id	sid	tid	token	lemma	pos
cana-1336	1	1	communications	communication	NOUN
cana-1336	1	2	on	on	ADP
cana-1336	1	3	applied	apply	VERB
cana-1336	1	4	nonlinear	nonlinear	ADJ
cana-1336	1	5	analysis	analysis	NOUN
cana-1336	1	6	issn	issn	NOUN
cana-1336	1	7	:	:	PUNCT
cana-1336	1	8	1074	1074	NUM
cana-1336	1	9	-	-	PUNCT
cana-1336	1	10	133x	133x	NUM
cana-1336	1	11	vol	vol	NOUN
cana-1336	1	12	31	31	NUM
cana-1336	1	13	no	no	NOUN
cana-1336	1	14	.	.	PUNCT
cana-1336	2	1	6s	6s	NUM
cana-1336	2	2	(	(	PUNCT
cana-1336	2	3	2024	2024	NUM
cana-1336	2	4	)	)	PUNCT
cana-1336	2	5	724	724	NUM
cana-1336	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1336	3	1	the	the	DET
cana-1336	3	2	carmichael	carmichael	PROPN
cana-1336	3	3	function	function	NOUN
cana-1336	3	4	in	in	ADP
cana-1336	3	5	graph	graph	NOUN
cana-1336	3	6	theory	theory	NOUN
cana-1336	3	7	nagham	nagham	NOUN
cana-1336	3	8	a.	a.	NOUN
cana-1336	3	9	hameed1	hameed1	PROPN
cana-1336	4	1	*	*	PUNCT
cana-1336	4	2	,	,	PUNCT
cana-1336	4	3	faez	faez	PROPN
cana-1336	4	4	a.	a.	PROPN
cana-1336	4	5	al	al	PROPN
cana-1336	4	6	-	-	PUNCT
cana-1336	4	7	maamori2	maamori2	PROPN
cana-1336	4	8	1department	1department	NUM
cana-1336	4	9	of	of	ADP
cana-1336	4	10	mathematics	mathematics	PROPN
cana-1336	4	11	,	,	PUNCT
cana-1336	4	12	college	college	NOUN
cana-1336	4	13	of	of	ADP
cana-1336	4	14	education	education	NOUN
cana-1336	4	15	for	for	ADP
cana-1336	4	16	pure	pure	ADJ
cana-1336	4	17	sciences	science	NOUN
cana-1336	4	18	,	,	PUNCT
cana-1336	4	19	university	university	NOUN
cana-1336	4	20	of	of	ADP
cana-1336	4	21	babylon	babylon	PROPN
cana-1336	4	22	,	,	PUNCT
cana-1336	4	23	babylon	babylon	PROPN
cana-1336	4	24	,	,	PUNCT
cana-1336	4	25	iraq	iraq	PROPN
cana-1336	4	26	.	.	PUNCT
cana-1336	5	1	1nagham.hameed.pure327@student.uobabylon.edu.iq	1nagham.hameed.pure327@student.uobabylon.edu.iq	NUM
cana-1336	5	2	2department	2department	NUM
cana-1336	5	3	of	of	ADP
cana-1336	5	4	security	security	NOUN
cana-1336	5	5	,	,	PUNCT
cana-1336	5	6	collage	collage	NOUN
cana-1336	5	7	of	of	ADP
cana-1336	5	8	information	information	NOUN
cana-1336	5	9	technology	technology	NOUN
cana-1336	5	10	,	,	PUNCT
cana-1336	5	11	university	university	NOUN
cana-1336	5	12	of	of	ADP
cana-1336	5	13	babylon	babylon	PROPN
cana-1336	5	14	,	,	PUNCT
cana-1336	5	15	babylon	babylon	PROPN
cana-1336	5	16	,	,	PUNCT
cana-1336	5	17	iraq	iraq	PROPN
cana-1336	5	18	.	.	PUNCT
cana-1336	6	1	2faez@itnetuobabylon.edu.iq	2faez@itnetuobabylon.edu.iq	NUM
cana-1336	6	2	article	article	NOUN
cana-1336	6	3	history	history	NOUN
cana-1336	6	4	:	:	PUNCT
cana-1336	6	5	received	receive	VERB
cana-1336	6	6	:	:	PUNCT
cana-1336	6	7	30	30	NUM
cana-1336	6	8	-	-	SYM
cana-1336	6	9	05	05	NUM
cana-1336	6	10	-	-	PUNCT
cana-1336	6	11	2024	2024	NUM
cana-1336	6	12	revised	revise	VERB
cana-1336	6	13	:	:	PUNCT
cana-1336	6	14	29	29	NUM
cana-1336	6	15	-	-	SYM
cana-1336	6	16	06	06	NUM
cana-1336	6	17	-	-	PUNCT
cana-1336	6	18	2024	2024	NUM
cana-1336	6	19	accepted	accept	VERB
cana-1336	6	20	:	:	PUNCT
cana-1336	6	21	20	20	NUM
cana-1336	6	22	-	-	SYM
cana-1336	6	23	07	07	NUM
cana-1336	6	24	-	-	PUNCT
cana-1336	6	25	2024	2024	NUM
cana-1336	6	26	abstract	abstract	NOUN
cana-1336	6	27	:	:	PUNCT
cana-1336	6	28	one	one	NUM
cana-1336	6	29	of	of	ADP
cana-1336	6	30	the	the	DET
cana-1336	6	31	most	most	ADV
cana-1336	6	32	flourishing	flourishing	ADJ
cana-1336	6	33	branches	branch	NOUN
cana-1336	6	34	of	of	ADP
cana-1336	6	35	modern	modern	ADJ
cana-1336	6	36	mathematics	mathematic	NOUN
cana-1336	6	37	is	be	AUX
cana-1336	6	38	the	the	DET
cana-1336	6	39	application	application	NOUN
cana-1336	6	40	of	of	ADP
cana-1336	6	41	graph	graph	NOUN
cana-1336	6	42	theory	theory	NOUN
cana-1336	6	43	in	in	ADP
cana-1336	6	44	graph	graph	NOUN
cana-1336	6	45	theory	theory	NOUN
cana-1336	6	46	.	.	PUNCT
cana-1336	7	1	this	this	DET
cana-1336	7	2	work	work	NOUN
cana-1336	7	3	presents	present	VERB
cana-1336	7	4	innovative	innovative	ADJ
cana-1336	7	5	graph	graph	NOUN
cana-1336	7	6	which	which	PRON
cana-1336	7	7	is	be	AUX
cana-1336	7	8	an	an	DET
cana-1336	7	9	application	application	NOUN
cana-1336	7	10	of	of	ADP
cana-1336	7	11	some	some	DET
cana-1336	7	12	arithmetical	arithmetical	ADJ
cana-1336	7	13	functions	function	NOUN
cana-1336	7	14	and	and	CCONJ
cana-1336	7	15	this	this	PRON
cana-1336	7	16	called	call	VERB
cana-1336	7	17	the	the	DET
cana-1336	7	18	carmichael	carmichael	PROPN
cana-1336	7	19	function	function	NOUN
cana-1336	7	20	graph	graph	NOUN
cana-1336	7	21	\lambda(g	\lambda(g	NOUN
cana-1336	7	22	)	)	PUNCT
cana-1336	7	23	.	.	PUNCT
cana-1336	8	1	using	use	VERB
cana-1336	8	2	a	a	DET
cana-1336	8	3	new	new	ADJ
cana-1336	8	4	technique	technique	NOUN
cana-1336	8	5	in	in	ADP
cana-1336	8	6	order	order	NOUN
cana-1336	8	7	to	to	PART
cana-1336	8	8	calculate	calculate	VERB
cana-1336	8	9	the	the	DET
cana-1336	8	10	results	result	NOUN
cana-1336	8	11	to	to	PART
cana-1336	8	12	be	be	AUX
cana-1336	8	13	found	find	VERB
cana-1336	8	14	in	in	ADP
cana-1336	8	15	this	this	DET
cana-1336	8	16	research	research	NOUN
cana-1336	8	17	.	.	PUNCT
cana-1336	9	1	moreover	moreover	ADV
cana-1336	9	2	this	this	DET
cana-1336	9	3	work	work	NOUN
cana-1336	9	4	considered	consider	VERB
cana-1336	9	5	a	a	DET
cana-1336	9	6	new	new	ADJ
cana-1336	9	7	kind	kind	NOUN
cana-1336	9	8	of	of	ADP
cana-1336	9	9	application	application	NOUN
cana-1336	9	10	of	of	ADP
cana-1336	9	11	some	some	DET
cana-1336	9	12	arithmetical	arithmetical	ADJ
cana-1336	9	13	functions	function	NOUN
cana-1336	9	14	in	in	ADP
cana-1336	9	15	graph	graph	NOUN
cana-1336	9	16	theory	theory	NOUN
cana-1336	9	17	.	.	PUNCT
cana-1336	10	1	so	so	ADV
cana-1336	10	2	for	for	ADP
cana-1336	10	3	this	this	DET
cana-1336	10	4	purpose	purpose	NOUN
cana-1336	10	5	,	,	PUNCT
cana-1336	10	6	many	many	ADJ
cana-1336	10	7	basic	basic	ADJ
cana-1336	10	8	properties	property	NOUN
cana-1336	10	9	in	in	ADP
cana-1336	10	10	graph	graph	NOUN
cana-1336	10	11	theory	theory	NOUN
cana-1336	10	12	,	,	PUNCT
cana-1336	10	13	including	include	VERB
cana-1336	10	14	finding	find	VERB
cana-1336	10	15	the	the	DET
cana-1336	10	16	characteristics	characteristic	NOUN
cana-1336	10	17	of	of	ADP
cana-1336	10	18	the	the	DET
cana-1336	10	19	independence	independence	NOUN
cana-1336	10	20	number	number	NOUN
cana-1336	10	21	,	,	PUNCT
cana-1336	10	22	domination	domination	NOUN
cana-1336	10	23	number	number	NOUN
cana-1336	10	24	,	,	PUNCT
cana-1336	10	25	clique	clique	ADJ
cana-1336	10	26	number	number	NOUN
cana-1336	10	27	and	and	CCONJ
cana-1336	10	28	chromatic	chromatic	ADJ
cana-1336	10	29	number	number	NOUN
cana-1336	10	30	of	of	ADP
cana-1336	10	31	this	this	DET
cana-1336	10	32	graph	graph	NOUN
cana-1336	10	33	have	have	AUX
cana-1336	10	34	been	be	AUX
cana-1336	10	35	calculated	calculate	VERB
cana-1336	10	36	.	.	PUNCT
cana-1336	11	1	keywords	keyword	NOUN
cana-1336	11	2	:	:	PUNCT
cana-1336	11	3	independence	independence	NOUN
cana-1336	11	4	number	number	NOUN
cana-1336	11	5	,	,	PUNCT
cana-1336	11	6	domination	domination	NOUN
cana-1336	11	7	number,\	number,\	ADP
cana-1336	11	8	clique	clique	NOUN
cana-1336	11	9	number	number	NOUN
cana-1336	11	10	and	and	CCONJ
cana-1336	11	11	chromatic	chromatic	ADJ
cana-1336	11	12	number	number	NOUN
cana-1336	11	13	,	,	PUNCT
cana-1336	11	14	carmichael	carmichael	PROPN
cana-1336	11	15	function	function	PROPN
cana-1336	11	16	graph\	graph\	PUNCT
cana-1336	11	17	g\	g\	PROPN
cana-1336	11	18	,	,	PUNCT
cana-1336	11	19	\	\	PROPN
cana-1336	11	20	clique	clique	NOUN
cana-1336	11	21	number	number	NOUN
cana-1336	11	22	and	and	CCONJ
cana-1336	11	23	chromatic	chromatic	ADJ
cana-1336	11	24	number	number	NOUN
cana-1336	11	25	.	.	PUNCT
cana-1336	12	1	1	1	X
cana-1336	12	2	.	.	X
cana-1336	12	3	introduction	introduction	NOUN
cana-1336	12	4	this	this	DET
cana-1336	12	5	paper	paper	NOUN
cana-1336	12	6	introduce	introduce	VERB
cana-1336	12	7	an	an	DET
cana-1336	12	8	application	application	NOUN
cana-1336	12	9	of	of	ADP
cana-1336	12	10	some	some	DET
cana-1336	12	11	arithmetical	arithmetical	ADJ
cana-1336	12	12	functions	function	NOUN
cana-1336	12	13	in	in	ADP
cana-1336	12	14	graph	graph	NOUN
cana-1336	12	15	theory	theory	NOUN
cana-1336	12	16	.	.	PUNCT
cana-1336	13	1	specially	specially	ADV
cana-1336	13	2	were	be	AUX
cana-1336	13	3	us	we	PRON
cana-1336	13	4	the	the	DET
cana-1336	13	5	application	application	NOUN
cana-1336	13	6	arithmetical	arithmetical	ADJ
cana-1336	13	7	functions	function	NOUN
cana-1336	13	8	has	have	AUX
cana-1336	13	9	been	be	AUX
cana-1336	13	10	applied	apply	VERB
cana-1336	13	11	on	on	ADP
cana-1336	13	12	the	the	DET
cana-1336	13	13	carmichael	carmichael	PROPN
cana-1336	13	14	function	function	PROPN
cana-1336	13	15	graph	graph	NOUN
cana-1336	13	16	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	13	17	)	)	PUNCT
cana-1336	13	18	in	in	ADP
cana-1336	13	19	simple	simple	ADJ
cana-1336	13	20	,	,	PUNCT
cana-1336	13	21	nontrivial	nontrivial	NOUN
cana-1336	13	22	,	,	PUNCT
cana-1336	13	23	finite	finite	VERB
