id	sid	tid	token	lemma	pos
ejpam-5274	1	1	european	european	PROPN
ejpam-5274	1	2	journal	journal	PROPN
ejpam-5274	1	3	of	of	ADP
ejpam-5274	1	4	pure	pure	ADJ
ejpam-5274	1	5	and	and	CCONJ
ejpam-5274	1	6	applied	apply	VERB
ejpam-5274	1	7	mathematics	mathematic	NOUN
ejpam-5274	1	8	vol	vol	NOUN
ejpam-5274	1	9	.	.	PROPN
ejpam-5274	2	1	17	17	NUM
ejpam-5274	2	2	,	,	PUNCT
ejpam-5274	2	3	no	no	INTJ
ejpam-5274	2	4	.	.	NOUN
ejpam-5274	2	5	3	3	NUM
ejpam-5274	2	6	,	,	PUNCT
ejpam-5274	2	7	2024	2024	NUM
ejpam-5274	2	8	,	,	PUNCT
ejpam-5274	2	9	1449	1449	NUM
ejpam-5274	2	10	-	-	SYM
ejpam-5274	2	11	1462	1462	NUM
ejpam-5274	2	12	issn	issn	PROPN
ejpam-5274	2	13	1307	1307	NUM
ejpam-5274	2	14	-	-	SYM
ejpam-5274	2	15	5543	5543	NUM
ejpam-5274	2	16	–	–	PUNCT
ejpam-5274	2	17	ejpam.com	ejpam.com	X
ejpam-5274	2	18	published	publish	VERB
ejpam-5274	2	19	by	by	ADP
ejpam-5274	2	20	new	new	PROPN
ejpam-5274	2	21	york	york	PROPN
ejpam-5274	2	22	business	business	PROPN
ejpam-5274	2	23	global	global	ADJ
ejpam-5274	2	24	polynomial	polynomial	ADJ
ejpam-5274	2	25	representations	representation	NOUN
ejpam-5274	2	26	and	and	CCONJ
ejpam-5274	2	27	degree	degree	NOUN
ejpam-5274	2	28	sequences	sequence	NOUN
ejpam-5274	2	29	of	of	ADP
ejpam-5274	2	30	graphs	graph	NOUN
ejpam-5274	2	31	resulting	result	VERB
ejpam-5274	2	32	from	from	ADP
ejpam-5274	2	33	some	some	DET
ejpam-5274	2	34	graph	graph	NOUN
ejpam-5274	2	35	operations	operation	NOUN
ejpam-5274	2	36	jayhan	jayhan	ADP
ejpam-5274	2	37	cruz1,∗	cruz1,∗	PROPN
ejpam-5274	2	38	,	,	PUNCT
ejpam-5274	2	39	gina	gina	PROPN
ejpam-5274	2	40	malacas1,2	malacas1,2	PROPN
ejpam-5274	2	41	,	,	PUNCT
ejpam-5274	2	42	sergio	sergio	PROPN
ejpam-5274	2	43	r.	r.	PROPN
ejpam-5274	2	44	canoy	canoy	PROPN
ejpam-5274	2	45	,	,	PUNCT
ejpam-5274	2	46	jr.1,2	jr.1,2	ADJ
ejpam-5274	2	47	1	1	NUM
ejpam-5274	2	48	department	department	NOUN
ejpam-5274	2	49	of	of	ADP
ejpam-5274	2	50	mathematics	mathematic	NOUN
ejpam-5274	2	51	and	and	CCONJ
ejpam-5274	2	52	statistics	statistic	NOUN
ejpam-5274	2	53	,	,	PUNCT
ejpam-5274	2	54	college	college	NOUN
ejpam-5274	2	55	of	of	ADP
ejpam-5274	2	56	science	science	NOUN
ejpam-5274	2	57	and	and	CCONJ
ejpam-5274	2	58	mathematics	mathematic	NOUN
ejpam-5274	2	59	msu	msu	PROPN
ejpam-5274	2	60	-	-	PUNCT
ejpam-5274	2	61	iligan	iligan	PROPN
ejpam-5274	2	62	institute	institute	PROPN
ejpam-5274	2	63	of	of	ADP
ejpam-5274	2	64	technology	technology	PROPN
ejpam-5274	2	65	,	,	PUNCT
ejpam-5274	2	66	9200	9200	NUM
ejpam-5274	2	67	iligan	iligan	ADJ
ejpam-5274	2	68	city	city	NOUN
ejpam-5274	2	69	,	,	PUNCT
ejpam-5274	2	70	philippines	philippine	NOUN
ejpam-5274	2	71	2	2	NUM
ejpam-5274	2	72	center	center	NOUN
ejpam-5274	2	73	for	for	ADP
ejpam-5274	2	74	mathematical	mathematical	ADJ
ejpam-5274	2	75	and	and	CCONJ
ejpam-5274	2	76	theoretical	theoretical	ADJ
ejpam-5274	2	77	physical	physical	ADJ
ejpam-5274	2	78	sciences	science	NOUN
ejpam-5274	2	79	-	-	PUNCT
ejpam-5274	2	80	prism	prism	NOUN
ejpam-5274	2	81	msu	msu	PROPN
ejpam-5274	2	82	-	-	PUNCT
ejpam-5274	2	83	iligan	iligan	PROPN
ejpam-5274	2	84	institute	institute	PROPN
ejpam-5274	2	85	of	of	ADP
ejpam-5274	2	86	technology	technology	PROPN
ejpam-5274	2	87	,	,	PUNCT
ejpam-5274	2	88	9200	9200	NUM
ejpam-5274	2	89	iligan	iligan	ADJ
ejpam-5274	2	90	city	city	NOUN
ejpam-5274	2	91	,	,	PUNCT
ejpam-5274	2	92	philippines	philippine	NOUN
ejpam-5274	2	93	abstract	abstract	ADJ
ejpam-5274	2	94	.	.	PUNCT
ejpam-5274	3	1	let	let	VERB
ejpam-5274	3	2	g	g	PROPN
ejpam-5274	3	3	=	=	SYM
ejpam-5274	3	4	(	(	PUNCT
ejpam-5274	3	5	v	v	NOUN
ejpam-5274	3	6	(	(	PUNCT
ejpam-5274	3	7	g	g	NOUN
ejpam-5274	3	8	)	)	PUNCT
ejpam-5274	3	9	,	,	PUNCT
ejpam-5274	3	10	e(g	e(g	PROPN
ejpam-5274	3	11	)	)	PUNCT
ejpam-5274	3	12	)	)	PUNCT
ejpam-5274	4	1	be	be	AUX
ejpam-5274	4	2	a	a	DET
ejpam-5274	4	3	graph	graph	NOUN
ejpam-5274	4	4	with	with	ADP
ejpam-5274	4	5	degree	degree	NOUN
ejpam-5274	4	6	sequence	sequence	NOUN
ejpam-5274	4	7	⟨d1	⟨d1	PROPN
ejpam-5274	4	8	,	,	PUNCT
ejpam-5274	4	9	d2	d2	PROPN
ejpam-5274	4	10	,	,	PUNCT
ejpam-5274	4	11	·	·	PUNCT
ejpam-5274	4	12	·	·	PUNCT
ejpam-5274	4	13	·	·	PUNCT
ejpam-5274	4	14	,	,	PUNCT
ejpam-5274	4	15	dn⟩	dn⟩	NOUN
ejpam-5274	4	16	,	,	PUNCT
ejpam-5274	4	17	where	where	SCONJ
ejpam-5274	4	18	d1	d1	PROPN
ejpam-5274	4	19	≥	≥	NOUN
ejpam-5274	4	20	d2	d2	PROPN
ejpam-5274	4	21	≥	≥	NUM
ejpam-5274	4	22	·	·	PUNCT
ejpam-5274	4	23	·	·	PUNCT
ejpam-5274	4	24	·	·	PUNCT
ejpam-5274	4	25	≥	≥	NUM
ejpam-5274	5	1	dn	dn	NOUN
ejpam-5274	5	2	.	.	PUNCT
ejpam-5274	6	1	the	the	DET
ejpam-5274	6	2	polynomial	polynomial	ADJ
ejpam-5274	6	3	representation	representation	NOUN
ejpam-5274	6	4	of	of	ADP
ejpam-5274	6	5	g	g	PROPN
ejpam-5274	6	6	is	be	AUX
ejpam-5274	6	7	given	give	VERB
ejpam-5274	6	8	by	by	ADP
ejpam-5274	6	9	fg(x	fg(x	PROPN
ejpam-5274	6	10	)	)	PUNCT
ejpam-5274	7	1	=	=	SYM
ejpam-5274	8	1	n∑	n∑	NOUN
ejpam-5274	8	2	i=1	i=1	X
ejpam-5274	8	3	xdi	xdi	PUNCT
ejpam-5274	9	1	=	=	PRON
ejpam-5274	10	1	∆(g)∑	∆(g)∑	PROPN
ejpam-5274	10	2	k=1	k=1	PROPN
ejpam-5274	10	3	akx	akx	PROPN
ejpam-5274	11	1	k	k	NOUN
ejpam-5274	11	2	,	,	PUNCT
ejpam-5274	11	3	where	where	SCONJ
ejpam-5274	11	4	ak	ak	PROPN
ejpam-5274	11	5	is	be	AUX
ejpam-5274	11	6	the	the	DET
ejpam-5274	11	7	number	number	NOUN
ejpam-5274	11	8	of	of	ADP
ejpam-5274	11	9	vertices	vertex	NOUN
ejpam-5274	11	10	of	of	ADP
ejpam-5274	11	11	g	g	NOUN
ejpam-5274	11	12	having	have	VERB
ejpam-5274	11	13	degree	degree	NOUN
ejpam-5274	11	14	k	k	PROPN
ejpam-5274	11	15	for	for	ADP
ejpam-5274	11	16	each	each	DET
ejpam-5274	11	17	i	i	NOUN
ejpam-5274	11	18	=	=	NOUN
ejpam-5274	11	19	1	1	NUM
ejpam-5274	11	20	,	,	PUNCT
ejpam-5274	11	21	2	2	NUM
ejpam-5274	11	22	,	,	PUNCT
ejpam-5274	11	23	·	·	PUNCT
ejpam-5274	11	24	·	·	PUNCT
ejpam-5274	11	25	·	·	PUNCT
ejpam-5274	11	26	n	n	SYM
ejpam-5274	11	27	=	=	SYM
ejpam-5274	11	28	∆(g	∆(g	PROPN
ejpam-5274	11	29	)	)	PUNCT
ejpam-5274	11	30	.	.	PUNCT
ejpam-5274	12	1	in	in	ADP
ejpam-5274	12	2	this	this	DET
ejpam-5274	12	3	paper	paper	NOUN
ejpam-5274	12	4	,	,	PUNCT
ejpam-5274	12	5	we	we	PRON
ejpam-5274	12	6	give	give	VERB
ejpam-5274	12	7	the	the	DET
ejpam-5274	12	8	polynomial	polynomial	ADJ
ejpam-5274	12	9	representation	representation	NOUN
ejpam-5274	12	10	of	of	ADP
ejpam-5274	12	11	the	the	DET
ejpam-5274	12	12	complement	complement	NOUN
ejpam-5274	12	13	and	and	CCONJ
ejpam-5274	12	14	line	line	NOUN
ejpam-5274	12	15	graph	graph	NOUN
ejpam-5274	12	16	of	of	ADP
ejpam-5274	12	17	a	a	DET
ejpam-5274	12	18	graph	graph	NOUN
ejpam-5274	12	19	,	,	PUNCT
ejpam-5274	12	20	the	the	DET
ejpam-5274	12	21	shadow	shadow	NOUN
ejpam-5274	12	22	graph	graph	NOUN
ejpam-5274	12	23	,	,	PUNCT
ejpam-5274	12	24	complementary	complementary	ADJ
ejpam-5274	12	25	prism	prism	NOUN
ejpam-5274	12	26	,	,	PUNCT
ejpam-5274	12	27	edge	edge	NOUN
ejpam-5274	12	28	corona	corona	NOUN
ejpam-5274	12	29	,	,	PUNCT
ejpam-5274	12	30	strong	strong	ADJ
ejpam-5274	12	31	product	product	NOUN
ejpam-5274	12	32	,	,	PUNCT
ejpam-5274	12	33	symmetric	symmetric	ADJ
ejpam-5274	12	34	product	product	NOUN
ejpam-5274	12	35	,	,	PUNCT
ejpam-5274	12	36	and	and	CCONJ
ejpam-5274	12	37	disjunction	disjunction	NOUN
ejpam-5274	12	38	of	of	ADP
ejpam-5274	12	39	two	two	NUM
ejpam-5274	12	40	graphs	graph	NOUN
ejpam-5274	12	41	.	.	PUNCT
ejpam-5274	13	1	2020	2020	NUM
ejpam-5274	13	2	mathematics	mathematic	NOUN
ejpam-5274	13	3	subject	subject	NOUN
ejpam-5274	13	4	classifications	classification	NOUN
ejpam-5274	13	5	:	:	PUNCT
ejpam-5274	13	6	05c69	05c69	X
ejpam-5274	13	7	key	key	ADJ
ejpam-5274	13	8	words	word	NOUN
ejpam-5274	13	9	and	and	CCONJ
ejpam-5274	13	10	phrases	phrase	NOUN
ejpam-5274	13	11	:	:	PUNCT
ejpam-5274	13	12	polynomial	polynomial	ADJ
ejpam-5274	13	13	representation	representation	NOUN
ejpam-5274	13	14	,	,	PUNCT
ejpam-5274	13	15	line	line	NOUN
ejpam-5274	13	16	graph	graph	NOUN
ejpam-5274	13	17	,	,	PUNCT
ejpam-5274	13	18	edge	edge	NOUN
ejpam-5274	13	19	corona	corona	NOUN
ejpam-5274	13	20	,	,	PUNCT
ejpam-5274	13	21	shadow	shadow	NOUN
ejpam-5274	13	22	graph	graph	NOUN
ejpam-5274	13	23	,	,	PUNCT
ejpam-5274	13	24	complementary	complementary	ADJ
ejpam-5274	13	25	prism	prism	NOUN
ejpam-5274	13	26	,	,	PUNCT
ejpam-5274	13	27	strong	strong	ADJ
ejpam-5274	13	28	product	product	NOUN
ejpam-5274	13	29	,	,	PUNCT
ejpam-5274	13	30	symmetric	symmetric	ADJ
ejpam-5274	13	31	product	product	NOUN
ejpam-5274	13	32	,	,	PUNCT
ejpam-5274	13	33	disjunction	disjunction	NOUN
ejpam-5274	13	34	1	1	NUM
ejpam-5274	13	35	.	.	PUNCT
ejpam-5274	14	1	introduction	introduction	NOUN
ejpam-5274	14	2	let	let	VERB
ejpam-5274	14	3	g	g	NOUN
ejpam-5274	14	4	=	=	SYM
ejpam-5274	14	5	(	(	PUNCT
ejpam-5274	14	6	v	v	NOUN
ejpam-5274	14	7	(	(	PUNCT
ejpam-5274	14	8	g	g	NOUN
ejpam-5274	14	9	)	)	PUNCT
ejpam-5274	14	10	,	,	PUNCT
ejpam-5274	14	11	e(g	e(g	PROPN
ejpam-5274	14	12	)	)	PUNCT
ejpam-5274	14	13	)	)	PUNCT
ejpam-5274	14	14	be	be	AUX
ejpam-5274	14	15	a	a	DET
ejpam-5274	14	16	graph	graph	NOUN
ejpam-5274	14	17	on	on	ADP
ejpam-5274	14	18	n	n	DET
ejpam-5274	14	19	vertices	vertex	NOUN
ejpam-5274	14	20	and	and	CCONJ
ejpam-5274	14	21	let	let	VERB
ejpam-5274	14	22	∆(g	∆(g	NOUN
ejpam-5274	14	23	)	)	PUNCT
ejpam-5274	14	24	be	be	AUX
ejpam-5274	14	25	the	the	DET
ejpam-5274	14	26	maximum	maximum	ADJ
ejpam-5274	14	27	degree	degree	NOUN
ejpam-5274	14	28	of	of	ADP
ejpam-5274	14	29	g.	g.	PROPN
ejpam-5274	15	1	if	if	SCONJ
ejpam-5274	15	2	s1	s1	PROPN
ejpam-5274	15	3	,	,	PUNCT
ejpam-5274	15	4	s2	s2	PROPN
ejpam-5274	15	5	,	,	PUNCT
ejpam-5274	15	6	·	·	PUNCT
ejpam-5274	15	7	·	·	PUNCT
ejpam-5274	15	8	·	·	PUNCT
ejpam-5274	15	9	,	,	PUNCT
ejpam-5274	15	10	sn	sn	PROPN
ejpam-5274	15	11	are	be	AUX
ejpam-5274	15	12	the	the	DET
ejpam-5274	15	13	degrees	degree	NOUN
ejpam-5274	15	14	of	of	ADP
ejpam-5274	15	15	the	the	DET
ejpam-5274	15	16	vertices	vertex	NOUN
ejpam-5274	15	17	of	of	ADP
ejpam-5274	15	18	g	g	NOUN
ejpam-5274	15	19	,	,	PUNCT
ejpam-5274	15	20	where	where	SCONJ
ejpam-5274	15	21	s1	s1	PROPN
ejpam-5274	15	22	≥	≥	X
ejpam-5274	15	23	s2	s2	PROPN
ejpam-5274	15	24	≥	≥	NUM
ejpam-5274	15	25	·	·	PUNCT
ejpam-5274	15	26	·	·	PUNCT
ejpam-5274	15	27	·	·	PUNCT
ejpam-5274	15	28	≥	≥	NUM
ejpam-5274	16	1	sn	sn	PROPN
ejpam-5274	16	2	,	,	PUNCT
ejpam-5274	16	3	then	then	ADV
ejpam-5274	16	4	the	the	DET
ejpam-5274	16	5	sequence	sequence	NOUN
ejpam-5274	16	6	⟨s1	⟨s1	PROPN
ejpam-5274	16	7	,	,	PUNCT
ejpam-5274	16	8	s2	s2	PROPN
ejpam-5274	16	9	,	,	PUNCT
ejpam-5274	16	10	·	·	PUNCT
ejpam-5274	16	11	·	·	PUNCT
ejpam-5274	16	12	·	·	PUNCT
ejpam-5274	16	13	,	,	PUNCT
ejpam-5274	16	14	sn⟩	sn⟩	PROPN
ejpam-5274	16	15	is	be	AUX
ejpam-5274	16	16	called	call	VERB
ejpam-5274	16	17	the	the	DET
ejpam-5274	16	18	degree	degree	NOUN
ejpam-5274	16	19	sequence	sequence	NOUN
ejpam-5274	16	20	of	of	ADP
ejpam-5274	16	21	g.	g.	PROPN
ejpam-5274	16	22	here	here	ADV
ejpam-5274	16	23	,	,	PUNCT
ejpam-5274	16	24	s1	s1	PROPN
ejpam-5274	16	25	=	=	SYM
ejpam-5274	16	26	∆(g	∆(g	PROPN
ejpam-5274	16	27	)	)	PUNCT
ejpam-5274	16	28	.	.	PUNCT
ejpam-5274	17	1	the	the	DET
ejpam-5274	17	2	polynomial	polynomial	ADJ
ejpam-5274	17	3	fg(x	fg(x	NUM
ejpam-5274	17	4	)	)	PUNCT
ejpam-5274	18	1	=	=	SYM
ejpam-5274	19	1	n∑	n∑	PROPN
ejpam-5274	19	2	i=1	i=1	PROPN
ejpam-5274	19	3	xsi	xsi	PROPN
ejpam-5274	19	4	is	be	AUX
ejpam-5274	19	5	called	call	VERB
ejpam-5274	19	6	the	the	DET
ejpam-5274	19	7	polynomial	polynomial	ADJ
ejpam-5274	19	8	representation	representation	NOUN
ejpam-5274	19	9	of	of	ADP
ejpam-5274	19	10	g.	g.	PROPN
ejpam-5274	19	11	a	a	DET
ejpam-5274	19	12	degree	degree	NOUN
ejpam-5274	19	13	sequence	sequence	NOUN
ejpam-5274	19	14	⟨s1	⟨s1	PROPN
ejpam-5274	19	15	,	,	PUNCT
ejpam-5274	19	16	s2	s2	PROPN
ejpam-5274	19	17	,	,	PUNCT
ejpam-5274	19	18	.	.	PUNCT
ejpam-5274	19	19	.	.	PUNCT
ejpam-5274	20	1	.	.	PUNCT
ejpam-5274	21	1	,	,	PUNCT
ejpam-5274	21	2	sn⟩	sn⟩	VERB
ejpam-5274	21	3	of	of	ADP
ejpam-5274	21	4	nonnegative	nonnegative	ADJ
ejpam-5274	21	5	integers	integer	NOUN
ejpam-5274	21	6	is	be	AUX
ejpam-5274	21	7	said	say	VERB
ejpam-5274	21	8	to	to	PART
ejpam-5274	21	9	be	be	AUX
ejpam-5274	21	10	graphic	graphic	ADJ
ejpam-5274	21	11	if	if	SCONJ
ejpam-5274	21	12	a	a	DET
ejpam-5274	21	13	simple	simple	ADJ
ejpam-5274	21	14	graph	graph	NOUN
ejpam-5274	21	15	g	g	NOUN
ejpam-5274	21	16	with	with	ADP
ejpam-5274	21	17	degree	degree	NOUN
ejpam-5274	21	18	sequence	sequence	NOUN
ejpam-5274	21	19	⟨s1	⟨s1	PROPN
ejpam-5274	21	20	,	,	PUNCT
ejpam-5274	21	21	s2	s2	PROPN
ejpam-5274	21	22	,	,	PUNCT
ejpam-5274	21	23	.	.	PUNCT
ejpam-5274	21	24	.	.	PUNCT
ejpam-5274	22	1	.	.	PUNCT
ejpam-5274	23	1	,	,	PUNCT
ejpam-5274	23	2	sn⟩	sn⟩	PROPN
ejpam-5274	23	3	can	can	AUX
ejpam-5274	23	4	be	be	AUX
ejpam-5274	23	5	found	find	VERB
ejpam-5274	23	6	(	(	PUNCT
ejpam-5274	23	7	see	see	VERB
ejpam-5274	23	8	[	[	X
ejpam-5274	23	9	1	1	NUM
ejpam-5274	23	10	]	]	PUNCT
ejpam-5274	23	11	)	)	PUNCT
ejpam-5274	23	12	.	.	PUNCT
ejpam-5274	24	1	using	use	VERB
ejpam-5274	24	2	the	the	DET
ejpam-5274	24	3	polynomial	polynomial	ADJ
ejpam-5274	24	4	representation	representation	NOUN
ejpam-5274	24	5	of	of	ADP
ejpam-5274	24	6	a	a	DET
ejpam-5274	24	7	graph	graph	NOUN
ejpam-5274	24	8	,	,	PUNCT
ejpam-5274	24	9	we	we	PRON
ejpam-5274	24	10	can	can	AUX
ejpam-5274	24	11	alternatively	alternatively	ADV
ejpam-5274	24	12	define	define	VERB
ejpam-5274	24	13	a	a	DET
ejpam-5274	24	14	polynomial	polynomial	ADJ
ejpam-5274	24	15	p	p	NOUN
ejpam-5274	24	16	(	(	PUNCT
ejpam-5274	24	17	x	x	X
ejpam-5274	24	18	)	)	PUNCT
ejpam-5274	24	19	to	to	PART
ejpam-5274	24	20	be	be	AUX
ejpam-5274	24	21	graphic	graphic	ADJ
ejpam-5274	24	22	if	if	SCONJ
ejpam-5274	24	23	there	there	PRON
ejpam-5274	24	24	exists	exist	VERB
ejpam-5274	24	25	a	a	DET
ejpam-5274	24	26	graph	graph	NOUN
ejpam-5274	24	27	g	g	ADP
ejpam-5274	25	1	such	such	ADJ
ejpam-5274	25	2	that	that	SCONJ
ejpam-5274	25	3	p	p	X
ejpam-5274	25	4	(	(	PUNCT
ejpam-5274	25	5	x	x	NOUN
ejpam-5274	25	6	)	)	PUNCT
ejpam-5274	25	7	=	=	PUNCT
ejpam-5274	25	8	fg(x	fg(x	X
ejpam-5274	25	9	)	)	PUNCT
ejpam-5274	25	10	.	.	PUNCT
ejpam-5274	26	1	it	it	PRON
ejpam-5274	26	2	is	be	AUX
ejpam-5274	26	3	easy	easy	ADJ
ejpam-5274	26	4	to	to	PART
ejpam-5274	26	5	verify	verify	VERB
ejpam-5274	26	6	not	not	PART
ejpam-5274	26	7	every	every	DET
ejpam-5274	26	8	polynomial	polynomial	NOUN
ejpam-5274	26	9	is	be	AUX
ejpam-5274	26	10	graphic	graphic	ADJ
ejpam-5274	26	11	.	.	PUNCT
ejpam-5274	27	1	the	the	DET
ejpam-5274	27	2	degree	degree	NOUN
ejpam-5274	27	3	sequence	sequence	NOUN
ejpam-5274	27	4	of	of	ADP
ejpam-5274	27	5	a	a	DET
ejpam-5274	27	6	graph	graph	NOUN
ejpam-5274	27	7	had	have	AUX
ejpam-5274	27	8	been	be	AUX
ejpam-5274	27	9	investigated	investigate	VERB
ejpam-5274	27	10	in	in	ADP
ejpam-5274	27	11	[	[	X
ejpam-5274	27	12	2	2	NUM
ejpam-5274	27	13	]	]	PUNCT
ejpam-5274	27	14	,	,	PUNCT
ejpam-5274	27	15	[	[	X
ejpam-5274	27	16	3	3	NUM
ejpam-5274	27	17	]	]	PUNCT
ejpam-5274	27	18	,	,	PUNCT
ejpam-5274	27	19	[	[	X
ejpam-5274	27	20	4	4	NUM
ejpam-5274	27	21	]	]	PUNCT
ejpam-5274	27	22	,	,	PUNCT
ejpam-5274	27	23	[	[	X
ejpam-5274	27	24	5	5	NUM
ejpam-5274	27	25	]	]	PUNCT
ejpam-5274	27	26	,	,	PUNCT
ejpam-5274	27	27	[	[	X
ejpam-5274	27	28	6	6	NUM
ejpam-5274	27	29	]	]	PUNCT
ejpam-5274	27	30	,	,	PUNCT
ejpam-5274	27	31	∗corresponding	∗corresponde	VERB
ejpam-5274	27	32	author	author	NOUN
ejpam-5274	27	33	.	.	PUNCT
ejpam-5274	28	1	doi	doi	NOUN
ejpam-5274	28	2	:	:	PUNCT
ejpam-5274	28	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5274	https://doi.org/10.29020/nybg.ejpam.v17i3.5274	ADJ
ejpam-5274	28	4	email	email	NOUN
ejpam-5274	28	5	addresses	address	VERB
ejpam-5274	28	6	:	:	PUNCT
ejpam-5274	29	1	jayhan.cruz@g.msuiit.edu.ph	jayhan.cruz@g.msuiit.edu.ph	PROPN
ejpam-5274	29	2	(	(	PUNCT
ejpam-5274	29	3	j.	j.	PROPN
ejpam-5274	29	4	cruz	cruz	PROPN
ejpam-5274	29	5	)	)	PUNCT
ejpam-5274	29	6	,	,	PUNCT
ejpam-5274	29	7	gina.malacas@g.msuiit.edu.ph	gina.malacas@g.msuiit.edu.ph	PROPN
ejpam-5274	29	8	(	(	PUNCT
ejpam-5274	29	9	g.	g.	PROPN
ejpam-5274	29	10	malacas	malacas	PROPN
ejpam-5274	29	11	)	)	PUNCT
ejpam-5274	29	12	,	,	PUNCT
ejpam-5274	29	13	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-5274	29	14	(	(	PUNCT
ejpam-5274	29	15	s.	s.	PROPN
ejpam-5274	29	16	canoy	canoy	PROPN
ejpam-5274	29	17	,	,	PUNCT
ejpam-5274	29	18	jr	jr	PROPN
ejpam-5274	29	19	.	.	PUNCT
ejpam-5274	29	20	)	)	PUNCT
ejpam-5274	29	21	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5274	30	1	1449	1449	NUM
ejpam-5274	31	1	©	©	ADP
ejpam-5274	31	2	2024	2024	NUM
ejpam-5274	31	3	ejpam	ejpam	NOUN
ejpam-5274	31	4	all	all	DET
ejpam-5274	31	5	rights	right	NOUN
ejpam-5274	31	6	reserved	reserve	VERB
ejpam-5274	31	7	.	.	PUNCT
ejpam-5274	32	1	jayhan	jayhan	PROPN
ejpam-5274	32	2	cruz	cruz	PROPN
ejpam-5274	32	3	,	,	PUNCT
ejpam-5274	32	4	g.	g.	PROPN
ejpam-5274	32	5	malacas	malacas	PROPN
ejpam-5274	32	6	,	,	PUNCT
ejpam-5274	32	7	s.	s.	PROPN
ejpam-5274	32	8	canoy	canoy	PROPN
ejpam-5274	32	9	,	,	PUNCT
ejpam-5274	32	10	jr	jr	PROPN
ejpam-5274	32	11	.	.	PROPN
ejpam-5274	32	12	/	/	SYM
ejpam-5274	32	13	eur	eur	PROPN
ejpam-5274	32	14	.	.	PUNCT
ejpam-5274	33	1	j.	j.	PROPN
ejpam-5274	33	2	pure	pure	PROPN
ejpam-5274	33	3	appl	appl	PROPN
ejpam-5274	33	4	.	.	PROPN
ejpam-5274	33	5	math	math	PROPN
ejpam-5274	33	6	,	,	PUNCT
ejpam-5274	33	7	17	17	NUM
ejpam-5274	33	8	(	(	PUNCT
ejpam-5274	33	9	3	3	NUM
ejpam-5274	33	10	)	)	PUNCT
ejpam-5274	33	11	(	(	PUNCT
ejpam-5274	33	12	2024	2024	NUM
ejpam-5274	33	13	)	)	PUNCT
ejpam-5274	33	14	,	,	PUNCT
ejpam-5274	33	15	1449	1449	NUM
ejpam-5274	33	16	-	-	SYM
ejpam-5274	33	17	1462	1462	NUM
ejpam-5274	33	18	1450	1450	NUM
ejpam-5274	34	1	[	[	X
ejpam-5274	34	2	9	9	NUM
ejpam-5274	34	3	]	]	PUNCT
ejpam-5274	34	4	,	,	PUNCT
ejpam-5274	34	5	and	and	CCONJ
ejpam-5274	34	6	[	[	X
ejpam-5274	34	7	12	12	NUM
ejpam-5274	34	8	]	]	PUNCT
ejpam-5274	34	9	)	)	PUNCT
ejpam-5274	34	10	.	.	PUNCT
ejpam-5274	35	1	erdős	erdős	NOUN
ejpam-5274	35	2	and	and	CCONJ
ejpam-5274	35	3	gallai	gallai	NOUN
ejpam-5274	35	4	in	in	ADP
ejpam-5274	35	5	[	[	X
ejpam-5274	35	6	4	4	NUM
ejpam-5274	35	7	]	]	PUNCT
ejpam-5274	35	8	obtained	obtain	VERB
ejpam-5274	35	9	a	a	DET
ejpam-5274	35	10	necessary	necessary	ADJ
ejpam-5274	35	11	and	and	CCONJ
ejpam-5274	35	12	sufficient	sufficient	ADJ
ejpam-5274	35	13	condition	condition	NOUN
ejpam-5274	35	14	for	for	ADP
ejpam-5274	35	15	a	a	DET
ejpam-5274	35	16	given	give	VERB
ejpam-5274	35	17	polynomial	polynomial	NOUN
ejpam-5274	35	18	to	to	PART
ejpam-5274	35	19	be	be	AUX
ejpam-5274	35	20	graphic	graphic	ADJ
ejpam-5274	35	21	.	.	PUNCT
ejpam-5274	36	1	the	the	DET
ejpam-5274	36	2	polynomial	polynomial	ADJ
ejpam-5274	36	3	representations	representation	NOUN
ejpam-5274	36	4	and	and	CCONJ
ejpam-5274	36	5	degree	degree	NOUN
ejpam-5274	36	6	sequences	sequence	NOUN
ejpam-5274	36	7	of	of	ADP
ejpam-5274	36	8	the	the	DET
ejpam-5274	36	9	the	the	DET
ejpam-5274	36	10	join	join	NOUN
ejpam-5274	36	11	,	,	PUNCT
ejpam-5274	36	12	corona	corona	PROPN
ejpam-5274	36	13	,	,	PUNCT
ejpam-5274	36	14	lexicographic	lexicographic	ADJ
ejpam-5274	36	15	product	product	NOUN
ejpam-5274	36	16	,	,	PUNCT
ejpam-5274	36	17	cartesian	cartesian	ADJ
ejpam-5274	36	18	product	product	NOUN
ejpam-5274	36	19	,	,	PUNCT
ejpam-5274	36	20	and	and	CCONJ
ejpam-5274	36	21	tensor	tensor	NOUN
ejpam-5274	36	22	product	product	NOUN
ejpam-5274	36	23	of	of	ADP
ejpam-5274	36	24	two	two	NUM
ejpam-5274	36	25	graphs	graph	NOUN
ejpam-5274	36	26	had	have	AUX
ejpam-5274	36	27	been	be	AUX
ejpam-5274	36	28	obtained	obtain	VERB
ejpam-5274	36	29	by	by	ADP
ejpam-5274	36	30	canoy	canoy	PROPN
ejpam-5274	36	31	et	et	PROPN
ejpam-5274	36	32	al	al	PROPN
ejpam-5274	36	33	.	.	PUNCT
ejpam-5274	37	1	in	in	ADP
ejpam-5274	37	2	[	[	X
ejpam-5274	37	3	8	8	NUM
ejpam-5274	37	4	]	]	PUNCT
ejpam-5274	37	5	.	.	PUNCT
ejpam-5274	38	1	these	these	DET
ejpam-5274	38	2	graphs	graph	NOUN
ejpam-5274	38	3	were	be	AUX
ejpam-5274	38	4	also	also	ADV
ejpam-5274	38	5	investigated	investigate	VERB
ejpam-5274	38	6	for	for	ADP
ejpam-5274	38	7	other	other	ADJ
ejpam-5274	38	8	graph	graph	NOUN
ejpam-5274	38	9	parameters	parameter	NOUN
ejpam-5274	38	10	in	in	ADP
ejpam-5274	38	11	previous	previous	ADJ
ejpam-5274	38	12	studies	study	NOUN
ejpam-5274	38	13	(	(	PUNCT
ejpam-5274	38	14	see	see	VERB
ejpam-5274	38	15	[	[	X
ejpam-5274	38	16	7	7	NUM
ejpam-5274	38	17	]	]	PUNCT
ejpam-5274	38	18	,	,	PUNCT
ejpam-5274	39	1	[	[	X
ejpam-5274	39	2	10	10	NUM
ejpam-5274	39	3	]	]	PUNCT
ejpam-5274	39	4	,	,	PUNCT
ejpam-5274	39	5	and	and	CCONJ
ejpam-5274	39	6	[	[	X
ejpam-5274	39	7	11	11	NUM
ejpam-5274	39	8	]	]	NUM
ejpam-5274	39	9	)	)	PUNCT
ejpam-5274	39	10	.	.	PUNCT
ejpam-5274	40	1	in	in	ADP
ejpam-5274	40	2	this	this	DET
ejpam-5274	40	3	present	present	ADJ
ejpam-5274	40	4	study	study	NOUN
ejpam-5274	40	5	,	,	PUNCT
ejpam-5274	40	6	the	the	DET
ejpam-5274	40	7	authors	author	NOUN
ejpam-5274	40	8	endeavored	endeavor	VERB
ejpam-5274	40	9	to	to	PART
ejpam-5274	40	10	determine	determine	VERB
ejpam-5274	40	11	expressions	expression	NOUN
ejpam-5274	40	12	for	for	ADP
ejpam-5274	40	13	the	the	DET
ejpam-5274	40	14	polynomial	polynomial	ADJ
ejpam-5274	40	15	representations	representation	NOUN
ejpam-5274	40	16	of	of	ADP
ejpam-5274	40	17	the	the	DET
ejpam-5274	40	18	complement	complement	NOUN
ejpam-5274	40	19	and	and	CCONJ
ejpam-5274	40	20	line	line	NOUN
ejpam-5274	40	21	graph	graph	NOUN
