id	sid	tid	token	lemma	pos
cana-788	1	1	communications	communication	NOUN
cana-788	1	2	on	on	ADP
cana-788	1	3	applied	apply	VERB
cana-788	1	4	nonlinear	nonlinear	ADJ
cana-788	1	5	analysis	analysis	NOUN
cana-788	1	6	issn	issn	NOUN
cana-788	1	7	:	:	PUNCT
cana-788	1	8	1074	1074	NUM
cana-788	1	9	-	-	PUNCT
cana-788	1	10	133x	133x	NUM
cana-788	1	11	vol	vol	NOUN
cana-788	1	12	31	31	NUM
cana-788	1	13	no	no	NOUN
cana-788	1	14	.	.	PUNCT
cana-788	2	1	3s	3s	NUM
cana-788	2	2	(	(	PUNCT
cana-788	2	3	2024	2024	NUM
cana-788	2	4	)	)	PUNCT
cana-788	2	5	377	377	NUM
cana-788	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	2	7	the	the	DET
cana-788	2	8	m	m	NOUN
cana-788	2	9	-	-	ADJ
cana-788	2	10	polynomial	polynomial	ADJ
cana-788	2	11	of	of	ADP
cana-788	2	12	schreier	schreier	NOUN
cana-788	2	13	graphs	graph	NOUN
cana-788	2	14	of	of	ADP
cana-788	2	15	the	the	DET
cana-788	2	16	basilica	basilica	NOUN
cana-788	2	17	and	and	CCONJ
cana-788	2	18	grigorchuk	grigorchuk	NOUN
cana-788	2	19	groups	group	NOUN
cana-788	2	20	:	:	PUNCT
cana-788	2	21	comparative	comparative	ADJ
cana-788	2	22	evaluation	evaluation	NOUN
cana-788	2	23	mr.v	mr.v	NOUN
cana-788	2	24	.	.	PUNCT
cana-788	3	1	rajkumar1	rajkumar1	PROPN
cana-788	3	2	,	,	PUNCT
cana-788	3	3	dr	dr	PROPN
cana-788	3	4	.	.	PROPN
cana-788	3	5	b.	b.	PROPN
cana-788	3	6	sivakumar2	sivakumar2	PROPN
cana-788	3	7	,	,	PUNCT
cana-788	3	8	dr	dr	PROPN
cana-788	3	9	.	.	PROPN
cana-788	3	10	nur	nur	PROPN
cana-788	3	11	idayu	idayu	PROPN
cana-788	3	12	alimon3	alimon3	PROPN
cana-788	3	13	1	1	NUM
cana-788	3	14	department	department	NOUN
cana-788	3	15	of	of	ADP
cana-788	3	16	mathematics	mathematic	NOUN
cana-788	3	17	,	,	PUNCT
cana-788	3	18	rajalakshmi	rajalakshmi	VERB
cana-788	3	19	engineering	engineering	NOUN
cana-788	3	20	college	college	NOUN
cana-788	3	21	,	,	PUNCT
cana-788	3	22	chennai	chennai	PROPN
cana-788	3	23	–	–	PUNCT
cana-788	3	24	602	602	NUM
cana-788	3	25	105	105	NUM
cana-788	3	26	,	,	PUNCT
cana-788	3	27	rajkumar.v@rajalakshmi.edu.in	rajkumar.v@rajalakshmi.edu.in	NOUN
cana-788	3	28	2	2	NUM
cana-788	3	29	department	department	NOUN
cana-788	3	30	of	of	ADP
cana-788	3	31	mathematics	mathematic	NOUN
cana-788	3	32	,	,	PUNCT
cana-788	3	33	sri	sri	PROPN
cana-788	3	34	sivasubramaniya	sivasubramaniya	PROPN
cana-788	3	35	nadar	nadar	PROPN
cana-788	3	36	college	college	PROPN
cana-788	3	37	of	of	ADP
cana-788	3	38	engineering	engineering	PROPN
cana-788	3	39	,	,	PUNCT
cana-788	3	40	chennai	chennai	PROPN
cana-788	3	41	-603	-603	PROPN
cana-788	3	42	110	110	NUM
cana-788	3	43	.	.	PUNCT
cana-788	4	1	sivakumarb@ssn.edu.in	sivakumarb@ssn.edu.in	PROPN
cana-788	4	2	3	3	NUM
cana-788	4	3	college	college	NOUN
cana-788	4	4	of	of	ADP
cana-788	4	5	computing	computing	NOUN
cana-788	4	6	,	,	PUNCT
cana-788	4	7	informatics	informatic	NOUN
cana-788	4	8	and	and	CCONJ
cana-788	4	9	mathematics	mathematic	NOUN
cana-788	4	10	,	,	PUNCT
cana-788	4	11	universiti	universiti	PROPN
cana-788	4	12	teknologi	teknologi	PROPN
cana-788	4	13	mara	mara	PROPN
cana-788	4	14	,	,	PUNCT
cana-788	4	15	johor	johor	PROPN
cana-788	4	16	branch	branch	PROPN
cana-788	4	17	,	,	PUNCT
cana-788	4	18	pasir	pasir	PROPN
cana-788	4	19	gudang	gudang	PROPN
cana-788	4	20	campus,81750	campus,81750	PROPN
cana-788	4	21	,	,	PUNCT
cana-788	4	22	johor	johor	PROPN
cana-788	4	23	,	,	PUNCT
cana-788	4	24	malaysia	malaysia	PROPN
cana-788	4	25	;	;	PUNCT
cana-788	4	26	idayualimon@uitm.edu.my	idayualimon@uitm.edu.my	NOUN
cana-788	4	27	article	article	NOUN
cana-788	4	28	history	history	NOUN
cana-788	4	29	:	:	PUNCT
cana-788	4	30	received	receive	VERB
cana-788	4	31	:	:	PUNCT
cana-788	4	32	10	10	NUM
cana-788	4	33	-	-	PUNCT
cana-788	4	34	04	04	NUM
cana-788	4	35	-	-	PUNCT
cana-788	4	36	2024	2024	NUM
cana-788	4	37	revised	revise	VERB
cana-788	4	38	:	:	PUNCT
cana-788	4	39	26	26	NUM
cana-788	4	40	-	-	SYM
cana-788	4	41	05	05	NUM
cana-788	4	42	-	-	PUNCT
cana-788	4	43	2024	2024	NUM
cana-788	4	44	accepted	accept	VERB
cana-788	4	45	:	:	PUNCT
cana-788	4	46	15	15	NUM
cana-788	4	47	-	-	SYM
cana-788	4	48	06	06	NUM
cana-788	4	49	-	-	PUNCT
cana-788	4	50	2024	2024	NUM
cana-788	4	51	abstract	abstract	NOUN
cana-788	4	52	:	:	PUNCT
cana-788	4	53	a	a	DET
cana-788	4	54	focus	focus	NOUN
cana-788	4	55	for	for	ADP
cana-788	4	56	comprehending	comprehend	VERB
cana-788	4	57	the	the	DET
cana-788	4	58	complex	complex	ADJ
cana-788	4	59	structures	structure	NOUN
cana-788	4	60	present	present	ADJ
cana-788	4	61	in	in	ADP
cana-788	4	62	self	self	NOUN
cana-788	4	63	-	-	PUNCT
cana-788	4	64	similar	similar	ADJ
cana-788	4	65	groups	group	NOUN
cana-788	4	66	has	have	AUX
cana-788	4	67	been	be	AUX
cana-788	4	68	the	the	DET
cana-788	4	69	study	study	NOUN
cana-788	4	70	of	of	ADP
cana-788	4	71	schreier	schreier	NOUN
cana-788	4	72	graphs	graph	NOUN
cana-788	4	73	related	relate	VERB
cana-788	4	74	to	to	ADP
cana-788	4	75	the	the	DET
cana-788	4	76	basilica	basilica	PROPN
cana-788	4	77	and	and	CCONJ
cana-788	4	78	grigorchuk	grigorchuk	PROPN
cana-788	4	79	group	group	NOUN
cana-788	4	80	in	in	ADP
cana-788	4	81	recent	recent	ADJ
cana-788	4	82	years	year	NOUN
cana-788	4	83	.	.	PUNCT
cana-788	5	1	basic	basic	ADJ
cana-788	5	2	to	to	ADP
cana-788	5	3	group	group	NOUN
cana-788	5	4	theory	theory	NOUN
cana-788	5	5	,	,	PUNCT
cana-788	5	6	schreier	schreier	NOUN
cana-788	5	7	graphs	graph	NOUN
cana-788	5	8	give	give	VERB
cana-788	5	9	a	a	DET
cana-788	5	10	geometric	geometric	ADJ
cana-788	5	11	picture	picture	NOUN
cana-788	5	12	of	of	ADP
cana-788	5	13	these	these	DET
cana-788	5	14	groups	group	NOUN
cana-788	5	15	’	'	PUNCT
cana-788	5	16	actions	action	NOUN
cana-788	5	17	on	on	ADP
cana-788	5	18	sets	set	NOUN
cana-788	5	19	and	and	CCONJ
cana-788	5	20	shed	shed	VERB
cana-788	5	21	light	light	NOUN
cana-788	5	22	on	on	ADP
cana-788	5	23	the	the	DET
cana-788	5	24	connectivity	connectivity	NOUN
cana-788	5	25	and	and	CCONJ
cana-788	5	26	symmetry	symmetry	NOUN
cana-788	5	27	characteristics	characteristic	NOUN
cana-788	5	28	of	of	ADP
cana-788	5	29	these	these	DET
cana-788	5	30	groups	group	NOUN
cana-788	5	31	.	.	PUNCT
cana-788	6	1	this	this	DET
cana-788	6	2	paper	paper	NOUN
cana-788	6	3	examines	examine	VERB
cana-788	6	4	the	the	DET
cana-788	6	5	m	m	NOUN
cana-788	6	6	-	-	ADJ
cana-788	6	7	polynomial	polynomial	ADJ
cana-788	6	8	of	of	ADP
cana-788	6	9	schreier	schreier	NOUN
cana-788	6	10	graphs	graph	NOUN
cana-788	6	11	of	of	ADP
cana-788	6	12	the	the	DET
cana-788	6	13	basilica	basilica	NOUN
cana-788	6	14	and	and	CCONJ
cana-788	6	15	grigorchuk	grigorchuk	NOUN
cana-788	6	16	groups	group	NOUN
cana-788	6	17	,	,	PUNCT
cana-788	6	18	examining	examine	VERB
cana-788	6	19	its	its	PRON
cana-788	6	20	consequences	consequence	NOUN
cana-788	6	21	for	for	ADP
cana-788	6	22	topological	topological	ADJ
cana-788	6	23	indices	index	NOUN
cana-788	6	24	and	and	CCONJ
cana-788	6	25	comparing	compare	VERB
cana-788	6	26	the	the	DET
cana-788	6	27	determined	determined	ADJ
cana-788	6	28	values	value	NOUN
cana-788	6	29	.	.	PUNCT
cana-788	7	1	keywords	keyword	NOUN
cana-788	7	2	:	:	PUNCT
cana-788	7	3	m	m	NOUN
cana-788	7	4	-	-	ADJ
cana-788	7	5	polynomial	polynomial	ADJ
cana-788	7	6	,	,	PUNCT
cana-788	7	7	topological	topological	ADJ
cana-788	7	8	indices	index	NOUN
cana-788	7	9	,	,	PUNCT
cana-788	7	10	schreier	schreier	NOUN
cana-788	7	11	graph	graph	NOUN
cana-788	7	12	,	,	PUNCT
cana-788	7	13	basilica	basilica	PROPN
cana-788	7	14	group	group	NOUN
cana-788	7	15	grigorchuk	grigorchuk	PROPN
cana-788	7	16	group	group	PROPN
cana-788	7	17	1	1	NUM
cana-788	7	18	.	.	PUNCT
cana-788	7	19	introduction	introduction	NOUN
cana-788	7	20	schreier	schreier	NOUN
cana-788	7	21	graphs	graph	NOUN
cana-788	7	22	emerge	emerge	VERB
cana-788	7	23	as	as	ADP
cana-788	7	24	a	a	DET
cana-788	7	25	fundamental	fundamental	ADJ
cana-788	7	26	representation	representation	NOUN
cana-788	7	27	of	of	ADP
cana-788	7	28	the	the	DET
cana-788	7	29	actions	action	NOUN
cana-788	7	30	performed	perform	VERB
cana-788	7	31	by	by	ADP
cana-788	7	32	finitely	finitely	ADV
cana-788	7	33	generated	generate	VERB
cana-788	7	34	groups	group	NOUN
cana-788	7	35	on	on	ADP
cana-788	7	36	sets	set	NOUN
cana-788	7	37	,	,	PUNCT
cana-788	7	38	offering	offer	VERB
cana-788	7	39	a	a	DET
cana-788	7	40	graphical	graphical	ADJ
cana-788	7	41	depiction	depiction	NOUN
cana-788	7	42	of	of	ADP
cana-788	7	43	these	these	DET
cana-788	7	44	group	group	NOUN
cana-788	7	45	actions	action	NOUN
cana-788	7	46	.	.	PUNCT
cana-788	8	1	particularly	particularly	ADV
cana-788	8	2	within	within	ADP
cana-788	8	3	the	the	DET
cana-788	8	4	monarchy	monarchy	NOUN
cana-788	8	5	of	of	ADP
cana-788	8	6	automaton	automaton	PROPN
cana-788	8	7	groups	group	NOUN
cana-788	8	8	,	,	PUNCT
cana-788	8	9	schreier	schreier	NOUN
cana-788	8	10	graphs	graph	NOUN
cana-788	8	11	stand	stand	VERB
cana-788	8	12	out	out	ADP
cana-788	8	13	as	as	ADP
cana-788	8	14	a	a	DET
cana-788	8	15	natural	natural	ADJ
cana-788	8	16	focus	focus	NOUN
cana-788	8	17	of	of	ADP
cana-788	8	18	investigation	investigation	NOUN
cana-788	8	19	.	.	PUNCT
cana-788	9	1	finite	finite	PROPN
cana-788	9	2	invertible	invertible	ADJ
cana-788	9	3	automata	automata	NOUN
cana-788	9	4	,	,	PUNCT
cana-788	9	5	functioning	function	VERB
cana-788	9	6	as	as	ADP
cana-788	9	7	finite	finite	ADJ
cana-788	9	8	input	input	NOUN
cana-788	9	9	/	/	SYM
cana-788	9	10	output	output	NOUN
cana-788	9	11	devices	device	NOUN
cana-788	9	12	,	,	PUNCT
cana-788	9	13	intricately	intricately	ADV
cana-788	9	14	capture	capture	VERB
cana-788	9	15	the	the	DET
cana-788	9	16	sequential	sequential	ADJ
cana-788	9	17	operations	operation	NOUN
cana-788	9	18	executed	execute	VERB
cana-788	9	19	by	by	ADP
cana-788	9	20	a	a	DET
cana-788	9	21	set	set	NOUN
cana-788	9	22	of	of	ADP
cana-788	9	23	states	state	NOUN
cana-788	9	24	on	on	ADP
cana-788	9	25	a	a	DET
cana-788	9	26	finite	finite	ADJ
cana-788	9	27	alphabet	alphabet	NOUN
cana-788	9	28	.	.	PUNCT
cana-788	10	1	these	these	DET
cana-788	10	2	automata	automata	NOUN
cana-788	10	3	,	,	PUNCT
cana-788	10	4	in	in	ADP
cana-788	10	5	turn	turn	NOUN
cana-788	10	6	,	,	PUNCT
cana-788	10	7	correspond	correspond	VERB
cana-788	10	8	to	to	ADP
cana-788	10	9	automorphisms	automorphisms	PROPN
cana-788	10	10	of	of	ADP
cana-788	10	11	rooted	rooted	ADJ
cana-788	10	12	regular	regular	ADJ
cana-788	10	13	trees	tree	NOUN
cana-788	10	14	,	,	PUNCT
cana-788	10	15	inducing	induce	VERB
cana-788	10	16	bijections	bijection	NOUN
cana-788	10	17	that	that	PRON
cana-788	10	18	collectively	collectively	ADV
cana-788	10	19	form	form	VERB
cana-788	10	20	what	what	PRON
cana-788	10	21	is	be	AUX
cana-788	10	22	known	know	VERB
cana-788	10	23	as	as	ADP
cana-788	10	24	an	an	DET
cana-788	10	25	automaton	automaton	NOUN
cana-788	10	26	group	group	NOUN
cana-788	10	27	(	(	PUNCT
cana-788	10	28	for	for	ADP
cana-788	10	29	more	more	ADJ
cana-788	10	30	details	detail	NOUN
cana-788	10	31	about	about	ADP
cana-788	10	32	automaton	automaton	PROPN
cana-788	10	33	group	group	NOUN
cana-788	10	34	please	please	INTJ
cana-788	10	35	see	see	VERB
cana-788	10	36	[	[	X
cana-788	10	37	1	1	NUM
cana-788	10	38	-	-	SYM
cana-788	10	39	2	2	NUM
cana-788	10	40	]	]	PUNCT
cana-788	10	41	.	.	PUNCT
cana-788	11	1	the	the	DET
cana-788	11	2	dynamic	dynamic	ADJ
cana-788	11	3	behaviour	behaviour	NOUN
cana-788	11	4	of	of	ADP
cana-788	11	5	an	an	DET
cana-788	11	6	automaton	automaton	NOUN
cana-788	11	7	group	group	NOUN
cana-788	11	8	when	when	SCONJ
cana-788	11	9	acting	act	VERB
cana-788	11	10	upon	upon	SCONJ
cana-788	11	11	a	a	DET
cana-788	11	12	rooted	rooted	ADJ
cana-788	11	13	tree	tree	NOUN
cana-788	11	14	manifests	manifest	NOUN
cana-788	11	15	in	in	ADP
cana-788	11	16	the	the	DET
cana-788	11	17	preservation	preservation	NOUN
cana-788	11	18	of	of	ADP
cana-788	11	19	the	the	DET
cana-788	11	20	hierarchical	hierarchical	ADJ
cana-788	11	21	structure	structure	NOUN
cana-788	11	22	inherent	inherent	ADJ
cana-788	11	23	in	in	ADP
cana-788	11	24	the	the	DET
cana-788	11	25	tree	tree	NOUN
cana-788	11	26	's	's	PART
cana-788	11	27	levels	level	NOUN
cana-788	11	28	.	.	PUNCT
cana-788	12	1	by	by	ADP
cana-788	12	2	restricting	restrict	VERB
cana-788	12	3	this	this	DET
cana-788	12	4	action	action	NOUN
cana-788	12	5	to	to	ADP
cana-788	12	6	a	a	DET
cana-788	12	7	specific	specific	ADJ
cana-788	12	8	level	level	NOUN
cana-788	12	9	of	of	ADP
cana-788	12	10	the	the	DET
cana-788	12	11	tree	tree	NOUN
cana-788	12	12	,	,	PUNCT
cana-788	12	13	one	one	PRON
cana-788	12	14	can	can	AUX
cana-788	12	15	construct	construct	VERB
cana-788	12	16	a	a	DET
cana-788	12	17	unique	unique	ADJ
cana-788	12	18	schreier	schreier	NOUN
cana-788	12	19	graph	graph	NOUN
cana-788	12	20	corresponding	correspond	VERB
cana-788	12	21	to	to	ADP
cana-788	12	22	that	that	DET
cana-788	12	23	level	level	NOUN
cana-788	12	24	.	.	PUNCT
cana-788	13	1	as	as	ADP
cana-788	13	2	a	a	DET
cana-788	13	3	result	result	NOUN
cana-788	13	4	,	,	PUNCT
cana-788	13	5	each	each	DET
cana-788	13	6	automata	automata	NOUN
cana-788	13	7	group	group	NOUN
cana-788	13	8	generates	generate	VERB
cana-788	13	9	an	an	DET
cana-788	13	10	unlimited	unlimited	ADJ
cana-788	13	11	number	number	NOUN
cana-788	13	12	of	of	ADP
cana-788	13	13	finite	finite	ADJ
cana-788	13	14	schreier	schreier	NOUN
cana-788	13	15	graphs	graph	NOUN
cana-788	13	16	,	,	PUNCT
cana-788	13	17	each	each	PRON
cana-788	13	18	of	of	ADP
cana-788	13	19	which	which	PRON
cana-788	13	20	replicates	replicate	VERB
cana-788	13	21	the	the	DET
cana-788	13	22	group	group	NOUN
cana-788	13	23	's	's	PART
cana-788	13	24	actions	action	NOUN
cana-788	13	25	on	on	ADP
cana-788	13	26	the	the	DET
cana-788	13	27	finite	finite	ADJ
cana-788	13	28	levels	level	NOUN
cana-788	13	29	of	of	ADP
cana-788	13	30	the	the	DET
cana-788	13	31	corresponding	corresponding	ADJ
cana-788	13	32	rooted	rooted	ADJ
cana-788	13	33	regular	regular	ADJ
cana-788	13	34	tree	tree	NOUN
cana-788	13	35	.	.	PUNCT
cana-788	14	1	within	within	ADP
cana-788	14	2	the	the	DET
cana-788	14	3	empire	empire	NOUN
cana-788	14	4	of	of	ADP
cana-788	14	5	automaton	automaton	NOUN
cana-788	14	6	groups	group	NOUN
cana-788	14	7	laid	lay	VERB
cana-788	14	8	captivating	captivate	VERB
cana-788	14	9	instances	instance	NOUN
cana-788	14	10	characterized	characterize	VERB
cana-788	14	11	by	by	ADP
cana-788	14	12	their	their	PRON
cana-788	14	13	unconventional	unconventional	ADJ
cana-788	14	14	and	and	CCONJ
cana-788	14	15	sometimes	sometimes	ADV
cana-788	14	16	enigmatic	enigmatic	ADJ
cana-788	14	17	properties	property	NOUN
cana-788	14	18	.	.	PUNCT
cana-788	15	1	notably	notably	ADV
cana-788	15	2	,	,	PUNCT
cana-788	15	3	among	among	ADP
cana-788	15	4	these	these	PRON
cana-788	15	5	is	be	AUX
cana-788	15	6	the	the	DET
cana-788	15	7	basilica	basilica	PROPN
cana-788	15	8	group	group	NOUN
cana-788	15	9	,	,	PUNCT
cana-788	15	10	a	a	DET
cana-788	15	11	pioneering	pioneering	ADJ
cana-788	15	12	example	example	NOUN
cana-788	15	13	showcasing	showcase	VERB
cana-788	15	14	intriguing	intriguing	ADJ
cana-788	15	15	traits	trait	NOUN
cana-788	15	16	.	.	PUNCT
cana-788	16	1	the	the	DET
cana-788	16	2	basilica	basilica	PROPN
cana-788	16	3	group	group	NOUN
cana-788	16	4	is	be	AUX
cana-788	16	5	a	a	DET
cana-788	16	6	unique	unique	ADJ
cana-788	16	7	entity	entity	NOUN
cana-788	16	8	that	that	PRON
cana-788	16	9	is	be	AUX
cana-788	16	10	produced	produce	VERB
cana-788	16	11	by	by	ADP
cana-788	16	12	a	a	DET
cana-788	16	13	three	three	NUM
cana-788	16	14	-	-	PUNCT
cana-788	16	15	state	state	NOUN
cana-788	16	16	automata	automata	NOUN
cana-788	16	17	and	and	CCONJ
cana-788	16	18	is	be	AUX
cana-788	16	19	derived	derive	VERB
cana-788	16	20	from	from	ADP
cana-788	16	21	the	the	DET
cana-788	16	22	work	work	NOUN
cana-788	16	23	of	of	ADP
cana-788	16	24	r.	r.	PROPN
cana-788	16	25	grigorchuk	grigorchuk	PROPN
cana-788	16	26	and	and	CCONJ
cana-788	16	27	a.	a.	NOUN
cana-788	16	28	zuk	zuk	PROPN
cana-788	17	1	[	[	X
cana-788	17	2	3	3	NUM
cana-788	17	3	]	]	PUNCT
cana-788	17	4	.	.	PUNCT
cana-788	18	1	this	this	DET
cana-788	18	2	group	group	NOUN
cana-788	18	3	serves	serve	VERB
cana-788	18	4	as	as	ADP
cana-788	18	5	a	a	DET
cana-788	18	6	noteworthy	noteworthy	ADJ
cana-788	18	7	illustration	illustration	NOUN
cana-788	18	8	within	within	ADP
cana-788	18	9	the	the	DET
cana-788	18	10	broader	broad	ADJ
cana-788	18	11	spectrum	spectrum	NOUN
cana-788	18	12	of	of	ADP
cana-788	18	13	automaton	automaton	PROPN
cana-788	18	14	groups	group	NOUN
cana-788	18	15	,	,	PUNCT
cana-788	18	16	highlighting	highlight	VERB
cana-788	18	17	the	the	DET
cana-788	18	18	richness	richness	NOUN
cana-788	18	19	and	and	CCONJ
cana-788	18	20	diversity	diversity	NOUN
cana-788	18	21	inherent	inherent	ADJ
cana-788	18	22	in	in	ADP
cana-788	18	23	their	their	PRON
cana-788	18	24	structures	structure	NOUN
cana-788	18	25	.	.	PUNCT
cana-788	19	1	mailto:rajkumar.v@rajalakshmi.edu.in	mailto:rajkumar.v@rajalakshmi.edu.in	NOUN
cana-788	19	2	communications	communication	NOUN
cana-788	19	3	on	on	ADP
cana-788	19	4	applied	apply	VERB
cana-788	19	5	nonlinear	nonlinear	ADJ
cana-788	19	6	analysis	analysis	NOUN
cana-788	19	7	issn	issn	NOUN
cana-788	19	8	:	:	PUNCT
cana-788	19	9	1074	1074	NUM
cana-788	19	10	-	-	PUNCT
cana-788	19	11	133x	133x	NUM
cana-788	19	12	vol	vol	NOUN
cana-788	19	13	31	31	NUM
cana-788	19	14	no	no	NOUN
cana-788	19	15	.	.	PUNCT
cana-788	20	1	3s	3s	NUM
cana-788	20	2	(	(	PUNCT
cana-788	20	3	2024	2024	NUM
cana-788	20	4	)	)	PUNCT
cana-788	20	5	378	378	NUM
cana-788	20	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	20	7	a	a	DET
cana-788	20	8	finitely	finitely	ADV
cana-788	20	9	produced	produce	VERB
cana-788	20	10	torsion	torsion	NOUN
cana-788	20	11	group	group	NOUN
cana-788	20	12	and	and	CCONJ
cana-788	20	13	the	the	DET
cana-788	20	14	first	first	ADJ
cana-788	20	15	known	know	VERB
cana-788	20	16	example	example	NOUN
cana-788	20	17	of	of	ADP
cana-788	20	18	a	a	DET
cana-788	20	19	finitely	finitely	ADV
cana-788	20	20	generated	generate	VERB
cana-788	20	21	group	group	NOUN
cana-788	20	22	of	of	ADP
cana-788	20	23	intermediate	intermediate	ADJ
cana-788	20	24	growth	growth	NOUN
cana-788	20	25	are	be	AUX
cana-788	20	26	obtained	obtain	VERB
cana-788	20	27	from	from	ADP
cana-788	20	28	the	the	DET
cana-788	20	29	grigorchuk	grigorchuk	NOUN
cana-788	20	30	group	group	NOUN
cana-788	20	31	,	,	PUNCT
cana-788	20	32	which	which	PRON
cana-788	20	33	also	also	ADV
cana-788	20	34	provides	provide	VERB
cana-788	20	35	the	the	DET
cana-788	20	36	simplest	simple	ADJ
cana-788	20	37	solution	solution	NOUN
cana-788	20	38	to	to	ADP
cana-788	20	39	the	the	DET
cana-788	20	40	burnside	burnside	NOUN
cana-788	20	41	question	question	NOUN
cana-788	20	42	.	.	PUNCT
cana-788	21	1	for	for	ADP
cana-788	21	2	more	more	ADJ
cana-788	21	3	details	detail	NOUN
cana-788	21	4	and	and	CCONJ
cana-788	21	5	additional	additional	ADJ
cana-788	21	6	references	reference	NOUN
cana-788	21	7	,	,	PUNCT
cana-788	21	8	see	see	VERB
cana-788	21	9	[	[	X
cana-788	21	10	4	4	X
cana-788	21	11	]	]	PUNCT
cana-788	21	12	and	and	CCONJ
cana-788	21	13	[	[	X
cana-788	21	14	5	5	NUM
cana-788	21	15	]	]	PUNCT
cana-788	21	16	.	.	PUNCT
cana-788	22	1	the	the	DET
cana-788	22	2	mpolynomial	mpolynomial	NOUN
cana-788	22	3	is	be	AUX
cana-788	22	4	defined	define	VERB
cana-788	22	5	by	by	ADP
cana-788	22	6	s.	s.	PROPN
cana-788	22	7	klavzar	klavzar	PROPN
cana-788	22	8	or	or	CCONJ
cana-788	22	9	e.	e.	PROPN
cana-788	22	10	deutsch	deutsch	PROPN
cana-788	22	11	in	in	ADP
cana-788	22	12	2015	2015	NUM
cana-788	22	13	.	.	PUNCT
cana-788	23	1	one	one	NUM
cana-788	23	2	important	important	ADJ
cana-788	23	3	component	component	NOUN
cana-788	23	4	in	in	ADP
cana-788	23	5	the	the	DET
cana-788	23	6	monarchy	monarchy	NOUN
cana-788	23	7	of	of	ADP
cana-788	23	8	degree	degree	NOUN
cana-788	23	9	-	-	PUNCT
cana-788	23	10	oriented	orient	VERB
cana-788	23	11	topological	topological	ADJ
cana-788	23	12	indices	index	NOUN
cana-788	23	13	is	be	AUX
cana-788	23	14	the	the	DET
cana-788	23	15	m	m	NOUN
cana-788	23	16	-	-	NOUN
cana-788	23	17	polynomial	polynomial	ADJ
cana-788	23	18	.	.	PUNCT
cana-788	24	1	it	it	PRON
cana-788	24	2	is	be	AUX
cana-788	24	3	the	the	DET
cana-788	24	4	most	most	ADV
cana-788	24	5	widely	widely	ADV
cana-788	24	6	used	use	VERB
cana-788	24	7	general	general	ADJ
cana-788	24	8	progressive	progressive	ADJ
cana-788	24	9	topological	topological	ADJ
cana-788	24	10	polynomial	polynomial	NOUN
cana-788	24	11	.	.	PUNCT
cana-788	25	1	additionally	additionally	ADV
cana-788	25	2	,	,	PUNCT
cana-788	25	3	it	it	PRON
cana-788	25	4	has	have	VERB
cana-788	25	5	many	many	ADJ
cana-788	25	6	potential	potential	ADJ
cana-788	25	7	applications	application	NOUN
cana-788	25	8	is	be	AUX
cana-788	25	9	various	various	ADJ
cana-788	25	10	field	field	NOUN
cana-788	25	11	,	,	PUNCT
cana-788	25	12	which	which	PRON
cana-788	25	13	they	they	PRON
cana-788	25	14	can	can	AUX
cana-788	25	15	be	be	AUX
cana-788	25	16	found	find	VERB
cana-788	25	17	in	in	ADP
cana-788	25	18	[	[	PUNCT
cana-788	25	19	6	6	NUM
cana-788	25	20	-	-	SYM
cana-788	25	21	10	10	NUM
cana-788	25	22	]	]	PUNCT
cana-788	25	23	.	.	PUNCT
cana-788	26	1	the	the	DET
cana-788	26	2	m	m	NOUN
cana-788	26	3	-	-	NOUN
cana-788	26	4	polynomial	polynomial	ADJ
cana-788	26	5	of	of	ADP
cana-788	26	6	the	the	DET
cana-788	26	7	schreier	schreier	NOUN
cana-788	26	8	graphs	graph	NOUN
cana-788	26	9	associated	associate	VERB
cana-788	26	10	with	with	ADP
cana-788	26	11	the	the	DET
cana-788	26	12	actions	action	NOUN
cana-788	26	13	of	of	ADP
cana-788	26	14	the	the	DET
cana-788	26	15	two	two	NUM
cana-788	26	16	well	well	ADV
cana-788	26	17	-	-	PUNCT
cana-788	26	18	known	know	VERB
cana-788	26	19	automorphism	automorphism	NOUN
cana-788	26	20	groups	group	NOUN
cana-788	26	21	of	of	ADP
cana-788	26	22	the	the	DET
cana-788	26	23	binary	binary	ADJ
cana-788	26	24	rooted	root	VERB
cana-788	26	25	tree	tree	NOUN
cana-788	26	26	,	,	PUNCT
cana-788	26	27	basilica	basilica	PROPN
cana-788	26	28	group	group	NOUN
cana-788	26	29	and	and	CCONJ
cana-788	26	30	grigorchuk	grigorchuk	PROPN
cana-788	26	31	group	group	NOUN
cana-788	26	32	is	be	AUX
cana-788	26	33	investigated	investigate	VERB
cana-788	26	34	in	in	ADP
cana-788	26	35	this	this	DET
cana-788	26	36	paper	paper	NOUN
cana-788	26	37	.	.	PUNCT
cana-788	27	1	these	these	DET
cana-788	27	2	groups	group	NOUN
cana-788	27	3	can	can	AUX
cana-788	27	4	be	be	AUX
cana-788	27	5	associated	associate	VERB
cana-788	27	6	with	with	ADP
cana-788	27	7	a	a	DET
cana-788	27	8	compact	compact	ADJ
cana-788	27	9	limit	limit	NOUN
cana-788	27	10	space	space	NOUN
cana-788	27	11	that	that	PRON
cana-788	27	12	is	be	AUX
cana-788	27	13	homeomorphic	homeomorphic	ADJ
cana-788	27	14	to	to	ADP
cana-788	27	15	the	the	DET
cana-788	27	16	basilica	basilica	PROPN
cana-788	27	17	fractal	fractal	NOUN
cana-788	27	18	since	since	SCONJ
cana-788	27	19	it	it	PRON
cana-788	27	20	can	can	AUX
cana-788	27	21	be	be	AUX
cana-788	27	22	described	describe	VERB
cana-788	27	23	as	as	ADP
cana-788	27	24	an	an	DET
cana-788	27	25	iterated	iterated	ADJ
cana-788	27	26	monodromy	monodromy	NOUN
cana-788	27	27	group	group	NOUN
cana-788	27	28	of	of	ADP
cana-788	27	29	the	the	DET
cana-788	27	30	complex	complex	ADJ
cana-788	27	31	polynomial	polynomial	ADJ
cana-788	27	32	z2	z2	NOUN
cana-788	27	33	–	–	PUNCT
cana-788	27	34	1	1	X
cana-788	27	35	.	.	PUNCT
cana-788	28	1	this	this	PRON
cana-788	28	2	is	be	AUX
cana-788	28	3	the	the	DET
cana-788	28	4	first	first	ADJ
cana-788	28	5	example	example	NOUN
cana-788	28	6	of	of	ADP
cana-788	28	7	an	an	DET
cana-788	28	8	amenable	amenable	ADJ
cana-788	28	9	group	group	NOUN
cana-788	28	10	that	that	PRON
cana-788	28	11	does	do	AUX
cana-788	28	12	not	not	PART
cana-788	28	13	belong	belong	VERB
cana-788	28	14	to	to	ADP
cana-788	28	15	a	a	DET
cana-788	28	16	subexponentially	subexponentially	ADV
cana-788	28	17	pliable	pliable	ADJ
cana-788	28	18	group	group	NOUN
cana-788	28	19	[	[	X
cana-788	28	20	11	11	NUM
cana-788	28	21	]	]	PUNCT
cana-788	28	22	.	.	PUNCT
cana-788	29	1	this	this	DET
cana-788	29	2	group	group	NOUN
cana-788	29	3	has	have	AUX
cana-788	29	4	been	be	AUX
cana-788	29	5	demonstrated	demonstrate	VERB
cana-788	29	6	to	to	PART
cana-788	29	7	have	have	VERB
cana-788	29	8	strong	strong	ADJ
cana-788	29	9	connections	connection	NOUN
cana-788	29	10	with	with	ADP
cana-788	29	11	complex	complex	ADJ
cana-788	29	12	dynamics	dynamic	NOUN
cana-788	29	13	and	and	CCONJ
cana-788	29	14	prophinite	prophinite	ADJ
cana-788	29	15	group	group	NOUN
cana-788	29	16	theory	theory	NOUN
cana-788	29	17	[	[	X
cana-788	29	18	12	12	NUM
cana-788	29	19	]	]	PUNCT
cana-788	29	20	.	.	PUNCT
cana-788	30	1	the	the	DET
cana-788	30	2	self	self	NOUN
cana-788	30	3	-	-	PUNCT
cana-788	30	4	similarity	similarity	NOUN
cana-788	30	5	value	value	NOUN
cana-788	30	6	of	of	ADP
cana-788	30	7	some	some	DET
cana-788	30	8	limit	limit	NOUN
cana-788	30	9	objects	object	NOUN
cana-788	30	10	related	relate	VERB
cana-788	30	11	to	to	ADP
cana-788	30	12	these	these	DET
cana-788	30	13	groups	group	NOUN
cana-788	30	14	reflects	reflect	VERB
cana-788	30	15	a	a	DET
cana-788	30	16	fractal	fractal	ADJ
cana-788	30	17	nature	nature	NOUN
cana-788	30	18	[	[	X
cana-788	30	19	8	8	NUM
cana-788	30	20	]	]	PUNCT
cana-788	30	21	.	.	PUNCT
cana-788	31	1	the	the	DET
cana-788	31	2	primary	primary	ADJ