cana-1336	13	24	an	an	DET
cana-1336	13	25	undirected	undirected	ADJ
cana-1336	13	26	.	.	PUNCT
cana-1336	14	1	in	in	ADP
cana-1336	14	2	general	general	ADJ
cana-1336	14	3	the	the	DET
cana-1336	14	4	application	application	NOUN
cana-1336	14	5	of	of	ADP
cana-1336	14	6	some	some	DET
cana-1336	14	7	arithmetical	arithmetical	ADJ
cana-1336	14	8	functions	function	NOUN
cana-1336	14	9	has	have	AUX
cana-1336	14	10	been	be	AUX
cana-1336	14	11	studied	study	VERB
cana-1336	14	12	from	from	ADP
cana-1336	14	13	several	several	ADJ
cana-1336	14	14	authors	author	NOUN
cana-1336	14	15	,	,	PUNCT
cana-1336	14	16	for	for	ADP
cana-1336	14	17	instant	instant	NOUN
cana-1336	14	18	the	the	DET
cana-1336	14	19	reader	reader	NOUN
cana-1336	14	20	can	can	AUX
cana-1336	14	21	see	see	VERB
cana-1336	14	22	(	(	PUNCT
cana-1336	14	23	m.	m.	NOUN
cana-1336	14	24	a.	a.	PROPN
cana-1336	14	25	seoud	seoud	PROPN
cana-1336	14	26	,	,	PUNCT
cana-1336	14	27	essam	essam	PROPN
cana-1336	14	28	el	el	PROPN
cana-1336	14	29	-	-	PUNCT
cana-1336	14	30	seidy	seidy	VERB
cana-1336	14	31	and	and	CCONJ
cana-1336	14	32	ahmed	ahmed	PROPN
cana-1336	14	33	a.	a.	PROPN
cana-1336	14	34	omran	omran	PROPN
cana-1336	14	35	studied	study	VERB
cana-1336	14	36	independence	independence	NOUN
cana-1336	14	37	in	in	ADP
cana-1336	14	38	isosceles	isoscele	NOUN
cana-1336	14	39	triangular	triangular	NOUN
cana-1336	14	40	chessboard	chessboard	NOUN
cana-1336	14	41	,	,	PUNCT
cana-1336	14	42	in	in	ADP
cana-1336	14	43	2012	2012	NUM
cana-1336	14	44	,	,	PUNCT
cana-1336	14	45	essam	essam	PROPN
cana-1336	14	46	elseidy	elseidy	PROPN
cana-1336	14	47	,	,	PUNCT
cana-1336	14	48	ahmed	ahmed	PROPN
cana-1336	14	49	a.	a.	PROPN
cana-1336	14	50	omran	omran	PROPN
cana-1336	14	51	studied	study	VERB
cana-1336	14	52	domination	domination	NOUN
cana-1336	14	53	in	in	ADP
cana-1336	14	54	rhombus	rhombus	ADJ
cana-1336	14	55	chessboard	chessboard	NOUN
cana-1336	14	56	in	in	ADP
cana-1336	14	57	2014	2014	NUM
cana-1336	14	58	and	and	CCONJ
cana-1336	15	1	sanaa	sanaa	PROPN
cana-1336	15	2	kadum	kadum	PROPN
cana-1336	15	3	kamel	kamel	PROPN
cana-1336	15	4	yaseen	yaseen	PROPN
cana-1336	15	5	,	,	PUNCT
cana-1336	15	6	faez	faez	PROPN
cana-1336	15	7	a.al	a.al	PROPN
cana-1336	15	8	-	-	NOUN
cana-1336	15	9	maamori	maamori	NOUN
cana-1336	15	10	and	and	CCONJ
cana-1336	15	11	ahmed	ahmed	PROPN
cana-1336	15	12	abed	abed	PROPN
cana-1336	15	13	ali	ali	PROPN
cana-1336	15	14	omran	omran	PROPN
cana-1336	15	15	studied	study	VERB
cana-1336	15	16	some	some	DET
cana-1336	15	17	kinds	kind	NOUN
cana-1336	15	18	of	of	ADP
cana-1336	15	19	mobius	mobius	NOUN
cana-1336	15	20	function	function	NOUN
cana-1336	15	21	graphs	graph	NOUN
cana-1336	15	22	in	in	ADP
cana-1336	15	23	2022	2022	NUM
cana-1336	15	24	)	)	PUNCT
cana-1336	15	25	.	.	PUNCT
cana-1336	16	1	the	the	DET
cana-1336	16	2	graph	graph	NOUN
cana-1336	16	3	g	g	NOUN
cana-1336	16	4	is	be	AUX
cana-1336	16	5	consists	consist	NOUN
cana-1336	16	6	of	of	ADP
cana-1336	16	7	a	a	DET
cana-1336	16	8	non	non	ADJ
cana-1336	16	9	-	-	ADJ
cana-1336	16	10	empty	empty	ADJ
cana-1336	16	11	finite	finite	NOUN
cana-1336	16	12	set	set	VERB
cana-1336	16	13	v(g	v(g	NOUN
cana-1336	16	14	)	)	PUNCT
cana-1336	16	15	of	of	ADP
cana-1336	16	16	elements	element	NOUN
cana-1336	16	17	called	call	VERB
cana-1336	16	18	vertices	vertex	NOUN
cana-1336	16	19	,	,	PUNCT
cana-1336	16	20	and	and	CCONJ
cana-1336	16	21	a	a	DET
cana-1336	16	22	finite	finite	ADJ
cana-1336	16	23	family	family	NOUN
cana-1336	16	24	e(g	e(g	PROPN
cana-1336	16	25	)	)	PUNCT
cana-1336	16	26	of	of	ADP
cana-1336	16	27	unordered	unordered	ADJ
cana-1336	16	28	pairs	pair	NOUN
cana-1336	16	29	of	of	ADP
cana-1336	16	30	(	(	PUNCT
cana-1336	16	31	not	not	PART
cana-1336	16	32	necessarily	necessarily	ADV
cana-1336	16	33	distinct	distinct	ADJ
cana-1336	16	34	)	)	PUNCT
cana-1336	16	35	elements	element	NOUN
cana-1336	16	36	of	of	ADP
cana-1336	16	37	v(g	v(g	NOUN
cana-1336	16	38	)	)	PUNCT
cana-1336	16	39	called	call	VERB
cana-1336	16	40	edges	edge	NOUN
cana-1336	16	41	[	[	X
cana-1336	16	42	6].we	6].we	PRON
cana-1336	16	43	called	call	VERB
cana-1336	16	44	that	that	SCONJ
cana-1336	16	45	a	a	DET
cana-1336	16	46	set	set	NOUN
cana-1336	16	47	d	d	NOUN
cana-1336	16	48	⊆	⊆	NUM
cana-1336	16	49	v	v	NOUN
cana-1336	16	50	is	be	AUX
cana-1336	16	51	a	a	DET
cana-1336	16	52	dominating	dominating	NOUN
cana-1336	16	53	set	set	NOUN
cana-1336	16	54	of	of	ADP
cana-1336	16	55	g	g	PROPN
cana-1336	16	56	if	if	SCONJ
cana-1336	16	57	every	every	DET
cana-1336	16	58	vertex	vertex	NOUN
cana-1336	16	59	in	in	ADP
cana-1336	16	60	v	v	NOUN
cana-1336	16	61	−	−	PROPN
cana-1336	16	62	d	d	NOUN
cana-1336	16	63	is	be	AUX
cana-1336	16	64	adjacent	adjacent	ADJ
cana-1336	16	65	to	to	ADP
cana-1336	16	66	a	a	DET
cana-1336	16	67	vertex	vertex	NOUN
cana-1336	16	68	in	in	ADP
cana-1336	16	69	d.	d.	PROPN
cana-1336	16	70	and	and	CCONJ
cana-1336	16	71	the	the	DET
cana-1336	16	72	domination	domination	NOUN
cana-1336	16	73	number	number	NOUN
cana-1336	16	74	of	of	ADP
cana-1336	16	75	graph	graph	NOUN
cana-1336	16	76	g	g	NOUN
cana-1336	16	77	,	,	PUNCT
cana-1336	16	78	denoted	denote	VERB
cana-1336	16	79	by	by	ADP
cana-1336	16	80	γ(g	γ(g	PROPN
cana-1336	16	81	)	)	PUNCT
cana-1336	16	82	,	,	PUNCT
cana-1336	16	83	is	be	AUX
cana-1336	16	84	the	the	DET
cana-1336	16	85	minimum	minimum	ADJ
cana-1336	16	86	cardinality	cardinality	NOUN
cana-1336	16	87	of	of	ADP
cana-1336	16	88	a	a	DET
cana-1336	16	89	dominating	dominating	NOUN
cana-1336	16	90	set	set	VERB
cana-1336	16	91	in	in	ADP
cana-1336	16	92	graph	graph	NOUN
cana-1336	16	93	g	g	PROPN
cana-1336	17	1	[	[	X
cana-1336	17	2	4	4	NUM
cana-1336	17	3	]	]	PUNCT
cana-1336	17	4	.	.	PUNCT
cana-1336	18	1	the	the	DET
cana-1336	18	2	independent	independent	ADJ
cana-1336	18	3	set	set	NOUN
cana-1336	18	4	is	be	AUX
cana-1336	18	5	a	a	DET
cana-1336	18	6	set	set	NOUN
cana-1336	18	7	of	of	ADP
cana-1336	18	8	vertices	vertex	NOUN
cana-1336	18	9	in	in	ADP
cana-1336	18	10	a	a	DET
cana-1336	18	11	graph	graph	NOUN
cana-1336	18	12	such	such	ADJ
cana-1336	18	13	that	that	PRON
cana-1336	18	14	are	be	AUX
cana-1336	18	15	said	say	VERB
cana-1336	18	16	to	to	PART
cana-1336	18	17	be	be	AUX
cana-1336	18	18	independent	independent	ADJ
cana-1336	18	19	if	if	SCONJ
cana-1336	18	20	no	no	DET
cana-1336	18	21	two	two	NUM
cana-1336	18	22	of	of	ADP
cana-1336	18	23	them	they	PRON
cana-1336	18	24	are	be	AUX
cana-1336	18	25	adjacent	adjacent	ADJ
cana-1336	18	26	.	.	PUNCT
cana-1336	19	1	the	the	DET
cana-1336	19	2	cardinality	cardinality	NOUN
cana-1336	19	3	of	of	ADP
cana-1336	19	4	such	such	DET
cana-1336	19	5	a	a	DET
cana-1336	19	6	biggest	big	ADJ
cana-1336	19	7	independent	independent	ADJ
cana-1336	19	8	set	set	NOUN
cana-1336	19	9	is	be	AUX
cana-1336	19	10	called	call	VERB
cana-1336	19	11	the	the	DET
cana-1336	19	12	independence	independence	NOUN
cana-1336	19	13	number	number	NOUN
cana-1336	19	14	of	of	ADP
cana-1336	19	15	the	the	DET
cana-1336	19	16	graph	graph	NOUN
cana-1336	19	17	and	and	CCONJ
cana-1336	19	18	is	be	AUX
cana-1336	19	19	denoted	denote	VERB
cana-1336	19	20	by	by	ADP
cana-1336	19	21	β[2	β[2	PROPN
cana-1336	19	22	]	]	PUNCT
cana-1336	19	23	.	.	PUNCT
cana-1336	20	1	we	we	PRON
cana-1336	20	2	called	call	VERB
cana-1336	20	3	that	that	SCONJ
cana-1336	20	4	a	a	DET
cana-1336	20	5	clique	clique	NOUN
cana-1336	20	6	of	of	ADP
cana-1336	20	7	a	a	DET
cana-1336	20	8	graph	graph	NOUN
cana-1336	20	9	is	be	AUX
cana-1336	20	10	its	its	PRON
cana-1336	20	11	maximal	maximal	ADJ
cana-1336	20	12	complete	complete	ADJ
cana-1336	20	13	sub	sub	NOUN
cana-1336	20	14	graph	graph	NOUN
cana-1336	20	15	.	.	PUNCT
cana-1336	21	1	the	the	DET
cana-1336	21	2	clique	clique	ADJ
cana-1336	21	3	number	number	NOUN
cana-1336	21	4	𝜔(𝐺	𝜔(𝐺	PROPN
cana-1336	21	5	)	)	PUNCT
cana-1336	21	6	of	of	ADP
cana-1336	21	7	a	a	DET
cana-1336	21	8	graph	graph	NOUN
cana-1336	21	9	is	be	AUX
cana-1336	21	10	the	the	DET
cana-1336	21	11	number	number	NOUN
cana-1336	21	12	of	of	ADP
cana-1336	21	13	graph	graph	NOUN
cana-1336	21	14	vertices	vertex	NOUN
cana-1336	21	15	in	in	ADP
cana-1336	21	16	the	the	DET
cana-1336	21	17	largest	large	ADJ
cana-1336	21	18	clique	clique	NOUN
cana-1336	21	19	of	of	ADP
cana-1336	21	20	g[3	g[3	PROPN
cana-1336	21	21	]	]	X
cana-1336	21	22	.	.	PUNCT
cana-1336	22	1	the	the	DET
cana-1336	22	2	set	set	NOUN
cana-1336	22	3	of	of	ADP
cana-1336	22	4	coloring	coloring	NOUN
cana-1336	22	5	is	be	AUX
cana-1336	22	6	called	call	VERB
cana-1336	22	7	the	the	DET
cana-1336	22	8	chromatic	chromatic	ADJ
cana-1336	22	9	number	number	NOUN
cana-1336	22	10	𝜒(𝐺	𝜒(𝐺	NOUN
cana-1336	22	11	)	)	PUNCT
cana-1336	22	12	of	of	ADP
cana-1336	22	13	a	a	DET
cana-1336	22	14	graph	graph	NOUN
cana-1336	22	15	is	be	AUX
cana-1336	22	16	the	the	DET
cana-1336	22	17	least	least	ADJ
cana-1336	22	18	number	number	NOUN
cana-1336	22	19	of	of	ADP
cana-1336	22	20	colors	color	NOUN
cana-1336	22	21	required	require	VERB
cana-1336	22	22	for	for	ADP
cana-1336	22	23	a	a	DET