ejpam-5274	40	22	of	of	ADP
ejpam-5274	40	23	a	a	DET
ejpam-5274	40	24	graph	graph	NOUN
ejpam-5274	40	25	,	,	PUNCT
ejpam-5274	40	26	shadow	shadow	NOUN
ejpam-5274	40	27	graph	graph	NOUN
ejpam-5274	40	28	,	,	PUNCT
ejpam-5274	40	29	complementary	complementary	ADJ
ejpam-5274	40	30	prism	prism	NOUN
ejpam-5274	40	31	,	,	PUNCT
ejpam-5274	40	32	edge	edge	NOUN
ejpam-5274	40	33	corona	corona	NOUN
ejpam-5274	40	34	,	,	PUNCT
ejpam-5274	40	35	strong	strong	ADJ
ejpam-5274	40	36	product	product	NOUN
ejpam-5274	40	37	,	,	PUNCT
ejpam-5274	40	38	symmetric	symmetric	ADJ
ejpam-5274	40	39	difference	difference	NOUN
ejpam-5274	40	40	,	,	PUNCT
ejpam-5274	40	41	and	and	CCONJ
ejpam-5274	40	42	disjunction	disjunction	NOUN
ejpam-5274	40	43	of	of	ADP
ejpam-5274	40	44	two	two	NUM
ejpam-5274	40	45	graphs	graph	NOUN
ejpam-5274	40	46	.	.	PUNCT
ejpam-5274	41	1	2	2	X
ejpam-5274	41	2	.	.	NOUN
ejpam-5274	41	3	terminologies	terminology	NOUN
ejpam-5274	41	4	and	and	CCONJ
ejpam-5274	41	5	notations	notation	NOUN
ejpam-5274	41	6	let	let	VERB
ejpam-5274	41	7	g	g	NOUN
ejpam-5274	41	8	=	=	SYM
ejpam-5274	41	9	(	(	PUNCT
ejpam-5274	41	10	v	v	NOUN
ejpam-5274	41	11	(	(	PUNCT
ejpam-5274	41	12	g	g	NOUN
ejpam-5274	41	13	)	)	PUNCT
ejpam-5274	41	14	,	,	PUNCT
ejpam-5274	41	15	e(g	e(g	PROPN
ejpam-5274	41	16	)	)	PUNCT
ejpam-5274	41	17	)	)	PUNCT
ejpam-5274	41	18	be	be	AUX
ejpam-5274	41	19	a	a	DET
ejpam-5274	41	20	simple	simple	ADJ
ejpam-5274	41	21	undirected	undirected	ADJ
ejpam-5274	41	22	graph	graph	NOUN
ejpam-5274	41	23	.	.	PUNCT
ejpam-5274	42	1	the	the	DET
ejpam-5274	42	2	distance	distance	NOUN
ejpam-5274	42	3	between	between	ADP
ejpam-5274	42	4	two	two	NUM
ejpam-5274	42	5	vertices	vertex	NOUN
ejpam-5274	42	6	u	u	NOUN
ejpam-5274	42	7	and	and	CCONJ
ejpam-5274	42	8	v	v	NOUN
ejpam-5274	42	9	of	of	ADP
ejpam-5274	42	10	g	g	NOUN
ejpam-5274	42	11	,	,	PUNCT
ejpam-5274	42	12	denoted	denote	VERB
ejpam-5274	42	13	by	by	ADP
ejpam-5274	42	14	dg(u	dg(u	NOUN
ejpam-5274	42	15	,	,	PUNCT
ejpam-5274	42	16	v	v	NOUN
ejpam-5274	42	17	)	)	PUNCT
ejpam-5274	42	18	,	,	PUNCT
ejpam-5274	42	19	is	be	AUX
ejpam-5274	42	20	equal	equal	ADJ
ejpam-5274	42	21	to	to	ADP
ejpam-5274	42	22	the	the	DET
ejpam-5274	42	23	length	length	NOUN
ejpam-5274	42	24	of	of	ADP
ejpam-5274	42	25	a	a	DET
ejpam-5274	42	26	shortest	short	ADJ
ejpam-5274	42	27	path	path	NOUN
ejpam-5274	42	28	connecting	connect	VERB
ejpam-5274	42	29	u	u	NOUN
ejpam-5274	42	30	and	and	CCONJ
ejpam-5274	42	31	v.	v.	ADP
ejpam-5274	42	32	any	any	DET
ejpam-5274	42	33	path	path	NOUN
ejpam-5274	42	34	connecting	connect	VERB
ejpam-5274	42	35	u	u	NOUN
ejpam-5274	42	36	and	and	CCONJ
ejpam-5274	42	37	v	v	NOUN
ejpam-5274	42	38	of	of	ADP
ejpam-5274	42	39	length	length	NOUN
ejpam-5274	42	40	dg(u	dg(u	ADJ
ejpam-5274	42	41	,	,	PUNCT
ejpam-5274	42	42	v	v	NOUN
ejpam-5274	42	43	)	)	PUNCT
ejpam-5274	42	44	is	be	AUX
ejpam-5274	42	45	called	call	VERB
ejpam-5274	42	46	a	a	DET
ejpam-5274	42	47	u	u	NOUN
ejpam-5274	42	48	-	-	NOUN
ejpam-5274	42	49	v	v	ADJ
ejpam-5274	42	50	geodesic	geodesic	NOUN
ejpam-5274	42	51	.	.	PUNCT
ejpam-5274	43	1	the	the	DET
ejpam-5274	43	2	open	open	ADJ
ejpam-5274	43	3	neighborhood	neighborhood	NOUN
ejpam-5274	43	4	of	of	ADP
ejpam-5274	43	5	a	a	DET
ejpam-5274	43	6	vertex	vertex	NOUN
ejpam-5274	43	7	v	v	NOUN
ejpam-5274	43	8	of	of	ADP
ejpam-5274	43	9	g	g	PROPN
ejpam-5274	43	10	is	be	AUX
ejpam-5274	43	11	the	the	DET
ejpam-5274	43	12	set	set	NOUN
ejpam-5274	43	13	ng(v	ng(v	PUNCT
ejpam-5274	43	14	)	)	PUNCT
ejpam-5274	43	15	=	=	SYM
ejpam-5274	44	1	{	{	PUNCT
ejpam-5274	44	2	u	u	NOUN
ejpam-5274	44	3	∈	∈	PROPN
ejpam-5274	44	4	v	v	NOUN
ejpam-5274	44	5	(	(	PUNCT
ejpam-5274	44	6	g	g	NOUN
ejpam-5274	44	7	)	)	PUNCT
ejpam-5274	44	8	:	:	PUNCT
ejpam-5274	44	9	uv	uv	PROPN
ejpam-5274	44	10	∈	∈	PROPN
ejpam-5274	44	11	e(g	e(g	PROPN
ejpam-5274	44	12	)	)	PUNCT
ejpam-5274	44	13	}	}	PUNCT
ejpam-5274	44	14	and	and	CCONJ
ejpam-5274	44	15	its	its	PRON
ejpam-5274	44	16	closed	closed	ADJ
ejpam-5274	44	17	neighborhood	neighborhood	NOUN
ejpam-5274	44	18	is	be	AUX
ejpam-5274	44	19	the	the	DET
ejpam-5274	44	20	set	set	NOUN
ejpam-5274	44	21	ng[v	ng[v	NOUN
ejpam-5274	44	22	]	]	X
ejpam-5274	44	23	=	=	SYM
ejpam-5274	44	24	ng(v	ng(v	X
ejpam-5274	44	25	)	)	PUNCT
ejpam-5274	44	26	∪	∪	ADP
ejpam-5274	44	27	{	{	PUNCT
ejpam-5274	44	28	v	v	NOUN
ejpam-5274	44	29	}	}	PUNCT
ejpam-5274	44	30	.	.	PUNCT
ejpam-5274	45	1	the	the	DET
ejpam-5274	45	2	open	open	ADJ
ejpam-5274	45	3	neighborhood	neighborhood	NOUN
ejpam-5274	45	4	of	of	ADP
ejpam-5274	45	5	a	a	DET
ejpam-5274	45	6	subset	subset	NOUN
ejpam-5274	45	7	s	s	NOUN
ejpam-5274	45	8	of	of	ADP
ejpam-5274	45	9	v	v	NOUN
ejpam-5274	45	10	(	(	PUNCT
ejpam-5274	45	11	g	g	NOUN
ejpam-5274	45	12	)	)	PUNCT
ejpam-5274	45	13	is	be	AUX
ejpam-5274	45	14	the	the	DET
ejpam-5274	45	15	set	set	NOUN
ejpam-5274	45	16	ng(s	ng(s	NOUN
ejpam-5274	45	17	)	)	PUNCT
ejpam-5274	45	18	=	=	SYM
ejpam-5274	45	19	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-5274	45	20	)	)	PUNCT
ejpam-5274	45	21	and	and	CCONJ
ejpam-5274	45	22	its	its	PRON
ejpam-5274	45	23	closed	closed	ADJ
ejpam-5274	45	24	neighborhood	neighborhood	NOUN
ejpam-5274	45	25	is	be	AUX
ejpam-5274	45	26	the	the	DET
ejpam-5274	45	27	set	set	VERB
ejpam-5274	45	28	ng[s	ng[	NOUN
ejpam-5274	45	29	]	]	PUNCT
ejpam-5274	45	30	=	=	SYM
ejpam-5274	45	31	ng(s	ng(s	X
ejpam-5274	45	32	)	)	PUNCT
ejpam-5274	45	33	∪	∪	ADP
ejpam-5274	45	34	s.	s.	PROPN
ejpam-5274	45	35	the	the	DET
ejpam-5274	45	36	degree	degree	NOUN
ejpam-5274	45	37	of	of	ADP
ejpam-5274	45	38	v	v	NOUN
ejpam-5274	45	39	,	,	PUNCT
ejpam-5274	45	40	denoted	denote	VERB
ejpam-5274	45	41	by	by	ADP
ejpam-5274	45	42	degg(v	degg(v	PROPN
ejpam-5274	45	43	)	)	PUNCT
ejpam-5274	45	44	,	,	PUNCT
ejpam-5274	45	45	is	be	AUX
ejpam-5274	45	46	equal	equal	ADJ
ejpam-5274	45	47	to	to	ADP
ejpam-5274	45	48	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	45	49	.	.	PUNCT
ejpam-5274	46	1	the	the	DET
ejpam-5274	46	2	maximum	maximum	ADJ
ejpam-5274	46	3	degree	degree	NOUN
ejpam-5274	46	4	of	of	ADP
ejpam-5274	46	5	g	g	NOUN
ejpam-5274	46	6	,	,	PUNCT
ejpam-5274	46	7	denoted	denote	VERB
ejpam-5274	46	8	by	by	ADP
ejpam-5274	46	9	∆(g	∆(g	PROPN
ejpam-5274	46	10	)	)	PUNCT
ejpam-5274	46	11	,	,	PUNCT
ejpam-5274	46	12	is	be	AUX
ejpam-5274	46	13	equal	equal	ADJ
ejpam-5274	46	14	to	to	ADP
ejpam-5274	46	15	max{degg(v	max{degg(v	NOUN
ejpam-5274	46	16	)	)	PUNCT
ejpam-5274	46	17	:	:	PUNCT
ejpam-5274	46	18	v	v	X
ejpam-5274	46	19	∈	∈	PROPN
ejpam-5274	46	20	v	v	NOUN
ejpam-5274	46	21	(	(	PUNCT
ejpam-5274	46	22	g	g	NOUN
ejpam-5274	46	23	)	)	PUNCT
ejpam-5274	46	24	}	}	PUNCT
ejpam-5274	46	25	.	.	PUNCT
ejpam-5274	47	1	suppose	suppose	VERB
ejpam-5274	47	2	∆(g	∆(g	NOUN
ejpam-5274	47	3	)	)	PUNCT
ejpam-5274	47	4	=	=	VERB
ejpam-5274	47	5	n.	n.	NOUN
ejpam-5274	47	6	for	for	ADP
ejpam-5274	47	7	each	each	DET
ejpam-5274	47	8	i	i	NOUN
ejpam-5274	47	9	=	=	NOUN
ejpam-5274	47	10	1	1	NUM
ejpam-5274	47	11	,	,	PUNCT
ejpam-5274	47	12	2	2	NUM
ejpam-5274	47	13	,	,	PUNCT
ejpam-5274	47	14	·	·	PUNCT
ejpam-5274	47	15	·	·	PUNCT
ejpam-5274	47	16	·	·	PUNCT
ejpam-5274	47	17	n	n	CCONJ
ejpam-5274	47	18	,	,	PUNCT
ejpam-5274	47	19	let	let	VERB
ejpam-5274	47	20	ai	ai	AUX
ejpam-5274	47	21	be	be	AUX
ejpam-5274	47	22	the	the	DET
ejpam-5274	47	23	number	number	NOUN
ejpam-5274	47	24	of	of	ADP
ejpam-5274	47	25	vertices	vertex	NOUN
ejpam-5274	47	26	of	of	ADP
ejpam-5274	47	27	g	g	NOUN
ejpam-5274	47	28	with	with	ADP
ejpam-5274	47	29	degree	degree	NOUN
ejpam-5274	47	30	i	i	PRON
ejpam-5274	47	31	≥	≥	NOUN
ejpam-5274	47	32	0	0	NUM
ejpam-5274	47	33	.	.	PUNCT
ejpam-5274	48	1	then	then	ADV
ejpam-5274	48	2	the	the	DET
ejpam-5274	48	3	polynomial	polynomial	ADJ
ejpam-5274	48	4	fg(x	fg(x	NUM
ejpam-5274	48	5	)	)	PUNCT
ejpam-5274	49	1	=	=	SYM
ejpam-5274	50	1	n∑	n∑	PROPN
ejpam-5274	50	2	i=1	i=1	PROPN
ejpam-5274	51	1	aix	aix	PROPN
ejpam-5274	51	2	i	i	PRON
ejpam-5274	51	3	is	be	AUX
ejpam-5274	51	4	called	call	VERB
ejpam-5274	51	5	the	the	DET
ejpam-5274	51	6	polynomial	polynomial	ADJ
ejpam-5274	51	7	representation	representation	NOUN
ejpam-5274	51	8	of	of	ADP
ejpam-5274	51	9	g.	g.	PROPN
ejpam-5274	51	10	equivalently	equivalently	ADV
ejpam-5274	51	11	,	,	PUNCT
ejpam-5274	51	12	fg(x	fg(x	NUM
ejpam-5274	51	13	)	)	PUNCT
ejpam-5274	51	14	=	=	SYM
ejpam-5274	52	1	∑	∑	PUNCT
ejpam-5274	52	2	v∈v	v∈v	NOUN
ejpam-5274	52	3	(	(	PUNCT
ejpam-5274	52	4	g	g	NOUN
ejpam-5274	52	5	)	)	PUNCT
ejpam-5274	52	6	x|ng(v)|	x|ng(v)|	PROPN
ejpam-5274	52	7	.	.	PUNCT
ejpam-5274	53	1	let	let	VERB
ejpam-5274	53	2	g	g	NOUN
ejpam-5274	53	3	and	and	CCONJ
ejpam-5274	53	4	h	h	PROPN
ejpam-5274	53	5	be	be	AUX
ejpam-5274	53	6	graphs	graph	NOUN
ejpam-5274	53	7	.	.	PUNCT
ejpam-5274	54	1	the	the	DET
ejpam-5274	54	2	complement	complement	NOUN
ejpam-5274	54	3	of	of	ADP
ejpam-5274	54	4	g	g	NOUN
ejpam-5274	54	5	,	,	PUNCT
ejpam-5274	54	6	denoted	denote	VERB
ejpam-5274	54	7	by	by	ADP
ejpam-5274	54	8	g	g	PROPN
ejpam-5274	54	9	is	be	AUX
ejpam-5274	54	10	the	the	DET
ejpam-5274	54	11	graph	graph	NOUN
ejpam-5274	54	12	with	with	ADP
ejpam-5274	54	13	v	v	NOUN
ejpam-5274	54	14	(	(	PUNCT
ejpam-5274	54	15	g	g	NOUN
ejpam-5274	54	16	)	)	PUNCT
ejpam-5274	54	17	=	=	NOUN
ejpam-5274	54	18	v	v	X
ejpam-5274	54	19	(	(	PUNCT
ejpam-5274	54	20	g	g	NOUN
ejpam-5274	54	21	)	)	PUNCT
ejpam-5274	54	22	and	and	CCONJ
ejpam-5274	54	23	vw	vw	PROPN
ejpam-5274	54	24	∈	∈	PROPN
ejpam-5274	54	25	e(g	e(g	PROPN
ejpam-5274	54	26	)	)	PUNCT
ejpam-5274	55	1	if	if	SCONJ
ejpam-5274	55	2	and	and	CCONJ
ejpam-5274	55	3	only	only	ADV
ejpam-5274	55	4	if	if	SCONJ
ejpam-5274	55	5	vw	vw	PROPN
ejpam-5274	55	6	/∈	/∈	PUNCT
ejpam-5274	55	7	e(g	e(g	PROPN
ejpam-5274	55	8	)	)	PUNCT
ejpam-5274	55	9	.	.	PUNCT
ejpam-5274	56	1	the	the	DET
ejpam-5274	56	2	line	line	NOUN
ejpam-5274	56	3	graph	graph	NOUN
ejpam-5274	56	4	l(g	l(g	NOUN
ejpam-5274	56	5	)	)	PUNCT
ejpam-5274	56	6	of	of	ADP
ejpam-5274	56	7	g	g	PROPN
ejpam-5274	56	8	is	be	AUX
ejpam-5274	56	9	the	the	DET
ejpam-5274	56	10	graph	graph	NOUN
ejpam-5274	56	11	with	with	ADP
ejpam-5274	56	12	v	v	NOUN
ejpam-5274	56	13	(	(	PUNCT
ejpam-5274	56	14	l(g	l(g	NOUN
ejpam-5274	56	15	)	)	PUNCT
ejpam-5274	56	16	)	)	PUNCT
ejpam-5274	57	1	=	=	SYM
ejpam-5274	57	2	e(g	e(g	PROPN
ejpam-5274	57	3	)	)	PUNCT
ejpam-5274	57	4	and	and	CCONJ
ejpam-5274	57	5	e1e2	e1e2	NOUN
ejpam-5274	57	6	∈	∈	NOUN
ejpam-5274	57	7	e((l(g	e((l(g	NOUN
ejpam-5274	57	8	)	)	PUNCT
ejpam-5274	57	9	)	)	PUNCT
ejpam-5274	58	1	if	if	SCONJ
ejpam-5274	58	2	and	and	CCONJ
ejpam-5274	58	3	only	only	ADV
ejpam-5274	58	4	if	if	SCONJ
ejpam-5274	58	5	e1	e1	PROPN
ejpam-5274	58	6	and	and	CCONJ
ejpam-5274	58	7	e2	e2	PROPN
ejpam-5274	58	8	have	have	AUX
ejpam-5274	58	9	a	a	DET
ejpam-5274	58	10	common	common	ADJ
ejpam-5274	58	11	vertex	vertex	NOUN
ejpam-5274	58	12	in	in	ADP
ejpam-5274	58	13	g.	g.	PROPN
ejpam-5274	58	14	the	the	DET
ejpam-5274	58	15	shadow	shadow	NOUN
ejpam-5274	58	16	graph	graph	VERB
ejpam-5274	58	17	d2(g	d2(g	PROPN
ejpam-5274	58	18	)	)	PUNCT
ejpam-5274	58	19	of	of	ADP
ejpam-5274	58	20	g	g	PROPN
ejpam-5274	58	21	is	be	AUX
ejpam-5274	58	22	the	the	DET
ejpam-5274	58	23	graph	graph	NOUN
ejpam-5274	58	24	obtained	obtain	VERB
ejpam-5274	58	25	by	by	ADP
ejpam-5274	58	26	taking	take	VERB
ejpam-5274	58	27	two	two	NUM
ejpam-5274	58	28	copies	copy	NOUN
ejpam-5274	58	29	of	of	ADP
ejpam-5274	58	30	g	g	NOUN
ejpam-5274	58	31	,	,	PUNCT
ejpam-5274	58	32	say	say	VERB
ejpam-5274	58	33	g1	g1	PROPN
ejpam-5274	58	34	and	and	CCONJ
ejpam-5274	58	35	g2	g2	PROPN
ejpam-5274	58	36	,	,	PUNCT
ejpam-5274	58	37	and	and	CCONJ
ejpam-5274	58	38	joining	join	VERB
ejpam-5274	58	39	each	each	DET
ejpam-5274	58	40	vertex	vertex	NOUN
ejpam-5274	58	41	u	u	NOUN
ejpam-5274	58	42	∈	∈	PROPN
ejpam-5274	58	43	v	v	NOUN
ejpam-5274	58	44	(	(	PUNCT
ejpam-5274	58	45	g1	g1	PROPN
ejpam-5274	58	46	)	)	PUNCT
ejpam-5274	58	47	to	to	ADP
ejpam-5274	58	48	the	the	DET
ejpam-5274	58	49	neighbors	neighbor	NOUN
ejpam-5274	58	50	of	of	ADP
ejpam-5274	58	51	the	the	DET
ejpam-5274	58	52	corresponding	corresponding	ADJ
ejpam-5274	58	53	vertex	vertex	NOUN
ejpam-5274	58	54	u′	u′	PROPN
ejpam-5274	58	55	∈	∈	PROPN
ejpam-5274	58	56	v	v	NOUN
ejpam-5274	58	57	(	(	PUNCT
ejpam-5274	58	58	g2	g2	PROPN
ejpam-5274	58	59	)	)	PUNCT
ejpam-5274	58	60	.	.	PUNCT
ejpam-5274	59	1	the	the	DET
ejpam-5274	59	2	complementary	complementary	ADJ
ejpam-5274	59	3	prism	prism	NOUN
ejpam-5274	59	4	gg	gg	NOUN
ejpam-5274	59	5	is	be	AUX
ejpam-5274	59	6	the	the	DET
ejpam-5274	59	7	graph	graph	NOUN
ejpam-5274	59	8	formed	form	VERB
ejpam-5274	59	9	from	from	ADP
ejpam-5274	59	10	the	the	DET
ejpam-5274	59	11	disjoint	disjoint	PROPN
ejpam-5274	59	12	union	union	NOUN
ejpam-5274	59	13	of	of	ADP
ejpam-5274	59	14	g	g	PROPN
ejpam-5274	59	15	and	and	CCONJ
ejpam-5274	59	16	its	its	PRON
ejpam-5274	59	17	complement	complement	NOUN
ejpam-5274	59	18	g	g	NOUN
ejpam-5274	59	19	by	by	ADP
ejpam-5274	59	20	adding	add	VERB
ejpam-5274	59	21	a	a	DET
ejpam-5274	59	22	perfect	perfect	ADJ
ejpam-5274	59	23	matching	matching	NOUN
ejpam-5274	59	24	between	between	ADP
ejpam-5274	59	25	corresponding	corresponding	ADJ
ejpam-5274	59	26	vertices	vertex	NOUN
ejpam-5274	59	27	of	of	ADP
ejpam-5274	59	28	g	g	PROPN
ejpam-5274	59	29	and	and	CCONJ
ejpam-5274	59	30	g.	g.	NOUN
ejpam-5274	59	31	for	for	ADP
ejpam-5274	59	32	each	each	DET
ejpam-5274	59	33	v	v	NUM
ejpam-5274	59	34	∈	∈	PROPN
ejpam-5274	59	35	v	v	NOUN
ejpam-5274	59	36	(	(	PUNCT
ejpam-5274	59	37	g	g	NOUN
ejpam-5274	59	38	)	)	PUNCT
ejpam-5274	59	39	,	,	PUNCT
ejpam-5274	59	40	let	let	VERB
ejpam-5274	59	41	v	v	PART
ejpam-5274	59	42	denote	denote	VERB
ejpam-5274	59	43	the	the	DET
ejpam-5274	59	44	vertex	vertex	NOUN
ejpam-5274	59	45	in	in	ADP
ejpam-5274	59	46	g	g	NOUN
ejpam-5274	59	47	corresponding	correspond	VERB
ejpam-5274	59	48	to	to	ADP
ejpam-5274	59	49	v.	v.	PROPN
ejpam-5274	59	50	in	in	ADP
ejpam-5274	59	51	simple	simple	ADJ
ejpam-5274	59	52	terms	term	NOUN
ejpam-5274	59	53	,	,	PUNCT
ejpam-5274	59	54	the	the	DET
ejpam-5274	59	55	graph	graph	NOUN
ejpam-5274	59	56	gg	gg	NOUN
ejpam-5274	59	57	is	be	AUX
ejpam-5274	59	58	formed	form	VERB
ejpam-5274	59	59	from	from	ADP
ejpam-5274	59	60	g	g	PROPN
ejpam-5274	59	61	∪	∪	ADP
ejpam-5274	59	62	g	g	NOUN
ejpam-5274	59	63	by	by	ADP
ejpam-5274	59	64	adding	add	VERB
ejpam-5274	59	65	the	the	DET
ejpam-5274	59	66	edge	edge	NOUN
ejpam-5274	59	67	vv	vv	NOUN
ejpam-5274	59	68	for	for	ADP
ejpam-5274	59	69	every	every	DET
ejpam-5274	59	70	vertex	vertex	NOUN
ejpam-5274	59	71	v	v	ADP
ejpam-5274	59	72	∈	∈	NOUN
ejpam-5274	59	73	v	v	NOUN
ejpam-5274	59	74	(	(	PUNCT
ejpam-5274	59	75	g	g	NOUN
ejpam-5274	59	76	)	)	PUNCT
ejpam-5274	59	77	.	.	PUNCT
ejpam-5274	60	1	the	the	DET
ejpam-5274	60	2	edge	edge	NOUN
ejpam-5274	60	3	corona	corona	PROPN
ejpam-5274	60	4	g	g	PROPN
ejpam-5274	60	5	⋄	⋄	PROPN
ejpam-5274	60	6	h	h	NOUN
ejpam-5274	60	7	of	of	ADP
ejpam-5274	60	8	graphs	graph	NOUN
ejpam-5274	60	9	g	g	NOUN
ejpam-5274	60	10	and	and	CCONJ
ejpam-5274	60	11	h	h	NOUN
ejpam-5274	60	12	is	be	AUX
ejpam-5274	60	13	the	the	DET
ejpam-5274	60	14	graph	graph	NOUN
ejpam-5274	60	15	obtained	obtain	VERB
ejpam-5274	60	16	by	by	ADP
ejpam-5274	60	17	taking	take	VERB
ejpam-5274	60	18	one	one	NUM
ejpam-5274	60	19	copy	copy	NOUN
ejpam-5274	60	20	of	of	ADP
ejpam-5274	60	21	g	g	PROPN
ejpam-5274	60	22	and	and	CCONJ
ejpam-5274	60	23	|e(g)|	|e(g)|	ADJ
ejpam-5274	60	24	copies	copy	NOUN
ejpam-5274	60	25	of	of	ADP
ejpam-5274	60	26	h	h	NOUN
ejpam-5274	60	27	and	and	CCONJ
ejpam-5274	60	28	joining	join	VERB
ejpam-5274	60	29	each	each	PRON
ejpam-5274	60	30	of	of	ADP
ejpam-5274	60	31	the	the	DET
ejpam-5274	60	32	end	end	NOUN
ejpam-5274	60	33	vertices	vertice	VERB
ejpam-5274	60	34	u	u	NOUN
ejpam-5274	60	35	and	and	CCONJ
ejpam-5274	60	36	v	v	NOUN
ejpam-5274	60	37	of	of	ADP
ejpam-5274	60	38	every	every	DET
ejpam-5274	60	39	edge	edge	NOUN
ejpam-5274	60	40	uv	uv	NOUN
ejpam-5274	60	41	in	in	ADP
ejpam-5274	60	42	g	g	NOUN
ejpam-5274	60	43	to	to	ADP
ejpam-5274	60	44	every	every	DET
ejpam-5274	60	45	vertex	vertex	NOUN
ejpam-5274	60	46	of	of	ADP
ejpam-5274	60	47	the	the	DET
ejpam-5274	60	48	copy	copy	NOUN
ejpam-5274	60	49	huv	huv	PROPN
ejpam-5274	60	50	of	of	ADP
ejpam-5274	60	51	h	h	PROPN
ejpam-5274	60	52	(	(	PUNCT
ejpam-5274	60	53	that	that	PRON
ejpam-5274	60	54	is	be	AUX
ejpam-5274	60	55	forming	form	VERB
ejpam-5274	60	56	the	the	DET
ejpam-5274	60	57	join	join	NOUN
ejpam-5274	60	58	⟨{u	⟨{u	PROPN
ejpam-5274	60	59	,	,	PUNCT
ejpam-5274	60	60	v}⟩+huv	v}⟩+huv	NOUN
ejpam-5274	60	61	for	for	ADP
ejpam-5274	60	62	each	each	DET
ejpam-5274	60	63	uv	uv	PROPN
ejpam-5274	60	64	∈	∈	PROPN
ejpam-5274	60	65	e(g	e(g	PROPN
ejpam-5274	60	66	)	)	PUNCT
ejpam-5274	60	67	)	)	PUNCT
ejpam-5274	60	68	.	.	PUNCT
ejpam-5274	61	1	the	the	DET
ejpam-5274	61	2	strong	strong	ADJ
ejpam-5274	61	3	product	product	NOUN
ejpam-5274	61	4	g⊠h	g⊠h	VERB
ejpam-5274	61	5	of	of	ADP
ejpam-5274	61	6	graphs	graph	NOUN
ejpam-5274	61	7	g	g	NOUN
ejpam-5274	62	1	and	and	CCONJ
ejpam-5274	62	2	h	h	NOUN
ejpam-5274	62	3	is	be	AUX
ejpam-5274	62	4	the	the	DET
ejpam-5274	62	5	graph	graph	NOUN
ejpam-5274	62	6	with	with	ADP
ejpam-5274	62	7	vertex	vertex	NOUN
ejpam-5274	62	8	set	set	VERB
ejpam-5274	62	9	v	v	NOUN
ejpam-5274	62	10	(	(	PUNCT
ejpam-5274	62	11	g)×	g)×	NOUN
ejpam-5274	62	12	v	v	NOUN
ejpam-5274	62	13	(	(	PUNCT
ejpam-5274	62	14	h	h	NOUN
ejpam-5274	62	15	)	)	PUNCT
ejpam-5274	62	16	and	and	CCONJ
ejpam-5274	62	17	(	(	PUNCT
ejpam-5274	62	18	u	u	NOUN
ejpam-5274	62	19	,	,	PUNCT
ejpam-5274	62	20	v	v	NOUN
ejpam-5274	62	21	)	)	PUNCT
ejpam-5274	62	22	is	be	AUX
ejpam-5274	62	23	adjacent	adjacent	ADJ
ejpam-5274	62	24	with	with	ADP
ejpam-5274	62	25	(	(	PUNCT
ejpam-5274	62	26	u′	u′	PROPN
ejpam-5274	62	27	,	,	PUNCT
ejpam-5274	62	28	v′	v′	PROPN
ejpam-5274	62	29	)	)	PUNCT
ejpam-5274	63	1	whenever	whenever	SCONJ
ejpam-5274	63	2	[	[	X
ejpam-5274	63	3	uu′	uu′	PROPN
ejpam-5274	63	4	∈	∈	PROPN
ejpam-5274	63	5	e(g	e(g	PROPN
ejpam-5274	63	6	)	)	PUNCT
ejpam-5274	63	7	and	and	CCONJ
ejpam-5274	63	8	v	v	NOUN
ejpam-5274	63	9	=	=	SYM
ejpam-5274	63	10	v′	v′	NOUN
ejpam-5274	63	11	]	]	PUNCT
ejpam-5274	63	12	or	or	CCONJ
ejpam-5274	63	13	[	[	X
ejpam-5274	63	14	vv′	vv′	NOUN
ejpam-5274	63	15	∈	∈	PROPN
ejpam-5274	63	16	e(h	e(h	PROPN
ejpam-5274	63	17	)	)	PUNCT
ejpam-5274	63	18	and	and	CCONJ
ejpam-5274	63	19	u	u	X
ejpam-5274	63	20	=	=	SYM
ejpam-5274	63	21	u′	u′	PROPN
ejpam-5274	63	22	]	]	PUNCT
ejpam-5274	63	23	or	or	CCONJ
ejpam-5274	63	24	[	[	X
ejpam-5274	63	25	uu′	uu′	PROPN
ejpam-5274	63	26	∈	∈	PROPN
ejpam-5274	63	27	e(g	e(g	PROPN
ejpam-5274	63	28	)	)	PUNCT
ejpam-5274	63	29	and	and	CCONJ
ejpam-5274	63	30	vv′	vv′	PROPN
ejpam-5274	63	31	∈	∈	PROPN
ejpam-5274	63	32	e(h	e(h	PROPN
ejpam-5274	63	33	)	)	PUNCT
ejpam-5274	63	34	]	]	PUNCT
ejpam-5274	63	35	.	.	PUNCT
ejpam-5274	64	1	the	the	DET
ejpam-5274	64	2	symmetric	symmetric	ADJ
ejpam-5274	64	3	difference	difference	NOUN
ejpam-5274	64	4	g⊕h	g⊕h	NOUN
ejpam-5274	64	5	of	of	ADP
ejpam-5274	64	6	graphs	graph	NOUN
ejpam-5274	64	7	g	g	PROPN
ejpam-5274	64	8	and	and	CCONJ
ejpam-5274	64	9	h	h	NOUN
ejpam-5274	64	10	is	be	AUX
ejpam-5274	64	11	the	the	DET
ejpam-5274	64	12	graph	graph	NOUN
ejpam-5274	64	13	with	with	ADP
ejpam-5274	64	14	vertex	vertex	NOUN
ejpam-5274	64	15	set	set	VERB
ejpam-5274	64	16	v	v	NOUN
ejpam-5274	64	17	(	(	PUNCT
ejpam-5274	64	18	g)×v	g)×v	PROPN
ejpam-5274	64	19	(	(	PUNCT
ejpam-5274	64	20	h	h	NOUN
ejpam-5274	64	21	)	)	PUNCT
ejpam-5274	64	22	and	and	CCONJ
ejpam-5274	64	23	(	(	PUNCT
ejpam-5274	64	24	u	u	NOUN
ejpam-5274	64	25	,	,	PUNCT
ejpam-5274	64	26	v	v	NOUN
ejpam-5274	64	27	)	)	PUNCT
ejpam-5274	64	28	is	be	AUX
ejpam-5274	64	29	adjacent	adjacent	ADJ
ejpam-5274	64	30	with	with	ADP
ejpam-5274	64	31	(	(	PUNCT
ejpam-5274	64	32	u′	u′	PROPN
ejpam-5274	64	33	,	,	PUNCT
ejpam-5274	64	34	v′	v′	PROPN
ejpam-5274	64	35	)	)	PUNCT
ejpam-5274	65	1	whenever	whenever	SCONJ
ejpam-5274	65	2	[	[	X
ejpam-5274	65	3	uu′	uu′	PROPN
ejpam-5274	65	4	∈	∈	PROPN
ejpam-5274	65	5	e(g	e(g	PROPN
ejpam-5274	65	6	)	)	PUNCT
ejpam-5274	65	7	]	]	PUNCT
ejpam-5274	65	8	or	or	CCONJ
ejpam-5274	65	9	[	[	X
ejpam-5274	65	10	vv′	vv′	PROPN
ejpam-5274	65	11	∈	∈	PROPN
ejpam-5274	65	12	e(h	e(h	PROPN
ejpam-5274	65	13	)	)	PUNCT
ejpam-5274	65	14	]	]	PUNCT
ejpam-5274	65	15	but	but	CCONJ
ejpam-5274	65	16	not	not	PART
ejpam-5274	65	17	both	both	PRON
ejpam-5274	65	18	.	.	PUNCT
ejpam-5274	66	1	jayhan	jayhan	PROPN
ejpam-5274	66	2	cruz	cruz	PROPN
ejpam-5274	66	3	,	,	PUNCT
ejpam-5274	66	4	g.	g.	PROPN
ejpam-5274	66	5	malacas	malacas	PROPN
ejpam-5274	66	6	,	,	PUNCT
ejpam-5274	66	7	s.	s.	PROPN
ejpam-5274	66	8	canoy	canoy	PROPN
ejpam-5274	66	9	,	,	PUNCT
ejpam-5274	66	10	jr	jr	PROPN
ejpam-5274	66	11	.	.	PROPN
ejpam-5274	66	12	/	/	SYM
ejpam-5274	66	13	eur	eur	PROPN
ejpam-5274	66	14	.	.	PUNCT
ejpam-5274	67	1	j.	j.	PROPN
ejpam-5274	67	2	pure	pure	PROPN
ejpam-5274	67	3	appl	appl	PROPN
ejpam-5274	67	4	.	.	PROPN
ejpam-5274	67	5	math	math	PROPN
ejpam-5274	67	6	,	,	PUNCT
ejpam-5274	67	7	17	17	NUM
ejpam-5274	67	8	(	(	PUNCT
ejpam-5274	67	9	3	3	NUM
ejpam-5274	67	10	)	)	PUNCT
ejpam-5274	67	11	(	(	PUNCT
ejpam-5274	67	12	2024	2024	NUM
ejpam-5274	67	13	)	)	PUNCT
ejpam-5274	67	14	,	,	PUNCT
ejpam-5274	67	15	1449	1449	NUM
ejpam-5274	67	16	-	-	SYM
ejpam-5274	67	17	1462	1462	NUM
ejpam-5274	67	18	1451	1451	NUM
ejpam-5274	67	19	the	the	DET
