id	sid	tid	token	lemma	pos
ejpam-4698	1	1	european	european	PROPN
ejpam-4698	1	2	journal	journal	PROPN
ejpam-4698	1	3	of	of	ADP
ejpam-4698	1	4	pure	pure	ADJ
ejpam-4698	1	5	and	and	CCONJ
ejpam-4698	1	6	applied	apply	VERB
ejpam-4698	1	7	mathematics	mathematic	NOUN
ejpam-4698	1	8	vol	vol	NOUN
ejpam-4698	1	9	.	.	PUNCT
ejpam-4698	2	1	16	16	NUM
ejpam-4698	2	2	,	,	PUNCT
ejpam-4698	2	3	no	no	INTJ
ejpam-4698	2	4	.	.	NOUN
ejpam-4698	2	5	2	2	NUM
ejpam-4698	2	6	,	,	PUNCT
ejpam-4698	2	7	2023	2023	NUM
ejpam-4698	2	8	,	,	PUNCT
ejpam-4698	2	9	773	773	NUM
ejpam-4698	2	10	-	-	SYM
ejpam-4698	2	11	783	783	NUM
ejpam-4698	2	12	issn	issn	PROPN
ejpam-4698	2	13	1307	1307	NUM
ejpam-4698	2	14	-	-	SYM
ejpam-4698	2	15	5543	5543	NUM
ejpam-4698	2	16	–	–	PUNCT
ejpam-4698	2	17	ejpam.com	ejpam.com	X
ejpam-4698	2	18	published	publish	VERB
ejpam-4698	2	19	by	by	ADP
ejpam-4698	2	20	new	new	PROPN
ejpam-4698	2	21	york	york	PROPN
ejpam-4698	2	22	business	business	PROPN
ejpam-4698	2	23	global	global	ADJ
ejpam-4698	2	24	relations	relation	NOUN
ejpam-4698	2	25	between	between	ADP
ejpam-4698	2	26	vertex	vertex	NOUN
ejpam-4698	2	27	–	–	PUNCT
ejpam-4698	2	28	edge	edge	NOUN
ejpam-4698	2	29	degree	degree	NOUN
ejpam-4698	2	30	based	base	VERB
ejpam-4698	2	31	topological	topological	ADJ
ejpam-4698	2	32	indices	index	NOUN
ejpam-4698	2	33	and	and	CCONJ
ejpam-4698	2	34	mve	mve	NOUN
ejpam-4698	2	35	-	-	PUNCT
ejpam-4698	2	36	polynomial	polynomial	NOUN
ejpam-4698	2	37	of	of	ADP
ejpam-4698	2	38	r−regular	r−regular	ADJ
ejpam-4698	2	39	simple	simple	ADJ
ejpam-4698	2	40	graph	graph	NOUN
ejpam-4698	2	41	kavi	kavi	PROPN
ejpam-4698	2	42	b.	b.	PROPN
ejpam-4698	2	43	rasool1,∗	rasool1,∗	PROPN
ejpam-4698	2	44	,	,	PUNCT
ejpam-4698	2	45	payman	payman	NOUN
ejpam-4698	2	46	a.	a.	PROPN
ejpam-4698	2	47	rashed2	rashed2	PROPN
ejpam-4698	2	48	,	,	PUNCT
ejpam-4698	2	49	ahmed	ahmed	PROPN
ejpam-4698	2	50	m.	m.	PROPN
ejpam-4698	2	51	ali3	ali3	PROPN
ejpam-4698	2	52	1	1	NUM
ejpam-4698	2	53	faculity	faculity	NOUN
ejpam-4698	2	54	of	of	ADP
ejpam-4698	2	55	science	science	NOUN
ejpam-4698	2	56	,	,	PUNCT
ejpam-4698	2	57	university	university	NOUN
ejpam-4698	2	58	of	of	ADP
ejpam-4698	2	59	zakho	zakho	PROPN
ejpam-4698	2	60	,	,	PUNCT
ejpam-4698	2	61	duhok	duhok	NOUN
ejpam-4698	2	62	,	,	PUNCT
ejpam-4698	2	63	kurdistan	kurdistan	PROPN
ejpam-4698	2	64	region	region	PROPN
ejpam-4698	2	65	-	-	PUNCT
ejpam-4698	2	66	iraq	iraq	PROPN
ejpam-4698	2	67	2	2	NUM
ejpam-4698	2	68	college	college	NOUN
ejpam-4698	2	69	of	of	ADP
ejpam-4698	2	70	basic	basic	ADJ
ejpam-4698	2	71	education	education	NOUN
ejpam-4698	2	72	,	,	PUNCT
ejpam-4698	2	73	university	university	NOUN
ejpam-4698	2	74	of	of	ADP
ejpam-4698	2	75	salahaddin	salahaddin	PROPN
ejpam-4698	2	76	,	,	PUNCT
ejpam-4698	2	77	erbil	erbil	PROPN
ejpam-4698	2	78	,	,	PUNCT
ejpam-4698	2	79	kurdistan	kurdistan	PROPN
ejpam-4698	2	80	region	region	PROPN
ejpam-4698	2	81	-	-	PUNCT
ejpam-4698	2	82	iraq	iraq	PROPN
ejpam-4698	2	83	3	3	NUM
ejpam-4698	2	84	college	college	NOUN
ejpam-4698	2	85	of	of	ADP
ejpam-4698	2	86	computer	computer	NOUN
ejpam-4698	2	87	science	science	NOUN
ejpam-4698	2	88	and	and	CCONJ
ejpam-4698	2	89	mathematics	mathematic	NOUN
ejpam-4698	2	90	,	,	PUNCT
ejpam-4698	2	91	university	university	NOUN
ejpam-4698	2	92	of	of	ADP
ejpam-4698	2	93	al	al	PROPN
ejpam-4698	2	94	mosul	mosul	PROPN
ejpam-4698	2	95	,	,	PUNCT
ejpam-4698	2	96	mosul	mosul	PROPN
ejpam-4698	2	97	,	,	PUNCT
ejpam-4698	2	98	iraq	iraq	PROPN
ejpam-4698	2	99	abstract	abstract	NOUN
ejpam-4698	2	100	.	.	PUNCT
ejpam-4698	3	1	one	one	NUM
ejpam-4698	3	2	of	of	ADP
ejpam-4698	3	3	the	the	DET
ejpam-4698	3	4	more	more	ADV
ejpam-4698	3	5	exciting	exciting	ADJ
ejpam-4698	3	6	polynomials	polynomial	NOUN
ejpam-4698	3	7	among	among	ADP
ejpam-4698	3	8	the	the	DET
ejpam-4698	3	9	newly	newly	ADV
ejpam-4698	3	10	presented	present	VERB
ejpam-4698	3	11	graph	graph	NOUN
ejpam-4698	3	12	algebraic	algebraic	ADJ
ejpam-4698	3	13	polynomials	polynomial	NOUN
ejpam-4698	3	14	is	be	AUX
ejpam-4698	3	15	them−polynomial	them−polynomial	ADJ
ejpam-4698	3	16	,	,	PUNCT
ejpam-4698	3	17	which	which	PRON
ejpam-4698	3	18	is	be	AUX
ejpam-4698	3	19	a	a	DET
ejpam-4698	3	20	standard	standard	ADJ
ejpam-4698	3	21	method	method	NOUN
ejpam-4698	3	22	for	for	ADP
ejpam-4698	3	23	calculating	calculate	VERB
ejpam-4698	3	24	degree−based	degree−base	VERB
ejpam-4698	3	25	topological	topological	ADJ
ejpam-4698	3	26	indices	index	NOUN
ejpam-4698	3	27	.	.	PUNCT
ejpam-4698	4	1	in	in	ADP
ejpam-4698	4	2	this	this	DET
ejpam-4698	4	3	paper	paper	NOUN
ejpam-4698	4	4	,	,	PUNCT
ejpam-4698	4	5	we	we	PRON
ejpam-4698	4	6	define	define	VERB
ejpam-4698	4	7	the	the	DET
ejpam-4698	4	8	mve−polynomials	mve−polynomial	NOUN
ejpam-4698	4	9	based	base	VERB
ejpam-4698	4	10	on	on	ADP
ejpam-4698	4	11	vertex	vertex	NOUN
ejpam-4698	4	12	–	–	PUNCT
ejpam-4698	4	13	edge	edge	NOUN
ejpam-4698	4	14	degree	degree	NOUN
ejpam-4698	4	15	and	and	CCONJ
ejpam-4698	4	16	derive	derive	VERB
ejpam-4698	4	17	various	various	ADJ
ejpam-4698	4	18	vertex	vertex	NOUN
ejpam-4698	4	19	–	–	PUNCT
ejpam-4698	4	20	edge	edge	NOUN
ejpam-4698	4	21	degree	degree	NOUN
ejpam-4698	4	22	based	base	VERB
ejpam-4698	4	23	topological	topological	ADJ
ejpam-4698	4	24	indices	index	NOUN
ejpam-4698	4	25	from	from	ADP
ejpam-4698	4	26	them	they	PRON
ejpam-4698	4	27	.	.	PUNCT
ejpam-4698	5	1	thus	thus	ADV
ejpam-4698	5	2	,	,	PUNCT
ejpam-4698	5	3	for	for	ADP
ejpam-4698	5	4	any	any	DET
ejpam-4698	5	5	graph	graph	NOUN
ejpam-4698	5	6	,	,	PUNCT
ejpam-4698	5	7	we	we	PRON
ejpam-4698	5	8	provide	provide	VERB
ejpam-4698	5	9	some	some	DET
ejpam-4698	5	10	relationships	relationship	NOUN
ejpam-4698	5	11	between	between	ADP
ejpam-4698	5	12	vertex	vertex	NOUN
ejpam-4698	5	13	–	–	PUNCT
ejpam-4698	5	14	edge	edge	NOUN
ejpam-4698	5	15	degree	degree	NOUN
ejpam-4698	5	16	topological	topological	ADJ
ejpam-4698	5	17	indices	index	NOUN
ejpam-4698	5	18	.	.	PUNCT
ejpam-4698	6	1	also	also	ADV
ejpam-4698	6	2	,	,	PUNCT
ejpam-4698	6	3	we	we	PRON
ejpam-4698	6	4	discuss	discuss	VERB
ejpam-4698	6	5	the	the	DET
ejpam-4698	6	6	general	general	PROPN
ejpam-4698	6	7	mve−polynomial	mve−polynomial	PROPN
ejpam-4698	6	8	of	of	ADP
ejpam-4698	6	9	r−regular	r−regular	ADJ
ejpam-4698	6	10	simple	simple	ADJ
ejpam-4698	6	11	graph	graph	NOUN
ejpam-4698	6	12	.	.	PUNCT
ejpam-4698	7	1	finally	finally	ADV
ejpam-4698	7	2	,	,	PUNCT
ejpam-4698	7	3	we	we	PRON
ejpam-4698	7	4	computed	compute	VERB
ejpam-4698	7	5	the	the	DET
ejpam-4698	7	6	mve−polynomial	mve−polynomial	PROPN
ejpam-4698	7	7	of	of	ADP
ejpam-4698	7	8	the	the	DET
ejpam-4698	7	9	2−ary	2−ary	ADJ
ejpam-4698	7	10	tree	tree	NOUN
ejpam-4698	7	11	graph	graph	NOUN
ejpam-4698	7	12	.	.	PUNCT
ejpam-4698	8	1	2020	2020	NUM
ejpam-4698	8	2	mathematics	mathematic	NOUN
ejpam-4698	8	3	subject	subject	NOUN
ejpam-4698	8	4	classifications	classification	NOUN
ejpam-4698	8	5	:	:	PUNCT
ejpam-4698	8	6	05c05	05c05	NOUN
ejpam-4698	8	7	,	,	PUNCT
ejpam-4698	8	8	05c07	05c07	NOUN
ejpam-4698	8	9	,	,	PUNCT
ejpam-4698	8	10	05c10	05c10	ADJ
ejpam-4698	8	11	,	,	PUNCT
ejpam-4698	8	12	94c15	94c15	NUM
ejpam-4698	8	13	key	key	ADJ
ejpam-4698	8	14	words	word	NOUN
ejpam-4698	8	15	and	and	CCONJ
ejpam-4698	8	16	phrases	phrase	NOUN
ejpam-4698	8	17	:	:	PUNCT
ejpam-4698	8	18	mve−polynomials	mve−polynomial	NOUN
ejpam-4698	8	19	,	,	PUNCT
ejpam-4698	8	20	mve−indices	mve−indice	NOUN
ejpam-4698	8	21	,	,	PUNCT
ejpam-4698	8	22	regular	regular	ADJ
ejpam-4698	8	23	graph	graph	NOUN
ejpam-4698	8	24	,	,	PUNCT
ejpam-4698	8	25	2−ary	2−ary	ADJ
ejpam-4698	8	26	tree	tree	NOUN
ejpam-4698	8	27	1	1	NUM
ejpam-4698	8	28	.	.	PUNCT
ejpam-4698	9	1	introduction	introduction	NOUN
ejpam-4698	9	2	let	let	VERB
ejpam-4698	9	3	g	g	PRON
ejpam-4698	9	4	be	be	AUX
ejpam-4698	9	5	a	a	DET
ejpam-4698	9	6	connected	connected	ADJ
ejpam-4698	9	7	simple	simple	ADJ
ejpam-4698	9	8	graph	graph	NOUN
ejpam-4698	9	9	and	and	CCONJ
ejpam-4698	9	10	let	let	VERB
ejpam-4698	9	11	w	w	NOUN
ejpam-4698	9	12	be	be	AUX
ejpam-4698	9	13	a	a	DET
ejpam-4698	9	14	weight	weight	NOUN
ejpam-4698	9	15	given	give	VERB
ejpam-4698	9	16	to	to	ADP
ejpam-4698	9	17	its	its	PRON
ejpam-4698	9	18	vertices	vertex	NOUN
ejpam-4698	9	19	.	.	PUNCT
ejpam-4698	10	1	that	that	PRON
ejpam-4698	10	2	is	be	AUX
ejpam-4698	10	3	,	,	PUNCT
ejpam-4698	10	4	for	for	ADP
ejpam-4698	10	5	each	each	DET
ejpam-4698	10	6	v	v	NUM
ejpam-4698	10	7	∈	∈	PROPN
ejpam-4698	10	8	v	v	NOUN
ejpam-4698	10	9	(	(	PUNCT
ejpam-4698	10	10	g	g	NOUN
ejpam-4698	10	11	)	)	PUNCT
ejpam-4698	10	12	”	"	PUNCT
ejpam-4698	10	13	v	v	NOUN
ejpam-4698	10	14	=	=	SYM
ejpam-4698	10	15	v	v	NOUN
ejpam-4698	10	16	(	(	PUNCT
ejpam-4698	10	17	g	g	NOUN
ejpam-4698	10	18	)	)	PUNCT
ejpam-4698	10	19	be	be	VERB
ejpam-4698	10	20	the	the	DET
ejpam-4698	10	21	vertex	vertex	NOUN
ejpam-4698	10	22	set	set	NOUN
ejpam-4698	10	23	”	"	PUNCT
ejpam-4698	10	24	,	,	PUNCT
ejpam-4698	10	25	w(v	w(v	PROPN
ejpam-4698	10	26	)	)	PUNCT
ejpam-4698	10	27	is	be	AUX
ejpam-4698	10	28	a	a	DET
ejpam-4698	10	29	positive	positive	ADJ
ejpam-4698	10	30	integer	integer	NOUN
ejpam-4698	10	31	.	.	PUNCT
ejpam-4698	11	1	a	a	DET
ejpam-4698	11	2	wve−polynomial	wve−polynomial	PROPN
ejpam-4698	11	3	of	of	ADP
ejpam-4698	11	4	g	g	PROPN
ejpam-4698	11	5	is	be	AUX
ejpam-4698	11	6	defined	define	VERB
ejpam-4698	11	7	by	by	ADP
ejpam-4698	11	8	:	:	PUNCT
ejpam-4698	11	9	wve(g;x	wve(g;x	PROPN
ejpam-4698	11	10	,	,	PUNCT
ejpam-4698	11	11	y	y	NOUN
ejpam-4698	11	12	)	)	PUNCT
ejpam-4698	11	13	=	=	SYM
ejpam-4698	12	1	σuv∈e(g)x	σuv∈e(g)x	NOUN
ejpam-4698	12	2	w(u)yw(v	w(u)yw(v	NOUN
ejpam-4698	12	3	)	)	PUNCT
ejpam-4698	12	4	,	,	PUNCT
ejpam-4698	12	5	w(u	w(u	PROPN
ejpam-4698	12	6	)	)	PUNCT
ejpam-4698	12	7	≤	≤	NUM
ejpam-4698	12	8	w(v	w(v	NOUN
ejpam-4698	12	9	)	)	PUNCT
ejpam-4698	12	10	,	,	PUNCT
ejpam-4698	12	11	(	(	PUNCT
ejpam-4698	12	12	1.1	1.1	NUM
ejpam-4698	12	13	)	)	PUNCT
ejpam-4698	12	14	where	where	SCONJ
ejpam-4698	12	15	”	"	PUNCT
ejpam-4698	12	16	e	e	PROPN
ejpam-4698	12	17	=	=	PROPN
ejpam-4698	12	18	e(g	e(g	PROPN
ejpam-4698	12	19	)	)	PUNCT
ejpam-4698	12	20	be	be	VERB
ejpam-4698	12	21	the	the	DET
ejpam-4698	12	22	edge	edge	NOUN
ejpam-4698	12	23	set	set	NOUN
ejpam-4698	12	24	”	"	PUNCT
ejpam-4698	12	25	.	.	PUNCT
ejpam-4698	13	1	collecting	collect	VERB
ejpam-4698	13	2	all	all	DET
ejpam-4698	13	3	similar	similar	ADJ
ejpam-4698	13	4	terms	term	NOUN
ejpam-4698	13	5	xiyj	xiyj	NOUN
ejpam-4698	13	6	we	we	PRON
ejpam-4698	13	7	can	can	AUX
ejpam-4698	13	8	rewrite	rewrite	VERB
ejpam-4698	13	9	this	this	DET
ejpam-4698	13	10	polynomial	polynomial	NOUN
ejpam-4698	13	11	as	as	ADP
ejpam-4698	13	12	:	:	PUNCT
ejpam-4698	13	13	wve(g;x	wve(g;x	PROPN
ejpam-4698	13	14	,	,	PUNCT
ejpam-4698	13	15	y	y	NOUN
ejpam-4698	13	16	)	)	PUNCT
ejpam-4698	13	17	=	=	PUNCT
ejpam-4698	14	1	σuv∈e(g)nijx	σuv∈e(g)nijx	PROPN
ejpam-4698	14	2	iyj	iyj	VERB
ejpam-4698	14	3	,	,	PUNCT
ejpam-4698	14	4	i	i	PROPN
ejpam-4698	14	5	≤	≤	PROPN
ejpam-4698	14	6	j	j	PROPN
ejpam-4698	14	7	,	,	PUNCT
ejpam-4698	14	8	(	(	PUNCT
ejpam-4698	14	9	1.2	1.2	NUM
ejpam-4698	14	10	)	)	PUNCT
ejpam-4698	14	11	∗corresponding	∗corresponde	VERB
ejpam-4698	14	12	author	author	NOUN
ejpam-4698	14	13	.	.	PUNCT
ejpam-4698	15	1	doi	doi	NOUN
ejpam-4698	15	2	:	:	PUNCT
ejpam-4698	15	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4698	https://doi.org/10.29020/nybg.ejpam.v16i2.4698	ADJ
ejpam-4698	15	4	email	email	NOUN
ejpam-4698	15	5	addresses	address	NOUN
ejpam-4698	15	6	:	:	PUNCT
ejpam-4698	15	7	kavi.rasool@uoz.edu.krd	kavi.rasool@uoz.edu.krd	PROPN
ejpam-4698	15	8	(	(	PUNCT
ejpam-4698	15	9	k.	k.	PROPN
ejpam-4698	15	10	rasool	rasool	PROPN
ejpam-4698	15	11	)	)	PUNCT
ejpam-4698	15	12	,	,	PUNCT
ejpam-4698	15	13	payman.rashed@su.edu.krd	payman.rashed@su.edu.krd	NOUN
ejpam-4698	15	14	(	(	PUNCT
ejpam-4698	15	15	p.	p.	NOUN
ejpam-4698	15	16	rashed	rashed	PROPN
ejpam-4698	15	17	)	)	PUNCT
ejpam-4698	15	18	,	,	PUNCT
ejpam-4698	15	19	ahmedgraph@uomosul.edu.iq	ahmedgraph@uomosul.edu.iq	NOUN
ejpam-4698	15	20	(	(	PUNCT
ejpam-4698	15	21	a.	a.	PROPN
ejpam-4698	15	22	ali	ali	PROPN
ejpam-4698	15	23	)	)	PUNCT
ejpam-4698	15	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4698	16	1	773	773	NUM
ejpam-4698	17	1	©	©	ADP
ejpam-4698	17	2	2023	2023	NUM
ejpam-4698	17	3	ejpam	ejpam	NOUN
ejpam-4698	17	4	all	all	DET
ejpam-4698	17	5	rights	right	NOUN
ejpam-4698	17	6	reserved	reserve	VERB
ejpam-4698	17	7	.	.	PUNCT
ejpam-4698	18	1	k.	k.	PROPN
ejpam-4698	18	2	rasool	rasool	PROPN
ejpam-4698	18	3	,	,	PUNCT
ejpam-4698	18	4	p.	p.	PROPN
ejpam-4698	18	5	rashed	rashed	PROPN
ejpam-4698	18	6	,	,	PUNCT
ejpam-4698	18	7	a.	a.	PROPN
ejpam-4698	18	8	ali	ali	PROPN
ejpam-4698	18	9	/	/	SYM
ejpam-4698	18	10	eur	eur	PROPN
ejpam-4698	18	11	.	.	PUNCT
ejpam-4698	19	1	j.	j.	PROPN
ejpam-4698	19	2	pure	pure	PROPN
ejpam-4698	19	3	appl	appl	PROPN
ejpam-4698	19	4	.	.	PROPN
ejpam-4698	19	5	math	math	PROPN
ejpam-4698	19	6	,	,	PUNCT
ejpam-4698	19	7	16	16	NUM
ejpam-4698	19	8	(	(	PUNCT
ejpam-4698	19	9	2	2	NUM
ejpam-4698	19	10	)	)	PUNCT
ejpam-4698	19	11	(	(	PUNCT
ejpam-4698	19	12	2023	2023	NUM
ejpam-4698	19	13	)	)	PUNCT
ejpam-4698	19	14	,	,	PUNCT
ejpam-4698	19	15	773	773	NUM
ejpam-4698	19	16	-	-	SYM
ejpam-4698	19	17	783	783	NUM
ejpam-4698	19	18	774	774	NUM
ejpam-4698	19	19	where	where	SCONJ
ejpam-4698	19	20	nij	nij	PROPN
ejpam-4698	19	21	is	be	AUX
ejpam-4698	19	22	the	the	DET
ejpam-4698	19	23	number	number	NOUN
ejpam-4698	19	24	of	of	ADP
ejpam-4698	19	25	all	all	DET
ejpam-4698	19	26	edges	edge	NOUN
ejpam-4698	19	27	uv	uv	INTJ
ejpam-4698	19	28	such	such	ADJ
ejpam-4698	19	29	that	that	SCONJ
ejpam-4698	19	30	w(u	w(u	PROPN
ejpam-4698	19	31	)	)	PUNCT
ejpam-4698	20	1	=	=	SYM
ejpam-4698	20	2	i	i	PROPN
ejpam-4698	20	3	and	and	CCONJ
ejpam-4698	20	4	w(v	w(v	PROPN
ejpam-4698	20	5	)	)	PUNCT
ejpam-4698	21	1	=	=	PRON
ejpam-4698	21	2	j.	j.	PROPN
ejpam-4698	22	1	the	the	DET
ejpam-4698	22	2	degree	degree	NOUN
ejpam-4698	22	3	of	of	ADP
ejpam-4698	22	4	a	a	DET
ejpam-4698	22	5	vertex	vertex	NOUN
ejpam-4698	22	6	u	u	NOUN
ejpam-4698	22	7	∈	∈	NOUN
ejpam-4698	22	8	v	v	NOUN
ejpam-4698	22	9	is	be	AUX
ejpam-4698	22	10	the	the	DET
ejpam-4698	22	11	number	number	NOUN
ejpam-4698	22	12	of	of	ADP
ejpam-4698	22	13	edges	edge	NOUN
ejpam-4698	22	14	incident	incident	NOUN
ejpam-4698	22	15	on	on	ADP
ejpam-4698	22	16	u	u	NOUN
ejpam-4698	22	17	,	,	PUNCT
ejpam-4698	22	18	denoted	denote	VERB
ejpam-4698	22	19	as	as	ADP
ejpam-4698	22	20	du	du	PROPN
ejpam-4698	22	21	.	.	PUNCT
ejpam-4698	23	1	the	the	DET
ejpam-4698	23	2	neighborhood	neighborhood	NOUN
ejpam-4698	23	3	of	of	ADP
ejpam-4698	23	4	a	a	DET
ejpam-4698	23	5	vertex	vertex	NOUN
ejpam-4698	23	6	u	u	NOUN
ejpam-4698	23	7	∈	∈	PROPN
ejpam-4698	23	8	v	v	NOUN
ejpam-4698	23	9	,	,	PUNCT
ejpam-4698	23	10	ng(u	ng(u	NOUN
ejpam-4698	23	11	)	)	PUNCT
ejpam-4698	23	12	is	be	AUX
ejpam-4698	23	13	a	a	DET
ejpam-4698	23	14	set	set	NOUN
ejpam-4698	23	15	of	of	ADP
ejpam-4698	23	16	all	all	DET
ejpam-4698	23	17	neighbors	neighbor	NOUN
ejpam-4698	23	18	of	of	ADP
ejpam-4698	23	19	u	u	NOUN
ejpam-4698	23	20	,	,	PUNCT
ejpam-4698	23	21	i.e.	i.e.	X
ejpam-4698	23	22	,	,	PUNCT
ejpam-4698	23	23	ng(u	ng(u	NOUN
ejpam-4698	23	24	)	)	PUNCT
ejpam-4698	23	25	=	=	PRON
ejpam-4698	23	26	{	{	PUNCT
ejpam-4698	23	27	v|uv	v|uv	PROPN
ejpam-4698	23	28	∈	∈	PROPN
ejpam-4698	23	29	e(g	e(g	PROPN
ejpam-4698	23	30	)	)	PUNCT
ejpam-4698	23	31	}	}	PUNCT
ejpam-4698	23	32	and	and	CCONJ
ejpam-4698	23	33	its	its	PRON
ejpam-4698	23	34	called	call	VERB
ejpam-4698	23	35	open	open	ADJ
ejpam-4698	23	36	neighborhood	neighborhood	NOUN
ejpam-4698	23	37	.	.	PUNCT
ejpam-4698	24	1	the	the	DET
ejpam-4698	24	2	closed	closed	ADJ
ejpam-4698	24	3	neighborhood	neighborhood	NOUN
ejpam-4698	24	4	of	of	ADP
ejpam-4698	24	5	a	a	DET
ejpam-4698	24	6	vertex	vertex	NOUN
ejpam-4698	24	7	u	u	NOUN
ejpam-4698	24	8	,	,	PUNCT
ejpam-4698	24	9	denoted	denote	VERB
ejpam-4698	24	10	by	by	ADP
ejpam-4698	24	11	ng[u	ng[u	PROPN
ejpam-4698	24	12	]	]	PUNCT
ejpam-4698	24	13	,	,	PUNCT
ejpam-4698	24	14	is	be	AUX
ejpam-4698	24	15	obtained	obtain	VERB
ejpam-4698	24	16	by	by	ADP
ejpam-4698	24	17	adding	add	VERB
ejpam-4698	24	18	a	a	DET
ejpam-4698	24	19	vertex	vertex	NOUN
ejpam-4698	24	20	u	u	NOUN
ejpam-4698	24	21	to	to	ADP
ejpam-4698	24	22	ng(u	ng(u	NOUN
ejpam-4698	24	23	)	)	PUNCT
ejpam-4698	24	24	,	,	PUNCT
ejpam-4698	24	25	that	that	ADV
ejpam-4698	24	26	is	is	ADV
ejpam-4698	24	27	,	,	PUNCT
ejpam-4698	24	28	ng[u	ng[u	PROPN
ejpam-4698	24	29	]	]	X
ejpam-4698	24	30	=	=	SYM
ejpam-4698	24	31	ng(u	ng(u	PROPN
ejpam-4698	24	32	)	)	PUNCT
ejpam-4698	24	33	∪	∪	NOUN
ejpam-4698	24	34	{	{	PUNCT
ejpam-4698	24	35	u	u	NOUN
ejpam-4698	24	36	}	}	PUNCT
ejpam-4698	24	37	.	.	PUNCT
ejpam-4698	25	1	the	the	DET
ejpam-4698	25	2	δu	δu	ADP
ejpam-4698	25	3	denotes	denote	NOUN
ejpam-4698	25	4	the	the	DET
ejpam-4698	25	5	degrees	degree	NOUN
ejpam-4698	25	6	sum	sum	NOUN
ejpam-4698	25	7	of	of	ADP
ejpam-4698	25	8	neighbors	neighbor	NOUN
ejpam-4698	25	9	of	of	ADP
ejpam-4698	25	10	u	u	PROPN
ejpam-4698	25	11	in	in	ADP
ejpam-4698	25	12	g.	g.	PROPN
ejpam-4698	25	13	it	it	PRON
ejpam-4698	25	14	is	be	AUX
