN
ejpam-4448	125	16	contains	contain	VERB
ejpam-4448	125	17	only	only	ADV
ejpam-4448	125	18	one	one	NUM
ejpam-4448	125	19	cycle	cycle	NOUN
ejpam-4448	125	20	.	.	PUNCT
ejpam-4448	126	1	moreover	moreover	ADV
ejpam-4448	126	2	,	,	PUNCT
ejpam-4448	126	3	hγ(c	hγ(c	NUM
ejpam-4448	126	4	)	)	PUNCT
ejpam-4448	126	5	∈	∈	PROPN
ejpam-4448	126	6	{	{	PUNCT
ejpam-4448	126	7	γ	γ	X
ejpam-4448	126	8	,	,	PUNCT
ejpam-4448	126	9	γ2	γ2	ADJ
ejpam-4448	126	10	}	}	PUNCT
ejpam-4448	126	11	and	and	CCONJ
ejpam-4448	126	12	hω(c	hω(c	NUM
ejpam-4448	126	13	)	)	PUNCT
ejpam-4448	126	14	∈	∈	PROPN
ejpam-4448	126	15	{	{	PUNCT
ejpam-4448	126	16	ω2	ω2	ADJ
ejpam-4448	126	17	,	,	PUNCT
ejpam-4448	126	18	ω2	ω2	ADJ
ejpam-4448	126	19	}	}	PUNCT
ejpam-4448	126	20	.	.	PUNCT
ejpam-4448	127	1	observing	observe	VERB
ejpam-4448	127	2	γ	γ	PROPN
ejpam-4448	127	3	=	=	PROPN
ejpam-4448	127	4	ω2	ω2	NOUN
ejpam-4448	127	5	we	we	PRON
ejpam-4448	127	6	have	have	VERB
ejpam-4448	127	7	χγ(d	χγ(d	NOUN
ejpam-4448	127	8	,	,	PUNCT
ejpam-4448	127	9	λ	λ	NOUN
ejpam-4448	127	10	)	)	PUNCT
ejpam-4448	127	11	=	=	SYM
ejpam-4448	127	12	χω(d	χω(d	NOUN
ejpam-4448	127	13	,	,	PUNCT
ejpam-4448	127	14	λ	λ	NOUN
ejpam-4448	127	15	)	)	PUNCT
ejpam-4448	127	16	.	.	PUNCT
ejpam-4448	128	1	in	in	ADP
ejpam-4448	128	2	contrast	contrast	NOUN
ejpam-4448	128	3	,	,	PUNCT
ejpam-4448	128	4	we	we	PRON
ejpam-4448	128	5	remark	remark	VERB
ejpam-4448	128	6	that	that	SCONJ
ejpam-4448	128	7	χα(d	χα(d	NOUN
ejpam-4448	128	8	,	,	PUNCT
ejpam-4448	128	9	λ	λ	NOUN
ejpam-4448	128	10	)	)	PUNCT
ejpam-4448	128	11	̸=	̸=	PROPN
ejpam-4448	128	12	χi(d	χi(d	NOUN
ejpam-4448	128	13	,	,	PUNCT
ejpam-4448	128	14	λ	λ	NOUN
ejpam-4448	128	15	)	)	PUNCT
ejpam-4448	128	16	,	,	PUNCT
ejpam-4448	128	17	hence	hence	ADV
ejpam-4448	128	18	σα(d	σα(d	NOUN
ejpam-4448	128	19	)	)	PUNCT
ejpam-4448	128	20	̸=	̸=	PROPN
ejpam-4448	128	21	σi(d	σi(d	NOUN
ejpam-4448	128	22	)	)	PUNCT
ejpam-4448	128	23	.	.	PUNCT
ejpam-4448	129	1	note	note	VERB
ejpam-4448	129	2	that	that	SCONJ
ejpam-4448	129	3	for	for	ADP
ejpam-4448	129	4	α	α	NOUN
ejpam-4448	129	5	=	=	SYM
ejpam-4448	129	6	1	1	NUM
ejpam-4448	129	7	we	we	PRON
ejpam-4448	129	8	have	have	VERB
ejpam-4448	129	9	hα	hα	NOUN
ejpam-4448	129	10	=	=	PUNCT
ejpam-4448	129	11	a(γ(d	a(γ(d	PROPN
ejpam-4448	129	12	)	)	PUNCT
ejpam-4448	129	13	)	)	PUNCT
ejpam-4448	129	14	,	,	PUNCT
ejpam-4448	129	15	so	so	CCONJ
ejpam-4448	129	16	the	the	DET
ejpam-4448	129	17	special	special	ADJ
ejpam-4448	129	18	case	case	NOUN
ejpam-4448	129	19	of	of	ADP
ejpam-4448	129	20	α-1	α-1	NOUN
ejpam-4448	129	21	-	-	PUNCT
ejpam-4448	129	22	cospectrality	cospectrality	NOUN
ejpam-4448	129	23	is	be	AUX
ejpam-4448	129	24	equivalent	equivalent	ADJ
ejpam-4448	129	25	to	to	ADP
ejpam-4448	129	26	asking	ask	VERB
ejpam-4448	129	27	whether	whether	SCONJ
ejpam-4448	129	28	the	the	DET
ejpam-4448	129	29	α	α	NOUN
ejpam-4448	129	30	-	-	NOUN
ejpam-4448	129	31	spectrum	spectrum	NOUN
ejpam-4448	129	32	of	of	ADP
ejpam-4448	129	33	a	a	DET
ejpam-4448	129	34	mixed	mixed	ADJ
ejpam-4448	129	35	graph	graph	NOUN
ejpam-4448	129	36	d	d	NOUN
ejpam-4448	129	37	coincides	coincide	VERB
ejpam-4448	129	38	with	with	ADP
ejpam-4448	129	39	the	the	DET
ejpam-4448	129	40	traditional	traditional	ADJ
ejpam-4448	129	41	spectrum	spectrum	NOUN
ejpam-4448	129	42	of	of	ADP
ejpam-4448	129	43	its	its	PRON
ejpam-4448	129	44	undirected	undirected	ADJ
ejpam-4448	129	45	counterpart	counterpart	NOUN
ejpam-4448	129	46	γ(d	γ(d	NOUN
ejpam-4448	129	47	)	)	PUNCT
ejpam-4448	129	48	.	.	PUNCT
ejpam-4448	130	1	thus	thus	ADV
ejpam-4448	130	2	corollary	corollary	ADJ
ejpam-4448	130	3	1	1	NUM
ejpam-4448	130	4	immediately	immediately	ADV
ejpam-4448	130	5	gives	give	VERB
ejpam-4448	130	6	rise	rise	NOUN
ejpam-4448	130	7	to	to	ADP
ejpam-4448	130	8	the	the	DET
ejpam-4448	130	9	following	following	ADJ
ejpam-4448	130	10	result	result	NOUN
ejpam-4448	130	11	:	:	PUNCT
ejpam-4448	130	12	m.	m.	NOUN
ejpam-4448	130	13	abudayah	abudayah	PROPN
ejpam-4448	130	14	,	,	PUNCT
ejpam-4448	130	15	o.	o.	PROPN
ejpam-4448	130	16	alomari	alomari	PROPN
ejpam-4448	130	17	,	,	PUNCT
ejpam-4448	130	18	t.	t.	PROPN
ejpam-4448	130	19	sander	sander	PROPN
ejpam-4448	130	20	/	/	SYM
ejpam-4448	130	21	eur	eur	PROPN
ejpam-4448	130	22	.	.	PUNCT
ejpam-4448	131	1	j.	j.	PROPN
ejpam-4448	131	2	pure	pure	PROPN
ejpam-4448	131	3	appl	appl	PROPN
ejpam-4448	131	4	.	.	PROPN
ejpam-4448	131	5	math	math	PROPN
ejpam-4448	131	6	,	,	PUNCT
ejpam-4448	131	7	15	15	NUM
ejpam-4448	131	8	(	(	PUNCT
ejpam-4448	131	9	3	3	NUM
ejpam-4448	131	10	)	)	PUNCT
ejpam-4448	131	11	(	(	PUNCT
ejpam-4448	131	12	2022	2022	NUM
ejpam-4448	131	13	)	)	PUNCT
ejpam-4448	131	14	,	,	PUNCT
ejpam-4448	131	15	841	841	NUM
ejpam-4448	131	16	-	-	SYM
ejpam-4448	131	17	855	855	NUM
ejpam-4448	131	18	846	846	NUM
ejpam-4448	131	19	1	1	NUM
ejpam-4448	131	20	2	2	NUM
ejpam-4448	131	21	3	3	NUM
ejpam-4448	131	22	4	4	NUM
ejpam-4448	131	23	5	5	NUM
ejpam-4448	131	24	6	6	NUM
ejpam-4448	131	25	7	7	NUM
ejpam-4448	131	26	8	8	NUM
ejpam-4448	131	27	9	9	NUM
ejpam-4448	131	28	10	10	NUM
ejpam-4448	131	29	11	11	NUM
ejpam-4448	131	30	figure	figure	NOUN
ejpam-4448	131	31	2	2	NUM
ejpam-4448	131	32	:	:	PUNCT
ejpam-4448	131	33	a	a	DET
ejpam-4448	131	34	γ	γ	PROPN
ejpam-4448	131	35	-	-	PUNCT
ejpam-4448	131	36	ω	ω	VERB
ejpam-4448	131	37	-	-	ADJ
ejpam-4448	131	38	cospectral	cospectral	ADJ
ejpam-4448	131	39	mixed	mixed	ADJ
ejpam-4448	131	40	graph	graph	NOUN
ejpam-4448	131	41	corollary	corollary	ADJ
ejpam-4448	131	42	2	2	NUM
ejpam-4448	131	43	.	.	PUNCT
ejpam-4448	132	1	let	let	VERB
ejpam-4448	132	2	t	t	PROPN
ejpam-4448	132	3	be	be	AUX
ejpam-4448	132	4	a	a	DET
ejpam-4448	132	5	mixed	mixed	ADJ
ejpam-4448	132	6	tree	tree	NOUN
ejpam-4448	132	7	.	.	PUNCT
ejpam-4448	133	1	then	then	ADV
ejpam-4448	133	2	σα(t	σα(t	PUNCT
ejpam-4448	133	3	)	)	PUNCT
ejpam-4448	134	1	=	=	PUNCT
ejpam-4448	134	2	σ(γ(t	σ(γ(t	NOUN
ejpam-4448	134	3	)	)	PUNCT
ejpam-4448	134	4	)	)	PUNCT
ejpam-4448	134	5	.	.	PUNCT
ejpam-4448	135	1	proof	proof	NOUN
ejpam-4448	135	2	.	.	PUNCT
ejpam-4448	136	1	trees	tree	NOUN
ejpam-4448	136	2	do	do	AUX
ejpam-4448	136	3	not	not	PART
ejpam-4448	136	4	contain	contain	VERB
ejpam-4448	136	5	cycles	cycle	NOUN
ejpam-4448	136	6	,	,	PUNCT
ejpam-4448	136	7	hence	hence	ADV
ejpam-4448	136	8	using	use	VERB
ejpam-4448	136	9	α	α	NOUN
ejpam-4448	136	10	=	=	SYM
ejpam-4448	136	11	1	1	NUM
ejpam-4448	136	12	in	in	ADP
ejpam-4448	136	13	(	(	PUNCT
ejpam-4448	136	14	7	7	NUM
ejpam-4448	136	15	)	)	PUNCT
ejpam-4448	136	16	instead	instead	ADV
ejpam-4448	136	17	of	of	ADP
ejpam-4448	136	18	the	the	DET
ejpam-4448	136	19	given	give	VERB
ejpam-4448	136	20	value	value	NOUN
ejpam-4448	136	21	does	do	AUX
ejpam-4448	136	22	not	not	PART
ejpam-4448	136	23	change	change	VERB
ejpam-4448	136	24	the	the	DET
ejpam-4448	136	25	result	result	NOUN
ejpam-4448	136	26	.	.	PUNCT
ejpam-4448	137	1	since	since	SCONJ
ejpam-4448	137	2	trees	tree	NOUN
ejpam-4448	137	3	are	be	AUX
ejpam-4448	137	4	α-1	α-1	NOUN
ejpam-4448	137	5	-	-	PUNCT
ejpam-4448	137	6	cospectral	cospectral	ADJ
ejpam-4448	137	7	for	for	ADP
ejpam-4448	137	8	any	any	DET
ejpam-4448	137	9	α	α	NOUN
ejpam-4448	137	10	we	we	PRON
ejpam-4448	137	11	see	see	VERB
ejpam-4448	137	12	that	that	SCONJ
ejpam-4448	137	13	they	they	PRON
ejpam-4448	137	14	are	be	AUX
ejpam-4448	137	15	α1	α1	NOUN
ejpam-4448	137	16	-	-	PUNCT
ejpam-4448	137	17	α2	α2	VERB
ejpam-4448	137	18	-	-	PUNCT
ejpam-4448	137	19	cospectral	cospectral	NOUN
ejpam-4448	137	20	for	for	ADP
ejpam-4448	137	21	arbitrary	arbitrary	ADJ
ejpam-4448	137	22	values	value	NOUN
ejpam-4448	137	23	α1,α2	α1,α2	PROPN
ejpam-4448	137	24	.	.	PUNCT
ejpam-4448	138	1	now	now	ADV
ejpam-4448	138	2	,	,	PUNCT
ejpam-4448	138	3	consider	consider	VERB
ejpam-4448	138	4	a	a	DET
ejpam-4448	138	5	mixed	mixed	ADJ
ejpam-4448	138	6	graph	graph	NOUN
ejpam-4448	138	7	that	that	PRON
ejpam-4448	138	8	contains	contain	VERB
ejpam-4448	138	9	cycles	cycle	NOUN
ejpam-4448	138	10	.	.	PUNCT
ejpam-4448	139	1	obviously	obviously	ADV
ejpam-4448	139	2	,	,	PUNCT
ejpam-4448	139	3	it	it	PRON
ejpam-4448	139	4	does	do	AUX
ejpam-4448	139	5	not	not	PART
ejpam-4448	139	6	matter	matter	VERB
ejpam-4448	139	7	for	for	ADP
ejpam-4448	139	8	equation	equation	NOUN
ejpam-4448	139	9	(	(	PUNCT
ejpam-4448	139	10	7	7	X
ejpam-4448	139	11	)	)	PUNCT
ejpam-4448	139	12	if	if	SCONJ
ejpam-4448	139	13	we	we	PRON
ejpam-4448	139	14	use	use	VERB
ejpam-4448	139	15	α	α	NOUN
ejpam-4448	139	16	=	=	SYM
ejpam-4448	139	17	1	1	NUM
ejpam-4448	139	18	or	or	CCONJ
ejpam-4448	139	19	some	some	DET
ejpam-4448	139	20	other	other	ADJ
ejpam-4448	139	21	specific	specific	ADJ
ejpam-4448	139	22	value	value	NOUN
ejpam-4448	139	23	as	as	ADV
ejpam-4448	139	24	long	long	ADV
ejpam-4448	139	25	as	as	ADP
ejpam-4448	139	26	(	(	PUNCT
ejpam-4448	139	27	with	with	ADP
ejpam-4448	139	28	respect	respect	NOUN
ejpam-4448	139	29	to	to	ADP
ejpam-4448	139	30	that	that	DET
ejpam-4448	139	31	other	other	ADJ
ejpam-4448	139	32	value	value	NOUN
ejpam-4448	139	33	)	)	PUNCT
ejpam-4448	139	34	all	all	DET
ejpam-4448	139	35	factors	factor	NOUN
ejpam-4448	139	36	in	in	ADP
ejpam-4448	139	37	the	the	DET
ejpam-4448	139	38	involved	involved	ADJ
ejpam-4448	139	39	products	product	NOUN
ejpam-4448	139	40	are	be	AUX
ejpam-4448	139	41	equal	equal	ADJ
ejpam-4448	139	42	to	to	ADP
ejpam-4448	139	43	one	one	NUM
ejpam-4448	139	44	.	.	PUNCT
ejpam-4448	140	1	this	this	PRON
ejpam-4448	140	2	motivates	motivate	VERB
ejpam-4448	140	3	the	the	DET
ejpam-4448	140	4	following	follow	VERB
ejpam-4448	140	5	definition	definition	NOUN
ejpam-4448	140	6	:	:	PUNCT
ejpam-4448	140	7	definition	definition	NOUN
ejpam-4448	140	8	4	4	NUM
ejpam-4448	140	9	.	.	PUNCT
ejpam-4448	140	10	a	a	DET
ejpam-4448	140	11	mixed	mixed	ADJ
ejpam-4448	140	12	graph	graph	NOUN
ejpam-4448	140	13	is	be	AUX
ejpam-4448	140	14	an	an	DET
ejpam-4448	140	15	α	α	NOUN
ejpam-4448	140	16	-	-	PUNCT
ejpam-4448	140	17	monograph	monograph	NOUN
ejpam-4448	140	18	(	(	PUNCT
ejpam-4448	140	19	of	of	ADP
ejpam-4448	140	20	1st	1st	ADJ
ejpam-4448	140	21	kind	kind	NOUN
ejpam-4448	140	22	)	)	PUNCT
ejpam-4448	140	23	if	if	SCONJ
ejpam-4448	140	24	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	140	25	)	)	PUNCT
ejpam-4448	140	26	=	=	SYM
ejpam-4448	140	27	1	1	NUM
ejpam-4448	140	28	for	for	ADP
ejpam-4448	140	29	all	all	DET
ejpam-4448	140	30	its	its	PRON
ejpam-4448	140	31	cycles	cycle	NOUN
ejpam-4448	140	32	c.	c.	NOUN
ejpam-4448	140	33	trivially	trivially	ADV
ejpam-4448	140	34	,	,	PUNCT
ejpam-4448	140	35	trees	tree	NOUN
ejpam-4448	140	36	are	be	AUX
ejpam-4448	140	37	α	α	PRON
ejpam-4448	140	38	-	-	PUNCT
ejpam-4448	140	39	monographs	monograph	NOUN
ejpam-4448	140	40	.	.	PUNCT
ejpam-4448	141	1	by	by	ADP
ejpam-4448	141	2	construction	construction	NOUN
ejpam-4448	141	3	,	,	PUNCT
ejpam-4448	141	4	corollary	corollary	ADJ
ejpam-4448	141	5	2	2	NUM
ejpam-4448	141	6	directly	directly	ADV
ejpam-4448	141	7	extends	extend	VERB
ejpam-4448	141	8	to	to	ADP
ejpam-4448	141	9	monographs	monograph	NOUN
ejpam-4448	141	10	:	:	PUNCT
ejpam-4448	141	11	theorem	theorem	NOUN
ejpam-4448	141	12	4	4	NUM
ejpam-4448	141	13	.	.	PUNCT
ejpam-4448	142	1	let	let	VERB
ejpam-4448	142	2	d	d	PRON
ejpam-4448	142	3	be	be	AUX
ejpam-4448	142	4	an	an	DET
ejpam-4448	142	5	α	α	NOUN
ejpam-4448	142	6	-	-	PUNCT
ejpam-4448	142	7	monograph	monograph	NOUN
ejpam-4448	142	8	(	(	PUNCT
ejpam-4448	142	9	of	of	ADP
ejpam-4448	142	10	1st	1st	ADJ
ejpam-4448	142	11	kind	kind	NOUN
ejpam-4448	142	12	)	)	PUNCT
ejpam-4448	142	13	.	.	PUNCT
ejpam-4448	143	1	then	then	ADV
ejpam-4448	143	2	,	,	PUNCT
ejpam-4448	143	3	σα(d	σα(d	NOUN
ejpam-4448	143	4	)	)	PUNCT
ejpam-4448	143	5	=	=	SYM
ejpam-4448	143	6	σ(γ(d	σ(γ(d	NOUN
ejpam-4448	143	7	)	)	PUNCT
ejpam-4448	143	8	)	)	PUNCT
ejpam-4448	143	9	.	.	PUNCT
ejpam-4448	144	1	regarding	regard	VERB
ejpam-4448	144	2	corollary	corollary	NOUN
ejpam-4448	144	3	1	1	NUM
ejpam-4448	144	4	and	and	CCONJ
ejpam-4448	144	5	(	(	PUNCT
ejpam-4448	144	6	7	7	NUM
ejpam-4448	144	7	)	)	PUNCT
ejpam-4448	144	8	,	,	PUNCT
ejpam-4448	144	9	note	note	VERB
ejpam-4448	144	10	that	that	SCONJ
ejpam-4448	144	11	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	144	12	)	)	PUNCT
ejpam-4448	144	13	=	=	SYM
ejpam-4448	145	1	αxᾱy	αxᾱy	NUM
ejpam-4448	145	2	=	=	SYM
ejpam-4448	145	3	αx−y	αx−y	NOUN
ejpam-4448	145	4	,	,	PUNCT
ejpam-4448	145	5	where	where	SCONJ
ejpam-4448	145	6	x	x	X
ejpam-4448	145	7	(	(	PUNCT
ejpam-4448	145	8	resp	resp	NOUN
ejpam-4448	145	9	.	.	PUNCT
ejpam-4448	146	1	y	y	X
ejpam-4448	146	2	)	)	PUNCT
ejpam-4448	146	3	is	be	AUX
ejpam-4448	146	4	the	the	DET
ejpam-4448	146	5	number	number	NOUN
ejpam-4448	146	6	of	of	ADP
ejpam-4448	146	7	forward	forward	ADV
ejpam-4448	146	8	(	(	PUNCT
ejpam-4448	146	9	resp	resp	NOUN
ejpam-4448	146	10	.	.	PUNCT
ejpam-4448	147	1	backward	backward	ADV
ejpam-4448	147	2	)	)	PUNCT
ejpam-4448	147	3	edges	edge	VERB
ejpam-4448	147	4	encounterd	encounterd	NOUN
ejpam-4448	147	5	while	while	SCONJ
ejpam-4448	147	6	traversing	traverse	VERB
ejpam-4448	147	7	c⃗.	c⃗.	NOUN
ejpam-4448	147	8	we	we	PRON
ejpam-4448	147	9	will	will	AUX
ejpam-4448	147	10	tacitly	tacitly	ADV
ejpam-4448	147	11	make	make	VERB
ejpam-4448	147	12	use	use	NOUN
ejpam-4448	147	13	of	of	ADP
ejpam-4448	147	14	this	this	DET
ejpam-4448	147	15	fact	fact	NOUN
ejpam-4448	147	16	hereafter	hereafter	ADV
ejpam-4448	147	17	.	.	PUNCT
ejpam-4448	148	1	corollary	corollary	ADJ
ejpam-4448	148	2	3	3	X
ejpam-4448	148	3	.	.	PUNCT
ejpam-4448	149	1	let	let	VERB
ejpam-4448	149	2	d	d	PRON
ejpam-4448	149	3	be	be	AUX
ejpam-4448	149	4	a	a	DET
ejpam-4448	149	5	connected	connected	ADJ
ejpam-4448	149	6	mixed	mixed	ADJ
ejpam-4448	149	7	graph	graph	NOUN
ejpam-4448	149	8	.	.	PUNCT
ejpam-4448	150	1	if	if	SCONJ
ejpam-4448	150	2	,	,	PUNCT
ejpam-4448	150	3	for	for	ADP
ejpam-4448	150	4	every	every	DET
ejpam-4448	150	5	cycle	cycle	NOUN
ejpam-4448	150	6	in	in	ADP
ejpam-4448	150	7	d	d	PROPN
ejpam-4448	150	8	,	,	PUNCT
ejpam-4448	150	9	the	the	DET
ejpam-4448	150	10	difference	difference	NOUN
ejpam-4448	150	11	between	between	ADP
ejpam-4448	150	12	the	the	DET
ejpam-4448	150	13	numbers	number	NOUN
ejpam-4448	150	14	of	of	ADP
ejpam-4448	150	15	encountered	encounter	VERB
ejpam-4448	150	16	forward	forward	ADJ
ejpam-4448	150	17	arcs	arc	NOUN
ejpam-4448	150	18	and	and	CCONJ
ejpam-4448	150	19	the	the	DET
ejpam-4448	150	20	number	number	NOUN
ejpam-4448	150	21	of	of	ADP
ejpam-4448	150	22	backward	backward	ADJ
ejpam-4448	150	23	arcs	arc	NOUN
ejpam-4448	150	24	(	(	PUNCT
ejpam-4448	150	25	w.r.t	w.r.t	NOUN
ejpam-4448	150	26	.	.	PUNCT
ejpam-4448	151	1	any	any	DET
ejpam-4448	151	2	traversal	traversal	NOUN
ejpam-4448	151	3	direction	direction	NOUN
ejpam-4448	151	4	)	)	PUNCT
ejpam-4448	151	5	is	be	AUX
ejpam-4448	151	6	a	a	DET
ejpam-4448	151	7	multiple	multiple	NOUN
ejpam-4448	151	8	of	of	ADP
ejpam-4448	151	9	the	the	DET
ejpam-4448	151	10	order	order	NOUN
ejpam-4448	151	11	of	of	ADP
ejpam-4448	151	12	α	α	NOUN
ejpam-4448	151	13	,	,	PUNCT
ejpam-4448	151	14	then	then	ADV
ejpam-4448	151	15	d	d	PROPN
ejpam-4448	151	16	is	be	AUX
ejpam-4448	151	17	an	an	DET
ejpam-4448	151	18	α	α	NOUN
ejpam-4448	151	19	-	-	PUNCT
ejpam-4448	151	20	monograph	monograph	NOUN
ejpam-4448	151	21	.	.	PUNCT
ejpam-4448	152	1	corollary	corollary	ADJ
ejpam-4448	152	2	4	4	NUM
ejpam-4448	152	3	.	.	PUNCT
ejpam-4448	153	1	a	a	DET
ejpam-4448	153	2	connected	connect	VERB
ejpam-4448	153	3	mixed	mixed	ADJ
ejpam-4448	153	4	graph	graph	NOUN
ejpam-4448	153	5	g	g	PROPN
ejpam-4448	153	6	is	be	AUX
ejpam-4448	153	7	an	an	DET
ejpam-4448	153	8	α	α	NOUN
ejpam-4448	153	9	-	-	PUNCT
ejpam-4448	153	10	monograph	monograph	NOUN
ejpam-4448	153	11	for	for	ADP
ejpam-4448	153	12	every	every	DET
ejpam-4448	153	13	value	value	NOUN
ejpam-4448	153	14	α	α	NOUN
ejpam-4448	153	15	if	if	SCONJ
ejpam-4448	154	1	and	and	CCONJ
ejpam-4448	154	2	only	only	ADV
ejpam-4448	154	3	if	if	SCONJ
ejpam-4448	154	4	every	every	DET
ejpam-4448	154	5	cycle	cycle	NOUN
ejpam-4448	154	6	in	in	ADP
ejpam-4448	154	7	d	d	PROPN
ejpam-4448	154	8	contains	contain	VERB
ejpam-4448	154	9	as	as	ADV
ejpam-4448	154	10	many	many	ADJ
ejpam-4448	154	11	forward	forward	ADJ
ejpam-4448	154	12	arcs	arc	NOUN
ejpam-4448	154	13	as	as	ADP
ejpam-4448	154	14	backward	backward	ADJ
ejpam-4448	154	15	arcs	arc	NOUN
ejpam-4448	154	16	.	.	PUNCT
ejpam-4448	155	1	proof	proof	NOUN
ejpam-4448	155	2	.	.	PUNCT
ejpam-4448	156	1	for	for	ADP
ejpam-4448	156	2	the	the	DET
ejpam-4448	156	3	forward	forward	ADJ
ejpam-4448	156	4	implication	implication	NOUN
ejpam-4448	156	5	note	note	NOUN
ejpam-4448	156	6	that	that	SCONJ
ejpam-4448	156	7	there	there	PRON
ejpam-4448	156	8	exist	exist	VERB
ejpam-4448	156	9	values	value	NOUN
ejpam-4448	156	10	α	α	PRON
ejpam-4448	156	11	of	of	ADP
ejpam-4448	156	12	infinite	infinite	ADJ
ejpam-4448	156	13	order	order	NOUN
ejpam-4448	156	14	(	(	PUNCT
ejpam-4448	156	15	i.e.	i.e.	X
ejpam-4448	156	16	αj	αj	X
ejpam-4448	156	17	=	=	PUNCT
ejpam-4448	156	18	αk	αk	PUNCT
ejpam-4448	156	19	only	only	ADV
ejpam-4448	156	20	if	if	SCONJ
ejpam-4448	156	21	j	j	PROPN
ejpam-4448	156	22	=	=	SYM
ejpam-4448	156	23	k	k	NOUN
ejpam-4448	156	24	)	)	PUNCT
ejpam-4448	156	25	.	.	PUNCT
ejpam-4448	157	1	the	the	DET
ejpam-4448	157	2	converse	converse	NOUN
ejpam-4448	157	3	follows	follow	VERB
ejpam-4448	157	4	directly	directly	ADV
ejpam-4448	157	5	from	from	ADP
ejpam-4448	157	6	corollary	corollary	ADJ
ejpam-4448	157	7	3	3	NUM
ejpam-4448	157	8	.	.	PUNCT
ejpam-4448	158	1	the	the	DET
ejpam-4448	158	2	properties	property	NOUN
ejpam-4448	158	3	of	of	ADP
ejpam-4448	158	4	α	α	NOUN
ejpam-4448	158	5	-	-	PUNCT
ejpam-4448	158	6	monographs	monograph	NOUN
ejpam-4448	158	7	deserve	deserve	VERB
ejpam-4448	158	8	further	further	ADJ
ejpam-4448	158	9	investigation	investigation	NOUN
ejpam-4448	158	10	.	.	PUNCT
ejpam-4448	159	1	but	but	CCONJ
ejpam-4448	159	2	beforehand	beforehand	ADV
ejpam-4448	159	3	,	,	PUNCT
ejpam-4448	159	4	we	we	PRON
ejpam-4448	159	5	will	will	AUX
ejpam-4448	159	6	elaborate	elaborate	VERB
ejpam-4448	159	7	more	more	ADV
ejpam-4448	159	8	on	on	ADP
ejpam-4448	159	9	cospectrality	cospectrality	NOUN
ejpam-4448	159	10	.	.	PUNCT
ejpam-4448	160	1	we	we	PRON
ejpam-4448	160	2	start	start	VERB
ejpam-4448	160	3	with	with	ADP
ejpam-4448	160	4	the	the	DET
ejpam-4448	160	5	question	question	NOUN
ejpam-4448	160	6	when	when	SCONJ
ejpam-4448	160	7	the	the	DET
ejpam-4448	160	8	α	α	NOUN
ejpam-4448	160	9	-	-	NOUN
ejpam-4448	160	10	spectra	spectra	NOUN
ejpam-4448	160	11	for	for	ADP
ejpam-4448	160	12	the	the	DET
ejpam-4448	160	13	special	special	ADJ
ejpam-4448	160	14	values	value	NOUN
ejpam-4448	160	15	γ	γ	X
ejpam-4448	160	16	and	and	CCONJ
ejpam-4448	160	17	ω	ω	PROPN
ejpam-4448	160	18	coincide	coincide	NOUN
ejpam-4448	160	19	:	:	PUNCT
ejpam-4448	160	20	m.	m.	NOUN
ejpam-4448	160	21	abudayah	abudayah	PROPN
ejpam-4448	160	22	,	,	PUNCT
ejpam-4448	160	23	o.	o.	PROPN
ejpam-4448	160	24	alomari	alomari	PROPN
ejpam-4448	160	25	,	,	PUNCT
ejpam-4448	160	26	t.	t.	PROPN
ejpam-4448	160	27	sander	sander	PROPN
ejpam-4448	160	28	/	/	SYM
ejpam-4448	160	29	eur	eur	PROPN
ejpam-4448	160	30	.	.	PUNCT
ejpam-4448	161	1	j.	j.	PROPN
