id	sid	tid	token	lemma	pos
cana-4517	1	1	communications	communication	NOUN
cana-4517	1	2	on	on	ADP
cana-4517	1	3	applied	apply	VERB
cana-4517	1	4	nonlinear	nonlinear	ADJ
cana-4517	1	5	analysis	analysis	NOUN
cana-4517	1	6	issn	issn	NOUN
cana-4517	1	7	:	:	PUNCT
cana-4517	1	8	1074	1074	NUM
cana-4517	1	9	-	-	PUNCT
cana-4517	1	10	133x	133x	NUM
cana-4517	1	11	vol	vol	NOUN
cana-4517	1	12	32	32	NUM
cana-4517	1	13	no	no	NOUN
cana-4517	1	14	.	.	PUNCT
cana-4517	2	1	9s	9s	NUM
cana-4517	2	2	(	(	PUNCT
cana-4517	2	3	2025	2025	NUM
cana-4517	2	4	)	)	PUNCT
cana-4517	2	5	2315	2315	NUM
cana-4517	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	2	7	a	a	DET
cana-4517	2	8	study	study	NOUN
cana-4517	2	9	on	on	ADP
cana-4517	2	10	perfect	perfect	ADJ
cana-4517	2	11	rings	ring	NOUN
cana-4517	2	12	dominating	dominate	VERB
cana-4517	2	13	energy	energy	NOUN
cana-4517	2	14	of	of	ADP
cana-4517	2	15	graphs	graph	NOUN
cana-4517	2	16	prema	prema	PROPN
cana-4517	2	17	m1	m1	PROPN
cana-4517	2	18	,	,	PUNCT
cana-4517	2	19	ruby	ruby	PROPN
cana-4517	2	20	selestina	selestina	PROPN
cana-4517	2	21	m2	m2	PROPN
cana-4517	2	22	and	and	CCONJ
cana-4517	2	23	purushothama	purushothama	PROPN
cana-4517	2	24	s3	s3	PROPN
cana-4517	2	25	1research	1research	NUM
cana-4517	2	26	scholar	scholar	NOUN
cana-4517	2	27	,	,	PUNCT
cana-4517	2	28	department	department	NOUN
cana-4517	2	29	of	of	ADP
cana-4517	2	30	mathematics	mathematics	PROPN
cana-4517	2	31	,	,	PUNCT
cana-4517	2	32	yuvaraja	yuvaraja	PROPN
cana-4517	2	33	’s	’s	PART
cana-4517	2	34	college	college	NOUN
cana-4517	2	35	,	,	PUNCT
cana-4517	2	36	university	university	NOUN
cana-4517	2	37	of	of	ADP
cana-4517	2	38	mysore	mysore	NOUN
cana-4517	2	39	,	,	PUNCT
cana-4517	2	40	mysuru	mysuru	NOUN
cana-4517	2	41	2	2	NUM
cana-4517	2	42	associate	associate	NOUN
cana-4517	2	43	professor	professor	NOUN
cana-4517	2	44	,	,	PUNCT
cana-4517	2	45	department	department	NOUN
cana-4517	2	46	of	of	ADP
cana-4517	2	47	mathematics	mathematics	PROPN
cana-4517	2	48	,	,	PUNCT
cana-4517	2	49	yuvaraja	yuvaraja	PROPN
cana-4517	2	50	’s	’s	PART
cana-4517	2	51	college	college	NOUN
cana-4517	2	52	,	,	PUNCT
cana-4517	2	53	university	university	NOUN
cana-4517	2	54	of	of	ADP
cana-4517	2	55	mysore	mysore	NOUN
cana-4517	2	56	,	,	PUNCT
cana-4517	2	57	mysuru	mysuru	NOUN
cana-4517	2	58	3associate	3associate	NUM
cana-4517	2	59	professor	professor	NOUN
cana-4517	2	60	,	,	PUNCT
cana-4517	2	61	department	department	NOUN
cana-4517	2	62	of	of	ADP
cana-4517	2	63	mathematics	mathematics	PROPN
cana-4517	2	64	,	,	PUNCT
cana-4517	2	65	mit	mit	NOUN
cana-4517	2	66	mysore	mysore	NOUN
cana-4517	2	67	,	,	PUNCT
cana-4517	2	68	mysuru	mysuru	NOUN
cana-4517	2	69	email	email	NOUN
cana-4517	2	70	:	:	PUNCT
cana-4517	2	71	mpremamallaiah@gmail.com	mpremamallaiah@gmail.com	X
cana-4517	3	1	ruby.salestina@gmail.com	ruby.salestina@gmail.com	X
cana-4517	3	2	psmandya@gmail.com	psmandya@gmail.com	X
cana-4517	4	1	article	article	NOUN
cana-4517	4	2	history	history	NOUN
cana-4517	4	3	:	:	PUNCT
cana-4517	4	4	received	receive	VERB
cana-4517	4	5	:	:	PUNCT
cana-4517	4	6	12	12	NUM
cana-4517	4	7	-	-	SYM
cana-4517	4	8	01	01	NUM
cana-4517	4	9	-	-	PUNCT
cana-4517	4	10	2025	2025	NUM
cana-4517	4	11	revised	revise	VERB
cana-4517	4	12	:	:	PUNCT
cana-4517	4	13	15	15	NUM
cana-4517	4	14	-	-	NUM
cana-4517	4	15	02	02	NUM
cana-4517	4	16	-	-	PUNCT
cana-4517	4	17	2025	2025	NUM
cana-4517	4	18	accepted	accept	VERB
cana-4517	4	19	:	:	PUNCT
cana-4517	4	20	01	01	NUM
cana-4517	4	21	-	-	SYM
cana-4517	4	22	03	03	NUM
cana-4517	4	23	-	-	PUNCT
cana-4517	4	24	2025	2025	NUM
cana-4517	4	25	abstract	abstract	NOUN
cana-4517	4	26	:	:	PUNCT
cana-4517	4	27	a	a	DET
cana-4517	4	28	novel	novel	ADJ
cana-4517	4	29	parameter	parameter	NOUN
cana-4517	4	30	called	call	VERB
cana-4517	4	31	perfect	perfect	ADJ
cana-4517	4	32	rings	ring	NOUN
cana-4517	4	33	domination	domination	NOUN
cana-4517	4	34	has	have	AUX
cana-4517	4	35	been	be	AUX
cana-4517	4	36	developed	develop	VERB
cana-4517	4	37	in	in	ADP
cana-4517	4	38	this	this	DET
cana-4517	4	39	study	study	NOUN
cana-4517	4	40	.	.	PUNCT
cana-4517	5	1	the	the	DET
cana-4517	5	2	corresponding	corresponding	ADJ
cana-4517	5	3	perfect	perfect	ADJ
cana-4517	5	4	rings	ring	NOUN
cana-4517	5	5	dominating	dominating	NOUN
cana-4517	5	6	matrix	matrix	NOUN
cana-4517	5	7	is	be	AUX
cana-4517	5	8	also	also	ADV
cana-4517	5	9	generated	generate	VERB
cana-4517	5	10	.	.	PUNCT
cana-4517	6	1	the	the	DET
cana-4517	6	2	energy	energy	NOUN
cana-4517	6	3	of	of	ADP
cana-4517	6	4	graphs	graph	NOUN
cana-4517	6	5	is	be	AUX
cana-4517	6	6	the	the	DET
cana-4517	6	7	sum	sum	NOUN
cana-4517	6	8	of	of	ADP
cana-4517	6	9	absolute	absolute	ADJ
cana-4517	6	10	value	value	NOUN
cana-4517	6	11	of	of	ADP
cana-4517	6	12	their	their	PRON
cana-4517	6	13	spectrum	spectrum	NOUN
cana-4517	6	14	.	.	PUNCT
cana-4517	7	1	this	this	DET
cana-4517	7	2	study	study	NOUN
cana-4517	7	3	investigates	investigate	VERB
cana-4517	7	4	the	the	DET
cana-4517	7	5	spectrum	spectrum	NOUN
cana-4517	7	6	and	and	CCONJ
cana-4517	7	7	energy	energy	NOUN
cana-4517	7	8	of	of	ADP
cana-4517	7	9	certain	certain	ADJ
cana-4517	7	10	family	family	NOUN
cana-4517	7	11	of	of	ADP
cana-4517	7	12	graphs	graph	NOUN
cana-4517	7	13	as	as	ADV
cana-4517	7	14	well	well	ADV
cana-4517	7	15	as	as	ADP
cana-4517	7	16	ϑ-obrazom	ϑ-obrazom	NOUN
cana-4517	7	17	of	of	ADP
cana-4517	7	18	graphs	graph	NOUN
cana-4517	7	19	corresponding	correspond	VERB
cana-4517	7	20	to	to	ADP
cana-4517	7	21	this	this	DET
cana-4517	7	22	matrix	matrix	NOUN
cana-4517	7	23	.	.	PUNCT
cana-4517	8	1	moreover	moreover	ADV
cana-4517	8	2	upper	upper	ADJ
cana-4517	8	3	and	and	CCONJ
cana-4517	8	4	lower	low	ADJ
cana-4517	8	5	bound	bind	VERB
cana-4517	8	6	are	be	AUX
cana-4517	8	7	also	also	ADV
cana-4517	8	8	established	establish	VERB
cana-4517	8	9	.	.	PUNCT
cana-4517	9	1	keywords	keyword	NOUN
cana-4517	9	2	:	:	PUNCT
cana-4517	9	3	pr	pr	NOUN
cana-4517	9	4	domination	domination	NOUN
cana-4517	9	5	,	,	PUNCT
cana-4517	9	6	prd	prd	NOUN
cana-4517	9	7	spectrum	spectrum	NOUN
cana-4517	9	8	,	,	PUNCT
cana-4517	9	9	prd	prd	NOUN
cana-4517	9	10	energy	energy	NOUN
cana-4517	9	11	1	1	NUM
cana-4517	9	12	.	.	PUNCT
cana-4517	10	1	introduction	introduction	NOUN
cana-4517	10	2	graphs	graph	NOUN
cana-4517	10	3	with	with	ADP
cana-4517	10	4	no	no	DET
cana-4517	10	5	isolated	isolated	ADJ
cana-4517	10	6	vertex	vertex	NOUN
cana-4517	10	7	are	be	AUX
cana-4517	10	8	considered	consider	VERB
cana-4517	10	9	for	for	ADP
cana-4517	10	10	this	this	DET
cana-4517	10	11	study	study	NOUN
cana-4517	10	12	.	.	PUNCT
cana-4517	11	1	graph	graph	NOUN
cana-4517	11	2	energy	energy	NOUN
cana-4517	11	3	is	be	AUX
cana-4517	11	4	a	a	DET
cana-4517	11	5	significant	significant	ADJ
cana-4517	11	6	topological	topological	ADJ
cana-4517	11	7	index	index	NOUN
cana-4517	11	8	in	in	ADP
cana-4517	11	9	the	the	DET
cana-4517	11	10	domain	domain	NOUN
cana-4517	11	11	of	of	ADP
cana-4517	11	12	chemical	chemical	NOUN
cana-4517	11	13	graph	graph	NOUN
cana-4517	11	14	theory	theory	NOUN
cana-4517	11	15	which	which	PRON
cana-4517	11	16	can	can	AUX
cana-4517	11	17	be	be	AUX
cana-4517	11	18	used	use	VERB
cana-4517	11	19	in	in	ADP
cana-4517	11	20	chemistry	chemistry	NOUN
cana-4517	11	21	.	.	PUNCT
cana-4517	12	1	the	the	DET
cana-4517	12	2	absicth	absicth	NOUN
cana-4517	12	3	of	of	ADP
cana-4517	12	4	graph	graph	NOUN
cana-4517	12	5	energy	energy	NOUN
cana-4517	12	6	was	be	AUX
cana-4517	12	7	debuted	debut	VERB
cana-4517	12	8	by	by	ADP
cana-4517	12	9	i.	i.	PROPN
cana-4517	12	10	gutman	gutman	PROPN
cana-4517	13	1	[	[	X
cana-4517	13	2	9	9	NUM
cana-4517	13	3	]	]	PUNCT
cana-4517	13	4	in	in	ADP
cana-4517	13	5	1978	1978	NUM
cana-4517	13	6	as	as	ADP
cana-4517	13	7	summation	summation	NOUN
cana-4517	13	8	of	of	ADP
cana-4517	13	9	modulus	modulus	ADJ
cana-4517	13	10	values	value	NOUN
cana-4517	13	11	of	of	ADP
cana-4517	13	12	spectrum	spectrum	NOUN
cana-4517	13	13	of	of	ADP
cana-4517	13	14	g	g	NOUN
cana-4517	13	15	,	,	PUNCT
cana-4517	13	16	corresponds	correspond	VERB
cana-4517	13	17	to	to	ADP
cana-4517	13	18	adjacency	adjacency	PROPN
cana-4517	13	19	matrix	matrix	NOUN
cana-4517	13	20	.	.	PUNCT
cana-4517	14	1	a	a	DET
cana-4517	14	2	vast	vast	ADJ
cana-4517	14	3	study	study	NOUN
cana-4517	14	4	of	of	ADP
cana-4517	14	5	utilization	utilization	NOUN
cana-4517	14	6	on	on	ADP
cana-4517	14	7	graph	graph	NOUN
cana-4517	14	8	energy	energy	NOUN
cana-4517	14	9	was	be	AUX
cana-4517	14	10	pursued	pursue	VERB
cana-4517	14	11	by	by	ADP
cana-4517	14	12	i.	i.	PROPN
cana-4517	14	13	gutman	gutman	PROPN
cana-4517	14	14	and	and	CCONJ
cana-4517	14	15	balakrishnan	balakrishnan	PROPN
cana-4517	15	1	[	[	X
cana-4517	15	2	2	2	NUM
cana-4517	15	3	,	,	PUNCT
cana-4517	15	4	10	10	NUM
cana-4517	15	5	]	]	PUNCT
cana-4517	15	6	.	.	PUNCT
cana-4517	16	1	the	the	DET
cana-4517	16	2	consummation	consummation	NOUN
cana-4517	16	3	of	of	ADP
cana-4517	16	4	energy	energy	NOUN
cana-4517	16	5	and	and	CCONJ
cana-4517	16	6	the	the	DET
cana-4517	16	7	equel	equel	NOUN
cana-4517	16	8	with	with	ADP
cana-4517	16	9	bounds	bound	NOUN
cana-4517	16	10	can	can	AUX
cana-4517	16	11	be	be	AUX
cana-4517	16	12	beholded	behold	VERB
cana-4517	16	13	in	in	ADP
cana-4517	16	14	vigous	vigous	ADJ
cana-4517	16	15	scrutinize	scrutinize	NOUN
cana-4517	16	16	of	of	ADP
cana-4517	16	17	graph	graph	NOUN
cana-4517	16	18	energy	energy	NOUN
cana-4517	16	19	[	[	X
cana-4517	16	20	12,13	12,13	NUM
cana-4517	16	21	]	]	PUNCT
cana-4517	16	22	.	.	PUNCT
cana-4517	17	1	the	the	DET
cana-4517	17	2	necessary	necessary	ADJ
cana-4517	17	3	properties	property	NOUN
cana-4517	17	4	and	and	CCONJ
cana-4517	17	5	vital	vital	ADJ
cana-4517	17	6	chemical	chemical	NOUN
cana-4517	17	7	utilizations	utilization	NOUN
cana-4517	17	8	were	be	AUX
cana-4517	17	9	evaluated	evaluate	VERB
cana-4517	17	10	in	in	ADP
cana-4517	17	11	the	the	DET
cana-4517	17	12	molecular	molecular	ADJ
cana-4517	17	13	orbital	orbital	ADJ
cana-4517	17	14	theory	theory	NOUN
cana-4517	17	15	of	of	ADP
cana-4517	17	16	conjugated	conjugate	VERB
cana-4517	17	17	molecules	molecule	NOUN
cana-4517	17	18	[	[	X
cana-4517	17	19	3	3	NUM
cana-4517	17	20	-	-	SYM
cana-4517	17	21	6	6	NUM
cana-4517	17	22	]	]	PUNCT
cana-4517	17	23	.	.	PUNCT
cana-4517	18	1	for	for	ADP
cana-4517	18	2	graph	graph	NOUN
cana-4517	18	3	theoretic	theoretic	ADJ
cana-4517	18	4	parlance	parlance	NOUN
cana-4517	18	5	one	one	PRON
cana-4517	18	6	may	may	AUX
cana-4517	18	7	refer	refer	VERB
cana-4517	18	8	harary	harary	NOUN
cana-4517	18	9	[	[	X
cana-4517	18	10	11	11	NUM
cana-4517	18	11	]	]	PUNCT
cana-4517	18	12	.	.	PUNCT
cana-4517	19	1	let	let	VERB
cana-4517	19	2	s	s	PRON
cana-4517	19	3	⊆	⊆	NUM
cana-4517	19	4	v.	v.	ADP
cana-4517	19	5	s	s	NOUN
cana-4517	19	6	is	be	AUX
cana-4517	19	7	a	a	DET
cana-4517	19	8	dominating	dominating	NOUN
cana-4517	19	9	set	set	NOUN
cana-4517	19	10	if	if	SCONJ
cana-4517	19	11	all	all	PRON
cana-4517	19	12	v	v	ADP
cana-4517	19	13	∈	∈	PRON
cana-4517	19	14	v	v	ADP
cana-4517	19	15	−	−	PROPN
cana-4517	19	16	s	s	PART
cana-4517	19	17	has	have	VERB
cana-4517	19	18	a	a	DET
cana-4517	19	19	neighbor	neighbor	NOUN
cana-4517	19	20	in	in	ADP
cana-4517	19	21	s	s	PROPN
cana-4517	19	22	,	,	PUNCT
cana-4517	19	23	minimum	minimum	ADJ
cana-4517	19	24	cardinality	cardinality	NOUN
cana-4517	19	25	among	among	ADP
cana-4517	19	26	such	such	ADJ
cana-4517	19	27	sets	set	NOUN
cana-4517	19	28	is	be	AUX
cana-4517	19	29	called	call	VERB
cana-4517	19	30	a	a	DET
cana-4517	19	31	minimum	minimum	ADJ
cana-4517	19	32	dominating	dominating	NOUN
cana-4517	19	33	set	set	NOUN
cana-4517	19	34	.	.	PUNCT
cana-4517	20	1	a	a	DET
cana-4517	20	2	dominating	dominating	NOUN
cana-4517	20	3	set	set	NOUN
cana-4517	20	4	s	s	VERB
cana-4517	20	5	is	be	AUX
cana-4517	20	6	perfect	perfect	ADJ
cana-4517	20	7	if	if	SCONJ
cana-4517	20	8	all	all	DET
cana-4517	20	9	v	v	VERB
cana-4517	20	10	in	in	ADP
cana-4517	20	11	g	g	PROPN
cana-4517	20	12	is	be	AUX
cana-4517	20	13	dominated	dominate	VERB
cana-4517	20	14	by	by	ADP
cana-4517	20	15	strictly	strictly	ADV
cana-4517	20	16	one	one	NUM
cana-4517	20	17	element	element	NOUN
cana-4517	20	18	of	of	ADP
cana-4517	20	19	s	s	NOUN
cana-4517	20	20	[	[	X
cana-4517	20	21	13	13	NUM
cana-4517	20	22	]	]	PUNCT
cana-4517	20	23	.	.	PUNCT
cana-4517	21	1	a	a	DET
cana-4517	21	2	dominating	dominating	NOUN
cana-4517	21	3	set	set	NOUN
cana-4517	21	4	s	s	VERB
cana-4517	21	5	is	be	AUX
cana-4517	21	6	rings	ring	NOUN
cana-4517	21	7	domination	domination	NOUN
cana-4517	21	8	if	if	SCONJ
cana-4517	21	9	each	each	DET
cana-4517	21	10	v	v	NOUN
cana-4517	21	11	in	in	ADP
cana-4517	21	12	v	v	NUM
cana-4517	21	13	–	–	PUNCT
cana-4517	21	14	s	s	VERB
cana-4517	21	15	is	be	AUX
cana-4517	21	16	adjacent	adjacent	ADJ
cana-4517	21	17	to	to	PART
cana-4517	21	18	minimum	minimum	VERB
cana-4517	21	19	two	two	NUM
cana-4517	21	20	elements	element	NOUN
cana-4517	21	21	in	in	ADP
cana-4517	21	22	v	v	NUM
cana-4517	21	23	–	–	PUNCT
cana-4517	21	24	s	s	X
cana-4517	22	1	[	[	X
cana-4517	22	2	1	1	NUM
cana-4517	22	3	]	]	PUNCT
cana-4517	22	4	.	.	PUNCT
cana-4517	23	1	a	a	DET
cana-4517	23	2	dominating	dominating	NOUN
cana-4517	23	3	set	set	NOUN
cana-4517	23	4	a	a	PRON
cana-4517	23	5	is	be	AUX
cana-4517	23	6	a	a	DET
cana-4517	23	7	perfect	perfect	ADJ
cana-4517	23	8	rings	ring	NOUN
cana-4517	23	9	dominating	dominating	NOUN
cana-4517	23	10	(	(	PUNCT
cana-4517	23	11	prd	prd	NOUN
cana-4517	23	12	)	)	PUNCT
cana-4517	23	13	set	set	VERB
cana-4517	23	14	if	if	SCONJ
cana-4517	23	15	(	(	PUNCT
cana-4517	23	16	i	i	NOUN
cana-4517	23	17	)	)	PUNCT
cana-4517	23	18	every	every	DET
cana-4517	23	19	v	v	NOUN
cana-4517	23	20	∈	∈	PROPN
cana-4517	23	21	g	g	NOUN
cana-4517	23	22	is	be	AUX
cana-4517	23	23	dominated	dominate	VERB
cana-4517	23	24	by	by	ADP
cana-4517	23	25	strictly	strictly	ADV
cana-4517	23	26	one	one	NUM
cana-4517	23	27	element	element	NOUN
cana-4517	23	28	of	of	ADP
cana-4517	23	29	a	a	DET
cana-4517	23	30	(	(	PUNCT
cana-4517	23	31	ii	ii	NOUN
cana-4517	23	32	)	)	PUNCT
cana-4517	23	33	∀	∀	NOUN
cana-4517	23	34	v	v	ADP
cana-4517	23	35	∈	∈	NOUN
cana-4517	23	36	v\a	v\a	PROPN
cana-4517	23	37	,	,	PUNCT
cana-4517	23	38	|n(v)⋂(v\a)|	|n(v)⋂(v\a)|	PROPN
cana-4517	23	39	≥	≥	NOUN
cana-4517	23	40	2	2	NUM
cana-4517	23	41	.	.	X
cana-4517	24	1	prd	prd	NOUN
cana-4517	24	2	set	set	VERB
cana-4517	24	3	with	with	ADP
cana-4517	24	4	minimum	minimum	ADJ
cana-4517	24	5	cardinality	cardinality	NOUN
cana-4517	24	6	is	be	AUX
cana-4517	24	7	the	the	DET
cana-4517	24	8	minimum	minimum	ADJ
cana-4517	24	9	prd	prd	NOUN
cana-4517	24	10	set	set	NOUN
cana-4517	24	11	of	of	ADP
cana-4517	24	12	g	g	PROPN
cana-4517	24	13	and	and	CCONJ
cana-4517	24	14	notate	notate	VERB
cana-4517	24	15	minimum	minimum	ADJ
cana-4517	24	16	prd	prd	NOUN
cana-4517	24	17	number	number	NOUN
cana-4517	24	18	by	by	ADP
cana-4517	24	19	‘	'	PUNCT
cana-4517	24	20	𝔭	𝔭	NOUN
cana-4517	24	21	’	'	PUNCT
cana-4517	24	22	.	.	PUNCT
cana-4517	25	1	this	this	DET
cana-4517	25	2	paper	paper	NOUN
cana-4517	25	3	scrutinize	scrutinize	VERB
cana-4517	25	4	the	the	DET
cana-4517	25	5	perfect	perfect	ADJ
cana-4517	25	6	rings	ring	NOUN
cana-4517	25	7	dominating	dominating	NOUN
cana-4517	25	8	spectrum	spectrum	NOUN
cana-4517	25	9	(	(	PUNCT
cana-4517	25	10	say	say	INTJ
cana-4517	25	11	𝔓-spectrum	𝔓-spectrum	PROPN
cana-4517	25	12	)	)	PUNCT
cana-4517	25	13	and	and	CCONJ
cana-4517	25	14	perfect	perfect	ADJ
cana-4517	25	15	rings	ring	NOUN
cana-4517	25	16	dominating	dominate	VERB
cana-4517	25	17	energy	energy	NOUN
cana-4517	25	18	(	(	PUNCT
cana-4517	25	19	say	say	INTJ
cana-4517	25	20	𝔓𝔈	𝔓𝔈	PROPN
cana-4517	25	21	)	)	PUNCT
cana-4517	25	22	of	of	ADP
cana-4517	25	23	few	few	ADJ
cana-4517	25	24	classes	class	NOUN
cana-4517	25	25	of	of	ADP
cana-4517	25	26	graphs	graph	NOUN
cana-4517	25	27	as	as	ADV
cana-4517	25	28	well	well	ADV
cana-4517	25	29	as	as	ADV
cana-4517	25	30	derive	derive	VERB
cana-4517	25	31	some	some	DET
cana-4517	25	32	bounds	bound	NOUN
cana-4517	25	33	on	on	ADP
cana-4517	25	34	𝔓𝔈.	𝔓𝔈.	NUM
cana-4517	25	35	1.1	1.1	NUM
cana-4517	25	36	lemma	lemma	PROPN
cana-4517	26	1	[	[	X
cana-4517	26	2	14	14	NUM
cana-4517	26	3	]	]	X
cana-4517	26	4	:	:	PUNCT
cana-4517	26	5	let	let	VERB
cana-4517	26	6	b	b	NOUN
cana-4517	26	7	=	=	PRON
cana-4517	26	8	[	[	PUNCT
cana-4517	26	9	b0	b0	VERB
cana-4517	26	10	b1	b1	NOUN
cana-4517	26	11	b1	b1	NOUN
cana-4517	26	12	b0	b0	PROPN
cana-4517	26	13	]	]	PUNCT
cana-4517	26	14	be	be	AUX
cana-4517	26	15	a	a	DET
cana-4517	26	16	symmetric	symmetric	ADJ
cana-4517	26	17	block	block	NOUN
cana-4517	26	18	matrix	matrix	NOUN
cana-4517	26	19	has	have	VERB
cana-4517	26	20	order	order	NOUN
cana-4517	26	21	2	2	NUM
cana-4517	26	22	with	with	ADP
cana-4517	26	23	b0	b0	NOUN
cana-4517	26	24	and	and	CCONJ
cana-4517	26	25	b1	b1	NOUN
cana-4517	26	26	are	be	AUX
cana-4517	26	27	square	square	ADJ
cana-4517	26	28	matrices	matrix	NOUN
cana-4517	26	29	of	of	ADP
cana-4517	26	30	same	same	ADJ
cana-4517	26	31	order	order	NOUN
cana-4517	26	32	.	.	PUNCT
cana-4517	27	1	then	then	ADV
cana-4517	27	2	spectrum	spectrum	NOUN
cana-4517	27	3	of	of	ADP
cana-4517	27	4	b	b	PROPN
cana-4517	27	5	is	be	AUX
cana-4517	27	6	the	the	DET
cana-4517	27	7	union	union	NOUN
cana-4517	27	8	of	of	ADP
cana-4517	27	9	spectrum	spectrum	NOUN
cana-4517	27	10	of	of	ADP
