id	sid	tid	token	lemma	pos
ejpam-6549	1	1	european	european	PROPN
ejpam-6549	1	2	journal	journal	PROPN
ejpam-6549	1	3	of	of	ADP
ejpam-6549	1	4	pure	pure	ADJ
ejpam-6549	1	5	and	and	CCONJ
ejpam-6549	1	6	applied	applied	ADJ
ejpam-6549	1	7	mathematics	mathematic	NOUN
ejpam-6549	1	8	2025	2025	NUM
ejpam-6549	1	9	,	,	PUNCT
ejpam-6549	1	10	vol	vol	NOUN
ejpam-6549	1	11	.	.	PROPN
ejpam-6549	1	12	18	18	NUM
ejpam-6549	1	13	,	,	PUNCT
ejpam-6549	1	14	issue	issue	NOUN
ejpam-6549	1	15	3	3	NUM
ejpam-6549	1	16	,	,	PUNCT
ejpam-6549	1	17	article	article	NOUN
ejpam-6549	1	18	number	number	NOUN
ejpam-6549	1	19	6549	6549	NUM
ejpam-6549	1	20	issn	issn	PROPN
ejpam-6549	1	21	1307	1307	NUM
ejpam-6549	1	22	-	-	SYM
ejpam-6549	1	23	5543	5543	NUM
ejpam-6549	1	24	–	–	PUNCT
ejpam-6549	1	25	ejpam.com	ejpam.com	X
ejpam-6549	1	26	published	publish	VERB
ejpam-6549	1	27	by	by	ADP
ejpam-6549	1	28	new	new	PROPN
ejpam-6549	1	29	york	york	PROPN
ejpam-6549	1	30	business	business	PROPN
ejpam-6549	1	31	global	global	PROPN
ejpam-6549	1	32	the	the	DET
ejpam-6549	1	33	energy	energy	NOUN
ejpam-6549	1	34	of	of	ADP
ejpam-6549	1	35	cameron	cameron	PROPN
ejpam-6549	1	36	-	-	PUNCT
ejpam-6549	1	37	walker	walker	PROPN
ejpam-6549	1	38	graphs	graphs	PROPN
ejpam-6549	1	39	alper	alper	PROPN
ejpam-6549	1	40	ülker1	ülker1	NOUN
ejpam-6549	1	41	,	,	PUNCT
ejpam-6549	1	42	tahsin	tahsin	NOUN
ejpam-6549	1	43	oner2	oner2	VERB
ejpam-6549	1	44	,	,	PUNCT
ejpam-6549	1	45	aiyared	aiyare	VERB
ejpam-6549	1	46	iampan3,∗	iampan3,∗	NOUN
ejpam-6549	1	47	,	,	PUNCT
ejpam-6549	1	48	burak	burak	PROPN
ejpam-6549	1	49	ordin2	ordin2	PROPN
ejpam-6549	1	50	1	1	NUM
ejpam-6549	1	51	department	department	NOUN
ejpam-6549	1	52	of	of	ADP
ejpam-6549	1	53	mathematics	mathematic	NOUN
ejpam-6549	1	54	and	and	CCONJ
ejpam-6549	1	55	computer	computer	NOUN
ejpam-6549	1	56	science	science	NOUN
ejpam-6549	1	57	,	,	PUNCT
ejpam-6549	1	58	istanbul	istanbul	PROPN
ejpam-6549	1	59	kültür	kültür	PROPN
ejpam-6549	1	60	university	university	PROPN
ejpam-6549	1	61	,	,	PUNCT
ejpam-6549	1	62	34156	34156	NUM
ejpam-6549	1	63	istanbul	istanbul	PROPN
ejpam-6549	1	64	,	,	PUNCT
ejpam-6549	1	65	turkey	turkey	PROPN
ejpam-6549	1	66	2	2	NUM
ejpam-6549	1	67	department	department	NOUN
ejpam-6549	1	68	of	of	ADP
ejpam-6549	1	69	mathematics	mathematic	NOUN
ejpam-6549	1	70	,	,	PUNCT
ejpam-6549	1	71	faculty	faculty	NOUN
ejpam-6549	1	72	of	of	ADP
ejpam-6549	1	73	science	science	NOUN
ejpam-6549	1	74	,	,	PUNCT
ejpam-6549	1	75	ege	ege	PROPN
ejpam-6549	1	76	university	university	NOUN
ejpam-6549	1	77	,	,	PUNCT
ejpam-6549	1	78	35100	35100	NUM
ejpam-6549	1	79	izmir	izmir	PROPN
ejpam-6549	1	80	,	,	PUNCT
ejpam-6549	1	81	turkey	turkey	PROPN
ejpam-6549	1	82	3	3	NUM
ejpam-6549	1	83	department	department	NOUN
ejpam-6549	1	84	of	of	ADP
ejpam-6549	1	85	mathematics	mathematic	NOUN
ejpam-6549	1	86	,	,	PUNCT
ejpam-6549	1	87	school	school	NOUN
ejpam-6549	1	88	of	of	ADP
ejpam-6549	1	89	science	science	NOUN
ejpam-6549	1	90	,	,	PUNCT
ejpam-6549	1	91	university	university	NOUN
ejpam-6549	1	92	of	of	ADP
ejpam-6549	1	93	phayao	phayao	NOUN
ejpam-6549	1	94	,	,	PUNCT
ejpam-6549	1	95	mae	mae	PROPN
ejpam-6549	1	96	ka	ka	PROPN
ejpam-6549	1	97	,	,	PUNCT
ejpam-6549	1	98	mueang	mueang	PROPN
ejpam-6549	1	99	,	,	PUNCT
ejpam-6549	1	100	phayao	phayao	NOUN
ejpam-6549	1	101	56000	56000	NUM
ejpam-6549	1	102	,	,	PUNCT
ejpam-6549	1	103	thailand	thailand	PROPN
ejpam-6549	1	104	abstract	abstract	NOUN
ejpam-6549	1	105	.	.	PUNCT
ejpam-6549	2	1	the	the	DET
ejpam-6549	2	2	cameron	cameron	PROPN
ejpam-6549	2	3	-	-	PUNCT
ejpam-6549	2	4	walker	walker	PROPN
ejpam-6549	2	5	graphs	graph	NOUN
ejpam-6549	2	6	are	be	AUX
ejpam-6549	2	7	the	the	DET
ejpam-6549	2	8	graphs	graph	NOUN
ejpam-6549	2	9	for	for	ADP
ejpam-6549	2	10	which	which	PRON
ejpam-6549	2	11	their	their	PRON
ejpam-6549	2	12	matching	match	VERB
ejpam-6549	2	13	number	number	NOUN
ejpam-6549	2	14	equals	equal	VERB
ejpam-6549	2	15	their	their	PRON
ejpam-6549	2	16	induced	induced	ADJ
ejpam-6549	2	17	matching	matching	NOUN
ejpam-6549	2	18	number	number	NOUN
ejpam-6549	2	19	.	.	PUNCT
ejpam-6549	3	1	in	in	ADP
ejpam-6549	3	2	this	this	DET
ejpam-6549	3	3	paper	paper	NOUN
ejpam-6549	3	4	,	,	PUNCT
ejpam-6549	3	5	we	we	PRON
ejpam-6549	3	6	study	study	VERB
ejpam-6549	3	7	lower	low	ADJ
ejpam-6549	3	8	bounds	bound	NOUN
ejpam-6549	3	9	for	for	ADP
ejpam-6549	3	10	the	the	DET
ejpam-6549	3	11	graph	graph	NOUN
ejpam-6549	3	12	energy	energy	NOUN
ejpam-6549	3	13	in	in	ADP
ejpam-6549	3	14	terms	term	NOUN
ejpam-6549	3	15	of	of	ADP
ejpam-6549	3	16	induced	induced	ADJ
ejpam-6549	3	17	matching	matching	NOUN
ejpam-6549	3	18	numbers	number	NOUN
ejpam-6549	3	19	for	for	ADP
ejpam-6549	3	20	the	the	DET
ejpam-6549	3	21	general	general	ADJ
ejpam-6549	3	22	graphs	graph	NOUN
ejpam-6549	3	23	with	with	ADP
ejpam-6549	3	24	equal	equal	ADJ
ejpam-6549	3	25	matching	matching	NOUN
ejpam-6549	3	26	and	and	CCONJ
ejpam-6549	3	27	induced	induced	ADJ
ejpam-6549	3	28	matching	matching	NOUN
ejpam-6549	3	29	number	number	NOUN
ejpam-6549	3	30	.	.	PUNCT
ejpam-6549	4	1	moreover	moreover	ADV
ejpam-6549	4	2	,	,	PUNCT
ejpam-6549	4	3	we	we	PRON
ejpam-6549	4	4	study	study	VERB
ejpam-6549	4	5	the	the	DET
ejpam-6549	4	6	lower	low	ADJ
ejpam-6549	4	7	bounds	bound	NOUN
ejpam-6549	4	8	of	of	ADP
ejpam-6549	4	9	the	the	DET
ejpam-6549	4	10	graph	graph	NOUN
ejpam-6549	4	11	energy	energy	NOUN
ejpam-6549	4	12	of	of	ADP
ejpam-6549	4	13	cameron	cameron	PROPN
ejpam-6549	4	14	-	-	PUNCT
ejpam-6549	4	15	walker	walker	PROPN
ejpam-6549	4	16	graphs	graph	NOUN
ejpam-6549	4	17	.	.	PUNCT
ejpam-6549	5	1	2020	2020	NUM
ejpam-6549	5	2	mathematics	mathematic	NOUN
ejpam-6549	5	3	subject	subject	NOUN
ejpam-6549	5	4	classifications	classification	NOUN
ejpam-6549	5	5	:	:	PUNCT
ejpam-6549	5	6	05c50	05c50	NUM
ejpam-6549	5	7	,	,	PUNCT
ejpam-6549	5	8	05c70	05c70	NUM
ejpam-6549	5	9	,	,	PUNCT
ejpam-6549	5	10	05c35	05c35	NUM
ejpam-6549	5	11	key	key	ADJ
ejpam-6549	5	12	words	word	NOUN
ejpam-6549	5	13	and	and	CCONJ
ejpam-6549	5	14	phrases	phrase	NOUN
ejpam-6549	5	15	:	:	PUNCT
ejpam-6549	5	16	cameron	cameron	PROPN
ejpam-6549	5	17	-	-	PUNCT
ejpam-6549	5	18	walker	walker	PROPN
ejpam-6549	5	19	graph	graph	NOUN
ejpam-6549	5	20	,	,	PUNCT
ejpam-6549	5	21	matching	match	VERB
ejpam-6549	5	22	number	number	NOUN
ejpam-6549	5	23	,	,	PUNCT
ejpam-6549	5	24	graph	graph	NOUN
ejpam-6549	5	25	energy	energy	NOUN
ejpam-6549	5	26	,	,	PUNCT
ejpam-6549	5	27	lower	lower	ADV
ejpam-6549	5	28	bound	bound	ADJ
ejpam-6549	5	29	1	1	NUM
ejpam-6549	5	30	.	.	PUNCT
ejpam-6549	6	1	introduction	introduction	NOUN
ejpam-6549	6	2	the	the	DET
ejpam-6549	6	3	energy	energy	NOUN
ejpam-6549	6	4	of	of	ADP
ejpam-6549	6	5	a	a	DET
ejpam-6549	6	6	graph	graph	NOUN
ejpam-6549	6	7	is	be	AUX
ejpam-6549	6	8	a	a	DET
ejpam-6549	6	9	measure	measure	NOUN
ejpam-6549	6	10	that	that	PRON
ejpam-6549	6	11	directly	directly	ADV
ejpam-6549	6	12	connects	connect	VERB
ejpam-6549	6	13	to	to	ADP
ejpam-6549	6	14	hückel	hückel	NOUN
ejpam-6549	6	15	theory	theory	NOUN
ejpam-6549	6	16	and	and	CCONJ
ejpam-6549	6	17	is	be	AUX
ejpam-6549	6	18	defined	define	VERB
ejpam-6549	6	19	as	as	ADP
ejpam-6549	6	20	the	the	DET
ejpam-6549	6	21	sum	sum	NOUN
ejpam-6549	6	22	of	of	ADP
ejpam-6549	6	23	the	the	DET
ejpam-6549	6	24	absolute	absolute	ADJ
ejpam-6549	6	25	values	value	NOUN
ejpam-6549	6	26	of	of	ADP
ejpam-6549	6	27	the	the	DET
ejpam-6549	6	28	eigenvalues	eigenvalue	NOUN
ejpam-6549	6	29	of	of	ADP
ejpam-6549	6	30	its	its	PRON
ejpam-6549	6	31	adjacency	adjacency	NOUN
ejpam-6549	6	32	matrix	matrix	NOUN
ejpam-6549	6	33	.	.	PUNCT
ejpam-6549	7	1	let	let	VERB
ejpam-6549	7	2	g	g	PRON
ejpam-6549	7	3	be	be	AUX
ejpam-6549	7	4	a	a	DET
ejpam-6549	7	5	simple	simple	ADJ
ejpam-6549	7	6	,	,	PUNCT
ejpam-6549	7	7	undirected	undirected	ADJ
ejpam-6549	7	8	graph	graph	NOUN
ejpam-6549	7	9	with	with	ADP
ejpam-6549	7	10	vertex	vertex	NOUN
ejpam-6549	7	11	set	set	VERB
ejpam-6549	7	12	v	v	NOUN
ejpam-6549	7	13	(	(	PUNCT
ejpam-6549	7	14	g	g	NOUN
ejpam-6549	7	15	)	)	PUNCT
ejpam-6549	7	16	and	and	CCONJ
ejpam-6549	7	17	edge	edge	VERB
ejpam-6549	7	18	set	set	VERB
ejpam-6549	7	19	e(g	e(g	PROPN
ejpam-6549	7	20	)	)	PUNCT
ejpam-6549	7	21	.	.	PUNCT
ejpam-6549	8	1	a	a	DET
ejpam-6549	8	2	matching	matching	NOUN
ejpam-6549	8	3	in	in	ADP
ejpam-6549	8	4	g	g	PROPN
ejpam-6549	8	5	is	be	AUX
ejpam-6549	8	6	a	a	DET
ejpam-6549	8	7	subset	subset	NOUN
ejpam-6549	8	8	m	m	VERB
ejpam-6549	8	9	⊆	⊆	NUM
ejpam-6549	8	10	e(g	e(g	NOUN
ejpam-6549	8	11	)	)	PUNCT
ejpam-6549	8	12	such	such	ADJ
ejpam-6549	8	13	that	that	PRON
ejpam-6549	8	14	for	for	ADP
ejpam-6549	8	15	any	any	DET
ejpam-6549	8	16	e1	e1	NOUN
ejpam-6549	8	17	,	,	PUNCT
ejpam-6549	8	18	e2	e2	PROPN
ejpam-6549	8	19	∈	∈	PROPN
ejpam-6549	8	20	m	m	NOUN
ejpam-6549	8	21	,	,	PUNCT
ejpam-6549	8	22	we	we	PRON
ejpam-6549	8	23	have	have	VERB
ejpam-6549	8	24	e1	e1	NOUN
ejpam-6549	8	25	∩	∩	ADJ
ejpam-6549	8	26	e2	e2	NOUN
ejpam-6549	8	27	=	=	PUNCT
ejpam-6549	8	28	∅.	∅.	ADP
ejpam-6549	8	29	the	the	DET
ejpam-6549	8	30	matching	matching	ADJ
ejpam-6549	8	31	number	number	NOUN
ejpam-6549	8	32	,	,	PUNCT
ejpam-6549	8	33	denoted	denote	VERB
ejpam-6549	8	34	by	by	ADP
ejpam-6549	8	35	m(g	m(g	PROPN
ejpam-6549	8	36	)	)	PUNCT
ejpam-6549	8	37	,	,	PUNCT
ejpam-6549	8	38	is	be	AUX
ejpam-6549	8	39	the	the	DET
ejpam-6549	8	40	maximum	maximum	ADJ
ejpam-6549	8	41	size	size	NOUN
ejpam-6549	8	42	of	of	ADP
ejpam-6549	8	43	a	a	DET
ejpam-6549	8	44	matching	matching	NOUN
ejpam-6549	8	45	in	in	ADP
ejpam-6549	8	46	g.	g.	PROPN
ejpam-6549	8	47	a	a	DET
ejpam-6549	8	48	matching	match	VERB
ejpam-6549	8	49	m	m	NOUN
ejpam-6549	8	50	in	in	ADP
ejpam-6549	8	51	a	a	DET
ejpam-6549	8	52	graph	graph	NOUN
ejpam-6549	8	53	g	g	NOUN
ejpam-6549	8	54	is	be	AUX
ejpam-6549	8	55	called	call	VERB
ejpam-6549	8	56	an	an	DET
ejpam-6549	8	57	induced	induced	ADJ
ejpam-6549	8	58	matching	matching	NOUN
ejpam-6549	8	59	if	if	SCONJ
ejpam-6549	8	60	,	,	PUNCT
ejpam-6549	8	61	for	for	ADP
ejpam-6549	8	62	any	any	DET
ejpam-6549	8	63	distinct	distinct	ADJ
ejpam-6549	8	64	e1	e1	NOUN
ejpam-6549	8	65	,	,	PUNCT
ejpam-6549	8	66	e2	e2	PROPN
ejpam-6549	8	67	∈	∈	PROPN
ejpam-6549	8	68	m	m	NOUN
ejpam-6549	8	69	,	,	PUNCT
ejpam-6549	8	70	there	there	PRON
ejpam-6549	8	71	exists	exist	VERB
ejpam-6549	8	72	no	no	DET
ejpam-6549	8	73	e	e	PROPN
ejpam-6549	8	74	∈	∈	PROPN
ejpam-6549	8	75	e(g	e(g	PROPN
ejpam-6549	8	76	)	)	PUNCT
ejpam-6549	8	77	such	such	ADJ
ejpam-6549	8	78	that	that	SCONJ
ejpam-6549	8	79	e∩e1	e∩e1	PROPN
ejpam-6549	8	80	̸=	̸=	PROPN
ejpam-6549	8	81	∅	∅	NOUN
ejpam-6549	8	82	and	and	CCONJ
ejpam-6549	8	83	e∩e2	e∩e2	NUM
ejpam-6549	8	84	̸=	̸=	PROPN
ejpam-6549	8	85	∅.	∅.	ADP
ejpam-6549	8	86	the	the	DET
ejpam-6549	8	87	induced	induced	ADJ
ejpam-6549	8	88	matching	matching	NOUN
ejpam-6549	8	89	number	number	NOUN
ejpam-6549	8	90	of	of	ADP
ejpam-6549	8	91	g	g	NOUN
ejpam-6549	8	92	,	,	PUNCT
ejpam-6549	8	93	denoted	denote	VERB
ejpam-6549	8	94	by	by	ADP
ejpam-6549	8	95	im(g	im(g	NOUN
ejpam-6549	8	96	)	)	PUNCT
ejpam-6549	8	97	,	,	PUNCT
ejpam-6549	8	98	is	be	AUX
ejpam-6549	8	99	the	the	DET
ejpam-6549	8	100	maximum	maximum	ADJ
ejpam-6549	8	101	size	size	NOUN
ejpam-6549	8	102	of	of	ADP
ejpam-6549	8	103	an	an	DET
ejpam-6549	8	104	induced	induced	ADJ
ejpam-6549	8	105	matching	matching	NOUN
ejpam-6549	8	106	in	in	ADP
ejpam-6549	8	107	g.	g.	PROPN
ejpam-6549	8	108	the	the	DET
ejpam-6549	8	109	adjacency	adjacency	PROPN
ejpam-6549	8	110	matrix	matrix	NOUN
ejpam-6549	8	111	a(g	a(g	PROPN
ejpam-6549	8	112	)	)	PUNCT
ejpam-6549	8	113	of	of	ADP
ejpam-6549	8	114	a	a	DET
ejpam-6549	8	115	graph	graph	NOUN
ejpam-6549	8	116	g	g	NOUN
ejpam-6549	8	117	=	=	PUNCT
ejpam-6549	8	118	(	(	PUNCT
ejpam-6549	8	119	v	v	NOUN
ejpam-6549	8	120	(	(	PUNCT
ejpam-6549	8	121	g	g	NOUN
ejpam-6549	8	122	)	)	PUNCT
ejpam-6549	8	123	,	,	PUNCT
ejpam-6549	8	124	e(g	e(g	PROPN
ejpam-6549	8	125	)	)	PUNCT
ejpam-6549	8	126	)	)	PUNCT
ejpam-6549	8	127	is	be	AUX
ejpam-6549	8	128	a	a	DET
ejpam-6549	8	129	symmetric	symmetric	ADJ
ejpam-6549	8	130	matrix	matrix	NOUN
ejpam-6549	8	131	with	with	ADP
ejpam-6549	8	132	entries	entry	NOUN
ejpam-6549	8	133	0	0	NUM
ejpam-6549	8	134	and	and	CCONJ
ejpam-6549	8	135	1	1	NUM
ejpam-6549	8	136	.	.	X
ejpam-6549	9	1	the	the	DET
ejpam-6549	9	2	energy	energy	NOUN
ejpam-6549	9	3	of	of	ADP
ejpam-6549	9	4	a	a	DET
ejpam-6549	9	5	graph	graph	NOUN
ejpam-6549	9	6	g	g	NOUN
ejpam-6549	9	7	with	with	ADP
ejpam-6549	9	8	|v	|v	PROPN
ejpam-6549	9	9	(	(	PUNCT
ejpam-6549	9	10	g)|	g)|	NOUN
ejpam-6549	9	11	=	=	PUNCT
ejpam-6549	9	12	n	n	NOUN
ejpam-6549	9	13	is	be	AUX
ejpam-6549	9	14	defined	define	VERB
ejpam-6549	9	15	as	as	ADP
ejpam-6549	9	16	ε(g	ε(g	NOUN
ejpam-6549	9	17	)	)	PUNCT
ejpam-6549	10	1	=	=	PUNCT
ejpam-6549	10	2	n∑	n∑	NOUN
ejpam-6549	10	3	i=1	i=1	X
ejpam-6549	10	4	|λi|	|λi|	PROPN
ejpam-6549	10	5	where	where	SCONJ
ejpam-6549	10	6	λ1	λ1	ADJ
ejpam-6549	10	7	,	,	PUNCT
ejpam-6549	10	8	λ2	λ2	NOUN
ejpam-6549	10	9	,	,	PUNCT
ejpam-6549	10	10	.	.	PUNCT
ejpam-6549	10	11	.	.	PUNCT
ejpam-6549	11	1	.	.	PUNCT
ejpam-6549	12	1	,	,	PUNCT
ejpam-6549	12	2	λn	λn	PROPN
ejpam-6549	12	3	are	be	AUX
ejpam-6549	12	4	the	the	DET
ejpam-6549	12	5	eigenvalues	eigenvalue	NOUN
ejpam-6549	12	6	of	of	ADP
ejpam-6549	12	7	the	the	DET
ejpam-6549	12	8	adjacency	adjacency	NOUN
ejpam-6549	12	9	matrix	matrix	NOUN
ejpam-6549	12	10	of	of	ADP
ejpam-6549	12	11	g.	g.	PROPN
ejpam-6549	12	12	∗corresponding	∗corresponde	VERB
ejpam-6549	12	13	author	author	NOUN
ejpam-6549	12	14	.	.	PUNCT
ejpam-6549	13	1	doi	doi	NOUN
ejpam-6549	13	2	:	:	PUNCT
ejpam-6549	13	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6549	https://doi.org/10.29020/nybg.ejpam.v18i3.6549	PROPN
ejpam-6549	13	4	email	email	NOUN
ejpam-6549	13	5	addresses	address	NOUN
ejpam-6549	13	6	:	:	PUNCT
ejpam-6549	13	7	a.ulker@iku.edu.tr	a.ulker@iku.edu.tr	PRON
ejpam-6549	13	8	(	(	PUNCT
ejpam-6549	13	9	a.	a.	NOUN
ejpam-6549	13	10	ülker	ülker	PROPN
ejpam-6549	13	11	)	)	PUNCT
ejpam-6549	13	12	,	,	PUNCT
ejpam-6549	13	13	tahsin.oner@ege.edu.tr	tahsin.oner@ege.edu.tr	NOUN
ejpam-6549	13	14	(	(	PUNCT
ejpam-6549	13	15	t.	t.	NOUN
ejpam-6549	13	16	oner	oner	PROPN
ejpam-6549	13	17	)	)	PUNCT
ejpam-6549	13	18	,	,	PUNCT
ejpam-6549	13	19	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-6549	13	20	(	(	PUNCT
ejpam-6549	13	21	a.	a.	NOUN
ejpam-6549	13	22	iampan	iampan	PROPN
ejpam-6549	13	23	)	)	PUNCT
ejpam-6549	13	24	,	,	PUNCT
ejpam-6549	13	25	burak.ordin@ege.edu.tr	burak.ordin@ege.edu.tr	PROPN
ejpam-6549	13	26	(	(	PUNCT
ejpam-6549	13	27	b.	b.	PROPN
ejpam-6549	13	28	ordin	ordin	PROPN
ejpam-6549	13	29	)	)	PUNCT
ejpam-6549	13	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6549	13	31	1	1	NUM
ejpam-6549	13	32	copyright	copyright	NOUN
ejpam-6549	13	33	:	:	PUNCT
ejpam-6549	14	1	©	©	PROPN
ejpam-6549	14	2	2025	2025	NUM
ejpam-6549	14	3	the	the	DET
ejpam-6549	14	4	author(s	author(s	NOUN
ejpam-6549	14	5	)	)	PUNCT
ejpam-6549	14	6	.	.	PUNCT
ejpam-6549	15	1	(	(	PUNCT
ejpam-6549	15	2	cc	cc	NOUN
ejpam-6549	15	3	by	by	ADP
ejpam-6549	15	4	-	-	PUNCT
ejpam-6549	15	5	nc	nc	PROPN
ejpam-6549	15	6	4.0	4.0	NUM
ejpam-6549	15	7	)	)	PUNCT
ejpam-6549	15	8	a.	a.	NOUN