ejpam-4448	161	2	pure	pure	PROPN
ejpam-4448	161	3	appl	appl	PROPN
ejpam-4448	161	4	.	.	PROPN
ejpam-4448	161	5	math	math	PROPN
ejpam-4448	161	6	,	,	PUNCT
ejpam-4448	161	7	15	15	NUM
ejpam-4448	161	8	(	(	PUNCT
ejpam-4448	161	9	3	3	NUM
ejpam-4448	161	10	)	)	PUNCT
ejpam-4448	161	11	(	(	PUNCT
ejpam-4448	161	12	2022	2022	NUM
ejpam-4448	161	13	)	)	PUNCT
ejpam-4448	161	14	,	,	PUNCT
ejpam-4448	161	15	841	841	NUM
ejpam-4448	161	16	-	-	SYM
ejpam-4448	161	17	855	855	NUM
ejpam-4448	161	18	847	847	NUM
ejpam-4448	161	19	theorem	theorem	NOUN
ejpam-4448	161	20	5	5	NUM
ejpam-4448	161	21	.	.	PUNCT
ejpam-4448	162	1	let	let	VERB
ejpam-4448	162	2	d	d	PRON
ejpam-4448	162	3	be	be	AUX
ejpam-4448	162	4	a	a	DET
ejpam-4448	162	5	mixed	mixed	ADJ
ejpam-4448	162	6	graph	graph	NOUN
ejpam-4448	162	7	.	.	PUNCT
ejpam-4448	163	1	if	if	SCONJ
ejpam-4448	163	2	,	,	PUNCT
ejpam-4448	163	3	for	for	ADP
ejpam-4448	163	4	any	any	DET
ejpam-4448	163	5	cycle	cycle	NOUN
ejpam-4448	163	6	in	in	ADP
ejpam-4448	163	7	d	d	PROPN
ejpam-4448	163	8	,	,	PUNCT
ejpam-4448	163	9	the	the	DET
ejpam-4448	163	10	difference	difference	NOUN
ejpam-4448	163	11	between	between	ADP
ejpam-4448	163	12	the	the	DET
ejpam-4448	163	13	numbers	number	NOUN
ejpam-4448	163	14	of	of	ADP
ejpam-4448	163	15	encountered	encounter	VERB
ejpam-4448	163	16	forward	forward	ADJ
ejpam-4448	163	17	arcs	arc	NOUN
ejpam-4448	163	18	and	and	CCONJ
ejpam-4448	163	19	the	the	DET
ejpam-4448	163	20	number	number	NOUN
ejpam-4448	163	21	of	of	ADP
ejpam-4448	163	22	backward	backward	ADJ
ejpam-4448	163	23	arcs	arc	NOUN
ejpam-4448	163	24	(	(	PUNCT
ejpam-4448	163	25	w.r.t	w.r.t	NOUN
ejpam-4448	163	26	.	.	PUNCT
ejpam-4448	164	1	any	any	DET
ejpam-4448	164	2	traversal	traversal	NOUN
ejpam-4448	164	3	direction	direction	NOUN
ejpam-4448	164	4	)	)	PUNCT
ejpam-4448	164	5	is	be	AUX
ejpam-4448	164	6	even	even	ADV
ejpam-4448	164	7	,	,	PUNCT
ejpam-4448	164	8	then	then	ADV
ejpam-4448	164	9	d	d	PROPN
ejpam-4448	164	10	is	be	AUX
ejpam-4448	164	11	γ	γ	PROPN
ejpam-4448	164	12	-	-	PUNCT
ejpam-4448	164	13	ω	ω	NOUN
ejpam-4448	164	14	-	-	PUNCT
ejpam-4448	164	15	cospectral	cospectral	NOUN
ejpam-4448	164	16	.	.	PUNCT
ejpam-4448	165	1	proof	proof	NOUN
ejpam-4448	165	2	.	.	PUNCT
ejpam-4448	166	1	under	under	ADP
ejpam-4448	166	2	the	the	DET
ejpam-4448	166	3	given	give	VERB
ejpam-4448	166	4	assumptions	assumption	NOUN
ejpam-4448	166	5	there	there	PRON
ejpam-4448	166	6	will	will	AUX
ejpam-4448	166	7	be	be	AUX
ejpam-4448	166	8	only	only	ADV
ejpam-4448	166	9	even	even	ADV
ejpam-4448	166	10	powers	power	NOUN
ejpam-4448	166	11	of	of	ADP
ejpam-4448	166	12	α	α	NOUN
ejpam-4448	166	13	in	in	ADP
ejpam-4448	166	14	(	(	PUNCT
ejpam-4448	166	15	7	7	NUM
ejpam-4448	166	16	)	)	PUNCT
ejpam-4448	166	17	.	.	PUNCT
ejpam-4448	167	1	but	but	CCONJ
ejpam-4448	167	2	re(γ2k	re(γ2k	PROPN
ejpam-4448	167	3	)	)	PUNCT
ejpam-4448	168	1	=	=	PUNCT
ejpam-4448	168	2	re(ω2k	re(ω2k	PROPN
ejpam-4448	168	3	)	)	PUNCT
ejpam-4448	168	4	.	.	PUNCT
ejpam-4448	169	1	corollary	corollary	ADJ
ejpam-4448	169	2	5	5	NUM
ejpam-4448	169	3	.	.	PUNCT
ejpam-4448	170	1	let	let	VERB
ejpam-4448	170	2	d	d	PRON
ejpam-4448	170	3	be	be	AUX
ejpam-4448	170	4	a	a	DET
ejpam-4448	170	5	mixed	mixed	ADJ
ejpam-4448	170	6	bipartite	bipartite	NOUN
ejpam-4448	170	7	graph	graph	NOUN
ejpam-4448	170	8	.	.	PUNCT
ejpam-4448	171	1	suppose	suppose	VERB
ejpam-4448	171	2	that	that	SCONJ
ejpam-4448	171	3	every	every	DET
ejpam-4448	171	4	cycle	cycle	NOUN
ejpam-4448	171	5	in	in	ADP
ejpam-4448	171	6	d	d	PROPN
ejpam-4448	171	7	contains	contain	VERB
ejpam-4448	171	8	an	an	DET
ejpam-4448	171	9	even	even	ADJ
ejpam-4448	171	10	number	number	NOUN
ejpam-4448	171	11	of	of	ADP
ejpam-4448	171	12	digons	digon	NOUN
ejpam-4448	171	13	.	.	PUNCT
ejpam-4448	172	1	then	then	ADV
ejpam-4448	172	2	d	d	PROPN
ejpam-4448	172	3	is	be	AUX
ejpam-4448	172	4	γ	γ	PROPN
ejpam-4448	172	5	-	-	PUNCT
ejpam-4448	172	6	ω	ω	NOUN
ejpam-4448	172	7	-	-	PUNCT
ejpam-4448	172	8	cospectral	cospectral	NOUN
ejpam-4448	172	9	.	.	PUNCT
ejpam-4448	173	1	proof	proof	NOUN
ejpam-4448	173	2	.	.	PUNCT
ejpam-4448	174	1	a	a	DET
ejpam-4448	174	2	bipartite	bipartite	ADJ
ejpam-4448	174	3	graph	graph	NOUN
ejpam-4448	174	4	contains	contain	VERB
ejpam-4448	174	5	only	only	ADV
ejpam-4448	174	6	even	even	ADV
ejpam-4448	174	7	cycles	cycle	NOUN
ejpam-4448	174	8	.	.	PUNCT
ejpam-4448	175	1	consider	consider	VERB
ejpam-4448	175	2	such	such	DET
ejpam-4448	175	3	a	a	DET
ejpam-4448	175	4	cycle	cycle	NOUN
ejpam-4448	175	5	c.	c.	NOUN
ejpam-4448	175	6	subtracting	subtract	VERB
ejpam-4448	175	7	an	an	DET
ejpam-4448	175	8	even	even	ADJ
ejpam-4448	175	9	number	number	NOUN
ejpam-4448	175	10	of	of	ADP
ejpam-4448	175	11	digons	digon	NOUN
ejpam-4448	175	12	,	,	PUNCT
ejpam-4448	175	13	we	we	PRON
ejpam-4448	175	14	conclude	conclude	VERB
ejpam-4448	175	15	that	that	SCONJ
ejpam-4448	175	16	c	c	PROPN
ejpam-4448	175	17	contains	contain	VERB
ejpam-4448	175	18	an	an	DET
ejpam-4448	175	19	even	even	ADJ
ejpam-4448	175	20	number	number	NOUN
ejpam-4448	175	21	of	of	ADP
ejpam-4448	175	22	arcs	arc	NOUN
ejpam-4448	175	23	.	.	PUNCT
ejpam-4448	176	1	traversing	traverse	VERB
ejpam-4448	176	2	c	c	X
ejpam-4448	176	3	,	,	PUNCT
ejpam-4448	176	4	these	these	DET
ejpam-4448	176	5	arcs	arc	NOUN
ejpam-4448	176	6	either	either	CCONJ
ejpam-4448	176	7	consist	consist	VERB
ejpam-4448	176	8	of	of	ADP
ejpam-4448	176	9	an	an	DET
ejpam-4448	176	10	even	even	ADJ
ejpam-4448	176	11	number	number	NOUN
ejpam-4448	176	12	of	of	ADP
ejpam-4448	176	13	forward	forward	ADJ
ejpam-4448	176	14	arcs	arc	NOUN
ejpam-4448	176	15	and	and	CCONJ
ejpam-4448	176	16	an	an	DET
ejpam-4448	176	17	even	even	ADJ
ejpam-4448	176	18	number	number	NOUN
ejpam-4448	176	19	of	of	ADP
ejpam-4448	176	20	backward	backward	ADJ
ejpam-4448	176	21	arcs	arc	NOUN
ejpam-4448	176	22	or	or	CCONJ
ejpam-4448	176	23	consist	consist	VERB
ejpam-4448	176	24	of	of	ADP
ejpam-4448	176	25	an	an	DET
ejpam-4448	176	26	odd	odd	ADJ
ejpam-4448	176	27	number	number	NOUN
ejpam-4448	176	28	of	of	ADP
ejpam-4448	176	29	forward	forward	ADJ
ejpam-4448	176	30	arcs	arc	NOUN
ejpam-4448	176	31	and	and	CCONJ
ejpam-4448	176	32	an	an	DET
ejpam-4448	176	33	odd	odd	ADJ
ejpam-4448	176	34	number	number	NOUN
ejpam-4448	176	35	of	of	ADP
ejpam-4448	176	36	backward	backward	ADJ
ejpam-4448	176	37	arcs	arc	NOUN
ejpam-4448	176	38	.	.	PUNCT
ejpam-4448	177	1	hence	hence	ADV
ejpam-4448	177	2	theorem	theorem	VERB
ejpam-4448	177	3	5	5	NUM
ejpam-4448	177	4	can	can	AUX
ejpam-4448	177	5	be	be	AUX
ejpam-4448	177	6	applied	apply	VERB
ejpam-4448	177	7	.	.	PUNCT
ejpam-4448	178	1	corollary	corollary	ADJ
ejpam-4448	178	2	6	6	NUM
ejpam-4448	178	3	.	.	PUNCT
ejpam-4448	179	1	every	every	DET
ejpam-4448	179	2	oriented	orient	VERB
ejpam-4448	179	3	bipartite	bipartite	NOUN
ejpam-4448	179	4	graph	graph	NOUN
ejpam-4448	179	5	is	be	AUX
ejpam-4448	179	6	γ	γ	PROPN
ejpam-4448	179	7	-	-	PUNCT
ejpam-4448	179	8	ω	ω	NOUN
ejpam-4448	179	9	-	-	PUNCT
ejpam-4448	179	10	cospectral	cospectral	NOUN
ejpam-4448	179	11	.	.	PUNCT
ejpam-4448	180	1	as	as	SCONJ
ejpam-4448	180	2	already	already	ADV
ejpam-4448	180	3	mentioned	mention	VERB
ejpam-4448	180	4	in	in	ADP
ejpam-4448	180	5	example	example	NOUN
ejpam-4448	180	6	2	2	NUM
ejpam-4448	180	7	,	,	PUNCT
ejpam-4448	180	8	the	the	DET
ejpam-4448	180	9	graph	graph	NOUN
ejpam-4448	180	10	in	in	ADP
ejpam-4448	180	11	figure	figure	NOUN
ejpam-4448	180	12	2	2	NUM
ejpam-4448	180	13	is	be	AUX
ejpam-4448	180	14	γ	γ	PROPN
ejpam-4448	180	15	-	-	PUNCT
ejpam-4448	180	16	ω	ω	NOUN
ejpam-4448	180	17	-	-	NOUN
ejpam-4448	180	18	cospectral	cospectral	NOUN
ejpam-4448	180	19	.	.	PUNCT
ejpam-4448	181	1	this	this	PRON
ejpam-4448	181	2	follows	follow	VERB
ejpam-4448	181	3	from	from	ADP
ejpam-4448	181	4	corollary	corollary	ADJ
ejpam-4448	181	5	6	6	NUM
ejpam-4448	181	6	since	since	SCONJ
ejpam-4448	181	7	it	it	PRON
ejpam-4448	181	8	is	be	AUX
ejpam-4448	181	9	an	an	DET
ejpam-4448	181	10	oriented	orient	VERB
ejpam-4448	181	11	bipartite	bipartite	NOUN
ejpam-4448	181	12	graph	graph	NOUN
ejpam-4448	181	13	.	.	PUNCT
ejpam-4448	182	1	using	use	VERB
ejpam-4448	182	2	the	the	DET
ejpam-4448	182	3	ideas	idea	NOUN
ejpam-4448	182	4	from	from	ADP
ejpam-4448	182	5	the	the	DET
ejpam-4448	182	6	proof	proof	NOUN
ejpam-4448	182	7	of	of	ADP
ejpam-4448	182	8	corollary	corollary	ADJ
ejpam-4448	182	9	5	5	NUM
ejpam-4448	182	10	,	,	PUNCT
ejpam-4448	182	11	one	one	PRON
ejpam-4448	182	12	can	can	AUX
ejpam-4448	182	13	generalize	generalize	VERB
ejpam-4448	182	14	as	as	SCONJ
ejpam-4448	182	15	follows	follow	VERB
ejpam-4448	182	16	:	:	PUNCT
ejpam-4448	182	17	theorem	theorem	NOUN
ejpam-4448	182	18	6	6	NUM
ejpam-4448	182	19	.	.	PUNCT
ejpam-4448	183	1	let	let	VERB
ejpam-4448	183	2	d	d	PRON
ejpam-4448	183	3	be	be	AUX
ejpam-4448	183	4	a	a	DET
ejpam-4448	183	5	mixed	mixed	ADJ
ejpam-4448	183	6	graph	graph	NOUN
ejpam-4448	183	7	.	.	PUNCT
ejpam-4448	184	1	suppose	suppose	VERB
ejpam-4448	184	2	that	that	SCONJ
ejpam-4448	184	3	every	every	DET
ejpam-4448	184	4	even	even	ADJ
ejpam-4448	184	5	cycle	cycle	NOUN
ejpam-4448	184	6	in	in	ADP
ejpam-4448	184	7	d	d	PROPN
ejpam-4448	184	8	contains	contain	VERB
ejpam-4448	184	9	an	an	DET
ejpam-4448	184	10	even	even	ADJ
ejpam-4448	184	11	number	number	NOUN
ejpam-4448	184	12	of	of	ADP
ejpam-4448	184	13	digons	digon	NOUN
ejpam-4448	184	14	and	and	CCONJ
ejpam-4448	184	15	every	every	DET
ejpam-4448	184	16	odd	odd	ADJ
ejpam-4448	184	17	cycle	cycle	NOUN
ejpam-4448	184	18	in	in	ADP
ejpam-4448	184	19	d	d	PROPN
ejpam-4448	184	20	contains	contain	VERB
ejpam-4448	184	21	an	an	DET
ejpam-4448	184	22	odd	odd	ADJ
ejpam-4448	184	23	number	number	NOUN
ejpam-4448	184	24	of	of	ADP
ejpam-4448	184	25	digons	digon	NOUN
ejpam-4448	184	26	.	.	PUNCT
ejpam-4448	185	1	then	then	ADV
ejpam-4448	185	2	d	d	PROPN
ejpam-4448	185	3	is	be	AUX
ejpam-4448	185	4	γ	γ	PROPN
ejpam-4448	185	5	-	-	PUNCT
ejpam-4448	185	6	ω	ω	NOUN
ejpam-4448	185	7	-	-	NOUN
ejpam-4448	185	8	cospectral	cospectral	NOUN
ejpam-4448	185	9	.	.	PUNCT
ejpam-4448	186	1	we	we	PRON
ejpam-4448	186	2	have	have	AUX
ejpam-4448	186	3	shown	show	VERB
ejpam-4448	186	4	how	how	SCONJ
ejpam-4448	186	5	corollary	corollary	ADJ
ejpam-4448	186	6	1	1	NUM
ejpam-4448	186	7	can	can	AUX
ejpam-4448	186	8	be	be	AUX
ejpam-4448	186	9	used	use	VERB
ejpam-4448	186	10	as	as	ADP
ejpam-4448	186	11	a	a	DET
ejpam-4448	186	12	tool	tool	NOUN
ejpam-4448	186	13	investigating	investigate	VERB
ejpam-4448	186	14	cospectrality	cospectrality	NOUN
ejpam-4448	186	15	.	.	PUNCT
ejpam-4448	187	1	in	in	ADP
ejpam-4448	187	2	view	view	NOUN
ejpam-4448	187	3	of	of	ADP
ejpam-4448	187	4	α1	α1	PROPN
ejpam-4448	187	5	-	-	PUNCT
ejpam-4448	187	6	α2	α2	VERB
ejpam-4448	187	7	-	-	PUNCT
ejpam-4448	187	8	cospectrality	cospectrality	NOUN
ejpam-4448	187	9	it	it	PRON
ejpam-4448	187	10	is	be	AUX
ejpam-4448	187	11	sufficient	sufficient	ADJ
ejpam-4448	187	12	to	to	PART
ejpam-4448	187	13	require	require	VERB
ejpam-4448	187	14	that	that	SCONJ
ejpam-4448	187	15	,	,	PUNCT
ejpam-4448	187	16	for	for	ADP
ejpam-4448	187	17	every	every	DET
ejpam-4448	187	18	elementary	elementary	ADJ
ejpam-4448	187	19	mixed	mixed	ADJ
ejpam-4448	187	20	subgraph	subgraph	NOUN
ejpam-4448	187	21	d′	d′	PROPN
ejpam-4448	187	22	,	,	PUNCT
ejpam-4448	187	23	the	the	DET
ejpam-4448	187	24	real	real	ADJ
ejpam-4448	187	25	part	part	NOUN
ejpam-4448	187	26	of	of	ADP
ejpam-4448	187	27	the	the	DET
ejpam-4448	187	28	associated	associated	ADJ
ejpam-4448	187	29	product	product	NOUN
ejpam-4448	187	30	in	in	ADP
ejpam-4448	187	31	(	(	PUNCT
ejpam-4448	187	32	7	7	NUM
ejpam-4448	187	33	)	)	PUNCT
ejpam-4448	187	34	is	be	AUX
ejpam-4448	187	35	the	the	DET
ejpam-4448	187	36	same	same	ADJ
ejpam-4448	187	37	for	for	ADP
ejpam-4448	187	38	both	both	DET
ejpam-4448	187	39	values	value	NOUN
ejpam-4448	187	40	α1	α1	PROPN
ejpam-4448	187	41	and	and	CCONJ
ejpam-4448	187	42	α2	α2	ADJ
ejpam-4448	187	43	.	.	PUNCT
ejpam-4448	188	1	as	as	ADP
ejpam-4448	188	2	a	a	DET
ejpam-4448	188	3	slightly	slightly	ADV
ejpam-4448	188	4	coarser	coarse	ADJ
ejpam-4448	188	5	condition	condition	NOUN
ejpam-4448	188	6	one	one	PRON
ejpam-4448	188	7	could	could	AUX
ejpam-4448	188	8	require	require	VERB
ejpam-4448	188	9	that	that	DET
ejpam-4448	188	10	hα1(c⃗	hα1(c⃗	NOUN
ejpam-4448	188	11	)	)	PUNCT
ejpam-4448	189	1	=	=	SYM
ejpam-4448	189	2	hα2(c⃗	hα2(c⃗	PROPN
ejpam-4448	189	3	)	)	PUNCT
ejpam-4448	189	4	for	for	ADP
ejpam-4448	189	5	all	all	DET
ejpam-4448	189	6	cycles	cycle	NOUN
ejpam-4448	189	7	c	c	PROPN
ejpam-4448	189	8	of	of	ADP
ejpam-4448	189	9	d′.	d′.	NOUN
ejpam-4448	189	10	in	in	ADP
ejpam-4448	189	11	view	view	NOUN
ejpam-4448	189	12	of	of	ADP
ejpam-4448	189	13	this	this	PRON
ejpam-4448	189	14	,	,	PUNCT
ejpam-4448	189	15	requiring	require	VERB
ejpam-4448	189	16	uniform	uniform	ADJ
ejpam-4448	189	17	cospectrality	cospectrality	NOUN
ejpam-4448	189	18	for	for	ADP
ejpam-4448	189	19	all	all	DET
ejpam-4448	189	20	values	value	NOUN
ejpam-4448	189	21	of	of	ADP
ejpam-4448	189	22	α1,α2	α1,α2	PROPN
ejpam-4448	189	23	(	(	PUNCT
ejpam-4448	189	24	hence	hence	ADV
ejpam-4448	189	25	including	include	VERB
ejpam-4448	189	26	value	value	NOUN
ejpam-4448	189	27	1	1	NUM
ejpam-4448	189	28	)	)	PUNCT
ejpam-4448	189	29	amounts	amount	VERB
ejpam-4448	189	30	to	to	ADP
ejpam-4448	189	31	the	the	DET
ejpam-4448	189	32	condition	condition	NOUN
ejpam-4448	189	33	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	189	34	)	)	PUNCT
ejpam-4448	189	35	=	=	SYM
ejpam-4448	190	1	1	1	NUM
ejpam-4448	190	2	stated	state	VERB
ejpam-4448	190	3	in	in	ADP
ejpam-4448	190	4	definition	definition	NOUN
ejpam-4448	190	5	4	4	NUM
ejpam-4448	190	6	.	.	PUNCT
ejpam-4448	191	1	in	in	ADP
ejpam-4448	191	2	view	view	NOUN
ejpam-4448	191	3	of	of	ADP
ejpam-4448	191	4	this	this	PRON
ejpam-4448	191	5	,	,	PUNCT
ejpam-4448	191	6	one	one	PRON
ejpam-4448	191	7	can	can	AUX
ejpam-4448	191	8	devise	devise	VERB
ejpam-4448	191	9	modifications	modification	NOUN
ejpam-4448	191	10	which	which	PRON
ejpam-4448	191	11	,	,	PUNCT
ejpam-4448	191	12	given	give	VERB
ejpam-4448	191	13	some	some	DET
ejpam-4448	191	14	α	α	NOUN
ejpam-4448	191	15	-	-	PUNCT
ejpam-4448	191	16	monograph	monograph	NOUN
ejpam-4448	191	17	d	d	NOUN
ejpam-4448	191	18	,	,	PUNCT
ejpam-4448	191	19	can	can	AUX
ejpam-4448	191	20	be	be	AUX
ejpam-4448	191	21	used	use	VERB
ejpam-4448	191	22	to	to	PART
ejpam-4448	191	23	construct	construct	VERB
ejpam-4448	191	24	arbitrarily	arbitrarily	ADV
ejpam-4448	191	25	many	many	ADJ
ejpam-4448	191	26	derived	derive	VERB
ejpam-4448	191	27	α	α	NUM
ejpam-4448	191	28	-	-	PUNCT
ejpam-4448	191	29	monographs	monograph	NOUN
ejpam-4448	191	30	containing	contain	VERB
ejpam-4448	191	31	d	d	PROPN
ejpam-4448	191	32	as	as	ADP
ejpam-4448	191	33	a	a	DET
ejpam-4448	191	34	subgraph	subgraph	NOUN
ejpam-4448	191	35	:	:	PUNCT
ejpam-4448	191	36	theorem	theorem	NOUN
ejpam-4448	191	37	7	7	NUM
ejpam-4448	191	38	.	.	PUNCT
ejpam-4448	192	1	let	let	VERB
ejpam-4448	192	2	d	d	PRON
ejpam-4448	192	3	be	be	AUX
ejpam-4448	192	4	an	an	DET
ejpam-4448	192	5	α	α	NOUN
ejpam-4448	192	6	-	-	PUNCT
ejpam-4448	192	7	monograph	monograph	NOUN
ejpam-4448	192	8	.	.	PUNCT
ejpam-4448	193	1	let	let	VERB
ejpam-4448	193	2	u	u	PRON
ejpam-4448	193	3	be	be	AUX
ejpam-4448	193	4	a	a	DET
ejpam-4448	193	5	connected	connected	ADJ
ejpam-4448	193	6	undirected	undirected	ADJ
ejpam-4448	193	7	subgraph	subgraph	NOUN
ejpam-4448	193	8	of	of	ADP
ejpam-4448	193	9	d.	d.	PROPN
ejpam-4448	193	10	fix	fix	VERB
ejpam-4448	193	11	a	a	DET
ejpam-4448	193	12	set	set	NOUN
ejpam-4448	193	13	m	m	NOUN
ejpam-4448	193	14	of	of	ADP
ejpam-4448	193	15	new	new	ADJ
ejpam-4448	193	16	vertices	vertex	NOUN
ejpam-4448	193	17	and	and	CCONJ
ejpam-4448	193	18	subsets	subset	NOUN
ejpam-4448	193	19	vx	vx	PROPN
ejpam-4448	193	20	⊂	⊂	PROPN
ejpam-4448	193	21	v	v	X
ejpam-4448	193	22	(	(	PUNCT
ejpam-4448	193	23	u	u	NOUN
ejpam-4448	193	24	)	)	PUNCT
ejpam-4448	193	25	,	,	PUNCT
ejpam-4448	193	26	for	for	ADP
ejpam-4448	193	27	x	x	PROPN
ejpam-4448	193	28	∈	∈	PROPN
ejpam-4448	193	29	m	m	PROPN
ejpam-4448	193	30	.	.	PUNCT
ejpam-4448	194	1	connect	connect	VERB
ejpam-4448	194	2	each	each	DET
ejpam-4448	194	3	vertex	vertex	NOUN
ejpam-4448	194	4	x	x	PUNCT
ejpam-4448	194	5	∈m	∈m	NOUN
ejpam-4448	194	6	to	to	ADP
ejpam-4448	194	7	vx	vx	ADP
ejpam-4448	194	8	such	such	ADJ
ejpam-4448	194	9	that	that	SCONJ
ejpam-4448	194	10	either	either	PRON
ejpam-4448	194	11	n+	n+	PUNCT
ejpam-4448	195	1	d	d	X
ejpam-4448	195	2	(	(	PUNCT
ejpam-4448	195	3	x	x	NOUN
ejpam-4448	195	4	)	)	PUNCT
ejpam-4448	195	5	=	=	SYM
ejpam-4448	195	6	vx	vx	PROPN
ejpam-4448	195	7	or	or	CCONJ
ejpam-4448	195	8	n−	n−	PROPN
ejpam-4448	195	9	d	d	X
ejpam-4448	195	10	(	(	PUNCT
ejpam-4448	195	11	x	x	NOUN
ejpam-4448	195	12	)	)	PUNCT
ejpam-4448	195	13	=	=	SYM
ejpam-4448	196	1	vx	vx	PROPN
ejpam-4448	196	2	.	.	PROPN
ejpam-4448	196	3	then	then	ADV
ejpam-4448	196	4	the	the	DET
ejpam-4448	196	5	resulting	result	VERB
ejpam-4448	196	6	mixed	mixed	ADJ
ejpam-4448	196	7	graph	graph	NOUN
ejpam-4448	196	8	d̃	d̃	PROPN
ejpam-4448	196	9	is	be	AUX
ejpam-4448	196	10	an	an	DET
ejpam-4448	196	11	α	α	NOUN
ejpam-4448	196	12	-	-	PUNCT
ejpam-4448	196	13	monograph	monograph	NOUN
ejpam-4448	196	14	.	.	PUNCT
ejpam-4448	197	1	proof	proof	NOUN
ejpam-4448	197	2	.	.	PUNCT
ejpam-4448	198	1	the	the	DET
ejpam-4448	198	2	newly	newly	ADV
ejpam-4448	198	3	added	add	VERB
ejpam-4448	198	4	vertices	vertex	NOUN
ejpam-4448	198	5	and	and	CCONJ
ejpam-4448	198	6	their	their	PRON
ejpam-4448	198	7	adjacent	adjacent	ADJ
ejpam-4448	198	8	edges	edge	NOUN
ejpam-4448	198	9	may	may	AUX
ejpam-4448	198	10	introduce	introduce	VERB
ejpam-4448	198	11	new	new	ADJ
ejpam-4448	198	12	cycles	cycle	NOUN
ejpam-4448	198	13	.	.	PUNCT
ejpam-4448	199	1	let	let	VERB
ejpam-4448	199	2	c	c	PRON
ejpam-4448	199	3	be	be	AUX
ejpam-4448	199	4	such	such	DET
ejpam-4448	199	5	a	a	DET
ejpam-4448	199	6	cycle	cycle	NOUN
ejpam-4448	199	7	in	in	ADP
ejpam-4448	199	8	d̃	d̃	PROPN
ejpam-4448	199	9	and	and	CCONJ
ejpam-4448	199	10	c⃗	c⃗	NOUN
ejpam-4448	199	11	any	any	DET
ejpam-4448	199	12	traversal	traversal	NOUN
ejpam-4448	199	13	of	of	ADP
ejpam-4448	199	14	c.	c.	NOUN
ejpam-4448	199	15	by	by	ADP
ejpam-4448	199	16	construction	construction	NOUN
ejpam-4448	199	17	,	,	PUNCT
ejpam-4448	199	18	the	the	DET
ejpam-4448	199	19	predecessor	predecessor	NOUN
ejpam-4448	199	20	vx	vx	PROPN
ejpam-4448	199	21	of	of	ADP
ejpam-4448	199	22	x	x	PROPN
ejpam-4448	199	23	and	and	CCONJ
ejpam-4448	199	24	its	its	PRON
ejpam-4448	199	25	successor	successor	NOUN
ejpam-4448	199	26	wx	wx	PROPN
ejpam-4448	199	27	(	(	PUNCT
ejpam-4448	199	28	w.r.t	w.r.t	NOUN
ejpam-4448	199	29	.	.	PUNCT
ejpam-4448	200	1	c⃗	c⃗	PROPN
ejpam-4448	200	2	)	)	PUNCT
ejpam-4448	200	3	belong	belong	VERB
ejpam-4448	200	4	to	to	ADP
ejpam-4448	200	5	u	u	PROPN
ejpam-4448	200	6	.	.	PUNCT
ejpam-4448	201	1	consider	consider	VERB
ejpam-4448	201	2	the	the	DET
ejpam-4448	201	3	residual	residual	ADJ
ejpam-4448	201	4	graph	graph	NOUN
ejpam-4448	201	5	r	r	NOUN
ejpam-4448	201	6	obtained	obtain	VERB
ejpam-4448	201	7	by	by	ADP
ejpam-4448	201	8	removing	remove	VERB
ejpam-4448	201	9	all	all	DET