ejpam-5274	67	20	disjunction	disjunction	NOUN
ejpam-5274	67	21	g∨h	g∨h	NOUN
ejpam-5274	67	22	of	of	ADP
ejpam-5274	67	23	graphs	graph	NOUN
ejpam-5274	67	24	g	g	PROPN
ejpam-5274	67	25	and	and	CCONJ
ejpam-5274	67	26	h	h	NOUN
ejpam-5274	67	27	is	be	AUX
ejpam-5274	67	28	the	the	DET
ejpam-5274	67	29	graph	graph	NOUN
ejpam-5274	67	30	with	with	ADP
ejpam-5274	67	31	vertex	vertex	NOUN
ejpam-5274	67	32	set	set	VERB
ejpam-5274	67	33	v	v	NOUN
ejpam-5274	67	34	(	(	PUNCT
ejpam-5274	67	35	g)×v	g)×v	PROPN
ejpam-5274	67	36	(	(	PUNCT
ejpam-5274	67	37	h	h	NOUN
ejpam-5274	67	38	)	)	PUNCT
ejpam-5274	67	39	and	and	CCONJ
ejpam-5274	67	40	(	(	PUNCT
ejpam-5274	67	41	u	u	NOUN
ejpam-5274	67	42	,	,	PUNCT
ejpam-5274	67	43	v	v	NOUN
ejpam-5274	67	44	)	)	PUNCT
ejpam-5274	67	45	is	be	AUX
ejpam-5274	67	46	adjacent	adjacent	ADJ
ejpam-5274	67	47	with	with	ADP
ejpam-5274	67	48	(	(	PUNCT
ejpam-5274	67	49	u	u	NOUN
ejpam-5274	67	50	′	′	NOUN
ejpam-5274	67	51	,	,	PUNCT
ejpam-5274	67	52	v	v	NOUN
ejpam-5274	67	53	′	′	NOUN
ejpam-5274	67	54	)	)	PUNCT
ejpam-5274	68	1	whenever	whenever	SCONJ
ejpam-5274	68	2	uu	uu	ADP
ejpam-5274	68	3	′	′	NUM
ejpam-5274	68	4	∈	∈	PROPN
ejpam-5274	68	5	e(g	e(g	PROPN
ejpam-5274	68	6	)	)	PUNCT
ejpam-5274	68	7	or	or	CCONJ
ejpam-5274	68	8	vv	vv	INTJ
ejpam-5274	68	9	′	′	NUM
ejpam-5274	68	10	∈	∈	PROPN
ejpam-5274	68	11	e(h	e(h	PROPN
ejpam-5274	68	12	)	)	PUNCT
ejpam-5274	68	13	.	.	PUNCT
ejpam-5274	69	1	3	3	X
ejpam-5274	69	2	.	.	NOUN
ejpam-5274	69	3	results	result	VERB
ejpam-5274	69	4	the	the	DET
ejpam-5274	69	5	first	first	ADJ
ejpam-5274	69	6	result	result	NOUN
ejpam-5274	69	7	gives	give	VERB
ejpam-5274	69	8	the	the	DET
ejpam-5274	69	9	polynomial	polynomial	ADJ
ejpam-5274	69	10	representation	representation	NOUN
ejpam-5274	69	11	of	of	ADP
ejpam-5274	69	12	the	the	DET
ejpam-5274	69	13	complement	complement	NOUN
ejpam-5274	69	14	of	of	ADP
ejpam-5274	69	15	a	a	DET
ejpam-5274	69	16	graph	graph	NOUN
ejpam-5274	69	17	.	.	PUNCT
ejpam-5274	70	1	theorem	theorem	NOUN
ejpam-5274	70	2	1	1	NUM
ejpam-5274	70	3	.	.	PUNCT
ejpam-5274	71	1	let	let	VERB
ejpam-5274	71	2	g	g	PRON
ejpam-5274	71	3	be	be	AUX
ejpam-5274	71	4	a	a	DET
ejpam-5274	71	5	non	non	ADJ
ejpam-5274	71	6	-	-	ADJ
ejpam-5274	71	7	trivial	trivial	ADJ
ejpam-5274	71	8	graph	graph	NOUN
ejpam-5274	71	9	of	of	ADP
ejpam-5274	71	10	order	order	NOUN
ejpam-5274	71	11	n.	n.	NOUN
ejpam-5274	71	12	then	then	ADV
ejpam-5274	71	13	fg(x	fg(x	NUM
ejpam-5274	71	14	)	)	PUNCT
ejpam-5274	71	15	=	=	PUNCT
ejpam-5274	72	1	xn−1fg	xn−1fg	NUM
ejpam-5274	72	2	(	(	PUNCT
ejpam-5274	72	3	1	1	NUM
ejpam-5274	72	4	x	x	NOUN
ejpam-5274	72	5	)	)	PUNCT
ejpam-5274	72	6	.	.	PUNCT
ejpam-5274	73	1	proof	proof	NOUN
ejpam-5274	73	2	.	.	PUNCT
ejpam-5274	74	1	let	let	VERB
ejpam-5274	74	2	v	v	NUM
ejpam-5274	74	3	∈	∈	PROPN
ejpam-5274	74	4	v	v	NOUN
ejpam-5274	74	5	(	(	PUNCT
ejpam-5274	74	6	g	g	NOUN
ejpam-5274	74	7	)	)	PUNCT
ejpam-5274	74	8	.	.	PUNCT
ejpam-5274	75	1	then	then	ADV
ejpam-5274	75	2	ng(v	ng(v	PUNCT
ejpam-5274	75	3	)	)	PUNCT
ejpam-5274	76	1	=	=	PRON
ejpam-5274	76	2	{	{	PUNCT
ejpam-5274	76	3	w	w	NOUN
ejpam-5274	76	4	∈	∈	PROPN
ejpam-5274	76	5	v	v	ADP
ejpam-5274	76	6	(	(	PUNCT
ejpam-5274	76	7	g	g	NOUN
ejpam-5274	76	8	)	)	PUNCT
ejpam-5274	76	9	:	:	PUNCT
ejpam-5274	76	10	w	w	X
ejpam-5274	76	11	∈	∈	PROPN
ejpam-5274	76	12	v	v	ADP
ejpam-5274	76	13	(	(	PUNCT
ejpam-5274	76	14	g	g	NOUN
ejpam-5274	76	15	)	)	PUNCT
ejpam-5274	76	16	\	\	PUNCT
ejpam-5274	77	1	ng[v	ng[v	ADV
ejpam-5274	77	2	]	]	PUNCT
ejpam-5274	77	3	}	}	PUNCT
ejpam-5274	77	4	.	.	PUNCT
ejpam-5274	78	1	this	this	PRON
ejpam-5274	78	2	implies	imply	VERB
ejpam-5274	78	3	that	that	SCONJ
ejpam-5274	78	4	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	78	5	=	=	SYM
ejpam-5274	78	6	n−	n−	NOUN
ejpam-5274	78	7	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	78	8	−	−	NOUN
ejpam-5274	78	9	1	1	NUM
ejpam-5274	78	10	.	.	PUNCT
ejpam-5274	79	1	it	it	PRON
ejpam-5274	79	2	follows	follow	VERB
ejpam-5274	79	3	that	that	PRON
ejpam-5274	79	4	fg(x	fg(x	X
ejpam-5274	79	5	)	)	PUNCT
ejpam-5274	79	6	=	=	SYM
ejpam-5274	79	7	∑	∑	PUNCT
ejpam-5274	79	8	v∈v	v∈v	NOUN
ejpam-5274	79	9	(	(	PUNCT
ejpam-5274	79	10	g	g	NOUN
ejpam-5274	79	11	)	)	PUNCT
ejpam-5274	79	12	x|ng(v)|	x|ng(v)|	X
ejpam-5274	80	1	=	=	PUNCT
ejpam-5274	80	2	∑	∑	PUNCT
ejpam-5274	80	3	v∈v	v∈v	PROPN
ejpam-5274	80	4	(	(	PUNCT
ejpam-5274	80	5	g	g	NOUN
ejpam-5274	80	6	)	)	PUNCT
ejpam-5274	80	7	xn−|ng(v)|−1	xn−|ng(v)|−1	NOUN
ejpam-5274	81	1	=	=	PUNCT
ejpam-5274	81	2	xn−1	xn−1	PROPN
ejpam-5274	81	3	∑	∑	PUNCT
ejpam-5274	81	4	v∈v	v∈v	PROPN
ejpam-5274	81	5	(	(	PUNCT
ejpam-5274	81	6	g	g	NOUN
ejpam-5274	81	7	)	)	PUNCT
ejpam-5274	81	8	x−|ng(v)|	x−|ng(v)|	X
ejpam-5274	81	9	=	=	SYM
ejpam-5274	81	10	xn−1fg	xn−1fg	PROPN
ejpam-5274	81	11	(	(	PUNCT
ejpam-5274	81	12	1	1	NUM
ejpam-5274	81	13	x	x	NOUN
ejpam-5274	81	14	)	)	PUNCT
ejpam-5274	81	15	.	.	PUNCT
ejpam-5274	82	1	theorem	theorem	NOUN
ejpam-5274	82	2	2	2	NUM
ejpam-5274	82	3	.	.	PUNCT
ejpam-5274	83	1	let	let	VERB
ejpam-5274	83	2	g	g	PRON
ejpam-5274	83	3	be	be	AUX
ejpam-5274	83	4	a	a	DET
ejpam-5274	83	5	non	non	ADJ
ejpam-5274	83	6	-	-	ADJ
ejpam-5274	83	7	trivial	trivial	ADJ
ejpam-5274	83	8	graph	graph	NOUN
ejpam-5274	83	9	of	of	ADP
ejpam-5274	83	10	order	order	NOUN
ejpam-5274	83	11	n.	n.	NOUN
ejpam-5274	83	12	if	if	SCONJ
ejpam-5274	83	13	the	the	DET
ejpam-5274	83	14	degree	degree	NOUN
ejpam-5274	83	15	sequence	sequence	NOUN
ejpam-5274	83	16	of	of	ADP
ejpam-5274	83	17	g	g	PROPN
ejpam-5274	83	18	is	be	AUX
ejpam-5274	83	19	⟨d1	⟨d1	PROPN
ejpam-5274	83	20	,	,	PUNCT
ejpam-5274	83	21	d2	d2	PROPN
ejpam-5274	83	22	,	,	PUNCT
ejpam-5274	83	23	·	·	PUNCT
ejpam-5274	83	24	·	·	PUNCT
ejpam-5274	83	25	·	·	PUNCT
ejpam-5274	83	26	dn⟩	dn⟩	NOUN
ejpam-5274	83	27	,	,	PUNCT
ejpam-5274	83	28	then	then	ADV
ejpam-5274	83	29	the	the	DET
ejpam-5274	83	30	degree	degree	NOUN
ejpam-5274	83	31	sequence	sequence	NOUN
ejpam-5274	83	32	of	of	ADP
ejpam-5274	83	33	g	g	PROPN
ejpam-5274	83	34	is	be	AUX
ejpam-5274	83	35	⟨n−	⟨n−	VERB
ejpam-5274	83	36	dn	dn	ADP
ejpam-5274	83	37	−	−	NOUN
ejpam-5274	83	38	1	1	NUM
ejpam-5274	83	39	,	,	PUNCT
ejpam-5274	83	40	n−	n−	NOUN
ejpam-5274	83	41	dn−1	dn−1	ADV
ejpam-5274	83	42	−	−	PROPN
ejpam-5274	83	43	1	1	NUM
ejpam-5274	83	44	,	,	PUNCT
ejpam-5274	83	45	·	·	PUNCT
ejpam-5274	83	46	·	·	PUNCT
ejpam-5274	83	47	·	·	PUNCT
ejpam-5274	83	48	,	,	PUNCT
ejpam-5274	83	49	n−	n−	PROPN
ejpam-5274	83	50	d2	d2	VERB
ejpam-5274	83	51	−	−	PROPN
ejpam-5274	83	52	1	1	NUM
ejpam-5274	83	53	,	,	PUNCT
ejpam-5274	83	54	n−	n−	NOUN
ejpam-5274	83	55	d1	d1	NOUN
ejpam-5274	83	56	−	−	PROPN
ejpam-5274	83	57	1⟩.	1⟩.	PROPN
ejpam-5274	83	58	proof	proof	NOUN
ejpam-5274	83	59	.	.	PUNCT
ejpam-5274	84	1	by	by	ADP
ejpam-5274	84	2	theorem	theorem	NOUN
ejpam-5274	84	3	1	1	NUM
ejpam-5274	84	4	,	,	PUNCT
ejpam-5274	84	5	fg(x	fg(x	NUM
ejpam-5274	84	6	)	)	PUNCT
ejpam-5274	85	1	=	=	SYM
ejpam-5274	85	2	n∑	n∑	NOUN
ejpam-5274	85	3	j=1	j=1	PROPN
ejpam-5274	85	4	x|ng(v)|	x|ng(v)|	PROPN
ejpam-5274	86	1	=	=	SYM
ejpam-5274	86	2	n∑	n∑	PROPN
ejpam-5274	86	3	j=1	j=1	PROPN
ejpam-5274	86	4	xn−dj−1	xn−dj−1	NOUN
ejpam-5274	86	5	.	.	PUNCT
ejpam-5274	87	1	hence	hence	ADV
ejpam-5274	87	2	,	,	PUNCT
ejpam-5274	87	3	the	the	DET
ejpam-5274	87	4	degree	degree	NOUN
ejpam-5274	87	5	sequence	sequence	NOUN
ejpam-5274	87	6	of	of	ADP
ejpam-5274	87	7	g	g	PROPN
ejpam-5274	87	8	is	be	AUX
ejpam-5274	87	9	⟨n−	⟨n−	VERB
ejpam-5274	87	10	dn	dn	ADP
ejpam-5274	87	11	−	−	NOUN
ejpam-5274	87	12	1	1	NUM
ejpam-5274	87	13	,	,	PUNCT
ejpam-5274	87	14	n−	n−	NOUN
ejpam-5274	87	15	dn−1	dn−1	ADV
ejpam-5274	87	16	−	−	PROPN
ejpam-5274	87	17	1	1	NUM
ejpam-5274	87	18	,	,	PUNCT
ejpam-5274	87	19	·	·	PUNCT
ejpam-5274	87	20	·	·	PUNCT
ejpam-5274	87	21	·	·	PUNCT
ejpam-5274	87	22	,	,	PUNCT
ejpam-5274	87	23	n−	n−	NOUN
ejpam-5274	87	24	d1	d1	NOUN
ejpam-5274	87	25	−	−	PROPN
ejpam-5274	87	26	1⟩.	1⟩.	PROPN
ejpam-5274	87	27	this	this	PRON
ejpam-5274	87	28	proves	prove	VERB
ejpam-5274	87	29	the	the	DET
ejpam-5274	87	30	assertion	assertion	NOUN
ejpam-5274	87	31	.	.	PUNCT
ejpam-5274	88	1	next	next	ADV
ejpam-5274	88	2	,	,	PUNCT
ejpam-5274	88	3	we	we	PRON
ejpam-5274	88	4	give	give	VERB
ejpam-5274	88	5	the	the	DET
ejpam-5274	88	6	polynomial	polynomial	ADJ
ejpam-5274	88	7	representation	representation	NOUN
ejpam-5274	88	8	of	of	ADP
ejpam-5274	88	9	the	the	DET
ejpam-5274	88	10	line	line	NOUN
ejpam-5274	88	11	graph	graph	NOUN
ejpam-5274	88	12	of	of	ADP
ejpam-5274	88	13	a	a	DET
ejpam-5274	88	14	graph	graph	NOUN
ejpam-5274	88	15	.	.	PUNCT
ejpam-5274	89	1	theorem	theorem	NOUN
ejpam-5274	89	2	3	3	X
ejpam-5274	89	3	.	.	PUNCT
ejpam-5274	90	1	let	let	VERB
ejpam-5274	90	2	g	g	PRON
ejpam-5274	90	3	be	be	AUX
ejpam-5274	90	4	a	a	DET
ejpam-5274	90	5	non	non	ADJ
ejpam-5274	90	6	-	-	ADJ
ejpam-5274	90	7	trivial	trivial	ADJ
ejpam-5274	90	8	connected	connected	ADJ
ejpam-5274	90	9	graph	graph	NOUN
ejpam-5274	90	10	.	.	PUNCT
ejpam-5274	91	1	then	then	ADV
ejpam-5274	91	2	fl(g)(x	fl(g)(x	NUM
ejpam-5274	91	3	)	)	PUNCT
ejpam-5274	92	1	=	=	PUNCT
ejpam-5274	92	2	1	1	NUM
ejpam-5274	92	3	x2	x2	PROPN
ejpam-5274	92	4	∑	∑	PUNCT
ejpam-5274	92	5	uv∈e(g	uv∈e(g	NOUN
ejpam-5274	92	6	)	)	PUNCT
ejpam-5274	93	1	x|ng(u)|+|ng(v)|	x|ng(u)|+|ng(v)|	PROPN
ejpam-5274	93	2	.	.	PROPN
ejpam-5274	93	3	jayhan	jayhan	PROPN
ejpam-5274	93	4	cruz	cruz	PROPN
ejpam-5274	93	5	,	,	PUNCT
ejpam-5274	93	6	g.	g.	PROPN
ejpam-5274	93	7	malacas	malacas	PROPN
ejpam-5274	93	8	,	,	PUNCT
ejpam-5274	93	9	s.	s.	PROPN
ejpam-5274	93	10	canoy	canoy	PROPN
ejpam-5274	93	11	,	,	PUNCT
ejpam-5274	93	12	jr	jr	PROPN
ejpam-5274	93	13	.	.	PROPN
ejpam-5274	93	14	/	/	SYM
ejpam-5274	93	15	eur	eur	PROPN
ejpam-5274	93	16	.	.	PUNCT
ejpam-5274	94	1	j.	j.	PROPN
ejpam-5274	94	2	pure	pure	PROPN
ejpam-5274	94	3	appl	appl	PROPN
ejpam-5274	94	4	.	.	PROPN
ejpam-5274	94	5	math	math	PROPN
ejpam-5274	94	6	,	,	PUNCT
ejpam-5274	94	7	17	17	NUM
ejpam-5274	94	8	(	(	PUNCT
ejpam-5274	94	9	3	3	NUM
ejpam-5274	94	10	)	)	PUNCT
ejpam-5274	94	11	(	(	PUNCT
ejpam-5274	94	12	2024	2024	NUM
ejpam-5274	94	13	)	)	PUNCT
ejpam-5274	94	14	,	,	PUNCT
ejpam-5274	94	15	1449	1449	NUM
ejpam-5274	94	16	-	-	SYM
ejpam-5274	94	17	1462	1462	NUM
ejpam-5274	94	18	1452	1452	NUM
ejpam-5274	94	19	proof	proof	NOUN
ejpam-5274	94	20	.	.	PUNCT
ejpam-5274	95	1	let	let	VERB
ejpam-5274	95	2	e	e	NOUN
ejpam-5274	95	3	=	=	PUNCT
ejpam-5274	95	4	uv	uv	PROPN
ejpam-5274	95	5	∈	∈	PROPN
ejpam-5274	95	6	v	v	NOUN
ejpam-5274	95	7	(	(	PUNCT
ejpam-5274	95	8	l(g	l(g	NOUN
ejpam-5274	95	9	)	)	PUNCT
ejpam-5274	95	10	)	)	PUNCT
ejpam-5274	95	11	.	.	PUNCT
ejpam-5274	96	1	then	then	ADV
ejpam-5274	96	2	|nl(g)(e)|	|nl(g)(e)|	VERB
ejpam-5274	97	1	=	=	SYM
ejpam-5274	97	2	|ng(u)|	|ng(u)|	NOUN
ejpam-5274	97	3	+	+	CCONJ
ejpam-5274	97	4	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	97	5	−	−	NOUN
ejpam-5274	97	6	2	2	NUM
ejpam-5274	97	7	.	.	PUNCT
ejpam-5274	98	1	it	it	PRON
ejpam-5274	98	2	follows	follow	VERB
ejpam-5274	98	3	that	that	SCONJ
ejpam-5274	98	4	fl(g)(x	fl(g)(x	NOUN
ejpam-5274	98	5	)	)	PUNCT
ejpam-5274	99	1	=	=	PUNCT
ejpam-5274	99	2	∑	∑	PUNCT
ejpam-5274	99	3	e∈v	e∈v	NOUN
ejpam-5274	99	4	(	(	PUNCT
ejpam-5274	99	5	l(g	l(g	PROPN
ejpam-5274	99	6	)	)	PUNCT
ejpam-5274	99	7	)	)	PUNCT
ejpam-5274	99	8	x|nl(g)(e)|	x|nl(g)(e)|	PUNCT
ejpam-5274	100	1	=	=	PUNCT
ejpam-5274	100	2	∑	∑	PUNCT
ejpam-5274	100	3	uv∈v	uv∈v	X
ejpam-5274	100	4	(	(	PUNCT
ejpam-5274	100	5	l(g	l(g	NOUN
ejpam-5274	100	6	)	)	PUNCT
ejpam-5274	100	7	)	)	PUNCT
ejpam-5274	101	1	x|ng(u)|+|ng(v)|−2	x|ng(u)|+|ng(v)|−2	PROPN
ejpam-5274	101	2	=	=	SYM
ejpam-5274	101	3	1	1	NUM
ejpam-5274	101	4	x2	x2	PROPN
ejpam-5274	101	5	∑	∑	PUNCT
ejpam-5274	101	6	uv∈e(g	uv∈e(g	NOUN
ejpam-5274	101	7	)	)	PUNCT
ejpam-5274	102	1	x|ng(u)|+|ng(v)|	x|ng(u)|+|ng(v)|	PROPN
ejpam-5274	102	2	.	.	PUNCT
ejpam-5274	102	3	corollary	corollary	ADJ
ejpam-5274	102	4	1	1	NUM
ejpam-5274	102	5	.	.	PUNCT
ejpam-5274	103	1	let	let	VERB
ejpam-5274	103	2	g	g	PRON
ejpam-5274	103	3	be	be	AUX
ejpam-5274	103	4	a	a	DET
ejpam-5274	103	5	non	non	ADJ
ejpam-5274	103	6	-	-	ADJ
ejpam-5274	103	7	trivial	trivial	ADJ
ejpam-5274	103	8	r	r	NOUN
ejpam-5274	103	9	-	-	ADJ
ejpam-5274	103	10	regular	regular	ADJ
ejpam-5274	103	11	connected	connected	ADJ
ejpam-5274	103	12	graph	graph	NOUN
ejpam-5274	103	13	of	of	ADP
ejpam-5274	103	14	size	size	NOUN
ejpam-5274	104	1	p.	p.	NOUN
ejpam-5274	104	2	then	then	ADV
ejpam-5274	104	3	fl(g)(x	fl(g)(x	NUM
ejpam-5274	104	4	)	)	PUNCT
ejpam-5274	105	1	=	=	SYM
ejpam-5274	105	2	px2r−2	px2r−2	X
ejpam-5274	105	3	.	.	PUNCT
ejpam-5274	106	1	proof	proof	NOUN
ejpam-5274	106	2	.	.	PUNCT
ejpam-5274	107	1	since	since	SCONJ
ejpam-5274	107	2	g	g	PROPN
ejpam-5274	107	3	is	be	AUX
ejpam-5274	107	4	r	r	NOUN
ejpam-5274	107	5	-	-	ADJ
ejpam-5274	107	6	regular	regular	ADJ
ejpam-5274	107	7	,	,	PUNCT
ejpam-5274	107	8	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	107	9	=	=	SYM
ejpam-5274	107	10	r	r	NOUN
ejpam-5274	107	11	for	for	ADP
ejpam-5274	107	12	all	all	DET
ejpam-5274	107	13	v	v	ADP
ejpam-5274	107	14	∈	∈	NOUN
ejpam-5274	107	15	v	v	NOUN
ejpam-5274	107	16	(	(	PUNCT
ejpam-5274	107	17	g	g	NOUN
ejpam-5274	107	18	)	)	PUNCT
ejpam-5274	107	19	.	.	PUNCT
ejpam-5274	108	1	by	by	ADP
ejpam-5274	108	2	theorem	theorem	NOUN
ejpam-5274	108	3	3	3	NUM
ejpam-5274	108	4	,	,	PUNCT
ejpam-5274	108	5	we	we	PRON
ejpam-5274	108	6	have	have	VERB
ejpam-5274	108	7	fl(g)(x	fl(g)(x	NUM
ejpam-5274	108	8	)	)	PUNCT
ejpam-5274	109	1	=	=	SYM
ejpam-5274	110	1	1	1	NUM
ejpam-5274	110	2	x2	x2	PROPN
ejpam-5274	110	3	∑	∑	PUNCT
ejpam-5274	110	4	uv∈e(g	uv∈e(g	NOUN
ejpam-5274	110	5	)	)	PUNCT
ejpam-5274	111	1	x|ng(u)|+|ng(v)|	x|ng(u)|+|ng(v)|	PROPN
ejpam-5274	111	2	=	=	SYM
ejpam-5274	111	3	1	1	NUM
ejpam-5274	111	4	x2	x2	PROPN
ejpam-5274	111	5	∑	∑	PUNCT
ejpam-5274	111	6	uv∈e(g	uv∈e(g	NUM
ejpam-5274	111	7	)	)	PUNCT
ejpam-5274	111	8	x2r	x2r	PUNCT
ejpam-5274	112	1	=	=	SYM
ejpam-5274	112	2	px2r−2	px2r−2	X
ejpam-5274	112	3	.	.	PUNCT
ejpam-5274	112	4	theorem	theorem	NOUN
ejpam-5274	112	5	4	4	NUM
ejpam-5274	112	6	.	.	PUNCT
ejpam-5274	113	1	let	let	VERB
ejpam-5274	113	2	g	g	PRON
ejpam-5274	113	3	be	be	AUX
ejpam-5274	113	4	a	a	DET
ejpam-5274	113	5	connected	connected	ADJ
ejpam-5274	113	6	graph	graph	NOUN
ejpam-5274	113	7	of	of	ADP
ejpam-5274	113	8	size	size	NOUN
ejpam-5274	113	9	p	p	NOUN
ejpam-5274	113	10	and	and	CCONJ
ejpam-5274	113	11	let	let	VERB
ejpam-5274	113	12	h	h	PRON
ejpam-5274	113	13	any	any	DET
ejpam-5274	113	14	graph	graph	NOUN
ejpam-5274	113	15	of	of	ADP
ejpam-5274	113	16	order	order	NOUN
ejpam-5274	113	17	n.	n.	NOUN
ejpam-5274	113	18	then	then	ADV
ejpam-5274	113	19	fg⋄h(x	fg⋄h(x	PROPN
ejpam-5274	113	20	)	)	PUNCT
ejpam-5274	113	21	=	=	NUM
ejpam-5274	113	22	fg(x	fg(x	NUM
ejpam-5274	113	23	n+1	n+1	NUM
ejpam-5274	113	24	)	)	PUNCT
ejpam-5274	114	1	+	+	NUM
ejpam-5274	114	2	px2fh(x	px2fh(x	PROPN
ejpam-5274	114	3	)	)	PUNCT
ejpam-5274	114	4	.	.	PUNCT
ejpam-5274	115	1	proof	proof	NOUN
ejpam-5274	115	2	.	.	PUNCT
ejpam-5274	116	1	let	let	VERB
ejpam-5274	116	2	v	v	NUM
ejpam-5274	116	3	∈	∈	PROPN
ejpam-5274	116	4	v	v	NOUN
ejpam-5274	116	5	(	(	PUNCT
ejpam-5274	116	6	g	g	PROPN
ejpam-5274	116	7	⋄h	⋄h	PROPN
ejpam-5274	116	8	)	)	PUNCT
ejpam-5274	116	9	.	.	PUNCT
ejpam-5274	117	1	if	if	SCONJ
ejpam-5274	117	2	v	v	NUM
ejpam-5274	117	3	∈	∈	PROPN
ejpam-5274	117	4	v	v	NOUN
ejpam-5274	117	5	(	(	PUNCT
ejpam-5274	117	6	g	g	NOUN
ejpam-5274	117	7	)	)	PUNCT
ejpam-5274	117	8	,	,	PUNCT
ejpam-5274	117	9	then	then	ADV
ejpam-5274	117	10	ng⋄h(v	ng⋄h(v	NUM
ejpam-5274	117	11	)	)	PUNCT
ejpam-5274	117	12	=	=	PUNCT
ejpam-5274	117	13	ng(v	ng(v	X
ejpam-5274	117	14	)	)	PUNCT
ejpam-5274	117	15	∪	∪	ADP
ejpam-5274	117	16	[	[	X
ejpam-5274	117	17	∪u∈ng(v)v	∪u∈ng(v)v	ADJ
ejpam-5274	117	18	(	(	PUNCT
ejpam-5274	117	19	huv	huv	PROPN
ejpam-5274	117	20	)	)	PUNCT
ejpam-5274	117	21	]	]	PUNCT
ejpam-5274	117	22	.	.	PUNCT
ejpam-5274	118	1	if	if	SCONJ
ejpam-5274	118	2	v	v	NUM
ejpam-5274	118	3	∈	∈	NOUN
ejpam-5274	118	4	v	v	NOUN
ejpam-5274	118	5	(	(	PUNCT
ejpam-5274	118	6	he	he	PRON
ejpam-5274	118	7	)	)	PUNCT
ejpam-5274	118	8	for	for	ADP
ejpam-5274	118	9	e	e	PROPN
ejpam-5274	118	10	=	=	SYM
ejpam-5274	118	11	uw	uw	PROPN
ejpam-5274	118	12	∈	∈	PROPN
ejpam-5274	118	13	e(g	e(g	PROPN
ejpam-5274	118	14	)	)	PUNCT
ejpam-5274	118	15	,	,	PUNCT
ejpam-5274	118	16	then	then	ADV
ejpam-5274	118	17	ng⋄h(v	ng⋄h(v	NUM
ejpam-5274	118	18	)	)	PUNCT
ejpam-5274	118	19	=	=	PUNCT
ejpam-5274	118	20	nhe(v	nhe(v	PROPN
ejpam-5274	118	21	)	)	PUNCT
ejpam-5274	118	22	∪	∪	NOUN
ejpam-5274	118	23	{	{	PUNCT
ejpam-5274	118	24	u	u	NOUN
ejpam-5274	118	25	,	,	PUNCT
ejpam-5274	118	26	w	w	NOUN
ejpam-5274	118	27	}	}	PUNCT
ejpam-5274	118	28	.	.	PUNCT
ejpam-5274	119	1	thus	thus	ADV
ejpam-5274	119	2	,	,	PUNCT
ejpam-5274	119	3	fg⋄h(x	fg⋄h(x	PROPN
ejpam-5274	119	4	)	)	PUNCT
ejpam-5274	119	5	=	=	SYM
ejpam-5274	119	6	∑	∑	PUNCT
ejpam-5274	119	7	v∈v	v∈v	PROPN
ejpam-5274	119	8	(	(	PUNCT
ejpam-5274	119	9	g⋄h	g⋄h	X
ejpam-5274	119	10	)	)	PUNCT
ejpam-5274	119	11	x|ng⋄h(v)|	x|ng⋄h(v)|	PUNCT
ejpam-5274	120	1	=	=	PUNCT
ejpam-5274	120	2	∑	∑	PUNCT
ejpam-5274	120	3	v∈v	v∈v	NOUN
ejpam-5274	120	4	(	(	PUNCT
ejpam-5274	120	5	g	g	NOUN
ejpam-5274	120	6	)	)	PUNCT
ejpam-5274	120	7	x|ng⋄h(v)|	x|ng⋄h(v)|	PUNCT
ejpam-5274	121	1	+	+	CCONJ
ejpam-5274	121	2	∑	∑	PUNCT
ejpam-5274	121	3	v∈v	v∈v	PROPN
ejpam-5274	121	4	(	(	PUNCT
ejpam-5274	121	5	g⋄h)\v	g⋄h)\v	PROPN
ejpam-5274	121	6	(	(	PUNCT
ejpam-5274	121	7	g	g	NOUN
ejpam-5274	121	8	)	)	PUNCT
ejpam-5274	121	9	x|ng⋄h(v)|	x|ng⋄h(v)|	PUNCT
ejpam-5274	122	1	=	=	PUNCT
ejpam-5274	122	2	∑	∑	PUNCT
ejpam-5274	122	3	v∈v	v∈v	PROPN
ejpam-5274	122	4	(	(	PUNCT
ejpam-5274	122	5	g	g	NOUN
ejpam-5274	122	6	)	)	PUNCT
ejpam-5274	122	7	x|ng(v)|+n|ng(v)|	x|ng(v)|+n|ng(v)|	PUNCT
ejpam-5274	123	1	+	+	CCONJ
ejpam-5274	123	2	∑	∑	PUNCT
ejpam-5274	123	3	e∈e(g	e∈e(g	PROPN
ejpam-5274	123	4	)	)	PUNCT
ejpam-5274	123	5	∑	∑	PUNCT
ejpam-5274	123	6	v∈v	v∈v	PROPN
ejpam-5274	123	7	(	(	PUNCT
ejpam-5274	123	8	he	he	PRON
ejpam-5274	123	9	)	)	PUNCT
ejpam-5274	123	10	x|nhe	x|nhe	PROPN
ejpam-5274	123	11	(	(	PUNCT
ejpam-5274	123	12	v)|+2	v)|+2	NOUN
ejpam-5274	123	13	=	=	PUNCT
ejpam-5274	123	14	∑	∑	PUNCT
ejpam-5274	123	15	v∈v	v∈v	PROPN
ejpam-5274	123	16	(	(	PUNCT
ejpam-5274	123	17	g	g	NOUN
ejpam-5274	123	18	)	)	PUNCT
ejpam-5274	123	19	x(n+1)|ng(v)|	x(n+1)|ng(v)|	PUNCT
ejpam-5274	124	1	+	+	CCONJ
ejpam-5274	124	2	px2	px2	PROPN
ejpam-5274	124	3	∑	∑	PUNCT
ejpam-5274	124	4	v∈v	v∈v	PROPN
ejpam-5274	124	5	(	(	PUNCT
ejpam-5274	124	6	h	h	NOUN
ejpam-5274	124	7	)	)	PUNCT
ejpam-5274	124	8	x|nh(v)|	x|nh(v)|	PROPN
ejpam-5274	125	1	=	=	AUX
ejpam-5274	125	2	fg(x	fg(x	NUM
ejpam-5274	125	3	n+1	n+1	NUM
ejpam-5274	125	4	)	)	PUNCT
ejpam-5274	125	5	+	+	NUM
ejpam-5274	125	6	px2fh(x	px2fh(x	PROPN
ejpam-5274	125	7	)	)	PUNCT
ejpam-5274	125	8	.	.	PUNCT
ejpam-5274	126	1	jayhan	jayhan	PROPN
ejpam-5274	126	2	cruz	cruz	PROPN
ejpam-5274	126	3	,	,	PUNCT
ejpam-5274	126	4	g.	g.	PROPN
ejpam-5274	126	5	malacas	malacas	PROPN
ejpam-5274	126	6	,	,	PUNCT
ejpam-5274	126	7	s.	s.	PROPN
ejpam-5274	126	8	canoy	canoy	PROPN
ejpam-5274	126	9	,	,	PUNCT
ejpam-5274	126	10	jr	jr	PROPN
ejpam-5274	126	11	.	.	PROPN
ejpam-5274	126	12	/	/	SYM
ejpam-5274	126	13	eur	eur	PROPN
ejpam-5274	126	14	.	.	PUNCT
ejpam-5274	127	1	j.	j.	PROPN
ejpam-5274	127	2	pure	pure	PROPN
ejpam-5274	127	3	appl	appl	PROPN
ejpam-5274	127	4	.	.	PROPN
ejpam-5274	127	5	math	math	PROPN
ejpam-5274	127	6	,	,	PUNCT
ejpam-5274	127	7	17	17	NUM
ejpam-5274	127	8	(	(	PUNCT
ejpam-5274	127	9	3	3	NUM
ejpam-5274	127	10	)	)	PUNCT
ejpam-5274	127	11	(	(	PUNCT
ejpam-5274	127	12	2024	2024	NUM
ejpam-5274	127	13	)	)	PUNCT
ejpam-5274	127	14	,	,	PUNCT
ejpam-5274	127	15	1449	1449	NUM
ejpam-5274	127	16	-	-	SYM
ejpam-5274	127	17	1462	1462	NUM
ejpam-5274	127	18	1453	1453	NUM
ejpam-5274	127	19	the	the	DET
ejpam-5274	127	20	following	following	ADJ
ejpam-5274	127	21	result	result	NOUN
ejpam-5274	127	22	is	be	AUX
ejpam-5274	127	23	immediate	immediate	ADJ
ejpam-5274	127	24	from	from	ADP
ejpam-5274	127	25	the	the	DET
ejpam-5274	127	26	above	above	ADJ
ejpam-5274	127	27	result	result	NOUN
ejpam-5274	127	28	:	:	PUNCT
ejpam-5274	127	29	corollary	corollary	ADJ
ejpam-5274	127	30	2	2	X
ejpam-5274	127	31	.	.	PUNCT
ejpam-5274	128	1	let	let	VERB
ejpam-5274	128	2	g	g	PRON
ejpam-5274	128	3	be	be	AUX
ejpam-5274	128	4	a	a	DET
ejpam-5274	128	5	connected	connected	ADJ
ejpam-5274	128	6	graph	graph	NOUN
ejpam-5274	128	7	of	of	ADP
ejpam-5274	128	8	size	size	NOUN
ejpam-5274	128	9	p	p	NOUN
ejpam-5274	128	10	and	and	CCONJ
ejpam-5274	128	11	let	let	VERB
ejpam-5274	128	12	h	h	NOUN
ejpam-5274	128	13	be	be	AUX
ejpam-5274	128	14	an	an	DET
ejpam-5274	128	15	r	r	NOUN
ejpam-5274	128	16	-	-	PUNCT
ejpam-5274	128	17	regular	regular	ADJ
ejpam-5274	128	18	graph	graph	NOUN
ejpam-5274	128	19	of	of	ADP
ejpam-5274	128	20	order	order	NOUN
ejpam-5274	128	21	n.	n.	NOUN
ejpam-5274	128	22	then	then	ADV
ejpam-5274	128	23	fg⋄h(x	fg⋄h(x	PROPN
ejpam-5274	128	24	)	)	PUNCT
ejpam-5274	128	25	=	=	NUM
ejpam-5274	128	26	fg(x	fg(x	NUM
ejpam-5274	128	27	n+1	n+1	NUM
ejpam-5274	128	28	)	)	PUNCT
ejpam-5274	129	1	+	+	NUM
ejpam-5274	129	2	pnxr+2	pnxr+2	NOUN
ejpam-5274	129	3	.	.	PUNCT
ejpam-5274	130	1	in	in	ADP
ejpam-5274	130	2	particular	particular	ADJ
ejpam-5274	130	3	,	,	PUNCT
ejpam-5274	130	4	fg⋄kn(x	fg⋄kn(x	X
ejpam-5274	130	5	)	)	PUNCT
ejpam-5274	130	6	=	=	SYM
