id	sid	tid	token	lemma	pos
cana-5776	1	1	connected	connect	VERB
cana-5776	1	2	edge	edge	NOUN
cana-5776	1	3	and	and	CCONJ
cana-5776	1	4	entire	entire	ADJ
cana-5776	1	5	domination	domination	NOUN
cana-5776	1	6	of	of	ADP
cana-5776	1	7	involutory	involutory	ADJ
cana-5776	1	8	addition	addition	NOUN
cana-5776	1	9	cayley	cayley	NOUN
cana-5776	1	10	graph	graph	NOUN
cana-5776	1	11	1e.lavanya	1e.lavanya	NOUN
cana-5776	1	12	2	2	NUM
cana-5776	1	13	,	,	PUNCT
cana-5776	1	14	*	*	PUNCT
cana-5776	1	15	m.siva	m.siva	PROPN
cana-5776	1	16	parvathi	parvathi	PROPN
cana-5776	1	17	1,2,3department	1,2,3department	NUM
cana-5776	1	18	of	of	ADP
cana-5776	1	19	applied	apply	VERB
cana-5776	1	20	mathematics	mathematic	NOUN
cana-5776	1	21	,	,	PUNCT
cana-5776	1	22	sri	sri	PROPN
cana-5776	1	23	padmavati	padmavati	PROPN
cana-5776	1	24	mahila	mahila	PROPN
cana-5776	2	1	visvavidyalayam	visvavidyalayam	PROPN
cana-5776	2	2	,	,	PUNCT
cana-5776	2	3	tirupati	tirupati	PROPN
cana-5776	2	4	,	,	PUNCT
cana-5776	2	5	andhra	andhra	PROPN
cana-5776	2	6	pradesh	pradesh	PROPN
cana-5776	2	7	,	,	PUNCT
cana-5776	2	8	india	india	PROPN
cana-5776	2	9	corresponding	corresponding	PROPN
cana-5776	2	10	author	author	NOUN
cana-5776	2	11	:	:	PUNCT
cana-5776	2	12	m.siva	m.siva	PROPN
cana-5776	2	13	parvathi	parvathi	PROPN
cana-5776	2	14	article	article	PROPN
cana-5776	2	15	history	history	NOUN
cana-5776	2	16	:	:	PUNCT
cana-5776	2	17	received	receive	VERB
cana-5776	2	18	20.09.2024	20.09.2024	NUM
cana-5776	2	19	revised	revise	VERB
cana-5776	2	20	:	:	PUNCT
cana-5776	2	21	24.10.2024	24.10.2024	NUM
cana-5776	2	22	accepted	accept	VERB
cana-5776	2	23	:	:	PUNCT
cana-5776	2	24	30.11.2024	30.11.2024	NUM
cana-5776	2	25	abstract	abstract	ADJ
cana-5776	2	26	graph	graph	NOUN
cana-5776	2	27	theory	theory	NOUN
cana-5776	2	28	is	be	AUX
cana-5776	2	29	one	one	NUM
cana-5776	2	30	of	of	ADP
cana-5776	2	31	the	the	DET
cana-5776	2	32	most	most	ADV
cana-5776	2	33	advanced	advanced	ADJ
cana-5776	2	34	branches	branch	NOUN
cana-5776	2	35	of	of	ADP
cana-5776	2	36	discrete	discrete	ADJ
cana-5776	2	37	mathematics	mathematic	NOUN
cana-5776	2	38	with	with	ADP
cana-5776	2	39	variety	variety	NOUN
cana-5776	2	40	of	of	ADP
cana-5776	2	41	applications	application	NOUN
cana-5776	2	42	to	to	ADP
cana-5776	2	43	different	different	ADJ
cana-5776	2	44	branches	branch	NOUN
cana-5776	2	45	of	of	ADP
cana-5776	2	46	science	science	NOUN
cana-5776	2	47	and	and	CCONJ
cana-5776	2	48	technology	technology	NOUN
cana-5776	2	49	.	.	PUNCT
cana-5776	3	1	for	for	ADP
cana-5776	3	2	a	a	DET
cana-5776	3	3	positive	positive	ADJ
cana-5776	3	4	integer	integer	NOUN
cana-5776	3	5	𝑛	𝑛	ADP
cana-5776	3	6	>	>	SYM
cana-5776	3	7	1	1	NUM
cana-5776	3	8	,	,	PUNCT
cana-5776	3	9	the	the	DET
cana-5776	3	10	involutory	involutory	NOUN
cana-5776	3	11	addition	addition	NOUN
cana-5776	3	12	cayley	cayley	PROPN
cana-5776	3	13	graph	graph	NOUN
cana-5776	3	14	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	3	15	,	,	PUNCT
cana-5776	3	16	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	3	17	)	)	PUNCT
cana-5776	3	18	,	,	PUNCT
cana-5776	3	19	is	be	AUX
cana-5776	3	20	the	the	DET
cana-5776	3	21	graph	graph	NOUN
cana-5776	3	22	whose	whose	DET
cana-5776	3	23	vertex	vertex	NOUN
cana-5776	3	24	set	set	NOUN
cana-5776	3	25	is	be	AUX
cana-5776	3	26	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	3	27	=	=	PUNCT
cana-5776	3	28	{	{	PUNCT
cana-5776	3	29	0,1,2,3	0,1,2,3	NUM
cana-5776	3	30	,	,	PUNCT
cana-5776	3	31	…	…	PUNCT
cana-5776	3	32	,	,	PUNCT
cana-5776	3	33	𝑛	𝑛	DET
cana-5776	3	34	−	−	PROPN
cana-5776	3	35	1	1	NUM
cana-5776	3	36	}	}	PUNCT
cana-5776	3	37	and	and	CCONJ
cana-5776	3	38	edge	edge	VERB
cana-5776	3	39	set	set	VERB
cana-5776	3	40	𝐸(𝐺𝑛	𝐸(𝐺𝑛	NOUN
cana-5776	3	41	)	)	PUNCT
cana-5776	3	42	=	=	NOUN
cana-5776	3	43	{	{	PUNCT
cana-5776	3	44	𝑥𝑦	𝑥𝑦	NOUN
cana-5776	3	45	/𝑥	/𝑥	INTJ
cana-5776	3	46	,	,	PUNCT
cana-5776	3	47	𝑦	𝑦	NOUN
cana-5776	3	48	𝜖	𝜖	X
cana-5776	3	49	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	3	50	,	,	PUNCT
cana-5776	3	51	𝑥	𝑥	PROPN
cana-5776	3	52	+	+	SYM
cana-5776	3	53	𝑦	𝑦	X
cana-5776	3	54	𝜖	𝜖	X
cana-5776	3	55	𝐼𝑣	𝐼𝑣	NOUN
cana-5776	3	56	}	}	PUNCT
cana-5776	3	57	,	,	PUNCT
cana-5776	3	58	where	where	SCONJ
cana-5776	3	59	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	3	60	=	=	PRON
cana-5776	3	61	{	{	PUNCT
cana-5776	3	62	𝑥	𝑥	X
cana-5776	3	63	𝜖	𝜖	X
cana-5776	3	64	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	3	65	∶	∶	NOUN
cana-5776	3	66	𝑥2	𝑥2	NOUN
cana-5776	3	67	≡	≡	PROPN
cana-5776	3	68	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-5776	3	69	𝑛	𝑛	NOUN
cana-5776	3	70	)	)	PUNCT
cana-5776	3	71	}	}	PUNCT
cana-5776	3	72	is	be	AUX
cana-5776	3	73	the	the	DET
cana-5776	3	74	set	set	NOUN
cana-5776	3	75	of	of	ADP
cana-5776	3	76	involutory	involutory	ADJ
cana-5776	3	77	elements	element	NOUN
cana-5776	3	78	of	of	ADP
cana-5776	3	79	𝑍𝑛.by	𝑍𝑛.by	PROPN
cana-5776	3	80	taking	take	VERB
cana-5776	3	81	involutory	involutory	NOUN
cana-5776	3	82	addition	addition	NOUN
cana-5776	3	83	cayley	cayley	NOUN
cana-5776	3	84	graphs	graph	VERB
cana-5776	3	85	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	3	86	,	,	PUNCT
cana-5776	3	87	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	3	88	)	)	PUNCT
cana-5776	3	89	the	the	DET
cana-5776	3	90	author	author	NOUN
cana-5776	3	91	’s	’s	PART
cana-5776	3	92	evaluated	evaluate	VERB
cana-5776	3	93	graph	graph	NOUN
cana-5776	3	94	related	relate	VERB
cana-5776	3	95	connected	connected	ADJ
cana-5776	3	96	edge	edge	NOUN
cana-5776	3	97	domination	domination	NOUN
cana-5776	3	98	numbers	number	NOUN
cana-5776	3	99	and	and	CCONJ
cana-5776	3	100	entire	entire	ADJ
cana-5776	3	101	domination	domination	NOUN
cana-5776	3	102	numbers	number	NOUN
cana-5776	3	103	.	.	PUNCT
cana-5776	4	1	in	in	ADP
cana-5776	4	2	this	this	DET
cana-5776	4	3	paper	paper	NOUN
cana-5776	4	4	,	,	PUNCT
cana-5776	4	5	connected	connected	ADJ
cana-5776	4	6	edge	edge	NOUN
cana-5776	4	7	domination	domination	NOUN
cana-5776	4	8	number	number	NOUN
cana-5776	4	9	,	,	PUNCT
cana-5776	4	10	entire	entire	ADJ
cana-5776	4	11	domination	domination	NOUN
cana-5776	4	12	number	number	NOUN
cana-5776	4	13	of	of	ADP
cana-5776	4	14	the	the	DET
cana-5776	4	15	involutory	involutory	NOUN
cana-5776	4	16	addition	addition	NOUN
cana-5776	4	17	cayley	cayley	NOUN
cana-5776	4	18	graphs	graph	VERB
cana-5776	4	19	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	4	20	,	,	PUNCT
cana-5776	4	21	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	4	22	)	)	PUNCT
cana-5776	4	23	were	be	AUX
cana-5776	4	24	discussed	discuss	VERB
cana-5776	4	25	.	.	PUNCT
cana-5776	5	1	keywords	keyword	NOUN
cana-5776	5	2	:	:	PUNCT
cana-5776	5	3	involutory	involutory	ADJ
cana-5776	5	4	addition	addition	NOUN
cana-5776	5	5	cayley	cayley	NOUN
cana-5776	5	6	graphs	graph	NOUN
cana-5776	5	7	,	,	PUNCT
cana-5776	5	8	connected	connected	ADJ
cana-5776	5	9	edge	edge	NOUN
cana-5776	5	10	dominating	dominating	NOUN
cana-5776	5	11	sets	set	NOUN
cana-5776	5	12	,	,	PUNCT
cana-5776	5	13	entire	entire	ADJ
cana-5776	5	14	dominating	dominating	NOUN
cana-5776	5	15	sets	set	NOUN
cana-5776	5	16	,	,	PUNCT
cana-5776	5	17	connected	connected	ADJ
cana-5776	5	18	edge	edge	NOUN
cana-5776	5	19	domination	domination	NOUN
cana-5776	5	20	number	number	NOUN
cana-5776	5	21	,	,	PUNCT
cana-5776	5	22	entire	entire	ADJ
cana-5776	5	23	domination	domination	NOUN
cana-5776	5	24	number	number	NOUN
cana-5776	5	25	.	.	PUNCT
cana-5776	6	1	ams	am	NOUN
cana-5776	6	2	subject	subject	ADJ
cana-5776	6	3	classification	classification	NOUN
cana-5776	6	4	:	:	PUNCT
cana-5776	6	5	05c40	05c40	NUM
cana-5776	6	6	,	,	PUNCT
cana-5776	6	7	05c19	05c19	NOUN
cana-5776	6	8	.	.	PUNCT
cana-5776	7	1	1	1	X
cana-5776	7	2	.	.	X
cana-5776	7	3	introduction	introduction	NOUN
cana-5776	7	4	a	a	DET
cana-5776	7	5	graph	graph	NOUN
cana-5776	7	6	𝐺(𝑉	𝐺(𝑉	PRON
cana-5776	7	7	,	,	PUNCT
cana-5776	7	8	𝐸	𝐸	PROPN
cana-5776	7	9	)	)	PUNCT
cana-5776	7	10	is	be	AUX
cana-5776	7	11	a	a	DET
cana-5776	7	12	mathematical	mathematical	ADJ
cana-5776	7	13	object	object	NOUN
cana-5776	7	14	that	that	PRON
cana-5776	7	15	may	may	AUX
cana-5776	7	16	be	be	AUX
cana-5776	7	17	thought	think	VERB
cana-5776	7	18	of	of	ADP
cana-5776	7	19	as	as	ADP
cana-5776	7	20	a	a	DET
cana-5776	7	21	collection	collection	NOUN
cana-5776	7	22	of	of	ADP
cana-5776	7	23	edges	edge	NOUN
cana-5776	7	24	and	and	CCONJ
cana-5776	7	25	a	a	DET
cana-5776	7	26	set	set	NOUN
cana-5776	7	27	of	of	ADP
cana-5776	7	28	vertices	vertex	NOUN
cana-5776	7	29	that	that	PRON
cana-5776	7	30	connect	connect	VERB
cana-5776	7	31	any	any	PRON
cana-5776	7	32	or	or	CCONJ
cana-5776	7	33	all	all	PRON
cana-5776	7	34	of	of	ADP
cana-5776	7	35	the	the	DET
cana-5776	7	36	vertices	vertex	NOUN
cana-5776	7	37	.	.	PUNCT
cana-5776	8	1	in	in	ADP
cana-5776	8	2	a	a	DET
cana-5776	8	3	graph	graph	NOUN
cana-5776	8	4	𝐺	𝐺	NOUN
cana-5776	8	5	,	,	PUNCT
cana-5776	8	6	two	two	NUM
cana-5776	8	7	vertices	vertex	NOUN
cana-5776	8	8	are	be	AUX
cana-5776	8	9	considered	consider	VERB
cana-5776	8	10	neighboring	neighbor	VERB
cana-5776	8	11	if	if	SCONJ
cana-5776	8	12	an	an	DET
cana-5776	8	13	edge	edge	NOUN
cana-5776	8	14	joins	join	VERB
cana-5776	8	15	them	they	PRON
cana-5776	8	16	;	;	PUNCT
cana-5776	8	17	otherwise	otherwise	ADV
cana-5776	8	18	,	,	PUNCT
cana-5776	8	19	the	the	DET
cana-5776	8	20	edge	edge	NOUN
cana-5776	8	21	is	be	AUX
cana-5776	8	22	considered	consider	VERB
cana-5776	8	23	non	non	ADJ
cana-5776	8	24	-	-	ADJ
cana-5776	8	25	adjacent	adjacent	ADJ
cana-5776	8	26	.	.	PUNCT
cana-5776	9	1	we	we	PRON
cana-5776	9	2	indicate	indicate	VERB
cana-5776	9	3	that	that	SCONJ
cana-5776	9	4	a	a	DET
cana-5776	9	5	graph	graph	NOUN
cana-5776	9	6	𝐺	𝐺	NOUN
cana-5776	9	7	has	have	VERB
cana-5776	9	8	𝑉	𝑉	PROPN
cana-5776	9	9	(	(	PUNCT
cana-5776	9	10	𝐺	𝐺	NOUN
cana-5776	9	11	)	)	PUNCT
cana-5776	9	12	vertices	vertex	NOUN
cana-5776	9	13	and	and	CCONJ
cana-5776	9	14	𝐸(𝐺	𝐸(𝐺	NOUN
cana-5776	9	15	)	)	PUNCT
cana-5776	9	16	edges	edge	NOUN
cana-5776	9	17	,	,	PUNCT
cana-5776	9	18	accordingly	accordingly	ADV
cana-5776	9	19	.	.	PUNCT
cana-5776	10	1	the	the	DET
cana-5776	10	2	cardinality	cardinality	NOUN
cana-5776	10	3	of	of	ADP
cana-5776	10	4	𝑉	𝑉	PROPN
cana-5776	10	5	(	(	PUNCT
cana-5776	10	6	𝐺	𝐺	NOUN
cana-5776	10	7	)	)	PUNCT
cana-5776	10	8	is	be	AUX
cana-5776	10	9	the	the	DET
cana-5776	10	10	definition	definition	NOUN
cana-5776	10	11	of	of	ADP
cana-5776	10	12	the	the	DET
cana-5776	10	13	order	order	NOUN
cana-5776	10	14	of	of	ADP
cana-5776	10	15	𝐺.	𝐺.	NOUN
cana-5776	10	16	the	the	DET
cana-5776	10	17	cardinality	cardinality	NOUN
cana-5776	10	18	of	of	ADP
cana-5776	10	19	𝐸(𝐺	𝐸(𝐺	PROPN
cana-5776	10	20	)	)	PUNCT
cana-5776	10	21	is	be	AUX
cana-5776	10	22	represented	represent	VERB
cana-5776	10	23	by	by	ADP
cana-5776	10	24	|𝐸|	|𝐸|	NOUN
cana-5776	10	25	,	,	PUNCT
cana-5776	10	26	and	and	CCONJ
cana-5776	10	27	that	that	PRON
cana-5776	10	28	of	of	ADP
cana-5776	10	29	𝑉	𝑉	PROPN
cana-5776	10	30	(	(	PUNCT
cana-5776	10	31	𝐺	𝐺	NOUN
cana-5776	10	32	)	)	PUNCT
cana-5776	10	33	by	by	ADP
cana-5776	10	34	|𝑉	|𝑉	NOUN
cana-5776	10	35	|	|	NOUN
cana-5776	10	36	.	.	PUNCT
cana-5776	11	1	the	the	DET
cana-5776	11	2	number	number	NOUN
cana-5776	11	3	of	of	ADP
cana-5776	11	4	edges	edge	NOUN
cana-5776	11	5	that	that	PRON
cana-5776	11	6	occur	occur	VERB
cana-5776	11	7	with	with	ADP
cana-5776	11	8	a	a	DET
cana-5776	11	9	vertex	vertex	NOUN
cana-5776	11	10	𝑣	𝑣	X
cana-5776	11	11	in	in	ADP
cana-5776	11	12	a	a	DET
cana-5776	11	13	graph	graph	NOUN
cana-5776	11	14	𝐺	𝐺	NOUN
cana-5776	11	15	is	be	AUX
cana-5776	11	16	known	know	VERB
cana-5776	11	17	as	as	ADP
cana-5776	11	18	its	its	PRON
cana-5776	11	19	degree	degree	NOUN
cana-5776	11	20	,	,	PUNCT
cana-5776	11	21	or	or	CCONJ
cana-5776	11	22	deg(𝑣	deg(𝑣	PROPN
cana-5776	11	23	)	)	PUNCT
cana-5776	11	24	.	.	PUNCT
cana-5776	12	1	involutory	involutory	ADJ
cana-5776	12	2	cayley	cayley	NOUN
cana-5776	12	3	graphs	graph	NOUN
cana-5776	12	4	were	be	AUX
cana-5776	12	5	introduced	introduce	VERB
cana-5776	12	6	by	by	ADP
cana-5776	12	7	venkata	venkata	PROPN
cana-5776	12	8	anusha	anusha	PROPN
cana-5776	12	9	et	et	PROPN
cana-5776	12	10	al	al	PROPN
cana-5776	12	11	.	.	PUNCT
cana-5776	13	1	[	[	X
cana-5776	13	2	8	8	NUM
cana-5776	13	3	]	]	PUNCT
cana-5776	13	4	,	,	PUNCT
cana-5776	13	5	who	who	PRON
cana-5776	13	6	also	also	ADV
cana-5776	13	7	looked	look	VERB
cana-5776	13	8	into	into	ADP
cana-5776	13	9	some	some	PRON
cana-5776	13	10	of	of	ADP
cana-5776	13	11	its	its	PRON
cana-5776	13	12	characteristics	characteristic	NOUN
cana-5776	13	13	.	.	PUNCT
cana-5776	14	1	involutory	involutory	NOUN
cana-5776	14	2	addition	addition	NOUN
cana-5776	14	3	cayley	cayley	NOUN
cana-5776	14	4	graphs	graph	NOUN
cana-5776	14	5	were	be	AUX
cana-5776	14	6	first	first	ADV
cana-5776	14	7	shown	show	VERB
cana-5776	14	8	by	by	ADP
cana-5776	14	9	shanmuga	shanmuga	PROPN
cana-5776	14	10	priya	priya	PROPN
cana-5776	14	11	et	et	PROPN
cana-5776	14	12	al	al	PROPN
cana-5776	14	13	.	.	PUNCT
cana-5776	15	1	[	[	X
cana-5776	15	2	7	7	NUM
cana-5776	15	3	]	]	PUNCT
cana-5776	15	4	,	,	PUNCT
cana-5776	15	5	who	who	PRON
cana-5776	15	6	also	also	ADV
cana-5776	15	7	examined	examine	VERB
cana-5776	15	8	several	several	ADJ
cana-5776	15	9	of	of	ADP
cana-5776	15	10	their	their	PRON
cana-5776	15	11	characteristics.in	characteristics.in	NOUN
cana-5776	15	12	addition	addition	NOUN
cana-5776	15	13	to	to	ADP
cana-5776	15	14	studying	study	VERB
cana-5776	15	15	the	the	DET
cana-5776	15	16	characterization	characterization	NOUN
cana-5776	15	17	of	of	ADP
cana-5776	15	18	the	the	DET
cana-5776	15	19	set	set	NOUN
cana-5776	15	20	of	of	ADP
cana-5776	15	21	involutory	involutory	ADJ
cana-5776	15	22	elements	element	NOUN
cana-5776	15	23	of	of	ADP
cana-5776	15	24	(	(	PUNCT
cana-5776	15	25	𝑍𝑛,⨁,⨀	𝑍𝑛,⨁,⨀	NOUN
cana-5776	15	26	)	)	PUNCT
cana-5776	15	27	,	,	PUNCT
cana-5776	15	28	prameela	prameela	PROPN
cana-5776	15	29	rani	rani	PROPN
cana-5776	15	30	et	et	PROPN
cana-5776	15	31	al	al	PROPN
cana-5776	15	32	.	.	PUNCT
cana-5776	16	1	[	[	X
cana-5776	16	2	3,4	3,4	NUM
cana-5776	16	3	]	]	PUNCT
cana-5776	16	4	communications	communication	NOUN
cana-5776	16	5	on	on	ADP
cana-5776	16	6	applied	apply	VERB
cana-5776	16	7	nonlinear	nonlinear	ADJ
cana-5776	16	8	analysis	analysis	NOUN
cana-5776	16	9	issn	issn	NOUN
cana-5776	16	10	:	:	PUNCT
cana-5776	16	11	1074	1074	NUM
cana-5776	16	12	-	-	PUNCT
cana-5776	16	13	133x	133x	NUM
cana-5776	16	14	vol	vol	NOUN
cana-5776	16	15	31	31	NUM
cana-5776	16	16	no	no	NOUN
cana-5776	16	17	.	.	PUNCT
cana-5776	17	1	7s	7	NOUN
cana-5776	17	2	(	(	PUNCT
cana-5776	17	3	2024	2024	NUM
cana-5776	17	4	)	)	PUNCT
cana-5776	17	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	18	1	781	781	NUM
cana-5776	18	2	investigated	investigate	VERB
cana-5776	18	3	certain	certain	ADJ
cana-5776	18	4	properties	property	NOUN
cana-5776	18	5	of	of	ADP
cana-5776	18	6	the	the	DET
cana-5776	18	7	domatic	domatic	ADJ
cana-5776	18	8	number	number	NOUN
cana-5776	18	9	and	and	CCONJ
cana-5776	18	10	dominance	dominance	NOUN
cana-5776	18	11	of	of	ADP
cana-5776	18	12	the	the	DET
cana-5776	18	13	involutory	involutory	ADJ
cana-5776	18	14	cayley	cayley	NOUN
cana-5776	18	15	graph	graph	NOUN
cana-5776	18	16	.	.	PUNCT
cana-5776	19	1	lavanya	lavanya	NOUN
cana-5776	19	2	et	et	PROPN
cana-5776	19	3	al	al	PROPN
cana-5776	19	4	.	.	PUNCT
cana-5776	20	1	[	[	X
cana-5776	20	2	6	6	NUM
cana-5776	20	3	]	]	PUNCT
cana-5776	20	4	investigated	investigate	VERB
cana-5776	20	5	the	the	DET