ejpam-4448	201	10	edges	edge	NOUN
ejpam-4448	201	11	of	of	ADP
ejpam-4448	201	12	u	u	NOUN
ejpam-4448	201	13	from	from	ADP
ejpam-4448	201	14	d.	d.	PROPN
ejpam-4448	201	15	every	every	DET
ejpam-4448	201	16	path	path	NOUN
ejpam-4448	201	17	w	w	NOUN
ejpam-4448	201	18	in	in	ADP
ejpam-4448	201	19	r	r	NOUN
ejpam-4448	201	20	joining	join	VERB
ejpam-4448	201	21	two	two	NUM
ejpam-4448	201	22	vertices	vertex	NOUN
ejpam-4448	201	23	u1	u1	NOUN
ejpam-4448	201	24	,	,	PUNCT
ejpam-4448	201	25	u2	u2	NOUN
ejpam-4448	201	26	that	that	PRON
ejpam-4448	201	27	originally	originally	ADV
ejpam-4448	201	28	belong	belong	VERB
ejpam-4448	201	29	to	to	ADP
ejpam-4448	201	30	the	the	DET
ejpam-4448	201	31	subgraph	subgraph	NOUN
ejpam-4448	201	32	u	u	NOUN
ejpam-4448	201	33	in	in	ADP
ejpam-4448	201	34	d	d	PROPN
ejpam-4448	201	35	satisfies	satisfie	NOUN
ejpam-4448	201	36	hα(w	hα(w	X
ejpam-4448	201	37	)	)	PUNCT
ejpam-4448	201	38	=	=	PUNCT
ejpam-4448	202	1	1	1	X
ejpam-4448	202	2	.	.	PUNCT
ejpam-4448	202	3	to	to	PART
ejpam-4448	202	4	see	see	VERB
ejpam-4448	202	5	this	this	PRON
ejpam-4448	202	6	,	,	PUNCT
ejpam-4448	202	7	add	add	VERB
ejpam-4448	202	8	any	any	DET
ejpam-4448	202	9	path	path	NOUN
ejpam-4448	202	10	w	w	ADP
ejpam-4448	202	11	′	′	NOUN
ejpam-4448	202	12	between	between	ADP
ejpam-4448	202	13	u1	u1	NOUN
ejpam-4448	202	14	and	and	CCONJ
ejpam-4448	202	15	u2	u2	NOUN
ejpam-4448	202	16	in	in	ADP
ejpam-4448	202	17	u	u	NOUN
ejpam-4448	202	18	to	to	PART
ejpam-4448	202	19	obtain	obtain	VERB
ejpam-4448	202	20	a	a	DET
ejpam-4448	202	21	cycle	cycle	NOUN
ejpam-4448	202	22	cw	cw	NOUN
ejpam-4448	202	23	ind	ind	NOUN
ejpam-4448	202	24	.	.	PUNCT
ejpam-4448	203	1	choosing	choose	VERB
ejpam-4448	203	2	matching	matching	NOUN
ejpam-4448	203	3	traversals	traversal	NOUN
ejpam-4448	203	4	forw	forw	ADJ
ejpam-4448	203	5	m.	m.	NOUN
ejpam-4448	203	6	abudayah	abudayah	PROPN
ejpam-4448	203	7	,	,	PUNCT
ejpam-4448	203	8	o.	o.	PROPN
ejpam-4448	203	9	alomari	alomari	PROPN
ejpam-4448	203	10	,	,	PUNCT
ejpam-4448	203	11	t.	t.	PROPN
ejpam-4448	203	12	sander	sander	PROPN
ejpam-4448	203	13	/	/	SYM
ejpam-4448	203	14	eur	eur	PROPN
ejpam-4448	203	15	.	.	PUNCT
ejpam-4448	204	1	j.	j.	PROPN
ejpam-4448	204	2	pure	pure	PROPN
ejpam-4448	204	3	appl	appl	PROPN
ejpam-4448	204	4	.	.	PROPN
ejpam-4448	204	5	math	math	PROPN
ejpam-4448	204	6	,	,	PUNCT
ejpam-4448	204	7	15	15	NUM
ejpam-4448	204	8	(	(	PUNCT
ejpam-4448	204	9	3	3	NUM
ejpam-4448	204	10	)	)	PUNCT
ejpam-4448	204	11	(	(	PUNCT
ejpam-4448	204	12	2022	2022	NUM
ejpam-4448	204	13	)	)	PUNCT
ejpam-4448	204	14	,	,	PUNCT
ejpam-4448	204	15	841	841	NUM
ejpam-4448	204	16	-	-	SYM
ejpam-4448	204	17	855	855	NUM
ejpam-4448	204	18	848	848	NUM
ejpam-4448	204	19	1	1	NUM
ejpam-4448	204	20	2	2	NUM
ejpam-4448	204	21	3	3	NUM
ejpam-4448	204	22	4	4	NUM
ejpam-4448	204	23	5	5	NUM
ejpam-4448	204	24	6	6	NUM
ejpam-4448	204	25	7	7	NUM
ejpam-4448	204	26	12	12	NUM
ejpam-4448	204	27	8	8	NUM
ejpam-4448	204	28	9	9	NUM
ejpam-4448	204	29	10	10	NUM
ejpam-4448	204	30	11	11	NUM
ejpam-4448	204	31	13	13	NUM
ejpam-4448	204	32	figure	figure	NOUN
ejpam-4448	204	33	3	3	NUM
ejpam-4448	204	34	:	:	PUNCT
ejpam-4448	204	35	extending	extend	VERB
ejpam-4448	204	36	an	an	DET
ejpam-4448	204	37	α	α	NOUN
ejpam-4448	204	38	-	-	PUNCT
ejpam-4448	204	39	monograph	monograph	NOUN
ejpam-4448	204	40	and	and	CCONJ
ejpam-4448	204	41	w	w	PROPN
ejpam-4448	204	42	′	′	NOUN
ejpam-4448	204	43	,	,	PUNCT
ejpam-4448	204	44	we	we	PRON
ejpam-4448	204	45	get	get	VERB
ejpam-4448	204	46	hα(c⃗w	hα(c⃗w	NOUN
ejpam-4448	204	47	)	)	PUNCT
ejpam-4448	205	1	=	=	SYM
ejpam-4448	205	2	hα(w⃗	hα(w⃗	PROPN
ejpam-4448	205	3	)	)	PUNCT
ejpam-4448	205	4	hα(w⃗	hα(w⃗	PROPN
ejpam-4448	205	5	′	′	NOUN
ejpam-4448	205	6	)	)	PUNCT
ejpam-4448	205	7	.	.	PUNCT
ejpam-4448	206	1	observe	observe	VERB
ejpam-4448	206	2	hα(w⃗	hα(w⃗	PROPN
ejpam-4448	206	3	′	′	NOUN
ejpam-4448	206	4	)	)	PUNCT
ejpam-4448	206	5	=	=	SYM
ejpam-4448	207	1	1	1	NUM
ejpam-4448	207	2	since	since	SCONJ
ejpam-4448	207	3	u	u	NOUN
ejpam-4448	207	4	is	be	AUX
ejpam-4448	207	5	an	an	DET
ejpam-4448	207	6	undirected	undirected	ADJ
ejpam-4448	207	7	subgraph	subgraph	NOUN
ejpam-4448	207	8	of	of	ADP
ejpam-4448	207	9	d.	d.	PROPN
ejpam-4448	207	10	d	d	PROPN
ejpam-4448	207	11	is	be	AUX
ejpam-4448	207	12	an	an	DET
ejpam-4448	207	13	α	α	NOUN
ejpam-4448	207	14	-	-	PUNCT
ejpam-4448	207	15	monograph	monograph	NOUN
ejpam-4448	207	16	,	,	PUNCT
ejpam-4448	207	17	so	so	ADV
ejpam-4448	207	18	hα(c⃗w	hα(c⃗w	NOUN
ejpam-4448	207	19	)	)	PUNCT
ejpam-4448	208	1	=	=	SYM
ejpam-4448	208	2	1	1	NUM
ejpam-4448	208	3	and	and	CCONJ
ejpam-4448	208	4	therefore	therefore	ADV
ejpam-4448	208	5	hα(w⃗	hα(w⃗	PROPN
ejpam-4448	208	6	)	)	PUNCT
ejpam-4448	209	1	=	=	PUNCT
ejpam-4448	209	2	1	1	X
ejpam-4448	209	3	.	.	X
ejpam-4448	209	4	c	c	NOUN
ejpam-4448	209	5	can	can	AUX
ejpam-4448	209	6	be	be	AUX
ejpam-4448	209	7	segmented	segment	VERB
ejpam-4448	209	8	into	into	ADP
ejpam-4448	209	9	paths	path	NOUN
ejpam-4448	209	10	of	of	ADP
ejpam-4448	209	11	three	three	NUM
ejpam-4448	209	12	possible	possible	ADJ
ejpam-4448	209	13	types	type	NOUN
ejpam-4448	209	14	as	as	SCONJ
ejpam-4448	209	15	follows	follow	VERB
ejpam-4448	209	16	:	:	PUNCT
ejpam-4448	209	17	paths	path	NOUN
ejpam-4448	209	18	within	within	ADP
ejpam-4448	209	19	u	u	NOUN
ejpam-4448	209	20	,	,	PUNCT
ejpam-4448	209	21	paths	path	NOUN
ejpam-4448	209	22	within	within	ADP
ejpam-4448	209	23	r	r	NOUN
ejpam-4448	209	24	and	and	CCONJ
ejpam-4448	209	25	the	the	DET
ejpam-4448	209	26	segment	segment	NOUN
ejpam-4448	209	27	from	from	ADP
ejpam-4448	209	28	vx	vx	PROPN
ejpam-4448	209	29	via	via	ADP
ejpam-4448	209	30	x	x	X
ejpam-4448	209	31	to	to	ADP
ejpam-4448	209	32	wx	wx	PROPN
ejpam-4448	209	33	.	.	PUNCT
ejpam-4448	210	1	the	the	DET
ejpam-4448	210	2	latter	latter	ADJ
ejpam-4448	210	3	segment	segment	NOUN
ejpam-4448	210	4	can	can	AUX
ejpam-4448	210	5	only	only	ADV
ejpam-4448	210	6	be	be	AUX
ejpam-4448	210	7	vx→x←wx	vx→x←wx	PROPN
ejpam-4448	210	8	or	or	CCONJ
ejpam-4448	210	9	vx←x→wx	vx←x→wx	VERB
ejpam-4448	210	10	.	.	PUNCT
ejpam-4448	211	1	both	both	PRON
ejpam-4448	211	2	contribute	contribute	VERB
ejpam-4448	211	3	a	a	DET
ejpam-4448	211	4	factor	factor	NOUN
ejpam-4448	211	5	of	of	ADP
ejpam-4448	211	6	1	1	NUM
ejpam-4448	211	7	to	to	ADP
ejpam-4448	211	8	the	the	DET
ejpam-4448	211	9	product	product	NOUN
ejpam-4448	211	10	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	211	11	)	)	PUNCT
ejpam-4448	211	12	,	,	PUNCT
ejpam-4448	211	13	as	as	SCONJ
ejpam-4448	211	14	do	do	VERB
ejpam-4448	211	15	the	the	DET
ejpam-4448	211	16	two	two	NUM
ejpam-4448	211	17	segment	segment	NOUN
ejpam-4448	211	18	types	type	NOUN
ejpam-4448	211	19	mentioned	mention	VERB
ejpam-4448	211	20	first	first	ADV
ejpam-4448	211	21	.	.	PUNCT
ejpam-4448	212	1	overall	overall	ADJ
ejpam-4448	212	2	,	,	PUNCT
ejpam-4448	212	3	hα(c⃗	hα(c⃗	PROPN
ejpam-4448	212	4	)	)	PUNCT
ejpam-4448	212	5	=	=	SYM
ejpam-4448	213	1	1	1	X
ejpam-4448	213	2	.	.	NOUN
ejpam-4448	213	3	example	example	NOUN
ejpam-4448	213	4	3	3	NUM
ejpam-4448	213	5	.	.	X
ejpam-4448	213	6	figure	figure	NOUN
ejpam-4448	213	7	3	3	NUM
ejpam-4448	213	8	illustrates	illustrate	VERB
ejpam-4448	213	9	the	the	DET
ejpam-4448	213	10	construction	construction	NOUN
ejpam-4448	213	11	mentioned	mention	VERB
ejpam-4448	213	12	in	in	ADP
ejpam-4448	213	13	theorem	theorem	NOUN
ejpam-4448	213	14	7	7	NUM
ejpam-4448	213	15	.	.	PUNCT
ejpam-4448	214	1	the	the	DET
ejpam-4448	214	2	vertices	vertex	NOUN
ejpam-4448	214	3	no	no	INTJ
ejpam-4448	214	4	.	.	NOUN
ejpam-4448	214	5	12	12	NUM
ejpam-4448	214	6	und	und	NOUN
ejpam-4448	214	7	13	13	NUM
ejpam-4448	214	8	have	have	AUX
ejpam-4448	214	9	been	be	AUX
ejpam-4448	214	10	newly	newly	ADV
ejpam-4448	214	11	added	add	VERB
ejpam-4448	214	12	.	.	PUNCT
ejpam-4448	215	1	5	5	X
ejpam-4448	215	2	.	.	X
ejpam-4448	215	3	monograph	monograph	NOUN
ejpam-4448	215	4	structure	structure	NOUN
ejpam-4448	215	5	definition	definition	NOUN
ejpam-4448	215	6	4	4	NUM
ejpam-4448	215	7	characterizes	characterize	VERB
ejpam-4448	215	8	α	α	PRON
ejpam-4448	215	9	-	-	PUNCT
ejpam-4448	215	10	monographs	monograph	NOUN
ejpam-4448	215	11	by	by	ADP
ejpam-4448	215	12	a	a	DET
ejpam-4448	215	13	condition	condition	NOUN
ejpam-4448	215	14	concerning	concern	VERB
ejpam-4448	215	15	the	the	DET
ejpam-4448	215	16	traversal	traversal	NOUN
ejpam-4448	215	17	of	of	ADP
ejpam-4448	215	18	cycles	cycle	NOUN
ejpam-4448	215	19	.	.	PUNCT
ejpam-4448	216	1	in	in	ADP
ejpam-4448	216	2	the	the	DET
ejpam-4448	216	3	following	following	NOUN
ejpam-4448	216	4	,	,	PUNCT
ejpam-4448	216	5	we	we	PRON
ejpam-4448	216	6	will	will	AUX
ejpam-4448	216	7	render	render	VERB
ejpam-4448	216	8	the	the	DET
ejpam-4448	216	9	implications	implication	NOUN
ejpam-4448	216	10	of	of	ADP
ejpam-4448	216	11	this	this	DET
ejpam-4448	216	12	condition	condition	NOUN
ejpam-4448	216	13	more	more	ADV
ejpam-4448	216	14	tangible	tangible	ADJ
ejpam-4448	216	15	,	,	PUNCT
ejpam-4448	216	16	by	by	ADP
ejpam-4448	216	17	way	way	NOUN
ejpam-4448	216	18	of	of	ADP
ejpam-4448	216	19	studing	stud	VERB
ejpam-4448	216	20	mixed	mixed	ADJ
ejpam-4448	216	21	walks	walk	NOUN
ejpam-4448	216	22	.	.	PUNCT
ejpam-4448	217	1	to	to	ADP
ejpam-4448	217	2	this	this	DET
ejpam-4448	217	3	end	end	NOUN
ejpam-4448	217	4	,	,	PUNCT
ejpam-4448	217	5	let	let	VERB
ejpam-4448	217	6	d	d	PRON
ejpam-4448	217	7	be	be	AUX
ejpam-4448	217	8	a	a	DET
ejpam-4448	217	9	connected	connected	ADJ
ejpam-4448	217	10	mixed	mixed	ADJ
ejpam-4448	217	11	graph	graph	NOUN
ejpam-4448	217	12	.	.	PUNCT
ejpam-4448	218	1	fix	fix	VERB
ejpam-4448	218	2	any	any	DET
ejpam-4448	218	3	u	u	NOUN
ejpam-4448	218	4	∈	∈	PROPN
ejpam-4448	218	5	v	v	NOUN
ejpam-4448	218	6	(	(	PUNCT
ejpam-4448	218	7	d	d	NOUN
ejpam-4448	218	8	)	)	PUNCT
ejpam-4448	218	9	and	and	CCONJ
ejpam-4448	218	10	consider	consider	VERB
ejpam-4448	218	11	some	some	DET
ejpam-4448	218	12	mixed	mixed	ADJ
ejpam-4448	218	13	walk	walk	NOUN
ejpam-4448	218	14	w	w	NOUN
ejpam-4448	218	15	in	in	ADP
ejpam-4448	218	16	d	d	PROPN
ejpam-4448	218	17	,	,	PUNCT
ejpam-4448	218	18	say	say	VERB
ejpam-4448	218	19	u	u	PROPN
ejpam-4448	218	20	=	=	PROPN
ejpam-4448	218	21	r1	r1	PROPN
ejpam-4448	218	22	,	,	PUNCT
ejpam-4448	218	23	.	.	PUNCT
ejpam-4448	218	24	.	.	PUNCT
ejpam-4448	219	1	.	.	PUNCT
ejpam-4448	220	1	,	,	PUNCT
ejpam-4448	220	2	rk	rk	NOUN
ejpam-4448	220	3	.	.	PROPN
ejpam-4448	220	4	denote	denote	VERB
ejpam-4448	220	5	the	the	DET
ejpam-4448	220	6	contained	contain	VERB
ejpam-4448	220	7	partial	partial	ADJ
ejpam-4448	220	8	walks	walk	NOUN
ejpam-4448	220	9	r1	r1	NOUN
ejpam-4448	220	10	,	,	PUNCT
ejpam-4448	220	11	.	.	PUNCT
ejpam-4448	220	12	.	.	PUNCT
ejpam-4448	221	1	.	.	PUNCT
ejpam-4448	222	1	,	,	PUNCT
ejpam-4448	222	2	rj	rj	PROPN
ejpam-4448	222	3	by	by	ADP
ejpam-4448	222	4	wj	wj	PROPN
ejpam-4448	222	5	(	(	PUNCT
ejpam-4448	222	6	for	for	ADP
ejpam-4448	222	7	j	j	PROPN
ejpam-4448	222	8	=	=	SYM
ejpam-4448	222	9	1	1	PROPN
ejpam-4448	222	10	,	,	PUNCT
ejpam-4448	222	11	.	.	PUNCT
ejpam-4448	222	12	.	.	PUNCT
ejpam-4448	222	13	.	.	PUNCT
ejpam-4448	223	1	,	,	PUNCT
ejpam-4448	223	2	k	k	NOUN
ejpam-4448	223	3	)	)	PUNCT
ejpam-4448	223	4	.	.	PUNCT
ejpam-4448	224	1	consequently	consequently	ADV
ejpam-4448	224	2	,	,	PUNCT
ejpam-4448	224	3	hα(w1	hα(w1	ADJ
ejpam-4448	224	4	)	)	PUNCT
ejpam-4448	224	5	=	=	SYM
ejpam-4448	224	6	1	1	NUM
ejpam-4448	224	7	(	(	PUNCT
ejpam-4448	224	8	8)	8)	NUM
ejpam-4448	224	9	and	and	CCONJ
ejpam-4448	224	10	hα(wj+1	hα(wj+1	NOUN
ejpam-4448	224	11	)	)	PUNCT
ejpam-4448	224	12	=	=	SYM
ejpam-4448	225	1			PROPN
ejpam-4448	225	2	hα(wj	hα(wj	PROPN
ejpam-4448	225	3	)	)	PUNCT
ejpam-4448	225	4	if	if	SCONJ
ejpam-4448	225	5	rj∼rj+1	rj∼rj+1	NOUN
ejpam-4448	225	6	αhα(wj	αhα(wj	ADV
ejpam-4448	225	7	)	)	PUNCT
ejpam-4448	225	8	if	if	SCONJ
ejpam-4448	225	9	rj→rj+1	rj→rj+1	NOUN
ejpam-4448	225	10	ᾱhα(wj	ᾱhα(wj	NOUN
ejpam-4448	225	11	)	)	PUNCT
ejpam-4448	225	12	if	if	SCONJ
ejpam-4448	225	13	rj←rj+1	rj←rj+1	NOUN
ejpam-4448	225	14	(	(	PUNCT
ejpam-4448	225	15	9	9	NUM
ejpam-4448	225	16	)	)	PUNCT
ejpam-4448	225	17	for	for	ADP
ejpam-4448	225	18	j	j	PROPN
ejpam-4448	225	19	=	=	SYM
ejpam-4448	225	20	1	1	PROPN
ejpam-4448	225	21	,	,	PUNCT
ejpam-4448	225	22	.	.	PUNCT
ejpam-4448	225	23	.	.	PUNCT
ejpam-4448	225	24	.	.	PUNCT
ejpam-4448	226	1	,	,	PUNCT
ejpam-4448	227	1	k	k	PROPN
ejpam-4448	227	2	−	−	PROPN
ejpam-4448	227	3	1	1	X
ejpam-4448	227	4	.	.	PUNCT
ejpam-4448	227	5	m.	m.	NOUN
ejpam-4448	227	6	abudayah	abudayah	PROPN
ejpam-4448	227	7	,	,	PUNCT
ejpam-4448	227	8	o.	o.	PROPN
ejpam-4448	227	9	alomari	alomari	PROPN
ejpam-4448	227	10	,	,	PUNCT
ejpam-4448	227	11	t.	t.	PROPN
ejpam-4448	227	12	sander	sander	PROPN
ejpam-4448	227	13	/	/	SYM
ejpam-4448	227	14	eur	eur	PROPN
ejpam-4448	227	15	.	.	PUNCT
ejpam-4448	228	1	j.	j.	PROPN
ejpam-4448	228	2	pure	pure	PROPN
ejpam-4448	228	3	appl	appl	PROPN
ejpam-4448	228	4	.	.	PROPN
ejpam-4448	228	5	math	math	PROPN
ejpam-4448	228	6	,	,	PUNCT
ejpam-4448	228	7	15	15	NUM
ejpam-4448	228	8	(	(	PUNCT
ejpam-4448	228	9	3	3	NUM
ejpam-4448	228	10	)	)	PUNCT
ejpam-4448	228	11	(	(	PUNCT
ejpam-4448	228	12	2022	2022	NUM
ejpam-4448	228	13	)	)	PUNCT
ejpam-4448	228	14	,	,	PUNCT
ejpam-4448	228	15	841	841	NUM
ejpam-4448	228	16	-	-	SYM
ejpam-4448	228	17	855	855	NUM
ejpam-4448	228	18	849	849	NUM
ejpam-4448	228	19	1	1	NUM
ejpam-4448	228	20	2	2	NUM
ejpam-4448	228	21	3	3	NUM
ejpam-4448	228	22	γ	γ	X
ejpam-4448	228	23	4	4	NUM
ejpam-4448	228	24	1	1	NUM
ejpam-4448	228	25	5	5	NUM
ejpam-4448	228	26	1	1	NUM
ejpam-4448	228	27	1	1	NUM
ejpam-4448	228	28	6	6	NUM
ejpam-4448	228	29	1	1	NUM
ejpam-4448	228	30	7	7	NUM
ejpam-4448	228	31	γ	γ	X
ejpam-4448	228	32	(	(	PUNCT
ejpam-4448	228	33	a	a	NOUN
ejpam-4448	228	34	)	)	PUNCT
ejpam-4448	228	35	walk	walk	NOUN
ejpam-4448	228	36	from	from	ADP
ejpam-4448	228	37	/	/	SYM
ejpam-4448	228	38	to	to	PART
ejpam-4448	228	39	vertex	vertex	VERB
ejpam-4448	228	40	no	no	NOUN
ejpam-4448	228	41	.	.	NOUN
ejpam-4448	228	42	5	5	NUM
ejpam-4448	228	43	1	1	NUM
ejpam-4448	228	44	γ	γ	SYM
ejpam-4448	228	45	2	2	NUM
ejpam-4448	228	46	1	1	NUM
ejpam-4448	228	47	3	3	NUM
ejpam-4448	228	48	1	1	NUM
ejpam-4448	228	49	γ	γ	SYM
ejpam-4448	228	50	4	4	NUM
ejpam-4448	228	51	γ2	γ2	NOUN
ejpam-4448	228	52	5	5	NUM
ejpam-4448	228	53	1	1	NUM
ejpam-4448	228	54	6	6	NUM
ejpam-4448	228	55	7	7	NUM
ejpam-4448	228	56	γ2	γ2	NOUN
ejpam-4448	228	57	(	(	PUNCT
ejpam-4448	228	58	b	b	NOUN
ejpam-4448	228	59	)	)	PUNCT
ejpam-4448	228	60	walk	walk	VERB
ejpam-4448	228	61	from	from	ADP
ejpam-4448	228	62	/	/	SYM
ejpam-4448	228	63	to	to	PART
ejpam-4448	228	64	vertex	vertex	VERB
ejpam-4448	229	1	no	no	NOUN
ejpam-4448	229	2	.	.	NOUN
ejpam-4448	229	3	3	3	NUM
ejpam-4448	229	4	1	1	NUM
ejpam-4448	229	5	γ2	γ2	NOUN
ejpam-4448	229	6	2	2	NUM
ejpam-4448	229	7	γ	γ	NOUN
ejpam-4448	229	8	3	3	NUM
ejpam-4448	229	9	γ	γ	X
ejpam-4448	229	10	γ2	γ2	PROPN
ejpam-4448	229	11	4	4	NUM
ejpam-4448	229	12	1	1	NUM
ejpam-4448	229	13	5	5	NUM
ejpam-4448	229	14	1	1	NUM
ejpam-4448	229	15	γ	γ	PROPN
ejpam-4448	229	16	6	6	NUM
ejpam-4448	229	17	7	7	NUM
ejpam-4448	229	18	(	(	PUNCT
ejpam-4448	229	19	c	c	NOUN
ejpam-4448	229	20	)	)	PUNCT
ejpam-4448	229	21	another	another	PRON
ejpam-4448	229	22	walk	walk	VERB
ejpam-4448	229	23	from	from	ADP
ejpam-4448	229	24	/	/	SYM
ejpam-4448	229	25	to	to	PART
ejpam-4448	229	26	vertex	vertex	VERB
ejpam-4448	229	27	no	no	NOUN
ejpam-4448	229	28	.	.	NOUN
ejpam-4448	229	29	5	5	NUM
ejpam-4448	229	30	figure	figure	NOUN
ejpam-4448	229	31	4	4	NUM
ejpam-4448	229	32	:	:	PUNCT
ejpam-4448	229	33	values	value	NOUN
ejpam-4448	229	34	of	of	ADP
ejpam-4448	229	35	hγ(wj	hγ(wj	PROPN
ejpam-4448	229	36	)	)	PUNCT
ejpam-4448	229	37	for	for	ADP
ejpam-4448	229	38	three	three	NUM
ejpam-4448	229	39	closed	closed	ADJ
ejpam-4448	229	40	mixed	mixed	ADJ
ejpam-4448	229	41	walks	walk	NOUN
ejpam-4448	229	42	w	w	PROPN
ejpam-4448	229	43	example	example	NOUN
ejpam-4448	229	44	4	4	NUM
ejpam-4448	229	45	.	.	PUNCT
ejpam-4448	229	46	figure	figure	VERB
ejpam-4448	229	47	4	4	NUM
ejpam-4448	229	48	shows	show	VERB
ejpam-4448	229	49	the	the	DET
ejpam-4448	229	50	values	value	NOUN
ejpam-4448	229	51	hγ(wj	hγ(wj	PROPN
ejpam-4448	229	52	)	)	PUNCT
ejpam-4448	229	53	–	–	PUNCT
ejpam-4448	229	54	each	each	DET
ejpam-4448	229	55	value	value	NOUN
ejpam-4448	229	56	written	write	VERB
ejpam-4448	229	57	near	near	ADP
ejpam-4448	229	58	the	the	DET
ejpam-4448	229	59	respective	respective	ADJ
ejpam-4448	229	60	j	j	PROPN
ejpam-4448	229	61	-	-	PUNCT
ejpam-4448	229	62	th	th	VERB
ejpam-4448	229	63	vertex	vertex	NOUN
ejpam-4448	229	64	along	along	ADP
ejpam-4448	229	65	the	the	DET
ejpam-4448	229	66	walk	walk	NOUN
ejpam-4448	229	67	–	–	PUNCT
ejpam-4448	229	68	for	for	ADP
ejpam-4448	229	69	three	three	NUM
ejpam-4448	229	70	different	different	ADJ
ejpam-4448	229	71	mixed	mixed	ADJ
ejpam-4448	229	72	walks	walk	NOUN
ejpam-4448	229	73	w	w	NOUN
ejpam-4448	229	74	in	in	ADP
ejpam-4448	229	75	a	a	DET
ejpam-4448	229	76	mixed	mixed	ADJ
ejpam-4448	229	77	graph	graph	NOUN
ejpam-4448	229	78	.	.	PUNCT
ejpam-4448	230	1	next	next	ADV
ejpam-4448	230	2	,	,	PUNCT
ejpam-4448	230	3	we	we	PRON
ejpam-4448	230	4	state	state	VERB
ejpam-4448	230	5	three	three	NUM
ejpam-4448	230	6	useful	useful	ADJ
ejpam-4448	230	7	basic	basic	ADJ
ejpam-4448	230	8	properties	property	NOUN
ejpam-4448	230	9	of	of	ADP
ejpam-4448	230	10	hγ	hγ	PRON
ejpam-4448	230	11	with	with	ADP
ejpam-4448	230	12	respect	respect	NOUN
ejpam-4448	230	13	to	to	ADP
ejpam-4448	230	14	mixed	mixed	ADJ
ejpam-4448	230	15	walks	walk	NOUN
ejpam-4448	230	16	(	(	PUNCT
ejpam-4448	230	17	some	some	PRON
ejpam-4448	230	18	of	of	ADP
ejpam-4448	230	19	which	which	PRON
ejpam-4448	230	20	already	already	ADV
ejpam-4448	230	21	implicit	implicit	ADJ
ejpam-4448	230	22	in	in	ADP
ejpam-4448	230	23	the	the	DET
ejpam-4448	230	24	previous	previous	ADJ
ejpam-4448	230	25	section	section	NOUN
ejpam-4448	230	26	)	)	PUNCT
ejpam-4448	230	27	.	.	PUNCT
ejpam-4448	231	1	proposition	proposition	NOUN
ejpam-4448	231	2	2	2	NUM
ejpam-4448	231	3	.	.	PUNCT
ejpam-4448	232	1	let	let	VERB
ejpam-4448	232	2	w	w	NOUN
ejpam-4448	232	3	be	be	AUX
ejpam-4448	232	4	a	a	DET
ejpam-4448	232	5	mixed	mixed	ADJ
ejpam-4448	232	6	walk	walk	NOUN
ejpam-4448	232	7	containing	contain	VERB
ejpam-4448	232	8	r	r	NOUN
ejpam-4448	232	9	forward	forward	ADJ
ejpam-4448	232	10	arcs	arc	NOUN
ejpam-4448	232	11	and	and	CCONJ
ejpam-4448	232	12	s	s	VERB
ejpam-4448	232	13	backward	backward	ADJ
ejpam-4448	232	14	arcs	arc	NOUN
ejpam-4448	232	15	.	.	PUNCT
ejpam-4448	233	1	then	then	ADV
ejpam-4448	233	2	hα(w	hα(w	VERB
ejpam-4448	233	3	)	)	PUNCT
ejpam-4448	234	1	=	=	SYM
ejpam-4448	234	2	αrᾱs	αrᾱs	PROPN
ejpam-4448	234	3	.	.	PUNCT
ejpam-4448	234	4	proposition	proposition	NOUN
ejpam-4448	234	5	3	3	X
ejpam-4448	234	6	.	.	PUNCT
ejpam-4448	235	1	let	let	VERB
ejpam-4448	235	2	w	w	PRON
ejpam-4448	235	3	′	′	PROPN
ejpam-4448	235	4	be	be	AUX
ejpam-4448	235	5	a	a	DET
ejpam-4448	235	6	mixed	mixed	ADJ
ejpam-4448	235	7	walk	walk	NOUN
ejpam-4448	235	8	and	and	CCONJ
ejpam-4448	235	9	w	w	ADP
ejpam-4448	235	10	′′	′′	NOUN
ejpam-4448	235	11	the	the	DET
ejpam-4448	235	12	corresponding	correspond	VERB