ejpam-4698	25	15	clear	clear	ADJ
ejpam-4698	25	16	that	that	SCONJ
ejpam-4698	25	17	(	(	PUNCT
ejpam-4698	25	18	1.1	1.1	NUM
ejpam-4698	25	19	)	)	PUNCT
ejpam-4698	25	20	is	be	AUX
ejpam-4698	25	21	a	a	DET
ejpam-4698	25	22	general	general	ADJ
ejpam-4698	25	23	wve−polynomial	wve−polynomial	NOUN
ejpam-4698	25	24	.	.	PUNCT
ejpam-4698	26	1	if	if	SCONJ
ejpam-4698	26	2	w(u	w(u	VERB
ejpam-4698	26	3	)	)	PUNCT
ejpam-4698	26	4	=	=	PUNCT
ejpam-4698	26	5	du	du	VERB
ejpam-4698	26	6	for	for	ADP
ejpam-4698	26	7	all	all	DET
ejpam-4698	26	8	u	u	PROPN
ejpam-4698	26	9	∈	∈	PROPN
ejpam-4698	26	10	v	v	NOUN
ejpam-4698	26	11	(	(	PUNCT
ejpam-4698	26	12	g	g	NOUN
ejpam-4698	26	13	)	)	PUNCT
ejpam-4698	26	14	,	,	PUNCT
ejpam-4698	26	15	then	then	ADV
ejpam-4698	26	16	(	(	PUNCT
ejpam-4698	26	17	1.1	1.1	NUM
ejpam-4698	26	18	)	)	PUNCT
ejpam-4698	26	19	is	be	AUX
ejpam-4698	26	20	m−polynomial	m−polynomial	ADJ
ejpam-4698	26	21	of	of	ADP
ejpam-4698	26	22	g	g	NOUN
ejpam-4698	26	23	,	,	PUNCT
ejpam-4698	26	24	which	which	PRON
ejpam-4698	26	25	may	may	AUX
ejpam-4698	26	26	simplified	simplify	VERB
ejpam-4698	26	27	as	as	ADP
ejpam-4698	26	28	:	:	PUNCT
ejpam-4698	26	29	m(g;x	m(g;x	PROPN
ejpam-4698	26	30	,	,	PUNCT
ejpam-4698	26	31	y	y	PROPN
ejpam-4698	26	32	)	)	PUNCT
ejpam-4698	27	1	=	=	PUNCT
ejpam-4698	28	1	σuv∈e(g)mijx	σuv∈e(g)mijx	PROPN
ejpam-4698	28	2	iyj	iyj	VERB
ejpam-4698	28	3	,	,	PUNCT
ejpam-4698	28	4	i	i	PROPN
ejpam-4698	28	5	≤	≤	PROPN
ejpam-4698	28	6	j	j	PROPN
ejpam-4698	28	7	,	,	PUNCT
ejpam-4698	28	8	(	(	PUNCT
ejpam-4698	28	9	1.3	1.3	NUM
ejpam-4698	28	10	)	)	PUNCT
ejpam-4698	28	11	where	where	SCONJ
ejpam-4698	28	12	mij	mij	NOUN
ejpam-4698	28	13	is	be	AUX
ejpam-4698	28	14	the	the	DET
ejpam-4698	28	15	number	number	NOUN
ejpam-4698	28	16	of	of	ADP
ejpam-4698	28	17	edges	edge	NOUN
ejpam-4698	28	18	uv	uv	PROPN
ejpam-4698	28	19	∈	∈	PROPN
ejpam-4698	28	20	e(g	e(g	PROPN
ejpam-4698	28	21	)	)	PUNCT
ejpam-4698	28	22	such	such	ADJ
ejpam-4698	28	23	that	that	SCONJ
ejpam-4698	28	24	{	{	PUNCT
ejpam-4698	28	25	du	du	PROPN
ejpam-4698	28	26	,	,	PUNCT
ejpam-4698	28	27	dv	dv	PROPN
ejpam-4698	28	28	}	}	PUNCT
ejpam-4698	28	29	=	=	SYM
ejpam-4698	28	30	{	{	PUNCT
ejpam-4698	28	31	i	i	PROPN
ejpam-4698	28	32	,	,	PUNCT
ejpam-4698	28	33	j	j	PROPN
ejpam-4698	28	34	}	}	PUNCT
ejpam-4698	28	35	.	.	PUNCT
ejpam-4698	29	1	this	this	DET
ejpam-4698	29	2	polynomial	polynomial	NOUN
ejpam-4698	29	3	was	be	AUX
ejpam-4698	29	4	first	first	ADV
ejpam-4698	29	5	introduced	introduce	VERB
ejpam-4698	29	6	by	by	ADP
ejpam-4698	29	7	deutsch	deutsch	NOUN
ejpam-4698	29	8	and	and	CCONJ
ejpam-4698	29	9	klavžar	klavžar	PROPN
ejpam-4698	29	10	[	[	X
ejpam-4698	29	11	6	6	NUM
ejpam-4698	29	12	]	]	PUNCT
ejpam-4698	29	13	.	.	PUNCT
ejpam-4698	30	1	and	and	CCONJ
ejpam-4698	30	2	,	,	PUNCT
ejpam-4698	30	3	if	if	SCONJ
ejpam-4698	30	4	w(u	w(u	VERB
ejpam-4698	30	5	)	)	PUNCT
ejpam-4698	30	6	=	=	SYM
ejpam-4698	30	7	δu	δu	X
ejpam-4698	30	8	for	for	ADP
ejpam-4698	30	9	all	all	DET
ejpam-4698	30	10	u	u	NOUN
ejpam-4698	30	11	∈	∈	PROPN
ejpam-4698	30	12	v	v	NOUN
ejpam-4698	30	13	(	(	PUNCT
ejpam-4698	30	14	g	g	NOUN
ejpam-4698	30	15	)	)	PUNCT
ejpam-4698	30	16	,	,	PUNCT
ejpam-4698	30	17	then	then	ADV
ejpam-4698	30	18	from	from	ADP
ejpam-4698	30	19	(	(	PUNCT
ejpam-4698	30	20	1.1	1.1	NUM
ejpam-4698	30	21	)	)	PUNCT
ejpam-4698	30	22	,	,	PUNCT
ejpam-4698	30	23	we	we	PRON
ejpam-4698	30	24	get	get	VERB
ejpam-4698	30	25	nm−polynomial	nm−polynomial	ADJ
ejpam-4698	30	26	of	of	ADP
ejpam-4698	30	27	g	g	NOUN
ejpam-4698	30	28	:	:	PUNCT
ejpam-4698	30	29	nm(g;x	nm(g;x	PROPN
ejpam-4698	30	30	,	,	PUNCT
ejpam-4698	30	31	y	y	NOUN
ejpam-4698	30	32	)	)	PUNCT
ejpam-4698	30	33	=	=	PUNCT
ejpam-4698	31	1	σuv∈e(g)m	σuv∈e(g)m	ADP
ejpam-4698	31	2	′	′	NUM
ejpam-4698	31	3	ijx	ijx	NOUN
ejpam-4698	31	4	iyj	iyj	VERB
ejpam-4698	31	5	,	,	PUNCT
ejpam-4698	31	6	i	i	PROPN
ejpam-4698	31	7	≤	≤	PROPN
ejpam-4698	31	8	j	j	PROPN
ejpam-4698	31	9	,	,	PUNCT
ejpam-4698	31	10	(	(	PUNCT
ejpam-4698	31	11	1.4	1.4	NUM
ejpam-4698	31	12	)	)	PUNCT
ejpam-4698	31	13	where	where	SCONJ
ejpam-4698	31	14	m′	m′	NOUN
ejpam-4698	31	15	ij	ij	NOUN
ejpam-4698	31	16	is	be	AUX
ejpam-4698	31	17	the	the	DET
ejpam-4698	31	18	total	total	ADJ
ejpam-4698	31	19	number	number	NOUN
ejpam-4698	31	20	of	of	ADP
ejpam-4698	31	21	edges	edge	NOUN
ejpam-4698	31	22	uv	uv	PROPN
ejpam-4698	31	23	∈	∈	PROPN
ejpam-4698	31	24	e(g	e(g	PROPN
ejpam-4698	31	25	)	)	PUNCT
ejpam-4698	31	26	such	such	ADJ
ejpam-4698	31	27	that	that	SCONJ
ejpam-4698	31	28	{	{	PUNCT
ejpam-4698	31	29	δu	δu	NOUN
ejpam-4698	31	30	,	,	PUNCT
ejpam-4698	31	31	δv	δv	ADV
ejpam-4698	31	32	}	}	PUNCT
ejpam-4698	31	33	=	=	SYM
ejpam-4698	31	34	{	{	PUNCT
ejpam-4698	31	35	i	i	PROPN
ejpam-4698	31	36	,	,	PUNCT
ejpam-4698	31	37	j	j	PROPN
ejpam-4698	31	38	}	}	PUNCT
ejpam-4698	31	39	.	.	PUNCT
ejpam-4698	32	1	mondal	mondal	PROPN
ejpam-4698	32	2	and	and	CCONJ
ejpam-4698	32	3	others	other	NOUN
ejpam-4698	32	4	[	[	X
ejpam-4698	32	5	13	13	NUM
ejpam-4698	32	6	]	]	PUNCT
ejpam-4698	32	7	developed	develop	VERB
ejpam-4698	32	8	a	a	DET
ejpam-4698	32	9	m−polynomial	m−polynomial	NOUN
ejpam-4698	32	10	into	into	ADP
ejpam-4698	32	11	a	a	DET
ejpam-4698	32	12	nm−polynomial	nm−polynomial	NOUN
ejpam-4698	32	13	of	of	ADP
ejpam-4698	32	14	a	a	DET
ejpam-4698	32	15	graph	graph	NOUN
ejpam-4698	32	16	g.	g.	NOUN
ejpam-4698	32	17	now	now	ADV
ejpam-4698	32	18	,	,	PUNCT
ejpam-4698	32	19	for	for	SCONJ
ejpam-4698	32	20	each	each	DET
ejpam-4698	32	21	u	u	PROPN
ejpam-4698	32	22	∈	∈	PROPN
ejpam-4698	32	23	v	v	NOUN
ejpam-4698	32	24	(	(	PUNCT
ejpam-4698	32	25	g	g	NOUN
ejpam-4698	32	26	)	)	PUNCT
ejpam-4698	32	27	,	,	PUNCT
ejpam-4698	32	28	τu	τu	PUNCT
ejpam-4698	32	29	is	be	AUX
ejpam-4698	32	30	defined	define	VERB
ejpam-4698	32	31	as	as	ADP
ejpam-4698	32	32	the	the	DET
ejpam-4698	32	33	number	number	NOUN
ejpam-4698	32	34	of	of	ADP
ejpam-4698	32	35	all	all	DET
ejpam-4698	32	36	edges	edge	NOUN
ejpam-4698	32	37	in	in	ADP
ejpam-4698	32	38	g	g	PROPN
ejpam-4698	32	39	incident	incident	NOUN
ejpam-4698	32	40	to	to	ADP
ejpam-4698	32	41	a	a	DET
ejpam-4698	32	42	vertex	vertex	NOUN
ejpam-4698	32	43	of	of	ADP
ejpam-4698	32	44	ng[u	ng[u	PROPN
ejpam-4698	32	45	]	]	PUNCT
ejpam-4698	32	46	.	.	PUNCT
ejpam-4698	33	1	from	from	ADP
ejpam-4698	33	2	(	(	PUNCT
ejpam-4698	33	3	1.1	1.1	NUM
ejpam-4698	33	4	)	)	PUNCT
ejpam-4698	33	5	,	,	PUNCT
ejpam-4698	33	6	substituting	substitute	VERB
ejpam-4698	33	7	w(u	w(u	PROPN
ejpam-4698	33	8	)	)	PUNCT
ejpam-4698	33	9	=	=	SYM
ejpam-4698	33	10	τu	τu	PROPN
ejpam-4698	33	11	,	,	PUNCT
ejpam-4698	33	12	we	we	PRON
ejpam-4698	33	13	get	get	VERB
ejpam-4698	33	14	mve−polynomial	mve−polynomial	ADJ
ejpam-4698	33	15	:	:	PUNCT
ejpam-4698	33	16	mve(g	mve(g	NOUN
ejpam-4698	33	17	)	)	PUNCT
ejpam-4698	33	18	=	=	SYM
ejpam-4698	34	1	σuv∈e(g)x	σuv∈e(g)x	NOUN
ejpam-4698	34	2	τuyτv	τuyτv	NOUN
ejpam-4698	34	3	,	,	PUNCT
ejpam-4698	34	4	τu	τu	ADP
ejpam-4698	34	5	≤	≤	ADJ
ejpam-4698	34	6	τv	τv	NOUN
ejpam-4698	34	7	,	,	PUNCT
ejpam-4698	34	8	(	(	PUNCT
ejpam-4698	34	9	1.5	1.5	NUM
ejpam-4698	34	10	)	)	PUNCT
ejpam-4698	34	11	simplifying	simplifying	NOUN
ejpam-4698	34	12	(	(	PUNCT
ejpam-4698	34	13	1.5	1.5	NUM
ejpam-4698	34	14	)	)	PUNCT
ejpam-4698	34	15	by	by	ADP
ejpam-4698	34	16	collecting	collect	VERB
ejpam-4698	34	17	similar	similar	ADJ
ejpam-4698	34	18	terms	term	NOUN
ejpam-4698	34	19	,	,	PUNCT
ejpam-4698	34	20	we	we	PRON
ejpam-4698	34	21	get	get	VERB
ejpam-4698	34	22	mve(g	mve(g	NOUN
ejpam-4698	34	23	)	)	PUNCT
ejpam-4698	35	1	=	=	SYM
ejpam-4698	36	1	σuv∈e(g)cijx	σuv∈e(g)cijx	PROPN
ejpam-4698	36	2	τuyτv	τuyτv	NOUN
ejpam-4698	36	3	,	,	PUNCT
ejpam-4698	36	4	τu	τu	ADP
ejpam-4698	36	5	≤	≤	ADJ
ejpam-4698	36	6	τv	τv	NOUN
ejpam-4698	36	7	,	,	PUNCT
ejpam-4698	36	8	in	in	ADP
ejpam-4698	36	9	which	which	PRON
ejpam-4698	36	10	cij	cij	PROPN
ejpam-4698	36	11	is	be	AUX
ejpam-4698	36	12	the	the	DET
ejpam-4698	36	13	number	number	NOUN
ejpam-4698	36	14	of	of	ADP
ejpam-4698	36	15	all	all	DET
ejpam-4698	36	16	edges	edge	NOUN
ejpam-4698	36	17	uv	uv	PROPN
ejpam-4698	36	18	∈	∈	PROPN
ejpam-4698	36	19	e(g	e(g	PROPN
ejpam-4698	36	20	)	)	PUNCT
ejpam-4698	36	21	such	such	ADJ
ejpam-4698	36	22	that	that	SCONJ
ejpam-4698	36	23	{	{	PUNCT
ejpam-4698	36	24	τu	τu	PROPN
ejpam-4698	36	25	,	,	PUNCT
ejpam-4698	36	26	τv	τv	NOUN
ejpam-4698	36	27	}	}	PUNCT
ejpam-4698	36	28	=	=	SYM
ejpam-4698	36	29	{	{	PUNCT
ejpam-4698	36	30	i	i	PROPN
ejpam-4698	36	31	,	,	PUNCT
ejpam-4698	36	32	j	j	PROPN
ejpam-4698	36	33	}	}	PUNCT
ejpam-4698	36	34	.	.	PUNCT
ejpam-4698	37	1	the	the	DET
ejpam-4698	37	2	terms	term	NOUN
ejpam-4698	37	3	vertex	vertex	X
ejpam-4698	37	4	–	–	PUNCT
ejpam-4698	37	5	edge	edge	NOUN
ejpam-4698	37	6	degree	degree	NOUN
ejpam-4698	37	7	of	of	ADP
ejpam-4698	37	8	the	the	DET
ejpam-4698	37	9	graphs	graph	NOUN
ejpam-4698	37	10	,	,	PUNCT
ejpam-4698	37	11	τu	τu	PART
ejpam-4698	37	12	were	be	AUX
ejpam-4698	37	13	first	first	ADV
ejpam-4698	37	14	introduced	introduce	VERB
ejpam-4698	37	15	by	by	ADP
ejpam-4698	37	16	chellali	chellali	NOUN
ejpam-4698	37	17	and	and	CCONJ
ejpam-4698	37	18	others	other	NOUN
ejpam-4698	38	1	[	[	X
ejpam-4698	38	2	4	4	NUM
ejpam-4698	38	3	]	]	PUNCT
ejpam-4698	38	4	.	.	PUNCT
ejpam-4698	39	1	the	the	DET
ejpam-4698	39	2	authors	author	NOUN
ejpam-4698	39	3	defined	define	VERB
ejpam-4698	39	4	these	these	DET
ejpam-4698	39	5	novel	novel	ADJ
ejpam-4698	39	6	degree	degree	NOUN
ejpam-4698	39	7	concepts	concept	NOUN
ejpam-4698	39	8	in	in	ADP
ejpam-4698	39	9	relation	relation	NOUN
ejpam-4698	39	10	to	to	ADP
ejpam-4698	39	11	the	the	DET
ejpam-4698	39	12	vertex−edge	vertex−edge	NOUN
ejpam-4698	39	13	domination	domination	NOUN
ejpam-4698	39	14	parameters	parameter	NOUN
ejpam-4698	39	15	[	[	X
ejpam-4698	39	16	4	4	NUM
ejpam-4698	39	17	,	,	PUNCT
ejpam-4698	39	18	10	10	NUM
ejpam-4698	39	19	]	]	PUNCT
ejpam-4698	39	20	.	.	PUNCT
ejpam-4698	40	1	the	the	DET
ejpam-4698	40	2	vertex	vertex	NOUN
ejpam-4698	40	3	–	–	PUNCT
ejpam-4698	40	4	edge	edge	NOUN
ejpam-4698	40	5	degree	degree	NOUN
ejpam-4698	40	6	(	(	PUNCT
ejpam-4698	40	7	ve−degree	ve−degree	NUM
ejpam-4698	40	8	)	)	PUNCT
ejpam-4698	40	9	concepts	concept	NOUN
ejpam-4698	40	10	of	of	ADP
ejpam-4698	40	11	the	the	DET
ejpam-4698	40	12	graphs	graph	NOUN
ejpam-4698	40	13	were	be	AUX
ejpam-4698	40	14	extensively	extensively	ADV
ejpam-4698	40	15	used	use	VERB
ejpam-4698	40	16	in	in	ADP
ejpam-4698	40	17	chemical	chemical	NOUN
ejpam-4698	40	18	graph	graph	NOUN
ejpam-4698	40	19	theory	theory	NOUN
ejpam-4698	40	20	[	[	X
ejpam-4698	40	21	3	3	NUM
ejpam-4698	40	22	,	,	PUNCT
ejpam-4698	40	23	7	7	NUM
ejpam-4698	40	24	,	,	PUNCT
ejpam-4698	40	25	8	8	NUM
ejpam-4698	40	26	]	]	PUNCT
ejpam-4698	40	27	and	and	CCONJ
ejpam-4698	40	28	[	[	X
ejpam-4698	40	29	19	19	NUM
ejpam-4698	40	30	]	]	PUNCT
ejpam-4698	40	31	.	.	PUNCT
ejpam-4698	41	1	the	the	DET
ejpam-4698	41	2	m−polynomial	m−polynomial	ADJ
ejpam-4698	41	3	is	be	AUX
ejpam-4698	41	4	the	the	DET
ejpam-4698	41	5	most	most	ADV
ejpam-4698	41	6	general	general	ADJ
ejpam-4698	41	7	polynomial	polynomial	NOUN
ejpam-4698	41	8	that	that	PRON
ejpam-4698	41	9	may	may	AUX
ejpam-4698	41	10	generate	generate	VERB
ejpam-4698	41	11	a	a	DET
ejpam-4698	41	12	wide	wide	ADJ
ejpam-4698	41	13	range	range	NOUN
ejpam-4698	41	14	of	of	ADP
ejpam-4698	41	15	degree−based	degree−base	VERB
ejpam-4698	41	16	topological	topological	ADJ
ejpam-4698	41	17	indices	index	NOUN
ejpam-4698	41	18	[	[	X
ejpam-4698	41	19	5	5	NUM
ejpam-4698	41	20	,	,	PUNCT
ejpam-4698	41	21	6	6	NUM
ejpam-4698	41	22	,	,	PUNCT
ejpam-4698	41	23	9	9	NUM
ejpam-4698	41	24	,	,	PUNCT
ejpam-4698	41	25	11–13	11–13	NUM
ejpam-4698	41	26	,	,	PUNCT
ejpam-4698	41	27	15	15	NUM
ejpam-4698	41	28	,	,	PUNCT
ejpam-4698	41	29	17	17	NUM
ejpam-4698	41	30	]	]	PUNCT
ejpam-4698	41	31	and	and	CCONJ
ejpam-4698	41	32	[	[	X
ejpam-4698	41	33	18	18	NUM
ejpam-4698	41	34	]	]	PUNCT
ejpam-4698	41	35	.	.	PUNCT
ejpam-4698	42	1	k.	k.	PROPN
ejpam-4698	42	2	rasool	rasool	PROPN
ejpam-4698	42	3	,	,	PUNCT
ejpam-4698	42	4	p.	p.	PROPN
ejpam-4698	42	5	rashed	rashed	PROPN
ejpam-4698	42	6	,	,	PUNCT
ejpam-4698	42	7	a.	a.	PROPN
ejpam-4698	42	8	ali	ali	PROPN
ejpam-4698	42	9	/	/	SYM
ejpam-4698	42	10	eur	eur	PROPN
ejpam-4698	42	11	.	.	PUNCT
ejpam-4698	43	1	j.	j.	PROPN
ejpam-4698	43	2	pure	pure	PROPN
ejpam-4698	43	3	appl	appl	PROPN
ejpam-4698	43	4	.	.	PROPN
ejpam-4698	43	5	math	math	PROPN
ejpam-4698	43	6	,	,	PUNCT
ejpam-4698	43	7	16	16	NUM
ejpam-4698	43	8	(	(	PUNCT
ejpam-4698	43	9	2	2	NUM
ejpam-4698	43	10	)	)	PUNCT
ejpam-4698	43	11	(	(	PUNCT
ejpam-4698	43	12	2023	2023	NUM
ejpam-4698	43	13	)	)	PUNCT
ejpam-4698	43	14	,	,	PUNCT
ejpam-4698	43	15	773	773	NUM
ejpam-4698	43	16	-	-	SYM
ejpam-4698	43	17	783	783	NUM
ejpam-4698	43	18	775	775	NUM
ejpam-4698	43	19	from	from	ADP
ejpam-4698	43	20	the	the	DET
ejpam-4698	43	21	above	above	ADJ
ejpam-4698	43	22	three	three	NUM
ejpam-4698	43	23	definitions	definition	NOUN
ejpam-4698	43	24	,	,	PUNCT
ejpam-4698	43	25	we	we	PRON
ejpam-4698	43	26	get	get	VERB
ejpam-4698	43	27	some	some	DET
ejpam-4698	43	28	properties	property	NOUN
ejpam-4698	43	29	:	:	PUNCT
ejpam-4698	43	30	(	(	PUNCT
ejpam-4698	43	31	i	i	NOUN
ejpam-4698	43	32	)	)	PUNCT
ejpam-4698	44	1	σi≤jmijx	σi≤jmijx	PROPN
ejpam-4698	44	2	iyj	iyj	VERB
ejpam-4698	44	3	|x	|x	NOUN
ejpam-4698	44	4	=	=	SYM
ejpam-4698	44	5	y=1	y=1	NOUN
ejpam-4698	44	6	=	=	PUNCT
ejpam-4698	44	7	σi≤jnijx	σi≤jnijx	PROPN
ejpam-4698	44	8	iyj	iyj	VERB
ejpam-4698	44	9	|x	|x	PROPN
ejpam-4698	44	10	=	=	SYM
ejpam-4698	44	11	y=1	y=1	NOUN
ejpam-4698	44	12	=	=	SYM
ejpam-4698	44	13	σi≤jcijx	σi≤jcijx	PROPN
ejpam-4698	44	14	iyj	iyj	VERB
ejpam-4698	44	15	|x	|x	PROPN
ejpam-4698	44	16	=	=	SYM
ejpam-4698	44	17	y=1	y=1	NOUN
ejpam-4698	44	18	=	=	SYM
ejpam-4698	44	19	q.	q.	PROPN
ejpam-4698	44	20	(	(	PUNCT
ejpam-4698	44	21	ii	ii	NOUN
ejpam-4698	44	22	)	)	PUNCT
ejpam-4698	44	23	for	for	ADP
ejpam-4698	44	24	any	any	DET
ejpam-4698	44	25	vertex	vertex	NOUN
ejpam-4698	44	26	u	u	NOUN
ejpam-4698	44	27	in	in	ADP
ejpam-4698	44	28	a	a	DET
ejpam-4698	44	29	graph	graph	NOUN
ejpam-4698	44	30	g	g	NOUN
ejpam-4698	44	31	,	,	PUNCT
ejpam-4698	44	32	we	we	PRON
ejpam-4698	44	33	have	have	AUX
ejpam-4698	44	34	:	:	PUNCT
ejpam-4698	44	35	du	du	PROPN
ejpam-4698	44	36	≤	≤	PROPN
ejpam-4698	44	37	τu	τu	ADP
ejpam-4698	44	38	≤	≤	ADJ
ejpam-4698	44	39	δu	δu	X
ejpam-4698	44	40	.	.	PUNCT
ejpam-4698	45	1	(	(	PUNCT
ejpam-4698	45	2	iii	iii	NOUN
ejpam-4698	45	3	)	)	PUNCT
ejpam-4698	45	4	let	let	VERB
ejpam-4698	45	5	f(hu	f(hu	NUM
ejpam-4698	45	6	,	,	PUNCT
ejpam-4698	45	7	hv	hv	PROPN
ejpam-4698	45	8	)	)	PUNCT
ejpam-4698	45	9	be	be	VERB
ejpam-4698	45	10	the	the	DET
ejpam-4698	45	11	index	index	NOUN
ejpam-4698	45	12	function	function	NOUN
ejpam-4698	45	13	,	,	PUNCT
ejpam-4698	45	14	where	where	SCONJ
ejpam-4698	45	15	hz	hz	ADP
ejpam-4698	45	16	∈	∈	PROPN
ejpam-4698	45	17	{	{	PUNCT
ejpam-4698	45	18	dz	dz	X
ejpam-4698	45	19	,	,	PUNCT
ejpam-4698	45	20	τz	τz	ADP
ejpam-4698	45	21	,	,	PUNCT
ejpam-4698	45	22	δz	δz	X
ejpam-4698	45	23	}	}	PUNCT
ejpam-4698	45	24	.	.	PUNCT
ejpam-4698	46	1	then	then	ADV
ejpam-4698	46	2	f(du	f(du	PROPN
ejpam-4698	46	3	,	,	PUNCT
ejpam-4698	46	4	dv	dv	PROPN
ejpam-4698	46	5	)	)	PUNCT
ejpam-4698	46	6	≤	≤	NOUN
ejpam-4698	46	7	f(τu	f(τu	PROPN
ejpam-4698	46	8	,	,	PUNCT
ejpam-4698	46	9	τv	τv	NOUN
ejpam-4698	46	10	)	)	PUNCT
ejpam-4698	46	11	≤	≤	NOUN
ejpam-4698	46	12	f(δu	f(δu	PROPN
ejpam-4698	46	13	,	,	PUNCT
ejpam-4698	46	14	δv	δv	ADV
ejpam-4698	46	15	)	)	PUNCT
ejpam-4698	46	16	,	,	PUNCT
ejpam-4698	46	17	where	where	SCONJ
ejpam-4698	46	18	f	f	PROPN
ejpam-4698	46	19	∈	∈	PROPN
ejpam-4698	46	20	[	[	X
ejpam-4698	46	21	first	first	ADJ
ejpam-4698	46	22	and	and	CCONJ
ejpam-4698	46	23	second	second	ADJ
ejpam-4698	46	24	zagreb	zagreb	PROPN
ejpam-4698	46	25	,	,	PUNCT
ejpam-4698	46	26	reduced	reduce	VERB
ejpam-4698	46	27	first	first	ADV
ejpam-4698	46	28	and	and	CCONJ
ejpam-4698	46	29	second	second	ADJ
ejpam-4698	46	30	zagreb	zagreb	PROPN
ejpam-4698	46	31	,	,	PUNCT
ejpam-4698	46	32	hyper	hyper	PROPN
ejpam-4698	46	33	zagreb	zagreb	PROPN
ejpam-4698	46	34	index	index	PROPN
ejpam-4698	46	35	,	,	PUNCT
ejpam-4698	46	36	forgotten	forget	VERB
ejpam-4698	46	37	index	index	PROPN
ejpam-4698	46	38	,	,	PUNCT
ejpam-4698	46	39	albertson	albertson	PROPN
ejpam-4698	46	40	index	index	PROPN
ejpam-4698	46	41	,	,	PUNCT
ejpam-4698	46	42	sigma	sigma	VERB
ejpam-4698	46	43	index	index	PROPN
ejpam-4698	46	44	]	]	PUNCT
ejpam-4698	46	45	.	.	PUNCT
ejpam-4698	47	1	finally	finally	ADV
ejpam-4698	47	2	,	,	PUNCT
ejpam-4698	47	3	there	there	PRON
ejpam-4698	47	4	are	be	VERB
ejpam-4698	47	5	many	many	ADJ
ejpam-4698	47	6	polynomials	polynomial	NOUN
ejpam-4698	47	7	that	that	PRON
ejpam-4698	47	8	have	have	AUX
ejpam-4698	47	9	been	be	AUX
ejpam-4698	47	10	found	find	VERB
ejpam-4698	47	11	over	over	ADP
ejpam-4698	47	12	the	the	DET
ejpam-4698	47	13	current	current	ADJ
ejpam-4698	47	14	century	century	NOUN
ejpam-4698	47	15	that	that	PRON
ejpam-4698	47	16	have	have	VERB
ejpam-4698	47	17	chemical	chemical	ADJ
ejpam-4698	47	18	applications	application	NOUN
ejpam-4698	47	19	;	;	PUNCT
ejpam-4698	47	20	see	see	VERB
ejpam-4698	47	21	[	[	X