cana-5776	20	6	involutory	involutory	NOUN
cana-5776	20	7	addition	addition	NOUN
cana-5776	20	8	cayley	cayley	NOUN
cana-5776	20	9	graphs	graph	NOUN
cana-5776	20	10	edge	edge	VERB
cana-5776	20	11	domination	domination	NOUN
cana-5776	20	12	number	number	NOUN
cana-5776	20	13	and	and	CCONJ
cana-5776	20	14	total	total	ADJ
cana-5776	20	15	edge	edge	NOUN
cana-5776	20	16	domination	domination	NOUN
cana-5776	20	17	number	number	NOUN
cana-5776	20	18	.	.	PUNCT
cana-5776	21	1	for	for	ADP
cana-5776	21	2	a	a	DET
cana-5776	21	3	positive	positive	ADJ
cana-5776	21	4	integer	integer	NOUN
cana-5776	21	5	𝑛	𝑛	ADP
cana-5776	21	6	>	>	SYM
cana-5776	21	7	1	1	NUM
cana-5776	21	8	,	,	PUNCT
cana-5776	21	9	the	the	DET
cana-5776	21	10	involutory	involutory	NOUN
cana-5776	21	11	addition	addition	NOUN
cana-5776	21	12	cayley	cayley	PROPN
cana-5776	21	13	graph	graph	NOUN
cana-5776	21	14	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	21	15	,	,	PUNCT
cana-5776	21	16	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	21	17	)	)	PUNCT
cana-5776	21	18	,	,	PUNCT
cana-5776	21	19	is	be	AUX
cana-5776	21	20	the	the	DET
cana-5776	21	21	graph	graph	NOUN
cana-5776	21	22	whose	whose	DET
cana-5776	21	23	vertex	vertex	NOUN
cana-5776	21	24	set	set	NOUN
cana-5776	21	25	is	be	AUX
cana-5776	21	26	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	21	27	=	=	PUNCT
cana-5776	21	28	{	{	PUNCT
cana-5776	21	29	0,1,2,3	0,1,2,3	NUM
cana-5776	21	30	,	,	PUNCT
cana-5776	21	31	…	…	PUNCT
cana-5776	21	32	,	,	PUNCT
cana-5776	21	33	𝑛	𝑛	DET
cana-5776	21	34	−	−	PROPN
cana-5776	21	35	1	1	NUM
cana-5776	21	36	}	}	PUNCT
cana-5776	21	37	and	and	CCONJ
cana-5776	21	38	edge	edge	VERB
cana-5776	21	39	set	set	VERB
cana-5776	21	40	𝐸(𝐺𝑛	𝐸(𝐺𝑛	NOUN
cana-5776	21	41	)	)	PUNCT
cana-5776	21	42	=	=	NOUN
cana-5776	21	43	{	{	PUNCT
cana-5776	21	44	𝑥𝑦	𝑥𝑦	NOUN
cana-5776	21	45	/𝑥	/𝑥	INTJ
cana-5776	21	46	,	,	PUNCT
cana-5776	21	47	𝑦	𝑦	NOUN
cana-5776	21	48	𝜖	𝜖	X
cana-5776	21	49	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	21	50	,	,	PUNCT
cana-5776	21	51	𝑥	𝑥	PROPN
cana-5776	22	1	+	+	SYM
cana-5776	22	2	𝑦	𝑦	X
cana-5776	22	3	𝜖	𝜖	X
cana-5776	22	4	𝐼𝑣	𝐼𝑣	NOUN
cana-5776	22	5	}	}	PUNCT
cana-5776	22	6	,	,	PUNCT
cana-5776	22	7	where	where	SCONJ
cana-5776	22	8	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	22	9	=	=	PRON
cana-5776	22	10	{	{	PUNCT
cana-5776	22	11	𝑥	𝑥	X
cana-5776	22	12	𝜖	𝜖	X
cana-5776	22	13	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	22	14	∶	∶	NOUN
cana-5776	22	15	𝑥	𝑥	DET
cana-5776	22	16	2	2	NUM
cana-5776	22	17	≡	≡	PROPN
cana-5776	22	18	1(𝑚𝑜𝑑	1(𝑚𝑜𝑑	ADJ
cana-5776	22	19	𝑛	𝑛	NOUN
cana-5776	22	20	)	)	PUNCT
cana-5776	22	21	}	}	PUNCT
cana-5776	22	22	is	be	AUX
cana-5776	22	23	the	the	DET
cana-5776	22	24	set	set	NOUN
cana-5776	22	25	of	of	ADP
cana-5776	22	26	involutory	involutory	ADJ
cana-5776	22	27	elements	element	NOUN
cana-5776	22	28	of	of	ADP
cana-5776	22	29	𝑍𝑛.	𝑍𝑛.	PROPN
cana-5776	22	30	the	the	DET
cana-5776	22	31	theory	theory	NOUN
cana-5776	22	32	of	of	ADP
cana-5776	22	33	domination	domination	NOUN
cana-5776	22	34	in	in	ADP
cana-5776	22	35	graphs	graph	NOUN
cana-5776	22	36	was	be	AUX
cana-5776	22	37	introduced	introduce	VERB
cana-5776	22	38	in	in	ADP
cana-5776	22	39	1958	1958	NUM
cana-5776	22	40	by	by	ADP
cana-5776	22	41	claude	claude	PROPN
cana-5776	22	42	berge[1	berge[1	PROPN
cana-5776	22	43	]	]	PUNCT
cana-5776	22	44	in	in	ADP
cana-5776	22	45	which	which	PRON
cana-5776	22	46	he	he	PRON
cana-5776	22	47	used	use	VERB
cana-5776	22	48	the	the	DET
cana-5776	22	49	concept	concept	NOUN
cana-5776	22	50	‘	'	PUNCT
cana-5776	22	51	coefficient	coefficient	NOUN
cana-5776	22	52	of	of	ADP
cana-5776	22	53	external	external	ADJ
cana-5776	22	54	stability	stability	NOUN
cana-5776	22	55	’	'	PUNCT
cana-5776	22	56	to	to	PART
cana-5776	22	57	refer	refer	VERB
cana-5776	22	58	the	the	DET
cana-5776	22	59	domination	domination	NOUN
cana-5776	22	60	number	number	NOUN
cana-5776	22	61	of	of	ADP
cana-5776	22	62	a	a	DET
cana-5776	22	63	graph	graph	NOUN
cana-5776	22	64	.	.	PUNCT
cana-5776	23	1	in	in	ADP
cana-5776	23	2	1962.dominating	1962.dominate	VERB
cana-5776	23	3	sets	set	NOUN
cana-5776	23	4	of	of	ADP
cana-5776	23	5	edges	edge	NOUN
cana-5776	23	6	were	be	AUX
cana-5776	23	7	studied	study	VERB
cana-5776	23	8	by	by	ADP
cana-5776	23	9	mitchell	mitchell	PROPN
cana-5776	23	10	and	and	CCONJ
cana-5776	23	11	hedetniemi[3].connected	hedetniemi[3].connecte	VERB
cana-5776	23	12	edge	edge	NOUN
cana-5776	23	13	domination	domination	NOUN
cana-5776	23	14	was	be	AUX
cana-5776	23	15	introduced	introduce	VERB
cana-5776	23	16	by	by	ADP
cana-5776	23	17	kulli	kulli	PROPN
cana-5776	23	18	and	and	CCONJ
cana-5776	23	19	sigarkanti[5].entire	sigarkanti[5].entire	PROPN
cana-5776	23	20	domination	domination	NOUN
cana-5776	23	21	was	be	AUX
cana-5776	23	22	introduced	introduce	VERB
cana-5776	23	23	by	by	ADP
cana-5776	23	24	kulli[6	kulli[6	PROPN
cana-5776	23	25	]	]	PUNCT
cana-5776	23	26	.	.	PUNCT
cana-5776	24	1	kulli	kulli	PROPN
cana-5776	24	2	sigarkanti	sigarkanti	PROPN
cana-5776	24	3	and	and	CCONJ
cana-5776	24	4	soner	soner	NOUN
cana-5776	24	5	established	establish	VERB
cana-5776	24	6	a	a	DET
cana-5776	24	7	relationship	relationship	NOUN
cana-5776	24	8	between	between	ADP
cana-5776	24	9	the	the	DET
cana-5776	24	10	domination	domination	NOUN
cana-5776	24	11	,	,	PUNCT
cana-5776	24	12	edge	edge	NOUN
cana-5776	24	13	domination	domination	NOUN
cana-5776	24	14	and	and	CCONJ
cana-5776	24	15	entire	entire	ADJ
cana-5776	24	16	domination	domination	NOUN
cana-5776	24	17	number	number	NOUN
cana-5776	24	18	.	.	PUNCT
cana-5776	25	1	2	2	NUM
cana-5776	25	2	connected	connect	VERB
cana-5776	25	3	edge	edge	NOUN
cana-5776	25	4	domination	domination	NOUN
cana-5776	25	5	the	the	DET
cana-5776	25	6	theory	theory	NOUN
cana-5776	25	7	of	of	ADP
cana-5776	25	8	domination	domination	NOUN
cana-5776	25	9	in	in	ADP
cana-5776	25	10	graphs	graph	NOUN
cana-5776	25	11	was	be	AUX
cana-5776	25	12	introduced	introduce	VERB
cana-5776	25	13	in	in	ADP
cana-5776	25	14	1958	1958	NUM
cana-5776	25	15	by	by	ADP
cana-5776	25	16	claude	claude	PROPN
cana-5776	25	17	berge[1	berge[1	PROPN
cana-5776	25	18	]	]	PUNCT
cana-5776	25	19	in	in	ADP
cana-5776	25	20	which	which	PRON
cana-5776	25	21	he	he	PRON
cana-5776	25	22	used	use	VERB
cana-5776	25	23	the	the	DET
cana-5776	25	24	concept	concept	NOUN
cana-5776	25	25	‘	'	PUNCT
cana-5776	25	26	coefficient	coefficient	NOUN
cana-5776	25	27	of	of	ADP
cana-5776	25	28	external	external	ADJ
cana-5776	25	29	stability	stability	NOUN
cana-5776	25	30	’	'	PUNCT
cana-5776	25	31	to	to	PART
cana-5776	25	32	refer	refer	VERB
cana-5776	25	33	the	the	DET
cana-5776	25	34	domination	domination	NOUN
cana-5776	25	35	number	number	NOUN
cana-5776	25	36	of	of	ADP
cana-5776	25	37	a	a	DET
cana-5776	25	38	graph	graph	NOUN
cana-5776	25	39	.	.	PUNCT
cana-5776	26	1	in	in	ADP
cana-5776	26	2	1962	1962	NUM
cana-5776	26	3	,	,	PUNCT
cana-5776	26	4	oystein	oystein	PROPN
cana-5776	26	5	ore[4	ore[4	X
cana-5776	26	6	]	]	PUNCT
cana-5776	26	7	wrote	write	VERB
cana-5776	26	8	another	another	DET
cana-5776	26	9	book	book	NOUN
cana-5776	26	10	on	on	ADP
cana-5776	26	11	graph	graph	NOUN
cana-5776	26	12	theory	theory	NOUN
cana-5776	26	13	,	,	PUNCT
cana-5776	26	14	in	in	ADP
cana-5776	26	15	which	which	PRON
cana-5776	26	16	he	he	PRON
cana-5776	26	17	studied	study	VERB
cana-5776	26	18	the	the	DET
cana-5776	26	19	concept	concept	NOUN
cana-5776	26	20	of	of	ADP
cana-5776	26	21	domination	domination	NOUN
cana-5776	26	22	using	use	VERB
cana-5776	26	23	the	the	DET
cana-5776	26	24	terms	term	NOUN
cana-5776	26	25	‘	'	PUNCT
cana-5776	26	26	dominating	dominating	NOUN
cana-5776	26	27	set	set	NOUN
cana-5776	26	28	’	'	PUNCT
cana-5776	26	29	and	and	CCONJ
cana-5776	26	30	‘	'	PUNCT
cana-5776	26	31	domination	domination	NOUN
cana-5776	26	32	number	number	NOUN
cana-5776	26	33	’	'	PUNCT
cana-5776	26	34	with	with	ADP
cana-5776	26	35	notation	notation	NOUN
cana-5776	26	36	𝑑(𝐺	𝑑(𝐺	NUM
cana-5776	26	37	)	)	PUNCT
cana-5776	26	38	for	for	ADP
cana-5776	26	39	the	the	DET
cana-5776	26	40	first	first	ADJ
cana-5776	26	41	time	time	NOUN
cana-5776	26	42	.	.	PUNCT
cana-5776	27	1	cockayne	cockayne	NOUN
cana-5776	27	2	et	et	PROPN
cana-5776	27	3	al[2	al[2	PROPN
cana-5776	27	4	]	]	PUNCT
cana-5776	27	5	discussed	discuss	VERB
cana-5776	27	6	the	the	DET
cana-5776	27	7	review	review	NOUN
cana-5776	27	8	of	of	ADP
cana-5776	27	9	results	result	NOUN
cana-5776	27	10	and	and	CCONJ
cana-5776	27	11	applications	application	NOUN
cana-5776	27	12	concerning	concern	VERB
cana-5776	27	13	dominating	dominating	NOUN
cana-5776	27	14	sets	set	NOUN
cana-5776	27	15	in	in	ADP
cana-5776	27	16	graphs.dominating	graphs.dominate	VERB
cana-5776	27	17	sets	set	NOUN
cana-5776	27	18	of	of	ADP
cana-5776	27	19	edges	edge	NOUN
cana-5776	27	20	were	be	AUX
cana-5776	27	21	studied	study	VERB
cana-5776	27	22	by	by	ADP
cana-5776	27	23	mitchell	mitchell	PROPN
cana-5776	27	24	and	and	CCONJ
cana-5776	27	25	hedetniemi[3].connected	hedetniemi[3].connecte	VERB
cana-5776	27	26	edge	edge	NOUN
cana-5776	27	27	domination	domination	NOUN
cana-5776	27	28	was	be	AUX
cana-5776	27	29	introduced	introduce	VERB
cana-5776	27	30	by	by	ADP
cana-5776	27	31	kulli	kulli	PROPN
cana-5776	27	32	and	and	CCONJ
cana-5776	27	33	sigarkanti[5	sigarkanti[5	PROPN
cana-5776	27	34	]	]	X
cana-5776	27	35	.	.	PUNCT
cana-5776	28	1	an	an	DET
cana-5776	28	2	edge	edge	NOUN
cana-5776	28	3	domination	domination	NOUN
cana-5776	28	4	set	set	VERB
cana-5776	28	5	𝐹	𝐹	PROPN
cana-5776	28	6	of	of	ADP
cana-5776	28	7	a	a	DET
cana-5776	28	8	graph	graph	NOUN
cana-5776	28	9	𝐺	𝐺	NOUN
cana-5776	28	10	is	be	AUX
cana-5776	28	11	a	a	DET
cana-5776	28	12	connected	connected	ADJ
cana-5776	28	13	edge	edge	NOUN
cana-5776	28	14	domination	domination	NOUN
cana-5776	28	15	set	set	VERB
cana-5776	28	16	if	if	SCONJ
cana-5776	28	17	the	the	DET
cana-5776	28	18	induced	induced	ADJ
cana-5776	28	19	subgraph	subgraph	NOUN
cana-5776	28	20	<	<	X
cana-5776	28	21	𝐹	𝐹	PROPN
cana-5776	28	22	>	>	X
cana-5776	28	23	is	be	AUX
cana-5776	28	24	connected	connect	VERB
cana-5776	28	25	.	.	PUNCT
cana-5776	29	1	the	the	DET
cana-5776	29	2	connected	connected	ADJ
cana-5776	29	3	edge	edge	NOUN
cana-5776	29	4	domination	domination	NOUN
cana-5776	29	5	number	number	NOUN
cana-5776	29	6	𝛾𝑐	𝛾𝑐	ADP
cana-5776	29	7	′(𝐺	′(𝐺	NOUN
cana-5776	29	8	)	)	PUNCT
cana-5776	29	9	of	of	ADP
cana-5776	29	10	𝐺	𝐺	PROPN
cana-5776	29	11	is	be	AUX
cana-5776	29	12	the	the	DET
cana-5776	29	13	minimum	minimum	ADJ
cana-5776	29	14	cardinality	cardinality	NOUN
cana-5776	29	15	of	of	ADP
cana-5776	29	16	a	a	DET
cana-5776	29	17	connected	connected	ADJ
cana-5776	29	18	edge	edge	NOUN
cana-5776	29	19	dominating	dominating	NOUN
cana-5776	29	20	set	set	NOUN
cana-5776	29	21	.	.	PUNCT
cana-5776	30	1	theorem	theorem	VERB
cana-5776	30	2	2.1	2.1	NUM
cana-5776	30	3	:	:	PUNCT
cana-5776	30	4	for	for	ADP
cana-5776	30	5	the	the	DET
cana-5776	30	6	involutory	involutory	NOUN
cana-5776	30	7	addition	addition	NOUN
cana-5776	30	8	cayley	cayley	NOUN
cana-5776	30	9	graph	graph	NOUN
cana-5776	30	10	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	30	11	,	,	PUNCT
cana-5776	30	12	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	30	13	)	)	PUNCT
cana-5776	30	14	,	,	PUNCT
cana-5776	30	15	if	if	SCONJ
cana-5776	30	16	n	n	PRON
cana-5776	30	17	is	be	AUX
cana-5776	30	18	even	even	ADV
cana-5776	30	19	,	,	PUNCT
cana-5776	30	20	𝑛	𝑛	PROPN
cana-5776	30	21	>	>	X
cana-5776	30	22	2	2	NUM
cana-5776	30	23	then	then	ADV
cana-5776	30	24	the	the	DET
cana-5776	30	25	connected	connected	ADJ
cana-5776	30	26	edge	edge	NOUN
cana-5776	30	27	domination	domination	NOUN
cana-5776	30	28	number	number	NOUN
cana-5776	30	29	is	be	AUX
cana-5776	30	30	𝛾𝑐	𝛾𝑐	ADP
cana-5776	30	31	′(𝐺+(𝑍𝑛	′(𝐺+(𝑍𝑛	PROPN
cana-5776	30	32	,	,	PUNCT
cana-5776	30	33	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	30	34	)	)	PUNCT
cana-5776	30	35	)	)	PUNCT
cana-5776	31	1	=	=	SYM
cana-5776	32	1	𝑛	𝑛	DET
cana-5776	32	2	−	−	NOUN
cana-5776	32	3	2	2	NUM
cana-5776	32	4	proof	proof	NOUN
cana-5776	32	5	:	:	PUNCT
cana-5776	32	6	consider	consider	VERB
cana-5776	32	7	a	a	DET
cana-5776	32	8	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	32	9	,	,	PUNCT
cana-5776	32	10	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	32	11	)	)	PUNCT
cana-5776	32	12	with	with	ADP
cana-5776	32	13	vertex	vertex	NOUN
cana-5776	32	14	set	set	VERB
cana-5776	33	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	33	2	=	=	PUNCT
cana-5776	33	3	{	{	PUNCT
cana-5776	33	4	0,1,2,3	0,1,2,3	NUM
cana-5776	33	5	,	,	PUNCT
cana-5776	33	6	…	…	PUNCT
cana-5776	33	7	,	,	PUNCT
cana-5776	33	8	𝑛	𝑛	DET
cana-5776	33	9	−	−	NOUN
cana-5776	33	10	1	1	NUM
cana-5776	33	11	}	}	PUNCT
cana-5776	33	12	where	where	SCONJ
cana-5776	33	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	33	14	denotes	denote	VERB
cana-5776	33	15	the	the	DET
cana-5776	33	16	set	set	NOUN
cana-5776	33	17	of	of	ADP
cana-5776	33	18	involutory	involutory	ADJ
cana-5776	33	19	elements	element	NOUN
cana-5776	33	20	in	in	ADP
cana-5776	33	21	𝑍𝑛.	𝑍𝑛.	PROPN
cana-5776	33	22	communications	communication	NOUN
cana-5776	33	23	on	on	ADP
cana-5776	33	24	applied	apply	VERB
cana-5776	33	25	nonlinear	nonlinear	ADJ
cana-5776	33	26	analysis	analysis	NOUN
cana-5776	33	27	issn	issn	NOUN
cana-5776	33	28	:	:	PUNCT
cana-5776	33	29	1074	1074	NUM
cana-5776	33	30	-	-	PUNCT
cana-5776	33	31	133x	133x	NUM
cana-5776	33	32	vol	vol	NOUN
cana-5776	33	33	31	31	NUM
cana-5776	33	34	no	no	NOUN
cana-5776	33	35	.	.	PUNCT
cana-5776	34	1	7s	7	NOUN
cana-5776	34	2	(	(	PUNCT
cana-5776	34	3	2024	2024	NUM
cana-5776	34	4	)	)	PUNCT
cana-5776	34	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	34	6	782	782	NUM
cana-5776	34	7	let	let	VERB
cana-5776	34	8	𝑛	𝑛	PRON
cana-5776	34	9	be	be	AUX
cana-5776	34	10	even	even	ADV
cana-5776	34	11	and	and	CCONJ
cana-5776	34	12	𝑛	𝑛	ADJ
cana-5776	34	13	>	>	X
cana-5776	34	14	2	2	X
cana-5776	34	15	.	.	PUNCT
cana-5776	34	16	|𝐸|	|𝐸|	NOUN
cana-5776	34	17	=	=	SYM
cana-5776	34	18	𝑛	𝑛	PROPN
cana-5776	34	19	or	or	CCONJ
cana-5776	34	20	2𝑛	2𝑛	PROPN
cana-5776	34	21	if	if	SCONJ
cana-5776	34	22	|𝐸|	|𝐸|	NOUN
cana-5776	34	23	=	=	SYM
cana-5776	34	24	𝑛	𝑛	NOUN
cana-5776	34	25	,	,	PUNCT
cana-5776	34	26	graph	graph	NOUN
cana-5776	34	27	is	be	AUX
cana-5776	34	28	a	a	DET
cana-5776	34	29	hamilton	hamilton	PROPN
cana-5776	34	30	cycle	cycle	NOUN
cana-5776	34	31	.	.	PUNCT
cana-5776	35	1	if	if	SCONJ
cana-5776	35	2	|𝐸|	|𝐸|	NOUN
cana-5776	35	3	=	=	SYM
cana-5776	35	4	2𝑛	2𝑛	NUM
cana-5776	35	5	,	,	PUNCT
cana-5776	35	6	graph	graph	NOUN
cana-5776	35	7	contains	contain	VERB
cana-5776	35	8	two	two	NUM
cana-5776	35	9	hamilton	hamilton	PROPN
cana-5776	35	10	cycles	cycle	NOUN
cana-5776	35	11	,	,	PUNCT
cana-5776	35	12	each	each	PRON
cana-5776	35	13	contains	contain	VERB
cana-5776	35	14	`	`	PUNCT
cana-5776	35	15	𝑛′	𝑛′	NOUN
cana-5776	35	16	number	number	NOUN
cana-5776	35	17	of	of	ADP
cana-5776	35	18	edges	edge	NOUN
cana-5776	35	19	.	.	PUNCT
cana-5776	36	1	from	from	ADP
cana-5776	36	2	first	first	PROPN
cana-5776	36	3	hamilton	hamilton	PROPN
cana-5776	36	4	cycle	cycle	NOUN
cana-5776	36	5	,	,	PUNCT
cana-5776	36	6	consider	consider	VERB
cana-5776	36	7	𝐹	𝐹	PROPN
cana-5776	36	8	=	=	PRON
cana-5776	36	9	{	{	PUNCT
cana-5776	36	10	𝑒𝑖/𝑒𝑖	𝑒𝑖/𝑒𝑖	NUM
cana-5776	36	11	≠	≠	PROPN
cana-5776	36	12	𝑒1	𝑒1	NOUN
cana-5776	36	13	,	,	PUNCT
cana-5776	36	14	𝑒2	𝑒2	NOUN
cana-5776	36	15	}	}	PUNCT
cana-5776	36	16	.	.	PUNCT
cana-5776	37	1	now	now	ADV
cana-5776	37	2	every	every	DET
cana-5776	37	3	edge	edge	NOUN