cana-4517	27	11	b0	b0	NOUN
cana-4517	27	12	+	+	CCONJ
cana-4517	27	13	b1	b1	NOUN
cana-4517	27	14	and	and	CCONJ
cana-4517	27	15	b0	b0	NOUN
cana-4517	27	16	−	−	PROPN
cana-4517	27	17	b1	b1	NOUN
cana-4517	27	18	.	.	PUNCT
cana-4517	28	1	2	2	X
cana-4517	28	2	.	.	X
cana-4517	28	3	perfect	perfect	ADJ
cana-4517	28	4	rings	ring	NOUN
cana-4517	28	5	dominating	dominate	VERB
cana-4517	28	6	energy	energy	NOUN
cana-4517	28	7	2.1	2.1	NUM
cana-4517	28	8	definition	definition	NOUN
cana-4517	28	9	:	:	PUNCT
cana-4517	28	10	considerg	considerg	PROPN
cana-4517	28	11	=	=	PRON
cana-4517	28	12	(	(	PUNCT
cana-4517	28	13	p	p	X
cana-4517	28	14	,	,	PUNCT
cana-4517	28	15	q	q	NOUN
cana-4517	28	16	)	)	PUNCT
cana-4517	28	17	.	.	PUNCT
cana-4517	29	1	let	let	VERB
cana-4517	29	2	a	a	DET
cana-4517	29	3	⊆	⊆	NUM
cana-4517	29	4	v(g	v(g	NUM
cana-4517	29	5	)	)	PUNCT
cana-4517	29	6	be	be	AUX
cana-4517	29	7	its	its	PRON
cana-4517	29	8	minimum	minimum	ADJ
cana-4517	29	9	perfect	perfect	ADJ
cana-4517	29	10	rings	ring	NOUN
cana-4517	29	11	dominating	dominating	NOUN
cana-4517	29	12	(	(	PUNCT
cana-4517	29	13	prd	prd	NOUN
cana-4517	29	14	)	)	PUNCT
cana-4517	29	15	set	set	NOUN
cana-4517	29	16	.	.	PUNCT
cana-4517	30	1	then	then	ADV
cana-4517	30	2	the	the	DET
cana-4517	30	3	perfect	perfect	ADJ
cana-4517	30	4	rings	ring	NOUN
cana-4517	30	5	dominating	dominating	NOUN
cana-4517	30	6	matrix	matrix	NOUN
cana-4517	30	7	of	of	ADP
cana-4517	30	8	g	g	PROPN
cana-4517	30	9	corresponding	correspond	VERB
cana-4517	30	10	to	to	ADP
cana-4517	30	11	a	a	PRON
cana-4517	30	12	is	be	AUX
cana-4517	30	13	a	a	DET
cana-4517	30	14	matrix	matrix	NOUN
cana-4517	30	15	𝔓a	𝔓a	PROPN
cana-4517	30	16	has	have	VERB
cana-4517	30	17	order	order	NOUN
cana-4517	30	18	p	p	PRON
cana-4517	30	19	,	,	PUNCT
cana-4517	30	20	defined	define	VERB
cana-4517	30	21	as	as	ADP
cana-4517	30	22	𝔓a	𝔓a	PROPN
cana-4517	30	23	=	=	SYM
cana-4517	30	24	{	{	PUNCT
cana-4517	30	25	1	1	NUM
cana-4517	30	26	if	if	SCONJ
cana-4517	30	27	vivj	vivj	NOUN
cana-4517	30	28	∈	∈	NOUN
cana-4517	30	29	e	e	NOUN
cana-4517	30	30	1	1	NUM
cana-4517	30	31	if	if	SCONJ
cana-4517	30	32	i	i	PRON
cana-4517	30	33	=	=	SYM
cana-4517	30	34	j	j	PROPN
cana-4517	30	35	,	,	PUNCT
cana-4517	30	36	vi	vi	PROPN
cana-4517	30	37	∈	∈	PROPN
cana-4517	30	38	a	a	DET
cana-4517	30	39	0	0	NUM
cana-4517	30	40	otherwise	otherwise	ADV
cana-4517	30	41	mailto:mpremamallaiah@gmail.com	mailto:mpremamallaiah@gmail.com	X
cana-4517	30	42	mailto:ruby.salestina@gmail.com	mailto:ruby.salestina@gmail.com	X
cana-4517	30	43	mailto:psmandya@gmail.com	mailto:psmandya@gmail.com	X
cana-4517	30	44	communications	communication	NOUN
cana-4517	30	45	on	on	ADP
cana-4517	30	46	applied	apply	VERB
cana-4517	30	47	nonlinear	nonlinear	ADJ
cana-4517	30	48	analysis	analysis	NOUN
cana-4517	30	49	issn	issn	NOUN
cana-4517	30	50	:	:	PUNCT
cana-4517	30	51	1074	1074	NUM
cana-4517	30	52	-	-	PUNCT
cana-4517	30	53	133x	133x	NUM
cana-4517	30	54	vol	vol	NOUN
cana-4517	30	55	32	32	NUM
cana-4517	30	56	no	no	NOUN
cana-4517	30	57	.	.	PUNCT
cana-4517	31	1	9s	9s	NUM
cana-4517	31	2	(	(	PUNCT
cana-4517	31	3	2025	2025	NUM
cana-4517	31	4	)	)	PUNCT
cana-4517	31	5	2316	2316	NUM
cana-4517	31	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	32	1	the	the	DET
cana-4517	32	2	characteristic	characteristic	ADJ
cana-4517	32	3	polynomial	polynomial	NOUN
cana-4517	32	4	of	of	ADP
cana-4517	32	5	𝔓a	𝔓a	PROPN
cana-4517	32	6	is	be	AUX
cana-4517	32	7	φ(𝔓a	φ(𝔓a	NOUN
cana-4517	32	8	,	,	PUNCT
cana-4517	32	9	λ	λ	X
cana-4517	32	10	)	)	PUNCT
cana-4517	32	11	=	=	SYM
cana-4517	32	12	det	det	NOUN
cana-4517	32	13	(	(	PUNCT
cana-4517	32	14	𝔓a	𝔓a	PROPN
cana-4517	32	15	−	−	PROPN
cana-4517	32	16	λi	λi	NOUN
cana-4517	32	17	)	)	PUNCT
cana-4517	32	18	.	.	PUNCT
cana-4517	33	1	the	the	DET
cana-4517	33	2	𝔓-spectrum	𝔓-spectrum	PROPN
cana-4517	33	3	of	of	ADP
cana-4517	33	4	g	g	PROPN
cana-4517	33	5	is	be	AUX
cana-4517	33	6	the	the	DET
cana-4517	33	7	eigenvalues	eigenvalue	NOUN
cana-4517	33	8	of	of	ADP
cana-4517	33	9	the	the	DET
cana-4517	33	10	matrix𝔓a	matrix𝔓a	PROPN
cana-4517	33	11	.	.	PUNCT
cana-4517	34	1	let	let	VERB
cana-4517	34	2	λ1	λ1	ADJ
cana-4517	34	3	,	,	PUNCT
cana-4517	34	4	λ2	λ2	NOUN
cana-4517	34	5	,	,	PUNCT
cana-4517	34	6	…	…	PUNCT
cana-4517	34	7	λp	λp	X
cana-4517	34	8	be	be	AUX
cana-4517	34	9	the	the	DET
cana-4517	34	10	spectrum	spectrum	NOUN
cana-4517	34	11	of	of	ADP
cana-4517	34	12	𝔓a	𝔓a	PROPN
cana-4517	34	13	.	.	PROPN
cana-4517	35	1	then	then	ADV
cana-4517	35	2	the	the	DET
cana-4517	35	3	perfect	perfect	ADJ
cana-4517	35	4	rings	ring	NOUN
cana-4517	35	5	dominating	dominate	VERB
cana-4517	35	6	energy	energy	NOUN
cana-4517	35	7	𝔓𝔈	𝔓𝔈	PROPN
cana-4517	35	8	of	of	ADP
cana-4517	35	9	g	g	PROPN
cana-4517	35	10	corresponding	correspond	VERB
cana-4517	35	11	to	to	ADP
cana-4517	35	12	a	a	PRON
cana-4517	35	13	is	be	AUX
cana-4517	35	14	defined	define	VERB
cana-4517	35	15	as	as	ADP
cana-4517	35	16	𝔓𝔈a(g	𝔓𝔈a(g	PROPN
cana-4517	35	17	)	)	PUNCT
cana-4517	35	18	=	=	PUNCT
cana-4517	36	1	∑	∑	PUNCT
cana-4517	36	2	|λi|	|λi|	PROPN
cana-4517	36	3	p	p	X
cana-4517	36	4	i=1	i=1	X
cana-4517	36	5	.	.	PUNCT
cana-4517	37	1	let	let	VERB
cana-4517	37	2	λ1	λ1	ADJ
cana-4517	37	3	,	,	PUNCT
cana-4517	37	4	λ2	λ2	PROPN
cana-4517	37	5	,	,	PUNCT
cana-4517	37	6	λ3	λ3	PROPN
cana-4517	37	7	,	,	PUNCT
cana-4517	37	8	…	…	PUNCT
cana-4517	37	9	λp	λp	X
cana-4517	37	10	be	be	AUX
cana-4517	37	11	the	the	DET
cana-4517	37	12	spectrum	spectrum	NOUN
cana-4517	37	13	of	of	ADP
cana-4517	37	14	𝔓a	𝔓a	PROPN
cana-4517	37	15	and	and	CCONJ
cana-4517	37	16	they	they	PRON
cana-4517	37	17	can	can	AUX
cana-4517	37	18	be	be	AUX
cana-4517	37	19	notated	notate	VERB
cana-4517	37	20	as	as	ADP
cana-4517	37	21	speca(g	speca(g	NOUN
cana-4517	37	22	)	)	PUNCT
cana-4517	38	1	=	=	PRON
cana-4517	38	2	{	{	PUNCT
cana-4517	38	3	λ1	λ1	PROPN
cana-4517	38	4	λ2	λ2	PROPN
cana-4517	38	5	λ3	λ3	PROPN
cana-4517	38	6	…	…	PUNCT
cana-4517	38	7	λp	λp	PROPN
cana-4517	38	8	m1	m1	PROPN
cana-4517	38	9	m2	m2	PROPN
cana-4517	38	10	m3	m3	PROPN
cana-4517	38	11	…	…	PUNCT
cana-4517	38	12	mp	mp	PROPN
cana-4517	38	13	}	}	PUNCT
cana-4517	38	14	where	where	SCONJ
cana-4517	38	15	mi	mi	PROPN
cana-4517	38	16	is	be	AUX
cana-4517	38	17	the	the	DET
cana-4517	38	18	algebraic	algebraic	ADJ
cana-4517	38	19	multiplicity	multiplicity	NOUN
cana-4517	38	20	of	of	ADP
cana-4517	38	21	eigenvaluesλi	eigenvaluesλi	NOUN
cana-4517	38	22	,	,	PUNCT
cana-4517	38	23	for	for	ADP
cana-4517	38	24	1	1	NUM
cana-4517	38	25	≤	≤	NUM
cana-4517	39	1	i	i	PRON
cana-4517	39	2	≤	≤	ADJ
cana-4517	40	1	p.	p.	NOUN
cana-4517	40	2	2.2	2.2	NUM
cana-4517	40	3	remark	remark	NOUN
cana-4517	40	4	:	:	PUNCT
cana-4517	40	5	though	though	SCONJ
cana-4517	40	6	all	all	DET
cana-4517	40	7	the	the	DET
cana-4517	40	8	minimum	minimum	ADJ
cana-4517	40	9	prd	prd	NOUN
cana-4517	40	10	sets	set	NOUN
cana-4517	40	11	are	be	AUX
cana-4517	40	12	of	of	ADP
cana-4517	40	13	same	same	ADJ
cana-4517	40	14	cardinality	cardinality	NOUN
cana-4517	40	15	,	,	PUNCT
cana-4517	41	1	the	the	DET
cana-4517	41	2	perfect	perfect	ADJ
cana-4517	41	3	rings	ring	NOUN
cana-4517	41	4	dominating	dominate	VERB
cana-4517	41	5	energy	energy	NOUN
cana-4517	41	6	𝔓𝔈(g	𝔓𝔈(g	NOUN
cana-4517	41	7	)	)	PUNCT
cana-4517	41	8	need	need	AUX
cana-4517	41	9	not	not	PART
cana-4517	41	10	be	be	AUX
cana-4517	41	11	same	same	ADJ
cana-4517	41	12	for	for	ADP
cana-4517	41	13	all	all	DET
cana-4517	41	14	prd	prd	NOUN
cana-4517	41	15	set	set	NOUN
cana-4517	41	16	.	.	PUNCT
cana-4517	42	1	2.3	2.3	NUM
cana-4517	42	2	remark	remark	NOUN
cana-4517	42	3	:	:	PUNCT
cana-4517	42	4	if	if	SCONJ
cana-4517	42	5	g	g	PROPN
cana-4517	42	6	has	have	VERB
cana-4517	42	7	a	a	DET
cana-4517	42	8	unique	unique	ADJ
cana-4517	42	9	minimum	minimum	NOUN
cana-4517	42	10	prd	prd	NOUN
cana-4517	42	11	set	set	NOUN
cana-4517	42	12	,	,	PUNCT
cana-4517	42	13	then	then	ADV
cana-4517	42	14	𝔓𝔈a(g	𝔓𝔈a(g	PROPN
cana-4517	42	15	)	)	PUNCT
cana-4517	42	16	can	can	AUX
cana-4517	42	17	be	be	AUX
cana-4517	42	18	denoted	denote	VERB
cana-4517	42	19	as	as	ADP
cana-4517	42	20	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	42	21	)	)	PUNCT
cana-4517	42	22	.	.	PUNCT
cana-4517	43	1	3	3	X
cana-4517	43	2	.	.	X
cana-4517	43	3	prd	prd	NOUN
cana-4517	43	4	energy	energy	NOUN
cana-4517	43	5	of	of	ADP
cana-4517	43	6	graphs	graph	NOUN
cana-4517	43	7	3.1	3.1	NUM
cana-4517	43	8	theorem	theorem	NOUN
cana-4517	43	9	:	:	PUNCT
cana-4517	43	10	for	for	ADP
cana-4517	43	11	any	any	DET
cana-4517	43	12	complete	complete	ADJ
cana-4517	43	13	graph	graph	NOUN
cana-4517	43	14	kp	kp	PROPN
cana-4517	43	15	with	with	ADP
cana-4517	43	16	p	p	PRON
cana-4517	43	17	≥	≥	NUM
cana-4517	43	18	3	3	NUM
cana-4517	43	19	,	,	PUNCT
cana-4517	43	20	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	43	21	)	)	PUNCT
cana-4517	44	1	=	=	PUNCT
cana-4517	45	1	p	p	NOUN
cana-4517	45	2	−	−	PROPN
cana-4517	45	3	2	2	NUM
cana-4517	46	1	+	+	CCONJ
cana-4517	46	2	√p(p	√p(p	NUM
cana-4517	46	3	−	−	NOUN
cana-4517	46	4	2	2	NUM
cana-4517	46	5	)	)	PUNCT
cana-4517	46	6	+	+	CCONJ
cana-4517	46	7	5	5	NUM
cana-4517	46	8	proof	proof	NOUN
cana-4517	46	9	:	:	PUNCT
cana-4517	46	10	let	let	VERB
cana-4517	46	11	v(kp	v(kp	NUM
cana-4517	46	12	)	)	PUNCT
cana-4517	46	13	=	=	PRON
cana-4517	46	14	{	{	PUNCT
cana-4517	46	15	v1	v1	PROPN
cana-4517	46	16	,	,	PUNCT
cana-4517	46	17	v2	v2	PROPN
cana-4517	46	18	,	,	PUNCT
cana-4517	46	19	.	.	PUNCT
cana-4517	46	20	.	.	PUNCT
cana-4517	46	21	.	.	PUNCT
cana-4517	47	1	,	,	PUNCT
cana-4517	47	2	vp	vp	PROPN
cana-4517	47	3	}	}	PUNCT
cana-4517	47	4	.	.	PUNCT
cana-4517	48	1	any	any	DET
cana-4517	48	2	arbitrary	arbitrary	ADJ
cana-4517	48	3	element	element	NOUN
cana-4517	48	4	of	of	ADP
cana-4517	48	5	g	g	PROPN
cana-4517	48	6	will	will	AUX
cana-4517	48	7	be	be	AUX
cana-4517	48	8	a	a	DET
cana-4517	48	9	prd	prd	NOUN
cana-4517	48	10	-	-	PUNCT
cana-4517	48	11	set	set	NOUN
cana-4517	48	12	.	.	PUNCT
cana-4517	49	1	consider	consider	VERB
cana-4517	49	2	a	a	PRON
cana-4517	49	3	=	=	SYM
cana-4517	49	4	{	{	PUNCT
cana-4517	49	5	v1	v1	NOUN
cana-4517	49	6	}	}	PUNCT
cana-4517	49	7	as	as	ADP
cana-4517	49	8	a	a	DET
cana-4517	49	9	prd	prd	NOUN
cana-4517	49	10	-	-	PUNCT
cana-4517	49	11	set	set	NOUN
cana-4517	49	12	.	.	PUNCT
cana-4517	50	1	therefore	therefore	ADV
cana-4517	50	2	the	the	DET
cana-4517	50	3	prd	prd	NOUN
cana-4517	50	4	-	-	PUNCT
cana-4517	50	5	matrix	matrix	NOUN
cana-4517	50	6	has	have	VERB
cana-4517	50	7	the	the	DET
cana-4517	50	8	structure	structure	NOUN
cana-4517	50	9	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	50	10	)	)	PUNCT
cana-4517	50	11	=	=	PUNCT
cana-4517	51	1	(	(	PUNCT
cana-4517	51	2	1	1	NUM
cana-4517	51	3	j1×p−1	j1×p−1	NOUN
cana-4517	51	4	jp−1×1	jp−1×1	NOUN
cana-4517	51	5	jp−1	jp−1	PROPN
cana-4517	51	6	−	−	PROPN
cana-4517	51	7	ip−1	ip−1	PROPN
cana-4517	51	8	)	)	PUNCT
cana-4517	51	9	where	where	SCONJ
cana-4517	51	10	j	j	PROPN
cana-4517	51	11	and	and	CCONJ
cana-4517	51	12	i	i	PRON
cana-4517	51	13	represents	represent	VERB
cana-4517	51	14	the	the	DET
cana-4517	51	15	matrix	matrix	NOUN
cana-4517	51	16	of	of	ADP
cana-4517	51	17	1	1	NUM
cana-4517	51	18	’s	’s	PART
cana-4517	51	19	and	and	CCONJ
cana-4517	51	20	identity	identity	NOUN
cana-4517	51	21	matrix	matrix	NOUN
cana-4517	51	22	.	.	PUNCT
cana-4517	52	1	and	and	CCONJ
cana-4517	52	2	the	the	DET
cana-4517	52	3	corresponding	corresponding	ADJ
cana-4517	52	4	characteristic	characteristic	ADJ
cana-4517	52	5	polynomial	polynomial	NOUN
cana-4517	52	6	is	be	AUX
cana-4517	52	7	φ(kp	φ(kp	PROPN
cana-4517	52	8	,	,	PUNCT
cana-4517	52	9	λ	λ	NOUN
cana-4517	52	10	)	)	PUNCT
cana-4517	52	11	=	=	SYM
cana-4517	52	12	det	det	PROPN
cana-4517	52	13	(	(	PUNCT
cana-4517	52	14	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	52	15	)	)	PUNCT
cana-4517	52	16	−	−	PROPN
cana-4517	53	1	λi	λi	NOUN
cana-4517	53	2	)	)	PUNCT
cana-4517	53	3	⟹	⟹	X
cana-4517	53	4	φ(kp	φ(kp	PROPN
cana-4517	53	5	,	,	PUNCT
cana-4517	53	6	λ	λ	NOUN
cana-4517	53	7	)	)	PUNCT
cana-4517	53	8	=	=	SYM
cana-4517	53	9	(	(	PUNCT
cana-4517	53	10	−1	−1	NOUN
cana-4517	53	11	)	)	PUNCT
cana-4517	53	12	p(λ	p(λ	NOUN
cana-4517	53	13	+	+	CCONJ
cana-4517	53	14	1)p−2(λ2	1)p−2(λ2	PROPN
cana-4517	53	15	−	−	NOUN
cana-4517	54	1	(	(	PUNCT
cana-4517	54	2	p	p	NOUN
cana-4517	54	3	−	−	PROPN
cana-4517	54	4	1)λ	1)λ	NUM
cana-4517	54	5	−	−	PROPN
cana-4517	54	6	1	1	NUM
cana-4517	54	7	)	)	PUNCT
cana-4517	54	8	then	then	ADV
cana-4517	54	9	,	,	PUNCT
cana-4517	54	10	speca(kp	speca(kp	NOUN
cana-4517	54	11	)	)	PUNCT
cana-4517	54	12	=	=	PRON
cana-4517	54	13	{	{	PUNCT
cana-4517	54	14	−1	−1	NOUN
cana-4517	54	15	(	(	PUNCT
cana-4517	54	16	p	p	NOUN
cana-4517	54	17	−	−	PROPN
cana-4517	54	18	2	2	NUM
cana-4517	54	19	)	)	PUNCT
cana-4517	54	20	+	+	CCONJ
cana-4517	54	21	√p2	√p2	VERB
cana-4517	54	22	−	−	NUM
cana-4517	54	23	2p	2p	NUM
cana-4517	54	24	+	+	CCONJ
cana-4517	54	25	5	5	NUM
cana-4517	54	26	2	2	NUM
cana-4517	54	27	(	(	PUNCT
cana-4517	54	28	p	p	NOUN
cana-4517	54	29	−	−	PROPN
cana-4517	54	30	2	2	NUM
cana-4517	54	31	)	)	PUNCT
cana-4517	54	32	−	−	PROPN
cana-4517	55	1	√p2	√p2	PUNCT
cana-4517	55	2	−	−	NOUN
cana-4517	55	3	2p	2p	NUM
cana-4517	55	4	+	+	CCONJ
cana-4517	55	5	5	5	NUM
cana-4517	55	6	2	2	NUM
cana-4517	55	7	p	p	NOUN
cana-4517	55	8	−	−	NUM
cana-4517	55	9	2	2	NUM
cana-4517	55	10	1	1	NUM
cana-4517	55	11	1	1	NUM
cana-4517	55	12	}	}	PUNCT
cana-4517	55	13	thus	thus	ADV
cana-4517	55	14	the	the	DET
cana-4517	55	15	prd	prd	NOUN
cana-4517	55	16	-	-	PUNCT
cana-4517	55	17	energy	energy	NOUN
cana-4517	55	18	is	be	AUX
cana-4517	55	19	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	55	20	)	)	PUNCT
cana-4517	56	1	=	=	PUNCT
cana-4517	57	1	(	(	PUNCT
cana-4517	57	2	p	p	X
cana-4517	57	3	−	−	PROPN
cana-4517	57	4	2)|−1|	2)|−1|	NUM
cana-4517	58	1	+	+	CCONJ
cana-4517	58	2	|	|	ADV
cana-4517	58	3	(	(	PUNCT
cana-4517	58	4	p	p	NOUN
cana-4517	58	5	−	−	PROPN
cana-4517	58	6	2	2	NUM
cana-4517	58	7	)	)	PUNCT
cana-4517	58	8	+	+	CCONJ
cana-4517	58	9	√p2	√p2	VERB
cana-4517	58	10	−	−	NUM
cana-4517	58	11	2p	2p	NUM
cana-4517	58	12	+	+	CCONJ
cana-4517	58	13	5	5	NUM
cana-4517	58	14	2	2	NUM
cana-4517	59	1	|	|	ADV
cana-4517	60	1	+	+	CCONJ
cana-4517	60	2	|	|	ADV
cana-4517	60	3	(	(	PUNCT
cana-4517	60	4	p	p	NOUN
cana-4517	60	5	−	−	PROPN
cana-4517	60	6	2	2	NUM
cana-4517	60	7	)	)	PUNCT
cana-4517	60	8	−	−	PROPN
cana-4517	61	1	√p2	√p2	PUNCT
cana-4517	61	2	−	−	NOUN
cana-4517	61	3	2p	2p	NUM
cana-4517	61	4	+	+	CCONJ
cana-4517	61	5	5	5	NUM
cana-4517	61	6	2	2	NUM
cana-4517	61	7	|	|	NOUN
cana-4517	61	8	=	=	PUNCT
cana-4517	62	1	p	p	NOUN
cana-4517	62	2	−	−	PROPN
cana-4517	62	3	2	2	NUM
cana-4517	62	4	+	+	CCONJ
cana-4517	62	5	√p2	√p2	PROPN
cana-4517	62	6	−	−	NUM
cana-4517	62	7	2p	2p	NUM
cana-4517	62	8	+	+	SYM
cana-4517	62	9	5	5	NUM
cana-4517	62	10	𝔓𝔈a(kp	𝔓𝔈a(kp	NUM
cana-4517	62	11	)	)	PUNCT
cana-4517	62	12	=	=	PUNCT
cana-4517	63	1	p	p	NOUN
cana-4517	63	2	−	−	PROPN
cana-4517	63	3	2	2	NUM
cana-4517	64	1	+	+	CCONJ
cana-4517	64	2	√p(p	√p(p	NUM
cana-4517	64	3	−	−	NOUN
cana-4517	64	4	2	2	NUM
cana-4517	64	5	)	)	PUNCT
cana-4517	64	6	+	+	CCONJ
cana-4517	64	7	5	5	NUM
cana-4517	64	8	3.2	3.2	NUM
cana-4517	64	9	theorem	theorem	VERB
cana-4517	64	10	:	:	PUNCT
cana-4517	64	11	for	for	ADP
cana-4517	64	12	kp	kp	PROPN
cana-4517	64	13	,	,	PUNCT
cana-4517	64	14	q	q	NOUN
cana-4517	64	15	with	with	ADP
cana-4517	64	16	p	p	NOUN
cana-4517	64	17	,	,	PUNCT
cana-4517	64	18	q	q	X
cana-4517	64	19	≥	≥	NOUN
cana-4517	64	20	2	2	NUM
cana-4517	64	21	,	,	PUNCT
cana-4517	64	22	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	64	23	,	,	PUNCT
cana-4517	64	24	q	q	NOUN
cana-4517	64	25	)	)	PUNCT
cana-4517	64	26	=	=	PRON
cana-4517	64	27	{	{	PUNCT
cana-4517	64	28	√(p	√(p	NOUN
cana-4517	64	29	−	−	PROPN
cana-4517	65	1	1)2	1)2	NUM
cana-4517	65	2	+	+	CCONJ
cana-4517	65	3	3	3	NUM
cana-4517	65	4	+	+	NOUN
cana-4517	65	5	√(p	√(p	NOUN
cana-4517	65	6	+	+	CCONJ
cana-4517	65	7	1)2	1)2	NUM
cana-4517	65	8	−	−	NOUN
cana-4517	65	9	4	4	NUM
cana-4517	65	10	if	if	SCONJ
cana-4517	65	11	p	p	NOUN
cana-4517	65	12	=	=	X
cana-4517	65	13	q	q	X
cana-4517	65	14	2(1	2(1	NUM
cana-4517	65	15	+	+	CCONJ
cana-4517	65	16	√pq	√pq	NOUN
cana-4517	65	17	)	)	PUNCT
cana-4517	65	18	if	if	SCONJ
cana-4517	65	19	p	p	PROPN
cana-4517	65	20	<	<	X
cana-4517	65	21	q	q	X
cana-4517	66	1	&	&	CCONJ
cana-4517	66	2	if	if	SCONJ
cana-4517	66	3	p	p	X
cana-4517	66	4	>	>	X
cana-4517	66	5	q	q	PUNCT
cana-4517	66	6	proof	proof	NOUN
cana-4517	66	7	:	:	PUNCT
cana-4517	66	8	let	let	VERB
cana-4517	66	9	v(kp	v(kp	NOUN
cana-4517	66	10	,	,	PUNCT
cana-4517	66	11	q	q	NOUN
cana-4517	66	12	)	)	PUNCT
cana-4517	66	13	=	=	SYM
cana-4517	66	14	{	{	PUNCT
cana-4517	66	15	v1	v1	PROPN
cana-4517	66	16	,	,	PUNCT
cana-4517	66	17	.	.	PUNCT
cana-4517	66	18	.	.	PUNCT
cana-4517	66	19	.	.	PUNCT
cana-4517	67	1	,	,	PUNCT
cana-4517	67	2	vp	vp	PROPN
cana-4517	67	3	,	,	PUNCT
cana-4517	67	4	v1′	v1′	NOUN
cana-4517	67	5	,	,	PUNCT
cana-4517	67	6	.	.	PUNCT
cana-4517	67	7	.	.	PUNCT
cana-4517	68	1	.	.	PUNCT
cana-4517	69	1	,	,	PUNCT
cana-4517	69	2	vq′	vq′	VERB
cana-4517	69	3	}	}	PUNCT
cana-4517	69	4	.	.	PUNCT
cana-4517	70	1	consider	consider	VERB
cana-4517	70	2	a	a	DET
cana-4517	70	3	=	=	SYM
cana-4517	70	4	{	{	PUNCT
cana-4517	70	5	v1	v1	NOUN
cana-4517	70	6	,	,	PUNCT
cana-4517	70	7	v1′	v1′	NOUN
cana-4517	70	8	}	}	PUNCT
cana-4517	70	9	as	as	ADP