ejpam-6549	15	9	ülker	ülker	PROPN
ejpam-6549	15	10	et	et	PROPN
ejpam-6549	15	11	al	al	PROPN
ejpam-6549	15	12	.	.	PUNCT
ejpam-6549	15	13	/	/	SYM
ejpam-6549	15	14	eur	eur	PROPN
ejpam-6549	15	15	.	.	PUNCT
ejpam-6549	16	1	j.	j.	PROPN
ejpam-6549	16	2	pure	pure	PROPN
ejpam-6549	16	3	appl	appl	PROPN
ejpam-6549	16	4	.	.	PROPN
ejpam-6549	16	5	math	math	PROPN
ejpam-6549	16	6	,	,	PUNCT
ejpam-6549	16	7	18	18	NUM
ejpam-6549	16	8	(	(	PUNCT
ejpam-6549	16	9	3	3	NUM
ejpam-6549	16	10	)	)	PUNCT
ejpam-6549	16	11	(	(	PUNCT
ejpam-6549	16	12	2025	2025	NUM
ejpam-6549	16	13	)	)	PUNCT
ejpam-6549	16	14	,	,	PUNCT
ejpam-6549	16	15	6549	6549	NUM
ejpam-6549	16	16	2	2	NUM
ejpam-6549	16	17	of	of	ADP
ejpam-6549	16	18	8	8	NUM
ejpam-6549	16	19	the	the	DET
ejpam-6549	16	20	matching	match	VERB
ejpam-6549	16	21	polynomial	polynomial	NOUN
ejpam-6549	16	22	of	of	ADP
ejpam-6549	16	23	a	a	DET
ejpam-6549	16	24	graph	graph	NOUN
ejpam-6549	16	25	g	g	NOUN
ejpam-6549	16	26	is	be	AUX
ejpam-6549	16	27	given	give	VERB
ejpam-6549	16	28	by	by	ADP
ejpam-6549	16	29	µ(g	µ(g	NOUN
ejpam-6549	16	30	)	)	PUNCT
ejpam-6549	16	31	=	=	SYM
ejpam-6549	16	32	µ(g	µ(g	PROPN
ejpam-6549	16	33	,	,	PUNCT
ejpam-6549	16	34	λ	λ	NOUN
ejpam-6549	16	35	)	)	PUNCT
ejpam-6549	16	36	=	=	SYM
ejpam-6549	16	37	∑	∑	PUNCT
ejpam-6549	16	38	k≥0	k≥0	PROPN
ejpam-6549	16	39	(	(	PUNCT
ejpam-6549	16	40	−1)km(g	−1)km(g	ADV
ejpam-6549	16	41	,	,	PUNCT
ejpam-6549	16	42	k)λn−2k	k)λn−2k	PROPN
ejpam-6549	16	43	where	where	SCONJ
ejpam-6549	16	44	m(g	m(g	PROPN
ejpam-6549	16	45	,	,	PUNCT
ejpam-6549	16	46	k	k	NOUN
ejpam-6549	16	47	)	)	PUNCT
ejpam-6549	16	48	is	be	AUX
ejpam-6549	16	49	the	the	DET
ejpam-6549	16	50	number	number	NOUN
ejpam-6549	16	51	of	of	ADP
ejpam-6549	16	52	k	k	NOUN
ejpam-6549	16	53	-	-	NOUN
ejpam-6549	16	54	matchings	matching	NOUN
ejpam-6549	16	55	in	in	ADP
ejpam-6549	16	56	g.	g.	PROPN
ejpam-6549	16	57	the	the	DET
ejpam-6549	16	58	matching	match	VERB
ejpam-6549	16	59	energy	energy	NOUN
ejpam-6549	16	60	(	(	PUNCT
ejpam-6549	16	61	me	i	PRON
ejpam-6549	16	62	)	)	PUNCT
ejpam-6549	16	63	of	of	ADP
ejpam-6549	16	64	g	g	PROPN
ejpam-6549	16	65	is	be	AUX
ejpam-6549	16	66	defined	define	VERB
ejpam-6549	16	67	as	as	ADP
ejpam-6549	16	68	the	the	DET
ejpam-6549	16	69	sum	sum	NOUN
ejpam-6549	16	70	of	of	ADP
ejpam-6549	16	71	the	the	DET
ejpam-6549	16	72	absolute	absolute	ADJ
ejpam-6549	16	73	values	value	NOUN
ejpam-6549	16	74	of	of	ADP
ejpam-6549	16	75	the	the	DET
ejpam-6549	16	76	zeros	zero	NOUN
ejpam-6549	16	77	of	of	ADP
ejpam-6549	16	78	its	its	PRON
ejpam-6549	16	79	matching	matching	NOUN
ejpam-6549	16	80	polynomial	polynomial	NOUN
ejpam-6549	16	81	.	.	PUNCT
ejpam-6549	17	1	this	this	DET
ejpam-6549	17	2	topic	topic	NOUN
ejpam-6549	17	3	has	have	AUX
ejpam-6549	17	4	been	be	AUX
ejpam-6549	17	5	extensively	extensively	ADV
ejpam-6549	17	6	studied	study	VERB
ejpam-6549	17	7	[	[	PUNCT
ejpam-6549	17	8	1–5	1–5	X
ejpam-6549	17	9	]	]	X
ejpam-6549	17	10	.	.	PUNCT
ejpam-6549	18	1	a	a	DET
ejpam-6549	18	2	fundamental	fundamental	ADJ
ejpam-6549	18	3	question	question	NOUN
ejpam-6549	18	4	in	in	ADP
ejpam-6549	18	5	spectral	spectral	ADJ
ejpam-6549	18	6	graph	graph	NOUN
ejpam-6549	18	7	theory	theory	NOUN
ejpam-6549	18	8	concerns	concern	NOUN
ejpam-6549	18	9	lower	low	ADJ
ejpam-6549	18	10	bounds	bound	NOUN
ejpam-6549	18	11	on	on	ADP
ejpam-6549	18	12	the	the	DET
ejpam-6549	18	13	energy	energy	NOUN
ejpam-6549	18	14	of	of	ADP
ejpam-6549	18	15	a	a	DET
ejpam-6549	18	16	graph	graph	NOUN
ejpam-6549	18	17	in	in	ADP
ejpam-6549	18	18	terms	term	NOUN
ejpam-6549	18	19	of	of	ADP
ejpam-6549	18	20	its	its	PRON
ejpam-6549	18	21	matching	matching	ADJ
ejpam-6549	18	22	number	number	NOUN
ejpam-6549	18	23	.	.	PUNCT
ejpam-6549	19	1	in	in	ADP
ejpam-6549	19	2	[	[	X
ejpam-6549	19	3	6	6	NUM
ejpam-6549	19	4	]	]	PUNCT
ejpam-6549	19	5	,	,	PUNCT
ejpam-6549	19	6	wong	wong	PROPN
ejpam-6549	19	7	et	et	PROPN
ejpam-6549	19	8	al	al	PROPN
ejpam-6549	19	9	.	.	PROPN
ejpam-6549	19	10	proved	prove	VERB
ejpam-6549	19	11	that	that	SCONJ
ejpam-6549	19	12	ε(g	ε(g	NOUN
ejpam-6549	19	13	)	)	PUNCT
ejpam-6549	19	14	≥	≥	NOUN
ejpam-6549	19	15	2im(g	2im(g	NUM
ejpam-6549	19	16	)	)	PUNCT
ejpam-6549	19	17	for	for	ADP
ejpam-6549	19	18	any	any	DET
ejpam-6549	19	19	graph	graph	NOUN
ejpam-6549	19	20	g.	g.	PROPN
ejpam-6549	19	21	moreover	moreover	ADV
ejpam-6549	19	22	,	,	PUNCT
ejpam-6549	19	23	they	they	PRON
ejpam-6549	19	24	established	establish	VERB
ejpam-6549	19	25	the	the	DET
ejpam-6549	19	26	refined	refined	ADJ
ejpam-6549	19	27	bound	bind	VERB
ejpam-6549	19	28	ε(g	ε(g	NOUN
ejpam-6549	19	29	)	)	PUNCT
ejpam-6549	19	30	≥	≥	NOUN
ejpam-6549	19	31	2im(g	2im(g	NUM
ejpam-6549	19	32	)	)	PUNCT
ejpam-6549	20	1	+	+	CCONJ
ejpam-6549	20	2	√	√	NUM
ejpam-6549	20	3	5	5	NUM
ejpam-6549	20	4	5	5	NUM
ejpam-6549	20	5	c1(g	c1(g	PROPN
ejpam-6549	20	6	)	)	PUNCT
ejpam-6549	20	7	,	,	PUNCT
ejpam-6549	20	8	where	where	SCONJ
ejpam-6549	20	9	c1(g	c1(g	NOUN
ejpam-6549	20	10	)	)	PUNCT
ejpam-6549	20	11	denotes	denote	VERB
ejpam-6549	20	12	the	the	DET
ejpam-6549	20	13	number	number	NOUN
ejpam-6549	20	14	of	of	ADP
ejpam-6549	20	15	disjoint	disjoint	ADJ
ejpam-6549	20	16	odd	odd	ADJ
ejpam-6549	20	17	cycles	cycle	NOUN
ejpam-6549	20	18	in	in	ADP
ejpam-6549	20	19	g.	g.	PROPN
ejpam-6549	20	20	later	later	ADV
ejpam-6549	20	21	,	,	PUNCT
ejpam-6549	20	22	in	in	ADP
ejpam-6549	20	23	[	[	PUNCT
ejpam-6549	20	24	7	7	NUM
ejpam-6549	20	25	]	]	PUNCT
ejpam-6549	20	26	,	,	PUNCT
ejpam-6549	20	27	ashraf	ashraf	NOUN
ejpam-6549	20	28	improved	improve	VERB
ejpam-6549	20	29	this	this	DET
ejpam-6549	20	30	result	result	NOUN
ejpam-6549	20	31	by	by	ADP
ejpam-6549	20	32	showing	show	VERB
ejpam-6549	20	33	that	that	SCONJ
ejpam-6549	20	34	ε(g	ε(g	NOUN
ejpam-6549	20	35	)	)	PUNCT
ejpam-6549	20	36	≥	≥	NOUN
ejpam-6549	20	37	2im(g	2im(g	NUM
ejpam-6549	20	38	)	)	PUNCT
ejpam-6549	21	1	+	+	NUM
ejpam-6549	22	1	c0(g	c0(g	NOUN
ejpam-6549	22	2	)	)	PUNCT
ejpam-6549	22	3	,	,	PUNCT
ejpam-6549	22	4	where	where	SCONJ
ejpam-6549	22	5	c0(g	c0(g	NOUN
ejpam-6549	22	6	)	)	PUNCT
ejpam-6549	22	7	represents	represent	VERB
ejpam-6549	22	8	the	the	DET
ejpam-6549	22	9	number	number	NOUN
ejpam-6549	22	10	of	of	ADP
ejpam-6549	22	11	disjoint	disjoint	ADJ
ejpam-6549	22	12	odd	odd	ADJ
ejpam-6549	22	13	cycles	cycle	NOUN
ejpam-6549	22	14	of	of	ADP
ejpam-6549	22	15	length	length	NOUN
ejpam-6549	22	16	at	at	ADV
ejpam-6549	22	17	least	least	ADV
ejpam-6549	22	18	5	5	NUM
ejpam-6549	22	19	.	.	PUNCT
ejpam-6549	23	1	in	in	ADP
ejpam-6549	23	2	this	this	DET
ejpam-6549	23	3	paper	paper	NOUN
ejpam-6549	23	4	,	,	PUNCT
ejpam-6549	23	5	we	we	PRON
ejpam-6549	23	6	extend	extend	VERB
ejpam-6549	23	7	these	these	DET
ejpam-6549	23	8	results	result	NOUN
ejpam-6549	23	9	to	to	ADP
ejpam-6549	23	10	the	the	DET
ejpam-6549	23	11	class	class	NOUN
ejpam-6549	23	12	of	of	ADP
ejpam-6549	23	13	graphs	graph	NOUN
ejpam-6549	23	14	known	know	VERB
ejpam-6549	23	15	as	as	ADP
ejpam-6549	23	16	cameron	cameron	PROPN
ejpam-6549	23	17	-	-	PUNCT
ejpam-6549	23	18	walker	walker	PROPN
ejpam-6549	23	19	graphs	graph	NOUN
ejpam-6549	23	20	in	in	ADP
ejpam-6549	23	21	which	which	PRON
ejpam-6549	23	22	the	the	DET
ejpam-6549	23	23	matching	matching	NOUN
ejpam-6549	23	24	number	number	NOUN
ejpam-6549	23	25	equals	equal	VERB
ejpam-6549	23	26	the	the	DET
ejpam-6549	23	27	induced	induced	ADJ
ejpam-6549	23	28	matching	matching	NOUN
ejpam-6549	23	29	number	number	NOUN
ejpam-6549	23	30	.	.	PUNCT
ejpam-6549	24	1	in	in	ADP
ejpam-6549	24	2	section	section	NOUN
ejpam-6549	24	3	3	3	NUM
ejpam-6549	24	4	,	,	PUNCT
ejpam-6549	24	5	for	for	ADP
ejpam-6549	24	6	a	a	DET
ejpam-6549	24	7	given	give	VERB
ejpam-6549	24	8	constant	constant	ADJ
ejpam-6549	24	9	induced	induce	VERB
ejpam-6549	24	10	matching	matching	NOUN
ejpam-6549	24	11	number	number	NOUN
ejpam-6549	24	12	,	,	PUNCT
ejpam-6549	24	13	we	we	PRON
ejpam-6549	24	14	determine	determine	VERB
ejpam-6549	24	15	the	the	DET
ejpam-6549	24	16	graph	graph	NOUN
ejpam-6549	24	17	with	with	ADP
ejpam-6549	24	18	minimum	minimum	ADJ
ejpam-6549	24	19	energy	energy	NOUN
ejpam-6549	24	20	using	use	VERB
ejpam-6549	24	21	matching	match	VERB
ejpam-6549	24	22	energy	energy	NOUN
ejpam-6549	24	23	.	.	PUNCT
ejpam-6549	25	1	in	in	ADP
ejpam-6549	25	2	section	section	NOUN
ejpam-6549	25	3	4	4	NUM
ejpam-6549	25	4	,	,	PUNCT
ejpam-6549	25	5	we	we	PRON
ejpam-6549	25	6	establish	establish	VERB
ejpam-6549	25	7	new	new	ADJ
ejpam-6549	25	8	lower	low	ADJ
ejpam-6549	25	9	bounds	bound	NOUN
ejpam-6549	25	10	on	on	ADP
ejpam-6549	25	11	the	the	DET
ejpam-6549	25	12	energy	energy	NOUN
ejpam-6549	25	13	of	of	ADP
ejpam-6549	25	14	graphs	graph	NOUN
ejpam-6549	25	15	where	where	SCONJ
ejpam-6549	25	16	the	the	DET
ejpam-6549	25	17	matching	matching	NOUN
ejpam-6549	25	18	and	and	CCONJ
ejpam-6549	25	19	induced	induced	ADJ
ejpam-6549	25	20	matching	matching	NOUN
ejpam-6549	25	21	numbers	number	NOUN
ejpam-6549	25	22	are	be	AUX
ejpam-6549	25	23	equal	equal	ADJ
ejpam-6549	25	24	.	.	PUNCT
ejpam-6549	26	1	2	2	X
ejpam-6549	26	2	.	.	X
ejpam-6549	26	3	preliminaries	preliminary	NOUN
ejpam-6549	26	4	this	this	DET
ejpam-6549	26	5	chapter	chapter	NOUN
ejpam-6549	26	6	is	be	AUX
ejpam-6549	26	7	devoted	devote	VERB
ejpam-6549	26	8	to	to	ADP
ejpam-6549	26	9	the	the	DET
ejpam-6549	26	10	definitions	definition	NOUN
ejpam-6549	26	11	and	and	CCONJ
ejpam-6549	26	12	previous	previous	ADJ
ejpam-6549	26	13	results	result	NOUN
ejpam-6549	26	14	that	that	PRON
ejpam-6549	26	15	will	will	AUX
ejpam-6549	26	16	be	be	AUX
ejpam-6549	26	17	used	use	VERB
ejpam-6549	26	18	in	in	ADP
ejpam-6549	26	19	the	the	DET
ejpam-6549	26	20	rest	rest	NOUN
ejpam-6549	26	21	of	of	ADP
ejpam-6549	26	22	the	the	DET
ejpam-6549	26	23	paper	paper	NOUN
ejpam-6549	26	24	.	.	PUNCT
ejpam-6549	27	1	definition	definition	NOUN
ejpam-6549	27	2	1	1	NUM
ejpam-6549	27	3	.	.	PUNCT
ejpam-6549	28	1	let	let	VERB
ejpam-6549	28	2	g	g	PRON
ejpam-6549	28	3	be	be	AUX
ejpam-6549	28	4	a	a	DET
ejpam-6549	28	5	simple	simple	ADJ
ejpam-6549	28	6	undirected	undirected	ADJ
ejpam-6549	28	7	graph	graph	NOUN
ejpam-6549	28	8	of	of	ADP
ejpam-6549	28	9	order	order	NOUN
ejpam-6549	28	10	n.	n.	NOUN
ejpam-6549	28	11	if	if	SCONJ
ejpam-6549	28	12	mk(g	mk(g	NOUN
ejpam-6549	28	13	)	)	PUNCT
ejpam-6549	28	14	is	be	AUX
ejpam-6549	28	15	the	the	DET
ejpam-6549	28	16	number	number	NOUN
ejpam-6549	28	17	of	of	ADP
ejpam-6549	28	18	k	k	NOUN
ejpam-6549	28	19	-	-	NOUN
ejpam-6549	28	20	matchings	matching	NOUN
ejpam-6549	28	21	of	of	ADP
ejpam-6549	28	22	g	g	NOUN
ejpam-6549	28	23	,	,	PUNCT
ejpam-6549	28	24	k	k	PROPN
ejpam-6549	28	25	∈	∈	PROPN
ejpam-6549	28	26	{	{	PUNCT
ejpam-6549	28	27	0	0	NUM
ejpam-6549	28	28	,	,	PUNCT
ejpam-6549	28	29	1	1	NUM
ejpam-6549	28	30	,	,	PUNCT
ejpam-6549	28	31	2	2	NUM
ejpam-6549	28	32	,	,	PUNCT
ejpam-6549	28	33	...	...	PUNCT
ejpam-6549	28	34	,	,	PUNCT
ejpam-6549	28	35	⌊	⌊	VERB
ejpam-6549	28	36	n	n	ADV
ejpam-6549	28	37	2	2	NUM
ejpam-6549	28	38	⌋	⌋	NOUN
ejpam-6549	28	39	}	}	PUNCT
ejpam-6549	28	40	,	,	PUNCT
ejpam-6549	28	41	then	then	ADV
ejpam-6549	28	42	the	the	DET
ejpam-6549	28	43	matching	match	VERB
ejpam-6549	28	44	energy	energy	NOUN
ejpam-6549	28	45	of	of	ADP
ejpam-6549	28	46	g	g	PROPN
ejpam-6549	28	47	is	be	AUX
ejpam-6549	28	48	me(g	me(g	PRON
ejpam-6549	28	49	)	)	PUNCT
ejpam-6549	29	1	=	=	SYM
ejpam-6549	30	1	2	2	NUM
ejpam-6549	30	2	π	π	PROPN
ejpam-6549	30	3	∞∫	∞∫	NOUN
ejpam-6549	30	4	0	0	NUM
ejpam-6549	30	5	1	1	NUM
ejpam-6549	30	6	x2	x2	NOUN
ejpam-6549	30	7	ln	ln	ADJ
ejpam-6549	30	8	[	[	PUNCT
ejpam-6549	30	9	∑	∑	INTJ
ejpam-6549	30	10	k≥0	k≥0	PROPN
ejpam-6549	30	11	mk(g)x2k]dx	mk(g)x2k]dx	PROPN
ejpam-6549	30	12	.	.	PUNCT
ejpam-6549	31	1	by	by	ADP
ejpam-6549	31	2	the	the	DET
ejpam-6549	31	3	monotonicity	monotonicity	NOUN
ejpam-6549	31	4	of	of	ADP
ejpam-6549	31	5	the	the	DET
ejpam-6549	31	6	logarithm	logarithm	NOUN
ejpam-6549	31	7	,	,	PUNCT
ejpam-6549	31	8	one	one	PRON
ejpam-6549	31	9	can	can	AUX
ejpam-6549	31	10	define	define	VERB
ejpam-6549	31	11	a	a	DET
ejpam-6549	31	12	quasi	quasi	ADJ
ejpam-6549	31	13	-	-	NOUN
ejpam-6549	31	14	order	order	NOUN
ejpam-6549	31	15	relation	relation	NOUN
ejpam-6549	31	16	“	"	PUNCT
ejpam-6549	31	17	⪰	⪰	NOUN
ejpam-6549	31	18	”	"	PUNCT
ejpam-6549	31	19	as	as	ADP
ejpam-6549	31	20	the	the	DET
ejpam-6549	31	21	following	following	NOUN
ejpam-6549	31	22	:	:	PUNCT
ejpam-6549	31	23	let	let	VERB
ejpam-6549	31	24	g1	g1	PROPN
ejpam-6549	31	25	and	and	CCONJ
ejpam-6549	31	26	g2	g2	PROPN
ejpam-6549	31	27	be	be	AUX
ejpam-6549	31	28	two	two	NUM
ejpam-6549	31	29	graphs	graph	NOUN
ejpam-6549	31	30	.	.	PUNCT
ejpam-6549	32	1	then	then	ADV
ejpam-6549	32	2	g1	g1	VERB
ejpam-6549	32	3	⪰	⪰	NOUN
ejpam-6549	32	4	g2	g2	PROPN
ejpam-6549	32	5	⇐	⇐	ADJ
ejpam-6549	32	6	⇒	⇒	PROPN
ejpam-6549	32	7	mk(g1	mk(g1	NOUN
ejpam-6549	32	8	)	)	PUNCT
ejpam-6549	32	9	≥	≥	X
ejpam-6549	32	10	mk(g2	mk(g2	NUM
ejpam-6549	32	11	)	)	PUNCT
ejpam-6549	32	12	for	for	ADP
ejpam-6549	32	13	all	all	DET
ejpam-6549	32	14	k.	k.	PROPN
ejpam-6549	32	15	the	the	DET
ejpam-6549	32	16	following	follow	VERB
ejpam-6549	32	17	theorem	theorem	NOUN
ejpam-6549	32	18	can	can	AUX
ejpam-6549	32	19	provide	provide	VERB
ejpam-6549	32	20	the	the	DET
ejpam-6549	32	21	matching	match	VERB
ejpam-6549	32	22	energy	energy	NOUN
ejpam-6549	32	23	of	of	ADP
ejpam-6549	32	24	a	a	DET
ejpam-6549	32	25	graph	graph	NOUN
ejpam-6549	32	26	.	.	PUNCT
ejpam-6549	32	27	a.	a.	NOUN
ejpam-6549	32	28	ülker	ülker	PROPN
ejpam-6549	32	29	et	et	PROPN
ejpam-6549	32	30	al	al	PROPN
ejpam-6549	32	31	.	.	PUNCT
ejpam-6549	32	32	/	/	SYM
ejpam-6549	32	33	eur	eur	PROPN
ejpam-6549	32	34	.	.	PUNCT
ejpam-6549	33	1	j.	j.	PROPN
ejpam-6549	33	2	pure	pure	PROPN
ejpam-6549	33	3	appl	appl	PROPN
ejpam-6549	33	4	.	.	PROPN
ejpam-6549	33	5	math	math	PROPN
ejpam-6549	33	6	,	,	PUNCT
ejpam-6549	33	7	18	18	NUM
ejpam-6549	33	8	(	(	PUNCT
ejpam-6549	33	9	3	3	NUM
ejpam-6549	33	10	)	)	PUNCT
ejpam-6549	33	11	(	(	PUNCT
ejpam-6549	33	12	2025	2025	NUM
ejpam-6549	33	13	)	)	PUNCT
ejpam-6549	33	14	,	,	PUNCT
ejpam-6549	33	15	6549	6549	NUM
ejpam-6549	33	16	3	3	NUM
ejpam-6549	33	17	of	of	ADP
ejpam-6549	33	18	8	8	NUM
ejpam-6549	33	19	theorem	theorem	NOUN
ejpam-6549	33	20	1	1	NUM
ejpam-6549	33	21	.	.	PUNCT
ejpam-6549	34	1	(	(	PUNCT
ejpam-6549	34	2	[	[	X
ejpam-6549	34	3	1	1	NUM
ejpam-6549	34	4	]	]	PUNCT
ejpam-6549	34	5	,	,	PUNCT
ejpam-6549	34	6	theorem	theorem	VERB
ejpam-6549	34	7	1	1	NUM
ejpam-6549	34	8	)	)	PUNCT
ejpam-6549	34	9	let	let	VERB
ejpam-6549	34	10	g	g	NOUN
ejpam-6549	34	11	be	be	AUX
ejpam-6549	34	12	a	a	DET
ejpam-6549	34	13	simple	simple	ADJ
ejpam-6549	34	14	graph	graph	NOUN
ejpam-6549	34	15	and	and	CCONJ
ejpam-6549	34	16	let	let	VERB
ejpam-6549	34	17	σ1	σ1	PROPN
ejpam-6549	34	18	,	,	PUNCT
ejpam-6549	34	19	σ2	σ2	PROPN
ejpam-6549	34	20	,	,	PUNCT
ejpam-6549	34	21	...	...	PUNCT
ejpam-6549	34	22	,	,	PUNCT
ejpam-6549	34	23	σn	σn	PROPN
ejpam-6549	34	24	be	be	VERB
ejpam-6549	34	25	the	the	DET
ejpam-6549	34	26	zeros	zero	NOUN
ejpam-6549	34	27	of	of	ADP
ejpam-6549	34	28	the	the	DET
ejpam-6549	34	29	matching	match	VERB
ejpam-6549	34	30	polynomial	polynomial	NOUN
ejpam-6549	34	31	of	of	ADP
ejpam-6549	34	32	g.	g.	PROPN
