id	sid	tid	token	lemma	pos
cana-1725	1	1	communications	communication	NOUN
cana-1725	1	2	on	on	ADP
cana-1725	1	3	applied	apply	VERB
cana-1725	1	4	nonlinear	nonlinear	ADJ
cana-1725	1	5	analysis	analysis	NOUN
cana-1725	1	6	issn	issn	NOUN
cana-1725	1	7	:	:	PUNCT
cana-1725	1	8	1074	1074	NUM
cana-1725	1	9	-	-	PUNCT
cana-1725	1	10	133x	133x	NUM
cana-1725	1	11	vol	vol	NOUN
cana-1725	1	12	32	32	NUM
cana-1725	1	13	no	no	NOUN
cana-1725	1	14	.	.	NOUN
cana-1725	1	15	2	2	NUM
cana-1725	1	16	(	(	PUNCT
cana-1725	1	17	2025	2025	NUM
cana-1725	1	18	)	)	PUNCT
cana-1725	1	19	123	123	NUM
cana-1725	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	1	21	bounds	bound	VERB
cana-1725	1	22	on	on	ADP
cana-1725	1	23	the	the	DET
cana-1725	1	24	reduced	reduce	VERB
cana-1725	1	25	sombor	sombor	NOUN
cana-1725	1	26	index	index	NOUN
cana-1725	1	27	of	of	ADP
cana-1725	1	28	graphs	graph	NOUN
cana-1725	1	29	s.	s.	PROPN
cana-1725	1	30	nagarajan¹	nagarajan¹	PROPN
cana-1725	1	31	,	,	PUNCT
cana-1725	1	32	b.	b.	PROPN
cana-1725	1	33	aswini²	aswini²	PROPN
cana-1725	2	1	¹department	¹department	NUM
cana-1725	2	2	of	of	ADP
cana-1725	2	3	mathematics	mathematic	NOUN
cana-1725	2	4	,	,	PUNCT
cana-1725	2	5	kongu	kongu	PROPN
cana-1725	2	6	arts	art	NOUN
cana-1725	2	7	and	and	CCONJ
cana-1725	2	8	science	science	PROPN
cana-1725	2	9	college	college	PROPN
cana-1725	2	10	(	(	PUNCT
cana-1725	2	11	autonomous	autonomous	ADJ
cana-1725	2	12	)	)	PUNCT
cana-1725	2	13	,	,	PUNCT
cana-1725	2	14	erode	erode	VERB
cana-1725	2	15	,	,	PUNCT
cana-1725	2	16	tamilnadu-638	tamilnadu-638	ADJ
cana-1725	2	17	107	107	NUM
cana-1725	2	18	profnagarajan.s@gmail.com	profnagarajan.s@gmail.com	PRON
cana-1725	2	19	²school	²school	PROPN
cana-1725	2	20	of	of	ADP
cana-1725	2	21	mathematics	mathematic	NOUN
cana-1725	2	22	,	,	PUNCT
cana-1725	2	23	a.v.p	a.v.p	NOUN
cana-1725	2	24	.	.	PUNCT
cana-1725	3	1	college	college	PROPN
cana-1725	3	2	of	of	ADP
cana-1725	3	3	arts	art	NOUN
cana-1725	3	4	and	and	CCONJ
cana-1725	3	5	science	science	NOUN
cana-1725	3	6	,	,	PUNCT
cana-1725	3	7	tirupur-641	tirupur-641	PROPN
cana-1725	3	8	652	652	NUM
cana-1725	3	9	.	.	PUNCT
cana-1725	4	1	aswiniprasaad@gmail.com	aswiniprasaad@gmail.com	X
cana-1725	4	2	article	article	NOUN
cana-1725	4	3	history	history	NOUN
cana-1725	4	4	:	:	PUNCT
cana-1725	4	5	received	receive	VERB
cana-1725	4	6	:	:	PUNCT
cana-1725	4	7	28	28	NUM
cana-1725	4	8	-	-	SYM
cana-1725	4	9	07	07	NUM
cana-1725	4	10	-	-	PUNCT
cana-1725	4	11	2024	2024	NUM
cana-1725	4	12	revised	revise	VERB
cana-1725	4	13	:	:	PUNCT
cana-1725	4	14	07	07	NUM
cana-1725	4	15	-	-	PUNCT
cana-1725	4	16	09	09	NUM
cana-1725	4	17	-	-	PUNCT
cana-1725	4	18	2024	2024	NUM
cana-1725	4	19	accepted	accept	VERB
cana-1725	4	20	:	:	PUNCT
cana-1725	4	21	17	17	NUM
cana-1725	4	22	-	-	SYM
cana-1725	4	23	09	09	NUM
cana-1725	4	24	-	-	PUNCT
cana-1725	4	25	2024	2024	NUM
cana-1725	4	26	abstract	abstract	NOUN
cana-1725	4	27	:	:	PUNCT
cana-1725	4	28	the	the	DET
cana-1725	4	29	reduced	reduce	VERB
cana-1725	4	30	sombor	sombor	NOUN
cana-1725	4	31	index	index	NOUN
cana-1725	4	32	is	be	AUX
cana-1725	4	33	a	a	DET
cana-1725	4	34	modified	modify	VERB
cana-1725	4	35	version	version	NOUN
cana-1725	4	36	of	of	ADP
cana-1725	4	37	the	the	DET
cana-1725	4	38	very	very	ADV
cana-1725	4	39	famous	famous	ADJ
cana-1725	4	40	sombor	sombor	NOUN
cana-1725	4	41	index	index	NOUN
cana-1725	4	42	for	for	ADP
cana-1725	4	43	a	a	DET
cana-1725	4	44	graph	graph	NOUN
cana-1725	4	45	𝐺.	𝐺.	NOUN
cana-1725	4	46	in	in	ADP
cana-1725	4	47	this	this	DET
cana-1725	4	48	article	article	NOUN
cana-1725	4	49	,	,	PUNCT
cana-1725	4	50	the	the	DET
cana-1725	4	51	reduced	reduce	VERB
cana-1725	4	52	sombor	sombor	NOUN
cana-1725	4	53	index	index	NOUN
cana-1725	4	54	is	be	AUX
cana-1725	4	55	studied	study	VERB
cana-1725	4	56	on	on	ADP
cana-1725	4	57	various	various	ADJ
cana-1725	4	58	classes	class	NOUN
cana-1725	4	59	of	of	ADP
cana-1725	4	60	graphs	graph	NOUN
cana-1725	4	61	and	and	CCONJ
cana-1725	4	62	novel	novel	ADJ
cana-1725	4	63	results	result	NOUN
cana-1725	4	64	on	on	ADP
cana-1725	4	65	the	the	DET
cana-1725	4	66	bounds	bound	NOUN
cana-1725	4	67	of	of	ADP
cana-1725	4	68	the	the	DET
cana-1725	4	69	reduced	reduce	VERB
cana-1725	4	70	sombor	sombor	NOUN
cana-1725	4	71	index	index	NOUN
cana-1725	4	72	of	of	ADP
cana-1725	4	73	graphs	graph	NOUN
cana-1725	4	74	are	be	AUX
cana-1725	4	75	obtained	obtain	VERB
cana-1725	4	76	.	.	PUNCT
cana-1725	5	1	the	the	DET
cana-1725	5	2	graphs	graph	NOUN
cana-1725	5	3	with	with	ADP
cana-1725	5	4	minimum	minimum	NOUN
cana-1725	5	5	reduced	reduce	VERB
cana-1725	5	6	sombor	sombor	NOUN
cana-1725	5	7	index	index	NOUN
cana-1725	5	8	are	be	AUX
cana-1725	5	9	studied	study	VERB
cana-1725	5	10	,	,	PUNCT
cana-1725	5	11	and	and	CCONJ
cana-1725	5	12	the	the	DET
cana-1725	5	13	graphs	graph	NOUN
cana-1725	5	14	achieving	achieve	VERB
cana-1725	5	15	certain	certain	ADJ
cana-1725	5	16	bounds	bound	NOUN
cana-1725	5	17	are	be	AUX
cana-1725	5	18	found	find	VERB
cana-1725	5	19	.	.	PUNCT
cana-1725	6	1	keywords	keyword	NOUN
cana-1725	6	2	:	:	PUNCT
cana-1725	6	3	reduced	reduce	VERB
cana-1725	6	4	sombor	sombor	NOUN
cana-1725	6	5	index	index	NOUN
cana-1725	6	6	;	;	PUNCT
cana-1725	6	7	topological	topological	ADJ
cana-1725	6	8	index	index	NOUN
cana-1725	6	9	;	;	PUNCT
cana-1725	6	10	graph	graph	NOUN
cana-1725	6	11	invariant	invariant	ADJ
cana-1725	6	12	;	;	PUNCT
cana-1725	6	13	trees	tree	NOUN
cana-1725	6	14	;	;	PUNCT
cana-1725	6	15	extremal	extremal	ADJ
cana-1725	6	16	problem	problem	NOUN
cana-1725	6	17	;	;	PUNCT
cana-1725	6	18	characterization	characterization	NOUN
cana-1725	6	19	.	.	PUNCT
cana-1725	7	1	1	1	X
cana-1725	7	2	.	.	X
cana-1725	7	3	introduction	introduction	NOUN
cana-1725	7	4	by	by	ADP
cana-1725	7	5	a	a	DET
cana-1725	7	6	graph	graph	NOUN
cana-1725	7	7	g	g	NOUN
cana-1725	7	8	,	,	PUNCT
cana-1725	7	9	in	in	ADP
cana-1725	7	10	this	this	DET
cana-1725	7	11	article	article	NOUN
cana-1725	7	12	,	,	PUNCT
cana-1725	7	13	we	we	PRON
cana-1725	7	14	mean	mean	VERB
cana-1725	7	15	an	an	DET
cana-1725	7	16	ordered	order	VERB
cana-1725	7	17	pair	pair	NOUN
cana-1725	7	18	(	(	PUNCT
cana-1725	7	19	𝑉𝐺	𝑉𝐺	PROPN
cana-1725	7	20	,	,	PUNCT
cana-1725	7	21	𝐸𝐺	𝐸𝐺	PROPN
cana-1725	7	22	)	)	PUNCT
cana-1725	7	23	,	,	PUNCT
cana-1725	7	24	and	and	CCONJ
cana-1725	7	25	the	the	DET
cana-1725	7	26	members	member	NOUN
cana-1725	7	27	of	of	ADP
cana-1725	7	28	the	the	DET
cana-1725	7	29	sets	set	NOUN
cana-1725	7	30	𝑉𝐺	𝑉𝐺	PROPN
cana-1725	7	31	and	and	CCONJ
cana-1725	7	32	𝐸𝐺	𝐸𝐺	PROPN
cana-1725	7	33	respectively	respectively	ADV
cana-1725	7	34	are	be	AUX
cana-1725	7	35	the	the	DET
cana-1725	7	36	vertices	vertex	NOUN
cana-1725	7	37	and	and	CCONJ
cana-1725	7	38	edges	edge	NOUN
cana-1725	7	39	of	of	ADP
cana-1725	7	40	the	the	DET
cana-1725	7	41	graph	graph	NOUN
cana-1725	7	42	.	.	PUNCT
cana-1725	8	1	the	the	DET
cana-1725	8	2	set	set	NOUN
cana-1725	8	3	of	of	ADP
cana-1725	8	4	vertices	vertex	NOUN
cana-1725	8	5	that	that	PRON
cana-1725	8	6	are	be	AUX
cana-1725	8	7	adjacent	adjacent	ADJ
cana-1725	8	8	to	to	ADP
cana-1725	8	9	a	a	DET
cana-1725	8	10	vertex	vertex	NOUN
cana-1725	8	11	𝑢	𝑢	NOUN
cana-1725	8	12	in	in	ADP
cana-1725	8	13	𝐺	𝐺	PROPN
cana-1725	8	14	is	be	AUX
cana-1725	8	15	denoted	denote	VERB
cana-1725	8	16	as	as	ADP
cana-1725	8	17	𝑁𝐺(𝑢	𝑁𝐺(𝑢	PROPN
cana-1725	8	18	)	)	PUNCT
cana-1725	8	19	and	and	CCONJ
cana-1725	8	20	called	call	VERB
cana-1725	8	21	by	by	ADP
cana-1725	8	22	“	"	PUNCT
cana-1725	8	23	the	the	DET
cana-1725	8	24	open	open	ADJ
cana-1725	8	25	neighbourhood	neighbourhood	NOUN
cana-1725	8	26	”	"	PUNCT
cana-1725	8	27	of	of	ADP
cana-1725	8	28	𝑢	𝑢	NOUN
cana-1725	8	29	in	in	ADP
cana-1725	8	30	𝐺.	𝐺.	NOUN
cana-1725	8	31	the	the	DET
cana-1725	8	32	term	term	NOUN
cana-1725	8	33	“	"	PUNCT
cana-1725	8	34	closed	close	VERB
cana-1725	8	35	neighbourhood	neighbourhood	NOUN
cana-1725	8	36	”	"	PUNCT
cana-1725	8	37	is	be	AUX
cana-1725	8	38	𝑁𝐺[𝑣	𝑁𝐺[𝑣	NOUN
cana-1725	8	39	]	]	X
cana-1725	8	40	=	=	PUNCT
cana-1725	8	41	𝑁𝐺(𝑣	𝑁𝐺(𝑣	X
cana-1725	8	42	)	)	PUNCT
cana-1725	8	43	∪	∪	ADP
cana-1725	8	44	{	{	PUNCT
cana-1725	8	45	𝑣	𝑣	NOUN
cana-1725	8	46	}	}	PUNCT
cana-1725	8	47	and	and	CCONJ
cana-1725	8	48	by	by	ADP
cana-1725	8	49	a	a	DET
cana-1725	8	50	(	(	PUNCT
cana-1725	8	51	𝑣	𝑣	NOUN
cana-1725	8	52	,	,	PUNCT
cana-1725	8	53	𝑤)-path	𝑤)-path	NOUN
cana-1725	8	54	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1725	8	55	.	.	PUNCT
cana-1725	8	56	.	.	PUNCT
cana-1725	8	57	.	.	PUNCT
cana-1725	9	1	𝑤	𝑤	X
cana-1725	9	2	is	be	AUX
cana-1725	9	3	a	a	DET
cana-1725	9	4	sequence	sequence	NOUN
cana-1725	9	5	of	of	ADP
cana-1725	9	6	distinct	distinct	ADJ
cana-1725	9	7	members	member	NOUN
cana-1725	9	8	of	of	ADP
cana-1725	9	9	the	the	DET
cana-1725	9	10	set	set	ADJ
cana-1725	9	11	𝑉𝐺	𝑉𝐺	NOUN
cana-1725	9	12	and	and	CCONJ
cana-1725	9	13	the	the	DET
cana-1725	9	14	vertices	vertex	NOUN
cana-1725	9	15	𝑣	𝑣	NOUN
cana-1725	9	16	,	,	PUNCT
cana-1725	9	17	𝑤	𝑤	X
cana-1725	9	18	are	be	AUX
cana-1725	9	19	usually	usually	ADV
cana-1725	9	20	known	know	VERB
cana-1725	9	21	as	as	ADP
cana-1725	9	22	the	the	DET
cana-1725	9	23	origin	origin	NOUN
cana-1725	9	24	and	and	CCONJ
cana-1725	9	25	the	the	DET
cana-1725	9	26	terminus	terminus	NOUN
cana-1725	9	27	of	of	ADP
cana-1725	9	28	the	the	DET
cana-1725	9	29	path	path	NOUN
cana-1725	9	30	𝑃	𝑃	NOUN
cana-1725	9	31	respectively	respectively	ADV
cana-1725	9	32	.	.	PUNCT
cana-1725	10	1	the	the	DET
cana-1725	10	2	concept	concept	NOUN
cana-1725	10	3	of	of	ADP
cana-1725	10	4	distance	distance	NOUN
cana-1725	10	5	between	between	ADP
cana-1725	10	6	any	any	DET
cana-1725	10	7	two	two	NUM
cana-1725	10	8	vertices	vertex	NOUN
cana-1725	10	9	𝑥	𝑥	NOUN
cana-1725	10	10	,	,	PUNCT
cana-1725	10	11	𝑦	𝑦	PRON
cana-1725	10	12	∈	∈	NOUN
cana-1725	10	13	𝑉𝐺	𝑉𝐺	PROPN
cana-1725	10	14	is	be	AUX
cana-1725	10	15	usually	usually	ADV
cana-1725	10	16	defined	define	VERB
cana-1725	10	17	as	as	ADP
cana-1725	10	18	the	the	DET
cana-1725	10	19	length	length	NOUN
cana-1725	10	20	of	of	ADP
cana-1725	10	21	the	the	DET
cana-1725	10	22	smallest	small	ADJ
cana-1725	10	23	(	(	PUNCT
cana-1725	10	24	𝑥	𝑥	NOUN
cana-1725	10	25	,	,	PUNCT
cana-1725	10	26	𝑦)-path	𝑦)-path	PUNCT
cana-1725	10	27	that	that	PRON
cana-1725	10	28	exists	exist	VERB
cana-1725	10	29	in	in	ADP
cana-1725	10	30	𝐺.	𝐺.	PROPN
cana-1725	10	31	if	if	SCONJ
cana-1725	10	32	𝑑𝐺(𝑢	𝑑𝐺(𝑢	NUM
cana-1725	10	33	)	)	PUNCT
cana-1725	11	1	=	=	SYM
cana-1725	11	2	1	1	NUM
cana-1725	11	3	,	,	PUNCT
cana-1725	11	4	then	then	ADV
cana-1725	11	5	v	v	NOUN
cana-1725	11	6	is	be	AUX
cana-1725	11	7	a	a	DET
cana-1725	11	8	pendant	pendant	ADJ
cana-1725	11	9	vertex	vertex	NOUN
cana-1725	11	10	and	and	CCONJ
cana-1725	11	11	it	it	PRON
cana-1725	11	12	is	be	AUX
cana-1725	11	13	adjacent	adjacent	ADJ
cana-1725	11	14	to	to	ADP
cana-1725	11	15	a	a	DET
cana-1725	11	16	unique	unique	ADJ
cana-1725	11	17	vertex	vertex	NOUN
cana-1725	11	18	in	in	ADP
cana-1725	11	19	𝐺	𝐺	PROPN
cana-1725	11	20	,	,	PUNCT
cana-1725	11	21	say	say	VERB
cana-1725	11	22	𝑢	𝑢	PRON
cana-1725	11	23	which	which	PRON
cana-1725	11	24	is	be	AUX
cana-1725	11	25	called	call	VERB
cana-1725	11	26	a	a	DET
cana-1725	11	27	support	support	NOUN
cana-1725	11	28	vertex	vertex	NOUN
cana-1725	11	29	.	.	PUNCT
cana-1725	12	1	for	for	ADP
cana-1725	12	2	more	more	ADJ
cana-1725	12	3	on	on	ADP
cana-1725	12	4	graphs	graph	NOUN
cana-1725	12	5	and	and	CCONJ
cana-1725	12	6	related	related	ADJ
cana-1725	12	7	works	work	NOUN
cana-1725	12	8	,	,	PUNCT
cana-1725	12	9	the	the	DET
cana-1725	12	10	reader	reader	NOUN
cana-1725	12	11	is	be	AUX
cana-1725	12	12	referred	refer	VERB
cana-1725	12	13	to	to	ADP
cana-1725	12	14	[	[	X
cana-1725	12	15	1	1	NUM
cana-1725	12	16	–	–	PUNCT
cana-1725	12	17	3	3	NUM
cana-1725	12	18	]	]	PUNCT
cana-1725	12	19	.	.	PUNCT
cana-1725	13	1	the	the	DET
cana-1725	13	2	topological	topological	ADJ
cana-1725	13	3	indices	index	NOUN
cana-1725	13	4	(	(	PUNCT
cana-1725	13	5	also	also	ADV
cana-1725	13	6	known	know	VERB
cana-1725	13	7	as	as	ADP
cana-1725	13	8	graph	graph	NOUN
cana-1725	13	9	invariants	invariant	NOUN
cana-1725	13	10	)	)	PUNCT
cana-1725	13	11	play	play	VERB
cana-1725	13	12	a	a	DET
cana-1725	13	13	major	major	ADJ
cana-1725	13	14	role	role	NOUN
cana-1725	13	15	in	in	ADP
cana-1725	13	16	the	the	DET
cana-1725	13	17	chemical	chemical	NOUN
cana-1725	13	18	graph	graph	NOUN
cana-1725	13	19	theory	theory	NOUN
cana-1725	13	20	because	because	SCONJ
cana-1725	13	21	they	they	PRON
cana-1725	13	22	are	be	AUX
cana-1725	13	23	used	use	VERB
cana-1725	13	24	to	to	PART
cana-1725	13	25	analyze	analyze	VERB
cana-1725	13	26	the	the	DET
cana-1725	13	27	behaviour	behaviour	NOUN
cana-1725	13	28	of	of	ADP
cana-1725	13	29	the	the	DET
cana-1725	13	30	molecule	molecule	NOUN
cana-1725	13	31	structures	structure	NOUN
cana-1725	13	32	and	and	CCONJ
cana-1725	13	33	their	their	PRON
cana-1725	13	34	interrelationships	interrelationship	NOUN
cana-1725	13	35	.	.	PUNCT
cana-1725	14	1	there	there	PRON
cana-1725	14	2	are	be	VERB
cana-1725	14	3	numerous	numerous	ADJ
cana-1725	14	4	topological	topological	ADJ
cana-1725	14	5	indices	index	NOUN
cana-1725	14	6	available	available	ADJ
cana-1725	14	7	in	in	ADP
cana-1725	14	8	the	the	DET
cana-1725	14	9	literature	literature	NOUN
cana-1725	14	10	;	;	PUNCT
cana-1725	14	11	a	a	DET
cana-1725	14	12	few	few	ADJ
cana-1725	14	13	of	of	ADP
cana-1725	14	14	them	they	PRON
cana-1725	14	15	are	be	AUX
cana-1725	14	16	the	the	DET
cana-1725	14	17	sombor	sombor	NOUN
cana-1725	14	18	index	index	NOUN
cana-1725	14	19	,	,	PUNCT
cana-1725	14	20	zagreb	zagreb	PROPN
cana-1725	14	21	index	index	PROPN
cana-1725	14	22	,	,	PUNCT
cana-1725	14	23	and	and	CCONJ
cana-1725	14	24	so	so	ADV
cana-1725	14	25	on	on	ADV
cana-1725	14	26	.	.	PUNCT
cana-1725	15	1	the	the	DET
cana-1725	15	2	topological	topological	ADJ
cana-1725	15	3	indices	index	NOUN
cana-1725	15	4	were	be	AUX
cana-1725	15	5	defined	define	VERB
cana-1725	15	6	with	with	ADP
cana-1725	15	7	minor	minor	ADJ
cana-1725	15	8	and	and	CCONJ
cana-1725	15	9	major	major	ADJ
cana-1725	15	10	modifications	modification	NOUN
cana-1725	15	11	in	in	ADP
cana-1725	15	12	the	the	DET
cana-1725	15	13	past	past	NOUN
cana-1725	15	14	and	and	CCONJ
cana-1725	15	15	several	several	ADJ
cana-1725	15	16	classes	class	NOUN
cana-1725	15	17	of	of	ADP
cana-1725	15	18	topological	topological	ADJ
cana-1725	15	19	indices	index	NOUN
cana-1725	15	20	are	be	AUX
cana-1725	15	21	available	available	ADJ
cana-1725	15	22	for	for	ADP
cana-1725	15	23	the	the	DET
cana-1725	15	24	sombor	sombor	NOUN
cana-1725	15	25	index	index	NOUN
cana-1725	15	26	and	and	CCONJ
cana-1725	15	27	zagreb	zagreb	PROPN
cana-1725	15	28	index	index	NOUN
