id	sid	tid	token	lemma	pos
cana-834	1	1	communications	communication	NOUN
cana-834	1	2	on	on	ADP
cana-834	1	3	applied	apply	VERB
cana-834	1	4	nonlinear	nonlinear	ADJ
cana-834	1	5	analysis	analysis	NOUN
cana-834	1	6	issn	issn	NOUN
cana-834	1	7	:	:	PUNCT
cana-834	1	8	1074	1074	NUM
cana-834	1	9	-	-	PUNCT
cana-834	1	10	133x	133x	NUM
cana-834	1	11	vol	vol	NOUN
cana-834	1	12	31	31	NUM
cana-834	1	13	no	no	NOUN
cana-834	1	14	.	.	PUNCT
cana-834	2	1	4s	4s	NUM
cana-834	2	2	(	(	PUNCT
cana-834	2	3	2024	2024	NUM
cana-834	2	4	)	)	PUNCT
cana-834	2	5	164	164	NUM
cana-834	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	2	7	optimizing	optimize	VERB
cana-834	2	8	graph	graph	NOUN
cana-834	2	9	theory	theory	NOUN
cana-834	2	10	algorithms	algorithm	NOUN
cana-834	2	11	for	for	ADP
cana-834	2	12	social	social	ADJ
cana-834	2	13	network	network	NOUN
cana-834	2	14	analysis	analysis	NOUN
cana-834	2	15	susanta	susanta	VERB
cana-834	2	16	kumar	kumar	PROPN
cana-834	2	17	sahoo1	sahoo1	PROPN
cana-834	2	18	,	,	PUNCT
cana-834	2	19	sasmita	sasmita	PROPN
cana-834	2	20	mishra2	mishra2	PROPN
cana-834	2	21	1department	1department	NUM
cana-834	2	22	of	of	ADP
cana-834	2	23	computer	computer	NOUN
cana-834	2	24	science	science	NOUN
cana-834	2	25	engineering	engineering	PROPN
cana-834	2	26	&	&	CCONJ
cana-834	2	27	applications	application	NOUN
cana-834	2	28	,	,	PUNCT
cana-834	2	29	igit	igit	PROPN
cana-834	2	30	sarang	sarang	PROPN
cana-834	2	31	,	,	PUNCT
cana-834	2	32	odisha	odisha	PROPN
cana-834	2	33	,	,	PUNCT
cana-834	2	34	india	india	PROPN
cana-834	2	35	susantasahoo79@gmail.com	susantasahoo79@gmail.com	PROPN
cana-834	2	36	2department	2department	NUM
cana-834	2	37	of	of	ADP
cana-834	2	38	computer	computer	NOUN
cana-834	2	39	science	science	NOUN
cana-834	2	40	engineering	engineering	PROPN
cana-834	2	41	&	&	CCONJ
cana-834	2	42	applications	application	NOUN
cana-834	2	43	,	,	PUNCT
cana-834	2	44	igit	igit	PROPN
cana-834	2	45	sarang	sarang	PROPN
cana-834	2	46	,	,	PUNCT
cana-834	2	47	odisha	odisha	PROPN
cana-834	2	48	,	,	PUNCT
cana-834	2	49	india	india	PROPN
cana-834	2	50	sasmita.mishra.csea@gmail.com	sasmita.mishra.csea@gmail.com	PROPN
cana-834	2	51	article	article	NOUN
cana-834	2	52	history	history	NOUN
cana-834	2	53	:	:	PUNCT
cana-834	2	54	received	receive	VERB
cana-834	2	55	:	:	PUNCT
cana-834	2	56	16	16	NUM
cana-834	2	57	-	-	PUNCT
cana-834	2	58	04	04	NUM
cana-834	2	59	-	-	PUNCT
cana-834	2	60	2024	2024	NUM
cana-834	2	61	revised	revise	VERB
cana-834	2	62	:	:	PUNCT
cana-834	2	63	10	10	NUM
cana-834	2	64	-	-	SYM
cana-834	2	65	06	06	NUM
cana-834	2	66	-	-	PUNCT
cana-834	2	67	2024	2024	NUM
cana-834	2	68	accepted	accept	VERB
cana-834	2	69	:	:	PUNCT
cana-834	2	70	24	24	NUM
cana-834	2	71	-	-	PUNCT
cana-834	2	72	06	06	NUM
cana-834	2	73	-	-	PUNCT
cana-834	2	74	2024	2024	NUM
cana-834	2	75	abstract	abstract	ADJ
cana-834	2	76	social	social	ADJ
cana-834	2	77	network	network	NOUN
cana-834	2	78	analysis	analysis	NOUN
cana-834	2	79	(	(	PUNCT
cana-834	2	80	sna	sna	NOUN
cana-834	2	81	)	)	PUNCT
cana-834	2	82	leverages	leverage	NOUN
cana-834	2	83	graph	graph	NOUN
cana-834	2	84	theory	theory	NOUN
cana-834	2	85	to	to	PART
cana-834	2	86	understand	understand	VERB
cana-834	2	87	and	and	CCONJ
cana-834	2	88	visualize	visualize	VERB
cana-834	2	89	the	the	DET
cana-834	2	90	complex	complex	ADJ
cana-834	2	91	relationships	relationship	NOUN
cana-834	2	92	and	and	CCONJ
cana-834	2	93	structures	structure	NOUN
cana-834	2	94	within	within	ADP
cana-834	2	95	social	social	ADJ
cana-834	2	96	networks	network	NOUN
cana-834	2	97	.	.	PUNCT
cana-834	3	1	this	this	DET
cana-834	3	2	research	research	NOUN
cana-834	3	3	paper	paper	NOUN
cana-834	3	4	explores	explore	VERB
cana-834	3	5	the	the	DET
cana-834	3	6	optimization	optimization	NOUN
cana-834	3	7	of	of	ADP
cana-834	3	8	graph	graph	NOUN
cana-834	3	9	theory	theory	NOUN
cana-834	3	10	algorithms	algorithm	NOUN
cana-834	3	11	tailored	tailor	VERB
cana-834	3	12	for	for	ADP
cana-834	3	13	sna	sna	PROPN
cana-834	3	14	,	,	PUNCT
cana-834	3	15	focusing	focus	VERB
cana-834	3	16	on	on	ADP
cana-834	3	17	efficiency	efficiency	NOUN
cana-834	3	18	improvements	improvement	NOUN
cana-834	3	19	in	in	ADP
cana-834	3	20	handling	handle	VERB
cana-834	3	21	large	large	ADJ
cana-834	3	22	-	-	PUNCT
cana-834	3	23	scale	scale	NOUN
cana-834	3	24	networks	network	NOUN
cana-834	3	25	.	.	PUNCT
cana-834	4	1	the	the	DET
cana-834	4	2	study	study	NOUN
cana-834	4	3	reviews	review	VERB
cana-834	4	4	key	key	ADJ
cana-834	4	5	graph	graph	NOUN
cana-834	4	6	theory	theory	NOUN
cana-834	4	7	concepts	concept	NOUN
cana-834	4	8	,	,	PUNCT
cana-834	4	9	identifies	identify	VERB
cana-834	4	10	common	common	ADJ
cana-834	4	11	challenges	challenge	NOUN
cana-834	4	12	in	in	ADP
cana-834	4	13	sna	sna	PROPN
cana-834	4	14	,	,	PUNCT
cana-834	4	15	and	and	CCONJ
cana-834	4	16	evaluates	evaluate	VERB
cana-834	4	17	various	various	ADJ
cana-834	4	18	optimization	optimization	NOUN
cana-834	4	19	techniques	technique	NOUN
cana-834	4	20	.	.	PUNCT
cana-834	5	1	practical	practical	ADJ
cana-834	5	2	applications	application	NOUN
cana-834	5	3	and	and	CCONJ
cana-834	5	4	case	case	NOUN
cana-834	5	5	studies	study	NOUN
cana-834	5	6	are	be	AUX
cana-834	5	7	presented	present	VERB
cana-834	5	8	to	to	PART
cana-834	5	9	demonstrate	demonstrate	VERB
cana-834	5	10	the	the	DET
cana-834	5	11	impact	impact	NOUN
cana-834	5	12	of	of	ADP
cana-834	5	13	these	these	DET
cana-834	5	14	optimizations	optimization	NOUN
cana-834	5	15	in	in	ADP
cana-834	5	16	real	real	ADJ
cana-834	5	17	-	-	PUNCT
cana-834	5	18	world	world	NOUN
cana-834	5	19	scenarios	scenario	NOUN
cana-834	5	20	.	.	PUNCT
cana-834	6	1	keywords	keyword	NOUN
cana-834	6	2	:	:	PUNCT
cana-834	6	3	graph	graph	NOUN
cana-834	6	4	theory	theory	NOUN
cana-834	6	5	,	,	PUNCT
cana-834	6	6	algorithms	algorithms	PROPN
cana-834	6	7	,	,	PUNCT
cana-834	6	8	sna	sna	PROPN
cana-834	6	9	.	.	PROPN
cana-834	6	10	1	1	NUM
cana-834	6	11	.	.	X
cana-834	6	12	introduction	introduction	NOUN
cana-834	6	13	the	the	DET
cana-834	6	14	dissemination	dissemination	NOUN
cana-834	6	15	of	of	ADP
cana-834	6	16	information	information	NOUN
cana-834	6	17	and	and	CCONJ
cana-834	6	18	even	even	ADV
cana-834	6	19	political	political	ADJ
cana-834	6	20	movements	movement	NOUN
cana-834	6	21	are	be	AUX
cana-834	6	22	all	all	PRON
cana-834	6	23	shaped	shape	VERB
cana-834	6	24	by	by	ADP
cana-834	6	25	social	social	ADJ
cana-834	6	26	networks	network	NOUN
cana-834	6	27	,	,	PUNCT
cana-834	6	28	which	which	PRON
cana-834	6	29	influence	influence	VERB
cana-834	6	30	everything	everything	PRON
cana-834	6	31	from	from	ADP
cana-834	6	32	personal	personal	ADJ
cana-834	6	33	relationships	relationship	NOUN
cana-834	6	34	to	to	ADP
cana-834	6	35	professional	professional	ADJ
cana-834	6	36	connections	connection	NOUN
cana-834	6	37	.	.	PUNCT
cana-834	7	1	individuals	individual	NOUN
cana-834	7	2	or	or	CCONJ
cana-834	7	3	entities	entity	NOUN
cana-834	7	4	(	(	PUNCT
cana-834	7	5	nodes	node	NOUN
cana-834	7	6	)	)	PUNCT
cana-834	7	7	and	and	CCONJ
cana-834	7	8	their	their	PRON
cana-834	7	9	interactions	interaction	NOUN
cana-834	7	10	(	(	PUNCT
cana-834	7	11	edges	edge	NOUN
cana-834	7	12	)	)	PUNCT
cana-834	7	13	make	make	VERB
cana-834	7	14	up	up	ADP
cana-834	7	15	these	these	DET
cana-834	7	16	networks	network	NOUN
cana-834	7	17	,	,	PUNCT
cana-834	7	18	which	which	PRON
cana-834	7	19	include	include	VERB
cana-834	7	20	offline	offline	ADJ
cana-834	7	21	social	social	ADJ
cana-834	7	22	structures	structure	NOUN
cana-834	7	23	as	as	ADV
cana-834	7	24	well	well	ADV
cana-834	7	25	as	as	ADP
cana-834	7	26	online	online	ADJ
cana-834	7	27	social	social	ADJ
cana-834	7	28	platforms	platform	NOUN
cana-834	7	29	like	like	ADP
cana-834	7	30	facebook	facebook	NOUN
cana-834	7	31	,	,	PUNCT
cana-834	7	32	twitter	twitter	NOUN
cana-834	7	33	,	,	PUNCT
cana-834	7	34	and	and	CCONJ
cana-834	7	35	linkedin	linkedin	NOUN
cana-834	7	36	.	.	PUNCT
cana-834	8	1	the	the	DET
cana-834	8	2	investigation	investigation	NOUN
cana-834	8	3	of	of	ADP
cana-834	8	4	these	these	DET
cana-834	8	5	organizations	organization	NOUN
cana-834	8	6	,	,	PUNCT
cana-834	8	7	known	know	VERB
cana-834	8	8	as	as	ADP
cana-834	8	9	informal	informal	ADJ
cana-834	8	10	community	community	NOUN
cana-834	8	11	examination	examination	NOUN
cana-834	8	12	(	(	PUNCT
cana-834	8	13	sna	sna	PROPN
cana-834	8	14	)	)	PUNCT
cana-834	8	15	,	,	PUNCT
cana-834	8	16	uses	use	VERB
cana-834	8	17	chart	chart	NOUN
cana-834	8	18	hypothesis	hypothesis	NOUN
cana-834	8	19	to	to	PART
cana-834	8	20	show	show	VERB
cana-834	8	21	and	and	CCONJ
cana-834	8	22	break	break	VERB
cana-834	8	23	down	down	ADP
cana-834	8	24	these	these	DET
cana-834	8	25	complicated	complicated	ADJ
cana-834	8	26	frameworks	framework	NOUN
cana-834	8	27	[	[	X
cana-834	8	28	1	1	NUM
cana-834	8	29	]	]	PUNCT
cana-834	8	30	.	.	PUNCT
cana-834	9	1	the	the	DET
cana-834	9	2	mathematical	mathematical	ADJ
cana-834	9	3	foundation	foundation	NOUN
cana-834	9	4	required	require	VERB
cana-834	9	5	to	to	PART
cana-834	9	6	represent	represent	VERB
cana-834	9	7	and	and	CCONJ
cana-834	9	8	investigate	investigate	VERB
cana-834	9	9	the	the	DET
cana-834	9	10	intricate	intricate	ADJ
cana-834	9	11	web	web	NOUN
cana-834	9	12	of	of	ADP
cana-834	9	13	connections	connection	NOUN
cana-834	9	14	that	that	PRON
cana-834	9	15	makes	make	VERB
cana-834	9	16	up	up	ADP
cana-834	9	17	social	social	ADJ
cana-834	9	18	networks	network	NOUN
cana-834	9	19	is	be	AUX
cana-834	9	20	provided	provide	VERB
cana-834	9	21	by	by	ADP
cana-834	9	22	graph	graph	NOUN
cana-834	9	23	theory	theory	NOUN
cana-834	9	24	.	.	PUNCT
cana-834	10	1	researchers	researcher	NOUN
cana-834	10	2	and	and	CCONJ
cana-834	10	3	practitioners	practitioner	NOUN
cana-834	10	4	can	can	AUX
cana-834	10	5	discover	discover	VERB
cana-834	10	6	community	community	NOUN
cana-834	10	7	structures	structure	NOUN
cana-834	10	8	,	,	PUNCT
cana-834	10	9	predict	predict	VERB
cana-834	10	10	future	future	ADJ
cana-834	10	11	trends	trend	NOUN
cana-834	10	12	,	,	PUNCT
cana-834	10	13	and	and	CCONJ
cana-834	10	14	gain	gain	VERB
cana-834	10	15	insight	insight	NOUN
cana-834	10	16	into	into	ADP
cana-834	10	17	the	the	DET
cana-834	10	18	dynamics	dynamic	NOUN
cana-834	10	19	of	of	ADP
cana-834	10	20	networks	network	NOUN
cana-834	10	21	by	by	ADP
cana-834	10	22	utilizing	utilize	VERB
cana-834	10	23	graph	graph	NOUN
cana-834	10	24	theory	theory	NOUN
cana-834	10	25	.	.	PUNCT
cana-834	11	1	in	in	ADP
cana-834	11	2	any	any	DET
cana-834	11	3	case	case	NOUN
cana-834	11	4	,	,	PUNCT
cana-834	11	5	the	the	DET
cana-834	11	6	sheer	sheer	ADJ
cana-834	11	7	scale	scale	NOUN
cana-834	11	8	and	and	CCONJ
cana-834	11	9	intricacy	intricacy	NOUN
cana-834	11	10	of	of	ADP
cana-834	11	11	contemporary	contemporary	ADJ
cana-834	11	12	informal	informal	ADJ
cana-834	11	13	communities	community	NOUN
cana-834	11	14	’	'	PUNCT
cana-834	11	15	present	present	ADJ
cana-834	11	16	huge	huge	ADJ
cana-834	11	17	difficulties	difficulty	NOUN
cana-834	11	18	to	to	ADP
cana-834	11	19	customary	customary	ADJ
cana-834	11	20	diagram	diagram	NOUN
cana-834	11	21	hypothesis	hypothesis	NOUN
cana-834	11	22	calculations	calculation	NOUN
cana-834	11	23	.	.	PUNCT
cana-834	12	1	optimizing	optimize	VERB
cana-834	12	2	graph	graph	NOUN
cana-834	12	3	theory	theory	NOUN
cana-834	12	4	algorithms	algorithms	NOUN
cana-834	12	5	is	be	AUX
cana-834	12	6	now	now	ADV
cana-834	12	7	a	a	DET
cana-834	12	8	necessity	necessity	NOUN
cana-834	12	9	as	as	SCONJ
cana-834	12	10	the	the	DET
cana-834	12	11	amount	amount	NOUN
cana-834	12	12	of	of	ADP
cana-834	12	13	data	datum	NOUN
cana-834	12	14	from	from	ADP
cana-834	12	15	social	social	ADJ
cana-834	12	16	networks	network	NOUN
cana-834	12	17	grows	grow	VERB
cana-834	12	18	at	at	ADP
cana-834	12	19	an	an	DET
cana-834	12	20	exponential	exponential	ADJ
cana-834	12	21	rate	rate	NOUN
cana-834	12	22	.	.	PUNCT
cana-834	13	1	many	many	ADJ
cana-834	13	2	modern	modern	ADJ
cana-834	13	3	social	social	ADJ
cana-834	13	4	networks	network	NOUN
cana-834	13	5	have	have	VERB
cana-834	13	6	millions	million	NOUN
cana-834	13	7	of	of	ADP
cana-834	13	8	nodes	node	NOUN
cana-834	13	9	and	and	CCONJ
cana-834	13	10	billions	billion	NOUN
cana-834	13	11	of	of	ADP
cana-834	13	12	edges	edge	NOUN
cana-834	13	13	,	,	PUNCT
cana-834	13	14	requiring	require	VERB
cana-834	13	15	a	a	DET
cana-834	13	16	lot	lot	NOUN
cana-834	13	17	of	of	ADP
cana-834	13	18	memory	memory	NOUN
cana-834	13	19	and	and	CCONJ
cana-834	13	20	computation	computation	NOUN
cana-834	13	21	.	.	PUNCT
cana-834	14	1	the	the	DET
cana-834	14	2	dynamic	dynamic	ADJ
cana-834	14	3	nature	nature	NOUN
cana-834	14	4	of	of	ADP
cana-834	14	5	social	social	ADJ
cana-834	14	6	networks	network	NOUN
cana-834	14	7	,	,	PUNCT
cana-834	14	8	where	where	SCONJ
cana-834	14	9	edges	edge	NOUN
cana-834	14	10	and	and	CCONJ
cana-834	14	11	nodes	node	NOUN
cana-834	14	12	are	be	AUX
cana-834	14	13	constantly	constantly	ADV
cana-834	14	14	added	add	VERB
cana-834	14	15	or	or	CCONJ
cana-834	14	16	removed	remove	VERB
cana-834	14	17	,	,	PUNCT
cana-834	14	18	makes	make	VERB
cana-834	14	19	it	it	PRON
cana-834	14	20	even	even	ADV
cana-834	14	21	more	more	ADV
cana-834	14	22	difficult	difficult	ADJ
cana-834	14	23	to	to	PART
cana-834	14	24	solve	solve	VERB
cana-834	14	25	these	these	DET
cana-834	14	26	problems	problem	NOUN
cana-834	14	27	[	[	X
cana-834	14	28	2	2	NUM
cana-834	14	29	]	]	PUNCT
cana-834	14	30	.	.	PUNCT
cana-834	15	1	several	several	ADJ
cana-834	15	2	real	real	ADJ
cana-834	15	3	-	-	PUNCT
cana-834	15	4	time	time	NOUN
cana-834	15	5	applications	application	NOUN
cana-834	15	6	,	,	PUNCT
cana-834	15	7	such	such	ADJ
cana-834	15	8	as	as	ADP
cana-834	15	9	recommendation	recommendation	NOUN
cana-834	15	10	systems	system	NOUN
cana-834	15	11	,	,	PUNCT
cana-834	15	12	fraud	fraud	NOUN
cana-834	15	13	detection	detection	NOUN
cana-834	15	14	,	,	PUNCT
cana-834	15	15	targeted	target	VERB
cana-834	15	16	marketing	marketing	NOUN
cana-834	15	17	,	,	PUNCT
cana-834	15	18	and	and	CCONJ
cana-834	15	19	public	public	ADJ
cana-834	15	20	health	health	NOUN
cana-834	15	21	monitoring	monitoring	NOUN
cana-834	15	22	,	,	PUNCT
cana-834	15	23	require	require	VERB
cana-834	15	24	optimized	optimize	VERB
cana-834	15	25	graph	graph	NOUN
cana-834	15	26	theory	theory	NOUN
cana-834	15	27	algorithms	algorithm	NOUN
cana-834	15	28	.	.	PUNCT
cana-834	16	1	we	we	PRON
cana-834	16	2	can	can	AUX
cana-834	16	3	guarantee	guarantee	VERB
cana-834	16	4	timely	timely	ADJ
cana-834	16	5	and	and	CCONJ
cana-834	16	6	accurate	accurate	ADJ
cana-834	16	7	analysis	analysis	NOUN
cana-834	16	8	,	,	PUNCT
cana-834	16	9	facilitate	facilitate	VERB
cana-834	16	10	better	well	ADJ
cana-834	16	11	decision	decision	NOUN
cana-834	16	12	-	-	PUNCT
cana-834	16	13	making	making	NOUN
cana-834	16	14	,	,	PUNCT
cana-834	16	15	and	and	CCONJ
cana-834	16	16	encourage	encourage	VERB
cana-834	16	17	innovation	innovation	NOUN
cana-834	16	18	in	in	ADP
cana-834	16	19	a	a	DET
cana-834	16	20	variety	variety	NOUN
cana-834	16	21	of	of	ADP
cana-834	16	22	fields	field	NOUN
cana-834	16	23	by	by	ADP
cana-834	16	24	increasing	increase	VERB
cana-834	16	25	these	these	DET
cana-834	16	26	algorithms	algorithm	NOUN
cana-834	16	27	'	'	PART
cana-834	16	28	efficiency	efficiency	NOUN
cana-834	16	29	and	and	CCONJ
cana-834	16	30	scalability	scalability	NOUN
cana-834	16	31	[	[	X
cana-834	16	32	3	3	NUM
cana-834	16	33	]	]	PUNCT
cana-834	16	34	.	.	PUNCT
cana-834	17	1	the	the	DET
cana-834	17	2	goal	goal	NOUN
cana-834	17	3	of	of	ADP
cana-834	17	4	this	this	DET
cana-834	17	5	paper	paper	NOUN
cana-834	17	6	is	be	AUX
cana-834	17	7	to	to	PART
cana-834	17	8	:	:	PUNCT
cana-834	17	9	•	•	NUM
cana-834	17	10	go	go	VERB
cana-834	17	11	over	over	ADP
cana-834	17	12	the	the	DET
cana-834	17	13	fundamental	fundamental	ADJ
cana-834	17	14	ideas	idea	NOUN
cana-834	17	15	of	of	ADP
cana-834	17	16	graph	graph	NOUN
cana-834	17	17	theory	theory	NOUN
cana-834	17	18	that	that	PRON
cana-834	17	19	are	be	AUX
cana-834	17	20	relevant	relevant	ADJ
cana-834	17	21	to	to	AUX
cana-834	17	22	sna	sna	PROPN
cana-834	17	23	.	.	PROPN
cana-834	17	24	•	•	NUM
cana-834	17	25	identify	identify	VERB
cana-834	17	26	obstacles	obstacle	NOUN
cana-834	17	27	that	that	PRON
cana-834	17	28	arise	arise	VERB
cana-834	17	29	when	when	SCONJ
cana-834	17	30	analyzing	analyze	VERB
cana-834	17	31	massive	massive	ADJ
cana-834	17	32	social	social	ADJ
cana-834	17	33	networks	network	NOUN
cana-834	17	34	.	.	PUNCT
cana-834	18	1	communications	communication	NOUN
cana-834	18	2	on	on	ADP
cana-834	18	3	applied	apply	VERB
cana-834	18	4	nonlinear	nonlinear	ADJ
cana-834	18	5	analysis	analysis	NOUN
cana-834	18	6	issn	issn	NOUN
cana-834	18	7	:	:	PUNCT
cana-834	18	8	1074	1074	NUM
cana-834	18	9	-	-	PUNCT
cana-834	18	10	133x	133x	NUM
cana-834	18	11	vol	vol	NOUN
cana-834	18	12	31	31	NUM
cana-834	18	13	no	no	NOUN
cana-834	18	14	.	.	PUNCT
cana-834	19	1	4s	4s	NUM
cana-834	19	2	(	(	PUNCT
cana-834	19	3	2024	2024	NUM
cana-834	19	4	)	)	PUNCT
cana-834	19	5	165	165	NUM
cana-834	19	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-834	19	7	•	•	NOUN
cana-834	19	8	assess	assess	NOUN
cana-834	19	9	streamlining	streamlining	NOUN
cana-834	19	10	procedures	procedure	NOUN
cana-834	19	11	for	for	ADP
cana-834	19	12	diagram	diagram	NOUN
cana-834	19	13	hypothesis	hypothesis	NOUN
cana-834	19	14	calculations	calculation	NOUN
cana-834	19	15	.	.	PUNCT
cana-834	20	1	•	•	NUM
cana-834	20	2	provide	provide	VERB
cana-834	20	3	examples	example	NOUN
cana-834	20	4	of	of	ADP
cana-834	20	5	real	real	ADJ
cana-834	20	6	-	-	PUNCT
cana-834	20	7	world	world	NOUN
cana-834	20	8	uses	use	VERB
cana-834	20	9	for	for	ADP
cana-834	20	10	optimized	optimize	VERB
cana-834	20	11	algorithms	algorithm	NOUN
cana-834	20	12	in	in	ADP
cana-834	20	13	case	case	NOUN
cana-834	20	14	studies	study	NOUN
cana-834	20	15	.	.	PUNCT
cana-834	21	1	the	the	DET
cana-834	21	2	theoretical	theoretical	ADJ
cana-834	21	3	and	and	CCONJ
cana-834	21	4	practical	practical	ADJ
cana-834	21	5	aspects	aspect	NOUN
cana-834	21	6	of	of	ADP
cana-834	21	7	optimizing	optimize	VERB
cana-834	21	8	graph	graph	NOUN
cana-834	21	9	theory	theory	NOUN
cana-834	21	10	algorithms	algorithm	NOUN
cana-834	21	11	for	for	ADP
cana-834	21	12	sna	sna	PROPN
cana-834	21	13	are	be	AUX
cana-834	21	14	covered	cover	VERB
cana-834	21	15	in	in	ADP
cana-834	21	16	this	this	DET
cana-834	21	17	paper	paper	NOUN
cana-834	21	18	[	[	X
cana-834	21	19	4	4	NUM
cana-834	21	20	]	]	PUNCT
cana-834	21	21	.	.	PUNCT
cana-834	22	1	we	we	PRON
cana-834	22	2	begin	begin	VERB
cana-834	22	3	by	by	ADP
cana-834	22	4	looking	look	VERB
cana-834	22	5	at	at	ADP
cana-834	22	6	fundamental	fundamental	ADJ
cana-834	22	7	ideas	idea	NOUN
cana-834	22	8	in	in	ADP
cana-834	22	9	graph	graph	NOUN
cana-834	22	10	theory	theory	NOUN
cana-834	22	11	and	and	CCONJ
cana-834	22	12	how	how	SCONJ
cana-834	22	13	they	they	PRON
cana-834	22	14	apply	apply	VERB
cana-834	22	15	to	to	ADP
cana-834	22	16	social	social	ADJ
cana-834	22	17	networks	network	NOUN
cana-834	22	18	.	.	PUNCT
cana-834	23	1	the	the	DET
cana-834	23	2	unique	unique	ADJ
cana-834	23	3	difficulties	difficulty	NOUN
cana-834	23	4	that	that	PRON
cana-834	23	5	large	large	ADJ
cana-834	23	6	-	-	PUNCT
cana-834	23	7	scale	scale	NOUN
cana-834	23	8	social	social	ADJ
cana-834	23	9	network	network	NOUN
cana-834	23	10	analysis	analysis	NOUN
cana-834	23	11	presents	present	NOUN
cana-834	23	12	,	,	PUNCT
cana-834	23	13	such	such	ADJ
cana-834	23	14	as	as	ADP
cana-834	23	15	scalability	scalability	NOUN
cana-834	23	16	,	,	PUNCT
cana-834	23	17	dynamic	dynamic	ADJ
cana-834	23	18	nature	nature	NOUN
cana-834	23	19	,	,	PUNCT
cana-834	23	20	and	and	CCONJ
cana-834	23	21	computational	computational	ADJ
cana-834	23	22	complexity	complexity	NOUN
cana-834	23	23	[	[	X
cana-834	23	24	5	5	NUM
cana-834	23	25	]	]	PUNCT
cana-834	23	26	,	,	PUNCT
cana-834	23	27	are	be	AUX
cana-834	23	28	then	then	ADV
cana-834	23	29	discussed	discuss	VERB
cana-834	23	30	.	.	PUNCT
cana-834	24	1	after	after	ADP
cana-834	24	2	that	that	PRON
cana-834	24	3	,	,	PUNCT
cana-834	24	4	we	we	PRON
cana-834	24	5	delve	delve	VERB
cana-834	24	6	into	into	ADP
cana-834	24	7	a	a	DET
cana-834	24	8	variety	variety	NOUN
cana-834	24	9	of	of	ADP
cana-834	24	10	optimization	optimization	NOUN
cana-834	24	11	strategies	strategy	NOUN
cana-834	24	12	,	,	PUNCT
cana-834	24	13	including	include	VERB
cana-834	24	14	enhancements	enhancement	NOUN
cana-834	24	15	to	to	ADP
cana-834	24	16	algorithms	algorithm	NOUN
cana-834	24	17	,	,	PUNCT
cana-834	24	18	cutting	cut	VERB
cana-834	24	19	-	-	PUNCT
cana-834	24	20	edge	edge	NOUN
cana-834	24	21	data	datum	NOUN
cana-834	24	22	structures	structure	NOUN
cana-834	24	23	,	,	PUNCT
cana-834	24	24	and	and	CCONJ
cana-834	24	25	specialized	specialized	ADJ
cana-834	24	26	hardware	hardware	NOUN
cana-834	24	27	.	.	PUNCT
cana-834	25	1	the	the	DET
cana-834	25	2	efficacy	efficacy	NOUN
cana-834	25	3	of	of	ADP
cana-834	25	4	these	these	DET
cana-834	25	5	methods	method	NOUN
cana-834	25	6	in	in	ADP
cana-834	25	7	improving	improve	VERB
cana-834	25	8	the	the	DET
cana-834	25	9	performance	performance	NOUN
cana-834	25	10	and	and	CCONJ
cana-834	25	11	scalability	scalability	NOUN
cana-834	25	12	of	of	ADP
cana-834	25	13	graph	graph	NOUN
cana-834	25	14	algorithms	algorithms	NOUN
cana-834	25	15	is	be	AUX
cana-834	25	16	the	the	DET
cana-834	25	17	basis	basis	NOUN
cana-834	25	18	for	for	ADP
cana-834	25	19	their	their	PRON
cana-834	25	20	evaluation	evaluation	NOUN
cana-834	25	21	[	[	X
cana-834	25	22	6	6	NUM
cana-834	25	23	]	]	PUNCT
cana-834	25	24	.	.	PUNCT
cana-834	26	1	finally	finally	ADV
cana-834	26	2	,	,	PUNCT
cana-834	26	3	we	we	PRON
cana-834	26	4	present	present	VERB
cana-834	26	5	case	case	NOUN
cana-834	26	6	studies	study	NOUN
cana-834	26	7	from	from	ADP
cana-834	26	8	a	a	DET
cana-834	26	9	variety	variety	NOUN
cana-834	26	10	of	of	ADP
cana-834	26	11	application	application	NOUN
cana-834	26	12	areas	area	NOUN
cana-834	26	13	to	to	PART
cana-834	26	14	show	show	VERB
cana-834	26	15	how	how	SCONJ
cana-834	26	16	optimized	optimize	VERB
cana-834	26	17	graph	graph	NOUN
cana-834	26	18	theory	theory	NOUN
cana-834	26	19	algorithms	algorithm	NOUN
cana-834	26	20	can	can	AUX
cana-834	26	21	be	be	AUX
cana-834	26	22	used	use	VERB
cana-834	26	23	in	in	ADP
cana-834	26	24	real	real	ADJ
cana-834	26	25	-	-	PUNCT
cana-834	26	26	world	world	NOUN
cana-834	26	27	situations	situation	NOUN
cana-834	26	28	.	.	PUNCT
cana-834	27	1	these	these	DET
cana-834	27	2	case	case	NOUN
cana-834	27	3	studies	study	NOUN
cana-834	27	4	demonstrate	demonstrate	VERB
cana-834	27	5	the	the	DET
cana-834	27	6	transformative	transformative	ADJ
cana-834	27	7	power	power	NOUN
cana-834	27	8	of	of	ADP
cana-834	27	9	optimization	optimization	NOUN
cana-834	27	10	methods	method	NOUN
cana-834	27	11	for	for	ADP
cana-834	27	12	dealing	deal	VERB
cana-834	27	13	with	with	ADP
cana-834	27	14	current	current	ADJ
cana-834	27	15	sna	sna	PROPN
cana-834	27	16	issues	issue	NOUN
cana-834	27	17	[	[	X
cana-834	27	18	7	7	NUM
cana-834	27	19	]	]	PUNCT
cana-834	27	20	.	.	PUNCT
cana-834	28	1	this	this	DET
cana-834	28	2	paper	paper	NOUN
cana-834	28	3	aims	aim	VERB
cana-834	28	4	to	to	PART
cana-834	28	5	contribute	contribute	VERB
cana-834	28	6	to	to	ADP
cana-834	28	7	ongoing	ongoing	ADJ
cana-834	28	8	efforts	effort	NOUN
cana-834	28	9	to	to	PART
cana-834	28	10	optimize	optimize	VERB
cana-834	28	11	graph	graph	NOUN
cana-834	28	12	theory	theory	NOUN
cana-834	28	13	algorithms	algorithm	NOUN
cana-834	28	14	for	for	ADP
cana-834	28	15	social	social	ADJ
cana-834	28	16	network	network	NOUN
cana-834	28	17	analysis	analysis	NOUN
cana-834	28	18	by	by	ADP
cana-834	28	19	providing	provide	VERB
cana-834	28	20	a	a	DET
cana-834	28	21	comprehensive	comprehensive	ADJ
cana-834	28	22	overview	overview	NOUN
cana-834	28	23	of	of	ADP
cana-834	28	24	the	the	DET
cana-834	28	25	current	current	ADJ
cana-834	28	26	state	state	NOUN
cana-834	28	27	of	of	ADP
cana-834	28	28	research	research	NOUN
cana-834	28	29	and	and	CCONJ
cana-834	28	30	practical	practical	ADJ
cana-834	28	31	applications	application	NOUN
cana-834	28	32	.	.	PUNCT
cana-834	29	1	future	future	ADJ
cana-834	29	2	research	research	NOUN
cana-834	29	3	and	and	CCONJ
cana-834	29	4	development	development	NOUN
cana-834	29	5	in	in	ADP
cana-834	29	6	this	this	DET
cana-834	29	7	crucial	crucial	ADJ
cana-834	29	8	field	field	NOUN
cana-834	29	9	will	will	AUX
cana-834	29	10	benefit	benefit	VERB
cana-834	29	11	from	from	ADP
cana-834	29	12	the	the	DET
cana-834	29	13	insights	insight	NOUN
cana-834	29	14	and	and	CCONJ
cana-834	29	15	findings	finding	NOUN
cana-834	29	16	presented	present	VERB
cana-834	29	17	here	here	ADV
cana-834	29	18	,	,	PUNCT
cana-834	29	19	allowing	allow	VERB
cana-834	29	20	for	for	ADP
cana-834	29	21	advancements	advancement	NOUN
cana-834	29	22	that	that	PRON
cana-834	29	23	can	can	AUX
cana-834	29	24	keep	keep	VERB
cana-834	29	25	up	up	ADP
cana-834	29	26	with	with	ADP
cana-834	29	27	the	the	DET
cana-834	29	28	ever	ever	ADV
cana-834	29	29	-	-	PUNCT
cana-834	29	30	changing	change	VERB
cana-834	29	31	social	social	ADJ
cana-834	29	32	network	network	NOUN
cana-834	29	33	landscape	landscape	NOUN
cana-834	29	34	.	.	PUNCT
cana-834	30	1	2	2	X
cana-834	30	2	.	.	X
cana-834	30	3	graph	graph	NOUN
cana-834	30	4	theory	theory	NOUN
cana-834	30	5	fundamentals	fundamental	VERB
cana-834	30	6	the	the	DET
cana-834	30	7	study	study	NOUN
cana-834	30	8	of	of	ADP
cana-834	30	9	graphs	graph	NOUN
cana-834	30	10	,	,	PUNCT
cana-834	30	11	mathematical	mathematical	ADJ
cana-834	30	12	structures	structure	NOUN
cana-834	30	13	used	use	VERB
cana-834	30	14	to	to	PART
cana-834	30	15	model	model	VERB
cana-834	30	16	pairwise	pairwise	NOUN
cana-834	30	17	relationships	relationship	NOUN
cana-834	30	18	between	between	ADP
cana-834	30	19	objects	object	NOUN
cana-834	30	20	,	,	PUNCT
cana-834	30	21	is	be	AUX
cana-834	30	22	known	know	VERB
cana-834	30	23	as	as	ADP
cana-834	30	24	graph	graph	NOUN
cana-834	30	25	theory	theory	NOUN
cana-834	30	26	.	.	PUNCT
cana-834	31	1	these	these	DET
cana-834	31	2	objects	object	NOUN
cana-834	31	3	are	be	AUX
cana-834	31	4	typically	typically	ADV
cana-834	31	5	individuals	individual	NOUN
cana-834	31	6	or	or	CCONJ
cana-834	31	7	entities	entity	NOUN
cana-834	31	8	in	in	ADP
cana-834	31	9	the	the	DET
cana-834	31	10	context	context	NOUN
cana-834	31	11	of	of	ADP
cana-834	31	12	social	social	ADJ
cana-834	31	13	network	network	NOUN
cana-834	31	14	analysis	analysis	NOUN
cana-834	31	15	(	(	PUNCT
cana-834	31	16	sna	sna	PROPN
cana-834	31	17	)	)	PUNCT
cana-834	31	18	,	,	PUNCT
cana-834	31	19	and	and	CCONJ
cana-834	31	20	the	the	DET
cana-834	31	21	relations	relation	NOUN
cana-834	31	22	are	be	AUX
cana-834	31	23	the	the	DET
cana-834	31	24	interactions	interaction	NOUN
cana-834	31	25	or	or	CCONJ
cana-834	31	26	connections	connection	NOUN
cana-834	31	27	that	that	PRON
cana-834	31	28	exist	exist	VERB
cana-834	31	29	between	between	ADP
cana-834	31	30	them	they	PRON
cana-834	31	31	[	[	X
cana-834	31	32	8	8	NUM
cana-834	31	33	]	]	PUNCT
cana-834	31	34	.	.	PUNCT
cana-834	32	1	the	the	DET
cana-834	32	2	fundamental	fundamental	ADJ
cana-834	32	3	ideas	idea	NOUN
cana-834	32	4	of	of	ADP
cana-834	32	5	graph	graph	NOUN
cana-834	32	6	theory	theory	NOUN
cana-834	32	7	,	,	PUNCT
cana-834	32	8	such	such	ADJ
cana-834	32	9	as	as	ADP
cana-834	32	10	the	the	DET
cana-834	32	11	fundamental	fundamental	ADJ
cana-834	32	12	metrics	metric	NOUN
cana-834	32	13	and	and	CCONJ
cana-834	32	14	common	common	ADJ
cana-834	32	15	representations	representation	NOUN
cana-834	32	16	that	that	PRON
cana-834	32	17	are	be	AUX
cana-834	32	18	necessary	necessary	ADJ
cana-834	32	19	for	for	ADP
cana-834	32	20	comprehending	comprehend	VERB
cana-834	32	21	and	and	CCONJ
cana-834	32	22	analyzing	analyze	VERB
cana-834	32	23	social	social	ADJ
cana-834	32	24	networks	network	NOUN
cana-834	32	25	,	,	PUNCT
cana-834	32	26	are	be	AUX
cana-834	32	27	discussed	discuss	VERB
cana-834	32	28	in	in	ADP
cana-834	32	29	this	this	DET
cana-834	32	30	section	section	NOUN
cana-834	32	31	.	.	PUNCT
cana-834	33	1	a	a	DET
cana-834	33	2	graph	graph	NOUN
cana-834	33	3	g	g	NOUN
cana-834	33	4	is	be	AUX
cana-834	33	5	made	make	VERB
cana-834	33	6	up	up	ADP
cana-834	33	7	of	of	ADP
cana-834	33	8	a	a	DET
cana-834	33	9	set	set	NOUN
cana-834	33	10	of	of	ADP
cana-834	33	11	edges	edge	NOUN
cana-834	33	12	e	e	NOUN
cana-834	33	13	and	and	CCONJ
cana-834	33	14	a	a	DET
cana-834	33	15	set	set	NOUN
cana-834	33	16	of	of	ADP
cana-834	33	17	nodes	node	NOUN
cana-834	33	18	v	v	ADP
cana-834	33	19	,	,	PUNCT
cana-834	33	20	which	which	PRON
cana-834	33	21	are	be	AUX
cana-834	33	22	also	also	ADV
cana-834	33	23	known	know	VERB
cana-834	33	24	as	as	ADP
cana-834	33	25	vertices	vertex	NOUN
cana-834	33	26	in	in	ADP
cana-834	33	27	graph	graph	NOUN
cana-834	33	28	theory	theory	NOUN
cana-834	33	29	.	.	PUNCT
cana-834	34	1	a	a	DET
cana-834	34	2	pair	pair	NOUN
cana-834	34	3	of	of	ADP
cana-834	34	4	nodes	node	NOUN
cana-834	34	5	in	in	ADP
cana-834	34	6	v	v	NUM
cana-834	34	7	are	be	AUX
cana-834	34	8	connected	connect	VERB
cana-834	34	9	by	by	ADP
cana-834	34	10	each	each	DET
cana-834	34	11	edge	edge	NOUN
cana-834	34	12	e	e	NOUN
cana-834	34	13	in	in	ADP
cana-834	34	14	e.	e.	PROPN
cana-834	34	15	g	g	PROPN
cana-834	34	16	=	=	PUNCT
cana-834	34	17	(	(	PUNCT
cana-834	34	18	v	v	NOUN
cana-834	34	19	,	,	PUNCT
cana-834	34	20	e	e	NOUN
cana-834	34	21	)	)	PUNCT
cana-834	34	22	is	be	AUX
cana-834	34	23	the	the	DET
cana-834	34	24	formal	formal	ADJ
cana-834	34	25	representation	representation	NOUN
cana-834	34	26	of	of	ADP
cana-834	34	27	a	a	DET
cana-834	34	28	graph	graph	NOUN
cana-834	34	29	.	.	PUNCT
cana-834	35	1	-nodes	-node	NOUN
cana-834	35	2	(	(	PUNCT
cana-834	35	3	vertices	vertex	NOUN
cana-834	35	4	):	):	PUNCT
cana-834	35	5	these	these	PRON
cana-834	35	6	represent	represent	VERB
cana-834	35	7	the	the	DET
cana-834	35	8	network	network	NOUN
cana-834	35	9	's	's	PART
cana-834	35	10	constituent	constituent	NOUN
cana-834	35	11	parts	part	NOUN
cana-834	35	12	.	.	PUNCT
cana-834	36	1	nodes	node	NOUN
cana-834	36	2	typically	typically	ADV
cana-834	36	3	represent	represent	VERB
cana-834	36	4	individuals	individual	NOUN
cana-834	36	5	in	in	ADP
cana-834	36	6	a	a	DET
cana-834	36	7	social	social	ADJ
cana-834	36	8	network	network	NOUN
cana-834	36	9	.	.	PUNCT
cana-834	37	1	edges	edge	NOUN
cana-834	37	2	:	:	PUNCT
cana-834	37	3	depict	depict	VERB
cana-834	37	4	the	the	DET
cana-834	37	5	relationships	relationship	NOUN
cana-834	37	6	or	or	CCONJ
cana-834	37	7	connections	connection	NOUN
cana-834	37	8	that	that	PRON
cana-834	37	9	exist	exist	VERB
cana-834	37	10	between	between	ADP
cana-834	37	11	nodes	node	NOUN
cana-834	37	12	.	.	PUNCT
cana-834	38	1	edges	edge	NOUN
cana-834	38	2	can	can	AUX
cana-834	38	3	represent	represent	VERB
cana-834	38	4	friendships	friendship	NOUN
cana-834	38	5	,	,	PUNCT
cana-834	38	6	communications	communication	NOUN
cana-834	38	7	,	,	PUNCT
cana-834	38	8	or	or	CCONJ
cana-834	38	9	other	other	ADJ
cana-834	38	10	interactions	interaction	NOUN
cana-834	38	11	in	in	ADP
cana-834	38	12	a	a	DET
cana-834	38	13	social	social	ADJ
cana-834	38	14	network	network	NOUN
cana-834	38	15	.	.	PUNCT
cana-834	39	1	directed	direct	VERB
cana-834	39	2	graph	graph	NOUN
cana-834	39	3	:	:	PUNCT
cana-834	39	4	edges	edge	NOUN
cana-834	39	5	in	in	ADP
cana-834	39	6	this	this	DET
cana-834	39	7	graph	graph	NOUN
cana-834	39	8	have	have	AUX
cana-834	39	9	a	a	DET
cana-834	39	10	direction	direction	NOUN
cana-834	39	11	,	,	PUNCT
cana-834	39	12	indicating	indicate	VERB
cana-834	39	13	a	a	DET
cana-834	39	14	one	one	NUM
cana-834	39	15	-	-	PUNCT
cana-834	39	16	way	way	NOUN
cana-834	39	17	relationship	relationship	NOUN
cana-834	39	18	(	(	PUNCT
cana-834	39	19	such	such	ADJ
cana-834	39	20	as	as	ADP
cana-834	39	21	a	a	DET
cana-834	39	22	relationship	relationship	NOUN
cana-834	39	23	between	between	ADP
cana-834	39	24	twitter	twitter	NOUN
cana-834	39	25	followers	follower	NOUN
cana-834	39	26	)	)	PUNCT
cana-834	39	27	.	.	PUNCT
cana-834	40	1	undirected	undirected	ADJ
cana-834	40	2	graph	graph	NOUN
cana-834	40	3	:	:	PUNCT
cana-834	40	4	edges	edge	NOUN
cana-834	40	5	in	in	ADP
cana-834	40	6	this	this	DET
cana-834	40	7	graph	graph	NOUN
cana-834	40	8	do	do	AUX
cana-834	40	9	not	not	PART
cana-834	40	10	have	have	AUX
cana-834	40	11	a	a	DET
cana-834	40	12	direction	direction	NOUN
cana-834	40	13	,	,	PUNCT
cana-834	40	14	indicating	indicate	VERB
cana-834	40	15	a	a	DET
cana-834	40	16	relationship	relationship	NOUN
cana-834	40	17	(	(	PUNCT
cana-834	40	18	like	like	ADP
cana-834	40	19	facebook	facebook	NOUN
cana-834	40	20	friendships	friendship	NOUN
cana-834	40	21	,	,	PUNCT
cana-834	40	22	for	for	ADP
cana-834	40	23	instance	instance	NOUN
cana-834	40	24	)	)	PUNCT
cana-834	40	25	.	.	PUNCT
cana-834	41	1	degree	degree	NOUN
cana-834	41	2	centrality	centrality	NOUN
cana-834	41	3	is	be	AUX
cana-834	41	4	a	a	DET
cana-834	41	5	proportion	proportion	NOUN
cana-834	41	6	of	of	ADP
cana-834	41	7	the	the	DET
cana-834	41	8	quantity	quantity	NOUN
cana-834	41	9	of	of	ADP
cana-834	41	10	direct	direct	ADJ