ejpam-6549	34	33	then	then	ADV
ejpam-6549	34	34	,	,	PUNCT
ejpam-6549	34	35	me(g	me(g	X
ejpam-6549	34	36	)	)	PUNCT
ejpam-6549	35	1	=	=	PUNCT
ejpam-6549	35	2	n∑	n∑	PROPN
ejpam-6549	35	3	i=1	i=1	PROPN
ejpam-6549	36	1	|σi|	|σi|	PROPN
ejpam-6549	36	2	.	.	PUNCT
ejpam-6549	37	1	if	if	SCONJ
ejpam-6549	37	2	e	e	NOUN
ejpam-6549	37	3	=	=	NOUN
ejpam-6549	37	4	uv	uv	NOUN
ejpam-6549	37	5	is	be	AUX
ejpam-6549	37	6	an	an	DET
ejpam-6549	37	7	edge	edge	NOUN
ejpam-6549	37	8	in	in	ADP
ejpam-6549	37	9	g	g	NOUN
ejpam-6549	37	10	,	,	PUNCT
ejpam-6549	37	11	then	then	ADV
ejpam-6549	37	12	the	the	DET
ejpam-6549	37	13	equality	equality	NOUN
ejpam-6549	37	14	m(g	m(g	PROPN
ejpam-6549	37	15	;	;	PUNCT
ejpam-6549	37	16	k	k	X
ejpam-6549	37	17	)	)	PUNCT
ejpam-6549	37	18	=	=	SYM
ejpam-6549	37	19	m(g−e	m(g−e	PROPN
ejpam-6549	37	20	,	,	PUNCT
ejpam-6549	37	21	k)+m(g−u−v	k)+m(g−u−v	PROPN
ejpam-6549	37	22	;	;	PUNCT
ejpam-6549	37	23	k−1	k−1	PROPN
ejpam-6549	37	24	)	)	PUNCT
ejpam-6549	37	25	holds	hold	VERB
ejpam-6549	37	26	.	.	PUNCT
ejpam-6549	38	1	so	so	ADV
ejpam-6549	38	2	m(g	m(g	PROPN
ejpam-6549	38	3	;	;	PUNCT
ejpam-6549	38	4	k	k	X
ejpam-6549	38	5	)	)	PUNCT
ejpam-6549	38	6	increases	increase	VERB
ejpam-6549	38	7	whenever	whenever	SCONJ
ejpam-6549	38	8	an	an	DET
ejpam-6549	38	9	edge	edge	NOUN
ejpam-6549	38	10	is	be	AUX
ejpam-6549	38	11	added	add	VERB
ejpam-6549	38	12	to	to	ADP
ejpam-6549	38	13	the	the	DET
ejpam-6549	38	14	graph	graph	NOUN
ejpam-6549	38	15	.	.	PUNCT
ejpam-6549	39	1	hence	hence	ADV
ejpam-6549	39	2	,	,	PUNCT
ejpam-6549	39	3	we	we	PRON
ejpam-6549	39	4	can	can	AUX
ejpam-6549	39	5	give	give	VERB
ejpam-6549	39	6	the	the	DET
ejpam-6549	39	7	following	follow	VERB
ejpam-6549	39	8	result	result	NOUN
ejpam-6549	39	9	.	.	PUNCT
ejpam-6549	40	1	theorem	theorem	NOUN
ejpam-6549	40	2	2	2	NUM
ejpam-6549	40	3	.	.	PUNCT
ejpam-6549	41	1	(	(	PUNCT
ejpam-6549	41	2	[	[	X
ejpam-6549	41	3	1	1	NUM
ejpam-6549	41	4	]	]	PUNCT
ejpam-6549	41	5	,	,	PUNCT
ejpam-6549	41	6	theorem	theorem	VERB
ejpam-6549	41	7	3	3	NUM
ejpam-6549	41	8	)	)	PUNCT
ejpam-6549	41	9	let	let	VERB
ejpam-6549	41	10	g	g	NOUN
ejpam-6549	41	11	be	be	AUX
ejpam-6549	41	12	a	a	DET
ejpam-6549	41	13	graph	graph	NOUN
ejpam-6549	41	14	and	and	CCONJ
ejpam-6549	41	15	e	e	AUX
ejpam-6549	41	16	be	be	AUX
ejpam-6549	41	17	its	its	PRON
ejpam-6549	41	18	edge	edge	NOUN
ejpam-6549	41	19	.	.	PUNCT
ejpam-6549	42	1	if	if	SCONJ
ejpam-6549	42	2	g−	g−	PROPN
ejpam-6549	42	3	e	e	NOUN
ejpam-6549	42	4	is	be	AUX
ejpam-6549	42	5	obtained	obtain	VERB
ejpam-6549	42	6	by	by	ADP
ejpam-6549	42	7	removing	remove	VERB
ejpam-6549	42	8	e	e	NOUN
ejpam-6549	42	9	from	from	ADP
ejpam-6549	42	10	g	g	NOUN
ejpam-6549	42	11	and	and	CCONJ
ejpam-6549	42	12	keeping	keep	VERB
ejpam-6549	42	13	all	all	DET
ejpam-6549	42	14	the	the	DET
ejpam-6549	42	15	vertices	vertex	NOUN
ejpam-6549	42	16	of	of	ADP
ejpam-6549	42	17	g	g	NOUN
ejpam-6549	42	18	remaining	remain	VERB
ejpam-6549	42	19	,	,	PUNCT
ejpam-6549	42	20	then	then	ADV
ejpam-6549	42	21	me(g−e	me(g−e	NUM
ejpam-6549	42	22	)	)	PUNCT
ejpam-6549	42	23	<	<	X
ejpam-6549	42	24	me(g	me(g	X
ejpam-6549	42	25	)	)	PUNCT
ejpam-6549	42	26	.	.	PUNCT
ejpam-6549	43	1	the	the	DET
ejpam-6549	43	2	following	follow	VERB
ejpam-6549	43	3	theorem	theorem	ADJ
ejpam-6549	43	4	states	state	NOUN
ejpam-6549	43	5	that	that	SCONJ
ejpam-6549	43	6	energy	energy	NOUN
ejpam-6549	43	7	and	and	CCONJ
ejpam-6549	43	8	matching	match	VERB
ejpam-6549	43	9	energy	energy	NOUN
ejpam-6549	43	10	coincide	coincide	NOUN
ejpam-6549	43	11	for	for	ADP
ejpam-6549	43	12	trees	tree	NOUN
ejpam-6549	43	13	.	.	PUNCT
ejpam-6549	44	1	theorem	theorem	NOUN
ejpam-6549	44	2	3	3	NUM
ejpam-6549	44	3	.	.	PUNCT
ejpam-6549	45	1	(	(	PUNCT
ejpam-6549	45	2	[	[	X
ejpam-6549	45	3	1	1	NUM
ejpam-6549	45	4	]	]	PUNCT
ejpam-6549	45	5	,	,	PUNCT
ejpam-6549	45	6	theorem	theorem	VERB
ejpam-6549	45	7	2	2	NUM
ejpam-6549	45	8	)	)	PUNCT
ejpam-6549	45	9	if	if	SCONJ
ejpam-6549	45	10	a	a	DET
ejpam-6549	45	11	graph	graph	NOUN
ejpam-6549	45	12	g	g	NOUN
ejpam-6549	45	13	has	have	VERB
ejpam-6549	45	14	no	no	DET
ejpam-6549	45	15	cycle	cycle	NOUN
ejpam-6549	45	16	,	,	PUNCT
ejpam-6549	45	17	then	then	ADV
ejpam-6549	45	18	its	its	PRON
ejpam-6549	45	19	matching	match	VERB
ejpam-6549	45	20	energy	energy	NOUN
ejpam-6549	45	21	and	and	CCONJ
ejpam-6549	45	22	energy	energy	NOUN
ejpam-6549	45	23	are	be	AUX
ejpam-6549	45	24	equal	equal	ADJ
ejpam-6549	45	25	.	.	PUNCT
ejpam-6549	46	1	in	in	ADP
ejpam-6549	46	2	[	[	X
ejpam-6549	46	3	8	8	NUM
ejpam-6549	46	4	]	]	PUNCT
ejpam-6549	46	5	,	,	PUNCT
ejpam-6549	46	6	arizmendi	arizmendi	PROPN
ejpam-6549	46	7	et	et	PROPN
ejpam-6549	46	8	al	al	PROPN
ejpam-6549	46	9	.	.	PROPN
ejpam-6549	46	10	studied	study	VERB
ejpam-6549	46	11	the	the	DET
ejpam-6549	46	12	vertex	vertex	NOUN
ejpam-6549	46	13	energy	energy	NOUN
ejpam-6549	46	14	in	in	ADP
ejpam-6549	46	15	a	a	DET
ejpam-6549	46	16	graph	graph	NOUN
ejpam-6549	46	17	energy	energy	NOUN
ejpam-6549	46	18	.	.	PUNCT
ejpam-6549	47	1	the	the	DET
ejpam-6549	47	2	following	follow	VERB
ejpam-6549	47	3	theorem	theorem	NOUN
ejpam-6549	47	4	gives	give	VERB
ejpam-6549	47	5	the	the	DET
ejpam-6549	47	6	lower	low	ADJ
ejpam-6549	47	7	bound	bind	VERB
ejpam-6549	47	8	of	of	ADP
ejpam-6549	47	9	a	a	DET
ejpam-6549	47	10	vertex	vertex	NOUN
ejpam-6549	47	11	energy	energy	NOUN
ejpam-6549	47	12	in	in	ADP
ejpam-6549	47	13	terms	term	NOUN
ejpam-6549	47	14	of	of	ADP
ejpam-6549	47	15	its	its	PRON
ejpam-6549	47	16	degree	degree	NOUN
ejpam-6549	47	17	.	.	PUNCT
ejpam-6549	48	1	theorem	theorem	ADJ
ejpam-6549	48	2	4	4	NUM
ejpam-6549	48	3	.	.	PUNCT
ejpam-6549	49	1	(	(	PUNCT
ejpam-6549	49	2	[	[	X
ejpam-6549	49	3	8	8	NUM
ejpam-6549	49	4	]	]	PUNCT
ejpam-6549	49	5	,	,	PUNCT
ejpam-6549	49	6	theorem	theorem	VERB
ejpam-6549	49	7	3.3	3.3	NUM
ejpam-6549	49	8	)	)	PUNCT
ejpam-6549	49	9	let	let	VERB
ejpam-6549	49	10	g	g	NOUN
ejpam-6549	49	11	be	be	AUX
ejpam-6549	49	12	a	a	DET
ejpam-6549	49	13	connected	connected	ADJ
ejpam-6549	49	14	graph	graph	NOUN
ejpam-6549	49	15	with	with	ADP
ejpam-6549	49	16	at	at	ADV
ejpam-6549	49	17	least	least	ADV
ejpam-6549	49	18	one	one	NUM
ejpam-6549	49	19	edge	edge	NOUN
ejpam-6549	49	20	.	.	PUNCT
ejpam-6549	50	1	then	then	ADV
ejpam-6549	50	2	for	for	ADP
ejpam-6549	50	3	all	all	DET
ejpam-6549	50	4	vi	vi	NOUN
ejpam-6549	50	5	∈	∈	NOUN
ejpam-6549	50	6	v	v	NOUN
ejpam-6549	50	7	(	(	PUNCT
ejpam-6549	50	8	g	g	NOUN
ejpam-6549	50	9	)	)	PUNCT
ejpam-6549	50	10	εg(vi	εg(vi	PROPN
ejpam-6549	50	11	)	)	PUNCT
ejpam-6549	50	12	≥	≥	NOUN
ejpam-6549	50	13	di	di	NOUN
ejpam-6549	50	14	∆	∆	PROPN
ejpam-6549	50	15	.	.	PUNCT
ejpam-6549	51	1	equality	equality	NOUN
ejpam-6549	51	2	holds	hold	VERB
ejpam-6549	51	3	if	if	SCONJ
ejpam-6549	51	4	and	and	CCONJ
ejpam-6549	51	5	only	only	ADV
ejpam-6549	51	6	if	if	SCONJ
ejpam-6549	51	7	g	g	PROPN
ejpam-6549	51	8	is	be	AUX
ejpam-6549	51	9	isomorphic	isomorphic	ADJ
ejpam-6549	51	10	to	to	ADP
ejpam-6549	51	11	the	the	DET
ejpam-6549	51	12	complete	complete	ADJ
ejpam-6549	51	13	bipartite	bipartite	PROPN
ejpam-6549	51	14	graph	graph	NOUN
ejpam-6549	51	15	kd	kd	PROPN
ejpam-6549	51	16	,	,	PUNCT
ejpam-6549	51	17	d.	d.	PROPN
ejpam-6549	51	18	3	3	NUM
ejpam-6549	51	19	.	.	PUNCT
ejpam-6549	51	20	graph	graph	NOUN
ejpam-6549	51	21	parameters	parameter	NOUN
ejpam-6549	51	22	and	and	CCONJ
ejpam-6549	51	23	energy	energy	NOUN
ejpam-6549	51	24	in	in	ADP
ejpam-6549	51	25	this	this	DET
ejpam-6549	51	26	section	section	NOUN
ejpam-6549	51	27	,	,	PUNCT
ejpam-6549	51	28	we	we	PRON
ejpam-6549	51	29	study	study	VERB
ejpam-6549	51	30	the	the	DET
ejpam-6549	51	31	graphs	graph	NOUN
ejpam-6549	51	32	with	with	ADP
ejpam-6549	51	33	minimal	minimal	ADJ
ejpam-6549	51	34	energy	energy	NOUN
ejpam-6549	51	35	with	with	ADP
ejpam-6549	51	36	induced	induced	ADJ
ejpam-6549	51	37	matching	matching	NOUN
ejpam-6549	51	38	number	number	NOUN
ejpam-6549	51	39	im(g	im(g	PUNCT
ejpam-6549	51	40	)	)	PUNCT
ejpam-6549	51	41	.	.	PUNCT
ejpam-6549	52	1	moreover	moreover	ADV
ejpam-6549	52	2	,	,	PUNCT
ejpam-6549	52	3	we	we	PRON
ejpam-6549	52	4	provide	provide	VERB
ejpam-6549	52	5	lower	low	ADJ
ejpam-6549	52	6	bounds	bound	NOUN
ejpam-6549	52	7	on	on	ADP
ejpam-6549	52	8	graph	graph	NOUN
ejpam-6549	52	9	energy	energy	NOUN
ejpam-6549	52	10	with	with	ADP
ejpam-6549	52	11	respect	respect	NOUN
ejpam-6549	52	12	to	to	ADP
ejpam-6549	52	13	the	the	DET
ejpam-6549	52	14	induced	induced	ADJ
ejpam-6549	52	15	matching	matching	NOUN
ejpam-6549	52	16	number	number	NOUN
ejpam-6549	52	17	.	.	PUNCT
ejpam-6549	53	1	lemma	lemma	PROPN
ejpam-6549	53	2	1	1	X
ejpam-6549	53	3	.	.	PUNCT
ejpam-6549	54	1	let	let	VERB
ejpam-6549	54	2	g	g	NOUN
ejpam-6549	54	3	be	be	AUX
ejpam-6549	54	4	any	any	DET
ejpam-6549	54	5	graph	graph	NOUN
ejpam-6549	54	6	and	and	CCONJ
ejpam-6549	54	7	h	h	NOUN
ejpam-6549	54	8	be	be	AUX
ejpam-6549	54	9	its	its	PRON
ejpam-6549	54	10	induced	induced	ADJ
ejpam-6549	54	11	subgraph	subgraph	NOUN
ejpam-6549	54	12	.	.	PUNCT
ejpam-6549	55	1	then	then	ADV
ejpam-6549	55	2	ε(h	ε(h	NOUN
ejpam-6549	55	3	)	)	PUNCT
ejpam-6549	55	4	≤	≤	NUM
ejpam-6549	55	5	ε(g	ε(g	NOUN
ejpam-6549	55	6	)	)	PUNCT
ejpam-6549	55	7	.	.	PUNCT
ejpam-6549	56	1	proof	proof	NOUN
ejpam-6549	56	2	.	.	PUNCT
ejpam-6549	57	1	since	since	SCONJ
ejpam-6549	57	2	h	h	NOUN
ejpam-6549	57	3	is	be	AUX
ejpam-6549	57	4	an	an	DET
ejpam-6549	57	5	induced	induced	ADJ
ejpam-6549	57	6	subgraph	subgraph	NOUN
ejpam-6549	57	7	of	of	ADP
ejpam-6549	57	8	g	g	NOUN
ejpam-6549	57	9	,	,	PUNCT
ejpam-6549	57	10	then	then	ADV
ejpam-6549	57	11	adjacency	adjacency	NOUN
ejpam-6549	57	12	matrix	matrix	NOUN
ejpam-6549	57	13	of	of	ADP
ejpam-6549	57	14	g	g	PROPN
ejpam-6549	57	15	is	be	AUX
ejpam-6549	57	16	a(g	a(g	PROPN
ejpam-6549	57	17	)	)	PUNCT
ejpam-6549	58	1	=	=	PUNCT
ejpam-6549	58	2	[	[	PUNCT
ejpam-6549	58	3	a(h	a(h	PROPN
ejpam-6549	58	4	)	)	PUNCT
ejpam-6549	58	5	x	x	SYM
ejpam-6549	58	6	xt	xt	PROPN
ejpam-6549	58	7	a(g\h	a(g\h	PROPN
ejpam-6549	58	8	)	)	PUNCT
ejpam-6549	58	9	]	]	PUNCT
ejpam-6549	58	10	which	which	PRON
ejpam-6549	58	11	implies	imply	VERB
ejpam-6549	58	12	that	that	SCONJ
ejpam-6549	58	13	a(h	a(h	PROPN
ejpam-6549	58	14	)	)	PUNCT
ejpam-6549	58	15	is	be	AUX
ejpam-6549	58	16	the	the	DET
ejpam-6549	58	17	principal	principal	ADJ
ejpam-6549	58	18	submatrix	submatrix	NOUN
ejpam-6549	58	19	of	of	ADP
ejpam-6549	58	20	a(g	a(g	PROPN
ejpam-6549	58	21	)	)	PUNCT
ejpam-6549	58	22	.	.	PUNCT
ejpam-6549	59	1	thus	thus	ADV
ejpam-6549	59	2	,	,	PUNCT
ejpam-6549	59	3	the	the	DET
ejpam-6549	59	4	eigenvalues	eigenvalue	NOUN
ejpam-6549	59	5	of	of	ADP
ejpam-6549	59	6	a(h	a(h	PROPN
ejpam-6549	59	7	)	)	PUNCT
ejpam-6549	59	8	are	be	AUX
ejpam-6549	59	9	less	less	ADJ
ejpam-6549	59	10	than	than	ADP
ejpam-6549	59	11	those	those	PRON
ejpam-6549	59	12	of	of	ADP
ejpam-6549	59	13	a(g	a(g	PROPN
ejpam-6549	59	14	)	)	PUNCT
ejpam-6549	59	15	.	.	PUNCT
ejpam-6549	60	1	therefore	therefore	ADV
ejpam-6549	60	2	,	,	PUNCT
ejpam-6549	60	3	ε(h	ε(h	NOUN
ejpam-6549	60	4	)	)	PUNCT
ejpam-6549	60	5	<	<	X
ejpam-6549	60	6	ε(g	ε(g	NOUN
ejpam-6549	60	7	)	)	PUNCT
ejpam-6549	60	8	.	.	PUNCT
ejpam-6549	61	1	if	if	SCONJ
ejpam-6549	61	2	g	g	PROPN
ejpam-6549	61	3	=	=	SYM
ejpam-6549	61	4	h	h	NOUN
ejpam-6549	61	5	,	,	PUNCT
ejpam-6549	61	6	then	then	ADV
ejpam-6549	61	7	a(g\h	a(g\h	PROPN
ejpam-6549	61	8	)	)	PUNCT
ejpam-6549	61	9	=	=	SYM
ejpam-6549	61	10	0	0	NUM
ejpam-6549	61	11	,	,	PUNCT
ejpam-6549	61	12	thus	thus	ADV
ejpam-6549	61	13	eigenvalues	eigenvalue	VERB
ejpam-6549	61	14	of	of	ADP
ejpam-6549	61	15	a(h	a(h	PROPN
ejpam-6549	61	16	)	)	PUNCT
ejpam-6549	61	17	and	and	CCONJ
ejpam-6549	61	18	a(g	a(g	PROPN
ejpam-6549	61	19	)	)	PUNCT
ejpam-6549	61	20	are	be	AUX
ejpam-6549	61	21	same	same	ADJ
ejpam-6549	61	22	and	and	CCONJ
ejpam-6549	61	23	ε(h	ε(h	ADJ
ejpam-6549	61	24	)	)	PUNCT
ejpam-6549	61	25	=	=	SYM
ejpam-6549	61	26	ε(g	ε(g	NOUN
ejpam-6549	61	27	)	)	PUNCT
ejpam-6549	61	28	.	.	PUNCT
ejpam-6549	62	1	next	next	ADV
ejpam-6549	62	2	,	,	PUNCT
ejpam-6549	62	3	we	we	PRON
ejpam-6549	62	4	define	define	VERB
ejpam-6549	62	5	a	a	DET
ejpam-6549	62	6	new	new	ADJ
ejpam-6549	62	7	graph	graph	NOUN
ejpam-6549	62	8	called	call	VERB
ejpam-6549	62	9	rake	rake	NOUN
ejpam-6549	62	10	-	-	PUNCT
ejpam-6549	62	11	graph	graph	NOUN
ejpam-6549	62	12	rgn	rgn	NOUN
ejpam-6549	62	13	over	over	ADP
ejpam-6549	62	14	n	n	PART
ejpam-6549	62	15	vertices	vertex	NOUN
ejpam-6549	62	16	,	,	PUNCT
ejpam-6549	62	17	and	and	CCONJ
ejpam-6549	62	18	we	we	PRON
ejpam-6549	62	19	show	show	VERB
ejpam-6549	62	20	that	that	SCONJ
ejpam-6549	62	21	this	this	DET
ejpam-6549	62	22	graph	graph	NOUN
ejpam-6549	62	23	has	have	VERB
ejpam-6549	62	24	the	the	DET
ejpam-6549	62	25	minimum	minimum	ADJ
ejpam-6549	62	26	energy	energy	NOUN
ejpam-6549	62	27	among	among	ADP
ejpam-6549	62	28	the	the	DET
ejpam-6549	62	29	graphs	graph	NOUN
ejpam-6549	62	30	with	with	ADP
ejpam-6549	62	31	induced	induced	ADJ
ejpam-6549	62	32	matching	matching	NOUN
ejpam-6549	62	33	number	number	NOUN
ejpam-6549	62	34	n.	n.	NOUN
ejpam-6549	62	35	a.	a.	NOUN
ejpam-6549	62	36	ülker	ülker	PROPN
ejpam-6549	62	37	et	et	PROPN
ejpam-6549	62	38	al	al	PROPN
ejpam-6549	62	39	.	.	PUNCT
ejpam-6549	62	40	/	/	SYM
ejpam-6549	62	41	eur	eur	PROPN
ejpam-6549	62	42	.	.	PUNCT
ejpam-6549	63	1	j.	j.	PROPN
ejpam-6549	63	2	pure	pure	PROPN
ejpam-6549	63	3	appl	appl	PROPN
ejpam-6549	63	4	.	.	PROPN
ejpam-6549	63	5	math	math	PROPN
ejpam-6549	63	6	,	,	PUNCT
ejpam-6549	63	7	18	18	NUM
ejpam-6549	63	8	(	(	PUNCT
ejpam-6549	63	9	3	3	NUM
ejpam-6549	63	10	)	)	PUNCT
ejpam-6549	63	11	(	(	PUNCT
ejpam-6549	63	12	2025	2025	NUM
ejpam-6549	63	13	)	)	PUNCT
ejpam-6549	63	14	,	,	PUNCT
ejpam-6549	63	15	6549	6549	NUM
ejpam-6549	63	16	4	4	NUM
ejpam-6549	63	17	of	of	ADP
ejpam-6549	63	18	8	8	NUM
ejpam-6549	63	19	definition	definition	NOUN
ejpam-6549	63	20	2	2	NUM
ejpam-6549	63	21	.	.	PUNCT
ejpam-6549	64	1	let	let	VERB
ejpam-6549	64	2	n	n	PRON
ejpam-6549	64	3	≥	≥	X
ejpam-6549	64	4	2	2	NUM
ejpam-6549	64	5	be	be	AUX
ejpam-6549	64	6	an	an	DET
ejpam-6549	64	7	integer	integer	NOUN
ejpam-6549	64	8	.	.	PUNCT
ejpam-6549	65	1	a	a	DET
ejpam-6549	65	2	graph	graph	NOUN
ejpam-6549	65	3	rgn	rgn	PROPN
ejpam-6549	65	4	,	,	PUNCT
ejpam-6549	65	5	called	call	VERB
ejpam-6549	65	6	a	a	DET
ejpam-6549	65	7	rake	rake	NOUN
ejpam-6549	65	8	graph	graph	NOUN
ejpam-6549	65	9	,	,	PUNCT
ejpam-6549	65	10	is	be	AUX
ejpam-6549	65	11	a	a	DET
ejpam-6549	65	12	graph	graph	NOUN
ejpam-6549	65	13	on	on	ADP
ejpam-6549	65	14	2n	2n	NUM
ejpam-6549	66	1	+	+	CCONJ
ejpam-6549	66	2	1	1	NUM
ejpam-6549	66	3	vertices	vertex	NOUN
ejpam-6549	66	4	constructed	construct	VERB
ejpam-6549	66	5	as	as	SCONJ
ejpam-6549	66	6	follows	follow	VERB
ejpam-6549	66	7	:	:	PUNCT
ejpam-6549	66	8	take	take	VERB
ejpam-6549	66	9	n	n	PRON
ejpam-6549	66	10	disjoint	disjoint	NOUN
ejpam-6549	66	11	copies	copy	NOUN
ejpam-6549	66	12	of	of	ADP
ejpam-6549	66	13	the	the	DET
ejpam-6549	66	14	complete	complete	ADJ
ejpam-6549	66	15	graph	graph	NOUN
ejpam-6549	66	16	k2	k2	NOUN
ejpam-6549	66	17	,	,	PUNCT
ejpam-6549	66	18	and	and	CCONJ
ejpam-6549	66	19	connect	connect	VERB
ejpam-6549	66	20	each	each	PRON
ejpam-6549	66	21	of	of	ADP