cana-1725	15	29	.	.	PUNCT
cana-1725	16	1	given	give	VERB
cana-1725	16	2	a	a	DET
cana-1725	16	3	graph	graph	NOUN
cana-1725	16	4	,	,	PUNCT
cana-1725	16	5	the	the	DET
cana-1725	16	6	sombor	sombor	NOUN
cana-1725	16	7	(	(	PUNCT
cana-1725	16	8	so	so	ADV
cana-1725	16	9	)	)	PUNCT
cana-1725	16	10	index	index	NOUN
cana-1725	16	11	is	be	AUX
cana-1725	16	12	defined	define	VERB
cana-1725	16	13	(	(	PUNCT
cana-1725	16	14	by	by	ADP
cana-1725	16	15	gutman[4	gutman[4	NOUN
cana-1725	16	16	]	]	PUNCT
cana-1725	16	17	)	)	PUNCT
cana-1725	16	18	as	as	ADP
cana-1725	16	19	𝑆𝑂(𝐺	𝑆𝑂(𝐺	NOUN
cana-1725	16	20	)	)	PUNCT
cana-1725	16	21	=	=	SYM
cana-1725	16	22	∑	∑	PUNCT
cana-1725	16	23	√𝑑𝐺(𝑢)2	√𝑑𝐺(𝑢)2	PROPN
cana-1725	16	24	+	+	PROPN
cana-1725	16	25	𝑑𝐺(𝑣)2	𝑑𝐺(𝑣)2	PROPN
cana-1725	16	26	𝑈,𝑉∈𝐸𝐺	𝑈,𝑉∈𝐸𝐺	PROPN
cana-1725	16	27	the	the	DET
cana-1725	16	28	sombor	sombor	NOUN
cana-1725	16	29	index	index	NOUN
cana-1725	16	30	,	,	PUNCT
cana-1725	16	31	in	in	ADP
cana-1725	16	32	recent	recent	ADJ
cana-1725	16	33	years	year	NOUN
cana-1725	16	34	,	,	PUNCT
cana-1725	16	35	received	receive	VERB
cana-1725	16	36	numerous	numerous	ADJ
cana-1725	16	37	attentions	attention	NOUN
cana-1725	16	38	from	from	ADP
cana-1725	16	39	academics	academic	NOUN
cana-1725	16	40	and	and	CCONJ
cana-1725	16	41	researchers	researcher	NOUN
cana-1725	16	42	throughout	throughout	ADP
cana-1725	16	43	the	the	DET
cana-1725	16	44	globe	globe	NOUN
cana-1725	17	1	[	[	X
cana-1725	17	2	8–11	8–11	X
cana-1725	17	3	]	]	PUNCT
cana-1725	17	4	.	.	PUNCT
cana-1725	18	1	for	for	ADP
cana-1725	18	2	some	some	DET
cana-1725	18	3	recent	recent	ADJ
cana-1725	18	4	surveys	survey	NOUN
cana-1725	18	5	in	in	ADP
cana-1725	18	6	sombor	sombor	NOUN
cana-1725	18	7	index	index	NOUN
cana-1725	18	8	,	,	PUNCT
cana-1725	18	9	one	one	PRON
cana-1725	18	10	can	can	AUX
cana-1725	18	11	refer	refer	VERB
cana-1725	18	12	to	to	ADP
cana-1725	18	13	the	the	DET
cana-1725	18	14	articles	article	NOUN
cana-1725	18	15	[	[	X
cana-1725	18	16	6	6	NUM
cana-1725	18	17	,	,	PUNCT
cana-1725	18	18	7	7	NUM
cana-1725	18	19	]	]	PUNCT
cana-1725	18	20	.	.	PUNCT
cana-1725	19	1	chemical	chemical	NOUN
cana-1725	19	2	applications	application	NOUN
cana-1725	19	3	have	have	AUX
cana-1725	19	4	been	be	AUX
cana-1725	19	5	carried	carry	VERB
cana-1725	19	6	out	out	ADP
cana-1725	19	7	in	in	ADP
cana-1725	19	8	the	the	DET
cana-1725	19	9	articles	article	NOUN
cana-1725	19	10	[	[	X
cana-1725	19	11	12	12	NUM
cana-1725	19	12	,	,	PUNCT
cana-1725	19	13	13	13	NUM
cana-1725	19	14	]	]	PUNCT
cana-1725	19	15	.	.	PUNCT
cana-1725	20	1	for	for	ADP
cana-1725	20	2	various	various	ADJ
cana-1725	20	3	results	result	NOUN
cana-1725	20	4	and	and	CCONJ
cana-1725	20	5	versions	version	NOUN
cana-1725	20	6	of	of	ADP
cana-1725	20	7	sombor	sombor	NOUN
cana-1725	20	8	index	index	NOUN
cana-1725	20	9	,	,	PUNCT
cana-1725	20	10	one	one	PRON
cana-1725	20	11	can	can	AUX
cana-1725	20	12	refer	refer	VERB
cana-1725	20	13	[	[	X
cana-1725	20	14	5	5	NUM
cana-1725	20	15	,	,	PUNCT
cana-1725	20	16	14–17	14–17	NUM
cana-1725	20	17	]	]	PUNCT
cana-1725	20	18	.	.	PUNCT
cana-1725	21	1	communications	communication	NOUN
cana-1725	21	2	on	on	ADP
cana-1725	21	3	applied	apply	VERB
cana-1725	21	4	nonlinear	nonlinear	ADJ
cana-1725	21	5	analysis	analysis	NOUN
cana-1725	21	6	issn	issn	NOUN
cana-1725	21	7	:	:	PUNCT
cana-1725	21	8	1074	1074	NUM
cana-1725	21	9	-	-	PUNCT
cana-1725	21	10	133x	133x	NUM
cana-1725	21	11	vol	vol	NOUN
cana-1725	21	12	32	32	NUM
cana-1725	21	13	no	no	NOUN
cana-1725	21	14	.	.	NOUN
cana-1725	21	15	2	2	NUM
cana-1725	21	16	(	(	PUNCT
cana-1725	21	17	2025	2025	NUM
cana-1725	21	18	)	)	PUNCT
cana-1725	21	19	124	124	NUM
cana-1725	21	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	22	1	the	the	DET
cana-1725	22	2	reduced	reduce	VERB
cana-1725	22	3	sombor	sombor	NOUN
cana-1725	22	4	index	index	NOUN
cana-1725	22	5	is	be	AUX
cana-1725	22	6	defined	define	VERB
cana-1725	22	7	as	as	ADP
cana-1725	22	8	𝑆𝑂(𝐺	𝑆𝑂(𝐺	NOUN
cana-1725	22	9	)	)	PUNCT
cana-1725	22	10	=	=	SYM
cana-1725	23	1	∑	∑	PUNCT
cana-1725	23	2	√(𝑑𝐺(𝑢	√(𝑑𝐺(𝑢	NUM
cana-1725	23	3	)	)	PUNCT
cana-1725	23	4	−	−	PROPN
cana-1725	24	1	1)2	1)2	NUM
cana-1725	24	2	+	+	CCONJ
cana-1725	24	3	(	(	PUNCT
cana-1725	24	4	𝑑𝐺(𝑣	𝑑𝐺(𝑣	PROPN
cana-1725	24	5	)	)	PUNCT
cana-1725	24	6	−	−	PROPN
cana-1725	25	1	1)2	1)2	NUM
cana-1725	25	2	𝑢,𝑣∈𝐸𝐺	𝑢,𝑣∈𝐸𝐺	PROPN
cana-1725	25	3	the	the	DET
cana-1725	25	4	reduced	reduce	VERB
cana-1725	25	5	sombor	sombor	NOUN
cana-1725	25	6	index	index	NOUN
cana-1725	25	7	is	be	AUX
cana-1725	25	8	a	a	DET
cana-1725	25	9	recently	recently	ADV
cana-1725	25	10	introduced	introduce	VERB
cana-1725	25	11	term	term	NOUN
cana-1725	25	12	and	and	CCONJ
cana-1725	25	13	some	some	PRON
cana-1725	25	14	of	of	ADP
cana-1725	25	15	the	the	DET
cana-1725	25	16	works	work	NOUN
cana-1725	25	17	can	can	AUX
cana-1725	25	18	be	be	AUX
cana-1725	25	19	found	find	VERB
cana-1725	25	20	in	in	ADP
cana-1725	25	21	[	[	X
cana-1725	25	22	18	18	NUM
cana-1725	25	23	,	,	PUNCT
cana-1725	25	24	19	19	NUM
cana-1725	25	25	]	]	PUNCT
cana-1725	25	26	.	.	PUNCT
cana-1725	26	1	the	the	DET
cana-1725	26	2	reduced	reduce	VERB
cana-1725	26	3	sombor	sombor	NOUN
cana-1725	26	4	index	index	NOUN
cana-1725	26	5	,	,	PUNCT
cana-1725	26	6	for	for	ADP
cana-1725	26	7	a	a	DET
cana-1725	26	8	wide	wide	ADJ
cana-1725	26	9	collection	collection	NOUN
cana-1725	26	10	of	of	ADP
cana-1725	26	11	graphs	graph	NOUN
cana-1725	26	12	,	,	PUNCT
cana-1725	26	13	is	be	AUX
cana-1725	26	14	studied	study	VERB
cana-1725	26	15	in	in	ADP
cana-1725	26	16	this	this	DET
cana-1725	26	17	article	article	NOUN
cana-1725	26	18	and	and	CCONJ
cana-1725	26	19	characterized	characterize	VERB
cana-1725	26	20	the	the	DET
cana-1725	26	21	graphs	graph	NOUN
cana-1725	26	22	with	with	ADP
cana-1725	26	23	minimum	minimum	NOUN
cana-1725	26	24	reduced	reduce	VERB
cana-1725	26	25	sombor	sombor	NOUN
cana-1725	26	26	index	index	NOUN
cana-1725	26	27	of	of	ADP
cana-1725	26	28	connected	connected	ADJ
cana-1725	26	29	and	and	CCONJ
cana-1725	26	30	disconnected	disconnected	ADJ
cana-1725	26	31	graphs	graph	NOUN
cana-1725	26	32	.	.	PUNCT
cana-1725	27	1	2	2	X
cana-1725	27	2	.	.	X
cana-1725	27	3	objectives	objective	NOUN
cana-1725	27	4	in	in	ADP
cana-1725	27	5	this	this	DET
cana-1725	27	6	paper	paper	NOUN
cana-1725	27	7	,	,	PUNCT
cana-1725	27	8	the	the	DET
cana-1725	27	9	emphasis	emphasis	NOUN
cana-1725	27	10	is	be	AUX
cana-1725	27	11	on	on	ADP
cana-1725	27	12	identifying	identify	VERB
cana-1725	27	13	the	the	DET
cana-1725	27	14	limits	limit	NOUN
cana-1725	27	15	of	of	ADP
cana-1725	27	16	the	the	DET
cana-1725	27	17	reduced	reduce	VERB
cana-1725	27	18	sombor	sombor	NOUN
cana-1725	27	19	index	index	NOUN
cana-1725	27	20	,	,	PUNCT
cana-1725	27	21	a	a	DET
cana-1725	27	22	newly	newly	ADV
cana-1725	27	23	introduced	introduce	VERB
cana-1725	27	24	topological	topological	ADJ
cana-1725	27	25	measure	measure	NOUN
cana-1725	27	26	applied	apply	VERB
cana-1725	27	27	in	in	ADP
cana-1725	27	28	the	the	DET
cana-1725	27	29	exploration	exploration	NOUN
cana-1725	27	30	of	of	ADP
cana-1725	27	31	chemical	chemical	NOUN
cana-1725	27	32	graph	graph	NOUN
cana-1725	27	33	theory	theory	NOUN
cana-1725	27	34	and	and	CCONJ
cana-1725	27	35	related	related	ADJ
cana-1725	27	36	disciplines	discipline	NOUN
cana-1725	27	37	.	.	PUNCT
cana-1725	28	1	the	the	DET
cana-1725	28	2	sombor	sombor	NOUN
cana-1725	28	3	index	index	NOUN
cana-1725	28	4	is	be	AUX
cana-1725	28	5	derived	derive	VERB
cana-1725	28	6	from	from	ADP
cana-1725	28	7	the	the	DET
cana-1725	28	8	degrees	degree	NOUN
cana-1725	28	9	of	of	ADP
cana-1725	28	10	nodes	node	NOUN
cana-1725	28	11	in	in	ADP
cana-1725	28	12	a	a	DET
cana-1725	28	13	graph	graph	NOUN
cana-1725	28	14	,	,	PUNCT
cana-1725	28	15	and	and	CCONJ
cana-1725	28	16	the	the	DET
cana-1725	28	17	reduced	reduce	VERB
cana-1725	28	18	sombor	sombor	NOUN
cana-1725	28	19	index	index	NOUN
cana-1725	28	20	further	far	ADV
cana-1725	28	21	elaborates	elaborate	VERB
cana-1725	28	22	on	on	ADP
cana-1725	28	23	this	this	DET
cana-1725	28	24	idea	idea	NOUN
cana-1725	28	25	.	.	PUNCT
cana-1725	29	1	the	the	DET
cana-1725	29	2	researchers	researcher	NOUN
cana-1725	29	3	investigate	investigate	VERB
cana-1725	29	4	the	the	DET
cana-1725	29	5	smallest	small	ADJ
cana-1725	29	6	and	and	CCONJ
cana-1725	29	7	largest	large	ADJ
cana-1725	29	8	values	value	NOUN
cana-1725	29	9	of	of	ADP
cana-1725	29	10	the	the	DET
cana-1725	29	11	reduced	reduce	VERB
cana-1725	29	12	sombor	sombor	NOUN
cana-1725	29	13	index	index	NOUN
cana-1725	29	14	over	over	ADP
cana-1725	29	15	a	a	DET
cana-1725	29	16	wide	wide	ADJ
cana-1725	29	17	range	range	NOUN
cana-1725	29	18	of	of	ADP
cana-1725	29	19	graphs	graph	NOUN
cana-1725	29	20	.	.	PUNCT
cana-1725	30	1	by	by	ADP
cana-1725	30	2	examining	examine	VERB
cana-1725	30	3	various	various	ADJ
cana-1725	30	4	graph	graph	NOUN
cana-1725	30	5	configurations	configuration	NOUN
cana-1725	30	6	,	,	PUNCT
cana-1725	30	7	they	they	PRON
cana-1725	30	8	are	be	AUX
cana-1725	30	9	able	able	ADJ
cana-1725	30	10	to	to	PART
cana-1725	30	11	define	define	VERB
cana-1725	30	12	the	the	DET
cana-1725	30	13	specific	specific	ADJ
cana-1725	30	14	types	type	NOUN
cana-1725	30	15	of	of	ADP
cana-1725	30	16	graphs	graph	NOUN
cana-1725	30	17	that	that	PRON
cana-1725	30	18	reach	reach	VERB
cana-1725	30	19	these	these	DET
cana-1725	30	20	boundary	boundary	ADJ
cana-1725	30	21	values	value	NOUN
cana-1725	30	22	.	.	PUNCT
cana-1725	31	1	this	this	DET
cana-1725	31	2	study	study	NOUN
cana-1725	31	3	enhances	enhance	VERB
cana-1725	31	4	the	the	DET
cana-1725	31	5	understanding	understanding	NOUN
cana-1725	31	6	of	of	ADP
cana-1725	31	7	how	how	SCONJ
cana-1725	31	8	graph	graph	NOUN
cana-1725	31	9	structure	structure	NOUN
cana-1725	31	10	impacts	impact	VERB
cana-1725	31	11	the	the	DET
cana-1725	31	12	reduced	reduce	VERB
cana-1725	31	13	sombor	sombor	NOUN
cana-1725	31	14	index	index	NOUN
cana-1725	31	15	,	,	PUNCT
cana-1725	31	16	offering	offer	VERB
cana-1725	31	17	valuable	valuable	ADJ
cana-1725	31	18	insights	insight	NOUN
cana-1725	31	19	into	into	ADP
cana-1725	31	20	its	its	PRON
cana-1725	31	21	behaviour	behaviour	NOUN
cana-1725	31	22	and	and	CCONJ
cana-1725	31	23	usage	usage	NOUN
cana-1725	31	24	in	in	ADP
cana-1725	31	25	both	both	CCONJ
cana-1725	31	26	theoretical	theoretical	ADJ
cana-1725	31	27	and	and	CCONJ
cana-1725	31	28	practical	practical	ADJ
cana-1725	31	29	graph	graph	NOUN
cana-1725	31	30	analyses	analysis	NOUN
cana-1725	31	31	.	.	PUNCT
cana-1725	32	1	this	this	DET
cana-1725	32	2	structured	structured	ADJ
cana-1725	32	3	approach	approach	NOUN
cana-1725	32	4	not	not	PART
cana-1725	32	5	only	only	ADV
cana-1725	32	6	provides	provide	VERB
cana-1725	32	7	thorough	thorough	ADJ
cana-1725	32	8	proofs	proof	NOUN
cana-1725	32	9	for	for	ADP
cana-1725	32	10	the	the	DET
cana-1725	32	11	claims	claim	NOUN
cana-1725	32	12	and	and	CCONJ
cana-1725	32	13	theorems	theorem	NOUN
cana-1725	32	14	but	but	CCONJ
cana-1725	32	15	also	also	ADV
cana-1725	32	16	uses	use	VERB
cana-1725	32	17	logical	logical	ADJ
cana-1725	32	18	reasoning	reasoning	NOUN
cana-1725	32	19	to	to	PART
cana-1725	32	20	eliminate	eliminate	VERB
cana-1725	32	21	other	other	ADJ
cana-1725	32	22	possibilities	possibility	NOUN
cana-1725	32	23	,	,	PUNCT
cana-1725	32	24	ensuring	ensure	VERB
cana-1725	32	25	the	the	DET
cana-1725	32	26	reduced	reduce	VERB
cana-1725	32	27	sombor	sombor	NOUN
cana-1725	32	28	index	index	NOUN
cana-1725	32	29	bounds	bound	NOUN
cana-1725	32	30	are	be	AUX
cana-1725	32	31	well	well	ADV
cana-1725	32	32	established	establish	VERB
cana-1725	32	33	for	for	ADP
cana-1725	32	34	various	various	ADJ
cana-1725	32	35	graph	graph	NOUN
cana-1725	32	36	types	type	NOUN
cana-1725	32	37	.	.	PUNCT
cana-1725	33	1	in	in	ADP
cana-1725	33	2	future	future	ADJ
cana-1725	33	3	studies	study	NOUN
cana-1725	33	4	,	,	PUNCT
cana-1725	33	5	attention	attention	NOUN
cana-1725	33	6	can	can	AUX
cana-1725	33	7	be	be	AUX
cana-1725	33	8	directed	direct	VERB
cana-1725	33	9	toward	toward	ADP
cana-1725	33	10	trees	tree	NOUN
cana-1725	33	11	showing	show	VERB
cana-1725	33	12	either	either	CCONJ
cana-1725	33	13	the	the	DET
cana-1725	33	14	minimal	minimal	ADJ
cana-1725	33	15	or	or	CCONJ
cana-1725	33	16	maximal	maximal	ADJ
cana-1725	33	17	reduced	reduce	VERB
cana-1725	33	18	sombor	sombor	NOUN
cana-1725	33	19	index	index	NOUN
cana-1725	33	20	.	.	PUNCT
cana-1725	34	1	analyses	analysis	NOUN
cana-1725	34	2	may	may	AUX
cana-1725	34	3	also	also	ADV
cana-1725	34	4	investigate	investigate	VERB
cana-1725	34	5	graphs	graph	NOUN
cana-1725	34	6	with	with	ADP
cana-1725	34	7	a	a	DET
cana-1725	34	8	secondary	secondary	ADJ
cana-1725	34	9	minimum	minimum	NOUN
cana-1725	34	10	or	or	CCONJ
cana-1725	34	11	maximum	maximum	NOUN
cana-1725	34	12	.	.	PUNCT
cana-1725	35	1	furthermore	furthermore	ADV
cana-1725	35	2	,	,	PUNCT
cana-1725	35	3	the	the	DET
cana-1725	35	4	constraints	constraint	NOUN
cana-1725	35	5	for	for	ADP
cana-1725	35	6	trees	tree	NOUN
cana-1725	35	7	based	base	VERB
cana-1725	35	8	on	on	ADP
cana-1725	35	9	the	the	DET
cana-1725	35	10	number	number	NOUN
cana-1725	35	11	of	of	ADP
cana-1725	35	12	vertices	vertex	NOUN
cana-1725	35	13	and	and	CCONJ
cana-1725	35	14	leaves	leave	NOUN
cana-1725	35	15	could	could	AUX
cana-1725	35	16	be	be	AUX
cana-1725	35	17	assessed	assess	VERB
cana-1725	35	18	.	.	PUNCT
cana-1725	36	1	3	3	X
cana-1725	36	2	.	.	X
cana-1725	36	3	methods	method	NOUN
cana-1725	36	4	the	the	DET
cana-1725	36	5	initial	initial	ADJ
cana-1725	36	6	phase	phase	NOUN
cana-1725	36	7	of	of	ADP
cana-1725	36	8	our	our	PRON
cana-1725	36	9	study	study	NOUN
cana-1725	36	10	establishes	establish	VERB
cana-1725	36	11	a	a	DET
cana-1725	36	12	universal	universal	ADJ
cana-1725	36	13	bound	bind	VERB
cana-1725	36	14	for	for	ADP
cana-1725	36	15	any	any	DET
cana-1725	36	16	graph	graph	NOUN
cana-1725	36	17	,	,	PUNCT
cana-1725	36	18	proving	prove	VERB
cana-1725	36	19	that	that	SCONJ
cana-1725	36	20	the	the	DET
cana-1725	36	21	reduced	reduce	VERB
cana-1725	36	22	sombor	sombor	NOUN
cana-1725	36	23	index	index	NOUN
cana-1725	36	24	(	(	PUNCT
cana-1725	36	25	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	36	26	)	)	PUNCT
cana-1725	36	27	is	be	AUX
cana-1725	36	28	always	always	ADV
cana-1725	36	29	non	non	ADJ
cana-1725	36	30	-	-	ADJ
cana-1725	36	31	negative	negative	ADJ
cana-1725	36	32	,	,	PUNCT
cana-1725	36	33	with	with	ADP
cana-1725	36	34	equality	equality	NOUN
cana-1725	36	35	occurring	occur	VERB
cana-1725	36	36	only	only	ADV
cana-1725	36	37	in	in	ADP
cana-1725	36	38	specific	specific	ADJ
cana-1725	36	39	cases	case	NOUN
cana-1725	36	40	like	like	ADP