ejpam-5274	130	7	fg(x	fg(x	NUM
ejpam-5274	130	8	n+1	n+1	NUM
ejpam-5274	130	9	)	)	PUNCT
ejpam-5274	130	10	+	+	CCONJ
ejpam-5274	130	11	pnxn+1	pnxn+1	NOUN
ejpam-5274	130	12	.	.	PUNCT
ejpam-5274	130	13	corollary	corollary	ADJ
ejpam-5274	130	14	3	3	X
ejpam-5274	130	15	.	.	PUNCT
ejpam-5274	131	1	let	let	VERB
ejpam-5274	131	2	m	m	PRON
ejpam-5274	131	3	and	and	CCONJ
ejpam-5274	131	4	n	n	ADV
ejpam-5274	131	5	be	be	AUX
ejpam-5274	131	6	positive	positive	ADJ
ejpam-5274	131	7	integers	integer	NOUN
ejpam-5274	131	8	.	.	PUNCT
ejpam-5274	132	1	then	then	ADV
ejpam-5274	132	2	(	(	PUNCT
ejpam-5274	132	3	i	i	NOUN
ejpam-5274	132	4	)	)	PUNCT
ejpam-5274	132	5	fpm⋄pn(x	fpm⋄pn(x	X
ejpam-5274	132	6	)	)	PUNCT
ejpam-5274	132	7	=	=	SYM
ejpam-5274	132	8	(	(	PUNCT
ejpam-5274	132	9	m−	m−	PROPN
ejpam-5274	132	10	2)x2n+2	2)x2n+2	PROPN
ejpam-5274	133	1	+	+	CCONJ
ejpam-5274	134	1	2xn+1	2xn+1	NUM
ejpam-5274	135	1	+	+	CCONJ
ejpam-5274	135	2	(	(	PUNCT
ejpam-5274	135	3	m−	m−	PROPN
ejpam-5274	135	4	1)[(n−	1)[(n−	NUM
ejpam-5274	135	5	2)x4	2)x4	NUM
ejpam-5274	135	6	+	+	CCONJ
ejpam-5274	135	7	2x3	2x3	NUM
ejpam-5274	135	8	]	]	PUNCT
ejpam-5274	135	9	for	for	ADP
ejpam-5274	135	10	m	m	PROPN
ejpam-5274	135	11	,	,	PUNCT
ejpam-5274	135	12	n	n	PRON
ejpam-5274	135	13	≥	≥	NOUN
ejpam-5274	135	14	2	2	NUM
ejpam-5274	135	15	;	;	PUNCT
ejpam-5274	135	16	(	(	PUNCT
ejpam-5274	135	17	ii	ii	NOUN
ejpam-5274	135	18	)	)	PUNCT
ejpam-5274	135	19	fpm⋄cn(x	fpm⋄cn(x	X
ejpam-5274	135	20	)	)	PUNCT
ejpam-5274	135	21	=	=	PUNCT
ejpam-5274	136	1	(	(	PUNCT
ejpam-5274	136	2	m−	m−	PROPN
ejpam-5274	136	3	2)x2n+2	2)x2n+2	PROPN
ejpam-5274	137	1	+	+	CCONJ
ejpam-5274	138	1	2xn+1	2xn+1	NUM
ejpam-5274	139	1	+	+	CCONJ
ejpam-5274	139	2	(	(	PUNCT
ejpam-5274	139	3	m−	m−	PROPN
ejpam-5274	139	4	1)nx4	1)nx4	NUM
ejpam-5274	139	5	for	for	ADP
ejpam-5274	139	6	m	m	PROPN
ejpam-5274	139	7	≥	≥	NUM
ejpam-5274	139	8	2	2	NUM
ejpam-5274	139	9	and	and	CCONJ
ejpam-5274	139	10	n	n	PRON
ejpam-5274	139	11	≥	≥	NOUN
ejpam-5274	139	12	3	3	NUM
ejpam-5274	139	13	;	;	PUNCT
ejpam-5274	139	14	and	and	CCONJ
ejpam-5274	139	15	(	(	PUNCT
ejpam-5274	139	16	iii	iii	NOUN
ejpam-5274	139	17	)	)	PUNCT
ejpam-5274	139	18	fcm⋄cn(x	fcm⋄cn(x	PROPN
ejpam-5274	139	19	)	)	PUNCT
ejpam-5274	140	1	=	=	SYM
ejpam-5274	140	2	mx2n+2	mx2n+2	PROPN
ejpam-5274	141	1	+	+	NOUN
ejpam-5274	141	2	mnx4	mnx4	PROPN
ejpam-5274	141	3	for	for	ADP
ejpam-5274	141	4	m	m	PROPN
ejpam-5274	141	5	,	,	PUNCT
ejpam-5274	141	6	n	n	PRON
ejpam-5274	141	7	≥	≥	NOUN
ejpam-5274	141	8	3	3	NUM
ejpam-5274	141	9	.	.	PUNCT
ejpam-5274	142	1	proof	proof	NOUN
ejpam-5274	142	2	.	.	PUNCT
ejpam-5274	143	1	clearly	clearly	ADV
ejpam-5274	143	2	,	,	PUNCT
ejpam-5274	143	3	|e(pr)|	|e(pr)|	NOUN
ejpam-5274	143	4	=	=	SYM
ejpam-5274	143	5	r−	r−	PROPN
ejpam-5274	143	6	1	1	NUM
ejpam-5274	143	7	,	,	PUNCT
ejpam-5274	143	8	|e(cs)|	|e(cs)|	X
ejpam-5274	143	9	=	=	SYM
ejpam-5274	143	10	s	s	PROPN
ejpam-5274	143	11	,	,	PUNCT
ejpam-5274	143	12	fpr(x	fpr(x	PROPN
ejpam-5274	143	13	)	)	PUNCT
ejpam-5274	143	14	=	=	SYM
ejpam-5274	143	15	2x+(r−	2x+(r−	NUM
ejpam-5274	143	16	2)x2	2)x2	NUM
ejpam-5274	143	17	and	and	CCONJ
ejpam-5274	143	18	fcs(x	fcs(x	NUM
ejpam-5274	143	19	)	)	PUNCT
ejpam-5274	144	1	=	=	SYM
ejpam-5274	144	2	sx2	sx2	NOUN
ejpam-5274	144	3	for	for	ADP
ejpam-5274	144	4	positive	positive	ADJ
ejpam-5274	144	5	integers	integer	NOUN
ejpam-5274	144	6	r	r	NOUN
ejpam-5274	144	7	≥	≥	NUM
ejpam-5274	144	8	2	2	NUM
ejpam-5274	144	9	and	and	CCONJ
ejpam-5274	144	10	s	s	PRON
ejpam-5274	144	11	≥	≥	NOUN
ejpam-5274	144	12	3	3	NUM
ejpam-5274	144	13	.	.	PUNCT
ejpam-5274	144	14	by	by	ADP
ejpam-5274	144	15	theorem	theorem	NOUN
ejpam-5274	144	16	4	4	NUM
ejpam-5274	144	17	,	,	PUNCT
ejpam-5274	144	18	we	we	PRON
ejpam-5274	144	19	find	find	VERB
ejpam-5274	144	20	that	that	SCONJ
ejpam-5274	144	21	(	(	PUNCT
ejpam-5274	144	22	i	i	NOUN
ejpam-5274	144	23	)	)	PUNCT
ejpam-5274	144	24	,	,	PUNCT
ejpam-5274	144	25	(	(	PUNCT
ejpam-5274	144	26	ii	ii	NOUN
ejpam-5274	144	27	)	)	PUNCT
ejpam-5274	144	28	,	,	PUNCT
ejpam-5274	144	29	and	and	CCONJ
ejpam-5274	144	30	(	(	PUNCT
ejpam-5274	144	31	iii	iii	NOUN
ejpam-5274	144	32	)	)	PUNCT
ejpam-5274	144	33	hold	hold	NOUN
ejpam-5274	144	34	.	.	PUNCT
ejpam-5274	145	1	theorem	theorem	NOUN
ejpam-5274	145	2	5	5	NUM
ejpam-5274	145	3	.	.	PUNCT
ejpam-5274	146	1	let	let	VERB
ejpam-5274	146	2	g	g	PRON
ejpam-5274	146	3	be	be	AUX
ejpam-5274	146	4	a	a	DET
ejpam-5274	146	5	connected	connected	ADJ
ejpam-5274	146	6	graph	graph	NOUN
ejpam-5274	146	7	of	of	ADP
ejpam-5274	146	8	size	size	NOUN
ejpam-5274	146	9	p	p	NOUN
ejpam-5274	146	10	and	and	CCONJ
ejpam-5274	146	11	let	let	VERB
ejpam-5274	146	12	h	h	NOUN
ejpam-5274	146	13	be	be	AUX
ejpam-5274	146	14	any	any	DET
ejpam-5274	146	15	graph	graph	NOUN
ejpam-5274	146	16	with	with	ADP
ejpam-5274	146	17	degree	degree	NOUN
ejpam-5274	146	18	sequences	sequence	NOUN
ejpam-5274	146	19	⟨d1	⟨d1	PROPN
ejpam-5274	146	20	,	,	PUNCT
ejpam-5274	146	21	d2	d2	PROPN
ejpam-5274	146	22	,	,	PUNCT
ejpam-5274	146	23	·	·	PUNCT
ejpam-5274	146	24	·	·	PUNCT
ejpam-5274	146	25	·	·	PUNCT
ejpam-5274	146	26	dm⟩	dm⟩	PROPN
ejpam-5274	146	27	and	and	CCONJ
ejpam-5274	146	28	⟨r1	⟨r1	PROPN
ejpam-5274	146	29	,	,	PUNCT
ejpam-5274	146	30	r2	r2	PROPN
ejpam-5274	146	31	,	,	PUNCT
ejpam-5274	146	32	·	·	PUNCT
ejpam-5274	146	33	·	·	PUNCT
ejpam-5274	146	34	·	·	PUNCT
ejpam-5274	146	35	rn⟩	rn⟩	PROPN
ejpam-5274	146	36	,	,	PUNCT
ejpam-5274	146	37	respectively	respectively	ADV
ejpam-5274	146	38	.	.	PUNCT
ejpam-5274	147	1	then	then	ADV
ejpam-5274	147	2	the	the	DET
ejpam-5274	147	3	terms	term	NOUN
ejpam-5274	147	4	of	of	ADP
ejpam-5274	147	5	the	the	DET
ejpam-5274	147	6	degree	degree	NOUN
ejpam-5274	147	7	sequence	sequence	NOUN
ejpam-5274	147	8	of	of	ADP
ejpam-5274	147	9	g⋄h	g⋄h	PROPN
ejpam-5274	147	10	are	be	AUX
ejpam-5274	147	11	the	the	DET
ejpam-5274	147	12	elements	element	NOUN
ejpam-5274	147	13	of	of	ADP
ejpam-5274	147	14	the	the	DET
ejpam-5274	147	15	set	set	NOUN
ejpam-5274	147	16	{	{	PUNCT
ejpam-5274	147	17	(	(	PUNCT
ejpam-5274	147	18	n+1)di	n+1)di	NOUN
ejpam-5274	147	19	:	:	PUNCT
ejpam-5274	147	20	1	1	NUM
ejpam-5274	147	21	≤	≤	NUM
ejpam-5274	147	22	i	i	PRON
ejpam-5274	147	23	≤	≤	NOUN
ejpam-5274	147	24	m}∪{rj+2	m}∪{rj+2	NOUN
ejpam-5274	147	25	:	:	PUNCT
ejpam-5274	147	26	1	1	NUM
ejpam-5274	147	27	≤	≤	NUM
ejpam-5274	147	28	j	j	PROPN
ejpam-5274	147	29	≤	≤	PROPN
ejpam-5274	147	30	n	n	CCONJ
ejpam-5274	147	31	}	}	PUNCT
ejpam-5274	147	32	,	,	PUNCT
ejpam-5274	147	33	where	where	SCONJ
ejpam-5274	147	34	p	p	DET
ejpam-5274	147	35	consecutive	consecutive	ADJ
ejpam-5274	147	36	terms	term	NOUN
ejpam-5274	147	37	of	of	ADP
ejpam-5274	147	38	the	the	DET
ejpam-5274	147	39	degree	degree	NOUN
ejpam-5274	147	40	sequence	sequence	NOUN
ejpam-5274	147	41	are	be	AUX
ejpam-5274	147	42	rj	rj	PROPN
ejpam-5274	147	43	+	+	ADP
ejpam-5274	147	44	2	2	NUM
ejpam-5274	147	45	for	for	ADP
ejpam-5274	147	46	each	each	DET
ejpam-5274	147	47	j	j	NOUN
ejpam-5274	147	48	with	with	ADP
ejpam-5274	147	49	1	1	NUM
ejpam-5274	147	50	≤	≤	NUM
ejpam-5274	147	51	i	i	PRON
ejpam-5274	147	52	≤	≤	ADJ
ejpam-5274	147	53	n.	n.	NOUN
ejpam-5274	147	54	proof	proof	NOUN
ejpam-5274	147	55	.	.	PUNCT
ejpam-5274	148	1	the	the	DET
ejpam-5274	148	2	polynomial	polynomial	ADJ
ejpam-5274	148	3	representations	representation	NOUN
ejpam-5274	148	4	of	of	ADP
ejpam-5274	148	5	g	g	NOUN
ejpam-5274	148	6	and	and	CCONJ
ejpam-5274	148	7	h	h	NOUN
ejpam-5274	148	8	are	be	AUX
ejpam-5274	148	9	,	,	PUNCT
ejpam-5274	148	10	respectively	respectively	ADV
ejpam-5274	148	11	,	,	PUNCT
ejpam-5274	148	12	fg(x	fg(x	NUM
ejpam-5274	148	13	)	)	PUNCT
ejpam-5274	149	1	=	=	PUNCT
ejpam-5274	149	2	m∑	m∑	INTJ
ejpam-5274	149	3	i=1	i=1	PROPN
ejpam-5274	149	4	xdi	xdi	PROPN
ejpam-5274	149	5	and	and	CCONJ
ejpam-5274	149	6	fh(x	fh(x	PUNCT
ejpam-5274	149	7	)	)	PUNCT
ejpam-5274	150	1	=	=	SYM
ejpam-5274	150	2	n∑	n∑	NOUN
ejpam-5274	150	3	j=1	j=1	PROPN
ejpam-5274	150	4	xrj	xrj	PROPN
ejpam-5274	150	5	.	.	PUNCT
ejpam-5274	151	1	by	by	ADP
ejpam-5274	151	2	theorem	theorem	NOUN
ejpam-5274	151	3	4	4	NUM
ejpam-5274	151	4	,	,	PUNCT
ejpam-5274	151	5	fg⋄h(x	fg⋄h(x	PROPN
ejpam-5274	151	6	)	)	PUNCT
ejpam-5274	151	7	=	=	PUNCT
ejpam-5274	151	8	m∑	m∑	NOUN
ejpam-5274	151	9	i=1	i=1	PROPN
ejpam-5274	151	10	x(n+1)di	x(n+1)di	PROPN
ejpam-5274	152	1	+	+	CCONJ
ejpam-5274	152	2	px2	px2	PROPN
ejpam-5274	152	3	n∑	n∑	PROPN
ejpam-5274	152	4	j=1	j=1	PROPN
ejpam-5274	152	5	xrj	xrj	PUNCT
ejpam-5274	153	1	=	=	PUNCT
ejpam-5274	153	2	m∑	m∑	CCONJ
ejpam-5274	153	3	i=1	i=1	PROPN
ejpam-5274	153	4	x(n+1)di	x(n+1)di	PROPN
ejpam-5274	154	1	+	+	CCONJ
ejpam-5274	154	2	p	p	X
ejpam-5274	154	3	n∑	n∑	NOUN
ejpam-5274	154	4	j=1	j=1	NOUN
ejpam-5274	154	5	xrj+2	xrj+2	PROPN
ejpam-5274	154	6	.	.	PUNCT
ejpam-5274	155	1	it	it	PRON
ejpam-5274	155	2	follows	follow	VERB
ejpam-5274	155	3	that	that	SCONJ
ejpam-5274	155	4	the	the	DET
ejpam-5274	155	5	terms	term	NOUN
ejpam-5274	155	6	of	of	ADP
ejpam-5274	155	7	the	the	DET
ejpam-5274	155	8	degree	degree	NOUN
ejpam-5274	155	9	sequence	sequence	NOUN
ejpam-5274	155	10	of	of	ADP
ejpam-5274	155	11	g	g	PROPN
ejpam-5274	155	12	⋄	⋄	PROPN
ejpam-5274	155	13	h	h	NOUN
ejpam-5274	155	14	are	be	AUX
ejpam-5274	155	15	the	the	DET
ejpam-5274	155	16	elements	element	NOUN
ejpam-5274	155	17	of	of	ADP
ejpam-5274	155	18	the	the	DET
ejpam-5274	155	19	set	set	NOUN
ejpam-5274	155	20	{	{	PUNCT
ejpam-5274	155	21	(	(	PUNCT
ejpam-5274	155	22	n	n	PROPN
ejpam-5274	155	23	+	+	NUM
ejpam-5274	155	24	1)di	1)di	NUM
ejpam-5274	155	25	:	:	PUNCT
ejpam-5274	155	26	1	1	NUM
ejpam-5274	155	27	≤	≤	NUM
ejpam-5274	155	28	i	i	X
ejpam-5274	155	29	≤	≤	NOUN
ejpam-5274	155	30	m	m	VERB
ejpam-5274	155	31	}	}	PUNCT
ejpam-5274	155	32	∪	∪	X
ejpam-5274	155	33	{	{	PUNCT
ejpam-5274	155	34	rj	rj	NOUN
ejpam-5274	155	35	+	+	PROPN
ejpam-5274	155	36	2	2	NUM
ejpam-5274	155	37	:	:	SYM
ejpam-5274	155	38	1	1	NUM
ejpam-5274	155	39	≤	≤	NUM
ejpam-5274	155	40	j	j	PROPN
ejpam-5274	155	41	≤	≤	PROPN
ejpam-5274	155	42	n	n	CCONJ
ejpam-5274	155	43	}	}	PUNCT
ejpam-5274	155	44	.	.	PUNCT
ejpam-5274	156	1	moreover	moreover	ADV
ejpam-5274	156	2	,	,	PUNCT
ejpam-5274	156	3	p	p	PRON
ejpam-5274	156	4	consecutive	consecutive	ADJ
ejpam-5274	156	5	terms	term	NOUN
ejpam-5274	156	6	of	of	ADP
ejpam-5274	156	7	the	the	DET
ejpam-5274	156	8	degree	degree	NOUN
ejpam-5274	156	9	sequence	sequence	NOUN
ejpam-5274	156	10	are	be	AUX
ejpam-5274	156	11	rj	rj	PROPN
ejpam-5274	156	12	+	+	ADP
ejpam-5274	156	13	2	2	NUM
ejpam-5274	156	14	for	for	ADP
ejpam-5274	156	15	each	each	DET
ejpam-5274	156	16	j	j	NOUN
ejpam-5274	156	17	with	with	ADP
ejpam-5274	156	18	1	1	NUM
ejpam-5274	156	19	≤	≤	NUM
ejpam-5274	156	20	j	j	PROPN
ejpam-5274	156	21	≤	≤	PROPN
ejpam-5274	156	22	n.	n.	PROPN
ejpam-5274	156	23	theorem	theorem	VERB
ejpam-5274	156	24	6	6	NUM
ejpam-5274	156	25	.	.	PUNCT
ejpam-5274	157	1	let	let	VERB
ejpam-5274	157	2	g	g	PRON
ejpam-5274	157	3	be	be	AUX
ejpam-5274	157	4	a	a	DET
ejpam-5274	157	5	non	non	ADJ
ejpam-5274	157	6	-	-	ADJ
ejpam-5274	157	7	trivial	trivial	ADJ
ejpam-5274	157	8	connected	connected	ADJ
ejpam-5274	157	9	graph	graph	NOUN
ejpam-5274	157	10	and	and	CCONJ
ejpam-5274	157	11	let	let	VERB
ejpam-5274	157	12	g1	g1	PROPN
ejpam-5274	157	13	and	and	CCONJ
ejpam-5274	157	14	g2	g2	PROPN
ejpam-5274	157	15	be	be	VERB
ejpam-5274	157	16	copies	copy	NOUN
ejpam-5274	157	17	of	of	ADP
ejpam-5274	157	18	g	g	NOUN
ejpam-5274	157	19	in	in	ADP
ejpam-5274	157	20	the	the	DET
ejpam-5274	157	21	shadow	shadow	NOUN
ejpam-5274	157	22	graph	graph	VERB
ejpam-5274	157	23	d2(g	d2(g	PROPN
ejpam-5274	157	24	)	)	PUNCT
ejpam-5274	157	25	.	.	PUNCT
ejpam-5274	158	1	then	then	ADV
ejpam-5274	158	2	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	158	3	)	)	PUNCT
ejpam-5274	158	4	=	=	PUNCT
ejpam-5274	159	1	2fg(x	2fg(x	NUM
ejpam-5274	159	2	2	2	NUM
ejpam-5274	159	3	)	)	PUNCT
ejpam-5274	159	4	.	.	PUNCT
ejpam-5274	160	1	jayhan	jayhan	PROPN
ejpam-5274	160	2	cruz	cruz	PROPN
ejpam-5274	160	3	,	,	PUNCT
ejpam-5274	160	4	g.	g.	PROPN
ejpam-5274	160	5	malacas	malacas	PROPN
ejpam-5274	160	6	,	,	PUNCT
ejpam-5274	160	7	s.	s.	PROPN
ejpam-5274	160	8	canoy	canoy	PROPN
ejpam-5274	160	9	,	,	PUNCT
ejpam-5274	160	10	jr	jr	PROPN
ejpam-5274	160	11	.	.	PROPN
ejpam-5274	160	12	/	/	SYM
ejpam-5274	160	13	eur	eur	PROPN
ejpam-5274	160	14	.	.	PUNCT
ejpam-5274	161	1	j.	j.	PROPN
ejpam-5274	161	2	pure	pure	PROPN
ejpam-5274	161	3	appl	appl	PROPN
ejpam-5274	161	4	.	.	PROPN
ejpam-5274	161	5	math	math	PROPN
ejpam-5274	161	6	,	,	PUNCT
ejpam-5274	161	7	17	17	NUM
ejpam-5274	161	8	(	(	PUNCT
ejpam-5274	161	9	3	3	NUM
ejpam-5274	161	10	)	)	PUNCT
ejpam-5274	161	11	(	(	PUNCT
ejpam-5274	161	12	2024	2024	NUM
ejpam-5274	161	13	)	)	PUNCT
ejpam-5274	161	14	,	,	PUNCT
ejpam-5274	161	15	1449	1449	NUM
ejpam-5274	161	16	-	-	SYM
ejpam-5274	161	17	1462	1462	NUM
ejpam-5274	161	18	1454	1454	NUM
ejpam-5274	161	19	proof	proof	NOUN
ejpam-5274	161	20	.	.	PUNCT
ejpam-5274	162	1	let	let	VERB
ejpam-5274	162	2	v	v	NUM
ejpam-5274	162	3	∈	∈	PROPN
ejpam-5274	162	4	v	v	NOUN
ejpam-5274	162	5	(	(	PUNCT
ejpam-5274	162	6	g1	g1	PROPN
ejpam-5274	162	7	)	)	PUNCT
ejpam-5274	162	8	and	and	CCONJ
ejpam-5274	162	9	let	let	VERB
ejpam-5274	162	10	v′	v′	NOUN
ejpam-5274	162	11	be	be	AUX
ejpam-5274	162	12	the	the	DET
ejpam-5274	162	13	vertex	vertex	NOUN
ejpam-5274	162	14	of	of	ADP
ejpam-5274	162	15	g2	g2	PROPN
ejpam-5274	162	16	corresponding	correspond	VERB
ejpam-5274	162	17	to	to	ADP
ejpam-5274	162	18	v.	v.	ADP
ejpam-5274	162	19	then	then	ADV
ejpam-5274	162	20	nd2(g)(v	nd2(g)(v	PROPN
ejpam-5274	162	21	)	)	PUNCT
ejpam-5274	162	22	=	=	SYM
ejpam-5274	162	23	ng1(v	ng1(v	X
ejpam-5274	162	24	)	)	PUNCT
ejpam-5274	162	25	∪ng2(v	∪ng2(v	NOUN
ejpam-5274	162	26	′	′	NOUN
ejpam-5274	162	27	)	)	PUNCT
ejpam-5274	163	1	=	=	SYM
ejpam-5274	163	2	nd2(g)(v	nd2(g)(v	NOUN
ejpam-5274	163	3	′	′	NUM
ejpam-5274	163	4	)	)	PUNCT
ejpam-5274	163	5	.	.	PUNCT
ejpam-5274	164	1	this	this	PRON
ejpam-5274	164	2	implies	imply	VERB
ejpam-5274	164	3	that	that	SCONJ
ejpam-5274	164	4	|nd2(g)(v)|	|nd2(g)(v)|	PROPN
ejpam-5274	164	5	=	=	SYM
ejpam-5274	164	6	|nd2(g)(v	|nd2(g)(v	NOUN
ejpam-5274	164	7	′)|	′)|	NOUN
ejpam-5274	164	8	=	=	PUNCT
ejpam-5274	164	9	2|ng(v)|	2|ng(v)|	NUM
ejpam-5274	164	10	.	.	PUNCT
ejpam-5274	165	1	thus	thus	ADV
ejpam-5274	165	2	,	,	PUNCT
ejpam-5274	165	3	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	165	4	)	)	PUNCT
ejpam-5274	165	5	=	=	PUNCT
ejpam-5274	165	6	∑	∑	PUNCT
ejpam-5274	165	7	v∈v	v∈v	PROPN
ejpam-5274	165	8	(	(	PUNCT
ejpam-5274	165	9	d2(g	d2(g	PROPN
ejpam-5274	165	10	)	)	PUNCT
ejpam-5274	165	11	)	)	PUNCT
ejpam-5274	165	12	x|nd2(g)(v)|	x|nd2(g)(v)|	X
ejpam-5274	166	1	=	=	PUNCT
ejpam-5274	166	2	∑	∑	PUNCT
ejpam-5274	166	3	v∈v	v∈v	PROPN
ejpam-5274	166	4	(	(	PUNCT
ejpam-5274	166	5	g1	g1	PROPN
ejpam-5274	166	6	)	)	PUNCT
ejpam-5274	166	7	x|nd2(g)(v)|	x|nd2(g)(v)|	PROPN
ejpam-5274	167	1	+	+	CCONJ
ejpam-5274	167	2	∑	∑	PROPN
ejpam-5274	167	3	v′∈v	v′∈v	PROPN
ejpam-5274	167	4	(	(	PUNCT
ejpam-5274	167	5	g2	g2	PROPN
ejpam-5274	167	6	)	)	PUNCT
ejpam-5274	167	7	x|nd2(g)(v	x|nd2(g)(v	PROPN
ejpam-5274	167	8	′)|	′)|	PRON
ejpam-5274	167	9	=	=	SYM
ejpam-5274	167	10	2	2	NUM
ejpam-5274	167	11	∑	∑	NOUN
ejpam-5274	167	12	v∈v	v∈v	NOUN
ejpam-5274	167	13	(	(	PUNCT
ejpam-5274	167	14	g	g	NOUN
ejpam-5274	167	15	)	)	PUNCT
ejpam-5274	167	16	x2|ng(v)|	x2|ng(v)|	NOUN
ejpam-5274	167	17	=	=	NOUN
ejpam-5274	168	1	2fg(x	2fg(x	NUM
ejpam-5274	168	2	2	2	NUM
ejpam-5274	168	3	)	)	PUNCT
ejpam-5274	168	4	.	.	PUNCT
ejpam-5274	169	1	corollary	corollary	ADJ
ejpam-5274	169	2	4	4	NUM
ejpam-5274	169	3	.	.	PUNCT
ejpam-5274	170	1	let	let	VERB
ejpam-5274	170	2	m	m	PRON
ejpam-5274	170	3	and	and	CCONJ
ejpam-5274	170	4	n	n	ADV
ejpam-5274	170	5	be	be	VERB
ejpam-5274	170	6	positive	positive	ADJ
ejpam-5274	170	7	integers	integer	NOUN
ejpam-5274	171	1	such	such	ADJ
ejpam-5274	171	2	that	that	SCONJ
ejpam-5274	171	3	m	m	PROPN
ejpam-5274	171	4	≥	≥	NOUN
ejpam-5274	171	5	2	2	NUM
ejpam-5274	171	6	and	and	CCONJ
ejpam-5274	171	7	n	n	PRON
ejpam-5274	171	8	≥	≥	NOUN
ejpam-5274	171	9	3	3	NUM
ejpam-5274	171	10	.	.	PUNCT
ejpam-5274	172	1	then	then	ADV
ejpam-5274	172	2	(	(	PUNCT
ejpam-5274	172	3	i	i	NOUN
ejpam-5274	172	4	)	)	PUNCT
ejpam-5274	172	5	fd2(pm)(x	fd2(pm)(x	NOUN
ejpam-5274	172	6	)	)	PUNCT
ejpam-5274	172	7	=	=	PUNCT
ejpam-5274	173	1	2(m−	2(m−	NUM
ejpam-5274	173	2	2)x4	2)x4	NUM
ejpam-5274	173	3	+	+	CCONJ
ejpam-5274	173	4	4x2	4x2	NUM
ejpam-5274	173	5	;	;	PUNCT
ejpam-5274	173	6	(	(	PUNCT
ejpam-5274	173	7	ii	ii	NOUN
ejpam-5274	173	8	)	)	PUNCT
ejpam-5274	173	9	fd2(cn)(x	fd2(cn)(x	PROPN
ejpam-5274	173	10	)	)	PUNCT
ejpam-5274	173	11	=	=	PUNCT
ejpam-5274	174	1	2nx4	2nx4	NUM
ejpam-5274	174	2	;	;	PUNCT
ejpam-5274	174	3	and	and	CCONJ
ejpam-5274	174	4	(	(	PUNCT
ejpam-5274	174	5	iii	iii	X
ejpam-5274	174	6	)	)	PUNCT
ejpam-5274	174	7	fd2(sm)(x	fd2(sm)(x	PROPN
ejpam-5274	174	8	)	)	PUNCT
ejpam-5274	174	9	=	=	PUNCT
ejpam-5274	175	1	2x2	2x2	NUM
ejpam-5274	175	2	m	m	VERB
ejpam-5274	175	3	+	+	NOUN
ejpam-5274	175	4	2mx2	2mx2	NUM
ejpam-5274	175	5	,	,	PUNCT
ejpam-5274	175	6	where	where	SCONJ
ejpam-5274	175	7	sm	sm	PROPN
ejpam-5274	175	8	is	be	AUX
ejpam-5274	175	9	a	a	DET
ejpam-5274	175	10	star	star	NOUN
ejpam-5274	175	11	of	of	ADP
ejpam-5274	175	12	order	order	NOUN
ejpam-5274	175	13	m+	m+	NOUN
ejpam-5274	175	14	1	1	NUM
ejpam-5274	175	15	.	.	PUNCT
ejpam-5274	176	1	proof	proof	NOUN
ejpam-5274	176	2	.	.	PUNCT
ejpam-5274	177	1	the	the	DET
ejpam-5274	177	2	polynomial	polynomial	ADJ
ejpam-5274	177	3	representations	representation	NOUN
ejpam-5274	177	4	of	of	ADP
ejpam-5274	177	5	pm	pm	NOUN
ejpam-5274	177	6	,	,	PUNCT
ejpam-5274	177	7	cn	cn	INTJ
ejpam-5274	177	8	,	,	PUNCT
ejpam-5274	177	9	and	and	CCONJ
ejpam-5274	177	10	sm	sm	PROPN
ejpam-5274	177	11	are	be	AUX
ejpam-5274	177	12	,	,	PUNCT
ejpam-5274	177	13	respectively	respectively	ADV
ejpam-5274	177	14	,	,	PUNCT
ejpam-5274	177	15	fpm(x	fpm(x	PROPN
ejpam-5274	177	16	)	)	PUNCT
ejpam-5274	177	17	=	=	SYM
ejpam-5274	177	18	(	(	PUNCT
ejpam-5274	177	19	m−	m−	PROPN
ejpam-5274	177	20	2)x2	2)x2	NUM
ejpam-5274	177	21	+	+	CCONJ
ejpam-5274	177	22	2x	2x	NUM
ejpam-5274	177	23	,	,	PUNCT
ejpam-5274	177	24	fcn(x	fcn(x	X
ejpam-5274	177	25	)	)	PUNCT
ejpam-5274	177	26	=	=	SYM
ejpam-5274	177	27	nx2	nx2	PROPN
ejpam-5274	177	28	,	,	PUNCT
ejpam-5274	177	29	and	and	CCONJ
ejpam-5274	177	30	fsm(x	fsm(x	PROPN
ejpam-5274	177	31	)	)	PUNCT
ejpam-5274	177	32	=	=	SYM
ejpam-5274	177	33	xm	xm	PROPN
ejpam-5274	178	1	+	+	PROPN
ejpam-5274	178	2	mx	mx	PROPN
ejpam-5274	178	3	.	.	PUNCT
ejpam-5274	179	1	it	it	PRON
ejpam-5274	179	2	follows	follow	VERB
ejpam-5274	179	3	from	from	ADP
ejpam-5274	179	4	theorem	theorem	NOUN
ejpam-5274	179	5	6	6	NUM
ejpam-5274	179	6	that	that	DET
ejpam-5274	179	7	fd2(pm)(x	fd2(pm)(x	NOUN
ejpam-5274	179	8	)	)	PUNCT
ejpam-5274	179	9	=	=	PUNCT
ejpam-5274	180	1	2fpm(x	2fpm(x	NUM
ejpam-5274	180	2	2	2	X
ejpam-5274	180	3	)	)	PUNCT
ejpam-5274	180	4	=	=	SYM
ejpam-5274	181	1	2(m−	2(m−	NUM
ejpam-5274	181	2	2)x4	2)x4	NUM
ejpam-5274	181	3	+	+	CCONJ
ejpam-5274	181	4	4x2	4x2	NUM
ejpam-5274	181	5	,	,	PUNCT
ejpam-5274	181	6	fd2(cn)(x	fd2(cn)(x	PROPN
ejpam-5274	181	7	)	)	PUNCT
ejpam-5274	181	8	=	=	PUNCT
ejpam-5274	182	1	2fcn(x	2fcn(x	NUM
ejpam-5274	182	2	2	2	NUM
ejpam-5274	182	3	)	)	PUNCT
ejpam-5274	182	4	=	=	SYM
ejpam-5274	182	5	2nx4	2nx4	NUM
ejpam-5274	182	6	,	,	PUNCT
ejpam-5274	182	7	and	and	CCONJ
ejpam-5274	182	8	fd2(sm)(x	fd2(sm)(x	PROPN
ejpam-5274	182	9	)	)	PUNCT
ejpam-5274	182	10	=	=	PUNCT
ejpam-5274	183	1	2fsm(x	2fsm(x	NUM
ejpam-5274	183	2	2	2	NUM
ejpam-5274	183	3	)	)	PUNCT
ejpam-5274	183	4	=	=	SYM
ejpam-5274	184	1	2x2	2x2	NUM
ejpam-5274	184	2	m	m	VERB
ejpam-5274	184	3	+	+	NOUN
ejpam-5274	184	4	2mx2	2mx2	NUM
ejpam-5274	184	5	.	.	PUNCT
ejpam-5274	185	1	this	this	PRON
ejpam-5274	185	2	proves	prove	VERB
ejpam-5274	185	3	the	the	DET
ejpam-5274	185	4	assertion	assertion	NOUN
ejpam-5274	185	5	.	.	PUNCT
ejpam-5274	186	1	theorem	theorem	ADJ
ejpam-5274	186	2	7	7	NUM
ejpam-5274	186	3	.	.	PUNCT
ejpam-5274	187	1	let	let	VERB
ejpam-5274	187	2	g	g	PRON
ejpam-5274	187	3	be	be	AUX
ejpam-5274	187	4	a	a	DET
ejpam-5274	187	5	non	non	ADJ
ejpam-5274	187	6	-	-	ADJ
ejpam-5274	187	7	trivial	trivial	ADJ
ejpam-5274	187	8	connected	connected	ADJ
ejpam-5274	187	9	graph	graph	NOUN
ejpam-5274	187	10	.	.	PUNCT
ejpam-5274	188	1	then	then	ADV
ejpam-5274	188	2	a	a	DET
ejpam-5274	188	3	∈	∈	NOUN
ejpam-5274	188	4	r	r	NOUN
ejpam-5274	188	5	is	be	AUX
ejpam-5274	188	6	a	a	DET
ejpam-5274	188	7	zero	zero	NUM
ejpam-5274	188	8	of	of	ADP
ejpam-5274	188	9	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	188	10	)	)	PUNCT
ejpam-5274	189	1	if	if	SCONJ
ejpam-5274	189	2	and	and	CCONJ
ejpam-5274	189	3	only	only	ADV
ejpam-5274	189	4	if	if	SCONJ
ejpam-5274	189	5	a2	a2	PROPN
ejpam-5274	189	6	is	be	AUX
ejpam-5274	189	7	a	a	DET
ejpam-5274	189	8	zero	zero	NUM
ejpam-5274	189	9	of	of	ADP
ejpam-5274	189	10	fg(x	fg(x	NUM
ejpam-5274	189	11	)	)	PUNCT
ejpam-5274	189	12	.	.	PUNCT
ejpam-5274	190	1	proof	proof	NOUN
ejpam-5274	190	2	.	.	PUNCT
ejpam-5274	191	1	by	by	ADP
ejpam-5274	191	2	theorem	theorem	NOUN
ejpam-5274	191	3	6	6	NUM
ejpam-5274	191	4	,	,	PUNCT
ejpam-5274	191	5	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	191	6	)	)	PUNCT
ejpam-5274	191	7	=	=	PUNCT
ejpam-5274	192	1	2fg(x	2fg(x	NUM
ejpam-5274	192	2	2	2	NUM
