id	sid	tid	token	lemma	pos
ma-166	1	1	2023	2023	NUM
ma-166	1	2	ada	ada	PROPN
ma-166	1	3	academica	academica	PROPN
ma-166	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-166	1	5	.	.	PUNCT
ma-166	2	1	j.	j.	PROPN
ma-166	2	2	math	math	PROPN
ma-166	2	3	.	.	PUNCT
ma-166	3	1	anal	anal	ADJ
ma-166	3	2	.	.	PUNCT
ma-166	4	1	3	3	NUM
ma-166	4	2	(	(	PUNCT
ma-166	4	3	2023	2023	NUM
ma-166	4	4	)	)	PUNCT
ma-166	4	5	20doi	20doi	NOUN
ma-166	4	6	:	:	PUNCT
ma-166	4	7	10.28924	10.28924	NUM
ma-166	4	8	/	/	SYM
ma-166	4	9	ada	ada	PROPN
ma-166	4	10	/	/	SYM
ma-166	4	11	ma.3.20	ma.3.20	PROPN
ma-166	5	1	on	on	ADP
ma-166	5	2	degree	degree	NOUN
ma-166	5	3	-	-	PUNCT
ma-166	5	4	based	base	VERB
ma-166	5	5	topological	topological	ADJ
ma-166	5	6	indices	index	NOUN
ma-166	5	7	of	of	ADP
ma-166	5	8	petersen	petersen	PROPN
ma-166	5	9	subdivision	subdivision	NOUN
ma-166	5	10	graph	graph	NOUN
ma-166	5	11	mukhtar	mukhtar	PROPN
ma-166	5	12	ahmad1	ahmad1	PROPN
ma-166	5	13	,	,	PUNCT
ma-166	5	14	saddam	saddam	PROPN
ma-166	5	15	hussain2	hussain2	PROPN
ma-166	5	16	,	,	PUNCT
ma-166	5	17	ulfat	ulfat	PROPN
ma-166	5	18	parveen3	parveen3	PROPN
ma-166	5	19	,	,	PUNCT
ma-166	5	20	iqra	iqra	NOUN
ma-166	5	21	zahid3	zahid3	PROPN
ma-166	5	22	,	,	PUNCT
ma-166	5	23	muhammad	muhammad	X
ma-166	6	1	sultan3,ather	sultan3,ather	SCONJ
ma-166	6	2	qayyum3,∗	qayyum3,∗	PRON
ma-166	6	3	1department	1department	NUM
ma-166	6	4	of	of	ADP
ma-166	6	5	mathematics	mathematic	NOUN
ma-166	6	6	,	,	PUNCT
ma-166	6	7	khawaja	khawaja	PROPN
ma-166	6	8	fareed	fareed	PROPN
ma-166	6	9	university	university	PROPN
ma-166	6	10	of	of	ADP
ma-166	6	11	engineering	engineering	NOUN
ma-166	6	12	and	and	CCONJ
ma-166	6	13	information	information	NOUN
ma-166	6	14	technology	technology	PROPN
ma-166	6	15	rahim	rahim	PROPN
ma-166	6	16	yar	yar	PROPN
ma-166	6	17	khan	khan	PROPN
ma-166	6	18	,	,	PUNCT
ma-166	6	19	pakistan	pakistan	PROPN
ma-166	6	20	itxmemuktar@gmail.com	itxmemuktar@gmail.com	PROPN
ma-166	7	1	2department	2department	NUM
ma-166	7	2	of	of	ADP
ma-166	7	3	statistics	statistic	NOUN
ma-166	7	4	,	,	PUNCT
ma-166	7	5	university	university	NOUN
ma-166	7	6	of	of	ADP
ma-166	7	7	mian	mian	PROPN
ma-166	7	8	wali	wali	PROPN
ma-166	7	9	,	,	PUNCT
ma-166	7	10	pakistan	pakistan	PROPN
ma-166	7	11	saddamhussain.stat885@gmail.com	saddamhussain.stat885@gmail.com	PROPN
ma-166	7	12	3department	3department	NUM
ma-166	7	13	of	of	ADP
ma-166	7	14	mathematics	mathematic	NOUN
ma-166	7	15	,	,	PUNCT
ma-166	7	16	institute	institute	NOUN
ma-166	7	17	of	of	ADP
ma-166	7	18	southern	southern	PROPN
ma-166	7	19	punjab	punjab	PROPN
ma-166	7	20	multan	multan	PROPN
ma-166	7	21	,	,	PUNCT
ma-166	7	22	pakistan	pakistan	PROPN
ma-166	7	23	uulfat05@gmail.com	uulfat05@gmail.com	PROPN
ma-166	7	24	,	,	PUNCT
ma-166	7	25	iqraimran57@gmail.com	iqraimran57@gmail.com	PROPN
ma-166	7	26	,	,	PUNCT
ma-166	7	27	sultan.sadeeq7866127@gmail.com	sultan.sadeeq7866127@gmail.com	X
ma-166	7	28	,	,	PUNCT
ma-166	7	29	atherqayyum@isp.edu.pk	atherqayyum@isp.edu.pk	VERB
ma-166	7	30	∗correspondence	∗correspondence	NOUN
ma-166	7	31	:	:	PUNCT
ma-166	7	32	atherqayyum@isp.edu.pk	atherqayyum@isp.edu.pk	NOUN
ma-166	7	33	abstract	abstract	ADJ
ma-166	7	34	.	.	PUNCT
ma-166	8	1	in	in	ADP
ma-166	8	2	this	this	DET
ma-166	8	3	paper	paper	NOUN
ma-166	8	4	,	,	PUNCT
ma-166	8	5	we	we	PRON
ma-166	8	6	adequately	adequately	ADV
ma-166	8	7	describe	describe	VERB
ma-166	8	8	the	the	DET
ma-166	8	9	generalised	generalise	VERB
ma-166	8	10	petersen	petersen	NOUN
ma-166	8	11	graph	graph	NOUN
ma-166	8	12	,	,	PUNCT
ma-166	8	13	expanding	expand	VERB
ma-166	8	14	to	to	ADP
ma-166	8	15	thecategories	thecategorie	NOUN
ma-166	8	16	of	of	ADP
ma-166	8	17	graphs	graph	NOUN
ma-166	8	18	.	.	PUNCT
ma-166	9	1	we	we	PRON
ma-166	9	2	created	create	VERB
ma-166	9	3	a	a	DET
ma-166	9	4	petersen	petersen	NOUN
ma-166	9	5	graph	graph	NOUN
ma-166	9	6	,	,	PUNCT
ma-166	9	7	which	which	PRON
ma-166	9	8	is	be	AUX
ma-166	9	9	cyclic	cyclic	ADJ
ma-166	9	10	and	and	CCONJ
ma-166	9	11	has	have	VERB
ma-166	9	12	vertices	vertex	NOUN
ma-166	9	13	that	that	PRON
ma-166	9	14	are	be	AUX
ma-166	9	15	arrangedin	arrangedin	VERB
ma-166	9	16	the	the	DET
ma-166	9	17	centre	centre	NOUN
ma-166	9	18	and	and	CCONJ
ma-166	9	19	nine	nine	NUM
ma-166	9	20	gons	gon	NOUN
ma-166	9	21	plus	plus	CCONJ
ma-166	9	22	one	one	NUM
ma-166	9	23	vertex	vertex	NOUN
ma-166	9	24	,	,	PUNCT
ma-166	9	25	leading	lead	VERB
ma-166	9	26	to	to	ADP
ma-166	9	27	the	the	DET
ma-166	9	28	factorization	factorization	NOUN
ma-166	9	29	of	of	ADP
ma-166	9	30	regular	regular	ADJ
ma-166	9	31	graphs	graph	NOUN
ma-166	9	32	.	.	PUNCT
ma-166	10	1	petersengraph	petersengraph	NOUN
ma-166	10	2	is	be	AUX
ma-166	10	3	still	still	ADV
ma-166	10	4	shown	show	VERB
ma-166	10	5	in	in	ADP
ma-166	10	6	graph	graph	NOUN
ma-166	10	7	theory	theory	NOUN
ma-166	10	8	literature	literature	NOUN
ma-166	10	9	,	,	PUNCT
ma-166	10	10	nevertheless	nevertheless	ADV
ma-166	10	11	.	.	PUNCT
ma-166	11	1	1	1	X
ma-166	11	2	.	.	X
ma-166	11	3	introduction	introduction	NOUN
ma-166	11	4	named	name	VERB
ma-166	11	5	after	after	ADP
ma-166	11	6	julius	julius	PROPN
ma-166	11	7	petersen	petersen	PROPN
ma-166	11	8	,	,	PUNCT
ma-166	11	9	a	a	DET
ma-166	11	10	danish	danish	ADJ
ma-166	11	11	mathematician	mathematician	NOUN
ma-166	11	12	,	,	PUNCT
ma-166	11	13	the	the	DET
ma-166	11	14	graph	graph	NOUN
ma-166	11	15	of	of	ADP
ma-166	11	16	petersen	petersen	PROPN
ma-166	11	17	is(from	is(from	ADP
ma-166	11	18	1839	1839	NUM
ma-166	11	19	to1910	to1910	NOUN
ma-166	11	20	)	)	PUNCT
ma-166	11	21	.	.	PUNCT
ma-166	12	1	petersen	petersen	PROPN
ma-166	12	2	researched	research	VERB
ma-166	12	3	factorizations	factorization	NOUN
ma-166	12	4	of	of	ADP
ma-166	12	5	normal	normal	ADJ
ma-166	12	6	factorizations	factorization	NOUN
ma-166	12	7	during	during	ADP
ma-166	12	8	the	the	DET
ma-166	12	9	1890s	1890	NOUN
ma-166	12	10	.	.	PUNCT
ma-166	13	1	in	in	ADP
ma-166	13	2	1891	1891	NUM
ma-166	13	3	,	,	PUNCT
ma-166	13	4	asignificant	asignificant	ADJ
ma-166	13	5	paper	paper	NOUN
ma-166	13	6	of	of	ADP
ma-166	13	7	graphs	graph	NOUN
ma-166	13	8	was	be	AUX
ma-166	13	9	published	publish	VERB
ma-166	13	10	which	which	PRON
ma-166	13	11	is	be	AUX
ma-166	13	12	commemorated	commemorate	VERB
ma-166	13	13	in	in	ADP
ma-166	13	14	that	that	DET
ma-166	13	15	volume	volume	NOUN
ma-166	13	16	.	.	PUNCT
ma-166	14	1	petersen	petersen	PROPN
ma-166	14	2	provedthat	provedthat	VERB
ma-166	14	3	any	any	DET
ma-166	14	4	graph	graph	NOUN
ma-166	14	5	of	of	ADP
ma-166	14	6	3	3	NUM
ma-166	14	7	-	-	NOUN
ma-166	14	8	regular	regular	NOUN
ma-166	14	9	with	with	ADP
ma-166	14	10	at	at	ADP
ma-166	14	11	a	a	DET
ma-166	14	12	i	i	NOUN
ma-166	14	13	-	-	PUNCT
ma-166	14	14	factor	factor	NOUN
ma-166	14	15	includes	include	VERB
ma-166	14	16	much	much	ADJ
ma-166	14	17	of	of	ADP
ma-166	14	18	the	the	DET
ma-166	14	19	two	two	NUM
ma-166	14	20	bridges	bridge	NOUN
ma-166	14	21	.	.	PUNCT
ma-166	15	1	tait	tait	PROPN
ma-166	15	2	had	have	AUX
ma-166	15	3	writtena	writtena	VERB
ma-166	15	4	few	few	ADJ
ma-166	15	5	years	year	NOUN
ma-166	15	6	ago	ago	ADV
ma-166	15	7	that	that	SCONJ
ma-166	15	8	he	he	PRON
ma-166	15	9	had	have	AUX
ma-166	15	10	shown	show	VERB
ma-166	15	11	i	i	PRON
ma-166	15	12	-	-	NOUN
ma-166	15	13	factorable	factorable	ADJ
ma-166	15	14	for	for	ADP
ma-166	15	15	each	each	DET
ma-166	15	16	3	3	NUM
ma-166	15	17	-	-	PUNCT
ma-166	15	18	regular	regular	ADJ
ma-166	15	19	graph	graph	NOUN
ma-166	15	20	,	,	PUNCT
ma-166	15	21	but	but	CCONJ
ma-166	15	22	that	that	SCONJ
ma-166	15	23	this	this	DET
ma-166	15	24	outcomeit	outcomeit	NOUN
ma-166	15	25	was	be	AUX
ma-166	15	26	not	not	PART
ma-166	15	27	valid	valid	ADJ
ma-166	15	28	without	without	ADP
ma-166	15	29	restriction	restriction	NOUN
ma-166	15	30	.	.	PUNCT
ma-166	16	1	but	but	CCONJ
ma-166	16	2	tait	tait	PROPN
ma-166	16	3	’s	’s	PART
ma-166	16	4	comment	comment	NOUN
ma-166	16	5	in	in	ADP
ma-166	16	6	1898	1898	NUM
ma-166	16	7	was	be	AUX
ma-166	16	8	interpreted	interpret	VERB
ma-166	16	9	by	by	ADP
ma-166	16	10	petersen	petersen	PROPN
ma-166	16	11	toimply	toimply	ADV
ma-166	16	12	that	that	SCONJ
ma-166	16	13	each	each	DET
ma-166	16	14	3	3	NUM
ma-166	16	15	-	-	PUNCT
ma-166	16	16	regular	regular	ADJ
ma-166	16	17	bridge	bridge	NOUN
ma-166	16	18	less	less	ADJ
ma-166	16	19	graph	graph	NOUN
ma-166	16	20	is	be	AUX
ma-166	16	21	l	l	NOUN
ma-166	16	22	-	-	ADJ
ma-166	16	23	factorable	factorable	ADJ
ma-166	16	24	.	.	PUNCT
ma-166	17	1	if	if	SCONJ
ma-166	17	2	this	this	DET
ma-166	17	3	outcome	outcome	NOUN
ma-166	17	4	were	be	AUX
ma-166	17	5	valid	valid	ADJ
ma-166	17	6	,	,	PUNCT
ma-166	17	7	then	then	ADV
ma-166	17	8	itwould	itwould	AUX
ma-166	17	9	have	have	AUX
ma-166	17	10	been	be	AUX
ma-166	17	11	stronger	strong	ADJ
ma-166	17	12	than	than	SCONJ
ma-166	17	13	theorem	theorem	VERB
ma-166	17	14	for	for	ADP
ma-166	17	15	petersen	petersen	PROPN
ma-166	17	16	.	.	PUNCT
ma-166	18	1	the	the	DET
ma-166	18	2	key	key	ADJ
ma-166	18	3	characteristics	characteristic	NOUN
ma-166	18	4	of	of	ADP
ma-166	18	5	the	the	DET
ma-166	18	6	petersengraph	petersengraph	NOUN
ma-166	18	7	were	be	AUX
ma-166	18	8	examined	examine	VERB
ma-166	18	9	in	in	ADP
ma-166	18	10	detail	detail	NOUN
ma-166	18	11	in	in	ADP
ma-166	18	12	1985	1985	NUM
ma-166	18	13	.	.	PUNCT
ma-166	19	1	the	the	DET
ma-166	19	2	graph	graph	NOUN
ma-166	19	3	of	of	ADP
ma-166	19	4	petersen	petersen	NOUN
ma-166	19	5	continously	continously	ADV
ma-166	19	6	to	to	PART
ma-166	19	7	express	express	VERB
ma-166	19	8	in	in	ADP
ma-166	19	9	the	the	DET
ma-166	19	10	entiregraph	entiregraph	NOUN
ma-166	19	11	-	-	PUNCT
ma-166	19	12	theory	theory	NOUN
ma-166	19	13	education	education	NOUN
ma-166	19	14	.	.	PUNCT
ma-166	20	1	we	we	PRON
ma-166	20	2	update	update	VERB
ma-166	20	3	our	our	PRON
ma-166	20	4	previous	previous	ADJ
ma-166	20	5	analysis	analysis	NOUN
ma-166	20	6	in	in	ADP
ma-166	20	7	the	the	DET
ma-166	20	8	present	present	ADJ
ma-166	20	9	article	article	NOUN
ma-166	20	10	by	by	ADP
ma-166	20	11	denoting	denote	VERB
ma-166	20	12	extrarecently	extrarecently	ADJ
ma-166	20	13	findings	finding	NOUN
ma-166	20	14	concerning	concern	VERB
ma-166	20	15	the	the	DET
ma-166	20	16	petersen	petersen	PROPN
ma-166	20	17	graph.julius	graph.julius	PROPN
ma-166	20	18	petersen	petersen	PROPN
ma-166	20	19	’s	’s	PART
ma-166	20	20	’	'	PUNCT
ma-166	20	21	die	die	PROPN
ma-166	20	22	theorie	theorie	PROPN
ma-166	20	23	der	der	PROPN
ma-166	20	24	regulken	regulken	ADJ
ma-166	20	25	graphs	graph	NOUN
ma-166	20	26	’	'	PUNCT
ma-166	20	27	is	be	AUX
ma-166	20	28	an	an	DET
ma-166	20	29	exceptional	exceptional	ADJ
ma-166	20	30	paper	paper	NOUN
ma-166	20	31	that	that	PRON
ma-166	20	32	developed	develop	VERB
ma-166	20	33	a	a	DET
ma-166	20	34	new	new	ADJ
ma-166	20	35	received	receive	VERB
ma-166	20	36	:	:	PUNCT
ma-166	20	37	15	15	NUM
ma-166	20	38	apr	apr	NOUN
ma-166	20	39	2023	2023	NUM
ma-166	20	40	.	.	PUNCT
ma-166	21	1	key	key	ADJ
ma-166	21	2	words	word	NOUN
ma-166	21	3	and	and	CCONJ
ma-166	21	4	phrases	phrase	NOUN
ma-166	21	5	.	.	PUNCT
ma-166	22	1	atom	atom	NOUN
ma-166	22	2	-	-	PUNCT
ma-166	22	3	bond	bond	NOUN
ma-166	22	4	connectivity	connectivity	NOUN
ma-166	22	5	index	index	NOUN
ma-166	22	6	;	;	PUNCT
ma-166	22	7	reduced	reduced	ADJ
ma-166	22	8	zagreb	zagreb	PROPN
ma-166	22	9	;	;	PUNCT
ma-166	22	10	randic	randic	ADJ
ma-166	22	11	indices	index	NOUN
ma-166	22	12	;	;	PUNCT
ma-166	22	13	general	general	ADJ
ma-166	22	14	connectivity	connectivity	NOUN
ma-166	22	15	index;petersen	index;petersen	PROPN
ma-166	22	16	graph	graph	NOUN
ma-166	22	17	.	.	PUNCT
ma-166	23	1	1	1	NUM
ma-166	23	2	https://adac.ee	https://adac.ee	PROPN
ma-166	23	3	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	23	4	eur	eur	PROPN
ma-166	23	5	.	.	PUNCT
ma-166	24	1	j.	j.	PROPN
ma-166	24	2	math	math	PROPN
ma-166	24	3	.	.	PUNCT
ma-166	25	1	anal	anal	PROPN
ma-166	25	2	.	.	PUNCT
ma-166	26	1	10.28924	10.28924	NUM
ma-166	26	2	/	/	SYM
ma-166	26	3	ada	ada	PROPN
ma-166	26	4	/	/	SYM
ma-166	26	5	ma.3.20	ma.3.20	PROPN
ma-166	27	1	2theory	2theory	NUM
ma-166	27	2	in	in	ADP
ma-166	27	3	graph	graph	NOUN
ma-166	27	4	theory	theory	NOUN
ma-166	27	5	,	,	PUNCT
ma-166	27	6	based	base	VERB
ma-166	27	7	on	on	ADP
ma-166	27	8	the	the	DET
ma-166	27	9	exchange	exchange	NOUN
ma-166	27	10	property	property	NOUN
ma-166	27	11	of	of	ADP
ma-166	27	12	trees	tree	NOUN
ma-166	27	13	spanning	span	VERB
ma-166	27	14	and	and	CCONJ
ma-166	27	15	the	the	DET
ma-166	27	16	cyclomaticnumber	cyclomaticnumber	NOUN
ma-166	27	17	of	of	ADP
ma-166	27	18	trees	tree	NOUN
ma-166	27	19	spanningresently	spanningresently	ADV
ma-166	27	20	resently	resently	ADV
ma-166	27	21	zaib	zaib	PROPN
ma-166	27	22	hassan	hassan	PROPN
ma-166	27	23	niazi	niazi	PROPN
ma-166	27	24	et.al[15	et.al[15	PROPN
ma-166	27	25	]	]	PUNCT
ma-166	27	26	.	.	PUNCT
ma-166	28	1	1.1	1.1	NUM
ma-166	28	2	.	.	PUNCT
ma-166	29	1	the	the	DET
ma-166	29	2	graph	graph	NOUN
ma-166	29	3	of	of	ADP
ma-166	29	4	petersen	petersen	PROPN
ma-166	29	5	.	.	PUNCT
ma-166	30	1	every	every	DET
ma-166	30	2	petersen	petersen	NOUN
ma-166	30	3	graph	graph	NOUN
ma-166	30	4	is	be	AUX
ma-166	30	5	cyclic	cyclic	ADJ
ma-166	30	6	graph	graph	NOUN
ma-166	30	7	and	and	CCONJ
ma-166	30	8	the	the	DET
ma-166	30	9	graph	graph	NOUN
ma-166	30	10	g′	g′	NOUN
ma-166	30	11	in	in	ADP
ma-166	30	12	general	general	ADJ
ma-166	30	13	formconsists	formconsist	NOUN
ma-166	30	14	v	v	ADP
ma-166	30	15	having	having	AUX
ma-166	30	16	set	set	VERB
ma-166	30	17	of	of	ADP
ma-166	30	18	vertex	vertex	NOUN
ma-166	30	19	and	and	CCONJ
ma-166	30	20	e	e	NOUN
ma-166	30	21	having	having	AUX
ma-166	30	22	set	set	VERB
ma-166	30	23	of	of	ADP
ma-166	30	24	edge	edge	NOUN
ma-166	30	25	,	,	PUNCT
ma-166	30	26	if	if	SCONJ
ma-166	30	27	the	the	DET
ma-166	30	28	natural	natural	ADJ
ma-166	30	29	number	number	NOUN
ma-166	30	30	,	,	PUNCT
ma-166	30	31	there	there	PRON
ma-166	30	32	exist	exist	VERB
ma-166	30	33	n	n	PRON
ma-166	30	34	thegraph	thegraph	NOUN
ma-166	30	35	with	with	ADP
ma-166	30	36	vertices	vertex	NOUN
ma-166	30	37	are	be	AUX
ma-166	30	38	v	v	ADJ
ma-166	30	39	(	(	PUNCT
ma-166	30	40	g′	g′	NOUN
ma-166	30	41	)	)	PUNCT
ma-166	31	1	=	=	SYM
ma-166	31	2	4n	4n	NOUN
ma-166	31	3	,	,	PUNCT
ma-166	31	4	edges	edge	NOUN
ma-166	31	5	are	be	AUX
ma-166	31	6	e(g′	e(g′	PRON
ma-166	31	7	)	)	PUNCT
ma-166	32	1	=	=	SYM
ma-166	32	2	6n	6n	NOUN
ma-166	32	3	,	,	PUNCT
ma-166	32	4	and	and	CCONJ
ma-166	32	5	the	the	DET
ma-166	32	6	specific	specific	NOUN
ma-166	32	7	of	of	ADP
ma-166	32	8	this	this	DET
ma-166	32	9	graph	graph	NOUN
ma-166	32	10	is	be	AUX
ma-166	32	11	thatabout	thatabout	ADP
ma-166	32	12	degree	degree	NOUN
ma-166	32	13	of	of	ADP
ma-166	32	14	every	every	PRON
ma-166	32	15	each	each	DET
ma-166	32	16	vertex	vertex	NOUN
ma-166	32	17	is	be	AUX
ma-166	32	18	p(k	p(k	NOUN
ma-166	32	19	,	,	PUNCT
ma-166	32	20	t	t	NOUN
ma-166	32	21	)	)	PUNCT
ma-166	32	22	=	=	PUNCT
ma-166	33	1	[	[	X
ma-166	33	2	d(x1	d(x1	NOUN
ma-166	33	3	)	)	PUNCT
ma-166	33	4	,	,	PUNCT
ma-166	33	5	d(x2	d(x2	NOUN
ma-166	33	6	)	)	PUNCT
ma-166	33	7	]	]	PUNCT
ma-166	34	1	=	=	PUNCT
ma-166	34	2	3	3	X
ma-166	34	3	.	.	PUNCT
ma-166	34	4	then	then	ADV
ma-166	34	5	this	this	DET
ma-166	34	6	graphic	graphic	NOUN
ma-166	34	7	which	which	PRON
ma-166	34	8	is	be	AUX
ma-166	34	9	said	say	VERB
ma-166	34	10	tobe	tobe	PROPN
ma-166	34	11	petersen	petersen	PROPN
ma-166	34	12	graphic	graphic	PROPN
ma-166	34	13	.	.	PUNCT
ma-166	35	1	then	then	ADV
ma-166	35	2	petersen	petersen	PROPN
ma-166	35	3	graphic	graphic	PROPN
ma-166	35	4	is	be	AUX
ma-166	35	5	denoted	denote	VERB
ma-166	35	6	by	by	ADP
ma-166	35	7	p[v	p[v	PROPN
ma-166	35	8	(	(	PUNCT
ma-166	35	9	g′),e(g′	g′),e(g′	PROPN
ma-166	35	10	)	)	PUNCT
ma-166	35	11	]	]	PUNCT
ma-166	36	1	=	=	PUNCT
ma-166	36	2	(	(	PUNCT
ma-166	36	3	4n	4n	X
ma-166	36	4	,	,	PUNCT
ma-166	36	5	6n	6n	NUM
ma-166	36	6	)	)	PUNCT
ma-166	36	7	petersen	petersen	NOUN
ma-166	36	8	mapexplored	mapexplore	VERB
ma-166	36	9	by	by	ADP
ma-166	36	10	1985	1985	NUM
ma-166	36	11	,	,	PUNCT
ma-166	36	12	updated	update	VERB
ma-166	36	13	by	by	ADP
ma-166	36	14	new	new	ADJ
ma-166	36	15	analysis.sylvester	analysis.sylvester	PROPN
ma-166	36	16	’s	’s	PART
ma-166	36	17	association	association	NOUN
ma-166	36	18	to	to	ADP
ma-166	36	19	graphs	graph	NOUN
ma-166	36	20	of	of	ADP
ma-166	36	21	invariants	invariant	NOUN
ma-166	36	22	and	and	CCONJ
ma-166	36	23	covariants	covariant	NOUN
ma-166	36	24	requires	require	VERB
ma-166	36	25	interpretation	interpretation	NOUN
ma-166	36	26	of	of	ADP
ma-166	36	27	principleof	principleof	ADJ
ma-166	36	28	invariants	invariant	NOUN
ma-166	36	29	in	in	ADP
ma-166	36	30	1880s	1880	NOUN
ma-166	36	31	.	.	PUNCT
ma-166	37	1	1.2	1.2	NUM
ma-166	37	2	.	.	PUNCT
ma-166	37	3	graphical	graphical	ADJ
ma-166	37	4	idea	idea	NOUN
ma-166	37	5	of	of	ADP
ma-166	37	6	petersen	petersen	PROPN
ma-166	37	7	graph	graph	NOUN
ma-166	37	8	.	.	PUNCT
ma-166	38	1	if	if	SCONJ
ma-166	38	2	the	the	DET
ma-166	38	3	set	set	NOUN
ma-166	38	4	of	of	ADP
ma-166	38	5	natural	natural	ADJ
ma-166	38	6	number	number	NOUN
ma-166	38	7	is	be	AUX
ma-166	38	8	tn	tn	NOUN
ma-166	38	9	=	=	SYM
ma-166	38	10	{	{	PUNCT
ma-166	38	11	1	1	NUM
ma-166	38	12	,	,	PUNCT
ma-166	38	13	2	2	NUM
ma-166	38	14	,	,	PUNCT
ma-166	38	15	3	3	NUM
ma-166	38	16	,	,	PUNCT
ma-166	38	17	...	...	PUNCT
ma-166	38	18	}	}	PUNCT
ma-166	38	19	,	,	PUNCT
ma-166	38	20	if	if	SCONJ
ma-166	38	21	thereexists	thereexist	NOUN
ma-166	38	22	n	n	CCONJ
ma-166	38	23	then	then	ADV
ma-166	38	24	graph	graph	VERB
ma-166	38	25	with	with	ADP
ma-166	38	26	vertices	vertex	NOUN
ma-166	38	27	are	be	AUX
ma-166	38	28	v	v	ADJ
ma-166	38	29	(	(	PUNCT
ma-166	38	30	g′	g′	NOUN
ma-166	38	31	)	)	PUNCT
ma-166	38	32	=	=	PUNCT
ma-166	38	33	4n	4n	NOUN
ma-166	38	34	,	,	PUNCT
ma-166	38	35	and	and	CCONJ
ma-166	38	36	edges	edge	VERB
ma-166	38	37	e(g′	e(g′	PRON
ma-166	38	38	)	)	PUNCT
ma-166	39	1	=	=	SYM
ma-166	39	2	6n	6n	NOUN
ma-166	39	3	,	,	PUNCT
ma-166	39	4	in	in	ADP
ma-166	39	5	general	general	ADJ
ma-166	39	6	form	form	NOUN
ma-166	39	7	ofpetersen	ofpetersen	NOUN
ma-166	39	8	expressed	express	VERB
ma-166	39	9	by	by	ADP
ma-166	39	10	p[v	p[v	PROPN
ma-166	39	11	(	(	PUNCT
ma-166	39	12	g′),e(g′	g′),e(g′	PROPN
ma-166	39	13	)	)	PUNCT
ma-166	39	14	]	]	PUNCT
ma-166	40	1	=	=	PUNCT
ma-166	40	2	(	(	PUNCT
ma-166	40	3	4n	4n	NOUN
ma-166	40	4	,	,	PUNCT
ma-166	40	5	6n	6n	PROPN
ma-166	40	6	)	)	PUNCT
ma-166	40	7	.	.	PUNCT
ma-166	41	1	this	this	DET
ma-166	41	2	graph	graph	NOUN
ma-166	41	3	having	have	VERB
ma-166	41	4	a	a	DET
ma-166	41	5	specification	specification	NOUN
ma-166	41	6	,	,	PUNCT
ma-166	41	7	that	that	DET
ma-166	41	8	degree	degree	NOUN
ma-166	41	9	ofevery	ofevery	NOUN
ma-166	41	10	each	each	DET
ma-166	41	11	vertex	vertex	NOUN
ma-166	41	12	is	be	AUX
ma-166	41	13	p(k	p(k	NOUN
ma-166	41	14	,	,	PUNCT
ma-166	41	15	t	t	NOUN
ma-166	41	16	)	)	PUNCT
ma-166	41	17	=	=	PUNCT
ma-166	42	1	[	[	X
ma-166	42	2	d(x1	d(x1	NOUN
ma-166	42	3	)	)	PUNCT
ma-166	42	4	,	,	PUNCT
ma-166	42	5	d(x2	d(x2	NOUN
ma-166	42	6	)	)	PUNCT
ma-166	42	7	]	]	PUNCT
ma-166	43	1	=	=	PUNCT
ma-166	43	2	3.now	3.now	INTJ
ma-166	43	3	we	we	PRON
ma-166	43	4	write	write	VERB
ma-166	43	5	;	;	PUNCT
ma-166	43	6	tn	tn	NOUN
ma-166	43	7	=	=	SYM
ma-166	43	8	1	1	NUM
ma-166	43	9	,	,	PUNCT
ma-166	43	10	2	2	NUM
ma-166	43	11	,	,	PUNCT
ma-166	43	12	3	3	NUM
ma-166	43	13	,	,	PUNCT
ma-166	43	14	....	....	PUNCT
ma-166	43	15	v	v	X
ma-166	43	16	(	(	PUNCT
ma-166	43	17	g′	g′	NOUN
ma-166	43	18	)	)	PUNCT
ma-166	43	19	=	=	NOUN
ma-166	44	1	4n	4n	X
ma-166	44	2	→	→	PUNCT
ma-166	44	3	[	[	X
ma-166	44	4	1	1	NUM
ma-166	44	5	]	]	PUNCT
ma-166	44	6	e(g′	e(g′	X
ma-166	44	7	)	)	PUNCT
ma-166	45	1	=	=	SYM
ma-166	46	1	6n	6n	NUM
ma-166	46	2	→	→	PUNCT
ma-166	46	3	[	[	X
ma-166	46	4	2	2	NUM
ma-166	46	5	]	]	X
ma-166	46	6	k	k	NOUN
ma-166	46	7	=	=	SYM
ma-166	46	8	d(x1	d(x1	NOUN
ma-166	46	9	)	)	PUNCT
ma-166	46	10	=	=	SYM
ma-166	46	11	3	3	NUM
ma-166	46	12	t	t	NOUN
ma-166	46	13	=	=	PUNCT
ma-166	46	14	d(x2	d(x2	NOUN
ma-166	46	15	)	)	PUNCT
ma-166	46	16	=	=	SYM
ma-166	46	17	3	3	NUM
ma-166	46	18	then	then	ADV
ma-166	46	19	p(k	p(k	NOUN
ma-166	46	20	,	,	PUNCT
ma-166	46	21	t	t	NOUN
ma-166	46	22	)	)	PUNCT
ma-166	46	23	=	=	PUNCT
ma-166	47	1	[	[	X
ma-166	47	2	d(x1	d(x1	NOUN
ma-166	47	3	)	)	PUNCT
ma-166	47	4	,	,	PUNCT
ma-166	47	5	d(x2	d(x2	NOUN
ma-166	47	6	)	)	PUNCT
ma-166	47	7	]	]	PUNCT
ma-166	48	1	=	=	SYM
ma-166	48	2	3	3	NUM
ma-166	48	3	p[v	p[v	NUM
ma-166	48	4	(	(	PUNCT
ma-166	48	5	g′),e(g′	g′),e(g′	PROPN
ma-166	48	6	)	)	PUNCT
ma-166	48	7	]	]	PUNCT
ma-166	49	1	=	=	PUNCT
ma-166	49	2	(	(	PUNCT
ma-166	49	3	4n	4n	NOUN
ma-166	49	4	,	,	PUNCT
ma-166	49	5	6n	6n	PROPN
ma-166	49	6	)	)	PUNCT
ma-166	49	7	.	.	PUNCT
ma-166	50	1	next	next	ADV
ma-166	50	2	we	we	PRON
ma-166	50	3	discuss	discuss	VERB
ma-166	50	4	the	the	DET
ma-166	50	5	topological	topological	ADJ
ma-166	50	6	indiceszagreb	indiceszagreb	NOUN
ma-166	50	7	indices	index	NOUN
ma-166	50	8	of	of	ADP
ma-166	50	9	the	the	DET
ma-166	50	10	group	group	NOUN
ma-166	50	11	were	be	AUX
ma-166	50	12	recognized	recognize	VERB
ma-166	50	13	in	in	ADP
ma-166	50	14	the	the	DET
ma-166	50	15	early	early	ADJ
ma-166	50	16	1980s	1980	NOUN
ma-166	50	17	and	and	CCONJ
ma-166	50	18	are	be	AUX
ma-166	50	19	now	now	ADV
ma-166	50	20	known	know	VERB
ma-166	50	21	as	as	ADP
ma-166	50	22	thefirst	thefirst	ADJ
ma-166	50	23	and	and	CCONJ
ma-166	50	24	second	second	ADJ
ma-166	50	25	zagreb	zagreb	PROPN
ma-166	50	26	indices	index	NOUN
ma-166	50	27	.	.	PUNCT
ma-166	51	1	they	they	PRON
ma-166	51	2	are	be	AUX
ma-166	51	3	important	important	ADJ
ma-166	51	4	molecular	molecular	ADJ
ma-166	51	5	descriptors	descriptor	NOUN
ma-166	51	6	and	and	CCONJ
ma-166	51	7	have	have	AUX
ma-166	51	8	been	be	AUX
ma-166	51	9	closelycorrelated	closelycorrelate	VERB
ma-166	51	10	with	with	ADP
ma-166	51	11	chemical	chemical	NOUN
ma-166	51	12	properties.[degree	properties.[degree	NOUN
ma-166	51	13	based	base	VERB
ma-166	51	14	topological	topological	ADJ
ma-166	51	15	indices]the	indices]the	DET
ma-166	51	16	first	first	PROPN
ma-166	51	17	zagreb	zagreb	PROPN
ma-166	51	18	index	index	PROPN
ma-166	51	19	m1(g	m1(g	NOUN
ma-166	51	20	)	)	PUNCT
ma-166	51	21	is	be	AUX
ma-166	51	22	equal	equal	ADJ
ma-166	51	23	to	to	ADP
ma-166	51	24	the	the	DET
ma-166	51	25	sum	sum	NOUN
ma-166	51	26	of	of	ADP
ma-166	51	27	the	the	DET
ma-166	51	28	squares	square	NOUN
ma-166	51	29	of	of	ADP
ma-166	51	30	the	the	DET
ma-166	51	31	degrees	degree	NOUN
ma-166	51	32	of	of	ADP
ma-166	51	33	the	the	DET
ma-166	51	34	vertices	vertex	NOUN
ma-166	51	35	for	for	ADP
ma-166	51	36	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	PROPN
ma-166	51	37	eur	eur	PROPN
ma-166	51	38	.	.	PUNCT
ma-166	52	1	j.	j.	PROPN
ma-166	52	2	math	math	PROPN
ma-166	52	3	.	.	PUNCT
ma-166	53	1	anal	anal	PROPN
ma-166	53	2	.	.	PUNCT
ma-166	54	1	10.28924	10.28924	NUM
ma-166	54	2	/	/	SYM
ma-166	54	3	ada	ada	PROPN
ma-166	54	4	/	/	SYM
ma-166	54	5	ma.3.20	ma.3.20	NOUN
ma-166	54	6	3the	3the	DET
ma-166	54	7	(	(	PUNCT
ma-166	54	8	molecular	molecular	ADJ
ma-166	54	9	)	)	PUNCT
ma-166	54	10	graph	graph	NOUN
ma-166	54	11	g[1	g[1	PROPN
ma-166	54	12	]	]	PUNCT
ma-166	54	13	.	.	PUNCT
ma-166	55	1	it	it	PRON
ma-166	55	2	can	can	AUX
ma-166	55	3	also	also	ADV
ma-166	55	4	be	be	AUX
ma-166	55	5	considered	consider	VERB
ma-166	55	6	as	as	ADP
ma-166	55	7	the	the	DET
ma-166	55	8	sum	sum	NOUN
ma-166	55	9	over	over	ADP
ma-166	55	10	the	the	DET
ma-166	55	11	edges	edge	NOUN
ma-166	55	12	of	of	ADP