ejpam-4448	235	13	reverse	reverse	ADJ
ejpam-4448	235	14	walk	walk	NOUN
ejpam-4448	235	15	.	.	PUNCT
ejpam-4448	236	1	let	let	VERB
ejpam-4448	236	2	w	w	NOUN
ejpam-4448	236	3	be	be	AUX
ejpam-4448	236	4	the	the	DET
ejpam-4448	236	5	walk	walk	NOUN
ejpam-4448	236	6	resulting	result	VERB
ejpam-4448	236	7	from	from	ADP
ejpam-4448	236	8	concatenating	concatenate	VERB
ejpam-4448	236	9	w	w	PROPN
ejpam-4448	236	10	′	′	NOUN
ejpam-4448	236	11	and	and	CCONJ
ejpam-4448	236	12	w	w	NOUN
ejpam-4448	236	13	′′.	′′.	PROPN
ejpam-4448	236	14	then	then	ADV
ejpam-4448	236	15	hα(w	hα(w	VERB
ejpam-4448	236	16	)	)	PUNCT
ejpam-4448	236	17	=	=	SYM
ejpam-4448	237	1	1	1	X
ejpam-4448	237	2	.	.	X
ejpam-4448	237	3	proposition	proposition	NOUN
ejpam-4448	237	4	4	4	NUM
ejpam-4448	237	5	.	.	PUNCT
ejpam-4448	238	1	let	let	VERB
ejpam-4448	238	2	w	w	NOUN
ejpam-4448	238	3	′,w	′,w	NOUN
ejpam-4448	238	4	′′	′′	PROPN
ejpam-4448	238	5	be	be	AUX
ejpam-4448	238	6	two	two	NUM
ejpam-4448	238	7	mixed	mixed	ADJ
ejpam-4448	238	8	walks	walk	NOUN
ejpam-4448	238	9	such	such	ADJ
ejpam-4448	238	10	that	that	SCONJ
ejpam-4448	238	11	the	the	DET
ejpam-4448	238	12	final	final	ADJ
ejpam-4448	238	13	vertex	vertex	NOUN
ejpam-4448	238	14	of	of	ADP
ejpam-4448	238	15	w	w	PROPN
ejpam-4448	238	16	′	′	NUM
ejpam-4448	238	17	is	be	AUX
ejpam-4448	238	18	the	the	DET
ejpam-4448	238	19	start	start	ADJ
ejpam-4448	238	20	vertex	vertex	NOUN
ejpam-4448	238	21	of	of	ADP
ejpam-4448	238	22	w	w	PROPN
ejpam-4448	238	23	′′.	′′.	PROPN
ejpam-4448	238	24	let	let	VERB
ejpam-4448	238	25	w	w	NOUN
ejpam-4448	238	26	be	be	AUX
ejpam-4448	238	27	the	the	DET
ejpam-4448	238	28	walk	walk	NOUN
ejpam-4448	238	29	resulting	result	VERB
ejpam-4448	238	30	from	from	ADP
ejpam-4448	238	31	concatenating	concatenate	VERB
ejpam-4448	238	32	w	w	PROPN
ejpam-4448	238	33	′	′	NOUN
ejpam-4448	238	34	and	and	CCONJ
ejpam-4448	238	35	w	w	NOUN
ejpam-4448	238	36	′′.	′′.	PROPN
ejpam-4448	238	37	then	then	ADV
ejpam-4448	238	38	hα(w	hα(w	VERB
ejpam-4448	238	39	)	)	PUNCT
ejpam-4448	238	40	=	=	PUNCT
ejpam-4448	238	41	hα(w	hα(w	NOUN
ejpam-4448	238	42	′)hα(w	′)hα(w	PUNCT
ejpam-4448	238	43	′′	′′	PROPN
ejpam-4448	238	44	)	)	PUNCT
ejpam-4448	238	45	.	.	PUNCT
ejpam-4448	239	1	as	as	SCONJ
ejpam-4448	239	2	can	can	AUX
ejpam-4448	239	3	be	be	AUX
ejpam-4448	239	4	seen	see	VERB
ejpam-4448	239	5	from	from	ADP
ejpam-4448	239	6	figure	figure	NOUN
ejpam-4448	239	7	4	4	NUM
ejpam-4448	239	8	,	,	PUNCT
ejpam-4448	239	9	hα	hα	VERB
ejpam-4448	239	10	can	can	AUX
ejpam-4448	239	11	be	be	AUX
ejpam-4448	239	12	used	use	VERB
ejpam-4448	239	13	to	to	PART
ejpam-4448	239	14	assign	assign	VERB
ejpam-4448	239	15	(	(	PUNCT
ejpam-4448	239	16	possibly	possibly	ADV
ejpam-4448	239	17	multiple	multiple	ADJ
ejpam-4448	239	18	)	)	PUNCT
ejpam-4448	239	19	complex	complex	ADJ
ejpam-4448	239	20	numbers	number	NOUN
ejpam-4448	239	21	to	to	ADP
ejpam-4448	239	22	each	each	PRON
ejpam-4448	239	23	of	of	ADP
ejpam-4448	239	24	the	the	DET
ejpam-4448	239	25	vertices	vertex	NOUN
ejpam-4448	239	26	of	of	ADP
ejpam-4448	239	27	a	a	DET
ejpam-4448	239	28	mixed	mixed	ADJ
ejpam-4448	239	29	graph	graph	NOUN
ejpam-4448	239	30	.	.	PUNCT
ejpam-4448	240	1	in	in	ADP
ejpam-4448	240	2	particular	particular	ADJ
ejpam-4448	240	3	,	,	PUNCT
ejpam-4448	240	4	we	we	PRON
ejpam-4448	240	5	are	be	AUX
ejpam-4448	240	6	concerned	concerned	ADJ
ejpam-4448	240	7	about	about	ADP
ejpam-4448	240	8	the	the	DET
ejpam-4448	240	9	possible	possible	ADJ
ejpam-4448	240	10	values	value	NOUN
ejpam-4448	240	11	that	that	PRON
ejpam-4448	240	12	get	get	AUX
ejpam-4448	240	13	assigned	assign	VERB
ejpam-4448	240	14	to	to	ADP
ejpam-4448	240	15	the	the	DET
ejpam-4448	240	16	start	start	NOUN
ejpam-4448	240	17	/	/	SYM
ejpam-4448	240	18	end	end	NOUN
ejpam-4448	240	19	vertices	vertex	NOUN
ejpam-4448	240	20	of	of	ADP
ejpam-4448	240	21	closed	closed	ADJ
ejpam-4448	240	22	walks	walk	NOUN
ejpam-4448	240	23	:	:	PUNCT
ejpam-4448	240	24	definition	definition	NOUN
ejpam-4448	240	25	5	5	NUM
ejpam-4448	240	26	.	.	PUNCT
ejpam-4448	241	1	let	let	VERB
ejpam-4448	241	2	d	d	PRON
ejpam-4448	241	3	be	be	AUX
ejpam-4448	241	4	a	a	DET
ejpam-4448	241	5	mixed	mixed	ADJ
ejpam-4448	241	6	graph	graph	NOUN
ejpam-4448	241	7	.	.	PUNCT
ejpam-4448	242	1	the	the	DET
ejpam-4448	242	2	α	α	NOUN
ejpam-4448	242	3	-	-	PUNCT
ejpam-4448	242	4	store	store	NOUN
ejpam-4448	242	5	sα(u	sα(u	NOUN
ejpam-4448	242	6	)	)	PUNCT
ejpam-4448	242	7	of	of	ADP
ejpam-4448	242	8	u	u	PROPN
ejpam-4448	242	9	∈	∈	PROPN
ejpam-4448	242	10	v	v	ADP
ejpam-4448	242	11	(	(	PUNCT
ejpam-4448	242	12	d	d	NOUN
ejpam-4448	242	13	)	)	PUNCT
ejpam-4448	242	14	is	be	AUX
ejpam-4448	242	15	defined	define	VERB
ejpam-4448	242	16	as	as	ADP
ejpam-4448	242	17	sα(u	sα(u	NUM
ejpam-4448	242	18	)	)	PUNCT
ejpam-4448	243	1	=	=	PRON
ejpam-4448	243	2	{	{	PUNCT
ejpam-4448	243	3	hα(w	hα(w	PROPN
ejpam-4448	243	4	)	)	PUNCT
ejpam-4448	243	5	:	:	PUNCT
ejpam-4448	243	6	w	w	NOUN
ejpam-4448	243	7	is	be	AUX
ejpam-4448	243	8	a	a	DET
ejpam-4448	243	9	closed	closed	ADJ
ejpam-4448	243	10	walk	walk	NOUN
ejpam-4448	243	11	in	in	ADP
ejpam-4448	243	12	d	d	PROPN
ejpam-4448	243	13	from	from	ADP
ejpam-4448	243	14	/	/	SYM
ejpam-4448	243	15	to	to	PART
ejpam-4448	243	16	u	u	NOUN
ejpam-4448	243	17	}	}	PUNCT
ejpam-4448	243	18	.	.	PUNCT
ejpam-4448	244	1	let	let	VERB
ejpam-4448	244	2	sα(u	sα(u	PRON
ejpam-4448	244	3	)	)	PUNCT
ejpam-4448	244	4	=	=	SYM
ejpam-4448	244	5	|sα(u)|	|sα(u)|	PROPN
ejpam-4448	244	6	denote	denote	VERB
ejpam-4448	244	7	the	the	DET
ejpam-4448	244	8	associated	associated	ADJ
ejpam-4448	244	9	store	store	NOUN
ejpam-4448	244	10	size	size	NOUN
ejpam-4448	244	11	.	.	PUNCT
ejpam-4448	245	1	m.	m.	NOUN
ejpam-4448	245	2	abudayah	abudayah	PROPN
ejpam-4448	245	3	,	,	PUNCT
ejpam-4448	245	4	o.	o.	PROPN
ejpam-4448	245	5	alomari	alomari	PROPN
ejpam-4448	245	6	,	,	PUNCT
ejpam-4448	245	7	t.	t.	PROPN
ejpam-4448	245	8	sander	sander	PROPN
ejpam-4448	245	9	/	/	SYM
ejpam-4448	245	10	eur	eur	PROPN
ejpam-4448	245	11	.	.	PUNCT
ejpam-4448	246	1	j.	j.	PROPN
ejpam-4448	246	2	pure	pure	PROPN
ejpam-4448	246	3	appl	appl	PROPN
ejpam-4448	246	4	.	.	PROPN
ejpam-4448	246	5	math	math	PROPN
ejpam-4448	246	6	,	,	PUNCT
ejpam-4448	246	7	15	15	NUM
ejpam-4448	246	8	(	(	PUNCT
ejpam-4448	246	9	3	3	NUM
ejpam-4448	246	10	)	)	PUNCT
ejpam-4448	246	11	(	(	PUNCT
ejpam-4448	246	12	2022	2022	NUM
ejpam-4448	246	13	)	)	PUNCT
ejpam-4448	246	14	,	,	PUNCT
ejpam-4448	246	15	841	841	NUM
ejpam-4448	246	16	-	-	SYM
ejpam-4448	246	17	855	855	NUM
ejpam-4448	246	18	850	850	NUM
ejpam-4448	246	19	1	1	NUM
ejpam-4448	246	20	2	2	NUM
ejpam-4448	246	21	3	3	NUM
ejpam-4448	246	22	γ	γ	X
ejpam-4448	246	23	4	4	NUM
ejpam-4448	246	24	1	1	NUM
ejpam-4448	246	25	5	5	NUM
ejpam-4448	246	26	1	1	NUM
ejpam-4448	246	27	1	1	NUM
ejpam-4448	246	28	6	6	NUM
ejpam-4448	246	29	1	1	NUM
ejpam-4448	246	30	7	7	NUM
ejpam-4448	246	31	γ	γ	X
ejpam-4448	246	32	(	(	PUNCT
ejpam-4448	246	33	a	a	NOUN
ejpam-4448	246	34	)	)	PUNCT
ejpam-4448	246	35	walk	walk	NOUN
ejpam-4448	246	36	#	#	SYM
ejpam-4448	246	37	1	1	NUM
ejpam-4448	246	38	from	from	ADP
ejpam-4448	246	39	/	/	SYM
ejpam-4448	246	40	to	to	PART
ejpam-4448	246	41	vertex	vertex	VERB
ejpam-4448	246	42	no	no	NOUN
ejpam-4448	246	43	.	.	NOUN
ejpam-4448	246	44	5	5	NUM
ejpam-4448	246	45	1	1	NUM
ejpam-4448	246	46	γ2	γ2	NOUN
ejpam-4448	246	47	2	2	NUM
ejpam-4448	246	48	γ	γ	NOUN
ejpam-4448	246	49	3	3	NUM
ejpam-4448	246	50	γ	γ	X
ejpam-4448	246	51	γ	γ	X
ejpam-4448	246	52	4	4	NUM
ejpam-4448	246	53	1	1	NUM
ejpam-4448	246	54	1	1	NUM
ejpam-4448	246	55	5	5	NUM
ejpam-4448	246	56	1	1	NUM
ejpam-4448	246	57	1	1	NUM
ejpam-4448	246	58	6	6	NUM
ejpam-4448	246	59	1	1	NUM
ejpam-4448	246	60	7	7	NUM
ejpam-4448	246	61	γ	γ	X
ejpam-4448	246	62	(	(	PUNCT
ejpam-4448	246	63	b	b	NOUN
ejpam-4448	246	64	)	)	PUNCT
ejpam-4448	246	65	walk	walk	VERB
ejpam-4448	246	66	#	#	SYM
ejpam-4448	246	67	2	2	NUM
ejpam-4448	246	68	from	from	ADP
ejpam-4448	246	69	/	/	SYM
ejpam-4448	246	70	to	to	PART
ejpam-4448	246	71	vertex	vertex	VERB
ejpam-4448	247	1	no	no	NOUN
ejpam-4448	247	2	.	.	NOUN
ejpam-4448	247	3	5	5	NUM
ejpam-4448	247	4	figure	figure	NOUN
ejpam-4448	247	5	5	5	NUM
ejpam-4448	247	6	:	:	PUNCT
ejpam-4448	247	7	values	value	NOUN
ejpam-4448	247	8	of	of	ADP
ejpam-4448	247	9	hγ(wj	hγ(wj	PROPN
ejpam-4448	247	10	)	)	PUNCT
ejpam-4448	247	11	for	for	ADP
ejpam-4448	247	12	two	two	NUM
ejpam-4448	247	13	closed	closed	ADJ
ejpam-4448	247	14	walks	walk	NOUN
ejpam-4448	247	15	in	in	ADP
ejpam-4448	247	16	a	a	DET
ejpam-4448	247	17	γ	γ	NOUN
ejpam-4448	247	18	-	-	PUNCT
ejpam-4448	247	19	monograph	monograph	NOUN
ejpam-4448	247	20	trivially	trivially	ADV
ejpam-4448	247	21	,	,	PUNCT
ejpam-4448	247	22	1	1	NUM
ejpam-4448	247	23	∈	∈	NOUN
ejpam-4448	247	24	sα(u	sα(u	PUNCT
ejpam-4448	247	25	)	)	PUNCT
ejpam-4448	247	26	and	and	CCONJ
ejpam-4448	247	27	so	so	ADV
ejpam-4448	247	28	sα(u	sα(u	PUNCT
ejpam-4448	247	29	)	)	PUNCT
ejpam-4448	247	30	≥	≥	NOUN
ejpam-4448	247	31	1	1	NUM
ejpam-4448	247	32	.	.	PUNCT
ejpam-4448	248	1	as	as	ADP
ejpam-4448	248	2	the	the	DET
ejpam-4448	248	3	following	follow	VERB
ejpam-4448	248	4	lemma	lemma	PROPN
ejpam-4448	248	5	shows	show	NOUN
ejpam-4448	248	6	,	,	PUNCT
ejpam-4448	248	7	the	the	DET
ejpam-4448	248	8	store	store	NOUN
ejpam-4448	248	9	content	content	NOUN
ejpam-4448	248	10	is	be	AUX
ejpam-4448	248	11	independent	independent	ADJ
ejpam-4448	248	12	of	of	ADP
ejpam-4448	248	13	u	u	NOUN
ejpam-4448	248	14	,	,	PUNCT
ejpam-4448	248	15	hence	hence	ADV
ejpam-4448	248	16	we	we	PRON
ejpam-4448	248	17	may	may	AUX
ejpam-4448	248	18	speak	speak	VERB
ejpam-4448	248	19	of	of	ADP
ejpam-4448	248	20	‘	'	PUNCT
ejpam-4448	248	21	the	the	DET
ejpam-4448	248	22	’	'	PUNCT
ejpam-4448	248	23	α	α	NOUN
ejpam-4448	248	24	-	-	PUNCT
ejpam-4448	248	25	store	store	NOUN
ejpam-4448	248	26	of	of	ADP
ejpam-4448	248	27	a	a	DET
ejpam-4448	248	28	mixed	mixed	ADJ
ejpam-4448	248	29	graph	graph	NOUN
ejpam-4448	248	30	:	:	PUNCT
ejpam-4448	248	31	lemma	lemma	PROPN
ejpam-4448	248	32	1	1	X
ejpam-4448	248	33	.	.	PUNCT
ejpam-4448	249	1	let	let	VERB
ejpam-4448	249	2	u	u	NOUN
ejpam-4448	249	3	,	,	PUNCT
ejpam-4448	249	4	v	v	PROPN
ejpam-4448	249	5	∈	∈	PROPN
ejpam-4448	249	6	v	v	NOUN
ejpam-4448	249	7	(	(	PUNCT
ejpam-4448	249	8	d	d	NOUN
ejpam-4448	249	9	)	)	PUNCT
ejpam-4448	249	10	.	.	PUNCT
ejpam-4448	250	1	then	then	ADV
ejpam-4448	250	2	sα(u	sα(u	PUNCT
ejpam-4448	250	3	)	)	PUNCT
ejpam-4448	250	4	=	=	SYM
ejpam-4448	250	5	sα(v	sα(v	X
ejpam-4448	250	6	)	)	PUNCT
ejpam-4448	250	7	.	.	PUNCT
ejpam-4448	251	1	proof	proof	NOUN
ejpam-4448	251	2	.	.	PUNCT
ejpam-4448	252	1	let	let	VERB
ejpam-4448	252	2	w	w	X
ejpam-4448	252	3	′′	′′	PROPN
ejpam-4448	252	4	be	be	AUX
ejpam-4448	252	5	a	a	DET
ejpam-4448	252	6	closed	closed	ADJ
ejpam-4448	252	7	mixed	mixed	ADJ
ejpam-4448	252	8	walk	walk	NOUN
ejpam-4448	252	9	from	from	ADP
ejpam-4448	252	10	/	/	SYM
ejpam-4448	252	11	to	to	ADP
ejpam-4448	252	12	v.	v.	NOUN
ejpam-4448	252	13	clearly	clearly	ADV
ejpam-4448	252	14	,	,	PUNCT
ejpam-4448	252	15	hα(w	hα(w	VERB
ejpam-4448	252	16	′′	′′	PROPN
ejpam-4448	252	17	)	)	PUNCT
ejpam-4448	252	18	∈	∈	PROPN
ejpam-4448	252	19	sα(v	sα(v	NOUN
ejpam-4448	252	20	)	)	PUNCT
ejpam-4448	252	21	.	.	PUNCT
ejpam-4448	253	1	now	now	ADV
ejpam-4448	253	2	let	let	VERB
ejpam-4448	253	3	w	w	PRON
ejpam-4448	253	4	′	′	PROPN
ejpam-4448	253	5	be	be	AUX
ejpam-4448	253	6	a	a	DET
ejpam-4448	253	7	mixed	mixed	ADJ
ejpam-4448	253	8	walk	walk	NOUN
ejpam-4448	253	9	from	from	ADP
ejpam-4448	253	10	u	u	NOUN
ejpam-4448	253	11	to	to	ADP
ejpam-4448	253	12	v	v	NOUN
ejpam-4448	253	13	and	and	CCONJ
ejpam-4448	253	14	let	let	VERB
ejpam-4448	253	15	w	w	PART
ejpam-4448	253	16	′′′	′′′	VERB
ejpam-4448	253	17	its	its	PRON
ejpam-4448	253	18	reverse	reverse	ADJ
ejpam-4448	253	19	walk	walk	NOUN
ejpam-4448	253	20	.	.	PUNCT
ejpam-4448	254	1	concatenating	concatenate	VERB
ejpam-4448	254	2	w	w	PROPN
ejpam-4448	254	3	′	′	PROPN
ejpam-4448	254	4	,	,	PUNCT
ejpam-4448	254	5	w	w	PROPN
ejpam-4448	254	6	′′	′′	PROPN
ejpam-4448	254	7	,	,	PUNCT
ejpam-4448	254	8	w	w	PROPN
ejpam-4448	254	9	′′′	′′′	PROPN
ejpam-4448	254	10	one	one	NUM
ejpam-4448	254	11	obtains	obtain	VERB
ejpam-4448	254	12	a	a	DET
ejpam-4448	254	13	closed	closed	ADJ
ejpam-4448	254	14	mixed	mixed	ADJ
ejpam-4448	254	15	walk	walk	NOUN
ejpam-4448	254	16	w	w	NOUN
ejpam-4448	254	17	from	from	ADP
ejpam-4448	254	18	/	/	SYM
ejpam-4448	254	19	to	to	ADP
ejpam-4448	254	20	v.	v.	NOUN
ejpam-4448	254	21	using	use	VERB
ejpam-4448	254	22	proposition	proposition	NOUN
ejpam-4448	254	23	3	3	NUM
ejpam-4448	254	24	and	and	CCONJ
ejpam-4448	254	25	proposition	proposition	NOUN
ejpam-4448	254	26	4	4	NUM
ejpam-4448	254	27	we	we	PRON
ejpam-4448	254	28	get	get	VERB
ejpam-4448	254	29	hα(w	hα(w	NOUN
ejpam-4448	254	30	)	)	PUNCT
ejpam-4448	255	1	=	=	PUNCT
ejpam-4448	255	2	hα(w	hα(w	X
ejpam-4448	255	3	′)hα(w	′)hα(w	PROPN
ejpam-4448	255	4	′′)hα(w	′′)hα(w	X
ejpam-4448	255	5	′′′	′′′	NOUN
ejpam-4448	255	6	)	)	PUNCT
ejpam-4448	255	7	=	=	PRON
ejpam-4448	255	8	hα(w	hα(w	VERB
ejpam-4448	255	9	′′	′′	PROPN
ejpam-4448	255	10	)	)	PUNCT
ejpam-4448	255	11	,	,	PUNCT
ejpam-4448	255	12	so	so	SCONJ
ejpam-4448	255	13	that	that	PRON
ejpam-4448	255	14	hα(w	hα(w	VERB
ejpam-4448	255	15	′′	′′	PROPN
ejpam-4448	255	16	)	)	PUNCT
ejpam-4448	255	17	∈	∈	PROPN
ejpam-4448	255	18	sα(u	sα(u	PUNCT
ejpam-4448	255	19	)	)	PUNCT
ejpam-4448	255	20	.	.	PUNCT
ejpam-4448	256	1	repeating	repeat	VERB
ejpam-4448	256	2	the	the	DET
ejpam-4448	256	3	argument	argument	NOUN
ejpam-4448	256	4	with	with	ADP
ejpam-4448	256	5	the	the	DET
ejpam-4448	256	6	roles	role	NOUN
ejpam-4448	256	7	of	of	ADP
ejpam-4448	256	8	u	u	NOUN
ejpam-4448	256	9	and	and	CCONJ
ejpam-4448	256	10	v	v	ADP
ejpam-4448	256	11	swapped	swap	VERB
ejpam-4448	256	12	yields	yield	NOUN
ejpam-4448	256	13	sα(v	sα(v	NUM
ejpam-4448	256	14	)	)	PUNCT
ejpam-4448	256	15	=	=	SYM
ejpam-4448	256	16	sα(u	sα(u	X
ejpam-4448	256	17	)	)	PUNCT
ejpam-4448	256	18	.	.	PUNCT
ejpam-4448	257	1	theorem	theorem	ADJ
ejpam-4448	257	2	8	8	NUM
ejpam-4448	257	3	.	.	PUNCT
ejpam-4448	258	1	let	let	VERB
ejpam-4448	258	2	d	d	PRON
ejpam-4448	258	3	be	be	AUX
ejpam-4448	258	4	a	a	DET
ejpam-4448	258	5	connected	connected	ADJ
ejpam-4448	258	6	mixed	mixed	ADJ
ejpam-4448	258	7	graph	graph	NOUN
ejpam-4448	258	8	.	.	PUNCT
ejpam-4448	259	1	then	then	ADV
ejpam-4448	259	2	the	the	DET
ejpam-4448	259	3	following	follow	VERB
ejpam-4448	259	4	statements	statement	NOUN
ejpam-4448	259	5	are	be	AUX
ejpam-4448	259	6	equivalent	equivalent	ADJ
ejpam-4448	259	7	:	:	PUNCT
ejpam-4448	259	8	(	(	PUNCT
ejpam-4448	259	9	i	i	NOUN
ejpam-4448	259	10	)	)	PUNCT
ejpam-4448	259	11	sα(u	sα(u	PUNCT
ejpam-4448	259	12	)	)	PUNCT
ejpam-4448	259	13	=	=	SYM
ejpam-4448	259	14	1	1	NUM
ejpam-4448	259	15	for	for	ADP
ejpam-4448	259	16	at	at	ADV
ejpam-4448	259	17	least	least	ADV
ejpam-4448	259	18	one	one	NUM
ejpam-4448	259	19	u	u	NOUN
ejpam-4448	259	20	∈	∈	PROPN
ejpam-4448	259	21	v	v	NOUN
ejpam-4448	259	22	(	(	PUNCT
ejpam-4448	259	23	d	d	NOUN
ejpam-4448	259	24	)	)	PUNCT
ejpam-4448	259	25	.	.	PUNCT
ejpam-4448	260	1	(	(	PUNCT
ejpam-4448	260	2	ii	ii	NOUN
ejpam-4448	260	3	)	)	PUNCT
ejpam-4448	260	4	sα(u	sα(u	PUNCT
ejpam-4448	260	5	)	)	PUNCT
ejpam-4448	260	6	=	=	SYM
ejpam-4448	260	7	1	1	NUM
ejpam-4448	260	8	for	for	ADP
ejpam-4448	260	9	every	every	DET
ejpam-4448	260	10	u	u	PROPN
ejpam-4448	260	11	∈	∈	PROPN
ejpam-4448	260	12	v	v	NOUN
ejpam-4448	260	13	(	(	PUNCT
ejpam-4448	260	14	d	d	NOUN
ejpam-4448	260	15	)	)	PUNCT
ejpam-4448	260	16	.	.	PUNCT
ejpam-4448	261	1	(	(	PUNCT
ejpam-4448	261	2	iii	iii	X
ejpam-4448	261	3	)	)	PUNCT
ejpam-4448	261	4	hα(w	hα(w	NOUN
ejpam-4448	261	5	′	′	NUM
ejpam-4448	261	6	)	)	PUNCT
ejpam-4448	262	1	=	=	PRON
ejpam-4448	262	2	hα(w	hα(w	VERB
ejpam-4448	262	3	′′	′′	PROPN
ejpam-4448	262	4	)	)	PUNCT
ejpam-4448	262	5	for	for	ADP
ejpam-4448	262	6	every	every	DET
ejpam-4448	262	7	pair	pair	NOUN
ejpam-4448	262	8	w	w	ADP
ejpam-4448	262	9	′,w	′,w	NOUN
ejpam-4448	263	1	′′	′′	PROPN
ejpam-4448	263	2	of	of	ADP
ejpam-4448	263	3	mixed	mixed	ADJ
ejpam-4448	263	4	walks	walk	NOUN
ejpam-4448	263	5	sharing	share	VERB
ejpam-4448	263	6	the	the	DET
ejpam-4448	263	7	same	same	ADJ
ejpam-4448	263	8	start	start	NOUN
ejpam-4448	263	9	and	and	CCONJ
ejpam-4448	263	10	end	end	NOUN
ejpam-4448	263	11	vertices	vertex	NOUN
ejpam-4448	263	12	.	.	PUNCT
ejpam-4448	264	1	(	(	PUNCT
ejpam-4448	264	2	iv	iv	X
ejpam-4448	264	3	)	)	PUNCT
ejpam-4448	264	4	d	d	NOUN
ejpam-4448	264	5	is	be	AUX
ejpam-4448	264	6	an	an	DET
ejpam-4448	264	7	α	α	NOUN
ejpam-4448	264	8	-	-	PUNCT
ejpam-4448	264	9	monograph	monograph	NOUN
ejpam-4448	264	10	(	(	PUNCT
ejpam-4448	264	11	of	of	ADP
ejpam-4448	264	12	1st	1st	ADJ
ejpam-4448	264	13	kind	kind	NOUN
ejpam-4448	264	14	)	)	PUNCT
ejpam-4448	264	15	.	.	PUNCT
ejpam-4448	265	1	proof	proof	NOUN
ejpam-4448	265	2	.	.	PUNCT
ejpam-4448	266	1	observing	observe	VERB
ejpam-4448	266	2	definition	definition	NOUN
ejpam-4448	266	3	4	4	NUM
ejpam-4448	266	4	and	and	CCONJ
ejpam-4448	266	5	definition	definition	NOUN
ejpam-4448	266	6	5	5	NUM
ejpam-4448	266	7	,	,	PUNCT
ejpam-4448	266	8	this	this	PRON
ejpam-4448	266	9	follows	follow	VERB
ejpam-4448	266	10	from	from	ADP
ejpam-4448	266	11	proposition	proposition	NOUN
ejpam-4448	266	12	3	3	NUM
ejpam-4448	266	13	,	,	PUNCT
ejpam-4448	266	14	proposition	proposition	NOUN
ejpam-4448	266	15	4	4	NUM
ejpam-4448	266	16	and	and	CCONJ
ejpam-4448	266	17	lemma	lemma	PROPN
ejpam-4448	266	18	1	1	NUM
ejpam-4448	266	19	.	.	PUNCT
ejpam-4448	266	20	example	example	NOUN
ejpam-4448	266	21	5	5	NUM
ejpam-4448	266	22	.	.	PUNCT
ejpam-4448	267	1	the	the	DET
ejpam-4448	267	2	mixed	mixed	ADJ
ejpam-4448	267	3	graph	graph	NOUN
ejpam-4448	267	4	shown	show	VERB
ejpam-4448	267	5	in	in	ADP
ejpam-4448	267	6	figure	figure	NOUN
ejpam-4448	267	7	4	4	NUM
ejpam-4448	267	8	is	be	AUX
ejpam-4448	267	9	not	not	PART
ejpam-4448	267	10	a	a	DET
ejpam-4448	267	11	γ	γ	NOUN
ejpam-4448	267	12	-	-	PUNCT
ejpam-4448	267	13	monograph	monograph	NOUN
ejpam-4448	267	14	,	,	PUNCT
ejpam-4448	267	15	as	as	SCONJ
ejpam-4448	267	16	opposed	oppose	VERB
ejpam-4448	267	17	to	to	ADP
ejpam-4448	267	18	the	the	DET
ejpam-4448	267	19	slightly	slightly	ADV
ejpam-4448	267	20	different	different	ADJ
ejpam-4448	267	21	graph	graph	NOUN
ejpam-4448	267	22	depicted	depict	VERB
ejpam-4448	267	23	in	in	ADP
ejpam-4448	267	24	figure	figure	NOUN
ejpam-4448	267	25	5	5	NUM
ejpam-4448	267	26	.	.	PUNCT
ejpam-4448	267	27	notice	notice	VERB
ejpam-4448	267	28	how	how	SCONJ
ejpam-4448	267	29	,	,	PUNCT
ejpam-4448	267	30	in	in	ADP
ejpam-4448	267	31	view	view	NOUN
ejpam-4448	267	32	of	of	ADP
ejpam-4448	267	33	theorem	theorem	NOUN
ejpam-4448	267	34	8	8	NUM
ejpam-4448	267	35	,	,	PUNCT
ejpam-4448	267	36	every	every	DET
ejpam-4448	267	37	mixed	mixed	ADJ
ejpam-4448	267	38	walk	walk	NOUN
ejpam-4448	267	39	with	with	ADP
ejpam-4448	267	40	the	the	DET
ejpam-4448	267	41	same	same	ADJ
ejpam-4448	267	42	start	start	NOUN
ejpam-4448	267	43	/	/	SYM
ejpam-4448	267	44	end	end	NOUN
ejpam-4448	267	45	vertex	vertex	NOUN
ejpam-4448	267	46	creates	create	VERB
ejpam-4448	267	47	exactly	exactly	ADV
ejpam-4448	267	48	the	the	DET