cana-788	31	3	conclusion	conclusion	NOUN
cana-788	31	4	of	of	ADP
cana-788	31	5	section	section	NOUN
cana-788	31	6	2	2	NUM
cana-788	31	7	is	be	AUX
cana-788	31	8	the	the	DET
cana-788	31	9	m	m	NOUN
cana-788	31	10	-	-	ADJ
cana-788	31	11	polynomial	polynomial	ADJ
cana-788	31	12	of	of	ADP
cana-788	31	13	the	the	DET
cana-788	31	14	basilica	basilica	PROPN
cana-788	31	15	group	group	NOUN
cana-788	31	16	's	's	PART
cana-788	31	17	and	and	CCONJ
cana-788	31	18	grigorchuk	grigorchuk	NOUN
cana-788	31	19	group	group	NOUN
cana-788	31	20	's	's	PART
cana-788	31	21	schreier	schreier	NOUN
cana-788	31	22	graphs	graph	NOUN
cana-788	31	23	,	,	PUNCT
cana-788	31	24	from	from	ADP
cana-788	31	25	which	which	PRON
cana-788	31	26	various	various	ADJ
cana-788	31	27	degree	degree	NOUN
cana-788	31	28	-	-	PUNCT
cana-788	31	29	based	base	VERB
cana-788	31	30	topological	topological	ADJ
cana-788	31	31	indices	index	NOUN
cana-788	31	32	are	be	AUX
cana-788	31	33	constructed	construct	VERB
cana-788	31	34	.	.	PUNCT
cana-788	32	1	in	in	ADP
cana-788	32	2	section	section	NOUN
cana-788	32	3	3	3	NUM
cana-788	32	4	,	,	PUNCT
cana-788	32	5	the	the	DET
cana-788	32	6	selected	select	VERB
cana-788	32	7	topological	topological	NOUN
cana-788	32	8	are	be	AUX
cana-788	32	9	calculated	calculate	VERB
cana-788	32	10	and	and	CCONJ
cana-788	32	11	we	we	PRON
cana-788	32	12	make	make	VERB
cana-788	32	13	a	a	DET
cana-788	32	14	comparison	comparison	NOUN
cana-788	32	15	between	between	ADP
cana-788	32	16	the	the	DET
cana-788	32	17	topological	topological	ADJ
cana-788	32	18	indices	index	NOUN
cana-788	32	19	obtained	obtain	VERB
cana-788	32	20	from	from	ADP
cana-788	32	21	the	the	DET
cana-788	32	22	m	m	NOUN
cana-788	32	23	-	-	ADJ
cana-788	32	24	polynomial	polynomial	ADJ
cana-788	32	25	and	and	CCONJ
cana-788	32	26	the	the	DET
cana-788	32	27	estimated	estimate	VERB
cana-788	32	28	values	value	NOUN
cana-788	32	29	of	of	ADP
cana-788	32	30	a	a	DET
cana-788	32	31	few	few	ADJ
cana-788	32	32	selected	select	VERB
cana-788	32	33	topological	topological	ADJ
cana-788	32	34	indices	index	NOUN
cana-788	32	35	.	.	PUNCT
cana-788	33	1	finally	finally	ADV
cana-788	33	2	,	,	PUNCT
cana-788	33	3	the	the	DET
cana-788	33	4	conclusion	conclusion	NOUN
cana-788	33	5	of	of	ADP
cana-788	33	6	this	this	DET
cana-788	33	7	work	work	NOUN
cana-788	33	8	is	be	AUX
cana-788	33	9	done	do	VERB
cana-788	33	10	in	in	ADP
cana-788	33	11	section	section	NOUN
cana-788	33	12	4	4	NUM
cana-788	33	13	.	.	NOUN
cana-788	33	14	2	2	NUM
cana-788	33	15	.	.	X
cana-788	33	16	main	main	ADJ
cana-788	33	17	results	result	NOUN
cana-788	33	18	in	in	ADP
cana-788	33	19	this	this	DET
cana-788	33	20	section	section	NOUN
cana-788	33	21	the	the	DET
cana-788	33	22	m	m	NOUN
cana-788	33	23	-	-	ADJ
cana-788	33	24	polynomial	polynomial	ADJ
cana-788	33	25	of	of	ADP
cana-788	33	26	schreier	schreier	NOUN
cana-788	33	27	graphs	graph	NOUN
cana-788	33	28	of	of	ADP
cana-788	33	29	the	the	DET
cana-788	33	30	basilica	basilica	NOUN
cana-788	33	31	group	group	NOUN
cana-788	33	32	is	be	AUX
cana-788	33	33	derived	derive	VERB
cana-788	33	34	.	.	PUNCT
cana-788	34	1	using	use	VERB
cana-788	34	2	that	that	SCONJ
cana-788	34	3	some	some	DET
cana-788	34	4	degree	degree	NOUN
cana-788	34	5	based	base	VERB
cana-788	34	6	topological	topological	ADJ
cana-788	34	7	indices	index	NOUN
cana-788	34	8	is	be	AUX
cana-788	34	9	derived	derive	VERB
cana-788	34	10	.	.	PUNCT
cana-788	35	1	2.1	2.1	NUM
cana-788	35	2	the	the	DET
cana-788	35	3	m	m	NOUN
cana-788	35	4	-	-	ADJ
cana-788	35	5	polynomial	polynomial	ADJ
cana-788	35	6	of	of	ADP
cana-788	35	7	schreier	schreier	NOUN
cana-788	35	8	graphs	graph	NOUN
cana-788	35	9	(	(	PUNCT
cana-788	35	10	without	without	ADP
cana-788	35	11	loops	loop	NOUN
cana-788	35	12	)	)	PUNCT
cana-788	35	13	of	of	ADP
cana-788	35	14	the	the	DET
cana-788	35	15	basilica	basilica	PROPN
cana-788	35	16	group	group	NOUN
cana-788	35	17	the	the	DET
cana-788	35	18	basilica	basilica	PROPN
cana-788	35	19	group	group	NOUN
cana-788	35	20	is	be	AUX
cana-788	35	21	an	an	DET
cana-788	35	22	automorphism	automorphism	NOUN
cana-788	35	23	collection	collection	NOUN
cana-788	35	24	that	that	PRON
cana-788	35	25	is	be	AUX
cana-788	35	26	self	self	NOUN
cana-788	35	27	-	-	PUNCT
cana-788	35	28	similar	similar	ADJ
cana-788	35	29	and	and	CCONJ
cana-788	35	30	generated	generate	VERB
cana-788	35	31	by	by	ADP
cana-788	35	32	the	the	DET
cana-788	35	33	elements	element	NOUN
cana-788	35	34	𝑎	𝑎	X
cana-788	35	35	=	=	SYM
cana-788	35	36	𝑒(𝑏	𝑒(𝑏	PROPN
cana-788	35	37	,	,	PUNCT
cana-788	35	38	𝑖𝑑	𝑖𝑑	ADP
cana-788	35	39	)	)	PUNCT
cana-788	35	40	and	and	CCONJ
cana-788	35	41	𝑏	𝑏	PRON
cana-788	35	42	=	=	PROPN
cana-788	35	43	∈	∈	PROPN
cana-788	35	44	(	(	PUNCT
cana-788	35	45	𝑎	𝑎	NOUN
cana-788	35	46	,	,	PUNCT
cana-788	35	47	𝑖𝑑	𝑖𝑑	NOUN
cana-788	35	48	)	)	PUNCT
cana-788	35	49	of	of	ADP
cana-788	35	50	the	the	DET
cana-788	35	51	rooted	rooted	ADJ
cana-788	35	52	binary	binary	NOUN
cana-788	35	53	tree	tree	NOUN
cana-788	35	54	.	.	PUNCT
cana-788	36	1	you	you	PRON
cana-788	36	2	may	may	AUX
cana-788	36	3	find	find	VERB
cana-788	36	4	the	the	DET
cana-788	36	5	substitution	substitution	NOUN
cana-788	36	6	rules	rule	NOUN
cana-788	36	7	in	in	ADP
cana-788	36	8	[	[	X
cana-788	36	9	13	13	NUM
cana-788	36	10	]	]	PUNCT
cana-788	36	11	,	,	PUNCT
cana-788	36	12	which	which	PRON
cana-788	36	13	are	be	AUX
cana-788	36	14	used	use	VERB
cana-788	36	15	to	to	PART
cana-788	36	16	generate	generate	VERB
cana-788	36	17	the	the	DET
cana-788	36	18	corresponding	corresponding	ADJ
cana-788	36	19	schreier	schreier	NOUN
cana-788	36	20	graphs	graph	NOUN
cana-788	36	21	iteratively	iteratively	ADV
cana-788	36	22	.	.	PUNCT
cana-788	37	1	for	for	ADP
cana-788	37	2	n>1	n>1	PROPN
cana-788	37	3	,	,	PUNCT
cana-788	37	4	let	let	VERB
cana-788	37	5	𝐵𝑛	𝐵𝑛	PROPN
cana-788	37	6	be	be	AUX
cana-788	37	7	the	the	DET
cana-788	37	8	notation	notation	NOUN
cana-788	37	9	for	for	ADP
cana-788	37	10	the	the	DET
cana-788	37	11	schreier	schreier	ADJ
cana-788	37	12	graphs	graph	NOUN
cana-788	37	13	of	of	ADP
cana-788	37	14	the	the	DET
cana-788	37	15	basilica	basilica	PROPN
cana-788	37	16	group	group	NOUN
cana-788	37	17	,	,	PUNCT
cana-788	37	18	which	which	PRON
cana-788	37	19	are	be	AUX
cana-788	37	20	thought	think	VERB
cana-788	37	21	to	to	PART
cana-788	37	22	be	be	AUX
cana-788	37	23	loop	loop	NOUN
cana-788	37	24	-	-	PUNCT
cana-788	37	25	free	free	ADJ
cana-788	37	26	because	because	SCONJ
cana-788	37	27	the	the	DET
cana-788	37	28	calculations	calculation	NOUN
cana-788	37	29	for	for	ADP
cana-788	37	30	graphs	graph	NOUN
cana-788	37	31	with	with	ADP
cana-788	37	32	loops	loop	NOUN
cana-788	37	33	are	be	AUX
cana-788	37	34	trivial	trivial	ADJ
cana-788	37	35	.	.	PUNCT
cana-788	38	1	figure	figure	NOUN
cana-788	38	2	1	1	NUM
cana-788	38	3	display	display	VERB
cana-788	38	4	these	these	DET
cana-788	38	5	graphs	graph	NOUN
cana-788	38	6	,	,	PUNCT
cana-788	38	7	as	as	SCONJ
cana-788	38	8	shown	show	VERB
cana-788	38	9	in	in	ADP
cana-788	38	10	[	[	X
cana-788	38	11	14	14	NUM
cana-788	38	12	]	]	PUNCT
cana-788	38	13	.	.	PUNCT
cana-788	39	1	definition	definition	NOUN
cana-788	39	2	2.1	2.1	NUM
cana-788	39	3	let	let	VERB
cana-788	39	4	𝐵𝑛	𝐵𝑛	PROPN
cana-788	39	5	=	=	SYM
cana-788	39	6	𝐵𝑛	𝐵𝑛	PROPN
cana-788	39	7	(	(	PUNCT
cana-788	39	8	ɣ	ɣ	PROPN
cana-788	39	9	,	,	PUNCT
cana-788	39	10	ȅ	ȅ	X
cana-788	39	11	)	)	PUNCT
cana-788	39	12	be	be	AUX
cana-788	39	13	a	a	DET
cana-788	39	14	connected	connected	ADJ
cana-788	39	15	graph	graph	NOUN
cana-788	39	16	where	where	SCONJ
cana-788	39	17	ɣ	ɣ	PROPN
cana-788	39	18	is	be	AUX
cana-788	39	19	the	the	DET
cana-788	39	20	set	set	NOUN
cana-788	39	21	of	of	ADP
cana-788	39	22	vertices	vertex	NOUN
cana-788	39	23	and	and	CCONJ
cana-788	39	24	ȅ	ȅ	PROPN
cana-788	39	25	is	be	AUX
cana-788	39	26	the	the	DET
cana-788	39	27	set	set	NOUN
cana-788	39	28	of	of	ADP
cana-788	39	29	edges	edge	NOUN
cana-788	39	30	of	of	ADP
cana-788	39	31	𝐵𝑛.	𝐵𝑛.	PROPN
cana-788	39	32	the	the	DET
cana-788	39	33	m	m	ADJ
cana-788	39	34	–	–	PUNCT
cana-788	39	35	polynomial	polynomial	ADJ
cana-788	39	36	of	of	ADP
cana-788	39	37	graph	graph	NOUN
cana-788	39	38	𝐵𝑛	𝐵𝑛	PROPN
cana-788	39	39	is	be	AUX
cana-788	39	40	defined	define	VERB
cana-788	39	41	as	as	ADP
cana-788	39	42	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	39	43	,	,	PUNCT
cana-788	39	44	𝑧	𝑧	NOUN
cana-788	39	45	)	)	PUNCT
cana-788	39	46	=	=	SYM
cana-788	39	47	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	39	48	,	,	PUNCT
cana-788	39	49	w	w	PROPN
cana-788	39	50	,	,	PUNCT
cana-788	39	51	z	z	NOUN
cana-788	39	52	)	)	PUNCT
cana-788	39	53	=	=	PUNCT
cana-788	40	1	∑	∑	PUNCT
cana-788	40	2	|ȅ𝑖𝑗(𝐵𝑛)|𝑤𝑖𝑧𝑗	|ȅ𝑖𝑗(𝐵𝑛)|𝑤𝑖𝑧𝑗	PROPN
cana-788	40	3	𝕀≤𝑖≤𝑗≤𝕁	𝕀≤𝑖≤𝑗≤𝕁	PROPN
cana-788	40	4	,	,	PUNCT
cana-788	40	5	where	where	SCONJ
cana-788	40	6	𝕀	𝕀	PROPN
cana-788	40	7	=	=	SYM
cana-788	40	8	𝑀𝑖𝑛{𝑑(𝓋	𝑀𝑖𝑛{𝑑(𝓋	PROPN
cana-788	40	9	)	)	PUNCT
cana-788	40	10	𝓋𝜖ɣ(𝐵𝑛)⁄	𝓋𝜖ɣ(𝐵𝑛)⁄	VERB
cana-788	40	11	}	}	PUNCT
cana-788	40	12	;	;	PUNCT
cana-788	40	13	𝕁	𝕁	PROPN
cana-788	40	14	=	=	PUNCT
cana-788	40	15	𝑀𝑎𝑥{𝑑(𝓋	𝑀𝑎𝑥{𝑑(𝓋	PROPN
cana-788	40	16	)	)	PUNCT
cana-788	40	17	𝓋𝜖ɣ(𝐵𝑛)⁄	𝓋𝜖ɣ(𝐵𝑛)⁄	NOUN
cana-788	40	18	}	}	PUNCT
cana-788	40	19	,	,	PUNCT
cana-788	40	20	and	and	CCONJ
cana-788	40	21	ȅ𝑖𝑗(𝐵𝑛	ȅ𝑖𝑗(𝐵𝑛	NUM
cana-788	40	22	)	)	PUNCT
cana-788	40	23	is	be	AUX
cana-788	40	24	the	the	DET
cana-788	40	25	edge	edge	NOUN
cana-788	40	26	𝓋𝓊	𝓋𝓊	ADP
cana-788	40	27	𝜖	𝜖	X
cana-788	40	28	ȅ	ȅ	X
cana-788	40	29	for	for	ADP
cana-788	40	30	which	which	PRON
cana-788	40	31	{	{	PUNCT
cana-788	40	32	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	40	33	)	)	PUNCT
cana-788	40	34	,	,	PUNCT
cana-788	40	35	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	40	36	)	)	PUNCT
cana-788	40	37	}	}	PUNCT
cana-788	40	38	=	=	SYM
cana-788	40	39	{	{	PUNCT
cana-788	40	40	𝑖	𝑖	SYM
cana-788	40	41	,	,	PUNCT
cana-788	40	42	𝑗	𝑗	NOUN
cana-788	40	43	}	}	PUNCT
cana-788	40	44	.	.	PUNCT
cana-788	41	1	where	where	SCONJ
cana-788	41	2	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	41	3	)	)	PUNCT
cana-788	41	4	and	and	CCONJ
cana-788	41	5	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	41	6	)	)	PUNCT
cana-788	41	7	are	be	AUX
cana-788	41	8	degree	degree	NOUN
cana-788	41	9	of	of	ADP
cana-788	41	10	a	a	DET
cana-788	41	11	vertices	vertex	NOUN
cana-788	41	12	𝓋	𝓋	NOUN
cana-788	41	13	and	and	CCONJ
cana-788	41	14	𝓊	𝓊	ADP
cana-788	41	15	respectively	respectively	ADV
cana-788	41	16	.	.	PUNCT
cana-788	42	1	on	on	ADP
cana-788	42	2	the	the	DET
cana-788	42	3	bases	basis	NOUN
cana-788	42	4	of	of	ADP
cana-788	42	5	growth	growth	NOUN
cana-788	42	6	of	of	ADP
cana-788	42	7	the	the	DET
cana-788	42	8	schreier	schreier	NOUN
cana-788	42	9	graphs	graph	NOUN
cana-788	42	10	(	(	PUNCT
cana-788	42	11	without	without	ADP
cana-788	42	12	loops	loop	NOUN
cana-788	42	13	)	)	PUNCT
cana-788	42	14	of	of	ADP
cana-788	42	15	basilica	basilica	NOUN
cana-788	42	16	groups	group	NOUN
cana-788	42	17	having	have	VERB
cana-788	42	18	edges	edge	NOUN
cana-788	42	19	partition	partition	NOUN
cana-788	42	20	ȅ𝑖𝑗	ȅ𝑖𝑗	NOUN
cana-788	42	21	=	=	SYM
cana-788	42	22	{	{	PUNCT
cana-788	42	23	𝓋𝓊	𝓋𝓊	ADP
cana-788	42	24	𝜖	𝜖	PROPN
cana-788	42	25	ȅ(ɠ	ȅ(ɠ	PROPN
cana-788	42	26	)	)	PUNCT
cana-788	42	27	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	42	28	)	)	PUNCT
cana-788	42	29	=	=	SYM
cana-788	42	30	𝑖	𝑖	SYM
cana-788	42	31	,	,	PUNCT
cana-788	42	32	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	42	33	)	)	PUNCT
cana-788	42	34	=	=	PUNCT
cana-788	43	1	𝑗⁄	𝑗⁄	PROPN
cana-788	43	2	}	}	PUNCT
cana-788	43	3	is	be	AUX
cana-788	43	4	the	the	DET
cana-788	43	5	set	set	NOUN
cana-788	43	6	of	of	ADP
cana-788	43	7	all	all	DET
cana-788	43	8	edges	edge	NOUN
cana-788	43	9	with	with	ADP
cana-788	43	10	end	end	NOUN
cana-788	43	11	degrees	degree	NOUN
cana-788	43	12	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	43	13	)	)	PUNCT
cana-788	43	14	=	=	SYM
cana-788	43	15	𝑖	𝑖	SYM
cana-788	43	16	,	,	PUNCT
cana-788	43	17	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	43	18	)	)	PUNCT
cana-788	43	19	=	=	SYM
cana-788	43	20	𝑗	𝑗	NOUN
cana-788	43	21	)	)	PUNCT
cana-788	43	22	.	.	PUNCT
cana-788	44	1	on	on	ADP
cana-788	44	2	the	the	DET
cana-788	44	3	bases	basis	NOUN
cana-788	44	4	of	of	ADP
cana-788	44	5	degrees	degree	NOUN
cana-788	44	6	,	,	PUNCT
cana-788	44	7	we	we	PRON
cana-788	44	8	divide	divide	VERB
cana-788	44	9	the	the	DET
cana-788	44	10	edges	edge	NOUN
cana-788	44	11	into	into	ADP
cana-788	44	12	two	two	NUM
cana-788	44	13	partitions	partition	NOUN
cana-788	44	14	for	for	ADP
cana-788	44	15	𝐵𝑛	𝐵𝑛	PROPN
cana-788	44	16	,	,	PUNCT
cana-788	44	17	𝑛	𝑛	PROPN
cana-788	44	18	>	>	X
cana-788	44	19	1	1	X
cana-788	44	20	.	.	PUNCT
cana-788	44	21	communications	communication	NOUN
cana-788	44	22	on	on	ADP
cana-788	44	23	applied	apply	VERB
cana-788	44	24	nonlinear	nonlinear	ADJ
cana-788	44	25	analysis	analysis	NOUN
cana-788	44	26	issn	issn	NOUN
cana-788	44	27	:	:	PUNCT
cana-788	44	28	1074	1074	NUM
cana-788	44	29	-	-	PUNCT
cana-788	44	30	133x	133x	NUM
cana-788	44	31	vol	vol	NOUN
cana-788	44	32	31	31	NUM
cana-788	44	33	no	no	NOUN
cana-788	44	34	.	.	PUNCT
cana-788	45	1	3s	3s	NUM
cana-788	45	2	(	(	PUNCT
cana-788	45	3	2024	2024	NUM
cana-788	45	4	)	)	PUNCT
cana-788	45	5	379	379	NUM
cana-788	45	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	45	7	ȅ24	ȅ24	NUM
cana-788	45	8	=	=	SYM
cana-788	45	9	{	{	PUNCT
cana-788	45	10	𝓋𝓊	𝓋𝓊	ADP
cana-788	45	11	𝜖	𝜖	PROPN
cana-788	45	12	ȅ(ɠ	ȅ(ɠ	PROPN
cana-788	45	13	)	)	PUNCT
cana-788	45	14	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	45	15	)	)	PUNCT
cana-788	45	16	=	=	SYM
cana-788	45	17	2	2	NUM
cana-788	45	18	,	,	PUNCT
cana-788	45	19	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	45	20	)	)	PUNCT
cana-788	45	21	=	=	SYM
cana-788	46	1	4⁄	4⁄	NUM
cana-788	46	2	}	}	PUNCT
cana-788	46	3	ȅ44	ȅ44	NOUN
cana-788	46	4	=	=	SYM
cana-788	46	5	{	{	PUNCT
cana-788	46	6	𝓋𝓊	𝓋𝓊	ADP
cana-788	46	7	𝜖	𝜖	PROPN
cana-788	46	8	ȅ(ɠ	ȅ(ɠ	PROPN
cana-788	46	9	)	)	PUNCT
cana-788	46	10	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	46	11	)	)	PUNCT
cana-788	46	12	=	=	SYM
cana-788	46	13	4	4	NUM
cana-788	46	14	,	,	PUNCT
cana-788	46	15	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	46	16	)	)	PUNCT
cana-788	46	17	=	=	SYM
cana-788	47	1	4⁄	4⁄	NUM
cana-788	47	2	}	}	PUNCT
cana-788	47	3	.	.	PUNCT
cana-788	48	1	fig	fig	NOUN
cana-788	48	2	.	.	PUNCT
cana-788	49	1	1	1	NUM
cana-788	49	2	structure	structure	NOUN
cana-788	49	3	of	of	ADP
cana-788	49	4	𝐵𝑛	𝐵𝑛	PROPN
cana-788	49	5	for	for	ADP
cana-788	49	6	some	some	DET
cana-788	49	7	𝑛	𝑛	DET
cana-788	49	8	communications	communication	NOUN
cana-788	49	9	on	on	ADP
cana-788	49	10	applied	apply	VERB
cana-788	49	11	nonlinear	nonlinear	ADJ
cana-788	49	12	analysis	analysis	NOUN
cana-788	49	13	issn	issn	NOUN
cana-788	49	14	:	:	PUNCT
cana-788	49	15	1074	1074	NUM
cana-788	49	16	-	-	PUNCT
cana-788	49	17	133x	133x	NUM
cana-788	49	18	vol	vol	NOUN
cana-788	49	19	31	31	NUM
cana-788	49	20	no	no	NOUN
cana-788	49	21	.	.	PUNCT
cana-788	50	1	3s	3s	NUM
cana-788	50	2	(	(	PUNCT
cana-788	50	3	2024	2024	NUM
cana-788	50	4	)	)	PUNCT
cana-788	50	5	380	380	NUM
cana-788	50	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	50	7	the	the	DET
cana-788	50	8	following	follow	VERB
cana-788	50	9	table	table	NOUN
cana-788	50	10	1	1	NUM
cana-788	50	11	depicts	depict	VERB
cana-788	50	12	the	the	DET
cana-788	50	13	selected	select	VERB
cana-788	50	14	topological	topological	ADJ
cana-788	50	15	indices	index	NOUN
cana-788	50	16	for	for	ADP
cana-788	50	17	this	this	DET
cana-788	50	18	study	study	NOUN
cana-788	50	19	and	and	CCONJ
cana-788	50	20	its	its	PRON
cana-788	50	21	derivation	derivation	NOUN
cana-788	50	22	formulas	formula	NOUN
cana-788	51	1	[	[	X
cana-788	51	2	6	6	NUM
cana-788	51	3	]	]	PUNCT
cana-788	51	4	through	through	ADP
cana-788	51	5	m	m	ADJ
cana-788	51	6	-	-	ADJ
cana-788	51	7	polynomial	polynomial	ADJ
cana-788	51	8	.	.	PUNCT
cana-788	52	1	table	table	NOUN
cana-788	52	2	1	1	NUM
cana-788	52	3	.	.	PUNCT
cana-788	52	4	degree	degree	NOUN
cana-788	52	5	based	base	VERB
cana-788	52	6	m	m	PROPN
cana-788	52	7	–	–	PUNCT
cana-788	53	1	polynomial	polynomial	ADJ
cana-788	53	2	topological	topological	ADJ
cana-788	53	3	index	index	NOUN
cana-788	53	4	degree	degree	NOUN
cana-788	53	5	based	base	VERB
cana-788	53	6	derivation	derivation	NOUN
cana-788	53	7	first	first	PROPN
cana-788	53	8	zagreb	zagreb	PROPN
cana-788	53	9	index	index	NOUN
cana-788	53	10	(	(	PUNCT
cana-788	53	11	𝑀1	𝑀1	PROPN
cana-788	53	12	)	)	PUNCT
cana-788	53	13	∑	∑	ADV
cana-788	53	14	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	53	15	)	)	PUNCT
cana-788	54	1	+	+	CCONJ
cana-788	54	2	𝑑(𝓊	𝑑(𝓊	X
cana-788	54	3	)	)	PUNCT
cana-788	54	4	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	54	5	(	(	PUNCT
cana-788	54	6	𝐷𝑤	𝐷𝑤	PROPN
cana-788	54	7	+	+	CCONJ
cana-788	54	8	𝐷𝑧	𝐷𝑧	PROPN
cana-788	54	9	)	)	PUNCT
cana-788	54	10	(	(	PUNCT
cana-788	54	11	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	54	12	,	,	PUNCT
cana-788	54	13	w	w	PROPN
cana-788	54	14	,	,	PUNCT
cana-788	54	15	z))|	z))|	PROPN
cana-788	54	16	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	54	17	second	second	ADJ
cana-788	54	18	zagreb	zagreb	PROPN
cana-788	54	19	index	index	NOUN
cana-788	54	20	(	(	PUNCT
cana-788	54	21	𝑀2	𝑀2	PROPN
cana-788	54	22	)	)	PUNCT
cana-788	54	23	∑	∑	ADV
cana-788	54	24	𝑑(𝓋)𝑑(𝓊	𝑑(𝓋)𝑑(𝓊	X
cana-788	54	25	)	)	PUNCT
cana-788	54	26	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	54	27	(	(	PUNCT
cana-788	54	28	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	54	29	)	)	PUNCT
cana-788	54	30	(	(	PUNCT
cana-788	54	31	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	54	32	,	,	PUNCT
cana-788	54	33	w	w	PROPN
cana-788	54	34	,	,	PUNCT
cana-788	54	35	z))|	z))|	PROPN
cana-788	54	36	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	54	37	modified	modify	VERB
cana-788	54	38	second	second	ADJ
cana-788	54	39	zagreb	zagreb	PROPN
cana-788	54	40	index	index	NOUN
cana-788	54	41	(	(	PUNCT
cana-788	54	42	𝑀2	𝑀2	PROPN
cana-788	54	43	𝑚	𝑚	PROPN
cana-788	54	44	)	)	PUNCT
cana-788	54	45	∑	∑	PROPN
cana-788	54	46	1	1	NUM
cana-788	54	47	𝑑(𝓋)𝑑(𝓊	𝑑(𝓋)𝑑(𝓊	NOUN
cana-788	54	48	)	)	PUNCT
cana-788	54	49	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	54	50	(	(	PUNCT
cana-788	54	51	𝑆𝑤𝑆𝑧	𝑆𝑤𝑆𝑧	PROPN
cana-788	54	52	)	)	PUNCT
cana-788	54	53	(	(	PUNCT
cana-788	54	54	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	54	55	,	,	PUNCT
cana-788	54	56	w	w	PROPN
cana-788	54	57	,	,	PUNCT
cana-788	54	58	z))|	z))|	PROPN
cana-788	54	59	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	54	60	hyper	hyper	ADJ
cana-788	54	61	zagreb	zagreb	PROPN
cana-788	54	62	index	index	PROPN
cana-788	54	63	(	(	PUNCT
cana-788	54	64	hm	hm	INTJ
cana-788	54	65	)	)	PUNCT
cana-788	54	66	∑	∑	PUNCT
cana-788	55	1	[	[	X
cana-788	55	2	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	55	3	)	)	PUNCT
cana-788	55	4	+	+	CCONJ
cana-788	55	5	𝑑(𝓊)]2	𝑑(𝓊)]2	ADV
cana-788	55	6	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	55	7	(	(	PUNCT
cana-788	55	8	𝐷𝑤	𝐷𝑤	PROPN
cana-788	55	9	+	+	PROPN
cana-788	55	10	𝐷𝑧)2	𝐷𝑧)2	PROPN
cana-788	55	11	(	(	PUNCT
cana-788	55	12	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	55	13	,	,	PUNCT
cana-788	55	14	w	w	PROPN
cana-788	55	15	,	,	PUNCT
cana-788	55	16	z))|	z))|	PROPN
cana-788	55	17	𝑤=𝑧=1	𝑤=𝑧=1	NOUN
cana-788	55	18	forgotten	forget	VERB
cana-788	55	19	index	index	NOUN
cana-788	55	20	(	(	PUNCT
cana-788	55	21	f	f	X
cana-788	55	22	)	)	PUNCT
cana-788	55	23	∑	∑	PUNCT
cana-788	55	24	𝑑(𝓋)2	𝑑(𝓋)2	PROPN
cana-788	55	25	+	+	CCONJ
cana-788	55	26	𝑑(𝓊)2	𝑑(𝓊)2	PROPN
cana-788	55	27	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	55	28	(	(	PUNCT
cana-788	55	29	𝐷𝑤	𝐷𝑤	PROPN
cana-788	55	30	2	2	NUM
cana-788	55	31	+	+	CCONJ
cana-788	55	32	𝐷𝑧	𝐷𝑧	PROPN
cana-788	55	33	2	2	NUM
cana-788	55	34	)	)	PUNCT
cana-788	55	35	(	(	PUNCT
cana-788	55	36	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	55	37	,	,	PUNCT
cana-788	55	38	w	w	PROPN
cana-788	55	39	,	,	PUNCT
cana-788	55	40	z))|	z))|	PROPN
cana-788	55	41	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	55	42	inverse	inverse	NOUN
cana-788	55	43	sum	sum	NOUN
cana-788	55	44	index	index	NOUN
cana-788	55	45	(	(	PUNCT
cana-788	55	46	i	i	NOUN
cana-788	55	47	)	)	PUNCT
cana-788	55	48	∑	∑	ADV
cana-788	55	49	𝑑(𝓋)𝑑(𝓊	𝑑(𝓋)𝑑(𝓊	X
cana-788	55	50	)	)	PUNCT
cana-788	55	51	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	55	52	)	)	PUNCT
cana-788	56	1	+	+	CCONJ
cana-788	56	2	𝑑(𝓊	𝑑(𝓊	X
cana-788	56	3	)	)	PUNCT
cana-788	56	4	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	56	5	(	(	PUNCT
cana-788	56	6	𝑆𝑤𝐽	𝑆𝑤𝐽	PROPN
cana-788	56	7	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	56	8	)	)	PUNCT
cana-788	56	9	(	(	PUNCT
cana-788	56	10	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	56	11	,	,	PUNCT
cana-788	56	12	w	w	PROPN
cana-788	56	13	,	,	PUNCT
cana-788	56	14	z))|	z))|	PROPN
cana-788	56	15	𝑤=1	𝑤=1	PUNCT
cana-788	56	16	augmented	augment	VERB
cana-788	56	17	zagreb	zagreb	PROPN
cana-788	56	18	index	index	NOUN
cana-788	56	19	(	(	PUNCT
cana-788	56	20	a	a	NOUN
cana-788	56	21	)	)	PUNCT
cana-788	56	22	∑	∑	PROPN
cana-788	56	23	(	(	PUNCT
cana-788	56	24	𝑑(𝓋)𝑑(𝓊	𝑑(𝓋)𝑑(𝓊	NOUN
cana-788	56	25	)	)	PUNCT
cana-788	56	26	𝑑(𝓋	𝑑(𝓋	NOUN
cana-788	56	27	)	)	PUNCT
cana-788	57	1	+	+	CCONJ
cana-788	57	2	𝑑(𝓊	𝑑(𝓊	NUM
cana-788	57	3	)	)	PUNCT
cana-788	57	4	−	−	PROPN
cana-788	57	5	2	2	NUM
cana-788	57	6	)	)	PUNCT
cana-788	57	7	3	3	NUM
cana-788	57	8	𝑒=𝓋𝓊∈ȅ	𝑒=𝓋𝓊∈ȅ	PROPN
cana-788	57	9	(	(	PUNCT
cana-788	57	10	𝑆𝑤𝑄−2𝐽	𝑆𝑤𝑄−2𝐽	ADP
cana-788	57	11	𝐷𝑤	𝐷𝑤	PROPN
cana-788	57	12	3𝐷𝑧	3𝐷𝑧	NUM
cana-788	57	13	3	3	NUM
cana-788	57	14	)	)	PUNCT
cana-788	57	15	(	(	PUNCT
cana-788	57	16	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	57	17	,	,	PUNCT
cana-788	57	18	w	w	PROPN
cana-788	57	19	,	,	PUNCT
cana-788	57	20	z))|	z))|	PROPN
cana-788	57	21	𝑤=1=11	𝑤=1=11	NOUN
cana-788	57	22	note	note	VERB
cana-788	57	23	that	that	SCONJ
cana-788	57	24	𝐷𝑤	𝐷𝑤	PROPN
cana-788	57	25	=	=	SYM
cana-788	57	26	𝑤	𝑤	ADP
cana-788	57	27	𝜕	𝜕	NOUN
cana-788	57	28	𝜕𝑤	𝜕𝑤	NOUN
cana-788	57	29	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	57	30	,	,	PUNCT
cana-788	57	31	𝑧	𝑧	NOUN
cana-788	57	32	)	)	PUNCT
cana-788	57	33	,	,	PUNCT
cana-788	57	34	𝐷𝑧	𝐷𝑧	PROPN
cana-788	57	35	=	=	PUNCT
cana-788	57	36	𝑧	𝑧	PRON
cana-788	57	37	𝜕	𝜕	NOUN
cana-788	57	38	𝜕𝑧	𝜕𝑧	PROPN
cana-788	57	39	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	57	40	,	,	PUNCT
cana-788	57	41	𝑧	𝑧	NOUN
cana-788	57	42	)	)	PUNCT
cana-788	57	43	,	,	PUNCT
cana-788	57	44	𝑆𝑤	𝑆𝑤	PROPN
cana-788	57	45	=	=	SYM
cana-788	57	46	∫	∫	PROPN
cana-788	57	47	𝑓(𝑡,𝑧	𝑓(𝑡,𝑧	NUM
cana-788	57	48	)	)	PUNCT
cana-788	57	49	𝑡	𝑡	PROPN
cana-788	57	50	𝑑𝑡	𝑑𝑡	ADP
cana-788	57	51	𝑤	𝑤	ADP
cana-788	57	52	0	0	NUM
cana-788	57	53	,	,	PUNCT
cana-788	57	54	𝑆𝑧	𝑆𝑧	PROPN
cana-788	57	55	=	=	PUNCT
cana-788	57	56	∫	∫	PROPN
cana-788	57	57	𝑓(𝑤,𝑡	𝑓(𝑤,𝑡	NUM
cana-788	57	58	)	)	PUNCT
cana-788	57	59	𝑡	𝑡	PROPN
cana-788	57	60	𝑑𝑡	𝑑𝑡	ADP
cana-788	57	61	𝑧	𝑧	PROPN
cana-788	57	62	0	0	NUM
cana-788	57	63	,	,	PUNCT
cana-788	57	64	𝐽(𝑓(𝑤	𝐽(𝑓(𝑤	PROPN
cana-788	57	65	,	,	PUNCT
cana-788	57	66	𝑧	𝑧	NOUN
cana-788	57	67	)	)	PUNCT
cana-788	57	68	)	)	PUNCT
cana-788	58	1	=	=	SYM
cana-788	58	2	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	58	3	,	,	PUNCT
cana-788	58	4	𝑤),𝑄∝(𝑓(𝑤	𝑤),𝑄∝(𝑓(𝑤	NOUN
cana-788	58	5	,	,	PUNCT
cana-788	58	6	𝑧	𝑧	NOUN
cana-788	58	7	)	)	PUNCT
cana-788	58	8	)	)	PUNCT
cana-788	58	9	=	=	PUNCT
cana-788	58	10	𝑤∝𝑓(𝑤	𝑤∝𝑓(𝑤	ADJ
cana-788	58	11	,	,	PUNCT
cana-788	58	12	𝑧	𝑧	NOUN
cana-788	58	13	)	)	PUNCT
cana-788	58	14	.	.	PUNCT
cana-788	59	1	theorem	theorem	VERB
cana-788	59	2	2.1	2.1	NUM
cana-788	59	3	.	.	PUNCT
cana-788	60	1	let	let	VERB
cana-788	60	2	𝐵𝑛	𝐵𝑛	PROPN
cana-788	60	3	be	be	AUX
cana-788	60	4	schreier	schreier	ADJ
cana-788	60	5	graph	graph	NOUN
cana-788	60	6	(	(	PUNCT
cana-788	60	7	without	without	ADP
cana-788	60	8	loops	loop	NOUN
cana-788	60	9	)	)	PUNCT
cana-788	60	10	of	of	ADP
cana-788	60	11	basilica	basilica	PROPN
cana-788	60	12	group	group	NOUN
cana-788	60	13	.	.	PUNCT
cana-788	61	1	then	then	ADV
cana-788	61	2	the	the	DET
cana-788	61	3	m	m	ADJ
cana-788	61	4	–	–	PUNCT
cana-788	61	5	polynomial	polynomial	ADJ
cana-788	61	6	of	of	ADP
cana-788	61	7	𝐵𝑛	𝐵𝑛	PROPN
cana-788	61	8	is	be	AUX
cana-788	61	9	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	61	10	,	,	PUNCT
cana-788	61	11	𝑧	𝑧	NOUN
cana-788	61	12	)	)	PUNCT
cana-788	61	13	=	=	SYM
cana-788	61	14	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	61	15	,	,	PUNCT
cana-788	61	16	w	w	PROPN
cana-788	61	17	,	,	PUNCT
cana-788	61	18	z	z	NOUN
cana-788	61	19	)	)	PUNCT
cana-788	61	20	=	=	PUNCT
cana-788	62	1	(	(	PUNCT
cana-788	62	2	2𝑛)𝑤2𝑧4	2𝑛)𝑤2𝑧4	NUM
cana-788	62	3	+	+	CCONJ
cana-788	62	4	(	(	PUNCT
cana-788	62	5	2𝑛−1)𝑤4𝑧4	2𝑛−1)𝑤4𝑧4	NOUN
cana-788	62	6	.	.	PUNCT
cana-788	62	7	proof	proof	NOUN
cana-788	62	8	.	.	PUNCT
cana-788	63	1	from	from	ADP
cana-788	63	2	the	the	DET
cana-788	63	3	following	follow	VERB
cana-788	63	4	table	table	NOUN
cana-788	63	5	2	2	NUM
cana-788	63	6	we	we	PRON
cana-788	63	7	can	can	AUX
cana-788	63	8	understand	understand	VERB
cana-788	63	9	the	the	DET
cana-788	63	10	edge	edge	NOUN
cana-788	63	11	growth	growth	NOUN
cana-788	63	12	pattern	pattern	NOUN
cana-788	63	13	and	and	CCONJ
cana-788	63	14	edge	edge	NOUN