ma-166	55	13	g	g	NOUN
ma-166	55	14	,	,	PUNCT
ma-166	55	15	and	and	CCONJ
ma-166	55	16	m1(g)is	m1(g)is	NUM
ma-166	55	17	defined	define	VERB
ma-166	55	18	as:[the	as:[the	DET
ma-166	55	19	first	first	ADJ
ma-166	55	20	and	and	CCONJ
ma-166	55	21	second	second	ADJ
ma-166	55	22	zagreb	zagreb	PROPN
ma-166	55	23	indices	index	NOUN
ma-166	55	24	of	of	ADP
ma-166	55	25	some	some	DET
ma-166	55	26	graph	graph	NOUN
ma-166	55	27	operations	operation	NOUN
ma-166	55	28	]	]	PUNCT
ma-166	55	29	m1(g	m1(g	NOUN
ma-166	55	30	,	,	PUNCT
ma-166	55	31	x	x	NOUN
ma-166	55	32	)	)	PUNCT
ma-166	55	33	=	=	PUNCT
ma-166	56	1	∑	∑	PUNCT
ma-166	57	1	[	[	X
ma-166	57	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	57	3	)	)	PUNCT
ma-166	57	4	]	]	PUNCT
ma-166	58	1	[	[	X
ma-166	58	2	d(x1	d(x1	NOUN
ma-166	58	3	)	)	PUNCT
ma-166	58	4	+	+	NUM
ma-166	58	5	d(x2	d(x2	NOUN
ma-166	58	6	)	)	PUNCT
ma-166	58	7	]	]	PUNCT
ma-166	59	1	(	(	PUNCT
ma-166	59	2	1	1	X
ma-166	59	3	)	)	PUNCT
ma-166	59	4	the	the	DET
ma-166	59	5	second	second	ADJ
ma-166	59	6	zagreb	zagreb	PROPN
ma-166	59	7	index	index	NOUN
ma-166	59	8	m2(g	m2(g	NOUN
ma-166	59	9	)	)	PUNCT
ma-166	59	10	is	be	AUX
ma-166	59	11	equal	equal	ADJ
ma-166	59	12	to	to	ADP
ma-166	59	13	the	the	DET
ma-166	59	14	sum	sum	NOUN
ma-166	59	15	of	of	ADP
ma-166	59	16	the	the	DET
ma-166	59	17	products	product	NOUN
ma-166	59	18	of	of	ADP
ma-166	59	19	the	the	DET
ma-166	59	20	degrees	degree	NOUN
ma-166	59	21	of	of	ADP
ma-166	59	22	the	the	DET
ma-166	59	23	adjacentvertices	adjacentvertice	NOUN
ma-166	59	24	for	for	ADP
ma-166	59	25	the	the	DET
ma-166	59	26	pair	pair	NOUN
ma-166	59	27	of	of	ADP
ma-166	59	28	vertices	vertex	NOUN
ma-166	59	29	for	for	ADP
ma-166	59	30	the	the	DET
ma-166	59	31	(	(	PUNCT
ma-166	59	32	molecular	molecular	ADJ
ma-166	59	33	)	)	PUNCT
ma-166	59	34	graph	graph	NOUN
ma-166	59	35	g	g	NOUN
ma-166	59	36	,	,	PUNCT
ma-166	59	37	and	and	CCONJ
ma-166	59	38	m2(g	m2(g	NOUN
ma-166	59	39	)	)	PUNCT
ma-166	59	40	is	be	AUX
ma-166	59	41	defined	define	VERB
ma-166	59	42	as:[the	as:[the	DET
ma-166	59	43	first	first	ADJ
ma-166	59	44	andsecond	andsecond	ADJ
ma-166	59	45	zagreb	zagreb	PROPN
ma-166	59	46	indices	index	NOUN
ma-166	59	47	of	of	ADP
ma-166	59	48	some	some	DET
ma-166	59	49	graph	graph	NOUN
ma-166	59	50	operations	operation	NOUN
ma-166	59	51	]	]	PUNCT
ma-166	59	52	m2(g	m2(g	NOUN
ma-166	59	53	,	,	PUNCT
ma-166	59	54	x	x	NOUN
ma-166	59	55	)	)	PUNCT
ma-166	59	56	=	=	PUNCT
ma-166	60	1	∑	∑	PUNCT
ma-166	61	1	[	[	X
ma-166	61	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	61	3	)	)	PUNCT
ma-166	61	4	]	]	PUNCT
ma-166	62	1	[	[	X
ma-166	62	2	d(x1)d(x2	d(x1)d(x2	NOUN
ma-166	62	3	)	)	PUNCT
ma-166	62	4	]	]	PUNCT
ma-166	62	5	(	(	PUNCT
ma-166	62	6	2	2	X
ma-166	62	7	)	)	PUNCT
ma-166	62	8	in	in	ADP
ma-166	62	9	1972	1972	NUM
ma-166	62	10	,	,	PUNCT
ma-166	62	11	the	the	DET
ma-166	62	12	first	first	PROPN
ma-166	62	13	zagreb	zagreb	PROPN
ma-166	62	14	index	index	PROPN
ma-166	62	15	,	,	PUNCT
ma-166	62	16	a	a	DET
ma-166	62	17	very	very	ADV
ma-166	62	18	old	old	ADJ
ma-166	62	19	topological	topological	ADJ
ma-166	62	20	index	index	NOUN
ma-166	62	21	,	,	PUNCT
ma-166	62	22	was	be	AUX
ma-166	62	23	launched	launch	VERB
ma-166	62	24	and	and	CCONJ
ma-166	62	25	several	several	ADJ
ma-166	62	26	variants	variant	NOUN
ma-166	62	27	ofthe	ofthe	PROPN
ma-166	62	28	zagreb	zagreb	PROPN
ma-166	62	29	index	index	NOUN
ma-166	62	30	were	be	AUX
ma-166	62	31	subsequently	subsequently	ADV
ma-166	62	32	proposed	propose	VERB
ma-166	62	33	,	,	PUNCT
ma-166	62	34	e.g.	e.g.	ADV
ma-166	62	35	shirdel	shirdel	VERB
ma-166	62	36	et	et	PROPN
ma-166	62	37	al	al	PROPN
ma-166	62	38	.	.	PROPN
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ma-166	62	40	a	a	DET
ma-166	62	41	novel	novel	ADJ
ma-166	62	42	index	index	NOUN
ma-166	62	43	in	in	ADP
ma-166	62	44	2013under	2013under	NUM
ma-166	62	45	the	the	DET
ma-166	62	46	title	title	NOUN
ma-166	62	47	of	of	ADP
ma-166	62	48	’	'	PUNCT
ma-166	62	49	hyper	hyper	ADJ
ma-166	62	50	-	-	PROPN
ma-166	62	51	zagreb	zagreb	PROPN
ma-166	62	52	index	index	NOUN
ma-166	62	53	’	'	PUNCT
ma-166	62	54	and	and	CCONJ
ma-166	62	55	then	then	ADV
ma-166	62	56	it	it	PRON
ma-166	62	57	was	be	AUX
ma-166	62	58	identified	identify	VERB
ma-166	62	59	as[2	as[2	ADV
ma-166	62	60	]	]	PUNCT
ma-166	62	61	:	:	PUNCT
ma-166	63	1	[	[	X
ma-166	63	2	a	a	DET
ma-166	63	3	note	note	NOUN
ma-166	63	4	on	on	ADP
ma-166	63	5	hyper	hyper	NOUN
ma-166	63	6	-	-	NOUN
ma-166	63	7	zagrebindex	zagrebindex	NOUN
ma-166	63	8	of	of	ADP
ma-166	63	9	graph	graph	NOUN
ma-166	63	10	operations	operation	NOUN
ma-166	63	11	]	]	PUNCT
ma-166	63	12	hm1(g	hm1(g	X
ma-166	63	13	)	)	PUNCT
ma-166	63	14	=	=	PUNCT
ma-166	63	15	∑	∑	PUNCT
ma-166	64	1	[	[	X
ma-166	64	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	64	3	)	)	PUNCT
ma-166	64	4	]	]	PUNCT
ma-166	65	1	[	[	X
ma-166	65	2	d(x1	d(x1	NOUN
ma-166	65	3	)	)	PUNCT
ma-166	65	4	+	+	NUM
ma-166	65	5	d(x2	d(x2	NOUN
ma-166	65	6	)	)	PUNCT
ma-166	65	7	]	]	PUNCT
ma-166	65	8	2	2	NUM
ma-166	65	9	(	(	PUNCT
ma-166	65	10	3	3	NUM
ma-166	65	11	)	)	PUNCT
ma-166	65	12	e.	e.	PROPN
ma-166	65	13	deutshi	deutshi	PROPN
ma-166	65	14	and	and	CCONJ
ma-166	65	15	s.	s.	PROPN
ma-166	65	16	klavzar	klavzar	PROPN
ma-166	65	17	,	,	PUNCT
ma-166	65	18	in	in	ADP
ma-166	65	19	2015	2015	NUM
ma-166	65	20	,	,	PUNCT
ma-166	65	21	defined	define	VERB
ma-166	65	22	a	a	DET
ma-166	65	23	new	new	ADJ
ma-166	65	24	polynomial	polynomial	ADJ
ma-166	65	25	,	,	PUNCT
ma-166	65	26	m	m	NOUN
ma-166	65	27	-	-	NOUN
ma-166	65	28	polynomial	polynomial	ADJ
ma-166	65	29	in	in	ADP
ma-166	65	30	the	the	DET
ma-166	65	31	followingway	followingway	NOUN
ma-166	65	32	,	,	PUNCT
ma-166	65	33	based	base	VERB
ma-166	65	34	on	on	ADP
ma-166	65	35	the	the	DET
ma-166	65	36	degree	degree	NOUN
ma-166	65	37	of	of	ADP
ma-166	65	38	the	the	DET
ma-166	65	39	vertex[3]:[computing	vertex[3]:[compute	VERB
ma-166	65	40	hyper	hyper	ADJ
ma-166	65	41	zagreb	zagreb	PROPN
ma-166	65	42	index	index	PROPN
ma-166	65	43	and	and	CCONJ
ma-166	65	44	m	m	NOUN
ma-166	65	45	-	-	NOUN
ma-166	65	46	polynomials	polynomial	NOUN
ma-166	65	47	]	]	X
ma-166	65	48	m1(g	m1(g	NOUN
ma-166	65	49	,	,	PUNCT
ma-166	65	50	y	y	PROPN
ma-166	65	51	,	,	PUNCT
ma-166	65	52	z	z	X
ma-166	65	53	)	)	PUNCT
ma-166	65	54	=	=	PUNCT
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ma-166	67	1	[	[	X
ma-166	67	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	67	3	)	)	PUNCT
ma-166	67	4	]	]	PUNCT
ma-166	68	1	y	y	PROPN
ma-166	69	1	[	[	X
ma-166	69	2	d(x1)]z	d(x1)]z	INTJ
ma-166	69	3	[	[	X
ma-166	69	4	d(x2	d(x2	NOUN
ma-166	69	5	)	)	PUNCT
ma-166	69	6	]	]	PUNCT
ma-166	69	7	(	(	PUNCT
ma-166	69	8	4	4	X
ma-166	69	9	)	)	PUNCT
ma-166	69	10	in	in	ADP
ma-166	69	11	shuxian	shuxian	PROPN
ma-166	69	12	defined	define	VERB
ma-166	69	13	two	two	NUM
ma-166	69	14	polynomials	polynomial	NOUN
ma-166	69	15	related	relate	VERB
ma-166	69	16	to	to	ADP
ma-166	69	17	the	the	DET
ma-166	69	18	first	first	ADJ
ma-166	69	19	zagreb	zagreb	PROPN
ma-166	69	20	index	index	NOUN
ma-166	69	21	as	as	ADP
ma-166	69	22	in	in	ADP
ma-166	69	23	the	the	DET
ma-166	69	24	form	form	NOUN
ma-166	69	25	:	:	PUNCT
ma-166	69	26	m∗1(g	m∗1(g	PROPN
ma-166	69	27	,	,	PUNCT
ma-166	69	28	x	x	NOUN
ma-166	69	29	)	)	PUNCT
ma-166	69	30	=	=	PUNCT
ma-166	70	1	∑	∑	PUNCT
ma-166	71	1	[	[	X
ma-166	71	2	xi∈v	xi∈v	X
ma-166	71	3	(	(	PUNCT
ma-166	71	4	g	g	NOUN
ma-166	71	5	)	)	PUNCT
ma-166	71	6	]	]	PUNCT
ma-166	72	1	[	[	X
ma-166	72	2	d(xi)][x	d(xi)][x	NOUN
ma-166	72	3	(	(	PUNCT
ma-166	72	4	xi	xi	PROPN
ma-166	72	5	)	)	PUNCT
ma-166	72	6	]	]	PUNCT
ma-166	72	7	(	(	PUNCT
ma-166	72	8	5	5	X
ma-166	72	9	)	)	PUNCT
ma-166	72	10	m0(g	m0(g	NOUN
ma-166	72	11	,	,	PUNCT
ma-166	72	12	x	x	X
ma-166	72	13	)	)	PUNCT
ma-166	72	14	=	=	PUNCT
ma-166	72	15	∑	∑	PUNCT
ma-166	73	1	[	[	X
ma-166	73	2	xi∈v	xi∈v	X
ma-166	73	3	(	(	PUNCT
ma-166	73	4	g	g	NOUN
ma-166	73	5	)	)	PUNCT
ma-166	73	6	]	]	PUNCT
ma-166	73	7	(	(	PUNCT
ma-166	73	8	x)[d(xi	x)[d(xi	PROPN
ma-166	73	9	)	)	PUNCT
ma-166	73	10	]	]	PUNCT
ma-166	73	11	(	(	PUNCT
ma-166	73	12	6	6	X
ma-166	73	13	)	)	PUNCT
ma-166	73	14	two	two	NUM
ma-166	73	15	zagreb	zagreb	PROPN
ma-166	73	16	type	type	NOUN
ma-166	73	17	polynomials	polynomial	NOUN
ma-166	73	18	are	be	AUX
ma-166	73	19	defined	define	VERB
ma-166	73	20	as	as	ADP
ma-166	73	21	follow	follow	NOUN
ma-166	73	22	:	:	PUNCT
ma-166	73	23	ma	ma	PROPN
ma-166	73	24	,	,	PUNCT
ma-166	73	25	b(g	b(g	PROPN
ma-166	73	26	,	,	PUNCT
ma-166	73	27	x	x	X
ma-166	73	28	)	)	PUNCT
ma-166	73	29	=	=	PUNCT
ma-166	74	1	∑	∑	PUNCT
ma-166	75	1	[	[	X
ma-166	75	2	xi	xi	X
ma-166	75	3	,	,	PUNCT
ma-166	75	4	xj∈e(g	xj∈e(g	PROPN
ma-166	75	5	)	)	PUNCT
ma-166	75	6	]	]	PUNCT
ma-166	76	1	(	(	PUNCT
ma-166	76	2	x)[a{d(xi	x)[a{d(xi	NUM
ma-166	76	3	)	)	PUNCT
ma-166	76	4	}	}	PUNCT
ma-166	77	1	+	+	ADJ
ma-166	77	2	b{d(xi	b{d(xi	NOUN
ma-166	77	3	)	)	PUNCT
ma-166	77	4	}	}	PUNCT
ma-166	77	5	]	]	PUNCT
ma-166	77	6	(	(	PUNCT
ma-166	77	7	7	7	X
ma-166	77	8	)	)	PUNCT
ma-166	77	9	m	m	VERB
ma-166	77	10	′a	′a	NOUN
ma-166	77	11	,	,	PUNCT
ma-166	77	12	b(g	b(g	PROPN
ma-166	77	13	,	,	PUNCT
ma-166	77	14	x	x	X
ma-166	77	15	)	)	PUNCT
ma-166	77	16	=	=	PUNCT
ma-166	77	17	∑	∑	PUNCT
ma-166	77	18	[	[	X
ma-166	77	19	xi	xi	X
ma-166	77	20	,	,	PUNCT
ma-166	77	21	xj∈e(g	xj∈e(g	PROPN
ma-166	77	22	)	)	PUNCT
ma-166	77	23	]	]	PUNCT
ma-166	78	1	(	(	PUNCT
ma-166	78	2	x)([a+{d(xi	x)([a+{d(xi	PROPN
ma-166	78	3	)	)	PUNCT
ma-166	78	4	}	}	PUNCT
ma-166	78	5	]	]	PUNCT
ma-166	79	1	[	[	X
ma-166	79	2	b+{d(xi	b+{d(xi	NOUN
ma-166	79	3	)	)	PUNCT
ma-166	79	4	}	}	PUNCT
ma-166	79	5	]	]	PUNCT
ma-166	79	6	)	)	PUNCT
ma-166	79	7	(	(	PUNCT
ma-166	79	8	8)	8)	NUM
ma-166	79	9	todeshine	todeshine	NOUN
ma-166	79	10	et	et	PROPN
ma-166	79	11	al	al	PROPN
ma-166	79	12	.	.	PROPN
ma-166	79	13	introduced	introduce	VERB
ma-166	79	14	two	two	NUM
ma-166	79	15	updated	update	VERB
ma-166	79	16	models	model	NOUN
ma-166	79	17	of	of	ADP
ma-166	79	18	the	the	DET
ma-166	79	19	zagreb	zagreb	PROPN
ma-166	79	20	index	index	NOUN
ma-166	79	21	for	for	ADP
ma-166	79	22	moleculargraphs[4]:[multiplicative	moleculargraphs[4]:[multiplicative	ADJ
ma-166	79	23	zagreb	zagreb	PROPN
ma-166	79	24	indices	index	NOUN
ma-166	79	25	of	of	ADP
ma-166	79	26	trees	tree	NOUN
ma-166	79	27	]	]	PUNCT
ma-166	80	1	first	first	PROPN
ma-166	80	2	multiplicative	multiplicative	ADJ
ma-166	80	3	zagreb	zagreb	PROPN
ma-166	80	4	index	index	NOUN
ma-166	80	5	formolecular	formolecular	ADJ
ma-166	80	6	graph	graph	NOUN
ma-166	80	7	g	g	PROPN
ma-166	80	8	defined	define	VERB
ma-166	80	9	as	as	SCONJ
ma-166	80	10	follows	follow	VERB
ma-166	80	11	:	:	PUNCT
ma-166	80	12	pm1(g	pm1(g	NOUN
ma-166	80	13	)	)	PUNCT
ma-166	81	1	=	=	SYM
ma-166	81	2	∏	∏	PROPN
ma-166	82	1	[	[	X
ma-166	82	2	x1,x2∈e(g	x1,x2∈e(g	NOUN
ma-166	82	3	)	)	PUNCT
ma-166	82	4	]	]	PUNCT
ma-166	83	1	[	[	X
ma-166	83	2	d(x1	d(x1	NOUN
ma-166	83	3	)	)	PUNCT
ma-166	83	4	+	+	NUM
ma-166	83	5	d(x2	d(x2	NOUN
ma-166	83	6	)	)	PUNCT
ma-166	83	7	]	]	PUNCT
ma-166	83	8	(	(	PUNCT
ma-166	83	9	9	9	X
ma-166	83	10	)	)	PUNCT
ma-166	83	11	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	83	12	eur	eur	PROPN
ma-166	83	13	.	.	PUNCT
ma-166	84	1	j.	j.	PROPN
ma-166	84	2	math	math	PROPN
ma-166	84	3	.	.	PUNCT
ma-166	85	1	anal	anal	PROPN
ma-166	85	2	.	.	PUNCT
ma-166	86	1	10.28924	10.28924	NUM
ma-166	86	2	/	/	SYM
ma-166	86	3	ada	ada	PROPN
ma-166	86	4	/	/	SYM
ma-166	86	5	ma.3.20	ma.3.20	PROPN
ma-166	87	1	4second	4second	NUM
ma-166	87	2	multiplicative	multiplicative	ADJ
ma-166	87	3	zagreb	zagreb	PROPN
ma-166	87	4	index	index	NOUN
ma-166	87	5	for	for	ADP
ma-166	87	6	molecular	molecular	ADJ
ma-166	87	7	graph	graph	NOUN
ma-166	87	8	g	g	PROPN
ma-166	87	9	defined	define	VERB
ma-166	87	10	as	as	SCONJ
ma-166	87	11	follows	follow	VERB
ma-166	87	12	:	:	PUNCT
ma-166	87	13	pm2(g	pm2(g	X
ma-166	87	14	)	)	PUNCT
ma-166	87	15	=	=	SYM
ma-166	87	16	∏	∏	PROPN
ma-166	88	1	[	[	X
ma-166	88	2	x1,x2∈e(g	x1,x2∈e(g	NOUN
ma-166	88	3	)	)	PUNCT
ma-166	88	4	]	]	PUNCT
ma-166	89	1	[	[	X
ma-166	89	2	d(x1)×	d(x1)×	NOUN
ma-166	89	3	d(x2	d(x2	NOUN
ma-166	89	4	)	)	PUNCT
ma-166	89	5	]	]	PUNCT
ma-166	90	1	(	(	PUNCT
ma-166	90	2	10	10	NUM
ma-166	90	3	)	)	PUNCT
ma-166	90	4	first	first	ADV
ma-166	90	5	multiplicative	multiplicative	ADJ
ma-166	90	6	zagreb	zagreb	PROPN
ma-166	90	7	polynomial	polynomial	PROPN
ma-166	90	8	for	for	ADP
ma-166	90	9	molecular	molecular	ADJ
ma-166	90	10	graph	graph	NOUN
ma-166	90	11	g	g	PROPN
ma-166	90	12	defined	define	VERB
ma-166	90	13	as	as	ADP
ma-166	90	14	follows	follow	VERB
ma-166	90	15	:	:	PUNCT
ma-166	90	16	pm1(g	pm1(g	NOUN
ma-166	90	17	,	,	PUNCT
ma-166	90	18	x	x	NOUN
ma-166	90	19	)	)	PUNCT
ma-166	90	20	=	=	SYM
ma-166	90	21	∏	∏	PROPN
ma-166	91	1	[	[	X
ma-166	91	2	x1,x2∈e(g	x1,x2∈e(g	NOUN
ma-166	91	3	)	)	PUNCT
ma-166	91	4	]	]	PUNCT
ma-166	92	1	x[d(x1)+d(x2	x[d(x1)+d(x2	PROPN
ma-166	92	2	)	)	PUNCT
ma-166	92	3	]	]	PUNCT
ma-166	92	4	(	(	PUNCT
ma-166	92	5	11	11	NUM
ma-166	92	6	)	)	PUNCT
ma-166	92	7	second	second	ADJ
ma-166	92	8	multiplicative	multiplicative	ADJ
ma-166	92	9	zagreb	zagreb	PROPN
ma-166	92	10	polynomial	polynomial	PROPN
ma-166	92	11	for	for	ADP
ma-166	92	12	molecular	molecular	ADJ
ma-166	92	13	graph	graph	NOUN
ma-166	92	14	g	g	PROPN
ma-166	92	15	defined	define	VERB
ma-166	92	16	as	as	ADP
ma-166	92	17	follows	follow	VERB
ma-166	92	18	:	:	PUNCT
ma-166	92	19	pm2(g	pm2(g	NUM
ma-166	92	20	,	,	PUNCT
ma-166	92	21	x	x	X
ma-166	92	22	)	)	PUNCT
ma-166	92	23	=	=	SYM
ma-166	92	24	∏	∏	PROPN
ma-166	93	1	[	[	X
ma-166	93	2	x1,x2∈e(g	x1,x2∈e(g	NOUN
ma-166	93	3	)	)	PUNCT
ma-166	93	4	]	]	PUNCT
ma-166	93	5	x[d(x1)d(x2	x[d(x1)d(x2	NOUN
ma-166	93	6	)	)	PUNCT
ma-166	93	7	]	]	PUNCT
ma-166	94	1	(	(	PUNCT
ma-166	94	2	12	12	NUM
ma-166	94	3	)	)	PUNCT
ma-166	94	4	the	the	DET
ma-166	94	5	first	first	ADJ
ma-166	94	6	degree	degree	NOUN
ma-166	94	7	-	-	PUNCT
ma-166	94	8	based	base	VERB
ma-166	94	9	topological	topological	ADJ
ma-166	94	10	index	index	NOUN
ma-166	94	11	was	be	AUX
ma-166	94	12	proposed	propose	VERB
ma-166	94	13	by	by	ADP
ma-166	94	14	milan	milan	PROPN
ma-166	94	15	randic	randic	NOUN
ma-166	94	16	in	in	ADP
ma-166	94	17	1975[5]:[degree	1975[5]:[degree	NUM
ma-166	94	18	-	-	PUNCT
ma-166	94	19	basedtopological	basedtopological	ADJ
ma-166	94	20	indices	index	NOUN
ma-166	94	21	]	]	X
ma-166	94	22	r1(α)(g	r1(α)(g	NOUN
ma-166	94	23	)	)	PUNCT
ma-166	94	24	=	=	PUNCT
ma-166	95	1	∑	∑	PUNCT
ma-166	96	1	[	[	X
ma-166	96	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	96	3	)	)	PUNCT
ma-166	96	4	]	]	PUNCT
ma-166	97	1	[	[	X
ma-166	97	2	d(x1	d(x1	NOUN
ma-166	97	3	)	)	PUNCT
ma-166	97	4	+	+	NUM
ma-166	97	5	d(x2	d(x2	NOUN
ma-166	97	6	)	)	PUNCT
ma-166	97	7	]	]	PUNCT
ma-166	98	1	α	α	PRON
ma-166	98	2	(	(	PUNCT
ma-166	98	3	13	13	NUM
ma-166	98	4	)	)	PUNCT
ma-166	98	5	atom	atom	NOUN
ma-166	98	6	-	-	PUNCT
ma-166	98	7	bond	bond	NOUN
ma-166	98	8	connectivity	connectivity	NOUN
ma-166	98	9	index	index	NOUN
ma-166	98	10	(	(	PUNCT
ma-166	98	11	abc	abc	PROPN
ma-166	98	12	)	)	PUNCT
ma-166	98	13	is	be	AUX
ma-166	98	14	a	a	DET
ma-166	98	15	topological	topological	ADJ
ma-166	98	16	index	index	NOUN
ma-166	98	17	used	use	VERB
ma-166	98	18	in	in	ADP
ma-166	98	19	chemistry	chemistry	NOUN
ma-166	98	20	,	,	PUNCT
ma-166	98	21	environmental	environmental	ADJ
ma-166	98	22	sci	sci	PROPN
ma-166	98	23	-	-	PUNCT
ma-166	98	24	ences	ence	NOUN
ma-166	98	25	and	and	CCONJ
ma-166	98	26	pharmacology[6	pharmacology[6	NOUN
ma-166	98	27	]	]	X
ma-166	98	28	:	:	PUNCT
ma-166	99	1	[	[	X
ma-166	99	2	estrada	estrada	PROPN
ma-166	99	3	,	,	PUNCT
ma-166	99	4	torres	torre	NOUN
ma-166	99	5	,	,	PUNCT
ma-166	99	6	rodriguez	rodriguez	NOUN
ma-166	99	7	,	,	PUNCT
ma-166	99	8	and	and	CCONJ
ma-166	99	9	gutman	gutman	NOUN
ma-166	99	10	,	,	PUNCT
ma-166	99	11	1998b	1998b	NUM
ma-166	99	12	]	]	PUNCT
ma-166	99	13	abc(g	abc(g	X
ma-166	99	14	)	)	PUNCT
ma-166	99	15	=	=	PUNCT
ma-166	99	16	∑	∑	PUNCT
ma-166	100	1	[	[	X
ma-166	100	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	100	3	)	)	PUNCT
ma-166	100	4	]	]	PUNCT
ma-166	100	5	√	√	PROPN
ma-166	101	1	[	[	X
ma-166	101	2	d(x1	d(x1	NOUN
ma-166	101	3	)	)	PUNCT
ma-166	101	4	+	+	CCONJ
ma-166	101	5	d(x2)]−	d(x2)]−	PROPN
ma-166	101	6	2	2	NUM
ma-166	101	7	d(x1)×	d(x1)×	PROPN
ma-166	101	8	d(x2	d(x2	NOUN
ma-166	101	9	)	)	PUNCT
ma-166	101	10	(	(	PUNCT
ma-166	101	11	14	14	NUM
ma-166	101	12	)	)	PUNCT
ma-166	101	13	first	first	ADV
ma-166	101	14	,	,	PUNCT
ma-166	101	15	second	second	ADJ
ma-166	101	16	and	and	CCONJ
ma-166	101	17	third	third	ADV
ma-166	101	18	reduced	reduce	VERB
ma-166	101	19	zagreb	zagreb	PROPN
ma-166	101	20	indices[7	indices[7	NOUN
ma-166	101	21	]	]	PUNCT
ma-166	101	22	are	be	AUX
ma-166	101	23	described	describe	VERB
ma-166	101	24	as	as	ADP
ma-166	101	25	follow	follow	NOUN
ma-166	101	26	:	:	PUNCT
ma-166	101	27	mr1(g	mr1(g	PROPN
ma-166	101	28	)	)	PUNCT
ma-166	101	29	=	=	PUNCT
ma-166	102	1	∑	∑	PUNCT
ma-166	103	1	[	[	X
ma-166	103	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	103	3	)	)	PUNCT
ma-166	103	4	]	]	PUNCT
ma-166	103	5	|(d(x1)−	|(d(x1)−	PROPN
ma-166	103	6	1	1	NUM
ma-166	103	7	)	)	PUNCT
ma-166	103	8	+	+	CCONJ
ma-166	103	9	(	(	PUNCT
ma-166	103	10	d(x2)−	d(x2)−	PROPN
ma-166	103	11	1)|	1)|	NUM
ma-166	103	12	(	(	PUNCT
ma-166	103	13	15	15	NUM
ma-166	103	14	)	)	PUNCT
ma-166	103	15	mr2(g	mr2(g	NOUN
ma-166	103	16	)	)	PUNCT
ma-166	103	17	=	=	PUNCT
ma-166	103	18	∑	∑	PUNCT
ma-166	104	1	[	[	X
ma-166	104	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	104	3	)	)	PUNCT
ma-166	104	4	]	]	PUNCT
ma-166	105	1	[	[	X
ma-166	105	2	(	(	PUNCT
ma-166	105	3	d(x1)−	d(x1)−	PROPN
ma-166	105	4	1)(d(x2)−	1)(d(x2)−	ADJ
ma-166	105	5	1	1	NUM
ma-166	105	6	)	)	PUNCT
ma-166	105	7	]	]	PUNCT
ma-166	105	8	(	(	PUNCT
ma-166	105	9	16	16	NUM
ma-166	105	10	)	)	PUNCT
ma-166	105	11	mr3(g	mr3(g	PROPN
ma-166	105	12	)	)	PUNCT
ma-166	105	13	=	=	PUNCT
ma-166	105	14	∑	∑	PUNCT
ma-166	105	15	[	[	X
ma-166	105	16	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	105	17	)	)	PUNCT
ma-166	105	18	]	]	PUNCT
ma-166	106	1	|(d(x1)−	|(d(x1)−	PROPN
ma-166	106	2	1)−	1)−	PROPN
ma-166	106	3	(	(	PUNCT
ma-166	106	4	d(x2)−	d(x2)−	PROPN
ma-166	106	5	1)|	1)|	NUM
ma-166	106	6	(	(	PUNCT
ma-166	106	7	17	17	NUM
ma-166	106	8	)	)	PUNCT
ma-166	106	9	rr(g	rr(g	NUM
ma-166	106	10	)	)	PUNCT
ma-166	107	1	=	=	PUNCT
ma-166	107	2	∑	∑	PUNCT
ma-166	108	1	[	[	X
ma-166	108	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	108	3	)	)	PUNCT
ma-166	108	4	]	]	PUNCT
ma-166	108	5	√	√	PROPN
ma-166	108	6	d(x1)×	d(x1)×	PROPN
ma-166	108	7	d(x2	d(x2	NOUN
ma-166	108	8	)	)	PUNCT
ma-166	108	9	(	(	PUNCT
ma-166	108	10	18	18	NUM
ma-166	108	11	)	)	PUNCT
ma-166	108	12	the	the	DET
ma-166	108	13	reduced	reduce	VERB
ma-166	108	14	reciprocal	reciprocal	ADJ
ma-166	108	15	randic	randic	ADJ
ma-166	108	16	index	index	NOUN
ma-166	108	17	is	be	AUX
ma-166	108	18	defined	define	VERB
ma-166	108	19	as[8	as[8	PROPN
ma-166	108	20	]	]	X
ma-166	108	21	:	:	PUNCT
ma-166	108	22	rrr(g	rrr(g	PROPN
ma-166	108	23	)	)	PUNCT
ma-166	109	1	=	=	PUNCT
ma-166	109	2	∑	∑	PUNCT
ma-166	110	1	[	[	X
ma-166	110	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	110	3	)	)	PUNCT
ma-166	110	4	]	]	PUNCT
ma-166	111	1	√	√	PROPN
ma-166	112	1	[	[	X
ma-166	112	2	d(x1)−	d(x1)−	PROPN
ma-166	112	3	1]×	1]×	PROPN
ma-166	112	4	[	[	X
ma-166	112	5	d(x2)−	d(x2)−	NOUN
ma-166	112	6	1	1	NUM
ma-166	112	7	]	]	PUNCT
ma-166	112	8	(	(	PUNCT
ma-166	112	9	19	19	NUM
ma-166	112	10	)	)	PUNCT
ma-166	112	11	recently	recently	ADV
ma-166	112	12	in	in	ADP
ma-166	112	13	2015	2015	NUM
ma-166	112	14	furtula	furtula	NOUN
ma-166	112	15	and	and	CCONJ
ma-166	112	16	gutman	gutman	NOUN
ma-166	112	17	[	[	X
ma-166	112	18	8	8	NUM
ma-166	112	19	]	]	PUNCT
ma-166	112	20	introduced	introduce	VERB
ma-166	112	21	another	another	DET
ma-166	112	22	topological	topological	ADJ
ma-166	112	23	index	index	NOUN
ma-166	112	24	known	know	VERB
ma-166	112	25	as	as	ADP
ma-166	112	26	forgottenindex	forgottenindex	NOUN
ma-166	112	27	or	or	CCONJ
ma-166	112	28	f	f	PROPN
ma-166	112	29	−	−	PROPN
ma-166	112	30	index	index	NOUN
ma-166	112	31	.	.	PUNCT
ma-166	113	1	for	for	ADP
ma-166	113	2	more	more	ADJ
ma-166	113	3	detail	detail	NOUN
ma-166	113	4	on	on	ADP
ma-166	113	5	the	the	DET
ma-166	113	6	f	f	PROPN
ma-166	113	7	−	−	PROPN
ma-166	113	8	index	index	NOUN
ma-166	113	9	,	,	PUNCT
ma-166	113	10	we	we	PRON
ma-166	113	11	refer	refer	VERB
ma-166	113	12	to	to	ADP
ma-166	113	13	the	the	DET
ma-166	113	14	articles	article	NOUN
ma-166	113	15	[	[	X
ma-166	113	16	9].the	9].the	DET
ma-166	113	17	forgottenindex	forgottenindex	NOUN
ma-166	113	18	of	of	ADP
ma-166	113	19	a	a	DET
ma-166	113	20	graph	graph	NOUN
ma-166	113	21	g	g	NOUN
ma-166	113	22	is	be	AUX
ma-166	113	23	defined	define	VERB
ma-166	113	24	as[10	as[10	PROPN
ma-166	113	25	,	,	PUNCT
ma-166	113	26	11	11	NUM
ma-166	113	27	,	,	PUNCT
ma-166	113	28	12	12	NUM
ma-166	113	29	]	]	PUNCT
ma-166	113	30	.	.	PUNCT
ma-166	114	1	f	f	PROPN
ma-166	114	2	(	(	PUNCT
ma-166	114	3	g	g	NOUN
ma-166	114	4	)	)	PUNCT
ma-166	114	5	=	=	PUNCT
ma-166	115	1	∑	∑	PUNCT
ma-166	116	1	[	[	X
ma-166	116	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	116	3	)	)	PUNCT
ma-166	116	4	]	]	PUNCT
ma-166	117	1	[	[	X
ma-166	117	2	(	(	PUNCT
ma-166	117	3	dx1	dx1	PROPN
ma-166	117	4	)	)	PUNCT
ma-166	117	5	2	2	NUM
ma-166	117	6	+	+	CCONJ
ma-166	117	7	(	(	PUNCT
ma-166	117	8	dx2	dx2	PROPN
ma-166	117	9	)	)	PUNCT
ma-166	117	10	2	2	NUM
ma-166	117	11	]	]	PUNCT
ma-166	117	12	(	(	PUNCT
ma-166	117	13	20	20	NUM
ma-166	117	14	)	)	PUNCT
ma-166	117	15	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	117	16	eur	eur	PROPN
ma-166	117	17	.	.	PUNCT
ma-166	118	1	j.	j.	PROPN
ma-166	118	2	math	math	PROPN
ma-166	118	3	.	.	PUNCT
ma-166	119	1	anal	anal	PROPN
ma-166	119	2	.	.	PUNCT
ma-166	120	1	10.28924	10.28924	NUM
ma-166	120	2	/	/	SYM
ma-166	120	3	ada	ada	PROPN
ma-166	120	4	/	/	SYM
ma-166	120	5	ma.3.20	ma.3.20	NOUN
ma-166	120	6	5the	5the	DET
ma-166	120	7	forgotten	forget	VERB
ma-166	120	8	polynomial	polynomial	NOUN
ma-166	120	9	of	of	ADP
ma-166	120	10	a	a	DET
ma-166	120	11	graph	graph	NOUN
ma-166	120	12	g	g	NOUN
ma-166	120	13	is	be	AUX
ma-166	120	14	defined	define	VERB
ma-166	120	15	as	as	ADP
ma-166	120	16	:	:	PUNCT
ma-166	120	17	f	f	PROPN
ma-166	120	18	(	(	PUNCT
ma-166	120	19	g	g	PROPN
ma-166	120	20	,	,	PUNCT
ma-166	120	21	x	x	NOUN
ma-166	120	22	)	)	PUNCT
ma-166	120	23	=	=	PUNCT
ma-166	121	1	∑	∑	PUNCT
ma-166	122	1	[	[	X
ma-166	122	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	122	3	)	)	PUNCT
ma-166	122	4	]	]	PUNCT
ma-166	122	5	(	(	PUNCT
ma-166	122	6	x)[(dx1	x)[(dx1	PROPN
ma-166	122	7	)	)	PUNCT
ma-166	122	8	2+(dx2	2+(dx2	NOUN
ma-166	122	9	)	)	PUNCT
ma-166	122	10	2	2	NUM
ma-166	122	11	]	]	PUNCT
ma-166	122	12	(	(	PUNCT
ma-166	122	13	21	21	NUM
ma-166	122	14	)	)	PUNCT
ma-166	122	15	the	the	DET
ma-166	122	16	symmetric	symmetric	ADJ
ma-166	122	17	division	division	NOUN
ma-166	122	18	degree	degree	NOUN
ma-166	122	19	index	index	NOUN
ma-166	122	20	of	of	ADP
ma-166	122	21	a	a	DET
ma-166	122	22	connected	connected	ADJ
ma-166	122	23	graph	graph	NOUN
ma-166	122	24	g	g	NOUN
ma-166	122	25	is	be	AUX
ma-166	122	26	defined	define	VERB
ma-166	122	27	as	as	ADP
ma-166	122	28	:	:	PUNCT
ma-166	122	29	sdd(g	sdd(g	PROPN
ma-166	122	30	)	)	PUNCT
ma-166	122	31	=	=	PUNCT
ma-166	123	1	∑	∑	PUNCT
ma-166	124	1	[	[	X
ma-166	124	2	x1,x2∈e(g	x1,x2∈e(g	NOUN
ma-166	124	3	)	)	PUNCT
ma-166	124	4	]	]	PUNCT
ma-166	125	1	mini(d(x1	mini(d(x1	PROPN
ma-166	125	2	)	)	PUNCT
ma-166	125	3	,	,	PUNCT
