id	sid	tid	token	lemma	pos
cana-1710	1	1	communications	communication	NOUN
cana-1710	1	2	on	on	ADP
cana-1710	1	3	applied	apply	VERB
cana-1710	1	4	nonlinear	nonlinear	ADJ
cana-1710	1	5	analysis	analysis	NOUN
cana-1710	1	6	issn	issn	NOUN
cana-1710	1	7	:	:	PUNCT
cana-1710	1	8	1074	1074	NUM
cana-1710	1	9	-	-	PUNCT
cana-1710	1	10	133x	133x	NUM
cana-1710	1	11	vol	vol	NOUN
cana-1710	1	12	32	32	NUM
cana-1710	1	13	no	no	NOUN
cana-1710	1	14	.	.	NOUN
cana-1710	1	15	2	2	NUM
cana-1710	1	16	(	(	PUNCT
cana-1710	1	17	2025	2025	NUM
cana-1710	1	18	)	)	PUNCT
cana-1710	1	19	64	64	NUM
cana-1710	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1710	1	21	bounds	bound	VERB
cana-1710	1	22	on	on	ADP
cana-1710	1	23	dominating	dominate	VERB
cana-1710	1	24	energy	energy	NOUN
cana-1710	1	25	of	of	ADP
cana-1710	1	26	graph	graph	NOUN
cana-1710	1	27	shivakumar	shivakumar	PROPN
cana-1710	1	28	swamy	swamy	PROPN
cana-1710	1	29	c	c	PROPN
cana-1710	1	30	s1	s1	PROPN
cana-1710	1	31	,	,	PUNCT
cana-1710	1	32	ashwini	ashwini	NOUN
cana-1710	1	33	g	g	PROPN
cana-1710	1	34	2	2	NUM
cana-1710	1	35	,	,	PUNCT
cana-1710	1	36	ramesha	ramesha	VERB
cana-1710	1	37	m	m	PROPN
cana-1710	1	38	s	s	NOUN
cana-1710	1	39	3	3	NUM
cana-1710	1	40	,	,	PUNCT
cana-1710	1	41	nanjundaswamy	nanjundaswamy	NOUN
cana-1710	1	42	n	n	CCONJ
cana-1710	1	43	4	4	NUM
cana-1710	1	44	1,2,3	1,2,3	NUM
cana-1710	1	45	department	department	NOUN
cana-1710	1	46	of	of	ADP
cana-1710	1	47	mathematics	mathematic	NOUN
cana-1710	1	48	,	,	PUNCT
cana-1710	1	49	government	government	NOUN
cana-1710	1	50	college	college	NOUN
cana-1710	1	51	for	for	ADP
cana-1710	1	52	women	woman	NOUN
cana-1710	1	53	(	(	PUNCT
cana-1710	1	54	autonomous	autonomous	ADJ
cana-1710	1	55	)	)	PUNCT
cana-1710	1	56	,	,	PUNCT
cana-1710	1	57	mandya-571401	mandya-571401	NOUN
cana-1710	1	58	,	,	PUNCT
cana-1710	1	59	india	india	PROPN
cana-1710	1	60	.	.	PROPN
cana-1710	2	1	4	4	NUM
cana-1710	2	2	department	department	NOUN
cana-1710	2	3	of	of	ADP
cana-1710	2	4	mathematics	mathematic	NOUN
cana-1710	2	5	,	,	PUNCT
cana-1710	2	6	government	government	NOUN
cana-1710	2	7	first	first	ADJ
cana-1710	2	8	grade	grade	NOUN
cana-1710	2	9	college	college	NOUN
cana-1710	2	10	for	for	ADP
cana-1710	2	11	women	woman	NOUN
cana-1710	2	12	,	,	PUNCT
cana-1710	2	13	byrapura	byrapura	NOUN
cana-1710	2	14	,	,	PUNCT
cana-1710	2	15	t.narasipura	t.narasipura	NOUN
cana-1710	2	16	taluk	taluk	NOUN
cana-1710	2	17	,	,	PUNCT
cana-1710	2	18	mysore	mysore	NOUN
cana-1710	2	19	district-571124	district-571124	NOUN
cana-1710	2	20	,	,	PUNCT
cana-1710	2	21	india	india	PROPN
cana-1710	2	22	.	.	PUNCT
cana-1710	3	1	article	article	PROPN
cana-1710	3	2	history	history	NOUN
cana-1710	3	3	:	:	PUNCT
cana-1710	3	4	received	receive	VERB
cana-1710	3	5	:	:	PUNCT
cana-1710	3	6	23	23	NUM
cana-1710	3	7	-	-	SYM
cana-1710	3	8	07	07	NUM
cana-1710	3	9	-	-	PUNCT
cana-1710	3	10	2024	2024	NUM
cana-1710	3	11	revised	revise	VERB
cana-1710	3	12	:	:	PUNCT
cana-1710	3	13	01	01	NUM
cana-1710	3	14	-	-	SYM
cana-1710	3	15	09	09	NUM
cana-1710	3	16	-	-	PUNCT
cana-1710	3	17	2024	2024	NUM
cana-1710	3	18	accepted	accept	VERB
cana-1710	3	19	:	:	PUNCT
cana-1710	3	20	15	15	NUM
cana-1710	3	21	-	-	SYM
cana-1710	3	22	09	09	NUM
cana-1710	3	23	-	-	PUNCT
cana-1710	3	24	2024	2024	NUM
cana-1710	3	25	abstract	abstract	NOUN
cana-1710	3	26	:	:	PUNCT
cana-1710	3	27	the	the	DET
cana-1710	3	28	minimum	minimum	ADJ
cana-1710	3	29	dominating	dominating	NOUN
cana-1710	3	30	energy	energy	NOUN
cana-1710	3	31	(	(	PUNCT
cana-1710	3	32	mde	mde	PROPN
cana-1710	3	33	)	)	PUNCT
cana-1710	3	34	of	of	ADP
cana-1710	3	35	a	a	DET
cana-1710	3	36	graph𝔄	graph𝔄	PROPN
cana-1710	3	37	,	,	PUNCT
cana-1710	3	38	denoted	denote	VERB
cana-1710	3	39	by	by	ADP
cana-1710	3	40	𝐸𝑀	𝐸𝑀	PROPN
cana-1710	3	41	𝐷	𝐷	PROPN
cana-1710	3	42	(	(	PUNCT
cana-1710	3	43	𝔄	𝔄	PROPN
cana-1710	3	44	)	)	PUNCT
cana-1710	3	45	,	,	PUNCT
cana-1710	3	46	is	be	AUX
cana-1710	3	47	nothing	nothing	PRON
cana-1710	3	48	but	but	SCONJ
cana-1710	3	49	the	the	DET
cana-1710	3	50	sum	sum	NOUN
cana-1710	3	51	of	of	ADP
cana-1710	3	52	absolute	absolute	ADJ
cana-1710	3	53	values	value	NOUN
cana-1710	3	54	of	of	ADP
cana-1710	3	55	all	all	DET
cana-1710	3	56	minimum	minimum	ADJ
cana-1710	3	57	dominating	dominating	NOUN
cana-1710	3	58	eigenvalues	eigenvalue	NOUN
cana-1710	3	59	of	of	ADP
cana-1710	3	60	𝔄.	𝔄.	PROPN
cana-1710	3	61	in	in	ADP
cana-1710	3	62	this	this	DET
cana-1710	3	63	study	study	NOUN
cana-1710	3	64	,	,	PUNCT
cana-1710	3	65	few	few	ADJ
cana-1710	3	66	upper	upper	ADJ
cana-1710	3	67	and	and	CCONJ
cana-1710	3	68	lower	low	ADJ
cana-1710	3	69	constraints	constraint	NOUN
cana-1710	3	70	on	on	ADP
cana-1710	3	71	the	the	DET
cana-1710	3	72	minimum	minimum	ADJ
cana-1710	3	73	dominating	dominating	NOUN
cana-1710	3	74	energy	energy	NOUN
cana-1710	3	75	are	be	AUX
cana-1710	3	76	obtained	obtain	VERB
cana-1710	3	77	.	.	PUNCT
cana-1710	4	1	keywords	keyword	NOUN
cana-1710	4	2	:	:	PUNCT
cana-1710	4	3	minimum	minimum	ADJ
cana-1710	4	4	dominating	dominating	NOUN
cana-1710	4	5	matrix	matrix	NOUN
cana-1710	4	6	,	,	PUNCT
cana-1710	4	7	minimum	minimum	ADJ
cana-1710	4	8	dominating	dominating	NOUN
cana-1710	4	9	eigenvalues	eigenvalue	NOUN
cana-1710	4	10	,	,	PUNCT
cana-1710	4	11	minimum	minimum	ADJ
cana-1710	4	12	dominating	dominating	NOUN
cana-1710	4	13	energy	energy	NOUN
cana-1710	4	14	.	.	PUNCT
cana-1710	5	1	2000	2000	NUM
cana-1710	5	2	ams	am	NOUN
cana-1710	5	3	subject	subject	ADJ
cana-1710	5	4	classification	classification	NOUN
cana-1710	5	5	:	:	PUNCT
cana-1710	5	6	05𝐶50	05𝐶50	PUNCT
cana-1710	5	7	.	.	X
cana-1710	6	1	1	1	X
cana-1710	6	2	.	.	X
cana-1710	6	3	introduction	introduction	NOUN
cana-1710	6	4	let	let	VERB
cana-1710	6	5	𝔈	𝔈	PROPN
cana-1710	6	6	be	be	AUX
cana-1710	6	7	the	the	DET
cana-1710	6	8	edge	edge	NOUN
cana-1710	6	9	set	set	VERB
cana-1710	6	10	and	and	CCONJ
cana-1710	6	11	𝔙	𝔙	PROPN
cana-1710	6	12	,	,	PUNCT
cana-1710	6	13	set	set	VERB
cana-1710	6	14	of	of	ADP
cana-1710	6	15	vertices	vertex	NOUN
cana-1710	6	16	of	of	ADP
cana-1710	6	17	a	a	DET
cana-1710	6	18	simple	simple	ADJ
cana-1710	6	19	graph	graph	NOUN
cana-1710	6	20	𝔄.	𝔄.	PROPN
cana-1710	6	21	let	let	VERB
cana-1710	6	22	|𝔈|	|𝔈|	NOUN
cana-1710	6	23	=	=	PUNCT
cana-1710	6	24	𝑚	𝑚	PROPN
cana-1710	6	25	and	and	CCONJ
cana-1710	6	26	|𝔙|	|𝔙|	PROPN
cana-1710	6	27	=	=	SYM
cana-1710	7	1	𝑛.	𝑛.	NOUN
cana-1710	7	2	ℇ(𝔄	ℇ(𝔄	PROPN
cana-1710	7	3	):	):	PUNCT
cana-1710	7	4	=	=	SYM
cana-1710	7	5	∑|𝒱𝑘|	∑|𝒱𝑘|	PUNCT
cana-1710	7	6	𝑛	𝑛	PRON
cana-1710	7	7	𝑘=1	𝑘=1	X
cana-1710	7	8	where	where	SCONJ
cana-1710	7	9	𝒱𝑘	𝒱𝑘	PROPN
cana-1710	7	10	,	,	PUNCT
cana-1710	7	11	𝑘	𝑘	X
cana-1710	7	12	=	=	SYM
cana-1710	7	13	1	1	NUM
cana-1710	7	14	,	,	PUNCT
cana-1710	7	15	2	2	NUM
cana-1710	7	16	,	,	PUNCT
cana-1710	7	17	3	3	NUM
cana-1710	7	18	,	,	PUNCT
cana-1710	7	19	…	…	PUNCT
cana-1710	7	20	,	,	PUNCT
cana-1710	7	21	𝑛	𝑛	PROPN
cana-1710	7	22	are	be	AUX
cana-1710	7	23	the	the	DET
cana-1710	7	24	eigenvalues	eigenvalue	NOUN
cana-1710	7	25	(	(	PUNCT
cana-1710	7	26	characteristic	characteristic	ADJ
cana-1710	7	27	roots	root	NOUN
cana-1710	7	28	)	)	PUNCT
cana-1710	7	29	of	of	ADP
cana-1710	7	30	the	the	DET
cana-1710	7	31	adjacency	adjacency	NOUN
cana-1710	7	32	matrix	matrix	NOUN
cana-1710	7	33	(	(	PUNCT
cana-1710	7	34	𝐴𝑀	𝐴𝑀	PROPN
cana-1710	7	35	)	)	PUNCT
cana-1710	7	36	of	of	ADP
cana-1710	7	37	a	a	DET
cana-1710	7	38	graph	graph	NOUN
cana-1710	7	39	𝔄	𝔄	PROPN
cana-1710	7	40	,	,	PUNCT
cana-1710	7	41	ivan	ivan	PROPN
cana-1710	7	42	gutman	gutman	PROPN
cana-1710	7	43	[	[	X
cana-1710	7	44	12	12	NUM
cana-1710	7	45	]	]	PUNCT
cana-1710	7	46	conducted	conduct	VERB
cana-1710	7	47	this	this	DET
cana-1710	7	48	study	study	NOUN
cana-1710	7	49	on	on	ADP
cana-1710	7	50	𝔄	𝔄	PROPN
cana-1710	7	51	for	for	ADP
cana-1710	7	52	the	the	DET
cana-1710	7	53	first	first	ADJ
cana-1710	7	54	time	time	NOUN
cana-1710	7	55	in	in	ADP
cana-1710	7	56	1978	1978	NUM
cana-1710	7	57	,	,	PUNCT
cana-1710	7	58	and	and	CCONJ
cana-1710	7	59	named	name	VERB
cana-1710	7	60	it	it	PRON
cana-1710	7	61	as	as	ADP
cana-1710	7	62	energy	energy	NOUN
cana-1710	7	63	of	of	ADP
cana-1710	7	64	a	a	DET
cana-1710	7	65	graph	graph	NOUN
cana-1710	7	66	𝔄	𝔄	NOUN
cana-1710	7	67	,	,	PUNCT
cana-1710	7	68	from	from	ADP
cana-1710	7	69	then	then	ADV
cana-1710	7	70	numerous	numerous	ADJ
cana-1710	7	71	research	research	NOUN
cana-1710	7	72	has	have	AUX
cana-1710	7	73	been	be	AUX
cana-1710	7	74	conducted	conduct	VERB
cana-1710	7	75	on	on	ADP
cana-1710	7	76	𝐴𝑀	𝐴𝑀	PROPN
cana-1710	7	77	,	,	PUNCT
cana-1710	7	78	with	with	ADP
cana-1710	7	79	inspiration	inspiration	NOUN
cana-1710	7	80	drawn	draw	VERB
cana-1710	7	81	by	by	ADP
cana-1710	7	82	this	this	PRON
cana-1710	7	83	,	,	PUNCT
cana-1710	7	84	different	different	ADJ
cana-1710	7	85	matrix	matrix	NOUN
cana-1710	7	86	types	type	NOUN
cana-1710	7	87	for	for	ADP
cana-1710	7	88	a	a	DET
cana-1710	7	89	graph	graph	NOUN
cana-1710	7	90	𝔄	𝔄	NOUN
cana-1710	7	91	[	[	X
cana-1710	7	92	18	18	NUM
cana-1710	7	93	,	,	PUNCT
cana-1710	7	94	17	17	NUM
cana-1710	7	95	,	,	PUNCT
cana-1710	7	96	13	13	NUM
cana-1710	7	97	,	,	PUNCT
cana-1710	7	98	2	2	NUM
cana-1710	7	99	]	]	PUNCT
cana-1710	7	100	are	be	AUX
cana-1710	7	101	defined	define	VERB
cana-1710	7	102	and	and	CCONJ
cana-1710	7	103	studied	study	VERB
cana-1710	7	104	.	.	PUNCT
cana-1710	8	1	for	for	ADP
cana-1710	8	2	basic	basic	ADJ
cana-1710	8	3	mathematical	mathematical	ADJ
cana-1710	8	4	properties	property	NOUN
cana-1710	8	5	of	of	ADP
cana-1710	8	6	the	the	DET
cana-1710	8	7	theory	theory	NOUN
cana-1710	8	8	of	of	ADP
cana-1710	8	9	graph	graph	NOUN