cana-1336	22	24	proper	proper	ADJ
cana-1336	22	25	vertex	vertex	NOUN
cana-1336	22	26	coloring	coloring	NOUN
cana-1336	22	27	of	of	ADP
cana-1336	22	28	g	g	PROPN
cana-1336	23	1	[	[	X
cana-1336	23	2	1	1	NUM
cana-1336	23	3	]	]	PUNCT
cana-1336	23	4	.	.	PUNCT
cana-1336	24	1	the	the	DET
cana-1336	24	2	carmichael	carmichael	PROPN
cana-1336	24	3	function	function	NOUN
cana-1336	24	4	in	in	ADP
cana-1336	24	5	is	be	AUX
cana-1336	24	6	a	a	DET
cana-1336	24	7	well	well	ADV
cana-1336	24	8	-	-	PUNCT
cana-1336	24	9	known	know	VERB
cana-1336	24	10	function	function	NOUN
cana-1336	24	11	in	in	ADP
cana-1336	24	12	number	number	NOUN
cana-1336	24	13	theory	theory	NOUN
cana-1336	24	14	and	and	CCONJ
cana-1336	24	15	it	it	PRON
cana-1336	24	16	is	be	AUX
cana-1336	24	17	defined	define	VERB
cana-1336	24	18	as	as	ADP
cana-1336	24	19	𝜆(1	𝜆(1	NOUN
cana-1336	24	20	)	)	PUNCT
cana-1336	24	21	=	=	SYM
cana-1336	25	1	1	1	NUM
cana-1336	25	2	and	and	CCONJ
cana-1336	25	3	if	if	SCONJ
cana-1336	25	4	𝑛	𝑛	PROPN
cana-1336	25	5	>	>	X
cana-1336	25	6	1	1	NUM
cana-1336	25	7	,	,	PUNCT
cana-1336	25	8	we	we	PRON
cana-1336	25	9	write	write	VERB
cana-1336	25	10	if	if	SCONJ
cana-1336	25	11	mailto:2faez@itnetuobabylon.edu.iq	mailto:2faez@itnetuobabylon.edu.iq	NOUN
cana-1336	25	12	communications	communication	NOUN
cana-1336	25	13	on	on	ADP
cana-1336	25	14	applied	apply	VERB
cana-1336	25	15	nonlinear	nonlinear	ADJ
cana-1336	25	16	analysis	analysis	NOUN
cana-1336	25	17	issn	issn	NOUN
cana-1336	25	18	:	:	PUNCT
cana-1336	25	19	1074	1074	NUM
cana-1336	25	20	-	-	PUNCT
cana-1336	25	21	133x	133x	NUM
cana-1336	25	22	vol	vol	NOUN
cana-1336	25	23	31	31	NUM
cana-1336	25	24	no	no	NOUN
cana-1336	25	25	.	.	PUNCT
cana-1336	26	1	6s	6s	NUM
cana-1336	26	2	(	(	PUNCT
cana-1336	26	3	2024	2024	NUM
cana-1336	26	4	)	)	PUNCT
cana-1336	27	1	725	725	NUM
cana-1336	27	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1336	27	3	𝑛	𝑛	PROPN
cana-1336	27	4	=	=	SYM
cana-1336	27	5	𝑝1	𝑝1	NOUN
cana-1336	27	6	𝛼1	𝛼1	NOUN
cana-1336	27	7	,	,	PUNCT
cana-1336	27	8	𝑝2	𝑝2	NOUN
cana-1336	27	9	𝛼2	𝛼2	PROPN
cana-1336	27	10	…	…	PUNCT
cana-1336	27	11	.	.	PUNCT
cana-1336	28	1	𝑝𝑘	𝑝𝑘	ADP
cana-1336	29	1	𝛼𝑘	𝛼𝑘	NOUN
cana-1336	29	2	then	then	ADV
cana-1336	29	3	𝜆(𝑝𝛼	𝜆(𝑝𝛼	VERB
cana-1336	29	4	)	)	PUNCT
cana-1336	29	5	=	=	PRON
cana-1336	29	6	{	{	PUNCT
cana-1336	29	7	𝑝𝛼	𝑝𝛼	X
cana-1336	29	8	(	(	PUNCT
cana-1336	29	9	𝑝	𝑝	NOUN
cana-1336	29	10	−	−	NOUN
cana-1336	29	11	1	1	NUM
cana-1336	29	12	)	)	PUNCT
cana-1336	29	13	,	,	PUNCT
cana-1336	29	14	𝑖𝑓	𝑖𝑓	PROPN
cana-1336	29	15	𝑝	𝑝	NUM
cana-1336	29	16	≥	≥	NUM
cana-1336	29	17	3	3	NUM
cana-1336	29	18	𝑜𝑟	𝑜𝑟	ADP
cana-1336	29	19	𝛼	𝛼	PROPN
cana-1336	29	20	≤	≤	NUM
cana-1336	29	21	2	2	NUM
cana-1336	29	22	2𝛼−2	2𝛼−2	NUM
cana-1336	29	23	,	,	PUNCT
cana-1336	29	24	𝑖𝑓	𝑖𝑓	NOUN
cana-1336	29	25	𝑝	𝑝	NOUN
cana-1336	29	26	=	=	SYM
cana-1336	29	27	2	2	NUM
cana-1336	29	28	,	,	PUNCT
cana-1336	29	29	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1336	29	30	𝛼	𝛼	PROPN
cana-1336	29	31	≥	≥	NUM
cana-1336	29	32	3	3	NUM
cana-1336	29	33	we	we	PRON
cana-1336	29	34	called	call	VERB
cana-1336	29	35	the	the	DET
cana-1336	29	36	graph	graph	NOUN
cana-1336	29	37	g	g	PROPN
cana-1336	29	38	(	(	PUNCT
cana-1336	29	39	v	v	NOUN
cana-1336	29	40	,	,	PUNCT
cana-1336	29	41	e	e	NOUN
cana-1336	29	42	)	)	PUNCT
cana-1336	29	43	by	by	ADP
cana-1336	29	44	defining	define	VERB
cana-1336	29	45	on	on	ADP
cana-1336	29	46	the	the	DET
cana-1336	29	47	solid	solid	ADJ
cana-1336	29	48	and	and	CCONJ
cana-1336	29	49	tight	tight	ADJ
cana-1336	29	50	arithmetic	arithmetic	ADJ
cana-1336	29	51	function	function	NOUN
cana-1336	29	52	called	call	VERB
cana-1336	29	53	carmichael	carmichael	PROPN
cana-1336	29	54	function	function	PROPN
cana-1336	29	55	graph	graph	NOUN
cana-1336	29	56	𝜆(𝐺)[7	𝜆(𝐺)[7	PROPN
cana-1336	29	57	]	]	X
cana-1336	29	58	.the	.the	PUNCT
cana-1336	30	1	features	feature	NOUN
cana-1336	30	2	of	of	ADP
cana-1336	30	3	number	number	NOUN
cana-1336	30	4	theory	theory	NOUN
cana-1336	30	5	were	be	AUX
cana-1336	30	6	applied	apply	VERB
cana-1336	30	7	in	in	ADP
cana-1336	30	8	graph	graph	NOUN
cana-1336	30	9	theory	theory	NOUN
cana-1336	30	10	to	to	PART
cana-1336	30	11	design	design	VERB
cana-1336	30	12	a	a	DET
cana-1336	30	13	graph	graph	NOUN
cana-1336	30	14	is	be	AUX
cana-1336	30	15	introduced	introduce	VERB
cana-1336	30	16	by	by	ADP
cana-1336	30	17	nathonson	nathonson	PROPN
cana-1336	30	18	in	in	ADP
cana-1336	30	19	1980	1980	NUM
cana-1336	30	20	[	[	X
cana-1336	30	21	5	5	NUM
cana-1336	30	22	]	]	PUNCT
cana-1336	30	23	.	.	PUNCT
cana-1336	31	1	this	this	DET
cana-1336	31	2	section	section	NOUN
cana-1336	31	3	stats	stat	VERB
cana-1336	31	4	the	the	DET
cana-1336	31	5	main	main	ADJ
cana-1336	31	6	results	result	NOUN
cana-1336	31	7	and	and	CCONJ
cana-1336	31	8	started	start	VERB
cana-1336	31	9	with	with	ADP
cana-1336	31	10	:	:	PUNCT
cana-1336	31	11	2	2	X
cana-1336	31	12	.	.	PUNCT
cana-1336	31	13	materials	material	NOUN
cana-1336	31	14	and	and	CCONJ
cana-1336	31	15	methods	method	NOUN
cana-1336	31	16	theorem	theorem	VERB
cana-1336	31	17	2.1	2.1	NUM
cana-1336	31	18	if	if	SCONJ
cana-1336	31	19	𝐺	𝐺	PROPN
cana-1336	31	20	be	be	VERB
cana-1336	31	21	a	a	DET
cana-1336	31	22	carmichael	carmichael	PROPN
cana-1336	31	23	function	function	NOUN
cana-1336	31	24	graph	graph	NOUN
cana-1336	31	25	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	31	26	)	)	PUNCT
cana-1336	31	27	of	of	ADP
cana-1336	31	28	order	order	NOUN
cana-1336	31	29	n	n	NOUN
cana-1336	31	30	and	and	CCONJ
cana-1336	31	31	𝑛(𝐺	𝑛(𝐺	NUM
cana-1336	31	32	)	)	PUNCT
cana-1336	31	33	is	be	AUX
cana-1336	31	34	the	the	DET
cana-1336	31	35	number	number	NOUN
cana-1336	31	36	of	of	ADP
cana-1336	31	37	components	component	NOUN
cana-1336	31	38	,	,	PUNCT
cana-1336	31	39	then	then	ADV
cana-1336	31	40	each	each	DET
cana-1336	31	41	component	component	NOUN
cana-1336	31	42	in	in	ADP
cana-1336	31	43	graph	graph	NOUN
cana-1336	31	44	𝐺	𝐺	PROPN
cana-1336	31	45	is	be	AUX
cana-1336	31	46	complete	complete	ADJ
cana-1336	31	47	.	.	PUNCT
cana-1336	32	1	proof	proof	NOUN
cana-1336	32	2	.	.	PUNCT
cana-1336	33	1	since	since	SCONJ
cana-1336	33	2	every	every	DET
cana-1336	33	3	vertex	vertex	NOUN
cana-1336	33	4	is	be	AUX
cana-1336	33	5	adjacent	adjacent	ADJ
cana-1336	33	6	to	to	ADP
cana-1336	33	7	all	all	DET
cana-1336	33	8	vertices	vertex	NOUN
cana-1336	33	9	in	in	ADP
cana-1336	33	10	every	every	DET
cana-1336	33	11	component	component	NOUN
cana-1336	33	12	,	,	PUNCT
cana-1336	33	13	and	and	CCONJ
cana-1336	33	14	this	this	DET
cana-1336	33	15	graph	graph	NOUN
cana-1336	33	16	is	be	AUX
cana-1336	33	17	divided	divide	VERB
cana-1336	33	18	to	to	ADP
cana-1336	33	19	many	many	ADJ
cana-1336	33	20	components	component	NOUN
cana-1336	33	21	,	,	PUNCT
cana-1336	33	22	and	and	CCONJ
cana-1336	33	23	every	every	DET
cana-1336	33	24	part	part	NOUN
cana-1336	33	25	in	in	ADP
cana-1336	33	26	this	this	DET
cana-1336	33	27	graph	graph	NOUN
cana-1336	33	28	is	be	AUX
cana-1336	33	29	complete	complete	ADJ
cana-1336	33	30	then	then	ADV
cana-1336	33	31	we	we	PRON
cana-1336	33	32	can	can	AUX
cana-1336	33	33	say	say	VERB
cana-1336	33	34	that	that	SCONJ
cana-1336	33	35	each	each	DET
cana-1336	33	36	component	component	NOUN
cana-1336	33	37	in	in	ADP
cana-1336	33	38	graph	graph	NOUN
cana-1336	33	39	g	g	PROPN
cana-1336	33	40	is	be	AUX
cana-1336	33	41	complete	complete	ADJ
cana-1336	33	42	.	.	PUNCT
cana-1336	34	1	this	this	DET
cana-1336	34	2	graph	graph	NOUN
cana-1336	34	3	is	be	AUX
cana-1336	34	4	not	not	PART
cana-1336	34	5	divided	divide	VERB
cana-1336	34	6	where	where	SCONJ
cana-1336	34	7	the	the	DET
cana-1336	34	8	numbers	number	NOUN
cana-1336	34	9	of	of	ADP
cana-1336	34	10	vertices	vertex	NOUN
cana-1336	34	11	one	one	NUM
cana-1336	34	12	or	or	CCONJ
cana-1336	34	13	two	two	NUM
cana-1336	34	14	.	.	PUNCT
cana-1336	35	1	corollary	corollary	ADJ
cana-1336	35	2	2.2	2.2	NUM
cana-1336	35	3	if	if	SCONJ
cana-1336	35	4	𝐺	𝐺	PROPN
cana-1336	35	5	be	be	VERB
cana-1336	35	6	a	a	DET
cana-1336	35	7	carmichael	carmichael	PROPN
cana-1336	35	8	function	function	NOUN
cana-1336	35	9	graph	graph	NOUN
cana-1336	35	10	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	35	11	)	)	PUNCT
cana-1336	35	12	of	of	ADP
cana-1336	35	13	order	order	NOUN
cana-1336	35	14	n	n	CCONJ
cana-1336	35	15	,	,	PUNCT
cana-1336	35	16	and	and	CCONJ
cana-1336	35	17	𝑛(𝐺	𝑛(𝐺	NUM
cana-1336	35	18	)	)	PUNCT
cana-1336	35	19	is	be	AUX
cana-1336	35	20	the	the	DET
cana-1336	35	21	number	number	NOUN
cana-1336	35	22	of	of	ADP
cana-1336	35	23	components	component	NOUN
cana-1336	35	24	,	,	PUNCT
cana-1336	35	25	then	then	ADV
cana-1336	35	26	the	the	DET
cana-1336	35	27	dominations	domination	NOUN
cana-1336	35	28	numbers	number	NOUN
cana-1336	35	29	as	as	ADP
cana-1336	35	30	following	follow	VERB
cana-1336	35	31	:	:	PUNCT
cana-1336	35	32	𝛾	𝛾	X
cana-1336	35	33	(	(	PUNCT
cana-1336	35	34	𝐺	𝐺	NOUN
cana-1336	35	35	)	)	PUNCT
cana-1336	35	36	=	=	SYM
cana-1336	35	37	𝑛(𝐺	𝑛(𝐺	NOUN
cana-1336	35	38	)	)	PUNCT
