id	sid	tid	token	lemma	pos
m-1071	1	1	ijo	ijo	PROPN
m-1071	1	2	international	international	PROPN
m-1071	1	3	journal	journal	PROPN
m-1071	1	4	of	of	ADP
m-1071	1	5	mathematics	mathematics	PROPN
m-1071	1	6	(	(	PUNCT
m-1071	1	7	issn	issn	PROPN
m-1071	1	8	:	:	PUNCT
m-1071	1	9	2992	2992	NUM
m-1071	1	10	-	-	SYM
m-1071	1	11	4421	4421	NUM
m-1071	1	12	)	)	PUNCT
m-1071	2	1	jyothi.mj	jyothi.mj	X
m-1071	2	2	1	1	NUM
m-1071	2	3	*	*	PUNCT
m-1071	2	4	https://ijojournals.com/	https://ijojournals.com/	NOUN
m-1071	2	5	volume	volume	NOUN
m-1071	2	6	08	08	NUM
m-1071	2	7	||	||	PROPN
m-1071	2	8	issue	issue	NOUN
m-1071	2	9	04	04	NUM
m-1071	2	10	||	||	NOUN
m-1071	2	11	april	april	PROPN
m-1071	2	12	,	,	PUNCT
m-1071	2	13	2025	2025	NUM
m-1071	2	14	||	||	X
m-1071	3	1	*	*	PUNCT
m-1071	3	2	on	on	ADP
m-1071	3	3	sio4	sio4	PROPN
m-1071	3	4	molecular	molecular	ADJ
m-1071	3	5	topological	topological	ADJ
m-1071	3	6	characterization	characterization	NOUN
m-1071	3	7	of	of	ADP
m-1071	3	8	chemical	chemical	NOUN
m-1071	3	9	structures	structure	NOUN
m-1071	3	10	*	*	PUNCT
m-1071	3	11	on	on	ADP
m-1071	3	12	sio4	sio4	PROPN
m-1071	3	13	molecular	molecular	ADJ
m-1071	3	14	topological	topological	ADJ
m-1071	3	15	characterization	characterization	NOUN
m-1071	3	16	of	of	ADP
m-1071	3	17	chemical	chemical	NOUN
m-1071	3	18	structures	structure	NOUN
m-1071	3	19	jyothi	jyothi	PROPN
m-1071	3	20	.	.	PUNCT
m-1071	4	1	mj	mj	PROPN
m-1071	4	2	*	*	PROPN
m-1071	4	3	,	,	PUNCT
m-1071	5	1	k.	k.	PROPN
m-1071	5	2	shivashankara	shivashankara	PROPN
m-1071	6	1	*	*	PUNCT
m-1071	6	2	*	*	PUNCT
m-1071	6	3	*	*	PUNCT
m-1071	6	4	department	department	NOUN
m-1071	6	5	of	of	ADP
m-1071	6	6	mathematics	mathematic	NOUN
m-1071	6	7	,	,	PUNCT
m-1071	6	8	maharanis	maharanis	PROPN
m-1071	6	9	science	science	PROPN
m-1071	6	10	college	college	PROPN
m-1071	6	11	for	for	ADP
m-1071	6	12	women	woman	NOUN
m-1071	6	13	,	,	PUNCT
m-1071	6	14	mysore	mysore	NOUN
m-1071	6	15	570005	570005	NUM
m-1071	6	16	,	,	PUNCT
m-1071	6	17	india	india	PROPN
m-1071	6	18	*	*	PUNCT
m-1071	6	19	*	*	PROPN
m-1071	6	20	department	department	PROPN
m-1071	6	21	of	of	ADP
m-1071	6	22	mathematics	mathematic	NOUN
m-1071	6	23	,	,	PUNCT
m-1071	6	24	yuvaraja	yuvaraja	PROPN
m-1071	6	25	’s	’s	PART
m-1071	6	26	college	college	NOUN
m-1071	6	27	,	,	PUNCT
m-1071	6	28	university	university	NOUN
m-1071	6	29	of	of	ADP
m-1071	6	30	mysore	mysore	NOUN
m-1071	6	31	,	,	PUNCT
m-1071	6	32	mysore	mysore	NOUN
m-1071	6	33	570005	570005	NUM
m-1071	6	34	,	,	PUNCT
m-1071	6	35	india	india	PROPN
m-1071	6	36	abstract	abstract	NOUN
m-1071	6	37	in	in	ADP
m-1071	6	38	this	this	DET
m-1071	6	39	paper	paper	NOUN
m-1071	6	40	,	,	PUNCT
m-1071	6	41	our	our	PRON
m-1071	6	42	aim	aim	NOUN
m-1071	6	43	is	be	AUX
m-1071	6	44	to	to	PART
m-1071	6	45	study	study	VERB
m-1071	6	46	valency	valency	NOUN
m-1071	6	47	-	-	PUNCT
m-1071	6	48	based	base	VERB
m-1071	6	49	molecular	molecular	ADJ
m-1071	6	50	invariants	invariant	NOUN
m-1071	6	51	for	for	ADP
m-1071	6	52	sio4	sio4	NOUN
m-1071	6	53	in	in	ADP
m-1071	6	54	a	a	DET
m-1071	6	55	chain	chain	NOUN
m-1071	6	56	network	network	NOUN
m-1071	6	57	.	.	PUNCT
m-1071	7	1	we	we	PRON
m-1071	7	2	compute	compute	VERB
m-1071	7	3	the	the	DET
m-1071	7	4	harmonic	harmonic	ADJ
m-1071	7	5	polynomial	polynomial	ADJ
m-1071	7	6	,	,	PUNCT
m-1071	7	7	atom	atom	NOUN
m-1071	7	8	bond	bond	NOUN
m-1071	7	9	connectivity	connectivity	NOUN
m-1071	7	10	polynomial	polynomial	ADJ
m-1071	7	11	,	,	PUNCT
m-1071	7	12	forgotten	forget	VERB
m-1071	7	13	polynomial	polynomial	ADJ
m-1071	7	14	,	,	PUNCT
m-1071	7	15	geometric	geometric	ADJ
m-1071	7	16	arithmetic	arithmetic	ADJ
m-1071	7	17	polynomial	polynomial	ADJ
m-1071	7	18	,	,	PUNCT
m-1071	7	19	randic	randic	ADJ
m-1071	7	20	polynomial	polynomial	ADJ
m-1071	7	21	,	,	PUNCT
m-1071	7	22	reciprocal	reciprocal	ADJ
m-1071	7	23	randic	randic	ADJ
m-1071	7	24	polynomial	polynomial	ADJ
m-1071	7	25	,	,	PUNCT
m-1071	7	26	symmetric	symmetric	ADJ
m-1071	7	27	division	division	NOUN
m-1071	7	28	polynomial	polynomial	ADJ
m-1071	7	29	,	,	PUNCT
m-1071	7	30	inverse	inverse	ADJ
m-1071	7	31	symmetric	symmetric	ADJ
m-1071	7	32	division	division	NOUN
m-1071	7	33	polynomial	polynomial	NOUN
m-1071	7	34	,	,	PUNCT
m-1071	7	35	sigma	sigma	NOUN
m-1071	7	36	polynomial	polynomial	ADJ
m-1071	7	37	,	,	PUNCT
m-1071	7	38	sombor	sombor	NOUN
m-1071	7	39	polynomial	polynomial	NOUN
m-1071	7	40	,	,	PUNCT
m-1071	7	41	and	and	CCONJ
m-1071	7	42	their	their	PRON
m-1071	7	43	degree	degree	NOUN
m-1071	7	44	-	-	PUNCT
m-1071	7	45	base	base	NOUN
m-1071	7	46	topological	topological	ADJ
m-1071	7	47	indices	index	NOUN
m-1071	7	48	for	for	ADP
m-1071	7	49	sio4	sio4	NOUN
m-1071	7	50	embedded	embed	VERB
m-1071	7	51	in	in	ADP
m-1071	7	52	a	a	DET
m-1071	7	53	silicate	silicate	ADJ
m-1071	7	54	chain	chain	NOUN
m-1071	7	55	network	network	NOUN
m-1071	7	56	for	for	ADP
m-1071	7	57	various	various	ADJ
m-1071	7	58	conditions	condition	NOUN
m-1071	7	59	.	.	PUNCT
m-1071	8	1	physio	physio	NOUN
m-1071	8	2	-	-	PUNCT
m-1071	8	3	chemical	chemical	NOUN
m-1071	8	4	properties	property	NOUN
m-1071	8	5	of	of	ADP
m-1071	8	6	chemical	chemical	NOUN
m-1071	8	7	compounds	compound	NOUN
m-1071	8	8	,	,	PUNCT
m-1071	8	9	such	such	ADJ
m-1071	8	10	as	as	ADP
m-1071	8	11	formation	formation	NOUN
m-1071	8	12	enthalpies	enthalpy	NOUN
m-1071	8	13	,	,	PUNCT
m-1071	8	14	boiling	boiling	NOUN
m-1071	8	15	points	point	NOUN
m-1071	8	16	,	,	PUNCT
m-1071	8	17	chromatographic	chromatographic	ADJ
m-1071	8	18	retention	retention	NOUN
m-1071	8	19	times	time	NOUN
m-1071	8	20	,	,	PUNCT
m-1071	8	21	vapour	vapour	NOUN
m-1071	8	22	pressure	pressure	NOUN
m-1071	8	23	,	,	PUNCT
m-1071	8	24	and	and	CCONJ
m-1071	8	25	surface	surface	NOUN
m-1071	8	26	areas	area	NOUN
m-1071	8	27	,	,	PUNCT
m-1071	8	28	can	can	AUX
m-1071	8	29	be	be	AUX
m-1071	8	30	determined	determine	VERB
m-1071	8	31	using	use	VERB
m-1071	8	32	our	our	PRON
m-1071	8	33	investigated	investigate	VERB
m-1071	8	34	results	result	NOUN
m-1071	8	35	,	,	PUNCT
m-1071	8	36	such	such	ADJ
m-1071	8	37	as	as	ADP
m-1071	8	38	the	the	DET
m-1071	8	39	h	h	NOUN
m-1071	8	40	-	-	PUNCT
m-1071	8	41	index	index	NOUN
m-1071	8	42	,	,	PUNCT
m-1071	8	43	abc	abc	PROPN
m-1071	8	44	-	-	PUNCT
m-1071	8	45	index	index	NOUN
m-1071	8	46	,	,	PUNCT
m-1071	8	47	f	f	PROPN
m-1071	8	48	-	-	PUNCT
m-1071	8	49	index	index	NOUN
m-1071	8	50	,	,	PUNCT
m-1071	8	51	ga	ga	PROPN
m-1071	8	52	-	-	PUNCT
m-1071	8	53	index	index	NOUN
m-1071	8	54	,	,	PUNCT
m-1071	8	55	r	r	NOUN
m-1071	8	56	-	-	PUNCT
m-1071	8	57	index	index	NOUN
m-1071	8	58	,	,	PUNCT
m-1071	8	59	rr	rr	NOUN
m-1071	8	60	-	-	PUNCT
m-1071	8	61	index	index	NOUN
m-1071	8	62	,	,	PUNCT
m-1071	8	63	sdd	sdd	NOUN
m-1071	8	64	-	-	PUNCT
m-1071	8	65	index	index	NOUN
m-1071	8	66	,	,	PUNCT
m-1071	8	67	isddindex	isddindex	NOUN
m-1071	8	68	,	,	PUNCT
m-1071	8	69	s	s	NOUN
m-1071	8	70	-	-	NOUN
m-1071	8	71	index	index	NOUN
m-1071	8	72	,	,	PUNCT
m-1071	8	73	and	and	CCONJ
m-1071	8	74	so	so	ADV
m-1071	8	75	-	-	PUNCT
m-1071	8	76	index	index	NOUN
m-1071	8	77	.	.	PUNCT
m-1071	9	1	we	we	PRON
m-1071	9	2	also	also	ADV
m-1071	9	3	create	create	VERB
m-1071	9	4	graphical	graphical	ADJ
m-1071	9	5	representations	representation	NOUN
m-1071	9	6	of	of	ADP
m-1071	9	7	the	the	DET
m-1071	9	8	results	result	NOUN
m-1071	9	9	that	that	PRON
m-1071	9	10	describe	describe	VERB
m-1071	9	11	the	the	DET
m-1071	9	12	dependence	dependence	NOUN
m-1071	9	13	of	of	ADP
m-1071	9	14	topological	topological	ADJ
m-1071	9	15	indices	index	NOUN
m-1071	9	16	on	on	ADP
m-1071	9	17	polynomial	polynomial	ADJ
m-1071	9	18	structure	structure	NOUN
m-1071	9	19	parameters	parameter	NOUN
m-1071	9	20	.	.	PUNCT
m-1071	10	1	keywords	keyword	NOUN
m-1071	10	2	:	:	PUNCT
m-1071	10	3	sio4	sio4	NOUN
m-1071	10	4	in	in	ADP
m-1071	10	5	a	a	DET
m-1071	10	6	chain	chain	NOUN
m-1071	10	7	,	,	PUNCT
m-1071	10	8	abc	abc	PROPN
m-1071	10	9	polynomial	polynomial	PROPN
m-1071	10	10	and	and	CCONJ
m-1071	10	11	abc	abc	PROPN
m-1071	10	12	index	index	PROPN
m-1071	10	13	,	,	PUNCT
m-1071	10	14	geometric	geometric	ADJ
m-1071	10	15	arithmetic	arithmetic	ADJ
m-1071	10	16	polynomial	polynomial	ADJ
m-1071	10	17	,	,	PUNCT
m-1071	10	18	randic	randic	ADJ
m-1071	10	19	index	index	NOUN
m-1071	10	20	and	and	CCONJ
m-1071	10	21	reciprocal	reciprocal	ADJ
m-1071	10	22	randic	randic	ADJ
m-1071	10	23	polynomial	polynomial	ADJ
m-1071	10	24	,	,	PUNCT
m-1071	10	25	sigma	sigma	NOUN
m-1071	10	26	and	and	CCONJ
m-1071	10	27	sombor	sombor	NOUN
m-1071	10	28	index	index	NOUN
m-1071	10	29	2020	2020	NUM
m-1071	10	30	mathematics	mathematic	NOUN
m-1071	10	31	subject	subject	ADJ
m-1071	10	32	classification	classification	NOUN
m-1071	10	33	:	:	PUNCT
m-1071	10	34	05c07	05c07	NOUN
m-1071	10	35	,	,	PUNCT
m-1071	10	36	05c09	05c09	NUM
m-1071	10	37	,	,	PUNCT
m-1071	10	38	05c31	05c31	NUM
m-1071	10	39	,	,	PUNCT
m-1071	10	40	05c76	05c76	NUM
m-1071	10	41	,	,	PUNCT
m-1071	10	42	05	05	NUM
m-1071	10	43	c	c	NOUN
m-1071	10	44	99	99	NUM
m-1071	10	45	ijo	ijo	PROPN
m-1071	10	46	international	international	PROPN
m-1071	10	47	journal	journal	PROPN
m-1071	10	48	of	of	ADP
m-1071	10	49	mathematics	mathematics	PROPN
m-1071	10	50	(	(	PUNCT
m-1071	10	51	issn	issn	PROPN
m-1071	10	52	:	:	PUNCT
m-1071	10	53	2992	2992	NUM
m-1071	10	54	-	-	SYM
m-1071	10	55	4421	4421	NUM
m-1071	10	56	)	)	PUNCT
m-1071	10	57	ijo	ijo	PROPN
m-1071	10	58	journals	journal	NOUN
m-1071	10	59	volume	volume	NOUN
m-1071	10	60	08	08	NUM
m-1071	10	61	|	|	ADV
m-1071	10	62	issue	issue	VERB
m-1071	10	63	04	04	NUM
m-1071	11	1	|	|	CCONJ
m-1071	11	2	april	april	PROPN
m-1071	11	3	2025	2025	NUM
m-1071	12	1	|	|	ADV
m-1071	12	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	12	3	1	1	NUM
m-1071	12	4	introduction	introduction	NOUN
m-1071	12	5	one	one	NUM
m-1071	12	6	of	of	ADP
m-1071	12	7	the	the	DET
m-1071	12	8	standard	standard	ADJ
m-1071	12	9	procedures	procedure	NOUN
m-1071	12	10	used	use	VERB
m-1071	12	11	in	in	ADP
m-1071	12	12	the	the	DET
m-1071	12	13	study	study	NOUN
m-1071	12	14	of	of	ADP
m-1071	12	15	structure	structure	NOUN
m-1071	12	16	-	-	PUNCT
m-1071	12	17	property	property	NOUN
m-1071	12	18	relations	relation	NOUN
m-1071	12	19	is	be	AUX
m-1071	12	20	the	the	DET
m-1071	12	21	use	use	NOUN
m-1071	12	22	of	of	ADP
m-1071	12	23	structure	structure	NOUN
m-1071	12	24	descriptors	descriptor	NOUN
m-1071	12	25	.	.	PUNCT
m-1071	13	1	the	the	DET
m-1071	13	2	ability	ability	NOUN
m-1071	13	3	to	to	PART
m-1071	13	4	correlate	correlate	VERB
m-1071	13	5	and	and	CCONJ
m-1071	13	6	predict	predict	VERB
m-1071	13	7	physical	physical	ADJ
m-1071	13	8	,	,	PUNCT
m-1071	13	9	chemical	chemical	NOUN
m-1071	13	10	,	,	PUNCT
m-1071	13	11	and	and	CCONJ
m-1071	13	12	biological	biological	ADJ
m-1071	13	13	activity	activity	NOUN
m-1071	13	14	(	(	PUNCT
m-1071	13	15	property	property	NOUN
m-1071	13	16	)	)	PUNCT
m-1071	13	17	from	from	ADP
m-1071	13	18	a	a	DET
m-1071	13	19	molecular	molecular	ADJ
m-1071	13	20	structure	structure	NOUN
m-1071	13	21	is	be	AUX
m-1071	13	22	a	a	DET
m-1071	13	23	challenging	challenging	ADJ
m-1071	13	24	problem	problem	NOUN
m-1071	13	25	in	in	ADP
m-1071	13	26	theoretical	theoretical	ADJ
m-1071	13	27	and	and	CCONJ
m-1071	13	28	computational	computational	ADJ
m-1071	13	29	chemistry	chemistry	NOUN
m-1071	13	30	[	[	X
m-1071	13	31	1	1	NUM
m-1071	13	32	,	,	PUNCT
m-1071	13	33	2	2	NUM
m-1071	13	34	]	]	PUNCT
m-1071	13	35	.	.	PUNCT
m-1071	14	1	a	a	DET
m-1071	14	2	topological	topological	ADJ
m-1071	14	3	index	index	NOUN
m-1071	14	4	is	be	AUX
m-1071	14	5	a	a	DET
m-1071	14	6	number	number	NOUN
m-1071	14	7	that	that	PRON
m-1071	14	8	describes	describe	VERB
m-1071	14	9	the	the	DET
m-1071	14	10	graph	graph	NOUN
m-1071	14	11	’s	’s	PART
m-1071	14	12	topology	topology	NOUN
m-1071	14	13	.	.	PUNCT
m-1071	15	1	it	it	PRON
m-1071	15	2	is	be	AUX
m-1071	15	3	one	one	NUM
m-1071	15	4	of	of	ADP
m-1071	15	5	the	the	DET
m-1071	15	6	best	good	ADJ
m-1071	15	7	quantification	quantification	NOUN
m-1071	15	8	methods	method	NOUN
m-1071	15	9	because	because	SCONJ
m-1071	15	10	it	it	PRON
m-1071	15	11	can	can	AUX
m-1071	15	12	be	be	AUX
m-1071	15	13	computed	compute	VERB
m-1071	15	14	quickly	quickly	ADV
m-1071	15	15	for	for	ADP
m-1071	15	16	a	a	DET
m-1071	15	17	large	large	ADJ
m-1071	15	18	number	number	NOUN
m-1071	15	19	of	of	ADP
m-1071	15	20	molecules	molecule	NOUN
m-1071	15	21	and	and	CCONJ
m-1071	15	22	can	can	AUX
m-1071	15	23	be	be	AUX
m-1071	15	24	obtained	obtain	VERB
m-1071	15	25	directly	directly	ADV
m-1071	15	26	from	from	ADP
m-1071	15	27	molecular	molecular	ADJ
m-1071	15	28	structures	structure	NOUN
m-1071	15	29	.	.	PUNCT
m-1071	16	1	wiener	wiener	NOUN
m-1071	16	2	,	,	PUNCT
m-1071	16	3	a	a	DET
m-1071	16	4	chemist	chemist	NOUN
m-1071	16	5	,	,	PUNCT
m-1071	16	6	used	use	VERB
m-1071	16	7	a	a	DET
m-1071	16	8	topological	topological	ADJ
m-1071	16	9	index	index	NOUN
m-1071	16	10	for	for	ADP
m-1071	16	11	the	the	DET
m-1071	16	12	first	first	ADJ
m-1071	16	13	time	time	NOUN
m-1071	16	14	in	in	ADP
m-1071	16	15	1947	1947	NUM
m-1071	16	16	while	while	SCONJ
m-1071	16	17	studying	study	VERB
m-1071	16	18	the	the	DET
m-1071	16	19	relationship	relationship	NOUN
m-1071	16	20	between	between	ADP
m-1071	16	21	molecular	molecular	ADJ
m-1071	16	22	structure	structure	NOUN
m-1071	16	23	and	and	CCONJ
m-1071	16	24	the	the	DET
m-1071	16	25	physical	physical	ADJ
m-1071	16	26	and	and	CCONJ
m-1071	16	27	chemical	chemical	NOUN
m-1071	16	28	properties	property	NOUN
m-1071	16	29	of	of	ADP
m-1071	16	30	certain	certain	ADJ
m-1071	16	31	hydrocarbon	hydrocarbon	NOUN
m-1071	16	32	compounds	compound	NOUN
m-1071	16	33	[	[	X
m-1071	16	34	3	3	NUM
m-1071	16	35	,	,	PUNCT
m-1071	16	36	17	17	NUM
m-1071	16	37	]	]	PUNCT
m-1071	16	38	.	.	PUNCT
m-1071	17	1	liu	liu	PROPN
m-1071	17	2	et	et	PROPN
m-1071	17	3	al	al	PROPN
m-1071	17	4	.	.	PROPN
m-1071	17	5	discussed	discuss	VERB
m-1071	17	6	several	several	ADJ
m-1071	17	7	aspects	aspect	NOUN
m-1071	17	8	of	of	ADP
m-1071	17	9	graph	graph	NOUN
m-1071	17	10	theory	theory	NOUN
m-1071	17	11	in	in	ADP
m-1071	17	12	[	[	X
m-1071	17	13	5]-[12	5]-[12	NOUN
m-1071	17	14	]	]	PUNCT
m-1071	17	15	.	.	PUNCT
m-1071	18	1	mathematical	mathematical	ADJ
m-1071	18	2	chemistry	chemistry	NOUN
m-1071	18	3	describes	describe	VERB
m-1071	18	4	how	how	SCONJ
m-1071	18	5	to	to	PART
m-1071	18	6	use	use	VERB
m-1071	18	7	polynomials	polynomial	NOUN
m-1071	18	8	and	and	CCONJ
m-1071	18	9	functions	function	NOUN
m-1071	18	10	to	to	PART
m-1071	18	11	offer	offer	VERB
m-1071	18	12	instructions	instruction	NOUN
m-1071	18	13	concealed	conceal	VERB
m-1071	18	14	in	in	ADP
m-1071	18	15	the	the	DET
m-1071	18	16	symmetry	symmetry	NOUN
m-1071	18	17	of	of	ADP
m-1071	18	18	molecular	molecular	ADJ
m-1071	18	19	graphs	graph	NOUN
m-1071	18	20	,	,	PUNCT
m-1071	18	21	and	and	CCONJ
m-1071	18	22	graph	graph	NOUN
m-1071	18	23	theory	theory	NOUN
m-1071	18	24	has	have	VERB
m-1071	18	25	many	many	ADJ
m-1071	18	26	applications	application	NOUN
m-1071	18	27	in	in	ADP
m-1071	18	28	modern	modern	ADJ
m-1071	18	29	chemistry	chemistry	NOUN
m-1071	18	30	,	,	PUNCT
m-1071	18	31	particularly	particularly	ADV
m-1071	18	32	organic	organic	ADJ
m-1071	18	33	chemistry	chemistry	NOUN
m-1071	18	34	.	.	PUNCT
m-1071	19	1	the	the	DET
m-1071	19	2	atoms	atom	NOUN
m-1071	19	3	and	and	CCONJ
m-1071	19	4	bonds	bond	NOUN
m-1071	19	5	of	of	ADP
m-1071	19	6	a	a	DET
m-1071	19	7	molecular	molecular	ADJ
m-1071	19	8	structure	structure	NOUN
m-1071	19	9	are	be	AUX
m-1071	19	10	represented	represent	VERB
m-1071	19	11	by	by	ADP
m-1071	19	12	vertices	vertex	NOUN
m-1071	19	13	and	and	CCONJ
m-1071	19	14	edges	edge	NOUN
m-1071	19	15	,	,	PUNCT
m-1071	19	16	respectively	respectively	ADV
m-1071	19	17	,	,	PUNCT
m-1071	19	18	in	in	ADP
m-1071	19	19	chemical	chemical	NOUN
m-1071	19	20	graph	graph	NOUN
m-1071	19	21	theory	theory	NOUN
m-1071	19	22	.	.	PUNCT
m-1071	20	1	many	many	ADJ
m-1071	20	2	applications	application	NOUN
m-1071	20	3	of	of	ADP
m-1071	20	4	topological	topological	ADJ
m-1071	20	5	indices	index	NOUN
m-1071	20	6	are	be	AUX
m-1071	20	7	employed	employ	VERB
m-1071	20	8	in	in	ADP
m-1071	20	9	theoretical	theoretical	ADJ
m-1071	20	10	chemistry	chemistry	NOUN
m-1071	20	11	,	,	PUNCT
m-1071	20	12	[	[	X
m-1071	20	13	13	13	NUM
m-1071	20	14	,	,	PUNCT
m-1071	20	15	14	14	NUM
m-1071	20	16	]	]	PUNCT
m-1071	20	17	,	,	PUNCT
m-1071	20	18	particularly	particularly	ADV
m-1071	20	19	qspr	qspr	NOUN
m-1071	20	20	/	/	SYM
m-1071	20	21	qsar	qsar	NOUN
m-1071	20	22	research	research	NOUN
m-1071	20	23	.	.	PUNCT
m-1071	21	1	many	many	ADJ
m-1071	21	2	famous	famous	ADJ
m-1071	21	3	researchers	researcher	NOUN
m-1071	21	4	have	have	AUX
m-1071	21	5	studied	study	VERB
m-1071	21	6	topological	topological	ADJ
m-1071	21	7	indices	index	NOUN
m-1071	21	8	to	to	PART
m-1071	21	9	get	get	VERB
m-1071	21	10	information	information	NOUN
m-1071	21	11	about	about	ADP
m-1071	21	12	different	different	ADJ
m-1071	21	13	families	family	NOUN
m-1071	21	14	of	of	ADP
m-1071	21	15	graphs	graph	NOUN
m-1071	21	16	[	[	X
m-1071	21	17	4	4	NUM
m-1071	21	18	,	,	PUNCT
m-1071	21	19	15	15	NUM
m-1071	21	20	]	]	PUNCT
m-1071	21	21	.	.	PUNCT
m-1071	22	1	in	in	ADP
m-1071	22	2	qualitative	qualitative	ADJ
m-1071	22	3	structure	structure	NOUN
m-1071	22	4	-	-	PUNCT
m-1071	22	5	property	property	NOUN
m-1071	22	6	relationships	relationship	NOUN
m-1071	22	7	(	(	PUNCT
m-1071	22	8	qspr	qspr	PROPN
m-1071	22	9	)	)	PUNCT
m-1071	22	10	and	and	CCONJ
m-1071	22	11	qualitative	qualitative	ADJ
m-1071	22	12	structure	structure	NOUN
m-1071	22	13	-	-	PUNCT
m-1071	22	14	activity	activity	NOUN
m-1071	22	15	relationships	relationship	NOUN
m-1071	22	16	(	(	PUNCT
m-1071	22	17	qsar	qsar	NOUN
m-1071	22	18	)	)	PUNCT
m-1071	22	19	,	,	PUNCT
m-1071	22	20	topological	topological	ADJ
m-1071	22	21	indices	index	NOUN
m-1071	22	22	are	be	AUX
m-1071	22	23	used	use	VERB
m-1071	22	24	directly	directly	ADV
m-1071	22	25	as	as	ADP
m-1071	22	26	simple	simple	ADJ
m-1071	22	27	numerical	numerical	ADJ
m-1071	22	28	descriptors	descriptor	NOUN
m-1071	22	29	in	in	ADP
m-1071	22	30	comparison	comparison	NOUN
m-1071	22	31	with	with	ADP
m-1071	22	32	physical	physical	ADJ
m-1071	22	33	,	,	PUNCT
m-1071	22	34	biological	biological	ADJ
m-1071	22	35	,	,	PUNCT
m-1071	22	36	or	or	CCONJ
m-1071	22	37	chemical	chemical	NOUN
m-1071	22	38	characteristics	characteristic	NOUN
m-1071	22	39	of	of	ADP
m-1071	22	40	molecules	molecule	NOUN
m-1071	22	41	,	,	PUNCT
m-1071	22	42	which	which	PRON
m-1071	22	43	is	be	AUX
m-1071	22	44	a	a	DET
m-1071	22	45	benefit	benefit	NOUN
m-1071	22	46	.	.	PUNCT
m-1071	23	1	many	many	ADJ
m-1071	23	2	researchers	researcher	NOUN
m-1071	23	3	have	have	AUX
m-1071	23	4	worked	work	VERB
m-1071	23	5	on	on	ADP
m-1071	23	6	various	various	ADJ
m-1071	23	7	chemical	chemical	NOUN
m-1071	23	8	compounds	compound	NOUN
m-1071	23	9	and	and	CCONJ
m-1071	23	10	computed	compute	VERB
m-1071	23	11	topological	topological	ADJ
m-1071	23	12	descriptors	descriptor	NOUN
m-1071	23	13	of	of	ADP
m-1071	23	14	various	various	ADJ
m-1071	23	15	molecular	molecular	ADJ
m-1071	23	16	graphs	graph	NOUN
m-1071	23	17	during	during	ADP
m-1071	23	18	the	the	DET
m-1071	23	19	last	last	ADJ
m-1071	23	20	few	few	ADJ
m-1071	23	21	decades	decade	NOUN
m-1071	23	22	[	[	X
m-1071	23	23	18	18	NUM
m-1071	23	24	]	]	PUNCT
m-1071	23	25	.	.	PUNCT
m-1071	24	1	in	in	ADP
m-1071	24	2	chemical	chemical	NOUN
m-1071	24	3	graph	graph	NOUN
m-1071	24	4	theory	theory	NOUN
m-1071	24	5	,	,	PUNCT
m-1071	24	6	a	a	DET
m-1071	24	7	molecular	molecular	ADJ
m-1071	24	8	graph	graph	NOUN
m-1071	24	9	is	be	AUX
m-1071	24	10	a	a	DET
m-1071	24	11	simple	simple	ADJ
m-1071	24	12	connected	connected	ADJ
m-1071	24	13	graph	graph	NOUN
m-1071	24	14	that	that	PRON
m-1071	24	15	contains	contain	VERB
m-1071	24	16	chemical	chemical	ADJ
m-1071	24	17	atoms	atom	NOUN
m-1071	24	18	and	and	CCONJ
m-1071	24	19	bonds	bond	NOUN
m-1071	24	20	,	,	PUNCT
m-1071	24	21	which	which	PRON
m-1071	24	22	are	be	AUX
m-1071	24	23	often	often	ADV
m-1071	24	24	referred	refer	VERB
m-1071	24	25	to	to	ADP
m-1071	24	26	as	as	ADP
m-1071	24	27	vertices	vertex	NOUN
m-1071	24	28	and	and	CCONJ
m-1071	24	29	edges	edge	NOUN
m-1071	24	30	,	,	PUNCT
m-1071	24	31	respectively	respectively	ADV
m-1071	24	32	,	,	PUNCT
m-1071	24	33	and	and	CCONJ
m-1071	24	34	there	there	PRON
m-1071	24	35	must	must	AUX
m-1071	24	36	be	be	AUX
m-1071	24	37	a	a	DET
m-1071	24	38	linkage	linkage	NOUN
m-1071	24	39	between	between	ADP
m-1071	24	40	the	the	DET
m-1071	24	41	vertices	vertex	NOUN
m-1071	24	42	set	set	VERB
m-1071	24	43	vg	vg	NOUN
m-1071	24	44	and	and	CCONJ
m-1071	24	45	edges	edge	NOUN
m-1071	24	46	set	set	VERB
m-1071	24	47	eg.if	eg.if	PROPN
m-1071	24	48	two	two	NUM
m-1071	24	49	atoms	atom	NOUN
m-1071	24	50	have	have	VERB
m-1071	24	51	an	an	DET
m-1071	24	52	atom	atom	NOUN
m-1071	24	53	-	-	PUNCT
m-1071	24	54	bond	bond	NOUN
m-1071	24	55	,	,	PUNCT
m-1071	24	56	then	then	ADV
m-1071	24	57	it	it	PRON
m-1071	24	58	is	be	AUX
m-1071	24	59	denoted	denote	VERB
m-1071	24	60	by	by	ADP
m-1071	24	61	e	e	NOUN
m-1071	24	62	∼	∼	NOUN
m-1071	24	63	f	f	NOUN
m-1071	24	64	,	,	PUNCT
m-1071	24	65	the	the	DET
m-1071	24	66	valency	valency	NOUN
m-1071	24	67	of	of	ADP
m-1071	24	68	every	every	DET
m-1071	24	69	atom	atom	NOUN
m-1071	24	70	of	of	ADP
m-1071	24	71	g	g	PROPN
m-1071	24	72	is	be	AUX
m-1071	24	73	actually	actually	ADV
m-1071	24	74	the	the	DET
m-1071	24	75	total	total	ADJ
m-1071	24	76	number	number	NOUN
m-1071	24	77	of	of	ADP
m-1071	24	78	atoms	atom	NOUN
m-1071	24	79	connected	connect	VERB
m-1071	24	80	to	to	ADP
m-1071	24	81	f	f	PROPN
m-1071	24	82	of	of	ADP
m-1071	24	83	g	g	PROPN
m-1071	24	84	and	and	CCONJ
m-1071	24	85	it	it	PRON
m-1071	24	86	is	be	AUX
m-1071	24	87	denoted	denote	VERB
m-1071	24	88	by	by	ADP
m-1071	24	89	df	df	PROPN
m-1071	24	90	,	,	PUNCT
m-1071	24	91	[	[	X
m-1071	24	92	16	16	NUM
m-1071	24	93	,	,	PUNCT
m-1071	24	94	19	19	NUM
m-1071	24	95	]	]	PUNCT
m-1071	24	96	.	.	PUNCT
m-1071	25	1	several	several	ADJ
m-1071	25	2	polynomials	polynomial	NOUN
m-1071	25	3	closely	closely	ADV
m-1071	25	4	related	relate	VERB
m-1071	25	5	to	to	ADP
m-1071	25	6	degree	degree	NOUN
m-1071	25	7	-	-	PUNCT
m-1071	25	8	based	base	VERB
m-1071	25	9	indices	index	NOUN
m-1071	25	10	are	be	AUX
m-1071	25	11	also	also	ADV
m-1071	25	12	introduced	introduce	VERB
m-1071	25	13	.	.	PUNCT
m-1071	26	1	ijo	ijo	PROPN
m-1071	26	2	international	international	PROPN
m-1071	26	3	journal	journal	PROPN
m-1071	26	4	of	of	ADP
m-1071	26	5	mathematics	mathematics	PROPN
m-1071	26	6	(	(	PUNCT
m-1071	26	7	issn	issn	PROPN
m-1071	26	8	:	:	PUNCT
m-1071	26	9	2992	2992	NUM
m-1071	26	10	-	-	SYM
m-1071	26	11	4421	4421	NUM
m-1071	26	12	)	)	PUNCT
m-1071	26	13	ijo	ijo	PROPN
m-1071	26	14	journals	journal	NOUN
m-1071	26	15	volume	volume	NOUN
m-1071	26	16	08	08	NUM
m-1071	27	1	|	|	ADV
m-1071	27	2	issue	issue	VERB
m-1071	27	3	04	04	NUM
m-1071	28	1	|	|	CCONJ
m-1071	28	2	april	april	PROPN
m-1071	28	3	2025	2025	NUM
m-1071	28	4	|	|	ADV
m-1071	28	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	28	6	2	2	NUM
m-1071	28	7	in	in	ADP
m-1071	28	8	2012	2012	NUM
m-1071	28	9	,	,	PUNCT
m-1071	28	10	zhang	zhang	PROPN
m-1071	28	11	introduced	introduce	VERB
m-1071	28	12	harmonic	harmonic	ADJ
m-1071	28	13	index	index	NOUN
m-1071	28	14	[	[	X
m-1071	28	15	31	31	NUM
m-1071	28	16	]	]	PUNCT
m-1071	28	17	.	.	PUNCT
m-1071	29	1	the	the	DET
m-1071	29	2	harmonic	harmonic	ADJ
m-1071	29	3	polynomial	polynomial	NOUN
m-1071	29	4	corresponding	corresponding	NOUN
m-1071	29	5	to	to	ADP
m-1071	29	6	harmonic	harmonic	ADJ
m-1071	29	7	index	index	NOUN
m-1071	29	8	is	be	AUX
m-1071	29	9	defined	define	VERB
m-1071	29	10	as	as	ADP
m-1071	29	11	�	�	PROPN
m-1071	29	12	(	(	PUNCT
m-1071	29	13	�	�	PROPN
m-1071	29	14	,	,	PUNCT
m-1071	29	15	�	�	PROPN
m-1071	29	16	)	)	PUNCT
m-1071	29	17	=	=	SYM
m-1071	29	18	∑	∑	PUNCT
m-1071	29	19	�	�	PROPN
m-1071	29	20	�	�	PROPN
m-1071	29	21	�	�	PROPN
m-1071	29	22	�	�	PROPN
m-1071	29	23	�	�	PROPN
m-1071	29	24	�	�	PROPN
m-1071	29	25	�	�	PROPN
m-1071	29	26	�	�	PROPN
m-1071	29	27	�	�	PROPN
m-1071	29	28	∈	∈	PROPN
m-1071	29	29	�	�	PROPN
m-1071	29	30	(	(	PUNCT
m-1071	29	31	�	�	PROPN
m-1071	29	32	)	)	PUNCT
m-1071	29	33	&	&	CCONJ
m-1071	29	34	�	�	PROPN
m-1071	29	35	(	(	PUNCT
m-1071	29	36	�	�	PROPN
m-1071	29	37	)	)	PUNCT
m-1071	29	38	=	=	SYM
m-1071	29	39	∑	∑	PUNCT
m-1071	29	40	�	�	PROPN
m-1071	29	41	�	�	PROPN
m-1071	29	42	�	�	PROPN
m-1071	29	43	�	�	PROPN
m-1071	29	44	�	�	PROPN
m-1071	29	45	�	�	PROPN
m-1071	29	46	�	�	PROPN
m-1071	29	47	�	�	PROPN
m-1071	29	48	∈	∈	PROPN
m-1071	29	49	�	�	PROPN
m-1071	29	50	(	(	PUNCT
m-1071	29	51	�	�	PROPN
m-1071	29	52	)	)	PUNCT
m-1071	29	53	(	(	PUNCT
m-1071	29	54	1	1	X
m-1071	29	55	)	)	PUNCT
m-1071	29	56	in	in	ADP
m-1071	29	57	1998	1998	NUM
m-1071	29	58	,	,	PUNCT
m-1071	29	59	estrada	estrada	PROPN
m-1071	29	60	et.al	et.al	PROPN
m-1071	29	61	introduced	introduce	VERB
m-1071	29	62	atom	atom	NOUN
m-1071	29	63	bond	bond	NOUN
m-1071	29	64	connectivity	connectivity	NOUN
m-1071	29	65	index	index	NOUN
m-1071	29	66	[	[	X
m-1071	29	67	21	21	NUM
m-1071	29	68	]	]	PUNCT
m-1071	29	69	.	.	PUNCT
m-1071	30	1	the	the	DET
m-1071	30	2	abc	abc	PROPN
m-1071	30	3	polynomial	polynomial	PROPN
m-1071	30	4	corresponding	corresponding	NOUN
m-1071	30	5	to	to	ADP
m-1071	30	6	the	the	DET
m-1071	30	7	abc	abc	PROPN
m-1071	30	8	indices	index	NOUN
m-1071	30	9	is	be	AUX
m-1071	30	10	expressed	express	VERB
m-1071	30	11	as	as	ADP
m-1071	30	12	�	�	PROPN
m-1071	30	13	�	�	PROPN
m-1071	30	14	�	�	PROPN
m-1071	30	15	(	(	PUNCT
m-1071	30	16	�	�	PROPN
m-1071	30	17	,	,	PUNCT
m-1071	30	18	�	�	PROPN
m-1071	30	19	)	)	PUNCT
m-1071	31	1	=	=	SYM
m-1071	31	2	�	�	PROPN
m-1071	31	3	�	�	PROPN
m-1071	31	4	�	�	PROPN
m-1071	31	5	�	�	PROPN
m-1071	31	6	�	�	PROPN
m-1071	31	7	+	+	NOUN
m-1071	31	8	�	�	PROPN
m-1071	31	9	�	�	PROPN
m-1071	31	10	−2	−2	PROPN
m-1071	31	11	�	�	PROPN
m-1071	31	12	�	�	PROPN
m-1071	31	13	+	+	PROPN
m-1071	31	14	�	�	PROPN
m-1071	31	15	�	�	PROPN
m-1071	31	16	�	�	PROPN
m-1071	31	17	�	�	PROPN
m-1071	31	18	∈	∈	PROPN
m-1071	31	19	�	�	PROPN
m-1071	31	20	(	(	PUNCT
m-1071	31	21	�	�	PROPN
m-1071	31	22	)	)	PUNCT
m-1071	31	23	&	&	CCONJ
m-1071	31	24	�	�	PROPN
m-1071	31	25	�	�	PROPN
m-1071	31	26	�	�	PROPN
m-1071	31	27	(	(	PUNCT
m-1071	31	28	�	�	PROPN
m-1071	31	29	)	)	PUNCT
m-1071	31	30	=	=	SYM
m-1071	31	31	�	�	PROPN
m-1071	31	32	�	�	PROPN
m-1071	31	33	�	�	PROPN
m-1071	31	34	�	�	PROPN
m-1071	31	35	+	+	NOUN
m-1071	31	36	�	�	PROPN
m-1071	31	37	�	�	PROPN
m-1071	31	38	−2	−2	PROPN
m-1071	31	39	�	�	PROPN
m-1071	31	40	�	�	PROPN
m-1071	31	41	+	+	PROPN
m-1071	31	42	�	�	PROPN
m-1071	31	43	�	�	PROPN
m-1071	31	44	(	(	PUNCT
m-1071	31	45	2	2	NUM
m-1071	31	46	)	)	PUNCT
m-1071	31	47	�	�	PROPN
m-1071	31	48	�	�	PROPN
m-1071	31	49	∈	∈	PROPN
m-1071	31	50	�	�	PROPN
m-1071	31	51	(	(	PUNCT
m-1071	31	52	�	�	PROPN
m-1071	31	53	)	)	PUNCT
m-1071	31	54	in	in	ADP
m-1071	31	55	2015	2015	NUM
m-1071	31	56	,	,	PUNCT
m-1071	31	57	formula	formula	NOUN
m-1071	31	58	and	and	CCONJ
m-1071	31	59	gutmann	gutmann	PROPN
m-1071	31	60	introduced	introduce	VERB
m-1071	31	61	forgotten	forget	VERB
m-1071	31	62	topological	topological	ADJ
m-1071	31	63	index	index	NOUN
m-1071	31	64	or	or	CCONJ
m-1071	31	65	f	f	NOUN
m-1071	31	66	-	-	PUNCT
m-1071	31	67	index	index	NOUN
m-1071	31	68	[	[	X
m-1071	31	69	22	22	NUM
m-1071	31	70	]	]	PUNCT
m-1071	31	71	.	.	PUNCT
m-1071	32	1	the	the	DET
m-1071	32	2	forgotten	forget	VERB
m-1071	32	3	polynomial	polynomial	ADJ
m-1071	32	4	and	and	CCONJ
m-1071	32	5	index	index	NOUN
m-1071	32	6	are	be	AUX
m-1071	32	7	defined	define	VERB
m-1071	32	8	as	as	ADP
m-1071	32	9	�	�	PROPN
m-1071	32	10	(	(	PUNCT
m-1071	32	11	�	�	PROPN
m-1071	32	12	,	,	PUNCT
m-1071	32	13	�	�	PROPN
m-1071	32	14	)	)	PUNCT
m-1071	32	15	=	=	SYM
m-1071	32	16	�	�	PROPN
m-1071	32	17	�	�	PROPN
m-1071	32	18	[	[	PUNCT
m-1071	32	19	(	(	PUNCT
m-1071	32	20	�	�	PROPN
m-1071	32	21	�	�	PROPN
m-1071	32	22	)	)	PUNCT
m-1071	32	23	�	�	PROPN
m-1071	32	24	�	�	PROPN
m-1071	32	25	[	[	X
m-1071	32	26	(	(	PUNCT
m-1071	32	27	�	�	PROPN
m-1071	32	28	�	�	PROPN
m-1071	32	29	)	)	PUNCT
m-1071	32	30	�	�	PROPN
m-1071	32	31	]	]	PUNCT
m-1071	32	32	�	�	PROPN
m-1071	32	33	�	�	PROPN
m-1071	32	34	∈	∈	PROPN
m-1071	32	35	�	�	PROPN
m-1071	32	36	(	(	PUNCT
m-1071	32	37	�	�	PROPN
m-1071	32	38	)	)	PUNCT
m-1071	32	39	&	&	CCONJ
m-1071	32	40	�	�	PROPN
m-1071	32	41	(	(	PUNCT
m-1071	32	42	�	�	PROPN
m-1071	32	43	)	)	PUNCT
m-1071	32	44	=	=	SYM
m-1071	32	45	�	�	PROPN
m-1071	32	46	[	[	PUNCT
m-1071	32	47	(	(	PUNCT
m-1071	32	48	�	�	PROPN
m-1071	32	49	�	�	NOUN
m-1071	32	50	)	)	PUNCT
m-1071	32	51	�	�	PROPN
m-1071	32	52	+	+	CCONJ
m-1071	33	1	[	[	X
m-1071	33	2	(	(	PUNCT
m-1071	33	3	�	�	PROPN
m-1071	33	4	�	�	PROPN
m-1071	33	5	)	)	PUNCT
m-1071	33	6	�	�	PROPN
m-1071	33	7	]	]	PUNCT
m-1071	33	8	(	(	PUNCT
m-1071	33	9	3	3	X
m-1071	33	10	)	)	PUNCT
m-1071	33	11	�	�	PROPN
m-1071	33	12	�	�	PROPN
m-1071	33	13	∈	∈	PROPN
m-1071	33	14	�	�	PROPN
m-1071	33	15	(	(	PUNCT
m-1071	33	16	�	�	PROPN
m-1071	33	17	)	)	PUNCT
m-1071	33	18	the	the	DET
m-1071	33	19	first	first	PROPN
m-1071	33	20	ga	ga	PROPN
m-1071	33	21	-	-	PUNCT
m-1071	33	22	index	index	NOUN
m-1071	33	23	was	be	AUX
m-1071	33	24	proposed	propose	VERB
m-1071	33	25	by	by	ADP
m-1071	33	26	vukicevic	vukicevic	NOUN
m-1071	33	27	[	[	X
m-1071	33	28	23	23	NUM
m-1071	33	29	]	]	PUNCT
m-1071	33	30	.	.	PUNCT
m-1071	34	1	the	the	DET
m-1071	34	2	geometric	geometric	ADJ
m-1071	34	3	arithmetic	arithmetic	ADJ
m-1071	34	4	polynomial	polynomial	ADJ
m-1071	34	5	and	and	CCONJ
m-1071	34	6	index	index	NOUN
m-1071	34	7	are	be	AUX
m-1071	34	8	defined	define	VERB
m-1071	34	9	as	as	ADP
m-1071	34	10	�	�	PROPN
m-1071	34	11	�	�	PROPN
m-1071	34	12	(	(	PUNCT
m-1071	34	13	�	�	PROPN
m-1071	34	14	,	,	PUNCT
m-1071	34	15	�	�	PROPN
m-1071	34	16	)	)	PUNCT
m-1071	34	17	=	=	SYM
m-1071	34	18	�	�	PROPN
m-1071	34	19	�	�	PROPN
m-1071	34	20	�	�	PROPN
m-1071	34	21	�	�	PROPN
m-1071	34	22	�	�	PROPN
m-1071	34	23	+	+	PROPN
m-1071	34	24	�	�	PROPN
m-1071	34	25	�	�	PROPN
m-1071	34	26	�	�	PROPN
m-1071	34	27	�	�	PROPN
m-1071	34	28	�	�	PROPN
m-1071	34	29	+	+	PROPN
m-1071	34	30	�	�	PROPN
m-1071	34	31	�	�	PROPN
m-1071	34	32	�	�	PROPN
m-1071	34	33	�	�	PROPN
m-1071	34	34	∈	∈	PROPN
m-1071	34	35	�	�	PROPN
m-1071	34	36	(	(	PUNCT
m-1071	34	37	�	�	PROPN
m-1071	34	38	)	)	PUNCT
m-1071	34	39	&	&	CCONJ
m-1071	34	40	�	�	PROPN
m-1071	34	41	�	�	PROPN
m-1071	34	42	(	(	PUNCT
m-1071	34	43	�	�	PROPN
m-1071	34	44	)	)	PUNCT
m-1071	34	45	=	=	SYM
m-1071	34	46	�	�	PROPN
m-1071	34	47	�	�	PROPN
m-1071	34	48	�	�	PROPN
m-1071	34	49	�	�	PROPN
m-1071	34	50	+	+	PROPN
m-1071	34	51	�	�	PROPN
m-1071	34	52	�	�	PROPN
m-1071	34	53	�	�	PROPN
m-1071	34	54	�	�	PROPN
m-1071	34	55	�	�	PROPN
m-1071	34	56	+	+	PROPN
m-1071	34	57	�	�	PROPN
m-1071	34	58	�	�	PROPN
m-1071	34	59	(	(	PUNCT
m-1071	34	60	4	4	NUM
m-1071	34	61	)	)	PUNCT
m-1071	34	62	�	�	PROPN
m-1071	34	63	�	�	PROPN
m-1071	34	64	∈	∈	PROPN
m-1071	34	65	�	�	PROPN
m-1071	34	66	(	(	PUNCT
m-1071	34	67	�	�	PROPN
m-1071	34	68	)	)	PUNCT
m-1071	34	69	the	the	DET
m-1071	34	70	randic	randic	ADJ
m-1071	34	71	polynomial	polynomial	ADJ
m-1071	34	72	and	and	CCONJ
m-1071	34	73	index	index	NOUN
m-1071	34	74	,	,	PUNCT
m-1071	34	75	[	[	X
m-1071	34	76	24	24	NUM
m-1071	34	77	]	]	PUNCT
m-1071	34	78	are	be	AUX
m-1071	34	79	defined	define	VERB
m-1071	34	80	as	as	ADP
m-1071	34	81	�	�	PROPN
m-1071	34	82	(	(	PUNCT
m-1071	34	83	�	�	PROPN
m-1071	34	84	,	,	PUNCT
m-1071	34	85	�	�	PROPN
m-1071	34	86	)	)	PUNCT
m-1071	34	87	=	=	SYM
m-1071	34	88	�	�	PROPN
m-1071	34	89	�	�	PROPN
m-1071	34	90	�	�	PROPN
m-1071	34	91	�	�	PROPN
m-1071	34	92	�	�	PROPN
m-1071	34	93	�	�	PROPN
m-1071	34	94	+	+	PROPN
m-1071	34	95	�	�	PROPN
m-1071	34	96	�	�	PROPN
m-1071	34	97	�	�	PROPN
m-1071	34	98	�	�	PROPN
m-1071	34	99	∈	∈	PROPN
m-1071	34	100	�	�	PROPN