cana-1725	36	41	𝐾2	𝐾2	NOUN
cana-1725	36	42	or	or	CCONJ
cana-1725	36	43	a	a	DET
cana-1725	36	44	disjoint	disjoint	NOUN
cana-1725	36	45	union	union	NOUN
cana-1725	36	46	of	of	ADP
cana-1725	36	47	𝐾2	𝐾2	PROPN
cana-1725	36	48	.	.	PUNCT
cana-1725	37	1	for	for	ADP
cana-1725	37	2	paths	path	NOUN
cana-1725	37	3	and	and	CCONJ
cana-1725	37	4	stars	star	NOUN
cana-1725	37	5	with	with	ADP
cana-1725	37	6	𝑛	𝑛	PROPN
cana-1725	37	7	vertices	vertex	NOUN
cana-1725	37	8	,	,	PUNCT
cana-1725	37	9	we	we	PRON
cana-1725	37	10	calculate	calculate	VERB
cana-1725	37	11	the	the	DET
cana-1725	37	12	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	37	13	by	by	ADP
cana-1725	37	14	analyzing	analyze	VERB
cana-1725	37	15	vertex	vertex	NOUN
cana-1725	37	16	degrees	degree	NOUN
cana-1725	37	17	and	and	CCONJ
cana-1725	37	18	structural	structural	ADJ
cana-1725	37	19	properties	property	NOUN
cana-1725	37	20	.	.	PUNCT
cana-1725	38	1	in	in	ADP
cana-1725	38	2	paths	path	NOUN
cana-1725	38	3	,	,	PUNCT
cana-1725	38	4	the	the	DET
cana-1725	38	5	linear	linear	ADJ
cana-1725	38	6	structure	structure	NOUN
cana-1725	38	7	simplifies	simplify	VERB
cana-1725	38	8	the	the	DET
cana-1725	38	9	degree	degree	NOUN
cana-1725	38	10	summation	summation	NOUN
cana-1725	38	11	,	,	PUNCT
cana-1725	38	12	while	while	SCONJ
cana-1725	38	13	for	for	ADP
cana-1725	38	14	stars	star	NOUN
cana-1725	38	15	,	,	PUNCT
cana-1725	38	16	the	the	DET
cana-1725	38	17	central	central	ADJ
cana-1725	38	18	and	and	CCONJ
cana-1725	38	19	leaf	leaf	NOUN
cana-1725	38	20	vertices	vertex	NOUN
cana-1725	38	21	have	have	VERB
cana-1725	38	22	distinct	distinct	ADJ
cana-1725	38	23	degree	degree	NOUN
cana-1725	38	24	contributions	contribution	NOUN
cana-1725	38	25	to	to	ADP
cana-1725	38	26	the	the	DET
cana-1725	38	27	𝑅𝑆𝑂.	𝑅𝑆𝑂.	PUNCT
cana-1725	38	28	for	for	ADP
cana-1725	38	29	wheel	wheel	NOUN
cana-1725	38	30	graphs	graph	NOUN
cana-1725	38	31	,	,	PUNCT
cana-1725	38	32	which	which	PRON
cana-1725	38	33	consist	consist	VERB
cana-1725	38	34	of	of	ADP
cana-1725	38	35	an	an	DET
cana-1725	38	36	outer	outer	ADJ
cana-1725	38	37	cycle	cycle	NOUN
cana-1725	38	38	and	and	CCONJ
cana-1725	38	39	a	a	DET
cana-1725	38	40	central	central	ADJ
cana-1725	38	41	hub	hub	NOUN
cana-1725	38	42	vertex	vertex	NOUN
cana-1725	38	43	connected	connect	VERB
cana-1725	38	44	to	to	ADP
cana-1725	38	45	all	all	DET
cana-1725	38	46	others	other	NOUN
cana-1725	38	47	,	,	PUNCT
cana-1725	38	48	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	38	49	calculation	calculation	NOUN
cana-1725	38	50	is	be	AUX
cana-1725	38	51	based	base	VERB
cana-1725	38	52	on	on	ADP
cana-1725	38	53	the	the	DET
cana-1725	38	54	interaction	interaction	NOUN
cana-1725	38	55	between	between	ADP
cana-1725	38	56	the	the	DET
cana-1725	38	57	central	central	ADJ
cana-1725	38	58	hub	hub	NOUN
cana-1725	38	59	and	and	CCONJ
cana-1725	38	60	the	the	DET
cana-1725	38	61	outer	outer	ADJ
cana-1725	38	62	cycle	cycle	NOUN
cana-1725	38	63	,	,	PUNCT
cana-1725	38	64	by	by	ADP
cana-1725	38	65	summing	sum	VERB
cana-1725	38	66	the	the	DET
cana-1725	38	67	degree	degree	NOUN
cana-1725	38	68	contributions	contribution	NOUN
cana-1725	38	69	from	from	ADP
cana-1725	38	70	both	both	CCONJ
cana-1725	38	71	the	the	DET
cana-1725	38	72	hub	hub	NOUN
cana-1725	38	73	and	and	CCONJ
cana-1725	38	74	outer	outer	ADJ
cana-1725	38	75	vertices	vertex	NOUN
cana-1725	38	76	.	.	PUNCT
cana-1725	39	1	in	in	ADP
cana-1725	39	2	graphs	graph	NOUN
cana-1725	39	3	with	with	ADP
cana-1725	39	4	cycles	cycle	NOUN
cana-1725	39	5	,	,	PUNCT
cana-1725	39	6	the	the	DET
cana-1725	39	7	lower	lower	ADV
cana-1725	39	8	bound	bind	VERB
cana-1725	39	9	of	of	ADP
cana-1725	39	10	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	39	11	is	be	AUX
cana-1725	39	12	achieved	achieve	VERB
cana-1725	39	13	when	when	SCONJ
cana-1725	39	14	all	all	DET
cana-1725	39	15	vertices	vertex	NOUN
cana-1725	39	16	are	be	AUX
cana-1725	39	17	on	on	ADP
cana-1725	39	18	a	a	DET
cana-1725	39	19	single	single	ADJ
cana-1725	39	20	unique	unique	ADJ
cana-1725	39	21	cycle	cycle	NOUN
cana-1725	39	22	,	,	PUNCT
cana-1725	39	23	derived	derive	VERB
cana-1725	39	24	from	from	ADP
cana-1725	39	25	analyzing	analyze	VERB
cana-1725	39	26	vertex	vertex	NOUN
cana-1725	39	27	degree	degree	NOUN
cana-1725	39	28	distributions	distribution	NOUN
cana-1725	39	29	and	and	CCONJ
cana-1725	39	30	their	their	PRON
cana-1725	39	31	cyclical	cyclical	ADJ
cana-1725	39	32	nature	nature	NOUN
cana-1725	39	33	.	.	PUNCT
cana-1725	40	1	additionally	additionally	ADV
cana-1725	40	2	,	,	PUNCT
cana-1725	40	3	we	we	PRON
cana-1725	40	4	provide	provide	VERB
cana-1725	40	5	exact	exact	ADJ
cana-1725	40	6	formulas	formula	NOUN
cana-1725	40	7	for	for	ADP
cana-1725	40	8	the	the	DET
cana-1725	40	9	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	40	10	in	in	ADP
cana-1725	40	11	complete	complete	ADJ
cana-1725	40	12	graphs	graph	NOUN
cana-1725	40	13	and	and	CCONJ
cana-1725	40	14	complete	complete	ADJ
cana-1725	40	15	bipartite	bipartite	NOUN
cana-1725	40	16	graphs	graph	NOUN
cana-1725	40	17	,	,	PUNCT
cana-1725	40	18	depending	depend	VERB
cana-1725	40	19	on	on	ADP
cana-1725	40	20	the	the	DET
cana-1725	40	21	number	number	NOUN
cana-1725	40	22	of	of	ADP
cana-1725	40	23	vertices	vertex	NOUN
cana-1725	40	24	and	and	CCONJ
cana-1725	40	25	their	their	PRON
cana-1725	40	26	partitions	partition	NOUN
cana-1725	40	27	.	.	PUNCT
cana-1725	41	1	for	for	ADP
cana-1725	41	2	a	a	DET
cana-1725	41	3	complete	complete	ADJ
cana-1725	41	4	graph	graph	NOUN
cana-1725	41	5	with	with	ADP
cana-1725	41	6	𝑛	𝑛	PROPN
cana-1725	41	7	vertices	vertex	NOUN
cana-1725	41	8	,	,	PUNCT
cana-1725	41	9	the	the	DET
cana-1725	41	10	reduced	reduce	VERB
cana-1725	41	11	sombor	sombor	NOUN
cana-1725	41	12	index	index	NOUN
cana-1725	41	13	(	(	PUNCT
cana-1725	41	14	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	41	15	)	)	PUNCT
cana-1725	41	16	is	be	AUX
cana-1725	41	17	given	give	VERB
cana-1725	41	18	by	by	ADP
cana-1725	41	19	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	41	20	)	)	PUNCT
cana-1725	41	21	=	=	PUNCT
cana-1725	42	1	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1725	42	2	−	−	PROPN
cana-1725	43	1	1)(𝑛	1)(𝑛	NUM
cana-1725	44	1	−	−	PROPN
cana-1725	44	2	2)√2	2)√2	NOUN
cana-1725	44	3	.	.	PUNCT
cana-1725	45	1	for	for	ADP
cana-1725	45	2	a	a	DET
cana-1725	45	3	complete	complete	ADJ
cana-1725	45	4	bipartite	bipartite	NOUN
cana-1725	45	5	graph	graph	NOUN
cana-1725	45	6	,	,	PUNCT
cana-1725	45	7	communications	communication	NOUN
cana-1725	45	8	on	on	ADP
cana-1725	45	9	applied	apply	VERB
cana-1725	45	10	nonlinear	nonlinear	ADJ
cana-1725	45	11	analysis	analysis	NOUN
cana-1725	45	12	issn	issn	NOUN
cana-1725	45	13	:	:	PUNCT
cana-1725	45	14	1074	1074	NUM
cana-1725	45	15	-	-	PUNCT
cana-1725	45	16	133x	133x	NUM
cana-1725	45	17	vol	vol	NOUN
cana-1725	45	18	32	32	NUM
cana-1725	45	19	no	no	NOUN
cana-1725	45	20	.	.	NOUN
cana-1725	45	21	2	2	NUM
cana-1725	45	22	(	(	PUNCT
cana-1725	45	23	2025	2025	NUM
cana-1725	45	24	)	)	PUNCT
cana-1725	45	25	125	125	NUM
cana-1725	45	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	45	27	the	the	DET
cana-1725	45	28	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	45	29	is	be	AUX
cana-1725	45	30	calculated	calculate	VERB
cana-1725	45	31	as	as	ADP
cana-1725	45	32	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	45	33	)	)	PUNCT
cana-1725	45	34	=	=	SYM
cana-1725	46	1	𝑝𝑞√(𝑝	𝑝𝑞√(𝑝	PROPN
cana-1725	47	1	−	−	PROPN
cana-1725	47	2	1)2	1)2	NUM
cana-1725	47	3	+	+	CCONJ
cana-1725	47	4	(	(	PUNCT
cana-1725	47	5	𝑞	𝑞	X
cana-1725	47	6	−	−	PROPN
cana-1725	47	7	1)2	1)2	NUM
cana-1725	47	8	,	,	PUNCT
cana-1725	47	9	where	where	SCONJ
cana-1725	47	10	∣	∣	ADJ
cana-1725	47	11	𝑋	𝑋	PROPN
cana-1725	47	12	∣=	∣=	PROPN
cana-1725	47	13	𝑝	𝑝	PROPN
cana-1725	47	14	and	and	CCONJ
cana-1725	47	15	∣	∣	ADJ
cana-1725	47	16	𝑌	𝑌	PROPN
cana-1725	47	17	∣=	∣=	PROPN
cana-1725	47	18	𝑞	𝑞	PROPN
cana-1725	47	19	represent	represent	VERB
cana-1725	47	20	the	the	DET
cana-1725	47	21	two	two	NUM
cana-1725	47	22	vertex	vertex	NOUN
cana-1725	47	23	partitions	partition	NOUN
cana-1725	47	24	.	.	PUNCT
cana-1725	48	1	these	these	DET
cana-1725	48	2	results	result	NOUN
cana-1725	48	3	are	be	AUX
cana-1725	48	4	derived	derive	VERB
cana-1725	48	5	from	from	ADP
cana-1725	48	6	analyzing	analyze	VERB
cana-1725	48	7	the	the	DET
cana-1725	48	8	degree	degree	NOUN
cana-1725	48	9	distributions	distribution	NOUN
cana-1725	48	10	in	in	ADP
cana-1725	48	11	these	these	DET
cana-1725	48	12	regular	regular	ADJ
cana-1725	48	13	structures	structure	NOUN
cana-1725	48	14	,	,	PUNCT
cana-1725	48	15	which	which	PRON
cana-1725	48	16	facilitate	facilitate	VERB
cana-1725	48	17	the	the	DET
cana-1725	48	18	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	48	19	calculations	calculation	NOUN
cana-1725	48	20	.	.	PUNCT
cana-1725	49	1	for	for	ADP
cana-1725	49	2	connected	connected	ADJ
cana-1725	49	3	graphs	graph	NOUN
cana-1725	49	4	with	with	ADP
cana-1725	49	5	weak	weak	ADJ
cana-1725	49	6	support	support	NOUN
cana-1725	49	7	vertices	vertex	NOUN
cana-1725	49	8	,	,	PUNCT
cana-1725	49	9	we	we	PRON
cana-1725	49	10	demonstrate	demonstrate	VERB
cana-1725	49	11	that	that	SCONJ
cana-1725	49	12	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	49	13	)	)	PUNCT
cana-1725	49	14	≥	≥	NOUN
cana-1725	49	15	𝑚	𝑚	NOUN
cana-1725	49	16	,	,	PUNCT
cana-1725	49	17	where	where	SCONJ
cana-1725	49	18	𝑚	𝑚	PROPN
cana-1725	49	19	is	be	AUX
cana-1725	49	20	the	the	DET
cana-1725	49	21	number	number	NOUN
cana-1725	49	22	of	of	ADP
cana-1725	49	23	edges	edge	NOUN
cana-1725	49	24	and	and	CCONJ
cana-1725	49	25	𝑞	𝑞	X
cana-1725	49	26	is	be	AUX
cana-1725	49	27	the	the	DET
cana-1725	49	28	number	number	NOUN
cana-1725	49	29	of	of	ADP
cana-1725	49	30	weak	weak	ADJ
cana-1725	49	31	support	support	NOUN
cana-1725	49	32	vertices	vertex	NOUN
cana-1725	49	33	.	.	PUNCT
cana-1725	50	1	this	this	DET
cana-1725	50	2	finding	finding	NOUN
cana-1725	50	3	stems	stem	VERB
cana-1725	50	4	from	from	ADP
cana-1725	50	5	analyzing	analyze	VERB
cana-1725	50	6	how	how	SCONJ
cana-1725	50	7	pendant	pendant	ADJ
cana-1725	50	8	vertices	vertex	NOUN
cana-1725	50	9	next	next	ADV
cana-1725	50	10	to	to	ADP
cana-1725	50	11	weak	weak	ADJ
cana-1725	50	12	support	support	NOUN
cana-1725	50	13	vertices	vertex	NOUN
cana-1725	50	14	affect	affect	VERB
cana-1725	50	15	the	the	DET
cana-1725	50	16	graph	graph	NOUN
cana-1725	50	17	's	's	PART
cana-1725	50	18	structure	structure	NOUN
cana-1725	50	19	and	and	CCONJ
cana-1725	50	20	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	50	21	value	value	NOUN
cana-1725	50	22	.	.	PUNCT
cana-1725	51	1	additionally	additionally	ADV
cana-1725	51	2	,	,	PUNCT
cana-1725	51	3	we	we	PRON
cana-1725	51	4	establish	establish	VERB
cana-1725	51	5	a	a	DET
cana-1725	51	6	theorem	theorem	ADJ
cana-1725	51	7	identifying	identify	VERB
cana-1725	51	8	graphs	graph	NOUN
cana-1725	51	9	with	with	ADP
cana-1725	51	10	an	an	DET
cana-1725	51	11	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	51	12	of	of	ADP
cana-1725	51	13	exactly	exactly	ADV
cana-1725	51	14	zero	zero	NUM
cana-1725	51	15	through	through	ADP
cana-1725	51	16	a	a	DET
cana-1725	51	17	caseby	caseby	ADJ
cana-1725	51	18	-	-	PUNCT
cana-1725	51	19	case	case	NOUN
cana-1725	51	20	analysis	analysis	NOUN
cana-1725	51	21	of	of	ADP
cana-1725	51	22	connected	connected	ADJ
cana-1725	51	23	and	and	CCONJ
cana-1725	51	24	disconnected	disconnected	ADJ
cana-1725	51	25	graphs	graph	NOUN
cana-1725	51	26	,	,	PUNCT
cana-1725	51	27	utilizing	utilize	VERB
cana-1725	51	28	vertex	vertex	NOUN
cana-1725	51	29	degree	degree	NOUN
cana-1725	51	30	properties	property	NOUN
cana-1725	51	31	.	.	PUNCT
cana-1725	52	1	we	we	PRON
cana-1725	52	2	also	also	ADV
cana-1725	52	3	prove	prove	VERB
cana-1725	52	4	that	that	SCONJ
cana-1725	52	5	for	for	ADP
cana-1725	52	6	graphs	graph	NOUN
cana-1725	52	7	with	with	ADP
cana-1725	52	8	𝑛	𝑛	PRON
cana-1725	52	9	≥	≥	NUM
cana-1725	52	10	3	3	NUM
cana-1725	52	11	vertices	vertex	NOUN
cana-1725	52	12	,	,	PUNCT
cana-1725	52	13	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	52	14	)	)	PUNCT
cana-1725	52	15	≥	≥	NOUN
cana-1725	52	16	𝑛	𝑛	DET
cana-1725	52	17	−	−	NUM
cana-1725	52	18	1	1	NUM
cana-1725	52	19	,	,	PUNCT
cana-1725	52	20	achieving	achieve	VERB
cana-1725	52	21	equality	equality	NOUN
cana-1725	52	22	only	only	ADV
cana-1725	52	23	if	if	SCONJ
cana-1725	52	24	the	the	DET
cana-1725	52	25	graph	graph	NOUN
cana-1725	52	26	is	be	AUX
cana-1725	52	27	a	a	DET
cana-1725	52	28	path	path	NOUN
cana-1725	52	29	or	or	CCONJ
cana-1725	52	30	a	a	DET
cana-1725	52	31	star	star	NOUN
cana-1725	52	32	,	,	PUNCT
cana-1725	52	33	based	base	VERB
cana-1725	52	34	on	on	ADP
cana-1725	52	35	graph	graph	NOUN
cana-1725	52	36	diameters	diameter	NOUN
cana-1725	52	37	.	.	PUNCT
cana-1725	53	1	to	to	PART
cana-1725	53	2	establish	establish	VERB
cana-1725	53	3	the	the	DET
cana-1725	53	4	results	result	NOUN
cana-1725	53	5	and	and	CCONJ
cana-1725	53	6	bounds	bound	NOUN
cana-1725	53	7	,	,	PUNCT
cana-1725	53	8	we	we	PRON
cana-1725	53	9	utilized	utilize	VERB
cana-1725	53	10	several	several	ADJ
cana-1725	53	11	key	key	ADJ
cana-1725	53	12	graph	graph	NOUN
cana-1725	53	13	theory	theory	NOUN
cana-1725	53	14	techniques	technique	NOUN
cana-1725	53	15	.	.	PUNCT
cana-1725	54	1	the	the	DET
cana-1725	54	2	degree	degree	NOUN
cana-1725	54	3	of	of	ADP
cana-1725	54	4	each	each	DET
cana-1725	54	5	vertex	vertex	NOUN
cana-1725	54	6	is	be	AUX
cana-1725	54	7	crucial	crucial	ADJ
cana-1725	54	8	for	for	ADP
cana-1725	54	9	determining	determine	VERB
cana-1725	54	10	the	the	DET
cana-1725	54	11	reduced	reduce	VERB
cana-1725	54	12	sombor	sombor	NOUN
cana-1725	54	13	index	index	NOUN
cana-1725	54	14	(	(	PUNCT
cana-1725	54	15	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	54	16	)	)	PUNCT
cana-1725	54	17	.	.	PUNCT
cana-1725	55	1	by	by	ADP
cana-1725	55	2	analyzing	analyze	VERB
cana-1725	55	3	vertex	vertex	NOUN
cana-1725	55	4	degrees	degree	NOUN
cana-1725	55	5	across	across	ADP
cana-1725	55	6	different	different	ADJ
cana-1725	55	7	graph	graph	NOUN
cana-1725	55	8	structures	structure	NOUN
cana-1725	55	9	,	,	PUNCT
cana-1725	55	10	we	we	PRON
cana-1725	55	11	derived	derive	VERB
cana-1725	55	12	precise	precise	ADJ
cana-1725	55	13	bounds	bound	NOUN
cana-1725	55	14	and	and	CCONJ
cana-1725	55	15	values	value	NOUN
cana-1725	55	16	for	for	ADP
cana-1725	55	17	the	the	PRON
cana-1725	55	18	𝑅𝑆𝑂.	𝑅𝑆𝑂.	PUNCT
cana-1725	55	19	we	we	PRON
cana-1725	55	20	conducted	conduct	VERB
cana-1725	55	21	detailed	detailed	ADJ
cana-1725	55	22	case	case	NOUN
cana-1725	55	23	-	-	PUNCT
cana-1725	55	24	by	by	ADP