cana-788	63	15	partition	partition	NOUN
cana-788	63	16	of	of	ADP
cana-788	63	17	schreier	schreier	ADJ
cana-788	63	18	graph	graph	NOUN
cana-788	63	19	of	of	ADP
cana-788	63	20	basilica	basilica	PROPN
cana-788	63	21	group	group	NOUN
cana-788	63	22	.	.	PUNCT
cana-788	64	1	table	table	NOUN
cana-788	64	2	2	2	NUM
cana-788	64	3	.	.	PUNCT
cana-788	64	4	edge	edge	NOUN
cana-788	64	5	growth	growth	NOUN
cana-788	64	6	pattern	pattern	NOUN
cana-788	64	7	and	and	CCONJ
cana-788	64	8	edge	edge	NOUN
cana-788	64	9	partition	partition	NOUN
cana-788	64	10	of	of	ADP
cana-788	64	11	schreier	schreier	ADJ
cana-788	64	12	graph	graph	NOUN
cana-788	64	13	of	of	ADP
cana-788	64	14	basilica	basilica	PROPN
cana-788	64	15	group	group	PROPN
cana-788	64	16	basilica	basilica	PROPN
cana-788	64	17	group	group	PROPN
cana-788	64	18	(	(	PUNCT
cana-788	64	19	𝑩𝒏	𝑩𝒏	PROPN
cana-788	64	20	)	)	PUNCT
cana-788	64	21	number	number	NOUN
cana-788	64	22	of	of	ADP
cana-788	64	23	vertices	vertex	NOUN
cana-788	64	24	number	number	NOUN
cana-788	64	25	of	of	ADP
cana-788	64	26	edges	edge	NOUN
cana-788	64	27	edge	edge	NOUN
cana-788	64	28	growth	growth	NOUN
cana-788	64	29	pattern	pattern	NOUN
cana-788	64	30	edge	edge	NOUN
cana-788	64	31	partitions	partition	NOUN
cana-788	64	32	(	(	PUNCT
cana-788	64	33	2,4	2,4	NUM
cana-788	64	34	)	)	PUNCT
cana-788	64	35	(	(	PUNCT
cana-788	64	36	4,4	4,4	NOUN
cana-788	64	37	)	)	PUNCT
cana-788	64	38	b2	b2	NOUN
cana-788	64	39	4	4	NUM
cana-788	64	40	6	6	NUM
cana-788	64	41	(	(	PUNCT
cana-788	64	42	22	22	NUM
cana-788	64	43	)	)	PUNCT
cana-788	65	1	+	+	CCONJ
cana-788	65	2	(	(	PUNCT
cana-788	65	3	21	21	NUM
cana-788	65	4	)	)	SYM
cana-788	65	5	4	4	NUM
cana-788	65	6	2	2	NUM
cana-788	65	7	b3	b3	NOUN
cana-788	65	8	8	8	NUM
cana-788	65	9	12	12	NUM
cana-788	65	10	(	(	PUNCT
cana-788	65	11	23	23	NUM
cana-788	65	12	)	)	PUNCT
cana-788	65	13	+	+	CCONJ
cana-788	65	14	(	(	PUNCT
cana-788	65	15	22	22	NUM
cana-788	65	16	)	)	PUNCT
cana-788	65	17	8	8	NUM
cana-788	65	18	4	4	NUM
cana-788	65	19	b4	b4	NOUN
cana-788	65	20	16	16	NUM
cana-788	65	21	24	24	NUM
cana-788	65	22	(	(	PUNCT
cana-788	65	23	24	24	NUM
cana-788	65	24	)	)	PUNCT
cana-788	65	25	+	+	CCONJ
cana-788	65	26	(	(	PUNCT
cana-788	65	27	23	23	NUM
cana-788	65	28	)	)	PUNCT
cana-788	65	29	16	16	NUM
cana-788	65	30	8	8	NUM
cana-788	65	31	b5	b5	PROPN
cana-788	65	32	32	32	NUM
cana-788	65	33	48	48	NUM
cana-788	65	34	(	(	PUNCT
cana-788	65	35	25	25	NUM
cana-788	65	36	)	)	PUNCT
cana-788	65	37	+	+	CCONJ
cana-788	65	38	(	(	PUNCT
cana-788	65	39	24	24	NUM
cana-788	65	40	)	)	PUNCT
cana-788	65	41	32	32	NUM
cana-788	65	42	16	16	NUM
cana-788	65	43	b6	b6	NOUN
cana-788	65	44	64	64	NUM
cana-788	65	45	96	96	NUM
cana-788	65	46	(	(	PUNCT
cana-788	65	47	26	26	NUM
cana-788	65	48	)	)	PUNCT
cana-788	65	49	+	+	CCONJ
cana-788	65	50	(	(	PUNCT
cana-788	65	51	25	25	NUM
cana-788	65	52	)	)	PUNCT
cana-788	65	53	64	64	NUM
cana-788	65	54	32	32	NUM
cana-788	65	55	.	.	PUNCT
cana-788	65	56	.	.	PUNCT
cana-788	65	57	.	.	PUNCT
cana-788	65	58	.	.	PUNCT
cana-788	65	59	.	.	PUNCT
cana-788	65	60	.	.	PUNCT
cana-788	65	61	.	.	PUNCT
cana-788	65	62	.	.	PUNCT
cana-788	65	63	.	.	PUNCT
cana-788	65	64	.	.	PUNCT
cana-788	65	65	.	.	PUNCT
cana-788	65	66	.	.	PUNCT
cana-788	66	1	bn	bn	PROPN
cana-788	66	2	2𝑛	2𝑛	PROPN
cana-788	66	3	(	(	PUNCT
cana-788	66	4	2𝑛	2𝑛	NUM
cana-788	66	5	)	)	PUNCT
cana-788	67	1	+	+	CCONJ
cana-788	67	2	(	(	PUNCT
cana-788	67	3	2𝑛−1	2𝑛−1	NUM
cana-788	67	4	)	)	PUNCT
cana-788	67	5	2𝑛	2𝑛	PROPN
cana-788	67	6	2𝑛−1	2𝑛−1	NUM
cana-788	67	7	from	from	ADP
cana-788	67	8	table	table	NOUN
cana-788	67	9	1	1	NUM
cana-788	67	10	,	,	PUNCT
cana-788	67	11	it	it	PRON
cana-788	67	12	is	be	AUX
cana-788	67	13	clear	clear	ADJ
cana-788	67	14	that	that	SCONJ
cana-788	67	15	the	the	DET
cana-788	67	16	number	number	NOUN
cana-788	67	17	of	of	ADP
cana-788	67	18	edges	edge	NOUN
cana-788	67	19	in	in	ADP
cana-788	67	20	the	the	DET
cana-788	67	21	above	above	ADJ
cana-788	67	22	two	two	NUM
cana-788	67	23	categories	category	NOUN
cana-788	67	24	are	be	AUX
cana-788	67	25	|ȅ24|	|ȅ24|	PROPN
cana-788	67	26	=	=	SYM
cana-788	67	27	2𝑛	2𝑛	PROPN
cana-788	67	28	and	and	CCONJ
cana-788	67	29	|ȅ44|	|ȅ44|	NOUN
cana-788	67	30	=	=	SYM
cana-788	67	31	2𝑛−1	2𝑛−1	NOUN
cana-788	67	32	.	.	PUNCT
cana-788	68	1	now	now	ADV
cana-788	68	2	,	,	PUNCT
cana-788	68	3	by	by	ADP
cana-788	68	4	the	the	DET
cana-788	68	5	definition	definition	NOUN
cana-788	68	6	of	of	ADP
cana-788	68	7	m	m	NOUN
cana-788	68	8	-	-	ADJ
cana-788	68	9	polynomial	polynomial	ADJ
cana-788	68	10	,	,	PUNCT
cana-788	68	11	we	we	PRON
cana-788	68	12	have	have	VERB
cana-788	68	13	communications	communication	NOUN
cana-788	68	14	on	on	ADP
cana-788	68	15	applied	apply	VERB
cana-788	68	16	nonlinear	nonlinear	ADJ
cana-788	68	17	analysis	analysis	NOUN
cana-788	68	18	issn	issn	NOUN
cana-788	68	19	:	:	PUNCT
cana-788	68	20	1074	1074	NUM
cana-788	68	21	-	-	PUNCT
cana-788	68	22	133x	133x	NUM
cana-788	68	23	vol	vol	NOUN
cana-788	68	24	31	31	NUM
cana-788	68	25	no	no	NOUN
cana-788	68	26	.	.	PUNCT
cana-788	69	1	3s	3s	NUM
cana-788	69	2	(	(	PUNCT
cana-788	69	3	2024	2024	NUM
cana-788	69	4	)	)	PUNCT
cana-788	69	5	381	381	NUM
cana-788	69	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	69	7	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	69	8	,	,	PUNCT
cana-788	69	9	𝑧	𝑧	NOUN
cana-788	69	10	)	)	PUNCT
cana-788	69	11	=	=	SYM
cana-788	69	12	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	69	13	,	,	PUNCT
cana-788	69	14	w	w	PROPN
cana-788	69	15	,	,	PUNCT
cana-788	69	16	z	z	NOUN
cana-788	69	17	)	)	PUNCT
cana-788	69	18	=	=	PUNCT
cana-788	69	19	∑	∑	PUNCT
cana-788	69	20	|ȅ𝑖𝑗(𝐵𝑛)|𝑤𝑖𝑧𝑗	|ȅ𝑖𝑗(𝐵𝑛)|𝑤𝑖𝑧𝑗	PRON
cana-788	69	21	𝕀≤𝑖≤𝑗≤𝕁	𝕀≤𝑖≤𝑗≤𝕁	PROPN
cana-788	69	22	=	=	SYM
cana-788	69	23	|ȅ24|𝑤2𝑧4	|ȅ24|𝑤2𝑧4	PROPN
cana-788	69	24	+	+	CCONJ
cana-788	69	25	|ȅ44|𝑤4𝑧4	|ȅ44|𝑤4𝑧4	PROPN
cana-788	69	26	∴	∴	PROPN
cana-788	69	27	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	69	28	,	,	PUNCT
cana-788	69	29	𝑧	𝑧	NOUN
cana-788	69	30	)	)	PUNCT
cana-788	69	31	=	=	SYM
cana-788	69	32	(	(	PUNCT
cana-788	69	33	2𝑛)𝑤2𝑧4	2𝑛)𝑤2𝑧4	NUM
cana-788	69	34	+	+	CCONJ
cana-788	69	35	(	(	PUNCT
cana-788	69	36	2𝑛−1)𝑤4𝑧4	2𝑛−1)𝑤4𝑧4	NUM
cana-788	69	37	.	.	PUNCT
cana-788	70	1	□	□	PUNCT
cana-788	70	2	fig	fig	NOUN
cana-788	70	3	.	.	NOUN
cana-788	70	4	2	2	NUM
cana-788	70	5	surface	surface	NOUN
cana-788	70	6	plot	plot	NOUN
cana-788	70	7	of	of	ADP
cana-788	70	8	m	m	NOUN
cana-788	70	9	-	-	ADJ
cana-788	70	10	polynomial	polynomial	ADJ
cana-788	70	11	of	of	ADP
cana-788	70	12	𝐵𝑛.	𝐵𝑛.	PROPN
cana-788	70	13	theorem	theorem	ADJ
cana-788	70	14	2.2	2.2	NUM
cana-788	70	15	let	let	VERB
cana-788	70	16	𝐵𝑛	𝐵𝑛	PROPN
cana-788	70	17	be	be	AUX
cana-788	70	18	the	the	DET
cana-788	70	19	schreier	schreier	ADJ
cana-788	70	20	graph	graph	NOUN
cana-788	70	21	of	of	ADP
cana-788	70	22	basilica	basilica	PROPN
cana-788	70	23	group	group	NOUN
cana-788	70	24	.	.	PUNCT
cana-788	71	1	then	then	ADV
cana-788	71	2	the	the	DET
cana-788	71	3	m	m	NOUN
cana-788	71	4	-	-	PUNCT
cana-788	71	5	polynomials	polynomial	NOUN
cana-788	71	6	for	for	ADP
cana-788	71	7	the	the	DET
cana-788	71	8	topological	topological	ADJ
cana-788	71	9	indices	index	NOUN
cana-788	71	10	of	of	ADP
cana-788	71	11	𝐵𝑛	𝐵𝑛	PROPN
cana-788	71	12	for	for	ADP
cana-788	71	13	𝑛	𝑛	ADP
cana-788	71	14	>	>	SYM
cana-788	71	15	1	1	NUM
cana-788	71	16	are	be	AUX
cana-788	71	17	a	a	DET
cana-788	71	18	)	)	PUNCT
cana-788	71	19	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	PROPN
cana-788	71	20	)	)	PUNCT
cana-788	71	21	=	=	SYM
cana-788	71	22	5	5	NUM
cana-788	71	23	×	×	NOUN
cana-788	71	24	2𝑛+1	2𝑛+1	NUM
cana-788	71	25	b	b	NOUN
cana-788	71	26	)	)	PUNCT
cana-788	71	27	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	NOUN
cana-788	71	28	)	)	PUNCT
cana-788	71	29	=	=	PUNCT
cana-788	72	1	2𝑛+4	2𝑛+4	NUM
cana-788	72	2	c	c	NOUN
cana-788	72	3	)	)	PUNCT
cana-788	72	4	𝑀2	𝑀2	PROPN
cana-788	72	5	𝑚(𝐵𝑛	𝑚(𝐵𝑛	NUM
cana-788	72	6	)	)	PUNCT
cana-788	72	7	=	=	SYM
cana-788	72	8	5	5	NUM
cana-788	72	9	×	×	NOUN
cana-788	72	10	2𝑛−5	2𝑛−5	NUM
cana-788	72	11	d	d	NOUN
cana-788	72	12	)	)	PUNCT
cana-788	72	13	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	PROPN
cana-788	72	14	)	)	PUNCT
cana-788	72	15	=	=	SYM
cana-788	72	16	3	3	NUM
cana-788	72	17	×	×	NOUN
cana-788	72	18	2𝑛+1	2𝑛+1	NUM
cana-788	72	19	e	e	NOUN
cana-788	72	20	)	)	PUNCT
cana-788	72	21	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	PROPN
cana-788	72	22	)	)	PUNCT
cana-788	72	23	=	=	SYM
cana-788	73	1	17	17	NUM
cana-788	73	2	×	×	NOUN
cana-788	73	3	2𝑛+2	2𝑛+2	PROPN
cana-788	73	4	f	f	X
cana-788	73	5	)	)	PUNCT
cana-788	73	6	𝐹(𝐵𝑛	𝐹(𝐵𝑛	NUM
cana-788	73	7	)	)	PUNCT
cana-788	73	8	=	=	SYM
cana-788	73	9	3	3	NUM
cana-788	73	10	×	×	NOUN
cana-788	73	11	2𝑛+3	2𝑛+3	PROPN
cana-788	73	12	g	g	NOUN
cana-788	73	13	)	)	PUNCT
cana-788	73	14	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NOUN
cana-788	73	15	)	)	PUNCT
cana-788	73	16	=	=	SYM
cana-788	73	17	7	7	NUM
cana-788	73	18	3	3	NUM
cana-788	73	19	2𝑛.	2𝑛.	NUM
cana-788	73	20	proof	proof	NOUN
cana-788	73	21	.	.	PUNCT
cana-788	74	1	for	for	ADP
cana-788	74	2	(	(	PUNCT
cana-788	74	3	a	a	NOUN
cana-788	74	4	)	)	PUNCT
cana-788	74	5	,	,	PUNCT
cana-788	74	6	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	PROPN
cana-788	74	7	)	)	PUNCT
cana-788	74	8	=	=	PUNCT
cana-788	75	1	(	(	PUNCT
cana-788	75	2	𝐷𝑤	𝐷𝑤	PROPN
cana-788	75	3	+	+	CCONJ
cana-788	75	4	𝐷𝑧	𝐷𝑧	PROPN
cana-788	75	5	)	)	PUNCT
cana-788	75	6	(	(	PUNCT
cana-788	75	7	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	75	8	,	,	PUNCT
cana-788	75	9	w	w	PROPN
cana-788	75	10	,	,	PUNCT
cana-788	75	11	z))|	z))|	PROPN
cana-788	75	12	𝑤=𝑧=1	𝑤=𝑧=1	PROPN
cana-788	75	13	=	=	SYM
cana-788	75	14	|(2𝑛	|(2𝑛	PROPN
cana-788	75	15	)	)	PUNCT
cana-788	75	16	.	.	PUNCT
cana-788	76	1	2𝑤2𝑧4	2𝑤2𝑧4	NUM
cana-788	77	1	+	+	CCONJ
cana-788	77	2	(	(	PUNCT
cana-788	77	3	2𝑛	2𝑛	NUM
cana-788	77	4	)	)	PUNCT
cana-788	77	5	.	.	PUNCT
cana-788	78	1	4𝑤2𝑧4	4𝑤2𝑧4	NOUN
cana-788	78	2	+	+	X
cana-788	78	3	(	(	PUNCT
cana-788	78	4	2𝑛−1)4𝑤4𝑧4	2𝑛−1)4𝑤4𝑧4	NUM
cana-788	78	5	+	+	NUM
cana-788	78	6	(	(	PUNCT
cana-788	78	7	2𝑛−1)4𝑤4𝑧4|𝑤=𝑧=1	2𝑛−1)4𝑤4𝑧4|𝑤=𝑧=1	NUM
cana-788	78	8	=	=	SYM
cana-788	78	9	5	5	NUM
cana-788	78	10	×	×	PROPN
cana-788	78	11	2𝑛+1	2𝑛+1	NUM
cana-788	78	12	.	.	PUNCT
cana-788	79	1	for	for	ADP
cana-788	79	2	(	(	PUNCT
cana-788	79	3	b	b	NOUN
cana-788	79	4	)	)	PUNCT
cana-788	79	5	,	,	PUNCT
cana-788	79	6	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	X
cana-788	79	7	)	)	PUNCT
cana-788	79	8	=	=	SYM
cana-788	79	9	(	(	PUNCT
cana-788	79	10	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	79	11	)	)	PUNCT
cana-788	79	12	(	(	PUNCT
cana-788	79	13	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	79	14	,	,	PUNCT
cana-788	79	15	w	w	PROPN
cana-788	79	16	,	,	PUNCT
cana-788	79	17	z))|	z))|	PROPN
cana-788	79	18	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	79	19	first	first	ADJ
cana-788	79	20	𝐷𝑍	𝐷𝑍	PROPN
cana-788	79	21	(	(	PUNCT
cana-788	79	22	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	79	23	,	,	PUNCT
cana-788	79	24	w	w	PROPN
cana-788	79	25	,	,	PUNCT
cana-788	79	26	z	z	NOUN
cana-788	79	27	)	)	PUNCT
cana-788	79	28	)	)	PUNCT
cana-788	80	1	=	=	PUNCT
cana-788	80	2	(	(	PUNCT
cana-788	80	3	2𝑛)4𝑤2𝑧4	2𝑛)4𝑤2𝑧4	NUM
cana-788	80	4	+	+	CCONJ
cana-788	80	5	(	(	PUNCT
cana-788	80	6	2𝑛−1)4𝑤4𝑧4	2𝑛−1)4𝑤4𝑧4	NUM
cana-788	80	7	now	now	ADV
cana-788	80	8	,	,	PUNCT
cana-788	80	9	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	80	10	(	(	PUNCT
cana-788	80	11	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	80	12	,	,	PUNCT
cana-788	80	13	w	w	PROPN
cana-788	80	14	,	,	PUNCT
cana-788	80	15	z	z	NOUN
cana-788	80	16	)	)	PUNCT
cana-788	80	17	)	)	PUNCT
cana-788	81	1	=	=	SYM
cana-788	81	2	𝐷𝑤[(2𝑛	𝐷𝑤[(2𝑛	NOUN
cana-788	81	3	)	)	PUNCT
cana-788	81	4	×	×	NOUN
cana-788	81	5	4	4	NUM
cana-788	81	6	×	×	NOUN
cana-788	81	7	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	81	8	+	+	CCONJ
cana-788	81	9	(	(	PUNCT
cana-788	81	10	2𝑛−1	2𝑛−1	NUM
cana-788	81	11	)	)	PUNCT
cana-788	81	12	×	×	NOUN
cana-788	81	13	4	4	NUM
cana-788	81	14	×	×	NOUN
cana-788	81	15	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	81	16	]	]	X
cana-788	81	17	=	=	SYM
cana-788	81	18	(	(	PUNCT
cana-788	81	19	2𝑛	2𝑛	NUM
cana-788	81	20	)	)	PUNCT
cana-788	81	21	×	×	NOUN
cana-788	81	22	4	4	NUM
cana-788	81	23	×	×	NOUN
cana-788	81	24	2	2	NUM
cana-788	81	25	×	×	NOUN
cana-788	81	26	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	81	27	+	+	CCONJ
cana-788	81	28	(	(	PUNCT
cana-788	81	29	2𝑛−1	2𝑛−1	NUM
cana-788	81	30	)	)	PUNCT
cana-788	81	31	×	×	NOUN
cana-788	81	32	4	4	NUM
cana-788	81	33	×	×	NOUN
cana-788	81	34	4	4	NUM
cana-788	81	35	×	×	NOUN
cana-788	81	36	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	81	37	=	=	NOUN
cana-788	81	38	2𝑛+4	2𝑛+4	NUM
cana-788	81	39	.	.	PUNCT
cana-788	82	1	communications	communication	NOUN
cana-788	82	2	on	on	ADP
cana-788	82	3	applied	apply	VERB
cana-788	82	4	nonlinear	nonlinear	ADJ
cana-788	82	5	analysis	analysis	NOUN
cana-788	82	6	issn	issn	NOUN
cana-788	82	7	:	:	PUNCT
cana-788	82	8	1074	1074	NUM
cana-788	82	9	-	-	PUNCT
cana-788	82	10	133x	133x	NUM
cana-788	82	11	vol	vol	NOUN
cana-788	82	12	31	31	NUM
cana-788	82	13	no	no	NOUN
cana-788	82	14	.	.	PUNCT
cana-788	83	1	3s	3s	NUM
cana-788	83	2	(	(	PUNCT
cana-788	83	3	2024	2024	NUM
cana-788	83	4	)	)	PUNCT
cana-788	83	5	382	382	NUM
cana-788	83	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	83	7	for	for	ADP
cana-788	83	8	(	(	PUNCT
cana-788	83	9	c	c	NOUN
cana-788	83	10	)	)	PUNCT
cana-788	83	11	,	,	PUNCT
cana-788	83	12	𝑀2	𝑀2	PROPN
cana-788	83	13	𝑚(𝐵𝑛	𝑚(𝐵𝑛	NUM
cana-788	83	14	)	)	PUNCT
cana-788	83	15	=	=	SYM
cana-788	83	16	(	(	PUNCT
cana-788	83	17	𝑆𝑤𝑆𝑧	𝑆𝑤𝑆𝑧	PROPN
cana-788	83	18	)	)	PUNCT
cana-788	83	19	(	(	PUNCT
cana-788	83	20	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	83	21	,	,	PUNCT
cana-788	83	22	w	w	PROPN
cana-788	83	23	,	,	PUNCT
cana-788	83	24	z))|	z))|	PROPN
cana-788	83	25	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	83	26	,	,	PUNCT
cana-788	83	27	𝑆𝑤	𝑆𝑤	PROPN
cana-788	83	28	=	=	PUNCT
cana-788	83	29	∫	∫	PROPN
cana-788	83	30	𝑓(𝑡,𝑧	𝑓(𝑡,𝑧	NUM
cana-788	83	31	)	)	PUNCT
cana-788	83	32	𝑡	𝑡	PROPN
cana-788	83	33	𝑑𝑡	𝑑𝑡	ADP
cana-788	83	34	𝑤	𝑤	ADP
cana-788	83	35	0	0	NUM
cana-788	83	36	=	=	SYM
cana-788	83	37	∫	∫	PROPN
cana-788	83	38	(	(	PUNCT
cana-788	83	39	2𝑛	2𝑛	PROPN
cana-788	83	40	)	)	PUNCT
cana-788	83	41	𝑡2𝑧4+(2𝑛−1)𝑡4𝑧4	𝑡2𝑧4+(2𝑛−1)𝑡4𝑧4	PROPN
cana-788	83	42	𝑡	𝑡	NOUN
cana-788	84	1	𝑑𝑡	𝑑𝑡	ADP
cana-788	84	2	𝑤	𝑤	ADP
cana-788	84	3	0	0	NUM
cana-788	84	4	∴	∴	NOUN
cana-788	84	5	𝑆𝑤	𝑆𝑤	PROPN
cana-788	84	6	=	=	SYM
cana-788	84	7	(	(	PUNCT
cana-788	84	8	2𝑛	2𝑛	NOUN
cana-788	84	9	)	)	PUNCT
cana-788	84	10	𝑧4	𝑧4	PROPN
cana-788	84	11	𝑤2	𝑤2	NOUN
cana-788	84	12	2	2	NUM
cana-788	84	13	+	+	CCONJ
cana-788	84	14	(	(	PUNCT
cana-788	84	15	2𝑛−1)𝑧4	2𝑛−1)𝑧4	NUM
cana-788	84	16	𝑤4	𝑤4	NOUN
cana-788	84	17	4	4	NUM
cana-788	84	18	,	,	PUNCT
cana-788	84	19	𝑆𝑤𝑆𝑧	𝑆𝑤𝑆𝑧	PROPN
cana-788	84	20	=	=	SYM
cana-788	84	21	∫	∫	PROPN
cana-788	84	22	𝑓(𝑤,𝑡	𝑓(𝑤,𝑡	NUM
cana-788	84	23	)	)	PUNCT
cana-788	84	24	𝑡	𝑡	PROPN
cana-788	84	25	𝑑𝑡	𝑑𝑡	ADP
cana-788	84	26	𝑧	𝑧	NOUN
cana-788	84	27	0	0	NUM
cana-788	85	1	=	=	SYM
cana-788	85	2	∫	∫	PROPN
cana-788	86	1	[	[	X
cana-788	86	2	(	(	PUNCT
cana-788	86	3	2𝑛	2𝑛	PROPN
cana-788	86	4	)	)	PUNCT
cana-788	86	5	𝑡4𝑤2	𝑡4𝑤2	PROPN
cana-788	86	6	2	2	NUM
cana-788	86	7	+	+	NOUN
cana-788	86	8	(	(	PUNCT
cana-788	86	9	2𝑛−1)𝑡4𝑤4	2𝑛−1)𝑡4𝑤4	NUM
cana-788	86	10	4	4	NUM
cana-788	86	11	]	]	PUNCT
cana-788	86	12	𝑡	𝑡	VERB
cana-788	86	13	𝑧	𝑧	PROPN
cana-788	86	14	0	0	NUM
cana-788	86	15	𝑑𝑡	𝑑𝑡	ADP
cana-788	86	16	=	=	SYM
cana-788	86	17	(	(	PUNCT
cana-788	86	18	2𝑛	2𝑛	PROPN
cana-788	86	19	)	)	PUNCT
cana-788	87	1	𝑧4	𝑧4	PROPN
cana-788	87	2	4	4	NUM
cana-788	87	3	𝑤2	𝑤2	NOUN
cana-788	87	4	2	2	NUM
cana-788	87	5	+	+	CCONJ
cana-788	87	6	(	(	PUNCT
cana-788	87	7	2𝑛−1	2𝑛−1	NUM
cana-788	87	8	)	)	PUNCT
cana-788	87	9	𝑧4	𝑧4	PROPN
cana-788	87	10	4	4	NUM
cana-788	87	11	𝑤4	𝑤4	NOUN
cana-788	87	12	4	4	NUM
cana-788	87	13	∴	∴	PROPN
cana-788	87	14	𝑀2	𝑀2	NOUN
cana-788	87	15	𝑚(𝐵𝑛	𝑚(𝐵𝑛	X
cana-788	87	16	)	)	PUNCT
cana-788	87	17	=	=	SYM
cana-788	87	18	(	(	PUNCT
cana-788	87	19	𝑆𝑤𝑆𝑧	𝑆𝑤𝑆𝑧	PROPN
cana-788	87	20	)	)	PUNCT
cana-788	87	21	(	(	PUNCT
cana-788	87	22	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	87	23	,	,	PUNCT
cana-788	87	24	w	w	PROPN
cana-788	87	25	,	,	PUNCT
cana-788	87	26	z))|	z))|	PROPN
cana-788	87	27	𝑤=𝑧=1	𝑤=𝑧=1	NOUN
cana-788	87	28	=	=	SYM
cana-788	87	29	2𝑛	2𝑛	PROPN
cana-788	87	30	8	8	NUM
cana-788	87	31	+	+	NUM
cana-788	87	32	2𝑛−1	2𝑛−1	NUM
cana-788	87	33	16	16	NUM
cana-788	87	34	=	=	SYM
cana-788	87	35	2𝑛−3	2𝑛−3	NOUN
cana-788	87	36	+	+	CCONJ
cana-788	87	37	2𝑛−5	2𝑛−5	NUM
cana-788	87	38	=	=	SYM
cana-788	87	39	5	5	NUM
cana-788	87	40	×	×	NOUN
cana-788	87	41	2𝑛−5	2𝑛−5	NUM
cana-788	87	42	for	for	ADP
cana-788	87	43	(	(	PUNCT
cana-788	87	44	d	d	NOUN
cana-788	87	45	)	)	PUNCT
cana-788	87	46	,	,	PUNCT
cana-788	87	47	to	to	PART
cana-788	87	48	find	find	VERB
cana-788	87	49	(	(	PUNCT
cana-788	87	50	𝑆𝑤	𝑆𝑤	PROPN
cana-788	87	51	3𝑄−2𝐽	3𝑄−2𝐽	NUM
cana-788	87	52	𝐷𝑤	𝐷𝑤	PROPN
cana-788	87	53	3𝐷𝑧	3𝐷𝑧	NUM
cana-788	87	54	3	3	NUM
cana-788	87	55	)	)	PUNCT
cana-788	87	56	(	(	PUNCT
cana-788	87	57	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	87	58	,	,	PUNCT
cana-788	87	59	w	w	PROPN
cana-788	87	60	,	,	PUNCT
cana-788	87	61	z))|	z))|	PROPN
cana-788	87	62	𝑤=1	𝑤=1	PUNCT
cana-788	87	63	𝐷𝑍	𝐷𝑍	PROPN
cana-788	87	64	(	(	PUNCT
cana-788	87	65	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	87	66	,	,	PUNCT
cana-788	87	67	w	w	PROPN
cana-788	87	68	,	,	PUNCT
cana-788	87	69	z	z	NOUN
cana-788	87	70	)	)	PUNCT
cana-788	87	71	)	)	PUNCT
cana-788	88	1	=	=	PUNCT
cana-788	88	2	(	(	PUNCT
cana-788	88	3	2𝑛)4𝑤2𝑧4	2𝑛)4𝑤2𝑧4	NUM
cana-788	88	4	+	+	CCONJ
cana-788	88	5	(	(	PUNCT
cana-788	88	6	2𝑛−1)4𝑤4𝑧4	2𝑛−1)4𝑤4𝑧4	NUM
cana-788	88	7	,	,	PUNCT
cana-788	88	8	𝐷𝑍	𝐷𝑍	PROPN
cana-788	88	9	2	2	NUM
cana-788	88	10	(	(	PUNCT
cana-788	88	11	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	88	12	,	,	PUNCT
cana-788	88	13	w	w	PROPN
cana-788	88	14	,	,	PUNCT
cana-788	88	15	z	z	NOUN
cana-788	88	16	)	)	PUNCT
cana-788	88	17	)	)	PUNCT
cana-788	89	1	=	=	PUNCT
cana-788	90	1	[	[	X
cana-788	90	2	(	(	PUNCT
cana-788	90	3	2𝑛	2𝑛	PROPN
cana-788	90	4	)	)	PUNCT
cana-788	90	5	×	×	NOUN
cana-788	90	6	16	16	NUM
cana-788	90	7	×	×	NOUN
cana-788	90	8	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	90	9	]	]	PUNCT
cana-788	90	10	+	+	CCONJ
cana-788	90	11	[	[	X
cana-788	90	12	(	(	PUNCT
cana-788	90	13	2𝑛−1	2𝑛−1	NUM
cana-788	90	14	)	)	PUNCT
cana-788	90	15	×	×	NOUN
cana-788	90	16	16	16	NUM
cana-788	90	17	×	×	NOUN
cana-788	90	18	𝑤4𝑧4	𝑤4𝑧4	PROPN
cana-788	90	19	]	]	PUNCT
cana-788	90	20	,	,	PUNCT
cana-788	90	21	𝐷𝑍	𝐷𝑍	PROPN
cana-788	90	22	3	3	NUM
cana-788	90	23	(	(	PUNCT
cana-788	90	24	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	90	25	,	,	PUNCT
cana-788	90	26	w	w	PROPN
cana-788	90	27	,	,	PUNCT
cana-788	90	28	z	z	NOUN
cana-788	90	29	)	)	PUNCT
cana-788	90	30	)	)	PUNCT
cana-788	91	1	=	=	PUNCT
cana-788	92	1	[	[	X
cana-788	92	2	(	(	PUNCT
cana-788	92	3	2𝑛	2𝑛	PROPN
cana-788	92	4	)	)	PUNCT
cana-788	92	5	×	×	NOUN
cana-788	92	6	64	64	NUM
cana-788	92	7	×	×	NOUN
cana-788	92	8	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	92	9	]	]	PUNCT
cana-788	92	10	+	+	CCONJ
cana-788	92	11	[	[	X
cana-788	92	12	(	(	PUNCT
cana-788	92	13	2𝑛−1	2𝑛−1	NUM
cana-788	92	14	)	)	PUNCT
cana-788	92	15	×	×	NOUN
cana-788	92	16	64	64	NUM
cana-788	92	17	×	×	NOUN
cana-788	92	18	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	92	19	]	]	PUNCT
cana-788	92	20	𝐷𝑤𝐷𝑍	𝐷𝑤𝐷𝑍	NOUN
cana-788	92	21	3	3	NUM
cana-788	92	22	(	(	PUNCT
cana-788	92	23	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	92	24	,	,	PUNCT
cana-788	92	25	w	w	PROPN
cana-788	92	26	,	,	PUNCT
cana-788	92	27	z	z	NOUN
cana-788	92	28	)	)	PUNCT
cana-788	92	29	)	)	PUNCT
cana-788	93	1	=	=	PUNCT
cana-788	94	1	[	[	X
cana-788	94	2	(	(	PUNCT
cana-788	94	3	2𝑛	2𝑛	PROPN
cana-788	94	4	)	)	PUNCT
cana-788	94	5	×	×	NOUN
cana-788	94	6	128	128	NUM
cana-788	94	7	×	×	NOUN
cana-788	94	8	𝑤2𝑧4	𝑤2𝑧4	PUNCT
cana-788	94	9	]	]	PUNCT
cana-788	94	10	+	+	CCONJ
cana-788	94	11	[	[	X
cana-788	94	12	(	(	PUNCT
cana-788	94	13	2𝑛−1	2𝑛−1	NUM
cana-788	94	14	)	)	PUNCT
cana-788	94	15	×	×	NOUN
cana-788	94	16	256	256	NUM
cana-788	94	17	×	×	NOUN
cana-788	94	18	𝑤4𝑧4	𝑤4𝑧4	SYM
cana-788	94	19	]	]	X
cana-788	94	20	𝐷𝑤	𝐷𝑤	PROPN
cana-788	94	21	2𝐷𝑍	2𝐷𝑍	NUM
cana-788	94	22	3	3	NUM
cana-788	94	23	(	(	PUNCT
cana-788	94	24	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	94	25	,	,	PUNCT
cana-788	94	26	w	w	PROPN
cana-788	94	27	,	,	PUNCT
cana-788	94	28	z	z	NOUN
cana-788	94	29	)	)	PUNCT
cana-788	94	30	)	)	PUNCT
cana-788	95	1	=	=	PUNCT
cana-788	96	1	[	[	X
cana-788	96	2	(	(	PUNCT
cana-788	96	3	2𝑛	2𝑛	PROPN
cana-788	96	4	)	)	PUNCT
cana-788	96	5	×	×	NOUN
cana-788	96	6	256	256	NUM
cana-788	96	7	×	×	NOUN
cana-788	96	8	𝑤2𝑧4	𝑤2𝑧4	PUNCT
cana-788	96	9	]	]	PUNCT
cana-788	96	10	+	+	CCONJ
cana-788	96	11	[	[	X
cana-788	96	12	(	(	PUNCT
cana-788	96	13	2𝑛−1	2𝑛−1	NUM
cana-788	96	14	)	)	PUNCT
cana-788	96	15	×	×	PROPN
cana-788	96	16	1024	1024	NUM
cana-788	96	17	×	×	NOUN
cana-788	96	18	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	96	19	]	]	X
cana-788	96	20	𝐷𝑤	𝐷𝑤	PROPN
cana-788	96	21	3𝐷𝑍	3𝐷𝑍	NUM
cana-788	96	22	3	3	NUM
cana-788	96	23	(	(	PUNCT
cana-788	96	24	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	96	25	,	,	PUNCT
cana-788	96	26	w	w	PROPN
cana-788	96	27	,	,	PUNCT
cana-788	96	28	z	z	NOUN
cana-788	96	29	)	)	PUNCT
cana-788	96	30	)	)	PUNCT
cana-788	97	1	=	=	PUNCT
cana-788	98	1	[	[	X
cana-788	98	2	(	(	PUNCT
cana-788	98	3	2𝑛	2𝑛	PROPN
cana-788	98	4	)	)	PUNCT
cana-788	98	5	×	×	PROPN
cana-788	98	6	512	512	NUM
cana-788	98	7	×	×	NOUN
cana-788	98	8	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	98	9	]	]	PUNCT
cana-788	98	10	+	+	CCONJ
cana-788	99	1	[	[	X
cana-788	99	2	(	(	PUNCT
cana-788	99	3	2𝑛−1	2𝑛−1	NUM
cana-788	99	4	)	)	PUNCT
cana-788	99	5	×	×	NOUN
cana-788	99	6	4096	4096	NUM
cana-788	99	7	×	×	NOUN
cana-788	99	8	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	99	9	]	]	PUNCT
cana-788	99	10	𝐽𝐷𝑤	𝐽𝐷𝑤	NOUN
cana-788	99	11	3𝐷𝑍	3𝐷𝑍	NUM
cana-788	99	12	3	3	NUM
cana-788	99	13	(	(	PUNCT
cana-788	99	14	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	99	15	,	,	PUNCT
cana-788	99	16	w	w	PROPN
cana-788	99	17	,	,	PUNCT
cana-788	99	18	z	z	NOUN
cana-788	99	19	)	)	PUNCT
cana-788	99	20	)	)	PUNCT
cana-788	100	1	=	=	PUNCT
cana-788	101	1	[	[	X
cana-788	101	2	(	(	PUNCT
cana-788	101	3	2𝑛	2𝑛	PROPN
cana-788	101	4	)	)	PUNCT
cana-788	101	5	×	×	NOUN
cana-788	101	6	512	512	NUM
cana-788	101	7	×	×	NOUN
cana-788	101	8	𝑤6	𝑤6	NOUN
cana-788	101	9	]	]	PUNCT
cana-788	101	10	+	+	CCONJ
cana-788	101	11	[	[	X
cana-788	101	12	(	(	PUNCT
cana-788	101	13	2𝑛−1	2𝑛−1	NUM
cana-788	101	14	)	)	PUNCT
cana-788	101	15	×	×	NOUN
cana-788	101	16	4096	4096	NUM
cana-788	101	17	×	×	NOUN
cana-788	101	18	𝑤8	𝑤8	NUM
cana-788	101	19	]	]	PUNCT
cana-788	101	20	𝑄−2𝐽𝐷𝑤	𝑄−2𝐽𝐷𝑤	NOUN
cana-788	102	1	3𝐷𝑍	3𝐷𝑍	NUM
cana-788	102	2	3	3	NUM
cana-788	102	3	(	(	PUNCT
cana-788	102	4	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	102	5	,	,	PUNCT
cana-788	102	6	w	w	PROPN
cana-788	102	7	,	,	PUNCT
cana-788	102	8	z	z	NOUN
cana-788	102	9	)	)	PUNCT
cana-788	102	10	)	)	PUNCT
cana-788	103	1	=	=	PUNCT
cana-788	103	2	𝑤−2	𝑤−2	NOUN
cana-788	103	3	{	{	PUNCT
cana-788	103	4	[	[	X
cana-788	103	5	(	(	PUNCT
cana-788	103	6	2𝑛	2𝑛	PROPN
cana-788	103	7	)	)	PUNCT
cana-788	103	8	×	×	NOUN
cana-788	103	9	512	512	NUM
cana-788	103	10	×	×	NOUN
cana-788	103	11	𝑤6	𝑤6	NOUN
cana-788	103	12	]	]	PUNCT
cana-788	103	13	+	+	CCONJ
cana-788	103	14	[	[	X
cana-788	103	15	(	(	PUNCT
cana-788	103	16	2𝑛−1	2𝑛−1	NUM
cana-788	103	17	)	)	PUNCT
cana-788	103	18	×	×	NOUN
cana-788	103	19	4096	4096	NUM
cana-788	103	20	×	×	NOUN
cana-788	103	21	𝑤8	𝑤8	NUM
cana-788	103	22	]	]	PUNCT
cana-788	103	23	}	}	PUNCT
cana-788	103	24	𝑆𝑤𝑄−2𝐽𝐷𝑤	𝑆𝑤𝑄−2𝐽𝐷𝑤	PROPN
cana-788	103	25	3𝐷𝑍	3𝐷𝑍	NUM
cana-788	103	26	3	3	NUM
cana-788	103	27	(	(	PUNCT
cana-788	103	28	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	103	29	,	,	PUNCT
cana-788	103	30	w	w	PROPN
cana-788	103	31	,	,	PUNCT
cana-788	103	32	z	z	NOUN
cana-788	103	33	)	)	PUNCT
cana-788	103	34	)	)	PUNCT
cana-788	104	1	=	=	SYM
cana-788	104	2	∫	∫	PROPN
cana-788	104	3	(	(	PUNCT
cana-788	104	4	[	[	X
cana-788	104	5	(	(	PUNCT
cana-788	104	6	2𝑛	2𝑛	NOUN
cana-788	104	7	)	)	PUNCT
cana-788	104	8	×512×𝑡6]+[(2𝑛−1	×512×𝑡6]+[(2𝑛−1	X
cana-788	104	9	)	)	PUNCT
cana-788	104	10	×4096×𝑡8	×4096×𝑡8	X
cana-788	104	11	]	]	PUNCT
cana-788	104	12	𝑡	𝑡	PROPN
cana-788	104	13	)	)	PUNCT
cana-788	104	14	𝑑𝑡	𝑑𝑡	ADP
cana-788	104	15	𝑤	𝑤	ADP
cana-788	104	16	0	0	NUM
cana-788	104	17	=	=	SYM
cana-788	104	18	(	(	PUNCT
cana-788	104	19	2𝑛	2𝑛	PROPN
cana-788	104	20	)	)	PUNCT
cana-788	104	21	×	×	NOUN
cana-788	105	1	512	512	NUM
cana-788	105	2	×	×	NOUN
cana-788	105	3	𝑤6	𝑤6	NOUN
cana-788	105	4	6	6	NUM
cana-788	105	5	+	+	CCONJ
cana-788	105	6	(	(	PUNCT
cana-788	105	7	2𝑛−1	2𝑛−1	NUM
cana-788	105	8	)	)	PUNCT
cana-788	105	9	×	×	NOUN
cana-788	105	10	4096	4096	NUM
cana-788	105	11	×	×	NOUN
cana-788	105	12	𝑤8	𝑤8	NOUN
cana-788	105	13	8	8	NUM
cana-788	105	14	.	.	PUNCT
cana-788	106	1	𝑆𝑤	𝑆𝑤	PROPN
cana-788	106	2	2𝑄−2𝐽𝐷𝑤	2𝑄−2𝐽𝐷𝑤	NUM
cana-788	106	3	3	3	NUM
cana-788	106	4	𝐷𝑍	𝐷𝑍	PROPN
cana-788	106	5	3	3	NUM
cana-788	106	6	(	(	PUNCT
cana-788	106	7	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	106	8	,	,	PUNCT