m-1071	34	101	(	(	PUNCT
m-1071	34	102	�	�	PROPN
m-1071	34	103	)	)	PUNCT
m-1071	34	104	&	&	CCONJ
m-1071	34	105	�	�	PROPN
m-1071	34	106	(	(	PUNCT
m-1071	34	107	�	�	PROPN
m-1071	34	108	)	)	PUNCT
m-1071	34	109	=	=	SYM
m-1071	34	110	�	�	PROPN
m-1071	34	111	1	1	NUM
m-1071	34	112	�	�	PROPN
m-1071	34	113	�	�	PROPN
m-1071	34	114	�	�	PROPN
m-1071	34	115	+	+	PROPN
m-1071	34	116	�	�	PROPN
m-1071	34	117	�	�	PROPN
m-1071	34	118	(	(	PUNCT
m-1071	34	119	5	5	NUM
m-1071	34	120	)	)	PUNCT
m-1071	34	121	�	�	PROPN
m-1071	34	122	�	�	PROPN
m-1071	34	123	∈	∈	PROPN
m-1071	34	124	�	�	PROPN
m-1071	34	125	(	(	PUNCT
m-1071	34	126	�	�	PROPN
m-1071	34	127	)	)	PUNCT
m-1071	34	128	the	the	DET
m-1071	34	129	reciprocal	reciprocal	ADJ
m-1071	34	130	randic	randic	ADJ
m-1071	34	131	polynomial	polynomial	ADJ
m-1071	34	132	and	and	CCONJ
m-1071	34	133	index	index	NOUN
m-1071	34	134	[	[	X
m-1071	34	135	25	25	NUM
m-1071	34	136	]	]	PUNCT
m-1071	34	137	are	be	AUX
m-1071	34	138	defined	define	VERB
m-1071	34	139	as	as	ADP
m-1071	34	140	�	�	PROPN
m-1071	34	141	�	�	PROPN
m-1071	34	142	(	(	PUNCT
m-1071	34	143	�	�	PROPN
m-1071	34	144	,	,	PUNCT
m-1071	34	145	�	�	PROPN
m-1071	34	146	)	)	PUNCT
m-1071	35	1	=	=	SYM
m-1071	35	2	�	�	PROPN
m-1071	35	3	�	�	PROPN
m-1071	35	4	�	�	PROPN
m-1071	35	5	�	�	PROPN
m-1071	35	6	�	�	PROPN
m-1071	35	7	�	�	PROPN
m-1071	35	8	�	�	PROPN
m-1071	35	9	�	�	PROPN
m-1071	35	10	�	�	PROPN
m-1071	35	11	∈	∈	PROPN
m-1071	35	12	�	�	PROPN
m-1071	35	13	(	(	PUNCT
m-1071	35	14	�	�	PROPN
m-1071	35	15	)	)	PUNCT
m-1071	35	16	&	&	CCONJ
m-1071	35	17	�	�	PROPN
m-1071	35	18	�	�	PROPN
m-1071	35	19	(	(	PUNCT
m-1071	35	20	�	�	PROPN
m-1071	35	21	)	)	PUNCT
m-1071	35	22	=	=	SYM
m-1071	35	23	�	�	PROPN
m-1071	35	24	�	�	PROPN
m-1071	35	25	�	�	PROPN
m-1071	35	26	�	�	PROPN
m-1071	35	27	�	�	PROPN
m-1071	35	28	�	�	PROPN
m-1071	35	29	(	(	PUNCT
m-1071	35	30	6	6	NUM
m-1071	35	31	)	)	PUNCT
m-1071	35	32	�	�	PROPN
m-1071	35	33	�	�	PROPN
m-1071	35	34	∈	∈	PROPN
m-1071	35	35	�	�	PROPN
m-1071	35	36	(	(	PUNCT
m-1071	35	37	�	�	PROPN
m-1071	35	38	)	)	PUNCT
m-1071	35	39	the	the	DET
m-1071	35	40	symmetric	symmetric	ADJ
m-1071	35	41	division	division	NOUN
m-1071	35	42	degree	degree	NOUN
m-1071	35	43	polynomial	polynomial	ADJ
m-1071	35	44	and	and	CCONJ
m-1071	35	45	index	index	NOUN
m-1071	35	46	[	[	X
m-1071	35	47	26	26	NUM
m-1071	35	48	]	]	PUNCT
m-1071	35	49	are	be	AUX
m-1071	35	50	defined	define	VERB
m-1071	35	51	as	as	ADP
m-1071	35	52	�	�	PROPN
m-1071	35	53	�	�	PROPN
m-1071	35	54	�	�	PROPN
m-1071	35	55	(	(	PUNCT
m-1071	35	56	�	�	PROPN
m-1071	35	57	,	,	PUNCT
m-1071	35	58	�	�	PROPN
m-1071	35	59	)	)	PUNCT
m-1071	35	60	=	=	SYM
m-1071	35	61	�	�	PROPN
m-1071	35	62	�	�	PROPN
m-1071	35	63	[	[	PUNCT
m-1071	35	64	(	(	PUNCT
m-1071	35	65	�	�	PROPN
m-1071	35	66	�	�	PROPN
m-1071	35	67	)	)	PUNCT
m-1071	35	68	�	�	PROPN
m-1071	35	69	�	�	PROPN
m-1071	35	70	[	[	X
m-1071	35	71	(	(	PUNCT
m-1071	35	72	�	�	PROPN
m-1071	35	73	�	�	PROPN
m-1071	35	74	)	)	PUNCT
m-1071	35	75	�	�	PROPN
m-1071	35	76	]	]	PUNCT
m-1071	35	77	(	(	PUNCT
m-1071	35	78	�	�	PROPN
m-1071	35	79	�	�	PROPN
m-1071	35	80	)	)	PUNCT
m-1071	35	81	(	(	PUNCT
m-1071	35	82	�	�	PROPN
m-1071	35	83	�	�	PROPN
m-1071	35	84	)	)	PUNCT
m-1071	35	85	�	�	PROPN
m-1071	35	86	�	�	PROPN
m-1071	35	87	∈	∈	PROPN
m-1071	35	88	�	�	PROPN
m-1071	35	89	(	(	PUNCT
m-1071	35	90	�	�	PROPN
m-1071	35	91	)	)	PUNCT
m-1071	35	92	&	&	CCONJ
m-1071	35	93	�	�	PROPN
m-1071	35	94	�	�	PROPN
m-1071	35	95	�	�	PROPN
m-1071	35	96	(	(	PUNCT
m-1071	35	97	�	�	PROPN
m-1071	35	98	)	)	PUNCT
m-1071	35	99	=	=	SYM
m-1071	35	100	�	�	PROPN
m-1071	35	101	[	[	PUNCT
m-1071	35	102	(	(	PUNCT
m-1071	35	103	�	�	PROPN
m-1071	35	104	�	�	NOUN
m-1071	35	105	)	)	PUNCT
m-1071	35	106	�	�	PROPN
m-1071	35	107	+	+	CCONJ
m-1071	36	1	[	[	X
m-1071	36	2	(	(	PUNCT
m-1071	36	3	�	�	PROPN
m-1071	36	4	�	�	PROPN
m-1071	36	5	)	)	PUNCT
m-1071	36	6	�	�	PROPN
m-1071	36	7	]	]	PUNCT
m-1071	36	8	(	(	PUNCT
m-1071	36	9	�	�	PROPN
m-1071	36	10	�	�	PROPN
m-1071	36	11	)	)	PUNCT
m-1071	36	12	(	(	PUNCT
m-1071	36	13	�	�	PROPN
m-1071	36	14	�	�	PROPN
m-1071	36	15	)	)	PUNCT
m-1071	36	16	(	(	PUNCT
m-1071	36	17	7	7	X
m-1071	36	18	)	)	PUNCT
m-1071	36	19	�	�	PROPN
m-1071	36	20	�	�	PROPN
m-1071	36	21	∈	∈	PROPN
m-1071	36	22	�	�	PROPN
m-1071	36	23	(	(	PUNCT
m-1071	36	24	�	�	PROPN
m-1071	36	25	)	)	PUNCT
m-1071	36	26	the	the	DET
m-1071	36	27	inverse	inverse	ADJ
m-1071	36	28	symmetric	symmetric	ADJ
m-1071	36	29	division	division	NOUN
m-1071	36	30	degree	degree	NOUN
m-1071	36	31	polynomial	polynomial	ADJ
m-1071	36	32	and	and	CCONJ
m-1071	36	33	index	index	NOUN
m-1071	36	34	[	[	X
m-1071	36	35	27	27	NUM
m-1071	36	36	]	]	PUNCT
m-1071	36	37	are	be	AUX
m-1071	36	38	defined	define	VERB
m-1071	36	39	as	as	ADP
m-1071	36	40	ijo	ijo	PROPN
m-1071	36	41	international	international	PROPN
m-1071	36	42	journal	journal	PROPN
m-1071	36	43	of	of	ADP
m-1071	36	44	mathematics	mathematics	PROPN
m-1071	36	45	(	(	PUNCT
m-1071	36	46	issn	issn	PROPN
m-1071	36	47	:	:	PUNCT
m-1071	36	48	2992	2992	NUM
m-1071	36	49	-	-	SYM
m-1071	36	50	4421	4421	NUM
m-1071	36	51	)	)	PUNCT
m-1071	36	52	ijo	ijo	PROPN
m-1071	36	53	journals	journal	NOUN
m-1071	36	54	volume	volume	NOUN
m-1071	36	55	08	08	NUM
m-1071	37	1	|	|	ADV
m-1071	37	2	issue	issue	VERB
m-1071	37	3	04	04	NUM
m-1071	38	1	|	|	CCONJ
m-1071	38	2	april	april	PROPN
m-1071	38	3	2025	2025	NUM
m-1071	38	4	|	|	ADV
m-1071	38	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	38	6	3	3	NUM
m-1071	38	7	�	�	PROPN
m-1071	38	8	�	�	PROPN
m-1071	38	9	�	�	PROPN
m-1071	38	10	�	�	PROPN
m-1071	38	11	(	(	PUNCT
m-1071	38	12	�	�	PROPN
m-1071	38	13	,	,	PUNCT
m-1071	38	14	�	�	PROPN
m-1071	38	15	)	)	PUNCT
m-1071	38	16	=	=	SYM
m-1071	38	17	�	�	PROPN
m-1071	38	18	�	�	PROPN
m-1071	38	19	(	(	PUNCT
m-1071	38	20	�	�	PROPN
m-1071	38	21	�	�	PROPN
m-1071	38	22	)	)	PUNCT
m-1071	38	23	(	(	PUNCT
m-1071	38	24	�	�	PROPN
m-1071	38	25	�	�	PROPN
m-1071	38	26	)	)	PUNCT
m-1071	38	27	[	[	PUNCT
m-1071	38	28	(	(	PUNCT
m-1071	38	29	�	�	PROPN
m-1071	38	30	�	�	PROPN
m-1071	38	31	)	)	PUNCT
m-1071	38	32	�	�	PROPN
m-1071	38	33	�	�	PROPN
m-1071	38	34	[	[	X
m-1071	38	35	(	(	PUNCT
m-1071	38	36	�	�	PROPN
m-1071	38	37	�	�	PROPN
m-1071	38	38	)	)	PUNCT
m-1071	38	39	�	�	PROPN
m-1071	38	40	]	]	PUNCT
m-1071	38	41	�	�	PROPN
m-1071	38	42	�	�	PROPN
m-1071	38	43	∈	∈	PROPN
m-1071	38	44	�	�	PROPN
m-1071	38	45	(	(	PUNCT
m-1071	38	46	�	�	PROPN
m-1071	38	47	)	)	PUNCT
m-1071	38	48	&	&	CCONJ
m-1071	38	49	�	�	PROPN
m-1071	38	50	�	�	PROPN
m-1071	38	51	�	�	PROPN
m-1071	38	52	�	�	PROPN
m-1071	38	53	(	(	PUNCT
m-1071	38	54	�	�	PROPN
m-1071	38	55	)	)	PUNCT
m-1071	38	56	=	=	SYM
m-1071	38	57	�	�	PROPN
m-1071	38	58	(	(	PUNCT
m-1071	38	59	�	�	PROPN
m-1071	38	60	�	�	PROPN
m-1071	38	61	)	)	PUNCT
m-1071	38	62	(	(	PUNCT
m-1071	38	63	�	�	PROPN
m-1071	38	64	�	�	PROPN
m-1071	38	65	)	)	PUNCT
m-1071	38	66	[	[	PUNCT
m-1071	38	67	(	(	PUNCT
m-1071	38	68	�	�	PROPN
m-1071	38	69	�	�	NOUN
m-1071	38	70	)	)	PUNCT
m-1071	38	71	�	�	PROPN
m-1071	38	72	+	+	CCONJ
m-1071	38	73	[	[	X
m-1071	38	74	(	(	PUNCT
m-1071	38	75	�	�	PROPN
m-1071	38	76	�	�	PROPN
m-1071	38	77	)	)	PUNCT
m-1071	38	78	�	�	PROPN
m-1071	38	79	]	]	PUNCT
m-1071	38	80	(	(	PUNCT
m-1071	38	81	8)	8)	NUM
m-1071	38	82	�	�	PROPN
m-1071	38	83	�	�	PROPN
m-1071	38	84	∈	∈	PROPN
m-1071	38	85	�	�	PROPN
m-1071	38	86	(	(	PUNCT
m-1071	38	87	�	�	PROPN
m-1071	38	88	)	)	PUNCT
m-1071	38	89	sigma	sigma	VERB
m-1071	38	90	polynomial	polynomial	ADJ
m-1071	38	91	and	and	CCONJ
m-1071	38	92	index	index	NOUN
m-1071	38	93	[	[	X
m-1071	38	94	28	28	NUM
m-1071	38	95	]	]	PUNCT
m-1071	38	96	are	be	AUX
m-1071	38	97	defined	define	VERB
m-1071	38	98	as	as	ADP
m-1071	38	99	�	�	PROPN
m-1071	38	100	(	(	PUNCT
m-1071	38	101	�	�	PROPN
m-1071	38	102	,	,	PUNCT
m-1071	38	103	�	�	PROPN
m-1071	38	104	)	)	PUNCT
m-1071	38	105	=	=	SYM
m-1071	38	106	�	�	PROPN
m-1071	38	107	�	�	PROPN
m-1071	38	108	(	(	PUNCT
m-1071	38	109	�	�	PROPN
m-1071	38	110	�	�	PROPN
m-1071	38	111	�	�	PROPN
m-1071	38	112	�	�	PROPN
m-1071	38	113	�	�	PROPN
m-1071	38	114	)	)	PUNCT
m-1071	38	115	�	�	PROPN
m-1071	38	116	�	�	PROPN
m-1071	38	117	�	�	PROPN
m-1071	38	118	∈	∈	PROPN
m-1071	38	119	�	�	PROPN
m-1071	38	120	(	(	PUNCT
m-1071	38	121	�	�	PROPN
m-1071	38	122	)	)	PUNCT
m-1071	38	123	&	&	CCONJ
m-1071	38	124	�	�	PROPN
m-1071	38	125	(	(	PUNCT
m-1071	38	126	�	�	PROPN
m-1071	38	127	)	)	PUNCT
m-1071	38	128	=	=	SYM
m-1071	38	129	�	�	PROPN
m-1071	38	130	(	(	PUNCT
m-1071	38	131	�	�	PROPN
m-1071	38	132	�	�	PROPN
m-1071	38	133	−	−	PROPN
m-1071	38	134	�	�	PROPN
m-1071	38	135	�	�	PROPN
m-1071	38	136	)	)	PUNCT
m-1071	38	137	�	�	PROPN
m-1071	38	138	(	(	PUNCT
m-1071	38	139	9	9	NUM
m-1071	38	140	)	)	PUNCT
m-1071	38	141	�	�	PROPN
m-1071	38	142	�	�	PROPN
m-1071	38	143	∈	∈	PROPN
m-1071	38	144	�	�	PROPN
m-1071	38	145	(	(	PUNCT
m-1071	38	146	�	�	PROPN
m-1071	38	147	)	)	PUNCT
m-1071	38	148	the	the	DET
m-1071	38	149	concept	concept	NOUN
m-1071	38	150	of	of	ADP
m-1071	38	151	sombor	sombor	NOUN
m-1071	38	152	index	index	NOUN
m-1071	38	153	was	be	AUX
m-1071	38	154	recently	recently	ADV
m-1071	38	155	introduced	introduce	VERB
m-1071	38	156	by	by	ADP
m-1071	38	157	gutman	gutman	NOUN
m-1071	38	158	[	[	X
m-1071	38	159	29	29	NUM
m-1071	38	160	]	]	PUNCT
m-1071	38	161	.	.	PUNCT
m-1071	39	1	the	the	DET
m-1071	39	2	sombor	sombor	NOUN
m-1071	39	3	polynomial	polynomial	ADJ
m-1071	39	4	and	and	CCONJ
m-1071	39	5	index	index	NOUN
m-1071	39	6	are	be	AUX
m-1071	39	7	defined	define	VERB
m-1071	39	8	as	as	ADP
m-1071	39	9	�	�	PROPN
m-1071	39	10	(	(	PUNCT
m-1071	39	11	�	�	PROPN
m-1071	39	12	,	,	PUNCT
m-1071	39	13	�	�	PROPN
m-1071	39	14	)	)	PUNCT
m-1071	39	15	=	=	SYM
m-1071	39	16	�	�	PROPN
m-1071	39	17	�	�	PROPN
m-1071	39	18	�	�	PROPN
m-1071	39	19	[	[	PUNCT
m-1071	39	20	(	(	PUNCT
m-1071	39	21	�	�	PROPN
m-1071	39	22	�	�	PROPN
m-1071	39	23	)	)	PUNCT
m-1071	39	24	�	�	PROPN
m-1071	39	25	�	�	PROPN
m-1071	39	26	[	[	X
m-1071	39	27	(	(	PUNCT
m-1071	39	28	�	�	PROPN
m-1071	39	29	�	�	PROPN
m-1071	39	30	)	)	PUNCT
m-1071	39	31	�	�	PROPN
m-1071	39	32	]	]	PUNCT
m-1071	39	33	�	�	PROPN
m-1071	39	34	�	�	PROPN
m-1071	39	35	∈	∈	PROPN
m-1071	39	36	�	�	PROPN
m-1071	39	37	(	(	PUNCT
m-1071	39	38	�	�	PROPN
m-1071	39	39	)	)	PUNCT
m-1071	39	40	&	&	CCONJ
m-1071	39	41	�	�	PROPN
m-1071	39	42	�	�	PROPN
m-1071	39	43	(	(	PUNCT
m-1071	39	44	�	�	PROPN
m-1071	39	45	)	)	PUNCT
m-1071	39	46	=	=	SYM
m-1071	39	47	�	�	PROPN
m-1071	39	48	�	�	PROPN
m-1071	39	49	[	[	PUNCT
m-1071	39	50	(	(	PUNCT
m-1071	39	51	�	�	PROPN
m-1071	39	52	�	�	NOUN
m-1071	39	53	)	)	PUNCT
m-1071	39	54	�	�	PROPN
m-1071	39	55	+	+	CCONJ
m-1071	40	1	[	[	X
m-1071	40	2	(	(	PUNCT
m-1071	40	3	�	�	PROPN
m-1071	40	4	�	�	PROPN
m-1071	40	5	)	)	PUNCT
m-1071	40	6	�	�	PROPN
m-1071	40	7	]	]	PUNCT
m-1071	40	8	(	(	PUNCT
m-1071	40	9	10	10	NUM
m-1071	40	10	)	)	PUNCT
m-1071	40	11	�	�	PROPN
m-1071	40	12	�	�	PROPN
m-1071	40	13	∈	∈	PROPN
m-1071	40	14	�	�	PROPN
m-1071	40	15	(	(	PUNCT
m-1071	40	16	�	�	PROPN
m-1071	40	17	)	)	PUNCT
m-1071	40	18	in	in	ADP
m-1071	40	19	this	this	DET
m-1071	40	20	study	study	NOUN
m-1071	40	21	,	,	PUNCT
m-1071	40	22	the	the	DET
m-1071	40	23	atom	atom	NOUN
m-1071	40	24	-	-	PUNCT
m-1071	40	25	bond	bond	NOUN
m-1071	40	26	partition	partition	NOUN
m-1071	40	27	set	set	NOUN
m-1071	40	28	of	of	ADP
m-1071	40	29	sio4	sio4	NOUN
m-1071	40	30	in	in	ADP
m-1071	40	31	a	a	DET
m-1071	40	32	chain	chain	NOUN
m-1071	40	33	network	network	NOUN
m-1071	40	34	,	,	PUNCT
m-1071	40	35	which	which	PRON
m-1071	40	36	is	be	AUX
m-1071	40	37	partitioned	partition	VERB
m-1071	40	38	according	accord	VERB
m-1071	40	39	to	to	ADP
m-1071	40	40	the	the	DET
m-1071	40	41	valencies	valency	NOUN
m-1071	40	42	of	of	ADP
m-1071	40	43	their	their	PRON
m-1071	40	44	si	si	NOUN
m-1071	40	45	and	and	CCONJ
m-1071	40	46	o2	o2	PROPN
m-1071	40	47	atoms	atom	NOUN
m-1071	40	48	,	,	PUNCT
m-1071	40	49	is	be	AUX
m-1071	40	50	used	use	VERB
m-1071	40	51	to	to	PART
m-1071	40	52	generate	generate	VERB
m-1071	40	53	the	the	DET
m-1071	40	54	ten	ten	NUM
m-1071	40	55	polynomials	polynomial	NOUN
m-1071	40	56	mentioned	mention	VERB
m-1071	40	57	above	above	ADV
m-1071	40	58	and	and	CCONJ
m-1071	40	59	their	their	PRON
m-1071	40	60	corresponding	corresponding	ADJ
m-1071	40	61	indices	index	NOUN
m-1071	40	62	.	.	PUNCT
m-1071	41	1	1	1	NUM
m-1071	41	2	chain	chain	NOUN
m-1071	41	3	of	of	ADP
m-1071	41	4	sio4	sio4	PROPN
m-1071	41	5	a	a	DET
m-1071	41	6	sio4	sio4	NOUN
m-1071	41	7	tetrahedron	tetrahedron	NOUN
m-1071	41	8	,	,	PUNCT
m-1071	41	9	the	the	DET
m-1071	41	10	fundamental	fundamental	ADJ
m-1071	41	11	building	building	NOUN
m-1071	41	12	block	block	NOUN
m-1071	41	13	of	of	ADP
m-1071	41	14	silicates	silicate	NOUN
m-1071	41	15	,	,	PUNCT
m-1071	41	16	is	be	AUX
m-1071	41	17	created	create	VERB
m-1071	41	18	by	by	ADP
m-1071	41	19	fusing	fuse	VERB
m-1071	41	20	metal	metal	NOUN
m-1071	41	21	oxides	oxide	NOUN
m-1071	41	22	or	or	CCONJ
m-1071	41	23	mixing	mix	VERB
m-1071	41	24	metal	metal	NOUN
m-1071	41	25	carbonates	carbonate	NOUN
m-1071	41	26	with	with	ADP
m-1071	41	27	sand	sand	NOUN
m-1071	41	28	.	.	PUNCT
m-1071	42	1	the	the	DET
m-1071	42	2	sio4	sio4	NOUN
m-1071	42	3	tetrahedron	tetrahedron	NOUN
m-1071	42	4	is	be	AUX
m-1071	42	5	present	present	ADJ
m-1071	42	6	in	in	ADP
m-1071	42	7	almost	almost	ADV
m-1071	42	8	all	all	PRON
m-1071	42	9	silicates	silicate	NOUN
m-1071	42	10	.	.	PUNCT
m-1071	43	1	as	as	SCONJ
m-1071	43	2	shown	show	VERB
m-1071	43	3	in	in	ADP
m-1071	43	4	figure	figure	NOUN
m-1071	43	5	1	1	NUM
m-1071	43	6	,	,	PUNCT
m-1071	43	7	a	a	DET
m-1071	43	8	tetrahedron	tetrahedron	NOUN
m-1071	43	9	sio4	sio4	NOUN
m-1071	43	10	is	be	AUX
m-1071	43	11	a	a	DET
m-1071	43	12	pyramid	pyramid	NOUN
m-1071	43	13	with	with	ADP
m-1071	43	14	a	a	DET
m-1071	43	15	triangular	triangular	NOUN
m-1071	43	16	base	base	NOUN
m-1071	43	17	(	(	PUNCT
m-1071	43	18	a	a	DET
m-1071	43	19	single	single	ADJ
m-1071	43	20	tetrahedron	tetrahedron	NOUN
m-1071	43	21	sio4	sio4	NOUN
m-1071	43	22	)	)	PUNCT
m-1071	43	23	,	,	PUNCT
m-1071	43	24	and	and	CCONJ
m-1071	43	25	the	the	DET
m-1071	43	26	silicon	silicon	NOUN
m-1071	43	27	atom	atom	NOUN
m-1071	43	28	si	si	X
m-1071	43	29	is	be	AUX
m-1071	43	30	bonded	bond	VERB
m-1071	43	31	with	with	ADP
m-1071	43	32	evenly	evenly	ADV
m-1071	43	33	spaced	spaced	ADJ
m-1071	43	34	oxygen	oxygen	NOUN
m-1071	43	35	atoms	atom	NOUN
m-1071	43	36	.	.	PUNCT
m-1071	44	1	the	the	DET
m-1071	44	2	resulting	result	VERB
m-1071	44	3	sio4	sio4	NOUN
m-1071	44	4	,	,	PUNCT
m-1071	44	5	a	a	DET
m-1071	44	6	silicate	silicate	ADJ
m-1071	44	7	tetrahedron	tetrahedron	NOUN
m-1071	44	8	that	that	PRON
m-1071	44	9	connects	connect	VERB
m-1071	44	10	with	with	ADP
m-1071	44	11	other	other	ADJ
m-1071	44	12	sio4	sio4	NOUN
m-1071	44	13	horizontally	horizontally	ADV
m-1071	44	14	,	,	PUNCT
m-1071	44	15	forms	form	VERB
m-1071	44	16	a	a	DET
m-1071	44	17	single	single	ADJ
m-1071	44	18	chain	chain	NOUN
m-1071	44	19	.	.	PUNCT
m-1071	45	1	similar	similar	ADJ
m-1071	45	2	to	to	ADP
m-1071	45	3	this	this	PRON
m-1071	45	4	,	,	PUNCT
m-1071	45	5	when	when	SCONJ
m-1071	45	6	two	two	NUM
m-1071	45	7	sio4	sio4	NOUN
m-1071	45	8	molecules	molecule	NOUN
m-1071	45	9	join	join	VERB
m-1071	45	10	corner	corner	NOUN
m-1071	45	11	to	to	PART
m-1071	45	12	corner	corner	VERB
m-1071	45	13	,	,	PUNCT
m-1071	45	14	each	each	DET
m-1071	45	15	one	one	NUM
m-1071	45	16	shares	share	VERB
m-1071	45	17	its	its	PRON
m-1071	45	18	o2	o2	ADJ
m-1071	45	19	atoms	atom	NOUN
m-1071	45	20	with	with	ADP
m-1071	45	21	the	the	DET
m-1071	45	22	other	other	ADJ
m-1071	45	23	,	,	PUNCT
m-1071	45	24	as	as	SCONJ
m-1071	45	25	shown	show	VERB
m-1071	45	26	in	in	ADP
m-1071	45	27	figure	figure	NOUN
m-1071	45	28	1	1	NUM
m-1071	45	29	.	.	PUNCT
m-1071	46	1	these	these	DET
m-1071	46	2	two	two	NUM
m-1071	46	3	molecules	molecule	NOUN
m-1071	46	4	of	of	ADP
m-1071	46	5	sio4	sio4	NOUN
m-1071	46	6	can	can	AUX
m-1071	46	7	be	be	AUX
m-1071	46	8	joined	join	VERB
m-1071	46	9	with	with	ADP
m-1071	46	10	two	two	NUM
m-1071	46	11	other	other	ADJ
m-1071	46	12	molecules	molecule	NOUN
m-1071	46	13	once	once	SCONJ
m-1071	46	14	this	this	DET
m-1071	46	15	sharing	sharing	NOUN
m-1071	46	16	process	process	NOUN
m-1071	46	17	is	be	AUX
m-1071	46	18	finished	finish	VERB
m-1071	46	19	.	.	PUNCT
m-1071	47	1	we	we	PRON
m-1071	47	2	now	now	ADV
m-1071	47	3	have	have	VERB
m-1071	47	4	a	a	DET
m-1071	47	5	silicate	silicate	ADJ
m-1071	47	6	chain	chain	NOUN
m-1071	47	7	,	,	PUNCT
m-1071	47	8	scqp	scqp	NOUN
m-1071	47	9	,	,	PUNCT
m-1071	47	10	where	where	SCONJ
m-1071	47	11	p	p	NOUN
m-1071	47	12	and	and	CCONJ
m-1071	47	13	q	q	NOUN
m-1071	47	14	stand	stand	NOUN
m-1071	47	15	for	for	ADP
m-1071	47	16	the	the	DET
m-1071	47	17	total	total	ADJ
m-1071	47	18	number	number	NOUN
m-1071	47	19	of	of	ADP
m-1071	47	20	sio4	sio4	NOUN
m-1071	47	21	atoms	atom	NOUN
m-1071	47	22	in	in	ADP
m-1071	47	23	one	one	NUM
m-1071	47	24	silicate	silicate	NOUN
m-1071	47	25	chain	chain	NOUN
m-1071	47	26	and	and	CCONJ
m-1071	47	27	the	the	DET
m-1071	47	28	number	number	NOUN
m-1071	47	29	of	of	ADP
m-1071	47	30	silicate	silicate	ADJ
m-1071	47	31	chains	chain	NOUN
m-1071	47	32	that	that	PRON
m-1071	47	33	were	be	AUX
m-1071	47	34	formed	form	VERB
m-1071	47	35	,	,	PUNCT
m-1071	47	36	respectively	respectively	ADV
m-1071	47	37	.	.	PUNCT
m-1071	48	1	the	the	DET
m-1071	48	2	pq	pq	NOUN
m-1071	48	3	number	number	NOUN
m-1071	48	4	of	of	ADP
m-1071	48	5	sio4	sio4	NOUN
m-1071	48	6	tetrahedrons	tetrahedron	NOUN
m-1071	48	7	used	use	VERB
m-1071	48	8	in	in	ADP
m-1071	48	9	the	the	DET
m-1071	48	10	chain	chain	NOUN
m-1071	48	11	of	of	ADP
m-1071	48	12	sio4	sio4	PROPN
m-1071	48	13	scqp	scqp	PROPN
m-1071	48	14	is	be	AUX
m-1071	48	15	shown	show	VERB
m-1071	48	16	in	in	ADP
m-1071	48	17	figure	figure	NOUN
m-1071	48	18	1	1	NUM
m-1071	48	19	.	.	PUNCT
m-1071	48	20	ijo	ijo	PROPN
m-1071	48	21	international	international	PROPN
m-1071	48	22	journal	journal	PROPN
m-1071	48	23	of	of	ADP
m-1071	48	24	mathematics	mathematics	PROPN
m-1071	48	25	(	(	PUNCT
m-1071	48	26	issn	issn	PROPN
m-1071	48	27	:	:	PUNCT
m-1071	48	28	2992	2992	NUM
m-1071	48	29	-	-	SYM
m-1071	48	30	4421	4421	NUM
m-1071	48	31	)	)	PUNCT
m-1071	48	32	ijo	ijo	PROPN
m-1071	48	33	journals	journal	NOUN
m-1071	48	34	volume	volume	NOUN
m-1071	48	35	08	08	NUM
m-1071	49	1	|	|	ADV
m-1071	49	2	issue	issue	VERB
m-1071	49	3	04	04	NUM
m-1071	50	1	|	|	CCONJ
m-1071	50	2	april	april	PROPN
m-1071	50	3	2025	2025	NUM
m-1071	50	4	|	|	ADV
m-1071	50	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	50	6	4	4	NUM
m-1071	50	7	1.1	1.1	NUM
m-1071	50	8	result	result	NOUN
m-1071	50	9	and	and	CCONJ
m-1071	50	10	discussion	discussion	NOUN
m-1071	50	11	here	here	ADV
m-1071	50	12	,	,	PUNCT
m-1071	50	13	we	we	PRON
m-1071	50	14	have	have	AUX
m-1071	50	15	observed	observe	VERB
m-1071	50	16	that	that	SCONJ
m-1071	50	17	there	there	PRON
m-1071	50	18	are	be	VERB
m-1071	50	19	three	three	NUM
m-1071	50	20	types	type	NOUN
m-1071	50	21	of	of	ADP
m-1071	50	22	atom	atom	NOUN
m-1071	50	23	bonds	bond	NOUN
m-1071	50	24	on	on	ADP
m-1071	50	25	the	the	DET
m-1071	50	26	bases	basis	NOUN
m-1071	50	27	of	of	ADP
m-1071	50	28	the	the	DET
m-1071	50	29	valency	valency	NOUN
m-1071	50	30	of	of	ADP
m-1071	50	31	each	each	DET
m-1071	50	32	atom	atom	NOUN
m-1071	50	33	of	of	ADP
m-1071	50	34	scqp	scqp	NOUN
m-1071	50	35	in	in	ADP
m-1071	50	36	a	a	DET
m-1071	50	37	chain	chain	NOUN
m-1071	50	38	of	of	ADP
m-1071	50	39	types	type	NOUN
m-1071	50	40	of	of	ADP
m-1071	50	41	atoms	atom	NOUN
m-1071	50	42	,	,	PUNCT
m-1071	50	43	vi	vi	PROPN
m-1071	50	44	and	and	CCONJ
m-1071	50	45	vj	vj	PROPN
m-1071	50	46	,	,	PUNCT
m-1071	50	47	with	with	ADP
m-1071	50	48	valencies	valency	NOUN
m-1071	50	49	of	of	ADP
m-1071	50	50	respectively	respectively	ADV
m-1071	50	51	.	.	PUNCT
m-1071	51	1	three	three	NUM
m-1071	51	2	different	different	ADJ
m-1071	51	3	types	type	NOUN
m-1071	51	4	of	of	ADP
m-1071	51	5	atom	atom	NOUN
m-1071	51	6	based	base	VERB
m-1071	51	7	on	on	ADP
m-1071	51	8	the	the	DET
m-1071	51	9	valencies	valency	NOUN
m-1071	51	10	(	(	PUNCT
m-1071	51	11	3	3	NUM
m-1071	51	12	and	and	CCONJ
m-1071	51	13	6	6	NUM
m-1071	51	14	)	)	PUNCT
m-1071	51	15	of	of	ADP
m-1071	51	16	atoms	atom	NOUN
m-1071	51	17	.	.	PUNCT
m-1071	52	1	table	table	NOUN
m-1071	52	2	1	1	NUM
m-1071	52	3	provides	provide	VERB
m-1071	52	4	the	the	DET
m-1071	52	5	division	division	NOUN
m-1071	52	6	of	of	ADP
m-1071	52	7	the	the	DET
m-1071	52	8	set	set	NOUN
m-1071	52	9	of	of	ADP
m-1071	52	10	atom	atom	NOUN
m-1071	52	11	bonds	bond	NOUN
m-1071	52	12	based	base	VERB
m-1071	52	13	on	on	ADP
m-1071	52	14	valency	valency	NOUN
m-1071	52	15	.	.	PUNCT
m-1071	53	1	table	table	NOUN
m-1071	53	2	1	1	NUM
m-1071	53	3	:	:	PUNCT
m-1071	53	4	atom	atom	NOUN
m-1071	53	5	type	type	NOUN
m-1071	53	6	of	of	ADP
m-1071	53	7	atom	atom	NOUN
m-1071	53	8	-	-	PUNCT
m-1071	53	9	bond	bond	NOUN
m-1071	53	10	number	number	NOUN
m-1071	53	11	of	of	ADP
m-1071	53	12	atom	atom	NOUN
m-1071	53	13	bonds	bond	NOUN
m-1071	53	14	theorem	theorem	VERB
m-1071	53	15	2.1	2.1	NUM
m-1071	53	16	.	.	PUNCT
m-1071	54	1	for	for	ADP
m-1071	54	2	p	p	PROPN
m-1071	54	3	>	>	PROPN
m-1071	54	4	1	1	NUM
m-1071	54	5	and	and	CCONJ
m-1071	54	6	p	p	X
m-1071	54	7	(	(	PUNCT
m-1071	54	8	3	3	NUM
m-1071	54	9	�	�	PROPN
m-1071	54	10	�	�	PROPN
m-1071	54	11	+	+	CCONJ
m-1071	54	12	3	3	NUM
m-1071	54	13	�	�	NOUN
m-1071	54	14	−	−	NUM
m-1071	54	15	4	4	NUM
m-1071	54	16	)	)	PUNCT
m-1071	54	17	�	�	PROPN
m-1071	54	18	�	�	PROPN
m-1071	54	19	�	�	PROPN
m-1071	54	20	+	+	CCONJ
m-1071	54	21	(	(	PUNCT
m-1071	54	22	3	3	NUM
m-1071	54	23	�	�	PROPN
m-1071	54	24	�	�	PROPN
m-1071	54	25	−	−	NUM
m-1071	54	26	6	6	NUM
m-1071	54	27	�	�	PROPN
m-1071	54	28	figure	figure	NOUN
m-1071	54	29	1	1	NUM
m-1071	54	30	:	:	SYM
m-1071	54	31	1	1	NUM
m-1071	54	32	e	e	NOUN
m-1071	54	33	,	,	PUNCT
m-1071	54	34	we	we	PRON
m-1071	54	35	have	have	AUX
m-1071	54	36	observed	observe	VERB
m-1071	54	37	that	that	SCONJ
m-1071	54	38	there	there	PRON
m-1071	54	39	are	be	VERB
m-1071	54	40	three	three	NUM
m-1071	54	41	types	type	NOUN
m-1071	54	42	of	of	ADP
m-1071	54	43	atom	atom	NOUN
m-1071	54	44	bonds	bond	NOUN
m-1071	54	45	on	on	ADP
m-1071	54	46	the	the	DET
m-1071	54	47	bases	basis	NOUN
m-1071	54	48	of	of	ADP
m-1071	54	49	the	the	PRON
m-1071	54	50	in	in	ADP
m-1071	54	51	a	a	DET
m-1071	54	52	chain	chain	NOUN
m-1071	54	53	of	of	ADP
m-1071	54	54	sio4	sio4	PROPN
m-1071	54	55	scqp	scqp	PROPN
m-1071	54	56	.	.	PUNCT
m-1071	55	1	as	as	ADP
m-1071	55	2	a	a	DET
m-1071	55	3	result	result	NOUN
m-1071	55	4	,	,	PUNCT
m-1071	55	5	there	there	PRON
m-1071	55	6	are	be	VERB
m-1071	55	7	two	two	NUM
m-1071	55	8	different	different	ADJ
m-1071	55	9	,	,	PUNCT
m-1071	55	10	with	with	ADP
m-1071	55	11	valencies	valency	NOUN
m-1071	55	12	of	of	ADP
m-1071	55	13	and	and	CCONJ
m-1071	55	14	dvi	dvi	PROPN
m-1071	55	15	=	=	PROPN
m-1071	55	16	3	3	NUM
m-1071	55	17	and	and	CCONJ
m-1071	55	18	three	three	NUM
m-1071	55	19	different	different	ADJ
m-1071	55	20	types	type	NOUN
m-1071	55	21	of	of	ADP
m-1071	55	22	atom	atom	NOUN
m-1071	55	23	-	-	PUNCT
m-1071	55	24	bonds	bond	NOUN
m-1071	55	25	(	(	PUNCT
m-1071	55	26	3	3	NUM
m-1071	55	27	∼	∼	NOUN
m-1071	55	28	3	3	NUM
m-1071	55	29	)	)	PUNCT
m-1071	55	30	,	,	PUNCT
m-1071	55	31	(	(	PUNCT
m-1071	55	32	3	3	NUM
m-1071	55	33	∼	∼	NOUN
m-1071	55	34	6	6	NUM
m-1071	55	35	)	)	PUNCT
m-1071	55	36	,	,	PUNCT
m-1071	55	37	and	and	CCONJ
m-1071	55	38	(	(	PUNCT
m-1071	55	39	6	6	NUM
m-1071	55	40	∼	∼	NOUN
m-1071	55	41	6	6	NUM
m-1071	55	42	)	)	PUNCT
m-1071	55	43	in	in	ADP
m-1071	55	44	sc	sc	PROPN
m-1071	55	45	based	base	VERB
m-1071	55	46	on	on	ADP
m-1071	55	47	the	the	DET
m-1071	55	48	valencies	valency	NOUN
m-1071	55	49	(	(	PUNCT
m-1071	55	50	3	3	NUM
m-1071	55	51	and	and	CCONJ
m-1071	55	52	6	6	NUM
m-1071	55	53	)	)	PUNCT
m-1071	55	54	of	of	ADP
m-1071	55	55	atoms	atom	NOUN
m-1071	55	56	.	.	PUNCT
m-1071	56	1	table	table	NOUN
m-1071	56	2	1	1	NUM
m-1071	56	3	provides	provide	VERB
m-1071	56	4	the	the	DET
m-1071	56	5	division	division	NOUN
m-1071	56	6	of	of	ADP
m-1071	56	7	the	the	DET
m-1071	56	8	set	set	NOUN
m-1071	56	9	of	of	ADP
m-1071	56	10	table	table	NOUN
m-1071	56	11	1	1	NUM
m-1071	56	12	:	:	PUNCT
m-1071	56	13	atom	atom	NOUN
m-1071	56	14	-	-	PUNCT
m-1071	56	15	bond	bond	NOUN
m-1071	56	16	partition	partition	NOUN
m-1071	56	17	of	of	ADP
m-1071	56	18	scqp	scqp	NOUN
m-1071	56	19	,	,	PUNCT
m-1071	56	20	for	for	ADP
m-1071	56	21	p	p	NOUN
m-1071	56	22	=	=	X
m-1071	56	23	q	q	PROPN
m-1071	56	24	3	3	NUM
m-1071	56	25	=	=	SYM
m-1071	56	26	de	de	X
m-1071	56	27	∼	∼	X
m-1071	56	28	df	df	NOUN
m-1071	56	29	=	=	SYM
m-1071	56	30	3	3	NUM
m-1071	56	31	3	3	NUM
m-1071	56	32	=	=	SYM
m-1071	56	33	de	de	X
m-1071	56	34	∼	∼	X
m-1071	56	35	df	df	NOUN
m-1071	56	36	=	=	NUM
m-1071	56	37	6	6	NUM
m-1071	56	38	6	6	NUM
m-1071	56	39	=	=	SYM
m-1071	56	40	de	de	X
m-1071	56	41	∼	∼	X
m-1071	56	42	d	d	NOUN
m-1071	56	43	3p	3p	NUM
m-1071	56	44	+	+	CCONJ
m-1071	56	45	2	2	NUM
m-1071	56	46	3(pq	3(pq	NUM
m-1071	56	47	+	+	CCONJ
m-1071	56	48	q	q	X
m-1071	56	49	)	)	PUNCT
m-1071	56	50	−	−	PROPN
m-1071	56	51	4	4	NUM
m-1071	56	52	3(pq	3(pq	NUM
m-1071	56	53	−	−	NOUN
m-1071	56	54	2q	2q	NOUN
m-1071	56	55	and	and	CCONJ
m-1071	56	56	p	p	NOUN
m-1071	56	57	=	=	ADJ
m-1071	56	58	q	q	X
m-1071	56	59	,	,	PUNCT
m-1071	56	60	the	the	DET
m-1071	56	61	harmonic	harmonic	ADJ
m-1071	56	62	polynomial	polynomial	ADJ
m-1071	56	63	ofscqp	ofscqp	NOUN
m-1071	56	64	,	,	PUNCT
m-1071	56	65	is	be	AUX
m-1071	56	66	(	(	PUNCT
m-1071	56	67	+	+	CCONJ
m-1071	56	68	2	2	X
m-1071	56	69	)	)	PUNCT
m-1071	56	70	�	�	PROPN
m-1071	56	71	�	�	PROPN
m-1071	56	72	�	�	PROPN
m-1071	56	73	e	e	PROPN
m-1071	56	74	,	,	PUNCT
m-1071	56	75	we	we	PRON
m-1071	56	76	have	have	AUX
m-1071	56	77	observed	observe	VERB
m-1071	56	78	that	that	SCONJ
m-1071	56	79	there	there	PRON
m-1071	56	80	are	be	VERB
m-1071	56	81	three	three	NUM
m-1071	56	82	types	type	NOUN
m-1071	56	83	of	of	ADP
m-1071	56	84	atom	atom	NOUN
m-1071	56	85	bonds	bond	NOUN
m-1071	56	86	on	on	ADP
m-1071	56	87	the	the	DET
m-1071	56	88	bases	basis	NOUN
m-1071	56	89	of	of	ADP
m-1071	56	90	the	the	PRON
m-1071	56	91	.	.	PUNCT
m-1071	57	1	as	as	ADP
m-1071	57	2	a	a	DET
m-1071	57	3	result	result	NOUN
m-1071	57	4	,	,	PUNCT
m-1071	57	5	there	there	PRON
m-1071	57	6	are	be	VERB
m-1071	57	7	two	two	NUM
m-1071	57	8	different	different	ADJ
m-1071	57	9	=	=	NOUN
m-1071	57	10	3	3	NUM
m-1071	57	11	and	and	CCONJ
m-1071	57	12	dvj	dvj	VERB
m-1071	57	13	=	=	SYM
m-1071	57	14	6	6	NUM
m-1071	57	15	,	,	PUNCT
m-1071	57	16	∼	∼	NOUN
m-1071	57	17	3	3	NUM
m-1071	57	18	)	)	PUNCT
m-1071	57	19	,	,	PUNCT
m-1071	57	20	(	(	PUNCT
m-1071	57	21	3	3	NUM
m-1071	57	22	∼	∼	NOUN
m-1071	57	23	6	6	NUM
m-1071	57	24	)	)	PUNCT
m-1071	57	25	,	,	PUNCT
m-1071	57	26	and	and	CCONJ
m-1071	57	27	(	(	PUNCT
m-1071	57	28	6	6	NUM
m-1071	57	29	∼	∼	NOUN
m-1071	57	30	6	6	NUM
m-1071	57	31	)	)	PUNCT
m-1071	57	32	in	in	ADP
m-1071	57	33	scqp	scqp	NOUN
m-1071	57	34	are	be	AUX
m-1071	57	35	based	base	VERB
m-1071	57	36	on	on	ADP
m-1071	57	37	the	the	DET
m-1071	57	38	valencies	valency	NOUN
m-1071	57	39	(	(	PUNCT
m-1071	57	40	3	3	NUM
m-1071	57	41	and	and	CCONJ
m-1071	57	42	6	6	NUM
m-1071	57	43	)	)	PUNCT
m-1071	57	44	of	of	ADP
m-1071	57	45	atoms	atom	NOUN
m-1071	57	46	.	.	PUNCT
m-1071	58	1	table	table	NOUN
m-1071	58	2	1	1	NUM
m-1071	58	3	provides	provide	VERB
m-1071	58	4	the	the	DET
m-1071	58	5	division	division	NOUN
m-1071	58	6	of	of	ADP
m-1071	58	7	the	the	DET
m-1071	58	8	set	set	NOUN
m-1071	58	9	of	of	ADP
m-1071	58	10	df	df	NOUN
m-1071	58	11	=	=	SYM
m-1071	58	12	6	6	NUM
m-1071	58	13	q	q	NOUN
m-1071	58	14	)	)	PUNCT
m-1071	58	15	+	+	CCONJ
m-1071	58	16	2	2	NUM
m-1071	58	17	(	(	PUNCT
m-1071	58	18	3	3	NUM
m-1071	58	19	�	�	NOUN
m-1071	58	20	+	+	CCONJ
m-1071	58	21	2	2	NUM
m-1071	58	22	)	)	PUNCT
m-1071	58	23	�	�	PROPN
m-1071	58	24	�	�	PROPN
m-1071	58	25	�	�	PROPN
m-1071	58	26	+	+	CCONJ
m-1071	58	27	ijo	ijo	PROPN
m-1071	58	28	international	international	ADJ
m-1071	58	29	journal	journal	PROPN
m-1071	58	30	of	of	ADP
m-1071	58	31	mathematics	mathematics	PROPN
m-1071	58	32	(	(	PUNCT
m-1071	58	33	issn	issn	PROPN
m-1071	58	34	:	:	PUNCT
m-1071	58	35	2992	2992	NUM
m-1071	58	36	-	-	SYM
m-1071	58	37	4421	4421	NUM
m-1071	58	38	)	)	PUNCT
m-1071	58	39	ijo	ijo	PROPN
m-1071	58	40	journals	journal	NOUN
m-1071	58	41	volume	volume	NOUN
m-1071	58	42	08	08	NUM
m-1071	59	1	|	|	ADV
m-1071	59	2	issue	issue	VERB
m-1071	59	3	04	04	NUM
m-1071	60	1	|	|	CCONJ
m-1071	60	2	april	april	PROPN
m-1071	60	3	2025	2025	NUM
m-1071	61	1	|	|	ADV
m-1071	61	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	61	3	5	5	NUM
m-1071	61	4	proof	proof	NOUN
m-1071	61	5	.	.	PUNCT
m-1071	62	1	using	use	VERB
m-1071	62	2	table”1	table”1	ADV
m-1071	62	3	”	"	PUNCT
m-1071	62	4	enter	enter	VERB
m-1071	62	5	the	the	DET
m-1071	62	6	following	follow	VERB
m-1071	62	7	formula	formula	NOUN
m-1071	62	8	harmonic	harmonic	ADJ
m-1071	62	9	polynomial	polynomial	ADJ
m-1071	62	10	(	(	PUNCT
m-1071	62	11	1	1	NUM
m-1071	62	12	)	)	PUNCT
m-1071	62	13	,	,	PUNCT
m-1071	62	14	we	we	PRON
m-1071	62	15	get	get	VERB
m-1071	62	16	�	�	PROPN
m-1071	62	17	�	�	PROPN
m-1071	62	18	sc	sc	PROPN
m-1071	62	19	�	�	PROPN
m-1071	62	20	�	�	PROPN
m-1071	62	21	,	,	PUNCT
m-1071	62	22	�	�	PROPN
m-1071	63	1	�	�	PROPN
m-1071	63	2	=	=	SYM
m-1071	63	3	�	�	PROPN
m-1071	63	4	�	�	PROPN
m-1071	63	5	�	�	PROPN
m-1071	63	6	�	�	PROPN
m-1071	63	7	�	�	PROPN
m-1071	63	8	�	�	PROPN
m-1071	63	9	�	�	PROPN
m-1071	63	10	�	�	PROPN
m-1071	63	11	�	�	PROPN
m-1071	63	12	�	�	PROPN
m-1071	63	13	~	~	SYM
m-1071	63	14	�	�	PROPN
m-1071	63	15	�	�	PROPN
m-1071	63	16	�	�	PROPN
m-1071	63	17	�	�	PROPN
m-1071	63	18	+	+	CCONJ
m-1071	63	19	�	�	PROPN
m-1071	63	20	�	�	PROPN
m-1071	63	21	�	�	PROPN
m-1071	63	22	�	�	PROPN
m-1071	63	23	�	�	PROPN
m-1071	63	24	�	�	PROPN
m-1071	63	25	�	�	PROPN
m-1071	63	26	�	�	PROPN
m-1071	63	27	�	�	PROPN
m-1071	63	28	�	�	PROPN
m-1071	63	29	~	~	SYM
m-1071	63	30	�	�	PROPN
m-1071	63	31	�	�	PROPN
m-1071	63	32	�	�	PROPN
m-1071	63	33	�	�	PROPN
m-1071	63	34	+	+	CCONJ
m-1071	63	35	�	�	PROPN
m-1071	63	36	�	�	PROPN
m-1071	63	37	�	�	PROPN
m-1071	63	38	�	�	PROPN
m-1071	63	39	�	�	PROPN
m-1071	63	40	�	�	PROPN
m-1071	63	41	�	�	PROPN
m-1071	63	42	�	�	PROPN
m-1071	63	43	�	�	PROPN
m-1071	63	44	�	�	PROPN
m-1071	63	45	~	~	SYM
m-1071	63	46	�	�	PROPN
m-1071	63	47	�	�	PROPN
m-1071	63	48	�	�	PROPN
m-1071	63	49	�	�	PROPN
m-1071	63	50	this	this	PRON
m-1071	63	51	gives	give	VERB
m-1071	63	52	�	�	PROPN
m-1071	63	53	�	�	PROPN
m-1071	63	54	sc	sc	PROPN
m-1071	63	55	�	�	PROPN
m-1071	63	56	�	�	PROPN
m-1071	63	57	,	,	PUNCT
m-1071	63	58	�	�	PROPN
m-1071	63	59	�	�	PROPN
m-1071	63	60	=	=	SYM
m-1071	63	61	(	(	PUNCT
m-1071	63	62	3	3	NUM
m-1071	63	63	�	�	NOUN
m-1071	63	64	+	+	CCONJ
m-1071	63	65	2	2	NUM
m-1071	63	66	)	)	PUNCT
m-1071	63	67	�	�	PROPN
m-1071	63	68	�	�	PROPN
m-1071	63	69	�	�	PROPN
m-1071	63	70	+	+	CCONJ
m-1071	63	71	(	(	PUNCT
m-1071	63	72	3	3	NUM
m-1071	63	73	�	�	PROPN
m-1071	63	74	�	�	PROPN
m-1071	63	75	+	+	CCONJ
m-1071	63	76	3	3	NUM
m-1071	63	77	�	�	NOUN
m-1071	63	78	−	−	NUM
m-1071	63	79	4	4	NUM
m-1071	63	80	)	)	PUNCT
m-1071	63	81	�	�	PROPN
m-1071	63	82	�	�	PROPN
m-1071	63	83	�	�	PROPN
m-1071	64	1	+	+	CCONJ
m-1071	64	2	(	(	PUNCT
m-1071	64	3	3	3	NUM
m-1071	64	4	�	�	PROPN
m-1071	64	5	�	�	PROPN
m-1071	64	6	−	−	NUM
m-1071	64	7	6	6	NUM
m-1071	64	8	�	�	PROPN
m-1071	64	9	+	+	CCONJ
m-1071	64	10	2	2	NUM
m-1071	64	11	)	)	PUNCT
m-1071	64	12	�	�	PROPN
m-1071	64	13	�	�	PROPN
m-1071	64	14	�	�	PROPN
m-1071	64	15	by	by	ADP
m-1071	64	16	taking	take	VERB
m-1071	64	17	the	the	DET
m-1071	64	18	first	first	ADJ
m-1071	64	19	derivative	derivative	NOUN
m-1071	64	20	of	of	ADP
m-1071	64	21	the	the	DET
m-1071	64	22	polynomial	polynomial	NOUN
m-1071	64	23	in	in	ADP
m-1071	64	24	theorem	theorem	ADJ
m-1071	64	25	2.1	2.1	NUM
m-1071	64	26	at	at	ADP