cana-5776	37	4	in	in	ADP
cana-5776	37	5	𝐸	𝐸	PROPN
cana-5776	37	6	−	−	NOUN
cana-5776	37	7	𝐹	𝐹	PROPN
cana-5776	37	8	is	be	AUX
cana-5776	37	9	adjacent	adjacent	ADJ
cana-5776	37	10	to	to	PART
cana-5776	37	11	atleast	atleast	VERB
cana-5776	37	12	one	one	NUM
cana-5776	37	13	edge	edge	NOUN
cana-5776	37	14	in	in	ADP
cana-5776	37	15	𝐹	𝐹	PROPN
cana-5776	37	16	,	,	PUNCT
cana-5776	37	17	𝐹	𝐹	PROPN
cana-5776	37	18	is	be	AUX
cana-5776	37	19	induced	induce	VERB
cana-5776	37	20	subgraph	subgraph	NOUN
cana-5776	37	21	of	of	ADP
cana-5776	37	22	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	37	23	,	,	PUNCT
cana-5776	37	24	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	37	25	)	)	PUNCT
cana-5776	37	26	and	and	CCONJ
cana-5776	37	27	connected	connect	VERB
cana-5776	37	28	.	.	PUNCT
cana-5776	38	1	then	then	ADV
cana-5776	38	2	𝐹	𝐹	PROPN
cana-5776	38	3	becomes	become	VERB
cana-5776	38	4	connected	connected	ADJ
cana-5776	38	5	edge	edge	NOUN
cana-5776	38	6	dominating	dominating	NOUN
cana-5776	38	7	set	set	NOUN
cana-5776	38	8	of	of	ADP
cana-5776	38	9	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	38	10	,	,	PUNCT
cana-5776	38	11	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	38	12	)	)	PUNCT
cana-5776	38	13	and	and	CCONJ
cana-5776	38	14	it	it	PRON
cana-5776	38	15	is	be	AUX
cana-5776	38	16	minimum	minimum	ADJ
cana-5776	38	17	.	.	PUNCT
cana-5776	39	1	therefore	therefore	ADV
cana-5776	39	2	|𝐹|	|𝐹|	NOUN
cana-5776	39	3	=	=	NOUN
cana-5776	39	4	|𝐸|	|𝐸|	NOUN
cana-5776	39	5	−	−	PROPN
cana-5776	39	6	2	2	NUM
cana-5776	39	7	(	(	PUNCT
cana-5776	39	8	𝐸	𝐸	NOUN
cana-5776	39	9	=	=	PUNCT
cana-5776	39	10	set	set	NOUN
cana-5776	39	11	of	of	ADP
cana-5776	39	12	edges	edge	NOUN
cana-5776	39	13	in	in	ADP
cana-5776	39	14	one	one	NUM
cana-5776	39	15	path	path	NOUN
cana-5776	39	16	)	)	PUNCT
cana-5776	39	17	=	=	SYM
cana-5776	40	1	𝑛	𝑛	DET
cana-5776	40	2	−	−	NUM
cana-5776	40	3	2	2	NUM
cana-5776	40	4	.	.	PUNCT
cana-5776	41	1	hence	hence	ADV
cana-5776	41	2	,	,	PUNCT
cana-5776	41	3	𝛾𝑐	𝛾𝑐	ADP
cana-5776	41	4	′(𝐺+(𝑍𝑛	′(𝐺+(𝑍𝑛	PROPN
cana-5776	41	5	,	,	PUNCT
cana-5776	41	6	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	41	7	)	)	PUNCT
cana-5776	41	8	)	)	PUNCT
cana-5776	42	1	=	=	SYM
cana-5776	43	1	𝑛	𝑛	PRON
cana-5776	43	2	−	−	NUM
cana-5776	43	3	2	2	NUM
cana-5776	43	4	.	.	PUNCT
cana-5776	43	5	theorem	theorem	VERB
cana-5776	43	6	2.2	2.2	NUM
cana-5776	43	7	:	:	PUNCT
cana-5776	43	8	for	for	ADP
cana-5776	43	9	the	the	DET
cana-5776	43	10	involutory	involutory	NOUN
cana-5776	43	11	addition	addition	NOUN
cana-5776	43	12	cayley	cayley	NOUN
cana-5776	43	13	graph	graph	NOUN
cana-5776	43	14	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	43	15	,	,	PUNCT
cana-5776	43	16	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	43	17	)	)	PUNCT
cana-5776	43	18	,	,	PUNCT
cana-5776	43	19	if	if	SCONJ
cana-5776	43	20	𝑛	𝑛	PROPN
cana-5776	43	21	is	be	AUX
cana-5776	43	22	odd	odd	ADJ
cana-5776	43	23	,	,	PUNCT
cana-5776	43	24	𝑛	𝑛	PROPN
cana-5776	43	25	>	>	X
cana-5776	43	26	3	3	NUM
cana-5776	43	27	then	then	ADV
cana-5776	43	28	the	the	DET
cana-5776	43	29	connected	connected	ADJ
cana-5776	43	30	edge	edge	NOUN
cana-5776	43	31	domination	domination	NOUN
cana-5776	43	32	number	number	NOUN
cana-5776	43	33	is	be	AUX
cana-5776	43	34	𝛾𝑐	𝛾𝑐	ADP
cana-5776	43	35	′(𝐺+(𝑍𝑛	′(𝐺+(𝑍𝑛	PROPN
cana-5776	43	36	,	,	PUNCT
cana-5776	43	37	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	43	38	)	)	PUNCT
cana-5776	43	39	)	)	PUNCT
cana-5776	44	1	=	=	SYM
cana-5776	44	2	𝑛	𝑛	PRON
cana-5776	44	3	−	−	NOUN
cana-5776	44	4	3	3	NUM
cana-5776	44	5	proof	proof	NOUN
cana-5776	44	6	:	:	PUNCT
cana-5776	44	7	consider	consider	VERB
cana-5776	44	8	a	a	DET
cana-5776	44	9	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	44	10	,	,	PUNCT
cana-5776	44	11	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	44	12	)	)	PUNCT
cana-5776	44	13	with	with	ADP
cana-5776	44	14	vertex	vertex	NOUN
cana-5776	44	15	set	set	VERB
cana-5776	45	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	45	2	=	=	PUNCT
cana-5776	45	3	{	{	PUNCT
cana-5776	45	4	0,1,2,3	0,1,2,3	NUM
cana-5776	45	5	,	,	PUNCT
cana-5776	45	6	…	…	PUNCT
cana-5776	45	7	,	,	PUNCT
cana-5776	45	8	𝑛	𝑛	DET
cana-5776	45	9	−	−	NOUN
cana-5776	45	10	1	1	NUM
cana-5776	45	11	}	}	PUNCT
cana-5776	45	12	where	where	SCONJ
cana-5776	45	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	45	14	denotes	denote	VERB
cana-5776	45	15	the	the	DET
cana-5776	45	16	set	set	NOUN
cana-5776	45	17	of	of	ADP
cana-5776	45	18	involutory	involutory	ADJ
cana-5776	45	19	elements	element	NOUN
cana-5776	45	20	in	in	ADP
cana-5776	45	21	𝑍𝑛.	𝑍𝑛.	PROPN
cana-5776	45	22	let	let	VERB
cana-5776	45	23	𝑛	𝑛	PART
cana-5776	45	24	be	be	AUX
cana-5776	45	25	odd	odd	ADJ
cana-5776	45	26	,	,	PUNCT
cana-5776	45	27	𝑛	𝑛	PROPN
cana-5776	45	28	>	>	X
cana-5776	45	29	3	3	X
cana-5776	45	30	.	.	PUNCT
cana-5776	45	31	|𝐸|	|𝐸|	NOUN
cana-5776	45	32	=	=	SYM
cana-5776	45	33	𝑛	𝑛	PRON
cana-5776	45	34	−	−	NUM
cana-5776	45	35	1	1	NUM
cana-5776	45	36	or	or	CCONJ
cana-5776	45	37	𝑛	𝑛	PRON
cana-5776	45	38	−	−	NOUN
cana-5776	45	39	2	2	NUM
cana-5776	45	40	.	.	PUNCT
cana-5776	46	1	if	if	SCONJ
cana-5776	46	2	|𝐸|	|𝐸|	NOUN
cana-5776	46	3	=	=	X
cana-5776	46	4	𝑛	𝑛	PRON
cana-5776	46	5	−	−	PROPN
cana-5776	46	6	1	1	NUM
cana-5776	46	7	,	,	PUNCT
cana-5776	46	8	graph	graph	NOUN
cana-5776	46	9	is	be	AUX
cana-5776	46	10	a	a	DET
cana-5776	46	11	path	path	NOUN
cana-5776	46	12	.	.	PUNCT
cana-5776	47	1	if	if	SCONJ
cana-5776	47	2	|𝐸|	|𝐸|	NOUN
cana-5776	47	3	=	=	SYM
cana-5776	47	4	2𝑛	2𝑛	NOUN
cana-5776	47	5	−	−	PROPN
cana-5776	47	6	2	2	NUM
cana-5776	47	7	,	,	PUNCT
cana-5776	47	8	graph	graph	NOUN
cana-5776	47	9	contains	contain	VERB
cana-5776	47	10	two	two	NUM
cana-5776	47	11	paths	path	NOUN
cana-5776	47	12	and	and	CCONJ
cana-5776	47	13	each	each	DET
cana-5776	47	14	path	path	NOUN
cana-5776	47	15	contains	contain	VERB
cana-5776	47	16	𝑛	𝑛	DET
cana-5776	47	17	−	−	NUM
cana-5776	47	18	1	1	NUM
cana-5776	47	19	number	number	NOUN
cana-5776	47	20	of	of	ADP
cana-5776	47	21	edges	edge	NOUN
cana-5776	47	22	.	.	PUNCT
cana-5776	48	1	from	from	ADP
cana-5776	48	2	first	first	ADJ
cana-5776	48	3	path	path	NOUN
cana-5776	48	4	,	,	PUNCT
cana-5776	48	5	consider	consider	VERB
cana-5776	48	6	𝐹	𝐹	PROPN
cana-5776	48	7	=	=	PRON
cana-5776	48	8	{	{	PUNCT
cana-5776	48	9	𝑒𝑖/𝑒𝑖	𝑒𝑖/𝑒𝑖	NUM
cana-5776	48	10	≠	≠	PROPN
cana-5776	48	11	𝑒1	𝑒1	NOUN
cana-5776	48	12	,	,	PUNCT
cana-5776	48	13	𝑒𝑛	𝑒𝑛	NOUN
cana-5776	48	14	}	}	PUNCT
cana-5776	48	15	.	.	PUNCT
cana-5776	49	1	now	now	ADV
cana-5776	49	2	every	every	DET
cana-5776	49	3	edge	edge	NOUN
cana-5776	49	4	in	in	ADP
cana-5776	49	5	𝐸	𝐸	PROPN
cana-5776	49	6	−	−	NOUN
cana-5776	49	7	𝐹	𝐹	PROPN
cana-5776	49	8	is	be	AUX
cana-5776	49	9	adjacent	adjacent	ADJ
cana-5776	49	10	to	to	ADP
cana-5776	49	11	at	at	ADV
cana-5776	49	12	least	least	ADV
cana-5776	49	13	one	one	NUM
cana-5776	49	14	edge	edge	NOUN
cana-5776	49	15	in	in	ADP
cana-5776	49	16	𝐹	𝐹	PROPN
cana-5776	49	17	,	,	PUNCT
cana-5776	49	18	𝐹	𝐹	PROPN
cana-5776	49	19	is	be	AUX
cana-5776	49	20	induced	induce	VERB
cana-5776	49	21	subgraph	subgraph	NOUN
cana-5776	49	22	of	of	ADP
cana-5776	49	23	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	49	24	,	,	PUNCT
cana-5776	49	25	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	49	26	)	)	PUNCT
cana-5776	49	27	and	and	CCONJ
cana-5776	49	28	connected	connect	VERB
cana-5776	49	29	.	.	PUNCT
cana-5776	50	1	then	then	ADV
cana-5776	50	2	𝐹	𝐹	PROPN
cana-5776	50	3	becomes	become	VERB
cana-5776	50	4	connected	connected	ADJ
cana-5776	50	5	edge	edge	NOUN
cana-5776	50	6	dominating	dominating	NOUN
cana-5776	50	7	set	set	NOUN
cana-5776	50	8	of	of	ADP
cana-5776	50	9	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	50	10	,	,	PUNCT
cana-5776	50	11	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	50	12	)	)	PUNCT
cana-5776	50	13	and	and	CCONJ
cana-5776	50	14	it	it	PRON
cana-5776	50	15	is	be	AUX
cana-5776	50	16	minimum	minimum	ADJ
cana-5776	50	17	.	.	PUNCT
cana-5776	51	1	therefore	therefore	ADV
cana-5776	51	2	|𝐹|	|𝐹|	NOUN
cana-5776	51	3	=	=	NOUN
cana-5776	51	4	|𝐸|	|𝐸|	NOUN
cana-5776	51	5	−	−	PROPN
cana-5776	51	6	2	2	NUM
cana-5776	51	7	(	(	PUNCT
cana-5776	51	8	𝐸	𝐸	NOUN
cana-5776	51	9	=	=	PUNCT
cana-5776	51	10	set	set	NOUN
cana-5776	51	11	of	of	ADP
cana-5776	51	12	edges	edge	NOUN
cana-5776	51	13	in	in	ADP
cana-5776	51	14	one	one	NUM
cana-5776	51	15	path)=	path)=	NOUN
cana-5776	51	16	𝑛	𝑛	PRON
cana-5776	51	17	−	−	NOUN
cana-5776	51	18	3	3	NUM
cana-5776	51	19	.	.	X
cana-5776	51	20	hence	hence	ADV
cana-5776	51	21	𝛾𝑐	𝛾𝑐	ADP
cana-5776	51	22	′(𝐺+(𝑍𝑛	′(𝐺+(𝑍𝑛	PROPN
cana-5776	51	23	,	,	PUNCT
cana-5776	51	24	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	51	25	)	)	PUNCT
cana-5776	51	26	)	)	PUNCT
cana-5776	52	1	=	=	SYM
cana-5776	52	2	𝑛	𝑛	PRON
cana-5776	52	3	−	−	NUM
cana-5776	52	4	3	3	NUM
cana-5776	52	5	3	3	NUM
cana-5776	52	6	.	.	PUNCT
cana-5776	52	7	entire	entire	ADJ
cana-5776	52	8	domination	domination	NOUN
cana-5776	52	9	the	the	DET
cana-5776	52	10	term	term	NOUN
cana-5776	52	11	domination	domination	NOUN
cana-5776	52	12	was	be	AUX
cana-5776	52	13	first	first	ADV
cana-5776	52	14	used	use	VERB
cana-5776	52	15	by	by	ADP
cana-5776	52	16	ore[4	ore[4	NOUN
cana-5776	52	17	]	]	PUNCT
cana-5776	52	18	.	.	PUNCT
cana-5776	53	1	cockayne	cockayne	NOUN
cana-5776	53	2	et	et	PROPN
cana-5776	53	3	al[2	al[2	PROPN
cana-5776	53	4	]	]	PUNCT
cana-5776	53	5	discussed	discuss	VERB
cana-5776	53	6	the	the	DET
cana-5776	53	7	review	review	NOUN
cana-5776	53	8	of	of	ADP
cana-5776	53	9	results	result	NOUN
cana-5776	53	10	and	and	CCONJ
cana-5776	53	11	applications	application	NOUN
cana-5776	53	12	concerning	concern	VERB
cana-5776	53	13	dominating	dominating	NOUN
cana-5776	53	14	sets	set	NOUN
cana-5776	53	15	in	in	ADP
cana-5776	53	16	graphs	graph	NOUN
cana-5776	53	17	.	.	PUNCT
cana-5776	54	1	dominating	dominating	NOUN
cana-5776	54	2	sets	set	NOUN
cana-5776	54	3	of	of	ADP
cana-5776	54	4	edges	edge	NOUN
cana-5776	54	5	were	be	AUX
cana-5776	54	6	studied	study	VERB
cana-5776	54	7	by	by	ADP
cana-5776	54	8	mitchell	mitchell	PROPN
cana-5776	54	9	and	and	CCONJ
cana-5776	54	10	hedetniemi[3	hedetniemi[3	NOUN
cana-5776	54	11	]	]	PUNCT
cana-5776	54	12	.	.	PUNCT
cana-5776	55	1	entire	entire	ADJ
cana-5776	55	2	domination	domination	NOUN
cana-5776	55	3	was	be	AUX
cana-5776	55	4	introduced	introduce	VERB
cana-5776	55	5	by	by	ADP
cana-5776	55	6	kulli[6	kulli[6	PROPN
cana-5776	55	7	]	]	PUNCT
cana-5776	55	8	.	.	PUNCT
cana-5776	56	1	kulli	kulli	PROPN
cana-5776	56	2	singarkanti	singarkanti	VERB
cana-5776	56	3	and	and	CCONJ
cana-5776	56	4	soner	soner	NOUN
cana-5776	56	5	established	establish	VERB
cana-5776	56	6	a	a	DET
cana-5776	56	7	relationship	relationship	NOUN
cana-5776	56	8	between	between	ADP
cana-5776	56	9	the	the	DET
cana-5776	56	10	domination	domination	NOUN
cana-5776	56	11	,	,	PUNCT
cana-5776	56	12	edge	edge	NOUN
cana-5776	56	13	domination	domination	NOUN
cana-5776	56	14	and	and	CCONJ
cana-5776	56	15	entire	entire	ADJ
cana-5776	56	16	domination	domination	NOUN
cana-5776	56	17	number	number	NOUN
cana-5776	56	18	.	.	PUNCT
cana-5776	57	1	the	the	DET
cana-5776	57	2	vertices	vertex	NOUN
cana-5776	57	3	and	and	CCONJ
cana-5776	57	4	edges	edge	NOUN
cana-5776	57	5	of	of	ADP
cana-5776	57	6	a	a	DET
cana-5776	57	7	graph	graph	NOUN
cana-5776	57	8	𝐺	𝐺	NOUN
cana-5776	57	9	are	be	AUX
cana-5776	57	10	called	call	VERB
cana-5776	57	11	the	the	DET
cana-5776	57	12	elements	element	NOUN
cana-5776	57	13	of	of	ADP
cana-5776	57	14	𝐺.	𝐺.	NOUN
cana-5776	57	15	let	let	VERB
cana-5776	57	16	𝑥	𝑥	NOUN
cana-5776	57	17	and	and	CCONJ
cana-5776	57	18	𝑦	𝑦	NOUN
cana-5776	57	19	be	be	AUX
cana-5776	57	20	two	two	NUM
cana-5776	57	21	elements	element	NOUN
cana-5776	57	22	of	of	ADP
cana-5776	57	23	𝐺.	𝐺.	NOUN
cana-5776	57	24	we	we	PRON
cana-5776	57	25	say	say	VERB
cana-5776	57	26	𝑥	𝑥	PUNCT
cana-5776	57	27	dominates	dominate	VERB
cana-5776	57	28	𝑦	𝑦	NOUN
cana-5776	57	29	if	if	SCONJ
cana-5776	57	30	𝑥	𝑥	ADP
cana-5776	57	31	=	=	SYM
cana-5776	57	32	𝑦	𝑦	NOUN
cana-5776	57	33	or	or	CCONJ
cana-5776	57	34	if	if	SCONJ
cana-5776	57	35	𝑥	𝑥	PROPN
cana-5776	57	36	and	and	CCONJ
cana-5776	57	37	𝑦	𝑦	NOUN
cana-5776	57	38	are	be	AUX
cana-5776	57	39	adjacent	adjacent	ADJ
cana-5776	57	40	or	or	CCONJ
cana-5776	57	41	incident	incident	NOUN
cana-5776	57	42	.	.	PUNCT
cana-5776	58	1	thus	thus	ADV
cana-5776	58	2	a	a	DET
cana-5776	58	3	vertex	vertex	NOUN
cana-5776	58	4	𝑣	𝑣	ADP
cana-5776	58	5	of	of	ADP
cana-5776	58	6	𝐺	𝐺	PROPN
cana-5776	58	7	is	be	AUX
cana-5776	58	8	said	say	VERB
cana-5776	58	9	to	to	PART
cana-5776	58	10	dominate	dominate	VERB
cana-5776	58	11	itself	itself	PRON
cana-5776	58	12	,	,	PUNCT
cana-5776	58	13	all	all	PRON
cana-5776	58	14	vertices	vertice	VERB
cana-5776	58	15	adjacent	adjacent	ADJ
cana-5776	58	16	to	to	ADP
cana-5776	58	17	𝑣	𝑣	ADP
cana-5776	58	18	communications	communication	NOUN
cana-5776	58	19	on	on	ADP
cana-5776	58	20	applied	apply	VERB
cana-5776	58	21	nonlinear	nonlinear	ADJ
cana-5776	58	22	analysis	analysis	NOUN
cana-5776	58	23	issn	issn	NOUN
cana-5776	58	24	:	:	PUNCT
cana-5776	58	25	1074	1074	NUM
cana-5776	58	26	-	-	PUNCT
cana-5776	58	27	133x	133x	NUM
cana-5776	58	28	vol	vol	NOUN
cana-5776	58	29	31	31	NUM
cana-5776	58	30	no	no	NOUN
cana-5776	58	31	.	.	PUNCT
cana-5776	59	1	7s	7	NOUN
cana-5776	59	2	(	(	PUNCT
cana-5776	59	3	2024	2024	NUM
cana-5776	59	4	)	)	PUNCT
cana-5776	59	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	59	6	783	783	NUM
cana-5776	59	7	and	and	CCONJ
cana-5776	59	8	all	all	DET
cana-5776	59	9	edges	edge	NOUN
cana-5776	59	10	incident	incident	NOUN
cana-5776	59	11	to	to	ADP
cana-5776	59	12	it	it	PRON
cana-5776	59	13	.	.	PUNCT
cana-5776	60	1	similarly	similarly	ADV
cana-5776	60	2	,	,	PUNCT
cana-5776	60	3	an	an	DET
cana-5776	60	4	edge	edge	NOUN
cana-5776	60	5	𝑒	𝑒	PROPN
cana-5776	60	6	of	of	ADP
cana-5776	60	7	𝐺	𝐺	PROPN
cana-5776	60	8	dominates	dominate	VERB
cana-5776	60	9	itself	itself	PRON
cana-5776	60	10	,	,	PUNCT
cana-5776	60	11	the	the	DET
cana-5776	60	12	two	two	NUM
cana-5776	60	13	end	end	NOUN
cana-5776	60	14	vertices	vertex	NOUN
cana-5776	60	15	of	of	ADP
cana-5776	60	16	𝑒	𝑒	PROPN
cana-5776	60	17	and	and	CCONJ
cana-5776	60	18	all	all	DET
cana-5776	60	19	edges	edge	NOUN
cana-5776	60	20	adjacent	adjacent	ADJ
cana-5776	60	21	to	to	PART
cana-5776	60	22	𝑒.	𝑒.	VERB
cana-5776	60	23	a	a	DET
cana-5776	60	24	set	set	ADJ
cana-5776	60	25	𝑆	𝑆	PROPN
cana-5776	60	26	of	of	ADP
cana-5776	60	27	elements	element	NOUN
cana-5776	60	28	of	of	ADP
cana-5776	60	29	𝐺	𝐺	PROPN
cana-5776	60	30	is	be	AUX
cana-5776	60	31	an	an	DET
cana-5776	60	32	entire	entire	ADJ
cana-5776	60	33	dominating	dominating	NOUN
cana-5776	60	34	set	set	NOUN
cana-5776	60	35	if	if	SCONJ
cana-5776	60	36	every	every	DET
cana-5776	60	37	element	element	NOUN
cana-5776	60	38	not	not	PART
cana-5776	60	39	in	in	ADP
cana-5776	60	40	𝑆	𝑆	PROPN
cana-5776	60	41	is	be	AUX
cana-5776	60	42	either	either	CCONJ
cana-5776	60	43	adjacent	adjacent	ADJ
cana-5776	60	44	or	or	CCONJ
cana-5776	60	45	incident	incident	NOUN
cana-5776	60	46	to	to	ADP
cana-5776	60	47	atleast	atleast	VERB
cana-5776	60	48	one	one	NUM
cana-5776	60	49	element	element	NOUN
cana-5776	60	50	in	in	ADP
cana-5776	60	51	𝑆.	𝑆.	PROPN
cana-5776	60	52	equivalently	equivalently	ADV
cana-5776	60	53	,	,	PUNCT
cana-5776	60	54	a	a	DET
cana-5776	60	55	set	set	ADJ
cana-5776	60	56	𝑆	𝑆	PROPN
cana-5776	60	57	of	of	ADP
cana-5776	60	58	elements	element	NOUN
cana-5776	60	59	of	of	ADP
cana-5776	60	60	𝐺	𝐺	PROPN