cana-4517	70	10	a	a	DET
cana-4517	70	11	prd	prd	NOUN
cana-4517	70	12	-	-	PUNCT
cana-4517	70	13	set	set	NOUN
cana-4517	70	14	.	.	PUNCT
cana-4517	71	1	therefore	therefore	ADV
cana-4517	71	2	the	the	DET
cana-4517	71	3	prd	prd	NOUN
cana-4517	71	4	-	-	PUNCT
cana-4517	71	5	matrix	matrix	NOUN
cana-4517	71	6	has	have	VERB
cana-4517	71	7	the	the	DET
cana-4517	71	8	structure	structure	NOUN
cana-4517	71	9	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	71	10	,	,	PUNCT
cana-4517	71	11	q	q	NOUN
cana-4517	71	12	)	)	PUNCT
cana-4517	71	13	=	=	SYM
cana-4517	71	14	(	(	PUNCT
cana-4517	71	15	1	1	NUM
cana-4517	71	16	j1×p−1	j1×p−1	NOUN
cana-4517	71	17	jp−1×1	jp−1×1	NOUN
cana-4517	71	18	jp−1	jp−1	PROPN
cana-4517	71	19	−	−	PROPN
cana-4517	71	20	ip−1	ip−1	PROPN
cana-4517	71	21	)	)	PUNCT
cana-4517	72	1	i	i	PRON
cana-4517	72	2	)	)	PUNCT
cana-4517	72	3	suppose	suppose	VERB
cana-4517	72	4	p	p	X
cana-4517	72	5	=	=	ADJ
cana-4517	72	6	q	q	NOUN
cana-4517	72	7	,	,	PUNCT
cana-4517	72	8	then	then	ADV
cana-4517	72	9	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	72	10	,	,	PUNCT
cana-4517	72	11	p	p	NOUN
cana-4517	72	12	)	)	PUNCT
cana-4517	72	13	has	have	VERB
cana-4517	72	14	the	the	DET
cana-4517	72	15	structure	structure	NOUN
cana-4517	72	16	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	72	17	,	,	PUNCT
cana-4517	72	18	p	p	NOUN
cana-4517	72	19	)	)	PUNCT
cana-4517	72	20	=	=	SYM
cana-4517	72	21	(	(	PUNCT
cana-4517	72	22	b0	b0	VERB
cana-4517	72	23	b1	b1	NOUN
cana-4517	72	24	b1	b1	NOUN
cana-4517	72	25	b0	b0	PROPN
cana-4517	72	26	)	)	PUNCT
cana-4517	72	27	communications	communication	NOUN
cana-4517	72	28	on	on	ADP
cana-4517	72	29	applied	apply	VERB
cana-4517	72	30	nonlinear	nonlinear	ADJ
cana-4517	72	31	analysis	analysis	NOUN
cana-4517	72	32	issn	issn	NOUN
cana-4517	72	33	:	:	PUNCT
cana-4517	72	34	1074	1074	NUM
cana-4517	72	35	-	-	PUNCT
cana-4517	72	36	133x	133x	NUM
cana-4517	72	37	vol	vol	NOUN
cana-4517	72	38	32	32	NUM
cana-4517	72	39	no	no	NOUN
cana-4517	72	40	.	.	PUNCT
cana-4517	73	1	9s	9s	NUM
cana-4517	73	2	(	(	PUNCT
cana-4517	73	3	2025	2025	NUM
cana-4517	73	4	)	)	PUNCT
cana-4517	73	5	2317	2317	NUM
cana-4517	73	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	73	7	where	where	SCONJ
cana-4517	73	8	b0	b0	NOUN
cana-4517	73	9	=	=	SYM
cana-4517	73	10	(	(	PUNCT
cana-4517	73	11	1	1	NUM
cana-4517	73	12	o1×p−1	o1×p−1	PROPN
cana-4517	73	13	op−1×1	op−1×1	NUM
cana-4517	73	14	op−1	op−1	PROPN
cana-4517	73	15	)	)	PUNCT
cana-4517	73	16	and	and	CCONJ
cana-4517	73	17	b1	b1	NOUN
cana-4517	73	18	=	=	SYM
cana-4517	73	19	(	(	PUNCT
cana-4517	73	20	jp	jp	NOUN
cana-4517	73	21	)	)	PUNCT
cana-4517	73	22	,	,	PUNCT
cana-4517	73	23	here	here	ADV
cana-4517	73	24	j	j	PROPN
cana-4517	73	25	and	and	CCONJ
cana-4517	73	26	o	o	PROPN
cana-4517	73	27	represents	represent	VERB
cana-4517	73	28	the	the	DET
cana-4517	73	29	matrix	matrix	NOUN
cana-4517	73	30	of	of	ADP
cana-4517	73	31	1	1	NUM
cana-4517	73	32	’s	’s	NOUN
cana-4517	73	33	and	and	CCONJ
cana-4517	73	34	0	0	NUM
cana-4517	73	35	’s	’s	NOUN
cana-4517	73	36	.	.	PUNCT
cana-4517	74	1	therefore	therefore	ADV
cana-4517	74	2	by	by	ADP
cana-4517	74	3	lemma	lemma	PROPN
cana-4517	74	4	1.1	1.1	NUM
cana-4517	74	5	,	,	PUNCT
cana-4517	74	6	speca(kp	speca(kp	NOUN
cana-4517	74	7	,	,	PUNCT
cana-4517	74	8	p	p	NOUN
cana-4517	74	9	)	)	PUNCT
cana-4517	74	10	=	=	NOUN
cana-4517	74	11	spec(b0	spec(b0	NOUN
cana-4517	74	12	+	+	CCONJ
cana-4517	74	13	b1)⋃	b1)⋃	PROPN
cana-4517	74	14	spec(b0	spec(b0	NOUN
cana-4517	74	15	−	−	PROPN
cana-4517	74	16	b1	b1	PROPN
cana-4517	74	17	)	)	PUNCT
cana-4517	74	18	.	.	PUNCT
cana-4517	75	1	consider	consider	VERB
cana-4517	75	2	,	,	PUNCT
cana-4517	75	3	b0	b0	NOUN
cana-4517	75	4	+	+	CCONJ
cana-4517	75	5	b1	b1	NOUN
cana-4517	75	6	:	:	PUNCT
cana-4517	75	7	b0	b0	NOUN
cana-4517	75	8	+	+	CCONJ
cana-4517	75	9	b1	b1	NOUN
cana-4517	75	10	=	=	SYM
cana-4517	75	11	(	(	PUNCT
cana-4517	75	12	2	2	NUM
cana-4517	75	13	j1×p−1	j1×p−1	NOUN
cana-4517	75	14	jp−1×1	jp−1×1	PROPN
cana-4517	75	15	jp−1	jp−1	PROPN
cana-4517	75	16	)	)	PUNCT
cana-4517	75	17	|(b0	|(b0	ADV
cana-4517	75	18	+	+	CCONJ
cana-4517	75	19	b1	b1	NOUN
cana-4517	75	20	)	)	PUNCT
cana-4517	75	21	−	−	PUNCT
cana-4517	76	1	λi|	λi|	NOUN
cana-4517	76	2	=	=	SYM
cana-4517	76	3	(	(	PUNCT
cana-4517	76	4	−1	−1	NOUN
cana-4517	76	5	)	)	PUNCT
cana-4517	76	6	pλp−2(λ2	pλp−2(λ2	NOUN
cana-4517	77	1	−	−	PROPN
cana-4517	77	2	(	(	PUNCT
cana-4517	77	3	p	p	X
cana-4517	77	4	+	+	NUM
cana-4517	77	5	1)λ	1)λ	NUM
cana-4517	77	6	+	+	CCONJ
cana-4517	77	7	(	(	PUNCT
cana-4517	77	8	p	p	NOUN
cana-4517	77	9	−	−	PROPN
cana-4517	77	10	1	1	NUM
cana-4517	77	11	)	)	PUNCT
cana-4517	77	12	)	)	PUNCT
cana-4517	77	13	therefore	therefore	ADV
cana-4517	77	14	,	,	PUNCT
cana-4517	77	15	speca(b0	speca(b0	NOUN
cana-4517	77	16	+	+	CCONJ
cana-4517	77	17	b1	b1	NOUN
cana-4517	77	18	)	)	PUNCT
cana-4517	77	19	=	=	PRON
cana-4517	77	20	{	{	PUNCT
cana-4517	77	21	0	0	NUM
cana-4517	77	22	(	(	PUNCT
cana-4517	77	23	p	p	NOUN
cana-4517	78	1	+	+	NOUN
cana-4517	78	2	1	1	NUM
cana-4517	78	3	)	)	PUNCT
cana-4517	78	4	+	+	NOUN
cana-4517	78	5	√(p	√(p	NOUN
cana-4517	78	6	−	−	X
cana-4517	79	1	1)2	1)2	NUM
cana-4517	79	2	+	+	CCONJ
cana-4517	79	3	3	3	NUM
cana-4517	79	4	2	2	NUM
cana-4517	79	5	(	(	PUNCT
cana-4517	79	6	p	p	NOUN
cana-4517	79	7	+	+	NOUN
cana-4517	79	8	1	1	NUM
cana-4517	79	9	)	)	PUNCT
cana-4517	79	10	−	−	NOUN
cana-4517	79	11	√(p	√(p	NOUN
cana-4517	79	12	−	−	X
cana-4517	80	1	1)2	1)2	NUM
cana-4517	80	2	+	+	CCONJ
cana-4517	80	3	3	3	NUM
cana-4517	80	4	2	2	NUM
cana-4517	80	5	p	p	NOUN
cana-4517	80	6	−	−	NUM
cana-4517	80	7	2	2	NUM
cana-4517	80	8	1	1	NUM
cana-4517	80	9	1	1	NUM
cana-4517	80	10	}	}	PUNCT
cana-4517	80	11	consider	consider	VERB
cana-4517	80	12	,	,	PUNCT
cana-4517	80	13	b0	b0	VERB
cana-4517	80	14	−	−	PROPN
cana-4517	80	15	b1	b1	NOUN
cana-4517	80	16	:	:	PUNCT
cana-4517	80	17	b0	b0	NOUN
cana-4517	80	18	−	−	NOUN
cana-4517	80	19	b1	b1	NOUN
cana-4517	80	20	=	=	SYM
cana-4517	80	21	(	(	PUNCT
cana-4517	80	22	0	0	NUM
cana-4517	80	23	−j1×p−1	−j1×p−1	VERB
cana-4517	80	24	−jp−1×1	−jp−1×1	INTJ
cana-4517	80	25	−jp−1	−jp−1	NUM
cana-4517	80	26	)	)	PUNCT
cana-4517	80	27	|(b0	|(b0	ADV
cana-4517	80	28	−	−	PROPN
cana-4517	80	29	b1	b1	NOUN
cana-4517	80	30	)	)	PUNCT
cana-4517	80	31	−	−	PROPN
cana-4517	81	1	λi|	λi|	NOUN
cana-4517	81	2	=	=	SYM
cana-4517	81	3	(	(	PUNCT
cana-4517	81	4	−1	−1	NOUN
cana-4517	81	5	)	)	PUNCT
cana-4517	81	6	pλp−2(λ2	pλp−2(λ2	NOUN
cana-4517	82	1	+	+	CCONJ
cana-4517	82	2	(	(	PUNCT
cana-4517	82	3	p	p	X
cana-4517	82	4	−	−	PROPN
cana-4517	82	5	1)λ	1)λ	NUM
cana-4517	82	6	−	−	PROPN
cana-4517	83	1	(	(	PUNCT
cana-4517	83	2	p	p	NOUN
cana-4517	83	3	−	−	PROPN
cana-4517	83	4	1	1	NUM
cana-4517	83	5	)	)	PUNCT
cana-4517	83	6	)	)	PUNCT
cana-4517	83	7	therefore	therefore	ADV
cana-4517	83	8	,	,	PUNCT
cana-4517	83	9	speca(b0	speca(b0	NOUN
cana-4517	83	10	−	−	PROPN
cana-4517	83	11	b1	b1	NOUN
cana-4517	83	12	)	)	PUNCT
cana-4517	83	13	=	=	PRON
cana-4517	83	14	{	{	PUNCT
cana-4517	83	15	0	0	NUM
cana-4517	83	16	(	(	PUNCT
cana-4517	83	17	1	1	NUM
cana-4517	83	18	−	−	PROPN
cana-4517	83	19	p	p	X
cana-4517	83	20	)	)	PUNCT
cana-4517	84	1	+	+	NOUN
cana-4517	84	2	√(p	√(p	NOUN
cana-4517	84	3	+	+	CCONJ
cana-4517	84	4	1)2	1)2	NUM
cana-4517	84	5	−	−	NOUN
cana-4517	84	6	4	4	NUM
cana-4517	84	7	2	2	NUM
cana-4517	84	8	(	(	PUNCT
cana-4517	84	9	1	1	NUM
cana-4517	84	10	−	−	PROPN
cana-4517	84	11	p	p	X
cana-4517	84	12	)	)	PUNCT
cana-4517	84	13	−	−	NOUN
cana-4517	84	14	√(p	√(p	NOUN
cana-4517	84	15	+	+	CCONJ
cana-4517	84	16	1)2	1)2	NUM
cana-4517	84	17	−	−	NOUN
cana-4517	84	18	4	4	NUM
cana-4517	84	19	2	2	NUM
cana-4517	84	20	p	p	NOUN
cana-4517	84	21	−	−	NUM
cana-4517	84	22	2	2	NUM
cana-4517	84	23	1	1	NUM
cana-4517	84	24	1	1	NUM
cana-4517	84	25	}	}	PUNCT
cana-4517	84	26	hence	hence	ADV
cana-4517	84	27	speca(kp	speca(kp	NOUN
cana-4517	84	28	,	,	PUNCT
cana-4517	84	29	p	p	NOUN
cana-4517	84	30	)	)	PUNCT
cana-4517	84	31	=	=	SYM
cana-4517	84	32	{	{	PUNCT
cana-4517	84	33	0	0	NUM
cana-4517	84	34	(	(	PUNCT
cana-4517	84	35	p	p	NOUN
cana-4517	84	36	+	+	NOUN
cana-4517	84	37	1	1	NUM
cana-4517	84	38	)	)	PUNCT
cana-4517	84	39	+	+	NOUN
cana-4517	84	40	√(p	√(p	NOUN
cana-4517	84	41	−	−	X
cana-4517	85	1	1)2	1)2	NUM
cana-4517	85	2	+	+	CCONJ
cana-4517	85	3	3	3	NUM
cana-4517	85	4	2	2	NUM
cana-4517	85	5	(	(	PUNCT
cana-4517	85	6	p	p	NOUN
cana-4517	85	7	+	+	NOUN
cana-4517	85	8	1	1	NUM
cana-4517	85	9	)	)	PUNCT
cana-4517	85	10	−	−	NOUN
cana-4517	85	11	√(p	√(p	NOUN
cana-4517	85	12	−	−	X
cana-4517	86	1	1)2	1)2	NUM
cana-4517	86	2	+	+	CCONJ
cana-4517	86	3	3	3	NUM
cana-4517	86	4	2	2	NUM
cana-4517	86	5	2p	2p	NUM
cana-4517	86	6	−	−	NOUN
cana-4517	86	7	4	4	NUM
cana-4517	86	8	1	1	NUM
cana-4517	86	9	1	1	NUM
cana-4517	86	10	(	(	PUNCT
cana-4517	86	11	1	1	NUM
cana-4517	86	12	−	−	PROPN
cana-4517	86	13	p	p	X
cana-4517	86	14	)	)	PUNCT
cana-4517	86	15	+	+	NOUN
cana-4517	86	16	√(p	√(p	NOUN
cana-4517	86	17	+	+	CCONJ
cana-4517	86	18	1)2	1)2	NUM
cana-4517	86	19	−	−	NOUN
cana-4517	86	20	4	4	NUM
cana-4517	86	21	2	2	NUM
cana-4517	86	22	(	(	PUNCT
cana-4517	86	23	1	1	NUM
cana-4517	86	24	−	−	PROPN
cana-4517	86	25	p	p	X
cana-4517	86	26	)	)	PUNCT
cana-4517	86	27	−	−	NOUN
cana-4517	86	28	√(p	√(p	NOUN
cana-4517	86	29	+	+	CCONJ
cana-4517	86	30	1)2	1)2	NUM
cana-4517	86	31	−	−	NOUN
cana-4517	86	32	4	4	NUM
cana-4517	86	33	2	2	NUM
cana-4517	86	34	1	1	NUM
cana-4517	86	35	1	1	NUM
cana-4517	86	36	}	}	PUNCT
cana-4517	86	37	now	now	ADV
cana-4517	86	38	,	,	PUNCT
cana-4517	86	39	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	86	40	,	,	PUNCT
cana-4517	86	41	p	p	NOUN
cana-4517	86	42	)	)	PUNCT
cana-4517	86	43	=	=	SYM
cana-4517	87	1	|	|	ADV
cana-4517	87	2	(	(	PUNCT
cana-4517	87	3	p	p	NOUN
cana-4517	87	4	+	+	NOUN
cana-4517	87	5	1	1	NUM
cana-4517	87	6	)	)	PUNCT
cana-4517	87	7	+	+	NOUN
cana-4517	87	8	√(p	√(p	NOUN
cana-4517	87	9	−	−	X
cana-4517	88	1	1)2	1)2	NUM
cana-4517	88	2	+	+	CCONJ
cana-4517	88	3	3	3	NUM
cana-4517	88	4	2	2	NUM
cana-4517	89	1	|	|	ADV
cana-4517	90	1	+	+	CCONJ
cana-4517	90	2	|	|	ADV
cana-4517	90	3	(	(	PUNCT
cana-4517	90	4	p	p	NOUN
cana-4517	90	5	+	+	NOUN
cana-4517	90	6	1	1	NUM
cana-4517	90	7	)	)	PUNCT
cana-4517	90	8	−	−	NOUN
cana-4517	90	9	√(p	√(p	NOUN
cana-4517	90	10	−	−	X
cana-4517	91	1	1)2	1)2	NUM
cana-4517	91	2	+	+	CCONJ
cana-4517	91	3	3	3	NUM
cana-4517	91	4	2	2	NUM
cana-4517	92	1	|	|	ADV
cana-4517	93	1	+	+	CCONJ
cana-4517	93	2	|	|	ADV
cana-4517	93	3	(	(	PUNCT
cana-4517	93	4	1	1	NUM
cana-4517	93	5	−	−	PROPN
cana-4517	93	6	p	p	X
cana-4517	93	7	)	)	PUNCT
cana-4517	93	8	+	+	NOUN
cana-4517	93	9	√(p	√(p	NOUN
cana-4517	93	10	+	+	CCONJ
cana-4517	94	1	1)2	1)2	NUM
cana-4517	94	2	−	−	NOUN
cana-4517	94	3	4	4	NUM
cana-4517	94	4	2	2	NUM
cana-4517	95	1	|	|	ADV
cana-4517	96	1	+	+	CCONJ
cana-4517	96	2	|	|	ADV
cana-4517	96	3	(	(	PUNCT
cana-4517	96	4	1	1	NUM
cana-4517	96	5	−	−	PROPN
cana-4517	96	6	p	p	X
cana-4517	96	7	)	)	PUNCT
cana-4517	96	8	−	−	NOUN
cana-4517	96	9	√(p	√(p	NOUN
cana-4517	96	10	+	+	CCONJ
cana-4517	97	1	1)2	1)2	NUM
cana-4517	97	2	−	−	NOUN
cana-4517	97	3	4	4	NUM
cana-4517	97	4	2	2	NUM
cana-4517	97	5	|	|	ADV
cana-4517	97	6	thus	thus	ADV
cana-4517	97	7	,	,	PUNCT
cana-4517	97	8	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	97	9	,	,	PUNCT
cana-4517	97	10	p	p	NOUN
cana-4517	97	11	)	)	PUNCT
cana-4517	97	12	=	=	NOUN
cana-4517	98	1	2√(p	2√(p	NUM
cana-4517	98	2	−	−	NUM
cana-4517	98	3	1)2	1)2	NUM
cana-4517	98	4	+	+	CCONJ
cana-4517	98	5	3	3	NUM
cana-4517	98	6	2	2	NUM
cana-4517	98	7	+	+	NUM
cana-4517	98	8	2√(p	2√(p	NUM
cana-4517	98	9	+	+	CCONJ
cana-4517	98	10	1)2	1)2	NUM
cana-4517	98	11	−	−	NOUN
cana-4517	98	12	4	4	NUM
cana-4517	98	13	2	2	NUM
cana-4517	98	14	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	98	15	,	,	PUNCT
cana-4517	98	16	p	p	NOUN
cana-4517	98	17	)	)	PUNCT
cana-4517	98	18	=	=	NOUN
cana-4517	98	19	√(p	√(p	NOUN
cana-4517	98	20	−	−	X
cana-4517	99	1	1)2	1)2	NUM
cana-4517	99	2	+	+	CCONJ
cana-4517	99	3	3	3	NUM
cana-4517	99	4	+	+	NOUN
cana-4517	99	5	√(p	√(p	NOUN
cana-4517	99	6	+	+	CCONJ
cana-4517	99	7	1)2	1)2	NUM
cana-4517	99	8	−	−	PROPN
cana-4517	99	9	4	4	NUM
cana-4517	99	10	ii	ii	NOUN
cana-4517	99	11	)	)	PUNCT
cana-4517	99	12	suppose	suppose	VERB
cana-4517	99	13	p	p	PRON
cana-4517	99	14	>	>	X
cana-4517	99	15	q	q	X
cana-4517	99	16	,	,	PUNCT
cana-4517	99	17	then	then	ADV
cana-4517	99	18	the	the	DET
cana-4517	99	19	prd	prd	NOUN
cana-4517	99	20	matrix	matrix	NOUN
cana-4517	99	21	is	be	AUX
cana-4517	99	22	of	of	ADP
cana-4517	99	23	the	the	DET
cana-4517	99	24	form	form	NOUN
cana-4517	99	25	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	99	26	,	,	PUNCT
cana-4517	99	27	q	q	NOUN
cana-4517	99	28	)	)	PUNCT
cana-4517	100	1	=	=	SYM
cana-4517	100	2	(	(	PUNCT
cana-4517	100	3	a	a	DET
cana-4517	100	4	b	b	NOUN
cana-4517	100	5	c	c	NOUN
cana-4517	100	6	d	d	NOUN
cana-4517	100	7	)	)	PUNCT
cana-4517	100	8	where	where	SCONJ
cana-4517	100	9	a	a	PRON
cana-4517	100	10	=	=	X
cana-4517	100	11	(	(	PUNCT
cana-4517	100	12	1	1	NUM
cana-4517	100	13	o1×p−1	o1×p−1	PROPN
cana-4517	100	14	op−1×1	op−1×1	NUM
cana-4517	100	15	op−1	op−1	PROPN
cana-4517	100	16	)	)	PUNCT
cana-4517	100	17	;	;	PUNCT
cana-4517	100	18	b	b	X
cana-4517	100	19	=	=	SYM
cana-4517	100	20	(	(	PUNCT
cana-4517	100	21	jp×q	jp×q	NOUN
cana-4517	100	22	)	)	PUNCT
cana-4517	100	23	c	c	NOUN
cana-4517	100	24	=	=	SYM
cana-4517	100	25	(	(	PUNCT
cana-4517	100	26	jq×p	jq×p	PROPN
cana-4517	100	27	)	)	PUNCT
cana-4517	100	28	;	;	PUNCT
cana-4517	100	29	d	d	X
cana-4517	100	30	=	=	SYM
cana-4517	100	31	(	(	PUNCT
cana-4517	100	32	1	1	NUM
cana-4517	100	33	o1×q−1	o1×q−1	NOUN
cana-4517	100	34	oq−1×1	oq−1×1	NUM
cana-4517	100	35	oq−1	oq−1	NOUN
cana-4517	100	36	)	)	PUNCT
cana-4517	100	37	,	,	PUNCT
cana-4517	100	38	here	here	ADV
cana-4517	100	39	j	j	PROPN
cana-4517	100	40	and	and	CCONJ
cana-4517	100	41	o	o	PROPN
cana-4517	100	42	represents	represent	VERB
cana-4517	100	43	the	the	DET
cana-4517	100	44	matrix	matrix	NOUN
cana-4517	100	45	of	of	ADP
cana-4517	100	46	1	1	NUM
cana-4517	100	47	’s	’s	NOUN
cana-4517	100	48	and	and	CCONJ
cana-4517	100	49	0	0	NUM
cana-4517	100	50	’s	’s	NOUN
cana-4517	100	51	.	.	PUNCT
cana-4517	101	1	communications	communication	NOUN
cana-4517	101	2	on	on	ADP
cana-4517	101	3	applied	apply	VERB
cana-4517	101	4	nonlinear	nonlinear	ADJ
cana-4517	101	5	analysis	analysis	NOUN
cana-4517	101	6	issn	issn	NOUN
cana-4517	101	7	:	:	PUNCT
cana-4517	101	8	1074	1074	NUM
cana-4517	101	9	-	-	PUNCT
cana-4517	101	10	133x	133x	NUM
cana-4517	101	11	vol	vol	NOUN
cana-4517	101	12	32	32	NUM
cana-4517	101	13	no	no	NOUN
cana-4517	101	14	.	.	PUNCT
cana-4517	102	1	9s	9s	NUM
cana-4517	102	2	(	(	PUNCT
cana-4517	102	3	2025	2025	NUM
cana-4517	102	4	)	)	PUNCT
cana-4517	102	5	2318	2318	NUM
cana-4517	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	102	7	and	and	CCONJ
cana-4517	102	8	the	the	DET
cana-4517	102	9	corresponding	corresponding	ADJ
cana-4517	102	10	characteristic	characteristic	ADJ
cana-4517	102	11	polynomial	polynomial	NOUN
cana-4517	102	12	is	be	AUX
cana-4517	102	13	,	,	PUNCT
cana-4517	102	14	φ(kp	φ(kp	PROPN
cana-4517	102	15	,	,	PUNCT
cana-4517	102	16	q	q	NOUN
cana-4517	102	17	,	,	PUNCT
cana-4517	102	18	λ	λ	NOUN
cana-4517	102	19	)	)	PUNCT
cana-4517	102	20	=	=	SYM
cana-4517	102	21	det(𝔓a(kp	det(𝔓a(kp	NOUN
cana-4517	102	22	,	,	PUNCT
cana-4517	102	23	q	q	NOUN
cana-4517	102	24	)	)	PUNCT
cana-4517	102	25	−	−	PROPN
cana-4517	102	26	λi	λi	NOUN
cana-4517	102	27	)	)	PUNCT
cana-4517	102	28	φ(kp	φ(kp	PROPN
cana-4517	102	29	,	,	PUNCT
cana-4517	102	30	q	q	NOUN
cana-4517	102	31	,	,	PUNCT
cana-4517	102	32	λ	λ	NOUN
cana-4517	102	33	)	)	PUNCT
cana-4517	102	34	=	=	SYM
cana-4517	102	35	(	(	PUNCT
cana-4517	102	36	−1	−1	NOUN
cana-4517	102	37	)	)	PUNCT
cana-4517	102	38	pλp+q−4	pλp+q−4	NOUN
cana-4517	103	1	[	[	X
cana-4517	103	2	λ4	λ4	ADJ
cana-4517	103	3	−	−	PROPN
cana-4517	103	4	2λ3	2λ3	NUM
cana-4517	103	5	−	−	PROPN
cana-4517	103	6	(	(	PUNCT
cana-4517	103	7	pq	pq	INTJ
cana-4517	103	8	−	−	NOUN
cana-4517	104	1	1)λ2	1)λ2	PROPN
cana-4517	105	1	+	+	CCONJ
cana-4517	105	2	(	(	PUNCT
cana-4517	105	3	2pq	2pq	ADJ
cana-4517	105	4	−	−	PROPN
cana-4517	105	5	(	(	PUNCT
cana-4517	105	6	p	p	X
cana-4517	105	7	+	+	PROPN
cana-4517	105	8	q))λ	q))λ	NOUN
cana-4517	105	9	−	−	PROPN
cana-4517	106	1	[	[	X
cana-4517	106	2	pq	pq	INTJ
cana-4517	106	3	−	−	PROPN
cana-4517	107	1	(	(	PUNCT
cana-4517	107	2	p	p	X
cana-4517	107	3	+	+	CCONJ
cana-4517	107	4	q	q	X
cana-4517	107	5	)	)	PUNCT
cana-4517	107	6	+	+	CCONJ
cana-4517	107	7	1	1	NUM
cana-4517	107	8	]	]	X
cana-4517	107	9	]	]	X
cana-4517	108	1	≈	≈	PROPN
cana-4517	108	2	λp+q−4(λ	λp+q−4(λ	PROPN
cana-4517	108	3	−	−	PROPN
cana-4517	108	4	1)[λ3	1)[λ3	NUM
cana-4517	108	5	−	−	PROPN
cana-4517	108	6	λ2	λ2	PROPN
cana-4517	108	7	−	−	PROPN
cana-4517	108	8	pqλ	pqλ	NOUN
cana-4517	108	9	−	−	PROPN
cana-4517	108	10	(	(	PUNCT
cana-4517	108	11	pq−	pq−	X
cana-4517	108	12	(	(	PUNCT
cana-4517	108	13	p	p	NOUN
cana-4517	108	14	+	+	NOUN
cana-4517	108	15	q	q	NOUN
cana-4517	108	16	)	)	PUNCT
cana-4517	108	17	)	)	PUNCT
cana-4517	108	18	]	]	PUNCT
cana-4517	109	1	φ(kp	φ(kp	PROPN
cana-4517	109	2	,	,	PUNCT
cana-4517	109	3	q	q	NOUN
cana-4517	109	4	,	,	PUNCT
cana-4517	109	5	λ	λ	NOUN
cana-4517	109	6	)	)	PUNCT
cana-4517	109	7	≈	≈	PROPN
cana-4517	109	8	λ	λ	PROPN
cana-4517	109	9	p+q−4(λ	p+q−4(λ	NOUN