ejpam-5274	192	3	)	)	PUNCT
ejpam-5274	192	4	.	.	PUNCT
ejpam-5274	193	1	hence	hence	ADV
ejpam-5274	193	2	,	,	PUNCT
ejpam-5274	193	3	if	if	SCONJ
ejpam-5274	193	4	a	a	PRON
ejpam-5274	193	5	is	be	AUX
ejpam-5274	193	6	a	a	DET
ejpam-5274	193	7	zero	zero	NUM
ejpam-5274	193	8	of	of	ADP
ejpam-5274	193	9	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	193	10	)	)	PUNCT
ejpam-5274	193	11	,	,	PUNCT
ejpam-5274	193	12	then	then	ADV
ejpam-5274	193	13	2fg(a	2fg(a	NUM
ejpam-5274	193	14	2	2	NUM
ejpam-5274	193	15	)	)	PUNCT
ejpam-5274	193	16	=	=	SYM
ejpam-5274	193	17	0	0	X
ejpam-5274	193	18	.	.	PUNCT
ejpam-5274	194	1	this	this	PRON
ejpam-5274	194	2	implies	imply	VERB
ejpam-5274	194	3	that	that	SCONJ
ejpam-5274	194	4	a2	a2	PROPN
ejpam-5274	194	5	is	be	AUX
ejpam-5274	194	6	a	a	DET
ejpam-5274	194	7	zero	zero	NUM
ejpam-5274	194	8	of	of	ADP
ejpam-5274	194	9	fg(x	fg(x	NUM
ejpam-5274	194	10	)	)	PUNCT
ejpam-5274	194	11	.	.	PUNCT
ejpam-5274	195	1	conversely	conversely	ADV
ejpam-5274	195	2	,	,	PUNCT
ejpam-5274	195	3	if	if	SCONJ
ejpam-5274	195	4	a2	a2	PROPN
ejpam-5274	195	5	is	be	AUX
ejpam-5274	195	6	a	a	DET
ejpam-5274	195	7	zero	zero	NUM
ejpam-5274	195	8	of	of	ADP
ejpam-5274	195	9	fg(x	fg(x	NUM
ejpam-5274	195	10	)	)	PUNCT
ejpam-5274	195	11	,	,	PUNCT
ejpam-5274	195	12	then	then	ADV
ejpam-5274	195	13	fd2(g)(a	fd2(g)(a	NOUN
ejpam-5274	195	14	)	)	PUNCT
ejpam-5274	195	15	=	=	PUNCT
ejpam-5274	196	1	2fg(a	2fg(a	NUM
ejpam-5274	196	2	2	2	NUM
ejpam-5274	196	3	)	)	PUNCT
ejpam-5274	196	4	=	=	SYM
ejpam-5274	196	5	0	0	X
ejpam-5274	196	6	.	.	PUNCT
ejpam-5274	197	1	thus	thus	ADV
ejpam-5274	197	2	,	,	PUNCT
ejpam-5274	197	3	a	a	PRON
ejpam-5274	197	4	is	be	AUX
ejpam-5274	197	5	a	a	DET
ejpam-5274	197	6	zero	zero	NUM
ejpam-5274	197	7	of	of	ADP
ejpam-5274	197	8	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	197	9	)	)	PUNCT
ejpam-5274	197	10	.	.	PUNCT
ejpam-5274	198	1	theorem	theorem	ADJ
ejpam-5274	198	2	8	8	NUM
ejpam-5274	198	3	.	.	PUNCT
ejpam-5274	199	1	let	let	VERB
ejpam-5274	199	2	g	g	PRON
ejpam-5274	199	3	be	be	AUX
ejpam-5274	199	4	a	a	DET
ejpam-5274	199	5	non	non	ADJ
ejpam-5274	199	6	-	-	ADJ
ejpam-5274	199	7	trivial	trivial	ADJ
ejpam-5274	199	8	connected	connected	ADJ
ejpam-5274	199	9	graph	graph	NOUN
ejpam-5274	199	10	with	with	ADP
ejpam-5274	199	11	degree	degree	NOUN
ejpam-5274	199	12	sequence	sequence	NOUN
ejpam-5274	199	13	⟨d1	⟨d1	PROPN
ejpam-5274	199	14	,	,	PUNCT
ejpam-5274	199	15	d2	d2	PROPN
ejpam-5274	199	16	,	,	PUNCT
ejpam-5274	199	17	·	·	PUNCT
ejpam-5274	199	18	·	·	PUNCT
ejpam-5274	199	19	·	·	PUNCT
ejpam-5274	200	1	dn⟩.	dn⟩.	NOUN
ejpam-5274	200	2	then	then	ADV
ejpam-5274	200	3	the	the	DET
ejpam-5274	200	4	degree	degree	NOUN
ejpam-5274	200	5	sequence	sequence	NOUN
ejpam-5274	200	6	of	of	ADP
ejpam-5274	200	7	d2(g	d2(g	PROPN
ejpam-5274	200	8	)	)	PUNCT
ejpam-5274	200	9	is	be	AUX
ejpam-5274	200	10	⟨2d1	⟨2d1	PROPN
ejpam-5274	200	11	,	,	PUNCT
ejpam-5274	200	12	2d1	2d1	NUM
ejpam-5274	200	13	,	,	PUNCT
ejpam-5274	200	14	2d2	2d2	NUM
ejpam-5274	200	15	,	,	PUNCT
ejpam-5274	200	16	2d2	2d2	NUM
ejpam-5274	200	17	,	,	PUNCT
ejpam-5274	200	18	·	·	PUNCT
ejpam-5274	200	19	·	·	PUNCT
ejpam-5274	200	20	·	·	PUNCT
ejpam-5274	200	21	,	,	PUNCT
ejpam-5274	200	22	2dn	2dn	X
ejpam-5274	200	23	,	,	PUNCT
ejpam-5274	200	24	2dn⟩.	2dn⟩.	NUM
ejpam-5274	200	25	jayhan	jayhan	PROPN
ejpam-5274	200	26	cruz	cruz	PROPN
ejpam-5274	200	27	,	,	PUNCT
ejpam-5274	200	28	g.	g.	PROPN
ejpam-5274	200	29	malacas	malacas	PROPN
ejpam-5274	200	30	,	,	PUNCT
ejpam-5274	200	31	s.	s.	PROPN
ejpam-5274	200	32	canoy	canoy	PROPN
ejpam-5274	200	33	,	,	PUNCT
ejpam-5274	200	34	jr	jr	PROPN
ejpam-5274	200	35	.	.	PROPN
ejpam-5274	200	36	/	/	SYM
ejpam-5274	200	37	eur	eur	PROPN
ejpam-5274	200	38	.	.	PUNCT
ejpam-5274	201	1	j.	j.	PROPN
ejpam-5274	201	2	pure	pure	PROPN
ejpam-5274	201	3	appl	appl	PROPN
ejpam-5274	201	4	.	.	PROPN
ejpam-5274	201	5	math	math	PROPN
ejpam-5274	201	6	,	,	PUNCT
ejpam-5274	201	7	17	17	NUM
ejpam-5274	201	8	(	(	PUNCT
ejpam-5274	201	9	3	3	NUM
ejpam-5274	201	10	)	)	PUNCT
ejpam-5274	201	11	(	(	PUNCT
ejpam-5274	201	12	2024	2024	NUM
ejpam-5274	201	13	)	)	PUNCT
ejpam-5274	201	14	,	,	PUNCT
ejpam-5274	201	15	1449	1449	NUM
ejpam-5274	201	16	-	-	SYM
ejpam-5274	201	17	1462	1462	NUM
ejpam-5274	201	18	1455	1455	NUM
ejpam-5274	201	19	proof	proof	NOUN
ejpam-5274	201	20	.	.	PUNCT
ejpam-5274	202	1	given	give	VERB
ejpam-5274	202	2	the	the	DET
ejpam-5274	202	3	degree	degree	NOUN
ejpam-5274	202	4	sequence	sequence	NOUN
ejpam-5274	202	5	⟨d1	⟨d1	PROPN
ejpam-5274	202	6	,	,	PUNCT
ejpam-5274	202	7	d2	d2	PROPN
ejpam-5274	202	8	,	,	PUNCT
ejpam-5274	202	9	·	·	PUNCT
ejpam-5274	202	10	·	·	PUNCT
ejpam-5274	202	11	·	·	PUNCT
ejpam-5274	202	12	dn⟩	dn⟩	NOUN
ejpam-5274	202	13	of	of	ADP
ejpam-5274	202	14	g	g	PROPN
ejpam-5274	202	15	,	,	PUNCT
ejpam-5274	202	16	it	it	PRON
ejpam-5274	202	17	follows	follow	VERB
ejpam-5274	202	18	that	that	PRON
ejpam-5274	202	19	fg(x	fg(x	VERB
ejpam-5274	202	20	)	)	PUNCT
ejpam-5274	202	21	=	=	SYM
ejpam-5274	203	1	∑n	∑n	NUM
ejpam-5274	203	2	i=1	i=1	PROPN
ejpam-5274	203	3	x	x	SYM
ejpam-5274	203	4	di	di	NOUN
ejpam-5274	203	5	.	.	PUNCT
ejpam-5274	204	1	hence	hence	ADV
ejpam-5274	204	2	,	,	PUNCT
ejpam-5274	204	3	by	by	ADP
ejpam-5274	204	4	theorem	theorem	NOUN
ejpam-5274	204	5	6	6	NUM
ejpam-5274	204	6	,	,	PUNCT
ejpam-5274	204	7	fd2(g)(x	fd2(g)(x	PROPN
ejpam-5274	204	8	)	)	PUNCT
ejpam-5274	204	9	=	=	PUNCT
ejpam-5274	205	1	2fg(x	2fg(x	NUM
ejpam-5274	205	2	2	2	NUM
ejpam-5274	205	3	)	)	PUNCT
ejpam-5274	205	4	=	=	SYM
ejpam-5274	205	5	2	2	NUM
ejpam-5274	205	6	n∑	n∑	NOUN
ejpam-5274	205	7	i=1	i=1	PROPN
ejpam-5274	205	8	x2di	x2di	PROPN
ejpam-5274	205	9	.	.	PUNCT
ejpam-5274	206	1	it	it	PRON
ejpam-5274	206	2	follows	follow	VERB
ejpam-5274	206	3	that	that	SCONJ
ejpam-5274	206	4	the	the	DET
ejpam-5274	206	5	degree	degree	NOUN
ejpam-5274	206	6	sequence	sequence	NOUN
ejpam-5274	206	7	of	of	ADP
ejpam-5274	206	8	d2(g	d2(g	PROPN
ejpam-5274	206	9	)	)	PUNCT
ejpam-5274	206	10	is	be	AUX
ejpam-5274	206	11	⟨2d1	⟨2d1	PROPN
ejpam-5274	206	12	,	,	PUNCT
ejpam-5274	206	13	2d1	2d1	NUM
ejpam-5274	206	14	,	,	PUNCT
ejpam-5274	206	15	2d2	2d2	NUM
ejpam-5274	206	16	,	,	PUNCT
ejpam-5274	206	17	2d2	2d2	NUM
ejpam-5274	206	18	,	,	PUNCT
ejpam-5274	206	19	·	·	PUNCT
ejpam-5274	206	20	·	·	PUNCT
ejpam-5274	206	21	·	·	PUNCT
ejpam-5274	206	22	,	,	PUNCT
ejpam-5274	206	23	2dn	2dn	X
ejpam-5274	206	24	,	,	PUNCT
ejpam-5274	206	25	2dn⟩.	2dn⟩.	NUM
ejpam-5274	206	26	theorem	theorem	VERB
ejpam-5274	206	27	9	9	NUM
ejpam-5274	206	28	.	.	PUNCT
ejpam-5274	207	1	let	let	VERB
ejpam-5274	207	2	g	g	PRON
ejpam-5274	207	3	be	be	AUX
ejpam-5274	207	4	a	a	DET
ejpam-5274	207	5	non	non	ADJ
ejpam-5274	207	6	-	-	ADJ
ejpam-5274	207	7	trivial	trivial	ADJ
ejpam-5274	207	8	connected	connected	ADJ
ejpam-5274	207	9	graph	graph	NOUN
ejpam-5274	207	10	of	of	ADP
ejpam-5274	207	11	order	order	NOUN
ejpam-5274	207	12	n.	n.	NOUN
ejpam-5274	207	13	then	then	ADV
ejpam-5274	207	14	fgg(x	fgg(x	PRON
ejpam-5274	207	15	)	)	PUNCT
ejpam-5274	207	16	=	=	SYM
ejpam-5274	208	1	xfg(x	xfg(x	X
ejpam-5274	208	2	)	)	PUNCT
ejpam-5274	209	1	+	+	CCONJ
ejpam-5274	209	2	xnfg	xnfg	PROPN
ejpam-5274	209	3	(	(	PUNCT
ejpam-5274	209	4	1	1	NUM
ejpam-5274	209	5	x	x	NOUN
ejpam-5274	209	6	)	)	PUNCT
ejpam-5274	209	7	.	.	PUNCT
ejpam-5274	210	1	proof	proof	NOUN
ejpam-5274	210	2	.	.	PUNCT
ejpam-5274	211	1	let	let	VERB
ejpam-5274	211	2	v	v	NUM
ejpam-5274	211	3	∈	∈	PROPN
ejpam-5274	211	4	v	v	NOUN
ejpam-5274	211	5	(	(	PUNCT
ejpam-5274	211	6	g	g	NOUN
ejpam-5274	211	7	)	)	PUNCT
ejpam-5274	211	8	.	.	PUNCT
ejpam-5274	212	1	then	then	ADV
ejpam-5274	212	2	ngg(v	ngg(v	PROPN
ejpam-5274	212	3	)	)	PUNCT
ejpam-5274	212	4	=	=	PUNCT
ejpam-5274	212	5	ng(v	ng(v	X
ejpam-5274	212	6	)	)	PUNCT
ejpam-5274	212	7	∪	∪	ADP
ejpam-5274	212	8	{	{	PUNCT
ejpam-5274	212	9	v	v	NOUN
ejpam-5274	212	10	}	}	PUNCT
ejpam-5274	212	11	and	and	CCONJ
ejpam-5274	212	12	ngg(v	ngg(v	NOUN
ejpam-5274	212	13	)	)	PUNCT
ejpam-5274	212	14	=	=	PUNCT
ejpam-5274	212	15	ng(v	ng(v	X
ejpam-5274	212	16	)	)	PUNCT
ejpam-5274	212	17	∪	∪	ADP
ejpam-5274	212	18	{	{	PUNCT
ejpam-5274	212	19	v	v	NOUN
ejpam-5274	212	20	}	}	PUNCT
ejpam-5274	212	21	=	=	PUNCT
ejpam-5274	212	22	{	{	PUNCT
ejpam-5274	212	23	z	z	NOUN
ejpam-5274	212	24	∈	∈	PROPN
ejpam-5274	212	25	v	v	NOUN
ejpam-5274	212	26	(	(	PUNCT
ejpam-5274	212	27	g	g	NOUN
ejpam-5274	212	28	)	)	PUNCT
ejpam-5274	212	29	:	:	PUNCT
ejpam-5274	213	1	z	z	PROPN
ejpam-5274	213	2	∈	∈	PROPN
ejpam-5274	213	3	v	v	ADP
ejpam-5274	213	4	(	(	PUNCT
ejpam-5274	213	5	g	g	NOUN
ejpam-5274	213	6	)	)	PUNCT
ejpam-5274	213	7	\ng[v	\ng[v	NOUN
ejpam-5274	213	8	]	]	PUNCT
ejpam-5274	213	9	}	}	PUNCT
ejpam-5274	213	10	∪	∪	X
ejpam-5274	213	11	{	{	PUNCT
ejpam-5274	213	12	v	v	NOUN
ejpam-5274	213	13	}	}	PUNCT
ejpam-5274	213	14	.	.	PUNCT
ejpam-5274	214	1	thus	thus	ADV
ejpam-5274	214	2	,	,	PUNCT
ejpam-5274	214	3	|ngg(v)|	|ngg(v)|	NOUN
ejpam-5274	214	4	=	=	SYM
ejpam-5274	214	5	|ng(v)|+1	|ng(v)|+1	NOUN
ejpam-5274	214	6	and	and	CCONJ
ejpam-5274	214	7	|ngg(v)|	|ngg(v)|	NOUN
ejpam-5274	214	8	=	=	SYM
ejpam-5274	214	9	(	(	PUNCT
ejpam-5274	214	10	n−|ng[v]|)+1	n−|ng[v]|)+1	NOUN
ejpam-5274	214	11	=	=	SYM
ejpam-5274	214	12	n−|ng(v)|	n−|ng(v)|	PROPN
ejpam-5274	214	13	.	.	PUNCT
ejpam-5274	215	1	therefore	therefore	ADV
ejpam-5274	215	2	,	,	PUNCT
ejpam-5274	215	3	fgg(x	fgg(x	NOUN
ejpam-5274	215	4	)	)	PUNCT
ejpam-5274	215	5	=	=	SYM
ejpam-5274	215	6	∑	∑	PUNCT
ejpam-5274	215	7	p∈v	p∈v	NOUN
ejpam-5274	215	8	(	(	PUNCT
ejpam-5274	215	9	gg	gg	NOUN
ejpam-5274	215	10	)	)	PUNCT
ejpam-5274	215	11	x|ngg(p)|	x|ngg(p)|	PUNCT
ejpam-5274	216	1	=	=	PUNCT
ejpam-5274	216	2	∑	∑	PUNCT
ejpam-5274	216	3	v∈v	v∈v	NOUN
ejpam-5274	216	4	(	(	PUNCT
ejpam-5274	216	5	g	g	NOUN
ejpam-5274	216	6	)	)	PUNCT
ejpam-5274	216	7	x|ngg(v)|	x|ngg(v)|	PUNCT
ejpam-5274	217	1	+	+	CCONJ
ejpam-5274	217	2	∑	∑	PUNCT
ejpam-5274	217	3	v∈v	v∈v	NOUN
ejpam-5274	217	4	(	(	PUNCT
ejpam-5274	217	5	g	g	NOUN
ejpam-5274	217	6	)	)	PUNCT
ejpam-5274	217	7	x|ngg(v)|	x|ngg(v)|	NUM
ejpam-5274	218	1	=	=	X
ejpam-5274	218	2	∑	∑	PUNCT
ejpam-5274	218	3	v∈v	v∈v	PROPN
ejpam-5274	218	4	(	(	PUNCT
ejpam-5274	218	5	g	g	NOUN
ejpam-5274	218	6	)	)	PUNCT
ejpam-5274	218	7	x|ng(v)|+1	x|ng(v)|+1	PROPN
ejpam-5274	218	8	+	+	CCONJ
ejpam-5274	218	9	∑	∑	PUNCT
ejpam-5274	218	10	v∈v	v∈v	NOUN
ejpam-5274	218	11	(	(	PUNCT
ejpam-5274	218	12	g	g	NOUN
ejpam-5274	218	13	)	)	PUNCT
ejpam-5274	218	14	xn−|ng(v)|	xn−|ng(v)|	PROPN
ejpam-5274	218	15	=	=	PUNCT
ejpam-5274	218	16	x	x	PUNCT
ejpam-5274	218	17	∑	∑	PUNCT
ejpam-5274	218	18	v∈v	v∈v	NOUN
ejpam-5274	218	19	(	(	PUNCT
ejpam-5274	218	20	g	g	NOUN
ejpam-5274	218	21	)	)	PUNCT
ejpam-5274	218	22	x|ng(v)|	x|ng(v)|	PROPN
ejpam-5274	219	1	+	+	CCONJ
ejpam-5274	219	2	xn	xn	PROPN
ejpam-5274	219	3	∑	∑	PUNCT
ejpam-5274	219	4	v∈v	v∈v	PROPN
ejpam-5274	219	5	(	(	PUNCT
ejpam-5274	219	6	g	g	NOUN
ejpam-5274	219	7	)	)	PUNCT
ejpam-5274	219	8	x−|ng(v)|	x−|ng(v)|	X
ejpam-5274	219	9	=	=	PUNCT
ejpam-5274	219	10	xfg(x	xfg(x	PROPN
ejpam-5274	219	11	)	)	PUNCT
ejpam-5274	220	1	+	+	CCONJ
ejpam-5274	220	2	xnfg	xnfg	PROPN
ejpam-5274	220	3	(	(	PUNCT
ejpam-5274	220	4	1	1	NUM
ejpam-5274	220	5	x	x	NOUN
ejpam-5274	220	6	)	)	PUNCT
ejpam-5274	220	7	.	.	PUNCT
ejpam-5274	221	1	corollary	corollary	ADJ
ejpam-5274	221	2	5	5	NUM
ejpam-5274	221	3	.	.	PUNCT
ejpam-5274	222	1	let	let	VERB
ejpam-5274	222	2	n	n	PRON
ejpam-5274	222	3	be	be	AUX
ejpam-5274	222	4	a	a	DET
ejpam-5274	222	5	positive	positive	ADJ
ejpam-5274	222	6	integer	integer	NOUN
ejpam-5274	222	7	.	.	PUNCT
ejpam-5274	223	1	then	then	ADV
ejpam-5274	223	2	(	(	PUNCT
ejpam-5274	223	3	i	i	NOUN
ejpam-5274	223	4	)	)	PUNCT
ejpam-5274	223	5	fpnpn	fpnpn	NOUN
ejpam-5274	223	6	(	(	PUNCT
ejpam-5274	223	7	x	x	NOUN
ejpam-5274	223	8	)	)	PUNCT
ejpam-5274	223	9	=	=	SYM
ejpam-5274	223	10	2xn−1	2xn−1	PROPN
ejpam-5274	223	11	+	+	CCONJ
ejpam-5274	223	12	(	(	PUNCT
ejpam-5274	223	13	n−	n−	NOUN
ejpam-5274	223	14	2)xn−2	2)xn−2	NUM
ejpam-5274	223	15	+	+	CCONJ
ejpam-5274	223	16	(	(	PUNCT
ejpam-5274	223	17	n−	n−	NOUN
ejpam-5274	223	18	2)x3	2)x3	NUM
ejpam-5274	223	19	+	+	CCONJ
ejpam-5274	223	20	2x2	2x2	NUM
ejpam-5274	223	21	for	for	ADP
ejpam-5274	223	22	n	n	PRON
ejpam-5274	223	23	≥	≥	NOUN
ejpam-5274	223	24	2	2	NUM
ejpam-5274	223	25	;	;	PUNCT
ejpam-5274	223	26	(	(	PUNCT
ejpam-5274	223	27	ii	ii	NOUN
ejpam-5274	223	28	)	)	PUNCT
ejpam-5274	223	29	fcncn	fcncn	NOUN
ejpam-5274	223	30	(	(	PUNCT
ejpam-5274	223	31	x	x	NOUN
ejpam-5274	223	32	)	)	PUNCT
ejpam-5274	223	33	=	=	SYM
ejpam-5274	223	34	nxn−2	nxn−2	PROPN
ejpam-5274	223	35	+	+	CCONJ
ejpam-5274	223	36	nx3	nx3	ADJ
ejpam-5274	223	37	for	for	ADP
ejpam-5274	223	38	n	n	X
ejpam-5274	223	39	≥	≥	NOUN
ejpam-5274	223	40	3	3	NUM
ejpam-5274	223	41	;	;	PUNCT
ejpam-5274	223	42	and	and	CCONJ
ejpam-5274	223	43	(	(	PUNCT
ejpam-5274	223	44	iii	iii	NOUN
ejpam-5274	223	45	)	)	PUNCT
ejpam-5274	223	46	fsnsn	fsnsn	NOUN
ejpam-5274	223	47	(	(	PUNCT
ejpam-5274	223	48	x	x	X
ejpam-5274	223	49	)	)	PUNCT
ejpam-5274	223	50	=	=	SYM
ejpam-5274	223	51	xn+1	xn+1	PROPN
ejpam-5274	224	1	+	+	CCONJ
ejpam-5274	224	2	nxn	nxn	PROPN
ejpam-5274	224	3	+	+	CCONJ
ejpam-5274	224	4	nx2	nx2	PROPN
ejpam-5274	224	5	+	+	CCONJ
ejpam-5274	224	6	x	x	X
ejpam-5274	224	7	for	for	ADP
ejpam-5274	224	8	n	n	PRON
ejpam-5274	224	9	≥	≥	NUM
ejpam-5274	224	10	2	2	NUM
ejpam-5274	224	11	.	.	PUNCT
ejpam-5274	224	12	proof	proof	NOUN
ejpam-5274	224	13	.	.	PUNCT
ejpam-5274	225	1	from	from	ADP
ejpam-5274	225	2	theorem	theorem	NOUN
ejpam-5274	225	3	9	9	NUM
ejpam-5274	225	4	and	and	CCONJ
ejpam-5274	225	5	the	the	DET
ejpam-5274	225	6	polynomial	polynomial	ADJ
ejpam-5274	225	7	representations	representation	NOUN
ejpam-5274	225	8	fpn(x	fpn(x	PROPN
ejpam-5274	225	9	)	)	PUNCT
ejpam-5274	225	10	=	=	SYM
ejpam-5274	225	11	(	(	PUNCT
ejpam-5274	225	12	n−	n−	NOUN
ejpam-5274	225	13	2)x2	2)x2	NUM
ejpam-5274	225	14	+	+	CCONJ
ejpam-5274	225	15	2x	2x	NUM
ejpam-5274	225	16	,	,	PUNCT
ejpam-5274	225	17	fcn(x	fcn(x	X
ejpam-5274	225	18	)	)	PUNCT
ejpam-5274	225	19	=	=	SYM
ejpam-5274	225	20	nx2	nx2	PROPN
ejpam-5274	225	21	,	,	PUNCT
ejpam-5274	225	22	and	and	CCONJ
ejpam-5274	225	23	fsn(x	fsn(x	X
ejpam-5274	225	24	)	)	PUNCT
ejpam-5274	225	25	=	=	SYM
ejpam-5274	225	26	xn	xn	PROPN
ejpam-5274	226	1	+	+	NUM
ejpam-5274	226	2	nx	nx	NOUN
ejpam-5274	226	3	of	of	ADP
ejpam-5274	226	4	pn	pn	PROPN
ejpam-5274	226	5	,	,	PUNCT
ejpam-5274	226	6	cn	cn	PROPN
ejpam-5274	226	7	,	,	PUNCT
ejpam-5274	226	8	and	and	CCONJ
ejpam-5274	226	9	sn	sn	PROPN
ejpam-5274	226	10	,	,	PUNCT
ejpam-5274	226	11	respectively	respectively	ADV
ejpam-5274	226	12	,	,	PUNCT
ejpam-5274	226	13	we	we	PRON
ejpam-5274	226	14	have	have	VERB
ejpam-5274	226	15	fpnpn	fpnpn	NOUN
ejpam-5274	226	16	(	(	PUNCT
ejpam-5274	226	17	x	x	NOUN
ejpam-5274	226	18	)	)	PUNCT
ejpam-5274	226	19	=	=	PUNCT
ejpam-5274	227	1	x[(n−2)x2	x[(n−2)x2	PROPN
ejpam-5274	227	2	+	+	PROPN
ejpam-5274	227	3	2x]+xn[(n−2	2x]+xn[(n−2	NUM
ejpam-5274	227	4	)	)	PUNCT
ejpam-5274	227	5	1	1	NUM
ejpam-5274	228	1	x2	x2	NOUN
ejpam-5274	228	2	+2	+2	ADV
ejpam-5274	228	3	1	1	NUM
ejpam-5274	228	4	x	x	SYM
ejpam-5274	228	5	]	]	X
ejpam-5274	228	6	=	=	PUNCT
ejpam-5274	229	1	2xn−1+(n−2)xn−2+(n−2)x3	2xn−1+(n−2)xn−2+(n−2)x3	NUM
ejpam-5274	229	2	+	+	NOUN
ejpam-5274	229	3	2x2	2x2	NUM
ejpam-5274	229	4	,	,	PUNCT
ejpam-5274	229	5	jayhan	jayhan	PROPN
ejpam-5274	229	6	cruz	cruz	PROPN
ejpam-5274	229	7	,	,	PUNCT
ejpam-5274	229	8	g.	g.	PROPN
ejpam-5274	229	9	malacas	malacas	PROPN
ejpam-5274	229	10	,	,	PUNCT
ejpam-5274	229	11	s.	s.	PROPN
ejpam-5274	229	12	canoy	canoy	PROPN
ejpam-5274	229	13	,	,	PUNCT
ejpam-5274	229	14	jr	jr	PROPN
ejpam-5274	229	15	.	.	PROPN
ejpam-5274	229	16	/	/	SYM
ejpam-5274	229	17	eur	eur	PROPN
ejpam-5274	229	18	.	.	PUNCT
ejpam-5274	230	1	j.	j.	PROPN
ejpam-5274	230	2	pure	pure	PROPN
ejpam-5274	230	3	appl	appl	PROPN
ejpam-5274	230	4	.	.	PROPN
ejpam-5274	230	5	math	math	PROPN
ejpam-5274	230	6	,	,	PUNCT
ejpam-5274	230	7	17	17	NUM
ejpam-5274	230	8	(	(	PUNCT
ejpam-5274	230	9	3	3	NUM
ejpam-5274	230	10	)	)	PUNCT
ejpam-5274	230	11	(	(	PUNCT
ejpam-5274	230	12	2024	2024	NUM
ejpam-5274	230	13	)	)	PUNCT
ejpam-5274	230	14	,	,	PUNCT
ejpam-5274	230	15	1449	1449	NUM
ejpam-5274	230	16	-	-	SYM
ejpam-5274	230	17	1462	1462	NUM
ejpam-5274	230	18	1456	1456	NUM
ejpam-5274	230	19	fcncn	fcncn	NOUN
ejpam-5274	230	20	(	(	PUNCT
ejpam-5274	230	21	x	x	NOUN
ejpam-5274	230	22	)	)	PUNCT
ejpam-5274	230	23	=	=	SYM
ejpam-5274	230	24	x(nx2	x(nx2	PROPN
ejpam-5274	230	25	)	)	PUNCT
ejpam-5274	231	1	+	+	CCONJ
ejpam-5274	231	2	xn(n	xn(n	NUM
ejpam-5274	231	3	1	1	NUM
ejpam-5274	231	4	x2	x2	NOUN
ejpam-5274	231	5	)	)	PUNCT
ejpam-5274	232	1	=	=	PUNCT
ejpam-5274	232	2	nxn−2	nxn−2	PROPN
ejpam-5274	232	3	+	+	CCONJ
ejpam-5274	232	4	nx3	nx3	ADJ
ejpam-5274	232	5	,	,	PUNCT
ejpam-5274	232	6	and	and	CCONJ
ejpam-5274	232	7	fsnsn	fsnsn	X
ejpam-5274	232	8	(	(	PUNCT
ejpam-5274	232	9	x	x	X
ejpam-5274	232	10	)	)	PUNCT
ejpam-5274	232	11	=	=	SYM
ejpam-5274	232	12	x(xn	x(xn	PROPN
ejpam-5274	233	1	+	+	CCONJ
ejpam-5274	233	2	nx	nx	X
ejpam-5274	233	3	)	)	PUNCT
ejpam-5274	234	1	+	+	CCONJ
ejpam-5274	234	2	xn+1	xn+1	NUM
ejpam-5274	234	3	(	(	PUNCT
ejpam-5274	234	4	1	1	NUM
ejpam-5274	234	5	xn	xn	NUM
ejpam-5274	234	6	+	+	NUM
ejpam-5274	234	7	n	n	CCONJ
ejpam-5274	234	8	1	1	NUM
ejpam-5274	234	9	x	x	X
ejpam-5274	234	10	)	)	PUNCT
ejpam-5274	234	11	=	=	SYM
ejpam-5274	234	12	xn+1	xn+1	PROPN
ejpam-5274	235	1	+	+	CCONJ
ejpam-5274	235	2	nxn	nxn	PROPN
ejpam-5274	235	3	+	+	CCONJ
ejpam-5274	235	4	nx2	nx2	PROPN
ejpam-5274	236	1	+	+	CCONJ
ejpam-5274	236	2	x.	x.	NOUN
ejpam-5274	236	3	this	this	PRON
ejpam-5274	236	4	proves	prove	VERB
ejpam-5274	236	5	the	the	DET
ejpam-5274	236	6	assertion	assertion	NOUN
ejpam-5274	236	7	.	.	PUNCT
ejpam-5274	237	1	theorem	theorem	ADJ
ejpam-5274	237	2	10	10	NUM
ejpam-5274	237	3	.	.	PUNCT
ejpam-5274	238	1	let	let	VERB
ejpam-5274	238	2	g	g	PRON
ejpam-5274	238	3	be	be	AUX
ejpam-5274	238	4	a	a	DET
ejpam-5274	238	5	non	non	ADJ
ejpam-5274	238	6	-	-	ADJ
ejpam-5274	238	7	trivial	trivial	ADJ
ejpam-5274	238	8	connected	connected	ADJ
ejpam-5274	238	9	graph	graph	NOUN
ejpam-5274	238	10	with	with	ADP
ejpam-5274	238	11	degree	degree	NOUN
ejpam-5274	238	12	sequence	sequence	NOUN
ejpam-5274	238	13	⟨d1	⟨d1	PROPN
ejpam-5274	238	14	,	,	PUNCT
ejpam-5274	238	15	d2	d2	PROPN
ejpam-5274	238	16	,	,	PUNCT
ejpam-5274	238	17	·	·	PUNCT
ejpam-5274	238	18	·	·	PUNCT
ejpam-5274	238	19	·	·	PUNCT
ejpam-5274	239	1	dn⟩.	dn⟩.	NOUN
ejpam-5274	239	2	then	then	ADV
ejpam-5274	239	3	the	the	DET
ejpam-5274	239	4	terms	term	NOUN
ejpam-5274	239	5	of	of	ADP
ejpam-5274	239	6	the	the	DET
ejpam-5274	239	7	degree	degree	NOUN
ejpam-5274	239	8	sequence	sequence	NOUN
ejpam-5274	239	9	of	of	ADP
ejpam-5274	239	10	gg	gg	PROPN
ejpam-5274	239	11	are	be	AUX
ejpam-5274	239	12	the	the	DET
ejpam-5274	239	13	elements	element	NOUN
ejpam-5274	239	14	of	of	ADP
ejpam-5274	239	15	the	the	DET
ejpam-5274	239	16	set	set	NOUN
ejpam-5274	239	17	{	{	PUNCT
ejpam-5274	239	18	di	di	NOUN
ejpam-5274	239	19	+	+	NOUN
ejpam-5274	239	20	1	1	NUM
ejpam-5274	239	21	:	:	SYM
ejpam-5274	239	22	1	1	NUM
ejpam-5274	239	23	≤	≤	NUM
ejpam-5274	239	24	i	i	PRON
ejpam-5274	239	25	≤	≤	NOUN
ejpam-5274	239	26	n	n	CCONJ
ejpam-5274	239	27	}	}	PUNCT
ejpam-5274	239	28	∪	∪	ADJ
ejpam-5274	239	29	{	{	PUNCT
ejpam-5274	239	30	n−	n−	NOUN
ejpam-5274	239	31	di	di	NOUN
ejpam-5274	239	32	:	:	PUNCT
ejpam-5274	239	33	1	1	NUM
ejpam-5274	239	34	≤	≤	NUM
ejpam-5274	239	35	i	i	PRON
ejpam-5274	239	36	≤	≤	NOUN
ejpam-5274	239	37	n	n	CCONJ
ejpam-5274	239	38	}	}	PUNCT
ejpam-5274	239	39	.	.	PUNCT
ejpam-5274	240	1	proof	proof	NOUN
ejpam-5274	240	2	.	.	PUNCT
ejpam-5274	241	1	from	from	ADP
ejpam-5274	241	2	the	the	DET
ejpam-5274	241	3	polynomial	polynomial	ADJ
ejpam-5274	241	4	representation	representation	NOUN
ejpam-5274	241	5	fg(x	fg(x	NUM
ejpam-5274	241	6	)	)	PUNCT
ejpam-5274	242	1	=	=	PUNCT
ejpam-5274	242	2	p∑	p∑	X
ejpam-5274	243	1	i=1	i=1	PROPN
ejpam-5274	243	2	xdi	xdi	PROPN
ejpam-5274	243	3	of	of	ADP
ejpam-5274	243	4	g	g	PROPN
ejpam-5274	243	5	and	and	CCONJ
ejpam-5274	243	6	from	from	ADP
ejpam-5274	243	7	theorem	theorem	ADJ
ejpam-5274	243	8	9	9	NUM
ejpam-5274	243	9	,	,	PUNCT
ejpam-5274	243	10	we	we	PRON
ejpam-5274	243	11	find	find	VERB
ejpam-5274	243	12	that	that	SCONJ
ejpam-5274	243	13	fgg(x	fgg(x	NOUN
ejpam-5274	243	14	)	)	PUNCT
ejpam-5274	243	15	=	=	SYM
ejpam-5274	244	1	xfg(x	xfg(x	X
ejpam-5274	244	2	)	)	PUNCT
ejpam-5274	245	1	+	+	CCONJ
ejpam-5274	245	2	xnfg	xnfg	PROPN
ejpam-5274	245	3	1	1	NUM
ejpam-5274	245	4	x	x	NOUN
ejpam-5274	245	5	)	)	PUNCT
ejpam-5274	246	1	=	=	PUNCT
ejpam-5274	247	1	x	x	PUNCT
ejpam-5274	247	2	n∑	n∑	NOUN
ejpam-5274	247	3	i=1	i=1	PROPN
ejpam-5274	247	4	xdi	xdi	PROPN
ejpam-5274	248	1	+	+	CCONJ
ejpam-5274	248	2	xn	xn	PROPN
ejpam-5274	248	3	n∑	n∑	PROPN
ejpam-5274	249	1	i=1	i=1	PROPN
ejpam-5274	249	2	x−di	x−di	PROPN
ejpam-5274	250	1	=	=	SYM
ejpam-5274	251	1	n∑	n∑	NOUN
ejpam-5274	251	2	i=1	i=1	PROPN
ejpam-5274	251	3	xdi+1	xdi+1	PROPN
ejpam-5274	252	1	+	+	X
ejpam-5274	253	1	n∑	n∑	X
ejpam-5274	253	2	i=1	i=1	X
ejpam-5274	253	3	xn−di	xn−di	PROPN
ejpam-5274	253	4	.	.	PUNCT
ejpam-5274	254	1	therefore	therefore	ADV
ejpam-5274	254	2	,	,	PUNCT
ejpam-5274	254	3	the	the	DET
ejpam-5274	254	4	terms	term	NOUN
ejpam-5274	254	5	of	of	ADP
ejpam-5274	254	6	the	the	DET
ejpam-5274	254	7	degree	degree	NOUN