cana-5776	60	61	is	be	AUX
cana-5776	60	62	an	an	DET
cana-5776	60	63	entire	entire	ADJ
cana-5776	60	64	dominating	dominating	NOUN
cana-5776	60	65	set	set	NOUN
cana-5776	60	66	if	if	SCONJ
cana-5776	60	67	each	each	DET
cana-5776	60	68	element	element	NOUN
cana-5776	60	69	in	in	ADP
cana-5776	60	70	𝐺	𝐺	PROPN
cana-5776	60	71	is	be	AUX
cana-5776	60	72	dominated	dominate	VERB
cana-5776	60	73	by	by	ADP
cana-5776	60	74	some	some	DET
cana-5776	60	75	element	element	NOUN
cana-5776	60	76	in	in	ADP
cana-5776	60	77	𝑆.	𝑆.	PROPN
cana-5776	60	78	the	the	DET
cana-5776	60	79	entire	entire	ADJ
cana-5776	60	80	domination	domination	NOUN
cana-5776	60	81	number	number	NOUN
cana-5776	60	82	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	NUM
cana-5776	60	83	)	)	PUNCT
cana-5776	60	84	is	be	AUX
cana-5776	60	85	the	the	DET
cana-5776	60	86	cardinality	cardinality	NOUN
cana-5776	60	87	of	of	ADP
cana-5776	60	88	a	a	DET
cana-5776	60	89	smallest	small	ADJ
cana-5776	60	90	entire	entire	ADJ
cana-5776	60	91	dominating	dominating	NOUN
cana-5776	60	92	set	set	NOUN
cana-5776	60	93	.	.	PUNCT
cana-5776	61	1	theorem	theorem	VERB
cana-5776	61	2	3.1	3.1	NUM
cana-5776	61	3	:	:	PUNCT
cana-5776	61	4	for	for	ADP
cana-5776	61	5	the	the	DET
cana-5776	61	6	involutory	involutory	NOUN
cana-5776	61	7	addition	addition	NOUN
cana-5776	61	8	cayley	cayley	NOUN
cana-5776	61	9	graph	graph	NOUN
cana-5776	61	10	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	61	11	,	,	PUNCT
cana-5776	61	12	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	61	13	)	)	PUNCT
cana-5776	61	14	,	,	PUNCT
cana-5776	61	15	if	if	SCONJ
cana-5776	61	16	n	n	PRON
cana-5776	61	17	is	be	AUX
cana-5776	61	18	even	even	ADV
cana-5776	61	19	|𝐼𝑣|	|𝐼𝑣|	PROPN
cana-5776	61	20	=	=	SYM
cana-5776	61	21	2	2	NUM
cana-5776	61	22	,	,	PUNCT
cana-5776	61	23	𝑛	𝑛	PROPN
cana-5776	61	24	>	>	X
cana-5776	61	25	2	2	NUM
cana-5776	61	26	,	,	PUNCT
cana-5776	61	27	then	then	ADV
cana-5776	61	28	the	the	DET
cana-5776	61	29	entire	entire	ADJ
cana-5776	61	30	domination	domination	NOUN
cana-5776	61	31	number	number	NOUN
cana-5776	61	32	is	be	AUX
cana-5776	61	33	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	61	34	+	+	PROPN
cana-5776	61	35	(	(	PUNCT
cana-5776	61	36	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	61	37	,	,	PUNCT
cana-5776	61	38	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	61	39	)	)	PUNCT
cana-5776	61	40	)	)	PUNCT
cana-5776	62	1	=	=	PRON
cana-5776	62	2	{	{	PUNCT
cana-5776	62	3	2𝑛	2𝑛	NUM
cana-5776	62	4	5	5	NUM
cana-5776	62	5	𝑖𝑓	𝑖𝑓	NUM
cana-5776	62	6	2𝑛	2𝑛	PROPN
cana-5776	62	7	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	NOUN
cana-5776	62	8	𝑏𝑦	𝑏𝑦	PROPN
cana-5776	62	9	5	5	NUM
cana-5776	62	10	2𝑛	2𝑛	PROPN
cana-5776	62	11	−	−	NOUN
cana-5776	62	12	𝑟	𝑟	SYM
cana-5776	62	13	5	5	NUM
cana-5776	62	14	+	+	SYM
cana-5776	62	15	1	1	NUM
cana-5776	62	16	𝑖𝑓	𝑖𝑓	NUM
cana-5776	62	17	2𝑛	2𝑛	PROPN
cana-5776	62	18	𝑛𝑜𝑡	𝑛𝑜𝑡	PROPN
cana-5776	62	19	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	PROPN
cana-5776	62	20	𝑏𝑦	𝑏𝑦	NOUN
cana-5776	62	21	5	5	NUM
cana-5776	62	22	where	where	SCONJ
cana-5776	62	23	𝑟	𝑟	NOUN
cana-5776	62	24	is	be	AUX
cana-5776	62	25	the	the	DET
cana-5776	62	26	remainder	remainder	NOUN
cana-5776	62	27	when	when	SCONJ
cana-5776	62	28	2𝑛	2𝑛	PROPN
cana-5776	62	29	divided	divide	VERB
cana-5776	62	30	by	by	ADP
cana-5776	62	31	5	5	NUM
cana-5776	62	32	.	.	PUNCT
cana-5776	63	1	proof	proof	NOUN
cana-5776	63	2	:	:	PUNCT
cana-5776	63	3	consider	consider	VERB
cana-5776	63	4	a	a	DET
cana-5776	63	5	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	63	6	,	,	PUNCT
cana-5776	64	1	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	64	2	)	)	PUNCT
cana-5776	64	3	with	with	ADP
cana-5776	64	4	vertex	vertex	NOUN
cana-5776	64	5	set	set	VERB
cana-5776	65	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	65	2	=	=	PUNCT
cana-5776	65	3	{	{	PUNCT
cana-5776	65	4	0,1,2,3	0,1,2,3	NUM
cana-5776	65	5	,	,	PUNCT
cana-5776	65	6	…	…	PUNCT
cana-5776	65	7	,	,	PUNCT
cana-5776	65	8	𝑛	𝑛	DET
cana-5776	65	9	−	−	NOUN
cana-5776	65	10	1	1	NUM
cana-5776	65	11	}	}	PUNCT
cana-5776	65	12	where	where	SCONJ
cana-5776	65	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	65	14	denotes	denote	VERB
cana-5776	65	15	the	the	DET
cana-5776	65	16	set	set	NOUN
cana-5776	65	17	of	of	ADP
cana-5776	65	18	involutory	involutory	ADJ
cana-5776	65	19	elements	element	NOUN
cana-5776	65	20	in	in	ADP
cana-5776	65	21	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	65	22	and	and	CCONJ
cana-5776	65	23	𝑛	𝑛	PROPN
cana-5776	65	24	is	be	AUX
cana-5776	65	25	even	even	ADV
cana-5776	65	26	,	,	PUNCT
cana-5776	65	27	𝑛	𝑛	PROPN
cana-5776	65	28	>	>	X
cana-5776	65	29	2	2	NUM
cana-5776	65	30	,	,	PUNCT
cana-5776	65	31	|𝑒|	|𝑒|	PROPN
cana-5776	65	32	=	=	SYM
cana-5776	65	33	𝑛.	𝑛.	NOUN
cana-5776	65	34	case	case	NOUN
cana-5776	65	35	1	1	NUM
cana-5776	65	36	:	:	PUNCT
cana-5776	65	37	if	if	SCONJ
cana-5776	65	38	2𝑛	2𝑛	PROPN
cana-5776	65	39	divisible	divisible	ADJ
cana-5776	65	40	by	by	ADP
cana-5776	65	41	5	5	NUM
cana-5776	65	42	(	(	PUNCT
cana-5776	65	43	start	start	VERB
cana-5776	65	44	entire	entire	ADJ
cana-5776	65	45	dominating	dominating	NOUN
cana-5776	65	46	set	set	VERB
cana-5776	65	47	with	with	ADP
cana-5776	65	48	edge	edge	NOUN
cana-5776	65	49	)	)	PUNCT
cana-5776	65	50	communications	communication	NOUN
cana-5776	65	51	on	on	ADP
cana-5776	65	52	applied	apply	VERB
cana-5776	65	53	nonlinear	nonlinear	ADJ
cana-5776	65	54	analysis	analysis	NOUN
cana-5776	65	55	issn	issn	NOUN
cana-5776	65	56	:	:	PUNCT
cana-5776	65	57	1074	1074	NUM
cana-5776	65	58	-	-	PUNCT
cana-5776	65	59	133x	133x	NUM
cana-5776	65	60	vol	vol	NOUN
cana-5776	65	61	31	31	NUM
cana-5776	65	62	no	no	NOUN
cana-5776	65	63	.	.	PUNCT
cana-5776	66	1	7s	7	NOUN
cana-5776	66	2	(	(	PUNCT
cana-5776	66	3	2024	2024	NUM
cana-5776	66	4	)	)	PUNCT
cana-5776	66	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	66	6	784	784	NUM
cana-5776	66	7	then	then	ADV
cana-5776	66	8	consider	consider	VERB
cana-5776	66	9	𝑆	𝑆	PROPN
cana-5776	66	10	=	=	SYM
cana-5776	66	11	{	{	PUNCT
cana-5776	66	12	𝑒𝑖	𝑒𝑖	PROPN
cana-5776	66	13	,	,	PUNCT
cana-5776	66	14	𝑣𝑗/𝑖	𝑣𝑗/𝑖	NOUN
cana-5776	66	15	=	=	SYM
cana-5776	66	16	0,5,10	0,5,10	PROPN
cana-5776	66	17	,	,	PUNCT
cana-5776	66	18	…	…	PUNCT
cana-5776	66	19	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-5776	66	20	𝑗	𝑗	NOUN
cana-5776	66	21	=	=	SYM
cana-5776	66	22	3,8,13	3,8,13	NUM
cana-5776	66	23	,	,	PUNCT
cana-5776	66	24	…	…	PUNCT
cana-5776	66	25	}	}	PUNCT
cana-5776	66	26	now	now	ADV
cana-5776	66	27	every	every	DET
cana-5776	66	28	edge	edge	NOUN
cana-5776	66	29	in	in	ADP
cana-5776	66	30	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	66	31	,	,	PUNCT
cana-5776	66	32	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	66	33	)	)	PUNCT
cana-5776	66	34	adjacent	adjacent	ADJ
cana-5776	66	35	to	to	AUX
cana-5776	66	36	atleast	atleast	VERB
cana-5776	66	37	one	one	NUM
cana-5776	66	38	edge	edge	NOUN
cana-5776	66	39	in	in	ADP
cana-5776	66	40	𝑆	𝑆	PROPN
cana-5776	66	41	or	or	CCONJ
cana-5776	66	42	incident	incident	NOUN
cana-5776	66	43	with	with	ADP
cana-5776	66	44	atleast	atleast	ADJ
cana-5776	66	45	one	one	NUM
cana-5776	66	46	vertex	vertex	NOUN
cana-5776	66	47	in	in	ADP
cana-5776	66	48	𝑆	𝑆	PROPN
cana-5776	66	49	also	also	ADV
cana-5776	66	50	every	every	DET
cana-5776	66	51	vertex	vertex	NOUN
cana-5776	66	52	in	in	ADP
cana-5776	66	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	66	54	,	,	PUNCT
cana-5776	66	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	66	56	)	)	PUNCT
cana-5776	66	57	either	either	DET
cana-5776	66	58	incident	incident	NOUN
cana-5776	66	59	with	with	ADP
cana-5776	66	60	atleast	atleast	ADJ
cana-5776	66	61	one	one	NUM
cana-5776	66	62	edge	edge	NOUN
cana-5776	66	63	or	or	CCONJ
cana-5776	66	64	adjacent	adjacent	ADJ
cana-5776	66	65	to	to	PART
cana-5776	66	66	atleast	atleast	VERB
cana-5776	66	67	one	one	NUM
cana-5776	66	68	vertex	vertex	NOUN
cana-5776	66	69	in	in	ADP
cana-5776	66	70	𝑆.	𝑆.	PROPN
cana-5776	66	71	therefore	therefore	ADV
cana-5776	66	72	𝑆	𝑆	PROPN
cana-5776	66	73	becomes	become	VERB
cana-5776	66	74	entire	entire	ADJ
cana-5776	66	75	domination	domination	NOUN
cana-5776	66	76	set	set	NOUN
cana-5776	66	77	of	of	ADP
cana-5776	66	78	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	66	79	,	,	PUNCT
cana-5776	66	80	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	66	81	)	)	PUNCT
cana-5776	66	82	and	and	CCONJ
cana-5776	66	83	it	it	PRON
cana-5776	66	84	is	be	AUX
cana-5776	66	85	minimum	minimum	ADJ
cana-5776	66	86	.	.	PUNCT
cana-5776	67	1	|𝑆|	|𝑆|	VERB
cana-5776	67	2	=	=	PUNCT
cana-5776	67	3	|𝐸|	|𝐸|	NOUN
cana-5776	67	4	+	+	CCONJ
cana-5776	67	5	|𝑉|	|𝑉|	NOUN
cana-5776	67	6	5	5	NUM
cana-5776	67	7	=	=	SYM
cana-5776	67	8	𝑛	𝑛	PROPN
cana-5776	67	9	+	+	NOUN
cana-5776	67	10	𝑛	𝑛	PRON
cana-5776	67	11	5	5	NUM
cana-5776	67	12	=	=	SYM
cana-5776	67	13	2𝑛	2𝑛	PROPN
cana-5776	67	14	5	5	NUM
cana-5776	67	15	therefore	therefore	ADV
cana-5776	67	16	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	67	17	+	+	ADJ
cana-5776	67	18	(	(	PUNCT
cana-5776	67	19	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	67	20	,	,	PUNCT
cana-5776	67	21	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	67	22	)	)	PUNCT
cana-5776	67	23	)	)	PUNCT
cana-5776	68	1	=	=	SYM
cana-5776	68	2	2𝑛	2𝑛	NOUN
cana-5776	68	3	5	5	NUM
cana-5776	68	4	case	case	NOUN
cana-5776	68	5	2	2	NUM
cana-5776	68	6	:	:	PUNCT
cana-5776	68	7	if	if	SCONJ
cana-5776	68	8	2𝑛	2𝑛	NUM
cana-5776	68	9	not	not	PART
cana-5776	68	10	divisible	divisible	ADJ
cana-5776	68	11	by	by	ADP
cana-5776	68	12	5	5	NUM
cana-5776	68	13	then	then	ADV
cana-5776	68	14	consider	consider	VERB
cana-5776	68	15	𝑆1	𝑆1	NOUN
cana-5776	68	16	=	=	SYM
cana-5776	68	17	{	{	PUNCT
cana-5776	68	18	𝑒𝑖	𝑒𝑖	PROPN
cana-5776	68	19	,	,	PUNCT
cana-5776	68	20	𝑣𝑗/𝑖	𝑣𝑗/𝑖	NOUN
cana-5776	68	21	=	=	SYM
cana-5776	68	22	0,5,10	0,5,10	PROPN
cana-5776	68	23	,	,	PUNCT
cana-5776	68	24	…	…	PUNCT
cana-5776	68	25	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-5776	68	26	𝑗	𝑗	NOUN
cana-5776	68	27	=	=	SYM
cana-5776	68	28	3,8,13	3,8,13	NUM
cana-5776	68	29	,	,	PUNCT
cana-5776	68	30	…	…	PUNCT
cana-5776	68	31	}	}	PUNCT
cana-5776	68	32	take	take	VERB
cana-5776	68	33	𝑆	𝑆	NOUN
cana-5776	68	34	=	=	PUNCT
cana-5776	68	35	𝑆1	𝑆1	NOUN
cana-5776	68	36	∪	∪	NOUN
cana-5776	68	37	(	(	PUNCT
cana-5776	68	38	𝐺	𝐺	NOUN
cana-5776	68	39	−	−	PROPN
cana-5776	68	40	𝑆1	𝑆1	NOUN
cana-5776	68	41	)	)	PUNCT
cana-5776	68	42	.	.	PUNCT
cana-5776	69	1	now	now	ADV
cana-5776	69	2	every	every	DET
cana-5776	69	3	edge	edge	NOUN
cana-5776	69	4	in	in	ADP
cana-5776	69	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	69	6	,	,	PUNCT
cana-5776	69	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	69	8	)	)	PUNCT
cana-5776	69	9	adjacent	adjacent	ADJ
cana-5776	69	10	to	to	AUX
cana-5776	69	11	atleast	atleast	VERB
cana-5776	69	12	one	one	NUM
cana-5776	69	13	edge	edge	NOUN
cana-5776	69	14	in	in	ADP
cana-5776	69	15	𝑆	𝑆	PROPN
cana-5776	69	16	or	or	CCONJ
cana-5776	69	17	incident	incident	NOUN
cana-5776	69	18	with	with	ADP
cana-5776	69	19	atleast	atleast	ADJ
cana-5776	69	20	one	one	NUM
cana-5776	69	21	vertex	vertex	NOUN
cana-5776	69	22	in	in	ADP
cana-5776	69	23	𝑆	𝑆	PROPN
cana-5776	69	24	also	also	ADV
cana-5776	69	25	every	every	DET
cana-5776	69	26	vertex	vertex	NOUN
cana-5776	69	27	in	in	ADP
cana-5776	69	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	69	29	,	,	PUNCT
cana-5776	69	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	69	31	)	)	PUNCT
cana-5776	69	32	either	either	DET
cana-5776	69	33	incident	incident	NOUN
cana-5776	69	34	with	with	ADP
cana-5776	69	35	atleast	atleast	ADJ
cana-5776	69	36	one	one	NUM
cana-5776	69	37	edge	edge	NOUN
cana-5776	69	38	or	or	CCONJ
cana-5776	69	39	adjacent	adjacent	ADJ
cana-5776	69	40	to	to	PART
cana-5776	69	41	atleast	atleast	VERB
cana-5776	69	42	one	one	NUM
cana-5776	69	43	vertex	vertex	NOUN
cana-5776	69	44	in	in	ADP
cana-5776	69	45	𝑆.	𝑆.	PROPN
cana-5776	69	46	therefore	therefore	ADV
cana-5776	69	47	𝑆	𝑆	PROPN
cana-5776	69	48	becomes	become	VERB
cana-5776	69	49	entire	entire	ADJ
cana-5776	69	50	domination	domination	NOUN
cana-5776	69	51	set	set	NOUN
cana-5776	69	52	of	of	ADP
cana-5776	69	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	69	54	,	,	PUNCT
cana-5776	69	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	69	56	)	)	PUNCT
cana-5776	69	57	and	and	CCONJ
cana-5776	69	58	it	it	PRON
cana-5776	69	59	is	be	AUX
cana-5776	69	60	minimum	minimum	ADJ
cana-5776	69	61	.	.	PUNCT
cana-5776	70	1	consider	consider	VERB
cana-5776	70	2	𝑟	𝑟	NOUN
cana-5776	70	3	is	be	AUX
cana-5776	70	4	the	the	DET
cana-5776	70	5	remainder	remainder	NOUN
cana-5776	70	6	when	when	SCONJ
cana-5776	70	7	2𝑛	2𝑛	PROPN
cana-5776	70	8	divisible	divisible	ADJ
cana-5776	70	9	by	by	ADP
cana-5776	70	10	5	5	NUM
cana-5776	70	11	|𝑆|	|𝑆|	VERB
cana-5776	70	12	=	=	NOUN
cana-5776	70	13	|𝐸|	|𝐸|	NOUN
cana-5776	70	14	+	+	X
cana-5776	70	15	|𝑉|	|𝑉|	NOUN
cana-5776	70	16	−	−	NOUN
cana-5776	70	17	𝑟	𝑟	SYM
cana-5776	70	18	5	5	NUM
cana-5776	70	19	+	+	SYM
cana-5776	70	20	1	1	NUM
cana-5776	70	21	=	=	SYM
cana-5776	70	22	𝑛	𝑛	PROPN
cana-5776	70	23	+	+	NOUN
cana-5776	70	24	𝑛	𝑛	DET
cana-5776	70	25	−	−	NOUN
cana-5776	70	26	𝑟	𝑟	SYM
cana-5776	70	27	5	5	NUM
cana-5776	70	28	+	+	SYM
cana-5776	70	29	1	1	NUM
cana-5776	70	30	=	=	SYM
cana-5776	70	31	2𝑛	2𝑛	PROPN
cana-5776	71	1	−	−	NOUN
cana-5776	71	2	𝑟	𝑟	SYM
cana-5776	71	3	5	5	NUM
cana-5776	71	4	+	+	CCONJ
cana-5776	71	5	1	1	X
cana-5776	71	6	.	.	X
cana-5776	71	7	therefore	therefore	ADV
cana-5776	71	8	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	71	9	+	+	ADJ
cana-5776	71	10	(	(	PUNCT
cana-5776	71	11	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	71	12	,	,	PUNCT
cana-5776	71	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	71	14	)	)	PUNCT
cana-5776	71	15	)	)	PUNCT
cana-5776	72	1	=	=	SYM
cana-5776	72	2	2𝑛−𝑟	2𝑛−𝑟	NUM
cana-5776	72	3	5	5	NUM
cana-5776	72	4	+	+	SYM
cana-5776	72	5	1	1	NUM
cana-5776	72	6	theorem	theorem	VERB
cana-5776	72	7	3.2	3.2	NUM
cana-5776	72	8	:	:	PUNCT
cana-5776	72	9	for	for	ADP
cana-5776	72	10	the	the	DET
cana-5776	72	11	involutory	involutory	NOUN
cana-5776	72	12	addition	addition	NOUN
cana-5776	72	13	cayley	cayley	NOUN
cana-5776	72	14	graph	graph	NOUN
cana-5776	72	15	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	72	16	,	,	PUNCT
cana-5776	72	17	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	72	18	)	)	PUNCT
cana-5776	72	19	,	,	PUNCT
cana-5776	72	20	if	if	SCONJ
cana-5776	72	21	𝑛	𝑛	PROPN
cana-5776	72	22	is	be	AUX
cana-5776	72	23	odd	odd	ADJ
cana-5776	72	24	|𝐼𝑣|	|𝐼𝑣|	NOUN
cana-5776	72	25	=	=	SYM
cana-5776	72	26	2	2	NUM
cana-5776	72	27	,	,	PUNCT
cana-5776	72	28	𝑛	𝑛	PROPN
cana-5776	72	29	>	>	X
cana-5776	72	30	2	2	NUM
cana-5776	72	31	then	then	ADV
cana-5776	72	32	the	the	DET
cana-5776	72	33	entire	entire	ADJ
cana-5776	72	34	domination	domination	NOUN
cana-5776	72	35	number	number	NOUN
cana-5776	72	36	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	72	37	+	+	PROPN
cana-5776	72	38	(	(	PUNCT
cana-5776	72	39	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	72	40	,	,	PUNCT
cana-5776	72	41	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	72	42	)	)	PUNCT
cana-5776	72	43	)	)	PUNCT
cana-5776	73	1	=	=	PRON
cana-5776	73	2	{	{	PUNCT
cana-5776	73	3	2𝑛	2𝑛	NUM
cana-5776	73	4	−	−	NOUN
cana-5776	73	5	1	1	NUM
cana-5776	73	6	5	5	NUM
cana-5776	73	7	𝑖𝑓	𝑖𝑓	NUM
cana-5776	73	8	2𝑛	2𝑛	PROPN
cana-5776	73	9	−	−	NOUN
cana-5776	73	10	1	1	NUM