cana-1710	8	10	energy	energy	NOUN
cana-1710	8	11	including	include	VERB
cana-1710	8	12	its	its	PRON
cana-1710	8	13	upper	upper	ADJ
cana-1710	8	14	and	and	CCONJ
cana-1710	8	15	lower	low	ADJ
cana-1710	8	16	bounds	bound	NOUN
cana-1710	8	17	one	one	PRON
cana-1710	8	18	can	can	AUX
cana-1710	8	19	see	see	VERB
cana-1710	8	20	[	[	PUNCT
cana-1710	8	21	20	20	NUM
cana-1710	8	22	,	,	PUNCT
cana-1710	8	23	21	21	NUM
cana-1710	8	24	,	,	PUNCT
cana-1710	8	25	22	22	NUM
cana-1710	8	26	]	]	PUNCT
cana-1710	8	27	.	.	PUNCT
cana-1710	9	1	erich	erich	PROPN
cana-1710	9	2	huckle	huckle	PROPN
cana-1710	10	1	[	[	X
cana-1710	10	2	3	3	NUM
cana-1710	10	3	]	]	PUNCT
cana-1710	10	4	,	,	PUNCT
cana-1710	10	5	employed	employ	VERB
cana-1710	10	6	the	the	DET
cana-1710	10	7	energy	energy	NOUN
cana-1710	10	8	of	of	ADP
cana-1710	10	9	graphs	graph	NOUN
cana-1710	10	10	technique	technique	NOUN
cana-1710	10	11	in	in	ADP
cana-1710	10	12	the	the	DET
cana-1710	10	13	early	early	ADJ
cana-1710	10	14	1930s	1930	NOUN
cana-1710	10	15	to	to	PART
cana-1710	10	16	develop	develop	VERB
cana-1710	10	17	approximations	approximation	NOUN
cana-1710	10	18	solutions	solution	NOUN
cana-1710	10	19	for	for	ADP
cana-1710	10	20	a	a	DET
cana-1710	10	21	family	family	NOUN
cana-1710	10	22	of	of	ADP
cana-1710	10	23	organic	organic	ADJ
cana-1710	10	24	molecules	molecule	NOUN
cana-1710	10	25	known	know	VERB
cana-1710	10	26	as	as	ADP
cana-1710	10	27	conjugated	conjugate	VERB
cana-1710	10	28	hydro	hydro	NOUN
cana-1710	10	29	carbons	carbon	NOUN
cana-1710	10	30	.	.	PUNCT
cana-1710	11	1	let	let	VERB
cana-1710	11	2	𝒟	𝒟	PRON
cana-1710	11	3	⊆	⊆	NUM
cana-1710	11	4	𝔙	𝔙	PROPN
cana-1710	11	5	(	(	PUNCT
cana-1710	11	6	𝔄	𝔄	PROPN
cana-1710	11	7	)	)	PUNCT
cana-1710	11	8	,	,	PUNCT
cana-1710	12	1	if	if	SCONJ
cana-1710	12	2	every	every	DET
cana-1710	12	3	vertex	vertex	NOUN
cana-1710	12	4	of	of	ADP
cana-1710	12	5	𝔙	𝔙	PROPN
cana-1710	12	6	−	−	PROPN
cana-1710	13	1	𝒟	𝒟	NOUN
cana-1710	13	2	is	be	AUX
cana-1710	13	3	adjacent	adjacent	ADJ
cana-1710	13	4	to	to	ADP
cana-1710	13	5	some	some	DET
cana-1710	13	6	vertex	vertex	NOUN
cana-1710	13	7	in	in	ADP
cana-1710	13	8	𝒟	𝒟	PROPN
cana-1710	13	9	,	,	PUNCT
cana-1710	13	10	then	then	ADV
cana-1710	13	11	𝒟	𝒟	PROPN
cana-1710	13	12	is	be	AUX
cana-1710	13	13	referred	refer	VERB
cana-1710	13	14	to	to	ADP
cana-1710	13	15	as	as	ADP
cana-1710	13	16	a	a	DET
cana-1710	13	17	dominating	dominating	NOUN
cana-1710	13	18	set	set	NOUN
cana-1710	13	19	of	of	ADP
cana-1710	13	20	𝔄.	𝔄.	PROPN
cana-1710	13	21	a	a	DET
cana-1710	13	22	minimum	minimum	ADJ
cana-1710	13	23	dominating	dominating	NOUN
cana-1710	13	24	set	set	NOUN
cana-1710	13	25	(	(	PUNCT
cana-1710	13	26	mds	mds	NOUN
cana-1710	13	27	)	)	PUNCT
cana-1710	13	28	𝒟	𝒟	NOUN
cana-1710	13	29	of	of	ADP
cana-1710	13	30	𝔄	𝔄	PROPN
cana-1710	13	31	is	be	AUX
cana-1710	13	32	a	a	DET
cana-1710	13	33	dominating	dominating	NOUN
cana-1710	13	34	set	set	NOUN
cana-1710	13	35	of	of	ADP
cana-1710	13	36	𝔄	𝔄	PROPN
cana-1710	13	37	with	with	ADP
cana-1710	13	38	minimum	minimum	ADJ
cana-1710	13	39	cardinality	cardinality	NOUN
cana-1710	13	40	.	.	PUNCT
cana-1710	14	1	let	let	VERB
cana-1710	14	2	𝒟	𝒟	PRON
cana-1710	14	3	be	be	AUX
cana-1710	14	4	a	a	DET
cana-1710	14	5	mds	mds	NOUN
cana-1710	14	6	of	of	ADP
cana-1710	14	7	𝔄.	𝔄.	PROPN
cana-1710	14	8	the	the	DET
cana-1710	14	9	following	follow	VERB
cana-1710	14	10	kind	kind	NOUN
cana-1710	14	11	of	of	ADP
cana-1710	14	12	matrix	matrix	NOUN
cana-1710	14	13	,	,	PUNCT
cana-1710	14	14	known	know	VERB
cana-1710	14	15	as	as	ADP
cana-1710	14	16	the	the	DET
cana-1710	14	17	minimum	minimum	ADJ
cana-1710	14	18	dominating	dominating	NOUN
cana-1710	14	19	matrix	matrix	NOUN
cana-1710	14	20	(	(	PUNCT
cana-1710	14	21	mdm	mdm	PROPN
cana-1710	14	22	)	)	PUNCT
cana-1710	14	23	of	of	ADP
cana-1710	14	24	𝔄	𝔄	PROPN
cana-1710	14	25	,	,	PUNCT
cana-1710	14	26	introduced	introduce	VERB
cana-1710	14	27	by	by	ADP
cana-1710	14	28	m.r.rajesh	m.r.rajesh	PROPN
cana-1710	14	29	kanna	kanna	PROPN
cana-1710	14	30	et.al	et.al	PROPN
cana-1710	14	31	.	.	PUNCT
cana-1710	15	1	in	in	ADP
cana-1710	15	2	[	[	X
cana-1710	15	3	22	22	NUM
cana-1710	15	4	]	]	PUNCT
cana-1710	15	5	:	:	PUNCT
cana-1710	15	6	the	the	DET
cana-1710	15	7	𝑛	𝑛	PRON
cana-1710	15	8	×	×	NOUN
cana-1710	15	9	𝑛	𝑛	PRON
cana-1710	15	10	matrix	matrix	NOUN
cana-1710	15	11	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	15	12	)	)	PUNCT
cana-1710	15	13	=	=	PUNCT
cana-1710	16	1	[	[	X
cana-1710	16	2	𝑑𝑘	𝑑𝑘	ADP
cana-1710	16	3	𝑗	𝑗	X
cana-1710	16	4	]	]	X
cana-1710	16	5	,	,	PUNCT
cana-1710	16	6	is	be	AUX
cana-1710	16	7	the	the	DET
cana-1710	16	8	mdm	mdm	PROPN
cana-1710	16	9	of	of	ADP
cana-1710	16	10	𝔄	𝔄	PROPN
cana-1710	16	11	,	,	PUNCT
cana-1710	16	12	whose	whose	DET
cana-1710	16	13	𝑘𝑗	𝑘𝑗	NOUN
cana-1710	16	14	−	−	PROPN
cana-1710	16	15	𝑡ℎ	𝑡ℎ	PROPN
cana-1710	16	16	element	element	NOUN
cana-1710	16	17	is	be	AUX
cana-1710	16	18	given	give	VERB
cana-1710	16	19	by	by	ADP
cana-1710	16	20	𝑑𝑘𝑗	𝑑𝑘𝑗	NOUN
cana-1710	16	21	=	=	SYM
cana-1710	16	22	{	{	PUNCT
cana-1710	16	23	1	1	NUM
cana-1710	16	24	,	,	PUNCT
cana-1710	16	25	𝑖𝑓	𝑖𝑓	ADP
cana-1710	16	26	𝑘	𝑘	NOUN
cana-1710	16	27	=	=	PUNCT
cana-1710	16	28	𝑗	𝑗	X
cana-1710	16	29	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1710	16	30	𝑣𝑗	𝑣𝑗	ADP
cana-1710	16	31	𝑖𝑛	𝑖𝑛	PROPN
cana-1710	16	32	𝒟	𝒟	NOUN
cana-1710	16	33	;	;	PUNCT
cana-1710	16	34	1	1	NUM
cana-1710	16	35	,	,	PUNCT
cana-1710	16	36	𝑖𝑓	𝑖𝑓	ADV
cana-1710	16	37	𝑣𝑘	𝑣𝑘	INTJ
cana-1710	16	38	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1710	16	39	𝑣𝑗	𝑣𝑗	ADP
cana-1710	16	40	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
cana-1710	16	41	𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡	𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡	NOUN
cana-1710	16	42	;	;	PUNCT
cana-1710	16	43	0	0	NUM
cana-1710	16	44	,	,	PUNCT
cana-1710	16	45	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.	NOUN
cana-1710	16	46	communications	communication	NOUN
cana-1710	16	47	on	on	ADP
cana-1710	16	48	applied	apply	VERB
cana-1710	16	49	nonlinear	nonlinear	ADJ
cana-1710	16	50	analysis	analysis	NOUN
cana-1710	16	51	issn	issn	NOUN
cana-1710	16	52	:	:	PUNCT
cana-1710	16	53	1074	1074	NUM
cana-1710	16	54	-	-	PUNCT
cana-1710	16	55	133x	133x	NUM
cana-1710	16	56	vol	vol	NOUN
cana-1710	16	57	32	32	NUM
cana-1710	16	58	no	no	NOUN
cana-1710	16	59	.	.	NOUN
cana-1710	16	60	2	2	NUM
cana-1710	16	61	(	(	PUNCT
cana-1710	16	62	2025	2025	NUM
cana-1710	16	63	)	)	PUNCT
cana-1710	16	64	65	65	NUM
cana-1710	17	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1710	17	2	γ(𝔄	γ(𝔄	PROPN
cana-1710	17	3	∶	∶	NOUN
cana-1710	17	4	𝜒	𝜒	NOUN
cana-1710	17	5	)	)	PUNCT
cana-1710	17	6	=	=	PUNCT
cana-1710	17	7	𝑑𝑒𝑡(𝜒𝐼	𝑑𝑒𝑡(𝜒𝐼	ADJ
cana-1710	17	8	−	−	PROPN
cana-1710	17	9	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	17	10	)	)	PUNCT
cana-1710	17	11	)	)	PUNCT
cana-1710	17	12	is	be	AUX
cana-1710	17	13	the	the	DET
cana-1710	17	14	characteristic	characteristic	ADJ
cana-1710	17	15	polynomial	polynomial	NOUN
cana-1710	17	16	of	of	ADP
cana-1710	17	17	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	17	18	)	)	PUNCT
cana-1710	17	19	.	.	PUNCT
cana-1710	18	1	the	the	DET
cana-1710	18	2	minimum	minimum	ADJ
cana-1710	18	3	dominating	dominating	NOUN
cana-1710	18	4	eigenvalues	eigenvalue	NOUN
cana-1710	18	5	of	of	ADP
cana-1710	18	6	𝔄	𝔄	PROPN
cana-1710	18	7	are	be	AUX
cana-1710	18	8	the	the	DET
cana-1710	18	9	eigenvalues	eigenvalue	NOUN
cana-1710	18	10	𝜒1	𝜒1	NOUN
cana-1710	18	11	,	,	PUNCT
cana-1710	18	12	𝜒2	𝜒2	NOUN
cana-1710	18	13	,	,	PUNCT
cana-1710	18	14	.	.	PUNCT
cana-1710	18	15	.	.	PUNCT
cana-1710	18	16	.	.	PUNCT
cana-1710	19	1	,	,	PUNCT
cana-1710	19	2	𝜒𝑛of	𝜒𝑛of	NOUN
cana-1710	19	3	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	19	4	)	)	PUNCT
cana-1710	19	5	.	.	PUNCT
cana-1710	20	1	the	the	DET
cana-1710	20	2	matrix	matrix	NOUN
cana-1710	20	3	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	20	4	)	)	PUNCT
cana-1710	20	5	is	be	AUX
cana-1710	20	6	real	real	ADJ
cana-1710	20	7	as	as	ADV
cana-1710	20	8	well	well	ADV
cana-1710	20	9	as	as	ADP
cana-1710	20	10	symmetric	symmetric	ADJ
cana-1710	20	11	.	.	PUNCT
cana-1710	21	1	the	the	DET
cana-1710	21	2	real	real	ADJ
cana-1710	21	3	numbers	number	NOUN
cana-1710	21	4	that	that	PRON
cana-1710	21	5	make	make	VERB
cana-1710	21	6	up	up	ADP
cana-1710	21	7	the	the	DET
cana-1710	21	8	eigenvalues	eigenvalue	NOUN
cana-1710	21	9	of	of	ADP
cana-1710	21	10	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	21	11	)	)	PUNCT
cana-1710	21	12	are	be	AUX
cana-1710	21	13	arranged	arrange	VERB
cana-1710	21	14	to	to	PART
cana-1710	21	15	be	be	AUX
cana-1710	21	16	:	:	PUNCT
cana-1710	21	17	𝜒1	𝜒1	VERB
cana-1710	21	18	≥	≥	PRON
cana-1710	21	19	𝜒2	𝜒2	ADP
cana-1710	21	20	≥	≥	PRON
cana-1710	21	21	.	.	PUNCT
cana-1710	21	22	.	.	PUNCT
cana-1710	22	1	.	.	PUNCT
cana-1710	23	1	≥	≥	PRON
cana-1710	23	2	𝜒𝑛.	𝜒𝑛.	VERB
cana-1710	23	3	the	the	DET
cana-1710	23	4	formula	formula	NOUN
cana-1710	23	5	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	23	6	)	)	PUNCT
cana-1710	23	7	=	=	SYM
cana-1710	24	1	∑|𝜒𝑘|	∑|𝜒𝑘|	ADP
cana-1710	24	2	𝑛	𝑛	PRON
cana-1710	24	3	𝑘=1	𝑘=1	NOUN
cana-1710	24	4	defines	define	VERB
cana-1710	24	5	𝔄′𝑠	𝔄′𝑠	VERB
cana-1710	24	6	minimum	minimum	ADJ
cana-1710	24	7	dominating	dominating	NOUN
cana-1710	24	8	energy	energy	NOUN
cana-1710	24	9	(	(	PUNCT
cana-1710	24	10	mde	mde	PROPN
cana-1710	24	11	)	)	PUNCT
cana-1710	24	12	.	.	PUNCT
cana-1710	25	1	note	note	VERB
cana-1710	25	2	that	that	SCONJ
cana-1710	25	3	𝑀𝒟(𝔄	𝑀𝒟(𝔄	NOUN
cana-1710	25	4	)	)	PUNCT
cana-1710	25	5	has	have	VERB
cana-1710	25	6	trace	trace	NOUN