cana-1336	35	39	.	.	PUNCT
cana-1336	36	1	proof	proof	NOUN
cana-1336	36	2	:	:	PUNCT
cana-1336	36	3	depend	depend	VERB
cana-1336	36	4	on	on	ADP
cana-1336	36	5	the	the	DET
cana-1336	36	6	previse	previse	NOUN
cana-1336	36	7	theorem	theorem	VERB
cana-1336	36	8	that	that	SCONJ
cana-1336	36	9	each	each	DET
cana-1336	36	10	component	component	NOUN
cana-1336	36	11	in	in	ADP
cana-1336	36	12	graph	graph	NOUN
cana-1336	36	13	𝐺	𝐺	PROPN
cana-1336	36	14	is	be	AUX
cana-1336	36	15	complete	complete	ADJ
cana-1336	36	16	then	then	ADV
cana-1336	36	17	we	we	PRON
cana-1336	36	18	can	can	AUX
cana-1336	36	19	say	say	VERB
cana-1336	36	20	that	that	SCONJ
cana-1336	36	21	the	the	DET
cana-1336	36	22	domination	domination	NOUN
cana-1336	36	23	set	set	VERB
cana-1336	36	24	in	in	ADP
cana-1336	36	25	this	this	DET
cana-1336	36	26	graph	graph	NOUN
cana-1336	36	27	is	be	AUX
cana-1336	36	28	the	the	DET
cana-1336	36	29	number	number	NOUN
cana-1336	36	30	of	of	ADP
cana-1336	36	31	components	component	NOUN
cana-1336	36	32	.	.	PUNCT
cana-1336	37	1	corollary	corollary	ADJ
cana-1336	37	2	2.3	2.3	NUM
cana-1336	37	3	if	if	SCONJ
cana-1336	37	4	𝐺	𝐺	PROPN
cana-1336	37	5	be	be	VERB
cana-1336	37	6	a	a	DET
cana-1336	37	7	carmichael	carmichael	PROPN
cana-1336	37	8	function	function	NOUN
cana-1336	37	9	graph	graph	NOUN
cana-1336	37	10	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	37	11	)	)	PUNCT
cana-1336	37	12	of	of	ADP
cana-1336	37	13	order	order	NOUN
cana-1336	37	14	n	n	NOUN
cana-1336	37	15	and	and	CCONJ
cana-1336	37	16	𝑛(𝐺	𝑛(𝐺	NUM
cana-1336	37	17	)	)	PUNCT
cana-1336	37	18	is	be	AUX
cana-1336	37	19	the	the	DET
cana-1336	37	20	number	number	NOUN
cana-1336	37	21	of	of	ADP
cana-1336	37	22	components	component	NOUN
cana-1336	37	23	,	,	PUNCT
cana-1336	37	24	then	then	ADV
cana-1336	37	25	the	the	DET
cana-1336	37	26	independence	independence	NOUN
cana-1336	37	27	number	number	NOUN
cana-1336	37	28	as	as	ADP
cana-1336	37	29	following	follow	VERB
cana-1336	37	30	:	:	PUNCT
cana-1336	37	31	𝛽(𝐺	𝛽(𝐺	NOUN
cana-1336	37	32	)	)	PUNCT
cana-1336	37	33	=	=	SYM
cana-1336	37	34	𝑛(𝐺	𝑛(𝐺	NOUN
cana-1336	37	35	)	)	PUNCT
cana-1336	37	36	.	.	PUNCT
cana-1336	38	1	proof	proof	NOUN
cana-1336	38	2	:	:	PUNCT
cana-1336	38	3	depend	depend	VERB
cana-1336	38	4	on	on	ADP
cana-1336	38	5	the	the	DET
cana-1336	38	6	previse	previse	NOUN
cana-1336	38	7	theorem	theorem	VERB
cana-1336	38	8	that	that	SCONJ
cana-1336	38	9	each	each	DET
cana-1336	38	10	component	component	NOUN
cana-1336	38	11	in	in	ADP
cana-1336	38	12	graph	graph	NOUN
cana-1336	38	13	𝐺	𝐺	PROPN
cana-1336	38	14	is	be	AUX
cana-1336	38	15	complete	complete	ADJ
cana-1336	38	16	then	then	ADV
cana-1336	38	17	we	we	PRON
cana-1336	38	18	can	can	AUX
cana-1336	38	19	say	say	VERB
cana-1336	38	20	that	that	SCONJ
cana-1336	38	21	the	the	DET
cana-1336	38	22	independence	independence	NOUN
cana-1336	38	23	set	set	VERB
cana-1336	38	24	in	in	ADP
cana-1336	38	25	this	this	DET
cana-1336	38	26	graph	graph	NOUN
cana-1336	38	27	is	be	AUX
cana-1336	38	28	the	the	DET
cana-1336	38	29	number	number	NOUN
cana-1336	38	30	of	of	ADP
cana-1336	38	31	components	component	NOUN
cana-1336	38	32	.	.	PUNCT
cana-1336	39	1	theorem	theorem	VERB
cana-1336	39	2	2.4	2.4	NUM
cana-1336	39	3	if	if	SCONJ
cana-1336	39	4	𝐺	𝐺	PROPN
cana-1336	39	5	be	be	VERB
cana-1336	39	6	a	a	DET
cana-1336	39	7	carmichael	carmichael	PROPN
cana-1336	39	8	function	function	NOUN
cana-1336	39	9	graph	graph	NOUN
cana-1336	39	10	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	39	11	)	)	PUNCT
cana-1336	39	12	of	of	ADP
cana-1336	39	13	order	order	NOUN
cana-1336	39	14	n	n	CCONJ
cana-1336	39	15	,	,	PUNCT
cana-1336	39	16	then	then	ADV
cana-1336	39	17	the	the	DET
cana-1336	39	18	clique	clique	ADJ
cana-1336	39	19	number	number	NOUN
cana-1336	39	20	as	as	ADP
cana-1336	39	21	following	follow	VERB
cana-1336	39	22	:	:	PUNCT
cana-1336	39	23	𝜔(𝐺	𝜔(𝐺	X
cana-1336	39	24	)	)	PUNCT
cana-1336	39	25	=	=	SYM
cana-1336	40	1	|𝑆	|𝑆	PROPN
cana-1336	40	2	|	|	ADV
cana-1336	40	3	,	,	PUNCT
cana-1336	40	4	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1336	40	5	𝑆	𝑆	PROPN
cana-1336	40	6	=	=	SYM
cana-1336	40	7	{	{	PUNCT
cana-1336	40	8	𝑢	𝑢	PRON
cana-1336	40	9	∶	∶	NOUN
cana-1336	40	10	𝜆(𝑢	𝜆(𝑢	ADJ
cana-1336	40	11	)	)	PUNCT
cana-1336	40	12	=	=	SYM
cana-1336	40	13	2	2	X
cana-1336	40	14	}	}	PUNCT
cana-1336	40	15	proof	proof	NOUN
cana-1336	40	16	:	:	PUNCT
cana-1336	40	17	1	1	NUM
cana-1336	40	18	2	2	NUM
cana-1336	40	19	1	1	NUM
cana-1336	40	20	figure	figure	NOUN
cana-1336	40	21	2.1	2.1	NUM
cana-1336	40	22	the	the	DET
cana-1336	40	23	function	function	NOUN
cana-1336	40	24	carmichael	carmichael	PROPN
cana-1336	40	25	graph	graph	NOUN
cana-1336	40	26	(	(	PUNCT
cana-1336	40	27	g	g	NOUN
cana-1336	40	28	)	)	PUNCT
cana-1336	40	29	of	of	ADP
cana-1336	40	30	order	order	NOUN
cana-1336	40	31	10	10	NUM
cana-1336	40	32	.	.	NOUN
cana-1336	40	33	1	1	NUM
cana-1336	40	34	2	2	NUM
cana-1336	40	35	figure	figure	NOUN
cana-1336	40	36	2.2	2.2	NUM
cana-1336	40	37	the	the	DET
cana-1336	40	38	components	component	NOUN
cana-1336	40	39	of	of	ADP
cana-1336	40	40	function	function	NOUN
cana-1336	40	41	carmichael	carmichael	PROPN
cana-1336	40	42	graph	graph	NOUN
cana-1336	40	43	(	(	PUNCT
cana-1336	40	44	g	g	NOUN
cana-1336	40	45	)	)	PUNCT
cana-1336	40	46	of	of	ADP
cana-1336	40	47	order	order	NOUN
cana-1336	40	48	10	10	NUM
cana-1336	40	49	.	.	NOUN
cana-1336	40	50	1	1	NUM
cana-1336	40	51	communications	communication	NOUN
cana-1336	40	52	on	on	ADP
cana-1336	40	53	applied	apply	VERB
cana-1336	40	54	nonlinear	nonlinear	ADJ
cana-1336	40	55	analysis	analysis	NOUN
cana-1336	40	56	issn	issn	NOUN
cana-1336	40	57	:	:	PUNCT
cana-1336	40	58	1074	1074	NUM
cana-1336	40	59	-	-	PUNCT
cana-1336	40	60	133x	133x	NUM
cana-1336	40	61	vol	vol	NOUN
cana-1336	40	62	31	31	NUM
cana-1336	40	63	no	no	NOUN
cana-1336	40	64	.	.	PUNCT
cana-1336	41	1	6s	6s	NUM
cana-1336	41	2	(	(	PUNCT
cana-1336	41	3	2024	2024	NUM
cana-1336	41	4	)	)	PUNCT
cana-1336	41	5	726	726	NUM
cana-1336	41	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1336	41	7	since	since	SCONJ
cana-1336	41	8	every	every	DET
cana-1336	41	9	component	component	NOUN
cana-1336	41	10	is	be	AUX
cana-1336	41	11	a	a	DET
cana-1336	41	12	complete	complete	ADJ
cana-1336	41	13	sub	sub	NOUN
cana-1336	41	14	graph	graph	NOUN
cana-1336	41	15	and	and	CCONJ
cana-1336	41	16	this	this	DET
cana-1336	41	17	graph	graph	NOUN
cana-1336	41	18	has	have	VERB
cana-1336	41	19	many	many	ADJ
cana-1336	41	20	of	of	ADP
cana-1336	41	21	components	component	NOUN
cana-1336	41	22	such	such	ADJ
cana-1336	41	23	that	that	SCONJ
cana-1336	41	24	every	every	DET
cana-1336	41	25	component	component	NOUN
cana-1336	41	26	has	have	VERB
cana-1336	41	27	vertices	vertex	NOUN
cana-1336	41	28	are	be	AUX
cana-1336	41	29	adjacent	adjacent	ADJ
cana-1336	41	30	and	and	CCONJ
cana-1336	41	31	the	the	DET
cana-1336	41	32	clique	clique	ADJ
cana-1336	41	33	number	number	NOUN
cana-1336	41	34	dependent	dependent	ADJ
cana-1336	41	35	on	on	ADP
cana-1336	41	36	the	the	DET
cana-1336	41	37	largest	large	ADJ
cana-1336	41	38	component	component	NOUN
cana-1336	41	39	complete	complete	ADJ
cana-1336	41	40	in	in	ADP
cana-1336	41	41	this	this	DET
cana-1336	41	42	graph	graph	NOUN
cana-1336	41	43	.	.	PUNCT
cana-1336	42	1	the	the	DET
cana-1336	42	2	component	component	NOUN
cana-1336	42	3	of	of	ADP
cana-1336	42	4	the	the	DET
cana-1336	42	5	image	image	NOUN
cana-1336	42	6	is	be	AUX
cana-1336	42	7	equal	equal	ADJ
cana-1336	42	8	two	two	NUM
cana-1336	42	9	is	be	AUX
cana-1336	42	10	the	the	DET
cana-1336	42	11	largest	large	ADJ
cana-1336	42	12	complete	complete	ADJ
cana-1336	42	13	sub	sub	NOUN
cana-1336	42	14	graph	graph	NOUN
cana-1336	42	15	in	in	ADP
cana-1336	42	16	this	this	DET
cana-1336	42	17	graph	graph	NOUN
cana-1336	42	18	.	.	PUNCT
cana-1336	43	1	theorem	theorem	VERB
cana-1336	43	2	2.5	2.5	NUM
cana-1336	43	3	if	if	SCONJ
cana-1336	43	4	𝐺	𝐺	PROPN
cana-1336	43	5	be	be	VERB
cana-1336	43	6	a	a	DET
cana-1336	43	7	carmichael	carmichael	PROPN
cana-1336	43	8	function	function	NOUN
cana-1336	43	9	graph	graph	NOUN
cana-1336	43	10	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	43	11	)	)	PUNCT
cana-1336	43	12	of	of	ADP
cana-1336	43	13	order	order	NOUN
cana-1336	43	14	n	n	CCONJ
cana-1336	43	15	,	,	PUNCT
cana-1336	43	16	then	then	ADV
cana-1336	43	17	the	the	DET
cana-1336	43	18	chromatic	chromatic	ADJ
cana-1336	43	19	number	number	NOUN
cana-1336	43	20	as	as	ADP
cana-1336	43	21	following	follow	VERB
cana-1336	43	22	:	:	PUNCT
cana-1336	43	23	𝜒(𝐺	𝜒(𝐺	NOUN
cana-1336	43	24	)	)	PUNCT
cana-1336	43	25	=	=	SYM
cana-1336	44	1	|𝑆	|𝑆	PROPN
cana-1336	44	2	|	|	ADV
cana-1336	44	3	,	,	PUNCT
cana-1336	44	4	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1336	44	5	𝑆	𝑆	PROPN
cana-1336	44	6	=	=	SYM
cana-1336	44	7	{	{	PUNCT
cana-1336	44	8	𝑢	𝑢	PRON
cana-1336	44	9	∶	∶	NOUN
cana-1336	44	10	𝜆(𝑢	𝜆(𝑢	ADJ
cana-1336	44	11	)	)	PUNCT
cana-1336	44	12	=	=	SYM
cana-1336	44	13	2	2	X
cana-1336	44	14	}	}	PUNCT