ejpam-6549	66	22	their	their	PRON
ejpam-6549	66	23	vertices	vertex	NOUN
ejpam-6549	66	24	to	to	ADP
ejpam-6549	66	25	a	a	DET
ejpam-6549	66	26	single	single	ADJ
ejpam-6549	66	27	additional	additional	ADJ
ejpam-6549	66	28	vertex	vertex	NOUN
ejpam-6549	66	29	.	.	PUNCT
ejpam-6549	67	1	formally	formally	ADV
ejpam-6549	67	2	,	,	PUNCT
ejpam-6549	67	3	v	v	PROPN
ejpam-6549	67	4	(	(	PUNCT
ejpam-6549	67	5	rgn	rgn	PROPN
ejpam-6549	67	6	)	)	PUNCT
ejpam-6549	67	7	=	=	PRON
ejpam-6549	67	8	{	{	PUNCT
ejpam-6549	67	9	v0	v0	NOUN
ejpam-6549	67	10	}	}	PUNCT
ejpam-6549	67	11	∪	∪	NOUN
ejpam-6549	67	12	n⋃	n⋃	VERB
ejpam-6549	67	13	i=1	i=1	PRON
ejpam-6549	67	14	{	{	PUNCT
ejpam-6549	67	15	ui	ui	PROPN
ejpam-6549	67	16	,	,	PUNCT
ejpam-6549	67	17	vi	vi	PROPN
ejpam-6549	67	18	}	}	PUNCT
ejpam-6549	67	19	,	,	PUNCT
ejpam-6549	67	20	e(rgn	e(rgn	PROPN
ejpam-6549	67	21	)	)	PUNCT
ejpam-6549	67	22	=	=	PUNCT
ejpam-6549	68	1	n⋃	n⋃	VERB
ejpam-6549	68	2	i=1	i=1	PRON
ejpam-6549	68	3	{	{	PUNCT
ejpam-6549	68	4	(	(	PUNCT
ejpam-6549	68	5	ui	ui	PROPN
ejpam-6549	68	6	,	,	PUNCT
ejpam-6549	68	7	vi	vi	PROPN
ejpam-6549	68	8	)	)	PUNCT
ejpam-6549	68	9	,	,	PUNCT
ejpam-6549	68	10	(	(	PUNCT
ejpam-6549	68	11	v0	v0	NOUN
ejpam-6549	68	12	,	,	PUNCT
ejpam-6549	68	13	vi	vi	NOUN
ejpam-6549	68	14	)	)	PUNCT
ejpam-6549	68	15	}	}	PUNCT
ejpam-6549	68	16	.	.	PUNCT
ejpam-6549	69	1	equivalently	equivalently	ADV
ejpam-6549	69	2	,	,	PUNCT
ejpam-6549	69	3	the	the	DET
ejpam-6549	69	4	rake	rake	NOUN
ejpam-6549	69	5	graph	graph	NOUN
ejpam-6549	69	6	can	can	AUX
ejpam-6549	69	7	be	be	AUX
ejpam-6549	69	8	viewed	view	VERB
ejpam-6549	69	9	as	as	ADP
ejpam-6549	69	10	n	n	NUM
ejpam-6549	69	11	disjoint	disjoint	NOUN
ejpam-6549	69	12	edges	edge	NOUN
ejpam-6549	69	13	(	(	PUNCT
ejpam-6549	69	14	i.e.	i.e.	X
ejpam-6549	69	15	,	,	PUNCT
ejpam-6549	69	16	k2	k2	ADJ
ejpam-6549	69	17	components	component	NOUN
ejpam-6549	69	18	)	)	PUNCT
ejpam-6549	69	19	,	,	PUNCT
ejpam-6549	69	20	each	each	PRON
ejpam-6549	69	21	of	of	ADP
ejpam-6549	69	22	which	which	PRON
ejpam-6549	69	23	is	be	AUX
ejpam-6549	69	24	attached	attach	VERB
ejpam-6549	69	25	to	to	ADP
ejpam-6549	69	26	a	a	DET
ejpam-6549	69	27	single	single	ADJ
ejpam-6549	69	28	central	central	ADJ
ejpam-6549	69	29	vertex	vertex	NOUN
ejpam-6549	69	30	v0	v0	NOUN
ejpam-6549	69	31	via	via	ADP
ejpam-6549	69	32	one	one	NUM
ejpam-6549	69	33	of	of	ADP
ejpam-6549	69	34	its	its	PRON
ejpam-6549	69	35	endpoints	endpoint	NOUN
ejpam-6549	69	36	.	.	PUNCT
ejpam-6549	70	1	it	it	PRON
ejpam-6549	70	2	is	be	AUX
ejpam-6549	70	3	easy	easy	ADJ
ejpam-6549	70	4	to	to	PART
ejpam-6549	70	5	verify	verify	VERB
ejpam-6549	70	6	that	that	SCONJ
ejpam-6549	70	7	both	both	CCONJ
ejpam-6549	70	8	the	the	DET
ejpam-6549	70	9	matching	matching	NOUN
ejpam-6549	70	10	number	number	NOUN
ejpam-6549	70	11	and	and	CCONJ
ejpam-6549	70	12	the	the	DET
ejpam-6549	70	13	induced	induced	ADJ
ejpam-6549	70	14	matching	matching	NOUN
ejpam-6549	70	15	number	number	NOUN
ejpam-6549	70	16	of	of	ADP
ejpam-6549	70	17	rgn	rgn	PROPN
ejpam-6549	70	18	are	be	AUX
ejpam-6549	70	19	equal	equal	ADJ
ejpam-6549	70	20	to	to	PART
ejpam-6549	70	21	n.	n.	VERB
ejpam-6549	70	22	example	example	NOUN
ejpam-6549	70	23	1	1	X
ejpam-6549	70	24	.	.	PUNCT
ejpam-6549	70	25	figure	figure	VERB
ejpam-6549	70	26	1	1	NUM
ejpam-6549	70	27	depicts	depict	VERB
ejpam-6549	70	28	the	the	DET
ejpam-6549	70	29	graph	graph	NOUN
ejpam-6549	70	30	rg4	rg4	NOUN
ejpam-6549	70	31	.	.	PUNCT
ejpam-6549	71	1	the	the	DET
ejpam-6549	71	2	matching	match	VERB
ejpam-6549	71	3	number	number	NOUN
ejpam-6549	71	4	and	and	CCONJ
ejpam-6549	71	5	the	the	DET
ejpam-6549	71	6	induced	induced	ADJ
ejpam-6549	71	7	matching	matching	NOUN
ejpam-6549	71	8	number	number	NOUN
ejpam-6549	71	9	of	of	ADP
ejpam-6549	71	10	this	this	DET
ejpam-6549	71	11	graph	graph	NOUN
ejpam-6549	71	12	are	be	AUX
ejpam-6549	71	13	4	4	NUM
ejpam-6549	71	14	.	.	PUNCT
ejpam-6549	72	1	this	this	DET
ejpam-6549	72	2	graph	graph	NOUN
ejpam-6549	72	3	has	have	VERB
ejpam-6549	72	4	minimal	minimal	ADJ
ejpam-6549	72	5	energy	energy	NOUN
ejpam-6549	72	6	among	among	ADP
ejpam-6549	72	7	the	the	DET
ejpam-6549	72	8	connected	connected	ADJ
ejpam-6549	72	9	graphs	graph	NOUN
ejpam-6549	72	10	with	with	ADP
ejpam-6549	72	11	induced	induced	ADJ
ejpam-6549	72	12	matching	matching	NOUN
ejpam-6549	72	13	number	number	NOUN
ejpam-6549	72	14	4	4	NUM
ejpam-6549	72	15	(	(	PUNCT
ejpam-6549	72	16	see	see	VERB
ejpam-6549	72	17	theorem	theorem	NOUN
ejpam-6549	72	18	5	5	NUM
ejpam-6549	72	19	)	)	PUNCT
ejpam-6549	72	20	.	.	PUNCT
ejpam-6549	73	1	figure	figure	VERB
ejpam-6549	73	2	1	1	NUM
ejpam-6549	73	3	:	:	PUNCT
ejpam-6549	73	4	graph	graph	NOUN
ejpam-6549	73	5	rg4	rg4	NOUN
ejpam-6549	73	6	.	.	PUNCT
ejpam-6549	74	1	if	if	SCONJ
ejpam-6549	74	2	im(g	im(g	PUNCT
ejpam-6549	74	3	)	)	PUNCT
ejpam-6549	74	4	=	=	SYM
ejpam-6549	74	5	1	1	NUM
ejpam-6549	74	6	,	,	PUNCT
ejpam-6549	74	7	then	then	ADV
ejpam-6549	74	8	it	it	PRON
ejpam-6549	74	9	is	be	AUX
ejpam-6549	74	10	clear	clear	ADJ
ejpam-6549	74	11	that	that	SCONJ
ejpam-6549	74	12	k2	k2	PROPN
ejpam-6549	74	13	is	be	AUX
ejpam-6549	74	14	the	the	DET
ejpam-6549	74	15	graph	graph	NOUN
ejpam-6549	74	16	with	with	ADP
ejpam-6549	74	17	minimum	minimum	ADJ
ejpam-6549	74	18	energy	energy	NOUN
ejpam-6549	74	19	among	among	ADP
ejpam-6549	74	20	the	the	DET
ejpam-6549	74	21	graphs	graph	NOUN
ejpam-6549	74	22	with	with	ADP
ejpam-6549	74	23	im(g	im(g	PUNCT
ejpam-6549	74	24	)	)	PUNCT
ejpam-6549	74	25	=	=	SYM
ejpam-6549	75	1	1	1	X
ejpam-6549	75	2	.	.	PUNCT
ejpam-6549	75	3	in	in	ADP
ejpam-6549	75	4	the	the	DET
ejpam-6549	75	5	following	following	NOUN
ejpam-6549	75	6	theorem	theorem	NOUN
ejpam-6549	75	7	,	,	PUNCT
ejpam-6549	75	8	we	we	PRON
ejpam-6549	75	9	study	study	VERB
ejpam-6549	75	10	the	the	DET
ejpam-6549	75	11	graphs	graph	NOUN
ejpam-6549	75	12	with	with	ADP
ejpam-6549	75	13	minimum	minimum	ADJ
ejpam-6549	75	14	energy	energy	NOUN
ejpam-6549	75	15	among	among	ADP
ejpam-6549	75	16	those	those	PRON
ejpam-6549	75	17	with	with	ADP
ejpam-6549	75	18	a	a	DET
ejpam-6549	75	19	constant	constant	ADJ
ejpam-6549	75	20	induced	induce	VERB
ejpam-6549	75	21	matching	matching	NOUN
ejpam-6549	75	22	number	number	NOUN
ejpam-6549	75	23	im(g	im(g	PUNCT
ejpam-6549	75	24	)	)	PUNCT
ejpam-6549	75	25	>	>	X
ejpam-6549	76	1	2	2	X
ejpam-6549	76	2	.	.	X
ejpam-6549	76	3	theorem	theorem	NOUN
ejpam-6549	76	4	5	5	NUM
ejpam-6549	76	5	.	.	PUNCT
ejpam-6549	77	1	let	let	VERB
ejpam-6549	77	2	g	g	PRON
ejpam-6549	77	3	be	be	AUX
ejpam-6549	77	4	a	a	DET
ejpam-6549	77	5	connected	connected	ADJ
ejpam-6549	77	6	graph	graph	NOUN
ejpam-6549	77	7	with	with	ADP
ejpam-6549	77	8	induced	induced	ADJ
ejpam-6549	77	9	matching	matching	NOUN
ejpam-6549	77	10	number	number	NOUN
ejpam-6549	77	11	im(g	im(g	PUNCT
ejpam-6549	77	12	)	)	PUNCT
ejpam-6549	77	13	>	>	X
ejpam-6549	78	1	2	2	X
ejpam-6549	78	2	.	.	PUNCT
ejpam-6549	78	3	then	then	ADV
ejpam-6549	78	4	the	the	DET
ejpam-6549	78	5	energy	energy	NOUN
ejpam-6549	78	6	and	and	CCONJ
ejpam-6549	78	7	the	the	DET
ejpam-6549	78	8	matching	match	VERB
ejpam-6549	78	9	energy	energy	NOUN
ejpam-6549	78	10	of	of	ADP
ejpam-6549	78	11	g	g	PROPN
ejpam-6549	78	12	are	be	AUX
ejpam-6549	78	13	minimized	minimize	VERB
ejpam-6549	78	14	by	by	ADP
ejpam-6549	78	15	the	the	DET
ejpam-6549	78	16	graph	graph	NOUN
ejpam-6549	78	17	rgim(g	rgim(g	NOUN
ejpam-6549	78	18	)	)	PUNCT
ejpam-6549	78	19	.	.	PUNCT
ejpam-6549	79	1	proof	proof	NOUN
ejpam-6549	79	2	.	.	PUNCT
ejpam-6549	80	1	in	in	ADP
ejpam-6549	80	2	the	the	DET
ejpam-6549	80	3	view	view	NOUN
ejpam-6549	80	4	of	of	ADP
ejpam-6549	80	5	theorem	theorem	NOUN
ejpam-6549	80	6	2	2	NUM
ejpam-6549	80	7	,	,	PUNCT
ejpam-6549	80	8	the	the	DET
ejpam-6549	80	9	graph	graph	NOUN
ejpam-6549	80	10	g	g	PROPN
ejpam-6549	80	11	must	must	AUX
ejpam-6549	80	12	be	be	AUX
ejpam-6549	80	13	a	a	DET
ejpam-6549	80	14	tree	tree	NOUN
ejpam-6549	80	15	since	since	SCONJ
ejpam-6549	80	16	it	it	PRON
ejpam-6549	80	17	has	have	VERB
ejpam-6549	80	18	a	a	DET
ejpam-6549	80	19	minimal	minimal	ADJ
ejpam-6549	80	20	matching	matching	NOUN
ejpam-6549	80	21	energy	energy	NOUN
ejpam-6549	80	22	.	.	PUNCT
ejpam-6549	81	1	by	by	ADP
ejpam-6549	81	2	theorem	theorem	NOUN
ejpam-6549	81	3	3	3	NUM
ejpam-6549	81	4	,	,	PUNCT
ejpam-6549	81	5	it	it	PRON
ejpam-6549	81	6	is	be	AUX
ejpam-6549	81	7	enough	enough	ADJ
ejpam-6549	81	8	to	to	PART
ejpam-6549	81	9	show	show	VERB
ejpam-6549	81	10	that	that	SCONJ
ejpam-6549	81	11	this	this	DET
ejpam-6549	81	12	graph	graph	NOUN
ejpam-6549	81	13	has	have	VERB
ejpam-6549	81	14	minimal	minimal	ADJ
ejpam-6549	81	15	matching	matching	NOUN
ejpam-6549	81	16	energy	energy	NOUN
ejpam-6549	81	17	with	with	ADP
ejpam-6549	81	18	the	the	DET
ejpam-6549	81	19	given	give	VERB
ejpam-6549	81	20	induced	induce	VERB
ejpam-6549	81	21	matching	matching	NOUN
ejpam-6549	81	22	number	number	NOUN
ejpam-6549	81	23	.	.	PUNCT
ejpam-6549	82	1	let	let	VERB
ejpam-6549	82	2	m	m	PRON
ejpam-6549	82	3	be	be	AUX
ejpam-6549	82	4	the	the	DET
ejpam-6549	82	5	set	set	NOUN
ejpam-6549	82	6	of	of	ADP
ejpam-6549	82	7	induced	induce	VERB
ejpam-6549	82	8	matchings	matching	NOUN
ejpam-6549	82	9	.	.	PUNCT
ejpam-6549	83	1	then	then	ADV
ejpam-6549	83	2	g[m	g[m	VERB
ejpam-6549	83	3	]	]	PUNCT
ejpam-6549	83	4	forms	form	VERB
ejpam-6549	83	5	a	a	DET
ejpam-6549	83	6	disjoint	disjoint	NOUN
ejpam-6549	83	7	union	union	NOUN
ejpam-6549	83	8	of	of	ADP
ejpam-6549	83	9	graphs	graph	NOUN
ejpam-6549	83	10	k2	k2	ADJ
ejpam-6549	83	11	with	with	ADP
ejpam-6549	83	12	the	the	DET
ejpam-6549	83	13	number	number	NOUN
ejpam-6549	83	14	of	of	ADP
ejpam-6549	83	15	such	such	ADJ
ejpam-6549	83	16	a.	a.	NOUN
ejpam-6549	83	17	ülker	ülker	PROPN
ejpam-6549	83	18	et	et	PROPN
ejpam-6549	83	19	al	al	PROPN
ejpam-6549	83	20	.	.	PUNCT
ejpam-6549	83	21	/	/	SYM
ejpam-6549	83	22	eur	eur	PROPN
ejpam-6549	83	23	.	.	PUNCT
ejpam-6549	84	1	j.	j.	PROPN
ejpam-6549	84	2	pure	pure	PROPN
ejpam-6549	84	3	appl	appl	PROPN
ejpam-6549	84	4	.	.	PROPN
ejpam-6549	84	5	math	math	PROPN
ejpam-6549	84	6	,	,	PUNCT
ejpam-6549	84	7	18	18	NUM
ejpam-6549	84	8	(	(	PUNCT
ejpam-6549	84	9	3	3	NUM
ejpam-6549	84	10	)	)	PUNCT
ejpam-6549	84	11	(	(	PUNCT
ejpam-6549	84	12	2025	2025	NUM
ejpam-6549	84	13	)	)	PUNCT
ejpam-6549	84	14	,	,	PUNCT
ejpam-6549	84	15	6549	6549	NUM
ejpam-6549	84	16	5	5	NUM
ejpam-6549	84	17	of	of	ADP
ejpam-6549	84	18	8	8	NUM
ejpam-6549	84	19	graphs	graph	NOUN
ejpam-6549	84	20	equal	equal	ADJ
ejpam-6549	84	21	to	to	ADP
ejpam-6549	84	22	im(g	im(g	NOUN
ejpam-6549	84	23	)	)	PUNCT
ejpam-6549	84	24	.	.	PUNCT
ejpam-6549	85	1	since	since	SCONJ
ejpam-6549	85	2	m	m	PROPN
ejpam-6549	85	3	is	be	AUX
ejpam-6549	85	4	a	a	DET
ejpam-6549	85	5	set	set	NOUN
ejpam-6549	85	6	of	of	ADP
ejpam-6549	85	7	induced	induced	ADJ
ejpam-6549	85	8	matchings	matching	NOUN
ejpam-6549	85	9	,	,	PUNCT
ejpam-6549	85	10	this	this	PRON
ejpam-6549	85	11	implies	imply	VERB
ejpam-6549	85	12	that	that	SCONJ
ejpam-6549	85	13	for	for	ADP
ejpam-6549	85	14	each	each	PRON
ejpam-6549	85	15	ei	ei	X
ejpam-6549	85	16	∈	∈	NOUN
ejpam-6549	85	17	m	m	NOUN
ejpam-6549	85	18	,	,	PUNCT
ejpam-6549	85	19	there	there	PRON
ejpam-6549	85	20	exists	exist	VERB
ejpam-6549	85	21	at	at	ADV
ejpam-6549	85	22	least	least	ADV
ejpam-6549	85	23	one	one	NUM
ejpam-6549	85	24	edge	edge	NOUN
ejpam-6549	85	25	e′i	e′i	NOUN
ejpam-6549	85	26	adjacent	adjacent	ADJ
ejpam-6549	85	27	to	to	PART
ejpam-6549	85	28	ei	ei	VERB
ejpam-6549	85	29	with	with	ADP
ejpam-6549	85	30	e′i	e′i	PROPN
ejpam-6549	85	31	/∈	/∈	PUNCT
ejpam-6549	86	1	m	m	INTJ
ejpam-6549	86	2	.	.	PUNCT
ejpam-6549	87	1	moreover	moreover	ADV
ejpam-6549	87	2	,	,	PUNCT
ejpam-6549	87	3	e′i	e′i	PROPN
ejpam-6549	87	4	is	be	AUX
ejpam-6549	87	5	adjacent	adjacent	ADJ
ejpam-6549	87	6	to	to	ADP
ejpam-6549	87	7	only	only	ADV
ejpam-6549	87	8	ei	ei	PROPN
ejpam-6549	87	9	,	,	PUNCT
ejpam-6549	87	10	otherwise	otherwise	ADV
ejpam-6549	87	11	e′i	e′i	PROPN
ejpam-6549	87	12	would	would	AUX
ejpam-6549	87	13	be	be	AUX
ejpam-6549	87	14	a	a	DET
ejpam-6549	87	15	common	common	ADJ
ejpam-6549	87	16	edge	edge	NOUN
ejpam-6549	87	17	of	of	ADP
ejpam-6549	87	18	two	two	NUM
ejpam-6549	87	19	edges	edge	NOUN
ejpam-6549	87	20	in	in	ADP
ejpam-6549	87	21	m	m	PROPN
ejpam-6549	87	22	and	and	CCONJ
ejpam-6549	87	23	a	a	DET
ejpam-6549	87	24	contradiction	contradiction	NOUN
ejpam-6549	87	25	.	.	PUNCT
ejpam-6549	88	1	by	by	ADP
ejpam-6549	88	2	minimality	minimality	NOUN
ejpam-6549	88	3	,	,	PUNCT
ejpam-6549	88	4	we	we	PRON
ejpam-6549	88	5	can	can	AUX
ejpam-6549	88	6	consider	consider	VERB
ejpam-6549	88	7	one	one	NUM
ejpam-6549	88	8	of	of	ADP
ejpam-6549	88	9	the	the	DET
ejpam-6549	88	10	vertices	vertex	NOUN
ejpam-6549	88	11	of	of	ADP
ejpam-6549	88	12	the	the	DET
ejpam-6549	88	13	edges	edge	NOUN
ejpam-6549	88	14	ei	ei	NOUN
ejpam-6549	88	15	=	=	PUNCT
ejpam-6549	88	16	uivi	uivi	NOUN
ejpam-6549	88	17	of	of	ADP
ejpam-6549	88	18	m	m	PROPN
ejpam-6549	88	19	is	be	AUX
ejpam-6549	88	20	a	a	DET
ejpam-6549	88	21	pendant	pendant	ADJ
ejpam-6549	88	22	vertex	vertex	NOUN
ejpam-6549	88	23	,	,	PUNCT
ejpam-6549	88	24	say	say	VERB
ejpam-6549	88	25	ui	ui	PROPN
ejpam-6549	88	26	.	.	PUNCT
ejpam-6549	89	1	since	since	SCONJ
ejpam-6549	89	2	m(g	m(g	PROPN
ejpam-6549	89	3	;	;	PUNCT
ejpam-6549	89	4	k	k	X
ejpam-6549	89	5	)	)	PUNCT
ejpam-6549	89	6	=	=	SYM
ejpam-6549	89	7	m(g	m(g	PROPN
ejpam-6549	89	8	−	−	PROPN
ejpam-6549	89	9	ei	ei	NOUN
ejpam-6549	89	10	;	;	PUNCT
ejpam-6549	89	11	k	k	X
ejpam-6549	89	12	)	)	PUNCT
ejpam-6549	89	13	+	+	ADJ
ejpam-6549	89	14	m(g	m(g	ADJ
ejpam-6549	89	15	−	−	PROPN
ejpam-6549	89	16	ui	ui	NOUN
ejpam-6549	89	17	−	−	PROPN
ejpam-6549	89	18	vi	vi	PROPN
ejpam-6549	89	19	;	;	PUNCT
ejpam-6549	89	20	k	k	PROPN
ejpam-6549	89	21	−	−	PROPN
ejpam-6549	89	22	1	1	NUM
ejpam-6549	89	23	)	)	PUNCT
ejpam-6549	89	24	for	for	ADP
ejpam-6549	89	25	all	all	PRON
ejpam-6549	89	26	ei	ei	NOUN
ejpam-6549	89	27	=	=	PUNCT
ejpam-6549	89	28	uivi	uivi	NOUN
ejpam-6549	89	29	∈	∈	PROPN
ejpam-6549	89	30	m	m	NOUN
ejpam-6549	89	31	,	,	PUNCT
ejpam-6549	89	32	then	then	ADV
ejpam-6549	89	33	the	the	DET
ejpam-6549	89	34	subgraph	subgraph	NOUN
ejpam-6549	89	35	after	after	ADP
ejpam-6549	89	36	removing	remove	VERB
ejpam-6549	89	37	edges	edge	NOUN
ejpam-6549	89	38	ei	ei	ADP
ejpam-6549	89	39	∈	∈	NOUN
ejpam-6549	89	40	m	m	VERB
ejpam-6549	89	41	,	,	PUNCT
ejpam-6549	89	42	we	we	PRON
ejpam-6549	89	43	have	have	VERB
ejpam-6549	89	44	a	a	DET
ejpam-6549	89	45	cut	cut	VERB
ejpam-6549	89	46	-	-	PUNCT
ejpam-6549	89	47	vertex	vertex	NOUN
ejpam-6549	89	48	left	leave	VERB
ejpam-6549	89	49	for	for	ADP
ejpam-6549	89	50	minimality	minimality	NOUN
ejpam-6549	89	51	.	.	PUNCT
ejpam-6549	90	1	since	since	SCONJ
ejpam-6549	90	2	the	the	DET
ejpam-6549	90	3	matching	match	VERB
ejpam-6549	90	4	energy	energy	NOUN
ejpam-6549	90	5	of	of	ADP
ejpam-6549	90	6	a	a	DET
ejpam-6549	90	7	tree	tree	NOUN
ejpam-6549	90	8	coincides	coincide	VERB
ejpam-6549	90	9	with	with	ADP
ejpam-6549	90	10	its	its	PRON
ejpam-6549	90	11	energy	energy	NOUN
ejpam-6549	90	12	,	,	PUNCT
ejpam-6549	90	13	the	the	DET
ejpam-6549	90	14	graph	graph	NOUN
ejpam-6549	90	15	rgim(g	rgim(g	NOUN