ejpam-4698	47	22	1	1	NUM
ejpam-4698	47	23	,	,	PUNCT
ejpam-4698	47	24	2	2	NUM
ejpam-4698	47	25	]	]	PUNCT
ejpam-4698	47	26	and	and	CCONJ
ejpam-4698	47	27	[	[	X
ejpam-4698	47	28	14	14	NUM
ejpam-4698	47	29	]	]	SYM
ejpam-4698	47	30	2	2	NUM
ejpam-4698	47	31	.	.	X
ejpam-4698	48	1	some	some	DET
ejpam-4698	48	2	relations	relation	NOUN
ejpam-4698	48	3	between	between	ADP
ejpam-4698	48	4	vertex	vertex	NOUN
ejpam-4698	48	5	–	–	PUNCT
ejpam-4698	48	6	edge	edge	NOUN
ejpam-4698	48	7	degree	degree	NOUN
ejpam-4698	48	8	topological	topological	ADJ
ejpam-4698	48	9	indices	index	NOUN
ejpam-4698	48	10	in	in	ADP
ejpam-4698	48	11	this	this	DET
ejpam-4698	48	12	section	section	NOUN
ejpam-4698	48	13	,	,	PUNCT
ejpam-4698	48	14	we	we	PRON
ejpam-4698	48	15	give	give	VERB
ejpam-4698	48	16	some	some	DET
ejpam-4698	48	17	relations	relation	NOUN
ejpam-4698	48	18	between	between	ADP
ejpam-4698	48	19	vertex	vertex	NOUN
ejpam-4698	48	20	–	–	PUNCT
ejpam-4698	48	21	edge	edge	NOUN
ejpam-4698	48	22	degree	degree	NOUN
ejpam-4698	48	23	topological	topological	ADJ
ejpam-4698	48	24	indices	index	NOUN
ejpam-4698	48	25	for	for	ADP
ejpam-4698	48	26	any	any	DET
ejpam-4698	48	27	graph	graph	NOUN
ejpam-4698	48	28	g.	g.	PROPN
ejpam-4698	48	29	also	also	ADV
ejpam-4698	48	30	,	,	PUNCT
ejpam-4698	48	31	the	the	DET
ejpam-4698	48	32	lower	low	ADJ
ejpam-4698	48	33	and	and	CCONJ
ejpam-4698	48	34	upper	upper	ADJ
ejpam-4698	48	35	bounds	bound	NOUN
ejpam-4698	48	36	for	for	ADP
ejpam-4698	48	37	vertex	vertex	NOUN
ejpam-4698	48	38	–	–	PUNCT
ejpam-4698	48	39	edge	edge	NOUN
ejpam-4698	48	40	degree	degree	NOUN
ejpam-4698	48	41	topological	topological	ADJ
ejpam-4698	48	42	indices	index	NOUN
ejpam-4698	48	43	via	via	ADP
ejpam-4698	48	44	wve−polynomial	wve−polynomial	PROPN
ejpam-4698	48	45	are	be	AUX
ejpam-4698	48	46	determined	determine	VERB
ejpam-4698	48	47	;	;	PUNCT
ejpam-4698	48	48	some	some	PRON
ejpam-4698	48	49	of	of	ADP
ejpam-4698	48	50	them	they	PRON
ejpam-4698	48	51	are	be	AUX
ejpam-4698	48	52	shown	show	VERB
ejpam-4698	48	53	in	in	ADP
ejpam-4698	48	54	table	table	NOUN
ejpam-4698	48	55	1	1	NUM
ejpam-4698	48	56	.	.	PUNCT
ejpam-4698	49	1	for	for	ADP
ejpam-4698	49	2	any	any	DET
ejpam-4698	49	3	graph	graph	NOUN
ejpam-4698	49	4	g.	g.	NOUN
ejpam-4698	49	5	table	table	NOUN
ejpam-4698	49	6	1	1	NUM
ejpam-4698	49	7	:	:	PUNCT
ejpam-4698	49	8	some	some	DET
ejpam-4698	49	9	vertex	vertex	NOUN
ejpam-4698	49	10	–	–	PUNCT
ejpam-4698	49	11	edge	edge	NOUN
ejpam-4698	49	12	–	–	PUNCT
ejpam-4698	49	13	degree	degree	NOUN
ejpam-4698	49	14	based	base	VERB
ejpam-4698	49	15	topological	topological	ADJ
ejpam-4698	49	16	indices	index	NOUN
ejpam-4698	49	17	for	for	ADP
ejpam-4698	49	18	wve−polynomial	wve−polynomial	PROPN
ejpam-4698	49	19	.	.	PUNCT
ejpam-4698	50	1	topological	topological	ADJ
ejpam-4698	50	2	index	index	NOUN
ejpam-4698	50	3	symbol	symbol	NOUN
ejpam-4698	50	4	index	index	NOUN
ejpam-4698	50	5	formula	formula	NOUN
ejpam-4698	50	6	f(w(u	f(w(u	NOUN
ejpam-4698	50	7	)	)	PUNCT
ejpam-4698	50	8	,	,	PUNCT
ejpam-4698	50	9	w(v	w(v	NOUN
ejpam-4698	50	10	)	)	PUNCT
ejpam-4698	50	11	)	)	PUNCT
ejpam-4698	51	1	derivation	derivation	NOUN
ejpam-4698	51	2	from	from	ADP
ejpam-4698	51	3	wve(g;x	wve(g;x	PROPN
ejpam-4698	51	4	,	,	PUNCT
ejpam-4698	51	5	y	y	PROPN
ejpam-4698	51	6	)	)	PUNCT
ejpam-4698	51	7	first	first	ADV
ejpam-4698	51	8	zagreb	zagreb	PROPN
ejpam-4698	51	9	w	w	PROPN
ejpam-4698	51	10	1	1	NUM
ejpam-4698	51	11	ve(g	ve(g	NUM
ejpam-4698	51	12	)	)	PUNCT
ejpam-4698	51	13	σuv∈e(g)(w(u	σuv∈e(g)(w(u	VERB
ejpam-4698	51	14	)	)	PUNCT
ejpam-4698	51	15	+	+	CCONJ
ejpam-4698	52	1	w(v	w(v	NOUN
ejpam-4698	52	2	)	)	PUNCT
ejpam-4698	52	3	)	)	PUNCT
ejpam-4698	53	1	(	(	PUNCT
ejpam-4698	53	2	dx	dx	PROPN
ejpam-4698	53	3	+	+	PROPN
ejpam-4698	53	4	dy)wve(g;x	dy)wve(g;x	PROPN
ejpam-4698	53	5	,	,	PUNCT
ejpam-4698	53	6	y)|x	y)|x	NOUN
ejpam-4698	53	7	=	=	SYM
ejpam-4698	53	8	y=1	y=1	X
ejpam-4698	53	9	second	second	ADJ
ejpam-4698	53	10	zagreb	zagreb	PROPN
ejpam-4698	53	11	w	w	PROPN
ejpam-4698	53	12	2	2	NUM
ejpam-4698	53	13	ve(g	ve(g	NUM
ejpam-4698	53	14	)	)	PUNCT
ejpam-4698	53	15	σuv∈e(g)(w(u)w(v	σuv∈e(g)(w(u)w(v	NOUN
ejpam-4698	53	16	)	)	PUNCT
ejpam-4698	53	17	)	)	PUNCT
ejpam-4698	53	18	(	(	PUNCT
ejpam-4698	53	19	dxdy)wve(g;x	dxdy)wve(g;x	ADJ
ejpam-4698	53	20	,	,	PUNCT
ejpam-4698	53	21	y)|x	y)|x	NOUN
ejpam-4698	53	22	=	=	SYM
ejpam-4698	53	23	y=1	y=1	NOUN
ejpam-4698	53	24	reduced	reduce	VERB
ejpam-4698	53	25	first	first	ADJ
ejpam-4698	53	26	zagreb	zagreb	PROPN
ejpam-4698	53	27	rw	rw	PROPN
ejpam-4698	53	28	1	1	NUM
ejpam-4698	53	29	ve(g	ve(g	NUM
ejpam-4698	53	30	)	)	PUNCT
ejpam-4698	53	31	σuv∈e(g)(w(u	σuv∈e(g)(w(u	VERB
ejpam-4698	53	32	)	)	PUNCT
ejpam-4698	53	33	+	+	CCONJ
ejpam-4698	53	34	w(v)−	w(v)−	PROPN
ejpam-4698	53	35	2	2	NUM
ejpam-4698	53	36	)	)	PUNCT
ejpam-4698	53	37	(	(	PUNCT
ejpam-4698	53	38	dx	dx	PROPN
ejpam-4698	54	1	+	+	PROPN
ejpam-4698	54	2	dy	dy	NOUN
ejpam-4698	54	3	−	−	PROPN
ejpam-4698	54	4	2)wve(g;x	2)wve(g;x	NUM
ejpam-4698	54	5	,	,	PUNCT
ejpam-4698	54	6	y)|x	y)|x	NOUN
ejpam-4698	54	7	=	=	SYM
ejpam-4698	54	8	y=1	y=1	NOUN
ejpam-4698	54	9	reduced	reduce	VERB
ejpam-4698	54	10	second	second	ADJ
ejpam-4698	54	11	zagreb	zagreb	PROPN
ejpam-4698	54	12	rw	rw	PROPN
ejpam-4698	54	13	2	2	NUM
ejpam-4698	54	14	ve(g	ve(g	NUM
ejpam-4698	54	15	)	)	PUNCT
ejpam-4698	54	16	σuv∈e(g)(w(u)−	σuv∈e(g)(w(u)−	PROPN
ejpam-4698	55	1	1)(w(v))−	1)(w(v))−	NUM
ejpam-4698	55	2	1	1	NUM
ejpam-4698	55	3	)	)	PUNCT
ejpam-4698	55	4	(	(	PUNCT
ejpam-4698	55	5	dx	dx	PROPN
ejpam-4698	55	6	−	−	PROPN
ejpam-4698	55	7	1)(dy	1)(dy	NUM
ejpam-4698	55	8	−	−	PROPN
ejpam-4698	55	9	1)wve(g;x	1)wve(g;x	NUM
ejpam-4698	55	10	,	,	PUNCT
ejpam-4698	55	11	y)|x	y)|x	NOUN
ejpam-4698	55	12	=	=	SYM
ejpam-4698	55	13	y=1	y=1	PROPN
ejpam-4698	55	14	hyper	hyper	PROPN
ejpam-4698	55	15	zagreb	zagreb	PROPN
ejpam-4698	55	16	index	index	PROPN
ejpam-4698	55	17	hypwve(g	hypwve(g	PROPN
ejpam-4698	55	18	)	)	PUNCT
ejpam-4698	55	19	σuv∈e(g)(w(u	σuv∈e(g)(w(u	VERB
ejpam-4698	55	20	)	)	PUNCT
ejpam-4698	56	1	+	+	CCONJ
ejpam-4698	56	2	w(v))2	w(v))2	PROPN
ejpam-4698	56	3	d2	d2	PROPN
ejpam-4698	56	4	xjwve(g;x	xjwve(g;x	PROPN
ejpam-4698	56	5	,	,	PUNCT
ejpam-4698	56	6	y)|x	y)|x	NOUN
ejpam-4698	56	7	=	=	SYM
ejpam-4698	56	8	y=1	y=1	NOUN
ejpam-4698	56	9	forgotten	forget	VERB
ejpam-4698	56	10	index	index	NOUN
ejpam-4698	56	11	fwve(g	fwve(g	NOUN
ejpam-4698	56	12	)	)	PUNCT
ejpam-4698	56	13	σuv∈e(g)((w(u	σuv∈e(g)((w(u	NUM
ejpam-4698	56	14	)	)	PUNCT
ejpam-4698	56	15	)	)	PUNCT
ejpam-4698	57	1	2	2	NUM
ejpam-4698	58	1	+	+	CCONJ
ejpam-4698	58	2	(	(	PUNCT
ejpam-4698	58	3	w(v))2	w(v))2	PROPN
ejpam-4698	58	4	)	)	PUNCT
ejpam-4698	58	5	(	(	PUNCT
ejpam-4698	58	6	d2	d2	PROPN
ejpam-4698	58	7	x	x	SYM
ejpam-4698	58	8	+	+	PROPN
ejpam-4698	58	9	d2	d2	PROPN
ejpam-4698	58	10	y)wve(g;x	y)wve(g;x	PROPN
ejpam-4698	58	11	,	,	PUNCT
ejpam-4698	58	12	y)|x	y)|x	NOUN
ejpam-4698	58	13	=	=	SYM
ejpam-4698	58	14	y=1	y=1	PROPN
ejpam-4698	58	15	albertson	albertson	NOUN
ejpam-4698	58	16	index	index	PROPN
ejpam-4698	58	17	albwve(g	albwve(g	PROPN
ejpam-4698	58	18	)	)	PUNCT
ejpam-4698	58	19	σuv∈e(g)|w(u)−	σuv∈e(g)|w(u)−	PROPN
ejpam-4698	58	20	w(v)|	w(v)|	ADJ
ejpam-4698	58	21	dxiwve(g;x	dxiwve(g;x	NOUN
ejpam-4698	58	22	,	,	PUNCT
ejpam-4698	58	23	y)|x	y)|x	NOUN
ejpam-4698	58	24	=	=	SYM
ejpam-4698	58	25	y=1	y=1	NOUN
ejpam-4698	58	26	sigma	sigma	VERB
ejpam-4698	58	27	index	index	NOUN
ejpam-4698	58	28	σwve(g	σwve(g	PROPN
ejpam-4698	58	29	)	)	PUNCT
ejpam-4698	58	30	σuv∈e(g)(w(u)−	σuv∈e(g)(w(u)−	PROPN
ejpam-4698	58	31	w(v))2	w(v))2	PROPN
ejpam-4698	58	32	d2	d2	PROPN
ejpam-4698	58	33	xiwve(g;x	xiwve(g;x	PROPN
ejpam-4698	58	34	,	,	PUNCT
ejpam-4698	58	35	y)|x	y)|x	NOUN
ejpam-4698	58	36	=	=	NOUN
ejpam-4698	58	37	y=1	y=1	NOUN
ejpam-4698	58	38	where	where	SCONJ
ejpam-4698	58	39	the	the	DET
ejpam-4698	58	40	operator	operator	NOUN
ejpam-4698	58	41	dx	dx	PROPN
ejpam-4698	58	42	and	and	CCONJ
ejpam-4698	58	43	dy	dy	VERB
ejpam-4698	58	44	on	on	ADP
ejpam-4698	58	45	wve(g;x	wve(g;x	PROPN
ejpam-4698	58	46	,	,	PUNCT
ejpam-4698	58	47	y	y	NOUN
ejpam-4698	58	48	)	)	PUNCT
ejpam-4698	58	49	are	be	AUX
ejpam-4698	58	50	defined	define	VERB
ejpam-4698	58	51	as	as	ADP
ejpam-4698	58	52	:	:	PUNCT
ejpam-4698	58	53	dxwve(g;x	dxwve(g;x	PROPN
ejpam-4698	58	54	,	,	PUNCT
ejpam-4698	58	55	y	y	NOUN
ejpam-4698	58	56	)	)	PUNCT
ejpam-4698	58	57	=	=	PUNCT
ejpam-4698	59	1	x	x	PUNCT
ejpam-4698	59	2	∂wve(g;x	∂wve(g;x	PROPN
ejpam-4698	59	3	,	,	PUNCT
ejpam-4698	59	4	y	y	NOUN
ejpam-4698	59	5	)	)	PUNCT
ejpam-4698	59	6	∂x	∂x	PROPN
ejpam-4698	59	7	,	,	PUNCT
ejpam-4698	59	8	dywve(g;x	dywve(g;x	PROPN
ejpam-4698	59	9	,	,	PUNCT
ejpam-4698	59	10	y	y	NOUN
ejpam-4698	59	11	)	)	PUNCT
ejpam-4698	60	1	=	=	SYM
ejpam-4698	60	2	y	y	PROPN
ejpam-4698	60	3	∂wve(g;x	∂wve(g;x	PROPN
ejpam-4698	60	4	,	,	PUNCT
ejpam-4698	60	5	y	y	PROPN
ejpam-4698	60	6	)	)	PUNCT
ejpam-4698	60	7	∂y	∂y	PROPN
ejpam-4698	60	8	,	,	PUNCT
ejpam-4698	60	9	jwve(g;x	jwve(g;x	PROPN
ejpam-4698	60	10	,	,	PUNCT
ejpam-4698	60	11	y	y	PROPN
ejpam-4698	60	12	)	)	PUNCT
ejpam-4698	60	13	=	=	SYM
ejpam-4698	61	1	wve(g;x	wve(g;x	PROPN
ejpam-4698	61	2	,	,	PUNCT
ejpam-4698	61	3	x	x	NOUN
ejpam-4698	61	4	)	)	PUNCT
ejpam-4698	61	5	and	and	CCONJ
ejpam-4698	61	6	iwve(g;x	iwve(g;x	PROPN
ejpam-4698	61	7	,	,	PUNCT
ejpam-4698	61	8	y	y	NOUN
ejpam-4698	61	9	)	)	PUNCT
ejpam-4698	61	10	=	=	SYM
ejpam-4698	62	1	wve(g;x	wve(g;x	PROPN
ejpam-4698	62	2	,	,	PUNCT
ejpam-4698	62	3	x−1	x−1	PROPN
ejpam-4698	62	4	)	)	PUNCT
ejpam-4698	62	5	.	.	PUNCT
ejpam-4698	63	1	theorem	theorem	VERB
ejpam-4698	63	2	2.1	2.1	NUM
ejpam-4698	63	3	:	:	PUNCT
ejpam-4698	63	4	let	let	VERB
ejpam-4698	63	5	g	g	NOUN
ejpam-4698	63	6	be	be	AUX
ejpam-4698	63	7	any	any	DET
ejpam-4698	63	8	graph	graph	NOUN
ejpam-4698	63	9	with	with	ADP
ejpam-4698	63	10	order	order	NOUN
ejpam-4698	63	11	p	p	X
ejpam-4698	63	12	=	=	X
ejpam-4698	63	13	|v	|v	X
ejpam-4698	63	14	(	(	PUNCT
ejpam-4698	63	15	g)|	g)|	NOUN
ejpam-4698	63	16	and	and	CCONJ
ejpam-4698	63	17	size	size	NOUN
ejpam-4698	63	18	q	q	PROPN
ejpam-4698	63	19	=	=	SYM
ejpam-4698	63	20	|e(g)|	|e(g)|	NOUN
ejpam-4698	63	21	.	.	PROPN
ejpam-4698	64	1	then	then	ADV
ejpam-4698	64	2	1	1	X
ejpam-4698	64	3	.	.	PUNCT
ejpam-4698	64	4	rw	rw	PROPN
ejpam-4698	64	5	1	1	NUM
ejpam-4698	64	6	ve(g	ve(g	NUM
ejpam-4698	64	7	)	)	PUNCT
ejpam-4698	65	1	=	=	PUNCT
ejpam-4698	65	2	w	w	PROPN
ejpam-4698	65	3	1	1	NUM
ejpam-4698	65	4	ve(g)−	ve(g)−	NOUN
ejpam-4698	65	5	2q	2q	NUM
ejpam-4698	65	6	.	.	PUNCT
ejpam-4698	66	1	2	2	X
ejpam-4698	66	2	.	.	X
ejpam-4698	66	3	rw	rw	PROPN
ejpam-4698	66	4	2	2	NUM
ejpam-4698	66	5	ve(g	ve(g	NUM
ejpam-4698	66	6	)	)	PUNCT
ejpam-4698	67	1	=	=	SYM
ejpam-4698	67	2	w	w	PROPN
ejpam-4698	67	3	2	2	NUM
ejpam-4698	67	4	ve(g)−w	ve(g)−w	NOUN
ejpam-4698	67	5	1	1	NUM
ejpam-4698	67	6	ve(g	ve(g	NUM
ejpam-4698	67	7	)	)	PUNCT
ejpam-4698	68	1	+	+	CCONJ
ejpam-4698	68	2	q.	q.	PROPN
ejpam-4698	68	3	3	3	X
ejpam-4698	68	4	.	.	X
ejpam-4698	68	5	fwve(g	fwve(g	NOUN
ejpam-4698	68	6	)	)	PUNCT
ejpam-4698	69	1	=	=	PUNCT
ejpam-4698	69	2	hypwve(g)−	hypwve(g)−	ADJ
ejpam-4698	69	3	2w	2w	NUM
ejpam-4698	69	4	2	2	NUM
ejpam-4698	69	5	ve(g	ve(g	NUM
ejpam-4698	69	6	)	)	PUNCT
ejpam-4698	69	7	.	.	PUNCT
ejpam-4698	70	1	4	4	X
ejpam-4698	70	2	.	.	X
ejpam-4698	70	3	σwve(g	σwve(g	NOUN
ejpam-4698	70	4	)	)	PUNCT
ejpam-4698	71	1	=	=	PUNCT
ejpam-4698	71	2	hypwve(g)−	hypwve(g)−	ADJ
ejpam-4698	71	3	4w	4w	NUM
ejpam-4698	71	4	2	2	NUM
ejpam-4698	71	5	ve(g	ve(g	NUM
ejpam-4698	71	6	)	)	PUNCT
ejpam-4698	71	7	.	.	PUNCT
ejpam-4698	72	1	5	5	X
ejpam-4698	72	2	.	.	X
ejpam-4698	72	3	σwve(g	σwve(g	NOUN
ejpam-4698	72	4	)	)	PUNCT
ejpam-4698	72	5	=	=	SYM
ejpam-4698	73	1	2fwve(g)−hypwve(g	2fwve(g)−hypwve(g	NUM
ejpam-4698	73	2	)	)	PUNCT
ejpam-4698	73	3	.	.	PUNCT
ejpam-4698	74	1	k.	k.	PROPN
ejpam-4698	74	2	rasool	rasool	PROPN
ejpam-4698	74	3	,	,	PUNCT
ejpam-4698	74	4	p.	p.	PROPN
ejpam-4698	74	5	rashed	rashed	PROPN
ejpam-4698	74	6	,	,	PUNCT
ejpam-4698	74	7	a.	a.	PROPN
ejpam-4698	74	8	ali	ali	PROPN
ejpam-4698	74	9	/	/	SYM
ejpam-4698	74	10	eur	eur	PROPN
ejpam-4698	74	11	.	.	PUNCT
ejpam-4698	75	1	j.	j.	PROPN
ejpam-4698	75	2	pure	pure	PROPN
ejpam-4698	75	3	appl	appl	PROPN
ejpam-4698	75	4	.	.	PROPN
ejpam-4698	75	5	math	math	PROPN
ejpam-4698	75	6	,	,	PUNCT
ejpam-4698	75	7	16	16	NUM
ejpam-4698	75	8	(	(	PUNCT
ejpam-4698	75	9	2	2	NUM
ejpam-4698	75	10	)	)	PUNCT
ejpam-4698	75	11	(	(	PUNCT
ejpam-4698	75	12	2023	2023	NUM
ejpam-4698	75	13	)	)	PUNCT
ejpam-4698	75	14	,	,	PUNCT
ejpam-4698	75	15	773	773	NUM
ejpam-4698	75	16	-	-	SYM
ejpam-4698	75	17	783	783	NUM
ejpam-4698	75	18	776	776	NUM
ejpam-4698	75	19	proof	proof	NOUN
ejpam-4698	75	20	:	:	PUNCT
ejpam-4698	75	21	1	1	X
ejpam-4698	75	22	.	.	X
ejpam-4698	75	23	from	from	ADP
ejpam-4698	75	24	the	the	DET
ejpam-4698	75	25	definition	definition	NOUN
ejpam-4698	75	26	of	of	ADP
ejpam-4698	75	27	the	the	DET
ejpam-4698	75	28	reduced	reduce	VERB
ejpam-4698	75	29	first	first	ADJ
ejpam-4698	75	30	zagreb	zagreb	PROPN
ejpam-4698	75	31	,	,	PUNCT
ejpam-4698	75	32	we	we	PRON
ejpam-4698	75	33	have	have	VERB
ejpam-4698	75	34	:	:	PUNCT
ejpam-4698	75	35	rw	rw	PROPN
ejpam-4698	75	36	1	1	NUM
ejpam-4698	75	37	ve(g	ve(g	NUM
ejpam-4698	75	38	)	)	PUNCT
ejpam-4698	75	39	=	=	PRON
ejpam-4698	76	1	σuv∈e(g)(w(u	σuv∈e(g)(w(u	NOUN
ejpam-4698	76	2	)	)	PUNCT
ejpam-4698	76	3	+	+	CCONJ
ejpam-4698	76	4	w(v)−	w(v)−	PROPN
ejpam-4698	76	5	2	2	NUM
ejpam-4698	76	6	)	)	PUNCT
ejpam-4698	76	7	=	=	PRON
ejpam-4698	76	8	σuv∈e(g)(w(u	σuv∈e(g)(w(u	NOUN
ejpam-4698	76	9	)	)	PUNCT
ejpam-4698	76	10	+	+	CCONJ
ejpam-4698	76	11	w(v))−	w(v))−	ADJ
ejpam-4698	76	12	σuv∈e(g)2	σuv∈e(g)2	NOUN
ejpam-4698	76	13	=	=	SYM
ejpam-4698	76	14	w	w	PROPN
ejpam-4698	76	15	1	1	NUM
ejpam-4698	76	16	ve(g)−	ve(g)−	NOUN
ejpam-4698	76	17	2q	2q	NUM
ejpam-4698	76	18	.	.	PUNCT
ejpam-4698	77	1	2	2	X
ejpam-4698	77	2	.	.	X
ejpam-4698	77	3	from	from	ADP
ejpam-4698	77	4	the	the	DET
ejpam-4698	77	5	definition	definition	NOUN
ejpam-4698	77	6	of	of	ADP
ejpam-4698	77	7	the	the	DET
ejpam-4698	77	8	reduced	reduced	ADJ
ejpam-4698	77	9	second	second	ADJ
ejpam-4698	77	10	zagreb	zagreb	PROPN
ejpam-4698	77	11	,	,	PUNCT
ejpam-4698	77	12	we	we	PRON
ejpam-4698	77	13	have	have	VERB
ejpam-4698	77	14	:	:	PUNCT
ejpam-4698	77	15	rw	rw	PROPN
ejpam-4698	77	16	2	2	NUM
ejpam-4698	77	17	ve(g	ve(g	NUM
ejpam-4698	77	18	)	)	PUNCT
ejpam-4698	78	1	=	=	PUNCT
ejpam-4698	79	1	σuv∈e(g)(w(u)−	σuv∈e(g)(w(u)−	NOUN
ejpam-4698	79	2	1)(w(v)−	1)(w(v)−	NUM
ejpam-4698	79	3	1	1	NUM
ejpam-4698	79	4	)	)	PUNCT
ejpam-4698	79	5	=	=	VERB
ejpam-4698	79	6	σuv∈e(g)w(u)w(v)−	σuv∈e(g)w(u)w(v)−	NOUN
ejpam-4698	79	7	σuv∈e(g)(w(u	σuv∈e(g)(w(u	ADJ
ejpam-4698	79	8	)	)	PUNCT
ejpam-4698	79	9	+	+	ADJ
ejpam-4698	79	10	w(v	w(v	NOUN
ejpam-4698	79	11	)	)	PUNCT
ejpam-4698	79	12	)	)	PUNCT
ejpam-4698	80	1	+	+	CCONJ
ejpam-4698	80	2	σuv∈e(g)1	σuv∈e(g)1	NOUN
ejpam-4698	80	3	=	=	SYM
ejpam-4698	80	4	r2	r2	PROPN
ejpam-4698	80	5	ve(g)−r1	ve(g)−r1	NUM
ejpam-4698	80	6	ve(g	ve(g	PUNCT
ejpam-4698	80	7	)	)	PUNCT
ejpam-4698	81	1	+	+	CCONJ
ejpam-4698	81	2	q.	q.	PROPN
ejpam-4698	81	3	3	3	NUM
ejpam-4698	81	4	.	.	PUNCT
ejpam-4698	81	5	from	from	ADP
ejpam-4698	81	6	the	the	DET
ejpam-4698	81	7	definition	definition	NOUN
ejpam-4698	81	8	of	of	ADP
ejpam-4698	81	9	the	the	DET
ejpam-4698	81	10	forgotten	forget	VERB
ejpam-4698	81	11	index	index	NOUN
ejpam-4698	81	12	,	,	PUNCT
ejpam-4698	81	13	we	we	PRON
ejpam-4698	81	14	have	have	AUX
ejpam-4698	81	15	:	:	PUNCT
ejpam-4698	81	16	fwve(g	fwve(g	ADJ
ejpam-4698	81	17	)	)	PUNCT
ejpam-4698	81	18	=	=	SYM
ejpam-4698	81	19	σuv∈e(g)(w(u	σuv∈e(g)(w(u	NOUN
ejpam-4698	81	20	)	)	PUNCT
ejpam-4698	81	21	2	2	NUM
ejpam-4698	81	22	+	+	CCONJ
ejpam-4698	81	23	w(v)2	w(v)2	ADJ
ejpam-4698	81	24	)	)	PUNCT
ejpam-4698	81	25	=	=	PRON
ejpam-4698	81	26	σuv∈e(g)(w(u	σuv∈e(g)(w(u	NOUN
ejpam-4698	81	27	)	)	PUNCT
ejpam-4698	81	28	+	+	CCONJ
ejpam-4698	81	29	w(v))2	w(v))2	PROPN
ejpam-4698	81	30	−	−	PROPN
ejpam-4698	81	31	2σuv∈e(g)w(u)w(v	2σuv∈e(g)w(u)w(v	NUM
ejpam-4698	81	32	)	)	PUNCT
ejpam-4698	81	33	=	=	SYM
ejpam-4698	82	1	hypwve(g)−	hypwve(g)−	ADJ
ejpam-4698	82	2	2w	2w	NUM
ejpam-4698	82	3	2	2	NUM
ejpam-4698	82	4	ve(g	ve(g	NUM
ejpam-4698	82	5	)	)	PUNCT
ejpam-4698	82	6	.	.	PUNCT
ejpam-4698	83	1	4	4	X
ejpam-4698	83	2	.	.	X
ejpam-4698	83	3	from	from	ADP
ejpam-4698	83	4	the	the	DET
ejpam-4698	83	5	definition	definition	NOUN
ejpam-4698	83	6	of	of	ADP
ejpam-4698	83	7	the	the	DET
ejpam-4698	83	8	sigma	sigma	PROPN
ejpam-4698	83	9	index	index	PROPN
ejpam-4698	83	10	,	,	PUNCT
ejpam-4698	83	11	we	we	PRON
ejpam-4698	83	12	have	have	VERB
ejpam-4698	83	13	:	:	PUNCT
ejpam-4698	83	14	σwve(g	σwve(g	NUM
ejpam-4698	83	15	)	)	PUNCT
ejpam-4698	84	1	=	=	PUNCT
ejpam-4698	84	2	σuv∈e(g)(w(u)−	σuv∈e(g)(w(u)−	NOUN
ejpam-4698	84	3	w(v))2	w(v))2	NOUN
ejpam-4698	84	4	=	=	PUNCT
ejpam-4698	84	5	σuv∈e(g)((w(u	σuv∈e(g)((w(u	NUM
ejpam-4698	84	6	)	)	PUNCT
ejpam-4698	84	7	)	)	PUNCT
ejpam-4698	84	8	2	2	NUM
ejpam-4698	85	1	+	+	CCONJ
ejpam-4698	85	2	(	(	PUNCT
ejpam-4698	85	3	w(v))2)−	w(v))2)−	X
ejpam-4698	85	4	2σuv∈e(g)w(u)w(v	2σuv∈e(g)w(u)w(v	NUM
ejpam-4698	85	5	)	)	PUNCT