ejpam-4448	267	49	same	same	ADJ
ejpam-4448	267	50	walk	walk	NOUN
ejpam-4448	267	51	values	value	NOUN
ejpam-4448	267	52	along	along	ADP
ejpam-4448	267	53	the	the	DET
ejpam-4448	267	54	way	way	NOUN
ejpam-4448	267	55	.	.	PUNCT
ejpam-4448	268	1	in	in	ADP
ejpam-4448	268	2	view	view	NOUN
ejpam-4448	268	3	of	of	ADP
ejpam-4448	268	4	theorem	theorem	NOUN
ejpam-4448	268	5	8	8	NUM
ejpam-4448	268	6	,	,	PUNCT
ejpam-4448	268	7	one	one	PRON
ejpam-4448	268	8	can	can	AUX
ejpam-4448	268	9	characterize	characterize	VERB
ejpam-4448	268	10	α	α	NOUN
ejpam-4448	268	11	-	-	NOUN
ejpam-4448	268	12	monographs	monograph	NOUN
ejpam-4448	268	13	by	by	ADP
ejpam-4448	268	14	the	the	DET
ejpam-4448	268	15	way	way	NOUN
ejpam-4448	268	16	their	their	PRON
ejpam-4448	268	17	vertices	vertex	NOUN
ejpam-4448	268	18	can	can	AUX
ejpam-4448	268	19	be	be	AUX
ejpam-4448	268	20	partitioned	partition	VERB
ejpam-4448	268	21	:	:	PUNCT
ejpam-4448	268	22	m.	m.	NOUN
ejpam-4448	268	23	abudayah	abudayah	PROPN
ejpam-4448	268	24	,	,	PUNCT
ejpam-4448	268	25	o.	o.	PROPN
ejpam-4448	268	26	alomari	alomari	PROPN
ejpam-4448	268	27	,	,	PUNCT
ejpam-4448	268	28	t.	t.	PROPN
ejpam-4448	268	29	sander	sander	PROPN
ejpam-4448	268	30	/	/	SYM
ejpam-4448	268	31	eur	eur	PROPN
ejpam-4448	268	32	.	.	PUNCT
ejpam-4448	269	1	j.	j.	PROPN
ejpam-4448	269	2	pure	pure	PROPN
ejpam-4448	269	3	appl	appl	PROPN
ejpam-4448	269	4	.	.	PROPN
ejpam-4448	269	5	math	math	PROPN
ejpam-4448	269	6	,	,	PUNCT
ejpam-4448	269	7	15	15	NUM
ejpam-4448	269	8	(	(	PUNCT
ejpam-4448	269	9	3	3	NUM
ejpam-4448	269	10	)	)	PUNCT
ejpam-4448	269	11	(	(	PUNCT
ejpam-4448	269	12	2022	2022	NUM
ejpam-4448	269	13	)	)	PUNCT
ejpam-4448	269	14	,	,	PUNCT
ejpam-4448	269	15	841	841	NUM
ejpam-4448	269	16	-	-	SYM
ejpam-4448	269	17	855	855	NUM
ejpam-4448	269	18	851	851	NUM
ejpam-4448	269	19	theorem	theorem	VERB
ejpam-4448	269	20	9	9	NUM
ejpam-4448	269	21	.	.	PUNCT
ejpam-4448	270	1	a	a	DET
ejpam-4448	270	2	connected	connect	VERB
ejpam-4448	270	3	mixed	mixed	ADJ
ejpam-4448	270	4	graph	graph	NOUN
ejpam-4448	270	5	d	d	NOUN
ejpam-4448	270	6	is	be	AUX
ejpam-4448	270	7	an	an	DET
ejpam-4448	270	8	α	α	NOUN
ejpam-4448	270	9	-	-	PUNCT
ejpam-4448	270	10	monograph	monograph	NOUN
ejpam-4448	270	11	(	(	PUNCT
ejpam-4448	270	12	of	of	ADP
ejpam-4448	270	13	1st	1st	ADJ
ejpam-4448	270	14	kind	kind	NOUN
ejpam-4448	270	15	)	)	PUNCT
ejpam-4448	270	16	if	if	SCONJ
ejpam-4448	270	17	and	and	CCONJ
ejpam-4448	270	18	only	only	ADV
ejpam-4448	270	19	if	if	SCONJ
ejpam-4448	270	20	v	v	INTJ
ejpam-4448	270	21	(	(	PUNCT
ejpam-4448	270	22	d	d	NOUN
ejpam-4448	270	23	)	)	PUNCT
ejpam-4448	270	24	can	can	AUX
ejpam-4448	270	25	be	be	AUX
ejpam-4448	270	26	partitioned	partition	VERB
ejpam-4448	270	27	into	into	ADP
ejpam-4448	270	28	sets	set	NOUN
ejpam-4448	270	29	vα0	vα0	NOUN
ejpam-4448	270	30	,	,	PUNCT
ejpam-4448	270	31	vα1	vα1	NOUN
ejpam-4448	270	32	,	,	PUNCT
ejpam-4448	270	33	vα2	vα2	NOUN
ejpam-4448	270	34	,	,	PUNCT
ejpam-4448	270	35	.	.	PUNCT
ejpam-4448	270	36	.	.	PUNCT
ejpam-4448	270	37	.	.	PUNCT
ejpam-4448	271	1	(	(	PUNCT
ejpam-4448	271	2	some	some	PRON
ejpam-4448	271	3	of	of	ADP
ejpam-4448	271	4	which	which	PRON
ejpam-4448	271	5	possibly	possibly	ADV
ejpam-4448	271	6	empty	empty	ADJ
ejpam-4448	271	7	)	)	PUNCT
ejpam-4448	271	8	such	such	ADJ
ejpam-4448	271	9	that	that	SCONJ
ejpam-4448	271	10	there	there	PRON
ejpam-4448	271	11	are	be	VERB
ejpam-4448	271	12	no	no	DET
ejpam-4448	271	13	digons	digon	NOUN
ejpam-4448	271	14	between	between	ADP
ejpam-4448	271	15	any	any	DET
ejpam-4448	271	16	two	two	NUM
ejpam-4448	271	17	sets	set	NOUN
ejpam-4448	271	18	vαj1	vαj1	NOUN
ejpam-4448	271	19	,	,	PUNCT
ejpam-4448	271	20	vαj2	vαj2	PROPN
ejpam-4448	271	21	with	with	ADP
ejpam-4448	271	22	j1	j1	PROPN
ejpam-4448	271	23	̸=	̸=	PROPN
ejpam-4448	271	24	j2	j2	NOUN
ejpam-4448	271	25	and	and	CCONJ
ejpam-4448	271	26	every	every	DET
ejpam-4448	271	27	arc	arc	NOUN
ejpam-4448	271	28	starting	start	VERB
ejpam-4448	271	29	in	in	ADP
ejpam-4448	271	30	a	a	DET
ejpam-4448	271	31	set	set	NOUN
ejpam-4448	271	32	vαj	vαj	ADJ
ejpam-4448	271	33	ends	end	NOUN
ejpam-4448	271	34	in	in	ADP
ejpam-4448	271	35	vαj−1	vαj−1	NOUN
ejpam-4448	271	36	.	.	PUNCT
ejpam-4448	272	1	proof	proof	NOUN
ejpam-4448	272	2	.	.	PUNCT
ejpam-4448	273	1	given	give	VERB
ejpam-4448	273	2	an	an	DET
ejpam-4448	273	3	α	α	NOUN
ejpam-4448	273	4	-	-	PUNCT
ejpam-4448	273	5	monograph	monograph	NOUN
ejpam-4448	273	6	d	d	NOUN
ejpam-4448	273	7	,	,	PUNCT
ejpam-4448	273	8	use	use	NOUN
ejpam-4448	273	9	theorem	theorem	ADJ
ejpam-4448	273	10	8	8	NUM
ejpam-4448	273	11	(	(	PUNCT
ejpam-4448	273	12	iii	iii	NOUN
ejpam-4448	273	13	)	)	PUNCT
ejpam-4448	273	14	as	as	SCONJ
ejpam-4448	273	15	follows	follow	VERB
ejpam-4448	273	16	.	.	PUNCT
ejpam-4448	274	1	fix	fix	VERB
ejpam-4448	274	2	any	any	DET
ejpam-4448	274	3	vertex	vertex	NOUN
ejpam-4448	274	4	u	u	NOUN
ejpam-4448	274	5	∈	∈	PROPN
ejpam-4448	274	6	v	v	NOUN
ejpam-4448	274	7	(	(	PUNCT
ejpam-4448	274	8	d	d	NOUN
ejpam-4448	274	9	)	)	PUNCT
ejpam-4448	274	10	and	and	CCONJ
ejpam-4448	274	11	assign	assign	VERB
ejpam-4448	274	12	each	each	DET
ejpam-4448	274	13	vertex	vertex	NOUN
ejpam-4448	274	14	v	v	ADP
ejpam-4448	274	15	∈	∈	PROPN
ejpam-4448	274	16	v	v	NOUN
ejpam-4448	274	17	(	(	PUNCT
ejpam-4448	274	18	d	d	NOUN
ejpam-4448	274	19	)	)	PUNCT
ejpam-4448	274	20	to	to	ADP
ejpam-4448	274	21	the	the	DET
ejpam-4448	274	22	set	set	NOUN
ejpam-4448	274	23	vαj	vαj	INTJ
ejpam-4448	274	24	,	,	PUNCT
ejpam-4448	274	25	where	where	SCONJ
ejpam-4448	274	26	αj	αj	NOUN
ejpam-4448	274	27	=	=	NOUN
ejpam-4448	274	28	hα(w	hα(w	NOUN
ejpam-4448	274	29	)	)	PUNCT
ejpam-4448	274	30	for	for	ADP
ejpam-4448	274	31	an	an	DET
ejpam-4448	274	32	arbitrary	arbitrary	ADJ
ejpam-4448	274	33	walk	walk	NOUN
ejpam-4448	274	34	from	from	ADP
ejpam-4448	274	35	u	u	PRON
ejpam-4448	274	36	to	to	ADP
ejpam-4448	274	37	v.	v.	ADP
ejpam-4448	274	38	conversely	conversely	ADV
ejpam-4448	274	39	,	,	PUNCT
ejpam-4448	274	40	consider	consider	VERB
ejpam-4448	274	41	a	a	DET
ejpam-4448	274	42	mixed	mixed	ADJ
ejpam-4448	274	43	graph	graph	NOUN
ejpam-4448	274	44	with	with	ADP
ejpam-4448	274	45	a	a	DET
ejpam-4448	274	46	vertex	vertex	NOUN
ejpam-4448	274	47	partition	partition	NOUN
ejpam-4448	274	48	as	as	SCONJ
ejpam-4448	274	49	supposed	suppose	VERB
ejpam-4448	274	50	.	.	PUNCT
ejpam-4448	275	1	we	we	PRON
ejpam-4448	275	2	may	may	AUX
ejpam-4448	275	3	assume	assume	VERB
ejpam-4448	275	4	v0	v0	PROPN
ejpam-4448	275	5	̸=	̸=	PROPN
ejpam-4448	275	6	∅.	∅.	ADP
ejpam-4448	275	7	fix	fix	VERB
ejpam-4448	275	8	any	any	DET
ejpam-4448	275	9	v	v	ADP
ejpam-4448	275	10	∈	∈	PROPN
ejpam-4448	275	11	v0	v0	NOUN
ejpam-4448	275	12	.	.	PUNCT
ejpam-4448	276	1	considering	consider	VERB
ejpam-4448	276	2	some	some	DET
ejpam-4448	276	3	vertex	vertex	NOUN
ejpam-4448	276	4	w	w	PROPN
ejpam-4448	276	5	∈	∈	PROPN
ejpam-4448	276	6	vαk	vαk	NOUN
ejpam-4448	276	7	and	and	CCONJ
ejpam-4448	276	8	an	an	DET
ejpam-4448	276	9	arbitrary	arbitrary	ADJ
ejpam-4448	276	10	mixed	mixed	ADJ
ejpam-4448	276	11	walk	walk	NOUN
ejpam-4448	276	12	w	w	NOUN
ejpam-4448	276	13	from	from	ADP
ejpam-4448	276	14	v	v	NUM
ejpam-4448	276	15	to	to	ADP
ejpam-4448	276	16	w	w	PROPN
ejpam-4448	276	17	,	,	PUNCT
ejpam-4448	276	18	we	we	PRON
ejpam-4448	276	19	see	see	VERB
ejpam-4448	276	20	that	that	SCONJ
ejpam-4448	276	21	the	the	DET
ejpam-4448	276	22	partition	partition	NOUN
ejpam-4448	276	23	structure	structure	NOUN
ejpam-4448	276	24	aligns	align	VERB
ejpam-4448	276	25	with	with	ADP
ejpam-4448	276	26	(	(	PUNCT
ejpam-4448	276	27	8)	8)	NUM
ejpam-4448	276	28	and	and	CCONJ
ejpam-4448	276	29	(	(	PUNCT
ejpam-4448	276	30	9	9	NUM
ejpam-4448	276	31	)	)	PUNCT
ejpam-4448	276	32	,	,	PUNCT
ejpam-4448	276	33	so	so	SCONJ
ejpam-4448	276	34	that	that	SCONJ
ejpam-4448	276	35	inductively	inductively	ADV
ejpam-4448	276	36	we	we	PRON
ejpam-4448	276	37	conclude	conclude	VERB
ejpam-4448	276	38	αk	αk	ADP
ejpam-4448	276	39	=	=	NOUN
ejpam-4448	276	40	hα(w	hα(w	NOUN
ejpam-4448	276	41	)	)	PUNCT
ejpam-4448	276	42	.	.	PUNCT
ejpam-4448	277	1	thus	thus	ADV
ejpam-4448	277	2	,	,	PUNCT
ejpam-4448	277	3	condition	condition	NOUN
ejpam-4448	277	4	(	(	PUNCT
ejpam-4448	277	5	iii	iii	NOUN
ejpam-4448	277	6	)	)	PUNCT
ejpam-4448	277	7	of	of	ADP
ejpam-4448	277	8	theorem	theorem	ADJ
ejpam-4448	277	9	8	8	NUM
ejpam-4448	277	10	is	be	AUX
ejpam-4448	277	11	satisfied	satisfied	ADJ
ejpam-4448	277	12	.	.	PUNCT
ejpam-4448	278	1	furthermore	furthermore	ADV
ejpam-4448	278	2	,	,	PUNCT
ejpam-4448	278	3	the	the	DET
ejpam-4448	278	4	store	store	NOUN
ejpam-4448	278	5	values	value	NOUN
ejpam-4448	278	6	of	of	ADP
ejpam-4448	278	7	an	an	DET
ejpam-4448	278	8	α	α	NOUN
ejpam-4448	278	9	-	-	PUNCT
ejpam-4448	278	10	monograph	monograph	NOUN
ejpam-4448	278	11	d	d	PROPN
ejpam-4448	278	12	permit	permit	VERB
ejpam-4448	278	13	us	we	PRON
ejpam-4448	278	14	to	to	PART
ejpam-4448	278	15	convert	convert	VERB
ejpam-4448	278	16	the	the	DET
ejpam-4448	278	17	eigenvectors	eigenvector	NOUN
ejpam-4448	278	18	of	of	ADP
ejpam-4448	278	19	γ(d	γ(d	PROPN
ejpam-4448	278	20	)	)	PUNCT
ejpam-4448	278	21	into	into	ADP
ejpam-4448	278	22	eigenvectors	eigenvector	NOUN
ejpam-4448	278	23	of	of	ADP
ejpam-4448	278	24	d	d	NOUN
ejpam-4448	278	25	:	:	PUNCT
ejpam-4448	278	26	theorem	theorem	NOUN
ejpam-4448	278	27	10	10	NUM
ejpam-4448	278	28	.	.	PUNCT
ejpam-4448	279	1	let	let	VERB
ejpam-4448	279	2	d	d	PRON
ejpam-4448	279	3	be	be	AUX
ejpam-4448	279	4	an	an	DET
ejpam-4448	279	5	α	α	NOUN
ejpam-4448	279	6	-	-	PUNCT
ejpam-4448	279	7	monograph	monograph	NOUN
ejpam-4448	279	8	and	and	CCONJ
ejpam-4448	279	9	x	x	PUNCT
ejpam-4448	279	10	=	=	PUNCT
ejpam-4448	280	1	[	[	X
ejpam-4448	280	2	xu]u∈v	xu]u∈v	PROPN
ejpam-4448	280	3	(	(	PUNCT
ejpam-4448	280	4	d	d	NOUN
ejpam-4448	280	5	)	)	PUNCT
ejpam-4448	280	6	an	an	DET
ejpam-4448	280	7	eigenvector	eigenvector	NOUN
ejpam-4448	280	8	of	of	ADP
ejpam-4448	280	9	γ(d	γ(d	PROPN
ejpam-4448	280	10	)	)	PUNCT
ejpam-4448	280	11	for	for	ADP
ejpam-4448	280	12	eigenvalue	eigenvalue	PROPN
ejpam-4448	280	13	λ	λ	PROPN
ejpam-4448	280	14	.	.	PUNCT
ejpam-4448	280	15	fixing	fix	VERB
ejpam-4448	280	16	a	a	DET
ejpam-4448	280	17	reference	reference	NOUN
ejpam-4448	280	18	vertex	vertex	NOUN
ejpam-4448	280	19	v	v	ADP
ejpam-4448	280	20	∈	∈	PROPN
ejpam-4448	280	21	v	v	NOUN
ejpam-4448	280	22	(	(	PUNCT
ejpam-4448	280	23	d	d	NOUN
ejpam-4448	280	24	)	)	PUNCT
ejpam-4448	280	25	,	,	PUNCT
ejpam-4448	280	26	define	define	VERB
ejpam-4448	280	27	the	the	DET
ejpam-4448	280	28	vector	vector	NOUN
ejpam-4448	280	29	y	y	PROPN
ejpam-4448	281	1	=	=	PUNCT
ejpam-4448	282	1	[	[	X
ejpam-4448	282	2	yr]r∈v	yr]r∈v	X
ejpam-4448	282	3	(	(	PUNCT
ejpam-4448	282	4	d	d	NOUN
ejpam-4448	282	5	)	)	PUNCT
ejpam-4448	282	6	=	=	NOUN
ejpam-4448	283	1	[	[	X
ejpam-4448	283	2	hα(v	hα(v	NOUN
ejpam-4448	283	3	⇝	⇝	NOUN
ejpam-4448	283	4	r)xr]r∈v	r)xr]r∈v	NOUN
ejpam-4448	283	5	(	(	PUNCT
ejpam-4448	283	6	d	d	NOUN
ejpam-4448	283	7	)	)	PUNCT
ejpam-4448	283	8	,	,	PUNCT
ejpam-4448	283	9	(	(	PUNCT
ejpam-4448	283	10	10	10	NUM
ejpam-4448	283	11	)	)	PUNCT
ejpam-4448	283	12	where	where	SCONJ
ejpam-4448	283	13	v	v	NOUN
ejpam-4448	283	14	⇝	⇝	NOUN
ejpam-4448	283	15	r	r	NOUN
ejpam-4448	283	16	is	be	AUX
ejpam-4448	283	17	an	an	DET
ejpam-4448	283	18	arbitrary	arbitrary	ADJ
ejpam-4448	283	19	mixed	mixed	ADJ
ejpam-4448	283	20	walk	walk	NOUN
ejpam-4448	283	21	from	from	ADP
ejpam-4448	283	22	v	v	NUM
ejpam-4448	283	23	to	to	ADP
ejpam-4448	283	24	r	r	NOUN
ejpam-4448	283	25	in	in	ADP
ejpam-4448	283	26	d	d	PROPN
ejpam-4448	283	27	(	(	PUNCT
ejpam-4448	283	28	cf	cf	NOUN
ejpam-4448	283	29	.	.	PUNCT
ejpam-4448	284	1	theorem	theorem	ADJ
ejpam-4448	284	2	8	8	NUM
ejpam-4448	284	3	(	(	PUNCT
ejpam-4448	284	4	iii	iii	NOUN
ejpam-4448	284	5	)	)	PUNCT
ejpam-4448	284	6	)	)	PUNCT
ejpam-4448	284	7	.	.	PUNCT
ejpam-4448	285	1	then	then	ADV
ejpam-4448	285	2	y	y	PROPN
ejpam-4448	285	3	is	be	AUX
ejpam-4448	285	4	an	an	DET
ejpam-4448	285	5	α	α	NOUN
ejpam-4448	285	6	-	-	NOUN
ejpam-4448	285	7	eigenvector	eigenvector	NOUN
ejpam-4448	285	8	of	of	ADP
ejpam-4448	285	9	d	d	PROPN
ejpam-4448	285	10	for	for	ADP
ejpam-4448	285	11	eigenvalue	eigenvalue	PROPN
ejpam-4448	285	12	λ	λ	PROPN
ejpam-4448	285	13	.	.	PUNCT
ejpam-4448	285	14	proof	proof	NOUN
ejpam-4448	285	15	.	.	PUNCT
ejpam-4448	286	1	clearly	clearly	ADV
ejpam-4448	286	2	,	,	PUNCT
ejpam-4448	286	3	for	for	ADP
ejpam-4448	286	4	every	every	DET
ejpam-4448	286	5	vertex	vertex	NOUN
ejpam-4448	286	6	u	u	NOUN
ejpam-4448	286	7	,	,	PUNCT
ejpam-4448	286	8	the	the	DET
ejpam-4448	286	9	vector	vector	NOUN
ejpam-4448	286	10	x	x	VERB
ejpam-4448	286	11	satisfies	satisfy	VERB
ejpam-4448	286	12	the	the	DET
ejpam-4448	286	13	summation	summation	NOUN
ejpam-4448	286	14	rule	rule	VERB
ejpam-4448	286	15	λxu	λxu	X
ejpam-4448	286	16	=	=	SYM
ejpam-4448	286	17	∑	∑	SYM
ejpam-4448	286	18	r∈nγ(d)(u	r∈nγ(d)(u	ADJ
ejpam-4448	286	19	)	)	PUNCT
ejpam-4448	286	20	xr	xr	PROPN
ejpam-4448	286	21	.	.	PUNCT
ejpam-4448	287	1	(	(	PUNCT
ejpam-4448	287	2	11	11	NUM
ejpam-4448	287	3	)	)	PUNCT
ejpam-4448	287	4	using	use	VERB
ejpam-4448	287	5	the	the	DET
ejpam-4448	287	6	recursion	recursion	NOUN
ejpam-4448	287	7	(	(	PUNCT
ejpam-4448	287	8	9	9	NUM
ejpam-4448	287	9	)	)	PUNCT
ejpam-4448	287	10	as	as	ADV
ejpam-4448	287	11	well	well	ADV
ejpam-4448	287	12	as	as	ADP
ejpam-4448	287	13	equations	equation	NOUN
ejpam-4448	287	14	(	(	PUNCT
ejpam-4448	287	15	10	10	NUM
ejpam-4448	287	16	)	)	PUNCT
ejpam-4448	287	17	and	and	CCONJ
ejpam-4448	287	18	(	(	PUNCT
ejpam-4448	287	19	11	11	NUM
ejpam-4448	287	20	)	)	PUNCT
ejpam-4448	287	21	,	,	PUNCT
ejpam-4448	287	22	we	we	PRON
ejpam-4448	287	23	deduce	deduce	VERB
ejpam-4448	287	24	:	:	PUNCT
ejpam-4448	287	25	λyu	λyu	X
ejpam-4448	287	26	=	=	PUNCT
ejpam-4448	288	1	λhα(v	λhα(v	PRON
ejpam-4448	288	2	⇝	⇝	NOUN
ejpam-4448	288	3	u)xu	u)xu	PROPN
ejpam-4448	288	4	(	(	PUNCT
ejpam-4448	288	5	12	12	NUM
ejpam-4448	288	6	)	)	PUNCT
ejpam-4448	288	7	=	=	SYM
ejpam-4448	288	8	∑	∑	SYM
ejpam-4448	288	9	r∈nγ(d)(u	r∈nγ(d)(u	ADJ
ejpam-4448	288	10	)	)	PUNCT
ejpam-4448	288	11	hα(v	hα(v	NOUN
ejpam-4448	288	12	⇝	⇝	ADP
ejpam-4448	288	13	u)xr	u)xr	PROPN
ejpam-4448	288	14	(	(	PUNCT
ejpam-4448	288	15	13	13	NUM
ejpam-4448	288	16	)	)	PUNCT
ejpam-4448	288	17	=	=	SYM
ejpam-4448	288	18	∑	∑	PUNCT
ejpam-4448	288	19	r∈nd(u	r∈nd(u	PROPN
ejpam-4448	288	20	)	)	PUNCT
ejpam-4448	288	21	hα(v	hα(v	PUNCT
ejpam-4448	288	22	⇝	⇝	NOUN
ejpam-4448	288	23	u)xr	u)xr	PROPN
ejpam-4448	288	24	+	+	CCONJ
ejpam-4448	288	25	∑	∑	PROPN
ejpam-4448	288	26	r∈n+	r∈n+	NOUN
ejpam-4448	288	27	d(u	d(u	PROPN
ejpam-4448	288	28	)	)	PUNCT
ejpam-4448	288	29	hα(v	hα(v	VERB
ejpam-4448	288	30	⇝	⇝	VERB
ejpam-4448	288	31	u)xr	u)xr	PROPN
ejpam-4448	288	32	+	+	NUM
ejpam-4448	288	33	∑	∑	PUNCT
ejpam-4448	288	34	r∈n−	r∈n−	PROPN
ejpam-4448	288	35	d	d	PROPN
ejpam-4448	288	36	(	(	PUNCT
ejpam-4448	288	37	u	u	NOUN
ejpam-4448	288	38	)	)	PUNCT
ejpam-4448	288	39	hα(v	hα(v	VERB
ejpam-4448	288	40	⇝	⇝	ADP
ejpam-4448	288	41	u)xr	u)xr	PROPN
ejpam-4448	288	42	(	(	PUNCT
ejpam-4448	288	43	14	14	NUM
ejpam-4448	288	44	)	)	PUNCT
ejpam-4448	288	45	=	=	SYM
ejpam-4448	288	46	∑	∑	PUNCT
ejpam-4448	288	47	r∈nd(u	r∈nd(u	PROPN
ejpam-4448	288	48	)	)	PUNCT
ejpam-4448	288	49	hα(v	hα(v	PUNCT
ejpam-4448	288	50	⇝	⇝	NOUN
ejpam-4448	288	51	r)xr	r)xr	PROPN
ejpam-4448	288	52	+	+	PROPN
ejpam-4448	288	53	α	α	PROPN
ejpam-4448	288	54	∑	∑	PROPN
ejpam-4448	288	55	r∈n+	r∈n+	NOUN
ejpam-4448	288	56	d(u	d(u	PROPN
ejpam-4448	288	57	)	)	PUNCT
ejpam-4448	288	58	hα(v	hα(v	PUNCT
ejpam-4448	288	59	⇝	⇝	NOUN
ejpam-4448	288	60	r)xr	r)xr	PROPN
ejpam-4448	288	61	+	+	PROPN
ejpam-4448	288	62	ᾱ	ᾱ	NOUN
ejpam-4448	288	63	∑	∑	ADP
ejpam-4448	288	64	r∈n−	r∈n−	PROPN
ejpam-4448	288	65	d	d	PROPN
ejpam-4448	288	66	(	(	PUNCT
ejpam-4448	288	67	u	u	NOUN
ejpam-4448	288	68	)	)	PUNCT
ejpam-4448	288	69	hα(v	hα(v	NOUN
ejpam-4448	288	70	⇝	⇝	PROPN
ejpam-4448	288	71	r)xr	r)xr	PROPN
ejpam-4448	288	72	(	(	PUNCT
ejpam-4448	288	73	15	15	NUM
ejpam-4448	288	74	)	)	PUNCT
ejpam-4448	288	75	=	=	SYM
ejpam-4448	288	76	∑	∑	PUNCT
ejpam-4448	288	77	r∈nd(u	r∈nd(u	PROPN
ejpam-4448	288	78	)	)	PUNCT
ejpam-4448	288	79	yr	yr	NOUN
ejpam-4448	289	1	+	+	CCONJ
ejpam-4448	289	2	α	α	PROPN
ejpam-4448	289	3	∑	∑	NOUN
ejpam-4448	289	4	r∈n+	r∈n+	NOUN
ejpam-4448	289	5	d(u	d(u	PROPN
ejpam-4448	289	6	)	)	PUNCT
ejpam-4448	289	7	yr	yr	NOUN
ejpam-4448	290	1	+	+	CCONJ
ejpam-4448	290	2	ᾱ	ᾱ	NOUN
ejpam-4448	290	3	∑	∑	ADP
ejpam-4448	290	4	r∈n−	r∈n−	PROPN
ejpam-4448	290	5	d	d	PROPN
ejpam-4448	290	6	(	(	PUNCT
ejpam-4448	290	7	u	u	NOUN
ejpam-4448	290	8	)	)	PUNCT
ejpam-4448	290	9	yr	yr	NOUN
ejpam-4448	290	10	.	.	PUNCT
ejpam-4448	291	1	(	(	PUNCT
ejpam-4448	291	2	16	16	NUM
ejpam-4448	291	3	)	)	PUNCT
ejpam-4448	291	4	comparing	compare	VERB
ejpam-4448	291	5	this	this	PRON
ejpam-4448	291	6	to	to	ADP
ejpam-4448	291	7	the	the	DET
ejpam-4448	291	8	mixed	mixed	ADJ
ejpam-4448	291	9	summation	summation	NOUN
ejpam-4448	291	10	rule	rule	NOUN
ejpam-4448	291	11	(	(	PUNCT
ejpam-4448	291	12	2	2	NUM
ejpam-4448	291	13	)	)	PUNCT
ejpam-4448	291	14	from	from	ADP
ejpam-4448	291	15	proposition	proposition	NOUN
ejpam-4448	291	16	1	1	NUM
ejpam-4448	291	17	,	,	PUNCT
ejpam-4448	291	18	we	we	PRON
ejpam-4448	291	19	see	see	VERB
ejpam-4448	291	20	that	that	SCONJ
ejpam-4448	291	21	y	y	PROPN
ejpam-4448	291	22	is	be	AUX
ejpam-4448	291	23	as	as	SCONJ
ejpam-4448	291	24	claimed	claim	VERB
ejpam-4448	291	25	.	.	PUNCT
ejpam-4448	292	1	regarding	regard	VERB
ejpam-4448	292	2	theorem	theorem	ADJ
ejpam-4448	292	3	10	10	NUM
ejpam-4448	292	4	,	,	PUNCT
ejpam-4448	292	5	note	note	VERB
ejpam-4448	292	6	that	that	SCONJ
ejpam-4448	292	7	the	the	DET
ejpam-4448	292	8	construction	construction	NOUN
ejpam-4448	292	9	(	(	PUNCT
ejpam-4448	292	10	10	10	NUM
ejpam-4448	292	11	)	)	PUNCT
ejpam-4448	292	12	retains	retain	VERB
ejpam-4448	292	13	linear	linear	ADJ
ejpam-4448	292	14	independence	independence	NOUN
ejpam-4448	292	15	,	,	PUNCT
ejpam-4448	292	16	i.e.	i.e.	X
ejpam-4448	292	17	,	,	PUNCT
ejpam-4448	292	18	every	every	DET
ejpam-4448	292	19	basis	basis	NOUN
ejpam-4448	292	20	of	of	ADP
ejpam-4448	292	21	eigenvectors	eigenvector	NOUN
ejpam-4448	292	22	of	of	ADP
ejpam-4448	292	23	γ(d	γ(d	NOUN
ejpam-4448	292	24	)	)	PUNCT
ejpam-4448	292	25	can	can	AUX
ejpam-4448	292	26	be	be	AUX
ejpam-4448	292	27	converted	convert	VERB
ejpam-4448	292	28	into	into	ADP
ejpam-4448	292	29	a	a	DET
ejpam-4448	292	30	basis	basis	NOUN
ejpam-4448	292	31	of	of	ADP
ejpam-4448	292	32	α	α	NOUN
ejpam-4448	292	33	-	-	PUNCT
ejpam-4448	292	34	eigenvectors	eigenvector	NOUN
ejpam-4448	292	35	of	of	ADP
ejpam-4448	292	36	d	d	PROPN
ejpam-4448	292	37	(	(	PUNCT
ejpam-4448	292	38	using	use	VERB
ejpam-4448	292	39	the	the	DET
ejpam-4448	292	40	same	same	ADJ
ejpam-4448	292	41	reference	reference	NOUN
ejpam-4448	292	42	vertex	vertex	NOUN
ejpam-4448	292	43	v	v	NOUN
ejpam-4448	292	44	throughout	throughout	NOUN
ejpam-4448	292	45	)	)	PUNCT
ejpam-4448	292	46	.	.	PUNCT
ejpam-4448	293	1	m.	m.	PROPN
ejpam-4448	293	2	abudayah	abudayah	PROPN
ejpam-4448	293	3	,	,	PUNCT
ejpam-4448	293	4	o.	o.	PROPN
ejpam-4448	293	5	alomari	alomari	PROPN