m-1071	64	27	y	y	PROPN
m-1071	64	28	=	=	SYM
m-1071	64	29	1	1	NUM
m-1071	64	30	,	,	PUNCT
m-1071	64	31	we	we	PRON
m-1071	64	32	get	get	VERB
m-1071	64	33	the	the	DET
m-1071	64	34	harmonic	harmonic	ADJ
m-1071	64	35	index	index	NOUN
m-1071	64	36	of	of	ADP
m-1071	64	37	silicate	silicate	PROPN
m-1071	64	38	network	network	PROPN
m-1071	64	39	sc	sc	PROPN
m-1071	64	40	�	�	PROPN
m-1071	64	41	�	�	PROPN
m-1071	64	42	as	as	SCONJ
m-1071	64	43	follows	follow	VERB
m-1071	64	44	:	:	PUNCT
m-1071	64	45	corollary	corollary	ADJ
m-1071	64	46	2.2	2.2	NUM
m-1071	64	47	.	.	PUNCT
m-1071	65	1	for	for	ADP
m-1071	65	2	p	p	PROPN
m-1071	65	3	>	>	PROPN
m-1071	65	4	1	1	NUM
m-1071	65	5	and	and	CCONJ
m-1071	65	6	p	p	NOUN
m-1071	65	7	=	=	ADJ
m-1071	65	8	q	q	X
m-1071	65	9	,	,	PUNCT
m-1071	65	10	the	the	DET
m-1071	65	11	harmonic	harmonic	ADJ
m-1071	65	12	index	index	NOUN
m-1071	65	13	of	of	ADP
m-1071	65	14	sc	sc	PROPN
m-1071	65	15	�	�	PROPN
m-1071	65	16	�	�	PROPN
m-1071	65	17	is	be	AUX
m-1071	65	18	�	�	PROPN
m-1071	65	19	�	�	PROPN
m-1071	65	20	�	�	PROPN
m-1071	65	21	�	�	PROPN
m-1071	65	22	�	�	PROPN
m-1071	65	23	�	�	PROPN
m-1071	65	24	�	�	PROPN
m-1071	65	25	�	�	PROPN
m-1071	65	26	�	�	PROPN
m-1071	65	27	�	�	PROPN
m-1071	65	28	�	�	PROPN
m-1071	65	29	�	�	PROPN
m-1071	65	30	theorem	theorem	VERB
m-1071	65	31	2.3	2.3	NUM
m-1071	65	32	.	.	PUNCT
m-1071	66	1	for	for	ADP
m-1071	66	2	p	p	PROPN
m-1071	66	3	>	>	PROPN
m-1071	66	4	1	1	NUM
m-1071	66	5	and	and	CCONJ
m-1071	66	6	p	p	NOUN
m-1071	66	7	=	=	ADJ
m-1071	66	8	q	q	NOUN
m-1071	66	9	,	,	PUNCT
m-1071	66	10	the	the	DET
m-1071	66	11	abs	ab	NOUN
m-1071	66	12	polynomial	polynomial	ADJ
m-1071	66	13	ofsc	ofsc	PROPN
m-1071	66	14	�	�	PROPN
m-1071	66	15	�	�	PROPN
m-1071	66	16	is	be	AUX
m-1071	66	17	(	(	PUNCT
m-1071	66	18	3	3	NUM
m-1071	66	19	�	�	NOUN
m-1071	66	20	+	+	CCONJ
m-1071	66	21	2	2	NUM
m-1071	66	22	)	)	PUNCT
m-1071	66	23	�	�	PROPN
m-1071	66	24	�	�	PROPN
m-1071	66	25	�	�	PROPN
m-1071	66	26	+	+	CCONJ
m-1071	66	27	3	3	NUM
m-1071	66	28	�	�	PROPN
m-1071	66	29	�	�	PROPN
m-1071	66	30	+	+	CCONJ
m-1071	66	31	3	3	NUM
m-1071	66	32	�	�	NOUN
m-1071	66	33	−	−	NUM
m-1071	66	34	4	4	NUM
m-1071	66	35	)	)	PUNCT
m-1071	66	36	�	�	PROPN
m-1071	66	37	�	�	PROPN
m-1071	66	38	�	�	PROPN
m-1071	66	39	�	�	PROPN
m-1071	66	40	+	+	CCONJ
m-1071	66	41	(	(	PUNCT
m-1071	66	42	3	3	NUM
m-1071	66	43	�	�	PROPN
m-1071	66	44	�	�	PROPN
m-1071	66	45	−	−	NUM
m-1071	66	46	6	6	NUM
m-1071	66	47	�	�	PROPN
m-1071	66	48	+	+	CCONJ
m-1071	66	49	2	2	NUM
m-1071	66	50	)	)	PUNCT
m-1071	66	51	�	�	PROPN
m-1071	66	52	�	�	PROPN
m-1071	66	53	�	�	PROPN
m-1071	66	54	�	�	PROPN
m-1071	66	55	proof	proof	NOUN
m-1071	66	56	.	.	PUNCT
m-1071	67	1	using	use	VERB
m-1071	67	2	table”1	table”1	ADV
m-1071	67	3	”	"	PUNCT
m-1071	67	4	enter	enter	VERB
m-1071	67	5	the	the	DET
m-1071	67	6	following	follow	VERB
m-1071	67	7	formula	formula	NOUN
m-1071	67	8	abc	abc	PROPN
m-1071	67	9	polynomial	polynomial	PROPN
m-1071	67	10	(	(	PUNCT
m-1071	67	11	2	2	NUM
m-1071	67	12	)	)	PUNCT
m-1071	67	13	,	,	PUNCT
m-1071	67	14	we	we	PRON
m-1071	67	15	get	get	VERB
m-1071	67	16	�	�	PROPN
m-1071	67	17	�	�	PROPN
m-1071	67	18	�	�	PROPN
m-1071	67	19	�	�	PROPN
m-1071	67	20	sc	sc	PROPN
m-1071	67	21	�	�	PROPN
m-1071	67	22	�	�	PROPN
m-1071	67	23	,	,	PUNCT
m-1071	67	24	�	�	PROPN
m-1071	68	1	�	�	PROPN
m-1071	68	2	=	=	SYM
m-1071	68	3	�	�	PROPN
m-1071	68	4	�	�	PROPN
m-1071	68	5	�	�	PROPN
m-1071	68	6	�	�	PROPN
m-1071	68	7	�	�	PROPN
m-1071	68	8	�	�	PROPN
m-1071	68	9	�	�	PROPN
m-1071	68	10	�	�	PROPN
m-1071	68	11	(	(	PUNCT
m-1071	68	12	�	�	PROPN
m-1071	68	13	)	)	PUNCT
m-1071	68	14	(	(	PUNCT
m-1071	68	15	�	�	PROPN
m-1071	68	16	)	)	PUNCT
m-1071	68	17	�	�	PROPN
m-1071	68	18	�	�	PROPN
m-1071	68	19	�	�	PROPN
m-1071	68	20	�	�	PROPN
m-1071	68	21	~	~	SYM
m-1071	68	22	�	�	PROPN
m-1071	68	23	�	�	PROPN
m-1071	68	24	�	�	PROPN
m-1071	68	25	�	�	PROPN
m-1071	68	26	+	+	CCONJ
m-1071	68	27	�	�	PROPN
m-1071	68	28	�	�	PROPN
m-1071	68	29	�	�	PROPN
m-1071	68	30	�	�	PROPN
m-1071	68	31	�	�	PROPN
m-1071	68	32	�	�	PROPN
m-1071	68	33	�	�	PROPN
m-1071	68	34	�	�	PROPN
m-1071	68	35	(	(	PUNCT
m-1071	68	36	�	�	PROPN
m-1071	68	37	)	)	PUNCT
m-1071	68	38	(	(	PUNCT
m-1071	68	39	�	�	PROPN
m-1071	68	40	)	)	PUNCT
m-1071	68	41	�	�	PROPN
m-1071	68	42	�	�	PROPN
m-1071	68	43	�	�	PROPN
m-1071	68	44	�	�	PROPN
m-1071	68	45	~	~	SYM
m-1071	68	46	�	�	PROPN
m-1071	68	47	�	�	PROPN
m-1071	68	48	�	�	PROPN
m-1071	68	49	�	�	PROPN
m-1071	68	50	+	+	CCONJ
m-1071	68	51	�	�	PROPN
m-1071	68	52	�	�	PROPN
m-1071	68	53	�	�	PROPN
m-1071	68	54	�	�	PROPN
m-1071	68	55	�	�	PROPN
m-1071	68	56	�	�	PROPN
m-1071	68	57	�	�	PROPN
m-1071	68	58	�	�	PROPN
m-1071	68	59	(	(	PUNCT
m-1071	68	60	�	�	PROPN
m-1071	68	61	)	)	PUNCT
m-1071	68	62	(	(	PUNCT
m-1071	68	63	�	�	PROPN
m-1071	68	64	)	)	PUNCT
m-1071	68	65	�	�	PROPN
m-1071	68	66	�	�	PROPN
m-1071	68	67	�	�	PROPN
m-1071	68	68	�	�	PROPN
m-1071	68	69	~	~	SYM
m-1071	68	70	�	�	PROPN
m-1071	68	71	�	�	PROPN
m-1071	68	72	�	�	PROPN
m-1071	68	73	�	�	PROPN
m-1071	68	74	this	this	PRON
m-1071	68	75	gives	give	VERB
m-1071	68	76	�	�	PROPN
m-1071	68	77	�	�	PROPN
m-1071	68	78	�	�	PROPN
m-1071	68	79	�	�	PROPN
m-1071	68	80	sc	sc	PROPN
m-1071	68	81	�	�	PROPN
m-1071	68	82	�	�	PROPN
m-1071	68	83	,	,	PUNCT
m-1071	68	84	�	�	PROPN
m-1071	68	85	�	�	PROPN
m-1071	68	86	=	=	SYM
m-1071	68	87	(	(	PUNCT
m-1071	68	88	3	3	NUM
m-1071	68	89	�	�	NOUN
m-1071	68	90	+	+	CCONJ
m-1071	68	91	2	2	NUM
m-1071	68	92	)	)	PUNCT
m-1071	68	93	�	�	PROPN
m-1071	68	94	�	�	PROPN
m-1071	68	95	�	�	PROPN
m-1071	68	96	+	+	CCONJ
m-1071	68	97	(	(	PUNCT
m-1071	68	98	3	3	NUM
m-1071	68	99	�	�	PROPN
m-1071	68	100	�	�	PROPN
m-1071	68	101	+	+	CCONJ
m-1071	68	102	3	3	NUM
m-1071	68	103	�	�	NOUN
m-1071	68	104	−	−	NUM
m-1071	68	105	4	4	NUM
m-1071	68	106	)	)	PUNCT
m-1071	68	107	�	�	PROPN
m-1071	68	108	�	�	PROPN
m-1071	68	109	�	�	PROPN
m-1071	68	110	�	�	PROPN
m-1071	68	111	+	+	CCONJ
m-1071	68	112	(	(	PUNCT
m-1071	68	113	3	3	NUM
m-1071	68	114	�	�	PROPN
m-1071	68	115	�	�	PROPN
m-1071	68	116	−	−	NUM
m-1071	68	117	6	6	NUM
m-1071	68	118	�	�	PROPN
m-1071	68	119	+	+	CCONJ
m-1071	68	120	2	2	NUM
m-1071	68	121	)	)	PUNCT
m-1071	68	122	�	�	PROPN
m-1071	68	123	�	�	PROPN
m-1071	68	124	�	�	PROPN
m-1071	68	125	�	�	PROPN
m-1071	68	126	by	by	ADP
m-1071	68	127	taking	take	VERB
m-1071	68	128	the	the	DET
m-1071	68	129	first	first	ADJ
m-1071	68	130	derivative	derivative	NOUN
m-1071	68	131	of	of	ADP
m-1071	68	132	the	the	DET
m-1071	68	133	polynomial	polynomial	NOUN
m-1071	68	134	in	in	ADP
m-1071	68	135	theorem	theorem	ADJ
m-1071	68	136	2.3	2.3	NUM
m-1071	68	137	at	at	ADP
m-1071	68	138	y	y	PROPN
m-1071	68	139	=	=	SYM
m-1071	68	140	1	1	NUM
m-1071	69	1	,	,	PUNCT
m-1071	69	2	we	we	PRON
m-1071	69	3	get	get	VERB
m-1071	69	4	the	the	DET
m-1071	69	5	abc	abc	PROPN
m-1071	69	6	index	index	NOUN
m-1071	69	7	of	of	ADP
m-1071	69	8	chain	chain	PROPN
m-1071	69	9	�	�	PROPN
m-1071	69	10	�	�	PROPN
m-1071	69	11	�	�	PROPN
m-1071	69	12	�	�	PROPN
m-1071	69	13	(	(	PUNCT
m-1071	69	14	sc	sc	PROPN
m-1071	69	15	�	�	PROPN
m-1071	69	16	�	�	PROPN
m-1071	69	17	)	)	PUNCT
m-1071	69	18	of	of	ADP
m-1071	69	19	as	as	SCONJ
m-1071	69	20	follows	follow	VERB
m-1071	69	21	:	:	PUNCT
m-1071	69	22	corollary	corollary	ADJ
m-1071	69	23	2.4	2.4	NUM
m-1071	69	24	.	.	PUNCT
m-1071	70	1	for	for	ADP
m-1071	70	2	p	p	PROPN
m-1071	70	3	>	>	PROPN
m-1071	70	4	1	1	NUM
m-1071	70	5	and	and	CCONJ
m-1071	70	6	p	p	NOUN
m-1071	70	7	=	=	ADJ
m-1071	70	8	q	q	NOUN
m-1071	70	9	,	,	PUNCT
m-1071	70	10	the	the	DET
m-1071	70	11	abc	abc	PROPN
m-1071	70	12	index	index	NOUN
m-1071	70	13	of	of	ADP
m-1071	70	14	�	�	PROPN
m-1071	70	15	�	�	PROPN
m-1071	70	16	�	�	PROPN
m-1071	70	17	�	�	PROPN
m-1071	70	18	�	�	PROPN
m-1071	70	19	�	�	PROPN
m-1071	70	20	�	�	PROPN
m-1071	70	21	�	�	PROPN
m-1071	70	22	�	�	PROPN
m-1071	70	23	�	�	PROPN
m-1071	70	24	�	�	PROPN
m-1071	70	25	�	�	PROPN
m-1071	70	26	theorem	theorem	VERB
m-1071	70	27	2.5	2.5	NUM
m-1071	70	28	.	.	PUNCT
m-1071	71	1	for	for	ADP
m-1071	71	2	p	p	PROPN
m-1071	71	3	>	>	PROPN
m-1071	71	4	1	1	NUM
m-1071	71	5	and	and	CCONJ
m-1071	71	6	p	p	NOUN
m-1071	71	7	=	=	ADJ
m-1071	71	8	q	q	NOUN
m-1071	71	9	,	,	PUNCT
m-1071	71	10	the	the	DET
m-1071	71	11	forgotten	forget	VERB
m-1071	71	12	topological	topological	ADJ
m-1071	71	13	polynomial	polynomial	NOUN
m-1071	71	14	of	of	ADP
m-1071	71	15	sc	sc	PROPN
m-1071	71	16	�	�	PROPN
m-1071	71	17	�	�	PROPN
m-1071	71	18	is	be	AUX
m-1071	71	19	(	(	PUNCT
m-1071	71	20	3p	3p	NUM
m-1071	71	21	+	+	CCONJ
m-1071	71	22	2	2	NUM
m-1071	71	23	)	)	PUNCT
m-1071	71	24	y18	y18	NOUN
m-1071	71	25	+	+	CCONJ
m-1071	71	26	(	(	PUNCT
m-1071	71	27	3p2	3p2	NUM
m-1071	71	28	+	+	NUM
m-1071	71	29	3p	3p	NUM
m-1071	71	30	−	−	NOUN
m-1071	71	31	4	4	NUM
m-1071	71	32	)	)	PUNCT
m-1071	71	33	y45	y45	NOUN
m-1071	72	1	+	+	CCONJ
m-1071	72	2	(	(	PUNCT
m-1071	72	3	3p2	3p2	NUM
m-1071	72	4	−	−	PROPN
m-1071	72	5	6p	6p	NUM
m-1071	72	6	+	+	CCONJ
m-1071	72	7	2	2	NUM
m-1071	72	8	)	)	PUNCT
m-1071	72	9	y72	y72	NOUN
m-1071	72	10	.	.	PUNCT
m-1071	73	1	ijo	ijo	PROPN
m-1071	73	2	international	international	PROPN
m-1071	73	3	journal	journal	PROPN
m-1071	73	4	of	of	ADP
m-1071	73	5	mathematics	mathematics	PROPN
m-1071	73	6	(	(	PUNCT
m-1071	73	7	issn	issn	PROPN
m-1071	73	8	:	:	PUNCT
m-1071	73	9	2992	2992	NUM
m-1071	73	10	-	-	SYM
m-1071	73	11	4421	4421	NUM
m-1071	73	12	)	)	PUNCT
m-1071	73	13	ijo	ijo	PROPN
m-1071	73	14	journals	journal	NOUN
m-1071	73	15	volume	volume	NOUN
m-1071	73	16	08	08	NUM
m-1071	74	1	|	|	ADV
m-1071	74	2	issue	issue	VERB
m-1071	74	3	04	04	NUM
m-1071	75	1	|	|	CCONJ
m-1071	75	2	april	april	PROPN
m-1071	75	3	2025	2025	NUM
m-1071	76	1	|	|	ADV
m-1071	76	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	76	3	6	6	NUM
m-1071	76	4	proof	proof	NOUN
m-1071	76	5	.	.	PUNCT
m-1071	77	1	using	use	VERB
m-1071	77	2	table”1	table”1	ADV
m-1071	77	3	”	"	PUNCT
m-1071	77	4	enter	enter	VERB
m-1071	77	5	the	the	DET
m-1071	77	6	following	follow	VERB
m-1071	77	7	formula	formula	NOUN
m-1071	77	8	forgotten	forget	VERB
m-1071	77	9	topological	topological	ADJ
m-1071	77	10	polynomial	polynomial	NOUN
m-1071	77	11	(	(	PUNCT
m-1071	77	12	3	3	NUM
m-1071	77	13	)	)	PUNCT
m-1071	77	14	,	,	PUNCT
m-1071	77	15	weget	weget	PROPN
m-1071	77	16	�	�	PROPN
m-1071	77	17	�	�	PROPN
m-1071	77	18	sc	sc	PROPN
m-1071	77	19	�	�	PROPN
m-1071	77	20	�	�	PROPN
m-1071	77	21	,	,	PUNCT
m-1071	77	22	�	�	PROPN
m-1071	77	23	�	�	PROPN
m-1071	77	24	=	=	SYM
m-1071	77	25	�	�	PROPN
m-1071	77	26	�	�	PROPN
m-1071	77	27	[	[	SYM
m-1071	77	28	�	�	PROPN
m-1071	77	29	�	�	PROPN
m-1071	77	30	�	�	PROPN
m-1071	77	31	�	�	PROPN
m-1071	77	32	�	�	PROPN
m-1071	77	33	]	]	PUNCT
m-1071	77	34	�	�	PROPN
m-1071	77	35	�	�	PROPN
m-1071	77	36	�	�	PROPN
m-1071	77	37	�	�	PROPN
m-1071	77	38	~	~	SYM
m-1071	77	39	�	�	PROPN
m-1071	77	40	�	�	PROPN
m-1071	77	41	�	�	PROPN
m-1071	77	42	�	�	PROPN
m-1071	77	43	+	+	CCONJ
m-1071	77	44	�	�	PROPN
m-1071	77	45	�	�	PROPN
m-1071	77	46	[	[	SYM
m-1071	77	47	�	�	PROPN
m-1071	77	48	�	�	PROPN
m-1071	77	49	�	�	PROPN
m-1071	77	50	�	�	PROPN
m-1071	77	51	�	�	PROPN
m-1071	77	52	]	]	PUNCT
m-1071	77	53	�	�	PROPN
m-1071	77	54	�	�	PROPN
m-1071	77	55	�	�	PROPN
m-1071	77	56	�	�	PROPN
m-1071	77	57	~	~	SYM
m-1071	77	58	�	�	PROPN
m-1071	77	59	�	�	PROPN
m-1071	77	60	�	�	PROPN
m-1071	77	61	�	�	PROPN
m-1071	77	62	+	+	CCONJ
m-1071	77	63	�	�	PROPN
m-1071	77	64	�	�	PROPN
m-1071	77	65	[	[	SYM
m-1071	77	66	�	�	PROPN
m-1071	77	67	�	�	PROPN
m-1071	77	68	�	�	PROPN
m-1071	77	69	�	�	PROPN
m-1071	77	70	�	�	PROPN
m-1071	77	71	]	]	PUNCT
m-1071	77	72	�	�	PROPN
m-1071	77	73	�	�	PROPN
m-1071	77	74	�	�	PROPN
m-1071	77	75	�	�	PROPN
m-1071	77	76	~	~	SYM
m-1071	77	77	�	�	PROPN
m-1071	77	78	�	�	PROPN
m-1071	77	79	�	�	PROPN
m-1071	77	80	�	�	PROPN
m-1071	77	81	this	this	PRON
m-1071	77	82	gives	give	VERB
m-1071	77	83	�	�	PROPN
m-1071	77	84	�	�	PROPN
m-1071	77	85	sc	sc	PROPN
m-1071	77	86	�	�	PROPN
m-1071	77	87	�	�	PROPN
m-1071	77	88	,	,	PUNCT
m-1071	77	89	�	�	PROPN
m-1071	77	90	�	�	PROPN
m-1071	77	91	=	=	SYM
m-1071	77	92	(	(	PUNCT
m-1071	77	93	3	3	NUM
m-1071	77	94	�	�	NOUN
m-1071	77	95	+	+	CCONJ
m-1071	77	96	2	2	NUM
m-1071	77	97	)	)	PUNCT
m-1071	77	98	�	�	PROPN
m-1071	77	99	�	�	PROPN
m-1071	77	100	�	�	PROPN
m-1071	77	101	+	+	CCONJ
m-1071	77	102	(	(	PUNCT
m-1071	77	103	3	3	NUM
m-1071	77	104	�	�	PROPN
m-1071	77	105	�	�	PROPN
m-1071	77	106	+	+	CCONJ
m-1071	77	107	3	3	NUM
m-1071	77	108	�	�	NOUN
m-1071	77	109	−	−	NUM
m-1071	77	110	4	4	NUM
m-1071	77	111	)	)	PUNCT
m-1071	77	112	�	�	PROPN
m-1071	77	113	�	�	PROPN
m-1071	77	114	�	�	PROPN
m-1071	77	115	+	+	CCONJ
m-1071	77	116	(	(	PUNCT
m-1071	77	117	3	3	NUM
m-1071	77	118	�	�	PROPN
m-1071	77	119	�	�	PROPN
m-1071	77	120	−	−	NUM
m-1071	77	121	6	6	NUM
m-1071	77	122	�	�	PROPN
m-1071	77	123	+	+	CCONJ
m-1071	77	124	2	2	NUM
m-1071	77	125	)	)	PUNCT
m-1071	77	126	�	�	PROPN
m-1071	77	127	�	�	PROPN
m-1071	77	128	�	�	PROPN
m-1071	77	129	by	by	ADP
m-1071	77	130	taking	take	VERB
m-1071	77	131	the	the	DET
m-1071	77	132	first	first	ADJ
m-1071	77	133	derivative	derivative	NOUN
m-1071	77	134	of	of	ADP
m-1071	77	135	the	the	DET
m-1071	77	136	polynomial	polynomial	NOUN
m-1071	77	137	in	in	ADP
m-1071	77	138	theorem	theorem	ADJ
m-1071	77	139	2.5	2.5	NUM
m-1071	77	140	at	at	ADP
m-1071	77	141	y	y	PROPN
m-1071	77	142	=	=	SYM
m-1071	77	143	1	1	NUM
m-1071	77	144	,	,	PUNCT
m-1071	77	145	we	we	PRON
m-1071	77	146	get	get	VERB
m-1071	77	147	the	the	DET
m-1071	77	148	forgotten	forget	VERB
m-1071	77	149	index	index	NOUN
m-1071	77	150	of	of	ADP
m-1071	77	151	chain	chain	NOUN
m-1071	77	152	of	of	ADP
m-1071	77	153	sio4	sio4	NOUN
m-1071	77	154	(	(	PUNCT
m-1071	77	155	scpp	scpp	NOUN
m-1071	77	156	)	)	PUNCT
m-1071	77	157	as	as	SCONJ
m-1071	77	158	follows	follow	VERB
m-1071	77	159	:	:	PUNCT
m-1071	77	160	corollary	corollary	ADJ
m-1071	77	161	2.6	2.6	NUM
m-1071	77	162	.	.	PUNCT
m-1071	78	1	for	for	ADP
m-1071	78	2	p	p	PROPN
m-1071	78	3	>	>	PROPN
m-1071	78	4	1	1	NUM
m-1071	78	5	and	and	CCONJ
m-1071	78	6	p	p	NOUN
m-1071	78	7	=	=	ADJ
m-1071	78	8	q	q	NOUN
m-1071	78	9	,	,	PUNCT
m-1071	78	10	the	the	DET
m-1071	78	11	forgotten	forget	VERB
m-1071	78	12	topological	topological	ADJ
m-1071	78	13	index	index	NOUN
m-1071	78	14	of	of	ADP
m-1071	78	15	sc	sc	PROPN
m-1071	78	16	�	�	PROPN
m-1071	78	17	�	�	PROPN
m-1071	78	18	is	be	AUX
m-1071	78	19	351p2	351p2	NUM
m-1071	78	20	−243p	−243p	NOUN
m-1071	78	21	.	.	PUNCT
m-1071	79	1	theorem	theorem	PROPN
m-1071	79	2	2.7	2.7	NUM
m-1071	79	3	.	.	PUNCT
m-1071	80	1	for	for	ADP
m-1071	80	2	p	p	PROPN
m-1071	80	3	>	>	PROPN
m-1071	80	4	1	1	NUM
m-1071	80	5	and	and	CCONJ
m-1071	80	6	p	p	NOUN
m-1071	80	7	=	=	ADJ
m-1071	80	8	q	q	NOUN
m-1071	80	9	,	,	PUNCT
m-1071	80	10	the	the	DET
m-1071	80	11	geometric	geometric	ADJ
m-1071	80	12	arithmetic	arithmetic	ADJ
m-1071	80	13	polynomial	polynomial	NOUN
m-1071	80	14	of	of	ADP
m-1071	80	15	sc	sc	PROPN
m-1071	80	16	�	�	PROPN
m-1071	80	17	�	�	PROPN
m-1071	80	18	(	(	PUNCT
m-1071	80	19	3	3	NUM
m-1071	80	20	�	�	NOUN
m-1071	80	21	+	+	CCONJ
m-1071	80	22	2	2	NUM
m-1071	80	23	)	)	PUNCT
m-1071	80	24	�	�	PROPN
m-1071	80	25	√	√	NUM
m-1071	80	26	�	�	PROPN
m-1071	80	27	�	�	PROPN
m-1071	80	28	+	+	CCONJ
m-1071	80	29	(	(	PUNCT
m-1071	80	30	3	3	NUM
m-1071	80	31	�	�	PROPN
m-1071	80	32	�	�	PROPN
m-1071	80	33	+	+	CCONJ
m-1071	80	34	3	3	NUM
m-1071	80	35	�	�	NOUN
m-1071	80	36	−	−	NUM
m-1071	80	37	4	4	NUM
m-1071	80	38	)	)	PUNCT
m-1071	80	39	�	�	PROPN
m-1071	80	40	�	�	PROPN
m-1071	80	41	�	�	PROPN
m-1071	80	42	+	+	CCONJ
m-1071	80	43	(	(	PUNCT
m-1071	80	44	3	3	NUM
m-1071	80	45	�	�	PROPN
m-1071	80	46	�	�	PROPN
m-1071	80	47	−	−	NUM
m-1071	80	48	6	6	NUM
m-1071	80	49	�	�	PROPN
m-1071	80	50	+	+	CCONJ
m-1071	80	51	2	2	NUM
m-1071	80	52	)	)	PUNCT
m-1071	80	53	�	�	PROPN
m-1071	80	54	�	�	PROPN
m-1071	80	55	√	√	NUM
m-1071	80	56	�	�	PROPN
m-1071	80	57	proof	proof	NOUN
m-1071	80	58	.	.	PUNCT
m-1071	81	1	using	use	VERB
m-1071	81	2	table”1	table”1	ADV
m-1071	81	3	”	"	PUNCT
m-1071	81	4	enter	enter	VERB
m-1071	81	5	the	the	DET
m-1071	81	6	following	follow	VERB
m-1071	81	7	formula	formula	NOUN
m-1071	81	8	geometric	geometric	ADJ
m-1071	81	9	arithmetic	arithmetic	ADJ
m-1071	81	10	polynomial	polynomial	NOUN
m-1071	81	11	(	(	PUNCT
m-1071	81	12	4	4	NUM
m-1071	81	13	)	)	PUNCT
m-1071	81	14	,	,	PUNCT
m-1071	81	15	we	we	PRON
m-1071	81	16	get	get	VERB
m-1071	81	17	�	�	PROPN
m-1071	81	18	�	�	PROPN
m-1071	81	19	�	�	PROPN
m-1071	81	20	sc	sc	PROPN
m-1071	81	21	�	�	PROPN
m-1071	81	22	�	�	PROPN
m-1071	81	23	,	,	PUNCT
m-1071	81	24	�	�	PROPN
m-1071	81	25	�	�	PROPN
m-1071	81	26	=	=	SYM
m-1071	81	27	�	�	PROPN
m-1071	81	28	�	�	PROPN
m-1071	81	29	√	√	NUM
m-1071	81	30	�	�	PROPN
m-1071	81	31	�	�	PROPN
m-1071	81	32	�	�	PROPN
m-1071	81	33	�	�	PROPN
m-1071	81	34	�	�	PROPN
m-1071	81	35	�	�	PROPN
m-1071	81	36	�	�	PROPN
m-1071	81	37	�	�	PROPN
m-1071	81	38	�	�	PROPN
m-1071	81	39	�	�	PROPN
m-1071	81	40	�	�	PROPN
m-1071	81	41	~	~	SYM
m-1071	81	42	�	�	PROPN
m-1071	81	43	�	�	PROPN
m-1071	81	44	�	�	PROPN
m-1071	81	45	�	�	PROPN
m-1071	81	46	+	+	CCONJ
m-1071	81	47	�	�	PROPN
m-1071	81	48	�	�	PROPN
m-1071	81	49	√	√	NUM
m-1071	81	50	�	�	PROPN
m-1071	81	51	�	�	PROPN
m-1071	81	52	�	�	PROPN
m-1071	81	53	�	�	PROPN
m-1071	81	54	�	�	PROPN
m-1071	81	55	�	�	PROPN
m-1071	81	56	�	�	PROPN
m-1071	81	57	�	�	PROPN
m-1071	81	58	�	�	PROPN
m-1071	81	59	~	~	SYM
m-1071	81	60	�	�	PROPN
m-1071	81	61	+	+	CCONJ
m-1071	81	62	�	�	PROPN
m-1071	81	63	�	�	PROPN
m-1071	81	64	√	√	NUM
m-1071	81	65	�	�	PROPN
m-1071	81	66	�	�	PROPN
m-1071	81	67	�	�	PROPN
m-1071	81	68	�	�	PROPN
m-1071	81	69	�	�	PROPN
m-1071	81	70	�	�	PROPN
m-1071	81	71	�	�	PROPN
m-1071	81	72	�	�	PROPN
m-1071	81	73	�	�	PROPN
m-1071	81	74	�	�	PROPN
m-1071	81	75	�	�	PROPN
m-1071	81	76	~	~	SYM
m-1071	81	77	�	�	PROPN
m-1071	81	78	�	�	PROPN
m-1071	81	79	�	�	PROPN
m-1071	81	80	�	�	PROPN
m-1071	81	81	this	this	PRON
m-1071	81	82	gives	give	VERB
m-1071	81	83	�	�	PROPN
m-1071	81	84	�	�	PROPN
m-1071	81	85	�	�	PROPN
m-1071	81	86	sc	sc	PROPN
m-1071	81	87	�	�	PROPN
m-1071	81	88	�	�	PROPN
m-1071	81	89	,	,	PUNCT
m-1071	81	90	�	�	PROPN
m-1071	81	91	�	�	PROPN
m-1071	81	92	=	=	SYM
m-1071	81	93	(	(	PUNCT
m-1071	81	94	3	3	NUM
m-1071	81	95	�	�	NOUN
m-1071	81	96	+	+	CCONJ
m-1071	81	97	2	2	NUM
m-1071	81	98	)	)	PUNCT
m-1071	81	99	�	�	PROPN
m-1071	81	100	√	√	NUM
m-1071	81	101	�	�	PROPN
m-1071	81	102	�	�	PROPN
m-1071	81	103	+	+	CCONJ
m-1071	81	104	(	(	PUNCT
m-1071	81	105	3	3	NUM
m-1071	81	106	�	�	PROPN
m-1071	81	107	�	�	PROPN
m-1071	81	108	+	+	CCONJ
m-1071	81	109	3	3	NUM
m-1071	81	110	�	�	NOUN
m-1071	81	111	−	−	NUM
m-1071	81	112	4	4	NUM
m-1071	81	113	)	)	PUNCT
m-1071	81	114	�	�	PROPN
m-1071	81	115	�	�	PROPN
m-1071	81	116	�	�	PROPN
m-1071	81	117	+	+	CCONJ
m-1071	81	118	(	(	PUNCT
m-1071	81	119	3	3	NUM
m-1071	81	120	�	�	PROPN
m-1071	81	121	�	�	PROPN
m-1071	81	122	−	−	NUM
m-1071	81	123	6	6	NUM
m-1071	81	124	�	�	PROPN
m-1071	81	125	+	+	CCONJ
m-1071	81	126	2	2	NUM
m-1071	81	127	)	)	PUNCT
m-1071	81	128	�	�	PROPN
m-1071	81	129	�	�	PROPN
m-1071	81	130	√	√	NOUN
m-1071	81	131	�	�	PROPN
m-1071	81	132	by	by	ADP
m-1071	81	133	taking	take	VERB
m-1071	81	134	the	the	DET
m-1071	81	135	first	first	ADJ
m-1071	81	136	derivative	derivative	NOUN
m-1071	81	137	of	of	ADP
m-1071	81	138	the	the	DET
m-1071	81	139	polynomial	polynomial	NOUN
m-1071	81	140	in	in	ADP
m-1071	81	141	theorem	theorem	ADJ
m-1071	81	142	2.7	2.7	NUM
m-1071	81	143	at	at	ADP
m-1071	81	144	y	y	PROPN
m-1071	81	145	=	=	SYM
m-1071	81	146	1	1	NUM
m-1071	81	147	,	,	PUNCT
m-1071	81	148	we	we	PRON
m-1071	81	149	get	get	VERB
m-1071	81	150	the	the	DET
m-1071	81	151	geometric	geometric	ADJ
m-1071	81	152	arithmetic	arithmetic	ADJ
m-1071	81	153	index	index	NOUN
m-1071	81	154	of	of	ADP
m-1071	81	155	chain	chain	NOUN
m-1071	81	156	of	of	ADP
m-1071	81	157	sio4	sio4	PROPN
m-1071	81	158	(	(	PUNCT
m-1071	81	159	sc	sc	PROPN
m-1071	81	160	�	�	PROPN
m-1071	81	161	�	�	PROPN
m-1071	81	162	)	)	PUNCT
m-1071	81	163	as	as	SCONJ
m-1071	81	164	follows	follow	VERB
m-1071	81	165	:	:	PUNCT
m-1071	81	166	corollary	corollary	ADJ
m-1071	81	167	2.8	2.8	NUM
m-1071	81	168	.	.	PUNCT
m-1071	82	1	for	for	ADP
m-1071	82	2	p	p	PROPN
m-1071	82	3	>	>	PROPN
m-1071	82	4	1	1	NUM
m-1071	82	5	and	and	CCONJ
m-1071	82	6	p	p	NOUN
m-1071	82	7	=	=	ADJ
m-1071	82	8	q	q	NOUN
m-1071	82	9	,	,	PUNCT
m-1071	82	10	the	the	DET
m-1071	82	11	geometric	geometric	ADJ
m-1071	82	12	arithmetic	arithmetic	ADJ
m-1071	82	13	index	index	NOUN
m-1071	82	14	ofsc	ofsc	PROPN
m-1071	82	15	�	�	PROPN
m-1071	82	16	�	�	PROPN
m-1071	82	17	�	�	PROPN
m-1071	82	18	�	�	PROPN
m-1071	82	19	(	(	PUNCT
m-1071	82	20	3	3	NUM
m-1071	82	21	�	�	NOUN
m-1071	82	22	+	+	CCONJ
m-1071	82	23	2	2	NUM
m-1071	82	24	)	)	PUNCT
m-1071	82	25	�	�	PROPN
m-1071	82	26	√	√	NUM
m-1071	82	27	�	�	PROPN
m-1071	82	28	�	�	PROPN
m-1071	82	29	+	+	CCONJ
m-1071	82	30	(	(	PUNCT
m-1071	82	31	3	3	NUM
m-1071	82	32	�	�	PROPN
m-1071	82	33	�	�	PROPN
m-1071	82	34	+	+	CCONJ
m-1071	82	35	3	3	NUM
m-1071	82	36	�	�	NOUN
m-1071	82	37	−	−	NUM
m-1071	82	38	4	4	NUM
m-1071	82	39	)	)	PUNCT
m-1071	82	40	�	�	PROPN
m-1071	82	41	�	�	PROPN
m-1071	82	42	�	�	PROPN
m-1071	82	43	+	+	CCONJ
m-1071	82	44	(	(	PUNCT
m-1071	82	45	3	3	NUM
m-1071	82	46	�	�	PROPN
m-1071	82	47	�	�	PROPN
m-1071	82	48	−	−	NUM
m-1071	82	49	6	6	NUM
m-1071	82	50	�	�	PROPN
m-1071	82	51	+	+	CCONJ
m-1071	82	52	2	2	NUM
m-1071	82	53	)	)	PUNCT
m-1071	82	54	�	�	PROPN
m-1071	82	55	�	�	PROPN
m-1071	82	56	√	√	NUM
m-1071	82	57	�	�	PROPN
m-1071	82	58	theorem	theorem	VERB
m-1071	82	59	2.9	2.9	NUM
m-1071	82	60	.	.	PUNCT
m-1071	83	1	for	for	ADP
m-1071	83	2	p	p	PROPN
m-1071	83	3	>	>	PROPN
m-1071	83	4	1	1	NUM
m-1071	83	5	and	and	CCONJ
m-1071	83	6	p	p	NOUN
m-1071	83	7	=	=	ADJ
m-1071	83	8	q	q	NOUN
m-1071	83	9	,	,	PUNCT
m-1071	83	10	the	the	DET
m-1071	83	11	randic	randic	ADJ
m-1071	83	12	polynomial	polynomial	NOUN
m-1071	83	13	of	of	ADP
m-1071	83	14	sc	sc	PROPN
m-1071	83	15	�	�	PROPN
m-1071	83	16	�	�	PROPN
m-1071	83	17	�	�	PROPN
m-1071	83	18	�	�	PROPN
m-1071	83	19	(	(	PUNCT
m-1071	83	20	3	3	NUM
m-1071	83	21	�	�	NOUN
m-1071	83	22	+	+	CCONJ
m-1071	83	23	2	2	NUM
m-1071	83	24	)	)	PUNCT
m-1071	83	25	�	�	PROPN
m-1071	83	26	�	�	PROPN
m-1071	83	27	�	�	PROPN
m-1071	83	28	+	+	CCONJ
m-1071	83	29	(	(	PUNCT
m-1071	83	30	3	3	NUM
m-1071	83	31	�	�	PROPN
m-1071	83	32	�	�	PROPN
m-1071	83	33	+	+	CCONJ
m-1071	83	34	3	3	NUM
m-1071	83	35	�	�	NOUN
m-1071	83	36	−	−	NUM
m-1071	83	37	4	4	NUM
m-1071	83	38	)	)	PUNCT
m-1071	83	39	�	�	PROPN
m-1071	83	40	�	�	PROPN
m-1071	83	41	√	√	NUM
m-1071	83	42	�	�	PROPN
m-1071	83	43	�	�	PROPN
m-1071	83	44	+	+	CCONJ
m-1071	83	45	(	(	PUNCT
m-1071	83	46	3	3	NUM
m-1071	83	47	�	�	PROPN
m-1071	83	48	�	�	PROPN
m-1071	83	49	−	−	NUM
m-1071	83	50	6	6	NUM
m-1071	83	51	�	�	PROPN
m-1071	83	52	+	+	CCONJ
m-1071	83	53	2	2	NUM
m-1071	83	54	)	)	PUNCT
m-1071	83	55	�	�	PROPN
m-1071	83	56	�	�	PROPN
m-1071	83	57	�	�	PROPN
m-1071	83	58	ijo	ijo	PROPN
m-1071	83	59	international	international	PROPN
m-1071	83	60	journal	journal	PROPN
m-1071	83	61	of	of	ADP
m-1071	83	62	mathematics	mathematics	PROPN
m-1071	83	63	(	(	PUNCT
m-1071	83	64	issn	issn	PROPN
m-1071	83	65	:	:	PUNCT
m-1071	83	66	2992	2992	NUM
m-1071	83	67	-	-	SYM
m-1071	83	68	4421	4421	NUM
m-1071	83	69	)	)	PUNCT
m-1071	83	70	ijo	ijo	PROPN
m-1071	83	71	journals	journal	NOUN
m-1071	83	72	volume	volume	NOUN
m-1071	83	73	08	08	NUM
m-1071	83	74	|	|	ADV
m-1071	83	75	issue	issue	VERB
m-1071	83	76	04	04	NUM
m-1071	84	1	|	|	CCONJ
m-1071	84	2	april	april	PROPN
m-1071	84	3	2025	2025	NUM
m-1071	85	1	|	|	ADV
m-1071	85	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	85	3	7	7	NUM
m-1071	85	4	proof	proof	NOUN
m-1071	85	5	.	.	PUNCT
m-1071	86	1	using	use	VERB
m-1071	86	2	table”1	table”1	ADV
m-1071	86	3	”	"	PUNCT
m-1071	86	4	enter	enter	VERB
m-1071	86	5	the	the	DET
m-1071	86	6	following	follow	VERB
m-1071	86	7	formula	formula	NOUN
m-1071	86	8	randic	randic	ADJ
m-1071	86	9	polynomial	polynomial	ADJ
m-1071	86	10	polynomial	polynomial	ADJ
m-1071	86	11	(	(	PUNCT
m-1071	86	12	5	5	NUM
m-1071	86	13	)	)	PUNCT
m-1071	86	14	,	,	PUNCT
m-1071	86	15	we	we	PRON
m-1071	86	16	get	get	VERB
m-1071	86	17	�	�	PROPN
m-1071	86	18	�	�	PROPN
m-1071	86	19	sc	sc	PROPN
m-1071	86	20	�	�	PROPN
m-1071	86	21	�	�	PROPN
m-1071	86	22	,	,	PUNCT
m-1071	86	23	�	�	PROPN
m-1071	86	24	�	�	PROPN
m-1071	86	25	=	=	SYM
m-1071	86	26	�	�	PROPN
m-1071	86	27	�	�	PROPN
m-1071	86	28	�	�	PROPN
m-1071	86	29	�	�	PROPN
m-1071	86	30	(	(	PUNCT
m-1071	86	31	�	�	PROPN
m-1071	86	32	)	)	PUNCT
m-1071	86	33	(	(	PUNCT
m-1071	86	34	�	�	PROPN
m-1071	86	35	)	)	PUNCT
m-1071	86	36	�	�	PROPN
m-1071	86	37	�	�	PROPN
m-1071	86	38	�	�	PROPN
m-1071	86	39	�	�	PROPN
m-1071	86	40	~	~	SYM
m-1071	86	41	�	�	PROPN
m-1071	86	42	�	�	PROPN
m-1071	86	43	�	�	PROPN
m-1071	86	44	�	�	PROPN
m-1071	86	45	+	+	CCONJ
m-1071	86	46	�	�	PROPN
m-1071	86	47	�	�	PROPN
m-1071	86	48	�	�	PROPN
m-1071	86	49	�	�	PROPN
m-1071	86	50	(	(	PUNCT
m-1071	86	51	�	�	PROPN
m-1071	86	52	)	)	PUNCT
m-1071	86	53	(	(	PUNCT
m-1071	86	54	�	�	PROPN
m-1071	86	55	)	)	PUNCT
m-1071	86	56	�	�	PROPN
m-1071	86	57	�	�	PROPN
m-1071	86	58	�	�	PROPN
m-1071	86	59	�	�	PROPN
m-1071	86	60	~	~	SYM
m-1071	86	61	�	�	PROPN
m-1071	86	62	�	�	PROPN
m-1071	86	63	�	�	PROPN
m-1071	86	64	�	�	PROPN
m-1071	86	65	+	+	CCONJ
m-1071	86	66	�	�	PROPN
m-1071	86	67	�	�	PROPN
m-1071	86	68	�	�	PROPN
m-1071	86	69	�	�	PROPN
m-1071	86	70	(	(	PUNCT
m-1071	86	71	�	�	PROPN
m-1071	86	72	)	)	PUNCT
m-1071	86	73	(	(	PUNCT
m-1071	86	74	�	�	PROPN
m-1071	86	75	)	)	PUNCT
m-1071	86	76	�	�	PROPN
m-1071	86	77	�	�	PROPN
m-1071	86	78	�	�	PROPN
m-1071	86	79	�	�	PROPN
m-1071	86	80	~	~	SYM
m-1071	86	81	�	�	PROPN
m-1071	86	82	�	�	PROPN
m-1071	86	83	�	�	PROPN
m-1071	86	84	�	�	PROPN
m-1071	86	85	this	this	PRON
m-1071	86	86	gives	give	VERB
m-1071	86	87	�	�	PROPN
m-1071	86	88	�	�	PROPN
m-1071	86	89	sc	sc	PROPN
m-1071	86	90	�	�	PROPN
m-1071	86	91	�	�	PROPN
m-1071	86	92	,	,	PUNCT
m-1071	86	93	�	�	PROPN
m-1071	86	94	�	�	PROPN
m-1071	86	95	=	=	SYM
m-1071	86	96	(	(	PUNCT
m-1071	86	97	3	3	NUM
m-1071	86	98	�	�	NOUN
m-1071	86	99	+	+	CCONJ
m-1071	86	100	2	2	NUM
m-1071	86	101	)	)	PUNCT
m-1071	86	102	�	�	PROPN
m-1071	86	103	�	�	PROPN
m-1071	86	104	�	�	PROPN
m-1071	86	105	+	+	CCONJ
m-1071	86	106	(	(	PUNCT
m-1071	86	107	3	3	NUM
m-1071	86	108	�	�	PROPN
m-1071	86	109	�	�	PROPN
m-1071	86	110	+	+	CCONJ
m-1071	86	111	3	3	NUM
m-1071	86	112	�	�	NOUN
m-1071	86	113	−	−	NUM
m-1071	86	114	4	4	NUM
m-1071	86	115	)	)	PUNCT
m-1071	86	116	�	�	PROPN
m-1071	86	117	�	�	PROPN
m-1071	86	118	√	√	NUM
m-1071	86	119	�	�	PROPN
m-1071	86	120	�	�	PROPN
m-1071	86	121	+	+	CCONJ
m-1071	86	122	(	(	PUNCT
m-1071	86	123	3	3	NUM
m-1071	86	124	�	�	PROPN
m-1071	86	125	�	�	PROPN
m-1071	86	126	−	−	NUM
m-1071	86	127	6	6	NUM
m-1071	86	128	�	�	PROPN
m-1071	86	129	+	+	CCONJ
m-1071	86	130	2	2	NUM
m-1071	86	131	)	)	PUNCT
m-1071	86	132	�	�	PROPN
m-1071	86	133	�	�	PROPN
m-1071	86	134	�	�	PROPN
m-1071	86	135	by	by	ADP
m-1071	86	136	taking	take	VERB
m-1071	86	137	the	the	DET
m-1071	86	138	first	first	ADJ
m-1071	86	139	derivative	derivative	NOUN
m-1071	86	140	of	of	ADP
m-1071	86	141	the	the	DET
m-1071	86	142	polynomial	polynomial	NOUN
m-1071	86	143	in	in	ADP
m-1071	86	144	theorem	theorem	ADJ
m-1071	86	145	2.9	2.9	NUM
m-1071	86	146	at	at	ADP
m-1071	86	147	y	y	PROPN
m-1071	86	148	=	=	SYM
m-1071	86	149	1	1	NUM
m-1071	86	150	,	,	PUNCT
m-1071	86	151	we	we	PRON
m-1071	86	152	get	get	VERB
m-1071	86	153	the	the	DET
m-1071	86	154	randic	randic	ADJ
m-1071	86	155	polynomial	polynomial	ADJ
m-1071	86	156	index	index	NOUN
m-1071	86	157	of	of	ADP
m-1071	86	158	chain	chain	NOUN
m-1071	86	159	of	of	ADP
m-1071	86	160	sio4	sio4	PROPN
m-1071	86	161	(	(	PUNCT
m-1071	86	162	sc	sc	PROPN
m-1071	86	163	�	�	PROPN
m-1071	86	164	�	�	PROPN
m-1071	86	165	)	)	PUNCT
m-1071	86	166	as	as	SCONJ
m-1071	86	167	follows	follow	VERB
m-1071	86	168	:	:	PUNCT
m-1071	86	169	corollary	corollary	ADJ
m-1071	86	170	2.10	2.10	NUM
m-1071	86	171	.	.	PUNCT
m-1071	87	1	for	for	ADP
m-1071	87	2	p	p	PROPN
m-1071	87	3	>	>	PROPN
m-1071	87	4	1	1	NUM
m-1071	87	5	and	and	CCONJ
m-1071	87	6	p	p	NOUN
m-1071	87	7	=	=	ADJ
m-1071	87	8	q	q	NOUN
m-1071	87	9	,	,	PUNCT
m-1071	87	10	the	the	DET
m-1071	87	11	randic	randic	ADJ
m-1071	87	12	polynomial	polynomial	ADJ
m-1071	87	13	index	index	NOUN
m-1071	87	14	ofsc	ofsc	PROPN
m-1071	87	15	�	�	PROPN
m-1071	87	16	�	�	PROPN
m-1071	87	17	is	be	AUX
m-1071	87	18	√	√	NUM
m-1071	87	19	�	�	PROPN
m-1071	87	20	�	�	PROPN
m-1071	87	21	�	�	PROPN
m-1071	87	22	�	�	PROPN
m-1071	87	23	�	�	PROPN
m-1071	87	24	�	�	PROPN
m-1071	87	25	�	�	PROPN
m-1071	87	26	�	�	PROPN
m-1071	87	27	�	�	PROPN
m-1071	87	28	.√	.√	PROPN
m-1071	87	29	�	�	PROPN
m-1071	87	30	�	�	PROPN
m-1071	87	31	�	�	PROPN
m-1071	87	32	�	�	PROPN
m-1071	87	33	�	�	PROPN
m-1071	87	34	�	�	PROPN
m-1071	87	35	.√	.√	NOUN
m-1071	87	36	�	�	PROPN
m-1071	87	37	�	�	PROPN
m-1071	87	38	theorem	theorem	VERB
m-1071	87	39	2.11	2.11	NUM
m-1071	87	40	.	.	PUNCT
m-1071	88	1	for	for	ADP
m-1071	88	2	p	p	PROPN
m-1071	88	3	>	>	PROPN
m-1071	88	4	1	1	NUM
m-1071	88	5	and	and	CCONJ
m-1071	88	6	p	p	NOUN
m-1071	88	7	=	=	ADJ
m-1071	88	8	q	q	NOUN
m-1071	88	9	,	,	PUNCT
m-1071	88	10	the	the	DET
m-1071	88	11	reciprocal	reciprocal	ADJ
m-1071	88	12	randic	randic	ADJ
m-1071	88	13	polynomial	polynomial	PROPN
m-1071	88	14	ofsc	ofsc	PROPN
m-1071	88	15	�	�	PROPN
m-1071	88	16	�	�	PROPN
m-1071	88	17	is(3	is(3	PROPN
m-1071	88	18	�	�	PROPN
m-1071	88	19	+	+	CCONJ
m-1071	88	20	2	2	NUM
m-1071	88	21	)	)	PUNCT
m-1071	88	22	�	�	PROPN
m-1071	88	23	�	�	PROPN
m-1071	88	24	+	+	CCONJ
m-1071	88	25	(	(	PUNCT
m-1071	88	26	3	3	NUM
m-1071	88	27	�	�	PROPN
m-1071	88	28	�	�	PROPN
m-1071	88	29	+	+	CCONJ
m-1071	88	30	3	3	NUM
m-1071	88	31	�	�	PROPN
m-1071	88	32	−	−	PROPN
m-1071	88	33	4)	4)	NUM
m-1071	88	34	�	�	PROPN
m-1071	88	35	�	�	NOUN
m-1071	88	36	√	√	NUM
m-1071	88	37	�	�	PROPN
m-1071	88	38	+	+	CCONJ
m-1071	88	39	(	(	PUNCT
m-1071	88	40	3	3	NUM
m-1071	88	41	�	�	PROPN
m-1071	88	42	�	�	PROPN
m-1071	88	43	−	−	NUM
m-1071	88	44	6	6	NUM
m-1071	88	45	�	�	PROPN
m-1071	88	46	+	+	CCONJ
m-1071	88	47	2)	2)	NUM
m-1071	88	48	�	�	PROPN
m-1071	88	49	√	√	NOUN
m-1071	88	50	�	�	PROPN
m-1071	88	51	.	.	PUNCT
m-1071	88	52	proof	proof	NOUN
m-1071	88	53	.	.	PUNCT
m-1071	89	1	using	use	VERB
m-1071	89	2	table	table	NOUN
m-1071	89	3	“	"	PUNCT
m-1071	89	4	1	1	NUM
m-1071	89	5	”	"	PUNCT
m-1071	89	6	enter	enter	VERB
m-1071	89	7	the	the	DET
m-1071	89	8	following	follow	VERB
m-1071	89	9	formula	formula	NOUN
m-1071	89	10	reciprocal	reciprocal	ADJ
m-1071	89	11	randic	randic	ADJ
m-1071	89	12	polynomial	polynomial	ADJ
m-1071	89	13	polynomial	polynomial	ADJ
m-1071	89	14	(	(	PUNCT
m-1071	89	15	6	6	NUM
m-1071	89	16	)	)	PUNCT
m-1071	89	17	,	,	PUNCT
m-1071	89	18	we	we	PRON
m-1071	89	19	get	get	VERB
m-1071	89	20	�	�	PROPN
m-1071	89	21	�	�	PROPN
m-1071	89	22	�	�	PROPN
m-1071	89	23	sc	sc	PROPN
m-1071	89	24	�	�	PROPN
m-1071	89	25	�	�	PROPN
m-1071	89	26	,	,	PUNCT
m-1071	89	27	�	�	PROPN
m-1071	89	28	�	�	PROPN
m-1071	89	29	=	=	SYM
m-1071	89	30	�	�	PROPN
m-1071	89	31	�	�	PROPN
m-1071	89	32	�	�	PROPN
m-1071	89	33	(	(	PUNCT
m-1071	89	34	�	�	PROPN
m-1071	89	35	)	)	PUNCT
m-1071	89	36	(	(	PUNCT
m-1071	89	37	�	�	PROPN
m-1071	89	38	)	)	PUNCT
m-1071	89	39	�	�	PROPN
m-1071	89	40	�	�	PROPN
m-1071	89	41	�	�	PROPN
m-1071	89	42	�	�	PROPN
m-1071	89	43	~	~	SYM
m-1071	89	44	�	�	PROPN
m-1071	89	45	�	�	PROPN
m-1071	89	46	�	�	PROPN
m-1071	89	47	�	�	PROPN
m-1071	89	48	+	+	CCONJ
m-1071	89	49	�	�	PROPN
m-1071	89	50	�	�	PROPN
m-1071	89	51	�	�	PROPN
m-1071	89	52	(	(	PUNCT
m-1071	89	53	�	�	PROPN
m-1071	89	54	)	)	PUNCT
m-1071	89	55	(	(	PUNCT
m-1071	89	56	�	�	NOUN
m-1071	89	57	)	)	PUNCT
m-1071	89	58	�	�	PROPN
m-1071	89	59	�	�	PROPN
m-1071	89	60	~	~	SYM
m-1071	89	61	�	�	PROPN
m-1071	89	62	+	+	CCONJ
m-1071	89	63	�	�	PROPN
m-1071	89	64	�	�	PROPN
m-1071	89	65	�	�	PROPN
m-1071	89	66	(	(	PUNCT
m-1071	89	67	�	�	PROPN