cana-834	41	11	associations	association	NOUN
cana-834	41	12	a	a	DET
cana-834	41	13	hub	hub	NOUN
cana-834	41	14	has	have	VERB
cana-834	41	15	.	.	PUNCT
cana-834	42	1	the	the	DET
cana-834	42	2	most	most	ADV
cana-834	42	3	influential	influential	ADJ
cana-834	42	4	or	or	CCONJ
cana-834	42	5	connected	connected	ADJ
cana-834	42	6	nodes	node	NOUN
cana-834	42	7	in	in	ADP
cana-834	42	8	a	a	DET
cana-834	42	9	network	network	NOUN
cana-834	42	10	can	can	AUX
cana-834	42	11	be	be	AUX
cana-834	42	12	identified	identify	VERB
cana-834	42	13	using	use	VERB
cana-834	42	14	this	this	DET
cana-834	42	15	straightforward	straightforward	ADJ
cana-834	42	16	but	but	CCONJ
cana-834	42	17	effective	effective	ADJ
cana-834	42	18	metric	metric	ADJ
cana-834	43	1	[	[	X
cana-834	43	2	9	9	NUM
cana-834	43	3	]	]	PUNCT
cana-834	43	4	.	.	PUNCT
cana-834	44	1	level	level	NOUN
cana-834	44	2	of	of	ADP
cana-834	44	3	a	a	DET
cana-834	44	4	hub	hub	NOUN
cana-834	44	5	v	v	ADP
cana-834	44	6	:	:	PUNCT
cana-834	44	7	the	the	DET
cana-834	44	8	quantity	quantity	NOUN
cana-834	44	9	of	of	ADP
cana-834	44	10	edges	edge	NOUN
cana-834	44	11	episode	episode	NOUN
cana-834	44	12	to	to	ADP
cana-834	44	13	v.	v.	ADP
cana-834	44	14	in	in	ADP
cana-834	44	15	-	-	PUNCT
cana-834	44	16	degree	degree	NOUN
cana-834	44	17	:	:	PUNCT
cana-834	44	18	the	the	DET
cana-834	44	19	quantity	quantity	NOUN
cana-834	44	20	of	of	ADP
cana-834	44	21	edges	edge	NOUN
cana-834	44	22	that	that	PRON
cana-834	44	23	enter	enter	VERB
cana-834	44	24	a	a	DET
cana-834	44	25	node	node	NOUN
cana-834	44	26	(	(	PUNCT
cana-834	44	27	important	important	ADJ
cana-834	44	28	for	for	ADP
cana-834	44	29	directed	direct	VERB
cana-834	44	30	graphs	graph	NOUN
cana-834	44	31	)	)	PUNCT
cana-834	44	32	.	.	PUNCT
cana-834	45	1	communications	communication	NOUN
cana-834	45	2	on	on	ADP
cana-834	45	3	applied	apply	VERB
cana-834	45	4	nonlinear	nonlinear	ADJ
cana-834	45	5	analysis	analysis	NOUN
cana-834	45	6	issn	issn	NOUN
cana-834	45	7	:	:	PUNCT
cana-834	45	8	1074	1074	NUM
cana-834	45	9	-	-	PUNCT
cana-834	45	10	133x	133x	NUM
cana-834	45	11	vol	vol	NOUN
cana-834	45	12	31	31	NUM
cana-834	45	13	no	no	NOUN
cana-834	45	14	.	.	PUNCT
cana-834	46	1	4s	4s	NUM
cana-834	46	2	(	(	PUNCT
cana-834	46	3	2024	2024	NUM
cana-834	46	4	)	)	PUNCT
cana-834	46	5	166	166	NUM
cana-834	46	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	46	7	out	out	ADJ
cana-834	46	8	-	-	PUNCT
cana-834	46	9	degree	degree	NOUN
cana-834	46	10	:	:	PUNCT
cana-834	46	11	the	the	DET
cana-834	46	12	number	number	NOUN
cana-834	46	13	of	of	ADP
cana-834	46	14	node	node	NOUN
cana-834	46	15	-	-	PUNCT
cana-834	46	16	to	to	ADP
cana-834	46	17	-	-	PUNCT
cana-834	46	18	edge	edge	NOUN
cana-834	46	19	outgoing	outgoing	ADJ
cana-834	46	20	edges	edge	NOUN
cana-834	46	21	(	(	PUNCT
cana-834	46	22	important	important	ADJ
cana-834	46	23	for	for	ADP
cana-834	46	24	directed	direct	VERB
cana-834	46	25	graphs	graph	NOUN
cana-834	46	26	)	)	PUNCT
cana-834	46	27	.	.	PUNCT
cana-834	47	1	for	for	ADP
cana-834	47	2	a	a	DET
cana-834	47	3	node	node	NOUN
cana-834	47	4	v	v	NOUN
cana-834	47	5	in	in	ADP
cana-834	47	6	formal	formal	ADJ
cana-834	47	7	terms	term	NOUN
cana-834	47	8	:	:	PUNCT
cana-834	47	9	degree	degree	NOUN
cana-834	47	10	centrality(v	centrality(v	NOUN
cana-834	47	11	)	)	PUNCT
cana-834	47	12	=	=	SYM
cana-834	47	13	deg(v	deg(v	X
cana-834	47	14	)	)	PUNCT
cana-834	47	15	centrality	centrality	NOUN
cana-834	47	16	and	and	CCONJ
cana-834	47	17	betweenness	betweenness	VERB
cana-834	47	18	the	the	DET
cana-834	47	19	degree	degree	NOUN
cana-834	47	20	to	to	PART
cana-834	47	21	which	which	PRON
cana-834	47	22	a	a	DET
cana-834	47	23	node	node	NOUN
cana-834	47	24	is	be	AUX
cana-834	47	25	located	locate	VERB
cana-834	47	26	on	on	ADP
cana-834	47	27	the	the	DET
cana-834	47	28	shortest	short	ADJ
cana-834	47	29	paths	path	NOUN
cana-834	47	30	between	between	ADP
cana-834	47	31	other	other	ADJ
cana-834	47	32	nodes	node	NOUN
cana-834	47	33	is	be	AUX
cana-834	47	34	measured	measure	VERB
cana-834	47	35	by	by	ADP
cana-834	47	36	betweenness	betweenness	ADJ
cana-834	47	37	centrality	centrality	NOUN
cana-834	47	38	.	.	PUNCT
cana-834	48	1	it	it	PRON
cana-834	48	2	gives	give	VERB
cana-834	48	3	a	a	DET
cana-834	48	4	number	number	NOUN
cana-834	48	5	to	to	ADP
cana-834	48	6	how	how	SCONJ
cana-834	48	7	important	important	ADJ
cana-834	48	8	a	a	DET
cana-834	48	9	node	node	NOUN
cana-834	48	10	is	be	AUX
cana-834	48	11	in	in	ADP
cana-834	48	12	terms	term	NOUN
cana-834	48	13	of	of	ADP
cana-834	48	14	how	how	SCONJ
cana-834	48	15	it	it	PRON
cana-834	48	16	controls	control	VERB
cana-834	48	17	how	how	SCONJ
cana-834	48	18	information	information	NOUN
cana-834	48	19	moves	move	VERB
cana-834	48	20	through	through	ADP
cana-834	48	21	the	the	DET
cana-834	48	22	network	network	NOUN
cana-834	48	23	.	.	PUNCT
cana-834	49	1	a	a	DET
cana-834	49	2	general	general	ADJ
cana-834	49	3	idea	idea	NOUN
cana-834	49	4	of	of	ADP
cana-834	49	5	how	how	SCONJ
cana-834	49	6	clustered	clustered	ADJ
cana-834	49	7	the	the	DET
cana-834	49	8	network	network	NOUN
cana-834	49	9	is	be	AUX
cana-834	49	10	is	be	AUX
cana-834	49	11	provided	provide	VERB
cana-834	49	12	by	by	ADP
cana-834	49	13	the	the	DET
cana-834	49	14	average	average	ADJ
cana-834	49	15	clustering	clustering	ADJ
cana-834	49	16	coefficient	coefficient	NOUN
cana-834	49	17	of	of	ADP
cana-834	49	18	the	the	DET
cana-834	49	19	graph	graph	NOUN
cana-834	49	20	.	.	PUNCT
cana-834	50	1	effective	effective	ADJ
cana-834	50	2	portrayal	portrayal	NOUN
cana-834	50	3	of	of	ADP
cana-834	50	4	charts	chart	NOUN
cana-834	50	5	is	be	AUX
cana-834	50	6	basic	basic	ADJ
cana-834	50	7	for	for	ADP
cana-834	50	8	upgrading	upgrade	VERB
cana-834	50	9	diagram	diagram	NOUN
cana-834	50	10	calculations	calculation	NOUN
cana-834	50	11	.	.	PUNCT
cana-834	51	1	adjacency	adjacency	NOUN
cana-834	51	2	matrices	matrix	NOUN
cana-834	51	3	and	and	CCONJ
cana-834	51	4	adjacency	adjacency	NOUN
cana-834	51	5	lists	list	NOUN
cana-834	51	6	are	be	AUX
cana-834	51	7	the	the	DET
cana-834	51	8	two	two	NUM
cana-834	51	9	most	most	ADV
cana-834	51	10	prevalent	prevalent	ADJ
cana-834	51	11	representations	representation	NOUN
cana-834	51	12	.	.	PUNCT
cana-834	52	1	simple	simple	ADJ
cana-834	52	2	adjacency	adjacency	NOUN
cana-834	52	3	matrices	matrix	NOUN
cana-834	52	4	can	can	AUX
cana-834	52	5	be	be	AUX
cana-834	52	6	space	space	NOUN
cana-834	52	7	-	-	PUNCT
cana-834	52	8	inefficient	inefficient	ADJ
cana-834	52	9	for	for	ADP
cana-834	52	10	large	large	ADJ
cana-834	52	11	,	,	PUNCT
cana-834	52	12	sparse	sparse	ADJ
cana-834	52	13	graphs	graph	NOUN
cana-834	52	14	,	,	PUNCT
cana-834	52	15	though	though	ADV
cana-834	52	16	.	.	PUNCT
cana-834	53	1	a	a	DET
cana-834	53	2	graph	graph	NOUN
cana-834	53	3	is	be	AUX
cana-834	53	4	represented	represent	VERB
cana-834	53	5	by	by	ADP
cana-834	53	6	an	an	DET
cana-834	53	7	array	array	NOUN
cana-834	53	8	of	of	ADP
cana-834	53	9	lists	list	NOUN
cana-834	53	10	in	in	ADP
cana-834	53	11	an	an	DET
cana-834	53	12	adjacency	adjacency	NOUN
cana-834	53	13	list	list	NOUN
cana-834	53	14	.	.	PUNCT
cana-834	54	1	a	a	DET
cana-834	54	2	list	list	NOUN
cana-834	54	3	of	of	ADP
cana-834	54	4	adjacent	adjacent	ADJ
cana-834	54	5	nodes	node	NOUN
cana-834	54	6	to	to	ADP
cana-834	54	7	each	each	DET
cana-834	54	8	node	node	NOUN
cana-834	54	9	is	be	AUX
cana-834	54	10	included	include	VERB
cana-834	54	11	in	in	ADP
cana-834	54	12	each	each	DET
cana-834	54	13	list	list	NOUN
cana-834	54	14	,	,	PUNCT
cana-834	54	15	which	which	PRON
cana-834	54	16	corresponds	correspond	VERB
cana-834	54	17	to	to	ADP
cana-834	54	18	a	a	DET
cana-834	54	19	node	node	NOUN
cana-834	54	20	.	.	PUNCT
cana-834	55	1	for	for	ADP
cana-834	55	2	sparse	sparse	ADJ
cana-834	55	3	graphs	graph	NOUN
cana-834	55	4	,	,	PUNCT
cana-834	55	5	this	this	DET
cana-834	55	6	representation	representation	NOUN
cana-834	55	7	uses	use	VERB
cana-834	55	8	less	less	ADJ
cana-834	55	9	space	space	NOUN
cana-834	55	10	.	.	PUNCT
cana-834	56	1	for	for	ADP
cana-834	56	2	instance	instance	NOUN
cana-834	56	3	,	,	PUNCT
cana-834	56	4	suppose	suppose	VERB
cana-834	56	5	a	a	DET
cana-834	56	6	graph	graph	NOUN
cana-834	56	7	has	have	VERB
cana-834	56	8	nodes	node	NOUN
cana-834	56	9	v	v	ADP
cana-834	56	10	equal	equal	ADJ
cana-834	56	11	to	to	ADP
cana-834	56	12	1	1	NUM
cana-834	56	13	,	,	PUNCT
cana-834	56	14	2	2	NUM
cana-834	56	15	,	,	PUNCT
cana-834	56	16	3	3	NUM
cana-834	56	17	,	,	PUNCT
cana-834	56	18	and	and	CCONJ
cana-834	56	19	edges	edge	NOUN
cana-834	56	20	e	e	NOUN
cana-834	56	21	equal	equal	ADJ
cana-834	56	22	to	to	ADP
cana-834	56	23	(	(	PUNCT
cana-834	56	24	1	1	NUM
cana-834	56	25	,	,	PUNCT
cana-834	56	26	2	2	NUM
cana-834	56	27	)	)	PUNCT
cana-834	56	28	,	,	PUNCT
cana-834	56	29	(	(	PUNCT
cana-834	56	30	1	1	NUM
cana-834	56	31	,	,	PUNCT
cana-834	56	32	3	3	NUM
cana-834	56	33	):	):	PUNCT
cana-834	56	34	node	node	NOUN
cana-834	56	35	1	1	NUM
cana-834	56	36	:	:	PUNCT
cana-834	56	37	[	[	X
cana-834	56	38	2	2	NUM
cana-834	56	39	,	,	PUNCT
cana-834	56	40	3	3	NUM
cana-834	56	41	]	]	X
cana-834	56	42	node	node	NOUN
cana-834	56	43	2	2	NUM
cana-834	56	44	:	:	PUNCT
cana-834	56	45	[	[	X
cana-834	56	46	1	1	X
cana-834	56	47	]	]	PUNCT
cana-834	56	48	node	node	NOUN
cana-834	56	49	3	3	NUM
cana-834	56	50	:	:	PUNCT
cana-834	56	51	[	[	X
cana-834	56	52	1	1	X
cana-834	56	53	]	]	PUNCT
cana-834	56	54	the	the	DET
cana-834	56	55	foundation	foundation	NOUN
cana-834	56	56	for	for	ADP
cana-834	56	57	analyzing	analyze	VERB
cana-834	56	58	and	and	CCONJ
cana-834	56	59	optimizing	optimize	VERB
cana-834	56	60	social	social	ADJ
cana-834	56	61	networks	network	NOUN
cana-834	56	62	is	be	AUX
cana-834	56	63	an	an	DET
cana-834	56	64	understanding	understanding	NOUN
cana-834	56	65	of	of	ADP
cana-834	56	66	the	the	DET
cana-834	56	67	fundamental	fundamental	ADJ
cana-834	56	68	concepts	concept	NOUN
cana-834	56	69	and	and	CCONJ
cana-834	56	70	graph	graph	VERB
cana-834	56	71	representations	representation	NOUN
cana-834	56	72	[	[	X
cana-834	56	73	10	10	NUM
cana-834	56	74	]	]	PUNCT
cana-834	56	75	.	.	PUNCT
cana-834	57	1	measurements	measurement	NOUN
cana-834	57	2	like	like	ADP
cana-834	57	3	degree	degree	NOUN
cana-834	57	4	centrality	centrality	NOUN
cana-834	57	5	,	,	PUNCT
cana-834	57	6	betweenness	betweenness	NOUN
cana-834	57	7	centrality	centrality	NOUN
cana-834	57	8	,	,	PUNCT
cana-834	57	9	and	and	CCONJ
cana-834	57	10	grouping	group	VERB
cana-834	57	11	coefficient	coefficient	NOUN
cana-834	57	12	give	give	VERB
cana-834	57	13	experiences	experience	NOUN
cana-834	57	14	into	into	ADP
cana-834	57	15	the	the	DET
cana-834	57	16	design	design	NOUN
cana-834	57	17	and	and	CCONJ
cana-834	57	18	elements	element	NOUN
cana-834	57	19	of	of	ADP
cana-834	57	20	organizations	organization	NOUN
cana-834	57	21	.	.	PUNCT
cana-834	58	1	adjacency	adjacency	PROPN
cana-834	58	2	matrices	matrix	NOUN
cana-834	58	3	and	and	CCONJ
cana-834	58	4	adjacency	adjacency	NOUN
cana-834	58	5	lists	list	NOUN
cana-834	58	6	are	be	AUX
cana-834	58	7	effective	effective	ADJ
cana-834	58	8	graph	graph	NOUN
cana-834	58	9	representations	representation	NOUN
cana-834	58	10	that	that	PRON
cana-834	58	11	are	be	AUX
cana-834	58	12	necessary	necessary	ADJ
cana-834	58	13	for	for	ADP
cana-834	58	14	the	the	DET
cana-834	58	15	implementation	implementation	NOUN
cana-834	58	16	and	and	CCONJ
cana-834	58	17	optimization	optimization	NOUN
cana-834	58	18	of	of	ADP
cana-834	58	19	graph	graph	NOUN
cana-834	58	20	algorithms	algorithm	NOUN
cana-834	58	21	,	,	PUNCT
cana-834	58	22	particularly	particularly	ADV
cana-834	58	23	when	when	SCONJ
cana-834	58	24	dealing	deal	VERB
cana-834	58	25	with	with	ADP
cana-834	58	26	large	large	ADJ
cana-834	58	27	-	-	PUNCT
cana-834	58	28	scale	scale	NOUN
cana-834	58	29	social	social	ADJ
cana-834	58	30	networks	network	NOUN
cana-834	58	31	.	.	PUNCT
cana-834	59	1	we	we	PRON
cana-834	59	2	will	will	AUX
cana-834	59	3	discuss	discuss	VERB
cana-834	59	4	the	the	DET
cana-834	59	5	specific	specific	ADJ
cana-834	59	6	difficulties	difficulty	NOUN
cana-834	59	7	of	of	ADP
cana-834	59	8	social	social	ADJ
cana-834	59	9	network	network	NOUN
cana-834	59	10	analysis	analysis	NOUN
cana-834	59	11	and	and	CCONJ
cana-834	59	12	the	the	DET
cana-834	59	13	optimization	optimization	NOUN
cana-834	59	14	methods	method	NOUN
cana-834	59	15	used	use	VERB
cana-834	59	16	to	to	PART
cana-834	59	17	solve	solve	VERB
cana-834	59	18	them	they	PRON
cana-834	59	19	in	in	ADP
cana-834	59	20	the	the	DET
cana-834	59	21	following	follow	VERB
cana-834	59	22	sections	section	NOUN
cana-834	59	23	.	.	PUNCT
cana-834	60	1	main	main	ADJ
cana-834	60	2	results	result	NOUN
cana-834	60	3	:	:	PUNCT
cana-834	60	4	theorem	theorem	VERB
cana-834	60	5	2.1	2.1	NUM
cana-834	60	6	.	.	PUNCT
cana-834	61	1	let	let	VERB
cana-834	61	2	1	1	NUM
cana-834	61	3	g	g	NOUN
cana-834	61	4	and	and	CCONJ
cana-834	61	5	2	2	NUM
cana-834	61	6	g	g	NOUN
cana-834	61	7	be	be	VERB
cana-834	61	8	sna	sna	PROPN
cana-834	61	9	graphs	graph	VERB
cana-834	61	10	then	then	ADV
cana-834	61	11	(	(	PUNCT
cana-834	61	12	)	)	PUNCT
cana-834	61	13	21	21	NUM
cana-834	61	14	ggvd	ggvd	NOUN
cana-834	61	15			PUNCT
cana-834	61	16	is	be	AUX
cana-834	61	17	a	a	DET
cana-834	61	18	sna	sna	PROPN
cana-834	61	19	-	-	PUNCT
cana-834	61	20	set	set	NOUN
cana-834	61	21	in	in	ADP
cana-834	61	22	21	21	NUM
cana-834	61	23	gg	gg	NOUN
cana-834	61	24			PROPN
cana-834	61	25	if	if	SCONJ
cana-834	62	1	and	and	CCONJ
cana-834	62	2	only	only	ADV
cana-834	62	3	if	if	SCONJ
cana-834	62	4	one	one	NUM
cana-834	62	5	of	of	ADP
cana-834	62	6	the	the	DET
cana-834	62	7	following	follow	VERB
cana-834	62	8	conditions	condition	NOUN
cana-834	62	9	holds	hold	VERB
cana-834	62	10	.	.	PUNCT
cana-834	63	1	(	(	PUNCT
cana-834	63	2	i	i	NOUN
cana-834	63	3	)	)	PUNCT
cana-834	63	4	for	for	ADP
cana-834	63	5	each	each	PRON
cana-834	63	6	(	(	PUNCT
cana-834	63	7	)	)	PUNCT
cana-834	63	8	(	(	PUNCT
cana-834	63	9	)	)	PUNCT
cana-834	63	10	dgvgvv	dgvgvv	NOUN
cana-834	63	11	v	v	PART
cana-834	63	12			PROPN
cana-834	63	13	21	21	NUM
cana-834	63	14	,	,	PUNCT
cana-834	63	15	is	be	AUX
cana-834	63	16	a	a	DET
cana-834	63	17	dominating	dominating	NOUN
cana-834	63	18	in	in	ADP
cana-834	63	19	vg2	vg2	PROPN
cana-834	63	20	and	and	CCONJ
cana-834	63	21	(	(	PUNCT
cana-834	63	22	)	)	PUNCT
cana-834	63	23	(	(	PUNCT
cana-834	63	24	)	)	PUNCT
cana-834	63	25			PROPN
cana-834	63	26	gvu	gvu	PROPN
cana-834	63	27	ugvd	ugvd	ADJ
cana-834	63	28			PROPN
cana-834	63	29			PROPN
cana-834	63	30	2	2	NUM
cana-834	63	31	.	.	PUNCT
cana-834	64	1	(	(	PUNCT
cana-834	64	2	ii	ii	NOUN
cana-834	64	3	)	)	PUNCT
cana-834	64	4	(	(	PUNCT
cana-834	64	5	)	)	PUNCT
cana-834	64	6	dgv	dgv	PROPN
cana-834	64	7	1	1	PROPN
cana-834	64	8	is	be	AUX
cana-834	64	9	a	a	DET
cana-834	64	10	complementary	complementary	ADJ
cana-834	64	11	tree	tree	NOUN
cana-834	64	12	dominating	dominating	NOUN
cana-834	64	13	in	in	ADP
cana-834	64	14	1	1	NUM
cana-834	64	15	g	g	NOUN
cana-834	64	16	and	and	CCONJ
cana-834	64	17	(	(	PUNCT
cana-834	64	18	)	)	PUNCT
cana-834	64	19	dgv	dgv	PROPN
cana-834	64	20	v	v	PROPN
cana-834	64	21	2	2	NOUN
cana-834	64	22	whenever	whenever	SCONJ
cana-834	64	23	(	(	PUNCT
cana-834	64	24	)	)	PUNCT
cana-834	64	25	dgvv	dgvv	ADV
cana-834	64	26			ADP
cana-834	64	27	1	1	NUM
cana-834	64	28	and	and	CCONJ
cana-834	64	29	(	(	PUNCT
cana-834	64	30	)	)	PUNCT
cana-834	64	31	dgv	dgv	PROPN
cana-834	64	32	v	v	NUM
cana-834	64	33	2	2	NOUN
cana-834	64	34	is	be	AUX
cana-834	64	35	dominating	dominate	VERB
cana-834	64	36	in	in	ADP
cana-834	64	37	vg2	vg2	NOUN
cana-834	64	38	whenever	whenever	SCONJ
cana-834	64	39	(	(	PUNCT
cana-834	64	40	)	)	PUNCT
cana-834	64	41	.1	.1	NUM
cana-834	65	1	dgvv	dgvv	VERB
cana-834	65	2	−	−	PROPN
cana-834	65	3	proof	proof	NOUN
cana-834	65	4	.	.	PUNCT
cana-834	66	1	suppose	suppose	VERB
cana-834	66	2	(	(	PUNCT
cana-834	66	3	)	)	PUNCT
cana-834	66	4	.1	.1	NUM
cana-834	66	5	=	=	PROPN
cana-834	67	1	dgv	dgv	PROPN
cana-834	67	2	let	let	VERB
cana-834	67	3	(	(	PUNCT
cana-834	67	4	)	)	PUNCT
cana-834	67	5	1gvv	1gvv	NUM
cana-834	67	6	and	and	CCONJ
cana-834	67	7	(	(	PUNCT
cana-834	67	8	)	)	PUNCT
cana-834	67	9	.2	.2	NUM
cana-834	67	10	dgvx	dgvx	PROPN
cana-834	67	11	v	v	NOUN
cana-834	67	12	−	−	NOUN
cana-834	67	13	hence	hence	ADV
cana-834	67	14	(	(	PUNCT
cana-834	67	15	)	)	PUNCT
cana-834	67	16	.21	.21	NUM
cana-834	67	17	dggvx	dggvx	NOUN
cana-834	67	18	−	−	PROPN
cana-834	68	1			PROPN
cana-834	68	2	since	since	SCONJ
cana-834	68	3	d	d	PROPN
cana-834	68	4	is	be	AUX
cana-834	68	5	a	a	DET
cana-834	68	6	ctd	ctd	NOUN
cana-834	68	7	-	-	PUNCT
cana-834	68	8	set	set	NOUN
cana-834	68	9	of	of	ADP
cana-834	68	10	21	21	NUM
cana-834	68	11	gg	gg	NOUN
cana-834	68	12			PROPN
cana-834	68	13	there	there	PRON
cana-834	68	14	exists	exist	VERB
cana-834	68	15	dy	dy	NOUN
cana-834	68	16			NOUN
cana-834	68	17	such	such	ADJ
cana-834	68	18	that	that	SCONJ
cana-834	68	19	(	(	PUNCT
cana-834	68	20	)	)	PUNCT
cana-834	68	21	.1	.1	PROPN
cana-834	68	22	,	,	PUNCT
cana-834	68	23	21	21	NUM
cana-834	68	24	=	=	NUM
cana-834	68	25	yxd	yxd	NOUN
cana-834	68	26	gg	gg	NOUN
cana-834	68	27			PROPN
cana-834	68	28	since	since	SCONJ
cana-834	68	29	(	(	PUNCT
cana-834	68	30	)	)	PUNCT
cana-834	68	31	vgvx	vgvx	NOUN
cana-834	68	32	2	2	NUM
cana-834	68	33	and	and	CCONJ
cana-834	68	34	(	(	PUNCT
cana-834	68	35	)	)	PUNCT
cana-834	68	36	1	1	NUM
cana-834	68	37	,	,	PUNCT
cana-834	68	38	21	21	NUM
cana-834	68	39	=	=	NUM
cana-834	68	40	yxd	yxd	NOUN
cana-834	68	41	gg	gg	NOUN
cana-834	68	42			PROPN
cana-834	68	43	either	either	CCONJ
cana-834	68	44	(	(	PUNCT
cana-834	68	45	)	)	PUNCT
cana-834	68	46	vgvy	vgvy	NOUN
cana-834	68	47	2	2	NUM
cana-834	68	48	or	or	CCONJ
cana-834	68	49	vy	vy	X
cana-834	68	50	=	=	SYM
cana-834	68	51	.if	.if	PROPN
cana-834	69	1	vy	vy	NOUN
cana-834	69	2	=	=	PUNCT
cana-834	69	3	then	then	ADV
cana-834	69	4	(	(	PUNCT
cana-834	69	5	)	)	PUNCT
cana-834	69	6	,	,	PUNCT
cana-834	69	7	1	1	NUM
cana-834	69	8	dgvy	dgvy	NOUN
cana-834	69	9			ADP
cana-834	69	10	a	a	DET
cana-834	69	11	contradiction	contradiction	NOUN
cana-834	69	12	.	.	PUNCT
cana-834	70	1	therefore	therefore	ADV
cana-834	70	2	(	(	PUNCT
cana-834	70	3	)	)	PUNCT
cana-834	70	4	dgvy	dgvy	NOUN
cana-834	70	5	v	v	ADP
cana-834	70	6			PROPN
cana-834	70	7	2	2	NUM
cana-834	70	8	is	be	AUX
cana-834	70	9	a	a	DET
cana-834	70	10	dominating	dominating	NOUN
cana-834	70	11	set	set	NOUN
cana-834	70	12	of	of	ADP
cana-834	70	13	vg2	vg2	PROPN
cana-834	70	14	since	since	SCONJ
cana-834	70	15	(	(	PUNCT
cana-834	70	16	)	)	PUNCT
cana-834	70	17	,	,	PUNCT
cana-834	70	18	1	1	NUM
cana-834	70	19	=	=	NOUN
cana-834	70	20	dgv	dgv	PROPN
cana-834	70	21	(	(	PUNCT
cana-834	70	22	)	)	PUNCT
cana-834	70	23	(	(	PUNCT
cana-834	70	24	)	)	PUNCT
cana-834	71	1			VERB
cana-834	71	2	1	1	NUM
cana-834	71	3	2	2	NUM
cana-834	71	4	gvu	gvu	ADJ
cana-834	71	5	ugvd	ugvd	ADJ
cana-834	71	6			PROPN
cana-834	71	7			PROPN
cana-834	71	8	.	.	PUNCT
cana-834	72	1	hence	hence	ADV
cana-834	72	2	(	(	PUNCT
cana-834	72	3	i	i	NOUN
cana-834	72	4	)	)	PUNCT
cana-834	72	5	holds	hold	VERB
cana-834	72	6	.	.	PUNCT
cana-834	73	1	suppose	suppose	VERB
cana-834	73	2	(	(	PUNCT
cana-834	73	3	)	)	PUNCT
cana-834	73	4			PROPN
cana-834	73	5	dgv	dgv	PROPN
cana-834	73	6	1	1	NUM
cana-834	73	7	and	and	CCONJ
cana-834	73	8	(	(	PUNCT
cana-834	73	9	)	)	PUNCT
cana-834	73	10	(	(	PUNCT
cana-834	73	11	)	)	PUNCT
cana-834	73	12	11	11	NUM
cana-834	73	13	gvdgv	gvdgv	NOUN
cana-834	73	14			X
cana-834	73	15	.	.	PUNCT
cana-834	74	1	communications	communication	NOUN
cana-834	74	2	on	on	ADP
cana-834	74	3	applied	apply	VERB
cana-834	74	4	nonlinear	nonlinear	ADJ
cana-834	74	5	analysis	analysis	NOUN
cana-834	74	6	issn	issn	NOUN
cana-834	74	7	:	:	PUNCT
cana-834	74	8	1074	1074	NUM
cana-834	74	9	-	-	PUNCT
cana-834	74	10	133x	133x	NUM
cana-834	74	11	vol	vol	NOUN
cana-834	74	12	31	31	NUM
cana-834	74	13	no	no	NOUN
cana-834	74	14	.	.	PUNCT
cana-834	75	1	4s	4s	NUM
cana-834	75	2	(	(	PUNCT
cana-834	75	3	2024	2024	NUM
cana-834	75	4	)	)	PUNCT
cana-834	75	5	167	167	NUM
cana-834	75	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	75	7	let	let	VERB
cana-834	75	8	(	(	PUNCT
cana-834	75	9	)	)	PUNCT
cana-834	75	10	dgvx	dgvx	NOUN
cana-834	75	11	−	−	PROPN
cana-834	76	1	1	1	NUM
cana-834	77	1	it	it	PRON
cana-834	77	2	follows	follow	VERB
cana-834	77	3	(	(	PUNCT
cana-834	77	4	)	)	PUNCT
cana-834	77	5	dggvx	dggvx	NOUN
cana-834	77	6	−	−	PROPN
cana-834	77	7	21	21	NUM
cana-834	77	8			PROPN
cana-834	77	9	.	.	PUNCT
cana-834	78	1	let	let	VERB
cana-834	78	2	d	d	NOUN
cana-834	78	3	is	be	AUX
cana-834	78	4	a	a	DET
cana-834	78	5	dominating	dominating	NOUN
cana-834	78	6	set	set	NOUN
cana-834	78	7	of	of	ADP
cana-834	78	8	21	21	NUM
cana-834	78	9	gg	gg	NOUN
cana-834	78	10			PROPN
cana-834	78	11	there	there	PRON
cana-834	78	12	exists	exist	VERB
cana-834	78	13	dy	dy	NOUN
cana-834	78	14			NOUN
cana-834	78	15	such	such	ADJ
cana-834	78	16	that	that	SCONJ
cana-834	78	17	(	(	PUNCT
cana-834	78	18	)	)	PUNCT
cana-834	78	19	.1	.1	PROPN
cana-834	78	20	,	,	PUNCT
cana-834	78	21	21	21	NUM
cana-834	78	22	=	=	NUM
cana-834	78	23	yxd	yxd	NOUN
cana-834	78	24	gg	gg	NOUN
cana-834	78	25			PROPN
cana-834	78	26	if	if	SCONJ
cana-834	78	27	(	(	PUNCT
cana-834	78	28	)	)	PUNCT
cana-834	78	29	1gvy	1gvy	NUM
cana-834	78	30	then	then	ADV
cana-834	78	31	(	(	PUNCT
cana-834	78	32	)	)	PUNCT
cana-834	78	33	dgvy	dgvy	PROPN
cana-834	78	34			ADP
cana-834	78	35	1	1	NUM
cana-834	78	36	then	then	ADV
cana-834	78	37	(	(	PUNCT
cana-834	78	38	)	)	PUNCT
cana-834	78	39	.1	.1	NUM
cana-834	78	40	,	,	PUNCT
cana-834	78	41	=	=	NOUN
cana-834	78	42	yxdg	yxdg	NOUN
cana-834	78	43	hence	hence	ADV
cana-834	78	44	(	(	PUNCT
cana-834	78	45	)	)	PUNCT
cana-834	78	46	dgv	dgv	PROPN
cana-834	78	47	1	1	PROPN
cana-834	78	48	is	be	AUX
cana-834	78	49	a	a	DET
cana-834	78	50	dominating	dominating	NOUN
cana-834	78	51	set	set	VERB
cana-834	78	52	in	in	ADP
cana-834	78	53	.1	.1	PROPN
cana-834	78	54	g	g	PROPN
cana-834	78	55	now	now	ADV
cana-834	78	56	we	we	PRON
cana-834	78	57	have	have	VERB
cana-834	78	58	to	to	PART
cana-834	78	59	prove	prove	VERB
cana-834	78	60	(	(	PUNCT
cana-834	78	61	)	)	PUNCT
cana-834	78	62	dgv	dgv	PROPN
cana-834	78	63	1	1	PROPN
cana-834	78	64	is	be	AUX
cana-834	78	65	a	a	DET
cana-834	78	66	complementary	complementary	ADJ
cana-834	78	67	tree	tree	NOUN
cana-834	78	68	dominating	dominating	NOUN
cana-834	78	69	set	set	VERB
cana-834	78	70	in	in	ADP
cana-834	78	71	1	1	NUM
cana-834	78	72	g	g	NOUN
cana-834	78	73	it	it	PRON
cana-834	78	74	is	be	AUX
cana-834	78	75	enough	enough	ADJ
cana-834	78	76	to	to	PART
cana-834	78	77	prove	prove	VERB
cana-834	78	78	(	(	PUNCT
cana-834	78	79	)	)	PUNCT
cana-834	78	80	−	−	PROPN
cana-834	78	81	dgv	dgv	PROPN
cana-834	78	82	1	1	NUM
cana-834	78	83	is	be	AUX
cana-834	78	84	a	a	DET
cana-834	78	85	tree	tree	NOUN
cana-834	78	86	.	.	PUNCT
cana-834	79	1	let	let	VERB
cana-834	79	2	(	(	PUNCT
cana-834	79	3	)	)	PUNCT
cana-834	79	4	.	.	PUNCT
cana-834	80	1	,	,	PUNCT
cana-834	80	2	1	1	NUM
cana-834	80	3	dgvyx	dgvyx	NOUN
cana-834	80	4	−	−	X
cana-834	80	5	since	since	SCONJ
cana-834	80	6	−	−	ADJ
cana-834	80	7	dggv	dggv	PROPN
cana-834	80	8	)	)	PUNCT
cana-834	80	9	(	(	PUNCT
cana-834	80	10	21	21	NUM
cana-834	80	11			PROPN
cana-834	80	12	is	be	AUX
cana-834	80	13	a	a	DET
cana-834	80	14	tree	tree	NOUN
cana-834	80	15	t.	t.	NOUN
cana-834	80	16	therefore	therefore	ADV
cana-834	80	17	)	)	PUNCT
cana-834	80	18	(	(	PUNCT
cana-834	80	19	,	,	PUNCT
cana-834	80	20	1gvyx	1gvyx	NUM
cana-834	80	21			NOUN
cana-834	80	22	there	there	ADV
cana-834	80	23	exist	exist	VERB
cana-834	80	24	(	(	PUNCT
cana-834	80	25	)	)	PUNCT
cana-834	80	26	xgvu	xgvu	PROPN
cana-834	80	27	2	2	PROPN
cana-834	80	28	and	and	CCONJ
cana-834	80	29	v	v	X
cana-834	80	30	(	(	PUNCT
cana-834	80	31	)	)	PUNCT
cana-834	80	32	ygv	ygv	PROPN
cana-834	80	33	2	2	NUM
cana-834	80	34	.	.	PUNCT
cana-834	81	1	since	since	SCONJ
cana-834	81	2	t	t	PROPN
cana-834	81	3	is	be	AUX
cana-834	81	4	sna	sna	PROPN
cana-834	81	5	and	and	CCONJ
cana-834	81	6	acyclic	acyclic	ADJ
cana-834	81	7	.	.	PUNCT
cana-834	82	1	there	there	PRON
cana-834	82	2	exists	exist	VERB
cana-834	82	3	a	a	DET
cana-834	82	4	unique	unique	ADJ
cana-834	82	5	path	path	NOUN
cana-834	82	6	between	between	ADP
cana-834	82	7	u	u	NOUN
cana-834	82	8	and	and	CCONJ
cana-834	82	9	v	v	NOUN
cana-834	82	10	.hence	.hence	NOUN
cana-834	82	11	x	x	NOUN
cana-834	82	12	and	and	CCONJ
cana-834	82	13	y	y	PROPN
cana-834	82	14	are	be	AUX
cana-834	82	15	the	the	DET
cana-834	82	16	vertices	vertex	NOUN
cana-834	82	17	transverse	transverse	NOUN
cana-834	82	18	from	from	ADP
cana-834	82	19	u	u	NOUN
cana-834	82	20	and	and	CCONJ
cana-834	82	21	v	v	NOUN
cana-834	82	22	.	.	PUNCT
cana-834	83	1	hence	hence	ADV
cana-834	83	2	𝑉(𝐺1	𝑉(𝐺1	NOUN
cana-834	83	3	)	)	PUNCT
cana-834	84	1	∩	∩	NOUN
cana-834	84	2	𝐷is	𝐷is	PROPN
cana-834	84	3	a	a	DET
cana-834	84	4	ctd	ctd	NOUN
cana-834	84	5	-	-	PUNCT
cana-834	84	6	set	set	NOUN
cana-834	84	7	in	in	ADP
cana-834	84	8	g1	g1	PROPN
cana-834	84	9	.	.	PUNCT
cana-834	84	10	suppose	suppose	VERB
cana-834	85	1	dgvv	dgvv	ADV
cana-834	85	2			VERB
cana-834	85	3	)	)	PUNCT
cana-834	85	4	(	(	PUNCT
cana-834	85	5	1	1	NUM
cana-834	85	6	and	and	CCONJ
cana-834	85	7	(	(	PUNCT
cana-834	85	8	)	)	PUNCT
cana-834	85	9	−	−	PROPN
cana-834	85	10	dgv	dgv	PROPN
cana-834	85	11	v	v	NUM
cana-834	85	12	2	2	NUM
cana-834	85	13	.	.	PUNCT
cana-834	86	1	let	let	VERB
cana-834	86	2	dgvu	dgvu	NOUN
cana-834	86	3	v	v	ADP
cana-834	86	4	−	−	PROPN
cana-834	86	5	)	)	PUNCT
cana-834	87	1	(	(	PUNCT
cana-834	87	2	2	2	X
cana-834	87	3	.	.	PUNCT
cana-834	88	1	since	since	SCONJ
cana-834	88	2	)	)	PUNCT
cana-834	88	3	(	(	PUNCT
cana-834	88	4	)	)	PUNCT
cana-834	88	5	(	(	PUNCT
cana-834	88	6	11	11	NUM
cana-834	88	7	gvdgv	gvdgv	NOUN
cana-834	88	8			X
cana-834	88	9	there	there	ADV
cana-834	88	10	exists	exist	VERB
cana-834	88	11	.	.	PUNCT
cana-834	88	12	)	)	PUNCT
cana-834	89	1	(	(	PUNCT
cana-834	89	2	1	1	NUM
cana-834	89	3	dgvw	dgvw	NOUN
cana-834	89	4	−	−	X
cana-834	90	1	since	since	SCONJ
cana-834	90	2	−	−	ADJ
cana-834	90	3	dggv	dggv	PROPN
cana-834	90	4	)	)	PUNCT
cana-834	90	5	(	(	PUNCT
cana-834	90	6	21	21	NUM
cana-834	90	7			PROPN
cana-834	90	8	is	be	AUX
cana-834	90	9	a	a	DET
cana-834	90	10	tree	tree	NOUN
cana-834	90	11	.	.	PUNCT
cana-834	91	1	there	there	PRON
cana-834	91	2	exists	exist	VERB
cana-834	91	3	a	a	DET
cana-834	91	4	unique	unique	ADJ
cana-834	91	5	path	path	NOUN
cana-834	91	6	between	between	ADP
cana-834	91	7	u	u	NOUN
cana-834	91	8	w	w	VERB
cana-834	91	9	with	with	ADP
cana-834	91	10	vertices	vertex	NOUN
cana-834	91	11	from	from	ADP
cana-834	91	12	.	.	PUNCT
cana-834	91	13	)	)	PUNCT
cana-834	92	1	(	(	PUNCT
cana-834	92	2	21	21	NUM
cana-834	92	3	dggv	dggv	NOUN
cana-834	92	4	−	−	NOUN
cana-834	92	5	however	however	ADV
cana-834	92	6	any	any	DET
cana-834	92	7	u	u	PROPN
cana-834	92	8	-	-	PROPN
cana-834	92	9	w	w	ADJ
cana-834	92	10	path	path	NOUN
cana-834	92	11	must	must	AUX
cana-834	92	12	contain	contain	VERB
cana-834	92	13	a	a	DET
cana-834	92	14	vertex	vertex	NOUN
cana-834	92	15	v	v	NOUN
cana-834	92	16	which	which	PRON
cana-834	92	17	is	be	AUX
cana-834	92	18	impossible	impossible	ADJ
cana-834	92	19	.	.	PUNCT
cana-834	93	1	hence	hence	ADV
cana-834	93	2	.	.	PUNCT
cana-834	93	3	)	)	PUNCT
cana-834	94	1	(	(	PUNCT
cana-834	94	2	2	2	NUM
cana-834	94	3	=−	=−	NOUN
cana-834	94	4	dgv	dgv	PROPN
cana-834	94	5	v	v	NOUN
cana-834	94	6	hence	hence	ADV
cana-834	94	7	.	.	PUNCT
cana-834	94	8	)	)	PUNCT
cana-834	95	1	(	(	PUNCT
cana-834	95	2	2	2	NUM
cana-834	95	3	dgv	dgv	PROPN
cana-834	95	4	v	v	PROPN
cana-834	95	5			PROPN
cana-834	95	6	suppose	suppose	VERB
cana-834	95	7	dgvv	dgvv	ADV
cana-834	95	8	−	−	PROPN
cana-834	95	9	)	)	PUNCT
cana-834	96	1	(	(	PUNCT
cana-834	96	2	1	1	X
cana-834	96	3	.	.	PUNCT
cana-834	97	1	let	let	VERB
cana-834	97	2	dgvx	dgvx	VERB
cana-834	97	3	v	v	NOUN
cana-834	97	4	−	−	PROPN
cana-834	97	5	)	)	PUNCT
cana-834	98	1	(	(	PUNCT
cana-834	98	2	2	2	X
cana-834	98	3	.	.	PUNCT
cana-834	99	1	this	this	PRON
cana-834	99	2	implies	imply	VERB
cana-834	99	3	that	that	PRON
cana-834	99	4	.	.	PUNCT
cana-834	99	5	)	)	PUNCT
cana-834	100	1	(	(	PUNCT
cana-834	100	2	21	21	NUM
cana-834	100	3	−	−	NOUN
cana-834	100	4	dggvx	dggvx	NOUN
cana-834	100	5			PROPN
cana-834	100	6	since	since	SCONJ
cana-834	100	7	d	d	PROPN
cana-834	100	8	is	be	AUX
cana-834	100	9	a	a	DET
cana-834	100	10	ctd	ctd	NOUN
cana-834	100	11	-	-	PUNCT
cana-834	100	12	set	set	NOUN
cana-834	100	13	in	in	ADP
cana-834	100	14	21	21	NUM
cana-834	100	15	gg	gg	NOUN
cana-834	100	16			PROPN
cana-834	100	17	there	there	PRON
cana-834	100	18	exists	exist	VERB
cana-834	100	19	dy	dy	NOUN
cana-834	100	20			NOUN
cana-834	100	21	such	such	ADJ
cana-834	100	22	that	that	SCONJ
cana-834	100	23	(	(	PUNCT
cana-834	100	24	)	)	PUNCT
cana-834	100	25	.1	.1	PROPN
cana-834	100	26	,	,	PUNCT
cana-834	100	27	21	21	NUM
cana-834	100	28	=	=	NUM
cana-834	100	29	yxd	yxd	NOUN
cana-834	100	30	gg	gg	NOUN
cana-834	100	31			PROPN
cana-834	100	32	consequently	consequently	ADV
cana-834	100	33	vy	vy	VERB
cana-834	100	34	=	=	PUNCT
cana-834	100	35	or	or	CCONJ
cana-834	100	36	)	)	PUNCT
cana-834	100	37	.	.	PUNCT
cana-834	101	1	(	(	PUNCT
cana-834	101	2	2	2	X
cana-834	101	3	vgvy	vgvy	NOUN
cana-834	101	4	since	since	SCONJ
cana-834	101	5	dy	dy	PROPN
cana-834	101	6	and	and	CCONJ
cana-834	101	7	.	.	PUNCT
cana-834	101	8	)	)	PUNCT
cana-834	102	1	(	(	PUNCT
cana-834	102	2	1	1	X
cana-834	102	3	dgvv	dgvv	NOUN
cana-834	102	4	−	−	PROPN
cana-834	103	1	hence	hence	ADV
cana-834	103	2	vy	vy	PROPN
cana-834	103	3			PROPN
cana-834	103	4	then	then	ADV
cana-834	103	5	it	it	PRON
cana-834	103	6	follows	follow	VERB
cana-834	103	7	)	)	PUNCT
cana-834	103	8	(	(	PUNCT
cana-834	103	9	2	2	X
cana-834	103	10	vgvy	vgvy	VERB
cana-834	103	11	now	now	ADV
cana-834	103	12	(	(	PUNCT
cana-834	103	13	)	)	PUNCT
cana-834	103	14	1	1	NUM
cana-834	103	15	,	,	PUNCT
cana-834	103	16	21	21	NUM
cana-834	103	17	=	=	NUM
cana-834	103	18	yxd	yxd	NOUN
cana-834	103	19	gg	gg	NOUN
cana-834	103	20			PROPN
cana-834	103	21	implies	imply	VERB
cana-834	103	22	(	(	PUNCT
cana-834	103	23	)	)	PUNCT
cana-834	103	24	.1	.1	NOUN
cana-834	103	25	,	,	PUNCT
cana-834	103	26	2	2	NUM
cana-834	103	27	=	=	NOUN
cana-834	103	28	yxd	yxd	NOUN
cana-834	103	29	vg	vg	NOUN
cana-834	103	30	hence	hence	ADV
cana-834	103	31	dgv	dgv	PROPN
cana-834	103	32	v	v	NOUN
cana-834	103	33			X
cana-834	103	34	)	)	PUNCT
cana-834	103	35	(	(	PUNCT
cana-834	103	36	2	2	NUM
cana-834	103	37	is	be	AUX
cana-834	103	38	a	a	DET
cana-834	103	39	dominating	dominating	NOUN
cana-834	103	40	set	set	VERB
cana-834	103	41	in	in	ADP
cana-834	103	42	vg	vg	ADP
cana-834	103	43	2	2	NUM
cana-834	103	44	.therefore	.therefore	NOUN
cana-834	103	45	(	(	PUNCT
cana-834	103	46	iii	iii	NOUN
cana-834	103	47	)	)	PUNCT
cana-834	103	48	holds	hold	VERB
cana-834	103	49	.	.	PUNCT
cana-834	104	1	conversely	conversely	ADV
cana-834	104	2	,	,	PUNCT
cana-834	104	3	suppose	suppose	VERB
cana-834	104	4	(	(	PUNCT
cana-834	104	5	i	i	NOUN
cana-834	104	6	)	)	PUNCT
cana-834	104	7	holds	hold	VERB
cana-834	104	8	.	.	PUNCT
cana-834	105	1	let	let	VERB
cana-834	105	2	.	.	PUNCT
cana-834	105	3	)	)	PUNCT
cana-834	106	1	(	(	PUNCT
cana-834	106	2	21	21	NUM
cana-834	106	3	dggvx	dggvx	NOUN
cana-834	106	4	−	−	PROPN
cana-834	107	1			PRON
cana-834	107	2	suppose	suppose	VERB
cana-834	107	3	)	)	PUNCT
cana-834	107	4	(	(	PUNCT
cana-834	107	5	1gvx	1gvx	NUM
cana-834	107	6	.since	.since	NOUN
cana-834	108	1	dgv	dgv	PROPN
cana-834	108	2	x	x	INTJ
cana-834	108	3			X
cana-834	108	4	)	)	PUNCT
cana-834	108	5	(	(	PUNCT
cana-834	108	6	2	2	NUM
cana-834	108	7	is	be	AUX
cana-834	108	8	dominating	dominate	VERB
cana-834	108	9	set	set	VERB
cana-834	108	10	in	in	ADP
cana-834	108	11	xg2	xg2	PROPN
cana-834	108	12	and	and	CCONJ
cana-834	108	13	.	.	PUNCT
cana-834	108	14	)	)	PUNCT
cana-834	109	1	(	(	PUNCT
cana-834	109	2	2	2	NUM
cana-834	109	3			NOUN
cana-834	109	4	dgv	dgv	PROPN
cana-834	109	5	x	x	PRON
cana-834	109	6	let	let	VERB
cana-834	109	7	.	.	PUNCT
cana-834	109	8	)	)	PUNCT
cana-834	110	1	(	(	PUNCT
cana-834	110	2	2	2	NUM
cana-834	110	3	dgvu	dgvu	NOUN
cana-834	110	4	x	x	PUNCT
cana-834	111	1			VERB
cana-834	111	2	then	then	ADV
cana-834	111	3	1	1	NUM
cana-834	111	4	)	)	PUNCT
cana-834	111	5	,	,	PUNCT
cana-834	111	6	(	(	PUNCT
cana-834	111	7	21	21	NUM
cana-834	111	8	=	=	NUM
cana-834	111	9	yxd	yxd	NOUN
cana-834	111	10	gg	gg	NOUN
cana-834	111	11			PROPN
cana-834	111	12	.	.	PUNCT
cana-834	112	1	suppose	suppose	VERB
cana-834	112	2	)	)	PUNCT
cana-834	112	3	(	(	PUNCT
cana-834	112	4	1gvx	1gvx	NUM
cana-834	112	5	then	then	ADV
cana-834	112	6	there	there	PRON
cana-834	112	7	exist	exist	VERB
cana-834	112	8	)	)	PUNCT
cana-834	112	9	(	(	PUNCT
cana-834	112	10	1gvy	1gvy	NUM
cana-834	112	11	such	such	ADJ
cana-834	112	12	that	that	PRON
cana-834	112	13	(	(	PUNCT
cana-834	112	14	)	)	PUNCT
cana-834	112	15	.2	.2	NUM
cana-834	112	16	ygvx	ygvx	PROPN
cana-834	112	17	since	since	SCONJ
cana-834	112	18	dgv	dgv	PROPN
cana-834	112	19	y	y	PROPN
cana-834	112	20			PROPN
cana-834	112	21	)	)	PUNCT
cana-834	112	22	(	(	PUNCT
cana-834	112	23	2	2	NUM
cana-834	112	24	is	be	AUX
cana-834	112	25	a	a	DET
cana-834	112	26	dominating	dominating	NOUN
cana-834	112	27	set	set	VERB
cana-834	112	28	in	in	ADP