cana-5776	73	11	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	NOUN
cana-5776	73	12	𝑏𝑦	𝑏𝑦	NOUN
cana-5776	73	13	5	5	NUM
cana-5776	73	14	2𝑛	2𝑛	PROPN
cana-5776	73	15	−	−	PROPN
cana-5776	74	1	𝑟	𝑟	NOUN
cana-5776	75	1	+	+	CCONJ
cana-5776	75	2	4	4	NUM
cana-5776	75	3	5	5	NUM
cana-5776	75	4	𝑖𝑓	𝑖𝑓	NUM
cana-5776	75	5	2𝑛	2𝑛	PROPN
cana-5776	75	6	−	−	PROPN
cana-5776	75	7	1	1	NUM
cana-5776	75	8	𝑛𝑜𝑡	𝑛𝑜𝑡	NOUN
cana-5776	75	9	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	NOUN
cana-5776	75	10	𝑏𝑦	𝑏𝑦	NOUN
cana-5776	75	11	5	5	NUM
cana-5776	75	12	where	where	SCONJ
cana-5776	75	13	𝑟	𝑟	NOUN
cana-5776	75	14	is	be	AUX
cana-5776	75	15	the	the	DET
cana-5776	75	16	remainder	remainder	NOUN
cana-5776	75	17	when	when	SCONJ
cana-5776	75	18	2𝑛	2𝑛	PROPN
cana-5776	75	19	−	−	PROPN
cana-5776	75	20	1	1	NUM
cana-5776	75	21	divided	divide	VERB
cana-5776	75	22	by	by	ADP
cana-5776	75	23	5	5	NUM
cana-5776	75	24	.	.	PUNCT
cana-5776	76	1	proof	proof	NOUN
cana-5776	76	2	:	:	PUNCT
cana-5776	76	3	consider	consider	VERB
cana-5776	76	4	a	a	DET
cana-5776	76	5	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	76	6	,	,	PUNCT
cana-5776	77	1	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	77	2	)	)	PUNCT
cana-5776	77	3	with	with	ADP
cana-5776	77	4	vertex	vertex	NOUN
cana-5776	77	5	set	set	VERB
cana-5776	78	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	78	2	=	=	PUNCT
cana-5776	78	3	{	{	PUNCT
cana-5776	78	4	0,1,2,3	0,1,2,3	NUM
cana-5776	78	5	,	,	PUNCT
cana-5776	78	6	…	…	PUNCT
cana-5776	78	7	,	,	PUNCT
cana-5776	78	8	𝑛	𝑛	DET
cana-5776	78	9	−	−	NOUN
cana-5776	78	10	1	1	NUM
cana-5776	78	11	}	}	PUNCT
cana-5776	78	12	where	where	SCONJ
cana-5776	78	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	78	14	denotes	denote	VERB
cana-5776	78	15	the	the	DET
cana-5776	78	16	set	set	NOUN
cana-5776	78	17	of	of	ADP
cana-5776	78	18	involutory	involutory	ADJ
cana-5776	78	19	elements	element	NOUN
cana-5776	78	20	in	in	ADP
cana-5776	78	21	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	78	22	and	and	CCONJ
cana-5776	78	23	𝑛	𝑛	PROPN
cana-5776	78	24	is	be	AUX
cana-5776	78	25	odd	odd	ADJ
cana-5776	78	26	,	,	PUNCT
cana-5776	78	27	𝑛	𝑛	PROPN
cana-5776	78	28	>	>	X
cana-5776	78	29	2	2	NUM
cana-5776	78	30	,	,	PUNCT
cana-5776	78	31	|𝑒|	|𝑒|	PROPN
cana-5776	78	32	=	=	SYM
cana-5776	78	33	𝑛	𝑛	DET
cana-5776	78	34	−	−	NUM
cana-5776	78	35	1	1	NUM
cana-5776	78	36	case	case	NOUN
cana-5776	78	37	1	1	NUM
cana-5776	78	38	:	:	PUNCT
cana-5776	78	39	if	if	SCONJ
cana-5776	78	40	2𝑛	2𝑛	PROPN
cana-5776	78	41	−	−	PROPN
cana-5776	78	42	1	1	NUM
cana-5776	78	43	divisible	divisible	ADJ
cana-5776	78	44	by	by	ADP
cana-5776	78	45	5	5	NUM
cana-5776	78	46	.	.	PUNCT
cana-5776	79	1	consider	consider	VERB
cana-5776	79	2	𝑆	𝑆	PROPN
cana-5776	79	3	=	=	SYM
cana-5776	79	4	{	{	PUNCT
cana-5776	79	5	𝑒𝑖	𝑒𝑖	PROPN
cana-5776	79	6	,	,	PUNCT
cana-5776	79	7	𝑣𝑗/𝑖	𝑣𝑗/𝑖	NOUN
cana-5776	79	8	=	=	SYM
cana-5776	79	9	0,5,10	0,5,10	PROPN
cana-5776	79	10	,	,	PUNCT
cana-5776	79	11	…	…	PUNCT
cana-5776	79	12	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-5776	79	13	𝑗	𝑗	NOUN
cana-5776	79	14	=	=	SYM
cana-5776	79	15	3,8,13	3,8,13	NUM
cana-5776	79	16	,	,	PUNCT
cana-5776	79	17	…	…	PUNCT
cana-5776	79	18	}	}	PUNCT
cana-5776	79	19	.	.	PUNCT
cana-5776	80	1	communications	communication	NOUN
cana-5776	80	2	on	on	ADP
cana-5776	80	3	applied	apply	VERB
cana-5776	80	4	nonlinear	nonlinear	ADJ
cana-5776	80	5	analysis	analysis	NOUN
cana-5776	80	6	issn	issn	NOUN
cana-5776	80	7	:	:	PUNCT
cana-5776	80	8	1074	1074	NUM
cana-5776	80	9	-	-	PUNCT
cana-5776	80	10	133x	133x	NUM
cana-5776	80	11	vol	vol	NOUN
cana-5776	80	12	31	31	NUM
cana-5776	80	13	no	no	NOUN
cana-5776	80	14	.	.	PUNCT
cana-5776	81	1	7s	7	NOUN
cana-5776	81	2	(	(	PUNCT
cana-5776	81	3	2024	2024	NUM
cana-5776	81	4	)	)	PUNCT
cana-5776	81	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	81	6	785	785	NUM
cana-5776	81	7	now	now	ADV
cana-5776	81	8	every	every	DET
cana-5776	81	9	edge	edge	NOUN
cana-5776	81	10	in	in	ADP
cana-5776	81	11	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	81	12	,	,	PUNCT
cana-5776	81	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	81	14	)	)	PUNCT
cana-5776	81	15	adjacent	adjacent	ADJ
cana-5776	81	16	to	to	AUX
cana-5776	81	17	atleast	atleast	VERB
cana-5776	81	18	one	one	NUM
cana-5776	81	19	edge	edge	NOUN
cana-5776	81	20	in	in	ADP
cana-5776	81	21	𝑆	𝑆	PROPN
cana-5776	81	22	or	or	CCONJ
cana-5776	81	23	incident	incident	NOUN
cana-5776	81	24	with	with	ADP
cana-5776	81	25	atleast	atleast	ADJ
cana-5776	81	26	one	one	NUM
cana-5776	81	27	vertex	vertex	NOUN
cana-5776	81	28	in	in	ADP
cana-5776	81	29	𝑆	𝑆	PROPN
cana-5776	81	30	also	also	ADV
cana-5776	81	31	every	every	DET
cana-5776	81	32	vertex	vertex	NOUN
cana-5776	81	33	in	in	ADP
cana-5776	81	34	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	81	35	,	,	PUNCT
cana-5776	81	36	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	81	37	)	)	PUNCT
cana-5776	81	38	either	either	DET
cana-5776	81	39	incident	incident	NOUN
cana-5776	81	40	with	with	ADP
cana-5776	81	41	atleast	atleast	ADJ
cana-5776	81	42	one	one	NUM
cana-5776	81	43	edge	edge	NOUN
cana-5776	81	44	or	or	CCONJ
cana-5776	81	45	adjacent	adjacent	ADJ
cana-5776	81	46	to	to	PART
cana-5776	81	47	atleast	atleast	VERB
cana-5776	81	48	one	one	NUM
cana-5776	81	49	vertex	vertex	NOUN
cana-5776	81	50	in	in	ADP
cana-5776	81	51	𝑆.	𝑆.	PROPN
cana-5776	81	52	therefore	therefore	ADV
cana-5776	81	53	𝑆	𝑆	PROPN
cana-5776	81	54	becomes	become	VERB
cana-5776	81	55	entire	entire	ADJ
cana-5776	81	56	domination	domination	NOUN
cana-5776	81	57	set	set	NOUN
cana-5776	81	58	of	of	ADP
cana-5776	81	59	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	81	60	,	,	PUNCT
cana-5776	81	61	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	81	62	)	)	PUNCT
cana-5776	81	63	and	and	CCONJ
cana-5776	81	64	it	it	PRON
cana-5776	81	65	is	be	AUX
cana-5776	81	66	minimum	minimum	ADJ
cana-5776	81	67	.	.	PUNCT
cana-5776	82	1	|𝑆|	|𝑆|	VERB
cana-5776	82	2	=	=	PUNCT
cana-5776	82	3	|𝐸|	|𝐸|	NOUN
cana-5776	82	4	+	+	CCONJ
cana-5776	82	5	|𝑉|	|𝑉|	NOUN
cana-5776	82	6	5	5	NUM
cana-5776	82	7	=	=	SYM
cana-5776	82	8	𝑛	𝑛	PROPN
cana-5776	82	9	+	+	NOUN
cana-5776	82	10	𝑛	𝑛	DET
cana-5776	82	11	−	−	NUM
cana-5776	82	12	1	1	NUM
cana-5776	82	13	5	5	NUM
cana-5776	82	14	=	=	SYM
cana-5776	82	15	2𝑛	2𝑛	PROPN
cana-5776	82	16	−	−	NOUN
cana-5776	82	17	1	1	NUM
cana-5776	82	18	5	5	NUM
cana-5776	82	19	.	.	PUNCT
cana-5776	83	1	therefore	therefore	ADV
cana-5776	83	2	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	83	3	+	+	ADJ
cana-5776	83	4	(	(	PUNCT
cana-5776	83	5	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	83	6	,	,	PUNCT
cana-5776	83	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	83	8	)	)	PUNCT
cana-5776	83	9	)	)	PUNCT
cana-5776	84	1	=	=	SYM
cana-5776	84	2	2𝑛−1	2𝑛−1	NUM
cana-5776	84	3	5	5	NUM
cana-5776	84	4	case	case	NOUN
cana-5776	84	5	2	2	NUM
cana-5776	84	6	:	:	PUNCT
cana-5776	84	7	if	if	SCONJ
cana-5776	84	8	2𝑛	2𝑛	PROPN
cana-5776	84	9	−	−	PROPN
cana-5776	84	10	1	1	NUM
cana-5776	84	11	not	not	PART
cana-5776	84	12	divisible	divisible	ADJ
cana-5776	84	13	by	by	ADP
cana-5776	84	14	5	5	NUM
cana-5776	84	15	.	.	PUNCT
cana-5776	84	16	then	then	ADV
cana-5776	84	17	consider	consider	VERB
cana-5776	84	18	𝑆1	𝑆1	NOUN
cana-5776	84	19	=	=	SYM
cana-5776	84	20	{	{	PUNCT
cana-5776	84	21	𝑒𝑖	𝑒𝑖	PROPN
cana-5776	84	22	,	,	PUNCT
cana-5776	84	23	𝑣𝑗/𝑖	𝑣𝑗/𝑖	NOUN
cana-5776	84	24	=	=	SYM
cana-5776	84	25	0,5,10	0,5,10	PROPN
cana-5776	84	26	,	,	PUNCT
cana-5776	84	27	…	…	PUNCT
cana-5776	84	28	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-5776	84	29	𝑗	𝑗	NOUN
cana-5776	84	30	=	=	SYM
cana-5776	84	31	3,8,13	3,8,13	NUM
cana-5776	84	32	,	,	PUNCT
cana-5776	84	33	…	…	PUNCT
cana-5776	84	34	}	}	PUNCT
cana-5776	84	35	.	.	PUNCT
cana-5776	85	1	take	take	VERB
cana-5776	85	2	𝑆	𝑆	NOUN
cana-5776	85	3	=	=	PUNCT
cana-5776	85	4	𝑆1	𝑆1	NOUN
cana-5776	85	5	∪	∪	NOUN
cana-5776	85	6	(	(	PUNCT
cana-5776	85	7	𝐺	𝐺	NOUN
cana-5776	85	8	−	−	PROPN
cana-5776	85	9	𝑆1	𝑆1	NOUN
cana-5776	85	10	)	)	PUNCT
cana-5776	85	11	.	.	PUNCT
cana-5776	86	1	now	now	ADV
cana-5776	86	2	every	every	DET
cana-5776	86	3	edge	edge	NOUN
cana-5776	86	4	in	in	ADP
cana-5776	86	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	86	6	,	,	PUNCT
cana-5776	86	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	86	8	)	)	PUNCT
cana-5776	86	9	adjacent	adjacent	ADJ
cana-5776	86	10	to	to	AUX
cana-5776	86	11	atleast	atleast	VERB
cana-5776	86	12	one	one	NUM
cana-5776	86	13	edge	edge	NOUN
cana-5776	86	14	in	in	ADP
cana-5776	86	15	𝑆	𝑆	PROPN
cana-5776	86	16	or	or	CCONJ
cana-5776	86	17	incident	incident	NOUN
cana-5776	86	18	with	with	ADP
cana-5776	86	19	atleast	atleast	ADJ
cana-5776	86	20	one	one	NUM
cana-5776	86	21	vertex	vertex	NOUN
cana-5776	86	22	in	in	ADP
cana-5776	86	23	𝑆	𝑆	PROPN
cana-5776	86	24	also	also	ADV
cana-5776	86	25	every	every	DET
cana-5776	86	26	vertex	vertex	NOUN
cana-5776	86	27	in	in	ADP
cana-5776	86	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	86	29	,	,	PUNCT
cana-5776	86	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	86	31	)	)	PUNCT
cana-5776	86	32	either	either	DET
cana-5776	86	33	incident	incident	NOUN
cana-5776	86	34	with	with	ADP
cana-5776	86	35	atleast	atleast	ADJ
cana-5776	86	36	one	one	NUM
cana-5776	86	37	edge	edge	NOUN
cana-5776	86	38	or	or	CCONJ
cana-5776	86	39	adjacent	adjacent	ADJ
cana-5776	86	40	to	to	PART
cana-5776	86	41	atleast	atleast	VERB
cana-5776	86	42	one	one	NUM
cana-5776	86	43	vertex	vertex	NOUN
cana-5776	86	44	in	in	ADP
cana-5776	86	45	𝑆	𝑆	PROPN
cana-5776	86	46	therefore	therefore	ADV
cana-5776	86	47	𝑆	𝑆	PROPN
cana-5776	86	48	becomes	become	VERB
cana-5776	86	49	entire	entire	ADJ
cana-5776	86	50	domination	domination	NOUN
cana-5776	86	51	set	set	NOUN
cana-5776	86	52	of	of	ADP
cana-5776	86	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	86	54	,	,	PUNCT
cana-5776	86	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	86	56	)	)	PUNCT
cana-5776	86	57	and	and	CCONJ
cana-5776	86	58	it	it	PRON
cana-5776	86	59	is	be	AUX
cana-5776	86	60	minimum	minimum	ADJ
cana-5776	86	61	.	.	PUNCT
cana-5776	87	1	|𝑆|	|𝑆|	VERB
cana-5776	87	2	=	=	PUNCT
cana-5776	87	3	|𝐸|	|𝐸|	NOUN
cana-5776	87	4	+	+	CCONJ
cana-5776	87	5	|𝑉|	|𝑉|	NOUN
cana-5776	87	6	5	5	NUM
cana-5776	87	7	+	+	CCONJ
cana-5776	87	8	1	1	NUM
cana-5776	87	9	=	=	SYM
cana-5776	87	10	(	(	PUNCT
cana-5776	87	11	𝑛	𝑛	PROPN
cana-5776	87	12	+	+	NOUN
cana-5776	87	13	𝑛	𝑛	PRON
cana-5776	87	14	−	−	NUM
cana-5776	87	15	1	1	NUM
cana-5776	87	16	)	)	PUNCT
cana-5776	87	17	−	−	NOUN
cana-5776	87	18	𝑟	𝑟	SYM
cana-5776	87	19	5	5	NUM
cana-5776	87	20	+	+	SYM
cana-5776	87	21	1	1	NUM
cana-5776	87	22	=	=	SYM
cana-5776	87	23	2𝑛	2𝑛	PROPN
cana-5776	88	1	−	−	PROPN
cana-5776	88	2	𝑟	𝑟	DET
cana-5776	88	3	−	−	PROPN
cana-5776	88	4	1	1	NUM
cana-5776	88	5	5	5	NUM
cana-5776	88	6	+	+	CCONJ
cana-5776	88	7	1	1	NUM
cana-5776	88	8	=	=	SYM
cana-5776	88	9	2𝑛	2𝑛	PROPN
cana-5776	89	1	−	−	PROPN
cana-5776	89	2	𝑟	𝑟	NOUN
cana-5776	89	3	+	+	CCONJ
cana-5776	89	4	4	4	NUM
cana-5776	89	5	5	5	NUM
cana-5776	89	6	.	.	PUNCT
cana-5776	90	1	therefore	therefore	ADV
cana-5776	90	2	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	90	3	+	+	ADJ
cana-5776	90	4	(	(	PUNCT
cana-5776	90	5	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	90	6	,	,	PUNCT
cana-5776	90	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	90	8	)	)	PUNCT
cana-5776	90	9	)	)	PUNCT
cana-5776	91	1	=	=	PUNCT
cana-5776	91	2	2𝑛−𝑟+4	2𝑛−𝑟+4	NUM
cana-5776	91	3	5	5	NUM
cana-5776	91	4	theorem	theorem	VERB
cana-5776	91	5	3.3	3.3	NUM
cana-5776	91	6	:	:	PUNCT
cana-5776	91	7	for	for	ADP
cana-5776	91	8	the	the	DET
cana-5776	91	9	involutory	involutory	NOUN
cana-5776	91	10	addition	addition	NOUN
cana-5776	91	11	cayley	cayley	NOUN
cana-5776	91	12	graph	graph	NOUN
cana-5776	91	13	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	91	14	,	,	PUNCT
cana-5776	91	15	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	91	16	)	)	PUNCT
cana-5776	91	17	,	,	PUNCT
cana-5776	91	18	if	if	SCONJ
cana-5776	91	19	n	n	PRON
cana-5776	91	20	is	be	AUX
cana-5776	91	21	even	even	ADV
cana-5776	91	22	and𝑛	and𝑛	ADJ
cana-5776	91	23	>	>	X
cana-5776	91	24	2	2	NUM
cana-5776	91	25	,	,	PUNCT
cana-5776	91	26	|𝐼𝑣|	|𝐼𝑣|	NOUN
cana-5776	91	27	=	=	NOUN
cana-5776	91	28	4	4	NUM
cana-5776	91	29	,	,	PUNCT
cana-5776	91	30	𝑛	𝑛	NOUN
cana-5776	91	31	=	=	SYM
cana-5776	91	32	2	2	NUM
cana-5776	91	33	𝑞	𝑞	PROPN
cana-5776	91	34	,	,	PUNCT
cana-5776	91	35	𝑞	𝑞	X
cana-5776	91	36	=	=	PROPN
cana-5776	91	37	3,4,5	3,4,5	NUM
cana-5776	91	38	,	,	PUNCT
cana-5776	91	39	…	…	PUNCT
cana-5776	91	40	then	then	ADV
cana-5776	91	41	the	the	DET
cana-5776	91	42	entire	entire	ADJ
cana-5776	91	43	domination	domination	NOUN
cana-5776	91	44	number	number	NOUN
cana-5776	91	45	is	be	AUX
cana-5776	91	46	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	91	47	+	+	ADJ
cana-5776	91	48	(	(	PUNCT
cana-5776	91	49	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	91	50	,	,	PUNCT
cana-5776	91	51	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	91	52	)	)	PUNCT
cana-5776	91	53	)	)	PUNCT
cana-5776	92	1	=	=	SYM
cana-5776	92	2	𝑛	𝑛	DET
cana-5776	92	3	2	2	NUM
cana-5776	92	4	.	.	PUNCT
cana-5776	93	1	proof	proof	NOUN
cana-5776	93	2	:	:	PUNCT
cana-5776	93	3	consider	consider	VERB
cana-5776	93	4	a	a	DET
cana-5776	93	5	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	93	6	,	,	PUNCT
cana-5776	93	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	93	8	)	)	PUNCT
cana-5776	93	9	with	with	ADP
cana-5776	93	10	vertex	vertex	NOUN
cana-5776	93	11	set	set	VERB
cana-5776	94	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	94	2	=	=	PUNCT
cana-5776	94	3	{	{	PUNCT
cana-5776	94	4	0,1,2,3	0,1,2,3	NUM
cana-5776	94	5	,	,	PUNCT
cana-5776	94	6	…	…	PUNCT
cana-5776	94	7	,	,	PUNCT
cana-5776	94	8	𝑛	𝑛	DET
cana-5776	94	9	−	−	NOUN
cana-5776	94	10	1	1	NUM
cana-5776	94	11	}	}	PUNCT
cana-5776	94	12	where	where	SCONJ
cana-5776	94	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	94	14	denotes	denote	VERB
cana-5776	94	15	the	the	DET
cana-5776	94	16	set	set	NOUN
cana-5776	94	17	of	of	ADP
cana-5776	94	18	involutory	involutory	ADJ
cana-5776	94	19	elements	element	NOUN
cana-5776	94	20	in	in	ADP
cana-5776	94	21	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	94	22	and	and	CCONJ
cana-5776	94	23	𝑛	𝑛	PROPN
cana-5776	94	24	is	be	AUX
cana-5776	94	25	even	even	ADV
cana-5776	94	26	,	,	PUNCT
cana-5776	94	27	𝑛	𝑛	PROPN
cana-5776	94	28	>	>	X
cana-5776	94	29	2	2	NUM
cana-5776	94	30	,	,	PUNCT
cana-5776	94	31	|𝑒|	|𝑒|	PROPN
cana-5776	94	32	=	=	SYM
cana-5776	94	33	2𝑛.	2𝑛.	NUM
cana-5776	94	34	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	94	35	,	,	PUNCT
cana-5776	94	36	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	94	37	)	)	PUNCT
cana-5776	94	38	contains	contain	VERB
cana-5776	94	39	two	two	NUM
cana-5776	94	40	hamiltonian	hamiltonian	ADJ
cana-5776	94	41	cycles	cycle	NOUN
cana-5776	94	42	.	.	PUNCT
cana-5776	95	1	from	from	ADP
cana-5776	95	2	first	first	ADJ
cana-5776	95	3	hamiltonian	hamiltonian	ADJ
cana-5776	95	4	cycle	cycle	NOUN
cana-5776	95	5	.	.	PUNCT
cana-5776	96	1	then	then	ADV
cana-5776	96	2	consider	consider	VERB
cana-5776	96	3	𝑆	𝑆	PROPN
cana-5776	96	4	=	=	PRON
cana-5776	96	5	{	{	PUNCT
cana-5776	96	6	𝑒𝑖𝑗/0	𝑒𝑖𝑗/0	NOUN
cana-5776	96	7	≤	≤	NOUN
cana-5776	96	8	𝑖	𝑖	SYM
cana-5776	96	9	≤	≤	NUM
cana-5776	96	10	𝑛	𝑛	DET
cana-5776	96	11	2	2	NUM
cana-5776	96	12	−	−	NOUN
cana-5776	96	13	1	1	NUM
cana-5776	96	14	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5776	96	15	𝑛	𝑛	DET