ejpam-5274	254	8	sequence	sequence	NOUN
ejpam-5274	254	9	of	of	ADP
ejpam-5274	254	10	gg	gg	PROPN
ejpam-5274	254	11	are	be	AUX
ejpam-5274	254	12	exactly	exactly	ADV
ejpam-5274	254	13	the	the	DET
ejpam-5274	254	14	elements	element	NOUN
ejpam-5274	254	15	of	of	ADP
ejpam-5274	254	16	the	the	DET
ejpam-5274	254	17	set	set	NOUN
ejpam-5274	254	18	{	{	PUNCT
ejpam-5274	254	19	di	di	NOUN
ejpam-5274	254	20	+	+	NOUN
ejpam-5274	254	21	1	1	NUM
ejpam-5274	254	22	:	:	SYM
ejpam-5274	254	23	1	1	NUM
ejpam-5274	254	24	≤	≤	NUM
ejpam-5274	254	25	i	i	PRON
ejpam-5274	254	26	≤	≤	NOUN
ejpam-5274	254	27	n	n	CCONJ
ejpam-5274	254	28	}	}	PUNCT
ejpam-5274	254	29	∪	∪	ADJ
ejpam-5274	254	30	{	{	PUNCT
ejpam-5274	254	31	n−	n−	NOUN
ejpam-5274	254	32	di	di	NOUN
ejpam-5274	254	33	:	:	PUNCT
ejpam-5274	254	34	1	1	NUM
ejpam-5274	254	35	≤	≤	NUM
ejpam-5274	254	36	i	i	PRON
ejpam-5274	254	37	≤	≤	NOUN
ejpam-5274	254	38	n	n	CCONJ
ejpam-5274	254	39	}	}	PUNCT
ejpam-5274	254	40	.	.	PUNCT
ejpam-5274	255	1	theorem	theorem	NOUN
ejpam-5274	255	2	11	11	NUM
ejpam-5274	255	3	.	.	PUNCT
ejpam-5274	256	1	let	let	VERB
ejpam-5274	256	2	g	g	NOUN
ejpam-5274	256	3	and	and	CCONJ
ejpam-5274	256	4	h	h	PROPN
ejpam-5274	256	5	be	be	VERB
ejpam-5274	256	6	non	non	ADJ
ejpam-5274	256	7	-	-	ADJ
ejpam-5274	256	8	trivial	trivial	ADJ
ejpam-5274	256	9	connected	connected	ADJ
ejpam-5274	256	10	graphs	graph	NOUN
ejpam-5274	256	11	.	.	PUNCT
ejpam-5274	257	1	then	then	ADV
ejpam-5274	257	2	fg⊠h(x	fg⊠h(x	PROPN
ejpam-5274	257	3	)	)	PUNCT
ejpam-5274	257	4	=	=	SYM
ejpam-5274	257	5	∑	∑	PUNCT
ejpam-5274	257	6	v∈v	v∈v	PROPN
ejpam-5274	257	7	(	(	PUNCT
ejpam-5274	257	8	g	g	NOUN
ejpam-5274	257	9	)	)	PUNCT
ejpam-5274	257	10	fh(x|ng(v)|+1)x|ng(v)|	fh(x|ng(v)|+1)x|ng(v)|	PUNCT
ejpam-5274	258	1	=	=	PUNCT
ejpam-5274	258	2	∑	∑	PUNCT
ejpam-5274	258	3	p∈v	p∈v	NOUN
ejpam-5274	258	4	(	(	PUNCT
ejpam-5274	258	5	h	h	NOUN
ejpam-5274	258	6	)	)	PUNCT
ejpam-5274	258	7	fg(x	fg(x	NUM
ejpam-5274	258	8	|nh(p)|+1)x|nh(p)|	|nh(p)|+1)x|nh(p)|	NOUN
ejpam-5274	258	9	.	.	PUNCT
ejpam-5274	258	10	proof	proof	NOUN
ejpam-5274	258	11	.	.	PUNCT
ejpam-5274	259	1	let	let	VERB
ejpam-5274	259	2	(	(	PUNCT
ejpam-5274	259	3	v	v	NOUN
ejpam-5274	259	4	,	,	PUNCT
ejpam-5274	259	5	p	p	NOUN
ejpam-5274	259	6	)	)	PUNCT
ejpam-5274	259	7	∈	∈	PROPN
ejpam-5274	259	8	v	v	NOUN
ejpam-5274	259	9	(	(	PUNCT
ejpam-5274	259	10	g	g	PROPN
ejpam-5274	259	11	⊠	⊠	PROPN
ejpam-5274	259	12	h	h	NOUN
ejpam-5274	259	13	)	)	PUNCT
ejpam-5274	259	14	.	.	PUNCT
ejpam-5274	260	1	let	let	VERB
ejpam-5274	260	2	d1	d1	PROPN
ejpam-5274	260	3	=	=	SYM
ejpam-5274	260	4	ng(v	ng(v	X
ejpam-5274	260	5	)	)	PUNCT
ejpam-5274	260	6	×	×	NOUN
ejpam-5274	260	7	{	{	PUNCT
ejpam-5274	260	8	p	p	NOUN
ejpam-5274	260	9	}	}	PUNCT
ejpam-5274	260	10	,	,	PUNCT
ejpam-5274	260	11	d2	d2	PROPN
ejpam-5274	260	12	=	=	SYM
ejpam-5274	260	13	{	{	PUNCT
ejpam-5274	260	14	v	v	NOUN
ejpam-5274	260	15	}	}	PUNCT
ejpam-5274	260	16	×	×	NOUN
ejpam-5274	260	17	nh(p	nh(p	NUM
ejpam-5274	260	18	)	)	PUNCT
ejpam-5274	260	19	,	,	PUNCT
ejpam-5274	260	20	and	and	CCONJ
ejpam-5274	260	21	d3	d3	PROPN
ejpam-5274	260	22	=	=	SYM
ejpam-5274	260	23	ng(v	ng(v	X
ejpam-5274	260	24	)	)	PUNCT
ejpam-5274	260	25	×	×	NOUN
ejpam-5274	260	26	nh(p	nh(p	NUM
ejpam-5274	260	27	)	)	PUNCT
ejpam-5274	260	28	.	.	PUNCT
ejpam-5274	261	1	by	by	ADP
ejpam-5274	261	2	definition	definition	NOUN
ejpam-5274	261	3	of	of	ADP
ejpam-5274	261	4	strong	strong	ADJ
ejpam-5274	261	5	product	product	NOUN
ejpam-5274	261	6	of	of	ADP
ejpam-5274	261	7	two	two	NUM
ejpam-5274	261	8	graphs	graph	NOUN
ejpam-5274	261	9	,	,	PUNCT
ejpam-5274	261	10	it	it	PRON
ejpam-5274	261	11	follows	follow	VERB
ejpam-5274	261	12	that	that	SCONJ
ejpam-5274	261	13	ng⊠h((v	ng⊠h((v	NOUN
ejpam-5274	261	14	,	,	PUNCT
ejpam-5274	261	15	p	p	NOUN
ejpam-5274	261	16	)	)	PUNCT
ejpam-5274	261	17	)	)	PUNCT
ejpam-5274	262	1	=	=	PUNCT
ejpam-5274	262	2	d1	d1	PROPN
ejpam-5274	262	3	∪d2	∪d2	ADJ
ejpam-5274	262	4	∪d3	∪d3	X
ejpam-5274	262	5	.	.	PUNCT
ejpam-5274	263	1	hence	hence	ADV
ejpam-5274	263	2	,	,	PUNCT
ejpam-5274	263	3	|ng⊠h((v	|ng⊠h((v	PROPN
ejpam-5274	263	4	,	,	PUNCT
ejpam-5274	263	5	p))|	p))|	PROPN
ejpam-5274	263	6	=	=	PUNCT
ejpam-5274	263	7	|ng(v)|+	|ng(v)|+	PROPN
ejpam-5274	263	8	|nh(p)|+	|nh(p)|+	PROPN
ejpam-5274	263	9	|ng(v)||nh(p)|	|ng(v)||nh(p)|	PROPN
ejpam-5274	263	10	.	.	PUNCT
ejpam-5274	264	1	thus	thus	ADV
ejpam-5274	264	2	,	,	PUNCT
ejpam-5274	264	3	fg⊠h(x	fg⊠h(x	PROPN
ejpam-5274	264	4	)	)	PUNCT
ejpam-5274	264	5	=	=	SYM
ejpam-5274	265	1	∑	∑	PUNCT
ejpam-5274	265	2	(	(	PUNCT
ejpam-5274	265	3	v	v	NOUN
ejpam-5274	265	4	,	,	PUNCT
ejpam-5274	265	5	p)∈v	p)∈v	X
ejpam-5274	265	6	(	(	PUNCT
ejpam-5274	265	7	g⊠h	g⊠h	PROPN
ejpam-5274	265	8	)	)	PUNCT
ejpam-5274	265	9	x|ng⊠h(v	x|ng⊠h(v	PROPN
ejpam-5274	265	10	,	,	PUNCT
ejpam-5274	265	11	p)|	p)|	NOUN
ejpam-5274	265	12	=	=	SYM
ejpam-5274	265	13	∑	∑	PROPN
ejpam-5274	265	14	(	(	PUNCT
ejpam-5274	265	15	v	v	NOUN
ejpam-5274	265	16	,	,	PUNCT
ejpam-5274	265	17	p)∈v	p)∈v	X
ejpam-5274	265	18	(	(	PUNCT
ejpam-5274	265	19	g⊠h	g⊠h	PROPN
ejpam-5274	265	20	)	)	PUNCT
ejpam-5274	265	21	x|ng(v)|+|nh(p)|+|ng(v)||nh(p)|	x|ng(v)|+|nh(p)|+|ng(v)||nh(p)|	NOUN
ejpam-5274	265	22	=	=	SYM
ejpam-5274	265	23	∑	∑	PUNCT
ejpam-5274	265	24	v∈v	v∈v	PROPN
ejpam-5274	265	25	(	(	PUNCT
ejpam-5274	265	26	g	g	NOUN
ejpam-5274	265	27	)	)	PUNCT
ejpam-5274	265	28	x|ng(v)|	x|ng(v)|	PROPN
ejpam-5274	265	29	∑	∑	PUNCT
ejpam-5274	266	1	p∈v	p∈v	NOUN
ejpam-5274	266	2	(	(	PUNCT
ejpam-5274	266	3	h	h	NOUN
ejpam-5274	266	4	)	)	PUNCT
ejpam-5274	266	5	x(|ng(v)|+1)|ng(p)|	x(|ng(v)|+1)|ng(p)|	NOUN
ejpam-5274	267	1	jayhan	jayhan	PROPN
ejpam-5274	267	2	cruz	cruz	PROPN
ejpam-5274	267	3	,	,	PUNCT
ejpam-5274	267	4	g.	g.	PROPN
ejpam-5274	267	5	malacas	malacas	PROPN
ejpam-5274	267	6	,	,	PUNCT
ejpam-5274	267	7	s.	s.	PROPN
ejpam-5274	267	8	canoy	canoy	PROPN
ejpam-5274	267	9	,	,	PUNCT
ejpam-5274	267	10	jr	jr	PROPN
ejpam-5274	267	11	.	.	PROPN
ejpam-5274	267	12	/	/	SYM
ejpam-5274	267	13	eur	eur	PROPN
ejpam-5274	267	14	.	.	PUNCT
ejpam-5274	268	1	j.	j.	PROPN
ejpam-5274	268	2	pure	pure	PROPN
ejpam-5274	268	3	appl	appl	PROPN
ejpam-5274	268	4	.	.	PROPN
ejpam-5274	268	5	math	math	PROPN
ejpam-5274	268	6	,	,	PUNCT
ejpam-5274	268	7	17	17	NUM
ejpam-5274	268	8	(	(	PUNCT
ejpam-5274	268	9	3	3	NUM
ejpam-5274	268	10	)	)	PUNCT
ejpam-5274	268	11	(	(	PUNCT
ejpam-5274	268	12	2024	2024	NUM
ejpam-5274	268	13	)	)	PUNCT
ejpam-5274	268	14	,	,	PUNCT
ejpam-5274	268	15	1449	1449	NUM
ejpam-5274	268	16	-	-	SYM
ejpam-5274	268	17	1462	1462	NUM
ejpam-5274	268	18	1457	1457	NUM
ejpam-5274	268	19	=	=	SYM
ejpam-5274	268	20	∑	∑	PUNCT
ejpam-5274	268	21	v∈v	v∈v	PROPN
ejpam-5274	268	22	(	(	PUNCT
ejpam-5274	268	23	g	g	NOUN
ejpam-5274	268	24	)	)	PUNCT
ejpam-5274	268	25	x|ng(v)|fh(x|ng(v)|+1	x|ng(v)|fh(x|ng(v)|+1	PROPN
ejpam-5274	268	26	)	)	PUNCT
ejpam-5274	268	27	.	.	PUNCT
ejpam-5274	269	1	since	since	SCONJ
ejpam-5274	269	2	∑	∑	PROPN
ejpam-5274	269	3	(	(	PUNCT
ejpam-5274	269	4	v	v	NOUN
ejpam-5274	269	5	,	,	PUNCT
ejpam-5274	269	6	p)∈v	p)∈v	X
ejpam-5274	269	7	(	(	PUNCT
ejpam-5274	269	8	g⊠h	g⊠h	PROPN
ejpam-5274	269	9	)	)	PUNCT
ejpam-5274	269	10	x|ng(v)|+|nh(p)|+|ng(v)||nh(p)|	x|ng(v)|+|nh(p)|+|ng(v)||nh(p)|	NOUN
ejpam-5274	269	11	=	=	SYM
ejpam-5274	269	12	∑	∑	PUNCT
ejpam-5274	269	13	p∈v	p∈v	NOUN
ejpam-5274	269	14	(	(	PUNCT
ejpam-5274	269	15	h	h	NOUN
ejpam-5274	269	16	)	)	PUNCT
ejpam-5274	269	17	x|nh(p)|	x|nh(p)|	PROPN
ejpam-5274	269	18	∑	∑	PUNCT
ejpam-5274	269	19	v∈v	v∈v	PROPN
ejpam-5274	269	20	(	(	PUNCT
ejpam-5274	269	21	g	g	NOUN
ejpam-5274	269	22	)	)	PUNCT
ejpam-5274	269	23	x(|nh(p)|+1)|ng(v)|	x(|nh(p)|+1)|ng(v)|	PROPN
ejpam-5274	269	24	=	=	PRON
ejpam-5274	270	1	∑	∑	PUNCT
ejpam-5274	270	2	p∈v	p∈v	NOUN
ejpam-5274	270	3	(	(	PUNCT
ejpam-5274	270	4	h	h	NOUN
ejpam-5274	270	5	)	)	PUNCT
ejpam-5274	270	6	x|nh(p)|fg(x	x|nh(p)|fg(x	PROPN
ejpam-5274	270	7	|nh(p)|+1	|nh(p)|+1	PROPN
ejpam-5274	270	8	)	)	PUNCT
ejpam-5274	271	1	,	,	PUNCT
ejpam-5274	271	2	it	it	PRON
ejpam-5274	271	3	follows	follow	VERB
ejpam-5274	271	4	that	that	SCONJ
ejpam-5274	271	5	fg⊠h(x	fg⊠h(x	NOUN
ejpam-5274	271	6	)	)	PUNCT
ejpam-5274	271	7	=	=	SYM
ejpam-5274	271	8	∑	∑	PUNCT
ejpam-5274	271	9	v∈v	v∈v	PROPN
ejpam-5274	271	10	(	(	PUNCT
ejpam-5274	271	11	g	g	NOUN
ejpam-5274	271	12	)	)	PUNCT
ejpam-5274	271	13	fh(x|ng(v)|+1)x|ng(v)|	fh(x|ng(v)|+1)x|ng(v)|	PUNCT
ejpam-5274	272	1	=	=	PUNCT
ejpam-5274	272	2	∑	∑	PUNCT
ejpam-5274	272	3	p∈v	p∈v	NOUN
ejpam-5274	272	4	(	(	PUNCT
ejpam-5274	272	5	h	h	NOUN
ejpam-5274	272	6	)	)	PUNCT
ejpam-5274	272	7	fg(x	fg(x	NUM
ejpam-5274	272	8	|nh(p)|+1)x|nh(p)|	|nh(p)|+1)x|nh(p)|	PROPN
ejpam-5274	272	9	.	.	PUNCT
ejpam-5274	272	10	corollary	corollary	ADJ
ejpam-5274	272	11	6	6	NUM
ejpam-5274	272	12	.	.	PUNCT
ejpam-5274	273	1	let	let	VERB
ejpam-5274	273	2	g	g	NOUN
ejpam-5274	273	3	and	and	CCONJ
ejpam-5274	273	4	h	h	PROPN
ejpam-5274	273	5	be	be	VERB
ejpam-5274	273	6	non	non	ADJ
ejpam-5274	273	7	-	-	ADJ
ejpam-5274	273	8	trivial	trivial	ADJ
ejpam-5274	273	9	r1	r1	NOUN
ejpam-5274	273	10	-	-	PUNCT
ejpam-5274	273	11	regular	regular	ADJ
ejpam-5274	273	12	and	and	CCONJ
ejpam-5274	273	13	r2	r2	NOUN
ejpam-5274	273	14	-	-	PUNCT
ejpam-5274	273	15	regular	regular	ADJ
ejpam-5274	273	16	connected	connected	ADJ
ejpam-5274	273	17	graphs	graph	NOUN
ejpam-5274	273	18	of	of	ADP
ejpam-5274	273	19	orders	order	NOUN
ejpam-5274	273	20	m	m	VERB
ejpam-5274	273	21	and	and	CCONJ
ejpam-5274	273	22	n	n	CCONJ
ejpam-5274	273	23	,	,	PUNCT
ejpam-5274	273	24	respectively	respectively	ADV
ejpam-5274	273	25	.	.	PUNCT
ejpam-5274	274	1	then	then	ADV
ejpam-5274	274	2	fg⊠h(x	fg⊠h(x	PROPN
ejpam-5274	274	3	)	)	PUNCT
ejpam-5274	274	4	=	=	PRON
ejpam-5274	274	5	mnxr1+r2+r1r2	mnxr1+r2+r1r2	X
ejpam-5274	274	6	.	.	PUNCT
ejpam-5274	275	1	proof	proof	NOUN
ejpam-5274	275	2	.	.	PUNCT
ejpam-5274	276	1	since	since	SCONJ
ejpam-5274	276	2	g	g	PROPN
ejpam-5274	276	3	andh	andh	NOUN
ejpam-5274	276	4	are	be	AUX
ejpam-5274	276	5	,	,	PUNCT
ejpam-5274	276	6	respectively	respectively	ADV
ejpam-5274	276	7	,	,	PUNCT
ejpam-5274	276	8	r1	r1	NOUN
ejpam-5274	276	9	-	-	PUNCT
ejpam-5274	276	10	regular	regular	ADJ
ejpam-5274	276	11	and	and	CCONJ
ejpam-5274	276	12	r2	r2	NOUN
ejpam-5274	276	13	-	-	PUNCT
ejpam-5274	276	14	regular	regular	ADJ
ejpam-5274	276	15	graphs	graph	NOUN
ejpam-5274	276	16	,	,	PUNCT
ejpam-5274	276	17	fg(x	fg(x	NUM
ejpam-5274	276	18	)	)	PUNCT
ejpam-5274	277	1	=	=	SYM
ejpam-5274	277	2	mxr1	mxr1	NOUN
ejpam-5274	277	3	and	and	CCONJ
ejpam-5274	277	4	fh(x	fh(x	PUNCT
ejpam-5274	277	5	)	)	PUNCT
ejpam-5274	278	1	=	=	SYM
ejpam-5274	278	2	nxr2	nxr2	PROPN
ejpam-5274	278	3	.	.	PUNCT
ejpam-5274	279	1	it	it	PRON
ejpam-5274	279	2	follows	follow	VERB
ejpam-5274	279	3	that	that	SCONJ
ejpam-5274	279	4	fh(xr1	fh(xr1	PROPN
ejpam-5274	280	1	+	+	NOUN
ejpam-5274	280	2	1	1	NUM
ejpam-5274	280	3	)	)	PUNCT
ejpam-5274	280	4	=	=	PUNCT
ejpam-5274	280	5	nxr2(r1	nxr2(r1	PROPN
ejpam-5274	280	6	+	+	NOUN
ejpam-5274	280	7	1	1	NUM
ejpam-5274	280	8	)	)	PUNCT
ejpam-5274	280	9	.	.	PUNCT
ejpam-5274	281	1	thus	thus	ADV
ejpam-5274	281	2	,	,	PUNCT
ejpam-5274	281	3	by	by	ADP
ejpam-5274	281	4	theorem	theorem	NOUN
ejpam-5274	281	5	6	6	NUM
ejpam-5274	281	6	,	,	PUNCT
ejpam-5274	281	7	fg⊠h(x	fg⊠h(x	NOUN
ejpam-5274	281	8	)	)	PUNCT
ejpam-5274	281	9	=	=	SYM
ejpam-5274	281	10	∑	∑	PUNCT
ejpam-5274	281	11	v∈v	v∈v	PROPN
ejpam-5274	281	12	(	(	PUNCT
ejpam-5274	281	13	g	g	NOUN
ejpam-5274	281	14	)	)	PUNCT
ejpam-5274	281	15	x|ng(v)|fh(x|ng(v)|+1	x|ng(v)|fh(x|ng(v)|+1	PROPN
ejpam-5274	281	16	)	)	PUNCT
ejpam-5274	281	17	=	=	PUNCT
ejpam-5274	281	18	∑	∑	PUNCT
ejpam-5274	281	19	v∈v	v∈v	NOUN
ejpam-5274	281	20	(	(	PUNCT
ejpam-5274	281	21	g	g	NOUN
ejpam-5274	281	22	)	)	PUNCT
ejpam-5274	281	23	xr1nxr1r2+r2	xr1nxr1r2+r2	X
ejpam-5274	282	1	=	=	SYM
ejpam-5274	282	2	n	n	PROPN
ejpam-5274	282	3	∑	∑	ADV
ejpam-5274	282	4	v∈v	v∈v	NOUN
ejpam-5274	282	5	(	(	PUNCT
ejpam-5274	282	6	g	g	NOUN
ejpam-5274	282	7	)	)	PUNCT
ejpam-5274	282	8	xr1+r2+r1r2	xr1+r2+r1r2	X
ejpam-5274	282	9	=	=	PUNCT
ejpam-5274	282	10	mnxr1+r2+r1r2	mnxr1+r2+r1r2	X
ejpam-5274	282	11	.	.	PUNCT
ejpam-5274	283	1	theorem	theorem	NOUN
ejpam-5274	283	2	12	12	NUM
ejpam-5274	283	3	.	.	PUNCT
ejpam-5274	284	1	let	let	VERB
ejpam-5274	284	2	g	g	NOUN
ejpam-5274	284	3	and	and	CCONJ
ejpam-5274	284	4	h	h	PROPN
ejpam-5274	284	5	be	be	VERB
ejpam-5274	284	6	non	non	ADJ
ejpam-5274	284	7	-	-	ADJ
ejpam-5274	284	8	trivial	trivial	ADJ
ejpam-5274	284	9	connected	connected	ADJ
ejpam-5274	284	10	graphs	graph	NOUN
ejpam-5274	284	11	of	of	ADP
ejpam-5274	284	12	orders	order	NOUN
ejpam-5274	284	13	m	m	VERB
ejpam-5274	284	14	and	and	CCONJ
ejpam-5274	284	15	n	n	CCONJ
ejpam-5274	284	16	,	,	PUNCT
ejpam-5274	284	17	respectively	respectively	ADV
ejpam-5274	284	18	.	.	PUNCT
ejpam-5274	285	1	then	then	ADV
ejpam-5274	285	2	fg⊕h(x	fg⊕h(x	NOUN
ejpam-5274	285	3	)	)	PUNCT
ejpam-5274	285	4	=	=	PUNCT
ejpam-5274	285	5	∑	∑	PUNCT
ejpam-5274	285	6	v∈v	v∈v	PROPN
ejpam-5274	285	7	(	(	PUNCT
ejpam-5274	285	8	g	g	NOUN
ejpam-5274	285	9	)	)	PUNCT
ejpam-5274	285	10	xn|ng(v)|fh(xm−2|ng(v)|	xn|ng(v)|fh(xm−2|ng(v)|	PROPN
ejpam-5274	285	11	)	)	PUNCT
ejpam-5274	285	12	.	.	PUNCT
ejpam-5274	286	1	proof	proof	NOUN
ejpam-5274	286	2	.	.	PUNCT
ejpam-5274	287	1	let	let	VERB
ejpam-5274	287	2	(	(	PUNCT
ejpam-5274	287	3	v	v	NOUN
ejpam-5274	287	4	,	,	PUNCT
ejpam-5274	287	5	p	p	NOUN
ejpam-5274	287	6	)	)	PUNCT
ejpam-5274	287	7	∈	∈	PROPN
ejpam-5274	287	8	v	v	NOUN
ejpam-5274	287	9	(	(	PUNCT
ejpam-5274	287	10	g⊕h	g⊕h	NOUN
ejpam-5274	287	11	)	)	PUNCT
ejpam-5274	287	12	.	.	PUNCT
ejpam-5274	288	1	from	from	ADP
ejpam-5274	288	2	the	the	DET
ejpam-5274	288	3	definition	definition	NOUN
ejpam-5274	288	4	of	of	ADP
ejpam-5274	288	5	g⊕h	g⊕h	NOUN
ejpam-5274	288	6	,	,	PUNCT
ejpam-5274	288	7	it	it	PRON
ejpam-5274	288	8	follows	follow	VERB
ejpam-5274	288	9	that	that	SCONJ
ejpam-5274	288	10	ng⊕h((v	ng⊕h((v	NOUN
ejpam-5274	288	11	,	,	PUNCT
ejpam-5274	288	12	p	p	NOUN
ejpam-5274	288	13	)	)	PUNCT
ejpam-5274	288	14	)	)	PUNCT
ejpam-5274	289	1	=	=	PUNCT
ejpam-5274	290	1	[	[	X
ejpam-5274	290	2	ng(v)×	ng(v)×	PROPN
ejpam-5274	290	3	(	(	PUNCT
ejpam-5274	290	4	v	v	NOUN
ejpam-5274	290	5	(	(	PUNCT
ejpam-5274	290	6	h	h	NOUN
ejpam-5274	290	7	)	)	PUNCT
ejpam-5274	290	8	\nh(p	\nh(p	NOUN
ejpam-5274	290	9	)	)	PUNCT
ejpam-5274	290	10	]	]	PUNCT
ejpam-5274	290	11	∪	∪	ADP
ejpam-5274	290	12	[	[	X
ejpam-5274	290	13	(	(	PUNCT
ejpam-5274	290	14	v	v	NOUN
ejpam-5274	290	15	(	(	PUNCT
ejpam-5274	290	16	g	g	NOUN
ejpam-5274	290	17	)	)	PUNCT
ejpam-5274	290	18	\ng(v))×nh(p	\ng(v))×nh(p	NOUN
ejpam-5274	290	19	)	)	PUNCT
ejpam-5274	290	20	]	]	PUNCT
ejpam-5274	290	21	.	.	PUNCT
ejpam-5274	291	1	hence	hence	ADV
ejpam-5274	291	2	,	,	PUNCT
ejpam-5274	291	3	|ng⊕h((v	|ng⊕h((v	PROPN
ejpam-5274	291	4	,	,	PUNCT
ejpam-5274	291	5	p))|	p))|	PROPN
ejpam-5274	291	6	=	=	PUNCT
ejpam-5274	291	7	|ng(v)|(n−	|ng(v)|(n−	ADP
ejpam-5274	291	8	|nh(p)|	|nh(p)|	NOUN
ejpam-5274	291	9	)	)	PUNCT
ejpam-5274	292	1	+	+	X
ejpam-5274	292	2	|nh(p)|(m−	|nh(p)|(m−	X
ejpam-5274	292	3	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	292	4	)	)	PUNCT
ejpam-5274	292	5	.	.	PUNCT
ejpam-5274	293	1	jayhan	jayhan	PROPN
ejpam-5274	293	2	cruz	cruz	PROPN
ejpam-5274	293	3	,	,	PUNCT
ejpam-5274	293	4	g.	g.	PROPN
ejpam-5274	293	5	malacas	malacas	PROPN
ejpam-5274	293	6	,	,	PUNCT
ejpam-5274	293	7	s.	s.	PROPN
ejpam-5274	293	8	canoy	canoy	PROPN
ejpam-5274	293	9	,	,	PUNCT
ejpam-5274	293	10	jr	jr	PROPN
ejpam-5274	293	11	.	.	PROPN
ejpam-5274	293	12	/	/	SYM
ejpam-5274	293	13	eur	eur	PROPN
ejpam-5274	293	14	.	.	PUNCT
ejpam-5274	294	1	j.	j.	PROPN
ejpam-5274	294	2	pure	pure	PROPN
ejpam-5274	294	3	appl	appl	PROPN
ejpam-5274	294	4	.	.	PROPN
ejpam-5274	294	5	math	math	PROPN
ejpam-5274	294	6	,	,	PUNCT
ejpam-5274	294	7	17	17	NUM
ejpam-5274	294	8	(	(	PUNCT
ejpam-5274	294	9	3	3	NUM
ejpam-5274	294	10	)	)	PUNCT
ejpam-5274	294	11	(	(	PUNCT
ejpam-5274	294	12	2024	2024	NUM
ejpam-5274	294	13	)	)	PUNCT
ejpam-5274	294	14	,	,	PUNCT
ejpam-5274	294	15	1449	1449	NUM
ejpam-5274	294	16	-	-	SYM
ejpam-5274	294	17	1462	1462	NUM
ejpam-5274	294	18	1458	1458	NUM
ejpam-5274	294	19	therefore	therefore	ADV
ejpam-5274	294	20	,	,	PUNCT
ejpam-5274	294	21	fg⊕h(x	fg⊕h(x	NOUN
ejpam-5274	294	22	)	)	PUNCT
ejpam-5274	294	23	=	=	PUNCT
ejpam-5274	294	24	∑	∑	PUNCT
ejpam-5274	294	25	(	(	PUNCT
ejpam-5274	294	26	v	v	NOUN
ejpam-5274	294	27	,	,	PUNCT
ejpam-5274	294	28	p)∈v	p)∈v	X
ejpam-5274	294	29	(	(	PUNCT
ejpam-5274	294	30	g⊕h	g⊕h	PROPN
ejpam-5274	294	31	)	)	PUNCT
ejpam-5274	294	32	x|ng⊕h(v	x|ng⊕h(v	NUM
ejpam-5274	294	33	,	,	PUNCT
ejpam-5274	294	34	p)|	p)|	NOUN
ejpam-5274	294	35	=	=	SYM
ejpam-5274	294	36	∑	∑	PROPN
ejpam-5274	294	37	(	(	PUNCT
ejpam-5274	294	38	v	v	NOUN
ejpam-5274	294	39	,	,	PUNCT
ejpam-5274	294	40	p)∈v	p)∈v	X
ejpam-5274	294	41	(	(	PUNCT
ejpam-5274	294	42	g⊕h	g⊕h	NOUN
ejpam-5274	294	43	)	)	PUNCT
ejpam-5274	294	44	x|ng(v)|(n−|nh(p)|)+|nh(p)|(m−|ng(v)|	x|ng(v)|(n−|nh(p)|)+|nh(p)|(m−|ng(v)|	PUNCT
ejpam-5274	294	45	)	)	PUNCT
ejpam-5274	294	46	=	=	SYM
ejpam-5274	294	47	∑	∑	PUNCT
ejpam-5274	294	48	v∈v	v∈v	NOUN
ejpam-5274	294	49	(	(	PUNCT
ejpam-5274	294	50	g	g	NOUN
ejpam-5274	294	51	)	)	PUNCT
ejpam-5274	294	52	∑	∑	PUNCT
ejpam-5274	294	53	p∈v	p∈v	NOUN
ejpam-5274	294	54	(	(	PUNCT
ejpam-5274	294	55	h	h	NOUN
ejpam-5274	294	56	)	)	PUNCT
ejpam-5274	294	57	x|ng(v)|(n−|nh(p)|)+|nh(p)|(m−|ng(v)|	x|ng(v)|(n−|nh(p)|)+|nh(p)|(m−|ng(v)|	PUNCT
ejpam-5274	294	58	)	)	PUNCT
ejpam-5274	294	59	=	=	SYM
ejpam-5274	294	60	∑	∑	PUNCT
ejpam-5274	294	61	v∈v	v∈v	NOUN
ejpam-5274	294	62	(	(	PUNCT
ejpam-5274	294	63	g	g	NOUN
ejpam-5274	294	64	)	)	PUNCT
ejpam-5274	294	65	xn|ng(v)|	xn|ng(v)|	PROPN
ejpam-5274	294	66	∑	∑	PUNCT
ejpam-5274	294	67	p∈v	p∈v	NOUN
ejpam-5274	294	68	(	(	PUNCT
ejpam-5274	294	69	h	h	NOUN
ejpam-5274	294	70	)	)	PUNCT
ejpam-5274	294	71	x(m−2|ng(v)|)|nh(p)|	x(m−2|ng(v)|)|nh(p)|	NOUN
ejpam-5274	295	1	=	=	PUNCT
ejpam-5274	295	2	∑	∑	PUNCT
ejpam-5274	295	3	v∈v	v∈v	PROPN
ejpam-5274	295	4	(	(	PUNCT
ejpam-5274	295	5	g	g	NOUN
ejpam-5274	295	6	)	)	PUNCT
ejpam-5274	295	7	xn|ng(v)|fh(xm−2|ng(v)|	xn|ng(v)|fh(xm−2|ng(v)|	PROPN
ejpam-5274	295	8	)	)	PUNCT
ejpam-5274	295	9	.	.	PUNCT
ejpam-5274	296	1	corollary	corollary	ADJ
ejpam-5274	296	2	7	7	NUM
ejpam-5274	296	3	.	.	PUNCT
ejpam-5274	297	1	let	let	VERB
ejpam-5274	297	2	g	g	NOUN
ejpam-5274	297	3	and	and	CCONJ
ejpam-5274	297	4	h	h	PROPN
ejpam-5274	297	5	be	be	AUX
ejpam-5274	297	6	non	non	ADJ
ejpam-5274	297	7	-	-	ADJ
ejpam-5274	297	8	trivial	trivial	ADJ
ejpam-5274	297	9	of	of	ADP
ejpam-5274	297	10	orders	order	NOUN
ejpam-5274	297	11	m	m	PROPN
ejpam-5274	297	12	and	and	CCONJ
ejpam-5274	297	13	n	n	CCONJ
ejpam-5274	297	14	,	,	PUNCT
ejpam-5274	297	15	respectively	respectively	ADV
ejpam-5274	297	16	.	.	PUNCT
ejpam-5274	298	1	if	if	SCONJ
ejpam-5274	298	2	g	g	PROPN
ejpam-5274	298	3	is	be	AUX
ejpam-5274	298	4	r	r	NOUN
ejpam-5274	298	5	-	-	ADJ
ejpam-5274	298	6	regular	regular	ADJ
ejpam-5274	298	7	,	,	PUNCT
ejpam-5274	298	8	then	then	ADV
ejpam-5274	298	9	fg⊕h(x	fg⊕h(x	NOUN
ejpam-5274	298	10	)	)	PUNCT
ejpam-5274	298	11	=	=	SYM
ejpam-5274	298	12	mxnrfh(xm−2r	mxnrfh(xm−2r	NOUN
ejpam-5274	298	13	)	)	PUNCT
ejpam-5274	298	14	.	.	PUNCT
ejpam-5274	299	1	proof	proof	NOUN
ejpam-5274	299	2	.	.	PUNCT
ejpam-5274	300	1	since	since	SCONJ
ejpam-5274	300	2	g	g	PROPN
ejpam-5274	300	3	is	be	AUX
ejpam-5274	300	4	r	r	NOUN
ejpam-5274	300	5	-	-	ADJ
ejpam-5274	300	6	regular	regular	ADJ
ejpam-5274	300	7	,	,	PUNCT
ejpam-5274	300	8	|ng(v)|	|ng(v)|	NOUN
ejpam-5274	300	9	=	=	SYM
ejpam-5274	300	10	r	r	NOUN
ejpam-5274	300	11	for	for	ADP
ejpam-5274	300	12	all	all	DET
ejpam-5274	300	13	v	v	ADP
ejpam-5274	300	14	∈	∈	NOUN
ejpam-5274	300	15	v	v	NOUN
ejpam-5274	300	16	(	(	PUNCT
ejpam-5274	300	17	g	g	NOUN
ejpam-5274	300	18	)	)	PUNCT
ejpam-5274	300	19	.	.	PUNCT
ejpam-5274	301	1	thus	thus	ADV
ejpam-5274	301	2	,	,	PUNCT
ejpam-5274	301	3	from	from	ADP
ejpam-5274	301	4	theorem	theorem	NOUN
ejpam-5274	301	5	12	12	NUM
ejpam-5274	301	6	,	,	PUNCT
ejpam-5274	301	7	we	we	PRON
ejpam-5274	301	8	have	have	VERB
ejpam-5274	301	9	fg⊕h(x	fg⊕h(x	NOUN
ejpam-5274	301	10	)	)	PUNCT
ejpam-5274	301	11	=	=	PUNCT
ejpam-5274	301	12	∑	∑	PUNCT
ejpam-5274	301	13	v∈v	v∈v	PROPN
ejpam-5274	301	14	(	(	PUNCT
ejpam-5274	301	15	g	g	NOUN
ejpam-5274	301	16	)	)	PUNCT
ejpam-5274	301	17	xn|ng(v)|fh(xm−2|ng(v)|	xn|ng(v)|fh(xm−2|ng(v)|	PROPN
ejpam-5274	301	18	)	)	PUNCT
ejpam-5274	302	1	=	=	PUNCT
ejpam-5274	302	2	∑	∑	PUNCT
ejpam-5274	302	3	v∈v	v∈v	NOUN
ejpam-5274	302	4	(	(	PUNCT
ejpam-5274	302	5	g	g	NOUN
ejpam-5274	302	6	)	)	PUNCT
ejpam-5274	302	7	xnrfh(xm−2r	xnrfh(xm−2r	PROPN
ejpam-5274	302	8	)	)	PUNCT
ejpam-5274	302	9	=	=	SYM
ejpam-5274	302	10	mxnrfh(xm−2r	mxnrfh(xm−2r	NOUN
ejpam-5274	302	11	)	)	PUNCT
ejpam-5274	302	12	.	.	PUNCT
ejpam-5274	303	1	theorem	theorem	NOUN
ejpam-5274	303	2	13	13	NUM
ejpam-5274	303	3	.	.	PUNCT
ejpam-5274	304	1	let	let	VERB
ejpam-5274	304	2	g	g	NOUN
ejpam-5274	304	3	and	and	CCONJ
ejpam-5274	304	4	h	h	PROPN
ejpam-5274	304	5	be	be	VERB
ejpam-5274	304	6	non	non	ADJ
ejpam-5274	304	7	-	-	ADJ
ejpam-5274	304	8	trivial	trivial	ADJ
ejpam-5274	304	9	connected	connected	ADJ
ejpam-5274	304	10	graphs	graph	NOUN
ejpam-5274	304	11	of	of	ADP
ejpam-5274	304	12	orders	order	NOUN
ejpam-5274	304	13	m	m	VERB
ejpam-5274	304	14	and	and	CCONJ
ejpam-5274	304	15	n	n	CCONJ
ejpam-5274	304	16	,	,	PUNCT