ejpam-4698	85	6	=	=	PUNCT
ejpam-4698	85	7	fwve(g)−	fwve(g)−	PROPN
ejpam-4698	85	8	2w	2w	NUM
ejpam-4698	85	9	2	2	NUM
ejpam-4698	85	10	ve(g	ve(g	NUM
ejpam-4698	85	11	)	)	PUNCT
ejpam-4698	86	1	=	=	PUNCT
ejpam-4698	86	2	hypwve(g)−	hypwve(g)−	ADJ
ejpam-4698	86	3	4w	4w	NUM
ejpam-4698	86	4	2	2	NUM
ejpam-4698	86	5	ve(g	ve(g	NUM
ejpam-4698	86	6	)	)	PUNCT
ejpam-4698	86	7	.	.	PUNCT
ejpam-4698	87	1	5	5	X
ejpam-4698	87	2	.	.	X
ejpam-4698	87	3	from	from	ADP
ejpam-4698	87	4	3	3	NUM
ejpam-4698	87	5	and	and	CCONJ
ejpam-4698	87	6	4	4	NUM
ejpam-4698	87	7	,	,	PUNCT
ejpam-4698	87	8	we	we	PRON
ejpam-4698	87	9	have	have	VERB
ejpam-4698	87	10	:	:	PUNCT
ejpam-4698	87	11	σwve(g	σwve(g	NUM
ejpam-4698	87	12	)	)	PUNCT
ejpam-4698	87	13	=	=	SYM
ejpam-4698	87	14	2fwve(g)−hypwve(g	2fwve(g)−hypwve(g	NUM
ejpam-4698	87	15	)	)	PUNCT
ejpam-4698	87	16	.	.	PUNCT
ejpam-4698	87	17	.	.	PUNCT
ejpam-4698	88	1	in	in	ADP
ejpam-4698	88	2	the	the	DET
ejpam-4698	88	3	next	next	ADJ
ejpam-4698	88	4	theorem	theorem	NOUN
ejpam-4698	88	5	,	,	PUNCT
ejpam-4698	88	6	the	the	DET
ejpam-4698	88	7	upper	upper	ADJ
ejpam-4698	88	8	and	and	CCONJ
ejpam-4698	88	9	lower	low	ADJ
ejpam-4698	88	10	boundaries	boundary	NOUN
ejpam-4698	88	11	will	will	AUX
ejpam-4698	88	12	be	be	AUX
ejpam-4698	88	13	found	find	VERB
ejpam-4698	88	14	using	use	VERB
ejpam-4698	88	15	the	the	DET
ejpam-4698	88	16	vertex−degree−based	vertex−degree−based	ADJ
ejpam-4698	88	17	topological	topological	ADJ
ejpam-4698	88	18	indices	index	NOUN
ejpam-4698	88	19	τu	τu	ADP
ejpam-4698	88	20	,	,	PUNCT
ejpam-4698	88	21	u	u	PROPN
ejpam-4698	88	22	∈	∈	PROPN
ejpam-4698	88	23	v	v	NOUN
ejpam-4698	88	24	(	(	PUNCT
ejpam-4698	88	25	g	g	NOUN
ejpam-4698	88	26	)	)	PUNCT
ejpam-4698	88	27	.	.	PUNCT
ejpam-4698	89	1	theorem	theorem	VERB
ejpam-4698	89	2	2.2	2.2	NUM
ejpam-4698	89	3	:	:	PUNCT
ejpam-4698	89	4	let	let	VERB
ejpam-4698	89	5	g	g	NOUN
ejpam-4698	89	6	be	be	AUX
ejpam-4698	89	7	any	any	DET
ejpam-4698	89	8	graph	graph	NOUN
ejpam-4698	89	9	with	with	ADP
ejpam-4698	89	10	order	order	NOUN
ejpam-4698	89	11	p	p	NOUN
ejpam-4698	89	12	and	and	CCONJ
ejpam-4698	89	13	size	size	NOUN
ejpam-4698	89	14	q.	q.	PROPN
ejpam-4698	89	15	then	then	ADV
ejpam-4698	89	16	1	1	NUM
ejpam-4698	89	17	.	.	X
ejpam-4698	89	18	2q	2q	NUM
ejpam-4698	89	19	≤	≤	ADJ
ejpam-4698	89	20	m1	m1	PROPN
ejpam-4698	89	21	ve(g	ve(g	PUNCT
ejpam-4698	89	22	)	)	PUNCT
ejpam-4698	89	23	≤	≤	NOUN
ejpam-4698	89	24	2q2	2q2	NUM
ejpam-4698	89	25	.	.	PUNCT
ejpam-4698	90	1	2	2	X
ejpam-4698	90	2	.	.	X
ejpam-4698	90	3	q	q	PROPN
ejpam-4698	90	4	≤	≤	NUM
ejpam-4698	90	5	m2	m2	PROPN
ejpam-4698	90	6	ve(g	ve(g	NUM
ejpam-4698	90	7	)	)	PUNCT
ejpam-4698	90	8	≤	≤	NUM
ejpam-4698	90	9	q3	q3	NOUN
ejpam-4698	90	10	.	.	PUNCT
ejpam-4698	91	1	3	3	NUM
ejpam-4698	91	2	.	.	NOUN
ejpam-4698	91	3	0	0	NUM
ejpam-4698	91	4	≤	≤	PROPN
ejpam-4698	91	5	rm1	rm1	PROPN
ejpam-4698	91	6	ve(g	ve(g	PROPN
ejpam-4698	91	7	)	)	PUNCT
ejpam-4698	91	8	≤	≤	ADV
ejpam-4698	91	9	2q(q	2q(q	NUM
ejpam-4698	91	10	−	−	NOUN
ejpam-4698	91	11	1	1	NUM
ejpam-4698	91	12	)	)	PUNCT
ejpam-4698	91	13	.	.	PUNCT
ejpam-4698	92	1	4	4	NUM
ejpam-4698	92	2	.	.	NOUN
ejpam-4698	92	3	0	0	NUM
ejpam-4698	92	4	≤	≤	NUM
ejpam-4698	92	5	rm2	rm2	PROPN
ejpam-4698	92	6	ve(g	ve(g	PROPN
ejpam-4698	92	7	)	)	PUNCT
ejpam-4698	92	8	≤	≤	NUM
ejpam-4698	92	9	q(q	q(q	PROPN
ejpam-4698	92	10	−	−	PROPN
ejpam-4698	92	11	1)2	1)2	NUM
ejpam-4698	92	12	.	.	PUNCT
ejpam-4698	93	1	5	5	NUM
ejpam-4698	93	2	.	.	NUM
ejpam-4698	93	3	4q	4q	NOUN
ejpam-4698	93	4	≤	≤	NUM
ejpam-4698	93	5	hypmve(g	hypmve(g	NOUN
ejpam-4698	93	6	)	)	PUNCT
ejpam-4698	93	7	≤	≤	NOUN
ejpam-4698	93	8	4q3	4q3	NUM
ejpam-4698	93	9	.	.	PUNCT
ejpam-4698	94	1	k.	k.	PROPN
ejpam-4698	94	2	rasool	rasool	PROPN
ejpam-4698	94	3	,	,	PUNCT
ejpam-4698	94	4	p.	p.	PROPN
ejpam-4698	94	5	rashed	rashed	PROPN
ejpam-4698	94	6	,	,	PUNCT
ejpam-4698	94	7	a.	a.	PROPN
ejpam-4698	94	8	ali	ali	PROPN
ejpam-4698	94	9	/	/	SYM
ejpam-4698	94	10	eur	eur	PROPN
ejpam-4698	94	11	.	.	PUNCT
ejpam-4698	95	1	j.	j.	PROPN
ejpam-4698	95	2	pure	pure	PROPN
ejpam-4698	95	3	appl	appl	PROPN
ejpam-4698	95	4	.	.	PROPN
ejpam-4698	95	5	math	math	PROPN
ejpam-4698	95	6	,	,	PUNCT
ejpam-4698	95	7	16	16	NUM
ejpam-4698	95	8	(	(	PUNCT
ejpam-4698	95	9	2	2	NUM
ejpam-4698	95	10	)	)	PUNCT
ejpam-4698	95	11	(	(	PUNCT
ejpam-4698	95	12	2023	2023	NUM
ejpam-4698	95	13	)	)	PUNCT
ejpam-4698	95	14	,	,	PUNCT
ejpam-4698	95	15	773	773	NUM
ejpam-4698	95	16	-	-	SYM
ejpam-4698	95	17	783	783	NUM
ejpam-4698	95	18	777	777	NUM
ejpam-4698	95	19	6	6	NUM
ejpam-4698	95	20	.	.	PUNCT
ejpam-4698	96	1	2q	2q	NOUN
ejpam-4698	96	2	≤	≤	NUM
ejpam-4698	96	3	fmve(g	fmve(g	NOUN
ejpam-4698	96	4	)	)	PUNCT
ejpam-4698	96	5	≤	≤	NOUN
ejpam-4698	96	6	2q3	2q3	NUM
ejpam-4698	96	7	.	.	PUNCT
ejpam-4698	97	1	proof	proof	NOUN
ejpam-4698	97	2	:	:	PUNCT
ejpam-4698	97	3	since	since	SCONJ
ejpam-4698	97	4	1	1	NUM
ejpam-4698	97	5	≤	≤	NUM
ejpam-4698	97	6	τu	τu	ADP
ejpam-4698	97	7	≤	≤	ADJ
ejpam-4698	97	8	q	q	NOUN
ejpam-4698	97	9	,	,	PUNCT
ejpam-4698	97	10	for	for	ADP
ejpam-4698	97	11	all	all	DET
ejpam-4698	97	12	u	u	NOUN
ejpam-4698	97	13	in	in	ADP
ejpam-4698	97	14	g	g	PROPN
ejpam-4698	97	15	,	,	PUNCT
ejpam-4698	97	16	then	then	ADV
ejpam-4698	97	17	1	1	NUM
ejpam-4698	97	18	.	.	SYM
ejpam-4698	97	19	2	2	NUM
ejpam-4698	97	20	≤	≤	NOUN
ejpam-4698	97	21	τu	τu	ADP
ejpam-4698	97	22	+	+	CCONJ
ejpam-4698	97	23	τv	τv	ADP
ejpam-4698	97	24	≤	≤	ADJ
ejpam-4698	97	25	2q	2q	NOUN
ejpam-4698	97	26	,	,	PUNCT
ejpam-4698	97	27	this	this	PRON
ejpam-4698	97	28	implies	imply	VERB
ejpam-4698	97	29	that	that	SCONJ
ejpam-4698	97	30	σuv∈e(g)2	σuv∈e(g)2	VERB
ejpam-4698	97	31	≤	≤	NOUN
ejpam-4698	97	32	σuv∈e(g)(τu	σuv∈e(g)(τu	PUNCT
ejpam-4698	97	33	+	+	NUM
ejpam-4698	97	34	τv	τv	NOUN
ejpam-4698	97	35	)	)	PUNCT
ejpam-4698	97	36	≤	≤	NOUN
ejpam-4698	97	37	σuv∈e(g)2q	σuv∈e(g)2q	ADP
ejpam-4698	97	38	,	,	PUNCT
ejpam-4698	97	39	then	then	ADV
ejpam-4698	97	40	2q	2q	NUM
ejpam-4698	97	41	≤	≤	ADJ
ejpam-4698	97	42	m1	m1	PROPN
ejpam-4698	97	43	ve(g	ve(g	PUNCT
ejpam-4698	97	44	)	)	PUNCT
ejpam-4698	97	45	≤	≤	NOUN
ejpam-4698	97	46	2q2	2q2	NUM
ejpam-4698	97	47	.	.	PUNCT
ejpam-4698	98	1	2	2	NUM
ejpam-4698	98	2	.	.	SYM
ejpam-4698	98	3	1	1	NUM
ejpam-4698	98	4	≤	≤	NOUN
ejpam-4698	98	5	τuτv	τuτv	VERB
ejpam-4698	98	6	≤	≤	NUM
ejpam-4698	98	7	q2	q2	NOUN
ejpam-4698	98	8	,	,	PUNCT
ejpam-4698	98	9	this	this	PRON
ejpam-4698	98	10	implies	imply	VERB
ejpam-4698	98	11	that	that	DET
ejpam-4698	98	12	σuv∈e(g)1	σuv∈e(g)1	VERB
ejpam-4698	98	13	≤	≤	NOUN
ejpam-4698	98	14	σuv∈e(g)(τuτv	σuv∈e(g)(τuτv	NOUN
ejpam-4698	98	15	)	)	PUNCT
ejpam-4698	98	16	≤	≤	NUM
ejpam-4698	98	17	σuv∈e(g)q	σuv∈e(g)q	NOUN
ejpam-4698	98	18	2	2	NUM
ejpam-4698	98	19	,	,	PUNCT
ejpam-4698	98	20	then	then	ADV
ejpam-4698	98	21	q	q	PROPN
ejpam-4698	98	22	≤	≤	PROPN
ejpam-4698	98	23	m2	m2	PROPN
ejpam-4698	98	24	ve(g	ve(g	NUM
ejpam-4698	98	25	)	)	PUNCT
ejpam-4698	98	26	≤	≤	NUM
ejpam-4698	98	27	q3	q3	NOUN
ejpam-4698	98	28	.	.	PUNCT
ejpam-4698	99	1	3	3	NUM
ejpam-4698	99	2	.	.	NOUN
ejpam-4698	99	3	0	0	NUM
ejpam-4698	99	4	≤	≤	NOUN
ejpam-4698	99	5	τu	τu	ADP
ejpam-4698	99	6	+	+	CCONJ
ejpam-4698	99	7	τv	τv	NOUN
ejpam-4698	99	8	−	−	PROPN
ejpam-4698	99	9	2	2	NUM
ejpam-4698	99	10	≤	≤	NOUN
ejpam-4698	99	11	2(q	2(q	NUM
ejpam-4698	99	12	−	−	NOUN
ejpam-4698	99	13	1	1	NUM
ejpam-4698	99	14	)	)	PUNCT
ejpam-4698	100	1	,	,	PUNCT
ejpam-4698	100	2	this	this	PRON
ejpam-4698	100	3	implies	imply	VERB
ejpam-4698	100	4	that	that	SCONJ
ejpam-4698	100	5	σuv∈e(g)0	σuv∈e(g)0	ADJ
ejpam-4698	100	6	≤	≤	NOUN
ejpam-4698	100	7	σuv∈e(g)(τu	σuv∈e(g)(τu	PUNCT
ejpam-4698	100	8	+	+	NOUN
ejpam-4698	100	9	τv	τv	NOUN
ejpam-4698	100	10	−	−	PROPN
ejpam-4698	100	11	2	2	NUM
ejpam-4698	100	12	)	)	PUNCT
ejpam-4698	100	13	≤	≤	NOUN
ejpam-4698	100	14	σuv∈e(g)2(q	σuv∈e(g)2(q	NOUN
ejpam-4698	100	15	−	−	NOUN
ejpam-4698	100	16	1	1	NUM
ejpam-4698	100	17	)	)	PUNCT
ejpam-4698	100	18	,	,	PUNCT
ejpam-4698	100	19	then	then	ADV
ejpam-4698	100	20	0	0	NUM
ejpam-4698	100	21	≤	≤	PROPN
ejpam-4698	100	22	rm1	rm1	PROPN
ejpam-4698	100	23	ve(g	ve(g	PROPN
ejpam-4698	100	24	)	)	PUNCT
ejpam-4698	100	25	≤	≤	ADV
ejpam-4698	100	26	2q(q	2q(q	NUM
ejpam-4698	100	27	−	−	NOUN
ejpam-4698	100	28	1	1	NUM
ejpam-4698	100	29	)	)	PUNCT
ejpam-4698	100	30	.	.	PUNCT
ejpam-4698	101	1	4	4	NUM
ejpam-4698	101	2	.	.	NOUN
ejpam-4698	101	3	0	0	NUM
ejpam-4698	101	4	≤	≤	NOUN
ejpam-4698	101	5	(	(	PUNCT
ejpam-4698	101	6	τu	τu	ADP
ejpam-4698	101	7	−	−	PROPN
ejpam-4698	101	8	1)(τv	1)(τv	NUM
ejpam-4698	101	9	−	−	PROPN
ejpam-4698	101	10	1	1	NUM
ejpam-4698	101	11	)	)	PUNCT
ejpam-4698	101	12	≤	≤	NOUN
ejpam-4698	101	13	(	(	PUNCT
ejpam-4698	101	14	q	q	NOUN
ejpam-4698	101	15	−	−	PROPN
ejpam-4698	101	16	1)2	1)2	NUM
ejpam-4698	101	17	,	,	PUNCT
ejpam-4698	101	18	this	this	PRON
ejpam-4698	101	19	implies	imply	VERB
ejpam-4698	101	20	that	that	SCONJ
ejpam-4698	101	21	σuv∈e(g)0	σuv∈e(g)0	ADJ
ejpam-4698	101	22	≤	≤	NOUN
ejpam-4698	101	23	σuv∈e(g)(τu	σuv∈e(g)(τu	PUNCT
ejpam-4698	101	24	−	−	PROPN
ejpam-4698	101	25	1)(τv	1)(τv	NUM
ejpam-4698	101	26	−	−	NOUN
ejpam-4698	101	27	1	1	NUM
ejpam-4698	101	28	)	)	PUNCT
ejpam-4698	101	29	≤	≤	NOUN
ejpam-4698	101	30	σuv∈e(g)(q	σuv∈e(g)(q	NOUN
ejpam-4698	101	31	−	−	PROPN
ejpam-4698	102	1	1)2	1)2	NUM
ejpam-4698	102	2	,	,	PUNCT
ejpam-4698	102	3	then	then	ADV
ejpam-4698	102	4	0	0	NUM
ejpam-4698	102	5	≤	≤	PROPN
ejpam-4698	102	6	rm2	rm2	PROPN
ejpam-4698	102	7	ve(g	ve(g	PROPN
ejpam-4698	102	8	)	)	PUNCT
ejpam-4698	102	9	≤	≤	NUM
ejpam-4698	103	1	q(q	q(q	PROPN
ejpam-4698	103	2	−	−	PROPN
ejpam-4698	103	3	1)2	1)2	NUM
ejpam-4698	103	4	.	.	PUNCT
ejpam-4698	104	1	5	5	NUM
ejpam-4698	104	2	.	.	NOUN
ejpam-4698	104	3	4	4	NUM
ejpam-4698	104	4	≤	≤	NUM
ejpam-4698	104	5	(	(	PUNCT
ejpam-4698	104	6	τu	τu	ADP
ejpam-4698	104	7	+	+	X
ejpam-4698	104	8	τv	τv	X
ejpam-4698	104	9	)	)	PUNCT
ejpam-4698	104	10	2	2	NUM
ejpam-4698	104	11	≤	≤	NOUN
ejpam-4698	104	12	4q2	4q2	NUM
ejpam-4698	104	13	,	,	PUNCT
ejpam-4698	104	14	this	this	PRON
ejpam-4698	104	15	implies	imply	VERB
ejpam-4698	104	16	that	that	SCONJ
ejpam-4698	104	17	σuv∈e(g)4	σuv∈e(g)4	ADJ
ejpam-4698	104	18	≤	≤	NOUN
ejpam-4698	104	19	σuv∈e(g)(τu	σuv∈e(g)(τu	PUNCT
ejpam-4698	104	20	+	+	SYM
ejpam-4698	104	21	τv	τv	X
ejpam-4698	104	22	)	)	PUNCT
ejpam-4698	104	23	2	2	NUM
ejpam-4698	104	24	≤	≤	NOUN
ejpam-4698	104	25	σuv∈e(g)4q	σuv∈e(g)4q	PROPN
ejpam-4698	104	26	2	2	NUM
ejpam-4698	104	27	,	,	PUNCT
ejpam-4698	104	28	then	then	ADV
ejpam-4698	104	29	4q	4q	NOUN
ejpam-4698	104	30	≤	≤	PROPN
ejpam-4698	104	31	hypmve(g	hypmve(g	NOUN
ejpam-4698	104	32	)	)	PUNCT
ejpam-4698	104	33	≤	≤	NOUN
ejpam-4698	104	34	4q3	4q3	NUM
ejpam-4698	104	35	.	.	PUNCT
ejpam-4698	105	1	6	6	NUM
ejpam-4698	105	2	.	.	SYM
ejpam-4698	105	3	2	2	NUM
ejpam-4698	105	4	≤	≤	NOUN
ejpam-4698	105	5	(	(	PUNCT
ejpam-4698	105	6	τu	τu	SYM
ejpam-4698	105	7	)	)	PUNCT
ejpam-4698	105	8	2	2	NUM
ejpam-4698	106	1	+	+	CCONJ
ejpam-4698	106	2	(	(	PUNCT
ejpam-4698	106	3	τv	τv	NOUN
ejpam-4698	106	4	)	)	PUNCT
ejpam-4698	106	5	2	2	NUM
ejpam-4698	106	6	≤	≤	NOUN
ejpam-4698	106	7	2q2	2q2	NUM
ejpam-4698	106	8	,	,	PUNCT
ejpam-4698	106	9	this	this	PRON
ejpam-4698	106	10	implies	imply	VERB
ejpam-4698	106	11	that	that	SCONJ
ejpam-4698	106	12	σuv∈e(g)2	σuv∈e(g)2	VERB
ejpam-4698	106	13	≤	≤	X
ejpam-4698	106	14	σuv∈e(g)((τu	σuv∈e(g)((τu	NOUN
ejpam-4698	106	15	)	)	PUNCT
ejpam-4698	106	16	2	2	NUM
ejpam-4698	106	17	+	+	CCONJ
ejpam-4698	106	18	(	(	PUNCT
ejpam-4698	106	19	τv	τv	NOUN
ejpam-4698	106	20	)	)	PUNCT
ejpam-4698	106	21	2	2	NUM
ejpam-4698	106	22	)	)	PUNCT
ejpam-4698	106	23	≤	≤	NOUN
ejpam-4698	106	24	σuv∈e(g)2q	σuv∈e(g)2q	ADP
ejpam-4698	106	25	2	2	NUM
ejpam-4698	106	26	,	,	PUNCT
ejpam-4698	106	27	then	then	ADV
ejpam-4698	106	28	2q	2q	ADJ
ejpam-4698	106	29	≤	≤	ADJ
ejpam-4698	106	30	fmve(g	fmve(g	NOUN
ejpam-4698	106	31	)	)	PUNCT
ejpam-4698	106	32	≤	≤	NOUN
ejpam-4698	106	33	2q3	2q3	NUM
ejpam-4698	106	34	.	.	PUNCT
ejpam-4698	107	1	remark	remark	NOUN
ejpam-4698	107	2	:	:	PUNCT
ejpam-4698	107	3	we	we	PRON
ejpam-4698	107	4	note	note	VERB
ejpam-4698	107	5	that	that	SCONJ
ejpam-4698	107	6	equality	equality	NOUN
ejpam-4698	107	7	exists	exist	VERB
ejpam-4698	107	8	for	for	ADP
ejpam-4698	107	9	all	all	DET
ejpam-4698	107	10	topological	topological	ADJ
ejpam-4698	107	11	indices	index	NOUN
ejpam-4698	107	12	in	in	ADP
ejpam-4698	107	13	theorem	theorem	ADJ
ejpam-4698	107	14	2.2	2.2	NUM
ejpam-4698	107	15	,	,	PUNCT
ejpam-4698	107	16	if	if	SCONJ
ejpam-4698	107	17	g	g	PROPN
ejpam-4698	107	18	=	=	SYM
ejpam-4698	107	19	k2	k2	PROPN
ejpam-4698	107	20	.	.	PUNCT
ejpam-4698	108	1	3	3	NUM
ejpam-4698	108	2	.	.	NUM
ejpam-4698	108	3	mve−polynomials	mve−polynomial	NOUN
ejpam-4698	108	4	of	of	ADP
ejpam-4698	108	5	r−regular	r−regular	ADJ
ejpam-4698	108	6	graphs	graph	NOUN
ejpam-4698	108	7	in	in	ADP
ejpam-4698	108	8	the	the	DET
ejpam-4698	108	9	next	next	ADJ
ejpam-4698	108	10	theorems	theorem	NOUN
ejpam-4698	108	11	,	,	PUNCT
ejpam-4698	108	12	we	we	PRON
ejpam-4698	108	13	discuss	discuss	VERB
ejpam-4698	108	14	the	the	DET
ejpam-4698	108	15	general	general	PROPN
ejpam-4698	108	16	mve−polynomial	mve−polynomial	PROPN
ejpam-4698	108	17	of	of	ADP
ejpam-4698	108	18	r−regular	r−regular	ADJ
ejpam-4698	108	19	simple	simple	ADJ
ejpam-4698	108	20	graph	graph	NOUN
ejpam-4698	108	21	of	of	ADP
ejpam-4698	108	22	size	size	NOUN
ejpam-4698	108	23	q.	q.	PROPN
ejpam-4698	108	24	theorem	theorem	VERB
ejpam-4698	108	25	3.1	3.1	NUM
ejpam-4698	108	26	:	:	PUNCT
ejpam-4698	108	27	let	let	VERB
ejpam-4698	108	28	g	g	PROPN
ejpam-4698	108	29	is	be	AUX
ejpam-4698	108	30	an	an	DET
ejpam-4698	108	31	r−regular	r−regular	ADJ
ejpam-4698	108	32	simple	simple	ADJ
ejpam-4698	108	33	graph	graph	NOUN
ejpam-4698	108	34	g	g	NOUN
ejpam-4698	108	35	of	of	ADP
ejpam-4698	108	36	size	size	NOUN
ejpam-4698	108	37	q	q	PROPN
ejpam-4698	108	38	,	,	PUNCT
ejpam-4698	108	39	then	then	ADV
ejpam-4698	109	1	k.	k.	PROPN
ejpam-4698	109	2	rasool	rasool	PROPN
ejpam-4698	109	3	,	,	PUNCT
ejpam-4698	109	4	p.	p.	PROPN
ejpam-4698	109	5	rashed	rashed	PROPN
ejpam-4698	109	6	,	,	PUNCT
ejpam-4698	109	7	a.	a.	PROPN
ejpam-4698	109	8	ali	ali	PROPN
ejpam-4698	109	9	/	/	SYM
ejpam-4698	109	10	eur	eur	PROPN
ejpam-4698	109	11	.	.	PUNCT
ejpam-4698	110	1	j.	j.	PROPN
ejpam-4698	110	2	pure	pure	PROPN
ejpam-4698	110	3	appl	appl	PROPN
ejpam-4698	110	4	.	.	PROPN
ejpam-4698	110	5	math	math	PROPN
ejpam-4698	110	6	,	,	PUNCT
ejpam-4698	110	7	16	16	NUM
ejpam-4698	110	8	(	(	PUNCT
ejpam-4698	110	9	2	2	NUM
ejpam-4698	110	10	)	)	PUNCT
ejpam-4698	110	11	(	(	PUNCT
ejpam-4698	110	12	2023	2023	NUM
ejpam-4698	110	13	)	)	PUNCT
ejpam-4698	110	14	,	,	PUNCT
ejpam-4698	110	15	773	773	NUM
ejpam-4698	110	16	-	-	SYM
ejpam-4698	110	17	783	783	NUM
ejpam-4698	110	18	778	778	NUM
ejpam-4698	110	19	1	1	NUM
ejpam-4698	110	20	.	.	PUNCT
ejpam-4698	111	1	m(g;x	m(g;x	PROPN
ejpam-4698	111	2	,	,	PUNCT
ejpam-4698	111	3	y	y	NOUN
ejpam-4698	111	4	)	)	PUNCT
ejpam-4698	111	5	=	=	SYM
ejpam-4698	112	1	qxryr	qxryr	NOUN
ejpam-4698	112	2	.	.	PUNCT
ejpam-4698	113	1	2	2	X
ejpam-4698	113	2	.	.	X
ejpam-4698	113	3	nm(g;x	nm(g;x	PROPN
ejpam-4698	113	4	,	,	PUNCT
ejpam-4698	113	5	y	y	NOUN
ejpam-4698	113	6	)	)	PUNCT
ejpam-4698	113	7	=	=	PRON
ejpam-4698	113	8	qxr	qxr	NOUN
ejpam-4698	113	9	2	2	NUM
ejpam-4698	113	10	yr	yr	NOUN
ejpam-4698	113	11	2	2	NUM
ejpam-4698	113	12	.	.	PUNCT
ejpam-4698	114	1	3	3	X
ejpam-4698	114	2	.	.	X
ejpam-4698	115	1	if	if	SCONJ
ejpam-4698	115	2	the	the	DET
ejpam-4698	115	3	graph	graph	NOUN
ejpam-4698	115	4	g	g	PROPN
ejpam-4698	115	5	without	without	ADP
ejpam-4698	115	6	triangle	triangle	NOUN
ejpam-4698	115	7	cycles	cycle	NOUN
ejpam-4698	115	8	,	,	PUNCT
ejpam-4698	115	9	then	then	ADV
ejpam-4698	115	10	mve(g;x	mve(g;x	PROPN
ejpam-4698	115	11	,	,	PUNCT
ejpam-4698	115	12	y	y	NOUN
ejpam-4698	115	13	)	)	PUNCT
ejpam-4698	115	14	=	=	PRON
ejpam-4698	115	15	qxr	qxr	NOUN
ejpam-4698	115	16	2	2	NUM
ejpam-4698	115	17	yr	yr	NOUN
ejpam-4698	115	18	2	2	NUM
ejpam-4698	115	19	.	.	PUNCT
ejpam-4698	116	1	proof	proof	NOUN
ejpam-4698	116	2	:	:	PUNCT
ejpam-4698	116	3	1	1	X
ejpam-4698	116	4	.	.	X
ejpam-4698	116	5	let	let	VERB
ejpam-4698	116	6	g	g	PROPN
ejpam-4698	116	7	is	be	AUX
ejpam-4698	116	8	r−regular	r−regular	NUM
ejpam-4698	116	9	graph	graph	NOUN
ejpam-4698	116	10	with	with	ADP
ejpam-4698	116	11	q	q	NOUN
ejpam-4698	116	12	edges	edge	NOUN
ejpam-4698	116	13	,	,	PUNCT
ejpam-4698	116	14	then	then	ADV
ejpam-4698	116	15	every	every	DET
ejpam-4698	116	16	edge	edge	NOUN
ejpam-4698	116	17	in	in	ADP
ejpam-4698	116	18	g	g	PROPN
ejpam-4698	116	19	is	be	AUX
ejpam-4698	116	20	an	an	DET
ejpam-4698	116	21	incident	incident	NOUN
ejpam-4698	116	22	on	on	ADP
ejpam-4698	116	23	two	two	NUM
ejpam-4698	116	24	vertices	vertex	NOUN
ejpam-4698	116	25	which	which	PRON
ejpam-4698	116	26	have	have	VERB
ejpam-4698	116	27	r	r	NOUN
ejpam-4698	116	28	degree	degree	NOUN
ejpam-4698	116	29	.	.	PUNCT
ejpam-4698	117	1	hence	hence	ADV
ejpam-4698	117	2	,	,	PUNCT
ejpam-4698	117	3	m(g;x	m(g;x	PROPN
ejpam-4698	117	4	,	,	PUNCT
ejpam-4698	117	5	y	y	NOUN
ejpam-4698	117	6	)	)	PUNCT
ejpam-4698	117	7	=	=	SYM
ejpam-4698	118	1	qxryr	qxryr	NOUN