cana-834	112	29	yg2	yg2	NOUN
cana-834	112	30	and	and	CCONJ
cana-834	112	31	dgvx	dgvx	VERB
cana-834	112	32	y	y	PROPN
cana-834	112	33	−	−	PROPN
cana-834	112	34	)	)	PUNCT
cana-834	112	35	(	(	PUNCT
cana-834	112	36	2	2	NUM
cana-834	112	37	there	there	PRON
cana-834	112	38	exist	exist	VERB
cana-834	112	39	dgvt	dgvt	NOUN
cana-834	112	40	y	y	PROPN
cana-834	112	41			PROPN
cana-834	112	42	)	)	PUNCT
cana-834	112	43	(	(	PUNCT
cana-834	112	44	2	2	NUM
cana-834	112	45	such	such	ADJ
cana-834	112	46	that	that	DET
cana-834	112	47	1	1	NUM
cana-834	112	48	)	)	PUNCT
cana-834	112	49	,	,	PUNCT
cana-834	112	50	(	(	PUNCT
cana-834	112	51	21	21	NUM
cana-834	112	52	=	=	NUM
cana-834	112	53	txd	txd	NOUN
cana-834	112	54	gg	gg	NOUN
cana-834	112	55			PROPN
cana-834	112	56	then	then	ADV
cana-834	112	57	d	d	PROPN
cana-834	112	58	is	be	AUX
cana-834	112	59	a	a	DET
cana-834	112	60	dominating	dominating	NOUN
cana-834	112	61	set	set	NOUN
cana-834	112	62	in	in	ADP
cana-834	112	63	21	21	NUM
cana-834	112	64	gg	gg	NOUN
cana-834	112	65			PROPN
cana-834	112	66	.	.	PUNCT
cana-834	113	1	since	since	SCONJ
cana-834	113	2	(	(	PUNCT
cana-834	113	3	)	)	PUNCT
cana-834	113	4	(	(	PUNCT
cana-834	113	5	)	)	PUNCT
cana-834	113	6	(	(	PUNCT
cana-834	113	7	)	)	PUNCT
cana-834	113	8	.12	.12	NUM
cana-834	113	9	1	1	NUM
cana-834	113	10	=	=	NUM
cana-834	113	11			NOUN
cana-834	113	12	dgvandgvd	dgvandgvd	VERB
cana-834	113	13	gvu	gvu	PROPN
cana-834	113	14	u	u	PROPN
cana-834	113	15	consequently	consequently	ADV
cana-834	113	16	,	,	PUNCT
cana-834	113	17	)	)	PUNCT
cana-834	113	18	(	(	PUNCT
cana-834	113	19	)	)	PUNCT
cana-834	113	20	(	(	PUNCT
cana-834	113	21	)	)	PUNCT
cana-834	113	22	(	(	PUNCT
cana-834	113	23	1	1	NUM
cana-834	113	24	)	)	PUNCT
cana-834	113	25	(	(	PUNCT
cana-834	113	26	221	221	NUM
cana-834	113	27	gvdgvdggv	gvdgvdggv	NOUN
cana-834	113	28	gvv	gvv	NOUN
cana-834	113	29	v	v	ADP
cana-834	113	30	−=−	−=−	ADP
cana-834	113	31			NOUN
cana-834	113	32			NOUN
cana-834	113	33	.let	.let	PUNCT
cana-834	114	1	qpdggvqp	qpdggvqp	PROPN
cana-834	114	2	−	−	PROPN
cana-834	114	3	,	,	PUNCT
cana-834	114	4	)	)	PUNCT
cana-834	114	5	(	(	PUNCT
cana-834	114	6	,	,	PUNCT
cana-834	114	7	21	21	NUM
cana-834	114	8			PROPN
cana-834	114	9	.if	.if	PUNCT
cana-834	114	10	dgvqp	dgvqp	VERB
cana-834	114	11	v	v	ADP
cana-834	114	12	−	−	PROPN
cana-834	114	13	)	)	PUNCT
cana-834	115	1	(	(	PUNCT
cana-834	115	2	,	,	PUNCT
cana-834	115	3	2	2	NUM
cana-834	115	4	for	for	ADP
cana-834	115	5	some	some	PRON
cana-834	115	6	)	)	PUNCT
cana-834	115	7	(	(	PUNCT
cana-834	115	8	1gvv	1gvv	NUM
cana-834	115	9	then	then	ADV
cana-834	115	10	there	there	PRON
cana-834	115	11	is	be	VERB
cana-834	115	12	a	a	DET
cana-834	115	13	path	path	NOUN
cana-834	115	14	with	with	ADP
cana-834	115	15	vertices	vertex	NOUN
cana-834	115	16	qvp	qvp	NOUN
cana-834	115	17	,	,	PUNCT
cana-834	115	18	,	,	PUNCT
cana-834	115	19	in	in	ADP
cana-834	115	20	.	.	PUNCT
cana-834	115	21	)	)	PUNCT
cana-834	116	1	(	(	PUNCT
cana-834	116	2	21	21	NUM
cana-834	116	3	dggv	dggv	NOUN
cana-834	116	4	−	−	NOUN
cana-834	116	5	if	if	SCONJ
cana-834	116	6	)	)	PUNCT
cana-834	116	7	(	(	PUNCT
cana-834	116	8	,	,	PUNCT
cana-834	116	9	1gvqp	1gvqp	NUM
cana-834	116	10			NOUN
cana-834	116	11	.then	.then	PUNCT
cana-834	116	12	there	there	PRON
cana-834	116	13	is	be	VERB
cana-834	116	14	a	a	DET
cana-834	116	15	tree	tree	NOUN
cana-834	116	16	which	which	PRON
cana-834	116	17	contains	contain	VERB
cana-834	116	18	a	a	DET
cana-834	116	19	p	p	NOUN
cana-834	116	20	-	-	PUNCT
cana-834	116	21	q	q	NOUN
cana-834	116	22	path	path	NOUN
cana-834	116	23	in	in	ADP
cana-834	116	24	dggv	dggv	PROPN
cana-834	116	25	−	−	PROPN
cana-834	116	26	)	)	PUNCT
cana-834	116	27	(	(	PUNCT
cana-834	116	28	21	21	NUM
cana-834	116	29			PROPN
cana-834	116	30	.since	.since	NOUN
cana-834	116	31	g1	g1	PROPN
cana-834	116	32	is	be	AUX
cana-834	116	33	sna	sna	PROPN
cana-834	116	34	.	.	PUNCT
cana-834	117	1	if	if	SCONJ
cana-834	117	2	(	(	PUNCT
cana-834	117	3	)	)	PUNCT
cana-834	117	4	1gvp	1gvp	NUM
cana-834	117	5	and	and	CCONJ
cana-834	117	6	(	(	PUNCT
cana-834	117	7	)	)	PUNCT
cana-834	117	8	dgvq	dgvq	VERB
cana-834	117	9	v	v	ADP
cana-834	117	10	−	−	PROPN
cana-834	117	11	2	2	NUM
cana-834	117	12	for	for	ADP
cana-834	117	13	some	some	PRON
cana-834	117	14	(	(	PUNCT
cana-834	117	15	)	)	PUNCT
cana-834	117	16	1gvv	1gvv	NUM
cana-834	117	17	then	then	ADV
cana-834	117	18	dggv	dggv	VERB
cana-834	117	19	−	−	PROPN
cana-834	117	20	)	)	PUNCT
cana-834	117	21	(	(	PUNCT
cana-834	117	22	21	21	NUM
cana-834	117	23			PROPN
cana-834	117	24	contains	contain	VERB
cana-834	117	25	a	a	DET
cana-834	117	26	tree	tree	NOUN
cana-834	117	27	with	with	ADP
cana-834	117	28	vertices	vertex	NOUN
cana-834	117	29	p	p	NOUN
cana-834	117	30	and	and	CCONJ
cana-834	117	31	v	v	NOUN
cana-834	117	32	.	.	PUNCT
cana-834	118	1	suppose	suppose	VERB
cana-834	118	2	vp	vp	PROPN
cana-834	118	3			PROPN
cana-834	118	4	since	since	SCONJ
cana-834	118	5	g1	g1	PROPN
cana-834	118	6	is	be	AUX
cana-834	118	7	connected	connect	VERB
cana-834	118	8	then	then	ADV
cana-834	118	9	dggv	dggv	VERB
cana-834	118	10	−	−	PROPN
cana-834	118	11	)	)	PUNCT
cana-834	118	12	(	(	PUNCT
cana-834	118	13	21	21	NUM
cana-834	118	14			PROPN
cana-834	118	15	contains	contain	VERB
cana-834	118	16	a	a	DET
cana-834	118	17	tree	tree	NOUN
cana-834	118	18	with	with	ADP
cana-834	118	19	vertices	vertex	NOUN
cana-834	118	20	qvp	qvp	NOUN
cana-834	118	21	,	,	PUNCT
cana-834	118	22	,	,	PUNCT
cana-834	118	23	.suppose	.suppose	X
cana-834	118	24	(	(	PUNCT
cana-834	118	25	)	)	PUNCT
cana-834	118	26	dgvp	dgvp	VERB
cana-834	118	27	v	v	ADP
cana-834	118	28	−	−	PROPN
cana-834	118	29	2	2	NUM
cana-834	118	30	and	and	CCONJ
cana-834	118	31	(	(	PUNCT
cana-834	118	32	)	)	PUNCT
cana-834	118	33	wgvq	wgvq	NOUN
cana-834	118	34	2	2	NUM
cana-834	118	35	for	for	ADP
cana-834	118	36	some	some	PRON
cana-834	118	37	)	)	PUNCT
cana-834	118	38	.	.	PUNCT
cana-834	119	1	(	(	PUNCT
cana-834	119	2	,	,	PUNCT
cana-834	119	3	1gvwv	1gvwv	NUM
cana-834	119	4			NOUN
cana-834	119	5	since	since	SCONJ
cana-834	119	6	g1	g1	PROPN
cana-834	119	7	is	be	AUX
cana-834	119	8	connected	connect	VERB
cana-834	119	9	communications	communication	NOUN
cana-834	119	10	on	on	ADP
cana-834	119	11	applied	apply	VERB
cana-834	119	12	nonlinear	nonlinear	ADJ
cana-834	119	13	analysis	analysis	NOUN
cana-834	119	14	issn	issn	NOUN
cana-834	119	15	:	:	PUNCT
cana-834	119	16	1074	1074	NUM
cana-834	119	17	-	-	PUNCT
cana-834	119	18	133x	133x	NUM
cana-834	119	19	vol	vol	NOUN
cana-834	119	20	31	31	NUM
cana-834	119	21	no	no	NOUN
cana-834	119	22	.	.	PUNCT
cana-834	120	1	4s	4s	NUM
cana-834	120	2	(	(	PUNCT
cana-834	120	3	2024	2024	NUM
cana-834	120	4	)	)	PUNCT
cana-834	120	5	168	168	NUM
cana-834	120	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	120	7	then	then	ADV
cana-834	120	8	there	there	PRON
cana-834	120	9	exist	exist	VERB
cana-834	120	10	a	a	DET
cana-834	120	11	tree	tree	NOUN
cana-834	120	12	with	with	ADP
cana-834	120	13	vertices	vertex	NOUN
cana-834	120	14	qwvp	qwvp	NOUN
cana-834	120	15	,	,	PUNCT
cana-834	120	16	,	,	PUNCT
cana-834	120	17	,	,	PUNCT
cana-834	120	18	in	in	ADP
cana-834	120	19	dggv	dggv	PROPN
cana-834	120	20	−	−	PROPN
cana-834	120	21	)	)	PUNCT
cana-834	120	22	(	(	PUNCT
cana-834	120	23	21	21	NUM
cana-834	120	24			PROPN
cana-834	120	25	.hence	.hence	ADP
cana-834	120	26	dggv	dggv	PROPN
cana-834	120	27	−	−	PROPN
cana-834	120	28	)	)	PUNCT
cana-834	120	29	(	(	PUNCT
cana-834	120	30	21	21	NUM
cana-834	120	31			PROPN
cana-834	120	32	is	be	AUX
cana-834	120	33	sna	sna	PROPN
cana-834	120	34	and	and	CCONJ
cana-834	120	35	acyclic	acyclic	ADJ
cana-834	120	36	.	.	PUNCT
cana-834	121	1	therefore	therefore	ADV
cana-834	121	2	dggv	dggv	PROPN
cana-834	121	3	−	−	PROPN
cana-834	121	4	)	)	PUNCT
cana-834	121	5	(	(	PUNCT
cana-834	121	6	21	21	NUM
cana-834	121	7			PROPN
cana-834	121	8	is	be	AUX
cana-834	121	9	a	a	DET
cana-834	121	10	tree	tree	NOUN
cana-834	121	11	.	.	PUNCT
cana-834	122	1	hence	hence	ADV
cana-834	122	2	d	d	PROPN
cana-834	122	3	is	be	AUX
cana-834	122	4	a	a	DET
cana-834	122	5	complementary	complementary	ADJ
cana-834	122	6	tree	tree	NOUN
cana-834	122	7	dominating	dominating	NOUN
cana-834	122	8	in	in	ADP
cana-834	122	9	21	21	NUM
cana-834	122	10	gg	gg	NOUN
cana-834	122	11			PROPN
cana-834	122	12	suppose	suppose	VERB
cana-834	122	13	(	(	PUNCT
cana-834	122	14	ii	ii	NOUN
cana-834	122	15	)	)	PUNCT
cana-834	122	16	holds	hold	VERB
cana-834	122	17	.	.	PUNCT
cana-834	123	1	let	let	VERB
cana-834	123	2	.	.	PUNCT
cana-834	123	3	)	)	PUNCT
cana-834	124	1	(	(	PUNCT
cana-834	124	2	21	21	NUM
cana-834	124	3	dggvx	dggvx	NOUN
cana-834	124	4	−	−	PROPN
cana-834	125	1			PRON
cana-834	125	2	suppose	suppose	VERB
cana-834	125	3	.	.	PUNCT
cana-834	125	4	)	)	PUNCT
cana-834	125	5	(	(	PUNCT
cana-834	125	6	)	)	PUNCT
cana-834	125	7	.	.	PUNCT
cana-834	126	1	(	(	PUNCT
cana-834	126	2	)	)	PUNCT
cana-834	126	3	(	(	PUNCT
cana-834	126	4	11	11	NUM
cana-834	126	5	dgvxeigvx	dgvxeigvx	NOUN
cana-834	126	6	−	−	NOUN
cana-834	126	7	since	since	SCONJ
cana-834	126	8	dgv	dgv	PROPN
cana-834	126	9			X
cana-834	126	10	)	)	PUNCT
cana-834	126	11	(	(	PUNCT
cana-834	126	12	1	1	NUM
cana-834	126	13	is	be	AUX
cana-834	126	14	a	a	DET
cana-834	126	15	dominating	dominating	NOUN
cana-834	126	16	in	in	ADP
cana-834	126	17	g1	g1	NOUN
cana-834	126	18	,	,	PUNCT
cana-834	126	19	there	there	PRON
cana-834	126	20	exist	exist	VERB
cana-834	126	21	dgvy	dgvy	ADJ
cana-834	126	22			NOUN
cana-834	126	23	)	)	PUNCT
cana-834	127	1	(	(	PUNCT
cana-834	127	2	1	1	NUM
cana-834	127	3	such	such	ADJ
cana-834	127	4	that	that	SCONJ
cana-834	127	5	(	(	PUNCT
cana-834	127	6	)	)	PUNCT
cana-834	127	7	1	1	NUM
cana-834	127	8	,	,	PUNCT
cana-834	127	9	=	=	NOUN
cana-834	127	10	txdg	txdg	NOUN
cana-834	127	11	for	for	ADP
cana-834	127	12	some	some	PRON
cana-834	127	13	)	)	PUNCT
cana-834	127	14	(	(	PUNCT
cana-834	127	15	1gvt	1gvt	NUM
cana-834	127	16			NOUN
cana-834	127	17	is	be	AUX
cana-834	127	18	follows	follow	VERB
cana-834	127	19	that	that	SCONJ
cana-834	127	20	(	(	PUNCT
cana-834	127	21	)	)	PUNCT
cana-834	127	22	.1	.1	PROPN
cana-834	127	23	,	,	PUNCT
cana-834	127	24	21	21	NUM
cana-834	127	25	=	=	NUM
cana-834	127	26	yxd	yxd	NOUN
cana-834	127	27	gg	gg	NOUN
cana-834	127	28			PROPN
cana-834	127	29	suppose	suppose	VERB
cana-834	127	30	(	(	PUNCT
cana-834	127	31	)	)	PUNCT
cana-834	127	32	vgvx	vgvx	NOUN
cana-834	127	33	2	2	NUM
cana-834	127	34	for	for	ADP
cana-834	127	35	some	some	PRON
cana-834	127	36	)	)	PUNCT
cana-834	127	37	(	(	PUNCT
cana-834	127	38	1gvv	1gvv	NUM
cana-834	127	39	.1	.1	NUM
cana-834	127	40	)	)	PUNCT
cana-834	127	41	,	,	PUNCT
cana-834	127	42	(	(	PUNCT
cana-834	127	43	)	)	PUNCT
cana-834	127	44	.	.	PUNCT
cana-834	128	1	(	(	PUNCT
cana-834	128	2	)	)	PUNCT
cana-834	128	3	(	(	PUNCT
cana-834	128	4	)	)	PUNCT
cana-834	128	5	.	.	PUNCT
cana-834	129	1	(	(	PUNCT
cana-834	129	2	212	212	NUM
cana-834	129	3	=	=	NOUN
cana-834	129	4	−	−	PROPN
cana-834	129	5	vxdeidgvxei	vxdeidgvxei	PROPN
cana-834	129	6	gg	gg	PROPN
cana-834	129	7	v	v	NUM
cana-834	129	8			PROPN
cana-834	129	9	if	if	SCONJ
cana-834	129	10	dv	dv	ADP
cana-834	129	11	which	which	PRON
cana-834	129	12	is	be	AUX
cana-834	129	13	a	a	DET
cana-834	129	14	contradiction	contradiction	NOUN
cana-834	129	15	to	to	ADP
cana-834	129	16	(	(	PUNCT
cana-834	129	17	)	)	PUNCT
cana-834	129	18	.2	.2	NUM
cana-834	129	19	dgvx	dgvx	PROPN
cana-834	129	20	v	v	NOUN
cana-834	129	21	−	−	PROPN
cana-834	130	1	hence	hence	ADV
cana-834	130	2	dv	dv	NOUN
cana-834	130	3	is	be	AUX
cana-834	130	4	(	(	PUNCT
cana-834	130	5	)	)	PUNCT
cana-834	131	1	dgvv	dgvv	ADP
cana-834	131	2	−	−	PROPN
cana-834	132	1	1	1	X
cana-834	132	2	.	.	PUNCT
cana-834	133	1	in	in	ADP
cana-834	133	2	this	this	DET
cana-834	133	3	case	case	NOUN
cana-834	133	4	(	(	PUNCT
cana-834	133	5	)	)	PUNCT
cana-834	133	6	dgv	dgv	PROPN
cana-834	133	7	v	v	NUM
cana-834	133	8	2	2	NOUN
cana-834	133	9	is	be	AUX
cana-834	133	10	dominating	dominate	VERB
cana-834	133	11	in	in	ADP
cana-834	133	12	vg2	vg2	PROPN
cana-834	133	13	(	(	PUNCT
cana-834	133	14	𝑖.	𝑖.	NOUN
cana-834	133	15	𝑒	𝑒	X
cana-834	133	16	)	)	PUNCT
cana-834	133	17	there	there	PRON
cana-834	133	18	exist	exist	VERB
cana-834	133	19	(	(	PUNCT
cana-834	133	20	)	)	PUNCT
cana-834	133	21	dgvw	dgvw	PROPN
cana-834	133	22	w	w	PROPN
cana-834	133	23			PROPN
cana-834	133	24	2	2	NUM
cana-834	133	25	such	such	ADJ
cana-834	133	26	that	that	DET
cana-834	133	27	.1	.1	NOUN
cana-834	133	28	)	)	PUNCT
cana-834	133	29	,	,	PUNCT
cana-834	133	30	(	(	PUNCT
cana-834	133	31	2	2	NUM
cana-834	133	32	=	=	NUM
cana-834	133	33	wxd	wxd	NOUN
cana-834	133	34	vg	vg	ADV
cana-834	133	35	it	it	PRON
cana-834	133	36	follows	follow	VERB
cana-834	133	37	that	that	PRON
cana-834	133	38	(	(	PUNCT
cana-834	133	39	)	)	PUNCT
cana-834	133	40	.1	.1	PROPN
cana-834	133	41	,	,	PUNCT
cana-834	133	42	21	21	NUM
cana-834	133	43	=	=	NUM
cana-834	133	44	wxd	wxd	ADJ
cana-834	133	45	gg	gg	NOUN
cana-834	133	46			PROPN
cana-834	133	47	hence	hence	ADV
cana-834	133	48	d	d	PROPN
cana-834	133	49	is	be	AUX
cana-834	133	50	a	a	DET
cana-834	133	51	dominating	dominating	NOUN
cana-834	133	52	set	set	NOUN
cana-834	133	53	in	in	ADP
cana-834	133	54	21	21	NUM
cana-834	133	55	gg	gg	NOUN
cana-834	133	56			PROPN
cana-834	133	57	.	.	PUNCT
cana-834	134	1	let	let	VERB
cana-834	134	2	(	(	PUNCT
cana-834	134	3	)	)	PUNCT
cana-834	134	4	21	21	NUM
cana-834	134	5	,	,	PUNCT
cana-834	134	6	ggvyx	ggvyx	NOUN
cana-834	134	7			AUX
cana-834	134	8	suppose	suppose	VERB
cana-834	134	9	(	(	PUNCT
cana-834	134	10	)	)	PUNCT
cana-834	134	11	1	1	NUM
cana-834	134	12	,	,	PUNCT
cana-834	134	13	gvyx	gvyx	VERB
cana-834	134	14			NOUN
cana-834	134	15	)	)	PUNCT
cana-834	134	16	.	.	PUNCT
cana-834	135	1	(	(	PUNCT
cana-834	135	2	ei	ei	X
cana-834	135	3	(	(	PUNCT
cana-834	135	4	)	)	PUNCT
cana-834	135	5	dgvyx	dgvyx	NOUN
cana-834	135	6	−	−	PROPN
cana-834	135	7	,	,	PUNCT
cana-834	135	8	.	.	PUNCT
cana-834	136	1	since	since	SCONJ
cana-834	136	2	dgv	dgv	PROPN
cana-834	136	3			NUM
cana-834	136	4	)	)	PUNCT
cana-834	136	5	(	(	PUNCT
cana-834	136	6	is	be	AUX
cana-834	136	7	a	a	DET
cana-834	136	8	complementary	complementary	ADJ
cana-834	136	9	tree	tree	NOUN
cana-834	136	10	dominating	dominating	NOUN
cana-834	136	11	set	set	VERB
cana-834	136	12	in	in	ADP
cana-834	136	13	g	g	NOUN
cana-834	136	14	,	,	PUNCT
cana-834	136	15	and	and	CCONJ
cana-834	136	16	−	−	VERB
cana-834	136	17	dgv	dgv	PROPN
cana-834	136	18	)	)	PUNCT
cana-834	136	19	(	(	PUNCT
cana-834	136	20	1	1	NUM
cana-834	136	21	is	be	AUX
cana-834	136	22	a	a	DET
cana-834	136	23	tree	tree	NOUN
cana-834	136	24	.	.	PUNCT
cana-834	137	1	from	from	ADP
cana-834	137	2	this	this	DET
cana-834	137	3	−	−	VERB
cana-834	137	4	dggv	dggv	PROPN
cana-834	137	5	)	)	PUNCT
cana-834	137	6	(	(	PUNCT
cana-834	137	7	21	21	NUM
cana-834	137	8			PROPN
cana-834	137	9	is	be	AUX
cana-834	137	10	a	a	DET
cana-834	137	11	tree	tree	NOUN
cana-834	137	12	which	which	PRON
cana-834	137	13	contains	contain	VERB
cana-834	137	14	a	a	DET
cana-834	137	15	path	path	NOUN
cana-834	137	16	x	x	NOUN
cana-834	137	17	-	-	NOUN
cana-834	137	18	y	y	NOUN
cana-834	137	19	in	in	ADP
cana-834	137	20	.	.	PUNCT
cana-834	137	21	)	)	PUNCT
cana-834	138	1	(	(	PUNCT
cana-834	138	2	dgv	dgv	PROPN
cana-834	138	3	−	−	PROPN
cana-834	138	4	suppose	suppose	VERB
cana-834	138	5	(	(	PUNCT
cana-834	138	6	)	)	PUNCT
cana-834	138	7	1gvx	1gvx	NUM
cana-834	138	8	and	and	CCONJ
cana-834	138	9	(	(	PUNCT
cana-834	138	10	)	)	PUNCT
cana-834	138	11	vgvy	vgvy	NOUN
cana-834	138	12	2	2	NUM
cana-834	138	13	for	for	ADP
cana-834	138	14	some	some	PRON
cana-834	138	15	(	(	PUNCT
cana-834	138	16	)	)	PUNCT
cana-834	138	17	(	(	PUNCT
cana-834	138	18	)	)	PUNCT
cana-834	138	19	dgvxeigvv	dgvxeigvv	VERB
cana-834	138	20	−	−	ADP
cana-834	138	21	1	1	NUM
cana-834	138	22	)	)	PUNCT
cana-834	138	23	.	.	PUNCT
cana-834	139	1	(	(	PUNCT
cana-834	139	2	and	and	CCONJ
cana-834	139	3	(	(	PUNCT
cana-834	139	4	)	)	PUNCT
cana-834	139	5	.2	.2	NUM
cana-834	139	6	dgvy	dgvy	NOUN
cana-834	139	7	v	v	INTJ
cana-834	139	8	−	−	PROPN
cana-834	140	1	if	if	SCONJ
cana-834	140	2	vx=	vx=	NOUN
cana-834	140	3	there	there	PRON
cana-834	140	4	exists	exist	VERB
cana-834	140	5	between	between	ADP
cana-834	140	6	x	x	PROPN
cana-834	140	7	-	-	PUNCT
cana-834	140	8	y.	y.	NOUN
cana-834	140	9	suppose	suppose	PUNCT
cana-834	140	10	vx	vx	PROPN
cana-834	140	11	if	if	SCONJ
cana-834	140	12	(	(	PUNCT
cana-834	140	13	)	)	PUNCT
cana-834	140	14	dgvv	dgvv	ADV
cana-834	140	15			ADP
cana-834	140	16	1	1	NUM
cana-834	140	17	then	then	ADV
cana-834	140	18	(	(	PUNCT
cana-834	140	19	)	)	PUNCT
cana-834	140	20	.2	.2	NUM
cana-834	140	21	dgv	dgv	PROPN
cana-834	140	22	v	v	NUM
cana-834	140	23			PROPN
cana-834	140	24	this	this	PRON
cana-834	140	25	contradicts	contradict	VERB
cana-834	140	26	the	the	DET
cana-834	140	27	fact	fact	NOUN
cana-834	140	28	that	that	SCONJ
cana-834	140	29	(	(	PUNCT
cana-834	140	30	)	)	PUNCT
cana-834	140	31	.02	.02	NUM
cana-834	141	1	−	−	ADP
cana-834	141	2	dgv	dgv	PROPN
cana-834	141	3	v	v	NOUN
cana-834	141	4	then	then	ADV
cana-834	141	5	(	(	PUNCT
cana-834	141	6	)	)	PUNCT
cana-834	141	7	,	,	PUNCT
cana-834	141	8	1	1	NUM
cana-834	141	9	dgvv	dgvv	NOUN
cana-834	141	10	−	−	VERB
cana-834	142	1	consequently	consequently	ADV
cana-834	142	2	,	,	PUNCT
cana-834	142	3	(	(	PUNCT
cana-834	142	4	)	)	PUNCT
cana-834	142	5	dgv	dgv	PROPN
cana-834	142	6	v	v	INTJ
cana-834	142	7	−2	−2	PROPN
cana-834	142	8	is	be	AUX
cana-834	142	9	a	a	DET
cana-834	142	10	dominating	dominating	NOUN
cana-834	142	11	set	set	VERB
cana-834	142	12	in	in	ADP
cana-834	142	13	vg2	vg2	PROPN
cana-834	142	14	)	)	PUNCT
cana-834	142	15	.	.	PUNCT
cana-834	143	1	(	(	PUNCT
cana-834	143	2	ei	ei	X
cana-834	143	3	(	(	PUNCT
cana-834	143	4	)	)	PUNCT
cana-834	143	5	.2	.2	NUM
cana-834	143	6			PROPN
cana-834	143	7	dgv	dgv	PROPN
cana-834	143	8	v	v	PROPN
cana-834	143	9	suppose	suppose	VERB
cana-834	143	10	(	(	PUNCT
cana-834	143	11	)	)	PUNCT
cana-834	143	12	.2	.2	NUM
cana-834	143	13	dgvx	dgvx	PROPN
cana-834	143	14	v	v	PROPN
cana-834	143	15			PROPN
cana-834	143	16	since	since	SCONJ
cana-834	143	17	(	(	PUNCT
cana-834	143	18	)	)	PUNCT
cana-834	143	19	dgvvx	dgvvx	NOUN
cana-834	143	20	−	−	NOUN
cana-834	143	21	1	1	NUM
cana-834	143	22	,	,	PUNCT
cana-834	143	23	and	and	CCONJ
cana-834	143	24	(	(	PUNCT
cana-834	143	25	)	)	PUNCT
cana-834	143	26	−	−	VERB
cana-834	143	27	dgv	dgv	PROPN
cana-834	143	28	1	1	NUM
cana-834	143	29	contains	contain	VERB
cana-834	143	30	a	a	DET
cana-834	143	31	path	path	NOUN
cana-834	143	32	with	with	ADP
cana-834	143	33	vertices	vertex	NOUN
cana-834	143	34	.vx−	.vx−	PUNCT
cana-834	144	1	thus	thus	ADV
cana-834	144	2	(	(	PUNCT
cana-834	144	3	)	)	PUNCT
cana-834	144	4	−	−	VERB
cana-834	144	5	dggv	dggv	NOUN
cana-834	144	6	21	21	NUM
cana-834	144	7			PROPN
cana-834	144	8	contains	contain	VERB
cana-834	144	9	a	a	DET
cana-834	144	10	tree	tree	NOUN
cana-834	144	11	with	with	ADP
cana-834	144	12	vertices	vertex	NOUN
cana-834	144	13	.vx−	.vx−	PUNCT
cana-834	144	14	suppose	suppose	VERB
cana-834	144	15	(	(	PUNCT
cana-834	144	16	)	)	PUNCT
cana-834	144	17	yxdgvyx	yxdgvyx	NOUN
cana-834	144	18	v	v	NUM
cana-834	144	19	−	−	PROPN
cana-834	144	20	,	,	PUNCT
cana-834	144	21	,	,	PUNCT
cana-834	144	22	2	2	NUM
cana-834	144	23	for	for	ADP
cana-834	144	24	some	some	PRON
cana-834	144	25	(	(	PUNCT
cana-834	144	26	)	)	PUNCT
cana-834	144	27	.gvv	.gvv	NOUN
cana-834	144	28	if	if	SCONJ
cana-834	144	29	,	,	PUNCT
cana-834	144	30	dv	dv	ADP
cana-834	144	31	then	then	ADV
cana-834	144	32	(	(	PUNCT
cana-834	144	33	)	)	PUNCT
cana-834	144	34	dgv	dgv	PROPN
cana-834	144	35	v	v	PROPN
cana-834	144	36	2	2	NOUN
cana-834	144	37	.this	.this	PRON
cana-834	144	38	contradicts	contradict	VERB
cana-834	144	39	the	the	DET
cana-834	144	40	fact	fact	NOUN
cana-834	144	41	that	that	SCONJ
cana-834	144	42	(	(	PUNCT
cana-834	144	43	)	)	PUNCT
cana-834	144	44	.2	.2	NUM
cana-834	144	45	−	−	PROPN
cana-834	144	46	dgv	dgv	PROPN
cana-834	144	47	v	v	NOUN
cana-834	144	48	thus	thus	ADV
cana-834	144	49	dv	dv	NOUN
cana-834	144	50	(	(	PUNCT
cana-834	144	51	i.e	i.e	X
cana-834	144	52	)	)	PUNCT
cana-834	144	53	.	.	PUNCT
cana-834	144	54	)	)	PUNCT
cana-834	145	1	(	(	PUNCT
cana-834	145	2	1	1	X
cana-834	145	3	dgvv	dgvv	ADV
cana-834	145	4	−	−	VERB
cana-834	146	1	now	now	ADV
cana-834	146	2	there	there	PRON
cana-834	146	3	exists	exist	VERB
cana-834	146	4	a	a	DET
cana-834	146	5	path	path	NOUN
cana-834	146	6	with	with	ADP
cana-834	146	7	vertices	vertex	NOUN
cana-834	146	8	yvx	yvx	PROPN
cana-834	146	9	,	,	PUNCT
cana-834	146	10	,	,	PUNCT
cana-834	146	11	in	in	ADP
cana-834	146	12	(	(	PUNCT
cana-834	146	13	)	)	PUNCT
cana-834	146	14	.21	.21	NUM
cana-834	146	15	dggv	dggv	NOUN
cana-834	146	16	−	−	NOUN
cana-834	146	17	suppose	suppose	VERB
cana-834	146	18	(	(	PUNCT
cana-834	146	19	)	)	PUNCT
cana-834	146	20	vgvx	vgvx	NOUN
cana-834	146	21	2	2	NUM
cana-834	146	22	and	and	CCONJ
cana-834	146	23	(	(	PUNCT
cana-834	146	24	)	)	PUNCT
cana-834	146	25	wgvy	wgvy	NOUN
cana-834	146	26	2	2	NUM
cana-834	146	27	for	for	ADP
cana-834	146	28	some	some	PRON
cana-834	146	29	(	(	PUNCT
cana-834	146	30	)	)	PUNCT
cana-834	146	31	.	.	PUNCT
cana-834	147	1	,	,	PUNCT
cana-834	147	2	,	,	PUNCT
cana-834	147	3	wvgvwv	wvgvwv	PROPN
cana-834	147	4			PROPN
cana-834	147	5	then	then	ADV
cana-834	147	6	(	(	PUNCT
cana-834	147	7	)	)	PUNCT
cana-834	147	8	dgvx	dgvx	NOUN
cana-834	147	9	v	v	NOUN
cana-834	147	10	−	−	PROPN
cana-834	147	11	2	2	NUM
cana-834	147	12	and	and	CCONJ
cana-834	147	13	(	(	PUNCT
cana-834	147	14	)	)	PUNCT
cana-834	148	1	dgvy	dgvy	INTJ
cana-834	148	2	w	w	NOUN
cana-834	148	3	−	−	NOUN
cana-834	148	4	2	2	NUM
cana-834	148	5	if	if	SCONJ
cana-834	148	6	dv	dv	NOUN
cana-834	148	7	or	or	CCONJ
cana-834	148	8	dw	dw	NOUN
cana-834	148	9	then	then	ADV
cana-834	148	10	(	(	PUNCT
cana-834	148	11	)	)	PUNCT
cana-834	148	12	dgv	dgv	PROPN
cana-834	148	13	v	v	PROPN
cana-834	148	14	2	2	NOUN
cana-834	148	15	and	and	CCONJ
cana-834	148	16	(	(	PUNCT
cana-834	148	17	)	)	PUNCT
cana-834	149	1	.2	.2	NUM
cana-834	149	2	dgv	dgv	PROPN
cana-834	149	3	w	w	PROPN
cana-834	149	4			PROPN
cana-834	149	5	this	this	PRON
cana-834	149	6	contradicts	contradict	VERB
cana-834	149	7	the	the	DET
cana-834	149	8	facts	fact	NOUN
cana-834	149	9	that	that	SCONJ
cana-834	149	10	(	(	PUNCT
cana-834	149	11	)	)	PUNCT
cana-834	149	12	−	−	PROPN
cana-834	149	13	dgv	dgv	PROPN
cana-834	149	14	v	v	NUM
cana-834	149	15	2	2	NUM
cana-834	149	16	and	and	CCONJ
cana-834	149	17	(	(	PUNCT
cana-834	149	18	)	)	PUNCT
cana-834	149	19	−	−	PROPN
cana-834	150	1	dgv	dgv	PROPN
cana-834	150	2	w	w	PROPN
cana-834	150	3	2	2	NUM
cana-834	150	4	.	.	PUNCT
cana-834	151	1	thus	thus	ADV
cana-834	151	2	dwv	dwv	PROPN
cana-834	151	3			PROPN
cana-834	151	4	,	,	PUNCT
cana-834	151	5	that	that	ADV
cana-834	151	6	is	is	ADV
cana-834	151	7	(	(	PUNCT
cana-834	151	8	)	)	PUNCT
cana-834	151	9	(	(	PUNCT
cana-834	151	10	)	)	PUNCT
cana-834	151	11	.	.	PUNCT
cana-834	152	1	,	,	PUNCT
cana-834	152	2	211	211	NUM
cana-834	152	3	dggvdgvwv	dggvdgvwv	NOUN
cana-834	152	4	−−	−−	PROPN
cana-834	152	5			PROPN
cana-834	152	6	since	since	SCONJ
cana-834	152	7	(	(	PUNCT
cana-834	152	8	)	)	PUNCT
cana-834	152	9	−	−	VERB
cana-834	152	10	dgv	dgv	PROPN
cana-834	152	11	1	1	NUM
cana-834	152	12	is	be	AUX
cana-834	152	13	a	a	DET
cana-834	152	14	tree	tree	NOUN
cana-834	152	15	.	.	PUNCT
cana-834	153	1	there	there	PRON
cana-834	153	2	is	be	VERB
cana-834	153	3	a	a	DET
cana-834	153	4	tree	tree	NOUN
cana-834	153	5	with	with	ADP
cana-834	153	6	support	support	NOUN
cana-834	153	7	vertices	vertice	VERB
cana-834	153	8	v	v	NOUN
cana-834	153	9	and	and	CCONJ
cana-834	153	10	w	w	NOUN
cana-834	153	11	in	in	ADP
cana-834	153	12	(	(	PUNCT
cana-834	153	13	)	)	PUNCT
cana-834	153	14	dggv	dggv	PROPN
cana-834	153	15	−21	−21	PROPN
cana-834	154	1			PROPN
cana-834	154	2	.hence	.hence	ADP
cana-834	154	3	(	(	PUNCT
cana-834	154	4	)	)	PUNCT
cana-834	154	5	−	−	VERB
cana-834	154	6	dggv	dggv	NOUN
cana-834	154	7	21	21	NUM
cana-834	154	8			PROPN
cana-834	154	9	is	be	AUX
cana-834	154	10	a	a	DET
cana-834	154	11	tree	tree	NOUN
cana-834	154	12	.	.	PUNCT
cana-834	155	1	hence	hence	ADV
cana-834	155	2	d	d	PROPN
cana-834	155	3	is	be	AUX
cana-834	155	4	a	a	DET
cana-834	155	5	sna	sna	PROPN
cana-834	155	6	complementary	complementary	ADJ
cana-834	155	7	tree	tree	NOUN
cana-834	155	8	dominating	dominate	VERB
cana-834	155	9	in	in	ADP
cana-834	155	10	21	21	NUM
cana-834	155	11	gg	gg	PROPN
cana-834	155	12			PROPN
cana-834	155	13	.	.	PUNCT
cana-834	156	1	theorem	theorem	VERB
cana-834	156	2	2.2	2.2	NUM
cana-834	156	3	.	.	PUNCT
cana-834	157	1	let	let	VERB
cana-834	157	2	1	1	NUM
cana-834	157	3	g	g	NOUN
cana-834	157	4	and	and	CCONJ
cana-834	157	5	2	2	NUM
cana-834	157	6	g	g	NOUN
cana-834	157	7	be	be	VERB
cana-834	157	8	any	any	DET
cana-834	157	9	sna	sna	PROPN
cana-834	157	10	graph	graph	NOUN
cana-834	157	11	.	.	PUNCT
cana-834	158	1	then	then	ADV
cana-834	158	2	(	(	PUNCT
cana-834	158	3	)	)	PUNCT
cana-834	158	4	(	(	PUNCT
cana-834	158	5	)	)	PUNCT
cana-834	158	6	(	(	PUNCT
cana-834	158	7	)	)	SYM
cana-834	158	8	1	1	NUM
cana-834	158	9	2	2	NUM
cana-834	158	10	1	1	NUM
cana-834	158	11	22	22	NUM
cana-834	158	12	1	1	NUM
cana-834	158	13	.sna	.sna	NOUN
cana-834	158	14	g	g	PROPN
cana-834	158	15	g	g	PROPN
cana-834	158	16	g	g	PROPN
cana-834	158	17	g	g	PROPN
cana-834	158	18			VERB
cana-834	158	19	−	−	PROPN
cana-834	158	20	g1	g1	PROPN
cana-834	158	21	is	be	AUX
cana-834	158	22	not	not	PART
cana-834	158	23	a	a	DET
cana-834	158	24	tree	tree	NOUN
cana-834	158	25	.	.	PUNCT
cana-834	159	1	proof	proof	NOUN
cana-834	159	2	.	.	PUNCT
cana-834	160	1	let	let	VERB
cana-834	160	2	.1	.1	NUM
cana-834	160	3	ng	ng	PROPN
cana-834	161	1	=	=	PRON
cana-834	161	2	suppose	suppose	VERB
cana-834	161	3	there	there	PRON
cana-834	161	4	exist	exist	VERB
cana-834	161	5	d	d	PRON
cana-834	161	6	such	such	ADJ
cana-834	161	7	that	that	SCONJ
cana-834	161	8	:	:	PUNCT
cana-834	161	9	communications	communication	NOUN
cana-834	161	10	on	on	ADP
cana-834	161	11	applied	apply	VERB
cana-834	161	12	nonlinear	nonlinear	ADJ
cana-834	161	13	analysis	analysis	NOUN
cana-834	161	14	issn	issn	NOUN
cana-834	161	15	:	:	PUNCT
cana-834	161	16	1074	1074	NUM
cana-834	161	17	-	-	PUNCT
cana-834	161	18	133x	133x	NUM
cana-834	161	19	vol	vol	NOUN
cana-834	161	20	31	31	NUM
cana-834	161	21	no	no	NOUN
cana-834	161	22	.	.	PUNCT
cana-834	162	1	4s	4s	NUM
cana-834	162	2	(	(	PUNCT
cana-834	162	3	2024	2024	NUM
cana-834	162	4	)	)	PUNCT
cana-834	162	5	169	169	NUM
cana-834	162	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	162	7	(	(	PUNCT
cana-834	162	8	)	)	PUNCT
cana-834	162	9	(	(	PUNCT
cana-834	162	10	)	)	PUNCT
cana-834	162	11	(	(	PUNCT
cana-834	162	12	)	)	SYM
cana-834	162	13	1	1	NUM
cana-834	162	14	2	2	NUM
cana-834	162	15	1	1	NUM
cana-834	162	16	22	22	NUM
cana-834	162	17	1	1	NUM
cana-834	162	18	.sna	.sna	NOUN
cana-834	162	19	g	g	PROPN
cana-834	162	20	g	g	PROPN
cana-834	162	21	g	g	PROPN
cana-834	162	22	g	g	PROPN
cana-834	162	23			AUX
cana-834	162	24	−	−	PROPN
cana-834	162	25	theorem	theorem	VERB
cana-834	162	26	2.3	2.3	NUM
cana-834	162	27	.	.	PUNCT
cana-834	163	1	let	let	VERB
cana-834	163	2	g1be	g1be	VERB
cana-834	163	3	a	a	DET
cana-834	163	4	tree	tree	NOUN
cana-834	163	5	and	and	CCONJ
cana-834	163	6	g2	g2	PROPN
cana-834	163	7	be	be	VERB
cana-834	163	8	any	any	DET
cana-834	163	9	sna	sna	PROPN
cana-834	163	10	graph	graph	NOUN
cana-834	163	11	respectively	respectively	ADV
cana-834	163	12	.	.	PUNCT
cana-834	164	1	then	then	ADV
cana-834	164	2	(	(	PUNCT
cana-834	164	3	)	)	PUNCT
cana-834	164	4	(	(	PUNCT
cana-834	164	5	)	)	SYM
cana-834	164	6	1	1	NUM
cana-834	164	7	2	2	NUM
cana-834	164	8	1	1	NUM
cana-834	164	9	2	2	NUM
cana-834	164	10	.sna	.sna	NOUN
cana-834	164	11	g	g	PROPN
cana-834	164	12	g	g	PROPN
cana-834	164	13	g	g	PROPN
cana-834	164	14	g	g	NOUN
cana-834	164	15	=	=	NOUN
cana-834	164	16	proof	proof	NOUN
cana-834	164	17	.	.	PUNCT
cana-834	165	1	for	for	ADP
cana-834	165	2	each	each	PRON
cana-834	165	3	(	(	PUNCT
cana-834	165	4	)	)	PUNCT
cana-834	165	5	.1gvv	.1gvv	PUNCT
cana-834	165	6	let	let	VERB
cana-834	165	7	v	v	AUX
cana-834	165	8	g2	g2	PROPN
cana-834	165	9	be	be	AUX
cana-834	165	10	a	a	DET
cana-834	165	11	copy	copy	NOUN
cana-834	165	12	of	of	ADP
cana-834	165	13	g2	g2	PROPN
cana-834	165	14	corresponding	correspond	VERB
cana-834	165	15	to	to	ADP
cana-834	165	16	vertex	vertex	NOUN
cana-834	165	17	.v	.v	ADV
cana-834	165	18	further	far	ADV
cana-834	165	19	,	,	PUNCT
cana-834	165	20	for	for	ADP
cana-834	165	21	each	each	PRON
cana-834	165	22	(	(	PUNCT
cana-834	165	23	)	)	PUNCT
cana-834	165	24	.1gvv	.1gvv	PUNCT
cana-834	166	1	let	let	VERB
cana-834	166	2	vd	vd	PART
cana-834	166	3	be	be	AUX
cana-834	166	4	a	a	DET
cana-834	166	5	minimum	minimum	ADJ
cana-834	166	6	dominating	dominating	NOUN
cana-834	166	7	set	set	VERB
cana-834	166	8	in	in	ADP
cana-834	166	9	.2	.2	NUM
cana-834	166	10	vg	vg	NOUN
cana-834	166	11	by	by	ADP
cana-834	166	12	definition	definition	NOUN
cana-834	166	13	(	(	PUNCT
cana-834	166	14	)	)	PUNCT
cana-834	166	15			PROPN
cana-834	166	16	1gvv	1gvv	NUM
cana-834	166	17	vdd	vdd	PROPN
cana-834	166	18			NOUN
cana-834	166	19	=	=	PUNCT
cana-834	166	20	is	be	AUX
cana-834	166	21	a	a	DET
cana-834	166	22	complementary	complementary	ADJ
cana-834	166	23	tree	tree	NOUN
cana-834	166	24	dominating	dominating	NOUN
cana-834	166	25	set	set	VERB
cana-834	166	26	in	in	ADP
cana-834	166	27	.21	.21	NUM
cana-834	166	28	gg	gg	NOUN
cana-834	166	29			PRON
cana-834	166	30	thus	thus	ADV
cana-834	166	31	(	(	PUNCT
cana-834	166	32	)	)	PUNCT
cana-834	166	33	(	(	PUNCT
cana-834	166	34	)	)	PUNCT
cana-834	166	35	(	(	PUNCT
cana-834	166	36	)	)	PUNCT
cana-834	166	37	(	(	PUNCT
cana-834	166	38	)	)	PUNCT
cana-834	166	39	1	1	NUM
cana-834	166	40	1	1	NUM
cana-834	166	41	1	1	NUM
cana-834	166	42	2	2	NUM
cana-834	166	43	1	1	NUM
cana-834	166	44	2	2	NUM
cana-834	166	45	sna	sna	PROPN
cana-834	166	46	v	v	NOUN
cana-834	166	47	v	v	NOUN
cana-834	166	48	v	v	ADP
cana-834	166	49	g	g	NOUN
cana-834	166	50	v	v	NUM
cana-834	166	51	v	v	NOUN
cana-834	166	52	v	v	NOUN
cana-834	166	53	g	g	NOUN
cana-834	166	54	g	g	PROPN
cana-834	166	55	g	g	PROPN
cana-834	167	1	d	d	PROPN
cana-834	167	2	d	d	PROPN
cana-834	167	3	d	d	X
cana-834	167	4	g	g	PROPN
cana-834	167	5	g	g	PROPN
cana-834	167	6			NUM
cana-834	167	7			NUM
cana-834	167	8			NOUN
cana-834	167	9			NOUN
cana-834	167	10			NUM
cana-834	167	11	=	=	PUNCT
cana-834	168	1	=	=	PUNCT
cana-834	168	2	=	=	SYM
cana-834	168	3			X
cana-834	168	4	therefore	therefore	ADV
cana-834	168	5	,	,	PUNCT
cana-834	168	6	(	(	PUNCT
cana-834	168	7	)	)	PUNCT
cana-834	168	8	(	(	PUNCT
cana-834	168	9	)	)	SYM
cana-834	168	10	1	1	NUM
cana-834	168	11	2	2	NUM
cana-834	168	12	1	1	NUM
cana-834	168	13	2	2	NUM
cana-834	168	14	.sna	.sna	NOUN
cana-834	168	15	g	g	PROPN
cana-834	168	16	g	g	PROPN
cana-834	168	17	g	g	PROPN
cana-834	168	18	g	g	PROPN
cana-834	168	19			X
cana-834	169	1	3	3	X
cana-834	169	2	.	.	X
cana-834	169	3	challenges	challenge	NOUN
cana-834	169	4	in	in	ADP
cana-834	169	5	social	social	ADJ
cana-834	169	6	network	network	NOUN
cana-834	169	7	analysis	analysis	NOUN
cana-834	169	8	to	to	PART
cana-834	169	9	discover	discover	VERB
cana-834	169	10	patterns	pattern	NOUN
cana-834	169	11	,	,	PUNCT
cana-834	169	12	relationships	relationship	NOUN
cana-834	169	13	,	,	PUNCT
cana-834	169	14	and	and	CCONJ
cana-834	169	15	structures	structure	NOUN
cana-834	169	16	within	within	ADP
cana-834	169	17	social	social	ADJ
cana-834	169	18	networks	network	NOUN
cana-834	169	19	,	,	PUNCT
cana-834	169	20	social	social	ADJ
cana-834	169	21	network	network	NOUN
cana-834	169	22	analysis	analysis	NOUN
cana-834	169	23	(	(	PUNCT
cana-834	169	24	sna	sna	PROPN
cana-834	169	25	)	)	PUNCT
cana-834	169	26	uses	use	VERB
cana-834	169	27	intricate	intricate	ADJ
cana-834	169	28	data	datum	NOUN
cana-834	169	29	sets	set	NOUN
cana-834	169	30	and	and	CCONJ
cana-834	169	31	sophisticated	sophisticated	ADJ
cana-834	169	32	computational	computational	ADJ
cana-834	169	33	techniques	technique	NOUN
cana-834	170	1	[	[	X
cana-834	170	2	11	11	NUM
cana-834	170	3	]	]	PUNCT
cana-834	170	4	.	.	PUNCT
cana-834	171	1	scalability	scalability	NOUN
cana-834	171	2	,	,	PUNCT
cana-834	171	3	the	the	DET
cana-834	171	4	dynamic	dynamic	ADJ
cana-834	171	5	nature	nature	NOUN
cana-834	171	6	of	of	ADP
cana-834	171	7	large	large	ADJ
cana-834	171	8	-	-	PUNCT
cana-834	171	9	scale	scale	NOUN
cana-834	171	10	social	social	ADJ
cana-834	171	11	networks	network	NOUN
cana-834	171	12	,	,	PUNCT
cana-834	171	13	and	and	CCONJ
cana-834	171	14	the	the	DET
cana-834	171	15	computational	computational	ADJ
cana-834	171	16	complexity	complexity	NOUN
cana-834	171	17	of	of	ADP
cana-834	171	18	key	key	ADJ
cana-834	171	19	metrics	metric	NOUN
cana-834	171	20	are	be	AUX
cana-834	171	21	some	some	PRON
cana-834	171	22	of	of	ADP
cana-834	171	23	the	the	DET
cana-834	171	24	primary	primary	ADJ
cana-834	171	25	obstacles	obstacle	NOUN
cana-834	171	26	encountered	encounter	VERB
cana-834	171	27	during	during	ADP
cana-834	171	28	the	the	DET
cana-834	171	29	analysis	analysis	NOUN
cana-834	171	30	process	process	NOUN
cana-834	171	31	.	.	PUNCT
cana-834	172	1	with	with	ADP
cana-834	172	2	millions	million	NOUN
cana-834	172	3	of	of	ADP
cana-834	172	4	nodes	node	NOUN
cana-834	172	5	and	and	CCONJ