ma-166	125	4	d(x2	d(x2	NOUN
ma-166	125	5	)	)	PUNCT
ma-166	125	6	)	)	PUNCT
ma-166	126	1	max(d(x1	max(d(x1	PROPN
ma-166	126	2	)	)	PUNCT
ma-166	126	3	,	,	PUNCT
ma-166	126	4	d(x2	d(x2	NOUN
ma-166	126	5	)	)	PUNCT
ma-166	126	6	)	)	PUNCT
ma-166	127	1	+	+	CCONJ
ma-166	127	2	maxi(d(x1	maxi(d(x1	ADJ
ma-166	127	3	)	)	PUNCT
ma-166	127	4	,	,	PUNCT
ma-166	127	5	d(x2	d(x2	NOUN
ma-166	127	6	)	)	PUNCT
ma-166	127	7	)	)	PUNCT
ma-166	128	1	mini(d(x1	mini(d(x1	PROPN
ma-166	128	2	)	)	PUNCT
ma-166	128	3	,	,	PUNCT
ma-166	128	4	d(x2	d(x2	NOUN
ma-166	128	5	)	)	PUNCT
ma-166	128	6	)	)	PUNCT
ma-166	129	1	(	(	PUNCT
ma-166	129	2	22	22	X
ma-166	129	3	)	)	PUNCT
ma-166	129	4	there	there	PRON
ma-166	129	5	are	be	VERB
ma-166	129	6	two	two	NUM
ma-166	129	7	types	type	NOUN
ma-166	129	8	of	of	ADP
ma-166	129	9	general	general	ADJ
ma-166	129	10	connectivity	connectivity	NOUN
ma-166	129	11	index	index	NOUN
ma-166	129	12	.	.	PUNCT
ma-166	130	1	the	the	DET
ma-166	130	2	general	general	PROPN
ma-166	130	3	randic	randic	ADJ
ma-166	130	4	index	index	NOUN
ma-166	130	5	(	(	PUNCT
ma-166	130	6	or	or	CCONJ
ma-166	130	7	product	product	NOUN
ma-166	130	8	-	-	PUNCT
ma-166	130	9	connectivityindex	connectivityindex	NOUN
ma-166	130	10	)	)	PUNCT
ma-166	130	11	was	be	AUX
ma-166	130	12	proposed	propose	VERB
ma-166	130	13	by	by	ADP
ma-166	130	14	bolloba	bolloba	NOUN
ma-166	130	15	and	and	CCONJ
ma-166	130	16	erdos	erdo	NOUN
ma-166	130	17	and	and	CCONJ
ma-166	130	18	is	be	AUX
ma-166	130	19	defined	define	VERB
ma-166	130	20	as	as	SCONJ
ma-166	130	21	follows	follow	VERB
ma-166	130	22	:	:	PUNCT
ma-166	130	23	m1(g	m1(g	NOUN
ma-166	130	24	)	)	PUNCT
ma-166	130	25	=	=	PUNCT
ma-166	130	26	∑	∑	PUNCT
ma-166	131	1	[	[	X
ma-166	131	2	x2∈v	x2∈v	X
ma-166	131	3	(	(	PUNCT
ma-166	131	4	g	g	NOUN
ma-166	131	5	)	)	PUNCT
ma-166	131	6	]	]	PUNCT
ma-166	132	1	[	[	X
ma-166	132	2	dg(x2	dg(x2	NOUN
ma-166	132	3	)	)	PUNCT
ma-166	132	4	]	]	PUNCT
ma-166	132	5	2	2	NUM
ma-166	132	6	(	(	PUNCT
ma-166	132	7	23	23	NUM
ma-166	132	8	)	)	PUNCT
ma-166	132	9	where	where	SCONJ
ma-166	132	10	α	α	NOUN
ma-166	132	11	is	be	AUX
ma-166	132	12	a	a	DET
ma-166	132	13	real	real	ADJ
ma-166	132	14	number	number	NOUN
ma-166	132	15	.	.	PUNCT
ma-166	133	1	if	if	SCONJ
ma-166	133	2	α	α	PRON
ma-166	133	3	=	=	X
ma-166	133	4	−12	−12	PROPN
ma-166	133	5	,	,	PUNCT
ma-166	133	6	then	then	ADV
ma-166	133	7	it	it	PRON
ma-166	133	8	becomes	become	VERB
ma-166	133	9	the	the	DET
ma-166	133	10	randic	randic	ADJ
ma-166	133	11	index	index	NOUN
ma-166	133	12	and	and	CCONJ
ma-166	133	13	if	if	SCONJ
ma-166	133	14	α	α	PRON
ma-166	133	15	=	=	NOUN
ma-166	133	16	1	1	NUM
ma-166	133	17	then	then	ADV
ma-166	133	18	it	it	PRON
ma-166	133	19	becomesthe	becomesthe	VERB
ma-166	133	20	second	second	PROPN
ma-166	133	21	zagreb	zagreb	PROPN
ma-166	133	22	index	index	PROPN
ma-166	133	23	.	.	PUNCT
ma-166	134	1	zhou	zhou	PROPN
ma-166	134	2	and	and	CCONJ
ma-166	134	3	trinajstic	trinajstic	PROPN
ma-166	134	4	developed	develop	VERB
ma-166	134	5	the	the	DET
ma-166	134	6	general	general	ADJ
ma-166	134	7	sum	sum	NOUN
ma-166	134	8	-	-	PUNCT
ma-166	134	9	connectivity	connectivity	NOUN
ma-166	134	10	index	index	NOUN
ma-166	134	11	:	:	PUNCT
ma-166	135	1	[	[	X
ma-166	135	2	onthe	onthe	ADJ
ma-166	135	3	general	general	ADJ
ma-166	135	4	sum	sum	NOUN
ma-166	135	5	-	-	PUNCT
ma-166	135	6	connectivity	connectivity	NOUN
ma-166	135	7	index	index	NOUN
ma-166	135	8	of	of	ADP
ma-166	135	9	trees	tree	NOUN
ma-166	135	10	]	]	PUNCT
ma-166	135	11	m1(g	m1(g	NOUN
ma-166	135	12	)	)	PUNCT
ma-166	135	13	=	=	PUNCT
ma-166	135	14	∑	∑	PUNCT
ma-166	136	1	[	[	X
ma-166	136	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	136	3	)	)	PUNCT
ma-166	136	4	]	]	PUNCT
ma-166	137	1	[	[	X
ma-166	137	2	d(x1	d(x1	NOUN
ma-166	137	3	)	)	PUNCT
ma-166	137	4	+	+	NUM
ma-166	137	5	d(x2	d(x2	NOUN
ma-166	137	6	)	)	PUNCT
ma-166	137	7	]	]	PUNCT
ma-166	138	1	α	α	PRON
ma-166	138	2	(	(	PUNCT
ma-166	138	3	24	24	NUM
ma-166	138	4	)	)	PUNCT
ma-166	138	5	where	where	SCONJ
ma-166	138	6	α	α	NOUN
ma-166	138	7	is	be	AUX
ma-166	138	8	a	a	DET
ma-166	138	9	real	real	ADJ
ma-166	138	10	number	number	NOUN
ma-166	138	11	.	.	PUNCT
ma-166	139	1	if	if	SCONJ
ma-166	139	2	α	α	PRON
ma-166	139	3	=	=	SYM
ma-166	139	4	1	1	NUM
ma-166	139	5	,	,	PUNCT
ma-166	139	6	then	then	ADV
ma-166	139	7	the	the	DET
ma-166	139	8	general	general	ADJ
ma-166	139	9	sum	sum	NOUN
ma-166	139	10	connectivity	connectivity	NOUN
ma-166	139	11	index	index	NOUN
ma-166	139	12	becomes	become	VERB
ma-166	139	13	the	the	DET
ma-166	139	14	firstzagreb	firstzagreb	NOUN
ma-166	139	15	index	index	NOUN
ma-166	139	16	resently	resently	ADV
ma-166	139	17	asghar	asghar	PROPN
ma-166	139	18	et.al[14	et.al[14	PROPN
ma-166	139	19	]	]	PUNCT
ma-166	139	20	.	.	PUNCT
ma-166	140	1	2	2	X
ma-166	140	2	.	.	X
ma-166	140	3	main	main	ADJ
ma-166	140	4	results	result	NOUN
ma-166	140	5	in	in	ADP
ma-166	140	6	this	this	DET
ma-166	140	7	section	section	NOUN
ma-166	140	8	,	,	PUNCT
ma-166	140	9	we	we	PRON
ma-166	140	10	established	establish	VERB
ma-166	140	11	some	some	DET
ma-166	140	12	results	result	NOUN
ma-166	140	13	on	on	ADP
ma-166	140	14	degree	degree	NOUN
ma-166	140	15	based	base	VERB
ma-166	140	16	topological	topological	ADJ
ma-166	140	17	indices	index	NOUN
ma-166	140	18	of	of	ADP
ma-166	140	19	petersengraph	petersengraph	NOUN
ma-166	140	20	.	.	PUNCT
ma-166	141	1	theorem	theorem	VERB
ma-166	141	2	2.1	2.1	NUM
ma-166	141	3	let	let	VERB
ma-166	141	4	p(k	p(k	NOUN
ma-166	141	5	,	,	PUNCT
ma-166	141	6	t	t	NUM
ma-166	141	7	)	)	PUNCT
ma-166	141	8	be	be	AUX
ma-166	141	9	petersen	petersen	NOUN
ma-166	141	10	subdivision	subdivision	NOUN
ma-166	141	11	graph	graph	NOUN
ma-166	141	12	.	.	PUNCT
ma-166	142	1	then	then	ADV
ma-166	142	2	,	,	PUNCT
ma-166	142	3	for	for	ADP
ma-166	142	4	tn	tn	NOUN
ma-166	142	5	=	=	SYM
ma-166	142	6	{	{	PUNCT
ma-166	142	7	1	1	NUM
ma-166	142	8	,	,	PUNCT
ma-166	142	9	2	2	NUM
ma-166	142	10	,	,	PUNCT
ma-166	142	11	3	3	NUM
ma-166	142	12	,	,	PUNCT
ma-166	142	13	...	...	PUNCT
ma-166	142	14	}	}	PUNCT
ma-166	142	15	,	,	PUNCT
ma-166	142	16	first	first	ADJ
ma-166	142	17	zagrebpolynomials	zagrebpolynomial	NOUN
ma-166	142	18	indices	index	NOUN
ma-166	142	19	are	be	AUX
ma-166	142	20	,	,	PUNCT
ma-166	142	21	m1(g	m1(g	NOUN
ma-166	142	22	,	,	PUNCT
ma-166	142	23	x	x	NOUN
ma-166	142	24	)	)	PUNCT
ma-166	142	25	=[	=[	NOUN
ma-166	142	26	|6n|](x)(6	|6n|](x)(6	NOUN
ma-166	142	27	)	)	PUNCT
ma-166	142	28	proof	proof	NOUN
ma-166	142	29	:	:	PUNCT
ma-166	142	30	the	the	DET
ma-166	142	31	petersen	petersen	PROPN
ma-166	142	32	graph	graph	VERB
ma-166	142	33	tn	tn	PROPN
ma-166	143	1	=	=	PUNCT
ma-166	143	2	{	{	PUNCT
ma-166	143	3	1	1	NUM
ma-166	143	4	,	,	PUNCT
ma-166	143	5	2	2	NUM
ma-166	143	6	,	,	PUNCT
ma-166	143	7	3	3	NUM
ma-166	143	8	,	,	PUNCT
ma-166	143	9	...	...	PUNCT
ma-166	143	10	}	}	PUNCT
ma-166	143	11	appears	appear	VERB
ma-166	143	12	in	in	ADP
ma-166	143	13	figure(graph	figure(graph	PROPN
ma-166	143	14	)	)	PUNCT
ma-166	143	15	.	.	PUNCT
ma-166	144	1	the	the	DET
ma-166	144	2	petersen	petersen	PROPN
ma-166	144	3	graph	graph	NOUN
ma-166	144	4	tn=	tn=	PROPN
ma-166	144	5	{	{	PUNCT
ma-166	144	6	1	1	NUM
ma-166	144	7	,	,	PUNCT
ma-166	144	8	2	2	NUM
ma-166	144	9	,	,	PUNCT
ma-166	144	10	3	3	NUM
ma-166	144	11	,	,	PUNCT
ma-166	144	12	...	...	PUNCT
ma-166	144	13	}	}	PUNCT
ma-166	144	14	contains	contain	VERB
ma-166	144	15	v	v	NOUN
ma-166	144	16	(	(	PUNCT
ma-166	144	17	g′	g′	NOUN
ma-166	144	18	)	)	PUNCT
ma-166	144	19	=	=	NOUN
ma-166	145	1	4n	4n	NOUN
ma-166	145	2	no	no	PRON
ma-166	145	3	of	of	ADP
ma-166	145	4	vertices	vertex	NOUN
ma-166	145	5	and	and	CCONJ
ma-166	145	6	e(g′	e(g′	PRON
ma-166	145	7	)	)	PUNCT
ma-166	146	1	=	=	SYM
ma-166	146	2	6n	6n	NUM
ma-166	146	3	no	no	PRON
ma-166	146	4	of	of	ADP
ma-166	146	5	edges	edge	NOUN
ma-166	146	6	.	.	PUNCT
ma-166	147	1	the	the	DET
ma-166	147	2	degree	degree	NOUN
ma-166	147	3	ofeach	ofeach	NOUN
ma-166	147	4	vertex	vertex	NOUN
ma-166	147	5	in	in	ADP
ma-166	147	6	p(k	p(k	NOUN
ma-166	147	7	,	,	PUNCT
ma-166	147	8	t	t	NUM
ma-166	147	9	)	)	PUNCT
ma-166	147	10	is	be	AUX
ma-166	147	11	3	3	NUM
ma-166	147	12	and	and	CCONJ
ma-166	147	13	now	now	ADV
ma-166	147	14	first	first	ADJ
ma-166	147	15	zagreb	zagreb	PROPN
ma-166	147	16	polynomials	polynomial	NOUN
ma-166	147	17	indices	index	NOUN
ma-166	147	18	are	be	AUX
ma-166	147	19	i.e.	i.e.	X
ma-166	147	20	,	,	PUNCT
ma-166	147	21	⇒	⇒	NOUN
ma-166	147	22	ga(r	ga(r	PROPN
ma-166	147	23	)	)	PUNCT
ma-166	148	1	=	=	SYM
ma-166	148	2	∑	∑	PUNCT
ma-166	148	3	y1,y2∈e(r	y1,y2∈e(r	NOUN
ma-166	148	4	)	)	PUNCT
ma-166	148	5	2	2	NUM
ma-166	148	6	√	√	NUM
ma-166	148	7	dy1dy2	dy1dy2	INTJ
ma-166	149	1	dy1+dy2now	dy1+dy2now	INTJ
ma-166	149	2	we	we	PRON
ma-166	149	3	suppose	suppose	VERB
ma-166	149	4	vertices	vertex	NOUN
ma-166	149	5	are	be	AUX
ma-166	149	6	v	v	ADP
ma-166	149	7	(	(	PUNCT
ma-166	149	8	g	g	NOUN
ma-166	149	9	)	)	PUNCT
ma-166	149	10	=	=	SYM
ma-166	149	11	4n	4n	NOUN
ma-166	149	12	,	,	PUNCT
ma-166	149	13	edges	edge	NOUN
ma-166	149	14	are	be	AUX
ma-166	149	15	e(g	e(g	NOUN
ma-166	149	16	)	)	PUNCT
ma-166	150	1	=	=	SYM
ma-166	150	2	6n	6n	NOUN
ma-166	150	3	and	and	CCONJ
ma-166	150	4	degree	degree	NOUN
ma-166	150	5	of	of	ADP
ma-166	150	6	petersen	petersen	PROPN
ma-166	150	7	graphabout	graphabout	PROPN
ma-166	150	8	every	every	DET
ma-166	150	9	each	each	DET
ma-166	150	10	vertices	vertex	NOUN
ma-166	150	11	is	be	AUX
ma-166	150	12	p(k	p(k	NOUN
ma-166	150	13	,	,	PUNCT
ma-166	150	14	t	t	NOUN
ma-166	150	15	)	)	PUNCT
ma-166	150	16	=[	=[	NOUN
ma-166	150	17	d(x1	d(x1	NOUN
ma-166	150	18	)	)	PUNCT
ma-166	150	19	,	,	PUNCT
ma-166	150	20	d(x2	d(x2	NOUN
ma-166	150	21	)	)	PUNCT
ma-166	150	22	]	]	PUNCT
ma-166	151	1	=	=	PUNCT
ma-166	151	2	3	3	X
ma-166	151	3	.	.	PUNCT
ma-166	151	4	now	now	ADV
ma-166	151	5	putting	put	VERB
ma-166	151	6	the	the	DET
ma-166	151	7	values	value	NOUN
ma-166	151	8	in	in	ADP
ma-166	151	9	first	first	ADJ
ma-166	151	10	zagrebtopological	zagrebtopological	ADJ
ma-166	151	11	index	index	NOUN
ma-166	151	12	of	of	ADP
ma-166	151	13	the	the	DET
ma-166	151	14	general	general	ADJ
ma-166	151	15	form	form	NOUN
ma-166	151	16	,	,	PUNCT
ma-166	151	17	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	151	18	eur	eur	PROPN
ma-166	151	19	.	.	PUNCT
ma-166	152	1	j.	j.	PROPN
ma-166	152	2	math	math	PROPN
ma-166	152	3	.	.	PUNCT
ma-166	153	1	anal	anal	PROPN
ma-166	153	2	.	.	PUNCT
ma-166	154	1	10.28924	10.28924	NUM
ma-166	154	2	/	/	SYM
ma-166	154	3	ada	ada	PROPN
ma-166	154	4	/	/	SYM
ma-166	154	5	ma.3.20	ma.3.20	PROPN
ma-166	154	6	6	6	NUM
ma-166	154	7	⇒	⇒	PROPN
ma-166	154	8	m1(g	m1(g	NOUN
ma-166	154	9	,	,	PUNCT
ma-166	154	10	x	x	NOUN
ma-166	154	11	)	)	PUNCT
ma-166	154	12	=[	=[	NOUN
ma-166	154	13	|e(g)|](x)[(3)+(3	|e(g)|](x)[(3)+(3	NOUN
ma-166	154	14	)	)	PUNCT
ma-166	154	15	]	]	PUNCT
ma-166	154	16	⇒	⇒	PROPN
ma-166	154	17	m1(g	m1(g	PROPN
ma-166	154	18	,	,	PUNCT
ma-166	154	19	x	x	NOUN
ma-166	154	20	)	)	PUNCT
ma-166	154	21	=(	=(	ADJ
ma-166	154	22	6n)(x)(6	6n)(x)(6	NOUN
ma-166	154	23	)	)	PUNCT
ma-166	154	24	⇒	⇒	PROPN
ma-166	154	25	m1(g	m1(g	NOUN
ma-166	154	26	,	,	PUNCT
ma-166	154	27	x	x	NOUN
ma-166	154	28	)	)	PUNCT
ma-166	154	29	=	=	SYM
ma-166	154	30	(	(	PUNCT
ma-166	154	31	6n)(x)6	6n)(x)6	NOUN
ma-166	154	32	.	.	PUNCT
ma-166	155	1	m1(g	m1(g	NOUN
ma-166	155	2	,	,	PUNCT
ma-166	155	3	x	x	NOUN
ma-166	155	4	)	)	PUNCT
ma-166	155	5	=	=	SYM
ma-166	155	6	(	(	PUNCT
ma-166	155	7	x)6×	x)6×	PUNCT
ma-166	156	1	[	[	X
ma-166	156	2	general	general	ADJ
ma-166	156	3	edges	edge	NOUN
ma-166	156	4	of	of	ADP
ma-166	156	5	petersen	petersen	PROPN
ma-166	156	6	graph	graph	NOUN
ma-166	156	7	]	]	PUNCT
ma-166	156	8	theorem	theorem	VERB
ma-166	156	9	2.2	2.2	NUM
ma-166	156	10	let	let	VERB
ma-166	156	11	p(k	p(k	NOUN
ma-166	156	12	,	,	PUNCT
ma-166	156	13	t	t	NUM
ma-166	156	14	)	)	PUNCT
ma-166	156	15	be	be	AUX
ma-166	156	16	petersen	petersen	NOUN
ma-166	156	17	subdivision	subdivision	NOUN
ma-166	156	18	graph	graph	NOUN
ma-166	156	19	.	.	PUNCT
ma-166	157	1	then	then	ADV
ma-166	157	2	,	,	PUNCT
ma-166	157	3	for	for	ADP
ma-166	157	4	tn	tn	NOUN
ma-166	157	5	=	=	SYM
ma-166	157	6	{	{	PUNCT
ma-166	157	7	1	1	NUM
ma-166	157	8	,	,	PUNCT
ma-166	157	9	2	2	NUM
ma-166	157	10	,	,	PUNCT
ma-166	157	11	3	3	NUM
ma-166	157	12	,	,	PUNCT
ma-166	157	13	...	...	PUNCT
ma-166	157	14	}	}	PUNCT
ma-166	157	15	,	,	PUNCT
ma-166	157	16	second	second	ADJ
ma-166	157	17	zagrebpolynomials	zagrebpolynomial	NOUN
ma-166	157	18	indices	index	NOUN
ma-166	157	19	are	be	AUX
ma-166	157	20	,	,	PUNCT
ma-166	157	21	m2(g	m2(g	PROPN
ma-166	157	22	,	,	PUNCT
ma-166	157	23	x	x	NOUN
ma-166	157	24	)	)	PUNCT
ma-166	157	25	=	=	SYM
ma-166	158	1	(	(	PUNCT
ma-166	158	2	6n)(x)9	6n)(x)9	NUM
ma-166	158	3	proof	proof	NOUN
ma-166	158	4	:	:	PUNCT
ma-166	158	5	the	the	DET
ma-166	158	6	petersen	petersen	PROPN
ma-166	158	7	graph	graph	VERB
ma-166	158	8	tn	tn	PROPN
ma-166	158	9	=	=	PUNCT
ma-166	158	10	{	{	PUNCT
ma-166	158	11	1	1	NUM
ma-166	158	12	,	,	PUNCT
ma-166	158	13	2	2	NUM
ma-166	158	14	,	,	PUNCT
ma-166	158	15	3	3	NUM
ma-166	158	16	,	,	PUNCT
ma-166	158	17	...	...	PUNCT
ma-166	158	18	}	}	PUNCT
ma-166	158	19	appears	appear	VERB
ma-166	158	20	in	in	ADP
ma-166	158	21	figure(graph	figure(graph	PROPN
ma-166	158	22	)	)	PUNCT
ma-166	158	23	.	.	PUNCT
ma-166	159	1	the	the	DET
ma-166	159	2	petersen	petersen	PROPN
ma-166	159	3	graph	graph	NOUN
ma-166	159	4	tn=	tn=	PROPN
ma-166	159	5	{	{	PUNCT
ma-166	159	6	1	1	NUM
ma-166	159	7	,	,	PUNCT
ma-166	159	8	2	2	NUM
ma-166	159	9	,	,	PUNCT
ma-166	159	10	3	3	NUM
ma-166	159	11	,	,	PUNCT
ma-166	159	12	...	...	PUNCT
ma-166	159	13	}	}	PUNCT
ma-166	159	14	contains	contain	VERB
ma-166	159	15	v	v	NOUN
ma-166	159	16	(	(	PUNCT
ma-166	159	17	g′	g′	NOUN
ma-166	159	18	)	)	PUNCT
ma-166	159	19	=	=	NOUN
ma-166	160	1	4n	4n	NOUN
ma-166	160	2	no	no	PRON
ma-166	160	3	of	of	ADP
ma-166	160	4	vertices	vertex	NOUN
ma-166	160	5	and	and	CCONJ
ma-166	160	6	e(g′	e(g′	PRON
ma-166	160	7	)	)	PUNCT
ma-166	161	1	=	=	SYM
ma-166	161	2	6n	6n	NUM
ma-166	161	3	no	no	PRON
ma-166	161	4	of	of	ADP
ma-166	161	5	edges	edge	NOUN
ma-166	161	6	.	.	PUNCT
ma-166	162	1	the	the	DET
ma-166	162	2	degree	degree	NOUN
ma-166	162	3	ofeach	ofeach	NOUN
ma-166	162	4	vertex	vertex	NOUN
ma-166	162	5	in	in	ADP
ma-166	162	6	p(k	p(k	NOUN
ma-166	162	7	,	,	PUNCT
ma-166	162	8	t	t	NUM
ma-166	162	9	)	)	PUNCT
ma-166	162	10	is	be	AUX
ma-166	162	11	3	3	NUM
ma-166	162	12	and	and	CCONJ
ma-166	162	13	now	now	ADV
ma-166	162	14	second	second	ADJ
ma-166	162	15	zagreb	zagreb	PROPN
ma-166	162	16	polynomials	polynomial	NOUN
ma-166	162	17	indices	index	NOUN
ma-166	162	18	are	be	AUX
ma-166	162	19	i.e.	i.e.	X
ma-166	162	20	,	,	PUNCT
ma-166	162	21	m2(g	m2(g	PROPN
ma-166	162	22	,	,	PUNCT
ma-166	162	23	x	x	NOUN
ma-166	162	24	)	)	PUNCT
ma-166	162	25	=	=	PUNCT
ma-166	163	1	∑	∑	PUNCT
ma-166	164	1	[	[	X
ma-166	164	2	x1,x2∈e(g)](x	x1,x2∈e(g)](x	X
ma-166	164	3	)	)	PUNCT
ma-166	165	1	[	[	X
ma-166	165	2	d(x1)×d(x2	d(x1)×d(x2	X
ma-166	165	3	)	)	PUNCT
ma-166	165	4	]	]	PUNCT
ma-166	165	5	→	→	PUNCT
ma-166	165	6	[	[	X
ma-166	165	7	1	1	X
ma-166	165	8	]	]	PUNCT
ma-166	165	9	now	now	ADV
ma-166	165	10	we	we	PRON
ma-166	165	11	suppose	suppose	VERB
ma-166	165	12	vertices	vertex	NOUN
ma-166	165	13	are	be	AUX
ma-166	165	14	v	v	ADP
ma-166	165	15	(	(	PUNCT
ma-166	165	16	g	g	NOUN
ma-166	165	17	)	)	PUNCT
ma-166	165	18	=	=	SYM
ma-166	165	19	4n	4n	NOUN
ma-166	165	20	,	,	PUNCT
ma-166	165	21	edges	edge	NOUN
ma-166	165	22	are	be	AUX
ma-166	165	23	e(g	e(g	NOUN
ma-166	165	24	)	)	PUNCT
ma-166	166	1	=	=	SYM
ma-166	166	2	6n	6n	NOUN
ma-166	166	3	and	and	CCONJ
ma-166	166	4	degree	degree	NOUN
ma-166	166	5	of	of	ADP
ma-166	166	6	petersen	petersen	PROPN
ma-166	166	7	graphabout	graphabout	PROPN
ma-166	166	8	every	every	DET
ma-166	166	9	each	each	DET
ma-166	166	10	vertices	vertex	NOUN
ma-166	166	11	is	be	AUX
ma-166	166	12	p(k	p(k	NOUN
ma-166	166	13	,	,	PUNCT
ma-166	166	14	t	t	NOUN
ma-166	166	15	)	)	PUNCT
ma-166	166	16	=[	=[	NOUN
ma-166	166	17	d(x1	d(x1	NOUN
ma-166	166	18	)	)	PUNCT
ma-166	166	19	,	,	PUNCT
ma-166	166	20	d(x2	d(x2	NOUN
ma-166	166	21	)	)	PUNCT
ma-166	166	22	]	]	PUNCT
ma-166	167	1	=	=	PUNCT
ma-166	167	2	3	3	X
ma-166	167	3	.	.	PUNCT
ma-166	167	4	now	now	ADV
ma-166	167	5	putting	put	VERB
ma-166	167	6	the	the	DET
ma-166	167	7	values	value	NOUN
ma-166	167	8	in	in	ADP
ma-166	167	9	second	second	ADJ
ma-166	167	10	zagrebtopological	zagrebtopological	ADJ
ma-166	167	11	index	index	NOUN
ma-166	167	12	of	of	ADP
ma-166	167	13	the	the	DET
ma-166	167	14	general	general	ADJ
ma-166	167	15	form	form	NOUN
ma-166	167	16	,	,	PUNCT
ma-166	167	17	⇒	⇒	VERB
ma-166	167	18	m2(g	m2(g	NOUN
ma-166	167	19	,	,	PUNCT
ma-166	167	20	x	x	NOUN
ma-166	167	21	)	)	PUNCT
ma-166	167	22	=	=	NOUN
ma-166	167	23	∑	∑	NOUN
ma-166	167	24	x1,x2∈e(g)(x	x1,x2∈e(g)(x	PROPN
ma-166	167	25	)	)	PUNCT
ma-166	168	1	[	[	X
ma-166	168	2	(	(	PUNCT
ma-166	168	3	3)(3	3)(3	NUM
ma-166	168	4	)	)	PUNCT
ma-166	168	5	]	]	PUNCT
ma-166	168	6	⇒	⇒	VERB
ma-166	168	7	m2(g	m2(g	NOUN
ma-166	168	8	,	,	PUNCT
ma-166	168	9	x	x	X
ma-166	168	10	)	)	PUNCT
ma-166	168	11	=[	=[	VERB
ma-166	168	12	|e(g)|](x)9	|e(g)|](x)9	PROPN
ma-166	168	13	⇒	⇒	PROPN
ma-166	168	14	m2(g	m2(g	NOUN
ma-166	168	15	,	,	PUNCT
ma-166	168	16	x	x	NOUN
ma-166	168	17	)	)	PUNCT
ma-166	168	18	=	=	SYM
ma-166	168	19	(	(	PUNCT
ma-166	168	20	6n)(x)9	6n)(x)9	PROPN
ma-166	168	21	.	.	PUNCT
ma-166	169	1	m2(g	m2(g	NOUN
ma-166	169	2	,	,	PUNCT
ma-166	169	3	x	x	NOUN
ma-166	169	4	)	)	PUNCT
ma-166	169	5	=	=	SYM
ma-166	169	6	(	(	PUNCT
ma-166	169	7	x)9×	x)9×	PROPN
ma-166	170	1	[	[	X
ma-166	170	2	general	general	ADJ
ma-166	170	3	edges	edge	NOUN
ma-166	170	4	of	of	ADP
ma-166	170	5	petersen	petersen	PROPN
ma-166	170	6	graph	graph	NOUN
ma-166	170	7	]	]	PUNCT
ma-166	170	8	theorem	theorem	VERB
ma-166	170	9	2.3	2.3	NUM
ma-166	170	10	let	let	VERB
ma-166	170	11	p(k	p(k	NOUN
ma-166	170	12	,	,	PUNCT
ma-166	170	13	t	t	NUM
ma-166	170	14	)	)	PUNCT
ma-166	170	15	be	be	AUX
ma-166	170	16	petersen	petersen	NOUN
ma-166	170	17	subdivision	subdivision	NOUN
ma-166	170	18	graph	graph	NOUN
ma-166	170	19	.	.	PUNCT
ma-166	171	1	then	then	ADV
ma-166	171	2	,	,	PUNCT
ma-166	171	3	for	for	ADP
ma-166	171	4	tn	tn	NOUN
ma-166	171	5	=	=	SYM
ma-166	171	6	{	{	PUNCT
ma-166	171	7	1	1	NUM
ma-166	171	8	,	,	PUNCT
ma-166	171	9	2	2	NUM
ma-166	171	10	,	,	PUNCT
ma-166	171	11	3	3	NUM
ma-166	171	12	,	,	PUNCT
ma-166	171	13	...	...	PUNCT
ma-166	171	14	}	}	PUNCT
ma-166	171	15	,	,	PUNCT
ma-166	171	16	randic	randic	ADJ
ma-166	171	17	indicesare	indicesare	PROPN
ma-166	171	18	,	,	PUNCT
ma-166	171	19	r1(α)(g	r1(α)(g	NOUN
ma-166	171	20	)	)	PUNCT
ma-166	171	21	=	=	PUNCT
ma-166	172	1	(	(	PUNCT
ma-166	172	2	6n)[6]α	6n)[6]α	NUM
ma-166	172	3	proof	proof	NOUN
ma-166	172	4	:	:	PUNCT
ma-166	172	5	the	the	DET
ma-166	172	6	petersen	petersen	PROPN
ma-166	172	7	graph	graph	VERB
ma-166	172	8	tn	tn	PROPN
ma-166	172	9	=	=	PUNCT
ma-166	172	10	{	{	PUNCT
ma-166	172	11	1	1	NUM
ma-166	172	12	,	,	PUNCT
ma-166	172	13	2	2	NUM
ma-166	172	14	,	,	PUNCT
ma-166	172	15	3	3	NUM
ma-166	172	16	,	,	PUNCT
ma-166	172	17	...	...	PUNCT
ma-166	172	18	}	}	PUNCT
ma-166	172	19	appears	appear	VERB
ma-166	172	20	in	in	ADP
ma-166	172	21	figure(graph	figure(graph	PROPN
ma-166	172	22	)	)	PUNCT
ma-166	172	23	.	.	PUNCT
ma-166	173	1	the	the	DET
ma-166	173	2	petersen	petersen	PROPN
ma-166	173	3	graph	graph	NOUN
ma-166	173	4	tn=	tn=	PROPN
ma-166	173	5	{	{	PUNCT
ma-166	173	6	1	1	NUM
ma-166	173	7	,	,	PUNCT
ma-166	173	8	2	2	NUM
ma-166	173	9	,	,	PUNCT
ma-166	173	10	3	3	NUM
ma-166	173	11	,	,	PUNCT
ma-166	173	12	...	...	PUNCT
ma-166	173	13	}	}	PUNCT
ma-166	173	14	contains	contain	VERB
ma-166	173	15	v	v	NOUN
ma-166	173	16	(	(	PUNCT
ma-166	173	17	g′	g′	NOUN
ma-166	173	18	)	)	PUNCT
ma-166	173	19	=	=	NOUN
ma-166	174	1	4n	4n	NOUN
ma-166	174	2	no	no	PRON
ma-166	174	3	of	of	ADP
ma-166	174	4	vertices	vertex	NOUN
ma-166	174	5	and	and	CCONJ
ma-166	174	6	e(g′	e(g′	PRON
ma-166	174	7	)	)	PUNCT
ma-166	175	1	=	=	SYM
ma-166	175	2	6n	6n	NUM
ma-166	175	3	no	no	PRON
ma-166	175	4	of	of	ADP
ma-166	175	5	edges	edge	NOUN
ma-166	175	6	.	.	PUNCT
ma-166	176	1	the	the	DET
ma-166	176	2	degree	degree	NOUN
ma-166	176	3	ofeach	ofeach	NOUN
ma-166	176	4	vertex	vertex	NOUN
ma-166	176	5	in	in	ADP
ma-166	176	6	p(k	p(k	NOUN
ma-166	176	7	,	,	PUNCT
ma-166	176	8	t	t	NUM
ma-166	176	9	)	)	PUNCT
ma-166	176	10	is	be	AUX
ma-166	176	11	3	3	NUM
ma-166	176	12	and	and	CCONJ
ma-166	176	13	now	now	ADV
ma-166	176	14	randic	randic	ADJ
ma-166	176	15	indices	index	NOUN
ma-166	176	16	are	be	AUX
ma-166	176	17	i.e.	i.e.	X
ma-166	176	18	,	,	PUNCT
ma-166	176	19	r1(α)(g	r1(α)(g	NOUN
ma-166	176	20	)	)	PUNCT
ma-166	176	21	=	=	PUNCT
ma-166	177	1	∑	∑	PUNCT
ma-166	177	2	[	[	X
ma-166	177	3	x1,x2∈e(g)][d(x1	x1,x2∈e(g)][d(x1	X
ma-166	177	4	)	)	PUNCT
ma-166	177	5	+	+	NUM
ma-166	177	6	d(x2	d(x2	NOUN
ma-166	177	7	)	)	PUNCT
ma-166	177	8	]	]	PUNCT
ma-166	178	1	α	α	PRON
ma-166	178	2	now	now	ADV
ma-166	178	3	we	we	PRON
ma-166	178	4	suppose	suppose	VERB
ma-166	178	5	vertices	vertex	NOUN
ma-166	178	6	are	be	AUX
ma-166	178	7	v	v	ADP
ma-166	178	8	(	(	PUNCT
ma-166	178	9	g	g	NOUN
ma-166	178	10	)	)	PUNCT
ma-166	178	11	=	=	SYM
ma-166	178	12	4n	4n	NOUN
ma-166	178	13	,	,	PUNCT
ma-166	178	14	edges	edge	NOUN
ma-166	178	15	are	be	AUX
ma-166	178	16	e(g	e(g	NOUN
ma-166	178	17	)	)	PUNCT
ma-166	179	1	=	=	SYM
ma-166	179	2	6n	6n	NOUN
ma-166	179	3	and	and	CCONJ
ma-166	179	4	degree	degree	NOUN
ma-166	179	5	of	of	ADP
ma-166	179	6	petersen	petersen	PROPN
ma-166	179	7	graphabout	graphabout	PROPN
ma-166	179	8	every	every	DET
ma-166	179	9	each	each	DET
ma-166	179	10	vertices	vertex	NOUN
ma-166	179	11	is	be	AUX
ma-166	179	12	p(k	p(k	NOUN
ma-166	179	13	,	,	PUNCT
ma-166	179	14	t	t	NOUN
ma-166	179	15	)	)	PUNCT
ma-166	179	16	=[	=[	NOUN
ma-166	179	17	d(x1	d(x1	NOUN
ma-166	179	18	)	)	PUNCT
ma-166	179	19	,	,	PUNCT
ma-166	179	20	d(x2	d(x2	NOUN
ma-166	179	21	)	)	PUNCT
ma-166	179	22	]	]	PUNCT
ma-166	180	1	=	=	PUNCT
ma-166	180	2	3	3	X
ma-166	180	3	.	.	PUNCT
ma-166	180	4	now	now	ADV
ma-166	180	5	putting	put	VERB
ma-166	180	6	the	the	DET
ma-166	180	7	values	value	NOUN
ma-166	180	8	in	in	ADP
ma-166	180	9	randic	randic	ADJ
ma-166	180	10	indices	index	NOUN
ma-166	180	11	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	PROPN
ma-166	180	12	eur	eur	PROPN
ma-166	180	13	.	.	PUNCT
ma-166	181	1	j.	j.	PROPN
ma-166	181	2	math	math	PROPN
ma-166	181	3	.	.	PUNCT
ma-166	182	1	anal	anal	PROPN
ma-166	182	2	.	.	PUNCT
ma-166	183	1	10.28924	10.28924	NUM
ma-166	183	2	/	/	SYM
ma-166	183	3	ada	ada	PROPN
ma-166	183	4	/	/	SYM
ma-166	183	5	ma.3.20	ma.3.20	PROPN
ma-166	183	6	7topological	7topological	NUM
ma-166	183	7	index	index	NOUN
ma-166	183	8	of	of	ADP
ma-166	183	9	the	the	DET
ma-166	183	10	general	general	ADJ
ma-166	183	11	form	form	NOUN
ma-166	183	12	,	,	PUNCT
ma-166	183	13	r1(α)(g	r1(α)(g	NOUN
ma-166	183	14	)	)	PUNCT
ma-166	183	15	=	=	PUNCT
ma-166	184	1	∑	∑	PUNCT
ma-166	184	2	[	[	X
ma-166	184	3	x1,x2∈e(g)][d(x1	x1,x2∈e(g)][d(x1	X
ma-166	184	4	)	)	PUNCT
ma-166	184	5	+	+	NUM
ma-166	184	6	d(x2	d(x2	NOUN
ma-166	184	7	)	)	PUNCT
ma-166	184	8	]	]	PUNCT
ma-166	185	1	α	α	PROPN
ma-166	185	2	in	in	ADP
ma-166	185	3	general	general	ADJ
ma-166	185	4	form	form	NOUN
ma-166	185	5	of	of	ADP
ma-166	185	6	topological	topological	ADJ
ma-166	185	7	index	index	NOUN
ma-166	185	8	becomes	become	VERB
ma-166	185	9	;	;	PUNCT
ma-166	185	10	⇒	⇒	PROPN
ma-166	185	11	r1(α)(g	r1(α)(g	PROPN
ma-166	185	12	)	)	PUNCT
ma-166	185	13	=	=	PUNCT