cana-5776	96	16	2	2	NUM
cana-5776	96	17	≤	≤	NUM
cana-5776	96	18	𝑗	𝑗	PRON
cana-5776	96	19	≤	≤	NUM
cana-5776	96	20	𝑛	𝑛	PRON
cana-5776	96	21	−	−	NOUN
cana-5776	96	22	1	1	NUM
cana-5776	96	23	}	}	PUNCT
cana-5776	96	24	.	.	PUNCT
cana-5776	97	1	now	now	ADV
cana-5776	97	2	every	every	DET
cana-5776	97	3	edge	edge	NOUN
cana-5776	97	4	in	in	ADP
cana-5776	97	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	97	6	,	,	PUNCT
cana-5776	97	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	97	8	)	)	PUNCT
cana-5776	97	9	adjacent	adjacent	ADJ
cana-5776	97	10	to	to	AUX
cana-5776	97	11	atleast	atleast	VERB
cana-5776	97	12	one	one	NUM
cana-5776	97	13	edge	edge	NOUN
cana-5776	97	14	in	in	ADP
cana-5776	97	15	𝑆	𝑆	PROPN
cana-5776	97	16	or	or	CCONJ
cana-5776	97	17	incident	incident	NOUN
cana-5776	97	18	with	with	ADP
cana-5776	97	19	atleast	atleast	ADJ
cana-5776	97	20	one	one	NUM
cana-5776	97	21	vertex	vertex	NOUN
cana-5776	97	22	in	in	ADP
cana-5776	97	23	𝑆	𝑆	PROPN
cana-5776	97	24	also	also	ADV
cana-5776	97	25	every	every	DET
cana-5776	97	26	vertex	vertex	NOUN
cana-5776	97	27	in	in	ADP
cana-5776	97	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	97	29	,	,	PUNCT
cana-5776	97	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	97	31	)	)	PUNCT
cana-5776	97	32	either	either	DET
cana-5776	97	33	incident	incident	NOUN
cana-5776	97	34	with	with	ADP
cana-5776	97	35	atleast	atleast	ADJ
cana-5776	97	36	one	one	NUM
cana-5776	97	37	edge	edge	NOUN
cana-5776	97	38	or	or	CCONJ
cana-5776	97	39	adjacent	adjacent	ADJ
cana-5776	97	40	to	to	PART
cana-5776	97	41	atleast	atleast	VERB
cana-5776	97	42	one	one	NUM
cana-5776	97	43	vertex	vertex	NOUN
cana-5776	97	44	in	in	ADP
cana-5776	97	45	𝑆.	𝑆.	PROPN
cana-5776	97	46	therefore	therefore	ADV
cana-5776	97	47	𝑆	𝑆	PROPN
cana-5776	97	48	becomes	become	VERB
cana-5776	97	49	entire	entire	ADJ
cana-5776	97	50	domination	domination	NOUN
cana-5776	97	51	set	set	NOUN
cana-5776	97	52	of	of	ADP
cana-5776	97	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	97	54	,	,	PUNCT
cana-5776	97	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	97	56	)	)	PUNCT
cana-5776	97	57	and	and	CCONJ
cana-5776	97	58	it	it	PRON
cana-5776	97	59	is	be	AUX
cana-5776	97	60	minimum	minimum	ADJ
cana-5776	97	61	,	,	PUNCT
cana-5776	97	62	|𝑆|	|𝑆|	VERB
cana-5776	97	63	=	=	PUNCT
cana-5776	97	64	|𝑉|	|𝑉|	NOUN
cana-5776	97	65	2	2	NUM
cana-5776	97	66	=	=	SYM
cana-5776	97	67	𝑛	𝑛	DET
cana-5776	97	68	2	2	NUM
cana-5776	97	69	therefore	therefore	ADV
cana-5776	97	70	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	97	71	+	+	ADJ
cana-5776	97	72	(	(	PUNCT
cana-5776	97	73	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	97	74	,	,	PUNCT
cana-5776	97	75	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	97	76	)	)	PUNCT
cana-5776	97	77	)	)	PUNCT
cana-5776	98	1	=	=	SYM
cana-5776	98	2	𝑛	𝑛	DET
cana-5776	98	3	2	2	NUM
cana-5776	98	4	.	.	PUNCT
cana-5776	99	1	communications	communication	NOUN
cana-5776	99	2	on	on	ADP
cana-5776	99	3	applied	apply	VERB
cana-5776	99	4	nonlinear	nonlinear	ADJ
cana-5776	99	5	analysis	analysis	NOUN
cana-5776	99	6	issn	issn	NOUN
cana-5776	99	7	:	:	PUNCT
cana-5776	99	8	1074	1074	NUM
cana-5776	99	9	-	-	PUNCT
cana-5776	99	10	133x	133x	NUM
cana-5776	99	11	vol	vol	NOUN
cana-5776	99	12	31	31	NUM
cana-5776	99	13	no	no	NOUN
cana-5776	99	14	.	.	PUNCT
cana-5776	100	1	7s	7	NOUN
cana-5776	100	2	(	(	PUNCT
cana-5776	100	3	2024	2024	NUM
cana-5776	100	4	)	)	PUNCT
cana-5776	100	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	100	6	786	786	NUM
cana-5776	100	7	theorem	theorem	VERB
cana-5776	100	8	3.4	3.4	NUM
cana-5776	100	9	:	:	PUNCT
cana-5776	100	10	for	for	ADP
cana-5776	100	11	the	the	DET
cana-5776	100	12	involutory	involutory	NOUN
cana-5776	100	13	addition	addition	NOUN
cana-5776	100	14	cayley	cayley	NOUN
cana-5776	100	15	graph	graph	NOUN
cana-5776	100	16	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	100	17	,	,	PUNCT
cana-5776	100	18	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	100	19	)	)	PUNCT
cana-5776	100	20	,	,	PUNCT
cana-5776	100	21	if	if	SCONJ
cana-5776	100	22	n	n	PRON
cana-5776	100	23	is	be	AUX
cana-5776	100	24	even	even	ADV
cana-5776	100	25	and	and	CCONJ
cana-5776	100	26	𝑛	𝑛	ADJ
cana-5776	100	27	>	>	X
cana-5776	100	28	2	2	NUM
cana-5776	100	29	,	,	PUNCT
cana-5776	100	30	|𝐼𝑣|	|𝐼𝑣|	NOUN
cana-5776	100	31	=	=	NOUN
cana-5776	100	32	4	4	NUM
cana-5776	100	33	,	,	PUNCT
cana-5776	100	34	𝑛	𝑛	PROPN
cana-5776	100	35	=	=	SYM
cana-5776	100	36	4𝑝	4𝑝	PROPN
cana-5776	100	37	,	,	PUNCT
cana-5776	100	38	𝑝	𝑝	PROPN
cana-5776	100	39	>	>	X
cana-5776	100	40	2	2	NUM
cana-5776	100	41	,	,	PUNCT
cana-5776	101	1	𝑝	𝑝	NOUN
cana-5776	101	2	𝑖𝑠	𝑖𝑠	NOUN
cana-5776	101	3	𝑝𝑟𝑖𝑚𝑒	𝑝𝑟𝑖𝑚𝑒	NOUN
cana-5776	101	4	then	then	ADV
cana-5776	101	5	the	the	DET
cana-5776	101	6	entire	entire	ADJ
cana-5776	101	7	domination	domination	NOUN
cana-5776	101	8	number	number	NOUN
cana-5776	101	9	is	be	AUX
cana-5776	101	10	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	101	11	+	+	ADJ
cana-5776	101	12	(	(	PUNCT
cana-5776	101	13	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	101	14	,	,	PUNCT
cana-5776	101	15	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	101	16	)	)	PUNCT
cana-5776	101	17	)	)	PUNCT
cana-5776	102	1	=	=	PRON
cana-5776	102	2	{	{	PUNCT
cana-5776	102	3	𝑛	𝑛	DET
cana-5776	102	4	2	2	NUM
cana-5776	102	5	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-5776	102	6	𝑝	𝑝	NOUN
cana-5776	102	7	=	=	SYM
cana-5776	102	8	3	3	NUM
cana-5776	102	9	2𝑛	2𝑛	PROPN
cana-5776	102	10	5	5	NUM
cana-5776	102	11	𝑖𝑓	𝑖𝑓	NUM
cana-5776	102	12	2𝑛	2𝑛	PROPN
cana-5776	102	13	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	NOUN
cana-5776	102	14	𝑏𝑦	𝑏𝑦	PROPN
cana-5776	102	15	5(𝑝	5(𝑝	NUM
cana-5776	102	16	>	>	X
cana-5776	102	17	3	3	NUM
cana-5776	102	18	)	)	PUNCT
cana-5776	102	19	2𝑛	2𝑛	NOUN
cana-5776	103	1	−	−	PROPN
cana-5776	103	2	𝑟	𝑟	NOUN
cana-5776	103	3	+	+	CCONJ
cana-5776	103	4	5	5	NUM
cana-5776	103	5	5	5	NUM
cana-5776	103	6	𝑖𝑓	𝑖𝑓	NUM
cana-5776	103	7	2𝑛	2𝑛	PROPN
cana-5776	103	8	𝑛𝑜𝑡	𝑛𝑜𝑡	PROPN
cana-5776	103	9	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒	VERB
cana-5776	103	10	𝑏𝑦	𝑏𝑦	PROPN
cana-5776	103	11	5(𝑝	5(𝑝	NUM
cana-5776	103	12	>	>	X
cana-5776	103	13	3	3	NUM
cana-5776	103	14	)	)	PUNCT
cana-5776	103	15	where	where	SCONJ
cana-5776	103	16	𝑟	𝑟	PRON
cana-5776	103	17	is	be	AUX
cana-5776	103	18	the	the	DET
cana-5776	103	19	remainder	remainder	NOUN
cana-5776	103	20	when	when	SCONJ
cana-5776	103	21	2𝑛	2𝑛	PROPN
cana-5776	103	22	divided	divide	VERB
cana-5776	103	23	by	by	ADP
cana-5776	103	24	5	5	NUM
cana-5776	103	25	.	.	PUNCT
cana-5776	104	1	proof	proof	NOUN
cana-5776	104	2	:	:	PUNCT
cana-5776	104	3	consider	consider	VERB
cana-5776	104	4	a	a	DET
cana-5776	104	5	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	104	6	,	,	PUNCT
cana-5776	105	1	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	105	2	)	)	PUNCT
cana-5776	105	3	with	with	ADP
cana-5776	105	4	vertex	vertex	NOUN
cana-5776	105	5	set	set	VERB
cana-5776	106	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	106	2	=	=	PUNCT
cana-5776	106	3	{	{	PUNCT
cana-5776	106	4	0,1,2,3	0,1,2,3	NUM
cana-5776	106	5	,	,	PUNCT
cana-5776	106	6	…	…	PUNCT
cana-5776	106	7	,	,	PUNCT
cana-5776	106	8	𝑛	𝑛	DET
cana-5776	106	9	−	−	NOUN
cana-5776	106	10	1	1	NUM
cana-5776	106	11	}	}	PUNCT
cana-5776	106	12	where	where	SCONJ
cana-5776	106	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	106	14	denotes	denote	VERB
cana-5776	106	15	the	the	DET
cana-5776	106	16	set	set	NOUN
cana-5776	106	17	of	of	ADP
cana-5776	106	18	involutory	involutory	ADJ
cana-5776	106	19	elements	element	NOUN
cana-5776	106	20	in	in	ADP
cana-5776	106	21	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	106	22	and	and	CCONJ
cana-5776	106	23	𝑛	𝑛	PROPN
cana-5776	106	24	is	be	AUX
cana-5776	106	25	even	even	ADV
cana-5776	106	26	,	,	PUNCT
cana-5776	106	27	𝑛	𝑛	PROPN
cana-5776	106	28	>	>	X
cana-5776	106	29	2	2	NUM
cana-5776	106	30	,	,	PUNCT
cana-5776	106	31	|𝐼𝑣|	|𝐼𝑣|	NOUN
cana-5776	106	32	=	=	NOUN
cana-5776	106	33	4	4	NUM
cana-5776	106	34	,	,	PUNCT
cana-5776	106	35	𝑛	𝑛	PROPN
cana-5776	106	36	=	=	SYM
cana-5776	106	37	4𝑝	4𝑝	PROPN
cana-5776	106	38	,	,	PUNCT
cana-5776	106	39	𝑝	𝑝	PROPN
cana-5776	106	40	>	>	X
cana-5776	106	41	2	2	NUM
cana-5776	106	42	,	,	PUNCT
cana-5776	106	43	𝑝	𝑝	NOUN
cana-5776	106	44	is	be	AUX
cana-5776	106	45	prime|𝑒|	prime|𝑒|	NOUN
cana-5776	106	46	=	=	PUNCT
cana-5776	106	47	2𝑛.	2𝑛.	NUM
cana-5776	106	48	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	106	49	,	,	PUNCT
cana-5776	106	50	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	106	51	)	)	PUNCT
cana-5776	106	52	contains	contain	VERB
cana-5776	106	53	two	two	NUM
cana-5776	106	54	hamiltonian	hamiltonian	ADJ
cana-5776	106	55	cycles	cycle	NOUN
cana-5776	106	56	,	,	PUNCT
cana-5776	106	57	each	each	DET
cana-5776	106	58	hamiltonian	hamiltonian	ADJ
cana-5776	106	59	cycle	cycle	NOUN
cana-5776	106	60	contains	contain	VERB
cana-5776	106	61	𝑛	𝑛	DET
cana-5776	106	62	number	number	NOUN
cana-5776	106	63	of	of	ADP
cana-5776	106	64	elements	element	NOUN
cana-5776	106	65	.	.	PUNCT
cana-5776	107	1	case	case	NOUN
cana-5776	107	2	1	1	NUM
cana-5776	107	3	:	:	PUNCT
cana-5776	107	4	from	from	ADP
cana-5776	107	5	first	first	ADJ
cana-5776	107	6	hamiltonian	hamiltonian	ADJ
cana-5776	107	7	cycle	cycle	NOUN
cana-5776	107	8	,	,	PUNCT
cana-5776	107	9	for	for	ADP
cana-5776	107	10	𝑝	𝑝	NOUN
cana-5776	107	11	=	=	SYM
cana-5776	107	12	3	3	NUM
cana-5776	107	13	.	.	PUNCT
cana-5776	107	14	then	then	ADV
cana-5776	107	15	consider	consider	VERB
cana-5776	107	16	𝑆	𝑆	PROPN
cana-5776	107	17	=	=	PRON
cana-5776	107	18	{	{	PUNCT
cana-5776	107	19	𝑒𝑖𝑗	𝑒𝑖𝑗	NOUN
cana-5776	107	20	0	0	NUM
cana-5776	107	21	≤	≤	NUM
cana-5776	107	22	𝑖	𝑖	SYM
cana-5776	107	23	≤	≤	NUM
cana-5776	107	24	𝑛	𝑛	DET
cana-5776	107	25	2	2	NUM
cana-5776	107	26	−	−	NOUN
cana-5776	107	27	1	1	NUM
cana-5776	107	28	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5776	107	29	𝑛	𝑛	DET
cana-5776	107	30	2	2	NUM
cana-5776	107	31	≤	≤	NUM
cana-5776	107	32	𝑗	𝑗	PRON
cana-5776	107	33	≤	≤	NUM
cana-5776	107	34	𝑛	𝑛	PRON
cana-5776	107	35	−	−	NOUN
cana-5776	107	36	1	1	NUM
cana-5776	107	37	}	}	PUNCT
cana-5776	107	38	.	.	PUNCT
cana-5776	108	1	now	now	ADV
cana-5776	108	2	every	every	DET
cana-5776	108	3	edge	edge	NOUN
cana-5776	108	4	in	in	ADP
cana-5776	108	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	108	6	,	,	PUNCT
cana-5776	108	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	108	8	)	)	PUNCT
cana-5776	108	9	adjacent	adjacent	ADJ
cana-5776	108	10	to	to	AUX
cana-5776	108	11	atleast	atleast	VERB
cana-5776	108	12	one	one	NUM
cana-5776	108	13	edge	edge	NOUN
cana-5776	108	14	in	in	ADP
cana-5776	108	15	𝑆	𝑆	PROPN
cana-5776	108	16	or	or	CCONJ
cana-5776	108	17	incident	incident	NOUN
cana-5776	108	18	with	with	ADP
cana-5776	108	19	atleast	atleast	ADJ
cana-5776	108	20	one	one	NUM
cana-5776	108	21	vertex	vertex	NOUN
cana-5776	108	22	in	in	ADP
cana-5776	108	23	𝑆	𝑆	PROPN
cana-5776	108	24	also	also	ADV
cana-5776	108	25	every	every	DET
cana-5776	108	26	vertex	vertex	NOUN
cana-5776	108	27	in	in	ADP
cana-5776	108	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	108	29	,	,	PUNCT
cana-5776	108	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	108	31	)	)	PUNCT
cana-5776	108	32	either	either	DET
cana-5776	108	33	incident	incident	NOUN
cana-5776	108	34	with	with	ADP
cana-5776	108	35	atleast	atleast	ADJ
cana-5776	108	36	one	one	NUM
cana-5776	108	37	edge	edge	NOUN
cana-5776	108	38	or	or	CCONJ
cana-5776	108	39	adjacent	adjacent	ADJ
cana-5776	108	40	to	to	PART
cana-5776	108	41	atleast	atleast	VERB
cana-5776	108	42	one	one	NUM
cana-5776	108	43	vertex	vertex	NOUN
cana-5776	108	44	in	in	ADP
cana-5776	108	45	𝑆.	𝑆.	PROPN
cana-5776	108	46	therefore	therefore	ADV
cana-5776	108	47	𝑆	𝑆	PROPN
cana-5776	108	48	becomes	become	VERB
cana-5776	108	49	entire	entire	ADJ
cana-5776	108	50	domination	domination	NOUN
cana-5776	108	51	set	set	NOUN
cana-5776	108	52	of	of	ADP
cana-5776	108	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	108	54	,	,	PUNCT
cana-5776	108	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	108	56	)	)	PUNCT
cana-5776	108	57	and	and	CCONJ
cana-5776	108	58	it	it	PRON
cana-5776	108	59	is	be	AUX
cana-5776	108	60	minimum	minimum	ADJ
cana-5776	108	61	,	,	PUNCT
cana-5776	108	62	|𝑆|	|𝑆|	VERB
cana-5776	108	63	=	=	PUNCT
cana-5776	108	64	|𝑉|	|𝑉|	NOUN
cana-5776	108	65	2	2	NUM
cana-5776	108	66	=	=	SYM
cana-5776	108	67	𝑛	𝑛	DET
cana-5776	108	68	2	2	NUM
cana-5776	108	69	.	.	PUNCT
cana-5776	109	1	therefore	therefore	ADV
cana-5776	109	2	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	109	3	+	+	ADJ
cana-5776	109	4	(	(	PUNCT
cana-5776	109	5	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	109	6	,	,	PUNCT
cana-5776	109	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	109	8	)	)	PUNCT
cana-5776	109	9	)	)	PUNCT
cana-5776	110	1	=	=	SYM
cana-5776	110	2	𝑛	𝑛	DET
cana-5776	110	3	2	2	NUM
cana-5776	110	4	case	case	NOUN
cana-5776	110	5	2	2	NUM
cana-5776	110	6	:	:	PUNCT
cana-5776	110	7	from	from	ADP
cana-5776	110	8	first	first	ADJ
cana-5776	110	9	hamiltonian	hamiltonian	ADJ
cana-5776	110	10	cycle	cycle	NOUN
cana-5776	110	11	,	,	PUNCT
cana-5776	110	12	for	for	ADP
cana-5776	110	13	𝑝	𝑝	PROPN
cana-5776	110	14	>	>	SYM
cana-5776	110	15	3	3	NUM
cana-5776	110	16	,	,	PUNCT
cana-5776	110	17	if	if	SCONJ
cana-5776	110	18	2𝑛	2𝑛	PROPN
cana-5776	110	19	divisible	divisible	ADJ
cana-5776	110	20	by	by	ADP
cana-5776	110	21	5	5	NUM
cana-5776	110	22	.	.	PUNCT
cana-5776	110	23	then	then	ADV
cana-5776	110	24	consider	consider	VERB
cana-5776	110	25	𝑆	𝑆	PROPN
cana-5776	110	26	=	=	SYM
cana-5776	110	27	{	{	PUNCT
cana-5776	110	28	𝑒𝑖	𝑒𝑖	PROPN
cana-5776	110	29	,	,	PUNCT
cana-5776	110	30	𝑣𝑗/𝑖	𝑣𝑗/𝑖	NOUN
cana-5776	110	31	=	=	SYM
cana-5776	110	32	0,5,10	0,5,10	PROPN
cana-5776	110	33	,	,	PUNCT
cana-5776	110	34	…	…	PUNCT
cana-5776	110	35	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-5776	110	36	𝑗	𝑗	NOUN
cana-5776	110	37	=	=	SYM
cana-5776	110	38	3,8,13	3,8,13	NUM
cana-5776	110	39	,	,	PUNCT
cana-5776	110	40	…	…	PUNCT
cana-5776	110	41	}	}	PUNCT
cana-5776	110	42	.	.	PUNCT
cana-5776	111	1	now	now	ADV
cana-5776	111	2	every	every	DET
cana-5776	111	3	edge	edge	NOUN
cana-5776	111	4	in	in	ADP
cana-5776	111	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	111	6	,	,	PUNCT
cana-5776	111	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	111	8	)	)	PUNCT
cana-5776	111	9	adjacent	adjacent	ADJ
cana-5776	111	10	to	to	AUX
cana-5776	111	11	atleast	atleast	VERB
cana-5776	111	12	one	one	NUM
cana-5776	111	13	edge	edge	NOUN
cana-5776	111	14	in	in	ADP
cana-5776	111	15	𝑆	𝑆	PROPN
cana-5776	111	16	or	or	CCONJ
cana-5776	111	17	incident	incident	NOUN
cana-5776	111	18	with	with	ADP
cana-5776	111	19	atleast	atleast	ADJ
cana-5776	111	20	one	one	NUM
cana-5776	111	21	vertex	vertex	NOUN
cana-5776	111	22	in	in	ADP
cana-5776	111	23	𝑆	𝑆	PROPN
cana-5776	111	24	also	also	ADV
cana-5776	111	25	every	every	DET
cana-5776	111	26	vertex	vertex	NOUN
cana-5776	111	27	in	in	ADP
cana-5776	111	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	111	29	,	,	PUNCT
cana-5776	111	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	111	31	)	)	PUNCT
cana-5776	111	32	either	either	DET
cana-5776	111	33	incident	incident	NOUN
cana-5776	111	34	with	with	ADP
cana-5776	111	35	atleast	atleast	ADJ
cana-5776	111	36	one	one	NUM
cana-5776	111	37	edge	edge	NOUN
cana-5776	111	38	or	or	CCONJ