cana-1725	55	25	-	-	PUNCT
cana-1725	55	26	case	case	NOUN
cana-1725	55	27	examinations	examination	NOUN
cana-1725	55	28	of	of	ADP
cana-1725	55	29	specific	specific	ADJ
cana-1725	55	30	graph	graph	NOUN
cana-1725	55	31	types	type	NOUN
cana-1725	55	32	—	—	PUNCT
cana-1725	55	33	such	such	ADJ
cana-1725	55	34	as	as	ADP
cana-1725	55	35	paths	path	NOUN
cana-1725	55	36	,	,	PUNCT
cana-1725	55	37	cycles	cycle	NOUN
cana-1725	55	38	,	,	PUNCT
cana-1725	55	39	trees	tree	NOUN
cana-1725	55	40	,	,	PUNCT
cana-1725	55	41	and	and	CCONJ
cana-1725	55	42	stars	star	NOUN
cana-1725	55	43	to	to	PART
cana-1725	55	44	understand	understand	VERB
cana-1725	55	45	how	how	SCONJ
cana-1725	55	46	their	their	PRON
cana-1725	55	47	unique	unique	ADJ
cana-1725	55	48	properties	property	NOUN
cana-1725	55	49	influence	influence	VERB
cana-1725	55	50	the	the	PRON
cana-1725	55	51	𝑅𝑆𝑂.	𝑅𝑆𝑂.	ADP
cana-1725	55	52	proof	proof	NOUN
cana-1725	55	53	by	by	ADP
cana-1725	55	54	contradiction	contradiction	NOUN
cana-1725	55	55	helped	help	VERB
cana-1725	55	56	us	we	PRON
cana-1725	55	57	eliminate	eliminate	VERB
cana-1725	55	58	graph	graph	NOUN
cana-1725	55	59	configurations	configuration	NOUN
cana-1725	55	60	that	that	PRON
cana-1725	55	61	did	do	AUX
cana-1725	55	62	not	not	PART
cana-1725	55	63	meet	meet	VERB
cana-1725	55	64	specific	specific	ADJ
cana-1725	55	65	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	55	66	conditions	condition	NOUN
cana-1725	55	67	,	,	PUNCT
cana-1725	55	68	proving	prove	VERB
cana-1725	55	69	the	the	DET
cana-1725	55	70	uniqueness	uniqueness	NOUN
cana-1725	55	71	of	of	ADP
cana-1725	55	72	certain	certain	ADJ
cana-1725	55	73	structures	structure	NOUN
cana-1725	55	74	for	for	ADP
cana-1725	55	75	given	give	VERB
cana-1725	55	76	bounds	bound	NOUN
cana-1725	55	77	.	.	PUNCT
cana-1725	56	1	additionally	additionally	ADV
cana-1725	56	2	,	,	PUNCT
cana-1725	56	3	we	we	PRON
cana-1725	56	4	applied	apply	VERB
cana-1725	56	5	logical	logical	ADJ
cana-1725	56	6	reasoning	reasoning	NOUN
cana-1725	56	7	regarding	regard	VERB
cana-1725	56	8	graph	graph	NOUN
cana-1725	56	9	connectivity	connectivity	NOUN
cana-1725	56	10	and	and	CCONJ
cana-1725	56	11	diameter	diameter	NOUN
cana-1725	56	12	to	to	PART
cana-1725	56	13	identify	identify	VERB
cana-1725	56	14	necessary	necessary	ADJ
cana-1725	56	15	and	and	CCONJ
cana-1725	56	16	sufficient	sufficient	ADJ
cana-1725	56	17	conditions	condition	NOUN
cana-1725	56	18	for	for	ADP
cana-1725	56	19	specific	specific	ADJ
cana-1725	56	20	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	56	21	values	value	NOUN
cana-1725	56	22	.	.	PUNCT
cana-1725	57	1	these	these	DET
cana-1725	57	2	techniques	technique	NOUN
cana-1725	57	3	allowed	allow	VERB
cana-1725	57	4	us	we	PRON
cana-1725	57	5	to	to	PART
cana-1725	57	6	establish	establish	VERB
cana-1725	57	7	rigorous	rigorous	ADJ
cana-1725	57	8	bounds	bound	NOUN
cana-1725	57	9	on	on	ADP
cana-1725	57	10	the	the	DET
cana-1725	57	11	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	57	12	for	for	ADP
cana-1725	57	13	a	a	DET
cana-1725	57	14	wide	wide	ADJ
cana-1725	57	15	variety	variety	NOUN
cana-1725	57	16	of	of	ADP
cana-1725	57	17	graphs	graph	NOUN
cana-1725	57	18	.	.	PUNCT
cana-1725	58	1	4	4	X
cana-1725	58	2	.	.	NOUN
cana-1725	58	3	results	result	NOUN
cana-1725	58	4	finding	find	VERB
cana-1725	58	5	bounds	bound	NOUN
cana-1725	58	6	of	of	ADP
cana-1725	58	7	topological	topological	ADJ
cana-1725	58	8	indices	index	NOUN
cana-1725	58	9	are	be	AUX
cana-1725	58	10	always	always	ADV
cana-1725	58	11	has	have	AUX
cana-1725	58	12	been	be	AUX
cana-1725	58	13	interesting	interesting	ADJ
cana-1725	58	14	and	and	CCONJ
cana-1725	58	15	studied	study	VERB
cana-1725	58	16	by	by	ADP
cana-1725	58	17	researchers	researcher	NOUN
cana-1725	58	18	in	in	ADP
cana-1725	58	19	the	the	DET
cana-1725	58	20	literature	literature	NOUN
cana-1725	58	21	.	.	PUNCT
cana-1725	59	1	for	for	ADP
cana-1725	59	2	results	result	NOUN
cana-1725	59	3	on	on	ADP
cana-1725	59	4	the	the	DET
cana-1725	59	5	bounds	bound	NOUN
cana-1725	59	6	of	of	ADP
cana-1725	59	7	various	various	ADJ
cana-1725	59	8	topological	topological	ADJ
cana-1725	59	9	indices	index	NOUN
cana-1725	59	10	,	,	PUNCT
cana-1725	59	11	the	the	DET
cana-1725	59	12	reader	reader	NOUN
cana-1725	59	13	may	may	AUX
cana-1725	59	14	refer	refer	VERB
cana-1725	59	15	[	[	NOUN
cana-1725	59	16	4	4	NUM
cana-1725	59	17	]	]	PUNCT
cana-1725	59	18	.	.	PUNCT
cana-1725	60	1	for	for	ADP
cana-1725	60	2	a	a	DET
cana-1725	60	3	wide	wide	ADJ
cana-1725	60	4	collection	collection	NOUN
cana-1725	60	5	of	of	ADP
cana-1725	60	6	graphs	graph	NOUN
cana-1725	60	7	,	,	PUNCT
cana-1725	60	8	the	the	DET
cana-1725	60	9	bounds	bound	NOUN
cana-1725	60	10	on	on	ADP
cana-1725	60	11	the	the	DET
cana-1725	60	12	reduced	reduce	VERB
cana-1725	60	13	sombor	sombor	NOUN
cana-1725	60	14	index	index	NOUN
cana-1725	60	15	are	be	AUX
cana-1725	60	16	observed	observe	VERB
cana-1725	60	17	in	in	ADP
cana-1725	60	18	this	this	DET
cana-1725	60	19	section	section	NOUN
cana-1725	60	20	.	.	PUNCT
cana-1725	61	1	these	these	DET
cana-1725	61	2	results	result	NOUN
cana-1725	61	3	are	be	AUX
cana-1725	61	4	also	also	ADV
cana-1725	61	5	used	use	VERB
cana-1725	61	6	in	in	ADP
cana-1725	61	7	the	the	DET
cana-1725	61	8	subsequent	subsequent	ADJ
cana-1725	61	9	results	result	NOUN
cana-1725	61	10	where	where	SCONJ
cana-1725	61	11	we	we	PRON
cana-1725	61	12	found	find	VERB
cana-1725	61	13	the	the	DET
cana-1725	61	14	maximum	maximum	ADJ
cana-1725	61	15	and	and	CCONJ
cana-1725	61	16	minimum	minimum	ADJ
cana-1725	61	17	values	value	NOUN
cana-1725	61	18	of	of	ADP
cana-1725	61	19	the	the	DET
cana-1725	61	20	reduced	reduce	VERB
cana-1725	61	21	sombor	sombor	NOUN
cana-1725	61	22	index	index	NOUN
cana-1725	61	23	of	of	ADP
cana-1725	61	24	graphs	graph	NOUN
cana-1725	61	25	.	.	PUNCT
cana-1725	62	1	let	let	VERB
cana-1725	62	2	us	we	PRON
cana-1725	62	3	start	start	VERB
cana-1725	62	4	with	with	ADP
cana-1725	62	5	the	the	DET
cana-1725	62	6	following	follow	VERB
cana-1725	62	7	proposition	proposition	NOUN
cana-1725	62	8	:	:	PUNCT
cana-1725	62	9	observation	observation	NOUN
cana-1725	62	10	1	1	NUM
cana-1725	62	11	for	for	ADP
cana-1725	62	12	any	any	DET
cana-1725	62	13	graph	graph	NOUN
cana-1725	62	14	𝐺	𝐺	NOUN
cana-1725	62	15	,	,	PUNCT
cana-1725	62	16	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	62	17	≥	≥	NUM
cana-1725	62	18	0	0	NUM
cana-1725	62	19	.	.	PUNCT
cana-1725	63	1	proposition	proposition	NOUN
cana-1725	63	2	2	2	NUM
cana-1725	63	3	if	if	SCONJ
cana-1725	63	4	𝐺	𝐺	PROPN
cana-1725	63	5	has	have	VERB
cana-1725	63	6	a	a	DET
cana-1725	63	7	path	path	NOUN
cana-1725	63	8	on	on	ADP
cana-1725	63	9	𝑛	𝑛	DET
cana-1725	63	10	vertices	vertex	NOUN
cana-1725	63	11	,	,	PUNCT
cana-1725	63	12	then	then	ADV
cana-1725	63	13	,	,	PUNCT
cana-1725	63	14	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	63	15	)	)	PUNCT
cana-1725	63	16	=	=	SYM
cana-1725	63	17	𝑛	𝑛	PRON
cana-1725	63	18	−	−	NUM
cana-1725	63	19	1	1	X
cana-1725	63	20	.	.	PUNCT
cana-1725	63	21	proposition	proposition	NOUN
cana-1725	63	22	3	3	NUM
cana-1725	63	23	if	if	SCONJ
cana-1725	63	24	𝐺	𝐺	PROPN
cana-1725	63	25	is	be	AUX
cana-1725	63	26	a	a	DET
cana-1725	63	27	star	star	NOUN
cana-1725	63	28	on	on	ADP
cana-1725	63	29	𝑛	𝑛	DET
cana-1725	63	30	vertices	vertex	NOUN
cana-1725	63	31	where	where	SCONJ
cana-1725	63	32	𝑛	𝑛	PRON
cana-1725	63	33	≥	≥	NOUN
cana-1725	63	34	4	4	NUM
cana-1725	63	35	,	,	PUNCT
cana-1725	63	36	then	then	ADV
cana-1725	63	37	,	,	PUNCT
cana-1725	63	38	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	63	39	)	)	PUNCT
cana-1725	63	40	=	=	SYM
cana-1725	64	1	𝑛	𝑛	PRON
cana-1725	64	2	−	−	NUM
cana-1725	65	1	1	1	X
cana-1725	65	2	.	.	PUNCT
cana-1725	65	3	proposition	proposition	NOUN
cana-1725	65	4	4	4	NUM
cana-1725	65	5	if	if	SCONJ
cana-1725	65	6	𝐺	𝐺	PROPN
cana-1725	65	7	is	be	AUX
cana-1725	65	8	a	a	DET
cana-1725	65	9	wheel	wheel	NOUN
cana-1725	65	10	on	on	ADP
cana-1725	65	11	𝑛	𝑛	DET
cana-1725	65	12	vertices	vertex	NOUN
cana-1725	65	13	where	where	SCONJ
cana-1725	65	14	𝑛	𝑛	PRON
cana-1725	65	15	≥	≥	NOUN
cana-1725	65	16	4	4	NUM
cana-1725	65	17	,	,	PUNCT
cana-1725	65	18	then	then	ADV
cana-1725	65	19	,	,	PUNCT
cana-1725	65	20	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	65	21	)	)	PUNCT
cana-1725	65	22	=	=	PUNCT
cana-1725	65	23	(	(	PUNCT
cana-1725	65	24	𝑛	𝑛	PRON
cana-1725	65	25	−	−	NUM
cana-1725	65	26	1)(1	1)(1	NUM
cana-1725	65	27	+	+	CCONJ
cana-1725	65	28	√2	√2	NOUN
cana-1725	65	29	)	)	PUNCT
cana-1725	65	30	proposition	proposition	NOUN
cana-1725	65	31	5	5	NUM
cana-1725	65	32	if	if	SCONJ
cana-1725	65	33	𝐺	𝐺	PROPN
cana-1725	65	34	is	be	AUX
cana-1725	65	35	a	a	DET
cana-1725	65	36	graph	graph	NOUN
cana-1725	65	37	with	with	ADP
cana-1725	65	38	at	at	ADV
cana-1725	65	39	least	least	ADV
cana-1725	65	40	one	one	NUM
cana-1725	65	41	cycle	cycle	NOUN
cana-1725	65	42	,	,	PUNCT
cana-1725	65	43	then	then	ADV
cana-1725	65	44	,	,	PUNCT
cana-1725	65	45	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	65	46	)	)	PUNCT
cana-1725	65	47	≥	≥	NOUN
cana-1725	65	48	3√2	3√2	NUM
cana-1725	65	49	.	.	PUNCT
cana-1725	66	1	proposition	proposition	NOUN
cana-1725	66	2	6	6	NUM
cana-1725	66	3	if	if	SCONJ
cana-1725	66	4	𝐺	𝐺	PROPN
cana-1725	66	5	is	be	AUX
cana-1725	66	6	a	a	DET
cana-1725	66	7	graph	graph	NOUN
cana-1725	66	8	with	with	ADP
cana-1725	66	9	a	a	DET
cana-1725	66	10	cycle	cycle	NOUN
cana-1725	66	11	of	of	ADP
cana-1725	66	12	length	length	NOUN
cana-1725	66	13	𝑘.	𝑘.	NOUN
cana-1725	66	14	then	then	ADV
cana-1725	66	15	,	,	PUNCT
cana-1725	66	16	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	66	17	)	)	PUNCT
cana-1725	66	18	≥	≥	NOUN
cana-1725	67	1	𝑘√2	𝑘√2	VERB
cana-1725	67	2	with	with	ADP
cana-1725	67	3	equality	equality	NOUN
cana-1725	67	4	if	if	SCONJ
cana-1725	67	5	and	and	CCONJ
cana-1725	67	6	only	only	ADV
cana-1725	67	7	if	if	SCONJ
cana-1725	67	8	every	every	DET
cana-1725	67	9	vertex	vertex	NOUN
cana-1725	67	10	of	of	ADP
cana-1725	67	11	𝐺	𝐺	PROPN
cana-1725	67	12	lies	lie	VERB
cana-1725	67	13	in	in	ADP
cana-1725	67	14	the	the	DET
cana-1725	67	15	unique	unique	ADJ
cana-1725	67	16	cycle	cycle	NOUN
cana-1725	67	17	of	of	ADP
cana-1725	67	18	cycle	cycle	NOUN
cana-1725	67	19	.	.	PUNCT
cana-1725	68	1	proposition	proposition	NOUN
cana-1725	68	2	7	7	NUM
cana-1725	68	3	if	if	SCONJ
cana-1725	68	4	𝐺	𝐺	PROPN
cana-1725	68	5	is	be	AUX
cana-1725	68	6	a	a	DET
cana-1725	68	7	complete	complete	ADJ
cana-1725	68	8	graph	graph	NOUN
cana-1725	68	9	,	,	PUNCT
cana-1725	68	10	then	then	ADV
cana-1725	68	11	,	,	PUNCT
cana-1725	68	12	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	68	13	)	)	PUNCT
cana-1725	68	14	=	=	PUNCT
cana-1725	69	1	𝑛(𝑛	𝑛(𝑛	PROPN
cana-1725	69	2	−	−	PROPN
cana-1725	70	1	1)(𝑛	1)(𝑛	NUM
cana-1725	70	2	−	−	X
cana-1725	70	3	2)√2	2)√2	NUM
cana-1725	70	4	..	..	PUNCT
cana-1725	70	5	proposition	proposition	NOUN
cana-1725	70	6	8	8	NUM
cana-1725	70	7	if	if	SCONJ
cana-1725	70	8	𝐺	𝐺	PROPN
cana-1725	70	9	=	=	SYM
cana-1725	70	10	(	(	PUNCT
cana-1725	70	11	𝑋	𝑋	PROPN
cana-1725	70	12	,	,	PUNCT
cana-1725	70	13	𝑌	𝑌	PROPN
cana-1725	70	14	)	)	PUNCT
cana-1725	70	15	is	be	AUX
cana-1725	70	16	a	a	DET
cana-1725	70	17	complete	complete	ADJ
cana-1725	70	18	bipartite	bipartite	NOUN
cana-1725	70	19	graph	graph	NOUN
cana-1725	70	20	with	with	ADP
cana-1725	70	21	|𝑋|	|𝑋|	NOUN
cana-1725	70	22	=	=	SYM
cana-1725	70	23	𝑝	𝑝	PROPN
cana-1725	70	24	and	and	CCONJ
cana-1725	70	25	|𝑌	|𝑌	PROPN
cana-1725	70	26	|	|	NOUN
cana-1725	70	27	=	=	SYM
cana-1725	70	28	𝑞	𝑞	PROPN
cana-1725	70	29	,	,	PUNCT
cana-1725	70	30	then	then	ADV
cana-1725	70	31	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	70	32	)	)	PUNCT
cana-1725	70	33	=	=	SYM
cana-1725	70	34	𝑝𝑞√(𝑝	𝑝𝑞√(𝑝	PROPN
cana-1725	71	1	−	−	PROPN
cana-1725	71	2	1)2	1)2	NUM
cana-1725	71	3	+	+	CCONJ
cana-1725	71	4	(	(	PUNCT
cana-1725	71	5	𝑞	𝑞	X
cana-1725	71	6	−	−	PROPN
cana-1725	71	7	1)2	1)2	NUM
cana-1725	71	8	.	.	PUNCT
cana-1725	72	1	communications	communication	NOUN
cana-1725	72	2	on	on	ADP
cana-1725	72	3	applied	apply	VERB
cana-1725	72	4	nonlinear	nonlinear	ADJ
cana-1725	72	5	analysis	analysis	NOUN
cana-1725	72	6	issn	issn	NOUN
cana-1725	72	7	:	:	PUNCT
cana-1725	72	8	1074	1074	NUM
cana-1725	72	9	-	-	PUNCT
cana-1725	72	10	133x	133x	NUM
cana-1725	72	11	vol	vol	NOUN
cana-1725	72	12	32	32	NUM
cana-1725	72	13	no	no	NOUN
cana-1725	72	14	.	.	NOUN
cana-1725	72	15	2	2	NUM
cana-1725	72	16	(	(	PUNCT
cana-1725	72	17	2025	2025	NUM
cana-1725	72	18	)	)	PUNCT
cana-1725	73	1	126	126	NUM
cana-1725	73	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	73	3	proposition	proposition	NOUN
cana-1725	73	4	9	9	NUM
cana-1725	73	5	if	if	SCONJ
cana-1725	73	6	𝐺	𝐺	PROPN
cana-1725	73	7	is	be	AUX
cana-1725	73	8	a	a	DET
cana-1725	73	9	connected	connected	ADJ
cana-1725	73	10	graph	graph	NOUN
cana-1725	73	11	with	with	ADP
cana-1725	73	12	𝑞	𝑞	X
cana-1725	73	13	weak	weak	ADJ
cana-1725	73	14	support	support	NOUN
cana-1725	73	15	vertices	vertex	NOUN
cana-1725	73	16	,	,	PUNCT
cana-1725	73	17	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	73	18	)	)	PUNCT
cana-1725	73	19	≥	≥	NOUN
cana-1725	73	20	(	(	PUNCT
cana-1725	73	21	𝑚	𝑚	PROPN
cana-1725	73	22	−	−	PROPN
cana-1725	73	23	𝑞	𝑞	PROPN
cana-1725	73	24	)	)	PUNCT
cana-1725	73	25	,	,	PUNCT
cana-1725	73	26	where	where	SCONJ
cana-1725	73	27	𝑚	𝑚	PROPN
cana-1725	73	28	is	be	AUX
cana-1725	73	29	the	the	DET
cana-1725	73	30	number	number	NOUN
cana-1725	73	31	of	of	ADP
cana-1725	73	32	edges	edge	NOUN
cana-1725	73	33	in	in	ADP
cana-1725	73	34	𝐺.	𝐺.	NOUN
cana-1725	73	35	proof	proof	NOUN
cana-1725	73	36	:	:	PUNCT
cana-1725	73	37	let	let	VERB
cana-1725	73	38	𝐺	𝐺	PRON
cana-1725	73	39	be	be	AUX
cana-1725	73	40	a	a	DET
cana-1725	73	41	connected	connected	ADJ
cana-1725	73	42	graph	graph	NOUN
cana-1725	73	43	with	with	ADP
cana-1725	73	44	𝑞	𝑞	X
cana-1725	73	45	weak	weak	ADJ
cana-1725	73	46	support	support	NOUN
cana-1725	73	47	vertices	vertex	NOUN
cana-1725	73	48	,	,	PUNCT
cana-1725	73	49	say	say	VERB
cana-1725	73	50	𝑢1	𝑢1	PROPN
cana-1725	73	51	,	,	PUNCT
cana-1725	73	52	𝑢2	𝑢2	PROPN
cana-1725	73	53	,	,	PUNCT
cana-1725	73	54	.	.	PUNCT
cana-1725	73	55	.	.	PUNCT
cana-1725	74	1	.	.	PUNCT
cana-1725	75	1	,	,	PUNCT
cana-1725	75	2	𝑢𝑞.	𝑢𝑞.	VERB
cana-1725	75	3	then	then	ADV
cana-1725	75	4	,	,	PUNCT
cana-1725	75	5	there	there	PRON
cana-1725	75	6	are	be	VERB
cana-1725	75	7	𝑞	𝑞	DET
cana-1725	75	8	pendant	pendant	ADJ
cana-1725	75	9	vertices	vertex	NOUN
cana-1725	75	10	adjacent	adjacent	ADJ
cana-1725	75	11	to	to	ADP
cana-1725	75	12	these	these	DET
cana-1725	75	13	weak	weak	ADJ
cana-1725	75	14	support	support	NOUN
cana-1725	75	15	vertices	vertex	NOUN
cana-1725	75	16	,	,	PUNCT