cana-834	172	6	billions	billion	NOUN
cana-834	172	7	of	of	ADP
cana-834	172	8	edges	edge	NOUN
cana-834	172	9	in	in	ADP
cana-834	172	10	modern	modern	ADJ
cana-834	172	11	social	social	ADJ
cana-834	172	12	networks	network	NOUN
cana-834	172	13	,	,	PUNCT
cana-834	172	14	data	datum	NOUN
cana-834	172	15	storage	storage	NOUN
cana-834	172	16	,	,	PUNCT
cana-834	172	17	processing	processing	NOUN
cana-834	172	18	,	,	PUNCT
cana-834	172	19	and	and	CCONJ
cana-834	172	20	analysis	analysis	NOUN
cana-834	172	21	present	present	ADJ
cana-834	172	22	significant	significant	ADJ
cana-834	172	23	challenges	challenge	NOUN
cana-834	172	24	.	.	PUNCT
cana-834	173	1	in	in	ADP
cana-834	173	2	these	these	DET
cana-834	173	3	networks	network	NOUN
cana-834	173	4	,	,	PUNCT
cana-834	173	5	the	the	DET
cana-834	173	6	volume	volume	NOUN
cana-834	173	7	of	of	ADP
cana-834	173	8	data	datum	NOUN
cana-834	173	9	is	be	AUX
cana-834	173	10	often	often	ADV
cana-834	173	11	too	too	ADV
cana-834	173	12	much	much	ADJ
cana-834	173	13	for	for	SCONJ
cana-834	173	14	traditional	traditional	ADJ
cana-834	173	15	graph	graph	NOUN
cana-834	173	16	algorithms	algorithm	NOUN
cana-834	173	17	to	to	PART
cana-834	173	18	handle	handle	VERB
cana-834	173	19	,	,	PUNCT
cana-834	173	20	which	which	PRON
cana-834	173	21	results	result	VERB
cana-834	173	22	in	in	ADP
cana-834	173	23	high	high	ADJ
cana-834	173	24	memory	memory	NOUN
cana-834	173	25	and	and	CCONJ
cana-834	173	26	computational	computational	ADJ
cana-834	173	27	costs	cost	NOUN
cana-834	173	28	[	[	X
cana-834	173	29	12	12	NUM
cana-834	173	30	]	]	PUNCT
cana-834	173	31	.	.	PUNCT
cana-834	174	1	effective	effective	ADJ
cana-834	174	2	data	data	NOUN
cana-834	174	3	management	management	NOUN
cana-834	174	4	strategies	strategy	NOUN
cana-834	174	5	are	be	AUX
cana-834	174	6	required	require	VERB
cana-834	174	7	due	due	ADP
cana-834	174	8	to	to	ADP
cana-834	174	9	the	the	DET
cana-834	174	10	sheer	sheer	ADJ
cana-834	174	11	volume	volume	NOUN
cana-834	174	12	of	of	ADP
cana-834	174	13	data	datum	NOUN
cana-834	174	14	in	in	ADP
cana-834	174	15	large	large	ADJ
cana-834	174	16	social	social	ADJ
cana-834	174	17	networks	network	NOUN
cana-834	174	18	.	.	PUNCT
cana-834	175	1	keeping	keep	VERB
cana-834	175	2	and	and	CCONJ
cana-834	175	3	retrieving	retrieve	VERB
cana-834	175	4	such	such	ADJ
cana-834	175	5	huge	huge	ADJ
cana-834	175	6	amounts	amount	NOUN
cana-834	175	7	of	of	ADP
cana-834	175	8	data	datum	NOUN
cana-834	175	9	can	can	AUX
cana-834	175	10	take	take	VERB
cana-834	175	11	a	a	DET
cana-834	175	12	lot	lot	NOUN
cana-834	175	13	of	of	ADP
cana-834	175	14	resources	resource	NOUN
cana-834	175	15	.	.	PUNCT
cana-834	176	1	large	large	ADJ
cana-834	176	2	-	-	PUNCT
cana-834	176	3	scale	scale	NOUN
cana-834	176	4	network	network	NOUN
cana-834	176	5	analysis	analysis	NOUN
cana-834	176	6	necessitates	necessitate	VERB
cana-834	176	7	a	a	DET
cana-834	176	8	lot	lot	NOUN
cana-834	176	9	of	of	ADP
cana-834	176	10	(	(	PUNCT
cana-834	176	11	)	)	PUNCT
cana-834	176	12	(	(	PUNCT
cana-834	176	13	)	)	PUNCT
cana-834	176	14	(	(	PUNCT
cana-834	176	15	)	)	PUNCT
cana-834	176	16	(	(	PUNCT
cana-834	176	17	)	)	PUNCT
cana-834	176	18	(	(	PUNCT
cana-834	176	19	)	)	PUNCT
cana-834	176	20	(	(	PUNCT
cana-834	176	21	)	)	PUNCT
cana-834	176	22	(	(	PUNCT
cana-834	176	23	)	)	PUNCT
cana-834	176	24	(	(	PUNCT
cana-834	176	25	)	)	PUNCT
cana-834	176	26	(	(	PUNCT
cana-834	176	27	)	)	PUNCT
cana-834	176	28	(	(	PUNCT
cana-834	176	29	)	)	PUNCT
cana-834	176	30	21	21	NUM
cana-834	176	31	2	2	NUM
cana-834	176	32	22	22	NUM
cana-834	176	33	22	22	NUM
cana-834	176	34	2	2	NUM
cana-834	176	35	2	2	NUM
cana-834	176	36	2	2	NUM
cana-834	176	37	12	12	NUM
cana-834	176	38	12	12	NUM
cana-834	176	39	222	222	NUM
cana-834	176	40	2	2	NUM
cana-834	176	41	)	)	PUNCT
cana-834	176	42	.	.	PUNCT
cana-834	177	1	(	(	PUNCT
cana-834	177	2	2	2	NUM
cana-834	177	3	since.2)2	since.2)2	NOUN
cana-834	177	4	(	(	PUNCT
cana-834	177	5	gg	gg	PROPN
cana-834	177	6	gn	gn	PROPN
cana-834	177	7	ggnd	ggnd	PROPN
cana-834	177	8	ggei	ggei	NOUN
cana-834	177	9	g	g	PROPN
cana-834	177	10	g	g	PROPN
cana-834	177	11	ggnd	ggnd	NOUN
cana-834	177	12			NUM
cana-834	177	13			NUM
cana-834	177	14			PUNCT
cana-834	177	15			PRON
cana-834	177	16			NUM
cana-834	178	1	−	−	PROPN
cana-834	178	2	−=	−=	ADJ
cana-834	179	1	+	+	NOUN
cana-834	179	2	−	−	ADJ
cana-834	179	3			NOUN
cana-834	179	4	+	+	NOUN
cana-834	179	5	−	−	ADJ
cana-834	179	6	communications	communication	NOUN
cana-834	179	7	on	on	ADP
cana-834	179	8	applied	apply	VERB
cana-834	179	9	nonlinear	nonlinear	ADJ
cana-834	179	10	analysis	analysis	NOUN
cana-834	179	11	issn	issn	NOUN
cana-834	179	12	:	:	PUNCT
cana-834	179	13	1074	1074	NUM
cana-834	179	14	-	-	PUNCT
cana-834	179	15	133x	133x	NUM
cana-834	179	16	vol	vol	NOUN
cana-834	179	17	31	31	NUM
cana-834	179	18	no	no	NOUN
cana-834	179	19	.	.	PUNCT
cana-834	180	1	4s	4s	NUM
cana-834	180	2	(	(	PUNCT
cana-834	180	3	2024	2024	NUM
cana-834	180	4	)	)	PUNCT
cana-834	180	5	170	170	NUM
cana-834	180	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	180	7	processing	processing	NOUN
cana-834	180	8	power	power	NOUN
cana-834	180	9	.	.	PUNCT
cana-834	181	1	to	to	PART
cana-834	181	2	run	run	VERB
cana-834	181	3	effectively	effectively	ADV
cana-834	181	4	on	on	ADP
cana-834	181	5	high	high	ADJ
cana-834	181	6	-	-	PUNCT
cana-834	181	7	performance	performance	NOUN
cana-834	181	8	computing	computing	NOUN
cana-834	181	9	infrastructures	infrastructure	NOUN
cana-834	181	10	,	,	PUNCT
cana-834	181	11	algorithms	algorithm	NOUN
cana-834	181	12	must	must	AUX
cana-834	181	13	be	be	AUX
cana-834	181	14	optimized	optimize	VERB
cana-834	181	15	[	[	PUNCT
cana-834	181	16	13	13	NUM
cana-834	181	17	]	]	PUNCT
cana-834	181	18	.	.	PUNCT
cana-834	182	1	large	large	ADJ
cana-834	182	2	graphs	graph	NOUN
cana-834	182	3	can	can	AUX
cana-834	182	4	use	use	VERB
cana-834	182	5	a	a	DET
cana-834	182	6	lot	lot	NOUN
cana-834	182	7	of	of	ADP
cana-834	182	8	memory	memory	NOUN
cana-834	182	9	,	,	PUNCT
cana-834	182	10	so	so	SCONJ
cana-834	182	11	algorithms	algorithm	NOUN
cana-834	182	12	and	and	CCONJ
cana-834	182	13	data	datum	NOUN
cana-834	182	14	structures	structure	NOUN
cana-834	182	15	that	that	PRON
cana-834	182	16	use	use	VERB
cana-834	182	17	less	less	ADJ
cana-834	182	18	memory	memory	NOUN
cana-834	182	19	are	be	AUX
cana-834	182	20	needed	need	VERB
cana-834	182	21	.	.	PUNCT
cana-834	183	1	nodes	node	NOUN
cana-834	183	2	and	and	CCONJ
cana-834	183	3	edges	edge	NOUN
cana-834	183	4	in	in	ADP
cana-834	183	5	social	social	ADJ
cana-834	183	6	networks	network	NOUN
cana-834	183	7	are	be	AUX
cana-834	183	8	always	always	ADV
cana-834	183	9	being	be	AUX
cana-834	183	10	added	add	VERB
cana-834	183	11	,	,	PUNCT
cana-834	183	12	changed	change	VERB
cana-834	183	13	,	,	PUNCT
cana-834	183	14	or	or	CCONJ
cana-834	183	15	taken	take	VERB
cana-834	183	16	out	out	ADP
cana-834	183	17	of	of	ADP
cana-834	183	18	sync	sync	NOUN
cana-834	183	19	.	.	PUNCT
cana-834	184	1	the	the	DET
cana-834	184	2	dynamic	dynamic	ADJ
cana-834	184	3	nature	nature	NOUN
cana-834	184	4	poses	pose	VERB
cana-834	184	5	several	several	ADJ
cana-834	184	6	difficulties	difficulty	NOUN
cana-834	184	7	for	for	ADP
cana-834	184	8	analysis	analysis	NOUN
cana-834	184	9	[	[	X
cana-834	184	10	14	14	NUM
cana-834	184	11	]	]	SYM
cana-834	184	12	:	:	PUNCT
cana-834	184	13	•	•	NOUN
cana-834	184	14	for	for	ADP
cana-834	184	15	an	an	DET
cana-834	184	16	accurate	accurate	ADJ
cana-834	184	17	and	and	CCONJ
cana-834	184	18	current	current	ADJ
cana-834	184	19	representation	representation	NOUN
cana-834	184	20	of	of	ADP
cana-834	184	21	the	the	DET
cana-834	184	22	network	network	NOUN
cana-834	184	23	to	to	PART
cana-834	184	24	be	be	AUX
cana-834	184	25	maintained	maintain	VERB
cana-834	184	26	,	,	PUNCT
cana-834	184	27	algorithms	algorithm	NOUN
cana-834	184	28	must	must	AUX
cana-834	184	29	be	be	AUX
cana-834	184	30	able	able	ADJ
cana-834	184	31	to	to	PART
cana-834	184	32	handle	handle	VERB
cana-834	184	33	updates	update	NOUN
cana-834	184	34	in	in	ADP
cana-834	184	35	real	real	ADJ
cana-834	184	36	time	time	NOUN
cana-834	184	37	.	.	PUNCT
cana-834	185	1	•	•	NUM
cana-834	185	2	methods	method	NOUN
cana-834	185	3	based	base	VERB
cana-834	185	4	on	on	ADP
cana-834	185	5	batch	batch	NOUN
cana-834	185	6	processing	processing	NOUN
cana-834	185	7	are	be	AUX
cana-834	185	8	frequently	frequently	ADV
cana-834	185	9	incompatible	incompatible	ADJ
cana-834	185	10	with	with	ADP
cana-834	185	11	dynamic	dynamic	ADJ
cana-834	185	12	networks	network	NOUN
cana-834	185	13	.	.	PUNCT
cana-834	186	1	it	it	PRON
cana-834	186	2	is	be	AUX
cana-834	186	3	necessary	necessary	ADJ
cana-834	186	4	to	to	PART
cana-834	186	5	use	use	VERB
cana-834	186	6	incremental	incremental	ADJ
cana-834	186	7	algorithms	algorithm	NOUN
cana-834	186	8	that	that	PRON
cana-834	186	9	can	can	AUX
cana-834	186	10	update	update	VERB
cana-834	186	11	results	result	NOUN
cana-834	186	12	based	base	VERB
cana-834	186	13	on	on	ADP
cana-834	186	14	network	network	NOUN
cana-834	186	15	changes	change	NOUN
cana-834	186	16	.	.	PUNCT
cana-834	187	1	•	•	NOUN
cana-834	187	2	understanding	understand	VERB
cana-834	187	3	how	how	SCONJ
cana-834	187	4	organizations	organization	NOUN
cana-834	187	5	develop	develop	VERB
cana-834	187	6	over	over	ADP
cana-834	187	7	the	the	DET
cana-834	187	8	long	long	ADJ
cana-834	187	9	haul	haul	NOUN
cana-834	187	10	requires	require	VERB
cana-834	187	11	transient	transient	ADJ
cana-834	187	12	investigation	investigation	NOUN
cana-834	187	13	procedures	procedure	NOUN
cana-834	187	14	that	that	PRON
cana-834	187	15	can	can	AUX
cana-834	187	16	follow	follow	VERB
cana-834	187	17	and	and	CCONJ
cana-834	187	18	examine	examine	VERB
cana-834	187	19	changes	change	NOUN
cana-834	187	20	across	across	ADP
cana-834	187	21	different	different	ADJ
cana-834	187	22	time	time	NOUN
cana-834	187	23	focuses	focus	VERB
cana-834	187	24	.	.	PUNCT
cana-834	188	1	computation	computation	NOUN
cana-834	188	2	is	be	AUX
cana-834	188	3	required	require	VERB
cana-834	188	4	for	for	ADP
cana-834	188	5	some	some	PRON
cana-834	188	6	of	of	ADP
cana-834	188	7	sna	sna	PROPN
cana-834	188	8	's	's	PART
cana-834	188	9	most	most	ADV
cana-834	188	10	useful	useful	ADJ
cana-834	188	11	metrics	metric	NOUN
cana-834	188	12	,	,	PUNCT
cana-834	188	13	such	such	ADJ
cana-834	188	14	as	as	ADP
cana-834	188	15	betweenness	betweenness	NOUN
cana-834	188	16	centrality	centrality	NOUN
cana-834	188	17	and	and	CCONJ
cana-834	188	18	clustering	clustering	ADJ
cana-834	188	19	coefficients	coefficient	NOUN
cana-834	188	20	.	.	PUNCT
cana-834	189	1	these	these	DET
cana-834	189	2	metrics	metric	NOUN
cana-834	189	3	can	can	AUX
cana-834	189	4	be	be	AUX
cana-834	189	5	prohibitively	prohibitively	ADV
cana-834	189	6	expensive	expensive	ADJ
cana-834	189	7	to	to	PART
cana-834	189	8	calculate	calculate	VERB
cana-834	189	9	for	for	ADP
cana-834	189	10	massive	massive	ADJ
cana-834	189	11	networks	network	NOUN
cana-834	189	12	[	[	X
cana-834	189	13	15	15	NUM
cana-834	189	14	]	]	PUNCT
cana-834	189	15	.	.	PUNCT
cana-834	190	1	requires	require	VERB
cana-834	190	2	the	the	DET
cana-834	190	3	calculation	calculation	NOUN
cana-834	190	4	of	of	ADP
cana-834	190	5	the	the	DET
cana-834	190	6	shortest	short	ADJ
cana-834	190	7	paths	path	NOUN
cana-834	190	8	between	between	ADP
cana-834	190	9	each	each	DET
cana-834	190	10	pair	pair	NOUN
cana-834	190	11	of	of	ADP
cana-834	190	12	nodes	node	NOUN
cana-834	190	13	,	,	PUNCT
cana-834	190	14	which	which	PRON
cana-834	190	15	results	result	VERB
cana-834	190	16	in	in	ADP
cana-834	190	17	a	a	DET
cana-834	190	18	high	high	ADJ
cana-834	190	19	level	level	NOUN
cana-834	190	20	of	of	ADP
cana-834	190	21	computational	computational	ADJ
cana-834	190	22	complexity	complexity	NOUN
cana-834	190	23	.	.	PUNCT
cana-834	191	1	involves	involve	VERB
cana-834	191	2	looking	look	VERB
cana-834	191	3	at	at	ADP
cana-834	191	4	the	the	DET
cana-834	191	5	area	area	NOUN
cana-834	191	6	around	around	ADP
cana-834	191	7	each	each	DET
cana-834	191	8	node	node	NOUN
cana-834	191	9	,	,	PUNCT
cana-834	191	10	which	which	PRON
cana-834	191	11	can	can	AUX
cana-834	191	12	be	be	AUX
cana-834	191	13	expensive	expensive	ADJ
cana-834	191	14	in	in	ADP
cana-834	191	15	large	large	ADJ
cana-834	191	16	networks	network	NOUN
cana-834	191	17	.	.	PUNCT
cana-834	192	1	complex	complex	ADJ
cana-834	192	2	algorithms	algorithm	NOUN
cana-834	192	3	that	that	PRON
cana-834	192	4	do	do	AUX
cana-834	192	5	n't	not	PART
cana-834	192	6	scale	scale	VERB
cana-834	192	7	well	well	ADV
cana-834	192	8	with	with	ADP
cana-834	192	9	network	network	NOUN
cana-834	192	10	size	size	NOUN
cana-834	192	11	are	be	AUX
cana-834	192	12	often	often	ADV
cana-834	192	13	used	use	VERB
cana-834	192	14	to	to	PART
cana-834	192	15	find	find	VERB
cana-834	192	16	communities	community	NOUN
cana-834	192	17	within	within	ADP
cana-834	192	18	a	a	DET
cana-834	192	19	network	network	NOUN
cana-834	192	20	.	.	PUNCT
cana-834	193	1	the	the	DET
cana-834	193	2	accuracy	accuracy	NOUN
cana-834	193	3	and	and	CCONJ
cana-834	193	4	dependability	dependability	NOUN
cana-834	193	5	of	of	ADP
cana-834	193	6	the	the	DET
cana-834	193	7	analysis	analysis	NOUN
cana-834	193	8	can	can	AUX
cana-834	193	9	be	be	AUX
cana-834	193	10	impacted	impact	VERB
cana-834	193	11	by	by	ADP
cana-834	193	12	the	the	DET
cana-834	193	13	incomplete	incomplete	ADJ
cana-834	193	14	or	or	CCONJ
cana-834	193	15	noisy	noisy	ADJ
cana-834	193	16	nature	nature	NOUN
cana-834	193	17	of	of	ADP
cana-834	193	18	the	the	DET
cana-834	193	19	data	datum	NOUN
cana-834	193	20	from	from	ADP
cana-834	193	21	social	social	ADJ
cana-834	193	22	networks	network	NOUN
cana-834	193	23	[	[	X
cana-834	193	24	16	16	NUM
cana-834	193	25	]	]	PUNCT
cana-834	193	26	.	.	PUNCT
cana-834	194	1	a	a	DET
cana-834	194	2	few	few	ADJ
cana-834	194	3	connections	connection	NOUN
cana-834	194	4	or	or	CCONJ
cana-834	194	5	connections	connection	NOUN
cana-834	194	6	probably	probably	ADV
cana-834	194	7	wo	will	AUX
cana-834	194	8	n't	not	PART
cana-834	194	9	be	be	AUX
cana-834	194	10	caught	catch	VERB
cana-834	194	11	,	,	PUNCT
cana-834	194	12	prompting	prompt	VERB
cana-834	194	13	deficient	deficient	ADJ
cana-834	194	14	charts	chart	NOUN
cana-834	194	15	.	.	PUNCT
cana-834	195	1	the	the	DET
cana-834	195	2	network	network	NOUN
cana-834	195	3	may	may	AUX
cana-834	195	4	contain	contain	VERB
cana-834	195	5	inaccurate	inaccurate	ADJ
cana-834	195	6	or	or	CCONJ
cana-834	195	7	false	false	ADJ
cana-834	195	8	information	information	NOUN
cana-834	195	9	,	,	PUNCT
cana-834	195	10	skewing	skew	VERB
cana-834	195	11	the	the	DET
cana-834	195	12	analysis	analysis	NOUN
cana-834	195	13	's	's	PART
cana-834	195	14	findings	finding	NOUN
cana-834	195	15	.	.	PUNCT
cana-834	196	1	to	to	PART
cana-834	196	2	guarantee	guarantee	VERB
cana-834	196	3	the	the	DET
cana-834	196	4	analysis	analysis	NOUN
cana-834	196	5	's	's	PART
cana-834	196	6	robustness	robustness	NOUN
cana-834	196	7	,	,	PUNCT
cana-834	196	8	it	it	PRON
cana-834	196	9	is	be	AUX
cana-834	196	10	essential	essential	ADJ
cana-834	196	11	to	to	PART
cana-834	196	12	identify	identify	VERB
cana-834	196	13	and	and	CCONJ
cana-834	196	14	deal	deal	VERB
cana-834	196	15	with	with	ADP
cana-834	196	16	anomalies	anomaly	NOUN
cana-834	196	17	(	(	PUNCT
cana-834	196	18	such	such	ADJ
cana-834	196	19	as	as	ADP
cana-834	196	20	outliers	outlier	NOUN
cana-834	196	21	or	or	CCONJ
cana-834	196	22	unexpected	unexpected	ADJ
cana-834	196	23	changes	change	NOUN
cana-834	196	24	)	)	PUNCT
cana-834	196	25	.	.	PUNCT
cana-834	197	1	privacy	privacy	NOUN
cana-834	197	2	and	and	CCONJ
cana-834	197	3	ethical	ethical	ADJ
cana-834	197	4	concerns	concern	NOUN
cana-834	197	5	are	be	AUX
cana-834	197	6	raised	raise	VERB
cana-834	197	7	by	by	ADP
cana-834	197	8	the	the	DET
cana-834	197	9	fact	fact	NOUN
cana-834	197	10	that	that	SCONJ
cana-834	197	11	social	social	ADJ
cana-834	197	12	network	network	NOUN
cana-834	197	13	data	datum	NOUN
cana-834	197	14	frequently	frequently	ADV
cana-834	197	15	contains	contain	VERB
cana-834	197	16	sensitive	sensitive	ADJ
cana-834	197	17	information	information	NOUN
cana-834	197	18	about	about	ADP
cana-834	197	19	individuals	individual	NOUN
cana-834	197	20	[	[	X
cana-834	197	21	17	17	NUM
cana-834	197	22	]	]	PUNCT
cana-834	197	23	.	.	PUNCT
cana-834	198	1	maintaining	maintain	VERB
cana-834	198	2	user	user	NOUN
cana-834	198	3	privacy	privacy	NOUN
cana-834	198	4	necessitates	necessitate	VERB
cana-834	198	5	ensuring	ensure	VERB
cana-834	198	6	that	that	SCONJ
cana-834	198	7	personal	personal	ADJ
cana-834	198	8	information	information	NOUN
cana-834	198	9	is	be	AUX
cana-834	198	10	anonymized	anonymize	VERB
cana-834	198	11	and	and	CCONJ
cana-834	198	12	protected	protect	VERB
cana-834	198	13	.	.	PUNCT
cana-834	199	1	specialists	specialist	NOUN
cana-834	199	2	and	and	CCONJ
cana-834	199	3	experts	expert	NOUN
cana-834	199	4	should	should	AUX
cana-834	199	5	comply	comply	VERB
cana-834	199	6	with	with	ADP
cana-834	199	7	moral	moral	ADJ
cana-834	199	8	rules	rule	NOUN
cana-834	199	9	to	to	PART
cana-834	199	10	stay	stay	VERB
cana-834	199	11	away	away	ADV
cana-834	199	12	from	from	ADP
cana-834	199	13	abuse	abuse	NOUN
cana-834	199	14	of	of	ADP
cana-834	199	15	informal	informal	ADJ
cana-834	199	16	organization	organization	NOUN
cana-834	199	17	information	information	NOUN
cana-834	199	18	.	.	PUNCT
cana-834	200	1	when	when	SCONJ
cana-834	200	2	handling	handle	VERB
cana-834	200	3	data	datum	NOUN
cana-834	200	4	from	from	ADP
cana-834	200	5	social	social	ADJ
cana-834	200	6	networks	network	NOUN
cana-834	200	7	,	,	PUNCT
cana-834	200	8	it	it	PRON
cana-834	200	9	is	be	AUX
cana-834	200	10	essential	essential	ADJ
cana-834	200	11	to	to	PART
cana-834	200	12	adhere	adhere	VERB
cana-834	200	13	to	to	ADP
cana-834	200	14	data	datum	NOUN
cana-834	200	15	protection	protection	NOUN
cana-834	200	16	laws	law	NOUN
cana-834	200	17	like	like	ADP
cana-834	200	18	the	the	DET
cana-834	200	19	gdpr	gdpr	NOUN
cana-834	200	20	.	.	PUNCT
cana-834	201	1	it	it	PRON
cana-834	201	2	is	be	AUX
cana-834	201	3	difficult	difficult	ADJ
cana-834	201	4	to	to	PART
cana-834	201	5	create	create	VERB
cana-834	201	6	effective	effective	ADJ
cana-834	201	7	algorithms	algorithm	NOUN
cana-834	201	8	that	that	PRON
cana-834	201	9	can	can	AUX
cana-834	201	10	adapt	adapt	VERB
cana-834	201	11	to	to	ADP
cana-834	201	12	the	the	DET
cana-834	201	13	network	network	NOUN
cana-834	201	14	's	's	PART
cana-834	201	15	size	size	NOUN
cana-834	201	16	.	.	PUNCT
cana-834	202	1	utilizing	utilize	VERB
cana-834	202	2	equal	equal	ADJ
cana-834	202	3	and	and	CCONJ
cana-834	202	4	dispersed	disperse	VERB
cana-834	202	5	processing	processing	NOUN
cana-834	202	6	systems	system	NOUN
cana-834	202	7	can	can	AUX
cana-834	202	8	upgrade	upgrade	VERB
cana-834	202	9	the	the	DET
cana-834	202	10	exhibition	exhibition	NOUN
cana-834	202	11	of	of	ADP
cana-834	202	12	diagram	diagram	NOUN
cana-834	202	13	calculations	calculation	NOUN
cana-834	202	14	.	.	PUNCT
cana-834	203	1	with	with	ADP
cana-834	203	2	lower	low	ADJ
cana-834	203	3	computational	computational	ADJ
cana-834	203	4	costs	cost	NOUN
cana-834	203	5	,	,	PUNCT
cana-834	203	6	solutions	solution	NOUN
cana-834	203	7	that	that	PRON
cana-834	203	8	are	be	AUX
cana-834	203	9	close	close	ADJ
cana-834	203	10	to	to	ADP
cana-834	203	11	optimal	optimal	ADJ
cana-834	203	12	can	can	AUX
cana-834	203	13	be	be	AUX
cana-834	203	14	obtained	obtain	VERB
cana-834	203	15	by	by	ADP
cana-834	203	16	employing	employ	VERB
cana-834	203	17	approximation	approximation	NOUN
cana-834	203	18	methods	method	NOUN
cana-834	203	19	.	.	PUNCT
cana-834	204	1	using	use	VERB
cana-834	204	2	specific	specific	ADJ
cana-834	204	3	equipment	equipment	NOUN
cana-834	204	4	,	,	PUNCT
cana-834	204	5	like	like	ADP
cana-834	204	6	gpus	gpu	NOUN
cana-834	204	7	,	,	PUNCT
cana-834	204	8	can	can	AUX
cana-834	204	9	speed	speed	VERB
cana-834	204	10	up	up	ADP
cana-834	204	11	diagram	diagram	NOUN
cana-834	204	12	calculations	calculation	NOUN
cana-834	204	13	.	.	PUNCT
cana-834	205	1	•	•	NOUN
cana-834	205	2	ensuring	ensure	VERB
cana-834	205	3	that	that	SCONJ
cana-834	205	4	algorithms	algorithm	NOUN
cana-834	205	5	scale	scale	VERB
cana-834	205	6	effectively	effectively	ADV
cana-834	205	7	as	as	SCONJ
cana-834	205	8	the	the	DET
cana-834	205	9	size	size	NOUN
cana-834	205	10	of	of	ADP
cana-834	205	11	the	the	DET
cana-834	205	12	network	network	NOUN
cana-834	205	13	grows	grow	VERB
cana-834	205	14	.	.	PUNCT
cana-834	206	1	•	•	NUM
cana-834	206	2	algorithm	algorithm	NOUN
cana-834	206	3	design	design	NOUN
cana-834	206	4	must	must	AUX
cana-834	206	5	strike	strike	VERB
cana-834	206	6	a	a	DET
cana-834	206	7	balance	balance	NOUN
cana-834	206	8	between	between	ADP
cana-834	206	9	accuracy	accuracy	NOUN
cana-834	206	10	and	and	CCONJ
cana-834	206	11	computational	computational	ADJ
cana-834	206	12	efficiency	efficiency	NOUN
cana-834	206	13	.	.	PUNCT
cana-834	207	1	•	•	NUM
cana-834	207	2	creating	create	VERB
cana-834	207	3	algorithms	algorithm	NOUN
cana-834	207	4	that	that	PRON
cana-834	207	5	are	be	AUX
cana-834	207	6	adaptable	adaptable	ADJ
cana-834	207	7	to	to	ADP
cana-834	207	8	a	a	DET
cana-834	207	9	variety	variety	NOUN
cana-834	207	10	of	of	ADP
cana-834	207	11	network	network	NOUN
cana-834	207	12	types	type	NOUN
cana-834	207	13	and	and	CCONJ
cana-834	207	14	analysis	analysis	NOUN
cana-834	207	15	tasks	task	NOUN
cana-834	207	16	.	.	PUNCT
cana-834	208	1	the	the	DET
cana-834	208	2	large	large	ADJ
cana-834	208	3	scale	scale	NOUN
cana-834	208	4	and	and	CCONJ
cana-834	208	5	dynamic	dynamic	ADJ
cana-834	208	6	nature	nature	NOUN
cana-834	208	7	of	of	ADP
cana-834	208	8	contemporary	contemporary	ADJ
cana-834	208	9	social	social	ADJ
cana-834	208	10	networks	network	NOUN
cana-834	208	11	,	,	PUNCT
cana-834	208	12	the	the	DET
cana-834	208	13	computational	computational	ADJ
cana-834	208	14	complexity	complexity	NOUN
cana-834	208	15	of	of	ADP
cana-834	208	16	key	key	ADJ
cana-834	208	17	metrics	metric	NOUN
cana-834	208	18	,	,	PUNCT
cana-834	208	19	and	and	CCONJ
cana-834	208	20	concerns	concern	NOUN
cana-834	208	21	regarding	regard	VERB
cana-834	208	22	data	data	NOUN
cana-834	208	23	quality	quality	NOUN
cana-834	208	24	and	and	CCONJ
cana-834	208	25	privacy	privacy	NOUN
cana-834	208	26	present	present	ADJ
cana-834	208	27	multiple	multiple	ADJ
cana-834	208	28	obstacles	obstacle	NOUN
cana-834	208	29	to	to	ADP
cana-834	208	30	social	social	ADJ
cana-834	208	31	network	network	NOUN
cana-834	208	32	analysis	analysis	NOUN
cana-834	208	33	[	[	X
cana-834	208	34	18	18	NUM
cana-834	208	35	]	]	PUNCT
cana-834	208	36	.	.	PUNCT
cana-834	209	1	effective	effective	ADJ
cana-834	209	2	algorithm	algorithm	NOUN
cana-834	209	3	design	design	NOUN
cana-834	209	4	,	,	PUNCT
cana-834	209	5	cutting	cut	VERB
cana-834	209	6	-	-	PUNCT
cana-834	209	7	edge	edge	NOUN
cana-834	209	8	data	datum	NOUN
cana-834	209	9	structures	structure	NOUN
cana-834	209	10	,	,	PUNCT
cana-834	209	11	parallel	parallel	ADJ
cana-834	209	12	and	and	CCONJ
cana-834	209	13	distributed	distributed	ADJ
cana-834	209	14	computing	computing	NOUN
cana-834	209	15	,	,	PUNCT
cana-834	209	16	and	and	CCONJ
cana-834	209	17	solid	solid	ADJ
cana-834	209	18	data	datum	NOUN
cana-834	209	19	handling	handling	NOUN
cana-834	209	20	practices	practice	NOUN
cana-834	209	21	are	be	AUX
cana-834	209	22	all	all	PRON
cana-834	209	23	needed	need	VERB
cana-834	209	24	to	to	PART
cana-834	209	25	meet	meet	VERB
cana-834	209	26	these	these	DET
cana-834	209	27	challenges	challenge	NOUN
cana-834	209	28	.	.	PUNCT
cana-834	210	1	communications	communication	NOUN
cana-834	210	2	on	on	ADP
cana-834	210	3	applied	apply	VERB
cana-834	210	4	nonlinear	nonlinear	ADJ
cana-834	210	5	analysis	analysis	NOUN
cana-834	210	6	issn	issn	NOUN
cana-834	210	7	:	:	PUNCT
cana-834	210	8	1074	1074	NUM
cana-834	210	9	-	-	PUNCT
cana-834	210	10	133x	133x	NUM
cana-834	210	11	vol	vol	NOUN
cana-834	210	12	31	31	NUM
cana-834	210	13	no	no	NOUN
cana-834	210	14	.	.	PUNCT
cana-834	211	1	4s	4s	NUM
cana-834	211	2	(	(	PUNCT
cana-834	211	3	2024	2024	NUM
cana-834	211	4	)	)	PUNCT
cana-834	211	5	171	171	NUM
cana-834	211	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	211	7	we	we	PRON
cana-834	211	8	'll	will	AUX
cana-834	211	9	look	look	VERB
cana-834	211	10	at	at	ADP
cana-834	211	11	optimization	optimization	NOUN
cana-834	211	12	techniques	technique	NOUN
cana-834	211	13	and	and	CCONJ
cana-834	211	14	case	case	NOUN
cana-834	211	15	studies	study	NOUN
cana-834	211	16	that	that	PRON
cana-834	211	17	show	show	VERB
cana-834	211	18	how	how	SCONJ
cana-834	211	19	to	to	PART
cana-834	211	20	deal	deal	VERB
cana-834	211	21	with	with	ADP
cana-834	211	22	these	these	DET
cana-834	211	23	problems	problem	NOUN
cana-834	211	24	in	in	ADP
cana-834	211	25	the	the	DET
cana-834	211	26	real	real	ADJ
cana-834	211	27	world	world	NOUN
cana-834	211	28	in	in	ADP
cana-834	211	29	the	the	DET
cana-834	211	30	following	follow	VERB
cana-834	211	31	sections	section	NOUN
cana-834	211	32	.	.	PUNCT
cana-834	212	1	other	other	ADJ
cana-834	212	2	results	result	NOUN
cana-834	212	3	:	:	PUNCT
cana-834	212	4	theorem	theorem	VERB
cana-834	212	5	3.1	3.1	NUM
cana-834	212	6	.	.	PUNCT
cana-834	213	1			NUM
cana-834	213	2	𝑠𝑛𝑎	𝑠𝑛𝑎	INTJ
cana-834	213	3	(	(	PUNCT
cana-834	213	4	𝑇	𝑇	PROPN
cana-834	213	5	∘	∘	NOUN
cana-834	213	6	𝐾1	𝐾1	PROPN
cana-834	213	7	)	)	PUNCT
cana-834	213	8	=	=	NOUN
cana-834	213	9	𝑛.	𝑛.	NOUN
cana-834	213	10	proof	proof	NOUN
cana-834	213	11	.	.	PUNCT
cana-834	214	1	let	let	VERB
cana-834	214	2	𝐺	𝐺	PROPN
cana-834	214	3	=	=	PROPN
cana-834	214	4	𝑇	𝑇	PROPN
cana-834	214	5	∘	∘	NOUN
cana-834	214	6	𝐾1	𝐾1	PROPN
cana-834	214	7	.	.	PUNCT
cana-834	215	1	let	let	VERB
cana-834	215	2	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	215	3	)	)	PUNCT
cana-834	216	1	=	=	PRON
cana-834	216	2	{	{	PUNCT
cana-834	216	3	𝑣1	𝑣1	PROPN
cana-834	216	4	,	,	PUNCT
cana-834	216	5	𝑣2	𝑣2	PROPN
cana-834	216	6	,	,	PUNCT
cana-834	216	7	.	.	PUNCT
cana-834	216	8	.	.	PUNCT
cana-834	216	9	.	.	PUNCT
cana-834	216	10	.	.	PUNCT
cana-834	216	11	.	.	PUNCT
cana-834	216	12	.	.	PUNCT
cana-834	216	13	.	.	PUNCT
cana-834	217	1	𝑣𝑛	𝑣𝑛	X
cana-834	217	2	}	}	PUNCT
cana-834	217	3	and	and	CCONJ
cana-834	217	4	vertex	vertex	NOUN
cana-834	217	5	𝑢𝑖	𝑢𝑖	NOUN
cana-834	217	6	be	be	AUX
cana-834	217	7	the	the	DET
cana-834	217	8	ith	ith	PROPN
cana-834	217	9	copy	copy	NOUN
cana-834	217	10	of	of	ADP
cana-834	217	11	𝐾1	𝐾1	PROPN
cana-834	217	12	attached	attach	VERB
cana-834	217	13	to	to	ADP
cana-834	217	14	the	the	DET
cana-834	217	15	vertex	vertex	NOUN
cana-834	217	16	𝑣𝑖.then	𝑣𝑖.then	ADP
cana-834	217	17	𝑉(𝐺	𝑉(𝐺	VERB
cana-834	217	18	)	)	PUNCT
cana-834	217	19	=	=	PRON
cana-834	217	20	{	{	PUNCT
cana-834	217	21	𝑣𝑖	𝑣𝑖	NOUN
cana-834	217	22	,	,	PUNCT
cana-834	217	23	𝑢𝑖/1	𝑢𝑖/1	ADJ
cana-834	217	24	≤	≤	NOUN
cana-834	217	25	𝑖	𝑖	SYM
cana-834	217	26	≤	≤	NOUN
cana-834	217	27	𝑛}.here	𝑛}.here	NUM
cana-834	217	28	𝑢1	𝑢1	PROPN
cana-834	217	29	,	,	PUNCT
cana-834	217	30	𝑢2	𝑢2	PROPN
cana-834	217	31	.	.	PUNCT
cana-834	217	32	.	.	PUNCT
cana-834	217	33	.	.	PUNCT
cana-834	217	34	.	.	PUNCT
cana-834	217	35	.	.	PUNCT
cana-834	217	36	.	.	PUNCT
cana-834	217	37	.	.	PUNCT
cana-834	218	1	𝑢𝑛	𝑢𝑛	PROPN
cana-834	218	2	are	be	AUX
cana-834	218	3	the	the	DET
cana-834	218	4	pendant	pendant	ADJ
cana-834	218	5	vertices	vertex	NOUN
cana-834	218	6	of𝐺.	of𝐺.	PRON
cana-834	218	7	we	we	PRON
cana-834	218	8	have	have	VERB
cana-834	218	9	pendant	pendant	ADJ
cana-834	218	10	vertices	vertex	NOUN
cana-834	218	11	are	be	AUX
cana-834	218	12	members	member	NOUN
cana-834	218	13	of	of	ADP
cana-834	218	14	ctd	ctd	NOUN
cana-834	218	15	-	-	PUNCT
cana-834	218	16	set	set	NOUN
cana-834	218	17	of	of	ADP
cana-834	218	18	g[3	g[3	PROPN
cana-834	218	19	]	]	X
cana-834	218	20	.	.	PUNCT
cana-834	219	1	hence,𝐷	hence,𝐷	PUNCT
cana-834	220	1	=	=	SYM
cana-834	220	2	{	{	PUNCT
cana-834	220	3	𝑢1	𝑢1	PROPN
cana-834	220	4	,	,	PUNCT
cana-834	220	5	𝑢2	𝑢2	PROPN
cana-834	220	6	,	,	PUNCT
cana-834	220	7	.	.	PUNCT
cana-834	220	8	.	.	PUNCT
cana-834	220	9	.	.	PUNCT
cana-834	221	1	𝑢𝑛	𝑢𝑛	X
cana-834	221	2	}	}	PUNCT
cana-834	221	3	is	be	AUX
cana-834	221	4	a	a	DET
cana-834	221	5	minimum	minimum	ADJ
cana-834	221	6	sna	sna	NOUN
cana-834	221	7	-	-	PUNCT
cana-834	221	8	set	set	NOUN
cana-834	221	9	of	of	ADP
cana-834	221	10	𝐺.	𝐺.	NOUN
cana-834	221	11	hence	hence	ADV
cana-834	221	12	,	,	PUNCT
cana-834	221	13	|𝐷|	|𝐷|	X
cana-834	221	14	  	  	SPACE
cana-834	221	15	=	=	SYM
cana-834	221	16	  	  	SPACE
cana-834	221	17	𝛾𝑠𝑛𝑎(𝐺	𝛾𝑠𝑛𝑎(𝐺	NOUN
cana-834	221	18	)	)	PUNCT
cana-834	222	1	=	=	SYM
cana-834	222	2	𝑛.	𝑛.	NOUN
cana-834	222	3	theorem	theorem	VERB
cana-834	222	4	3.2	3.2	NUM
cana-834	222	5	.	.	PUNCT
cana-834	223	1	for	for	ADP
cana-834	223	2	,	,	PUNCT
cana-834	223	3	𝑚	𝑚	X
cana-834	223	4	≥	≥	NUM
cana-834	223	5	4	4	NUM
cana-834	223	6	,	,	PUNCT
cana-834	223	7	then	then	ADV
cana-834	223	8	𝛾𝑠𝑛𝑎(𝑇	𝛾𝑠𝑛𝑎(𝑇	PUNCT
cana-834	223	9	∘	∘	PROPN
cana-834	223	10	𝑃𝑚	𝑃𝑚	PROPN
cana-834	223	11	)	)	PUNCT
cana-834	223	12	  	  	SPACE
cana-834	223	13	=	=	PUNCT
cana-834	223	14	𝑛	𝑛	PART
cana-834	223	15	⌊	⌊	VERB
cana-834	223	16	𝑚	𝑚	ADP
cana-834	223	17	2	2	NUM
cana-834	223	18	⌋.	⌋.	NOUN
cana-834	223	19	proof	proof	NOUN
cana-834	223	20	.	.	PUNCT
cana-834	224	1	let	let	VERB
cana-834	224	2	𝐺	𝐺	NOUN
cana-834	224	3	=	=	PUNCT
cana-834	224	4	𝑇	𝑇	PROPN
cana-834	224	5	∘	∘	PROPN
cana-834	224	6	𝑃𝑚.	𝑃𝑚.	PROPN
cana-834	224	7	let	let	VERB
cana-834	224	8	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	224	9	)	)	PUNCT
cana-834	225	1	=	=	PRON
cana-834	225	2	{	{	PUNCT
cana-834	225	3	𝑣1	𝑣1	PROPN
cana-834	225	4	,	,	PUNCT
cana-834	225	5	𝑣2	𝑣2	PROPN
cana-834	225	6	,	,	PUNCT
cana-834	225	7	.	.	PUNCT
cana-834	225	8	.	.	PUNCT
cana-834	225	9	.	.	PUNCT
cana-834	225	10	.	.	PUNCT
cana-834	225	11	.	.	PUNCT
cana-834	225	12	.	.	PUNCT
cana-834	226	1	𝑣𝑛	𝑣𝑛	X
cana-834	226	2	}	}	PUNCT
cana-834	226	3	and	and	CCONJ
cana-834	226	4	{	{	PUNCT
cana-834	226	5	𝑢𝑗/1	𝑢𝑗/1	NOUN
cana-834	226	6	≤	≤	NUM
cana-834	226	7	𝑗	𝑗	PRON
cana-834	226	8	≤	≤	NUM
cana-834	226	9	𝑚	𝑚	NOUN
cana-834	226	10	}	}	PUNCT
cana-834	226	11	be	be	VERB
cana-834	226	12	the	the	DET
cana-834	226	13	vertex	vertex	NOUN
cana-834	226	14	set	set	NOUN
cana-834	226	15	of	of	ADP
cana-834	226	16	ith	ith	PROPN
cana-834	226	17	copy	copy	NOUN
cana-834	226	18	of	of	ADP
cana-834	226	19	𝑃𝑚	𝑃𝑚	PROPN
cana-834	226	20	is	be	AUX
cana-834	226	21	adjacent	adjacent	ADJ
cana-834	226	22	to	to	ADP
cana-834	226	23	the	the	DET
cana-834	226	24	vertex	vertex	NOUN
cana-834	226	25	𝑣𝑖	𝑣𝑖	ADV
cana-834	226	26	in	in	ADP
cana-834	226	27	𝑇.	𝑇.	PROPN
cana-834	226	28	therefore	therefore	ADV
cana-834	226	29	𝑉(𝐺	𝑉(𝐺	NOUN
cana-834	226	30	)	)	PUNCT
cana-834	226	31	=	=	SYM
cana-834	227	1	{	{	PUNCT
cana-834	227	2	𝑣𝑖/1	𝑣𝑖/1	NOUN
cana-834	227	3	≤	≤	NUM
cana-834	227	4	𝑖	𝑖	SYM
cana-834	227	5	≤	≤	NUM
cana-834	227	6	𝑛	𝑛	NOUN
cana-834	227	7	}	}	PUNCT
cana-834	227	8	∪	∪	X
cana-834	227	9	{	{	PUNCT
cana-834	227	10	𝑢𝑖𝑗/1	𝑢𝑖𝑗/1	PROPN
cana-834	227	11	≤	≤	NUM
cana-834	227	12	𝑖	𝑖	SYM
cana-834	227	13	≤	≤	NUM
cana-834	227	14	𝑛	𝑛	NOUN
cana-834	227	15	,	,	PUNCT
cana-834	227	16	1	1	NUM
cana-834	227	17	≤	≤	NUM
cana-834	227	18	𝑗	𝑗	PRON
cana-834	227	19	≤	≤	NUM
cana-834	227	20	𝑚	𝑚	NOUN
cana-834	227	21	}	}	PUNCT
cana-834	227	22	.	.	PUNCT
cana-834	228	1	we	we	PRON
cana-834	228	2	have	have	VERB
cana-834	228	3	𝛾𝑠𝑛𝑎(𝑃𝑛	𝛾𝑠𝑛𝑎(𝑃𝑛	NOUN
cana-834	228	4	)	)	PUNCT
cana-834	228	5	=	=	SYM
cana-834	229	1	𝑛	𝑛	DET
cana-834	229	2	−	−	PROPN
cana-834	229	3	2	2	NUM
cana-834	229	4	,	,	PUNCT
cana-834	229	5	𝑛	𝑛	DET
cana-834	229	6	≥	≥	NOUN
cana-834	229	7	4	4	NUM
cana-834	229	8	.	.	PUNCT
cana-834	229	9	therefore	therefore	ADV
cana-834	229	10	by	by	ADP
cana-834	229	11	choosing	choose	VERB
cana-834	229	12	n	n	PRON
cana-834	229	13	copies	copy	NOUN
cana-834	229	14	of	of	ADP
cana-834	229	15	𝑚	𝑚	NOUN
cana-834	229	16	−	−	NUM
cana-834	229	17	2	2	NUM
cana-834	229	18	vertices	vertex	NOUN
cana-834	229	19	of	of	ADP
cana-834	229	20	𝑃𝑚.	𝑃𝑚.	PROPN
cana-834	229	21	let	let	VERB
cana-834	229	22	d={𝑢𝑖𝑗/1	d={𝑢𝑖𝑗/1	NOUN
cana-834	229	23	≤	≤	PROPN
cana-834	229	24	𝑖	𝑖	SYM
cana-834	229	25	≤	≤	NUM
cana-834	229	26	𝑛	𝑛	NOUN
cana-834	229	27	,	,	PUNCT
cana-834	229	28	1	1	NUM
cana-834	229	29	≤	≤	NUM
cana-834	229	30	𝑗	𝑗	PRON
cana-834	229	31	≤	≤	NOUN
cana-834	229	32	𝑚	𝑚	ADP
cana-834	229	33	−	−	PROPN
cana-834	229	34	2	2	NUM
cana-834	229	35	}	}	PUNCT
cana-834	229	36	which	which	PRON
cana-834	229	37	dominates	dominate	VERB
cana-834	229	38	all	all	DET
cana-834	229	39	the	the	DET
cana-834	229	40	vertices	vertex	NOUN
cana-834	229	41	of	of	ADP
cana-834	229	42	g	g	NOUN
cana-834	229	43	and	and	CCONJ
cana-834	229	44	⟨𝑉(𝐺	⟨𝑉(𝐺	NOUN
cana-834	229	45	)	)	PUNCT
cana-834	229	46	−	−	PROPN
cana-834	230	1	𝐷⟩	𝐷⟩	PROPN
cana-834	230	2	is	be	AUX
cana-834	230	3	a	a	DET
cana-834	230	4	network	network	NOUN
cana-834	230	5	.	.	PUNCT
cana-834	231	1	case	case	NOUN
cana-834	231	2	(	(	PUNCT
cana-834	231	3	i	i	NOUN
cana-834	231	4	)	)	PUNCT
cana-834	231	5	.	.	PUNCT
cana-834	232	1	m	m	PROPN
cana-834	232	2	is	be	AUX
cana-834	232	3	even	even	ADV
cana-834	232	4	let	let	VERB
cana-834	232	5	𝐷	𝐷	NOUN
cana-834	232	6	=	=	PUNCT
cana-834	232	7	{	{	PUNCT