cana-788	106	9	w	w	PROPN
cana-788	106	10	,	,	PUNCT
cana-788	106	11	z	z	NOUN
cana-788	106	12	)	)	PUNCT
cana-788	106	13	)	)	PUNCT
cana-788	107	1	=	=	SYM
cana-788	107	2	∫	∫	PROPN
cana-788	107	3	(	(	PUNCT
cana-788	107	4	2𝑛	2𝑛	PROPN
cana-788	107	5	)	)	PUNCT
cana-788	107	6	×512×𝑡6	×512×𝑡6	NOUN
cana-788	107	7	6	6	NUM
cana-788	107	8	+	+	ADJ
cana-788	107	9	(	(	PUNCT
cana-788	107	10	2𝑛−1	2𝑛−1	NUM
cana-788	107	11	)	)	PUNCT
cana-788	107	12	×4096×𝑡8	×4096×𝑡8	NOUN
cana-788	107	13	8	8	NUM
cana-788	107	14	𝑡	𝑡	NOUN
cana-788	107	15	𝑤	𝑤	ADP
cana-788	107	16	0	0	NUM
cana-788	107	17	𝑑𝑡	𝑑𝑡	ADP
cana-788	107	18	=	=	SYM
cana-788	107	19	(	(	PUNCT
cana-788	107	20	2𝑛	2𝑛	PROPN
cana-788	107	21	)	)	PUNCT
cana-788	107	22	×	×	NOUN
cana-788	107	23	512	512	NUM
cana-788	107	24	×	×	NOUN
cana-788	107	25	𝑤6	𝑤6	NOUN
cana-788	107	26	36	36	NUM
cana-788	108	1	+	+	CCONJ
cana-788	108	2	(	(	PUNCT
cana-788	108	3	2𝑛−1	2𝑛−1	NUM
cana-788	108	4	)	)	PUNCT
cana-788	108	5	×	×	NOUN
cana-788	108	6	4096	4096	NUM
cana-788	108	7	×	×	NOUN
cana-788	108	8	𝑤8	𝑤8	NOUN
cana-788	108	9	64	64	NUM
cana-788	108	10	.	.	PUNCT
cana-788	109	1	𝑆𝑤	𝑆𝑤	NOUN
cana-788	109	2	3𝑄−2𝐽𝐷𝑤	3𝑄−2𝐽𝐷𝑤	NUM
cana-788	109	3	3	3	NUM
cana-788	109	4	𝐷𝑍	𝐷𝑍	PROPN
cana-788	109	5	3	3	NUM
cana-788	109	6	(	(	PUNCT
cana-788	109	7	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	109	8	,	,	PUNCT
cana-788	109	9	w	w	PROPN
cana-788	109	10	,	,	PUNCT
cana-788	109	11	z	z	NOUN
cana-788	109	12	)	)	PUNCT
cana-788	109	13	)	)	PUNCT
cana-788	110	1	=	=	SYM
cana-788	110	2	∫	∫	PROPN
cana-788	110	3	(	(	PUNCT
cana-788	110	4	2𝑛	2𝑛	PROPN
cana-788	110	5	)	)	PUNCT
cana-788	110	6	×512×𝑡6	×512×𝑡6	NOUN
cana-788	110	7	36	36	NUM
cana-788	110	8	+	+	ADJ
cana-788	110	9	(	(	PUNCT
cana-788	110	10	2𝑛−1	2𝑛−1	NUM
cana-788	110	11	)	)	PUNCT
cana-788	110	12	×4096×𝑡8	×4096×𝑡8	PROPN
cana-788	110	13	64	64	NUM
cana-788	110	14	𝑡	𝑡	NOUN
cana-788	110	15	𝑤	𝑤	ADP
cana-788	110	16	0	0	NUM
cana-788	110	17	𝑑𝑡	𝑑𝑡	ADP
cana-788	110	18	=	=	SYM
cana-788	110	19	(	(	PUNCT
cana-788	110	20	2𝑛	2𝑛	PROPN
cana-788	110	21	)	)	PUNCT
cana-788	110	22	×	×	NOUN
cana-788	110	23	512	512	NUM
cana-788	110	24	×	×	NOUN
cana-788	110	25	𝑤6	𝑤6	NOUN
cana-788	110	26	216	216	NUM
cana-788	111	1	+	+	CCONJ
cana-788	111	2	(	(	PUNCT
cana-788	111	3	2𝑛−1	2𝑛−1	NUM
cana-788	111	4	)	)	PUNCT
cana-788	111	5	×	×	NOUN
cana-788	111	6	4096	4096	NUM
cana-788	111	7	×	×	NOUN
cana-788	111	8	𝑤8	𝑤8	NOUN
cana-788	111	9	512	512	NUM
cana-788	111	10	.	.	PUNCT
cana-788	112	1	∴	∴	PROPN
cana-788	112	2	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	PROPN
cana-788	112	3	)	)	PUNCT
cana-788	113	1	=	=	NOUN
cana-788	113	2	(	(	PUNCT
cana-788	113	3	𝑆𝑤	𝑆𝑤	PROPN
cana-788	113	4	3𝑄−2𝐽	3𝑄−2𝐽	NUM
cana-788	113	5	𝐷𝑤	𝐷𝑤	PROPN
cana-788	113	6	3𝐷𝑧	3𝐷𝑧	NUM
cana-788	113	7	3	3	NUM
cana-788	113	8	)	)	PUNCT
cana-788	113	9	(	(	PUNCT
cana-788	113	10	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	113	11	,	,	PUNCT
cana-788	113	12	w	w	PROPN
cana-788	113	13	,	,	PUNCT
cana-788	113	14	z))|	z))|	PROPN
cana-788	113	15	𝑤=1	𝑤=1	PUNCT
cana-788	113	16	=	=	PUNCT
cana-788	114	1	[	[	X
cana-788	114	2	(	(	PUNCT
cana-788	114	3	2𝑛	2𝑛	PROPN
cana-788	114	4	)	)	PUNCT
cana-788	114	5	×	×	NOUN
cana-788	114	6	2	2	NUM
cana-788	114	7	]	]	PUNCT
cana-788	114	8	+	+	CCONJ
cana-788	114	9	[	[	X
cana-788	114	10	(	(	PUNCT
cana-788	114	11	2𝑛−1	2𝑛−1	NUM
cana-788	114	12	)	)	PUNCT
cana-788	114	13	×	×	NOUN
cana-788	114	14	8	8	NUM
cana-788	114	15	]	]	PUNCT
cana-788	114	16	=	=	SYM
cana-788	114	17	3	3	NUM
cana-788	114	18	×	×	NOUN
cana-788	114	19	2𝑛+1	2𝑛+1	PROPN
cana-788	114	20	for	for	ADP
cana-788	114	21	(	(	PUNCT
cana-788	114	22	e	e	NOUN
cana-788	114	23	)	)	PUNCT
cana-788	114	24	,	,	PUNCT
cana-788	114	25	to	to	PART
cana-788	114	26	find	find	VERB
cana-788	114	27	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	NUM
cana-788	114	28	)	)	PUNCT
cana-788	114	29	=	=	NOUN
cana-788	115	1	(	(	PUNCT
cana-788	115	2	𝐷𝑤	𝐷𝑤	PROPN
cana-788	115	3	+	+	PROPN
cana-788	115	4	𝐷𝑧)2	𝐷𝑧)2	PROPN
cana-788	115	5	(	(	PUNCT
cana-788	115	6	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	115	7	,	,	PUNCT
cana-788	115	8	w	w	PROPN
cana-788	115	9	,	,	PUNCT
cana-788	115	10	z))|	z))|	PROPN
cana-788	115	11	𝑤=𝑧=1	𝑤=𝑧=1	NOUN
cana-788	115	12	,	,	PUNCT
cana-788	115	13	(	(	PUNCT
cana-788	115	14	𝐷𝑤	𝐷𝑤	PROPN
cana-788	115	15	+	+	PROPN
cana-788	115	16	𝐷𝑧)2	𝐷𝑧)2	NOUN
cana-788	115	17	=	=	PUNCT
cana-788	116	1	(	(	PUNCT
cana-788	116	2	𝐷𝑤	𝐷𝑤	PROPN
cana-788	116	3	+	+	CCONJ
cana-788	116	4	𝐷𝑧)(𝐷𝑤	𝐷𝑧)(𝐷𝑤	PROPN
cana-788	116	5	+	+	CCONJ
cana-788	116	6	𝐷𝑧)𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝐷𝑧)𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	ADJ
cana-788	116	7	,	,	PUNCT
cana-788	116	8	w	w	PROPN
cana-788	116	9	,	,	PUNCT
cana-788	116	10	z	z	NOUN
cana-788	116	11	)	)	PUNCT
cana-788	116	12	=	=	SYM
cana-788	116	13	(	(	PUNCT
cana-788	116	14	𝐷𝑤	𝐷𝑤	PROPN
cana-788	116	15	+	+	CCONJ
cana-788	116	16	𝐷𝑧	𝐷𝑧	PROPN
cana-788	116	17	)	)	PUNCT
cana-788	116	18	{	{	PUNCT
cana-788	116	19	(	(	PUNCT
cana-788	116	20	2𝑛	2𝑛	PROPN
cana-788	116	21	)	)	PUNCT
cana-788	116	22	.	.	PUNCT
cana-788	117	1	2𝑤2𝑧4	2𝑤2𝑧4	NUM
cana-788	118	1	+	+	CCONJ
cana-788	118	2	(	(	PUNCT
cana-788	118	3	2𝑛	2𝑛	PROPN
cana-788	118	4	)	)	PUNCT
cana-788	118	5	.	.	PUNCT
cana-788	119	1	4𝑤2𝑧4	4𝑤2𝑧4	NOUN
cana-788	119	2	+	+	X
cana-788	119	3	(	(	PUNCT
cana-788	119	4	2𝑛−1)4𝑤4𝑧4	2𝑛−1)4𝑤4𝑧4	NUM
cana-788	119	5	+	+	NUM
cana-788	119	6	(	(	PUNCT
cana-788	119	7	2𝑛−1)4𝑤4𝑧4	2𝑛−1)4𝑤4𝑧4	NUM
cana-788	119	8	}	}	PUNCT
cana-788	119	9	(	(	PUNCT
cana-788	119	10	𝐷𝑤	𝐷𝑤	PROPN
cana-788	119	11	+	+	PROPN
cana-788	119	12	𝐷𝑧)2	𝐷𝑧)2	PROPN
cana-788	119	13	=	=	PUNCT
cana-788	120	1	|9𝑤2𝑧42𝑛+2	|9𝑤2𝑧42𝑛+2	PROPN
cana-788	121	1	+	+	CCONJ
cana-788	121	2	𝑤4𝑧42𝑛+5	𝑤4𝑧42𝑛+5	ADJ
cana-788	121	3	|	|	ADV
cana-788	121	4	𝑤=1	𝑤=1	PROPN
cana-788	121	5	,	,	PUNCT
cana-788	121	6	∴	∴	PROPN
cana-788	121	7	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	PROPN
cana-788	121	8	)	)	PUNCT
cana-788	121	9	=	=	SYM
cana-788	121	10	17	17	NUM
cana-788	121	11	×	×	PROPN
cana-788	121	12	2𝑛+2	2𝑛+2	PROPN
cana-788	121	13	.	.	PUNCT
cana-788	122	1	for	for	ADP
cana-788	122	2	(	(	PUNCT
cana-788	122	3	f	f	X
cana-788	122	4	)	)	PUNCT
cana-788	122	5	to	to	PART
cana-788	122	6	find	find	VERB
cana-788	122	7	𝐹(ɠ	𝐹(ɠ	NOUN
cana-788	122	8	)	)	PUNCT
cana-788	122	9	=	=	SYM
cana-788	122	10	(	(	PUNCT
cana-788	122	11	𝐷𝑤	𝐷𝑤	PROPN
cana-788	122	12	2	2	NUM
cana-788	122	13	+	+	CCONJ
cana-788	122	14	𝐷𝑧	𝐷𝑧	PROPN
cana-788	122	15	2	2	NUM
cana-788	122	16	)	)	PUNCT
cana-788	122	17	(	(	PUNCT
cana-788	122	18	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	122	19	,	,	PUNCT
cana-788	122	20	w	w	PROPN
cana-788	122	21	,	,	PUNCT
cana-788	122	22	z))|	z))|	PROPN
cana-788	122	23	𝑤=𝑧=1	𝑤=𝑧=1	VERB
cana-788	122	24	communications	communication	NOUN
cana-788	122	25	on	on	ADP
cana-788	122	26	applied	apply	VERB
cana-788	122	27	nonlinear	nonlinear	ADJ
cana-788	122	28	analysis	analysis	NOUN
cana-788	122	29	issn	issn	NOUN
cana-788	122	30	:	:	PUNCT
cana-788	122	31	1074	1074	NUM
cana-788	122	32	-	-	PUNCT
cana-788	122	33	133x	133x	NUM
cana-788	122	34	vol	vol	NOUN
cana-788	122	35	31	31	NUM
cana-788	122	36	no	no	NOUN
cana-788	122	37	.	.	PUNCT
cana-788	123	1	3s	3s	NUM
cana-788	123	2	(	(	PUNCT
cana-788	123	3	2024	2024	NUM
cana-788	123	4	)	)	PUNCT
cana-788	123	5	383	383	NUM
cana-788	123	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	124	1	𝐷𝑤	𝐷𝑤	PROPN
cana-788	124	2	(	(	PUNCT
cana-788	124	3	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	PROPN
cana-788	124	4	,	,	PUNCT
cana-788	124	5	w	w	PROPN
cana-788	124	6	,	,	PUNCT
cana-788	124	7	z	z	NOUN
cana-788	124	8	)	)	PUNCT
cana-788	124	9	)	)	PUNCT
cana-788	124	10	=	=	PUNCT
cana-788	125	1	𝐷𝑤[(2𝑛)𝑤2𝑧4	𝐷𝑤[(2𝑛)𝑤2𝑧4	NOUN
cana-788	125	2	+	+	CCONJ
cana-788	125	3	(	(	PUNCT
cana-788	125	4	2𝑛−1)𝑤4𝑧4	2𝑛−1)𝑤4𝑧4	NUM
cana-788	125	5	]	]	X
cana-788	125	6	=	=	PUNCT
cana-788	126	1	[	[	X
cana-788	126	2	2𝑛+1	2𝑛+1	NUM
cana-788	126	3	×	×	NOUN
cana-788	126	4	𝑤2𝑧4	𝑤2𝑧4	PUNCT
cana-788	126	5	]	]	PUNCT
cana-788	126	6	+	+	CCONJ
cana-788	127	1	[	[	X
cana-788	127	2	2𝑛+1	2𝑛+1	NUM
cana-788	127	3	×	×	NOUN
cana-788	127	4	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	127	5	]	]	X
cana-788	127	6	𝐷𝑤	𝐷𝑤	PROPN
cana-788	127	7	2	2	NUM
cana-788	127	8	(	(	PUNCT
cana-788	127	9	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	127	10	,	,	PUNCT
cana-788	127	11	w	w	PROPN
cana-788	127	12	,	,	PUNCT
cana-788	127	13	z	z	NOUN
cana-788	127	14	)	)	PUNCT
cana-788	127	15	)	)	PUNCT
cana-788	128	1	=	=	PUNCT
cana-788	129	1	[	[	X
cana-788	129	2	2𝑛+2	2𝑛+2	NUM
cana-788	129	3	×	×	NOUN
cana-788	129	4	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	129	5	]	]	PUNCT
cana-788	129	6	+	+	CCONJ
cana-788	130	1	[	[	X
cana-788	130	2	2𝑛+3	2𝑛+3	PROPN
cana-788	130	3	×	×	NOUN
cana-788	130	4	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	130	5	]	]	PUNCT
cana-788	130	6	,	,	PUNCT
cana-788	130	7	𝐷𝑍	𝐷𝑍	PROPN
cana-788	130	8	(	(	PUNCT
cana-788	130	9	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	130	10	,	,	PUNCT
cana-788	130	11	w	w	PROPN
cana-788	130	12	,	,	PUNCT
cana-788	130	13	z	z	NOUN
cana-788	130	14	)	)	PUNCT
cana-788	130	15	)	)	PUNCT
cana-788	131	1	=	=	PUNCT
cana-788	132	1	[	[	X
cana-788	132	2	2𝑛+2	2𝑛+2	NUM
cana-788	132	3	×	×	NOUN
cana-788	132	4	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	132	5	]	]	PUNCT
cana-788	132	6	+	+	CCONJ
cana-788	133	1	[	[	X
cana-788	133	2	2𝑛+1	2𝑛+1	NUM
cana-788	133	3	×	×	NOUN
cana-788	133	4	𝑤4𝑧4	𝑤4𝑧4	NOUN
cana-788	133	5	]	]	X
cana-788	133	6	𝐷𝑍	𝐷𝑍	PROPN
cana-788	133	7	2	2	NUM
cana-788	133	8	(	(	PUNCT
cana-788	133	9	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	133	10	,	,	PUNCT
cana-788	133	11	w	w	PROPN
cana-788	133	12	,	,	PUNCT
cana-788	133	13	z	z	NOUN
cana-788	133	14	)	)	PUNCT
cana-788	133	15	)	)	PUNCT
cana-788	134	1	=	=	PUNCT
cana-788	135	1	[	[	X
cana-788	135	2	2𝑛+4	2𝑛+4	NUM
cana-788	135	3	×	×	NOUN
cana-788	135	4	𝑤2𝑧4	𝑤2𝑧4	X
cana-788	135	5	]	]	PUNCT
cana-788	135	6	+	+	CCONJ
cana-788	136	1	[	[	X
cana-788	136	2	2𝑛+3	2𝑛+3	PROPN
cana-788	136	3	×	×	NOUN
cana-788	136	4	𝑤4𝑧4	𝑤4𝑧4	NOUN
cana-788	136	5	]	]	X
cana-788	136	6	∴	∴	NOUN
cana-788	136	7	(	(	PUNCT
cana-788	136	8	𝐷𝑤	𝐷𝑤	PROPN
cana-788	136	9	2	2	NUM
cana-788	136	10	+	+	CCONJ
cana-788	136	11	𝐷𝑧	𝐷𝑧	PROPN
cana-788	136	12	2	2	NUM
cana-788	136	13	)	)	PUNCT
cana-788	136	14	(	(	PUNCT
cana-788	136	15	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	136	16	,	,	PUNCT
cana-788	136	17	w	w	PROPN
cana-788	136	18	,	,	PUNCT
cana-788	136	19	z	z	NOUN
cana-788	136	20	)	)	PUNCT
cana-788	136	21	)	)	PUNCT
cana-788	137	1	=	=	SYM
cana-788	137	2	3	3	NUM
cana-788	137	3	×	×	NOUN
cana-788	137	4	2𝑛+3	2𝑛+3	PROPN
cana-788	137	5	∴	∴	PROPN
cana-788	137	6	𝐹(𝐵𝑛	𝐹(𝐵𝑛	NUM
cana-788	137	7	)	)	PUNCT
cana-788	137	8	=	=	SYM
cana-788	137	9	3	3	NUM
cana-788	137	10	×	×	NOUN
cana-788	137	11	2𝑛+3	2𝑛+3	PROPN
cana-788	137	12	.	.	PUNCT
cana-788	138	1	for	for	ADP
cana-788	138	2	(	(	PUNCT
cana-788	138	3	g	g	NOUN
cana-788	138	4	)	)	PUNCT
cana-788	138	5	to	to	PART
cana-788	138	6	find	find	VERB
cana-788	138	7	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NOUN
cana-788	138	8	)	)	PUNCT
cana-788	138	9	=	=	PUNCT
cana-788	138	10	(	(	PUNCT
cana-788	138	11	𝑆𝑤𝐽	𝑆𝑤𝐽	PROPN
cana-788	138	12	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	138	13	)	)	PUNCT
cana-788	138	14	(	(	PUNCT
cana-788	138	15	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	138	16	,	,	PUNCT
cana-788	138	17	w	w	PROPN
cana-788	138	18	,	,	PUNCT
cana-788	138	19	z))|	z))|	PROPN
cana-788	138	20	𝑤=1	𝑤=1	PUNCT
cana-788	138	21	from	from	ADP
cana-788	138	22	(	(	PUNCT
cana-788	138	23	ii	ii	NOUN
cana-788	138	24	)	)	PUNCT
cana-788	138	25	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	138	26	(	(	PUNCT
cana-788	138	27	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	138	28	,	,	PUNCT
cana-788	138	29	w	w	PROPN
cana-788	138	30	,	,	PUNCT
cana-788	138	31	z	z	NOUN
cana-788	138	32	)	)	PUNCT
cana-788	138	33	)	)	PUNCT
cana-788	138	34	=	=	PRON
cana-788	138	35	(	(	PUNCT
cana-788	138	36	2𝑛	2𝑛	NUM
cana-788	138	37	)	)	PUNCT
cana-788	138	38	×	×	NOUN
cana-788	138	39	4	4	NUM
cana-788	138	40	×	×	NOUN
cana-788	138	41	2	2	NUM
cana-788	138	42	×	×	NOUN
cana-788	138	43	𝑤2𝑧4	𝑤2𝑧4	ADP
cana-788	138	44	+	+	CCONJ
cana-788	138	45	(	(	PUNCT
cana-788	138	46	2𝑛−1	2𝑛−1	NUM
cana-788	138	47	)	)	PUNCT
cana-788	138	48	×	×	NOUN
cana-788	138	49	4	4	NUM
cana-788	138	50	×	×	NOUN
cana-788	138	51	4	4	NUM
cana-788	138	52	×	×	NOUN
cana-788	138	53	𝑤4𝑧4	𝑤4𝑧4	X
cana-788	138	54	𝐽𝐷𝑤𝐷𝑧	𝐽𝐷𝑤𝐷𝑧	PROPN
cana-788	138	55	(	(	PUNCT
cana-788	138	56	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	X
cana-788	138	57	,	,	PUNCT
cana-788	138	58	w	w	PROPN
cana-788	138	59	,	,	PUNCT
cana-788	138	60	z	z	NOUN
cana-788	138	61	)	)	PUNCT
cana-788	138	62	)	)	PUNCT
cana-788	139	1	=	=	PUNCT
cana-788	140	1	[	[	X
cana-788	140	2	8	8	NUM
cana-788	140	3	×	×	NOUN
cana-788	140	4	2𝑛𝑤6	2𝑛𝑤6	NUM
cana-788	140	5	]	]	PUNCT
cana-788	140	6	+	+	CCONJ
cana-788	140	7	[	[	X
cana-788	140	8	16	16	NUM
cana-788	140	9	×	×	NOUN
cana-788	140	10	2𝑛−1𝑤8	2𝑛−1𝑤8	NUM
cana-788	140	11	]	]	PUNCT
cana-788	140	12	,	,	PUNCT
cana-788	140	13	(	(	PUNCT
cana-788	140	14	𝑆𝑤𝐽	𝑆𝑤𝐽	PROPN
cana-788	140	15	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	AUX
cana-788	140	16	)	)	PUNCT
cana-788	140	17	(	(	PUNCT
cana-788	140	18	𝑀𝑝𝑜𝑙𝑦(ɠ	𝑀𝑝𝑜𝑙𝑦(ɠ	VERB
cana-788	140	19	,	,	PUNCT
cana-788	140	20	w	w	PROPN
cana-788	140	21	,	,	PUNCT
cana-788	140	22	z	z	NOUN
cana-788	140	23	)	)	PUNCT
cana-788	140	24	)	)	PUNCT
cana-788	141	1	=	=	SYM
cana-788	141	2	∫	∫	PROPN
cana-788	142	1	[	[	PUNCT
cana-788	142	2	[	[	X
cana-788	142	3	8×2𝑛𝑡6]+[16×2𝑛−1𝑡8	8×2𝑛𝑡6]+[16×2𝑛−1𝑡8	NUM
cana-788	142	4	]	]	X
cana-788	142	5	𝑡	𝑡	X
cana-788	142	6	]	]	PUNCT
cana-788	142	7	𝑑𝑡	𝑑𝑡	ADP
cana-788	142	8	𝑤	𝑤	ADP
cana-788	142	9	0	0	PUNCT
cana-788	142	10	=	=	SYM
cana-788	142	11	8×2𝑛𝑤6	8×2𝑛𝑤6	NOUN
cana-788	142	12	6	6	NUM
cana-788	142	13	+	+	SYM
cana-788	142	14	16×2𝑛−1𝑤8	16×2𝑛−1𝑤8	NUM
cana-788	142	15	8	8	NUM
cana-788	142	16	∴	∴	PROPN
cana-788	142	17	𝐼(ɠ	𝐼(ɠ	NUM
cana-788	142	18	)	)	PUNCT
cana-788	142	19	=	=	PUNCT
cana-788	142	20	(	(	PUNCT
cana-788	142	21	𝑆𝑤𝐽	𝑆𝑤𝐽	PROPN
cana-788	142	22	𝐷𝑤𝐷𝑧	𝐷𝑤𝐷𝑧	PROPN
cana-788	142	23	)	)	PUNCT
cana-788	142	24	(	(	PUNCT
cana-788	142	25	𝑀𝑝𝑜𝑙𝑦(ɠ	𝑀𝑝𝑜𝑙𝑦(ɠ	VERB
cana-788	142	26	,	,	PUNCT
cana-788	142	27	w	w	PROPN
cana-788	142	28	,	,	PUNCT
cana-788	142	29	z))|	z))|	PROPN
cana-788	142	30	𝑤=1	𝑤=1	PUNCT
cana-788	142	31	=	=	SYM
cana-788	143	1	7	7	NUM
cana-788	143	2	3	3	NUM
cana-788	143	3	2𝑛.	2𝑛.	NUM
cana-788	143	4	□	□	SYM
cana-788	143	5	2.2	2.2	NUM
cana-788	143	6	the	the	DET
cana-788	143	7	m	m	NOUN
cana-788	143	8	-	-	ADJ
cana-788	143	9	polynomial	polynomial	ADJ
cana-788	143	10	of	of	ADP
cana-788	143	11	schreier	schreier	NOUN
cana-788	143	12	graphs	graph	NOUN
cana-788	143	13	(	(	PUNCT
cana-788	143	14	with	with	ADP
cana-788	143	15	loops	loop	NOUN
cana-788	143	16	)	)	PUNCT
cana-788	143	17	of	of	ADP
cana-788	143	18	the	the	DET
cana-788	143	19	basilica	basilica	PROPN
cana-788	143	20	group	group	NOUN
cana-788	143	21	in	in	ADP
cana-788	143	22	this	this	DET
cana-788	143	23	section	section	NOUN
cana-788	143	24	the	the	DET
cana-788	143	25	schreier	schreier	NOUN
cana-788	143	26	graph	graph	NOUN
cana-788	143	27	of	of	ADP
cana-788	143	28	the	the	DET
cana-788	143	29	basilica	basilica	PROPN
cana-788	143	30	group	group	NOUN
cana-788	143	31	with	with	ADP
cana-788	143	32	loops	loop	NOUN
cana-788	143	33	considered	consider	VERB
cana-788	143	34	and	and	CCONJ
cana-788	143	35	the	the	DET
cana-788	143	36	mpolynomial	mpolynomial	NOUN
cana-788	143	37	for	for	ADP
cana-788	143	38	the	the	DET
cana-788	143	39	graph	graph	NOUN
cana-788	143	40	is	be	AUX
cana-788	143	41	derived	derive	VERB
cana-788	143	42	in	in	ADP
cana-788	143	43	the	the	DET
cana-788	143	44	following	following	NOUN
cana-788	143	45	theorem	theorem	VERB
cana-788	143	46	2.3	2.3	NUM
cana-788	143	47	.	.	PUNCT
cana-788	144	1	the	the	DET
cana-788	144	2	surface	surface	NOUN
cana-788	144	3	plot	plot	NOUN
cana-788	144	4	of	of	ADP
cana-788	144	5	this	this	DET
cana-788	144	6	mpolynomial	mpolynomial	NOUN
cana-788	144	7	can	can	AUX
cana-788	144	8	be	be	AUX
cana-788	144	9	seen	see	VERB
cana-788	144	10	in	in	ADP
cana-788	144	11	figure	figure	NOUN
cana-788	144	12	4	4	NUM
cana-788	144	13	.	.	PUNCT
cana-788	145	1	moreover	moreover	ADV
cana-788	145	2	,	,	PUNCT
cana-788	145	3	by	by	ADP
cana-788	145	4	the	the	DET
cana-788	145	5	use	use	NOUN
cana-788	145	6	of	of	ADP
cana-788	145	7	this	this	DET
cana-788	145	8	m	m	NOUN
cana-788	145	9	-	-	ADJ
cana-788	145	10	polynomial	polynomial	ADJ
cana-788	145	11	the	the	DET
cana-788	145	12	chosen	choose	VERB
cana-788	145	13	topological	topological	ADJ
cana-788	145	14	indices	index	NOUN
cana-788	145	15	are	be	AUX
cana-788	145	16	derived	derive	VERB
cana-788	145	17	in	in	ADP
cana-788	145	18	theorem	theorem	ADJ
cana-788	145	19	2.4	2.4	NUM
cana-788	145	20	.	.	PUNCT
cana-788	146	1	theorem	theorem	VERB
cana-788	146	2	2.3	2.3	NUM
cana-788	146	3	.	.	PUNCT
cana-788	147	1	let	let	VERB
cana-788	147	2	𝐵𝑛	𝐵𝑛	PROPN
cana-788	147	3	∗	∗	NOUN
cana-788	147	4	be	be	AUX
cana-788	147	5	schreier	schreier	ADJ
cana-788	147	6	graph	graph	NOUN
cana-788	147	7	(	(	PUNCT
cana-788	147	8	with	with	ADP
cana-788	147	9	loops	loop	NOUN
cana-788	147	10	)	)	PUNCT
cana-788	147	11	of	of	ADP
cana-788	147	12	basilica	basilica	PROPN
cana-788	147	13	group	group	NOUN
cana-788	147	14	.	.	PUNCT
cana-788	148	1	then	then	ADV
cana-788	148	2	the	the	DET
cana-788	148	3	m	m	ADJ
cana-788	148	4	–	–	PUNCT
cana-788	148	5	polynomial	polynomial	ADJ
cana-788	148	6	of	of	ADP
cana-788	148	7	𝐵𝑛	𝐵𝑛	PROPN
cana-788	148	8	∗	∗	NOUN
cana-788	148	9	is	be	AUX
cana-788	148	10	𝑓(𝑤	𝑓(𝑤	PROPN
cana-788	148	11	,	,	PUNCT
cana-788	148	12	𝑧	𝑧	NOUN
cana-788	148	13	)	)	PUNCT
cana-788	148	14	=	=	SYM
cana-788	149	1	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	149	2	∗	∗	NOUN
cana-788	149	3	,	,	PUNCT
cana-788	149	4	w	w	PROPN
cana-788	149	5	,	,	PUNCT
cana-788	149	6	z	z	NOUN
cana-788	149	7	)	)	PUNCT
cana-788	149	8	=	=	SYM
cana-788	149	9	(	(	PUNCT
cana-788	149	10	2𝑛+1)𝑤4𝑧4	2𝑛+1)𝑤4𝑧4	ADV
cana-788	149	11	.	.	PUNCT
cana-788	150	1	proof	proof	NOUN
cana-788	150	2	.	.	PUNCT
cana-788	151	1	from	from	ADP
cana-788	151	2	figure	figure	NOUN
cana-788	151	3	1	1	NUM
cana-788	151	4	,	,	PUNCT
cana-788	151	5	degree	degree	NOUN
cana-788	151	6	of	of	ADP
cana-788	151	7	each	each	DET
cana-788	151	8	vertex	vertex	NOUN
cana-788	151	9	in	in	ADP
cana-788	151	10	𝐵𝑛	𝐵𝑛	PROPN
cana-788	151	11	∗	∗	NOUN
cana-788	151	12	is	be	AUX
cana-788	151	13	four	four	NUM
cana-788	151	14	if	if	SCONJ
cana-788	151	15	we	we	PRON
cana-788	151	16	consider	consider	VERB
cana-788	151	17	the	the	DET
cana-788	151	18	loops	loop	NOUN
cana-788	151	19	.	.	PUNCT
cana-788	152	1	hence	hence	ADV
cana-788	152	2	the	the	DET
cana-788	152	3	schreier	schreier	ADJ
cana-788	152	4	graph	graph	NOUN
cana-788	152	5	of	of	ADP
cana-788	152	6	basilica	basilica	PROPN
cana-788	152	7	group	group	NOUN
cana-788	152	8	is	be	AUX
cana-788	152	9	a	a	DET
cana-788	152	10	4regular	4regular	NUM
cana-788	152	11	graph	graph	NOUN
cana-788	152	12	.	.	PUNCT
cana-788	153	1	so	so	ADV
cana-788	153	2	,	,	PUNCT
cana-788	153	3	the	the	DET
cana-788	153	4	number	number	NOUN
cana-788	153	5	vertices	vertice	VERB
cana-788	153	6	in	in	ADP
cana-788	153	7	the	the	DET
cana-788	153	8	graph	graph	NOUN
cana-788	153	9	are	be	AUX
cana-788	153	10	2𝑛	2𝑛	NUM
cana-788	153	11	and	and	CCONJ
cana-788	153	12	the	the	DET
cana-788	153	13	number	number	NOUN
cana-788	153	14	of	of	ADP
cana-788	153	15	edges	edge	NOUN
cana-788	153	16	is	be	AUX
cana-788	153	17	2𝑛+1	2𝑛+1	NUM
cana-788	153	18	.	.	PUNCT
cana-788	154	1	that	that	PRON
cana-788	154	2	is	be	AUX
cana-788	154	3	,	,	PUNCT
cana-788	154	4	|𝑉(𝐵𝑛	|𝑉(𝐵𝑛	PROPN
cana-788	154	5	⬚	⬚	PROPN
cana-788	154	6	∗	∗	NOUN
cana-788	154	7	)	)	PUNCT
cana-788	154	8	|	|	ADV
cana-788	154	9	=	=	SYM
cana-788	154	10	2𝑛	2𝑛	NUM
cana-788	154	11	,	,	PUNCT
cana-788	154	12	|𝐸(𝐵𝑛	|𝐸(𝐵𝑛	PROPN
cana-788	154	13	∗)|	∗)|	PROPN
cana-788	154	14	=	=	SYM
cana-788	154	15	2𝑛+1	2𝑛+1	PROPN
cana-788	154	16	.	.	PUNCT
cana-788	155	1	since	since	SCONJ
cana-788	155	2	𝐵𝑛	𝐵𝑛	PROPN
cana-788	155	3	∗	∗	NOUN
cana-788	155	4	is	be	AUX
cana-788	155	5	a	a	DET
cana-788	155	6	4	4	NUM
cana-788	155	7	-	-	PUNCT
cana-788	155	8	regular	regular	ADJ
cana-788	155	9	graph	graph	NOUN
cana-788	155	10	,	,	PUNCT
cana-788	155	11	there	there	PRON
cana-788	155	12	will	will	AUX
cana-788	155	13	be	be	AUX
cana-788	155	14	only	only	ADV
cana-788	155	15	one	one	NUM
cana-788	155	16	edge	edge	NOUN
cana-788	155	17	partition	partition	NOUN
cana-788	155	18	which	which	PRON
cana-788	155	19	is	be	AUX
cana-788	155	20	(	(	PUNCT
cana-788	155	21	4	4	NUM
cana-788	155	22	,	,	PUNCT
cana-788	155	23	4	4	NUM
cana-788	155	24	)	)	PUNCT
cana-788	155	25	and	and	CCONJ
cana-788	155	26	therefore	therefore	ADV
cana-788	155	27	|ȅ44|	|ȅ44|	VERB
cana-788	156	1	=	=	SYM
cana-788	157	1	2𝑛+1	2𝑛+1	NUM
cana-788	157	2	therefore	therefore	ADV
cana-788	157	3	,	,	PUNCT
cana-788	157	4	we	we	PRON
cana-788	157	5	have	have	VERB
cana-788	157	6	by	by	ADP
cana-788	157	7	the	the	DET
cana-788	157	8	definition	definition	NOUN
cana-788	157	9	2.1	2.1	NUM
cana-788	157	10	,	,	PUNCT
cana-788	157	11	𝑓(𝐵𝑛	𝑓(𝐵𝑛	X
cana-788	157	12	∗	∗	NOUN
cana-788	157	13	,	,	PUNCT
cana-788	157	14	𝑤	𝑤	ADP
cana-788	157	15	,	,	PUNCT
cana-788	157	16	𝑧	𝑧	NOUN
cana-788	157	17	)	)	PUNCT
cana-788	157	18	=	=	SYM
cana-788	158	1	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	𝑀𝑝𝑜𝑙𝑦(𝐵𝑛	NUM
cana-788	158	2	∗	∗	NOUN
cana-788	158	3	,	,	PUNCT
cana-788	158	4	w	w	PROPN
cana-788	158	5	,	,	PUNCT
cana-788	158	6	z	z	NOUN
cana-788	158	7	)	)	PUNCT
cana-788	158	8	=	=	SYM
cana-788	158	9	∑	∑	PROPN
cana-788	158	10	|ȅ𝑖𝑗(𝐵𝑛	|ȅ𝑖𝑗(𝐵𝑛	PROPN
cana-788	158	11	∗)|𝑤𝑖𝑧𝑗	∗)|𝑤𝑖𝑧𝑗	PROPN
cana-788	158	12	𝕀≤𝑖≤𝑗≤𝕁	𝕀≤𝑖≤𝑗≤𝕁	PROPN
cana-788	158	13	=	=	PRON
cana-788	159	1	|ȅ44|𝑤4𝑧4	|ȅ44|𝑤4𝑧4	ADV
cana-788	159	2	communications	communication	NOUN
cana-788	159	3	on	on	ADP
cana-788	159	4	applied	apply	VERB
cana-788	159	5	nonlinear	nonlinear	ADJ
cana-788	159	6	analysis	analysis	NOUN
cana-788	159	7	issn	issn	NOUN
cana-788	159	8	:	:	PUNCT
cana-788	159	9	1074	1074	NUM
cana-788	159	10	-	-	PUNCT
cana-788	159	11	133x	133x	NUM
cana-788	159	12	vol	vol	NOUN
cana-788	159	13	31	31	NUM
cana-788	159	14	no	no	NOUN
cana-788	159	15	.	.	PUNCT
cana-788	160	1	3s	3s	NUM
cana-788	160	2	(	(	PUNCT
cana-788	160	3	2024	2024	NUM
cana-788	160	4	)	)	PUNCT
cana-788	160	5	384	384	NUM
cana-788	160	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	160	7	∴	∴	PROPN
cana-788	160	8	𝑓(𝐵𝑛	𝑓(𝐵𝑛	X
cana-788	160	9	∗	∗	PROPN
cana-788	160	10	,	,	PUNCT
cana-788	160	11	𝑤	𝑤	ADP
cana-788	160	12	,	,	PUNCT
cana-788	160	13	𝑧	𝑧	NOUN
cana-788	160	14	)	)	PUNCT
cana-788	160	15	=	=	SYM
cana-788	160	16	(	(	PUNCT
cana-788	160	17	2𝑛+1)𝑤4𝑧4	2𝑛+1)𝑤4𝑧4	NUM
cana-788	160	18	.	.	PUNCT
cana-788	160	19	fig	fig	NOUN
cana-788	160	20	.	.	PUNCT
cana-788	161	1	3	3	NUM
cana-788	161	2	surface	surface	NOUN
cana-788	161	3	plot	plot	NOUN
cana-788	161	4	of	of	ADP
cana-788	161	5	m	m	NOUN
cana-788	161	6	-	-	ADJ
cana-788	161	7	polynomial	polynomial	ADJ
cana-788	161	8	of	of	ADP
cana-788	161	9	𝐵𝑛	𝐵𝑛	PROPN
cana-788	161	10	∗	∗	NOUN
cana-788	161	11	theorem	theorem	VERB
cana-788	161	12	2.4	2.4	NUM
cana-788	161	13	let	let	VERB
cana-788	161	14	𝐵𝑛	𝐵𝑛	PROPN
cana-788	161	15	∗	∗	NOUN
cana-788	161	16	be	be	AUX
cana-788	161	17	the	the	DET
cana-788	161	18	schreier	schreier	NOUN
cana-788	161	19	graph	graph	NOUN
cana-788	161	20	(	(	PUNCT
cana-788	161	21	including	include	VERB
cana-788	161	22	loops	loop	NOUN
cana-788	161	23	)	)	PUNCT
cana-788	161	24	of	of	ADP
cana-788	161	25	basilica	basilica	PROPN
cana-788	161	26	group	group	NOUN
cana-788	161	27	.	.	PUNCT
cana-788	162	1	then	then	ADV
cana-788	162	2	the	the	DET
cana-788	162	3	topological	topological	ADJ
cana-788	162	4	indices	index	NOUN
cana-788	162	5	of	of	ADP
cana-788	162	6	𝐵𝑛	𝐵𝑛	PROPN
cana-788	162	7	∗	∗	NOUN
cana-788	162	8	,	,	PUNCT
cana-788	162	9	𝑛	𝑛	PROPN
cana-788	162	10	>	>	X
cana-788	162	11	1	1	NUM
cana-788	162	12	using	use	VERB
cana-788	162	13	the	the	DET
cana-788	162	14	m	m	NOUN
cana-788	162	15	-	-	ADJ
cana-788	162	16	polynomial	polynomial	ADJ
cana-788	162	17	are	be	AUX
cana-788	162	18	a	a	DET
cana-788	162	19	)	)	PUNCT
cana-788	162	20	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	NOUN
cana-788	162	21	∗	∗	NOUN
cana-788	162	22	)	)	PUNCT
cana-788	163	1	=	=	PUNCT
cana-788	163	2	2𝑛+4	2𝑛+4	NUM
cana-788	163	3	b	b	X
cana-788	163	4	)	)	PUNCT
cana-788	163	5	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	PROPN
cana-788	163	6	∗	∗	NOUN
cana-788	163	7	)	)	PUNCT
cana-788	164	1	=	=	SYM
cana-788	164	2	2𝑛+5	2𝑛+5	PROPN
cana-788	164	3	c	c	NOUN
cana-788	164	4	)	)	PUNCT
cana-788	164	5	𝑀2	𝑀2	PROPN
cana-788	164	6	𝑚(𝐵𝑛	𝑚(𝐵𝑛	X
cana-788	164	7	∗	∗	NOUN
cana-788	164	8	)	)	PUNCT
cana-788	165	1	=	=	NOUN
cana-788	165	2	2𝑛−3	2𝑛−3	NUM
cana-788	165	3	d	d	NOUN
cana-788	165	4	)	)	PUNCT
cana-788	165	5	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	PROPN
cana-788	165	6	∗	∗	NOUN
cana-788	165	7	)	)	PUNCT
cana-788	165	8	=	=	SYM
cana-788	166	1	2𝑛+4	2𝑛+4	NUM
cana-788	166	2	e	e	NOUN
cana-788	166	3	)	)	PUNCT
cana-788	166	4	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	PROPN
cana-788	166	5	∗	∗	NOUN
cana-788	166	6	)	)	PUNCT
cana-788	167	1	=	=	PUNCT
cana-788	167	2	2𝑛+7	2𝑛+7	NOUN