cana-1336	44	15	proof	proof	NOUN
cana-1336	44	16	:	:	PUNCT
cana-1336	44	17	according	accord	VERB
cana-1336	44	18	to	to	ADP
cana-1336	44	19	the	the	DET
cana-1336	44	20	theorem	theorem	NOUN
cana-1336	44	21	(	(	PUNCT
cana-1336	44	22	2.4	2.4	NUM
cana-1336	44	23	)	)	PUNCT
cana-1336	44	24	the	the	DET
cana-1336	44	25	clique	clique	ADJ
cana-1336	44	26	number	number	NOUN
cana-1336	44	27	is	be	AUX
cana-1336	44	28	equal	equal	ADJ
cana-1336	44	29	|𝑆	|𝑆	ADV
cana-1336	44	30	|	|	ADV
cana-1336	44	31	,	,	PUNCT
cana-1336	44	32	{	{	PUNCT
cana-1336	44	33	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1336	44	34	𝑆	𝑆	PROPN
cana-1336	44	35	=	=	SYM
cana-1336	44	36	{	{	PUNCT
cana-1336	44	37	𝑢	𝑢	PRON
cana-1336	44	38	∶	∶	NOUN
cana-1336	44	39	𝜆(𝑢	𝜆(𝑢	ADJ
cana-1336	44	40	)	)	PUNCT
cana-1336	44	41	=	=	SYM
cana-1336	44	42	2	2	X
cana-1336	44	43	}	}	PUNCT
cana-1336	44	44	since	since	SCONJ
cana-1336	44	45	this	this	DET
cana-1336	44	46	component	component	NOUN
cana-1336	44	47	is	be	AUX
cana-1336	44	48	the	the	DET
cana-1336	44	49	largest	large	ADJ
cana-1336	44	50	component	component	NOUN
cana-1336	44	51	in	in	ADP
cana-1336	44	52	this	this	DET
cana-1336	44	53	graph	graph	NOUN
cana-1336	44	54	is	be	AUX
cana-1336	44	55	complete	complete	ADJ
cana-1336	44	56	such	such	ADJ
cana-1336	44	57	that	that	SCONJ
cana-1336	44	58	all	all	DET
cana-1336	44	59	vertices	vertex	NOUN
cana-1336	44	60	are	be	AUX
cana-1336	44	61	adjacent	adjacent	ADJ
cana-1336	44	62	.	.	PUNCT
cana-1336	45	1	theorem	theorem	VERB
cana-1336	45	2	2.6	2.6	NUM
cana-1336	45	3	if	if	SCONJ
cana-1336	45	4	g	g	PROPN
cana-1336	45	5	be	be	VERB
cana-1336	45	6	a	a	DET
cana-1336	45	7	carmichael	carmichael	PROPN
cana-1336	45	8	function	function	NOUN
cana-1336	45	9	graph	graph	NOUN
cana-1336	45	10	λ(gc	λ(gc	PROPN
cana-1336	45	11	)	)	PUNCT
cana-1336	45	12	of	of	ADP
cana-1336	45	13	order	order	NOUN
cana-1336	45	14	n	n	CCONJ
cana-1336	45	15	,	,	PUNCT
cana-1336	45	16	then	then	ADV
cana-1336	45	17	the	the	DET
cana-1336	45	18	complement	complement	NOUN
cana-1336	45	19	of	of	ADP
cana-1336	45	20	domination	domination	NOUN
cana-1336	45	21	number	number	NOUN
cana-1336	45	22	as	as	ADP
cana-1336	45	23	:	:	PUNCT
cana-1336	45	24	γ(gc	γ(gc	NUM
cana-1336	45	25	)	)	PUNCT
cana-1336	46	1	=	=	PRON
cana-1336	46	2	{	{	PUNCT
cana-1336	46	3	1	1	NUM
cana-1336	46	4	,	,	PUNCT
cana-1336	46	5	𝑖𝑓	𝑖𝑓	ADP
cana-1336	46	6	𝑛	𝑛	PRON
cana-1336	46	7	=	=	SYM
cana-1336	46	8	𝑝	𝑝	PROPN
cana-1336	46	9	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	PROPN
cana-1336	46	10	𝑝	𝑝	PROPN
cana-1336	46	11	𝑖𝑠	𝑖𝑠	PROPN
cana-1336	46	12	𝑝𝑟𝑖𝑚𝑒	𝑝𝑟𝑖𝑚𝑒	VERB
cana-1336	46	13	2	2	NUM
cana-1336	46	14	,	,	PUNCT
cana-1336	46	15	𝑖𝑓	𝑖𝑓	NUM
cana-1336	46	16	𝑡ℎ𝑒𝑟𝑒	𝑡ℎ𝑒𝑟𝑒	NOUN
cana-1336	46	17	𝑖𝑠	𝑖𝑠	PROPN
cana-1336	46	18	𝑛𝑜𝑡	𝑛𝑜𝑡	PROPN
cana-1336	46	19	𝑖𝑠𝑜𝑙𝑎𝑡𝑒	𝑖𝑠𝑜𝑙𝑎𝑡𝑒	NOUN
cana-1336	46	20	𝑣𝑒𝑟𝑡𝑒𝑥	𝑣𝑒𝑟𝑡𝑒𝑥	NOUN
cana-1336	46	21	proof	proof	NOUN
cana-1336	46	22	.	.	PUNCT
cana-1336	47	1	if	if	SCONJ
cana-1336	47	2	f	f	PROPN
cana-1336	47	3	n	n	ADV
cana-1336	47	4	=	=	NOUN
cana-1336	47	5	p	p	NOUN
cana-1336	47	6	for	for	ADP
cana-1336	47	7	some	some	DET
cana-1336	47	8	p	p	NOUN
cana-1336	47	9	,	,	PUNCT
cana-1336	47	10	where	where	SCONJ
cana-1336	47	11	p	p	NOUN
cana-1336	47	12	is	be	AUX
cana-1336	47	13	prime	prime	ADJ
cana-1336	47	14	then	then	ADV
cana-1336	47	15	the	the	DET
cana-1336	47	16	graph	graph	NOUN
cana-1336	47	17	λ(g	λ(g	PROPN
cana-1336	47	18	)	)	PUNCT
cana-1336	47	19	has	have	VERB
cana-1336	47	20	an	an	DET
cana-1336	47	21	isolate	isolate	ADJ
cana-1336	47	22	vertex	vertex	NOUN
cana-1336	47	23	.	.	PUNCT
cana-1336	48	1	if	if	SCONJ
cana-1336	48	2	there	there	PRON
cana-1336	48	3	exist	exist	VERB
cana-1336	48	4	isolate	isolate	ADJ
cana-1336	48	5	vertex	vertex	NOUN
cana-1336	48	6	in	in	ADP
cana-1336	48	7	this	this	DET
cana-1336	48	8	graph	graph	NOUN
cana-1336	48	9	then	then	ADV
cana-1336	48	10	there	there	PRON
cana-1336	48	11	is	be	VERB
cana-1336	48	12	vertex	vertex	NOUN
cana-1336	48	13	is	be	AUX
cana-1336	48	14	called	call	VERB
cana-1336	48	15	dominating	dominate	VERB
cana-1336	48	16	vertex	vertex	NOUN
cana-1336	48	17	.	.	PUNCT
cana-1336	49	1	if	if	SCONJ
cana-1336	49	2	the	the	DET
cana-1336	49	3	graph	graph	NOUN
cana-1336	49	4	has	have	VERB
cana-1336	49	5	no	no	DET
cana-1336	49	6	dominating	dominating	NOUN
cana-1336	49	7	vertex	vertex	NOUN
cana-1336	49	8	then	then	ADV
cana-1336	49	9	there	there	PRON
cana-1336	49	10	is	be	VERB
cana-1336	49	11	not	not	PART
cana-1336	49	12	isolate	isolate	VERB
cana-1336	49	13	vertex	vertex	NOUN
cana-1336	49	14	.	.	PUNCT
cana-1336	50	1	if	if	SCONJ
cana-1336	50	2	there	there	PRON
cana-1336	50	3	is	be	VERB
cana-1336	50	4	not	not	PART
cana-1336	50	5	isolate	isolate	VERB
cana-1336	50	6	vertex	vertex	NOUN
cana-1336	50	7	then	then	ADV
cana-1336	50	8	the	the	DET
cana-1336	50	9	complement	complement	NOUN
cana-1336	50	10	of	of	ADP
cana-1336	50	11	domination	domination	NOUN
cana-1336	50	12	number	number	NOUN
cana-1336	50	13	is	be	AUX
cana-1336	50	14	2	2	NUM
cana-1336	50	15	.	.	PUNCT
cana-1336	51	1	this	this	DET
cana-1336	51	2	graph	graph	NOUN
cana-1336	51	3	is	be	AUX
cana-1336	51	4	connected	connect	VERB
cana-1336	51	5	in	in	ADP
cana-1336	51	6	every	every	DET
cana-1336	51	7	vertices	vertex	NOUN
cana-1336	51	8	except	except	SCONJ
cana-1336	51	9	when	when	SCONJ
cana-1336	51	10	the	the	DET
cana-1336	51	11	numbers	number	NOUN
cana-1336	51	12	of	of	ADP
cana-1336	51	13	vertices	vertex	NOUN
cana-1336	51	14	is	be	AUX
cana-1336	51	15	two	two	NUM
cana-1336	51	16	be	be	AUX
cana-1336	51	17	dis	dis	PRON
cana-1336	51	18	connected	connect	VERB
cana-1336	51	19	.	.	PUNCT
cana-1336	52	1	theorem	theorem	VERB
cana-1336	52	2	2.7	2.7	NUM
cana-1336	52	3	if	if	SCONJ
cana-1336	52	4	𝐺	𝐺	PROPN
cana-1336	52	5	be	be	VERB
cana-1336	52	6	a	a	DET
cana-1336	52	7	carmichael	carmichael	PROPN
cana-1336	52	8	function	function	NOUN
cana-1336	52	9	graph	graph	NOUN
cana-1336	52	10	𝜆(𝐺𝑐	𝜆(𝐺𝑐	NOUN
cana-1336	52	11	)	)	PUNCT
cana-1336	52	12	of	of	ADP
cana-1336	52	13	order	order	NOUN
cana-1336	52	14	n	n	CCONJ
cana-1336	52	15	,	,	PUNCT
cana-1336	52	16	then	then	ADV
cana-1336	52	17	the	the	DET
cana-1336	52	18	complement	complement	NOUN
cana-1336	52	19	of	of	ADP
cana-1336	52	20	independence	independence	NOUN
cana-1336	52	21	number	number	NOUN
cana-1336	52	22	as	as	ADP
cana-1336	52	23	:	:	PUNCT
cana-1336	52	24	𝛽(𝐺𝑐	𝛽(𝐺𝑐	X
cana-1336	52	25	)	)	PUNCT
cana-1336	52	26	=	=	PUNCT
cana-1336	53	1	|𝑆	|𝑆	PROPN
cana-1336	53	2	|	|	ADV
cana-1336	53	3	,	,	PUNCT
cana-1336	53	4	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1336	53	5	𝑆	𝑆	PROPN
cana-1336	53	6	=	=	SYM
cana-1336	53	7	{	{	PUNCT
cana-1336	53	8	𝑢	𝑢	PRON
cana-1336	53	9	∶	∶	NOUN
cana-1336	53	10	𝜆(𝑢	𝜆(𝑢	ADJ
cana-1336	53	11	)	)	PUNCT
cana-1336	53	12	=	=	SYM
cana-1336	53	13	2	2	X
cana-1336	53	14	}	}	PUNCT
cana-1336	53	15	.	.	PUNCT
cana-1336	54	1	proof	proof	NOUN
cana-1336	54	2	:	:	PUNCT
cana-1336	54	3	communications	communication	NOUN
cana-1336	54	4	on	on	ADP
cana-1336	54	5	applied	apply	VERB
cana-1336	54	6	nonlinear	nonlinear	ADJ
cana-1336	54	7	analysis	analysis	NOUN
cana-1336	54	8	issn	issn	NOUN
cana-1336	54	9	:	:	PUNCT
cana-1336	54	10	1074	1074	NUM
cana-1336	54	11	-	-	PUNCT
cana-1336	54	12	133x	133x	NUM
cana-1336	54	13	vol	vol	NOUN
cana-1336	54	14	31	31	NUM
cana-1336	54	15	no	no	NOUN
cana-1336	54	16	.	.	PUNCT
cana-1336	55	1	6s	6s	NUM
cana-1336	55	2	(	(	PUNCT
cana-1336	55	3	2024	2024	NUM
cana-1336	55	4	)	)	PUNCT
cana-1336	55	5	727	727	NUM
cana-1336	55	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1336	55	7	since	since	SCONJ
cana-1336	55	8	every	every	DET
cana-1336	55	9	vertex	vertex	NOUN
cana-1336	55	10	is	be	AUX
cana-1336	55	11	adjacent	adjacent	ADJ
cana-1336	55	12	to	to	ADP
cana-1336	55	13	all	all	DET
cana-1336	55	14	vertices	vertex	NOUN
cana-1336	55	15	in	in	ADP
cana-1336	55	16	other	other	ADJ
cana-1336	55	17	components	component	NOUN
cana-1336	55	18	,	,	PUNCT
cana-1336	55	19	and	and	CCONJ
cana-1336	55	20	these	these	DET
cana-1336	55	21	vertices	vertex	NOUN
cana-1336	55	22	are	be	AUX
cana-1336	55	23	not	not	PART
cana-1336	55	24	adjacent	adjacent	ADJ
cana-1336	55	25	in	in	ADP
cana-1336	55	26	the	the	DET
cana-1336	55	27	same	same	ADJ
cana-1336	55	28	component	component	NOUN
cana-1336	55	29	.	.	PUNCT
cana-1336	56	1	we	we	PRON
cana-1336	56	2	will	will	AUX
cana-1336	56	3	take	take	VERB
cana-1336	56	4	the	the	DET
cana-1336	56	5	largest	large	ADJ
cana-1336	56	6	component	component	NOUN
cana-1336	56	7	in	in	ADP
cana-1336	56	8	this	this	DET
cana-1336	56	9	graph	graph	NOUN
cana-1336	56	10	that	that	PRON
cana-1336	56	11	represent	represent	VERB
cana-1336	56	12	the	the	DET
cana-1336	56	13	image	image	NOUN
cana-1336	56	14	of	of	ADP
cana-1336	56	15	vertices	vertex	NOUN
cana-1336	56	16	is	be	AUX
cana-1336	56	17	equal	equal	ADJ
cana-1336	56	18	two	two	NUM
cana-1336	56	19	in	in	ADP
cana-1336	56	20	this	this	DET
cana-1336	56	21	graph	graph	NOUN
cana-1336	56	22	.	.	PUNCT
cana-1336	57	1	theorem	theorem	VERB
cana-1336	57	2	2.8	2.8	NUM