cana-834	232	8	𝑢𝑖2	𝑢𝑖2	NOUN
cana-834	232	9	,	,	PUNCT
cana-834	232	10	𝑢𝑖4	𝑢𝑖4	NOUN
cana-834	232	11	,	,	PUNCT
cana-834	232	12	.	.	PUNCT
cana-834	232	13	.	.	PUNCT
cana-834	232	14	.	.	PUNCT
cana-834	232	15	.	.	PUNCT
cana-834	232	16	.	.	PUNCT
cana-834	232	17	.	.	PUNCT
cana-834	233	1	𝑢𝑖𝑚}is	𝑢𝑖𝑚}is	VERB
cana-834	233	2	a	a	DET
cana-834	233	3	minimum	minimum	ADJ
cana-834	233	4	ctd	ctd	NOUN
cana-834	233	5	-	-	PUNCT
cana-834	233	6	set	set	NOUN
cana-834	233	7	of	of	ADP
cana-834	233	8	g	g	PROPN
cana-834	233	9	and〈𝑉(𝐺	and〈𝑉(𝐺	NOUN
cana-834	233	10	)	)	PUNCT
cana-834	233	11	−	−	PROPN
cana-834	234	1	𝐷	𝐷	NOUN
cana-834	234	2	〉	〉	NOUN
cana-834	234	3	≅	≅	NOUN
cana-834	234	4	𝑃𝑛	𝑃𝑛	PROPN
cana-834	234	5	∘	∘	NOUN
cana-834	235	1	𝐾𝑚	𝐾𝑚	PROPN
cana-834	235	2	2	2	NUM
cana-834	235	3	is	be	AUX
cana-834	235	4	a	a	DET
cana-834	235	5	tree	tree	NOUN
cana-834	235	6	.	.	PUNCT
cana-834	236	1	therefore	therefore	ADV
cana-834	236	2	|𝐷|	|𝐷|	X
cana-834	236	3	=	=	PUNCT
cana-834	236	4			NUM
cana-834	236	5	𝑠𝑛𝑎	𝑠𝑛𝑎	X
cana-834	236	6	(	(	PUNCT
cana-834	236	7	𝐺	𝐺	NOUN
cana-834	236	8	)	)	PUNCT
cana-834	236	9	=	=	SYM
cana-834	236	10	𝑛	𝑛	PROPN
cana-834	236	11	(	(	PUNCT
cana-834	236	12	𝑚	𝑚	PROPN
cana-834	236	13	2	2	NUM
cana-834	236	14	)	)	PUNCT
cana-834	236	15	(	(	PUNCT
cana-834	236	16	1	1	X
cana-834	236	17	)	)	PUNCT
cana-834	236	18	case	case	NOUN
cana-834	236	19	(	(	PUNCT
cana-834	236	20	ii	ii	NOUN
cana-834	236	21	)	)	PUNCT
cana-834	236	22	.	.	PUNCT
cana-834	237	1	m	m	PROPN
cana-834	237	2	is	be	AUX
cana-834	237	3	odd	odd	ADJ
cana-834	237	4	let	let	VERB
cana-834	237	5	𝐷	𝐷	NOUN
cana-834	237	6	=	=	PUNCT
cana-834	237	7	{	{	PUNCT
cana-834	237	8	𝑢𝑖2	𝑢𝑖2	NOUN
cana-834	237	9	,	,	PUNCT
cana-834	237	10	𝑢𝑖4	𝑢𝑖4	NOUN
cana-834	237	11	,	,	PUNCT
cana-834	237	12	.	.	PUNCT
cana-834	237	13	.	.	PUNCT
cana-834	237	14	.	.	PUNCT
cana-834	237	15	.	.	PUNCT
cana-834	237	16	.	.	PUNCT
cana-834	237	17	.	.	PUNCT
cana-834	237	18	.	.	PUNCT
cana-834	237	19	.	.	PUNCT
cana-834	238	1	𝑢𝑖𝑚−1}is	𝑢𝑖𝑚−1}is	ADP
cana-834	238	2	a	a	DET
cana-834	238	3	minimum	minimum	ADJ
cana-834	238	4	sna	sna	NOUN
cana-834	238	5	-	-	PUNCT
cana-834	238	6	set	set	NOUN
cana-834	238	7	of	of	ADP
cana-834	238	8	g	g	PROPN
cana-834	238	9	and〈𝑉(𝐺	and〈𝑉(𝐺	NOUN
cana-834	238	10	)	)	PUNCT
cana-834	238	11	−	−	PROPN
cana-834	238	12	𝐷	𝐷	NOUN
cana-834	238	13	〉	〉	NOUN
cana-834	238	14	≅	≅	NOUN
cana-834	238	15	𝑃𝑛	𝑃𝑛	PROPN
cana-834	238	16	∘	∘	ADJ
cana-834	238	17	𝐾𝑚−1	𝐾𝑚−1	PROPN
cana-834	238	18	2	2	NUM
cana-834	238	19	is	be	AUX
cana-834	238	20	a	a	DET
cana-834	238	21	tree	tree	NOUN
cana-834	238	22	.	.	PUNCT
cana-834	239	1	therefore	therefore	ADV
cana-834	239	2	|𝐷|	|𝐷|	X
cana-834	239	3	=	=	PUNCT
cana-834	239	4			NUM
cana-834	239	5	𝑠𝑛𝑎	𝑠𝑛𝑎	X
cana-834	239	6	(	(	PUNCT
cana-834	239	7	𝐺	𝐺	NOUN
cana-834	239	8	)	)	PUNCT
cana-834	239	9	=	=	SYM
cana-834	239	10	𝑛	𝑛	PROPN
cana-834	239	11	(	(	PUNCT
cana-834	239	12	𝑚−1	𝑚−1	PROPN
cana-834	239	13	2	2	NUM
cana-834	239	14	)	)	PUNCT
cana-834	239	15	(	(	PUNCT
cana-834	239	16	2	2	NUM
cana-834	239	17	)	)	PUNCT
cana-834	239	18	from	from	ADP
cana-834	239	19	(	(	PUNCT
cana-834	239	20	1	1	NUM
cana-834	239	21	)	)	PUNCT
cana-834	239	22	and	and	CCONJ
cana-834	239	23	(	(	PUNCT
cana-834	239	24	2	2	X
cana-834	239	25	)	)	PUNCT
cana-834	239	26	|𝐷|	|𝐷|	NOUN
cana-834	239	27	=	=	SYM
cana-834	239	28	𝛾𝑠𝑛𝑎(𝑇	𝛾𝑠𝑛𝑎(𝑇	PROPN
cana-834	239	29	∘	∘	X
cana-834	239	30	𝑃𝑚	𝑃𝑚	PROPN
cana-834	239	31	)	)	PUNCT
cana-834	239	32	  	  	SPACE
cana-834	239	33	=	=	PUNCT
cana-834	240	1	𝑛	𝑛	PART
cana-834	240	2	⌊	⌊	VERB
cana-834	240	3	𝑚	𝑚	ADP
cana-834	240	4	2	2	NUM
cana-834	240	5	⌋	⌋	NOUN
cana-834	240	6	 	 	SPACE
cana-834	240	7	.	.	PUNCT
cana-834	241	1	communications	communication	NOUN
cana-834	241	2	on	on	ADP
cana-834	241	3	applied	apply	VERB
cana-834	241	4	nonlinear	nonlinear	ADJ
cana-834	241	5	analysis	analysis	NOUN
cana-834	241	6	issn	issn	NOUN
cana-834	241	7	:	:	PUNCT
cana-834	241	8	1074	1074	NUM
cana-834	241	9	-	-	PUNCT
cana-834	241	10	133x	133x	NUM
cana-834	241	11	vol	vol	NOUN
cana-834	241	12	31	31	NUM
cana-834	241	13	no	no	NOUN
cana-834	241	14	.	.	PUNCT
cana-834	242	1	4s	4s	NUM
cana-834	242	2	(	(	PUNCT
cana-834	242	3	2024	2024	NUM
cana-834	242	4	)	)	PUNCT
cana-834	242	5	172	172	NUM
cana-834	242	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	242	7	theorem	theorem	VERB
cana-834	242	8	3.3	3.3	NUM
cana-834	242	9	.	.	PUNCT
cana-834	243	1	for	for	ADP
cana-834	243	2	𝑚	𝑚	PROPN
cana-834	243	3	≥	≥	NUM
cana-834	243	4	3	3	NUM
cana-834	243	5	then	then	ADV
cana-834	243	6	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	PROPN
cana-834	243	7	∘	∘	PROPN
cana-834	243	8	𝐶𝑚	𝐶𝑚	NOUN
cana-834	243	9	)	)	PUNCT
cana-834	243	10	=	=	SYM
cana-834	243	11	𝑛	𝑛	DET
cana-834	243	12	⌈	⌈	NUM
cana-834	243	13	𝑚	𝑚	ADP
cana-834	243	14	2	2	NUM
cana-834	243	15	⌉.	⌉.	ADJ
cana-834	243	16	proof	proof	NOUN
cana-834	243	17	.	.	PUNCT
cana-834	244	1	take𝐺	take𝐺	NOUN
cana-834	244	2	=	=	PUNCT
cana-834	244	3	𝑇	𝑇	PROPN
cana-834	244	4	∘	∘	NOUN
cana-834	244	5	𝐶𝑚.let	𝐶𝑚.let	NOUN
cana-834	244	6	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	244	7	)	)	PUNCT
cana-834	244	8	  	  	SPACE
cana-834	244	9	=	=	PRON
cana-834	244	10	{	{	PUNCT
cana-834	244	11	𝑣𝑖	𝑣𝑖	ADP
cana-834	244	12	∶	∶	NOUN
cana-834	244	13	1	1	NUM
cana-834	244	14	≤	≤	NOUN
cana-834	244	15	𝑖	𝑖	SYM
cana-834	244	16	≤	≤	NUM
cana-834	244	17	𝑛	𝑛	NOUN
cana-834	244	18	}	}	PUNCT
cana-834	244	19	and	and	CCONJ
cana-834	244	20	)	)	PUNCT
cana-834	244	21	(	(	PUNCT
cana-834	244	22	mcv	mcv	NOUN
cana-834	244	23	=	=	SYM
cana-834	244	24	{	{	PUNCT
cana-834	244	25	𝑢1	𝑢1	PROPN
cana-834	244	26	,	,	PUNCT
cana-834	244	27	𝑢2	𝑢2	PROPN
cana-834	244	28	,	,	PUNCT
cana-834	244	29	.	.	PUNCT
cana-834	244	30	.	.	PUNCT
cana-834	244	31	.	.	PUNCT
cana-834	244	32	.	.	PUNCT
cana-834	244	33	.	.	PUNCT
cana-834	245	1	𝑢𝑚	𝑢𝑚	X
cana-834	245	2	}	}	PUNCT
cana-834	245	3	.	.	PUNCT
cana-834	246	1	then𝑉(𝑇	then𝑉(𝑇	NOUN
cana-834	246	2	∘	∘	NUM
cana-834	246	3	𝐶𝑚	𝐶𝑚	NOUN
cana-834	246	4	)	)	PUNCT
cana-834	246	5	=	=	NOUN
cana-834	246	6	{	{	PUNCT
cana-834	246	7	𝑣𝑖	𝑣𝑖	ADP
cana-834	246	8	:	:	PUNCT
cana-834	246	9	1	1	NUM
cana-834	246	10	≤	≤	NUM
cana-834	246	11	𝑖	𝑖	SYM
cana-834	246	12	≤	≤	NUM
cana-834	246	13	𝑛	𝑛	NOUN
cana-834	246	14	}	}	PUNCT
cana-834	246	15	∪	∪	X
cana-834	246	16	{	{	PUNCT
cana-834	246	17	𝑢𝑖𝑗/1	𝑢𝑖𝑗/1	PROPN
cana-834	246	18	≤	≤	NUM
cana-834	246	19	𝑖	𝑖	SYM
cana-834	246	20	≤	≤	NUM
cana-834	246	21	𝑛	𝑛	NOUN
cana-834	246	22	,	,	PUNCT
cana-834	246	23	1	1	NUM
cana-834	246	24	≤	≤	NUM
cana-834	246	25	𝑗	𝑗	PRON
cana-834	246	26	≤	≤	NUM
cana-834	246	27	𝑚	𝑚	NOUN
cana-834	246	28	}	}	PUNCT
cana-834	246	29	we	we	PRON
cana-834	246	30	have𝛾𝑆𝑁𝐴(𝐶𝑛	have𝛾𝑆𝑁𝐴(𝐶𝑛	VERB
cana-834	246	31	)	)	PUNCT
cana-834	246	32	=	=	SYM
cana-834	246	33	𝑛	𝑛	DET
cana-834	246	34	−	−	PROPN
cana-834	246	35	2[3	2[3	NUM
cana-834	246	36	]	]	PUNCT
cana-834	246	37	,	,	PUNCT
cana-834	246	38	therefore	therefore	ADV
cana-834	246	39	by	by	ADP
cana-834	246	40	choosing	choose	VERB
cana-834	246	41	n	n	PRON
cana-834	246	42	copies	copy	NOUN
cana-834	246	43	of	of	ADP
cana-834	246	44	(	(	PUNCT
cana-834	246	45	m-2	m-2	NOUN
cana-834	246	46	)	)	PUNCT
cana-834	246	47	vertices	vertex	NOUN
cana-834	246	48	of	of	ADP
cana-834	246	49	𝐶𝑚.let	𝐶𝑚.let	NOUN
cana-834	246	50	𝐷	𝐷	PROPN
cana-834	246	51	=	=	PUNCT
cana-834	246	52	{	{	PUNCT
cana-834	246	53	𝑢𝑖𝑗/1	𝑢𝑖𝑗/1	PROPN
cana-834	246	54	≤	≤	NUM
cana-834	246	55	𝑖	𝑖	SYM
cana-834	246	56	≤	≤	NUM
cana-834	246	57	𝑛	𝑛	NOUN
cana-834	246	58	,	,	PUNCT
cana-834	246	59	1	1	NUM
cana-834	246	60	≤	≤	NUM
cana-834	246	61	𝑗	𝑗	PRON
cana-834	246	62	≤	≤	NOUN
cana-834	246	63	𝑚	𝑚	ADP
cana-834	246	64	−	−	PROPN
cana-834	246	65	2	2	NUM
cana-834	246	66	}	}	PUNCT
cana-834	246	67	which	which	PRON
cana-834	246	68	dominates	dominate	VERB
cana-834	246	69	all	all	DET
cana-834	246	70	the	the	DET
cana-834	246	71	vertices	vertex	NOUN
cana-834	246	72	of	of	ADP
cana-834	246	73	g	g	NOUN
cana-834	246	74	and	and	CCONJ
cana-834	246	75	⟨𝑉(𝐺	⟨𝑉(𝐺	NOUN
cana-834	246	76	)	)	PUNCT
cana-834	246	77	−	−	PROPN
cana-834	246	78	𝐷⟩is	𝐷⟩is	ADP
cana-834	246	79	a	a	DET
cana-834	246	80	tree	tree	NOUN
cana-834	246	81	.	.	PUNCT
cana-834	247	1	which	which	PRON
cana-834	247	2	contradicts	contradict	VERB
cana-834	247	3	the	the	DET
cana-834	247	4	minimality	minimality	NOUN
cana-834	247	5	of	of	ADP
cana-834	247	6	the	the	DET
cana-834	247	7	sna	sna	PROPN
cana-834	247	8	-	-	PUNCT
cana-834	247	9	set	set	PROPN
cana-834	247	10	.	.	PUNCT
cana-834	248	1	therefore	therefore	ADV
cana-834	248	2	,	,	PUNCT
cana-834	248	3	by	by	ADP
cana-834	248	4	choosing	choose	VERB
cana-834	248	5	nonadjacent	nonadjacent	ADJ
cana-834	248	6	vertices	vertex	NOUN
cana-834	248	7	in	in	ADP
cana-834	248	8	n	n	ADP
cana-834	248	9	copies	copy	NOUN
cana-834	248	10	of	of	ADP
cana-834	248	11	𝐶𝑚.	𝐶𝑚.	NOUN
cana-834	248	12	here	here	ADV
cana-834	248	13	two	two	NUM
cana-834	248	14	parts	part	NOUN
cana-834	248	15	arise	arise	VERB
cana-834	248	16	.	.	PUNCT
cana-834	249	1	part	part	NOUN
cana-834	249	2	(	(	PUNCT
cana-834	249	3	i	i	NOUN
cana-834	249	4	)	)	PUNCT
cana-834	249	5	.	.	PUNCT
cana-834	250	1	m	m	PROPN
cana-834	250	2	is	be	AUX
cana-834	250	3	even	even	ADV
cana-834	250	4	let	let	VERB
cana-834	250	5	𝐷	𝐷	NOUN
cana-834	250	6	=	=	PRON
cana-834	250	7	{	{	PUNCT
cana-834	250	8	𝑢𝑖1	𝑢𝑖1	VERB
cana-834	250	9	,	,	PUNCT
cana-834	250	10	𝑢𝑖3	𝑢𝑖3	NOUN
cana-834	250	11	.	.	PUNCT
cana-834	250	12	.	.	PUNCT
cana-834	250	13	.	.	PUNCT
cana-834	250	14	.	.	PUNCT
cana-834	250	15	.	.	PUNCT
cana-834	251	1	𝑢𝑖𝑚−1	𝑢𝑖𝑚−1	NOUN
cana-834	251	2	}	}	PUNCT
cana-834	251	3	is	be	AUX
cana-834	251	4	a	a	DET
cana-834	251	5	minimum	minimum	ADJ
cana-834	251	6	ctd	ctd	NOUN
cana-834	251	7	-	-	PUNCT
cana-834	251	8	set	set	NOUN
cana-834	251	9	of	of	ADP
cana-834	251	10	g	g	NOUN
cana-834	251	11	which	which	PRON
cana-834	251	12	dominates	dominate	VERB
cana-834	251	13	all	all	DET
cana-834	251	14	the	the	DET
cana-834	251	15	vertices	vertex	NOUN
cana-834	251	16	of	of	ADP
cana-834	251	17	g	g	NOUN
cana-834	251	18	and	and	CCONJ
cana-834	251	19	⟨𝑉(𝐺	⟨𝑉(𝐺	NOUN
cana-834	251	20	)	)	PUNCT
cana-834	251	21	−	−	PROPN
cana-834	251	22	𝐷⟩	𝐷⟩	PROPN
cana-834	251	23	≅	≅	PROPN
cana-834	251	24	𝑇	𝑇	PROPN
cana-834	251	25	∘	∘	NOUN
cana-834	252	1	𝐾𝑚	𝐾𝑚	PROPN
cana-834	252	2	2	2	NUM
cana-834	252	3	is	be	AUX
cana-834	252	4	a	a	DET
cana-834	252	5	tree	tree	NOUN
cana-834	252	6	.	.	PUNCT
cana-834	253	1	therefore	therefore	ADV
cana-834	253	2	|𝐷|	|𝐷|	X
cana-834	253	3	=	=	SYM
cana-834	253	4	𝛾𝑐𝑡𝑑(𝐺	𝛾𝑐𝑡𝑑(𝐺	X
cana-834	253	5	)	)	PUNCT
cana-834	253	6	=	=	SYM
cana-834	253	7	𝑛	𝑛	PROPN
cana-834	253	8	(	(	PUNCT
cana-834	253	9	𝑚	𝑚	PROPN
cana-834	253	10	2	2	NUM
cana-834	253	11	)	)	PUNCT
cana-834	253	12	(	(	PUNCT
cana-834	253	13	3	3	X
cana-834	253	14	)	)	PUNCT
cana-834	253	15	part	part	NOUN
cana-834	253	16	(	(	PUNCT
cana-834	253	17	ii	ii	NOUN
cana-834	253	18	)	)	PUNCT
cana-834	253	19	.	.	PUNCT
cana-834	254	1	m	m	PROPN
cana-834	254	2	is	be	AUX
cana-834	254	3	odd	odd	ADJ
cana-834	254	4	let	let	VERB
cana-834	254	5	𝐷	𝐷	NOUN
cana-834	254	6	=	=	PRON
cana-834	254	7	{	{	PUNCT
cana-834	254	8	𝑢𝑖1	𝑢𝑖1	VERB
cana-834	254	9	,	,	PUNCT
cana-834	254	10	𝑢𝑖3	𝑢𝑖3	NOUN
cana-834	254	11	.	.	PUNCT
cana-834	254	12	.	.	PUNCT
cana-834	254	13	.	.	PUNCT
cana-834	254	14	.	.	PUNCT
cana-834	254	15	.	.	PUNCT
cana-834	255	1	𝑢𝑖𝑚−2	𝑢𝑖𝑚−2	NOUN
cana-834	255	2	}	}	PUNCT
cana-834	255	3	is	be	AUX
cana-834	255	4	a	a	DET
cana-834	255	5	minimum	minimum	ADJ
cana-834	255	6	sna	sna	NOUN
cana-834	255	7	-	-	PUNCT
cana-834	255	8	set	set	NOUN
cana-834	255	9	of	of	ADP
cana-834	255	10	g	g	NOUN
cana-834	255	11	which	which	PRON
cana-834	255	12	dominates	dominate	VERB
cana-834	255	13	all	all	DET
cana-834	255	14	the	the	DET
cana-834	255	15	vertices	vertex	NOUN
cana-834	255	16	of	of	ADP
cana-834	255	17	g	g	NOUN
cana-834	255	18	and⟨𝑉(𝐺	and⟨𝑉(𝐺	VERB
cana-834	255	19	)	)	PUNCT
cana-834	256	1	−	−	PROPN
cana-834	256	2	𝐷⟩	𝐷⟩	PROPN
cana-834	256	3	≅	≅	PROPN
cana-834	256	4	𝑇	𝑇	PROPN
cana-834	256	5	∘	∘	NOUN
cana-834	256	6	𝐾𝑚+1	𝐾𝑚+1	NUM
cana-834	256	7	2	2	NUM
cana-834	256	8	is	be	AUX
cana-834	256	9	a	a	DET
cana-834	256	10	tree	tree	NOUN
cana-834	256	11	.	.	PUNCT
cana-834	257	1	therefore	therefore	ADV
cana-834	257	2	|𝐷|	|𝐷|	X
cana-834	257	3	=	=	SYM
cana-834	257	4	𝛾𝑠𝑛𝑎(𝐺	𝛾𝑠𝑛𝑎(𝐺	X
cana-834	257	5	)	)	PUNCT
cana-834	257	6	=	=	SYM
cana-834	257	7	𝑛	𝑛	PROPN
cana-834	257	8	(	(	PUNCT
cana-834	257	9	𝑚+1	𝑚+1	NUM
cana-834	257	10	2	2	X
cana-834	257	11	)	)	PUNCT
cana-834	257	12	(	(	PUNCT
cana-834	257	13	4	4	NUM
cana-834	257	14	)	)	PUNCT
cana-834	257	15	from	from	ADP
cana-834	257	16	(	(	PUNCT
cana-834	257	17	3	3	NUM
cana-834	257	18	)	)	PUNCT
cana-834	257	19	and	and	CCONJ
cana-834	257	20	(	(	PUNCT
cana-834	257	21	4	4	X
cana-834	257	22	)	)	PUNCT
cana-834	257	23	|𝐷|	|𝐷|	NOUN
cana-834	257	24	=	=	SYM
cana-834	257	25	𝛾𝑠𝑛𝑎(𝑇	𝛾𝑠𝑛𝑎(𝑇	PROPN
cana-834	257	26	∘	∘	PROPN
cana-834	257	27	𝐶𝑚	𝐶𝑚	NOUN
cana-834	257	28	)	)	PUNCT
cana-834	257	29	=	=	SYM
cana-834	257	30	𝑛	𝑛	PRON
cana-834	257	31	⌈	⌈	NUM
cana-834	257	32	𝑚	𝑚	ADP
cana-834	257	33	2	2	NUM
cana-834	257	34	⌉	⌉	PRON
cana-834	257	35	 	 	SPACE
cana-834	257	36	where	where	SCONJ
cana-834	257	37	𝑛	𝑛	PRON
cana-834	257	38	≥	≥	NOUN
cana-834	257	39	2	2	NUM
cana-834	257	40	,	,	PUNCT
cana-834	257	41	𝑚	𝑚	X
cana-834	257	42	≥	≥	NUM
cana-834	257	43	3	3	NUM
cana-834	257	44	.	.	PUNCT
cana-834	257	45	theorem	theorem	VERB
cana-834	257	46	3.4	3.4	NUM
cana-834	257	47	.	.	PUNCT
cana-834	258	1	for	for	ADP
cana-834	258	2	𝑚	𝑚	PROPN
cana-834	258	3	≥	≥	NUM
cana-834	258	4	4	4	NUM
cana-834	258	5	,	,	PUNCT
cana-834	258	6	then	then	ADV
cana-834	258	7	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	PROPN
cana-834	258	8	∘	∘	PROPN
cana-834	258	9	𝐾𝑚	𝐾𝑚	PROPN
cana-834	258	10	)	)	PUNCT
cana-834	258	11	=	=	PUNCT
cana-834	258	12	𝑛(𝑚	𝑛(𝑚	NOUN
cana-834	258	13	−	−	NOUN
cana-834	258	14	1	1	NUM
cana-834	258	15	)	)	PUNCT
cana-834	258	16	.	.	PUNCT
cana-834	259	1	proof	proof	NOUN
cana-834	259	2	.	.	PUNCT
cana-834	260	1	take	take	VERB
cana-834	260	2	𝐺	𝐺	NOUN
cana-834	260	3	=	=	PUNCT
cana-834	260	4	𝑇	𝑇	PROPN
cana-834	260	5	∘	∘	PROPN
cana-834	261	1	𝐾𝑚.	𝐾𝑚.	PROPN
cana-834	261	2	let	let	VERB
cana-834	261	3	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	261	4	)	)	PUNCT
cana-834	261	5	=	=	SYM
cana-834	262	1	{	{	PUNCT
cana-834	262	2	𝑣𝑖/1	𝑣𝑖/1	NOUN
cana-834	262	3	≤	≤	NUM
cana-834	262	4	𝑖	𝑖	SYM
cana-834	262	5	≤	≤	NUM
cana-834	262	6	𝑛	𝑛	NOUN
cana-834	262	7	}	}	PUNCT
cana-834	262	8	and	and	CCONJ
cana-834	262	9	𝑉(𝐾𝑚)={𝑢1	𝑉(𝐾𝑚)={𝑢1	ADJ
cana-834	262	10	,	,	PUNCT
cana-834	262	11	𝑢2	𝑢2	PROPN
cana-834	262	12	,	,	PUNCT
cana-834	262	13	.	.	PUNCT
cana-834	262	14	.	.	PUNCT
cana-834	262	15	.	.	PUNCT
cana-834	262	16	.	.	PUNCT
cana-834	262	17	.	.	PUNCT
cana-834	263	1	𝑢𝑚}.then	𝑢𝑚}.then	ADV
cana-834	263	2	𝑉(𝐺	𝑉(𝐺	NOUN
cana-834	263	3	)	)	PUNCT
cana-834	263	4	=	=	SYM
cana-834	264	1	{	{	PUNCT
cana-834	264	2	𝑣𝑖/1	𝑣𝑖/1	NOUN
cana-834	264	3	≤	≤	NUM
cana-834	264	4	𝑖	𝑖	SYM
cana-834	264	5	≤	≤	NUM
cana-834	264	6	𝑛	𝑛	NOUN
cana-834	264	7	}	}	PUNCT
cana-834	264	8	∪	∪	X
cana-834	264	9	{	{	PUNCT
cana-834	264	10	𝑢𝑖𝑗/1	𝑢𝑖𝑗/1	PROPN
cana-834	264	11	≤	≤	NUM
cana-834	264	12	𝑖	𝑖	SYM
cana-834	264	13	≤	≤	NUM
cana-834	264	14	𝑛	𝑛	NOUN
cana-834	264	15	,	,	PUNCT
cana-834	264	16	1	1	NUM
cana-834	264	17	≤	≤	NUM
cana-834	264	18	𝑗	𝑗	PRON
cana-834	264	19	≤	≤	NUM
cana-834	264	20	𝑚}.we	𝑚}.we	NOUN
cana-834	264	21	have	have	VERB
cana-834	264	22	𝛾𝑆𝑁𝐴(𝐾𝑛	𝛾𝑆𝑁𝐴(𝐾𝑛	NOUN
cana-834	264	23	)	)	PUNCT
cana-834	265	1	=	=	SYM
cana-834	265	2	𝑛	𝑛	PRON
cana-834	265	3	−	−	NUM
cana-834	265	4	2	2	NUM
cana-834	266	1	[	[	X
cana-834	266	2	3	3	NUM
cana-834	266	3	]	]	PUNCT
cana-834	266	4	.	.	PUNCT
cana-834	267	1	therefore	therefore	ADV
cana-834	267	2	by	by	ADP
cana-834	267	3	choosing	choose	VERB
cana-834	267	4	n	n	PRON
cana-834	267	5	copies	copy	NOUN
cana-834	267	6	of	of	ADP
cana-834	267	7	(	(	PUNCT
cana-834	267	8	𝑚	𝑚	PROPN
cana-834	267	9	−	−	PROPN
cana-834	267	10	2)vertices	2)vertice	NOUN
cana-834	267	11	of	of	ADP
cana-834	267	12	𝐾𝑚.let	𝐾𝑚.let	NOUN
cana-834	267	13	𝐷	𝐷	PROPN
cana-834	267	14	1	1	NUM
cana-834	267	15	=	=	SYM
cana-834	267	16	{	{	PUNCT
cana-834	267	17	21,1/	21,1/	PROPN
cana-834	267	18	−	−	PROPN
cana-834	267	19	mjniu	mjniu	VERB
cana-834	267	20	ji	ji	PROPN
cana-834	267	21	}	}	PUNCT
cana-834	267	22	which	which	PRON
cana-834	267	23	dominates	dominate	VERB
cana-834	267	24	all	all	DET
cana-834	267	25	the	the	DET
cana-834	267	26	vertices	vertex	NOUN
cana-834	267	27	of	of	ADP
cana-834	267	28	g.	g.	PROPN
cana-834	267	29	but	but	CCONJ
cana-834	267	30	⟨𝑉(𝐺	⟨𝑉(𝐺	NOUN
cana-834	267	31	)	)	PUNCT
cana-834	267	32	−	−	PROPN
cana-834	267	33	𝐷	𝐷	NOUN
cana-834	267	34	1	1	NUM
cana-834	267	35	⟩	⟩	NOUN
cana-834	267	36	contains	contain	VERB
cana-834	267	37	a	a	DET
cana-834	267	38	cycle	cycle	NOUN
cana-834	267	39	which	which	PRON
cana-834	267	40	contradicts	contradict	VERB
cana-834	267	41	the	the	DET
cana-834	267	42	condition	condition	NOUN
cana-834	267	43	of	of	ADP
cana-834	267	44	ctd	ctd	NOUN
cana-834	267	45	-	-	PUNCT
cana-834	267	46	set	set	NOUN
cana-834	267	47	.	.	PUNCT
cana-834	268	1	let	let	VERB
cana-834	268	2			PRON
cana-834	268	3	niudd	niudd	PROPN
cana-834	268	4	mi	mi	PROPN
cana-834	268	5	=	=	PROPN
cana-834	268	6	−	−	PROPN
cana-834	269	1	1/1,1	1/1,1	NUM
cana-834	269	2	which	which	PRON
cana-834	269	3	dominates	dominate	VERB
cana-834	269	4	all	all	DET
cana-834	269	5	the	the	DET
cana-834	269	6	vertices	vertex	NOUN
cana-834	269	7	of	of	ADP
cana-834	269	8	g	g	NOUN
cana-834	269	9	and	and	CCONJ
cana-834	269	10	⟨𝑉(𝐺	⟨𝑉(𝐺	NOUN
cana-834	269	11	)	)	PUNCT
cana-834	269	12	−	−	PROPN
cana-834	269	13	𝐷⟩	𝐷⟩	PROPN
cana-834	269	14	≅	≅	PROPN
cana-834	269	15	𝑇	𝑇	PROPN
cana-834	269	16	∘	∘	PROPN
cana-834	269	17	communications	communication	NOUN
cana-834	269	18	on	on	ADP
cana-834	269	19	applied	apply	VERB
cana-834	269	20	nonlinear	nonlinear	ADJ
cana-834	269	21	analysis	analysis	NOUN
cana-834	269	22	issn	issn	NOUN
cana-834	269	23	:	:	PUNCT
cana-834	269	24	1074	1074	NUM
cana-834	269	25	-	-	PUNCT
cana-834	269	26	133x	133x	NUM
cana-834	269	27	vol	vol	NOUN
cana-834	269	28	31	31	NUM
cana-834	269	29	no	no	NOUN
cana-834	269	30	.	.	PUNCT
cana-834	270	1	4s	4s	NUM
cana-834	270	2	(	(	PUNCT
cana-834	270	3	2024	2024	NUM
cana-834	270	4	)	)	PUNCT
cana-834	270	5	173	173	NUM
cana-834	270	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	270	7	𝐾1	𝐾1	PROPN
cana-834	270	8	is	be	AUX
cana-834	270	9	a	a	DET
cana-834	270	10	tree	tree	NOUN
cana-834	270	11	.	.	PUNCT
cana-834	271	1	hence,𝐷	hence,𝐷	PUNCT
cana-834	272	1	=	=	PUNCT
cana-834	272	2	{	{	PUNCT
cana-834	272	3	𝑢𝑖𝑗/1	𝑢𝑖𝑗/1	PROPN
cana-834	272	4	≤	≤	NUM
cana-834	272	5	𝑖	𝑖	SYM
cana-834	272	6	≤	≤	NUM
cana-834	272	7	𝑛	𝑛	NOUN
cana-834	272	8	,	,	PUNCT
cana-834	272	9	 	 	SPACE
cana-834	272	10	1	1	NUM
cana-834	272	11	≤	≤	NUM
cana-834	272	12	𝑗	𝑗	PRON
cana-834	272	13	≤	≤	NOUN
cana-834	272	14	𝑚	𝑚	ADP
cana-834	272	15	−	−	PROPN
cana-834	272	16	1	1	NUM
cana-834	272	17	}	}	PUNCT
cana-834	272	18	is	be	AUX
cana-834	272	19	a	a	DET
cana-834	272	20	minimum	minimum	ADJ
cana-834	272	21	ctd	ctd	NOUN
cana-834	272	22	-	-	PUNCT
cana-834	272	23	set	set	NOUN
cana-834	272	24	of	of	ADP
cana-834	272	25	g.	g.	PROPN
cana-834	272	26	therefore	therefore	ADV
cana-834	272	27	,	,	PUNCT
cana-834	272	28	|𝐷|	|𝐷|	X
cana-834	272	29	=	=	PUNCT
cana-834	273	1	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	PROPN
cana-834	273	2	∘	∘	NUM
cana-834	273	3	𝐾𝑚	𝐾𝑚	PROPN
cana-834	273	4	)	)	PUNCT
cana-834	273	5	=	=	PUNCT
cana-834	273	6	𝑛(𝑚	𝑛(𝑚	NOUN
cana-834	273	7	−	−	NOUN
cana-834	273	8	1	1	NUM
cana-834	273	9	)	)	PUNCT
cana-834	273	10	,	,	PUNCT
cana-834	273	11	𝑛	𝑛	PRON
cana-834	273	12	≥	≥	NOUN
cana-834	273	13	2	2	NUM
cana-834	273	14	,	,	PUNCT
cana-834	273	15	𝑚	𝑚	X
cana-834	273	16	≥	≥	NUM
cana-834	273	17	4	4	NUM
cana-834	273	18	.	.	PUNCT
cana-834	273	19	theorem	theorem	VERB
cana-834	273	20	3.5	3.5	NUM
cana-834	273	21	.	.	PUNCT
cana-834	274	1	for	for	ADP
cana-834	274	2	𝑚	𝑚	PROPN
cana-834	274	3	≥	≥	NUM
cana-834	274	4	3	3	NUM
cana-834	274	5	,	,	PUNCT
cana-834	274	6	then	then	ADV
cana-834	274	7	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	PROPN
cana-834	274	8	∘	∘	PROPN
cana-834	274	9	𝑊𝑚	𝑊𝑚	PROPN
cana-834	274	10	)	)	PUNCT
cana-834	274	11	=	=	SYM
cana-834	274	12	𝑛	𝑛	PRON
cana-834	274	13			ADJ
cana-834	274	14			NOUN
cana-834	274	15			X
cana-834	274	16			NOUN
cana-834	274	17			X
cana-834	274	18			PROPN
cana-834	274	19	+	+	CCONJ
cana-834	274	20	2	2	NUM
cana-834	274	21	1	1	NUM
cana-834	274	22	m	m	NOUN
cana-834	274	23	.	.	PUNCT
cana-834	275	1	proof	proof	NOUN
cana-834	275	2	.	.	PUNCT
cana-834	276	1	take	take	VERB
cana-834	276	2	𝐺	𝐺	NOUN
cana-834	276	3	=	=	PUNCT
cana-834	276	4	𝑇	𝑇	PROPN
cana-834	276	5	∘	∘	NOUN
cana-834	276	6	𝑊𝑚.	𝑊𝑚.	PROPN
cana-834	276	7	where	where	SCONJ
cana-834	276	8	t	t	PROPN
cana-834	276	9	is	be	AUX
cana-834	276	10	a	a	DET
cana-834	276	11	tree	tree	NOUN
cana-834	276	12	with	with	ADP
cana-834	276	13	2n	2n	NUM
cana-834	276	14	vertices	vertex	NOUN
cana-834	276	15	.let	.let	PUNCT
cana-834	276	16	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	276	17	)	)	PUNCT
cana-834	276	18	=	=	SYM
cana-834	277	1	{	{	PUNCT
cana-834	277	2	𝑣𝑖/1	𝑣𝑖/1	NOUN
cana-834	277	3	≤	≤	NUM
cana-834	277	4	𝑖	𝑖	SYM
cana-834	277	5	≤	≤	NUM
cana-834	277	6	𝑛	𝑛	NOUN
cana-834	277	7	}	}	PUNCT
cana-834	277	8	and	and	CCONJ
cana-834	277	9	𝑉(𝑊𝑚	𝑉(𝑊𝑚	ADJ
cana-834	277	10	)	)	PUNCT
cana-834	277	11	=	=	SYM
cana-834	277	12	{	{	PUNCT
cana-834	277	13	𝑐	𝑐	NOUN
cana-834	277	14	,	,	PUNCT
cana-834	277	15	𝑢1	𝑢1	NOUN
cana-834	277	16	,	,	PUNCT
cana-834	277	17	𝑢2	𝑢2	PROPN
cana-834	277	18	,	,	PUNCT
cana-834	277	19	…	…	PUNCT
cana-834	277	20	𝑢𝑚−1	𝑢𝑚−1	PROPN
cana-834	277	21	}	}	PUNCT
cana-834	277	22	where	where	SCONJ
cana-834	277	23	c	c	PROPN
cana-834	277	24	is	be	AUX
cana-834	277	25	the	the	DET
cana-834	277	26	centre	centre	ADJ
cana-834	277	27	vertex	vertex	NOUN
cana-834	277	28	and	and	CCONJ
cana-834	277	29	remaining	remain	VERB
cana-834	277	30	vertices	vertex	NOUN
cana-834	277	31	are	be	AUX
cana-834	277	32	in	in	ADP
cana-834	277	33	1−mc	1−mc	PROPN
cana-834	277	34	.then𝑉(𝑇	.then𝑉(𝑇	PROPN
cana-834	277	35	∘	∘	PROPN
cana-834	277	36	𝑊𝑚	𝑊𝑚	PROPN
cana-834	277	37	)	)	PUNCT
cana-834	277	38	=	=	SYM
cana-834	278	1	{	{	PUNCT
cana-834	278	2	𝑣𝑖/1	𝑣𝑖/1	NOUN
cana-834	278	3	≤	≤	NUM
cana-834	278	4	𝑖	𝑖	SYM
cana-834	278	5	≤	≤	NUM
cana-834	278	6	𝑛	𝑛	NOUN
cana-834	278	7	}	}	PUNCT
cana-834	278	8	∪	∪	ADJ
cana-834	278	9	{	{	PUNCT
cana-834	278	10	𝑐𝑖	𝑐𝑖	NOUN
cana-834	278	11	,	,	PUNCT
cana-834	278	12	𝑢𝑖𝑗	𝑢𝑖𝑗	NOUN
cana-834	278	13	 	 	SPACE
cana-834	278	14	:	:	PUNCT
cana-834	278	15	1	1	NUM
cana-834	278	16	≤	≤	NUM
cana-834	278	17	𝑖	𝑖	SYM
cana-834	278	18	≤	≤	NOUN
cana-834	278	19	𝑛	𝑛	PRON
cana-834	278	20	 	 	SPACE
cana-834	278	21	;	;	PUNCT
cana-834	278	22	1	1	NUM
cana-834	278	23	≤	≤	NUM
cana-834	278	24	𝑗	𝑗	PRON
cana-834	278	25	≤	≤	NOUN
cana-834	278	26	𝑚	𝑚	ADP
cana-834	278	27	−	−	PROPN
cana-834	278	28	1	1	NUM
cana-834	278	29	}	}	PUNCT
cana-834	278	30	where	where	SCONJ
cana-834	278	31	𝑐𝑖is	𝑐𝑖is	NOUN
cana-834	278	32	the	the	DET
cana-834	278	33	ith	ith	PROPN
cana-834	278	34	copy	copy	NOUN
cana-834	278	35	of	of	ADP
cana-834	278	36	wm	wm	PROPN
cana-834	278	37	,	,	PUNCT
cana-834	278	38	𝑢𝑖𝑗	𝑢𝑖𝑗	PROPN
cana-834	278	39	is	be	AUX
cana-834	278	40	the	the	DET
cana-834	278	41	ith	ith	PROPN
cana-834	278	42	copy	copy	NOUN
cana-834	278	43	of	of	ADP
cana-834	278	44	𝑊𝑚	𝑊𝑚	PROPN
cana-834	278	45	is	be	AUX
cana-834	278	46	adjacent	adjacent	ADJ
cana-834	278	47	to	to	ADP
cana-834	278	48	the	the	DET
cana-834	278	49	vertex	vertex	NOUN
cana-834	278	50	𝑣𝑖	𝑣𝑖	ADV
cana-834	278	51	in	in	ADP
cana-834	278	52	𝑇.	𝑇.	PROPN
cana-834	278	53	let	let	VERB
cana-834	278	54			PRON
cana-834	278	55	nicd	nicd	VERB
cana-834	278	56	i	i	PRON
cana-834	278	57	=	=	VERB
cana-834	278	58	1/1	1/1	NUM
cana-834	278	59	which	which	PRON
cana-834	278	60	dominates	dominate	VERB
cana-834	278	61	all	all	DET
cana-834	278	62	the	the	DET
cana-834	278	63	vertices	vertex	NOUN
cana-834	278	64	of	of	ADP
cana-834	278	65	g	g	NOUN
cana-834	278	66	and	and	CCONJ
cana-834	278	67	11	11	NUM
cana-834	278	68	)	)	PUNCT
cana-834	278	69	(	(	PUNCT
cana-834	278	70	−−	−−	PROPN
cana-834	278	71	mctdgv	mctdgv	PROPN
cana-834	278	72			PROPN
cana-834	278	73	which	which	PRON
cana-834	278	74	contradicts	contradict	VERB
cana-834	278	75	sna	sna	PROPN
cana-834	278	76	-	-	PUNCT
cana-834	278	77	set	set	NOUN
cana-834	278	78	.	.	PUNCT
cana-834	279	1	let	let	VERB
cana-834	279	2	(	(	PUNCT
cana-834	279	3	)	)	PUNCT
cana-834	279	4			PROPN
cana-834	279	5			NOUN
cana-834	279	6			X
cana-834	279	7			NOUN
cana-834	279	8			VERB
cana-834	279	9			NOUN
cana-834	279	10	−	−	NOUN
cana-834	279	11	=	=	NOUN
cana-834	279	12	=	=	SYM
cana-834	280	1	−	−	NUM
cana-834	280	2	2	2	NUM
cana-834	280	3	1	1	NUM
cana-834	280	4	12	12	NUM
cana-834	280	5	m	m	VERB
cana-834	280	6	nctd	nctd	VERB
cana-834	280	7	mctd	mctd	VERB
cana-834	280	8			NOUN
cana-834	280	9	.hence	.hence	PUNCT
cana-834	280	10	21	21	NUM
cana-834	280	11	ddd	ddd	NOUN
cana-834	280	12	=	=	X
cana-834	280	13	is	be	AUX
cana-834	280	14	a	a	DET
cana-834	280	15	sna	sna	PROPN
cana-834	280	16	-	-	PUNCT
cana-834	280	17	set	set	NOUN
cana-834	280	18	of	of	ADP
cana-834	280	19	g.	g.	PROPN
cana-834	280	20	|𝐷|	|𝐷|	PROPN
cana-834	280	21	=	=	PUNCT
cana-834	280	22	n	n	CCONJ
cana-834	280	23	m	m	NOUN
cana-834	280	24	n	n	ADJ
cana-834	280	25	+	+	ADJ
cana-834	280	26			ADJ
cana-834	280	27			NOUN
cana-834	280	28			X
cana-834	280	29			NOUN
cana-834	280	30			VERB
cana-834	280	31			NOUN
cana-834	280	32	−	−	NUM
cana-834	280	33	2	2	NUM
cana-834	280	34	1	1	NUM
cana-834	280	35			NOUN
cana-834	280	36			NOUN
cana-834	280	37			X
cana-834	280	38			NOUN
cana-834	280	39			X
cana-834	280	40			PROPN
cana-834	280	41	+	+	CCONJ
cana-834	280	42	=	=	SYM
cana-834	280	43	2	2	NUM
cana-834	280	44	1	1	NUM
cana-834	280	45	m	m	NOUN
cana-834	280	46	n	n	NUM
cana-834	280	47			NUM
cana-834	280	48	𝑆𝑁𝐴	𝑆𝑁𝐴	PROPN
cana-834	280	49	(	(	PUNCT
cana-834	280	50	𝑇	𝑇	PROPN
cana-834	280	51	∘	∘	PROPN
cana-834	280	52	𝑊𝑚	𝑊𝑚	PROPN
cana-834	280	53	)	)	PUNCT
cana-834	280	54	=	=	SYM
cana-834	280	55	𝑛	𝑛	PRON
cana-834	280	56			ADJ
cana-834	280	57			NOUN
cana-834	280	58			X
cana-834	280	59			NOUN
cana-834	280	60			X
cana-834	280	61			PROPN
cana-834	280	62	+	+	CCONJ
cana-834	280	63	2	2	NUM
cana-834	280	64	1	1	NUM
cana-834	280	65	m	m	NOUN
cana-834	280	66	.	.	PUNCT
cana-834	281	1	theorem	theorem	VERB
cana-834	281	2	3.6	3.6	NUM
cana-834	281	3	.	.	PUNCT
cana-834	282	1	for	for	ADP
cana-834	282	2	𝑚1	𝑚1	NOUN
cana-834	282	3	,	,	PUNCT
cana-834	282	4	𝑚2	𝑚2	NOUN
cana-834	282	5	≥	≥	NOUN
cana-834	282	6	2	2	NUM
cana-834	282	7	,	,	PUNCT
cana-834	282	8	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	NOUN
cana-834	282	9	∘	∘	NOUN
cana-834	282	10	𝐾𝑚1,𝑚2	𝐾𝑚1,𝑚2	PROPN
cana-834	282	11	)	)	PUNCT
cana-834	282	12	=	=	SYM
cana-834	282	13	𝑛	𝑛	PRON
cana-834	282	14	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-834	282	15	(	(	PUNCT
cana-834	282	16	)	)	PUNCT
cana-834	282	17	21,mm	21,mm	X
cana-834	282	18	.	.	PUNCT
cana-834	283	1	proof	proof	NOUN
cana-834	283	2	.	.	PUNCT
cana-834	284	1	take	take	VERB
cana-834	284	2	𝐺	𝐺	NOUN
cana-834	284	3	=	=	PUNCT
cana-834	284	4	𝑇	𝑇	PROPN
cana-834	284	5	∘	∘	ADJ
cana-834	285	1	𝐾𝑚	𝐾𝑚	PROPN
cana-834	285	2	 	 	SPACE
cana-834	285	3	1,𝑚	1,𝑚	NUM
cana-834	285	4	 	 	SPACE
cana-834	285	5	2	2	NUM
cana-834	285	6	.let	.let	NOUN
cana-834	285	7	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	285	8	)	)	PUNCT
cana-834	285	9	=	=	PRON
cana-834	285	10	{	{	PUNCT
cana-834	285	11	𝑣𝑖	𝑣𝑖	ADP
cana-834	285	12	:	:	PUNCT
cana-834	285	13	1	1	NUM
cana-834	285	14	≤	≤	NUM
cana-834	285	15	𝑖	𝑖	SYM
cana-834	285	16	≤	≤	NUM
cana-834	285	17	𝑛	𝑛	NOUN
cana-834	285	18	}	}	PUNCT
cana-834	285	19	and	and	CCONJ
cana-834	285	20	𝑉(𝐾𝑚1,𝑚2	𝑉(𝐾𝑚1,𝑚2	PROPN
cana-834	285	21	)	)	PUNCT
cana-834	286	1	=	=	PRON
cana-834	286	2	{	{	PUNCT
cana-834	286	3	𝑢𝑗	𝑢𝑗	NOUN
cana-834	286	4	:	:	PUNCT
cana-834	286	5	1	1	NUM
cana-834	286	6	≤	≤	NUM
cana-834	286	7	𝑗	𝑗	PRON
cana-834	286	8	≤	≤	NUM
cana-834	286	9	𝑚1	𝑚1	NOUN
cana-834	286	10	}	}	PUNCT
cana-834	286	11	∪	∪	X
cana-834	286	12	{	{	PUNCT
cana-834	286	13	𝑤𝑗	𝑤𝑗	ADJ
cana-834	286	14	:	:	SYM
cana-834	286	15	1	1	NUM
cana-834	286	16	≤	≤	NUM
cana-834	286	17	𝑗	𝑗	PRON
cana-834	286	18	≤	≤	NUM
cana-834	286	19	𝑚2	𝑚2	NOUN
cana-834	286	20	}	}	PUNCT
cana-834	286	21	.	.	PUNCT
cana-834	287	1	then	then	ADV
cana-834	287	2	𝑉(𝑇	𝑉(𝑇	VERB
cana-834	287	3	∘	∘	NOUN
cana-834	287	4	𝐾𝑚1𝑚2	𝐾𝑚1𝑚2	ADV
cana-834	287	5	)	)	PUNCT
cana-834	288	1	=	=	PUNCT
cana-834	289	1	⋃	⋃	NOUN
cana-834	289	2	𝑣𝑖	𝑣𝑖	NOUN
cana-834	289	3	𝑛	𝑛	PRON
cana-834	289	4	𝑖=1	𝑖=1	PROPN
cana-834	289	5	∪	∪	X
cana-834	289	6	{	{	PUNCT
cana-834	289	7	𝑢𝑖𝑗	𝑢𝑖𝑗	NOUN
cana-834	289	8	 	 	SPACE
cana-834	289	9	:	:	PUNCT
cana-834	289	10	1	1	NUM
cana-834	289	11	≤	≤	NUM
cana-834	289	12	𝑖	𝑖	SYM
cana-834	289	13	≤	≤	NOUN
cana-834	289	14	𝑛	𝑛	NOUN
cana-834	289	15	;	;	PUNCT
cana-834	289	16	1	1	NUM
cana-834	289	17	≤	≤	NUM
cana-834	290	1	𝑗	𝑗	PRON
cana-834	290	2	≤	≤	NUM
cana-834	290	3	𝑚1	𝑚1	NOUN
cana-834	290	4	}	}	PUNCT
cana-834	290	5	∪	∪	NOUN
cana-834	290	6	{	{	PUNCT
cana-834	290	7	𝑤𝑖𝑗	𝑤𝑖𝑗	PROPN
cana-834	290	8	:	:	PUNCT
cana-834	290	9	1	1	NUM
cana-834	290	10	≤	≤	NUM
cana-834	290	11	𝑖	𝑖	SYM
cana-834	290	12	≤	≤	NOUN
cana-834	290	13	𝑛	𝑛	NOUN
cana-834	290	14	;	;	PUNCT
cana-834	290	15	1	1	NUM
cana-834	290	16	≤	≤	NUM
cana-834	290	17	𝑗	𝑗	PRON
cana-834	290	18	≤	≤	NUM
cana-834	290	19	𝑚2	𝑚2	NOUN
cana-834	290	20	}	}	PUNCT
cana-834	290	21	case	case	NOUN
cana-834	290	22	(	(	PUNCT
cana-834	290	23	i	i	NOUN
cana-834	290	24	)	)	PUNCT
cana-834	290	25	.	.	PUNCT
cana-834	291	1	𝑚1	𝑚1	NOUN
cana-834	291	2	<	<	X
cana-834	291	3	𝑚2	𝑚2	PROPN
cana-834	291	4	let	let	VERB
cana-834	291	5	𝐷	𝐷	NOUN
cana-834	291	6	=	=	SYM
cana-834	291	7	{	{	PUNCT
cana-834	291	8	𝑢𝑖𝑗	𝑢𝑖𝑗	ADJ
cana-834	291	9	:	:	PUNCT
cana-834	291	10	1	1	NUM
cana-834	291	11	≤	≤	NUM
cana-834	291	12	𝑖	𝑖	SYM
cana-834	291	13	≤	≤	NUM
cana-834	291	14	𝑛	𝑛	NOUN
cana-834	291	15	,	,	PUNCT
cana-834	291	16	1	1	NUM
cana-834	291	17	≤	≤	NUM
cana-834	291	18	𝑗	𝑗	PRON
cana-834	291	19	≤	≤	NUM
cana-834	291	20	𝑚1	𝑚1	NOUN
cana-834	291	21	}	}	PUNCT
cana-834	291	22	which	which	PRON
cana-834	291	23	dominates	dominate	VERB
cana-834	291	24	all	all	DET
cana-834	291	25	the	the	DET
cana-834	291	26	vertices	vertex	NOUN
cana-834	291	27	of	of	ADP
cana-834	291	28	v(g	v(g	NOUN
cana-834	291	29	)	)	PUNCT
cana-834	291	30	and	and	CCONJ
cana-834	291	31	〈	〈	NOUN
cana-834	291	32	𝑉(𝐺	𝑉(𝐺	NOUN
cana-834	291	33	)	)	PUNCT
cana-834	291	34	−	−	PROPN
cana-834	291	35	𝐷	𝐷	NOUN
cana-834	291	36	〉	〉	NOUN
cana-834	291	37	≅	≅	PROPN
cana-834	291	38	𝑇	𝑇	PROPN
cana-834	291	39	∘	∘	PROPN
cana-834	291	40	𝐾𝑚2	𝐾𝑚2	PROPN
cana-834	291	41	.	.	PUNCT
cana-834	292	1	hence	hence	ADV
cana-834	292	2	d	d	PRON
cana-834	292	3	is	be	AUX
cana-834	292	4	a	a	DET
cana-834	292	5	minimum	minimum	ADJ
cana-834	292	6	sna	sna	NOUN
cana-834	292	7	-	-	PUNCT
cana-834	292	8	set	set	NOUN
cana-834	292	9	of	of	ADP
cana-834	292	10	g.	g.	PROPN
cana-834	292	11	therefore	therefore	ADV
cana-834	292	12	,	,	PUNCT
cana-834	292	13	communications	communication	NOUN
cana-834	292	14	on	on	ADP
cana-834	292	15	applied	apply	VERB
cana-834	292	16	nonlinear	nonlinear	ADJ
cana-834	292	17	analysis	analysis	NOUN
cana-834	292	18	issn	issn	NOUN
cana-834	292	19	:	:	PUNCT
cana-834	292	20	1074	1074	NUM
cana-834	292	21	-	-	PUNCT
cana-834	292	22	133x	133x	NUM
cana-834	292	23	vol	vol	NOUN
cana-834	292	24	31	31	NUM
cana-834	292	25	no	no	NOUN
cana-834	292	26	.	.	PUNCT
cana-834	293	1	4s	4s	NUM
cana-834	293	2	(	(	PUNCT
cana-834	293	3	2024	2024	NUM
cana-834	293	4	)	)	PUNCT
cana-834	293	5	174	174	NUM
cana-834	293	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	293	7	|𝐷|	|𝐷|	NOUN
cana-834	293	8	=	=	SYM
cana-834	293	9	𝑛𝑚1	𝑛𝑚1	NOUN
cana-834	293	10	(	(	PUNCT
cana-834	293	11	5	5	NUM
cana-834	293	12	)	)	PUNCT
cana-834	293	13	case	case	NOUN
cana-834	293	14	(	(	PUNCT
cana-834	293	15	ii).𝑚2	ii).𝑚2	X
cana-834	293	16	<	<	X
cana-834	293	17	𝑚1	𝑚1	X
cana-834	293	18	.	.	PUNCT
cana-834	294	1	let	let	VERB
cana-834	294	2	𝐷	𝐷	NOUN
cana-834	294	3	=	=	SYM
cana-834	294	4	{	{	PUNCT
cana-834	294	5	𝑤𝑖𝑗	𝑤𝑖𝑗	PROPN
cana-834	294	6	:	:	PUNCT
cana-834	294	7	1	1	NUM
cana-834	294	8	≤	≤	NUM
cana-834	294	9	𝑖	𝑖	SYM
cana-834	294	10	≤	≤	NUM
cana-834	294	11	𝑛	𝑛	NOUN
cana-834	294	12	,	,	PUNCT
cana-834	294	13	1	1	NUM
cana-834	294	14	≤	≤	NUM
cana-834	294	15	𝑗	𝑗	PRON
cana-834	294	16	≤	≤	NUM
cana-834	294	17	𝑚2	𝑚2	PROPN
cana-834	294	18	}	}	PUNCT
cana-834	294	19	which	which	PRON
cana-834	294	20	dominates	dominate	VERB
cana-834	294	21	all	all	DET
cana-834	294	22	the	the	DET
cana-834	294	23	vertices	vertex	NOUN
cana-834	294	24	of	of	ADP
cana-834	294	25	𝑉(𝐺	𝑉(𝐺	NOUN
cana-834	294	26	)	)	PUNCT
cana-834	294	27	and	and	CCONJ
cana-834	294	28	〈	〈	NOUN
cana-834	294	29	𝑉(𝐺	𝑉(𝐺	NOUN
cana-834	294	30	)	)	PUNCT
cana-834	294	31	−	−	PROPN
cana-834	294	32	𝐷	𝐷	NOUN
cana-834	294	33	〉	〉	NOUN
cana-834	294	34	≅	≅	PROPN
cana-834	294	35	𝑇	𝑇	PROPN
cana-834	294	36	∘	∘	PROPN
cana-834	294	37	𝐾𝑚1	𝐾𝑚1	NOUN
cana-834	294	38	.	.	PUNCT
cana-834	295	1	hence	hence	ADV
cana-834	295	2	d	d	PRON
cana-834	295	3	is	be	AUX
cana-834	295	4	a	a	DET
cana-834	295	5	minimum	minimum	ADJ
cana-834	295	6	sna	sna	NOUN
cana-834	295	7	-	-	PUNCT
cana-834	295	8	set	set	NOUN
cana-834	295	9	of	of	ADP
cana-834	295	10	g.	g.	PROPN
cana-834	295	11	therefore	therefore	ADV
cana-834	295	12	,	,	PUNCT
cana-834	295	13	|𝐷|	|𝐷|	X
cana-834	295	14	=	=	SYM