cana-1710	25	7	=	=	NOUN
cana-1710	25	8	domination	domination	NOUN
cana-1710	25	9	number	number	NOUN
cana-1710	25	10	=	=	SYM
cana-1710	25	11	d	d	NOUN
cana-1710	25	12	,	,	PUNCT
cana-1710	25	13	and	and	CCONJ
cana-1710	25	14	∑	∑	ADP
cana-1710	25	15	𝜒𝑘	𝜒𝑘	NOUN
cana-1710	25	16	2	2	NUM
cana-1710	25	17	=	=	SYM
cana-1710	25	18	2|𝔈|	2|𝔈|	NUM
cana-1710	25	19	+	+	CCONJ
cana-1710	25	20	|𝒟|	|𝒟|	NOUN
cana-1710	25	21	=	=	SYM
cana-1710	25	22	2𝑚	2𝑚	NOUN
cana-1710	26	1	+	+	CCONJ
cana-1710	26	2	|𝑑|𝑛	|𝑑|𝑛	PROPN
cana-1710	26	3	𝑘=1	𝑘=1	NOUN
cana-1710	26	4	we	we	PRON
cana-1710	26	5	derive	derive	VERB
cana-1710	26	6	some	some	DET
cana-1710	26	7	upper	upper	ADJ
cana-1710	26	8	and	and	CCONJ
cana-1710	26	9	lower	low	ADJ
cana-1710	26	10	bounds	bound	NOUN
cana-1710	26	11	for	for	ADP
cana-1710	26	12	the	the	DET
cana-1710	26	13	mde	mde	NOUN
cana-1710	26	14	,	,	PUNCT
cana-1710	26	15	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	PROPN
cana-1710	26	16	)	)	PUNCT
cana-1710	26	17	,	,	PUNCT
cana-1710	26	18	in	in	ADP
cana-1710	26	19	this	this	DET
cana-1710	26	20	study	study	NOUN
cana-1710	26	21	.	.	PUNCT
cana-1710	27	1	2	2	X
cana-1710	27	2	.	.	X
cana-1710	27	3	upper	upper	ADJ
cana-1710	27	4	bounds	bound	NOUN
cana-1710	27	5	for	for	ADP
cana-1710	27	6	mde	mde	NOUN
cana-1710	27	7	throughout	throughout	ADP
cana-1710	27	8	this	this	DET
cana-1710	27	9	series	series	NOUN
cana-1710	27	10	𝔄	𝔄	PROPN
cana-1710	27	11	denotes	denote	VERB
cana-1710	27	12	a	a	DET
cana-1710	27	13	simple	simple	ADJ
cana-1710	27	14	graph	graph	NOUN
cana-1710	27	15	.	.	PUNCT
cana-1710	28	1	this	this	DET
cana-1710	28	2	section	section	NOUN
cana-1710	28	3	is	be	AUX
cana-1710	28	4	aimed	aim	VERB
cana-1710	28	5	to	to	PART
cana-1710	28	6	discuss	discuss	VERB
cana-1710	28	7	upper	upper	ADJ
cana-1710	28	8	bounds	bound	NOUN
cana-1710	28	9	for	for	ADP
cana-1710	28	10	mde	mde	PROPN
cana-1710	28	11	of	of	ADP
cana-1710	28	12	𝔄.	𝔄.	PROPN
cana-1710	28	13	theorem	theorem	VERB
cana-1710	28	14	2.1	2.1	NUM
cana-1710	28	15	let	let	VERB
cana-1710	28	16	𝔄	𝔄	PROPN
cana-1710	28	17	be	be	AUX
cana-1710	28	18	graph	graph	NOUN
cana-1710	28	19	of	of	ADP
cana-1710	28	20	order	order	NOUN
cana-1710	28	21	𝑛	𝑛	NOUN
cana-1710	28	22	and	and	CCONJ
cana-1710	28	23	size	size	VERB
cana-1710	28	24	𝑚.	𝑚.	ADV
cana-1710	28	25	then	then	ADV
cana-1710	28	26	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	PROPN
cana-1710	28	27	)	)	PUNCT
cana-1710	28	28	≤	≤	NOUN
cana-1710	28	29	√	√	NUM
cana-1710	28	30	(	(	PUNCT
cana-1710	28	31	1	1	NUM
cana-1710	28	32	2	2	NUM
cana-1710	28	33	(	(	PUNCT
cana-1710	28	34	𝑛2	𝑛2	NOUN
cana-1710	28	35	+	+	CCONJ
cana-1710	28	36	|𝑑|2	|𝑑|2	PROPN
cana-1710	28	37	)	)	PUNCT
cana-1710	29	1	+	+	NUM
cana-1710	30	1	2𝑚(𝑚	2𝑚(𝑚	NUM
cana-1710	30	2	+	+	CCONJ
cana-1710	30	3	|𝑑|	|𝑑|	NOUN
cana-1710	30	4	)	)	PUNCT
cana-1710	30	5	)	)	PUNCT
cana-1710	31	1	proof	proof	NOUN
cana-1710	31	2	:	:	PUNCT
cana-1710	31	3	we	we	PRON
cana-1710	31	4	recall	recall	VERB
cana-1710	31	5	the	the	DET
cana-1710	31	6	following	follow	VERB
cana-1710	31	7	well	well	ADV
cana-1710	31	8	-	-	PUNCT
cana-1710	31	9	known	know	VERB
cana-1710	31	10	inequality	inequality	NOUN
cana-1710	31	11	from	from	ADP
cana-1710	31	12	[	[	X
cana-1710	31	13	11	11	NUM
cana-1710	31	14	]	]	NUM
cana-1710	31	15	:	:	PUNCT
cana-1710	31	16	(	(	PUNCT
cana-1710	31	17	∑	∑	PROPN
cana-1710	31	18	𝑝𝑘𝑒𝑘	𝑝𝑘𝑒𝑘	NOUN
cana-1710	31	19	2	2	NUM
cana-1710	31	20	𝑛	𝑛	DET
cana-1710	31	21	𝑘=1	𝑘=1	NOUN
cana-1710	31	22	)	)	PUNCT
cana-1710	31	23	(	(	PUNCT
cana-1710	31	24	∑	∑	PUNCT
cana-1710	31	25	𝑞𝑘𝑓𝑘	𝑞𝑘𝑓𝑘	PROPN
cana-1710	31	26	2	2	NUM
cana-1710	31	27	𝑛	𝑛	NOUN
cana-1710	31	28	𝑘=1	𝑘=1	PROPN
cana-1710	31	29	)	)	PUNCT
cana-1710	32	1	+	+	CCONJ
cana-1710	32	2	(	(	PUNCT
cana-1710	32	3	∑	∑	ADP
cana-1710	32	4	𝑝𝑘𝑔𝑘	𝑝𝑘𝑔𝑘	VERB
cana-1710	32	5	2	2	NUM
cana-1710	32	6	𝑛	𝑛	DET
cana-1710	32	7	𝑘=1	𝑘=1	NOUN
cana-1710	32	8	)	)	PUNCT
cana-1710	32	9	(	(	PUNCT
cana-1710	32	10	∑	∑	PUNCT
cana-1710	32	11	𝑞𝑘ℎ𝑘	𝑞𝑘ℎ𝑘	PROPN
cana-1710	32	12	2	2	NUM
cana-1710	32	13	𝑛	𝑛	PRON
cana-1710	32	14	𝑘=1	𝑘=1	PROPN
cana-1710	32	15	)	)	PUNCT
cana-1710	32	16	≥	≥	NOUN
cana-1710	32	17	2	2	NUM
cana-1710	32	18	(	(	PUNCT
cana-1710	32	19	∑	∑	PART
cana-1710	32	20	𝑝𝑘𝑒𝑘𝑔𝑘	𝑝𝑘𝑒𝑘𝑔𝑘	VERB
cana-1710	32	21	𝑛	𝑛	DET
cana-1710	32	22	𝑘=1	𝑘=1	NOUN
cana-1710	32	23	)	)	PUNCT
cana-1710	32	24	(	(	PUNCT
cana-1710	32	25	∑	∑	PUNCT
cana-1710	32	26	𝑞𝑘𝑓𝑘ℎ𝑘	𝑞𝑘𝑓𝑘ℎ𝑘	VERB
cana-1710	32	27	𝑛	𝑛	DET
cana-1710	32	28	𝑘=1	𝑘=1	NOUN
cana-1710	32	29	)	)	PUNCT
cana-1710	32	30	(	(	PUNCT
cana-1710	32	31	2.1	2.1	NUM
cana-1710	32	32	)	)	PUNCT
cana-1710	32	33	where	where	SCONJ
cana-1710	32	34	𝑒𝑘	𝑒𝑘	X
cana-1710	32	35	,	,	PUNCT
cana-1710	32	36	𝑓𝑘	𝑓𝑘	NOUN
cana-1710	32	37	,	,	PUNCT
cana-1710	32	38	𝑔𝑘	𝑔𝑘	ADP
cana-1710	32	39	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1710	32	40	ℎ𝑘denote	ℎ𝑘denote	NOUN
cana-1710	32	41	sequence	sequence	NOUN
cana-1710	32	42	of	of	ADP
cana-1710	32	43	real	real	ADJ
cana-1710	32	44	numbers	number	NOUN
cana-1710	32	45	;	;	PUNCT
cana-1710	32	46	𝑝𝑘	𝑝𝑘	PART
cana-1710	32	47	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-1710	32	48	𝑞𝑘	𝑞𝑘	ADP
cana-1710	32	49	denote	denote	VERB
cana-1710	32	50	non	non	ADJ
cana-1710	32	51	-	-	ADJ
cana-1710	32	52	negative	negative	ADJ
cana-1710	32	53	numbers	number	NOUN
cana-1710	32	54	for	for	ADP
cana-1710	32	55	1	1	NUM
cana-1710	32	56	≤	≤	NOUN
cana-1710	32	57	𝑘	𝑘	DET
cana-1710	32	58	≤	≤	ADJ
cana-1710	32	59	𝑛.	𝑛.	NOUN
cana-1710	32	60	for	for	ADP
cana-1710	32	61	𝑝𝑘	𝑝𝑘	NOUN
cana-1710	32	62	=	=	SYM
cana-1710	32	63	𝑞𝑘	𝑞𝑘	ADP
cana-1710	32	64	=	=	PUNCT
cana-1710	32	65	𝑒𝑘	𝑒𝑘	NOUN
cana-1710	32	66	=	=	PUNCT
cana-1710	32	67	𝑓𝑘	𝑓𝑘	NOUN
cana-1710	32	68	=	=	SYM
cana-1710	32	69	1	1	NUM
cana-1710	32	70	and	and	CCONJ
cana-1710	32	71	𝑔𝑘	𝑔𝑘	ADP
cana-1710	32	72	=	=	PUNCT
cana-1710	32	73	ℎ𝑘	ℎ𝑘	PROPN
cana-1710	32	74	=	=	SYM
cana-1710	32	75	|𝜒𝑘|	|𝜒𝑘|	PROPN
cana-1710	32	76	,	,	PUNCT
cana-1710	32	77	1	1	NUM
cana-1710	32	78	≤	≤	NOUN
cana-1710	32	79	𝑘	𝑘	DET
cana-1710	32	80	≤	≤	NUM
cana-1710	32	81	𝑛	𝑛	PROPN
cana-1710	32	82	,	,	PUNCT
cana-1710	32	83	the	the	DET
cana-1710	32	84	inequality	inequality	NOUN
cana-1710	32	85	(	(	PUNCT
cana-1710	32	86	2.1	2.1	NUM
cana-1710	32	87	)	)	PUNCT
cana-1710	32	88	reduces	reduce	VERB
cana-1710	32	89	to	to	PART
cana-1710	32	90	(	(	PUNCT
cana-1710	32	91	∑	∑	PROPN
cana-1710	32	92	1	1	NUM
cana-1710	32	93	𝑛	𝑛	PRON
cana-1710	32	94	𝑘=1	𝑘=1	NOUN
cana-1710	32	95	)	)	PUNCT
cana-1710	32	96	(	(	PUNCT
cana-1710	32	97	∑	∑	PROPN
cana-1710	32	98	1	1	NUM
cana-1710	32	99	𝑛	𝑛	PRON
cana-1710	32	100	𝑘=1	𝑘=1	NOUN
cana-1710	32	101	)	)	PUNCT
cana-1710	33	1	+	+	CCONJ
cana-1710	33	2	(	(	PUNCT
cana-1710	33	3	∑|𝜒𝑘|2	∑|𝜒𝑘|2	ADJ
cana-1710	33	4	𝑛	𝑛	PROPN
cana-1710	33	5	𝑘=1	𝑘=1	PROPN
cana-1710	33	6	)	)	PUNCT
cana-1710	33	7	(	(	PUNCT
cana-1710	33	8	∑|𝜒𝑘|2	∑|𝜒𝑘|2	NOUN
cana-1710	33	9	𝑛	𝑛	PROPN
cana-1710	33	10	𝑘=1	𝑘=1	PROPN
cana-1710	33	11	)	)	PUNCT
cana-1710	33	12	≥	≥	NOUN
cana-1710	33	13	2	2	NUM
cana-1710	33	14	(	(	PUNCT
cana-1710	33	15	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	33	16	𝑛	𝑛	PRON
cana-1710	33	17	𝑘=1	𝑘=1	PROPN
cana-1710	33	18	)	)	PUNCT
cana-1710	33	19	(	(	PUNCT
cana-1710	33	20	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	33	21	𝑛	𝑛	PRON
cana-1710	33	22	𝑘=1	𝑘=1	NOUN
cana-1710	33	23	)	)	PUNCT
cana-1710	33	24	.	.	PUNCT
cana-1710	34	1	using	use	VERB
cana-1710	34	2	,	,	PUNCT
cana-1710	34	3	∑|𝜒𝑘|2	∑|𝜒𝑘|2	ADJ
cana-1710	34	4	𝑛	𝑛	PRON
cana-1710	34	5	𝑘=1	𝑘=1	PROPN
cana-1710	34	6	=	=	PUNCT
cana-1710	34	7	∑	∑	PUNCT
cana-1710	34	8	𝜒𝑘	𝜒𝑘	NOUN
cana-1710	34	9	2	2	NUM
cana-1710	34	10	𝑛	𝑛	PRON
cana-1710	34	11	𝑘=1	𝑘=1	NOUN
cana-1710	34	12	=	=	SYM
cana-1710	34	13	2𝑚	2𝑚	NOUN
cana-1710	34	14	+	+	CCONJ
cana-1710	34	15	|𝑑|	|𝑑|	NOUN
cana-1710	34	16	in	in	ADP
cana-1710	34	17	the	the	DET
cana-1710	34	18	above	above	ADJ
cana-1710	34	19	inequality	inequality	NOUN
cana-1710	34	20	,	,	PUNCT
cana-1710	34	21	we	we	PRON
cana-1710	34	22	deduce	deduce	VERB
cana-1710	34	23	that	that	DET
cana-1710	34	24	𝑛.	𝑛.	NOUN
cana-1710	34	25	𝑛	𝑛	ADP
cana-1710	34	26	+	+	CCONJ
cana-1710	34	27	(	(	PUNCT
cana-1710	34	28	|𝑑|	|𝑑|	NOUN
cana-1710	34	29	+	+	CCONJ
cana-1710	34	30	2𝑚)(|𝑑|	2𝑚)(|𝑑|	NUM
cana-1710	34	31	+	+	NUM
cana-1710	34	32	2𝑚	2𝑚	NOUN
cana-1710	34	33	)	)	PUNCT
cana-1710	34	34	≥	≥	NOUN
cana-1710	34	35	2	2	NUM
cana-1710	34	36	.	.	PUNCT
cana-1710	34	37	ε𝑀𝒟(𝔄).ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄).ε𝑀𝒟(𝔄	NOUN
cana-1710	34	38	)	)	PUNCT
cana-1710	34	39	which	which	PRON
cana-1710	34	40	gives	give	VERB
cana-1710	34	41	2	2	NUM
cana-1710	34	42	.	.	PUNCT
cana-1710	35	1	ε𝑀𝒟(𝔄)2	ε𝑀𝒟(𝔄)2	VERB
cana-1710	35	2	≤	≤	PROPN
cana-1710	35	3	𝑛2	𝑛2	NOUN
cana-1710	35	4	+	+	CCONJ
cana-1710	35	5	(	(	PUNCT
cana-1710	35	6	|𝑑|	|𝑑|	NOUN
cana-1710	35	7	+	+	CCONJ
cana-1710	35	8	2𝑚)2	2𝑚)2	NUM
cana-1710	35	9	.	.	PUNCT
cana-1710	36	1	communications	communication	NOUN
cana-1710	36	2	on	on	ADP
cana-1710	36	3	applied	apply	VERB
cana-1710	36	4	nonlinear	nonlinear	ADJ
cana-1710	36	5	analysis	analysis	NOUN
cana-1710	36	6	issn	issn	NOUN
cana-1710	36	7	:	:	PUNCT
cana-1710	36	8	1074	1074	NUM
cana-1710	36	9	-	-	PUNCT
cana-1710	36	10	133x	133x	NUM
cana-1710	36	11	vol	vol	NOUN
cana-1710	36	12	32	32	NUM
cana-1710	36	13	no	no	NOUN
cana-1710	36	14	.	.	NOUN
cana-1710	36	15	2	2	NUM
cana-1710	36	16	(	(	PUNCT
cana-1710	36	17	2025	2025	NUM
cana-1710	36	18	)	)	PUNCT
cana-1710	36	19	66	66	NUM
cana-1710	36	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1710	36	21	hence	hence	ADV
cana-1710	36	22	,	,	PUNCT
cana-1710	36	23	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	36	24	)	)	PUNCT
cana-1710	36	25	≤	≤	NOUN
cana-1710	36	26	√	√	NUM