ejpam-6549	90	16	)	)	PUNCT
ejpam-6549	90	17	has	have	VERB
ejpam-6549	90	18	the	the	DET
ejpam-6549	90	19	minimum	minimum	ADJ
ejpam-6549	90	20	energy	energy	NOUN
ejpam-6549	90	21	among	among	ADP
ejpam-6549	90	22	the	the	DET
ejpam-6549	90	23	graphs	graph	NOUN
ejpam-6549	90	24	with	with	ADP
ejpam-6549	90	25	induced	induced	ADJ
ejpam-6549	90	26	matching	matching	NOUN
ejpam-6549	90	27	number	number	NOUN
ejpam-6549	90	28	im(g	im(g	PUNCT
ejpam-6549	90	29	)	)	PUNCT
ejpam-6549	90	30	≥	≥	NOUN
ejpam-6549	90	31	2	2	NUM
ejpam-6549	90	32	.	.	PUNCT
ejpam-6549	90	33	theorem	theorem	NOUN
ejpam-6549	90	34	6	6	NUM
ejpam-6549	90	35	.	.	PUNCT
ejpam-6549	91	1	let	let	VERB
ejpam-6549	91	2	g	g	PROPN
ejpam-6549	91	3	=	=	SYM
ejpam-6549	91	4	(	(	PUNCT
ejpam-6549	91	5	v	v	NOUN
ejpam-6549	91	6	,	,	PUNCT
ejpam-6549	91	7	e	e	NOUN
ejpam-6549	91	8	)	)	PUNCT
ejpam-6549	91	9	be	be	AUX
ejpam-6549	91	10	a	a	DET
ejpam-6549	91	11	graph	graph	NOUN
ejpam-6549	91	12	with	with	ADP
ejpam-6549	91	13	induced	induced	ADJ
ejpam-6549	91	14	matching	matching	NOUN
ejpam-6549	91	15	number	number	NOUN
ejpam-6549	91	16	im(g	im(g	PUNCT
ejpam-6549	91	17	)	)	PUNCT
ejpam-6549	91	18	.	.	PUNCT
ejpam-6549	92	1	then	then	ADV
ejpam-6549	92	2	the	the	DET
ejpam-6549	92	3	followings	following	NOUN
ejpam-6549	92	4	hold	hold	VERB
ejpam-6549	92	5	:	:	PUNCT
ejpam-6549	92	6	(	(	PUNCT
ejpam-6549	92	7	1	1	X
ejpam-6549	92	8	)	)	PUNCT
ejpam-6549	92	9	for	for	ADP
ejpam-6549	92	10	any	any	DET
ejpam-6549	92	11	graph	graph	NOUN
ejpam-6549	92	12	g	g	NOUN
ejpam-6549	92	13	,	,	PUNCT
ejpam-6549	92	14	ε(g	ε(g	PROPN
ejpam-6549	92	15	)	)	PUNCT
ejpam-6549	92	16	≥	≥	NOUN
ejpam-6549	92	17	2im(g	2im(g	NUM
ejpam-6549	92	18	)	)	PUNCT
ejpam-6549	92	19	,	,	PUNCT
ejpam-6549	92	20	and	and	CCONJ
ejpam-6549	92	21	equality	equality	NOUN
ejpam-6549	92	22	holds	hold	VERB
ejpam-6549	92	23	if	if	SCONJ
ejpam-6549	92	24	and	and	CCONJ
ejpam-6549	92	25	only	only	ADV
ejpam-6549	92	26	if	if	SCONJ
ejpam-6549	92	27	g	g	PROPN
ejpam-6549	92	28	is	be	AUX
ejpam-6549	92	29	isomorphic	isomorphic	ADJ
ejpam-6549	92	30	to	to	ADP
ejpam-6549	92	31	the	the	DET
ejpam-6549	92	32	disjoint	disjoint	NOUN
ejpam-6549	92	33	union	union	NOUN
ejpam-6549	92	34	of	of	ADP
ejpam-6549	92	35	graphs	graph	NOUN
ejpam-6549	92	36	k2	k2	PROPN
ejpam-6549	92	37	.	.	PUNCT
ejpam-6549	93	1	(	(	PUNCT
ejpam-6549	93	2	2	2	X
ejpam-6549	93	3	)	)	PUNCT
ejpam-6549	93	4	if	if	SCONJ
ejpam-6549	93	5	g	g	PROPN
ejpam-6549	93	6	is	be	AUX
ejpam-6549	93	7	connected	connect	VERB
ejpam-6549	93	8	and	and	CCONJ
ejpam-6549	93	9	im(g	im(g	PUNCT
ejpam-6549	93	10	)	)	PUNCT
ejpam-6549	93	11	≥	≥	NOUN
ejpam-6549	93	12	2	2	NUM
ejpam-6549	93	13	,	,	PUNCT
ejpam-6549	93	14	then	then	ADV
ejpam-6549	93	15	ε(g	ε(g	NOUN
ejpam-6549	93	16	)	)	PUNCT
ejpam-6549	93	17	≥	≥	NOUN
ejpam-6549	93	18	ε(rgim(g	ε(rgim(g	NOUN
ejpam-6549	93	19	)	)	PUNCT
ejpam-6549	93	20	)	)	PUNCT
ejpam-6549	93	21	.	.	PUNCT
ejpam-6549	94	1	proof	proof	NOUN
ejpam-6549	94	2	.	.	PUNCT
ejpam-6549	95	1	(	(	PUNCT
ejpam-6549	95	2	1	1	X
ejpam-6549	95	3	)	)	PUNCT
ejpam-6549	95	4	assume	assume	VERB
ejpam-6549	95	5	that	that	SCONJ
ejpam-6549	95	6	m	m	VERB
ejpam-6549	95	7	⊆	⊆	NUM
ejpam-6549	95	8	e	e	AUX
ejpam-6549	95	9	be	be	AUX
ejpam-6549	95	10	an	an	DET
ejpam-6549	95	11	induced	induced	ADJ
ejpam-6549	95	12	matching	matching	NOUN
ejpam-6549	95	13	set	set	NOUN
ejpam-6549	95	14	.	.	PUNCT
ejpam-6549	96	1	the	the	DET
ejpam-6549	96	2	induced	induced	ADJ
ejpam-6549	96	3	subgraph	subgraph	NOUN
ejpam-6549	96	4	on	on	ADP
ejpam-6549	96	5	g[m	g[m	NOUN
ejpam-6549	96	6	]	]	PUNCT
ejpam-6549	96	7	forms	form	VERB
ejpam-6549	96	8	a	a	DET
ejpam-6549	96	9	disjoint	disjoint	NOUN
ejpam-6549	96	10	union	union	NOUN
ejpam-6549	96	11	of	of	ADP
ejpam-6549	96	12	graphs	graph	NOUN
ejpam-6549	96	13	k2	k2	PROPN
ejpam-6549	96	14	.	.	PUNCT
ejpam-6549	97	1	since	since	SCONJ
ejpam-6549	97	2	each	each	DET
ejpam-6549	97	3	k2	k2	PROPN
ejpam-6549	97	4	contributes	contribute	VERB
ejpam-6549	97	5	2	2	NUM
ejpam-6549	97	6	to	to	ADP
ejpam-6549	97	7	the	the	DET
ejpam-6549	97	8	ε(g	ε(g	NOUN
ejpam-6549	97	9	)	)	PUNCT
ejpam-6549	97	10	,	,	PUNCT
ejpam-6549	97	11	then	then	ADV
ejpam-6549	97	12	by	by	ADP
ejpam-6549	97	13	lemma	lemma	PROPN
ejpam-6549	97	14	1	1	NUM
ejpam-6549	97	15	,	,	PUNCT
ejpam-6549	97	16	we	we	PRON
ejpam-6549	97	17	get	get	VERB
ejpam-6549	97	18	ε(g	ε(g	NOUN
ejpam-6549	97	19	)	)	PUNCT
ejpam-6549	97	20	≥	≥	NOUN
ejpam-6549	97	21	2im(g	2im(g	NUM
ejpam-6549	97	22	)	)	PUNCT
ejpam-6549	97	23	.	.	PUNCT
ejpam-6549	98	1	if	if	SCONJ
ejpam-6549	98	2	we	we	PRON
ejpam-6549	98	3	assume	assume	VERB
ejpam-6549	98	4	g	g	PROPN
ejpam-6549	98	5	is	be	AUX
ejpam-6549	98	6	isomorphic	isomorphic	ADJ
ejpam-6549	98	7	to	to	ADP
ejpam-6549	98	8	the	the	DET
ejpam-6549	98	9	disjoint	disjoint	NOUN
ejpam-6549	98	10	union	union	NOUN
ejpam-6549	98	11	of	of	ADP
ejpam-6549	98	12	graphs	graph	NOUN
ejpam-6549	98	13	k2	k2	ADJ
ejpam-6549	98	14	,	,	PUNCT
ejpam-6549	98	15	i.e.	i.e.	X
ejpam-6549	98	16	,	,	PUNCT
ejpam-6549	98	17	g	g	PROPN
ejpam-6549	98	18	∼=	∼=	PROPN
ejpam-6549	98	19	nk2	nk2	NOUN
ejpam-6549	98	20	,	,	PUNCT
ejpam-6549	98	21	this	this	PRON
ejpam-6549	98	22	implies	imply	VERB
ejpam-6549	98	23	that	that	SCONJ
ejpam-6549	98	24	im(g	im(g	PUNCT
ejpam-6549	98	25	)	)	PUNCT
ejpam-6549	98	26	=	=	VERB
ejpam-6549	98	27	n.	n.	NOUN
ejpam-6549	98	28	since	since	SCONJ
ejpam-6549	98	29	ε(g	ε(g	NOUN
ejpam-6549	98	30	)	)	PUNCT
ejpam-6549	98	31	=	=	SYM
ejpam-6549	98	32	2n	2n	NUM
ejpam-6549	98	33	,	,	PUNCT
ejpam-6549	98	34	then	then	ADV
ejpam-6549	98	35	equality	equality	NOUN
ejpam-6549	98	36	holds	hold	VERB
ejpam-6549	98	37	.	.	PUNCT
ejpam-6549	99	1	(	(	PUNCT
ejpam-6549	99	2	2	2	X
ejpam-6549	99	3	)	)	PUNCT
ejpam-6549	99	4	assume	assume	VERB
ejpam-6549	99	5	that	that	SCONJ
ejpam-6549	99	6	g	g	PROPN
ejpam-6549	99	7	is	be	AUX
ejpam-6549	99	8	a	a	DET
ejpam-6549	99	9	connected	connected	ADJ
ejpam-6549	99	10	graph	graph	NOUN
ejpam-6549	99	11	.	.	PUNCT
ejpam-6549	100	1	since	since	SCONJ
ejpam-6549	100	2	the	the	DET
ejpam-6549	100	3	minimal	minimal	ADJ
ejpam-6549	100	4	energy	energy	NOUN
ejpam-6549	100	5	graph	graph	NOUN
ejpam-6549	100	6	with	with	ADP
ejpam-6549	100	7	induced	induced	ADJ
ejpam-6549	100	8	matching	matching	NOUN
ejpam-6549	100	9	number	number	NOUN
ejpam-6549	100	10	im(g	im(g	PUNCT
ejpam-6549	100	11	)	)	PUNCT
ejpam-6549	100	12	≥	≥	X
ejpam-6549	101	1	2	2	NUM
ejpam-6549	101	2	is	be	AUX
ejpam-6549	101	3	rgim(g	rgim(g	NOUN
ejpam-6549	101	4	)	)	PUNCT
ejpam-6549	101	5	by	by	ADP
ejpam-6549	101	6	theorem	theorem	NOUN
ejpam-6549	101	7	5	5	NUM
ejpam-6549	101	8	,	,	PUNCT
ejpam-6549	101	9	then	then	ADV
ejpam-6549	101	10	it	it	PRON
ejpam-6549	101	11	is	be	AUX
ejpam-6549	101	12	clear	clear	ADJ
ejpam-6549	101	13	that	that	SCONJ
ejpam-6549	101	14	ε(g	ε(g	NOUN
ejpam-6549	101	15	)	)	PUNCT
ejpam-6549	101	16	≥	≥	NOUN
ejpam-6549	101	17	ε(rgim(g	ε(rgim(g	NUM
ejpam-6549	101	18	)	)	PUNCT
ejpam-6549	101	19	)	)	PUNCT
ejpam-6549	101	20	for	for	ADP
ejpam-6549	101	21	any	any	DET
ejpam-6549	101	22	connected	connected	ADJ
ejpam-6549	101	23	graph	graph	NOUN
ejpam-6549	101	24	g.	g.	NOUN
ejpam-6549	101	25	4	4	NUM
ejpam-6549	101	26	.	.	PUNCT
ejpam-6549	101	27	energy	energy	NOUN
ejpam-6549	101	28	of	of	ADP
ejpam-6549	101	29	graphs	graph	NOUN
ejpam-6549	101	30	with	with	ADP
ejpam-6549	101	31	equal	equal	ADJ
ejpam-6549	101	32	matching	matching	NOUN
ejpam-6549	101	33	and	and	CCONJ
ejpam-6549	101	34	induced	induced	ADJ
ejpam-6549	101	35	matching	matching	NOUN
ejpam-6549	101	36	number	number	NOUN
ejpam-6549	101	37	cameron	cameron	PROPN
ejpam-6549	101	38	and	and	CCONJ
ejpam-6549	101	39	walker	walker	PROPN
ejpam-6549	101	40	gave	give	VERB
ejpam-6549	101	41	a	a	DET
ejpam-6549	101	42	characterization	characterization	NOUN
ejpam-6549	101	43	of	of	ADP
ejpam-6549	101	44	undirected	undirected	ADJ
ejpam-6549	101	45	connected	connected	ADJ
ejpam-6549	101	46	finite	finite	ADJ
ejpam-6549	101	47	simple	simple	ADJ
ejpam-6549	101	48	graphs	graph	NOUN
ejpam-6549	101	49	that	that	PRON
ejpam-6549	101	50	satisfy	satisfy	VERB
ejpam-6549	101	51	m(g	m(g	PROPN
ejpam-6549	101	52	)	)	PUNCT
ejpam-6549	101	53	=	=	SYM
ejpam-6549	102	1	im(g	im(g	X
ejpam-6549	102	2	)	)	PUNCT
ejpam-6549	103	1	in	in	ADP
ejpam-6549	103	2	[	[	X
ejpam-6549	103	3	9	9	NUM
ejpam-6549	103	4	]	]	PUNCT
ejpam-6549	103	5	.	.	PUNCT
ejpam-6549	104	1	these	these	DET
ejpam-6549	104	2	graphs	graph	NOUN
ejpam-6549	104	3	are	be	AUX
ejpam-6549	104	4	known	know	VERB
ejpam-6549	104	5	as	as	ADP
ejpam-6549	104	6	cameron	cameron	PROPN
ejpam-6549	104	7	-	-	PUNCT
ejpam-6549	104	8	walker	walker	PROPN
ejpam-6549	104	9	graphs	graph	NOUN
ejpam-6549	104	10	and	and	CCONJ
ejpam-6549	104	11	have	have	AUX
ejpam-6549	104	12	previously	previously	ADV
ejpam-6549	104	13	been	be	AUX
ejpam-6549	104	14	studied	study	VERB
ejpam-6549	104	15	from	from	ADP
ejpam-6549	104	16	a	a	DET
ejpam-6549	104	17	commutative	commutative	ADJ
ejpam-6549	104	18	algebra	algebra	NOUN
ejpam-6549	104	19	perspective	perspective	NOUN
ejpam-6549	104	20	,	,	PUNCT
ejpam-6549	104	21	particularly	particularly	ADV
ejpam-6549	104	22	in	in	ADP
ejpam-6549	104	23	relation	relation	NOUN
ejpam-6549	104	24	to	to	PART
ejpam-6549	104	25	edge	edge	VERB
ejpam-6549	104	26	ideals	ideal	NOUN
ejpam-6549	104	27	and	and	CCONJ
ejpam-6549	104	28	their	their	PRON
ejpam-6549	104	29	algebraic	algebraic	ADJ
ejpam-6549	104	30	invariants	invariant	NOUN
ejpam-6549	105	1	[	[	X
ejpam-6549	105	2	10	10	NUM
ejpam-6549	105	3	]	]	PUNCT
ejpam-6549	105	4	.	.	PUNCT
ejpam-6549	106	1	the	the	DET
ejpam-6549	106	2	authors	author	NOUN
ejpam-6549	106	3	gave	give	VERB
ejpam-6549	106	4	a	a	DET
ejpam-6549	106	5	slightly	slightly	ADV
ejpam-6549	106	6	modified	modify	VERB
ejpam-6549	106	7	definition	definition	NOUN
ejpam-6549	106	8	of	of	ADP
ejpam-6549	106	9	these	these	DET
ejpam-6549	106	10	graphs	graph	NOUN
ejpam-6549	106	11	.	.	PUNCT
ejpam-6549	107	1	next	next	ADV
ejpam-6549	107	2	,	,	PUNCT
ejpam-6549	107	3	we	we	PRON
ejpam-6549	107	4	provide	provide	VERB
ejpam-6549	107	5	the	the	DET
ejpam-6549	107	6	definitions	definition	NOUN
ejpam-6549	107	7	of	of	ADP
ejpam-6549	107	8	this	this	DET
ejpam-6549	107	9	graph	graph	NOUN
ejpam-6549	107	10	in	in	ADP
ejpam-6549	107	11	the	the	DET
ejpam-6549	107	12	light	light	NOUN
ejpam-6549	107	13	of	of	ADP
ejpam-6549	107	14	[	[	X
ejpam-6549	107	15	10	10	NUM
ejpam-6549	107	16	]	]	PUNCT
ejpam-6549	107	17	.	.	PUNCT
ejpam-6549	108	1	definition	definition	NOUN
ejpam-6549	108	2	3	3	NUM
ejpam-6549	108	3	.	.	PUNCT
ejpam-6549	109	1	a	a	DET
ejpam-6549	109	2	finite	finite	ADJ
ejpam-6549	109	3	simple	simple	ADJ
ejpam-6549	109	4	and	and	CCONJ
ejpam-6549	109	5	connected	connected	ADJ
ejpam-6549	109	6	graph	graph	NOUN
ejpam-6549	109	7	g	g	NOUN
ejpam-6549	109	8	satisfies	satisfie	NOUN
ejpam-6549	109	9	m(g	m(g	NOUN
ejpam-6549	109	10	)	)	PUNCT
ejpam-6549	109	11	=	=	SYM
ejpam-6549	110	1	im(g	im(g	X
ejpam-6549	110	2	)	)	PUNCT
ejpam-6549	110	3	if	if	SCONJ
ejpam-6549	110	4	g	g	PROPN
ejpam-6549	110	5	is	be	AUX
ejpam-6549	110	6	one	one	NUM
ejpam-6549	110	7	of	of	ADP
ejpam-6549	110	8	the	the	DET
ejpam-6549	110	9	following	following	NOUN
ejpam-6549	110	10	:	:	PUNCT
ejpam-6549	110	11	(	(	PUNCT
ejpam-6549	110	12	i	i	NOUN
ejpam-6549	110	13	)	)	PUNCT
ejpam-6549	110	14	g	g	PROPN
ejpam-6549	110	15	is	be	AUX
ejpam-6549	110	16	a	a	DET
ejpam-6549	110	17	star	star	NOUN
ejpam-6549	110	18	graph	graph	NOUN
ejpam-6549	110	19	,	,	PUNCT
ejpam-6549	110	20	(	(	PUNCT
ejpam-6549	110	21	ii	ii	NOUN
ejpam-6549	110	22	)	)	PUNCT
ejpam-6549	110	23	g	g	NOUN
ejpam-6549	110	24	is	be	AUX
ejpam-6549	110	25	a	a	DET
ejpam-6549	110	26	star	star	NOUN
ejpam-6549	110	27	triangle	triangle	NOUN
ejpam-6549	110	28	,	,	PUNCT
ejpam-6549	110	29	(	(	PUNCT
ejpam-6549	110	30	iii	iii	X
ejpam-6549	110	31	)	)	PUNCT
ejpam-6549	110	32	g	g	NOUN
ejpam-6549	110	33	is	be	AUX
ejpam-6549	110	34	a	a	DET
ejpam-6549	110	35	finite	finite	ADJ
ejpam-6549	110	36	graph	graph	NOUN
ejpam-6549	110	37	consisting	consist	VERB
ejpam-6549	110	38	of	of	ADP
ejpam-6549	110	39	a	a	DET
ejpam-6549	110	40	connected	connected	ADJ
ejpam-6549	110	41	supporting	support	VERB
ejpam-6549	110	42	bipartite	bipartite	NOUN
ejpam-6549	110	43	graph	graph	NOUN
ejpam-6549	110	44	with	with	ADP
ejpam-6549	110	45	vertex	vertex	NOUN
ejpam-6549	110	46	partition	partition	NOUN
ejpam-6549	110	47	u	u	NOUN
ejpam-6549	110	48	⊔v	⊔v	ADP
ejpam-6549	110	49	such	such	ADJ
ejpam-6549	110	50	that	that	SCONJ
ejpam-6549	110	51	there	there	PRON
ejpam-6549	110	52	is	be	VERB
ejpam-6549	110	53	at	at	ADV
ejpam-6549	110	54	least	least	ADJ
ejpam-6549	110	55	one	one	NUM
ejpam-6549	110	56	leaf	leaf	NOUN
ejpam-6549	110	57	edge	edge	NOUN
ejpam-6549	110	58	attached	attach	VERB
ejpam-6549	110	59	to	to	ADP
ejpam-6549	110	60	each	each	DET
ejpam-6549	110	61	vertex	vertex	NOUN
ejpam-6549	110	62	u	u	NOUN
ejpam-6549	110	63	∈	∈	PROPN
ejpam-6549	110	64	u	u	NOUN
ejpam-6549	110	65	and	and	CCONJ
ejpam-6549	110	66	that	that	SCONJ
ejpam-6549	110	67	there	there	PRON
ejpam-6549	110	68	may	may	AUX
ejpam-6549	110	69	be	be	AUX
ejpam-6549	110	70	some	some	DET
ejpam-6549	110	71	pendant	pendant	ADJ
ejpam-6549	110	72	triangles	triangle	NOUN
ejpam-6549	110	73	connected	connect	VERB
ejpam-6549	110	74	to	to	ADP
ejpam-6549	110	75	each	each	DET
ejpam-6549	110	76	vertex	vertex	NOUN
ejpam-6549	110	77	v	v	ADP
ejpam-6549	110	78	∈	∈	PROPN
ejpam-6549	110	79	v	v	NOUN
ejpam-6549	110	80	.	.	PUNCT
ejpam-6549	111	1	a.	a.	NOUN
ejpam-6549	111	2	ülker	ülker	PROPN
ejpam-6549	111	3	et	et	PROPN
ejpam-6549	111	4	al	al	PROPN
ejpam-6549	111	5	.	.	PUNCT
ejpam-6549	111	6	/	/	SYM
ejpam-6549	111	7	eur	eur	PROPN
ejpam-6549	111	8	.	.	PUNCT
ejpam-6549	112	1	j.	j.	PROPN
ejpam-6549	112	2	pure	pure	PROPN
ejpam-6549	112	3	appl	appl	PROPN
ejpam-6549	112	4	.	.	PROPN
ejpam-6549	112	5	math	math	PROPN
ejpam-6549	112	6	,	,	PUNCT
ejpam-6549	112	7	18	18	NUM
ejpam-6549	112	8	(	(	PUNCT
ejpam-6549	112	9	3	3	NUM
ejpam-6549	112	10	)	)	PUNCT
ejpam-6549	112	11	(	(	PUNCT
ejpam-6549	112	12	2025	2025	NUM
ejpam-6549	112	13	)	)	PUNCT
ejpam-6549	112	14	,	,	PUNCT
ejpam-6549	112	15	6549	6549	NUM
ejpam-6549	112	16	6	6	NUM
ejpam-6549	112	17	of	of	ADP
ejpam-6549	112	18	8	8	NUM
ejpam-6549	112	19	definition	definition	NOUN
ejpam-6549	112	20	4	4	NUM
ejpam-6549	112	21	.	.	PUNCT
ejpam-6549	113	1	a	a	DET
ejpam-6549	113	2	finite	finite	ADJ
ejpam-6549	113	3	connected	connect	VERB
ejpam-6549	113	4	simple	simple	ADJ
ejpam-6549	113	5	graph	graph	NOUN
ejpam-6549	113	6	g	g	PROPN
ejpam-6549	113	7	is	be	AUX
ejpam-6549	113	8	called	call	VERB
ejpam-6549	113	9	a	a	DET
ejpam-6549	113	10	cameron	cameron	PROPN
ejpam-6549	113	11	–	–	PUNCT
ejpam-6549	113	12	walker	walker	PROPN
ejpam-6549	113	13	graph	graph	NOUN
ejpam-6549	113	14	if	if	SCONJ
ejpam-6549	113	15	im(g	im(g	PUNCT
ejpam-6549	113	16	)	)	PUNCT
ejpam-6549	113	17	=	=	SYM
ejpam-6549	113	18	m(g	m(g	PROPN
ejpam-6549	113	19	)	)	PUNCT
ejpam-6549	113	20	and	and	CCONJ
ejpam-6549	113	21	if	if	SCONJ
ejpam-6549	113	22	g	g	PROPN
ejpam-6549	113	23	is	be	AUX
ejpam-6549	113	24	neither	neither	CCONJ
ejpam-6549	113	25	a	a	DET
ejpam-6549	113	26	star	star	NOUN