ejpam-4698	118	2	.	.	PUNCT
ejpam-4698	119	1	2	2	X
ejpam-4698	119	2	.	.	X
ejpam-4698	119	3	since	since	SCONJ
ejpam-4698	119	4	every	every	DET
ejpam-4698	119	5	vertex	vertex	NOUN
ejpam-4698	119	6	u	u	NOUN
ejpam-4698	119	7	in	in	ADP
ejpam-4698	119	8	r−regular	r−regular	ADJ
ejpam-4698	119	9	graph	graph	NOUN
ejpam-4698	119	10	g	g	NOUN
ejpam-4698	119	11	is	be	AUX
ejpam-4698	119	12	adjacent	adjacent	ADJ
ejpam-4698	119	13	r	r	NOUN
ejpam-4698	119	14	vertices	vertex	NOUN
ejpam-4698	119	15	,	,	PUNCT
ejpam-4698	119	16	then	then	ADV
ejpam-4698	119	17	δu	δu	ADP
ejpam-4698	119	18	=	=	SYM
ejpam-4698	119	19	r2	r2	PROPN
ejpam-4698	119	20	.	.	PUNCT
ejpam-4698	120	1	hence	hence	ADV
ejpam-4698	120	2	,	,	PUNCT
ejpam-4698	120	3	nm(g;x	nm(g;x	PROPN
ejpam-4698	120	4	,	,	PUNCT
ejpam-4698	120	5	y	y	NOUN
ejpam-4698	120	6	)	)	PUNCT
ejpam-4698	120	7	=	=	PRON
ejpam-4698	121	1	qxr	qxr	NOUN
ejpam-4698	121	2	2	2	NUM
ejpam-4698	121	3	yr	yr	NOUN
ejpam-4698	121	4	2	2	NUM
ejpam-4698	121	5	.	.	PUNCT
ejpam-4698	122	1	3	3	X
ejpam-4698	122	2	.	.	X
ejpam-4698	122	3	if	if	SCONJ
ejpam-4698	122	4	g	g	PROPN
ejpam-4698	122	5	is	be	AUX
ejpam-4698	122	6	an	an	DET
ejpam-4698	122	7	r−regular	r−regular	ADJ
ejpam-4698	122	8	simple	simple	ADJ
ejpam-4698	122	9	graph	graph	NOUN
ejpam-4698	122	10	without	without	ADP
ejpam-4698	122	11	triangle	triangle	NOUN
ejpam-4698	122	12	cycles	cycle	NOUN
ejpam-4698	122	13	,	,	PUNCT
ejpam-4698	122	14	then	then	ADV
ejpam-4698	122	15	τu	τu	ADP
ejpam-4698	122	16	=	=	NOUN
ejpam-4698	122	17	τv	τv	NOUN
ejpam-4698	122	18	=	=	PROPN
ejpam-4698	122	19	r2	r2	PROPN
ejpam-4698	122	20	,	,	PUNCT
ejpam-4698	122	21	for	for	ADP
ejpam-4698	122	22	every	every	DET
ejpam-4698	122	23	edge	edge	NOUN
ejpam-4698	122	24	e	e	NOUN
ejpam-4698	122	25	=	=	NOUN
ejpam-4698	122	26	uv	uv	NOUN
ejpam-4698	122	27	where	where	SCONJ
ejpam-4698	122	28	u	u	NOUN
ejpam-4698	122	29	,	,	PUNCT
ejpam-4698	122	30	v	v	PROPN
ejpam-4698	122	31	∈	∈	PROPN
ejpam-4698	122	32	v	v	NOUN
ejpam-4698	122	33	(	(	PUNCT
ejpam-4698	122	34	g	g	NOUN
ejpam-4698	122	35	)	)	PUNCT
ejpam-4698	122	36	.	.	PUNCT
ejpam-4698	123	1	hence	hence	ADV
ejpam-4698	123	2	mve(g;x	mve(g;x	PROPN
ejpam-4698	123	3	,	,	PUNCT
ejpam-4698	123	4	y	y	NOUN
ejpam-4698	123	5	)	)	PUNCT
ejpam-4698	123	6	=	=	PRON
ejpam-4698	123	7	qxr	qxr	NOUN
ejpam-4698	123	8	2	2	NUM
ejpam-4698	123	9	yr	yr	NOUN
ejpam-4698	123	10	2	2	NUM
ejpam-4698	123	11	.	.	PUNCT
ejpam-4698	123	12	example	example	NOUN
ejpam-4698	123	13	3.2	3.2	NUM
ejpam-4698	123	14	:	:	PUNCT
ejpam-4698	123	15	let	let	VERB
ejpam-4698	123	16	hi	hi	INTJ
ejpam-4698	123	17	be	be	AUX
ejpam-4698	123	18	3−regular	3−regular	NUM
ejpam-4698	123	19	graph	graph	NOUN
ejpam-4698	123	20	for	for	ADP
ejpam-4698	123	21	all	all	PRON
ejpam-4698	123	22	i	i	PRON
ejpam-4698	123	23	=	=	NOUN
ejpam-4698	123	24	1	1	NUM
ejpam-4698	123	25	,	,	PUNCT
ejpam-4698	123	26	2	2	NUM
ejpam-4698	123	27	,	,	PUNCT
ejpam-4698	123	28	3	3	NUM
ejpam-4698	123	29	.	.	X
ejpam-4698	124	1	see	see	VERB
ejpam-4698	124	2	figure	figure	NOUN
ejpam-4698	124	3	1	1	NUM
ejpam-4698	124	4	.	.	PUNCT
ejpam-4698	124	5	figure	figure	NOUN
ejpam-4698	124	6	1	1	NUM
ejpam-4698	124	7	:	:	PUNCT
ejpam-4698	124	8	3−regular	3−regular	NUM
ejpam-4698	124	9	graphs	graph	NOUN
ejpam-4698	124	10	.	.	PUNCT
ejpam-4698	125	1	we	we	PRON
ejpam-4698	125	2	note	note	VERB
ejpam-4698	125	3	that	that	SCONJ
ejpam-4698	125	4	:	:	PUNCT
ejpam-4698	125	5	m(h1;x	m(h1;x	PROPN
ejpam-4698	125	6	,	,	PUNCT
ejpam-4698	125	7	y	y	NOUN
ejpam-4698	125	8	)	)	PUNCT
ejpam-4698	125	9	=	=	SYM
ejpam-4698	125	10	9x3y3	9x3y3	NUM
ejpam-4698	125	11	,	,	PUNCT
ejpam-4698	125	12	nm(h1;x	nm(h1;x	PROPN
ejpam-4698	125	13	,	,	PUNCT
ejpam-4698	125	14	y	y	NOUN
ejpam-4698	125	15	)	)	PUNCT
ejpam-4698	125	16	=	=	SYM
ejpam-4698	125	17	9x9y9	9x9y9	PROPN
ejpam-4698	125	18	and	and	CCONJ
ejpam-4698	125	19	mve(h1;x	mve(h1;x	PROPN
ejpam-4698	125	20	,	,	PUNCT
ejpam-4698	125	21	y	y	PROPN
ejpam-4698	125	22	)	)	PUNCT
ejpam-4698	125	23	=	=	SYM
ejpam-4698	126	1	9x8y8	9x8y8	X
ejpam-4698	126	2	.	.	PUNCT
ejpam-4698	127	1	m(h2;x	m(h2;x	PROPN
ejpam-4698	127	2	,	,	PUNCT
ejpam-4698	127	3	y	y	NOUN
ejpam-4698	127	4	)	)	PUNCT
ejpam-4698	127	5	=	=	SYM
ejpam-4698	127	6	12x3y3	12x3y3	NOUN
ejpam-4698	127	7	and	and	CCONJ
ejpam-4698	127	8	nm(h2;x	nm(h2;x	PROPN
ejpam-4698	127	9	,	,	PUNCT
ejpam-4698	127	10	y	y	NOUN
ejpam-4698	127	11	)	)	PUNCT
ejpam-4698	127	12	=	=	SYM
ejpam-4698	127	13	mve(h2;x	mve(h2;x	PROPN
ejpam-4698	127	14	,	,	PUNCT
ejpam-4698	127	15	y	y	NOUN
ejpam-4698	127	16	)	)	PUNCT
ejpam-4698	127	17	=	=	NOUN
ejpam-4698	127	18	12x9y9	12x9y9	NOUN
ejpam-4698	127	19	.	.	PUNCT
ejpam-4698	128	1	m(h3;x	m(h3;x	NOUN
ejpam-4698	128	2	,	,	PUNCT
ejpam-4698	128	3	y	y	NOUN
ejpam-4698	128	4	)	)	PUNCT
ejpam-4698	128	5	=	=	SYM
ejpam-4698	128	6	15x3y3	15x3y3	NUM
ejpam-4698	128	7	,	,	PUNCT
ejpam-4698	128	8	nm(h3;x	nm(h3;x	PROPN
ejpam-4698	128	9	,	,	PUNCT
ejpam-4698	128	10	y	y	NOUN
ejpam-4698	128	11	)	)	PUNCT
ejpam-4698	128	12	=	=	SYM
ejpam-4698	128	13	15x9y9	15x9y9	NOUN
ejpam-4698	128	14	and	and	CCONJ
ejpam-4698	128	15	mve(h3;x	mve(h3;x	PROPN
ejpam-4698	128	16	,	,	PUNCT
ejpam-4698	128	17	y	y	NOUN
ejpam-4698	128	18	)	)	PUNCT
ejpam-4698	128	19	=	=	SYM
ejpam-4698	129	1	7x8y8	7x8y8	NUM
ejpam-4698	129	2	+	+	NUM
ejpam-4698	129	3	4x8y9	4x8y9	NOUN
ejpam-4698	129	4	+	+	CCONJ
ejpam-4698	129	5	4x9y9	4x9y9	NOUN
ejpam-4698	129	6	.	.	PUNCT
ejpam-4698	130	1	it	it	PRON
ejpam-4698	130	2	is	be	AUX
ejpam-4698	130	3	very	very	ADV
ejpam-4698	130	4	difficult	difficult	ADJ
ejpam-4698	130	5	to	to	PART
ejpam-4698	130	6	find	find	VERB
ejpam-4698	130	7	a	a	DET
ejpam-4698	130	8	general	general	ADJ
ejpam-4698	130	9	formula	formula	NOUN
ejpam-4698	130	10	for	for	ADP
ejpam-4698	130	11	mve−polynomial	mve−polynomial	PROPN
ejpam-4698	130	12	of	of	ADP
ejpam-4698	130	13	a	a	DET
ejpam-4698	130	14	regular	regular	ADJ
ejpam-4698	130	15	graph	graph	NOUN
ejpam-4698	130	16	,	,	PUNCT
ejpam-4698	130	17	so	so	SCONJ
ejpam-4698	130	18	some	some	DET
ejpam-4698	130	19	conditions	condition	NOUN
ejpam-4698	130	20	were	be	AUX
ejpam-4698	130	21	given	give	VERB
ejpam-4698	130	22	on	on	ADP
ejpam-4698	130	23	the	the	DET
ejpam-4698	130	24	regular	regular	ADJ
ejpam-4698	130	25	graphs	graph	NOUN
ejpam-4698	130	26	in	in	ADP
ejpam-4698	130	27	order	order	NOUN
ejpam-4698	130	28	to	to	PART
ejpam-4698	130	29	obtain	obtain	VERB
ejpam-4698	130	30	mve−polynomial	mve−polynomial	PROPN
ejpam-4698	130	31	.	.	PUNCT
ejpam-4698	131	1	theorem	theorem	VERB
ejpam-4698	131	2	3.3	3.3	NUM
ejpam-4698	131	3	:	:	PUNCT
ejpam-4698	131	4	let	let	VERB
ejpam-4698	131	5	g	g	PROPN
ejpam-4698	131	6	is	be	AUX
ejpam-4698	131	7	an	an	DET
ejpam-4698	131	8	r−regular	r−regular	ADJ
ejpam-4698	131	9	simple	simple	ADJ
ejpam-4698	131	10	graph	graph	NOUN
ejpam-4698	131	11	such	such	ADJ
ejpam-4698	131	12	that	that	SCONJ
ejpam-4698	131	13	every	every	DET
ejpam-4698	131	14	vertex	vertex	NOUN
ejpam-4698	131	15	in	in	ADP
ejpam-4698	131	16	g	g	PROPN
ejpam-4698	131	17	lies	lie	VERB
ejpam-4698	131	18	on	on	ADP
ejpam-4698	131	19	only	only	ADV
ejpam-4698	131	20	one	one	NUM
ejpam-4698	131	21	triangle	triangle	NOUN
ejpam-4698	131	22	cycle	cycle	NOUN
ejpam-4698	131	23	,	,	PUNCT
ejpam-4698	131	24	then	then	ADV
ejpam-4698	132	1	k.	k.	PROPN
ejpam-4698	132	2	rasool	rasool	PROPN
ejpam-4698	132	3	,	,	PUNCT
ejpam-4698	132	4	p.	p.	PROPN
ejpam-4698	132	5	rashed	rashed	PROPN
ejpam-4698	132	6	,	,	PUNCT
ejpam-4698	132	7	a.	a.	PROPN
ejpam-4698	132	8	ali	ali	PROPN
ejpam-4698	132	9	/	/	SYM
ejpam-4698	132	10	eur	eur	PROPN
ejpam-4698	132	11	.	.	PUNCT
ejpam-4698	133	1	j.	j.	PROPN
ejpam-4698	133	2	pure	pure	PROPN
ejpam-4698	133	3	appl	appl	PROPN
ejpam-4698	133	4	.	.	PROPN
ejpam-4698	133	5	math	math	PROPN
ejpam-4698	133	6	,	,	PUNCT
ejpam-4698	133	7	16	16	NUM
ejpam-4698	133	8	(	(	PUNCT
ejpam-4698	133	9	2	2	NUM
ejpam-4698	133	10	)	)	PUNCT
ejpam-4698	133	11	(	(	PUNCT
ejpam-4698	133	12	2023	2023	NUM
ejpam-4698	133	13	)	)	PUNCT
ejpam-4698	133	14	,	,	PUNCT
ejpam-4698	133	15	773	773	NUM
ejpam-4698	133	16	-	-	SYM
ejpam-4698	133	17	783	783	NUM
ejpam-4698	133	18	779	779	NUM
ejpam-4698	133	19	mve(g;x	mve(g;x	PROPN
ejpam-4698	133	20	,	,	PUNCT
ejpam-4698	133	21	y	y	NOUN
ejpam-4698	133	22	)	)	PUNCT
ejpam-4698	133	23	=	=	PRON
ejpam-4698	133	24	qxr	qxr	VERB
ejpam-4698	133	25	2−1yr	2−1yr	NUM
ejpam-4698	133	26	2−1	2−1	NUM
ejpam-4698	133	27	,	,	PUNCT
ejpam-4698	133	28	where	where	SCONJ
ejpam-4698	133	29	q	q	NOUN
ejpam-4698	133	30	=	=	SYM
ejpam-4698	133	31	|e(g)|	|e(g)|	NOUN
ejpam-4698	133	32	.	.	PUNCT
ejpam-4698	133	33	proof	proof	NOUN
ejpam-4698	133	34	:	:	PUNCT
ejpam-4698	133	35	let	let	VERB
ejpam-4698	133	36	e	e	NOUN
ejpam-4698	133	37	=	=	NOUN
ejpam-4698	133	38	uv	uv	NOUN
ejpam-4698	133	39	be	be	AUX
ejpam-4698	133	40	any	any	DET
ejpam-4698	133	41	edge	edge	NOUN
ejpam-4698	133	42	in	in	ADP
ejpam-4698	133	43	g	g	PROPN
ejpam-4698	134	1	where	where	SCONJ
ejpam-4698	134	2	u	u	NOUN
ejpam-4698	134	3	,	,	PUNCT
ejpam-4698	134	4	v	v	PROPN
ejpam-4698	134	5	∈	∈	PROPN
ejpam-4698	134	6	v	v	NOUN
ejpam-4698	134	7	(	(	PUNCT
ejpam-4698	134	8	g	g	NOUN
ejpam-4698	134	9	)	)	PUNCT
ejpam-4698	134	10	,	,	PUNCT
ejpam-4698	134	11	then	then	ADV
ejpam-4698	134	12	the	the	DET
ejpam-4698	134	13	two	two	NUM
ejpam-4698	134	14	vertices	vertex	NOUN
ejpam-4698	134	15	u	u	NOUN
ejpam-4698	134	16	and	and	CCONJ
ejpam-4698	134	17	v	v	NOUN
ejpam-4698	134	18	are	be	AUX
ejpam-4698	134	19	adjacent	adjacent	ADJ
ejpam-4698	134	20	with	with	ADP
ejpam-4698	134	21	third	third	ADJ
ejpam-4698	134	22	vertex	vertex	NOUN
ejpam-4698	134	23	,	,	PUNCT
ejpam-4698	134	24	then	then	ADV
ejpam-4698	134	25	τu	τu	ADP
ejpam-4698	134	26	=	=	NOUN
ejpam-4698	134	27	τv	τv	NOUN
ejpam-4698	134	28	=	=	PROPN
ejpam-4698	134	29	r2	r2	PROPN
ejpam-4698	134	30	−	−	PROPN
ejpam-4698	134	31	1	1	NUM
ejpam-4698	134	32	.	.	PUNCT
ejpam-4698	135	1	hence	hence	ADV
ejpam-4698	135	2	mve(g;x	mve(g;x	PROPN
ejpam-4698	135	3	,	,	PUNCT
ejpam-4698	135	4	y	y	NOUN
ejpam-4698	135	5	)	)	PUNCT
ejpam-4698	136	1	=	=	PRON
ejpam-4698	136	2	qxr	qxr	VERB
ejpam-4698	136	3	2−1yr	2−1yr	NUM
ejpam-4698	136	4	2−1	2−1	NUM
ejpam-4698	136	5	,	,	PUNCT
ejpam-4698	136	6	where	where	SCONJ
ejpam-4698	136	7	q	q	NOUN
ejpam-4698	136	8	=	=	SYM
ejpam-4698	136	9	|e(g)|	|e(g)|	NOUN
ejpam-4698	136	10	.	.	PROPN
ejpam-4698	136	11	theorem	theorem	VERB
ejpam-4698	136	12	3.4	3.4	NUM
ejpam-4698	136	13	:	:	PUNCT
ejpam-4698	136	14	let	let	VERB
ejpam-4698	136	15	g	g	PROPN
ejpam-4698	136	16	is	be	AUX
ejpam-4698	136	17	a	a	DET
ejpam-4698	136	18	r−regular	r−regular	ADJ
ejpam-4698	136	19	simple	simple	ADJ
ejpam-4698	136	20	graph	graph	NOUN
ejpam-4698	136	21	such	such	ADJ
ejpam-4698	136	22	that	that	SCONJ
ejpam-4698	136	23	any	any	DET
ejpam-4698	136	24	vertex	vertex	NOUN
ejpam-4698	136	25	in	in	ADP
ejpam-4698	136	26	g	g	PROPN
ejpam-4698	136	27	lies	lie	VERB
ejpam-4698	136	28	at	at	ADP
ejpam-4698	136	29	most	most	ADV
ejpam-4698	136	30	on	on	ADP
ejpam-4698	136	31	one	one	NUM
ejpam-4698	136	32	triangular	triangular	NOUN
ejpam-4698	136	33	cycle	cycle	NOUN
ejpam-4698	136	34	.	.	PUNCT
ejpam-4698	137	1	if	if	SCONJ
ejpam-4698	137	2	h	h	NOUN
ejpam-4698	137	3	be	be	VERB
ejpam-4698	137	4	the	the	DET
ejpam-4698	137	5	number	number	NOUN
ejpam-4698	137	6	of	of	ADP
ejpam-4698	137	7	edges	edge	NOUN
ejpam-4698	137	8	lies	lie	VERB
ejpam-4698	137	9	between	between	ADP
ejpam-4698	137	10	any	any	DET
ejpam-4698	137	11	two	two	NUM
ejpam-4698	137	12	vertices	vertex	NOUN
ejpam-4698	137	13	belong	belong	VERB
ejpam-4698	137	14	to	to	ADP
ejpam-4698	137	15	triangular	triangular	NOUN
ejpam-4698	137	16	cycle	cycle	NOUN
ejpam-4698	137	17	(	(	PUNCT
ejpam-4698	137	18	or	or	CCONJ
ejpam-4698	137	19	cycles	cycle	NOUN
ejpam-4698	137	20	)	)	PUNCT
ejpam-4698	137	21	and	and	CCONJ
ejpam-4698	137	22	k	k	PROPN
ejpam-4698	137	23	be	be	AUX
ejpam-4698	137	24	the	the	DET
ejpam-4698	137	25	number	number	NOUN
ejpam-4698	137	26	of	of	ADP
ejpam-4698	137	27	edges	edge	NOUN
ejpam-4698	137	28	which	which	DET
ejpam-4698	137	29	one	one	NUM
ejpam-4698	137	30	of	of	ADP
ejpam-4698	137	31	whose	whose	DET
ejpam-4698	137	32	ends	end	VERB
ejpam-4698	137	33	,	,	PUNCT
ejpam-4698	137	34	but	but	CCONJ
ejpam-4698	137	35	not	not	PART
ejpam-4698	137	36	both	both	PRON
ejpam-4698	137	37	lies	lie	VERB
ejpam-4698	137	38	on	on	ADP
ejpam-4698	137	39	a	a	DET
ejpam-4698	137	40	triangular	triangular	NOUN
ejpam-4698	137	41	cycle	cycle	NOUN
ejpam-4698	137	42	.	.	PUNCT
ejpam-4698	138	1	then	then	ADV
ejpam-4698	138	2	mve(g;x	mve(g;x	PROPN
ejpam-4698	138	3	,	,	PUNCT
ejpam-4698	138	4	y	y	NOUN
ejpam-4698	138	5	)	)	PUNCT
ejpam-4698	138	6	=	=	SYM
ejpam-4698	138	7	hxr	hxr	NOUN
ejpam-4698	138	8	2−1yr	2−1yr	NUM
ejpam-4698	138	9	2−1	2−1	NUM
ejpam-4698	139	1	+	+	CCONJ
ejpam-4698	140	1	kxr	kxr	NOUN
ejpam-4698	140	2	2−1yr	2−1yr	NUM
ejpam-4698	140	3	2	2	NUM
ejpam-4698	140	4	+	+	CCONJ
ejpam-4698	140	5	(	(	PUNCT
ejpam-4698	140	6	q	q	NOUN
ejpam-4698	140	7	−	−	PROPN
ejpam-4698	140	8	h−	h−	PROPN
ejpam-4698	140	9	k)xr	k)xr	PROPN
ejpam-4698	140	10	2	2	NUM
ejpam-4698	140	11	yr	yr	NOUN
ejpam-4698	140	12	2	2	NUM
ejpam-4698	140	13	,	,	PUNCT
ejpam-4698	140	14	where	where	SCONJ
ejpam-4698	140	15	q	q	NOUN
ejpam-4698	140	16	=	=	SYM
ejpam-4698	140	17	|e(g)|	|e(g)|	NOUN
ejpam-4698	140	18	.	.	PUNCT
ejpam-4698	140	19	proof	proof	NOUN
ejpam-4698	140	20	:	:	PUNCT
ejpam-4698	140	21	obvious	obvious	ADJ
ejpam-4698	140	22	.	.	PUNCT
ejpam-4698	141	1	definition	definition	NOUN
ejpam-4698	141	2	3.5	3.5	NUM
ejpam-4698	141	3	:	:	PUNCT
ejpam-4698	141	4	let	let	VERB
ejpam-4698	141	5	g	g	PRON
ejpam-4698	141	6	be	be	AUX
ejpam-4698	141	7	a	a	DET
ejpam-4698	141	8	simple	simple	ADJ
ejpam-4698	141	9	graph	graph	NOUN
ejpam-4698	141	10	and	and	CCONJ
ejpam-4698	141	11	a	a	DET
ejpam-4698	141	12	vertex	vertex	NOUN
ejpam-4698	141	13	u	u	NOUN
ejpam-4698	141	14	∈	∈	PROPN
ejpam-4698	141	15	v	v	NOUN
ejpam-4698	141	16	(	(	PUNCT
ejpam-4698	141	17	g	g	NOUN
ejpam-4698	141	18	)	)	PUNCT
ejpam-4698	141	19	,	,	PUNCT
ejpam-4698	141	20	we	we	PRON
ejpam-4698	141	21	can	can	AUX
ejpam-4698	141	22	rewrite	rewrite	VERB
ejpam-4698	141	23	δu	δu	PRON
ejpam-4698	141	24	as	as	ADP
ejpam-4698	141	25	:	:	PUNCT
ejpam-4698	141	26	δu	δu	ADP
ejpam-4698	141	27	=	=	SYM
ejpam-4698	141	28	σz∈n(u)dz	σz∈n(u)dz	PROPN
ejpam-4698	141	29	=	=	SYM
ejpam-4698	141	30	du	du	NOUN
ejpam-4698	141	31	+	+	NUM
ejpam-4698	141	32	2ε1	2ε1	NUM
ejpam-4698	141	33	+	+	CCONJ
ejpam-4698	141	34	ε2	ε2	ADJ
ejpam-4698	141	35	,	,	PUNCT
ejpam-4698	141	36	where	where	SCONJ
ejpam-4698	141	37	ε1	ε1	PROPN
ejpam-4698	141	38	be	be	AUX
ejpam-4698	141	39	the	the	DET
ejpam-4698	141	40	number	number	NOUN
ejpam-4698	141	41	of	of	ADP
ejpam-4698	141	42	edges	edge	NOUN
ejpam-4698	141	43	of	of	ADP
ejpam-4698	141	44	which	which	PRON
ejpam-4698	141	45	both	both	DET
ejpam-4698	141	46	its	its	PRON
ejpam-4698	141	47	ends	end	NOUN
ejpam-4698	141	48	belong	belong	VERB
ejpam-4698	141	49	to	to	ADP
ejpam-4698	141	50	a	a	DET
ejpam-4698	141	51	triangular	triangular	NOUN
ejpam-4698	141	52	cycle	cycle	NOUN
ejpam-4698	141	53	and	and	CCONJ
ejpam-4698	141	54	ε2	ε2	NOUN
ejpam-4698	141	55	be	be	AUX
ejpam-4698	141	56	the	the	DET
ejpam-4698	141	57	number	number	NOUN
ejpam-4698	141	58	of	of	ADP
ejpam-4698	141	59	edges	edge	NOUN
ejpam-4698	141	60	which	which	PRON
ejpam-4698	141	61	lies	lie	VERB
ejpam-4698	141	62	on	on	ADP
ejpam-4698	141	63	the	the	DET
ejpam-4698	141	64	vertices	vertex	NOUN
ejpam-4698	141	65	are	be	AUX
ejpam-4698	141	66	neighbors	neighbor	NOUN
ejpam-4698	141	67	of	of	ADP
ejpam-4698	141	68	a	a	DET
ejpam-4698	141	69	vertex	vertex	NOUN
ejpam-4698	141	70	u.	u.	NOUN
ejpam-4698	141	71	theorem	theorem	VERB
ejpam-4698	141	72	3.6	3.6	NUM
ejpam-4698	141	73	:	:	PUNCT
ejpam-4698	141	74	let	let	VERB
ejpam-4698	141	75	g	g	PRON
ejpam-4698	141	76	be	be	AUX
ejpam-4698	141	77	a	a	DET
ejpam-4698	141	78	any	any	DET
ejpam-4698	141	79	graph	graph	NOUN
ejpam-4698	141	80	without	without	ADP
ejpam-4698	141	81	triangular	triangular	NOUN
ejpam-4698	141	82	cycle	cycle	NOUN
ejpam-4698	141	83	,	,	PUNCT
ejpam-4698	141	84	then	then	ADV
ejpam-4698	141	85	,	,	PUNCT
ejpam-4698	141	86	nm(g;x	nm(g;x	PROPN
ejpam-4698	141	87	,	,	PUNCT
ejpam-4698	141	88	y	y	NOUN
ejpam-4698	141	89	)	)	PUNCT
ejpam-4698	141	90	=	=	SYM
ejpam-4698	142	1	mve(g;x	mve(g;x	PROPN
ejpam-4698	142	2	,	,	PUNCT
ejpam-4698	142	3	y	y	PROPN
ejpam-4698	142	4	)	)	PUNCT
ejpam-4698	142	5	.	.	PUNCT
ejpam-4698	143	1	proof	proof	NOUN
ejpam-4698	143	2	:	:	PUNCT
ejpam-4698	143	3	for	for	ADP
ejpam-4698	143	4	edge	edge	NOUN
ejpam-4698	143	5	e	e	NOUN
ejpam-4698	143	6	=	=	NOUN
ejpam-4698	143	7	uv	uv	NOUN
ejpam-4698	143	8	where	where	SCONJ
ejpam-4698	143	9	u	u	NOUN
ejpam-4698	143	10	,	,	PUNCT
ejpam-4698	143	11	v	v	PROPN
ejpam-4698	143	12	∈	∈	PROPN
ejpam-4698	143	13	v	v	NOUN
ejpam-4698	143	14	(	(	PUNCT
ejpam-4698	143	15	g	g	NOUN
ejpam-4698	143	16	)	)	PUNCT
ejpam-4698	143	17	,	,	PUNCT
ejpam-4698	143	18	then	then	ADV
ejpam-4698	143	19	m′	m′	NOUN
ejpam-4698	143	20	δuδv	δuδv	VERB
ejpam-4698	143	21	xδuyδv	xδuyδv	PROPN
ejpam-4698	144	1	=	=	SYM
ejpam-4698	144	2	m′	m′	X
ejpam-4698	144	3	δuδv	δuδv	VERB