cana-1725	75	17	say	say	VERB
cana-1725	75	18	𝑣1	𝑣1	PROPN
cana-1725	75	19	,	,	PUNCT
cana-1725	75	20	𝑣2	𝑣2	PROPN
cana-1725	75	21	,	,	PUNCT
cana-1725	75	22	.	.	PUNCT
cana-1725	75	23	.	.	PUNCT
cana-1725	76	1	.	.	PUNCT
cana-1725	77	1	,	,	PUNCT
cana-1725	77	2	𝑣𝑞.	𝑣𝑞.	X
cana-1725	77	3	then	then	ADV
cana-1725	77	4	,	,	PUNCT
cana-1725	77	5	out	out	ADP
cana-1725	77	6	of	of	ADP
cana-1725	77	7	the	the	DET
cana-1725	77	8	𝑚-edges	𝑚-edge	NOUN
cana-1725	77	9	of	of	ADP
cana-1725	77	10	𝐺	𝐺	PROPN
cana-1725	77	11	,	,	PUNCT
cana-1725	77	12	these	these	DET
cana-1725	77	13	𝑞	𝑞	PROPN
cana-1725	77	14	edges	edge	VERB
cana-1725	77	15	𝑢1𝑣1	𝑢1𝑣1	NOUN
cana-1725	77	16	,	,	PUNCT
cana-1725	77	17	𝑢2𝑣2	𝑢2𝑣2	NOUN
cana-1725	77	18	,	,	PUNCT
cana-1725	77	19	.	.	PUNCT
cana-1725	77	20	.	.	PUNCT
cana-1725	77	21	.	.	PUNCT
cana-1725	78	1	𝑢𝑞𝑣𝑞	𝑢𝑞𝑣𝑞	PROPN
cana-1725	78	2	are	be	AUX
cana-1725	78	3	adding	add	VERB
cana-1725	78	4	a	a	DET
cana-1725	78	5	zero	zero	NUM
cana-1725	78	6	for	for	ADP
cana-1725	78	7	each	each	PRON
cana-1725	78	8	,	,	PUNCT
cana-1725	78	9	3	3	NUM
cana-1725	78	10	and	and	CCONJ
cana-1725	78	11	the	the	DET
cana-1725	78	12	remaining	remain	VERB
cana-1725	78	13	(	(	PUNCT
cana-1725	78	14	𝑚	𝑚	PROPN
cana-1725	78	15	−	−	PROPN
cana-1725	78	16	𝑞	𝑞	NOUN
cana-1725	78	17	)	)	PUNCT
cana-1725	78	18	edges	edge	NOUN
cana-1725	78	19	add	add	VERB
cana-1725	78	20	at	at	ADV
cana-1725	78	21	least	least	ADJ
cana-1725	78	22	one	one	NUM
cana-1725	78	23	for	for	ADP
cana-1725	78	24	each	each	PRON
cana-1725	78	25	implies	imply	VERB
cana-1725	78	26	that	that	SCONJ
cana-1725	78	27	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	78	28	)	)	PUNCT
cana-1725	78	29	=	=	PUNCT
cana-1725	78	30	𝑚	𝑚	PROPN
cana-1725	78	31	−	−	PROPN
cana-1725	78	32	𝑞.	𝑞.	NOUN
cana-1725	78	33	but	but	CCONJ
cana-1725	78	34	,	,	PUNCT
cana-1725	78	35	there	there	PRON
cana-1725	78	36	may	may	AUX
cana-1725	78	37	be	be	AUX
cana-1725	78	38	edges	edge	NOUN
cana-1725	78	39	(	(	PUNCT
cana-1725	78	40	out	out	ADP
cana-1725	78	41	of	of	ADP
cana-1725	78	42	the	the	DET
cana-1725	78	43	(	(	PUNCT
cana-1725	78	44	𝑚	𝑚	PROPN
cana-1725	78	45	−	−	PROPN
cana-1725	78	46	𝑞	𝑞	NOUN
cana-1725	78	47	)	)	PUNCT
cana-1725	78	48	−	−	ADP
cana-1725	78	49	𝑒𝑑𝑔𝑒𝑠	𝑒𝑑𝑔𝑒𝑠	NOUN
cana-1725	78	50	)	)	PUNCT
cana-1725	78	51	adding	add	VERB
cana-1725	78	52	more	more	ADJ
cana-1725	78	53	than	than	ADP
cana-1725	78	54	1	1	NUM
cana-1725	78	55	also	also	ADV
cana-1725	78	56	to	to	PART
cana-1725	78	57	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	VERB
cana-1725	78	58	)	)	PUNCT
cana-1725	78	59	,	,	PUNCT
cana-1725	78	60	for	for	ADP
cana-1725	78	61	example	example	NOUN
cana-1725	78	62	,	,	PUNCT
cana-1725	78	63	if	if	SCONJ
cana-1725	78	64	g	g	PROPN
cana-1725	78	65	have	have	VERB
cana-1725	78	66	a	a	DET
cana-1725	78	67	consecutive	consecutive	ADJ
cana-1725	78	68	3	3	NUM
cana-1725	78	69	adjacent	adjacent	ADJ
cana-1725	78	70	vertices	vertex	NOUN
cana-1725	78	71	𝑢	𝑢	PROPN
cana-1725	78	72	,	,	PUNCT
cana-1725	78	73	𝑣	𝑣	NOUN
cana-1725	78	74	,	,	PUNCT
cana-1725	78	75	𝑤	𝑤	ADP
cana-1725	78	76	such	such	ADJ
cana-1725	78	77	that	that	SCONJ
cana-1725	78	78	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1725	78	79	)	)	PUNCT
cana-1725	78	80	≥	≥	NOUN
cana-1725	78	81	3	3	NUM
cana-1725	78	82	,	,	PUNCT
cana-1725	78	83	then	then	ADV
cana-1725	78	84	the	the	DET
cana-1725	78	85	edges	edge	NOUN
cana-1725	78	86	𝑢𝑣	𝑢𝑣	NOUN
cana-1725	78	87	,	,	PUNCT
cana-1725	78	88	𝑣𝑤	𝑣𝑤	ADV
cana-1725	78	89	both	both	PRON
cana-1725	78	90	adds	add	VERB
cana-1725	78	91	at	at	ADP
cana-1725	78	92	least	least	ADJ
cana-1725	78	93	2	2	NUM
cana-1725	78	94	to	to	PART
cana-1725	78	95	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	78	96	)	)	PUNCT
cana-1725	78	97	.	.	PUNCT
cana-1725	79	1	thus	thus	ADV
cana-1725	79	2	,	,	PUNCT
cana-1725	79	3	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	79	4	)	)	PUNCT
cana-1725	79	5	≥	≥	NOUN
cana-1725	79	6	(	(	PUNCT
cana-1725	79	7	𝑚	𝑚	PROPN
cana-1725	79	8	−	−	PROPN
cana-1725	79	9	𝑞	𝑞	NOUN
cana-1725	79	10	)	)	PUNCT
cana-1725	79	11	.	.	PUNCT
cana-1725	80	1	□	□	PUNCT
cana-1725	80	2	theorem	theorem	NOUN
cana-1725	80	3	10	10	NUM
cana-1725	80	4	if	if	SCONJ
cana-1725	80	5	𝐺	𝐺	PROPN
cana-1725	80	6	is	be	AUX
cana-1725	80	7	a	a	DET
cana-1725	80	8	graph	graph	NOUN
cana-1725	80	9	with	with	ADP
cana-1725	80	10	the	the	DET
cana-1725	80	11	number	number	NOUN
cana-1725	80	12	of	of	ADP
cana-1725	80	13	vertices	vertex	NOUN
cana-1725	80	14	𝑛	𝑛	PRON
cana-1725	80	15	≥	≥	NUM
cana-1725	80	16	2	2	NUM
cana-1725	80	17	,	,	PUNCT
cana-1725	80	18	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	80	19	)	)	PUNCT
cana-1725	80	20	≥	≥	NOUN
cana-1725	80	21	0	0	NUM
cana-1725	80	22	and	and	CCONJ
cana-1725	80	23	the	the	DET
cana-1725	80	24	equality	equality	NOUN
cana-1725	80	25	is	be	AUX
cana-1725	80	26	true	true	ADJ
cana-1725	80	27	if	if	SCONJ
cana-1725	80	28	and	and	CCONJ
cana-1725	80	29	only	only	ADV
cana-1725	80	30	if	if	SCONJ
cana-1725	80	31	𝐺	𝐺	PROPN
cana-1725	80	32	=	=	NOUN
cana-1725	80	33	𝐾2	𝐾2	NOUN
cana-1725	80	34	𝑜𝑟	𝑜𝑟	PROPN
cana-1725	80	35	𝐺	𝐺	PROPN
cana-1725	80	36	=	=	SYM
cana-1725	80	37	𝑚𝐾2	𝑚𝐾2	PROPN
cana-1725	80	38	.	.	PUNCT
cana-1725	81	1	proof	proof	NOUN
cana-1725	81	2	:	:	PUNCT
cana-1725	81	3	the	the	DET
cana-1725	81	4	converse	converse	NOUN
cana-1725	81	5	of	of	ADP
cana-1725	81	6	the	the	DET
cana-1725	81	7	result	result	NOUN
cana-1725	81	8	is	be	AUX
cana-1725	81	9	true	true	ADJ
cana-1725	81	10	as	as	SCONJ
cana-1725	81	11	𝐺	𝐺	NOUN
cana-1725	81	12	=	=	NOUN
cana-1725	81	13	𝐾2	𝐾2	NOUN
cana-1725	81	14	𝑜𝑟	𝑜𝑟	PROPN
cana-1725	81	15	𝐺	𝐺	PROPN
cana-1725	81	16	=	=	PUNCT
cana-1725	81	17	𝑚𝐾2	𝑚𝐾2	NOUN
cana-1725	81	18	are	be	AUX
cana-1725	81	19	graphs	graph	NOUN
cana-1725	81	20	with	with	ADP
cana-1725	81	21	rso(g	rso(g	NOUN
cana-1725	81	22	)	)	PUNCT
cana-1725	81	23	=	=	SYM
cana-1725	82	1	0	0	X
cana-1725	82	2	.	.	PUNCT
cana-1725	82	3	to	to	PART
cana-1725	82	4	prove	prove	VERB
cana-1725	82	5	the	the	DET
cana-1725	82	6	necessary	necessary	ADJ
cana-1725	82	7	part	part	NOUN
cana-1725	82	8	,	,	PUNCT
cana-1725	82	9	first	first	ADV
cana-1725	82	10	assume	assume	VERB
cana-1725	82	11	that	that	SCONJ
cana-1725	82	12	rso(g	rso(g	AUX
cana-1725	82	13	)	)	PUNCT
cana-1725	82	14	=	=	SYM
cana-1725	83	1	0	0	X
cana-1725	83	2	.	.	PUNCT
cana-1725	84	1	let	let	VERB
cana-1725	84	2	𝑥	𝑥	PRON
cana-1725	84	3	be	be	AUX
cana-1725	84	4	a	a	DET
cana-1725	84	5	vertex	vertex	NOUN
cana-1725	84	6	of	of	ADP
cana-1725	84	7	𝐺	𝐺	NOUN
cana-1725	84	8	such	such	ADJ
cana-1725	84	9	that	that	SCONJ
cana-1725	84	10	𝑑(𝑥	𝑑(𝑥	PROPN
cana-1725	84	11	)	)	PUNCT
cana-1725	84	12	≥	≥	NOUN
cana-1725	84	13	2	2	X
cana-1725	84	14	.	.	PUNCT
cana-1725	85	1	let	let	VERB
cana-1725	85	2	𝑦	𝑦	PRON
cana-1725	85	3	be	be	AUX
cana-1725	85	4	a	a	DET
cana-1725	85	5	vertex	vertex	NOUN
cana-1725	85	6	incident	incident	NOUN
cana-1725	85	7	with	with	ADP
cana-1725	85	8	𝑥	𝑥	PROPN
cana-1725	85	9	in	in	ADP
cana-1725	85	10	𝐺.	𝐺.	NOUN
cana-1725	85	11	irrespective	irrespective	ADV
cana-1725	85	12	of	of	ADP
cana-1725	85	13	the	the	DET
cana-1725	85	14	degree	degree	NOUN
cana-1725	85	15	of	of	ADP
cana-1725	85	16	𝑦	𝑦	NOUN
cana-1725	85	17	,	,	PUNCT
cana-1725	85	18	the	the	DET
cana-1725	85	19	edge	edge	NOUN
cana-1725	85	20	𝑥𝑦	𝑥𝑦	VERB
cana-1725	85	21	adds	add	VERB
cana-1725	85	22	up	up	ADP
cana-1725	85	23	a	a	DET
cana-1725	85	24	1	1	NUM
cana-1725	85	25	to	to	ADP
cana-1725	85	26	the	the	DET
cana-1725	85	27	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	85	28	)	)	PUNCT
cana-1725	85	29	,	,	PUNCT
cana-1725	85	30	which	which	PRON
cana-1725	85	31	implies	imply	VERB
cana-1725	85	32	that	that	DET
cana-1725	85	33	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	85	34	)	)	PUNCT
cana-1725	85	35	≥	≥	NOUN
cana-1725	85	36	1	1	NUM
cana-1725	85	37	,	,	PUNCT
cana-1725	85	38	which	which	PRON
cana-1725	85	39	is	be	AUX
cana-1725	85	40	a	a	DET
cana-1725	85	41	contradiction	contradiction	NOUN
cana-1725	85	42	to	to	ADP
cana-1725	85	43	our	our	PRON
cana-1725	85	44	assumption	assumption	NOUN
cana-1725	85	45	.	.	PUNCT
cana-1725	86	1	thus	thus	ADV
cana-1725	86	2	,	,	PUNCT
cana-1725	86	3	𝑑(𝑥	𝑑(𝑥	PROPN
cana-1725	86	4	)	)	PUNCT
cana-1725	86	5	can	can	AUX
cana-1725	86	6	not	not	PART
cana-1725	86	7	be	be	AUX
cana-1725	86	8	more	more	ADJ
cana-1725	86	9	than	than	ADP
cana-1725	86	10	one	one	NUM
cana-1725	86	11	.	.	PUNCT
cana-1725	87	1	thus	thus	ADV
cana-1725	87	2	,	,	PUNCT
cana-1725	87	3	any	any	DET
cana-1725	87	4	vertex	vertex	NOUN
cana-1725	87	5	of	of	ADP
cana-1725	87	6	𝐺	𝐺	PROPN
cana-1725	87	7	can	can	AUX
cana-1725	87	8	have	have	VERB
cana-1725	87	9	only	only	ADV
cana-1725	87	10	degree	degree	VERB
cana-1725	87	11	one	one	NUM
cana-1725	87	12	.	.	PUNCT
cana-1725	88	1	case-1	case-1	NUM
cana-1725	88	2	:	:	PUNCT
cana-1725	89	1	𝐺	𝐺	PROPN
cana-1725	89	2	is	be	AUX
cana-1725	89	3	connected	connect	VERB
cana-1725	89	4	.	.	PUNCT
cana-1725	90	1	claim	claim	NOUN
cana-1725	90	2	:	:	PUNCT
cana-1725	90	3	𝐺	𝐺	NOUN
cana-1725	90	4	=	=	PUNCT
cana-1725	90	5	𝐾2	𝐾2	NOUN
cana-1725	90	6	.	.	PUNCT
cana-1725	91	1	it	it	PRON
cana-1725	91	2	is	be	AUX
cana-1725	91	3	enough	enough	ADJ
cana-1725	91	4	to	to	PART
cana-1725	91	5	prove	prove	VERB
cana-1725	91	6	that	that	SCONJ
cana-1725	91	7	𝐺	𝐺	PROPN
cana-1725	91	8	has	have	VERB
cana-1725	91	9	only	only	ADV
cana-1725	91	10	two	two	NUM
cana-1725	91	11	vertices	vertex	NOUN
cana-1725	91	12	.	.	PUNCT
cana-1725	92	1	suppose	suppose	VERB
cana-1725	92	2	that	that	SCONJ
cana-1725	92	3	g	g	PROPN
cana-1725	92	4	let	let	VERB
cana-1725	92	5	𝑢	𝑢	NOUN
cana-1725	92	6	,	,	PUNCT
cana-1725	92	7	𝑣	𝑣	NOUN
cana-1725	92	8	,	,	PUNCT
cana-1725	92	9	𝑤	𝑤	PART
cana-1725	92	10	be	be	AUX
cana-1725	92	11	3	3	NUM
cana-1725	92	12	arbitrary	arbitrary	ADJ
cana-1725	92	13	vertices	vertex	NOUN
cana-1725	92	14	of	of	ADP
cana-1725	92	15	𝐺	𝐺	PROPN
cana-1725	92	16	such	such	ADJ
cana-1725	92	17	that	that	SCONJ
cana-1725	92	18	𝑢	𝑢	NOUN
cana-1725	92	19	and	and	CCONJ
cana-1725	92	20	𝑣	𝑣	X
cana-1725	92	21	are	be	AUX
cana-1725	92	22	adjacent	adjacent	ADJ
cana-1725	92	23	and	and	CCONJ
cana-1725	92	24	𝑣	𝑣	X
cana-1725	92	25	and	and	CCONJ
cana-1725	92	26	𝑤	𝑤	PROPN
cana-1725	92	27	are	be	AUX
cana-1725	92	28	adjacent	adjacent	ADJ
cana-1725	92	29	.	.	PUNCT
cana-1725	93	1	since	since	SCONJ
cana-1725	93	2	𝐺	𝐺	PROPN
cana-1725	93	3	is	be	AUX
cana-1725	93	4	a	a	DET
cana-1725	93	5	connected	connected	ADJ
cana-1725	93	6	graph	graph	NOUN
cana-1725	93	7	,	,	PUNCT
cana-1725	93	8	this	this	DET
cana-1725	93	9	assumption	assumption	NOUN
cana-1725	93	10	is	be	AUX
cana-1725	93	11	possible	possible	ADJ
cana-1725	93	12	,	,	PUNCT
cana-1725	93	13	otherwise	otherwise	ADV
cana-1725	93	14	𝐺	𝐺	PROPN
cana-1725	93	15	would	would	AUX
cana-1725	93	16	be	be	AUX
cana-1725	93	17	disconnected	disconnect	VERB
cana-1725	93	18	.	.	PUNCT
cana-1725	94	1	thus	thus	ADV
cana-1725	94	2	,	,	PUNCT
cana-1725	94	3	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1725	94	4	)	)	PUNCT
cana-1725	94	5	≥	≥	NOUN
cana-1725	94	6	2	2	NUM
cana-1725	94	7	,	,	PUNCT
cana-1725	94	8	implying	imply	VERB
cana-1725	94	9	that	that	SCONJ
cana-1725	94	10	𝑅𝑆𝑂	𝑅𝑆𝑂	PROPN
cana-1725	94	11	≥	≥	NUM
cana-1725	94	12	2	2	NUM
cana-1725	94	13	,	,	PUNCT
cana-1725	94	14	a	a	DET
cana-1725	94	15	contradiction	contradiction	NOUN
cana-1725	94	16	.	.	PUNCT
cana-1725	95	1	thus	thus	ADV
cana-1725	95	2	,	,	PUNCT
cana-1725	95	3	𝐺	𝐺	PROPN
cana-1725	95	4	has	have	VERB
cana-1725	95	5	only	only	ADV
cana-1725	95	6	two	two	NUM
cana-1725	95	7	vertices	vertex	NOUN
cana-1725	95	8	,	,	PUNCT
cana-1725	95	9	which	which	PRON
cana-1725	95	10	means	mean	VERB
cana-1725	95	11	,	,	PUNCT
cana-1725	96	1	𝐺	𝐺	PROPN
cana-1725	96	2	=	=	PUNCT
cana-1725	96	3	𝐾2	𝐾2	PROPN
cana-1725	96	4	.	.	PUNCT
cana-1725	97	1	figure	figure	NOUN
cana-1725	97	2	1	1	NUM
cana-1725	97	3	:	:	PUNCT
cana-1725	97	4	graphs	graph	VERB
cana-1725	97	5	g	g	NOUN
cana-1725	97	6	=	=	PUNCT
cana-1725	97	7	𝐾2	𝐾2	NOUN
cana-1725	97	8	and	and	CCONJ
cana-1725	97	9	g	g	NOUN
cana-1725	97	10	=	=	PROPN
cana-1725	97	11	m𝐾2	m𝐾2	PROPN
cana-1725	97	12	case-2	case-2	NUM
cana-1725	97	13	:	:	PUNCT
cana-1725	97	14	𝐺	𝐺	PROPN
cana-1725	97	15	is	be	AUX
cana-1725	97	16	disconnected	disconnect	VERB
cana-1725	97	17	.	.	PUNCT
cana-1725	98	1	claim	claim	NOUN
cana-1725	98	2	:	:	PUNCT
cana-1725	98	3	𝐺	𝐺	PROPN
cana-1725	98	4	=	=	SYM
cana-1725	98	5	𝑚𝐾2	𝑚𝐾2	PROPN
cana-1725	98	6	.	.	PUNCT
cana-1725	99	1	since	since	SCONJ
cana-1725	99	2	𝐺	𝐺	PROPN
cana-1725	99	3	is	be	AUX
cana-1725	99	4	disconnected	disconnect	VERB
cana-1725	99	5	,	,	PUNCT
cana-1725	99	6	g	g	PROPN
cana-1725	99	7	must	must	AUX
cana-1725	99	8	be	be	AUX
cana-1725	99	9	the	the	DET
cana-1725	99	10	union	union	NOUN
cana-1725	99	11	of	of	ADP
cana-1725	99	12	a	a	DET
cana-1725	99	13	finite	finite	ADJ
cana-1725	99	14	number	number	NOUN
cana-1725	99	15	of	of	ADP
cana-1725	99	16	connected	connected	ADJ
cana-1725	99	17	components	component	NOUN
cana-1725	99	18	,	,	PUNCT
cana-1725	99	19	say	say	VERB
cana-1725	99	20	𝐺1	𝐺1	NOUN
cana-1725	99	21	,	,	PUNCT
cana-1725	99	22	𝐺2	𝐺2	ADV
cana-1725	99	23	,	,	PUNCT
cana-1725	99	24	.	.	PUNCT
cana-1725	99	25	.	.	PUNCT
cana-1725	100	1	.	.	PUNCT
cana-1725	101	1	,	,	PUNCT
cana-1725	101	2	𝐺𝑚.	𝐺𝑚.	PROPN
cana-1725	101	3	for	for	ADP
cana-1725	101	4	each	each	PRON
cana-1725	101	5	of	of	ADP
cana-1725	101	6	these	these	DET
cana-1725	101	7	connected	connected	ADJ
cana-1725	101	8	component	component	NOUN
cana-1725	101	9	,	,	PUNCT
cana-1725	101	10	by	by	ADP
cana-1725	101	11	case-1	case-1	NUM
cana-1725	101	12	,	,	PUNCT
cana-1725	101	13	we	we	PRON
cana-1725	101	14	have	have	VERB
cana-1725	101	15	that	that	PRON
cana-1725	101	16	𝐺𝑖	𝐺𝑖	PROPN
cana-1725	101	17	=	=	SYM
cana-1725	101	18	𝐾2.for	𝐾2.for	PROPN
cana-1725	101	19	𝑖	𝑖	SYM
cana-1725	101	20	=	=	SYM
cana-1725	101	21	1	1	NUM
cana-1725	101	22	,	,	PUNCT
cana-1725	101	23	2	2	NUM
cana-1725	101	24	,	,	PUNCT
cana-1725	101	25	.	.	PUNCT
cana-1725	101	26	.	.	PUNCT
cana-1725	101	27	.	.	PUNCT
cana-1725	102	1	,	,	PUNCT
cana-1725	102	2	𝑚.	𝑚.	ADV
cana-1725	102	3	thus	thus	ADV
cana-1725	102	4	,	,	PUNCT
cana-1725	102	5	𝐺	𝐺	PROPN
cana-1725	102	6	=	=	SYM
cana-1725	102	7	𝑚𝐾2	𝑚𝐾2	PROPN
cana-1725	102	8	.	.	PUNCT
cana-1725	103	1	by	by	ADP
cana-1725	103	2	case-1	case-1	PROPN
cana-1725	103	3	and	and	CCONJ
cana-1725	103	4	case-2	case-2	NUM
cana-1725	103	5	,	,	PUNCT
cana-1725	103	6	the	the	DET
cana-1725	103	7	theorem	theorem	NOUN
cana-1725	103	8	is	be	AUX
cana-1725	103	9	proved	prove	VERB
cana-1725	103	10	.	.	PUNCT
cana-1725	104	1	communications	communication	NOUN
cana-1725	104	2	on	on	ADP
cana-1725	104	3	applied	apply	VERB
cana-1725	104	4	nonlinear	nonlinear	ADJ
cana-1725	104	5	analysis	analysis	NOUN