ma-166	185	14	∑	∑	PUNCT
ma-166	186	1	[	[	X
ma-166	186	2	x1,x2∈e(g)][d(x1	x1,x2∈e(g)][d(x1	X
ma-166	186	3	)	)	PUNCT
ma-166	186	4	+	+	NUM
ma-166	186	5	d(x2	d(x2	NOUN
ma-166	186	6	)	)	PUNCT
ma-166	186	7	]	]	PUNCT
ma-166	187	1	αnow	αnow	NOUN
ma-166	187	2	putting	put	VERB
ma-166	187	3	values	value	NOUN
ma-166	187	4	in	in	ADP
ma-166	187	5	above	above	ADP
ma-166	187	6	equation	equation	NOUN
ma-166	187	7	,	,	PUNCT
ma-166	187	8	⇒	⇒	PROPN
ma-166	187	9	r1(α)(g	r1(α)(g	PROPN
ma-166	187	10	)	)	PUNCT
ma-166	187	11	=[	=[	PROPN
ma-166	187	12	|e(g)|][(3	|e(g)|][(3	NUM
ma-166	187	13	)	)	PUNCT
ma-166	188	1	+	+	CCONJ
ma-166	188	2	(	(	PUNCT
ma-166	188	3	3)]α	3)]α	NUM
ma-166	188	4	⇒	⇒	NOUN
ma-166	188	5	r1(α)(g	r1(α)(g	PROPN
ma-166	188	6	)	)	PUNCT
ma-166	188	7	=[	=[	NOUN
ma-166	188	8	|e(g)|][6]α	|e(g)|][6]α	PROPN
ma-166	188	9	⇒	⇒	PROPN
ma-166	188	10	r1(α)(g	r1(α)(g	PROPN
ma-166	188	11	)	)	PUNCT
ma-166	188	12	=[	=[	NOUN
ma-166	188	13	|6n|][6]α	|6n|][6]α	PROPN
ma-166	188	14	⇒	⇒	NOUN
ma-166	188	15	r1(α)(g	r1(α)(g	NOUN
ma-166	188	16	)	)	PUNCT
ma-166	188	17	=	=	PUNCT
ma-166	189	1	(	(	PUNCT
ma-166	189	2	6n)[6]α	6n)[6]α	NOUN
ma-166	189	3	.	.	PUNCT
ma-166	189	4	r1(α)(g	r1(α)(g	NOUN
ma-166	189	5	)	)	PUNCT
ma-166	189	6	=	=	PUNCT
ma-166	190	1	[	[	X
ma-166	190	2	6α]×	6α]×	X
ma-166	190	3	[	[	PUNCT
ma-166	190	4	the	the	DET
ma-166	190	5	general	general	ADJ
ma-166	190	6	edges	edge	NOUN
ma-166	190	7	of	of	ADP
ma-166	190	8	petersen	petersen	PROPN
ma-166	190	9	graph	graph	NOUN
ma-166	190	10	]	]	PUNCT
ma-166	190	11	theorem	theorem	VERB
ma-166	190	12	2.4	2.4	NUM
ma-166	190	13	let	let	VERB
ma-166	190	14	p(k	p(k	NOUN
ma-166	190	15	,	,	PUNCT
ma-166	190	16	t	t	NUM
ma-166	190	17	)	)	PUNCT
ma-166	190	18	be	be	AUX
ma-166	190	19	petersen	petersen	NOUN
ma-166	190	20	subdivision	subdivision	NOUN
ma-166	190	21	graph	graph	NOUN
ma-166	190	22	.	.	PUNCT
ma-166	191	1	then	then	ADV
ma-166	191	2	,	,	PUNCT
ma-166	191	3	for	for	ADP
ma-166	191	4	tn	tn	NOUN
ma-166	191	5	=	=	SYM
ma-166	191	6	{	{	PUNCT
ma-166	191	7	1	1	NUM
ma-166	191	8	,	,	PUNCT
ma-166	191	9	2	2	NUM
ma-166	191	10	,	,	PUNCT
ma-166	191	11	3	3	NUM
ma-166	191	12	,	,	PUNCT
ma-166	191	13	...	...	PUNCT
ma-166	191	14	}	}	PUNCT
ma-166	191	15	,	,	PUNCT
ma-166	191	16	reducedreciprocal	reducedreciprocal	ADJ
ma-166	191	17	randic	randic	NOUN
ma-166	191	18	are	be	AUX
ma-166	191	19	,	,	PUNCT
ma-166	191	20	rrr(g	rrr(g	PROPN
ma-166	191	21	)	)	PUNCT
ma-166	191	22	=	=	SYM
ma-166	191	23	12n	12n	NOUN
ma-166	191	24	.	.	PUNCT
ma-166	192	1	proof	proof	NOUN
ma-166	192	2	:	:	PUNCT
ma-166	192	3	the	the	DET
ma-166	192	4	petersen	petersen	PROPN
ma-166	192	5	graph	graph	VERB
ma-166	192	6	tn	tn	PROPN
ma-166	193	1	=	=	PUNCT
ma-166	193	2	{	{	PUNCT
ma-166	193	3	1	1	NUM
ma-166	193	4	,	,	PUNCT
ma-166	193	5	2	2	NUM
ma-166	193	6	,	,	PUNCT
ma-166	193	7	3	3	NUM
ma-166	193	8	,	,	PUNCT
ma-166	193	9	...	...	PUNCT
ma-166	193	10	}	}	PUNCT
ma-166	193	11	appears	appear	VERB
ma-166	193	12	in	in	ADP
ma-166	193	13	figure(graph	figure(graph	PROPN
ma-166	193	14	)	)	PUNCT
ma-166	193	15	.	.	PUNCT
ma-166	194	1	the	the	DET
ma-166	194	2	petersen	petersen	PROPN
ma-166	194	3	graph	graph	NOUN
ma-166	194	4	tn=	tn=	PROPN
ma-166	194	5	{	{	PUNCT
ma-166	194	6	1	1	NUM
ma-166	194	7	,	,	PUNCT
ma-166	194	8	2	2	NUM
ma-166	194	9	,	,	PUNCT
ma-166	194	10	3	3	NUM
ma-166	194	11	,	,	PUNCT
ma-166	194	12	...	...	PUNCT
ma-166	194	13	}	}	PUNCT
ma-166	194	14	contains	contain	VERB
ma-166	194	15	v	v	NOUN
ma-166	194	16	(	(	PUNCT
ma-166	194	17	g′	g′	NOUN
ma-166	194	18	)	)	PUNCT
ma-166	194	19	=	=	NOUN
ma-166	195	1	4n	4n	NOUN
ma-166	195	2	no	no	PRON
ma-166	195	3	of	of	ADP
ma-166	195	4	vertices	vertex	NOUN
ma-166	195	5	and	and	CCONJ
ma-166	195	6	e(g′	e(g′	PRON
ma-166	195	7	)	)	PUNCT
ma-166	196	1	=	=	SYM
ma-166	196	2	6n	6n	NUM
ma-166	196	3	no	no	PRON
ma-166	196	4	of	of	ADP
ma-166	196	5	edges	edge	NOUN
ma-166	196	6	.	.	PUNCT
ma-166	197	1	the	the	DET
ma-166	197	2	degree	degree	NOUN
ma-166	197	3	ofeach	ofeach	NOUN
ma-166	197	4	vertex	vertex	NOUN
ma-166	197	5	in	in	ADP
ma-166	197	6	p(k	p(k	NOUN
ma-166	197	7	,	,	PUNCT
ma-166	197	8	t	t	NUM
ma-166	197	9	)	)	PUNCT
ma-166	197	10	is	be	AUX
ma-166	197	11	3	3	NUM
ma-166	197	12	and	and	CCONJ
ma-166	197	13	now	now	ADV
ma-166	197	14	reduced	reduce	VERB
ma-166	197	15	reciprocal	reciprocal	ADJ
ma-166	197	16	randic	randic	ADJ
ma-166	197	17	are	be	AUX
ma-166	197	18	i.e.	i.e.	X
ma-166	197	19	,	,	PUNCT
ma-166	197	20	rr(g	rr(g	NUM
ma-166	197	21	)	)	PUNCT
ma-166	197	22	=	=	PUNCT
ma-166	198	1	∑	∑	PUNCT
ma-166	199	1	[	[	X
ma-166	199	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	199	3	)	)	PUNCT
ma-166	199	4	]	]	PUNCT
ma-166	199	5	√	√	PROPN
ma-166	199	6	d(x1)×	d(x1)×	PROPN
ma-166	199	7	d(x2	d(x2	NOUN
ma-166	199	8	)	)	PUNCT
ma-166	200	1	now	now	ADV
ma-166	200	2	we	we	PRON
ma-166	200	3	suppose	suppose	VERB
ma-166	200	4	vertices	vertex	NOUN
ma-166	200	5	are	be	AUX
ma-166	200	6	v	v	ADP
ma-166	200	7	(	(	PUNCT
ma-166	200	8	g	g	NOUN
ma-166	200	9	)	)	PUNCT
ma-166	200	10	=	=	SYM
ma-166	200	11	4n	4n	NOUN
ma-166	200	12	,	,	PUNCT
ma-166	200	13	edges	edge	NOUN
ma-166	200	14	are	be	AUX
ma-166	200	15	e(g	e(g	NOUN
ma-166	200	16	)	)	PUNCT
ma-166	201	1	=	=	SYM
ma-166	201	2	6n	6n	NOUN
ma-166	201	3	and	and	CCONJ
ma-166	201	4	degree	degree	NOUN
ma-166	201	5	of	of	ADP
ma-166	201	6	petersengraph	petersengraph	NOUN
ma-166	201	7	about	about	ADP
ma-166	201	8	every	every	DET
ma-166	201	9	each	each	DET
ma-166	201	10	vertices	vertex	NOUN
ma-166	201	11	is	be	AUX
ma-166	201	12	p(k	p(k	NOUN
ma-166	201	13	,	,	PUNCT
ma-166	201	14	t	t	NOUN
ma-166	201	15	)	)	PUNCT
ma-166	201	16	=[	=[	NOUN
ma-166	201	17	d(x1	d(x1	NOUN
ma-166	201	18	)	)	PUNCT
ma-166	201	19	,	,	PUNCT
ma-166	201	20	d(x2	d(x2	NOUN
ma-166	201	21	)	)	PUNCT
ma-166	201	22	]	]	PUNCT
ma-166	202	1	=	=	PUNCT
ma-166	202	2	3	3	X
ma-166	202	3	.	.	PUNCT
ma-166	202	4	now	now	ADV
ma-166	202	5	putting	put	VERB
ma-166	202	6	the	the	DET
ma-166	202	7	values	value	NOUN
ma-166	202	8	in	in	ADP
ma-166	202	9	reducedreciprocal	reducedreciprocal	ADJ
ma-166	202	10	randic	randic	ADJ
ma-166	202	11	topological	topological	ADJ
ma-166	202	12	index	index	NOUN
ma-166	202	13	of	of	ADP
ma-166	202	14	the	the	DET
ma-166	202	15	general	general	ADJ
ma-166	202	16	form	form	NOUN
ma-166	202	17	,	,	PUNCT
ma-166	202	18	⇒	⇒	VERB
ma-166	202	19	rrr(g	rrr(g	PROPN
ma-166	202	20	)	)	PUNCT
ma-166	202	21	=	=	PUNCT
ma-166	203	1	∑	∑	PUNCT
ma-166	204	1	[	[	X
ma-166	204	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	204	3	)	)	PUNCT
ma-166	204	4	]	]	PUNCT
ma-166	205	1	√	√	PROPN
ma-166	206	1	[	[	X
ma-166	206	2	d(x1)−	d(x1)−	PROPN
ma-166	206	3	1]×	1]×	PROPN
ma-166	207	1	[	[	X
ma-166	207	2	d(x2)−	d(x2)−	VERB
ma-166	207	3	1]now	1]now	PROPN
ma-166	207	4	puttings	putting	NOUN
ma-166	207	5	the	the	DET
ma-166	207	6	values	value	NOUN
ma-166	207	7	then	then	ADV
ma-166	207	8	;	;	PUNCT
ma-166	207	9	⇒	⇒	PROPN
ma-166	207	10	rrr(g	rrr(g	PROPN
ma-166	207	11	)	)	PUNCT
ma-166	207	12	=	=	PUNCT
ma-166	208	1	∑	∑	PUNCT
ma-166	209	1	[	[	X
ma-166	209	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	209	3	)	)	PUNCT
ma-166	209	4	]	]	PUNCT
ma-166	210	1	√	√	INTJ
ma-166	210	2	(	(	PUNCT
ma-166	210	3	3−	3−	NUM
ma-166	210	4	1)×	1)×	NUM
ma-166	210	5	(	(	PUNCT
ma-166	210	6	3−	3−	NUM
ma-166	210	7	1	1	NUM
ma-166	210	8	)	)	PUNCT
ma-166	210	9	⇒	⇒	NOUN
ma-166	210	10	rrr(g	rrr(g	PROPN
ma-166	210	11	)	)	PUNCT
ma-166	210	12	=	=	PUNCT
ma-166	211	1	[	[	X
ma-166	211	2	|e(g)|]√(4	|e(g)|]√(4	NUM
ma-166	211	3	)	)	PUNCT
ma-166	211	4	⇒	⇒	NOUN
ma-166	211	5	rrr(g	rrr(g	PROPN
ma-166	211	6	)	)	PUNCT
ma-166	211	7	=	=	PUNCT
ma-166	212	1	[	[	X
ma-166	212	2	|6n|]√(4	|6n|]√(4	NOUN
ma-166	212	3	)	)	PUNCT
ma-166	212	4	⇒	⇒	NOUN
ma-166	212	5	rrr(g	rrr(g	PROPN
ma-166	212	6	)	)	PUNCT
ma-166	213	1	=	=	SYM
ma-166	213	2	(	(	PUNCT
ma-166	213	3	6n)√(4	6n)√(4	NOUN
ma-166	213	4	)	)	PUNCT
ma-166	213	5	⇒	⇒	NOUN
ma-166	213	6	rrr(g	rrr(g	PROPN
ma-166	213	7	)	)	PUNCT
ma-166	213	8	=	=	SYM
ma-166	213	9	6n(2	6n(2	NUM
ma-166	213	10	)	)	PUNCT
ma-166	213	11	⇒	⇒	NOUN
ma-166	213	12	rrr(g	rrr(g	PROPN
ma-166	213	13	)	)	PUNCT
ma-166	213	14	=	=	SYM
ma-166	213	15	12n	12n	NOUN
ma-166	213	16	.	.	PUNCT
ma-166	214	1	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	PROPN
ma-166	214	2	eur	eur	PROPN
ma-166	214	3	.	.	PUNCT
ma-166	215	1	j.	j.	PROPN
ma-166	215	2	math	math	PROPN
ma-166	215	3	.	.	PUNCT
ma-166	216	1	anal	anal	PROPN
ma-166	216	2	.	.	PUNCT
ma-166	217	1	10.28924	10.28924	NUM
ma-166	217	2	/	/	SYM
ma-166	217	3	ada	ada	PROPN
ma-166	217	4	/	/	SYM
ma-166	217	5	ma.3.20	ma.3.20	PROPN
ma-166	217	6	8	8	NUM
ma-166	217	7	theorem	theorem	VERB
ma-166	217	8	2.5	2.5	NUM
ma-166	217	9	let	let	VERB
ma-166	217	10	p(k	p(k	NOUN
ma-166	217	11	,	,	PUNCT
ma-166	217	12	t	t	NUM
ma-166	217	13	)	)	PUNCT
ma-166	217	14	be	be	AUX
ma-166	217	15	petersen	petersen	NOUN
ma-166	217	16	subdivision	subdivision	NOUN
ma-166	217	17	graph	graph	NOUN
ma-166	217	18	.	.	PUNCT
ma-166	218	1	then	then	ADV
ma-166	218	2	,	,	PUNCT
ma-166	218	3	for	for	ADP
ma-166	218	4	tn	tn	NOUN
ma-166	218	5	=	=	SYM
ma-166	218	6	{	{	PUNCT
ma-166	218	7	1	1	NUM
ma-166	218	8	,	,	PUNCT
ma-166	218	9	2	2	NUM
ma-166	218	10	,	,	PUNCT
ma-166	218	11	3	3	NUM
ma-166	218	12	,	,	PUNCT
ma-166	218	13	...	...	PUNCT
ma-166	218	14	}	}	PUNCT
ma-166	218	15	,	,	PUNCT
ma-166	218	16	hyper	hyper	ADJ
ma-166	218	17	zagrebindex	zagrebindex	NOUN
ma-166	218	18	are	be	AUX
ma-166	218	19	,	,	PUNCT
ma-166	218	20	hm1(g	hm1(g	NOUN
ma-166	218	21	)	)	PUNCT
ma-166	218	22	=	=	SYM
ma-166	218	23	216n	216n	NOUN
ma-166	218	24	proof	proof	NOUN
ma-166	218	25	:	:	PUNCT
ma-166	218	26	the	the	DET
ma-166	218	27	petersen	petersen	PROPN
ma-166	218	28	graph	graph	VERB
ma-166	218	29	tn	tn	PROPN
ma-166	218	30	=	=	PUNCT
ma-166	218	31	{	{	PUNCT
ma-166	218	32	1	1	NUM
ma-166	218	33	,	,	PUNCT
ma-166	218	34	2	2	NUM
ma-166	218	35	,	,	PUNCT
ma-166	218	36	3	3	NUM
ma-166	218	37	,	,	PUNCT
ma-166	218	38	...	...	PUNCT
ma-166	218	39	}	}	PUNCT
ma-166	218	40	appears	appear	VERB
ma-166	218	41	in	in	ADP
ma-166	218	42	figure(graph	figure(graph	PROPN
ma-166	218	43	)	)	PUNCT
ma-166	218	44	.	.	PUNCT
ma-166	219	1	the	the	DET
ma-166	219	2	petersen	petersen	PROPN
ma-166	219	3	graph	graph	NOUN
ma-166	219	4	tn=	tn=	PROPN
ma-166	219	5	{	{	PUNCT
ma-166	219	6	1	1	NUM
ma-166	219	7	,	,	PUNCT
ma-166	219	8	2	2	NUM
ma-166	219	9	,	,	PUNCT
ma-166	219	10	3	3	NUM
ma-166	219	11	,	,	PUNCT
ma-166	219	12	...	...	PUNCT
ma-166	219	13	}	}	PUNCT
ma-166	219	14	contains	contain	VERB
ma-166	219	15	v	v	NOUN
ma-166	219	16	(	(	PUNCT
ma-166	219	17	g′	g′	NOUN
ma-166	219	18	)	)	PUNCT
ma-166	219	19	=	=	NOUN
ma-166	220	1	4n	4n	NOUN
ma-166	220	2	no	no	PRON
ma-166	220	3	of	of	ADP
ma-166	220	4	vertices	vertex	NOUN
ma-166	220	5	and	and	CCONJ
ma-166	220	6	e(g′	e(g′	PRON
ma-166	220	7	)	)	PUNCT
ma-166	221	1	=	=	SYM
ma-166	221	2	6n	6n	NUM
ma-166	221	3	no	no	PRON
ma-166	221	4	of	of	ADP
ma-166	221	5	edges	edge	NOUN
ma-166	221	6	.	.	PUNCT
ma-166	222	1	the	the	DET
ma-166	222	2	degree	degree	NOUN
ma-166	222	3	ofeach	ofeach	NOUN
ma-166	222	4	vertex	vertex	NOUN
ma-166	222	5	in	in	ADP
ma-166	222	6	p(k	p(k	NOUN
ma-166	222	7	,	,	PUNCT
ma-166	222	8	t	t	NUM
ma-166	222	9	)	)	PUNCT
ma-166	222	10	is	be	AUX
ma-166	222	11	3	3	NUM
ma-166	222	12	and	and	CCONJ
ma-166	222	13	now	now	ADV
ma-166	222	14	hyper	hyper	PROPN
ma-166	222	15	zagreb	zagreb	PROPN
ma-166	222	16	index	index	NOUN
ma-166	222	17	are	be	AUX
ma-166	222	18	i.e.	i.e.	X
ma-166	222	19	,	,	PUNCT
ma-166	222	20	hm1(g)=	hm1(g)=	VERB
ma-166	222	21	∑	∑	PROPN
ma-166	222	22	[	[	X
ma-166	222	23	x1,x2∈e(g)][d(x1	x1,x2∈e(g)][d(x1	X
ma-166	222	24	)	)	PUNCT
ma-166	222	25	+	+	NUM
ma-166	222	26	d(x2	d(x2	NOUN
ma-166	222	27	)	)	PUNCT
ma-166	222	28	]	]	PUNCT
ma-166	223	1	2	2	NUM
ma-166	223	2	→	→	SYM
ma-166	223	3	[	[	X
ma-166	223	4	1	1	X
ma-166	223	5	]	]	PUNCT
ma-166	223	6	now	now	ADV
ma-166	223	7	we	we	PRON
ma-166	223	8	suppose	suppose	VERB
ma-166	223	9	vertices	vertex	NOUN
ma-166	223	10	are	be	AUX
ma-166	223	11	v	v	ADP
ma-166	223	12	(	(	PUNCT
ma-166	223	13	g	g	NOUN
ma-166	223	14	)	)	PUNCT
ma-166	223	15	=	=	SYM
ma-166	223	16	4n	4n	NOUN
ma-166	223	17	,	,	PUNCT
ma-166	223	18	edges	edge	NOUN
ma-166	223	19	are	be	AUX
ma-166	223	20	e(g	e(g	NOUN
ma-166	223	21	)	)	PUNCT
ma-166	224	1	=	=	SYM
ma-166	224	2	6n	6n	NOUN
ma-166	224	3	and	and	CCONJ
ma-166	224	4	degree	degree	NOUN
ma-166	224	5	of	of	ADP
ma-166	224	6	petersen	petersen	PROPN
ma-166	224	7	graphabout	graphabout	PROPN
ma-166	224	8	every	every	DET
ma-166	224	9	each	each	DET
ma-166	224	10	vertices	vertex	NOUN
ma-166	224	11	is	be	AUX
ma-166	224	12	p(k	p(k	NOUN
ma-166	224	13	,	,	PUNCT
ma-166	224	14	t	t	NOUN
ma-166	224	15	)	)	PUNCT
ma-166	224	16	=[	=[	NOUN
ma-166	224	17	d(x1	d(x1	NOUN
ma-166	224	18	)	)	PUNCT
ma-166	224	19	,	,	PUNCT
ma-166	224	20	d(x2	d(x2	NOUN
ma-166	224	21	)	)	PUNCT
ma-166	224	22	]	]	PUNCT
ma-166	225	1	=	=	PUNCT
ma-166	225	2	3	3	X
ma-166	225	3	.	.	PUNCT
ma-166	225	4	now	now	ADV
ma-166	225	5	putting	put	VERB
ma-166	225	6	the	the	DET
ma-166	225	7	values	value	NOUN
ma-166	225	8	in	in	ADP
ma-166	225	9	hyper	hyper	ADJ
ma-166	225	10	zagrebindex	zagrebindex	PROPN
ma-166	225	11	topological	topological	PROPN
ma-166	225	12	index	index	NOUN
ma-166	225	13	of	of	ADP
ma-166	225	14	the	the	DET
ma-166	225	15	general	general	ADJ
ma-166	225	16	form	form	NOUN
ma-166	225	17	,	,	PUNCT
ma-166	225	18	⇒	⇒	PROPN
ma-166	225	19	hm1(g	hm1(g	NOUN
ma-166	225	20	)	)	PUNCT
ma-166	225	21	=[	=[	NOUN
ma-166	225	22	|e(g)|][(3	|e(g)|][(3	PROPN
ma-166	225	23	+	+	NUM
ma-166	225	24	3)]2	3)]2	NUM
ma-166	225	25	⇒	⇒	NOUN
ma-166	225	26	hm1(g	hm1(g	NOUN
ma-166	225	27	)	)	PUNCT
ma-166	225	28	=[	=[	PROPN
ma-166	225	29	|6n|](6)2	|6n|](6)2	PROPN
ma-166	225	30	⇒	⇒	PROPN
ma-166	225	31	hm1(g	hm1(g	PROPN
ma-166	225	32	)	)	PUNCT
ma-166	225	33	=	=	SYM
ma-166	226	1	(	(	PUNCT
ma-166	226	2	6n)(6)2	6n)(6)2	ADJ
ma-166	226	3	⇒	⇒	NOUN
ma-166	226	4	hm1(g	hm1(g	NOUN
ma-166	226	5	)	)	PUNCT
ma-166	226	6	=	=	PUNCT
ma-166	226	7	(	(	PUNCT
ma-166	226	8	6n)(36	6n)(36	NOUN
ma-166	226	9	)	)	PUNCT
ma-166	226	10	⇒	⇒	NOUN
ma-166	226	11	hm1(g	hm1(g	NOUN
ma-166	226	12	)	)	PUNCT
ma-166	226	13	=	=	SYM
ma-166	226	14	216n	216n	NUM
ma-166	226	15	.	.	PUNCT
ma-166	227	1	hm1(g	hm1(g	X
ma-166	227	2	)	)	PUNCT
ma-166	227	3	=	=	NOUN
ma-166	227	4	thirty	thirty	NUM
ma-166	227	5	six	six	NUM
ma-166	227	6	times	time	NOUN
ma-166	227	7	to	to	ADP
ma-166	227	8	general	general	ADJ
ma-166	227	9	edges	edge	NOUN
ma-166	227	10	of	of	ADP
ma-166	227	11	petersen	petersen	PROPN
ma-166	227	12	graph	graph	NOUN
ma-166	227	13	.	.	PUNCT
ma-166	228	1	theorem	theorem	VERB
ma-166	228	2	2.6	2.6	NUM
ma-166	228	3	let	let	VERB
ma-166	228	4	p(k	p(k	NOUN
ma-166	228	5	,	,	PUNCT
ma-166	228	6	t	t	NUM
ma-166	228	7	)	)	PUNCT
ma-166	228	8	be	be	AUX
ma-166	228	9	petersen	petersen	NOUN
ma-166	228	10	subdivision	subdivision	NOUN
ma-166	228	11	graph	graph	NOUN
ma-166	228	12	.	.	PUNCT
ma-166	229	1	then	then	ADV
ma-166	229	2	,	,	PUNCT
ma-166	229	3	for	for	ADP
ma-166	229	4	tn	tn	NOUN
ma-166	229	5	=	=	SYM
ma-166	229	6	{	{	PUNCT
ma-166	229	7	1	1	NUM
ma-166	229	8	,	,	PUNCT
ma-166	229	9	2	2	NUM
ma-166	229	10	,	,	PUNCT
ma-166	229	11	3	3	NUM
ma-166	229	12	,	,	PUNCT
ma-166	229	13	...	...	PUNCT
ma-166	229	14	}	}	PUNCT
ma-166	229	15	,	,	PUNCT
ma-166	229	16	two	two	NUM
ma-166	229	17	polynomialrelated	polynomialrelate	VERB
ma-166	229	18	to	to	ADP
ma-166	229	19	the	the	DET
ma-166	229	20	first	first	PROPN
ma-166	229	21	zagreb	zagreb	PROPN
ma-166	229	22	index	index	NOUN
ma-166	229	23	are	be	AUX
ma-166	229	24	,	,	PUNCT
ma-166	229	25	m∗1(g	m∗1(g	PROPN
ma-166	229	26	,	,	PUNCT
ma-166	229	27	x	x	NOUN
ma-166	229	28	)	)	PUNCT
ma-166	229	29	=(	=(	NOUN
ma-166	229	30	12n)x4n	12n)x4n	NUM
ma-166	229	31	m0(g	m0(g	NOUN
ma-166	229	32	,	,	PUNCT
ma-166	229	33	x	x	X
ma-166	229	34	)	)	PUNCT
ma-166	230	1	=	=	NOUN
ma-166	230	2	4nx3	4nx3	NUM
ma-166	230	3	proof	proof	NOUN
ma-166	230	4	:	:	PUNCT
ma-166	230	5	the	the	DET
ma-166	230	6	petersen	petersen	PROPN
ma-166	230	7	graph	graph	VERB
ma-166	230	8	tn	tn	PROPN
ma-166	231	1	=	=	PUNCT
ma-166	232	1	{	{	PUNCT
ma-166	232	2	1	1	NUM
ma-166	232	3	,	,	PUNCT
ma-166	232	4	2	2	NUM
ma-166	232	5	,	,	PUNCT
ma-166	232	6	3	3	NUM
ma-166	232	7	,	,	PUNCT
ma-166	232	8	...	...	PUNCT
ma-166	232	9	}	}	PUNCT
ma-166	232	10	appears	appear	VERB
ma-166	232	11	in	in	ADP
ma-166	232	12	figure(graph	figure(graph	PROPN
ma-166	232	13	)	)	PUNCT
ma-166	232	14	.	.	PUNCT
ma-166	233	1	the	the	DET
ma-166	233	2	petersengraph	petersengraph	PROPN
ma-166	233	3	tn	tn	PROPN
ma-166	233	4	=	=	SYM
ma-166	233	5	{	{	PUNCT
ma-166	233	6	1	1	NUM
ma-166	233	7	,	,	PUNCT
ma-166	233	8	2	2	NUM
ma-166	233	9	,	,	PUNCT
ma-166	233	10	3	3	NUM
ma-166	233	11	,	,	PUNCT
ma-166	233	12	...	...	PUNCT
ma-166	233	13	}	}	PUNCT
ma-166	233	14	contains	contain	VERB
ma-166	233	15	v	v	NOUN
ma-166	233	16	(	(	PUNCT
ma-166	233	17	g′	g′	NOUN
ma-166	233	18	)	)	PUNCT
ma-166	234	1	=	=	NOUN
ma-166	235	1	4n	4n	NOUN
ma-166	235	2	no	no	PRON
ma-166	235	3	of	of	ADP
ma-166	235	4	vertices	vertex	NOUN
ma-166	235	5	and	and	CCONJ
ma-166	235	6	e(g′	e(g′	PRON
ma-166	235	7	)	)	PUNCT
ma-166	236	1	=	=	SYM
ma-166	236	2	6n	6n	NUM
ma-166	236	3	no	no	PRON
ma-166	236	4	of	of	ADP
ma-166	236	5	edges	edge	NOUN
ma-166	236	6	.	.	PUNCT
ma-166	237	1	thedegree	thedegree	NOUN
ma-166	237	2	of	of	ADP
ma-166	237	3	each	each	DET
ma-166	237	4	vertex	vertex	NOUN
ma-166	237	5	in	in	ADP
ma-166	237	6	p(k	p(k	NOUN
ma-166	237	7	,	,	PUNCT
ma-166	237	8	t	t	NUM
ma-166	237	9	)	)	PUNCT
ma-166	237	10	is	be	AUX
ma-166	237	11	3	3	NUM
ma-166	237	12	and	and	CCONJ
ma-166	237	13	now	now	ADV
ma-166	237	14	two	two	NUM
ma-166	237	15	polynomial	polynomial	ADJ
ma-166	237	16	related	relate	VERB
ma-166	237	17	to	to	ADP
ma-166	237	18	the	the	DET
ma-166	237	19	first	first	ADJ
ma-166	237	20	zagreb	zagreb	PROPN
ma-166	237	21	index	index	NOUN
ma-166	237	22	are	be	AUX
ma-166	237	23	i.e.	i.e.	X
ma-166	237	24	,	,	PUNCT
ma-166	237	25	m∗1(g	m∗1(g	PROPN
ma-166	237	26	,	,	PUNCT
ma-166	237	27	x	x	NOUN
ma-166	237	28	)	)	PUNCT
ma-166	238	1	=	=	NOUN
ma-166	238	2	∑	∑	PUNCT
ma-166	238	3	[	[	X
ma-166	238	4	xi∈v	xi∈v	X
ma-166	238	5	(	(	PUNCT
ma-166	238	6	g)][d(xi)][x	g)][d(xi)][x	X
ma-166	238	7	[	[	X
ma-166	238	8	xi	xi	X
ma-166	238	9	]	]	X
ma-166	238	10	]	]	X
ma-166	238	11	m0(g	m0(g	NOUN
ma-166	238	12	,	,	PUNCT
ma-166	238	13	x	x	X
ma-166	238	14	)	)	PUNCT
ma-166	239	1	=	=	NOUN
ma-166	239	2	∑	∑	PUNCT
ma-166	240	1	[	[	X
ma-166	240	2	xi∈v	xi∈v	X
ma-166	240	3	(	(	PUNCT
ma-166	240	4	g)](x	g)](x	NOUN
ma-166	240	5	)	)	PUNCT
ma-166	240	6	[	[	X
ma-166	240	7	d(xi	d(xi	X
ma-166	240	8	)	)	PUNCT
ma-166	240	9	]	]	PUNCT
ma-166	240	10	now	now	ADV
ma-166	240	11	we	we	PRON
ma-166	240	12	suppose	suppose	VERB
ma-166	240	13	vertices	vertex	NOUN
ma-166	240	14	are	be	AUX
ma-166	240	15	v	v	ADP
ma-166	240	16	(	(	PUNCT
ma-166	240	17	g	g	NOUN
ma-166	240	18	)	)	PUNCT
ma-166	240	19	=	=	SYM
ma-166	240	20	4n	4n	NOUN
ma-166	240	21	,	,	PUNCT
ma-166	240	22	edges	edge	NOUN
ma-166	240	23	are	be	AUX
ma-166	240	24	e(g	e(g	NOUN
ma-166	240	25	)	)	PUNCT
ma-166	241	1	=	=	SYM
ma-166	241	2	6n	6n	NOUN
ma-166	241	3	and	and	CCONJ
ma-166	241	4	degree	degree	NOUN
ma-166	241	5	of	of	ADP
ma-166	241	6	petersen	petersen	PROPN
ma-166	241	7	graphabout	graphabout	PROPN
ma-166	241	8	every	every	DET
ma-166	241	9	each	each	DET
ma-166	241	10	vertices	vertex	NOUN
ma-166	241	11	is	be	AUX
ma-166	241	12	p(k	p(k	NOUN
ma-166	241	13	,	,	PUNCT
ma-166	241	14	t	t	NOUN
ma-166	241	15	)	)	PUNCT
ma-166	241	16	=[	=[	NOUN
ma-166	241	17	d(x1	d(x1	NOUN
ma-166	241	18	)	)	PUNCT
ma-166	241	19	,	,	PUNCT
ma-166	241	20	d(x2	d(x2	NOUN
ma-166	241	21	)	)	PUNCT
ma-166	241	22	]	]	PUNCT
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ma-166	249	5	of	of	ADP
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ma-166	249	21	)	)	PUNCT
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ma-166	250	1	=	=	PUNCT
ma-166	250	2	3	3	X
ma-166	250	3	.	.	PUNCT
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ma-166	250	5	putting	put	VERB
ma-166	250	6	the	the	DET
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ma-166	250	8	in	in	ADP
ma-166	250	9	two	two	NUM
ma-166	250	10	polynomialrelated	polynomialrelate	VERB
ma-166	250	11	to	to	ADP
ma-166	250	12	the	the	DET
ma-166	250	13	first	first	ADJ
ma-166	250	14	zagreb	zagreb	PROPN
ma-166	250	15	index	index	PROPN
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ma-166	250	17	index	index	NOUN
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ma-166	251	2	∑	∑	PUNCT
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ma-166	251	21	)	)	PUNCT
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ma-166	253	1	[	[	X
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ma-166	254	2	,	,	PUNCT
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ma-166	257	2	:	:	PUNCT
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ma-166	257	4	petersen	petersen	PROPN
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ma-166	259	7	,	,	PUNCT
ma-166	259	8	...	...	PUNCT
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ma-166	265	4	xj∈e(g)](x	xj∈e(g)](x	X
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ma-166	268	5	,	,	PUNCT
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ma-166	269	1	[	[	X
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ma-166	269	19	we	we	PRON
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ma-166	270	9	each	each	DET
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ma-166	271	12	index	index	NOUN
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ma-166	271	15	general	general	ADJ
ma-166	271	16	form	form	NOUN
ma-166	271	17	,	,	PUNCT
ma-166	271	18	⇒	⇒	PROPN
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ma-166	282	2	.	.	PUNCT
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ma-166	283	14	)	)	PUNCT
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ma-166	284	10	(	(	PUNCT
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ma-166	284	20	b(g	b(g	PROPN
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ma-166	284	22	x	x	X
ma-166	284	23	)	)	PUNCT
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ma-166	284	25	|e(g)|](x)[(a+(3))(b+(3	|e(g)|](x)[(a+(3))(b+(3	NOUN
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ma-166	285	7	x	x	X
ma-166	285	8	)	)	PUNCT
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ma-166	285	11	)	)	PUNCT
ma-166	285	12	]	]	PUNCT
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ma-166	285	16	b(g	b(g	PROPN
ma-166	285	17	,	,	PUNCT
ma-166	285	18	x	x	X
ma-166	285	19	)	)	PUNCT
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ma-166	285	21	6n)(x)[(a+3)(b+3	6n)(x)[(a+3)(b+3	PROPN
ma-166	285	22	)	)	PUNCT
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ma-166	286	2	′a	′a	PROPN
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ma-166	286	4	b(g	b(g	PROPN
ma-166	286	5	,	,	PUNCT
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ma-166	286	7	)	)	PUNCT
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ma-166	286	9	(	(	PUNCT
ma-166	286	10	x)[(a+3)(b+3)]]×	x)[(a+3)(b+3)]]×	PUNCT
ma-166	287	1	[	[	X
ma-166	287	2	general	general	ADJ
ma-166	287	3	edges	edge	NOUN
ma-166	287	4	of	of	ADP
ma-166	287	5	petersen	petersen	PROPN
ma-166	287	6	graph	graph	NOUN
ma-166	287	7	]	]	PUNCT
ma-166	287	8	theorem	theorem	VERB
ma-166	287	9	2.8	2.8	NUM
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ma-166	287	11	p(k	p(k	NOUN
ma-166	287	12	,	,	PUNCT
ma-166	287	13	t	t	NUM
ma-166	287	14	)	)	PUNCT