ejpam-4698	144	4	xdu+2ε1+ε2ydv+2ε1+ε2	xdu+2ε1+ε2ydv+2ε1+ε2	PROPN
ejpam-4698	144	5	,	,	PUNCT
ejpam-4698	144	6	by	by	ADP
ejpam-4698	144	7	definition	definition	NOUN
ejpam-4698	144	8	3.5	3.5	NUM
ejpam-4698	144	9	.	.	PUNCT
ejpam-4698	145	1	since	since	SCONJ
ejpam-4698	145	2	g	g	PROPN
ejpam-4698	145	3	is	be	AUX
ejpam-4698	145	4	a	a	DET
ejpam-4698	145	5	graph	graph	NOUN
ejpam-4698	145	6	without	without	ADP
ejpam-4698	145	7	triangular	triangular	NOUN
ejpam-4698	145	8	cycles	cycle	NOUN
ejpam-4698	145	9	,	,	PUNCT
ejpam-4698	145	10	then	then	ADV
ejpam-4698	145	11	ε1	ε1	VERB
ejpam-4698	145	12	=	=	SYM
ejpam-4698	145	13	0	0	X
ejpam-4698	145	14	.	.	PUNCT
ejpam-4698	146	1	hence	hence	ADV
ejpam-4698	146	2	m′	m′	NUM
ejpam-4698	146	3	δuδv	δuδv	VERB
ejpam-4698	146	4	xδuyδv	xδuyδv	PROPN
ejpam-4698	146	5	=	=	SYM
ejpam-4698	146	6	m′	m′	X
ejpam-4698	146	7	δuδv	δuδv	VERB
ejpam-4698	146	8	xdu+ε2ydv+ε2	xdu+ε2ydv+ε2	PROPN
ejpam-4698	147	1	=	=	PUNCT
ejpam-4698	147	2	cδuδvx	cδuδvx	ADP
ejpam-4698	147	3	τuyτv	τuyτv	NOUN
ejpam-4698	147	4	,	,	PUNCT
ejpam-4698	147	5	by	by	ADP
ejpam-4698	147	6	definition	definition	NOUN
ejpam-4698	147	7	vertex	vertex	NOUN
ejpam-4698	147	8	–	–	PUNCT
ejpam-4698	147	9	edge	edge	NOUN
ejpam-4698	147	10	degree	degree	NOUN
ejpam-4698	147	11	of	of	ADP
ejpam-4698	147	12	the	the	DET
ejpam-4698	147	13	graph	graph	NOUN
ejpam-4698	147	14	.	.	PUNCT
ejpam-4698	148	1	hence	hence	ADV
ejpam-4698	148	2	,	,	PUNCT
ejpam-4698	148	3	nm(g;x	nm(g;x	PROPN
ejpam-4698	148	4	,	,	PUNCT
ejpam-4698	148	5	y	y	NOUN
ejpam-4698	148	6	)	)	PUNCT
ejpam-4698	148	7	=	=	SYM
ejpam-4698	149	1	mve(g;x	mve(g;x	PROPN
ejpam-4698	149	2	,	,	PUNCT
ejpam-4698	149	3	y	y	PROPN
ejpam-4698	149	4	)	)	PUNCT
ejpam-4698	149	5	.	.	PUNCT
ejpam-4698	150	1	4	4	X
ejpam-4698	150	2	.	.	NUM
ejpam-4698	150	3	mve−polynomials	mve−polynomial	NOUN
ejpam-4698	150	4	of	of	ADP
ejpam-4698	150	5	2−ary	2−ary	ADJ
ejpam-4698	150	6	tree	tree	NOUN
ejpam-4698	150	7	graph	graph	NOUN
ejpam-4698	150	8	definition	definition	NOUN
ejpam-4698	150	9	4.1	4.1	NUM
ejpam-4698	150	10	:	:	PUNCT
ejpam-4698	151	1	[	[	X
ejpam-4698	151	2	16	16	NUM
ejpam-4698	151	3	]	]	PUNCT
ejpam-4698	151	4	a	a	DET
ejpam-4698	151	5	rooted	rooted	ADJ
ejpam-4698	151	6	tree	tree	NOUN
ejpam-4698	151	7	g	g	PROPN
ejpam-4698	151	8	is	be	AUX
ejpam-4698	151	9	an	an	DET
ejpam-4698	151	10	acyclic	acyclic	ADJ
ejpam-4698	151	11	connected	connect	VERB
ejpam-4698	151	12	graph	graph	NOUN
ejpam-4698	151	13	with	with	ADP
ejpam-4698	151	14	a	a	DET
ejpam-4698	151	15	special	special	ADJ
ejpam-4698	151	16	node	node	NOUN
ejpam-4698	151	17	that	that	PRON
ejpam-4698	151	18	is	be	AUX
ejpam-4698	151	19	called	call	VERB
ejpam-4698	151	20	the	the	DET
ejpam-4698	151	21	root	root	NOUN
ejpam-4698	151	22	of	of	ADP
ejpam-4698	151	23	the	the	DET
ejpam-4698	151	24	tree	tree	NOUN
ejpam-4698	151	25	and	and	CCONJ
ejpam-4698	151	26	every	every	DET
ejpam-4698	151	27	edge	edge	NOUN
ejpam-4698	151	28	directly	directly	ADV
ejpam-4698	151	29	or	or	CCONJ
ejpam-4698	151	30	indirectly	indirectly	ADV
ejpam-4698	151	31	originates	originate	NOUN
ejpam-4698	151	32	from	from	ADP
ejpam-4698	151	33	the	the	DET
ejpam-4698	151	34	k.	k.	PROPN
ejpam-4698	151	35	rasool	rasool	PROPN
ejpam-4698	151	36	,	,	PUNCT
ejpam-4698	151	37	p.	p.	PROPN
ejpam-4698	151	38	rashed	rashed	PROPN
ejpam-4698	151	39	,	,	PUNCT
ejpam-4698	151	40	a.	a.	PROPN
ejpam-4698	151	41	ali	ali	PROPN
ejpam-4698	151	42	/	/	SYM
ejpam-4698	151	43	eur	eur	PROPN
ejpam-4698	151	44	.	.	PUNCT
ejpam-4698	152	1	j.	j.	PROPN
ejpam-4698	152	2	pure	pure	PROPN
ejpam-4698	152	3	appl	appl	PROPN
ejpam-4698	152	4	.	.	PROPN
ejpam-4698	152	5	math	math	PROPN
ejpam-4698	152	6	,	,	PUNCT
ejpam-4698	152	7	16	16	NUM
ejpam-4698	152	8	(	(	PUNCT
ejpam-4698	152	9	2	2	NUM
ejpam-4698	152	10	)	)	PUNCT
ejpam-4698	152	11	(	(	PUNCT
ejpam-4698	152	12	2023	2023	NUM
ejpam-4698	152	13	)	)	PUNCT
ejpam-4698	152	14	,	,	PUNCT
ejpam-4698	152	15	773	773	NUM
ejpam-4698	152	16	-	-	SYM
ejpam-4698	152	17	783	783	NUM
ejpam-4698	152	18	780	780	NUM
ejpam-4698	152	19	root	root	NOUN
ejpam-4698	152	20	.	.	PUNCT
ejpam-4698	153	1	an	an	DET
ejpam-4698	153	2	ordered	order	VERB
ejpam-4698	153	3	rooted	rooted	ADJ
ejpam-4698	153	4	tree	tree	NOUN
ejpam-4698	153	5	is	be	AUX
ejpam-4698	153	6	a	a	DET
ejpam-4698	153	7	rooted	rooted	ADJ
ejpam-4698	153	8	tree	tree	NOUN
ejpam-4698	153	9	where	where	SCONJ
ejpam-4698	153	10	the	the	DET
ejpam-4698	153	11	children	child	NOUN
ejpam-4698	153	12	of	of	ADP
ejpam-4698	153	13	each	each	DET
ejpam-4698	153	14	internal	internal	ADJ
ejpam-4698	153	15	vertex	vertex	NOUN
ejpam-4698	153	16	is	be	AUX
ejpam-4698	153	17	ordered	order	VERB
ejpam-4698	153	18	.	.	PUNCT
ejpam-4698	154	1	if	if	SCONJ
ejpam-4698	154	2	every	every	DET
ejpam-4698	154	3	internal	internal	ADJ
ejpam-4698	154	4	vertex	vertex	NOUN
ejpam-4698	154	5	of	of	ADP
ejpam-4698	154	6	a	a	DET
ejpam-4698	154	7	rooted	rooted	ADJ
ejpam-4698	154	8	tree	tree	NOUN
ejpam-4698	154	9	has	have	VERB
ejpam-4698	154	10	not	not	PART
ejpam-4698	154	11	more	more	ADJ
ejpam-4698	154	12	than	than	ADP
ejpam-4698	154	13	m	m	PROPN
ejpam-4698	154	14	children	child	NOUN
ejpam-4698	154	15	,	,	PUNCT
ejpam-4698	154	16	it	it	PRON
ejpam-4698	154	17	is	be	AUX
ejpam-4698	154	18	called	call	VERB
ejpam-4698	154	19	an	an	DET
ejpam-4698	154	20	m−ary	m−ary	ADJ
ejpam-4698	154	21	tree	tree	NOUN
ejpam-4698	154	22	.	.	PUNCT
ejpam-4698	155	1	in	in	ADP
ejpam-4698	155	2	this	this	DET
ejpam-4698	155	3	section	section	NOUN
ejpam-4698	155	4	,	,	PUNCT
ejpam-4698	155	5	we	we	PRON
ejpam-4698	155	6	determine	determine	VERB
ejpam-4698	155	7	the	the	DET
ejpam-4698	155	8	mve−polynomial	mve−polynomial	ADJ
ejpam-4698	155	9	of	of	ADP
ejpam-4698	155	10	special	special	ADJ
ejpam-4698	155	11	case	case	NOUN
ejpam-4698	155	12	of	of	ADP
ejpam-4698	155	13	m−ary	m−ary	ADJ
ejpam-4698	155	14	tree	tree	NOUN
ejpam-4698	155	15	is	be	AUX
ejpam-4698	155	16	2−ary	2−ary	ADJ
ejpam-4698	155	17	tree	tree	NOUN
ejpam-4698	155	18	of	of	ADP
ejpam-4698	155	19	n	n	NOUN
ejpam-4698	155	20	levels	level	NOUN
ejpam-4698	155	21	denoted	denote	VERB
ejpam-4698	155	22	by	by	ADP
ejpam-4698	155	23	ℵn	ℵn	NOUN
ejpam-4698	155	24	and	and	CCONJ
ejpam-4698	155	25	as	as	SCONJ
ejpam-4698	155	26	shown	show	VERB
ejpam-4698	155	27	in	in	ADP
ejpam-4698	155	28	figure	figure	NOUN
ejpam-4698	155	29	2	2	NUM
ejpam-4698	155	30	.	.	PUNCT
ejpam-4698	155	31	figure	figure	NOUN
ejpam-4698	155	32	2	2	NUM
ejpam-4698	155	33	:	:	PUNCT
ejpam-4698	155	34	2−ary	2−ary	ADJ
ejpam-4698	155	35	tree	tree	NOUN
ejpam-4698	155	36	graph	graph	NOUN
ejpam-4698	155	37	ℵn	ℵn	NOUN
ejpam-4698	155	38	.	.	PUNCT
ejpam-4698	156	1	some	some	DET
ejpam-4698	156	2	properties	property	NOUN
ejpam-4698	156	3	of	of	ADP
ejpam-4698	156	4	a	a	DET
ejpam-4698	156	5	2−ary	2−ary	ADJ
ejpam-4698	156	6	tree	tree	NOUN
ejpam-4698	156	7	graph	graph	NOUN
ejpam-4698	156	8	ℵn	ℵn	NOUN
ejpam-4698	156	9	:	:	PUNCT
ejpam-4698	156	10	•	•	NOUN
ejpam-4698	156	11	at	at	ADP
ejpam-4698	156	12	each	each	DET
ejpam-4698	156	13	level	level	NOUN
ejpam-4698	156	14	of	of	ADP
ejpam-4698	156	15	i	i	PRON
ejpam-4698	156	16	,	,	PUNCT
ejpam-4698	156	17	the	the	DET
ejpam-4698	156	18	number	number	NOUN
ejpam-4698	156	19	of	of	ADP
ejpam-4698	156	20	vertices	vertex	NOUN
ejpam-4698	156	21	are	be	AUX
ejpam-4698	156	22	2i	2i	NUM
ejpam-4698	156	23	,	,	PUNCT
ejpam-4698	156	24	for	for	ADP
ejpam-4698	156	25	0	0	NUM
ejpam-4698	156	26	≤	≤	NUM
ejpam-4698	156	27	i	i	PRON
ejpam-4698	156	28	≤	≤	ADJ
ejpam-4698	156	29	n.	n.	NOUN
ejpam-4698	156	30	•	•	ADP
ejpam-4698	156	31	the	the	DET
ejpam-4698	156	32	order	order	NOUN
ejpam-4698	156	33	and	and	CCONJ
ejpam-4698	156	34	the	the	DET
ejpam-4698	156	35	size	size	NOUN
ejpam-4698	156	36	are	be	AUX
ejpam-4698	156	37	p(ℵn	p(ℵn	NOUN
ejpam-4698	156	38	)	)	PUNCT
ejpam-4698	156	39	=	=	SYM
ejpam-4698	156	40	2n+1	2n+1	NOUN
ejpam-4698	156	41	−	−	NOUN
ejpam-4698	156	42	1	1	NUM
ejpam-4698	156	43	and	and	CCONJ
ejpam-4698	156	44	q(ℵn	q(ℵn	NOUN
ejpam-4698	156	45	)	)	PUNCT
ejpam-4698	156	46	=	=	SYM
ejpam-4698	156	47	2n+1	2n+1	NOUN
ejpam-4698	156	48	−	−	NOUN
ejpam-4698	156	49	2	2	NUM
ejpam-4698	156	50	,	,	PUNCT
ejpam-4698	156	51	respectively	respectively	ADV
ejpam-4698	156	52	.	.	PUNCT
ejpam-4698	157	1	•	•	NUM
ejpam-4698	157	2	the	the	DET
ejpam-4698	157	3	rooted	root	VERB
ejpam-4698	157	4	vertex	vertex	NOUN
ejpam-4698	157	5	of	of	ADP
ejpam-4698	157	6	degree	degree	NOUN
ejpam-4698	157	7	2	2	NUM
ejpam-4698	157	8	,	,	PUNCT
ejpam-4698	157	9	the	the	DET
ejpam-4698	157	10	degrees	degree	NOUN
ejpam-4698	157	11	of	of	ADP
ejpam-4698	157	12	vertices	vertex	NOUN
ejpam-4698	157	13	at	at	ADP
ejpam-4698	157	14	each	each	DET
ejpam-4698	157	15	level	level	NOUN
ejpam-4698	157	16	of	of	ADP
ejpam-4698	157	17	i	i	PRON
ejpam-4698	157	18	,	,	PUNCT
ejpam-4698	157	19	1	1	NUM
ejpam-4698	157	20	≤	≤	NUM
ejpam-4698	157	21	i	i	X
ejpam-4698	157	22	≤	≤	PROPN
ejpam-4698	157	23	n−1	n−1	PROPN
ejpam-4698	157	24	are	be	AUX
ejpam-4698	157	25	3	3	NUM
ejpam-4698	157	26	which	which	PRON
ejpam-4698	157	27	represent	represent	VERB
ejpam-4698	157	28	the	the	DET
ejpam-4698	157	29	maximum	maximum	ADJ
ejpam-4698	157	30	degree	degree	NOUN
ejpam-4698	157	31	”	"	PUNCT
ejpam-4698	157	32	△	△	X
ejpam-4698	157	33	(	(	PUNCT
ejpam-4698	157	34	ℵn	ℵn	NOUN
ejpam-4698	157	35	)	)	PUNCT
ejpam-4698	157	36	=	=	SYM
ejpam-4698	157	37	3	3	X
ejpam-4698	157	38	”	"	PUNCT
ejpam-4698	157	39	and	and	CCONJ
ejpam-4698	157	40	the	the	DET
ejpam-4698	157	41	degrees	degree	NOUN
ejpam-4698	157	42	of	of	ADP
ejpam-4698	157	43	the	the	DET
ejpam-4698	157	44	last	last	ADJ
ejpam-4698	157	45	level	level	NOUN
ejpam-4698	157	46	are	be	AUX
ejpam-4698	157	47	1	1	NUM
ejpam-4698	157	48	which	which	PRON
ejpam-4698	157	49	represent	represent	VERB
ejpam-4698	157	50	the	the	DET
ejpam-4698	157	51	minimum	minimum	NOUN
ejpam-4698	157	52	degree	degree	NOUN
ejpam-4698	157	53	”	"	PUNCT
ejpam-4698	157	54	δ(ℵn	δ(ℵn	NOUN
ejpam-4698	157	55	)	)	PUNCT
ejpam-4698	157	56	=	=	SYM
ejpam-4698	157	57	1	1	NUM
ejpam-4698	157	58	”	"	PUNCT
ejpam-4698	157	59	.	.	PUNCT
ejpam-4698	158	1	•	•	NUM
ejpam-4698	158	2	the	the	DET
ejpam-4698	158	3	maximum	maximum	ADJ
ejpam-4698	158	4	and	and	CCONJ
ejpam-4698	158	5	minimum	minimum	NOUN
ejpam-4698	158	6	ve−degree	ve−degree	NUM
ejpam-4698	158	7	are	be	AUX
ejpam-4698	158	8	△	△	X
ejpam-4698	158	9	ve	ve	X
ejpam-4698	158	10	(	(	PUNCT
ejpam-4698	158	11	ℵn	ℵn	NOUN
ejpam-4698	158	12	)	)	PUNCT
ejpam-4698	158	13	=	=	SYM
ejpam-4698	158	14	9	9	NUM
ejpam-4698	158	15	and	and	CCONJ
ejpam-4698	158	16	δve(ℵn	δve(ℵn	NOUN
ejpam-4698	158	17	)	)	PUNCT
ejpam-4698	158	18	=	=	SYM
ejpam-4698	158	19	3	3	NUM
ejpam-4698	158	20	,	,	PUNCT
ejpam-4698	158	21	respectively	respectively	ADV
ejpam-4698	158	22	.	.	PUNCT
ejpam-4698	159	1	theorem	theorem	VERB
ejpam-4698	159	2	4.2	4.2	NUM
ejpam-4698	159	3	:	:	PUNCT
ejpam-4698	159	4	let	let	VERB
ejpam-4698	159	5	ℵn	ℵn	PART
ejpam-4698	159	6	be	be	AUX
ejpam-4698	159	7	the	the	DET
ejpam-4698	159	8	2−ary	2−ary	ADJ
ejpam-4698	159	9	tree	tree	NOUN
ejpam-4698	159	10	graph	graph	NOUN
ejpam-4698	159	11	of	of	ADP
ejpam-4698	159	12	order	order	NOUN
ejpam-4698	159	13	2n+1	2n+1	NOUN
ejpam-4698	159	14	−	−	NOUN
ejpam-4698	159	15	1	1	NUM
ejpam-4698	159	16	,	,	PUNCT
ejpam-4698	159	17	n	n	PRON
ejpam-4698	159	18	≥	≥	NOUN
ejpam-4698	159	19	4	4	NUM
ejpam-4698	159	20	.	.	PUNCT
ejpam-4698	160	1	then	then	ADV
ejpam-4698	160	2	,	,	PUNCT
ejpam-4698	160	3	mve(ℵn;x	mve(ℵn;x	PROPN
ejpam-4698	160	4	,	,	PUNCT
ejpam-4698	160	5	y	y	NOUN
ejpam-4698	160	6	)	)	PUNCT
ejpam-4698	160	7	=	=	PUNCT
ejpam-4698	161	1	2x6y8	2x6y8	NUM
ejpam-4698	162	1	+	+	CCONJ
ejpam-4698	162	2	4x8y9	4x8y9	NOUN
ejpam-4698	162	3	+	+	CCONJ
ejpam-4698	162	4	2nx3y5	2nx3y5	NUM
ejpam-4698	162	5	+	+	NUM
ejpam-4698	162	6	2n−1x5y9	2n−1x5y9	NUM
ejpam-4698	162	7	+	+	CCONJ
ejpam-4698	162	8	(	(	PUNCT
ejpam-4698	162	9	2n−1	2n−1	NUM
ejpam-4698	162	10	−	−	PROPN
ejpam-4698	162	11	8)x9y9	8)x9y9	NOUN
ejpam-4698	162	12	.	.	PUNCT
ejpam-4698	163	1	proof	proof	NOUN
ejpam-4698	163	2	:	:	PUNCT
ejpam-4698	163	3	from	from	ADP
ejpam-4698	163	4	the	the	DET
ejpam-4698	163	5	definition	definition	NOUN
ejpam-4698	163	6	4.1	4.1	NUM
ejpam-4698	163	7	and	and	CCONJ
ejpam-4698	163	8	figure	figure	VERB
ejpam-4698	163	9	2	2	NUM
ejpam-4698	163	10	of	of	ADP
ejpam-4698	163	11	2−ary	2−ary	ADJ
ejpam-4698	163	12	tree	tree	NOUN
ejpam-4698	163	13	graph	graph	NOUN
ejpam-4698	163	14	ℵn	ℵn	NOUN
ejpam-4698	163	15	,	,	PUNCT
ejpam-4698	163	16	we	we	PRON
ejpam-4698	163	17	can	can	AUX
ejpam-4698	163	18	observe	observe	VERB
ejpam-4698	163	19	that	that	SCONJ
ejpam-4698	163	20	the	the	DET
ejpam-4698	163	21	vertices	vertex	NOUN
ejpam-4698	163	22	are	be	AUX
ejpam-4698	163	23	divided	divide	VERB
ejpam-4698	163	24	into	into	ADP
ejpam-4698	163	25	five	five	NUM
ejpam-4698	163	26	partitions	partition	NOUN
ejpam-4698	163	27	:	:	PUNCT
ejpam-4698	163	28	|v1|	|v1|	NOUN
ejpam-4698	163	29	=	=	PUNCT
ejpam-4698	163	30	|v	|v	PROPN
ejpam-4698	163	31	∈	∈	PROPN
ejpam-4698	163	32	v	v	ADP
ejpam-4698	163	33	(	(	PUNCT
ejpam-4698	163	34	ℵn	ℵn	NOUN
ejpam-4698	163	35	)	)	PUNCT
ejpam-4698	163	36	:	:	PUNCT
ejpam-4698	163	37	τv	τv	X
ejpam-4698	163	38	=	=	SYM
ejpam-4698	163	39	3|	3|	NUM
ejpam-4698	163	40	=	=	SYM
ejpam-4698	163	41	2n	2n	NUM
ejpam-4698	163	42	,	,	PUNCT
ejpam-4698	163	43	|v2|	|v2|	NOUN
ejpam-4698	163	44	=	=	PUNCT
ejpam-4698	163	45	|v	|v	PROPN
ejpam-4698	163	46	∈	∈	PROPN
ejpam-4698	163	47	v	v	ADP
ejpam-4698	163	48	(	(	PUNCT
ejpam-4698	163	49	ℵn	ℵn	NOUN
ejpam-4698	163	50	)	)	PUNCT
ejpam-4698	163	51	:	:	PUNCT
ejpam-4698	163	52	τv	τv	X
ejpam-4698	163	53	=	=	SYM
ejpam-4698	163	54	5|	5|	NUM
ejpam-4698	163	55	=	=	SYM
ejpam-4698	163	56	2n−1	2n−1	NUM
ejpam-4698	163	57	,	,	PUNCT
ejpam-4698	163	58	|v3|	|v3|	NOUN
ejpam-4698	163	59	=	=	SYM
ejpam-4698	163	60	|v	|v	PROPN
ejpam-4698	163	61	∈	∈	PROPN
ejpam-4698	163	62	v	v	ADP
ejpam-4698	163	63	(	(	PUNCT
ejpam-4698	163	64	ℵn	ℵn	NOUN
ejpam-4698	163	65	)	)	PUNCT
ejpam-4698	163	66	:	:	PUNCT
ejpam-4698	163	67	τv	τv	X
ejpam-4698	163	68	=	=	SYM
ejpam-4698	163	69	6|	6|	NUM
ejpam-4698	163	70	=	=	SYM
ejpam-4698	163	71	20	20	NUM
ejpam-4698	163	72	,	,	PUNCT
ejpam-4698	163	73	|v4|	|v4|	NOUN
ejpam-4698	163	74	=	=	SYM
ejpam-4698	163	75	|v	|v	PROPN
ejpam-4698	163	76	∈	∈	PROPN
ejpam-4698	163	77	v	v	ADP
ejpam-4698	163	78	(	(	PUNCT
ejpam-4698	163	79	ℵn	ℵn	NOUN
ejpam-4698	163	80	)	)	PUNCT
ejpam-4698	163	81	:	:	PUNCT
ejpam-4698	163	82	τv	τv	X
ejpam-4698	163	83	=	=	SYM
ejpam-4698	163	84	8|	8|	NUM
ejpam-4698	163	85	=	=	SYM
ejpam-4698	163	86	21	21	NUM
ejpam-4698	163	87	,	,	PUNCT
ejpam-4698	163	88	|v5|	|v5|	NOUN
ejpam-4698	163	89	=	=	SYM
ejpam-4698	163	90	|v	|v	PROPN
ejpam-4698	163	91	∈	∈	PROPN
ejpam-4698	163	92	v	v	ADP
ejpam-4698	163	93	(	(	PUNCT
ejpam-4698	163	94	ℵn	ℵn	NOUN
ejpam-4698	163	95	)	)	PUNCT
ejpam-4698	163	96	:	:	PUNCT
ejpam-4698	164	1	τv	τv	X
ejpam-4698	164	2	=	=	SYM
ejpam-4698	164	3	9|	9|	NUM
ejpam-4698	165	1	=	=	SYM
ejpam-4698	165	2	2n−1	2n−1	NUM
ejpam-4698	165	3	−	−	NOUN
ejpam-4698	165	4	4	4	NUM
ejpam-4698	165	5	.	.	PUNCT
ejpam-4698	165	6	k.	k.	PROPN
ejpam-4698	165	7	rasool	rasool	PROPN
ejpam-4698	165	8	,	,	PUNCT
ejpam-4698	165	9	p.	p.	PROPN
ejpam-4698	165	10	rashed	rashed	PROPN
ejpam-4698	165	11	,	,	PUNCT
ejpam-4698	165	12	a.	a.	PROPN
ejpam-4698	165	13	ali	ali	PROPN
ejpam-4698	165	14	/	/	SYM
ejpam-4698	165	15	eur	eur	PROPN
ejpam-4698	165	16	.	.	PUNCT
ejpam-4698	166	1	j.	j.	PROPN
ejpam-4698	166	2	pure	pure	PROPN
ejpam-4698	166	3	appl	appl	PROPN
ejpam-4698	166	4	.	.	PROPN
ejpam-4698	166	5	math	math	PROPN
ejpam-4698	166	6	,	,	PUNCT
ejpam-4698	166	7	16	16	NUM
ejpam-4698	166	8	(	(	PUNCT
ejpam-4698	166	9	2	2	NUM
ejpam-4698	166	10	)	)	PUNCT
ejpam-4698	166	11	(	(	PUNCT
ejpam-4698	166	12	2023	2023	NUM
ejpam-4698	166	13	)	)	PUNCT
ejpam-4698	166	14	,	,	PUNCT
ejpam-4698	166	15	773	773	NUM
ejpam-4698	166	16	-	-	SYM
ejpam-4698	166	17	783	783	NUM
ejpam-4698	166	18	781	781	NUM
ejpam-4698	166	19	the	the	DET
ejpam-4698	166	20	edges	edge	NOUN
ejpam-4698	166	21	set	set	VERB
ejpam-4698	166	22	of	of	ADP
ejpam-4698	166	23	2	2	NUM
ejpam-4698	166	24	−	−	PROPN
ejpam-4698	166	25	ary	ary	PROPN
ejpam-4698	166	26	tree	tree	NOUN
ejpam-4698	166	27	graph	graph	NOUN
ejpam-4698	166	28	ℵn	ℵn	NOUN
ejpam-4698	166	29	can	can	AUX
ejpam-4698	166	30	be	be	AUX
ejpam-4698	166	31	partitioned	partition	VERB
ejpam-4698	166	32	as	as	ADP
ejpam-4698	166	33	|e3,5|	|e3,5|	PROPN
ejpam-4698	166	34	=	=	PUNCT
ejpam-4698	166	35	|uv	|uv	PART
ejpam-4698	166	36	∈	∈	PROPN
ejpam-4698	166	37	e(ℵn	e(ℵn	NOUN
ejpam-4698	166	38	)	)	PUNCT
ejpam-4698	166	39	:	:	PUNCT
ejpam-4698	166	40	τu	τu	ADP
ejpam-4698	166	41	=	=	SYM
ejpam-4698	166	42	3	3	NUM
ejpam-4698	166	43	and	and	CCONJ
ejpam-4698	166	44	τv	τv	NOUN
ejpam-4698	166	45	=	=	SYM
ejpam-4698	166	46	5|	5|	NUM
ejpam-4698	166	47	=	=	SYM
ejpam-4698	166	48	2n	2n	NUM
ejpam-4698	166	49	,	,	PUNCT
ejpam-4698	166	50	|e5,9|	|e5,9|	NOUN
ejpam-4698	166	51	=	=	SYM
ejpam-4698	166	52	|uv	|uv	PART
ejpam-4698	166	53	∈	∈	PROPN
ejpam-4698	166	54	e(ℵn	e(ℵn	NOUN
ejpam-4698	166	55	)	)	PUNCT
ejpam-4698	166	56	:	:	PUNCT
ejpam-4698	166	57	τu	τu	ADP
ejpam-4698	166	58	=	=	SYM