cana-4517	109	10	−	−	PROPN
cana-4517	109	11	1)2(λ2	1)2(λ2	NUM
cana-4517	109	12	−	−	PROPN
cana-4517	109	13	pq	pq	PROPN
cana-4517	109	14	)	)	PUNCT
cana-4517	109	15	iii	iii	PROPN
cana-4517	109	16	)	)	PUNCT
cana-4517	109	17	suppose	suppose	VERB
cana-4517	109	18	p	p	PRON
cana-4517	109	19	>	>	X
cana-4517	109	20	q	q	X
cana-4517	109	21	,	,	PUNCT
cana-4517	109	22	then	then	ADV
cana-4517	109	23	the	the	DET
cana-4517	109	24	prd	prd	NOUN
cana-4517	109	25	matrix	matrix	NOUN
cana-4517	109	26	is	be	AUX
cana-4517	109	27	of	of	ADP
cana-4517	109	28	the	the	DET
cana-4517	109	29	form	form	NOUN
cana-4517	109	30	𝔓a(kp	𝔓a(kp	NOUN
cana-4517	109	31	,	,	PUNCT
cana-4517	109	32	q	q	NOUN
cana-4517	109	33	)	)	PUNCT
cana-4517	109	34	=	=	SYM
cana-4517	110	1	(	(	PUNCT
cana-4517	110	2	p	p	X
cana-4517	110	3	q	q	X
cana-4517	110	4	r	r	NOUN
cana-4517	110	5	s	s	NOUN
cana-4517	110	6	)	)	PUNCT
cana-4517	111	1	where	where	SCONJ
cana-4517	111	2	p	p	NOUN
cana-4517	111	3	=	=	X
cana-4517	111	4	(	(	PUNCT
cana-4517	111	5	1	1	NUM
cana-4517	111	6	o1×p−1	o1×p−1	PROPN
cana-4517	111	7	op−1×1	op−1×1	NUM
cana-4517	111	8	op−1	op−1	PROPN
cana-4517	111	9	)	)	PUNCT
cana-4517	111	10	;	;	PUNCT
cana-4517	111	11	q	q	SYM
cana-4517	111	12	=	=	SYM
cana-4517	111	13	(	(	PUNCT
cana-4517	111	14	jp×q	jp×q	NOUN
cana-4517	111	15	)	)	PUNCT
cana-4517	111	16	r	r	NOUN
cana-4517	111	17	=	=	PUNCT
cana-4517	111	18	(	(	PUNCT
cana-4517	111	19	jq×p	jq×p	PROPN
cana-4517	111	20	)	)	PUNCT
cana-4517	111	21	;	;	PUNCT
cana-4517	111	22	s	s	X
cana-4517	111	23	=	=	PUNCT
cana-4517	111	24	(	(	PUNCT
cana-4517	111	25	1	1	NUM
cana-4517	111	26	o1×q−1	o1×q−1	NOUN
cana-4517	111	27	oq−1×1	oq−1×1	NUM
cana-4517	111	28	oq−1	oq−1	NOUN
cana-4517	111	29	)	)	PUNCT
cana-4517	111	30	here	here	ADV
cana-4517	111	31	j	j	PROPN
cana-4517	111	32	and	and	CCONJ
cana-4517	111	33	o	o	PROPN
cana-4517	111	34	represents	represent	VERB
cana-4517	111	35	the	the	DET
cana-4517	111	36	matrix	matrix	NOUN
cana-4517	111	37	of	of	ADP
cana-4517	111	38	1	1	NUM
cana-4517	111	39	’s	’s	NOUN
cana-4517	111	40	and	and	CCONJ
cana-4517	111	41	0	0	NUM
cana-4517	111	42	’s	’s	NOUN
cana-4517	111	43	.	.	PUNCT
cana-4517	112	1	therefore	therefore	ADV
cana-4517	112	2	the	the	DET
cana-4517	112	3	corresponding	corresponding	ADJ
cana-4517	112	4	characteristic	characteristic	ADJ
cana-4517	112	5	polynomial	polynomial	NOUN
cana-4517	112	6	is	be	AUX
cana-4517	112	7	,	,	PUNCT
cana-4517	112	8	φ(kp	φ(kp	PROPN
cana-4517	112	9	,	,	PUNCT
cana-4517	112	10	q	q	NOUN
cana-4517	112	11	,	,	PUNCT
cana-4517	112	12	λ	λ	NOUN
cana-4517	112	13	)	)	PUNCT
cana-4517	112	14	=	=	SYM
cana-4517	112	15	det(𝔓a(kp	det(𝔓a(kp	NOUN
cana-4517	112	16	,	,	PUNCT
cana-4517	112	17	q	q	NOUN
cana-4517	112	18	)	)	PUNCT
cana-4517	112	19	−	−	PROPN
cana-4517	112	20	λi	λi	NOUN
cana-4517	112	21	)	)	PUNCT
cana-4517	112	22	φ(kp	φ(kp	PROPN
cana-4517	112	23	,	,	PUNCT
cana-4517	112	24	q	q	NOUN
cana-4517	112	25	,	,	PUNCT
cana-4517	112	26	λ	λ	NOUN
cana-4517	112	27	)	)	PUNCT
cana-4517	112	28	=	=	SYM
cana-4517	112	29	(	(	PUNCT
cana-4517	112	30	−1	−1	NOUN
cana-4517	112	31	)	)	PUNCT
cana-4517	112	32	pλp+q−4	pλp+q−4	NOUN
cana-4517	113	1	[	[	X
cana-4517	113	2	λ4	λ4	ADJ
cana-4517	113	3	−	−	PROPN
cana-4517	113	4	2λ3	2λ3	NUM
cana-4517	113	5	−	−	PROPN
cana-4517	113	6	(	(	PUNCT
cana-4517	113	7	pq	pq	INTJ
cana-4517	113	8	−	−	NOUN
cana-4517	114	1	1)λ2	1)λ2	PROPN
cana-4517	115	1	+	+	CCONJ
cana-4517	115	2	(	(	PUNCT
cana-4517	115	3	2pq	2pq	ADJ
cana-4517	115	4	−	−	PROPN
cana-4517	115	5	(	(	PUNCT
cana-4517	115	6	p	p	X
cana-4517	115	7	+	+	PROPN
cana-4517	115	8	q))λ	q))λ	NOUN
cana-4517	115	9	−	−	PROPN
cana-4517	116	1	[	[	X
cana-4517	116	2	pq	pq	INTJ
cana-4517	116	3	−	−	PROPN
cana-4517	117	1	(	(	PUNCT
cana-4517	117	2	p	p	X
cana-4517	117	3	+	+	CCONJ
cana-4517	117	4	q	q	X
cana-4517	117	5	)	)	PUNCT
cana-4517	117	6	+	+	CCONJ
cana-4517	117	7	1	1	NUM
cana-4517	117	8	]	]	X
cana-4517	117	9	]	]	X
cana-4517	117	10	⟹φ(kp	⟹φ(kp	NUM
cana-4517	117	11	,	,	PUNCT
cana-4517	117	12	q	q	NOUN
cana-4517	117	13	,	,	PUNCT
cana-4517	117	14	λ	λ	NOUN
cana-4517	117	15	)	)	PUNCT
cana-4517	117	16	≈	≈	PROPN
cana-4517	117	17	λ	λ	PROPN
cana-4517	117	18	p+q−4(λ	p+q−4(λ	NOUN
cana-4517	117	19	−	−	PROPN
cana-4517	117	20	1)2(λ2	1)2(λ2	NUM
cana-4517	117	21	−	−	PROPN
cana-4517	117	22	pq	pq	NOUN
cana-4517	117	23	)	)	PUNCT
cana-4517	117	24	for	for	ADP
cana-4517	117	25	the	the	DET
cana-4517	117	26	both	both	DET
cana-4517	117	27	cases	case	NOUN
cana-4517	117	28	ii	ii	NOUN
cana-4517	117	29	)	)	PUNCT
cana-4517	117	30	,	,	PUNCT
cana-4517	117	31	iii	iii	X
cana-4517	117	32	)	)	PUNCT
cana-4517	117	33	the	the	DET
cana-4517	117	34	characteristic	characteristic	ADJ
cana-4517	117	35	polynomial	polynomial	NOUN
cana-4517	117	36	remains	remain	VERB
cana-4517	117	37	same	same	ADJ
cana-4517	117	38	.	.	PUNCT
cana-4517	118	1	hence	hence	ADV
cana-4517	118	2	,	,	PUNCT
cana-4517	118	3	speca(kp	speca(kp	NOUN
cana-4517	118	4	,	,	PUNCT
cana-4517	118	5	q	q	NOUN
cana-4517	118	6	)	)	PUNCT
cana-4517	118	7	=	=	SYM
cana-4517	118	8	{	{	PUNCT
cana-4517	118	9	0	0	NUM
cana-4517	118	10	1	1	NUM
cana-4517	118	11	√pq	√pq	PROPN
cana-4517	118	12	−√pq	−√pq	NUM
cana-4517	118	13	p	p	NOUN
cana-4517	118	14	+	+	NOUN
cana-4517	118	15	q	q	NOUN
cana-4517	118	16	−	−	PROPN
cana-4517	118	17	4	4	NUM
cana-4517	118	18	2	2	NUM
cana-4517	118	19	1	1	NUM
cana-4517	118	20	1	1	NUM
cana-4517	118	21	}	}	PUNCT
cana-4517	118	22	𝔓𝔈a(kp	𝔓𝔈a(kp	PROPN
cana-4517	118	23	,	,	PUNCT
cana-4517	118	24	p	p	NOUN
cana-4517	118	25	)	)	PUNCT
cana-4517	118	26	=	=	SYM
cana-4517	118	27	2(1	2(1	NUM
cana-4517	118	28	+	+	CCONJ
cana-4517	118	29	√pq	√pq	ADJ
cana-4517	118	30	)	)	PUNCT
cana-4517	118	31	3.3	3.3	NUM
cana-4517	118	32	theorem	theorem	NOUN
cana-4517	118	33	:	:	PUNCT
cana-4517	118	34	for	for	ADP
cana-4517	118	35	a	a	DET
cana-4517	118	36	crown	crown	NOUN
cana-4517	118	37	graph	graph	NOUN
cana-4517	118	38	sp	sp	ADP
cana-4517	118	39	0	0	NUM
cana-4517	118	40	with	with	ADP
cana-4517	118	41	p	p	PRON
cana-4517	118	42	≥	≥	NUM
cana-4517	118	43	2	2	NUM
cana-4517	118	44	,	,	PUNCT
cana-4517	118	45	𝔓𝔈a(sp	𝔓𝔈a(sp	PROPN
cana-4517	118	46	0	0	NUM
cana-4517	118	47	)	)	PUNCT
cana-4517	118	48	=	=	PUNCT
cana-4517	119	1	2(p	2(p	NUM
cana-4517	120	1	−	−	NOUN
cana-4517	120	2	2	2	NUM
cana-4517	120	3	)	)	PUNCT
cana-4517	120	4	+	+	CCONJ
cana-4517	120	5	√p2	√p2	VERB
cana-4517	120	6	−	−	NUM
cana-4517	120	7	2p	2p	NOUN
cana-4517	120	8	+	+	CCONJ
cana-4517	120	9	5	5	NUM
cana-4517	120	10	+	+	CCONJ
cana-4517	120	11	√p2	√p2	PROPN
cana-4517	120	12	+	+	CCONJ
cana-4517	120	13	2p	2p	NUM
cana-4517	120	14	−	−	NOUN
cana-4517	120	15	3	3	NUM
cana-4517	120	16	proof	proof	NOUN
cana-4517	120	17	:	:	PUNCT
cana-4517	120	18	let	let	VERB
cana-4517	120	19	v(sp	v(sp	NOUN
cana-4517	120	20	0	0	NUM
cana-4517	120	21	)	)	PUNCT
cana-4517	120	22	=	=	PRON
cana-4517	120	23	{	{	PUNCT
cana-4517	120	24	v1	v1	PROPN
cana-4517	120	25	,	,	PUNCT
cana-4517	120	26	.	.	PUNCT
cana-4517	120	27	.	.	PUNCT
cana-4517	120	28	.	.	PUNCT
cana-4517	121	1	,	,	PUNCT
cana-4517	121	2	vp	vp	INTJ
cana-4517	121	3	,	,	PUNCT
cana-4517	121	4	v1′	v1′	NOUN
cana-4517	121	5	,	,	PUNCT
cana-4517	121	6	.	.	PUNCT
cana-4517	121	7	.	.	PUNCT
cana-4517	122	1	.	.	PUNCT
cana-4517	123	1	,	,	PUNCT
cana-4517	123	2	vp′	vp′	VERB
cana-4517	123	3	}	}	PUNCT
cana-4517	123	4	,	,	PUNCT
cana-4517	123	5	let	let	VERB
cana-4517	123	6	a	a	DET
cana-4517	123	7	=	=	X
cana-4517	123	8	{	{	PUNCT
cana-4517	123	9	v1	v1	NOUN
cana-4517	123	10	,	,	PUNCT
cana-4517	123	11	v1′	v1′	NOUN
cana-4517	123	12	}	}	PUNCT
cana-4517	123	13	be	be	AUX
cana-4517	123	14	a	a	DET
cana-4517	123	15	prd	prd	NOUN
cana-4517	123	16	set	set	NOUN
cana-4517	123	17	.	.	PUNCT
cana-4517	124	1	then	then	ADV
cana-4517	124	2	𝔓a(sp	𝔓a(sp	PROPN
cana-4517	124	3	0	0	NUM
cana-4517	124	4	)	)	PUNCT
cana-4517	124	5	has	have	VERB
cana-4517	124	6	the	the	DET
cana-4517	124	7	form	form	NOUN
cana-4517	124	8	(	(	PUNCT
cana-4517	124	9	b0	b0	VERB
cana-4517	124	10	b1	b1	NOUN
cana-4517	124	11	b1	b1	NOUN
cana-4517	124	12	b0	b0	PROPN
cana-4517	124	13	)	)	PUNCT
cana-4517	124	14	,	,	PUNCT
cana-4517	124	15	where	where	SCONJ
cana-4517	124	16	b0	b0	NOUN
cana-4517	124	17	=	=	SYM
cana-4517	124	18	(	(	PUNCT
cana-4517	124	19	1	1	NUM
cana-4517	124	20	o1×p−1	o1×p−1	PROPN
cana-4517	124	21	op−1×1	op−1×1	NUM
cana-4517	124	22	op−1	op−1	PROPN
cana-4517	124	23	)	)	PUNCT
cana-4517	124	24	and	and	CCONJ
cana-4517	124	25	b1	b1	NOUN
cana-4517	124	26	=	=	SYM
cana-4517	124	27	(	(	PUNCT
cana-4517	124	28	jp	jp	INTJ
cana-4517	124	29	−	−	NOUN
cana-4517	124	30	ip	ip	NOUN
cana-4517	124	31	)	)	PUNCT
cana-4517	124	32	,	,	PUNCT
cana-4517	124	33	here	here	ADV
cana-4517	124	34	j	j	PROPN
cana-4517	124	35	,	,	PUNCT
cana-4517	125	1	o	o	NOUN
cana-4517	126	1	and	and	CCONJ
cana-4517	126	2	i	i	PRON
cana-4517	126	3	represents	represent	VERB
cana-4517	126	4	the	the	DET
cana-4517	126	5	matrix	matrix	NOUN
cana-4517	126	6	of	of	ADP
cana-4517	126	7	1	1	NUM
cana-4517	126	8	’s	’s	PART
cana-4517	126	9	,	,	PUNCT
cana-4517	126	10	0	0	NUM
cana-4517	126	11	’s	’s	PART
cana-4517	126	12	and	and	CCONJ
cana-4517	126	13	identity	identity	NOUN
cana-4517	126	14	matrix	matrix	NOUN
cana-4517	126	15	.	.	PUNCT
cana-4517	127	1	therefore	therefore	ADV
cana-4517	127	2	by	by	ADP
cana-4517	127	3	lemma	lemma	PROPN
cana-4517	127	4	1.1	1.1	NUM
cana-4517	127	5	,	,	PUNCT
cana-4517	127	6	speca(sp	speca(sp	ADP
cana-4517	127	7	0	0	NUM
cana-4517	127	8	)	)	PUNCT
cana-4517	127	9	=	=	NOUN
cana-4517	127	10	spec(b0	spec(b0	NOUN
cana-4517	127	11	+	+	CCONJ
cana-4517	127	12	b1)⋃spec(b0	b1)⋃spec(b0	ADP
cana-4517	127	13	−	−	PROPN
cana-4517	127	14	b1	b1	NOUN
cana-4517	127	15	)	)	PUNCT
cana-4517	127	16	.	.	PUNCT
cana-4517	128	1	consider	consider	VERB
cana-4517	128	2	,	,	PUNCT
cana-4517	128	3	b0	b0	NOUN
cana-4517	128	4	+	+	CCONJ
cana-4517	128	5	b1	b1	NOUN
cana-4517	128	6	:	:	PUNCT
cana-4517	128	7	b0	b0	NOUN
cana-4517	128	8	+	+	CCONJ
cana-4517	128	9	b1	b1	NOUN
cana-4517	128	10	=	=	SYM
cana-4517	128	11	(	(	PUNCT
cana-4517	128	12	1	1	NUM
cana-4517	128	13	j1×p−1	j1×p−1	NOUN
cana-4517	128	14	jp−1×1	jp−1×1	NOUN
cana-4517	128	15	jp−1	jp−1	PROPN
cana-4517	128	16	−	−	PROPN
cana-4517	128	17	ip−1	ip−1	PROPN
cana-4517	128	18	)	)	PUNCT
cana-4517	128	19	|(b0	|(b0	ADV
cana-4517	128	20	+	+	CCONJ
cana-4517	128	21	b1	b1	NOUN
cana-4517	128	22	)	)	PUNCT
cana-4517	128	23	−	−	PUNCT
cana-4517	129	1	λi|	λi|	NOUN
cana-4517	129	2	=	=	SYM
cana-4517	129	3	(	(	PUNCT
cana-4517	129	4	−1	−1	NOUN
cana-4517	129	5	)	)	PUNCT
cana-4517	129	6	p(λ	p(λ	NOUN
cana-4517	129	7	+	+	CCONJ
cana-4517	129	8	1)p−2(λ2	1)p−2(λ2	PROPN
cana-4517	129	9	−	−	NOUN
cana-4517	129	10	(	(	PUNCT
cana-4517	129	11	p	p	NOUN
cana-4517	129	12	−	−	PROPN
cana-4517	129	13	1)λ	1)λ	NUM
cana-4517	129	14	−	−	PROPN
cana-4517	129	15	1	1	NUM
cana-4517	129	16	)	)	PUNCT
cana-4517	129	17	therefore	therefore	ADV
cana-4517	129	18	,	,	PUNCT
cana-4517	129	19	speca(b0	speca(b0	NOUN
cana-4517	129	20	+	+	CCONJ
cana-4517	129	21	b1	b1	NOUN
cana-4517	129	22	)	)	PUNCT
cana-4517	129	23	=	=	PRON
cana-4517	129	24	{	{	PUNCT
cana-4517	129	25	−1	−1	NOUN
cana-4517	129	26	(	(	PUNCT
cana-4517	129	27	p	p	NOUN
cana-4517	129	28	−	−	PROPN
cana-4517	129	29	1	1	NUM
cana-4517	129	30	)	)	PUNCT
cana-4517	129	31	+	+	CCONJ
cana-4517	129	32	√p2	√p2	VERB
cana-4517	129	33	−	−	NUM
cana-4517	129	34	2p	2p	NUM
cana-4517	129	35	+	+	CCONJ
cana-4517	129	36	5	5	NUM
cana-4517	129	37	2	2	NUM
cana-4517	129	38	(	(	PUNCT
cana-4517	129	39	p	p	NOUN
cana-4517	129	40	−	−	PROPN
cana-4517	129	41	1	1	NUM
cana-4517	129	42	)	)	PUNCT
cana-4517	129	43	−	−	PROPN
cana-4517	130	1	√p2	√p2	PUNCT
cana-4517	130	2	−	−	NOUN
cana-4517	130	3	2p	2p	NUM
cana-4517	130	4	+	+	CCONJ
cana-4517	130	5	5	5	NUM
cana-4517	130	6	2	2	NUM
cana-4517	130	7	p	p	NOUN
cana-4517	130	8	−	−	NUM
cana-4517	130	9	2	2	NUM
cana-4517	130	10	1	1	NUM
cana-4517	130	11	1	1	NUM
cana-4517	130	12	}	}	PUNCT
cana-4517	130	13	consider	consider	VERB
cana-4517	130	14	,	,	PUNCT
cana-4517	130	15	b0	b0	VERB
cana-4517	130	16	−	−	PROPN
cana-4517	130	17	b1	b1	NOUN
cana-4517	130	18	:	:	PUNCT
cana-4517	130	19	b0	b0	NOUN
cana-4517	130	20	−	−	NOUN
cana-4517	130	21	b1	b1	NOUN
cana-4517	130	22	=	=	PUNCT
cana-4517	130	23	(	(	PUNCT
cana-4517	130	24	1	1	NUM
cana-4517	130	25	−j1×p−1	−j1×p−1	PROPN
cana-4517	130	26	−jp−1×1	−jp−1×1	ADV
cana-4517	130	27	−(jp−1	−(jp−1	PROPN
cana-4517	130	28	−	−	PROPN
cana-4517	130	29	ip−1	ip−1	PROPN
cana-4517	130	30	)	)	PUNCT
cana-4517	130	31	)	)	PUNCT
cana-4517	130	32	|(b0	|(b0	ADV
cana-4517	130	33	−	−	PROPN
cana-4517	130	34	b1	b1	NOUN
cana-4517	130	35	)	)	PUNCT
cana-4517	130	36	−	−	PROPN
cana-4517	131	1	λi|	λi|	NOUN
cana-4517	131	2	=	=	SYM
cana-4517	131	3	(	(	PUNCT
cana-4517	131	4	−1	−1	NOUN
cana-4517	131	5	)	)	PUNCT
cana-4517	131	6	p(λ	p(λ	NOUN
cana-4517	131	7	−	−	PROPN
cana-4517	131	8	1)p−2(λ2	1)p−2(λ2	PROPN
cana-4517	131	9	+	+	CCONJ
cana-4517	131	10	(	(	PUNCT
cana-4517	131	11	p	p	X
cana-4517	131	12	−	−	PROPN
cana-4517	131	13	3)λ	3)λ	NUM
cana-4517	131	14	−	−	PROPN
cana-4517	131	15	(	(	PUNCT
cana-4517	131	16	2p	2p	NUM
cana-4517	131	17	−	−	NOUN
cana-4517	131	18	3	3	NUM
cana-4517	131	19	)	)	PUNCT
cana-4517	131	20	)	)	PUNCT
cana-4517	132	1	therefore	therefore	ADV
cana-4517	132	2	,	,	PUNCT
cana-4517	132	3	communications	communication	NOUN
cana-4517	132	4	on	on	ADP
cana-4517	132	5	applied	apply	VERB
cana-4517	132	6	nonlinear	nonlinear	ADJ
cana-4517	132	7	analysis	analysis	NOUN
cana-4517	132	8	issn	issn	NOUN
cana-4517	132	9	:	:	PUNCT
cana-4517	132	10	1074	1074	NUM
cana-4517	132	11	-	-	PUNCT
cana-4517	132	12	133x	133x	NUM
cana-4517	132	13	vol	vol	NOUN
cana-4517	132	14	32	32	NUM
cana-4517	132	15	no	no	NOUN
cana-4517	132	16	.	.	PUNCT
cana-4517	133	1	9s	9s	NUM
cana-4517	133	2	(	(	PUNCT
cana-4517	133	3	2025	2025	NUM
cana-4517	133	4	)	)	PUNCT
cana-4517	133	5	2319	2319	NUM
cana-4517	133	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	133	7	speca(b0	speca(b0	NOUN
cana-4517	133	8	−	−	PROPN
cana-4517	133	9	b1	b1	NOUN
cana-4517	133	10	)	)	PUNCT
cana-4517	133	11	=	=	PRON
cana-4517	133	12	{	{	PUNCT
cana-4517	133	13	1	1	NUM
cana-4517	133	14	(	(	PUNCT
cana-4517	133	15	3	3	NUM
cana-4517	133	16	−	−	PROPN
cana-4517	133	17	p	p	X
cana-4517	133	18	)	)	PUNCT
cana-4517	133	19	+	+	CCONJ
cana-4517	133	20	√p2	√p2	PUNCT
cana-4517	134	1	+	+	NUM
cana-4517	134	2	2p	2p	NUM
cana-4517	134	3	−	−	NOUN
cana-4517	134	4	3	3	NUM
cana-4517	134	5	2	2	NUM
cana-4517	134	6	(	(	PUNCT
cana-4517	134	7	3	3	NUM
cana-4517	134	8	−	−	PROPN
cana-4517	134	9	p	p	NOUN
cana-4517	134	10	)	)	PUNCT
cana-4517	134	11	−	−	PROPN
cana-4517	134	12	√p2	√p2	PUNCT
cana-4517	135	1	+	+	CCONJ
cana-4517	135	2	2p	2p	NUM
cana-4517	135	3	−	−	NOUN
cana-4517	135	4	3	3	NUM
cana-4517	135	5	2	2	NUM
cana-4517	135	6	p	p	NOUN
cana-4517	135	7	−	−	NUM
cana-4517	135	8	2	2	NUM
cana-4517	135	9	1	1	NUM
cana-4517	135	10	1	1	NUM
cana-4517	135	11	}	}	PUNCT
cana-4517	135	12	hence	hence	ADV
cana-4517	135	13	speca(sp	speca(sp	PROPN
cana-4517	135	14	0	0	NUM
cana-4517	135	15	)	)	PUNCT
cana-4517	135	16	=	=	PRON
cana-4517	135	17	{	{	PUNCT
cana-4517	135	18	−1	−1	NOUN
cana-4517	135	19	1	1	NUM
cana-4517	135	20	(	(	PUNCT
cana-4517	135	21	p	p	NOUN
cana-4517	135	22	−	−	PROPN
cana-4517	135	23	1	1	NUM
cana-4517	135	24	)	)	PUNCT
cana-4517	135	25	+	+	CCONJ
cana-4517	135	26	√p2	√p2	VERB
cana-4517	135	27	−	−	NUM
cana-4517	135	28	2p	2p	NUM
cana-4517	135	29	+	+	CCONJ
cana-4517	135	30	5	5	NUM
cana-4517	135	31	2	2	NUM
cana-4517	135	32	(	(	PUNCT
cana-4517	135	33	p	p	NOUN
cana-4517	135	34	−	−	PROPN
cana-4517	135	35	1	1	NUM
cana-4517	135	36	)	)	PUNCT
cana-4517	135	37	−	−	PROPN
cana-4517	136	1	√p2	√p2	PUNCT
cana-4517	136	2	−	−	NOUN
cana-4517	136	3	2p	2p	NUM
cana-4517	136	4	+	+	CCONJ
cana-4517	136	5	5	5	NUM
cana-4517	136	6	2	2	NUM
cana-4517	136	7	p	p	NOUN
cana-4517	136	8	−	−	PROPN
cana-4517	136	9	2	2	NUM
cana-4517	136	10	p	p	NOUN
cana-4517	136	11	−	−	PROPN
cana-4517	136	12	2	2	NUM
cana-4517	136	13	1	1	NUM
cana-4517	136	14	1	1	NUM
cana-4517	136	15	(	(	PUNCT
cana-4517	136	16	3	3	NUM
cana-4517	136	17	−	−	PROPN
cana-4517	136	18	p	p	X
cana-4517	136	19	)	)	PUNCT
cana-4517	136	20	+	+	CCONJ
cana-4517	136	21	√p2	√p2	PUNCT
cana-4517	136	22	+	+	NUM
cana-4517	136	23	2p	2p	NUM
cana-4517	136	24	−	−	NOUN
cana-4517	136	25	3	3	NUM
cana-4517	136	26	2	2	NUM
cana-4517	136	27	(	(	PUNCT
cana-4517	136	28	3	3	NUM
cana-4517	136	29	−	−	PROPN
cana-4517	136	30	p	p	NOUN
cana-4517	136	31	)	)	PUNCT
cana-4517	136	32	−	−	PROPN
cana-4517	137	1	√p2	√p2	PUNCT
cana-4517	137	2	+	+	CCONJ
cana-4517	137	3	2p	2p	NUM
cana-4517	137	4	−	−	NOUN
cana-4517	137	5	3	3	NUM
cana-4517	137	6	2	2	NUM
cana-4517	137	7	1	1	NUM
cana-4517	137	8	1	1	NUM
cana-4517	137	9	}	}	PUNCT
cana-4517	137	10	now	now	ADV
cana-4517	137	11	,	,	PUNCT
cana-4517	137	12	𝔓𝔈a(sp	𝔓𝔈a(sp	PROPN
cana-4517	137	13	0	0	NUM
cana-4517	137	14	)	)	PUNCT
cana-4517	138	1	=	=	NOUN
cana-4517	138	2	(	(	PUNCT
cana-4517	138	3	p	p	X
cana-4517	138	4	−	−	PROPN
cana-4517	138	5	2)|−1|	2)|−1|	NUM
cana-4517	138	6	+	+	CCONJ
cana-4517	138	7	(	(	PUNCT
cana-4517	138	8	p	p	X
cana-4517	138	9	−	−	PROPN
cana-4517	138	10	2)|1|	2)|1|	PROPN
cana-4517	139	1	+	+	CCONJ
cana-4517	139	2	|	|	ADV
cana-4517	139	3	(	(	PUNCT
cana-4517	139	4	p	p	NOUN
cana-4517	139	5	−	−	PROPN
cana-4517	139	6	1	1	NUM
cana-4517	139	7	)	)	PUNCT
cana-4517	139	8	+	+	CCONJ
cana-4517	139	9	√p2	√p2	VERB