cana-1725	104	6	issn	issn	NOUN
cana-1725	104	7	:	:	PUNCT
cana-1725	104	8	1074	1074	NUM
cana-1725	104	9	-	-	PUNCT
cana-1725	104	10	133x	133x	NUM
cana-1725	104	11	vol	vol	NOUN
cana-1725	104	12	32	32	NUM
cana-1725	104	13	no	no	NOUN
cana-1725	104	14	.	.	NOUN
cana-1725	104	15	2	2	NUM
cana-1725	104	16	(	(	PUNCT
cana-1725	104	17	2025	2025	NUM
cana-1725	104	18	)	)	PUNCT
cana-1725	104	19	127	127	NUM
cana-1725	104	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	104	21	theorem	theorem	VERB
cana-1725	104	22	11	11	NUM
cana-1725	104	23	if	if	SCONJ
cana-1725	104	24	𝐺	𝐺	PROPN
cana-1725	104	25	is	be	AUX
cana-1725	104	26	a	a	DET
cana-1725	104	27	connected	connected	ADJ
cana-1725	104	28	graph	graph	NOUN
cana-1725	104	29	with	with	ADP
cana-1725	104	30	the	the	DET
cana-1725	104	31	number	number	NOUN
cana-1725	104	32	of	of	ADP
cana-1725	104	33	vertices	vertex	NOUN
cana-1725	104	34	𝑛	𝑛	PRON
cana-1725	104	35	≥	≥	NUM
cana-1725	104	36	3	3	NUM
cana-1725	104	37	,	,	PUNCT
cana-1725	104	38	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	104	39	)	)	PUNCT
cana-1725	104	40	≥	≥	NOUN
cana-1725	104	41	𝑛	𝑛	DET
cana-1725	104	42	−	−	NUM
cana-1725	104	43	1	1	NUM
cana-1725	104	44	and	and	CCONJ
cana-1725	104	45	the	the	DET
cana-1725	104	46	equality	equality	NOUN
cana-1725	104	47	is	be	AUX
cana-1725	104	48	true	true	ADJ
cana-1725	104	49	if	if	SCONJ
cana-1725	104	50	and	and	CCONJ
cana-1725	104	51	only	only	ADV
cana-1725	104	52	if	if	SCONJ
cana-1725	104	53	𝐺	𝐺	PROPN
cana-1725	104	54	is	be	AUX
cana-1725	104	55	a	a	DET
cana-1725	104	56	path	path	NOUN
cana-1725	104	57	on	on	ADP
cana-1725	104	58	𝑛	𝑛	DET
cana-1725	104	59	vertices	vertex	NOUN
cana-1725	104	60	,	,	PUNCT
cana-1725	104	61	or	or	CCONJ
cana-1725	104	62	𝐺	𝐺	NOUN
cana-1725	104	63	is	be	AUX
cana-1725	104	64	a	a	DET
cana-1725	104	65	star	star	NOUN
cana-1725	104	66	.	.	PUNCT
cana-1725	105	1	proof	proof	NOUN
cana-1725	105	2	:	:	PUNCT
cana-1725	105	3	let	let	VERB
cana-1725	105	4	𝐺	𝐺	PRON
cana-1725	105	5	be	be	AUX
cana-1725	105	6	a	a	DET
cana-1725	105	7	graph	graph	NOUN
cana-1725	105	8	with	with	ADP
cana-1725	105	9	at	at	ADV
cana-1725	105	10	least	least	ADJ
cana-1725	105	11	3	3	NUM
cana-1725	105	12	vertices	vertex	NOUN
cana-1725	105	13	.	.	PUNCT
cana-1725	106	1	if	if	SCONJ
cana-1725	106	2	𝐺	𝐺	PROPN
cana-1725	106	3	is	be	AUX
cana-1725	106	4	a	a	DET
cana-1725	106	5	path	path	NOUN
cana-1725	106	6	on	on	ADP
cana-1725	106	7	𝑛	𝑛	DET
cana-1725	106	8	vertices	vertex	NOUN
cana-1725	106	9	,	,	PUNCT
cana-1725	106	10	then	then	ADV
cana-1725	106	11	by	by	ADP
cana-1725	106	12	proposition-2	proposition-2	NUM
cana-1725	106	13	,	,	PUNCT
cana-1725	106	14	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	106	15	)	)	PUNCT
cana-1725	106	16	=	=	SYM
cana-1725	107	1	𝑛	𝑛	DET
cana-1725	107	2	−	−	NUM
cana-1725	107	3	1	1	NUM
cana-1725	107	4	,	,	PUNCT
cana-1725	107	5	or	or	CCONJ
cana-1725	107	6	if	if	SCONJ
cana-1725	107	7	𝐺	𝐺	PROPN
cana-1725	107	8	is	be	AUX
cana-1725	107	9	a	a	DET
cana-1725	107	10	star	star	NOUN
cana-1725	107	11	on	on	ADP
cana-1725	107	12	𝑛	𝑛	DET
cana-1725	107	13	vertices	vertex	NOUN
cana-1725	107	14	,	,	PUNCT
cana-1725	107	15	then	then	ADV
cana-1725	107	16	by	by	ADP
cana-1725	107	17	proposition-3	proposition-3	PROPN
cana-1725	107	18	,	,	PUNCT
cana-1725	107	19	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	107	20	)	)	PUNCT
cana-1725	107	21	=	=	SYM
cana-1725	107	22	𝑛	𝑛	PRON
cana-1725	107	23	−	−	NUM
cana-1725	107	24	1	1	NUM
cana-1725	107	25	.	.	PUNCT
cana-1725	108	1	thus	thus	ADV
cana-1725	108	2	,	,	PUNCT
cana-1725	108	3	the	the	DET
cana-1725	108	4	sufficient	sufficient	ADJ
cana-1725	108	5	part	part	NOUN
cana-1725	108	6	is	be	AUX
cana-1725	108	7	true	true	ADJ
cana-1725	108	8	.	.	PUNCT
cana-1725	109	1	to	to	PART
cana-1725	109	2	prove	prove	VERB
cana-1725	109	3	the	the	DET
cana-1725	109	4	other	other	ADJ
cana-1725	109	5	part	part	NOUN
cana-1725	109	6	,	,	PUNCT
cana-1725	109	7	let	let	VERB
cana-1725	109	8	𝐺	𝐺	PRON
cana-1725	109	9	be	be	AUX
cana-1725	109	10	a	a	DET
cana-1725	109	11	graph	graph	NOUN
cana-1725	109	12	with	with	ADP
cana-1725	109	13	n	n	ADP
cana-1725	109	14	vertices	vertex	NOUN
cana-1725	109	15	and	and	CCONJ
cana-1725	109	16	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	109	17	)	)	PUNCT
cana-1725	109	18	=	=	SYM
cana-1725	110	1	𝑛	𝑛	PRON
cana-1725	110	2	−	−	NUM
cana-1725	111	1	1	1	NUM
cana-1725	111	2	.	.	PUNCT
cana-1725	112	1	if	if	SCONJ
cana-1725	112	2	𝐺	𝐺	PROPN
cana-1725	112	3	is	be	AUX
cana-1725	112	4	a	a	DET
cana-1725	112	5	cycle	cycle	NOUN
cana-1725	112	6	,	,	PUNCT
cana-1725	112	7	then	then	ADV
cana-1725	112	8	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	112	9	)	)	PUNCT
cana-1725	112	10	≥	≥	NOUN
cana-1725	112	11	𝑛	𝑛	DET
cana-1725	112	12	√2	√2	PROPN
cana-1725	112	13	>	>	X
cana-1725	112	14	𝑛	𝑛	NOUN
cana-1725	112	15	−	−	NOUN
cana-1725	112	16	1	1	NUM
cana-1725	112	17	.	.	PUNCT
cana-1725	113	1	thus	thus	ADV
cana-1725	113	2	,	,	PUNCT
cana-1725	113	3	𝐺	𝐺	PROPN
cana-1725	113	4	can	can	AUX
cana-1725	113	5	not	not	PART
cana-1725	113	6	even	even	ADV
cana-1725	113	7	contain	contain	VERB
cana-1725	113	8	a	a	DET
cana-1725	113	9	cycle	cycle	NOUN
cana-1725	113	10	.	.	PUNCT
cana-1725	114	1	since	since	SCONJ
cana-1725	114	2	𝐺	𝐺	PROPN
cana-1725	114	3	is	be	AUX
cana-1725	114	4	a	a	DET
cana-1725	114	5	connected	connected	ADJ
cana-1725	114	6	graph	graph	NOUN
cana-1725	114	7	,	,	PUNCT
cana-1725	114	8	we	we	PRON
cana-1725	114	9	must	must	AUX
cana-1725	114	10	have	have	VERB
cana-1725	114	11	that	that	SCONJ
cana-1725	114	12	𝐺	𝐺	PROPN
cana-1725	114	13	is	be	AUX
cana-1725	114	14	a	a	DET
cana-1725	114	15	tree	tree	NOUN
cana-1725	114	16	.	.	PUNCT
cana-1725	115	1	we	we	PRON
cana-1725	115	2	shall	shall	AUX
cana-1725	115	3	prove	prove	VERB
cana-1725	115	4	the	the	DET
cana-1725	115	5	result	result	NOUN
cana-1725	115	6	based	base	VERB
cana-1725	115	7	on	on	ADP
cana-1725	115	8	the	the	DET
cana-1725	115	9	diameter	diameter	NOUN
cana-1725	115	10	of	of	ADP
cana-1725	115	11	𝐺.	𝐺.	PROPN
cana-1725	115	12	claim-1	claim-1	PROPN
cana-1725	115	13	:	:	PUNCT
cana-1725	115	14	if	if	SCONJ
cana-1725	115	15	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	PRON
cana-1725	115	16	)	)	PUNCT
cana-1725	115	17	=	=	SYM
cana-1725	115	18	2	2	NUM
cana-1725	115	19	,	,	PUNCT
cana-1725	115	20	then	then	ADV
cana-1725	115	21	𝐺	𝐺	PROPN
cana-1725	115	22	is	be	AUX
cana-1725	115	23	a	a	DET
cana-1725	115	24	star	star	NOUN
cana-1725	115	25	.	.	PUNCT
cana-1725	116	1	assume	assume	VERB
cana-1725	116	2	that	that	SCONJ
cana-1725	116	3	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	PRON
cana-1725	116	4	)	)	PUNCT
cana-1725	116	5	=	=	SYM
cana-1725	116	6	2	2	X
cana-1725	116	7	.	.	PUNCT
cana-1725	116	8	then	then	ADV
cana-1725	116	9	,	,	PUNCT
cana-1725	116	10	the	the	DET
cana-1725	116	11	path	path	NOUN
cana-1725	116	12	joining	join	VERB
cana-1725	116	13	any	any	DET
cana-1725	116	14	two	two	NUM
cana-1725	116	15	vertices	vertex	NOUN
cana-1725	116	16	is	be	AUX
cana-1725	116	17	of	of	ADP
cana-1725	116	18	length	length	NOUN
cana-1725	116	19	two	two	NUM
cana-1725	116	20	.	.	PUNCT
cana-1725	117	1	since	since	SCONJ
cana-1725	117	2	the	the	DET
cana-1725	117	3	graph	graph	NOUN
cana-1725	117	4	is	be	AUX
cana-1725	117	5	connected	connect	VERB
cana-1725	117	6	,	,	PUNCT
cana-1725	117	7	there	there	PRON
cana-1725	117	8	must	must	AUX
cana-1725	117	9	be	be	AUX
cana-1725	117	10	a	a	DET
cana-1725	117	11	vertex	vertex	NOUN
cana-1725	117	12	𝑥	𝑥	NOUN
cana-1725	117	13	which	which	PRON
cana-1725	117	14	is	be	AUX
cana-1725	117	15	the	the	DET
cana-1725	117	16	center	center	NOUN
cana-1725	117	17	of	of	ADP
cana-1725	117	18	all	all	DET
cana-1725	117	19	paths	path	NOUN
cana-1725	117	20	of	of	ADP
cana-1725	117	21	length	length	NOUN
cana-1725	117	22	two	two	NUM
cana-1725	117	23	.	.	PUNCT
cana-1725	118	1	otherwise	otherwise	ADV
cana-1725	118	2	,	,	PUNCT
cana-1725	118	3	each	each	DET
cana-1725	118	4	pair	pair	NOUN
cana-1725	118	5	of	of	ADP
cana-1725	118	6	vertices	vertex	NOUN
cana-1725	118	7	are	be	AUX
cana-1725	118	8	connected	connect	VERB
cana-1725	118	9	by	by	ADP
cana-1725	118	10	a	a	DET
cana-1725	118	11	distinct	distinct	ADJ
cana-1725	118	12	path	path	NOUN
cana-1725	118	13	of	of	ADP
cana-1725	118	14	length	length	NOUN
cana-1725	118	15	two	two	NUM
cana-1725	118	16	which	which	PRON
cana-1725	118	17	implies	imply	VERB
cana-1725	118	18	that	that	SCONJ
cana-1725	118	19	𝐺	𝐺	PROPN
cana-1725	118	20	is	be	AUX
cana-1725	118	21	the	the	DET
cana-1725	118	22	union	union	NOUN
cana-1725	118	23	of	of	ADP
cana-1725	118	24	copies	copy	NOUN
cana-1725	118	25	of	of	ADP
cana-1725	118	26	𝑃3	𝑃3	NOUN
cana-1725	118	27	.	.	PUNCT
cana-1725	119	1	then	then	ADV
cana-1725	119	2	,	,	PUNCT
cana-1725	119	3	𝐺	𝐺	PROPN
cana-1725	119	4	is	be	AUX
cana-1725	119	5	disconnected	disconnect	VERB
cana-1725	119	6	,	,	PUNCT
cana-1725	119	7	a	a	DET
cana-1725	119	8	contradiction	contradiction	NOUN
cana-1725	119	9	.	.	PUNCT
cana-1725	120	1	figure	figure	NOUN
cana-1725	120	2	2	2	NUM
cana-1725	120	3	:	:	PUNCT
cana-1725	120	4	a	a	DET
cana-1725	120	5	star	star	NOUN
cana-1725	120	6	thus	thus	ADV
cana-1725	120	7	,	,	PUNCT
cana-1725	120	8	there	there	PRON
cana-1725	120	9	exists	exist	VERB
cana-1725	120	10	a	a	DET
cana-1725	120	11	center	center	ADJ
cana-1725	120	12	vertex	vertex	NOUN
cana-1725	120	13	of	of	ADP
cana-1725	120	14	all	all	DET
cana-1725	120	15	paths	path	NOUN
cana-1725	120	16	of	of	ADP
cana-1725	120	17	length	length	NOUN
cana-1725	120	18	two	two	NUM
cana-1725	120	19	.	.	PUNCT
cana-1725	121	1	now	now	ADV
cana-1725	121	2	,	,	PUNCT
cana-1725	121	3	if	if	SCONJ
cana-1725	121	4	𝑥1	𝑥1	NOUN
cana-1725	121	5	,	,	PUNCT
cana-1725	121	6	𝑥2	𝑥2	NOUN
cana-1725	121	7	,	,	PUNCT
cana-1725	121	8	.	.	PUNCT
cana-1725	121	9	.	.	PUNCT
cana-1725	122	1	.	.	PUNCT
cana-1725	123	1	,	,	PUNCT
cana-1725	123	2	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-1725	123	3	are	be	AUX
cana-1725	123	4	the	the	DET
cana-1725	123	5	vertices	vertex	NOUN
cana-1725	123	6	connected	connect	VERB
cana-1725	123	7	to	to	ADP
cana-1725	123	8	𝑥	𝑥	PRON
cana-1725	123	9	,	,	PUNCT
cana-1725	123	10	we	we	PRON
cana-1725	123	11	claim	claim	VERB
cana-1725	123	12	that	that	SCONJ
cana-1725	123	13	none	none	NOUN
cana-1725	123	14	of	of	ADP
cana-1725	123	15	these	these	DET
cana-1725	123	16	vertices	vertex	NOUN
cana-1725	123	17	𝑥𝑖𝑥𝑗	𝑥𝑖𝑥𝑗	VERB
cana-1725	123	18	,	,	PUNCT
cana-1725	123	19	1	1	NUM
cana-1725	123	20	≤	≤	NUM
cana-1725	123	21	𝑖	𝑖	SYM
cana-1725	123	22	≤	≤	NOUN
cana-1725	123	23	(	(	PUNCT
cana-1725	123	24	𝑛	𝑛	PRON
cana-1725	123	25	−	−	PROPN
cana-1725	123	26	2	2	NUM
cana-1725	123	27	)	)	PUNCT
cana-1725	123	28	,	,	PUNCT
cana-1725	123	29	𝑗	𝑗	NOUN
cana-1725	123	30	=	=	SYM
cana-1725	123	31	𝑖	𝑖	NOUN
cana-1725	124	1	+	+	NOUN
cana-1725	124	2	1	1	NUM
cana-1725	124	3	are	be	AUX
cana-1725	124	4	connected	connect	VERB
cana-1725	124	5	.	.	PUNCT
cana-1725	125	1	otherwise	otherwise	ADV
cana-1725	125	2	,	,	PUNCT
cana-1725	125	3	it	it	PRON
cana-1725	125	4	forms	form	VERB
cana-1725	125	5	a	a	DET
cana-1725	125	6	wheel	wheel	NOUN
cana-1725	125	7	,	,	PUNCT
cana-1725	125	8	and	and	CCONJ
cana-1725	125	9	by	by	ADP
cana-1725	125	10	proposition-4	proposition-4	PROPN
cana-1725	125	11	,	,	PUNCT
cana-1725	125	12	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	125	13	)	)	PUNCT
cana-1725	125	14	≥	≥	NOUN
cana-1725	125	15	(	(	PUNCT
cana-1725	125	16	𝑛	𝑛	PRON
cana-1725	125	17	−	−	PROPN
cana-1725	125	18	1	1	NUM
cana-1725	125	19	)	)	PUNCT
cana-1725	125	20	,	,	PUNCT
cana-1725	125	21	a	a	DET
cana-1725	125	22	contradiction	contradiction	NOUN
cana-1725	125	23	.	.	PUNCT
cana-1725	126	1	thus	thus	ADV
cana-1725	126	2	,	,	PUNCT
cana-1725	126	3	𝑥	𝑥	PROPN
cana-1725	126	4	is	be	AUX
cana-1725	126	5	the	the	DET
cana-1725	126	6	center	center	ADJ
cana-1725	126	7	vertex	vertex	NOUN
cana-1725	126	8	and	and	CCONJ
cana-1725	126	9	all	all	DET
cana-1725	126	10	other	other	ADJ
cana-1725	126	11	𝑥𝑖′s	𝑥𝑖′s	NOUN
cana-1725	126	12	,	,	PUNCT
cana-1725	126	13	1	1	NUM
cana-1725	126	14	≤	≤	NUM
cana-1725	126	15	𝑖	𝑖	SYM
cana-1725	126	16	≤	≤	NOUN
cana-1725	126	17	(	(	PUNCT
cana-1725	126	18	𝑛	𝑛	PRON
cana-1725	126	19	−	−	NOUN
cana-1725	126	20	1	1	NUM
cana-1725	126	21	)	)	PUNCT
cana-1725	126	22	are	be	AUX
cana-1725	126	23	adjacent	adjacent	ADJ
cana-1725	126	24	to	to	ADP
cana-1725	126	25	𝑥.	𝑥.	NOUN
cana-1725	126	26	this	this	PRON
cana-1725	126	27	implies	imply	VERB
cana-1725	126	28	that	that	SCONJ
cana-1725	126	29	𝐺	𝐺	PROPN
cana-1725	126	30	is	be	AUX
cana-1725	126	31	a	a	DET
cana-1725	126	32	star	star	NOUN
cana-1725	126	33	.	.	PUNCT
cana-1725	127	1	claim-2	claim-2	NOUN
cana-1725	127	2	:	:	PUNCT
cana-1725	127	3	if	if	SCONJ
cana-1725	127	4	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	NOUN
cana-1725	127	5	)	)	PUNCT
cana-1725	127	6	≥	≥	NOUN
cana-1725	127	7	3	3	NUM
cana-1725	127	8	,	,	PUNCT
cana-1725	127	9	then	then	ADV
cana-1725	127	10	𝐺	𝐺	PROPN
cana-1725	127	11	is	be	AUX
cana-1725	127	12	a	a	DET
cana-1725	127	13	path	path	NOUN
cana-1725	127	14	.	.	PUNCT
cana-1725	128	1	assume	assume	VERB
cana-1725	128	2	now	now	ADV
cana-1725	128	3	that	that	SCONJ
cana-1725	128	4	𝑑𝑖𝑎𝑚(𝐺	𝑑𝑖𝑎𝑚(𝐺	VERB
cana-1725	128	5	)	)	PUNCT
cana-1725	128	6	≥	≥	NOUN
cana-1725	128	7	3	3	NUM
cana-1725	128	8	.	.	PUNCT
cana-1725	129	1	we	we	PRON
cana-1725	129	2	claim	claim	VERB
cana-1725	129	3	that	that	SCONJ
cana-1725	129	4	all	all	DET
cana-1725	129	5	vertices	vertex	NOUN
cana-1725	129	6	of	of	ADP
cana-1725	129	7	𝐺	𝐺	PROPN
cana-1725	129	8	belong	belong	VERB
cana-1725	129	9	to	to	ADP
cana-1725	129	10	a	a	DET
cana-1725	129	11	unique	unique	ADJ
cana-1725	129	12	diametrical	diametrical	ADJ
cana-1725	129	13	path	path	NOUN
cana-1725	129	14	𝑃	𝑃	PROPN
cana-1725	129	15	∶	∶	NOUN
cana-1725	129	16	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1725	129	17	,	,	PUNCT
cana-1725	129	18	.	.	PUNCT
cana-1725	129	19	.	.	PUNCT
cana-1725	129	20	.	.	PUNCT
cana-1725	130	1	,	,	PUNCT
cana-1725	130	2	𝑣𝑛	𝑣𝑛	NOUN
cana-1725	130	3	of	of	ADP
cana-1725	130	4	𝐺.	𝐺.	NOUN
cana-1725	130	5	suppose	suppose	VERB
cana-1725	130	6	there	there	PRON
cana-1725	130	7	exists	exist	VERB
cana-1725	130	8	a	a	DET
cana-1725	130	9	vertex	vertex	NOUN
cana-1725	130	10	,	,	PUNCT
cana-1725	130	11	say	say	VERB
cana-1725	130	12	𝑢	𝑢	PRON
cana-1725	130	13	which	which	PRON
cana-1725	130	14	is	be	AUX
cana-1725	130	15	not	not	PART
cana-1725	130	16	on	on	ADP
cana-1725	130	17	the	the	DET
cana-1725	130	18	diametrical	diametrical	ADJ
cana-1725	130	19	path	path	NOUN
cana-1725	130	20	but	but	CCONJ
cana-1725	130	21	adjacent	adjacent	ADJ
cana-1725	130	22	to	to	ADP
cana-1725	130	23	a	a	DET
cana-1725	130	24	vertex	vertex	NOUN