ejpam-4698	166	59	5	5	NUM
ejpam-4698	166	60	and	and	CCONJ
ejpam-4698	166	61	τv	τv	NOUN
ejpam-4698	166	62	=	=	NOUN
ejpam-4698	166	63	9|	9|	NUM
ejpam-4698	166	64	=	=	SYM
ejpam-4698	166	65	2n−1	2n−1	NUM
ejpam-4698	166	66	,	,	PUNCT
ejpam-4698	166	67	|e6,8|	|e6,8|	ADJ
ejpam-4698	166	68	=	=	SYM
ejpam-4698	166	69	|uv	|uv	PART
ejpam-4698	166	70	∈	∈	PROPN
ejpam-4698	166	71	e(ℵn	e(ℵn	NOUN
ejpam-4698	166	72	)	)	PUNCT
ejpam-4698	166	73	:	:	PUNCT
ejpam-4698	166	74	τu	τu	ADP
ejpam-4698	166	75	=	=	PUNCT
ejpam-4698	166	76	6	6	NUM
ejpam-4698	166	77	and	and	CCONJ
ejpam-4698	166	78	τv	τv	ADP
ejpam-4698	166	79	=	=	SYM
ejpam-4698	166	80	8|	8|	NUM
ejpam-4698	166	81	=	=	SYM
ejpam-4698	166	82	2	2	NUM
ejpam-4698	166	83	,	,	PUNCT
ejpam-4698	166	84	|e8,9|	|e8,9|	NOUN
ejpam-4698	166	85	=	=	SYM
ejpam-4698	166	86	|uv	|uv	PART
ejpam-4698	166	87	∈	∈	PROPN
ejpam-4698	166	88	e(ℵn	e(ℵn	NOUN
ejpam-4698	166	89	)	)	PUNCT
ejpam-4698	166	90	:	:	PUNCT
ejpam-4698	166	91	τu	τu	ADP
ejpam-4698	166	92	=	=	SYM
ejpam-4698	166	93	8	8	NUM
ejpam-4698	166	94	and	and	CCONJ
ejpam-4698	166	95	τv	τv	NOUN
ejpam-4698	166	96	=	=	SYM
ejpam-4698	166	97	9|	9|	NUM
ejpam-4698	166	98	=	=	SYM
ejpam-4698	166	99	22	22	NUM
ejpam-4698	166	100	,	,	PUNCT
ejpam-4698	166	101	|e9,9|	|e9,9|	NOUN
ejpam-4698	166	102	=	=	PUNCT
ejpam-4698	166	103	|uv	|uv	PART
ejpam-4698	166	104	∈	∈	PROPN
ejpam-4698	166	105	e(ℵn	e(ℵn	NOUN
ejpam-4698	166	106	)	)	PUNCT
ejpam-4698	166	107	:	:	PUNCT
ejpam-4698	166	108	τu	τu	ADP
ejpam-4698	166	109	=	=	NOUN
ejpam-4698	166	110	τv	τv	NOUN
ejpam-4698	166	111	=	=	SYM
ejpam-4698	166	112	9|	9|	NUM
ejpam-4698	166	113	=	=	NOUN
ejpam-4698	166	114	|e(ℵn)|	|e(ℵn)|	NOUN
ejpam-4698	166	115	−	−	PROPN
ejpam-4698	166	116	|e3,5|	|e3,5|	PROPN
ejpam-4698	166	117	−	−	PROPN
ejpam-4698	166	118	|e5,9|	|e5,9|	NOUN
ejpam-4698	166	119	−	−	PROPN
ejpam-4698	166	120	|e6,8|	|e6,8|	ADJ
ejpam-4698	166	121	−	−	NOUN
ejpam-4698	166	122	|e8,9|	|e8,9|	NOUN
ejpam-4698	166	123	=	=	NOUN
ejpam-4698	167	1	2n−1	2n−1	NUM
ejpam-4698	167	2	−	−	NOUN
ejpam-4698	167	3	8	8	NUM
ejpam-4698	167	4	.	.	PUNCT
ejpam-4698	168	1	thus	thus	ADV
ejpam-4698	168	2	,	,	PUNCT
ejpam-4698	168	3	the	the	DET
ejpam-4698	168	4	mve−polynomial	mve−polynomial	PROPN
ejpam-4698	168	5	of	of	ADP
ejpam-4698	168	6	2−ary	2−ary	ADJ
ejpam-4698	168	7	tree	tree	NOUN
ejpam-4698	168	8	graph	graph	NOUN
ejpam-4698	168	9	ℵn	ℵn	NOUN
ejpam-4698	168	10	is	be	AUX
ejpam-4698	168	11	mve(ℵn;x	mve(ℵn;x	ADJ
ejpam-4698	168	12	,	,	PUNCT
ejpam-4698	168	13	y	y	NOUN
ejpam-4698	168	14	)	)	PUNCT
ejpam-4698	168	15	=	=	PUNCT
ejpam-4698	169	1	σi≤jcijx	σi≤jcijx	PROPN
ejpam-4698	169	2	iyj	iyj	VERB
ejpam-4698	169	3	=	=	PUNCT
ejpam-4698	169	4	σ3≤5c35x	σ3≤5c35x	PROPN
ejpam-4698	169	5	3y5	3y5	NUM
ejpam-4698	169	6	+	+	ADJ
ejpam-4698	169	7	σ5≤9c59x	σ5≤9c59x	ADJ
ejpam-4698	169	8	5y9	5y9	NUM
ejpam-4698	169	9	+	+	NOUN
ejpam-4698	169	10	σ6≤8c68x	σ6≤8c68x	PROPN
ejpam-4698	169	11	6y8	6y8	NUM
ejpam-4698	170	1	+	+	ADJ
ejpam-4698	170	2	σ8≤9c89x	σ8≤9c89x	ADJ
ejpam-4698	170	3	8y9	8y9	NUM
ejpam-4698	170	4	+	+	ADJ
ejpam-4698	170	5	σ9≤9c99x	σ9≤9c99x	PROPN
ejpam-4698	170	6	9y9	9y9	NOUN
ejpam-4698	170	7	=	=	PUNCT
ejpam-4698	170	8	σuv∈e3,5c35x	σuv∈e3,5c35x	PROPN
ejpam-4698	170	9	3y5+σuv∈e5,9c59x	3y5+σuv∈e5,9c59x	NUM
ejpam-4698	170	10	5y9+σuv∈e6,8c68x	5y9+σuv∈e6,8c68x	NUM
ejpam-4698	171	1	6y8+σuv∈e8,9c89x	6y8+σuv∈e8,9c89x	NUM
ejpam-4698	171	2	8y9+σuv∈e9,9c99x	8y9+σuv∈e9,9c99x	NUM
ejpam-4698	171	3	9y9	9y9	NUM
ejpam-4698	171	4	.	.	PUNCT
ejpam-4698	172	1	=	=	PRON
ejpam-4698	172	2	|e3,5|x3y5	|e3,5|x3y5	PRON
ejpam-4698	172	3	+	+	CCONJ
ejpam-4698	172	4	|e5,9|x5y9	|e5,9|x5y9	VERB
ejpam-4698	172	5	+	+	CCONJ
ejpam-4698	172	6	|e6,8|x6y8	|e6,8|x6y8	NOUN
ejpam-4698	172	7	+	+	NUM
ejpam-4698	172	8	|e8,9|x8y9	|e8,9|x8y9	NOUN
ejpam-4698	172	9	+	+	CCONJ
ejpam-4698	172	10	|e9,9|x9y9	|e9,9|x9y9	NOUN
ejpam-4698	172	11	=	=	SYM
ejpam-4698	172	12	2nx3y5	2nx3y5	NUM
ejpam-4698	173	1	+	+	NUM
ejpam-4698	173	2	2n−1x5y9	2n−1x5y9	NUM
ejpam-4698	174	1	+	+	CCONJ
ejpam-4698	174	2	2x6y8	2x6y8	NUM
ejpam-4698	174	3	+	+	CCONJ
ejpam-4698	174	4	4x8y9	4x8y9	NOUN
ejpam-4698	174	5	+	+	CCONJ
ejpam-4698	174	6	(	(	PUNCT
ejpam-4698	174	7	2n−1	2n−1	NUM
ejpam-4698	174	8	−	−	PROPN
ejpam-4698	174	9	8)x9y9	8)x9y9	NOUN
ejpam-4698	174	10	.	.	PUNCT
ejpam-4698	175	1	corollary	corollary	NOUN
ejpam-4698	175	2	4.3	4.3	NUM
ejpam-4698	175	3	:	:	PUNCT
ejpam-4698	175	4	let	let	VERB
ejpam-4698	175	5	ℵn	ℵn	PART
ejpam-4698	175	6	be	be	AUX
ejpam-4698	175	7	the	the	DET
ejpam-4698	175	8	2−	2−	NUM
ejpam-4698	175	9	ary	ary	PROPN
ejpam-4698	175	10	tree	tree	NOUN
ejpam-4698	175	11	graph	graph	NOUN
ejpam-4698	175	12	of	of	ADP
ejpam-4698	175	13	order	order	NOUN
ejpam-4698	175	14	2n+1	2n+1	NOUN
ejpam-4698	175	15	−	−	NOUN
ejpam-4698	175	16	1	1	NUM
ejpam-4698	175	17	,	,	PUNCT
ejpam-4698	175	18	n	n	PRON
ejpam-4698	175	19	≥	≥	NOUN
ejpam-4698	175	20	4	4	NUM
ejpam-4698	175	21	.	.	PUNCT
ejpam-4698	176	1	then	then	ADV
ejpam-4698	176	2	1	1	X
ejpam-4698	176	3	.	.	PUNCT
ejpam-4698	176	4	m1	m1	PROPN
ejpam-4698	176	5	ve(ℵn	ve(ℵn	PROPN
ejpam-4698	176	6	)	)	PUNCT
ejpam-4698	177	1	=	=	SYM
ejpam-4698	177	2	8(2n+1	8(2n+1	NUM
ejpam-4698	178	1	+	+	NUM
ejpam-4698	178	2	2n	2n	NUM
ejpam-4698	178	3	−	−	ADP
ejpam-4698	178	4	6	6	NUM
ejpam-4698	178	5	)	)	PUNCT
ejpam-4698	178	6	.	.	PUNCT
ejpam-4698	179	1	2	2	X
ejpam-4698	179	2	.	.	PUNCT
ejpam-4698	179	3	m2	m2	PROPN
ejpam-4698	179	4	ve(ℵn	ve(ℵn	PROPN
ejpam-4698	179	5	)	)	PUNCT
ejpam-4698	179	6	=	=	NOUN
ejpam-4698	180	1	3(42×	3(42×	NUM
ejpam-4698	180	2	2n−1	2n−1	NUM
ejpam-4698	181	1	+	+	CCONJ
ejpam-4698	181	2	5×	5×	NUM
ejpam-4698	181	3	2n	2n	NUM
ejpam-4698	181	4	−	−	ADP
ejpam-4698	181	5	88	88	NUM
ejpam-4698	181	6	)	)	PUNCT
ejpam-4698	181	7	.	.	PUNCT
ejpam-4698	182	1	3	3	X
ejpam-4698	182	2	.	.	X
ejpam-4698	182	3	rm1	rm1	PROPN
ejpam-4698	182	4	ve(ℵn	ve(ℵn	PROPN
ejpam-4698	182	5	)	)	PUNCT
ejpam-4698	183	1	=	=	PUNCT
ejpam-4698	184	1	(	(	PUNCT
ejpam-4698	184	2	2n+4	2n+4	NOUN
ejpam-4698	184	3	+	+	CCONJ
ejpam-4698	185	1	2n+3	2n+3	PROPN
ejpam-4698	185	2	−	−	NOUN
ejpam-4698	186	1	2n+2	2n+2	NUM
ejpam-4698	186	2	−	−	NOUN
ejpam-4698	186	3	44	44	NUM
ejpam-4698	186	4	)	)	PUNCT
ejpam-4698	186	5	.	.	PUNCT
ejpam-4698	187	1	4	4	X
ejpam-4698	187	2	.	.	X
ejpam-4698	187	3	rm2	rm2	PROPN
ejpam-4698	187	4	ve(ℵn	ve(ℵn	PROPN
ejpam-4698	187	5	)	)	PUNCT
ejpam-4698	187	6	=	=	PUNCT
ejpam-4698	188	1	2(2n+2	2(2n+2	NUM
ejpam-4698	189	1	+	+	CCONJ
ejpam-4698	189	2	48×	48×	NUM
ejpam-4698	189	3	2n−1	2n−1	NUM
ejpam-4698	189	4	−	−	NUM
ejpam-4698	189	5	109	109	NUM
ejpam-4698	189	6	)	)	PUNCT
ejpam-4698	189	7	.	.	PUNCT
ejpam-4698	190	1	5	5	X
ejpam-4698	190	2	.	.	X
ejpam-4698	190	3	hypmve(ℵn	hypmve(ℵn	NOUN
ejpam-4698	190	4	)	)	PUNCT
ejpam-4698	190	5	=	=	SYM
ejpam-4698	191	1	4(2n+4	4(2n+4	NUM
ejpam-4698	192	1	+	+	CCONJ
ejpam-4698	192	2	130×	130×	NUM
ejpam-4698	192	3	2n−1	2n−1	NUM
ejpam-4698	192	4	−	−	NUM
ejpam-4698	192	5	261	261	NUM
ejpam-4698	192	6	)	)	PUNCT
ejpam-4698	192	7	.	.	PUNCT
ejpam-4698	193	1	6	6	X
ejpam-4698	193	2	.	.	X
ejpam-4698	193	3	fmve(ℵn	fmve(ℵn	NOUN
ejpam-4698	193	4	)	)	PUNCT
ejpam-4698	193	5	=	=	SYM
ejpam-4698	193	6	2(17×	2(17×	NUM
ejpam-4698	193	7	2n	2n	NUM
ejpam-4698	194	1	+	+	CCONJ
ejpam-4698	194	2	134×	134×	NUM
ejpam-4698	194	3	2n−1	2n−1	NUM
ejpam-4698	194	4	−	−	NOUN
ejpam-4698	194	5	258	258	NUM
ejpam-4698	194	6	)	)	PUNCT
ejpam-4698	194	7	.	.	PUNCT
ejpam-4698	195	1	7	7	X
ejpam-4698	195	2	.	.	X
ejpam-4698	195	3	albmve(ℵn	albmve(ℵn	NOUN
ejpam-4698	195	4	)	)	PUNCT
ejpam-4698	195	5	=	=	SYM
ejpam-4698	195	6	4(2n	4(2n	NUM
ejpam-4698	196	1	+	+	CCONJ
ejpam-4698	196	2	2	2	NUM
ejpam-4698	196	3	)	)	PUNCT
ejpam-4698	196	4	.	.	PUNCT
ejpam-4698	197	1	8	8	X
ejpam-4698	197	2	.	.	X
ejpam-4698	197	3	σmve(ℵn	σmve(ℵn	NOUN
ejpam-4698	197	4	)	)	PUNCT
ejpam-4698	197	5	=	=	SYM
ejpam-4698	197	6	4(2n+1	4(2n+1	PROPN
ejpam-4698	198	1	+	+	NUM
ejpam-4698	198	2	2n	2n	NUM
ejpam-4698	198	3	+	+	CCONJ
ejpam-4698	198	4	3	3	NUM
ejpam-4698	198	5	)	)	PUNCT
ejpam-4698	198	6	.	.	PUNCT
ejpam-4698	199	1	remark	remark	VERB
ejpam-4698	199	2	4.4	4.4	NUM
ejpam-4698	199	3	:	:	PUNCT
ejpam-4698	199	4	from	from	ADP
ejpam-4698	199	5	theorem	theorem	ADJ
ejpam-4698	199	6	3.5	3.5	NUM
ejpam-4698	199	7	,	,	PUNCT
ejpam-4698	199	8	we	we	PRON
ejpam-4698	199	9	get	get	VERB
ejpam-4698	199	10	mve(ℵn;x	mve(ℵn;x	ADV
ejpam-4698	199	11	,	,	PUNCT
ejpam-4698	199	12	y	y	NOUN
ejpam-4698	199	13	)	)	PUNCT
ejpam-4698	199	14	=	=	SYM
ejpam-4698	199	15	nm(ℵn;x	nm(ℵn;x	PROPN
ejpam-4698	199	16	,	,	PUNCT
ejpam-4698	199	17	y	y	NOUN
ejpam-4698	199	18	)	)	PUNCT
ejpam-4698	199	19	=	=	SYM
ejpam-4698	200	1	2x6y8	2x6y8	NUM
ejpam-4698	201	1	+	+	CCONJ
ejpam-4698	201	2	4x8y9	4x8y9	NOUN
ejpam-4698	201	3	+	+	CCONJ
ejpam-4698	201	4	2nx3y5	2nx3y5	NUM
ejpam-4698	201	5	+	+	NUM
ejpam-4698	201	6	2n−1x5y9	2n−1x5y9	NUM
ejpam-4698	201	7	+	+	CCONJ
ejpam-4698	201	8	(	(	PUNCT
ejpam-4698	201	9	2n−1	2n−1	NUM
ejpam-4698	201	10	−	−	PROPN
ejpam-4698	201	11	8)x9y9	8)x9y9	NOUN
ejpam-4698	201	12	.	.	PUNCT
ejpam-4698	202	1	5	5	X
ejpam-4698	202	2	.	.	X
ejpam-4698	202	3	conclusion	conclusion	NOUN
ejpam-4698	202	4	:	:	PUNCT
ejpam-4698	202	5	in	in	ADP
ejpam-4698	202	6	this	this	DET
ejpam-4698	202	7	paper	paper	NOUN
ejpam-4698	202	8	,	,	PUNCT
ejpam-4698	202	9	we	we	PRON
ejpam-4698	202	10	have	have	AUX
ejpam-4698	202	11	given	give	VERB
ejpam-4698	202	12	a	a	DET
ejpam-4698	202	13	new	new	ADJ
ejpam-4698	202	14	polynomial	polynomial	NOUN
ejpam-4698	202	15	based	base	VERB
ejpam-4698	202	16	on	on	ADP
ejpam-4698	202	17	the	the	DET
ejpam-4698	202	18	terms	term	NOUN
ejpam-4698	202	19	of	of	ADP
ejpam-4698	202	20	a	a	DET
ejpam-4698	202	21	vertex	vertex	NOUN
ejpam-4698	202	22	–	–	PUNCT
ejpam-4698	202	23	edge	edge	NOUN
ejpam-4698	202	24	degree	degree	NOUN
ejpam-4698	202	25	of	of	ADP
ejpam-4698	202	26	a	a	DET
ejpam-4698	202	27	graph	graph	NOUN
ejpam-4698	202	28	g.	g.	NOUN
ejpam-4698	202	29	from	from	ADP
ejpam-4698	202	30	this	this	DET
ejpam-4698	202	31	polynomial	polynomial	NOUN
ejpam-4698	202	32	we	we	PRON
ejpam-4698	202	33	proved	prove	VERB
ejpam-4698	202	34	many	many	ADJ
ejpam-4698	202	35	properties	property	NOUN
ejpam-4698	202	36	and	and	CCONJ
ejpam-4698	202	37	proved	prove	VERB
ejpam-4698	202	38	that	that	SCONJ
ejpam-4698	202	39	it	it	PRON
ejpam-4698	202	40	’s	’	VERB
ejpam-4698	202	41	equal	equal	ADJ
ejpam-4698	202	42	to	to	ADP
ejpam-4698	202	43	the	the	DET
ejpam-4698	202	44	neighbor	neighbor	NOUN
ejpam-4698	202	45	polynomial	polynomial	NOUN
ejpam-4698	202	46	when	when	SCONJ
ejpam-4698	202	47	g	g	PROPN
ejpam-4698	202	48	there	there	PRON
ejpam-4698	202	49	is	be	VERB
ejpam-4698	202	50	a	a	DET
ejpam-4698	202	51	graph	graph	NOUN
ejpam-4698	202	52	without	without	ADP
ejpam-4698	202	53	a	a	DET
ejpam-4698	202	54	triangular	triangular	NOUN
ejpam-4698	202	55	cycle	cycle	NOUN
ejpam-4698	202	56	.	.	PUNCT
ejpam-4698	203	1	we	we	PRON
ejpam-4698	203	2	also	also	ADV
ejpam-4698	203	3	discover	discover	VERB
ejpam-4698	203	4	a	a	DET
ejpam-4698	203	5	vertex−edge	vertex−edge	NOUN
ejpam-4698	203	6	degree	degree	NOUN
ejpam-4698	203	7	polynomial	polynomial	NOUN
ejpam-4698	203	8	for	for	ADP
ejpam-4698	203	9	an	an	DET
ejpam-4698	203	10	r−regular	r−regular	ADJ
ejpam-4698	203	11	graph	graph	NOUN
ejpam-4698	203	12	under	under	ADP
ejpam-4698	203	13	certain	certain	ADJ
ejpam-4698	203	14	conditions	condition	NOUN
ejpam-4698	203	15	.	.	PUNCT
ejpam-4698	204	1	references	reference	NOUN
ejpam-4698	204	2	782	782	NUM
ejpam-4698	204	3	references	reference	NOUN
ejpam-4698	204	4	[	[	X
ejpam-4698	204	5	1	1	NUM
ejpam-4698	204	6	]	]	PUNCT
ejpam-4698	204	7	haveen	haveen	NOUN
ejpam-4698	205	1	j	j	PROPN
ejpam-4698	205	2	ahmed	ahmed	PROPN
ejpam-4698	205	3	,	,	PUNCT
ejpam-4698	205	4	ahmed	ahmed	PROPN
ejpam-4698	205	5	m	m	PROPN
ejpam-4698	205	6	ali	ali	PROPN
ejpam-4698	205	7	,	,	PUNCT
ejpam-4698	205	8	and	and	CCONJ
ejpam-4698	205	9	gashaw	gashaw	NOUN
ejpam-4698	205	10	a	a	DET
ejpam-4698	205	11	mohammed	mohammed	PROPN
ejpam-4698	205	12	saleh	saleh	NOUN
ejpam-4698	205	13	.	.	PUNCT
ejpam-4698	206	1	detour	detour	NOUN
ejpam-4698	206	2	polynomials	polynomial	NOUN
ejpam-4698	206	3	of	of	ADP
ejpam-4698	206	4	vertex	vertex	NOUN
ejpam-4698	206	5	coalenscence	coalenscence	NOUN
ejpam-4698	206	6	and	and	CCONJ
ejpam-4698	206	7	bridges	bridge	NOUN
ejpam-4698	206	8	coalenscence	coalenscence	NOUN
ejpam-4698	206	9	graphs	graph	NOUN
ejpam-4698	206	10	.	.	PUNCT
ejpam-4698	207	1	asian	asian	ADJ
ejpam-4698	207	2	-	-	PUNCT
ejpam-4698	207	3	european	european	ADJ
ejpam-4698	207	4	journal	journal	NOUN
ejpam-4698	207	5	of	of	ADP
ejpam-4698	207	6	mathematics	mathematic	NOUN
ejpam-4698	207	7	,	,	PUNCT
ejpam-4698	207	8	15(02):2250025	15(02):2250025	NUM
ejpam-4698	207	9	,	,	PUNCT
ejpam-4698	207	10	2022	2022	NUM
ejpam-4698	207	11	.	.	PUNCT
ejpam-4698	208	1	[	[	X
ejpam-4698	208	2	2	2	X
ejpam-4698	208	3	]	]	PUNCT
ejpam-4698	208	4	ahmed	ahmed	PROPN
ejpam-4698	208	5	mohammed	mohammed	PROPN
ejpam-4698	208	6	ali	ali	PROPN
ejpam-4698	208	7	,	,	PUNCT
ejpam-4698	208	8	herish	herish	PROPN
ejpam-4698	208	9	omer	omer	PROPN
ejpam-4698	208	10	abdullah	abdullah	PROPN
ejpam-4698	208	11	,	,	PUNCT
ejpam-4698	208	12	and	and	CCONJ
ejpam-4698	208	13	gashaw	gashaw	PROPN
ejpam-4698	208	14	aziz	aziz	PROPN
ejpam-4698	208	15	mohammed	mohammed	PROPN
ejpam-4698	208	16	saleh	saleh	PROPN
ejpam-4698	208	17	.	.	PUNCT
ejpam-4698	209	1	hosoya	hosoya	PROPN
ejpam-4698	209	2	polynomials	polynomial	NOUN
ejpam-4698	209	3	and	and	CCONJ
ejpam-4698	209	4	wiener	wiener	NOUN
ejpam-4698	209	5	indices	index	NOUN
ejpam-4698	209	6	of	of	ADP
ejpam-4698	209	7	carbon	carbon	NOUN
ejpam-4698	209	8	nanotubes	nanotube	NOUN
ejpam-4698	209	9	using	use	VERB
ejpam-4698	209	10	mathematica	mathematica	PROPN
ejpam-4698	209	11	programming	programming	PROPN
ejpam-4698	209	12	.	.	PUNCT
ejpam-4698	210	1	journal	journal	PROPN
ejpam-4698	210	2	of	of	ADP
ejpam-4698	210	3	discrete	discrete	ADJ
ejpam-4698	210	4	mathematical	mathematical	ADJ
ejpam-4698	210	5	sciences	science	NOUN
ejpam-4698	210	6	and	and	CCONJ
ejpam-4698	210	7	cryptography	cryptography	NOUN
ejpam-4698	210	8	,	,	PUNCT
ejpam-4698	210	9	25(1):147	25(1):147	NUM
ejpam-4698	210	10	–	–	PUNCT
ejpam-4698	210	11	158	158	NUM
ejpam-4698	210	12	,	,	PUNCT
ejpam-4698	210	13	2022	2022	NUM
ejpam-4698	210	14	.	.	PUNCT
ejpam-4698	211	1	[	[	X
ejpam-4698	211	2	3	3	X
ejpam-4698	211	3	]	]	X
ejpam-4698	211	4	murat	murat	PROPN
ejpam-4698	211	5	cancan	cancan	PROPN
ejpam-4698	211	6	,	,	PUNCT
ejpam-4698	211	7	süleyman	süleyman	ADJ
ejpam-4698	211	8	ediz	ediz	NOUN
ejpam-4698	211	9	,	,	PUNCT
ejpam-4698	211	10	mehdi	mehdi	PROPN
ejpam-4698	211	11	alaeiyan	alaeiyan	PROPN
ejpam-4698	211	12	,	,	PUNCT
ejpam-4698	211	13	and	and	CCONJ
ejpam-4698	211	14	mohammad	mohammad	PROPN
ejpam-4698	211	15	reza	reza	PROPN
ejpam-4698	211	16	farahani	farahani	PROPN
ejpam-4698	211	17	.	.	PUNCT
ejpam-4698	212	1	on	on	ADP
ejpam-4698	212	2	ve	ve	NOUN
ejpam-4698	212	3	-	-	PUNCT
ejpam-4698	212	4	degree	degree	NOUN
ejpam-4698	212	5	molecular	molecular	ADJ
ejpam-4698	212	6	properties	property	NOUN
ejpam-4698	212	7	of	of	ADP
ejpam-4698	212	8	copper	copper	NOUN
ejpam-4698	212	9	oxide	oxide	NOUN
ejpam-4698	212	10	.	.	PUNCT
ejpam-4698	213	1	journal	journal	NOUN
ejpam-4698	213	2	of	of	ADP
ejpam-4698	213	3	information	information	NOUN
ejpam-4698	213	4	and	and	CCONJ
ejpam-4698	213	5	optimization	optimization	NOUN
ejpam-4698	213	6	sciences	science	NOUN
ejpam-4698	213	7	,	,	PUNCT
ejpam-4698	213	8	41(4):949–957	41(4):949–957	NOUN
ejpam-4698	213	9	,	,	PUNCT
ejpam-4698	213	10	2020	2020	NUM
ejpam-4698	213	11	.	.	PUNCT
ejpam-4698	214	1	[	[	X
ejpam-4698	214	2	4	4	X
ejpam-4698	214	3	]	]	X
ejpam-4698	214	4	mustapha	mustapha	PROPN
ejpam-4698	214	5	chellali	chellali	PROPN
ejpam-4698	214	6	,	,	PUNCT
ejpam-4698	214	7	teresa	teresa	PROPN
ejpam-4698	214	8	w	w	PROPN
ejpam-4698	214	9	haynes	haynes	PROPN
ejpam-4698	214	10	,	,	PUNCT
ejpam-4698	214	11	stephen	stephen	PROPN
ejpam-4698	214	12	t	t	PROPN
ejpam-4698	214	13	hedetniemi	hedetniemi	ADV
ejpam-4698	214	14	,	,	PUNCT
ejpam-4698	214	15	and	and	CCONJ
ejpam-4698	214	16	thomas	thomas	PROPN
ejpam-4698	214	17	m	m	PROPN
ejpam-4698	214	18	lewis	lewis	PROPN
ejpam-4698	214	19	.	.	PUNCT
ejpam-4698	215	1	on	on	ADP
ejpam-4698	215	2	ve	ve	NOUN
ejpam-4698	215	3	-	-	PUNCT
ejpam-4698	215	4	degrees	degree	NOUN
ejpam-4698	215	5	and	and	CCONJ
ejpam-4698	215	6	ev	ev	NOUN
ejpam-4698	215	7	-	-	PUNCT
ejpam-4698	215	8	degrees	degree	NOUN
ejpam-4698	215	9	in	in	ADP
ejpam-4698	215	10	graphs	graph	NOUN
ejpam-4698	215	11	.	.	PUNCT
ejpam-4698	216	1	discrete	discrete	ADJ
ejpam-4698	216	2	mathematics	mathematic	NOUN
ejpam-4698	216	3	,	,	PUNCT
ejpam-4698	216	4	340(2):31–38	340(2):31–38	NUM
ejpam-4698	216	5	,	,	PUNCT
ejpam-4698	216	6	2017	2017	NUM
ejpam-4698	216	7	.	.	PUNCT
ejpam-4698	217	1	[	[	X
ejpam-4698	217	2	5	5	NUM
ejpam-4698	217	3	]	]	X
ejpam-4698	217	4	yu	yu	PROPN
ejpam-4698	217	5	-	-	PROPN
ejpam-4698	217	6	ming	ming	PROPN
ejpam-4698	217	7	chu	chu	PROPN
ejpam-4698	217	8	,	,	PUNCT
ejpam-4698	217	9	mehwish	mehwish	PROPN
ejpam-4698	217	10	hussain	hussain	PROPN