cana-4517	139	10	−	−	NUM
cana-4517	139	11	2p	2p	NUM
cana-4517	139	12	+	+	CCONJ
cana-4517	139	13	5	5	NUM
cana-4517	139	14	2	2	NUM
cana-4517	140	1	|	|	ADV
cana-4517	141	1	+	+	CCONJ
cana-4517	141	2	|	|	ADV
cana-4517	141	3	(	(	PUNCT
cana-4517	141	4	p	p	NOUN
cana-4517	141	5	−	−	PROPN
cana-4517	141	6	1	1	NUM
cana-4517	141	7	)	)	PUNCT
cana-4517	141	8	−	−	PROPN
cana-4517	142	1	√p2	√p2	PUNCT
cana-4517	142	2	−	−	NOUN
cana-4517	142	3	2p	2p	NUM
cana-4517	142	4	+	+	CCONJ
cana-4517	142	5	5	5	NUM
cana-4517	142	6	2	2	NUM
cana-4517	143	1	|	|	ADV
cana-4517	144	1	+	+	CCONJ
cana-4517	144	2	|	|	ADV
cana-4517	144	3	(	(	PUNCT
cana-4517	144	4	3	3	NUM
cana-4517	144	5	−	−	PROPN
cana-4517	144	6	p	p	X
cana-4517	144	7	)	)	PUNCT
cana-4517	145	1	+	+	CCONJ
cana-4517	145	2	√p2	√p2	PUNCT
cana-4517	145	3	+	+	NUM
cana-4517	145	4	2p	2p	NUM
cana-4517	145	5	−	−	NOUN
cana-4517	145	6	3	3	NUM
cana-4517	145	7	2	2	NUM
cana-4517	146	1	|	|	ADV
cana-4517	147	1	+	+	CCONJ
cana-4517	147	2	|	|	ADV
cana-4517	147	3	(	(	PUNCT
cana-4517	147	4	3	3	NUM
cana-4517	147	5	−	−	PROPN
cana-4517	147	6	p	p	X
cana-4517	147	7	)	)	PUNCT
cana-4517	147	8	−	−	PROPN
cana-4517	147	9	√p2	√p2	PUNCT
cana-4517	148	1	+	+	CCONJ
cana-4517	148	2	2p	2p	NUM
cana-4517	148	3	−	−	NOUN
cana-4517	148	4	3	3	NUM
cana-4517	148	5	2	2	NUM
cana-4517	148	6	|	|	ADV
cana-4517	148	7	thus	thus	ADV
cana-4517	148	8	,	,	PUNCT
cana-4517	148	9	𝔓𝔈a(sp	𝔓𝔈a(sp	PROPN
cana-4517	148	10	0	0	NUM
cana-4517	148	11	)	)	PUNCT
cana-4517	149	1	=	=	PUNCT
cana-4517	149	2	2(p	2(p	NUM
cana-4517	150	1	−	−	NOUN
cana-4517	150	2	2	2	NUM
cana-4517	150	3	)	)	PUNCT
cana-4517	150	4	+	+	CCONJ
cana-4517	150	5	√p2	√p2	VERB
cana-4517	150	6	−	−	NUM
cana-4517	150	7	2p	2p	NOUN
cana-4517	150	8	+	+	CCONJ
cana-4517	150	9	5	5	NUM
cana-4517	150	10	+	+	CCONJ
cana-4517	150	11	√p2	√p2	PROPN
cana-4517	150	12	+	+	CCONJ
cana-4517	150	13	2p	2p	NUM
cana-4517	150	14	−	−	NOUN
cana-4517	150	15	3	3	NUM
cana-4517	150	16	3.4	3.4	NUM
cana-4517	150	17	theorem	theorem	VERB
cana-4517	150	18	:	:	PUNCT
cana-4517	150	19	for	for	ADP
cana-4517	150	20	a	a	DET
cana-4517	150	21	barbell	barbell	NOUN
cana-4517	150	22	graph	graph	NOUN
cana-4517	150	23	bp	bp	PROPN
cana-4517	150	24	,	,	PUNCT
cana-4517	150	25	p	p	NOUN
cana-4517	150	26	with	with	ADP
cana-4517	150	27	p	p	PROPN
cana-4517	150	28	>	>	X
cana-4517	150	29	3	3	NUM
cana-4517	150	30	,	,	PUNCT
cana-4517	150	31	𝔓𝔈a(bp	𝔓𝔈a(bp	PROPN
cana-4517	150	32	,	,	PUNCT
cana-4517	150	33	p	p	NOUN
cana-4517	150	34	)	)	PUNCT
cana-4517	150	35	=	=	SYM
cana-4517	150	36	3p	3p	NUM
cana-4517	150	37	−	−	NOUN
cana-4517	150	38	4	4	NUM
cana-4517	150	39	+	+	NOUN
cana-4517	150	40	√(p	√(p	NOUN
cana-4517	150	41	−	−	NOUN
cana-4517	150	42	2)2	2)2	NUM
cana-4517	150	43	+	+	SYM
cana-4517	150	44	8	8	NUM
cana-4517	150	45	proof	proof	NOUN
cana-4517	150	46	:	:	PUNCT
cana-4517	150	47	let	let	VERB
cana-4517	150	48	the	the	DET
cana-4517	150	49	vertex	vertex	NOUN
cana-4517	150	50	set	set	NOUN
cana-4517	150	51	v(bp	v(bp	PROPN
cana-4517	150	52	,	,	PUNCT
cana-4517	150	53	p	p	X
cana-4517	150	54	)	)	PUNCT
cana-4517	150	55	=	=	SYM
cana-4517	150	56	{	{	PUNCT
cana-4517	150	57	v1	v1	PROPN
cana-4517	150	58	,	,	PUNCT
cana-4517	150	59	.	.	PUNCT
cana-4517	150	60	.	.	PUNCT
cana-4517	150	61	.	.	PUNCT
cana-4517	151	1	,	,	PUNCT
cana-4517	151	2	vp	vp	PROPN
cana-4517	151	3	,	,	PUNCT
cana-4517	151	4	v1′	v1′	NOUN
cana-4517	151	5	,	,	PUNCT
cana-4517	151	6	.	.	PUNCT
cana-4517	151	7	.	.	PUNCT
cana-4517	152	1	.	.	PUNCT
cana-4517	153	1	,	,	PUNCT
cana-4517	153	2	vp′	vp′	VERB
cana-4517	153	3	}	}	PUNCT
cana-4517	153	4	.	.	PUNCT
cana-4517	154	1	consider	consider	VERB
cana-4517	154	2	the	the	DET
cana-4517	154	3	prd	prd	NOUN
cana-4517	154	4	set	set	VERB
cana-4517	154	5	a	a	DET
cana-4517	154	6	=	=	X
cana-4517	154	7	{	{	PUNCT
cana-4517	154	8	v1	v1	NOUN
cana-4517	154	9	,	,	PUNCT
cana-4517	154	10	v1′	v1′	NOUN
cana-4517	154	11	}	}	PUNCT
cana-4517	154	12	.	.	PUNCT
cana-4517	155	1	then	then	ADV
cana-4517	155	2	𝔓a(bp	𝔓a(bp	PROPN
cana-4517	155	3	,	,	PUNCT
cana-4517	155	4	p	p	NOUN
cana-4517	155	5	)	)	PUNCT
cana-4517	155	6	has	have	VERB
cana-4517	155	7	the	the	DET
cana-4517	155	8	form	form	NOUN
cana-4517	155	9	(	(	PUNCT
cana-4517	155	10	b0	b0	VERB
cana-4517	155	11	b1	b1	NOUN
cana-4517	155	12	b1	b1	NOUN
cana-4517	155	13	b0	b0	PROPN
cana-4517	155	14	)	)	PUNCT
cana-4517	155	15	,	,	PUNCT
cana-4517	155	16	where	where	SCONJ
cana-4517	155	17	b0	b0	NOUN
cana-4517	155	18	=	=	SYM
cana-4517	155	19	(	(	PUNCT
cana-4517	155	20	1	1	NUM
cana-4517	155	21	j1×p−1	j1×p−1	NOUN
cana-4517	155	22	jp−1×1	jp−1×1	NOUN
cana-4517	155	23	jp−1	jp−1	PROPN
cana-4517	155	24	−	−	PROPN
cana-4517	155	25	ip−1	ip−1	PROPN
cana-4517	155	26	)	)	PUNCT
cana-4517	155	27	and	and	CCONJ
cana-4517	155	28	b1	b1	NOUN
cana-4517	155	29	=	=	SYM
cana-4517	155	30	(	(	PUNCT
cana-4517	155	31	1	1	NUM
cana-4517	155	32	o1×p−1	o1×p−1	PROPN
cana-4517	155	33	op−1×1	op−1×1	NUM
cana-4517	155	34	op−1	op−1	PROPN
cana-4517	155	35	)	)	PUNCT
cana-4517	155	36	,	,	PUNCT
cana-4517	155	37	here	here	ADV
cana-4517	155	38	j	j	PROPN
cana-4517	155	39	and	and	CCONJ
cana-4517	155	40	o	o	PROPN
cana-4517	155	41	represents	represent	VERB
cana-4517	155	42	the	the	DET
cana-4517	155	43	matrix	matrix	NOUN
cana-4517	155	44	of	of	ADP
cana-4517	155	45	1	1	NUM
cana-4517	155	46	’s	’s	NOUN
cana-4517	155	47	and	and	CCONJ
cana-4517	155	48	0	0	NUM
cana-4517	155	49	’s	’s	NOUN
cana-4517	155	50	.	.	PUNCT
cana-4517	156	1	therefore	therefore	ADV
cana-4517	156	2	by	by	ADP
cana-4517	156	3	lemma	lemma	PROPN
cana-4517	156	4	1.1	1.1	NUM
cana-4517	156	5	,	,	PUNCT
cana-4517	156	6	speca(bp	speca(bp	PROPN
cana-4517	156	7	,	,	PUNCT
cana-4517	156	8	p	p	NOUN
cana-4517	156	9	)	)	PUNCT
cana-4517	156	10	=	=	NOUN
cana-4517	156	11	spec(b0	spec(b0	NOUN
cana-4517	157	1	+	+	CCONJ
cana-4517	157	2	b1)⋃spec(b0	b1)⋃spec(b0	ADP
cana-4517	157	3	−	−	PROPN
cana-4517	157	4	b1	b1	NOUN
cana-4517	157	5	)	)	PUNCT
cana-4517	157	6	.	.	PUNCT
cana-4517	158	1	consider	consider	VERB
cana-4517	158	2	,	,	PUNCT
cana-4517	158	3	b0	b0	NOUN
cana-4517	158	4	+	+	CCONJ
cana-4517	158	5	b1	b1	NOUN
cana-4517	158	6	:	:	PUNCT
cana-4517	158	7	b0	b0	NOUN
cana-4517	158	8	+	+	CCONJ
cana-4517	158	9	b1	b1	NOUN
cana-4517	158	10	=	=	SYM
cana-4517	158	11	(	(	PUNCT
cana-4517	158	12	2	2	NUM
cana-4517	158	13	j1×p−1	j1×p−1	NOUN
cana-4517	158	14	jp−1×1	jp−1×1	NOUN
cana-4517	158	15	jp−1	jp−1	PROPN
cana-4517	158	16	−	−	PROPN
cana-4517	158	17	ip−1	ip−1	PROPN
cana-4517	158	18	)	)	PUNCT
cana-4517	158	19	|(b0	|(b0	ADV
cana-4517	158	20	+	+	CCONJ
cana-4517	158	21	b1	b1	NOUN
cana-4517	158	22	)	)	PUNCT
cana-4517	158	23	−	−	PUNCT
cana-4517	159	1	λi|	λi|	NOUN
cana-4517	159	2	=	=	SYM
cana-4517	159	3	(	(	PUNCT
cana-4517	159	4	−1	−1	NOUN
cana-4517	159	5	)	)	PUNCT
cana-4517	159	6	p(λ	p(λ	NOUN
cana-4517	159	7	+	+	CCONJ
cana-4517	159	8	1)p−2(λ2	1)p−2(λ2	PROPN
cana-4517	159	9	−	−	PROPN
cana-4517	159	10	pλ	pλ	NOUN
cana-4517	160	1	+	+	CCONJ
cana-4517	160	2	(	(	PUNCT
cana-4517	160	3	p	p	X
cana-4517	160	4	−	−	PROPN
cana-4517	160	5	3	3	NUM
cana-4517	160	6	)	)	PUNCT
cana-4517	160	7	)	)	PUNCT
cana-4517	160	8	therefore	therefore	ADV
cana-4517	160	9	,	,	PUNCT
cana-4517	160	10	speca(b0	speca(b0	NOUN
cana-4517	160	11	+	+	CCONJ
cana-4517	160	12	b1	b1	NOUN
cana-4517	160	13	)	)	PUNCT
cana-4517	160	14	=	=	PRON
cana-4517	160	15	{	{	PUNCT
cana-4517	160	16	−1	−1	NOUN
cana-4517	160	17	p	p	X
cana-4517	160	18	+	+	NOUN
cana-4517	160	19	√(p	√(p	NOUN
cana-4517	160	20	−	−	NOUN
cana-4517	160	21	2)2	2)2	NUM
cana-4517	160	22	+	+	NUM
cana-4517	160	23	8	8	NUM
cana-4517	160	24	2	2	NUM
cana-4517	160	25	p	p	NOUN
cana-4517	160	26	−	−	PROPN
cana-4517	160	27	√(p	√(p	NOUN
cana-4517	160	28	−	−	ADP
cana-4517	160	29	2)2	2)2	NUM
cana-4517	160	30	+	+	NUM
cana-4517	160	31	8	8	NUM
cana-4517	160	32	2	2	NUM
cana-4517	160	33	p	p	NOUN
cana-4517	160	34	−	−	NUM
cana-4517	160	35	2	2	NUM
cana-4517	160	36	1	1	NUM
cana-4517	160	37	1	1	NUM
cana-4517	160	38	}	}	PUNCT
cana-4517	160	39	consider	consider	VERB
cana-4517	160	40	,	,	PUNCT
cana-4517	160	41	b0	b0	VERB
cana-4517	160	42	−	−	PROPN
cana-4517	160	43	b1	b1	NOUN
cana-4517	160	44	:	:	PUNCT
cana-4517	160	45	b0	b0	NOUN
cana-4517	160	46	−	−	NOUN
cana-4517	160	47	b1	b1	NOUN
cana-4517	160	48	=	=	SYM
cana-4517	160	49	(	(	PUNCT
cana-4517	160	50	0	0	NUM
cana-4517	160	51	j1×p−1	j1×p−1	NOUN
cana-4517	160	52	jp−1×1	jp−1×1	NOUN
cana-4517	160	53	jp−1	jp−1	PROPN
cana-4517	160	54	−	−	PROPN
cana-4517	160	55	ip−1	ip−1	PROPN
cana-4517	160	56	)	)	PUNCT
cana-4517	160	57	|(b0	|(b0	ADV
cana-4517	160	58	−	−	PROPN
cana-4517	160	59	b1	b1	NOUN
cana-4517	160	60	)	)	PUNCT
cana-4517	160	61	−	−	PROPN
cana-4517	161	1	λi|	λi|	NOUN
cana-4517	161	2	=	=	SYM
cana-4517	161	3	(	(	PUNCT
cana-4517	161	4	−1	−1	NOUN
cana-4517	161	5	)	)	PUNCT
cana-4517	161	6	p(λ	p(λ	NOUN
cana-4517	161	7	−	−	PROPN
cana-4517	162	1	(	(	PUNCT
cana-4517	162	2	p	p	NOUN
cana-4517	162	3	−	−	PROPN
cana-4517	162	4	1))(λ	1))(λ	PROPN
cana-4517	162	5	+	+	CCONJ
cana-4517	162	6	1)p−1	1)p−1	NUM
cana-4517	162	7	therefore	therefore	ADV
cana-4517	162	8	,	,	PUNCT
cana-4517	162	9	speca(b0	speca(b0	NOUN
cana-4517	162	10	−	−	PROPN
cana-4517	162	11	b1	b1	NOUN
cana-4517	162	12	)	)	PUNCT
cana-4517	162	13	=	=	PRON
cana-4517	162	14	{	{	PUNCT
cana-4517	163	1	p	p	NOUN
cana-4517	163	2	−	−	PROPN
cana-4517	163	3	1	1	NUM
cana-4517	163	4	−1	−1	NOUN
cana-4517	163	5	1	1	NUM
cana-4517	163	6	p	p	NOUN
cana-4517	163	7	−	−	PROPN
cana-4517	163	8	1	1	NUM
cana-4517	163	9	}	}	PUNCT
cana-4517	163	10	hence	hence	ADV
cana-4517	163	11	communications	communication	NOUN
cana-4517	163	12	on	on	ADP
cana-4517	163	13	applied	apply	VERB
cana-4517	163	14	nonlinear	nonlinear	ADJ
cana-4517	163	15	analysis	analysis	NOUN
cana-4517	163	16	issn	issn	NOUN
cana-4517	163	17	:	:	PUNCT
cana-4517	163	18	1074	1074	NUM
cana-4517	163	19	-	-	PUNCT
cana-4517	163	20	133x	133x	NUM
cana-4517	163	21	vol	vol	NOUN
cana-4517	163	22	32	32	NUM
cana-4517	163	23	no	no	NOUN
cana-4517	163	24	.	.	PUNCT
cana-4517	164	1	9s	9s	NUM
cana-4517	164	2	(	(	PUNCT
cana-4517	164	3	2025	2025	NUM
cana-4517	164	4	)	)	PUNCT
cana-4517	164	5	2320	2320	NUM
cana-4517	164	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	164	7	speca(bp	speca(bp	PROPN
cana-4517	164	8	,	,	PUNCT
cana-4517	164	9	p	p	NOUN
cana-4517	164	10	)	)	PUNCT
cana-4517	164	11	=	=	NOUN
cana-4517	164	12	{	{	PUNCT
cana-4517	164	13	−1	−1	NOUN
cana-4517	164	14	p	p	NOUN
cana-4517	164	15	−	−	PROPN
cana-4517	164	16	1	1	NUM
cana-4517	164	17	p	p	NOUN
cana-4517	164	18	+	+	NOUN
cana-4517	164	19	√(p	√(p	NOUN
cana-4517	164	20	−	−	NOUN
cana-4517	164	21	2)2	2)2	NUM
cana-4517	164	22	+	+	NUM
cana-4517	164	23	8	8	NUM
cana-4517	164	24	2	2	NUM
cana-4517	164	25	p	p	NOUN
cana-4517	164	26	−	−	PROPN
cana-4517	164	27	√(p	√(p	NOUN
cana-4517	164	28	−	−	ADP
cana-4517	164	29	2)2	2)2	NUM
cana-4517	164	30	+	+	NUM
cana-4517	164	31	8	8	NUM
cana-4517	164	32	2	2	NUM
cana-4517	164	33	2p	2p	NUM
cana-4517	164	34	−	−	NOUN
cana-4517	164	35	3	3	NUM
cana-4517	164	36	1	1	NUM
cana-4517	164	37	1	1	NUM
cana-4517	164	38	1	1	NUM
cana-4517	164	39	now	now	ADV
cana-4517	164	40	,	,	PUNCT
cana-4517	164	41	𝔓𝔈a(bp	𝔓𝔈a(bp	PROPN
cana-4517	164	42	,	,	PUNCT
cana-4517	164	43	p	p	NOUN
cana-4517	164	44	)	)	PUNCT
cana-4517	164	45	=	=	SYM
cana-4517	165	1	(	(	PUNCT
cana-4517	165	2	2p	2p	NUM
cana-4517	165	3	−	−	PROPN
cana-4517	165	4	3)|−1|	3)|−1|	NUM
cana-4517	165	5	+	+	NUM
cana-4517	165	6	|p	|p	PROPN
cana-4517	165	7	−	−	ADP
cana-4517	165	8	1|	1|	NUM
cana-4517	166	1	+	+	CCONJ
cana-4517	166	2	|	|	ADV
cana-4517	166	3	p	p	X
cana-4517	166	4	+	+	NOUN
cana-4517	166	5	√(p	√(p	NOUN
cana-4517	166	6	−	−	NOUN
cana-4517	166	7	2)2	2)2	NUM
cana-4517	166	8	+	+	NUM
cana-4517	166	9	8	8	NUM
cana-4517	166	10	2	2	NUM
cana-4517	167	1	|	|	ADV
cana-4517	168	1	+	+	CCONJ
cana-4517	168	2	|	|	ADV
cana-4517	168	3	p	p	NOUN
cana-4517	169	1	−	−	PROPN
cana-4517	169	2	√(p	√(p	NOUN
cana-4517	169	3	−	−	ADP
cana-4517	169	4	2)2	2)2	NUM
cana-4517	169	5	+	+	NUM
cana-4517	169	6	8	8	NUM
cana-4517	169	7	2	2	NUM
cana-4517	169	8	|	|	ADV
cana-4517	169	9	thus	thus	ADV
cana-4517	169	10	,	,	PUNCT
cana-4517	169	11	𝔓𝔈a(bp	𝔓𝔈a(bp	PROPN
cana-4517	169	12	,	,	PUNCT
cana-4517	169	13	p	p	NOUN
cana-4517	169	14	)	)	PUNCT
cana-4517	169	15	=	=	SYM
cana-4517	169	16	3p	3p	NUM
cana-4517	169	17	−	−	NOUN
cana-4517	169	18	4	4	NUM
cana-4517	169	19	+	+	NOUN
cana-4517	169	20	√(p	√(p	NOUN
cana-4517	169	21	−	−	NOUN
cana-4517	169	22	2)2	2)2	NUM
cana-4517	169	23	+	+	NUM
cana-4517	169	24	8	8	NUM
cana-4517	169	25	3.5	3.5	NUM
cana-4517	169	26	theorem	theorem	NOUN
cana-4517	169	27	:	:	PUNCT
cana-4517	169	28	let	let	VERB
cana-4517	169	29	g	g	PROPN
cana-4517	169	30	be	be	AUX
cana-4517	169	31	obtained	obtain	VERB
cana-4517	169	32	by	by	ADP
cana-4517	169	33	removing	remove	VERB
cana-4517	169	34	an	an	DET
cana-4517	169	35	edge	edge	NOUN
cana-4517	169	36	′e′	′e′	PUNCT
cana-4517	169	37	from	from	ADP
cana-4517	169	38	kp	kp	PROPN
cana-4517	169	39	,	,	PUNCT
cana-4517	169	40	p	p	X
cana-4517	169	41	>	>	X
cana-4517	169	42	4	4	NUM
cana-4517	169	43	.	.	PUNCT
cana-4517	170	1	then	then	ADV
cana-4517	170	2	g	g	PROPN
cana-4517	170	3	has	have	VERB
cana-4517	170	4	the	the	DET
cana-4517	170	5	spectrump	spectrump	NOUN
cana-4517	170	6	−	−	PROPN
cana-4517	170	7	1	1	NUM
cana-4517	170	8	,	,	PUNCT
cana-4517	170	9	-1	-1	PUNCT
cana-4517	170	10	and	and	CCONJ
cana-4517	170	11	0	0	NUM
cana-4517	170	12	with	with	ADP
cana-4517	170	13	multiplicities	multiplicity	NOUN
cana-4517	170	14	1,p	1,p	PROPN
cana-4517	170	15	−	−	PROPN
cana-4517	170	16	2	2	NUM
cana-4517	170	17	and	and	CCONJ
cana-4517	170	18	1	1	NUM
cana-4517	170	19	respectively	respectively	ADV
cana-4517	170	20	.	.	PUNCT
cana-4517	171	1	and	and	CCONJ
cana-4517	171	2	hence	hence	ADV
cana-4517	171	3	𝔓𝔈a(g	𝔓𝔈a(g	NUM
cana-4517	171	4	)	)	PUNCT
cana-4517	172	1	=	=	NOUN
cana-4517	172	2	2p	2p	NUM
cana-4517	172	3	−	−	NOUN
cana-4517	172	4	3	3	NUM
cana-4517	172	5	.	.	X
cana-4517	173	1	proof	proof	NOUN
cana-4517	173	2	:	:	PUNCT
cana-4517	173	3	let	let	VERB
cana-4517	173	4	g	g	PROPN
cana-4517	173	5	=	=	PUNCT
cana-4517	173	6	kp	kp	PROPN
cana-4517	173	7	−	−	PROPN
cana-4517	173	8	e.	e.	PROPN
cana-4517	173	9	let	let	VERB
cana-4517	173	10	v1	v1	PROPN
cana-4517	173	11	,	,	PUNCT
cana-4517	173	12	vp	vp	X
cana-4517	173	13	be	be	AUX
cana-4517	173	14	the	the	DET
cana-4517	173	15	non	non	ADJ
cana-4517	173	16	-	-	ADJ
cana-4517	173	17	adjacent	adjacent	ADJ
cana-4517	173	18	vertices	vertex	NOUN
cana-4517	173	19	of	of	ADP
cana-4517	173	20	g.	g.	PROPN
cana-4517	174	1	then	then	ADV
cana-4517	174	2	the	the	DET
cana-4517	174	3	vertices	vertex	NOUN
cana-4517	174	4	other	other	ADJ
cana-4517	174	5	than	than	ADP
cana-4517	174	6	v1	v1	NOUN
cana-4517	174	7	,	,	PUNCT
cana-4517	174	8	vp	vp	PROPN
cana-4517	174	9	would	would	AUX
cana-4517	174	10	be	be	AUX
cana-4517	174	11	act	act	NOUN
cana-4517	174	12	as	as	ADP
cana-4517	174	13	a	a	DET
cana-4517	174	14	pd	pd	NOUN
cana-4517	174	15	-	-	PUNCT
cana-4517	174	16	set	set	NOUN
cana-4517	174	17	.	.	PUNCT
cana-4517	175	1	since	since	SCONJ
cana-4517	175	2	p	p	PROPN
cana-4517	175	3	≥	≥	NUM
cana-4517	175	4	5	5	NUM
cana-4517	175	5	,	,	PUNCT
cana-4517	175	6	deg(v1	deg(v1	NOUN
cana-4517	175	7	)	)	PUNCT
cana-4517	175	8	=	=	SYM
cana-4517	175	9	deg(vp	deg(vp	X
cana-4517	175	10	)	)	PUNCT
cana-4517	176	1	=	=	PUNCT
cana-4517	177	1	p	p	NOUN
cana-4517	177	2	−	−	PROPN
cana-4517	177	3	2	2	NUM
cana-4517	177	4	,	,	PUNCT
cana-4517	177	5	any	any	DET
cana-4517	177	6	pd	pd	NOUN
cana-4517	177	7	-	-	PUNCT
cana-4517	177	8	set	set	NOUN
cana-4517	177	9	will	will	AUX
cana-4517	177	10	satisfy	satisfy	VERB
cana-4517	177	11	the	the	DET
cana-4517	177	12	constrain	constrain	NOUN
cana-4517	177	13	of	of	ADP
cana-4517	177	14	rd	rd	NOUN
cana-4517	177	15	-	-	PUNCT
cana-4517	177	16	set	set	NOUN
cana-4517	177	17	.	.	PUNCT
cana-4517	178	1	let	let	VERB
cana-4517	178	2	a	a	PRON
cana-4517	178	3	=	=	X
cana-4517	178	4	{	{	PUNCT
cana-4517	178	5	v2	v2	NOUN
cana-4517	178	6	}	}	PUNCT
cana-4517	178	7	is	be	AUX
cana-4517	178	8	the	the	DET
cana-4517	178	9	prd	prd	NOUN
cana-4517	178	10	-	-	PUNCT
cana-4517	178	11	set	set	NOUN
cana-4517	178	12	of	of	ADP
cana-4517	178	13	g.	g.	PROPN
cana-4517	178	14	then	then	ADV
cana-4517	178	15	the	the	DET
cana-4517	178	16	prd	prd	NOUN
cana-4517	178	17	matrix	matrix	NOUN
cana-4517	178	18	is	be	AUX
cana-4517	178	19	,	,	PUNCT
cana-4517	178	20	𝔓a(g	𝔓a(g	NOUN
cana-4517	178	21	)	)	PUNCT
cana-4517	178	22	=	=	PUNCT
cana-4517	179	1	(	(	PUNCT
cana-4517	179	2	p	p	X
cana-4517	179	3	q	q	X
cana-4517	179	4	r	r	NOUN
cana-4517	179	5	s	s	PROPN
cana-4517	179	6	)	)	PUNCT
cana-4517	179	7	where	where	SCONJ
cana-4517	179	8	a	a	DET
cana-4517	179	9	=	=	X
cana-4517	179	10	(	(	PUNCT
cana-4517	179	11	0	0	NUM
cana-4517	179	12	1	1	NUM
cana-4517	179	13	1	1	NUM
cana-4517	179	14	1	1	NUM
cana-4517	179	15	)	)	PUNCT
cana-4517	179	16	;	;	PUNCT
cana-4517	179	17	b	b	X
cana-4517	179	18	=	=	SYM
cana-4517	179	19	(	(	PUNCT
cana-4517	179	20	j1×p−3	j1×p−3	PROPN
cana-4517	179	21	0	0	NUM
cana-4517	179	22	j1×p−3	j1×p−3	ADJ
cana-4517	179	23	1	1	NUM
cana-4517	179	24	)	)	PUNCT
cana-4517	179	25	c	c	NOUN
cana-4517	180	1	=	=	PUNCT
cana-4517	181	1	(	(	PUNCT
cana-4517	181	2	jp−3×1	jp−3×1	ADV
cana-4517	181	3	jp−3×1	jp−3×1	ADV
cana-4517	181	4	0	0	NUM
cana-4517	181	5	1	1	NUM
cana-4517	181	6	)	)	PUNCT
cana-4517	181	7	;	;	PUNCT