m-1071	89	68	)	)	PUNCT
m-1071	89	69	(	(	PUNCT
m-1071	89	70	�	�	PROPN
m-1071	89	71	)	)	PUNCT
m-1071	89	72	�	�	PROPN
m-1071	89	73	�	�	PROPN
m-1071	89	74	�	�	PROPN
m-1071	89	75	�	�	PROPN
m-1071	89	76	~	~	SYM
m-1071	89	77	�	�	PROPN
m-1071	89	78	�	�	PROPN
m-1071	89	79	�	�	PROPN
m-1071	89	80	�	�	PROPN
m-1071	89	81	this	this	PRON
m-1071	89	82	gives	give	VERB
m-1071	89	83	�	�	PROPN
m-1071	89	84	�	�	PROPN
m-1071	89	85	�	�	PROPN
m-1071	89	86	sc	sc	PROPN
m-1071	89	87	�	�	PROPN
m-1071	89	88	�	�	PROPN
m-1071	89	89	,	,	PUNCT
m-1071	89	90	�	�	PROPN
m-1071	89	91	�	�	PROPN
m-1071	89	92	=	=	SYM
m-1071	89	93	(	(	PUNCT
m-1071	89	94	3	3	NUM
m-1071	89	95	�	�	NOUN
m-1071	89	96	+	+	CCONJ
m-1071	89	97	2	2	NUM
m-1071	89	98	)	)	PUNCT
m-1071	89	99	�	�	PROPN
m-1071	89	100	�	�	PROPN
m-1071	89	101	+	+	CCONJ
m-1071	89	102	(	(	PUNCT
m-1071	89	103	3	3	NUM
m-1071	89	104	�	�	PROPN
m-1071	89	105	�	�	PROPN
m-1071	89	106	+	+	CCONJ
m-1071	89	107	3	3	NUM
m-1071	89	108	�	�	PROPN
m-1071	89	109	−	−	PROPN
m-1071	89	110	4)	4)	NUM
m-1071	89	111	�	�	PROPN
m-1071	89	112	�	�	NOUN
m-1071	89	113	√	√	NUM
m-1071	89	114	�	�	PROPN
m-1071	89	115	+	+	CCONJ
m-1071	89	116	(	(	PUNCT
m-1071	89	117	3	3	NUM
m-1071	89	118	�	�	PROPN
m-1071	89	119	�	�	PROPN
m-1071	89	120	−	−	NUM
m-1071	89	121	6	6	NUM
m-1071	89	122	�	�	PROPN
m-1071	89	123	+	+	CCONJ
m-1071	89	124	2)	2)	NUM
m-1071	89	125	�	�	PROPN
m-1071	89	126	√	√	NUM
m-1071	89	127	�	�	PROPN
m-1071	89	128	.	.	PUNCT
m-1071	90	1	by	by	ADP
m-1071	90	2	taking	take	VERB
m-1071	90	3	the	the	DET
m-1071	90	4	first	first	ADJ
m-1071	90	5	derivative	derivative	NOUN
m-1071	90	6	of	of	ADP
m-1071	90	7	the	the	DET
m-1071	90	8	polynomial	polynomial	NOUN
m-1071	90	9	in	in	ADP
m-1071	90	10	theorem	theorem	ADJ
m-1071	90	11	2.11	2.11	NUM
m-1071	90	12	at	at	ADP
m-1071	90	13	y	y	PROPN
m-1071	90	14	=	=	SYM
m-1071	90	15	1	1	NUM
m-1071	90	16	,	,	PUNCT
m-1071	90	17	we	we	PRON
m-1071	90	18	get	get	VERB
m-1071	90	19	the	the	DET
m-1071	90	20	reciprocal	reciprocal	ADJ
m-1071	90	21	randic	randic	ADJ
m-1071	90	22	polynomial	polynomial	ADJ
m-1071	90	23	index	index	NOUN
m-1071	90	24	of	of	ADP
m-1071	90	25	chain	chain	NOUN
m-1071	90	26	of	of	ADP
m-1071	90	27	sio4	sio4	PROPN
m-1071	90	28	(	(	PUNCT
m-1071	90	29	sc	sc	PROPN
m-1071	90	30	�	�	PROPN
m-1071	90	31	�	�	PROPN
m-1071	90	32	)	)	PUNCT
m-1071	90	33	as	as	SCONJ
m-1071	90	34	follows	follow	VERB
m-1071	90	35	:	:	PUNCT
m-1071	90	36	corollary	corollary	ADJ
m-1071	90	37	2.12	2.12	NUM
m-1071	90	38	.	.	PUNCT
m-1071	91	1	for	for	ADP
m-1071	91	2	p	p	PROPN
m-1071	91	3	>	>	PROPN
m-1071	91	4	1	1	NUM
m-1071	91	5	and	and	CCONJ
m-1071	91	6	p	p	NOUN
m-1071	91	7	=	=	ADJ
m-1071	91	8	q	q	NOUN
m-1071	91	9	,	,	PUNCT
m-1071	91	10	the	the	DET
m-1071	91	11	reciprocal	reciprocal	ADJ
m-1071	91	12	randic	randic	ADJ
m-1071	91	13	polynomial	polynomial	ADJ
m-1071	91	14	index	index	NOUN
m-1071	91	15	of	of	ADP
m-1071	91	16	3(3	3(3	NUM
m-1071	91	17	�	�	NOUN
m-1071	91	18	+	+	CCONJ
m-1071	91	19	2	2	NUM
m-1071	91	20	)	)	PUNCT
m-1071	91	21	+	+	CCONJ
m-1071	91	22	3√2(3	3√2(3	NUM
m-1071	91	23	�	�	NOUN
m-1071	91	24	�	�	PROPN
m-1071	91	25	+	+	CCONJ
m-1071	91	26	3	3	NUM
m-1071	91	27	�	�	NOUN
m-1071	91	28	−	−	NUM
m-1071	91	29	4	4	NUM
m-1071	91	30	)	)	PUNCT
m-1071	91	31	+	+	CCONJ
m-1071	91	32	√6(3	√6(3	PROPN
m-1071	91	33	�	�	PROPN
m-1071	91	34	�	�	PROPN
m-1071	91	35	−	−	NUM
m-1071	91	36	6	6	NUM
m-1071	91	37	�	�	PROPN
m-1071	91	38	+	+	CCONJ
m-1071	91	39	2	2	NUM
m-1071	91	40	)	)	PUNCT
m-1071	91	41	.	.	PUNCT
m-1071	92	1	theorem	theorem	VERB
m-1071	92	2	2.13	2.13	NUM
m-1071	92	3	.	.	PUNCT
m-1071	93	1	for	for	ADP
m-1071	93	2	p>1	p>1	PROPN
m-1071	93	3	and	and	CCONJ
m-1071	93	4	p	p	X
m-1071	93	5	=	=	NOUN
m-1071	93	6	q.	q.	NOUN
m-1071	93	7	the	the	DET
m-1071	93	8	symmetric	symmetric	ADJ
m-1071	93	9	degree	degree	NOUN
m-1071	93	10	polynomial	polynomial	NOUN
m-1071	93	11	of	of	ADP
m-1071	93	12	sc	sc	PROPN
m-1071	93	13	�	�	PROPN
m-1071	93	14	�	�	PROPN
m-1071	93	15	is(3	is(3	PROPN
m-1071	93	16	�	�	PROPN
m-1071	93	17	�	�	PROPN
m-1071	93	18	−	−	PROPN
m-1071	93	19	3	3	NUM
m-1071	93	20	�	�	NOUN
m-1071	93	21	+	+	CCONJ
m-1071	93	22	4	4	NUM
m-1071	93	23	)	)	PUNCT
m-1071	93	24	�	�	PROPN
m-1071	93	25	�	�	PROPN
m-1071	93	26	+	+	CCONJ
m-1071	93	27	(	(	PUNCT
m-1071	93	28	3	3	NUM
m-1071	93	29	�	�	PROPN
m-1071	93	30	�	�	PROPN
m-1071	93	31	+	+	CCONJ
m-1071	93	32	3	3	NUM
m-1071	93	33	�	�	NOUN
m-1071	93	34	−	−	NUM
m-1071	93	35	4	4	NUM
m-1071	93	36	)	)	PUNCT
m-1071	93	37	�	�	PROPN
m-1071	93	38	�	�	PROPN
m-1071	93	39	�	�	PROPN
m-1071	93	40	�	�	PROPN
m-1071	93	41	ijo	ijo	PROPN
m-1071	93	42	international	international	PROPN
m-1071	93	43	journal	journal	PROPN
m-1071	93	44	of	of	ADP
m-1071	93	45	mathematics	mathematics	PROPN
m-1071	93	46	(	(	PUNCT
m-1071	93	47	issn	issn	PROPN
m-1071	93	48	:	:	PUNCT
m-1071	93	49	2992	2992	NUM
m-1071	93	50	-	-	SYM
m-1071	93	51	4421	4421	NUM
m-1071	93	52	)	)	PUNCT
m-1071	93	53	ijo	ijo	PROPN
m-1071	93	54	journals	journal	NOUN
m-1071	93	55	volume	volume	NOUN
m-1071	93	56	08	08	NUM
m-1071	94	1	|	|	ADV
m-1071	94	2	issue	issue	VERB
m-1071	94	3	04	04	NUM
m-1071	95	1	|	|	CCONJ
m-1071	95	2	april	april	PROPN
m-1071	95	3	2025	2025	NUM
m-1071	96	1	|	|	ADV
m-1071	96	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	96	3	8	8	NUM
m-1071	96	4	proof	proof	NOUN
m-1071	96	5	.	.	PUNCT
m-1071	97	1	using	use	VERB
m-1071	97	2	table	table	NOUN
m-1071	97	3	enter	enter	VERB
m-1071	97	4	the	the	DET
m-1071	97	5	following	follow	VERB
m-1071	97	6	formula	formula	NOUN
m-1071	97	7	symmetric	symmetric	ADJ
m-1071	97	8	division	division	NOUN
m-1071	97	9	degree	degree	NOUN
m-1071	97	10	polynomial	polynomial	ADJ
m-1071	97	11	(	(	PUNCT
m-1071	97	12	7	7	NUM
m-1071	97	13	)	)	PUNCT
m-1071	97	14	,	,	PUNCT
m-1071	97	15	we	we	PRON
m-1071	97	16	get	get	VERB
m-1071	97	17	�	�	PROPN
m-1071	97	18	�	�	PROPN
m-1071	97	19	�	�	PROPN
m-1071	97	20	�	�	PROPN
m-1071	97	21	sc	sc	PROPN
m-1071	97	22	�	�	PROPN
m-1071	97	23	�	�	PROPN
m-1071	97	24	,	,	PUNCT
m-1071	97	25	�	�	PROPN
m-1071	97	26	�	�	PROPN
m-1071	97	27	=	=	SYM
m-1071	97	28	�	�	PROPN
m-1071	97	29	�	�	PROPN
m-1071	97	30	[	[	X
m-1071	97	31	(	(	PUNCT
m-1071	97	32	�	�	NOUN
m-1071	97	33	)	)	PUNCT
m-1071	97	34	�	�	PROPN
m-1071	97	35	�	�	PROPN
m-1071	97	36	(	(	PUNCT
m-1071	97	37	�	�	PROPN
m-1071	97	38	)	)	PUNCT
m-1071	97	39	�	�	PROPN
m-1071	97	40	]	]	PUNCT
m-1071	97	41	(	(	PUNCT
m-1071	97	42	�	�	PROPN
m-1071	97	43	)	)	PUNCT
m-1071	97	44	(	(	PUNCT
m-1071	97	45	�	�	PROPN
m-1071	97	46	)	)	PUNCT
m-1071	97	47	�	�	PROPN
m-1071	97	48	�	�	PROPN
m-1071	97	49	�	�	PROPN
m-1071	97	50	�	�	PROPN
m-1071	97	51	~	~	SYM
m-1071	97	52	�	�	PROPN
m-1071	97	53	�	�	PROPN
m-1071	97	54	�	�	PROPN
m-1071	97	55	�	�	PROPN
m-1071	97	56	+	+	CCONJ
m-1071	97	57	�	�	PROPN
m-1071	97	58	�	�	PROPN
m-1071	97	59	[	[	X
m-1071	97	60	(	(	PUNCT
m-1071	97	61	�	�	NOUN
m-1071	97	62	)	)	PUNCT
m-1071	97	63	�	�	PROPN
m-1071	97	64	�	�	PROPN
m-1071	97	65	(	(	PUNCT
m-1071	97	66	�	�	PROPN
m-1071	97	67	)	)	PUNCT
m-1071	97	68	�	�	PROPN
m-1071	97	69	]	]	PUNCT
m-1071	97	70	(	(	PUNCT
m-1071	97	71	�	�	PROPN
m-1071	97	72	)	)	PUNCT
m-1071	97	73	(	(	PUNCT
m-1071	97	74	�	�	PROPN
m-1071	97	75	)	)	PUNCT
m-1071	97	76	�	�	PROPN
m-1071	97	77	�	�	PROPN
m-1071	97	78	�	�	PROPN
m-1071	97	79	�	�	PROPN
m-1071	97	80	~	~	SYM
m-1071	97	81	�	�	PROPN
m-1071	97	82	�	�	PROPN
m-1071	97	83	�	�	PROPN
m-1071	97	84	�	�	PROPN
m-1071	97	85	+	+	CCONJ
m-1071	97	86	�	�	PROPN
m-1071	97	87	�	�	PROPN
m-1071	97	88	[	[	X
m-1071	97	89	(	(	PUNCT
m-1071	97	90	�	�	NOUN
m-1071	97	91	)	)	PUNCT
m-1071	97	92	�	�	PROPN
m-1071	97	93	�	�	PROPN
m-1071	97	94	(	(	PUNCT
m-1071	97	95	�	�	PROPN
m-1071	97	96	)	)	PUNCT
m-1071	97	97	�	�	PROPN
m-1071	97	98	]	]	PUNCT
m-1071	97	99	(	(	PUNCT
m-1071	97	100	�	�	PROPN
m-1071	97	101	)	)	PUNCT
m-1071	97	102	(	(	PUNCT
m-1071	97	103	�	�	NOUN
m-1071	97	104	)	)	PUNCT
m-1071	97	105	�	�	PROPN
m-1071	97	106	�	�	PROPN
m-1071	97	107	~	~	NOUN
m-1071	97	108	�	�	PROPN
m-1071	97	109	this	this	PRON
m-1071	97	110	gives	give	VERB
m-1071	97	111	�	�	PROPN
m-1071	97	112	�	�	PROPN
m-1071	97	113	�	�	PROPN
m-1071	97	114	�	�	PROPN
m-1071	97	115	sc	sc	PROPN
m-1071	97	116	�	�	PROPN
m-1071	97	117	�	�	PROPN
m-1071	97	118	,	,	PUNCT
m-1071	97	119	�	�	PROPN
m-1071	97	120	�	�	PROPN
m-1071	97	121	=	=	SYM
m-1071	97	122	(	(	PUNCT
m-1071	97	123	3	3	NUM
m-1071	97	124	�	�	NOUN
m-1071	97	125	+	+	CCONJ
m-1071	97	126	2	2	NUM
m-1071	97	127	)	)	PUNCT
m-1071	97	128	�	�	PROPN
m-1071	97	129	�	�	PROPN
m-1071	97	130	+	+	CCONJ
m-1071	97	131	(	(	PUNCT
m-1071	97	132	3	3	NUM
m-1071	97	133	�	�	PROPN
m-1071	97	134	�	�	PROPN
m-1071	97	135	+	+	CCONJ
m-1071	97	136	3	3	NUM
m-1071	97	137	�	�	NOUN
m-1071	97	138	−	−	NUM
m-1071	97	139	4	4	NUM
m-1071	97	140	)	)	PUNCT
m-1071	97	141	�	�	PROPN
m-1071	97	142	�	�	PROPN
m-1071	97	143	�	�	PROPN
m-1071	97	144	�	�	PROPN
m-1071	97	145	+	+	CCONJ
m-1071	97	146	(	(	PUNCT
m-1071	97	147	3	3	NUM
m-1071	97	148	�	�	PROPN
m-1071	97	149	�	�	PROPN
m-1071	97	150	−	−	NUM
m-1071	97	151	6	6	NUM
m-1071	97	152	�	�	PROPN
m-1071	97	153	+	+	CCONJ
m-1071	97	154	2	2	NUM
m-1071	97	155	)	)	PUNCT
m-1071	97	156	�	�	PROPN
m-1071	97	157	�	�	PROPN
m-1071	97	158	=	=	SYM
m-1071	97	159	(	(	PUNCT
m-1071	97	160	3	3	NUM
m-1071	97	161	�	�	PROPN
m-1071	97	162	�	�	PROPN
m-1071	97	163	−	−	PROPN
m-1071	97	164	3	3	NUM
m-1071	97	165	�	�	NOUN
m-1071	97	166	+	+	CCONJ
m-1071	97	167	4	4	NUM
m-1071	97	168	)	)	PUNCT
m-1071	97	169	�	�	PROPN
m-1071	97	170	�	�	PROPN
m-1071	97	171	+	+	CCONJ
m-1071	97	172	(	(	PUNCT
m-1071	97	173	3	3	NUM
m-1071	97	174	�	�	PROPN
m-1071	97	175	�	�	PROPN
m-1071	97	176	+	+	CCONJ
m-1071	97	177	3	3	NUM
m-1071	97	178	�	�	NOUN
m-1071	97	179	−	−	NUM
m-1071	97	180	4	4	NUM
m-1071	97	181	)	)	PUNCT
m-1071	97	182	�	�	PROPN
m-1071	97	183	�	�	PROPN
m-1071	97	184	�	�	PROPN
m-1071	97	185	�	�	PROPN
m-1071	97	186	by	by	ADP
m-1071	97	187	taking	take	VERB
m-1071	97	188	the	the	DET
m-1071	97	189	first	first	ADJ
m-1071	97	190	derivative	derivative	NOUN
m-1071	97	191	of	of	ADP
m-1071	97	192	the	the	DET
m-1071	97	193	polynomial	polynomial	NOUN
m-1071	97	194	in	in	ADP
m-1071	97	195	theorem	theorem	NOUN
m-1071	97	196	2.19	2.19	NUM
m-1071	97	197	at	at	ADP
m-1071	97	198	y	y	PROPN
m-1071	97	199	=	=	SYM
m-1071	97	200	1	1	NUM
m-1071	97	201	,	,	PUNCT
m-1071	97	202	we	we	PRON
m-1071	97	203	get	get	VERB
m-1071	97	204	the	the	DET
m-1071	97	205	symmetric	symmetric	ADJ
m-1071	97	206	division	division	NOUN
m-1071	97	207	degree	degree	NOUN
m-1071	97	208	index	index	NOUN
m-1071	97	209	of	of	ADP
m-1071	97	210	chain	chain	NOUN
m-1071	97	211	of	of	ADP
m-1071	97	212	sio4	sio4	PROPN
m-1071	97	213	(	(	PUNCT
m-1071	97	214	sc	sc	PROPN
m-1071	97	215	�	�	PROPN
m-1071	97	216	�	�	PROPN
m-1071	97	217	as	as	SCONJ
m-1071	97	218	follows	follow	VERB
m-1071	97	219	:	:	PUNCT
m-1071	97	220	corollary	corollary	ADJ
m-1071	97	221	2.14	2.14	NUM
m-1071	97	222	.	.	PUNCT
m-1071	98	1	for	for	ADP
m-1071	98	2	p	p	PROPN
m-1071	98	3	>	>	PROPN
m-1071	98	4	1	1	NUM
m-1071	98	5	and	and	CCONJ
m-1071	98	6	p	p	NOUN
m-1071	98	7	=	=	ADJ
m-1071	98	8	q	q	NOUN
m-1071	98	9	,	,	PUNCT
m-1071	98	10	the	the	DET
m-1071	98	11	symmetric	symmetric	ADJ
m-1071	98	12	division	division	NOUN
m-1071	98	13	degree	degree	NOUN
m-1071	98	14	index	index	NOUN
m-1071	98	15	of	of	ADP
m-1071	98	16	sc	sc	PROPN
m-1071	98	17	�	�	PROPN
m-1071	98	18	�	�	PROPN
m-1071	98	19	is2(3	is2(3	PROPN
m-1071	98	20	�	�	PROPN
m-1071	98	21	�	�	PROPN
m-1071	98	22	−	−	PROPN
m-1071	98	23	3	3	NUM
m-1071	98	24	�	�	NOUN
m-1071	98	25	+	+	CCONJ
m-1071	98	26	4	4	NUM
m-1071	98	27	)	)	PUNCT
m-1071	98	28	+	+	CCONJ
m-1071	98	29	�	�	PROPN
m-1071	98	30	�	�	PROPN
m-1071	98	31	�	�	PROPN
m-1071	98	32	�	�	PROPN
m-1071	98	33	�	�	PROPN
m-1071	98	34	�	�	PROPN
m-1071	98	35	�	�	PROPN
m-1071	98	36	�	�	PROPN
m-1071	98	37	�	�	PROPN
m-1071	98	38	�	�	PROPN
m-1071	98	39	�	�	PROPN
m-1071	98	40	�	�	PROPN
m-1071	98	41	�	�	PROPN
m-1071	98	42	theorem	theorem	VERB
m-1071	98	43	2.15	2.15	NUM
m-1071	98	44	.	.	PUNCT
m-1071	99	1	for	for	ADP
m-1071	99	2	p	p	PROPN
m-1071	99	3	>	>	PROPN
m-1071	99	4	1	1	NUM
m-1071	99	5	and	and	CCONJ
m-1071	99	6	p	p	NOUN
m-1071	99	7	=	=	ADJ
m-1071	99	8	q	q	X
m-1071	99	9	,	,	PUNCT
m-1071	99	10	the	the	DET
m-1071	99	11	inverse	inverse	ADJ
m-1071	99	12	symmetric	symmetric	ADJ
m-1071	99	13	division	division	NOUN
m-1071	99	14	polynomial	polynomial	NOUN
m-1071	99	15	of	of	ADP
m-1071	99	16	sc	sc	PROPN
m-1071	99	17	�	�	PROPN
m-1071	99	18	�	�	PROPN
m-1071	99	19	is	be	AUX
m-1071	99	20	(	(	PUNCT
m-1071	99	21	3	3	NUM
m-1071	99	22	�	�	PROPN
m-1071	99	23	�	�	PROPN
m-1071	99	24	−	−	PROPN
m-1071	99	25	3	3	NUM
m-1071	99	26	�	�	NOUN
m-1071	99	27	+	+	CCONJ
m-1071	99	28	4	4	NUM
m-1071	99	29	)	)	PUNCT
m-1071	99	30	�	�	PROPN
m-1071	99	31	�	�	PROPN
m-1071	99	32	�	�	PROPN
m-1071	99	33	+	+	CCONJ
m-1071	99	34	(	(	PUNCT
m-1071	99	35	3	3	NUM
m-1071	99	36	�	�	PROPN
m-1071	99	37	�	�	PROPN
m-1071	99	38	+	+	CCONJ
m-1071	99	39	3	3	NUM
m-1071	99	40	�	�	NOUN
m-1071	99	41	−	−	NUM
m-1071	99	42	4	4	NUM
m-1071	99	43	)	)	PUNCT
m-1071	99	44	�	�	PROPN
m-1071	99	45	�	�	PROPN
m-1071	99	46	�	�	PROPN
m-1071	99	47	�	�	PROPN
m-1071	99	48	proof	proof	NOUN
m-1071	99	49	.	.	PUNCT
m-1071	100	1	using	use	VERB
m-1071	100	2	table	table	NOUN
m-1071	100	3	“	"	PUNCT
m-1071	100	4	1	1	NUM
m-1071	100	5	”	"	PUNCT
m-1071	100	6	enter	enter	VERB
m-1071	100	7	the	the	DET
m-1071	100	8	following	follow	VERB
m-1071	100	9	formula	formula	NOUN
m-1071	100	10	inverse	inverse	NOUN
m-1071	100	11	symmetric	symmetric	ADJ
m-1071	100	12	division	division	NOUN
m-1071	100	13	degree	degree	NOUN
m-1071	100	14	polynomial	polynomial	ADJ
m-1071	100	15	(	(	PUNCT
m-1071	100	16	8)	8)	NUM
m-1071	100	17	,	,	PUNCT
m-1071	100	18	we	we	PRON
m-1071	100	19	get	get	VERB
m-1071	100	20	�	�	PROPN
m-1071	100	21	�	�	PROPN
m-1071	100	22	�	�	PROPN
m-1071	100	23	�	�	PROPN
m-1071	100	24	�	�	PROPN
m-1071	100	25	sc	sc	PROPN
m-1071	100	26	�	�	PROPN
m-1071	100	27	�	�	PROPN
m-1071	100	28	,	,	PUNCT
m-1071	100	29	�	�	PROPN
m-1071	100	30	�	�	PROPN
m-1071	100	31	=	=	SYM
m-1071	100	32	�	�	PROPN
m-1071	100	33	�	�	PROPN
m-1071	100	34	[	[	X
m-1071	100	35	(	(	PUNCT
m-1071	100	36	�	�	PROPN
m-1071	100	37	)	)	PUNCT
m-1071	100	38	�	�	PROPN
m-1071	100	39	(	(	PUNCT
m-1071	100	40	�	�	PROPN
m-1071	100	41	)	)	PUNCT
m-1071	100	42	]	]	PUNCT
m-1071	100	43	(	(	PUNCT
m-1071	100	44	�	�	NOUN
m-1071	100	45	)	)	PUNCT
m-1071	100	46	�	�	PROPN
m-1071	100	47	�	�	PROPN
m-1071	100	48	(	(	PUNCT
m-1071	100	49	�	�	PROPN
m-1071	100	50	)	)	PUNCT
m-1071	100	51	�	�	PROPN
m-1071	100	52	�	�	PROPN
m-1071	100	53	�	�	PROPN
m-1071	100	54	�	�	PROPN
m-1071	100	55	�	�	PROPN
m-1071	100	56	~	~	SYM
m-1071	100	57	�	�	PROPN
m-1071	100	58	�	�	PROPN
m-1071	100	59	�	�	PROPN
m-1071	100	60	�	�	PROPN
m-1071	100	61	+	+	CCONJ
m-1071	100	62	�	�	PROPN
m-1071	100	63	�	�	PROPN
m-1071	100	64	[	[	X
m-1071	100	65	(	(	PUNCT
m-1071	100	66	�	�	NOUN
m-1071	100	67	)	)	PUNCT
m-1071	100	68	�	�	PROPN
m-1071	100	69	�	�	PROPN
m-1071	100	70	(	(	PUNCT
m-1071	100	71	�	�	PROPN
m-1071	100	72	)	)	PUNCT
m-1071	100	73	�	�	PROPN
m-1071	100	74	]	]	PUNCT
m-1071	100	75	(	(	PUNCT
m-1071	100	76	�	�	PROPN
m-1071	100	77	)	)	PUNCT
m-1071	100	78	(	(	PUNCT
m-1071	100	79	�	�	NOUN
m-1071	100	80	)	)	PUNCT
m-1071	100	81	�	�	PROPN
m-1071	100	82	�	�	PROPN
m-1071	100	83	~	~	SYM
m-1071	100	84	�	�	PROPN
m-1071	100	85	+	+	CCONJ
m-1071	100	86	�	�	PROPN
m-1071	100	87	�	�	PROPN
m-1071	100	88	[	[	X
m-1071	100	89	(	(	PUNCT
m-1071	100	90	�	�	NOUN
m-1071	100	91	)	)	PUNCT
m-1071	100	92	�	�	PROPN
m-1071	100	93	�	�	PROPN
m-1071	100	94	(	(	PUNCT
m-1071	100	95	�	�	PROPN
m-1071	100	96	)	)	PUNCT
m-1071	100	97	�	�	PROPN
m-1071	100	98	]	]	PUNCT
m-1071	100	99	(	(	PUNCT
m-1071	100	100	�	�	PROPN
m-1071	100	101	)	)	PUNCT
m-1071	100	102	(	(	PUNCT
m-1071	100	103	�	�	PROPN
m-1071	100	104	)	)	PUNCT
m-1071	100	105	�	�	PROPN
m-1071	100	106	�	�	PROPN
m-1071	100	107	�	�	PROPN
m-1071	100	108	�	�	PROPN
m-1071	100	109	~	~	SYM
m-1071	100	110	�	�	PROPN
m-1071	100	111	�	�	PROPN
m-1071	100	112	�	�	PROPN
m-1071	100	113	�	�	PROPN
m-1071	100	114	this	this	PRON
m-1071	100	115	gives	give	VERB
m-1071	100	116	�	�	PROPN
m-1071	100	117	�	�	PROPN
m-1071	100	118	�	�	PROPN
m-1071	100	119	�	�	PROPN
m-1071	100	120	sc	sc	PROPN
m-1071	100	121	�	�	PROPN
m-1071	100	122	�	�	PROPN
m-1071	100	123	,	,	PUNCT
m-1071	100	124	�	�	PROPN
m-1071	100	125	�	�	PROPN
m-1071	100	126	=	=	SYM
m-1071	100	127	(	(	PUNCT
m-1071	100	128	3	3	NUM
m-1071	100	129	�	�	NOUN
m-1071	100	130	+	+	CCONJ
m-1071	100	131	2	2	NUM
m-1071	100	132	)	)	PUNCT
m-1071	100	133	�	�	PROPN
m-1071	100	134	�	�	PROPN
m-1071	100	135	�	�	PROPN
m-1071	100	136	+	+	CCONJ
m-1071	100	137	(	(	PUNCT
m-1071	100	138	3	3	NUM
m-1071	100	139	�	�	PROPN
m-1071	100	140	�	�	PROPN
m-1071	100	141	+	+	CCONJ
m-1071	100	142	3	3	NUM
m-1071	100	143	�	�	NOUN
m-1071	100	144	−	−	NUM
m-1071	100	145	4	4	NUM
m-1071	100	146	)	)	PUNCT
m-1071	100	147	�	�	PROPN
m-1071	100	148	�	�	PROPN
m-1071	100	149	�	�	PROPN
m-1071	100	150	�	�	PROPN
m-1071	100	151	+	+	CCONJ
m-1071	100	152	(	(	PUNCT
m-1071	100	153	3	3	NUM
m-1071	100	154	�	�	PROPN
m-1071	100	155	�	�	PROPN
m-1071	100	156	−	−	NUM
m-1071	100	157	6	6	NUM
m-1071	100	158	�	�	PROPN
m-1071	100	159	+	+	CCONJ
m-1071	100	160	2	2	NUM
m-1071	100	161	)	)	PUNCT
m-1071	100	162	�	�	PROPN
m-1071	100	163	�	�	PROPN
m-1071	100	164	�	�	PROPN
m-1071	100	165	=	=	SYM
m-1071	100	166	(	(	PUNCT
m-1071	100	167	3	3	NUM
m-1071	100	168	�	�	PROPN
m-1071	100	169	�	�	PROPN
m-1071	100	170	−	−	PROPN
m-1071	100	171	3	3	NUM
m-1071	100	172	�	�	PROPN
m-1071	100	173	+	+	CCONJ
m-1071	100	174	2	2	NUM
m-1071	100	175	)	)	PUNCT
m-1071	100	176	�	�	PROPN
m-1071	100	177	�	�	PROPN
m-1071	100	178	�	�	PROPN
m-1071	100	179	+	+	CCONJ
m-1071	100	180	(	(	PUNCT
m-1071	100	181	3	3	NUM
m-1071	100	182	�	�	PROPN
m-1071	100	183	�	�	PROPN
m-1071	100	184	+	+	CCONJ
m-1071	100	185	3	3	NUM
m-1071	100	186	�	�	NOUN
m-1071	100	187	−	−	NUM
m-1071	100	188	4	4	NUM
m-1071	100	189	)	)	PUNCT
m-1071	100	190	�	�	PROPN
m-1071	100	191	�	�	PROPN
m-1071	100	192	�	�	PROPN
m-1071	100	193	�	�	PROPN
m-1071	100	194	by	by	ADP
m-1071	100	195	taking	take	VERB
m-1071	100	196	the	the	DET
m-1071	100	197	first	first	ADJ
m-1071	100	198	derivative	derivative	NOUN
m-1071	100	199	of	of	ADP
m-1071	100	200	the	the	DET
m-1071	100	201	polynomial	polynomial	NOUN
m-1071	100	202	in	in	ADP
m-1071	100	203	theorem	theorem	NOUN
m-1071	100	204	2.15	2.15	NUM
m-1071	100	205	at	at	ADP
m-1071	100	206	y	y	PROPN
m-1071	100	207	=	=	SYM
m-1071	100	208	1	1	NUM
m-1071	100	209	,	,	PUNCT
m-1071	100	210	we	we	PRON
m-1071	100	211	get	get	VERB
m-1071	100	212	the	the	DET
m-1071	100	213	inverse	inverse	ADJ
m-1071	100	214	symmetric	symmetric	ADJ
m-1071	100	215	division	division	NOUN
m-1071	100	216	degree	degree	NOUN
m-1071	100	217	index	index	NOUN
m-1071	100	218	of	of	ADP
m-1071	100	219	chain	chain	NOUN
m-1071	100	220	of	of	ADP
m-1071	100	221	sio4	sio4	PROPN
m-1071	100	222	sc	sc	PROPN
m-1071	100	223	�	�	PROPN
m-1071	100	224	�	�	PROPN
m-1071	100	225	as	as	SCONJ
m-1071	100	226	follows	follow	VERB
m-1071	100	227	:	:	PUNCT
m-1071	100	228	corollary	corollary	ADJ
m-1071	100	229	2.16	2.16	NUM
m-1071	100	230	.	.	PUNCT
m-1071	101	1	for	for	ADP
m-1071	101	2	p	p	PROPN
m-1071	101	3	>	>	PROPN
m-1071	101	4	1	1	NUM
m-1071	101	5	and	and	CCONJ
m-1071	101	6	p	p	NOUN
m-1071	101	7	=	=	ADJ
m-1071	101	8	q	q	X
m-1071	101	9	,	,	PUNCT
m-1071	101	10	the	the	DET
m-1071	101	11	inverse	inverse	ADJ
m-1071	101	12	symmetric	symmetric	ADJ
m-1071	101	13	division	division	NOUN
m-1071	101	14	degree	degree	NOUN
m-1071	101	15	index	index	NOUN
m-1071	101	16	of	of	ADP
m-1071	101	17	sc	sc	PROPN
m-1071	101	18	�	�	PROPN
m-1071	101	19	�	�	PROPN
m-1071	101	20	is	be	AUX
m-1071	101	21	�	�	PROPN
m-1071	101	22	�	�	PROPN
m-1071	101	23	�	�	PROPN
m-1071	101	24	�	�	PROPN
m-1071	101	25	�	�	PROPN
m-1071	101	26	�	�	PROPN
m-1071	101	27	�	�	PROPN
m-1071	101	28	�	�	PROPN
m-1071	101	29	�	�	PROPN
m-1071	101	30	�	�	PROPN
m-1071	101	31	�	�	PROPN
m-1071	101	32	�	�	PROPN
m-1071	101	33	�	�	PROPN
m-1071	101	34	.	.	PUNCT
m-1071	102	1	ijo	ijo	PROPN
m-1071	102	2	international	international	PROPN
m-1071	102	3	journal	journal	PROPN
m-1071	102	4	of	of	ADP
m-1071	102	5	mathematics	mathematics	PROPN
m-1071	102	6	(	(	PUNCT
m-1071	102	7	issn	issn	PROPN
m-1071	102	8	:	:	PUNCT
m-1071	102	9	2992	2992	NUM
m-1071	102	10	-	-	SYM
m-1071	102	11	4421	4421	NUM
m-1071	102	12	)	)	PUNCT
m-1071	102	13	ijo	ijo	PROPN
m-1071	102	14	journals	journal	NOUN
m-1071	102	15	volume	volume	NOUN
m-1071	102	16	08	08	NUM
m-1071	103	1	|	|	ADV
m-1071	103	2	issue	issue	VERB
m-1071	103	3	04	04	NUM
m-1071	104	1	|	|	CCONJ
m-1071	104	2	april	april	PROPN
m-1071	104	3	2025	2025	NUM
m-1071	104	4	|	|	ADV
m-1071	104	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	104	6	9	9	NUM
m-1071	104	7	theorem	theorem	VERB
m-1071	104	8	2.17	2.17	NUM
m-1071	104	9	.	.	PUNCT
m-1071	105	1	for	for	ADP
m-1071	105	2	p	p	PROPN
m-1071	105	3	>	>	PROPN
m-1071	105	4	1	1	NUM
m-1071	105	5	and	and	CCONJ
m-1071	105	6	p	p	ADJ
m-1071	105	7	3	3	NUM
m-1071	105	8	�	�	PROPN
m-1071	105	9	�	�	PROPN
m-1071	105	10	−	−	PROPN
m-1071	105	11	3	3	NUM
m-1071	105	12	�	�	NOUN
m-1071	105	13	+	+	CCONJ
m-1071	105	14	4	4	NUM
m-1071	105	15	.	.	X
m-1071	105	16	proof	proof	NOUN
m-1071	105	17	.	.	PUNCT
m-1071	106	1	using	use	VERB
m-1071	106	2	table	table	NOUN
m-1071	106	3	“	"	PUNCT
m-1071	106	4	1	1	NUM
m-1071	106	5	”	"	PUNCT
m-1071	106	6	enter	enter	VERB
m-1071	106	7	the	the	DET
m-1071	106	8	following	follow	VERB
m-1071	106	9	formula	formula	NOUN
m-1071	106	10	sigma	sigma	NOUN
m-1071	106	11	polynomial	polynomial	ADJ
m-1071	106	12	(	(	PUNCT
m-1071	106	13	7	7	NUM
m-1071	106	14	)	)	PUNCT
m-1071	106	15	,	,	PUNCT
m-1071	106	16	we	we	PRON
m-1071	106	17	get	get	VERB
m-1071	106	18	�	�	PROPN
m-1071	106	19	(	(	PUNCT
m-1071	106	20	g	g	PROPN
m-1071	106	21	,	,	PUNCT
m-1071	106	22	�	�	PROPN
m-1071	106	23	)	)	PUNCT
m-1071	106	24	�	�	PROPN
m-1071	106	25	sc	sc	PROPN
m-1071	106	26	�	�	PROPN
m-1071	106	27	�	�	PROPN
m-1071	106	28	,	,	PUNCT
m-1071	106	29	�	�	PROPN
m-1071	106	30	�	�	PROPN
m-1071	106	31	=	=	SYM
m-1071	106	32	�	�	PROPN
m-1071	106	33	this	this	PRON
m-1071	106	34	gives	give	VERB
m-1071	106	35	�	�	PROPN
m-1071	106	36	�	�	PROPN
m-1071	106	37	�	�	PROPN
m-1071	106	38	�	�	PROPN
m-1071	106	39	sc	sc	PROPN
m-1071	106	40	�	�	PROPN
m-1071	106	41	�	�	PROPN
m-1071	106	42	,	,	PUNCT
m-1071	106	43	�	�	PROPN
m-1071	106	44	�	�	PROPN
m-1071	106	45	=	=	PUNCT
m-1071	107	1	(	(	PUNCT
m-1071	107	2	=	=	SYM
m-1071	107	3	(	(	PUNCT
m-1071	107	4	by	by	ADP
m-1071	107	5	taking	take	VERB
m-1071	107	6	the	the	DET
m-1071	107	7	first	first	ADJ
m-1071	107	8	derivative	derivative	NOUN
m-1071	107	9	of	of	ADP
m-1071	107	10	the	the	DET
m-1071	107	11	polynomial	polynomial	NOUN
m-1071	107	12	in	in	ADP
m-1071	107	13	theorem	theorem	NOUN
m-1071	107	14	2.19	2.19	NUM
m-1071	107	15	at	at	ADP
m-1071	107	16	sigma	sigma	PROPN
m-1071	107	17	index	index	NOUN
m-1071	107	18	of	of	ADP
m-1071	107	19	chain	chain	NOUN
m-1071	107	20	of	of	ADP
m-1071	107	21	corollary	corollary	ADJ
m-1071	107	22	2.18	2.18	NUM
m-1071	107	23	.	.	PUNCT
m-1071	108	1	for	for	ADP
m-1071	108	2	p	p	PROPN
m-1071	108	3	>	>	PROPN
m-1071	108	4	1	1	NUM
m-1071	108	5	and	and	CCONJ
m-1071	108	6	p	p	NOUN
m-1071	108	7	theorem2.19	theorem2.19	X
m-1071	108	8	.	.	PUNCT
m-1071	109	1	for	for	ADP
m-1071	109	2	p	p	PROPN
m-1071	109	3	>	>	PROPN
m-1071	109	4	1	1	NUM
m-1071	109	5	and√	and√	PRON
m-1071	109	6	p	p	X
m-1071	109	7	(	(	PUNCT
m-1071	109	8	3	3	NUM
m-1071	109	9	�	�	PROPN
m-1071	109	10	�	�	PROPN
m-1071	109	11	+	+	CCONJ
m-1071	109	12	3	3	NUM
m-1071	109	13	�	�	PROPN
m-1071	109	14	−	−	PROPN
m-1071	109	15	4)	4)	NUM
m-1071	109	16	�	�	PROPN
m-1071	109	17	�	�	NOUN
m-1071	109	18	√	√	NUM
m-1071	109	19	�	�	PROPN
m-1071	109	20	+	+	CCONJ
m-1071	109	21	(	(	PUNCT
m-1071	109	22	3	3	NUM
m-1071	109	23	�	�	PROPN
m-1071	109	24	�	�	PROPN
m-1071	109	25	−	−	PROPN
m-1071	109	26	6	6	NUM
m-1071	109	27	proof	proof	NOUN
m-1071	109	28	.	.	PUNCT
m-1071	110	1	using	use	VERB
m-1071	110	2	table	table	NOUN
m-1071	110	3	“	"	PUNCT
m-1071	110	4	1	1	NUM
m-1071	110	5	”	"	PUNCT
m-1071	110	6	enter	enter	VERB
m-1071	110	7	the	the	DET
m-1071	110	8	following	follow	VERB
m-1071	110	9	formula	formula	NOUN
m-1071	110	10	somber	somber	ADJ
m-1071	110	11	polynomial	polynomial	NOUN
m-1071	110	12	(	(	PUNCT
m-1071	110	13	7	7	NUM
m-1071	110	14	)	)	PUNCT
m-1071	110	15	,	,	PUNCT
m-1071	110	16	we	we	PRON
m-1071	110	17	get	get	VERB
m-1071	110	18	�	�	PROPN
m-1071	110	19	�	�	PROPN
m-1071	110	20	�	�	PROPN
m-1071	110	21	sc	sc	PROPN
m-1071	110	22	�	�	PROPN
m-1071	110	23	�	�	PROPN
m-1071	110	24	,	,	PUNCT
m-1071	110	25	�	�	PROPN
m-1071	110	26	�	�	PROPN
m-1071	110	27	=	=	SYM
m-1071	110	28	�	�	PROPN
m-1071	110	29	�	�	PROPN
m-1071	110	30	�	�	PROPN
m-1071	110	31	�	�	PROPN
m-1071	110	32	�	�	PROPN
m-1071	110	33	~	~	SYM
m-1071	110	34	�	�	PROPN
m-1071	110	35	�	�	PROPN
m-1071	110	36	�	�	PROPN
m-1071	110	37	�	�	PROPN
m-1071	110	38	this	this	PRON
m-1071	110	39	gives	give	VERB
m-1071	110	40	�	�	PROPN
m-1071	110	41	�	�	PROPN
m-1071	110	42	�	�	PROPN
m-1071	110	43	sc	sc	PROPN
m-1071	110	44	�	�	PROPN
m-1071	110	45	�	�	PROPN
m-1071	110	46	,	,	PUNCT
m-1071	110	47	�	�	PROPN
m-1071	110	48	�	�	PROPN
m-1071	110	49	=	=	SYM
m-1071	110	50	(	(	PUNCT
m-1071	110	51	3	3	NUM
m-1071	110	52	�	�	NOUN
m-1071	110	53	+	+	CCONJ
m-1071	110	54	2	2	NUM
m-1071	110	55	by	by	ADP
m-1071	110	56	taking	take	VERB
m-1071	110	57	the	the	DET
m-1071	110	58	first	first	ADJ
m-1071	110	59	derivative	derivative	NOUN
m-1071	110	60	of	of	ADP
m-1071	110	61	the	the	DET
m-1071	110	62	polynomial	polynomial	NOUN
m-1071	110	63	in	in	ADP
m-1071	110	64	theore	theore	ADJ
m-1071	110	65	somber	somber	ADJ
m-1071	110	66	index	index	NOUN
m-1071	110	67	of	of	ADP
m-1071	110	68	chain	chain	NOUN
m-1071	110	69	of	of	ADP
m-1071	110	70	corollary	corollary	ADJ
m-1071	110	71	2.20	2.20	NUM
m-1071	110	72	.	.	PUNCT
m-1071	111	1	for	for	ADP
m-1071	111	2	p	p	PROPN
m-1071	111	3	>	>	PROPN
m-1071	111	4	1	1	NUM
m-1071	111	5	and	and	CCONJ
m-1071	111	6	p	p	NOUN
m-1071	111	7	=	=	ADJ
m-1071	111	8	q	q	NOUN
m-1071	111	9	,	,	PUNCT
m-1071	111	10	the	the	DET
m-1071	111	11	somber	somber	ADJ
m-1071	111	12	index	index	NOUN
m-1071	111	13	of	of	ADP
m-1071	111	14	3	3	NUM
m-1071	111	15	�	�	PROPN
m-1071	111	16	−	−	PROPN
m-1071	111	17	4)√5	4)√5	PROPN
m-1071	111	18	.	.	PUNCT
m-1071	112	1	and	and	CCONJ
m-1071	112	2	p	p	NOUN
m-1071	112	3	=	=	ADJ
m-1071	112	4	q	q	X
m-1071	112	5	,	,	PUNCT
m-1071	112	6	the	the	DET
m-1071	112	7	sigma	sigma	PROPN
m-1071	112	8	polynomial	polynomial	NOUN
m-1071	112	9	of	of	ADP
m-1071	112	10	sc	sc	PROPN
m-1071	112	11	�	�	PROPN
m-1071	112	12	�	�	PROPN
m-1071	112	13	�	�	PROPN
m-1071	112	14	�	�	PROPN
m-1071	112	15	(	(	PUNCT
m-1071	112	16	3	3	NUM
m-1071	112	17	�	�	PROPN
m-1071	112	18	�	�	PROPN
m-1071	112	19	+	+	CCONJ
m-1071	112	20	3	3	NUM
m-1071	112	21	�	�	NOUN
m-1071	112	22	1	1	NUM
m-1071	112	23	”	"	PUNCT
m-1071	112	24	enter	enter	VERB
m-1071	112	25	the	the	DET
m-1071	112	26	following	follow	VERB
m-1071	112	27	formula	formula	NOUN
m-1071	112	28	sigma	sigma	NOUN
m-1071	112	29	polynomial	polynomial	ADJ
m-1071	112	30	(	(	PUNCT
m-1071	112	31	7	7	NUM
m-1071	112	32	)	)	PUNCT
m-1071	112	33	,	,	PUNCT
m-1071	112	34	we	we	PRON
m-1071	112	35	get	get	VERB
m-1071	112	36	�	�	PROPN
m-1071	112	37	�	�	PROPN
m-1071	112	38	�	�	PROPN
m-1071	112	39	(	(	PUNCT
m-1071	112	40	�	�	PROPN
m-1071	112	41	�	�	PROPN
m-1071	112	42	�	�	PROPN
m-1071	112	43	)	)	PUNCT
m-1071	112	44	�	�	PROPN
m-1071	112	45	�	�	PROPN
m-1071	112	46	�	�	PROPN
m-1071	112	47	�	�	PROPN
m-1071	112	48	�	�	PROPN
m-1071	112	49	~	~	SYM
m-1071	112	50	�	�	PROPN
m-1071	112	51	�	�	PROPN
m-1071	112	52	�	�	PROPN
m-1071	112	53	�	�	PROPN
m-1071	112	54	+	+	CCONJ
m-1071	112	55	�	�	PROPN
m-1071	112	56	�	�	PROPN
m-1071	112	57	(	(	PUNCT
m-1071	112	58	�	�	PROPN
m-1071	112	59	�	�	PROPN
m-1071	112	60	�	�	PROPN
m-1071	112	61	)	)	PUNCT
m-1071	112	62	�	�	PROPN
m-1071	112	63	�	�	PROPN
m-1071	112	64	�	�	PROPN
m-1071	112	65	�	�	PROPN
m-1071	112	66	�	�	PROPN
m-1071	112	67	~	~	SYM
m-1071	112	68	�	�	PROPN
m-1071	112	69	�	�	PROPN
m-1071	112	70	�	�	PROPN
m-1071	112	71	�	�	PROPN
m-1071	112	72	+	+	CCONJ
m-1071	112	73	�	�	PROPN
m-1071	112	74	�	�	PROPN
m-1071	112	75	(	(	PUNCT
m-1071	112	76	�	�	PROPN
m-1071	112	77	�	�	PROPN
m-1071	112	78	~	~	SYM
m-1071	112	79	�	�	PROPN
m-1071	112	80	�	�	PROPN
m-1071	112	81	(	(	PUNCT
m-1071	112	82	3	3	NUM
m-1071	112	83	�	�	NOUN
m-1071	112	84	+	+	CCONJ
m-1071	112	85	2	2	NUM
m-1071	112	86	)	)	PUNCT
m-1071	113	1	+	+	CCONJ
m-1071	113	2	(	(	PUNCT
m-1071	113	3	3	3	NUM
m-1071	113	4	�	�	PROPN
m-1071	113	5	�	�	PROPN
m-1071	113	6	+	+	CCONJ
m-1071	113	7	3	3	NUM
m-1071	113	8	�	�	NOUN
m-1071	113	9	−	−	NUM
m-1071	113	10	4	4	NUM
m-1071	113	11	)	)	PUNCT
m-1071	113	12	�	�	PROPN
m-1071	113	13	�	�	PROPN
m-1071	113	14	+	+	CCONJ
m-1071	113	15	(	(	PUNCT
m-1071	113	16	3	3	NUM
m-1071	113	17	�	�	PROPN
m-1071	113	18	�	�	PROPN
m-1071	113	19	−	−	NUM
m-1071	113	20	6	6	NUM
m-1071	113	21	�	�	NOUN
m-1071	113	22	+	+	CCONJ
m-1071	113	23	2	2	NUM
m-1071	113	24	(	(	PUNCT
m-1071	113	25	3	3	NUM
m-1071	113	26	�	�	PROPN
m-1071	113	27	�	�	PROPN
m-1071	113	28	+	+	CCONJ
m-1071	113	29	3	3	NUM
m-1071	113	30	�	�	NOUN
m-1071	113	31	−	−	NUM
m-1071	113	32	4	4	NUM
m-1071	113	33	)	)	PUNCT
m-1071	113	34	�	�	PROPN
m-1071	113	35	�	�	PROPN
m-1071	113	36	+	+	CCONJ
m-1071	113	37	3	3	NUM
m-1071	113	38	�	�	PROPN
m-1071	113	39	�	�	PROPN
m-1071	113	40	−	−	PROPN
m-1071	113	41	3	3	NUM
m-1071	113	42	�	�	PROPN
m-1071	113	43	+	+	CCONJ
m-1071	113	44	4	4	NUM
m-1071	113	45	by	by	ADP
m-1071	113	46	taking	take	VERB
m-1071	113	47	the	the	DET
m-1071	113	48	first	first	ADJ
m-1071	113	49	derivative	derivative	NOUN
m-1071	113	50	of	of	ADP
m-1071	113	51	the	the	DET
m-1071	113	52	polynomial	polynomial	NOUN
m-1071	113	53	in	in	ADP
m-1071	113	54	theorem	theorem	NOUN
m-1071	113	55	2.19	2.19	NUM
m-1071	113	56	at	at	ADP
m-1071	113	57	y	y	PROPN
m-1071	113	58	)	)	PUNCT
m-1071	113	59	as	as	SCONJ
m-1071	113	60	follows	follow	VERB
m-1071	113	61	:	:	PUNCT
m-1071	113	62	and	and	CCONJ
m-1071	113	63	p	p	X
m-1071	113	64	=	=	ADJ
m-1071	113	65	q	q	X
m-1071	113	66	,	,	PUNCT
m-1071	113	67	the	the	DET
m-1071	113	68	sigma	sigma	PROPN
m-1071	113	69	index	index	NOUN
m-1071	113	70	of	of	ADP
m-1071	113	71	sc	sc	PROPN
m-1071	113	72	�	�	PROPN
m-1071	113	73	�	�	PROPN
m-1071	113	74	p	p	PROPN
m-1071	113	75	is	be	AUX
m-1071	113	76	9(3p2	9(3p2	NOUN
m-1071	113	77	+	+	CCONJ
m-1071	113	78	3p	3p	NUM
m-1071	113	79	−	−	NOUN
m-1071	113	80	4	4	NUM
m-1071	113	81	)	)	PUNCT
m-1071	113	82	.	.	PUNCT
m-1071	114	1	p	p	X
m-1071	115	1	=	=	PUNCT
m-1071	115	2	q	q	NOUN
m-1071	115	3	,	,	PUNCT
m-1071	115	4	the	the	DET
m-1071	115	5	somber	somber	ADJ
m-1071	115	6	polynomial	polynomial	NOUN
m-1071	115	7	of	of	ADP
m-1071	115	8	sc	sc	PROPN
m-1071	115	9	�	�	PROPN
m-1071	115	10	�	�	PROPN
m-1071	115	11	is	be	AUX
m-1071	115	12	(	(	PUNCT
m-1071	115	13	3	3	NUM
m-1071	115	14	�	�	NOUN
m-1071	115	15	+	+	CCONJ
m-1071	115	16	2	2	NUM
m-1071	115	17	)	)	PUNCT
m-1071	115	18	�	�	PROPN
m-1071	115	19	6	6	NUM
m-1071	115	20	�	�	PROPN
m-1071	115	21	+	+	CCONJ
m-1071	115	22	2)	2)	NUM
m-1071	115	23	�	�	PROPN
m-1071	115	24	�	�	NOUN
m-1071	115	25	√	√	NUM
m-1071	115	26	�	�	PROPN
m-1071	115	27	1	1	NUM
m-1071	115	28	”	"	PUNCT
m-1071	115	29	enter	enter	VERB
m-1071	115	30	the	the	DET
m-1071	115	31	following	follow	VERB
m-1071	115	32	formula	formula	NOUN
m-1071	115	33	somber	somber	ADJ
m-1071	115	34	polynomial	polynomial	NOUN
m-1071	115	35	(	(	PUNCT
m-1071	115	36	7	7	NUM
m-1071	115	37	)	)	PUNCT
m-1071	115	38	,	,	PUNCT
m-1071	115	39	we	we	PRON
m-1071	115	40	get	get	VERB
m-1071	115	41	�	�	PROPN
m-1071	115	42	�	�	PROPN
m-1071	115	43	(	(	PUNCT
m-1071	115	44	�	�	PROPN
m-1071	115	45	)	)	PUNCT
m-1071	115	46	�	�	PROPN
m-1071	115	47	�	�	PROPN