ejpam-4698	217	11	muhammad	muhammad	PROPN
ejpam-4698	217	12	,	,	PUNCT
ejpam-4698	217	13	abdul	abdul	PROPN
ejpam-4698	217	14	rauf	rauf	PROPN
ejpam-4698	217	15	,	,	PUNCT
ejpam-4698	217	16	muhammad	muhammad	PROPN
ejpam-4698	217	17	ishtiaq	ishtiaq	PROPN
ejpam-4698	217	18	,	,	PUNCT
ejpam-4698	217	19	and	and	CCONJ
ejpam-4698	217	20	muhammad	muhammad	PROPN
ejpam-4698	217	21	kamran	kamran	PROPN
ejpam-4698	217	22	siddiqui	siddiqui	PROPN
ejpam-4698	217	23	.	.	PUNCT
ejpam-4698	218	1	topological	topological	ADJ
ejpam-4698	218	2	study	study	NOUN
ejpam-4698	218	3	of	of	ADP
ejpam-4698	218	4	polycyclic	polycyclic	NOUN
ejpam-4698	218	5	graphite	graphite	NOUN
ejpam-4698	218	6	carbon	carbon	NOUN
ejpam-4698	218	7	nitride	nitride	NOUN
ejpam-4698	218	8	.	.	PUNCT
ejpam-4698	219	1	polycyclic	polycyclic	ADJ
ejpam-4698	219	2	aromatic	aromatic	ADJ
ejpam-4698	219	3	compounds	compound	NOUN
ejpam-4698	219	4	,	,	PUNCT
ejpam-4698	219	5	42(6):3203–3215	42(6):3203–3215	NUM
ejpam-4698	219	6	,	,	PUNCT
ejpam-4698	219	7	2020	2020	NUM
ejpam-4698	219	8	.	.	PUNCT
ejpam-4698	220	1	[	[	X
ejpam-4698	220	2	6	6	NUM
ejpam-4698	220	3	]	]	X
ejpam-4698	220	4	emeric	emeric	ADJ
ejpam-4698	220	5	deutsch	deutsch	NOUN
ejpam-4698	220	6	and	and	CCONJ
ejpam-4698	220	7	sandi	sandi	PROPN
ejpam-4698	220	8	klavžar	klavžar	PROPN
ejpam-4698	220	9	.	.	PUNCT
ejpam-4698	221	1	m	m	ADJ
ejpam-4698	221	2	-	-	ADJ
ejpam-4698	221	3	polynomial	polynomial	ADJ
ejpam-4698	221	4	and	and	CCONJ
ejpam-4698	221	5	degree	degree	NOUN
ejpam-4698	221	6	-	-	PUNCT
ejpam-4698	221	7	based	base	VERB
ejpam-4698	221	8	topological	topological	ADJ
ejpam-4698	221	9	indices	index	NOUN
ejpam-4698	221	10	.	.	PUNCT
ejpam-4698	222	1	arxiv	arxiv	PROPN
ejpam-4698	222	2	preprint	preprint	NOUN
ejpam-4698	222	3	arxiv:1407.1592	arxiv:1407.1592	NOUN
ejpam-4698	222	4	,	,	PUNCT
ejpam-4698	222	5	2014	2014	NUM
ejpam-4698	222	6	.	.	PUNCT
ejpam-4698	223	1	[	[	X
ejpam-4698	223	2	7	7	X
ejpam-4698	223	3	]	]	X
ejpam-4698	223	4	süleyman	süleyman	ADJ
ejpam-4698	223	5	ediz	ediz	NOUN
ejpam-4698	223	6	.	.	PUNCT
ejpam-4698	224	1	predicting	predict	VERB
ejpam-4698	224	2	some	some	DET
ejpam-4698	224	3	physicochemical	physicochemical	ADJ
ejpam-4698	224	4	properties	property	NOUN
ejpam-4698	224	5	of	of	ADP
ejpam-4698	224	6	octane	octane	NOUN
ejpam-4698	224	7	isomers	isomer	NOUN
ejpam-4698	224	8	:	:	PUNCT
ejpam-4698	224	9	a	a	DET
ejpam-4698	224	10	topological	topological	ADJ
ejpam-4698	224	11	approach	approach	NOUN
ejpam-4698	224	12	using	use	VERB
ejpam-4698	224	13	ev	ev	ADJ
ejpam-4698	224	14	-	-	PUNCT
ejpam-4698	224	15	degree	degree	NOUN
ejpam-4698	224	16	and	and	CCONJ
ejpam-4698	224	17	ve	ve	NOUN
ejpam-4698	224	18	-	-	PUNCT
ejpam-4698	224	19	degree	degree	NOUN
ejpam-4698	224	20	zagreb	zagreb	PROPN
ejpam-4698	224	21	indices	index	NOUN
ejpam-4698	224	22	.	.	PUNCT
ejpam-4698	225	1	arxiv	arxiv	PROPN
ejpam-4698	225	2	preprint	preprint	VERB
ejpam-4698	225	3	arxiv:1701.02859	arxiv:1701.02859	PROPN
ejpam-4698	225	4	,	,	PUNCT
ejpam-4698	225	5	2017	2017	NUM
ejpam-4698	225	6	.	.	PUNCT
ejpam-4698	226	1	[	[	X
ejpam-4698	226	2	8	8	NUM
ejpam-4698	226	3	]	]	PUNCT
ejpam-4698	226	4	vr	vr	PROPN
ejpam-4698	226	5	kulli	kulli	PROPN
ejpam-4698	226	6	.	.	PUNCT
ejpam-4698	227	1	on	on	ADP
ejpam-4698	227	2	ve	ve	NOUN
ejpam-4698	227	3	-	-	PUNCT
ejpam-4698	227	4	degree	degree	NOUN
ejpam-4698	227	5	indices	index	NOUN
ejpam-4698	227	6	and	and	CCONJ
ejpam-4698	227	7	their	their	PRON
ejpam-4698	227	8	polynomials	polynomial	NOUN
ejpam-4698	227	9	of	of	ADP
ejpam-4698	227	10	dominating	dominate	VERB
ejpam-4698	227	11	oxide	oxide	NOUN
ejpam-4698	227	12	networks	network	NOUN
ejpam-4698	227	13	.	.	PUNCT
ejpam-4698	228	1	annals	annal	NOUN
ejpam-4698	228	2	of	of	ADP
ejpam-4698	228	3	pure	pure	ADJ
ejpam-4698	228	4	and	and	CCONJ
ejpam-4698	228	5	applied	applied	ADJ
ejpam-4698	228	6	mathematics	mathematic	NOUN
ejpam-4698	228	7	,	,	PUNCT
ejpam-4698	228	8	18(1):1–7	18(1):1–7	NUM
ejpam-4698	228	9	,	,	PUNCT
ejpam-4698	228	10	2018	2018	NUM
ejpam-4698	228	11	.	.	PUNCT
ejpam-4698	229	1	[	[	X
ejpam-4698	229	2	9	9	NUM
ejpam-4698	229	3	]	]	X
ejpam-4698	229	4	young	young	ADJ
ejpam-4698	229	5	chel	chel	PROPN
ejpam-4698	229	6	kwun	kwun	PROPN
ejpam-4698	229	7	,	,	PUNCT
ejpam-4698	229	8	mobeen	mobeen	PROPN
ejpam-4698	229	9	munir	munir	PROPN
ejpam-4698	229	10	,	,	PUNCT
ejpam-4698	229	11	waqas	waqas	PROPN
ejpam-4698	229	12	nazeer	nazeer	PROPN
ejpam-4698	229	13	,	,	PUNCT
ejpam-4698	229	14	shazia	shazia	PROPN
ejpam-4698	229	15	rafique	rafique	PROPN
ejpam-4698	229	16	,	,	PUNCT
ejpam-4698	229	17	and	and	CCONJ
ejpam-4698	229	18	shin	shin	PROPN
ejpam-4698	229	19	min	min	PROPN
ejpam-4698	229	20	kang	kang	PROPN
ejpam-4698	229	21	.	.	PUNCT
ejpam-4698	230	1	m	m	NOUN
ejpam-4698	230	2	-	-	PUNCT
ejpam-4698	230	3	polynomials	polynomial	NOUN
ejpam-4698	230	4	and	and	CCONJ
ejpam-4698	230	5	topological	topological	ADJ
ejpam-4698	230	6	indices	index	NOUN
ejpam-4698	230	7	of	of	ADP
ejpam-4698	230	8	v	v	NOUN
ejpam-4698	230	9	-	-	PUNCT
ejpam-4698	230	10	phenylenic	phenylenic	ADJ
ejpam-4698	230	11	nanotubes	nanotube	NOUN
ejpam-4698	230	12	and	and	CCONJ
ejpam-4698	230	13	nanotori	nanotori	NOUN
ejpam-4698	230	14	.	.	PUNCT
ejpam-4698	231	1	scientific	scientific	ADJ
ejpam-4698	231	2	reports	report	NOUN
ejpam-4698	231	3	,	,	PUNCT
ejpam-4698	231	4	7(1):1–9	7(1):1–9	NUM
ejpam-4698	231	5	,	,	PUNCT
ejpam-4698	231	6	2017	2017	NUM
ejpam-4698	231	7	.	.	PUNCT
ejpam-4698	232	1	[	[	X
ejpam-4698	232	2	10	10	NUM
ejpam-4698	232	3	]	]	X
ejpam-4698	232	4	jason	jason	PROPN
ejpam-4698	232	5	robert	robert	PROPN
ejpam-4698	232	6	lewis	lewis	PROPN
ejpam-4698	232	7	.	.	PUNCT
ejpam-4698	232	8	vertex	vertex	NOUN
ejpam-4698	232	9	-	-	PUNCT
ejpam-4698	232	10	edge	edge	NOUN
ejpam-4698	232	11	and	and	CCONJ
ejpam-4698	232	12	edge	edge	NOUN
ejpam-4698	232	13	-	-	PUNCT
ejpam-4698	232	14	vertex	vertex	NOUN
ejpam-4698	232	15	parameters	parameter	NOUN
ejpam-4698	232	16	in	in	ADP
ejpam-4698	232	17	graphs	graph	NOUN
ejpam-4698	232	18	.	.	PUNCT
ejpam-4698	233	1	phd	phd	NOUN
ejpam-4698	233	2	thesis	thesis	PROPN
ejpam-4698	233	3	,	,	PUNCT
ejpam-4698	233	4	clemson	clemson	NOUN
ejpam-4698	233	5	university	university	NOUN
ejpam-4698	233	6	,	,	PUNCT
ejpam-4698	233	7	2007	2007	NUM
ejpam-4698	233	8	.	.	PUNCT
ejpam-4698	234	1	[	[	X
ejpam-4698	234	2	11	11	NUM
ejpam-4698	234	3	]	]	X
ejpam-4698	234	4	sourav	sourav	PROPN
ejpam-4698	234	5	mondal	mondal	PROPN
ejpam-4698	234	6	,	,	PUNCT
ejpam-4698	234	7	muhammad	muhammad	PROPN
ejpam-4698	234	8	imran	imran	PROPN
ejpam-4698	234	9	,	,	PUNCT
ejpam-4698	234	10	nilanjan	nilanjan	NOUN
ejpam-4698	234	11	de	de	PROPN
ejpam-4698	234	12	,	,	PUNCT
ejpam-4698	234	13	and	and	CCONJ
ejpam-4698	234	14	anita	anita	PROPN
ejpam-4698	234	15	pal	pal	PROPN
ejpam-4698	234	16	.	.	PUNCT
ejpam-4698	235	1	neighborhood	neighborhood	NOUN
ejpam-4698	235	2	mpolynomial	mpolynomial	NOUN
ejpam-4698	235	3	of	of	ADP
ejpam-4698	235	4	titanium	titanium	NOUN
ejpam-4698	235	5	compounds	compound	NOUN
ejpam-4698	235	6	.	.	PUNCT
ejpam-4698	236	1	arabian	arabian	ADJ
ejpam-4698	236	2	journal	journal	PROPN
ejpam-4698	236	3	of	of	ADP
ejpam-4698	236	4	chemistry	chemistry	NOUN
ejpam-4698	236	5	,	,	PUNCT
ejpam-4698	236	6	14(8):103244	14(8):103244	NUM
ejpam-4698	236	7	,	,	PUNCT
ejpam-4698	236	8	2021	2021	NUM
ejpam-4698	236	9	.	.	PUNCT
ejpam-4698	237	1	[	[	X
ejpam-4698	237	2	12	12	NUM
ejpam-4698	237	3	]	]	X
ejpam-4698	237	4	sourav	sourav	PROPN
ejpam-4698	237	5	mondal	mondal	PROPN
ejpam-4698	237	6	,	,	PUNCT
ejpam-4698	237	7	de	de	PROPN
ejpam-4698	237	8	nilanjan	nilanjan	NOUN
ejpam-4698	237	9	,	,	PUNCT
ejpam-4698	237	10	and	and	CCONJ
ejpam-4698	237	11	pal	pal	ADJ
ejpam-4698	237	12	anita	anita	PROPN
ejpam-4698	237	13	.	.	PUNCT
ejpam-4698	238	1	topological	topological	ADJ
ejpam-4698	238	2	properties	property	NOUN
ejpam-4698	238	3	of	of	ADP
ejpam-4698	238	4	networks	network	NOUN
ejpam-4698	238	5	using	use	VERB
ejpam-4698	238	6	m	m	ADJ
ejpam-4698	238	7	-	-	ADJ
ejpam-4698	238	8	polynomial	polynomial	ADJ
ejpam-4698	238	9	approach	approach	NOUN
ejpam-4698	238	10	.	.	PUNCT
ejpam-4698	239	1	konuralp	konuralp	PROPN
ejpam-4698	239	2	journal	journal	PROPN
ejpam-4698	239	3	of	of	ADP
ejpam-4698	239	4	mathematics	mathematic	NOUN
ejpam-4698	239	5	,	,	PUNCT
ejpam-4698	239	6	8(1):97–105	8(1):97–105	NUM
ejpam-4698	239	7	,	,	PUNCT
ejpam-4698	239	8	2020	2020	NUM
ejpam-4698	239	9	.	.	PUNCT
ejpam-4698	240	1	[	[	X
ejpam-4698	240	2	13	13	NUM
ejpam-4698	240	3	]	]	X
ejpam-4698	240	4	sourav	sourav	PROPN
ejpam-4698	240	5	mondal	mondal	PROPN
ejpam-4698	240	6	,	,	PUNCT
ejpam-4698	240	7	muhammad	muhammad	PROPN
ejpam-4698	240	8	kamran	kamran	PROPN
ejpam-4698	240	9	siddiqui	siddiqui	PROPN
ejpam-4698	240	10	,	,	PUNCT
ejpam-4698	240	11	nilanjan	nilanjan	NOUN
ejpam-4698	240	12	de	de	PROPN
ejpam-4698	240	13	,	,	PUNCT
ejpam-4698	240	14	and	and	CCONJ
ejpam-4698	240	15	anita	anita	PROPN
ejpam-4698	240	16	pal	pal	PROPN
ejpam-4698	240	17	.	.	PUNCT
ejpam-4698	241	1	neighborhood	neighborhood	NOUN
ejpam-4698	241	2	m	m	NOUN
ejpam-4698	241	3	-	-	PUNCT
ejpam-4698	241	4	polynomial	polynomial	ADJ
ejpam-4698	241	5	of	of	ADP
ejpam-4698	241	6	crystallographic	crystallographic	ADJ
ejpam-4698	241	7	structures	structure	NOUN
ejpam-4698	241	8	.	.	PUNCT
ejpam-4698	242	1	biointerface	biointerface	NOUN
ejpam-4698	242	2	res	re	NOUN
ejpam-4698	242	3	.	.	PUNCT
ejpam-4698	243	1	appl	appl	PROPN
ejpam-4698	243	2	.	.	PROPN
ejpam-4698	244	1	chem	chem	PROPN
ejpam-4698	244	2	,	,	PUNCT
ejpam-4698	244	3	11(2):9372–9381	11(2):9372–9381	NUM
ejpam-4698	244	4	,	,	PUNCT
ejpam-4698	244	5	2021	2021	NUM
ejpam-4698	244	6	.	.	PUNCT
ejpam-4698	245	1	references	reference	NOUN
ejpam-4698	245	2	783	783	NUM
ejpam-4698	246	1	[	[	X
ejpam-4698	246	2	14	14	NUM
ejpam-4698	246	3	]	]	PUNCT
ejpam-4698	246	4	raghad	raghad	VERB
ejpam-4698	246	5	a	a	DET
ejpam-4698	246	6	mustafa	mustafa	PROPN
ejpam-4698	246	7	,	,	PUNCT
ejpam-4698	247	1	ahmed	ahmed	PROPN
ejpam-4698	247	2	m	m	PROPN
ejpam-4698	247	3	ali	ali	PROPN
ejpam-4698	247	4	,	,	PUNCT
ejpam-4698	247	5	and	and	CCONJ
ejpam-4698	247	6	abdulsattar	abdulsattar	PROPN
ejpam-4698	247	7	m	m	PROPN
ejpam-4698	247	8	khidhir	khidhir	NOUN
ejpam-4698	247	9	.	.	PUNCT
ejpam-4698	248	1	mn	mn	PROPN
ejpam-4698	248	2	–	–	PUNCT
ejpam-4698	248	3	polynomials	polynomial	NOUN
ejpam-4698	248	4	of	of	ADP
ejpam-4698	248	5	some	some	DET
ejpam-4698	248	6	special	special	NOUN
ejpam-4698	248	7	for	for	ADP
ejpam-4698	248	8	cog	cog	NOUN
ejpam-4698	248	9	-	-	PUNCT
ejpam-4698	248	10	graphs	graph	NOUN
ejpam-4698	248	11	.	.	PUNCT
ejpam-4698	249	1	journal	journal	NOUN
ejpam-4698	249	2	of	of	ADP
ejpam-4698	249	3	discrete	discrete	ADJ
ejpam-4698	249	4	mathematical	mathematical	ADJ
ejpam-4698	249	5	sciences	science	NOUN
ejpam-4698	249	6	and	and	CCONJ
ejpam-4698	249	7	cryptography	cryptography	NOUN
ejpam-4698	249	8	,	,	PUNCT
ejpam-4698	249	9	pages	page	NOUN
ejpam-4698	249	10	1–16	1–16	PROPN
ejpam-4698	249	11	,	,	PUNCT
ejpam-4698	249	12	2022	2022	NUM
ejpam-4698	249	13	.	.	PUNCT
ejpam-4698	250	1	[	[	X
ejpam-4698	250	2	15	15	NUM
ejpam-4698	250	3	]	]	X
ejpam-4698	250	4	zahid	zahid	PROPN
ejpam-4698	250	5	raza	raza	PROPN
ejpam-4698	250	6	and	and	CCONJ
ejpam-4698	250	7	mark	mark	PROPN
ejpam-4698	250	8	essa	essa	PROPN
ejpam-4698	250	9	k.	k.	PROPN
ejpam-4698	250	10	sukaiti	sukaiti	PROPN
ejpam-4698	250	11	.	.	PUNCT
ejpam-4698	251	1	m	m	ADJ
ejpam-4698	251	2	-	-	ADJ
ejpam-4698	251	3	polynomial	polynomial	ADJ
ejpam-4698	251	4	and	and	CCONJ
ejpam-4698	251	5	degree	degree	NOUN
ejpam-4698	251	6	based	base	VERB
ejpam-4698	251	7	topological	topological	ADJ
ejpam-4698	251	8	indices	index	NOUN
ejpam-4698	251	9	of	of	ADP
ejpam-4698	251	10	some	some	DET
ejpam-4698	251	11	nanostructures	nanostructure	NOUN
ejpam-4698	251	12	.	.	PUNCT
ejpam-4698	252	1	symmetry	symmetry	PROPN
ejpam-4698	252	2	,	,	PUNCT
ejpam-4698	252	3	12(5):831	12(5):831	NUM
ejpam-4698	252	4	,	,	PUNCT
ejpam-4698	252	5	2020	2020	NUM
ejpam-4698	252	6	.	.	PUNCT
ejpam-4698	253	1	[	[	X
ejpam-4698	253	2	16	16	NUM
ejpam-4698	253	3	]	]	X
ejpam-4698	253	4	charles	charles	PROPN
ejpam-4698	253	5	semple	semple	PROPN
ejpam-4698	253	6	and	and	CCONJ
ejpam-4698	253	7	mike	mike	PROPN
ejpam-4698	253	8	steel	steel	PROPN
ejpam-4698	253	9	.	.	PUNCT
ejpam-4698	254	1	a	a	DET
ejpam-4698	254	2	supertree	supertree	ADJ
ejpam-4698	254	3	method	method	NOUN
ejpam-4698	254	4	for	for	ADP
ejpam-4698	254	5	rooted	rooted	ADJ
ejpam-4698	254	6	trees	tree	NOUN
ejpam-4698	254	7	.	.	PUNCT
ejpam-4698	255	1	discrete	discrete	ADJ
ejpam-4698	255	2	applied	apply	VERB
ejpam-4698	255	3	mathematics	mathematic	NOUN
ejpam-4698	255	4	,	,	PUNCT
ejpam-4698	255	5	105(1	105(1	PROPN
ejpam-4698	255	6	-	-	SYM
ejpam-4698	255	7	3):147–158	3):147–158	NUM
ejpam-4698	255	8	,	,	PUNCT
ejpam-4698	255	9	2000	2000	NUM
ejpam-4698	255	10	.	.	PUNCT
ejpam-4698	256	1	[	[	X
ejpam-4698	256	2	17	17	NUM
ejpam-4698	256	3	]	]	X
ejpam-4698	256	4	ashish	ashish	PROPN
ejpam-4698	256	5	verma	verma	PROPN
ejpam-4698	256	6	,	,	PUNCT
ejpam-4698	256	7	sourav	sourav	PROPN
ejpam-4698	256	8	mondal	mondal	PROPN
ejpam-4698	256	9	,	,	PUNCT
ejpam-4698	256	10	nilanjan	nilanjan	PROPN
ejpam-4698	256	11	de	de	PROPN
ejpam-4698	256	12	,	,	PUNCT
ejpam-4698	256	13	and	and	CCONJ
ejpam-4698	256	14	anita	anita	PROPN
ejpam-4698	256	15	pal	pal	NOUN
ejpam-4698	256	16	.	.	PUNCT
ejpam-4698	257	1	topological	topological	ADJ
ejpam-4698	257	2	properties	property	NOUN
ejpam-4698	257	3	of	of	ADP
ejpam-4698	257	4	bismuth	bismuth	NOUN
ejpam-4698	257	5	tri	tri	NOUN
ejpam-4698	257	6	-	-	NOUN
ejpam-4698	257	7	iodide	iodide	ADJ
ejpam-4698	257	8	using	use	VERB
ejpam-4698	257	9	neighborhood	neighborhood	NOUN
ejpam-4698	257	10	m	m	NOUN
ejpam-4698	257	11	-	-	PUNCT
ejpam-4698	257	12	polynomial	polynomial	ADJ
ejpam-4698	257	13	.	.	PUNCT
ejpam-4698	258	1	international	international	ADJ
ejpam-4698	258	2	journal	journal	PROPN
ejpam-4698	258	3	of	of	ADP
ejpam-4698	258	4	mathematics	mathematics	NOUN
ejpam-4698	258	5	trends	trend	NOUN
ejpam-4698	258	6	and	and	CCONJ
ejpam-4698	258	7	technology	technology	NOUN
ejpam-4698	258	8	,	,	PUNCT
ejpam-4698	258	9	67(10):83–90	67(10):83–90	NUM
ejpam-4698	258	10	,	,	PUNCT
ejpam-4698	258	11	2019	2019	NUM
ejpam-4698	258	12	.	.	PUNCT
ejpam-4698	259	1	[	[	X
ejpam-4698	259	2	18	18	NUM
ejpam-4698	259	3	]	]	X
ejpam-4698	259	4	satyanarayana	satyanarayana	PROPN
ejpam-4698	259	5	vollala	vollala	PROPN
ejpam-4698	259	6	and	and	CCONJ
ejpam-4698	259	7	indrajeet	indrajeet	PROPN
ejpam-4698	259	8	saravanan	saravanan	PROPN
ejpam-4698	259	9	.	.	PUNCT
ejpam-4698	260	1	vertex	vertex	NOUN
ejpam-4698	260	2	degree	degree	NOUN
ejpam-4698	260	3	-	-	PUNCT
ejpam-4698	260	4	based	base	VERB
ejpam-4698	260	5	topological	topological	ADJ
ejpam-4698	260	6	indices	index	NOUN
ejpam-4698	260	7	of	of	ADP
ejpam-4698	260	8	penta	penta	NOUN
ejpam-4698	260	9	-	-	PUNCT
ejpam-4698	260	10	chains	chain	NOUN
ejpam-4698	260	11	using	use	VERB
ejpam-4698	260	12	m	m	NOUN
ejpam-4698	260	13	-	-	ADJ
ejpam-4698	260	14	polynomial	polynomial	ADJ
ejpam-4698	260	15	.	.	PUNCT
ejpam-4698	261	1	international	international	ADJ
ejpam-4698	261	2	journal	journal	NOUN
ejpam-4698	261	3	of	of	ADP
ejpam-4698	261	4	advances	advance	NOUN
ejpam-4698	261	5	in	in	ADP
ejpam-4698	261	6	engineering	engineer	VERB
ejpam-4698	261	7	sciences	science	NOUN
ejpam-4698	261	8	and	and	CCONJ
ejpam-4698	261	9	applied	apply	VERB
ejpam-4698	261	10	mathematics	mathematic	NOUN
ejpam-4698	261	11	,	,	PUNCT
ejpam-4698	261	12	11(1):53–67	11(1):53–67	NUM
ejpam-4698	261	13	,	,	PUNCT
ejpam-4698	261	14	2019	2019	NUM
ejpam-4698	261	15	.	.	PUNCT
ejpam-4698	262	1	[	[	X
ejpam-4698	262	2	19	19	NUM
ejpam-4698	262	3	]	]	PUNCT
ejpam-4698	262	4	jing	jing	PROPN
ejpam-4698	262	5	zhang	zhang	PROPN
ejpam-4698	262	6	,	,	PUNCT
ejpam-4698	262	7	muhammad	muhammad	PROPN
ejpam-4698	262	8	kamran	kamran	PROPN
ejpam-4698	262	9	siddiqui	siddiqui	PROPN
ejpam-4698	262	10	,	,	PUNCT
ejpam-4698	262	11	abdul	abdul	PROPN
ejpam-4698	262	12	rauf	rauf	PROPN
ejpam-4698	262	13	,	,	PUNCT
ejpam-4698	262	14	and	and	CCONJ
ejpam-4698	262	15	muhammad	muhammad	PROPN
ejpam-4698	262	16	ishtiaq	ishtiaq	PROPN
ejpam-4698	262	17	.	.	PUNCT
ejpam-4698	263	1	on	on	ADP
ejpam-4698	263	2	ve	ve	NOUN
ejpam-4698	263	3	-	-	PUNCT
ejpam-4698	263	4	degree	degree	NOUN
ejpam-4698	263	5	and	and	CCONJ
ejpam-4698	263	6	ev	ev	ADJ
ejpam-4698	263	7	-	-	PUNCT
ejpam-4698	263	8	degree	degree	NOUN
ejpam-4698	263	9	based	base	VERB
ejpam-4698	263	10	topological	topological	ADJ
ejpam-4698	263	11	properties	property	NOUN
ejpam-4698	263	12	of	of	ADP
ejpam-4698	263	13	single	single	ADJ
ejpam-4698	263	14	walled	walled	ADJ
ejpam-4698	263	15	titanium	titanium	NOUN
ejpam-4698	263	16	dioxide	dioxide	NOUN
ejpam-4698	263	17	nanotube	nanotube	NOUN
ejpam-4698	263	18	.	.	PUNCT
ejpam-4698	264	1	journal	journal	PROPN
ejpam-4698	264	2	of	of	ADP
ejpam-4698	264	3	cluster	cluster	NOUN
ejpam-4698	264	4	science	science	NOUN
ejpam-4698	264	5	,	,	PUNCT
ejpam-4698	264	6	32(4):821–832	32(4):821–832	NUM
ejpam-4698	264	7	,	,	PUNCT
ejpam-4698	264	8	2021	2021	NUM
ejpam-4698	264	9	.	.	PUNCT