ma-166	287	15	be	be	AUX
ma-166	287	16	petersen	petersen	NOUN
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ma-166	287	18	graph	graph	NOUN
ma-166	287	19	.	.	PUNCT
ma-166	288	1	then	then	ADV
ma-166	288	2	,	,	PUNCT
ma-166	288	3	for	for	ADP
ma-166	288	4	tn	tn	NOUN
ma-166	288	5	=	=	SYM
ma-166	288	6	{	{	PUNCT
ma-166	288	7	1	1	NUM
ma-166	288	8	,	,	PUNCT
ma-166	288	9	2	2	NUM
ma-166	288	10	,	,	PUNCT
ma-166	288	11	3	3	NUM
ma-166	288	12	,	,	PUNCT
ma-166	288	13	...	...	PUNCT
ma-166	288	14	}	}	PUNCT
ma-166	288	15	,	,	PUNCT
ma-166	288	16	atomic	atomic	ADJ
ma-166	288	17	-	-	PUNCT
ma-166	288	18	bond	bond	NOUN
ma-166	288	19	-	-	PUNCT
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ma-166	288	21	(	(	PUNCT
ma-166	288	22	abc	abc	PROPN
ma-166	288	23	)	)	PUNCT
ma-166	288	24	index	index	NOUN
ma-166	288	25	are	be	AUX
ma-166	288	26	,	,	PUNCT
ma-166	288	27	abc(g	abc(g	PROPN
ma-166	288	28	)	)	PUNCT
ma-166	288	29	=	=	SYM
ma-166	288	30	(	(	PUNCT
ma-166	288	31	4n	4n	NOUN
ma-166	288	32	)	)	PUNCT
ma-166	288	33	.	.	PUNCT
ma-166	289	1	proof	proof	NOUN
ma-166	289	2	:	:	PUNCT
ma-166	289	3	the	the	DET
ma-166	289	4	petersen	petersen	PROPN
ma-166	289	5	graph	graph	VERB
ma-166	289	6	tn	tn	PROPN
ma-166	290	1	=	=	PUNCT
ma-166	290	2	{	{	PUNCT
ma-166	290	3	1	1	NUM
ma-166	290	4	,	,	PUNCT
ma-166	290	5	2	2	NUM
ma-166	290	6	,	,	PUNCT
ma-166	290	7	3	3	NUM
ma-166	290	8	,	,	PUNCT
ma-166	290	9	...	...	PUNCT
ma-166	290	10	}	}	PUNCT
ma-166	290	11	appears	appear	VERB
ma-166	290	12	in	in	ADP
ma-166	290	13	figure(graph	figure(graph	PROPN
ma-166	290	14	)	)	PUNCT
ma-166	290	15	.	.	PUNCT
ma-166	291	1	the	the	DET
ma-166	291	2	petersen	petersen	PROPN
ma-166	291	3	graph	graph	NOUN
ma-166	291	4	tn=	tn=	PROPN
ma-166	291	5	{	{	PUNCT
ma-166	291	6	1	1	NUM
ma-166	291	7	,	,	PUNCT
ma-166	291	8	2	2	NUM
ma-166	291	9	,	,	PUNCT
ma-166	291	10	3	3	NUM
ma-166	291	11	,	,	PUNCT
ma-166	291	12	...	...	PUNCT
ma-166	291	13	}	}	PUNCT
ma-166	291	14	contains	contain	VERB
ma-166	291	15	v	v	NOUN
ma-166	291	16	(	(	PUNCT
ma-166	291	17	g′	g′	NOUN
ma-166	291	18	)	)	PUNCT
ma-166	291	19	=	=	NOUN
ma-166	292	1	4n	4n	NOUN
ma-166	292	2	no	no	PRON
ma-166	292	3	of	of	ADP
ma-166	292	4	vertices	vertex	NOUN
ma-166	292	5	and	and	CCONJ
ma-166	292	6	e(g′	e(g′	PRON
ma-166	292	7	)	)	PUNCT
ma-166	293	1	=	=	SYM
ma-166	293	2	6n	6n	NUM
ma-166	293	3	no	no	PRON
ma-166	293	4	of	of	ADP
ma-166	293	5	edges	edge	NOUN
ma-166	293	6	.	.	PUNCT
ma-166	294	1	the	the	DET
ma-166	294	2	degree	degree	NOUN
ma-166	294	3	ofeach	ofeach	NOUN
ma-166	294	4	vertex	vertex	NOUN
ma-166	294	5	in	in	ADP
ma-166	294	6	p(k	p(k	NOUN
ma-166	294	7	,	,	PUNCT
ma-166	294	8	t	t	NUM
ma-166	294	9	)	)	PUNCT
ma-166	294	10	is	be	AUX
ma-166	294	11	3	3	NUM
ma-166	294	12	and	and	CCONJ
ma-166	294	13	now	now	ADV
ma-166	294	14	atomic	atomic	ADJ
ma-166	294	15	-	-	PUNCT
ma-166	294	16	bond	bond	NOUN
ma-166	294	17	-	-	PUNCT
ma-166	294	18	connectivity	connectivity	NOUN
ma-166	294	19	(	(	PUNCT
ma-166	294	20	abc	abc	PROPN
ma-166	294	21	)	)	PUNCT
ma-166	294	22	index	index	NOUN
ma-166	294	23	are	be	AUX
ma-166	294	24	i.e.	i.e.	X
ma-166	294	25	,	,	PUNCT
ma-166	294	26	abc(g	abc(g	PROPN
ma-166	294	27	)	)	PUNCT
ma-166	294	28	=	=	PUNCT
ma-166	295	1	∑	∑	PUNCT
ma-166	296	1	[	[	X
ma-166	296	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	296	3	)	)	PUNCT
ma-166	296	4	]	]	PUNCT
ma-166	297	1	√	√	PROPN
ma-166	298	1	[	[	X
ma-166	298	2	d(x1)+d(x2)]−2	d(x1)+d(x2)]−2	X
ma-166	298	3	d(x1)×d(x2	d(x1)×d(x2	NOUN
ma-166	298	4	)	)	PUNCT
ma-166	298	5	→	→	PUNCT
ma-166	299	1	[	[	X
ma-166	299	2	1	1	X
ma-166	299	3	]	]	PUNCT
ma-166	299	4	now	now	ADV
ma-166	299	5	we	we	PRON
ma-166	299	6	suppose	suppose	VERB
ma-166	299	7	vertices	vertex	NOUN
ma-166	299	8	are	be	AUX
ma-166	299	9	v	v	ADP
ma-166	299	10	(	(	PUNCT
ma-166	299	11	g	g	NOUN
ma-166	299	12	)	)	PUNCT
ma-166	299	13	=	=	SYM
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ma-166	299	15	,	,	PUNCT
ma-166	299	16	edges	edge	NOUN
ma-166	299	17	are	be	AUX
ma-166	299	18	e(g	e(g	NOUN
ma-166	299	19	)	)	PUNCT
ma-166	300	1	=	=	SYM
ma-166	300	2	6n	6n	NOUN
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ma-166	300	4	degree	degree	NOUN
ma-166	300	5	of	of	ADP
ma-166	300	6	petersengraph	petersengraph	NOUN
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ma-166	300	8	every	every	DET
ma-166	300	9	each	each	DET
ma-166	300	10	vertices	vertex	NOUN
ma-166	300	11	is	be	AUX
ma-166	300	12	p(k	p(k	NOUN
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ma-166	300	14	t	t	NOUN
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ma-166	300	21	)	)	PUNCT
ma-166	300	22	]	]	PUNCT
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ma-166	301	2	3	3	X
ma-166	301	3	.	.	PUNCT
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ma-166	301	5	putting	put	VERB
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ma-166	301	8	inatomic	inatomic	ADJ
ma-166	301	9	-	-	PUNCT
ma-166	301	10	bond	bond	NOUN
ma-166	301	11	-	-	PUNCT
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ma-166	301	13	(	(	PUNCT
ma-166	301	14	abc	abc	PROPN
ma-166	301	15	)	)	PUNCT
ma-166	301	16	index	index	NOUN
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ma-166	301	18	index	index	NOUN
ma-166	301	19	of	of	ADP
ma-166	301	20	the	the	DET
ma-166	301	21	general	general	ADJ
ma-166	301	22	form	form	NOUN
ma-166	301	23	,	,	PUNCT
ma-166	301	24	⇒	⇒	VERB
ma-166	301	25	abc(g	abc(g	PROPN
ma-166	301	26	)	)	PUNCT
ma-166	301	27	=	=	PUNCT
ma-166	301	28	∑	∑	PUNCT
ma-166	302	1	[	[	X
ma-166	302	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	302	3	)	)	PUNCT
ma-166	302	4	]	]	PUNCT
ma-166	303	1	√	√	PROPN
ma-166	304	1	[	[	X
ma-166	304	2	d(x1)+d(x2)]−2	d(x1)+d(x2)]−2	NUM
ma-166	304	3	d(x1)×d(x2)putting	d(x1)×d(x2)putte	VERB
ma-166	304	4	values	value	NOUN
ma-166	304	5	in	in	ADP
ma-166	304	6	above	above	ADP
ma-166	304	7	equation	equation	NOUN
ma-166	304	8	;	;	PUNCT
ma-166	304	9	⇒	⇒	PROPN
ma-166	304	10	abc(g	abc(g	PROPN
ma-166	304	11	)	)	PUNCT
ma-166	304	12	=[	=[	NOUN
ma-166	304	13	|e(g)|]√	|e(g)|]√	NOUN
ma-166	305	1	[	[	X
ma-166	305	2	(	(	PUNCT
ma-166	305	3	3)+(3)−2	3)+(3)−2	NOUN
ma-166	305	4	]	]	X
ma-166	305	5	(	(	PUNCT
ma-166	305	6	3)×(3	3)×(3	NUM
ma-166	305	7	)	)	PUNCT
ma-166	305	8	⇒	⇒	VERB
ma-166	305	9	abc(g	abc(g	PROPN
ma-166	305	10	)	)	PUNCT
ma-166	305	11	=[	=[	NOUN
ma-166	305	12	|6n|]√	|6n|]√	X
ma-166	305	13	(	(	PUNCT
ma-166	305	14	4	4	NUM
ma-166	305	15	)	)	PUNCT
ma-166	305	16	(	(	PUNCT
ma-166	305	17	3)2	3)2	NUM
ma-166	305	18	⇒	⇒	NOUN
ma-166	305	19	abc(g	abc(g	PROPN
ma-166	305	20	)	)	PUNCT
ma-166	305	21	=(	=(	NOUN
ma-166	305	22	6n)√4(3	6n)√4(3	NUM
ma-166	305	23	)	)	PUNCT
ma-166	305	24	⇒	⇒	VERB
ma-166	305	25	abc(g	abc(g	PROPN
ma-166	305	26	)	)	PUNCT
ma-166	305	27	=	=	SYM
ma-166	305	28	(	(	PUNCT
ma-166	305	29	4n	4n	NOUN
ma-166	305	30	)	)	PUNCT
ma-166	305	31	.	.	PUNCT
ma-166	306	1	abc(g	abc(g	X
ma-166	306	2	)	)	PUNCT
ma-166	306	3	=	=	SYM
ma-166	306	4	general	general	ADJ
ma-166	306	5	vertices	vertex	NOUN
ma-166	306	6	of	of	ADP
ma-166	306	7	petersen	petersen	PROPN
ma-166	306	8	graph	graph	NOUN
ma-166	306	9	theorem	theorem	VERB
ma-166	306	10	2.9	2.9	NUM
ma-166	306	11	let	let	VERB
ma-166	306	12	p(k	p(k	NOUN
ma-166	306	13	,	,	PUNCT
ma-166	306	14	t	t	NUM
ma-166	306	15	)	)	PUNCT
ma-166	306	16	be	be	AUX
ma-166	306	17	petersen	petersen	NOUN
ma-166	306	18	subdivision	subdivision	NOUN
ma-166	306	19	graph	graph	NOUN
ma-166	306	20	.	.	PUNCT
ma-166	307	1	then	then	ADV
ma-166	307	2	,	,	PUNCT
ma-166	307	3	for	for	ADP
ma-166	307	4	tn	tn	NOUN
ma-166	307	5	=	=	SYM
ma-166	307	6	{	{	PUNCT
ma-166	307	7	1	1	NUM
ma-166	307	8	,	,	PUNCT
ma-166	307	9	2	2	NUM
ma-166	307	10	,	,	PUNCT
ma-166	307	11	3	3	NUM
ma-166	307	12	,	,	PUNCT
ma-166	307	13	...	...	PUNCT
ma-166	307	14	}	}	PUNCT
ma-166	307	15	,	,	PUNCT
ma-166	307	16	geometricarithmetic(ga	geometricarithmetic(ga	ADJ
ma-166	307	17	)	)	PUNCT
ma-166	307	18	index	index	NOUN
ma-166	307	19	are	be	AUX
ma-166	307	20	,	,	PUNCT
ma-166	307	21	ga(g	ga(g	NOUN
ma-166	307	22	)	)	PUNCT
ma-166	308	1	=	=	NOUN
ma-166	308	2	6n	6n	NUM
ma-166	308	3	proof	proof	NOUN
ma-166	308	4	:	:	PUNCT
ma-166	308	5	the	the	DET
ma-166	308	6	petersen	petersen	PROPN
ma-166	308	7	graph	graph	VERB
ma-166	308	8	tn	tn	PROPN
ma-166	309	1	=	=	PUNCT
ma-166	310	1	{	{	PUNCT
ma-166	310	2	1	1	NUM
ma-166	310	3	,	,	PUNCT
ma-166	310	4	2	2	NUM
ma-166	310	5	,	,	PUNCT
ma-166	310	6	3	3	NUM
ma-166	310	7	,	,	PUNCT
ma-166	310	8	...	...	PUNCT
ma-166	310	9	}	}	PUNCT
ma-166	310	10	appears	appear	VERB
ma-166	310	11	in	in	ADP
ma-166	310	12	figure(graph	figure(graph	PROPN
ma-166	310	13	)	)	PUNCT
ma-166	310	14	.	.	PUNCT
ma-166	311	1	the	the	DET
ma-166	311	2	petersen	petersen	PROPN
ma-166	311	3	graph	graph	NOUN
ma-166	311	4	tn=	tn=	PROPN
ma-166	311	5	{	{	PUNCT
ma-166	311	6	1	1	NUM
ma-166	311	7	,	,	PUNCT
ma-166	311	8	2	2	NUM
ma-166	311	9	,	,	PUNCT
ma-166	311	10	3	3	NUM
ma-166	311	11	,	,	PUNCT
ma-166	311	12	...	...	PUNCT
ma-166	311	13	}	}	PUNCT
ma-166	311	14	contains	contain	VERB
ma-166	311	15	v	v	NOUN
ma-166	311	16	(	(	PUNCT
ma-166	311	17	g′	g′	NOUN
ma-166	311	18	)	)	PUNCT
ma-166	311	19	=	=	NOUN
ma-166	312	1	4n	4n	NOUN
ma-166	312	2	no	no	PRON
ma-166	312	3	of	of	ADP
ma-166	312	4	vertices	vertex	NOUN
ma-166	312	5	and	and	CCONJ
ma-166	312	6	e(g′	e(g′	PRON
ma-166	312	7	)	)	PUNCT
ma-166	313	1	=	=	SYM
ma-166	313	2	6n	6n	NUM
ma-166	313	3	no	no	PRON
ma-166	313	4	of	of	ADP
ma-166	313	5	edges	edge	NOUN
ma-166	313	6	.	.	PUNCT
ma-166	314	1	the	the	DET
ma-166	314	2	degree	degree	NOUN
ma-166	314	3	ofeach	ofeach	NOUN
ma-166	314	4	vertex	vertex	NOUN
ma-166	314	5	in	in	ADP
ma-166	314	6	p(k	p(k	NOUN
ma-166	314	7	,	,	PUNCT
ma-166	314	8	t	t	NUM
ma-166	314	9	)	)	PUNCT
ma-166	314	10	is	be	AUX
ma-166	314	11	3	3	NUM
ma-166	314	12	and	and	CCONJ
ma-166	314	13	now	now	ADV
ma-166	314	14	geometric	geometric	ADJ
ma-166	314	15	arithmetic(ga	arithmetic(ga	NOUN
ma-166	314	16	)	)	PUNCT
ma-166	314	17	index	index	NOUN
ma-166	314	18	are	be	AUX
ma-166	314	19	i.e.	i.e.	X
ma-166	314	20	,	,	PUNCT
ma-166	314	21	ga(g	ga(g	NOUN
ma-166	314	22	)	)	PUNCT
ma-166	314	23	=	=	PUNCT
ma-166	315	1	∑	∑	PUNCT
ma-166	316	1	[	[	X
ma-166	316	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	316	3	)	)	PUNCT
ma-166	316	4	]	]	PUNCT
ma-166	316	5	2	2	NUM
ma-166	316	6	√	√	NUM
ma-166	316	7	d(x1)×d(x2	d(x1)×d(x2	NOUN
ma-166	316	8	)	)	PUNCT
ma-166	316	9	d(x1)+d(x2	d(x1)+d(x2	NOUN
ma-166	316	10	)	)	PUNCT
ma-166	316	11	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	316	12	eur	eur	PROPN
ma-166	316	13	.	.	PUNCT
ma-166	317	1	j.	j.	PROPN
ma-166	317	2	math	math	PROPN
ma-166	317	3	.	.	PUNCT
ma-166	318	1	anal	anal	PROPN
ma-166	318	2	.	.	PUNCT
ma-166	319	1	10.28924	10.28924	NUM
ma-166	319	2	/	/	SYM
ma-166	319	3	ada	ada	PROPN
ma-166	319	4	/	/	SYM
ma-166	319	5	ma.3.20	ma.3.20	NOUN
ma-166	320	1	11now	11now	INTJ
ma-166	320	2	we	we	PRON
ma-166	320	3	suppose	suppose	VERB
ma-166	320	4	vertices	vertex	NOUN
ma-166	320	5	are	be	AUX
ma-166	320	6	v	v	ADP
ma-166	320	7	(	(	PUNCT
ma-166	320	8	g	g	NOUN
ma-166	320	9	)	)	PUNCT
ma-166	320	10	=	=	SYM
ma-166	320	11	4n	4n	NOUN
ma-166	320	12	,	,	PUNCT
ma-166	320	13	edges	edge	NOUN
ma-166	320	14	are	be	AUX
ma-166	320	15	e(g	e(g	NOUN
ma-166	320	16	)	)	PUNCT
ma-166	321	1	=	=	SYM
ma-166	321	2	6n	6n	NOUN
ma-166	321	3	and	and	CCONJ
ma-166	321	4	degree	degree	NOUN
ma-166	321	5	of	of	ADP
ma-166	321	6	petersen	petersen	PROPN
ma-166	321	7	graphabout	graphabout	PROPN
ma-166	321	8	every	every	DET
ma-166	321	9	each	each	DET
ma-166	321	10	vertices	vertex	NOUN
ma-166	321	11	is	be	AUX
ma-166	321	12	p(k	p(k	NOUN
ma-166	321	13	,	,	PUNCT
ma-166	321	14	t	t	NOUN
ma-166	321	15	)	)	PUNCT
ma-166	321	16	=[	=[	NOUN
ma-166	321	17	d(x1	d(x1	NOUN
ma-166	321	18	)	)	PUNCT
ma-166	321	19	,	,	PUNCT
ma-166	321	20	d(x2	d(x2	NOUN
ma-166	321	21	)	)	PUNCT
ma-166	321	22	]	]	PUNCT
ma-166	322	1	=	=	PUNCT
ma-166	322	2	3	3	X
ma-166	322	3	.	.	PUNCT
ma-166	322	4	now	now	ADV
ma-166	322	5	putting	put	VERB
ma-166	322	6	the	the	DET
ma-166	322	7	values	value	NOUN
ma-166	322	8	in	in	ADP
ma-166	322	9	geometricarithmetic(ga	geometricarithmetic(ga	ADJ
ma-166	322	10	)	)	PUNCT
ma-166	322	11	index	index	NOUN
ma-166	322	12	topological	topological	ADJ
ma-166	322	13	index	index	NOUN
ma-166	322	14	of	of	ADP
ma-166	322	15	the	the	DET
ma-166	322	16	general	general	ADJ
ma-166	322	17	form	form	NOUN
ma-166	322	18	,	,	PUNCT
ma-166	322	19	⇒	⇒	NOUN
ma-166	322	20	ga(g	ga(g	PROPN
ma-166	322	21	)	)	PUNCT
ma-166	322	22	=[	=[	PROPN
ma-166	322	23	|e(g)|]2√(3)×(3)(3)+(3	|e(g)|]2√(3)×(3)(3)+(3	PROPN
ma-166	322	24	)	)	PUNCT
ma-166	322	25	⇒	⇒	NOUN
ma-166	322	26	ga(g	ga(g	PROPN
ma-166	322	27	)	)	PUNCT
ma-166	322	28	=[	=[	NOUN
ma-166	322	29	|6n|]2√(3)2(6	|6n|]2√(3)2(6	PROPN
ma-166	322	30	)	)	PUNCT
ma-166	322	31	⇒	⇒	NOUN
ma-166	322	32	ga(g	ga(g	PROPN
ma-166	322	33	)	)	PUNCT
ma-166	322	34	=(	=(	NOUN
ma-166	322	35	6n)2√(3)26	6n)2√(3)26	NUM
ma-166	322	36	⇒	⇒	NOUN
ma-166	322	37	ga(g	ga(g	PROPN
ma-166	322	38	)	)	PUNCT
ma-166	322	39	=(	=(	NOUN
ma-166	322	40	6n)2(3)(6	6n)2(3)(6	NUM
ma-166	322	41	)	)	PUNCT
ma-166	322	42	⇒	⇒	NOUN
ma-166	322	43	ga(g	ga(g	PROPN
ma-166	322	44	)	)	PUNCT
ma-166	323	1	=	=	NOUN
ma-166	323	2	6n	6n	PROPN
ma-166	323	3	.	.	PUNCT
ma-166	323	4	ga(r	ga(r	X
ma-166	323	5	)	)	PUNCT
ma-166	324	1	=	=	SYM
ma-166	324	2	general	general	ADJ
ma-166	324	3	edges	edge	NOUN
ma-166	324	4	of	of	ADP
ma-166	324	5	petersen	petersen	PROPN
ma-166	324	6	graph	graph	NOUN
ma-166	324	7	.	.	PUNCT
ma-166	325	1	theorem	theorem	VERB
ma-166	325	2	2.10	2.10	NUM
ma-166	325	3	let	let	VERB
ma-166	325	4	p(k	p(k	NOUN
ma-166	325	5	,	,	PUNCT
ma-166	325	6	t	t	NUM
ma-166	325	7	)	)	PUNCT
ma-166	325	8	be	be	AUX
ma-166	325	9	petersen	petersen	NOUN
ma-166	325	10	subdivision	subdivision	NOUN
ma-166	325	11	graph	graph	NOUN
ma-166	325	12	.	.	PUNCT
ma-166	326	1	then	then	ADV
ma-166	326	2	,	,	PUNCT
ma-166	326	3	for	for	ADP
ma-166	326	4	tn	tn	NOUN
ma-166	326	5	=	=	SYM
ma-166	326	6	{	{	PUNCT
ma-166	326	7	1	1	NUM
ma-166	326	8	,	,	PUNCT
ma-166	326	9	2	2	NUM
ma-166	326	10	,	,	PUNCT
ma-166	326	11	3	3	NUM
ma-166	326	12	,	,	PUNCT
ma-166	326	13	...	...	PUNCT
ma-166	326	14	}	}	PUNCT
ma-166	326	15	,	,	PUNCT
ma-166	326	16	first	first	ADJ
ma-166	326	17	multiplezagreb	multiplezagreb	PROPN
ma-166	326	18	index	index	NOUN
ma-166	326	19	are	be	AUX
ma-166	326	20	,	,	PUNCT
ma-166	326	21	pm1(g	pm1(g	NOUN
ma-166	326	22	)	)	PUNCT
ma-166	326	23	=	=	PUNCT
ma-166	327	1	(	(	PUNCT
ma-166	327	2	6)6n	6)6n	NOUN
ma-166	327	3	proof	proof	VERB
ma-166	327	4	:	:	PUNCT
ma-166	327	5	the	the	DET
ma-166	327	6	petersen	petersen	PROPN
ma-166	327	7	graph	graph	VERB
ma-166	327	8	tn	tn	PROPN
ma-166	327	9	=	=	PUNCT
ma-166	327	10	{	{	PUNCT
ma-166	327	11	1	1	NUM
ma-166	327	12	,	,	PUNCT
ma-166	327	13	2	2	NUM
ma-166	327	14	,	,	PUNCT
ma-166	327	15	3	3	NUM
ma-166	327	16	,	,	PUNCT
ma-166	327	17	...	...	PUNCT
ma-166	327	18	}	}	PUNCT
ma-166	327	19	appears	appear	VERB
ma-166	327	20	in	in	ADP
ma-166	327	21	figure(graph	figure(graph	PROPN
ma-166	327	22	)	)	PUNCT
ma-166	327	23	.	.	PUNCT
ma-166	328	1	the	the	DET
ma-166	328	2	petersen	petersen	PROPN
ma-166	328	3	graph	graph	NOUN
ma-166	328	4	tn=	tn=	PROPN
ma-166	328	5	{	{	PUNCT
ma-166	328	6	1	1	NUM
ma-166	328	7	,	,	PUNCT
ma-166	328	8	2	2	NUM
ma-166	328	9	,	,	PUNCT
ma-166	328	10	3	3	NUM
ma-166	328	11	,	,	PUNCT
ma-166	328	12	...	...	PUNCT
ma-166	328	13	}	}	PUNCT
ma-166	328	14	contains	contain	VERB
ma-166	328	15	v	v	NOUN
ma-166	328	16	(	(	PUNCT
ma-166	328	17	g′	g′	NOUN
ma-166	328	18	)	)	PUNCT
ma-166	328	19	=	=	NOUN
ma-166	329	1	4n	4n	NOUN
ma-166	329	2	no	no	PRON
ma-166	329	3	of	of	ADP
ma-166	329	4	vertices	vertex	NOUN
ma-166	329	5	and	and	CCONJ
ma-166	329	6	e(g′	e(g′	PRON
ma-166	329	7	)	)	PUNCT
ma-166	330	1	=	=	SYM
ma-166	330	2	6n	6n	NUM
ma-166	330	3	no	no	PRON
ma-166	330	4	of	of	ADP
ma-166	330	5	edges	edge	NOUN
ma-166	330	6	.	.	PUNCT
ma-166	331	1	the	the	DET
ma-166	331	2	degree	degree	NOUN
ma-166	331	3	ofeach	ofeach	NOUN
ma-166	331	4	vertex	vertex	NOUN
ma-166	331	5	in	in	ADP
ma-166	331	6	p(k	p(k	NOUN
ma-166	331	7	,	,	PUNCT
ma-166	331	8	t	t	NUM
ma-166	331	9	)	)	PUNCT
ma-166	331	10	is	be	AUX
ma-166	331	11	3	3	NUM
ma-166	331	12	and	and	CCONJ
ma-166	331	13	now	now	ADV
ma-166	331	14	first	first	ADJ
ma-166	331	15	multiple	multiple	ADJ
ma-166	331	16	zagreb	zagreb	PROPN
ma-166	331	17	index	index	NOUN
ma-166	331	18	are	be	AUX
ma-166	331	19	i.e.	i.e.	X
ma-166	331	20	,	,	PUNCT
ma-166	331	21	pm1(g	pm1(g	NOUN
ma-166	331	22	)	)	PUNCT
ma-166	331	23	=	=	SYM
ma-166	332	1	∏	∏	PROPN
ma-166	332	2	[	[	X
ma-166	332	3	x1,x2∈e(g)][d(x1	x1,x2∈e(g)][d(x1	X
ma-166	332	4	)	)	PUNCT
ma-166	333	1	+	+	CCONJ
ma-166	333	2	d(x2)]now	d(x2)]now	NOUN
ma-166	333	3	we	we	PRON
ma-166	333	4	suppose	suppose	VERB
ma-166	333	5	vertices	vertex	NOUN
ma-166	333	6	are	be	AUX
ma-166	333	7	v	v	ADP
ma-166	333	8	(	(	PUNCT
ma-166	333	9	g	g	NOUN
ma-166	333	10	)	)	PUNCT
ma-166	333	11	=	=	SYM
ma-166	333	12	4n	4n	NOUN
ma-166	333	13	,	,	PUNCT
ma-166	333	14	edges	edge	NOUN
ma-166	333	15	are	be	AUX
ma-166	333	16	e(g	e(g	NOUN
ma-166	333	17	)	)	PUNCT
ma-166	334	1	=	=	SYM
ma-166	334	2	6n	6n	NOUN
ma-166	334	3	and	and	CCONJ
ma-166	334	4	degree	degree	NOUN
ma-166	334	5	of	of	ADP
ma-166	334	6	petersen	petersen	PROPN
ma-166	334	7	graphabout	graphabout	PROPN
ma-166	334	8	every	every	DET
ma-166	334	9	each	each	DET
ma-166	334	10	vertices	vertex	NOUN
ma-166	334	11	is	be	AUX
ma-166	334	12	p(k	p(k	NOUN
ma-166	334	13	,	,	PUNCT
ma-166	334	14	t	t	NOUN
ma-166	334	15	)	)	PUNCT
ma-166	334	16	=[	=[	NOUN
ma-166	334	17	d(x1	d(x1	NOUN
ma-166	334	18	)	)	PUNCT
ma-166	334	19	,	,	PUNCT
ma-166	334	20	d(x2	d(x2	NOUN
ma-166	334	21	)	)	PUNCT
ma-166	334	22	]	]	PUNCT
ma-166	335	1	=	=	PUNCT
ma-166	335	2	3	3	X
ma-166	335	3	.	.	PUNCT
ma-166	335	4	now	now	ADV
ma-166	335	5	putting	put	VERB
ma-166	335	6	the	the	DET
ma-166	335	7	values	value	NOUN
ma-166	335	8	in	in	ADP
ma-166	335	9	first	first	ADJ
ma-166	335	10	multiplezagreb	multiplezagreb	PROPN
ma-166	335	11	index	index	PROPN
ma-166	335	12	topological	topological	ADJ
ma-166	335	13	index	index	NOUN
ma-166	335	14	of	of	ADP
ma-166	335	15	the	the	DET
ma-166	335	16	general	general	ADJ
ma-166	335	17	form	form	NOUN
ma-166	335	18	,	,	PUNCT
ma-166	335	19	⇒	⇒	PROPN
ma-166	335	20	pm1(g	pm1(g	PROPN
ma-166	335	21	)	)	PUNCT
ma-166	335	22	=	=	SYM
ma-166	336	1	∏	∏	PROPN
ma-166	337	1	[	[	X
ma-166	337	2	x1,x2∈e(g)][(3	x1,x2∈e(g)][(3	X
ma-166	337	3	+	+	NUM
ma-166	337	4	3	3	NUM
ma-166	337	5	)	)	PUNCT
ma-166	337	6	]	]	PUNCT
ma-166	337	7	⇒	⇒	PROPN
ma-166	337	8	pm1(g	pm1(g	PROPN
ma-166	337	9	)	)	PUNCT
ma-166	337	10	=	=	SYM
ma-166	338	1	(	(	PUNCT
ma-166	338	2	6)[|e(g)|	6)[|e(g)|	PROPN
ma-166	338	3	]	]	PUNCT
ma-166	338	4	⇒	⇒	PROPN
ma-166	338	5	pm1(g	pm1(g	PROPN
ma-166	338	6	)	)	PUNCT
ma-166	339	1	=	=	PUNCT
ma-166	339	2	(	(	PUNCT
ma-166	339	3	6)[|6n|	6)[|6n|	PROPN
ma-166	339	4	]	]	PUNCT
ma-166	339	5	⇒	⇒	PROPN
ma-166	339	6	pm1(g	pm1(g	PROPN
ma-166	339	7	)	)	PUNCT
ma-166	339	8	=	=	PUNCT
ma-166	339	9	(	(	PUNCT
ma-166	339	10	6)6n	6)6n	NOUN
ma-166	339	11	.	.	PUNCT
ma-166	340	1	pm1(g	pm1(g	NOUN
ma-166	340	2	)	)	PUNCT
ma-166	341	1	=	=	SYM
ma-166	341	2	general	general	ADJ
ma-166	341	3	edges	edge	NOUN
ma-166	341	4	of	of	ADP
ma-166	341	5	petersen	petersen	PROPN
ma-166	341	6	graph	graph	NOUN
ma-166	341	7	to	to	ADP
ma-166	341	8	the	the	DET
ma-166	341	9	power	power	NOUN
ma-166	341	10	of	of	ADP
ma-166	341	11	six	six	NUM
ma-166	341	12	.	.	PUNCT
ma-166	342	1	theorem	theorem	VERB
ma-166	342	2	2.11	2.11	NUM
ma-166	342	3	let	let	VERB
ma-166	342	4	p(k	p(k	NOUN
ma-166	342	5	,	,	PUNCT
ma-166	342	6	t	t	NUM
ma-166	342	7	)	)	PUNCT
ma-166	342	8	be	be	AUX
ma-166	342	9	petersen	petersen	NOUN
ma-166	342	10	subdivision	subdivision	NOUN
ma-166	342	11	graph	graph	NOUN
ma-166	342	12	.	.	PUNCT
ma-166	343	1	then	then	ADV
ma-166	343	2	,	,	PUNCT
ma-166	343	3	for	for	ADP
ma-166	343	4	tn	tn	NOUN
ma-166	343	5	=	=	SYM
ma-166	343	6	{	{	PUNCT
ma-166	343	7	1	1	NUM
ma-166	343	8	,	,	PUNCT
ma-166	343	9	2	2	NUM
ma-166	343	10	,	,	PUNCT
ma-166	343	11	3	3	NUM
ma-166	343	12	,	,	PUNCT
ma-166	343	13	...	...	PUNCT
ma-166	343	14	}	}	PUNCT
ma-166	343	15	,	,	PUNCT
ma-166	343	16	secondmultiple	secondmultiple	PROPN
ma-166	343	17	zagreb	zagreb	PROPN
ma-166	343	18	index	index	PROPN
ma-166	343	19	are	be	AUX
ma-166	343	20	,	,	PUNCT
ma-166	343	21	pm2(g	pm2(g	X
ma-166	343	22	)	)	PUNCT
ma-166	343	23	=	=	SYM
ma-166	344	1	(	(	PUNCT
ma-166	344	2	9)[6n	9)[6n	X
ma-166	344	3	]	]	X
ma-166	344	4	proof	proof	NOUN
ma-166	344	5	:	:	PUNCT
ma-166	344	6	the	the	DET
ma-166	344	7	petersen	petersen	PROPN
ma-166	344	8	graph	graph	VERB
ma-166	344	9	tn	tn	PROPN
ma-166	344	10	=	=	PUNCT
ma-166	344	11	{	{	PUNCT
ma-166	344	12	1	1	NUM
ma-166	344	13	,	,	PUNCT
ma-166	344	14	2	2	NUM
ma-166	344	15	,	,	PUNCT
ma-166	344	16	3	3	NUM
ma-166	344	17	,	,	PUNCT
ma-166	344	18	...	...	PUNCT
ma-166	344	19	}	}	PUNCT
ma-166	344	20	appears	appear	VERB
ma-166	344	21	in	in	ADP
ma-166	344	22	figure(graph	figure(graph	PROPN
ma-166	344	23	)	)	PUNCT
ma-166	344	24	.	.	PUNCT
ma-166	345	1	the	the	DET
ma-166	345	2	petersen	petersen	PROPN
ma-166	345	3	graph	graph	NOUN
ma-166	345	4	tn=	tn=	PROPN
ma-166	345	5	{	{	PUNCT
ma-166	345	6	1	1	NUM
ma-166	345	7	,	,	PUNCT
ma-166	345	8	2	2	NUM
ma-166	345	9	,	,	PUNCT
ma-166	345	10	3	3	NUM
ma-166	345	11	,	,	PUNCT
ma-166	345	12	...	...	PUNCT
ma-166	345	13	}	}	PUNCT
ma-166	345	14	contains	contain	VERB
ma-166	345	15	v	v	NOUN
ma-166	345	16	(	(	PUNCT
ma-166	345	17	g′	g′	NOUN
ma-166	345	18	)	)	PUNCT
ma-166	345	19	=	=	NOUN
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ma-166	346	3	of	of	ADP
ma-166	346	4	vertices	vertex	NOUN
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ma-166	346	6	e(g′	e(g′	PRON
ma-166	346	7	)	)	PUNCT
ma-166	347	1	=	=	SYM
ma-166	347	2	6n	6n	NUM
ma-166	347	3	no	no	PRON
ma-166	347	4	of	of	ADP
ma-166	347	5	edges	edge	NOUN
ma-166	347	6	.	.	PUNCT
ma-166	348	1	the	the	DET
ma-166	348	2	degree	degree	NOUN
ma-166	348	3	ofeach	ofeach	NOUN
ma-166	348	4	vertex	vertex	NOUN
ma-166	348	5	in	in	ADP
ma-166	348	6	p(k	p(k	NOUN
ma-166	348	7	,	,	PUNCT
ma-166	348	8	t	t	NUM
ma-166	348	9	)	)	PUNCT
ma-166	348	10	is	be	AUX
ma-166	348	11	3	3	NUM
ma-166	348	12	and	and	CCONJ
ma-166	348	13	now	now	ADV
ma-166	348	14	second	second	ADJ
ma-166	348	15	multiple	multiple	ADJ
ma-166	348	16	zagreb	zagreb	PROPN
ma-166	348	17	index	index	NOUN
ma-166	348	18	are	be	AUX
ma-166	348	19	i.e.	i.e.	X
ma-166	348	20	,	,	PUNCT
ma-166	348	21	pm2(g	pm2(g	NUM
ma-166	348	22	)	)	PUNCT
ma-166	348	23	=	=	SYM
ma-166	348	24	∏	∏	PROPN
ma-166	349	1	[	[	X
ma-166	349	2	x1,x2∈e(g)][d(x1)×	x1,x2∈e(g)][d(x1)×	PROPN
ma-166	349	3	d(x2	d(x2	NOUN
ma-166	349	4	)	)	PUNCT
ma-166	349	5	]	]	PUNCT
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ma-166	349	7	eur	eur	PROPN
ma-166	349	8	.	.	PUNCT
ma-166	350	1	j.	j.	PROPN
ma-166	350	2	math	math	PROPN
ma-166	350	3	.	.	PUNCT
ma-166	351	1	anal	anal	PROPN
ma-166	351	2	.	.	PUNCT
ma-166	352	1	10.28924	10.28924	NUM
ma-166	352	2	/	/	SYM
ma-166	352	3	ada	ada	PROPN
ma-166	352	4	/	/	SYM
ma-166	352	5	ma.3.20	ma.3.20	PROPN
ma-166	353	1	12now	12now	NOUN
ma-166	353	2	we	we	PRON
ma-166	353	3	suppose	suppose	VERB
ma-166	353	4	vertices	vertex	NOUN
ma-166	353	5	are	be	AUX
ma-166	353	6	v	v	ADP
ma-166	353	7	(	(	PUNCT
ma-166	353	8	g	g	NOUN
ma-166	353	9	)	)	PUNCT
ma-166	353	10	=	=	SYM
ma-166	353	11	4n	4n	NOUN
ma-166	353	12	,	,	PUNCT
ma-166	353	13	edges	edge	NOUN
ma-166	353	14	are	be	AUX
ma-166	353	15	e(g	e(g	NOUN
ma-166	353	16	)	)	PUNCT
ma-166	354	1	=	=	SYM