ejpam-6549	113	27	nor	nor	CCONJ
ejpam-6549	113	28	a	a	DET
ejpam-6549	113	29	star	star	NOUN
ejpam-6549	113	30	triangle	triangle	NOUN
ejpam-6549	113	31	.	.	PUNCT
ejpam-6549	114	1	the	the	DET
ejpam-6549	114	2	following	follow	VERB
ejpam-6549	114	3	proposition	proposition	NOUN
ejpam-6549	114	4	provides	provide	VERB
ejpam-6549	114	5	a	a	DET
ejpam-6549	114	6	lower	low	ADJ
ejpam-6549	114	7	bound	bind	VERB
ejpam-6549	114	8	for	for	ADP
ejpam-6549	114	9	graphs	graph	NOUN
ejpam-6549	114	10	that	that	PRON
ejpam-6549	114	11	include	include	VERB
ejpam-6549	114	12	an	an	DET
ejpam-6549	114	13	induced	induced	ADJ
ejpam-6549	114	14	subgraph	subgraph	NOUN
ejpam-6549	114	15	with	with	ADP
ejpam-6549	114	16	im(g	im(g	PUNCT
ejpam-6549	114	17	)	)	PUNCT
ejpam-6549	114	18	=	=	SYM
ejpam-6549	114	19	m(g	m(g	PROPN
ejpam-6549	114	20	)	)	PUNCT
ejpam-6549	114	21	.	.	PUNCT
ejpam-6549	115	1	these	these	DET
ejpam-6549	115	2	graphs	graph	NOUN
ejpam-6549	115	3	do	do	AUX
ejpam-6549	115	4	not	not	PART
ejpam-6549	115	5	belong	belong	VERB
ejpam-6549	115	6	to	to	ADP
ejpam-6549	115	7	the	the	DET
ejpam-6549	115	8	class	class	NOUN
ejpam-6549	115	9	of	of	ADP
ejpam-6549	115	10	cameron	cameron	PROPN
ejpam-6549	115	11	-	-	PUNCT
ejpam-6549	115	12	walker	walker	PROPN
ejpam-6549	115	13	graphs	graph	NOUN
ejpam-6549	115	14	.	.	PUNCT
ejpam-6549	116	1	proposition	proposition	NOUN
ejpam-6549	116	2	1	1	NUM
ejpam-6549	116	3	.	.	PUNCT
ejpam-6549	117	1	let	let	VERB
ejpam-6549	117	2	g	g	PRON
ejpam-6549	117	3	be	be	AUX
ejpam-6549	117	4	a	a	DET
ejpam-6549	117	5	graph	graph	NOUN
ejpam-6549	117	6	containing	contain	VERB
ejpam-6549	117	7	an	an	DET
ejpam-6549	117	8	induced	induced	ADJ
ejpam-6549	117	9	subgraph	subgraph	NOUN
ejpam-6549	117	10	consisting	consisting	NOUN
ejpam-6549	117	11	of	of	ADP
ejpam-6549	117	12	n	n	DET
ejpam-6549	117	13	pendant	pendant	ADJ
ejpam-6549	117	14	triangles	triangle	NOUN
ejpam-6549	117	15	sharing	share	VERB
ejpam-6549	117	16	a	a	DET
ejpam-6549	117	17	common	common	ADJ
ejpam-6549	117	18	vertex	vertex	NOUN
ejpam-6549	117	19	.	.	PUNCT
ejpam-6549	118	1	then	then	ADV
ejpam-6549	118	2	the	the	DET
ejpam-6549	118	3	energy	energy	NOUN
ejpam-6549	118	4	of	of	ADP
ejpam-6549	118	5	g	g	PROPN
ejpam-6549	118	6	satisfies	satisfie	NOUN
ejpam-6549	118	7	ε(g	ε(g	PROPN
ejpam-6549	118	8	)	)	PUNCT
ejpam-6549	118	9	≥	≥	NOUN
ejpam-6549	118	10	2n+	2n+	NUM
ejpam-6549	118	11	1	1	NUM
ejpam-6549	118	12	.	.	PUNCT
ejpam-6549	119	1	proof	proof	NOUN
ejpam-6549	119	2	.	.	PUNCT
ejpam-6549	120	1	let	let	VERB
ejpam-6549	120	2	g′	g′	NOUN
ejpam-6549	120	3	be	be	AUX
ejpam-6549	120	4	the	the	DET
ejpam-6549	120	5	induced	induced	ADJ
ejpam-6549	120	6	subgraph	subgraph	NOUN
ejpam-6549	120	7	of	of	ADP
ejpam-6549	120	8	g	g	PROPN
ejpam-6549	120	9	consisting	consist	VERB
ejpam-6549	120	10	of	of	ADP
ejpam-6549	120	11	n	n	DET
ejpam-6549	120	12	pendant	pendant	ADJ
ejpam-6549	120	13	triangles	triangle	NOUN
ejpam-6549	120	14	sharing	share	VERB
ejpam-6549	120	15	a	a	DET
ejpam-6549	120	16	common	common	ADJ
ejpam-6549	120	17	vertex	vertex	NOUN
ejpam-6549	120	18	.	.	PUNCT
ejpam-6549	121	1	the	the	DET
ejpam-6549	121	2	adjacency	adjacency	PROPN
ejpam-6549	121	3	matrix	matrix	NOUN
ejpam-6549	121	4	a(g′	a(g′	PROPN
ejpam-6549	121	5	)	)	PUNCT
ejpam-6549	121	6	of	of	ADP
ejpam-6549	121	7	g′	g′	NOUN
ejpam-6549	121	8	is	be	AUX
ejpam-6549	121	9	[	[	PUNCT
ejpam-6549	121	10	0	0	NUM
ejpam-6549	121	11	1	1	NUM
ejpam-6549	121	12	1	1	NUM
ejpam-6549	121	13	t	t	NOUN
ejpam-6549	121	14	q	q	NOUN
ejpam-6549	121	15	]	]	PUNCT
ejpam-6549	121	16	.	.	PUNCT
ejpam-6549	122	1	since	since	SCONJ
ejpam-6549	122	2	rank(q	rank(q	NOUN
ejpam-6549	122	3	)	)	PUNCT
ejpam-6549	122	4	=	=	SYM
ejpam-6549	122	5	2n	2n	NUM
ejpam-6549	122	6	and	and	CCONJ
ejpam-6549	122	7	1	1	NUM
ejpam-6549	122	8	is	be	AUX
ejpam-6549	122	9	1×2n	1×2n	NUM
ejpam-6549	122	10	matrix	matrix	NOUN
ejpam-6549	122	11	with	with	ADP
ejpam-6549	122	12	all	all	DET
ejpam-6549	122	13	1	1	NUM
ejpam-6549	122	14	’s	’s	PART
ejpam-6549	122	15	,	,	PUNCT
ejpam-6549	122	16	it	it	PRON
ejpam-6549	122	17	is	be	AUX
ejpam-6549	122	18	clear	clear	ADJ
ejpam-6549	122	19	that	that	SCONJ
ejpam-6549	122	20	rank(a(g′	rank(a(g′	NOUN
ejpam-6549	122	21	)	)	PUNCT
ejpam-6549	122	22	)	)	PUNCT
ejpam-6549	123	1	=	=	SYM
ejpam-6549	123	2	2n+1	2n+1	PROPN
ejpam-6549	123	3	.	.	PUNCT
ejpam-6549	124	1	hence	hence	ADV
ejpam-6549	124	2	ε(g′	ε(g′	NUM
ejpam-6549	124	3	)	)	PUNCT
ejpam-6549	124	4	≥	≥	PART
ejpam-6549	125	1	2n+1	2n+1	NOUN
ejpam-6549	125	2	by	by	ADP
ejpam-6549	125	3	lemma	lemma	PROPN
ejpam-6549	125	4	2	2	NUM
ejpam-6549	125	5	in	in	ADP
ejpam-6549	125	6	[	[	X
ejpam-6549	125	7	11	11	NUM
ejpam-6549	125	8	]	]	PUNCT
ejpam-6549	125	9	.	.	PUNCT
ejpam-6549	126	1	thus	thus	ADV
ejpam-6549	126	2	,	,	PUNCT
ejpam-6549	126	3	by	by	ADP
ejpam-6549	126	4	lemma	lemma	PROPN
ejpam-6549	126	5	1	1	NUM
ejpam-6549	126	6	,	,	PUNCT
ejpam-6549	126	7	we	we	PRON
ejpam-6549	126	8	get	get	VERB
ejpam-6549	126	9	ε(g	ε(g	NOUN
ejpam-6549	126	10	)	)	PUNCT
ejpam-6549	126	11	≥	≥	NOUN
ejpam-6549	126	12	2n+	2n+	NUM
ejpam-6549	126	13	1	1	NUM
ejpam-6549	126	14	.	.	PUNCT
ejpam-6549	127	1	in	in	ADP
ejpam-6549	127	2	the	the	DET
ejpam-6549	127	3	next	next	ADJ
ejpam-6549	127	4	proposition	proposition	NOUN
ejpam-6549	127	5	,	,	PUNCT
ejpam-6549	127	6	we	we	PRON
ejpam-6549	127	7	give	give	VERB
ejpam-6549	127	8	a	a	DET
ejpam-6549	127	9	lower	low	ADJ
ejpam-6549	127	10	bound	bind	VERB
ejpam-6549	127	11	for	for	ADP
ejpam-6549	127	12	the	the	DET
ejpam-6549	127	13	energy	energy	NOUN
ejpam-6549	127	14	of	of	ADP
ejpam-6549	127	15	a	a	DET
ejpam-6549	127	16	cameron	cameron	PROPN
ejpam-6549	127	17	-	-	PUNCT
ejpam-6549	127	18	walker	walker	PROPN
ejpam-6549	127	19	graph	graph	NOUN
ejpam-6549	127	20	in	in	ADP
ejpam-6549	127	21	terms	term	NOUN
ejpam-6549	127	22	of	of	ADP
ejpam-6549	127	23	its	its	PRON
ejpam-6549	127	24	matching	matching	NOUN
ejpam-6549	127	25	number	number	NOUN
ejpam-6549	127	26	and	and	CCONJ
ejpam-6549	127	27	maximum	maximum	ADJ
ejpam-6549	127	28	degree	degree	NOUN
ejpam-6549	127	29	.	.	PUNCT
ejpam-6549	128	1	proposition	proposition	NOUN
ejpam-6549	128	2	2	2	NUM
ejpam-6549	128	3	.	.	PUNCT
ejpam-6549	129	1	the	the	DET
ejpam-6549	129	2	graph	graph	NOUN
ejpam-6549	129	3	rgim(g	rgim(g	NOUN
ejpam-6549	129	4	)	)	PUNCT
ejpam-6549	129	5	is	be	AUX
ejpam-6549	129	6	the	the	DET
ejpam-6549	129	7	minimal	minimal	ADJ
ejpam-6549	129	8	energy	energy	NOUN
ejpam-6549	129	9	cameron	cameron	PROPN
ejpam-6549	129	10	-	-	PUNCT
ejpam-6549	129	11	walker	walker	PROPN
ejpam-6549	129	12	graph	graph	NOUN
ejpam-6549	129	13	with	with	ADP
ejpam-6549	129	14	(	(	PUNCT
ejpam-6549	129	15	induced	induced	ADJ
ejpam-6549	129	16	)	)	PUNCT
ejpam-6549	129	17	matching	matching	NOUN
ejpam-6549	129	18	number	number	NOUN
ejpam-6549	129	19	im(g	im(g	PUNCT
ejpam-6549	129	20	)	)	PUNCT
ejpam-6549	129	21	≥	≥	NOUN
ejpam-6549	129	22	2	2	NUM
ejpam-6549	129	23	.	.	PUNCT
ejpam-6549	129	24	proof	proof	NOUN
ejpam-6549	129	25	.	.	PUNCT
ejpam-6549	130	1	the	the	DET
ejpam-6549	130	2	graph	graph	NOUN
ejpam-6549	130	3	rgim(g	rgim(g	NOUN
ejpam-6549	130	4	)	)	PUNCT
ejpam-6549	130	5	is	be	AUX
ejpam-6549	130	6	a	a	DET
ejpam-6549	130	7	cameron	cameron	PROPN
ejpam-6549	130	8	-	-	PUNCT
ejpam-6549	130	9	walker	walker	PROPN
ejpam-6549	130	10	graph	graph	NOUN
ejpam-6549	130	11	with	with	ADP
ejpam-6549	130	12	supporting	support	VERB
ejpam-6549	130	13	bipartite	bipartite	NOUN
ejpam-6549	130	14	graph	graph	NOUN
ejpam-6549	130	15	k1,im(g	k1,im(g	PRON
ejpam-6549	130	16	)	)	PUNCT
ejpam-6549	130	17	.	.	PUNCT
ejpam-6549	131	1	and	and	CCONJ
ejpam-6549	131	2	the	the	DET
ejpam-6549	131	3	number	number	NOUN
ejpam-6549	131	4	of	of	ADP
ejpam-6549	131	5	leaves	leave	NOUN
ejpam-6549	131	6	is	be	AUX
ejpam-6549	131	7	im(g	im(g	PUNCT
ejpam-6549	131	8	)	)	PUNCT
ejpam-6549	131	9	,	,	PUNCT
ejpam-6549	131	10	which	which	PRON
ejpam-6549	131	11	implies	imply	VERB
ejpam-6549	131	12	(	(	PUNCT
ejpam-6549	131	13	induced	induced	ADJ
ejpam-6549	131	14	)	)	PUNCT
ejpam-6549	131	15	matching	match	VERB
ejpam-6549	131	16	number	number	NOUN
ejpam-6549	131	17	is	be	AUX
ejpam-6549	131	18	im(g	im(g	PUNCT
ejpam-6549	131	19	)	)	PUNCT
ejpam-6549	131	20	.	.	PUNCT
ejpam-6549	132	1	thus	thus	ADV
ejpam-6549	132	2	,	,	PUNCT
ejpam-6549	132	3	by	by	ADP
ejpam-6549	132	4	theorem	theorem	NOUN
ejpam-6549	132	5	5	5	NUM
ejpam-6549	132	6	,	,	PUNCT
ejpam-6549	132	7	rgim(g	rgim(g	NOUN
ejpam-6549	132	8	)	)	PUNCT
ejpam-6549	132	9	is	be	AUX
ejpam-6549	132	10	a	a	DET
ejpam-6549	132	11	cameron	cameron	PROPN
ejpam-6549	132	12	-	-	PUNCT
ejpam-6549	132	13	walker	walker	PROPN
ejpam-6549	132	14	graph	graph	NOUN
ejpam-6549	132	15	of	of	ADP
ejpam-6549	132	16	minimal	minimal	ADJ
ejpam-6549	132	17	energy	energy	NOUN
ejpam-6549	132	18	.	.	PUNCT
ejpam-6549	133	1	proposition	proposition	NOUN
ejpam-6549	133	2	3	3	NUM
ejpam-6549	133	3	.	.	PUNCT
ejpam-6549	134	1	let	let	VERB
ejpam-6549	134	2	g	g	PRON
ejpam-6549	134	3	be	be	AUX
ejpam-6549	134	4	a	a	DET
ejpam-6549	134	5	cameron	cameron	PROPN
ejpam-6549	134	6	-	-	PUNCT
ejpam-6549	134	7	walker	walker	PROPN
ejpam-6549	134	8	graph	graph	NOUN
ejpam-6549	134	9	.	.	PUNCT
ejpam-6549	135	1	then	then	ADV
ejpam-6549	135	2	the	the	DET
ejpam-6549	135	3	followings	following	NOUN
ejpam-6549	135	4	hold	hold	VERB
ejpam-6549	135	5	:	:	PUNCT
ejpam-6549	135	6	(	(	PUNCT
ejpam-6549	135	7	1	1	X
ejpam-6549	135	8	)	)	PUNCT
ejpam-6549	135	9	if	if	SCONJ
ejpam-6549	135	10	g	g	PROPN
ejpam-6549	135	11	is	be	AUX
ejpam-6549	135	12	a	a	DET
ejpam-6549	135	13	c3	c3	NOUN
ejpam-6549	135	14	-	-	PUNCT
ejpam-6549	135	15	free	free	ADJ
ejpam-6549	135	16	graph	graph	NOUN
ejpam-6549	135	17	with	with	ADP
ejpam-6549	135	18	m	m	PROPN
ejpam-6549	135	19	disjoint	disjoint	ADJ
ejpam-6549	135	20	leaf	leaf	NOUN
ejpam-6549	135	21	edges	edge	NOUN
ejpam-6549	135	22	,	,	PUNCT
ejpam-6549	135	23	then	then	ADV
ejpam-6549	135	24	ε(g	ε(g	PROPN
ejpam-6549	135	25	)	)	PUNCT
ejpam-6549	135	26	≥	≥	NUM
ejpam-6549	135	27	ε(rgm	ε(rgm	NOUN
ejpam-6549	135	28	)	)	PUNCT
ejpam-6549	135	29	.	.	PUNCT
ejpam-6549	136	1	(	(	PUNCT
ejpam-6549	136	2	2	2	X
ejpam-6549	136	3	)	)	PUNCT
ejpam-6549	136	4	if	if	SCONJ
ejpam-6549	136	5	g	g	PROPN
ejpam-6549	136	6	has	have	VERB
ejpam-6549	136	7	m	m	PROPN
ejpam-6549	136	8	disjoint	disjoint	ADJ
ejpam-6549	136	9	leaf	leaf	NOUN
ejpam-6549	136	10	edges	edge	NOUN
ejpam-6549	136	11	and	and	CCONJ
ejpam-6549	136	12	n	n	DET
ejpam-6549	136	13	disjoint	disjoint	ADJ
ejpam-6549	136	14	pendant	pendant	ADJ
ejpam-6549	136	15	triangles	triangle	NOUN
ejpam-6549	136	16	,	,	PUNCT
ejpam-6549	136	17	then	then	ADV
ejpam-6549	136	18	ε(g	ε(g	PROPN
ejpam-6549	136	19	)	)	PUNCT
ejpam-6549	136	20	≥	≥	PROPN
ejpam-6549	136	21	2m+4n	2m+4n	NUM
ejpam-6549	136	22	.	.	PUNCT
ejpam-6549	137	1	(	(	PUNCT
ejpam-6549	137	2	3	3	X
ejpam-6549	137	3	)	)	PUNCT
ejpam-6549	137	4	if	if	SCONJ
ejpam-6549	137	5	g	g	PROPN
ejpam-6549	137	6	has	have	VERB
ejpam-6549	137	7	m	m	PROPN
ejpam-6549	137	8	disjoint	disjoint	ADJ
ejpam-6549	137	9	leaf	leaf	NOUN
ejpam-6549	137	10	edges	edge	NOUN
ejpam-6549	137	11	and	and	CCONJ
ejpam-6549	137	12	n	n	CCONJ
ejpam-6549	137	13	pendant	pendant	ADJ
ejpam-6549	137	14	triangles	triangle	NOUN
ejpam-6549	137	15	,	,	PUNCT
ejpam-6549	137	16	then	then	ADV
ejpam-6549	137	17	ε(g	ε(g	PROPN
ejpam-6549	137	18	)	)	PUNCT
ejpam-6549	137	19	≥	≥	NOUN
ejpam-6549	137	20	2m+	2m+	NUM
ejpam-6549	137	21	2n+	2n+	NUM
ejpam-6549	137	22	1	1	NUM
ejpam-6549	137	23	.	.	PUNCT
ejpam-6549	138	1	proof	proof	NOUN
ejpam-6549	138	2	.	.	PUNCT
ejpam-6549	139	1	(	(	PUNCT
ejpam-6549	139	2	1	1	X
ejpam-6549	139	3	)	)	PUNCT
ejpam-6549	139	4	a	a	DET
ejpam-6549	139	5	cameron	cameron	PROPN
ejpam-6549	139	6	-	-	PUNCT
ejpam-6549	139	7	walker	walker	PROPN
ejpam-6549	139	8	graph	graph	NOUN
ejpam-6549	139	9	is	be	AUX
ejpam-6549	139	10	a	a	DET
ejpam-6549	139	11	connected	connected	ADJ
ejpam-6549	139	12	graph	graph	NOUN
ejpam-6549	139	13	.	.	PUNCT
ejpam-6549	140	1	since	since	SCONJ
ejpam-6549	140	2	g	g	PROPN
ejpam-6549	140	3	is	be	AUX
ejpam-6549	140	4	c3	c3	NOUN
ejpam-6549	140	5	-	-	PUNCT
ejpam-6549	140	6	free	free	ADJ
ejpam-6549	140	7	and	and	CCONJ
ejpam-6549	140	8	possessesm	possessesm	NOUN
ejpam-6549	140	9	leaves	leave	NOUN
ejpam-6549	140	10	,	,	PUNCT
ejpam-6549	140	11	then	then	ADV
ejpam-6549	140	12	there	there	PRON
ejpam-6549	140	13	are	be	VERB
ejpam-6549	140	14	no	no	DET
ejpam-6549	140	15	pendant	pendant	ADJ
ejpam-6549	140	16	triangles	triangle	NOUN
ejpam-6549	140	17	and	and	CCONJ
ejpam-6549	140	18	the	the	DET
ejpam-6549	140	19	(	(	PUNCT
ejpam-6549	140	20	induced	induced	ADJ
ejpam-6549	140	21	)	)	PUNCT
ejpam-6549	140	22	matching	match	VERB
ejpam-6549	140	23	number	number	NOUN
ejpam-6549	140	24	of	of	ADP
ejpam-6549	140	25	g	g	PROPN
ejpam-6549	140	26	is	be	AUX
ejpam-6549	140	27	m.	m.	NOUN
ejpam-6549	140	28	hence	hence	ADV
ejpam-6549	140	29	,	,	PUNCT
ejpam-6549	140	30	by	by	ADP
ejpam-6549	140	31	proposition	proposition	NOUN
ejpam-6549	140	32	2	2	NUM
ejpam-6549	140	33	,	,	PUNCT
ejpam-6549	140	34	we	we	PRON
ejpam-6549	140	35	conclude	conclude	VERB
ejpam-6549	140	36	that	that	SCONJ
ejpam-6549	140	37	ε(g	ε(g	NOUN
ejpam-6549	140	38	)	)	PUNCT
ejpam-6549	140	39	≥	≥	NOUN
ejpam-6549	140	40	ε(rgm	ε(rgm	NOUN
ejpam-6549	140	41	)	)	PUNCT
ejpam-6549	140	42	.	.	PUNCT
ejpam-6549	141	1	(	(	PUNCT
ejpam-6549	141	2	2	2	X
ejpam-6549	141	3	)	)	PUNCT
ejpam-6549	141	4	let	let	VERB
ejpam-6549	141	5	l	l	NOUN
ejpam-6549	141	6	be	be	AUX
ejpam-6549	141	7	the	the	DET
ejpam-6549	141	8	set	set	NOUN
ejpam-6549	141	9	of	of	ADP
ejpam-6549	141	10	disjoint	disjoint	ADJ
ejpam-6549	141	11	leaf	leaf	NOUN
ejpam-6549	141	12	edges	edge	NOUN
ejpam-6549	141	13	and	and	CCONJ
ejpam-6549	141	14	let	let	VERB
ejpam-6549	141	15	t	t	PROPN
ejpam-6549	141	16	be	be	AUX
ejpam-6549	141	17	the	the	DET
ejpam-6549	141	18	set	set	NOUN
ejpam-6549	141	19	of	of	ADP
ejpam-6549	141	20	disjoint	disjoint	ADJ
ejpam-6549	141	21	pendant	pendant	ADJ
ejpam-6549	141	22	triangles	triangle	NOUN
ejpam-6549	141	23	.	.	PUNCT
ejpam-6549	142	1	in	in	ADP
ejpam-6549	142	2	a	a	DET
ejpam-6549	142	3	cameron	cameron	PROPN
ejpam-6549	142	4	-	-	PUNCT
ejpam-6549	142	5	walker	walker	PROPN
ejpam-6549	142	6	graph	graph	NOUN
ejpam-6549	142	7	,	,	PUNCT
ejpam-6549	142	8	it	it	PRON
ejpam-6549	142	9	is	be	AUX
ejpam-6549	142	10	clear	clear	ADJ
ejpam-6549	142	11	that	that	SCONJ
ejpam-6549	142	12	l∩t	l∩t	ADV
ejpam-6549	142	13	=	=	PUNCT
ejpam-6549	142	14	∅.	∅.	NOUN
ejpam-6549	142	15	since	since	SCONJ
ejpam-6549	142	16	each	each	DET
ejpam-6549	142	17	leaf	leaf	NOUN
ejpam-6549	142	18	edge	edge	NOUN
ejpam-6549	142	19	has	have	VERB
ejpam-6549	142	20	energy	energy	NOUN
ejpam-6549	142	21	equal	equal	ADJ
ejpam-6549	142	22	to	to	ADP
ejpam-6549	142	23	2	2	NUM
ejpam-6549	142	24	and	and	CCONJ
ejpam-6549	142	25	each	each	DET
ejpam-6549	142	26	pendant	pendant	ADJ
ejpam-6549	142	27	triangle	triangle	NOUN
ejpam-6549	142	28	has	have	VERB
ejpam-6549	142	29	energy	energy	NOUN
ejpam-6549	142	30	equal	equal	ADJ
ejpam-6549	142	31	to	to	ADP
ejpam-6549	142	32	4	4	NUM
ejpam-6549	142	33	,	,	PUNCT
ejpam-6549	142	34	by	by	ADP
ejpam-6549	142	35	lemma	lemma	PROPN