cana-1710	36	27	(	(	PUNCT
cana-1710	36	28	1	1	NUM
cana-1710	36	29	2	2	NUM
cana-1710	36	30	(	(	PUNCT
cana-1710	36	31	𝑛2	𝑛2	NOUN
cana-1710	36	32	+	+	CCONJ
cana-1710	36	33	|𝑑|2	|𝑑|2	PROPN
cana-1710	36	34	)	)	PUNCT
cana-1710	36	35	+	+	NUM
cana-1710	36	36	2𝑚(𝑚	2𝑚(𝑚	NUM
cana-1710	36	37	+	+	CCONJ
cana-1710	36	38	|𝑑|	|𝑑|	NOUN
cana-1710	36	39	)	)	PUNCT
cana-1710	36	40	)	)	PUNCT
cana-1710	36	41	.	.	PUNCT
cana-1710	37	1	theorem	theorem	VERB
cana-1710	37	2	2.2	2.2	NUM
cana-1710	37	3	let	let	VERB
cana-1710	37	4	𝔄	𝔄	PROPN
cana-1710	37	5	be	be	AUX
cana-1710	37	6	graph	graph	NOUN
cana-1710	37	7	of	of	ADP
cana-1710	37	8	order	order	NOUN
cana-1710	37	9	𝑛	𝑛	NOUN
cana-1710	37	10	and	and	CCONJ
cana-1710	37	11	size	size	VERB
cana-1710	37	12	𝑚.	𝑚.	ADV
cana-1710	37	13	then	then	ADV
cana-1710	37	14	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	PROPN
cana-1710	37	15	)	)	PUNCT
cana-1710	37	16	≤	≤	NOUN
cana-1710	37	17	𝑚	𝑚	ADP
cana-1710	37	18	+	+	NOUN
cana-1710	37	19	1	1	NUM
cana-1710	37	20	2	2	NUM
cana-1710	37	21	(	(	PUNCT
cana-1710	37	22	𝑛	𝑛	PROPN
cana-1710	37	23	+	+	X
cana-1710	37	24	|𝑑|	|𝑑|	NOUN
cana-1710	37	25	)	)	PUNCT
cana-1710	37	26	.	.	PUNCT
cana-1710	38	1	proof	proof	NOUN
cana-1710	38	2	:	:	PUNCT
cana-1710	38	3	we	we	PRON
cana-1710	38	4	recall	recall	VERB
cana-1710	38	5	the	the	DET
cana-1710	38	6	following	follow	VERB
cana-1710	38	7	well	well	ADV
cana-1710	38	8	-	-	PUNCT
cana-1710	38	9	known	know	VERB
cana-1710	38	10	inequality	inequality	NOUN
cana-1710	38	11	from	from	ADP
cana-1710	38	12	[	[	X
cana-1710	38	13	11	11	NUM
cana-1710	38	14	]	]	NUM
cana-1710	38	15	:	:	PUNCT
cana-1710	38	16	(	(	PUNCT
cana-1710	38	17	∑	∑	PROPN
cana-1710	38	18	𝑝𝑘𝑒𝑘	𝑝𝑘𝑒𝑘	NOUN
cana-1710	38	19	2	2	NUM
cana-1710	38	20	𝑛	𝑛	DET
cana-1710	38	21	𝑘=1	𝑘=1	NOUN
cana-1710	38	22	)	)	PUNCT
cana-1710	38	23	(	(	PUNCT
cana-1710	38	24	∑	∑	PUNCT
cana-1710	38	25	𝑞𝑘𝑓𝑘	𝑞𝑘𝑓𝑘	PROPN
cana-1710	38	26	2	2	NUM
cana-1710	38	27	𝑛	𝑛	NOUN
cana-1710	38	28	𝑘=1	𝑘=1	PROPN
cana-1710	38	29	)	)	PUNCT
cana-1710	39	1	+	+	CCONJ
cana-1710	39	2	(	(	PUNCT
cana-1710	39	3	∑	∑	ADP
cana-1710	39	4	𝑝𝑘𝑔𝑘	𝑝𝑘𝑔𝑘	VERB
cana-1710	39	5	2	2	NUM
cana-1710	39	6	𝑛	𝑛	DET
cana-1710	39	7	𝑘=1	𝑘=1	NOUN
cana-1710	39	8	)	)	PUNCT
cana-1710	39	9	(	(	PUNCT
cana-1710	39	10	∑	∑	PUNCT
cana-1710	39	11	𝑞𝑘ℎ𝑘	𝑞𝑘ℎ𝑘	PROPN
cana-1710	39	12	2	2	NUM
cana-1710	39	13	𝑛	𝑛	PRON
cana-1710	39	14	𝑘=1	𝑘=1	PROPN
cana-1710	39	15	)	)	PUNCT
cana-1710	39	16	≥	≥	NOUN
cana-1710	39	17	2	2	NUM
cana-1710	39	18	(	(	PUNCT
cana-1710	39	19	∑	∑	PART
cana-1710	39	20	𝑝𝑘𝑒𝑘𝑔𝑘	𝑝𝑘𝑒𝑘𝑔𝑘	VERB
cana-1710	39	21	𝑛	𝑛	DET
cana-1710	39	22	𝑘=1	𝑘=1	NOUN
cana-1710	39	23	)	)	PUNCT
cana-1710	39	24	(	(	PUNCT
cana-1710	39	25	∑	∑	PUNCT
cana-1710	39	26	𝑞𝑘𝑓𝑘ℎ𝑘	𝑞𝑘𝑓𝑘ℎ𝑘	VERB
cana-1710	39	27	𝑛	𝑛	DET
cana-1710	39	28	𝑘=1	𝑘=1	NOUN
cana-1710	39	29	)	)	PUNCT
cana-1710	39	30	(	(	PUNCT
cana-1710	39	31	2.2	2.2	NUM
cana-1710	39	32	)	)	PUNCT
cana-1710	39	33	where	where	SCONJ
cana-1710	39	34	𝑒𝑘	𝑒𝑘	X
cana-1710	39	35	,	,	PUNCT
cana-1710	39	36	𝑓𝑘	𝑓𝑘	NOUN
cana-1710	39	37	,	,	PUNCT
cana-1710	39	38	𝑔𝑘	𝑔𝑘	ADP
cana-1710	39	39	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1710	39	40	ℎ𝑘denote	ℎ𝑘denote	NOUN
cana-1710	39	41	sequence	sequence	NOUN
cana-1710	39	42	of	of	ADP
cana-1710	39	43	real	real	ADJ
cana-1710	39	44	numbers	number	NOUN
cana-1710	39	45	;	;	PUNCT
cana-1710	39	46	𝑝𝑘	𝑝𝑘	PART
cana-1710	39	47	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-1710	39	48	𝑞𝑘	𝑞𝑘	ADP
cana-1710	39	49	denote	denote	VERB
cana-1710	39	50	non	non	ADJ
cana-1710	39	51	-	-	ADJ
cana-1710	39	52	negative	negative	ADJ
cana-1710	39	53	numbers	number	NOUN
cana-1710	39	54	for	for	ADP
cana-1710	39	55	1	1	NUM
cana-1710	39	56	≤	≤	NOUN
cana-1710	39	57	𝑘	𝑘	DET
cana-1710	39	58	≤	≤	ADJ
cana-1710	39	59	𝑛.	𝑛.	NOUN
cana-1710	39	60	for	for	ADP
cana-1710	39	61	𝑝𝑘	𝑝𝑘	NOUN
cana-1710	39	62	=	=	SYM
cana-1710	39	63	𝑞𝑘	𝑞𝑘	ADP
cana-1710	39	64	=	=	PUNCT
cana-1710	39	65	𝑒𝑘	𝑒𝑘	NOUN
cana-1710	39	66	=	=	PUNCT
cana-1710	39	67	𝑓𝑘	𝑓𝑘	NOUN
cana-1710	39	68	=	=	PUNCT
cana-1710	39	69	ℎ𝑘	ℎ𝑘	PROPN
cana-1710	39	70	=	=	SYM
cana-1710	39	71	1	1	NUM
cana-1710	39	72	and	and	CCONJ
cana-1710	39	73	𝑔𝑘	𝑔𝑘	ADP
cana-1710	39	74	=	=	SYM
cana-1710	39	75	|𝜒𝑘|	|𝜒𝑘|	PROPN
cana-1710	39	76	,	,	PUNCT
cana-1710	39	77	1	1	NUM
cana-1710	39	78	≤	≤	NOUN
cana-1710	39	79	𝑘	𝑘	DET
cana-1710	39	80	≤	≤	NUM
cana-1710	39	81	𝑛	𝑛	PROPN
cana-1710	39	82	,	,	PUNCT
cana-1710	39	83	the	the	DET
cana-1710	39	84	inequality	inequality	NOUN
cana-1710	39	85	(	(	PUNCT
cana-1710	39	86	2.2	2.2	NUM
cana-1710	39	87	)	)	PUNCT
cana-1710	39	88	yields	yield	NOUN
cana-1710	39	89	(	(	PUNCT
cana-1710	39	90	∑	∑	PROPN
cana-1710	39	91	1	1	NUM
cana-1710	39	92	𝑛	𝑛	DET
cana-1710	39	93	𝑘=1	𝑘=1	NOUN
cana-1710	39	94	)	)	PUNCT
cana-1710	39	95	(	(	PUNCT
cana-1710	39	96	∑	∑	PROPN
cana-1710	39	97	1	1	NUM
cana-1710	39	98	𝑛	𝑛	PRON
cana-1710	39	99	𝑘=1	𝑘=1	NOUN
cana-1710	39	100	)	)	PUNCT
cana-1710	40	1	+	+	CCONJ
cana-1710	40	2	(	(	PUNCT
cana-1710	40	3	∑|𝜒𝑘|2	∑|𝜒𝑘|2	ADJ
cana-1710	40	4	𝑛	𝑛	PROPN
cana-1710	40	5	𝑘=1	𝑘=1	PROPN
cana-1710	40	6	)	)	PUNCT
cana-1710	40	7	(	(	PUNCT
cana-1710	40	8	∑	∑	PROPN
cana-1710	40	9	1	1	NUM
cana-1710	40	10	𝑛	𝑛	PRON
cana-1710	40	11	𝑘=1	𝑘=1	PROPN
cana-1710	40	12	)	)	PUNCT
cana-1710	40	13	≥	≥	NOUN
cana-1710	40	14	2	2	NUM
cana-1710	40	15	(	(	PUNCT
cana-1710	40	16	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	40	17	𝑛	𝑛	PRON
cana-1710	40	18	𝑘=1	𝑘=1	PROPN
cana-1710	40	19	)	)	PUNCT
cana-1710	40	20	(	(	PUNCT
cana-1710	40	21	∑	∑	PROPN
cana-1710	40	22	1	1	NUM
cana-1710	40	23	𝑛	𝑛	PRON
cana-1710	40	24	𝑘=1	𝑘=1	NOUN
cana-1710	40	25	)	)	PUNCT
cana-1710	40	26	.	.	PUNCT
cana-1710	41	1	that	that	PRON
cana-1710	41	2	is	be	AUX
cana-1710	41	3	,	,	PUNCT
cana-1710	41	4	𝑛2	𝑛2	NOUN
cana-1710	41	5	+	+	CCONJ
cana-1710	41	6	(	(	PUNCT
cana-1710	41	7	∑|𝜒𝑘|2	∑|𝜒𝑘|2	ADJ
cana-1710	41	8	𝑛	𝑛	PROPN
cana-1710	41	9	𝑘=1	𝑘=1	PROPN
cana-1710	41	10	)	)	PUNCT
cana-1710	42	1	𝑛	𝑛	PRON
cana-1710	42	2	≥	≥	NUM
cana-1710	42	3	2𝑛.	2𝑛.	NOUN
cana-1710	42	4	(	(	PUNCT
cana-1710	42	5	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	42	6	𝑛	𝑛	PRON
cana-1710	42	7	𝑘=1	𝑘=1	PROPN
cana-1710	42	8	)	)	PUNCT
cana-1710	42	9	which	which	PRON
cana-1710	42	10	gives	give	VERB
cana-1710	42	11	,	,	PUNCT
cana-1710	42	12	𝑛	𝑛	DET
cana-1710	42	13	+	+	NUM
cana-1710	42	14	2𝑚	2𝑚	NOUN
cana-1710	42	15	+	+	CCONJ
cana-1710	42	16	|𝑑|	|𝑑|	ADJ
cana-1710	42	17	≥	≥	NOUN
cana-1710	42	18	2ε𝑀𝒟(𝔄	2ε𝑀𝒟(𝔄	NUM
cana-1710	42	19	)	)	PUNCT
cana-1710	42	20	.	.	PUNCT
cana-1710	43	1	hence	hence	ADV
cana-1710	43	2	,	,	PUNCT
cana-1710	43	3	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	43	4	)	)	PUNCT
cana-1710	43	5	≤	≤	NOUN
cana-1710	43	6	𝑚	𝑚	ADP
cana-1710	43	7	+	+	NOUN
cana-1710	43	8	1	1	NUM
cana-1710	43	9	2	2	NUM
cana-1710	43	10	(	(	PUNCT
cana-1710	43	11	𝑛	𝑛	PROPN
cana-1710	43	12	+	+	X
cana-1710	43	13	|𝑑|	|𝑑|	NOUN
cana-1710	43	14	)	)	PUNCT
cana-1710	43	15	.	.	PUNCT
cana-1710	44	1	3	3	X
cana-1710	44	2	.	.	X
cana-1710	44	3	lower	low	ADJ
cana-1710	44	4	bounds	bound	NOUN
cana-1710	44	5	for	for	ADP
cana-1710	44	6	mde	mde	NOUN
cana-1710	44	7	throughout	throughout	ADP
cana-1710	44	8	this	this	DET
cana-1710	44	9	section	section	NOUN
cana-1710	44	10	𝔄	𝔄	PROPN
cana-1710	44	11	denotes	denote	VERB
cana-1710	44	12	a	a	DET
cana-1710	44	13	simple	simple	ADJ
cana-1710	44	14	graph	graph	NOUN
cana-1710	44	15	.	.	PUNCT
cana-1710	45	1	this	this	DET
cana-1710	45	2	section	section	NOUN
cana-1710	45	3	is	be	AUX
cana-1710	45	4	aimed	aim	VERB
cana-1710	45	5	to	to	PART
cana-1710	45	6	discuss	discuss	VERB
cana-1710	45	7	lower	low	ADJ
cana-1710	45	8	bounds	bound	NOUN
cana-1710	45	9	for	for	ADP
cana-1710	45	10	mde	mde	PROPN
cana-1710	45	11	of	of	ADP
cana-1710	45	12	𝔄.	𝔄.	PROPN
cana-1710	45	13	theorem	theorem	PROPN
cana-1710	45	14	3.1	3.1	NUM
cana-1710	45	15	let	let	VERB
cana-1710	45	16	𝔄	𝔄	PROPN
cana-1710	45	17	be	be	AUX
cana-1710	45	18	a	a	DET
cana-1710	45	19	bipartite	bipartite	ADJ
cana-1710	45	20	graph	graph	NOUN
cana-1710	45	21	of	of	ADP
cana-1710	45	22	order	order	NOUN
cana-1710	45	23	𝑛	𝑛	PRON
cana-1710	45	24	≥	≥	NUM
cana-1710	45	25	2	2	NUM
cana-1710	45	26	and	and	CCONJ
cana-1710	45	27	size	size	NOUN
cana-1710	45	28	𝑚	𝑚	PROPN
cana-1710	45	29	with	with	ADP
cana-1710	45	30	spectral	spectral	ADJ
cana-1710	45	31	radius	radius	NOUN
cana-1710	45	32	𝜒1	𝜒1	PROPN
cana-1710	45	33	.	.	PUNCT
cana-1710	46	1	then	then	ADV
cana-1710	46	2	|𝑑|+2𝑚	|𝑑|+2𝑚	PROPN
cana-1710	46	3	𝜒1	𝜒1	PROPN
cana-1710	46	4	≤	≤	NUM
cana-1710	46	5	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	46	6	)	)	PUNCT
cana-1710	46	7	.	.	PUNCT
cana-1710	47	1	proof	proof	NOUN
cana-1710	47	2	:	:	PUNCT
cana-1710	47	3	let	let	VERB
cana-1710	47	4	𝑒𝑘	𝑒𝑘	PRON
cana-1710	47	5	,	,	PUNCT
cana-1710	47	6	𝑓𝑘	𝑓𝑘	PRON
cana-1710	47	7	be	be	VERB
cana-1710	47	8	non	non	ADJ
cana-1710	47	9	-	-	ADJ
cana-1710	47	10	negative	negative	ADJ
cana-1710	47	11	decreasing	decrease	VERB
cana-1710	47	12	sequences	sequence	NOUN
cana-1710	47	13	where	where	SCONJ
cana-1710	47	14	𝑒𝑘	𝑒𝑘	X
cana-1710	47	15	,	,	PUNCT
cana-1710	47	16	𝑓𝑘	𝑓𝑘	ADP
cana-1710	47	17	≠	≠	PROPN
cana-1710	47	18	0	0	NUM
cana-1710	47	19	,	,	PUNCT
cana-1710	47	20	and	and	CCONJ
cana-1710	47	21	𝑗𝑘	𝑗𝑘	PART
cana-1710	47	22	be	be	AUX
cana-1710	47	23	a	a	DET
cana-1710	47	24	non	non	ADJ
cana-1710	47	25	-	-	ADJ
cana-1710	47	26	negative	negative	ADJ
cana-1710	47	27	sequence	sequence	NOUN
cana-1710	47	28	for	for	ADP
cana-1710	47	29	1	1	NUM
cana-1710	47	30	≤	≤	NOUN
cana-1710	47	31	𝑘	𝑘	DET
cana-1710	47	32	≤	≤	ADJ
cana-1710	47	33	𝑛.	𝑛.	NOUN
cana-1710	47	34	then	then	ADV
cana-1710	47	35	we	we	PRON
cana-1710	47	36	have	have	VERB
cana-1710	47	37	the	the	DET