ma-166	354	2	6n	6n	NOUN
ma-166	354	3	and	and	CCONJ
ma-166	354	4	degree	degree	NOUN
ma-166	354	5	of	of	ADP
ma-166	354	6	petersen	petersen	PROPN
ma-166	354	7	graphabout	graphabout	PROPN
ma-166	354	8	every	every	DET
ma-166	354	9	each	each	DET
ma-166	354	10	vertices	vertex	NOUN
ma-166	354	11	is	be	AUX
ma-166	354	12	p(k	p(k	NOUN
ma-166	354	13	,	,	PUNCT
ma-166	354	14	t	t	NOUN
ma-166	354	15	)	)	PUNCT
ma-166	354	16	=[	=[	NOUN
ma-166	354	17	d(x1	d(x1	NOUN
ma-166	354	18	)	)	PUNCT
ma-166	354	19	,	,	PUNCT
ma-166	354	20	d(x2	d(x2	NOUN
ma-166	354	21	)	)	PUNCT
ma-166	354	22	]	]	PUNCT
ma-166	355	1	=	=	PUNCT
ma-166	355	2	3	3	X
ma-166	355	3	.	.	PUNCT
ma-166	355	4	now	now	ADV
ma-166	355	5	putting	put	VERB
ma-166	355	6	the	the	DET
ma-166	355	7	values	value	NOUN
ma-166	355	8	in	in	ADP
ma-166	355	9	second	second	ADJ
ma-166	355	10	multiplezagreb	multiplezagreb	NOUN
ma-166	355	11	index	index	PROPN
ma-166	355	12	topological	topological	ADJ
ma-166	355	13	index	index	NOUN
ma-166	355	14	of	of	ADP
ma-166	355	15	the	the	DET
ma-166	355	16	general	general	ADJ
ma-166	355	17	form	form	NOUN
ma-166	355	18	,	,	PUNCT
ma-166	355	19	⇒	⇒	NOUN
ma-166	355	20	pm2(g	pm2(g	NUM
ma-166	355	21	)	)	PUNCT
ma-166	355	22	=	=	SYM
ma-166	355	23	∏	∏	PROPN
ma-166	356	1	[	[	X
ma-166	356	2	x1,x2∈e(r)][(3)×	x1,x2∈e(r)][(3)×	X
ma-166	356	3	(	(	PUNCT
ma-166	356	4	3	3	NUM
ma-166	356	5	)	)	PUNCT
ma-166	356	6	]	]	PUNCT
ma-166	356	7	⇒	⇒	VERB
ma-166	356	8	pm2(g	pm2(g	NUM
ma-166	356	9	)	)	PUNCT
ma-166	356	10	=	=	SYM
ma-166	356	11	(	(	PUNCT
ma-166	356	12	9)[|e(r)|	9)[|e(r)|	PROPN
ma-166	356	13	]	]	PUNCT
ma-166	356	14	⇒	⇒	NOUN
ma-166	356	15	pm2(g	pm2(g	NUM
ma-166	356	16	)	)	PUNCT
ma-166	356	17	=	=	SYM
ma-166	356	18	(	(	PUNCT
ma-166	356	19	9)[6n	9)[6n	NUM
ma-166	356	20	]	]	PUNCT
ma-166	356	21	.	.	PUNCT
ma-166	357	1	pm2(g	pm2(g	X
ma-166	357	2	)	)	PUNCT
ma-166	357	3	=	=	SYM
ma-166	357	4	(	(	PUNCT
ma-166	357	5	9)6n	9)6n	NOUN
ma-166	357	6	pm2(g	pm2(g	NUM
ma-166	357	7	)	)	PUNCT
ma-166	357	8	=	=	SYM
ma-166	357	9	general	general	ADJ
ma-166	357	10	edges	edge	NOUN
ma-166	357	11	of	of	ADP
ma-166	357	12	petersen	petersen	PROPN
ma-166	357	13	graph	graph	NOUN
ma-166	357	14	to	to	ADP
ma-166	357	15	the	the	DET
ma-166	357	16	power	power	NOUN
ma-166	357	17	of	of	ADP
ma-166	357	18	nine	nine	NUM
ma-166	357	19	.	.	PUNCT
ma-166	358	1	theorem	theorem	VERB
ma-166	358	2	2.12	2.12	NUM
ma-166	358	3	let	let	VERB
ma-166	358	4	p(k	p(k	NOUN
ma-166	358	5	,	,	PUNCT
ma-166	358	6	t	t	NUM
ma-166	358	7	)	)	PUNCT
ma-166	358	8	be	be	AUX
ma-166	358	9	petersen	petersen	NOUN
ma-166	358	10	subdivision	subdivision	NOUN
ma-166	358	11	graph	graph	NOUN
ma-166	358	12	.	.	PUNCT
ma-166	359	1	then	then	ADV
ma-166	359	2	,	,	PUNCT
ma-166	359	3	for	for	ADP
ma-166	359	4	tn	tn	NOUN
ma-166	359	5	=	=	SYM
ma-166	359	6	{	{	PUNCT
ma-166	359	7	1	1	NUM
ma-166	359	8	,	,	PUNCT
ma-166	359	9	2	2	NUM
ma-166	359	10	,	,	PUNCT
ma-166	359	11	3	3	NUM
ma-166	359	12	,	,	PUNCT
ma-166	359	13	...	...	PUNCT
ma-166	359	14	}	}	PUNCT
ma-166	359	15	,	,	PUNCT
ma-166	359	16	forgottenpolynomial	forgottenpolynomial	NOUN
ma-166	359	17	are	be	AUX
ma-166	359	18	,	,	PUNCT
ma-166	359	19	f	f	PROPN
ma-166	359	20	(	(	PUNCT
ma-166	359	21	r	r	NOUN
ma-166	359	22	)	)	PUNCT
ma-166	359	23	=	=	SYM
ma-166	359	24	(	(	PUNCT
ma-166	359	25	6n)x18	6n)x18	NUM
ma-166	359	26	proof	proof	NOUN
ma-166	359	27	:	:	PUNCT
ma-166	359	28	the	the	DET
ma-166	359	29	petersen	petersen	PROPN
ma-166	359	30	graph	graph	VERB
ma-166	359	31	tn	tn	PROPN
ma-166	359	32	=	=	PUNCT
ma-166	359	33	{	{	PUNCT
ma-166	359	34	1	1	NUM
ma-166	359	35	,	,	PUNCT
ma-166	359	36	2	2	NUM
ma-166	359	37	,	,	PUNCT
ma-166	359	38	3	3	NUM
ma-166	359	39	,	,	PUNCT
ma-166	359	40	...	...	PUNCT
ma-166	359	41	}	}	PUNCT
ma-166	359	42	appears	appear	VERB
ma-166	359	43	in	in	ADP
ma-166	359	44	figure(graph	figure(graph	PROPN
ma-166	359	45	)	)	PUNCT
ma-166	359	46	.	.	PUNCT
ma-166	360	1	the	the	DET
ma-166	360	2	petersen	petersen	PROPN
ma-166	360	3	graph	graph	NOUN
ma-166	360	4	tn=	tn=	PROPN
ma-166	360	5	{	{	PUNCT
ma-166	360	6	1	1	NUM
ma-166	360	7	,	,	PUNCT
ma-166	360	8	2	2	NUM
ma-166	360	9	,	,	PUNCT
ma-166	360	10	3	3	NUM
ma-166	360	11	,	,	PUNCT
ma-166	360	12	...	...	PUNCT
ma-166	360	13	}	}	PUNCT
ma-166	360	14	contains	contain	VERB
ma-166	360	15	v	v	NOUN
ma-166	360	16	(	(	PUNCT
ma-166	360	17	g′	g′	NOUN
ma-166	360	18	)	)	PUNCT
ma-166	360	19	=	=	NOUN
ma-166	361	1	4n	4n	NOUN
ma-166	361	2	no	no	PRON
ma-166	361	3	of	of	ADP
ma-166	361	4	vertices	vertex	NOUN
ma-166	361	5	and	and	CCONJ
ma-166	361	6	e(g′	e(g′	PRON
ma-166	361	7	)	)	PUNCT
ma-166	362	1	=	=	SYM
ma-166	362	2	6n	6n	NUM
ma-166	362	3	no	no	PRON
ma-166	362	4	of	of	ADP
ma-166	362	5	edges	edge	NOUN
ma-166	362	6	.	.	PUNCT
ma-166	363	1	the	the	DET
ma-166	363	2	degree	degree	NOUN
ma-166	363	3	ofeach	ofeach	NOUN
ma-166	363	4	vertex	vertex	NOUN
ma-166	363	5	in	in	ADP
ma-166	363	6	p(k	p(k	NOUN
ma-166	363	7	,	,	PUNCT
ma-166	363	8	t	t	NUM
ma-166	363	9	)	)	PUNCT
ma-166	363	10	is	be	AUX
ma-166	363	11	3	3	NUM
ma-166	363	12	and	and	CCONJ
ma-166	363	13	now	now	ADV
ma-166	363	14	forgotten	forget	VERB
ma-166	363	15	polynomial	polynomial	ADJ
ma-166	363	16	are	be	AUX
ma-166	363	17	i.e.	i.e.	X
ma-166	363	18	,	,	PUNCT
ma-166	363	19	f	f	PROPN
ma-166	363	20	(	(	PUNCT
ma-166	363	21	g	g	PROPN
ma-166	363	22	,	,	PUNCT
ma-166	363	23	x	x	NOUN
ma-166	363	24	)	)	PUNCT
ma-166	364	1	=	=	NOUN
ma-166	364	2	∑	∑	PUNCT
ma-166	364	3	[	[	X
ma-166	364	4	x1,x2∈e(g)](x	x1,x2∈e(g)](x	X
ma-166	364	5	)	)	PUNCT
ma-166	364	6	[	[	X
ma-166	364	7	(	(	PUNCT
ma-166	364	8	dx1	dx1	PROPN
ma-166	364	9	)	)	PUNCT
ma-166	364	10	2+(dx2	2+(dx2	PROPN
ma-166	364	11	)	)	PUNCT
ma-166	364	12	2]now	2]now	NOUN
ma-166	364	13	we	we	PRON
ma-166	364	14	suppose	suppose	VERB
ma-166	364	15	vertices	vertex	NOUN
ma-166	364	16	are	be	AUX
ma-166	364	17	v	v	ADP
ma-166	364	18	(	(	PUNCT
ma-166	364	19	g	g	NOUN
ma-166	364	20	)	)	PUNCT
ma-166	364	21	=	=	SYM
ma-166	364	22	4n	4n	NOUN
ma-166	364	23	,	,	PUNCT
ma-166	364	24	edges	edge	NOUN
ma-166	364	25	are	be	AUX
ma-166	364	26	e(g	e(g	NOUN
ma-166	364	27	)	)	PUNCT
ma-166	365	1	=	=	SYM
ma-166	365	2	6n	6n	NOUN
ma-166	365	3	and	and	CCONJ
ma-166	365	4	degree	degree	NOUN
ma-166	365	5	of	of	ADP
ma-166	365	6	petersen	petersen	PROPN
ma-166	365	7	graphabout	graphabout	PROPN
ma-166	365	8	every	every	DET
ma-166	365	9	each	each	DET
ma-166	365	10	vertices	vertex	NOUN
ma-166	365	11	is	be	AUX
ma-166	365	12	p(k	p(k	NOUN
ma-166	365	13	,	,	PUNCT
ma-166	365	14	t	t	NOUN
ma-166	365	15	)	)	PUNCT
ma-166	365	16	=[	=[	NOUN
ma-166	365	17	d(x1	d(x1	NOUN
ma-166	365	18	)	)	PUNCT
ma-166	365	19	,	,	PUNCT
ma-166	365	20	d(x2	d(x2	NOUN
ma-166	365	21	)	)	PUNCT
ma-166	365	22	]	]	PUNCT
ma-166	366	1	=	=	PUNCT
ma-166	366	2	3	3	X
ma-166	366	3	.	.	PUNCT
ma-166	366	4	now	now	ADV
ma-166	366	5	putting	put	VERB
ma-166	366	6	the	the	DET
ma-166	366	7	values	value	NOUN
ma-166	366	8	in	in	ADP
ma-166	366	9	forgottenpolynomial	forgottenpolynomial	ADJ
ma-166	366	10	topological	topological	ADJ
ma-166	366	11	index	index	NOUN
ma-166	366	12	of	of	ADP
ma-166	366	13	the	the	DET
ma-166	366	14	general	general	ADJ
ma-166	366	15	form	form	NOUN
ma-166	366	16	,	,	PUNCT
ma-166	366	17	⇒	⇒	PROPN
ma-166	366	18	f	f	PROPN
ma-166	366	19	(	(	PUNCT
ma-166	366	20	g	g	PROPN
ma-166	366	21	,	,	PUNCT
ma-166	366	22	x	x	NOUN
ma-166	366	23	)	)	PUNCT
ma-166	367	1	=	=	NOUN
ma-166	367	2	∑	∑	PUNCT
ma-166	367	3	[	[	X
ma-166	367	4	x1,x2∈e(g)](x	x1,x2∈e(g)](x	X
ma-166	367	5	)	)	PUNCT
ma-166	367	6	[	[	X
ma-166	367	7	(	(	PUNCT
ma-166	367	8	3)2+(3)2	3)2+(3)2	NOUN
ma-166	367	9	]	]	X
ma-166	367	10	⇒	⇒	X
ma-166	367	11	f	f	X
ma-166	367	12	(	(	PUNCT
ma-166	367	13	g	g	PROPN
ma-166	367	14	,	,	PUNCT
ma-166	367	15	x	x	NOUN
ma-166	367	16	)	)	PUNCT
ma-166	367	17	=[	=[	NOUN
ma-166	367	18	|e(g)|](x)[9	|e(g)|](x)[9	NOUN
ma-166	367	19	+	+	ADJ
ma-166	367	20	9	9	NUM
ma-166	367	21	]	]	PUNCT
ma-166	367	22	⇒	⇒	NOUN
ma-166	367	23	f	f	PROPN
ma-166	367	24	(	(	PUNCT
ma-166	367	25	g	g	NOUN
ma-166	367	26	)	)	PUNCT
ma-166	367	27	=[	=[	NOUN
ma-166	367	28	6n](x)18	6n](x)18	NUM
ma-166	367	29	⇒	⇒	PROPN
ma-166	367	30	f	f	PROPN
ma-166	367	31	(	(	PUNCT
ma-166	367	32	r	r	NOUN
ma-166	367	33	)	)	PUNCT
ma-166	367	34	=(	=(	NOUN
ma-166	367	35	6n)x18	6n)x18	NOUN
ma-166	367	36	.	.	PUNCT
ma-166	368	1	f	f	X
ma-166	368	2	(	(	PUNCT
ma-166	368	3	r	r	NOUN
ma-166	368	4	)	)	PUNCT
ma-166	368	5	=[	=[	NOUN
ma-166	368	6	x18]×	x18]×	PUNCT
ma-166	369	1	[	[	X
ma-166	369	2	general	general	ADJ
ma-166	369	3	edges	edge	NOUN
ma-166	369	4	of	of	ADP
ma-166	369	5	petersen	petersen	PROPN
ma-166	369	6	graph	graph	NOUN
ma-166	369	7	]	]	PUNCT
ma-166	369	8	theorem	theorem	VERB
ma-166	369	9	2.13	2.13	NUM
ma-166	369	10	let	let	VERB
ma-166	369	11	p(k	p(k	NOUN
ma-166	369	12	,	,	PUNCT
ma-166	369	13	t	t	NUM
ma-166	369	14	)	)	PUNCT
ma-166	369	15	be	be	AUX
ma-166	369	16	petersen	petersen	NOUN
ma-166	369	17	subdivision	subdivision	NOUN
ma-166	369	18	graph	graph	NOUN
ma-166	369	19	.	.	PUNCT
ma-166	370	1	then	then	ADV
ma-166	370	2	,	,	PUNCT
ma-166	370	3	for	for	ADP
ma-166	370	4	tn	tn	NOUN
ma-166	370	5	=	=	SYM
ma-166	370	6	{	{	PUNCT
ma-166	370	7	1	1	NUM
ma-166	370	8	,	,	PUNCT
ma-166	370	9	2	2	NUM
ma-166	370	10	,	,	PUNCT
ma-166	370	11	3	3	NUM
ma-166	370	12	,	,	PUNCT
ma-166	370	13	...	...	PUNCT
ma-166	370	14	}	}	PUNCT
ma-166	370	15	,	,	PUNCT
ma-166	370	16	symmetricdivision	symmetricdivision	PROPN
ma-166	370	17	deg	deg	NOUN
ma-166	370	18	.	.	PUNCT
ma-166	371	1	index	index	NOUN
ma-166	371	2	are	be	AUX
ma-166	371	3	,	,	PUNCT
ma-166	371	4	sdd(g	sdd(g	PROPN
ma-166	371	5	)	)	PUNCT
ma-166	371	6	=	=	SYM
ma-166	371	7	12n	12n	NOUN
ma-166	371	8	proof	proof	NOUN
ma-166	371	9	:	:	PUNCT
ma-166	371	10	the	the	DET
ma-166	371	11	petersen	petersen	PROPN
ma-166	371	12	graph	graph	VERB
ma-166	371	13	tn	tn	PROPN
ma-166	371	14	=	=	PUNCT
ma-166	371	15	{	{	PUNCT
ma-166	371	16	1	1	NUM
ma-166	371	17	,	,	PUNCT
ma-166	371	18	2	2	NUM
ma-166	371	19	,	,	PUNCT
ma-166	371	20	3	3	NUM
ma-166	371	21	,	,	PUNCT
ma-166	371	22	...	...	PUNCT
ma-166	371	23	}	}	PUNCT
ma-166	371	24	appears	appear	VERB
ma-166	371	25	in	in	ADP
ma-166	371	26	figure(graph	figure(graph	PROPN
ma-166	371	27	)	)	PUNCT
ma-166	371	28	.	.	PUNCT
ma-166	372	1	the	the	DET
ma-166	372	2	petersen	petersen	PROPN
ma-166	372	3	graph	graph	NOUN
ma-166	372	4	tn=	tn=	PROPN
ma-166	372	5	{	{	PUNCT
ma-166	372	6	1	1	NUM
ma-166	372	7	,	,	PUNCT
ma-166	372	8	2	2	NUM
ma-166	372	9	,	,	PUNCT
ma-166	372	10	3	3	NUM
ma-166	372	11	,	,	PUNCT
ma-166	372	12	...	...	PUNCT
ma-166	372	13	}	}	PUNCT
ma-166	372	14	contains	contain	VERB
ma-166	372	15	v	v	NOUN
ma-166	372	16	(	(	PUNCT
ma-166	372	17	g′	g′	NOUN
ma-166	372	18	)	)	PUNCT
ma-166	372	19	=	=	NOUN
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ma-166	373	3	of	of	ADP
ma-166	373	4	vertices	vertex	NOUN
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ma-166	373	6	e(g′	e(g′	PRON
ma-166	373	7	)	)	PUNCT
ma-166	374	1	=	=	SYM
ma-166	374	2	6n	6n	NUM
ma-166	374	3	no	no	PRON
ma-166	374	4	of	of	ADP
ma-166	374	5	edges	edge	NOUN
ma-166	374	6	.	.	PUNCT
ma-166	375	1	the	the	DET
ma-166	375	2	degree	degree	NOUN
ma-166	375	3	ofeach	ofeach	NOUN
ma-166	375	4	vertex	vertex	NOUN
ma-166	375	5	in	in	ADP
ma-166	375	6	p(k	p(k	NOUN
ma-166	375	7	,	,	PUNCT
ma-166	375	8	t	t	NUM
ma-166	375	9	)	)	PUNCT
ma-166	375	10	is	be	AUX
ma-166	375	11	3	3	NUM
ma-166	375	12	and	and	CCONJ
ma-166	375	13	now	now	ADV
ma-166	375	14	symmetric	symmetric	ADJ
ma-166	375	15	division	division	NOUN
ma-166	375	16	deg	deg	PROPN
ma-166	375	17	.	.	PUNCT
ma-166	376	1	index	index	NOUN
ma-166	376	2	are	be	AUX
ma-166	376	3	i.e.	i.e.	X
ma-166	376	4	,	,	PUNCT
ma-166	376	5	sdd(g	sdd(g	PROPN
ma-166	376	6	)	)	PUNCT
ma-166	376	7	=	=	PUNCT
ma-166	377	1	∑	∑	PUNCT
ma-166	378	1	[	[	X
ma-166	378	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	378	3	)	)	PUNCT
ma-166	378	4	]	]	X
ma-166	378	5	[	[	PUNCT
ma-166	378	6	d(x1	d(x1	ADJ
ma-166	378	7	)	)	PUNCT
ma-166	378	8	2+d(x2	2+d(x2	NOUN
ma-166	378	9	)	)	PUNCT
ma-166	378	10	2	2	NUM
ma-166	378	11	d(x1)d(x2	d(x1)d(x2	NOUN
ma-166	378	12	)	)	PUNCT
ma-166	378	13	]	]	PUNCT
ma-166	378	14	now	now	ADV
ma-166	378	15	we	we	PRON
ma-166	378	16	suppose	suppose	VERB
ma-166	378	17	vertices	vertex	NOUN
ma-166	378	18	are	be	AUX
ma-166	378	19	v	v	ADP
ma-166	378	20	(	(	PUNCT
ma-166	378	21	g	g	NOUN
ma-166	378	22	)	)	PUNCT
ma-166	378	23	=	=	SYM
ma-166	378	24	4n	4n	NOUN
ma-166	378	25	,	,	PUNCT
ma-166	378	26	edges	edge	NOUN
ma-166	378	27	are	be	AUX
ma-166	378	28	e(g	e(g	NOUN
ma-166	378	29	)	)	PUNCT
ma-166	379	1	=	=	SYM
ma-166	379	2	6n	6n	NOUN
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ma-166	379	5	of	of	ADP
ma-166	379	6	petersen	petersen	PROPN
ma-166	379	7	graphabout	graphabout	PROPN
ma-166	379	8	every	every	DET
ma-166	379	9	each	each	DET
ma-166	379	10	vertices	vertex	NOUN
ma-166	379	11	is	be	AUX
ma-166	379	12	p(k	p(k	NOUN
ma-166	379	13	,	,	PUNCT
ma-166	379	14	t	t	NOUN
ma-166	379	15	)	)	PUNCT
ma-166	379	16	=[	=[	NOUN
ma-166	379	17	d(x1	d(x1	NOUN
ma-166	379	18	)	)	PUNCT
ma-166	379	19	,	,	PUNCT
ma-166	379	20	d(x2	d(x2	NOUN
ma-166	379	21	)	)	PUNCT
ma-166	379	22	]	]	PUNCT
ma-166	380	1	=	=	PUNCT
ma-166	380	2	3	3	X
ma-166	380	3	.	.	PUNCT
ma-166	380	4	now	now	ADV
ma-166	380	5	putting	put	VERB
ma-166	380	6	the	the	DET
ma-166	380	7	values	value	NOUN
ma-166	380	8	in	in	ADP
ma-166	380	9	symmetric	symmetric	ADJ
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ma-166	380	11	eur	eur	PROPN
ma-166	380	12	.	.	PUNCT
ma-166	381	1	j.	j.	PROPN
ma-166	381	2	math	math	PROPN
ma-166	381	3	.	.	PUNCT
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ma-166	382	2	.	.	PUNCT
ma-166	383	1	10.28924	10.28924	NUM
ma-166	383	2	/	/	SYM
ma-166	383	3	ada	ada	PROPN
ma-166	383	4	/	/	SYM
ma-166	383	5	ma.3.20	ma.3.20	PROPN
ma-166	383	6	13division	13division	NUM
ma-166	383	7	deg	deg	PROPN
ma-166	383	8	.	.	PUNCT
ma-166	384	1	index	index	PROPN
ma-166	384	2	topological	topological	ADJ
ma-166	384	3	index	index	NOUN
ma-166	384	4	of	of	ADP
ma-166	384	5	the	the	DET
ma-166	384	6	general	general	ADJ
ma-166	384	7	form	form	NOUN
ma-166	384	8	,	,	PUNCT
ma-166	384	9	⇒	⇒	PROPN
ma-166	384	10	sdd(g	sdd(g	VERB
ma-166	384	11	)	)	PUNCT
ma-166	384	12	=	=	PUNCT
ma-166	385	1	∑	∑	PUNCT
ma-166	386	1	[	[	X
ma-166	386	2	x1,x2∈e(g	x1,x2∈e(g	X
ma-166	386	3	)	)	PUNCT
ma-166	386	4	]	]	PUNCT
ma-166	386	5	mini(3,3	mini(3,3	NOUN
ma-166	386	6	)	)	PUNCT
ma-166	386	7	max(3,3	max(3,3	PROPN
ma-166	386	8	)	)	PUNCT
ma-166	387	1	+	+	CCONJ
ma-166	387	2	maxi(3,3	maxi(3,3	ADJ
ma-166	387	3	)	)	PUNCT
ma-166	387	4	mini(3,3	mini(3,3	NOUN
ma-166	387	5	)	)	PUNCT
ma-166	387	6	⇒	⇒	NOUN
ma-166	387	7	sdd(g	sdd(g	PROPN
ma-166	387	8	)	)	PUNCT
ma-166	387	9	=[	=[	NOUN
ma-166	388	1	|e(g)|][33	|e(g)|][33	NOUN
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ma-166	388	3	33	33	NUM
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ma-166	388	5	⇒	⇒	PROPN
ma-166	388	6	sdd(g	sdd(g	PROPN
ma-166	388	7	)	)	PUNCT
ma-166	388	8	=[	=[	NOUN
ma-166	389	1	|e(g)|	|e(g)|	NOUN
ma-166	389	2	]	]	X
ma-166	389	3	[	[	PUNCT
ma-166	389	4	(	(	PUNCT
ma-166	389	5	3)+(3)3	3)+(3)3	NUM
ma-166	389	6	]	]	PUNCT
ma-166	389	7	⇒	⇒	PROPN
ma-166	389	8	sdd(g	sdd(g	PROPN
ma-166	389	9	)	)	PUNCT
ma-166	389	10	=[	=[	NOUN
ma-166	389	11	|6n|	|6n|	PROPN
ma-166	389	12	]	]	X
ma-166	389	13	[	[	PUNCT
ma-166	389	14	(	(	PUNCT
ma-166	389	15	6)3	6)3	NOUN
ma-166	389	16	]	]	PUNCT
ma-166	389	17	⇒	⇒	PROPN
ma-166	389	18	sdd(g	sdd(g	PROPN
ma-166	389	19	)	)	PUNCT
ma-166	389	20	=(	=(	NOUN
ma-166	389	21	6n)[2	6n)[2	NUM
ma-166	389	22	]	]	PUNCT
ma-166	389	23	⇒	⇒	PROPN
ma-166	389	24	sdd(g	sdd(g	VERB
ma-166	389	25	)	)	PUNCT
ma-166	389	26	=	=	NOUN
ma-166	389	27	12n	12n	NOUN
ma-166	389	28	.	.	PUNCT
ma-166	390	1	sdd(g	sdd(g	VERB
ma-166	390	2	)	)	PUNCT
ma-166	390	3	=	=	SYM
ma-166	390	4	two	two	NUM
ma-166	390	5	times	time	NOUN
ma-166	390	6	of	of	ADP
ma-166	390	7	general	general	ADJ
ma-166	390	8	edges	edge	NOUN
ma-166	390	9	of	of	ADP
ma-166	390	10	petersen	petersen	PROPN
ma-166	390	11	graph	graph	NOUN
ma-166	390	12	.	.	PUNCT
ma-166	391	1	theorem	theorem	VERB
ma-166	391	2	2.14	2.14	NUM
ma-166	391	3	let	let	VERB
ma-166	391	4	p(k	p(k	NOUN
ma-166	391	5	,	,	PUNCT
ma-166	391	6	t	t	NUM
ma-166	391	7	)	)	PUNCT
ma-166	391	8	be	be	AUX
ma-166	391	9	petersen	petersen	NOUN
ma-166	391	10	subdivision	subdivision	NOUN
ma-166	391	11	graph	graph	NOUN
ma-166	391	12	.	.	PUNCT
ma-166	392	1	then	then	ADV
ma-166	392	2	,	,	PUNCT
ma-166	392	3	for	for	ADP
ma-166	392	4	tn	tn	NOUN
ma-166	392	5	=	=	SYM
ma-166	392	6	{	{	PUNCT
ma-166	392	7	1	1	NUM
ma-166	392	8	,	,	PUNCT
ma-166	392	9	2	2	NUM
ma-166	392	10	,	,	PUNCT
ma-166	392	11	3	3	NUM
ma-166	392	12	,	,	PUNCT
ma-166	392	13	...	...	PUNCT
ma-166	392	14	}	}	PUNCT
ma-166	392	15	,	,	PUNCT
ma-166	392	16	generalconnectivity	generalconnectivity	NOUN
ma-166	392	17	index	index	NOUN
ma-166	392	18	are	be	AUX
ma-166	392	19	,	,	PUNCT
ma-166	392	20	sdd(g	sdd(g	PROPN
ma-166	392	21	)	)	PUNCT
ma-166	392	22	=	=	SYM
ma-166	392	23	12n	12n	NOUN
ma-166	392	24	proof	proof	NOUN
ma-166	392	25	:	:	PUNCT
ma-166	392	26	the	the	DET
ma-166	392	27	petersen	petersen	PROPN
ma-166	392	28	graph	graph	VERB
ma-166	392	29	tn	tn	PROPN
ma-166	392	30	=	=	PUNCT
ma-166	392	31	{	{	PUNCT
ma-166	392	32	1	1	NUM
ma-166	392	33	,	,	PUNCT
ma-166	392	34	2	2	NUM
ma-166	392	35	,	,	PUNCT
ma-166	392	36	3	3	NUM
ma-166	392	37	,	,	PUNCT
ma-166	392	38	...	...	PUNCT
ma-166	392	39	}	}	PUNCT
ma-166	392	40	appears	appear	VERB
ma-166	392	41	in	in	ADP
ma-166	392	42	figure(graph	figure(graph	PROPN
ma-166	392	43	)	)	PUNCT
ma-166	392	44	.	.	PUNCT
ma-166	393	1	the	the	DET
ma-166	393	2	petersen	petersen	PROPN
ma-166	393	3	graph	graph	NOUN
ma-166	393	4	tn=	tn=	PROPN
ma-166	393	5	{	{	PUNCT
ma-166	393	6	1	1	NUM
ma-166	393	7	,	,	PUNCT
ma-166	393	8	2	2	NUM
ma-166	393	9	,	,	PUNCT
ma-166	393	10	3	3	NUM
ma-166	393	11	,	,	PUNCT
ma-166	393	12	...	...	PUNCT
ma-166	393	13	}	}	PUNCT
ma-166	393	14	contains	contain	VERB
ma-166	393	15	v	v	NOUN
ma-166	393	16	(	(	PUNCT
ma-166	393	17	g′	g′	NOUN
ma-166	393	18	)	)	PUNCT
ma-166	393	19	=	=	NOUN
ma-166	394	1	4n	4n	NOUN
ma-166	394	2	no	no	PRON
ma-166	394	3	of	of	ADP
ma-166	394	4	vertices	vertex	NOUN
ma-166	394	5	and	and	CCONJ
ma-166	394	6	e(g′	e(g′	PRON
ma-166	394	7	)	)	PUNCT
ma-166	395	1	=	=	SYM
ma-166	395	2	6n	6n	NUM
ma-166	395	3	no	no	PRON
ma-166	395	4	of	of	ADP
ma-166	395	5	edges	edge	NOUN
ma-166	395	6	.	.	PUNCT
ma-166	396	1	the	the	DET
ma-166	396	2	degree	degree	NOUN
ma-166	396	3	ofeach	ofeach	NOUN
ma-166	396	4	vertex	vertex	NOUN
ma-166	396	5	in	in	ADP
ma-166	396	6	p(k	p(k	NOUN
ma-166	396	7	,	,	PUNCT
ma-166	396	8	t	t	NUM
ma-166	396	9	)	)	PUNCT
ma-166	396	10	is	be	AUX
ma-166	396	11	3	3	NUM
ma-166	396	12	and	and	CCONJ
ma-166	396	13	now	now	ADV
ma-166	396	14	general	general	ADJ
ma-166	396	15	connectivity	connectivity	NOUN
ma-166	396	16	index	index	NOUN
ma-166	396	17	are	be	AUX
ma-166	396	18	i.e.	i.e.	X
ma-166	396	19	,	,	PUNCT
ma-166	396	20	m1(g	m1(g	NOUN
ma-166	396	21	)	)	PUNCT
ma-166	396	22	=	=	PUNCT
ma-166	397	1	∑	∑	PUNCT
ma-166	397	2	[	[	X
ma-166	397	3	x2∈v	x2∈v	X
ma-166	397	4	(	(	PUNCT
ma-166	397	5	g)][dg(x2	g)][dg(x2	NOUN
ma-166	397	6	)	)	PUNCT
ma-166	397	7	]	]	PUNCT
ma-166	397	8	2	2	NUM
ma-166	397	9	m2(g	m2(g	NOUN
ma-166	397	10	)	)	PUNCT
ma-166	397	11	=	=	PUNCT
ma-166	397	12	∑	∑	PUNCT
ma-166	398	1	[	[	X
ma-166	398	2	x1,x2∈e(g)][(dg(x1))×	x1,x2∈e(g)][(dg(x1))×	PROPN
ma-166	398	3	(	(	PUNCT
ma-166	398	4	dg(x2	dg(x2	NOUN
ma-166	398	5	)	)	PUNCT
ma-166	398	6	)	)	PUNCT
ma-166	398	7	]	]	PUNCT
ma-166	399	1	→	→	PUNCT
ma-166	399	2	[	[	X
ma-166	399	3	1	1	X
ma-166	399	4	]	]	PUNCT
ma-166	399	5	now	now	ADV
ma-166	399	6	we	we	PRON
ma-166	399	7	suppose	suppose	VERB
ma-166	399	8	vertices	vertex	NOUN
ma-166	399	9	are	be	AUX
ma-166	399	10	v	v	ADP
ma-166	399	11	(	(	PUNCT
ma-166	399	12	g	g	NOUN
ma-166	399	13	)	)	PUNCT
ma-166	399	14	=	=	SYM
ma-166	399	15	4n	4n	NOUN
ma-166	399	16	,	,	PUNCT
ma-166	399	17	edges	edge	NOUN
ma-166	399	18	are	be	AUX
ma-166	399	19	e(g	e(g	NOUN
ma-166	399	20	)	)	PUNCT
ma-166	400	1	=	=	SYM
ma-166	400	2	6n	6n	NOUN
ma-166	400	3	and	and	CCONJ
ma-166	400	4	degree	degree	NOUN
ma-166	400	5	of	of	ADP
ma-166	400	6	petersengraph	petersengraph	NOUN
ma-166	400	7	about	about	ADP
ma-166	400	8	every	every	DET
ma-166	400	9	each	each	DET
ma-166	400	10	vertices	vertex	NOUN
ma-166	400	11	is	be	AUX
ma-166	400	12	p(k	p(k	NOUN
ma-166	400	13	,	,	PUNCT
ma-166	400	14	t	t	NOUN
ma-166	400	15	)	)	PUNCT
ma-166	400	16	=[	=[	NOUN
ma-166	400	17	d(x1	d(x1	NOUN
ma-166	400	18	)	)	PUNCT
ma-166	400	19	,	,	PUNCT
ma-166	400	20	d(x2	d(x2	NOUN
ma-166	400	21	)	)	PUNCT
ma-166	400	22	]	]	PUNCT
ma-166	401	1	=	=	PUNCT
ma-166	401	2	3	3	X
ma-166	401	3	.	.	PUNCT
ma-166	401	4	now	now	ADV
ma-166	401	5	putting	put	VERB
ma-166	401	6	the	the	DET
ma-166	401	7	values	value	NOUN
ma-166	401	8	in	in	ADP
ma-166	401	9	generalconnectivity	generalconnectivity	NOUN
ma-166	401	10	index	index	NOUN
ma-166	401	11	topological	topological	ADJ
ma-166	401	12	index	index	NOUN
ma-166	401	13	of	of	ADP
ma-166	401	14	the	the	DET
ma-166	401	15	general	general	ADJ
ma-166	401	16	form	form	NOUN
ma-166	401	17	,	,	PUNCT
ma-166	401	18	⇒	⇒	NOUN
ma-166	401	19	m1(g	m1(g	NOUN
ma-166	401	20	)	)	PUNCT
ma-166	401	21	=	=	PUNCT
ma-166	401	22	∑	∑	PUNCT
ma-166	402	1	[	[	X
ma-166	402	2	x2∈v	x2∈v	X
ma-166	402	3	(	(	PUNCT
ma-166	402	4	g	g	NOUN
ma-166	402	5	)	)	PUNCT
ma-166	402	6	]	]	PUNCT
ma-166	403	1	[	[	X
ma-166	403	2	(	(	PUNCT
ma-166	403	3	3	3	NUM
ma-166	403	4	)	)	PUNCT
ma-166	403	5	2	2	NUM
ma-166	403	6	]	]	PUNCT
ma-166	403	7	⇒	⇒	NOUN
ma-166	403	8	m1(g	m1(g	NOUN
ma-166	403	9	)	)	PUNCT
ma-166	403	10	=	=	PUNCT
ma-166	404	1	[	[	X
ma-166	404	2	|v	|v	X
ma-166	404	3	(	(	PUNCT
ma-166	404	4	g)|](3)2	g)|](3)2	PROPN
ma-166	404	5	⇒	⇒	NOUN
ma-166	404	6	m1(g	m1(g	NOUN
ma-166	404	7	)	)	PUNCT
ma-166	404	8	=	=	PUNCT
ma-166	404	9	(	(	PUNCT
ma-166	404	10	4n)(9	4n)(9	NOUN
ma-166	404	11	)	)	PUNCT
ma-166	404	12	⇒	⇒	NOUN
ma-166	404	13	m1(g	m1(g	NOUN
ma-166	404	14	)	)	PUNCT
ma-166	404	15	=	=	PUNCT
ma-166	404	16	36n	36n	X
ma-166	404	17	.	.	PUNCT
ma-166	405	1	m1(g	m1(g	NOUN
ma-166	405	2	)	)	PUNCT
ma-166	405	3	=	=	NUM
ma-166	405	4	nine	nine	NUM
ma-166	405	5	times	time	NOUN
ma-166	405	6	to	to	ADP
ma-166	405	7	vertices	vertex	NOUN
ma-166	405	8	of	of	ADP
ma-166	405	9	petersen	petersen	NOUN
ma-166	405	10	graph	graph	NOUN
ma-166	405	11	.	.	PUNCT
ma-166	406	1	in	in	ADP
ma-166	406	2	general	general	ADJ
ma-166	406	3	form	form	NOUN
ma-166	406	4	of	of	ADP
ma-166	406	5	real	real	ADJ
ma-166	406	6	number	number	NOUN
ma-166	406	7	index	index	NOUN
ma-166	406	8	is	be	AUX
ma-166	406	9	,	,	PUNCT
ma-166	406	10	in	in	ADP
ma-166	406	11	equation	equation	NOUN
ma-166	406	12	[	[	X
ma-166	406	13	1	1	X
ma-166	406	14	]	]	PUNCT
ma-166	406	15	becomes	become	VERB
ma-166	406	16	;	;	PUNCT
ma-166	406	17	⇒	⇒	NOUN
ma-166	406	18	m2(g	m2(g	NOUN
ma-166	406	19	)	)	PUNCT
ma-166	406	20	=	=	PUNCT
ma-166	406	21	∑	∑	PUNCT
ma-166	407	1	[	[	X
ma-166	407	2	x1,x2∈e(g)][dg(x1)×	x1,x2∈e(g)][dg(x1)×	AUX
ma-166	407	3	dg(x2)]now	dg(x2)]now	ADV
ma-166	407	4	putting	put	VERB
ma-166	407	5	values	value	NOUN
ma-166	407	6	in	in	ADP
ma-166	407	7	above	above	ADP
ma-166	407	8	equation	equation	NOUN
ma-166	407	9	.	.	PUNCT
ma-166	408	1	⇒	⇒	PROPN
ma-166	408	2	m2(g	m2(g	NOUN
ma-166	408	3	)	)	PUNCT
ma-166	408	4	=	=	PUNCT
ma-166	408	5	∑	∑	PUNCT
ma-166	408	6	[	[	X
ma-166	408	7	x1,x2∈e(g)](3)(3	x1,x2∈e(g)](3)(3	X
ma-166	408	8	)	)	PUNCT
ma-166	408	9	⇒	⇒	VERB
ma-166	408	10	m2(g	m2(g	NOUN