cana-1336	57	3	if	if	SCONJ
cana-1336	57	4	𝐺	𝐺	PROPN
cana-1336	57	5	be	be	VERB
cana-1336	57	6	a	a	DET
cana-1336	57	7	carmichael	carmichael	PROPN
cana-1336	57	8	function	function	NOUN
cana-1336	57	9	graph	graph	NOUN
cana-1336	57	10	(	(	PUNCT
cana-1336	57	11	𝐺𝑐	𝐺𝑐	PROPN
cana-1336	57	12	)	)	PUNCT
cana-1336	57	13	of	of	ADP
cana-1336	57	14	order	order	NOUN
cana-1336	57	15	n	n	CCONJ
cana-1336	57	16	,	,	PUNCT
cana-1336	57	17	then	then	ADV
cana-1336	57	18	the	the	DET
cana-1336	57	19	complement	complement	NOUN
cana-1336	57	20	of	of	ADP
cana-1336	57	21	clique	clique	ADJ
cana-1336	57	22	number	number	NOUN
cana-1336	57	23	as	as	ADP
cana-1336	57	24	:	:	PUNCT
cana-1336	57	25	𝜔(𝐺𝑐	𝜔(𝐺𝑐	X
cana-1336	57	26	)	)	PUNCT
cana-1336	57	27	=	=	SYM
cana-1336	57	28	𝐾𝑛(𝐺	𝐾𝑛(𝐺	NOUN
cana-1336	57	29	)	)	PUNCT
cana-1336	57	30	proof	proof	NOUN
cana-1336	57	31	:	:	PUNCT
cana-1336	57	32	the	the	DET
cana-1336	57	33	number	number	NOUN
cana-1336	57	34	of	of	ADP
cana-1336	57	35	components	component	NOUN
cana-1336	57	36	is	be	AUX
cana-1336	57	37	𝑛(𝐺	𝑛(𝐺	NOUN
cana-1336	57	38	)	)	PUNCT
cana-1336	57	39	and	and	CCONJ
cana-1336	57	40	the	the	DET
cana-1336	57	41	largest	large	ADJ
cana-1336	57	42	component	component	NOUN
cana-1336	57	43	complete	complete	ADJ
cana-1336	57	44	in	in	ADP
cana-1336	57	45	this	this	DET
cana-1336	57	46	graph	graph	NOUN
cana-1336	57	47	is	be	AUX
cana-1336	57	48	𝐾𝑛(𝐺	𝐾𝑛(𝐺	ADP
cana-1336	57	49	)	)	PUNCT
cana-1336	57	50	.	.	PUNCT
cana-1336	58	1	since	since	SCONJ
cana-1336	58	2	every	every	DET
cana-1336	58	3	vertex	vertex	NOUN
cana-1336	58	4	in	in	ADP
cana-1336	58	5	any	any	DET
cana-1336	58	6	component	component	NOUN
cana-1336	58	7	is	be	AUX
cana-1336	58	8	adjacent	adjacent	ADJ
cana-1336	58	9	to	to	ADP
cana-1336	58	10	all	all	DET
cana-1336	58	11	vertices	vertex	NOUN
cana-1336	58	12	in	in	ADP
cana-1336	58	13	other	other	ADJ
cana-1336	58	14	components	component	NOUN
cana-1336	58	15	and	and	CCONJ
cana-1336	58	16	these	these	DET
cana-1336	58	17	vertices	vertex	NOUN
cana-1336	58	18	are	be	AUX
cana-1336	58	19	not	not	PART
cana-1336	58	20	adjacent	adjacent	ADJ
cana-1336	58	21	in	in	ADP
cana-1336	58	22	the	the	DET
cana-1336	58	23	same	same	ADJ
cana-1336	58	24	component	component	NOUN
cana-1336	58	25	then	then	ADV
cana-1336	58	26	we	we	PRON
cana-1336	58	27	take	take	VERB
cana-1336	58	28	the	the	DET
cana-1336	58	29	complement	complement	NOUN
cana-1336	58	30	of	of	ADP
cana-1336	58	31	clique	clique	ADJ
cana-1336	58	32	number	number	NOUN
cana-1336	58	33	is	be	AUX
cana-1336	58	34	𝐾𝑛(𝐺	𝐾𝑛(𝐺	ADP
cana-1336	58	35	)	)	PUNCT
cana-1336	58	36	.	.	PUNCT
cana-1336	59	1	thus	thus	ADV
cana-1336	59	2	the	the	DET
cana-1336	59	3	vertices	vertex	NOUN
cana-1336	59	4	which	which	PRON
cana-1336	59	5	have	have	VERB
cana-1336	59	6	numbers	number	NOUN
cana-1336	59	7	constitute	constitute	VERB
cana-1336	59	8	an	an	DET
cana-1336	59	9	induced	induced	ADJ
cana-1336	59	10	sub	sub	NOUN
cana-1336	59	11	graph	graph	NOUN
cana-1336	59	12	isomorphic	isomorphic	ADJ
cana-1336	59	13	to	to	ADP
cana-1336	59	14	the	the	DET
cana-1336	59	15	complete	complete	ADJ
cana-1336	59	16	.	.	PUNCT
cana-1336	60	1	theorem	theorem	VERB
cana-1336	60	2	2.9	2.9	NUM
cana-1336	60	3	if	if	SCONJ
cana-1336	60	4	𝐺	𝐺	PROPN
cana-1336	60	5	be	be	VERB
cana-1336	60	6	a	a	DET
cana-1336	60	7	carmichael	carmichael	PROPN
cana-1336	60	8	function	function	NOUN
cana-1336	60	9	graph	graph	NOUN
cana-1336	60	10	of	of	ADP
cana-1336	60	11	order	order	NOUN
cana-1336	60	12	n	n	CCONJ
cana-1336	60	13	,	,	PUNCT
cana-1336	60	14	then	then	ADV
cana-1336	60	15	the	the	DET
cana-1336	60	16	complement	complement	NOUN
cana-1336	60	17	of	of	ADP
cana-1336	60	18	chromatic	chromatic	ADJ
cana-1336	60	19	number	number	NOUN
cana-1336	60	20	as	as	ADP
cana-1336	60	21	following	follow	VERB
cana-1336	60	22	:	:	PUNCT
cana-1336	60	23	𝜒(𝐺𝑐	𝜒(𝐺𝑐	X
cana-1336	60	24	)	)	PUNCT
cana-1336	60	25	=	=	SYM
cana-1336	60	26	𝐾𝑛(𝐺	𝐾𝑛(𝐺	NOUN
cana-1336	60	27	)	)	PUNCT
cana-1336	60	28	proof	proof	NOUN
cana-1336	60	29	:	:	PUNCT
cana-1336	60	30	according	accord	VERB
cana-1336	60	31	to	to	ADP
cana-1336	60	32	the	the	DET
cana-1336	60	33	theorem	theorem	NOUN
cana-1336	60	34	(	(	PUNCT
cana-1336	60	35	2.8	2.8	NUM
cana-1336	60	36	)	)	PUNCT
cana-1336	60	37	that	that	SCONJ
cana-1336	60	38	the	the	DET
cana-1336	60	39	complement	complement	NOUN
cana-1336	60	40	of	of	ADP
cana-1336	60	41	clique	clique	ADJ
cana-1336	60	42	number	number	NOUN
cana-1336	60	43	is	be	AUX
cana-1336	60	44	equal	equal	ADJ
cana-1336	60	45	𝜔(𝐺𝑐	𝜔(𝐺𝑐	NOUN
cana-1336	60	46	)	)	PUNCT
cana-1336	60	47	=	=	SYM
cana-1336	60	48	𝐾𝑛(𝐺	𝐾𝑛(𝐺	NOUN
cana-1336	60	49	)	)	PUNCT
cana-1336	60	50	since	since	SCONJ
cana-1336	60	51	this	this	DET
cana-1336	60	52	component	component	NOUN
cana-1336	60	53	is	be	AUX
cana-1336	60	54	the	the	DET
cana-1336	60	55	largest	large	ADJ
cana-1336	60	56	component	component	NOUN
cana-1336	60	57	in	in	ADP
cana-1336	60	58	this	this	DET
cana-1336	60	59	graph	graph	NOUN
cana-1336	60	60	is	be	AUX
cana-1336	60	61	complete	complete	ADJ
cana-1336	60	62	such	such	ADJ
cana-1336	60	63	that	that	SCONJ
cana-1336	60	64	all	all	DET
cana-1336	60	65	vertices	vertex	NOUN
cana-1336	60	66	are	be	AUX
cana-1336	60	67	adjacent	adjacent	ADJ
cana-1336	60	68	between	between	ADP
cana-1336	60	69	them	they	PRON
cana-1336	60	70	in	in	ADP
cana-1336	60	71	all	all	DET
cana-1336	60	72	components	component	NOUN
cana-1336	60	73	and	and	CCONJ
cana-1336	60	74	theses	theses	PRON
cana-1336	60	75	vertices	vertex	NOUN
cana-1336	60	76	are	be	AUX
cana-1336	60	77	not	not	PART
cana-1336	60	78	connected	connect	VERB
cana-1336	60	79	in	in	ADP
cana-1336	60	80	one	one	NUM
cana-1336	60	81	component	component	NOUN
cana-1336	60	82	.	.	PUNCT
cana-1336	61	1	theorem	theorem	VERB
cana-1336	61	2	2.10	2.10	NUM
cana-1336	61	3	if	if	SCONJ
cana-1336	61	4	𝐺	𝐺	PROPN
cana-1336	61	5	be	be	VERB
cana-1336	61	6	a	a	DET
cana-1336	61	7	carmichael	carmichael	PROPN
cana-1336	61	8	function	function	NOUN
cana-1336	61	9	graph	graph	NOUN
cana-1336	61	10	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	61	11	)	)	PUNCT
cana-1336	61	12	of	of	ADP
cana-1336	61	13	order	order	NOUN
cana-1336	61	14	n	n	CCONJ
cana-1336	61	15	,	,	PUNCT
cana-1336	61	16	then	then	ADV
cana-1336	61	17	𝛾−1(𝐺	𝛾−1(𝐺	NOUN
cana-1336	61	18	)	)	PUNCT
cana-1336	62	1	is	be	AUX
cana-1336	62	2	not	not	PART
cana-1336	62	3	exist	exist	VERB
cana-1336	62	4	if	if	SCONJ
cana-1336	62	5	g	g	PROPN
cana-1336	62	6	has	have	VERB
cana-1336	62	7	an	an	DET
cana-1336	62	8	isolated	isolated	ADJ
cana-1336	62	9	vertex	vertex	NOUN
cana-1336	62	10	,	,	PUNCT
cana-1336	62	11	otherwise𝛾−1(𝐺	otherwise𝛾−1(𝐺	NOUN
cana-1336	62	12	)	)	PUNCT
cana-1336	62	13	=	=	SYM
cana-1336	63	1	𝛾	𝛾	X
cana-1336	63	2	(	(	PUNCT
cana-1336	63	3	𝐺	𝐺	NOUN
cana-1336	63	4	)	)	PUNCT
cana-1336	63	5	=	=	SYM
cana-1336	63	6	𝑛(𝐺	𝑛(𝐺	NOUN
cana-1336	63	7	)	)	PUNCT
cana-1336	63	8	.	.	PUNCT
cana-1336	64	1	proof	proof	NOUN
cana-1336	64	2	:	:	PUNCT
cana-1336	64	3	depend	depend	VERB
cana-1336	64	4	on	on	ADP
cana-1336	64	5	the	the	DET
cana-1336	64	6	theorem	theorem	NOUN
cana-1336	64	7	(	(	PUNCT
cana-1336	64	8	2.1	2.1	NUM
cana-1336	64	9	)	)	PUNCT
cana-1336	64	10	that	that	SCONJ
cana-1336	64	11	each	each	DET
cana-1336	64	12	component	component	NOUN
cana-1336	64	13	in	in	ADP
cana-1336	64	14	graph	graph	NOUN
cana-1336	64	15	𝐺	𝐺	PROPN
cana-1336	64	16	is	be	AUX
cana-1336	64	17	complete	complete	ADJ
cana-1336	64	18	then	then	ADV
cana-1336	64	19	we	we	PRON
cana-1336	64	20	can	can	AUX
cana-1336	64	21	say	say	VERB
cana-1336	64	22	that	that	SCONJ
cana-1336	64	23	the	the	DET
cana-1336	64	24	inverse	inverse	NOUN
cana-1336	64	25	of	of	ADP
cana-1336	64	26	domination	domination	NOUN
cana-1336	64	27	set	set	VERB
cana-1336	64	28	in	in	ADP
cana-1336	64	29	this	this	DET
cana-1336	64	30	graph	graph	NOUN
cana-1336	64	31	is	be	AUX
cana-1336	64	32	the	the	DET
cana-1336	64	33	number	number	NOUN
cana-1336	64	34	of	of	ADP
cana-1336	64	35	components	component	NOUN
cana-1336	64	36	.	.	PUNCT
cana-1336	65	1	if	if	SCONJ
cana-1336	65	2	the	the	DET
cana-1336	65	3	graph	graph	NOUN
cana-1336	65	4	has	have	AUX
cana-1336	65	5	only	only	ADV
cana-1336	65	6	isolate	isolate	VERB
cana-1336	65	7	vertex	vertex	NOUN
cana-1336	65	8	then	then	ADV
cana-1336	65	9	there	there	PRON
cana-1336	65	10	is	be	VERB
cana-1336	65	11	not	not	PART
cana-1336	65	12	exist	exist	VERB
cana-1336	65	13	any	any	DET
cana-1336	65	14	inverse	inverse	NOUN
cana-1336	65	15	of	of	ADP
cana-1336	65	16	dominations	domination	NOUN
cana-1336	65	17	number	number	NOUN
cana-1336	65	18	of	of	ADP
cana-1336	65	19	this	this	DET
cana-1336	65	20	graph	graph	NOUN
cana-1336	65	21	.	.	PUNCT
cana-1336	66	1	theorem	theorem	VERB
cana-1336	66	2	2.11	2.11	NUM
cana-1336	66	3	if	if	SCONJ
cana-1336	66	4	𝐺	𝐺	PROPN
cana-1336	66	5	be	be	VERB
cana-1336	66	6	a	a	DET
cana-1336	66	7	carmichael	carmichael	PROPN
cana-1336	66	8	function	function	NOUN
cana-1336	66	9	graph	graph	NOUN
cana-1336	66	10	𝜆(𝐺	𝜆(𝐺	PROPN
cana-1336	66	11	)	)	PUNCT
cana-1336	66	12	of	of	ADP