cana-834	295	15	𝑛𝑚2	𝑛𝑚2	PROPN
cana-834	295	16	(	(	PUNCT
cana-834	295	17	6	6	NUM
cana-834	295	18	)	)	PUNCT
cana-834	295	19	from	from	ADP
cana-834	295	20	(	(	PUNCT
cana-834	295	21	5	5	NUM
cana-834	295	22	)	)	PUNCT
cana-834	295	23	and	and	CCONJ
cana-834	295	24	(	(	PUNCT
cana-834	295	25	6)|𝐷|	6)|𝐷|	NUM
cana-834	295	26	=	=	SYM
cana-834	295	27	𝑛	𝑛	PROPN
cana-834	295	28	  	  	SPACE
cana-834	295	29	𝑚𝑖𝑛(𝑚1	𝑚𝑖𝑛(𝑚1	PROPN
cana-834	295	30	,	,	PUNCT
cana-834	295	31	𝑚2	𝑚2	PROPN
cana-834	295	32	)	)	PUNCT
cana-834	295	33	.	.	PUNCT
cana-834	296	1	theorem	theorem	VERB
cana-834	296	2	3.7	3.7	NUM
cana-834	296	3	.	.	PUNCT
cana-834	297	1	for	for	ADP
cana-834	297	2	m	m	PROPN
cana-834	297	3	≥	≥	NUM
cana-834	297	4	3	3	NUM
cana-834	297	5	,	,	PUNCT
cana-834	297	6	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	PROPN
cana-834	297	7	∘	∘	PROPN
cana-834	297	8	𝐾1,𝑚−1	𝐾1,𝑚−1	PROPN
cana-834	297	9	)	)	PUNCT
cana-834	297	10	=	=	NOUN
cana-834	298	1	𝑛.	𝑛.	NOUN
cana-834	298	2	proof	proof	NOUN
cana-834	298	3	.	.	PUNCT
cana-834	299	1	let	let	VERB
cana-834	299	2	𝑉(𝑇	𝑉(𝑇	NOUN
cana-834	299	3	)	)	PUNCT
cana-834	299	4	=	=	PRON
cana-834	299	5	{	{	PUNCT
cana-834	299	6	𝑣1	𝑣1	PROPN
cana-834	299	7	,	,	PUNCT
cana-834	299	8	𝑣2	𝑣2	PROPN
cana-834	299	9	,	,	PUNCT
cana-834	299	10	.	.	PUNCT
cana-834	299	11	.	.	PUNCT
cana-834	299	12	.	.	PUNCT
cana-834	300	1	.	.	PUNCT
cana-834	301	1	𝑣𝑛}and	𝑣𝑛}and	PUNCT
cana-834	302	1	𝑉(𝐾1,𝑚−1	𝑉(𝐾1,𝑚−1	X
cana-834	302	2	)	)	PUNCT
cana-834	302	3	=	=	SYM
cana-834	302	4	{	{	PUNCT
cana-834	302	5	𝑐	𝑐	NOUN
cana-834	302	6	,	,	PUNCT
cana-834	302	7	𝑢1	𝑢1	NOUN
cana-834	302	8	,	,	PUNCT
cana-834	302	9	𝑢2	𝑢2	PROPN
cana-834	302	10	.	.	PUNCT
cana-834	302	11	.	.	PUNCT
cana-834	302	12	.	.	PUNCT
cana-834	302	13	.	.	PUNCT
cana-834	302	14	.	.	PUNCT
cana-834	302	15	.	.	PUNCT
cana-834	303	1	𝑢𝑚	𝑢𝑚	X
cana-834	303	2	}	}	PUNCT
cana-834	303	3	then	then	ADV
cana-834	303	4	𝑉(𝑇	𝑉(𝑇	VERB
cana-834	303	5	∘	∘	X
cana-834	303	6	𝐾1,𝑚	𝐾1,𝑚	ADV
cana-834	303	7	)	)	PUNCT
cana-834	304	1	=	=	PRON
cana-834	304	2	{	{	PUNCT
cana-834	304	3	𝑣𝑖	𝑣𝑖	ADP
cana-834	304	4	:	:	PUNCT
cana-834	304	5	1	1	NUM
cana-834	304	6	≤	≤	NUM
cana-834	304	7	𝑖	𝑖	SYM
cana-834	304	8	≤	≤	NUM
cana-834	304	9	𝑛	𝑛	PRON
cana-834	304	10	}	}	PUNCT
cana-834	304	11	∪	∪	ADJ
cana-834	304	12	{	{	PUNCT
cana-834	304	13	𝑐𝑖	𝑐𝑖	NOUN
cana-834	304	14	:	:	PUNCT
cana-834	304	15	1	1	NUM
cana-834	304	16	≤	≤	NUM
cana-834	304	17	𝑖	𝑖	SYM
cana-834	304	18	≤	≤	NUM
cana-834	304	19	𝑛	𝑛	PRON
cana-834	304	20	}	}	PUNCT
cana-834	304	21	∪	∪	ADJ
cana-834	304	22	{	{	PUNCT
cana-834	304	23	𝑢𝑖𝑗	𝑢𝑖𝑗	ADJ
cana-834	304	24	:	:	PUNCT
cana-834	304	25	1	1	NUM
cana-834	304	26	≤	≤	NUM
cana-834	304	27	𝑖	𝑖	SYM
cana-834	304	28	≤	≤	NOUN
cana-834	304	29	𝑛	𝑛	PRON
cana-834	304	30	 	 	SPACE
cana-834	304	31	;	;	PUNCT
cana-834	304	32	1	1	NUM
cana-834	304	33	≤	≤	NUM
cana-834	304	34	𝑗	𝑗	PRON
cana-834	304	35	≤	≤	NUM
cana-834	304	36	𝑚	𝑚	NOUN
cana-834	304	37	}	}	PUNCT
cana-834	304	38	.	.	PUNCT
cana-834	305	1	let	let	VERB
cana-834	305	2	𝐷	𝐷	NOUN
cana-834	305	3	=	=	PUNCT
cana-834	305	4	{	{	PUNCT
cana-834	305	5	𝑐𝑖	𝑐𝑖	NOUN
cana-834	305	6	:	:	PUNCT
cana-834	305	7	1	1	NUM
cana-834	305	8	≤	≤	NUM
cana-834	305	9	𝑖	𝑖	SYM
cana-834	305	10	≤	≤	NUM
cana-834	305	11	𝑛	𝑛	PRON
cana-834	305	12	}	}	PUNCT
cana-834	305	13	is	be	AUX
cana-834	305	14	a	a	DET
cana-834	305	15	minimum	minimum	ADJ
cana-834	305	16	sna	sna	NOUN
cana-834	305	17	-	-	PUNCT
cana-834	305	18	set	set	NOUN
cana-834	305	19	of	of	ADP
cana-834	305	20	𝑇	𝑇	PROPN
cana-834	305	21	∘	∘	PROPN
cana-834	305	22	𝐾1,𝑚−1	𝐾1,𝑚−1	VERB
cana-834	305	23	and	and	CCONJ
cana-834	305	24	⟨𝑉(𝑇	⟨𝑉(𝑇	VERB
cana-834	305	25	∘	∘	PROPN
cana-834	305	26	𝐾1,𝑚−1	𝐾1,𝑚−1	PROPN
cana-834	305	27	)	)	PUNCT
cana-834	306	1	−	−	PROPN
cana-834	307	1	𝐷⟩	𝐷⟩	PROPN
cana-834	307	2	≅	≅	PROPN
cana-834	307	3	𝑇	𝑇	PROPN
cana-834	307	4	∘	∘	PROPN
cana-834	307	5	𝐾𝑚−1	𝐾𝑚−1	PROPN
cana-834	307	6	is	be	AUX
cana-834	307	7	a	a	DET
cana-834	307	8	tree	tree	NOUN
cana-834	307	9	.	.	PUNCT
cana-834	308	1	hence	hence	ADV
cana-834	308	2	,	,	PUNCT
cana-834	308	3	|𝐷|	|𝐷|	X
cana-834	308	4	=	=	SYM
cana-834	308	5	𝑛	𝑛	NOUN
cana-834	308	6	=	=	SYM
cana-834	308	7	𝛾𝑆𝑁𝐴(𝑇	𝛾𝑆𝑁𝐴(𝑇	PROPN
cana-834	308	8	∘	∘	PROPN
cana-834	308	9	𝐾1,𝑚−1	𝐾1,𝑚−1	PROPN
cana-834	308	10	)	)	PUNCT
cana-834	308	11	.	.	PUNCT
cana-834	309	1	theorem	theorem	VERB
cana-834	309	2	3.8	3.8	NUM
cana-834	309	3	.	.	PUNCT
cana-834	310	1	for	for	ADP
cana-834	310	2	m≥5	m≥5	PROPN
cana-834	310	3	then	then	ADV
cana-834	310	4	γsna	γsna	VERB
cana-834	310	5	(	(	PUNCT
cana-834	310	6	t	t	NOUN
cana-834	310	7	◦	◦	NOUN
cana-834	310	8	tm)≤	tm)≤	NOUN
cana-834	310	9	n(m−p+1),where	n(m−p+1),where	X
cana-834	311	1	p≥2	p≥2	NOUN
cana-834	311	2	be	be	VERB
cana-834	311	3	the	the	DET
cana-834	311	4	number	number	NOUN
cana-834	311	5	of	of	ADP
cana-834	311	6	pendant	pendant	ADJ
cana-834	311	7	vertices	vertex	NOUN
cana-834	311	8	of	of	ADP
cana-834	311	9	tree	tree	NOUN
cana-834	311	10	tm	tm	NOUN
cana-834	311	11	.	.	PUNCT
cana-834	312	1	proof	proof	NOUN
cana-834	312	2	.	.	PUNCT
cana-834	313	1	take	take	VERB
cana-834	313	2	g	g	NOUN
cana-834	313	3	=	=	NOUN
cana-834	313	4	t∘tm	t∘tm	NOUN
cana-834	313	5	.	.	PUNCT
cana-834	314	1	let	let	VERB
cana-834	314	2	v(t)={vi/1≤i≤n}and	v(t)={vi/1≤i≤n}and	ADV
cana-834	314	3	v(tm)={uj/1≤j≤m	v(tm)={uj/1≤j≤m	ADJ
cana-834	314	4	}	}	PUNCT
cana-834	314	5	.	.	PUNCT
cana-834	315	1	then	then	ADV
cana-834	315	2	v(g)=v(t)∪	v(g)=v(t)∪	VERB
cana-834	315	3	v(⋃	v(⋃	PROPN
cana-834	315	4	𝑇𝑚	𝑇𝑚	PROPN
cana-834	315	5	𝑖𝑛	𝑖𝑛	NOUN
cana-834	315	6	𝑖=1	𝑖=1	PROPN
cana-834	315	7	)	)	PUNCT
cana-834	315	8	where	where	SCONJ
cana-834	315	9	i	i	PRON
cana-834	315	10	mt	mt	PROPN
cana-834	315	11	is	be	AUX
cana-834	315	12	i	i	PRON
cana-834	315	13	t	t	NOUN
cana-834	315	14	h	h	PROPN
cana-834	315	15	copy	copy	NOUN
cana-834	315	16	of	of	ADP
cana-834	315	17	tm	tm	PROPN
cana-834	315	18	is	be	AUX
cana-834	315	19	adjacent	adjacent	ADJ
cana-834	315	20	to	to	ADP
cana-834	315	21	the	the	DET
cana-834	315	22	vertex	vertex	NOUN
cana-834	315	23	vi	vi	PROPN
cana-834	315	24	in	in	ADP
cana-834	315	25	t.	t.	PROPN
cana-834	315	26	let	let	VERB
cana-834	315	27	di	di	NOUN
cana-834	315	28	=	=	NOUN
cana-834	315	29	v	v	NOUN
cana-834	315	30	(	(	PUNCT
cana-834	315	31	i	i	PROPN
cana-834	315	32	mt	mt	PROPN
cana-834	315	33	)	)	PUNCT
cana-834	315	34	−p	−p	NOUN
cana-834	315	35	which	which	PRON
cana-834	315	36	dominates	dominate	VERB
cana-834	315	37	all	all	DET
cana-834	315	38	the	the	DET
cana-834	315	39	vertices	vertex	NOUN
cana-834	315	40	of	of	ADP
cana-834	315	41	i	i	PRON
cana-834	315	42	mt	mt	PROPN
cana-834	315	43	and	and	CCONJ
cana-834	315	44	vi	vi	VERB
cana-834	315	45	in	in	ADP
cana-834	315	46	t.	t.	PROPN
cana-834	315	47	therefore	therefore	ADV
cana-834	315	48	,	,	PUNCT
cana-834	315	49	d	d	PROPN
cana-834	315	50	=	=	SYM
cana-834	315	51	v(t)∪di	v(t)∪di	X
cana-834	315	52	=	=	NOUN
cana-834	315	53	n(m+1−p	n(m+1−p	X
cana-834	315	54	)	)	PUNCT
cana-834	315	55	4	4	NUM
cana-834	315	56	.	.	PUNCT
cana-834	315	57	optimization	optimization	NOUN
cana-834	315	58	techniques	technique	NOUN
cana-834	315	59	for	for	ADP
cana-834	315	60	algorithmic	algorithmic	ADJ
cana-834	315	61	improvements	improvement	NOUN
cana-834	315	62	and	and	CCONJ
cana-834	315	63	data	datum	NOUN
cana-834	315	64	structures	structure	NOUN
cana-834	315	65	there	there	PRON
cana-834	315	66	are	be	VERB
cana-834	315	67	several	several	ADJ
cana-834	315	68	optimization	optimization	NOUN
cana-834	315	69	strategies	strategy	NOUN
cana-834	315	70	that	that	PRON
cana-834	315	71	can	can	AUX
cana-834	315	72	be	be	AUX
cana-834	315	73	utilized	utilize	VERB
cana-834	315	74	in	in	ADP
cana-834	315	75	order	order	NOUN
cana-834	315	76	to	to	PART
cana-834	315	77	deal	deal	VERB
cana-834	315	78	with	with	ADP
cana-834	315	79	the	the	DET
cana-834	315	80	difficulties	difficulty	NOUN
cana-834	315	81	posed	pose	VERB
cana-834	315	82	by	by	ADP
cana-834	315	83	large	large	ADJ
cana-834	315	84	-	-	PUNCT
cana-834	315	85	scale	scale	NOUN
cana-834	315	86	and	and	CCONJ
cana-834	315	87	dynamic	dynamic	ADJ
cana-834	315	88	social	social	ADJ
cana-834	315	89	networks	network	NOUN
cana-834	315	90	[	[	X
cana-834	315	91	19	19	NUM
cana-834	315	92	]	]	PUNCT
cana-834	315	93	.	.	PUNCT
cana-834	316	1	the	the	DET
cana-834	316	2	goal	goal	NOUN
cana-834	316	3	of	of	ADP
cana-834	316	4	these	these	DET
cana-834	316	5	methods	method	NOUN
cana-834	316	6	is	be	AUX
cana-834	316	7	to	to	PART
cana-834	316	8	improve	improve	VERB
cana-834	316	9	the	the	DET
cana-834	316	10	efficiency	efficiency	NOUN
cana-834	316	11	,	,	PUNCT
cana-834	316	12	scalability	scalability	NOUN
cana-834	316	13	,	,	PUNCT
cana-834	316	14	and	and	CCONJ
cana-834	316	15	accuracy	accuracy	NOUN
cana-834	316	16	of	of	ADP
cana-834	316	17	social	social	ADJ
cana-834	316	18	network	network	NOUN
cana-834	316	19	analysis	analysis	NOUN
cana-834	316	20	.	.	PUNCT
cana-834	317	1	they	they	PRON
cana-834	317	2	include	include	VERB
cana-834	317	3	improvements	improvement	NOUN
cana-834	317	4	to	to	ADP
cana-834	317	5	algorithms	algorithm	NOUN
cana-834	317	6	,	,	PUNCT
cana-834	317	7	cutting	cut	VERB
cana-834	317	8	-	-	PUNCT
cana-834	317	9	edge	edge	NOUN
cana-834	317	10	data	datum	NOUN
cana-834	317	11	structures	structure	NOUN
cana-834	317	12	,	,	PUNCT
cana-834	317	13	and	and	CCONJ
cana-834	317	14	specialized	specialized	ADJ
cana-834	317	15	hardware	hardware	NOUN
cana-834	317	16	implementations	implementation	NOUN
cana-834	317	17	.	.	PUNCT
cana-834	318	1	this	this	DET
cana-834	318	2	segment	segment	NOUN
cana-834	318	3	investigates	investigate	VERB
cana-834	318	4	these	these	DET
cana-834	318	5	advanced	advanced	ADJ
cana-834	318	6	systems	system	NOUN
cana-834	318	7	exhaustively	exhaustively	ADV
cana-834	318	8	.	.	PUNCT
cana-834	319	1	(	(	PUNCT
cana-834	319	2	i).parallel	i).parallel	PROPN
cana-834	319	3	computing	compute	VERB
cana-834	319	4	a	a	DET
cana-834	319	5	computational	computational	ADJ
cana-834	319	6	problem	problem	NOUN
cana-834	319	7	is	be	AUX
cana-834	319	8	broken	break	VERB
cana-834	319	9	down	down	ADP
cana-834	319	10	into	into	ADP
cana-834	319	11	smaller	small	ADJ
cana-834	319	12	tasks	task	NOUN
cana-834	319	13	that	that	PRON
cana-834	319	14	can	can	AUX
cana-834	319	15	be	be	AUX
cana-834	319	16	done	do	VERB
cana-834	319	17	simultaneously	simultaneously	ADV
cana-834	319	18	using	use	VERB
cana-834	319	19	parallel	parallel	ADJ
cana-834	319	20	computing	computing	NOUN
cana-834	319	21	.	.	PUNCT
cana-834	320	1	this	this	DET
cana-834	320	2	method	method	NOUN
cana-834	320	3	works	work	VERB
cana-834	320	4	best	good	ADJ
cana-834	320	5	for	for	ADP
cana-834	320	6	graph	graph	NOUN
cana-834	320	7	algorithms	algorithm	NOUN
cana-834	320	8	,	,	PUNCT
cana-834	320	9	which	which	PRON
cana-834	320	10	frequently	frequently	ADV
cana-834	320	11	require	require	VERB
cana-834	320	12	independent	independent	ADJ
cana-834	320	13	and	and	CCONJ
cana-834	320	14	repetitive	repetitive	ADJ
cana-834	320	15	computations	computation	NOUN
cana-834	320	16	[	[	X
cana-834	320	17	20	20	NUM
cana-834	320	18	]	]	PUNCT
cana-834	320	19	.	.	PUNCT
cana-834	321	1	multi	multi	ADJ
cana-834	321	2	-	-	ADJ
cana-834	321	3	threading	threading	NOUN
cana-834	321	4	:	:	PUNCT
cana-834	321	5	concurrently	concurrently	ADV
cana-834	321	6	using	use	VERB
cana-834	321	7	multiple	multiple	ADJ
cana-834	321	8	cpu	cpu	NOUN
cana-834	321	9	threads	thread	NOUN
cana-834	321	10	for	for	ADP
cana-834	321	11	computations	computation	NOUN
cana-834	321	12	the	the	DET
cana-834	321	13	shortest	short	ADJ
cana-834	321	14	path	path	NOUN
cana-834	321	15	calculation	calculation	NOUN
cana-834	321	16	,	,	PUNCT
cana-834	321	17	for	for	ADP
cana-834	321	18	instance	instance	NOUN
cana-834	321	19	,	,	PUNCT
cana-834	321	20	can	can	AUX
cana-834	321	21	be	be	AUX
cana-834	321	22	sped	speed	VERB
cana-834	321	23	up	up	ADP
cana-834	321	24	with	with	ADP
cana-834	321	25	parallel	parallel	ADJ
cana-834	321	26	bfs	bfs	NOUN
cana-834	321	27	(	(	PUNCT
cana-834	321	28	breadth	breadth	NOUN
cana-834	321	29	-	-	PUNCT
cana-834	321	30	first	first	ADJ
cana-834	321	31	search	search	NOUN
cana-834	321	32	)	)	PUNCT
cana-834	321	33	.	.	PUNCT
cana-834	322	1	distributed	distribute	VERB
cana-834	322	2	computing	computing	NOUN
cana-834	322	3	:	:	PUNCT
cana-834	322	4	communications	communication	NOUN
cana-834	322	5	on	on	ADP
cana-834	322	6	applied	apply	VERB
cana-834	322	7	nonlinear	nonlinear	ADJ
cana-834	322	8	analysis	analysis	NOUN
cana-834	322	9	issn	issn	NOUN
cana-834	322	10	:	:	PUNCT
cana-834	322	11	1074	1074	NUM
cana-834	322	12	-	-	PUNCT
cana-834	322	13	133x	133x	NUM
cana-834	322	14	vol	vol	NOUN
cana-834	322	15	31	31	NUM
cana-834	322	16	no	no	NOUN
cana-834	322	17	.	.	PUNCT
cana-834	323	1	4s	4s	NUM
cana-834	323	2	(	(	PUNCT
cana-834	323	3	2024	2024	NUM
cana-834	323	4	)	)	PUNCT
cana-834	323	5	175	175	NUM
cana-834	323	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	323	7	processing	process	VERB
cana-834	323	8	large	large	ADJ
cana-834	323	9	graphs	graph	NOUN
cana-834	323	10	on	on	ADP
cana-834	323	11	multiple	multiple	ADJ
cana-834	323	12	machines	machine	NOUN
cana-834	323	13	by	by	ADP
cana-834	323	14	utilizing	utilize	VERB
cana-834	323	15	distributed	distribute	VERB
cana-834	323	16	systems	system	NOUN
cana-834	323	17	and	and	CCONJ
cana-834	323	18	frameworks	framework	NOUN
cana-834	323	19	like	like	ADP
cana-834	323	20	apache	apache	PROPN
cana-834	323	21	hadoop	hadoop	PROPN
cana-834	323	22	and	and	CCONJ
cana-834	323	23	apache	apache	NOUN
cana-834	323	24	spark	spark	NOUN
cana-834	323	25	.	.	PUNCT
cana-834	324	1	in	in	ADP
cana-834	324	2	large	large	ADJ
cana-834	324	3	-	-	PUNCT
cana-834	324	4	scale	scale	NOUN
cana-834	324	5	networks	network	NOUN
cana-834	324	6	,	,	PUNCT
cana-834	324	7	this	this	PRON
cana-834	324	8	is	be	AUX
cana-834	324	9	useful	useful	ADJ
cana-834	324	10	for	for	ADP
cana-834	324	11	things	thing	NOUN
cana-834	324	12	like	like	ADP
cana-834	324	13	pagerank	pagerank	NOUN
cana-834	324	14	and	and	CCONJ
cana-834	324	15	community	community	NOUN
cana-834	324	16	detection	detection	NOUN
cana-834	324	17	.	.	PUNCT
cana-834	325	1	gpu	gpu	NOUN
cana-834	325	2	acceleration	acceleration	NOUN
cana-834	325	3	:	:	PUNCT
cana-834	325	4	using	use	VERB
cana-834	325	5	gpus	gpu	NOUN
cana-834	325	6	'	'	PART
cana-834	325	7	enormous	enormous	ADJ
cana-834	325	8	parallelism	parallelism	NOUN
cana-834	325	9	to	to	PART
cana-834	325	10	accelerate	accelerate	VERB
cana-834	325	11	graph	graph	NOUN
cana-834	325	12	algorithms	algorithm	NOUN
cana-834	325	13	gpu	gpu	NOUN
cana-834	325	14	-	-	PUNCT
cana-834	325	15	based	base	VERB
cana-834	325	16	executions	execution	NOUN
cana-834	325	17	of	of	ADP
cana-834	325	18	calculations	calculation	NOUN
cana-834	325	19	like	like	ADP
cana-834	325	20	bfs	bfs	NOUN
cana-834	325	21	,	,	PUNCT
cana-834	325	22	dfs	dfs	INTJ
cana-834	325	23	(	(	PUNCT
cana-834	325	24	profundity	profundity	NOUN
cana-834	325	25	first	first	ADJ
cana-834	325	26	hunt	hunt	NOUN
cana-834	325	27	)	)	PUNCT
cana-834	325	28	,	,	PUNCT
cana-834	325	29	and	and	CCONJ
cana-834	325	30	centrality	centrality	NOUN
cana-834	325	31	measures	measure	NOUN
cana-834	325	32	can	can	AUX
cana-834	325	33	altogether	altogether	ADV
cana-834	325	34	beat	beat	VERB
cana-834	325	35	computer	computer	NOUN
cana-834	325	36	chip	chip	NOUN
cana-834	325	37	-	-	PUNCT
cana-834	325	38	based	base	VERB
cana-834	325	39	adaptations	adaptation	NOUN
cana-834	325	40	.	.	PUNCT
cana-834	326	1	(	(	PUNCT
cana-834	326	2	ii).approximation	ii).approximation	NOUN
cana-834	326	3	algorithms	algorithm	VERB
cana-834	326	4	approximation	approximation	NOUN
cana-834	326	5	algorithms	algorithm	NOUN
cana-834	326	6	are	be	AUX
cana-834	326	7	suitable	suitable	ADJ
cana-834	326	8	for	for	ADP
cana-834	326	9	large	large	ADJ
cana-834	326	10	-	-	PUNCT
cana-834	326	11	scale	scale	NOUN
cana-834	326	12	networks	network	NOUN
cana-834	326	13	where	where	SCONJ
cana-834	326	14	exact	exact	ADJ
cana-834	326	15	solutions	solution	NOUN
cana-834	326	16	are	be	AUX
cana-834	326	17	computationally	computationally	ADV
cana-834	326	18	infeasible	infeasible	ADJ
cana-834	326	19	because	because	SCONJ
cana-834	326	20	they	they	PRON
cana-834	326	21	offer	offer	VERB
cana-834	326	22	near	near	ADP
cana-834	326	23	-	-	PUNCT
cana-834	326	24	optimal	optimal	ADJ
cana-834	326	25	solutions	solution	NOUN
cana-834	326	26	at	at	ADP
cana-834	326	27	lower	low	ADJ
cana-834	326	28	computational	computational	ADJ
cana-834	326	29	costs	cost	NOUN
cana-834	326	30	[	[	X
cana-834	326	31	21	21	NUM
cana-834	326	32	]	]	PUNCT
cana-834	326	33	.	.	PUNCT
cana-834	327	1	outlining	outline	VERB
cana-834	327	2	strategies	strategy	NOUN
cana-834	327	3	:	:	PUNCT
cana-834	327	4	utilizing	utilize	VERB
cana-834	327	5	information	information	NOUN
cana-834	327	6	portrayals	portrayal	NOUN
cana-834	327	7	to	to	PART
cana-834	327	8	make	make	VERB
cana-834	327	9	minimized	minimized	ADJ
cana-834	327	10	rundowns	rundown	NOUN
cana-834	327	11	of	of	ADP
cana-834	327	12	the	the	DET
cana-834	327	13	diagram	diagram	NOUN
cana-834	327	14	,	,	PUNCT
cana-834	327	15	which	which	PRON
cana-834	327	16	can	can	AUX
cana-834	327	17	then	then	ADV
cana-834	327	18	be	be	AUX
cana-834	327	19	utilized	utilize	VERB
cana-834	327	20	to	to	PART
cana-834	327	21	appraise	appraise	VERB
cana-834	327	22	measurements	measurement	NOUN
cana-834	327	23	like	like	ADP
cana-834	327	24	centrality	centrality	NOUN
cana-834	327	25	and	and	CCONJ
cana-834	327	26	availability	availability	NOUN
cana-834	327	27	with	with	ADP
cana-834	327	28	high	high	ADJ
cana-834	327	29	precision	precision	NOUN
cana-834	327	30	.	.	PUNCT
cana-834	328	1	sampling	sample	VERB
cana-834	328	2	methods	method	NOUN
cana-834	328	3	:	:	PUNCT
cana-834	328	4	using	use	VERB
cana-834	328	5	a	a	DET
cana-834	328	6	representative	representative	ADJ
cana-834	328	7	subset	subset	NOUN
cana-834	328	8	of	of	ADP
cana-834	328	9	the	the	DET
cana-834	328	10	network	network	NOUN
cana-834	328	11	to	to	PART
cana-834	328	12	draw	draw	VERB
cana-834	328	13	conclusions	conclusion	NOUN
cana-834	328	14	about	about	ADP
cana-834	328	15	its	its	PRON
cana-834	328	16	properties	property	NOUN
cana-834	328	17	.	.	PUNCT
cana-834	329	1	procedures	procedure	NOUN
cana-834	329	2	like	like	ADP
cana-834	329	3	irregular	irregular	ADJ
cana-834	329	4	hub	hub	NOUN
cana-834	329	5	testing	testing	NOUN
cana-834	329	6	or	or	CCONJ
cana-834	329	7	edge	edge	NOUN
cana-834	329	8	inspecting	inspect	VERB
cana-834	329	9	can	can	AUX
cana-834	329	10	decrease	decrease	VERB
cana-834	329	11	the	the	DET
cana-834	329	12	computational	computational	ADJ
cana-834	329	13	weight	weight	NOUN
cana-834	329	14	while	while	SCONJ
cana-834	329	15	saving	save	VERB
cana-834	329	16	key	key	ADJ
cana-834	329	17	underlying	underlying	ADJ
cana-834	329	18	properties	property	NOUN
cana-834	329	19	.	.	PUNCT
cana-834	330	1	heuristic	heuristic	ADJ
cana-834	330	2	algorithms	algorithm	NOUN
cana-834	330	3	:	:	PUNCT
cana-834	330	4	using	use	VERB
cana-834	330	5	heuristics	heuristic	NOUN
cana-834	330	6	to	to	PART
cana-834	330	7	quickly	quickly	ADV
cana-834	330	8	find	find	VERB
cana-834	330	9	solutions	solution	NOUN
cana-834	330	10	that	that	PRON
cana-834	330	11	are	be	AUX
cana-834	330	12	adequate	adequate	ADJ
cana-834	330	13	for	for	ADP
cana-834	330	14	instance	instance	NOUN
cana-834	330	15	,	,	PUNCT
cana-834	330	16	greedy	greedy	ADJ
cana-834	330	17	community	community	NOUN
cana-834	330	18	detection	detection	NOUN
cana-834	330	19	algorithms	algorithm	NOUN
cana-834	330	20	are	be	AUX
cana-834	330	21	capable	capable	ADJ
cana-834	330	22	of	of	ADP
cana-834	330	23	effectively	effectively	ADV
cana-834	330	24	identifying	identify	VERB
cana-834	330	25	dense	dense	ADJ
cana-834	330	26	subgraphs	subgraph	NOUN
cana-834	330	27	.	.	PUNCT
cana-834	331	1	sub	sub	VERB
cana-834	331	2	results	result	NOUN
cana-834	331	3	on	on	ADP
cana-834	331	4	sna	sna	PROPN
cana-834	331	5	for	for	ADP
cana-834	331	6	graph	graph	NOUN
cana-834	331	7	theory	theory	NOUN
cana-834	331	8	:	:	PUNCT
cana-834	331	9	theorem	theorem	VERB
cana-834	331	10	4.1	4.1	NUM
cana-834	331	11	.	.	PUNCT
cana-834	332	1	for	for	ADP
cana-834	332	2	any	any	DET
cana-834	332	3	sna	sna	PROPN
cana-834	332	4	graph	graph	NOUN
cana-834	332	5	g	g	PROPN
cana-834	332	6	with	with	ADP
cana-834	332	7	m	m	PROPN
cana-834	332	8	≥	≥	NUM
cana-834	332	9	2	2	NUM
cana-834	332	10	vertices	vertex	NOUN
cana-834	332	11	then	then	ADV
cana-834	332	12	n	n	NOUN
cana-834	332	13	≤	≤	NOUN
cana-834	332	14	γsna(t	γsna(t	ADP
cana-834	332	15	◦	◦	NOUN
cana-834	332	16	g	g	NOUN
cana-834	332	17	)	)	PUNCT
cana-834	332	18	≤	≤	NUM
cana-834	332	19	mn	mn	PROPN
cana-834	332	20	.	.	PUNCT
cana-834	333	1	the	the	DET
cana-834	333	2	lower	lower	ADV
cana-834	333	3	bound	bind	VERB
cana-834	333	4	is	be	AUX
cana-834	333	5	attained	attain	VERB
cana-834	333	6	if	if	SCONJ
cana-834	333	7	g	g	PROPN
cana-834	333	8	k1,m(m	k1,m(m	VERB
cana-834	333	9	≥	≥	NOUN
cana-834	333	10	1	1	NUM
cana-834	333	11	)	)	PUNCT
cana-834	333	12	and	and	CCONJ
cana-834	333	13	the	the	DET
cana-834	333	14	upper	upper	ADJ
cana-834	333	15	bound	bound	NOUN
cana-834	333	16	is	be	AUX
cana-834	333	17	attained	attain	VERB
cana-834	333	18	if	if	SCONJ
cana-834	333	19	g	g	PROPN
cana-834	333	20	mk	mk	VERB
cana-834	333	21	proof	proof	NOUN
cana-834	333	22	.	.	PUNCT
cana-834	334	1	let	let	VERB
cana-834	334	2	v	v	X
cana-834	334	3	(	(	PUNCT
cana-834	334	4	t	t	NOUN
cana-834	334	5	)	)	PUNCT
cana-834	334	6	=	=	PRON
cana-834	334	7	{	{	PUNCT
cana-834	334	8	ui	ui	PROPN
cana-834	334	9	/	/	SYM
cana-834	334	10	i	i	NOUN
cana-834	334	11	=	=	NOUN
cana-834	334	12	1	1	NUM
cana-834	334	13	,	,	PUNCT
cana-834	334	14	.	.	PUNCT
cana-834	334	15	.	.	PUNCT
cana-834	335	1	.	.	PUNCT
cana-834	336	1	,	,	PUNCT
cana-834	336	2	n}and	n}and	ADV
cana-834	336	3	v	v	ADP
cana-834	336	4	(	(	PUNCT
cana-834	336	5	g	g	NOUN
cana-834	336	6	)	)	PUNCT
cana-834	336	7	=	=	SYM
cana-834	336	8	{	{	PUNCT
cana-834	336	9	uj	uj	PROPN
cana-834	336	10	/	/	SYM
cana-834	336	11	j	j	PROPN
cana-834	336	12	=	=	SYM
cana-834	336	13	1	1	NUM
cana-834	336	14	,	,	PUNCT
cana-834	336	15	.	.	PUNCT
cana-834	336	16	.	.	PUNCT
cana-834	336	17	.	.	PUNCT
cana-834	337	1	,	,	PUNCT
cana-834	337	2	m}then	m}then	ADV
cana-834	337	3	v	v	INTJ
cana-834	337	4	(	(	PUNCT
cana-834	337	5	t	t	NOUN
cana-834	337	6	◦	◦	NOUN
cana-834	337	7	g	g	NOUN
cana-834	337	8	)	)	PUNCT
cana-834	337	9	=	=	PRON
cana-834	337	10	{	{	PUNCT
cana-834	337	11	ui	ui	NOUN
cana-834	337	12	:	:	PUNCT
cana-834	337	13	1	1	NUM
cana-834	337	14	≤	≤	NUM
cana-834	337	15	i	i	PRON
cana-834	337	16	≤	≤	NOUN
cana-834	337	17	n	n	CCONJ
cana-834	337	18	}	}	PUNCT
cana-834	337	19	∪	∪	X
cana-834	337	20	{	{	PUNCT
cana-834	337	21	uj	uj	NOUN
cana-834	337	22	:	:	PUNCT
cana-834	337	23	1	1	NUM
cana-834	337	24	≤	≤	NUM
cana-834	337	25	j	j	PROPN
cana-834	337	26	≤	≤	PROPN
cana-834	337	27	m	m	PRON
cana-834	337	28	}	}	PUNCT
cana-834	337	29	.	.	PUNCT
cana-834	338	1	assume	assume	VERB
cana-834	338	2	γsna(t	γsna(t	ADP
cana-834	338	3	◦	◦	NOUN
cana-834	338	4	g	g	NOUN
cana-834	338	5	)	)	PUNCT
cana-834	338	6	=	=	SYM
cana-834	338	7	nm	nm	NOUN
cana-834	339	1	+	+	CCONJ
cana-834	339	2	n.	n.	NOUN
cana-834	339	3	let	let	VERB
cana-834	339	4	d	d	PRON
cana-834	339	5	be	be	AUX
cana-834	339	6	a	a	DET
cana-834	339	7	γsna	γsna	NOUN
cana-834	339	8	-	-	PUNCT
cana-834	339	9	set	set	NOUN
cana-834	339	10	of	of	ADP
cana-834	339	11	t	t	PROPN
cana-834	339	12	◦	◦	NOUN
cana-834	339	13	g	g	NOUN
cana-834	339	14	having	have	VERB
cana-834	339	15	nm	nm	INTJ
cana-834	339	16	–	–	PUNCT
cana-834	339	17	n	n	DET
cana-834	339	18	vertices	vertex	NOUN
cana-834	339	19	and	and	CCONJ
cana-834	339	20	|v	|v	PROPN
cana-834	339	21	(	(	PUNCT
cana-834	339	22	t	t	NOUN
cana-834	339	23	◦	◦	NOUN
cana-834	339	24	g	g	NOUN
cana-834	339	25	)	)	PUNCT
cana-834	340	1	−	−	PROPN
cana-834	340	2	d|	d|	PROPN
cana-834	340	3	=	=	SYM
cana-834	340	4	mn	mn	PROPN
cana-834	341	1	+	+	CCONJ
cana-834	342	1	n	n	CCONJ
cana-834	342	2	−	−	PROPN
cana-834	342	3	nm	nm	NOUN
cana-834	342	4	+	+	CCONJ
cana-834	342	5	n	n	CCONJ
cana-834	342	6	=	=	SYM
cana-834	342	7	2n	2n	NUM
cana-834	342	8	vertices	vertex	NOUN
cana-834	342	9	.	.	PUNCT
cana-834	343	1	since	since	SCONJ
cana-834	343	2	g	g	PROPN
cana-834	343	3	is	be	AUX
cana-834	343	4	sna	sna	PROPN
cana-834	343	5	and	and	CCONJ
cana-834	343	6	each	each	DET
cana-834	343	7	vertex	vertex	NOUN
cana-834	343	8	of	of	ADP
cana-834	343	9	t	t	PROPN
cana-834	343	10	is	be	AUX
cana-834	343	11	a	a	DET
cana-834	343	12	member	member	NOUN
cana-834	343	13	of	of	ADP
cana-834	343	14	sna	sna	PROPN
cana-834	343	15	-	-	PUNCT
cana-834	343	16	set	set	PROPN
cana-834	343	17	.	.	PUNCT
cana-834	344	1	therefore	therefore	ADV
cana-834	344	2	,	,	PUNCT
cana-834	344	3	n	n	PRON
cana-834	344	4	vertices	vertex	NOUN
cana-834	344	5	of	of	ADP
cana-834	344	6	t	t	NOUN
cana-834	344	7	which	which	PRON
cana-834	344	8	are	be	AUX
cana-834	344	9	adjacent	adjacent	ADJ
cana-834	344	10	to	to	ADP
cana-834	344	11	the	the	DET
cana-834	344	12	vertex	vertex	NOUN
cana-834	344	13	of	of	ADP
cana-834	344	14	n	n	NOUN
cana-834	344	15	copies	copy	NOUN
cana-834	344	16	of	of	ADP
cana-834	344	17	g	g	PROPN
cana-834	344	18	are	be	AUX
cana-834	344	19	member	member	NOUN
cana-834	344	20	of	of	ADP
cana-834	344	21	⟨v	⟨v	PROPN
cana-834	344	22	(	(	PUNCT
cana-834	344	23	t	t	X
cana-834	344	24	◦	◦	NOUN
cana-834	344	25	g	g	NOUN
cana-834	344	26	)	)	PUNCT
cana-834	345	1	−	−	PROPN
cana-834	345	2	d⟩	d⟩	NOUN
cana-834	345	3	set	set	NOUN
cana-834	345	4	.	.	PUNCT
cana-834	346	1	let	let	VERB
cana-834	346	2	v	v	X
cana-834	346	3	(	(	PUNCT
cana-834	346	4	t	t	NOUN
cana-834	346	5	◦	◦	NOUN
cana-834	346	6	g	g	NOUN
cana-834	346	7	)	)	PUNCT
cana-834	346	8	−	−	PROPN
cana-834	347	1	d	d	NOUN
cana-834	347	2	=	=	PRON
cana-834	347	3	{	{	PUNCT
cana-834	347	4	ui	ui	NOUN
cana-834	347	5	:	:	PUNCT
cana-834	347	6	1	1	NUM
cana-834	347	7	≤	≤	NUM
cana-834	347	8	i	i	PRON
cana-834	347	9	≤	≤	NOUN
cana-834	347	10	n	n	CCONJ
cana-834	347	11	}	}	PUNCT
cana-834	347	12	∪	∪	NOUN
cana-834	347	13	{	{	PUNCT
cana-834	347	14	ui1/1	ui1/1	PROPN
cana-834	347	15	≤	≤	PROPN
cana-834	347	16	i	i	PROPN
cana-834	347	17	≤	≤	NOUN
cana-834	347	18	n	n	CCONJ
cana-834	347	19	}	}	PUNCT
cana-834	347	20	.	.	PUNCT
cana-834	348	1	hence	hence	ADV
cana-834	348	2	⟨v	⟨v	AUX
cana-834	348	3	(	(	PUNCT
cana-834	348	4	t	t	X
cana-834	348	5	◦	◦	NOUN
cana-834	348	6	g	g	NOUN
cana-834	348	7	)	)	PUNCT
cana-834	348	8	−	−	PROPN
cana-834	348	9	d⟩	d⟩	NOUN
cana-834	348	10	t	t	PROPN
cana-834	348	11	◦	◦	NOUN
cana-834	348	12	k1	k1	PROPN
cana-834	348	13	.	.	PUNCT
cana-834	349	1	since	since	SCONJ
cana-834	349	2	g	g	PROPN
cana-834	349	3	is	be	AUX
cana-834	349	4	sna	sna	PROPN
cana-834	349	5	graph	graph	NOUN
cana-834	349	6	with	with	ADP
cana-834	349	7	m	m	PROPN
cana-834	349	8	≥	≥	NUM
cana-834	349	9	2	2	NUM
cana-834	349	10	vertices	vertex	NOUN
cana-834	349	11	.	.	PUNCT
cana-834	350	1	case	case	NOUN
cana-834	350	2	(	(	PUNCT
cana-834	350	3	i).if	i).if	NOUN
cana-834	350	4	δ(g	δ(g	X
cana-834	350	5	)	)	PUNCT
cana-834	350	6	=	=	SYM
cana-834	350	7	0	0	NUM
cana-834	350	8	,	,	PUNCT
cana-834	350	9	then	then	ADV
cana-834	350	10	g	g	PROPN
cana-834	350	11	k1or	k1or	PROPN
cana-834	350	12	mk	mk	PROPN
cana-834	350	13	suppose	suppose	VERB
cana-834	350	14	g	g	PROPN
cana-834	350	15	mk	mk	PROPN
cana-834	350	16	.	.	PUNCT
cana-834	351	1	then	then	ADV
cana-834	351	2	γsna(t	γsna(t	ADP
cana-834	351	3	◦	◦	NOUN
cana-834	351	4	mk	mk	NOUN
cana-834	351	5	)	)	PUNCT
cana-834	352	1	=	=	SYM
cana-834	352	2	mn	mn	PROPN
cana-834	352	3	.	.	PUNCT
cana-834	353	1	the	the	DET
cana-834	353	2	upper	upper	ADJ
cana-834	353	3	bound	bind	VERB
cana-834	353	4	equality	equality	NOUN
cana-834	353	5	hold	hold	VERB
cana-834	353	6	.	.	PUNCT
cana-834	354	1	suppose	suppose	VERB
cana-834	354	2	g	g	PROPN
cana-834	354	3	k1	k1	PROPN
cana-834	354	4	then	then	ADV
cana-834	354	5	γsna(t	γsna(t	ADP
cana-834	354	6	◦	◦	NOUN
cana-834	354	7	k1	k1	NOUN
cana-834	354	8	)	)	PUNCT
cana-834	355	1	=	=	VERB
cana-834	355	2	n.	n.	NOUN
cana-834	355	3	the	the	DET
cana-834	355	4	lower	lower	ADV
cana-834	355	5	bound	bind	VERB
cana-834	355	6	equality	equality	NOUN
cana-834	355	7	holds	hold	VERB
cana-834	355	8	.	.	PUNCT
cana-834	356	1	case	case	NOUN
cana-834	356	2	(	(	PUNCT
cana-834	356	3	ii).if	ii).if	NOUN
cana-834	356	4	δ(g	δ(g	X
cana-834	356	5	)	)	PUNCT
cana-834	356	6	=	=	SYM
cana-834	356	7	1	1	NUM
cana-834	356	8	,	,	PUNCT
cana-834	356	9	then	then	ADV
cana-834	356	10	g	g	PROPN
cana-834	356	11			PROPN
cana-834	356	12	tm	tm	PROPN
cana-834	356	13	.	.	PROPN
cana-834	356	14	suppose	suppose	VERB
cana-834	356	15	g	g	PROPN
cana-834	356	16			PROPN
cana-834	356	17	tm	tm	PROPN
cana-834	356	18	then	then	ADV
cana-834	356	19	γsna(t	γsna(t	PROPN
cana-834	356	20	◦	◦	PROPN
cana-834	356	21	tm	tm	NOUN
cana-834	356	22	)	)	PUNCT
cana-834	356	23	≤	≤	PUNCT
cana-834	357	1	n(m	n(m	PROPN
cana-834	357	2	−	−	NOUN
cana-834	357	3	2	2	NUM
cana-834	357	4	)	)	PUNCT
cana-834	357	5	<	<	X
cana-834	357	6	mn	mn	PROPN
cana-834	357	7	.	.	PROPN
cana-834	357	8	case	case	NOUN
cana-834	357	9	(	(	PUNCT
cana-834	357	10	iii).δ(g	iii).δ(g	PROPN
cana-834	357	11	)	)	PUNCT
cana-834	357	12	≥	≥	NOUN
cana-834	357	13	2	2	NUM
cana-834	357	14	communications	communication	NOUN
cana-834	357	15	on	on	ADP
cana-834	357	16	applied	apply	VERB
cana-834	357	17	nonlinear	nonlinear	ADJ
cana-834	357	18	analysis	analysis	NOUN
cana-834	357	19	issn	issn	NOUN
cana-834	357	20	:	:	PUNCT
cana-834	357	21	1074	1074	NUM
cana-834	357	22	-	-	PUNCT
cana-834	357	23	133x	133x	NUM
cana-834	357	24	vol	vol	NOUN
cana-834	357	25	31	31	NUM
cana-834	357	26	no	no	NOUN
cana-834	357	27	.	.	PUNCT
cana-834	358	1	4s	4s	NUM
cana-834	358	2	(	(	PUNCT
cana-834	358	3	2024	2024	NUM
cana-834	358	4	)	)	PUNCT
cana-834	358	5	176	176	NUM
cana-834	358	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	358	7	we	we	PRON
cana-834	358	8	have	have	VERB
cana-834	358	9	,	,	PUNCT
cana-834	358	10	γsna(g	γsna(g	NOUN
cana-834	358	11	)	)	PUNCT
cana-834	358	12	≤	≤	PUNCT
cana-834	359	1	n(m	n(m	PROPN
cana-834	359	2	−	−	NOUN
cana-834	359	3	2	2	NUM
cana-834	359	4	)	)	PUNCT
cana-834	360	1	[	[	X
cana-834	360	2	3	3	NUM
cana-834	360	3	]	]	PUNCT
cana-834	360	4	.	.	PUNCT
cana-834	360	5	suppose	suppose	VERB
cana-834	360	6	γsna(g	γsna(g	NOUN
cana-834	360	7	)	)	PUNCT
cana-834	360	8	=	=	PUNCT
cana-834	360	9	m	m	VERB
cana-834	360	10	−	−	NOUN
cana-834	360	11	2	2	NUM
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cana-834	360	13	g	g	PROPN
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cana-834	360	15	,	,	PUNCT
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cana-834	360	21	γsna(t	γsna(t	ADP
cana-834	360	22	◦	◦	NOUN
cana-834	360	23	km	km	NOUN
cana-834	360	24	)	)	PUNCT
cana-834	360	25	=	=	SYM
cana-834	360	26	n(m−1	n(m−1	PROPN
cana-834	360	27	)	)	PUNCT
cana-834	360	28	and	and	CCONJ
cana-834	360	29	suppose	suppose	VERB
cana-834	360	30	γsna(t	γsna(t	ADP
cana-834	360	31	◦	◦	NOUN
cana-834	360	32	cm	cm	NOUN
cana-834	360	33	)	)	PUNCT
cana-834	361	1	=	=	NOUN
cana-834	361	2	n	n	NOUN
cana-834	361	3			NOUN
cana-834	361	4			NOUN
cana-834	361	5			X
cana-834	361	6			NOUN
cana-834	361	7			X
cana-834	361	8			PROPN
cana-834	361	9	2	2	NUM
cana-834	361	10	m	m	NOUN
cana-834	361	11	<	<	X
cana-834	361	12	n(m	n(m	PROPN
cana-834	361	13	−	−	NOUN
cana-834	361	14	1	1	NUM
cana-834	361	15	)	)	PUNCT
cana-834	361	16	<	<	X
cana-834	361	17	nm	nm	PROPN
cana-834	361	18	,	,	PUNCT
cana-834	361	19	γsna(t	γsna(t	NOUN
cana-834	361	20	◦	◦	NOUN
cana-834	361	21	wm	wm	NOUN
cana-834	361	22	)	)	PUNCT
cana-834	361	23	=	=	SYM
cana-834	361	24	n	n	PRON
cana-834	361	25			NOUN
cana-834	361	26			NOUN
cana-834	361	27			X
cana-834	361	28			NOUN
cana-834	361	29			VERB
cana-834	361	30			NOUN
cana-834	361	31	−	−	NUM
cana-834	361	32	2	2	NUM
cana-834	361	33	1	1	NUM
cana-834	361	34	m	m	PROPN
cana-834	361	35	≤	≤	NUM
cana-834	361	36	nm	nm	VERB
cana-834	361	37	.	.	PUNCT
cana-834	362	1	theorem	theorem	VERB
cana-834	362	2	4.2	4.2	NUM
cana-834	362	3	.	.	PUNCT
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cana-834	363	3	graph	graph	VERB
cana-834	363	4	g	g	NOUN
cana-834	363	5	with	with	ADP
cana-834	363	6	at	at	ADV
cana-834	363	7	least	least	ADV
cana-834	363	8	two	two	NUM
cana-834	363	9	vertices	vertex	NOUN
cana-834	363	10	,	,	PUNCT
cana-834	363	11	γsna(t	γsna(t	ADP
cana-834	363	12	◦	◦	NOUN
cana-834	363	13	g	g	NOUN
cana-834	363	14	)	)	PUNCT
cana-834	363	15	=	=	SYM
cana-834	363	16	n	n	X
cana-834	363	17	,	,	PUNCT
cana-834	363	18	n	n	PRON
cana-834	363	19	≥	≥	NOUN
cana-834	363	20	2	2	NUM
cana-834	363	21	if	if	SCONJ
cana-834	363	22	and	and	CCONJ
cana-834	363	23	only	only	ADV
cana-834	363	24	if	if	SCONJ
cana-834	363	25	g	g	PROPN
cana-834	363	26	k1,mm	k1,mm	PROPN
cana-834	363	27	≥	≥	NOUN
cana-834	363	28	2	2	NUM
cana-834	363	29	.	.	PUNCT
cana-834	363	30	proof	proof	NOUN
cana-834	363	31	.	.	PUNCT
cana-834	364	1	for	for	ADP
cana-834	364	2	all	all	DET
cana-834	364	3	graph	graph	NOUN
cana-834	364	4	given	give	VERB
cana-834	364	5	in	in	ADP
cana-834	364	6	the	the	DET
cana-834	364	7	theorem	theorem	NOUN
cana-834	364	8	,	,	PUNCT
cana-834	364	9	γctd(t	γctd(t	ADP
cana-834	364	10	◦	◦	NOUN
cana-834	364	11	g	g	NOUN
cana-834	364	12	)	)	PUNCT
cana-834	364	13	=	=	VERB
cana-834	364	14	n.	n.	NOUN
cana-834	364	15	conversely	conversely	ADV