ejpam-5274	304	17	respectively	respectively	ADV
ejpam-5274	304	18	.	.	PUNCT
ejpam-5274	305	1	if	if	SCONJ
ejpam-5274	305	2	⟨d1	⟨d1	PROPN
ejpam-5274	305	3	,	,	PUNCT
ejpam-5274	305	4	d2	d2	PROPN
ejpam-5274	305	5	,	,	PUNCT
ejpam-5274	305	6	·	·	PUNCT
ejpam-5274	305	7	·	·	PUNCT
ejpam-5274	305	8	·	·	PUNCT
ejpam-5274	305	9	dm⟩	dm⟩	PROPN
ejpam-5274	305	10	and	and	CCONJ
ejpam-5274	305	11	⟨q1	⟨q1	PROPN
ejpam-5274	305	12	,	,	PUNCT
ejpam-5274	305	13	q2	q2	NOUN
ejpam-5274	305	14	,	,	PUNCT
ejpam-5274	305	15	·	·	PUNCT
ejpam-5274	305	16	·	·	PUNCT
ejpam-5274	305	17	·	·	PUNCT
ejpam-5274	305	18	qn⟩	qn⟩	NOUN
ejpam-5274	305	19	are	be	AUX
ejpam-5274	305	20	the	the	DET
ejpam-5274	305	21	degree	degree	NOUN
ejpam-5274	305	22	sequences	sequence	NOUN
ejpam-5274	305	23	of	of	ADP
ejpam-5274	305	24	g	g	PROPN
ejpam-5274	305	25	and	and	CCONJ
ejpam-5274	305	26	h	h	NOUN
ejpam-5274	305	27	,	,	PUNCT
ejpam-5274	305	28	respectively	respectively	ADV
ejpam-5274	305	29	,	,	PUNCT
ejpam-5274	305	30	then	then	ADV
ejpam-5274	305	31	the	the	DET
ejpam-5274	305	32	terms	term	NOUN
ejpam-5274	305	33	of	of	ADP
ejpam-5274	305	34	the	the	DET
ejpam-5274	305	35	degree	degree	NOUN
ejpam-5274	305	36	sequence	sequence	NOUN
ejpam-5274	305	37	of	of	ADP
ejpam-5274	305	38	g	g	PROPN
ejpam-5274	305	39	⊕	⊕	PROPN
ejpam-5274	305	40	h	h	PROPN
ejpam-5274	305	41	are	be	AUX
ejpam-5274	305	42	the	the	DET
ejpam-5274	305	43	elements	element	NOUN
ejpam-5274	305	44	of	of	ADP
ejpam-5274	305	45	the	the	DET
ejpam-5274	305	46	set	set	NOUN
ejpam-5274	305	47	{	{	PUNCT
ejpam-5274	305	48	ndi	ndi	PROPN
ejpam-5274	305	49	+	+	CCONJ
ejpam-5274	305	50	(	(	PUNCT
ejpam-5274	305	51	m−	m−	PROPN
ejpam-5274	305	52	2di)qj	2di)qj	NUM
ejpam-5274	305	53	:	:	PUNCT
ejpam-5274	305	54	1	1	NUM
ejpam-5274	305	55	≤	≤	NUM
ejpam-5274	305	56	i	i	X
ejpam-5274	305	57	≤	≤	NOUN
ejpam-5274	305	58	m	m	VERB
ejpam-5274	305	59	and	and	CCONJ
ejpam-5274	305	60	1	1	NUM
ejpam-5274	305	61	≤	≤	NUM
ejpam-5274	305	62	j	j	PROPN
ejpam-5274	305	63	≤	≤	PROPN
ejpam-5274	305	64	n	n	CCONJ
ejpam-5274	305	65	}	}	PUNCT
ejpam-5274	305	66	.	.	PUNCT
ejpam-5274	306	1	proof	proof	NOUN
ejpam-5274	306	2	.	.	PUNCT
ejpam-5274	307	1	from	from	ADP
ejpam-5274	307	2	theorem	theorem	ADJ
ejpam-5274	307	3	12	12	NUM
ejpam-5274	307	4	and	and	CCONJ
ejpam-5274	307	5	the	the	DET
ejpam-5274	307	6	polynomial	polynomial	ADJ
ejpam-5274	307	7	representation	representation	NOUN
ejpam-5274	307	8	fh(x	fh(x	PUNCT
ejpam-5274	307	9	)	)	PUNCT
ejpam-5274	307	10	=	=	SYM
ejpam-5274	307	11	n∑	n∑	PROPN
ejpam-5274	307	12	j=2	j=2	PROPN
ejpam-5274	308	1	xqj	xqj	INTJ
ejpam-5274	308	2	,	,	PUNCT
ejpam-5274	308	3	we	we	PRON
ejpam-5274	308	4	have	have	VERB
ejpam-5274	308	5	fg⊕h(x	fg⊕h(x	NOUN
ejpam-5274	308	6	)	)	PUNCT
ejpam-5274	308	7	=	=	PUNCT
ejpam-5274	309	1	∑	∑	PUNCT
ejpam-5274	309	2	v∈v	v∈v	PROPN
ejpam-5274	309	3	(	(	PUNCT
ejpam-5274	309	4	g	g	NOUN
ejpam-5274	309	5	)	)	PUNCT
ejpam-5274	309	6	xn|ng(v)|fh(xm−2|ng(v)|	xn|ng(v)|fh(xm−2|ng(v)|	PROPN
ejpam-5274	309	7	)	)	PUNCT
ejpam-5274	310	1	=	=	PUNCT
ejpam-5274	310	2	m∑	m∑	ADP
ejpam-5274	310	3	i=1	i=1	PROPN
ejpam-5274	310	4	xndi	xndi	PROPN
ejpam-5274	311	1	n∑	n∑	PROPN
ejpam-5274	311	2	j=1	j=1	PROPN
ejpam-5274	311	3	x(m−2di)qj	x(m−2di)qj	PUNCT
ejpam-5274	312	1	jayhan	jayhan	PROPN
ejpam-5274	312	2	cruz	cruz	PROPN
ejpam-5274	312	3	,	,	PUNCT
ejpam-5274	312	4	g.	g.	PROPN
ejpam-5274	312	5	malacas	malacas	PROPN
ejpam-5274	312	6	,	,	PUNCT
ejpam-5274	312	7	s.	s.	PROPN
ejpam-5274	312	8	canoy	canoy	PROPN
ejpam-5274	312	9	,	,	PUNCT
ejpam-5274	312	10	jr	jr	PROPN
ejpam-5274	312	11	.	.	PROPN
ejpam-5274	312	12	/	/	SYM
ejpam-5274	312	13	eur	eur	PROPN
ejpam-5274	312	14	.	.	PUNCT
ejpam-5274	313	1	j.	j.	PROPN
ejpam-5274	313	2	pure	pure	PROPN
ejpam-5274	313	3	appl	appl	PROPN
ejpam-5274	313	4	.	.	PROPN
ejpam-5274	313	5	math	math	PROPN
ejpam-5274	313	6	,	,	PUNCT
ejpam-5274	313	7	17	17	NUM
ejpam-5274	313	8	(	(	PUNCT
ejpam-5274	313	9	3	3	NUM
ejpam-5274	313	10	)	)	PUNCT
ejpam-5274	313	11	(	(	PUNCT
ejpam-5274	313	12	2024	2024	NUM
ejpam-5274	313	13	)	)	PUNCT
ejpam-5274	313	14	,	,	PUNCT
ejpam-5274	313	15	1449	1449	NUM
ejpam-5274	313	16	-	-	SYM
ejpam-5274	313	17	1462	1462	NUM
ejpam-5274	313	18	1459	1459	NUM
ejpam-5274	313	19	=	=	SYM
ejpam-5274	313	20	m∑	m∑	INTJ
ejpam-5274	313	21	i=1	i=1	PROPN
ejpam-5274	313	22	n∑	n∑	PROPN
ejpam-5274	314	1	j=1	j=1	PROPN
ejpam-5274	314	2	xndix(m−2di)qj	xndix(m−2di)qj	PROPN
ejpam-5274	314	3	=	=	PUNCT
ejpam-5274	314	4	m∑	m∑	INTJ
ejpam-5274	314	5	i=1	i=1	PROPN
ejpam-5274	314	6	n∑	n∑	PROPN
ejpam-5274	314	7	j=1	j=1	PROPN
ejpam-5274	315	1	xndi+(m−2di)qj	xndi+(m−2di)qj	PROPN
ejpam-5274	315	2	.	.	PUNCT
ejpam-5274	316	1	it	it	PRON
ejpam-5274	316	2	follows	follow	VERB
ejpam-5274	316	3	that	that	SCONJ
ejpam-5274	316	4	the	the	DET
ejpam-5274	316	5	terms	term	NOUN
ejpam-5274	316	6	of	of	ADP
ejpam-5274	316	7	the	the	DET
ejpam-5274	316	8	degree	degree	NOUN
ejpam-5274	316	9	sequence	sequence	NOUN
ejpam-5274	316	10	of	of	ADP
ejpam-5274	316	11	g	g	PROPN
ejpam-5274	316	12	⊕	⊕	PROPN
ejpam-5274	316	13	h	h	PROPN
ejpam-5274	316	14	are	be	AUX
ejpam-5274	316	15	the	the	DET
ejpam-5274	316	16	elements	element	NOUN
ejpam-5274	316	17	of	of	ADP
ejpam-5274	316	18	the	the	DET
ejpam-5274	316	19	set	set	NOUN
ejpam-5274	316	20	{	{	PUNCT
ejpam-5274	316	21	ndi	ndi	PROPN
ejpam-5274	317	1	+	+	CCONJ
ejpam-5274	317	2	(	(	PUNCT
ejpam-5274	317	3	m−	m−	PROPN
ejpam-5274	317	4	2di)qj	2di)qj	NUM
ejpam-5274	317	5	:	:	PUNCT
ejpam-5274	317	6	1	1	NUM
ejpam-5274	317	7	≤	≤	NUM
ejpam-5274	317	8	i	i	X
ejpam-5274	318	1	≤	≤	NOUN
ejpam-5274	318	2	m	m	VERB
ejpam-5274	318	3	and	and	CCONJ
ejpam-5274	318	4	1	1	NUM
ejpam-5274	318	5	≤	≤	NUM
ejpam-5274	318	6	j	j	PROPN
ejpam-5274	318	7	≤	≤	PROPN
ejpam-5274	318	8	n	n	CCONJ
ejpam-5274	318	9	}	}	PUNCT
ejpam-5274	318	10	.	.	PUNCT
ejpam-5274	319	1	theorem	theorem	NOUN
ejpam-5274	319	2	14	14	NUM
ejpam-5274	319	3	.	.	PUNCT
ejpam-5274	320	1	let	let	VERB
ejpam-5274	320	2	g	g	NOUN
ejpam-5274	320	3	and	and	CCONJ
ejpam-5274	320	4	h	h	PROPN
ejpam-5274	320	5	be	be	VERB
ejpam-5274	320	6	non	non	ADJ
ejpam-5274	320	7	-	-	ADJ
ejpam-5274	320	8	trivial	trivial	ADJ
ejpam-5274	320	9	connected	connected	ADJ
ejpam-5274	320	10	graphs	graph	NOUN
ejpam-5274	320	11	of	of	ADP
ejpam-5274	320	12	orders	order	NOUN
ejpam-5274	320	13	m	m	VERB
ejpam-5274	320	14	and	and	CCONJ
ejpam-5274	320	15	n	n	CCONJ
ejpam-5274	320	16	,	,	PUNCT
ejpam-5274	320	17	respectively	respectively	ADV
ejpam-5274	320	18	.	.	PUNCT
ejpam-5274	321	1	then	then	ADV
ejpam-5274	321	2	fg∨h(x	fg∨h(x	X
ejpam-5274	321	3	)	)	PUNCT
ejpam-5274	321	4	=	=	SYM
ejpam-5274	321	5	fg(x	fg(x	NUM
ejpam-5274	321	6	n)fh(xm	n)fh(xm	NOUN
ejpam-5274	321	7	)	)	PUNCT
ejpam-5274	321	8	.	.	PUNCT
ejpam-5274	322	1	proof	proof	NOUN
ejpam-5274	322	2	.	.	PUNCT
ejpam-5274	323	1	let	let	VERB
ejpam-5274	323	2	(	(	PUNCT
ejpam-5274	323	3	v	v	NOUN
ejpam-5274	323	4	,	,	PUNCT
ejpam-5274	323	5	p	p	NOUN
ejpam-5274	323	6	)	)	PUNCT
ejpam-5274	323	7	∈	∈	PROPN
ejpam-5274	323	8	v	v	NOUN
ejpam-5274	323	9	(	(	PUNCT
ejpam-5274	323	10	g	g	NOUN
ejpam-5274	323	11	∨h	∨h	PROPN
ejpam-5274	323	12	)	)	PUNCT
ejpam-5274	323	13	.	.	PUNCT
ejpam-5274	324	1	from	from	ADP
ejpam-5274	324	2	the	the	DET
ejpam-5274	324	3	definition	definition	NOUN
ejpam-5274	324	4	of	of	ADP
ejpam-5274	324	5	g	g	PROPN
ejpam-5274	324	6	∨h	∨h	NOUN
ejpam-5274	324	7	,	,	PUNCT
ejpam-5274	324	8	it	it	PRON
ejpam-5274	324	9	follows	follow	VERB
ejpam-5274	324	10	that	that	SCONJ
ejpam-5274	324	11	ng∨h((v	ng∨h((v	NOUN
ejpam-5274	324	12	,	,	PUNCT
ejpam-5274	324	13	p	p	NOUN
ejpam-5274	324	14	)	)	PUNCT
ejpam-5274	324	15	)	)	PUNCT
ejpam-5274	325	1	=	=	PUNCT
ejpam-5274	326	1	(	(	PUNCT
ejpam-5274	326	2	ng(v)×	ng(v)×	PROPN
ejpam-5274	326	3	v	v	INTJ
ejpam-5274	326	4	(	(	PUNCT
ejpam-5274	326	5	h	h	NOUN
ejpam-5274	326	6	)	)	PUNCT
ejpam-5274	326	7	)	)	PUNCT
ejpam-5274	326	8	∪	∪	ADP
ejpam-5274	326	9	(	(	PUNCT
ejpam-5274	326	10	v	v	NOUN
ejpam-5274	326	11	(	(	PUNCT
ejpam-5274	326	12	g)×nh(p	g)×nh(p	NOUN
ejpam-5274	326	13	)	)	PUNCT
ejpam-5274	326	14	)	)	PUNCT
ejpam-5274	326	15	.	.	PUNCT
ejpam-5274	327	1	hence	hence	ADV
ejpam-5274	327	2	,	,	PUNCT
ejpam-5274	327	3	|ng∨h((v	|ng∨h((v	PROPN
ejpam-5274	327	4	,	,	PUNCT
ejpam-5274	327	5	p))|	p))|	PROPN
ejpam-5274	327	6	=	=	SYM
ejpam-5274	327	7	n|ng(v)|+m|nh(p)|	n|ng(v)|+m|nh(p)|	PROPN
ejpam-5274	327	8	.	.	PUNCT
ejpam-5274	328	1	therefore	therefore	ADV
ejpam-5274	328	2	,	,	PUNCT
ejpam-5274	328	3	fg∨h(x	fg∨h(x	X
ejpam-5274	328	4	)	)	PUNCT
ejpam-5274	328	5	=	=	SYM
ejpam-5274	328	6	∑	∑	PUNCT
ejpam-5274	328	7	(	(	PUNCT
ejpam-5274	328	8	v	v	NOUN
ejpam-5274	328	9	,	,	PUNCT
ejpam-5274	328	10	p)∈v	p)∈v	X
ejpam-5274	328	11	(	(	PUNCT
ejpam-5274	328	12	g∨h	g∨h	PROPN
ejpam-5274	328	13	)	)	PUNCT
ejpam-5274	328	14	x|ng∨h(v	x|ng∨h(v	PROPN
ejpam-5274	328	15	,	,	PUNCT
ejpam-5274	328	16	p)|	p)|	NOUN
ejpam-5274	328	17	=	=	SYM
ejpam-5274	328	18	∑	∑	PROPN
ejpam-5274	328	19	(	(	PUNCT
ejpam-5274	328	20	v	v	NOUN
ejpam-5274	328	21	,	,	PUNCT
ejpam-5274	328	22	p)∈v	p)∈v	X
ejpam-5274	328	23	(	(	PUNCT
ejpam-5274	328	24	g∨h	g∨h	PROPN
ejpam-5274	328	25	)	)	PUNCT
ejpam-5274	328	26	xn|ng(v)|+m|nh(p)|	xn|ng(v)|+m|nh(p)|	PUNCT
ejpam-5274	329	1	=	=	PUNCT
ejpam-5274	329	2	∑	∑	PUNCT
ejpam-5274	329	3	v∈v	v∈v	NOUN
ejpam-5274	329	4	(	(	PUNCT
ejpam-5274	329	5	g	g	NOUN
ejpam-5274	329	6	)	)	PUNCT
ejpam-5274	329	7	xn|ng(v)|	xn|ng(v)|	PROPN
ejpam-5274	329	8	∑	∑	PUNCT
ejpam-5274	329	9	p∈v	p∈v	NOUN
ejpam-5274	329	10	(	(	PUNCT
ejpam-5274	329	11	h	h	NOUN
ejpam-5274	329	12	)	)	PUNCT
ejpam-5274	329	13	xm|nh(p)|	xm|nh(p)|	PROPN
ejpam-5274	330	1	=	=	PUNCT
ejpam-5274	330	2	∑	∑	PUNCT
ejpam-5274	330	3	v∈v	v∈v	PROPN
ejpam-5274	330	4	(	(	PUNCT
ejpam-5274	330	5	g	g	NOUN
ejpam-5274	330	6	)	)	PUNCT
ejpam-5274	330	7	xn|ng(v)|fh(xm	xn|ng(v)|fh(xm	PUNCT
ejpam-5274	330	8	)	)	PUNCT
ejpam-5274	330	9	=	=	SYM
ejpam-5274	330	10	fg(x	fg(x	NUM
ejpam-5274	330	11	n)fh(xm	n)fh(xm	NOUN
ejpam-5274	330	12	)	)	PUNCT
ejpam-5274	330	13	.	.	PUNCT
ejpam-5274	331	1	corollary	corollary	ADJ
ejpam-5274	331	2	8	8	NUM
ejpam-5274	331	3	.	.	PUNCT
ejpam-5274	332	1	let	let	VERB
ejpam-5274	332	2	n	n	PRON
ejpam-5274	332	3	and	and	CCONJ
ejpam-5274	332	4	m	m	AUX
ejpam-5274	332	5	be	be	AUX
ejpam-5274	332	6	positive	positive	ADJ
ejpam-5274	332	7	integers	integer	NOUN
ejpam-5274	332	8	.	.	PUNCT
ejpam-5274	333	1	then	then	ADV
ejpam-5274	333	2	(	(	PUNCT
ejpam-5274	333	3	i	i	NOUN
ejpam-5274	333	4	)	)	PUNCT
ejpam-5274	333	5	fpm∨pn(x	fpm∨pn(x	PROPN
ejpam-5274	333	6	)	)	PUNCT
ejpam-5274	333	7	=	=	PUNCT
ejpam-5274	334	1	(	(	PUNCT
ejpam-5274	334	2	m−	m−	PROPN
ejpam-5274	334	3	2)(n−	2)(n−	NUM
ejpam-5274	334	4	2)x2m+2n	2)x2m+2n	NUM
ejpam-5274	335	1	+	+	CCONJ
ejpam-5274	335	2	2(m−	2(m−	NUM
ejpam-5274	335	3	2)xm+2n	2)xm+2n	NOUN
ejpam-5274	335	4	+	+	CCONJ
ejpam-5274	335	5	2(n−	2(n−	NUM
ejpam-5274	335	6	2)x2m+n	2)x2m+n	NUM
ejpam-5274	336	1	+	+	CCONJ
ejpam-5274	336	2	4xm+n	4xm+n	NOUN
ejpam-5274	336	3	for	for	ADP
ejpam-5274	336	4	m	m	PROPN
ejpam-5274	336	5	,	,	PUNCT
ejpam-5274	336	6	n	n	PRON
ejpam-5274	336	7	≥	≥	NOUN
ejpam-5274	336	8	2	2	NUM
ejpam-5274	336	9	;	;	PUNCT
ejpam-5274	336	10	(	(	PUNCT
ejpam-5274	336	11	ii	ii	NOUN
ejpam-5274	336	12	)	)	PUNCT
ejpam-5274	336	13	fpm∨cn(x	fpm∨cn(x	PROPN
ejpam-5274	336	14	)	)	PUNCT
ejpam-5274	337	1	=	=	SYM
ejpam-5274	337	2	n(m−	n(m−	PROPN
ejpam-5274	337	3	2)x2m+2n	2)x2m+2n	NUM
ejpam-5274	338	1	+	+	NUM
ejpam-5274	338	2	2nx2m+n	2nx2m+n	NUM
ejpam-5274	338	3	for	for	ADP
ejpam-5274	338	4	m	m	PROPN
ejpam-5274	338	5	≥	≥	NUM
ejpam-5274	338	6	2	2	NUM
ejpam-5274	338	7	and	and	CCONJ
ejpam-5274	338	8	n	n	PRON
ejpam-5274	338	9	≥	≥	NOUN
ejpam-5274	338	10	3	3	NUM
ejpam-5274	338	11	;	;	PUNCT
ejpam-5274	338	12	and	and	CCONJ
ejpam-5274	338	13	(	(	PUNCT
ejpam-5274	338	14	iii	iii	X
ejpam-5274	338	15	)	)	PUNCT
ejpam-5274	338	16	fcm∨cn(x	fcm∨cn(x	PROPN
ejpam-5274	338	17	)	)	PUNCT
ejpam-5274	338	18	=	=	PROPN
ejpam-5274	338	19	mnx2m+2n	mnx2m+2n	PROPN
ejpam-5274	338	20	for	for	ADP
ejpam-5274	338	21	m	m	PROPN
ejpam-5274	338	22	,	,	PUNCT
ejpam-5274	338	23	n	n	PRON
ejpam-5274	338	24	≥	≥	NOUN
ejpam-5274	338	25	3	3	NUM
ejpam-5274	338	26	.	.	PUNCT
ejpam-5274	339	1	proof	proof	NOUN
ejpam-5274	339	2	.	.	PUNCT
ejpam-5274	340	1	for	for	ADP
ejpam-5274	340	2	any	any	DET
ejpam-5274	340	3	k	k	PROPN
ejpam-5274	340	4	≥	≥	NUM
ejpam-5274	340	5	2	2	NUM
ejpam-5274	340	6	and	and	CCONJ
ejpam-5274	340	7	r	r	NOUN
ejpam-5274	340	8	≥	≥	NUM
ejpam-5274	340	9	3	3	NUM
ejpam-5274	340	10	,	,	PUNCT
ejpam-5274	340	11	fpk	fpk	X
ejpam-5274	340	12	(	(	PUNCT
ejpam-5274	340	13	x	x	X
ejpam-5274	340	14	)	)	PUNCT
ejpam-5274	340	15	=	=	SYM
ejpam-5274	340	16	(	(	PUNCT
ejpam-5274	340	17	k	k	NOUN
ejpam-5274	340	18	−	−	PROPN
ejpam-5274	340	19	2)x2	2)x2	NUM
ejpam-5274	340	20	+	+	CCONJ
ejpam-5274	340	21	2x	2x	NUM
ejpam-5274	340	22	and	and	CCONJ
ejpam-5274	340	23	fcr(x	fcr(x	PROPN
ejpam-5274	340	24	)	)	PUNCT
ejpam-5274	340	25	=	=	SYM
ejpam-5274	340	26	rx2	rx2	PROPN
ejpam-5274	340	27	.	.	PUNCT
ejpam-5274	340	28	by	by	ADP
ejpam-5274	340	29	theorem	theorem	NOUN
ejpam-5274	340	30	14	14	NUM
ejpam-5274	340	31	,	,	PUNCT
ejpam-5274	340	32	we	we	PRON
ejpam-5274	340	33	have	have	VERB
ejpam-5274	340	34	fpm∨pn(x	fpm∨pn(x	NOUN
ejpam-5274	340	35	)	)	PUNCT
ejpam-5274	341	1	=	=	SYM
ejpam-5274	342	1	fpm(x	fpm(x	PROPN
ejpam-5274	342	2	n)fpn(x	n)fpn(x	PROPN
ejpam-5274	342	3	m	m	NOUN
ejpam-5274	342	4	)	)	PUNCT
ejpam-5274	342	5	jayhan	jayhan	PROPN
ejpam-5274	342	6	cruz	cruz	PROPN
ejpam-5274	342	7	,	,	PUNCT
ejpam-5274	342	8	g.	g.	PROPN
ejpam-5274	342	9	malacas	malacas	PROPN
ejpam-5274	342	10	,	,	PUNCT
ejpam-5274	342	11	s.	s.	PROPN
ejpam-5274	342	12	canoy	canoy	PROPN
ejpam-5274	342	13	,	,	PUNCT
ejpam-5274	342	14	jr	jr	PROPN
ejpam-5274	342	15	.	.	PROPN
ejpam-5274	342	16	/	/	SYM
ejpam-5274	342	17	eur	eur	PROPN
ejpam-5274	342	18	.	.	PUNCT
ejpam-5274	343	1	j.	j.	PROPN
ejpam-5274	343	2	pure	pure	PROPN
ejpam-5274	343	3	appl	appl	PROPN
ejpam-5274	343	4	.	.	PROPN
ejpam-5274	343	5	math	math	PROPN
ejpam-5274	343	6	,	,	PUNCT
ejpam-5274	343	7	17	17	NUM
ejpam-5274	343	8	(	(	PUNCT
ejpam-5274	343	9	3	3	NUM
ejpam-5274	343	10	)	)	PUNCT
ejpam-5274	343	11	(	(	PUNCT
ejpam-5274	343	12	2024	2024	NUM
ejpam-5274	343	13	)	)	PUNCT
ejpam-5274	343	14	,	,	PUNCT
ejpam-5274	343	15	1449	1449	NUM
ejpam-5274	343	16	-	-	SYM
ejpam-5274	343	17	1462	1462	NUM
ejpam-5274	343	18	1460	1460	NUM
ejpam-5274	343	19	=	=	SYM
ejpam-5274	344	1	[	[	X
ejpam-5274	344	2	(	(	PUNCT
ejpam-5274	344	3	m−	m−	PROPN
ejpam-5274	344	4	2)(xn)2	2)(xn)2	NUM
ejpam-5274	344	5	+	+	CCONJ
ejpam-5274	344	6	2(xn)][(n−	2(xn)][(n−	NUM
ejpam-5274	344	7	2)(xm)2	2)(xm)2	NUM
ejpam-5274	344	8	+	+	CCONJ
ejpam-5274	344	9	2(xm	2(xm	NUM
ejpam-5274	344	10	)	)	PUNCT
ejpam-5274	344	11	]	]	PUNCT
ejpam-5274	345	1	=	=	PUNCT
ejpam-5274	345	2	(	(	PUNCT
ejpam-5274	345	3	m−	m−	PROPN
ejpam-5274	345	4	2)(n−	2)(n−	NUM
ejpam-5274	345	5	2)x2m+2n	2)x2m+2n	NUM
ejpam-5274	346	1	+	+	CCONJ
ejpam-5274	346	2	2(m−	2(m−	NUM
ejpam-5274	346	3	2)xm+2n	2)xm+2n	NOUN
ejpam-5274	346	4	+	+	CCONJ
ejpam-5274	346	5	2(n−	2(n−	NUM
ejpam-5274	346	6	2)x2m+n	2)x2m+n	NUM
ejpam-5274	347	1	+	+	CCONJ
ejpam-5274	348	1	4xm+n	4xm+n	NOUN
ejpam-5274	348	2	,	,	PUNCT
ejpam-5274	348	3	fpm∨cn(x	fpm∨cn(x	PROPN
ejpam-5274	348	4	)	)	PUNCT
ejpam-5274	348	5	=	=	PUNCT
ejpam-5274	349	1	fpm(x	fpm(x	X
ejpam-5274	349	2	n)fcn(x	n)fcn(x	NUM
ejpam-5274	349	3	m	m	NOUN
ejpam-5274	349	4	)	)	PUNCT
ejpam-5274	350	1	=	=	PUNCT
ejpam-5274	351	1	[	[	X
ejpam-5274	351	2	(	(	PUNCT
ejpam-5274	351	3	m−	m−	PROPN
ejpam-5274	351	4	2)(xn)2	2)(xn)2	NUM
ejpam-5274	351	5	+	+	CCONJ
ejpam-5274	351	6	2(xn)](n(xm)2	2(xn)](n(xm)2	NUM
ejpam-5274	351	7	)	)	PUNCT
ejpam-5274	351	8	=	=	SYM
ejpam-5274	351	9	n(m−	n(m−	PROPN
ejpam-5274	351	10	2)x2m+2n	2)x2m+2n	NUM
ejpam-5274	351	11	+	+	CCONJ
ejpam-5274	351	12	2nx2m+n	2nx2m+n	NUM
ejpam-5274	351	13	,	,	PUNCT
ejpam-5274	351	14	and	and	CCONJ
ejpam-5274	351	15	fcm∨cn(x	fcm∨cn(x	ADJ
ejpam-5274	351	16	)	)	PUNCT
ejpam-5274	351	17	=	=	PUNCT
ejpam-5274	352	1	fcm(x	fcm(x	VERB
ejpam-5274	352	2	n)fcn(x	n)fcn(x	NUM
ejpam-5274	352	3	m	m	NOUN
ejpam-5274	352	4	)	)	PUNCT
ejpam-5274	353	1	=	=	PRON
ejpam-5274	353	2	(	(	PUNCT
ejpam-5274	353	3	m(xn)2)(n(xm)2	m(xn)2)(n(xm)2	PROPN
ejpam-5274	353	4	)	)	PUNCT
ejpam-5274	353	5	=	=	SYM
ejpam-5274	353	6	mnx2m+2n	mnx2m+2n	PROPN
ejpam-5274	353	7	.	.	PUNCT
ejpam-5274	353	8	theorem	theorem	VERB
ejpam-5274	353	9	15	15	NUM
ejpam-5274	353	10	.	.	PUNCT
ejpam-5274	354	1	let	let	VERB
ejpam-5274	354	2	g	g	NOUN
ejpam-5274	354	3	and	and	CCONJ
ejpam-5274	354	4	h	h	PROPN
ejpam-5274	354	5	be	be	VERB
ejpam-5274	354	6	non	non	ADJ
ejpam-5274	354	7	-	-	ADJ
ejpam-5274	354	8	trivial	trivial	ADJ
ejpam-5274	354	9	connected	connected	ADJ
ejpam-5274	354	10	graphs	graph	NOUN
ejpam-5274	354	11	of	of	ADP
ejpam-5274	354	12	orders	order	NOUN
ejpam-5274	354	13	m	m	VERB
ejpam-5274	354	14	and	and	CCONJ
ejpam-5274	354	15	n	n	CCONJ
ejpam-5274	354	16	,	,	PUNCT
ejpam-5274	354	17	respectively	respectively	ADV
ejpam-5274	354	18	.	.	PUNCT
ejpam-5274	355	1	then	then	ADV
ejpam-5274	355	2	a	a	DET
ejpam-5274	355	3	∈	∈	NOUN
ejpam-5274	355	4	r	r	NOUN
ejpam-5274	355	5	is	be	AUX
ejpam-5274	355	6	a	a	DET
ejpam-5274	355	7	zero	zero	NUM
ejpam-5274	355	8	of	of	ADP
ejpam-5274	355	9	fg∨h(x	fg∨h(x	NOUN
ejpam-5274	355	10	)	)	PUNCT
ejpam-5274	355	11	if	if	SCONJ
ejpam-5274	355	12	and	and	CCONJ
ejpam-5274	355	13	only	only	ADV
ejpam-5274	355	14	if	if	SCONJ
ejpam-5274	355	15	an	an	PRON
ejpam-5274	355	16	is	be	AUX
ejpam-5274	355	17	a	a	DET
ejpam-5274	355	18	zero	zero	NUM
ejpam-5274	355	19	of	of	ADP
ejpam-5274	355	20	fg(x	fg(x	NUM
ejpam-5274	355	21	)	)	PUNCT
ejpam-5274	355	22	or	or	CCONJ
ejpam-5274	355	23	am	be	AUX
ejpam-5274	355	24	is	be	AUX
ejpam-5274	355	25	a	a	DET
ejpam-5274	355	26	zero	zero	NUM
ejpam-5274	355	27	of	of	ADP
ejpam-5274	355	28	fh(x	fh(x	NUM
ejpam-5274	355	29	)	)	PUNCT
ejpam-5274	355	30	.	.	PUNCT
ejpam-5274	356	1	proof	proof	NOUN
ejpam-5274	356	2	.	.	PUNCT
ejpam-5274	357	1	by	by	ADP
ejpam-5274	357	2	theorem	theorem	NOUN
ejpam-5274	357	3	14	14	NUM
ejpam-5274	357	4	,	,	PUNCT
ejpam-5274	357	5	fg∨h)(x	fg∨h)(x	PROPN
ejpam-5274	357	6	)	)	PUNCT
ejpam-5274	357	7	=	=	NUM
ejpam-5274	357	8	fg(x	fg(x	NUM
ejpam-5274	357	9	n)fh(xm	n)fh(xm	NOUN
ejpam-5274	357	10	)	)	PUNCT
ejpam-5274	357	11	.	.	PUNCT
ejpam-5274	358	1	suppose	suppose	VERB
ejpam-5274	358	2	a	a	PRON
ejpam-5274	358	3	is	be	AUX
ejpam-5274	358	4	a	a	DET
ejpam-5274	358	5	zero	zero	NUM
ejpam-5274	358	6	of	of	ADP
ejpam-5274	358	7	fg∨h)(x	fg∨h)(x	PROPN
ejpam-5274	358	8	)	)	PUNCT
ejpam-5274	358	9	.	.	PUNCT
ejpam-5274	359	1	then	then	ADV
ejpam-5274	359	2	fg∨h)(a	fg∨h)(a	PROPN
ejpam-5274	359	3	)	)	PUNCT
ejpam-5274	360	1	=	=	PRON
ejpam-5274	360	2	fg(a	fg(a	NOUN
ejpam-5274	360	3	n)fh(am	n)fh(am	NOUN
ejpam-5274	360	4	)	)	PUNCT
ejpam-5274	361	1	=	=	SYM
ejpam-5274	361	2	0	0	X
ejpam-5274	361	3	.	.	PUNCT
ejpam-5274	362	1	this	this	PRON
ejpam-5274	362	2	implies	imply	VERB
ejpam-5274	362	3	that	that	SCONJ
ejpam-5274	362	4	fg(a	fg(a	VERB
ejpam-5274	362	5	n	n	CCONJ
ejpam-5274	362	6	)	)	PUNCT
ejpam-5274	362	7	=	=	SYM
ejpam-5274	362	8	0	0	NUM
ejpam-5274	362	9	or	or	CCONJ
ejpam-5274	362	10	fh(am	fh(am	PROPN
ejpam-5274	362	11	)	)	PUNCT
ejpam-5274	363	1	=	=	SYM
ejpam-5274	363	2	0	0	X
ejpam-5274	363	3	.	.	PUNCT
ejpam-5274	364	1	hence	hence	ADV
ejpam-5274	364	2	,	,	PUNCT
ejpam-5274	364	3	an	an	PRON
ejpam-5274	364	4	is	be	AUX
ejpam-5274	364	5	a	a	DET
ejpam-5274	364	6	zero	zero	NUM
ejpam-5274	364	7	of	of	ADP
ejpam-5274	364	8	fg(x	fg(x	NUM
ejpam-5274	364	9	)	)	PUNCT
ejpam-5274	364	10	or	or	CCONJ
ejpam-5274	364	11	a	a	DET
ejpam-5274	364	12	m	m	NOUN
ejpam-5274	364	13	is	be	AUX
ejpam-5274	364	14	a	a	DET
ejpam-5274	364	15	zero	zero	NUM
ejpam-5274	364	16	of	of	ADP
ejpam-5274	364	17	fh(x	fh(x	NUM
ejpam-5274	364	18	)	)	PUNCT
ejpam-5274	364	19	.	.	PUNCT
ejpam-5274	365	1	conversely	conversely	ADV
ejpam-5274	365	2	,	,	PUNCT
ejpam-5274	365	3	suppose	suppose	VERB
ejpam-5274	365	4	an	an	PRON
ejpam-5274	365	5	is	be	AUX
ejpam-5274	365	6	a	a	DET
ejpam-5274	365	7	zero	zero	NUM
ejpam-5274	365	8	of	of	ADP
ejpam-5274	365	9	fg(x	fg(x	NUM
ejpam-5274	365	10	)	)	PUNCT
ejpam-5274	365	11	or	or	CCONJ
ejpam-5274	365	12	am	be	AUX
ejpam-5274	365	13	is	be	AUX
ejpam-5274	365	14	a	a	DET
ejpam-5274	365	15	zero	zero	NUM
ejpam-5274	365	16	of	of	ADP
ejpam-5274	365	17	fh(x	fh(x	NUM
ejpam-5274	365	18	)	)	PUNCT
ejpam-5274	365	19	.	.	PUNCT
ejpam-5274	366	1	then	then	ADV
ejpam-5274	366	2	,	,	PUNCT
ejpam-5274	366	3	clearly	clearly	ADV
ejpam-5274	366	4	,	,	PUNCT
ejpam-5274	366	5	fg∨h)(a	fg∨h)(a	PROPN
ejpam-5274	366	6	)	)	PUNCT
ejpam-5274	367	1	=	=	PRON
ejpam-5274	367	2	fg(a	fg(a	NOUN
ejpam-5274	367	3	n)fh(am	n)fh(am	NOUN
ejpam-5274	367	4	)	)	PUNCT
ejpam-5274	368	1	=	=	SYM
ejpam-5274	368	2	0	0	NUM
ejpam-5274	368	3	,	,	PUNCT
ejpam-5274	368	4	showing	show	VERB
ejpam-5274	368	5	that	that	SCONJ
ejpam-5274	368	6	a	a	PRON
ejpam-5274	368	7	is	be	AUX
ejpam-5274	368	8	a	a	DET
ejpam-5274	368	9	zero	zero	NUM
ejpam-5274	368	10	of	of	ADP
ejpam-5274	368	11	fg∨h)(x	fg∨h)(x	PROPN
ejpam-5274	368	12	)	)	PUNCT
ejpam-5274	368	13	.	.	PUNCT
ejpam-5274	369	1	theorem	theorem	VERB
ejpam-5274	369	2	16	16	NUM
ejpam-5274	369	3	.	.	PUNCT
ejpam-5274	370	1	let	let	VERB
ejpam-5274	370	2	g	g	NOUN
ejpam-5274	370	3	and	and	CCONJ
ejpam-5274	370	4	h	h	PROPN