ejpam-6549	142	36	1	1	NUM
ejpam-6549	142	37	it	it	PRON
ejpam-6549	142	38	follows	follow	VERB
ejpam-6549	142	39	that	that	SCONJ
ejpam-6549	142	40	ε(g	ε(g	NOUN
ejpam-6549	142	41	)	)	PUNCT
ejpam-6549	142	42	≥	≥	NOUN
ejpam-6549	142	43	2m+	2m+	NUM
ejpam-6549	142	44	4n	4n	X
ejpam-6549	142	45	.	.	PUNCT
ejpam-6549	143	1	(	(	PUNCT
ejpam-6549	143	2	3	3	X
ejpam-6549	143	3	)	)	PUNCT
ejpam-6549	143	4	now	now	ADV
ejpam-6549	143	5	,	,	PUNCT
ejpam-6549	143	6	without	without	ADP
ejpam-6549	143	7	loss	loss	NOUN
ejpam-6549	143	8	of	of	ADP
ejpam-6549	143	9	generality	generality	NOUN
ejpam-6549	143	10	,	,	PUNCT
ejpam-6549	143	11	we	we	PRON
ejpam-6549	143	12	assume	assume	VERB
ejpam-6549	143	13	that	that	SCONJ
ejpam-6549	143	14	all	all	DET
ejpam-6549	143	15	the	the	DET
ejpam-6549	143	16	pendant	pendant	ADJ
ejpam-6549	143	17	triangles	triangle	NOUN
ejpam-6549	143	18	of	of	ADP
ejpam-6549	143	19	g	g	PROPN
ejpam-6549	143	20	have	have	VERB
ejpam-6549	143	21	a	a	DET
ejpam-6549	143	22	common	common	ADJ
ejpam-6549	143	23	vertex	vertex	NOUN
ejpam-6549	143	24	.	.	PUNCT
ejpam-6549	144	1	therefore	therefore	ADV
ejpam-6549	144	2	,	,	PUNCT
ejpam-6549	144	3	the	the	DET
ejpam-6549	144	4	induced	induced	ADJ
ejpam-6549	144	5	subgraph	subgraph	NOUN
ejpam-6549	144	6	on	on	ADP
ejpam-6549	144	7	n	n	DET
ejpam-6549	144	8	pendant	pendant	ADJ
ejpam-6549	144	9	triangles	triangle	NOUN
ejpam-6549	144	10	has	have	VERB
ejpam-6549	144	11	energy	energy	NOUN
ejpam-6549	144	12	at	at	ADP
ejpam-6549	144	13	least	least	ADJ
ejpam-6549	144	14	2n	2n	NUM
ejpam-6549	144	15	+	+	CCONJ
ejpam-6549	144	16	1	1	NUM
ejpam-6549	144	17	by	by	ADP
ejpam-6549	144	18	proposition	proposition	NOUN
ejpam-6549	144	19	1	1	NUM
ejpam-6549	144	20	.	.	PUNCT
ejpam-6549	145	1	since	since	SCONJ
ejpam-6549	145	2	each	each	DET
ejpam-6549	145	3	leaf	leaf	NOUN
ejpam-6549	145	4	edge	edge	NOUN
ejpam-6549	145	5	contributes	contribute	VERB
ejpam-6549	145	6	2	2	NUM
ejpam-6549	145	7	to	to	ADP
ejpam-6549	145	8	the	the	DET
ejpam-6549	145	9	energy	energy	NOUN
ejpam-6549	145	10	,	,	PUNCT
ejpam-6549	145	11	then	then	ADV
ejpam-6549	145	12	we	we	PRON
ejpam-6549	145	13	get	get	VERB
ejpam-6549	145	14	that	that	DET
ejpam-6549	145	15	ε(g	ε(g	NOUN
ejpam-6549	145	16	)	)	PUNCT
ejpam-6549	145	17	≥	≥	NOUN
ejpam-6549	145	18	2m+	2m+	NUM
ejpam-6549	145	19	2n+	2n+	NUM
ejpam-6549	145	20	1	1	NUM
ejpam-6549	145	21	.	.	PUNCT
ejpam-6549	145	22	example	example	NOUN
ejpam-6549	146	1	2	2	NUM
ejpam-6549	146	2	.	.	PUNCT
ejpam-6549	146	3	the	the	DET
ejpam-6549	146	4	following	follow	VERB
ejpam-6549	146	5	graph	graph	NOUN
ejpam-6549	146	6	g	g	PROPN
ejpam-6549	146	7	is	be	AUX
ejpam-6549	146	8	a	a	DET
ejpam-6549	146	9	cameron	cameron	PROPN
ejpam-6549	146	10	-	-	PUNCT
ejpam-6549	146	11	walker	walker	PROPN
ejpam-6549	146	12	graph	graph	NOUN
ejpam-6549	146	13	with	with	ADP
ejpam-6549	146	14	4	4	NUM
ejpam-6549	146	15	leaf	leaf	NOUN
ejpam-6549	146	16	edges	edge	NOUN
ejpam-6549	146	17	and	and	CCONJ
ejpam-6549	146	18	2	2	NUM
ejpam-6549	146	19	pendant	pendant	ADJ
ejpam-6549	146	20	triangles	triangle	NOUN
ejpam-6549	146	21	.	.	PUNCT
ejpam-6549	147	1	the	the	DET
ejpam-6549	147	2	energy	energy	NOUN
ejpam-6549	147	3	ε(g	ε(g	NOUN
ejpam-6549	147	4	)	)	PUNCT
ejpam-6549	147	5	∼=	∼=	PROPN
ejpam-6549	147	6	19.55	19.55	NUM
ejpam-6549	147	7	and	and	CCONJ
ejpam-6549	147	8	(	(	PUNCT
ejpam-6549	147	9	induced	induced	ADJ
ejpam-6549	147	10	)	)	PUNCT
ejpam-6549	147	11	matching	matching	NOUN
ejpam-6549	147	12	number	number	NOUN
ejpam-6549	147	13	is	be	AUX
ejpam-6549	147	14	6	6	NUM
ejpam-6549	147	15	.	.	PUNCT
ejpam-6549	148	1	in	in	ADP
ejpam-6549	148	2	the	the	DET
ejpam-6549	148	3	following	following	NOUN
ejpam-6549	148	4	theorem	theorem	NOUN
ejpam-6549	148	5	,	,	PUNCT
ejpam-6549	148	6	we	we	PRON
ejpam-6549	148	7	give	give	VERB
ejpam-6549	148	8	a	a	DET
ejpam-6549	148	9	lower	low	ADJ
ejpam-6549	148	10	bound	bind	VERB
ejpam-6549	148	11	for	for	ADP
ejpam-6549	148	12	a	a	DET
ejpam-6549	148	13	cameron	cameron	PROPN
ejpam-6549	148	14	-	-	PUNCT
ejpam-6549	148	15	walker	walker	PROPN
ejpam-6549	148	16	graph	graph	NOUN
ejpam-6549	148	17	in	in	ADP
ejpam-6549	148	18	terms	term	NOUN
ejpam-6549	148	19	of	of	ADP
ejpam-6549	148	20	its	its	PRON
ejpam-6549	148	21	matching	matching	NOUN
ejpam-6549	148	22	number	number	NOUN
ejpam-6549	148	23	and	and	CCONJ
ejpam-6549	148	24	number	number	NOUN
ejpam-6549	148	25	of	of	ADP
ejpam-6549	148	26	triangles	triangle	NOUN
ejpam-6549	148	27	.	.	PUNCT
ejpam-6549	149	1	a.	a.	NOUN
ejpam-6549	149	2	ülker	ülker	PROPN
ejpam-6549	149	3	et	et	PROPN
ejpam-6549	149	4	al	al	PROPN
ejpam-6549	149	5	.	.	PUNCT
ejpam-6549	149	6	/	/	SYM
ejpam-6549	149	7	eur	eur	PROPN
ejpam-6549	149	8	.	.	PUNCT
ejpam-6549	150	1	j.	j.	PROPN
ejpam-6549	150	2	pure	pure	PROPN
ejpam-6549	150	3	appl	appl	PROPN
ejpam-6549	150	4	.	.	PROPN
ejpam-6549	150	5	math	math	PROPN
ejpam-6549	150	6	,	,	PUNCT
ejpam-6549	150	7	18	18	NUM
ejpam-6549	150	8	(	(	PUNCT
ejpam-6549	150	9	3	3	NUM
ejpam-6549	150	10	)	)	PUNCT
ejpam-6549	150	11	(	(	PUNCT
ejpam-6549	150	12	2025	2025	NUM
ejpam-6549	150	13	)	)	PUNCT
ejpam-6549	150	14	,	,	PUNCT
ejpam-6549	150	15	6549	6549	NUM
ejpam-6549	150	16	7	7	NUM
ejpam-6549	150	17	of	of	ADP
ejpam-6549	150	18	8	8	NUM
ejpam-6549	150	19	figure	figure	NOUN
ejpam-6549	150	20	2	2	NUM
ejpam-6549	150	21	:	:	PUNCT
ejpam-6549	150	22	a	a	DET
ejpam-6549	150	23	cameron	cameron	PROPN
ejpam-6549	150	24	-	-	PUNCT
ejpam-6549	150	25	walker	walker	PROPN
ejpam-6549	150	26	graph	graph	NOUN
ejpam-6549	150	27	with	with	ADP
ejpam-6549	150	28	2	2	NUM
ejpam-6549	150	29	pendant	pendant	ADJ
ejpam-6549	150	30	triangles	triangle	NOUN
ejpam-6549	150	31	theorem	theorem	VERB
ejpam-6549	150	32	7	7	NUM
ejpam-6549	150	33	.	.	PUNCT
ejpam-6549	151	1	if	if	SCONJ
ejpam-6549	151	2	g	g	PROPN
ejpam-6549	151	3	is	be	AUX
ejpam-6549	151	4	a	a	DET
ejpam-6549	151	5	cameron	cameron	PROPN
ejpam-6549	151	6	-	-	PUNCT
ejpam-6549	151	7	walker	walker	PROPN
ejpam-6549	151	8	graph	graph	NOUN
ejpam-6549	151	9	with	with	ADP
ejpam-6549	151	10	maximum	maximum	ADJ
ejpam-6549	151	11	degree	degree	NOUN
ejpam-6549	151	12	∆	∆	PROPN
ejpam-6549	151	13	,	,	PUNCT
ejpam-6549	151	14	then	then	ADV
ejpam-6549	151	15	ε(g	ε(g	PROPN
ejpam-6549	151	16	)	)	PUNCT
ejpam-6549	151	17	≥	≥	NOUN
ejpam-6549	151	18	2m(g	2m(g	NUM
ejpam-6549	151	19	)	)	PUNCT
ejpam-6549	152	1	+	+	CCONJ
ejpam-6549	152	2	3	3	NUM
ejpam-6549	152	3	∆	∆	NUM
ejpam-6549	152	4	c3(g	c3(g	NUM
ejpam-6549	152	5	)	)	PUNCT
ejpam-6549	152	6	,	,	PUNCT
ejpam-6549	152	7	where	where	SCONJ
ejpam-6549	152	8	m(g	m(g	NOUN
ejpam-6549	152	9	)	)	PUNCT
ejpam-6549	152	10	is	be	AUX
ejpam-6549	152	11	the	the	DET
ejpam-6549	152	12	matching	match	VERB
ejpam-6549	152	13	number	number	NOUN
ejpam-6549	152	14	and	and	CCONJ
ejpam-6549	152	15	c3(g	c3(g	NUM
ejpam-6549	152	16	)	)	PUNCT
ejpam-6549	152	17	is	be	AUX
ejpam-6549	152	18	the	the	DET
ejpam-6549	152	19	number	number	NOUN
ejpam-6549	152	20	of	of	ADP
ejpam-6549	152	21	disjoint	disjoint	ADJ
ejpam-6549	152	22	pendant	pendant	ADJ
ejpam-6549	152	23	triangles	triangle	NOUN
ejpam-6549	152	24	in	in	ADP
ejpam-6549	152	25	g.	g.	PROPN
ejpam-6549	152	26	proof	proof	PROPN
ejpam-6549	152	27	.	.	PUNCT
ejpam-6549	153	1	let	let	VERB
ejpam-6549	153	2	m	m	PRON
ejpam-6549	153	3	be	be	AUX
ejpam-6549	153	4	the	the	DET
ejpam-6549	153	5	maximum	maximum	ADJ
ejpam-6549	153	6	matching	matching	NOUN
ejpam-6549	153	7	set	set	NOUN
ejpam-6549	153	8	of	of	ADP
ejpam-6549	153	9	g.	g.	PROPN
ejpam-6549	153	10	and	and	CCONJ
ejpam-6549	153	11	let	let	VERB
ejpam-6549	153	12	c	c	NOUN
ejpam-6549	153	13	=	=	SYM
ejpam-6549	153	14	{	{	PUNCT
ejpam-6549	153	15	c1	c1	PROPN
ejpam-6549	153	16	,	,	PUNCT
ejpam-6549	153	17	c2	c2	PROPN
ejpam-6549	153	18	,	,	PUNCT
ejpam-6549	153	19	...	...	PUNCT
ejpam-6549	153	20	,	,	PUNCT
ejpam-6549	153	21	cr	cr	PART
ejpam-6549	153	22	}	}	PUNCT
ejpam-6549	153	23	be	be	AUX
ejpam-6549	153	24	the	the	DET
ejpam-6549	153	25	set	set	NOUN
ejpam-6549	153	26	of	of	ADP
ejpam-6549	153	27	disjoint	disjoint	NOUN
ejpam-6549	153	28	triangles	triangle	NOUN
ejpam-6549	153	29	of	of	ADP
ejpam-6549	153	30	g.	g.	PROPN
ejpam-6549	153	31	since	since	SCONJ
ejpam-6549	153	32	g	g	PROPN
ejpam-6549	153	33	is	be	AUX
ejpam-6549	153	34	a	a	DET
ejpam-6549	153	35	cameron	cameron	PROPN
ejpam-6549	153	36	-	-	PUNCT
ejpam-6549	153	37	walker	walker	PROPN
ejpam-6549	153	38	graph	graph	NOUN
ejpam-6549	153	39	,	,	PUNCT
ejpam-6549	153	40	the	the	DET
ejpam-6549	153	41	set	set	NOUN
ejpam-6549	153	42	m	m	VERB
ejpam-6549	153	43	consists	consist	VERB
ejpam-6549	153	44	of	of	ADP
ejpam-6549	153	45	disjoint	disjoint	ADJ
ejpam-6549	153	46	pendant	pendant	ADJ
ejpam-6549	153	47	edges	edge	NOUN
ejpam-6549	153	48	and	and	CCONJ
ejpam-6549	153	49	edges	edge	NOUN
ejpam-6549	153	50	of	of	ADP
ejpam-6549	153	51	triangles	triangle	NOUN
ejpam-6549	153	52	with	with	ADP
ejpam-6549	153	53	end	end	NOUN
ejpam-6549	153	54	vertices	vertex	NOUN
ejpam-6549	153	55	that	that	PRON
ejpam-6549	153	56	have	have	VERB
ejpam-6549	153	57	degree	degree	NOUN
ejpam-6549	153	58	2	2	NUM
ejpam-6549	153	59	.	.	NOUN
ejpam-6549	153	60	without	without	ADP
ejpam-6549	153	61	loss	loss	NOUN
ejpam-6549	153	62	of	of	ADP
ejpam-6549	153	63	generality	generality	NOUN
ejpam-6549	153	64	,	,	PUNCT
ejpam-6549	153	65	let	let	VERB
ejpam-6549	153	66	us	we	PRON
ejpam-6549	153	67	assume	assume	VERB
ejpam-6549	153	68	that	that	SCONJ
ejpam-6549	153	69	all	all	DET
ejpam-6549	153	70	the	the	DET
ejpam-6549	153	71	pendant	pendant	ADJ
ejpam-6549	153	72	triangles	triangle	NOUN
ejpam-6549	153	73	are	be	AUX
ejpam-6549	153	74	disjoint	disjoint	ADJ
ejpam-6549	153	75	in	in	ADP
ejpam-6549	153	76	g.	g.	PROPN
ejpam-6549	153	77	let	let	VERB
ejpam-6549	153	78	the	the	DET
ejpam-6549	153	79	sets	set	NOUN
ejpam-6549	153	80	v	v	ADP
ejpam-6549	153	81	(	(	PUNCT
ejpam-6549	153	82	g	g	NOUN
ejpam-6549	153	83	)	)	PUNCT
ejpam-6549	153	84	and	and	CCONJ
ejpam-6549	153	85	v	v	NOUN
ejpam-6549	153	86	(	(	PUNCT
ejpam-6549	153	87	g[m	g[m	NOUN
ejpam-6549	153	88	]	]	PUNCT
ejpam-6549	153	89	)	)	PUNCT
ejpam-6549	153	90	be	be	VERB
ejpam-6549	153	91	the	the	DET
ejpam-6549	153	92	vertex	vertex	NOUN
ejpam-6549	153	93	sets	set	NOUN
ejpam-6549	153	94	of	of	ADP
ejpam-6549	153	95	g	g	PROPN
ejpam-6549	153	96	and	and	CCONJ
ejpam-6549	153	97	the	the	DET
ejpam-6549	153	98	induced	induced	ADJ
ejpam-6549	153	99	graph	graph	NOUN
ejpam-6549	153	100	on	on	ADP
ejpam-6549	153	101	m	m	PROPN
ejpam-6549	153	102	,	,	PUNCT
ejpam-6549	153	103	respectively	respectively	ADV
ejpam-6549	153	104	.	.	PUNCT
ejpam-6549	154	1	thus	thus	ADV
ejpam-6549	154	2	,	,	PUNCT
ejpam-6549	154	3	ε(g	ε(g	PROPN
ejpam-6549	154	4	)	)	PUNCT
ejpam-6549	154	5	≥	≥	NOUN
ejpam-6549	154	6	ε(g[m	ε(g[m	NOUN
ejpam-6549	154	7	]	]	PUNCT
ejpam-6549	154	8	)	)	PUNCT
ejpam-6549	155	1	+	+	CCONJ
ejpam-6549	155	2	ε(v	ε(v	PROPN
ejpam-6549	155	3	(	(	PUNCT
ejpam-6549	155	4	g)\g[m	g)\g[m	NOUN
ejpam-6549	155	5	]	]	PUNCT
ejpam-6549	155	6	)	)	PUNCT
ejpam-6549	155	7	by	by	ADP
ejpam-6549	155	8	lemma	lemma	PROPN
ejpam-6549	155	9	1	1	NUM
ejpam-6549	155	10	.	.	PUNCT
ejpam-6549	156	1	it	it	PRON
ejpam-6549	156	2	is	be	AUX
ejpam-6549	156	3	clear	clear	ADJ
ejpam-6549	156	4	that	that	SCONJ
ejpam-6549	156	5	ε(g[m	ε(g[m	NOUN
ejpam-6549	156	6	]	]	PUNCT
ejpam-6549	156	7	)	)	PUNCT
ejpam-6549	156	8	=	=	SYM
ejpam-6549	156	9	2m(g	2m(g	NUM
ejpam-6549	156	10	)	)	PUNCT
ejpam-6549	156	11	.	.	PUNCT
ejpam-6549	157	1	and	and	CCONJ
ejpam-6549	157	2	from	from	ADP
ejpam-6549	157	3	theorem	theorem	ADJ
ejpam-6549	157	4	4	4	NUM
ejpam-6549	157	5	,	,	PUNCT
ejpam-6549	157	6	for	for	ADP
ejpam-6549	157	7	any	any	DET
ejpam-6549	157	8	vi	vi	PROPN
ejpam-6549	157	9	∈	∈	PROPN
ejpam-6549	157	10	v	v	NOUN
ejpam-6549	157	11	(	(	PUNCT
ejpam-6549	157	12	g)\v	g)\v	NOUN
ejpam-6549	157	13	(	(	PUNCT
ejpam-6549	157	14	g[m	g[m	NOUN
ejpam-6549	157	15	]	]	PUNCT
ejpam-6549	157	16	)	)	PUNCT
ejpam-6549	157	17	we	we	PRON
ejpam-6549	157	18	have	have	VERB
ejpam-6549	157	19	ε(vi	ε(vi	NOUN
ejpam-6549	157	20	)	)	PUNCT
ejpam-6549	157	21	≥	≥	NOUN
ejpam-6549	157	22	3	3	NUM
ejpam-6549	157	23	∆	∆	PROPN
ejpam-6549	157	24	since	since	SCONJ
ejpam-6549	157	25	the	the	DET
ejpam-6549	157	26	degree	degree	NOUN
ejpam-6549	157	27	of	of	ADP
ejpam-6549	157	28	a	a	DET
ejpam-6549	157	29	vertex	vertex	NOUN
ejpam-6549	157	30	in	in	ADP
ejpam-6549	157	31	v	v	NOUN
ejpam-6549	157	32	(	(	PUNCT
ejpam-6549	157	33	g)\v	g)\v	NOUN
ejpam-6549	157	34	(	(	PUNCT
ejpam-6549	157	35	g[m	g[m	NOUN
ejpam-6549	157	36	]	]	PUNCT
ejpam-6549	157	37	)	)	PUNCT
ejpam-6549	157	38	has	have	VERB
ejpam-6549	157	39	degree	degree	NOUN
ejpam-6549	157	40	at	at	ADV
ejpam-6549	157	41	least	least	ADJ
ejpam-6549	157	42	3	3	NUM
ejpam-6549	157	43	.	.	PUNCT
ejpam-6549	158	1	thus	thus	ADV
ejpam-6549	158	2	,	,	PUNCT
ejpam-6549	158	3	it	it	PRON
ejpam-6549	158	4	follows	follow	VERB
ejpam-6549	158	5	that	that	SCONJ
ejpam-6549	158	6	ε(g	ε(g	NOUN
ejpam-6549	158	7	)	)	PUNCT
ejpam-6549	158	8	≥	≥	NOUN
ejpam-6549	158	9	2m(g	2m(g	NUM
ejpam-6549	158	10	)	)	PUNCT
ejpam-6549	159	1	+	+	CCONJ
ejpam-6549	159	2	3	3	NUM
ejpam-6549	159	3	∆c3(g	∆c3(g	NOUN
ejpam-6549	159	4	)	)	PUNCT
ejpam-6549	159	5	.	.	PUNCT
ejpam-6549	160	1	5	5	X
ejpam-6549	160	2	.	.	X
ejpam-6549	160	3	conclusion	conclusion	NOUN
ejpam-6549	160	4	we	we	PRON
ejpam-6549	160	5	investigated	investigate	VERB
ejpam-6549	160	6	the	the	DET
ejpam-6549	160	7	graph	graph	NOUN
ejpam-6549	160	8	energy	energy	NOUN
ejpam-6549	160	9	of	of	ADP
ejpam-6549	160	10	a	a	DET
ejpam-6549	160	11	special	special	ADJ
ejpam-6549	160	12	class	class	NOUN
ejpam-6549	160	13	of	of	ADP
ejpam-6549	160	14	graphs	graph	NOUN
ejpam-6549	160	15	—	—	PUNCT
ejpam-6549	160	16	those	those	PRON
ejpam-6549	160	17	for	for	ADP
ejpam-6549	160	18	which	which	PRON
ejpam-6549	160	19	the	the	DET
ejpam-6549	160	20	matching	matching	NOUN
ejpam-6549	160	21	number	number	NOUN
ejpam-6549	160	22	equals	equal	VERB
ejpam-6549	160	23	the	the	DET
ejpam-6549	160	24	induced	induced	ADJ
ejpam-6549	160	25	matching	matching	NOUN
ejpam-6549	160	26	number	number	NOUN
ejpam-6549	160	27	,	,	PUNCT
ejpam-6549	160	28	with	with	ADP
ejpam-6549	160	29	particular	particular	ADJ
ejpam-6549	160	30	focus	focus	NOUN
ejpam-6549	160	31	on	on	ADP
ejpam-6549	160	32	cameronwalker	cameronwalker	NOUN
ejpam-6549	160	33	graphs	graph	NOUN
ejpam-6549	160	34	.	.	PUNCT
ejpam-6549	161	1	using	use	VERB
ejpam-6549	161	2	spectral	spectral	ADJ
ejpam-6549	161	3	graph	graph	NOUN
ejpam-6549	161	4	-	-	PUNCT
ejpam-6549	161	5	theoretic	theoretic	NOUN
ejpam-6549	161	6	techniques	technique	NOUN
ejpam-6549	161	7	,	,	PUNCT
ejpam-6549	161	8	we	we	PRON
ejpam-6549	161	9	established	establish	VERB
ejpam-6549	161	10	new	new	ADJ
ejpam-6549	161	11	lower	low	ADJ
ejpam-6549	161	12	bounds	bound	NOUN
ejpam-6549	161	13	for	for	ADP
ejpam-6549	161	14	the	the	DET
ejpam-6549	161	15	energy	energy	NOUN
ejpam-6549	161	16	in	in	ADP
ejpam-6549	161	17	terms	term	NOUN
ejpam-6549	161	18	of	of	ADP
ejpam-6549	161	19	the	the	DET
ejpam-6549	161	20	induced	induced	ADJ
ejpam-6549	161	21	matching	matching	NOUN
ejpam-6549	161	22	number	number	NOUN
ejpam-6549	161	23	and	and	CCONJ
ejpam-6549	161	24	provided	provide	VERB