ma-166	408	11	)	)	PUNCT
ma-166	408	12	=	=	PUNCT
ma-166	409	1	[	[	X
ma-166	409	2	|e(g)|](3)(3	|e(g)|](3)(3	NOUN
ma-166	409	3	)	)	PUNCT
ma-166	409	4	⇒	⇒	NOUN
ma-166	409	5	m2(g	m2(g	NOUN
ma-166	409	6	)	)	PUNCT
ma-166	409	7	=	=	SYM
ma-166	409	8	(	(	PUNCT
ma-166	409	9	6n)(9	6n)(9	NUM
ma-166	409	10	)	)	PUNCT
ma-166	409	11	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	409	12	eur	eur	PROPN
ma-166	409	13	.	.	PUNCT
ma-166	410	1	j.	j.	PROPN
ma-166	410	2	math	math	PROPN
ma-166	410	3	.	.	PUNCT
ma-166	411	1	anal	anal	PROPN
ma-166	411	2	.	.	PUNCT
ma-166	412	1	10.28924	10.28924	NUM
ma-166	412	2	/	/	SYM
ma-166	412	3	ada	ada	PROPN
ma-166	412	4	/	/	SYM
ma-166	412	5	ma.3.20	ma.3.20	PROPN
ma-166	412	6	14	14	NUM
ma-166	412	7	⇒	⇒	NOUN
ma-166	412	8	m2(g	m2(g	NOUN
ma-166	412	9	)	)	PUNCT
ma-166	412	10	=	=	NOUN
ma-166	412	11	54n	54n	PROPN
ma-166	412	12	.	.	PUNCT
ma-166	413	1	m2(g	m2(g	X
ma-166	413	2	)	)	PUNCT
ma-166	413	3	=	=	NUM
ma-166	413	4	nine	nine	NUM
ma-166	413	5	times	time	NOUN
ma-166	413	6	to	to	ADP
ma-166	413	7	edges	edge	NOUN
ma-166	413	8	of	of	ADP
ma-166	413	9	petersen	petersen	NOUN
ma-166	413	10	graph	graph	NOUN
ma-166	413	11	.	.	PUNCT
ma-166	414	1	3	3	X
ma-166	414	2	.	.	X
ma-166	414	3	numerical	numerical	ADJ
ma-166	414	4	examples	example	NOUN
ma-166	414	5	every	every	DET
ma-166	414	6	petersen	petersen	NOUN
ma-166	414	7	graph	graph	NOUN
ma-166	414	8	is	be	AUX
ma-166	414	9	cyclic	cyclic	ADJ
ma-166	414	10	graph	graph	NOUN
ma-166	414	11	and	and	CCONJ
ma-166	414	12	the	the	DET
ma-166	414	13	graph	graph	NOUN
ma-166	414	14	g′	g′	NOUN
ma-166	414	15	in	in	ADP
ma-166	414	16	general	general	ADJ
ma-166	414	17	form	form	NOUN
ma-166	414	18	consists	consist	VERB
ma-166	414	19	v	v	ADP
ma-166	414	20	having	having	AUX
ma-166	414	21	set	set	VERB
ma-166	414	22	ofvertex	ofvertex	NOUN
ma-166	414	23	and	and	CCONJ
ma-166	414	24	e	e	NOUN
ma-166	414	25	having	having	AUX
ma-166	414	26	set	set	VERB
ma-166	414	27	of	of	ADP
ma-166	414	28	edge	edge	NOUN
ma-166	414	29	,	,	PUNCT
ma-166	414	30	if	if	SCONJ
ma-166	414	31	the	the	DET
ma-166	414	32	natural	natural	ADJ
ma-166	414	33	number	number	NOUN
ma-166	414	34	,	,	PUNCT
ma-166	414	35	there	there	PRON
ma-166	414	36	exist	exist	VERB
ma-166	414	37	n	n	PRON
ma-166	414	38	the	the	DET
ma-166	414	39	graph	graph	NOUN
ma-166	414	40	with	with	ADP
ma-166	414	41	vertices	vertex	NOUN
ma-166	414	42	are	be	AUX
ma-166	414	43	v	v	ADJ
ma-166	414	44	(	(	PUNCT
ma-166	414	45	g′	g′	NOUN
ma-166	414	46	)	)	PUNCT
ma-166	415	1	=	=	SYM
ma-166	415	2	4n	4n	NOUN
ma-166	415	3	,	,	PUNCT
ma-166	415	4	edges	edge	NOUN
ma-166	415	5	are	be	AUX
ma-166	415	6	e(g′	e(g′	PRON
ma-166	415	7	)	)	PUNCT
ma-166	416	1	=	=	SYM
ma-166	416	2	6n	6n	NOUN
ma-166	416	3	,	,	PUNCT
ma-166	416	4	and	and	CCONJ
ma-166	416	5	the	the	DET
ma-166	416	6	specific	specific	NOUN
ma-166	416	7	of	of	ADP
ma-166	416	8	this	this	DET
ma-166	416	9	graph	graph	NOUN
ma-166	416	10	is	be	AUX
ma-166	416	11	that	that	SCONJ
ma-166	416	12	about	about	ADP
ma-166	416	13	degree	degree	NOUN
ma-166	416	14	of	of	ADP
ma-166	416	15	everyeach	everyeach	ADJ
ma-166	416	16	vertex	vertex	NOUN
ma-166	416	17	is	be	AUX
ma-166	416	18	p(k	p(k	NOUN
ma-166	416	19	,	,	PUNCT
ma-166	416	20	t	t	NOUN
ma-166	416	21	)	)	PUNCT
ma-166	416	22	=	=	PUNCT
ma-166	417	1	[	[	X
ma-166	417	2	d(x1	d(x1	NOUN
ma-166	417	3	)	)	PUNCT
ma-166	417	4	,	,	PUNCT
ma-166	417	5	d(x2	d(x2	NOUN
ma-166	417	6	)	)	PUNCT
ma-166	417	7	]	]	PUNCT
ma-166	418	1	=	=	PUNCT
ma-166	418	2	3	3	X
ma-166	418	3	.	.	PUNCT
ma-166	418	4	then	then	ADV
ma-166	418	5	this	this	DET
ma-166	418	6	graphic	graphic	NOUN
ma-166	418	7	which	which	PRON
ma-166	418	8	is	be	AUX
ma-166	418	9	said	say	VERB
ma-166	418	10	to	to	PART
ma-166	418	11	be	be	AUX
ma-166	418	12	petersen	petersen	PROPN
ma-166	418	13	graphic.then	graphic.then	PROPN
ma-166	418	14	petersen	petersen	PROPN
ma-166	418	15	graphic	graphic	NOUN
ma-166	418	16	is	be	AUX
ma-166	418	17	denoted	denote	VERB
ma-166	418	18	by	by	ADP
ma-166	418	19	p[v	p[v	PROPN
ma-166	418	20	(	(	PUNCT
ma-166	418	21	g′),e(g′	g′),e(g′	PROPN
ma-166	418	22	)	)	PUNCT
ma-166	418	23	]	]	PUNCT
ma-166	419	1	=	=	PUNCT
ma-166	419	2	(	(	PUNCT
ma-166	419	3	4n	4n	X
ma-166	419	4	,	,	PUNCT
ma-166	419	5	6n)the	6n)the	DET
ma-166	419	6	core	core	NOUN
ma-166	419	7	features	feature	NOUN
ma-166	419	8	of	of	ADP
ma-166	419	9	petersen	petersen	PROPN
ma-166	419	10	map	map	NOUN
ma-166	419	11	explored	explore	VERB
ma-166	419	12	by	by	ADP
ma-166	419	13	length	length	NOUN
ma-166	419	14	in	in	ADP
ma-166	419	15	1985	1985	NUM
ma-166	419	16	.	.	PUNCT
ma-166	420	1	however	however	ADV
ma-166	420	2	,	,	PUNCT
ma-166	420	3	the	the	DET
ma-166	420	4	petersen	petersen	PROPN
ma-166	420	5	line	line	NOUN
ma-166	420	6	contin	contin	NOUN
ma-166	420	7	-	-	PUNCT
ma-166	420	8	uously	uously	ADV
ma-166	420	9	arise	arise	NOUN
ma-166	420	10	in	in	ADP
ma-166	420	11	literature	literature	NOUN
ma-166	420	12	of	of	ADP
ma-166	420	13	the	the	DET
ma-166	420	14	theoretical	theoretical	ADJ
ma-166	420	15	graphing	graphing	NOUN
ma-166	420	16	.	.	PUNCT
ma-166	421	1	by	by	ADP
ma-166	421	2	this	this	DET
ma-166	421	3	article	article	NOUN
ma-166	421	4	,	,	PUNCT
ma-166	421	5	we	we	PRON
ma-166	421	6	update	update	VERB
ma-166	421	7	previous	previous	ADJ
ma-166	421	8	analysisto	analysisto	NOUN
ma-166	421	9	introduce	introduce	VERB
ma-166	421	10	additionally	additionally	ADV
ma-166	421	11	fresh	fresh	ADJ
ma-166	421	12	findings	finding	NOUN
ma-166	421	13	on	on	ADP
ma-166	421	14	the	the	DET
ma-166	421	15	petersen	petersen	PROPN
ma-166	421	16	mapping.sylvester	mapping.sylvester	PROPN
ma-166	421	17	’s	’s	PART
ma-166	421	18	association	association	NOUN
ma-166	421	19	to	to	ADP
ma-166	421	20	graphs	graph	NOUN
ma-166	421	21	of	of	ADP
ma-166	421	22	invariants	invariant	NOUN
ma-166	421	23	and	and	CCONJ
ma-166	421	24	covariants	covariant	NOUN
ma-166	421	25	requires	require	VERB
ma-166	421	26	interpretation	interpretation	NOUN
ma-166	421	27	of	of	ADP
ma-166	421	28	principleof	principleof	ADJ
ma-166	421	29	invariants	invariant	NOUN
ma-166	421	30	in	in	ADP
ma-166	421	31	1880s	1880	NOUN
ma-166	421	32	.	.	PUNCT
ma-166	421	33	example	example	NOUN
ma-166	421	34	3.1	3.1	NUM
ma-166	421	35	.	.	PUNCT
ma-166	422	1	if	if	SCONJ
ma-166	422	2	there	there	PRON
ma-166	422	3	exists	exist	VERB
ma-166	422	4	n	n	PRON
ma-166	422	5	is	be	AUX
ma-166	422	6	positive	positive	ADJ
ma-166	422	7	natural	natural	ADJ
ma-166	422	8	number	number	NOUN
ma-166	422	9	then	then	ADV
ma-166	422	10	tn	tn	PROPN
ma-166	423	1	=	=	SYM
ma-166	423	2	{	{	PUNCT
ma-166	423	3	1	1	NUM
ma-166	423	4	,	,	PUNCT
ma-166	423	5	2	2	NUM
ma-166	423	6	,	,	PUNCT
ma-166	423	7	3	3	NUM
ma-166	423	8	,	,	PUNCT
ma-166	423	9	...	...	PUNCT
ma-166	423	10	}	}	PUNCT
ma-166	423	11	,	,	PUNCT
ma-166	423	12	so	so	ADV
ma-166	423	13	graphwith	graphwith	PROPN
ma-166	423	14	vertices	vertex	NOUN
ma-166	423	15	are	be	AUX
ma-166	423	16	v	v	ADJ
ma-166	423	17	(	(	PUNCT
ma-166	423	18	g′	g′	NOUN
ma-166	423	19	)	)	PUNCT
ma-166	423	20	=	=	PUNCT
ma-166	423	21	4n	4n	NOUN
ma-166	423	22	,	,	PUNCT
ma-166	423	23	and	and	CCONJ
ma-166	423	24	edges	edge	VERB
ma-166	423	25	e(g′	e(g′	PRON
ma-166	423	26	)	)	PUNCT
ma-166	424	1	=	=	SYM
ma-166	424	2	6n	6n	NOUN
ma-166	424	3	,	,	PUNCT
ma-166	424	4	in	in	ADP
ma-166	424	5	general	general	ADJ
ma-166	424	6	form	form	NOUN
ma-166	424	7	of	of	ADP
ma-166	424	8	petersen	petersen	PROPN
ma-166	424	9	expressed	express	VERB
ma-166	424	10	by	by	ADP
ma-166	424	11	p[v	p[v	PROPN
ma-166	424	12	(	(	PUNCT
ma-166	424	13	g′),e(g′	g′),e(g′	PROPN
ma-166	424	14	)	)	PUNCT
ma-166	424	15	]	]	PUNCT
ma-166	425	1	=	=	PUNCT
ma-166	425	2	(	(	PUNCT
ma-166	425	3	4n	4n	NOUN
ma-166	425	4	,	,	PUNCT
ma-166	425	5	6n	6n	PROPN
ma-166	425	6	)	)	PUNCT
ma-166	425	7	.	.	PUNCT
ma-166	426	1	this	this	DET
ma-166	426	2	graph	graph	NOUN
ma-166	426	3	having	have	VERB
ma-166	426	4	a	a	DET
ma-166	426	5	specification	specification	NOUN
ma-166	426	6	,	,	PUNCT
ma-166	426	7	that	that	DET
ma-166	426	8	degree	degree	NOUN
ma-166	426	9	of	of	ADP
ma-166	426	10	every	every	PRON
ma-166	426	11	each	each	DET
ma-166	426	12	vertex	vertex	NOUN
ma-166	426	13	is	be	AUX
ma-166	426	14	p(k	p(k	NOUN
ma-166	426	15	,	,	PUNCT
ma-166	426	16	t	t	NOUN
ma-166	426	17	)	)	PUNCT
ma-166	426	18	=	=	PUNCT
ma-166	427	1	[	[	X
ma-166	427	2	d(x1	d(x1	NOUN
ma-166	427	3	)	)	PUNCT
ma-166	427	4	,	,	PUNCT
ma-166	427	5	d(x2	d(x2	NOUN
ma-166	427	6	)	)	PUNCT
ma-166	427	7	]	]	PUNCT
ma-166	428	1	=	=	PUNCT
ma-166	428	2	3.now	3.now	INTJ
ma-166	428	3	we	we	PRON
ma-166	428	4	write	write	VERB
ma-166	428	5	;	;	PUNCT
ma-166	428	6	tn	tn	NOUN
ma-166	428	7	=	=	SYM
ma-166	428	8	1	1	NUM
ma-166	428	9	,	,	PUNCT
ma-166	428	10	2	2	NUM
ma-166	428	11	,	,	PUNCT
ma-166	428	12	3	3	NUM
ma-166	428	13	,	,	PUNCT
ma-166	428	14	....	....	PUNCT
ma-166	428	15	v	v	X
ma-166	428	16	(	(	PUNCT
ma-166	428	17	g′	g′	NOUN
ma-166	428	18	)	)	PUNCT
ma-166	428	19	=	=	SYM
ma-166	429	1	4n	4n	NOUN
ma-166	429	2	.........	.........	PUNCT
ma-166	429	3	(	(	PUNCT
ma-166	429	4	1	1	X
ma-166	429	5	)	)	PUNCT
ma-166	429	6	e(g′	e(g′	NUM
ma-166	429	7	)	)	PUNCT
ma-166	430	1	=	=	SYM
ma-166	430	2	6n	6n	NOUN
ma-166	430	3	.........	.........	PUNCT
ma-166	430	4	(	(	PUNCT
ma-166	430	5	2	2	X
ma-166	430	6	)	)	PUNCT
ma-166	430	7	k	k	NOUN
ma-166	430	8	=	=	PUNCT
ma-166	430	9	d(x1	d(x1	NOUN
ma-166	430	10	)	)	PUNCT
ma-166	430	11	=	=	SYM
ma-166	430	12	3	3	NUM
ma-166	430	13	t	t	NOUN
ma-166	430	14	=	=	PUNCT
ma-166	430	15	d(x2	d(x2	NOUN
ma-166	430	16	)	)	PUNCT
ma-166	431	1	=	=	PUNCT
ma-166	432	1	3put	3put	NUM
ma-166	432	2	n	n	NOUN
ma-166	432	3	=	=	SYM
ma-166	432	4	3	3	NUM
ma-166	432	5	in	in	ADP
ma-166	432	6	equations	equation	NOUN
ma-166	432	7	(	(	PUNCT
ma-166	432	8	1	1	NUM
ma-166	432	9	)	)	PUNCT
ma-166	432	10	and	and	CCONJ
ma-166	432	11	(	(	PUNCT
ma-166	432	12	2	2	X
ma-166	432	13	)	)	PUNCT
ma-166	432	14	and	and	CCONJ
ma-166	432	15	these	these	DET
ma-166	432	16	equations	equation	NOUN
ma-166	432	17	become	become	VERB
ma-166	432	18	;	;	PUNCT
ma-166	432	19	t3	t3	PROPN
ma-166	432	20	=	=	SYM
ma-166	432	21	3	3	NUM
ma-166	432	22	v	v	NOUN
ma-166	432	23	(	(	PUNCT
ma-166	432	24	g′	g′	NOUN
ma-166	432	25	)	)	PUNCT
ma-166	432	26	=	=	NOUN
ma-166	432	27	12	12	NUM
ma-166	432	28	e(g′	e(g′	NUM
ma-166	432	29	)	)	PUNCT
ma-166	432	30	=	=	SYM
ma-166	432	31	18	18	NUM
ma-166	432	32	p(k	p(k	NOUN
ma-166	432	33	,	,	PUNCT
ma-166	432	34	t	t	NOUN
ma-166	432	35	)	)	PUNCT
ma-166	432	36	=	=	PUNCT
ma-166	433	1	[	[	X
ma-166	433	2	d(x1	d(x1	NOUN
ma-166	433	3	)	)	PUNCT
ma-166	433	4	,	,	PUNCT
ma-166	433	5	d(x2	d(x2	NOUN
ma-166	433	6	)	)	PUNCT
ma-166	433	7	]	]	PUNCT
ma-166	434	1	=	=	SYM
ma-166	434	2	3	3	NUM
ma-166	434	3	p[v	p[v	NUM
ma-166	434	4	(	(	PUNCT
ma-166	434	5	g′),e(g′	g′),e(g′	PROPN
ma-166	434	6	)	)	PUNCT
ma-166	434	7	]	]	PUNCT
ma-166	435	1	=	=	PUNCT
ma-166	435	2	(	(	PUNCT
ma-166	435	3	12	12	NUM
ma-166	435	4	,	,	PUNCT
ma-166	435	5	18)now	18)now	NUM
ma-166	435	6	figure	figure	NOUN
ma-166	435	7	is	be	AUX
ma-166	435	8	;	;	PUNCT
ma-166	435	9	example	example	NOUN
ma-166	435	10	3.2	3.2	NUM
ma-166	435	11	.	.	PUNCT
ma-166	436	1	if	if	SCONJ
ma-166	436	2	there	there	PRON
ma-166	436	3	exists	exist	VERB
ma-166	436	4	n	n	PRON
ma-166	436	5	is	be	AUX
ma-166	436	6	positive	positive	ADJ
ma-166	436	7	natural	natural	ADJ
ma-166	436	8	number	number	NOUN
ma-166	436	9	then	then	ADV
ma-166	436	10	tn	tn	PROPN
ma-166	437	1	=	=	SYM
ma-166	437	2	{	{	PUNCT
ma-166	437	3	1	1	NUM
ma-166	437	4	,	,	PUNCT
ma-166	437	5	2	2	NUM
ma-166	437	6	,	,	PUNCT
ma-166	437	7	3	3	NUM
ma-166	437	8	,	,	PUNCT
ma-166	437	9	...	...	PUNCT
ma-166	437	10	}	}	PUNCT
ma-166	437	11	,	,	PUNCT
ma-166	437	12	so	so	ADV
ma-166	437	13	graphwith	graphwith	PROPN
ma-166	437	14	vertices	vertex	NOUN
ma-166	437	15	are	be	AUX
ma-166	437	16	v	v	ADJ
ma-166	437	17	(	(	PUNCT
ma-166	437	18	g′	g′	NOUN
ma-166	437	19	)	)	PUNCT
ma-166	437	20	=	=	PUNCT
ma-166	437	21	4n	4n	NOUN
ma-166	437	22	,	,	PUNCT
ma-166	437	23	and	and	CCONJ
ma-166	437	24	edges	edge	VERB
ma-166	437	25	e(g′	e(g′	PRON
ma-166	437	26	)	)	PUNCT
ma-166	438	1	=	=	SYM
ma-166	438	2	6n	6n	NOUN
ma-166	438	3	,	,	PUNCT
ma-166	438	4	in	in	ADP
ma-166	438	5	general	general	ADJ
ma-166	438	6	form	form	NOUN
ma-166	438	7	of	of	ADP
ma-166	438	8	petersen	petersen	PROPN
ma-166	438	9	expressed	express	VERB
ma-166	438	10	by	by	ADP
ma-166	438	11	p[v	p[v	PROPN
ma-166	438	12	(	(	PUNCT
ma-166	438	13	g′),e(g′	g′),e(g′	PROPN
ma-166	438	14	)	)	PUNCT
ma-166	438	15	]	]	PUNCT
ma-166	439	1	=	=	PUNCT
ma-166	439	2	(	(	PUNCT
ma-166	439	3	4n	4n	NOUN
ma-166	439	4	,	,	PUNCT
ma-166	439	5	6n	6n	PROPN
ma-166	439	6	)	)	PUNCT
ma-166	439	7	.	.	PUNCT
ma-166	440	1	this	this	DET
ma-166	440	2	graph	graph	NOUN
ma-166	440	3	having	have	VERB
ma-166	440	4	a	a	DET
ma-166	440	5	specification	specification	NOUN
ma-166	440	6	,	,	PUNCT
ma-166	440	7	that	that	DET
ma-166	440	8	degree	degree	NOUN
ma-166	440	9	of	of	ADP
ma-166	440	10	every	every	PRON
ma-166	440	11	each	each	DET
ma-166	440	12	vertex	vertex	NOUN
ma-166	440	13	is	be	AUX
ma-166	440	14	p(k	p(k	NOUN
ma-166	440	15	,	,	PUNCT
ma-166	440	16	t	t	NOUN
ma-166	440	17	)	)	PUNCT
ma-166	440	18	=	=	PUNCT
ma-166	441	1	[	[	X
ma-166	441	2	d(x1	d(x1	NOUN
ma-166	441	3	)	)	PUNCT
ma-166	441	4	,	,	PUNCT
ma-166	441	5	d(x2	d(x2	NOUN
ma-166	441	6	)	)	PUNCT
ma-166	441	7	]	]	PUNCT
ma-166	442	1	=	=	PUNCT
ma-166	442	2	3	3	X
ma-166	442	3	.	.	X
ma-166	442	4	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	NUM
ma-166	442	5	eur	eur	PROPN
ma-166	442	6	.	.	PUNCT
ma-166	443	1	j.	j.	PROPN
ma-166	443	2	math	math	PROPN
ma-166	443	3	.	.	PUNCT
ma-166	444	1	anal	anal	PROPN
ma-166	444	2	.	.	PUNCT
ma-166	445	1	10.28924	10.28924	NUM
ma-166	445	2	/	/	SYM
ma-166	445	3	ada	ada	PROPN
ma-166	445	4	/	/	SYM
ma-166	445	5	ma.3.20	ma.3.20	NOUN
ma-166	445	6	15	15	NUM
ma-166	445	7	figure	figure	NOUN
ma-166	445	8	1	1	NUM
ma-166	445	9	.	.	PUNCT
ma-166	446	1	petersen	petersen	PROPN
ma-166	446	2	graph	graph	NOUN
ma-166	446	3	now	now	ADV
ma-166	446	4	we	we	PRON
ma-166	446	5	write	write	VERB
ma-166	446	6	;	;	PUNCT
ma-166	446	7	tn	tn	NOUN
ma-166	446	8	=	=	SYM
ma-166	446	9	1	1	NUM
ma-166	446	10	,	,	PUNCT
ma-166	446	11	2	2	NUM
ma-166	446	12	,	,	PUNCT
ma-166	446	13	3	3	NUM
ma-166	446	14	,	,	PUNCT
ma-166	446	15	....	....	PUNCT
ma-166	447	1	v	v	X
ma-166	447	2	(	(	PUNCT
ma-166	447	3	g′	g′	NOUN
ma-166	447	4	)	)	PUNCT
ma-166	447	5	=	=	SYM
ma-166	448	1	4n	4n	NOUN
ma-166	448	2	.........	.........	PUNCT
ma-166	448	3	(	(	PUNCT
ma-166	448	4	1	1	X
ma-166	448	5	)	)	PUNCT
ma-166	448	6	e(g′	e(g′	NUM
ma-166	448	7	)	)	PUNCT
ma-166	449	1	=	=	SYM
ma-166	449	2	6n	6n	NOUN
ma-166	449	3	.........	.........	PUNCT
ma-166	449	4	(	(	PUNCT
ma-166	449	5	2	2	X
ma-166	449	6	)	)	PUNCT
ma-166	449	7	k	k	NOUN
ma-166	449	8	=	=	PUNCT
ma-166	449	9	d(x1	d(x1	NOUN
ma-166	449	10	)	)	PUNCT
ma-166	449	11	=	=	SYM
ma-166	449	12	3	3	NUM
ma-166	449	13	t	t	NOUN
ma-166	449	14	=	=	PUNCT
ma-166	449	15	d(x2	d(x2	NOUN
ma-166	449	16	)	)	PUNCT
ma-166	450	1	=	=	PUNCT
ma-166	451	1	3put	3put	NUM
ma-166	451	2	n	n	NOUN
ma-166	451	3	=	=	SYM
ma-166	451	4	4	4	NUM
ma-166	451	5	in	in	ADP
ma-166	451	6	equations	equation	NOUN
ma-166	451	7	(	(	PUNCT
ma-166	451	8	1	1	NUM
ma-166	451	9	)	)	PUNCT
ma-166	451	10	and	and	CCONJ
ma-166	451	11	(	(	PUNCT
ma-166	451	12	2	2	X
ma-166	451	13	)	)	PUNCT
ma-166	451	14	and	and	CCONJ
ma-166	451	15	these	these	DET
ma-166	451	16	equations	equation	NOUN
ma-166	451	17	become	become	VERB
ma-166	451	18	;	;	PUNCT
ma-166	451	19	t4	t4	PROPN
ma-166	451	20	=	=	PROPN
ma-166	451	21	4	4	NUM
ma-166	451	22	v	v	NOUN
ma-166	451	23	(	(	PUNCT
ma-166	451	24	g′	g′	NOUN
ma-166	451	25	)	)	PUNCT
ma-166	451	26	=	=	NOUN
ma-166	451	27	16	16	NUM
ma-166	451	28	e(g′	e(g′	NUM
ma-166	451	29	)	)	PUNCT
ma-166	451	30	=	=	SYM
ma-166	451	31	24	24	NUM
ma-166	451	32	p(k	p(k	NOUN
ma-166	451	33	,	,	PUNCT
ma-166	451	34	t	t	NOUN
ma-166	451	35	)	)	PUNCT
ma-166	451	36	=	=	PUNCT
ma-166	452	1	[	[	X
ma-166	452	2	d(x1	d(x1	NOUN
ma-166	452	3	)	)	PUNCT
ma-166	452	4	,	,	PUNCT
ma-166	452	5	d(x2	d(x2	NOUN
ma-166	452	6	)	)	PUNCT
ma-166	452	7	]	]	PUNCT
ma-166	453	1	=	=	SYM
ma-166	453	2	3	3	NUM
ma-166	453	3	p[v	p[v	NUM
ma-166	453	4	(	(	PUNCT
ma-166	453	5	g′),e(g′	g′),e(g′	PROPN
ma-166	453	6	)	)	PUNCT
ma-166	453	7	]	]	PUNCT
ma-166	454	1	=	=	PUNCT
ma-166	454	2	(	(	PUNCT
ma-166	454	3	16	16	NUM
ma-166	454	4	,	,	PUNCT
ma-166	454	5	24)now	24)now	NUM
ma-166	454	6	figure	figure	NOUN
ma-166	454	7	is	be	AUX
ma-166	454	8	;	;	PUNCT
ma-166	454	9	figure	figure	VERB
ma-166	454	10	2	2	NUM
ma-166	454	11	.	.	PUNCT
ma-166	455	1	petersen	petersen	PROPN
ma-166	455	2	graph	graph	NOUN
ma-166	455	3	example	example	NOUN
ma-166	455	4	3.3	3.3	NUM
ma-166	455	5	.	.	PUNCT
ma-166	456	1	if	if	SCONJ
ma-166	456	2	there	there	PRON
ma-166	456	3	exists	exist	VERB
ma-166	456	4	n	n	PRON
ma-166	456	5	is	be	AUX
ma-166	456	6	positive	positive	ADJ
ma-166	456	7	natural	natural	ADJ
ma-166	456	8	number	number	NOUN
ma-166	456	9	then	then	ADV
ma-166	456	10	tn	tn	PROPN
ma-166	457	1	=	=	SYM
ma-166	457	2	{	{	PUNCT
ma-166	457	3	1	1	NUM
ma-166	457	4	,	,	PUNCT
ma-166	457	5	2	2	NUM
ma-166	457	6	,	,	PUNCT
ma-166	457	7	3	3	NUM
ma-166	457	8	,	,	PUNCT
ma-166	457	9	...	...	PUNCT
ma-166	457	10	}	}	PUNCT
ma-166	457	11	,	,	PUNCT
ma-166	457	12	so	so	ADV
ma-166	457	13	graphwith	graphwith	PROPN
ma-166	457	14	vertices	vertex	NOUN
ma-166	457	15	are	be	AUX
ma-166	457	16	v	v	ADJ
ma-166	457	17	(	(	PUNCT
ma-166	457	18	g′	g′	NOUN
ma-166	457	19	)	)	PUNCT
ma-166	457	20	=	=	PUNCT
ma-166	457	21	4n	4n	NOUN
ma-166	457	22	,	,	PUNCT
ma-166	457	23	and	and	CCONJ
ma-166	457	24	edges	edge	VERB
ma-166	457	25	e(g′	e(g′	PRON
ma-166	457	26	)	)	PUNCT
ma-166	458	1	=	=	SYM
ma-166	458	2	6n	6n	NOUN
ma-166	458	3	,	,	PUNCT
ma-166	458	4	in	in	ADP
ma-166	458	5	general	general	ADJ
ma-166	458	6	form	form	NOUN
ma-166	458	7	of	of	ADP
ma-166	458	8	petersen	petersen	PROPN
ma-166	458	9	expressed	express	VERB
ma-166	458	10	by	by	ADP
ma-166	458	11	p[v	p[v	PROPN
ma-166	458	12	(	(	PUNCT
ma-166	458	13	g′),e(g′	g′),e(g′	PROPN
ma-166	458	14	)	)	PUNCT
ma-166	458	15	]	]	PUNCT
ma-166	459	1	=	=	PUNCT
ma-166	459	2	(	(	PUNCT
ma-166	459	3	4n	4n	NOUN
ma-166	459	4	,	,	PUNCT
ma-166	459	5	6n	6n	PROPN
ma-166	459	6	)	)	PUNCT
ma-166	459	7	.	.	PUNCT
ma-166	460	1	this	this	DET
ma-166	460	2	graph	graph	NOUN
ma-166	460	3	having	have	VERB
ma-166	460	4	a	a	DET
ma-166	460	5	specification	specification	NOUN
ma-166	460	6	,	,	PUNCT
ma-166	460	7	that	that	DET
ma-166	460	8	degree	degree	NOUN
ma-166	460	9	of	of	ADP
ma-166	460	10	every	every	PRON
ma-166	460	11	each	each	DET
ma-166	460	12	vertex	vertex	NOUN
ma-166	460	13	is	be	AUX
ma-166	460	14	p(k	p(k	NOUN
ma-166	460	15	,	,	PUNCT
ma-166	460	16	t	t	NOUN
ma-166	460	17	)	)	PUNCT
ma-166	460	18	=	=	PUNCT
ma-166	461	1	[	[	X
ma-166	461	2	d(x1	d(x1	NOUN
ma-166	461	3	)	)	PUNCT
ma-166	461	4	,	,	PUNCT
ma-166	461	5	d(x2	d(x2	NOUN
ma-166	461	6	)	)	PUNCT
ma-166	461	7	]	]	PUNCT
ma-166	462	1	=	=	PUNCT
ma-166	462	2	3.now	3.now	INTJ
ma-166	462	3	we	we	PRON
ma-166	462	4	write	write	VERB
ma-166	462	5	;	;	PUNCT
ma-166	462	6	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	NUM
ma-166	462	7	eur	eur	PROPN
ma-166	462	8	.	.	PUNCT
ma-166	463	1	j.	j.	PROPN
ma-166	463	2	math	math	PROPN
ma-166	463	3	.	.	PUNCT
ma-166	464	1	anal	anal	PROPN
ma-166	464	2	.	.	PUNCT
ma-166	465	1	10.28924	10.28924	NUM
ma-166	465	2	/	/	SYM
ma-166	465	3	ada	ada	PROPN
ma-166	465	4	/	/	SYM
ma-166	465	5	ma.3.20	ma.3.20	PROPN
ma-166	465	6	16	16	NUM
ma-166	465	7	tn	tn	NOUN
ma-166	465	8	=	=	SYM
ma-166	465	9	1	1	NUM
ma-166	465	10	,	,	PUNCT
ma-166	465	11	2	2	NUM
ma-166	465	12	,	,	PUNCT
ma-166	465	13	3	3	NUM
ma-166	465	14	,	,	PUNCT
ma-166	465	15	....	....	PUNCT
ma-166	466	1	v	v	X
ma-166	466	2	(	(	PUNCT
ma-166	466	3	g′	g′	NOUN
ma-166	466	4	)	)	PUNCT
ma-166	466	5	=	=	SYM
ma-166	467	1	4n	4n	NOUN
ma-166	467	2	.........	.........	PUNCT
ma-166	467	3	(	(	PUNCT
ma-166	467	4	1	1	X
ma-166	467	5	)	)	PUNCT
ma-166	467	6	e(g′	e(g′	NUM
ma-166	467	7	)	)	PUNCT
ma-166	468	1	=	=	SYM
ma-166	468	2	6n	6n	NOUN
ma-166	468	3	.........	.........	PUNCT
ma-166	468	4	(	(	PUNCT
ma-166	468	5	2	2	X
ma-166	468	6	)	)	PUNCT
ma-166	468	7	k	k	NOUN
ma-166	468	8	=	=	PUNCT
ma-166	468	9	d(x1	d(x1	NOUN
ma-166	468	10	)	)	PUNCT
ma-166	468	11	=	=	SYM
ma-166	468	12	3	3	NUM
ma-166	468	13	t	t	NOUN
ma-166	468	14	=	=	PUNCT
ma-166	468	15	d(x2	d(x2	NOUN
ma-166	468	16	)	)	PUNCT
ma-166	469	1	=	=	PUNCT
ma-166	470	1	3put	3put	NUM
ma-166	470	2	n	n	NOUN
ma-166	470	3	=	=	SYM
ma-166	470	4	5	5	NUM
ma-166	470	5	in	in	ADP
ma-166	470	6	equations	equation	NOUN
ma-166	470	7	(	(	PUNCT
ma-166	470	8	1	1	NUM
ma-166	470	9	)	)	PUNCT
ma-166	470	10	and	and	CCONJ
ma-166	470	11	(	(	PUNCT
ma-166	470	12	2	2	X
ma-166	470	13	)	)	PUNCT
ma-166	470	14	and	and	CCONJ
ma-166	470	15	these	these	DET
ma-166	470	16	equations	equation	NOUN
ma-166	470	17	become	become	VERB
ma-166	470	18	;	;	PUNCT
ma-166	470	19	t5	t5	PROPN
ma-166	470	20	=	=	SYM
ma-166	470	21	5	5	NUM
ma-166	470	22	v	v	NOUN
ma-166	470	23	(	(	PUNCT
ma-166	470	24	g′	g′	NOUN
ma-166	470	25	)	)	PUNCT
ma-166	470	26	=	=	NOUN
ma-166	470	27	20	20	NUM
ma-166	470	28	e(g′	e(g′	NUM
ma-166	470	29	)	)	PUNCT
ma-166	470	30	=	=	SYM
ma-166	470	31	30	30	NUM
ma-166	470	32	p(k	p(k	NOUN
ma-166	470	33	,	,	PUNCT
ma-166	470	34	t	t	NOUN
ma-166	470	35	)	)	PUNCT
ma-166	470	36	=	=	PUNCT
ma-166	471	1	[	[	X
ma-166	471	2	d(x1	d(x1	NOUN
ma-166	471	3	)	)	PUNCT
ma-166	471	4	,	,	PUNCT
ma-166	471	5	d(x2	d(x2	NOUN
ma-166	471	6	)	)	PUNCT
ma-166	471	7	]	]	PUNCT
ma-166	472	1	=	=	SYM
ma-166	472	2	3	3	NUM
ma-166	472	3	p[v	p[v	NUM
ma-166	472	4	(	(	PUNCT
ma-166	472	5	g′),e(g′	g′),e(g′	PROPN
ma-166	472	6	)	)	PUNCT
ma-166	472	7	]	]	PUNCT
ma-166	473	1	=	=	PUNCT
ma-166	473	2	(	(	PUNCT
ma-166	473	3	20	20	NUM
ma-166	473	4	,	,	PUNCT
ma-166	473	5	30)now	30)now	NUM
ma-166	473	6	figure	figure	NOUN
ma-166	473	7	is	be	AUX
ma-166	473	8	;	;	PUNCT
ma-166	473	9	figure	figure	VERB
ma-166	473	10	3	3	NUM
ma-166	473	11	.	.	PUNCT
ma-166	474	1	petersen	petersen	PROPN
ma-166	474	2	graph	graph	NOUN
ma-166	474	3	example	example	NOUN
ma-166	474	4	3.4	3.4	NUM
ma-166	474	5	.	.	PUNCT
ma-166	475	1	if	if	SCONJ
ma-166	475	2	there	there	PRON
ma-166	475	3	exists	exist	VERB
ma-166	475	4	n	n	PRON
ma-166	475	5	is	be	AUX
ma-166	475	6	positive	positive	ADJ
ma-166	475	7	natural	natural	ADJ
ma-166	475	8	number	number	NOUN
ma-166	475	9	then	then	ADV
ma-166	475	10	tn	tn	PROPN
ma-166	476	1	=	=	SYM
ma-166	476	2	{	{	PUNCT
ma-166	476	3	1	1	NUM
ma-166	476	4	,	,	PUNCT
ma-166	476	5	2	2	NUM
ma-166	476	6	,	,	PUNCT
ma-166	476	7	3	3	NUM
ma-166	476	8	,	,	PUNCT
ma-166	476	9	...	...	PUNCT
ma-166	476	10	}	}	PUNCT
ma-166	476	11	,	,	PUNCT
ma-166	476	12	so	so	ADV
ma-166	476	13	graphwith	graphwith	PROPN
ma-166	476	14	vertices	vertex	NOUN
ma-166	476	15	are	be	AUX
ma-166	476	16	v	v	ADJ
ma-166	476	17	(	(	PUNCT
ma-166	476	18	g′	g′	NOUN
ma-166	476	19	)	)	PUNCT
ma-166	476	20	=	=	PUNCT
ma-166	476	21	4n	4n	NOUN
ma-166	476	22	,	,	PUNCT
ma-166	476	23	and	and	CCONJ
ma-166	476	24	edges	edge	VERB
ma-166	476	25	e(g′	e(g′	PRON
ma-166	476	26	)	)	PUNCT
ma-166	477	1	=	=	SYM
ma-166	477	2	6n	6n	NOUN
ma-166	477	3	,	,	PUNCT
ma-166	477	4	in	in	ADP
ma-166	477	5	general	general	ADJ
ma-166	477	6	form	form	NOUN
ma-166	477	7	of	of	ADP
ma-166	477	8	petersen	petersen	PROPN
ma-166	477	9	expressed	express	VERB
ma-166	477	10	by	by	ADP
ma-166	477	11	p[v	p[v	PROPN
ma-166	477	12	(	(	PUNCT
ma-166	477	13	g′),e(g′	g′),e(g′	PROPN
ma-166	477	14	)	)	PUNCT
ma-166	477	15	]	]	PUNCT
ma-166	478	1	=	=	PUNCT
ma-166	478	2	(	(	PUNCT