ejpam-5274	370	5	be	be	VERB
ejpam-5274	370	6	non	non	ADJ
ejpam-5274	370	7	-	-	ADJ
ejpam-5274	370	8	trivial	trivial	ADJ
ejpam-5274	370	9	connected	connected	ADJ
ejpam-5274	370	10	graphs	graph	NOUN
ejpam-5274	370	11	of	of	ADP
ejpam-5274	370	12	orders	order	NOUN
ejpam-5274	370	13	m	m	VERB
ejpam-5274	370	14	and	and	CCONJ
ejpam-5274	370	15	n	n	CCONJ
ejpam-5274	370	16	,	,	PUNCT
ejpam-5274	370	17	respectively	respectively	ADV
ejpam-5274	370	18	.	.	PUNCT
ejpam-5274	371	1	if	if	SCONJ
ejpam-5274	371	2	⟨d1	⟨d1	PROPN
ejpam-5274	371	3	,	,	PUNCT
ejpam-5274	371	4	d2	d2	PROPN
ejpam-5274	371	5	,	,	PUNCT
ejpam-5274	371	6	·	·	PUNCT
ejpam-5274	371	7	·	·	PUNCT
ejpam-5274	371	8	·	·	PUNCT
ejpam-5274	371	9	dm⟩	dm⟩	PROPN
ejpam-5274	371	10	and	and	CCONJ
ejpam-5274	371	11	⟨q1	⟨q1	PROPN
ejpam-5274	371	12	,	,	PUNCT
ejpam-5274	371	13	q2	q2	NOUN
ejpam-5274	371	14	,	,	PUNCT
ejpam-5274	371	15	·	·	PUNCT
ejpam-5274	371	16	·	·	PUNCT
ejpam-5274	371	17	·	·	PUNCT
ejpam-5274	371	18	qn⟩	qn⟩	NOUN
ejpam-5274	371	19	are	be	AUX
ejpam-5274	371	20	the	the	DET
ejpam-5274	371	21	degree	degree	NOUN
ejpam-5274	371	22	sequences	sequence	NOUN
ejpam-5274	371	23	of	of	ADP
ejpam-5274	371	24	g	g	PROPN
ejpam-5274	371	25	and	and	CCONJ
ejpam-5274	371	26	h	h	NOUN
ejpam-5274	371	27	,	,	PUNCT
ejpam-5274	371	28	respectively	respectively	ADV
ejpam-5274	371	29	,	,	PUNCT
ejpam-5274	371	30	then	then	ADV
ejpam-5274	371	31	the	the	DET
ejpam-5274	371	32	terms	term	NOUN
ejpam-5274	371	33	of	of	ADP
ejpam-5274	371	34	the	the	DET
ejpam-5274	371	35	degree	degree	NOUN
ejpam-5274	371	36	sequence	sequence	NOUN
ejpam-5274	371	37	of	of	ADP
ejpam-5274	371	38	g	g	PROPN
ejpam-5274	371	39	∨	∨	PROPN
ejpam-5274	371	40	h	h	NOUN
ejpam-5274	371	41	are	be	AUX
ejpam-5274	371	42	the	the	DET
ejpam-5274	371	43	elements	element	NOUN
ejpam-5274	371	44	of	of	ADP
ejpam-5274	371	45	the	the	DET
ejpam-5274	371	46	set	set	NOUN
ejpam-5274	371	47	{	{	PUNCT
ejpam-5274	371	48	ndi	ndi	PROPN
ejpam-5274	372	1	+	+	NUM
ejpam-5274	372	2	mqj	mqj	NOUN
ejpam-5274	372	3	:	:	PUNCT
ejpam-5274	372	4	1	1	NUM
ejpam-5274	372	5	≤	≤	NUM
ejpam-5274	372	6	i	i	X
ejpam-5274	373	1	≤	≤	NOUN
ejpam-5274	373	2	m	m	VERB
ejpam-5274	373	3	and	and	CCONJ
ejpam-5274	373	4	1	1	NUM
ejpam-5274	373	5	≤	≤	NUM
ejpam-5274	373	6	j	j	PROPN
ejpam-5274	373	7	≤	≤	PROPN
ejpam-5274	373	8	n	n	CCONJ
ejpam-5274	373	9	}	}	PUNCT
ejpam-5274	373	10	.	.	PUNCT
ejpam-5274	374	1	proof	proof	NOUN
ejpam-5274	374	2	.	.	PUNCT
ejpam-5274	375	1	from	from	ADP
ejpam-5274	375	2	theorem	theorem	ADJ
ejpam-5274	375	3	14	14	NUM
ejpam-5274	375	4	,	,	PUNCT
ejpam-5274	375	5	fg∨h(x	fg∨h(x	NOUN
ejpam-5274	375	6	)	)	PUNCT
ejpam-5274	375	7	=	=	SYM
ejpam-5274	375	8	fg(x	fg(x	NUM
ejpam-5274	375	9	n)fh(xm	n)fh(xm	NOUN
ejpam-5274	375	10	)	)	PUNCT
ejpam-5274	375	11	=	=	PUNCT
ejpam-5274	375	12	m∑	m∑	ADP
ejpam-5274	375	13	i=1	i=1	PROPN
ejpam-5274	375	14	xndi	xndi	PROPN
ejpam-5274	376	1	n∑	n∑	PROPN
ejpam-5274	376	2	j=1	j=1	PROPN
ejpam-5274	376	3	xmqj	xmqj	PROPN
ejpam-5274	377	1	=	=	PUNCT
ejpam-5274	377	2	m∑	m∑	ADP
ejpam-5274	377	3	i=1	i=1	PROPN
ejpam-5274	377	4	n∑	n∑	PROPN
ejpam-5274	378	1	j=1	j=1	PROPN
ejpam-5274	378	2	xndixmqj	xndixmqj	NOUN
ejpam-5274	378	3	=	=	SYM
ejpam-5274	378	4	m∑	m∑	INTJ
ejpam-5274	378	5	i=1	i=1	PROPN
ejpam-5274	378	6	n∑	n∑	PROPN
ejpam-5274	379	1	j=1	j=1	PROPN
ejpam-5274	379	2	xndi+mqj	xndi+mqj	PROPN
ejpam-5274	379	3	.	.	PUNCT
ejpam-5274	380	1	it	it	PRON
ejpam-5274	380	2	follows	follow	VERB
ejpam-5274	380	3	that	that	SCONJ
ejpam-5274	380	4	the	the	DET
ejpam-5274	380	5	terms	term	NOUN
ejpam-5274	380	6	of	of	ADP
ejpam-5274	380	7	the	the	DET
ejpam-5274	380	8	degree	degree	NOUN
ejpam-5274	380	9	sequence	sequence	NOUN
ejpam-5274	380	10	of	of	ADP
ejpam-5274	380	11	g	g	PROPN
ejpam-5274	380	12	∨	∨	PROPN
ejpam-5274	380	13	h	h	NOUN
ejpam-5274	380	14	are	be	AUX
ejpam-5274	380	15	the	the	DET
ejpam-5274	380	16	elements	element	NOUN
ejpam-5274	380	17	of	of	ADP
ejpam-5274	380	18	the	the	DET
ejpam-5274	380	19	set	set	NOUN
ejpam-5274	380	20	{	{	PUNCT
ejpam-5274	380	21	ndi	ndi	PROPN
ejpam-5274	381	1	+	+	NUM
ejpam-5274	381	2	mqj	mqj	NOUN
ejpam-5274	381	3	:	:	PUNCT
ejpam-5274	381	4	1	1	NUM
ejpam-5274	381	5	≤	≤	NUM
ejpam-5274	381	6	i	i	X
ejpam-5274	382	1	≤	≤	NOUN
ejpam-5274	382	2	m	m	VERB
ejpam-5274	382	3	and	and	CCONJ
ejpam-5274	382	4	1	1	NUM
ejpam-5274	382	5	≤	≤	NUM
ejpam-5274	382	6	j	j	PROPN
ejpam-5274	382	7	≤	≤	PROPN
ejpam-5274	382	8	n	n	CCONJ
ejpam-5274	382	9	}	}	PUNCT
ejpam-5274	382	10	.	.	PUNCT
ejpam-5274	383	1	references	reference	NOUN
ejpam-5274	383	2	1461	1461	NUM
ejpam-5274	383	3	4	4	NUM
ejpam-5274	383	4	.	.	PUNCT
ejpam-5274	383	5	conclusion	conclusion	NOUN
ejpam-5274	383	6	and	and	CCONJ
ejpam-5274	383	7	recommendation	recommendation	VERB
ejpam-5274	383	8	the	the	DET
ejpam-5274	383	9	polynomial	polynomial	ADJ
ejpam-5274	383	10	representations	representation	NOUN
ejpam-5274	383	11	of	of	ADP
ejpam-5274	383	12	some	some	DET
ejpam-5274	383	13	graphs	graph	NOUN
ejpam-5274	383	14	have	have	AUX
ejpam-5274	383	15	been	be	AUX
ejpam-5274	383	16	obtained	obtain	VERB
ejpam-5274	383	17	in	in	ADP
ejpam-5274	383	18	this	this	DET
ejpam-5274	383	19	study	study	NOUN
ejpam-5274	383	20	.	.	PUNCT
ejpam-5274	384	1	the	the	DET
ejpam-5274	384	2	authors	author	NOUN
ejpam-5274	384	3	were	be	AUX
ejpam-5274	384	4	not	not	PART
ejpam-5274	384	5	able	able	ADJ
ejpam-5274	384	6	to	to	PART
ejpam-5274	384	7	describe	describe	VERB
ejpam-5274	384	8	the	the	DET
ejpam-5274	384	9	degree	degree	NOUN
ejpam-5274	384	10	sequence	sequence	NOUN
ejpam-5274	384	11	of	of	ADP
ejpam-5274	384	12	some	some	DET
ejpam-5274	384	13	graphs	graph	NOUN
ejpam-5274	384	14	resulting	result	VERB
ejpam-5274	384	15	from	from	ADP
ejpam-5274	384	16	some	some	DET
ejpam-5274	384	17	operations	operation	NOUN
ejpam-5274	384	18	.	.	PUNCT
ejpam-5274	385	1	however	however	ADV
ejpam-5274	385	2	,	,	PUNCT
ejpam-5274	385	3	for	for	ADP
ejpam-5274	385	4	particular	particular	ADJ
ejpam-5274	385	5	graphs	graph	NOUN
ejpam-5274	385	6	,	,	PUNCT
ejpam-5274	385	7	the	the	DET
ejpam-5274	385	8	degree	degree	NOUN
ejpam-5274	385	9	sequence	sequence	NOUN
ejpam-5274	385	10	of	of	ADP
ejpam-5274	385	11	the	the	DET
ejpam-5274	385	12	graphs	graph	NOUN
ejpam-5274	385	13	may	may	AUX
ejpam-5274	385	14	be	be	AUX
ejpam-5274	385	15	obtained	obtain	VERB
ejpam-5274	385	16	.	.	PUNCT
ejpam-5274	385	17	determining	determine	VERB
ejpam-5274	385	18	real	real	ADJ
ejpam-5274	385	19	roots	root	NOUN
ejpam-5274	385	20	or	or	CCONJ
ejpam-5274	385	21	zeros	zero	NOUN
ejpam-5274	385	22	of	of	ADP
ejpam-5274	385	23	the	the	DET
ejpam-5274	385	24	polynomial	polynomial	ADJ
ejpam-5274	385	25	representation	representation	NOUN
ejpam-5274	385	26	of	of	ADP
ejpam-5274	385	27	a	a	DET
ejpam-5274	385	28	graph	graph	NOUN
ejpam-5274	385	29	,	,	PUNCT
ejpam-5274	385	30	if	if	SCONJ
ejpam-5274	385	31	they	they	PRON
ejpam-5274	385	32	exist	exist	VERB
ejpam-5274	385	33	,	,	PUNCT
ejpam-5274	385	34	can	can	AUX
ejpam-5274	385	35	be	be	AUX
ejpam-5274	385	36	an	an	DET
ejpam-5274	385	37	aspect	aspect	NOUN
ejpam-5274	385	38	for	for	ADP
ejpam-5274	385	39	further	further	ADJ
ejpam-5274	385	40	investigation	investigation	NOUN
ejpam-5274	385	41	.	.	PUNCT
ejpam-5274	386	1	acknowledgements	acknowledgement	NOUN
ejpam-5274	386	2	the	the	DET
ejpam-5274	386	3	authors	author	NOUN
ejpam-5274	386	4	would	would	AUX
ejpam-5274	386	5	like	like	VERB
ejpam-5274	386	6	to	to	PART
ejpam-5274	386	7	thank	thank	VERB
ejpam-5274	386	8	the	the	DET
ejpam-5274	386	9	referees	referee	NOUN
ejpam-5274	386	10	for	for	ADP
ejpam-5274	386	11	the	the	DET
ejpam-5274	386	12	suggestions	suggestion	NOUN
ejpam-5274	386	13	and	and	CCONJ
ejpam-5274	386	14	comments	comment	NOUN
ejpam-5274	386	15	that	that	PRON
ejpam-5274	386	16	helped	helped	AUX
ejpam-5274	386	17	improve	improve	VERB
ejpam-5274	386	18	the	the	DET
ejpam-5274	386	19	paper	paper	NOUN
ejpam-5274	386	20	.	.	PUNCT
ejpam-5274	387	1	the	the	DET
ejpam-5274	387	2	authors	author	NOUN
ejpam-5274	387	3	also	also	ADV
ejpam-5274	387	4	extend	extend	VERB
ejpam-5274	387	5	their	their	PRON
ejpam-5274	387	6	thankfulness	thankfulness	NOUN
ejpam-5274	387	7	to	to	ADP
ejpam-5274	387	8	the	the	DET
ejpam-5274	387	9	department	department	PROPN
ejpam-5274	387	10	of	of	ADP
ejpam-5274	387	11	science	science	NOUN
ejpam-5274	387	12	and	and	CCONJ
ejpam-5274	387	13	technology	technology	NOUN
ejpam-5274	387	14	accelerated	accelerate	VERB
ejpam-5274	387	15	science	science	NOUN
ejpam-5274	387	16	and	and	CCONJ
ejpam-5274	387	17	technology	technology	NOUN
ejpam-5274	387	18	human	human	ADJ
ejpam-5274	387	19	resource	resource	NOUN
ejpam-5274	387	20	development	development	NOUN
ejpam-5274	387	21	program	program	NOUN
ejpam-5274	387	22	(	(	PUNCT
ejpam-5274	387	23	dost	dost	NOUN
ejpam-5274	387	24	-	-	PUNCT
ejpam-5274	387	25	asthrdp)-philippines	asthrdp)-philippine	NOUN
ejpam-5274	387	26	,	,	PUNCT
ejpam-5274	387	27	and	and	CCONJ
ejpam-5274	387	28	msu	msu	PROPN
ejpam-5274	387	29	-	-	PUNCT
ejpam-5274	387	30	iligan	iligan	PROPN
ejpam-5274	387	31	institute	institute	PROPN
ejpam-5274	387	32	of	of	ADP
ejpam-5274	387	33	technology	technology	NOUN
ejpam-5274	387	34	for	for	ADP
ejpam-5274	387	35	funding	fund	VERB
ejpam-5274	387	36	this	this	DET
ejpam-5274	387	37	research	research	NOUN
ejpam-5274	387	38	.	.	PUNCT
ejpam-5274	388	1	references	reference	NOUN
ejpam-5274	388	2	[	[	X
ejpam-5274	388	3	1	1	NUM
ejpam-5274	388	4	]	]	PUNCT
ejpam-5274	388	5	g.	g.	PROPN
ejpam-5274	388	6	chen	chen	PROPN
ejpam-5274	388	7	,	,	PUNCT
ejpam-5274	388	8	m.	m.	NOUN
ejpam-5274	388	9	ferrara	ferrara	PROPN
ejpam-5274	388	10	,	,	PUNCT
ejpam-5274	388	11	r.	r.	PROPN
ejpam-5274	388	12	gould	gould	PROPN
ejpam-5274	388	13	,	,	PUNCT
ejpam-5274	388	14	and	and	CCONJ
ejpam-5274	388	15	j.	j.	PROPN
ejpam-5274	388	16	schmitt	schmitt	PROPN
ejpam-5274	388	17	.	.	PUNCT
ejpam-5274	389	1	graphic	graphic	ADJ
ejpam-5274	389	2	sequences	sequence	NOUN
ejpam-5274	389	3	with	with	ADP
ejpam-5274	389	4	a	a	DET
ejpam-5274	389	5	realization	realization	NOUN
ejpam-5274	389	6	containing	contain	VERB
ejpam-5274	389	7	a	a	DET
ejpam-5274	389	8	complete	complete	ADJ
ejpam-5274	389	9	multipartite	multipartite	ADJ
ejpam-5274	389	10	subgraph	subgraph	NOUN
ejpam-5274	389	11	.	.	PUNCT
ejpam-5274	390	1	discrete	discrete	ADJ
ejpam-5274	390	2	mathematics	mathematic	NOUN
ejpam-5274	390	3	,	,	PUNCT
ejpam-5274	390	4	308(23):5712	308(23):5712	PROPN
ejpam-5274	390	5	–	–	PUNCT
ejpam-5274	390	6	5721	5721	NUM
ejpam-5274	390	7	,	,	PUNCT
ejpam-5274	390	8	2008	2008	NUM
ejpam-5274	390	9	.	.	PUNCT
ejpam-5274	391	1	[	[	X
ejpam-5274	391	2	2	2	NUM
ejpam-5274	391	3	]	]	X
ejpam-5274	391	4	s.a	s.a	PROPN
ejpam-5274	391	5	.	.	PROPN
ejpam-5274	391	6	choudum	choudum	NOUN
ejpam-5274	391	7	.	.	PUNCT
ejpam-5274	392	1	a	a	DET
ejpam-5274	392	2	simple	simple	ADJ
ejpam-5274	392	3	proof	proof	NOUN
ejpam-5274	392	4	of	of	ADP
ejpam-5274	392	5	the	the	DET
ejpam-5274	392	6	erdös	erdös	ADJ
ejpam-5274	392	7	-	-	PUNCT
ejpam-5274	392	8	gallai	gallai	NOUN
ejpam-5274	392	9	theorem	theorem	NOUN
ejpam-5274	392	10	on	on	ADP
ejpam-5274	392	11	graph	graph	NOUN
ejpam-5274	392	12	sequences	sequence	NOUN
ejpam-5274	392	13	.	.	PUNCT
ejpam-5274	393	1	bull	bull	NOUN
ejpam-5274	393	2	.	.	PUNCT
ejpam-5274	394	1	austral	austral	PROPN
ejpam-5274	394	2	.	.	PUNCT
ejpam-5274	394	3	math	math	NOUN
ejpam-5274	394	4	.	.	PUNCT
ejpam-5274	395	1	soc	soc	PROPN
ejpam-5274	395	2	.	.	PROPN
ejpam-5274	395	3	,	,	PUNCT
ejpam-5274	395	4	33:67–70	33:67–70	NUM
ejpam-5274	395	5	,	,	PUNCT
ejpam-5274	395	6	1986	1986	NUM
ejpam-5274	395	7	.	.	PUNCT
ejpam-5274	396	1	[	[	X
ejpam-5274	396	2	3	3	X
ejpam-5274	396	3	]	]	X
ejpam-5274	396	4	r.b	r.b	PROPN
ejpam-5274	396	5	.	.	PROPN
ejpam-5274	396	6	eggleton	eggleton	PROPN
ejpam-5274	396	7	.	.	PUNCT
ejpam-5274	396	8	graphic	graphic	ADJ
ejpam-5274	396	9	sequences	sequence	NOUN
ejpam-5274	396	10	and	and	CCONJ
ejpam-5274	396	11	graphic	graphic	ADJ
ejpam-5274	396	12	polynomials	polynomial	NOUN
ejpam-5274	396	13	:	:	PUNCT
ejpam-5274	396	14	a	a	DET
ejpam-5274	396	15	reported	report	VERB
ejpam-5274	396	16	.	.	PUNCT
ejpam-5274	397	1	colloq	colloq	PROPN
ejpam-5274	397	2	.	.	PUNCT
ejpam-5274	398	1	math	math	PROPN
ejpam-5274	398	2	.	.	PUNCT
ejpam-5274	399	1	soc	soc	PROPN
ejpam-5274	399	2	.	.	PUNCT
ejpam-5274	400	1	j.	j.	PROPN
ejpam-5274	400	2	bolyai	bolyai	PROPN
ejpam-5274	400	3	,	,	PUNCT
ejpam-5274	400	4	10:385–392	10:385–392	NUM
ejpam-5274	400	5	,	,	PUNCT
ejpam-5274	400	6	1975	1975	NUM
ejpam-5274	400	7	.	.	PUNCT
ejpam-5274	401	1	[	[	X
ejpam-5274	401	2	4	4	X
ejpam-5274	401	3	]	]	PUNCT
ejpam-5274	401	4	p.	p.	NOUN
ejpam-5274	401	5	erdös	erdös	PROPN
ejpam-5274	401	6	and	and	CCONJ
ejpam-5274	401	7	t.	t.	NOUN
ejpam-5274	401	8	gallai	gallai	NOUN
ejpam-5274	401	9	.	.	PUNCT
ejpam-5274	402	1	graphs	graph	NOUN
ejpam-5274	402	2	with	with	ADP
ejpam-5274	402	3	prescribed	prescribed	ADJ
ejpam-5274	402	4	degree	degree	NOUN
ejpam-5274	402	5	of	of	ADP
ejpam-5274	402	6	vertices	vertex	NOUN
ejpam-5274	402	7	(	(	PUNCT
ejpam-5274	402	8	hungarian	hungarian	ADJ
ejpam-5274	402	9	)	)	PUNCT
ejpam-5274	402	10	.	.	PUNCT
ejpam-5274	403	1	mat	mat	NOUN
ejpam-5274	403	2	.	.	PUNCT
ejpam-5274	403	3	lapok	lapok	NOUN
ejpam-5274	403	4	,	,	PUNCT
ejpam-5274	403	5	11:264–274	11:264–274	NUM
ejpam-5274	403	6	,	,	PUNCT
ejpam-5274	403	7	1960	1960	NUM
ejpam-5274	403	8	.	.	PUNCT
ejpam-5274	404	1	[	[	X
ejpam-5274	404	2	5	5	NUM
ejpam-5274	404	3	]	]	X
ejpam-5274	404	4	s.l	s.l	PROPN
ejpam-5274	404	5	.	.	PROPN
ejpam-5274	404	6	hakimi	hakimi	PROPN
ejpam-5274	404	7	.	.	PUNCT
ejpam-5274	405	1	on	on	ADP
ejpam-5274	405	2	the	the	DET
ejpam-5274	405	3	realizability	realizability	NOUN
ejpam-5274	405	4	of	of	ADP
ejpam-5274	405	5	a	a	DET
ejpam-5274	405	6	set	set	NOUN
ejpam-5274	405	7	of	of	ADP
ejpam-5274	405	8	integers	integer	NOUN
ejpam-5274	405	9	as	as	ADP
ejpam-5274	405	10	degrees	degree	NOUN
ejpam-5274	405	11	of	of	ADP
ejpam-5274	405	12	the	the	DET
ejpam-5274	405	13	vertices	vertex	NOUN
ejpam-5274	405	14	of	of	ADP
ejpam-5274	405	15	a	a	DET
ejpam-5274	405	16	simple	simple	ADJ
ejpam-5274	405	17	graph	graph	NOUN
ejpam-5274	405	18	.	.	PUNCT
ejpam-5274	406	1	j.	j.	PROPN
ejpam-5274	406	2	siam	siam	PROPN
ejpam-5274	406	3	appl	appl	PROPN
ejpam-5274	406	4	.	.	PROPN
ejpam-5274	406	5	math	math	PROPN
ejpam-5274	406	6	.	.	PUNCT
ejpam-5274	406	7	,	,	PUNCT
ejpam-5274	406	8	10:496–506	10:496–506	NUM
ejpam-5274	406	9	,	,	PUNCT
ejpam-5274	406	10	1962	1962	NUM
ejpam-5274	406	11	.	.	PUNCT
ejpam-5274	407	1	[	[	X
ejpam-5274	407	2	6	6	NUM
ejpam-5274	407	3	]	]	PUNCT
ejpam-5274	407	4	v.	v.	X
ejpam-5274	407	5	havel	havel	NOUN
ejpam-5274	407	6	.	.	PUNCT
ejpam-5274	408	1	a	a	DET
ejpam-5274	408	2	remark	remark	NOUN
ejpam-5274	408	3	on	on	ADP
ejpam-5274	408	4	the	the	DET
ejpam-5274	408	5	existence	existence	NOUN
ejpam-5274	408	6	of	of	ADP
ejpam-5274	408	7	finite	finite	ADJ
ejpam-5274	408	8	graphs	graph	NOUN
ejpam-5274	408	9	(	(	PUNCT
ejpam-5274	408	10	czech	czech	PROPN
ejpam-5274	408	11	)	)	PUNCT
ejpam-5274	408	12	.	.	PUNCT
ejpam-5274	409	1	casopis	casopis	PROPN
ejpam-5274	409	2	pest	pest	NOUN
ejpam-5274	409	3	.	.	PUNCT
ejpam-5274	410	1	mat	mat	NOUN
ejpam-5274	410	2	.	.	PROPN
ejpam-5274	410	3	,	,	PUNCT
ejpam-5274	410	4	80(4):477–480	80(4):477–480	PROPN
ejpam-5274	410	5	,	,	PUNCT
ejpam-5274	410	6	1955	1955	NUM
ejpam-5274	410	7	.	.	PUNCT
ejpam-5274	411	1	[	[	X
ejpam-5274	411	2	7	7	X
ejpam-5274	411	3	]	]	X
ejpam-5274	411	4	s.	s.	PROPN
ejpam-5274	411	5	canoy	canoy	PROPN
ejpam-5274	411	6	jr	jr	PROPN
ejpam-5274	411	7	and	and	CCONJ
ejpam-5274	411	8	j.	j.	PROPN
ejpam-5274	411	9	hassan	hassan	PROPN
ejpam-5274	411	10	.	.	PUNCT
ejpam-5274	412	1	weakly	weakly	ADJ
ejpam-5274	412	2	convex	convex	VERB
ejpam-5274	412	3	hop	hop	NOUN
ejpam-5274	412	4	dominating	dominating	NOUN
ejpam-5274	412	5	sets	set	NOUN
ejpam-5274	412	6	in	in	ADP
ejpam-5274	412	7	graphs	graph	NOUN
ejpam-5274	412	8	.	.	PUNCT
ejpam-5274	413	1	european	european	ADJ
ejpam-5274	413	2	journal	journal	PROPN
ejpam-5274	413	3	of	of	ADP
ejpam-5274	413	4	pure	pure	ADJ
ejpam-5274	413	5	and	and	CCONJ
ejpam-5274	413	6	applied	applied	ADJ
ejpam-5274	413	7	mathematics	mathematic	NOUN
ejpam-5274	413	8	,	,	PUNCT
ejpam-5274	413	9	16(2):1196–1211	16(2):1196–1211	NUM
ejpam-5274	413	10	,	,	PUNCT
ejpam-5274	413	11	2023	2023	NUM
ejpam-5274	413	12	.	.	PUNCT
ejpam-5274	414	1	[	[	X
ejpam-5274	414	2	8	8	NUM
ejpam-5274	414	3	]	]	X
ejpam-5274	414	4	s.r	s.r	PROPN
ejpam-5274	414	5	.	.	PROPN
ejpam-5274	414	6	canoy	canoy	PROPN
ejpam-5274	414	7	jr	jr	PROPN
ejpam-5274	414	8	.	.	PROPN
ejpam-5274	414	9	and	and	CCONJ
ejpam-5274	414	10	m.	m.	NOUN
ejpam-5274	414	11	labendia	labendia	PROPN
ejpam-5274	414	12	.	.	PUNCT
ejpam-5274	415	1	polynomial	polynomial	ADJ
ejpam-5274	415	2	representation	representation	NOUN
ejpam-5274	415	3	and	and	CCONJ
ejpam-5274	415	4	degree	degree	NOUN
ejpam-5274	415	5	sequenc	sequenc	NOUN
ejpam-5274	415	6	of	of	ADP
ejpam-5274	415	7	a	a	DET
ejpam-5274	415	8	graph	graph	NOUN
ejpam-5274	415	9	.	.	PUNCT
ejpam-5274	416	1	int	int	NOUN
ejpam-5274	416	2	.	.	PUNCT
ejpam-5274	417	1	journal	journal	PROPN
ejpam-5274	417	2	of	of	ADP
ejpam-5274	417	3	math	math	NOUN
ejpam-5274	417	4	.	.	PUNCT
ejpam-5274	418	1	analysis	analysis	NOUN
ejpam-5274	418	2	.	.	PUNCT
ejpam-5274	418	3	,	,	PUNCT
ejpam-5274	418	4	8(29):1445–1455	8(29):1445–1455	NUM
ejpam-5274	418	5	,	,	PUNCT
ejpam-5274	418	6	2014	2014	NUM
ejpam-5274	418	7	.	.	PUNCT
ejpam-5274	419	1	[	[	X
ejpam-5274	419	2	9	9	NUM
ejpam-5274	419	3	]	]	X
ejpam-5274	419	4	s.y.r	s.y.r	PROPN
ejpam-5274	419	5	.	.	PUNCT
ejpam-5274	419	6	li	li	PROPN
ejpam-5274	419	7	.	.	PROPN
ejpam-5274	419	8	graphic	graphic	ADJ
ejpam-5274	419	9	sequences	sequence	NOUN
ejpam-5274	419	10	with	with	ADP
ejpam-5274	419	11	unique	unique	ADJ
ejpam-5274	419	12	realization	realization	NOUN
ejpam-5274	419	13	.	.	PUNCT
ejpam-5274	420	1	j.	j.	PROPN
ejpam-5274	420	2	combin	combin	PROPN
ejpam-5274	420	3	.	.	PUNCT
ejpam-5274	420	4	theory	theory	NOUN
ejpam-5274	420	5	ser	ser	PROPN
ejpam-5274	420	6	.	.	PUNCT
ejpam-5274	421	1	b.	b.	PROPN
ejpam-5274	421	2	,	,	PUNCT
ejpam-5274	421	3	19:42–68	19:42–68	NUM
ejpam-5274	421	4	,	,	PUNCT
ejpam-5274	421	5	1975	1975	NUM
ejpam-5274	421	6	.	.	PUNCT
ejpam-5274	422	1	references	reference	NOUN
ejpam-5274	422	2	1462	1462	NUM
ejpam-5274	422	3	[	[	X
ejpam-5274	422	4	10	10	NUM
ejpam-5274	422	5	]	]	PUNCT
ejpam-5274	422	6	b.	b.	NOUN
ejpam-5274	422	7	omamalin	omamalin	PROPN
ejpam-5274	422	8	,	,	PUNCT
ejpam-5274	422	9	s.	s.	PROPN
ejpam-5274	422	10	canoy	canoy	PROPN
ejpam-5274	422	11	jr	jr	PROPN
ejpam-5274	422	12	,	,	PUNCT
ejpam-5274	422	13	and	and	CCONJ
ejpam-5274	422	14	h.	h.	PROPN
ejpam-5274	422	15	rara	rara	PROPN
ejpam-5274	422	16	.	.	PUNCT
ejpam-5274	423	1	locating	locate	VERB
ejpam-5274	423	2	total	total	ADJ
ejpam-5274	423	3	dominating	dominating	NOUN
ejpam-5274	423	4	sets	set	NOUN
ejpam-5274	423	5	in	in	ADP
ejpam-5274	423	6	the	the	DET
ejpam-5274	423	7	join	join	NOUN
ejpam-5274	423	8	,	,	PUNCT
ejpam-5274	423	9	corona	corona	NOUN
ejpam-5274	423	10	and	and	CCONJ
ejpam-5274	423	11	composition	composition	NOUN
ejpam-5274	423	12	of	of	ADP
ejpam-5274	423	13	graphs	graph	NOUN
ejpam-5274	423	14	.	.	PUNCT
ejpam-5274	424	1	applied	apply	VERB
ejpam-5274	424	2	mathematical	mathematical	ADJ
ejpam-5274	424	3	sciences	science	NOUN
ejpam-5274	424	4	,	,	PUNCT
ejpam-5274	424	5	8(48):2363–2374	8(48):2363–2374	NUM
ejpam-5274	424	6	,	,	PUNCT
ejpam-5274	424	7	2014	2014	NUM
ejpam-5274	424	8	.	.	PUNCT
ejpam-5274	425	1	[	[	X
ejpam-5274	425	2	11	11	NUM
ejpam-5274	425	3	]	]	X
ejpam-5274	425	4	g.	g.	NOUN
ejpam-5274	425	5	salasalan	salasalan	NOUN
ejpam-5274	425	6	and	and	CCONJ
ejpam-5274	425	7	s.	s.	PROPN
ejpam-5274	425	8	canoy	canoy	PROPN
ejpam-5274	425	9	jr	jr	PROPN
ejpam-5274	425	10	.	.	PROPN
ejpam-5274	425	11	global	global	PROPN
ejpam-5274	425	12	hop	hop	PROPN
ejpam-5274	425	13	domination	domination	PROPN
ejpam-5274	425	14	numbers	number	NOUN
ejpam-5274	425	15	of	of	ADP
ejpam-5274	425	16	graphs	graph	NOUN
ejpam-5274	425	17	.	.	PUNCT
ejpam-5274	426	1	european	european	ADJ
ejpam-5274	426	2	journal	journal	PROPN
ejpam-5274	426	3	of	of	ADP
ejpam-5274	426	4	pure	pure	ADJ
ejpam-5274	426	5	and	and	CCONJ
ejpam-5274	426	6	applied	applied	ADJ
ejpam-5274	426	7	mathematics	mathematic	NOUN
ejpam-5274	426	8	,	,	PUNCT
ejpam-5274	426	9	14(1):112–125	14(1):112–125	NUM
ejpam-5274	426	10	,	,	PUNCT
ejpam-5274	426	11	2021	2021	NUM
ejpam-5274	426	12	.	.	PUNCT
ejpam-5274	427	1	[	[	X
ejpam-5274	427	2	12	12	NUM
ejpam-5274	427	3	]	]	PUNCT
ejpam-5274	427	4	i.e.	i.e.	X
ejpam-5274	427	5	zverovich	zverovich	PROPN
ejpam-5274	427	6	and	and	CCONJ
ejpam-5274	427	7	v.e	v.e	PROPN
ejpam-5274	427	8	.	.	PROPN
ejpam-5274	427	9	zverovich	zverovich	PROPN
ejpam-5274	427	10	.	.	PUNCT
ejpam-5274	428	1	contributions	contribution	NOUN
ejpam-5274	428	2	to	to	ADP
ejpam-5274	428	3	the	the	DET
ejpam-5274	428	4	theory	theory	NOUN
ejpam-5274	428	5	of	of	ADP
ejpam-5274	428	6	graphic	graphic	ADJ
ejpam-5274	428	7	sequences	sequence	NOUN
ejpam-5274	428	8	.	.	PUNCT
ejpam-5274	429	1	discrete	discrete	ADJ
ejpam-5274	429	2	mathematics	mathematic	NOUN
ejpam-5274	429	3	,	,	PUNCT
ejpam-5274	429	4	105:293–303	105:293–303	NUM
ejpam-5274	429	5	,	,	PUNCT
ejpam-5274	429	6	1992	1992	NUM
ejpam-5274	429	7	.	.	PUNCT