ejpam-6549	161	25	comparisons	comparison	NOUN
ejpam-6549	161	26	with	with	ADP
ejpam-6549	161	27	known	know	VERB
ejpam-6549	161	28	bounds	bound	NOUN
ejpam-6549	161	29	in	in	ADP
ejpam-6549	161	30	the	the	DET
ejpam-6549	161	31	literature	literature	NOUN
ejpam-6549	161	32	.	.	PUNCT
ejpam-6549	162	1	our	our	PRON
ejpam-6549	162	2	results	result	NOUN
ejpam-6549	162	3	demonstrate	demonstrate	VERB
ejpam-6549	162	4	that	that	SCONJ
ejpam-6549	162	5	the	the	DET
ejpam-6549	162	6	structural	structural	ADJ
ejpam-6549	162	7	constraints	constraint	NOUN
ejpam-6549	162	8	of	of	ADP
ejpam-6549	162	9	cameron	cameron	PROPN
ejpam-6549	162	10	-	-	PUNCT
ejpam-6549	162	11	walker	walker	PROPN
ejpam-6549	162	12	graphs	graphs	PROPN
ejpam-6549	162	13	yield	yield	VERB
ejpam-6549	162	14	meaningful	meaningful	ADJ
ejpam-6549	162	15	spectral	spectral	ADJ
ejpam-6549	162	16	restrictions	restriction	NOUN
ejpam-6549	162	17	,	,	PUNCT
ejpam-6549	162	18	thereby	thereby	ADV
ejpam-6549	162	19	influencing	influence	VERB
ejpam-6549	162	20	their	their	PRON
ejpam-6549	162	21	energy	energy	NOUN
ejpam-6549	162	22	in	in	ADP
ejpam-6549	162	23	predictable	predictable	ADJ
ejpam-6549	162	24	ways	way	NOUN
ejpam-6549	162	25	.	.	PUNCT
ejpam-6549	163	1	the	the	DET
ejpam-6549	163	2	cameron	cameron	PROPN
ejpam-6549	163	3	-	-	PUNCT
ejpam-6549	163	4	walker	walker	PROPN
ejpam-6549	163	5	graphs	graph	NOUN
ejpam-6549	163	6	can	can	AUX
ejpam-6549	163	7	have	have	VERB
ejpam-6549	163	8	unique	unique	ADJ
ejpam-6549	163	9	energy	energy	NOUN
ejpam-6549	163	10	properties	property	NOUN
ejpam-6549	163	11	due	due	ADJ
ejpam-6549	163	12	to	to	ADP
ejpam-6549	163	13	their	their	PRON
ejpam-6549	163	14	controlled	control	VERB
ejpam-6549	163	15	structure	structure	NOUN
ejpam-6549	163	16	.	.	PUNCT
ejpam-6549	164	1	further	further	ADJ
ejpam-6549	164	2	studies	study	NOUN
ejpam-6549	164	3	can	can	AUX
ejpam-6549	164	4	explore	explore	VERB
ejpam-6549	164	5	their	their	PRON
ejpam-6549	164	6	other	other	ADJ
ejpam-6549	164	7	spectral	spectral	ADJ
ejpam-6549	164	8	properties	property	NOUN
ejpam-6549	164	9	.	.	PUNCT
ejpam-6549	165	1	acknowledgements	acknowledgement	NOUN
ejpam-6549	165	2	the	the	DET
ejpam-6549	165	3	authors	author	NOUN
ejpam-6549	165	4	would	would	AUX
ejpam-6549	165	5	like	like	VERB
ejpam-6549	165	6	to	to	PART
ejpam-6549	165	7	thank	thank	VERB
ejpam-6549	165	8	the	the	DET
ejpam-6549	165	9	anonymous	anonymous	ADJ
ejpam-6549	165	10	referees	referee	NOUN
ejpam-6549	165	11	for	for	ADP
ejpam-6549	165	12	their	their	PRON
ejpam-6549	165	13	valuable	valuable	ADJ
ejpam-6549	165	14	comments	comment	NOUN
ejpam-6549	165	15	and	and	CCONJ
ejpam-6549	165	16	insightful	insightful	ADJ
ejpam-6549	165	17	feedback	feedback	NOUN
ejpam-6549	165	18	,	,	PUNCT
ejpam-6549	165	19	which	which	PRON
ejpam-6549	165	20	helped	help	VERB
ejpam-6549	165	21	improve	improve	VERB
ejpam-6549	165	22	the	the	DET
ejpam-6549	165	23	quality	quality	NOUN
ejpam-6549	165	24	and	and	CCONJ
ejpam-6549	165	25	clarity	clarity	NOUN
ejpam-6549	165	26	of	of	ADP
ejpam-6549	165	27	this	this	DET
ejpam-6549	165	28	work	work	NOUN
ejpam-6549	165	29	.	.	PUNCT
ejpam-6549	166	1	this	this	DET
ejpam-6549	166	2	a.	a.	NOUN
ejpam-6549	166	3	ülker	ülker	PROPN
ejpam-6549	166	4	et	et	PROPN
ejpam-6549	166	5	al	al	PROPN
ejpam-6549	166	6	.	.	PUNCT
ejpam-6549	166	7	/	/	SYM
ejpam-6549	166	8	eur	eur	PROPN
ejpam-6549	166	9	.	.	PUNCT
ejpam-6549	167	1	j.	j.	PROPN
ejpam-6549	167	2	pure	pure	PROPN
ejpam-6549	167	3	appl	appl	PROPN
ejpam-6549	167	4	.	.	PROPN
ejpam-6549	167	5	math	math	PROPN
ejpam-6549	167	6	,	,	PUNCT
ejpam-6549	167	7	18	18	NUM
ejpam-6549	167	8	(	(	PUNCT
ejpam-6549	167	9	3	3	NUM
ejpam-6549	167	10	)	)	PUNCT
ejpam-6549	167	11	(	(	PUNCT
ejpam-6549	167	12	2025	2025	NUM
ejpam-6549	167	13	)	)	PUNCT
ejpam-6549	167	14	,	,	PUNCT
ejpam-6549	167	15	6549	6549	NUM
ejpam-6549	167	16	8	8	NUM
ejpam-6549	167	17	of	of	ADP
ejpam-6549	167	18	8	8	NUM
ejpam-6549	167	19	research	research	NOUN
ejpam-6549	167	20	was	be	AUX
ejpam-6549	167	21	supported	support	VERB
ejpam-6549	167	22	by	by	ADP
ejpam-6549	167	23	university	university	NOUN
ejpam-6549	167	24	of	of	ADP
ejpam-6549	167	25	phayao	phayao	NOUN
ejpam-6549	167	26	and	and	CCONJ
ejpam-6549	167	27	thailand	thailand	PROPN
ejpam-6549	167	28	science	science	PROPN
ejpam-6549	167	29	research	research	PROPN
ejpam-6549	167	30	and	and	CCONJ
ejpam-6549	167	31	innovation	innovation	NOUN
ejpam-6549	167	32	fund	fund	NOUN
ejpam-6549	167	33	(	(	PUNCT
ejpam-6549	167	34	fundamental	fundamental	ADJ
ejpam-6549	167	35	fund	fund	NOUN
ejpam-6549	167	36	2025	2025	NUM
ejpam-6549	167	37	,	,	PUNCT
ejpam-6549	167	38	grant	grant	VERB
ejpam-6549	167	39	no	no	NOUN
ejpam-6549	167	40	.	.	PROPN
ejpam-6549	168	1	5027/2567	5027/2567	NUM
ejpam-6549	168	2	)	)	PUNCT
ejpam-6549	168	3	.	.	PUNCT
ejpam-6549	169	1	references	reference	NOUN
ejpam-6549	169	2	[	[	X
ejpam-6549	169	3	1	1	NUM
ejpam-6549	169	4	]	]	PUNCT
ejpam-6549	169	5	i.	i.	PROPN
ejpam-6549	169	6	gutman	gutman	PROPN
ejpam-6549	169	7	and	and	CCONJ
ejpam-6549	169	8	s.	s.	PROPN
ejpam-6549	169	9	wagner	wagner	PROPN
ejpam-6549	169	10	.	.	PUNCT
ejpam-6549	170	1	the	the	DET
ejpam-6549	170	2	matching	match	VERB
ejpam-6549	170	3	energy	energy	NOUN
ejpam-6549	170	4	of	of	ADP
ejpam-6549	170	5	a	a	DET
ejpam-6549	170	6	graph	graph	NOUN
ejpam-6549	170	7	.	.	PUNCT
ejpam-6549	171	1	discrete	discrete	ADJ
ejpam-6549	171	2	appl	appl	PROPN
ejpam-6549	171	3	.	.	PUNCT
ejpam-6549	171	4	math	math	PROPN
ejpam-6549	171	5	.	.	PUNCT
ejpam-6549	171	6	,	,	PUNCT
ejpam-6549	171	7	160:2177–2187	160:2177–2187	PROPN
ejpam-6549	171	8	,	,	PUNCT
ejpam-6549	171	9	2012	2012	NUM
ejpam-6549	171	10	.	.	PUNCT
ejpam-6549	172	1	[	[	X
ejpam-6549	172	2	2	2	X
ejpam-6549	172	3	]	]	PUNCT
ejpam-6549	172	4	s.	s.	PROPN
ejpam-6549	172	5	k.	k.	PROPN
ejpam-6549	172	6	ghezelahmad	ghezelahmad	PROPN
ejpam-6549	172	7	.	.	PUNCT
ejpam-6549	173	1	lower	low	ADJ
ejpam-6549	173	2	bounds	bound	NOUN
ejpam-6549	173	3	on	on	ADP
ejpam-6549	173	4	matching	match	VERB
ejpam-6549	173	5	energy	energy	NOUN
ejpam-6549	173	6	of	of	ADP
ejpam-6549	173	7	graphs	graph	NOUN
ejpam-6549	173	8	.	.	PUNCT
ejpam-6549	174	1	discrete	discrete	ADJ
ejpam-6549	174	2	appl	appl	PROPN
ejpam-6549	174	3	.	.	PUNCT
ejpam-6549	174	4	math	math	PROPN
ejpam-6549	174	5	.	.	PUNCT
ejpam-6549	174	6	,	,	PUNCT
ejpam-6549	174	7	307:153–159	307:153–159	NUM
ejpam-6549	174	8	,	,	PUNCT
ejpam-6549	174	9	2022	2022	NUM
ejpam-6549	174	10	.	.	PUNCT
ejpam-6549	175	1	[	[	X
ejpam-6549	175	2	3	3	X
ejpam-6549	175	3	]	]	PUNCT
ejpam-6549	175	4	l.	l.	PROPN
ejpam-6549	175	5	zou	zou	PROPN
ejpam-6549	175	6	and	and	CCONJ
ejpam-6549	175	7	h.-h	h.-h	NOUN
ejpam-6549	175	8	.	.	PUNCT
ejpam-6549	176	1	li	li	PROPN
ejpam-6549	176	2	.	.	PROPN
ejpam-6549	177	1	on	on	ADP
ejpam-6549	177	2	matching	match	VERB
ejpam-6549	177	3	energy	energy	NOUN
ejpam-6549	177	4	of	of	ADP
ejpam-6549	177	5	bicyclic	bicyclic	NOUN
ejpam-6549	177	6	graphs	graph	NOUN
ejpam-6549	177	7	.	.	PUNCT
ejpam-6549	178	1	int	int	NOUN
ejpam-6549	178	2	.	.	PUNCT
ejpam-6549	179	1	j.	j.	PROPN
ejpam-6549	179	2	graph	graph	PROPN
ejpam-6549	179	3	theory	theory	NOUN
ejpam-6549	179	4	appl	appl	PROPN
ejpam-6549	179	5	.	.	PROPN
ejpam-6549	179	6	,	,	PUNCT
ejpam-6549	179	7	1(2):97–110	1(2):97–110	NUM
ejpam-6549	179	8	,	,	PUNCT
ejpam-6549	179	9	2015	2015	NUM
ejpam-6549	179	10	.	.	PUNCT
ejpam-6549	180	1	[	[	X
ejpam-6549	180	2	4	4	X
ejpam-6549	180	3	]	]	PUNCT
ejpam-6549	180	4	x.	x.	PROPN
ejpam-6549	180	5	chen	chen	PROPN
ejpam-6549	180	6	,	,	PUNCT
ejpam-6549	180	7	x.	x.	PROPN
ejpam-6549	180	8	li	li	PROPN
ejpam-6549	180	9	,	,	PUNCT
ejpam-6549	180	10	and	and	CCONJ
ejpam-6549	180	11	h.	h.	PROPN
ejpam-6549	180	12	lian	lian	PROPN
ejpam-6549	180	13	.	.	PUNCT
ejpam-6549	181	1	the	the	DET
ejpam-6549	181	2	matching	match	VERB
ejpam-6549	181	3	energy	energy	NOUN
ejpam-6549	181	4	of	of	ADP
ejpam-6549	181	5	random	random	ADJ
ejpam-6549	181	6	graphs	graph	NOUN
ejpam-6549	181	7	.	.	PUNCT
ejpam-6549	182	1	discrete	discrete	ADJ
ejpam-6549	182	2	appl	appl	PROPN
ejpam-6549	182	3	.	.	PUNCT
ejpam-6549	182	4	math	math	PROPN
ejpam-6549	182	5	.	.	PUNCT
ejpam-6549	182	6	,	,	PUNCT
ejpam-6549	182	7	193:102–109	193:102–109	NUM
ejpam-6549	182	8	,	,	PUNCT
ejpam-6549	182	9	2015	2015	NUM
ejpam-6549	182	10	.	.	PUNCT
ejpam-6549	183	1	[	[	X
ejpam-6549	183	2	5	5	X
ejpam-6549	183	3	]	]	PUNCT
ejpam-6549	183	4	s.	s.	PROPN
ejpam-6549	183	5	ji	ji	PROPN
ejpam-6549	183	6	,	,	PUNCT
ejpam-6549	183	7	h.	h.	PROPN
ejpam-6549	183	8	ma	ma	PROPN
ejpam-6549	183	9	,	,	PUNCT
ejpam-6549	183	10	and	and	CCONJ
ejpam-6549	183	11	g.	g.	PROPN
ejpam-6549	183	12	ma	ma	PROPN
ejpam-6549	183	13	.	.	PUNCT
ejpam-6549	184	1	the	the	DET
ejpam-6549	184	2	matching	match	VERB
ejpam-6549	184	3	energy	energy	NOUN
ejpam-6549	184	4	of	of	ADP
ejpam-6549	184	5	graphs	graph	NOUN
ejpam-6549	184	6	with	with	ADP
ejpam-6549	184	7	given	give	VERB
ejpam-6549	184	8	edge	edge	NOUN
ejpam-6549	184	9	connectivity	connectivity	NOUN
ejpam-6549	184	10	.	.	PUNCT
ejpam-6549	185	1	j.	j.	PROPN
ejpam-6549	185	2	inequal	inequal	PROPN
ejpam-6549	185	3	.	.	PUNCT
ejpam-6549	186	1	appl	appl	PROPN
ejpam-6549	186	2	.	.	PROPN
ejpam-6549	186	3	,	,	PUNCT
ejpam-6549	186	4	2015:415	2015:415	NUM
ejpam-6549	186	5	,	,	PUNCT
ejpam-6549	186	6	2015	2015	NUM
ejpam-6549	186	7	.	.	PUNCT
ejpam-6549	187	1	[	[	X
ejpam-6549	187	2	6	6	NUM
ejpam-6549	187	3	]	]	X
ejpam-6549	187	4	d.	d.	PROPN
ejpam-6549	187	5	wong	wong	PROPN
ejpam-6549	187	6	,	,	PUNCT
ejpam-6549	187	7	x.	x.	PROPN
ejpam-6549	187	8	wang	wang	PROPN
ejpam-6549	187	9	,	,	PUNCT
ejpam-6549	187	10	and	and	CCONJ
ejpam-6549	187	11	r.	r.	PROPN
ejpam-6549	187	12	chu	chu	PROPN
ejpam-6549	187	13	.	.	PUNCT
ejpam-6549	188	1	lower	low	ADJ
ejpam-6549	188	2	bounds	bound	NOUN
ejpam-6549	188	3	of	of	ADP
ejpam-6549	188	4	graph	graph	NOUN
ejpam-6549	188	5	energy	energy	NOUN
ejpam-6549	188	6	in	in	ADP
ejpam-6549	188	7	terms	term	NOUN
ejpam-6549	188	8	of	of	ADP
ejpam-6549	188	9	matching	match	VERB
ejpam-6549	188	10	number	number	NOUN
ejpam-6549	188	11	.	.	PUNCT
ejpam-6549	189	1	linear	linear	PROPN
ejpam-6549	189	2	algebra	algebra	PROPN
ejpam-6549	189	3	appl	appl	NOUN
ejpam-6549	189	4	.	.	PROPN
ejpam-6549	189	5	,	,	PUNCT
ejpam-6549	190	1	549:276–286	549:276–286	NUM
ejpam-6549	190	2	,	,	PUNCT
ejpam-6549	190	3	2018	2018	NUM
ejpam-6549	190	4	.	.	PUNCT
ejpam-6549	191	1	[	[	X
ejpam-6549	191	2	7	7	X
ejpam-6549	191	3	]	]	X
ejpam-6549	191	4	f.	f.	PROPN
ejpam-6549	191	5	ashraf	ashraf	PROPN
ejpam-6549	191	6	.	.	PUNCT
ejpam-6549	192	1	energy	energy	NOUN
ejpam-6549	192	2	,	,	PUNCT
ejpam-6549	192	3	matching	match	VERB
ejpam-6549	192	4	number	number	NOUN
ejpam-6549	192	5	and	and	CCONJ
ejpam-6549	192	6	odd	odd	ADJ
ejpam-6549	192	7	cycles	cycle	NOUN
ejpam-6549	192	8	of	of	ADP
ejpam-6549	192	9	graphs	graph	NOUN
ejpam-6549	192	10	.	.	PUNCT
ejpam-6549	193	1	linear	linear	ADJ
ejpam-6549	193	2	algebra	algebra	PROPN
ejpam-6549	193	3	appl	appl	NOUN
ejpam-6549	193	4	.	.	PROPN
ejpam-6549	193	5	,	,	PUNCT
ejpam-6549	193	6	577:159–167	577:159–167	NUM
ejpam-6549	193	7	,	,	PUNCT
ejpam-6549	193	8	2019	2019	NUM
ejpam-6549	193	9	.	.	PUNCT
ejpam-6549	194	1	[	[	X
ejpam-6549	194	2	8	8	NUM
ejpam-6549	194	3	]	]	X
ejpam-6549	194	4	o.	o.	PROPN
ejpam-6549	194	5	arizmendi	arizmendi	PROPN
ejpam-6549	194	6	,	,	PUNCT
ejpam-6549	194	7	j.	j.	PROPN
ejpam-6549	194	8	f.	f.	PROPN
ejpam-6549	194	9	hidalgo	hidalgo	PROPN
ejpam-6549	194	10	,	,	PUNCT
ejpam-6549	194	11	and	and	CCONJ
ejpam-6549	194	12	o.	o.	PROPN
ejpam-6549	194	13	juarez	juarez	PROPN
ejpam-6549	194	14	-	-	PUNCT
ejpam-6549	194	15	romero	romero	PROPN
ejpam-6549	194	16	.	.	PUNCT
ejpam-6549	195	1	energy	energy	NOUN
ejpam-6549	195	2	of	of	ADP
ejpam-6549	195	3	a	a	DET
ejpam-6549	195	4	vertex	vertex	NOUN
ejpam-6549	195	5	.	.	PUNCT
ejpam-6549	196	1	linear	linear	PROPN
ejpam-6549	196	2	algebra	algebra	PROPN
ejpam-6549	196	3	appl	appl	NOUN
ejpam-6549	196	4	.	.	PROPN
ejpam-6549	196	5	,	,	PUNCT
ejpam-6549	196	6	557:464–495	557:464–495	NUM
ejpam-6549	196	7	,	,	PUNCT
ejpam-6549	196	8	2018	2018	NUM
ejpam-6549	196	9	.	.	PUNCT
ejpam-6549	197	1	[	[	X
ejpam-6549	197	2	9	9	NUM
ejpam-6549	197	3	]	]	PUNCT
ejpam-6549	197	4	k.	k.	PROPN
ejpam-6549	197	5	cameron	cameron	PROPN
ejpam-6549	197	6	and	and	CCONJ
ejpam-6549	197	7	t.	t.	PROPN
ejpam-6549	197	8	walker	walker	PROPN
ejpam-6549	197	9	.	.	PUNCT
ejpam-6549	198	1	the	the	DET
ejpam-6549	198	2	graphs	graph	NOUN
ejpam-6549	198	3	with	with	ADP
ejpam-6549	198	4	maximum	maximum	ADJ
ejpam-6549	198	5	induced	induced	ADJ
ejpam-6549	198	6	matching	matching	NOUN
ejpam-6549	198	7	and	and	CCONJ
ejpam-6549	198	8	maximum	maximum	ADJ
ejpam-6549	198	9	matching	match	VERB
ejpam-6549	198	10	the	the	DET
ejpam-6549	198	11	same	same	ADJ
ejpam-6549	198	12	size	size	NOUN
ejpam-6549	198	13	.	.	PUNCT
ejpam-6549	199	1	discrete	discrete	ADJ
ejpam-6549	199	2	math	math	NOUN
ejpam-6549	199	3	.	.	PUNCT
ejpam-6549	199	4	,	,	PUNCT
ejpam-6549	199	5	299:49–55	299:49–55	NUM
ejpam-6549	199	6	,	,	PUNCT
ejpam-6549	199	7	2005	2005	NUM
ejpam-6549	199	8	.	.	PUNCT
ejpam-6549	200	1	[	[	X
ejpam-6549	200	2	10	10	NUM
ejpam-6549	200	3	]	]	PUNCT
ejpam-6549	200	4	t.	t.	NOUN
ejpam-6549	200	5	hibi	hibi	PROPN
ejpam-6549	200	6	,	,	PUNCT
ejpam-6549	200	7	a.	a.	NOUN
ejpam-6549	200	8	higashitani	higashitani	PROPN
ejpam-6549	200	9	,	,	PUNCT
ejpam-6549	200	10	k.	k.	PROPN
ejpam-6549	200	11	kimura	kimura	PROPN
ejpam-6549	200	12	,	,	PUNCT
ejpam-6549	200	13	and	and	CCONJ
ejpam-6549	200	14	a.	a.	PROPN
ejpam-6549	200	15	b.	b.	PROPN
ejpam-6549	200	16	o’keefe	o’keefe	PROPN
ejpam-6549	200	17	.	.	PUNCT
ejpam-6549	201	1	algebraic	algebraic	ADJ
ejpam-6549	201	2	study	study	NOUN
ejpam-6549	201	3	on	on	ADP
ejpam-6549	201	4	cameronwalker	cameronwalker	NOUN
ejpam-6549	201	5	graphs	graph	NOUN
ejpam-6549	201	6	.	.	PUNCT
ejpam-6549	202	1	j.	j.	PROPN
ejpam-6549	202	2	algebra	algebra	PROPN
ejpam-6549	202	3	,	,	PUNCT
ejpam-6549	202	4	422:257–269	422:257–269	NUM
ejpam-6549	202	5	,	,	PUNCT
ejpam-6549	202	6	2015	2015	NUM
ejpam-6549	202	7	.	.	PUNCT
ejpam-6549	203	1	[	[	X
ejpam-6549	203	2	11	11	NUM
ejpam-6549	203	3	]	]	PUNCT
ejpam-6549	203	4	s.	s.	PROPN
ejpam-6549	203	5	akbari	akbari	PROPN
ejpam-6549	203	6	,	,	PUNCT
ejpam-6549	203	7	e.	e.	PROPN
ejpam-6549	203	8	ghorbani	ghorbani	PROPN
ejpam-6549	203	9	,	,	PUNCT
ejpam-6549	203	10	and	and	CCONJ
ejpam-6549	203	11	s.	s.	PROPN
ejpam-6549	203	12	zare	zare	PROPN
ejpam-6549	203	13	.	.	PUNCT
ejpam-6549	204	1	some	some	DET
ejpam-6549	204	2	relations	relation	NOUN
ejpam-6549	204	3	between	between	ADP
ejpam-6549	204	4	rank	rank	NOUN
ejpam-6549	204	5	,	,	PUNCT
ejpam-6549	204	6	chromatic	chromatic	ADJ
ejpam-6549	204	7	number	number	NOUN
ejpam-6549	204	8	and	and	CCONJ
ejpam-6549	204	9	energy	energy	NOUN
ejpam-6549	204	10	of	of	ADP
ejpam-6549	204	11	graphs	graph	NOUN
ejpam-6549	204	12	.	.	PUNCT
ejpam-6549	205	1	discrete	discrete	ADJ
ejpam-6549	205	2	math	math	NOUN
ejpam-6549	205	3	.	.	PUNCT
ejpam-6549	205	4	,	,	PUNCT
ejpam-6549	205	5	309:601–605	309:601–605	NUM
ejpam-6549	205	6	,	,	PUNCT
ejpam-6549	205	7	2009	2009	NUM
ejpam-6549	205	8	.	.	PUNCT