cana-5776	111	39	adjacent	adjacent	ADJ
cana-5776	111	40	to	to	PART
cana-5776	111	41	atleast	atleast	VERB
cana-5776	111	42	one	one	NUM
cana-5776	111	43	vertex	vertex	NOUN
cana-5776	111	44	in	in	ADP
cana-5776	111	45	𝑆.	𝑆.	PROPN
cana-5776	111	46	therefore	therefore	ADV
cana-5776	111	47	𝑆	𝑆	PROPN
cana-5776	111	48	becomes	become	VERB
cana-5776	111	49	entire	entire	ADJ
cana-5776	111	50	domination	domination	NOUN
cana-5776	111	51	set	set	NOUN
cana-5776	111	52	of	of	ADP
cana-5776	111	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	111	54	,	,	PUNCT
cana-5776	111	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	111	56	)	)	PUNCT
cana-5776	111	57	and	and	CCONJ
cana-5776	111	58	it	it	PRON
cana-5776	111	59	is	be	AUX
cana-5776	111	60	minimum	minimum	ADJ
cana-5776	111	61	,	,	PUNCT
cana-5776	111	62	|𝑆|	|𝑆|	VERB
cana-5776	111	63	=	=	PUNCT
cana-5776	111	64	|𝐸|	|𝐸|	NOUN
cana-5776	111	65	2	2	NUM
cana-5776	111	66	+	+	NUM
cana-5776	111	67	|𝑉|	|𝑉|	NOUN
cana-5776	111	68	5	5	NUM
cana-5776	111	69	=	=	SYM
cana-5776	111	70	2𝑛	2𝑛	PROPN
cana-5776	111	71	2	2	NUM
cana-5776	112	1	+	+	CCONJ
cana-5776	112	2	𝑛	𝑛	DET
cana-5776	112	3	5	5	NUM
cana-5776	112	4	=	=	SYM
cana-5776	112	5	𝑛	𝑛	PROPN
cana-5776	112	6	+	+	NOUN
cana-5776	112	7	𝑛	𝑛	PRON
cana-5776	112	8	5	5	NUM
cana-5776	112	9	=	=	SYM
cana-5776	112	10	2𝑛	2𝑛	PROPN
cana-5776	112	11	5	5	NUM
cana-5776	112	12	.	.	PUNCT
cana-5776	113	1	therefore	therefore	ADV
cana-5776	113	2	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	113	3	+	+	ADJ
cana-5776	113	4	(	(	PUNCT
cana-5776	113	5	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	113	6	,	,	PUNCT
cana-5776	113	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	113	8	)	)	PUNCT
cana-5776	113	9	)	)	PUNCT
cana-5776	114	1	=	=	SYM
cana-5776	114	2	2𝑛	2𝑛	NOUN
cana-5776	114	3	5	5	NUM
cana-5776	114	4	case	case	NOUN
cana-5776	114	5	3	3	NUM
cana-5776	114	6	:	:	PUNCT
cana-5776	114	7	from	from	ADP
cana-5776	114	8	first	first	ADJ
cana-5776	114	9	hamiltonian	hamiltonian	ADJ
cana-5776	114	10	cycle	cycle	NOUN
cana-5776	114	11	,	,	PUNCT
cana-5776	114	12	for	for	ADP
cana-5776	114	13	𝑝	𝑝	PROPN
cana-5776	114	14	>	>	SYM
cana-5776	114	15	3	3	NUM
cana-5776	114	16	,	,	PUNCT
cana-5776	114	17	if	if	SCONJ
cana-5776	114	18	2𝑛	2𝑛	NUM
cana-5776	114	19	not	not	PART
cana-5776	114	20	divisible	divisible	ADJ
cana-5776	114	21	by	by	ADP
cana-5776	114	22	5	5	NUM
cana-5776	114	23	.	.	PUNCT
cana-5776	114	24	then	then	ADV
cana-5776	114	25	consider	consider	VERB
cana-5776	114	26	𝑆1	𝑆1	NOUN
cana-5776	114	27	=	=	SYM
cana-5776	114	28	{	{	PUNCT
cana-5776	114	29	𝑒𝑖	𝑒𝑖	PROPN
cana-5776	114	30	,	,	PUNCT
cana-5776	114	31	𝑣𝑗/𝑖	𝑣𝑗/𝑖	NOUN
cana-5776	114	32	=	=	SYM
cana-5776	114	33	0,5,10	0,5,10	PROPN
cana-5776	114	34	,	,	PUNCT
cana-5776	114	35	…	…	PUNCT
cana-5776	114	36	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-5776	114	37	𝑗	𝑗	NOUN
cana-5776	114	38	=	=	SYM
cana-5776	114	39	3,8,13	3,8,13	NUM
cana-5776	114	40	,	,	PUNCT
cana-5776	114	41	…	…	PUNCT
cana-5776	114	42	}	}	PUNCT
cana-5776	114	43	communications	communication	NOUN
cana-5776	114	44	on	on	ADP
cana-5776	114	45	applied	apply	VERB
cana-5776	114	46	nonlinear	nonlinear	ADJ
cana-5776	114	47	analysis	analysis	NOUN
cana-5776	114	48	issn	issn	NOUN
cana-5776	114	49	:	:	PUNCT
cana-5776	114	50	1074	1074	NUM
cana-5776	114	51	-	-	PUNCT
cana-5776	114	52	133x	133x	NUM
cana-5776	114	53	vol	vol	NOUN
cana-5776	114	54	31	31	NUM
cana-5776	114	55	no	no	NOUN
cana-5776	114	56	.	.	PUNCT
cana-5776	115	1	7s	7	NOUN
cana-5776	115	2	(	(	PUNCT
cana-5776	115	3	2024	2024	NUM
cana-5776	115	4	)	)	PUNCT
cana-5776	115	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	115	6	787	787	NUM
cana-5776	115	7	take	take	VERB
cana-5776	115	8	𝑆	𝑆	NOUN
cana-5776	115	9	=	=	PUNCT
cana-5776	115	10	𝑆1	𝑆1	NOUN
cana-5776	115	11	∪	∪	NOUN
cana-5776	115	12	(	(	PUNCT
cana-5776	115	13	𝐺	𝐺	NOUN
cana-5776	115	14	−	−	PROPN
cana-5776	115	15	𝑆1	𝑆1	NOUN
cana-5776	115	16	)	)	PUNCT
cana-5776	115	17	.	.	PUNCT
cana-5776	116	1	now	now	ADV
cana-5776	116	2	every	every	DET
cana-5776	116	3	edge	edge	NOUN
cana-5776	116	4	in	in	ADP
cana-5776	116	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	116	6	,	,	PUNCT
cana-5776	116	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	116	8	)	)	PUNCT
cana-5776	116	9	adjacent	adjacent	ADJ
cana-5776	116	10	to	to	AUX
cana-5776	116	11	atleast	atleast	VERB
cana-5776	116	12	one	one	NUM
cana-5776	116	13	edge	edge	NOUN
cana-5776	116	14	in	in	ADP
cana-5776	116	15	𝑆	𝑆	PROPN
cana-5776	116	16	or	or	CCONJ
cana-5776	116	17	incident	incident	NOUN
cana-5776	116	18	with	with	ADP
cana-5776	116	19	atleast	atleast	ADJ
cana-5776	116	20	one	one	NUM
cana-5776	116	21	vertex	vertex	NOUN
cana-5776	116	22	in	in	ADP
cana-5776	116	23	𝑆	𝑆	PROPN
cana-5776	116	24	also	also	ADV
cana-5776	116	25	every	every	DET
cana-5776	116	26	vertex	vertex	NOUN
cana-5776	116	27	in	in	ADP
cana-5776	116	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	116	29	,	,	PUNCT
cana-5776	116	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	116	31	)	)	PUNCT
cana-5776	116	32	either	either	DET
cana-5776	116	33	incident	incident	NOUN
cana-5776	116	34	with	with	ADP
cana-5776	116	35	atleast	atleast	ADJ
cana-5776	116	36	one	one	NUM
cana-5776	116	37	edge	edge	NOUN
cana-5776	116	38	or	or	CCONJ
cana-5776	116	39	adjacent	adjacent	ADJ
cana-5776	116	40	to	to	PART
cana-5776	116	41	atleast	atleast	VERB
cana-5776	116	42	one	one	NUM
cana-5776	116	43	vertex	vertex	NOUN
cana-5776	116	44	in	in	ADP
cana-5776	116	45	𝑆.	𝑆.	PROPN
cana-5776	116	46	therefore	therefore	ADV
cana-5776	116	47	𝑆	𝑆	PROPN
cana-5776	116	48	becomes	become	VERB
cana-5776	116	49	entire	entire	ADJ
cana-5776	116	50	domination	domination	NOUN
cana-5776	116	51	set	set	NOUN
cana-5776	116	52	of	of	ADP
cana-5776	116	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	116	54	,	,	PUNCT
cana-5776	116	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	116	56	)	)	PUNCT
cana-5776	116	57	and	and	CCONJ
cana-5776	116	58	it	it	PRON
cana-5776	116	59	is	be	AUX
cana-5776	116	60	minimum	minimum	ADJ
cana-5776	116	61	,	,	PUNCT
cana-5776	116	62	|𝑆|	|𝑆|	VERB
cana-5776	116	63	=	=	SYM
cana-5776	116	64	(	(	PUNCT
cana-5776	116	65	|𝐸|	|𝐸|	X
cana-5776	116	66	2	2	NUM
cana-5776	116	67	+	+	NUM
cana-5776	116	68	|𝑉|	|𝑉|	NOUN
cana-5776	116	69	)	)	PUNCT
cana-5776	117	1	−	−	NOUN
cana-5776	117	2	𝑟	𝑟	SYM
cana-5776	117	3	5	5	NUM
cana-5776	117	4	+	+	SYM
cana-5776	117	5	1	1	NUM
cana-5776	117	6	=	=	SYM
cana-5776	117	7	(	(	PUNCT
cana-5776	117	8	2𝑛	2𝑛	PROPN
cana-5776	117	9	2	2	NUM
cana-5776	117	10	+	+	CCONJ
cana-5776	117	11	𝑛	𝑛	NOUN
cana-5776	117	12	)	)	PUNCT
cana-5776	117	13	−	−	NOUN
cana-5776	117	14	𝑟	𝑟	SYM
cana-5776	117	15	5	5	NUM
cana-5776	117	16	+	+	SYM
cana-5776	117	17	1	1	NUM
cana-5776	117	18	=	=	SYM
cana-5776	117	19	(	(	PUNCT
cana-5776	117	20	𝑛	𝑛	PROPN
cana-5776	117	21	+	+	NUM
cana-5776	117	22	𝑛	𝑛	NOUN
cana-5776	117	23	)	)	PUNCT
cana-5776	117	24	−	−	NOUN
cana-5776	117	25	𝑟	𝑟	SYM
cana-5776	117	26	5	5	NUM
cana-5776	117	27	+	+	SYM
cana-5776	117	28	1	1	NUM
cana-5776	117	29	=	=	SYM
cana-5776	117	30	2𝑛	2𝑛	PROPN
cana-5776	118	1	−	−	PROPN
cana-5776	118	2	𝑟	𝑟	NOUN
cana-5776	118	3	+	+	CCONJ
cana-5776	118	4	5	5	NUM
cana-5776	118	5	5	5	NUM
cana-5776	118	6	.	.	PUNCT
cana-5776	119	1	therefore	therefore	ADV
cana-5776	119	2	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	119	3	+	+	ADJ
cana-5776	119	4	(	(	PUNCT
cana-5776	119	5	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	119	6	,	,	PUNCT
cana-5776	119	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	119	8	)	)	PUNCT
cana-5776	119	9	)	)	PUNCT
cana-5776	120	1	=	=	SYM
cana-5776	120	2	2𝑛−𝑟+5	2𝑛−𝑟+5	NUM
cana-5776	120	3	5	5	NUM
cana-5776	120	4	theorem	theorem	VERB
cana-5776	120	5	3.5	3.5	NUM
cana-5776	120	6	:	:	PUNCT
cana-5776	120	7	for	for	ADP
cana-5776	120	8	the	the	DET
cana-5776	120	9	involutory	involutory	NOUN
cana-5776	120	10	addition	addition	NOUN
cana-5776	120	11	cayley	cayley	NOUN
cana-5776	120	12	graph	graph	NOUN
cana-5776	120	13	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	120	14	,	,	PUNCT
cana-5776	120	15	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	120	16	)	)	PUNCT
cana-5776	120	17	,	,	PUNCT
cana-5776	120	18	if	if	SCONJ
cana-5776	120	19	n	n	PRON
cana-5776	120	20	is	be	AUX
cana-5776	120	21	odd	odd	ADJ
cana-5776	120	22	and	and	CCONJ
cana-5776	120	23	𝑛	𝑛	ADP
cana-5776	120	24	>	>	X
cana-5776	120	25	2	2	NUM
cana-5776	120	26	,	,	PUNCT
cana-5776	120	27	|𝐼𝑣|	|𝐼𝑣|	NOUN
cana-5776	120	28	=	=	NOUN
cana-5776	120	29	4	4	NUM
cana-5776	120	30	,	,	PUNCT
cana-5776	120	31	𝑛	𝑛	PRON
cana-5776	120	32	=	=	VERB
cana-5776	120	33	3𝑝	3𝑝	NOUN
cana-5776	120	34	,	,	PUNCT
cana-5776	120	35	𝑝	𝑝	NOUN
cana-5776	120	36	>	>	SYM
cana-5776	120	37	3	3	NUM
cana-5776	120	38	then	then	ADV
cana-5776	120	39	the	the	DET
cana-5776	120	40	entire	entire	ADJ
cana-5776	120	41	domination	domination	NOUN
cana-5776	120	42	number	number	NOUN
cana-5776	120	43	is	be	AUX
cana-5776	120	44	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	120	45	+	+	PROPN
cana-5776	120	46	(	(	PUNCT
cana-5776	120	47	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	120	48	,	,	PUNCT
cana-5776	120	49	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	120	50	)	)	PUNCT
cana-5776	120	51	)	)	PUNCT
cana-5776	121	1	=	=	SYM
cana-5776	121	2	𝑛	𝑛	PROPN
cana-5776	122	1	+	+	NOUN
cana-5776	122	2	1	1	NUM
cana-5776	122	3	2	2	NUM
cana-5776	122	4	.	.	PUNCT
cana-5776	123	1	proof	proof	NOUN
cana-5776	123	2	:	:	PUNCT
cana-5776	123	3	consider	consider	VERB
cana-5776	123	4	a	a	DET
cana-5776	123	5	graph𝐺+(𝑍𝑛	graph𝐺+(𝑍𝑛	NOUN
cana-5776	123	6	,	,	PUNCT
cana-5776	123	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	123	8	)	)	PUNCT
cana-5776	123	9	with	with	ADP
cana-5776	123	10	vertex	vertex	NOUN
cana-5776	123	11	set	set	VERB
cana-5776	124	1	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	124	2	=	=	PUNCT
cana-5776	124	3	{	{	PUNCT
cana-5776	124	4	0,1,2,3	0,1,2,3	NUM
cana-5776	124	5	,	,	PUNCT
cana-5776	124	6	…	…	PUNCT
cana-5776	124	7	,	,	PUNCT
cana-5776	124	8	𝑛	𝑛	DET
cana-5776	124	9	−	−	NOUN
cana-5776	124	10	1	1	NUM
cana-5776	124	11	}	}	PUNCT
cana-5776	124	12	where	where	SCONJ
cana-5776	124	13	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	124	14	denotes	denote	VERB
cana-5776	124	15	the	the	DET
cana-5776	124	16	set	set	NOUN
cana-5776	124	17	of	of	ADP
cana-5776	124	18	involutory	involutory	ADJ
cana-5776	124	19	elements	element	NOUN
cana-5776	124	20	in	in	ADP
cana-5776	124	21	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	124	22	and	and	CCONJ
cana-5776	124	23	𝑛	𝑛	PROPN
cana-5776	124	24	is	be	AUX
cana-5776	124	25	odd	odd	ADJ
cana-5776	124	26	,	,	PUNCT
cana-5776	124	27	𝑛	𝑛	PROPN
cana-5776	124	28	>	>	X
cana-5776	124	29	2,|𝐼𝑣|	2,|𝐼𝑣|	NUM
cana-5776	124	30	=	=	SYM
cana-5776	124	31	4	4	NUM
cana-5776	124	32	,	,	PUNCT
cana-5776	124	33	𝑛	𝑛	PRON
cana-5776	125	1	=	=	VERB
cana-5776	125	2	3𝑝	3𝑝	NOUN
cana-5776	125	3	,	,	PUNCT
cana-5776	125	4	𝑝	𝑝	NOUN
cana-5776	125	5	>	>	X
cana-5776	125	6	3	3	NUM
cana-5776	125	7	,	,	PUNCT
cana-5776	125	8	𝑝	𝑝	NOUN
cana-5776	125	9	is	be	AUX
cana-5776	125	10	prime|𝑒|	prime|𝑒|	NOUN
cana-5776	125	11	=	=	SYM
cana-5776	125	12	2𝑛	2𝑛	PROPN
cana-5776	126	1	−	−	PROPN
cana-5776	126	2	2	2	X
cana-5776	126	3	.	.	X
cana-5776	126	4	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	126	5	,	,	PUNCT
cana-5776	126	6	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	126	7	)	)	PUNCT
cana-5776	126	8	contains	contain	VERB
cana-5776	126	9	two	two	NUM
cana-5776	126	10	paths	path	NOUN
cana-5776	126	11	and	and	CCONJ
cana-5776	126	12	each	each	DET
cana-5776	126	13	path	path	NOUN
cana-5776	126	14	contains	contain	VERB
cana-5776	126	15	𝑛	𝑛	DET
cana-5776	126	16	−	−	NUM
cana-5776	126	17	1	1	NUM
cana-5776	126	18	number	number	NOUN
cana-5776	126	19	of	of	ADP
cana-5776	126	20	edges	edge	NOUN
cana-5776	126	21	.	.	PUNCT
cana-5776	127	1	from	from	ADP
cana-5776	127	2	first	first	ADJ
cana-5776	127	3	hamiltonian	hamiltonian	ADJ
cana-5776	127	4	cycle	cycle	NOUN
cana-5776	127	5	,	,	PUNCT
cana-5776	127	6	consider	consider	VERB
cana-5776	127	7	𝑆	𝑆	PROPN
cana-5776	127	8	=	=	SYM
cana-5776	127	9	{	{	PUNCT
cana-5776	127	10	𝑒𝑖𝑗	𝑒𝑖𝑗	PROPN
cana-5776	127	11	,	,	PUNCT
cana-5776	127	12	𝑣𝑘	𝑣𝑘	ADV
cana-5776	127	13	0	0	NUM
cana-5776	127	14	≤	≤	NUM
cana-5776	127	15	𝑖	𝑖	PUNCT
cana-5776	127	16	≤	≤	NOUN
cana-5776	128	1	𝑛−3	𝑛−3	PROPN
cana-5776	128	2	2	2	NUM
cana-5776	128	3	,	,	PUNCT
cana-5776	128	4	𝑛+1	𝑛+1	PROPN
cana-5776	128	5	2	2	NUM
cana-5776	128	6	≤	≤	NUM
cana-5776	128	7	𝑗	𝑗	PRON
cana-5776	128	8	≤	≤	NUM
cana-5776	128	9	𝑛	𝑛	PRON
cana-5776	128	10	−	−	PROPN
cana-5776	128	11	1	1	NUM
cana-5776	128	12	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-5776	128	13	𝑘	𝑘	PROPN
cana-5776	128	14	=	=	SYM
cana-5776	128	15	𝑛−1	𝑛−1	NUM
cana-5776	128	16	2	2	NUM
cana-5776	128	17	}	}	PUNCT
cana-5776	128	18	.	.	PUNCT
cana-5776	129	1	now	now	ADV
cana-5776	129	2	every	every	DET
cana-5776	129	3	edge	edge	NOUN
cana-5776	129	4	in	in	ADP
cana-5776	129	5	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	129	6	,	,	PUNCT
cana-5776	129	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	129	8	)	)	PUNCT
cana-5776	129	9	adjacent	adjacent	ADJ
cana-5776	129	10	to	to	AUX
cana-5776	129	11	atleast	atleast	VERB
cana-5776	129	12	one	one	NUM
cana-5776	129	13	edge	edge	NOUN
cana-5776	129	14	in	in	ADP
cana-5776	129	15	𝑆	𝑆	PROPN
cana-5776	129	16	or	or	CCONJ
cana-5776	129	17	incident	incident	NOUN
cana-5776	129	18	with	with	ADP
cana-5776	129	19	atleast	atleast	ADJ
cana-5776	129	20	one	one	NUM
cana-5776	129	21	vertex	vertex	NOUN
cana-5776	129	22	in	in	ADP
cana-5776	129	23	𝑆	𝑆	PROPN
cana-5776	129	24	also	also	ADV
cana-5776	129	25	every	every	DET
cana-5776	129	26	vertex	vertex	NOUN
cana-5776	129	27	in	in	ADP
cana-5776	129	28	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	PROPN
cana-5776	129	29	,	,	PUNCT
cana-5776	129	30	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	129	31	)	)	PUNCT
cana-5776	129	32	either	either	DET
cana-5776	129	33	incident	incident	NOUN
cana-5776	129	34	with	with	ADP
cana-5776	129	35	atleast	atleast	ADJ
cana-5776	129	36	one	one	NUM
cana-5776	129	37	edge	edge	NOUN
cana-5776	129	38	or	or	CCONJ
cana-5776	129	39	adjacent	adjacent	ADJ
cana-5776	129	40	to	to	PART
cana-5776	129	41	atleast	atleast	VERB
cana-5776	129	42	one	one	NUM
cana-5776	129	43	vertex	vertex	NOUN
cana-5776	129	44	in	in	ADP
cana-5776	129	45	𝑆.	𝑆.	PROPN
cana-5776	129	46	therefore	therefore	ADV
cana-5776	129	47	𝑆	𝑆	PROPN
cana-5776	129	48	becomes	become	VERB
cana-5776	129	49	entire	entire	ADJ
cana-5776	129	50	domination	domination	NOUN
cana-5776	129	51	set	set	NOUN
cana-5776	129	52	of	of	ADP
cana-5776	129	53	𝐺+(𝑍𝑛	𝐺+(𝑍𝑛	NUM
cana-5776	129	54	,	,	PUNCT
cana-5776	129	55	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	129	56	)	)	PUNCT
cana-5776	129	57	and	and	CCONJ
cana-5776	129	58	it	it	PRON
cana-5776	129	59	is	be	AUX
cana-5776	129	60	minimum	minimum	ADJ
cana-5776	129	61	.	.	PUNCT
cana-5776	130	1	|𝑆|	|𝑆|	VERB
cana-5776	130	2	=	=	PUNCT
cana-5776	130	3	|𝑉|	|𝑉|	NOUN
cana-5776	130	4	−	−	NOUN
cana-5776	130	5	1	1	NUM
cana-5776	130	6	2	2	NUM
cana-5776	130	7	+	+	NUM
cana-5776	130	8	1	1	NUM
cana-5776	130	9	=	=	SYM
cana-5776	130	10	𝑛	𝑛	PRON
cana-5776	130	11	−	−	NUM
cana-5776	130	12	1	1	NUM
cana-5776	130	13	2	2	NUM
cana-5776	130	14	+	+	NUM
cana-5776	130	15	1	1	NUM
cana-5776	130	16	=	=	SYM
cana-5776	130	17	𝑛	𝑛	NOUN
cana-5776	130	18	+	+	NOUN
cana-5776	130	19	1	1	NUM
cana-5776	130	20	2	2	NUM
cana-5776	130	21	.	.	PUNCT
cana-5776	131	1	therefore	therefore	ADV