cana-834	364	16	,	,	PUNCT
cana-834	364	17	assume	assume	VERB
cana-834	364	18	γsna(t	γsna(t	ADP
cana-834	364	19	◦	◦	NOUN
cana-834	364	20	g	g	NOUN
cana-834	364	21	)	)	PUNCT
cana-834	365	1	=	=	VERB
cana-834	365	2	n.	n.	NOUN
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cana-834	365	4	d	d	PRON
cana-834	365	5	be	be	AUX
cana-834	365	6	a	a	DET
cana-834	365	7	sna	sna	PROPN
cana-834	365	8	-	-	PUNCT
cana-834	365	9	set	set	NOUN
cana-834	365	10	of	of	ADP
cana-834	365	11	t	t	PROPN
cana-834	365	12	◦	◦	NOUN
cana-834	365	13	g	g	NOUN
cana-834	365	14	containing	contain	VERB
cana-834	365	15	n	n	NOUN
cana-834	365	16	vertices	vertex	NOUN
cana-834	365	17	.	.	PUNCT
cana-834	366	1	then	then	ADV
cana-834	366	2	|v	|v	PROPN
cana-834	366	3	(	(	PUNCT
cana-834	366	4	t	t	NOUN
cana-834	366	5	◦	◦	NOUN
cana-834	366	6	g	g	NOUN
cana-834	366	7	)	)	PUNCT
cana-834	366	8	−	−	PROPN
cana-834	366	9	d|	d|	PROPN
cana-834	366	10	=	=	SYM
cana-834	366	11	|v	|v	PROPN
cana-834	366	12	(	(	PUNCT
cana-834	366	13	t	t	NOUN
cana-834	366	14	◦	◦	NOUN
cana-834	366	15	g)|	g)|	PROPN
cana-834	366	16	−	−	PROPN
cana-834	366	17	|d|	|d|	PROPN
cana-834	366	18	=	=	SYM
cana-834	366	19	mn	mn	PROPN
cana-834	366	20	+	+	CCONJ
cana-834	366	21	n	n	CCONJ
cana-834	366	22	−	−	PROPN
cana-834	366	23	n	n	PROPN
cana-834	366	24	=	=	PROPN
cana-834	366	25	mn	mn	PROPN
cana-834	366	26	.	.	PUNCT
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cana-834	367	2	⟨v	⟨v	PROPN
cana-834	367	3	(	(	PUNCT
cana-834	367	4	t	t	X
cana-834	367	5	◦	◦	NOUN
cana-834	367	6	g	g	NOUN
cana-834	367	7	)	)	PUNCT
cana-834	367	8	−	−	PROPN
cana-834	367	9	d⟩	d⟩	NOUN
cana-834	367	10	is	be	AUX
cana-834	367	11	a	a	DET
cana-834	367	12	tree	tree	NOUN
cana-834	367	13	contains	contain	VERB
cana-834	367	14	t.	t.	NOUN
cana-834	367	15	let	let	VERB
cana-834	367	16	d	d	X
cana-834	367	17	=	=	PUNCT
cana-834	367	18	{	{	PUNCT
cana-834	367	19	v11	v11	NOUN
cana-834	367	20	,	,	PUNCT
cana-834	367	21	v21	v21	NOUN
cana-834	367	22	,	,	PUNCT
cana-834	367	23	.	.	PUNCT
cana-834	367	24	.	.	PUNCT
cana-834	367	25	.	.	PUNCT
cana-834	368	1	vn1	vn1	NOUN
cana-834	368	2	}	}	PUNCT
cana-834	368	3	.	.	PUNCT
cana-834	369	1	then	then	ADV
cana-834	369	2	|d|	|d|	PROPN
cana-834	369	3	=	=	PUNCT
cana-834	369	4	n	n	PROPN
cana-834	369	5	is	be	AUX
cana-834	369	6	possible	possible	ADJ
cana-834	369	7	only	only	ADV
cana-834	369	8	if	if	SCONJ
cana-834	369	9	one	one	NUM
cana-834	369	10	vertex	vertex	NOUN
cana-834	369	11	of	of	ADP
cana-834	369	12	each	each	DET
cana-834	369	13	copies	copy	NOUN
cana-834	369	14	of	of	ADP
cana-834	369	15	g	g	PROPN
cana-834	369	16	dominate	dominate	VERB
cana-834	369	17	the	the	DET
cana-834	369	18	vertices	vertex	NOUN
cana-834	369	19	of	of	ADP
cana-834	369	20	g	g	NOUN
cana-834	369	21	and	and	CCONJ
cana-834	369	22	vertices	vertex	NOUN
cana-834	369	23	of	of	ADP
cana-834	369	24	t	t	NOUN
cana-834	369	25	i.e.	i.e.	X
cana-834	369	26	,	,	PUNCT
cana-834	369	27	{	{	PUNCT
cana-834	369	28	vi1	vi1	NOUN
cana-834	369	29	:	:	PUNCT
cana-834	369	30	1	1	NUM
cana-834	369	31	≤	≤	NUM
cana-834	369	32	i	i	X
cana-834	369	33	≤	≤	ADJ
cana-834	369	34	n}is	n}is	PROPN
cana-834	369	35	adjacent	adjacent	ADJ
cana-834	369	36	to	to	PART
cana-834	369	37	vertices	vertex	NOUN
cana-834	369	38	(	(	PUNCT
cana-834	369	39	v	v	NOUN
cana-834	369	40	(	(	PUNCT
cana-834	369	41	g))i	g))i	NOUN
cana-834	369	42	and	and	CCONJ
cana-834	369	43	uit	uit	ADJ
cana-834	369	44	.	.	PUNCT
cana-834	370	1	suppose	suppose	VERB
cana-834	370	2	vij(v	vij(v	NOUN
cana-834	370	3	(	(	PUNCT
cana-834	370	4	g	g	NOUN
cana-834	370	5	)	)	PUNCT
cana-834	370	6	)	)	PUNCT
cana-834	370	7	,	,	PUNCT
cana-834	370	8	i	i	PRON
cana-834	370	9	=	=	NOUN
cana-834	370	10	1	1	NUM
cana-834	370	11	,	,	PUNCT
cana-834	370	12	2	2	NUM
cana-834	370	13	,	,	PUNCT
cana-834	370	14	.	.	PUNCT
cana-834	370	15	.	.	PUNCT
cana-834	371	1	.	.	PUNCT
cana-834	372	1	,	,	PUNCT
cana-834	372	2	n	n	PROPN
cana-834	372	3	j	j	NOUN
cana-834	372	4	=	=	SYM
cana-834	372	5	1	1	NUM
cana-834	372	6	,	,	PUNCT
cana-834	372	7	2	2	NUM
cana-834	372	8	,	,	PUNCT
cana-834	372	9	.	.	PUNCT
cana-834	372	10	.	.	PUNCT
cana-834	373	1	.	.	PUNCT
cana-834	374	1	,	,	PUNCT
cana-834	374	2	m.	m.	NOUN
cana-834	374	3	since	since	SCONJ
cana-834	374	4	g	g	PROPN
cana-834	374	5	is	be	AUX
cana-834	374	6	sna	sna	PROPN
cana-834	374	7	,	,	PUNCT
cana-834	374	8	hence	hence	ADV
cana-834	374	9	vij	vij	PROPN
cana-834	374	10	’s	’s	PART
cana-834	374	11	are	be	AUX
cana-834	374	12	adjacent	adjacent	ADJ
cana-834	374	13	to	to	ADP
cana-834	374	14	each	each	DET
cana-834	374	15	other	other	ADJ
cana-834	374	16	in	in	ADP
cana-834	374	17	(	(	PUNCT
cana-834	374	18	v	v	NOUN
cana-834	374	19	(	(	PUNCT
cana-834	374	20	g))i	g))i	NOUN
cana-834	374	21	are	be	AUX
cana-834	374	22	adjacent	adjacent	ADJ
cana-834	374	23	to	to	ADP
cana-834	374	24	ui	ui	VERB
cana-834	374	25	in	in	ADP
cana-834	374	26	t.	t.	PROPN
cana-834	374	27	but	but	CCONJ
cana-834	374	28	⟨v	⟨v	PROPN
cana-834	374	29	(	(	PUNCT
cana-834	374	30	t	t	X
cana-834	374	31	◦	◦	NOUN
cana-834	374	32	g	g	NOUN
cana-834	374	33	)	)	PUNCT
cana-834	375	1	−	−	PROPN
cana-834	375	2	d⟩	d⟩	NOUN
cana-834	375	3	is	be	AUX
cana-834	375	4	not	not	PART
cana-834	375	5	a	a	DET
cana-834	375	6	tree	tree	NOUN
cana-834	375	7	,	,	PUNCT
cana-834	375	8	which	which	PRON
cana-834	375	9	contradicts	contradict	VERB
cana-834	375	10	snaset	snaset	NOUN
cana-834	375	11	.	.	PUNCT
cana-834	376	1	therefore	therefore	ADV
cana-834	376	2	vi1i	vi1i	PROPN
cana-834	376	3	=	=	SYM
cana-834	376	4	1	1	NUM
cana-834	376	5	,	,	PUNCT
cana-834	376	6	2	2	NUM
cana-834	376	7	,	,	PUNCT
cana-834	376	8	.	.	PUNCT
cana-834	376	9	.	.	PUNCT
cana-834	377	1	.	.	PUNCT
cana-834	378	1	,	,	PUNCT
cana-834	378	2	n	n	PRON
cana-834	378	3	is	be	AUX
cana-834	378	4	the	the	DET
cana-834	378	5	only	only	ADJ
cana-834	378	6	vertex	vertex	NOUN
cana-834	378	7	adjacent	adjacent	ADJ
cana-834	378	8	to	to	ADP
cana-834	378	9	vij	vij	PROPN
cana-834	378	10	’s	’s	PART
cana-834	378	11	i	i	NOUN
cana-834	378	12	=	=	NOUN
cana-834	378	13	1	1	NUM
cana-834	378	14	,	,	PUNCT
cana-834	378	15	2	2	NUM
cana-834	378	16	,	,	PUNCT
cana-834	378	17	.	.	PUNCT
cana-834	378	18	.	.	PUNCT
cana-834	379	1	.	.	PUNCT
cana-834	380	1	,	,	PUNCT
cana-834	380	2	n	n	PROPN
cana-834	380	3	j	j	NOUN
cana-834	380	4	=	=	SYM
cana-834	380	5	1	1	NUM
cana-834	380	6	,	,	PUNCT
cana-834	380	7	2	2	NUM
cana-834	380	8	,	,	PUNCT
cana-834	380	9	.	.	PUNCT
cana-834	380	10	.	.	PUNCT
cana-834	381	1	.	.	PUNCT
cana-834	382	1	,	,	PUNCT
cana-834	382	2	m.	m.	NOUN
cana-834	382	3	hence	hence	ADV
cana-834	382	4	,	,	PUNCT
cana-834	382	5	g	g	PROPN
cana-834	382	6			PROPN
cana-834	382	7	t	t	PROPN
cana-834	382	8	◦	◦	NOUN
cana-834	382	9	k1,m(m	k1,m(m	PROPN
cana-834	382	10	≥	≥	NUM
cana-834	382	11	1	1	NUM
cana-834	382	12	)	)	PUNCT
cana-834	382	13	.	.	PUNCT
cana-834	383	1	applications	application	NOUN
cana-834	383	2	of	of	ADP
cana-834	383	3	graph	graph	NOUN
cana-834	383	4	theory	theory	NOUN
cana-834	383	5	algorithms	algorithm	NOUN
cana-834	383	6	for	for	ADP
cana-834	383	7	social	social	ADJ
cana-834	383	8	network	network	NOUN
cana-834	383	9	analysis	analysis	NOUN
cana-834	383	10	proficient	proficient	ADJ
cana-834	383	11	information	information	NOUN
cana-834	383	12	structures	structure	NOUN
cana-834	383	13	are	be	AUX
cana-834	383	14	significant	significant	ADJ
cana-834	383	15	for	for	ADP
cana-834	383	16	taking	take	VERB
cana-834	383	17	care	care	NOUN
cana-834	383	18	of	of	ADP
cana-834	383	19	huge	huge	ADJ
cana-834	383	20	diagrams	diagram	NOUN
cana-834	383	21	,	,	PUNCT
cana-834	383	22	lessening	lessen	VERB
cana-834	383	23	memory	memory	NOUN
cana-834	383	24	use	use	NOUN
cana-834	383	25	,	,	PUNCT
cana-834	383	26	and	and	CCONJ
cana-834	383	27	further	far	ADV
cana-834	383	28	developing	develop	VERB
cana-834	383	29	access	access	NOUN
cana-834	383	30	times	time	NOUN
cana-834	383	31	[	[	X
cana-834	383	32	22	22	NUM
cana-834	383	33	]	]	PUNCT
cana-834	383	34	.	.	PUNCT
cana-834	384	1	since	since	SCONJ
cana-834	384	2	these	these	DET
cana-834	384	3	formats	format	NOUN
cana-834	384	4	only	only	ADV
cana-834	384	5	store	store	VERB
cana-834	384	6	entries	entry	NOUN
cana-834	384	7	that	that	PRON
cana-834	384	8	are	be	AUX
cana-834	384	9	not	not	PART
cana-834	384	10	zero	zero	NUM
cana-834	384	11	in	in	ADP
cana-834	384	12	adjacency	adjacency	NOUN
cana-834	384	13	matrices	matrix	NOUN
cana-834	384	14	,	,	PUNCT
cana-834	384	15	sparse	sparse	ADJ
cana-834	384	16	graph	graph	NOUN
cana-834	384	17	memory	memory	NOUN
cana-834	384	18	usage	usage	NOUN
cana-834	384	19	is	be	AUX
cana-834	384	20	significantly	significantly	ADV
cana-834	384	21	reduced	reduce	VERB
cana-834	384	22	.	.	PUNCT
cana-834	385	1	csr	csr	NOUN
cana-834	385	2	is	be	AUX
cana-834	385	3	ideal	ideal	ADJ
cana-834	385	4	for	for	ADP
cana-834	385	5	quick	quick	ADJ
cana-834	385	6	row	row	NOUN
cana-834	385	7	access	access	NOUN
cana-834	385	8	,	,	PUNCT
cana-834	385	9	whereas	whereas	SCONJ
cana-834	385	10	csc	csc	PROPN
cana-834	385	11	is	be	AUX
cana-834	385	12	preferable	preferable	ADJ
cana-834	385	13	for	for	ADP
cana-834	385	14	column	column	NOUN
cana-834	385	15	access	access	NOUN
cana-834	385	16	[	[	X
cana-834	385	17	23	23	NUM
cana-834	385	18	]	]	PUNCT
cana-834	385	19	.	.	PUNCT
cana-834	386	1	utilizing	utilize	VERB
cana-834	386	2	edge	edge	NOUN
cana-834	386	3	lists	list	NOUN
cana-834	386	4	and	and	CCONJ
cana-834	386	5	adjacency	adjacency	NOUN
cana-834	386	6	lists	list	NOUN
cana-834	386	7	in	in	ADP
cana-834	386	8	conjunction	conjunction	NOUN
cana-834	386	9	to	to	PART
cana-834	386	10	store	store	VERB
cana-834	386	11	and	and	CCONJ
cana-834	386	12	traverse	traverse	VERB
cana-834	386	13	large	large	ADJ
cana-834	386	14	graphs	graph	NOUN
cana-834	386	15	effectively	effectively	ADV
cana-834	386	16	.	.	PUNCT
cana-834	387	1	to	to	PART
cana-834	387	2	facilitate	facilitate	VERB
cana-834	387	3	quick	quick	ADJ
cana-834	387	4	access	access	NOUN
cana-834	387	5	and	and	CCONJ
cana-834	387	6	manipulation	manipulation	NOUN
cana-834	387	7	,	,	PUNCT
cana-834	387	8	edge	edge	NOUN
cana-834	387	9	lists	list	NOUN
cana-834	387	10	can	can	AUX
cana-834	387	11	be	be	AUX
cana-834	387	12	sorted	sort	VERB
cana-834	387	13	.	.	PUNCT
cana-834	388	1	a	a	DET
cana-834	388	2	library	library	NOUN
cana-834	388	3	that	that	PRON
cana-834	388	4	provides	provide	VERB
cana-834	388	5	sparse	sparse	ADJ
cana-834	388	6	matrix	matrix	NOUN
cana-834	388	7	operations	operation	NOUN
cana-834	388	8	-	-	PUNCT
cana-834	388	9	optimized	optimize	VERB
cana-834	388	10	building	building	NOUN
cana-834	388	11	blocks	block	NOUN
cana-834	388	12	for	for	ADP
cana-834	388	13	graph	graph	NOUN
cana-834	388	14	algorithms	algorithm	NOUN
cana-834	388	15	.	.	PUNCT
cana-834	389	1	to	to	PART
cana-834	389	2	boost	boost	VERB
cana-834	389	3	performance	performance	NOUN
cana-834	389	4	,	,	PUNCT
cana-834	389	5	it	it	PRON
cana-834	389	6	makes	make	VERB
cana-834	389	7	use	use	NOUN
cana-834	389	8	of	of	ADP
cana-834	389	9	cutting	cut	VERB
cana-834	389	10	-	-	PUNCT
cana-834	389	11	edge	edge	NOUN
cana-834	389	12	methods	method	NOUN
cana-834	389	13	from	from	ADP
cana-834	389	14	linear	linear	ADJ
cana-834	389	15	algebra	algebra	PROPN
cana-834	389	16	.	.	PUNCT
cana-834	390	1	making	make	VERB
cana-834	390	2	effective	effective	ADJ
cana-834	390	3	ordering	ordering	NOUN
cana-834	390	4	components	component	NOUN
cana-834	390	5	to	to	PART
cana-834	390	6	accelerate	accelerate	VERB
cana-834	390	7	diagram	diagram	NOUN
cana-834	390	8	activities	activity	NOUN
cana-834	390	9	,	,	PUNCT
cana-834	390	10	for	for	ADP
cana-834	390	11	example	example	NOUN
cana-834	390	12	,	,	PUNCT
cana-834	390	13	search	search	NOUN
cana-834	390	14	and	and	CCONJ
cana-834	390	15	crossing	crossing	NOUN
cana-834	390	16	[	[	X
cana-834	390	17	24	24	NUM
cana-834	390	18	]	]	PUNCT
cana-834	390	19	.	.	PUNCT
cana-834	391	1	hash	hash	NOUN
cana-834	391	2	maps	map	NOUN
cana-834	391	3	:	:	PUNCT
cana-834	391	4	hash	hash	NOUN
cana-834	391	5	maps	map	NOUN
cana-834	391	6	are	be	AUX
cana-834	391	7	used	use	VERB
cana-834	391	8	to	to	PART
cana-834	391	9	locate	locate	VERB
cana-834	391	10	edges	edge	NOUN
cana-834	391	11	and	and	CCONJ
cana-834	391	12	nodes	node	NOUN
cana-834	391	13	quickly	quickly	ADV
cana-834	391	14	.	.	PUNCT
cana-834	392	1	hash	hash	NOUN
cana-834	392	2	maps	map	NOUN
cana-834	392	3	,	,	PUNCT
cana-834	392	4	for	for	ADP
cana-834	392	5	instance	instance	NOUN
cana-834	392	6	,	,	PUNCT
cana-834	392	7	can	can	AUX
cana-834	392	8	be	be	AUX
cana-834	392	9	used	use	VERB
cana-834	392	10	to	to	PART
cana-834	392	11	store	store	VERB
cana-834	392	12	node	node	ADJ
cana-834	392	13	attributes	attribute	NOUN
cana-834	392	14	or	or	CCONJ
cana-834	392	15	adjacency	adjacency	NOUN
cana-834	392	16	lists	list	NOUN
cana-834	392	17	.	.	PUNCT
cana-834	393	1	b	b	X
cana-834	393	2	-	-	PUNCT
cana-834	393	3	trees	tree	NOUN
cana-834	393	4	and	and	CCONJ
cana-834	393	5	skip	skip	ADJ
cana-834	393	6	records	record	NOUN
cana-834	393	7	:	:	PUNCT
cana-834	393	8	information	information	NOUN
cana-834	393	9	structures	structure	NOUN
cana-834	393	10	that	that	PRON
cana-834	393	11	help	help	AUX
cana-834	393	12	proficient	proficient	ADJ
cana-834	393	13	reach	reach	VERB
cana-834	393	14	inquiries	inquiry	NOUN
cana-834	393	15	and	and	CCONJ
cana-834	393	16	updates	update	NOUN
cana-834	393	17	,	,	PUNCT
cana-834	393	18	valuable	valuable	ADJ
cana-834	393	19	for	for	ADP
cana-834	393	20	dynamic	dynamic	ADJ
cana-834	393	21	charts	chart	NOUN
cana-834	393	22	where	where	SCONJ
cana-834	393	23	hubs	hub	NOUN
cana-834	393	24	and	and	CCONJ
cana-834	393	25	edges	edge	NOUN
cana-834	393	26	are	be	AUX
cana-834	393	27	regularly	regularly	ADV
cana-834	393	28	added	add	VERB
cana-834	393	29	or	or	CCONJ
cana-834	393	30	eliminated	eliminate	VERB
cana-834	393	31	.	.	PUNCT
cana-834	394	1	because	because	SCONJ
cana-834	394	2	they	they	PRON
cana-834	394	3	can	can	AUX
cana-834	394	4	handle	handle	VERB
cana-834	394	5	thousands	thousand	NOUN
cana-834	394	6	of	of	ADP
cana-834	394	7	threads	thread	NOUN
cana-834	394	8	at	at	ADP
cana-834	394	9	once	once	ADV
cana-834	394	10	,	,	PUNCT
cana-834	394	11	gpus	gpus	PROPN
cana-834	394	12	are	be	AUX
cana-834	394	13	ideal	ideal	ADJ
cana-834	394	14	for	for	ADP
cana-834	394	15	parallel	parallel	ADJ
cana-834	394	16	graph	graph	NOUN
cana-834	394	17	processing	processing	NOUN
cana-834	394	18	[	[	X
cana-834	394	19	25	25	NUM
cana-834	394	20	]	]	PUNCT
cana-834	394	21	.	.	PUNCT
cana-834	395	1	the	the	DET
cana-834	395	2	frameworks	framework	NOUN
cana-834	395	3	are	be	AUX
cana-834	395	4	for	for	ADP
cana-834	395	5	programming	programming	NOUN
cana-834	395	6	graph	graph	NOUN
cana-834	395	7	algorithms	algorithm	NOUN
cana-834	395	8	that	that	PRON
cana-834	395	9	are	be	AUX
cana-834	395	10	gpu	gpu	PROPN
cana-834	395	11	-	-	PUNCT
cana-834	395	12	accelerated	accelerated	ADJ
cana-834	395	13	.	.	PUNCT
cana-834	396	1	parallel	parallel	ADJ
cana-834	396	2	versions	version	NOUN
cana-834	396	3	of	of	ADP
cana-834	396	4	graph	graph	NOUN
cana-834	396	5	algorithms	algorithm	NOUN
cana-834	396	6	are	be	AUX
cana-834	396	7	frequently	frequently	ADV
cana-834	396	8	implemented	implement	VERB
cana-834	396	9	with	with	ADP
cana-834	396	10	cuda	cuda	NOUN
cana-834	396	11	.	.	PUNCT
cana-834	397	1	hardware	hardware	NOUN
cana-834	397	2	that	that	PRON
cana-834	397	3	is	be	AUX
cana-834	397	4	made	make	VERB
cana-834	397	5	just	just	ADV
cana-834	397	6	for	for	ADP
cana-834	397	7	graph	graph	NOUN
cana-834	397	8	processing	processing	NOUN
cana-834	397	9	tasks	task	NOUN
cana-834	397	10	.	.	PUNCT
cana-834	398	1	models	model	NOUN
cana-834	398	2	incorporate	incorporate	VERB
cana-834	398	3	the	the	DET
cana-834	398	4	graphcore	graphcore	NOUN
cana-834	398	5	ipu	ipu	NOUN
cana-834	398	6	and	and	CCONJ
cana-834	398	7	google	google	PROPN
cana-834	398	8	's	's	PART
cana-834	398	9	tpu	tpu	NOUN
cana-834	398	10	,	,	PUNCT
cana-834	398	11	which	which	PRON
cana-834	398	12	can	can	AUX
cana-834	398	13	speed	speed	VERB
cana-834	398	14	up	up	ADP
cana-834	398	15	ai	ai	VERB
cana-834	398	16	communications	communication	NOUN
cana-834	398	17	on	on	ADP
cana-834	398	18	applied	apply	VERB
cana-834	398	19	nonlinear	nonlinear	ADJ
cana-834	398	20	analysis	analysis	NOUN
cana-834	398	21	issn	issn	NOUN
cana-834	398	22	:	:	PUNCT
cana-834	398	23	1074	1074	NUM
cana-834	398	24	-	-	PUNCT
cana-834	398	25	133x	133x	NUM
cana-834	398	26	vol	vol	NOUN
cana-834	398	27	31	31	NUM
cana-834	398	28	no	no	NOUN
cana-834	398	29	.	.	PUNCT
cana-834	399	1	4s	4s	NUM
cana-834	399	2	(	(	PUNCT
cana-834	399	3	2024	2024	NUM
cana-834	399	4	)	)	PUNCT
cana-834	399	5	177	177	NUM
cana-834	399	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	399	7	and	and	CCONJ
cana-834	399	8	chart	chart	NOUN
cana-834	399	9	handling	handle	VERB
cana-834	399	10	responsibilities	responsibility	NOUN
cana-834	399	11	.	.	PUNCT
cana-834	400	1	for	for	ADP
cana-834	400	2	graph	graph	NOUN
cana-834	400	3	processing	processing	NOUN
cana-834	400	4	,	,	PUNCT
cana-834	400	5	customizable	customizable	ADJ
cana-834	400	6	hardware	hardware	NOUN
cana-834	400	7	options	option	NOUN
cana-834	400	8	are	be	AUX
cana-834	400	9	provided	provide	VERB
cana-834	400	10	by	by	ADP
cana-834	400	11	application	application	NOUN
cana-834	400	12	-	-	PUNCT
cana-834	400	13	specific	specific	ADJ
cana-834	400	14	integrated	integrated	ADJ
cana-834	400	15	circuits	circuit	NOUN
cana-834	400	16	(	(	PUNCT
cana-834	400	17	asics	asic	NOUN
cana-834	400	18	)	)	PUNCT
cana-834	400	19	and	and	CCONJ
cana-834	400	20	field	field	NOUN
cana-834	400	21	-	-	PUNCT
cana-834	400	22	programmable	programmable	ADJ
cana-834	400	23	gate	gate	NOUN
cana-834	400	24	arrays	array	NOUN
cana-834	400	25	(	(	PUNCT
cana-834	400	26	fpgas	fpgas	NOUN
cana-834	400	27	)	)	PUNCT
cana-834	400	28	.	.	PUNCT
cana-834	401	1	allow	allow	VERB
cana-834	401	2	for	for	ADP
cana-834	401	3	custom	custom	NOUN
cana-834	401	4	implementations	implementation	NOUN
cana-834	401	5	that	that	PRON
cana-834	401	6	are	be	AUX
cana-834	401	7	quicker	quick	ADJ
cana-834	401	8	than	than	ADP
cana-834	401	9	general	general	ADJ
cana-834	401	10	-	-	PUNCT
cana-834	401	11	purpose	purpose	NOUN
cana-834	401	12	cpus	cpu	NOUN
cana-834	401	13	by	by	ADP
cana-834	401	14	providing	provide	VERB
cana-834	401	15	flexibility	flexibility	NOUN
cana-834	401	16	and	and	CCONJ
cana-834	401	17	high	high	ADJ
cana-834	401	18	performance	performance	NOUN
cana-834	401	19	for	for	ADP
cana-834	401	20	particular	particular	ADJ
cana-834	401	21	graph	graph	NOUN
cana-834	401	22	algorithms	algorithm	NOUN
cana-834	401	23	.	.	PUNCT
cana-834	402	1	designed	design	VERB
cana-834	402	2	specifically	specifically	ADV
cana-834	402	3	for	for	ADP
cana-834	402	4	use	use	NOUN
cana-834	402	5	in	in	ADP
cana-834	402	6	large	large	ADJ
cana-834	402	7	-	-	PUNCT
cana-834	402	8	scale	scale	NOUN
cana-834	402	9	graph	graph	NOUN
cana-834	402	10	processing	processing	NOUN
cana-834	402	11	tasks	task	NOUN
cana-834	402	12	,	,	PUNCT
cana-834	402	13	delivering	deliver	VERB
cana-834	402	14	the	the	DET
cana-834	402	15	highest	high	ADJ
cana-834	402	16	levels	level	NOUN
cana-834	402	17	of	of	ADP
cana-834	402	18	performance	performance	NOUN
cana-834	402	19	and	and	CCONJ
cana-834	402	20	efficiency	efficiency	NOUN
cana-834	402	21	.	.	PUNCT
cana-834	403	1	numerous	numerous	ADJ
cana-834	403	2	optimization	optimization	NOUN
cana-834	403	3	methods	method	NOUN
cana-834	403	4	can	can	AUX
cana-834	403	5	be	be	AUX
cana-834	403	6	combined	combine	VERB
cana-834	403	7	to	to	PART
cana-834	403	8	produce	produce	VERB
cana-834	403	9	significant	significant	ADJ
cana-834	403	10	performance	performance	NOUN
cana-834	403	11	enhancements	enhancement	NOUN
cana-834	403	12	.	.	PUNCT
cana-834	404	1	utilizing	utilize	VERB
cana-834	404	2	the	the	DET
cana-834	404	3	qualities	quality	NOUN
cana-834	404	4	of	of	ADP
cana-834	404	5	the	the	DET
cana-834	404	6	two	two	NUM
cana-834	404	7	computer	computer	NOUN
cana-834	404	8	processors	processor	NOUN
cana-834	404	9	and	and	CCONJ
cana-834	404	10	gpus	gpu	NOUN
cana-834	404	11	for	for	ADP
cana-834	404	12	various	various	ADJ
cana-834	404	13	pieces	piece	NOUN
cana-834	404	14	of	of	ADP
cana-834	404	15	the	the	DET
cana-834	404	16	diagram	diagram	NOUN
cana-834	404	17	calculation	calculation	NOUN
cana-834	404	18	.	.	PUNCT
cana-834	405	1	for	for	ADP
cana-834	405	2	instance	instance	NOUN
cana-834	405	3	,	,	PUNCT
cana-834	405	4	utilizing	utilize	VERB
cana-834	405	5	the	the	DET
cana-834	405	6	gpu	gpu	NOUN
cana-834	405	7	for	for	ADP
cana-834	405	8	parallel	parallel	ADJ
cana-834	405	9	computation	computation	NOUN
cana-834	405	10	and	and	CCONJ
cana-834	405	11	the	the	DET
cana-834	405	12	cpu	cpu	NOUN
cana-834	405	13	for	for	ADP
cana-834	405	14	control	control	NOUN
cana-834	405	15	flow	flow	NOUN
cana-834	405	16	.	.	PUNCT
cana-834	406	1	in	in	ADP
cana-834	406	2	a	a	DET
cana-834	406	3	distributed	distribute	VERB
cana-834	406	4	computing	computing	NOUN
cana-834	406	5	environment	environment	NOUN
cana-834	406	6	,	,	PUNCT
cana-834	406	7	using	use	VERB
cana-834	406	8	gpu	gpu	NOUN
cana-834	406	9	clusters	cluster	NOUN
cana-834	406	10	to	to	PART
cana-834	406	11	solve	solve	VERB
cana-834	406	12	extremely	extremely	ADV
cana-834	406	13	large	large	ADJ
cana-834	406	14	graph	graph	NOUN
cana-834	406	15	problems	problem	NOUN
cana-834	406	16	.	.	PUNCT
cana-834	407	1	distributed	distribute	VERB
cana-834	407	2	gpu	gpu	NOUN
cana-834	407	3	computation	computation	NOUN
cana-834	407	4	is	be	AUX
cana-834	407	5	supported	support	VERB
cana-834	407	6	by	by	ADP
cana-834	407	7	frameworks	framework	NOUN
cana-834	407	8	like	like	ADP
cana-834	407	9	tensorflow	tensorflow	NOUN
cana-834	407	10	and	and	CCONJ
cana-834	407	11	pytorch	pytorch	NOUN
cana-834	407	12	,	,	PUNCT
cana-834	407	13	making	make	VERB
cana-834	407	14	it	it	PRON
cana-834	407	15	simpler	simple	ADJ
cana-834	407	16	to	to	PART
cana-834	407	17	implement	implement	VERB
cana-834	407	18	and	and	CCONJ
cana-834	407	19	scale	scale	NOUN
cana-834	407	20	graph	graph	NOUN
cana-834	407	21	algorithms	algorithm	NOUN
cana-834	407	22	.	.	PUNCT
cana-834	408	1	real	real	ADJ
cana-834	408	2	-	-	PUNCT
cana-834	408	3	time	time	NOUN
cana-834	408	4	analysis	analysis	NOUN
cana-834	408	5	of	of	ADP
cana-834	408	6	user	user	NOUN
cana-834	408	7	interactions	interaction	NOUN
cana-834	408	8	and	and	CCONJ
cana-834	408	9	trending	trend	VERB
cana-834	408	10	topics	topic	NOUN
cana-834	408	11	is	be	AUX
cana-834	408	12	crucial	crucial	ADJ
cana-834	408	13	on	on	ADP
cana-834	408	14	social	social	ADJ
cana-834	408	15	media	medium	NOUN
cana-834	408	16	platforms	platform	NOUN
cana-834	408	17	like	like	ADP
cana-834	408	18	twitter	twitter	NOUN
cana-834	408	19	.	.	PUNCT
cana-834	409	1	streamlined	streamline	VERB
cana-834	409	2	chart	chart	NOUN
cana-834	409	3	calculations	calculation	NOUN
cana-834	409	4	are	be	AUX
cana-834	409	5	utilized	utilize	VERB
cana-834	409	6	to	to	PART
cana-834	409	7	recognize	recognize	VERB
cana-834	409	8	persuasive	persuasive	ADJ
cana-834	409	9	clients	client	NOUN
cana-834	409	10	and	and	CCONJ
cana-834	409	11	distinguish	distinguish	VERB
cana-834	409	12	arising	arise	VERB
cana-834	409	13	patterns	pattern	NOUN
cana-834	409	14	.	.	PUNCT
cana-834	410	1	when	when	SCONJ
cana-834	410	2	it	it	PRON
cana-834	410	3	comes	come	VERB
cana-834	410	4	to	to	ADP
cana-834	410	5	real	real	ADJ
cana-834	410	6	-	-	PUNCT
cana-834	410	7	time	time	NOUN
cana-834	410	8	processing	processing	NOUN
cana-834	410	9	of	of	ADP
cana-834	410	10	the	the	DET
cana-834	410	11	enormous	enormous	ADJ
cana-834	410	12	volume	volume	NOUN
cana-834	410	13	of	of	ADP
cana-834	410	14	tweets	tweet	NOUN
cana-834	410	15	and	and	CCONJ
cana-834	410	16	user	user	NOUN
cana-834	410	17	interactions	interaction	NOUN
cana-834	410	18	,	,	PUNCT
cana-834	410	19	gpu	gpu	NOUN
cana-834	410	20	acceleration	acceleration	NOUN
cana-834	410	21	and	and	CCONJ
cana-834	410	22	parallel	parallel	ADJ
cana-834	410	23	computing	computing	NOUN
cana-834	410	24	are	be	AUX
cana-834	410	25	particularly	particularly	ADV
cana-834	410	26	effective	effective	ADJ
cana-834	410	27	.	.	PUNCT
cana-834	411	1	by	by	ADP
cana-834	411	2	analyzing	analyze	VERB
cana-834	411	3	transaction	transaction	NOUN
cana-834	411	4	networks	network	NOUN
cana-834	411	5	,	,	PUNCT
cana-834	411	6	financial	financial	ADJ
cana-834	411	7	institutions	institution	NOUN
cana-834	411	8	employ	employ	VERB
cana-834	411	9	graph	graph	NOUN
cana-834	411	10	-	-	PUNCT
cana-834	411	11	based	base	VERB
cana-834	411	12	methods	method	NOUN
cana-834	411	13	to	to	PART
cana-834	411	14	identify	identify	VERB
cana-834	411	15	fraudulent	fraudulent	ADJ
cana-834	411	16	activities	activity	NOUN
cana-834	411	17	.	.	PUNCT
cana-834	412	1	by	by	ADP
cana-834	412	2	processing	process	VERB
cana-834	412	3	transaction	transaction	NOUN
cana-834	412	4	data	datum	NOUN
cana-834	412	5	in	in	ADP
cana-834	412	6	a	a	DET
cana-834	412	7	timely	timely	ADJ
cana-834	412	8	manner	manner	NOUN
cana-834	412	9	,	,	PUNCT
cana-834	412	10	optimized	optimize	VERB
cana-834	412	11	algorithms	algorithm	NOUN
cana-834	412	12	enable	enable	VERB
cana-834	412	13	anomaly	anomaly	NOUN
cana-834	412	14	detection	detection	NOUN
cana-834	412	15	.	.	PUNCT
cana-834	413	1	without	without	ADP
cana-834	413	2	requiring	require	VERB
cana-834	413	3	extensive	extensive	ADJ
cana-834	413	4	computation	computation	NOUN
cana-834	413	5	,	,	PUNCT
cana-834	413	6	suspicious	suspicious	ADJ
cana-834	413	7	patterns	pattern	NOUN
cana-834	413	8	can	can	AUX
cana-834	413	9	be	be	AUX
cana-834	413	10	identified	identify	VERB
cana-834	413	11	with	with	ADP
cana-834	413	12	the	the	DET
cana-834	413	13	help	help	NOUN
cana-834	413	14	of	of	ADP
cana-834	413	15	approximate	approximate	ADJ
cana-834	413	16	algorithms	algorithm	NOUN
cana-834	413	17	and	and	CCONJ
cana-834	413	18	effective	effective	ADJ
cana-834	413	19	indexing	indexing	NOUN
cana-834	413	20	methods	method	NOUN
cana-834	413	21	.	.	PUNCT
cana-834	414	1	understanding	understand	VERB
cana-834	414	2	how	how	SCONJ
cana-834	414	3	diseases	disease	NOUN
cana-834	414	4	spread	spread	VERB
cana-834	414	5	through	through	ADP
cana-834	414	6	social	social	ADJ
cana-834	414	7	contacts	contact	NOUN
cana-834	414	8	in	in	ADP
cana-834	414	9	healthcare	healthcare	NOUN
cana-834	414	10	networks	network	NOUN
cana-834	414	11	is	be	AUX
cana-834	414	12	made	make	VERB
cana-834	414	13	easier	easy	ADJ
cana-834	414	14	by	by	ADP
cana-834	414	15	graph	graph	NOUN
cana-834	414	16	theory	theory	NOUN
cana-834	414	17	.	.	PUNCT
cana-834	415	1	public	public	ADJ
cana-834	415	2	health	health	NOUN
cana-834	415	3	officials	official	NOUN
cana-834	415	4	can	can	AUX
cana-834	415	5	respond	respond	VERB
cana-834	415	6	more	more	ADV
cana-834	415	7	quickly	quickly	ADV
cana-834	415	8	thanks	thank	NOUN
cana-834	415	9	to	to	ADP
cana-834	415	10	optimized	optimize	VERB
cana-834	415	11	algorithms	algorithm	NOUN
cana-834	415	12	,	,	PUNCT
cana-834	415	13	which	which	PRON
cana-834	415	14	make	make	VERB
cana-834	415	15	it	it	PRON
cana-834	415	16	easier	easy	ADJ
cana-834	415	17	to	to	PART
cana-834	415	18	model	model	VERB
cana-834	415	19	and	and	CCONJ
cana-834	415	20	forecast	forecast	NOUN
cana-834	415	21	disease	disease	NOUN
cana-834	415	22	outbreaks	outbreak	NOUN
cana-834	415	23	.	.	PUNCT
cana-834	416	1	when	when	SCONJ
cana-834	416	2	dealing	deal	VERB
cana-834	416	3	with	with	ADP
cana-834	416	4	the	the	DET
cana-834	416	5	large	large	ADJ
cana-834	416	6	amounts	amount	NOUN
cana-834	416	7	of	of	ADP
cana-834	416	8	data	datum	NOUN
cana-834	416	9	that	that	PRON
cana-834	416	10	are	be	AUX
cana-834	416	11	generated	generate	VERB
cana-834	416	12	in	in	ADP
cana-834	416	13	healthcare	healthcare	NOUN
cana-834	416	14	scenarios	scenario	NOUN
cana-834	416	15	,	,	PUNCT
cana-834	416	16	distributed	distribute	VERB
cana-834	416	17	computing	computing	NOUN
cana-834	416	18	and	and	CCONJ
cana-834	416	19	compressed	compressed	ADJ
cana-834	416	20	representations	representation	NOUN
cana-834	416	21	are	be	AUX
cana-834	416	22	absolutely	absolutely	ADV
cana-834	416	23	necessary	necessary	ADJ
cana-834	416	24	.	.	PUNCT
cana-834	417	1	a	a	DET
cana-834	417	2	multifaceted	multifaceted	ADJ
cana-834	417	3	strategy	strategy	NOUN
cana-834	417	4	that	that	PRON
cana-834	417	5	incorporates	incorporate	VERB
cana-834	417	6	algorithmic	algorithmic	ADJ
cana-834	417	7	enhancements	enhancement	NOUN
cana-834	417	8	,	,	PUNCT
cana-834	417	9	sophisticated	sophisticated	ADJ
cana-834	417	10	data	datum	NOUN
cana-834	417	11	structures	structure	NOUN
cana-834	417	12	,	,	PUNCT
cana-834	417	13	and	and	CCONJ
cana-834	417	14	specialized	specialized	ADJ
cana-834	417	15	hardware	hardware	NOUN
cana-834	417	16	implementations	implementation	NOUN
cana-834	417	17	is	be	AUX
cana-834	417	18	required	require	VERB
cana-834	417	19	to	to	PART
cana-834	417	20	optimize	optimize	VERB
cana-834	417	21	graph	graph	NOUN
cana-834	417	22	theory	theory	NOUN
cana-834	417	23	algorithms	algorithm	NOUN
cana-834	417	24	for	for	ADP
cana-834	417	25	social	social	ADJ
cana-834	417	26	network	network	NOUN
cana-834	417	27	analysis	analysis	NOUN
cana-834	417	28	.	.	PUNCT
cana-834	418	1	boosting	boost	VERB
cana-834	418	2	graph	graph	NOUN
cana-834	418	3	algorithms	algorithm	NOUN
cana-834	418	4	'	'	PART
cana-834	418	5	efficiency	efficiency	NOUN
cana-834	418	6	and	and	CCONJ
cana-834	418	7	scalability	scalability	NOUN
cana-834	418	8	requires	require	VERB
cana-834	418	9	using	use	VERB
cana-834	418	10	compressed	compress	VERB
cana-834	418	11	representations	representation	NOUN
cana-834	418	12	,	,	PUNCT
cana-834	418	13	approximation	approximation	NOUN
cana-834	418	14	algorithms	algorithm	NOUN
cana-834	418	15	,	,	PUNCT
cana-834	418	16	and	and	CCONJ
cana-834	418	17	parallel	parallel	ADJ
cana-834	418	18	computing	computing	NOUN
cana-834	418	19	.	.	PUNCT
cana-834	419	1	the	the	DET
cana-834	419	2	practical	practical	ADJ
cana-834	419	3	impact	impact	NOUN
cana-834	419	4	of	of	ADP
cana-834	419	5	these	these	DET
cana-834	419	6	optimizations	optimization	NOUN
cana-834	419	7	is	be	AUX
cana-834	419	8	demonstrated	demonstrate	VERB
cana-834	419	9	by	by	ADP
cana-834	419	10	real	real	ADJ
cana-834	419	11	-	-	PUNCT
cana-834	419	12	world	world	NOUN
cana-834	419	13	applications	application	NOUN
cana-834	419	14	in	in	ADP
cana-834	419	15	healthcare	healthcare	PROPN
cana-834	419	16	,	,	PUNCT
cana-834	419	17	social	social	ADJ
cana-834	419	18	media	medium	NOUN
cana-834	419	19	analysis	analysis	NOUN
cana-834	419	20	,	,	PUNCT
cana-834	419	21	and	and	CCONJ
cana-834	419	22	fraud	fraud	NOUN
cana-834	419	23	detection	detection	NOUN
cana-834	419	24	.	.	PUNCT
cana-834	420	1	researchers	researcher	NOUN
cana-834	420	2	and	and	CCONJ
cana-834	420	3	practitioners	practitioner	NOUN
cana-834	420	4	can	can	AUX
cana-834	420	5	effectively	effectively	ADV
cana-834	420	6	address	address	VERB
cana-834	420	7	the	the	DET
cana-834	420	8	difficulties	difficulty	NOUN
cana-834	420	9	of	of	ADP
cana-834	420	10	analyzing	analyze	VERB
cana-834	420	11	large	large	ADJ
cana-834	420	12	-	-	PUNCT
cana-834	420	13	scale	scale	NOUN
cana-834	420	14	and	and	CCONJ
cana-834	420	15	dynamic	dynamic	ADJ
cana-834	420	16	social	social	ADJ
cana-834	420	17	networks	network	NOUN
cana-834	420	18	by	by	ADP
cana-834	420	19	continuing	continue	VERB
cana-834	420	20	to	to	PART
cana-834	420	21	develop	develop	VERB
cana-834	420	22	and	and	CCONJ
cana-834	420	23	integrate	integrate	VERB
cana-834	420	24	these	these	DET
cana-834	420	25	methods	method	NOUN
cana-834	420	26	.	.	PUNCT
cana-834	421	1	communications	communication	NOUN
cana-834	421	2	on	on	ADP
cana-834	421	3	applied	apply	VERB
cana-834	421	4	nonlinear	nonlinear	ADJ
cana-834	421	5	analysis	analysis	NOUN
cana-834	421	6	issn	issn	NOUN
cana-834	421	7	:	:	PUNCT
cana-834	421	8	1074	1074	NUM
cana-834	421	9	-	-	PUNCT
cana-834	421	10	133x	133x	NUM
cana-834	421	11	vol	vol	NOUN
cana-834	421	12	31	31	NUM
cana-834	421	13	no	no	NOUN
cana-834	421	14	.	.	PUNCT
cana-834	422	1	4s	4s	NUM
cana-834	422	2	(	(	PUNCT
cana-834	422	3	2024	2024	NUM
cana-834	422	4	)	)	PUNCT
cana-834	422	5	178	178	NUM
cana-834	422	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	422	7	fig	fig	NOUN
cana-834	422	8	1	1	NUM
cana-834	422	9	social	social	ADJ
cana-834	422	10	networking	networking	NOUN
cana-834	422	11	in	in	ADP
cana-834	422	12	friends	friend	NOUN
cana-834	422	13	session	session	NOUN
cana-834	422	14	fig	fig	NOUN
cana-834	422	15	2	2	NUM
cana-834	422	16	sybil	sybil	NOUN
cana-834	422	17	attack	attack	NOUN
cana-834	422	18	in	in	ADP
cana-834	422	19	social	social	ADJ
cana-834	422	20	networks	network	NOUN
cana-834	422	21	using	use	VERB
cana-834	422	22	graphs	graph	NOUN
cana-834	422	23	communications	communication	NOUN
cana-834	422	24	on	on	ADP
cana-834	422	25	applied	apply	VERB
cana-834	422	26	nonlinear	nonlinear	ADJ
cana-834	422	27	analysis	analysis	NOUN
cana-834	422	28	issn	issn	NOUN
cana-834	422	29	:	:	PUNCT
cana-834	422	30	1074	1074	NUM
cana-834	422	31	-	-	PUNCT
cana-834	422	32	133x	133x	NUM
cana-834	422	33	vol	vol	NOUN
cana-834	422	34	31	31	NUM
cana-834	422	35	no	no	NOUN
cana-834	422	36	.	.	PUNCT
cana-834	423	1	4s	4s	NUM
cana-834	423	2	(	(	PUNCT
cana-834	423	3	2024	2024	NUM