m-1071	115	48	(	(	PUNCT
m-1071	115	49	�	�	PROPN
m-1071	115	50	)	)	PUNCT
m-1071	115	51	�	�	PROPN
m-1071	115	52	�	�	PROPN
m-1071	115	53	+	+	CCONJ
m-1071	115	54	�	�	PROPN
m-1071	115	55	�	�	PROPN
m-1071	115	56	�	�	PROPN
m-1071	115	57	(	(	PUNCT
m-1071	115	58	�	�	PROPN
m-1071	115	59	)	)	PUNCT
m-1071	115	60	�	�	PROPN
m-1071	115	61	�	�	PROPN
m-1071	115	62	(	(	PUNCT
m-1071	115	63	�	�	PROPN
m-1071	115	64	)	)	PUNCT
m-1071	115	65	�	�	PROPN
m-1071	115	66	�	�	PROPN
m-1071	115	67	�	�	PROPN
m-1071	115	68	�	�	PROPN
m-1071	115	69	�	�	PROPN
m-1071	115	70	~	~	SYM
m-1071	115	71	�	�	PROPN
m-1071	115	72	�	�	PROPN
m-1071	115	73	�	�	PROPN
m-1071	115	74	�	�	PROPN
m-1071	115	75	+	+	CCONJ
m-1071	115	76	�	�	PROPN
m-1071	115	77	�	�	PROPN
m-1071	115	78	�	�	PROPN
m-1071	115	79	�	�	PROPN
m-1071	115	80	�	�	PROPN
m-1071	115	81	~	~	SYM
m-1071	115	82	�	�	PROPN
m-1071	115	83	2)	2)	NUM
m-1071	115	84	�	�	PROPN
m-1071	115	85	�	�	NOUN
m-1071	115	86	√	√	NUM
m-1071	115	87	�	�	PROPN
m-1071	115	88	+	+	CCONJ
m-1071	115	89	(	(	PUNCT
m-1071	115	90	3	3	NUM
m-1071	115	91	�	�	PROPN
m-1071	115	92	�	�	PROPN
m-1071	115	93	+	+	CCONJ
m-1071	115	94	3	3	NUM
m-1071	115	95	�	�	PROPN
m-1071	115	96	−	−	PROPN
m-1071	115	97	4)	4)	NUM
m-1071	115	98	�	�	PROPN
m-1071	115	99	�	�	NOUN
m-1071	115	100	√	√	NUM
m-1071	115	101	�	�	PROPN
m-1071	115	102	+	+	CCONJ
m-1071	115	103	(	(	PUNCT
m-1071	115	104	3	3	NUM
m-1071	115	105	�	�	PROPN
m-1071	115	106	�	�	PROPN
m-1071	115	107	−	−	NUM
m-1071	115	108	6	6	NUM
m-1071	115	109	�	�	NOUN
m-1071	115	110	+	+	CCONJ
m-1071	115	111	2	2	NUM
m-1071	115	112	by	by	ADP
m-1071	115	113	taking	take	VERB
m-1071	115	114	the	the	DET
m-1071	115	115	first	first	ADJ
m-1071	115	116	derivative	derivative	NOUN
m-1071	115	117	of	of	ADP
m-1071	115	118	the	the	DET
m-1071	115	119	polynomial	polynomial	NOUN
m-1071	115	120	in	in	ADP
m-1071	115	121	theorem	theorem	NOUN
m-1071	115	122	2.19	2.19	NUM
m-1071	115	123	at	at	ADP
m-1071	115	124	y	y	PROPN
m-1071	115	125	)	)	PUNCT
m-1071	115	126	as	as	SCONJ
m-1071	115	127	follows	follow	VERB
m-1071	115	128	:	:	PUNCT
m-1071	115	129	q	q	X
m-1071	115	130	,	,	PUNCT
m-1071	115	131	the	the	DET
m-1071	115	132	somber	somber	ADJ
m-1071	115	133	index	index	NOUN
m-1071	115	134	ofsc	ofsc	PROPN
m-1071	115	135	�	�	PROPN
m-1071	115	136	�	�	PROPN
m-1071	115	137	�	�	PROPN
m-1071	115	138	�	�	PROPN
m-1071	115	139	9(2	9(2	NUM
m-1071	115	140	�	�	PROPN
m-1071	115	141	�	�	PROPN
m-1071	115	142	−	−	PROPN
m-1071	115	143	3	3	NUM
m-1071	115	144	�	�	PROPN
m-1071	115	145	+	+	CCONJ
m-1071	115	146	2)√	2)√	PROPN
m-1071	115	147	�	�	NOUN
m-1071	115	148	−	−	ADP
m-1071	115	149	4	4	NUM
m-1071	115	150	)	)	PUNCT
m-1071	115	151	�	�	PROPN
m-1071	115	152	�	�	PROPN
m-1071	115	153	+	+	CCONJ
m-1071	115	154	1	1	NUM
m-1071	115	155	”	"	PUNCT
m-1071	115	156	enter	enter	VERB
m-1071	115	157	the	the	DET
m-1071	115	158	following	follow	VERB
m-1071	115	159	formula	formula	NOUN
m-1071	115	160	sigma	sigma	NOUN
m-1071	115	161	polynomial	polynomial	ADJ
m-1071	115	162	(	(	PUNCT
m-1071	115	163	7	7	NUM
m-1071	115	164	)	)	PUNCT
m-1071	115	165	,	,	PUNCT
m-1071	115	166	we	we	PRON
m-1071	115	167	get	get	VERB
m-1071	115	168	(	(	PUNCT
m-1071	115	169	�	�	PROPN
m-1071	115	170	�	�	PROPN
m-1071	115	171	�	�	PROPN
m-1071	115	172	)	)	PUNCT
m-1071	115	173	�	�	PROPN
m-1071	115	174	2	2	NUM
m-1071	115	175	)	)	PUNCT
m-1071	115	176	y	y	NOUN
m-1071	116	1	=	=	SYM
m-1071	116	2	1	1	NUM
m-1071	116	3	,	,	PUNCT
m-1071	116	4	we	we	PRON
m-1071	116	5	get	get	VERB
m-1071	116	6	the	the	DET
m-1071	116	7	.	.	PUNCT
m-1071	116	8	)	)	PUNCT
m-1071	116	9	�	�	PROPN
m-1071	116	10	�	�	PROPN
m-1071	116	11	√	√	NUM
m-1071	116	12	�	�	PROPN
m-1071	116	13	+	+	CCONJ
m-1071	116	14	1	1	NUM
m-1071	116	15	”	"	PUNCT
m-1071	116	16	enter	enter	VERB
m-1071	116	17	the	the	DET
m-1071	116	18	following	follow	VERB
m-1071	116	19	formula	formula	NOUN
m-1071	116	20	somber	somber	ADJ
m-1071	116	21	polynomial	polynomial	NOUN
m-1071	116	22	(	(	PUNCT
m-1071	116	23	7	7	NUM
m-1071	116	24	)	)	PUNCT
m-1071	116	25	,	,	PUNCT
m-1071	116	26	we	we	PRON
m-1071	116	27	get	get	VERB
m-1071	116	28	�	�	PROPN
m-1071	116	29	(	(	PUNCT
m-1071	116	30	�	�	NOUN
m-1071	116	31	)	)	PUNCT
m-1071	116	32	�	�	PROPN
m-1071	116	33	�	�	PROPN
m-1071	116	34	(	(	PUNCT
m-1071	116	35	�	�	PROPN
m-1071	116	36	)	)	PUNCT
m-1071	116	37	�	�	PROPN
m-1071	116	38	2)	2)	NUM
m-1071	116	39	�	�	PROPN
m-1071	116	40	�	�	NOUN
m-1071	116	41	√	√	NOUN
m-1071	116	42	�	�	PROPN
m-1071	116	43	y	y	NOUN
m-1071	116	44	=	=	SYM
m-1071	116	45	1	1	NUM
m-1071	116	46	,	,	PUNCT
m-1071	116	47	we	we	PRON
m-1071	116	48	get	get	VERB
m-1071	116	49	the	the	DET
m-1071	116	50	)	)	PUNCT
m-1071	116	51	√2	√2	PROPN
m-1071	116	52	+	+	NOUN
m-1071	116	53	3(3	3(3	NUM
m-1071	116	54	�	�	PROPN
m-1071	116	55	�	�	PROPN
m-1071	116	56	+	+	CCONJ
m-1071	116	57	ijo	ijo	PROPN
m-1071	116	58	international	international	ADJ
m-1071	116	59	journal	journal	PROPN
m-1071	116	60	of	of	ADP
m-1071	116	61	mathematics	mathematics	PROPN
m-1071	116	62	(	(	PUNCT
m-1071	116	63	issn	issn	PROPN
m-1071	116	64	:	:	PUNCT
m-1071	116	65	2992	2992	NUM
m-1071	116	66	-	-	SYM
m-1071	116	67	4421	4421	NUM
m-1071	116	68	)	)	PUNCT
m-1071	116	69	ijo	ijo	PROPN
m-1071	116	70	journals	journal	NOUN
m-1071	116	71	volume	volume	NOUN
m-1071	116	72	08	08	NUM
m-1071	117	1	|	|	ADV
m-1071	117	2	issue	issue	VERB
m-1071	117	3	04	04	NUM
m-1071	118	1	|	|	CCONJ
m-1071	118	2	april	april	PROPN
m-1071	118	3	2025	2025	NUM
m-1071	119	1	|	|	ADV
m-1071	119	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	119	3	10	10	NUM
m-1071	119	4	1.2	1.2	NUM
m-1071	119	5	results	result	NOUN
m-1071	119	6	for	for	ADP
m-1071	119	7	p	p	X
m-1071	119	8	<	<	X
m-1071	119	9	q	q	X
m-1071	119	10	and	and	CCONJ
m-1071	119	11	p	p	NOUN
m-1071	119	12	is	be	AUX
m-1071	119	13	odd	odd	ADJ
m-1071	119	14	here	here	ADV
m-1071	119	15	,	,	PUNCT
m-1071	119	16	in	in	ADP
m-1071	119	17	chain	chain	NOUN
m-1071	119	18	of	of	ADP
m-1071	119	19	sio4	sio4	PROPN
m-1071	119	20	(	(	PUNCT
m-1071	119	21	sc	sc	PROPN
m-1071	119	22	�	�	PROPN
m-1071	119	23	�	�	PROPN
m-1071	119	24	)	)	PUNCT
m-1071	119	25	,	,	PUNCT
m-1071	119	26	we	we	PRON
m-1071	119	27	observed	observe	VERB
m-1071	119	28	for	for	ADP
m-1071	119	29	p	p	NOUN
m-1071	119	30	<	<	X
m-1071	119	31	q	q	NOUN
m-1071	119	32	and	and	CCONJ
m-1071	119	33	p	p	NOUN
m-1071	119	34	is	be	AUX
m-1071	119	35	odd	odd	ADJ
m-1071	119	36	,	,	PUNCT
m-1071	119	37	atom	atom	NOUN
m-1071	119	38	-	-	PUNCT
m-1071	119	39	bonds	bond	NOUN
m-1071	119	40	on	on	ADP
m-1071	119	41	the	the	DET
m-1071	119	42	bases	basis	NOUN
m-1071	119	43	of	of	ADP
m-1071	119	44	valency	valency	NOUN
m-1071	119	45	of	of	ADP
m-1071	119	46	every	every	DET
m-1071	119	47	atom	atom	NOUN
m-1071	119	48	of	of	ADP
m-1071	119	49	sc	sc	PROPN
m-1071	119	50	�	�	PROPN
m-1071	119	51	�	�	PROPN
m-1071	119	52	changed	change	VERB
m-1071	119	53	.	.	PUNCT
m-1071	120	1	so	so	ADV
m-1071	120	2	,	,	PUNCT
m-1071	120	3	on	on	ADP
m-1071	120	4	the	the	DET
m-1071	120	5	base	base	NOUN
m-1071	120	6	of	of	ADP
m-1071	120	7	valency	valency	NOUN
m-1071	120	8	,	,	PUNCT
m-1071	120	9	table	table	NOUN
m-1071	120	10	2	2	NUM
m-1071	120	11	provides	provide	VERB
m-1071	120	12	the	the	DET
m-1071	120	13	partition	partition	NOUN
m-1071	120	14	of	of	ADP
m-1071	120	15	the	the	DET
m-1071	120	16	set	set	NOUN
m-1071	120	17	of	of	ADP
m-1071	120	18	atom	atom	NOUN
m-1071	120	19	-	-	PUNCT
m-1071	120	20	bonds	bond	NOUN
m-1071	120	21	.	.	PUNCT
m-1071	121	1	table	table	NOUN
m-1071	121	2	2	2	NUM
m-1071	121	3	:	:	PUNCT
m-1071	121	4	atom	atom	NOUN
m-1071	121	5	-	-	PUNCT
m-1071	121	6	bond	bond	NOUN
m-1071	121	7	partition	partition	NOUN
m-1071	121	8	of	of	ADP
m-1071	121	9	sc	sc	PROPN
m-1071	121	10	�	�	PROPN
m-1071	121	11	�	�	PROPN
m-1071	121	12	,	,	PUNCT
m-1071	121	13	for	for	ADP
m-1071	121	14	p	p	PROPN
m-1071	121	15	is	be	AUX
m-1071	121	16	odd	odd	ADJ
m-1071	121	17	and	and	CCONJ
m-1071	121	18	p	p	X
m-1071	121	19	<	<	X
m-1071	121	20	q	q	NOUN
m-1071	121	21	type	type	NOUN
m-1071	121	22	of	of	ADP
m-1071	121	23	atom	atom	NOUN
m-1071	121	24	-	-	PUNCT
m-1071	121	25	bond	bond	NOUN
m-1071	121	26	3	3	NUM
m-1071	121	27	=	=	SYM
m-1071	121	28	de	de	X
m-1071	121	29	∼	∼	X
m-1071	121	30	df	df	NOUN
m-1071	121	31	=	=	SYM
m-1071	121	32	3	3	NUM
m-1071	121	33	3	3	NUM
m-1071	121	34	=	=	SYM
m-1071	121	35	de	de	X
m-1071	121	36	∼	∼	X
m-1071	121	37	df	df	NOUN
m-1071	121	38	=	=	NUM
m-1071	121	39	6	6	NUM
m-1071	121	40	6	6	NUM
m-1071	121	41	=	=	SYM
m-1071	121	42	de	de	X
m-1071	121	43	∼	∼	X
m-1071	121	44	df	df	NOUN
m-1071	121	45	=	=	NUM
m-1071	121	46	6	6	NUM
m-1071	121	47	number	number	NOUN
m-1071	121	48	of	of	ADP
m-1071	121	49	atom	atom	NOUN
m-1071	121	50	bonds	bond	NOUN
m-1071	121	51	3(p	3(p	NUM
m-1071	121	52	+	+	SYM
m-1071	121	53	1	1	NUM
m-1071	121	54	)	)	PUNCT
m-1071	121	55	3pq	3pq	NOUN
m-1071	122	1	+	+	CCONJ
m-1071	123	1	p	p	X
m-1071	124	1	+	+	NOUN
m-1071	124	2	2q	2q	NUM
m-1071	124	3	−	−	NUM
m-1071	124	4	5	5	NUM
m-1071	124	5	3pq	3pq	NOUN
m-1071	124	6	−	−	NOUN
m-1071	125	1	2(2p	2(2p	NUM
m-1071	125	2	+	+	SYM
m-1071	125	3	q	q	ADJ
m-1071	125	4	−	−	PROPN
m-1071	125	5	1	1	NUM
m-1071	125	6	)	)	PUNCT
m-1071	125	7	theorem	theorem	VERB
m-1071	125	8	2.21	2.21	NUM
m-1071	125	9	.	.	PUNCT
m-1071	126	1	let	let	VERB
m-1071	126	2	p	p	PRON
m-1071	126	3	be	be	AUX
m-1071	126	4	odd	odd	ADJ
m-1071	126	5	and	and	CCONJ
m-1071	126	6	p	p	X
m-1071	126	7	<	<	X
m-1071	126	8	q.	q.	PROPN
m-1071	127	1	then	then	ADV
m-1071	127	2	the	the	DET
m-1071	127	3	harmonic	harmonic	ADJ
m-1071	127	4	polynomial	polynomial	ADJ
m-1071	127	5	ofsc	ofsc	PROPN
m-1071	127	6	�	�	PROPN
m-1071	127	7	�	�	PROPN
m-1071	127	8	�	�	PROPN
m-1071	127	9	�	�	PROPN
m-1071	127	10	3	3	NUM
m-1071	127	11	(	(	PUNCT
m-1071	127	12	�	�	PROPN
m-1071	127	13	+	+	CCONJ
m-1071	127	14	1	1	NUM
m-1071	127	15	)	)	PUNCT
m-1071	127	16	�	�	PROPN
m-1071	127	17	�	�	PROPN
m-1071	127	18	�	�	PROPN
m-1071	127	19	+	+	CCONJ
m-1071	127	20	(	(	PUNCT
m-1071	127	21	3	3	NUM
m-1071	127	22	�	�	PROPN
m-1071	127	23	�	�	PROPN
m-1071	127	24	+	+	CCONJ
m-1071	127	25	�	�	PROPN
m-1071	127	26	+	+	CCONJ
m-1071	127	27	2	2	NUM
m-1071	127	28	�	�	NOUN
m-1071	127	29	−	−	NUM
m-1071	127	30	5	5	NUM
m-1071	127	31	)	)	PUNCT
m-1071	127	32	�	�	PROPN
m-1071	127	33	�	�	PROPN
m-1071	127	34	�	�	PROPN
m-1071	127	35	+	+	CCONJ
m-1071	127	36	(	(	PUNCT
m-1071	127	37	3	3	NUM
m-1071	127	38	�	�	PROPN
m-1071	127	39	�	�	PROPN
m-1071	127	40	−	−	NUM
m-1071	127	41	2(2	2(2	NUM
m-1071	127	42	�	�	PROPN
m-1071	127	43	+	+	CCONJ
m-1071	127	44	�	�	PROPN
m-1071	127	45	−	−	ADP
m-1071	127	46	1	1	NUM
m-1071	127	47	)	)	PUNCT
m-1071	127	48	�	�	PROPN
m-1071	127	49	�	�	PROPN
m-1071	127	50	�	�	PROPN
m-1071	127	51	proof	proof	NOUN
m-1071	127	52	.	.	PUNCT
m-1071	128	1	using	use	VERB
m-1071	128	2	the	the	DET
m-1071	128	3	atom	atom	NOUN
m-1071	128	4	-	-	PUNCT
m-1071	128	5	bond	bond	NOUN
m-1071	128	6	partition	partition	NOUN
m-1071	128	7	from	from	ADP
m-1071	128	8	table	table	NOUN
m-1071	128	9	2	2	NUM
m-1071	128	10	,	,	PUNCT
m-1071	128	11	in	in	ADP
m-1071	128	12	the	the	DET
m-1071	128	13	formula	formula	NOUN
m-1071	128	14	of	of	ADP
m-1071	128	15	harmonic	harmonic	ADJ
m-1071	128	16	polynomial	polynomial	ADJ
m-1071	128	17	(	(	PUNCT
m-1071	128	18	1	1	NUM
m-1071	128	19	)	)	PUNCT
m-1071	128	20	,	,	PUNCT
m-1071	128	21	we	we	PRON
m-1071	128	22	get	get	VERB
m-1071	128	23	�	�	PROPN
m-1071	128	24	�	�	PROPN
m-1071	128	25	sc	sc	PROPN
m-1071	128	26	�	�	PROPN
m-1071	128	27	�	�	PROPN
m-1071	128	28	,	,	PUNCT
m-1071	128	29	�	�	PROPN
m-1071	128	30	�	�	PROPN
m-1071	128	31	=	=	SYM
m-1071	128	32	�	�	PROPN
m-1071	128	33	�	�	PROPN
m-1071	128	34	�	�	PROPN
m-1071	128	35	�	�	PROPN
m-1071	128	36	�	�	PROPN
m-1071	128	37	�	�	PROPN
m-1071	128	38	�	�	PROPN
m-1071	128	39	�	�	PROPN
m-1071	128	40	�	�	PROPN
m-1071	128	41	�	�	PROPN
m-1071	128	42	~	~	SYM
m-1071	128	43	�	�	PROPN
m-1071	128	44	�	�	PROPN
m-1071	128	45	�	�	PROPN
m-1071	128	46	�	�	PROPN
m-1071	128	47	+	+	CCONJ
m-1071	128	48	�	�	PROPN
m-1071	128	49	�	�	PROPN
m-1071	128	50	�	�	PROPN
m-1071	128	51	�	�	PROPN
m-1071	128	52	�	�	PROPN
m-1071	128	53	�	�	PROPN
m-1071	128	54	�	�	PROPN
m-1071	128	55	�	�	PROPN
m-1071	128	56	~	~	SYM
m-1071	128	57	�	�	PROPN
m-1071	128	58	+	+	CCONJ
m-1071	128	59	�	�	PROPN
m-1071	128	60	�	�	PROPN
m-1071	128	61	�	�	PROPN
m-1071	128	62	�	�	PROPN
m-1071	128	63	�	�	PROPN
m-1071	128	64	�	�	PROPN
m-1071	128	65	�	�	PROPN
m-1071	128	66	�	�	PROPN
m-1071	128	67	�	�	PROPN
m-1071	128	68	�	�	PROPN
m-1071	128	69	~	~	SYM
m-1071	128	70	�	�	PROPN
m-1071	128	71	�	�	PROPN
m-1071	128	72	�	�	PROPN
m-1071	128	73	�	�	PROPN
m-1071	128	74	this	this	PRON
m-1071	128	75	gives	give	VERB
m-1071	128	76	�	�	PROPN
m-1071	128	77	�	�	PROPN
m-1071	128	78	sc	sc	PROPN
m-1071	128	79	�	�	PROPN
m-1071	128	80	�	�	PROPN
m-1071	128	81	,	,	PUNCT
m-1071	128	82	�	�	PROPN
m-1071	128	83	�	�	PROPN
m-1071	128	84	=	=	SYM
m-1071	128	85	3	3	NUM
m-1071	128	86	(	(	PUNCT
m-1071	128	87	�	�	PROPN
m-1071	128	88	+	+	CCONJ
m-1071	128	89	1	1	NUM
m-1071	128	90	)	)	PUNCT
m-1071	128	91	�	�	PROPN
m-1071	128	92	�	�	PROPN
m-1071	128	93	�	�	PROPN
m-1071	128	94	+	+	CCONJ
m-1071	128	95	(	(	PUNCT
m-1071	128	96	3	3	NUM
m-1071	128	97	�	�	PROPN
m-1071	128	98	�	�	PROPN
m-1071	128	99	+	+	CCONJ
m-1071	128	100	�	�	PROPN
m-1071	128	101	+	+	CCONJ
m-1071	128	102	2	2	NUM
m-1071	128	103	�	�	NOUN
m-1071	128	104	−	−	NUM
m-1071	128	105	5	5	NUM
m-1071	128	106	)	)	PUNCT
m-1071	128	107	�	�	PROPN
m-1071	128	108	�	�	PROPN
m-1071	128	109	�	�	PROPN
m-1071	128	110	+	+	CCONJ
m-1071	128	111	(	(	PUNCT
m-1071	128	112	3	3	NUM
m-1071	128	113	�	�	PROPN
m-1071	128	114	�	�	PROPN
m-1071	128	115	−	−	NUM
m-1071	128	116	2(2	2(2	NUM
m-1071	128	117	�	�	PROPN
m-1071	128	118	+	+	CCONJ
m-1071	128	119	�	�	PROPN
m-1071	128	120	−	−	ADP
m-1071	128	121	1	1	NUM
m-1071	128	122	)	)	PUNCT
m-1071	128	123	�	�	PROPN
m-1071	128	124	�	�	PROPN
m-1071	128	125	�	�	PROPN
m-1071	128	126	by	by	ADP
m-1071	128	127	taking	take	VERB
m-1071	128	128	the	the	DET
m-1071	128	129	first	first	ADJ
m-1071	128	130	derivative	derivative	NOUN
m-1071	128	131	of	of	ADP
m-1071	128	132	the	the	DET
m-1071	128	133	polynomial	polynomial	NOUN
m-1071	128	134	in	in	ADP
m-1071	128	135	theorem	theorem	NOUN
m-1071	128	136	2.21	2.21	NUM
m-1071	128	137	at	at	ADP
m-1071	128	138	y	y	PROPN
m-1071	128	139	=	=	SYM
m-1071	128	140	1	1	NUM
m-1071	128	141	,	,	PUNCT
m-1071	128	142	we	we	PRON
m-1071	128	143	get	get	VERB
m-1071	128	144	the	the	DET
m-1071	128	145	harmonic	harmonic	ADJ
m-1071	128	146	index	index	NOUN
m-1071	128	147	of	of	ADP
m-1071	128	148	silicate	silicate	PROPN
m-1071	128	149	network	network	PROPN
m-1071	128	150	sc	sc	PROPN
m-1071	128	151	�	�	PROPN
m-1071	128	152	�	�	PROPN
m-1071	128	153	as	as	SCONJ
m-1071	128	154	follows	follow	VERB
m-1071	128	155	:	:	PUNCT
m-1071	128	156	corollary	corollary	ADJ
m-1071	128	157	2.22	2.22	NUM
m-1071	128	158	.	.	PUNCT
m-1071	129	1	let	let	VERB
m-1071	129	2	p	p	PRON
m-1071	129	3	be	be	AUX
m-1071	129	4	odd	odd	ADJ
m-1071	129	5	and	and	CCONJ
m-1071	129	6	p	p	X
m-1071	129	7	<	<	X
m-1071	129	8	q.	q.	PROPN
m-1071	130	1	then	then	ADV
m-1071	130	2	the	the	DET
m-1071	130	3	harmonic	harmonic	ADJ
m-1071	130	4	index	index	NOUN
m-1071	130	5	of	of	ADP
m-1071	130	6	sc	sc	PROPN
m-1071	130	7	�	�	PROPN
m-1071	130	8	�	�	PROPN
m-1071	130	9	�	�	PROPN
m-1071	130	10	�	�	PROPN
m-1071	130	11	�	�	PROPN
m-1071	130	12	+	+	CCONJ
m-1071	130	13	�	�	PROPN
m-1071	130	14	�	�	PROPN
m-1071	130	15	�	�	PROPN
m-1071	130	16	�	�	PROPN
m-1071	130	17	�	�	PROPN
m-1071	130	18	�	�	PROPN
m-1071	130	19	�	�	PROPN
m-1071	130	20	�	�	PROPN
m-1071	130	21	�	�	PROPN
m-1071	130	22	�	�	PROPN
m-1071	130	23	�	�	PROPN
m-1071	130	24	�	�	PROPN
m-1071	130	25	�	�	PROPN
m-1071	130	26	�	�	PROPN
m-1071	130	27	�	�	PROPN
m-1071	130	28	+	+	CCONJ
m-1071	130	29	1	1	NUM
m-1071	130	30	.	.	X
m-1071	130	31	theorem	theorem	VERB
m-1071	130	32	2.23	2.23	NUM
m-1071	130	33	.	.	PUNCT
m-1071	131	1	let	let	VERB
m-1071	131	2	p	p	PRON
m-1071	131	3	be	be	AUX
m-1071	131	4	odd	odd	ADJ
m-1071	131	5	and	and	CCONJ
m-1071	131	6	p	p	X
m-1071	131	7	<	<	X
m-1071	131	8	q.	q.	PROPN
m-1071	132	1	then	then	ADV
m-1071	132	2	the	the	DET
m-1071	132	3	abc	abc	PROPN
m-1071	132	4	polynomial	polynomial	PROPN
m-1071	132	5	ofsc	ofsc	PROPN
m-1071	132	6	�	�	PROPN
m-1071	132	7	�	�	PROPN
m-1071	132	8	�	�	PROPN
m-1071	132	9	�	�	PROPN
m-1071	132	10	3	3	NUM
m-1071	132	11	(	(	PUNCT
m-1071	132	12	�	�	PROPN
m-1071	132	13	+	+	CCONJ
m-1071	132	14	1	1	NUM
m-1071	132	15	)	)	PUNCT
m-1071	132	16	�	�	PROPN
m-1071	132	17	�	�	PROPN
m-1071	132	18	�	�	PROPN
m-1071	132	19	(	(	PUNCT
m-1071	132	20	3	3	NUM
m-1071	132	21	�	�	PROPN
m-1071	132	22	�	�	PROPN
m-1071	132	23	+	+	CCONJ
m-1071	132	24	�	�	PROPN
m-1071	132	25	+	+	CCONJ
m-1071	132	26	2	2	NUM
m-1071	132	27	�	�	NOUN
m-1071	132	28	−	−	NUM
m-1071	132	29	5	5	NUM
m-1071	132	30	)	)	PUNCT
m-1071	132	31	�	�	PROPN
m-1071	132	32	�	�	PROPN
m-1071	132	33	�	�	PROPN
m-1071	132	34	�	�	PROPN
m-1071	132	35	+	+	CCONJ
m-1071	132	36	(	(	PUNCT
m-1071	132	37	3	3	NUM
m-1071	132	38	�	�	PROPN
m-1071	132	39	�	�	PROPN
m-1071	132	40	−	−	NUM
m-1071	132	41	2(2	2(2	NUM
m-1071	132	42	�	�	PROPN
m-1071	132	43	+	+	CCONJ
m-1071	132	44	�	�	PROPN
m-1071	132	45	−	−	ADP
m-1071	132	46	1	1	NUM
m-1071	132	47	)	)	PUNCT
m-1071	132	48	�	�	PROPN
m-1071	132	49	�	�	PROPN
m-1071	132	50	�	�	PROPN
m-1071	132	51	�	�	PROPN
m-1071	132	52	proof	proof	NOUN
m-1071	132	53	.	.	PUNCT
m-1071	133	1	using	use	VERB
m-1071	133	2	the	the	DET
m-1071	133	3	atom	atom	NOUN
m-1071	133	4	-	-	PUNCT
m-1071	133	5	bond	bond	NOUN
m-1071	133	6	partition	partition	NOUN
m-1071	133	7	from	from	ADP
m-1071	133	8	table	table	NOUN
m-1071	133	9	2	2	NUM
m-1071	133	10	,	,	PUNCT
m-1071	133	11	in	in	ADP
m-1071	133	12	the	the	DET
m-1071	133	13	formula	formula	NOUN
m-1071	133	14	of	of	ADP
m-1071	133	15	abc	abc	PROPN
m-1071	133	16	polynomial	polynomial	PROPN
m-1071	133	17	(	(	PUNCT
m-1071	133	18	2	2	NUM
m-1071	133	19	)	)	PUNCT
m-1071	133	20	,	,	PUNCT
m-1071	133	21	we	we	PRON
m-1071	133	22	get	get	VERB
m-1071	133	23	�	�	PROPN
m-1071	133	24	�	�	PROPN
m-1071	133	25	�	�	PROPN
m-1071	133	26	�	�	PROPN
m-1071	133	27	sc	sc	PROPN
m-1071	133	28	�	�	PROPN
m-1071	133	29	�	�	PROPN
m-1071	133	30	,	,	PUNCT
m-1071	133	31	�	�	PROPN
m-1071	133	32	�	�	PROPN
m-1071	133	33	=	=	SYM
m-1071	134	1	�	�	PROPN
m-1071	134	2	�	�	PROPN
m-1071	134	3	�	�	PROPN
m-1071	134	4	�	�	PROPN
m-1071	134	5	�	�	PROPN
m-1071	134	6	�	�	PROPN
m-1071	134	7	�	�	PROPN
m-1071	134	8	�	�	PROPN
m-1071	134	9	(	(	PUNCT
m-1071	134	10	�	�	PROPN
m-1071	134	11	)	)	PUNCT
m-1071	134	12	(	(	PUNCT
m-1071	134	13	�	�	PROPN
m-1071	134	14	)	)	PUNCT
m-1071	134	15	�	�	PROPN
m-1071	134	16	�	�	PROPN
m-1071	134	17	�	�	PROPN
m-1071	134	18	�	�	PROPN
m-1071	134	19	~	~	SYM
m-1071	134	20	�	�	PROPN
m-1071	134	21	�	�	PROPN
m-1071	134	22	�	�	PROPN
m-1071	134	23	�	�	PROPN
m-1071	134	24	+	+	CCONJ
m-1071	134	25	�	�	PROPN
m-1071	134	26	�	�	PROPN
m-1071	134	27	�	�	PROPN
m-1071	134	28	�	�	PROPN
m-1071	134	29	�	�	PROPN
m-1071	134	30	�	�	PROPN
m-1071	134	31	�	�	PROPN
m-1071	134	32	�	�	PROPN
m-1071	134	33	(	(	PUNCT
m-1071	134	34	�	�	PROPN
m-1071	134	35	)	)	PUNCT
m-1071	134	36	(	(	PUNCT
m-1071	134	37	�	�	PROPN
m-1071	134	38	)	)	PUNCT
m-1071	134	39	�	�	PROPN
m-1071	134	40	�	�	PROPN
m-1071	134	41	�	�	PROPN
m-1071	134	42	�	�	PROPN
m-1071	134	43	~	~	SYM
m-1071	134	44	�	�	PROPN
m-1071	134	45	�	�	PROPN
m-1071	134	46	�	�	PROPN
m-1071	134	47	�	�	PROPN
m-1071	134	48	+	+	CCONJ
m-1071	134	49	�	�	PROPN
m-1071	134	50	�	�	PROPN
m-1071	134	51	�	�	PROPN
m-1071	134	52	�	�	PROPN
m-1071	134	53	�	�	PROPN
m-1071	134	54	�	�	PROPN
m-1071	134	55	�	�	PROPN
m-1071	134	56	�	�	PROPN
m-1071	134	57	(	(	PUNCT
m-1071	134	58	�	�	PROPN
m-1071	134	59	)	)	PUNCT
m-1071	134	60	(	(	PUNCT
m-1071	134	61	�	�	PROPN
m-1071	134	62	)	)	PUNCT
m-1071	134	63	�	�	PROPN
m-1071	134	64	�	�	PROPN
m-1071	134	65	�	�	PROPN
m-1071	134	66	�	�	PROPN
m-1071	134	67	~	~	SYM
m-1071	134	68	�	�	PROPN
m-1071	134	69	�	�	PROPN
m-1071	134	70	�	�	PROPN
m-1071	134	71	�	�	PROPN
m-1071	134	72	ijo	ijo	PROPN
m-1071	134	73	international	international	PROPN
m-1071	134	74	journal	journal	PROPN
m-1071	134	75	of	of	ADP
m-1071	134	76	mathematics	mathematics	PROPN
m-1071	134	77	(	(	PUNCT
m-1071	134	78	issn	issn	PROPN
m-1071	134	79	:	:	PUNCT
m-1071	134	80	2992	2992	NUM
m-1071	134	81	-	-	SYM
m-1071	134	82	4421	4421	NUM
m-1071	134	83	)	)	PUNCT
m-1071	134	84	ijo	ijo	PROPN
m-1071	134	85	journals	journal	NOUN
m-1071	134	86	volume	volume	NOUN
m-1071	134	87	08	08	NUM
m-1071	135	1	|	|	ADV
m-1071	135	2	issue	issue	VERB
m-1071	135	3	04	04	NUM
m-1071	136	1	|	|	CCONJ
m-1071	136	2	april	april	PROPN
m-1071	136	3	2025	2025	NUM
m-1071	137	1	|	|	ADV
m-1071	137	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	137	3	11	11	NUM
m-1071	137	4	this	this	PRON
m-1071	137	5	gives	give	VERB
m-1071	137	6	�	�	PROPN
m-1071	137	7	�	�	PROPN
m-1071	137	8	�	�	PROPN
m-1071	137	9	�	�	PROPN
m-1071	137	10	sc	sc	PROPN
m-1071	137	11	�	�	PROPN
m-1071	137	12	�	�	PROPN
m-1071	137	13	,	,	PUNCT
m-1071	137	14	�	�	PROPN
m-1071	137	15	�	�	PROPN
m-1071	137	16	=	=	SYM
m-1071	137	17	3	3	NUM
m-1071	137	18	(	(	PUNCT
m-1071	137	19	�	�	PROPN
m-1071	137	20	+	+	CCONJ
m-1071	137	21	1	1	NUM
m-1071	137	22	)	)	PUNCT
m-1071	137	23	�	�	NOUN
m-1071	137	24	by	by	ADP
m-1071	137	25	taking	take	VERB
m-1071	137	26	the	the	DET
m-1071	137	27	first	first	ADJ
m-1071	137	28	derivative	derivative	NOUN
m-1071	137	29	of	of	ADP
m-1071	137	30	the	the	DET
m-1071	137	31	polynomial	polynomial	NOUN
m-1071	137	32	in	in	ADP
m-1071	137	33	theorem	theorem	ADJ
m-1071	137	34	2.23	2.23	NUM
m-1071	137	35	at	at	ADP
m-1071	137	36	index	index	NOUN
m-1071	137	37	of	of	ADP
m-1071	137	38	chain	chain	NOUN
m-1071	137	39	of	of	ADP
m-1071	137	40	sio4	sio4	PROPN
m-1071	137	41	(	(	PUNCT
m-1071	137	42	sc	sc	PROPN
m-1071	137	43	�	�	PROPN
m-1071	137	44	�	�	PROPN
m-1071	137	45	)	)	PUNCT
m-1071	137	46	as	as	SCONJ
m-1071	137	47	follows	follow	VERB
m-1071	137	48	:	:	PUNCT
m-1071	137	49	corollary	corollary	ADJ
m-1071	137	50	2.24	2.24	NUM
m-1071	137	51	.	.	PUNCT
m-1071	138	1	let	let	VERB
m-1071	138	2	p	p	PRON
m-1071	138	3	be	be	AUX
m-1071	138	4	odd	odd	ADJ
m-1071	138	5	and	and	CCONJ
m-1071	138	6	p	p	X
m-1071	138	7	<	<	X
m-1071	138	8	q.	q.	PROPN
m-1071	139	1	then	then	ADV
m-1071	139	2	the	the	DET
m-1071	139	3	abc	abc	PROPN
m-1071	139	4	index	index	NOUN
m-1071	139	5	of	of	ADP
m-1071	139	6	theorem	theorem	PROPN
m-1071	139	7	2.25	2.25	NUM
m-1071	139	8	.	.	PUNCT
m-1071	140	1	let	let	VERB
m-1071	140	2	p	p	PRON
m-1071	140	3	be	be	AUX
m-1071	140	4	odd	odd	ADJ
m-1071	140	5	and	and	CCONJ
m-1071	140	6	p	p	X
m-1071	140	7	<	<	X
m-1071	140	8	q.	q.	PROPN
m-1071	140	9	then	then	ADV
m-1071	140	10	the	the	DET
m-1071	140	11	forgotten	forget	VERB
m-1071	140	12	topological	topological	ADJ
m-1071	140	13	polynomial	polynomial	NOUN
m-1071	140	14	of	of	ADP
m-1071	140	15	3(p	3(p	NUM
m-1071	140	16	+	+	CCONJ
m-1071	140	17	1)y18	1)y18	NUM
m-1071	140	18	+	+	CCONJ
m-1071	140	19	(	(	PUNCT
m-1071	140	20	3pq	3pq	ADJ
m-1071	141	1	+	+	CCONJ
m-1071	142	1	p	p	X
m-1071	142	2	+	+	NOUN
m-1071	142	3	2q	2q	NUM
m-1071	142	4	−	−	NUM
m-1071	142	5	5)y	5)y	NUM
m-1071	142	6	proof	proof	NOUN
m-1071	142	7	.	.	PUNCT
m-1071	143	1	using	use	VERB
m-1071	143	2	the	the	DET
m-1071	143	3	atom	atom	NOUN
m-1071	143	4	-	-	PUNCT
m-1071	143	5	bond	bond	NOUN
m-1071	143	6	partition	partition	NOUN
m-1071	143	7	from	from	ADP
m-1071	143	8	table	table	NOUN
m-1071	143	9	2	2	NUM
m-1071	143	10	,	,	PUNCT
m-1071	143	11	in	in	ADP
m-1071	143	12	the	the	DET
m-1071	143	13	formula	formula	NOUN
m-1071	143	14	of	of	ADP
m-1071	143	15	forgotten	forget	VERB
m-1071	143	16	topological	topological	ADJ
m-1071	143	17	polynomial	polynomial	NOUN
m-1071	143	18	(	(	PUNCT
m-1071	143	19	3	3	NUM
m-1071	143	20	)	)	PUNCT
m-1071	143	21	,	,	PUNCT
m-1071	143	22	we	we	PRON
m-1071	143	23	get	get	VERB
m-1071	143	24	�	�	PROPN
m-1071	143	25	�	�	PROPN
m-1071	143	26	sc	sc	PROPN
m-1071	143	27	�	�	PROPN
m-1071	143	28	�	�	PROPN
m-1071	143	29	,	,	PUNCT
m-1071	143	30	�	�	PROPN
m-1071	143	31	�	�	PROPN
m-1071	143	32	=	=	SYM
m-1071	143	33	�	�	PROPN
m-1071	143	34	�	�	PROPN
m-1071	143	35	�	�	PROPN
m-1071	143	36	�	�	PROPN
m-1071	143	37	�	�	PROPN
m-1071	143	38	~	~	SYM
m-1071	143	39	�	�	PROPN
m-1071	143	40	�	�	PROPN
m-1071	143	41	this	this	PRON
m-1071	143	42	gives	give	VERB
m-1071	143	43	�	�	PROPN
m-1071	143	44	�	�	PROPN
m-1071	143	45	sc	sc	PROPN
m-1071	143	46	�	�	PROPN
m-1071	143	47	�	�	PROPN
m-1071	143	48	,	,	PUNCT
m-1071	143	49	�	�	PROPN
m-1071	143	50	�	�	PROPN
m-1071	143	51	=	=	SYM
m-1071	143	52	3	3	NUM
m-1071	143	53	(	(	PUNCT
m-1071	143	54	�	�	PROPN
m-1071	143	55	+	+	CCONJ
m-1071	143	56	1	1	NUM
m-1071	143	57	)	)	PUNCT
m-1071	143	58	�	�	PROPN
m-1071	143	59	�	�	PROPN
m-1071	143	60	�	�	PROPN
m-1071	143	61	by	by	ADP
m-1071	143	62	taking	take	VERB
m-1071	143	63	the	the	DET
m-1071	143	64	first	first	ADJ
m-1071	143	65	derivative	derivative	NOUN
m-1071	143	66	of	of	ADP
m-1071	143	67	the	the	DET
m-1071	143	68	polynomial	polynomial	NOUN
m-1071	143	69	in	in	ADP
m-1071	143	70	theorem	theorem	ADJ
m-1071	143	71	2.25	2.25	NUM
m-1071	143	72	at	at	ADP
m-1071	143	73	forgotten	forget	VERB
m-1071	143	74	topological	topological	ADJ
m-1071	143	75	index	index	NOUN
m-1071	143	76	of	of	ADP
m-1071	143	77	chain	chain	NOUN
m-1071	143	78	of	of	ADP
m-1071	143	79	corollary	corollary	ADJ
m-1071	143	80	2.26	2.26	NUM
m-1071	143	81	.	.	PUNCT
m-1071	144	1	let	let	VERB
m-1071	144	2	p	p	PRON
m-1071	144	3	be	be	AUX
m-1071	144	4	odd	odd	ADJ
m-1071	144	5	and	and	CCONJ
m-1071	144	6	p	p	X
m-1071	144	7	<	<	X
m-1071	144	8	q.	q.	PROPN
m-1071	144	9	then	then	ADV
m-1071	144	10	the	the	DET
m-1071	144	11	forgotten	forget	VERB
m-1071	144	12	topological	topological	PROPN
m-1071	144	13	i	i	PROPN
m-1071	144	14	−	−	PROPN
m-1071	144	15	189p	189p	PROPN
m-1071	144	16	−	−	PROPN
m-1071	144	17	54q	54q	NOUN
m-1071	144	18	−	−	PROPN
m-1071	144	19	27	27	NUM
m-1071	144	20	.	.	PUNCT
m-1071	145	1	theorem	theorem	VERB
m-1071	145	2	2.27	2.27	NUM
m-1071	145	3	.	.	PUNCT
m-1071	146	1	let	let	VERB
m-1071	146	2	p	p	PRON
m-1071	146	3	be	be	AUX
m-1071	146	4	odd	odd	ADJ
m-1071	146	5	and	and	CCONJ
m-1071	146	6	p	p	X
m-1071	146	7	<	<	X
m-1071	146	8	q.	q.	PROPN
m-1071	147	1	then	then	ADV
m-1071	147	2	the	the	DET
m-1071	147	3	geometric	geometric	ADJ
m-1071	147	4	arithmetic	arithmetic	ADJ
m-1071	147	5	polynomial	polynomial	NOUN
m-1071	147	6	of	of	ADP
m-1071	147	7	3	3	NUM
m-1071	147	8	(	(	PUNCT
m-1071	147	9	�	�	PROPN
m-1071	147	10	+	+	CCONJ
m-1071	147	11	1	1	NUM
m-1071	147	12	)	)	PUNCT
m-1071	147	13	�	�	PROPN
m-1071	147	14	√	√	NUM
m-1071	147	15	�	�	PROPN
m-1071	147	16	�	�	PROPN
m-1071	147	17	+	+	CCONJ
m-1071	147	18	proof	proof	NOUN
m-1071	147	19	.	.	PUNCT
m-1071	148	1	using	use	VERB
m-1071	148	2	the	the	DET
m-1071	148	3	atom	atom	NOUN
m-1071	148	4	-	-	PUNCT
m-1071	148	5	bond	bond	NOUN
m-1071	148	6	partition	partition	NOUN
m-1071	148	7	from	from	ADP
m-1071	148	8	table	table	NOUN
m-1071	148	9	2	2	NUM
m-1071	148	10	,	,	PUNCT
m-1071	148	11	in	in	ADP
m-1071	148	12	the	the	DET
m-1071	148	13	formula	formula	NOUN
m-1071	148	14	of	of	ADP
m-1071	148	15	geometric	geometric	ADJ
m-1071	148	16	polynomial	polynomial	NOUN
m-1071	148	17	(	(	PUNCT
m-1071	148	18	4	4	NUM
m-1071	148	19	)	)	PUNCT
m-1071	148	20	,	,	PUNCT
m-1071	148	21	we	we	PRON
m-1071	148	22	get	get	VERB
m-1071	148	23	�	�	PROPN
m-1071	148	24	�	�	PROPN
m-1071	148	25	�	�	PROPN
m-1071	148	26	�	�	PROPN
m-1071	148	27	sc	sc	PROPN
m-1071	148	28	�	�	PROPN
m-1071	148	29	�	�	PROPN
m-1071	148	30	,	,	PUNCT
m-1071	148	31	�	�	PROPN
m-1071	148	32	�	�	PROPN
m-1071	148	33	=	=	SYM
m-1071	148	34	�	�	PROPN
m-1071	148	35	)	)	PUNCT
m-1071	148	36	�	�	PROPN
m-1071	148	37	�	�	PROPN
m-1071	148	38	�	�	PROPN
m-1071	148	39	+	+	CCONJ
m-1071	148	40	(	(	PUNCT
m-1071	148	41	3	3	NUM
m-1071	148	42	�	�	PROPN
m-1071	148	43	�	�	PROPN
m-1071	148	44	+	+	CCONJ
m-1071	148	45	�	�	PROPN
m-1071	148	46	+	+	CCONJ
m-1071	148	47	2	2	NUM
m-1071	148	48	�	�	NOUN
m-1071	148	49	−	−	NUM
m-1071	148	50	5	5	NUM
m-1071	148	51	)	)	PUNCT
m-1071	148	52	�	�	PROPN
m-1071	148	53	�	�	PROPN
m-1071	148	54	�	�	PROPN
m-1071	148	55	�	�	PROPN
m-1071	148	56	+	+	CCONJ
m-1071	148	57	(	(	PUNCT
m-1071	148	58	3	3	NUM
m-1071	148	59	�	�	PROPN
m-1071	148	60	�	�	PROPN
m-1071	148	61	−	−	NUM
m-1071	148	62	2(2	2(2	NUM
m-1071	148	63	�	�	PROPN
m-1071	148	64	+	+	CCONJ
m-1071	148	65	by	by	ADP
m-1071	148	66	taking	take	VERB
m-1071	148	67	the	the	DET
m-1071	148	68	first	first	ADJ
m-1071	148	69	derivative	derivative	NOUN
m-1071	148	70	of	of	ADP
m-1071	148	71	the	the	DET
m-1071	148	72	polynomial	polynomial	NOUN
m-1071	148	73	in	in	ADP
m-1071	148	74	theorem	theorem	NOUN
m-1071	148	75	2.23	2.23	NUM
m-1071	148	76	at	at	ADP
m-1071	148	77	y	y	PROPN
m-1071	148	78	=	=	SYM
m-1071	148	79	1	1	NUM
m-1071	148	80	,	,	PUNCT
m-1071	148	81	we	we	PRON
m-1071	148	82	get	get	VERB
m-1071	148	83	the	the	PRON
m-1071	148	84	)	)	PUNCT
m-1071	148	85	as	as	SCONJ
m-1071	148	86	follows	follow	VERB
m-1071	148	87	:	:	PUNCT
m-1071	148	88	let	let	VERB
m-1071	148	89	p	p	PRON
m-1071	148	90	be	be	AUX
m-1071	148	91	odd	odd	ADJ
m-1071	148	92	and	and	CCONJ
m-1071	148	93	p	p	X
m-1071	148	94	<	<	X
m-1071	148	95	q.	q.	PROPN
m-1071	149	1	then	then	ADV
m-1071	149	2	the	the	DET
m-1071	149	3	abc	abc	PROPN
m-1071	149	4	index	index	PROPN
m-1071	149	5	of(sc	of(sc	PROPN
m-1071	149	6	�	�	PROPN
m-1071	149	7	�	�	PROPN
m-1071	149	8	)	)	PUNCT
m-1071	149	9	is	be	AUX
m-1071	149	10	�	�	PROPN
m-1071	149	11	�	�	PROPN
m-1071	149	12	�	�	PROPN
m-1071	149	13	�	�	PROPN
m-1071	149	14	�	�	PROPN
m-1071	149	15	�	�	PROPN
m-1071	149	16	�	�	PROPN
m-1071	149	17	�	�	PROPN
m-1071	149	18	�	�	PROPN
m-1071	149	19	let	let	VERB
m-1071	149	20	p	p	PRON
m-1071	149	21	be	be	AUX
m-1071	149	22	odd	odd	ADJ
m-1071	149	23	and	and	CCONJ
m-1071	149	24	p	p	X
m-1071	149	25	<	<	X
m-1071	149	26	q.	q.	PROPN
m-1071	149	27	then	then	ADV
m-1071	149	28	the	the	DET
m-1071	149	29	forgotten	forget	VERB
m-1071	149	30	topological	topological	ADJ
m-1071	149	31	polynomial	polynomial	NOUN
m-1071	149	32	of	of	ADP
m-1071	149	33	y45	y45	NOUN
m-1071	149	34	+	+	CCONJ
m-1071	149	35	(	(	PUNCT
m-1071	149	36	3pq	3pq	ADJ
m-1071	149	37	−	−	PROPN
m-1071	149	38	2(2p	2(2p	NUM
m-1071	149	39	+	+	SYM
m-1071	149	40	q	q	NOUN
m-1071	149	41	−	−	NOUN
m-1071	149	42	1))y72	1))y72	NUM
m-1071	149	43	.	.	PUNCT
m-1071	150	1	bond	bond	NOUN
m-1071	150	2	partition	partition	NOUN
m-1071	150	3	from	from	ADP
m-1071	150	4	table	table	NOUN
m-1071	150	5	2	2	NUM
m-1071	150	6	,	,	PUNCT
m-1071	150	7	in	in	ADP
m-1071	150	8	the	the	DET
m-1071	150	9	formula	formula	NOUN
m-1071	150	10	of	of	ADP
m-1071	150	11	forgotten	forget	VERB
m-1071	150	12	topological	topological	PROPN
m-1071	150	13	�	�	PROPN
m-1071	150	14	[	[	SYM
m-1071	150	15	�	�	PROPN
m-1071	150	16	�	�	PROPN
m-1071	150	17	�	�	PROPN
m-1071	150	18	�	�	PROPN
m-1071	150	19	�	�	PROPN
m-1071	150	20	]	]	PUNCT
m-1071	150	21	�	�	PROPN
m-1071	150	22	�	�	PROPN
m-1071	150	23	+	+	CCONJ
m-1071	150	24	�	�	PROPN
m-1071	150	25	�	�	PROPN
m-1071	150	26	[	[	SYM
m-1071	150	27	�	�	PROPN
m-1071	150	28	�	�	PROPN
m-1071	150	29	�	�	PROPN
m-1071	150	30	�	�	PROPN
m-1071	150	31	�	�	PROPN
m-1071	150	32	]	]	PUNCT
m-1071	150	33	�	�	PROPN
m-1071	150	34	�	�	PROPN
m-1071	150	35	�	�	PROPN
m-1071	150	36	�	�	PROPN
m-1071	150	37	~	~	SYM
m-1071	150	38	�	�	PROPN
m-1071	150	39	�	�	PROPN
m-1071	150	40	�	�	PROPN
m-1071	150	41	�	�	PROPN
m-1071	150	42	+	+	CCONJ
m-1071	150	43	�	�	PROPN
m-1071	150	44	�	�	PROPN
m-1071	150	45	[	[	PUNCT
m-1071	150	46	�	�	PROPN
m-1071	150	47	�	�	PROPN
m-1071	150	48	�	�	PROPN
m-1071	150	49	�	�	PROPN
m-1071	150	50	~	~	SYM
m-1071	150	51	�	�	PROPN
m-1071	150	52	�	�	PROPN
m-1071	150	53	�	�	PROPN
m-1071	150	54	�	�	PROPN
m-1071	150	55	+	+	CCONJ
m-1071	150	56	(	(	PUNCT
m-1071	150	57	3	3	NUM
m-1071	150	58	�	�	PROPN
m-1071	150	59	�	�	PROPN
m-1071	150	60	+	+	CCONJ