ejpam-4448	293	6	,	,	PUNCT
ejpam-4448	293	7	t.	t.	PROPN
ejpam-4448	293	8	sander	sander	PROPN
ejpam-4448	293	9	/	/	SYM
ejpam-4448	293	10	eur	eur	PROPN
ejpam-4448	293	11	.	.	PUNCT
ejpam-4448	294	1	j.	j.	PROPN
ejpam-4448	294	2	pure	pure	PROPN
ejpam-4448	294	3	appl	appl	PROPN
ejpam-4448	294	4	.	.	PROPN
ejpam-4448	294	5	math	math	PROPN
ejpam-4448	294	6	,	,	PUNCT
ejpam-4448	294	7	15	15	NUM
ejpam-4448	294	8	(	(	PUNCT
ejpam-4448	294	9	3	3	NUM
ejpam-4448	294	10	)	)	PUNCT
ejpam-4448	294	11	(	(	PUNCT
ejpam-4448	294	12	2022	2022	NUM
ejpam-4448	294	13	)	)	PUNCT
ejpam-4448	294	14	,	,	PUNCT
ejpam-4448	294	15	841	841	NUM
ejpam-4448	294	16	-	-	SYM
ejpam-4448	294	17	855	855	NUM
ejpam-4448	294	18	852	852	NUM
ejpam-4448	294	19	1	1	NUM
ejpam-4448	294	20	−i	−i	ADJ
ejpam-4448	294	21	2	2	NUM
ejpam-4448	294	22	−1	−1	NOUN
ejpam-4448	294	23	31	31	NUM
ejpam-4448	294	24	1	1	NUM
ejpam-4448	294	25	4	4	NUM
ejpam-4448	294	26	i	i	PRON
ejpam-4448	294	27	figure	figure	VERB
ejpam-4448	294	28	6	6	NUM
ejpam-4448	294	29	:	:	PUNCT
ejpam-4448	294	30	an	an	DET
ejpam-4448	294	31	i	i	NOUN
ejpam-4448	294	32	-	-	PUNCT
ejpam-4448	294	33	monograph	monograph	NOUN
ejpam-4448	294	34	of	of	ADP
ejpam-4448	294	35	2nd	2nd	ADJ
ejpam-4448	294	36	kind	kind	ADJ
ejpam-4448	294	37	example	example	NOUN
ejpam-4448	295	1	6	6	NUM
ejpam-4448	295	2	.	.	PUNCT
ejpam-4448	296	1	figure	figure	NOUN
ejpam-4448	296	2	5b	5b	PROPN
ejpam-4448	296	3	depicts	depict	VERB
ejpam-4448	296	4	an	an	DET
ejpam-4448	296	5	α	α	NOUN
ejpam-4448	296	6	-	-	PUNCT
ejpam-4448	296	7	monograph	monograph	ADJ
ejpam-4448	296	8	d.	d.	NOUN
ejpam-4448	296	9	with	with	ADP
ejpam-4448	296	10	respect	respect	NOUN
ejpam-4448	296	11	to	to	ADP
ejpam-4448	296	12	the	the	DET
ejpam-4448	296	13	indicated	indicate	VERB
ejpam-4448	296	14	vertex	vertex	NOUN
ejpam-4448	296	15	order	order	NOUN
ejpam-4448	296	16	,	,	PUNCT
ejpam-4448	296	17	(	(	PUNCT
ejpam-4448	296	18	−1,−1	−1,−1	PROPN
ejpam-4448	296	19	,	,	PUNCT
ejpam-4448	296	20	1	1	NUM
ejpam-4448	296	21	,	,	PUNCT
ejpam-4448	296	22	1	1	NUM
ejpam-4448	296	23	,	,	PUNCT
ejpam-4448	296	24	0,−1	0,−1	PRON
ejpam-4448	296	25	,	,	PUNCT
ejpam-4448	296	26	1)t	1)t	PROPN
ejpam-4448	296	27	is	be	AUX
ejpam-4448	296	28	an	an	DET
ejpam-4448	296	29	eigenvector	eigenvector	NOUN
ejpam-4448	296	30	of	of	ADP
ejpam-4448	296	31	γ(d	γ(d	PROPN
ejpam-4448	296	32	)	)	PUNCT
ejpam-4448	296	33	for	for	ADP
ejpam-4448	296	34	eigenvalue	eigenvalue	PROPN
ejpam-4448	296	35	0	0	PUNCT
ejpam-4448	296	36	.	.	PUNCT
ejpam-4448	297	1	using	use	VERB
ejpam-4448	297	2	equation	equation	NOUN
ejpam-4448	297	3	(	(	PUNCT
ejpam-4448	297	4	10	10	NUM
ejpam-4448	297	5	)	)	PUNCT
ejpam-4448	297	6	and	and	CCONJ
ejpam-4448	297	7	the	the	DET
ejpam-4448	297	8	store	store	NOUN
ejpam-4448	297	9	values	value	NOUN
ejpam-4448	297	10	given	give	VERB
ejpam-4448	297	11	in	in	ADP
ejpam-4448	297	12	the	the	DET
ejpam-4448	297	13	figure	figure	NOUN
ejpam-4448	297	14	,	,	PUNCT
ejpam-4448	297	15	one	one	NOUN
ejpam-4448	297	16	obtains	obtain	VERB
ejpam-4448	297	17	the	the	DET
ejpam-4448	297	18	γ	γ	NOUN
ejpam-4448	297	19	-	-	PUNCT
ejpam-4448	297	20	eigenvector	eigenvector	NOUN
ejpam-4448	297	21	(	(	PUNCT
ejpam-4448	297	22	−γ,−γ2	−γ,−γ2	PROPN
ejpam-4448	297	23	,	,	PUNCT
ejpam-4448	297	24	γ2	γ2	ADJ
ejpam-4448	297	25	,	,	PUNCT
ejpam-4448	297	26	1	1	NUM
ejpam-4448	297	27	,	,	PUNCT
ejpam-4448	297	28	0,−1	0,−1	PRON
ejpam-4448	297	29	,	,	PUNCT
ejpam-4448	297	30	γ2)t	γ2)t	ADJ
ejpam-4448	297	31	for	for	ADP
ejpam-4448	297	32	eigenvalue	eigenvalue	PROPN
ejpam-4448	297	33	0	0	NUM
ejpam-4448	297	34	of	of	ADP
ejpam-4448	297	35	d.	d.	PROPN
ejpam-4448	297	36	in	in	ADP
ejpam-4448	297	37	preparation	preparation	NOUN
ejpam-4448	297	38	for	for	ADP
ejpam-4448	297	39	the	the	DET
ejpam-4448	297	40	following	follow	VERB
ejpam-4448	297	41	section	section	NOUN
ejpam-4448	297	42	we	we	PRON
ejpam-4448	297	43	define	define	VERB
ejpam-4448	297	44	a	a	DET
ejpam-4448	297	45	variant	variant	NOUN
ejpam-4448	297	46	of	of	ADP
ejpam-4448	297	47	the	the	DET
ejpam-4448	297	48	function	function	NOUN
ejpam-4448	297	49	hα	hα	VERB
ejpam-4448	297	50	.	.	PUNCT
ejpam-4448	298	1	as	as	ADP
ejpam-4448	298	2	before	before	ADV
ejpam-4448	298	3	,	,	PUNCT
ejpam-4448	298	4	let	let	VERB
ejpam-4448	298	5	d	d	PRON
ejpam-4448	298	6	be	be	AUX
ejpam-4448	298	7	a	a	DET
ejpam-4448	298	8	connected	connected	ADJ
ejpam-4448	298	9	mixed	mixed	ADJ
ejpam-4448	298	10	graph	graph	NOUN
ejpam-4448	298	11	.	.	PUNCT
ejpam-4448	299	1	fix	fix	VERB
ejpam-4448	299	2	u	u	NOUN
ejpam-4448	299	3	∈	∈	PROPN
ejpam-4448	299	4	v	v	NOUN
ejpam-4448	299	5	(	(	PUNCT
ejpam-4448	299	6	d	d	NOUN
ejpam-4448	299	7	)	)	PUNCT
ejpam-4448	299	8	and	and	CCONJ
ejpam-4448	299	9	let	let	VERB
ejpam-4448	299	10	w	w	NOUN
ejpam-4448	299	11	be	be	AUX
ejpam-4448	299	12	a	a	DET
ejpam-4448	299	13	mixed	mixed	ADJ
ejpam-4448	299	14	walk	walk	NOUN
ejpam-4448	299	15	u	u	NOUN
ejpam-4448	299	16	=	=	PROPN
ejpam-4448	299	17	r1	r1	PROPN
ejpam-4448	299	18	,	,	PUNCT
ejpam-4448	299	19	.	.	PUNCT
ejpam-4448	299	20	.	.	PUNCT
ejpam-4448	300	1	.	.	PUNCT
ejpam-4448	301	1	,	,	PUNCT
ejpam-4448	301	2	rk	rk	NOUN
ejpam-4448	301	3	in	in	ADP
ejpam-4448	301	4	d.	d.	PROPN
ejpam-4448	301	5	slightly	slightly	ADV
ejpam-4448	301	6	changing	change	VERB
ejpam-4448	301	7	equations	equation	NOUN
ejpam-4448	301	8	(	(	PUNCT
ejpam-4448	301	9	8)	8)	NUM
ejpam-4448	301	10	and	and	CCONJ
ejpam-4448	301	11	(	(	PUNCT
ejpam-4448	301	12	9	9	NUM
ejpam-4448	301	13	)	)	PUNCT
ejpam-4448	301	14	,	,	PUNCT
ejpam-4448	301	15	we	we	PRON
ejpam-4448	301	16	define	define	VERB
ejpam-4448	301	17	gα(wj	gα(wj	PROPN
ejpam-4448	301	18	)	)	PUNCT
ejpam-4448	301	19	=	=	PUNCT
ejpam-4448	301	20	(	(	PUNCT
ejpam-4448	301	21	−1)j+1hα(wj	−1)j+1hα(wj	NOUN
ejpam-4448	301	22	)	)	PUNCT
ejpam-4448	301	23	,	,	PUNCT
ejpam-4448	301	24	(	(	PUNCT
ejpam-4448	301	25	17	17	NUM
ejpam-4448	301	26	)	)	PUNCT
ejpam-4448	301	27	for	for	ADP
ejpam-4448	301	28	j	j	PROPN
ejpam-4448	301	29	=	=	SYM
ejpam-4448	301	30	1	1	PROPN
ejpam-4448	301	31	,	,	PUNCT
ejpam-4448	301	32	.	.	PUNCT
ejpam-4448	301	33	.	.	PUNCT
ejpam-4448	301	34	.	.	PUNCT
ejpam-4448	302	1	,	,	PUNCT
ejpam-4448	302	2	k.	k.	PROPN
ejpam-4448	303	1	the	the	DET
ejpam-4448	303	2	properties	property	NOUN
ejpam-4448	303	3	mentioned	mention	VERB
ejpam-4448	303	4	in	in	ADP
ejpam-4448	303	5	propositions	proposition	NOUN
ejpam-4448	303	6	2	2	NUM
ejpam-4448	303	7	to	to	PART
ejpam-4448	303	8	4	4	NUM
ejpam-4448	303	9	also	also	ADV
ejpam-4448	303	10	hold	hold	VERB
ejpam-4448	303	11	for	for	ADP
ejpam-4448	303	12	gα(w	gα(w	NOUN
ejpam-4448	303	13	)	)	PUNCT
ejpam-4448	303	14	,	,	PUNCT
ejpam-4448	303	15	thus	thus	ADV
ejpam-4448	303	16	justifying	justify	VERB
ejpam-4448	303	17	an	an	DET
ejpam-4448	303	18	alternative	alternative	ADJ
ejpam-4448	303	19	notion	notion	NOUN
ejpam-4448	303	20	of	of	ADP
ejpam-4448	303	21	α	α	NOUN
ejpam-4448	303	22	-	-	NOUN
ejpam-4448	303	23	store	store	NOUN
ejpam-4448	303	24	.	.	PUNCT
ejpam-4448	304	1	using	use	VERB
ejpam-4448	304	2	this	this	DET
ejpam-4448	304	3	notion	notion	NOUN
ejpam-4448	304	4	and	and	CCONJ
ejpam-4448	304	5	the	the	DET
ejpam-4448	304	6	following	follow	VERB
ejpam-4448	304	7	definition	definition	NOUN
ejpam-4448	304	8	instead	instead	ADV
ejpam-4448	304	9	of	of	ADP
ejpam-4448	304	10	definition	definition	NOUN
ejpam-4448	304	11	4	4	NUM
ejpam-4448	304	12	,	,	PUNCT
ejpam-4448	304	13	one	one	PRON
ejpam-4448	304	14	can	can	AUX
ejpam-4448	304	15	check	check	VERB
ejpam-4448	304	16	that	that	SCONJ
ejpam-4448	304	17	lemma	lemma	PROPN
ejpam-4448	304	18	1	1	NUM
ejpam-4448	304	19	and	and	CCONJ
ejpam-4448	304	20	theorem	theorem	VERB
ejpam-4448	304	21	8	8	NUM
ejpam-4448	304	22	remain	remain	VERB
ejpam-4448	304	23	valid	valid	ADJ
ejpam-4448	304	24	for	for	ADP
ejpam-4448	304	25	gα(w	gα(w	NOUN
ejpam-4448	304	26	)	)	PUNCT
ejpam-4448	304	27	as	as	ADV
ejpam-4448	304	28	well	well	ADV
ejpam-4448	304	29	.	.	PUNCT
ejpam-4448	305	1	definition	definition	NOUN
ejpam-4448	305	2	6	6	NUM
ejpam-4448	305	3	.	.	PUNCT
ejpam-4448	306	1	a	a	DET
ejpam-4448	306	2	mixed	mixed	ADJ
ejpam-4448	306	3	graph	graph	NOUN
ejpam-4448	306	4	is	be	AUX
ejpam-4448	306	5	an	an	DET
ejpam-4448	306	6	α	α	NOUN
ejpam-4448	306	7	-	-	PUNCT
ejpam-4448	306	8	monograph	monograph	NOUN
ejpam-4448	306	9	(	(	PUNCT
ejpam-4448	306	10	of	of	ADP
ejpam-4448	306	11	2nd	2nd	ADJ
ejpam-4448	306	12	kind	kind	NOUN
ejpam-4448	306	13	)	)	PUNCT
ejpam-4448	306	14	if	if	SCONJ
ejpam-4448	306	15	gα(c⃗	gα(c⃗	PROPN
ejpam-4448	306	16	)	)	PUNCT
ejpam-4448	307	1	=	=	PUNCT
ejpam-4448	307	2	1	1	NUM
ejpam-4448	307	3	for	for	ADP
ejpam-4448	307	4	all	all	DET
ejpam-4448	307	5	its	its	PRON
ejpam-4448	307	6	cycles	cycle	NOUN
ejpam-4448	307	7	c.	c.	NOUN
ejpam-4448	307	8	as	as	ADP
ejpam-4448	307	9	a	a	DET
ejpam-4448	307	10	result	result	NOUN
ejpam-4448	307	11	,	,	PUNCT
ejpam-4448	307	12	we	we	PRON
ejpam-4448	307	13	can	can	AUX
ejpam-4448	307	14	derive	derive	VERB
ejpam-4448	307	15	results	result	NOUN
ejpam-4448	307	16	analogous	analogous	ADJ
ejpam-4448	307	17	to	to	ADP
ejpam-4448	307	18	(	(	PUNCT
ejpam-4448	307	19	but	but	CCONJ
ejpam-4448	307	20	slightly	slightly	ADV
ejpam-4448	307	21	different	different	ADJ
ejpam-4448	307	22	from	from	ADP
ejpam-4448	307	23	)	)	PUNCT
ejpam-4448	307	24	theorem	theorem	ADJ
ejpam-4448	307	25	9	9	NUM
ejpam-4448	307	26	and	and	CCONJ
ejpam-4448	307	27	theorem	theorem	VERB
ejpam-4448	307	28	4	4	NUM
ejpam-4448	307	29	:	:	PUNCT
ejpam-4448	307	30	theorem	theorem	NOUN
ejpam-4448	307	31	11	11	NUM
ejpam-4448	307	32	.	.	PUNCT
ejpam-4448	308	1	a	a	DET
ejpam-4448	308	2	connected	connect	VERB
ejpam-4448	308	3	mixed	mixed	ADJ
ejpam-4448	308	4	graph	graph	NOUN
ejpam-4448	308	5	d	d	NOUN
ejpam-4448	308	6	is	be	AUX
ejpam-4448	308	7	an	an	DET
ejpam-4448	308	8	α	α	NOUN
ejpam-4448	308	9	-	-	PUNCT
ejpam-4448	308	10	monograph	monograph	NOUN
ejpam-4448	308	11	(	(	PUNCT
ejpam-4448	308	12	of	of	ADP
ejpam-4448	308	13	2nd	2nd	ADJ
ejpam-4448	308	14	kind	kind	NOUN
ejpam-4448	308	15	)	)	PUNCT
ejpam-4448	308	16	if	if	SCONJ
ejpam-4448	308	17	and	and	CCONJ
ejpam-4448	308	18	only	only	ADV
ejpam-4448	308	19	if	if	SCONJ
ejpam-4448	308	20	v	v	INTJ
ejpam-4448	308	21	(	(	PUNCT
ejpam-4448	308	22	d	d	NOUN
ejpam-4448	308	23	)	)	PUNCT
ejpam-4448	308	24	can	can	AUX
ejpam-4448	308	25	be	be	AUX
ejpam-4448	308	26	partitioned	partition	VERB
ejpam-4448	308	27	into	into	ADP
ejpam-4448	308	28	sets	set	NOUN
ejpam-4448	308	29	.	.	PUNCT
ejpam-4448	308	30	.	.	PUNCT
ejpam-4448	309	1	.	.	PUNCT
ejpam-4448	310	1	,	,	PUNCT
ejpam-4448	310	2	v−α2	v−α2	X
ejpam-4448	310	3	,	,	PUNCT
ejpam-4448	310	4	v−α1	v−α1	CCONJ
ejpam-4448	310	5	,	,	PUNCT
ejpam-4448	310	6	v−α0	v−α0	X
ejpam-4448	310	7	,	,	PUNCT
ejpam-4448	310	8	vα0	vα0	X
ejpam-4448	310	9	,	,	PUNCT
ejpam-4448	310	10	vα1	vα1	NOUN
ejpam-4448	310	11	,	,	PUNCT
ejpam-4448	310	12	vα2	vα2	NOUN
ejpam-4448	310	13	,	,	PUNCT
ejpam-4448	310	14	.	.	PUNCT
ejpam-4448	310	15	.	.	PUNCT
ejpam-4448	310	16	.	.	PUNCT
ejpam-4448	311	1	(	(	PUNCT
ejpam-4448	311	2	some	some	PRON
ejpam-4448	311	3	of	of	ADP
ejpam-4448	311	4	which	which	PRON
ejpam-4448	311	5	possibly	possibly	ADV
ejpam-4448	311	6	empty	empty	ADJ
ejpam-4448	311	7	)	)	PUNCT
ejpam-4448	311	8	such	such	ADJ
ejpam-4448	311	9	that	that	SCONJ
ejpam-4448	311	10	digons	digon	NOUN
ejpam-4448	311	11	occur	occur	VERB
ejpam-4448	311	12	only	only	ADV
ejpam-4448	311	13	between	between	ADP
ejpam-4448	311	14	pairs	pair	NOUN
ejpam-4448	311	15	of	of	ADP
ejpam-4448	311	16	sets	set	NOUN
ejpam-4448	311	17	vαj	vαj	ADJ
ejpam-4448	311	18	,	,	PUNCT
ejpam-4448	311	19	v−αj	v−αj	NOUN
ejpam-4448	311	20	and	and	CCONJ
ejpam-4448	311	21	any	any	DET
ejpam-4448	311	22	arc	arc	NOUN
ejpam-4448	311	23	starting	start	VERB
ejpam-4448	311	24	in	in	ADP
ejpam-4448	311	25	a	a	DET
ejpam-4448	311	26	set	set	NOUN
ejpam-4448	311	27	vαj	vαj	ADJ
ejpam-4448	311	28	ends	end	NOUN
ejpam-4448	311	29	in	in	ADP
ejpam-4448	311	30	vαj−1	vαj−1	PROPN
ejpam-4448	311	31	.	.	PUNCT
ejpam-4448	311	32	theorem	theorem	NOUN
ejpam-4448	311	33	12	12	NUM
ejpam-4448	311	34	.	.	PUNCT
ejpam-4448	312	1	let	let	VERB
ejpam-4448	312	2	d	d	PRON
ejpam-4448	312	3	be	be	AUX
ejpam-4448	312	4	an	an	DET
ejpam-4448	312	5	α	α	NOUN
ejpam-4448	312	6	-	-	PUNCT
ejpam-4448	312	7	monograph	monograph	NOUN
ejpam-4448	312	8	(	(	PUNCT
ejpam-4448	312	9	of	of	ADP
ejpam-4448	312	10	2nd	2nd	ADJ
ejpam-4448	312	11	kind	kind	NOUN
ejpam-4448	312	12	)	)	PUNCT
ejpam-4448	312	13	.	.	PUNCT
ejpam-4448	313	1	then	then	ADV
ejpam-4448	313	2	,	,	PUNCT
ejpam-4448	313	3	σα(d	σα(d	NOUN
ejpam-4448	313	4	)	)	PUNCT
ejpam-4448	313	5	=	=	PUNCT
ejpam-4448	314	1	−σ(γ(d	−σ(γ(d	NOUN
ejpam-4448	314	2	)	)	PUNCT
ejpam-4448	314	3	)	)	PUNCT
ejpam-4448	314	4	.	.	PUNCT
ejpam-4448	315	1	example	example	NOUN
ejpam-4448	316	1	7	7	NUM
ejpam-4448	316	2	.	.	PUNCT
ejpam-4448	317	1	the	the	DET
ejpam-4448	317	2	mixed	mixed	ADJ
ejpam-4448	317	3	graph	graph	NOUN
ejpam-4448	317	4	d	d	NOUN
ejpam-4448	317	5	shown	show	VERB
ejpam-4448	317	6	in	in	ADP
ejpam-4448	317	7	figure	figure	NOUN
ejpam-4448	317	8	6	6	NUM
ejpam-4448	317	9	is	be	AUX
ejpam-4448	317	10	an	an	DET
ejpam-4448	317	11	i	i	NOUN
ejpam-4448	317	12	-	-	PUNCT
ejpam-4448	317	13	monograph	monograph	NOUN
ejpam-4448	317	14	of	of	ADP
ejpam-4448	317	15	2nd	2nd	ADJ
ejpam-4448	317	16	kind	kind	NOUN
ejpam-4448	317	17	.	.	PUNCT
ejpam-4448	318	1	we	we	PRON
ejpam-4448	318	2	have	have	VERB
ejpam-4448	318	3	σi(d	σi(d	NOUN
ejpam-4448	318	4	)	)	PUNCT
ejpam-4448	318	5	=	=	PRON
ejpam-4448	318	6	{	{	PUNCT
ejpam-4448	318	7	−3	−3	PROPN
ejpam-4448	318	8	,	,	PUNCT
ejpam-4448	318	9	1(3	1(3	NUM
ejpam-4448	318	10	)	)	PUNCT
ejpam-4448	318	11	}	}	PUNCT
ejpam-4448	318	12	and	and	CCONJ
ejpam-4448	318	13	σ(γ(d	σ(γ(d	NOUN
ejpam-4448	318	14	)	)	PUNCT
ejpam-4448	318	15	)	)	PUNCT
ejpam-4448	319	1	=	=	PRON
ejpam-4448	319	2	{	{	PUNCT
ejpam-4448	319	3	3,−1(3	3,−1(3	NUM
ejpam-4448	319	4	)	)	PUNCT
ejpam-4448	319	5	}	}	PUNCT
ejpam-4448	319	6	.	.	PUNCT
ejpam-4448	320	1	clearly	clearly	ADV
ejpam-4448	320	2	,	,	PUNCT
ejpam-4448	320	3	d	d	PRON
ejpam-4448	320	4	is	be	AUX
ejpam-4448	320	5	not	not	PART
ejpam-4448	320	6	an	an	DET
ejpam-4448	320	7	i	i	NOUN
ejpam-4448	320	8	-	-	PUNCT
ejpam-4448	320	9	monograph	monograph	NOUN
ejpam-4448	320	10	of	of	ADP
ejpam-4448	320	11	1st	1st	ADJ
ejpam-4448	320	12	kind	kind	NOUN
ejpam-4448	320	13	.	.	PUNCT
ejpam-4448	321	1	6	6	X
ejpam-4448	321	2	.	.	X
ejpam-4448	321	3	spectral	spectral	ADJ
ejpam-4448	321	4	radius	radius	NOUN
ejpam-4448	321	5	the	the	DET
ejpam-4448	321	6	spectral	spectral	ADJ
ejpam-4448	321	7	radius	radius	PROPN
ejpam-4448	321	8	ρ(m	ρ(m	NUM
ejpam-4448	321	9	)	)	PUNCT
ejpam-4448	321	10	of	of	ADP
ejpam-4448	321	11	a	a	DET
ejpam-4448	321	12	complex	complex	ADJ
ejpam-4448	321	13	matrix	matrix	NOUN
ejpam-4448	321	14	m	m	VERB
ejpam-4448	321	15	is	be	AUX
ejpam-4448	321	16	defined	define	VERB
ejpam-4448	321	17	as	as	ADP
ejpam-4448	321	18	the	the	DET
ejpam-4448	321	19	largest	large	ADJ
ejpam-4448	321	20	modulus	modulus	NOUN
ejpam-4448	321	21	among	among	ADP
ejpam-4448	321	22	its	its	PRON
ejpam-4448	321	23	eigenvalues	eigenvalue	NOUN
ejpam-4448	321	24	.	.	PUNCT
ejpam-4448	322	1	if	if	SCONJ
ejpam-4448	322	2	∥·∥	∥·∥	PROPN
ejpam-4448	322	3	is	be	AUX
ejpam-4448	322	4	any	any	DET
ejpam-4448	322	5	matrix	matrix	NOUN
ejpam-4448	322	6	norm	norm	NOUN
ejpam-4448	322	7	,	,	PUNCT
ejpam-4448	322	8	we	we	PRON
ejpam-4448	322	9	have	have	VERB
ejpam-4448	322	10	ρ(m	ρ(m	NUM
ejpam-4448	322	11	)	)	PUNCT
ejpam-4448	322	12	≤	≤	NUM
ejpam-4448	322	13	∥m∥	∥m∥	VERB
ejpam-4448	322	14	(	(	PUNCT
ejpam-4448	322	15	cf	cf	NOUN
ejpam-4448	322	16	.	.	PUNCT
ejpam-4448	322	17	theorem	theorem	VERB
ejpam-4448	322	18	5.6.9	5.6.9	NUM
ejpam-4448	322	19	m.	m.	NOUN
ejpam-4448	322	20	abudayah	abudayah	NOUN
ejpam-4448	322	21	,	,	PUNCT
ejpam-4448	322	22	o.	o.	PROPN
ejpam-4448	322	23	alomari	alomari	PROPN
ejpam-4448	322	24	,	,	PUNCT
ejpam-4448	322	25	t.	t.	PROPN
ejpam-4448	322	26	sander	sander	PROPN
ejpam-4448	322	27	/	/	SYM
ejpam-4448	322	28	eur	eur	PROPN
ejpam-4448	322	29	.	.	PUNCT
ejpam-4448	323	1	j.	j.	PROPN
ejpam-4448	323	2	pure	pure	PROPN
ejpam-4448	323	3	appl	appl	PROPN
ejpam-4448	323	4	.	.	PROPN
ejpam-4448	323	5	math	math	PROPN
ejpam-4448	323	6	,	,	PUNCT
ejpam-4448	323	7	15	15	NUM
ejpam-4448	323	8	(	(	PUNCT
ejpam-4448	323	9	3	3	NUM
ejpam-4448	323	10	)	)	PUNCT
ejpam-4448	323	11	(	(	PUNCT
ejpam-4448	323	12	2022	2022	NUM
ejpam-4448	323	13	)	)	PUNCT
ejpam-4448	323	14	,	,	PUNCT
ejpam-4448	323	15	841	841	NUM
ejpam-4448	323	16	-	-	SYM
ejpam-4448	323	17	855	855	NUM
ejpam-4448	323	18	853	853	NUM
ejpam-4448	323	19	in	in	ADP
ejpam-4448	323	20	[	[	X
ejpam-4448	323	21	7	7	NUM
ejpam-4448	323	22	]	]	NUM
ejpam-4448	323	23	)	)	PUNCT
ejpam-4448	323	24	.	.	PUNCT
ejpam-4448	324	1	using	use	VERB
ejpam-4448	324	2	the	the	DET
ejpam-4448	324	3	maximum	maximum	ADJ
ejpam-4448	324	4	norm	norm	NOUN
ejpam-4448	324	5	∥	∥	X
ejpam-4448	324	6	·	·	SYM
ejpam-4448	324	7	∥∞	∥∞	PROPN
ejpam-4448	324	8	,	,	PUNCT
ejpam-4448	324	9	one	one	PRON
ejpam-4448	324	10	immediately	immediately	ADV
ejpam-4448	324	11	obtains	obtain	VERB
ejpam-4448	324	12	the	the	DET
ejpam-4448	324	13	classic	classic	ADJ
ejpam-4448	324	14	upper	upper	ADJ
ejpam-4448	324	15	bound	bind	VERB
ejpam-4448	324	16	ρ(g	ρ(g	ADP
ejpam-4448	324	17	)	)	PUNCT
ejpam-4448	324	18	≤	≤	NUM
ejpam-4448	324	19	∆g	∆g	PROPN
ejpam-4448	324	20	on	on	ADP
ejpam-4448	324	21	the	the	DET
ejpam-4448	324	22	spectral	spectral	ADJ
ejpam-4448	324	23	radius	radius	NOUN
ejpam-4448	324	24	ρ(g	ρ(g	ADP
ejpam-4448	324	25	)	)	PUNCT
ejpam-4448	324	26	:	:	PUNCT
ejpam-4448	324	27	=	=	PROPN
ejpam-4448	324	28	ρ(a(g	ρ(a(g	PROPN
ejpam-4448	324	29	)	)	PUNCT
ejpam-4448	324	30	)	)	PUNCT
ejpam-4448	324	31	of	of	ADP
ejpam-4448	324	32	an	an	DET
ejpam-4448	324	33	undirected	undirected	ADJ
ejpam-4448	324	34	graph	graph	NOUN
ejpam-4448	324	35	g.	g.	NOUN
ejpam-4448	324	36	supposing	suppose	VERB
ejpam-4448	324	37	g	g	PROPN
ejpam-4448	324	38	is	be	AUX
ejpam-4448	324	39	connected	connect	VERB
ejpam-4448	324	40	,	,	PUNCT
ejpam-4448	324	41	equality	equality	NOUN
ejpam-4448	324	42	holds	hold	VERB
ejpam-4448	324	43	if	if	SCONJ
ejpam-4448	324	44	and	and	CCONJ
ejpam-4448	324	45	only	only	ADV
ejpam-4448	324	46	if	if	SCONJ
ejpam-4448	324	47	g	g	PROPN
ejpam-4448	324	48	is	be	AUX
ejpam-4448	324	49	regular	regular	ADJ
ejpam-4448	324	50	.	.	PUNCT
ejpam-4448	325	1	considering	consider	VERB
ejpam-4448	325	2	a	a	DET
ejpam-4448	325	3	mixed	mixed	ADJ
ejpam-4448	325	4	graph	graph	NOUN
ejpam-4448	325	5	d	d	NOUN
ejpam-4448	325	6	and	and	CCONJ
ejpam-4448	325	7	its	its	PRON
ejpam-4448	325	8	α	α	NOUN
ejpam-4448	325	9	-	-	ADJ
ejpam-4448	325	10	hermitian	hermitian	ADJ
ejpam-4448	325	11	adjacency	adjacency	NOUN
ejpam-4448	325	12	matrix	matrix	NOUN
ejpam-4448	325	13	hα(d	hα(d	PRON
ejpam-4448	325	14	)	)	PUNCT
ejpam-4448	325	15	instead	instead	ADV
ejpam-4448	325	16	,	,	PUNCT
ejpam-4448	325	17	it	it	PRON
ejpam-4448	325	18	follows	follow	VERB
ejpam-4448	325	19	from	from	ADP
ejpam-4448	325	20	definition	definition	NOUN
ejpam-4448	325	21	1	1	NUM
ejpam-4448	325	22	that	that	SCONJ
ejpam-4448	325	23	∥hα(d)∥∞	∥hα(d)∥∞	PROPN