cana-1710	47	38	following	follow	VERB
cana-1710	47	39	inequality	inequality	NOUN
cana-1710	47	40	[	[	X
cana-1710	47	41	11	11	NUM
cana-1710	47	42	]	]	NUM
cana-1710	47	43	:	:	PUNCT
cana-1710	47	44	(	(	PUNCT
cana-1710	47	45	∑	∑	PUNCT
cana-1710	47	46	𝑗𝑘𝑒𝑘	𝑗𝑘𝑒𝑘	NOUN
cana-1710	47	47	2	2	NUM
cana-1710	47	48	𝑛	𝑛	PRON
cana-1710	47	49	𝑘=1	𝑘=1	NOUN
cana-1710	47	50	)	)	PUNCT
cana-1710	47	51	(	(	PUNCT
cana-1710	47	52	∑	∑	PUNCT
cana-1710	47	53	𝑗𝑘𝑓𝑘	𝑗𝑘𝑓𝑘	PROPN
cana-1710	47	54	2	2	NUM
cana-1710	47	55	𝑛	𝑛	PRON
cana-1710	47	56	𝑘=1	𝑘=1	NOUN
cana-1710	47	57	)	)	PUNCT
cana-1710	47	58	≤	≤	NUM
cana-1710	47	59	max	max	PROPN
cana-1710	47	60	{	{	PUNCT
cana-1710	47	61	𝑓1	𝑓1	PROPN
cana-1710	47	62	∑	∑	PUNCT
cana-1710	47	63	𝑗𝑘𝑒𝑘	𝑗𝑘𝑒𝑘	NOUN
cana-1710	47	64	2	2	NUM
cana-1710	47	65	𝑛	𝑛	PRON
cana-1710	47	66	𝑘=1	𝑘=1	ADJ
cana-1710	47	67	,	,	PUNCT
cana-1710	47	68	𝑒1	𝑒1	PROPN
cana-1710	47	69	∑	∑	PUNCT
cana-1710	47	70	𝑗𝑘𝑓𝑘	𝑗𝑘𝑓𝑘	PROPN
cana-1710	47	71	2	2	NUM
cana-1710	47	72	𝑛	𝑛	PRON
cana-1710	47	73	𝑘=1	𝑘=1	NOUN
cana-1710	47	74	}	}	PUNCT
cana-1710	47	75	(	(	PUNCT
cana-1710	47	76	∑	∑	PUNCT
cana-1710	47	77	𝑗𝑘𝑒𝑘𝑓𝑘	𝑗𝑘𝑒𝑘𝑓𝑘	NOUN
cana-1710	47	78	𝑛	𝑛	PRON
cana-1710	47	79	𝑘=1	𝑘=1	NOUN
cana-1710	47	80	)	)	PUNCT
cana-1710	47	81	(	(	PUNCT
cana-1710	47	82	3.1	3.1	NUM
cana-1710	47	83	)	)	PUNCT
cana-1710	47	84	for	for	ADP
cana-1710	47	85	𝑒𝑘	𝑒𝑘	NOUN
cana-1710	47	86	=	=	PUNCT
cana-1710	47	87	𝑓𝑘	𝑓𝑘	NOUN
cana-1710	47	88	=	=	PUNCT
cana-1710	47	89	|𝜒𝑘|	|𝜒𝑘|	PROPN
cana-1710	47	90	and	and	CCONJ
cana-1710	47	91	𝑗𝑘	𝑗𝑘	ADP
cana-1710	47	92	=	=	NOUN
cana-1710	47	93	1	1	NUM
cana-1710	47	94	,	,	PUNCT
cana-1710	47	95	1	1	NUM
cana-1710	47	96	≤	≤	NOUN
cana-1710	47	97	𝑘	𝑘	DET
cana-1710	47	98	≤	≤	NUM
cana-1710	47	99	𝑛	𝑛	PROPN
cana-1710	47	100	,	,	PUNCT
cana-1710	47	101	the	the	DET
cana-1710	47	102	inequality	inequality	NOUN
cana-1710	47	103	(	(	PUNCT
cana-1710	47	104	3.1	3.1	NUM
cana-1710	47	105	)	)	PUNCT
cana-1710	47	106	gives	give	VERB
cana-1710	47	107	,	,	PUNCT
cana-1710	47	108	communications	communication	NOUN
cana-1710	47	109	on	on	ADP
cana-1710	47	110	applied	apply	VERB
cana-1710	47	111	nonlinear	nonlinear	ADJ
cana-1710	47	112	analysis	analysis	NOUN
cana-1710	47	113	issn	issn	NOUN
cana-1710	47	114	:	:	PUNCT
cana-1710	47	115	1074	1074	NUM
cana-1710	47	116	-	-	PUNCT
cana-1710	47	117	133x	133x	NUM
cana-1710	47	118	vol	vol	NOUN
cana-1710	47	119	32	32	NUM
cana-1710	47	120	no	no	NOUN
cana-1710	47	121	.	.	NOUN
cana-1710	47	122	2	2	NUM
cana-1710	47	123	(	(	PUNCT
cana-1710	47	124	2025	2025	NUM
cana-1710	47	125	)	)	PUNCT
cana-1710	47	126	67	67	NUM
cana-1710	47	127	https://internationalpubls.com	https://internationalpubls.com	X
cana-1710	47	128	(	(	PUNCT
cana-1710	47	129	∑	∑	PROPN
cana-1710	47	130	1	1	NUM
cana-1710	47	131	.	.	PUNCT
cana-1710	47	132	|𝜒𝑘|2	|𝜒𝑘|2	PUNCT
cana-1710	47	133	𝑛	𝑛	PROPN
cana-1710	47	134	𝑘=1	𝑘=1	NOUN
cana-1710	47	135	)	)	PUNCT
cana-1710	47	136	(	(	PUNCT
cana-1710	47	137	∑	∑	PROPN
cana-1710	47	138	1	1	NUM
cana-1710	47	139	.	.	PUNCT
cana-1710	47	140	|𝜒𝑘|2	|𝜒𝑘|2	PUNCT
cana-1710	47	141	𝑛	𝑛	PROPN
cana-1710	47	142	𝑘=1	𝑘=1	NOUN
cana-1710	47	143	)	)	PUNCT
cana-1710	47	144	≤	≤	NUM
cana-1710	47	145	max	max	PROPN
cana-1710	47	146	{	{	PUNCT
cana-1710	47	147	𝜒1	𝜒1	PROPN
cana-1710	47	148	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	47	149	𝑛	𝑛	PRON
cana-1710	47	150	𝑘=1	𝑘=1	PROPN
cana-1710	47	151	,	,	PUNCT
cana-1710	47	152	𝜒1	𝜒1	VERB
cana-1710	47	153	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	47	154	𝑛	𝑛	PRON
cana-1710	47	155	𝑘=1	𝑘=1	NOUN
cana-1710	47	156	}	}	PUNCT
cana-1710	47	157	(	(	PUNCT
cana-1710	47	158	∑|𝜒𝑘|2	∑|𝜒𝑘|2	NOUN
cana-1710	47	159	𝑛	𝑛	PROPN
cana-1710	47	160	𝑘=1	𝑘=1	PROPN
cana-1710	47	161	)	)	PUNCT
cana-1710	47	162	that	that	PRON
cana-1710	47	163	is	be	AUX
cana-1710	47	164	,	,	PUNCT
cana-1710	47	165	∑|𝜒𝑘|2	∑|𝜒𝑘|2	ADJ
cana-1710	47	166	𝑛	𝑛	PRON
cana-1710	47	167	𝑘=1	𝑘=1	PROPN
cana-1710	47	168	≤	≤	NUM
cana-1710	47	169	𝜒1	𝜒1	VERB
cana-1710	47	170	∑|𝜒𝑘|	∑|𝜒𝑘|	NOUN
cana-1710	47	171	𝑛	𝑛	PRON
cana-1710	47	172	𝑘=1	𝑘=1	NOUN
cana-1710	47	173	which	which	PRON
cana-1710	47	174	implies	imply	VERB
cana-1710	47	175	𝜒1ε𝑀𝒟(𝔄	𝜒1ε𝑀𝒟(𝔄	NOUN
cana-1710	47	176	)	)	PUNCT
cana-1710	47	177	≥	≥	NOUN
cana-1710	47	178	∑|𝜒𝑘|2	∑|𝜒𝑘|2	PROPN
cana-1710	47	179	𝑛	𝑛	DET
cana-1710	47	180	𝑘=1	𝑘=1	PROPN
cana-1710	47	181	hence	hence	ADV
cana-1710	47	182	,	,	PUNCT
cana-1710	47	183	|𝑑|	|𝑑|	NOUN
cana-1710	47	184	+	+	CCONJ
cana-1710	47	185	2𝑚	2𝑚	NOUN
cana-1710	47	186	𝜒1	𝜒1	VERB
cana-1710	47	187	≤	≤	NUM
cana-1710	47	188	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	47	189	)	)	PUNCT
cana-1710	47	190	lemma	lemma	PROPN
cana-1710	47	191	1	1	NUM
cana-1710	48	1	[	[	X
cana-1710	48	2	22	22	NUM
cana-1710	48	3	]	]	PUNCT
cana-1710	48	4	let	let	VERB
cana-1710	48	5	𝑛	𝑛	PART
cana-1710	48	6	be	be	AUX
cana-1710	48	7	a	a	DET
cana-1710	48	8	positive	positive	ADJ
cana-1710	48	9	integer	integer	NOUN
cana-1710	48	10	.	.	PUNCT
cana-1710	49	1	if	if	SCONJ
cana-1710	49	2	𝑙1	𝑙1	PROPN
cana-1710	49	3	,	,	PUNCT
cana-1710	49	4	𝑙2	𝑙2	PROPN
cana-1710	49	5	,	,	PUNCT
cana-1710	49	6	…	…	PUNCT
cana-1710	49	7	,	,	PUNCT
cana-1710	49	8	𝑙𝑛	𝑙𝑛	NOUN
cana-1710	49	9	are	be	AUX
cana-1710	49	10	non	non	ADJ
cana-1710	49	11	-	-	ADJ
cana-1710	49	12	negative	negative	ADJ
cana-1710	49	13	numbers	number	NOUN
cana-1710	49	14	with	with	ADP
cana-1710	49	15	𝑙1	𝑙1	PROPN
cana-1710	49	16	≥	≥	PROPN
cana-1710	49	17	𝑙2	𝑙2	PROPN
cana-1710	49	18	≥	≥	PROPN
cana-1710	49	19	⋯	⋯	PROPN
cana-1710	49	20	≥	≥	PROPN
cana-1710	49	21	𝑙𝑛	𝑙𝑛	NOUN
cana-1710	49	22	,	,	PUNCT
cana-1710	49	23	then	then	ADV
cana-1710	49	24	(	(	PUNCT
cana-1710	49	25	𝑙1	𝑙1	PROPN
cana-1710	49	26	+	+	CCONJ
cana-1710	49	27	𝑙2	𝑙2	PROPN
cana-1710	49	28	+	+	CCONJ
cana-1710	49	29	…	…	PUNCT
cana-1710	49	30	+	+	NUM
cana-1710	49	31	𝑙𝑛)(𝑙1	𝑙𝑛)(𝑙1	SYM
cana-1710	49	32	+	+	NUM
cana-1710	49	33	𝑙𝑛	𝑙𝑛	NOUN
cana-1710	49	34	)	)	PUNCT
cana-1710	49	35	≥	≥	NOUN
cana-1710	49	36	𝑙1	𝑙1	NOUN
cana-1710	49	37	2	2	NUM
cana-1710	50	1	+	+	CCONJ
cana-1710	50	2	⋯	⋯	VERB
cana-1710	50	3	+	+	CCONJ
cana-1710	50	4	𝑙𝑛	𝑙𝑛	NOUN
cana-1710	50	5	2	2	NUM
cana-1710	50	6	+	+	NUM
cana-1710	50	7	𝑛𝑙1𝑙𝑛	𝑛𝑙1𝑙𝑛	NUM
cana-1710	50	8	(	(	PUNCT
cana-1710	50	9	3.2	3.2	NUM
cana-1710	50	10	)	)	PUNCT
cana-1710	50	11	further	far	ADV
cana-1710	50	12	,	,	PUNCT
cana-1710	50	13	equality	equality	NOUN
cana-1710	50	14	holds	hold	VERB
cana-1710	50	15	in	in	ADP
cana-1710	50	16	(	(	PUNCT
cana-1710	50	17	3.2	3.2	NUM
cana-1710	50	18	)	)	PUNCT
cana-1710	50	19	if	if	SCONJ
cana-1710	51	1	and	and	CCONJ
cana-1710	51	2	only	only	ADV
cana-1710	51	3	if	if	SCONJ
cana-1710	51	4	for	for	ADP
cana-1710	51	5	some	some	DET
cana-1710	51	6	𝑟	𝑟	NOUN
cana-1710	51	7	,	,	PUNCT
cana-1710	51	8	1	1	NUM
cana-1710	51	9	≤	≤	NUM
cana-1710	51	10	𝑟	𝑟	PRON
cana-1710	51	11	≤	≤	ADJ
cana-1710	51	12	𝑛	𝑛	PROPN
cana-1710	51	13	,	,	PUNCT
cana-1710	51	14	𝑙1	𝑙1	NOUN
cana-1710	51	15	=	=	SYM
cana-1710	51	16	∙∙∙	∙∙∙	NOUN
cana-1710	51	17	=	=	PUNCT
cana-1710	51	18	𝑙𝑟	𝑙𝑟	NOUN
cana-1710	51	19	and	and	CCONJ
cana-1710	51	20	𝑙𝑟+1	𝑙𝑟+1	NOUN
cana-1710	51	21	=	=	PUNCT
cana-1710	51	22	∙∙∙	∙∙∙	X
cana-1710	51	23	=	=	SYM
cana-1710	51	24	𝑙𝑛.	𝑙𝑛.	NOUN
cana-1710	51	25	theorem	theorem	NOUN
cana-1710	51	26	3.2	3.2	NUM
cana-1710	51	27	let	let	VERB
cana-1710	51	28	𝔄	𝔄	PROPN
cana-1710	51	29	be	be	AUX
cana-1710	51	30	a	a	DET
cana-1710	51	31	graph	graph	NOUN
cana-1710	51	32	with	with	ADP
cana-1710	51	33	order	order	NOUN
cana-1710	51	34	𝑛	𝑛	PRON
cana-1710	51	35	≥	≥	NUM
cana-1710	51	36	2	2	NUM
cana-1710	51	37	and	and	CCONJ
cana-1710	51	38	size	size	NOUN
cana-1710	51	39	𝑚	𝑚	PROPN
cana-1710	51	40	≥	≥	NUM
cana-1710	51	41	1	1	NUM
cana-1710	51	42	.	.	PUNCT
cana-1710	51	43	assume	assume	VERB
cana-1710	51	44	that	that	SCONJ
cana-1710	51	45	𝜒1	𝜒1	VERB
cana-1710	51	46	,	,	PUNCT
cana-1710	51	47	…	…	PUNCT
cana-1710	51	48	,	,	PUNCT
cana-1710	51	49	𝜒𝑛	𝜒𝑛	PROPN
cana-1710	51	50	are	be	AUX
cana-1710	51	51	all	all	PRON
cana-1710	51	52	eigenvalues	eigenvalue	NOUN
cana-1710	51	53	of	of	ADP
cana-1710	51	54	𝔄	𝔄	NOUN
cana-1710	51	55	,	,	PUNCT
cana-1710	51	56	such	such	ADJ
cana-1710	51	57	that	that	SCONJ
cana-1710	51	58	|𝜒𝑛|	|𝜒𝑛|	PROPN
cana-1710	51	59	≥	≥	NUM
cana-1710	51	60	⋯	⋯	NOUN
cana-1710	51	61	≥	≥	PROPN
cana-1710	51	62	|𝜒1|	|𝜒1|	NOUN
cana-1710	51	63	≥	≥	NOUN
cana-1710	51	64	0	0	NUM
cana-1710	51	65	,	,	PUNCT
cana-1710	51	66	then	then	ADV
cana-1710	51	67	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	PROPN
cana-1710	51	68	)	)	PUNCT
cana-1710	51	69	≥	≥	NOUN
cana-1710	51	70	2√(2𝑚	2√(2𝑚	NUM
cana-1710	51	71	+	+	NUM
cana-1710	51	72	|𝑑|)𝑛√|𝜒1𝜒𝑛|	|𝑑|)𝑛√|𝜒1𝜒𝑛|	NOUN
cana-1710	51	73	|𝜒1|	|𝜒1|	NOUN
cana-1710	51	74	+	+	CCONJ
cana-1710	51	75	|𝜒𝑛|	|𝜒𝑛|	NUM
cana-1710	51	76	proof	proof	NOUN
cana-1710	51	77	:	:	PUNCT
cana-1710	51	78	since	since	SCONJ
cana-1710	51	79	there	there	PRON
cana-1710	51	80	is	be	VERB
cana-1710	51	81	at	at	ADV
cana-1710	51	82	least	least	ADJ
cana-1710	51	83	one	one	NUM
cana-1710	51	84	edge	edge	NOUN
cana-1710	51	85	in	in	ADP
cana-1710	51	86	the	the	DET
cana-1710	51	87	graph	graph	NOUN
cana-1710	51	88	𝔄	𝔄	PROPN
cana-1710	51	89	,	,	PUNCT
cana-1710	51	90	it	it	PRON
cana-1710	51	91	follows	follow	VERB
cana-1710	51	92	that	that	SCONJ
cana-1710	51	93	𝔄	𝔄	PROPN
cana-1710	51	94	has	have	VERB
cana-1710	51	95	at	at	ADV
cana-1710	51	96	least	least	ADV
cana-1710	51	97	one	one	NUM
cana-1710	51	98	eigenvalue	eigenvalue	NOUN
cana-1710	51	99	different	different	ADJ
cana-1710	51	100	from	from	ADP
cana-1710	51	101	zero	zero	NUM
cana-1710	51	102	.	.	PUNCT
cana-1710	52	1	applying	apply	VERB
cana-1710	52	2	lemma	lemma	PROPN
cana-1710	52	3	1	1	NUM
cana-1710	52	4	,	,	PUNCT
cana-1710	52	5	we	we	PRON
cana-1710	52	6	get	get	AUX