cana-4517	181	8	d	d	X
cana-4517	181	9	=	=	SYM
cana-4517	181	10	(	(	PUNCT
cana-4517	181	11	jp−3	jp−3	PROPN
cana-4517	181	12	−	−	PROPN
cana-4517	181	13	ip−3	ip−3	NOUN
cana-4517	181	14	)	)	PUNCT
cana-4517	181	15	where	where	SCONJ
cana-4517	181	16	j	j	PROPN
cana-4517	181	17	and	and	CCONJ
cana-4517	181	18	i	i	PRON
cana-4517	181	19	represents	represent	VERB
cana-4517	181	20	the	the	DET
cana-4517	181	21	matrix	matrix	NOUN
cana-4517	181	22	of	of	ADP
cana-4517	181	23	1	1	NUM
cana-4517	181	24	’s	’s	PART
cana-4517	181	25	and	and	CCONJ
cana-4517	181	26	identity	identity	NOUN
cana-4517	181	27	matrix	matrix	NOUN
cana-4517	181	28	.	.	PUNCT
cana-4517	182	1	now	now	ADV
cana-4517	182	2	,	,	PUNCT
cana-4517	182	3	|𝔓a(g	|𝔓a(g	PROPN
cana-4517	182	4	)	)	PUNCT
cana-4517	182	5	−	−	PROPN
cana-4517	183	1	λi|	λi|	NOUN
cana-4517	183	2	=	=	SYM
cana-4517	183	3	(	(	PUNCT
cana-4517	183	4	−1	−1	NOUN
cana-4517	183	5	)	)	PUNCT
cana-4517	183	6	pλ(λ	pλ(λ	VERB
cana-4517	184	1	+	+	NUM
cana-4517	184	2	1)p−4(λ3	1)p−4(λ3	NUM
cana-4517	184	3	−	−	NOUN
cana-4517	184	4	(	(	PUNCT
cana-4517	184	5	p	p	NOUN
cana-4517	184	6	−	−	PROPN
cana-4517	184	7	3)λ2	3)λ2	NUM
cana-4517	184	8	−	−	NOUN
cana-4517	184	9	(	(	PUNCT
cana-4517	184	10	2p	2p	NUM
cana-4517	184	11	−	−	PROPN
cana-4517	184	12	3)λ	3)λ	NUM
cana-4517	184	13	−	−	PROPN
cana-4517	184	14	2	2	NUM
cana-4517	184	15	)	)	PUNCT
cana-4517	184	16	⟹φ(g	⟹φ(g	PROPN
cana-4517	184	17	,	,	PUNCT
cana-4517	184	18	λ	λ	X
cana-4517	184	19	)	)	PUNCT
cana-4517	184	20	≈	≈	PROPN
cana-4517	184	21	(	(	PUNCT
cana-4517	184	22	−1)pλ(λ	−1)pλ(λ	NOUN
cana-4517	184	23	+	+	NUM
cana-4517	184	24	1)p−4(λ3	1)p−4(λ3	NOUN
cana-4517	184	25	−	−	NOUN
cana-4517	184	26	(	(	PUNCT
cana-4517	184	27	p	p	NOUN
cana-4517	184	28	−	−	PROPN
cana-4517	184	29	3)λ2	3)λ2	NUM
cana-4517	184	30	−	−	NOUN
cana-4517	184	31	(	(	PUNCT
cana-4517	184	32	2p	2p	NUM
cana-4517	184	33	−	−	PROPN
cana-4517	184	34	3)λ	3)λ	NUM
cana-4517	184	35	−	−	PROPN
cana-4517	184	36	(	(	PUNCT
cana-4517	184	37	p	p	NOUN
cana-4517	184	38	−	−	PROPN
cana-4517	184	39	1	1	NUM
cana-4517	184	40	)	)	PUNCT
cana-4517	184	41	)	)	PUNCT
cana-4517	185	1	⟹φ(g	⟹φ(g	PROPN
cana-4517	185	2	,	,	PUNCT
cana-4517	185	3	λ	λ	X
cana-4517	185	4	)	)	PUNCT
cana-4517	185	5	≈	≈	PROPN
cana-4517	185	6	(	(	PUNCT
cana-4517	185	7	−1)pλ(λ	−1)pλ(λ	NOUN
cana-4517	185	8	+	+	NOUN
cana-4517	185	9	1)p−2(λ	1)p−2(λ	NUM
cana-4517	185	10	−	−	NOUN
cana-4517	186	1	(	(	PUNCT
cana-4517	186	2	p	p	NOUN
cana-4517	186	3	−	−	PROPN
cana-4517	186	4	1	1	NUM
cana-4517	186	5	)	)	PUNCT
cana-4517	186	6	)	)	PUNCT
cana-4517	186	7	therefore	therefore	ADV
cana-4517	186	8	,	,	PUNCT
cana-4517	186	9	speca(g	speca(g	NOUN
cana-4517	186	10	)	)	PUNCT
cana-4517	186	11	=	=	PRON
cana-4517	186	12	{	{	PUNCT
cana-4517	187	1	p	p	NOUN
cana-4517	187	2	−	−	PROPN
cana-4517	187	3	1	1	NUM
cana-4517	187	4	−1	−1	NOUN
cana-4517	187	5	0	0	NUM
cana-4517	187	6	1	1	NUM
cana-4517	187	7	p	p	NOUN
cana-4517	187	8	−	−	NUM
cana-4517	187	9	2	2	NUM
cana-4517	187	10	1	1	NUM
cana-4517	187	11	}	}	PUNCT
cana-4517	187	12	thus	thus	ADV
cana-4517	187	13	,	,	PUNCT
cana-4517	187	14	𝔓𝔈a(g	𝔓𝔈a(g	PROPN
cana-4517	187	15	)	)	PUNCT
cana-4517	187	16	=	=	PROPN
cana-4517	187	17	|p	|p	NOUN
cana-4517	187	18	−	−	NOUN
cana-4517	187	19	1|	1|	NUM
cana-4517	187	20	+	+	CCONJ
cana-4517	187	21	(	(	PUNCT
cana-4517	187	22	p	p	X
cana-4517	187	23	−	−	PROPN
cana-4517	187	24	2)|−1|	2)|−1|	NUM
cana-4517	187	25	=	=	PUNCT
cana-4517	188	1	p	p	NOUN
cana-4517	188	2	−	−	PROPN
cana-4517	188	3	1	1	NUM
cana-4517	189	1	+	+	CCONJ
cana-4517	189	2	p	p	NOUN
cana-4517	189	3	−	−	PROPN
cana-4517	189	4	2	2	NUM
cana-4517	189	5	𝔓𝔈a(g	𝔓𝔈a(g	PROPN
cana-4517	189	6	)	)	PUNCT
cana-4517	189	7	=	=	NOUN
cana-4517	189	8	2p	2p	NUM
cana-4517	189	9	−	−	NOUN
cana-4517	189	10	3	3	NUM
cana-4517	189	11	4	4	NUM
cana-4517	189	12	.	.	PUNCT
cana-4517	190	1	prd	prd	NOUN
cana-4517	190	2	energy	energy	NOUN
cana-4517	190	3	of	of	ADP
cana-4517	190	4	line	line	NOUN
cana-4517	190	5	graphs	graph	NOUN
cana-4517	190	6	of	of	ADP
cana-4517	190	7	star	star	NOUN
cana-4517	190	8	graphs	graph	VERB
cana-4517	190	9	4.1	4.1	NUM
cana-4517	190	10	theorem	theorem	VERB
cana-4517	190	11	:	:	PUNCT
cana-4517	190	12	for	for	ADP
cana-4517	190	13	the	the	DET
cana-4517	190	14	line	line	NOUN
cana-4517	190	15	graph	graph	NOUN
cana-4517	190	16	of	of	ADP
cana-4517	190	17	a	a	DET
cana-4517	190	18	star	star	NOUN
cana-4517	190	19	graph	graph	NOUN
cana-4517	190	20	l(k1,p−1	l(k1,p−1	NOUN
cana-4517	190	21	)	)	PUNCT
cana-4517	190	22	withp	withp	PROPN
cana-4517	190	23	>	>	X
cana-4517	190	24	4	4	NUM
cana-4517	190	25	,	,	PUNCT
cana-4517	190	26	𝔓𝔈a	𝔓𝔈a	NOUN
cana-4517	190	27	(	(	PUNCT
cana-4517	190	28	l(k1,p−1	l(k1,p−1	NOUN
cana-4517	190	29	)	)	PUNCT
cana-4517	190	30	)	)	PUNCT
cana-4517	191	1	=	=	PUNCT
cana-4517	192	1	p	p	NOUN
cana-4517	192	2	−	−	PROPN
cana-4517	192	3	3	3	NUM
cana-4517	192	4	+	+	NOUN
cana-4517	192	5	√(p	√(p	NOUN
cana-4517	192	6	−	−	NOUN
cana-4517	192	7	2)2	2)2	NUM
cana-4517	192	8	+	+	SYM
cana-4517	192	9	4	4	NUM
cana-4517	192	10	proof	proof	NOUN
cana-4517	192	11	:	:	PUNCT
cana-4517	192	12	let	let	VERB
cana-4517	192	13	e(k1,p−1	e(k1,p−1	NOUN
cana-4517	192	14	)	)	PUNCT
cana-4517	192	15	=	=	PRON
cana-4517	192	16	{	{	PUNCT
cana-4517	192	17	e1	e1	PROPN
cana-4517	192	18	,	,	PUNCT
cana-4517	192	19	.	.	PUNCT
cana-4517	192	20	.	.	PUNCT
cana-4517	192	21	.	.	PUNCT
cana-4517	193	1	,	,	PUNCT
cana-4517	193	2	ep−1	ep−1	PROPN
cana-4517	193	3	}	}	PUNCT
cana-4517	193	4	.	.	PUNCT
cana-4517	194	1	and	and	CCONJ
cana-4517	194	2	it	it	PRON
cana-4517	194	3	’s	’s	AUX
cana-4517	194	4	known	know	VERB
cana-4517	194	5	that	that	SCONJ
cana-4517	194	6	l(k1,p−1	l(k1,p−1	NOUN
cana-4517	194	7	)	)	PUNCT
cana-4517	194	8	≅	≅	PROPN
cana-4517	194	9	kp−1	kp−1	PROPN
cana-4517	194	10	.	.	PUNCT
cana-4517	195	1	therefore	therefore	ADV
cana-4517	195	2	l(k1,p−1	l(k1,p−1	VERB
cana-4517	195	3	)	)	PUNCT
cana-4517	195	4	and	and	CCONJ
cana-4517	195	5	kp−1	kp−1	PROPN
cana-4517	195	6	are	be	AUX
cana-4517	195	7	co	co	ADJ
cana-4517	195	8	-	-	ADJ
cana-4517	195	9	spectral	spectral	ADJ
cana-4517	195	10	and	and	CCONJ
cana-4517	195	11	hence	hence	ADV
cana-4517	195	12	equi	equi	NOUN
cana-4517	195	13	-	-	PUNCT
cana-4517	195	14	energetic	energetic	ADJ
cana-4517	195	15	.	.	PUNCT
cana-4517	196	1	thus	thus	ADV
cana-4517	196	2	𝔓𝔈a	𝔓𝔈a	NOUN
cana-4517	196	3	(	(	PUNCT
cana-4517	196	4	l(k1,p−1	l(k1,p−1	NOUN
cana-4517	196	5	)	)	PUNCT
cana-4517	196	6	)	)	PUNCT
cana-4517	197	1	≅	≅	PROPN
cana-4517	197	2	𝔓𝔈a(kp−1	𝔓𝔈a(kp−1	PROPN
cana-4517	197	3	)	)	PUNCT
cana-4517	198	1	=	=	PUNCT
cana-4517	199	1	p	p	NOUN
cana-4517	199	2	−	−	PROPN
cana-4517	199	3	3	3	NUM
cana-4517	199	4	+	+	NOUN
cana-4517	199	5	√(p	√(p	NOUN
cana-4517	199	6	−	−	ADP
cana-4517	199	7	2)2	2)2	NUM
cana-4517	199	8	+	+	SYM
cana-4517	199	9	4	4	NUM
cana-4517	199	10	.	.	X
cana-4517	199	11	hence	hence	ADV
cana-4517	199	12	𝔓𝔈a	𝔓𝔈a	NOUN
cana-4517	199	13	(	(	PUNCT
cana-4517	199	14	l(k1,p−1	l(k1,p−1	NOUN
cana-4517	199	15	)	)	PUNCT
cana-4517	199	16	)	)	PUNCT
cana-4517	200	1	=	=	PUNCT
cana-4517	201	1	p	p	NOUN
cana-4517	201	2	−	−	PROPN
cana-4517	201	3	3	3	NUM
cana-4517	201	4	+	+	NOUN
cana-4517	201	5	√(p	√(p	NOUN
cana-4517	201	6	−	−	NOUN
cana-4517	201	7	2)2	2)2	NUM
cana-4517	201	8	+	+	NUM
cana-4517	201	9	4	4	NUM
cana-4517	201	10	4.2	4.2	NUM
cana-4517	201	11	theorem	theorem	VERB
cana-4517	201	12	:	:	PUNCT
cana-4517	201	13	for	for	ADP
cana-4517	201	14	the	the	DET
cana-4517	201	15	line	line	NOUN
cana-4517	201	16	graph	graph	NOUN
cana-4517	201	17	of	of	ADP
cana-4517	201	18	a	a	DET
cana-4517	201	19	double	double	ADJ
cana-4517	201	20	star	star	NOUN
cana-4517	201	21	graph	graph	NOUN
cana-4517	201	22	l(sp	l(sp	PROPN
cana-4517	201	23	,	,	PUNCT
cana-4517	201	24	p	p	NOUN
cana-4517	201	25	)	)	PUNCT
cana-4517	201	26	with	with	ADP
cana-4517	201	27	p	p	PROPN
cana-4517	201	28	>	>	X
cana-4517	201	29	3	3	NUM
cana-4517	201	30	,	,	PUNCT
cana-4517	201	31	𝔓𝔈(l(sp	𝔓𝔈(l(sp	NOUN
cana-4517	201	32	,	,	PUNCT
cana-4517	201	33	p	p	NOUN
cana-4517	201	34	)	)	PUNCT
cana-4517	201	35	)	)	PUNCT
cana-4517	202	1	=	=	SYM
cana-4517	202	2	4p	4p	NOUN
cana-4517	202	3	−	−	NOUN
cana-4517	202	4	5	5	NUM
cana-4517	202	5	proof	proof	NOUN
cana-4517	202	6	:	:	PUNCT
cana-4517	202	7	let	let	VERB
cana-4517	202	8	v(l(sp	v(l(sp	INTJ
cana-4517	202	9	,	,	PUNCT
cana-4517	202	10	p	p	NOUN
cana-4517	202	11	)	)	PUNCT
cana-4517	202	12	)	)	PUNCT
cana-4517	203	1	=	=	PRON
cana-4517	203	2	{	{	PUNCT
cana-4517	203	3	e0	e0	PROPN
cana-4517	203	4	,	,	PUNCT
cana-4517	203	5	e1	e1	PROPN
cana-4517	203	6	,	,	PUNCT
cana-4517	203	7	.	.	PUNCT
cana-4517	203	8	.	.	PUNCT
cana-4517	204	1	.	.	PUNCT
cana-4517	205	1	,	,	PUNCT
cana-4517	205	2	ep−1	ep−1	PROPN
cana-4517	205	3	,	,	PUNCT
cana-4517	205	4	e1′	e1′	ADJ
cana-4517	205	5	,	,	PUNCT
cana-4517	205	6	.	.	PUNCT
cana-4517	205	7	.	.	PUNCT
cana-4517	205	8	.	.	PUNCT
cana-4517	206	1	,	,	PUNCT
cana-4517	206	2	ep−1′	ep−1′	PROPN
cana-4517	206	3	}	}	PUNCT
cana-4517	206	4	,	,	PUNCT
cana-4517	206	5	where	where	SCONJ
cana-4517	206	6	e0	e0	PROPN
cana-4517	206	7	is	be	AUX
cana-4517	206	8	the	the	DET
cana-4517	206	9	bridge	bridge	NOUN
cana-4517	206	10	connecting	connect	VERB
cana-4517	206	11	two	two	NUM
cana-4517	206	12	star	star	NOUN
cana-4517	206	13	graphs	graph	NOUN
cana-4517	206	14	.	.	PUNCT
cana-4517	207	1	then	then	ADV
cana-4517	207	2	the	the	DET
cana-4517	207	3	unique	unique	ADJ
cana-4517	207	4	prd	prd	NOUN
cana-4517	207	5	set	set	NOUN
cana-4517	207	6	of	of	ADP
cana-4517	207	7	l(sp	l(sp	PROPN
cana-4517	207	8	,	,	PUNCT
cana-4517	207	9	p	p	NOUN
cana-4517	207	10	)	)	PUNCT
cana-4517	207	11	is	be	AUX
cana-4517	207	12	a	a	DET
cana-4517	207	13	=	=	X
cana-4517	207	14	{	{	PUNCT
cana-4517	207	15	e0	e0	PROPN
cana-4517	207	16	}	}	PUNCT
cana-4517	207	17	.	.	PUNCT
cana-4517	208	1	therefore	therefore	ADV
cana-4517	208	2	𝔓a(l(sp	𝔓a(l(sp	NUM
cana-4517	208	3	,	,	PUNCT
cana-4517	208	4	p	p	NOUN
cana-4517	208	5	)	)	PUNCT
cana-4517	208	6	)	)	PUNCT
cana-4517	209	1	=	=	PUNCT
cana-4517	209	2	(	(	PUNCT
cana-4517	209	3	1	1	NUM
cana-4517	209	4	j1×p−1	j1×p−1	NOUN
cana-4517	209	5	j1×p−1	j1×p−1	NOUN
cana-4517	209	6	jp−1×1	jp−1×1	PROPN
cana-4517	209	7	jp−1	jp−1	PROPN
cana-4517	209	8	−	−	PROPN
cana-4517	209	9	ip−1	ip−1	PROPN
cana-4517	209	10	op−1	op−1	PROPN
cana-4517	209	11	jp−1×1	jp−1×1	NOUN
cana-4517	209	12	op−1	op−1	VERB
cana-4517	209	13	jp−1	jp−1	PROPN
cana-4517	209	14	−	−	PROPN
cana-4517	209	15	ip−1	ip−1	PROPN
cana-4517	209	16	)	)	PUNCT
cana-4517	209	17	communications	communication	NOUN
cana-4517	209	18	on	on	ADP
cana-4517	209	19	applied	apply	VERB
cana-4517	209	20	nonlinear	nonlinear	ADJ
cana-4517	209	21	analysis	analysis	NOUN
cana-4517	209	22	issn	issn	NOUN
cana-4517	209	23	:	:	PUNCT
cana-4517	209	24	1074	1074	NUM
cana-4517	209	25	-	-	PUNCT
cana-4517	209	26	133x	133x	NUM
cana-4517	209	27	vol	vol	NOUN
cana-4517	209	28	32	32	NUM
cana-4517	210	1	no	no	NOUN
cana-4517	210	2	.	.	PUNCT
cana-4517	211	1	9s	9s	NUM
cana-4517	211	2	(	(	PUNCT
cana-4517	211	3	2025	2025	NUM
cana-4517	211	4	)	)	PUNCT
cana-4517	211	5	2321	2321	NUM
cana-4517	211	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	211	7	where	where	SCONJ
cana-4517	211	8	j	j	PROPN
cana-4517	211	9	,	,	PUNCT
cana-4517	211	10	o	o	NOUN
cana-4517	212	1	and	and	CCONJ
cana-4517	212	2	i	i	PRON
cana-4517	212	3	represents	represent	VERB
cana-4517	212	4	the	the	DET
cana-4517	212	5	matrix	matrix	NOUN
cana-4517	212	6	of	of	ADP
cana-4517	212	7	1	1	NUM
cana-4517	212	8	’s	’s	PART
cana-4517	212	9	,	,	PUNCT
cana-4517	212	10	0	0	NUM
cana-4517	212	11	’s	’s	PART
cana-4517	212	12	and	and	CCONJ
cana-4517	212	13	identity	identity	NOUN
cana-4517	212	14	matrix	matrix	NOUN
cana-4517	212	15	.	.	PUNCT
cana-4517	213	1	and	and	CCONJ
cana-4517	213	2	the	the	DET
cana-4517	213	3	corresponding	corresponding	ADJ
cana-4517	213	4	characteristic	characteristic	ADJ
cana-4517	213	5	polynomial	polynomial	NOUN
cana-4517	213	6	is	be	AUX
cana-4517	213	7	,	,	PUNCT
cana-4517	213	8	φ(l(sp	φ(l(sp	ADP
cana-4517	213	9	,	,	PUNCT
cana-4517	213	10	p	p	NOUN
cana-4517	213	11	)	)	PUNCT
cana-4517	213	12	,	,	PUNCT
cana-4517	213	13	λ	λ	NOUN
cana-4517	213	14	)	)	PUNCT
cana-4517	213	15	=	=	SYM
cana-4517	213	16	|𝔓a(l(sp	|𝔓a(l(sp	NUM
cana-4517	213	17	,	,	PUNCT
cana-4517	213	18	p	p	NOUN
cana-4517	213	19	)	)	PUNCT
cana-4517	213	20	)	)	PUNCT
cana-4517	214	1	−	−	PROPN
cana-4517	215	1	λi)|	λi)|	X
cana-4517	215	2	φ(l(sp	φ(l(sp	ADP
cana-4517	215	3	,	,	PUNCT
cana-4517	215	4	p	p	NOUN
cana-4517	215	5	)	)	PUNCT
cana-4517	215	6	,	,	PUNCT
cana-4517	215	7	λ	λ	X
cana-4517	215	8	)	)	PUNCT
cana-4517	215	9	=	=	PUNCT
cana-4517	215	10	−(λ	−(λ	VERB
cana-4517	215	11	+	+	CCONJ
cana-4517	215	12	1	1	X
cana-4517	215	13	)	)	PUNCT
cana-4517	215	14	2p−3(λ	2p−3(λ	NUM
cana-4517	215	15	−	−	PROPN
cana-4517	216	1	(	(	PUNCT
cana-4517	216	2	p	p	NOUN
cana-4517	216	3	−	−	PROPN
cana-4517	216	4	2	2	NUM
cana-4517	216	5	)	)	PUNCT
cana-4517	216	6	)	)	PUNCT
cana-4517	216	7	(	(	PUNCT
cana-4517	216	8	λ	λ	X
cana-4517	216	9	−	−	PROPN
cana-4517	216	10	p	p	NOUN
cana-4517	216	11	)	)	PUNCT
cana-4517	216	12	hence	hence	ADV
cana-4517	216	13	speca(l(sp	speca(l(sp	PROPN
cana-4517	216	14	,	,	PUNCT
cana-4517	216	15	p	p	NOUN
cana-4517	216	16	)	)	PUNCT
cana-4517	216	17	)	)	PUNCT
cana-4517	217	1	=	=	PRON
cana-4517	217	2	{	{	PUNCT
cana-4517	217	3	1	1	NUM
cana-4517	217	4	2	2	NUM
cana-4517	217	5	−	−	NOUN
cana-4517	217	6	p	p	NOUN
cana-4517	217	7	−p	−p	ADJ
cana-4517	217	8	2p	2p	NUM
cana-4517	217	9	−	−	NOUN
cana-4517	217	10	3	3	NUM
cana-4517	217	11	1	1	NUM
cana-4517	217	12	1	1	NUM
cana-4517	217	13	}	}	PUNCT
cana-4517	217	14	now	now	ADV
cana-4517	217	15	,	,	PUNCT
cana-4517	217	16	𝔓𝔈(l(sp	𝔓𝔈(l(sp	NOUN
cana-4517	217	17	,	,	PUNCT
cana-4517	217	18	p	p	NOUN
cana-4517	217	19	)	)	PUNCT
cana-4517	217	20	)	)	PUNCT
cana-4517	218	1	=	=	SYM
cana-4517	218	2	(	(	PUNCT
cana-4517	218	3	2p	2p	NUM
cana-4517	218	4	−	−	PROPN
cana-4517	218	5	3)|−1|	3)|−1|	NUM
cana-4517	218	6	+	+	NUM
cana-4517	218	7	|2	|2	NUM
cana-4517	218	8	−	−	NUM
cana-4517	218	9	p|	p|	NOUN
cana-4517	218	10	+	+	CCONJ
cana-4517	218	11	|−p|	|−p|	ADJ
cana-4517	218	12	⟹𝔓𝔈(l(sp	⟹𝔓𝔈(l(sp	PROPN
cana-4517	218	13	,	,	PUNCT
cana-4517	218	14	p	p	NOUN
cana-4517	218	15	)	)	PUNCT
cana-4517	218	16	)	)	PUNCT
cana-4517	219	1	=	=	SYM
cana-4517	219	2	4p	4p	NOUN
cana-4517	219	3	−	−	NUM
cana-4517	219	4	5	5	NUM
cana-4517	219	5	5	5	NUM
cana-4517	219	6	.	.	PUNCT
cana-4517	220	1	characteristics	characteristic	NOUN
cana-4517	220	2	of	of	ADP
cana-4517	220	3	prd	prd	PROPN
cana-4517	220	4	spectrum	spectrum	NOUN
cana-4517	220	5	5.1	5.1	NUM
cana-4517	220	6	theorem	theorem	NOUN
cana-4517	220	7	:	:	PUNCT
cana-4517	220	8	consider	consider	VERB
cana-4517	220	9	graph	graph	NOUN
cana-4517	220	10	g.	g.	NOUN
cana-4517	220	11	if	if	SCONJ
cana-4517	220	12	λ1	λ1	PROPN
cana-4517	220	13	,	,	PUNCT
cana-4517	220	14	λ2	λ2	NOUN
cana-4517	220	15	,	,	PUNCT
cana-4517	220	16	…	…	PUNCT
cana-4517	220	17	λp	λp	X
cana-4517	220	18	are	be	AUX
cana-4517	220	19	the	the	DET
cana-4517	220	20	𝔓-spectrum	𝔓-spectrum	PROPN
cana-4517	220	21	of	of	ADP
cana-4517	220	22	𝔓a(g	𝔓a(g	PROPN
cana-4517	220	23	)	)	PUNCT
cana-4517	220	24	,	,	PUNCT
cana-4517	220	25	then	then	ADV
cana-4517	220	26	the	the	DET
cana-4517	220	27	following	follow	VERB
cana-4517	220	28	condition	condition	NOUN
cana-4517	220	29	holds	hold	VERB
cana-4517	220	30	.	.	PUNCT
cana-4517	221	1	(	(	PUNCT
cana-4517	221	2	i	i	NOUN
cana-4517	221	3	)	)	PUNCT
cana-4517	221	4	∑	∑	PUNCT
cana-4517	221	5	λi	λi	ADP
cana-4517	221	6	p	p	X
cana-4517	221	7	i=1	i=1	PROPN
cana-4517	221	8	=	=	SYM
cana-4517	221	9	𝔭.	𝔭.	NOUN
cana-4517	221	10	(	(	PUNCT
cana-4517	221	11	ii	ii	PROPN
cana-4517	221	12	)	)	PUNCT
cana-4517	221	13	∑	∑	PUNCT
cana-4517	221	14	λi	λi	ADP
cana-4517	221	15	2p	2p	NUM
cana-4517	221	16	i=1	i=1	ADP
cana-4517	221	17	=	=	X
cana-4517	221	18	2q	2q	NOUN
cana-4517	222	1	+	+	CCONJ
cana-4517	222	2	𝔭.	𝔭.	NOUN
cana-4517	222	3	proof	proof	NOUN
cana-4517	222	4	:	:	PUNCT
cana-4517	222	5	as	as	ADP
cana-4517	222	6	sum	sum	NOUN
cana-4517	222	7	of	of	ADP
cana-4517	222	8	spectrum	spectrum	NOUN
cana-4517	222	9	of	of	ADP
cana-4517	222	10	𝔓a(g	𝔓a(g	NOUN
cana-4517	222	11	)	)	PUNCT
cana-4517	222	12	is	be	AUX
cana-4517	222	13	same	same	ADJ
cana-4517	222	14	as	as	ADP
cana-4517	222	15	its	its	PRON
cana-4517	222	16	trace	trace	NOUN
cana-4517	222	17	,	,	PUNCT
cana-4517	222	18	∑λi	∑λi	PUNCT
cana-4517	223	1	p	p	X
cana-4517	223	2	i=1	i=1	X
cana-4517	223	3	=	=	NOUN
cana-4517	223	4	∑𝔭ii	∑𝔭ii	NOUN
cana-4517	223	5	p	p	X
cana-4517	223	6	i=1	i=1	X
cana-4517	223	7	=	=	PUNCT
cana-4517	223	8	|a|	|a|	PROPN
cana-4517	223	9	=	=	SYM
cana-4517	223	10	𝔭	𝔭	PROPN
cana-4517	223	11	(	(	PUNCT
cana-4517	223	12	i	i	NOUN
cana-4517	223	13	)	)	PUNCT
cana-4517	223	14	as	as	ADP
cana-4517	223	15	summation	summation	NOUN
cana-4517	223	16	of	of	ADP
cana-4517	223	17	squares	square	NOUN
cana-4517	223	18	of	of	ADP
cana-4517	223	19	the	the	DET
cana-4517	223	20	spectrum	spectrum	NOUN
cana-4517	223	21	of	of	ADP
cana-4517	223	22	𝔓a(g	𝔓a(g	PROPN
cana-4517	223	23	)	)	PUNCT
cana-4517	223	24	is	be	AUX
cana-4517	223	25	the	the	DET
cana-4517	223	26	trace	trace	NOUN
cana-4517	223	27	of	of	ADP
cana-4517	223	28	[	[	X
cana-4517	223	29	𝔓a(g	𝔓a(g	NOUN
cana-4517	223	30	)	)	PUNCT
cana-4517	223	31	2	2	NUM
cana-4517	223	32	]	]	PUNCT
cana-4517	223	33	,	,	PUNCT