ma-166	478	3	4n	4n	NOUN
ma-166	478	4	,	,	PUNCT
ma-166	478	5	6n	6n	PROPN
ma-166	478	6	)	)	PUNCT
ma-166	478	7	.	.	PUNCT
ma-166	479	1	this	this	DET
ma-166	479	2	graph	graph	NOUN
ma-166	479	3	having	have	VERB
ma-166	479	4	a	a	DET
ma-166	479	5	specification	specification	NOUN
ma-166	479	6	,	,	PUNCT
ma-166	479	7	that	that	DET
ma-166	479	8	degree	degree	NOUN
ma-166	479	9	of	of	ADP
ma-166	479	10	every	every	PRON
ma-166	479	11	each	each	DET
ma-166	479	12	vertex	vertex	NOUN
ma-166	479	13	is	be	AUX
ma-166	479	14	p(k	p(k	NOUN
ma-166	479	15	,	,	PUNCT
ma-166	479	16	t	t	NOUN
ma-166	479	17	)	)	PUNCT
ma-166	479	18	=	=	PUNCT
ma-166	480	1	[	[	X
ma-166	480	2	d(x1	d(x1	NOUN
ma-166	480	3	)	)	PUNCT
ma-166	480	4	,	,	PUNCT
ma-166	480	5	d(x2	d(x2	NOUN
ma-166	480	6	)	)	PUNCT
ma-166	480	7	]	]	PUNCT
ma-166	481	1	=	=	PUNCT
ma-166	481	2	3.now	3.now	INTJ
ma-166	481	3	we	we	PRON
ma-166	481	4	write	write	VERB
ma-166	481	5	;	;	PUNCT
ma-166	481	6	tn	tn	NOUN
ma-166	481	7	=	=	SYM
ma-166	481	8	1	1	NUM
ma-166	481	9	,	,	PUNCT
ma-166	481	10	2	2	NUM
ma-166	481	11	,	,	PUNCT
ma-166	481	12	3	3	NUM
ma-166	481	13	,	,	PUNCT
ma-166	481	14	....	....	PUNCT
ma-166	481	15	v	v	X
ma-166	481	16	(	(	PUNCT
ma-166	481	17	g′	g′	NOUN
ma-166	481	18	)	)	PUNCT
ma-166	481	19	=	=	SYM
ma-166	482	1	4n	4n	NOUN
ma-166	482	2	.........	.........	PUNCT
ma-166	482	3	(	(	PUNCT
ma-166	482	4	1	1	X
ma-166	482	5	)	)	PUNCT
ma-166	482	6	e(g′	e(g′	NUM
ma-166	482	7	)	)	PUNCT
ma-166	483	1	=	=	SYM
ma-166	483	2	6n	6n	NOUN
ma-166	483	3	.........	.........	PUNCT
ma-166	483	4	(	(	PUNCT
ma-166	483	5	2	2	X
ma-166	483	6	)	)	PUNCT
ma-166	483	7	k	k	NOUN
ma-166	483	8	=	=	PUNCT
ma-166	483	9	d(x1	d(x1	NOUN
ma-166	483	10	)	)	PUNCT
ma-166	483	11	=	=	SYM
ma-166	483	12	3	3	NUM
ma-166	483	13	t	t	NOUN
ma-166	483	14	=	=	PUNCT
ma-166	483	15	d(x2	d(x2	NOUN
ma-166	483	16	)	)	PUNCT
ma-166	484	1	=	=	PUNCT
ma-166	485	1	3put	3put	NUM
ma-166	485	2	n	n	NOUN
ma-166	485	3	=	=	SYM
ma-166	485	4	10	10	NUM
ma-166	485	5	in	in	ADP
ma-166	485	6	equations	equation	NOUN
ma-166	485	7	(	(	PUNCT
ma-166	485	8	1	1	NUM
ma-166	485	9	)	)	PUNCT
ma-166	485	10	and	and	CCONJ
ma-166	485	11	(	(	PUNCT
ma-166	485	12	2	2	X
ma-166	485	13	)	)	PUNCT
ma-166	485	14	and	and	CCONJ
ma-166	485	15	these	these	DET
ma-166	485	16	equations	equation	NOUN
ma-166	485	17	become	become	VERB
ma-166	485	18	;	;	PUNCT
ma-166	485	19	t10	t10	NUM
ma-166	485	20	=	=	SYM
ma-166	485	21	10	10	NUM
ma-166	485	22	v	v	NOUN
ma-166	485	23	(	(	PUNCT
ma-166	485	24	g′	g′	NOUN
ma-166	485	25	)	)	PUNCT
ma-166	485	26	=	=	SYM
ma-166	486	1	40	40	NUM
ma-166	486	2	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	NUM
ma-166	486	3	eur	eur	PROPN
ma-166	486	4	.	.	PUNCT
ma-166	487	1	j.	j.	PROPN
ma-166	487	2	math	math	PROPN
ma-166	487	3	.	.	PUNCT
ma-166	488	1	anal	anal	PROPN
ma-166	488	2	.	.	PUNCT
ma-166	489	1	10.28924	10.28924	NUM
ma-166	489	2	/	/	SYM
ma-166	489	3	ada	ada	PROPN
ma-166	489	4	/	/	SYM
ma-166	489	5	ma.3.20	ma.3.20	PROPN
ma-166	489	6	17	17	NUM
ma-166	489	7	e(g′	e(g′	NUM
ma-166	489	8	)	)	PUNCT
ma-166	490	1	=	=	SYM
ma-166	490	2	60	60	NUM
ma-166	490	3	p(k	p(k	NOUN
ma-166	490	4	,	,	PUNCT
ma-166	490	5	t	t	NOUN
ma-166	490	6	)	)	PUNCT
ma-166	490	7	=	=	PUNCT
ma-166	491	1	[	[	X
ma-166	491	2	d(x1	d(x1	NOUN
ma-166	491	3	)	)	PUNCT
ma-166	491	4	,	,	PUNCT
ma-166	491	5	d(x2	d(x2	NOUN
ma-166	491	6	)	)	PUNCT
ma-166	491	7	]	]	PUNCT
ma-166	492	1	=	=	SYM
ma-166	492	2	3	3	NUM
ma-166	492	3	p[v	p[v	NUM
ma-166	492	4	(	(	PUNCT
ma-166	492	5	g′),e(g′	g′),e(g′	PROPN
ma-166	492	6	)	)	PUNCT
ma-166	492	7	]	]	PUNCT
ma-166	493	1	=	=	PUNCT
ma-166	493	2	(	(	PUNCT
ma-166	493	3	40	40	NUM
ma-166	493	4	,	,	PUNCT
ma-166	493	5	60)now	60)now	NUM
ma-166	493	6	figures	figure	NOUN
ma-166	493	7	are	be	AUX
ma-166	493	8	;	;	PUNCT
ma-166	493	9	figure	figure	VERB
ma-166	493	10	4	4	NUM
ma-166	493	11	.	.	PUNCT
ma-166	494	1	petersen	petersen	PROPN
ma-166	494	2	graph	graph	NOUN
ma-166	494	3	4	4	NUM
ma-166	494	4	.	.	NOUN
ma-166	494	5	conclusion	conclusion	NOUN
ma-166	494	6	and	and	CCONJ
ma-166	494	7	future	future	ADJ
ma-166	494	8	studies	study	NOUN
ma-166	494	9	frequently	frequently	ADV
ma-166	494	10	,	,	PUNCT
ma-166	494	11	graph	graph	NOUN
ma-166	494	12	theory	theory	NOUN
ma-166	494	13	is	be	AUX
ma-166	494	14	refuted	refute	VERB
ma-166	494	15	using	use	VERB
ma-166	494	16	the	the	DET
ma-166	494	17	petersen	petersen	NOUN
ma-166	494	18	graph	graph	NOUN
ma-166	494	19	.	.	PUNCT
ma-166	495	1	in	in	ADP
ma-166	495	2	this	this	DET
ma-166	495	3	paper	paper	NOUN
ma-166	495	4	,	,	PUNCT
ma-166	495	5	the	the	DET
ma-166	495	6	general	general	ADJ
ma-166	495	7	petersengraph	petersengraph	NOUN
ma-166	495	8	was	be	AUX
ma-166	495	9	constructed	construct	VERB
ma-166	495	10	,	,	PUNCT
ma-166	495	11	and	and	CCONJ
ma-166	495	12	the	the	DET
ma-166	495	13	exact	exact	ADJ
ma-166	495	14	expressions	expression	NOUN
ma-166	495	15	of	of	ADP
ma-166	495	16	the	the	DET
ma-166	495	17	first	first	ADJ
ma-166	495	18	and	and	CCONJ
ma-166	495	19	second	second	ADJ
ma-166	495	20	zagreb	zagreb	PROPN
ma-166	495	21	indices	index	NOUN
ma-166	495	22	,	,	PUNCT
ma-166	495	23	the	the	DET
ma-166	495	24	forgottentopological	forgottentopological	ADJ
ma-166	495	25	index	index	NOUN
ma-166	495	26	,	,	PUNCT
ma-166	495	27	the	the	DET
ma-166	495	28	hyper	hyper	PROPN
ma-166	495	29	zagreb	zagreb	PROPN
ma-166	495	30	index	index	PROPN
ma-166	495	31	,	,	PUNCT
ma-166	495	32	the	the	DET
ma-166	495	33	reduced	reduced	ADJ
ma-166	495	34	second	second	ADJ
ma-166	495	35	zagreb	zagreb	PROPN
ma-166	495	36	index	index	NOUN
ma-166	495	37	and	and	CCONJ
ma-166	495	38	the	the	DET
ma-166	495	39	petersen	petersen	PROPN
ma-166	495	40	graphin	graphin	PROPN
ma-166	495	41	terms	term	NOUN
ma-166	495	42	of	of	ADP
ma-166	495	43	cyclic	cyclic	ADJ
ma-166	495	44	graph	graph	NOUN
ma-166	495	45	were	be	AUX
ma-166	495	46	then	then	ADV
ma-166	495	47	examined	examine	VERB
ma-166	495	48	.	.	PUNCT
ma-166	496	1	the	the	DET
ma-166	496	2	future	future	ADJ
ma-166	496	3	work	work	NOUN
ma-166	496	4	will	will	AUX
ma-166	496	5	concentrate	concentrate	VERB
ma-166	496	6	on	on	ADP
ma-166	496	7	topologicalindeces	topologicalindece	NOUN
ma-166	496	8	,	,	PUNCT
ma-166	496	9	then	then	ADV
ma-166	496	10	generalised	generalise	VERB
ma-166	496	11	petersen	petersen	NOUN
ma-166	496	12	via	via	ADP
ma-166	496	13	graph	graph	NOUN
ma-166	496	14	operations	operation	NOUN
ma-166	496	15	.	.	PUNCT
ma-166	497	1	references	reference	NOUN
ma-166	497	2	[	[	X
ma-166	497	3	1	1	X
ma-166	497	4	]	]	X
ma-166	497	5	m.h	m.h	PROPN
ma-166	497	6	.	.	PROPN
ma-166	497	7	khalifeh	khalifeh	PROPN
ma-166	497	8	,	,	PUNCT
ma-166	497	9	h.	h.	PROPN
ma-166	497	10	yousefi	yousefi	PROPN
ma-166	497	11	-	-	PUNCT
ma-166	497	12	azari	azari	PROPN
ma-166	497	13	,	,	PUNCT
ma-166	497	14	a.r	a.r	PROPN
ma-166	497	15	.	.	PROPN
ma-166	497	16	ashrafi	ashrafi	PROPN
ma-166	497	17	,	,	PUNCT
ma-166	497	18	the	the	DET
ma-166	497	19	first	first	ADJ
ma-166	497	20	and	and	CCONJ
ma-166	497	21	second	second	ADJ
ma-166	497	22	zagreb	zagreb	PROPN
ma-166	497	23	indices	index	NOUN
ma-166	497	24	of	of	ADP
ma-166	497	25	some	some	DET
ma-166	497	26	graph	graph	NOUN
ma-166	497	27	operations	operation	NOUN
ma-166	497	28	,	,	PUNCT
ma-166	497	29	discr.appl	discr.appl	NOUN
ma-166	497	30	.	.	PUNCT
ma-166	497	31	math	math	NOUN
ma-166	497	32	.	.	PUNCT
ma-166	498	1	157	157	NUM
ma-166	498	2	(	(	PUNCT
ma-166	498	3	2009	2009	NUM
ma-166	498	4	)	)	PUNCT
ma-166	498	5	804	804	NUM
ma-166	498	6	-	-	SYM
ma-166	498	7	811	811	NUM
ma-166	498	8	.	.	PUNCT
ma-166	499	1	https://doi.org/10.1016/j.dam.2008.06.015.[2	https://doi.org/10.1016/j.dam.2008.06.015.[2	X
ma-166	499	2	]	]	X
ma-166	500	1	v.	v.	CCONJ
ma-166	500	2	anandkumar	anandkumar	PROPN
ma-166	500	3	,	,	PUNCT
ma-166	500	4	r.r	r.r	PROPN
ma-166	500	5	.	.	PROPN
ma-166	500	6	iyer	iyer	PROPN
ma-166	500	7	,	,	PUNCT
ma-166	500	8	on	on	ADP
ma-166	500	9	the	the	DET
ma-166	500	10	hyper	hyper	ADJ
ma-166	500	11	-	-	ADJ
ma-166	500	12	zagreb	zagreb	PROPN
ma-166	500	13	index	index	NOUN
ma-166	500	14	of	of	ADP
ma-166	500	15	some	some	DET
ma-166	500	16	operations	operation	NOUN
ma-166	500	17	on	on	ADP
ma-166	500	18	graphs	graph	NOUN
ma-166	500	19	,	,	PUNCT
ma-166	500	20	int	int	NOUN
ma-166	500	21	.	.	PUNCT
ma-166	501	1	j.	j.	PROPN
ma-166	501	2	pure	pure	PROPN
ma-166	501	3	appl	appl	PROPN
ma-166	501	4	.	.	PUNCT
ma-166	501	5	math	math	NOUN
ma-166	501	6	.	.	PUNCT
ma-166	502	1	112(2017	112(2017	NUM
ma-166	502	2	)	)	PUNCT
ma-166	502	3	213	213	NUM
ma-166	502	4	-	-	SYM
ma-166	502	5	220	220	NUM
ma-166	502	6	.	.	PUNCT
ma-166	503	1	https://doi.org/10.12732/ijpam.v112i2.2.[3	https://doi.org/10.12732/ijpam.v112i2.2.[3	PRON
ma-166	503	2	]	]	X
ma-166	503	3	s.m	s.m	PROPN
ma-166	503	4	.	.	PROPN
ma-166	503	5	sankarraman	sankarraman	PROPN
ma-166	503	6	,	,	PUNCT
ma-166	503	7	a	a	DET
ma-166	503	8	computational	computational	ADJ
ma-166	503	9	approach	approach	NOUN
ma-166	503	10	on	on	ADP
ma-166	503	11	acetaminophen	acetaminophen	NOUN
ma-166	503	12	drug	drug	NOUN
ma-166	503	13	using	use	VERB
ma-166	503	14	degree	degree	NOUN
ma-166	503	15	-	-	PUNCT
ma-166	503	16	based	base	VERB
ma-166	503	17	topological	topological	ADJ
ma-166	503	18	indices	index	NOUN
ma-166	503	19	andm	andm	NOUN
ma-166	503	20	-	-	PUNCT
ma-166	503	21	polynomials	polynomials	PROPN
ma-166	503	22	,	,	PUNCT
ma-166	503	23	biointerface	biointerface	NOUN
ma-166	503	24	res	re	NOUN
ma-166	503	25	.	.	PUNCT
ma-166	504	1	appl	appl	PROPN
ma-166	504	2	.	.	PUNCT
ma-166	505	1	chem	chem	PROPN
ma-166	505	2	.	.	PUNCT
ma-166	506	1	12	12	NUM
ma-166	506	2	(	(	PUNCT
ma-166	506	3	2021	2021	NUM
ma-166	506	4	)	)	PUNCT
ma-166	506	5	7249	7249	NUM
ma-166	506	6	-	-	SYM
ma-166	506	7	7266	7266	NUM
ma-166	506	8	.	.	PUNCT
ma-166	507	1	https://doi.org/10.33263/briac126	https://doi.org/10.33263/briac126	NOUN
ma-166	507	2	.	.	PUNCT
ma-166	508	1	72497266.[4	72497266.[4	NUM
ma-166	508	2	]	]	X
ma-166	508	3	i.	i.	PROPN
ma-166	508	4	gutman	gutman	PROPN
ma-166	508	5	,	,	PUNCT
ma-166	508	6	multiplicative	multiplicative	PROPN
ma-166	508	7	zagreb	zagreb	PROPN
ma-166	508	8	indices	index	NOUN
ma-166	508	9	of	of	ADP
ma-166	508	10	trees	tree	NOUN
ma-166	508	11	,	,	PUNCT
ma-166	508	12	bull	bull	NOUN
ma-166	508	13	.	.	PUNCT
ma-166	509	1	soc	soc	PROPN
ma-166	509	2	.	.	PUNCT
ma-166	510	1	math	math	PROPN
ma-166	510	2	.	.	PUNCT
ma-166	511	1	banja	banja	PROPN
ma-166	511	2	luka	luka	PROPN
ma-166	511	3	.	.	PUNCT
ma-166	512	1	18	18	NUM
ma-166	512	2	(	(	PUNCT
ma-166	512	3	2011	2011	NUM
ma-166	512	4	)	)	PUNCT
ma-166	512	5	17	17	NUM
ma-166	512	6	-	-	SYM
ma-166	512	7	23.[5	23.[5	NUM
ma-166	512	8	]	]	X
ma-166	512	9	e.	e.	PROPN
ma-166	512	10	estrada	estrada	PROPN
ma-166	512	11	,	,	PUNCT
ma-166	512	12	l.	l.	PROPN
ma-166	512	13	torres	torres	PROPN
ma-166	512	14	,	,	PUNCT
ma-166	512	15	l.	l.	PROPN
ma-166	512	16	rodriguez	rodriguez	PROPN
ma-166	512	17	,	,	PUNCT
ma-166	512	18	i.	i.	PROPN
ma-166	512	19	gutman	gutman	PROPN
ma-166	512	20	,	,	PUNCT
ma-166	512	21	an	an	DET
ma-166	512	22	atom	atom	NOUN
ma-166	512	23	-	-	PUNCT
ma-166	512	24	bond	bond	NOUN
ma-166	512	25	connectivity	connectivity	NOUN
ma-166	512	26	index	index	NOUN
ma-166	512	27	:	:	PUNCT
ma-166	512	28	modelling	model	VERB
ma-166	512	29	the	the	DET
ma-166	512	30	enthalpy	enthalpy	NOUN
ma-166	512	31	of	of	ADP
ma-166	512	32	formationof	formationof	NOUN
ma-166	512	33	alkanes	alkane	NOUN
ma-166	512	34	,	,	PUNCT
ma-166	512	35	indian	indian	ADJ
ma-166	512	36	j.	j.	PROPN
ma-166	512	37	chem	chem	PROPN
ma-166	512	38	.	.	PUNCT
ma-166	513	1	37	37	NUM
ma-166	513	2	(	(	PUNCT
ma-166	513	3	1998	1998	NUM
ma-166	513	4	)	)	PUNCT
ma-166	513	5	849	849	NUM
ma-166	513	6	-	-	SYM
ma-166	513	7	855.[6	855.[6	NUM
ma-166	513	8	]	]	PUNCT
ma-166	513	9	x.	x.	NOUN
ma-166	513	10	ren	ren	PROPN
ma-166	513	11	,	,	PUNCT
ma-166	513	12	x.	x.	PROPN
ma-166	513	13	hu	hu	PROPN
ma-166	513	14	,	,	PUNCT
ma-166	513	15	b.	b.	PROPN
ma-166	513	16	zhao	zhao	PROPN
ma-166	513	17	,	,	PUNCT
ma-166	513	18	proving	prove	VERB
ma-166	513	19	a	a	DET
ma-166	513	20	conjecture	conjecture	NOUN
ma-166	513	21	concerning	concern	VERB
ma-166	513	22	trees	tree	NOUN
ma-166	513	23	with	with	ADP
ma-166	513	24	maximal	maximal	ADJ
ma-166	513	25	reduced	reduce	VERB
ma-166	513	26	reciprocal	reciprocal	ADJ
ma-166	513	27	randic	randic	ADJ
ma-166	513	28	index	index	NOUN
ma-166	513	29	,	,	PUNCT
ma-166	513	30	matchcommun	matchcommun	PROPN
ma-166	513	31	.	.	PUNCT
ma-166	513	32	math	math	NOUN
ma-166	513	33	.	.	PUNCT
ma-166	514	1	comput	comput	NOUN
ma-166	514	2	.	.	PUNCT
ma-166	515	1	chem	chem	NOUN
ma-166	515	2	.	.	PUNCT
ma-166	516	1	76	76	NUM
ma-166	516	2	(	(	PUNCT
ma-166	516	3	2016	2016	NUM
ma-166	516	4	)	)	PUNCT
ma-166	516	5	171	171	NUM
ma-166	516	6	-	-	SYM
ma-166	516	7	184.[7	184.[7	NUM
ma-166	516	8	]	]	X
ma-166	516	9	a.r	a.r	PROPN
ma-166	516	10	.	.	PROPN
ma-166	516	11	bindusree	bindusree	PROPN
ma-166	516	12	,	,	PUNCT
ma-166	516	13	n.	n.	PROPN
ma-166	516	14	cangul	cangul	PROPN
ma-166	516	15	i.	i.	PROPN
ma-166	516	16	,	,	PUNCT
ma-166	516	17	v.	v.	ADP
ma-166	516	18	lokesha	lokesha	PROPN
ma-166	516	19	,	,	PUNCT
ma-166	516	20	s.	s.	PROPN
ma-166	516	21	cevik	cevik	PROPN
ma-166	516	22	a.	a.	PROPN
ma-166	516	23	,	,	PUNCT
ma-166	516	24	zagreb	zagreb	PROPN
ma-166	516	25	polynomials	polynomial	NOUN
ma-166	516	26	of	of	ADP
ma-166	516	27	three	three	NUM
ma-166	516	28	graph	graph	NOUN
ma-166	516	29	operators	operator	NOUN
ma-166	516	30	,	,	PUNCT
ma-166	516	31	filomat	filomat	NOUN
ma-166	516	32	.	.	PUNCT
ma-166	517	1	30(2016	30(2016	NUM
ma-166	517	2	)	)	PUNCT
ma-166	517	3	1979	1979	NUM
ma-166	517	4	-	-	SYM
ma-166	517	5	1986	1986	NUM
ma-166	517	6	.	.	PUNCT
ma-166	518	1	https://doi.org/10.2298/fil1607979b	https://doi.org/10.2298/fil1607979b	PROPN
ma-166	518	2	.	.	PUNCT
ma-166	519	1	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	PROPN
ma-166	519	2	https://doi.org/10.1016/j.dam.2008.06.015	https://doi.org/10.1016/j.dam.2008.06.015	PROPN
ma-166	519	3	https://doi.org/10.12732/ijpam.v112i2.2	https://doi.org/10.12732/ijpam.v112i2.2	VERB
ma-166	519	4	https://doi.org/10.33263/briac126.72497266	https://doi.org/10.33263/briac126.72497266	NUM
ma-166	519	5	https://doi.org/10.33263/briac126.72497266	https://doi.org/10.33263/briac126.72497266	NUM
ma-166	519	6	https://doi.org/10.2298/fil1607979b	https://doi.org/10.2298/fil1607979b	PROPN
ma-166	519	7	eur	eur	PROPN
ma-166	519	8	.	.	PUNCT
ma-166	520	1	j.	j.	PROPN
ma-166	520	2	math	math	PROPN
ma-166	520	3	.	.	PUNCT
ma-166	521	1	anal	anal	PROPN
ma-166	521	2	.	.	PUNCT
ma-166	522	1	10.28924	10.28924	NUM
ma-166	522	2	/	/	SYM
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ma-166	522	4	/	/	SYM
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ma-166	523	1	[	[	SYM
ma-166	523	2	8	8	NUM
ma-166	523	3	]	]	X
ma-166	523	4	b.	b.	PROPN
ma-166	523	5	furtula	furtula	PROPN
ma-166	523	6	,	,	PUNCT
ma-166	523	7	i.	i.	PROPN
ma-166	523	8	gutman	gutman	PROPN
ma-166	523	9	,	,	PUNCT
ma-166	523	10	a	a	DET
ma-166	523	11	forgotten	forget	VERB
ma-166	523	12	topological	topological	ADJ
ma-166	523	13	index	index	NOUN
ma-166	523	14	,	,	PUNCT
ma-166	523	15	j.	j.	PROPN
ma-166	523	16	math	math	PROPN
ma-166	523	17	.	.	PUNCT
ma-166	524	1	chem	chem	NOUN
ma-166	524	2	.	.	PUNCT
ma-166	525	1	53	53	NUM
ma-166	525	2	(	(	PUNCT
ma-166	525	3	2015	2015	NUM
ma-166	525	4	)	)	PUNCT
ma-166	525	5	1184	1184	NUM
ma-166	525	6	-	-	SYM
ma-166	525	7	1190	1190	NUM
ma-166	525	8	.	.	PUNCT
ma-166	526	1	https://doi.org/10	https://doi.org/10	PROPN
ma-166	526	2	.	.	PUNCT
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ma-166	527	2	/	/	SYM
ma-166	527	3	s10910	s10910	ADP
ma-166	527	4	-	-	PUNCT
ma-166	527	5	015	015	NUM
ma-166	527	6	-	-	PUNCT
ma-166	527	7	0480	0480	NUM
ma-166	527	8	-	-	PUNCT
ma-166	527	9	z.[9	z.[9	NOUN
ma-166	527	10	]	]	X
ma-166	527	11	b.	b.	PROPN
ma-166	527	12	bollabas	bollabas	PROPN
ma-166	527	13	,	,	PUNCT
ma-166	527	14	p.	p.	PROPN
ma-166	527	15	erd	erd	PROPN
ma-166	527	16	,	,	PUNCT
ma-166	527	17	graphs	graph	NOUN
ma-166	527	18	of	of	ADP
ma-166	527	19	extremal	extremal	ADJ
ma-166	527	20	weights	weight	NOUN
ma-166	527	21	,	,	PUNCT
ma-166	527	22	ars	ar	NOUN
ma-166	527	23	comb	comb	VERB
ma-166	527	24	.	.	PUNCT
ma-166	528	1	50	50	NUM
ma-166	528	2	(	(	PUNCT
ma-166	528	3	1998	1998	NUM
ma-166	528	4	)	)	PUNCT
ma-166	528	5	225	225	NUM
ma-166	528	6	-	-	SYM
ma-166	528	7	233.[10	233.[10	NUM
ma-166	528	8	]	]	X
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ma-166	528	10	.	.	PROPN
ma-166	528	11	ashrafi	ashrafi	PROPN
ma-166	528	12	,	,	PUNCT
ma-166	528	13	m.	m.	NOUN
ma-166	528	14	mirzargar	mirzargar	NOUN
ma-166	528	15	,	,	PUNCT
ma-166	528	16	pi	pi	NOUN
ma-166	528	17	,	,	PUNCT
ma-166	528	18	szeged	szeged	PROPN
ma-166	528	19	and	and	CCONJ
ma-166	528	20	edge	edge	NOUN
ma-166	528	21	szeged	szeged	PROPN
ma-166	528	22	of	of	ADP
ma-166	528	23	an	an	DET
ma-166	528	24	infinite	infinite	ADJ
ma-166	528	25	family	family	NOUN
ma-166	528	26	of	of	ADP
ma-166	528	27	nanostardendrimers	nanostardendrimer	NOUN
ma-166	528	28	,	,	PUNCT
ma-166	528	29	indian	indian	PROPN
ma-166	528	30	j.	j.	PROPN
ma-166	528	31	chem.47	chem.47	PROPN
ma-166	528	32	(	(	PUNCT
ma-166	528	33	2008	2008	NUM
ma-166	528	34	)	)	PUNCT
ma-166	528	35	1656	1656	NUM
ma-166	528	36	-	-	SYM
ma-166	528	37	1660.[11	1660.[11	NUM
ma-166	528	38	]	]	PUNCT
ma-166	528	39	z.	z.	PROPN
ma-166	528	40	chen	chen	PROPN
ma-166	528	41	,	,	PUNCT
ma-166	528	42	m.	m.	NOUN
ma-166	528	43	dehmer	dehmer	NOUN
ma-166	528	44	,	,	PUNCT
ma-166	528	45	f.	f.	PROPN
ma-166	528	46	emmert	emmert	PROPN
ma-166	528	47	-	-	PUNCT
ma-166	528	48	streib	streib	PROPN
ma-166	528	49	,	,	PUNCT
ma-166	528	50	y.	y.	PROPN
ma-166	528	51	shi	shi	PROPN
ma-166	528	52	,	,	PUNCT
ma-166	528	53	entropy	entropy	PROPN
ma-166	528	54	boundsfor	boundsfor	ADP
ma-166	528	55	dendrimers	dendrimers	PROPN
ma-166	528	56	,	,	PUNCT
ma-166	528	57	appl	appl	PROPN
ma-166	528	58	.	.	PROPN
ma-166	528	59	math	math	PROPN
ma-166	528	60	.	.	PUNCT
ma-166	529	1	comput	comput	NOUN
ma-166	529	2	.	.	PUNCT
ma-166	530	1	242	242	NUM
ma-166	530	2	(	(	PUNCT
ma-166	530	3	2014)462	2014)462	PROPN
ma-166	530	4	-	-	PUNCT
ma-166	530	5	472.[12	472.[12	PROPN
ma-166	530	6	]	]	X
ma-166	530	7	m.v	m.v	PROPN
ma-166	530	8	.	.	PROPN
ma-166	530	9	diudea	diudea	PROPN
ma-166	530	10	,	,	PUNCT
ma-166	530	11	a.e	a.e	PROPN
ma-166	530	12	.	.	PROPN
ma-166	530	13	vizitiu	vizitiu	PROPN
ma-166	530	14	,	,	PUNCT
ma-166	530	15	m.	m.	NOUN
ma-166	530	16	mirzagar	mirzagar	NOUN
ma-166	530	17	,	,	PUNCT
ma-166	530	18	a.r	a.r	PROPN
ma-166	530	19	.	.	PROPN
ma-166	530	20	ashrafi	ashrafi	PROPN
ma-166	530	21	,	,	PUNCT
ma-166	530	22	sadhana	sadhana	NOUN
ma-166	530	23	polynomial	polynomial	NOUN
ma-166	530	24	in	in	ADP
ma-166	530	25	nano	nano	NOUN
ma-166	530	26	-	-	PUNCT
ma-166	530	27	dendrimers	dendrimer	NOUN
ma-166	530	28	,	,	PUNCT
ma-166	530	29	carpathian	carpathian	ADJ
ma-166	530	30	j.	j.	PROPN
ma-166	530	31	math.26	math.26	PROPN
ma-166	530	32	(	(	PUNCT
ma-166	530	33	2010	2010	NUM
ma-166	530	34	)	)	PUNCT
ma-166	530	35	59	59	NUM
ma-166	530	36	-	-	SYM
ma-166	530	37	66.[13	66.[13	PROPN
ma-166	530	38	]	]	PUNCT
ma-166	530	39	b.	b.	PROPN
ma-166	530	40	zhou	zhou	PROPN
ma-166	530	41	,	,	PUNCT
ma-166	530	42	n.	n.	PROPN
ma-166	530	43	trinajstic	trinajstic	NOUN
ma-166	530	44	,	,	PUNCT
ma-166	530	45	on	on	ADP
ma-166	530	46	general	general	ADJ
ma-166	530	47	sum	sum	NOUN
ma-166	530	48	-	-	PUNCT
ma-166	530	49	connectivity	connectivity	NOUN
ma-166	530	50	index	index	NOUN
ma-166	530	51	,	,	PUNCT
ma-166	530	52	j.	j.	PROPN
ma-166	530	53	math	math	PROPN
ma-166	530	54	.	.	PUNCT
ma-166	531	1	chem	chem	PROPN
ma-166	531	2	.	.	PUNCT
ma-166	532	1	47	47	NUM
ma-166	532	2	(	(	PUNCT
ma-166	532	3	2010	2010	NUM
ma-166	532	4	)	)	PUNCT
ma-166	532	5	210	210	NUM
ma-166	532	6	-	-	SYM
ma-166	532	7	218.[14	218.[14	NUM
ma-166	532	8	]	]	PUNCT
ma-166	532	9	a.	a.	NOUN
ma-166	532	10	asghar	asghar	PROPN
ma-166	532	11	,	,	PUNCT
ma-166	532	12	a.	a.	PROPN
ma-166	532	13	qayyum	qayyum	PROPN
ma-166	532	14	,	,	PUNCT
ma-166	532	15	n.	n.	PROPN
ma-166	532	16	muhammad	muhammad	PROPN
ma-166	532	17	,	,	PUNCT
ma-166	532	18	different	different	ADJ
ma-166	532	19	types	type	NOUN
ma-166	532	20	of	of	ADP
ma-166	532	21	topological	topological	ADJ
ma-166	532	22	structures	structure	NOUN
ma-166	532	23	by	by	ADP
ma-166	532	24	graphs	graph	NOUN
ma-166	532	25	,	,	PUNCT
ma-166	532	26	eur	eur	PROPN
ma-166	532	27	.	.	PUNCT
ma-166	533	1	j.	j.	PROPN
ma-166	533	2	math	math	PROPN
ma-166	533	3	.	.	PUNCT
ma-166	534	1	anal	anal	PROPN
ma-166	534	2	.	.	PUNCT
ma-166	535	1	3(2022	3(2022	NUM
ma-166	535	2	)	)	PUNCT
ma-166	535	3	3	3	NUM
ma-166	535	4	.	.	NUM
ma-166	535	5	https://doi.org/10.28924/ada/ma.3.3.[15	https://doi.org/10.28924/ada/ma.3.3.[15	NOUN
ma-166	535	6	]	]	X
ma-166	536	1	z.h	z.h	PROPN
ma-166	536	2	.	.	PROPN
ma-166	536	3	niazi	niazi	PROPN
ma-166	536	4	,	,	PUNCT
ma-166	536	5	m.a.t	m.a.t	NOUN
ma-166	536	6	.	.	PUNCT
ma-166	536	7	bhatti	bhatti	PROPN
ma-166	536	8	,	,	PUNCT
ma-166	536	9	m.	m.	NOUN
ma-166	536	10	aslam	aslam	PROPN
ma-166	536	11	,	,	PUNCT
ma-166	536	12	y.	y.	PROPN
ma-166	536	13	qayyum	qayyum	PROPN
ma-166	536	14	,	,	PUNCT
ma-166	536	15	m.	m.	PROPN
ma-166	536	16	ibrahim	ibrahim	PROPN
ma-166	536	17	,	,	PUNCT
ma-166	536	18	a.	a.	PROPN
ma-166	536	19	qayyum	qayyum	PROPN
ma-166	536	20	,	,	PUNCT
ma-166	536	21	d	d	ADJ
ma-166	536	22	-	-	PUNCT
ma-166	536	23	lucky	lucky	ADJ
ma-166	536	24	labelling	labelling	NOUN
ma-166	536	25	of	of	ADP
ma-166	536	26	some	some	DET
ma-166	536	27	special	special	ADJ
ma-166	536	28	graphs	graph	NOUN
ma-166	536	29	,	,	PUNCT
ma-166	536	30	amer	amer	PROPN
ma-166	536	31	.	.	PUNCT
ma-166	537	1	j.	j.	PROPN
ma-166	537	2	math	math	PROPN
ma-166	537	3	.	.	PUNCT
ma-166	538	1	anal	anal	ADJ
ma-166	538	2	.	.	PUNCT
ma-166	539	1	10	10	NUM
ma-166	539	2	(	(	PUNCT
ma-166	539	3	2022	2022	NUM
ma-166	539	4	)	)	PUNCT
ma-166	539	5	3	3	NUM
ma-166	539	6	-	-	SYM
ma-166	539	7	11	11	NUM
ma-166	539	8	.	.	PUNCT
ma-166	540	1	https://doi.org/10.12691/ajma-10-1-2	https://doi.org/10.12691/ajma-10-1-2	PROPN
ma-166	540	2	.	.	PUNCT
ma-166	541	1	https://doi.org/10.28924/ada/ma.3.20	https://doi.org/10.28924/ada/ma.3.20	X
ma-166	541	2	https://doi.org/10.1007/s10910-015-0480-z	https://doi.org/10.1007/s10910-015-0480-z	PROPN
ma-166	541	3	https://doi.org/10.1007/s10910-015-0480-z	https://doi.org/10.1007/s10910-015-0480-z	PROPN
ma-166	541	4	https://doi.org/10.28924/ada/ma.3.3	https://doi.org/10.28924/ada/ma.3.3	PROPN
ma-166	541	5	https://doi.org/10.12691/ajma-10-1-2	https://doi.org/10.12691/ajma-10-1-2	NOUN
ma-166	541	6	1	1	NUM
ma-166	541	7	.	.	PUNCT
ma-166	541	8	introduction	introduction	NOUN
ma-166	541	9	1.1	1.1	NUM
ma-166	541	10	.	.	PUNCT
ma-166	542	1	the	the	DET
ma-166	542	2	graph	graph	NOUN
ma-166	542	3	of	of	ADP
ma-166	542	4	petersen	petersen	PROPN
ma-166	542	5	1.2	1.2	NUM
ma-166	542	6	.	.	PUNCT
ma-166	542	7	graphical	graphical	ADJ
ma-166	542	8	idea	idea	NOUN
ma-166	542	9	of	of	ADP
ma-166	542	10	petersen	petersen	PROPN
ma-166	542	11	graph	graph	NOUN
ma-166	542	12	2	2	NUM
ma-166	542	13	.	.	PUNCT
ma-166	542	14	main	main	ADJ
ma-166	542	15	results	result	NOUN
ma-166	542	16	3	3	NUM
ma-166	542	17	.	.	PUNCT
ma-166	542	18	numerical	numerical	ADJ
ma-166	542	19	examples	example	NOUN
ma-166	542	20	4	4	NUM
ma-166	542	21	.	.	NOUN
ma-166	542	22	conclusion	conclusion	NOUN
ma-166	542	23	and	and	CCONJ
ma-166	542	24	future	future	ADJ
ma-166	542	25	studies	study	NOUN
ma-166	542	26	references	reference	NOUN