ejpam-4448	325	24	=	=	SYM
ejpam-4448	325	25	∥a(γ(d))∥∞	∥a(γ(d))∥∞	NOUN
ejpam-4448	325	26	,	,	PUNCT
ejpam-4448	325	27	since	since	SCONJ
ejpam-4448	325	28	all	all	DET
ejpam-4448	325	29	nonzero	nonzero	ADJ
ejpam-4448	325	30	entries	entry	NOUN
ejpam-4448	325	31	of	of	ADP
ejpam-4448	325	32	hα(d	hα(d	NOUN
ejpam-4448	325	33	)	)	PUNCT
ejpam-4448	325	34	have	have	VERB
ejpam-4448	325	35	modulus	modulus	NOUN
ejpam-4448	325	36	1	1	NUM
ejpam-4448	325	37	.	.	PUNCT
ejpam-4448	325	38	hence	hence	ADV
ejpam-4448	325	39	,	,	PUNCT
ejpam-4448	325	40	ρα(d	ρα(d	X
ejpam-4448	325	41	)	)	PUNCT
ejpam-4448	325	42	≤	≤	NUM
ejpam-4448	325	43	∆γ(d	∆γ(d	PROPN
ejpam-4448	325	44	)	)	PUNCT
ejpam-4448	325	45	.	.	PUNCT
ejpam-4448	326	1	interestingly	interestingly	ADV
ejpam-4448	326	2	,	,	PUNCT
ejpam-4448	326	3	α	α	X
ejpam-4448	326	4	-	-	PUNCT
ejpam-4448	326	5	monographs	monograph	NOUN
ejpam-4448	326	6	come	come	VERB
ejpam-4448	326	7	into	into	ADP
ejpam-4448	326	8	play	play	NOUN
ejpam-4448	326	9	if	if	SCONJ
ejpam-4448	326	10	one	one	PRON
ejpam-4448	326	11	wants	want	VERB
ejpam-4448	326	12	to	to	PART
ejpam-4448	326	13	characterize	characterize	VERB
ejpam-4448	326	14	when	when	SCONJ
ejpam-4448	326	15	equality	equality	NOUN
ejpam-4448	326	16	holds	hold	VERB
ejpam-4448	326	17	:	:	PUNCT
ejpam-4448	326	18	theorem	theorem	NOUN
ejpam-4448	326	19	13	13	NUM
ejpam-4448	326	20	.	.	PUNCT
ejpam-4448	327	1	let	let	VERB
ejpam-4448	327	2	d	d	PRON
ejpam-4448	327	3	be	be	AUX
ejpam-4448	327	4	a	a	DET
ejpam-4448	327	5	connected	connected	ADJ
ejpam-4448	327	6	mixed	mixed	ADJ
ejpam-4448	327	7	graph	graph	NOUN
ejpam-4448	327	8	.	.	PUNCT
ejpam-4448	328	1	then	then	ADV
ejpam-4448	328	2	,	,	PUNCT
ejpam-4448	328	3	ρα(d	ρα(d	X
ejpam-4448	328	4	)	)	PUNCT
ejpam-4448	328	5	=	=	SYM
ejpam-4448	328	6	∆γ(d	∆γ(d	PROPN
ejpam-4448	328	7	)	)	PUNCT
ejpam-4448	328	8	if	if	SCONJ
ejpam-4448	328	9	and	and	CCONJ
ejpam-4448	328	10	only	only	ADV
ejpam-4448	328	11	if	if	SCONJ
ejpam-4448	328	12	d	d	NOUN
ejpam-4448	328	13	is	be	AUX
ejpam-4448	328	14	a	a	DET
ejpam-4448	328	15	regular	regular	ADJ
ejpam-4448	328	16	α	α	NOUN
ejpam-4448	328	17	-	-	NOUN
ejpam-4448	328	18	monograph	monograph	NOUN
ejpam-4448	328	19	(	(	PUNCT
ejpam-4448	328	20	of	of	ADP
ejpam-4448	328	21	1st	1st	ADJ
ejpam-4448	328	22	or	or	CCONJ
ejpam-4448	328	23	2nd	2nd	ADJ
ejpam-4448	328	24	kind	kind	NOUN
ejpam-4448	328	25	)	)	PUNCT
ejpam-4448	328	26	.	.	PUNCT
ejpam-4448	329	1	proof	proof	NOUN
ejpam-4448	329	2	.	.	PUNCT
ejpam-4448	330	1	let	let	VERB
ejpam-4448	330	2	x	x	PUNCT
ejpam-4448	330	3	=	=	PUNCT
ejpam-4448	331	1	[	[	X
ejpam-4448	331	2	xu]u∈v	xu]u∈v	PROPN
ejpam-4448	331	3	(	(	PUNCT
ejpam-4448	331	4	d	d	X
ejpam-4448	331	5	)	)	PUNCT
ejpam-4448	331	6	be	be	AUX
ejpam-4448	331	7	an	an	DET
ejpam-4448	331	8	α	α	NOUN
ejpam-4448	331	9	-	-	NOUN
ejpam-4448	331	10	eigenvector	eigenvector	NOUN
ejpam-4448	331	11	of	of	ADP
ejpam-4448	331	12	d	d	PROPN
ejpam-4448	331	13	for	for	ADP
ejpam-4448	331	14	α	α	NOUN
ejpam-4448	331	15	-	-	PUNCT
ejpam-4448	331	16	eigenvalue	eigenvalue	ADJ
ejpam-4448	331	17	λ	λ	PROPN
ejpam-4448	331	18	.	.	PUNCT
ejpam-4448	332	1	choose	choose	VERB
ejpam-4448	332	2	v	v	NUM
ejpam-4448	332	3	∈	∈	PROPN
ejpam-4448	332	4	v	v	NOUN
ejpam-4448	332	5	(	(	PUNCT
ejpam-4448	332	6	d	d	NOUN
ejpam-4448	332	7	)	)	PUNCT
ejpam-4448	332	8	such	such	ADJ
ejpam-4448	332	9	that	that	SCONJ
ejpam-4448	332	10	|xv|	|xv|	NOUN
ejpam-4448	332	11	is	be	AUX
ejpam-4448	332	12	maximal	maximal	ADJ
ejpam-4448	332	13	.	.	PUNCT
ejpam-4448	333	1	we	we	PRON
ejpam-4448	333	2	may	may	AUX
ejpam-4448	333	3	assume	assume	VERB
ejpam-4448	333	4	|xv|	|xv|	NOUN
ejpam-4448	333	5	=	=	SYM
ejpam-4448	333	6	1	1	X
ejpam-4448	333	7	.	.	X
ejpam-4448	333	8	using	use	VERB
ejpam-4448	333	9	proposition	proposition	NOUN
ejpam-4448	333	10	1	1	NUM
ejpam-4448	333	11	,	,	PUNCT
ejpam-4448	333	12	we	we	PRON
ejpam-4448	333	13	can	can	AUX
ejpam-4448	333	14	deduce	deduce	VERB
ejpam-4448	333	15	that	that	DET
ejpam-4448	333	16	|λ|	|λ|	NOUN
ejpam-4448	333	17	=	=	PUNCT
ejpam-4448	334	1	|λxv|	|λxv|	NOUN
ejpam-4448	334	2	≤	≤	NOUN
ejpam-4448	334	3	∑	∑	ADP
ejpam-4448	334	4	u∈n(v	u∈n(v	PROPN
ejpam-4448	334	5	)	)	PUNCT
ejpam-4448	334	6	|xu|+	|xu|+	NOUN
ejpam-4448	334	7	∑	∑	PUNCT
ejpam-4448	334	8	u∈n+(v	u∈n+(v	ADJ
ejpam-4448	334	9	)	)	PUNCT
ejpam-4448	334	10	|αxu|+	|αxu|+	PROPN
ejpam-4448	334	11	∑	∑	PUNCT
ejpam-4448	334	12	u∈n−(v	u∈n−(v	PROPN
ejpam-4448	334	13	)	)	PUNCT
ejpam-4448	334	14	|αxu|	|αxu|	NOUN
ejpam-4448	334	15	(	(	PUNCT
ejpam-4448	334	16	18	18	NUM
ejpam-4448	334	17	)	)	PUNCT
ejpam-4448	334	18	=	=	SYM
ejpam-4448	334	19	∑	∑	PUNCT
ejpam-4448	334	20	u∈n(v	u∈n(v	ADJ
ejpam-4448	334	21	)	)	PUNCT
ejpam-4448	334	22	|xu|+	|xu|+	NOUN
ejpam-4448	334	23	∑	∑	PUNCT
ejpam-4448	334	24	u∈n+(v	u∈n+(v	PROPN
ejpam-4448	334	25	)	)	PUNCT
ejpam-4448	334	26	|xu|+	|xu|+	NOUN
ejpam-4448	334	27	∑	∑	PUNCT
ejpam-4448	334	28	u∈n−(v	u∈n−(v	PROPN
ejpam-4448	334	29	)	)	PUNCT
ejpam-4448	334	30	|xu|	|xu|	PROPN
ejpam-4448	334	31	(	(	PUNCT
ejpam-4448	334	32	19	19	NUM
ejpam-4448	334	33	)	)	PUNCT
ejpam-4448	334	34	≤	≤	NOUN
ejpam-4448	334	35	∑	∑	PUNCT
ejpam-4448	334	36	u∈n(v	u∈n(v	PROPN
ejpam-4448	334	37	)	)	PUNCT
ejpam-4448	334	38	|xv|+	|xv|+	NOUN
ejpam-4448	334	39	∑	∑	PUNCT
ejpam-4448	334	40	u∈n+(v	u∈n+(v	ADJ
ejpam-4448	334	41	)	)	PUNCT
ejpam-4448	334	42	|xv|+	|xv|+	NOUN
ejpam-4448	334	43	∑	∑	ADP
ejpam-4448	334	44	u∈n−(v	u∈n−(v	PROPN
ejpam-4448	334	45	)	)	PUNCT
ejpam-4448	334	46	|xv|	|xv|	NOUN
ejpam-4448	334	47	(	(	PUNCT
ejpam-4448	334	48	20	20	NUM
ejpam-4448	334	49	)	)	PUNCT
ejpam-4448	334	50	=	=	SYM
ejpam-4448	334	51	degγ(d)(v	degγ(d)(v	NOUN
ejpam-4448	334	52	)	)	PUNCT
ejpam-4448	334	53	(	(	PUNCT
ejpam-4448	334	54	21	21	NUM
ejpam-4448	334	55	)	)	PUNCT
ejpam-4448	334	56	≤	≤	NOUN
ejpam-4448	334	57	∆γ(d	∆γ(d	PROPN
ejpam-4448	334	58	)	)	PUNCT
ejpam-4448	334	59	.	.	PUNCT
ejpam-4448	335	1	(	(	PUNCT
ejpam-4448	335	2	22	22	X
ejpam-4448	335	3	)	)	PUNCT
ejpam-4448	335	4	suppose	suppose	VERB
ejpam-4448	335	5	that	that	SCONJ
ejpam-4448	335	6	λ	λ	PROPN
ejpam-4448	335	7	is	be	AUX
ejpam-4448	335	8	an	an	DET
ejpam-4448	335	9	α	α	NOUN
ejpam-4448	335	10	-	-	PUNCT
ejpam-4448	335	11	eigenvalue	eigenvalue	NOUN
ejpam-4448	335	12	of	of	ADP
ejpam-4448	335	13	d	d	PROPN
ejpam-4448	335	14	with	with	ADP
ejpam-4448	335	15	largest	large	ADJ
ejpam-4448	335	16	modulus	modulus	NOUN
ejpam-4448	335	17	.	.	PUNCT
ejpam-4448	336	1	the	the	DET
ejpam-4448	336	2	condition	condition	NOUN
ejpam-4448	336	3	ρα(d	ρα(d	X
ejpam-4448	336	4	)	)	PUNCT
ejpam-4448	337	1	=	=	SYM
ejpam-4448	337	2	∆γ(d	∆γ(d	NOUN
ejpam-4448	337	3	)	)	PUNCT
ejpam-4448	337	4	holds	hold	VERB
ejpam-4448	337	5	if	if	SCONJ
ejpam-4448	337	6	and	and	CCONJ
ejpam-4448	337	7	only	only	ADV
ejpam-4448	337	8	if	if	SCONJ
ejpam-4448	337	9	equality	equality	NOUN
ejpam-4448	337	10	holds	hold	VERB
ejpam-4448	337	11	in	in	ADP
ejpam-4448	337	12	all	all	DET
ejpam-4448	337	13	three	three	NUM
ejpam-4448	337	14	conditions	condition	NOUN
ejpam-4448	337	15	(	(	PUNCT
ejpam-4448	337	16	18	18	NUM
ejpam-4448	337	17	)	)	PUNCT
ejpam-4448	337	18	,	,	PUNCT
ejpam-4448	337	19	(	(	PUNCT
ejpam-4448	337	20	20	20	NUM
ejpam-4448	337	21	)	)	PUNCT
ejpam-4448	337	22	and	and	CCONJ
ejpam-4448	337	23	(	(	PUNCT
ejpam-4448	337	24	22	22	NUM
ejpam-4448	337	25	)	)	PUNCT
ejpam-4448	337	26	.	.	PUNCT
ejpam-4448	338	1	equality	equality	NOUN
ejpam-4448	338	2	in	in	ADP
ejpam-4448	338	3	(	(	PUNCT
ejpam-4448	338	4	22	22	NUM
ejpam-4448	338	5	)	)	PUNCT
ejpam-4448	338	6	is	be	AUX
ejpam-4448	338	7	achieved	achieve	VERB
ejpam-4448	338	8	if	if	SCONJ
ejpam-4448	338	9	any	any	DET
ejpam-4448	338	10	only	only	ADV
ejpam-4448	338	11	if	if	SCONJ
ejpam-4448	338	12	d	d	PROPN
ejpam-4448	338	13	(	(	PUNCT
ejpam-4448	338	14	resp	resp	NOUN
ejpam-4448	338	15	.	.	PUNCT
ejpam-4448	339	1	γ(d	γ(d	NOUN
ejpam-4448	339	2	)	)	PUNCT
ejpam-4448	339	3	)	)	PUNCT
ejpam-4448	339	4	is	be	AUX
ejpam-4448	339	5	regular	regular	ADJ
ejpam-4448	339	6	of	of	ADP
ejpam-4448	339	7	degree	degree	NOUN
ejpam-4448	339	8	∆γ(d	∆γ(d	PROPN
ejpam-4448	339	9	)	)	PUNCT
ejpam-4448	339	10	.	.	PUNCT
ejpam-4448	340	1	since	since	SCONJ
ejpam-4448	340	2	|xv|	|xv|	NOUN
ejpam-4448	340	3	is	be	AUX
ejpam-4448	340	4	maximal	maximal	ADJ
ejpam-4448	340	5	,	,	PUNCT
ejpam-4448	340	6	equality	equality	NOUN
ejpam-4448	340	7	in	in	ADP
ejpam-4448	340	8	(	(	PUNCT
ejpam-4448	340	9	20	20	NUM
ejpam-4448	340	10	)	)	PUNCT
ejpam-4448	340	11	occurs	occur	VERB
ejpam-4448	340	12	exactly	exactly	ADV
ejpam-4448	340	13	if	if	SCONJ
ejpam-4448	340	14	|xu|	|xu|	PROPN
ejpam-4448	340	15	=	=	SYM
ejpam-4448	340	16	|xv|	|xv|	NOUN
ejpam-4448	340	17	=	=	SYM
ejpam-4448	340	18	1	1	NUM
ejpam-4448	340	19	for	for	ADP
ejpam-4448	340	20	all	all	DET
ejpam-4448	340	21	u	u	PROPN
ejpam-4448	340	22	∈	∈	PROPN
ejpam-4448	340	23	nγ(d)(v	nγ(d)(v	NOUN
ejpam-4448	340	24	)	)	PUNCT
ejpam-4448	340	25	.	.	PUNCT
ejpam-4448	341	1	repeat	repeat	VERB
ejpam-4448	341	2	this	this	DET
ejpam-4448	341	3	argument	argument	NOUN
ejpam-4448	341	4	for	for	ADP
ejpam-4448	341	5	all	all	DET
ejpam-4448	341	6	vertices	vertex	NOUN
ejpam-4448	341	7	u	u	PROPN
ejpam-4448	341	8	∈	∈	PROPN
ejpam-4448	341	9	nγ(d	nγ(d	NOUN
ejpam-4448	341	10	)	)	PUNCT
ejpam-4448	341	11	,	,	PUNCT
ejpam-4448	341	12	each	each	DET
ejpam-4448	341	13	time	time	NOUN
ejpam-4448	341	14	taking	take	VERB
ejpam-4448	341	15	the	the	DET
ejpam-4448	341	16	role	role	NOUN
ejpam-4448	341	17	of	of	ADP
ejpam-4448	341	18	v.	v.	ADV
ejpam-4448	341	19	since	since	SCONJ
ejpam-4448	341	20	γ(d	γ(d	PROPN
ejpam-4448	341	21	)	)	PUNCT
ejpam-4448	341	22	is	be	AUX
ejpam-4448	341	23	connected	connect	VERB
ejpam-4448	341	24	,	,	PUNCT
ejpam-4448	341	25	we	we	PRON
ejpam-4448	341	26	successively	successively	ADV
ejpam-4448	341	27	prove	prove	VERB
ejpam-4448	341	28	|xu|	|xu|	PROPN
ejpam-4448	341	29	=	=	NOUN
ejpam-4448	341	30	1	1	NUM
ejpam-4448	341	31	for	for	ADP
ejpam-4448	341	32	all	all	PRON
ejpam-4448	341	33	u	u	NOUN
ejpam-4448	341	34	∈	∈	PROPN
ejpam-4448	341	35	v	v	NOUN
ejpam-4448	341	36	(	(	PUNCT
ejpam-4448	341	37	d	d	NOUN
ejpam-4448	341	38	)	)	PUNCT
ejpam-4448	341	39	.	.	PUNCT
ejpam-4448	342	1	equality	equality	NOUN
ejpam-4448	342	2	holds	hold	VERB
ejpam-4448	342	3	in	in	ADP
ejpam-4448	342	4	the	the	DET
ejpam-4448	342	5	complex	complex	ADJ
ejpam-4448	342	6	triangle	triangle	NOUN
ejpam-4448	342	7	inequality	inequality	NOUN
ejpam-4448	342	8	(	(	PUNCT
ejpam-4448	342	9	18	18	NUM
ejpam-4448	342	10	)	)	PUNCT
ejpam-4448	342	11	if	if	SCONJ
ejpam-4448	343	1	and	and	CCONJ
ejpam-4448	343	2	only	only	ADV
ejpam-4448	343	3	if	if	SCONJ
ejpam-4448	343	4	arg(λxv	arg(λxv	NOUN
ejpam-4448	343	5	)	)	PUNCT
ejpam-4448	344	1	=	=	SYM
ejpam-4448	344	2	arg(xu	arg(xu	NOUN
ejpam-4448	344	3	)	)	PUNCT
ejpam-4448	344	4	for	for	ADP
ejpam-4448	344	5	all	all	DET
ejpam-4448	344	6	u	u	PROPN
ejpam-4448	344	7	∈	∈	PROPN
ejpam-4448	344	8	nγ(d)(v	nγ(d)(v	NOUN
ejpam-4448	344	9	)	)	PUNCT
ejpam-4448	344	10	.	.	PUNCT
ejpam-4448	345	1	in	in	ADP
ejpam-4448	345	2	the	the	DET
ejpam-4448	345	3	following	following	NOUN
ejpam-4448	345	4	,	,	PUNCT
ejpam-4448	345	5	we	we	PRON
ejpam-4448	345	6	shall	shall	AUX
ejpam-4448	345	7	skip	skip	VERB
ejpam-4448	345	8	the	the	DET
ejpam-4448	345	9	trivial	trivial	ADJ
ejpam-4448	345	10	case	case	NOUN
ejpam-4448	345	11	λ	λ	X
ejpam-4448	345	12	=	=	SYM
ejpam-4448	345	13	0	0	NUM
ejpam-4448	345	14	=	=	SYM
ejpam-4448	345	15	ρα(d	ρα(d	NOUN
ejpam-4448	345	16	)	)	PUNCT
ejpam-4448	345	17	.	.	PUNCT
ejpam-4448	346	1	let	let	VERB
ejpam-4448	346	2	arg(α	arg(α	PRON
ejpam-4448	346	3	)	)	PUNCT
ejpam-4448	346	4	=	=	SYM
ejpam-4448	346	5	θ	θ	PROPN
ejpam-4448	346	6	∈	∈	PROPN
ejpam-4448	346	7	r.	r.	NOUN
ejpam-4448	346	8	consider	consider	VERB
ejpam-4448	346	9	the	the	DET
ejpam-4448	346	10	following	follow	VERB
ejpam-4448	346	11	cases	case	NOUN
ejpam-4448	346	12	:	:	PUNCT
ejpam-4448	346	13	(	(	PUNCT
ejpam-4448	346	14	i	i	NOUN
ejpam-4448	346	15	)	)	PUNCT
ejpam-4448	346	16	case	case	NOUN
ejpam-4448	346	17	λ	λ	X
ejpam-4448	346	18	>	>	X
ejpam-4448	346	19	0	0	NUM
ejpam-4448	346	20	:	:	PUNCT
ejpam-4448	346	21	•	•	NOUN
ejpam-4448	346	22	if	if	SCONJ
ejpam-4448	346	23	u	u	PROPN
ejpam-4448	346	24	∈	∈	PROPN
ejpam-4448	346	25	n(v	n(v	PROPN
ejpam-4448	346	26	)	)	PUNCT
ejpam-4448	346	27	,	,	PUNCT
ejpam-4448	346	28	then	then	ADV
ejpam-4448	346	29	arg(xu	arg(xu	ADJ
ejpam-4448	346	30	)	)	PUNCT
ejpam-4448	346	31	=	=	SYM
ejpam-4448	346	32	arg(λxv	arg(λxv	NOUN
ejpam-4448	346	33	)	)	PUNCT
ejpam-4448	346	34	,	,	PUNCT
ejpam-4448	346	35	so	so	SCONJ
ejpam-4448	346	36	that	that	SCONJ
ejpam-4448	346	37	xu	xu	PROPN
ejpam-4448	346	38	=	=	PUNCT
ejpam-4448	347	1	xv	xv	PROPN
ejpam-4448	347	2	.	.	PROPN
ejpam-4448	348	1	•	•	INTJ
ejpam-4448	348	2	if	if	SCONJ
ejpam-4448	348	3	u	u	PROPN
ejpam-4448	348	4	∈	∈	PROPN
ejpam-4448	348	5	n+(v	n+(v	PROPN
ejpam-4448	348	6	)	)	PUNCT
ejpam-4448	349	1	,	,	PUNCT
ejpam-4448	349	2	then	then	ADV
ejpam-4448	349	3	arg(αxu	arg(αxu	NOUN
ejpam-4448	349	4	)	)	PUNCT
ejpam-4448	349	5	=	=	SYM
ejpam-4448	349	6	arg(λxv	arg(λxv	NOUN
ejpam-4448	349	7	)	)	PUNCT
ejpam-4448	349	8	,	,	PUNCT
ejpam-4448	349	9	so	so	SCONJ
ejpam-4448	349	10	that	that	SCONJ
ejpam-4448	349	11	xu	xu	PROPN
ejpam-4448	349	12	=	=	SYM
ejpam-4448	349	13	ᾱxv	ᾱxv	PROPN
ejpam-4448	349	14	.	.	PROPN
ejpam-4448	350	1	•	•	INTJ
ejpam-4448	350	2	if	if	SCONJ
ejpam-4448	350	3	u	u	PROPN
ejpam-4448	350	4	∈	∈	PROPN
ejpam-4448	350	5	n−(v	n−(v	PROPN
ejpam-4448	350	6	)	)	PUNCT
ejpam-4448	350	7	,	,	PUNCT
ejpam-4448	350	8	then	then	ADV
ejpam-4448	350	9	arg(ᾱxu	arg(ᾱxu	NUM
ejpam-4448	350	10	)	)	PUNCT
ejpam-4448	350	11	=	=	SYM
ejpam-4448	350	12	arg(λxv	arg(λxv	NOUN
ejpam-4448	350	13	)	)	PUNCT
ejpam-4448	350	14	,	,	PUNCT
ejpam-4448	350	15	so	so	SCONJ
ejpam-4448	350	16	that	that	SCONJ
ejpam-4448	350	17	xu	xu	PROPN
ejpam-4448	351	1	=	=	SYM
ejpam-4448	351	2	αxv	αxv	ADJ
ejpam-4448	351	3	.	.	PUNCT
ejpam-4448	352	1	assigning	assign	VERB
ejpam-4448	352	2	each	each	DET
ejpam-4448	352	3	vertex	vertex	NOUN
ejpam-4448	352	4	u	u	NOUN
ejpam-4448	352	5	∈	∈	PROPN
ejpam-4448	352	6	v	v	NOUN
ejpam-4448	352	7	(	(	PUNCT
ejpam-4448	352	8	d	d	NOUN
ejpam-4448	352	9	)	)	PUNCT
ejpam-4448	352	10	to	to	ADP
ejpam-4448	352	11	a	a	DET
ejpam-4448	352	12	set	set	NOUN
ejpam-4448	352	13	vθ	vθ	NOUN
ejpam-4448	352	14	with	with	ADP
ejpam-4448	352	15	θ	θ	PROPN
ejpam-4448	352	16	=	=	SYM
ejpam-4448	352	17	xu	xu	PROPN
ejpam-4448	352	18	/	/	SYM
ejpam-4448	352	19	xv	xv	PROPN
ejpam-4448	352	20	,	,	PUNCT
ejpam-4448	352	21	we	we	PRON
ejpam-4448	352	22	obtain	obtain	VERB
ejpam-4448	352	23	a	a	DET
ejpam-4448	352	24	partition	partition	NOUN
ejpam-4448	352	25	as	as	SCONJ
ejpam-4448	352	26	mentioned	mention	VERB
ejpam-4448	352	27	in	in	ADP
ejpam-4448	352	28	theorem	theorem	NOUN
ejpam-4448	352	29	9	9	NUM
ejpam-4448	352	30	.	.	PUNCT
ejpam-4448	353	1	references	reference	NOUN
ejpam-4448	353	2	854	854	NUM
ejpam-4448	353	3	(	(	PUNCT
ejpam-4448	353	4	ii	ii	NOUN
ejpam-4448	353	5	)	)	PUNCT
ejpam-4448	353	6	case	case	NOUN
ejpam-4448	353	7	λ	λ	X
ejpam-4448	353	8	<	<	X
ejpam-4448	353	9	0	0	NUM
ejpam-4448	353	10	:	:	PUNCT
ejpam-4448	353	11	•	•	NOUN
ejpam-4448	353	12	if	if	SCONJ
ejpam-4448	353	13	u	u	PROPN
ejpam-4448	353	14	∈	∈	PROPN
ejpam-4448	353	15	n(v	n(v	PROPN
ejpam-4448	353	16	)	)	PUNCT
ejpam-4448	353	17	,	,	PUNCT
ejpam-4448	353	18	then	then	ADV
ejpam-4448	353	19	arg(xu	arg(xu	ADJ
ejpam-4448	353	20	)	)	PUNCT
ejpam-4448	353	21	=	=	SYM
ejpam-4448	353	22	arg(λxv	arg(λxv	NOUN
ejpam-4448	353	23	)	)	PUNCT
ejpam-4448	353	24	,	,	PUNCT
ejpam-4448	353	25	so	so	SCONJ
ejpam-4448	353	26	that	that	SCONJ
ejpam-4448	353	27	xu	xu	PROPN
ejpam-4448	354	1	=	=	NOUN
ejpam-4448	354	2	−xv	−xv	PROPN
ejpam-4448	354	3	.	.	NOUN
ejpam-4448	355	1	•	•	NOUN
ejpam-4448	355	2	if	if	SCONJ
ejpam-4448	355	3	u	u	PROPN
ejpam-4448	355	4	∈	∈	PROPN
ejpam-4448	355	5	n+(v	n+(v	PROPN
ejpam-4448	355	6	)	)	PUNCT
ejpam-4448	355	7	,	,	PUNCT
ejpam-4448	355	8	then	then	ADV
ejpam-4448	355	9	arg(αxu	arg(αxu	NOUN
ejpam-4448	355	10	)	)	PUNCT
ejpam-4448	355	11	=	=	SYM
ejpam-4448	355	12	arg(λxv	arg(λxv	NOUN
ejpam-4448	355	13	)	)	PUNCT
ejpam-4448	355	14	,	,	PUNCT
ejpam-4448	355	15	so	so	SCONJ
ejpam-4448	355	16	that	that	SCONJ
ejpam-4448	355	17	xu	xu	PUNCT
ejpam-4448	356	1	=	=	SYM
ejpam-4448	356	2	−ᾱxv	−ᾱxv	NOUN
ejpam-4448	356	3	.	.	PUNCT
ejpam-4448	357	1	•	•	INTJ
ejpam-4448	357	2	if	if	SCONJ
ejpam-4448	357	3	u	u	PROPN
ejpam-4448	357	4	∈	∈	PROPN
ejpam-4448	357	5	n−(v	n−(v	PROPN
ejpam-4448	357	6	)	)	PUNCT
ejpam-4448	357	7	,	,	PUNCT
ejpam-4448	357	8	then	then	ADV
ejpam-4448	357	9	arg(ᾱxu	arg(ᾱxu	NUM
ejpam-4448	357	10	)	)	PUNCT
ejpam-4448	357	11	=	=	SYM
ejpam-4448	357	12	arg(λxv	arg(λxv	NOUN
ejpam-4448	357	13	)	)	PUNCT
ejpam-4448	357	14	,	,	PUNCT
ejpam-4448	357	15	so	so	SCONJ
ejpam-4448	357	16	that	that	SCONJ
ejpam-4448	357	17	xu	xu	X
ejpam-4448	357	18	=	=	PUNCT
ejpam-4448	357	19	−αxv	−αxv	VERB
ejpam-4448	357	20	.	.	PUNCT
ejpam-4448	358	1	assigning	assign	VERB
ejpam-4448	358	2	each	each	DET
ejpam-4448	358	3	vertex	vertex	NOUN
ejpam-4448	358	4	u	u	NOUN
ejpam-4448	358	5	∈	∈	PROPN
ejpam-4448	358	6	v	v	NOUN
ejpam-4448	358	7	(	(	PUNCT
ejpam-4448	358	8	d	d	NOUN
ejpam-4448	358	9	)	)	PUNCT
ejpam-4448	358	10	to	to	ADP
ejpam-4448	358	11	a	a	DET
ejpam-4448	358	12	set	set	NOUN
ejpam-4448	358	13	vθ	vθ	NOUN
ejpam-4448	358	14	with	with	ADP
ejpam-4448	358	15	θ	θ	PROPN
ejpam-4448	358	16	=	=	SYM
ejpam-4448	358	17	xu	xu	PROPN
ejpam-4448	358	18	/	/	SYM
ejpam-4448	358	19	xv	xv	PROPN
ejpam-4448	358	20	,	,	PUNCT
ejpam-4448	358	21	we	we	PRON
ejpam-4448	358	22	obtain	obtain	VERB
ejpam-4448	358	23	a	a	DET
ejpam-4448	358	24	partition	partition	NOUN
ejpam-4448	358	25	as	as	SCONJ
ejpam-4448	358	26	mentioned	mention	VERB
ejpam-4448	358	27	in	in	ADP
ejpam-4448	358	28	theorem	theorem	NOUN
ejpam-4448	358	29	11	11	NUM
ejpam-4448	358	30	.	.	PUNCT
ejpam-4448	359	1	conversely	conversely	ADV
ejpam-4448	359	2	,	,	PUNCT
ejpam-4448	359	3	let	let	VERB
ejpam-4448	359	4	d	d	PRON
ejpam-4448	359	5	be	be	AUX
ejpam-4448	359	6	a	a	DET
ejpam-4448	359	7	connected	connect	VERB
ejpam-4448	359	8	mixed	mixed	ADJ
ejpam-4448	359	9	graph	graph	NOUN
ejpam-4448	359	10	having	have	VERB
ejpam-4448	359	11	a	a	DET
ejpam-4448	359	12	vertex	vertex	NOUN
ejpam-4448	359	13	partition	partition	NOUN
ejpam-4448	359	14	according	accord	VERB
ejpam-4448	359	15	to	to	ADP
ejpam-4448	359	16	one	one	NUM
ejpam-4448	359	17	of	of	ADP
ejpam-4448	359	18	the	the	DET
ejpam-4448	359	19	cases	case	NOUN
ejpam-4448	359	20	(	(	PUNCT
ejpam-4448	359	21	i	i	NOUN
ejpam-4448	359	22	)	)	PUNCT
ejpam-4448	359	23	or	or	CCONJ
ejpam-4448	359	24	(	(	PUNCT
ejpam-4448	359	25	ii	ii	NOUN
ejpam-4448	359	26	)	)	PUNCT
ejpam-4448	359	27	.	.	PUNCT
ejpam-4448	360	1	construct	construct	VERB
ejpam-4448	360	2	a	a	DET
ejpam-4448	360	3	vector	vector	NOUN
ejpam-4448	360	4	x	x	PUNCT
ejpam-4448	360	5	=	=	PUNCT
ejpam-4448	361	1	[	[	X
ejpam-4448	361	2	xu]u∈v	xu]u∈v	PROPN
ejpam-4448	361	3	(	(	PUNCT
ejpam-4448	361	4	d	d	NOUN
ejpam-4448	361	5	)	)	PUNCT
ejpam-4448	361	6	as	as	SCONJ
ejpam-4448	361	7	follows	follow	VERB
ejpam-4448	361	8	.	.	PUNCT
ejpam-4448	362	1	set	set	VERB