cana-1336	66	13	order	order	NOUN
cana-1336	66	14	n	n	CCONJ
cana-1336	66	15	,	,	PUNCT
cana-1336	66	16	then	then	ADV
cana-1336	66	17	the	the	DET
cana-1336	66	18	results	result	NOUN
cana-1336	66	19	of	of	ADP
cana-1336	66	20	𝜆(𝐺	𝜆(𝐺	NOUN
cana-1336	66	21	)	)	PUNCT
cana-1336	66	22	is	be	AUX
cana-1336	66	23	even	even	ADV
cana-1336	66	24	or	or	CCONJ
cana-1336	66	25	one	one	NUM
cana-1336	66	26	.	.	PUNCT
cana-1336	67	1	proof	proof	NOUN
cana-1336	67	2	:	:	PUNCT
cana-1336	67	3	there	there	PRON
cana-1336	67	4	are	be	VERB
cana-1336	67	5	two	two	NUM
cana-1336	67	6	cases	case	NOUN
cana-1336	67	7	as	as	ADP
cana-1336	67	8	following	follow	VERB
cana-1336	67	9	:	:	PUNCT
cana-1336	67	10	case	case	NOUN
cana-1336	67	11	1	1	NUM
cana-1336	67	12	:	:	PUNCT
cana-1336	67	13	if	if	SCONJ
cana-1336	67	14	p	p	PROPN
cana-1336	67	15	≥	≥	PUNCT
cana-1336	67	16	3	3	NUM
cana-1336	67	17	or	or	CCONJ
cana-1336	67	18	α	α	PRON
cana-1336	67	19	<	<	X
cana-1336	67	20	2	2	NUM
cana-1336	67	21	,	,	PUNCT
cana-1336	67	22	then	then	ADV
cana-1336	67	23	𝑝α−1(𝑝	𝑝α−1(𝑝	PROPN
cana-1336	67	24	−	−	NOUN
cana-1336	67	25	1	1	NUM
cana-1336	67	26	)	)	PUNCT
cana-1336	67	27	,	,	PUNCT
cana-1336	67	28	so	so	CCONJ
cana-1336	67	29	there	there	PRON
cana-1336	67	30	is	be	VERB
cana-1336	67	31	two	two	NUM
cana-1336	67	32	subcases	subcase	NOUN
cana-1336	67	33	as	as	ADP
cana-1336	67	34	fallows	fallow	NOUN
cana-1336	67	35	:	:	PUNCT
cana-1336	67	36	subcase	subcase	NOUN
cana-1336	67	37	1	1	NUM
cana-1336	67	38	:	:	PUNCT
cana-1336	68	1	if	if	SCONJ
cana-1336	68	2	p	p	NOUN
cana-1336	68	3	=	=	SYM
cana-1336	68	4	2	2	NUM
cana-1336	68	5	,	,	PUNCT
cana-1336	68	6	then	then	ADV
cana-1336	68	7	2α−1(1	2α−1(1	NUM
cana-1336	68	8	)	)	PUNCT
cana-1336	68	9	=	=	SYM
cana-1336	68	10	2α−1	2α−1	NUM
cana-1336	68	11	and	and	CCONJ
cana-1336	68	12	α	α	NOUN
cana-1336	68	13	=	=	NOUN
cana-1336	68	14	1	1	NUM
cana-1336	68	15	then	then	ADV
cana-1336	68	16	the	the	DET
cana-1336	68	17	result	result	NOUN
cana-1336	68	18	is	be	AUX
cana-1336	68	19	one	one	NUM
cana-1336	68	20	.	.	PUNCT
cana-1336	69	1	subcase	subcase	PROPN
cana-1336	69	2	2	2	NUM
cana-1336	69	3	:	:	PUNCT
cana-1336	69	4	if	if	SCONJ
cana-1336	69	5	p	p	NOUN
cana-1336	69	6	=	=	SYM
cana-1336	69	7	2	2	NUM
cana-1336	69	8	,	,	PUNCT
cana-1336	69	9	then	then	ADV
cana-1336	69	10	2α−1(1	2α−1(1	NUM
cana-1336	69	11	)	)	PUNCT
cana-1336	69	12	=	=	SYM
cana-1336	69	13	2α−1	2α−1	NUM
cana-1336	69	14	and	and	CCONJ
cana-1336	69	15	𝛼	𝛼	ADJ
cana-1336	69	16	>	>	X
cana-1336	69	17	1	1	NUM
cana-1336	69	18	then	then	ADV
cana-1336	69	19	the	the	DET
cana-1336	69	20	result	result	NOUN
cana-1336	69	21	is	be	AUX
cana-1336	69	22	even	even	ADV
cana-1336	69	23	.	.	PUNCT
cana-1336	70	1	case	case	NOUN
cana-1336	70	2	2	2	NUM
cana-1336	70	3	:	:	PUNCT
cana-1336	70	4	if	if	SCONJ
cana-1336	70	5	p	p	NOUN
cana-1336	70	6	=	=	SYM
cana-1336	70	7	2	2	NUM
cana-1336	70	8	and	and	CCONJ
cana-1336	70	9	α	α	PRON
cana-1336	70	10	≥	≥	NOUN
cana-1336	70	11	3	3	NUM
cana-1336	70	12	then	then	ADV
cana-1336	70	13	2α−2	2α−2	NUM
cana-1336	70	14	and	and	CCONJ
cana-1336	70	15	the	the	DET
cana-1336	70	16	power	power	NOUN
cana-1336	70	17	of	of	ADP
cana-1336	70	18	the	the	DET
cana-1336	70	19	number	number	NOUN
cana-1336	70	20	2	2	NUM
cana-1336	70	21	is	be	AUX
cana-1336	70	22	positive	positive	ADJ
cana-1336	70	23	and	and	CCONJ
cana-1336	70	24	hence	hence	ADV
cana-1336	70	25	the	the	DET
cana-1336	70	26	result	result	NOUN
cana-1336	70	27	is	be	AUX
cana-1336	70	28	even	even	ADV
cana-1336	70	29	.	.	PUNCT
cana-1336	71	1	3	3	X
cana-1336	71	2	.	.	X
cana-1336	71	3	acknowledgements	acknowledgement	NOUN
cana-1336	71	4	communications	communication	NOUN
cana-1336	71	5	on	on	ADP
cana-1336	71	6	applied	apply	VERB
cana-1336	71	7	nonlinear	nonlinear	ADJ
cana-1336	71	8	analysis	analysis	NOUN
cana-1336	71	9	issn	issn	NOUN
cana-1336	71	10	:	:	PUNCT
cana-1336	71	11	1074	1074	NUM
cana-1336	71	12	-	-	PUNCT
cana-1336	71	13	133x	133x	NUM
cana-1336	71	14	vol	vol	NOUN
cana-1336	71	15	31	31	NUM
cana-1336	71	16	no	no	NOUN
cana-1336	71	17	.	.	PUNCT
cana-1336	72	1	6s	6s	NUM
cana-1336	72	2	(	(	PUNCT
cana-1336	72	3	2024	2024	NUM
cana-1336	72	4	)	)	PUNCT
cana-1336	72	5	728	728	NUM
cana-1336	72	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1336	73	1	i	i	PRON
cana-1336	73	2	extend	extend	VERB
cana-1336	73	3	my	my	PRON
cana-1336	73	4	thanks	thank	NOUN
cana-1336	73	5	and	and	CCONJ
cana-1336	73	6	gratitude	gratitude	NOUN
cana-1336	73	7	to	to	PART
cana-1336	73	8	supervisor	supervisor	VERB
cana-1336	73	9	prof	prof	NOUN
cana-1336	73	10	.	.	PUNCT
cana-1336	74	1	dr	dr	PROPN
cana-1336	74	2	.	.	PROPN
cana-1336	74	3	faez	faez	PROPN
cana-1336	74	4	ali	ali	PROPN
cana-1336	74	5	rashid	rashid	PROPN
cana-1336	74	6	al	al	PROPN
cana-1336	74	7	-	-	PUNCT
cana-1336	74	8	mamouri	mamouri	PROPN
cana-1336	74	9	and	and	CCONJ
cana-1336	74	10	big	big	ADJ
cana-1336	74	11	thanks	thank	NOUN
cana-1336	74	12	to	to	ADP
cana-1336	74	13	support	support	NOUN
cana-1336	74	14	has	have	AUX
cana-1336	74	15	been	be	AUX
cana-1336	74	16	given	give	VERB
cana-1336	74	17	to	to	ADP
cana-1336	74	18	prof	prof	PROPN
cana-1336	74	19	.	.	PUNCT
cana-1336	75	1	dr	dr	PROPN
cana-1336	75	2	.	.	PROPN
cana-1336	75	3	ahmed	ahmed	PROPN
cana-1336	75	4	abid	abid	PROPN
cana-1336	75	5	ali	ali	PROPN
cana-1336	75	6	omran	omran	PROPN
cana-1336	75	7	in	in	ADP
cana-1336	75	8	our	our	PRON
cana-1336	75	9	work	work	NOUN
cana-1336	75	10	for	for	ADP
cana-1336	75	11	their	their	PRON
cana-1336	75	12	continuous	continuous	ADJ
cana-1336	75	13	support	support	NOUN
cana-1336	75	14	and	and	CCONJ
cana-1336	75	15	the	the	DET
cana-1336	75	16	kindness	kindness	NOUN
cana-1336	75	17	of	of	ADP
cana-1336	75	18	their	their	PRON
cana-1336	75	19	time	time	NOUN
cana-1336	75	20	,	,	PUNCT
cana-1336	75	21	as	as	ADV
cana-1336	75	22	well	well	ADV
cana-1336	75	23	as	as	ADP
cana-1336	75	24	their	their	PRON
cana-1336	75	25	assistance	assistance	NOUN
cana-1336	75	26	to	to	ADP
cana-1336	75	27	me	i	PRON
cana-1336	75	28	in	in	ADP
cana-1336	75	29	accomplishing	accomplish	VERB
cana-1336	75	30	this	this	DET
cana-1336	75	31	work	work	NOUN
cana-1336	75	32	and	and	CCONJ
cana-1336	75	33	completing	complete	VERB
cana-1336	75	34	it	it	PRON
cana-1336	75	35	to	to	ADP
cana-1336	75	36	the	the	DET
cana-1336	75	37	fullest	full	ADJ
cana-1336	75	38	,	,	PUNCT
cana-1336	75	39	god	god	PROPN
cana-1336	75	40	willing	willing	ADJ
cana-1336	75	41	.	.	PUNCT
cana-1336	76	1	4.conclusion	4.conclusion	NUM
cana-1336	76	2	in	in	ADP
cana-1336	76	3	this	this	DET
cana-1336	76	4	paper	paper	NOUN
cana-1336	76	5	we	we	PRON
cana-1336	76	6	gets	get	VERB
cana-1336	76	7	that	that	DET
cana-1336	76	8	distinct	distinct	ADJ
cana-1336	76	9	kind	kind	NOUN
cana-1336	76	10	of	of	ADP
cana-1336	76	11	the	the	DET
cana-1336	76	12	graphs	graph	NOUN
cana-1336	76	13	is	be	AUX
cana-1336	76	14	called	call	VERB
cana-1336	76	15	carmichael	carmichael	PROPN
cana-1336	76	16	function	function	NOUN
cana-1336	76	17	graph	graph	NOUN
cana-1336	76	18	𝜆(𝐺).dependence	𝜆(𝐺).dependence	NOUN
cana-1336	76	19	on	on	ADP
cana-1336	76	20	the	the	DET
cana-1336	76	21	results	result	NOUN
cana-1336	76	22	that	that	PRON
cana-1336	76	23	proved	prove	VERB
cana-1336	76	24	we	we	PRON
cana-1336	76	25	obtained	obtain	VERB
cana-1336	76	26	the	the	DET
cana-1336	76	27	dominance	dominance	NOUN
cana-1336	76	28	,	,	PUNCT
cana-1336	76	29	independence	independence	NOUN
cana-1336	76	30	,	,	PUNCT
cana-1336	76	31	and	and	CCONJ
cana-1336	76	32	clique	clique	ADJ
cana-1336	76	33	number	number	NOUN
cana-1336	76	34	are	be	AUX
cana-1336	76	35	determined	determine	VERB
cana-1336	76	36	.	.	PUNCT
cana-1336	77	1	also	also	ADV
cana-1336	77	2	we	we	PRON
cana-1336	77	3	calculated	calculate	VERB
cana-1336	77	4	the	the	DET
cana-1336	77	5	complement	complement	NOUN
cana-1336	77	6	the	the	DET
cana-1336	77	7	basic	basic	ADJ
cana-1336	77	8	elements	element	NOUN
cana-1336	77	9	in	in	ADP
cana-1336	77	10	this	this	DET
cana-1336	77	11	graph	graph	NOUN
cana-1336	77	12	as	as	ADP
cana-1336	77	13	domination	domination	NOUN
cana-1336	77	14	number	number	NOUN
cana-1336	77	15	,	,	PUNCT
cana-1336	77	16	independence	independence	NOUN
cana-1336	77	17	number	number	NOUN
cana-1336	77	18	and	and	CCONJ
cana-1336	77	19	clique	clique	ADJ
cana-1336	77	20	number	number	NOUN
cana-1336	77	21	.	.	PUNCT
cana-1336	78	1	moreover	moreover	ADV
cana-1336	78	2	,	,	PUNCT
cana-1336	78	3	the	the	DET
cana-1336	78	4	relation	relation	NOUN
cana-1336	78	5	between	between	ADP
cana-1336	78	6	the	the	DET
cana-1336	78	7	independence	independence	NOUN
cana-1336	78	8	number	number	NOUN
cana-1336	78	9	and	and	CCONJ
cana-1336	78	10	domination	domination	NOUN
cana-1336	78	11	number	number	NOUN
cana-1336	78	12	is	be	AUX
cana-1336	78	13	discussed	discuss	VERB
cana-1336	78	14	and	and	CCONJ
cana-1336	78	15	determined	determined	ADJ
cana-1336	78	16	references	reference	NOUN
cana-1336	78	17	[	[	X
cana-1336	78	18	1	1	NUM
cana-1336	78	19	]	]	X
cana-1336	78	20	vince	vince	NOUN