cana-1725	130	25	,	,	PUNCT
cana-1725	130	26	say	say	VERB
cana-1725	130	27	𝑣	𝑣	ADP
cana-1725	130	28	,	,	PUNCT
cana-1725	130	29	which	which	PRON
cana-1725	130	30	is	be	AUX
cana-1725	130	31	lying	lie	VERB
cana-1725	130	32	on	on	ADP
cana-1725	130	33	the	the	DET
cana-1725	130	34	diametrical	diametrical	ADJ
cana-1725	130	35	path	path	NOUN
cana-1725	130	36	.	.	PUNCT
cana-1725	131	1	without	without	ADP
cana-1725	131	2	loss	loss	NOUN
cana-1725	131	3	of	of	ADP
cana-1725	131	4	generality	generality	NOUN
cana-1725	131	5	,	,	PUNCT
cana-1725	131	6	let	let	VERB
cana-1725	131	7	𝑣	𝑣	PART
cana-1725	131	8	be	be	AUX
cana-1725	131	9	a	a	DET
cana-1725	131	10	support	support	NOUN
cana-1725	131	11	vertex	vertex	NOUN
cana-1725	131	12	of	of	ADP
cana-1725	131	13	𝐺.	𝐺.	NOUN
cana-1725	131	14	assume	assume	VERB
cana-1725	131	15	that	that	SCONJ
cana-1725	131	16	the	the	DET
cana-1725	131	17	diametrical	diametrical	ADJ
cana-1725	131	18	path	path	NOUN
cana-1725	131	19	is	be	AUX
cana-1725	131	20	given	give	VERB
cana-1725	131	21	by	by	ADP
cana-1725	131	22	𝑣1𝑣2	𝑣1𝑣2	NOUN
cana-1725	131	23	=	=	SYM
cana-1725	131	24	𝑣	𝑣	NOUN
cana-1725	131	25	,	,	PUNCT
cana-1725	131	26	.	.	PUNCT
cana-1725	131	27	.	.	PUNCT
cana-1725	131	28	.	.	PUNCT
cana-1725	132	1	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-1725	132	2	.	.	PUNCT
cana-1725	133	1	thus	thus	ADV
cana-1725	133	2	,	,	PUNCT
cana-1725	133	3	the	the	DET
cana-1725	133	4	vertex	vertex	NOUN
cana-1725	133	5	𝑣2	𝑣2	NOUN
cana-1725	133	6	=	=	PUNCT
cana-1725	133	7	𝑣	𝑣	X
cana-1725	133	8	is	be	AUX
cana-1725	133	9	the	the	DET
cana-1725	133	10	only	only	ADJ
cana-1725	133	11	vertex	vertex	NOUN
cana-1725	133	12	that	that	PRON
cana-1725	133	13	has	have	VERB
cana-1725	133	14	𝑑(𝑣	𝑑(𝑣	NOUN
cana-1725	133	15	)	)	PUNCT
cana-1725	133	16	=	=	SYM
cana-1725	133	17	3	3	NUM
cana-1725	133	18	and	and	CCONJ
cana-1725	133	19	all	all	DET
cana-1725	133	20	other	other	ADJ
cana-1725	133	21	vertices	vertex	NOUN
cana-1725	133	22	have	have	VERB
cana-1725	133	23	degree	degree	NOUN
cana-1725	133	24	either	either	CCONJ
cana-1725	133	25	one	one	NUM
cana-1725	133	26	or	or	CCONJ
cana-1725	133	27	two	two	NUM
cana-1725	133	28	.	.	PUNCT
cana-1725	134	1	communications	communication	NOUN
cana-1725	134	2	on	on	ADP
cana-1725	134	3	applied	apply	VERB
cana-1725	134	4	nonlinear	nonlinear	ADJ
cana-1725	134	5	analysis	analysis	NOUN
cana-1725	134	6	issn	issn	NOUN
cana-1725	134	7	:	:	PUNCT
cana-1725	134	8	1074	1074	NUM
cana-1725	134	9	-	-	PUNCT
cana-1725	134	10	133x	133x	NUM
cana-1725	134	11	vol	vol	NOUN
cana-1725	134	12	32	32	NUM
cana-1725	134	13	no	no	NOUN
cana-1725	134	14	.	.	NOUN
cana-1725	134	15	2	2	NUM
cana-1725	134	16	(	(	PUNCT
cana-1725	134	17	2025	2025	NUM
cana-1725	134	18	)	)	PUNCT
cana-1725	134	19	128	128	NUM
cana-1725	134	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	134	21	for	for	ADP
cana-1725	134	22	this	this	DET
cana-1725	134	23	structure	structure	NOUN
cana-1725	134	24	,	,	PUNCT
cana-1725	134	25	the	the	DET
cana-1725	134	26	reduced	reduce	VERB
cana-1725	134	27	sombor	sombor	NOUN
cana-1725	134	28	index	index	NOUN
cana-1725	134	29	is	be	AUX
cana-1725	134	30	given	give	VERB
cana-1725	134	31	by	by	ADP
cana-1725	134	32	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	134	33	)	)	PUNCT
cana-1725	134	34	=	=	SYM
cana-1725	134	35	√(𝑑(𝑣1	√(𝑑(𝑣1	PROPN
cana-1725	134	36	)	)	PUNCT
cana-1725	134	37	−	−	PROPN
cana-1725	135	1	1)2	1)2	NUM
cana-1725	135	2	+	+	CCONJ
cana-1725	135	3	(	(	PUNCT
cana-1725	135	4	𝑑(𝑣2	𝑑(𝑣2	NOUN
cana-1725	135	5	)	)	PUNCT
cana-1725	135	6	−	−	PROPN
cana-1725	136	1	1)2	1)2	NUM
cana-1725	136	2	+	+	NUM
cana-1725	136	3	√(𝑑(𝑣2	√(𝑑(𝑣2	NOUN
cana-1725	136	4	)	)	PUNCT
cana-1725	136	5	−	−	PROPN
cana-1725	137	1	1)2	1)2	NUM
cana-1725	137	2	+	+	CCONJ
cana-1725	137	3	(	(	PUNCT
cana-1725	137	4	𝑑(𝑢	𝑑(𝑢	NOUN
cana-1725	137	5	)	)	PUNCT
cana-1725	137	6	−	−	PROPN
cana-1725	138	1	1)2	1)2	NUM
cana-1725	138	2	+	+	NUM
cana-1725	138	3	√(𝑑(𝑣𝑛−2	√(𝑑(𝑣𝑛−2	NOUN
cana-1725	138	4	)	)	PUNCT
cana-1725	138	5	−	−	PROPN
cana-1725	139	1	1)2	1)2	NUM
cana-1725	139	2	+	+	CCONJ
cana-1725	139	3	(	(	PUNCT
cana-1725	139	4	𝑑(𝑣𝑛−1	𝑑(𝑣𝑛−1	NOUN
cana-1725	139	5	)	)	PUNCT
cana-1725	139	6	−	−	PROPN
cana-1725	140	1	1)2	1)2	NUM
cana-1725	140	2	+	+	NUM
cana-1725	140	3	√(𝑑(𝑣2	√(𝑑(𝑣2	NOUN
cana-1725	140	4	)	)	PUNCT
cana-1725	140	5	−	−	PROPN
cana-1725	141	1	1)2	1)2	NUM
cana-1725	141	2	+	+	CCONJ
cana-1725	141	3	(	(	PUNCT
cana-1725	141	4	𝑑(𝑣3	𝑑(𝑣3	PROPN
cana-1725	141	5	)	)	PUNCT
cana-1725	142	1	−	−	PROPN
cana-1725	143	1	1)2	1)2	NUM
cana-1725	143	2	+	+	CCONJ
cana-1725	143	3	∑	∑	PUNCT
cana-1725	143	4	√(𝑑(𝑣𝑖	√(𝑑(𝑣𝑖	ADJ
cana-1725	143	5	)	)	PUNCT
cana-1725	143	6	−	−	PROPN
cana-1725	144	1	1)2	1)2	NUM
cana-1725	144	2	+	+	CCONJ
cana-1725	144	3	(	(	PUNCT
cana-1725	144	4	𝑑(𝑣𝑖+1	𝑑(𝑣𝑖+1	PROPN
cana-1725	144	5	)	)	PUNCT
cana-1725	144	6	−	−	PROPN
cana-1725	145	1	1)2	1)2	NUM
cana-1725	145	2	𝑛−2	𝑛−2	NOUN
cana-1725	145	3	𝑖=3	𝑖=3	PUNCT
cana-1725	145	4	=	=	SYM
cana-1725	146	1	√(1	√(1	NOUN
cana-1725	146	2	−	−	PROPN
cana-1725	146	3	1)2	1)2	NUM
cana-1725	146	4	+	+	CCONJ
cana-1725	146	5	(	(	PUNCT
cana-1725	146	6	3	3	NUM
cana-1725	146	7	−	−	PROPN
cana-1725	146	8	1)2	1)2	NUM
cana-1725	147	1	+	+	CCONJ
cana-1725	147	2	√(3	√(3	PROPN
cana-1725	147	3	−	−	PROPN
cana-1725	147	4	1)2	1)2	NUM
cana-1725	148	1	+	+	CCONJ
cana-1725	148	2	(	(	PUNCT
cana-1725	148	3	1	1	NUM
cana-1725	148	4	−	−	PROPN
cana-1725	148	5	1)2	1)2	NUM
cana-1725	148	6	+	+	CCONJ
cana-1725	148	7	√(2	√(2	PRON
cana-1725	148	8	−	−	PROPN
cana-1725	148	9	1)2	1)2	NUM
cana-1725	148	10	+	+	CCONJ
cana-1725	148	11	(	(	PUNCT
cana-1725	148	12	1	1	NUM
cana-1725	148	13	−	−	PROPN
cana-1725	148	14	1)2	1)2	NUM
cana-1725	148	15	+	+	CCONJ
cana-1725	148	16	√(3	√(3	PROPN
cana-1725	148	17	−	−	PROPN
cana-1725	148	18	1)2	1)2	NUM
cana-1725	148	19	+	+	CCONJ
cana-1725	148	20	(	(	PUNCT
cana-1725	148	21	2	2	NUM
cana-1725	148	22	−	−	PROPN
cana-1725	148	23	1)2	1)2	NUM
cana-1725	148	24	+	+	CCONJ
cana-1725	148	25	√(2	√(2	PRON
cana-1725	148	26	−	−	PROPN
cana-1725	148	27	1)2	1)2	NUM
cana-1725	148	28	+	+	CCONJ
cana-1725	148	29	(	(	PUNCT
cana-1725	148	30	2	2	NUM
cana-1725	148	31	−	−	PROPN
cana-1725	148	32	1)2	1)2	NUM
cana-1725	148	33	=	=	SYM
cana-1725	148	34	√22	√22	PROPN
cana-1725	149	1	+	+	PROPN
cana-1725	149	2	√22	√22	PROPN
cana-1725	149	3	+	+	CCONJ
cana-1725	149	4	√12	√12	PROPN
cana-1725	150	1	+	+	CCONJ
cana-1725	150	2	√5	√5	PROPN
cana-1725	150	3	+	+	CCONJ
cana-1725	150	4	(	(	PUNCT
cana-1725	150	5	𝑛	𝑛	PROPN
cana-1725	150	6	−	−	PROPN
cana-1725	150	7	4)√12	4)√12	NUM
cana-1725	150	8	=	=	SYM
cana-1725	150	9	2	2	NUM
cana-1725	150	10	+	+	NUM
cana-1725	150	11	2	2	NUM
cana-1725	150	12	+	+	SYM
cana-1725	150	13	1	1	NUM
cana-1725	151	1	+	+	CCONJ
cana-1725	151	2	√5	√5	PROPN
cana-1725	151	3	+	+	CCONJ
cana-1725	151	4	(	(	PUNCT
cana-1725	151	5	𝑛	𝑛	PRON
cana-1725	151	6	−	−	PROPN
cana-1725	151	7	4	4	NUM
cana-1725	151	8	)	)	PUNCT
cana-1725	151	9	>	>	X
cana-1725	151	10	𝑛	𝑛	PROPN
cana-1725	151	11	–	–	PUNCT
cana-1725	151	12	1	1	NUM
cana-1725	151	13	the	the	DET
cana-1725	151	14	choice	choice	NOUN
cana-1725	151	15	of	of	ADP
cana-1725	151	16	𝑣	𝑣	PRON
cana-1725	151	17	being	be	AUX
cana-1725	151	18	adjacent	adjacent	ADJ
cana-1725	151	19	to	to	ADP
cana-1725	151	20	any	any	DET
cana-1725	151	21	other	other	ADJ
cana-1725	151	22	vertex	vertex	NOUN
cana-1725	151	23	on	on	ADP
cana-1725	151	24	the	the	DET
cana-1725	151	25	diametrical	diametrical	ADJ
cana-1725	151	26	path	path	NOUN
cana-1725	151	27	results	result	NOUN
cana-1725	151	28	in	in	ADP
cana-1725	151	29	the	the	DET
cana-1725	151	30	same	same	ADJ
cana-1725	151	31	𝑅𝑆𝑂(𝐺	𝑅𝑆𝑂(𝐺	NOUN
cana-1725	151	32	)	)	PUNCT
cana-1725	151	33	,	,	PUNCT
cana-1725	151	34	which	which	PRON
cana-1725	151	35	implies	imply	VERB
cana-1725	151	36	that	that	SCONJ
cana-1725	151	37	𝑣	𝑣	PRON
cana-1725	151	38	can	can	AUX
cana-1725	151	39	not	not	PART
cana-1725	151	40	be	be	AUX
cana-1725	151	41	adjacent	adjacent	ADJ
cana-1725	151	42	to	to	ADP
cana-1725	151	43	a	a	DET
cana-1725	151	44	vertex	vertex	NOUN
cana-1725	151	45	on	on	ADP
cana-1725	151	46	the	the	DET
cana-1725	151	47	diametrical	diametrical	ADJ
cana-1725	151	48	path	path	NOUN
cana-1725	151	49	.	.	PUNCT
cana-1725	152	1	so	so	ADV
cana-1725	152	2	,	,	PUNCT
cana-1725	152	3	there	there	PRON
cana-1725	152	4	is	be	VERB
cana-1725	152	5	no	no	DET
cana-1725	152	6	other	other	ADJ
cana-1725	152	7	vertex	vertex	NOUN
cana-1725	152	8	adjacent	adjacent	ADJ
cana-1725	152	9	to	to	ADP
cana-1725	152	10	a	a	DET
cana-1725	152	11	vertex	vertex	NOUN
cana-1725	152	12	𝑤	𝑤	PART
cana-1725	152	13	which	which	PRON
cana-1725	152	14	is	be	AUX
cana-1725	152	15	adjacent	adjacent	ADJ
cana-1725	152	16	to	to	ADP
cana-1725	152	17	a	a	DET
cana-1725	152	18	vertex	vertex	NOUN
cana-1725	152	19	on	on	ADP
cana-1725	152	20	the	the	DET
cana-1725	152	21	diametrical	diametrical	ADJ
cana-1725	152	22	path	path	NOUN
cana-1725	152	23	.	.	PUNCT
cana-1725	153	1	thus	thus	ADV
cana-1725	153	2	,	,	PUNCT
cana-1725	153	3	there	there	PRON
cana-1725	153	4	can	can	AUX
cana-1725	153	5	not	not	PART
cana-1725	153	6	be	be	AUX
cana-1725	153	7	two	two	NUM
cana-1725	153	8	different	different	ADJ
cana-1725	153	9	diametrical	diametrical	ADJ
cana-1725	153	10	paths	path	NOUN
cana-1725	153	11	in	in	ADP
cana-1725	153	12	𝐺.	𝐺.	NOUN
cana-1725	153	13	thus	thus	ADV
cana-1725	153	14	,	,	PUNCT
cana-1725	153	15	𝐺	𝐺	PROPN
cana-1725	153	16	itself	itself	PRON
cana-1725	153	17	is	be	AUX
cana-1725	153	18	a	a	DET
cana-1725	153	19	path	path	NOUN
cana-1725	153	20	.	.	PUNCT
cana-1725	154	1	thus	thus	ADV
cana-1725	154	2	,	,	PUNCT
cana-1725	154	3	the	the	DET
cana-1725	154	4	theorem	theorem	NOUN
cana-1725	154	5	is	be	AUX
cana-1725	154	6	proved	prove	VERB
cana-1725	154	7	.	.	PUNCT
cana-1725	155	1	references	reference	NOUN
cana-1725	155	2	[	[	X
cana-1725	155	3	1	1	X
cana-1725	155	4	]	]	PUNCT
cana-1725	155	5	j.	j.	PROPN
cana-1725	155	6	a.	a.	PROPN
cana-1725	155	7	bondy	bondy	PROPN
cana-1725	155	8	,	,	PUNCT
cana-1725	155	9	u.	u.	PROPN
cana-1725	155	10	s.	s.	PROPN
cana-1725	155	11	r.	r.	PROPN
cana-1725	155	12	murty	murty	PROPN
cana-1725	155	13	,	,	PUNCT
cana-1725	155	14	graph	graph	NOUN
cana-1725	155	15	theory	theory	NOUN
cana-1725	155	16	,	,	PUNCT
cana-1725	155	17	springer	springer	NOUN
cana-1725	155	18	,	,	PUNCT
cana-1725	155	19	2008	2008	NUM
cana-1725	155	20	.	.	PUNCT
cana-1725	156	1	[	[	X
cana-1725	156	2	2	2	NUM
cana-1725	156	3	]	]	PUNCT
cana-1725	156	4	haynes	hayne	NOUN
cana-1725	156	5	,	,	PUNCT
cana-1725	156	6	t.	t.	PROPN
cana-1725	156	7	,	,	PUNCT
cana-1725	156	8	hedetniemi	hedetniemi	PROPN
cana-1725	156	9	,	,	PUNCT
cana-1725	156	10	s.	s.	PROPN
cana-1725	156	11	,	,	PUNCT
cana-1725	156	12	slater	slater	PROPN
cana-1725	156	13	,	,	PUNCT
cana-1725	156	14	p.:fundamentals	p.:fundamental	NOUN
cana-1725	156	15	of	of	ADP
cana-1725	156	16	domination	domination	NOUN
cana-1725	156	17	in	in	ADP
cana-1725	156	18	graphs	graph	NOUN
cana-1725	156	19	.	.	PUNCT
cana-1725	157	1	marcel	marcel	PROPN
cana-1725	157	2	dekker	dekker	PROPN
cana-1725	157	3	,	,	PUNCT
cana-1725	157	4	new	new	PROPN
cana-1725	157	5	york	york	PROPN
cana-1725	157	6	(	(	PUNCT
cana-1725	157	7	1998	1998	NUM
cana-1725	157	8	)	)	PUNCT
cana-1725	157	9	.	.	PUNCT
cana-1725	158	1	[	[	X
cana-1725	158	2	3	3	X
cana-1725	158	3	]	]	X
cana-1725	158	4	haynes	haynes	PROPN
cana-1725	158	5	t.w	t.w	PROPN
cana-1725	158	6	.	.	PROPN
cana-1725	158	7	hedetniemi	hedetniemi	PROPN
cana-1725	158	8	s.	s.	PROPN
cana-1725	158	9	slater	slater	PROPN
cana-1725	158	10	p.	p.	PROPN
cana-1725	158	11	,	,	PUNCT
cana-1725	158	12	domination	domination	NOUN
cana-1725	158	13	in	in	ADP
cana-1725	158	14	graphs	graph	NOUN
cana-1725	158	15	:	:	PUNCT
cana-1725	158	16	advanced	advanced	ADJ
cana-1725	158	17	topics	topic	NOUN
cana-1725	158	18	,	,	PUNCT
cana-1725	158	19	marcel	marcel	PROPN
cana-1725	158	20	dekker	dekker	PROPN
cana-1725	158	21	,	,	PUNCT
cana-1725	158	22	1998	1998	NUM
cana-1725	158	23	.	.	PUNCT
cana-1725	159	1	[	[	X
cana-1725	159	2	4	4	NUM
cana-1725	159	3	]	]	X
cana-1725	159	4	i.	i.	PROPN
cana-1725	159	5	gutman	gutman	PROPN
cana-1725	159	6	,	,	PUNCT
cana-1725	159	7	geometric	geometric	ADJ
cana-1725	159	8	approach	approach	NOUN
cana-1725	159	9	to	to	ADP
cana-1725	159	10	degree	degree	NOUN
cana-1725	159	11	-	-	PUNCT
cana-1725	159	12	based	base	VERB
cana-1725	159	13	topological	topological	ADJ
cana-1725	159	14	indices	index	NOUN
cana-1725	159	15	:	:	PUNCT
cana-1725	159	16	sombor	sombor	NOUN
cana-1725	159	17	indices	index	NOUN
cana-1725	159	18	,	,	PUNCT
cana-1725	159	19	match	match	NOUN
cana-1725	159	20	commun	commun	PROPN
cana-1725	159	21	.	.	PUNCT
cana-1725	160	1	math	math	PROPN
cana-1725	160	2	.	.	PUNCT
cana-1725	161	1	comput	comput	NOUN
cana-1725	161	2	.	.	PUNCT
cana-1725	162	1	chem	chem	NOUN
cana-1725	162	2	.	.	PUNCT
cana-1725	163	1	86	86	NUM
cana-1725	163	2	(	(	PUNCT
cana-1725	163	3	2021	2021	NUM
cana-1725	163	4	)	)	PUNCT
cana-1725	163	5	11–16	11–16	NUM
cana-1725	163	6	.	.	PUNCT
cana-1725	164	1	[	[	X
cana-1725	164	2	5	5	X
cana-1725	164	3	]	]	PUNCT
cana-1725	164	4	h.	h.	PROPN
cana-1725	164	5	liu	liu	PROPN
cana-1725	164	6	,	,	PUNCT
cana-1725	164	7	i.	i.	PROPN
cana-1725	164	8	gutman	gutman	PROPN
cana-1725	164	9	,	,	PUNCT
cana-1725	164	10	l.	l.	PROPN
cana-1725	164	11	you	you	PRON
cana-1725	164	12	,	,	PUNCT
cana-1725	164	13	y.	y.	PROPN
cana-1725	164	14	huang	huang	PROPN
cana-1725	164	15	,	,	PUNCT
cana-1725	164	16	sombor	sombor	NOUN
cana-1725	164	17	index	index	NOUN
cana-1725	164	18	:	:	PUNCT
cana-1725	164	19	review	review	NOUN
cana-1725	164	20	of	of	ADP
cana-1725	164	21	extremal	extremal	ADJ
cana-1725	164	22	results	result	NOUN
cana-1725	164	23	and	and	CCONJ
cana-1725	164	24	bounds	bound	NOUN
cana-1725	164	25	,	,	PUNCT
cana-1725	164	26	j.	j.	PROPN
cana-1725	164	27	math	math	PROPN
cana-1725	164	28	.	.	PUNCT
cana-1725	165	1	chem	chem	NOUN
cana-1725	165	2	.	.	PUNCT
cana-1725	166	1	60	60	NUM
cana-1725	166	2	(	(	PUNCT
cana-1725	166	3	2022	2022	NUM
cana-1725	166	4	)	)	PUNCT
cana-1725	167	1	771–798	771–798	NUM
cana-1725	167	2	.	.	PUNCT
cana-1725	168	1	[	[	X
cana-1725	168	2	6	6	NUM
cana-1725	168	3	]	]	PUNCT
cana-1725	168	4	i.	i.	PROPN
cana-1725	168	5	gutman	gutman	PROPN
cana-1725	168	6	,	,	PUNCT
cana-1725	168	7	sombor	sombor	NOUN
cana-1725	168	8	index	index	NOUN
cana-1725	168	9	–	–	PUNCT
cana-1725	168	10	one	one	NUM
cana-1725	168	11	year	year	NOUN
cana-1725	168	12	later	later	ADV
cana-1725	168	13	,	,	PUNCT
cana-1725	168	14	bull	bull	NOUN
cana-1725	168	15	.	.	PUNCT
cana-1725	169	1	acad	acad	PROPN
cana-1725	169	2	.	.	PUNCT
cana-1725	170	1	serb	serb	PROPN
cana-1725	170	2	.	.	PUNCT
cana-1725	171	1	sci	sci	PROPN
cana-1725	171	2	.	.	PUNCT