cana-788	167	3	f	f	NOUN
cana-788	167	4	)	)	PUNCT
cana-788	167	5	𝐹(𝐵𝑛	𝐹(𝐵𝑛	NOUN
cana-788	167	6	∗	∗	NOUN
cana-788	167	7	)	)	PUNCT
cana-788	168	1	=	=	PUNCT
cana-788	168	2	2𝑛+6	2𝑛+6	NUM
cana-788	168	3	g	g	NOUN
cana-788	168	4	)	)	PUNCT
cana-788	168	5	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NOUN
cana-788	168	6	∗	∗	NOUN
cana-788	168	7	)	)	PUNCT
cana-788	168	8	=	=	SYM
cana-788	168	9	2𝑛+2	2𝑛+2	NOUN
cana-788	168	10	.	.	PUNCT
cana-788	169	1	proof	proof	NOUN
cana-788	169	2	.	.	PUNCT
cana-788	170	1	these	these	DET
cana-788	170	2	results	result	NOUN
cana-788	170	3	can	can	AUX
cana-788	170	4	be	be	AUX
cana-788	170	5	proved	prove	VERB
cana-788	170	6	using	use	VERB
cana-788	170	7	theorem	theorem	NOUN
cana-788	170	8	2.2	2.2	NUM
cana-788	170	9	.	.	PUNCT
cana-788	171	1	□	□	PUNCT
cana-788	171	2	2.3	2.3	NUM
cana-788	171	3	the	the	DET
cana-788	171	4	m	m	NOUN
cana-788	171	5	-	-	ADJ
cana-788	171	6	polynomial	polynomial	ADJ
cana-788	171	7	of	of	ADP
cana-788	171	8	schreier	schreier	NOUN
cana-788	171	9	graphs	graph	NOUN
cana-788	171	10	(	(	PUNCT
cana-788	171	11	without	without	ADP
cana-788	171	12	loops	loop	NOUN
cana-788	171	13	)	)	PUNCT
cana-788	171	14	of	of	ADP
cana-788	171	15	the	the	DET
cana-788	171	16	grigorchuk	grigorchuk	NOUN
cana-788	171	17	group	group	NOUN
cana-788	171	18	let	let	VERB
cana-788	171	19	γ𝑛	γ𝑛	PART
cana-788	171	20	be	be	AUX
cana-788	171	21	the	the	DET
cana-788	171	22	schreier	schreier	ADJ
cana-788	171	23	graph	graph	NOUN
cana-788	171	24	of	of	ADP
cana-788	171	25	the	the	DET
cana-788	171	26	grigorchuk	grigorchuk	NOUN
cana-788	171	27	group	group	NOUN
cana-788	171	28	.	.	PUNCT
cana-788	172	1	in	in	ADP
cana-788	172	2	finding	find	VERB
cana-788	172	3	the	the	DET
cana-788	172	4	m	m	NOUN
cana-788	172	5	-	-	NOUN
cana-788	172	6	polynomial	polynomial	ADJ
cana-788	172	7	,	,	PUNCT
cana-788	172	8	this	this	DET
cana-788	172	9	graph	graph	NOUN
cana-788	172	10	is	be	AUX
cana-788	172	11	considered	consider	VERB
cana-788	172	12	as	as	ADV
cana-788	172	13	unlabelled	unlabelled	ADJ
cana-788	172	14	and	and	CCONJ
cana-788	172	15	without	without	ADP
cana-788	172	16	loops	loop	NOUN
cana-788	172	17	.	.	PUNCT
cana-788	173	1	the	the	DET
cana-788	173	2	graph	graph	NOUN
cana-788	173	3	γ𝑛	γ𝑛	ADV
cana-788	173	4	for	for	ADP
cana-788	173	5	𝑛	𝑛	PROPN
cana-788	173	6	>	>	SYM
cana-788	173	7	1	1	NUM
cana-788	173	8	can	can	AUX
cana-788	173	9	be	be	AUX
cana-788	173	10	seen	see	VERB
cana-788	173	11	in	in	ADP
cana-788	173	12	the	the	DET
cana-788	173	13	following	follow	VERB
cana-788	173	14	figure	figure	NOUN
cana-788	173	15	4	4	NUM
cana-788	173	16	.	.	PUNCT
cana-788	173	17	figure	figure	VERB
cana-788	173	18	4	4	NUM
cana-788	173	19	.	.	PUNCT
cana-788	174	1	structure	structure	NOUN
cana-788	174	2	of	of	ADP
cana-788	174	3	γ𝑛	γ𝑛	PROPN
cana-788	174	4	for	for	ADP
cana-788	174	5	𝑛	𝑛	PRON
cana-788	174	6	=	=	SYM
cana-788	174	7	1,2,3	1,2,3	NOUN
cana-788	174	8	.	.	PUNCT
cana-788	174	9	theorem	theorem	VERB
cana-788	174	10	2.5	2.5	NUM
cana-788	174	11	.	.	PUNCT
cana-788	175	1	let	let	VERB
cana-788	175	2	γ𝑛	γ𝑛	PART
cana-788	175	3	be	be	AUX
cana-788	175	4	schreier	schreier	ADJ
cana-788	175	5	graph	graph	NOUN
cana-788	175	6	(	(	PUNCT
cana-788	175	7	without	without	ADP
cana-788	175	8	loops	loop	NOUN
cana-788	175	9	)	)	PUNCT
cana-788	175	10	of	of	ADP
cana-788	175	11	grigorchuk	grigorchuk	PROPN
cana-788	175	12	group	group	NOUN
cana-788	175	13	.	.	PUNCT
cana-788	176	1	then	then	ADV
cana-788	176	2	the	the	DET
cana-788	176	3	m	m	ADJ
cana-788	176	4	–	–	PUNCT
cana-788	176	5	polynomial	polynomial	ADJ
cana-788	176	6	of	of	ADP
cana-788	176	7	γ𝑛	γ𝑛	PROPN
cana-788	176	8	is	be	AUX
cana-788	176	9	𝑓(γ𝑛	𝑓(γ𝑛	PROPN
cana-788	176	10	,	,	PUNCT
cana-788	176	11	𝑤	𝑤	ADP
cana-788	176	12	,	,	PUNCT
cana-788	176	13	𝑧	𝑧	NOUN
cana-788	176	14	)	)	PUNCT
cana-788	176	15	=	=	SYM
cana-788	176	16	𝑀𝑝𝑜𝑙𝑦(γ𝑛	𝑀𝑝𝑜𝑙𝑦(γ𝑛	PROPN
cana-788	176	17	,	,	PUNCT
cana-788	176	18	w	w	PROPN
cana-788	176	19	,	,	PUNCT
cana-788	176	20	z	z	NOUN
cana-788	176	21	)	)	PUNCT
cana-788	176	22	=	=	SYM
cana-788	176	23	2𝑤1𝑧3	2𝑤1𝑧3	NUM
cana-788	176	24	+	+	CCONJ
cana-788	176	25	(	(	PUNCT
cana-788	176	26	2𝑛	2𝑛	NUM
cana-788	176	27	+	+	CCONJ
cana-788	176	28	2𝑛−1	2𝑛−1	NUM
cana-788	176	29	−	−	PROPN
cana-788	176	30	4)𝑤3𝑧3	4)𝑤3𝑧3	PROPN
cana-788	176	31	.	.	PUNCT
cana-788	177	1	communications	communication	NOUN
cana-788	177	2	on	on	ADP
cana-788	177	3	applied	apply	VERB
cana-788	177	4	nonlinear	nonlinear	ADJ
cana-788	177	5	analysis	analysis	NOUN
cana-788	177	6	issn	issn	NOUN
cana-788	177	7	:	:	PUNCT
cana-788	177	8	1074	1074	NUM
cana-788	177	9	-	-	PUNCT
cana-788	177	10	133x	133x	NUM
cana-788	177	11	vol	vol	NOUN
cana-788	177	12	31	31	NUM
cana-788	177	13	no	no	NOUN
cana-788	177	14	.	.	PUNCT
cana-788	178	1	3s	3s	NUM
cana-788	178	2	(	(	PUNCT
cana-788	178	3	2024	2024	NUM
cana-788	178	4	)	)	PUNCT
cana-788	178	5	385	385	NUM
cana-788	178	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	178	7	proof	proof	NOUN
cana-788	178	8	.	.	PUNCT
cana-788	179	1	from	from	ADP
cana-788	179	2	the	the	DET
cana-788	179	3	following	follow	VERB
cana-788	179	4	table	table	NOUN
cana-788	179	5	3	3	NUM
cana-788	179	6	,	,	PUNCT
cana-788	179	7	one	one	PRON
cana-788	179	8	can	can	AUX
cana-788	179	9	understand	understand	VERB
cana-788	179	10	the	the	DET
cana-788	179	11	edge	edge	NOUN
cana-788	179	12	growth	growth	NOUN
cana-788	179	13	pattern	pattern	NOUN
cana-788	179	14	and	and	CCONJ
cana-788	179	15	edge	edge	NOUN
cana-788	179	16	partition	partition	NOUN
cana-788	179	17	of	of	ADP
cana-788	179	18	schreier	schreier	ADJ
cana-788	179	19	graph	graph	NOUN
cana-788	179	20	of	of	ADP
cana-788	179	21	basilica	basilica	PROPN
cana-788	179	22	group	group	NOUN
cana-788	179	23	.	.	PUNCT
cana-788	180	1	table	table	NOUN
cana-788	180	2	3	3	NUM
cana-788	180	3	.	.	PUNCT
cana-788	180	4	edge	edge	NOUN
cana-788	180	5	growth	growth	NOUN
cana-788	180	6	pattern	pattern	NOUN
cana-788	180	7	and	and	CCONJ
cana-788	180	8	edge	edge	NOUN
cana-788	180	9	partition	partition	NOUN
cana-788	180	10	of	of	ADP
cana-788	180	11	schreier	schreier	ADJ
cana-788	180	12	graph	graph	NOUN
cana-788	180	13	of	of	ADP
cana-788	180	14	grigorchuk	grigorchuk	PROPN
cana-788	180	15	group	group	NOUN
cana-788	180	16	grigorchuk	grigorchuk	PROPN
cana-788	180	17	group	group	PROPN
cana-788	180	18	no	no	INTJ
cana-788	180	19	.	.	PUNCT
cana-788	180	20	of	of	ADP
cana-788	180	21	vertex	vertex	NOUN
cana-788	180	22	(	(	PUNCT
cana-788	180	23	x	x	X
cana-788	180	24	)	)	PUNCT
cana-788	180	25	no	no	NOUN
cana-788	180	26	of	of	ADP
cana-788	180	27	edges(y	edges(y	NOUN
cana-788	180	28	)	)	PUNCT
cana-788	180	29	edge	edge	NOUN
cana-788	180	30	growth	growth	NOUN
cana-788	180	31	pattern	pattern	NOUN
cana-788	180	32	edge	edge	NOUN
cana-788	180	33	partitions	partition	NOUN
cana-788	180	34	(	(	PUNCT
cana-788	180	35	1,3	1,3	NUM
cana-788	180	36	)	)	PUNCT
cana-788	180	37	(	(	PUNCT
cana-788	180	38	3,3	3,3	NUM
cana-788	180	39	)	)	PUNCT
cana-788	180	40	γ2	γ2	NOUN
cana-788	180	41	4	4	NUM
cana-788	180	42	4	4	NUM
cana-788	180	43	(	(	PUNCT
cana-788	180	44	21	21	NUM
cana-788	180	45	)	)	PUNCT
cana-788	181	1	+	+	CCONJ
cana-788	181	2	(	(	PUNCT
cana-788	181	3	21	21	NUM
cana-788	181	4	)	)	SYM
cana-788	181	5	2	2	NUM
cana-788	181	6	2	2	NUM
cana-788	181	7	γ3	γ3	NOUN
cana-788	181	8	8	8	NUM
cana-788	181	9	10	10	NUM
cana-788	181	10	(	(	PUNCT
cana-788	181	11	21)+	21)+	NUM
cana-788	181	12	(	(	PUNCT
cana-788	181	13	2)3	2)3	NUM
cana-788	181	14	2	2	NUM
cana-788	181	15	8	8	NUM
cana-788	181	16	γ4	γ4	NOUN
cana-788	181	17	16	16	NUM
cana-788	181	18	22	22	NUM
cana-788	181	19	(	(	PUNCT
cana-788	181	20	21	21	NUM
cana-788	181	21	)	)	PUNCT
cana-788	181	22	+	+	CCONJ
cana-788	181	23	(	(	PUNCT
cana-788	181	24	24	24	NUM
cana-788	181	25	+	+	CCONJ
cana-788	181	26	4	4	NUM
cana-788	181	27	)	)	SYM
cana-788	181	28	2	2	NUM
cana-788	181	29	20	20	NUM
cana-788	181	30	γ5	γ5	NOUN
cana-788	181	31	32	32	NUM
cana-788	181	32	46	46	NUM
cana-788	181	33	(	(	PUNCT
cana-788	181	34	21	21	NUM
cana-788	181	35	)	)	PUNCT
cana-788	181	36	+	+	CCONJ
cana-788	181	37	(	(	PUNCT
cana-788	181	38	25	25	NUM
cana-788	181	39	+	+	NOUN
cana-788	181	40	12	12	NUM
cana-788	181	41	)	)	PUNCT
cana-788	181	42	2	2	NUM
cana-788	181	43	44	44	NUM
cana-788	181	44	γ6	γ6	NOUN
cana-788	181	45	64	64	NUM
cana-788	181	46	94	94	NUM
cana-788	181	47	(	(	PUNCT
cana-788	181	48	21	21	NUM
cana-788	181	49	)	)	PUNCT
cana-788	181	50	+	+	CCONJ
cana-788	181	51	(	(	PUNCT
cana-788	181	52	26	26	NUM
cana-788	181	53	+	+	NUM
cana-788	181	54	28	28	NUM
cana-788	181	55	)	)	PUNCT
cana-788	181	56	2	2	NUM
cana-788	181	57	92	92	NUM
cana-788	181	58	.	.	PUNCT
cana-788	181	59	.	.	PUNCT
cana-788	181	60	.	.	PUNCT
cana-788	181	61	.	.	PUNCT
cana-788	181	62	.	.	PUNCT
cana-788	181	63	.	.	PUNCT
cana-788	181	64	.	.	PUNCT
cana-788	181	65	.	.	PUNCT
cana-788	181	66	.	.	PUNCT
cana-788	181	67	.	.	PUNCT
cana-788	181	68	.	.	PUNCT
cana-788	181	69	.	.	PUNCT
cana-788	182	1	γ𝑛	γ𝑛	PROPN
cana-788	182	2	2𝑛	2𝑛	PROPN
cana-788	182	3	3	3	X
cana-788	182	4	.	.	X
cana-788	182	5	2𝑛−1	2𝑛−1	NUM
cana-788	183	1	−	−	NOUN
cana-788	183	2	2	2	NUM
cana-788	183	3	(	(	PUNCT
cana-788	183	4	21	21	NUM
cana-788	183	5	)	)	PUNCT
cana-788	184	1	+	+	NUM
cana-788	184	2	2𝑛	2𝑛	PROPN
cana-788	184	3	+	+	CCONJ
cana-788	184	4	(	(	PUNCT
cana-788	184	5	2𝑛−1	2𝑛−1	NUM
cana-788	184	6	−	−	NOUN
cana-788	184	7	4	4	NUM
cana-788	184	8	)	)	PUNCT
cana-788	184	9	21	21	NUM
cana-788	184	10	2𝑛	2𝑛	NOUN
cana-788	184	11	+	+	CCONJ
cana-788	184	12	2𝑛−1	2𝑛−1	NUM
cana-788	184	13	−	−	NOUN
cana-788	184	14	4	4	NUM
cana-788	184	15	from	from	ADP
cana-788	184	16	table	table	NOUN
cana-788	184	17	3	3	NUM
cana-788	184	18	,	,	PUNCT
cana-788	184	19	it	it	PRON
cana-788	184	20	is	be	AUX
cana-788	184	21	clear	clear	ADJ
cana-788	184	22	that	that	SCONJ
cana-788	184	23	the	the	DET
cana-788	184	24	number	number	NOUN
cana-788	184	25	of	of	ADP
cana-788	184	26	edges	edge	NOUN
cana-788	184	27	in	in	ADP
cana-788	184	28	the	the	DET
cana-788	184	29	above	above	ADJ
cana-788	184	30	two	two	NUM
cana-788	184	31	category	category	NOUN
cana-788	184	32	are	be	AUX
cana-788	184	33	|ȅ13|	|ȅ13|	NOUN
cana-788	184	34	=	=	NOUN
cana-788	184	35	2	2	NUM
cana-788	184	36	and|ȅ33|	and|ȅ33|	PROPN
cana-788	184	37	=	=	SYM
cana-788	184	38	2𝑛	2𝑛	PROPN
cana-788	185	1	+	+	CCONJ
cana-788	185	2	2𝑛−1	2𝑛−1	NUM
cana-788	185	3	−	−	NOUN
cana-788	185	4	4	4	NUM
cana-788	185	5	.	.	PUNCT
cana-788	186	1	now	now	ADV
cana-788	186	2	,	,	PUNCT
cana-788	186	3	by	by	ADP
cana-788	186	4	the	the	DET
cana-788	186	5	definition	definition	NOUN
cana-788	186	6	2.1	2.1	NUM
cana-788	186	7	,	,	PUNCT
cana-788	186	8	we	we	PRON
cana-788	186	9	have	have	VERB
cana-788	186	10	𝑓(γ𝑛	𝑓(γ𝑛	PROPN
cana-788	186	11	,	,	PUNCT
cana-788	186	12	𝑤	𝑤	ADP
cana-788	186	13	,	,	PUNCT
cana-788	186	14	𝑧	𝑧	NOUN
cana-788	186	15	)	)	PUNCT
cana-788	186	16	=	=	SYM
cana-788	186	17	𝑀𝑝𝑜𝑙𝑦(γ𝑛	𝑀𝑝𝑜𝑙𝑦(γ𝑛	PROPN
cana-788	186	18	,	,	PUNCT
cana-788	186	19	w	w	PROPN
cana-788	186	20	,	,	PUNCT
cana-788	186	21	z	z	NOUN
cana-788	186	22	)	)	PUNCT
cana-788	186	23	=	=	SYM
cana-788	186	24	∑	∑	PUNCT
cana-788	186	25	|ȅ𝑖𝑗(γ𝑛)|𝑤𝑖𝑧𝑗	|ȅ𝑖𝑗(γ𝑛)|𝑤𝑖𝑧𝑗	ADJ
cana-788	186	26	𝕀≤𝑖≤𝑗≤𝕁	𝕀≤𝑖≤𝑗≤𝕁	NOUN
cana-788	186	27	=	=	SYM
cana-788	186	28	|ȅ13|𝑤1𝑧3	|ȅ13|𝑤1𝑧3	PROPN
cana-788	186	29	+	+	CCONJ
cana-788	186	30	|ȅ33|𝑤3𝑧3	|ȅ33|𝑤3𝑧3	PROPN
cana-788	186	31	∴	∴	PROPN
cana-788	186	32	𝑓(γ𝑛	𝑓(γ𝑛	PROPN
cana-788	186	33	,	,	PUNCT
cana-788	186	34	𝑤	𝑤	ADP
cana-788	186	35	,	,	PUNCT
cana-788	186	36	𝑧	𝑧	NOUN
cana-788	186	37	)	)	PUNCT
cana-788	186	38	=	=	SYM
cana-788	186	39	2𝑤1𝑧3	2𝑤1𝑧3	NUM
cana-788	186	40	+	+	CCONJ
cana-788	186	41	(	(	PUNCT
cana-788	186	42	2𝑛	2𝑛	NUM
cana-788	186	43	+	+	CCONJ
cana-788	186	44	2𝑛−1	2𝑛−1	NUM
cana-788	186	45	−	−	PROPN
cana-788	186	46	4)𝑤3𝑧3	4)𝑤3𝑧3	PROPN
cana-788	186	47	.	.	PUNCT
cana-788	187	1	□	□	PUNCT
cana-788	187	2	figure	figure	NOUN
cana-788	187	3	5	5	NUM
cana-788	187	4	.	.	PUNCT
cana-788	187	5	surface	surface	NOUN
cana-788	187	6	plot	plot	NOUN
cana-788	187	7	of	of	ADP
cana-788	187	8	m	m	NOUN
cana-788	187	9	-	-	ADJ
cana-788	187	10	polynomial	polynomial	ADJ
cana-788	187	11	of	of	ADP
cana-788	187	12	γ𝑛.	γ𝑛.	PROPN
cana-788	187	13	theorem	theorem	ADJ
cana-788	187	14	2.6	2.6	NUM
cana-788	187	15	let	let	VERB
cana-788	187	16	γ𝑛	γ𝑛	NOUN
cana-788	187	17	be	be	AUX
cana-788	187	18	the	the	DET
cana-788	187	19	schreier	schreier	ADJ
cana-788	187	20	graph	graph	NOUN
cana-788	187	21	of	of	ADP
cana-788	187	22	grigorchuk	grigorchuk	NOUN
cana-788	187	23	group	group	NOUN
cana-788	187	24	.	.	PUNCT
cana-788	188	1	then	then	ADV
cana-788	188	2	the	the	DET
cana-788	188	3	m	m	NOUN
cana-788	188	4	-	-	PUNCT
cana-788	188	5	polynomials	polynomial	NOUN
cana-788	188	6	for	for	ADP
cana-788	188	7	the	the	DET
cana-788	188	8	topological	topological	ADJ
cana-788	188	9	indices	index	NOUN
cana-788	188	10	of	of	ADP
cana-788	188	11	γ𝑛	γ𝑛	PROPN
cana-788	188	12	,	,	PUNCT
cana-788	188	13	𝑛	𝑛	PROPN
cana-788	188	14	>	>	SYM
cana-788	188	15	1	1	NUM
cana-788	188	16	are	be	AUX
cana-788	188	17	a	a	PRON
cana-788	188	18	)	)	PUNCT
cana-788	189	1	𝑀1(γ𝑛)=	𝑀1(γ𝑛)=	NOUN
cana-788	189	2	6	6	NUM
cana-788	189	3	(	(	PUNCT
cana-788	189	4	2𝑛	2𝑛	PROPN
cana-788	189	5	+	+	CCONJ
cana-788	189	6	2𝑛−1	2𝑛−1	NUM
cana-788	189	7	)	)	PUNCT
cana-788	189	8	−	−	PROPN
cana-788	190	1	16	16	NUM
cana-788	190	2	.	.	PUNCT
cana-788	191	1	b	b	X
cana-788	191	2	)	)	PUNCT
cana-788	191	3	𝑀2(γ𝑛	𝑀2(γ𝑛	PROPN
cana-788	191	4	)	)	PUNCT
cana-788	191	5	=	=	SYM
cana-788	192	1	9(2𝑛	9(2𝑛	NUM
cana-788	192	2	+	+	NUM
cana-788	192	3	2𝑛−1	2𝑛−1	NUM
cana-788	192	4	)	)	PUNCT
cana-788	192	5	−	−	PROPN
cana-788	192	6	30	30	NUM
cana-788	192	7	.	.	PUNCT
cana-788	193	1	communications	communication	NOUN
cana-788	193	2	on	on	ADP
cana-788	193	3	applied	apply	VERB
cana-788	193	4	nonlinear	nonlinear	ADJ
cana-788	193	5	analysis	analysis	NOUN
cana-788	193	6	issn	issn	NOUN
cana-788	193	7	:	:	PUNCT
cana-788	193	8	1074	1074	NUM
cana-788	193	9	-	-	PUNCT
cana-788	193	10	133x	133x	NUM
cana-788	193	11	vol	vol	NOUN
cana-788	193	12	31	31	NUM
cana-788	193	13	no	no	NOUN
cana-788	193	14	.	.	PUNCT
cana-788	194	1	3s	3s	NUM
cana-788	194	2	(	(	PUNCT
cana-788	194	3	2024	2024	NUM
cana-788	194	4	)	)	PUNCT
cana-788	194	5	386	386	NUM
cana-788	194	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	194	7	c	c	X
cana-788	194	8	)	)	PUNCT
cana-788	194	9	𝑀2	𝑀2	PROPN
cana-788	194	10	𝑚(γ𝑛	𝑚(γ𝑛	NUM
cana-788	194	11	)	)	PUNCT
cana-788	194	12	=	=	PUNCT
cana-788	195	1	2𝑛+2𝑛−1	2𝑛+2𝑛−1	NUM
cana-788	195	2	+	+	NOUN
cana-788	195	3	2	2	NUM
cana-788	195	4	9	9	NUM
cana-788	195	5	.	.	PUNCT
cana-788	196	1	d	d	X
cana-788	196	2	)	)	PUNCT
cana-788	196	3	𝐴𝑍(γ𝑛	𝐴𝑍(γ𝑛	PROPN
cana-788	196	4	)	)	PUNCT
cana-788	196	5	=	=	NOUN
cana-788	197	1	729(2𝑛+2𝑛−1)−2484	729(2𝑛+2𝑛−1)−2484	NOUN
cana-788	197	2	64	64	NUM
cana-788	197	3	.	.	PUNCT
cana-788	198	1	e	e	X
cana-788	198	2	)	)	PUNCT
cana-788	198	3	𝐻𝑀(γ𝑛	𝐻𝑀(γ𝑛	PROPN
cana-788	198	4	)	)	PUNCT
cana-788	199	1	=	=	SYM
cana-788	200	1	36(2𝑛	36(2𝑛	NUM
cana-788	200	2	+	+	NUM
cana-788	200	3	2𝑛−1	2𝑛−1	NUM
cana-788	200	4	)	)	PUNCT
cana-788	200	5	−	−	PROPN
cana-788	201	1	112	112	NUM
cana-788	201	2	.	.	PUNCT
cana-788	202	1	f	f	X
cana-788	202	2	)	)	PUNCT
cana-788	202	3	𝐹(γ𝑛	𝐹(γ𝑛	PROPN
cana-788	202	4	)	)	PUNCT
cana-788	203	1	=	=	SYM
cana-788	203	2	18(2𝑛	18(2𝑛	NUM
cana-788	203	3	+	+	CCONJ
cana-788	203	4	2𝑛−1	2𝑛−1	NUM
cana-788	203	5	)	)	PUNCT
cana-788	203	6	−	−	PROPN
cana-788	204	1	52	52	NUM
cana-788	204	2	.	.	PUNCT
cana-788	205	1	g	g	NOUN
cana-788	205	2	)	)	PUNCT
cana-788	205	3	𝐼(γ𝑛	𝐼(γ𝑛	NOUN
cana-788	205	4	)	)	PUNCT
cana-788	205	5	=	=	SYM
cana-788	206	1	3(2𝑛+2𝑛−1)−9	3(2𝑛+2𝑛−1)−9	NUM
cana-788	206	2	2	2	NUM
cana-788	206	3	.	.	PUNCT
cana-788	207	1	proof	proof	NOUN
cana-788	207	2	.	.	PUNCT
cana-788	208	1	it	it	PRON
cana-788	208	2	can	can	AUX
cana-788	208	3	be	be	AUX
cana-788	208	4	verified	verify	VERB
cana-788	208	5	using	use	VERB
cana-788	208	6	theorem	theorem	NOUN
cana-788	208	7	2.2	2.2	NUM
cana-788	208	8	.	.	PUNCT
cana-788	209	1	□	□	PUNCT
cana-788	209	2	2.4	2.4	NUM
cana-788	209	3	the	the	DET
cana-788	209	4	m	m	NOUN
cana-788	209	5	-	-	ADJ
cana-788	209	6	polynomial	polynomial	ADJ
cana-788	209	7	of	of	ADP
cana-788	209	8	schreier	schreier	NOUN
cana-788	209	9	graphs	graph	NOUN
cana-788	209	10	(	(	PUNCT
cana-788	209	11	with	with	ADP
cana-788	209	12	loops	loop	NOUN
cana-788	209	13	)	)	PUNCT
cana-788	209	14	of	of	ADP
cana-788	209	15	the	the	DET
cana-788	209	16	grigorchuk	grigorchuk	NOUN
cana-788	209	17	group	group	NOUN
cana-788	209	18	let	let	VERB
cana-788	209	19	γ𝑛	γ𝑛	PROPN
cana-788	209	20	∗	∗	NOUN
cana-788	209	21	be	be	AUX
cana-788	209	22	the	the	DET
cana-788	209	23	schreier	schreier	NOUN
cana-788	209	24	graph	graph	NOUN
cana-788	209	25	(	(	PUNCT
cana-788	209	26	including	include	VERB
cana-788	209	27	loops	loop	NOUN
cana-788	209	28	)	)	PUNCT
cana-788	209	29	of	of	ADP
cana-788	209	30	the	the	DET
cana-788	209	31	grigorchuk	grigorchuk	NOUN
cana-788	209	32	group	group	NOUN
cana-788	209	33	.	.	PUNCT
cana-788	210	1	in	in	ADP
cana-788	210	2	finding	find	VERB
cana-788	210	3	the	the	DET
cana-788	210	4	mpolynomial	mpolynomial	NOUN
cana-788	210	5	,	,	PUNCT
cana-788	210	6	this	this	DET
cana-788	210	7	graph	graph	NOUN
cana-788	210	8	is	be	AUX
cana-788	210	9	considered	consider	VERB
cana-788	210	10	as	as	ADV
cana-788	210	11	unlabelled	unlabelled	ADJ
cana-788	210	12	and	and	CCONJ
cana-788	210	13	with	with	ADP
cana-788	210	14	loops	loop	NOUN
cana-788	210	15	.	.	PUNCT
cana-788	211	1	from	from	ADP
cana-788	211	2	figure	figure	NOUN
cana-788	211	3	4	4	NUM
cana-788	211	4	,	,	PUNCT
cana-788	211	5	it	it	PRON
cana-788	211	6	is	be	AUX
cana-788	211	7	clear	clear	ADJ
cana-788	211	8	that	that	SCONJ
cana-788	211	9	the	the	DET
cana-788	211	10	number	number	NOUN
cana-788	211	11	of	of	ADP
cana-788	211	12	vertices	vertex	NOUN
cana-788	211	13	|𝑉(γ𝑛	|𝑉(γ𝑛	PROPN
cana-788	211	14	∗	∗	NOUN
cana-788	211	15	)	)	PUNCT
cana-788	211	16	|	|	ADV
cana-788	211	17	=	=	SYM
cana-788	211	18	2𝑛	2𝑛	PROPN
cana-788	211	19	and	and	CCONJ
cana-788	211	20	|𝐸(γ𝑛	|𝐸(γ𝑛	PROPN
cana-788	211	21	∗	∗	NOUN
cana-788	211	22	)	)	PUNCT
cana-788	212	1	|	|	ADV
cana-788	212	2	=	=	SYM
cana-788	212	3	5	5	X
cana-788	212	4	.	.	NUM
cana-788	212	5	2𝑛−1	2𝑛−1	NUM
cana-788	212	6	+	+	CCONJ
cana-788	212	7	2	2	NUM
cana-788	212	8	when	when	SCONJ
cana-788	212	9	we	we	PRON
cana-788	212	10	consider	consider	VERB
cana-788	212	11	the	the	DET
cana-788	212	12	loops	loop	NOUN
cana-788	212	13	.	.	PUNCT
cana-788	213	1	theorem	theorem	NOUN
cana-788	213	2	2.7	2.7	NUM
cana-788	213	3	.	.	PUNCT
cana-788	214	1	let	let	VERB
cana-788	214	2	γ𝑛	γ𝑛	PROPN
cana-788	214	3	∗	∗	NOUN
cana-788	214	4	be	be	AUX
cana-788	214	5	schreier	schreier	ADJ
cana-788	214	6	graph	graph	NOUN
cana-788	214	7	(	(	PUNCT
cana-788	214	8	with	with	ADP
cana-788	214	9	loops	loop	NOUN
cana-788	214	10	)	)	PUNCT
cana-788	214	11	of	of	ADP
cana-788	214	12	grigorchuk	grigorchuk	PROPN
cana-788	214	13	group	group	NOUN
cana-788	214	14	.	.	PUNCT
cana-788	215	1	then	then	ADV
cana-788	215	2	the	the	DET
cana-788	215	3	m	m	ADJ
cana-788	215	4	–	–	PUNCT
cana-788	215	5	polynomial	polynomial	ADJ
cana-788	215	6	of	of	ADP
cana-788	215	7	γ𝑛	γ𝑛	PROPN
cana-788	215	8	∗	∗	NOUN
cana-788	215	9	is	be	AUX
cana-788	215	10	𝑓(γ𝑛	𝑓(γ𝑛	PROPN
cana-788	215	11	∗	∗	NOUN
cana-788	215	12	,	,	PUNCT
cana-788	215	13	𝑤	𝑤	ADP
cana-788	215	14	,	,	PUNCT
cana-788	215	15	𝑧	𝑧	NOUN
cana-788	215	16	)	)	PUNCT
cana-788	215	17	=	=	SYM
cana-788	215	18	𝑀𝑝𝑜𝑙𝑦(γ𝑛	𝑀𝑝𝑜𝑙𝑦(γ𝑛	PROPN
cana-788	215	19	∗	∗	NOUN
cana-788	215	20	,	,	PUNCT
cana-788	215	21	w	w	PROPN
cana-788	215	22	,	,	PUNCT
cana-788	215	23	z	z	NOUN
cana-788	215	24	)	)	PUNCT
cana-788	215	25	=	=	SYM
cana-788	216	1	2𝑤5𝑧7	2𝑤5𝑧7	NUM
cana-788	217	1	+	+	CCONJ
cana-788	217	2	6𝑤7𝑧7	6𝑤7𝑧7	NUM
cana-788	217	3	+	+	CCONJ
cana-788	217	4	(	(	PUNCT
cana-788	217	5	5	5	X
cana-788	217	6	.	.	NUM
cana-788	217	7	2𝑛−1	2𝑛−1	NUM
cana-788	217	8	−	−	PROPN
cana-788	217	9	6)𝑤5𝑧5	6)𝑤5𝑧5	PROPN
cana-788	217	10	.	.	PUNCT
cana-788	218	1	proof	proof	NOUN
cana-788	218	2	.	.	PUNCT
cana-788	219	1	from	from	ADP
cana-788	219	2	the	the	DET
cana-788	219	3	figure	figure	NOUN
cana-788	219	4	4	4	NUM
cana-788	219	5	,	,	PUNCT
cana-788	219	6	it	it	PRON
cana-788	219	7	is	be	AUX
cana-788	219	8	clear	clear	ADJ
cana-788	219	9	that	that	SCONJ
cana-788	219	10	there	there	PRON
cana-788	219	11	are	be	VERB
cana-788	219	12	three	three	NUM
cana-788	219	13	different	different	ADJ
cana-788	219	14	categories	category	NOUN
cana-788	219	15	of	of	ADP
cana-788	219	16	edge	edge	NOUN
cana-788	219	17	partitions	partition	NOUN
cana-788	219	18	of	of	ADP
cana-788	219	19	schreier	schreier	NOUN
cana-788	219	20	graph	graph	NOUN
cana-788	219	21	γ𝑛	γ𝑛	ADP
cana-788	219	22	∗	∗	NOUN
cana-788	219	23	of	of	ADP
cana-788	219	24	grigorchuk	grigorchuk	PROPN
cana-788	219	25	group	group	NOUN
cana-788	219	26	and	and	CCONJ
cana-788	219	27	they	they	PRON
cana-788	219	28	are	be	AUX
cana-788	219	29	|ȅ57|	|ȅ57|	PROPN
cana-788	219	30	=	=	SYM
cana-788	219	31	2	2	NUM
cana-788	219	32	,	,	PUNCT
cana-788	219	33	|ȅ77|	|ȅ77|	NOUN
cana-788	219	34	=	=	NOUN
cana-788	219	35	6	6	NUM
cana-788	219	36	and|ȅ55|	and|ȅ55|	ADJ
cana-788	219	37	=	=	SYM
cana-788	219	38	5	5	X
cana-788	219	39	.	.	X
cana-788	219	40	2𝑛−1	2𝑛−1	NUM
cana-788	220	1	−	−	NOUN
cana-788	220	2	6	6	NUM
cana-788	220	3	.	.	PUNCT
cana-788	221	1	now	now	ADV
cana-788	221	2	,	,	PUNCT
cana-788	221	3	by	by	ADP
cana-788	221	4	the	the	DET
cana-788	221	5	definition	definition	NOUN
cana-788	221	6	of	of	ADP
cana-788	221	7	2.1	2.1	NUM
cana-788	221	8	,	,	PUNCT
cana-788	221	9	we	we	PRON
cana-788	221	10	have	have	VERB
cana-788	221	11	𝑓(γ𝑛	𝑓(γ𝑛	PROPN
cana-788	221	12	∗	∗	NOUN
cana-788	221	13	,	,	PUNCT
cana-788	221	14	𝑤	𝑤	ADP
cana-788	221	15	,	,	PUNCT
cana-788	221	16	𝑧	𝑧	NOUN
cana-788	221	17	)	)	PUNCT
cana-788	221	18	=	=	SYM
cana-788	221	19	𝑀𝑝𝑜𝑙𝑦(γ𝑛	𝑀𝑝𝑜𝑙𝑦(γ𝑛	PROPN
cana-788	221	20	∗	∗	NOUN
cana-788	221	21	,	,	PUNCT
cana-788	221	22	w	w	PROPN
cana-788	221	23	,	,	PUNCT
cana-788	221	24	z	z	NOUN
cana-788	221	25	)	)	PUNCT
cana-788	221	26	=	=	PUNCT
cana-788	222	1	∑	∑	PROPN
cana-788	222	2	|ȅ𝑖𝑗(γ𝑛	|ȅ𝑖𝑗(γ𝑛	PROPN
cana-788	222	3	∗	∗	NOUN
cana-788	222	4	)	)	PUNCT
cana-788	222	5	|𝑤𝑖𝑧𝑗	|𝑤𝑖𝑧𝑗	PROPN
cana-788	222	6	𝕀≤𝑖≤𝑗≤𝕁	𝕀≤𝑖≤𝑗≤𝕁	PROPN
cana-788	223	1	=	=	PUNCT
cana-788	224	1	|ȅ57|𝑤5𝑧7	|ȅ57|𝑤5𝑧7	PROPN
cana-788	225	1	+	+	CCONJ
cana-788	225	2	|ȅ55|𝑤5𝑧5	|ȅ55|𝑤5𝑧5	PROPN
cana-788	225	3	+	+	CCONJ
cana-788	225	4	|ȅ77|𝑤7𝑧7	|ȅ77|𝑤7𝑧7	ADJ
cana-788	225	5	∴	∴	PROPN
cana-788	225	6	𝑓(γ𝑛	𝑓(γ𝑛	PROPN
cana-788	225	7	∗	∗	NOUN
cana-788	225	8	,	,	PUNCT
cana-788	225	9	𝑤	𝑤	ADP
cana-788	225	10	,	,	PUNCT
cana-788	225	11	𝑧	𝑧	NOUN
cana-788	225	12	)	)	PUNCT
cana-788	225	13	=	=	SYM
cana-788	225	14	2𝑤5𝑧7	2𝑤5𝑧7	NUM
cana-788	226	1	+	+	CCONJ
cana-788	226	2	6𝑤7𝑧7	6𝑤7𝑧7	NUM
cana-788	226	3	+	+	CCONJ
cana-788	226	4	(	(	PUNCT
cana-788	226	5	5	5	X
cana-788	226	6	.	.	NUM
cana-788	226	7	2𝑛−1	2𝑛−1	NUM
cana-788	226	8	−	−	PROPN
cana-788	226	9	6)𝑤5𝑧5	6)𝑤5𝑧5	PROPN
cana-788	226	10	.	.	PUNCT
cana-788	227	1	□	□	PUNCT
cana-788	227	2	figure	figure	NOUN
cana-788	227	3	6	6	NUM
cana-788	227	4	.	.	NOUN
cana-788	227	5	3	3	NUM
cana-788	227	6	-	-	SYM
cana-788	227	7	d	d	NOUN
cana-788	227	8	graph	graph	NOUN
cana-788	227	9	of	of	ADP
cana-788	227	10	m	m	NOUN
cana-788	227	11	-	-	ADJ
cana-788	227	12	polynomial	polynomial	ADJ
cana-788	227	13	of	of	ADP
cana-788	227	14	γ𝑛	γ𝑛	PROPN
cana-788	227	15	∗.	∗.	PROPN
cana-788	227	16	communications	communication	NOUN
cana-788	227	17	on	on	ADP
cana-788	227	18	applied	apply	VERB
cana-788	227	19	nonlinear	nonlinear	ADJ
cana-788	227	20	analysis	analysis	NOUN
cana-788	227	21	issn	issn	NOUN
cana-788	227	22	:	:	PUNCT
cana-788	227	23	1074	1074	NUM
cana-788	227	24	-	-	PUNCT
cana-788	227	25	133x	133x	NUM
cana-788	227	26	vol	vol	NOUN
cana-788	227	27	31	31	NUM
cana-788	227	28	no	no	NOUN
cana-788	227	29	.	.	PUNCT
cana-788	228	1	3s	3s	NUM
cana-788	228	2	(	(	PUNCT
cana-788	228	3	2024	2024	NUM
cana-788	228	4	)	)	PUNCT
cana-788	228	5	387	387	NUM
cana-788	228	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	228	7	theorem	theorem	VERB
cana-788	228	8	2.8	2.8	NUM
cana-788	228	9	let	let	VERB
cana-788	228	10	γ𝑛	γ𝑛	PROPN
cana-788	228	11	∗	∗	NOUN
cana-788	228	12	be	be	AUX
cana-788	228	13	the	the	DET
cana-788	228	14	schreier	schreier	NOUN
cana-788	228	15	graph	graph	NOUN
cana-788	228	16	(	(	PUNCT
cana-788	228	17	including	include	VERB
cana-788	228	18	loops	loop	NOUN
cana-788	228	19	)	)	PUNCT
cana-788	228	20	of	of	ADP
cana-788	228	21	grigorchuk	grigorchuk	PROPN
cana-788	228	22	group	group	NOUN
cana-788	228	23	.	.	PUNCT
cana-788	229	1	then	then	ADV
cana-788	229	2	the	the	DET
cana-788	229	3	topological	topological	ADJ
cana-788	229	4	indices	index	NOUN
cana-788	229	5	of	of	ADP
cana-788	229	6	γ𝑛	γ𝑛	PROPN
cana-788	229	7	∗	∗	NOUN
cana-788	229	8	,	,	PUNCT
cana-788	229	9	𝑛	𝑛	PROPN
cana-788	229	10	>	>	SYM
cana-788	229	11	1	1	NUM
cana-788	229	12	are	be	AUX
cana-788	229	13	a	a	DET
cana-788	229	14	)	)	PUNCT
cana-788	229	15	𝑀1(γ𝑛	𝑀1(γ𝑛	PROPN
cana-788	229	16	∗	∗	NOUN
cana-788	229	17	)	)	PUNCT
cana-788	230	1	=	=	SYM
cana-788	230	2	(	(	PUNCT
cana-788	230	3	25	25	NUM
cana-788	230	4	×	×	PROPN
cana-788	230	5	2𝑛	2𝑛	NUM
cana-788	230	6	)	)	PUNCT
cana-788	231	1	−	−	ADP
cana-788	232	1	62	62	NUM
cana-788	232	2	.	.	PUNCT
cana-788	233	1	b	b	X
cana-788	233	2	)	)	PUNCT
cana-788	233	3	𝑀2(γ𝑛	𝑀2(γ𝑛	PROPN
cana-788	233	4	∗	∗	NOUN
cana-788	233	5	)	)	PUNCT
cana-788	234	1	=	=	PUNCT
cana-788	234	2	(	(	PUNCT