m-1071	150	61	�	�	PROPN
m-1071	150	62	+	+	CCONJ
m-1071	150	63	2	2	NUM
m-1071	150	64	�	�	NOUN
m-1071	150	65	−	−	NUM
m-1071	150	66	5	5	NUM
m-1071	150	67	)	)	PUNCT
m-1071	150	68	�	�	PROPN
m-1071	150	69	�	�	PROPN
m-1071	150	70	�	�	PROPN
m-1071	150	71	+	+	CCONJ
m-1071	150	72	2	2	NUM
m-1071	150	73	�	�	PROPN
m-1071	150	74	3	3	NUM
m-1071	150	75	�	�	PROPN
m-1071	150	76	�	�	PROPN
m-1071	150	77	−	−	NUM
m-1071	150	78	2(2	2(2	NUM
m-1071	150	79	�	�	PROPN
m-1071	150	80	+	+	CCONJ
m-1071	150	81	by	by	ADP
m-1071	150	82	taking	take	VERB
m-1071	150	83	the	the	DET
m-1071	150	84	first	first	ADJ
m-1071	150	85	derivative	derivative	NOUN
m-1071	150	86	of	of	ADP
m-1071	150	87	the	the	DET
m-1071	150	88	polynomial	polynomial	NOUN
m-1071	150	89	in	in	ADP
m-1071	150	90	theorem	theorem	ADJ
m-1071	150	91	2.25	2.25	NUM
m-1071	150	92	at	at	ADP
m-1071	150	93	y	y	PROPN
m-1071	150	94	forgotten	forget	VERB
m-1071	150	95	topological	topological	ADJ
m-1071	150	96	index	index	NOUN
m-1071	150	97	of	of	ADP
m-1071	150	98	chain	chain	NOUN
m-1071	150	99	of	of	ADP
m-1071	150	100	sio4	sio4	PROPN
m-1071	150	101	scqp	scqp	PROPN
m-1071	150	102	as	as	SCONJ
m-1071	150	103	follows	follow	VERB
m-1071	150	104	:	:	PUNCT
m-1071	150	105	let	let	VERB
m-1071	150	106	p	p	PRON
m-1071	150	107	be	be	AUX
m-1071	150	108	odd	odd	ADJ
m-1071	150	109	and	and	CCONJ
m-1071	150	110	p	p	X
m-1071	150	111	<	<	X
m-1071	150	112	q.	q.	PROPN
m-1071	150	113	then	then	ADV
m-1071	151	1	the	the	DET
m-1071	151	2	forgotten	forget	VERB
m-1071	151	3	topological	topological	ADJ
m-1071	151	4	index	index	NOUN
m-1071	151	5	of	of	ADP
m-1071	151	6	let	let	VERB
m-1071	151	7	p	p	PRON
m-1071	151	8	be	be	AUX
m-1071	151	9	odd	odd	ADJ
m-1071	151	10	and	and	CCONJ
m-1071	151	11	p	p	X
m-1071	151	12	<	<	X
m-1071	151	13	q.	q.	PROPN
m-1071	151	14	then	then	ADV
m-1071	151	15	the	the	DET
m-1071	151	16	geometric	geometric	ADJ
m-1071	151	17	arithmetic	arithmetic	ADJ
m-1071	151	18	polynomial	polynomial	NOUN
m-1071	151	19	of	of	ADP
m-1071	151	20	+	+	X
m-1071	151	21	(	(	PUNCT
m-1071	151	22	3	3	NUM
m-1071	151	23	�	�	PROPN
m-1071	151	24	�	�	PROPN
m-1071	151	25	+	+	CCONJ
m-1071	151	26	�	�	PROPN
m-1071	151	27	+	+	CCONJ
m-1071	151	28	2	2	NUM
m-1071	151	29	�	�	NOUN
m-1071	151	30	−	−	NUM
m-1071	151	31	5	5	NUM
m-1071	151	32	)	)	PUNCT
m-1071	151	33	�	�	PROPN
m-1071	151	34	�	�	PROPN
m-1071	151	35	�	�	PROPN
m-1071	151	36	+	+	CCONJ
m-1071	151	37	2(2	2(2	NUM
m-1071	151	38	�	�	PROPN
m-1071	151	39	+	+	CCONJ
m-1071	151	40	�	�	PROPN
m-1071	151	41	−	−	ADP
m-1071	151	42	1	1	NUM
m-1071	151	43	)	)	PUNCT
m-1071	151	44	�	�	PROPN
m-1071	151	45	�	�	PROPN
m-1071	151	46	√	√	NUM
m-1071	151	47	�	�	PROPN
m-1071	151	48	bond	bond	NOUN
m-1071	151	49	partition	partition	NOUN
m-1071	151	50	from	from	ADP
m-1071	151	51	table	table	NOUN
m-1071	151	52	2	2	NUM
m-1071	151	53	,	,	PUNCT
m-1071	151	54	in	in	ADP
m-1071	151	55	the	the	DET
m-1071	151	56	formula	formula	NOUN
m-1071	151	57	of	of	ADP
m-1071	151	58	geometric	geometric	ADJ
m-1071	151	59	�	�	PROPN
m-1071	151	60	�	�	PROPN
m-1071	151	61	�	�	PROPN
m-1071	151	62	�	�	PROPN
m-1071	151	63	√	√	PROPN
m-1071	151	64	�	�	PROPN
m-1071	151	65	�	�	PROPN
m-1071	151	66	�	�	PROPN
m-1071	151	67	�	�	PROPN
m-1071	151	68	�	�	PROPN
m-1071	151	69	�	�	PROPN
m-1071	151	70	�	�	PROPN
m-1071	151	71	�	�	PROPN
m-1071	151	72	�	�	PROPN
m-1071	151	73	�	�	PROPN
m-1071	151	74	~	~	SYM
m-1071	151	75	�	�	PROPN
m-1071	151	76	�	�	PROPN
m-1071	151	77	�	�	PROPN
m-1071	151	78	�	�	PROPN
m-1071	151	79	+	+	CCONJ
m-1071	151	80	�	�	PROPN
m-1071	151	81	�	�	PROPN
m-1071	151	82	�	�	PROPN
m-1071	151	83	√	√	PROPN
m-1071	151	84	�	�	PROPN
m-1071	151	85	�	�	PROPN
m-1071	151	86	�	�	PROPN
m-1071	151	87	�	�	PROPN
m-1071	151	88	�	�	PROPN
m-1071	151	89	�	�	PROPN
m-1071	151	90	�	�	PROPN
m-1071	151	91	�	�	PROPN
m-1071	151	92	~	~	SYM
m-1071	151	93	�	�	PROPN
m-1071	151	94	+	+	CCONJ
m-1071	151	95	�	�	PROPN
m-1071	151	96	�	�	PROPN
m-1071	151	97	�	�	PROPN
m-1071	151	98	√	√	NUM
m-1071	151	99	�	�	PROPN
m-1071	151	100	�	�	PROPN
m-1071	151	101	�	�	PROPN
m-1071	151	102	�	�	PROPN
m-1071	151	103	�	�	PROPN
m-1071	151	104	�	�	PROPN
m-1071	151	105	�	�	PROPN
m-1071	151	106	~	~	SYM
m-1071	151	107	�	�	PROPN
m-1071	151	108	�	�	PROPN
m-1071	151	109	�	�	PROPN
m-1071	151	110	�	�	PROPN
m-1071	151	111	+	+	CCONJ
m-1071	151	112	�	�	PROPN
m-1071	151	113	−	−	ADP
m-1071	151	114	1	1	NUM
m-1071	151	115	)	)	PUNCT
m-1071	151	116	�	�	PROPN
m-1071	151	117	�	�	PROPN
m-1071	151	118	�	�	PROPN
m-1071	151	119	�	�	PROPN
m-1071	151	120	=	=	SYM
m-1071	151	121	1	1	NUM
m-1071	151	122	,	,	PUNCT
m-1071	151	123	we	we	PRON
m-1071	151	124	get	get	VERB
m-1071	151	125	the	the	DET
m-1071	151	126	abc	abc	PROPN
m-1071	151	127	�	�	PROPN
m-1071	151	128	�	�	PROPN
m-1071	151	129	�	�	PROPN
m-1071	151	130	�	�	PROPN
m-1071	151	131	�	�	PROPN
m-1071	151	132	�	�	PROPN
m-1071	151	133	�	�	PROPN
m-1071	151	134	�	�	PROPN
m-1071	151	135	�	�	PROPN
m-1071	151	136	�	�	PROPN
m-1071	151	137	let	let	VERB
m-1071	151	138	p	p	PRON
m-1071	151	139	be	be	AUX
m-1071	151	140	odd	odd	ADJ
m-1071	151	141	and	and	CCONJ
m-1071	152	1	p	p	X
m-1071	152	2	<	<	X
m-1071	152	3	q.	q.	PROPN
m-1071	152	4	then	then	ADV
m-1071	152	5	the	the	DET
m-1071	152	6	forgotten	forget	VERB
m-1071	152	7	topological	topological	ADJ
m-1071	152	8	polynomial	polynomial	NOUN
m-1071	152	9	of	of	ADP
m-1071	152	10	scqp	scqp	PROPN
m-1071	152	11	is	be	AUX
m-1071	152	12	bond	bond	NOUN
m-1071	152	13	partition	partition	NOUN
m-1071	152	14	from	from	ADP
m-1071	152	15	table	table	NOUN
m-1071	152	16	2	2	NUM
m-1071	152	17	,	,	PUNCT
m-1071	152	18	in	in	ADP
m-1071	152	19	the	the	DET
m-1071	152	20	formula	formula	NOUN
m-1071	152	21	of	of	ADP
m-1071	152	22	forgotten	forget	VERB
m-1071	152	23	topological	topological	PROPN
m-1071	152	24	[	[	X
m-1071	152	25	�	�	PROPN
m-1071	152	26	�	�	PROPN
m-1071	152	27	�	�	PROPN
m-1071	152	28	�	�	PROPN
m-1071	152	29	�	�	PROPN
m-1071	152	30	]	]	X
m-1071	152	31	�	�	PROPN
m-1071	152	32	−	−	ADP
m-1071	152	33	1	1	NUM
m-1071	152	34	)	)	PUNCT
m-1071	152	35	�	�	PROPN
m-1071	152	36	�	�	PROPN
m-1071	152	37	�	�	PROPN
m-1071	152	38	y	y	PROPN
m-1071	152	39	=	=	SYM
m-1071	152	40	1	1	NUM
m-1071	152	41	,	,	PUNCT
m-1071	152	42	we	we	PRON
m-1071	152	43	get	get	VERB
m-1071	152	44	the	the	DET
m-1071	152	45	ndex	ndex	NOUN
m-1071	152	46	of	of	ADP
m-1071	152	47	is	be	AUX
m-1071	152	48	351pq	351pq	NOUN
m-1071	152	49	let	let	VERB
m-1071	152	50	p	p	PRON
m-1071	152	51	be	be	AUX
m-1071	152	52	odd	odd	ADJ
m-1071	152	53	and	and	CCONJ
m-1071	152	54	p	p	X
m-1071	152	55	<	<	X
m-1071	152	56	q.	q.	PROPN
m-1071	153	1	then	then	ADV
m-1071	153	2	the	the	DET
m-1071	153	3	geometric	geometric	ADJ
m-1071	153	4	arithmetic	arithmetic	ADJ
m-1071	153	5	polynomial	polynomial	NOUN
m-1071	153	6	of	of	ADP
m-1071	153	7	is	be	AUX
m-1071	153	8	bond	bond	NOUN
m-1071	153	9	partition	partition	NOUN
m-1071	153	10	from	from	ADP
m-1071	153	11	table	table	NOUN
m-1071	153	12	2	2	NUM
m-1071	153	13	,	,	PUNCT
m-1071	153	14	in	in	ADP
m-1071	153	15	the	the	DET
m-1071	153	16	formula	formula	NOUN
m-1071	153	17	of	of	ADP
m-1071	153	18	geometric	geometric	ADJ
m-1071	153	19	arithmetic	arithmetic	PROPN
m-1071	153	20	�	�	PROPN
m-1071	153	21	�	�	PROPN
m-1071	153	22	�	�	PROPN
m-1071	153	23	�	�	PROPN
m-1071	153	24	�	�	PROPN
m-1071	153	25	ijo	ijo	PROPN
m-1071	153	26	international	international	PROPN
m-1071	153	27	journal	journal	PROPN
m-1071	153	28	of	of	ADP
m-1071	153	29	mathematics	mathematics	PROPN
m-1071	153	30	(	(	PUNCT
m-1071	153	31	issn	issn	PROPN
m-1071	153	32	:	:	PUNCT
m-1071	153	33	2992	2992	NUM
m-1071	153	34	-	-	SYM
m-1071	153	35	4421	4421	NUM
m-1071	153	36	)	)	PUNCT
m-1071	153	37	ijo	ijo	PROPN
m-1071	153	38	journals	journal	NOUN
m-1071	153	39	volume	volume	NOUN
m-1071	153	40	08	08	NUM
m-1071	154	1	|	|	ADV
m-1071	154	2	issue	issue	VERB
m-1071	154	3	04	04	NUM
m-1071	155	1	|	|	CCONJ
m-1071	155	2	april	april	PROPN
m-1071	155	3	2025	2025	NUM
m-1071	155	4	|	|	ADV
m-1071	155	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	155	6	12	12	NUM
m-1071	155	7	this	this	PRON
m-1071	155	8	gives	give	VERB
m-1071	155	9	�	�	PROPN
m-1071	155	10	�	�	PROPN
m-1071	155	11	�	�	PROPN
m-1071	155	12	sc	sc	PROPN
m-1071	155	13	�	�	PROPN
m-1071	155	14	�	�	PROPN
m-1071	155	15	,	,	PUNCT
m-1071	155	16	�	�	PROPN
m-1071	155	17	�	�	PROPN
m-1071	155	18	=	=	SYM
m-1071	155	19	3	3	NUM
m-1071	155	20	(	(	PUNCT
m-1071	155	21	�	�	PROPN
m-1071	155	22	+	+	CCONJ
m-1071	155	23	1	1	NUM
m-1071	155	24	)	)	PUNCT
m-1071	155	25	�	�	PROPN
m-1071	155	26	√	√	NUM
m-1071	155	27	�	�	PROPN
m-1071	155	28	�	�	PROPN
m-1071	155	29	by	by	ADP
m-1071	155	30	taking	take	VERB
m-1071	155	31	the	the	DET
m-1071	155	32	first	first	ADJ
m-1071	155	33	derivative	derivative	NOUN
m-1071	155	34	of	of	ADP
m-1071	155	35	the	the	DET
m-1071	155	36	polynomial	polynomial	NOUN
m-1071	155	37	in	in	ADP
m-1071	155	38	theorem	theorem	NOUN
m-1071	155	39	2.27	2.27	NUM
m-1071	155	40	at	at	ADP
m-1071	155	41	geometric	geometric	ADJ
m-1071	155	42	arithmetic	arithmetic	ADJ
m-1071	155	43	index	index	NOUN
m-1071	155	44	of	of	ADP
m-1071	155	45	chain	chain	NOUN
m-1071	155	46	of	of	ADP
m-1071	155	47	corollary	corollary	ADJ
m-1071	155	48	2.28	2.28	NUM
m-1071	155	49	.	.	PUNCT
m-1071	156	1	let	let	VERB
m-1071	156	2	p	p	PRON
m-1071	156	3	be	be	AUX
m-1071	156	4	odd	odd	ADJ
m-1071	156	5	and	and	CCONJ
m-1071	156	6	p	p	X
m-1071	156	7	<	<	X
m-1071	156	8	q.	q.	PROPN
m-1071	157	1	then	then	ADV
m-1071	157	2	the	the	DET
m-1071	157	3	geometric	geometric	ADJ
m-1071	157	4	arithmetic	arithmetic	ADJ
m-1071	157	5	index	index	NOUN
m-1071	157	6	of	of	ADP
m-1071	157	7	2	2	NUM
m-1071	157	8	(	(	PUNCT
m-1071	157	9	�	�	PROPN
m-1071	157	10	+	+	CCONJ
m-1071	157	11	3	3	NUM
m-1071	157	12	�	�	PROPN
m-1071	157	13	�	�	PROPN
m-1071	157	14	+	+	CCONJ
m-1071	157	15	2	2	NUM
m-1071	157	16	�	�	NOUN
m-1071	157	17	−	−	NUM
m-1071	157	18	5	5	NUM
m-1071	157	19	)	)	PUNCT
m-1071	157	20	+	+	CCONJ
m-1071	158	1	√3	√3	PROPN
m-1071	158	2	�	�	PROPN
m-1071	158	3	3	3	NUM
m-1071	158	4	�	�	PROPN
m-1071	158	5	�	�	PROPN
m-1071	158	6	−	−	NUM
m-1071	158	7	2(2	2(2	NUM
m-1071	158	8	3	3	NUM
m-1071	158	9	.	.	PUNCT
m-1071	158	10	theorem	theorem	VERB
m-1071	158	11	2.29	2.29	NUM
m-1071	158	12	.	.	PUNCT
m-1071	159	1	let	let	VERB
m-1071	159	2	p	p	PRON
m-1071	159	3	be	be	AUX
m-1071	159	4	odd	odd	ADJ
m-1071	159	5	and	and	CCONJ
m-1071	159	6	p	p	X
m-1071	159	7	<	<	X
m-1071	159	8	q.	q.	PROPN
m-1071	160	1	then	then	ADV
m-1071	160	2	the	the	DET
m-1071	160	3	randic	randic	ADJ
m-1071	160	4	polynomial	polynomial	ADJ
m-1071	160	5	polynomial	polynomial	NOUN
m-1071	160	6	of	of	ADP
m-1071	160	7	3	3	NUM
m-1071	160	8	(	(	PUNCT
m-1071	160	9	�	�	PROPN
m-1071	160	10	+	+	CCONJ
m-1071	160	11	1	1	NUM
m-1071	160	12	)	)	PUNCT
m-1071	160	13	�	�	PROPN
m-1071	160	14	�	�	PROPN
m-1071	160	15	�	�	PROPN
m-1071	160	16	+	+	CCONJ
m-1071	160	17	(	(	PUNCT
m-1071	160	18	3	3	NUM
m-1071	160	19	�	�	PROPN
m-1071	160	20	�	�	PROPN
m-1071	160	21	+	+	CCONJ
m-1071	160	22	�	�	PROPN
m-1071	160	23	+	+	CCONJ
m-1071	160	24	2	2	NUM
m-1071	160	25	�	�	NOUN
m-1071	160	26	−	−	NOUN
m-1071	160	27	proof	proof	NOUN
m-1071	160	28	.	.	PUNCT
m-1071	161	1	using	use	VERB
m-1071	161	2	the	the	DET
m-1071	161	3	atom	atom	NOUN
m-1071	161	4	-	-	PUNCT
m-1071	161	5	bond	bond	NOUN
m-1071	161	6	partition	partition	NOUN
m-1071	161	7	from	from	ADP
m-1071	161	8	table	table	NOUN
m-1071	161	9	2	2	NUM
m-1071	161	10	,	,	PUNCT
m-1071	161	11	in	in	ADP
m-1071	161	12	the	the	DET
m-1071	161	13	formula	formula	NOUN
m-1071	161	14	of	of	ADP
m-1071	161	15	randic	randic	ADJ
m-1071	161	16	polynomial	polynomial	ADJ
m-1071	161	17	polynomial	polynomial	ADJ
m-1071	161	18	(	(	PUNCT
m-1071	161	19	5	5	NUM
m-1071	161	20	)	)	PUNCT
m-1071	161	21	,	,	PUNCT
m-1071	161	22	we	we	PRON
m-1071	161	23	get	get	VERB
m-1071	161	24	�	�	PROPN
m-1071	161	25	�	�	PROPN
m-1071	161	26	sc	sc	PROPN
m-1071	161	27	�	�	PROPN
m-1071	161	28	�	�	PROPN
m-1071	161	29	,	,	PUNCT
m-1071	161	30	�	�	PROPN
m-1071	161	31	�	�	PROPN
m-1071	161	32	=	=	SYM
m-1071	161	33	�	�	PROPN
m-1071	161	34	�	�	PROPN
m-1071	161	35	�	�	PROPN
m-1071	161	36	�	�	PROPN
m-1071	161	37	�	�	PROPN
m-1071	161	38	~	~	NOUN
m-1071	161	39	�	�	PROPN
m-1071	161	40	this	this	PRON
m-1071	161	41	gives	give	VERB
m-1071	161	42	�	�	PROPN
m-1071	161	43	�	�	PROPN
m-1071	161	44	sc	sc	PROPN
m-1071	161	45	�	�	PROPN
m-1071	161	46	�	�	PROPN
m-1071	161	47	,	,	PUNCT
m-1071	161	48	�	�	PROPN
m-1071	161	49	�	�	PROPN
m-1071	161	50	=	=	SYM
m-1071	161	51	3	3	NUM
m-1071	161	52	(	(	PUNCT
m-1071	161	53	�	�	PROPN
m-1071	161	54	+	+	CCONJ
m-1071	161	55	1	1	NUM
m-1071	161	56	)	)	PUNCT
m-1071	161	57	�	�	PROPN
m-1071	161	58	�	�	PROPN
m-1071	161	59	�	�	PROPN
m-1071	161	60	+	+	CCONJ
m-1071	161	61	(	(	PUNCT
m-1071	161	62	3	3	NUM
m-1071	161	63	�	�	NOUN
m-1071	161	64	�	�	NOUN
m-1071	161	65	by	by	ADP
m-1071	161	66	taking	take	VERB
m-1071	161	67	the	the	DET
m-1071	161	68	first	first	ADJ
m-1071	161	69	derivative	derivative	NOUN
m-1071	161	70	of	of	ADP
m-1071	161	71	the	the	DET
m-1071	161	72	polynomial	polynomial	NOUN
m-1071	161	73	in	in	ADP
m-1071	161	74	theorem	theorem	ADJ
m-1071	161	75	2.29	2.29	NUM
m-1071	161	76	at	at	ADP
m-1071	161	77	randic	randic	ADJ
m-1071	161	78	polynomial	polynomial	ADJ
m-1071	161	79	index	index	NOUN
m-1071	161	80	of	of	ADP
m-1071	161	81	chain	chain	NOUN
m-1071	161	82	of	of	ADP
m-1071	161	83	corollary	corollary	ADJ
m-1071	161	84	2.30	2.30	NUM
m-1071	161	85	.	.	PUNCT
m-1071	162	1	let	let	VERB
m-1071	162	2	p	p	PRON
m-1071	162	3	be	be	AUX
m-1071	162	4	odd	odd	ADJ
m-1071	162	5	and	and	CCONJ
m-1071	162	6	p	p	X
m-1071	162	7	<	<	X
m-1071	162	8	q.then	q.then	ADV
m-1071	162	9	the	the	DET
m-1071	162	10	randic	randic	ADJ
m-1071	162	11	polynomial	polynomial	ADJ
m-1071	162	12	index	index	NOUN
m-1071	162	13	o	o	NOUN
m-1071	162	14	(	(	PUNCT
m-1071	162	15	3	3	NUM
m-1071	162	16	�	�	PROPN
m-1071	162	17	�	�	PROPN
m-1071	162	18	+	+	CCONJ
m-1071	162	19	�	�	PROPN
m-1071	162	20	+	+	CCONJ
m-1071	162	21	2	2	NUM
m-1071	162	22	�	�	NOUN
m-1071	162	23	−	−	NUM
m-1071	162	24	5	5	NUM
m-1071	162	25	)	)	PUNCT
m-1071	162	26	�	�	PROPN
m-1071	162	27	�	�	PROPN
m-1071	162	28	�	�	PROPN
m-1071	162	29	√	√	NUM
m-1071	162	30	�	�	PROPN
m-1071	162	31	+	+	CCONJ
m-1071	162	32	2	2	NUM
m-1071	162	33	�	�	PROPN
m-1071	162	34	3	3	NUM
m-1071	162	35	�	�	PROPN
m-1071	162	36	�	�	PROPN
m-1071	162	37	theorem	theorem	VERB
m-1071	162	38	2.31	2.31	NUM
m-1071	162	39	.	.	PUNCT
m-1071	163	1	let	let	VERB
m-1071	163	2	p	p	PRON
m-1071	163	3	be	be	AUX
m-1071	163	4	odd	odd	ADJ
m-1071	163	5	and	and	CCONJ
m-1071	163	6	p	p	X
m-1071	163	7	<	<	X
m-1071	163	8	q.	q.	PROPN
m-1071	164	1	then	then	ADV
m-1071	164	2	the	the	DET
m-1071	164	3	reciprocal	reciprocal	ADJ
m-1071	164	4	randic	randic	ADJ
m-1071	164	5	polynomial	polynomial	ADJ
m-1071	164	6	polynomial	polynomial	NOUN
m-1071	164	7	of	of	ADP
m-1071	164	8	sc	sc	PROPN
m-1071	164	9	�	�	PROPN
m-1071	164	10	�	�	PROPN
m-1071	164	11	is	be	AUX
m-1071	164	12	3	3	NUM
m-1071	164	13	(	(	PUNCT
m-1071	164	14	�	�	X
m-1071	164	15	+	+	CCONJ
m-1071	164	16	1	1	NUM
m-1071	164	17	)	)	PUNCT
m-1071	164	18	�	�	PROPN
m-1071	164	19	�	�	PROPN
m-1071	164	20	+	+	CCONJ
m-1071	164	21	(	(	PUNCT
m-1071	164	22	3	3	NUM
m-1071	164	23	�	�	PROPN
m-1071	164	24	�	�	PROPN
m-1071	164	25	+	+	CCONJ
m-1071	164	26	�	�	PROPN
m-1071	164	27	proof	proof	NOUN
m-1071	164	28	.	.	PUNCT
m-1071	165	1	using	use	VERB
m-1071	165	2	the	the	DET
m-1071	165	3	atom	atom	NOUN
m-1071	165	4	-	-	PUNCT
m-1071	165	5	bond	bond	NOUN
m-1071	165	6	partition	partition	NOUN
m-1071	165	7	from	from	ADP
m-1071	165	8	table	table	NOUN
m-1071	165	9	2	2	NUM
m-1071	165	10	,	,	PUNCT
m-1071	165	11	in	in	ADP
m-1071	165	12	the	the	DET
m-1071	165	13	polynomial	polynomial	ADJ
m-1071	165	14	polynomial	polynomial	NOUN
m-1071	165	15	(	(	PUNCT
m-1071	165	16	6	6	NUM
m-1071	165	17	)	)	PUNCT
m-1071	165	18	,	,	PUNCT
m-1071	165	19	we	we	PRON
m-1071	165	20	get	get	VERB
m-1071	165	21	�	�	PROPN
m-1071	165	22	�	�	PROPN
m-1071	165	23	sc	sc	PROPN
m-1071	165	24	�	�	PROPN
m-1071	165	25	�	�	PROPN
m-1071	165	26	,	,	PUNCT
m-1071	165	27	�	�	PROPN
m-1071	165	28	�	�	PROPN
m-1071	165	29	=	=	SYM
m-1071	165	30	�	�	PROPN
m-1071	165	31	�	�	PROPN
m-1071	165	32	�	�	PROPN
m-1071	165	33	�	�	PROPN
m-1071	165	34	�	�	PROPN
m-1071	165	35	~	~	SYM
m-1071	165	36	)	)	PUNCT
m-1071	165	37	�	�	PROPN
m-1071	165	38	�	�	PROPN
m-1071	165	39	+	+	CCONJ
m-1071	165	40	(	(	PUNCT
m-1071	165	41	3	3	NUM
m-1071	165	42	�	�	PROPN
m-1071	165	43	�	�	PROPN
m-1071	165	44	+	+	CCONJ
m-1071	165	45	�	�	PROPN
m-1071	165	46	+	+	CCONJ
m-1071	165	47	2	2	NUM
m-1071	165	48	�	�	NOUN
m-1071	165	49	−	−	NUM
m-1071	165	50	5	5	NUM
m-1071	165	51	)	)	PUNCT
m-1071	165	52	�	�	PROPN
m-1071	165	53	�	�	PROPN
m-1071	165	54	�	�	PROPN
m-1071	165	55	+	+	CCONJ
m-1071	165	56	2	2	NUM
m-1071	165	57	�	�	PROPN
m-1071	165	58	3	3	NUM
m-1071	165	59	�	�	PROPN
m-1071	165	60	�	�	PROPN
m-1071	165	61	−	−	NUM
m-1071	165	62	2(2	2(2	NUM
m-1071	165	63	�	�	PROPN
m-1071	165	64	+	+	CCONJ
m-1071	165	65	by	by	ADP
m-1071	165	66	taking	take	VERB
m-1071	165	67	the	the	DET
m-1071	165	68	first	first	ADJ
m-1071	165	69	derivative	derivative	NOUN
m-1071	165	70	of	of	ADP
m-1071	165	71	the	the	DET
m-1071	165	72	polynomial	polynomial	NOUN
m-1071	165	73	in	in	ADP
m-1071	165	74	theorem	theorem	NOUN
m-1071	165	75	2.27	2.27	NUM
m-1071	165	76	at	at	ADP
m-1071	165	77	y	y	PROPN
m-1071	165	78	=	=	SYM
m-1071	165	79	1	1	NUM
m-1071	165	80	,	,	PUNCT
m-1071	165	81	we	we	PRON
m-1071	165	82	get	get	VERB
m-1071	165	83	the	the	DET
m-1071	165	84	geometric	geometric	ADJ
m-1071	165	85	arithmetic	arithmetic	ADJ
m-1071	165	86	index	index	NOUN
m-1071	165	87	of	of	ADP
m-1071	165	88	chain	chain	NOUN
m-1071	165	89	of	of	ADP
m-1071	165	90	sio4	sio4	PROPN
m-1071	165	91	sc	sc	PROPN
m-1071	165	92	�	�	PROPN
m-1071	165	93	�	�	PROPN
m-1071	165	94	as	as	SCONJ
m-1071	165	95	follows	follow	VERB
m-1071	165	96	:	:	PUNCT
m-1071	165	97	let	let	VERB
m-1071	165	98	p	p	PRON
m-1071	165	99	be	be	AUX
m-1071	165	100	odd	odd	ADJ
m-1071	165	101	and	and	CCONJ
m-1071	165	102	p	p	X
m-1071	165	103	<	<	X
m-1071	165	104	q.	q.	PROPN
m-1071	166	1	then	then	ADV
m-1071	166	2	the	the	DET
m-1071	166	3	geometric	geometric	ADJ
m-1071	166	4	arithmetic	arithmetic	ADJ
m-1071	166	5	index	index	NOUN
m-1071	166	6	of	of	ADP
m-1071	166	7	(	(	PUNCT
m-1071	166	8	2	2	NUM
m-1071	166	9	�	�	PROPN
m-1071	166	10	+	+	CCONJ
m-1071	166	11	�	�	PROPN
m-1071	166	12	−	−	ADP
m-1071	166	13	1	1	X
m-1071	166	14	)	)	PUNCT
m-1071	166	15	�	�	NOUN
m-1071	166	16	+	+	CCONJ
m-1071	166	17	3	3	NUM
m-1071	166	18	√6	√6	PROPN
m-1071	166	19	�	�	PROPN
m-1071	166	20	+	+	CCONJ
m-1071	166	21	3√6	3√6	PROPN
m-1071	166	22	let	let	VERB
m-1071	166	23	p	p	PRON
m-1071	166	24	be	be	AUX
m-1071	166	25	odd	odd	ADJ
m-1071	166	26	and	and	CCONJ
m-1071	166	27	p	p	X
m-1071	166	28	<	<	X
m-1071	166	29	q.	q.	PROPN
m-1071	167	1	then	then	ADV
m-1071	167	2	the	the	DET
m-1071	167	3	randic	randic	ADJ
m-1071	167	4	polynomial	polynomial	ADJ
m-1071	167	5	polynomial	polynomial	NOUN
m-1071	167	6	of	of	ADP
m-1071	167	7	5	5	NUM
m-1071	167	8	)	)	PUNCT
m-1071	167	9	�	�	PROPN
m-1071	167	10	�	�	PROPN
m-1071	167	11	�	�	PROPN
m-1071	167	12	√	√	NUM
m-1071	167	13	�	�	PROPN
m-1071	167	14	+	+	CCONJ
m-1071	167	15	2	2	NUM
m-1071	167	16	�	�	PROPN
m-1071	167	17	3	3	NUM
m-1071	167	18	�	�	PROPN
m-1071	167	19	�	�	PROPN
m-1071	167	20	−	−	NUM
m-1071	167	21	2(2	2(2	NUM
m-1071	167	22	�	�	PROPN
m-1071	167	23	+	+	CCONJ
m-1071	167	24	�	�	PROPN
m-1071	167	25	−	−	ADP
m-1071	167	26	1	1	NUM
m-1071	167	27	)	)	PUNCT
m-1071	167	28	�	�	PROPN
m-1071	167	29	�	�	PROPN
m-1071	167	30	�	�	PROPN
m-1071	167	31	�	�	PROPN
m-1071	167	32	bond	bond	NOUN
m-1071	167	33	partition	partition	NOUN
m-1071	167	34	from	from	ADP
m-1071	167	35	table	table	NOUN
m-1071	167	36	2	2	NUM
m-1071	167	37	,	,	PUNCT
m-1071	167	38	in	in	ADP
m-1071	167	39	the	the	DET
m-1071	167	40	formula	formula	NOUN
m-1071	167	41	of	of	ADP
m-1071	167	42	randic	randic	ADJ
m-1071	167	43	polynomial	polynomial	PROPN
m-1071	167	44	�	�	PROPN
m-1071	167	45	�	�	PROPN
m-1071	167	46	�	�	PROPN
m-1071	167	47	�	�	PROPN
m-1071	167	48	(	(	PUNCT
m-1071	167	49	�	�	PROPN
m-1071	167	50	)	)	PUNCT
m-1071	167	51	(	(	PUNCT
m-1071	167	52	�	�	PROPN
m-1071	167	53	)	)	PUNCT
m-1071	167	54	�	�	PROPN
m-1071	167	55	�	�	PROPN
m-1071	167	56	�	�	PROPN
m-1071	167	57	�	�	PROPN
m-1071	167	58	+	+	CCONJ
m-1071	167	59	�	�	PROPN
m-1071	167	60	�	�	PROPN
m-1071	167	61	�	�	PROPN
m-1071	167	62	�	�	PROPN
m-1071	167	63	(	(	PUNCT
m-1071	167	64	�	�	PROPN
m-1071	167	65	)	)	PUNCT
m-1071	167	66	(	(	PUNCT
m-1071	167	67	�	�	PROPN
m-1071	167	68	)	)	PUNCT
m-1071	167	69	�	�	PROPN
m-1071	167	70	�	�	PROPN
m-1071	167	71	�	�	PROPN
m-1071	167	72	�	�	PROPN
m-1071	167	73	~	~	SYM
m-1071	167	74	�	�	PROPN
m-1071	167	75	�	�	PROPN
m-1071	167	76	�	�	PROPN
m-1071	167	77	�	�	PROPN
m-1071	167	78	+	+	CCONJ
m-1071	167	79	�	�	PROPN
m-1071	167	80	�	�	PROPN
m-1071	167	81	�	�	PROPN
m-1071	167	82	�	�	PROPN
m-1071	167	83	�	�	PROPN
m-1071	167	84	�	�	PROPN
m-1071	167	85	�	�	PROPN
m-1071	167	86	~	~	SYM
m-1071	167	87	�	�	PROPN
m-1071	167	88	�	�	PROPN
m-1071	167	89	�	�	PROPN
m-1071	167	90	�	�	PROPN
m-1071	167	91	(	(	PUNCT
m-1071	167	92	�	�	PROPN
m-1071	167	93	�	�	PROPN
m-1071	167	94	+	+	CCONJ
m-1071	167	95	�	�	PROPN
m-1071	167	96	+	+	CCONJ
m-1071	167	97	2	2	NUM
m-1071	167	98	�	�	NOUN
m-1071	167	99	−	−	NUM
m-1071	167	100	5	5	NUM
m-1071	167	101	)	)	PUNCT
m-1071	167	102	�	�	PROPN
m-1071	167	103	�	�	PROPN
m-1071	167	104	�	�	PROPN
m-1071	167	105	√	√	NUM
m-1071	167	106	�	�	PROPN
m-1071	167	107	+	+	CCONJ
m-1071	167	108	2	2	NUM
m-1071	167	109	�	�	PROPN
m-1071	167	110	3	3	NUM
m-1071	167	111	�	�	PROPN
m-1071	167	112	�	�	PROPN
m-1071	167	113	−	−	NUM
m-1071	167	114	2(2	2(2	NUM
m-1071	167	115	�	�	PROPN
m-1071	167	116	+	+	CCONJ
m-1071	167	117	�	�	PROPN
m-1071	167	118	−	−	NOUN
m-1071	167	119	1	1	NUM
m-1071	167	120	)	)	PUNCT
m-1071	167	121	by	by	ADP
m-1071	167	122	taking	take	VERB
m-1071	167	123	the	the	DET
m-1071	167	124	first	first	ADJ
m-1071	167	125	derivative	derivative	NOUN
m-1071	167	126	of	of	ADP
m-1071	167	127	the	the	DET
m-1071	167	128	polynomial	polynomial	NOUN
m-1071	167	129	in	in	ADP
m-1071	167	130	theorem	theorem	ADJ
m-1071	167	131	2.29	2.29	NUM
m-1071	167	132	at	at	ADP
m-1071	167	133	y	y	PROPN
m-1071	167	134	randic	randic	ADJ
m-1071	167	135	polynomial	polynomial	ADJ
m-1071	167	136	index	index	NOUN
m-1071	167	137	of	of	ADP
m-1071	167	138	chain	chain	NOUN
m-1071	167	139	of	of	ADP
m-1071	167	140	sio4	sio4	PROPN
m-1071	167	141	(	(	PUNCT
m-1071	167	142	sc	sc	PROPN
m-1071	167	143	�	�	PROPN
m-1071	167	144	�	�	PROPN
m-1071	167	145	)	)	PUNCT
m-1071	167	146	as	as	SCONJ
m-1071	167	147	follows	follow	VERB
m-1071	167	148	:	:	PUNCT
m-1071	167	149	let	let	VERB
m-1071	167	150	p	p	PRON
m-1071	167	151	be	be	AUX
m-1071	167	152	odd	odd	ADJ
m-1071	167	153	and	and	CCONJ
m-1071	167	154	p	p	X
m-1071	167	155	<	<	X
m-1071	167	156	q.then	q.then	ADV
m-1071	167	157	the	the	DET
m-1071	167	158	randic	randic	ADJ
m-1071	167	159	polynomial	polynomial	ADJ
m-1071	167	160	index	index	NOUN
m-1071	167	161	of	of	ADP
m-1071	167	162	sc	sc	PROPN
m-1071	167	163	�	�	PROPN
m-1071	167	164	�	�	PROPN
m-1071	167	165	�	�	PROPN
m-1071	167	166	−	−	PROPN
m-1071	167	167	2(2	2(2	NUM
m-1071	167	168	�	�	PROPN
m-1071	167	169	+	+	CCONJ
m-1071	167	170	�	�	PROPN
m-1071	167	171	−	−	ADP
m-1071	167	172	1	1	NUM
m-1071	167	173	)	)	PUNCT
m-1071	167	174	�	�	PROPN
m-1071	167	175	�	�	PROPN
m-1071	167	176	�	�	PROPN
m-1071	167	177	�	�	PROPN
m-1071	167	178	.	.	PUNCT
m-1071	168	1	let	let	VERB
m-1071	168	2	p	p	PRON
m-1071	168	3	be	be	AUX
m-1071	168	4	odd	odd	ADJ
m-1071	168	5	and	and	CCONJ
m-1071	168	6	p	p	X
m-1071	168	7	<	<	X
m-1071	168	8	q.	q.	PROPN
m-1071	169	1	then	then	ADV
m-1071	169	2	the	the	DET
m-1071	169	3	reciprocal	reciprocal	ADJ
m-1071	169	4	randic	randic	ADJ
m-1071	169	5	polynomial	polynomial	ADJ
m-1071	169	6	polynomial	polynomial	ADJ
m-1071	169	7	�	�	PROPN
m-1071	169	8	+	+	CCONJ
m-1071	169	9	2	2	NUM
m-1071	169	10	�	�	PROPN
m-1071	169	11	−	−	PROPN
m-1071	169	12	5)	5)	NUM
m-1071	169	13	�	�	PROPN
m-1071	169	14	�	�	NOUN
m-1071	169	15	√	√	NUM
m-1071	169	16	�	�	PROPN
m-1071	169	17	+	+	CCONJ
m-1071	169	18	�	�	PROPN
m-1071	169	19	3	3	NUM
m-1071	169	20	�	�	PROPN
m-1071	169	21	�	�	PROPN
m-1071	169	22	−	−	NUM
m-1071	169	23	2(2	2(2	NUM
m-1071	169	24	�	�	PROPN
m-1071	169	25	+	+	CCONJ
m-1071	169	26	�	�	PROPN
m-1071	169	27	−	−	ADP
m-1071	169	28	1	1	NUM
m-1071	169	29	)	)	PUNCT
m-1071	169	30	�	�	PROPN
m-1071	169	31	�	�	PROPN
m-1071	169	32	�	�	PROPN
m-1071	169	33	bond	bond	NOUN
m-1071	169	34	partition	partition	NOUN
m-1071	169	35	from	from	ADP
m-1071	169	36	table	table	NOUN
m-1071	169	37	2	2	NUM
m-1071	169	38	,	,	PUNCT
m-1071	169	39	in	in	ADP
m-1071	169	40	the	the	DET
m-1071	169	41	formula	formula	NOUN
m-1071	169	42	of	of	ADP
m-1071	169	43	reciprocal	reciprocal	ADJ
m-1071	169	44	randic	randic	ADJ
m-1071	169	45	polynomial	polynomial	ADJ
m-1071	169	46	polynomial	polynomial	ADJ
m-1071	169	47	(	(	PUNCT
m-1071	169	48	6	6	NUM
m-1071	169	49	)	)	PUNCT
m-1071	169	50	,	,	PUNCT
m-1071	169	51	we	we	PRON
m-1071	169	52	get	get	VERB
m-1071	169	53	�	�	PROPN
m-1071	169	54	�	�	PROPN
m-1071	169	55	�	�	PROPN
m-1071	169	56	(	(	PUNCT
m-1071	169	57	�	�	PROPN
m-1071	169	58	)	)	PUNCT
m-1071	169	59	(	(	PUNCT
m-1071	169	60	�	�	PROPN
m-1071	169	61	)	)	PUNCT
m-1071	169	62	~	~	SYM
m-1071	169	63	�	�	PROPN
m-1071	169	64	�	�	PROPN
m-1071	169	65	�	�	PROPN
m-1071	169	66	�	�	PROPN
m-1071	169	67	+	+	CCONJ
m-1071	169	68	�	�	PROPN
m-1071	169	69	�	�	PROPN
m-1071	169	70	�	�	PROPN
m-1071	169	71	(	(	PUNCT
m-1071	169	72	�	�	PROPN
m-1071	169	73	)	)	PUNCT
m-1071	169	74	(	(	PUNCT
m-1071	169	75	�	�	NOUN
m-1071	169	76	)	)	PUNCT
m-1071	169	77	�	�	PROPN
m-1071	169	78	�	�	PROPN
m-1071	169	79	~	~	SYM
m-1071	169	80	�	�	PROPN
m-1071	169	81	+	+	CCONJ
m-1071	169	82	�	�	PROPN
m-1071	169	83	�	�	PROPN
m-1071	169	84	�	�	PROPN
m-1071	169	85	(	(	PUNCT
m-1071	169	86	�	�	PROPN
m-1071	169	87	)	)	PUNCT
m-1071	169	88	�	�	PROPN
m-1071	169	89	�	�	PROPN
m-1071	169	90	�	�	PROPN
m-1071	169	91	�	�	PROPN
m-1071	169	92	~	~	SYM
m-1071	169	93	�	�	PROPN
m-1071	169	94	�	�	PROPN
m-1071	169	95	�	�	PROPN
m-1071	169	96	�	�	PROPN
m-1071	169	97	�	�	PROPN
m-1071	169	98	−	−	ADP
m-1071	169	99	1	1	NUM
m-1071	169	100	)	)	PUNCT
m-1071	169	101	�	�	PROPN
m-1071	169	102	�	�	PROPN
m-1071	169	103	�	�	PROPN
m-1071	169	104	√	√	NUM
m-1071	169	105	�	�	PROPN
m-1071	169	106	=	=	SYM
m-1071	169	107	1	1	NUM
m-1071	169	108	,	,	PUNCT
m-1071	169	109	we	we	PRON
m-1071	169	110	get	get	VERB
m-1071	169	111	the	the	DET
m-1071	169	112	let	let	NOUN
m-1071	169	113	p	p	PRON
m-1071	169	114	be	be	AUX
m-1071	169	115	odd	odd	ADJ
m-1071	169	116	and	and	CCONJ
m-1071	169	117	p	p	X
m-1071	169	118	<	<	X
m-1071	169	119	q.	q.	PROPN
m-1071	170	1	then	then	ADV
m-1071	170	2	the	the	DET
m-1071	170	3	geometric	geometric	ADJ
m-1071	170	4	arithmetic	arithmetic	ADJ
m-1071	170	5	index	index	NOUN
m-1071	170	6	of	of	ADP
m-1071	170	7	sc	sc	PROPN
m-1071	170	8	�	�	PROPN
m-1071	170	9	�	�	PROPN
m-1071	170	10	is	be	AUX
m-1071	170	11	let	let	VERB
m-1071	170	12	p	p	PRON
m-1071	170	13	be	be	AUX
m-1071	170	14	odd	odd	ADJ
m-1071	170	15	and	and	CCONJ
m-1071	170	16	p	p	X
m-1071	170	17	<	<	X
m-1071	170	18	q.	q.	PROPN
m-1071	171	1	then	then	ADV
m-1071	171	2	the	the	DET
m-1071	171	3	randic	randic	ADJ
m-1071	171	4	polynomial	polynomial	ADJ
m-1071	171	5	polynomial	polynomial	NOUN
m-1071	171	6	of	of	ADP
m-1071	171	7	is	be	AUX
m-1071	171	8	bond	bond	NOUN
m-1071	171	9	partition	partition	NOUN
m-1071	171	10	from	from	ADP
m-1071	171	11	table	table	NOUN
m-1071	171	12	2	2	NUM
m-1071	171	13	,	,	PUNCT
m-1071	171	14	in	in	ADP
m-1071	171	15	the	the	DET
m-1071	171	16	formula	formula	NOUN
m-1071	171	17	of	of	ADP
m-1071	171	18	randic	randic	ADJ
m-1071	171	19	polynomial	polynomial	PROPN
m-1071	171	20	�	�	PROPN
m-1071	171	21	�	�	PROPN
m-1071	171	22	(	(	PUNCT
m-1071	171	23	�	�	PROPN
m-1071	171	24	)	)	PUNCT
m-1071	171	25	(	(	PUNCT
m-1071	171	26	�	�	PROPN
m-1071	171	27	)	)	PUNCT
m-1071	171	28	)	)	PUNCT
m-1071	171	29	�	�	PROPN
m-1071	171	30	�	�	PROPN
m-1071	171	31	�	�	PROPN
m-1071	171	32	�	�	PROPN
m-1071	171	33	y	y	PROPN
m-1071	171	34	=	=	SYM
m-1071	171	35	1	1	NUM
m-1071	171	36	,	,	PUNCT
m-1071	171	37	we	we	PRON
m-1071	171	38	get	get	VERB
m-1071	171	39	the	the	DET
m-1071	171	40	c	c	PROPN
m-1071	171	41	�	�	PROPN
m-1071	171	42	�	�	PROPN
m-1071	171	43	is	be	AUX
m-1071	171	44	let	let	VERB
m-1071	171	45	p	p	PRON
m-1071	171	46	be	be	AUX
m-1071	171	47	odd	odd	ADJ
m-1071	171	48	and	and	CCONJ
m-1071	171	49	p	p	X
m-1071	171	50	<	<	X
m-1071	171	51	q.	q.	PROPN
m-1071	172	1	then	then	ADV
m-1071	172	2	the	the	DET
m-1071	172	3	reciprocal	reciprocal	ADJ
m-1071	172	4	randic	randic	ADJ
m-1071	172	5	polynomial	polynomial	ADJ
m-1071	172	6	polynomial	polynomial	ADJ
m-1071	172	7	�	�	PROPN
m-1071	172	8	formula	formula	NOUN
m-1071	172	9	of	of	ADP
m-1071	172	10	reciprocal	reciprocal	ADJ
m-1071	172	11	randic	randic	ADJ
m-1071	172	12	�	�	PROPN
m-1071	172	13	)	)	PUNCT
m-1071	172	14	(	(	PUNCT
m-1071	172	15	�	�	PROPN
m-1071	172	16	)	)	PUNCT
m-1071	172	17	ijo	ijo	PROPN
m-1071	172	18	international	international	PROPN
m-1071	172	19	journal	journal	PROPN
m-1071	172	20	of	of	ADP
m-1071	172	21	mathematics	mathematics	PROPN
m-1071	172	22	(	(	PUNCT
m-1071	172	23	issn	issn	PROPN
m-1071	172	24	:	:	PUNCT
m-1071	172	25	2992	2992	NUM
m-1071	172	26	-	-	SYM
m-1071	172	27	4421	4421	NUM
m-1071	172	28	)	)	PUNCT
m-1071	173	1	ijo	ijo	PROPN
m-1071	173	2	journals	journal	NOUN
m-1071	173	3	volume	volume	NOUN
m-1071	173	4	08	08	NUM
m-1071	174	1	|	|	ADV
m-1071	174	2	issue	issue	VERB
m-1071	174	3	04	04	NUM
m-1071	175	1	|	|	CCONJ
m-1071	175	2	april	april	PROPN
m-1071	175	3	2025	2025	NUM
m-1071	176	1	|	|	ADV
m-1071	176	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	176	3	13	13	NUM
m-1071	176	4	�	�	PROPN
m-1071	176	5	�	�	PROPN
m-1071	176	6	sc	sc	PROPN
m-1071	176	7	�	�	PROPN
m-1071	176	8	�	�	PROPN
m-1071	176	9	,	,	PUNCT
m-1071	176	10	�	�	PROPN
m-1071	176	11	�	�	PROPN
m-1071	176	12	=	=	SYM
m-1071	176	13	3	3	NUM
m-1071	176	14	(	(	PUNCT
m-1071	176	15	�	�	PROPN
m-1071	176	16	+	+	CCONJ
m-1071	176	17	1	1	NUM
m-1071	176	18	)	)	PUNCT
m-1071	176	19	�	�	PROPN
m-1071	176	20	�	�	PROPN
m-1071	176	21	+	+	CCONJ
m-1071	176	22	(	(	PUNCT
m-1071	176	23	3	3	NUM
m-1071	176	24	�	�	PROPN
m-1071	176	25	�	�	PROPN
m-1071	176	26	+	+	CCONJ
m-1071	176	27	�	�	PROPN
m-1071	176	28	+	+	CCONJ
m-1071	176	29	2	2	NUM
m-1071	176	30	�	�	PROPN
m-1071	176	31	−	−	PROPN
m-1071	176	32	5)	5)	NUM
m-1071	176	33	�	�	PROPN
m-1071	176	34	�	�	NOUN
m-1071	176	35	√	√	NUM
m-1071	176	36	�	�	PROPN
m-1071	176	37	+	+	CCONJ
m-1071	176	38	�	�	PROPN
m-1071	176	39	3	3	NUM
m-1071	176	40	�	�	PROPN
m-1071	176	41	�	�	PROPN
m-1071	176	42	−	−	NUM
m-1071	176	43	2(2	2(2	NUM
m-1071	176	44	�	�	PROPN
m-1071	176	45	+	+	CCONJ
m-1071	176	46	�	�	PROPN
m-1071	176	47	−	−	ADP
m-1071	176	48	1	1	NUM
m-1071	176	49	)	)	PUNCT
m-1071	176	50	�	�	PROPN
m-1071	176	51	�	�	PROPN
m-1071	176	52	�	�	PROPN
m-1071	176	53	by	by	ADP
m-1071	176	54	taking	take	VERB
m-1071	176	55	the	the	DET
m-1071	176	56	first	first	ADJ
m-1071	176	57	derivative	derivative	NOUN
m-1071	176	58	of	of	ADP
m-1071	176	59	the	the	DET
m-1071	176	60	polynomial	polynomial	NOUN
m-1071	176	61	in	in	ADP
m-1071	176	62	theorem	theorem	NOUN
m-1071	176	63	2.31	2.31	NUM
m-1071	176	64	at	at	ADP
m-1071	176	65	y	y	PROPN
m-1071	176	66	=	=	SYM
m-1071	176	67	1	1	NUM
m-1071	176	68	,	,	PUNCT
m-1071	176	69	we	we	PRON
m-1071	176	70	get	get	VERB
m-1071	176	71	the	the	DET
m-1071	176	72	reciprocal	reciprocal	ADJ
m-1071	176	73	randic	randic	ADJ
m-1071	176	74	polynomial	polynomial	ADJ
m-1071	176	75	index	index	NOUN
m-1071	176	76	of	of	ADP
m-1071	176	77	chain	chain	NOUN
m-1071	176	78	of	of	ADP
m-1071	176	79	sio4	sio4	PROPN