cana-1725	171	3	arts	art	NOUN
cana-1725	171	4	153	153	NUM
cana-1725	171	5	(	(	PUNCT
cana-1725	171	6	2020	2020	NUM
cana-1725	171	7	)	)	PUNCT
cana-1725	171	8	43–55	43–55	NUM
cana-1725	171	9	.	.	PUNCT
cana-1725	172	1	[	[	X
cana-1725	172	2	7	7	X
cana-1725	172	3	]	]	X
cana-1725	172	4	h.	h.	PROPN
cana-1725	172	5	chen	chen	PROPN
cana-1725	172	6	,	,	PUNCT
cana-1725	172	7	w.	w.	PROPN
cana-1725	172	8	li	li	PROPN
cana-1725	172	9	,	,	PUNCT
cana-1725	172	10	j.	j.	PROPN
cana-1725	172	11	wang	wang	PROPN
cana-1725	172	12	,	,	PUNCT
cana-1725	172	13	extremal	extremal	ADJ
cana-1725	172	14	values	value	NOUN
cana-1725	172	15	on	on	ADP
cana-1725	172	16	the	the	DET
cana-1725	172	17	sombor	sombor	NOUN
cana-1725	172	18	index	index	NOUN
cana-1725	172	19	of	of	ADP
cana-1725	172	20	trees	tree	NOUN
cana-1725	172	21	,	,	PUNCT
cana-1725	172	22	match	match	NOUN
cana-1725	172	23	commun	commun	PROPN
cana-1725	172	24	.	.	PUNCT
cana-1725	172	25	math	math	PROPN
cana-1725	172	26	.	.	PUNCT
cana-1725	173	1	comput	comput	NOUN
cana-1725	173	2	.	.	PUNCT
cana-1725	174	1	chem	chem	NOUN
cana-1725	174	2	.	.	PUNCT
cana-1725	175	1	87	87	NUM
cana-1725	175	2	(	(	PUNCT
cana-1725	175	3	2022	2022	NUM
cana-1725	175	4	)	)	PUNCT
cana-1725	175	5	23–49	23–49	NUM
cana-1725	175	6	.	.	PUNCT
cana-1725	176	1	[	[	X
cana-1725	176	2	8	8	NUM
cana-1725	176	3	]	]	PUNCT
cana-1725	176	4	x.	x.	NOUN
cana-1725	176	5	sun	sun	PROPN
cana-1725	176	6	,	,	PUNCT
cana-1725	176	7	j.	j.	PROPN
cana-1725	176	8	du	du	PROPN
cana-1725	176	9	,	,	PUNCT
cana-1725	176	10	on	on	ADP
cana-1725	176	11	sombor	sombor	NOUN
cana-1725	176	12	index	index	NOUN
cana-1725	176	13	of	of	ADP
cana-1725	176	14	trees	tree	NOUN
cana-1725	176	15	with	with	ADP
cana-1725	176	16	fixed	fix	VERB
cana-1725	176	17	domination	domination	NOUN
cana-1725	176	18	number	number	NOUN
cana-1725	176	19	,	,	PUNCT
cana-1725	176	20	appl	appl	PROPN
cana-1725	176	21	.	.	PROPN
cana-1725	176	22	math	math	PROPN
cana-1725	176	23	.	.	PUNCT
cana-1725	177	1	comput	comput	NOUN
cana-1725	177	2	.	.	PUNCT
cana-1725	178	1	421	421	NUM
cana-1725	178	2	(	(	PUNCT
cana-1725	178	3	2022	2022	NUM
cana-1725	178	4	)	)	PUNCT
cana-1725	178	5	#	#	SYM
cana-1725	178	6	126946	126946	NUM
cana-1725	178	7	.	.	PUNCT
cana-1725	179	1	[	[	X
cana-1725	179	2	9	9	NUM
cana-1725	179	3	]	]	PUNCT
cana-1725	179	4	s.	s.	PROPN
cana-1725	179	5	li	li	PROPN
cana-1725	179	6	,	,	PUNCT
cana-1725	179	7	z.	z.	PROPN
cana-1725	179	8	wang	wang	PROPN
cana-1725	179	9	,	,	PUNCT
cana-1725	179	10	m.	m.	PROPN
cana-1725	179	11	zhang	zhang	PROPN
cana-1725	179	12	,	,	PUNCT
cana-1725	179	13	on	on	ADP
cana-1725	179	14	the	the	DET
cana-1725	179	15	extremal	extremal	ADJ
cana-1725	179	16	sombor	sombor	NOUN
cana-1725	179	17	index	index	NOUN
cana-1725	179	18	of	of	ADP
cana-1725	179	19	trees	tree	NOUN
cana-1725	179	20	with	with	ADP
cana-1725	179	21	a	a	DET
cana-1725	179	22	given	give	VERB
cana-1725	179	23	diameter	diameter	NOUN
cana-1725	179	24	,	,	PUNCT
cana-1725	179	25	appl	appl	PROPN
cana-1725	179	26	.	.	PROPN
cana-1725	179	27	math	math	NOUN
cana-1725	179	28	.	.	PUNCT
cana-1725	180	1	comput	comput	NOUN
cana-1725	180	2	.	.	PUNCT
cana-1725	181	1	416	416	NUM
cana-1725	181	2	(	(	PUNCT
cana-1725	181	3	2022	2022	NUM
cana-1725	181	4	)	)	PUNCT
cana-1725	181	5	#	#	SYM
cana-1725	181	6	126731	126731	NUM
cana-1725	181	7	.	.	PUNCT
cana-1725	182	1	[	[	X
cana-1725	182	2	10	10	NUM
cana-1725	182	3	]	]	X
cana-1725	182	4	i.	i.	PROPN
cana-1725	182	5	gutman	gutman	PROPN
cana-1725	182	6	,	,	PUNCT
cana-1725	182	7	v.	v.	PROPN
cana-1725	182	8	r.	r.	PROPN
cana-1725	182	9	kulli	kulli	PROPN
cana-1725	182	10	,	,	PUNCT
cana-1725	182	11	i.	i.	PROPN
cana-1725	182	12	redˇzepovi´c	redˇzepovi´c	PROPN
cana-1725	182	13	,	,	PUNCT
cana-1725	182	14	sombor	sombor	NOUN
cana-1725	182	15	index	index	NOUN
cana-1725	182	16	of	of	ADP
cana-1725	182	17	kragujevac	kragujevac	PROPN
cana-1725	182	18	trees	tree	NOUN
cana-1725	182	19	,	,	PUNCT
cana-1725	182	20	sci	sci	PROPN
cana-1725	182	21	.	.	PUNCT
cana-1725	183	1	publ.univ	publ.univ	PROPN
cana-1725	183	2	.	.	PROPN
cana-1725	183	3	novi	novi	PROPN
cana-1725	183	4	pazar	pazar	PROPN
cana-1725	183	5	ser	ser	PROPN
cana-1725	183	6	.	.	PUNCT
cana-1725	184	1	a	a	DET
cana-1725	184	2	13	13	NUM
cana-1725	184	3	(	(	PUNCT
cana-1725	184	4	2021	2021	NUM
cana-1725	184	5	)	)	PUNCT
cana-1725	184	6	61–70	61–70	NOUN
cana-1725	184	7	.	.	PUNCT
cana-1725	185	1	[	[	X
cana-1725	185	2	11	11	NUM
cana-1725	185	3	]	]	PUNCT
cana-1725	185	4	k.	k.	PROPN
cana-1725	185	5	c.	c.	PROPN
cana-1725	185	6	das	das	PROPN
cana-1725	185	7	,	,	PUNCT
cana-1725	185	8	i.	i.	NOUN
cana-1725	185	9	gutman	gutman	PROPN
cana-1725	185	10	,	,	PUNCT
cana-1725	185	11	on	on	ADP
cana-1725	185	12	sombor	sombor	NOUN
cana-1725	185	13	index	index	NOUN
cana-1725	185	14	of	of	ADP
cana-1725	185	15	trees	tree	NOUN
cana-1725	185	16	,	,	PUNCT
cana-1725	185	17	appl	appl	PROPN
cana-1725	185	18	.	.	PROPN
cana-1725	185	19	math	math	NOUN
cana-1725	185	20	.	.	PUNCT
cana-1725	186	1	comput	comput	NOUN
cana-1725	186	2	.	.	PUNCT
cana-1725	187	1	412	412	NUM
cana-1725	187	2	(	(	PUNCT
cana-1725	187	3	2022)#126575	2022)#126575	NUM
cana-1725	187	4	.	.	PUNCT
cana-1725	188	1	communications	communication	NOUN
cana-1725	188	2	on	on	ADP
cana-1725	188	3	applied	apply	VERB
cana-1725	188	4	nonlinear	nonlinear	ADJ
cana-1725	188	5	analysis	analysis	NOUN
cana-1725	188	6	issn	issn	NOUN
cana-1725	188	7	:	:	PUNCT
cana-1725	188	8	1074	1074	NUM
cana-1725	188	9	-	-	PUNCT
cana-1725	188	10	133x	133x	NUM
cana-1725	188	11	vol	vol	NOUN
cana-1725	188	12	32	32	NUM
cana-1725	188	13	no	no	NOUN
cana-1725	188	14	.	.	NOUN
cana-1725	188	15	2	2	NUM
cana-1725	188	16	(	(	PUNCT
cana-1725	188	17	2025	2025	NUM
cana-1725	188	18	)	)	PUNCT
cana-1725	188	19	129	129	NUM
cana-1725	188	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1725	189	1	[	[	X
cana-1725	189	2	12	12	NUM
cana-1725	189	3	]	]	PUNCT
cana-1725	189	4	i.	i.	PROPN
cana-1725	189	5	redˇzepovi´c	redˇzepovi´c	PROPN
cana-1725	189	6	,	,	PUNCT
cana-1725	189	7	chemical	chemical	NOUN
cana-1725	189	8	applicability	applicability	NOUN
cana-1725	189	9	of	of	ADP
cana-1725	189	10	sombor	sombor	NOUN
cana-1725	189	11	indices	index	NOUN
cana-1725	189	12	,	,	PUNCT
cana-1725	189	13	j.	j.	PROPN
cana-1725	189	14	serb	serb	PROPN
cana-1725	189	15	.	.	PUNCT
cana-1725	190	1	chem	chem	PROPN
cana-1725	190	2	.	.	PUNCT
cana-1725	191	1	soc	soc	PROPN
cana-1725	191	2	.	.	PUNCT
cana-1725	192	1	86	86	NUM
cana-1725	192	2	(	(	PUNCT
cana-1725	192	3	2021	2021	NUM
cana-1725	192	4	)	)	PUNCT
cana-1725	192	5	445–457	445–457	NUM
cana-1725	192	6	.	.	PUNCT
cana-1725	193	1	[	[	X
cana-1725	193	2	13	13	NUM
cana-1725	193	3	]	]	X
cana-1725	193	4	h.	h.	PROPN
cana-1725	193	5	deng	deng	PROPN
cana-1725	193	6	,	,	PUNCT
cana-1725	193	7	z.	z.	PROPN
cana-1725	193	8	tang	tang	PROPN
cana-1725	193	9	,	,	PUNCT
cana-1725	193	10	r.	r.	PROPN
cana-1725	193	11	wu	wu	PROPN
cana-1725	193	12	,	,	PUNCT
cana-1725	193	13	molecular	molecular	ADJ
cana-1725	193	14	trees	tree	NOUN
cana-1725	193	15	with	with	ADP
cana-1725	193	16	extremal	extremal	ADJ
cana-1725	193	17	values	value	NOUN
cana-1725	193	18	of	of	ADP
cana-1725	193	19	sombor	sombor	NOUN
cana-1725	193	20	indices	index	NOUN
cana-1725	193	21	,	,	PUNCT
cana-1725	193	22	int	int	NOUN
cana-1725	193	23	.	.	PUNCT
cana-1725	194	1	j.	j.	PROPN
cana-1725	194	2	quantum	quantum	PROPN
cana-1725	194	3	chem	chem	NOUN
cana-1725	194	4	.	.	PUNCT
cana-1725	195	1	121	121	NUM
cana-1725	195	2	(	(	PUNCT
cana-1725	195	3	2021	2021	NUM
cana-1725	195	4	)	)	PUNCT
cana-1725	196	1	#	#	SYM
cana-1725	196	2	e26622	e26622	NOUN
cana-1725	196	3	.	.	PUNCT
cana-1725	197	1	[	[	X
cana-1725	197	2	14	14	NUM
cana-1725	197	3	]	]	X
cana-1725	197	4	i.	i.	PROPN
cana-1725	197	5	gutman	gutman	PROPN
cana-1725	197	6	,	,	PUNCT
cana-1725	197	7	i.	i.	PROPN
cana-1725	197	8	redzepovic	redzepovic	PROPN
cana-1725	197	9	,	,	PUNCT
cana-1725	197	10	b.	b.	PROPN
cana-1725	197	11	furtula	furtula	PROPN
cana-1725	197	12	,	,	PUNCT
cana-1725	197	13	on	on	ADP
cana-1725	197	14	the	the	DET
cana-1725	197	15	product	product	NOUN
cana-1725	197	16	of	of	ADP
cana-1725	197	17	sombor	sombor	NOUN
cana-1725	197	18	and	and	CCONJ
cana-1725	197	19	modified	modified	ADJ
cana-1725	197	20	sombor	sombor	NOUN
cana-1725	197	21	index	index	NOUN
cana-1725	197	22	,	,	PUNCT
cana-1725	197	23	open	open	ADJ
cana-1725	197	24	journal	journal	NOUN
cana-1725	197	25	of	of	ADP
cana-1725	197	26	applied	apply	VERB
cana-1725	197	27	discrete	discrete	ADJ
cana-1725	197	28	mathematics	mathematic	NOUN
cana-1725	197	29	,	,	PUNCT
cana-1725	197	30	6(2	6(2	NUM
cana-1725	197	31	)	)	PUNCT
cana-1725	197	32	(	(	PUNCT
cana-1725	197	33	2023	2023	NUM
cana-1725	197	34	)	)	PUNCT
cana-1725	197	35	1	1	NUM
cana-1725	197	36	-	-	SYM
cana-1725	197	37	6	6	NUM
cana-1725	197	38	.	.	PUNCT
cana-1725	198	1	[	[	X
cana-1725	198	2	15	15	NUM
cana-1725	198	3	]	]	X
cana-1725	198	4	h.	h.	NOUN
cana-1725	198	5	shoostari	shoostari	PROPN
cana-1725	198	6	,	,	PUNCT
cana-1725	198	7	s.m	s.m	PROPN
cana-1725	198	8	.	.	PROPN
cana-1725	198	9	sheikholeslami	sheikholeslami	PROPN
cana-1725	198	10	,	,	PUNCT
cana-1725	198	11	j.	j.	PROPN
cana-1725	198	12	amjadi	amjadi	PROPN
cana-1725	198	13	,	,	PUNCT
cana-1725	198	14	modified	modify	VERB
cana-1725	198	15	sombor	sombor	NOUN
cana-1725	198	16	index	index	NOUN
cana-1725	198	17	of	of	ADP
cana-1725	198	18	unicyclic	unicyclic	ADJ
cana-1725	198	19	graphs	graph	NOUN
cana-1725	198	20	with	with	ADP
cana-1725	198	21	a	a	DET
cana-1725	198	22	given	give	VERB
cana-1725	198	23	diameter	diameter	NOUN
cana-1725	198	24	,	,	PUNCT
cana-1725	198	25	asian	asian	ADJ
cana-1725	198	26	-	-	PUNCT
cana-1725	198	27	european	european	ADJ
cana-1725	198	28	journal	journal	NOUN
cana-1725	198	29	of	of	ADP
cana-1725	198	30	mathematics	mathematic	NOUN
cana-1725	198	31	,	,	PUNCT
cana-1725	198	32	16(06	16(06	NUM
cana-1725	198	33	)	)	PUNCT
cana-1725	198	34	(	(	PUNCT
cana-1725	198	35	2023	2023	NUM
cana-1725	198	36	)	)	PUNCT
cana-1725	198	37	2350098	2350098	NUM
cana-1725	198	38	.	.	PUNCT
cana-1725	199	1	[	[	X
cana-1725	199	2	16	16	NUM
cana-1725	199	3	]	]	X
cana-1725	199	4	yufei	yufei	PROPN
cana-1725	199	5	huang	huang	PROPN
cana-1725	199	6	,	,	PUNCT
cana-1725	199	7	hechao	hechao	PROPN
cana-1725	199	8	liu	liu	PROPN
cana-1725	199	9	,	,	PUNCT
cana-1725	199	10	bounds	bound	NOUN
cana-1725	199	11	of	of	ADP
cana-1725	199	12	modified	modify	VERB
cana-1725	199	13	sombor	sombor	NOUN
cana-1725	199	14	index	index	NOUN
cana-1725	199	15	,	,	PUNCT
cana-1725	199	16	spectral	spectral	ADJ
cana-1725	199	17	radius	radius	NOUN
cana-1725	199	18	and	and	CCONJ
cana-1725	199	19	energy	energy	NOUN
cana-1725	199	20	,	,	PUNCT
cana-1725	199	21	aims	aim	VERB
cana-1725	199	22	mathematics	mathematic	NOUN
cana-1725	199	23	,	,	PUNCT
cana-1725	199	24	6(10	6(10	NUM
cana-1725	199	25	)	)	PUNCT
cana-1725	199	26	,	,	PUNCT
cana-1725	199	27	(	(	PUNCT
cana-1725	199	28	2021	2021	NUM
cana-1725	199	29	)	)	PUNCT
cana-1725	199	30	11263	11263	NUM
cana-1725	199	31	-	-	SYM
cana-1725	199	32	11274	11274	NUM
cana-1725	199	33	.	.	PUNCT
cana-1725	200	1	[	[	X
cana-1725	200	2	17	17	NUM
cana-1725	200	3	]	]	SYM
cana-1725	200	4	xuewe	xuewe	PROPN
cana-1725	200	5	zuo	zuo	PROPN
cana-1725	200	6	,	,	PUNCT
cana-1725	200	7	bilal	bilal	PROPN
cana-1725	200	8	ahmed	ahmed	PROPN
cana-1725	200	9	rathar	rathar	PROPN
cana-1725	200	10	,	,	PUNCT
cana-1725	200	11	muhammad	muhammad	PROPN
cana-1725	200	12	imran	imran	PROPN
cana-1725	200	13	,	,	PUNCT
cana-1725	200	14	akbar	akbar	PROPN
cana-1725	200	15	ali	ali	PROPN
cana-1725	200	16	,	,	PUNCT
cana-1725	200	17	on	on	ADP
cana-1725	200	18	some	some	DET
cana-1725	200	19	topological	topological	ADJ
cana-1725	200	20	indices	index	NOUN
cana-1725	200	21	defined	define	VERB
cana-1725	200	22	via	via	ADP
cana-1725	200	23	the	the	DET
cana-1725	200	24	modified	modify	VERB
cana-1725	200	25	sombor	sombor	NOUN
cana-1725	200	26	index	index	NOUN
cana-1725	200	27	,	,	PUNCT
cana-1725	200	28	molecules	molecule	NOUN
cana-1725	200	29	27(19	27(19	NUM
cana-1725	200	30	)	)	PUNCT
cana-1725	200	31	,	,	PUNCT
cana-1725	200	32	(	(	PUNCT
cana-1725	200	33	2022	2022	NUM
cana-1725	200	34	)	)	PUNCT
cana-1725	200	35	6772	6772	NUM
cana-1725	200	36	.	.	PUNCT
cana-1725	201	1	[	[	X
cana-1725	201	2	18	18	NUM
cana-1725	201	3	]	]	X
cana-1725	201	4	fangxia	fangxia	PROPN
cana-1725	201	5	wang	wang	PROPN
cana-1725	201	6	,	,	PUNCT
cana-1725	201	7	baoyindureng	baoyindureng	PROPN
cana-1725	201	8	wu	wu	PROPN
cana-1725	201	9	,	,	PUNCT
cana-1725	201	10	the	the	DET
cana-1725	201	11	proof	proof	NOUN
cana-1725	201	12	of	of	ADP
cana-1725	201	13	a	a	DET
cana-1725	201	14	conjecture	conjecture	NOUN
cana-1725	201	15	on	on	ADP
cana-1725	201	16	the	the	DET
cana-1725	201	17	reduced	reduce	VERB
cana-1725	201	18	sombor	sombor	NOUN
cana-1725	201	19	index	index	NOUN
cana-1725	201	20	,	,	PUNCT
cana-1725	201	21	match	match	NOUN
cana-1725	201	22	commun	commun	PROPN
cana-1725	201	23	.	.	PUNCT
cana-1725	201	24	math	math	PROPN
cana-1725	201	25	.	.	PUNCT
cana-1725	202	1	comput	comput	NOUN
cana-1725	202	2	.	.	PUNCT
cana-1725	203	1	chem	chem	NOUN
cana-1725	203	2	.	.	PUNCT
cana-1725	204	1	88	88	NUM
cana-1725	204	2	(	(	PUNCT
cana-1725	204	3	2022	2022	NUM
cana-1725	204	4	)	)	PUNCT
cana-1725	204	5	583	583	NUM
cana-1725	204	6	-	-	SYM
cana-1725	204	7	591	591	NUM
cana-1725	204	8	.	.	PUNCT
cana-1725	205	1	[	[	X
cana-1725	205	2	19	19	NUM
cana-1725	205	3	]	]	X
cana-1725	205	4	hechao	hechao	PROPN
cana-1725	205	5	liu	liu	PROPN
cana-1725	205	6	,	,	PUNCT
cana-1725	205	7	lihua	lihua	PROPN
cana-1725	205	8	you	you	PRON
cana-1725	205	9	,	,	PUNCT
cana-1725	205	10	zikai	zikai	PROPN
cana-1725	205	11	tang	tang	PROPN
cana-1725	205	12	,	,	PUNCT
cana-1725	205	13	jia	jia	PROPN
cana-1725	205	14	–	–	PUNCT
cana-1725	205	15	bao	bao	PROPN
cana-1725	205	16	liu	liu	PROPN
cana-1725	205	17	,	,	PUNCT
cana-1725	205	18	on	on	ADP
cana-1725	205	19	the	the	DET
cana-1725	205	20	reduced	reduce	VERB
cana-1725	205	21	sombor	sombor	NOUN
cana-1725	205	22	index	index	NOUN
cana-1725	205	23	and	and	CCONJ
cana-1725	205	24	its	its	PRON
cana-1725	205	25	applications	application	NOUN
cana-1725	205	26	,	,	PUNCT
cana-1725	205	27	match	match	NOUN
cana-1725	205	28	commun	commun	PROPN
cana-1725	205	29	.	.	PUNCT
cana-1725	205	30	math	math	PROPN
cana-1725	205	31	.	.	PUNCT
cana-1725	206	1	comput	comput	NOUN
cana-1725	206	2	.	.	PUNCT
cana-1725	207	1	chem	chem	NOUN
cana-1725	207	2	.	.	PUNCT
cana-1725	208	1	86	86	NUM
cana-1725	208	2	(	(	PUNCT
cana-1725	208	3	2021	2021	NUM
cana-1725	208	4	)	)	PUNCT
cana-1725	208	5	729	729	NUM
cana-1725	208	6	-	-	SYM
cana-1725	208	7	753	753	NUM
cana-1725	208	8	.	.	PUNCT