cana-834	423	4	)	)	PUNCT
cana-834	423	5	179	179	NUM
cana-834	423	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	423	7	fig	fig	NOUN
cana-834	423	8	3	3	NUM
cana-834	423	9	networks	network	NOUN
cana-834	423	10	via	via	ADP
cana-834	423	11	graph	graph	NOUN
cana-834	423	12	theory	theory	NOUN
cana-834	423	13	fig	fig	NOUN
cana-834	423	14	4	4	NUM
cana-834	423	15	bipartite	bipartite	NOUN
cana-834	423	16	graph	graph	NOUN
cana-834	423	17	relation	relation	NOUN
cana-834	423	18	between	between	ADP
cana-834	423	19	customers	customer	NOUN
cana-834	423	20	and	and	CCONJ
cana-834	423	21	movies	movie	NOUN
cana-834	423	22	fig	fig	VERB
cana-834	423	23	5	5	NUM
cana-834	423	24	modeling	modeling	NOUN
cana-834	423	25	of	of	ADP
cana-834	423	26	microblogs	microblog	NOUN
cana-834	423	27	communications	communication	NOUN
cana-834	423	28	on	on	ADP
cana-834	423	29	applied	apply	VERB
cana-834	423	30	nonlinear	nonlinear	ADJ
cana-834	423	31	analysis	analysis	NOUN
cana-834	423	32	issn	issn	NOUN
cana-834	423	33	:	:	PUNCT
cana-834	423	34	1074	1074	NUM
cana-834	423	35	-	-	PUNCT
cana-834	423	36	133x	133x	NUM
cana-834	423	37	vol	vol	NOUN
cana-834	423	38	31	31	NUM
cana-834	423	39	no	no	NOUN
cana-834	423	40	.	.	PUNCT
cana-834	424	1	4s	4s	NUM
cana-834	424	2	(	(	PUNCT
cana-834	424	3	2024	2024	NUM
cana-834	424	4	)	)	PUNCT
cana-834	424	5	180	180	NUM
cana-834	424	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	424	7	fig	fig	NOUN
cana-834	424	8	6	6	NUM
cana-834	424	9	eigen	eigen	PROPN
cana-834	424	10	values	value	NOUN
cana-834	424	11	in	in	ADP
cana-834	424	12	social	social	ADJ
cana-834	424	13	network	network	NOUN
cana-834	424	14	analysis	analysis	NOUN
cana-834	424	15	5	5	NUM
cana-834	424	16	.	.	PUNCT
cana-834	424	17	conclusion	conclusion	NOUN
cana-834	424	18	in	in	ADP
cana-834	424	19	conclusion	conclusion	NOUN
cana-834	424	20	,	,	PUNCT
cana-834	424	21	a	a	DET
cana-834	424	22	reliable	reliable	ADJ
cana-834	424	23	strategy	strategy	NOUN
cana-834	424	24	for	for	ADP
cana-834	424	25	comprehending	comprehend	VERB
cana-834	424	26	intricate	intricate	ADJ
cana-834	424	27	social	social	ADJ
cana-834	424	28	structures	structure	NOUN
cana-834	424	29	and	and	CCONJ
cana-834	424	30	dynamics	dynamic	NOUN
cana-834	424	31	is	be	AUX
cana-834	424	32	provided	provide	VERB
cana-834	424	33	by	by	ADP
cana-834	424	34	optimizing	optimize	VERB
cana-834	424	35	graph	graph	NOUN
cana-834	424	36	theory	theory	NOUN
cana-834	424	37	algorithms	algorithm	NOUN
cana-834	424	38	for	for	ADP
cana-834	424	39	social	social	ADJ
cana-834	424	40	network	network	NOUN
cana-834	424	41	analysis	analysis	NOUN
cana-834	424	42	.	.	PUNCT
cana-834	425	1	by	by	ADP
cana-834	425	2	upgrading	upgrade	VERB
cana-834	425	3	the	the	DET
cana-834	425	4	proficiency	proficiency	NOUN
cana-834	425	5	and	and	CCONJ
cana-834	425	6	versatility	versatility	NOUN
cana-834	425	7	of	of	ADP
cana-834	425	8	these	these	DET
cana-834	425	9	calculations	calculation	NOUN
cana-834	425	10	,	,	PUNCT
cana-834	425	11	specialists	specialist	NOUN
cana-834	425	12	and	and	CCONJ
cana-834	425	13	experts	expert	NOUN
cana-834	425	14	can	can	AUX
cana-834	425	15	all	all	DET
cana-834	425	16	the	the	DET
cana-834	425	17	more	more	ADV
cana-834	425	18	actually	actually	ADV
cana-834	425	19	handle	handle	VERB
cana-834	425	20	the	the	DET
cana-834	425	21	enormous	enormous	ADJ
cana-834	425	22	scope	scope	NOUN
cana-834	425	23	of	of	ADP
cana-834	425	24	information	information	NOUN
cana-834	425	25	inborn	inborn	ADJ
cana-834	425	26	in	in	ADP
cana-834	425	27	current	current	ADJ
cana-834	425	28	interpersonal	interpersonal	ADJ
cana-834	425	29	organizations	organization	NOUN
cana-834	425	30	.	.	PUNCT
cana-834	426	1	the	the	DET
cana-834	426	2	creation	creation	NOUN
cana-834	426	3	of	of	ADP
cana-834	426	4	more	more	ADV
cana-834	426	5	effective	effective	ADJ
cana-834	426	6	algorithms	algorithm	NOUN
cana-834	426	7	for	for	ADP
cana-834	426	8	community	community	NOUN
cana-834	426	9	detection	detection	NOUN
cana-834	426	10	,	,	PUNCT
cana-834	426	11	shortest	short	ADJ
cana-834	426	12	path	path	NOUN
cana-834	426	13	computations	computation	NOUN
cana-834	426	14	,	,	PUNCT
cana-834	426	15	and	and	CCONJ
cana-834	426	16	centrality	centrality	NOUN
cana-834	426	17	measures	measure	NOUN
cana-834	426	18	—	—	PUNCT
cana-834	426	19	all	all	PRON
cana-834	426	20	of	of	ADP
cana-834	426	21	which	which	PRON
cana-834	426	22	are	be	AUX
cana-834	426	23	essential	essential	ADJ
cana-834	426	24	for	for	ADP
cana-834	426	25	identifying	identify	VERB
cana-834	426	26	influential	influential	ADJ
cana-834	426	27	nodes	node	NOUN
cana-834	426	28	and	and	CCONJ
cana-834	426	29	comprehending	comprehend	VERB
cana-834	426	30	the	the	DET
cana-834	426	31	overall	overall	ADJ
cana-834	426	32	topology	topology	NOUN
cana-834	426	33	of	of	ADP
cana-834	426	34	the	the	DET
cana-834	426	35	network	network	NOUN
cana-834	426	36	—	—	PUNCT
cana-834	426	37	are	be	AUX
cana-834	426	38	among	among	ADP
cana-834	426	39	the	the	DET
cana-834	426	40	most	most	ADV
cana-834	426	41	significant	significant	ADJ
cana-834	426	42	advancements	advancement	NOUN
cana-834	426	43	.	.	PUNCT
cana-834	427	1	parallel	parallel	ADJ
cana-834	427	2	processing	processing	NOUN
cana-834	427	3	,	,	PUNCT
cana-834	427	4	distributed	distribute	VERB
cana-834	427	5	computing	computing	NOUN
cana-834	427	6	,	,	PUNCT
cana-834	427	7	and	and	CCONJ
cana-834	427	8	machine	machine	NOUN
cana-834	427	9	learning	learning	NOUN
cana-834	427	10	integration	integration	NOUN
cana-834	427	11	are	be	AUX
cana-834	427	12	just	just	ADV
cana-834	427	13	a	a	DET
cana-834	427	14	few	few	ADJ
cana-834	427	15	of	of	ADP
cana-834	427	16	the	the	DET
cana-834	427	17	cutting	cutting	NOUN
cana-834	427	18	-	-	PUNCT
cana-834	427	19	edge	edge	NOUN
cana-834	427	20	computational	computational	ADJ
cana-834	427	21	methods	method	NOUN
cana-834	427	22	that	that	PRON
cana-834	427	23	can	can	AUX
cana-834	427	24	significantly	significantly	ADV
cana-834	427	25	reduce	reduce	VERB
cana-834	427	26	the	the	DET
cana-834	427	27	time	time	NOUN
cana-834	427	28	complexity	complexity	NOUN
cana-834	427	29	and	and	CCONJ
cana-834	427	30	computational	computational	ADJ
cana-834	427	31	overhead	overhead	NOUN
cana-834	427	32	of	of	ADP
cana-834	427	33	traditional	traditional	ADJ
cana-834	427	34	graph	graph	NOUN
cana-834	427	35	algorithms	algorithm	NOUN
cana-834	427	36	.	.	PUNCT
cana-834	428	1	in	in	ADP
cana-834	428	2	addition	addition	NOUN
cana-834	428	3	to	to	ADP
cana-834	428	4	enabling	enable	VERB
cana-834	428	5	real	real	ADJ
cana-834	428	6	-	-	PUNCT
cana-834	428	7	time	time	NOUN
cana-834	428	8	analysis	analysis	NOUN
cana-834	428	9	and	and	CCONJ
cana-834	428	10	insights	insight	NOUN
cana-834	428	11	,	,	PUNCT
cana-834	428	12	these	these	DET
cana-834	428	13	optimizations	optimization	NOUN
cana-834	428	14	open	open	VERB
cana-834	428	15	new	new	ADJ
cana-834	428	16	possibilities	possibility	NOUN
cana-834	428	17	for	for	ADP
cana-834	428	18	predictive	predictive	ADJ
cana-834	428	19	modeling	modeling	NOUN
cana-834	428	20	and	and	CCONJ
cana-834	428	21	intervention	intervention	NOUN
cana-834	428	22	strategies	strategy	NOUN
cana-834	428	23	in	in	ADP
cana-834	428	24	a	a	DET
cana-834	428	25	variety	variety	NOUN
cana-834	428	26	of	of	ADP
cana-834	428	27	applications	application	NOUN
cana-834	428	28	,	,	PUNCT
cana-834	428	29	including	include	VERB
cana-834	428	30	cybersecurity	cybersecurity	NOUN
cana-834	428	31	and	and	CCONJ
cana-834	428	32	social	social	ADJ
cana-834	428	33	behavior	behavior	NOUN
cana-834	428	34	studies	study	NOUN
cana-834	428	35	.	.	PUNCT
cana-834	429	1	the	the	DET
cana-834	429	2	difficulties	difficulty	NOUN
cana-834	429	3	of	of	ADP
cana-834	429	4	dynamic	dynamic	ADJ
cana-834	429	5	network	network	NOUN
cana-834	429	6	analysis	analysis	NOUN
cana-834	429	7	,	,	PUNCT
cana-834	429	8	in	in	ADP
cana-834	429	9	which	which	PRON
cana-834	429	10	the	the	DET
cana-834	429	11	structure	structure	NOUN
cana-834	429	12	of	of	ADP
cana-834	429	13	the	the	DET
cana-834	429	14	network	network	NOUN
cana-834	429	15	changes	change	VERB
cana-834	429	16	over	over	ADP
cana-834	429	17	time	time	NOUN
cana-834	429	18	,	,	PUNCT
cana-834	429	19	and	and	CCONJ
cana-834	429	20	the	the	DET
cana-834	429	21	incorporation	incorporation	NOUN
cana-834	429	22	of	of	ADP
cana-834	429	23	multi	multi	ADJ
cana-834	429	24	-	-	ADJ
cana-834	429	25	layered	layered	ADJ
cana-834	429	26	network	network	NOUN
cana-834	429	27	perspectives	perspective	NOUN
cana-834	429	28	,	,	PUNCT
cana-834	429	29	which	which	PRON
cana-834	429	30	simultaneously	simultaneously	ADV
cana-834	429	31	consider	consider	VERB
cana-834	429	32	a	a	DET
cana-834	429	33	variety	variety	NOUN
cana-834	429	34	of	of	ADP
cana-834	429	35	relationships	relationship	NOUN
cana-834	429	36	and	and	CCONJ
cana-834	429	37	interactions	interaction	NOUN
cana-834	429	38	,	,	PUNCT
cana-834	429	39	ought	ought	AUX
cana-834	429	40	to	to	PART
cana-834	429	41	be	be	AUX
cana-834	429	42	the	the	DET
cana-834	429	43	primary	primary	ADJ
cana-834	429	44	areas	area	NOUN
cana-834	429	45	of	of	ADP
cana-834	429	46	focus	focus	NOUN
cana-834	429	47	in	in	ADP
cana-834	429	48	future	future	ADJ
cana-834	429	49	research	research	NOUN
cana-834	429	50	.	.	PUNCT
cana-834	430	1	in	in	ADP
cana-834	430	2	addition	addition	NOUN
cana-834	430	3	,	,	PUNCT
cana-834	430	4	to	to	PART
cana-834	430	5	guarantee	guarantee	VERB
cana-834	430	6	the	the	DET
cana-834	430	7	ethical	ethical	ADJ
cana-834	430	8	use	use	NOUN
cana-834	430	9	of	of	ADP
cana-834	430	10	these	these	DET
cana-834	430	11	potent	potent	ADJ
cana-834	430	12	analytical	analytical	ADJ
cana-834	430	13	tools	tool	NOUN
cana-834	430	14	,	,	PUNCT
cana-834	430	15	it	it	PRON
cana-834	430	16	is	be	AUX
cana-834	430	17	necessary	necessary	ADJ
cana-834	430	18	to	to	PART
cana-834	430	19	constantly	constantly	ADV
cana-834	430	20	address	address	VERB
cana-834	430	21	the	the	DET
cana-834	430	22	privacy	privacy	NOUN
cana-834	430	23	and	and	CCONJ
cana-834	430	24	data	datum	NOUN
cana-834	430	25	security	security	NOUN
cana-834	430	26	implications	implication	NOUN
cana-834	430	27	of	of	ADP
cana-834	430	28	social	social	ADJ
cana-834	430	29	network	network	NOUN
cana-834	430	30	analysis	analysis	NOUN
cana-834	430	31	.	.	PUNCT
cana-834	431	1	in	in	ADP
cana-834	431	2	general	general	ADJ
cana-834	431	3	,	,	PUNCT
cana-834	431	4	the	the	DET
cana-834	431	5	continuous	continuous	ADJ
cana-834	431	6	development	development	NOUN
cana-834	431	7	and	and	CCONJ
cana-834	431	8	adaptation	adaptation	NOUN
cana-834	431	9	of	of	ADP
cana-834	431	10	graph	graph	NOUN
cana-834	431	11	theory	theory	NOUN
cana-834	431	12	algorithms	algorithm	NOUN
cana-834	431	13	for	for	ADP
cana-834	431	14	social	social	ADJ
cana-834	431	15	network	network	NOUN
cana-834	431	16	analysis	analysis	NOUN
cana-834	431	17	has	have	VERB
cana-834	431	18	enormous	enormous	ADJ
cana-834	431	19	potential	potential	NOUN
cana-834	431	20	to	to	PART
cana-834	431	21	improve	improve	VERB
cana-834	431	22	decision	decision	NOUN
cana-834	431	23	-	-	PUNCT
cana-834	431	24	making	make	VERB
cana-834	431	25	processes	process	NOUN
cana-834	431	26	and	and	CCONJ
cana-834	431	27	our	our	PRON
cana-834	431	28	comprehension	comprehension	NOUN
cana-834	431	29	of	of	ADP
cana-834	431	30	social	social	ADJ
cana-834	431	31	systems	system	NOUN
cana-834	431	32	in	in	ADP
cana-834	431	33	a	a	DET
cana-834	431	34	variety	variety	NOUN
cana-834	431	35	of	of	ADP
cana-834	431	36	fields	field	NOUN
cana-834	431	37	.	.	PUNCT
cana-834	432	1	references	reference	NOUN
cana-834	432	2	[	[	X
cana-834	432	3	1	1	NUM
cana-834	432	4	]	]	PUNCT
cana-834	432	5	r.frucht	r.frucht	NOUN
cana-834	432	6	and	and	CCONJ
cana-834	432	7	f.harary	f.harary	ADJ
cana-834	432	8	,	,	PUNCT
cana-834	432	9	“	"	PUNCT
cana-834	432	10	on	on	ADP
cana-834	432	11	the	the	DET
cana-834	432	12	corona	corona	NOUN
cana-834	432	13	of	of	ADP
cana-834	432	14	two	two	NUM
cana-834	432	15	graphs	graph	NOUN
cana-834	432	16	”	"	PUNCT
cana-834	432	17	,	,	PUNCT
cana-834	432	18	aequationes	aequatione	VERB
cana-834	432	19	math	math	PROPN
cana-834	432	20	.	.	PUNCT
cana-834	433	1	,	,	PUNCT
cana-834	433	2	vol	vol	NOUN
cana-834	433	3	.	.	PUNCT
cana-834	434	1	4,pp	4,pp	NUM
cana-834	434	2	.	.	PUNCT
cana-834	435	1	322–325	322–325	NUM
cana-834	435	2	,	,	PUNCT
cana-834	435	3	1970	1970	NUM
cana-834	435	4	.	.	PUNCT
cana-834	436	1	[	[	X
cana-834	436	2	2	2	NUM
cana-834	436	3	]	]	X
cana-834	436	4	f.harary,“graphtheory	f.harary,“graphtheory	ADJ
cana-834	436	5	”	"	PUNCT
cana-834	436	6	,	,	PUNCT
cana-834	436	7	addison	addison	PROPN
cana-834	436	8	wesley	wesley	PROPN
cana-834	436	9	,	,	PUNCT
cana-834	436	10	reading	read	VERB
cana-834	436	11	mass,1972	mass,1972	NOUN
cana-834	436	12	.	.	PUNCT
cana-834	437	1	[	[	X
cana-834	437	2	3	3	NUM
cana-834	437	3	]	]	PUNCT
cana-834	437	4	o.ore	o.ore	ADV
cana-834	437	5	,	,	PUNCT
cana-834	437	6	“	"	PUNCT
cana-834	437	7	theory	theory	NOUN
cana-834	437	8	of	of	ADP
cana-834	437	9	graphs	graph	NOUN
cana-834	437	10	”	"	PUNCT
cana-834	437	11	,	,	PUNCT
cana-834	437	12	amer	amer	PROPN
cana-834	437	13	.	.	PROPN
cana-834	437	14	math	math	PROPN
cana-834	437	15	.	.	PUNCT
cana-834	438	1	soc	soc	PROPN
cana-834	438	2	.	.	PUNCT
cana-834	439	1	colloq	colloq	PROPN
cana-834	439	2	.	.	PUNCT
cana-834	440	1	publ	publ	PROPN
cana-834	440	2	.	.	PUNCT
cana-834	440	3	,	,	PUNCT
cana-834	440	4	v	v	ADP
cana-834	440	5	o	o	NOUN
cana-834	440	6	l	l	NOUN
cana-834	440	7	.	.	PUNCT
cana-834	441	1	38,1962	38,1962	NUM
cana-834	441	2	.	.	PUNCT
cana-834	442	1	[	[	X
cana-834	442	2	4	4	X
cana-834	442	3	]	]	X
cana-834	442	4	sergio	sergio	PROPN
cana-834	442	5	canoyjr	canoyjr	PROPN
cana-834	442	6	and	and	CCONJ
cana-834	442	7	carmelito	carmelito	PROPN
cana-834	442	8	e.go	e.go	PROPN
cana-834	442	9	,	,	PUNCT
cana-834	442	10	“	"	PUNCT
cana-834	442	11	domination	domination	NOUN
cana-834	442	12	in	in	ADP
cana-834	442	13	the	the	DET
cana-834	442	14	corona	corona	NOUN
cana-834	442	15	and	and	CCONJ
cana-834	442	16	join	join	VERB
cana-834	442	17	of	of	ADP
cana-834	442	18	graphs	graph	NOUN
cana-834	442	19	”	"	PUNCT
cana-834	442	20	,	,	PUNCT
cana-834	442	21	international	international	ADJ
cana-834	442	22	communications	communication	NOUN
cana-834	442	23	on	on	ADP
cana-834	442	24	applied	apply	VERB
cana-834	442	25	nonlinear	nonlinear	ADJ
cana-834	442	26	analysis	analysis	NOUN
cana-834	442	27	issn	issn	NOUN
cana-834	442	28	:	:	PUNCT
cana-834	442	29	1074	1074	NUM
cana-834	442	30	-	-	PUNCT
cana-834	442	31	133x	133x	NUM
cana-834	442	32	vol	vol	NOUN
cana-834	442	33	31	31	NUM
cana-834	442	34	no	no	NOUN
cana-834	442	35	.	.	PUNCT
cana-834	443	1	4s	4s	NUM
cana-834	443	2	(	(	PUNCT
cana-834	443	3	2024	2024	NUM
cana-834	443	4	)	)	PUNCT
cana-834	443	5	181	181	NUM
cana-834	444	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-834	444	2	mathematical	mathematical	ADJ
cana-834	444	3	forum	forum	NOUN
cana-834	444	4	,	,	PUNCT
cana-834	444	5	vol	vol	NOUN
cana-834	444	6	.	.	PROPN
cana-834	444	7	6	6	NUM
cana-834	444	8	,	,	PUNCT
cana-834	444	9	no	no	INTJ
cana-834	444	10	.	.	PUNCT
cana-834	444	11	13,2011	13,2011	NUM
cana-834	444	12	.	.	PUNCT
cana-834	445	1	[	[	X
cana-834	445	2	5	5	NUM
cana-834	445	3	]	]	X
cana-834	445	4	fortunato	fortunato	PROPN
cana-834	445	5	,	,	PUNCT
cana-834	445	6	s.	s.	PROPN
cana-834	445	7	(	(	PUNCT
cana-834	445	8	2010	2010	NUM
cana-834	445	9	)	)	PUNCT
cana-834	445	10	.	.	PUNCT
cana-834	446	1	community	community	NOUN
cana-834	446	2	detection	detection	NOUN
cana-834	446	3	in	in	ADP
cana-834	446	4	graphs	graph	NOUN
cana-834	446	5	.	.	PUNCT
cana-834	447	1	*	*	PUNCT
cana-834	447	2	physics	physics	NOUN
cana-834	447	3	reports	report	VERB
cana-834	447	4	*	*	NOUN
cana-834	447	5	,	,	PUNCT
cana-834	447	6	486(3	486(3	NUM
cana-834	447	7	-	-	SYM
cana-834	447	8	5	5	NUM
cana-834	447	9	)	)	PUNCT
cana-834	447	10	,	,	PUNCT
cana-834	447	11	75	75	NUM
cana-834	447	12	-	-	SYM
cana-834	447	13	174	174	NUM
cana-834	447	14	.	.	PUNCT
cana-834	448	1	https://doi.org/10.1016/j.physrep.2009.11.002	https://doi.org/10.1016/j.physrep.2009.11.002	NUM
cana-834	448	2	[	[	X
cana-834	448	3	6	6	NUM
cana-834	448	4	]	]	SYM
cana-834	448	5	blondel	blondel	NOUN
cana-834	448	6	,	,	PUNCT
cana-834	448	7	v.	v.	PROPN
cana-834	448	8	d.	d.	PROPN
cana-834	448	9	,	,	PUNCT
cana-834	448	10	guillaume	guillaume	PROPN
cana-834	448	11	,	,	PUNCT
cana-834	448	12	j.	j.	PROPN
cana-834	448	13	l.	l.	PROPN
cana-834	448	14	,	,	PUNCT
cana-834	448	15	lambiotte	lambiotte	PROPN
cana-834	448	16	,	,	PUNCT
cana-834	448	17	r.	r.	PROPN
cana-834	448	18	,	,	PUNCT
cana-834	448	19	&	&	CCONJ
cana-834	448	20	lefebvre	lefebvre	PROPN
cana-834	448	21	,	,	PUNCT
cana-834	448	22	e.	e.	PROPN
cana-834	448	23	(	(	PUNCT
cana-834	448	24	2008	2008	NUM
cana-834	448	25	)	)	PUNCT
cana-834	448	26	.	.	PUNCT
cana-834	449	1	fast	fast	ADJ
cana-834	449	2	unfolding	unfolding	NOUN
cana-834	449	3	of	of	ADP
cana-834	449	4	communities	community	NOUN
cana-834	449	5	in	in	ADP
cana-834	449	6	large	large	ADJ
cana-834	449	7	networks	network	NOUN
cana-834	449	8	.	.	PUNCT
cana-834	450	1	*	*	PUNCT
cana-834	450	2	journal	journal	NOUN
cana-834	450	3	of	of	ADP
cana-834	450	4	statistical	statistical	ADJ
cana-834	450	5	mechanics	mechanic	NOUN
cana-834	450	6	:	:	PUNCT
cana-834	450	7	theory	theory	NOUN
cana-834	450	8	and	and	CCONJ
cana-834	450	9	experiment	experiment	NOUN
cana-834	450	10	*	*	SYM
cana-834	450	11	,	,	PUNCT
cana-834	450	12	2008(10	2008(10	NUM
cana-834	450	13	)	)	PUNCT
cana-834	450	14	,	,	PUNCT
cana-834	450	15	p10008	p10008	PROPN
cana-834	450	16	.	.	PUNCT
cana-834	451	1	https://doi.org/10.1088/1742-5468/2008/10/p10008	https://doi.org/10.1088/1742-5468/2008/10/p10008	PROPN
cana-834	451	2	[	[	X
cana-834	451	3	7	7	NUM
cana-834	451	4	]	]	PUNCT
cana-834	451	5	brandes	brande	NOUN
cana-834	451	6	,	,	PUNCT
cana-834	451	7	u.	u.	PROPN
cana-834	451	8	(	(	PUNCT
cana-834	451	9	2001	2001	NUM
cana-834	451	10	)	)	PUNCT
cana-834	451	11	.	.	PUNCT
cana-834	452	1	a	a	DET
cana-834	452	2	faster	fast	ADJ
cana-834	452	3	algorithm	algorithm	NOUN
cana-834	452	4	for	for	ADP
cana-834	452	5	betweenness	betweenness	NOUN
cana-834	452	6	centrality	centrality	NOUN
cana-834	452	7	.	.	PUNCT
cana-834	453	1	*	*	PUNCT
cana-834	453	2	journal	journal	PROPN
cana-834	453	3	of	of	ADP
cana-834	453	4	mathematical	mathematical	ADJ
cana-834	453	5	sociology	sociology	NOUN
cana-834	453	6	*	*	NOUN
cana-834	453	7	,	,	PUNCT
cana-834	453	8	25(2	25(2	NUM
cana-834	453	9	)	)	PUNCT
cana-834	453	10	,	,	PUNCT
cana-834	453	11	163	163	NUM
cana-834	453	12	-	-	SYM
cana-834	453	13	177	177	NUM
cana-834	453	14	.	.	PUNCT
cana-834	454	1	https://doi.org/10.1080/0022250x.2001.9990249	https://doi.org/10.1080/0022250x.2001.9990249	PROPN
cana-834	454	2	[	[	X
cana-834	454	3	8	8	NUM
cana-834	454	4	]	]	X
cana-834	454	5	newman	newman	PROPN
cana-834	454	6	,	,	PUNCT
cana-834	454	7	m.	m.	PROPN
cana-834	454	8	e.	e.	PROPN
cana-834	454	9	j.	j.	PROPN
cana-834	454	10	(	(	PUNCT
cana-834	454	11	2005	2005	NUM
cana-834	454	12	)	)	PUNCT
cana-834	454	13	.	.	PUNCT
cana-834	455	1	a	a	DET
cana-834	455	2	measure	measure	NOUN
cana-834	455	3	of	of	ADP
cana-834	455	4	betweenness	betweenness	NOUN
cana-834	455	5	centrality	centrality	NOUN
cana-834	455	6	based	base	VERB
cana-834	455	7	on	on	ADP
cana-834	455	8	random	random	ADJ
cana-834	455	9	walks	walk	NOUN
cana-834	455	10	.	.	PUNCT
cana-834	456	1	*	*	PUNCT
cana-834	456	2	social	social	ADJ
cana-834	456	3	networks	network	NOUN
cana-834	456	4	*	*	PUNCT
cana-834	456	5	,	,	PUNCT
cana-834	456	6	27(1	27(1	NUM
cana-834	456	7	)	)	PUNCT
cana-834	456	8	,	,	PUNCT
cana-834	456	9	39	39	NUM
cana-834	456	10	-	-	SYM
cana-834	456	11	54	54	NUM
cana-834	456	12	.	.	PUNCT
cana-834	457	1	https://doi.org/10.1016/j.socnet.2004.11.009	https://doi.org/10.1016/j.socnet.2004.11.009	NOUN
cana-834	457	2	[	[	X
cana-834	457	3	9	9	NUM
cana-834	457	4	]	]	SYM
cana-834	457	5	lambiotte	lambiotte	NOUN
cana-834	457	6	,	,	PUNCT
cana-834	457	7	r.	r.	PROPN
cana-834	457	8	,	,	PUNCT
cana-834	457	9	&	&	CCONJ
cana-834	457	10	rosvall	rosvall	PROPN
cana-834	457	11	,	,	PUNCT
cana-834	457	12	m.	m.	NOUN
cana-834	457	13	(	(	PUNCT
cana-834	457	14	2012	2012	NUM
cana-834	457	15	)	)	PUNCT
cana-834	457	16	.	.	PUNCT
cana-834	458	1	ranking	rank	VERB
cana-834	458	2	and	and	CCONJ
cana-834	458	3	clustering	clustering	NOUN
cana-834	458	4	of	of	ADP
cana-834	458	5	nodes	node	NOUN
cana-834	458	6	in	in	ADP
cana-834	458	7	networks	network	NOUN
cana-834	458	8	with	with	ADP
cana-834	458	9	smart	smart	ADJ
cana-834	458	10	teleportation	teleportation	NOUN
cana-834	458	11	.	.	PUNCT
cana-834	459	1	*	*	PUNCT
cana-834	459	2	physical	physical	ADJ
cana-834	459	3	review	review	NOUN
cana-834	459	4	e	e	NOUN
cana-834	459	5	*	*	NOUN
cana-834	459	6	,	,	PUNCT
cana-834	459	7	85(5	85(5	NUM
cana-834	459	8	)	)	PUNCT
cana-834	459	9	,	,	PUNCT
cana-834	459	10	056107	056107	NUM
cana-834	459	11	.	.	PUNCT
cana-834	460	1	https://doi.org/10.1103/physreve.85.056107	https://doi.org/10.1103/physreve.85.056107	PUNCT
cana-834	461	1	[	[	X
cana-834	461	2	10	10	NUM
cana-834	461	3	]	]	PUNCT
cana-834	461	4	leskovec	leskovec	PROPN
cana-834	461	5	,	,	PUNCT
cana-834	461	6	j.	j.	PROPN
cana-834	461	7	,	,	PUNCT
cana-834	461	8	rajaraman	rajaraman	NOUN
cana-834	461	9	,	,	PUNCT
cana-834	461	10	a.	a.	NOUN
cana-834	461	11	,	,	PUNCT
cana-834	461	12	&	&	CCONJ
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cana-834	461	14	,	,	PUNCT
cana-834	461	15	j.	j.	PROPN
cana-834	461	16	d.	d.	PROPN
cana-834	461	17	(	(	PUNCT
cana-834	461	18	2014	2014	NUM
cana-834	461	19	)	)	PUNCT
cana-834	461	20	.	.	PUNCT
cana-834	462	1	mining	mining	NOUN
cana-834	462	2	of	of	ADP
cana-834	462	3	massive	massive	ADJ
cana-834	462	4	datasets	dataset	NOUN
cana-834	462	5	(	(	PUNCT
cana-834	462	6	2nd	2nd	ADJ
cana-834	462	7	ed	ed	NOUN
cana-834	462	8	.	.	PUNCT
cana-834	462	9	)	)	PUNCT
cana-834	462	10	.	.	PUNCT
cana-834	463	1	cambridge	cambridge	PROPN
cana-834	463	2	university	university	PROPN
cana-834	463	3	press	press	NOUN
cana-834	463	4	.	.	PUNCT
cana-834	464	1	https://doi.org/10.1017/cbo9781139924801	https://doi.org/10.1017/cbo9781139924801	NOUN
cana-834	465	1	[	[	X
cana-834	465	2	11	11	NUM
cana-834	465	3	]	]	X
cana-834	465	4	koutra	koutra	PROPN
cana-834	465	5	,	,	PUNCT
cana-834	465	6	d.	d.	PROPN
cana-834	465	7	,	,	PUNCT
cana-834	465	8	kang	kang	PROPN
cana-834	465	9	,	,	PUNCT
cana-834	465	10	u.	u.	PROPN
cana-834	465	11	,	,	PUNCT
cana-834	465	12	vreeken	vreeken	VERB
cana-834	465	13	,	,	PUNCT
cana-834	465	14	j.	j.	PROPN
cana-834	465	15	,	,	PUNCT
cana-834	465	16	&	&	CCONJ
cana-834	465	17	faloutsos	faloutsos	PROPN
cana-834	465	18	,	,	PUNCT
cana-834	465	19	c.	c.	PROPN
cana-834	465	20	(	(	PUNCT
cana-834	465	21	2014	2014	NUM
cana-834	465	22	)	)	PUNCT
cana-834	465	23	.	.	PUNCT
cana-834	466	1	vog	vog	PROPN
cana-834	466	2	:	:	PUNCT
cana-834	466	3	summarizing	summarizing	NOUN
cana-834	466	4	and	and	CCONJ
cana-834	466	5	understanding	understand	VERB
cana-834	466	6	large	large	ADJ
cana-834	466	7	graphs	graph	NOUN
cana-834	466	8	.	.	PUNCT
cana-834	467	1	*	*	PUNCT
cana-834	467	2	2014	2014	NUM
cana-834	467	3	siam	siam	ADJ
cana-834	467	4	international	international	ADJ
cana-834	467	5	conference	conference	NOUN
cana-834	467	6	on	on	ADP
cana-834	467	7	data	datum	NOUN
cana-834	467	8	mining	mining	NOUN
cana-834	467	9	(	(	PUNCT
cana-834	467	10	sdm	sdm	PROPN
cana-834	467	11	)	)	PUNCT
cana-834	467	12	*	*	PUNCT
cana-834	467	13	,	,	PUNCT
cana-834	467	14	91	91	NUM
cana-834	467	15	-	-	SYM
cana-834	467	16	99	99	NUM
cana-834	467	17	.	.	PUNCT
cana-834	468	1	https://doi.org/10.1137/1.9781611973440.11	https://doi.org/10.1137/1.9781611973440.11	PROPN
cana-834	468	2	[	[	X
cana-834	468	3	12	12	NUM
cana-834	468	4	]	]	PUNCT
cana-834	468	5	ahmed	ahmed	PROPN
cana-834	468	6	,	,	PUNCT
cana-834	468	7	n.	n.	PROPN
cana-834	468	8	k.	k.	PROPN
cana-834	468	9	,	,	PUNCT
cana-834	468	10	neville	neville	PROPN
cana-834	468	11	,	,	PUNCT
cana-834	468	12	j.	j.	PROPN
cana-834	468	13	,	,	PUNCT
cana-834	468	14	rossi	rossi	PROPN
cana-834	468	15	,	,	PUNCT
cana-834	468	16	r.	r.	PROPN
cana-834	468	17	a.	a.	PROPN
cana-834	468	18	,	,	PUNCT
cana-834	468	19	&	&	CCONJ
cana-834	468	20	duffield	duffield	PROPN
cana-834	468	21	,	,	PUNCT
cana-834	468	22	n.	n.	PROPN
cana-834	468	23	(	(	PUNCT
cana-834	468	24	2017	2017	NUM
cana-834	468	25	)	)	PUNCT
cana-834	468	26	.	.	PUNCT
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cana-834	469	2	graphlet	graphlet	ADJ
cana-834	469	3	counting	counting	NOUN
cana-834	469	4	for	for	ADP
cana-834	469	5	large	large	ADJ
cana-834	469	6	networks	network	NOUN
cana-834	469	7	.	.	PUNCT
cana-834	470	1	*	*	PUNCT
cana-834	470	2	2017	2017	NUM
cana-834	470	3	ieee	ieee	NOUN
cana-834	470	4	international	international	ADJ
cana-834	470	5	conference	conference	NOUN
cana-834	470	6	on	on	ADP
cana-834	470	7	data	datum	NOUN
cana-834	470	8	mining	mining	NOUN
cana-834	470	9	(	(	PUNCT
cana-834	470	10	icdm	icdm	NOUN
cana-834	470	11	)	)	PUNCT
cana-834	470	12	*	*	SYM
cana-834	470	13	,	,	PUNCT
cana-834	470	14	1	1	NUM
cana-834	470	15	-	-	SYM
cana-834	470	16	10	10	NUM
cana-834	470	17	.	.	PUNCT
cana-834	471	1	https://doi.org/10.1109/icdm.2017.20	https://doi.org/10.1109/icdm.2017.20	ADJ
cana-834	472	1	[	[	X
cana-834	472	2	13	13	NUM
cana-834	472	3	]	]	X
cana-834	472	4	al	al	PROPN
cana-834	472	5	-	-	PUNCT
cana-834	472	6	karaki	karaki	PROPN
cana-834	472	7	,	,	PUNCT
cana-834	472	8	j.	j.	PROPN
cana-834	472	9	n.	n.	PROPN
cana-834	472	10	,	,	PUNCT
cana-834	472	11	&	&	CCONJ
cana-834	472	12	kamal	kamal	PROPN
cana-834	472	13	,	,	PUNCT
cana-834	472	14	a.	a.	PROPN
cana-834	472	15	e.	e.	PROPN
cana-834	472	16	(	(	PUNCT
cana-834	472	17	2004	2004	NUM
cana-834	472	18	)	)	PUNCT
cana-834	472	19	.	.	PUNCT
cana-834	473	1	routing	route	VERB
cana-834	473	2	techniques	technique	NOUN
cana-834	473	3	in	in	ADP
cana-834	473	4	wireless	wireless	ADJ
cana-834	473	5	sensor	sensor	NOUN
cana-834	473	6	networks	network	NOUN
cana-834	473	7	:	:	PUNCT
cana-834	473	8	a	a	DET
cana-834	473	9	survey	survey	NOUN
cana-834	473	10	.	.	PUNCT
cana-834	474	1	*	*	PUNCT
cana-834	474	2	ieee	ieee	PROPN
cana-834	474	3	wireless	wireless	NOUN
cana-834	474	4	communications	communication	NOUN
cana-834	474	5	*	*	PUNCT
cana-834	474	6	,	,	PUNCT
cana-834	474	7	11(6	11(6	NUM
cana-834	474	8	)	)	PUNCT
cana-834	474	9	,	,	PUNCT
cana-834	474	10	6	6	NUM
cana-834	474	11	-	-	SYM
cana-834	474	12	28	28	NUM
cana-834	474	13	.	.	PUNCT
cana-834	475	1	https://doi.org/10.1109/mwc.2004.1368893	https://doi.org/10.1109/mwc.2004.1368893	PROPN
cana-834	475	2	[	[	X
cana-834	475	3	14	14	NUM
cana-834	475	4	]	]	X
cana-834	475	5	borgatti	borgatti	PROPN
cana-834	475	6	,	,	PUNCT
cana-834	475	7	s.	s.	PROPN
cana-834	475	8	p.	p.	PROPN
cana-834	475	9	,	,	PUNCT
cana-834	475	10	&	&	CCONJ
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cana-834	475	12	,	,	PUNCT
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cana-834	475	15	(	(	PUNCT
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cana-834	475	17	)	)	PUNCT
cana-834	475	18	.	.	PUNCT
cana-834	476	1	a	a	DET
cana-834	476	2	graph	graph	NOUN
cana-834	476	3	-	-	PUNCT
cana-834	476	4	theoretic	theoretic	NOUN
cana-834	476	5	perspective	perspective	NOUN
cana-834	476	6	on	on	ADP
cana-834	476	7	centrality	centrality	NOUN
cana-834	476	8	.	.	PUNCT
cana-834	477	1	*	*	PUNCT
cana-834	477	2	social	social	ADJ
cana-834	477	3	networks	network	NOUN
cana-834	477	4	*	*	PUNCT
cana-834	477	5	,	,	PUNCT
cana-834	477	6	28(4	28(4	NUM
cana-834	477	7	)	)	PUNCT
cana-834	477	8	,	,	PUNCT
cana-834	477	9	466	466	NUM
cana-834	477	10	-	-	SYM
cana-834	477	11	484	484	NUM
cana-834	477	12	.	.	PUNCT
cana-834	478	1	https://doi.org/10.1016/j.socnet.2005.11.005	https://doi.org/10.1016/j.socnet.2005.11.005	PROPN
cana-834	478	2	[	[	X
cana-834	478	3	15	15	NUM
cana-834	478	4	]	]	X
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cana-834	478	6	,	,	PUNCT
cana-834	478	7	b.	b.	PROPN
cana-834	478	8	;	;	PUNCT
cana-834	478	9	de	de	PROPN
cana-834	478	10	,	,	PUNCT
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cana-834	478	14	of	of	ADP
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cana-834	478	17	in	in	ADP
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cana-834	478	19	field	field	NOUN
cana-834	478	20	.	.	PUNCT
cana-834	479	1	int	int	NOUN
cana-834	479	2	.	.	PUNCT
cana-834	480	1	j.	j.	PROPN
cana-834	480	2	sci	sci	PROPN
cana-834	480	3	.	.	PUNCT
cana-834	481	1	res	res	PROPN
cana-834	481	2	.	.	PUNCT
cana-834	482	1	comput	comput	PROPN
cana-834	482	2	.	.	PUNCT
cana-834	483	1	sci	sci	PROPN
cana-834	483	2	.	.	PUNCT
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cana-834	483	4	.	.	PROPN
cana-834	483	5	inf	inf	PROPN
cana-834	483	6	.	.	PUNCT
cana-834	484	1	technol	technol	PROPN
cana-834	484	2	.	.	PROPN
cana-834	484	3	2017	2017	NUM
cana-834	484	4	,	,	PUNCT
cana-834	484	5	2	2	NUM
cana-834	484	6	,	,	PUNCT
cana-834	484	7	751–759	751–759	NUM
cana-834	484	8	.	.	PUNCT
cana-834	485	1	[	[	X
cana-834	485	2	16	16	NUM
cana-834	485	3	]	]	X
cana-834	485	4	bisen	bisen	NOUN
cana-834	485	5	,	,	PUNCT
cana-834	485	6	s.k	s.k	PROPN
cana-834	485	7	.	.	PROPN
cana-834	485	8	application	application	NOUN
cana-834	485	9	of	of	ADP
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cana-834	485	12	in	in	ADP
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cana-834	485	15	.	.	PUNCT
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cana-834	486	2	.	.	PUNCT
cana-834	487	1	j.	j.	PROPN
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cana-834	487	3	.	.	PUNCT
cana-834	488	1	res	res	PROPN
cana-834	488	2	.	.	PROPN
cana-834	489	1	manag	manag	PROPN
cana-834	489	2	.	.	PROPN
cana-834	490	1	2017	2017	NUM
cana-834	490	2	,	,	PUNCT
cana-834	490	3	5	5	NUM
cana-834	490	4	,	,	PUNCT
cana-834	490	5	10–12	10–12	NUM
cana-834	490	6	.	.	PUNCT
cana-834	491	1	[	[	X
cana-834	491	2	17	17	NUM
cana-834	491	3	]	]	PUNCT
cana-834	491	4	durand	durand	NOUN
cana-834	491	5	,	,	PUNCT
cana-834	491	6	g.	g.	PROPN
cana-834	491	7	;	;	PUNCT
cana-834	491	8	belacel	belacel	PROPN
cana-834	491	9	,	,	PUNCT
cana-834	491	10	n.	n.	NOUN
cana-834	491	11	;	;	PUNCT
cana-834	491	12	laplante	laplante	PROPN
cana-834	491	13	,	,	PUNCT
cana-834	491	14	f.	f.	PROPN
cana-834	491	15	graph	graph	PROPN
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cana-834	491	23	.	.	PUNCT
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cana-834	492	2	.	.	PUNCT
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cana-834	493	2	.	.	PROPN
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cana-834	493	4	,	,	PUNCT
cana-834	493	5	251	251	NUM
cana-834	493	6	,	,	PUNCT
cana-834	493	7	10–21	10–21	NUM
cana-834	493	8	.	.	PUNCT
cana-834	494	1	[	[	X
cana-834	494	2	18	18	NUM
cana-834	494	3	]	]	X
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cana-834	494	5	,	,	PUNCT
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cana-834	494	7	.	.	PROPN
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cana-834	494	10	+	+	CCONJ
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cana-834	494	12	cities	city	NOUN
cana-834	494	13	:	:	PUNCT
cana-834	494	14	graph	graph	NOUN
cana-834	494	15	signals	signal	NOUN
cana-834	494	16	of	of	ADP
cana-834	494	17	urban	urban	ADJ
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cana-834	494	19	pollution	pollution	NOUN
cana-834	494	20	.	.	PUNCT
cana-834	495	1	ieee	ieee	NOUN
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cana-834	495	4	.	.	PUNCT
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cana-834	496	2	.	.	PROPN
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cana-834	496	4	,	,	PUNCT
cana-834	496	5	31	31	NUM
cana-834	496	6	,	,	PUNCT
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cana-834	496	8	.	.	PUNCT
cana-834	497	1	[	[	X
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cana-834	497	24	:	:	PUNCT
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cana-834	501	44	:	:	PUNCT
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cana-834	504	10	:	:	PUNCT
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cana-834	506	2	.	.	PUNCT
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cana-834	512	19	;	;	PUNCT
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cana-834	513	12	:	:	PUNCT
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cana-834	514	4	,	,	PUNCT
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cana-834	514	8	.	.	PUNCT
cana-834	515	1	[	[	X
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cana-834	515	7	;	;	PUNCT
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cana-834	515	18	:	:	PUNCT
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cana-834	520	2	,	,	PUNCT
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cana-834	521	1	[	[	X
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cana-834	521	5	-	-	PUNCT
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cana-834	521	7	,	,	PUNCT
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cana-834	525	2	.	.	PUNCT
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cana-834	526	2	,	,	PUNCT
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cana-834	526	4	,	,	PUNCT
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