m-1071	176	80	sc	sc	PROPN
m-1071	176	81	�	�	PROPN
m-1071	176	82	�	�	PROPN
m-1071	176	83	as	as	SCONJ
m-1071	176	84	follows	follow	VERB
m-1071	176	85	:	:	PUNCT
m-1071	176	86	corollary	corollary	ADJ
m-1071	176	87	2.32	2.32	NUM
m-1071	176	88	.	.	PUNCT
m-1071	177	1	let	let	VERB
m-1071	177	2	p	p	PRON
m-1071	177	3	be	be	AUX
m-1071	177	4	odd	odd	ADJ
m-1071	177	5	and	and	CCONJ
m-1071	177	6	p	p	X
m-1071	177	7	<	<	X
m-1071	177	8	q.	q.	PROPN
m-1071	178	1	then	then	ADV
m-1071	178	2	the	the	DET
m-1071	178	3	reciprocal	reciprocal	ADJ
m-1071	178	4	randic	randic	ADJ
m-1071	178	5	polynomial	polynomial	ADJ
m-1071	178	6	index	index	NOUN
m-1071	178	7	of	of	ADP
m-1071	178	8	sc	sc	PROPN
m-1071	178	9	�	�	PROPN
m-1071	178	10	�	�	PROPN
m-1071	178	11	�	�	PROPN
m-1071	178	12	�	�	PROPN
m-1071	178	13	9	9	NUM
m-1071	178	14	�	�	PROPN
m-1071	178	15	2	2	NUM
m-1071	178	16	�	�	PROPN
m-1071	178	17	�	�	PROPN
m-1071	178	18	+	+	CCONJ
m-1071	178	19	18	18	NUM
m-1071	178	20	�	�	PROPN
m-1071	178	21	�	�	PROPN
m-1071	178	22	+	+	CCONJ
m-1071	178	23	3	3	NUM
m-1071	178	24	�	�	PROPN
m-1071	178	25	2	2	NUM
m-1071	178	26	�	�	NOUN
m-1071	178	27	−	−	NUM
m-1071	178	28	15	15	NUM
m-1071	178	29	�	�	PROPN
m-1071	178	30	+	+	SYM
m-1071	178	31	6√2	6√2	PROPN
m-1071	178	32	�	�	PROPN
m-1071	178	33	+	+	CCONJ
m-1071	178	34	21	21	NUM
m-1071	178	35	−	−	NUM
m-1071	178	36	12	12	NUM
m-1071	178	37	�	�	PROPN
m-1071	178	38	−	−	NOUN
m-1071	178	39	15√2	15√2	NUM
m-1071	178	40	.	.	PUNCT
m-1071	179	1	theorem	theorem	VERB
m-1071	179	2	2.33	2.33	NUM
m-1071	179	3	.	.	PUNCT
m-1071	180	1	let	let	VERB
m-1071	180	2	p	p	PRON
m-1071	180	3	be	be	AUX
m-1071	180	4	odd	odd	ADJ
m-1071	180	5	and	and	CCONJ
m-1071	180	6	p	p	X
m-1071	180	7	<	<	X
m-1071	180	8	q.	q.	PROPN
m-1071	181	1	then	then	ADV
m-1071	181	2	the	the	DET
m-1071	181	3	sigma	sigma	PROPN
m-1071	181	4	polynomial	polynomial	PROPN
m-1071	181	5	of	of	ADP
m-1071	181	6	sc	sc	PROPN
m-1071	181	7	�	�	PROPN
m-1071	181	8	�	�	PROPN
m-1071	181	9	is(3	is(3	PROPN
m-1071	181	10	�	�	PROPN
m-1071	181	11	�	�	PROPN
m-1071	181	12	−	−	PROPN
m-1071	181	13	�	�	PROPN
m-1071	181	14	−	−	PROPN
m-1071	181	15	2	2	NUM
m-1071	181	16	�	�	PROPN
m-1071	181	17	+	+	CCONJ
m-1071	181	18	1	1	NUM
m-1071	181	19	)	)	PUNCT
m-1071	181	20	�	�	PROPN
m-1071	181	21	�	�	PROPN
m-1071	181	22	+	+	CCONJ
m-1071	181	23	(	(	PUNCT
m-1071	181	24	4	4	NUM
m-1071	181	25	�	�	PROPN
m-1071	181	26	�	�	PROPN
m-1071	181	27	+	+	CCONJ
m-1071	181	28	�	�	PROPN
m-1071	181	29	+	+	CCONJ
m-1071	181	30	2	2	NUM
m-1071	181	31	�	�	NOUN
m-1071	181	32	−	−	NUM
m-1071	181	33	5	5	NUM
m-1071	181	34	)	)	PUNCT
m-1071	181	35	�	�	PROPN
m-1071	181	36	�	�	PROPN
m-1071	181	37	�	�	PROPN
m-1071	181	38	�	�	PROPN
m-1071	181	39	proof	proof	NOUN
m-1071	181	40	.	.	PUNCT
m-1071	182	1	using	use	VERB
m-1071	182	2	the	the	DET
m-1071	182	3	atom	atom	NOUN
m-1071	182	4	-	-	PUNCT
m-1071	182	5	bond	bond	NOUN
m-1071	182	6	partition	partition	NOUN
m-1071	182	7	from	from	ADP
m-1071	182	8	table	table	NOUN
m-1071	182	9	2	2	NUM
m-1071	182	10	,	,	PUNCT
m-1071	182	11	in	in	ADP
m-1071	182	12	the	the	DET
m-1071	182	13	formula	formula	NOUN
m-1071	182	14	of	of	ADP
m-1071	182	15	symmetric	symmetric	ADJ
m-1071	182	16	division	division	NOUN
m-1071	182	17	degree	degree	NOUN
m-1071	182	18	polynomial	polynomial	ADJ
m-1071	182	19	(	(	PUNCT
m-1071	182	20	7	7	NUM
m-1071	182	21	)	)	PUNCT
m-1071	182	22	,	,	PUNCT
m-1071	182	23	we	we	PRON
m-1071	182	24	get	get	VERB
m-1071	182	25	�	�	PROPN
m-1071	182	26	�	�	PROPN
m-1071	182	27	�	�	PROPN
m-1071	182	28	�	�	PROPN
m-1071	182	29	sc	sc	PROPN
m-1071	182	30	�	�	PROPN
m-1071	182	31	�	�	PROPN
m-1071	182	32	,	,	PUNCT
m-1071	182	33	�	�	PROPN
m-1071	182	34	�	�	PROPN
m-1071	182	35	=	=	SYM
m-1071	182	36	�	�	PROPN
m-1071	182	37	�	�	PROPN
m-1071	182	38	[	[	X
m-1071	182	39	(	(	PUNCT
m-1071	182	40	�	�	NOUN
m-1071	182	41	)	)	PUNCT
m-1071	182	42	�	�	PROPN
m-1071	182	43	�	�	PROPN
m-1071	182	44	(	(	PUNCT
m-1071	182	45	�	�	PROPN
m-1071	182	46	)	)	PUNCT
m-1071	182	47	�	�	PROPN
m-1071	182	48	]	]	PUNCT
m-1071	182	49	(	(	PUNCT
m-1071	182	50	�	�	PROPN
m-1071	182	51	)	)	PUNCT
m-1071	182	52	(	(	PUNCT
m-1071	182	53	�	�	PROPN
m-1071	182	54	)	)	PUNCT
m-1071	182	55	�	�	PROPN
m-1071	182	56	�	�	PROPN
m-1071	182	57	�	�	PROPN
m-1071	182	58	�	�	PROPN
m-1071	182	59	~	~	SYM
m-1071	182	60	�	�	PROPN
m-1071	182	61	�	�	PROPN
m-1071	182	62	�	�	PROPN
m-1071	182	63	�	�	PROPN
m-1071	182	64	+	+	CCONJ
m-1071	182	65	�	�	PROPN
m-1071	182	66	�	�	PROPN
m-1071	182	67	[	[	X
m-1071	182	68	(	(	PUNCT
m-1071	182	69	�	�	NOUN
m-1071	182	70	)	)	PUNCT
m-1071	182	71	�	�	PROPN
m-1071	182	72	�	�	PROPN
m-1071	182	73	(	(	PUNCT
m-1071	182	74	�	�	PROPN
m-1071	182	75	)	)	PUNCT
m-1071	182	76	�	�	PROPN
m-1071	182	77	]	]	PUNCT
m-1071	182	78	(	(	PUNCT
m-1071	182	79	�	�	PROPN
m-1071	182	80	)	)	PUNCT
m-1071	182	81	(	(	PUNCT
m-1071	182	82	�	�	PROPN
m-1071	182	83	)	)	PUNCT
m-1071	182	84	�	�	PROPN
m-1071	182	85	�	�	PROPN
m-1071	182	86	�	�	PROPN
m-1071	182	87	�	�	PROPN
m-1071	182	88	~	~	SYM
m-1071	182	89	�	�	PROPN
m-1071	182	90	�	�	PROPN
m-1071	182	91	�	�	PROPN
m-1071	182	92	�	�	PROPN
m-1071	182	93	+	+	CCONJ
m-1071	182	94	�	�	PROPN
m-1071	182	95	�	�	PROPN
m-1071	182	96	[	[	X
m-1071	182	97	(	(	PUNCT
m-1071	182	98	�	�	NOUN
m-1071	182	99	)	)	PUNCT
m-1071	182	100	�	�	PROPN
m-1071	182	101	�	�	PROPN
m-1071	182	102	(	(	PUNCT
m-1071	182	103	�	�	PROPN
m-1071	182	104	)	)	PUNCT
m-1071	182	105	�	�	PROPN
m-1071	182	106	]	]	PUNCT
m-1071	182	107	(	(	PUNCT
m-1071	182	108	�	�	PROPN
m-1071	182	109	)	)	PUNCT
m-1071	182	110	(	(	PUNCT
m-1071	182	111	�	�	NOUN
m-1071	182	112	)	)	PUNCT
m-1071	182	113	�	�	PROPN
m-1071	182	114	�	�	PROPN
m-1071	182	115	~	~	NOUN
m-1071	182	116	�	�	PROPN
m-1071	182	117	this	this	PRON
m-1071	182	118	gives	give	VERB
m-1071	182	119	�	�	PROPN
m-1071	182	120	�	�	PROPN
m-1071	182	121	�	�	PROPN
m-1071	182	122	�	�	PROPN
m-1071	182	123	sc	sc	PROPN
m-1071	182	124	�	�	PROPN
m-1071	182	125	�	�	PROPN
m-1071	182	126	,	,	PUNCT
m-1071	182	127	�	�	PROPN
m-1071	182	128	�	�	PROPN
m-1071	182	129	=	=	SYM
m-1071	182	130	3	3	NUM
m-1071	182	131	(	(	PUNCT
m-1071	182	132	�	�	PROPN
m-1071	182	133	+	+	CCONJ
m-1071	182	134	1	1	NUM
m-1071	182	135	)	)	PUNCT
m-1071	182	136	�	�	PROPN
m-1071	182	137	�	�	PROPN
m-1071	182	138	+	+	CCONJ
m-1071	182	139	(	(	PUNCT
m-1071	182	140	3	3	NUM
m-1071	182	141	�	�	PROPN
m-1071	182	142	�	�	PROPN
m-1071	182	143	+	+	CCONJ
m-1071	182	144	�	�	PROPN
m-1071	182	145	+	+	CCONJ
m-1071	182	146	2	2	NUM
m-1071	182	147	�	�	NOUN
m-1071	182	148	−	−	NUM
m-1071	182	149	5	5	NUM
m-1071	182	150	)	)	PUNCT
m-1071	182	151	�	�	PROPN
m-1071	182	152	�	�	PROPN
m-1071	182	153	�	�	PROPN
m-1071	182	154	�	�	PROPN
m-1071	182	155	+	+	CCONJ
m-1071	182	156	(	(	PUNCT
m-1071	182	157	3	3	NUM
m-1071	182	158	�	�	PROPN
m-1071	182	159	�	�	PROPN
m-1071	182	160	−	−	NUM
m-1071	182	161	2(2	2(2	NUM
m-1071	182	162	�	�	PROPN
m-1071	182	163	+	+	CCONJ
m-1071	182	164	�	�	PROPN
m-1071	182	165	−	−	NOUN
m-1071	182	166	1	1	NUM
m-1071	182	167	)	)	PUNCT
m-1071	182	168	)	)	PUNCT
m-1071	183	1	�	�	PROPN
m-1071	183	2	�	�	PROPN
m-1071	183	3	=	=	SYM
m-1071	183	4	(	(	PUNCT
m-1071	183	5	3	3	NUM
m-1071	183	6	�	�	PROPN
m-1071	183	7	�	�	PROPN
m-1071	183	8	−	−	PROPN
m-1071	183	9	�	�	PROPN
m-1071	183	10	−	−	PROPN
m-1071	183	11	2	2	NUM
m-1071	183	12	�	�	PROPN
m-1071	183	13	+	+	CCONJ
m-1071	183	14	1	1	NUM
m-1071	183	15	)	)	PUNCT
m-1071	183	16	�	�	PROPN
m-1071	183	17	�	�	PROPN
m-1071	183	18	+	+	CCONJ
m-1071	183	19	(	(	PUNCT
m-1071	183	20	3	3	NUM
m-1071	183	21	�	�	PROPN
m-1071	183	22	�	�	PROPN
m-1071	183	23	+	+	CCONJ
m-1071	183	24	�	�	PROPN
m-1071	183	25	+	+	CCONJ
m-1071	183	26	2	2	NUM
m-1071	183	27	�	�	NOUN
m-1071	183	28	−	−	NUM
m-1071	183	29	5	5	NUM
m-1071	183	30	)	)	PUNCT
m-1071	183	31	�	�	PROPN
m-1071	183	32	�	�	PROPN
m-1071	183	33	�	�	PROPN
m-1071	183	34	�	�	PROPN
m-1071	183	35	by	by	ADP
m-1071	183	36	taking	take	VERB
m-1071	183	37	the	the	DET
m-1071	183	38	first	first	ADJ
m-1071	183	39	derivative	derivative	NOUN
m-1071	183	40	of	of	ADP
m-1071	183	41	the	the	DET
m-1071	183	42	polynomial	polynomial	NOUN
m-1071	183	43	in	in	ADP
m-1071	183	44	theorem	theorem	ADJ
m-1071	183	45	2.33	2.33	NUM
m-1071	183	46	at	at	ADP
m-1071	183	47	y	y	PROPN
m-1071	183	48	=	=	SYM
m-1071	183	49	1	1	NUM
m-1071	183	50	,	,	PUNCT
m-1071	183	51	we	we	PRON
m-1071	183	52	get	get	VERB
m-1071	183	53	the	the	DET
m-1071	183	54	symmetric	symmetric	ADJ
m-1071	183	55	division	division	NOUN
m-1071	183	56	degree	degree	NOUN
m-1071	183	57	index	index	NOUN
m-1071	183	58	of	of	ADP
m-1071	183	59	chain	chain	NOUN
m-1071	183	60	of	of	ADP
m-1071	183	61	sio4	sio4	PROPN
m-1071	183	62	(	(	PUNCT
m-1071	183	63	sc	sc	PROPN
m-1071	183	64	�	�	PROPN
m-1071	183	65	�	�	PROPN
m-1071	183	66	)	)	PUNCT
m-1071	183	67	as	as	SCONJ
m-1071	183	68	follows	follow	VERB
m-1071	183	69	:	:	PUNCT
m-1071	183	70	corollary	corollary	ADJ
m-1071	183	71	2.34	2.34	NUM
m-1071	183	72	.	.	PUNCT
m-1071	184	1	let	let	VERB
m-1071	184	2	p	p	PRON
m-1071	184	3	be	be	AUX
m-1071	184	4	odd	odd	ADJ
m-1071	184	5	and	and	CCONJ
m-1071	184	6	p	p	X
m-1071	184	7	<	<	X
m-1071	184	8	q.	q.	PROPN
m-1071	185	1	then	then	ADV
m-1071	185	2	the	the	DET
m-1071	185	3	symmetric	symmetric	ADJ
m-1071	185	4	division	division	NOUN
m-1071	185	5	degree	degree	NOUN
m-1071	185	6	index	index	NOUN
m-1071	185	7	of	of	ADP
m-1071	185	8	sc	sc	PROPN
m-1071	185	9	�	�	PROPN
m-1071	185	10	�	�	PROPN
m-1071	185	11	is	be	AUX
m-1071	185	12	�	�	PROPN
m-1071	185	13	�	�	PROPN
m-1071	185	14	(	(	PUNCT
m-1071	185	15	69	69	NUM
m-1071	185	16	�	�	PROPN
m-1071	185	17	�	�	PROPN
m-1071	185	18	+	+	CCONJ
m-1071	185	19	7	7	NUM
m-1071	185	20	�	�	PROPN
m-1071	185	21	+	+	CCONJ
m-1071	185	22	14	14	NUM
m-1071	185	23	�	�	PROPN
m-1071	185	24	−	−	PROPN
m-1071	185	25	567	567	NUM
m-1071	185	26	)	)	PUNCT
m-1071	185	27	.	.	PUNCT
m-1071	186	1	theorem	theorem	NOUN
m-1071	186	2	2.35	2.35	NUM
m-1071	186	3	.	.	PUNCT
m-1071	187	1	let	let	VERB
m-1071	187	2	p	p	PRON
m-1071	187	3	be	be	AUX
m-1071	187	4	odd	odd	ADJ
m-1071	187	5	and	and	CCONJ
m-1071	187	6	p	p	X
m-1071	187	7	<	<	X
m-1071	187	8	q.	q.	PROPN
m-1071	188	1	then	then	ADV
m-1071	188	2	the	the	DET
m-1071	188	3	inverse	inverse	ADJ
m-1071	188	4	symmetric	symmetric	ADJ
m-1071	188	5	division	division	NOUN
m-1071	188	6	polynomial	polynomial	NOUN
m-1071	188	7	of	of	ADP
m-1071	188	8	sc	sc	PROPN
m-1071	188	9	�	�	PROPN
m-1071	188	10	�	�	PROPN
m-1071	188	11	is(3	is(3	PROPN
m-1071	188	12	�	�	PROPN
m-1071	188	13	�	�	PROPN
m-1071	188	14	−	−	PROPN
m-1071	188	15	�	�	PROPN
m-1071	188	16	−	−	PROPN
m-1071	188	17	2	2	NUM
m-1071	188	18	�	�	PROPN
m-1071	188	19	+	+	CCONJ
m-1071	188	20	1	1	NUM
m-1071	188	21	)	)	PUNCT
m-1071	188	22	�	�	PROPN
m-1071	188	23	�	�	PROPN
m-1071	188	24	�	�	PROPN
m-1071	188	25	+	+	CCONJ
m-1071	188	26	(	(	PUNCT
m-1071	188	27	3	3	NUM
m-1071	188	28	�	�	PROPN
m-1071	188	29	�	�	PROPN
m-1071	188	30	+	+	CCONJ
m-1071	188	31	�	�	PROPN
m-1071	188	32	+	+	CCONJ
m-1071	188	33	2	2	NUM
m-1071	188	34	�	�	NOUN
m-1071	188	35	−	−	NUM
m-1071	188	36	5	5	NUM
m-1071	188	37	)	)	PUNCT
m-1071	188	38	�	�	PROPN
m-1071	188	39	�	�	PROPN
m-1071	188	40	�	�	PROPN
m-1071	188	41	�	�	PROPN
m-1071	188	42	.	.	PUNCT
m-1071	189	1	ijo	ijo	PROPN
m-1071	189	2	international	international	PROPN
m-1071	189	3	journal	journal	PROPN
m-1071	189	4	of	of	ADP
m-1071	189	5	mathematics	mathematics	PROPN
m-1071	189	6	(	(	PUNCT
m-1071	189	7	issn	issn	PROPN
m-1071	189	8	:	:	PUNCT
m-1071	189	9	2992	2992	NUM
m-1071	189	10	-	-	SYM
m-1071	189	11	4421	4421	NUM
m-1071	189	12	)	)	PUNCT
m-1071	189	13	ijo	ijo	PROPN
m-1071	189	14	journals	journal	NOUN
m-1071	189	15	volume	volume	NOUN
m-1071	189	16	08	08	NUM
m-1071	190	1	|	|	ADV
m-1071	190	2	issue	issue	VERB
m-1071	190	3	04	04	NUM
m-1071	191	1	|	|	CCONJ
m-1071	191	2	april	april	PROPN
m-1071	191	3	2025	2025	NUM
m-1071	192	1	|	|	ADV
m-1071	192	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	192	3	14	14	NUM
m-1071	192	4	proof	proof	NOUN
m-1071	192	5	.	.	PUNCT
m-1071	193	1	using	use	VERB
m-1071	193	2	the	the	DET
m-1071	193	3	atom	atom	NOUN
m-1071	193	4	-	-	PUNCT
m-1071	193	5	bond	bond	NOUN
m-1071	193	6	partition	partition	NOUN
m-1071	193	7	from	from	ADP
m-1071	193	8	table	table	NOUN
m-1071	193	9	2	2	NUM
m-1071	193	10	,	,	PUNCT
m-1071	193	11	in	in	ADP
m-1071	193	12	the	the	DET
m-1071	193	13	formula	formula	NOUN
m-1071	193	14	of	of	ADP
m-1071	193	15	inverse	inverse	NOUN
m-1071	193	16	symmetric	symmetric	ADJ
m-1071	193	17	division	division	NOUN
m-1071	193	18	polynomial	polynomial	NOUN
m-1071	193	19	(	(	PUNCT
m-1071	193	20	8)	8)	NUM
m-1071	193	21	,	,	PUNCT
m-1071	193	22	we	we	PRON
m-1071	193	23	get	get	VERB
m-1071	193	24	�	�	PROPN
m-1071	193	25	�	�	PROPN
m-1071	193	26	�	�	PROPN
m-1071	193	27	�	�	PROPN
m-1071	193	28	�	�	PROPN
m-1071	193	29	sc	sc	PROPN
m-1071	193	30	�	�	PROPN
m-1071	193	31	�	�	PROPN
m-1071	193	32	,	,	PUNCT
m-1071	193	33	�	�	PROPN
m-1071	193	34	�	�	PROPN
m-1071	193	35	=	=	SYM
m-1071	193	36	�	�	PROPN
m-1071	193	37	�	�	PROPN
m-1071	193	38	�	�	PROPN
m-1071	193	39	�	�	PROPN
m-1071	193	40	�	�	PROPN
m-1071	193	41	~	~	PUNCT
m-1071	193	42	this	this	PRON
m-1071	193	43	gives	give	VERB
m-1071	193	44	�	�	PROPN
m-1071	193	45	�	�	PROPN
m-1071	193	46	�	�	PROPN
m-1071	193	47	�	�	PROPN
m-1071	193	48	�	�	PROPN
m-1071	193	49	sc	sc	PROPN
m-1071	193	50	�	�	PROPN
m-1071	193	51	�	�	PROPN
m-1071	193	52	,	,	PUNCT
m-1071	193	53	�	�	PROPN
m-1071	193	54	�	�	PROPN
m-1071	193	55	=	=	SYM
m-1071	193	56	3	3	NUM
m-1071	193	57	(	(	PUNCT
m-1071	193	58	�	�	PROPN
m-1071	193	59	+	+	CCONJ
m-1071	193	60	1	1	NUM
m-1071	193	61	)	)	PUNCT
m-1071	193	62	�	�	PROPN
m-1071	193	63	�	�	PROPN
m-1071	193	64	�	�	PROPN
m-1071	193	65	+	+	CCONJ
m-1071	193	66	=	=	SYM
m-1071	193	67	(	(	PUNCT
m-1071	193	68	3	3	NUM
m-1071	193	69	�	�	PROPN
m-1071	193	70	�	�	PROPN
m-1071	193	71	−	−	PROPN
m-1071	193	72	�	�	PROPN
m-1071	193	73	−	−	PROPN
m-1071	193	74	2	2	NUM
m-1071	193	75	by	by	ADP
m-1071	193	76	taking	take	VERB
m-1071	193	77	the	the	DET
m-1071	193	78	first	first	ADJ
m-1071	193	79	derivative	derivative	NOUN
m-1071	193	80	of	of	ADP
m-1071	193	81	the	the	DET
m-1071	193	82	polynomial	polynomial	NOUN
m-1071	193	83	in	in	ADP
m-1071	193	84	theorem	theorem	ADJ
m-1071	193	85	2.39	2.39	NUM
m-1071	193	86	at	at	ADP
m-1071	193	87	inverse	inverse	NOUN
m-1071	193	88	symmetric	symmetric	ADJ
m-1071	193	89	division	division	NOUN
m-1071	193	90	index	index	NOUN
m-1071	193	91	of	of	ADP
m-1071	193	92	chain	chain	NOUN
m-1071	193	93	of	of	ADP
m-1071	193	94	corollary	corollary	ADJ
m-1071	193	95	2.36	2.36	NUM
m-1071	193	96	.	.	PUNCT
m-1071	194	1	let	let	VERB
m-1071	194	2	p	p	PRON
m-1071	194	3	be	be	AUX
m-1071	194	4	odd	odd	ADJ
m-1071	194	5	and	and	CCONJ
m-1071	194	6	p	p	X
m-1071	194	7	<	<	X
m-1071	194	8	q.	q.	PROPN
m-1071	195	1	then	then	ADV
m-1071	195	2	the	the	DET
m-1071	195	3	inverse	inverse	ADJ
m-1071	195	4	symmetric	symmetric	ADJ
m-1071	195	5	division	division	NOUN
m-1071	195	6	index	index	NOUN
m-1071	195	7	of	of	ADP
m-1071	195	8	.	.	PUNCT
m-1071	195	9	theorem	theorem	VERB
m-1071	195	10	2.37	2.37	NUM
m-1071	195	11	.	.	PUNCT
m-1071	196	1	let	let	VERB
m-1071	196	2	p	p	PRON
m-1071	196	3	be	be	AUX
m-1071	196	4	odd	odd	ADJ
m-1071	196	5	and	and	CCONJ
m-1071	196	6	p	p	X
m-1071	196	7	<	<	X
m-1071	196	8	q.	q.	PROPN
m-1071	197	1	then	then	ADV
m-1071	197	2	the	the	DET
m-1071	197	3	sigma	sigma	PROPN
m-1071	197	4	polynomial	polynomial	NOUN
m-1071	197	5	of	of	ADP
m-1071	197	6	y9	y9	PROPN
m-1071	197	7	+	+	CCONJ
m-1071	197	8	3pq	3pq	ADJ
m-1071	197	9	−	−	PROPN
m-1071	197	10	p	p	NOUN
m-1071	198	1	−	−	PROPN
m-1071	198	2	2q	2q	NOUN
m-1071	198	3	+	+	NOUN
m-1071	198	4	1	1	X
m-1071	198	5	.	.	X
m-1071	198	6	proof	proof	NOUN
m-1071	198	7	.	.	PUNCT
m-1071	199	1	using	use	VERB
m-1071	199	2	the	the	DET
m-1071	199	3	atom	atom	NOUN
m-1071	199	4	-	-	PUNCT
m-1071	199	5	bond	bond	NOUN
m-1071	199	6	partition	partition	NOUN
m-1071	199	7	from	from	ADP
m-1071	199	8	table	table	NOUN
m-1071	199	9	2	2	NUM
m-1071	199	10	,	,	PUNCT
m-1071	199	11	in	in	ADP
m-1071	199	12	the	the	DET
m-1071	199	13	formul	formul	NOUN
m-1071	199	14	we	we	PRON
m-1071	199	15	get	get	VERB
m-1071	199	16	�	�	PROPN
m-1071	199	17	�	�	PROPN
m-1071	199	18	sc	sc	PROPN
m-1071	199	19	�	�	PROPN
m-1071	199	20	�	�	PROPN
m-1071	199	21	,	,	PUNCT
m-1071	199	22	�	�	PROPN
m-1071	199	23	�	�	PROPN
m-1071	199	24	=	=	SYM
m-1071	199	25	�	�	PROPN
m-1071	199	26	�	�	PROPN
m-1071	199	27	�	�	PROPN
m-1071	199	28	�	�	PROPN
m-1071	199	29	�	�	PROPN
m-1071	199	30	this	this	PRON
m-1071	199	31	gives	give	VERB
m-1071	199	32	�	�	PROPN
m-1071	199	33	�	�	PROPN
m-1071	199	34	�	�	PROPN
m-1071	199	35	�	�	PROPN
m-1071	199	36	sc	sc	PROPN
m-1071	199	37	�	�	PROPN
m-1071	199	38	�	�	PROPN
m-1071	199	39	,	,	PUNCT
m-1071	199	40	�	�	PROPN
m-1071	199	41	�	�	PROPN
m-1071	199	42	=	=	SYM
m-1071	199	43	3	3	NUM
m-1071	199	44	(	(	PUNCT
m-1071	199	45	�	�	PROPN
m-1071	199	46	+	+	CCONJ
m-1071	199	47	1	1	NUM
m-1071	199	48	)	)	PUNCT
m-1071	199	49	+	+	NOUN
m-1071	200	1	=	=	SYM
m-1071	200	2	(	(	PUNCT
m-1071	200	3	3	3	NUM
m-1071	200	4	�	�	NOUN
m-1071	200	5	�	�	NOUN
m-1071	200	6	by	by	ADP
m-1071	200	7	taking	take	VERB
m-1071	200	8	the	the	DET
m-1071	200	9	first	first	ADJ
m-1071	200	10	derivative	derivative	NOUN
m-1071	200	11	of	of	ADP
m-1071	200	12	the	the	DET
m-1071	200	13	polynomial	polynomial	NOUN
m-1071	200	14	in	in	ADP
m-1071	200	15	theorem	theorem	NOUN
m-1071	200	16	2.39	2.39	NUM
m-1071	200	17	at	at	ADP
m-1071	200	18	sigma	sigma	PROPN
m-1071	200	19	index	index	NOUN
m-1071	200	20	of	of	ADP
m-1071	200	21	chain	chain	NOUN
m-1071	200	22	of	of	ADP
m-1071	200	23	corollary	corollary	ADJ
m-1071	200	24	2.38	2.38	NUM
m-1071	200	25	.	.	PUNCT
m-1071	201	1	let	let	VERB
m-1071	201	2	p	p	PRON
m-1071	201	3	be	be	AUX
m-1071	201	4	odd	odd	ADJ
m-1071	201	5	and	and	CCONJ
m-1071	201	6	p	p	X
m-1071	201	7	<	<	X
m-1071	201	8	q.	q.	PROPN
m-1071	202	1	then	then	ADV
m-1071	202	2	the	the	DET
m-1071	202	3	sigma	sigma	PROPN
m-1071	202	4	index	index	NOUN
m-1071	202	5	of	of	ADP
m-1071	202	6	bond	bond	NOUN
m-1071	202	7	partition	partition	NOUN
m-1071	202	8	from	from	ADP
m-1071	202	9	table	table	NOUN
m-1071	202	10	2	2	NUM
m-1071	202	11	,	,	PUNCT
m-1071	202	12	in	in	ADP
m-1071	202	13	the	the	DET
m-1071	202	14	formula	formula	NOUN
m-1071	202	15	of	of	ADP
m-1071	202	16	inverse	inverse	NOUN
m-1071	202	17	symmetric	symmetric	ADJ
m-1071	202	18	polynomial	polynomial	NOUN
m-1071	202	19	(	(	PUNCT
m-1071	202	20	8)	8)	NUM
m-1071	202	21	,	,	PUNCT
m-1071	202	22	we	we	PRON
m-1071	202	23	get	get	VERB
m-1071	202	24	�	�	PROPN
m-1071	202	25	�	�	PROPN
m-1071	202	26	(	(	PUNCT
m-1071	202	27	�	�	PROPN
m-1071	202	28	)	)	PUNCT
m-1071	202	29	(	(	PUNCT
m-1071	202	30	�	�	PROPN
m-1071	202	31	)	)	PUNCT
m-1071	202	32	[	[	X
m-1071	202	33	(	(	PUNCT
m-1071	202	34	�	�	NOUN
m-1071	202	35	)	)	PUNCT
m-1071	202	36	�	�	PROPN
m-1071	202	37	�	�	PROPN
m-1071	202	38	(	(	PUNCT
m-1071	202	39	�	�	PROPN
m-1071	202	40	)	)	PUNCT
m-1071	202	41	�	�	PROPN
m-1071	202	42	]	]	PUNCT
m-1071	202	43	~	~	PROPN
m-1071	202	44	�	�	PROPN
m-1071	202	45	�	�	PROPN
m-1071	202	46	�	�	PROPN
m-1071	202	47	�	�	PROPN
m-1071	202	48	+	+	CCONJ
m-1071	202	49	�	�	PROPN
m-1071	202	50	�	�	PROPN
m-1071	202	51	(	(	PUNCT
m-1071	202	52	�	�	PROPN
m-1071	202	53	)	)	PUNCT
m-1071	202	54	(	(	PUNCT
m-1071	202	55	�	�	PROPN
m-1071	202	56	)	)	PUNCT
m-1071	202	57	[	[	X
m-1071	202	58	(	(	PUNCT
m-1071	202	59	�	�	NOUN
m-1071	202	60	)	)	PUNCT
m-1071	202	61	�	�	PROPN
m-1071	202	62	�	�	PROPN
m-1071	202	63	(	(	PUNCT
m-1071	202	64	�	�	PROPN
m-1071	202	65	)	)	PUNCT
m-1071	202	66	�	�	PROPN
m-1071	202	67	]	]	PUNCT
m-1071	202	68	�	�	PROPN
m-1071	202	69	�	�	PROPN
m-1071	202	70	~	~	SYM
m-1071	202	71	�	�	PROPN
m-1071	202	72	+	+	CCONJ
m-1071	202	73	�	�	PROPN
m-1071	202	74	�	�	PROPN
m-1071	202	75	[	[	PUNCT
m-1071	202	76	�	�	PROPN
m-1071	202	77	�	�	PROPN
m-1071	202	78	�	�	PROPN
m-1071	202	79	�	�	PROPN
m-1071	202	80	~	~	SYM
m-1071	202	81	�	�	PROPN
m-1071	202	82	�	�	PROPN
m-1071	202	83	�	�	PROPN
m-1071	202	84	�	�	PROPN
m-1071	202	85	+	+	CCONJ
m-1071	202	86	(	(	PUNCT
m-1071	202	87	3	3	NUM
m-1071	202	88	�	�	PROPN
m-1071	202	89	�	�	PROPN
m-1071	202	90	+	+	CCONJ
m-1071	202	91	�	�	PROPN
m-1071	202	92	+	+	CCONJ
m-1071	202	93	2	2	NUM
m-1071	202	94	�	�	NOUN
m-1071	202	95	−	−	NUM
m-1071	202	96	5	5	NUM
m-1071	202	97	)	)	PUNCT
m-1071	202	98	�	�	PROPN
m-1071	202	99	�	�	PROPN
m-1071	202	100	�	�	PROPN
m-1071	202	101	�	�	PROPN
m-1071	202	102	+	+	CCONJ
m-1071	202	103	(	(	PUNCT
m-1071	202	104	3	3	NUM
m-1071	202	105	�	�	PROPN
m-1071	202	106	�	�	PROPN
m-1071	202	107	−	−	PROPN
m-1071	202	108	2(2	2(2	NUM
m-1071	202	109	�	�	PROPN
m-1071	202	110	−	−	NUM
m-1071	202	111	2(2	2(2	NUM
m-1071	202	112	2	2	NUM
m-1071	202	113	�	�	NOUN
m-1071	202	114	+	+	CCONJ
m-1071	202	115	1	1	NUM
m-1071	202	116	)	)	PUNCT
m-1071	202	117	�	�	PROPN
m-1071	202	118	�	�	PROPN
m-1071	202	119	�	�	PROPN
m-1071	202	120	+	+	CCONJ
m-1071	202	121	(	(	PUNCT
m-1071	202	122	3	3	NUM
m-1071	202	123	�	�	PROPN
m-1071	202	124	�	�	PROPN
m-1071	202	125	+	+	CCONJ
m-1071	202	126	�	�	PROPN
m-1071	202	127	+	+	CCONJ
m-1071	202	128	2	2	NUM
m-1071	202	129	�	�	NOUN
m-1071	202	130	−	−	NUM
m-1071	202	131	5	5	NUM
m-1071	202	132	)	)	PUNCT
m-1071	202	133	�	�	PROPN
m-1071	202	134	�	�	PROPN
m-1071	202	135	�	�	PROPN
m-1071	202	136	�	�	PROPN
m-1071	202	137	.	.	PUNCT
m-1071	202	138	by	by	ADP
m-1071	202	139	taking	take	VERB
m-1071	202	140	the	the	DET
m-1071	202	141	first	first	ADJ
m-1071	202	142	derivative	derivative	NOUN
m-1071	202	143	of	of	ADP
m-1071	202	144	the	the	DET
m-1071	202	145	polynomial	polynomial	NOUN
m-1071	202	146	in	in	ADP
m-1071	202	147	theorem	theorem	ADJ
m-1071	202	148	2.39	2.39	NUM
m-1071	202	149	at	at	ADP
m-1071	202	150	y	y	PROPN
m-1071	202	151	inverse	inverse	NOUN
m-1071	202	152	symmetric	symmetric	ADJ
m-1071	202	153	division	division	NOUN
m-1071	202	154	index	index	NOUN
m-1071	202	155	of	of	ADP
m-1071	202	156	chain	chain	NOUN
m-1071	202	157	of	of	ADP
m-1071	202	158	sio4	sio4	PROPN
m-1071	202	159	sc	sc	PROPN
m-1071	202	160	�	�	PROPN
m-1071	202	161	�	�	PROPN
m-1071	202	162	as	as	SCONJ
m-1071	202	163	follows	follow	VERB
m-1071	202	164	:	:	PUNCT
m-1071	202	165	let	let	VERB
m-1071	202	166	p	p	PRON
m-1071	202	167	be	be	AUX
m-1071	202	168	odd	odd	ADJ
m-1071	202	169	and	and	CCONJ
m-1071	202	170	p	p	X
m-1071	202	171	<	<	X
m-1071	202	172	q.	q.	PROPN
m-1071	203	1	then	then	ADV
m-1071	203	2	the	the	DET
m-1071	203	3	inverse	inverse	ADJ
m-1071	203	4	symmetric	symmetric	ADJ
m-1071	203	5	division	division	NOUN
m-1071	203	6	index	index	NOUN
m-1071	203	7	of	of	ADP
m-1071	203	8	let	let	VERB
m-1071	203	9	p	p	PRON
m-1071	203	10	be	be	AUX
m-1071	203	11	odd	odd	ADJ
m-1071	203	12	and	and	CCONJ
m-1071	203	13	p	p	X
m-1071	203	14	<	<	X
m-1071	203	15	q.	q.	PROPN
m-1071	204	1	then	then	ADV
m-1071	204	2	the	the	DET
m-1071	204	3	sigma	sigma	PROPN
m-1071	204	4	polynomial	polynomial	PROPN
m-1071	204	5	of	of	ADP
m-1071	204	6	sc	sc	PROPN
m-1071	204	7	�	�	PROPN
m-1071	204	8	�	�	PROPN
m-1071	204	9	is	be	AUX
m-1071	204	10	(	(	PUNCT
m-1071	204	11	3	3	NUM
m-1071	204	12	bond	bond	NOUN
m-1071	204	13	partition	partition	NOUN
m-1071	204	14	from	from	ADP
m-1071	204	15	table	table	NOUN
m-1071	204	16	2	2	NUM
m-1071	204	17	,	,	PUNCT
m-1071	204	18	in	in	ADP
m-1071	204	19	the	the	DET
m-1071	204	20	formula	formula	NOUN
m-1071	204	21	of	of	ADP
m-1071	204	22	sigma	sigma	PROPN
m-1071	204	23	polynomial	polynomial	ADJ
m-1071	204	24	(	(	PUNCT
m-1071	204	25	8)	8)	NUM
m-1071	204	26	,	,	PUNCT
m-1071	204	27	�	�	PROPN
m-1071	204	28	�	�	PROPN
m-1071	204	29	(	(	PUNCT
m-1071	204	30	�	�	PROPN
m-1071	204	31	�	�	PROPN
m-1071	204	32	�	�	PROPN
m-1071	204	33	)	)	PUNCT
m-1071	204	34	�	�	PROPN
m-1071	204	35	�	�	PROPN
m-1071	204	36	~	~	PROPN
m-1071	204	37	�	�	PROPN
m-1071	204	38	�	�	PROPN
m-1071	204	39	�	�	PROPN
m-1071	204	40	�	�	PROPN
m-1071	204	41	+	+	CCONJ
m-1071	204	42	�	�	PROPN
m-1071	204	43	�	�	PROPN
m-1071	204	44	(	(	PUNCT
m-1071	204	45	�	�	PROPN
m-1071	204	46	�	�	PROPN
m-1071	204	47	�	�	PROPN
m-1071	204	48	)	)	PUNCT
m-1071	204	49	�	�	PROPN
m-1071	204	50	�	�	PROPN
m-1071	204	51	�	�	PROPN
m-1071	204	52	�	�	PROPN
m-1071	204	53	�	�	PROPN
m-1071	204	54	~	~	SYM
m-1071	204	55	�	�	PROPN
m-1071	204	56	�	�	PROPN
m-1071	204	57	�	�	PROPN
m-1071	204	58	�	�	PROPN
m-1071	204	59	+	+	CCONJ
m-1071	204	60	�	�	PROPN
m-1071	204	61	�	�	PROPN
m-1071	204	62	(	(	PUNCT
m-1071	204	63	�	�	PROPN
m-1071	204	64	�	�	PROPN
m-1071	204	65	�	�	PROPN
m-1071	204	66	�	�	PROPN
m-1071	204	67	�	�	PROPN
m-1071	204	68	~	~	SYM
m-1071	204	69	�	�	PROPN
m-1071	204	70	)	)	PUNCT
m-1071	205	1	+	+	CCONJ
m-1071	205	2	(	(	PUNCT
m-1071	205	3	3	3	NUM
m-1071	205	4	�	�	PROPN
m-1071	205	5	�	�	PROPN
m-1071	205	6	+	+	CCONJ
m-1071	205	7	�	�	PROPN
m-1071	205	8	+	+	CCONJ
m-1071	205	9	2	2	NUM
m-1071	205	10	�	�	NOUN
m-1071	205	11	−	−	NUM
m-1071	205	12	5	5	NUM
m-1071	205	13	)	)	PUNCT
m-1071	205	14	�	�	PROPN
m-1071	205	15	�	�	PROPN
m-1071	205	16	+	+	CCONJ
m-1071	205	17	(	(	PUNCT
m-1071	205	18	3	3	NUM
m-1071	205	19	�	�	PROPN
m-1071	205	20	�	�	PROPN
m-1071	205	21	−	−	PROPN
m-1071	205	22	2(2	2(2	NUM
m-1071	205	23	�	�	PROPN
m-1071	205	24	−	−	NUM
m-1071	205	25	2(2	2(2	NUM
m-1071	205	26	(	(	PUNCT
m-1071	205	27	+	+	NUM
m-1071	205	28	�	�	PROPN
m-1071	205	29	+	+	CCONJ
m-1071	205	30	2	2	NUM
m-1071	205	31	�	�	NOUN
m-1071	205	32	−	−	NUM
m-1071	205	33	5	5	NUM
m-1071	205	34	)	)	PUNCT
m-1071	205	35	�	�	PROPN
m-1071	205	36	�	�	PROPN
m-1071	205	37	+	+	CCONJ
m-1071	205	38	3	3	NUM
m-1071	205	39	�	�	PROPN
m-1071	205	40	�	�	PROPN
m-1071	205	41	−	−	PROPN
m-1071	205	42	�	�	PROPN
m-1071	205	43	−	−	PROPN
m-1071	205	44	2	2	NUM
m-1071	205	45	�	�	PROPN
m-1071	205	46	+	+	CCONJ
m-1071	205	47	1	1	X
m-1071	205	48	.	.	PUNCT
m-1071	206	1	by	by	ADP
m-1071	206	2	taking	take	VERB
m-1071	206	3	the	the	DET
m-1071	206	4	first	first	ADJ
m-1071	206	5	derivative	derivative	NOUN
m-1071	206	6	of	of	ADP
m-1071	206	7	the	the	DET
m-1071	206	8	polynomial	polynomial	NOUN
m-1071	206	9	in	in	ADP
m-1071	206	10	theorem	theorem	ADJ
m-1071	206	11	2.39	2.39	NUM
m-1071	206	12	at	at	ADP
m-1071	206	13	y	y	PROPN
m-1071	206	14	as	as	SCONJ
m-1071	206	15	follows	follow	VERB
m-1071	206	16	:	:	PUNCT
m-1071	206	17	let	let	VERB
m-1071	206	18	p	p	PRON
m-1071	206	19	be	be	AUX
m-1071	206	20	odd	odd	ADJ
m-1071	206	21	and	and	CCONJ
m-1071	206	22	p	p	X
m-1071	206	23	<	<	X
m-1071	206	24	q.	q.	PROPN
m-1071	207	1	then	then	ADV
m-1071	207	2	the	the	DET
m-1071	207	3	sigma	sigma	PROPN
m-1071	207	4	index	index	NOUN
m-1071	207	5	of	of	ADP
m-1071	207	6	sc	sc	PROPN
m-1071	207	7	�	�	PROPN
m-1071	207	8	�	�	PROPN
m-1071	207	9	is	be	AUX
m-1071	207	10	(	(	PUNCT
m-1071	207	11	3pq	3pq	ADJ
m-1071	207	12	+	+	CCONJ
m-1071	207	13	p	p	NOUN
m-1071	207	14	bond	bond	NOUN
m-1071	207	15	partition	partition	NOUN
m-1071	207	16	from	from	ADP
m-1071	207	17	table	table	NOUN
m-1071	207	18	2	2	NUM
m-1071	207	19	,	,	PUNCT
m-1071	207	20	in	in	ADP
m-1071	207	21	the	the	DET
m-1071	207	22	formula	formula	NOUN
m-1071	207	23	of	of	ADP
m-1071	207	24	inverse	inverse	NOUN
m-1071	207	25	symmetric	symmetric	NOUN
m-1071	207	26	(	(	PUNCT
m-1071	207	27	�	�	PROPN
m-1071	207	28	)	)	PUNCT
m-1071	207	29	(	(	PUNCT
m-1071	207	30	�	�	PROPN
m-1071	207	31	)	)	PUNCT
m-1071	208	1	[	[	X
m-1071	208	2	(	(	PUNCT
m-1071	208	3	�	�	NOUN
m-1071	208	4	)	)	PUNCT
m-1071	208	5	�	�	PROPN
m-1071	208	6	�	�	PROPN
m-1071	208	7	(	(	PUNCT
m-1071	208	8	�	�	PROPN
m-1071	208	9	)	)	PUNCT
m-1071	208	10	�	�	PROPN
m-1071	208	11	]	]	PUNCT
m-1071	208	12	(	(	PUNCT
m-1071	208	13	2	2	NUM
m-1071	208	14	�	�	PROPN
m-1071	208	15	+	+	CCONJ
m-1071	208	16	�	�	PROPN
m-1071	208	17	−	−	NOUN
m-1071	208	18	1	1	NUM
m-1071	208	19	)	)	PUNCT
m-1071	208	20	)	)	PUNCT
m-1071	208	21	�	�	PROPN
m-1071	208	22	�	�	PROPN
m-1071	208	23	�	�	PROPN
m-1071	208	24	y	y	PROPN
m-1071	208	25	=	=	SYM
m-1071	208	26	1	1	NUM
m-1071	208	27	,	,	PUNCT
m-1071	208	28	we	we	PRON
m-1071	208	29	get	get	VERB
m-1071	208	30	the	the	DET
m-1071	208	31	let	let	NOUN
m-1071	208	32	p	p	PRON
m-1071	208	33	be	be	AUX
m-1071	208	34	odd	odd	ADJ
m-1071	208	35	and	and	CCONJ
m-1071	208	36	p	p	X
m-1071	208	37	<	<	X
m-1071	208	38	q.	q.	PROPN
m-1071	209	1	then	then	ADV
m-1071	209	2	the	the	DET
m-1071	209	3	inverse	inverse	ADJ
m-1071	209	4	symmetric	symmetric	ADJ
m-1071	209	5	division	division	NOUN
m-1071	209	6	index	index	NOUN
m-1071	209	7	of	of	ADP
m-1071	209	8	is	be	AUX
m-1071	209	9	(	(	PUNCT
m-1071	209	10	3pq	3pq	ADJ
m-1071	210	1	+	+	CCONJ
m-1071	210	2	p	p	X
m-1071	210	3	+	+	NOUN
m-1071	210	4	2q	2q	NUM
m-1071	210	5	−	−	NUM
m-1071	210	6	5	5	NUM
m-1071	210	7	)	)	PUNCT
m-1071	210	8	a	a	PRON
m-1071	210	9	of	of	ADP
m-1071	210	10	sigma	sigma	PROPN
m-1071	210	11	polynomial	polynomial	ADJ
m-1071	210	12	(	(	PUNCT
m-1071	210	13	8)	8)	NUM
m-1071	210	14	,	,	PUNCT
m-1071	210	15	�	�	PROPN
m-1071	210	16	)	)	PUNCT
m-1071	210	17	�	�	PROPN
m-1071	210	18	(	(	PUNCT
m-1071	210	19	2	2	NUM
m-1071	210	20	�	�	PROPN
m-1071	210	21	+	+	CCONJ
m-1071	210	22	�	�	PROPN
m-1071	210	23	−	−	NOUN
m-1071	210	24	1	1	NUM
m-1071	210	25	)	)	PUNCT
m-1071	210	26	)	)	PUNCT
m-1071	211	1	y	y	NOUN
m-1071	211	2	=	=	SYM
m-1071	211	3	1	1	NUM
m-1071	211	4	,	,	PUNCT
m-1071	211	5	we	we	PRON
m-1071	211	6	get	get	VERB
m-1071	211	7	the	the	DET
m-1071	211	8	p	p	NOUN
m-1071	211	9	+	+	NOUN
m-1071	211	10	2q	2q	NUM
m-1071	211	11	−	−	NOUN
m-1071	211	12	5	5	NUM
m-1071	211	13	)	)	PUNCT
m-1071	211	14	.	.	PUNCT
m-1071	212	1	ijo	ijo	PROPN
m-1071	212	2	international	international	PROPN
m-1071	212	3	journal	journal	PROPN
m-1071	212	4	of	of	ADP
m-1071	212	5	mathematics	mathematics	PROPN
m-1071	212	6	(	(	PUNCT
m-1071	212	7	issn	issn	PROPN
m-1071	212	8	:	:	PUNCT
m-1071	212	9	2992	2992	NUM
m-1071	212	10	-	-	SYM
m-1071	212	11	4421	4421	NUM
m-1071	212	12	)	)	PUNCT
m-1071	212	13	ijo	ijo	PROPN
m-1071	212	14	journals	journal	NOUN
m-1071	212	15	volume	volume	NOUN
m-1071	212	16	08	08	NUM
m-1071	213	1	|	|	ADV
m-1071	213	2	issue	issue	VERB
m-1071	213	3	04	04	NUM
m-1071	214	1	|	|	CCONJ
m-1071	214	2	april	april	PROPN
m-1071	214	3	2025	2025	NUM
m-1071	214	4	|	|	ADV
m-1071	214	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	214	6	15	15	NUM
m-1071	214	7	theorem	theorem	VERB
m-1071	214	8	2.39	2.39	NUM
m-1071	214	9	.	.	PUNCT
m-1071	215	1	let	let	VERB
m-1071	215	2	p	p	PRON
m-1071	215	3	be	be	AUX
m-1071	215	4	odd	odd	ADJ
m-1071	215	5	and	and	CCONJ
m-1071	215	6	p	p	X
m-1071	215	7	<	<	X
m-1071	215	8	q.	q.	PROPN
m-1071	215	9	then	then	ADV
m-1071	215	10	the	the	DET
m-1071	215	11	somber	somber	ADJ
m-1071	215	12	polynomial	polynomial	NOUN
m-1071	215	13	of	of	ADP
m-1071	215	14	sc	sc	PROPN
m-1071	215	15	�	�	PROPN
m-1071	215	16	�	�	PROPN
m-1071	215	17	is	be	AUX
m-1071	215	18	3	3	NUM
m-1071	215	19	(	(	PUNCT
m-1071	215	20	�	�	PROPN
m-1071	215	21	+	+	CCONJ
m-1071	215	22	1)	1)	NUM
m-1071	215	23	�	�	NOUN
m-1071	215	24	�	�	NOUN
m-1071	215	25	√	√	NUM
m-1071	215	26	�	�	PROPN
m-1071	215	27	+	+	CCONJ
m-1071	215	28	(	(	PUNCT
m-1071	215	29	3	3	NUM
m-1071	215	30	�	�	PROPN
m-1071	215	31	�	�	PROPN
m-1071	215	32	+	+	CCONJ
m-1071	215	33	�	�	PROPN
m-1071	215	34	+	+	CCONJ
m-1071	215	35	2	2	NUM
m-1071	215	36	�	�	PROPN
m-1071	215	37	−	−	PROPN
m-1071	215	38	5)	5)	NUM
m-1071	215	39	�	�	PROPN
m-1071	215	40	�	�	NOUN
m-1071	215	41	√	√	NUM
m-1071	215	42	�	�	PROPN
m-1071	215	43	+	+	CCONJ
m-1071	215	44	(	(	PUNCT
m-1071	215	45	3	3	NUM
m-1071	215	46	�	�	PROPN
m-1071	215	47	�	�	PROPN
m-1071	215	48	−	−	NUM
m-1071	215	49	2(2	2(2	NUM
m-1071	215	50	�	�	PROPN
m-1071	215	51	+	+	CCONJ
m-1071	215	52	�	�	PROPN
m-1071	215	53	−	−	PROPN
m-1071	215	54	1)	1)	NUM
m-1071	215	55	�	�	PROPN
m-1071	215	56	�	�	NOUN
m-1071	215	57	√	√	NUM
m-1071	215	58	�	�	PROPN
m-1071	215	59	.	.	PUNCT
m-1071	216	1	proof	proof	NOUN
m-1071	216	2	.	.	PUNCT
m-1071	217	1	using	use	VERB
m-1071	217	2	the	the	DET
m-1071	217	3	atom	atom	NOUN
m-1071	217	4	-	-	PUNCT
m-1071	217	5	bond	bond	NOUN