cana-788	234	3	125	125	NUM
cana-788	234	4	×	×	PROPN
cana-788	234	5	2𝑛−1	2𝑛−1	NUM
cana-788	234	6	)	)	PUNCT
cana-788	234	7	+	+	CCONJ
cana-788	235	1	214	214	NUM
cana-788	235	2	.	.	PUNCT
cana-788	236	1	c	c	X
cana-788	236	2	)	)	PUNCT
cana-788	236	3	𝑀2	𝑀2	PROPN
cana-788	236	4	𝑚(γ𝑛	𝑚(γ𝑛	X
cana-788	236	5	∗	∗	NOUN
cana-788	236	6	)	)	PUNCT
cana-788	237	1	=	=	SYM
cana-788	237	2	2𝑛	2𝑛	PROPN
cana-788	237	3	10	10	NUM
cana-788	237	4	−	−	NOUN
cana-788	237	5	0.0605	0.0605	NUM
cana-788	237	6	.	.	PUNCT
cana-788	238	1	d	d	X
cana-788	238	2	)	)	PUNCT
cana-788	239	1	𝐴𝑍(γ𝑛	𝐴𝑍(γ𝑛	PROPN
cana-788	239	2	∗	∗	NOUN
cana-788	239	3	)	)	PUNCT
cana-788	240	1	=	=	PUNCT
cana-788	240	2	(	(	PUNCT
cana-788	240	3	152.5875	152.5875	NUM
cana-788	240	4	×	×	PROPN
cana-788	240	5	2𝑛−1	2𝑛−1	NUM
cana-788	240	6	)	)	PUNCT
cana-788	240	7	+	+	CCONJ
cana-788	240	8	311.148	311.148	NUM
cana-788	240	9	.	.	PUNCT
cana-788	241	1	e	e	X
cana-788	241	2	)	)	PUNCT
cana-788	241	3	𝐻𝑀(γ𝑛	𝐻𝑀(γ𝑛	PROPN
cana-788	241	4	∗	∗	NOUN
cana-788	241	5	)	)	PUNCT
cana-788	242	1	=	=	PUNCT
cana-788	242	2	(	(	PUNCT
cana-788	242	3	250	250	NUM
cana-788	242	4	×	×	PROPN
cana-788	242	5	2𝑛	2𝑛	NUM
cana-788	242	6	)	)	PUNCT
cana-788	243	1	+	+	CCONJ
cana-788	243	2	864	864	NUM
cana-788	243	3	.	.	PUNCT
cana-788	244	1	f	f	X
cana-788	244	2	)	)	PUNCT
cana-788	244	3	𝐹(γ𝑛	𝐹(γ𝑛	PROPN
cana-788	244	4	∗	∗	NOUN
cana-788	244	5	)	)	PUNCT
cana-788	245	1	=	=	PUNCT
cana-788	245	2	(	(	PUNCT
cana-788	245	3	125	125	NUM
cana-788	245	4	×	×	NOUN
cana-788	245	5	2𝑛	2𝑛	NUM
cana-788	245	6	)	)	PUNCT
cana-788	246	1	+	+	CCONJ
cana-788	247	1	436	436	NUM
cana-788	247	2	.	.	PUNCT
cana-788	248	1	g	g	NOUN
cana-788	248	2	)	)	PUNCT
cana-788	248	3	𝐼(γ𝑛	𝐼(γ𝑛	NOUN
cana-788	248	4	∗	∗	NOUN
cana-788	248	5	)	)	PUNCT
cana-788	249	1	=	=	PUNCT
cana-788	249	2	(	(	PUNCT
cana-788	249	3	12.5	12.5	NUM
cana-788	249	4	×	×	PROPN
cana-788	249	5	2𝑛−1	2𝑛−1	NUM
cana-788	249	6	)	)	PUNCT
cana-788	250	1	+	+	CCONJ
cana-788	250	2	864	864	NUM
cana-788	250	3	.	.	PUNCT
cana-788	251	1	proof	proof	NOUN
cana-788	251	2	.	.	PUNCT
cana-788	252	1	it	it	PRON
cana-788	252	2	can	can	AUX
cana-788	252	3	be	be	AUX
cana-788	252	4	verified	verify	VERB
cana-788	252	5	using	use	VERB
cana-788	252	6	theorem	theorem	NOUN
cana-788	252	7	2.2	2.2	NUM
cana-788	252	8	.	.	PUNCT
cana-788	253	1	□	□	SYM
cana-788	253	2	3	3	X
cana-788	253	3	.	.	X
cana-788	253	4	calculations	calculation	NOUN
cana-788	253	5	and	and	CCONJ
cana-788	253	6	comparative	comparative	ADJ
cana-788	253	7	analysis	analysis	NOUN
cana-788	253	8	in	in	ADP
cana-788	253	9	this	this	DET
cana-788	253	10	section	section	NOUN
cana-788	253	11	the	the	DET
cana-788	253	12	values	value	NOUN
cana-788	253	13	of	of	ADP
cana-788	253	14	the	the	DET
cana-788	253	15	selected	select	VERB
cana-788	253	16	topological	topological	ADJ
cana-788	253	17	indices	index	NOUN
cana-788	253	18	calculated	calculate	VERB
cana-788	253	19	by	by	ADP
cana-788	253	20	using	use	VERB
cana-788	253	21	the	the	DET
cana-788	253	22	results	result	NOUN
cana-788	253	23	in	in	ADP
cana-788	253	24	the	the	DET
cana-788	253	25	previous	previous	ADJ
cana-788	253	26	sections	section	NOUN
cana-788	253	27	and	and	CCONJ
cana-788	253	28	comparative	comparative	ADJ
cana-788	253	29	study	study	NOUN
cana-788	253	30	done	do	VERB
cana-788	253	31	based	base	VERB
cana-788	253	32	on	on	ADP
cana-788	253	33	the	the	DET
cana-788	253	34	determined	determined	ADJ
cana-788	253	35	results	result	NOUN
cana-788	253	36	.	.	PUNCT
cana-788	254	1	table	table	NOUN
cana-788	254	2	4	4	NUM
cana-788	254	3	presents	present	NOUN
cana-788	254	4	calculated	calculate	VERB
cana-788	254	5	values	value	NOUN
cana-788	254	6	of	of	ADP
cana-788	254	7	various	various	ADJ
cana-788	254	8	topological	topological	ADJ
cana-788	254	9	indices	index	NOUN
cana-788	254	10	for	for	ADP
cana-788	254	11	𝐵𝑛.	𝐵𝑛.	PROPN
cana-788	254	12	these	these	DET
cana-788	254	13	indices	index	NOUN
cana-788	254	14	provide	provide	VERB
cana-788	254	15	various	various	ADJ
cana-788	254	16	measures	measure	NOUN
cana-788	254	17	of	of	ADP
cana-788	254	18	the	the	DET
cana-788	254	19	structure	structure	NOUN
cana-788	254	20	and	and	CCONJ
cana-788	254	21	properties	property	NOUN
cana-788	254	22	of	of	ADP
cana-788	254	23	𝐵𝑛	𝐵𝑛	PROPN
cana-788	254	24	at	at	ADP
cana-788	254	25	different	different	ADJ
cana-788	254	26	levels	level	NOUN
cana-788	254	27	.	.	PUNCT
cana-788	255	1	for	for	ADP
cana-788	255	2	example	example	NOUN
cana-788	255	3	,	,	PUNCT
cana-788	255	4	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	PROPN
cana-788	255	5	)	)	PUNCT
cana-788	255	6	and	and	CCONJ
cana-788	255	7	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	NOUN
cana-788	255	8	)	)	PUNCT
cana-788	255	9	provide	provide	VERB
cana-788	255	10	basic	basic	ADJ
cana-788	255	11	information	information	NOUN
cana-788	255	12	about	about	ADP
cana-788	255	13	the	the	DET
cana-788	255	14	size	size	NOUN
cana-788	255	15	of	of	ADP
cana-788	255	16	the	the	DET
cana-788	255	17	graph	graph	NOUN
cana-788	255	18	and	and	CCONJ
cana-788	255	19	its	its	PRON
cana-788	255	20	connectivity	connectivity	NOUN
cana-788	255	21	,	,	PUNCT
cana-788	255	22	while	while	SCONJ
cana-788	255	23	indices	index	NOUN
cana-788	255	24	like	like	ADP
cana-788	255	25	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	PROPN
cana-788	255	26	)	)	PUNCT
cana-788	255	27	,	,	PUNCT
cana-788	255	28	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	PROPN
cana-788	255	29	)	)	PUNCT
cana-788	255	30	,	,	PUNCT
cana-788	255	31	and	and	CCONJ
cana-788	255	32	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NOUN
cana-788	255	33	)	)	PUNCT
cana-788	255	34	give	give	VERB
cana-788	255	35	understandings	understanding	NOUN
cana-788	255	36	into	into	ADP
cana-788	255	37	more	more	ADJ
cana-788	255	38	specific	specific	ADJ
cana-788	255	39	properties	property	NOUN
cana-788	255	40	related	relate	VERB
cana-788	255	41	to	to	ADP
cana-788	255	42	vertex	vertex	NOUN
cana-788	255	43	independence	independence	NOUN
cana-788	255	44	,	,	PUNCT
cana-788	255	45	distances	distance	NOUN
cana-788	255	46	,	,	PUNCT
cana-788	255	47	and	and	CCONJ
cana-788	255	48	autonomy	autonomy	NOUN
cana-788	255	49	.	.	PUNCT
cana-788	255	50	table	table	NOUN
cana-788	255	51	4	4	NUM
cana-788	255	52	.	.	PUNCT
cana-788	255	53	calculated	calculate	VERB
cana-788	255	54	values	value	NOUN
cana-788	255	55	of	of	ADP
cana-788	255	56	topological	topological	ADJ
cana-788	255	57	indices	index	NOUN
cana-788	255	58	of	of	ADP
cana-788	255	59	𝐵𝑛.	𝐵𝑛.	PROPN
cana-788	255	60	when	when	SCONJ
cana-788	255	61	we	we	PRON
cana-788	255	62	display	display	VERB
cana-788	255	63	these	these	DET
cana-788	255	64	numbers	number	NOUN
cana-788	255	65	on	on	ADP
cana-788	255	66	a	a	DET
cana-788	255	67	line	line	NOUN
cana-788	255	68	graph	graph	NOUN
cana-788	255	69	(	(	PUNCT
cana-788	255	70	see	see	VERB
cana-788	255	71	figure	figure	NOUN
cana-788	255	72	7	7	NUM
cana-788	255	73	)	)	PUNCT
cana-788	255	74	,	,	PUNCT
cana-788	255	75	we	we	PRON
cana-788	255	76	see	see	VERB
cana-788	255	77	that	that	SCONJ
cana-788	255	78	when	when	SCONJ
cana-788	255	79	the	the	DET
cana-788	255	80	level	level	NOUN
cana-788	255	81	of	of	ADP
cana-788	255	82	the	the	DET
cana-788	255	83	basilica	basilica	PROPN
cana-788	255	84	group	group	NOUN
cana-788	255	85	rises	rise	VERB
cana-788	255	86	,	,	PUNCT
cana-788	255	87	all	all	PRON
cana-788	255	88	of	of	ADP
cana-788	255	89	the	the	DET
cana-788	255	90	indices	index	NOUN
cana-788	255	91	exhibit	exhibit	VERB
cana-788	255	92	consistent	consistent	ADJ
cana-788	255	93	patterns	pattern	NOUN
cana-788	255	94	of	of	ADP
cana-788	255	95	exponential	exponential	ADJ
cana-788	255	96	growth	growth	NOUN
cana-788	255	97	.	.	PUNCT
cana-788	256	1	the	the	DET
cana-788	256	2	graph	graph	NOUN
cana-788	256	3	's	's	PART
cana-788	256	4	rising	rise	VERB
cana-788	256	5	complexity	complexity	NOUN
cana-788	256	6	is	be	AUX
cana-788	256	7	reflected	reflect	VERB
cana-788	256	8	in	in	ADP
cana-788	256	9	the	the	DET
cana-788	256	10	indices	index	NOUN
cana-788	256	11	𝑀1(𝐵𝑛),𝑀2(𝐵𝑛	𝑀1(𝐵𝑛),𝑀2(𝐵𝑛	NUM
cana-788	256	12	)	)	PUNCT
cana-788	256	13	and	and	CCONJ
cana-788	256	14	𝐹(𝐵𝑛	𝐹(𝐵𝑛	NUM
cana-788	256	15	)	)	PUNCT
cana-788	256	16	,	,	PUNCT
cana-788	256	17	which	which	PRON
cana-788	256	18	grow	grow	VERB
cana-788	256	19	quickly	quickly	ADV
cana-788	256	20	as	as	SCONJ
cana-788	256	21	the	the	DET
cana-788	256	22	number	number	NOUN
cana-788	256	23	of	of	ADP
cana-788	256	24	vertices	vertex	NOUN
cana-788	256	25	and	and	CCONJ
cana-788	256	26	edges	edge	NOUN
cana-788	256	27	increases	increase	NOUN
cana-788	256	28	exponentially	exponentially	ADV
cana-788	256	29	.	.	PUNCT
cana-788	257	1	similarly	similarly	ADV
cana-788	257	2	,	,	PUNCT
cana-788	257	3	indices	index	NOUN
cana-788	257	4	like	like	ADP
cana-788	257	5	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	PROPN
cana-788	257	6	)	)	PUNCT
cana-788	257	7	and	and	CCONJ
cana-788	257	8	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	NUM
cana-788	257	9	)	)	PUNCT
cana-788	257	10	also	also	ADV
cana-788	257	11	display	display	VERB
cana-788	257	12	significant	significant	ADJ
cana-788	257	13	growth	growth	NOUN
cana-788	257	14	,	,	PUNCT
cana-788	257	15	though	though	SCONJ
cana-788	257	16	at	at	ADP
cana-788	257	17	a	a	DET
cana-788	257	18	slightly	slightly	ADV
cana-788	257	19	slower	slow	ADJ
cana-788	257	20	pace	pace	NOUN
cana-788	257	21	,	,	PUNCT
cana-788	257	22	indicating	indicate	VERB
cana-788	257	23	the	the	DET
cana-788	257	24	increasing	increase	VERB
cana-788	257	25	autonomy	autonomy	NOUN
cana-788	257	26	and	and	CCONJ
cana-788	257	27	harmonic	harmonic	ADJ
cana-788	257	28	complexity	complexity	NOUN
cana-788	257	29	of	of	ADP
cana-788	257	30	the	the	DET
cana-788	257	31	graph	graph	NOUN
cana-788	257	32	structure	structure	NOUN
cana-788	257	33	.	.	PUNCT
cana-788	258	1	additionally	additionally	ADV
cana-788	258	2	,	,	PUNCT
cana-788	258	3	the	the	DET
cana-788	258	4	indices	index	NOUN
cana-788	258	5	𝑀2	𝑀2	PROPN
cana-788	258	6	𝑚(𝐵𝑛	𝑚(𝐵𝑛	NUM
cana-788	258	7	)	)	PUNCT
cana-788	258	8	and	and	CCONJ
cana-788	258	9	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NOUN
cana-788	258	10	)	)	PUNCT
cana-788	258	11	follow	follow	VERB
cana-788	258	12	the	the	DET
cana-788	258	13	general	general	ADJ
cana-788	258	14	trend	trend	NOUN
cana-788	258	15	of	of	ADP
cana-788	258	16	exponential	exponential	ADJ
cana-788	258	17	growth	growth	NOUN
cana-788	258	18	but	but	CCONJ
cana-788	258	19	with	with	ADP
cana-788	258	20	smaller	small	ADJ
cana-788	258	21	numerical	numerical	ADJ
cana-788	258	22	values	value	NOUN
cana-788	258	23	.	.	PUNCT
cana-788	259	1	overall	overall	ADV
cana-788	259	2	,	,	PUNCT
cana-788	259	3	the	the	DET
cana-788	259	4	line	line	NOUN
cana-788	259	5	chart	chart	NOUN
cana-788	259	6	illustrates	illustrate	VERB
cana-788	259	7	the	the	DET
cana-788	259	8	dynamic	dynamic	ADJ
cana-788	259	9	growth	growth	NOUN
cana-788	259	10	and	and	CCONJ
cana-788	259	11	complexity	complexity	NOUN
cana-788	259	12	of	of	ADP
cana-788	259	13	the	the	DET
cana-788	259	14	basilica	basilica	PROPN
cana-788	259	15	group	group	NOUN
cana-788	259	16	's	's	PART
cana-788	259	17	schreier	schreier	NOUN
cana-788	259	18	graph	graph	NOUN
cana-788	259	19	across	across	ADP
cana-788	259	20	different	different	ADJ
cana-788	259	21	topological	topological	ADJ
cana-788	259	22	index	index	NOUN
cana-788	259	23	basilica	basilica	PROPN
cana-788	259	24	group	group	PROPN
cana-788	259	25	bn	bn	PROPN
cana-788	259	26	𝑩𝟐	𝑩𝟐	PROPN
cana-788	259	27	𝑩𝟑	𝑩𝟑	PROPN
cana-788	259	28	𝑩𝟒	𝑩𝟒	PROPN
cana-788	259	29	𝑩𝟓	𝑩𝟓	PROPN
cana-788	259	30	𝑩𝟔	𝑩𝟔	PROPN
cana-788	259	31	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	PROPN
cana-788	259	32	)	)	PUNCT
cana-788	259	33	40	40	NUM
cana-788	259	34	80	80	NUM
cana-788	259	35	160	160	NUM
cana-788	259	36	320	320	NUM
cana-788	259	37	640	640	NUM
cana-788	259	38	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	NOUN
cana-788	259	39	)	)	PUNCT
cana-788	259	40	64	64	NUM
cana-788	259	41	128	128	NUM
cana-788	259	42	256	256	NUM
cana-788	259	43	512	512	NUM
cana-788	259	44	1024	1024	NUM
cana-788	259	45	𝑀2	𝑀2	PROPN
cana-788	259	46	𝑚(𝐵𝑛	𝑚(𝐵𝑛	NUM
cana-788	259	47	)	)	PUNCT
cana-788	259	48	0.62	0.62	NUM
cana-788	259	49	1.25	1.25	NUM
cana-788	259	50	2.5	2.5	NUM
cana-788	259	51	5	5	NUM
cana-788	259	52	10	10	NUM
cana-788	259	53	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	NUM
cana-788	259	54	)	)	PUNCT
cana-788	259	55	24	24	NUM
cana-788	259	56	48	48	NUM
cana-788	259	57	96	96	NUM
cana-788	259	58	192	192	NUM
cana-788	259	59	384	384	NUM
cana-788	259	60	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	NUM
cana-788	259	61	)	)	PUNCT
cana-788	259	62	272	272	NUM
cana-788	259	63	544	544	NUM
cana-788	259	64	1088	1088	NUM
cana-788	259	65	2176	2176	NUM
cana-788	259	66	4352	4352	NUM
cana-788	259	67	𝐹(𝐵𝑛	𝐹(𝐵𝑛	NUM
cana-788	259	68	)	)	PUNCT
cana-788	259	69	96	96	NUM
cana-788	259	70	192	192	NUM
cana-788	259	71	384	384	NUM
cana-788	259	72	768	768	NUM
cana-788	259	73	1536	1536	NUM
cana-788	259	74	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NOUN
cana-788	259	75	)	)	PUNCT
cana-788	259	76	9.33	9.33	NUM
cana-788	259	77	18.67	18.67	NUM
cana-788	259	78	37.33	37.33	NUM
cana-788	259	79	74.67	74.67	NUM
cana-788	259	80	149.33	149.33	NUM
cana-788	259	81	communications	communication	NOUN
cana-788	259	82	on	on	ADP
cana-788	259	83	applied	apply	VERB
cana-788	259	84	nonlinear	nonlinear	ADJ
cana-788	259	85	analysis	analysis	NOUN
cana-788	259	86	issn	issn	NOUN
cana-788	259	87	:	:	PUNCT
cana-788	259	88	1074	1074	NUM
cana-788	259	89	-	-	PUNCT
cana-788	259	90	133x	133x	NUM
cana-788	259	91	vol	vol	NOUN
cana-788	259	92	31	31	NUM
cana-788	259	93	no	no	NOUN
cana-788	259	94	.	.	PUNCT
cana-788	260	1	3s	3s	NUM
cana-788	260	2	(	(	PUNCT
cana-788	260	3	2024	2024	NUM
cana-788	260	4	)	)	PUNCT
cana-788	260	5	388	388	NUM
cana-788	260	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	260	7	topological	topological	ADJ
cana-788	260	8	indices	index	NOUN
cana-788	260	9	,	,	PUNCT
cana-788	260	10	highlighting	highlight	VERB
cana-788	260	11	the	the	DET
cana-788	260	12	convoluted	convoluted	ADJ
cana-788	260	13	interplay	interplay	NOUN
cana-788	260	14	between	between	ADP
cana-788	260	15	vertices	vertex	NOUN
cana-788	260	16	,	,	PUNCT
cana-788	260	17	edges	edge	NOUN
cana-788	260	18	,	,	PUNCT
cana-788	260	19	distances	distance	NOUN
cana-788	260	20	,	,	PUNCT
cana-788	260	21	and	and	CCONJ
cana-788	260	22	autonomy	autonomy	NOUN
cana-788	260	23	within	within	ADP
cana-788	260	24	the	the	DET
cana-788	260	25	graph	graph	NOUN
cana-788	260	26	structure	structure	NOUN
cana-788	260	27	as	as	ADP
cana-788	260	28	the	the	DET
cana-788	260	29	group	group	NOUN
cana-788	260	30	level	level	NOUN
cana-788	260	31	increases	increase	NOUN
cana-788	260	32	.	.	PUNCT
cana-788	261	1	fig	fig	NOUN
cana-788	261	2	.	.	PUNCT
cana-788	262	1	7	7	NUM
cana-788	262	2	line	line	NOUN
cana-788	262	3	chart	chart	NOUN
cana-788	262	4	for	for	ADP
cana-788	262	5	topological	topological	ADJ
cana-788	262	6	index	index	NOUN
cana-788	262	7	value	value	NOUN
cana-788	262	8	of	of	ADP
cana-788	262	9	bn	bn	PROPN
cana-788	262	10	.	.	PUNCT
cana-788	262	11	table	table	NOUN
cana-788	262	12	5	5	NUM
cana-788	262	13	.	.	PUNCT
cana-788	262	14	calculated	calculate	VERB
cana-788	262	15	values	value	NOUN
cana-788	262	16	of	of	ADP
cana-788	262	17	topological	topological	ADJ
cana-788	262	18	indices	index	NOUN
cana-788	262	19	of	of	ADP
cana-788	262	20	bn	bn	NOUN
cana-788	262	21	*	*	ADJ
cana-788	262	22	topological	topological	PROPN
cana-788	262	23	index	index	NOUN
cana-788	262	24	basilica	basilica	PROPN
cana-788	262	25	group	group	PROPN
cana-788	262	26	bn	bn	PROPN
cana-788	262	27	*	*	PROPN
cana-788	262	28	b2	b2	PROPN
cana-788	262	29	*	*	PUNCT
cana-788	262	30	b3	b3	PROPN
cana-788	262	31	*	*	PUNCT
cana-788	262	32	b4	b4	PROPN
cana-788	262	33	*	*	SYM
cana-788	262	34	b5	b5	PROPN
cana-788	262	35	*	*	PUNCT
cana-788	262	36	b6	b6	PROPN
cana-788	262	37	*	*	PUNCT
cana-788	262	38	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	NOUN
cana-788	262	39	∗	∗	NOUN
cana-788	262	40	)	)	PUNCT
cana-788	262	41	64	64	NUM
cana-788	262	42	128	128	NUM
cana-788	262	43	256	256	NUM
cana-788	262	44	512	512	NUM
cana-788	262	45	1024	1024	NUM
cana-788	262	46	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	PROPN
cana-788	262	47	∗	∗	PROPN
cana-788	262	48	)	)	PUNCT
cana-788	262	49	128	128	NUM
cana-788	262	50	256	256	NUM
cana-788	262	51	512	512	NUM
cana-788	262	52	1024	1024	NUM
cana-788	262	53	2048	2048	NUM
cana-788	262	54	𝑀2	𝑀2	PROPN
cana-788	262	55	𝑚(𝐵𝑛	𝑚(𝐵𝑛	X
cana-788	262	56	∗	∗	NOUN
cana-788	262	57	)	)	PUNCT
cana-788	262	58	0.5	0.5	NUM
cana-788	262	59	1	1	NUM
cana-788	262	60	2	2	NUM
cana-788	262	61	4	4	NUM
cana-788	262	62	8	8	NUM
cana-788	262	63	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	NUM
cana-788	262	64	∗	∗	NOUN
cana-788	262	65	)	)	PUNCT
cana-788	262	66	64	64	NUM
cana-788	262	67	128	128	NUM
cana-788	262	68	256	256	NUM
cana-788	262	69	512	512	NUM
cana-788	262	70	1024	1024	NUM
cana-788	262	71	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	PROPN
cana-788	262	72	∗	∗	NOUN
cana-788	262	73	)	)	PUNCT
cana-788	262	74	512	512	NUM
cana-788	262	75	1024	1024	NUM
cana-788	262	76	2048	2048	NUM
cana-788	262	77	4096	4096	NUM
cana-788	262	78	8192	8192	NUM
cana-788	262	79	𝐹(𝐵𝑛	𝐹(𝐵𝑛	PROPN
cana-788	262	80	∗	∗	NOUN
cana-788	262	81	)	)	PUNCT
cana-788	262	82	256	256	NUM
cana-788	262	83	512	512	NUM
cana-788	262	84	1024	1024	NUM
cana-788	262	85	2048	2048	NUM
cana-788	262	86	4096	4096	NUM
cana-788	262	87	𝐼(𝐵𝑛	𝐼(𝐵𝑛	NUM
cana-788	262	88	∗	∗	NOUN
cana-788	262	89	)	)	PUNCT
cana-788	262	90	16	16	NUM
cana-788	262	91	32	32	NUM
cana-788	262	92	64	64	NUM
cana-788	262	93	128	128	NUM
cana-788	262	94	256	256	NUM
cana-788	262	95	table	table	NOUN
cana-788	262	96	5	5	NUM
cana-788	262	97	presents	present	VERB
cana-788	262	98	the	the	DET
cana-788	262	99	computed	computed	ADJ
cana-788	262	100	values	value	NOUN
cana-788	262	101	of	of	ADP
cana-788	262	102	various	various	ADJ
cana-788	262	103	topological	topological	ADJ
cana-788	262	104	indices	index	NOUN
cana-788	262	105	for	for	ADP
cana-788	262	106	the	the	DET
cana-788	262	107	schreier	schreier	NOUN
cana-788	262	108	graph	graph	NOUN
cana-788	262	109	with	with	ADP
cana-788	262	110	loops	loop	NOUN
cana-788	262	111	in	in	ADP
cana-788	262	112	the	the	DET
cana-788	262	113	basilica	basilica	NOUN
cana-788	262	114	group	group	NOUN
cana-788	262	115	at	at	ADP
cana-788	262	116	different	different	ADJ
cana-788	262	117	levels	level	NOUN
cana-788	262	118	labelled	label	VERB
cana-788	262	119	as	as	ADP
cana-788	262	120	𝐵1	𝐵1	NOUN
cana-788	262	121	∗	∗	NOUN
cana-788	262	122	to	to	ADP
cana-788	262	123	𝐵6	𝐵6	PROPN
cana-788	262	124	∗.	∗.	PROPN
cana-788	262	125	a	a	DET
cana-788	262	126	thorough	thorough	ADJ
cana-788	262	127	understanding	understanding	NOUN
cana-788	262	128	of	of	ADP
cana-788	262	129	the	the	DET
cana-788	262	130	structural	structural	ADJ
cana-788	262	131	complexity	complexity	NOUN
cana-788	262	132	of	of	ADP
cana-788	262	133	the	the	DET
cana-788	262	134	graph	graph	NOUN
cana-788	262	135	at	at	ADP
cana-788	262	136	various	various	ADJ
cana-788	262	137	group	group	NOUN
cana-788	262	138	levels	level	NOUN
cana-788	262	139	is	be	AUX
cana-788	262	140	offered	offer	VERB
cana-788	262	141	by	by	ADP
cana-788	262	142	these	these	DET
cana-788	262	143	indices	index	NOUN
cana-788	262	144	.	.	PUNCT
cana-788	263	1	analysis	analysis	NOUN
cana-788	263	2	reveals	reveal	VERB
cana-788	263	3	that	that	SCONJ
cana-788	263	4	most	most	ADJ
cana-788	263	5	indicators	indicator	NOUN
cana-788	263	6	show	show	VERB
cana-788	263	7	a	a	DET
cana-788	263	8	consistent	consistent	ADJ
cana-788	263	9	exponential	exponential	ADJ
cana-788	263	10	development	development	NOUN
cana-788	263	11	trend	trend	NOUN
cana-788	263	12	as	as	ADP
cana-788	263	13	the	the	DET
cana-788	263	14	group	group	NOUN
cana-788	263	15	level	level	NOUN
cana-788	263	16	increases	increase	NOUN
cana-788	263	17	.	.	PUNCT
cana-788	264	1	this	this	PRON
cana-788	264	2	can	can	AUX
cana-788	264	3	be	be	AUX
cana-788	264	4	seen	see	VERB
cana-788	264	5	in	in	ADP
cana-788	264	6	figure	figure	NOUN
cana-788	264	7	8	8	NUM
cana-788	264	8	.	.	PUNCT
cana-788	265	1	interestingly	interestingly	ADV
cana-788	265	2	,	,	PUNCT
cana-788	265	3	indices	indice	VERB
cana-788	265	4	such	such	ADJ
cana-788	265	5	as	as	ADP
cana-788	265	6	𝑀1(𝐵𝑛	𝑀1(𝐵𝑛	PROPN
cana-788	265	7	∗	∗	NOUN
cana-788	265	8	)	)	PUNCT
cana-788	265	9	,	,	PUNCT
cana-788	265	10	𝑀2(𝐵𝑛	𝑀2(𝐵𝑛	PROPN
cana-788	265	11	∗	∗	PROPN
cana-788	265	12	)	)	PUNCT
cana-788	265	13	,	,	PUNCT
cana-788	265	14	𝐴𝑍(𝐵𝑛	𝐴𝑍(𝐵𝑛	PROPN
cana-788	265	15	∗	∗	NOUN
cana-788	265	16	)	)	PUNCT
cana-788	265	17	,	,	PUNCT
cana-788	265	18	and	and	CCONJ
cana-788	265	19	𝐹(𝐵𝑛	𝐹(𝐵𝑛	PROPN
cana-788	265	20	∗	∗	NOUN
cana-788	265	21	)	)	PUNCT
cana-788	265	22	show	show	VERB
cana-788	265	23	strong	strong	ADJ
cana-788	265	24	increases	increase	NOUN
cana-788	265	25	,	,	PUNCT
cana-788	265	26	reflecting	reflect	VERB
cana-788	265	27	the	the	DET
cana-788	265	28	graph	graph	NOUN
cana-788	265	29	's	's	PART
cana-788	265	30	growing	grow	VERB
cana-788	265	31	vertex	vertex	NOUN
cana-788	265	32	and	and	CCONJ
cana-788	265	33	edge	edge	NOUN
cana-788	265	34	network	network	NOUN
cana-788	265	35	.	.	PUNCT
cana-788	266	1	in	in	ADP
cana-788	266	2	addition	addition	NOUN
cana-788	266	3	,	,	PUNCT
cana-788	266	4	the	the	DET
cana-788	266	5	hyper	hyper	PROPN
cana-788	266	6	zagreb	zagreb	PROPN
cana-788	266	7	index	index	PROPN
cana-788	266	8	𝐻𝑀(𝐵𝑛	𝐻𝑀(𝐵𝑛	PROPN
cana-788	266	9	∗	∗	NOUN
cana-788	266	10	)	)	PUNCT
cana-788	266	11	shows	show	VERB
cana-788	266	12	significant	significant	ADJ
cana-788	266	13	increase	increase	NOUN
cana-788	266	14	,	,	PUNCT
cana-788	266	15	indicating	indicate	VERB
cana-788	266	16	increased	increase	VERB
cana-788	266	17	harmonic	harmonic	ADJ
cana-788	266	18	complexity	complexity	NOUN
cana-788	266	19	in	in	ADP
cana-788	266	20	the	the	DET
cana-788	266	21	graph	graph	NOUN
cana-788	266	22	structure	structure	NOUN
cana-788	266	23	.	.	PUNCT
cana-788	267	1	interestingly	interestingly	ADV
cana-788	267	2	,	,	PUNCT
cana-788	267	3	however	however	ADV
cana-788	267	4	,	,	PUNCT
cana-788	267	5	the	the	DET
cana-788	267	6	indices	index	NOUN
cana-788	267	7	𝐼(𝐵𝑛	𝐼(𝐵𝑛	ADJ
cana-788	267	8	∗	∗	NOUN
cana-788	267	9	)	)	PUNCT
cana-788	267	10	and	and	CCONJ
cana-788	267	11	𝑀2	𝑀2	PROPN
cana-788	267	12	𝑚(𝐵𝑛	𝑚(𝐵𝑛	X
cana-788	267	13	∗	∗	NOUN
cana-788	267	14	)	)	PUNCT
cana-788	267	15	exhibit	exhibit	VERB
cana-788	267	16	exponential	exponential	ADJ
cana-788	267	17	development	development	NOUN
cana-788	267	18	patterns	pattern	NOUN
cana-788	267	19	with	with	ADP
cana-788	267	20	much	much	ADV
cana-788	267	21	smaller	small	ADJ
cana-788	267	22	numerical	numerical	ADJ
cana-788	267	23	values	value	NOUN
cana-788	267	24	.	.	PUNCT
cana-788	268	1	200	200	NUM
cana-788	268	2	1200	1200	NUM
cana-788	268	3	2200	2200	NUM
cana-788	268	4	3200	3200	NUM
cana-788	268	5	4200	4200	NUM
cana-788	268	6	5200	5200	NUM
cana-788	268	7	b1	b1	NOUN
cana-788	268	8	b2	b2	NOUN
cana-788	268	9	b3	b3	PROPN
cana-788	268	10	b4	b4	PROPN
cana-788	268	11	b5	b5	PROPN
cana-788	268	12	b6	b6	PROPN
cana-788	268	13	basilica	basilica	PROPN
cana-788	268	14	group	group	PROPN
cana-788	268	15	bn	bn	PROPN
cana-788	268	16	topological	topological	ADJ
cana-788	268	17	indices	index	NOUN
cana-788	268	18	of	of	ADP
cana-788	268	19	bn	bn	NOUN
cana-788	268	20	communications	communication	NOUN
cana-788	268	21	on	on	ADP
cana-788	268	22	applied	apply	VERB
cana-788	268	23	nonlinear	nonlinear	ADJ
cana-788	268	24	analysis	analysis	NOUN
cana-788	268	25	issn	issn	NOUN
cana-788	268	26	:	:	PUNCT
cana-788	268	27	1074	1074	NUM
cana-788	268	28	-	-	PUNCT
cana-788	268	29	133x	133x	NUM
cana-788	268	30	vol	vol	NOUN
cana-788	268	31	31	31	NUM
cana-788	268	32	no	no	NOUN
cana-788	268	33	.	.	PUNCT
cana-788	269	1	3s	3s	NUM
cana-788	269	2	(	(	PUNCT
cana-788	269	3	2024	2024	NUM
cana-788	269	4	)	)	PUNCT
cana-788	269	5	389	389	NUM
cana-788	269	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	269	7	fig	fig	NOUN
cana-788	269	8	.	.	PUNCT
cana-788	270	1	8	8	NUM
cana-788	270	2	line	line	NOUN
cana-788	270	3	chart	chart	NOUN
cana-788	270	4	for	for	ADP
cana-788	270	5	calculated	calculate	VERB
cana-788	270	6	values	value	NOUN
cana-788	270	7	of	of	ADP
cana-788	270	8	topological	topological	ADJ
cana-788	270	9	indices	index	NOUN
cana-788	270	10	of	of	ADP
cana-788	270	11	𝐵𝑛	𝐵𝑛	PROPN
cana-788	270	12	∗	∗	VERB
cana-788	270	13	the	the	DET
cana-788	270	14	following	follow	VERB
cana-788	270	15	table	table	NOUN
cana-788	270	16	6	6	NUM
cana-788	270	17	provides	provide	VERB
cana-788	270	18	calculated	calculated	ADJ
cana-788	270	19	values	value	NOUN
cana-788	270	20	of	of	ADP
cana-788	270	21	various	various	ADJ
cana-788	270	22	topological	topological	ADJ
cana-788	270	23	indices	index	NOUN
cana-788	270	24	for	for	ADP
cana-788	270	25	schreier	schreier	ADJ
cana-788	270	26	graph	graph	NOUN
cana-788	270	27	of	of	ADP
cana-788	270	28	the	the	DET
cana-788	270	29	grigorchuk	grigorchuk	NOUN
cana-788	270	30	group	group	NOUN
cana-788	270	31	at	at	ADP
cana-788	270	32	different	different	ADJ
cana-788	270	33	levels	level	NOUN
cana-788	270	34	.	.	PUNCT
cana-788	271	1	table	table	NOUN