cana-1710	52	7	(	(	PUNCT
cana-1710	52	8	|𝜒1|	|𝜒1|	NOUN
cana-1710	52	9	+	+	CCONJ
cana-1710	52	10	⋯	⋯	VERB
cana-1710	52	11	+	+	PROPN
cana-1710	52	12	|𝜒𝑛|)(|𝜒1|	|𝜒𝑛|)(|𝜒1|	PROPN
cana-1710	53	1	+	+	CCONJ
cana-1710	53	2	|𝜒𝑛|	|𝜒𝑛|	NUM
cana-1710	53	3	)	)	PUNCT
cana-1710	53	4	≥	≥	NOUN
cana-1710	53	5	|𝜒1|2	|𝜒1|2	PUNCT
cana-1710	54	1	+	+	NUM
cana-1710	54	2	⋯	⋯	VERB
cana-1710	54	3	+	+	NOUN
cana-1710	54	4	|𝜒𝑛|2	|𝜒𝑛|2	PROPN
cana-1710	55	1	+	+	CCONJ
cana-1710	55	2	𝑛|𝜒1||𝜒𝑛|	𝑛|𝜒1||𝜒𝑛|	VERB
cana-1710	55	3	(	(	PUNCT
cana-1710	55	4	3.3	3.3	NUM
cana-1710	55	5	)	)	PUNCT
cana-1710	55	6	and	and	CCONJ
cana-1710	55	7	equality	equality	NOUN
cana-1710	55	8	holds	hold	VERB
cana-1710	55	9	in	in	ADP
cana-1710	55	10	(	(	PUNCT
cana-1710	55	11	3.3	3.3	NUM
cana-1710	55	12	)	)	PUNCT
cana-1710	55	13	if	if	SCONJ
cana-1710	55	14	and	and	CCONJ
cana-1710	55	15	only	only	ADV
cana-1710	55	16	if	if	SCONJ
cana-1710	55	17	|𝜒1|	|𝜒1|	NOUN
cana-1710	55	18	=	=	SYM
cana-1710	55	19	⋯	⋯	X
cana-1710	55	20	=	=	PUNCT
cana-1710	55	21	|𝜒𝑟|	|𝜒𝑟|	PROPN
cana-1710	55	22	and	and	CCONJ
cana-1710	55	23	|𝜒𝑟+1|	|𝜒𝑟+1|	X
cana-1710	55	24	=	=	SYM
cana-1710	55	25	⋯	⋯	NOUN
cana-1710	55	26	=	=	SYM
cana-1710	55	27	|𝜒𝑛|	|𝜒𝑛|	PROPN
cana-1710	55	28	for	for	ADP
cana-1710	55	29	some𝑟	some𝑟	NOUN
cana-1710	55	30	∈	∈	PROPN
cana-1710	55	31	1,∙∙∙	1,∙∙∙	PROPN
cana-1710	55	32	,	,	PUNCT
cana-1710	55	33	𝑛	𝑛	VERB
cana-1710	55	34	since	since	SCONJ
cana-1710	55	35	|𝜒1|2	|𝜒1|2	PROPN
cana-1710	55	36	+	+	NUM
cana-1710	55	37	⋯	⋯	VERB
cana-1710	55	38	+	+	CCONJ
cana-1710	55	39	|𝜒𝑛|2	|𝜒𝑛|2	NOUN
cana-1710	55	40	=	=	SYM
cana-1710	55	41	2𝑚	2𝑚	NOUN
cana-1710	55	42	+	+	CCONJ
cana-1710	55	43	|𝑑|	|𝑑|	NOUN
cana-1710	55	44	.	.	PUNCT
cana-1710	56	1	by	by	ADP
cana-1710	56	2	equation	equation	NOUN
cana-1710	56	3	(	(	PUNCT
cana-1710	56	4	3.3	3.3	NUM
cana-1710	56	5	)	)	PUNCT
cana-1710	56	6	we	we	PRON
cana-1710	56	7	get	get	VERB
cana-1710	56	8	,	,	PUNCT
cana-1710	56	9	ε𝑀𝒟(𝔄)(|𝜒1|	ε𝑀𝒟(𝔄)(|𝜒1|	PROPN
cana-1710	56	10	+	+	CCONJ
cana-1710	56	11	|𝜒𝑛|	|𝜒𝑛|	NUM
cana-1710	56	12	)	)	PUNCT
cana-1710	56	13	≥	≥	NOUN
cana-1710	56	14	2𝑚	2𝑚	NOUN
cana-1710	56	15	+	+	CCONJ
cana-1710	56	16	|𝑑|	|𝑑|	NOUN
cana-1710	56	17	+	+	CCONJ
cana-1710	56	18	𝑛|𝜒1||𝜒𝑛|	𝑛|𝜒1||𝜒𝑛|	VERB
cana-1710	56	19	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	56	20	)	)	PUNCT
cana-1710	56	21	≥	≥	NOUN
cana-1710	56	22	2𝑚	2𝑚	NOUN
cana-1710	56	23	+	+	CCONJ
cana-1710	56	24	|𝑑|	|𝑑|	NOUN
cana-1710	56	25	+	+	CCONJ
cana-1710	56	26	𝑛|𝜒1||𝜒𝑛|	𝑛|𝜒1||𝜒𝑛|	ADJ
cana-1710	56	27	|𝜒1|	|𝜒1|	NOUN
cana-1710	56	28	+	+	CCONJ
cana-1710	56	29	|𝜒𝑛|	|𝜒𝑛|	NUM
cana-1710	56	30	(	(	PUNCT
cana-1710	56	31	3.4	3.4	NUM
cana-1710	56	32	)	)	PUNCT
cana-1710	56	33	and	and	CCONJ
cana-1710	56	34	the	the	DET
cana-1710	56	35	equality	equality	NOUN
cana-1710	56	36	holds	hold	VERB
cana-1710	56	37	if	if	SCONJ
cana-1710	56	38	and	and	CCONJ
cana-1710	56	39	only	only	ADV
cana-1710	56	40	if	if	SCONJ
cana-1710	56	41	|𝜒1|	|𝜒1|	NOUN
cana-1710	56	42	=	=	SYM
cana-1710	57	1	⋯	⋯	X
cana-1710	57	2	=	=	PUNCT
cana-1710	57	3	|𝜒𝑟|	|𝜒𝑟|	PROPN
cana-1710	57	4	and	and	CCONJ
cana-1710	57	5	|𝜒𝑟+1|	|𝜒𝑟+1|	X
cana-1710	57	6	=	=	SYM
cana-1710	57	7	⋯	⋯	NOUN
cana-1710	57	8	=	=	SYM
cana-1710	57	9	|𝜒𝑛|	|𝜒𝑛|	PROPN
cana-1710	57	10	for	for	ADP
cana-1710	57	11	some	some	DET
cana-1710	57	12	𝑟	𝑟	NOUN
cana-1710	57	13	∈	∈	PROPN
cana-1710	57	14	1	1	NUM
cana-1710	57	15	,	,	PUNCT
cana-1710	57	16	⋯	⋯	NOUN
cana-1710	57	17	,	,	PUNCT
cana-1710	57	18	𝑛.	𝑛.	NOUN
cana-1710	57	19	we	we	PRON
cana-1710	57	20	all	all	PRON
cana-1710	57	21	know	know	VERB
cana-1710	57	22	that	that	SCONJ
cana-1710	57	23	for	for	ADP
cana-1710	57	24	every	every	DET
cana-1710	57	25	real	real	ADJ
cana-1710	57	26	number	number	NOUN
cana-1710	57	27	𝑎	𝑎	NOUN
cana-1710	57	28	≥	≥	NOUN
cana-1710	57	29	0	0	NUM
cana-1710	57	30	and	and	CCONJ
cana-1710	57	31	𝑏	𝑏	PRON
cana-1710	57	32	≥	≥	NOUN
cana-1710	57	33	0	0	NUM
cana-1710	57	34	,	,	PUNCT
cana-1710	57	35	𝑎	𝑎	PRON
cana-1710	57	36	+	+	X
cana-1710	57	37	𝑏	𝑏	PROPN
cana-1710	57	38	≥	≥	NOUN
cana-1710	57	39	2√𝑎𝑏	2√𝑎𝑏	NUM
cana-1710	57	40	and	and	CCONJ
cana-1710	57	41	equality	equality	NOUN
cana-1710	57	42	holds	hold	VERB
cana-1710	57	43	if	if	SCONJ
cana-1710	57	44	and	and	CCONJ
cana-1710	57	45	only	only	ADV
cana-1710	57	46	if	if	SCONJ
cana-1710	57	47	𝑎	𝑎	NOUN
cana-1710	57	48	=	=	NOUN
cana-1710	57	49	𝑏.	𝑏.	NOUN
cana-1710	57	50	using	use	VERB
cana-1710	57	51	this	this	DET
cana-1710	57	52	fact	fact	NOUN
cana-1710	57	53	in	in	ADP
cana-1710	57	54	(	(	PUNCT
cana-1710	57	55	3.4	3.4	NUM
cana-1710	57	56	)	)	PUNCT
cana-1710	57	57	,	,	PUNCT
cana-1710	57	58	we	we	PRON
cana-1710	57	59	get	get	VERB
cana-1710	57	60	communications	communication	NOUN
cana-1710	57	61	on	on	ADP
cana-1710	57	62	applied	apply	VERB
cana-1710	57	63	nonlinear	nonlinear	ADJ
cana-1710	57	64	analysis	analysis	NOUN
cana-1710	57	65	issn	issn	NOUN
cana-1710	57	66	:	:	PUNCT
cana-1710	57	67	1074	1074	NUM
cana-1710	57	68	-	-	PUNCT
cana-1710	57	69	133x	133x	NUM
cana-1710	57	70	vol	vol	NOUN
cana-1710	57	71	32	32	NUM
cana-1710	57	72	no	no	NOUN
cana-1710	57	73	.	.	NOUN
cana-1710	57	74	2	2	NUM
cana-1710	57	75	(	(	PUNCT
cana-1710	57	76	2025	2025	NUM
cana-1710	57	77	)	)	PUNCT
cana-1710	57	78	68	68	NUM
cana-1710	57	79	https://internationalpubls.com	https://internationalpubls.com	X
cana-1710	57	80	ε𝑀𝒟(𝔄	ε𝑀𝒟(𝔄	NOUN
cana-1710	57	81	)	)	PUNCT
cana-1710	57	82	≥	≥	NOUN
cana-1710	57	83	(	(	PUNCT
cana-1710	57	84	2𝑚	2𝑚	NOUN
cana-1710	57	85	+	+	CCONJ
cana-1710	57	86	|𝑑|	|𝑑|	NOUN
cana-1710	57	87	)	)	PUNCT
cana-1710	57	88	+	+	CCONJ
cana-1710	57	89	𝑛|𝜒1||𝜒𝑛|	𝑛|𝜒1||𝜒𝑛|	VERB
cana-1710	57	90	|𝜒1|	|𝜒1|	NOUN
cana-1710	57	91	+	+	CCONJ
cana-1710	57	92	|𝜒𝑛|	|𝜒𝑛|	PROPN
cana-1710	57	93	≥	≥	NUM
cana-1710	57	94	2√(2𝑚	2√(2𝑚	NUM
cana-1710	57	95	+	+	NUM
cana-1710	57	96	|𝑑|)𝑛|𝜒1𝜒𝑛|	|𝑑|)𝑛|𝜒1𝜒𝑛|	PROPN
cana-1710	57	97	|𝜒1|	|𝜒1|	NOUN
cana-1710	57	98	+	+	CCONJ
cana-1710	57	99	|𝜒𝑛|	|𝜒𝑛|	NUM
cana-1710	57	100	=	=	SYM
cana-1710	57	101	2√(2𝑚	2√(2𝑚	NUM
cana-1710	57	102	+	+	NUM
cana-1710	57	103	|𝑑|)𝑛√|𝜒1𝜒𝑛|	|𝑑|)𝑛√|𝜒1𝜒𝑛|	NOUN
cana-1710	57	104	|𝜒1|	|𝜒1|	NOUN
cana-1710	57	105	+	+	CCONJ
cana-1710	57	106	|𝜒𝑛|	|𝜒𝑛|	NOUN
cana-1710	57	107	this	this	PRON
cana-1710	57	108	proves	prove	VERB
cana-1710	57	109	the	the	DET
cana-1710	57	110	result	result	NOUN
cana-1710	57	111	.	.	PUNCT
cana-1710	58	1	acknowledgement	acknowledgement	NOUN
cana-1710	58	2	:	:	PUNCT
cana-1710	58	3	the	the	DET
cana-1710	58	4	authors	author	NOUN
cana-1710	58	5	are	be	AUX
cana-1710	58	6	thankful	thankful	ADJ
cana-1710	58	7	to	to	ADP
cana-1710	58	8	prof.chandrashekara	prof.chandrashekara	PROPN
cana-1710	58	9	adiga	adiga	PROPN
cana-1710	58	10	for	for	ADP
cana-1710	58	11	his	his	PRON
cana-1710	58	12	encouragement	encouragement	NOUN
cana-1710	58	13	and	and	CCONJ
cana-1710	58	14	suggestions	suggestion	NOUN
cana-1710	58	15	.	.	PUNCT
cana-1710	59	1	references	reference	NOUN
cana-1710	59	2	[	[	X
cana-1710	59	3	1	1	NUM
cana-1710	59	4	]	]	PUNCT
cana-1710	59	5	c.	c.	PROPN
cana-1710	59	6	adiga	adiga	PROPN
cana-1710	59	7	,	,	PUNCT
cana-1710	59	8	abdelmejid	abdelmejid	PROPN
cana-1710	59	9	bayad	bayad	PROPN
cana-1710	59	10	,	,	PUNCT
cana-1710	59	11	ivan	ivan	PROPN
cana-1710	59	12	gutman	gutman	PROPN
cana-1710	59	13	and	and	CCONJ
cana-1710	59	14	shrikanth	shrikanth	NOUN
cana-1710	59	15	a	a	DET
cana-1710	59	16	s	s	PROPN
cana-1710	59	17	,	,	PUNCT
cana-1710	59	18	the	the	DET
cana-1710	59	19	minimum	minimum	NOUN
cana-1710	59	20	covering	cover	VERB
cana-1710	59	21	energy	energy	NOUN
cana-1710	59	22	of	of	ADP
cana-1710	59	23	a	a	DET
cana-1710	59	24	graph	graph	NOUN
cana-1710	59	25	,	,	PUNCT
cana-1710	59	26	kragujevac	kragujevac	PROPN
cana-1710	59	27	j.	j.	PROPN
cana-1710	59	28	sci.vol	sci.vol	PROPN
cana-1710	59	29	.	.	PROPN
cana-1710	59	30	4	4	NUM
cana-1710	59	31	,	,	PUNCT
cana-1710	59	32	(	(	PUNCT
cana-1710	59	33	8)	8)	NUM
cana-1710	59	34	,	,	PUNCT
cana-1710	59	35	(	(	PUNCT
cana-1710	59	36	2009	2009	NUM
cana-1710	59	37	)	)	PUNCT
cana-1710	59	38	385	385	NUM
cana-1710	59	39	396	396	NUM
cana-1710	59	40	.	.	PUNCT
cana-1710	60	1	[	[	X
cana-1710	60	2	2	2	X
cana-1710	60	3	]	]	PUNCT
cana-1710	60	4	c.	c.	PROPN
cana-1710	60	5	adiga	adiga	PROPN
cana-1710	60	6	and	and	CCONJ
cana-1710	60	7	smith	smith	PROPN
cana-1710	60	8	m	m	PROPN
cana-1710	60	9	,	,	PUNCT
cana-1710	60	10	on	on	ADP
cana-1710	60	11	maximum	maximum	ADJ
cana-1710	60	12	degree	degree	NOUN
cana-1710	60	13	energy	energy	NOUN
cana-1710	60	14	of	of	ADP
cana-1710	60	15	a	a	DET
cana-1710	60	16	graph	graph	NOUN
cana-1710	60	17	,	,	PUNCT
cana-1710	60	18	int	int	NOUN
cana-1710	60	19	.	.	PUNCT
cana-1710	61	1	j.	j.	PROPN
cana-1710	61	2	contemp	contemp	PROPN
cana-1710	61	3	.	.	PUNCT
cana-1710	62	1	math	math	NOUN
cana-1710	62	2	.	.	PUNCT
cana-1710	63	1	sciences	science	NOUN
cana-1710	63	2	,	,	PUNCT
cana-1710	63	3	vol	vol	NOUN
cana-1710	63	4	.	.	PROPN
cana-1710	63	5	4	4	NUM
cana-1710	63	6	,	,	PUNCT
cana-1710	63	7	34,(2012	34,(2012	NUM
cana-1710	63	8	)	)	PUNCT
cana-1710	63	9	.	.	PUNCT
cana-1710	64	1	[	[	X
cana-1710	64	2	3	3	X
cana-1710	64	3	]	]	PUNCT
cana-1710	64	4	e.hükel	e.hükel	NOUN
cana-1710	64	5	,	,	PUNCT
cana-1710	64	6	quantentheoretische	quantentheoretische	NOUN
cana-1710	64	7	beiträge	beiträge	ADJ
cana-1710	64	8	zum	zum	PROPN
cana-1710	64	9	benzolproblem	benzolproblem	NOUN
cana-1710	64	10	i.	i.	PROPN
cana-1710	64	11	die	die	PROPN
cana-1710	64	12	elektronenkonfiguration	elektronenkonfiguration	PROPN
cana-1710	64	13	des	des	PROPN
cana-1710	64	14	benzols	benzol	NOUN
cana-1710	64	15	und	und	VERB
cana-1710	64	16	verwandter	verwandter	NOUN
cana-1710	64	17	vebindungen	vebindungen	NOUN
cana-1710	64	18	.	.	PUNCT