cana-4517	223	34	∑λi	∑λi	ADV
cana-4517	223	35	2	2	NUM
cana-4517	223	36	p	p	NOUN
cana-4517	223	37	i=1	i=1	X
cana-4517	224	1	=	=	NOUN
cana-4517	224	2	∑∑𝔭ij	∑∑𝔭ij	NOUN
cana-4517	224	3	𝔭ji	𝔭ji	NOUN
cana-4517	225	1	p	p	X
cana-4517	225	2	i=1	i=1	PROPN
cana-4517	226	1	p	p	X
cana-4517	226	2	i=1	i=1	PROPN
cana-4517	226	3	=	=	SYM
cana-4517	226	4	∑(𝔭ii	∑(𝔭ii	PROPN
cana-4517	226	5	)	)	PUNCT
cana-4517	226	6	2	2	NUM
cana-4517	226	7	p	p	NOUN
cana-4517	226	8	i=1	i=1	PROPN
cana-4517	226	9	+	+	NOUN
cana-4517	226	10	∑𝔭ij	∑𝔭ij	NUM
cana-4517	226	11	𝔭ji	𝔭ji	PROPN
cana-4517	226	12	i≠j	i≠j	PROPN
cana-4517	226	13	=	=	SYM
cana-4517	226	14	∑(𝔭ii	∑(𝔭ii	PROPN
cana-4517	226	15	)	)	PUNCT
cana-4517	227	1	2	2	NUM
cana-4517	227	2	p	p	NOUN
cana-4517	227	3	i=1	i=1	X
cana-4517	228	1	+	+	CCONJ
cana-4517	228	2	2∑(𝔭ij	2∑(𝔭ij	NUM
cana-4517	228	3	)	)	PUNCT
cana-4517	228	4	2	2	NUM
cana-4517	229	1	i	i	PRON
cana-4517	229	2	<	<	X
cana-4517	229	3	j	j	X
cana-4517	229	4	=	=	SYM
cana-4517	229	5	𝔭	𝔭	PROPN
cana-4517	229	6	+	+	X
cana-4517	229	7	2q	2q	NUM
cana-4517	229	8	∑	∑	PUNCT
cana-4517	229	9	λi	λi	ADP
cana-4517	229	10	2p	2p	NUM
cana-4517	229	11	i=1	i=1	ADP
cana-4517	230	1	=	=	X
cana-4517	230	2	2q	2q	NOUN
cana-4517	231	1	+	+	CCONJ
cana-4517	231	2	𝔭.	𝔭.	NOUN
cana-4517	231	3	6	6	NUM
cana-4517	231	4	.	.	PUNCT
cana-4517	231	5	bounds	bound	NOUN
cana-4517	231	6	on	on	ADP
cana-4517	231	7	prd	prd	PROPN
cana-4517	231	8	energy	energy	NOUN
cana-4517	231	9	6.1	6.1	NUM
cana-4517	231	10	theorem	theorem	NOUN
cana-4517	231	11	:	:	PUNCT
cana-4517	231	12	let	let	VERB
cana-4517	231	13	g	g	PRON
cana-4517	231	14	be	be	AUX
cana-4517	231	15	a	a	DET
cana-4517	231	16	(	(	PUNCT
cana-4517	231	17	p	p	X
cana-4517	231	18	,	,	PUNCT
cana-4517	231	19	q	q	ADJ
cana-4517	231	20	)	)	PUNCT
cana-4517	231	21	simple	simple	ADJ
cana-4517	231	22	graph	graph	NOUN
cana-4517	231	23	.	.	PUNCT
cana-4517	232	1	let	let	VERB
cana-4517	232	2	∆=	∆=	VERB
cana-4517	232	3	|det	|det	NOUN
cana-4517	232	4	gs(g)|	gs(g)|	NOUN
cana-4517	232	5	.	.	PUNCT
cana-4517	233	1	then	then	ADV
cana-4517	233	2	√(2q+	√(2q+	VERB
cana-4517	233	3	𝔭	𝔭	NOUN
cana-4517	233	4	)	)	PUNCT
cana-4517	234	1	+	+	CCONJ
cana-4517	234	2	p(p	p(p	ADP
cana-4517	234	3	−	−	PROPN
cana-4517	234	4	1)∆	1)∆	NUM
cana-4517	234	5	(	(	PUNCT
cana-4517	234	6	2	2	NUM
cana-4517	234	7	p⁄	p⁄	PROPN
cana-4517	234	8	)	)	PUNCT
cana-4517	234	9	≤	≤	NOUN
cana-4517	234	10	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	234	11	)	)	PUNCT
cana-4517	234	12	≤	≤	PUNCT
cana-4517	235	1	√p(2q	√p(2q	NUM
cana-4517	235	2	+	+	CCONJ
cana-4517	235	3	𝔭	𝔭	X
cana-4517	235	4	)	)	PUNCT
cana-4517	235	5	where	where	SCONJ
cana-4517	235	6	𝔭	𝔭	NOUN
cana-4517	235	7	is	be	AUX
cana-4517	235	8	the	the	DET
cana-4517	235	9	prd	prd	ADJ
cana-4517	235	10	number	number	NOUN
cana-4517	235	11	of	of	ADP
cana-4517	235	12	g.	g.	PROPN
cana-4517	235	13	proof	proof	NOUN
cana-4517	235	14	:	:	PUNCT
cana-4517	235	15	consider	consider	VERB
cana-4517	235	16	cauchy	cauchy	PROPN
cana-4517	235	17	schwarz	schwarz	PROPN
cana-4517	235	18	inequality	inequality	PROPN
cana-4517	235	19	:	:	PUNCT
cana-4517	235	20	(	(	PUNCT
cana-4517	236	1	∑aibi	∑aibi	PROPN
cana-4517	236	2	p	p	X
cana-4517	236	3	i=1	i=1	PROPN
cana-4517	236	4	)	)	PUNCT
cana-4517	236	5	2	2	NUM
cana-4517	236	6	≤	≤	NOUN
cana-4517	236	7	(	(	PUNCT
cana-4517	236	8	∑ai	∑ai	NUM
cana-4517	236	9	2	2	NUM
cana-4517	236	10	p	p	NOUN
cana-4517	236	11	i=1	i=1	PROPN
cana-4517	236	12	)	)	PUNCT
cana-4517	236	13	(	(	PUNCT
cana-4517	236	14	∑bi	∑bi	NOUN
cana-4517	236	15	2	2	NUM
cana-4517	236	16	p	p	NOUN
cana-4517	236	17	i=1	i=1	X
cana-4517	236	18	)	)	PUNCT
cana-4517	236	19	take	take	VERB
cana-4517	236	20	ai	ai	NOUN
cana-4517	236	21	=	=	NOUN
cana-4517	236	22	1	1	NUM
cana-4517	236	23	,	,	PUNCT
cana-4517	236	24	bi	bi	NOUN
cana-4517	236	25	=	=	PROPN
cana-4517	236	26	|λi|	|λi|	PROPN
cana-4517	236	27	,	,	PUNCT
cana-4517	236	28	gives	give	VERB
cana-4517	236	29	(	(	PUNCT
cana-4517	236	30	∑|λi|	∑|λi|	PROPN
cana-4517	236	31	p	p	PROPN
cana-4517	236	32	i=1	i=1	PROPN
cana-4517	236	33	)	)	PUNCT
cana-4517	236	34	2	2	NUM
cana-4517	236	35	≤	≤	NOUN
cana-4517	236	36	(	(	PUNCT
cana-4517	236	37	∑1	∑1	NOUN
cana-4517	236	38	p	p	NOUN
cana-4517	236	39	i=1	i=1	PROPN
cana-4517	236	40	)	)	PUNCT
cana-4517	236	41	(	(	PUNCT
cana-4517	236	42	∑λi	∑λi	ADP
cana-4517	237	1	2	2	NUM
cana-4517	237	2	p	p	NOUN
cana-4517	237	3	i=1	i=1	PROPN
cana-4517	237	4	)	)	PUNCT
cana-4517	237	5	⇒	⇒	NOUN
cana-4517	237	6	[	[	X
cana-4517	237	7	𝔓𝔈(g	𝔓𝔈(g	X
cana-4517	237	8	)	)	PUNCT
cana-4517	237	9	]	]	PUNCT
cana-4517	237	10	2	2	X
cana-4517	237	11	≤	≤	NOUN
cana-4517	237	12	p	p	NOUN
cana-4517	237	13	∑λi	∑λi	ADJ
cana-4517	237	14	2	2	NUM
cana-4517	237	15	p	p	NOUN
cana-4517	237	16	i=1	i=1	PROPN
cana-4517	237	17	communications	communication	NOUN
cana-4517	237	18	on	on	ADP
cana-4517	237	19	applied	apply	VERB
cana-4517	237	20	nonlinear	nonlinear	ADJ
cana-4517	237	21	analysis	analysis	NOUN
cana-4517	237	22	issn	issn	NOUN
cana-4517	237	23	:	:	PUNCT
cana-4517	237	24	1074	1074	NUM
cana-4517	237	25	-	-	PUNCT
cana-4517	237	26	133x	133x	NUM
cana-4517	237	27	vol	vol	NOUN
cana-4517	237	28	32	32	NUM
cana-4517	238	1	no	no	NOUN
cana-4517	238	2	.	.	PUNCT
cana-4517	239	1	9s	9s	NUM
cana-4517	239	2	(	(	PUNCT
cana-4517	239	3	2025	2025	NUM
cana-4517	239	4	)	)	PUNCT
cana-4517	239	5	2322	2322	NUM
cana-4517	239	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	240	1	=	=	PUNCT
cana-4517	240	2	p	p	X
cana-4517	240	3	(	(	PUNCT
cana-4517	240	4	2q	2q	NOUN
cana-4517	240	5	+	+	CCONJ
cana-4517	240	6	𝔭	𝔭	X
cana-4517	240	7	)	)	PUNCT
cana-4517	240	8	⇒	⇒	NOUN
cana-4517	240	9	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	240	10	)	)	PUNCT
cana-4517	240	11	≤	≤	NOUN
cana-4517	240	12	√p	√p	NOUN
cana-4517	240	13	(	(	PUNCT
cana-4517	240	14	2q	2q	X
cana-4517	240	15	+	+	CCONJ
cana-4517	240	16	𝔭	𝔭	X
cana-4517	240	17	)	)	PUNCT
cana-4517	240	18	…	…	PUNCT
cana-4517	240	19	……	……	NOUN
cana-4517	240	20	(	(	PUNCT
cana-4517	240	21	i	i	NOUN
cana-4517	240	22	)	)	PUNCT
cana-4517	240	23	consider	consider	VERB
cana-4517	240	24	[	[	X
cana-4517	240	25	𝔓𝔈(g	𝔓𝔈(g	X
cana-4517	240	26	)	)	PUNCT
cana-4517	240	27	]	]	PUNCT
cana-4517	240	28	2	2	X
cana-4517	240	29	=	=	SYM
cana-4517	240	30	(	(	PUNCT
cana-4517	240	31	∑	∑	PART
cana-4517	240	32	|λi|	|λi|	NOUN
cana-4517	240	33	2p	2p	NOUN
cana-4517	240	34	i=1	i=1	X
cana-4517	240	35	)	)	PUNCT
cana-4517	241	1	=	=	PUNCT
cana-4517	241	2	(	(	PUNCT
cana-4517	241	3	∑|λi|	∑|λi|	NOUN
cana-4517	241	4	2	2	NUM
cana-4517	241	5	p	p	NOUN
cana-4517	241	6	i=1	i=1	PROPN
cana-4517	241	7	)	)	PUNCT
cana-4517	242	1	+	+	ADV
cana-4517	242	2	∑|λi||λj|	∑|λi||λj|	X
cana-4517	242	3	i≠j	i≠j	NOUN
cana-4517	242	4	[	[	X
cana-4517	242	5	𝔓𝔈(g	𝔓𝔈(g	X
cana-4517	242	6	)	)	PUNCT
cana-4517	242	7	]	]	PUNCT
cana-4517	242	8	2	2	X
cana-4517	242	9	=	=	SYM
cana-4517	242	10	(	(	PUNCT
cana-4517	242	11	2q	2q	NOUN
cana-4517	242	12	+	+	CCONJ
cana-4517	242	13	𝔭	𝔭	X
cana-4517	242	14	)	)	PUNCT
cana-4517	243	1	+	+	NOUN
cana-4517	243	2	∑|λi||λj|	∑|λi||λj|	X
cana-4517	243	3	i≠j	i≠j	NOUN
cana-4517	243	4	…	…	PUNCT
cana-4517	243	5	……	……	PRON
cana-4517	243	6	(	(	PUNCT
cana-4517	243	7	ii	ii	NOUN
cana-4517	243	8	)	)	PUNCT
cana-4517	243	9	where	where	SCONJ
cana-4517	243	10	2q	2q	NOUN
cana-4517	243	11	+	+	CCONJ
cana-4517	243	12	𝔭	𝔭	X
cana-4517	243	13	=	=	SYM
cana-4517	243	14	∑	∑	PUNCT
cana-4517	243	15	|λi|	|λi|	NOUN
cana-4517	243	16	2p	2p	NOUN
cana-4517	243	17	i=1	i=1	PRON
cana-4517	243	18	.	.	PUNCT
cana-4517	244	1	the	the	DET
cana-4517	244	2	geometric	geometric	ADJ
cana-4517	244	3	mean	mean	NOUN
cana-4517	244	4	can	can	AUX
cana-4517	244	5	not	not	PART
cana-4517	244	6	exceed	exceed	VERB
cana-4517	244	7	the	the	DET
cana-4517	244	8	arithmetic	arithmetic	ADJ
cana-4517	244	9	mean	mean	NOUN
cana-4517	244	10	,	,	PUNCT
cana-4517	244	11	so	so	ADV
cana-4517	244	12	1	1	NUM
cana-4517	244	13	p(p	p(p	ADV
cana-4517	244	14	−	−	NOUN
cana-4517	244	15	1	1	NUM
cana-4517	244	16	)	)	PUNCT
cana-4517	244	17	∑|λi||λj|	∑|λi||λj|	PUNCT
cana-4517	244	18	i≠j	i≠j	X
cana-4517	244	19	≥	≥	X
cana-4517	245	1	[	[	X
cana-4517	245	2	∏|λi||λj|	∏|λi||λj|	X
cana-4517	245	3	i≠j	i≠j	NOUN
cana-4517	245	4	]	]	PUNCT
cana-4517	245	5	1	1	NUM
cana-4517	245	6	p(p−1	p(p−1	NOUN
cana-4517	245	7	)	)	PUNCT
cana-4517	245	8	=	=	PUNCT
cana-4517	246	1	[	[	X
cana-4517	246	2	∏|λi|	∏|λi|	PROPN
cana-4517	246	3	2(p−1	2(p−1	NOUN
cana-4517	246	4	)	)	PUNCT
cana-4517	246	5	p	p	X
cana-4517	246	6	i=1	i=1	X
cana-4517	246	7	]	]	PUNCT
cana-4517	246	8	1	1	NUM
cana-4517	246	9	p(p−1	p(p−1	NOUN
cana-4517	246	10	)	)	PUNCT
cana-4517	246	11	=	=	PUNCT
cana-4517	247	1	[	[	X
cana-4517	247	2	∏|λi|	∏|λi|	X
cana-4517	247	3	p	p	X
cana-4517	247	4	i=1	i=1	X
cana-4517	247	5	]	]	PUNCT
cana-4517	247	6	2	2	NUM
cana-4517	247	7	p⁄	p⁄	NOUN
cana-4517	247	8	=	=	PUNCT
cana-4517	248	1	|∏λi	|∏λi	NOUN
cana-4517	249	1	p	p	X
cana-4517	249	2	i=1	i=1	X
cana-4517	250	1	|	|	ADV
cana-4517	250	2	2	2	NUM
cana-4517	250	3	p⁄	p⁄	NOUN
cana-4517	250	4	=	=	SYM
cana-4517	250	5	|det(𝔓𝔈(g	|det(𝔓𝔈(g	NUM
cana-4517	250	6	)	)	PUNCT
cana-4517	250	7	)	)	PUNCT
cana-4517	251	1	|	|	ADV
cana-4517	251	2	2	2	NUM
cana-4517	251	3	p⁄	p⁄	X
cana-4517	251	4	=	=	SYM
cana-4517	251	5	∆	∆	X
cana-4517	251	6	(	(	PUNCT
cana-4517	251	7	2	2	NUM
cana-4517	251	8	p⁄	p⁄	PROPN
cana-4517	251	9	)	)	PUNCT
cana-4517	251	10	i.e	i.e	PROPN
cana-4517	251	11	,	,	PUNCT
cana-4517	251	12	1	1	NUM
cana-4517	251	13	p(p−1	p(p−1	PROPN
cana-4517	251	14	)	)	PUNCT
cana-4517	251	15	∑	∑	ADP
cana-4517	251	16	|λi||λj|i≠j	|λi||λj|i≠j	PROPN
cana-4517	251	17	≥	≥	NUM
cana-4517	251	18	∆	∆	PROPN
cana-4517	251	19	(	(	PUNCT
cana-4517	251	20	2	2	NUM
cana-4517	251	21	p⁄	p⁄	PROPN
cana-4517	251	22	)	)	PUNCT
cana-4517	251	23	⇒∑|λi||λj|	⇒∑|λi||λj|	PROPN
cana-4517	251	24	i≠j	i≠j	NOUN
cana-4517	251	25	≥	≥	NOUN
cana-4517	251	26	p(p	p(p	ADV
cana-4517	251	27	−	−	NUM
cana-4517	251	28	1	1	NUM
cana-4517	251	29	)	)	PUNCT
cana-4517	251	30	∆	∆	NOUN
cana-4517	251	31	(	(	PUNCT
cana-4517	251	32	2	2	NUM
cana-4517	251	33	p⁄	p⁄	NOUN
cana-4517	251	34	)	)	PUNCT
cana-4517	251	35	…	…	PUNCT
cana-4517	251	36	……	……	X
cana-4517	251	37	(	(	PUNCT
cana-4517	251	38	iii	iii	NOUN
cana-4517	251	39	)	)	PUNCT
cana-4517	251	40	put	put	NOUN
cana-4517	251	41	(	(	PUNCT
cana-4517	251	42	iii	iii	NOUN
cana-4517	251	43	)	)	PUNCT
cana-4517	251	44	in	in	ADP
cana-4517	251	45	(	(	PUNCT
cana-4517	251	46	ii	ii	NOUN
cana-4517	251	47	)	)	PUNCT
cana-4517	252	1	[	[	X
cana-4517	252	2	𝔓𝔈(g	𝔓𝔈(g	X
cana-4517	252	3	)	)	PUNCT
cana-4517	252	4	]	]	PUNCT
cana-4517	252	5	2	2	NUM
cana-4517	252	6	≥	≥	NOUN
cana-4517	252	7	(	(	PUNCT
cana-4517	252	8	2q	2q	NOUN
cana-4517	252	9	+	+	CCONJ
cana-4517	252	10	𝔭	𝔭	X
cana-4517	252	11	)	)	PUNCT
cana-4517	252	12	+	+	CCONJ
cana-4517	252	13	p(p	p(p	ADV
cana-4517	252	14	−	−	NOUN
cana-4517	252	15	1	1	NUM
cana-4517	252	16	)	)	PUNCT
cana-4517	252	17	∆	∆	X
cana-4517	252	18	2	2	NUM
cana-4517	252	19	p⁄	p⁄	NOUN
cana-4517	252	20	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	252	21	)	)	PUNCT
cana-4517	252	22	≥	≥	NOUN
cana-4517	252	23	√(2q	√(2q	NOUN
cana-4517	252	24	+	+	CCONJ
cana-4517	252	25	𝔭	𝔭	X
cana-4517	252	26	)	)	PUNCT
cana-4517	252	27	+	+	CCONJ
cana-4517	252	28	p(p	p(p	ADV
cana-4517	252	29	−	−	NOUN
cana-4517	252	30	1	1	NUM
cana-4517	252	31	)	)	PUNCT
cana-4517	252	32	∆	∆	X
cana-4517	252	33	2	2	NUM
cana-4517	252	34	p⁄	p⁄	NOUN
cana-4517	252	35	…	…	PUNCT
cana-4517	252	36	……	……	X
cana-4517	252	37	(	(	PUNCT
cana-4517	252	38	iv	iv	X
cana-4517	252	39	)	)	PUNCT
cana-4517	252	40	from	from	ADP
cana-4517	252	41	(	(	PUNCT
cana-4517	252	42	i	i	NOUN
cana-4517	252	43	)	)	PUNCT
cana-4517	252	44	and	and	CCONJ
cana-4517	252	45	(	(	PUNCT
cana-4517	252	46	iv	iv	X
cana-4517	252	47	)	)	PUNCT
cana-4517	252	48	√(2q+	√(2q+	NOUN
cana-4517	252	49	𝔭	𝔭	NOUN
cana-4517	252	50	)	)	PUNCT
cana-4517	253	1	+	+	CCONJ
cana-4517	253	2	p(p	p(p	ADV
cana-4517	253	3	−	−	NUM
cana-4517	253	4	1)∆	1)∆	NUM
cana-4517	253	5	2	2	NUM
cana-4517	253	6	p⁄	p⁄	NOUN
cana-4517	253	7	≤	≤	NOUN
cana-4517	253	8	𝔓𝔈(g	𝔓𝔈(g	X
cana-4517	253	9	)	)	PUNCT
cana-4517	253	10	≤	≤	PUNCT
cana-4517	254	1	√p(2q	√p(2q	NUM
cana-4517	254	2	+	+	CCONJ
cana-4517	254	3	𝔭	𝔭	X
cana-4517	254	4	)	)	PUNCT
cana-4517	254	5	6.2	6.2	NUM
cana-4517	254	6	theorem	theorem	VERB
cana-4517	254	7	:	:	PUNCT
cana-4517	254	8	for	for	ADP
cana-4517	254	9	any	any	DET
cana-4517	254	10	graph	graph	NOUN
cana-4517	254	11	g√2q	g√2q	NOUN
cana-4517	254	12	+	+	CCONJ
cana-4517	254	13	𝔭	𝔭	VERB
cana-4517	254	14	≤	≤	NUM
cana-4517	254	15	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	254	16	)	)	PUNCT
cana-4517	254	17	≤	≤	PUNCT
cana-4517	255	1	√p(2q	√p(2q	NUM
cana-4517	255	2	+	+	CCONJ
cana-4517	255	3	𝔭	𝔭	NOUN
cana-4517	255	4	)	)	PUNCT
cana-4517	255	5	.	.	PUNCT
cana-4517	256	1	proof	proof	NOUN
cana-4517	256	2	:	:	PUNCT
cana-4517	256	3	consider	consider	VERB
cana-4517	256	4	cauchy	cauchy	PROPN
cana-4517	256	5	schwarz	schwarz	PROPN
cana-4517	256	6	inequality	inequality	PROPN
cana-4517	256	7	,	,	PUNCT
cana-4517	256	8	(	(	PUNCT
cana-4517	256	9	∑aibi	∑aibi	PROPN
cana-4517	256	10	p	p	NOUN
cana-4517	256	11	i=1	i=1	PROPN
cana-4517	256	12	)	)	PUNCT
cana-4517	256	13	2	2	NUM
cana-4517	256	14	≤	≤	NOUN
cana-4517	256	15	(	(	PUNCT
cana-4517	256	16	∑ai	∑ai	NUM
cana-4517	256	17	2	2	NUM
cana-4517	256	18	p	p	NOUN
cana-4517	256	19	i=1	i=1	PROPN
cana-4517	256	20	)	)	PUNCT
cana-4517	256	21	(	(	PUNCT
cana-4517	256	22	∑bi	∑bi	NOUN
cana-4517	256	23	2	2	NUM
cana-4517	256	24	p	p	NOUN
cana-4517	256	25	i=1	i=1	X
cana-4517	256	26	)	)	PUNCT
cana-4517	256	27	take	take	VERB
cana-4517	256	28	ai	ai	NOUN
cana-4517	256	29	=	=	NOUN
cana-4517	256	30	1	1	NUM
cana-4517	256	31	,	,	PUNCT
cana-4517	256	32	bi	bi	NOUN
cana-4517	256	33	=	=	PROPN
cana-4517	256	34	|λi|	|λi|	PROPN
cana-4517	256	35	,	,	PUNCT
cana-4517	256	36	gives	give	VERB
cana-4517	256	37	(	(	PUNCT
cana-4517	256	38	∑|λi|	∑|λi|	PROPN
cana-4517	256	39	p	p	PROPN
cana-4517	256	40	i=1	i=1	PROPN
cana-4517	256	41	)	)	PUNCT
cana-4517	256	42	2	2	NUM
cana-4517	256	43	≤	≤	NOUN
cana-4517	256	44	(	(	PUNCT
cana-4517	256	45	∑1	∑1	NOUN
cana-4517	256	46	p	p	NOUN
cana-4517	256	47	i=1	i=1	PROPN
cana-4517	256	48	)	)	PUNCT
cana-4517	257	1	(	(	PUNCT
cana-4517	257	2	∑|λi|	∑|λi|	NOUN
cana-4517	257	3	2	2	NUM
cana-4517	257	4	p	p	NOUN
cana-4517	257	5	i=1	i=1	PROPN
cana-4517	257	6	)	)	PUNCT
cana-4517	257	7	⇒	⇒	NOUN
cana-4517	257	8	[	[	PUNCT
cana-4517	257	9	𝔓𝔈(g)]2	𝔓𝔈(g)]2	NOUN
cana-4517	257	10	≤	≤	NOUN
cana-4517	257	11	p∑λi	p∑λi	VERB
cana-4517	257	12	2	2	NUM
cana-4517	257	13	p	p	NOUN
cana-4517	257	14	i=1	i=1	PROPN
cana-4517	257	15	communications	communication	NOUN
cana-4517	257	16	on	on	ADP
cana-4517	257	17	applied	apply	VERB
cana-4517	257	18	nonlinear	nonlinear	ADJ
cana-4517	257	19	analysis	analysis	NOUN
cana-4517	257	20	issn	issn	NOUN
cana-4517	257	21	:	:	PUNCT
cana-4517	257	22	1074	1074	NUM
cana-4517	257	23	-	-	PUNCT
cana-4517	257	24	133x	133x	NUM
cana-4517	257	25	vol	vol	NOUN
cana-4517	257	26	32	32	NUM
cana-4517	258	1	no	no	NOUN
cana-4517	258	2	.	.	PUNCT
cana-4517	259	1	9s	9s	NUM
cana-4517	259	2	(	(	PUNCT
cana-4517	259	3	2025	2025	NUM
cana-4517	259	4	)	)	PUNCT
cana-4517	259	5	2323	2323	NUM
cana-4517	259	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4517	259	7	=	=	SYM
cana-4517	260	1	p(2q	p(2q	X
cana-4517	260	2	+	+	CCONJ
cana-4517	260	3	𝔭	𝔭	X
cana-4517	260	4	)	)	PUNCT
cana-4517	260	5	⇒	⇒	NOUN
cana-4517	260	6	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	260	7	)	)	PUNCT
cana-4517	260	8	≤	≤	NOUN
cana-4517	260	9	√p(2q+	√p(2q+	NUM
cana-4517	260	10	𝔭	𝔭	NOUN
cana-4517	260	11	)	)	PUNCT
cana-4517	260	12	…	…	PUNCT
cana-4517	260	13	……	……	NOUN
cana-4517	260	14	(	(	PUNCT
cana-4517	260	15	i	i	NOUN
cana-4517	260	16	)	)	PUNCT
cana-4517	260	17	consider	consider	VERB
cana-4517	260	18	(	(	PUNCT
cana-4517	260	19	𝔓𝔈(g	𝔓𝔈(g	ADJ
cana-4517	260	20	)	)	PUNCT
cana-4517	260	21	)	)	PUNCT
cana-4517	261	1	2	2	NUM
cana-4517	261	2	=	=	SYM
cana-4517	261	3	(	(	PUNCT
cana-4517	261	4	∑	∑	PROPN
cana-4517	261	5	|λi|	|λi|	X
cana-4517	261	6	p	p	X
cana-4517	261	7	i=1	i=1	PROPN
cana-4517	261	8	)	)	PUNCT
cana-4517	261	9	2	2	NUM
cana-4517	261	10	≥∑|λi|	≥∑|λi|	NOUN
cana-4517	261	11	2	2	NUM
cana-4517	261	12	p	p	NOUN
cana-4517	261	13	i=1	i=1	X
cana-4517	261	14	=	=	PUNCT
cana-4517	261	15	2q	2q	NOUN
cana-4517	262	1	+	+	CCONJ
cana-4517	262	2	𝔭	𝔭	X
cana-4517	262	3	⇒	⇒	NOUN
cana-4517	262	4	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	262	5	)	)	PUNCT
cana-4517	262	6	≥	≥	NOUN
cana-4517	262	7	√2q	√2q	NOUN
cana-4517	262	8	+	+	CCONJ
cana-4517	262	9	𝔭	𝔭	X
cana-4517	262	10	…	…	PUNCT
cana-4517	262	11	……	……	X
cana-4517	262	12	(	(	PUNCT
cana-4517	262	13	ii	ii	NOUN
cana-4517	262	14	)	)	PUNCT
cana-4517	262	15	from	from	ADP
cana-4517	262	16	(	(	PUNCT
cana-4517	262	17	i	i	NOUN
cana-4517	262	18	)	)	PUNCT
cana-4517	262	19	and	and	CCONJ
cana-4517	262	20	(	(	PUNCT
cana-4517	262	21	ii	ii	NOUN
cana-4517	262	22	)	)	PUNCT
cana-4517	262	23	,	,	PUNCT
cana-4517	262	24	√2q	√2q	NOUN
cana-4517	262	25	+	+	CCONJ
cana-4517	262	26	𝔭	𝔭	VERB
cana-4517	262	27	≤	≤	NUM
cana-4517	262	28	𝔓𝔈(g	𝔓𝔈(g	NUM
cana-4517	262	29	)	)	PUNCT
cana-4517	262	30	≤	≤	PUNCT
cana-4517	263	1	√p(2q	√p(2q	NUM
cana-4517	263	2	+	+	CCONJ
cana-4517	263	3	𝔭	𝔭	X
cana-4517	263	4	)	)	PUNCT
cana-4517	263	5	7	7	NUM
cana-4517	263	6	.	.	PUNCT
cana-4517	264	1	open	open	ADJ
cana-4517	264	2	problems	problem	NOUN