ejpam-4448	362	2	xu	xu	INTJ
ejpam-4448	363	1	:	:	PUNCT
ejpam-4448	363	2	=	=	PUNCT
ejpam-4448	363	3	q	q	X
ejpam-4448	363	4	for	for	ADP
ejpam-4448	363	5	any	any	DET
ejpam-4448	363	6	u	u	PROPN
ejpam-4448	363	7	∈	∈	PROPN
ejpam-4448	363	8	vq	vq	NOUN
ejpam-4448	363	9	.	.	PUNCT
ejpam-4448	364	1	it	it	PRON
ejpam-4448	364	2	is	be	AUX
ejpam-4448	364	3	straightforward	straightforward	ADJ
ejpam-4448	364	4	to	to	PART
ejpam-4448	364	5	show	show	VERB
ejpam-4448	364	6	that	that	SCONJ
ejpam-4448	364	7	x	x	PRON
ejpam-4448	364	8	is	be	AUX
ejpam-4448	364	9	an	an	DET
ejpam-4448	364	10	α	α	NOUN
ejpam-4448	364	11	-	-	NOUN
ejpam-4448	364	12	eigenvector	eigenvector	NOUN
ejpam-4448	364	13	for	for	ADP
ejpam-4448	364	14	an	an	DET
ejpam-4448	364	15	eigenvalue	eigenvalue	NOUN
ejpam-4448	364	16	of	of	ADP
ejpam-4448	364	17	modulus	modulus	ADJ
ejpam-4448	364	18	ρα(d	ρα(d	NOUN
ejpam-4448	364	19	)	)	PUNCT
ejpam-4448	364	20	.	.	PUNCT
ejpam-4448	365	1	corollary	corollary	ADJ
ejpam-4448	365	2	7	7	NUM
ejpam-4448	365	3	.	.	PUNCT
ejpam-4448	366	1	let	let	VERB
ejpam-4448	366	2	d	d	PRON
ejpam-4448	366	3	be	be	AUX
ejpam-4448	366	4	a	a	DET
ejpam-4448	366	5	connected	connected	ADJ
ejpam-4448	366	6	mixed	mixed	ADJ
ejpam-4448	366	7	graph	graph	NOUN
ejpam-4448	366	8	.	.	PUNCT
ejpam-4448	367	1	suppose	suppose	VERB
ejpam-4448	367	2	that	that	SCONJ
ejpam-4448	367	3	αk	αk	PRON
ejpam-4448	367	4	̸=	̸=	PROPN
ejpam-4448	367	5	−αl	−αl	X
ejpam-4448	367	6	for	for	ADP
ejpam-4448	367	7	all	all	DET
ejpam-4448	367	8	k	k	PROPN
ejpam-4448	367	9	,	,	PUNCT
ejpam-4448	367	10	l	l	PROPN
ejpam-4448	367	11	∈	∈	PROPN
ejpam-4448	367	12	z.	z.	PROPN
ejpam-4448	367	13	then	then	ADV
ejpam-4448	367	14	,	,	PUNCT
ejpam-4448	367	15	ρα(d	ρα(d	X
ejpam-4448	367	16	)	)	PUNCT
ejpam-4448	367	17	=	=	SYM
ejpam-4448	367	18	∆γ(d	∆γ(d	PROPN
ejpam-4448	367	19	)	)	PUNCT
ejpam-4448	367	20	if	if	SCONJ
ejpam-4448	367	21	and	and	CCONJ
ejpam-4448	367	22	only	only	ADV
ejpam-4448	367	23	if	if	SCONJ
ejpam-4448	367	24	d	d	NOUN
ejpam-4448	367	25	is	be	AUX
ejpam-4448	367	26	a	a	DET
ejpam-4448	367	27	regular	regular	ADJ
ejpam-4448	367	28	α	α	NOUN
ejpam-4448	367	29	-	-	NOUN
ejpam-4448	367	30	monograph	monograph	NOUN
ejpam-4448	367	31	of	of	ADP
ejpam-4448	367	32	1st	1st	ADJ
ejpam-4448	367	33	kind	kind	NOUN
ejpam-4448	367	34	.	.	PUNCT
ejpam-4448	368	1	proof	proof	NOUN
ejpam-4448	368	2	.	.	PUNCT
ejpam-4448	369	1	the	the	DET
ejpam-4448	369	2	given	give	VERB
ejpam-4448	369	3	condition	condition	NOUN
ejpam-4448	369	4	on	on	ADP
ejpam-4448	369	5	α	α	PRON
ejpam-4448	369	6	guarantees	guarantee	VERB
ejpam-4448	369	7	that	that	SCONJ
ejpam-4448	369	8	the	the	DET
ejpam-4448	369	9	partition	partition	NOUN
ejpam-4448	369	10	arising	arise	VERB
ejpam-4448	369	11	in	in	ADP
ejpam-4448	369	12	case	case	NOUN
ejpam-4448	369	13	(	(	PUNCT
ejpam-4448	369	14	ii	ii	NOUN
ejpam-4448	369	15	)	)	PUNCT
ejpam-4448	369	16	in	in	ADP
ejpam-4448	369	17	the	the	DET
ejpam-4448	369	18	proof	proof	NOUN
ejpam-4448	369	19	of	of	ADP
ejpam-4448	369	20	theorem	theorem	ADJ
ejpam-4448	369	21	13	13	NUM
ejpam-4448	369	22	can	can	AUX
ejpam-4448	369	23	be	be	AUX
ejpam-4448	369	24	converted	convert	VERB
ejpam-4448	369	25	into	into	ADP
ejpam-4448	369	26	a	a	DET
ejpam-4448	369	27	bipartition	bipartition	NOUN
ejpam-4448	369	28	v	v	ADP
ejpam-4448	369	29	(	(	PUNCT
ejpam-4448	369	30	d	d	NOUN
ejpam-4448	369	31	)	)	PUNCT
ejpam-4448	369	32	=	=	SYM
ejpam-4448	369	33	v	v	NOUN
ejpam-4448	369	34	′∪̇v	′∪̇v	NOUN
ejpam-4448	369	35	′′	′′	PROPN
ejpam-4448	369	36	,	,	PUNCT
ejpam-4448	369	37	with	with	ADP
ejpam-4448	369	38	v	v	NOUN
ejpam-4448	369	39	′	′	NUM
ejpam-4448	369	40	=	=	PUNCT
ejpam-4448	369	41	vα0	vα0	NOUN
ejpam-4448	369	42	∪	∪	VERB
ejpam-4448	369	43	vα1	vα1	NOUN
ejpam-4448	369	44	∪	∪	ADJ
ejpam-4448	369	45	.	.	PUNCT
ejpam-4448	369	46	.	.	PUNCT
ejpam-4448	370	1	.	.	PUNCT
ejpam-4448	371	1	and	and	CCONJ
ejpam-4448	371	2	v	v	ADP
ejpam-4448	371	3	′′	′′	PROPN
ejpam-4448	371	4	=	=	PRON
ejpam-4448	371	5	v−α0	v−α0	NOUN
ejpam-4448	371	6	∪	∪	VERB
ejpam-4448	371	7	v−α1	v−α1	NOUN
ejpam-4448	371	8	∪	∪	ADJ
ejpam-4448	371	9	.	.	PUNCT
ejpam-4448	371	10	.	.	PUNCT
ejpam-4448	372	1	.	.	PUNCT
ejpam-4448	373	1	,	,	PUNCT
ejpam-4448	373	2	such	such	ADJ
ejpam-4448	373	3	that	that	SCONJ
ejpam-4448	373	4	the	the	DET
ejpam-4448	373	5	subgraphs	subgraph	NOUN
ejpam-4448	373	6	induced	induce	VERB
ejpam-4448	373	7	by	by	ADP
ejpam-4448	373	8	v	v	NOUN
ejpam-4448	373	9	′	′	NUM
ejpam-4448	373	10	and	and	CCONJ
ejpam-4448	373	11	v	v	ADP
ejpam-4448	373	12	′′	′′	PROPN
ejpam-4448	373	13	have	have	VERB
ejpam-4448	373	14	no	no	DET
ejpam-4448	373	15	edges	edge	NOUN
ejpam-4448	373	16	.	.	PUNCT
ejpam-4448	374	1	so	so	ADV
ejpam-4448	374	2	d	d	NOUN
ejpam-4448	374	3	would	would	AUX
ejpam-4448	374	4	be	be	AUX
ejpam-4448	374	5	bipartite	bipartite	ADJ
ejpam-4448	374	6	in	in	ADP
ejpam-4448	374	7	this	this	DET
ejpam-4448	374	8	case	case	NOUN
ejpam-4448	374	9	.	.	PUNCT
ejpam-4448	375	1	forming	form	VERB
ejpam-4448	375	2	pairwise	pairwise	NOUN
ejpam-4448	375	3	unions	union	NOUN
ejpam-4448	375	4	vαk	vαk	ADP
ejpam-4448	375	5	∪	∪	ADJ
ejpam-4448	375	6	v−αk	v−αk	NOUN
ejpam-4448	375	7	,	,	PUNCT
ejpam-4448	375	8	it	it	PRON
ejpam-4448	375	9	becomes	become	VERB
ejpam-4448	375	10	apparent	apparent	ADJ
ejpam-4448	375	11	that	that	SCONJ
ejpam-4448	375	12	d	d	NOUN
ejpam-4448	375	13	must	must	AUX
ejpam-4448	375	14	be	be	AUX
ejpam-4448	375	15	an	an	DET
ejpam-4448	375	16	α	α	NOUN
ejpam-4448	375	17	-	-	PUNCT
ejpam-4448	375	18	monograph	monograph	NOUN
ejpam-4448	375	19	of	of	ADP
ejpam-4448	375	20	1st	1st	ADJ
ejpam-4448	375	21	kind	kind	NOUN
ejpam-4448	375	22	.	.	PUNCT
ejpam-4448	376	1	to	to	PART
ejpam-4448	376	2	conclude	conclude	VERB
ejpam-4448	376	3	,	,	PUNCT
ejpam-4448	376	4	let	let	VERB
ejpam-4448	376	5	us	we	PRON
ejpam-4448	376	6	briefly	briefly	ADV
ejpam-4448	376	7	revisit	revisit	VERB
ejpam-4448	376	8	the	the	DET
ejpam-4448	376	9	three	three	NUM
ejpam-4448	376	10	special	special	ADJ
ejpam-4448	376	11	choices	choice	NOUN
ejpam-4448	376	12	i	i	PRON
ejpam-4448	376	13	,	,	PUNCT
ejpam-4448	376	14	ω	ω	PROPN
ejpam-4448	376	15	and	and	CCONJ
ejpam-4448	376	16	γ	γ	PROPN
ejpam-4448	376	17	for	for	ADP
ejpam-4448	376	18	α	α	NOUN
ejpam-4448	376	19	.	.	PUNCT
ejpam-4448	377	1	choosing	choose	VERB
ejpam-4448	377	2	α	α	PROPN
ejpam-4448	377	3	=	=	SYM
ejpam-4448	377	4	γ	γ	X
ejpam-4448	377	5	,	,	PUNCT
ejpam-4448	377	6	the	the	DET
ejpam-4448	377	7	conditions	condition	NOUN
ejpam-4448	377	8	of	of	ADP
ejpam-4448	377	9	corollary	corollary	ADJ
ejpam-4448	377	10	7	7	NUM
ejpam-4448	377	11	are	be	AUX
ejpam-4448	377	12	met	meet	VERB
ejpam-4448	377	13	,	,	PUNCT
ejpam-4448	377	14	but	but	CCONJ
ejpam-4448	377	15	not	not	PART
ejpam-4448	377	16	for	for	ADP
ejpam-4448	377	17	α	α	DET
ejpam-4448	377	18	∈	∈	PROPN
ejpam-4448	377	19	{	{	PUNCT
ejpam-4448	377	20	i	i	PROPN
ejpam-4448	377	21	,	,	PUNCT
ejpam-4448	377	22	ω	ω	PROPN
ejpam-4448	377	23	}	}	PUNCT
ejpam-4448	377	24	.	.	PUNCT
ejpam-4448	378	1	in	in	ADP
ejpam-4448	378	2	view	view	NOUN
ejpam-4448	378	3	of	of	ADP
ejpam-4448	378	4	this	this	PRON
ejpam-4448	378	5	,	,	PUNCT
ejpam-4448	378	6	γ	γ	PROPN
ejpam-4448	378	7	appears	appear	VERB
ejpam-4448	378	8	to	to	PART
ejpam-4448	378	9	be	be	AUX
ejpam-4448	378	10	an	an	DET
ejpam-4448	378	11	interesting	interesting	ADJ
ejpam-4448	378	12	choice	choice	NOUN
ejpam-4448	378	13	for	for	ADP
ejpam-4448	378	14	α	α	NOUN
ejpam-4448	378	15	.	.	PUNCT
ejpam-4448	379	1	moreover	moreover	ADV
ejpam-4448	379	2	,	,	PUNCT
ejpam-4448	379	3	since	since	SCONJ
ejpam-4448	379	4	γ	γ	PROPN
ejpam-4448	379	5	is	be	AUX
ejpam-4448	379	6	a	a	DET
ejpam-4448	379	7	third	third	ADJ
ejpam-4448	379	8	root	root	NOUN
ejpam-4448	379	9	of	of	ADP
ejpam-4448	379	10	unity	unity	NOUN
ejpam-4448	379	11	,	,	PUNCT
ejpam-4448	379	12	among	among	ADP
ejpam-4448	379	13	all	all	DET
ejpam-4448	379	14	candidates	candidate	NOUN
ejpam-4448	379	15	satisfying	satisfy	VERB
ejpam-4448	379	16	the	the	DET
ejpam-4448	379	17	conditions	condition	NOUN
ejpam-4448	379	18	of	of	ADP
ejpam-4448	379	19	corollary	corollary	ADJ
ejpam-4448	379	20	7	7	NUM
ejpam-4448	379	21	,	,	PUNCT
ejpam-4448	379	22	choosing	choose	VERB
ejpam-4448	379	23	α	α	NOUN
ejpam-4448	379	24	=	=	X
ejpam-4448	379	25	γ	γ	X
ejpam-4448	379	26	will	will	AUX
ejpam-4448	379	27	yields	yield	VERB
ejpam-4448	379	28	a	a	DET
ejpam-4448	379	29	minimal	minimal	ADJ
ejpam-4448	379	30	number	number	NOUN
ejpam-4448	379	31	of	of	ADP
ejpam-4448	379	32	sets	set	NOUN
ejpam-4448	379	33	in	in	ADP
ejpam-4448	379	34	the	the	DET
ejpam-4448	379	35	monograph	monograph	NOUN
ejpam-4448	379	36	vertex	vertex	NOUN
ejpam-4448	379	37	partition	partition	NOUN
ejpam-4448	379	38	.	.	PUNCT
ejpam-4448	380	1	references	reference	NOUN
ejpam-4448	380	2	[	[	X
ejpam-4448	380	3	1	1	NUM
ejpam-4448	380	4	]	]	X
ejpam-4448	380	5	mohammad	mohammad	PROPN
ejpam-4448	380	6	abudayah	abudayah	PROPN
ejpam-4448	380	7	,	,	PUNCT
ejpam-4448	380	8	omar	omar	PROPN
ejpam-4448	380	9	alomari	alomari	PROPN
ejpam-4448	380	10	,	,	PUNCT
ejpam-4448	380	11	and	and	CCONJ
ejpam-4448	380	12	torsten	torsten	PROPN
ejpam-4448	380	13	sander	sander	PROPN
ejpam-4448	380	14	.	.	PUNCT
ejpam-4448	381	1	on	on	ADP
ejpam-4448	381	2	the	the	DET
ejpam-4448	381	3	n	n	NOUN
ejpam-4448	381	4	-	-	PUNCT
ejpam-4448	381	5	spectrum	spectrum	NOUN
ejpam-4448	381	6	of	of	ADP
ejpam-4448	381	7	oriented	orient	VERB
ejpam-4448	381	8	graphs	graph	NOUN
ejpam-4448	381	9	.	.	PUNCT
ejpam-4448	382	1	open	open	ADJ
ejpam-4448	382	2	mathematics	mathematic	NOUN
ejpam-4448	382	3	,	,	PUNCT
ejpam-4448	382	4	18(1):486–495	18(1):486–495	NUM
ejpam-4448	382	5	,	,	PUNCT
ejpam-4448	382	6	2020	2020	NUM
ejpam-4448	382	7	.	.	PUNCT
ejpam-4448	383	1	[	[	X
ejpam-4448	383	2	2	2	NUM
ejpam-4448	383	3	]	]	X
ejpam-4448	383	4	omar	omar	PROPN
ejpam-4448	383	5	alomari	alomari	PROPN
ejpam-4448	383	6	,	,	PUNCT
ejpam-4448	383	7	mohammad	mohammad	PROPN
ejpam-4448	383	8	abudayah	abudayah	PROPN
ejpam-4448	383	9	,	,	PUNCT
ejpam-4448	383	10	and	and	CCONJ
ejpam-4448	383	11	torsten	torsten	PROPN
ejpam-4448	383	12	sander	sander	PROPN
ejpam-4448	383	13	.	.	PUNCT
ejpam-4448	384	1	the	the	DET
ejpam-4448	384	2	non	non	ADJ
ejpam-4448	384	3	-	-	ADJ
ejpam-4448	384	4	negative	negative	ADJ
ejpam-4448	384	5	spectrum	spectrum	NOUN
ejpam-4448	384	6	of	of	ADP
ejpam-4448	384	7	a	a	DET
ejpam-4448	384	8	digraph	digraph	NOUN
ejpam-4448	384	9	.	.	PUNCT
ejpam-4448	385	1	open	open	ADJ
ejpam-4448	385	2	mathematics	mathematic	NOUN
ejpam-4448	385	3	,	,	PUNCT
ejpam-4448	385	4	18(1):22–35	18(1):22–35	NUM
ejpam-4448	385	5	,	,	PUNCT
ejpam-4448	385	6	2020	2020	NUM
ejpam-4448	385	7	.	.	PUNCT
ejpam-4448	386	1	[	[	X
ejpam-4448	386	2	3	3	X
ejpam-4448	386	3	]	]	X
ejpam-4448	386	4	norman	norman	PROPN
ejpam-4448	386	5	biggs	biggs	PROPN
ejpam-4448	386	6	.	.	PUNCT
ejpam-4448	387	1	algebraic	algebraic	PROPN
ejpam-4448	387	2	graph	graph	NOUN
ejpam-4448	387	3	theory	theory	NOUN
ejpam-4448	387	4	.	.	PUNCT
ejpam-4448	388	1	cambridge	cambridge	PROPN
ejpam-4448	388	2	university	university	PROPN
ejpam-4448	388	3	press	press	NOUN
ejpam-4448	388	4	,	,	PUNCT
ejpam-4448	388	5	2nd	2nd	PROPN
ejpam-4448	388	6	ed	ed	NOUN
ejpam-4448	388	7	.	.	PUNCT
ejpam-4448	388	8	edition	edition	PROPN
ejpam-4448	388	9	,	,	PUNCT
ejpam-4448	388	10	1994	1994	NUM
ejpam-4448	388	11	.	.	PUNCT
ejpam-4448	389	1	[	[	X
ejpam-4448	389	2	4	4	X
ejpam-4448	389	3	]	]	X
ejpam-4448	389	4	richard	richard	PROPN
ejpam-4448	389	5	a.	a.	PROPN
ejpam-4448	389	6	brualdi	brualdi	PROPN
ejpam-4448	389	7	.	.	PUNCT
ejpam-4448	390	1	spectra	spectra	PROPN
ejpam-4448	390	2	of	of	ADP
ejpam-4448	390	3	digraphs	digraph	NOUN
ejpam-4448	390	4	.	.	PUNCT
ejpam-4448	391	1	linear	linear	ADJ
ejpam-4448	391	2	algebra	algebra	PROPN
ejpam-4448	391	3	appl	appl	NOUN
ejpam-4448	391	4	.	.	PUNCT
ejpam-4448	391	5	,	,	PUNCT
ejpam-4448	392	1	432(9):2181–2213	432(9):2181–2213	PROPN
ejpam-4448	392	2	,	,	PUNCT
ejpam-4448	392	3	2010	2010	NUM
ejpam-4448	392	4	.	.	PUNCT
ejpam-4448	393	1	[	[	X
ejpam-4448	393	2	5	5	NUM
ejpam-4448	393	3	]	]	X
ejpam-4448	393	4	krystal	krystal	ADJ
ejpam-4448	393	5	guo	guo	PROPN
ejpam-4448	393	6	and	and	CCONJ
ejpam-4448	393	7	bojan	bojan	PROPN
ejpam-4448	393	8	mohar	mohar	PROPN
ejpam-4448	393	9	.	.	PUNCT
ejpam-4448	394	1	hermitian	hermitian	ADJ
ejpam-4448	394	2	adjacency	adjacency	PROPN
ejpam-4448	394	3	matrix	matrix	NOUN
ejpam-4448	394	4	of	of	ADP
ejpam-4448	394	5	digraphs	digraph	NOUN
ejpam-4448	394	6	and	and	CCONJ
ejpam-4448	394	7	mixed	mixed	ADJ
ejpam-4448	394	8	graphs	graph	NOUN
ejpam-4448	394	9	.	.	PUNCT
ejpam-4448	395	1	j.	j.	PROPN
ejpam-4448	395	2	graph	graph	PROPN
ejpam-4448	395	3	theory	theory	NOUN
ejpam-4448	395	4	,	,	PUNCT
ejpam-4448	395	5	85(1):217–248	85(1):217–248	PROPN
ejpam-4448	395	6	,	,	PUNCT
ejpam-4448	395	7	2017	2017	NUM
ejpam-4448	395	8	.	.	PUNCT
ejpam-4448	396	1	references	reference	NOUN
ejpam-4448	396	2	855	855	NUM
ejpam-4448	396	3	[	[	X
ejpam-4448	396	4	6	6	NUM
ejpam-4448	396	5	]	]	PUNCT
ejpam-4448	396	6	frank	frank	PROPN
ejpam-4448	396	7	harary	harary	PROPN
ejpam-4448	396	8	.	.	PUNCT
ejpam-4448	397	1	the	the	DET
ejpam-4448	397	2	determinant	determinant	NOUN
ejpam-4448	397	3	of	of	ADP
ejpam-4448	397	4	the	the	DET
ejpam-4448	397	5	adjacency	adjacency	NOUN
ejpam-4448	397	6	matrix	matrix	NOUN
ejpam-4448	397	7	of	of	ADP
ejpam-4448	397	8	a	a	DET
ejpam-4448	397	9	graph	graph	NOUN
ejpam-4448	397	10	.	.	PUNCT
ejpam-4448	398	1	siam	siam	PROPN
ejpam-4448	398	2	rev	rev	PROPN
ejpam-4448	398	3	.	.	PROPN
ejpam-4448	398	4	,	,	PUNCT
ejpam-4448	398	5	4:202–210	4:202–210	PROPN
ejpam-4448	398	6	,	,	PUNCT
ejpam-4448	398	7	1962	1962	NUM
ejpam-4448	398	8	.	.	PUNCT
ejpam-4448	399	1	[	[	X
ejpam-4448	399	2	7	7	X
ejpam-4448	399	3	]	]	X
ejpam-4448	399	4	roger	roger	PROPN
ejpam-4448	399	5	a.	a.	NOUN
ejpam-4448	399	6	horn	horn	PROPN
ejpam-4448	399	7	and	and	CCONJ
ejpam-4448	399	8	charles	charles	PROPN
ejpam-4448	399	9	r.	r.	PROPN
ejpam-4448	399	10	johnson	johnson	PROPN
ejpam-4448	399	11	.	.	PUNCT
ejpam-4448	400	1	matrix	matrix	NOUN
ejpam-4448	400	2	analysis	analysis	NOUN
ejpam-4448	400	3	.	.	PUNCT
ejpam-4448	401	1	reprinted	reprint	VERB
ejpam-4448	401	2	with	with	ADP
ejpam-4448	401	3	corrections	correction	NOUN
ejpam-4448	401	4	.	.	PUNCT
ejpam-4448	402	1	cambridge	cambridge	PROPN
ejpam-4448	402	2	university	university	PROPN
ejpam-4448	402	3	press	press	NOUN
ejpam-4448	402	4	,	,	PUNCT
ejpam-4448	402	5	1990	1990	NUM
ejpam-4448	402	6	.	.	PUNCT
ejpam-4448	403	1	[	[	X
ejpam-4448	403	2	8	8	NUM
ejpam-4448	403	3	]	]	X
ejpam-4448	403	4	irena	irena	NOUN
ejpam-4448	403	5	m.	m.	NOUN
ejpam-4448	403	6	jovanović.	jovanović.	PROPN
ejpam-4448	403	7	non	non	ADJ
ejpam-4448	403	8	-	-	ADJ
ejpam-4448	403	9	negative	negative	ADJ
ejpam-4448	403	10	spectrum	spectrum	NOUN
ejpam-4448	403	11	of	of	ADP
ejpam-4448	403	12	a	a	DET
ejpam-4448	403	13	digraph	digraph	NOUN
ejpam-4448	403	14	.	.	PUNCT
ejpam-4448	403	15	ars	ars	PROPN
ejpam-4448	403	16	math	math	PROPN
ejpam-4448	403	17	.	.	PUNCT
ejpam-4448	404	1	contemp	contemp	NOUN
ejpam-4448	404	2	.	.	PUNCT
ejpam-4448	405	1	,	,	PUNCT
ejpam-4448	405	2	12(1):167–182	12(1):167–182	PROPN
ejpam-4448	405	3	,	,	PUNCT
ejpam-4448	405	4	2017	2017	NUM
ejpam-4448	405	5	.	.	PUNCT
ejpam-4448	406	1	[	[	X
ejpam-4448	406	2	9	9	NUM
ejpam-4448	406	3	]	]	X
ejpam-4448	406	4	honghai	honghai	PROPN
ejpam-4448	406	5	li	li	PROPN
ejpam-4448	406	6	and	and	CCONJ
ejpam-4448	406	7	teng	teng	PROPN
ejpam-4448	406	8	yu	yu	PROPN
ejpam-4448	406	9	.	.	PROPN
ejpam-4448	406	10	hermitian	hermitian	PROPN
ejpam-4448	406	11	adjacency	adjacency	PROPN
ejpam-4448	406	12	spectrum	spectrum	NOUN
ejpam-4448	406	13	of	of	ADP
ejpam-4448	406	14	cayley	cayley	NOUN
ejpam-4448	406	15	digraphs	digraph	VERB
ejpam-4448	406	16	over	over	ADP
ejpam-4448	406	17	dihedral	dihedral	ADJ
ejpam-4448	406	18	group	group	NOUN
ejpam-4448	406	19	.	.	PUNCT
ejpam-4448	407	1	algebra	algebra	PROPN
ejpam-4448	407	2	colloq	colloq	PROPN
ejpam-4448	407	3	.	.	PUNCT
ejpam-4448	407	4	,	,	PUNCT
ejpam-4448	407	5	27(1):121–130	27(1):121–130	NUM
ejpam-4448	407	6	,	,	PUNCT
ejpam-4448	407	7	2020	2020	NUM
ejpam-4448	407	8	.	.	PUNCT
ejpam-4448	408	1	[	[	X
ejpam-4448	408	2	10	10	NUM
ejpam-4448	408	3	]	]	X
ejpam-4448	408	4	jianxi	jianxi	PROPN
ejpam-4448	408	5	liu	liu	PROPN
ejpam-4448	408	6	and	and	CCONJ
ejpam-4448	408	7	xueliang	xueliang	PROPN
ejpam-4448	408	8	li	li	PROPN
ejpam-4448	408	9	.	.	PUNCT
ejpam-4448	409	1	hermitian	hermitian	ADJ
ejpam-4448	409	2	-	-	PUNCT
ejpam-4448	409	3	adjacency	adjacency	NOUN
ejpam-4448	409	4	matrices	matrix	NOUN
ejpam-4448	409	5	and	and	CCONJ
ejpam-4448	409	6	hermitian	hermitian	ADJ
ejpam-4448	409	7	energies	energy	NOUN
ejpam-4448	409	8	of	of	ADP
ejpam-4448	409	9	mixed	mixed	ADJ
ejpam-4448	409	10	graphs	graph	NOUN
ejpam-4448	409	11	.	.	PUNCT
ejpam-4448	410	1	linear	linear	ADJ
ejpam-4448	410	2	algebra	algebra	PROPN
ejpam-4448	410	3	appl	appl	NOUN
ejpam-4448	410	4	.	.	PROPN
ejpam-4448	410	5	,	,	PUNCT
ejpam-4448	410	6	466:182–207	466:182–207	NUM
ejpam-4448	410	7	,	,	PUNCT
ejpam-4448	410	8	2015	2015	NUM
ejpam-4448	410	9	.	.	PUNCT
ejpam-4448	411	1	[	[	X
ejpam-4448	411	2	11	11	NUM
ejpam-4448	411	3	]	]	PUNCT
ejpam-4448	411	4	bojan	bojan	PROPN
ejpam-4448	411	5	mohar	mohar	PROPN
ejpam-4448	411	6	.	.	PUNCT
ejpam-4448	412	1	a	a	DET
ejpam-4448	412	2	new	new	ADJ
ejpam-4448	412	3	kind	kind	NOUN
ejpam-4448	412	4	of	of	ADP
ejpam-4448	412	5	hermitian	hermitian	ADJ
ejpam-4448	412	6	matrices	matrix	NOUN
ejpam-4448	412	7	for	for	ADP
ejpam-4448	412	8	digraphs	digraph	NOUN
ejpam-4448	412	9	.	.	PUNCT
ejpam-4448	413	1	linear	linear	ADJ
ejpam-4448	413	2	algebra	algebra	PROPN
ejpam-4448	413	3	appl	appl	NOUN
ejpam-4448	413	4	.	.	PROPN
ejpam-4448	413	5	,	,	PUNCT
ejpam-4448	413	6	584:343–352	584:343–352	NUM
ejpam-4448	413	7	,	,	PUNCT
ejpam-4448	413	8	2020	2020	NUM
ejpam-4448	413	9	.	.	PUNCT
ejpam-4448	414	1	[	[	X
ejpam-4448	414	2	12	12	NUM
ejpam-4448	414	3	]	]	X
ejpam-4448	414	4	bo	bo	PROPN
ejpam-4448	414	5	-	-	PUNCT
ejpam-4448	414	6	jun	jun	PROPN
ejpam-4448	414	7	yuan	yuan	PROPN
ejpam-4448	414	8	,	,	PUNCT
ejpam-4448	414	9	yi	yi	PROPN
ejpam-4448	414	10	wang	wang	PROPN
ejpam-4448	414	11	,	,	PUNCT
ejpam-4448	414	12	shi	shi	PROPN
ejpam-4448	414	13	-	-	PUNCT
ejpam-4448	414	14	cai	cai	PROPN
ejpam-4448	414	15	gong	gong	PROPN
ejpam-4448	414	16	,	,	PUNCT
ejpam-4448	414	17	and	and	CCONJ
ejpam-4448	414	18	yun	yun	PROPN
ejpam-4448	414	19	qiao	qiao	PROPN
ejpam-4448	414	20	.	.	PUNCT
ejpam-4448	415	1	on	on	ADP
ejpam-4448	415	2	mixed	mixed	ADJ
ejpam-4448	415	3	graphs	graph	NOUN
ejpam-4448	415	4	whose	whose	DET
ejpam-4448	415	5	hermitian	hermitian	ADJ
ejpam-4448	415	6	spectral	spectral	ADJ
ejpam-4448	415	7	radii	radius	NOUN
ejpam-4448	415	8	are	be	AUX
ejpam-4448	415	9	at	at	ADP
ejpam-4448	415	10	most	most	ADV
ejpam-4448	415	11	2	2	NUM
ejpam-4448	415	12	.	.	PUNCT
ejpam-4448	415	13	graphs	graph	NOUN
ejpam-4448	415	14	comb	comb	NOUN
ejpam-4448	415	15	.	.	PUNCT
ejpam-4448	415	16	,	,	PUNCT
ejpam-4448	415	17	36(5):1573–1584	36(5):1573–1584	NUM
ejpam-4448	415	18	,	,	PUNCT
ejpam-4448	415	19	2020	2020	NUM
ejpam-4448	415	20	.	.	PUNCT