cana-1710	65	1	z.phys.70(1931	z.phys.70(1931	X
cana-1710	65	2	)	)	PUNCT
cana-1710	66	1	204	204	NUM
cana-1710	66	2	-	-	SYM
cana-1710	66	3	286	286	NUM
cana-1710	66	4	.	.	PUNCT
cana-1710	67	1	[	[	X
cana-1710	67	2	4	4	NUM
cana-1710	67	3	]	]	PUNCT
cana-1710	67	4	d.babic̀	d.babic̀	PROPN
cana-1710	67	5	and	and	CCONJ
cana-1710	67	6	i.gutman	i.gutman	NOUN
cana-1710	67	7	,	,	PUNCT
cana-1710	67	8	more	more	ADV
cana-1710	67	9	lower	low	ADJ
cana-1710	67	10	bounds	bound	NOUN
cana-1710	67	11	for	for	ADP
cana-1710	67	12	the	the	DET
cana-1710	67	13	total	total	ADJ
cana-1710	67	14	π	π	PROPN
cana-1710	67	15	−electron	−electron	ADP
cana-1710	67	16	energy	energy	NOUN
cana-1710	67	17	of	of	ADP
cana-1710	67	18	alternant	alternant	ADJ
cana-1710	67	19	hydrocarbons	hydrocarbon	NOUN
cana-1710	67	20	,	,	PUNCT
cana-1710	67	21	macth	macth	PROPN
cana-1710	67	22	commun	commun	PROPN
cana-1710	67	23	.	.	PUNCT
cana-1710	68	1	comput	comput	PROPN
cana-1710	68	2	.	.	PUNCT
cana-1710	68	3	,	,	PUNCT
cana-1710	68	4	32	32	NUM
cana-1710	68	5	(	(	PUNCT
cana-1710	68	6	1995	1995	NUM
cana-1710	68	7	)	)	PUNCT
cana-1710	68	8	,	,	PUNCT
cana-1710	68	9	7	7	NUM
cana-1710	68	10	-	-	SYM
cana-1710	68	11	17	17	NUM
cana-1710	68	12	.	.	PUNCT
cana-1710	69	1	[	[	X
cana-1710	69	2	5	5	X
cana-1710	69	3	]	]	PUNCT
cana-1710	69	4	d.	d.	PROPN
cana-1710	69	5	cvetković	cvetković	PROPN
cana-1710	69	6	,	,	PUNCT
cana-1710	69	7	i.	i.	PROPN
cana-1710	69	8	gutman	gutman	PROPN
cana-1710	69	9	(	(	PUNCT
cana-1710	69	10	eds	eds	PROPN
cana-1710	69	11	.	.	PUNCT
cana-1710	69	12	)	)	PUNCT
cana-1710	69	13	,	,	PUNCT
cana-1710	69	14	selected	select	VERB
cana-1710	69	15	topics	topic	NOUN
cana-1710	69	16	on	on	ADP
cana-1710	69	17	applications	application	NOUN
cana-1710	69	18	of	of	ADP
cana-1710	69	19	graph	graph	NOUN
cana-1710	69	20	spectra	spectra	PROPN
cana-1710	69	21	,	,	PUNCT
cana-1710	69	22	math.inst	math.inst	PROPN
cana-1710	69	23	.	.	PROPN
cana-1710	69	24	,	,	PUNCT
cana-1710	69	25	belgrade	belgrade	PROPN
cana-1710	69	26	,	,	PUNCT
cana-1710	69	27	2011	2011	NUM
cana-1710	69	28	.	.	PUNCT
cana-1710	70	1	[	[	X
cana-1710	70	2	6	6	NUM
cana-1710	70	3	]	]	PUNCT
cana-1710	70	4	i.	i.	NOUN
cana-1710	70	5	gutman	gutman	PROPN
cana-1710	70	6	,	,	PUNCT
cana-1710	70	7	topological	topological	ADJ
cana-1710	70	8	studies	study	NOUN
cana-1710	70	9	on	on	ADP
cana-1710	70	10	heteroconjugated	heteroconjugate	VERB
cana-1710	70	11	molecules	molecule	NOUN
cana-1710	70	12	.	.	PUNCT
cana-1710	71	1	vi	vi	X
cana-1710	71	2	.	.	PROPN
cana-1710	71	3	alternant	alternant	PROPN
cana-1710	71	4	systems	system	NOUN
cana-1710	71	5	with	with	ADP
cana-1710	71	6	two	two	NUM
cana-1710	71	7	heteroatoms	heteroatom	NOUN
cana-1710	71	8	,	,	PUNCT
cana-1710	71	9	z.	z.	PROPN
cana-1710	71	10	naturforsch	naturforsch	PROPN
cana-1710	71	11	.	.	PUNCT
cana-1710	71	12	,	,	PUNCT
cana-1710	71	13	45a	45a	X
cana-1710	71	14	(	(	PUNCT
cana-1710	71	15	1990),1085–1089	1990),1085–1089	NOUN
cana-1710	71	16	.	.	PUNCT
cana-1710	72	1	[	[	X
cana-1710	72	2	7	7	NUM
cana-1710	72	3	]	]	X
cana-1710	72	4	i.	i.	NOUN
cana-1710	72	5	gutman	gutman	PROPN
cana-1710	72	6	,	,	PUNCT
cana-1710	72	7	total	total	ADJ
cana-1710	72	8	π	π	PROPN
cana-1710	72	9	−electron	−electron	ADP
cana-1710	72	10	energy	energy	NOUN
cana-1710	72	11	of	of	ADP
cana-1710	72	12	benzenoid	benzenoid	NOUN
cana-1710	72	13	hydrocarbons	hydrocarbon	NOUN
cana-1710	72	14	,	,	PUNCT
cana-1710	72	15	topics	topic	NOUN
cana-1710	72	16	curr	curr	X
cana-1710	72	17	.	.	PUNCT
cana-1710	72	18	chem	chem	PROPN
cana-1710	72	19	.	.	PUNCT
cana-1710	73	1	,162	,162	PUNCT
cana-1710	73	2	(	(	PUNCT
cana-1710	73	3	1992	1992	NUM
cana-1710	73	4	)	)	PUNCT
cana-1710	73	5	,	,	PUNCT
cana-1710	73	6	29	29	NUM
cana-1710	73	7	-	-	SYM
cana-1710	73	8	63	63	NUM
cana-1710	73	9	.	.	PUNCT
cana-1710	74	1	[	[	X
cana-1710	74	2	8	8	NUM
cana-1710	74	3	]	]	X
cana-1710	74	4	i.	i.	PROPN
cana-1710	74	5	gutman	gutman	PROPN
cana-1710	74	6	,	,	PUNCT
cana-1710	74	7	the	the	DET
cana-1710	74	8	energy	energy	NOUN
cana-1710	74	9	of	of	ADP
cana-1710	74	10	a	a	DET
cana-1710	74	11	graph	graph	NOUN
cana-1710	74	12	:	:	PUNCT
cana-1710	74	13	old	old	ADJ
cana-1710	74	14	and	and	CCONJ
cana-1710	74	15	new	new	ADJ
cana-1710	74	16	results	result	NOUN
cana-1710	74	17	,	,	PUNCT
cana-1710	74	18	combinatorics	combinatoric	NOUN
cana-1710	74	19	and	and	CCONJ
cana-1710	74	20	applications	application	NOUN
cana-1710	74	21	,	,	PUNCT
cana-1710	74	22	a.	a.	NOUN
cana-1710	74	23	betten	betten	NOUN
cana-1710	74	24	,	,	PUNCT
cana-1710	74	25	a.	a.	NOUN
cana-1710	74	26	khoner	khoner	PROPN
cana-1710	74	27	,	,	PUNCT
cana-1710	74	28	r.	r.	PROPN
cana-1710	74	29	laue	laue	PROPN
cana-1710	74	30	and	and	CCONJ
cana-1710	74	31	a.	a.	NOUN
cana-1710	74	32	wassermann	wassermann	PROPN
cana-1710	74	33	,	,	PUNCT
cana-1710	74	34	eds	eds	PROPN
cana-1710	74	35	.	.	PROPN
cana-1710	74	36	,	,	PUNCT
cana-1710	74	37	springer	springer	NOUN
cana-1710	74	38	,	,	PUNCT
cana-1710	74	39	berlin	berlin	PROPN
cana-1710	74	40	,	,	PUNCT
cana-1710	74	41	(	(	PUNCT
cana-1710	74	42	2001	2001	NUM
cana-1710	74	43	)	)	PUNCT
cana-1710	74	44	,	,	PUNCT
cana-1710	74	45	196	196	NUM
cana-1710	74	46	-	-	SYM
cana-1710	74	47	211	211	NUM
cana-1710	74	48	.	.	PUNCT
cana-1710	75	1	[	[	X
cana-1710	75	2	9	9	NUM
cana-1710	75	3	]	]	SYM
cana-1710	75	4	i.	i.	NOUN
cana-1710	75	5	gutman	gutman	PROPN
cana-1710	75	6	,	,	PUNCT
cana-1710	75	7	topology	topology	NOUN
cana-1710	75	8	and	and	CCONJ
cana-1710	75	9	stability	stability	NOUN
cana-1710	75	10	of	of	ADP
cana-1710	75	11	conjugated	conjugated	ADJ
cana-1710	75	12	hydrocarbons	hydrocarbon	NOUN
cana-1710	75	13	.	.	PUNCT
cana-1710	76	1	the	the	DET
cana-1710	76	2	dependence	dependence	NOUN
cana-1710	76	3	of	of	ADP
cana-1710	76	4	total	total	ADJ
cana-1710	76	5	π	π	PROPN
cana-1710	76	6	−electron	−electron	ADP
cana-1710	76	7	energy	energy	NOUN
cana-1710	76	8	on	on	ADP
cana-1710	76	9	molecular	molecular	ADJ
cana-1710	76	10	topology	topology	NOUN
cana-1710	76	11	,	,	PUNCT
cana-1710	76	12	j.	j.	PROPN
cana-1710	76	13	serb	serb	PROPN
cana-1710	76	14	.	.	PUNCT
cana-1710	77	1	chem	chem	PROPN
cana-1710	77	2	.	.	PUNCT
cana-1710	78	1	soc	soc	PROPN
cana-1710	78	2	.	.	PUNCT
cana-1710	78	3	,	,	PUNCT
cana-1710	78	4	70	70	NUM
cana-1710	78	5	(	(	PUNCT
cana-1710	78	6	2005	2005	NUM
cana-1710	78	7	)	)	PUNCT
cana-1710	78	8	,	,	PUNCT
cana-1710	78	9	441	441	NUM
cana-1710	78	10	-	-	NUM
cana-1710	78	11	456	456	NUM
cana-1710	78	12	.	.	PUNCT
cana-1710	79	1	[	[	X
cana-1710	79	2	10	10	NUM
cana-1710	79	3	]	]	X
cana-1710	79	4	s.s.dragomir	s.s.dragomir	NOUN
cana-1710	79	5	,	,	PUNCT
cana-1710	79	6	a	a	DET
cana-1710	79	7	survey	survey	NOUN
cana-1710	79	8	on	on	ADP
cana-1710	79	9	cauchy	cauchy	NOUN
cana-1710	79	10	-	-	PUNCT
cana-1710	79	11	bunyakovsky	bunyakovsky	NOUN
cana-1710	79	12	-	-	PUNCT
cana-1710	79	13	schwarz	schwarz	NOUN
cana-1710	79	14	type	type	NOUN
cana-1710	79	15	discreate	discreate	NOUN
cana-1710	79	16	inequalities	inequality	NOUN
cana-1710	79	17	,	,	PUNCT
cana-1710	79	18	j.inequal.pure	j.inequal.pure	PROPN
cana-1710	79	19	appl.math.4	appl.math.4	PROPN
cana-1710	79	20	(	(	PUNCT
cana-1710	79	21	2003	2003	NUM
cana-1710	79	22	)	)	PUNCT
cana-1710	79	23	,	,	PUNCT
cana-1710	79	24	no	no	INTJ
cana-1710	79	25	.	.	PUNCT
cana-1710	80	1	3,1	3,1	NUM
cana-1710	80	2	-	-	SYM
cana-1710	80	3	142	142	NUM
cana-1710	80	4	.	.	PUNCT
cana-1710	81	1	[	[	X
cana-1710	81	2	11	11	NUM
cana-1710	81	3	]	]	PUNCT
cana-1710	81	4	i.	i.	PROPN
cana-1710	81	5	gutman	gutman	PROPN
cana-1710	81	6	,	,	PUNCT
cana-1710	81	7	the	the	DET
cana-1710	81	8	energy	energy	NOUN
cana-1710	81	9	of	of	ADP
cana-1710	81	10	a	a	DET
cana-1710	81	11	graph	graph	NOUN
cana-1710	81	12	,	,	PUNCT
cana-1710	81	13	ber	ber	PROPN
cana-1710	81	14	.	.	PUNCT
cana-1710	81	15	math	math	NOUN
cana-1710	81	16	.	.	PUNCT
cana-1710	82	1	stat	stat	PROPN
cana-1710	82	2	.	.	PUNCT
cana-1710	83	1	sekt	sekt	PROPN
cana-1710	83	2	.	.	PUNCT
cana-1710	84	1	forschungsz	forschungsz	PROPN
cana-1710	84	2	.	.	PUNCT
cana-1710	85	1	graz	graz	PROPN
cana-1710	85	2	,	,	PUNCT
cana-1710	85	3	103(1978	103(1978	NUM
cana-1710	85	4	)	)	PUNCT
cana-1710	85	5	,	,	PUNCT
cana-1710	85	6	1	1	NUM
cana-1710	85	7	-	-	SYM
cana-1710	85	8	22	22	NUM
cana-1710	85	9	.	.	PUNCT
cana-1710	86	1	[	[	X
cana-1710	86	2	12	12	NUM
cana-1710	86	3	]	]	PUNCT
cana-1710	86	4	i.	i.	NOUN
cana-1710	86	5	gutman	gutman	PROPN
cana-1710	86	6	and	and	CCONJ
cana-1710	86	7	b.jhou	b.jhou	NOUN
cana-1710	86	8	,	,	PUNCT
cana-1710	86	9	laplacian	laplacian	ADJ
cana-1710	86	10	energy	energy	NOUN
cana-1710	86	11	of	of	ADP
cana-1710	86	12	a	a	DET
cana-1710	86	13	graph	graph	NOUN
cana-1710	86	14	,	,	PUNCT
cana-1710	86	15	lin	lin	PROPN
cana-1710	86	16	.	.	PUNCT
cana-1710	87	1	algebra	algebra	PROPN
cana-1710	87	2	appl,414	appl,414	PROPN
cana-1710	87	3	(	(	PUNCT
cana-1710	87	4	2006	2006	NUM
cana-1710	87	5	)	)	PUNCT
cana-1710	87	6	,	,	PUNCT
cana-1710	87	7	29	29	NUM
cana-1710	87	8	-	-	SYM
cana-1710	87	9	37	37	NUM
cana-1710	87	10	.	.	PUNCT
cana-1710	88	1	the	the	DET
cana-1710	88	2	energy	energy	NOUN
cana-1710	88	3	of	of	ADP
cana-1710	88	4	a	a	DET
cana-1710	88	5	graph	graph	NOUN
cana-1710	88	6	,	,	PUNCT
cana-1710	88	7	ber	ber	PROPN
cana-1710	88	8	.	.	PUNCT
cana-1710	88	9	math	math	NOUN
cana-1710	88	10	.	.	PUNCT
cana-1710	89	1	stat	stat	PROPN
cana-1710	89	2	.	.	PUNCT
cana-1710	90	1	sekt	sekt	PROPN
cana-1710	90	2	.	.	PUNCT
cana-1710	91	1	forschungsz	forschungsz	PROPN
cana-1710	91	2	.	.	PUNCT
cana-1710	92	1	graz	graz	PROPN
cana-1710	92	2	,	,	PUNCT
cana-1710	92	3	103(1978	103(1978	NUM
cana-1710	92	4	)	)	PUNCT
cana-1710	92	5	,	,	PUNCT