cana-4517	264	3	1	1	NUM
cana-4517	264	4	.	.	PUNCT
cana-4517	264	5	determine	determine	VERB
cana-4517	264	6	the	the	DET
cana-4517	264	7	class	class	NOUN
cana-4517	264	8	of	of	ADP
cana-4517	264	9	graphs	graph	NOUN
cana-4517	264	10	whose	whose	DET
cana-4517	264	11	pe	pe	NOUN
cana-4517	264	12	is	be	AUX
cana-4517	264	13	same	same	ADJ
cana-4517	264	14	as	as	ADP
cana-4517	264	15	to	to	ADP
cana-4517	264	16	number	number	NOUN
cana-4517	264	17	of	of	ADP
cana-4517	264	18	vertices	vertex	NOUN
cana-4517	264	19	.	.	PUNCT
cana-4517	265	1	2	2	X
cana-4517	265	2	.	.	X
cana-4517	265	3	determine	determine	VERB
cana-4517	265	4	the	the	DET
cana-4517	265	5	class	class	NOUN
cana-4517	265	6	of	of	ADP
cana-4517	265	7	graphs	graph	NOUN
cana-4517	265	8	whose	whose	DET
cana-4517	265	9	pe	pe	NOUN
cana-4517	265	10	is	be	AUX
cana-4517	265	11	same	same	ADJ
cana-4517	265	12	as	as	ADP
cana-4517	265	13	usual	usual	ADJ
cana-4517	265	14	energy	energy	NOUN
cana-4517	265	15	.	.	PUNCT
cana-4517	266	1	3	3	X
cana-4517	266	2	.	.	X
cana-4517	266	3	construct	construct	VERB
cana-4517	266	4	the	the	DET
cana-4517	266	5	pe	pe	PROPN
cana-4517	266	6	equi	equi	NOUN
cana-4517	266	7	-	-	PUNCT
cana-4517	266	8	energetic	energetic	ADJ
cana-4517	266	9	graphs	graph	NOUN
cana-4517	266	10	.	.	PUNCT
cana-4517	267	1	refrences	refrence	VERB
cana-4517	268	1	[	[	X
cana-4517	268	2	1	1	X
cana-4517	268	3	]	]	PUNCT
cana-4517	268	4	abed	abe	VERB
cana-4517	268	5	,	,	PUNCT
cana-4517	268	6	s.	s.	PROPN
cana-4517	268	7	s.	s.	PROPN
cana-4517	268	8	,	,	PUNCT
cana-4517	268	9	&	&	CCONJ
cana-4517	268	10	al	al	PROPN
cana-4517	268	11	-	-	PUNCT
cana-4517	268	12	harere	harere	PROPN
cana-4517	268	13	,	,	PUNCT
cana-4517	268	14	m.	m.	NOUN
cana-4517	268	15	n.	n.	PROPN
cana-4517	268	16	(	(	PUNCT
cana-4517	268	17	2022	2022	NUM
cana-4517	268	18	)	)	PUNCT
cana-4517	268	19	.	.	PUNCT
cana-4517	269	1	rings	ring	NOUN
cana-4517	269	2	domination	domination	NOUN
cana-4517	269	3	in	in	ADP
cana-4517	269	4	graphs	graph	NOUN
cana-4517	269	5	.	.	PUNCT
cana-4517	270	1	international	international	ADJ
cana-4517	270	2	journal	journal	PROPN
cana-4517	270	3	of	of	ADP
cana-4517	270	4	nonlinear	nonlinear	ADJ
cana-4517	270	5	analysis	analysis	NOUN
cana-4517	270	6	and	and	CCONJ
cana-4517	270	7	applications	application	NOUN
cana-4517	270	8	,	,	PUNCT
cana-4517	270	9	13(2	13(2	NOUN
cana-4517	270	10	)	)	PUNCT
cana-4517	270	11	,	,	PUNCT
cana-4517	270	12	1833	1833	NUM
cana-4517	270	13	-	-	SYM
cana-4517	270	14	1839	1839	NUM
cana-4517	270	15	.	.	PUNCT
cana-4517	271	1	[	[	X
cana-4517	271	2	2	2	NUM
cana-4517	271	3	]	]	X
cana-4517	271	4	balakrishnan	balakrishnan	PROPN
cana-4517	271	5	,	,	PUNCT
cana-4517	271	6	r.	r.	PROPN
cana-4517	271	7	(	(	PUNCT
cana-4517	271	8	2004	2004	NUM
cana-4517	271	9	)	)	PUNCT
cana-4517	271	10	.	.	PUNCT
cana-4517	272	1	the	the	DET
cana-4517	272	2	energy	energy	NOUN
cana-4517	272	3	of	of	ADP
cana-4517	272	4	a	a	DET
cana-4517	272	5	graph	graph	NOUN
cana-4517	272	6	.	.	PUNCT
cana-4517	273	1	linear	linear	ADJ
cana-4517	273	2	algebra	algebra	NOUN
cana-4517	273	3	and	and	CCONJ
cana-4517	273	4	its	its	PRON
cana-4517	273	5	applications	application	NOUN
cana-4517	273	6	,	,	PUNCT
cana-4517	273	7	387	387	NUM
cana-4517	273	8	,	,	PUNCT
cana-4517	273	9	287	287	NUM
cana-4517	273	10	-	-	SYM
cana-4517	273	11	295	295	NUM
cana-4517	273	12	.	.	PUNCT
cana-4517	274	1	[	[	X
cana-4517	274	2	3	3	NUM
cana-4517	274	3	]	]	PUNCT
cana-4517	274	4	biernacki	biernacki	NOUN
cana-4517	274	5	,	,	PUNCT
cana-4517	274	6	m.	m.	NOUN
cana-4517	274	7	(	(	PUNCT
cana-4517	274	8	1950	1950	NUM
cana-4517	274	9	)	)	PUNCT
cana-4517	274	10	.	.	PUNCT
cana-4517	275	1	surune	surune	PROPN
cana-4517	275	2	inégalité	inégalité	PROPN
cana-4517	275	3	entre	entre	PROPN
cana-4517	275	4	des	des	PROPN
cana-4517	275	5	intégrales	intégrales	PROPN
cana-4517	275	6	définies	définie	NOUN
cana-4517	275	7	.	.	PUNCT
cana-4517	276	1	ann	ann	PROPN
cana-4517	276	2	.	.	PROPN
cana-4517	276	3	univ	univ	PROPN
cana-4517	276	4	.	.	PUNCT
cana-4517	277	1	mariae	mariae	PROPN
cana-4517	277	2	curie	curie	PROPN
cana-4517	277	3	-	-	PUNCT
cana-4517	277	4	sklodowska	sklodowska	NOUN
cana-4517	277	5	,	,	PUNCT
cana-4517	277	6	4	4	NUM
cana-4517	277	7	,	,	PUNCT
cana-4517	277	8	14	14	NUM
cana-4517	277	9	.	.	PUNCT
cana-4517	278	1	[	[	X
cana-4517	278	2	4	4	NUM
cana-4517	278	3	]	]	X
cana-4517	278	4	cvetkovic	cvetkovic	ADJ
cana-4517	278	5	,	,	PUNCT
cana-4517	278	6	d.	d.	PROPN
cana-4517	278	7	m.	m.	PROPN
cana-4517	278	8	,	,	PUNCT
cana-4517	278	9	&	&	CCONJ
cana-4517	278	10	gutman	gutman	PROPN
cana-4517	278	11	,	,	PUNCT
cana-4517	278	12	i.	i.	PROPN
cana-4517	278	13	(	(	PUNCT
cana-4517	278	14	2011	2011	NUM
cana-4517	278	15	)	)	PUNCT
cana-4517	278	16	.	.	PUNCT
cana-4517	279	1	selected	select	VERB
cana-4517	279	2	topics	topic	NOUN
cana-4517	279	3	on	on	ADP
cana-4517	279	4	applications	application	NOUN
cana-4517	279	5	of	of	ADP
cana-4517	279	6	graph	graph	NOUN
cana-4517	279	7	spectra	spectra	PROPN
cana-4517	279	8	.	.	PUNCT
cana-4517	280	1	beograd	beograd	PROPN
cana-4517	280	2	:	:	PUNCT
cana-4517	280	3	matematicki	matematicki	PROPN
cana-4517	280	4	institut	institut	PROPN
cana-4517	280	5	sanu	sanu	PROPN
cana-4517	280	6	.	.	PUNCT
cana-4517	281	1	[	[	X
cana-4517	281	2	5	5	NUM
cana-4517	281	3	]	]	X
cana-4517	281	4	cvetkovic	cvetkovic	ADJ
cana-4517	281	5	,	,	PUNCT
cana-4517	281	6	d.	d.	PROPN
cana-4517	281	7	,	,	PUNCT
cana-4517	281	8	gutman	gutman	PROPN
cana-4517	281	9	,	,	PUNCT
cana-4517	281	10	i.	i.	PROPN
cana-4517	281	11	(	(	PUNCT
cana-4517	281	12	2009	2009	NUM
cana-4517	281	13	)	)	PUNCT
cana-4517	281	14	.	.	PUNCT
cana-4517	282	1	applications	application	NOUN
cana-4517	282	2	of	of	ADP
cana-4517	282	3	graph	graph	NOUN
cana-4517	282	4	spectra	spectra	PROPN
cana-4517	282	5	.	.	PROPN
cana-4517	282	6	mathematical	mathematical	ADJ
cana-4517	282	7	institution	institution	NOUN
cana-4517	282	8	,	,	PUNCT
cana-4517	282	9	belgrade	belgrade	PROPN
cana-4517	282	10	.	.	PUNCT
cana-4517	283	1	[	[	X
cana-4517	283	2	6	6	NUM
cana-4517	283	3	]	]	X
cana-4517	283	4	diaz	diaz	PROPN
cana-4517	283	5	,	,	PUNCT
cana-4517	283	6	j.	j.	PROPN
cana-4517	283	7	b.	b.	PROPN
cana-4517	283	8	,	,	PUNCT
cana-4517	283	9	&	&	CCONJ
cana-4517	283	10	metcalf	metcalf	PROPN
cana-4517	283	11	,	,	PUNCT
cana-4517	283	12	f.	f.	PROPN
cana-4517	283	13	t.	t.	PROPN
cana-4517	283	14	(	(	PUNCT
cana-4517	283	15	1963	1963	NUM
cana-4517	283	16	)	)	PUNCT
cana-4517	283	17	.	.	PUNCT
cana-4517	284	1	stronger	strong	ADJ
cana-4517	284	2	forms	form	NOUN
cana-4517	284	3	of	of	ADP
cana-4517	284	4	a	a	DET
cana-4517	284	5	class	class	NOUN
cana-4517	284	6	of	of	ADP
cana-4517	284	7	inequalities	inequality	NOUN
cana-4517	284	8	of	of	ADP
cana-4517	284	9	g.	g.	PROPN
cana-4517	284	10	pólya	pólya	PROPN
cana-4517	284	11	-	-	PUNCT
cana-4517	284	12	g.	g.	NOUN
cana-4517	284	13	szegö	szegö	NOUN
cana-4517	284	14	,	,	PUNCT
cana-4517	284	15	and	and	CCONJ
cana-4517	284	16	lv	lv	PROPN
cana-4517	284	17	kantorovich	kantorovich	PROPN
cana-4517	284	18	.	.	PUNCT
cana-4517	285	1	bulletin	bulletin	NOUN
cana-4517	285	2	of	of	ADP
cana-4517	285	3	the	the	DET
cana-4517	285	4	american	american	PROPN
cana-4517	285	5	mathematical	mathematical	PROPN
cana-4517	285	6	society	society	NOUN
cana-4517	285	7	,	,	PUNCT
cana-4517	285	8	69(3	69(3	NUM
cana-4517	285	9	)	)	PUNCT
cana-4517	285	10	,	,	PUNCT
cana-4517	285	11	415	415	NUM
cana-4517	285	12	-	-	SYM
cana-4517	285	13	418	418	NUM
cana-4517	285	14	.	.	PUNCT
cana-4517	286	1	[	[	X
cana-4517	286	2	7	7	X
cana-4517	286	3	]	]	X
cana-4517	286	4	gutman	gutman	NOUN
cana-4517	286	5	,	,	PUNCT
cana-4517	286	6	i.	i.	PROPN
cana-4517	286	7	(	(	PUNCT
cana-4517	286	8	1978	1978	NUM
cana-4517	286	9	)	)	PUNCT
cana-4517	286	10	.	.	PUNCT
cana-4517	287	1	the	the	DET
cana-4517	287	2	energy	energy	NOUN
cana-4517	287	3	of	of	ADP
cana-4517	287	4	a	a	DET
cana-4517	287	5	graph	graph	NOUN
cana-4517	287	6	.	.	PUNCT
cana-4517	288	1	ber	ber	NOUN
cana-4517	288	2	.	.	PUNCT
cana-4517	288	3	math.-stat	math.-stat	PROPN
cana-4517	288	4	.	.	PROPN
cana-4517	288	5	sekt	sekt	PROPN
cana-4517	288	6	.	.	PUNCT
cana-4517	289	1	forschungszent	forschungszent	PROPN
cana-4517	289	2	.	.	PUNCT
cana-4517	290	1	graz	graz	PROPN
cana-4517	290	2	103	103	NUM
cana-4517	290	3	,	,	PUNCT
cana-4517	290	4	1–22	1–22	NOUN
cana-4517	290	5	.	.	PUNCT
cana-4517	291	1	[	[	X
cana-4517	291	2	8	8	NUM
cana-4517	291	3	]	]	X
cana-4517	291	4	gutman	gutman	NOUN
cana-4517	291	5	,	,	PUNCT
cana-4517	291	6	i.	i.	PROPN
cana-4517	291	7	(	(	PUNCT
cana-4517	291	8	2001	2001	NUM
cana-4517	291	9	)	)	PUNCT
cana-4517	291	10	.	.	PUNCT
cana-4517	292	1	the	the	DET
cana-4517	292	2	energy	energy	NOUN
cana-4517	292	3	of	of	ADP
cana-4517	292	4	a	a	DET
cana-4517	292	5	graph	graph	NOUN
cana-4517	292	6	:	:	PUNCT
cana-4517	292	7	old	old	ADJ
cana-4517	292	8	and	and	CCONJ
cana-4517	292	9	new	new	ADJ
cana-4517	292	10	results	result	NOUN
cana-4517	292	11	.	.	PUNCT
cana-4517	293	1	in	in	ADP
cana-4517	293	2	algebraic	algebraic	ADJ
cana-4517	293	3	combinatorics	combinatoric	NOUN
cana-4517	293	4	and	and	CCONJ
cana-4517	293	5	applications	application	NOUN
cana-4517	293	6	:	:	PUNCT
cana-4517	293	7	proceedings	proceeding	NOUN
cana-4517	293	8	of	of	ADP
cana-4517	293	9	the	the	DET
cana-4517	293	10	euroconference	euroconference	NOUN
cana-4517	293	11	,	,	PUNCT
cana-4517	293	12	algebraic	algebraic	ADJ
cana-4517	293	13	combinatorics	combinatoric	NOUN
cana-4517	293	14	and	and	CCONJ
cana-4517	293	15	applications	application	NOUN
cana-4517	293	16	(	(	PUNCT
cana-4517	293	17	alcoma	alcoma	ADJ
cana-4517	293	18	)	)	PUNCT
cana-4517	293	19	,	,	PUNCT
cana-4517	293	20	held	hold	VERB
cana-4517	293	21	in	in	ADP
cana-4517	293	22	gößweinstein	gößweinstein	PROPN
cana-4517	293	23	,	,	PUNCT
cana-4517	293	24	germany	germany	PROPN
cana-4517	293	25	,	,	PUNCT
cana-4517	293	26	september	september	PROPN
cana-4517	293	27	12–19	12–19	PROPN
cana-4517	293	28	,	,	PUNCT
cana-4517	293	29	1999	1999	NUM
cana-4517	293	30	(	(	PUNCT
cana-4517	293	31	pp	pp	ADJ
cana-4517	293	32	.	.	PUNCT
cana-4517	293	33	196	196	NUM
cana-4517	293	34	-	-	SYM
cana-4517	293	35	211	211	NUM
cana-4517	293	36	)	)	PUNCT
cana-4517	293	37	.	.	PUNCT
cana-4517	294	1	berlin	berlin	PROPN
cana-4517	294	2	,	,	PUNCT
cana-4517	294	3	heidelberg	heidelberg	PROPN
cana-4517	294	4	:	:	PUNCT
cana-4517	294	5	springer	springer	PROPN
cana-4517	294	6	berlin	berlin	PROPN
cana-4517	294	7	heidelberg	heidelberg	PROPN
cana-4517	294	8	.	.	PUNCT
cana-4517	295	1	[	[	X
cana-4517	295	2	9	9	NUM
cana-4517	295	3	]	]	SYM
cana-4517	295	4	harary	harary	NOUN
cana-4517	295	5	,	,	PUNCT
cana-4517	295	6	f.	f.	PROPN
cana-4517	295	7	,	,	PUNCT
cana-4517	295	8	&	&	CCONJ
cana-4517	295	9	buckley	buckley	PROPN
cana-4517	295	10	,	,	PUNCT
cana-4517	295	11	f.	f.	PROPN
cana-4517	295	12	(	(	PUNCT
cana-4517	295	13	1990	1990	NUM
cana-4517	295	14	)	)	PUNCT
cana-4517	295	15	.	.	PUNCT
cana-4517	296	1	distance	distance	NOUN
cana-4517	296	2	in	in	ADP
cana-4517	296	3	graphs	graph	NOUN
cana-4517	296	4	.	.	PUNCT
cana-4517	297	1	new	new	PROPN
cana-4517	297	2	york	york	PROPN
cana-4517	297	3	:	:	PUNCT
cana-4517	297	4	addison	addison	PROPN
cana-4517	297	5	and	and	CCONJ
cana-4517	297	6	wesley	wesley	PROPN
cana-4517	297	7	.	.	PUNCT
cana-4517	298	1	[	[	X
cana-4517	298	2	10	10	NUM
cana-4517	298	3	]	]	X
cana-4517	298	4	indriati	indriati	NOUN
cana-4517	298	5	,	,	PUNCT
cana-4517	298	6	d.	d.	PROPN
cana-4517	298	7	,	,	PUNCT
cana-4517	298	8	wijayanti	wijayanti	PROPN
cana-4517	298	9	,	,	PUNCT
cana-4517	298	10	i.	i.	PROPN
cana-4517	298	11	e.	e.	PROPN
cana-4517	298	12	,	,	PUNCT
cana-4517	298	13	&	&	CCONJ
cana-4517	298	14	sugeng	sugeng	PROPN
cana-4517	298	15	,	,	PUNCT
cana-4517	298	16	k.	k.	PROPN
cana-4517	298	17	a.	a.	PROPN
cana-4517	298	18	(	(	PUNCT
cana-4517	298	19	2015	2015	NUM
cana-4517	298	20	)	)	PUNCT
cana-4517	298	21	.	.	PUNCT
cana-4517	299	1	on	on	ADP
cana-4517	299	2	total	total	ADJ
cana-4517	299	3	irregularity	irregularity	NOUN
cana-4517	299	4	strength	strength	NOUN
cana-4517	299	5	of	of	ADP
cana-4517	299	6	double	double	ADJ
cana-4517	299	7	-	-	PUNCT
cana-4517	299	8	star	star	NOUN
cana-4517	299	9	and	and	CCONJ
cana-4517	299	10	related	related	ADJ
cana-4517	299	11	graphs	graph	NOUN
cana-4517	299	12	.	.	PUNCT
cana-4517	300	1	procedia	procedia	NOUN
cana-4517	300	2	computer	computer	NOUN
cana-4517	300	3	science	science	NOUN
cana-4517	300	4	,	,	PUNCT
cana-4517	300	5	74	74	NUM
cana-4517	300	6	,	,	PUNCT
cana-4517	300	7	118	118	NUM
cana-4517	300	8	-	-	SYM
cana-4517	300	9	123	123	NUM
cana-4517	300	10	.	.	PUNCT
cana-4517	301	1	[	[	X
cana-4517	301	2	11	11	NUM
cana-4517	301	3	]	]	X
cana-4517	301	4	livingston	livingston	PROPN
cana-4517	301	5	,	,	PUNCT
cana-4517	301	6	m.	m.	NOUN
cana-4517	301	7	,	,	PUNCT
cana-4517	301	8	&	&	CCONJ
cana-4517	301	9	stout	stout	PROPN
cana-4517	301	10	,	,	PUNCT
cana-4517	301	11	q.	q.	PROPN
cana-4517	301	12	f.	f.	PROPN
cana-4517	301	13	(	(	PUNCT
cana-4517	301	14	1990	1990	NUM
cana-4517	301	15	)	)	PUNCT
cana-4517	301	16	.	.	PUNCT
cana-4517	302	1	perfect	perfect	ADJ
cana-4517	302	2	dominating	dominating	NOUN
cana-4517	302	3	sets	set	NOUN
cana-4517	302	4	.	.	PUNCT
cana-4517	303	1	university	university	NOUN
cana-4517	303	2	of	of	ADP
cana-4517	303	3	michigan	michigan	PROPN
cana-4517	303	4	,	,	PUNCT
cana-4517	303	5	computer	computer	NOUN
cana-4517	303	6	science	science	NOUN
cana-4517	303	7	and	and	CCONJ
cana-4517	303	8	engineering	engineering	NOUN
cana-4517	303	9	division	division	NOUN
cana-4517	303	10	,	,	PUNCT
cana-4517	303	11	department	department	NOUN
cana-4517	303	12	of	of	ADP
cana-4517	303	13	electrical	electrical	ADJ
cana-4517	303	14	engineering	engineering	NOUN
cana-4517	303	15	and	and	CCONJ
cana-4517	303	16	computer	computer	NOUN
cana-4517	303	17	science	science	NOUN
cana-4517	303	18	.	.	PUNCT
cana-4517	304	1	[	[	X
cana-4517	304	2	12	12	NUM
cana-4517	304	3	]	]	X
cana-4517	304	4	palani	palani	PROPN
cana-4517	304	5	,	,	PUNCT
cana-4517	304	6	k.	k.	PROPN
cana-4517	304	7	,	,	PUNCT
cana-4517	304	8	kumari	kumari	PROPN
cana-4517	304	9	,	,	PUNCT
cana-4517	304	10	m.	m.	NOUN
cana-4517	304	11	l.	l.	PROPN
cana-4517	304	12	,	,	PUNCT
cana-4517	304	13	&	&	CCONJ
cana-4517	304	14	suganya	suganya	PROPN
cana-4517	304	15	,	,	PUNCT
cana-4517	304	16	g.	g.	PROPN
cana-4517	304	17	(	(	PUNCT
cana-4517	304	18	2022	2022	NUM
cana-4517	304	19	)	)	PUNCT
cana-4517	304	20	.	.	PUNCT
cana-4517	305	1	locating	locate	VERB
cana-4517	305	2	geo	geo	PROPN
cana-4517	305	3	spectrum	spectrum	PROPN
cana-4517	305	4	and	and	CCONJ
cana-4517	305	5	geo	geo	PROPN
cana-4517	305	6	energy	energy	NOUN
cana-4517	305	7	of	of	ADP
cana-4517	305	8	more	more	ADJ
cana-4517	305	9	graphs	graph	NOUN
cana-4517	305	10	,	,	PUNCT
cana-4517	305	11	aams	aam	NOUN
cana-4517	305	12	.	.	PUNCT
cana-4517	306	1	[	[	X
cana-4517	306	2	13	13	NUM
cana-4517	306	3	]	]	SYM
cana-4517	306	4	ramane	ramane	PROPN
cana-4517	306	5	,	,	PUNCT
cana-4517	306	6	h.	h.	PROPN
cana-4517	306	7	s.	s.	PROPN
cana-4517	306	8	,	,	PUNCT
cana-4517	306	9	revankar	revankar	PROPN
cana-4517	306	10	,	,	PUNCT
cana-4517	306	11	d.	d.	PROPN
cana-4517	306	12	s.	s.	PROPN
cana-4517	306	13	,	,	PUNCT
cana-4517	306	14	gutman	gutman	PROPN
cana-4517	306	15	,	,	PUNCT
cana-4517	306	16	i.	i.	PROPN
cana-4517	306	17	,	,	PUNCT
cana-4517	306	18	rao	rao	PROPN
cana-4517	306	19	,	,	PUNCT
cana-4517	306	20	s.	s.	PROPN
cana-4517	306	21	b.	b.	PROPN
cana-4517	306	22	,	,	PUNCT
cana-4517	306	23	acharya	acharya	PROPN
cana-4517	306	24	,	,	PUNCT
cana-4517	306	25	b.	b.	PROPN
cana-4517	306	26	d.	d.	PROPN
cana-4517	306	27	,	,	PUNCT
cana-4517	306	28	&	&	CCONJ
cana-4517	306	29	walikar	walikar	PROPN
cana-4517	306	30	,	,	PUNCT
cana-4517	306	31	h.	h.	PROPN
cana-4517	306	32	b.	b.	PROPN
cana-4517	306	33	(	(	PUNCT
cana-4517	306	34	2008	2008	NUM
cana-4517	306	35	)	)	PUNCT
cana-4517	306	36	.	.	PUNCT
cana-4517	307	1	bounds	bound	VERB
cana-4517	307	2	for	for	ADP
cana-4517	307	3	the	the	DET
cana-4517	307	4	distance	distance	NOUN
cana-4517	307	5	energy	energy	NOUN
cana-4517	307	6	of	of	ADP
cana-4517	307	7	a	a	DET
cana-4517	307	8	graph	graph	NOUN
cana-4517	307	9	.	.	PUNCT
cana-4517	308	1	kragujevac	kragujevac	PROPN
cana-4517	308	2	journal	journal	PROPN
cana-4517	308	3	of	of	ADP
cana-4517	308	4	mathematics	mathematic	NOUN
cana-4517	308	5	,	,	PUNCT
cana-4517	308	6	31(31	31(31	NUM
cana-4517	308	7	)	)	PUNCT
cana-4517	308	8	,	,	PUNCT
cana-4517	308	9	59	59	NUM
cana-4517	308	10	-	-	SYM
cana-4517	308	11	68	68	NUM
cana-4517	308	12	.	.	PUNCT
cana-4517	309	1	[	[	X
cana-4517	309	2	14	14	NUM
cana-4517	309	3	]	]	X
cana-4517	309	4	roger	roger	PROPN
cana-4517	309	5	,	,	PUNCT
cana-4517	309	6	h.	h.	PROPN
cana-4517	309	7	,	,	PUNCT
cana-4517	309	8	&	&	CCONJ
cana-4517	309	9	charles	charles	PROPN
cana-4517	309	10	,	,	PUNCT
cana-4517	309	11	r.	r.	PROPN
cana-4517	309	12	j.	j.	PROPN
cana-4517	309	13	(	(	PUNCT
cana-4517	309	14	1994	1994	NUM
cana-4517	309	15	)	)	PUNCT
cana-4517	309	16	.	.	PUNCT
cana-4517	310	1	topics	topic	NOUN
cana-4517	310	2	in	in	ADP
cana-4517	310	3	matrix	matrix	NOUN
cana-4517	310	4	analysis	analysis	NOUN
cana-4517	310	5	.	.	PUNCT
cana-4517	311	1	cambridge	cambridge	PROPN
cana-4517	311	2	university	university	PROPN
cana-4517	311	3	press	press	PROPN
cana-4517	311	4	,	,	PUNCT
cana-4517	311	5	cambridge	cambridge	NOUN