cana-1336	78	21	,	,	PUNCT
cana-1336	78	22	star	star	NOUN
cana-1336	78	23	chromatic	chromatic	ADJ
cana-1336	78	24	number	number	NOUN
cana-1336	78	25	,	,	PUNCT
cana-1336	78	26	journal	journal	NOUN
cana-1336	78	27	of	of	ADP
cana-1336	78	28	graph	graph	NOUN
cana-1336	78	29	theory	theory	NOUN
cana-1336	78	30	,	,	PUNCT
cana-1336	78	31	vol	vol	NOUN
cana-1336	78	32	.	.	PROPN
cana-1336	79	1	12	12	NUM
cana-1336	79	2	,	,	PUNCT
cana-1336	79	3	no	no	INTJ
cana-1336	79	4	.	.	NOUN
cana-1336	79	5	4	4	NUM
cana-1336	79	6	,	,	PUNCT
cana-1336	79	7	551	551	NUM
cana-1336	79	8	-	-	SYM
cana-1336	79	9	559	559	NUM
cana-1336	79	10	(	(	PUNCT
cana-1336	79	11	1988	1988	NUM
cana-1336	79	12	)	)	PUNCT
cana-1336	79	13	,	,	PUNCT
cana-1336	80	1	0	0	NUM
cana-1336	80	2	1988	1988	NUM
cana-1336	80	3	by	by	ADP
cana-1336	80	4	john	john	PROPN
cana-1336	80	5	wiley	wiley	PROPN
cana-1336	80	6	&	&	CCONJ
cana-1336	80	7	sons	sons	PROPN
cana-1336	80	8	,	,	PUNCT
cana-1336	80	9	inc	inc	PROPN
cana-1336	80	10	.	.	PUNCT
cana-1336	81	1	[	[	X
cana-1336	81	2	2	2	X
cana-1336	81	3	]	]	X
cana-1336	81	4	gayathri	gayathri	PROPN
cana-1336	81	5	,	,	PUNCT
cana-1336	81	6	s.	s.	PROPN
cana-1336	81	7	kaspar	kaspar	PROPN
cana-1336	81	8	,	,	PUNCT
cana-1336	81	9	connected	connected	ADJ
cana-1336	81	10	co	co	ADJ
cana-1336	81	11	-	-	ADJ
cana-1336	81	12	independent	independent	ADJ
cana-1336	81	13	domination	domination	NOUN
cana-1336	81	14	of	of	ADP
cana-1336	81	15	a	a	DET
cana-1336	81	16	graph	graph	NOUN
cana-1336	81	17	,	,	PUNCT
cana-1336	81	18	int	int	NOUN
cana-1336	81	19	.	.	PUNCT
cana-1336	82	1	j.	j.	PROPN
cana-1336	82	2	contemp	contemp	PROPN
cana-1336	82	3	.	.	PUNCT
cana-1336	83	1	math	math	NOUN
cana-1336	83	2	.	.	PUNCT
cana-1336	84	1	sciences	science	NOUN
cana-1336	84	2	,	,	PUNCT
cana-1336	84	3	vol	vol	NOUN
cana-1336	84	4	.	.	PROPN
cana-1336	84	5	6	6	NUM
cana-1336	84	6	,	,	PUNCT
cana-1336	84	7	2011	2011	NUM
cana-1336	84	8	,	,	PUNCT
cana-1336	84	9	no	no	INTJ
cana-1336	84	10	.	.	NOUN
cana-1336	84	11	9	9	NUM
cana-1336	84	12	,	,	PUNCT
cana-1336	84	13	423	423	NUM
cana-1336	84	14	–	–	SYM
cana-1336	84	15	429	429	NUM
cana-1336	84	16	.	.	PUNCT
cana-1336	85	1	[	[	X
cana-1336	85	2	3	3	X
cana-1336	85	3	]	]	X
cana-1336	85	4	e.	e.	PROPN
cana-1336	85	5	el	el	PROPN
cana-1336	85	6	-	-	PUNCT
cana-1336	85	7	kholyn	kholyn	NOUN
cana-1336	85	8	.	.	PUNCT
cana-1336	86	1	el	el	NOUN
cana-1336	86	2	-	-	NOUN
cana-1336	86	3	sharkawey	sharkawey	PROPN
cana-1336	86	4	,	,	PUNCT
cana-1336	86	5	the	the	DET
cana-1336	86	6	chromatic	chromatic	ADJ
cana-1336	86	7	number	number	NOUN
cana-1336	86	8	and	and	CCONJ
cana-1336	86	9	graph	graph	NOUN
cana-1336	86	10	folding	folding	NOUN
cana-1336	86	11	,	,	PUNCT
cana-1336	86	12	european	european	ADJ
cana-1336	86	13	journal	journal	PROPN
cana-1336	86	14	of	of	ADP
cana-1336	86	15	scientific	scientific	ADJ
cana-1336	86	16	research	research	NOUN
cana-1336	86	17	issn	issn	PROPN
cana-1336	86	18	1450	1450	NUM
cana-1336	86	19	-	-	PUNCT
cana-1336	86	20	216x	216x	NUM
cana-1336	86	21	/	/	SYM
cana-1336	86	22	1450	1450	NUM
cana-1336	86	23	-	-	PUNCT
cana-1336	86	24	202x	202x	NUM
cana-1336	86	25	vol.120	vol.120	NOUN
cana-1336	86	26	no.1	no.1	VERB
cana-1336	86	27	(	(	PUNCT
cana-1336	86	28	2014	2014	NUM
cana-1336	86	29	)	)	PUNCT
cana-1336	86	30	,	,	PUNCT
cana-1336	86	31	pp.138	pp.138	PROPN
cana-1336	86	32	-	-	PUNCT
cana-1336	86	33	144	144	NUM
cana-1336	86	34	.	.	PUNCT
cana-1336	87	1	[	[	X
cana-1336	87	2	4	4	X
cana-1336	87	3	]	]	X
cana-1336	87	4	g.	g.	PROPN
cana-1336	87	5	kokilambal	kokilambal	PROPN
cana-1336	87	6	,	,	PUNCT
cana-1336	87	7	a	a	DET
cana-1336	87	8	study	study	NOUN
cana-1336	87	9	on	on	ADP
cana-1336	87	10	dominating	dominating	NOUN
cana-1336	87	11	sets	set	NOUN
cana-1336	87	12	,	,	PUNCT
cana-1336	87	13	g.	g.	PROPN
cana-1336	87	14	kokilambal	kokilambal	PROPN
cana-1336	87	15	(	(	PUNCT
cana-1336	87	16	reg	reg	NOUN
cana-1336	87	17	.	.	PUNCT
cana-1336	88	1	no	no	INTJ
cana-1336	88	2	.	.	PUNCT
cana-1336	89	1	f9380	f9380	NOUN
cana-1336	89	2	)	)	PUNCT
cana-1336	89	3	research	research	NOUN
cana-1336	89	4	scholar	scholar	NOUN
cana-1336	89	5	post	post	NOUN
cana-1336	89	6	graduate	graduate	NOUN
cana-1336	89	7	and	and	CCONJ
cana-1336	89	8	research	research	PROPN
cana-1336	89	9	department	department	PROPN
cana-1336	89	10	of	of	ADP
cana-1336	89	11	mathematics	mathematics	PROPN
cana-1336	89	12	thiagarajar	thiagarajar	PROPN
cana-1336	89	13	college	college	PROPN
cana-1336	89	14	,	,	PUNCT
cana-1336	89	15	madurai-625	madurai-625	NOUN
cana-1336	89	16	009	009	NUM
cana-1336	89	17	tamil	tamil	PROPN
cana-1336	89	18	nadu	nadu	NOUN
cana-1336	89	19	.	.	PUNCT
cana-1336	90	1	[	[	X
cana-1336	90	2	5	5	X
cana-1336	90	3	]	]	PUNCT
cana-1336	90	4	k.	k.	PROPN
cana-1336	90	5	k.	k.	PROPN
cana-1336	91	1	srimitra1	srimitra1	PROPN
cana-1336	91	2	,	,	PUNCT
cana-1336	91	3	shaik	shaik	PROPN
cana-1336	91	4	sajana2	sajana2	PROPN
cana-1336	91	5	,	,	PUNCT
cana-1336	91	6	d.	d.	PROPN
cana-1336	91	7	bharathi3	bharathi3	PROPN
cana-1336	91	8	,	,	PUNCT
cana-1336	91	9	some	some	DET
cana-1336	91	10	properties	property	NOUN
cana-1336	91	11	of	of	ADP
cana-1336	91	12	graph	graph	NOUN
cana-1336	91	13	of	of	ADP
cana-1336	91	14	mobius	mobius	PROPN
cana-1336	91	15	function	function	NOUN
cana-1336	91	16	for	for	ADP
cana-1336	91	17	‘	'	PUNCT
cana-1336	91	18	0	0	NUM
cana-1336	91	19	,	,	PUNCT
cana-1336	91	20	international	international	ADJ
cana-1336	91	21	journal	journal	NOUN
cana-1336	91	22	of	of	ADP
cana-1336	91	23	innovative	innovative	ADJ
cana-1336	91	24	research	research	NOUN
cana-1336	91	25	in	in	ADP
cana-1336	91	26	science	science	NOUN
cana-1336	91	27	,	,	PUNCT
cana-1336	91	28	engineering	engineering	NOUN
cana-1336	91	29	and	and	CCONJ
cana-1336	91	30	technology	technology	NOUN
cana-1336	91	31	(	(	PUNCT
cana-1336	91	32	an	an	DET
cana-1336	91	33	iso	iso	NOUN
cana-1336	91	34	3297	3297	NUM
cana-1336	91	35	:	:	PUNCT
cana-1336	91	36	2007	2007	NUM
cana-1336	91	37	certified	certified	ADJ
cana-1336	91	38	organization	organization	NOUN
cana-1336	91	39	)	)	PUNCT
cana-1336	91	40	website	website	NOUN
cana-1336	91	41	:	:	PUNCT
cana-1336	91	42	www.ijirset.com	www.ijirset.com	X
cana-1336	91	43	vol	vol	NOUN
cana-1336	91	44	.	.	PROPN
cana-1336	92	1	6	6	NUM
cana-1336	92	2	,	,	PUNCT
cana-1336	92	3	issue	issue	NOUN
cana-1336	92	4	8	8	NUM
cana-1336	92	5	,	,	PUNCT
cana-1336	92	6	august	august	PROPN
cana-1336	92	7	2017	2017	NUM
cana-1336	92	8	,	,	PUNCT
cana-1336	92	9	issn(online	issn(online	PROPN
cana-1336	92	10	):	):	PUNCT
cana-1336	92	11	2319	2319	NUM
cana-1336	92	12	-	-	SYM
cana-1336	92	13	8753	8753	NUM
cana-1336	92	14	,	,	PUNCT
cana-1336	92	15	issn	issn	PROPN
cana-1336	92	16	(	(	PUNCT
cana-1336	92	17	print	print	NOUN
cana-1336	92	18	):	):	PUNCT
cana-1336	92	19	2347	2347	NUM
cana-1336	92	20	-	-	SYM
cana-1336	92	21	6710	6710	NUM
cana-1336	92	22	.	.	PUNCT
cana-1336	93	1	[	[	X
cana-1336	93	2	6	6	NUM
cana-1336	93	3	]	]	X
cana-1336	93	4	robin	robin	PROPN
cana-1336	93	5	j.	j.	PROPN
cana-1336	93	6	wilson	wilson	PROPN
cana-1336	93	7	,	,	PUNCT
cana-1336	93	8	introduction	introduction	NOUN
cana-1336	93	9	to	to	ADP
cana-1336	93	10	graph	graph	NOUN
cana-1336	93	11	theory	theory	NOUN
cana-1336	93	12	,	,	PUNCT
cana-1336	93	13	addison	addison	PROPN
cana-1336	93	14	wesley	wesley	PROPN
cana-1336	93	15	longman	longman	PROPN
cana-1336	93	16	limited	limited	PROPN
cana-1336	93	17	,	,	PUNCT
cana-1336	93	18	edinburgh	edinburgh	PROPN
cana-1336	93	19	gate	gate	NOUN
cana-1336	93	20	,	,	PUNCT
cana-1336	93	21	harlow	harlow	NOUN
cana-1336	93	22	,	,	PUNCT
cana-1336	93	23	essex	essex	NOUN
cana-1336	93	24	cm20	cm20	PROPN
cana-1336	93	25	2je	2je	PROPN
cana-1336	93	26	,	,	PUNCT
cana-1336	93	27	england	england	PROPN
cana-1336	93	28	and	and	CCONJ
cana-1336	93	29	associated	associated	ADJ
cana-1336	93	30	companies	company	NOUN
cana-1336	93	31	throughout	throughout	ADP
cana-1336	93	32	the	the	DET
cana-1336	93	33	world	world	NOUN
cana-1336	93	34	.	.	PUNCT
cana-1336	93	35	,	,	PUNCT
cana-1336	93	36	robin	robin	PROPN
cana-1336	93	37	wilson	wilson	PROPN
cana-1336	93	38	1972	1972	NUM
cana-1336	93	39	,	,	PUNCT
cana-1336	93	40	1996	1996	NUM
cana-1336	93	41	,	,	PUNCT
cana-1336	93	42	fourth	fourth	PROPN
cana-1336	93	43	edition	edition	NOUN
cana-1336	93	44	,	,	PUNCT
cana-1336	93	45	1996	1996	NUM
cana-1336	93	46	.	.	PUNCT
cana-1336	94	1	[	[	X
cana-1336	94	2	7	7	X
cana-1336	94	3	]	]	PUNCT
cana-1336	94	4	tom	tom	PROPN
cana-1336	94	5	m.	m.	PROPN
cana-1336	94	6	apostol	apostol	PROPN
cana-1336	94	7	,	,	PUNCT
cana-1336	94	8	introductionto	introductionto	ADJ
cana-1336	94	9	analytic	analytic	ADJ
cana-1336	94	10	number	number	NOUN
cana-1336	94	11	theory	theory	NOUN
cana-1336	94	12	,	,	PUNCT
cana-1336	94	13	s.axler	s.axler	NOUN
cana-1336	94	14	,	,	PUNCT
cana-1336	94	15	f.	f.	PROPN
cana-1336	94	16	w.	w.	PROPN
cana-1336	94	17	gehring	gehring	PROPN
cana-1336	94	18	,	,	PUNCT
cana-1336	94	19	k.a	k.a	PROPN
cana-1336	94	20	.	.	PUNCT
cana-1336	94	21	ribet	ribet	PROPN
cana-1336	94	22	.	.	PUNCT