cana-1710	92	6	1	1	NUM
cana-1710	92	7	-	-	SYM
cana-1710	92	8	22	22	NUM
cana-1710	92	9	.	.	PUNCT
cana-1710	93	1	[	[	X
cana-1710	93	2	13	13	NUM
cana-1710	93	3	]	]	SYM
cana-1710	93	4	i.	i.	PROPN
cana-1710	93	5	gutman	gutman	PROPN
cana-1710	93	6	,	,	PUNCT
cana-1710	93	7	mcclelland	mcclelland	ADJ
cana-1710	93	8	-	-	PUNCT
cana-1710	93	9	type	type	NOUN
cana-1710	93	10	lower	lower	ADV
cana-1710	93	11	bound	bind	VERB
cana-1710	93	12	for	for	ADP
cana-1710	93	13	total	total	ADJ
cana-1710	93	14	π	π	PROPN
cana-1710	93	15	−electron	−electron	ADP
cana-1710	93	16	energy	energy	NOUN
cana-1710	93	17	,	,	PUNCT
cana-1710	93	18	j.	j.	PROPN
cana-1710	93	19	chem	chem	PROPN
cana-1710	93	20	.	.	PUNCT
cana-1710	94	1	soc	soc	PROPN
cana-1710	94	2	.	.	PUNCT
cana-1710	95	1	faraday	faraday	PROPN
cana-1710	95	2	trans	trans	PROPN
cana-1710	95	3	.	.	PROPN
cana-1710	95	4	,	,	PUNCT
cana-1710	95	5	86	86	NUM
cana-1710	95	6	(	(	PUNCT
cana-1710	95	7	1990	1990	NUM
cana-1710	95	8	)	)	PUNCT
cana-1710	95	9	,	,	PUNCT
cana-1710	95	10	3373	3373	NUM
cana-1710	95	11	-	-	SYM
cana-1710	95	12	3375	3375	NUM
cana-1710	95	13	.	.	PUNCT
cana-1710	96	1	[	[	X
cana-1710	96	2	14	14	NUM
cana-1710	96	3	]	]	X
cana-1710	96	4	i.	i.	PROPN
cana-1710	96	5	gutman	gutman	PROPN
cana-1710	96	6	,	,	PUNCT
cana-1710	96	7	total	total	ADJ
cana-1710	96	8	π	π	PROPN
cana-1710	96	9	−electron	−electron	ADP
cana-1710	96	10	energy	energy	NOUN
cana-1710	96	11	of	of	ADP
cana-1710	96	12	benzenoid	benzenoid	NOUN
cana-1710	96	13	hydrocarbons	hydrocarbon	NOUN
cana-1710	96	14	,	,	PUNCT
cana-1710	96	15	topics	topic	NOUN
cana-1710	96	16	curr	curr	X
cana-1710	96	17	.	.	PUNCT
cana-1710	96	18	chem	chem	PROPN
cana-1710	96	19	.	.	PUNCT
cana-1710	96	20	,	,	PUNCT
cana-1710	96	21	162	162	NUM
cana-1710	96	22	(	(	PUNCT
cana-1710	96	23	1992	1992	NUM
cana-1710	96	24	)	)	PUNCT
cana-1710	96	25	,	,	PUNCT
cana-1710	96	26	29	29	NUM
cana-1710	96	27	-	-	SYM
cana-1710	96	28	63	63	NUM
cana-1710	96	29	.	.	PUNCT
cana-1710	97	1	[	[	X
cana-1710	97	2	15	15	NUM
cana-1710	97	3	]	]	X
cana-1710	97	4	i.	i.	PROPN
cana-1710	97	5	gutman	gutman	PROPN
cana-1710	97	6	,	,	PUNCT
cana-1710	97	7	topology	topology	NOUN
cana-1710	97	8	and	and	CCONJ
cana-1710	97	9	stability	stability	NOUN
cana-1710	97	10	of	of	ADP
cana-1710	97	11	conjugated	conjugated	ADJ
cana-1710	97	12	hydrocarbons	hydrocarbon	NOUN
cana-1710	97	13	.	.	PUNCT
cana-1710	98	1	the	the	DET
cana-1710	98	2	dependence	dependence	NOUN
cana-1710	98	3	of	of	ADP
cana-1710	98	4	total	total	ADJ
cana-1710	98	5	π	π	PROPN
cana-1710	98	6	−electron	−electron	ADP
cana-1710	98	7	energy	energy	NOUN
cana-1710	98	8	on	on	ADP
cana-1710	98	9	molecular	molecular	ADJ
cana-1710	98	10	topology	topology	NOUN
cana-1710	98	11	,	,	PUNCT
cana-1710	98	12	j.	j.	PROPN
cana-1710	98	13	serb	serb	PROPN
cana-1710	98	14	.	.	PUNCT
cana-1710	99	1	chem	chem	PROPN
cana-1710	99	2	.	.	PUNCT
cana-1710	100	1	soc	soc	PROPN
cana-1710	100	2	.	.	PUNCT
cana-1710	100	3	,	,	PUNCT
cana-1710	100	4	70	70	NUM
cana-1710	100	5	(	(	PUNCT
cana-1710	100	6	2005	2005	NUM
cana-1710	100	7	)	)	PUNCT
cana-1710	100	8	,	,	PUNCT
cana-1710	100	9	441	441	NUM
cana-1710	100	10	-	-	NUM
cana-1710	100	11	456	456	NUM
cana-1710	100	12	.	.	PUNCT
cana-1710	101	1	[	[	X
cana-1710	101	2	16	16	NUM
cana-1710	101	3	]	]	PUNCT
cana-1710	101	4	g.indulal	g.indulal	PROPN
cana-1710	101	5	,	,	PUNCT
cana-1710	101	6	i.gutman	i.gutman	NOUN
cana-1710	101	7	,	,	PUNCT
cana-1710	101	8	a.vijaykumar	a.vijaykumar	NOUN
cana-1710	101	9	,	,	PUNCT
cana-1710	101	10	on	on	ADP
cana-1710	101	11	distance	distance	NOUN
cana-1710	101	12	energy	energy	NOUN
cana-1710	101	13	of	of	ADP
cana-1710	101	14	graphs	graph	NOUN
cana-1710	101	15	,	,	PUNCT
cana-1710	101	16	match	match	NOUN
cana-1710	101	17	commun	commun	PROPN
cana-1710	101	18	.	.	PUNCT
cana-1710	102	1	math.comput.chem	math.comput.chem	PROPN
cana-1710	102	2	.	.	PUNCT
cana-1710	103	1	60(2008	60(2008	NOUN
cana-1710	103	2	)	)	PUNCT
cana-1710	103	3	355	355	NUM
cana-1710	103	4	-	-	SYM
cana-1710	103	5	372	372	NUM
cana-1710	103	6	.	.	PUNCT
cana-1710	104	1	[	[	X
cana-1710	104	2	17	17	NUM
cana-1710	104	3	]	]	X
cana-1710	104	4	m	m	NOUN
cana-1710	104	5	r	r	NOUN
cana-1710	104	6	jooyandeh	jooyandeh	NOUN
cana-1710	104	7	,	,	PUNCT
cana-1710	104	8	d.kiani	d.kiani	NOUN
cana-1710	104	9	,	,	PUNCT
cana-1710	104	10	m.mirzakhah	m.mirzakhah	NOUN
cana-1710	104	11	,	,	PUNCT
cana-1710	104	12	incidence	incidence	ADJ
cana-1710	104	13	energy	energy	NOUN
cana-1710	104	14	of	of	ADP
cana-1710	104	15	graph	graph	NOUN
cana-1710	104	16	,	,	PUNCT
cana-1710	104	17	match	match	NOUN
cana-1710	104	18	commun	commun	PROPN
cana-1710	104	19	.	.	PUNCT
cana-1710	105	1	math.comput.chem	math.comput.chem	PROPN
cana-1710	105	2	.	.	PUNCT
cana-1710	106	1	bf60(2008	bf60(2008	NOUN
cana-1710	106	2	)	)	PUNCT
cana-1710	107	1	561	561	NUM
cana-1710	107	2	-	-	SYM
cana-1710	107	3	572	572	NUM
cana-1710	107	4	.	.	PUNCT
cana-1710	108	1	[	[	X
cana-1710	108	2	18	18	NUM
cana-1710	108	3	]	]	PUNCT
cana-1710	108	4	j.	j.	PROPN
cana-1710	108	5	h.	h.	PROPN
cana-1710	108	6	koolen	koolen	PROPN
cana-1710	108	7	and	and	CCONJ
cana-1710	108	8	v.	v.	ADP
cana-1710	108	9	moulton	moulton	PROPN
cana-1710	108	10	,	,	PUNCT
cana-1710	108	11	maximal	maximal	ADJ
cana-1710	108	12	energy	energy	NOUN
cana-1710	108	13	graphs	graph	NOUN
cana-1710	108	14	,	,	PUNCT
cana-1710	108	15	adv	adv	PROPN
cana-1710	108	16	.	.	PUNCT
cana-1710	108	17	in	in	ADP
cana-1710	108	18	appl	appl	PROPN
cana-1710	108	19	.	.	PUNCT
cana-1710	108	20	math	math	PROPN
cana-1710	108	21	.	.	PUNCT
cana-1710	109	1	,	,	PUNCT
cana-1710	109	2	26	26	NUM
cana-1710	109	3	(	(	PUNCT
cana-1710	109	4	2001	2001	NUM
cana-1710	109	5	)	)	PUNCT
cana-1710	109	6	,	,	PUNCT
cana-1710	109	7	47	47	NUM
cana-1710	109	8	-	-	SYM
cana-1710	109	9	52	52	NUM
cana-1710	109	10	.	.	PUNCT
cana-1710	110	1	[	[	X
cana-1710	110	2	19	19	NUM
cana-1710	110	3	]	]	X
cana-1710	110	4	b.j	b.j	PROPN
cana-1710	110	5	.	.	PROPN
cana-1710	110	6	mcclelland	mcclelland	PROPN
cana-1710	110	7	,	,	PUNCT
cana-1710	110	8	properties	property	NOUN
cana-1710	110	9	of	of	ADP
cana-1710	110	10	the	the	DET
cana-1710	110	11	latent	latent	NOUN
cana-1710	110	12	roots	root	NOUN
cana-1710	110	13	of	of	ADP
cana-1710	110	14	a	a	DET
cana-1710	110	15	matrix	matrix	NOUN
cana-1710	110	16	:	:	PUNCT
cana-1710	110	17	the	the	DET
cana-1710	110	18	estimation	estimation	NOUN
cana-1710	110	19	of	of	ADP
cana-1710	110	20	π	π	PROPN
cana-1710	110	21	−electron	−electron	ADP
cana-1710	110	22	energy	energy	NOUN
cana-1710	110	23	,	,	PUNCT
cana-1710	110	24	j.chem.phys	j.chem.phys	PROPN
cana-1710	110	25	.	.	PROPN
cana-1710	110	26	,	,	PUNCT
cana-1710	110	27	41	41	NUM
cana-1710	110	28	,	,	PUNCT
cana-1710	110	29	no	no	INTJ
cana-1710	110	30	.	.	NOUN
cana-1710	110	31	1	1	NUM
cana-1710	110	32	(	(	PUNCT
cana-1710	110	33	2007	2007	NUM
cana-1710	110	34	)	)	PUNCT
cana-1710	110	35	.	.	PUNCT
cana-1710	111	1	[	[	X
cana-1710	111	2	20	20	NUM
cana-1710	111	3	]	]	SYM
cana-1710	111	4	mitrinović	mitrinović	PROPN
cana-1710	111	5	,	,	PUNCT
cana-1710	111	6	d.s	d.s	PROPN
cana-1710	111	7	.	.	PROPN
cana-1710	111	8	,vasić.p.m	,vasić.p.m	PROPN
cana-1710	111	9	,	,	PUNCT
cana-1710	111	10	analytic	analytic	ADJ
cana-1710	111	11	inequalities	inequality	NOUN
cana-1710	111	12	.	.	PUNCT
cana-1710	111	13	,	,	PUNCT
cana-1710	111	14	springer	springer	NOUN
cana-1710	111	15	,	,	PUNCT
cana-1710	111	16	berlin,(1970	berlin,(1970	NOUN
cana-1710	111	17	)	)	PUNCT
cana-1710	111	18	.	.	PUNCT
cana-1710	112	1	[	[	X
cana-1710	112	2	21	21	NUM
cana-1710	112	3	]	]	X
cana-1710	112	4	mohammad	mohammad	PROPN
cana-1710	112	5	reza	reza	PROPN
cana-1710	112	6	oboudi	oboudi	PROPN
cana-1710	112	7	,	,	PUNCT
cana-1710	112	8	a	a	DET
cana-1710	112	9	new	new	ADJ
cana-1710	112	10	lower	lower	ADV
cana-1710	112	11	bound	bind	VERB
cana-1710	112	12	for	for	ADP
cana-1710	112	13	the	the	DET
cana-1710	112	14	energy	energy	NOUN
cana-1710	112	15	of	of	ADP
cana-1710	112	16	graphs	graph	NOUN
cana-1710	112	17	,	,	PUNCT
cana-1710	112	18	lin	lin	PROPN
cana-1710	112	19	.	.	PUNCT
cana-1710	113	1	algebra	algebra	PROPN
cana-1710	113	2	appl,580	appl,580	PROPN
cana-1710	113	3	(	(	PUNCT
cana-1710	113	4	2019),381	2019),381	NUM
cana-1710	113	5	-	-	SYM
cana-1710	113	6	395	395	NUM
cana-1710	113	7	.	.	PUNCT
cana-1710	114	1	[	[	X
cana-1710	114	2	22	22	NUM
cana-1710	114	3	]	]	X
cana-1710	114	4	m.r.rajesh	m.r.rajesh	PROPN
cana-1710	114	5	kanna	kanna	PROPN
cana-1710	114	6	,	,	PUNCT
cana-1710	114	7	b.n.dharmendra	b.n.dharmendra	NOUN
cana-1710	114	8	,	,	PUNCT
cana-1710	114	9	g.	g.	PROPN
cana-1710	114	10	sridhara	sridhara	PROPN
cana-1710	114	11	,	,	PUNCT
cana-1710	114	12	the	the	DET
cana-1710	114	13	minimum	minimum	ADJ
cana-1710	114	14	dominating	dominating	NOUN
cana-1710	114	15	energy	energy	NOUN
cana-1710	114	16	of	of	ADP
cana-1710	114	17	a	a	DET
cana-1710	114	18	graph	graph	NOUN
cana-1710	114	19	,	,	PUNCT
cana-1710	114	20	international	international	ADJ
cana-1710	114	21	journal	journal	NOUN
cana-1710	114	22	of	of	ADP
cana-1710	114	23	pure	pure	ADJ
cana-1710	114	24	and	and	CCONJ
cana-1710	114	25	applied	applied	ADJ
cana-1710	114	26	mathematics	mathematic	NOUN
cana-1710	114	27	.	.	PUNCT
cana-1710	114	28	,	,	PUNCT
cana-1710	114	29	vol	vol	NOUN
cana-1710	114	30	85	85	NUM
cana-1710	114	31	,	,	PUNCT
cana-1710	114	32	(	(	PUNCT
cana-1710	114	33	4	4	NUM
cana-1710	114	34	)	)	PUNCT
cana-1710	114	35	,	,	PUNCT
cana-1710	114	36	(	(	PUNCT
cana-1710	114	37	2013	2013	NUM
cana-1710	114	38	)	)	PUNCT
cana-1710	114	39	707	707	NUM
cana-1710	114	40	-	-	SYM
cana-1710	114	41	718	718	NUM
cana-1710	114	42	.	.	PUNCT