cana-5776	131	2	𝛾𝑒𝑛(𝐺	𝛾𝑒𝑛(𝐺	PUNCT
cana-5776	131	3	+	+	ADJ
cana-5776	131	4	(	(	PUNCT
cana-5776	131	5	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	131	6	,	,	PUNCT
cana-5776	131	7	𝐼𝑣	𝐼𝑣	PROPN
cana-5776	131	8	)	)	PUNCT
cana-5776	131	9	)	)	PUNCT
cana-5776	132	1	=	=	SYM
cana-5776	132	2	𝑛+1	𝑛+1	ADP
cana-5776	132	3	2	2	NUM
cana-5776	132	4	conclusion	conclusion	NOUN
cana-5776	132	5	using	use	VERB
cana-5776	132	6	involutory	involutory	NOUN
cana-5776	132	7	addition	addition	NOUN
cana-5776	132	8	cayley	cayley	NOUN
cana-5776	132	9	graphs	graph	NOUN
cana-5776	132	10	,	,	PUNCT
cana-5776	132	11	it	it	PRON
cana-5776	132	12	is	be	AUX
cana-5776	132	13	interesting	interesting	ADJ
cana-5776	132	14	to	to	PART
cana-5776	132	15	find	find	VERB
cana-5776	132	16	the	the	DET
cana-5776	132	17	connected	connected	ADJ
cana-5776	132	18	edge	edge	NOUN
cana-5776	132	19	domination	domination	NOUN
cana-5776	132	20	number	number	NOUN
cana-5776	132	21	and	and	CCONJ
cana-5776	132	22	entire	entire	ADJ
cana-5776	132	23	domination	domination	NOUN
cana-5776	132	24	number	number	NOUN
cana-5776	132	25	of	of	ADP
cana-5776	132	26	involutory	involutory	ADJ
cana-5776	132	27	addition	addition	NOUN
cana-5776	132	28	cayley	cayley	NOUN
cana-5776	132	29	graphs	graph	NOUN
cana-5776	132	30	and	and	CCONJ
cana-5776	132	31	the	the	DET
cana-5776	132	32	authors	author	NOUN
cana-5776	132	33	have	have	AUX
cana-5776	132	34	also	also	ADV
cana-5776	132	35	studied	study	VERB
cana-5776	132	36	this	this	DET
cana-5776	132	37	aspect	aspect	NOUN
cana-5776	132	38	.	.	PUNCT
cana-5776	133	1	communications	communication	NOUN
cana-5776	133	2	on	on	ADP
cana-5776	133	3	applied	apply	VERB
cana-5776	133	4	nonlinear	nonlinear	ADJ
cana-5776	133	5	analysis	analysis	NOUN
cana-5776	133	6	issn	issn	NOUN
cana-5776	133	7	:	:	PUNCT
cana-5776	133	8	1074	1074	NUM
cana-5776	133	9	-	-	PUNCT
cana-5776	133	10	133x	133x	NUM
cana-5776	133	11	vol	vol	NOUN
cana-5776	133	12	31	31	NUM
cana-5776	133	13	no	no	NOUN
cana-5776	133	14	.	.	PUNCT
cana-5776	134	1	7s	7	NOUN
cana-5776	134	2	(	(	PUNCT
cana-5776	134	3	2024	2024	NUM
cana-5776	134	4	)	)	PUNCT
cana-5776	134	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	134	6	788	788	NUM
cana-5776	134	7	references	reference	NOUN
cana-5776	134	8	1	1	NUM
cana-5776	134	9	.	.	PUNCT
cana-5776	135	1	berge	berge	NOUN
cana-5776	135	2	,	,	PUNCT
cana-5776	135	3	c.-the	c.-the	PRON
cana-5776	135	4	theory	theory	NOUN
cana-5776	135	5	of	of	ADP
cana-5776	135	6	graphs	graph	NOUN
cana-5776	135	7	and	and	CCONJ
cana-5776	135	8	its	its	PRON
cana-5776	135	9	applications	application	NOUN
cana-5776	135	10	,	,	PUNCT
cana-5776	135	11	methuen	methuen	PROPN
cana-5776	135	12	,	,	PUNCT
cana-5776	135	13	london(1962	london(1962	PROPN
cana-5776	135	14	)	)	PUNCT
cana-5776	135	15	.	.	PUNCT
cana-5776	136	1	2	2	X
cana-5776	136	2	.	.	X
cana-5776	136	3	cockayne	cockayne	NOUN
cana-5776	136	4	,	,	PUNCT
cana-5776	136	5	e.j	e.j	PROPN
cana-5776	136	6	,	,	PUNCT
cana-5776	136	7	hedetniemi	hedetniemi	PROPN
cana-5776	136	8	,	,	PUNCT
cana-5776	136	9	s.t	s.t	PROPN
cana-5776	136	10	-	-	PUNCT
cana-5776	136	11	towards	towards	ADP
cana-5776	136	12	a	a	DET
cana-5776	136	13	theory	theory	NOUN
cana-5776	136	14	of	of	ADP
cana-5776	136	15	domination	domination	NOUN
cana-5776	136	16	in	in	ADP
cana-5776	136	17	graphs	graph	NOUN
cana-5776	136	18	,	,	PUNCT
cana-5776	136	19	networks	network	NOUN
cana-5776	136	20	,	,	PUNCT
cana-5776	136	21	7(1977),247	7(1977),247	PROPN
cana-5776	136	22	-	-	SYM
cana-5776	136	23	261	261	NUM
cana-5776	136	24	.	.	PUNCT
cana-5776	137	1	3	3	X
cana-5776	137	2	.	.	X
cana-5776	137	3	c.prameela	c.prameela	NOUN
cana-5776	137	4	rani	rani	PROPN
cana-5776	137	5	and	and	CCONJ
cana-5776	137	6	m.siva	m.siva	PROPN
cana-5776	137	7	parvathi	parvathi	PROPN
cana-5776	137	8	,	,	PUNCT
cana-5776	137	9	characterization	characterization	NOUN
cana-5776	137	10	of	of	ADP
cana-5776	137	11	the	the	DET
cana-5776	137	12	set	set	NOUN
cana-5776	137	13	of	of	ADP
cana-5776	137	14	involutory	involutory	ADJ
cana-5776	137	15	elements	element	NOUN
cana-5776	137	16	of	of	ADP
cana-5776	137	17	(	(	PUNCT
cana-5776	137	18	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	137	19	,	,	PUNCT
cana-5776	137	20	⨁𝑛	⨁𝑛	PROPN
cana-5776	137	21	,	,	PUNCT
cana-5776	137	22	⨀𝑛),advances	⨀𝑛),advance	NOUN
cana-5776	137	23	in	in	ADP
cana-5776	137	24	mathematics	mathematic	NOUN
cana-5776	137	25	:	:	PUNCT
cana-5776	137	26	scientific	scientific	ADJ
cana-5776	137	27	journal	journal	NOUN
cana-5776	137	28	10	10	NUM
cana-5776	137	29	(	(	PUNCT
cana-5776	137	30	2021	2021	NUM
cana-5776	137	31	)	)	PUNCT
cana-5776	137	32	,	,	PUNCT
cana-5776	137	33	no.1	no.1	NUM
cana-5776	137	34	,	,	PUNCT
cana-5776	137	35	583	583	NUM
cana-5776	137	36	-	-	SYM
cana-5776	137	37	588	588	NUM
cana-5776	137	38	,	,	PUNCT
cana-5776	137	39	issn:1857	issn:1857	NOUN
cana-5776	137	40	-	-	PUNCT
cana-5776	137	41	8365(printed	8365(printed	NUM
cana-5776	137	42	)	)	PUNCT
cana-5776	137	43	;	;	PUNCT
cana-5776	137	44	1857	1857	NUM
cana-5776	137	45	-	-	SYM
cana-5776	137	46	8438(electronic	8438(electronic	NUM
cana-5776	137	47	)	)	PUNCT
cana-5776	137	48	4	4	NUM
cana-5776	137	49	.	.	X
cana-5776	137	50	c.prameela	c.prameela	NOUN
cana-5776	137	51	rani	rani	PROPN
cana-5776	137	52	,	,	PUNCT
cana-5776	137	53	m.siva	m.siva	PROPN
cana-5776	137	54	parvathi	parvathi	PROPN
cana-5776	137	55	and	and	CCONJ
cana-5776	137	56	r.lakshmi	r.lakshmi	NOUN
cana-5776	137	57	,	,	PUNCT
cana-5776	137	58	domination	domination	NOUN
cana-5776	137	59	and	and	CCONJ
cana-5776	137	60	domatic	domatic	ADJ
cana-5776	137	61	numbers	number	NOUN
cana-5776	137	62	of	of	ADP
cana-5776	137	63	involutory	involutory	ADJ
cana-5776	137	64	cayley	cayley	NOUN
cana-5776	137	65	graph	graph	NOUN
cana-5776	137	66	,	,	PUNCT
cana-5776	137	67	advances	advance	NOUN
cana-5776	137	68	and	and	CCONJ
cana-5776	137	69	applications	application	NOUN
cana-5776	137	70	in	in	ADP
cana-5776	137	71	discrete	discrete	ADJ
cana-5776	137	72	mathematics	mathematic	NOUN
cana-5776	137	73	©	©	PROPN
cana-5776	137	74	2021	2021	NUM
cana-5776	137	75	pushpa	pushpa	NOUN
cana-5776	137	76	publishing	publishing	PROPN
cana-5776	137	77	house	house	PROPN
cana-5776	137	78	,	,	PUNCT
cana-5776	137	79	prayagraj	prayagraj	PROPN
cana-5776	137	80	,	,	PUNCT
cana-5776	137	81	india	india	PROPN
cana-5776	137	82	,	,	PUNCT
cana-5776	137	83	volume	volume	NOUN
cana-5776	137	84	28	28	NUM
cana-5776	137	85	,	,	PUNCT
cana-5776	137	86	number	number	NOUN
cana-5776	137	87	2	2	NUM
cana-5776	137	88	,	,	PUNCT
cana-5776	137	89	2021	2021	NUM
cana-5776	137	90	,	,	PUNCT
cana-5776	137	91	pages	page	NOUN
cana-5776	137	92	335	335	NUM
cana-5776	137	93	-	-	SYM
cana-5776	137	94	350	350	NUM
cana-5776	137	95	.	.	PUNCT
cana-5776	138	1	5	5	NUM
cana-5776	138	2	.	.	X
cana-5776	139	1	dr	dr	PROPN
cana-5776	139	2	.	.	PROPN
cana-5776	139	3	b	b	PROPN
cana-5776	140	1	p	p	X
cana-5776	140	2	mallikarjunaswamy	mallikarjunaswamy	NOUN
cana-5776	140	3	,	,	PUNCT
cana-5776	140	4	venugeetha	venugeetha	VERB
cana-5776	140	5	y	y	PROPN
cana-5776	140	6	,	,	PUNCT
cana-5776	140	7	prof	prof	PROPN
cana-5776	140	8	.	.	PUNCT
cana-5776	141	1	v	v	NOUN
cana-5776	141	2	r	r	NOUN
cana-5776	141	3	kulli	kulli	PROPN
cana-5776	141	4	-	-	PUNCT
cana-5776	141	5	theory	theory	NOUN
cana-5776	141	6	of	of	ADP
cana-5776	141	7	edge	edge	NOUN
cana-5776	141	8	domination	domination	NOUN
cana-5776	141	9	in	in	ADP
cana-5776	141	10	graphs	graph	NOUN
cana-5776	141	11	-	-	PUNCT
cana-5776	141	12	a	a	DET
cana-5776	141	13	study	study	NOUN
cana-5776	141	14	-	-	PUNCT
cana-5776	141	15	iosr	iosr	ADJ
cana-5776	141	16	journal	journal	NOUN
cana-5776	141	17	of	of	ADP
cana-5776	141	18	engineering	engineering	NOUN
cana-5776	141	19	(	(	PUNCT
cana-5776	141	20	iosrjen	iosrjen	PROPN
cana-5776	141	21	)	)	PUNCT
cana-5776	141	22	,	,	PUNCT
cana-5776	141	23	issn	issn	PROPN
cana-5776	141	24	(	(	PUNCT
cana-5776	141	25	e	e	NOUN
cana-5776	141	26	):	):	PUNCT
cana-5776	141	27	2250	2250	NUM
cana-5776	141	28	-	-	SYM
cana-5776	141	29	3021	3021	NUM
cana-5776	141	30	,	,	PUNCT
cana-5776	141	31	issn	issn	PROPN
cana-5776	141	32	(	(	PUNCT
cana-5776	141	33	p	p	NOUN
cana-5776	141	34	):	):	PUNCT
cana-5776	141	35	2278	2278	NUM
cana-5776	141	36	-	-	SYM
cana-5776	141	37	8719	8719	NUM
cana-5776	141	38	vol	vol	NOUN
cana-5776	141	39	.	.	PROPN
cana-5776	142	1	08	08	NUM
cana-5776	143	1	,	,	PUNCT
cana-5776	143	2	issue	issue	VERB
cana-5776	143	3	7	7	NUM
cana-5776	143	4	(	(	PUNCT
cana-5776	143	5	july	july	PROPN
cana-5776	143	6	.	.	PROPN
cana-5776	143	7	2018	2018	NUM
cana-5776	143	8	)	)	PUNCT
cana-5776	143	9	,	,	PUNCT
cana-5776	143	10	||v	||v	NOUN
cana-5776	143	11	(	(	PUNCT
cana-5776	143	12	iv	iv	X
cana-5776	143	13	)	)	PUNCT
cana-5776	143	14	||	||	NOUN
cana-5776	144	1	84	84	NUM
cana-5776	144	2	-	-	SYM
cana-5776	144	3	92	92	NUM
cana-5776	144	4	.	.	PUNCT
cana-5776	145	1	6	6	NUM
cana-5776	145	2	.	.	X
cana-5776	145	3	e.lavanya	e.lavanya	NOUN
cana-5776	145	4	,	,	PUNCT
cana-5776	145	5	g.keerthi	g.keerthi	PUNCT
cana-5776	145	6	and	and	CCONJ
cana-5776	145	7	dr.m.siva	dr.m.siva	PROPN
cana-5776	145	8	parvathi	parvathi	NOUN
cana-5776	145	9	,	,	PUNCT
cana-5776	145	10	edge	edge	VERB
cana-5776	145	11	domination	domination	NOUN
cana-5776	145	12	of	of	ADP
cana-5776	145	13	an	an	DET
cana-5776	145	14	involutory	involutory	NOUN
cana-5776	145	15	addition	addition	NOUN
cana-5776	145	16	cayley	cayley	NOUN
cana-5776	145	17	graph	graph	NOUN
cana-5776	145	18	,	,	PUNCT
cana-5776	145	19	journal	journal	NOUN
cana-5776	145	20	of	of	ADP
cana-5776	145	21	computational	computational	ADJ
cana-5776	145	22	analysis	analysis	NOUN
cana-5776	145	23	and	and	CCONJ
cana-5776	145	24	applications	application	NOUN
cana-5776	145	25	,	,	PUNCT
cana-5776	145	26	vol.33	vol.33	PROPN
cana-5776	145	27	,	,	PUNCT
cana-5776	145	28	no.2	no.2	PROPN
cana-5776	145	29	,	,	PUNCT
cana-5776	145	30	2024	2024	NUM
cana-5776	145	31	.	.	PUNCT
cana-5776	146	1	7	7	X
cana-5776	146	2	.	.	X
cana-5776	146	3	g.s.shanmuga	g.s.shanmuga	PROPN
cana-5776	146	4	priya	priya	PROPN
cana-5776	146	5	,	,	PUNCT
cana-5776	146	6	m.siva	m.siva	PROPN
cana-5776	146	7	parvathi	parvathi	PROPN
cana-5776	146	8	and	and	CCONJ
cana-5776	146	9	k.manjula	k.manjula	PROPN
cana-5776	146	10	,	,	PUNCT
cana-5776	146	11	some	some	DET
cana-5776	146	12	properties	property	NOUN
cana-5776	146	13	of	of	ADP
cana-5776	146	14	involutory	involutory	ADJ
cana-5776	146	15	addition	addition	NOUN
cana-5776	146	16	cayley	cayley	NOUN
cana-5776	146	17	graph	graph	NOUN
cana-5776	146	18	,	,	PUNCT
cana-5776	146	19	advances	advance	NOUN
cana-5776	146	20	in	in	ADP
cana-5776	146	21	mathematics	mathematic	NOUN
cana-5776	146	22	:	:	PUNCT
cana-5776	146	23	scientific	scientific	ADJ
cana-5776	146	24	journal	journal	NOUN
cana-5776	146	25	9	9	NUM
cana-5776	146	26	(	(	PUNCT
cana-5776	146	27	2020	2020	NUM
cana-5776	146	28	)	)	PUNCT
cana-5776	146	29	,	,	PUNCT
cana-5776	146	30	no.12	no.12	VERB
cana-5776	146	31	,	,	PUNCT
cana-5776	146	32	11089	11089	NUM
cana-5776	146	33	-	-	SYM
cana-5776	146	34	11095	11095	NUM
cana-5776	146	35	,	,	PUNCT
cana-5776	146	36	issn:1857	issn:1857	NOUN
cana-5776	146	37	-	-	PUNCT
cana-5776	146	38	8365(printed	8365(printed	NUM
cana-5776	146	39	)	)	PUNCT
cana-5776	146	40	;	;	PUNCT
cana-5776	146	41	1857	1857	NUM
cana-5776	146	42	-	-	SYM
cana-5776	146	43	8438(electronic	8438(electronic	NUM
cana-5776	146	44	)	)	PUNCT
cana-5776	146	45	8	8	NUM
cana-5776	146	46	.	.	PUNCT
cana-5776	146	47	m.venkata	m.venkata	NOUN
cana-5776	146	48	anusha	anusha	NOUN
cana-5776	146	49	and	and	CCONJ
cana-5776	146	50	m.siva	m.siva	PROPN
cana-5776	146	51	parvathi	parvathi	PROPN
cana-5776	146	52	,	,	PUNCT
cana-5776	146	53	properties	property	NOUN
cana-5776	146	54	of	of	ADP
cana-5776	146	55	the	the	DET
cana-5776	146	56	involutory	involutory	ADJ
cana-5776	146	57	cayley	cayley	NOUN
cana-5776	146	58	graphs	graph	NOUN
cana-5776	146	59	of	of	ADP
cana-5776	146	60	(	(	PUNCT
cana-5776	146	61	𝑍𝑛	𝑍𝑛	PROPN
cana-5776	146	62	,	,	PUNCT
cana-5776	146	63	⨁,⨀	⨁,⨀	PROPN
cana-5776	146	64	)	)	PUNCT
cana-5776	146	65	,	,	PUNCT
cana-5776	146	66	aip	aip	PROPN
cana-5776	146	67	conference	conference	NOUN
cana-5776	146	68	proceedings	proceeding	NOUN
cana-5776	146	69	2246	2246	NUM
cana-5776	146	70	,	,	PUNCT
cana-5776	146	71	020065	020065	NUM
cana-5776	146	72	(	(	PUNCT
cana-5776	146	73	2020	2020	NUM
cana-5776	146	74	)	)	PUNCT
cana-5776	146	75	.	.	PUNCT
cana-5776	147	1	9	9	X
cana-5776	147	2	.	.	X
cana-5776	147	3	ore	ore	NOUN
cana-5776	147	4	,	,	PUNCT
cana-5776	147	5	o.theory	o.theory	NOUN
cana-5776	147	6	of	of	ADP
cana-5776	147	7	graphs	graph	NOUN
cana-5776	147	8	,	,	PUNCT
cana-5776	147	9	amer	amer	PROPN
cana-5776	147	10	.	.	PROPN
cana-5776	147	11	math	math	PROPN
cana-5776	147	12	.	.	PUNCT
cana-5776	148	1	soc	soc	PROPN
cana-5776	148	2	.	.	PUNCT
cana-5776	149	1	colloq	colloq	PROPN
cana-5776	149	2	.	.	PUNCT
cana-5776	150	1	publ	publ	PROPN
cana-5776	150	2	.	.	PROPN
cana-5776	150	3	,	,	PUNCT
cana-5776	150	4	1962	1962	NUM
cana-5776	150	5	.	.	PUNCT
cana-5776	151	1	10	10	NUM
cana-5776	151	2	.	.	PUNCT
cana-5776	152	1	v.	v.	PROPN
cana-5776	152	2	r.	r.	PROPN
cana-5776	152	3	kulli	kulli	PROPN
cana-5776	152	4	and	and	CCONJ
cana-5776	152	5	s.c	s.c	PROPN
cana-5776	152	6	.	.	PROPN
cana-5776	152	7	sigarkanti	sigarkanti	PROPN
cana-5776	152	8	,	,	PUNCT
cana-5776	152	9	the	the	DET
cana-5776	152	10	connected	connected	ADJ
cana-5776	152	11	edge	edge	NOUN
cana-5776	152	12	domination	domination	NOUN
cana-5776	152	13	number	number	NOUN
cana-5776	152	14	,	,	PUNCT
cana-5776	152	15	proc	proc	NOUN
cana-5776	152	16	.	.	PUNCT
cana-5776	153	1	r.c	r.c	PROPN
cana-5776	153	2	.	.	PROPN
cana-5776	153	3	bose	bose	PROPN
cana-5776	153	4	mem	mem	PROPN
cana-5776	153	5	.	.	PUNCT
cana-5776	153	6	conf	conf	NOUN
cana-5776	153	7	.	.	PUNCT
cana-5776	154	1	abstract	abstract	ADJ
cana-5776	154	2	1988	1988	NUM
cana-5776	154	3	.	.	PUNCT
cana-5776	155	1	11	11	NUM
cana-5776	155	2	.	.	PUNCT
cana-5776	155	3	v.r.kulli	v.r.kulli	NOUN
cana-5776	155	4	,	,	PUNCT
cana-5776	155	5	on	on	ADP
cana-5776	155	6	entire	entire	ADJ
cana-5776	155	7	domination	domination	NOUN
cana-5776	155	8	number	number	NOUN
cana-5776	155	9	,	,	PUNCT
cana-5776	155	10	second	second	ADJ
cana-5776	155	11	conf.ram.math.soc	conf.ram.math.soc	NOUN
cana-5776	155	12	.	.	PUNCT
cana-5776	155	13	madras	madras	PROPN
cana-5776	155	14	(	(	PUNCT
cana-5776	155	15	1987	1987	NUM
cana-5776	155	16	)	)	PUNCT
cana-5776	155	17	.	.	PUNCT
cana-5776	156	1	communications	communication	NOUN
cana-5776	156	2	on	on	ADP
cana-5776	156	3	applied	apply	VERB
cana-5776	156	4	nonlinear	nonlinear	ADJ
cana-5776	156	5	analysis	analysis	NOUN
cana-5776	156	6	issn	issn	NOUN
cana-5776	156	7	:	:	PUNCT
cana-5776	156	8	1074	1074	NUM
cana-5776	156	9	-	-	PUNCT
cana-5776	156	10	133x	133x	NUM
cana-5776	156	11	vol	vol	NOUN
cana-5776	156	12	31	31	NUM
cana-5776	156	13	no	no	NOUN
cana-5776	156	14	.	.	PUNCT
cana-5776	157	1	7s	7	NOUN
cana-5776	157	2	(	(	PUNCT
cana-5776	157	3	2024	2024	NUM
cana-5776	157	4	)	)	PUNCT
cana-5776	157	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5776	157	6	789	789	NUM
cana-5776	157	7	article	article	NOUN
cana-5776	157	8	history	history	NOUN
cana-5776	157	9	:	:	PUNCT
cana-5776	157	10	received	receive	VERB
cana-5776	157	11	20.09.2024	20.09.2024	NUM
cana-5776	157	12	revised	revise	VERB
cana-5776	157	13	:	:	PUNCT
cana-5776	157	14	24.10.2024	24.10.2024	NUM
cana-5776	157	15	accepted	accept	VERB
cana-5776	157	16	:	:	PUNCT
cana-5776	157	17	30.11.2024	30.11.2024	NUM