cana-788	271	2	6	6	NUM
cana-788	271	3	.	.	PUNCT
cana-788	271	4	calculated	calculate	VERB
cana-788	271	5	values	value	NOUN
cana-788	271	6	of	of	ADP
cana-788	271	7	topological	topological	ADJ
cana-788	271	8	indices	index	NOUN
cana-788	271	9	of	of	ADP
cana-788	271	10	γ𝑛	γ𝑛	ADP
cana-788	271	11	topological	topological	ADJ
cana-788	271	12	index	index	NOUN
cana-788	271	13	grigorchuk	grigorchuk	NOUN
cana-788	271	14	group	group	NOUN
cana-788	271	15	γ2	γ2	PROPN
cana-788	271	16	γ3	γ3	PROPN
cana-788	271	17	γ4	γ4	PROPN
cana-788	271	18	γ5	γ5	PROPN
cana-788	271	19	γ6	γ6	PROPN
cana-788	271	20	𝑀1(γ𝑛	𝑀1(γ𝑛	PROPN
cana-788	271	21	)	)	PUNCT
cana-788	271	22	20	20	NUM
cana-788	271	23	56	56	NUM
cana-788	271	24	128	128	NUM
cana-788	271	25	272	272	NUM
cana-788	271	26	560	560	NUM
cana-788	271	27	𝑀2(γ𝑛	𝑀2(γ𝑛	PROPN
cana-788	271	28	)	)	PUNCT
cana-788	271	29	24	24	NUM
cana-788	271	30	78	78	NUM
cana-788	271	31	186	186	NUM
cana-788	271	32	402	402	NUM
cana-788	271	33	834	834	NUM
cana-788	271	34	𝑀2	𝑀2	PROPN
cana-788	271	35	𝑚(γ𝑛	𝑚(γ𝑛	NOUN
cana-788	271	36	)	)	PUNCT
cana-788	271	37	0.8889	0.8889	NUM
cana-788	271	38	1.6	1.6	NUM
cana-788	271	39	2.88889	2.88889	NUM
cana-788	271	40	5.55556	5.55556	NUM
cana-788	271	41	10.8889	10.8889	NUM
cana-788	271	42	𝐴𝑍(γ𝑛	𝐴𝑍(γ𝑛	PROPN
cana-788	271	43	)	)	PUNCT
cana-788	272	1	29.531	29.531	NUM
cana-788	272	2	98	98	NUM
cana-788	272	3	234.563	234.563	NUM
cana-788	272	4	507.938	507.938	NUM
cana-788	272	5	1054.69	1054.69	NUM
cana-788	272	6	𝐻𝑀(γ𝑛	𝐻𝑀(γ𝑛	PROPN
cana-788	272	7	)	)	PUNCT
cana-788	272	8	104	104	NUM
cana-788	272	9	320	320	NUM
cana-788	272	10	752	752	NUM
cana-788	272	11	1616	1616	NUM
cana-788	272	12	3344	3344	NUM
cana-788	272	13	𝐹(γ𝑛	𝐹(γ𝑛	NOUN
cana-788	272	14	)	)	PUNCT
cana-788	272	15	-4	-4	PUNCT
cana-788	272	16	44	44	NUM
cana-788	272	17	140	140	NUM
cana-788	272	18	332	332	NUM
cana-788	272	19	716	716	NUM
cana-788	272	20	𝐼(γ𝑛	𝐼(γ𝑛	NOUN
cana-788	272	21	)	)	PUNCT
cana-788	272	22	4.5	4.5	NUM
cana-788	272	23	14	14	NUM
cana-788	272	24	31.5	31.5	NUM
cana-788	272	25	67.5	67.5	NUM
cana-788	272	26	139.5	139.5	NUM
cana-788	272	27	a	a	DET
cana-788	272	28	3d	3d	NOUN
cana-788	272	29	graph	graph	NOUN
cana-788	272	30	(	(	PUNCT
cana-788	272	31	see	see	VERB
cana-788	272	32	figure	figure	NOUN
cana-788	272	33	9	9	NUM
cana-788	272	34	)	)	PUNCT
cana-788	272	35	representing	represent	VERB
cana-788	272	36	these	these	DET
cana-788	272	37	patterns	pattern	NOUN
cana-788	272	38	would	would	AUX
cana-788	272	39	give	give	VERB
cana-788	272	40	a	a	DET
cana-788	272	41	clear	clear	ADJ
cana-788	272	42	picture	picture	NOUN
cana-788	272	43	of	of	ADP
cana-788	272	44	how	how	SCONJ
cana-788	272	45	each	each	DET
cana-788	272	46	topological	topological	ADJ
cana-788	272	47	index	index	NOUN
cana-788	272	48	varies	vary	VERB
cana-788	272	49	with	with	ADP
cana-788	272	50	the	the	DET
cana-788	272	51	grigorchuk	grigorchuk	NOUN
cana-788	272	52	group	group	NOUN
cana-788	272	53	level	level	NOUN
cana-788	272	54	,	,	PUNCT
cana-788	272	55	revealing	reveal	VERB
cana-788	272	56	details	detail	NOUN
cana-788	272	57	about	about	ADP
cana-788	272	58	the	the	DET
cana-788	272	59	structural	structural	ADJ
cana-788	272	60	characteristics	characteristic	NOUN
cana-788	272	61	and	and	CCONJ
cana-788	272	62	complexity	complexity	NOUN
cana-788	272	63	of	of	ADP
cana-788	272	64	the	the	DET
cana-788	272	65	related	relate	VERB
cana-788	272	66	graph	graph	NOUN
cana-788	272	67	at	at	ADP
cana-788	272	68	various	various	ADJ
cana-788	272	69	group	group	NOUN
cana-788	272	70	sizes	size	NOUN
cana-788	272	71	.	.	PUNCT
cana-788	273	1	200	200	NUM
cana-788	273	2	700	700	NUM
cana-788	273	3	1200	1200	NUM
cana-788	273	4	1700	1700	NUM
cana-788	273	5	2200	2200	NUM
cana-788	273	6	2700	2700	NUM
cana-788	273	7	3200	3200	NUM
cana-788	273	8	3700	3700	NUM
cana-788	273	9	4200	4200	NUM
cana-788	273	10	4700	4700	NUM
cana-788	273	11	5200	5200	NUM
cana-788	273	12	b1	b1	PROPN
cana-788	273	13	*	*	PUNCT
cana-788	273	14	b2	b2	PROPN
cana-788	273	15	*	*	PUNCT
cana-788	273	16	b3	b3	PROPN
cana-788	273	17	*	*	PUNCT
cana-788	273	18	b4	b4	PROPN
cana-788	273	19	*	*	SYM
cana-788	273	20	b5	b5	PROPN
cana-788	273	21	*	*	PUNCT
cana-788	273	22	b6	b6	PROPN
cana-788	273	23	*	*	PROPN
cana-788	273	24	basilica	basilica	PROPN
cana-788	273	25	group	group	PROPN
cana-788	273	26	bn	bn	PROPN
cana-788	273	27	*	*	PROPN
cana-788	273	28	topological	topological	ADJ
cana-788	273	29	indices	index	NOUN
cana-788	273	30	of	of	ADP
cana-788	273	31	bn	bn	NOUN
cana-788	273	32	*	*	NOUN
cana-788	273	33	communications	communication	NOUN
cana-788	273	34	on	on	ADP
cana-788	273	35	applied	apply	VERB
cana-788	273	36	nonlinear	nonlinear	ADJ
cana-788	273	37	analysis	analysis	NOUN
cana-788	273	38	issn	issn	NOUN
cana-788	273	39	:	:	PUNCT
cana-788	273	40	1074	1074	NUM
cana-788	273	41	-	-	PUNCT
cana-788	273	42	133x	133x	NUM
cana-788	273	43	vol	vol	NOUN
cana-788	273	44	31	31	NUM
cana-788	273	45	no	no	NOUN
cana-788	273	46	.	.	PUNCT
cana-788	274	1	3s	3s	NUM
cana-788	274	2	(	(	PUNCT
cana-788	274	3	2024	2024	NUM
cana-788	274	4	)	)	PUNCT
cana-788	274	5	390	390	NUM
cana-788	274	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	274	7	fig	fig	NOUN
cana-788	274	8	.	.	PUNCT
cana-788	275	1	9	9	NUM
cana-788	275	2	3	3	NUM
cana-788	275	3	-	-	PUNCT
cana-788	275	4	d	d	NOUN
cana-788	275	5	plot	plot	NOUN
cana-788	275	6	of	of	ADP
cana-788	275	7	topological	topological	ADJ
cana-788	275	8	indices	index	NOUN
cana-788	275	9	of	of	ADP
cana-788	275	10	γ𝑛.	γ𝑛.	NOUN
cana-788	275	11	table	table	NOUN
cana-788	275	12	7	7	NUM
cana-788	275	13	shows	show	VERB
cana-788	275	14	the	the	DET
cana-788	275	15	determined	determined	ADJ
cana-788	275	16	values	value	NOUN
cana-788	275	17	of	of	ADP
cana-788	275	18	several	several	ADJ
cana-788	275	19	topological	topological	ADJ
cana-788	275	20	indices	index	NOUN
cana-788	275	21	at	at	ADP
cana-788	275	22	different	different	ADJ
cana-788	275	23	levels	level	NOUN
cana-788	275	24	(	(	PUNCT
cana-788	275	25	represented	represent	VERB
cana-788	275	26	by	by	ADP
cana-788	275	27	γ2	γ2	PROPN
cana-788	275	28	∗	∗	NOUN
cana-788	275	29	to	to	ADP
cana-788	275	30	γ6	γ6	PROPN
cana-788	275	31	∗	∗	NOUN
cana-788	275	32	)	)	PUNCT
cana-788	275	33	for	for	ADP
cana-788	275	34	the	the	DET
cana-788	275	35	schreier	schreier	ADJ
cana-788	275	36	graph	graph	NOUN
cana-788	275	37	of	of	ADP
cana-788	275	38	the	the	DET
cana-788	275	39	grigorchuk	grigorchuk	NOUN
cana-788	275	40	group	group	NOUN
cana-788	275	41	,	,	PUNCT
cana-788	275	42	including	include	VERB
cana-788	275	43	loops	loop	NOUN
cana-788	275	44	.	.	PUNCT
cana-788	276	1	there	there	PRON
cana-788	276	2	are	be	VERB
cana-788	276	3	distinct	distinct	ADJ
cana-788	276	4	trends	trend	NOUN
cana-788	276	5	among	among	ADP
cana-788	276	6	the	the	DET
cana-788	276	7	indices	index	NOUN
cana-788	276	8	:	:	PUNCT
cana-788	276	9	when	when	SCONJ
cana-788	276	10	the	the	DET
cana-788	276	11	group	group	NOUN
cana-788	276	12	level	level	NOUN
cana-788	276	13	rises	rise	NOUN
cana-788	276	14	,	,	PUNCT
cana-788	276	15	the	the	DET
cana-788	276	16	indices	index	NOUN
cana-788	276	17	𝑀1(γ𝑛	𝑀1(γ𝑛	PROPN
cana-788	276	18	)	)	PUNCT
cana-788	276	19	and	and	CCONJ
cana-788	276	20	𝑀2(γ𝑛	𝑀2(γ𝑛	PROPN
cana-788	276	21	)	)	PUNCT
cana-788	276	22	show	show	VERB
cana-788	276	23	steady	steady	ADJ
cana-788	276	24	growth	growth	NOUN
cana-788	276	25	,	,	PUNCT
cana-788	276	26	signifying	signify	VERB
cana-788	276	27	an	an	DET
cana-788	276	28	increase	increase	NOUN
cana-788	276	29	in	in	ADP
cana-788	276	30	vertices	vertex	NOUN
cana-788	276	31	and	and	CCONJ
cana-788	276	32	edges	edge	NOUN
cana-788	276	33	in	in	ADP
cana-788	276	34	the	the	DET
cana-788	276	35	graph	graph	NOUN
cana-788	276	36	.	.	PUNCT
cana-788	277	1	although	although	SCONJ
cana-788	277	2	its	its	PRON
cana-788	277	3	numerical	numerical	ADJ
cana-788	277	4	values	value	NOUN
cana-788	277	5	are	be	AUX
cana-788	277	6	less	less	ADJ
cana-788	277	7	,	,	PUNCT
cana-788	277	8	the	the	DET
cana-788	277	9	modified	modify	VERB
cana-788	277	10	second	second	ADJ
cana-788	277	11	zagreb	zagreb	PROPN
cana-788	277	12	index	index	NOUN
cana-788	277	13	displays	display	VERB
cana-788	277	14	a	a	DET
cana-788	277	15	similar	similar	ADJ
cana-788	277	16	tendency	tendency	NOUN
cana-788	277	17	.	.	PUNCT
cana-788	278	1	especially	especially	ADV
cana-788	278	2	,	,	PUNCT
cana-788	278	3	the	the	DET
cana-788	278	4	hyper	hyper	NOUN
cana-788	278	5	and	and	CCONJ
cana-788	278	6	augmented	augment	VERB
cana-788	278	7	zagreb	zagreb	PROPN
cana-788	278	8	indices	index	NOUN
cana-788	278	9	show	show	VERB
cana-788	278	10	significant	significant	ADJ
cana-788	278	11	increases	increase	NOUN
cana-788	278	12	,	,	PUNCT
cana-788	278	13	indicating	indicate	VERB
cana-788	278	14	more	more	ADV
cana-788	278	15	intricacy	intricacy	ADJ
cana-788	278	16	and	and	CCONJ
cana-788	278	17	connection	connection	NOUN
cana-788	278	18	in	in	ADP
cana-788	278	19	the	the	DET
cana-788	278	20	network	network	NOUN
cana-788	278	21	structure	structure	NOUN
cana-788	278	22	.	.	PUNCT
cana-788	279	1	the	the	DET
cana-788	279	2	inverse	inverse	NOUN
cana-788	279	3	sum	sum	NOUN
cana-788	279	4	index	index	NOUN
cana-788	279	5	gradually	gradually	ADV
cana-788	279	6	increases	increase	VERB
cana-788	279	7	across	across	ADP
cana-788	279	8	several	several	ADJ
cana-788	279	9	group	group	NOUN
cana-788	279	10	levels	level	NOUN
cana-788	279	11	,	,	PUNCT
cana-788	279	12	whereas	whereas	SCONJ
cana-788	279	13	the	the	DET
cana-788	279	14	forgotten	forget	VERB
cana-788	279	15	index	index	NOUN
cana-788	279	16	consistently	consistently	ADV
cana-788	279	17	remains	remain	VERB
cana-788	279	18	positive	positive	ADJ
cana-788	279	19	.	.	PUNCT
cana-788	280	1	table	table	NOUN
cana-788	280	2	7	7	NUM
cana-788	280	3	.	.	PUNCT
cana-788	280	4	calculated	calculate	VERB
cana-788	280	5	values	value	NOUN
cana-788	280	6	of	of	ADP
cana-788	280	7	topological	topological	ADJ
cana-788	280	8	indices	index	NOUN
cana-788	280	9	of	of	ADP
cana-788	280	10	γ𝑛	γ𝑛	PROPN
cana-788	280	11	∗	∗	PROPN
cana-788	280	12	topological	topological	PROPN
cana-788	280	13	index	index	PROPN
cana-788	280	14	grigorchuk	grigorchuk	NOUN
cana-788	280	15	group	group	PROPN
cana-788	280	16	𝚪𝟐	𝚪𝟐	PROPN
cana-788	280	17	∗	∗	PROPN
cana-788	280	18	𝚪𝟑	𝚪𝟑	PROPN
cana-788	280	19	∗	∗	PROPN
cana-788	280	20	𝚪𝟒	𝚪𝟒	PROPN
cana-788	280	21	∗	∗	PROPN
cana-788	280	22	𝚪𝟓	𝚪𝟓	PROPN
cana-788	280	23	∗	∗	NOUN
cana-788	280	24	𝚪𝟔	𝚪𝟔	PROPN
cana-788	280	25	∗	∗	PROPN
cana-788	280	26	𝑀1(γ𝑛	𝑀1(γ𝑛	PROPN
cana-788	280	27	∗	∗	NOUN
cana-788	280	28	)	)	PUNCT
cana-788	280	29	38	38	NUM
cana-788	280	30	138	138	NUM
cana-788	280	31	338	338	NUM
cana-788	280	32	738	738	NUM
cana-788	280	33	1538	1538	NUM
cana-788	280	34	𝑀2(γ𝑛	𝑀2(γ𝑛	PROPN
cana-788	280	35	∗	∗	NOUN
cana-788	280	36	)	)	PUNCT
cana-788	280	37	464	464	NUM
cana-788	280	38	714	714	NUM
cana-788	280	39	1214	1214	NUM
cana-788	280	40	2214	2214	NUM
cana-788	280	41	4214	4214	NUM
cana-788	280	42	𝑀2	𝑀2	PROPN
cana-788	280	43	𝑚(γ𝑛	𝑚(γ𝑛	X
cana-788	280	44	∗	∗	NOUN
cana-788	280	45	)	)	PUNCT
cana-788	281	1	0.3395	0.3395	PRON
cana-788	281	2	0.7395	0.7395	NUM
cana-788	282	1	1.5395	1.5395	NUM
cana-788	282	2	3.1395	3.1395	NUM
cana-788	282	3	6.3395	6.3395	NUM
cana-788	282	4	𝐴𝑍(γ𝑛	𝐴𝑍(γ𝑛	PROPN
cana-788	282	5	∗	∗	NOUN
cana-788	282	6	)	)	PUNCT
cana-788	282	7	616.32	616.32	NUM
cana-788	282	8	921.498	921.498	NUM
cana-788	282	9	1531.848	1531.848	NUM
cana-788	282	10	2752.548	2752.548	NUM
cana-788	282	11	5193.948	5193.948	PROPN
cana-788	282	12	𝐻𝑀(γ𝑛	𝐻𝑀(γ𝑛	PROPN
cana-788	282	13	∗	∗	NOUN
cana-788	282	14	)	)	PUNCT
cana-788	282	15	1864	1864	NUM
cana-788	282	16	2864	2864	NUM
cana-788	282	17	4864	4864	NUM
cana-788	282	18	8864	8864	NUM
cana-788	282	19	16864	16864	NUM
cana-788	282	20	𝐹(γ𝑛	𝐹(γ𝑛	X
cana-788	283	1	∗	∗	NOUN
cana-788	283	2	)	)	PUNCT
cana-788	283	3	936	936	NUM
cana-788	283	4	1436	1436	NUM
cana-788	283	5	2436	2436	NUM
cana-788	283	6	4436	4436	NUM
cana-788	283	7	8436	8436	NUM
cana-788	283	8	𝐼(γ𝑛	𝐼(γ𝑛	PROPN
cana-788	283	9	∗	∗	NOUN
cana-788	283	10	)	)	PUNCT
cana-788	283	11	889	889	NUM
cana-788	283	12	914	914	NUM
cana-788	283	13	964	964	NUM
cana-788	283	14	1064	1064	NUM
cana-788	283	15	1264	1264	NUM
cana-788	284	1	the	the	DET
cana-788	284	2	graphical	graphical	ADJ
cana-788	284	3	representation	representation	NOUN
cana-788	284	4	of	of	ADP
cana-788	284	5	the	the	DET
cana-788	284	6	calculated	calculate	VERB
cana-788	284	7	values	value	NOUN
cana-788	284	8	for	for	ADP
cana-788	284	9	the	the	DET
cana-788	284	10	topological	topological	ADJ
cana-788	284	11	indices	index	NOUN
cana-788	284	12	of	of	ADP
cana-788	284	13	γ𝑛	γ𝑛	ADP
cana-788	284	14	∗is	∗is	ADJ
cana-788	284	15	presented	present	VERB
cana-788	284	16	in	in	ADP
cana-788	284	17	the	the	DET
cana-788	284	18	figure	figure	NOUN
cana-788	284	19	10	10	NUM
cana-788	284	20	.	.	PUNCT
cana-788	285	1	in	in	ADP
cana-788	285	2	this	this	DET
cana-788	285	3	plot	plot	NOUN
cana-788	285	4	,	,	PUNCT
cana-788	285	5	each	each	DET
cana-788	285	6	axis	axis	NOUN
cana-788	285	7	represent	represent	VERB
cana-788	285	8	a	a	DET
cana-788	285	9	different	different	ADJ
cana-788	285	10	aspect	aspect	NOUN
cana-788	285	11	:	:	PUNCT
cana-788	285	12	one	one	NUM
cana-788	285	13	axis	axis	NOUN
cana-788	285	14	would	would	AUX
cana-788	285	15	represent	represent	VERB
cana-788	285	16	the	the	DET
cana-788	285	17	group	group	NOUN
cana-788	285	18	level	level	NOUN
cana-788	285	19	(	(	PUNCT
cana-788	285	20	γ2	γ2	PROPN
cana-788	285	21	∗	∗	NOUN
cana-788	285	22	to	to	ADP
cana-788	285	23	γ6	γ6	PROPN
cana-788	285	24	∗	∗	NOUN
cana-788	285	25	)	)	PUNCT
cana-788	285	26	,	,	PUNCT
cana-788	285	27	another	another	DET
cana-788	285	28	axis	axis	NOUN
cana-788	285	29	would	would	AUX
cana-788	285	30	represent	represent	VERB
cana-788	285	31	the	the	DET
cana-788	285	32	specific	specific	ADJ
cana-788	285	33	topological	topological	ADJ
cana-788	285	34	index	index	NOUN
cana-788	285	35	being	be	AUX
cana-788	285	36	measured	measure	VERB
cana-788	285	37	,	,	PUNCT
cana-788	285	38	and	and	CCONJ
cana-788	285	39	the	the	DET
cana-788	285	40	third	third	ADJ
cana-788	285	41	axis	axis	NOUN
cana-788	285	42	would	would	AUX
cana-788	285	43	represent	represent	VERB
cana-788	285	44	the	the	DET
cana-788	285	45	corresponding	corresponding	ADJ
cana-788	285	46	calculated	calculate	VERB
cana-788	285	47	value	value	NOUN
cana-788	285	48	for	for	ADP
cana-788	285	49	that	that	DET
cana-788	285	50	index	index	NOUN
cana-788	285	51	.	.	PUNCT
cana-788	286	1	the	the	DET
cana-788	286	2	plot	plot	NOUN
cana-788	286	3	would	would	AUX
cana-788	286	4	consist	consist	VERB
cana-788	286	5	of	of	ADP
cana-788	286	6	multiple	multiple	ADJ
cana-788	286	7	lines	line	NOUN
cana-788	286	8	or	or	CCONJ
cana-788	286	9	surfaces	surface	NOUN
cana-788	286	10	,	,	PUNCT
cana-788	286	11	each	each	PRON
cana-788	286	12	representing	represent	VERB
cana-788	286	13	a	a	DET
cana-788	286	14	different	different	ADJ
cana-788	286	15	topological	topological	ADJ
cana-788	286	16	index	index	NOUN
cana-788	286	17	,	,	PUNCT
cana-788	286	18	with	with	ADP
cana-788	286	19	points	point	NOUN
cana-788	286	20	along	along	ADP
cana-788	286	21	each	each	DET
cana-788	286	22	line	line	NOUN
cana-788	286	23	or	or	CCONJ
cana-788	286	24	surface	surface	NOUN
cana-788	286	25	indicating	indicate	VERB
cana-788	286	26	the	the	DET
cana-788	286	27	calculated	calculated	ADJ
cana-788	286	28	values	value	NOUN
cana-788	286	29	at	at	ADP
cana-788	286	30	different	different	ADJ
cana-788	286	31	group	group	NOUN
cana-788	286	32	levels	level	NOUN
cana-788	286	33	.	.	PUNCT
cana-788	287	1	grigorchuk	grigorchuk	PROPN
cana-788	287	2	group	group	NOUN
cana-788	287	3	ⴈ1	ⴈ1	PROPN
cana-788	287	4	grigorchuk	grigorchuk	PROPN
cana-788	287	5	group	group	NOUN
cana-788	287	6	ⴈ2	ⴈ2	PROPN
cana-788	287	7	grigorchuk	grigorchuk	PROPN
cana-788	287	8	group	group	PROPN
cana-788	287	9	ⴈ3	ⴈ3	PROPN
cana-788	287	10	grigorchuk	grigorchuk	PROPN
cana-788	287	11	group	group	PROPN
cana-788	287	12	ⴈ4	ⴈ4	PROPN
cana-788	287	13	grigorchuk	grigorchuk	PROPN
cana-788	287	14	group	group	NOUN
cana-788	287	15	ⴈ6	ⴈ6	PROPN
cana-788	287	16	communications	communication	NOUN
cana-788	287	17	on	on	ADP
cana-788	287	18	applied	apply	VERB
cana-788	287	19	nonlinear	nonlinear	ADJ
cana-788	287	20	analysis	analysis	NOUN
cana-788	287	21	issn	issn	NOUN
cana-788	287	22	:	:	PUNCT
cana-788	287	23	1074	1074	NUM
cana-788	287	24	-	-	PUNCT
cana-788	287	25	133x	133x	NUM
cana-788	287	26	vol	vol	NOUN
cana-788	287	27	31	31	NUM
cana-788	287	28	no	no	NOUN
cana-788	287	29	.	.	PUNCT
cana-788	288	1	3s	3s	NUM
cana-788	288	2	(	(	PUNCT
cana-788	288	3	2024	2024	NUM
cana-788	288	4	)	)	PUNCT
cana-788	288	5	391	391	NUM
cana-788	288	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	288	7	this	this	DET
cana-788	288	8	3d	3d	NUM
cana-788	288	9	visualization	visualization	NOUN
cana-788	288	10	would	would	AUX
cana-788	288	11	provide	provide	VERB
cana-788	288	12	a	a	DET
cana-788	288	13	comprehensive	comprehensive	ADJ
cana-788	288	14	depiction	depiction	NOUN
cana-788	288	15	of	of	ADP
cana-788	288	16	how	how	SCONJ
cana-788	288	17	each	each	DET
cana-788	288	18	topological	topological	ADJ
cana-788	288	19	index	index	NOUN
cana-788	288	20	varies	vary	VERB
cana-788	288	21	with	with	ADP
cana-788	288	22	both	both	CCONJ
cana-788	288	23	the	the	DET
cana-788	288	24	group	group	NOUN
cana-788	288	25	level	level	NOUN
cana-788	288	26	and	and	CCONJ
cana-788	288	27	the	the	DET
cana-788	288	28	specific	specific	ADJ
cana-788	288	29	measurement	measurement	NOUN
cana-788	288	30	being	be	AUX
cana-788	288	31	considered	consider	VERB
cana-788	288	32	,	,	PUNCT
cana-788	288	33	offering	offer	VERB
cana-788	288	34	a	a	DET
cana-788	288	35	clear	clear	ADJ
cana-788	288	36	and	and	CCONJ
cana-788	288	37	intuitive	intuitive	ADJ
cana-788	288	38	understanding	understanding	NOUN
cana-788	288	39	of	of	ADP
cana-788	288	40	the	the	DET
cana-788	288	41	evolving	evolve	VERB
cana-788	288	42	characteristics	characteristic	NOUN
cana-788	288	43	of	of	ADP
cana-788	288	44	γ𝑛	γ𝑛	PROPN
cana-788	288	45	∗.	∗.	PROPN
cana-788	288	46	fig	fig	PROPN
cana-788	288	47	.	.	PUNCT
cana-788	289	1	10	10	NUM
cana-788	289	2	calculated	calculate	VERB
cana-788	289	3	values	value	NOUN
cana-788	289	4	of	of	ADP
cana-788	289	5	topological	topological	ADJ
cana-788	289	6	indices	index	NOUN
cana-788	289	7	of	of	ADP
cana-788	289	8	γ𝑛	γ𝑛	PROPN
cana-788	289	9	∗.	∗.	PROPN
cana-788	289	10	4	4	NUM
cana-788	289	11	.	.	PUNCT
cana-788	289	12	conclusion	conclusion	VERB
cana-788	289	13	the	the	DET
cana-788	289	14	study	study	NOUN
cana-788	289	15	explores	explore	NOUN
cana-788	289	16	into	into	ADP
cana-788	289	17	the	the	DET
cana-788	289	18	characterization	characterization	NOUN
cana-788	289	19	of	of	ADP
cana-788	289	20	self	self	NOUN
cana-788	289	21	-	-	PUNCT
cana-788	289	22	similar	similar	ADJ
cana-788	289	23	groups	group	NOUN
cana-788	289	24	through	through	ADP
cana-788	289	25	schreier	schreier	NOUN
cana-788	289	26	graphs	graph	NOUN
cana-788	289	27	,	,	PUNCT
cana-788	289	28	particularly	particularly	ADV
cana-788	289	29	focusing	focus	VERB
cana-788	289	30	on	on	ADP
cana-788	289	31	the	the	DET
cana-788	289	32	basilica	basilica	NOUN
cana-788	289	33	and	and	CCONJ
cana-788	289	34	grigorchuk	grigorchuk	NOUN
cana-788	289	35	groups	group	NOUN
cana-788	289	36	.	.	PUNCT
cana-788	290	1	by	by	ADP
cana-788	290	2	analyzing	analyze	VERB
cana-788	290	3	the	the	DET
cana-788	290	4	m	m	NOUN
cana-788	290	5	-	-	ADJ
cana-788	290	6	polynomial	polynomial	ADJ
cana-788	290	7	of	of	ADP
cana-788	290	8	these	these	DET
cana-788	290	9	graphs	graph	NOUN
cana-788	290	10	,	,	PUNCT
cana-788	290	11	the	the	DET
cana-788	290	12	paper	paper	NOUN
cana-788	290	13	explores	explore	VERB
cana-788	290	14	its	its	PRON
cana-788	290	15	implications	implication	NOUN
cana-788	290	16	on	on	ADP
cana-788	290	17	various	various	ADJ
cana-788	290	18	topological	topological	ADJ
cana-788	290	19	indices	index	NOUN
cana-788	290	20	,	,	PUNCT
cana-788	290	21	providing	provide	VERB
cana-788	290	22	understandings	understanding	NOUN
cana-788	290	23	into	into	ADP
cana-788	290	24	the	the	DET
cana-788	290	25	connectivity	connectivity	NOUN
cana-788	290	26	and	and	CCONJ
cana-788	290	27	symmetry	symmetry	NOUN
cana-788	290	28	properties	property	NOUN
cana-788	290	29	of	of	ADP
cana-788	290	30	these	these	DET
cana-788	290	31	groups	group	NOUN
cana-788	290	32	.	.	PUNCT
cana-788	291	1	similarly	similarly	ADV
cana-788	291	2	,	,	PUNCT
cana-788	291	3	the	the	DET
cana-788	291	4	calculated	calculated	ADJ
cana-788	291	5	values	value	NOUN
cana-788	291	6	of	of	ADP
cana-788	291	7	these	these	DET
cana-788	291	8	indices	index	NOUN
cana-788	291	9	for	for	ADP
cana-788	291	10	the	the	DET
cana-788	291	11	grigorchuk	grigorchuk	NOUN
cana-788	291	12	group	group	NOUN
cana-788	291	13	γn	γn	NOUN
cana-788	291	14	and	and	CCONJ
cana-788	291	15	its	its	PRON
cana-788	291	16	variant	variant	NOUN
cana-788	291	17	γn	γn	NOUN
cana-788	291	18	*	*	PUNCT
cana-788	291	19	,	,	PUNCT
cana-788	291	20	offering	offer	VERB
cana-788	291	21	a	a	DET
cana-788	291	22	comparative	comparative	ADJ
cana-788	291	23	analysis	analysis	NOUN
cana-788	291	24	between	between	ADP
cana-788	291	25	the	the	DET
cana-788	291	26	two	two	NUM
cana-788	291	27	groups	group	NOUN
cana-788	291	28	.	.	PUNCT
cana-788	292	1	from	from	ADP
cana-788	292	2	the	the	DET
cana-788	292	3	calculated	calculated	ADJ
cana-788	292	4	values	value	NOUN
cana-788	292	5	,	,	PUNCT
cana-788	292	6	it	it	PRON
cana-788	292	7	is	be	AUX
cana-788	292	8	evident	evident	ADJ
cana-788	292	9	that	that	SCONJ
cana-788	292	10	there	there	PRON
cana-788	292	11	are	be	VERB
cana-788	292	12	discernible	discernible	ADJ
cana-788	292	13	patterns	pattern	NOUN
cana-788	292	14	in	in	ADP
cana-788	292	15	the	the	DET
cana-788	292	16	values	value	NOUN
cana-788	292	17	of	of	ADP
cana-788	292	18	the	the	DET
cana-788	292	19	topological	topological	ADJ
cana-788	292	20	indices	index	NOUN
cana-788	292	21	across	across	ADP
cana-788	292	22	different	different	ADJ
cana-788	292	23	iterations	iteration	NOUN
cana-788	292	24	(	(	PUNCT
cana-788	292	25	n	n	CCONJ
cana-788	292	26	)	)	PUNCT
cana-788	292	27	of	of	ADP
cana-788	292	28	both	both	DET
cana-788	292	29	groups	group	NOUN
cana-788	292	30	.	.	PUNCT
cana-788	293	1	for	for	ADP
cana-788	293	2	instance	instance	NOUN
cana-788	293	3	,	,	PUNCT
cana-788	293	4	the	the	DET
cana-788	293	5	first	first	PROPN
cana-788	293	6	zagreb	zagreb	PROPN
cana-788	293	7	index	index	NOUN
cana-788	293	8	generally	generally	ADV
cana-788	293	9	exhibits	exhibit	VERB
cana-788	293	10	exponential	exponential	ADJ
cana-788	293	11	growth	growth	NOUN
cana-788	293	12	with	with	ADP
cana-788	293	13	increasing	increase	VERB
cana-788	293	14	n	n	CCONJ
cana-788	293	15	for	for	ADP
cana-788	293	16	both	both	PRON
cana-788	293	17	basilica	basilica	NOUN
cana-788	293	18	and	and	CCONJ
cana-788	293	19	grigorchuk	grigorchuk	NOUN
cana-788	293	20	groups	group	NOUN
cana-788	293	21	,	,	PUNCT
cana-788	293	22	reflecting	reflect	VERB
cana-788	293	23	their	their	PRON
cana-788	293	24	complex	complex	ADJ
cana-788	293	25	structural	structural	ADJ
cana-788	293	26	characteristics	characteristic	NOUN
cana-788	293	27	.	.	PUNCT
cana-788	294	1	moreover	moreover	ADV
cana-788	294	2	,	,	PUNCT
cana-788	294	3	comparing	compare	VERB
cana-788	294	4	the	the	DET
cana-788	294	5	indices	index	NOUN
cana-788	294	6	of	of	ADP
cana-788	294	7	the	the	DET
cana-788	294	8	original	original	ADJ
cana-788	294	9	groups	group	NOUN
cana-788	294	10	with	with	ADP
cana-788	294	11	their	their	PRON
cana-788	294	12	variants	variant	NOUN
cana-788	294	13	(	(	PUNCT
cana-788	294	14	bn	bn	X
cana-788	294	15	vs.	vs.	X
cana-788	294	16	bn	bn	PROPN
cana-788	294	17	*	*	PROPN
cana-788	294	18	and	and	CCONJ
cana-788	294	19	γn	γn	X
cana-788	294	20	vs.	vs.	ADP
cana-788	294	21	γn	γn	NOUN
cana-788	294	22	*	*	NOUN
cana-788	294	23	)	)	PUNCT
cana-788	294	24	,	,	PUNCT
cana-788	294	25	it	it	PRON
cana-788	294	26	is	be	AUX
cana-788	294	27	seeming	seem	VERB
cana-788	294	28	that	that	SCONJ
cana-788	294	29	certain	certain	ADJ
cana-788	294	30	transformations	transformation	NOUN
cana-788	294	31	or	or	CCONJ
cana-788	294	32	modifications	modification	NOUN
cana-788	294	33	lead	lead	VERB
cana-788	294	34	to	to	ADP
cana-788	294	35	significant	significant	ADJ
cana-788	294	36	alterations	alteration	NOUN
cana-788	294	37	in	in	ADP
cana-788	294	38	the	the	DET
cana-788	294	39	topological	topological	ADJ
cana-788	294	40	properties	property	NOUN
cana-788	294	41	,	,	PUNCT
cana-788	294	42	as	as	SCONJ
cana-788	294	43	evidenced	evidence	VERB
cana-788	294	44	by	by	ADP
cana-788	294	45	changes	change	NOUN
cana-788	294	46	in	in	ADP
cana-788	294	47	index	index	NOUN
cana-788	294	48	values	value	NOUN
cana-788	294	49	.	.	PUNCT
cana-788	295	1	this	this	DET
cana-788	295	2	analysis	analysis	NOUN
cana-788	295	3	provides	provide	VERB
cana-788	295	4	valuable	valuable	ADJ
cana-788	295	5	perceptions	perception	NOUN
cana-788	295	6	into	into	ADP
cana-788	295	7	the	the	DET
cana-788	295	8	topological	topological	ADJ
cana-788	295	9	characteristics	characteristic	NOUN
cana-788	295	10	of	of	ADP
cana-788	295	11	the	the	DET
cana-788	295	12	basilica	basilica	NOUN
cana-788	295	13	and	and	CCONJ
cana-788	295	14	grigorchuk	grigorchuk	NOUN
cana-788	295	15	groups	group	NOUN
cana-788	295	16	,	,	PUNCT
cana-788	295	17	illuminating	illuminate	VERB
cana-788	295	18	on	on	ADP
cana-788	295	19	their	their	PRON
cana-788	295	20	complex	complex	ADJ
cana-788	295	21	structures	structure	NOUN
cana-788	295	22	and	and	CCONJ
cana-788	295	23	behaviors	behavior	NOUN
cana-788	295	24	.	.	PUNCT
cana-788	296	1	such	such	ADJ
cana-788	296	2	investigations	investigation	NOUN
cana-788	296	3	contribute	contribute	VERB
cana-788	296	4	to	to	ADP
cana-788	296	5	a	a	DET
cana-788	296	6	deeper	deep	ADJ
cana-788	296	7	understanding	understanding	NOUN
cana-788	296	8	of	of	ADP
cana-788	296	9	self	self	NOUN
cana-788	296	10	-	-	PUNCT
cana-788	296	11	similar	similar	ADJ
cana-788	296	12	groups	group	NOUN
cana-788	296	13	and	and	CCONJ
cana-788	296	14	their	their	PRON
cana-788	296	15	geometric	geometric	ADJ
cana-788	296	16	representations	representation	NOUN
cana-788	296	17	,	,	PUNCT
cana-788	296	18	with	with	ADP
cana-788	296	19	potential	potential	ADJ
cana-788	296	20	implications	implication	NOUN
cana-788	296	21	across	across	ADP
cana-788	296	22	various	various	ADJ
cana-788	296	23	domains	domain	NOUN
cana-788	296	24	,	,	PUNCT
cana-788	296	25	including	include	VERB
cana-788	296	26	group	group	NOUN
cana-788	296	27	theory	theory	NOUN
cana-788	296	28	,	,	PUNCT
cana-788	296	29	topology	topology	NOUN
cana-788	296	30	,	,	PUNCT
cana-788	296	31	and	and	CCONJ
cana-788	296	32	computational	computational	ADJ
cana-788	296	33	mathematics	mathematic	NOUN
cana-788	296	34	.	.	PUNCT
cana-788	297	1	grigorchuk	grigorchuk	PROPN
cana-788	297	2	group	group	PROPN
cana-788	297	3	ⴈ2	ⴈ2	PROPN
cana-788	297	4	grigorchuk	grigorchuk	PROPN
cana-788	297	5	group	group	PROPN
cana-788	297	6	ⴈ3	ⴈ3	PROPN
cana-788	297	7	grigorchuk	grigorchuk	PROPN
cana-788	297	8	group	group	PROPN
cana-788	297	9	ⴈ4	ⴈ4	PROPN
cana-788	297	10	grigorchuk	grigorchuk	PROPN
cana-788	297	11	group	group	PROPN
cana-788	297	12	ⴈ5	ⴈ5	PROPN
cana-788	297	13	grigorchuk	grigorchuk	PROPN
cana-788	297	14	group	group	NOUN
cana-788	297	15	ⴈ6	ⴈ6	PROPN
cana-788	297	16	0	0	NUM
cana-788	297	17	2000	2000	NUM
cana-788	297	18	4000	4000	NUM
cana-788	297	19	6000	6000	NUM
cana-788	297	20	8000	8000	NUM
cana-788	297	21	10000	10000	NUM
cana-788	297	22	12000	12000	NUM
cana-788	297	23	14000	14000	NUM
cana-788	297	24	16000	16000	NUM
cana-788	297	25	18000	18000	NUM
cana-788	297	26	communications	communication	NOUN
cana-788	297	27	on	on	ADP
cana-788	297	28	applied	apply	VERB
cana-788	297	29	nonlinear	nonlinear	ADJ
cana-788	297	30	analysis	analysis	NOUN
cana-788	297	31	issn	issn	NOUN
cana-788	297	32	:	:	PUNCT
cana-788	297	33	1074	1074	NUM
cana-788	297	34	-	-	PUNCT
cana-788	297	35	133x	133x	NUM
cana-788	297	36	vol	vol	NOUN
cana-788	297	37	31	31	NUM
cana-788	297	38	no	no	NOUN
cana-788	297	39	.	.	PUNCT
cana-788	298	1	3s	3s	NUM
cana-788	298	2	(	(	PUNCT
cana-788	298	3	2024	2024	NUM
cana-788	298	4	)	)	PUNCT
cana-788	298	5	392	392	NUM
cana-788	298	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-788	298	7	references	reference	NOUN
cana-788	298	8	[	[	X
cana-788	298	9	1	1	NUM
cana-788	298	10	]	]	X
cana-788	298	11	bondarenko	bondarenko	NOUN
cana-788	298	12	.	.	PUNCT
cana-788	299	1	i	i	PRON
cana-788	299	2	,	,	PUNCT
cana-788	299	3	groups	group	NOUN
cana-788	299	4	generated	generate	VERB
cana-788	299	5	by	by	ADP
cana-788	299	6	bounded	bounded	ADJ
cana-788	299	7	automata	automata	NOUN
cana-788	299	8	and	and	CCONJ
cana-788	299	9	their	their	PRON
cana-788	299	10	schreier	schreier	NOUN
cana-788	299	11	graph	graph	NOUN
cana-788	299	12	,	,	PUNCT
cana-788	299	13	texas	texas	PROPN
cana-788	299	14	a&m	a&m	PROPN
cana-788	299	15	university	university	PROPN
cana-788	299	16	,	,	PUNCT
cana-788	299	17	pro	pro	X
cana-788	299	18	quest	quest	PROPN
cana-788	299	19	llc	llc	PROPN
cana-788	299	20	,	,	PUNCT
cana-788	299	21	ann	ann	PROPN
cana-788	299	22	arbor	arbor	PROPN
cana-788	299	23	,	,	PUNCT
cana-788	299	24	mi	mi	PROPN
cana-788	299	25	,	,	PUNCT
cana-788	299	26	pp	pp	ADJ
cana-788	299	27	.	.	PUNCT
cana-788	299	28	162	162	NUM
cana-788	299	29	,	,	PUNCT
cana-788	299	30	2007	2007	NUM
cana-788	299	31	.	.	PUNCT
cana-788	300	1	doi	doi	NOUN
cana-788	300	2	:	:	PUNCT
cana-788	301	1	ba0569bc2d7ff08ffb948eb8159108de/1?18750	ba0569bc2d7ff08ffb948eb8159108de/1?18750	PROPN
cana-788	301	2	.	.	PUNCT
cana-788	302	1	[	[	X
cana-788	302	2	2	2	NUM
cana-788	302	3	]	]	PUNCT
cana-788	302	4	nekrashevych	nekrashevych	NOUN
cana-788	302	5	.	.	PUNCT
cana-788	303	1	v	v	NOUN
cana-788	303	2	,	,	PUNCT
cana-788	303	3	self	self	NOUN
cana-788	303	4	-	-	PUNCT
cana-788	303	5	similar	similar	ADJ
cana-788	303	6	groups	group	NOUN
cana-788	303	7	,	,	PUNCT
cana-788	303	8	mathematical	mathematical	ADJ
cana-788	303	9	surveys	survey	NOUN
cana-788	303	10	and	and	CCONJ
cana-788	303	11	monographs	monograph	NOUN
cana-788	303	12	,	,	PUNCT
cana-788	303	13	american	american	PROPN
cana-788	303	14	mathematical	mathematical	ADJ
cana-788	303	15	society	society	NOUN
cana-788	303	16	,	,	PUNCT
cana-788	303	17	providence	providence	NOUN
cana-788	303	18	,	,	PUNCT
cana-788	303	19	ri	ri	NOUN
cana-788	303	20	,	,	PUNCT
cana-788	303	21	vol	vol	NOUN
cana-788	303	22	.	.	PUNCT
cana-788	304	1	no	no	INTJ
cana-788	304	2	.	.	NOUN
cana-788	304	3	117	117	NUM
cana-788	304	4	,	,	PUNCT
cana-788	304	5	pp.231	pp.231	PROPN
cana-788	304	6	,	,	PUNCT
cana-788	304	7	2005	2005	NUM
cana-788	304	8	.	.	PUNCT
cana-788	305	1	doi	doi	NOUN
cana-788	305	2	:	:	PUNCT
cana-788	305	3	hl	hl	X
cana-788	305	4	=	=	VERB
cana-788	305	5	en&lr=&id	en&lr=&id	NOUN
cana-788	305	6	=	=	SYM
cana-788	305	7	ol3zbwaaqbaj&oi	ol3zbwaaqbaj&oi	VERB
cana-788	305	8	=	=	NOUN
cana-788	305	9	fnd&pg	fnd&pg	NOUN
cana-788	305	10	=	=	NOUN
cana-788	305	11	pr5&dq	pr5&dq	NOUN
cana-788	305	12	.	.	PUNCT
cana-788	306	1	[	[	X
cana-788	306	2	3	3	NUM
cana-788	306	3	]	]	X
cana-788	306	4	grigorchuk	grigorchuk	NOUN
cana-788	306	5	.	.	PUNCT
cana-788	307	1	r.	r.	PROPN
cana-788	307	2	i	i	PRON
cana-788	307	3	,	,	PUNCT
cana-788	307	4	żuk	żuk	ADV
cana-788	307	5	.	.	PUNCT
cana-788	308	1	a	a	PRON
cana-788	308	2	,	,	PUNCT
cana-788	308	3	on	on	ADP
cana-788	308	4	a	a	DET
cana-788	308	5	torsion	torsion	NOUN
cana-788	308	6	-	-	PUNCT
cana-788	308	7	free	free	ADJ
cana-788	308	8	weakly	weakly	ADJ
cana-788	308	9	branch	branch	NOUN
cana-788	308	10	group	group	NOUN
cana-788	308	11	defined	define	VERB
cana-788	308	12	by	by	ADP
cana-788	308	13	a	a	DET
cana-788	308	14	three	three	NUM
cana-788	308	15	state	state	NOUN
cana-788	308	16	automaton	automaton	NOUN
cana-788	308	17	.	.	PUNCT
cana-788	309	1	international	international	ADJ
cana-788	309	2	journal	journal	PROPN
cana-788	309	3	of	of	ADP
cana-788	309	4	algebra	algebra	NOUN
cana-788	309	5	and	and	CCONJ
cana-788	309	6	computation	computation	NOUN
cana-788	309	7	,	,	PUNCT
cana-788	309	8	vol	vol	NOUN
cana-788	309	9	.	.	PUNCT
cana-788	310	1	no	no	INTJ
cana-788	310	2	.	.	NOUN
cana-788	310	3	12	12	NUM
cana-788	310	4	(	(	PUNCT
cana-788	310	5	01n02	01n02	NUM
cana-788	310	6	)	)	PUNCT
cana-788	310	7	,	,	PUNCT
cana-788	310	8	pp.223	pp.223	PROPN
cana-788	310	9	-	-	PUNCT
cana-788	310	10	246	246	NUM
cana-788	310	11	,	,	PUNCT
cana-788	310	12	2002	2002	NUM
cana-788	310	13	.	.	PUNCT
cana-788	311	1	doi	doi	NOUN
cana-788	311	2	:	:	PUNCT
cana-788	311	3	doi	doi	PROPN
cana-788	311	4	/	/	SYM
cana-788	311	5	abs/10.1142	abs/10.1142	PROPN
cana-788	311	6	/	/	SYM
cana-788	311	7	s0218196702001000	s0218196702001000	PROPN
cana-788	311	8	.	.	PUNCT
cana-788	312	1	[	[	X
cana-788	312	2	4	4	NUM
cana-788	312	3	]	]	X
cana-788	312	4	ceccherini	ceccherini	NOUN
cana-788	312	5	-	-	PUNCT
cana-788	312	6	silberstein	silberstein	NOUN
cana-788	312	7	.	.	PUNCT
cana-788	313	1	t	t	PROPN
cana-788	313	2	,	,	PUNCT
cana-788	313	3	mach`ı	mach`ı	PROPN
cana-788	313	4	.	.	PUNCT
cana-788	314	1	a	a	PRON
cana-788	314	2	and	and	CCONJ
cana-788	314	3	scarabotti	scarabotti	NOUN
cana-788	314	4	.	.	PUNCT
cana-788	315	1	f	f	X
cana-788	315	2	,	,	PUNCT
cana-788	315	3	ii	ii	PROPN
cana-788	315	4	gruppo	gruppo	X
cana-788	315	5	di	di	X
cana-788	315	6	grigorchuk	grigorchuk	PROPN
cana-788	315	7	di	di	PROPN
cana-788	315	8	crescita	crescita	PROPN
cana-788	315	9	intermedia	intermedia	PROPN
cana-788	315	10	.	.	PUNCT
cana-788	316	1	rend.circ	rend.circ	ADV
cana-788	316	2	.	.	PUNCT
cana-788	317	1	mat	mat	NOUN
cana-788	317	2	.	.	PUNCT
cana-788	317	3	palermo	palermo	NOUN
cana-788	317	4	,	,	PUNCT
cana-788	317	5	vol	vol	NOUN
cana-788	317	6	.	.	PUNCT
cana-788	318	1	no	no	INTJ
cana-788	318	2	.	.	NOUN
cana-788	319	1	50	50	NUM
cana-788	319	2	(	(	PUNCT
cana-788	319	3	1	1	NUM
cana-788	319	4	)	)	PUNCT
cana-788	319	5	,	,	PUNCT
cana-788	319	6	pp.67–102	pp.67–102	PROPN
cana-788	319	7	,	,	PUNCT
cana-788	319	8	2001	2001	NUM
cana-788	319	9	.	.	PUNCT
cana-788	320	1	doi	doi	NOUN
cana-788	320	2	:	:	PUNCT
cana-788	320	3	/article/10.1007	/article/10.1007	PUNCT
cana-788	320	4	/	/	SYM
cana-788	320	5	bf02843919	bf02843919	PROPN
cana-788	320	6	.	.	PUNCT
cana-788	321	1	[	[	X
cana-788	321	2	5	5	NUM
cana-788	321	3	]	]	PUNCT
cana-788	321	4	grigorchuk	grigorchuk	NOUN
cana-788	321	5	.	.	PUNCT
cana-788	322	1	r	r	NOUN
cana-788	322	2	,	,	PUNCT
cana-788	322	3	solved	solved	ADJ
cana-788	322	4	and	and	CCONJ
cana-788	322	5	unsolved	unsolved	ADJ
cana-788	322	6	problems	problem	NOUN
cana-788	322	7	around	around	ADP
cana-788	322	8	one	one	NUM
cana-788	322	9	group	group	NOUN
cana-788	322	10	,	,	PUNCT
cana-788	322	11	infinite	infinite	ADJ
cana-788	322	12	groups	group	NOUN
cana-788	322	13	:	:	PUNCT
cana-788	322	14	geometric	geometric	ADJ
cana-788	322	15	,	,	PUNCT
cana-788	322	16	combinatorial	combinatorial	ADJ
cana-788	322	17	and	and	CCONJ
cana-788	322	18	dynamical	dynamical	ADJ
cana-788	322	19	aspects	aspect	NOUN
cana-788	322	20	,	,	PUNCT
cana-788	322	21	vol	vol	NOUN
cana-788	322	22	.	.	PUNCT
cana-788	323	1	no	no	INTJ
cana-788	323	2	.	.	NOUN
cana-788	323	3	248	248	NUM
cana-788	323	4	,	,	PUNCT
cana-788	323	5	pp.117–218	pp.117–218	NOUN
cana-788	323	6	,	,	PUNCT
cana-788	323	7	2005	2005	NUM
cana-788	323	8	.	.	PUNCT
cana-788	324	1	doi	doi	NOUN
cana-788	324	2	:	:	PUNCT
cana-788	324	3	chapter/10.1007/3	chapter/10.1007/3	VERB
cana-788	324	4	-	-	PUNCT
cana-788	324	5	7643	7643	NUM
cana-788	324	6	-	-	PUNCT
cana-788	324	7	7447	7447	NUM
cana-788	324	8	-	-	PUNCT
cana-788	324	9	0_5	0_5	NOUN
cana-788	324	10	.	.	PUNCT
cana-788	325	1	[	[	X
cana-788	325	2	6	6	NUM
cana-788	325	3	]	]	X
cana-788	325	4	chu	chu	PROPN
cana-788	325	5	y.	y.	PROPN
cana-788	325	6	m	m	PROPN
cana-788	325	7	,	,	PUNCT
cana-788	325	8	imran	imran	PROPN
cana-788	325	9	.	.	PUNCT
cana-788	326	1	m	m	PROPN
cana-788	326	2	,	,	PUNCT
cana-788	326	3	baig	baig	PROPN
cana-788	326	4	a.q	a.q	PROPN
cana-788	326	5	,	,	PUNCT
cana-788	326	6	akhter	akhter	NOUN
cana-788	326	7	.	.	PUNCT
cana-788	327	1	s	s	PART
cana-788	327	2	and	and	CCONJ
cana-788	327	3	siddiqui	siddiqui	PROPN
cana-788	327	4	m.	m.	PROPN
cana-788	327	5	k	k	PROPN
cana-788	327	6	,	,	PUNCT
cana-788	327	7	on	on	ADP
cana-788	327	8	m	m	ADJ
cana-788	327	9	-	-	ADJ
cana-788	327	10	polynomial	polynomial	ADJ
cana-788	327	11	-	-	PUNCT
cana-788	327	12	based	base	VERB
cana-788	327	13	topological	topological	ADJ
cana-788	327	14	descriptors	descriptor	NOUN
cana-788	327	15	of	of	ADP
cana-788	327	16	chemical	chemical	NOUN
cana-788	327	17	crystal	crystal	NOUN
cana-788	327	18	structures	structure	NOUN
cana-788	327	19	and	and	CCONJ
cana-788	327	20	their	their	PRON
cana-788	327	21	applications	application	NOUN
cana-788	327	22	.	.	PUNCT
cana-788	328	1	the	the	DET
cana-788	328	2	euro	euro	PROPN
cana-788	328	3	.	.	PUNCT
cana-788	329	1	phys	phy	NOUN
cana-788	329	2	.	.	PUNCT
cana-788	330	1	j.	j.	PROPN
cana-788	330	2	plus	plus	PROPN
cana-788	330	3	,	,	PUNCT
cana-788	330	4	vol	vol	NOUN
cana-788	330	5	.	.	PUNCT
cana-788	331	1	no	no	INTJ
cana-788	331	2	.	.	NOUN
cana-788	331	3	135	135	NUM
cana-788	331	4	(	(	PUNCT
cana-788	331	5	11	11	NUM
cana-788	331	6	)	)	PUNCT
cana-788	331	7	,	,	PUNCT
cana-788	331	8	pp.1	pp.1	NOUN
cana-788	331	9	-	-	PUNCT
cana-788	331	10	19	19	NUM
cana-788	331	11	,	,	PUNCT
cana-788	331	12	2020	2020	NUM
cana-788	331	13	.	.	PUNCT
cana-788	332	1	doi	doi	NOUN
cana-788	332	2	:	:	PUNCT
cana-788	332	3	https://doi.org/10.1140/epjp/s13360-020-00893-9	https://doi.org/10.1140/epjp/s13360-020-00893-9	NOUN
cana-788	332	4	.	.	PUNCT
cana-788	333	1	[	[	X
cana-788	333	2	7	7	NUM
cana-788	333	3	]	]	PUNCT
cana-788	333	4	cancan	cancan	NOUN
cana-788	333	5	.	.	PUNCT
cana-788	334	1	m	m	PROPN
cana-788	334	2	,	,	PUNCT
cana-788	334	3	ediz	ediz	ADJ
cana-788	334	4	.	.	PUNCT
cana-788	335	1	s	s	X
cana-788	335	2	,	,	PUNCT
cana-788	335	3	mutee	mutee	NOUN
cana-788	335	4	-	-	PUNCT
cana-788	335	5	ur	ur	INTJ
cana-788	335	6	-	-	NOUN
cana-788	335	7	rehman	rehman	NOUN
cana-788	335	8	.	.	PUNCT
cana-788	336	1	h	h	NOUN
cana-788	336	2	,	,	PUNCT
cana-788	336	3	and	and	CCONJ
cana-788	336	4	afzal	afzal	PROPN
cana-788	336	5	.	.	PUNCT
cana-788	337	1	d	d	X
cana-788	337	2	,	,	PUNCT
cana-788	337	3	m	m	VERB
cana-788	337	4	polynomial	polynomial	ADJ
cana-788	337	5	and	and	CCONJ
cana-788	337	6	topological	topological	ADJ
cana-788	337	7	indices	index	NOUN
cana-788	337	8	poly	poly	ADJ
cana-788	337	9	(	(	PUNCT
cana-788	337	10	ethyleneamidoamine	ethyleneamidoamine	NOUN
cana-788	337	11	)	)	PUNCT
cana-788	337	12	dendrimers	dendrimer	NOUN
cana-788	337	13	,	,	PUNCT
cana-788	337	14	j.	j.	PROPN
cana-788	337	15	inform	inform	PROPN
cana-788	337	16	.	.	PUNCT
cana-788	338	1	opt	opt	PROPN
cana-788	338	2	.	.	PUNCT
cana-788	339	1	sci	sci	PROPN
cana-788	339	2	,	,	PUNCT
cana-788	339	3	vol	vol	NOUN
cana-788	339	4	.	.	PUNCT
cana-788	340	1	no	no	INTJ
cana-788	340	2	.	.	PUNCT
cana-788	341	1	41(4	41(4	NUM
cana-788	341	2	)	)	PUNCT
cana-788	341	3	pp	pp	ADV
cana-788	341	4	.	.	PUNCT
cana-788	342	1	1117–1131	1117–1131	NUM
cana-788	342	2	,	,	PUNCT
cana-788	342	3	2020	2020	NUM
cana-788	342	4	.	.	PUNCT
cana-788	343	1	[	[	X
cana-788	343	2	8	8	NUM
cana-788	343	3	]	]	PUNCT
cana-788	343	4	cancan	cancan	NOUN
cana-788	343	5	.	.	PUNCT
cana-788	344	1	m	m	PROPN
cana-788	344	2	,	,	PUNCT
cana-788	344	3	ediz	ediz	ADJ
cana-788	344	4	.	.	PUNCT
cana-788	345	1	s	s	X
cana-788	345	2	,	,	PUNCT
cana-788	345	3	and	and	CCONJ
cana-788	345	4	farahani	farahani	PROPN
cana-788	345	5	m.r	m.r	PROPN
cana-788	345	6	,	,	PUNCT
cana-788	345	7	on	on	ADP
cana-788	345	8	ve	ve	NOUN
cana-788	345	9	-	-	PUNCT
cana-788	345	10	degree	degree	NOUN
cana-788	345	11	atom	atom	NOUN
cana-788	345	12	-	-	PUNCT
cana-788	345	13	bond	bond	NOUN
cana-788	345	14	connectivity	connectivity	NOUN
cana-788	345	15	,	,	PUNCT
cana-788	345	16	sumconnectivity	sumconnectivity	NOUN
cana-788	345	17	,	,	PUNCT
cana-788	345	18	geometric	geometric	ADJ
cana-788	345	19	arithmetic	arithmetic	ADJ
cana-788	345	20	and	and	CCONJ
cana-788	345	21	harmonic	harmonic	ADJ
cana-788	345	22	indices	index	NOUN
cana-788	345	23	of	of	ADP
cana-788	345	24	copper	copper	NOUN
cana-788	345	25	oxide	oxide	NOUN
cana-788	345	26	,	,	PUNCT
cana-788	345	27	eurasian	eurasian	ADJ
cana-788	345	28	chem.comm	chem.comm	PROPN
cana-788	345	29	.	.	PROPN
cana-788	345	30	vol	vol	NOUN
cana-788	345	31	.	.	PUNCT
cana-788	346	1	no	no	INTJ
cana-788	346	2	.	.	PUNCT
cana-788	347	1	2(5	2(5	NUM
cana-788	347	2	)	)	PUNCT
cana-788	347	3	,	,	PUNCT
cana-788	347	4	pp	pp	ADJ
cana-788	347	5	.	.	PUNCT
cana-788	348	1	641–645	641–645	NUM
cana-788	348	2	,	,	PUNCT
cana-788	348	3	2020	2020	NUM
cana-788	348	4	.	.	PUNCT
cana-788	349	1	doi	doi	NOUN
cana-788	349	2	:	:	PUNCT
cana-788	349	3	publication/340027283_on_ve	publication/340027283_on_ve	PROPN
cana-788	349	4	-	-	PUNCT
cana-788	349	5	degree_atom	degree_atom	NOUN
cana-788	349	6	-	-	PUNCT
cana-788	349	7	bond_connecti	bond_connecti	NOUN
cana-788	349	8	.	.	PUNCT
cana-788	350	1	[	[	X
cana-788	350	2	9	9	NUM
cana-788	350	3	]	]	PUNCT
cana-788	350	4	afzal	afzal	PROPN
cana-788	350	5	.	.	PUNCT
cana-788	351	1	f	f	X
cana-788	351	2	,	,	PUNCT
cana-788	351	3	hussain	hussain	PROPN
cana-788	351	4	.	.	PUNCT
cana-788	352	1	s	s	X
cana-788	352	2	,	,	PUNCT
cana-788	352	3	afzal	afzal	PROPN
cana-788	352	4	.	.	PUNCT
cana-788	353	1	d	d	X
cana-788	353	2	,	,	PUNCT
cana-788	353	3	and	and	CCONJ
cana-788	353	4	razaq	razaq	ADJ
cana-788	353	5	.	.	PUNCT
cana-788	354	1	s	s	X
cana-788	354	2	,	,	PUNCT
cana-788	354	3	some	some	DET
cana-788	354	4	new	new	ADJ
cana-788	354	5	degree	degree	NOUN
cana-788	354	6	based	base	VERB
cana-788	354	7	topological	topological	ADJ
cana-788	354	8	indices	index	NOUN
cana-788	354	9	via	via	ADP
cana-788	354	10	m	m	NOUN
cana-788	354	11	-	-	ADJ
cana-788	354	12	polynomial	polynomial	ADJ
cana-788	354	13	,	,	PUNCT
cana-788	354	14	j.	j.	PROPN
cana-788	354	15	inform	inform	PROPN
cana-788	354	16	.	.	PUNCT
cana-788	355	1	opt	opt	PROPN
cana-788	355	2	.	.	PUNCT
cana-788	356	1	sci	sci	PROPN
cana-788	356	2	.	.	PUNCT
cana-788	356	3	vol	vol	NOUN
cana-788	356	4	.	.	PUNCT
cana-788	357	1	no	no	INTJ
cana-788	357	2	.	.	PUNCT
cana-788	358	1	41(4	41(4	NUM
cana-788	358	2	)	)	PUNCT
cana-788	358	3	,	,	PUNCT
cana-788	358	4	pp.1061–1076	pp.1061–1076	PROPN
cana-788	358	5	,	,	PUNCT
cana-788	358	6	2020	2020	NUM
cana-788	358	7	.	.	PUNCT
cana-788	359	1	doi	doi	NOUN
cana-788	359	2	:	:	PUNCT
cana-788	359	3	https://www.tandfonline.com	https://www.tandfonline.com	X
cana-788	359	4	/	/	SYM
cana-788	359	5	doi	doi	NOUN
cana-788	359	6	/	/	SYM
cana-788	359	7	abs/10.1080/02522667.2020.1744307	abs/10.1080/02522667.2020.1744307	NOUN
cana-788	359	8	.	.	PUNCT
cana-788	360	1	[	[	X
cana-788	360	2	10	10	NUM
cana-788	360	3	]	]	X
cana-788	360	4	gao	gao	PROPN
cana-788	360	5	.	.	PUNCT
cana-788	361	1	w	w	PROPN
cana-788	361	2	,	,	PUNCT
cana-788	361	3	iqbal	iqbal	PROPN
cana-788	361	4	.	.	PUNCT
cana-788	362	1	z	z	X
cana-788	362	2	,	,	PUNCT
cana-788	362	3	ishaq	ishaq	PROPN
cana-788	362	4	.	.	PUNCT
cana-788	363	1	m	m	PROPN
cana-788	363	2	,	,	PUNCT
cana-788	363	3	sarfraz	sarfraz	NOUN
cana-788	363	4	.	.	PUNCT
cana-788	364	1	r	r	NOUN
cana-788	364	2	,	,	PUNCT
cana-788	364	3	aamir	aamir	PROPN
cana-788	364	4	.	.	PUNCT
cana-788	364	5	m	m	PROPN
cana-788	364	6	,	,	PUNCT
cana-788	364	7	and	and	CCONJ
cana-788	364	8	aslam	aslam	PROPN
cana-788	364	9	.	.	PUNCT
cana-788	365	1	a	a	PRON
cana-788	365	2	,	,	PUNCT
cana-788	365	3	on	on	ADP
cana-788	365	4	eccentricity	eccentricity	NOUN
cana-788	365	5	-	-	PUNCT
cana-788	365	6	based	base	VERB
cana-788	365	7	topological	topological	ADJ
cana-788	365	8	indices	index	NOUN
cana-788	365	9	study	study	NOUN
cana-788	365	10	of	of	ADP
cana-788	365	11	a	a	DET
cana-788	365	12	class	class	NOUN
cana-788	365	13	of	of	ADP
cana-788	365	14	porphyrin	porphyrin	NOUN
cana-788	365	15	-	-	PUNCT
cana-788	365	16	cored	core	VERB
cana-788	365	17	dendrimers	dendrimer	NOUN
cana-788	365	18	,	,	PUNCT
cana-788	365	19	biomolecules	biomolecule	NOUN
cana-788	365	20	,	,	PUNCT
cana-788	365	21	vol	vol	NOUN
cana-788	365	22	.	.	PUNCT
cana-788	366	1	no	no	INTJ
cana-788	366	2	.	.	PUNCT
cana-788	367	1	8(3	8(3	NUM
cana-788	367	2	)	)	PUNCT
cana-788	368	1	,	,	PUNCT
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cana-788	368	3	,	,	PUNCT
cana-788	368	4	2018	2018	NUM
cana-788	368	5	.	.	PUNCT
cana-788	369	1	doi	doi	NOUN
cana-788	369	2	:	:	PUNCT
cana-788	370	1	https://www.mdpi.com/2218-273x/8/3/71	https://www.mdpi.com/2218-273x/8/3/71	X
cana-788	370	2	.	.	PUNCT
cana-788	371	1	[	[	X
cana-788	371	2	11	11	NUM
cana-788	371	3	]	]	PUNCT
cana-788	371	4	bartholdi	bartholdi	NOUN
cana-788	371	5	.	.	PUNCT
cana-788	372	1	l	l	NOUN
cana-788	372	2	,	,	PUNCT
cana-788	372	3	virag	virag	NOUN
cana-788	372	4	.	.	PUNCT
cana-788	373	1	b.	b.	PROPN
cana-788	373	2	amenability	amenability	PROPN
cana-788	373	3	via	via	ADP
cana-788	373	4	random	random	ADJ
cana-788	373	5	walks	walk	NOUN
cana-788	373	6	.	.	PUNCT
cana-788	374	1	duke	duke	PROPN
cana-788	374	2	math	math	PROPN
cana-788	374	3	journal	journal	PROPN
cana-788	374	4	.	.	PUNCT
cana-788	375	1	vol	vol	NOUN
cana-788	375	2	.	.	PUNCT
cana-788	376	1	no	no	INTJ
cana-788	376	2	.	.	NOUN
cana-788	377	1	130(1	130(1	NUM
cana-788	377	2	)	)	PUNCT
cana-788	377	3	,	,	PUNCT
cana-788	378	1	pp	pp	ADP
cana-788	378	2	.	.	PUNCT
cana-788	379	1	39	39	NUM
cana-788	379	2	-	-	SYM
cana-788	379	3	56	56	NUM
cana-788	379	4	,	,	PUNCT
cana-788	379	5	2005.doi	2005.doi	NUM
cana-788	379	6	:	:	PUNCT
cana-788	379	7	projecteuclid.org/journals/dukemathematicaljournal/volume-	projecteuclid.org/journals/dukemathematicaljournal/volume-	NOUN
cana-788	379	8	.	.	PUNCT
cana-788	380	1	[	[	X
cana-788	380	2	12	12	NUM
cana-788	380	3	]	]	PUNCT
cana-788	380	4	bartholdi	bartholdi	NOUN
cana-788	380	5	.	.	PUNCT
cana-788	381	1	l	l	NOUN
cana-788	381	2	,	,	PUNCT
cana-788	381	3	grigorchuk	grigorchuk	NOUN
cana-788	381	4	.	.	PUNCT
cana-788	382	1	r	r	NOUN
cana-788	382	2	,	,	PUNCT
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cana-788	382	4	.	.	PUNCT
cana-788	383	1	v.	v.	CCONJ
cana-788	383	2	from	from	ADP
cana-788	383	3	fractal	fractal	ADJ
cana-788	383	4	groups	group	NOUN
cana-788	383	5	to	to	ADP
cana-788	383	6	fractal	fractal	ADJ
cana-788	383	7	sets	set	NOUN
cana-788	383	8	,	,	PUNCT
cana-788	383	9	in	in	ADP
cana-788	383	10	:	:	PUNCT
cana-788	383	11	“	"	PUNCT
cana-788	383	12	fractals	fractal	NOUN
cana-788	383	13	in	in	ADP
cana-788	383	14	graz	graz	PROPN
cana-788	383	15	”	"	PUNCT
cana-788	383	16	(	(	PUNCT
cana-788	383	17	p.	p.	NOUN
cana-788	383	18	grabner	grabner	NOUN
cana-788	383	19	and	and	CCONJ
cana-788	383	20	w.	w.	PROPN
cana-788	383	21	woess	woess	NOUN
cana-788	383	22	editors	editor	NOUN
cana-788	383	23	)	)	PUNCT
cana-788	383	24	.	.	PUNCT
cana-788	384	1	trends	trend	NOUN
cana-788	384	2	in	in	ADP
cana-788	384	3	mathematics	mathematic	NOUN
cana-788	384	4	.	.	PUNCT
cana-788	385	1	birkauser	birkauser	PROPN
cana-788	385	2	verlag	verlag	PROPN
cana-788	385	3	,	,	PUNCT
cana-788	385	4	basel	basel	PROPN
cana-788	385	5	,	,	PUNCT
cana-788	385	6	pp	pp	X
cana-788	385	7	.	.	PUNCT
cana-788	386	1	25	25	NUM
cana-788	386	2	–	–	PUNCT
cana-788	386	3	118	118	NUM
cana-788	386	4	,	,	PUNCT
cana-788	386	5	2003	2003	NUM
cana-788	386	6	.	.	PUNCT
cana-788	387	1	doi	doi	NOUN
cana-788	387	2	:	:	PUNCT
cana-788	387	3	https://link.springer.com/chapter/10.1007/978-3-0348-8014-5_2	https://link.springer.com/chapter/10.1007/978-3-0348-8014-5_2	NOUN
cana-788	387	4	.	.	PUNCT
cana-788	388	1	[	[	X
cana-788	388	2	13	13	NUM
cana-788	388	3	]	]	X
cana-788	388	4	ceccherini	ceccherini	NOUN
cana-788	388	5	-	-	PUNCT
cana-788	388	6	silberstein	silberstein	NOUN
cana-788	388	7	.	.	PUNCT
cana-788	389	1	t	t	PROPN
cana-788	389	2	,	,	PUNCT
cana-788	389	3	donno	donno	VERB
cana-788	389	4	.	.	PUNCT
cana-788	390	1	a	a	DET
cana-788	390	2	,	,	PUNCT
cana-788	390	3	iacono	iacono	NOUN
cana-788	390	4	.	.	PUNCT
cana-788	391	1	d.	d.	PROPN
cana-788	391	2	the	the	DET
cana-788	391	3	tutte	tutte	PROPN
cana-788	391	4	polynomial	polynomial	NOUN
cana-788	391	5	of	of	ADP
cana-788	391	6	the	the	DET
cana-788	391	7	schreier	schreier	NOUN
cana-788	391	8	graphs	graph	NOUN
cana-788	391	9	for	for	ADP
cana-788	391	10	the	the	DET
cana-788	391	11	grigorchuk	grigorchuk	NOUN
cana-788	391	12	group	group	NOUN
cana-788	391	13	and	and	CCONJ
cana-788	391	14	the	the	DET
cana-788	391	15	basilica	basilica	PROPN
cana-788	391	16	group	group	NOUN
cana-788	391	17	.	.	PUNCT
cana-788	392	1	ischia	ischia	PROPN
cana-788	392	2	group	group	PROPN
cana-788	392	3	theory	theory	NOUN
cana-788	392	4	,	,	PUNCT
cana-788	392	5	pp	pp	X
cana-788	392	6	.	.	PUNCT
cana-788	393	1	245	245	NUM
cana-788	393	2	–	–	PUNCT
cana-788	393	3	68	68	NUM
cana-788	393	4	,	,	PUNCT
cana-788	393	5	2008	2008	NUM
cana-788	393	6	.	.	PUNCT
cana-788	394	1	https://www.worldscientific.com/doi/abs/10.1142/9789814350051_0006	https://www.worldscientific.com/doi/abs/10.1142/9789814350051_0006	PROPN
cana-788	394	2	.	.	PUNCT
cana-788	395	1	[	[	X
cana-788	395	2	14	14	NUM
cana-788	395	3	]	]	X
cana-788	395	4	sahin	sahin	PROPN
cana-788	395	5	.	.	PROPN
cana-788	395	6	b	b	PROPN
cana-788	395	7	,	,	PUNCT
cana-788	395	8	sahin	sahin	PROPN
cana-788	395	9	.	.	PROPN
cana-788	396	1	a	a	PRON
cana-788	396	2	,	,	PUNCT
cana-788	396	3	the	the	DET
cana-788	396	4	hosoya	hosoya	ADJ
cana-788	396	5	polynomial	polynomial	NOUN
cana-788	396	6	of	of	ADP
cana-788	396	7	the	the	DET
cana-788	396	8	schreier	schreier	NOUN
cana-788	396	9	graphs	graph	NOUN
cana-788	396	10	of	of	ADP
cana-788	396	11	the	the	DET
cana-788	396	12	grigorchuk	grigorchuk	NOUN
cana-788	396	13	group	group	NOUN
cana-788	396	14	and	and	CCONJ
cana-788	396	15	the	the	DET
cana-788	396	16	basilica	basilica	PROPN
cana-788	396	17	group	group	NOUN
cana-788	396	18	,	,	PUNCT
cana-788	396	19	turkish	turkish	ADJ
cana-788	396	20	journal	journal	NOUN
cana-788	396	21	of	of	ADP
cana-788	396	22	science	science	NOUN
cana-788	396	23	,	,	PUNCT
cana-788	396	24	vol	vol	NOUN
cana-788	396	25	.	.	PUNCT
cana-788	397	1	no	no	INTJ
cana-788	397	2	.	.	PUNCT
cana-788	398	1	5(3	5(3	NUM
cana-788	398	2	)	)	PUNCT
cana-788	398	3	,	,	PUNCT
cana-788	398	4	pp	pp	PROPN
cana-788	398	5	.	.	PUNCT
cana-788	399	1	262	262	NUM
cana-788	399	2	-	-	SYM
cana-788	399	3	267	267	NUM
cana-788	399	4	,	,	PUNCT
cana-788	399	5	2020	2020	NUM
cana-788	399	6	.	.	PUNCT
cana-788	400	1	doi	doi	NOUN
cana-788	400	2	:	:	PUNCT
cana-788	400	3	https://dergipark.org.tr/en/pub/tjos/issue/59057/846400	https://dergipark.org.tr/en/pub/tjos/issue/59057/846400	PROPN
cana-788	400	4	.	.	PUNCT