m-1071	217	6	partition	partition	NOUN
m-1071	217	7	from	from	ADP
m-1071	217	8	table	table	NOUN
m-1071	217	9	2	2	NUM
m-1071	217	10	,	,	PUNCT
m-1071	217	11	in	in	ADP
m-1071	217	12	the	the	DET
m-1071	217	13	formula	formula	NOUN
m-1071	217	14	of	of	ADP
m-1071	217	15	somber	somber	ADJ
m-1071	217	16	polynomial	polynomial	NOUN
m-1071	217	17	(	(	PUNCT
m-1071	217	18	8)	8)	NUM
m-1071	217	19	,	,	PUNCT
m-1071	217	20	we	we	PRON
m-1071	217	21	get	get	VERB
m-1071	217	22	�	�	PROPN
m-1071	217	23	�	�	PROPN
m-1071	217	24	�	�	PROPN
m-1071	217	25	sc	sc	PROPN
m-1071	217	26	�	�	PROPN
m-1071	217	27	�	�	PROPN
m-1071	217	28	,	,	PUNCT
m-1071	217	29	�	�	PROPN
m-1071	217	30	�	�	PROPN
m-1071	217	31	=	=	SYM
m-1071	217	32	�	�	PROPN
m-1071	217	33	�	�	PROPN
m-1071	217	34	�	�	PROPN
m-1071	217	35	(	(	PUNCT
m-1071	217	36	�	�	PROPN
m-1071	217	37	)	)	PUNCT
m-1071	217	38	�	�	PROPN
m-1071	217	39	�	�	PROPN
m-1071	217	40	(	(	PUNCT
m-1071	217	41	�	�	PROPN
m-1071	217	42	)	)	PUNCT
m-1071	217	43	�	�	PROPN
m-1071	217	44	�	�	PROPN
m-1071	217	45	�	�	PROPN
m-1071	217	46	�	�	PROPN
m-1071	217	47	�	�	PROPN
m-1071	217	48	~	~	SYM
m-1071	217	49	�	�	PROPN
m-1071	217	50	�	�	PROPN
m-1071	217	51	�	�	PROPN
m-1071	217	52	�	�	PROPN
m-1071	217	53	+	+	CCONJ
m-1071	217	54	�	�	PROPN
m-1071	217	55	�	�	PROPN
m-1071	217	56	�	�	PROPN
m-1071	217	57	(	(	PUNCT
m-1071	217	58	�	�	PROPN
m-1071	217	59	)	)	PUNCT
m-1071	217	60	�	�	PROPN
m-1071	217	61	�	�	PROPN
m-1071	217	62	(	(	PUNCT
m-1071	217	63	�	�	PROPN
m-1071	217	64	)	)	PUNCT
m-1071	217	65	�	�	PROPN
m-1071	217	66	�	�	PROPN
m-1071	217	67	�	�	PROPN
m-1071	217	68	�	�	PROPN
m-1071	217	69	�	�	PROPN
m-1071	217	70	~	~	SYM
m-1071	217	71	�	�	PROPN
m-1071	217	72	�	�	PROPN
m-1071	217	73	�	�	PROPN
m-1071	217	74	�	�	PROPN
m-1071	217	75	+	+	CCONJ
m-1071	217	76	�	�	PROPN
m-1071	217	77	�	�	PROPN
m-1071	217	78	�	�	PROPN
m-1071	217	79	(	(	PUNCT
m-1071	217	80	�	�	PROPN
m-1071	217	81	)	)	PUNCT
m-1071	217	82	�	�	PROPN
m-1071	217	83	�	�	PROPN
m-1071	217	84	(	(	PUNCT
m-1071	217	85	�	�	PROPN
m-1071	217	86	)	)	PUNCT
m-1071	217	87	�	�	PROPN
m-1071	217	88	�	�	PROPN
m-1071	217	89	�	�	PROPN
m-1071	217	90	~	~	NOUN
m-1071	217	91	�	�	PROPN
m-1071	217	92	this	this	PRON
m-1071	217	93	gives	give	VERB
m-1071	217	94	�	�	PROPN
m-1071	217	95	�	�	PROPN
m-1071	217	96	�	�	PROPN
m-1071	217	97	sc	sc	PROPN
m-1071	217	98	�	�	PROPN
m-1071	217	99	�	�	PROPN
m-1071	217	100	,	,	PUNCT
m-1071	217	101	�	�	PROPN
m-1071	217	102	�	�	PROPN
m-1071	217	103	=	=	SYM
m-1071	217	104	3	3	NUM
m-1071	217	105	(	(	PUNCT
m-1071	217	106	�	�	PROPN
m-1071	217	107	+	+	CCONJ
m-1071	217	108	1)	1)	NUM
m-1071	217	109	�	�	NOUN
m-1071	217	110	�	�	NOUN
m-1071	217	111	√	√	NUM
m-1071	217	112	�	�	PROPN
m-1071	217	113	+	+	CCONJ
m-1071	217	114	(	(	PUNCT
m-1071	217	115	3	3	NUM
m-1071	217	116	�	�	PROPN
m-1071	217	117	�	�	PROPN
m-1071	217	118	+	+	CCONJ
m-1071	217	119	�	�	PROPN
m-1071	217	120	+	+	CCONJ
m-1071	217	121	2	2	NUM
m-1071	217	122	�	�	PROPN
m-1071	217	123	−	−	PROPN
m-1071	217	124	5)	5)	NUM
m-1071	217	125	�	�	PROPN
m-1071	217	126	�	�	NOUN
m-1071	217	127	√	√	NUM
m-1071	217	128	�	�	PROPN
m-1071	217	129	+	+	CCONJ
m-1071	217	130	(	(	PUNCT
m-1071	217	131	3	3	NUM
m-1071	217	132	�	�	PROPN
m-1071	217	133	�	�	PROPN
m-1071	217	134	−	−	NUM
m-1071	217	135	2(2	2(2	NUM
m-1071	217	136	�	�	PROPN
m-1071	217	137	+	+	CCONJ
m-1071	217	138	�	�	PROPN
m-1071	218	1	−	−	PROPN
m-1071	218	2	1)	1)	NUM
m-1071	218	3	�	�	PROPN
m-1071	218	4	�	�	NOUN
m-1071	218	5	√	√	NUM
m-1071	218	6	�	�	NOUN
m-1071	218	7	by	by	ADP
m-1071	218	8	taking	take	VERB
m-1071	218	9	the	the	DET
m-1071	218	10	first	first	ADJ
m-1071	218	11	derivative	derivative	NOUN
m-1071	218	12	of	of	ADP
m-1071	218	13	the	the	DET
m-1071	218	14	polynomial	polynomial	NOUN
m-1071	218	15	in	in	ADP
m-1071	218	16	theorem	theorem	ADJ
m-1071	218	17	2.39	2.39	NUM
m-1071	218	18	at	at	ADP
m-1071	218	19	y	y	PROPN
m-1071	218	20	=	=	SYM
m-1071	218	21	1	1	NUM
m-1071	218	22	,	,	PUNCT
m-1071	218	23	we	we	PRON
m-1071	218	24	get	get	VERB
m-1071	218	25	the	the	DET
m-1071	218	26	somber	somber	ADJ
m-1071	218	27	index	index	NOUN
m-1071	218	28	of	of	ADP
m-1071	218	29	chain	chain	NOUN
m-1071	218	30	of	of	ADP
m-1071	218	31	sio4	sio4	PROPN
m-1071	218	32	(	(	PUNCT
m-1071	218	33	scqp	scqp	PROPN
m-1071	218	34	)	)	PUNCT
m-1071	218	35	as	as	SCONJ
m-1071	218	36	follows	follow	VERB
m-1071	218	37	:	:	PUNCT
m-1071	218	38	corollary	corollary	ADJ
m-1071	218	39	2.40	2.40	NUM
m-1071	218	40	.	.	PUNCT
m-1071	219	1	let	let	VERB
m-1071	219	2	p	p	PRON
m-1071	219	3	be	be	AUX
m-1071	219	4	odd	odd	ADJ
m-1071	219	5	and	and	CCONJ
m-1071	219	6	p	p	X
m-1071	219	7	<	<	X
m-1071	219	8	q.	q.	PROPN
m-1071	220	1	then	then	ADV
m-1071	220	2	the	the	DET
m-1071	220	3	somber	somber	ADJ
m-1071	220	4	index	index	NOUN
m-1071	220	5	of	of	ADP
m-1071	220	6	sc	sc	PROPN
m-1071	220	7	�	�	PROPN
m-1071	220	8	�	�	PROPN
m-1071	220	9	�	�	PROPN
m-1071	220	10	�	�	PROPN
m-1071	220	11	(	(	PUNCT
m-1071	220	12	3	3	NUM
m-1071	220	13	�	�	PROPN
m-1071	220	14	�	�	PROPN
m-1071	220	15	+	+	CCONJ
m-1071	220	16	�	�	PROPN
m-1071	220	17	+	+	CCONJ
m-1071	220	18	2	2	NUM
m-1071	220	19	�	�	NOUN
m-1071	220	20	−	−	ADP
m-1071	220	21	5)3√5	5)3√5	NUM
m-1071	220	22	+	+	CCONJ
m-1071	220	23	(	(	PUNCT
m-1071	220	24	6	6	NUM
m-1071	220	25	�	�	PROPN
m-1071	220	26	�	�	NOUN
m-1071	220	27	−	−	PROPN
m-1071	220	28	5	5	NUM
m-1071	220	29	�	�	PROPN
m-1071	220	30	−	−	NUM
m-1071	220	31	4	4	NUM
m-1071	220	32	�	�	PROPN
m-1071	220	33	+	+	CCONJ
m-1071	220	34	5))3√2	5))3√2	NUM
m-1071	220	35	conclusion	conclusion	NOUN
m-1071	220	36	in	in	ADP
m-1071	220	37	this	this	DET
m-1071	220	38	article	article	NOUN
m-1071	220	39	,	,	PUNCT
m-1071	220	40	two	two	NUM
m-1071	220	41	important	important	ADJ
m-1071	220	42	silicon	silicon	NOUN
m-1071	220	43	tetrahedron	tetrahedron	NOUN
m-1071	220	44	compound	compound	NOUN
m-1071	220	45	structures	structure	NOUN
m-1071	220	46	are	be	AUX
m-1071	220	47	considered	consider	VERB
m-1071	220	48	,	,	PUNCT
m-1071	220	49	and	and	CCONJ
m-1071	220	50	the	the	DET
m-1071	220	51	accurate	accurate	ADJ
m-1071	220	52	formulas	formula	NOUN
m-1071	220	53	of	of	ADP
m-1071	220	54	some	some	DET
m-1071	220	55	important	important	ADJ
m-1071	220	56	valency	valency	NOUN
m-1071	220	57	-	-	PUNCT
m-1071	220	58	based	base	VERB
m-1071	220	59	topological	topological	ADJ
m-1071	220	60	indices	index	NOUN
m-1071	220	61	are	be	AUX
m-1071	220	62	calculated	calculate	VERB
m-1071	220	63	using	use	VERB
m-1071	220	64	the	the	DET
m-1071	220	65	technique	technique	NOUN
m-1071	220	66	of	of	ADP
m-1071	220	67	atom	atom	NOUN
m-1071	220	68	-	-	PUNCT
m-1071	220	69	bonds	bond	NOUN
m-1071	220	70	partitioning	partitioning	NOUN
m-1071	220	71	of	of	ADP
m-1071	220	72	these	these	DET
m-1071	220	73	molecular	molecular	ADJ
m-1071	220	74	structures	structure	NOUN
m-1071	220	75	.	.	PUNCT
m-1071	221	1	our	our	PRON
m-1071	221	2	investigated	investigate	VERB
m-1071	221	3	results	result	NOUN
m-1071	221	4	,	,	PUNCT
m-1071	221	5	such	such	ADJ
m-1071	221	6	as	as	ADP
m-1071	221	7	the	the	DET
m-1071	221	8	h	h	NOUN
m-1071	221	9	-	-	PUNCT
m-1071	221	10	index	index	NOUN
m-1071	221	11	,	,	PUNCT
m-1071	221	12	abc	abc	PROPN
m-1071	221	13	-	-	PUNCT
m-1071	221	14	index	index	NOUN
m-1071	221	15	,	,	PUNCT
m-1071	221	16	f	f	PROPN
m-1071	221	17	-	-	PUNCT
m-1071	221	18	index	index	NOUN
m-1071	221	19	,	,	PUNCT
m-1071	221	20	ga	ga	PROPN
m-1071	221	21	-	-	PUNCT
m-1071	221	22	index	index	NOUN
m-1071	221	23	,	,	PUNCT
m-1071	221	24	r	r	NOUN
m-1071	221	25	-	-	PUNCT
m-1071	221	26	index	index	NOUN
m-1071	221	27	,	,	PUNCT
m-1071	221	28	rrindex	rrindex	NOUN
m-1071	221	29	,	,	PUNCT
m-1071	221	30	sdd	sdd	NOUN
m-1071	221	31	-	-	PUNCT
m-1071	221	32	index	index	NOUN
m-1071	221	33	,	,	PUNCT
m-1071	221	34	isdd	isdd	NOUN
m-1071	221	35	-	-	PUNCT
m-1071	221	36	index	index	NOUN
m-1071	221	37	,	,	PUNCT
m-1071	221	38	s	s	NOUN
m-1071	221	39	-	-	NOUN
m-1071	221	40	index	index	NOUN
m-1071	221	41	and	and	CCONJ
m-1071	221	42	so	so	ADV
m-1071	221	43	-	-	PUNCT
m-1071	221	44	index	index	NOUN
m-1071	221	45	are	be	AUX
m-1071	221	46	useful	useful	ADJ
m-1071	221	47	for	for	ADP
m-1071	221	48	determining	determine	VERB
m-1071	221	49	physio	physio	ADJ
m-1071	221	50	-	-	PUNCT
m-1071	221	51	chemical	chemical	NOUN
m-1071	221	52	properties	property	NOUN
m-1071	221	53	of	of	ADP
m-1071	221	54	chemical	chemical	ADJ
m-1071	221	55	compounds	compound	NOUN
m-1071	221	56	,	,	PUNCT
m-1071	221	57	as	as	ADP
m-1071	221	58	in	in	ADP
m-1071	221	59	2005	2005	NUM
m-1071	221	60	,	,	PUNCT
m-1071	221	61	zhou	zhou	PROPN
m-1071	221	62	explain	explain	VERB
m-1071	221	63	in	in	ADP
m-1071	221	64	[	[	X
m-1071	221	65	34	34	NUM
m-1071	221	66	]	]	X
m-1071	221	67	,	,	PUNCT
m-1071	221	68	such	such	ADJ
m-1071	221	69	as	as	ADP
m-1071	221	70	formation	formation	NOUN
m-1071	221	71	enthalpies	enthalpy	NOUN
m-1071	221	72	,	,	PUNCT
m-1071	221	73	boiling	boiling	NOUN
m-1071	221	74	points	point	NOUN
m-1071	221	75	,	,	PUNCT
m-1071	221	76	chromatographic	chromatographic	ADJ
m-1071	221	77	retention	retention	NOUN
m-1071	221	78	times	time	NOUN
m-1071	221	79	,	,	PUNCT
m-1071	221	80	vapour	vapour	NOUN
m-1071	221	81	pressure	pressure	NOUN
m-1071	221	82	,	,	PUNCT
m-1071	221	83	and	and	CCONJ
m-1071	221	84	surface	surface	NOUN
m-1071	221	85	areas	area	NOUN
m-1071	221	86	.	.	PUNCT
m-1071	222	1	the	the	DET
m-1071	222	2	obtained	obtain	VERB
m-1071	222	3	results	result	NOUN
m-1071	222	4	are	be	AUX
m-1071	222	5	also	also	ADV
m-1071	222	6	innovative	innovative	ADJ
m-1071	222	7	and	and	CCONJ
m-1071	222	8	noteworthy	noteworthy	ADJ
m-1071	222	9	contributions	contribution	NOUN
m-1071	222	10	to	to	ADP
m-1071	222	11	network	network	NOUN
m-1071	222	12	science	science	NOUN
m-1071	222	13	,	,	PUNCT
m-1071	222	14	providing	provide	VERB
m-1071	222	15	a	a	DET
m-1071	222	16	foundation	foundation	NOUN
m-1071	222	17	for	for	ADP
m-1071	222	18	understanding	understand	VERB
m-1071	222	19	the	the	DET
m-1071	222	20	deep	deep	ADJ
m-1071	222	21	topology	topology	NOUN
m-1071	222	22	of	of	ADP
m-1071	222	23	these	these	DET
m-1071	222	24	important	important	ADJ
m-1071	222	25	networks	network	NOUN
m-1071	222	26	.	.	PUNCT
m-1071	223	1	these	these	DET
m-1071	223	2	findings	finding	NOUN
m-1071	223	3	,	,	PUNCT
m-1071	223	4	may	may	AUX
m-1071	223	5	also	also	ADV
m-1071	223	6	be	be	AUX
m-1071	223	7	useful	useful	ADJ
m-1071	223	8	in	in	ADP
m-1071	223	9	determining	determine	VERB
m-1071	223	10	the	the	DET
m-1071	223	11	role	role	NOUN
m-1071	223	12	of	of	ADP
m-1071	223	13	silicon	silicon	NOUN
m-1071	223	14	-	-	PUNCT
m-1071	223	15	carbon	carbon	NOUN
m-1071	223	16	in	in	ADP
m-1071	223	17	electronics	electronic	NOUN
m-1071	223	18	and	and	CCONJ
m-1071	223	19	industry	industry	NOUN
m-1071	223	20	.	.	PUNCT
m-1071	224	1	we	we	PRON
m-1071	224	2	also	also	ADV
m-1071	224	3	present	present	VERB
m-1071	224	4	a	a	DET
m-1071	224	5	numerical	numerical	ADJ
m-1071	224	6	comparison	comparison	NOUN
m-1071	224	7	of	of	ADP
m-1071	224	8	topological	topological	ADJ
m-1071	224	9	characterizations	characterization	NOUN
m-1071	224	10	for	for	ADP
m-1071	224	11	p	p	NOUN
m-1071	224	12	=	=	NOUN
m-1071	224	13	q	q	PROPN
m-1071	224	14	for	for	ADP
m-1071	224	15	the	the	DET
m-1071	224	16	sio4	sio4	NOUN
m-1071	224	17	chain	chain	NOUN
m-1071	224	18	in	in	ADP
m-1071	224	19	(	(	PUNCT
m-1071	224	20	scqp	scqp	PROPN
m-1071	224	21	)	)	PUNCT
m-1071	224	22	in	in	ADP
m-1071	224	23	table	table	NOUN
m-1071	224	24	3	3	NUM
m-1071	224	25	and	and	CCONJ
m-1071	224	26	a	a	DET
m-1071	224	27	graphical	graphical	ADJ
m-1071	224	28	comparison	comparison	NOUN
m-1071	224	29	in	in	ADP
m-1071	224	30	figure	figure	NOUN
m-1071	224	31	2	2	NUM
m-1071	224	32	.	.	PUNCT
m-1071	224	33	ijo	ijo	PROPN
m-1071	224	34	international	international	PROPN
m-1071	224	35	journal	journal	PROPN
m-1071	224	36	of	of	ADP
m-1071	224	37	mathematics	mathematics	PROPN
m-1071	224	38	(	(	PUNCT
m-1071	224	39	issn	issn	PROPN
m-1071	224	40	:	:	PUNCT
m-1071	224	41	2992	2992	NUM
m-1071	224	42	-	-	SYM
m-1071	224	43	4421	4421	NUM
m-1071	224	44	)	)	PUNCT
m-1071	224	45	ijo	ijo	PROPN
m-1071	224	46	journals	journal	NOUN
m-1071	224	47	volume	volume	NOUN
m-1071	224	48	08	08	NUM
m-1071	225	1	|	|	ADV
m-1071	225	2	issue	issue	VERB
m-1071	225	3	04	04	NUM
m-1071	226	1	|	|	CCONJ
m-1071	226	2	april	april	PROPN
m-1071	226	3	2025	2025	NUM
m-1071	226	4	|	|	ADV
m-1071	226	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	226	6	16	16	NUM
m-1071	226	7	table	table	NOUN
m-1071	226	8	3	3	NUM
m-1071	226	9	:	:	PUNCT
m-1071	226	10	topological	topological	ADJ
m-1071	226	11	characterizations	characterization	NOUN
m-1071	226	12	of	of	ADP
m-1071	226	13	p	p	X
m-1071	226	14	=	=	NOUN
m-1071	226	15	q	q	NOUN
m-1071	226	16	h	h	NOUN
m-1071	227	1	abc	abc	PROPN
m-1071	227	2	f	f	PROPN
m-1071	227	3	2	2	NUM
m-1071	227	4	6.11	6.11	NUM
m-1071	227	5	9.56	9.56	NUM
m-1071	227	6	918	918	NUM
m-1071	227	7	3	3	NUM
m-1071	227	8	12.61	12.61	NUM
m-1071	227	9	20.39	20.39	NUM
m-1071	227	10	2430	2430	NUM
m-1071	227	11	4	4	NUM
m-1071	227	12	21.44	21.44	NUM
m-1071	227	13	35.22	35.22	NUM
m-1071	227	14	4644	4644	NUM
m-1071	227	15	5	5	NUM
m-1071	227	16	32.61	32.61	NUM
m-1071	227	17	54.06	54.06	NUM
m-1071	227	18	7560	7560	NUM
m-1071	227	19	6	6	NUM
m-1071	227	20	46.11	46.11	NUM
m-1071	227	21	76.89	76.89	NUM
m-1071	227	22	11178	11178	NUM
m-1071	227	23	7	7	NUM
m-1071	227	24	61.94	61.94	NUM
m-1071	227	25	103.72	103.72	NUM
m-1071	227	26	15498	15498	NUM
m-1071	227	27	8	8	NUM
m-1071	227	28	80.11	80.11	NUM
m-1071	227	29	134.56	134.56	NUM
m-1071	227	30	20520	20520	NUM
m-1071	227	31	9	9	NUM
m-1071	227	32	100.61	100.61	NUM
m-1071	227	33	169.39	169.39	NUM
m-1071	227	34	26244	26244	NUM
m-1071	227	35	10	10	NUM
m-1071	227	36	123.44	123.44	NUM
m-1071	227	37	208.22	208.22	NUM
m-1071	227	38	32670	32670	NUM
m-1071	227	39	figure	figure	NOUN
m-1071	227	40	2	2	NUM
m-1071	227	41	:	:	PUNCT
m-1071	227	42	graphical	graphical	ADJ
m-1071	227	43	comparison	comparison	NOUN
m-1071	227	44	open	open	ADJ
m-1071	227	45	problems	problem	NOUN
m-1071	227	46	for	for	ADP
m-1071	227	47	the	the	DET
m-1071	227	48	characterization	characterization	NOUN
m-1071	227	49	of	of	ADP
m-1071	227	50	the	the	DET
m-1071	227	51	chain	chain	NOUN
m-1071	227	52	of	of	ADP
m-1071	227	53	discuss	discuss	NOUN
m-1071	227	54	or	or	CCONJ
m-1071	227	55	research	research	VERB
m-1071	227	56	the	the	DET
m-1071	227	57	these	these	DET
m-1071	227	58	open	open	ADJ
m-1071	227	59	problems	problem	NOUN
m-1071	227	60	.	.	PUNCT
m-1071	228	1	table	table	NOUN
m-1071	228	2	3	3	NUM
m-1071	228	3	:	:	PUNCT
m-1071	228	4	topological	topological	ADJ
m-1071	228	5	characterizations	characterization	NOUN
m-1071	228	6	of	of	ADP
m-1071	228	7	and	and	CCONJ
m-1071	228	8	p	p	PROPN
m-1071	228	9	is	be	AUX
m-1071	228	10	odd	odd	ADJ
m-1071	228	11	ga	ga	PROPN
m-1071	228	12	r	r	NOUN
m-1071	228	13	rr	rr	NOUN
m-1071	228	14	sdd	sdd	NOUN
m-1071	228	15	isdd	isdd	NOUN
m-1071	228	16	17.02	17.02	NUM
m-1071	228	17	6.30	6.30	NUM
m-1071	228	18	88.30	88.30	NUM
m-1071	228	19	72.5	72.5	NUM
m-1071	228	20	8.73	8.73	NUM
m-1071	228	21	36.66	36.66	NUM
m-1071	228	22	13.04	13.04	NUM
m-1071	228	23	195.71	195.71	NUM
m-1071	228	24	164	164	NUM
m-1071	228	25	19.53	19.53	NUM
m-1071	228	26	63.78	63.78	NUM
m-1071	228	27	22.20	22.20	NUM
m-1071	228	28	343.27	343.27	NUM
m-1071	228	29	290	290	NUM
m-1071	228	30	34.93	34.93	NUM
m-1071	228	31	98.34	98.34	NUM
m-1071	228	32	33.77	33.77	NUM
m-1071	228	33	530.99	530.99	NUM
m-1071	228	34	450.5	450.5	NUM
m-1071	228	35	54.93	54.93	NUM
m-1071	228	36	11178	11178	NUM
m-1071	228	37	140.38	140.38	NUM
m-1071	228	38	47.76	47.76	NUM
m-1071	228	39	758.86	758.86	NUM
m-1071	228	40	645.5	645.5	NUM
m-1071	228	41	79.53	79.53	NUM
m-1071	228	42	15498	15498	NUM
m-1071	228	43	189.89	189.89	NUM
m-1071	228	44	64.16	64.16	NUM
m-1071	228	45	1026.89	1026.89	NUM
m-1071	228	46	875	875	NUM
m-1071	228	47	108.73	108.73	NUM
m-1071	228	48	20520	20520	NUM
m-1071	228	49	246.86	246.86	NUM
m-1071	228	50	82.97	82.97	NUM
m-1071	228	51	1335.07	1335.07	NUM
m-1071	228	52	1139	1139	NUM
m-1071	228	53	142.53	142.53	NUM
m-1071	228	54	26244	26244	NUM
m-1071	228	55	311.29	311.29	NUM
m-1071	228	56	104.20	104.20	NUM
m-1071	228	57	1683.39	1683.39	NUM
m-1071	228	58	1437.5	1437.5	NUM
m-1071	228	59	180.93	180.93	NUM
m-1071	228	60	32670	32670	NUM
m-1071	229	1	383.18	383.18	NUM
m-1071	229	2	127.84	127.84	NUM
m-1071	229	3	2071.88	2071.88	NUM
m-1071	229	4	1770.5	1770.5	NUM
m-1071	229	5	223.93	223.93	NUM
m-1071	229	6	figure	figure	NOUN
m-1071	229	7	2	2	NUM
m-1071	229	8	:	:	PUNCT
m-1071	229	9	graphical	graphical	ADJ
m-1071	229	10	comparison	comparison	NOUN
m-1071	229	11	for	for	ADP
m-1071	229	12	the	the	DET
m-1071	229	13	characterization	characterization	NOUN
m-1071	229	14	of	of	ADP
m-1071	229	15	the	the	DET
m-1071	229	16	chain	chain	NOUN
m-1071	229	17	of	of	ADP
m-1071	229	18	sio4	sio4	NOUN
m-1071	229	19	the	the	DET
m-1071	229	20	followers	follower	NOUN
m-1071	229	21	are	be	AUX
m-1071	229	22	invited	invite	VERB
m-1071	229	23	to	to	PART
m-1071	229	24	discuss	discuss	VERB
m-1071	229	25	or	or	CCONJ
m-1071	229	26	research	research	VERB
m-1071	229	27	the	the	DET
m-1071	229	28	these	these	DET
m-1071	229	29	open	open	ADJ
m-1071	229	30	problems	problem	NOUN
m-1071	229	31	.	.	PUNCT
m-1071	230	1	is	be	AUX
m-1071	230	2	odd	odd	ADJ
m-1071	230	3	isdd	isdd	NOUN
m-1071	230	4	so	so	SCONJ
m-1071	231	1	8.73	8.73	NUM
m-1071	231	2	144.83	144.83	NUM
m-1071	231	3	19.53	19.53	NUM
m-1071	231	4	354.67	354.67	NUM
m-1071	231	5	34.93	34.93	NUM
m-1071	231	6	655.67	655.67	NUM
m-1071	231	7	54.93	54.93	NUM
m-1071	231	8	1047.84	1047.84	NUM
m-1071	231	9	79.53	79.53	NUM
m-1071	231	10	1531.16	1531.16	NUM
m-1071	231	11	108.73	108.73	NUM
m-1071	231	12	2105.65	2105.65	NUM
m-1071	231	13	142.53	142.53	NUM
m-1071	231	14	2771.30	2771.30	NUM
m-1071	231	15	180.93	180.93	NUM
m-1071	231	16	3528.11	3528.11	NUM
m-1071	231	17	223.93	223.93	NUM
m-1071	231	18	4376.08	4376.08	NUM
m-1071	231	19	the	the	DET
m-1071	231	20	followers	follower	NOUN
m-1071	231	21	are	be	AUX
m-1071	231	22	invited	invite	VERB
m-1071	231	23	to	to	ADP
m-1071	231	24	ijo	ijo	PROPN
m-1071	231	25	international	international	PROPN
m-1071	231	26	journal	journal	PROPN
m-1071	231	27	of	of	ADP
m-1071	231	28	mathematics	mathematics	PROPN
m-1071	231	29	(	(	PUNCT
m-1071	231	30	issn	issn	PROPN
m-1071	231	31	:	:	PUNCT
m-1071	231	32	2992	2992	NUM
m-1071	231	33	-	-	SYM
m-1071	231	34	4421	4421	NUM
m-1071	231	35	)	)	PUNCT
m-1071	231	36	ijo	ijo	PROPN
m-1071	231	37	journals	journal	NOUN
m-1071	231	38	volume	volume	NOUN
m-1071	231	39	08	08	NUM
m-1071	232	1	|	|	ADV
m-1071	232	2	issue	issue	VERB
m-1071	232	3	04	04	NUM
m-1071	233	1	|	|	CCONJ
m-1071	233	2	april	april	PROPN
m-1071	233	3	2025	2025	NUM
m-1071	234	1	|	|	ADV
m-1071	234	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	234	3	17	17	NUM
m-1071	234	4	references	reference	NOUN
m-1071	234	5	[	[	X
m-1071	234	6	1	1	NUM
m-1071	234	7	]	]	X
m-1071	234	8	chu	chu	PROPN
m-1071	234	9	,	,	PUNCT
m-1071	234	10	yu	yu	PROPN
m-1071	234	11	-	-	PROPN
m-1071	234	12	ming	ming	PROPN
m-1071	234	13	,	,	PUNCT
m-1071	234	14	et	et	PROPN
m-1071	234	15	al	al	PROPN
m-1071	234	16	.	.	PUNCT
m-1071	234	17	”	"	PUNCT
m-1071	234	18	computation	computation	NOUN
m-1071	234	19	of	of	ADP
m-1071	234	20	zagreb	zagreb	PROPN
m-1071	234	21	polynomials	polynomial	NOUN
m-1071	234	22	and	and	CCONJ
m-1071	234	23	zagreb	zagreb	PROPN
m-1071	234	24	indices	index	NOUN
m-1071	234	25	for	for	ADP
m-1071	234	26	benzenoid	benzenoid	NOUN
m-1071	234	27	triangular	triangular	PROPN
m-1071	234	28	&	&	CCONJ
m-1071	234	29	hourglass	hourglass	PROPN
m-1071	234	30	system	system	NOUN
m-1071	234	31	.	.	PUNCT
m-1071	234	32	”	"	PUNCT
m-1071	235	1	polycyclic	polycyclic	NOUN
m-1071	235	2	aromatic	aromatic	ADJ
m-1071	235	3	compounds	compound	NOUN
m-1071	235	4	(	(	PUNCT
m-1071	235	5	2022	2022	NUM
m-1071	235	6	):	):	PUNCT
m-1071	235	7	110	110	NUM
m-1071	235	8	.	.	PUNCT
m-1071	236	1	[	[	X
m-1071	236	2	2	2	NUM
m-1071	236	3	]	]	X
m-1071	236	4	mekapati	mekapati	PROPN
m-1071	236	5	,	,	PUNCT
m-1071	236	6	suresh	suresh	PROPN
m-1071	236	7	babu	babu	PROPN
m-1071	236	8	,	,	PUNCT
m-1071	236	9	and	and	CCONJ
m-1071	236	10	corwin	corwin	PROPN
m-1071	236	11	hansch	hansch	PROPN
m-1071	236	12	.	.	PUNCT
m-1071	237	1	”	"	PUNCT
m-1071	237	2	comparative	comparative	ADJ
m-1071	237	3	qsar	qsar	NOUN
m-1071	237	4	studies	study	NOUN
m-1071	237	5	on	on	ADP
m-1071	237	6	bibenzimidazoles	bibenzimidazole	NOUN
m-1071	237	7	and	and	CCONJ
m-1071	237	8	terbenzimidazoles	terbenzimidazole	NOUN
m-1071	237	9	inhibiting	inhibit	VERB
m-1071	237	10	topoisomerase	topoisomerase	ADJ
m-1071	237	11	i.	i.	NOUN
m-1071	237	12	”	"	PUNCT
m-1071	237	13	bioorganic	bioorganic	PROPN
m-1071	237	14	&	&	CCONJ
m-1071	237	15	medicinal	medicinal	ADJ
m-1071	237	16	chemistry	chemistry	NOUN
m-1071	237	17	9.11	9.11	NUM
m-1071	237	18	(	(	PUNCT
m-1071	237	19	2001	2001	NUM
m-1071	237	20	):	):	PUNCT
m-1071	237	21	2885	2885	NUM
m-1071	237	22	-	-	SYM
m-1071	237	23	2893	2893	NUM
m-1071	237	24	.	.	PUNCT
m-1071	238	1	[	[	X
m-1071	238	2	3	3	NUM
m-1071	238	3	]	]	X
m-1071	238	4	wiener	wiener	NOUN
m-1071	238	5	,	,	PUNCT
m-1071	238	6	harry	harry	PROPN
m-1071	238	7	.	.	PUNCT
m-1071	238	8	”	"	PUNCT
m-1071	238	9	structural	structural	ADJ
m-1071	238	10	determination	determination	NOUN
m-1071	238	11	of	of	ADP
m-1071	238	12	paraffin	paraffin	NOUN
m-1071	238	13	boiling	boiling	NOUN
m-1071	238	14	points	point	NOUN
m-1071	238	15	.	.	PUNCT
m-1071	238	16	”	"	PUNCT
m-1071	239	1	journal	journal	NOUN
m-1071	239	2	of	of	ADP
m-1071	239	3	the	the	DET
m-1071	239	4	american	american	PROPN
m-1071	239	5	chemical	chemical	PROPN
m-1071	239	6	society	society	PROPN
m-1071	239	7	69.1	69.1	NUM
m-1071	239	8	(	(	PUNCT
m-1071	239	9	1947	1947	NUM
m-1071	239	10	):	):	PUNCT
m-1071	239	11	17	17	NUM
m-1071	239	12	-	-	SYM
m-1071	239	13	20	20	NUM
m-1071	239	14	.	.	PUNCT
m-1071	240	1	[	[	X
m-1071	240	2	4	4	NUM
m-1071	240	3	]	]	X
m-1071	240	4	costa	costa	PROPN
m-1071	240	5	,	,	PUNCT
m-1071	240	6	paulo	paulo	PROPN
m-1071	240	7	cs	cs	PROPN
m-1071	240	8	,	,	PUNCT
m-1071	240	9	et	et	PROPN
m-1071	240	10	al	al	PROPN
m-1071	240	11	.	.	PUNCT
m-1071	240	12	”	"	PUNCT
m-1071	240	13	chemical	chemical	NOUN
m-1071	240	14	graph	graph	NOUN
m-1071	240	15	theory	theory	NOUN
m-1071	240	16	for	for	ADP
m-1071	240	17	property	property	NOUN
m-1071	240	18	modeling	modeling	NOUN
m-1071	240	19	in	in	ADP
m-1071	240	20	qsar	qsar	NOUN
m-1071	240	21	and	and	CCONJ
m-1071	240	22	qsprcharming	qsprcharme	VERB
m-1071	240	23	qsar	qsar	NOUN
m-1071	240	24	&	&	CCONJ
m-1071	240	25	qspr	qspr	PROPN
m-1071	240	26	.	.	PUNCT
m-1071	240	27	”	"	PUNCT
m-1071	241	1	mathematics	mathematics	PROPN
m-1071	241	2	9.1	9.1	NUM
m-1071	241	3	(	(	PUNCT
m-1071	241	4	2020	2020	NUM
m-1071	241	5	):	):	PUNCT
m-1071	241	6	60	60	NUM
m-1071	241	7	.	.	PUNCT
m-1071	242	1	[	[	X
m-1071	242	2	5	5	NUM
m-1071	242	3	]	]	PUNCT
m-1071	242	4	j.-b	j.-b	NOUN
m-1071	242	5	.	.	PUNCT
m-1071	243	1	liu	liu	PROPN
m-1071	243	2	,	,	PUNCT
m-1071	243	3	c.	c.	PROPN
m-1071	243	4	wang	wang	PROPN
m-1071	243	5	,	,	PUNCT
m-1071	243	6	s.	s.	PROPN
m-1071	243	7	wang	wang	PROPN
m-1071	243	8	,	,	PUNCT
m-1071	243	9	and	and	CCONJ
m-1071	243	10	b.	b.	PROPN
m-1071	243	11	wei	wei	PROPN
m-1071	243	12	,	,	PUNCT
m-1071	243	13	zagreb	zagreb	PROPN
m-1071	243	14	indices	index	NOUN
m-1071	243	15	and	and	CCONJ
m-1071	243	16	multiplicative	multiplicative	PROPN
m-1071	243	17	zagreb	zagreb	PROPN
m-1071	243	18	indices	index	NOUN
m-1071	243	19	of	of	ADP
m-1071	243	20	eulerian	eulerian	ADJ
m-1071	243	21	graphs	graph	NOUN
m-1071	243	22	,	,	PUNCT
m-1071	243	23	bulletin	bulletin	NOUN
m-1071	243	24	of	of	ADP
m-1071	243	25	the	the	DET
m-1071	243	26	malaysian	malaysian	PROPN
m-1071	243	27	mathematical	mathematical	PROPN
m-1071	243	28	sciences	sciences	PROPN
m-1071	243	29	society	society	NOUN
m-1071	243	30	,	,	PUNCT
m-1071	243	31	vol	vol	NOUN
m-1071	243	32	.	.	PROPN
m-1071	243	33	42	42	NUM
m-1071	243	34	,	,	PUNCT
m-1071	243	35	no	no	INTJ
m-1071	243	36	.	.	NOUN
m-1071	243	37	1	1	NUM
m-1071	243	38	,	,	PUNCT
m-1071	243	39	pp	pp	ADJ
m-1071	243	40	.	.	PUNCT
m-1071	243	41	6778	6778	NUM
m-1071	243	42	,	,	PUNCT
m-1071	243	43	2019	2019	NUM
m-1071	243	44	.	.	PUNCT
m-1071	244	1	[	[	X
m-1071	244	2	6	6	NUM
m-1071	244	3	]	]	PUNCT
m-1071	244	4	j.	j.	PROPN
m-1071	244	5	b.	b.	PROPN
m-1071	244	6	liu	liu	PROPN
m-1071	244	7	,	,	PUNCT
m-1071	244	8	j.	j.	PROPN
m-1071	244	9	zhao	zhao	PROPN
m-1071	244	10	,	,	PUNCT
m-1071	244	11	j.	j.	PROPN
m-1071	244	12	min	min	PROPN
m-1071	244	13	,	,	PUNCT
m-1071	244	14	and	and	CCONJ
m-1071	244	15	j.	j.	PROPN
m-1071	244	16	cao	cao	PROPN
m-1071	244	17	,	,	PUNCT
m-1071	244	18	,	,	PUNCT
m-1071	244	19	e	e	PROPN
m-1071	244	20	hosoya	hosoya	NOUN
m-1071	244	21	index	index	NOUN
m-1071	244	22	of	of	ADP
m-1071	244	23	graphs	graph	NOUN
m-1071	244	24	formed	form	VERB
m-1071	244	25	by	by	ADP
m-1071	244	26	a	a	DET
m-1071	244	27	fractal	fractal	ADJ
m-1071	244	28	graph	graph	NOUN
m-1071	244	29	,	,	PUNCT
m-1071	244	30	fractals	fractal	NOUN
m-1071	244	31	,	,	PUNCT
m-1071	244	32	vol	vol	NOUN
m-1071	244	33	.	.	PROPN
m-1071	244	34	27	27	NUM
m-1071	244	35	,	,	PUNCT
m-1071	244	36	no	no	INTJ
m-1071	244	37	.	.	NOUN
m-1071	244	38	8	8	NUM
m-1071	244	39	,	,	PUNCT
m-1071	244	40	article	article	NOUN
m-1071	244	41	i	i	PROPN
m-1071	244	42	d	d	PROPN
m-1071	244	43	1950135	1950135	NUM
m-1071	244	44	,	,	PUNCT
m-1071	244	45	2019	2019	NUM
m-1071	244	46	.	.	PUNCT
m-1071	245	1	[	[	X
m-1071	245	2	7	7	X
m-1071	245	3	]	]	PUNCT
m-1071	245	4	j.-b	j.-b	NOUN
m-1071	245	5	.	.	PUNCT
m-1071	246	1	liu	liu	PROPN
m-1071	246	2	and	and	CCONJ
m-1071	246	3	s.	s.	PROPN
m-1071	246	4	n.	n.	PROPN
m-1071	246	5	daoud	daoud	PROPN
m-1071	246	6	,	,	PUNCT
m-1071	246	7	number	number	NOUN
m-1071	246	8	of	of	ADP
m-1071	246	9	spanning	span	VERB
m-1071	246	10	trees	tree	NOUN
m-1071	246	11	in	in	ADP
m-1071	246	12	the	the	DET
m-1071	246	13	sequence	sequence	NOUN
m-1071	246	14	of	of	ADP
m-1071	246	15	some	some	DET
m-1071	246	16	graphs	graph	NOUN
m-1071	246	17	,	,	PUNCT
m-1071	246	18	complexity	complexity	NOUN
m-1071	246	19	,	,	PUNCT
m-1071	246	20	vol	vol	NOUN
m-1071	246	21	.	.	PROPN
m-1071	246	22	2019	2019	NUM
m-1071	246	23	,	,	PUNCT
m-1071	246	24	article	article	NOUN
m-1071	246	25	i	i	PROPN
m-1071	246	26	d	d	PROPN
m-1071	246	27	4271783	4271783	NUM
m-1071	246	28	,	,	PUNCT
m-1071	246	29	22	22	NUM
m-1071	246	30	pages	page	NOUN
m-1071	246	31	,	,	PUNCT
m-1071	246	32	2019	2019	NUM
m-1071	246	33	.	.	PUNCT
m-1071	247	1	[	[	X
m-1071	247	2	8	8	X
m-1071	247	3	]	]	PUNCT
m-1071	247	4	j.	j.	PROPN
m-1071	247	5	b.	b.	PROPN
m-1071	247	6	liu	liu	PROPN
m-1071	247	7	,	,	PUNCT
m-1071	247	8	j.	j.	PROPN
m-1071	247	9	zhao	zhao	PROPN
m-1071	247	10	,	,	PUNCT
m-1071	247	11	and	and	CCONJ
m-1071	247	12	z.	z.	PROPN
m-1071	247	13	q.	q.	PROPN
m-1071	247	14	cai	cai	PROPN
m-1071	247	15	,	,	PUNCT
m-1071	247	16	on	on	ADP
m-1071	247	17	the	the	DET
m-1071	247	18	generalized	generalized	ADJ
m-1071	247	19	adjacency	adjacency	NOUN
m-1071	247	20	,	,	PUNCT
m-1071	247	21	laplacian	laplacian	ADJ
m-1071	247	22	and	and	CCONJ
m-1071	247	23	signless	signless	ADJ
m-1071	247	24	laplacian	laplacian	ADJ
m-1071	247	25	spectra	spectra	NOUN
m-1071	247	26	of	of	ADP
m-1071	247	27	the	the	DET
m-1071	247	28	weighted	weight	VERB
m-1071	247	29	edge	edge	NOUN
m-1071	247	30	corona	corona	NOUN
m-1071	247	31	networks	network	NOUN
m-1071	247	32	,	,	PUNCT
m-1071	247	33	physica	physica	VERB
m-1071	247	34	a	a	DET
m-1071	247	35	:	:	PUNCT
m-1071	247	36	statistical	statistical	ADJ
m-1071	247	37	mechanics	mechanic	NOUN
m-1071	247	38	and	and	CCONJ
m-1071	247	39	its	its	PRON
m-1071	247	40	applications	application	NOUN
m-1071	247	41	,	,	PUNCT
m-1071	247	42	vol	vol	NOUN
m-1071	247	43	.	.	PROPN
m-1071	247	44	540	540	NUM
m-1071	247	45	,	,	PUNCT
m-1071	247	46	2020	2020	NUM
m-1071	247	47	.	.	PUNCT
m-1071	248	1	[	[	X
m-1071	248	2	9	9	NUM
m-1071	248	3	]	]	PUNCT
m-1071	248	4	j.	j.	PROPN
m-1071	248	5	b.	b.	PROPN
m-1071	248	6	liu	liu	PROPN
m-1071	248	7	,	,	PUNCT
m-1071	248	8	z.	z.	PROPN
m-1071	248	9	y.	y.	PROPN
m-1071	248	10	shi	shi	PROPN
m-1071	248	11	,	,	PUNCT
m-1071	248	12	y.	y.	PROPN
m-1071	248	13	h.	h.	PROPN
m-1071	248	14	pan	pan	PROPN
m-1071	248	15	,	,	PUNCT
m-1071	248	16	j.	j.	PROPN
m-1071	248	17	cao	cao	PROPN
m-1071	248	18	,	,	PUNCT
m-1071	248	19	m.	m.	PROPN
m-1071	248	20	abdel	abdel	PROPN
m-1071	248	21	-	-	PUNCT
m-1071	248	22	aty	aty	PROPN
m-1071	248	23	,	,	PUNCT
m-1071	248	24	and	and	CCONJ
m-1071	248	25	u.	u.	PROPN
m-1071	248	26	al	al	PROPN
m-1071	248	27	-	-	PUNCT
m-1071	248	28	juboori	juboori	PROPN
m-1071	248	29	,	,	PUNCT
m-1071	248	30	computing	compute	VERB
m-1071	248	31	the	the	DET
m-1071	248	32	laplacian	laplacian	ADJ
m-1071	248	33	spectrum	spectrum	NOUN
m-1071	248	34	of	of	ADP
m-1071	248	35	linear	linear	ADJ
m-1071	248	36	octagonal	octagonal	ADJ
m-1071	248	37	-	-	PUNCT
m-1071	248	38	quadrilateral	quadrilateral	ADJ
m-1071	248	39	networks	network	NOUN
m-1071	248	40	and	and	CCONJ
m-1071	248	41	its	its	PRON
m-1071	248	42	applications	application	NOUN
m-1071	248	43	,	,	PUNCT
m-1071	248	44	polycyclic	polycyclic	NOUN
m-1071	248	45	aromatic	aromatic	ADJ
m-1071	248	46	compounds	compound	NOUN
m-1071	248	47	,	,	PUNCT
m-1071	248	48	pp	pp	ADJ
m-1071	248	49	.	.	PUNCT
m-1071	248	50	1	1	NUM
m-1071	248	51	-	-	SYM
m-1071	248	52	2	2	NUM
m-1071	248	53	,	,	PUNCT
m-1071	248	54	2020	2020	NUM
m-1071	248	55	.	.	PUNCT
m-1071	249	1	[	[	X
m-1071	249	2	10	10	NUM
m-1071	249	3	]	]	X
m-1071	249	4	j.	j.	PROPN
m-1071	249	5	b.	b.	PROPN
m-1071	249	6	liu	liu	PROPN
m-1071	249	7	,	,	PUNCT
m-1071	249	8	x.	x.	PROPN
m-1071	249	9	f.	f.	PROPN
m-1071	249	10	pan	pan	PROPN
m-1071	249	11	,	,	PUNCT
m-1071	249	12	j.	j.	PROPN
m-1071	249	13	cao	cao	PROPN
m-1071	249	14	,	,	PUNCT
m-1071	249	15	and	and	CCONJ
m-1071	249	16	f.	f.	PROPN
m-1071	249	17	f.	f.	PROPN
m-1071	249	18	hu	hu	PROPN
m-1071	249	19	,	,	PUNCT
m-1071	249	20	a	a	DET
m-1071	249	21	note	note	NOUN
m-1071	249	22	on	on	ADP
m-1071	249	23	some	some	DET
m-1071	249	24	physical	physical	ADJ
m-1071	249	25	and	and	CCONJ
m-1071	249	26	chemical	chemical	NOUN
m-1071	249	27	indices	index	NOUN
m-1071	249	28	of	of	ADP
m-1071	249	29	clique	clique	NOUN
m-1071	249	30	-	-	PUNCT
m-1071	249	31	inserted	insert	VERB
m-1071	249	32	lattices	lattice	NOUN
m-1071	249	33	,	,	PUNCT
m-1071	249	34	journal	journal	NOUN
m-1071	249	35	of	of	ADP
m-1071	249	36	statistical	statistical	ADJ
m-1071	249	37	mechanics	mechanic	NOUN
m-1071	249	38	:	:	PUNCT
m-1071	249	39	7eory	7eory	NUM
m-1071	249	40	and	and	CCONJ
m-1071	249	41	experiment	experiment	NOUN
m-1071	249	42	,	,	PUNCT
m-1071	249	43	vol	vol	NOUN
m-1071	249	44	.	.	PROPN
m-1071	249	45	2014	2014	NUM
m-1071	249	46	,	,	PUNCT
m-1071	249	47	no	no	INTJ
m-1071	249	48	.	.	NOUN
m-1071	249	49	6	6	NUM
m-1071	249	50	,	,	PUNCT
m-1071	249	51	article	article	NOUN
m-1071	249	52	i	i	PROPN
m-1071	249	53	d	d	PROPN
m-1071	249	54	p06006	p06006	PROPN
m-1071	249	55	,	,	PUNCT
m-1071	249	56	2014	2014	NUM
m-1071	249	57	.	.	PUNCT
m-1071	250	1	ijo	ijo	PROPN
m-1071	250	2	international	international	PROPN
m-1071	250	3	journal	journal	PROPN
m-1071	250	4	of	of	ADP
m-1071	250	5	mathematics	mathematics	PROPN
m-1071	250	6	(	(	PUNCT
m-1071	250	7	issn	issn	PROPN
m-1071	250	8	:	:	PUNCT
m-1071	250	9	2992	2992	NUM
m-1071	250	10	-	-	SYM
m-1071	250	11	4421	4421	NUM
m-1071	250	12	)	)	PUNCT
m-1071	250	13	ijo	ijo	PROPN
m-1071	250	14	journals	journal	NOUN
m-1071	250	15	volume	volume	NOUN
m-1071	250	16	08	08	NUM
m-1071	251	1	|	|	ADV
m-1071	251	2	issue	issue	VERB
m-1071	251	3	04	04	NUM
m-1071	252	1	|	|	CCONJ
m-1071	252	2	april	april	PROPN
m-1071	252	3	2025	2025	NUM
m-1071	253	1	|	|	ADV
m-1071	253	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	253	3	18	18	NUM
m-1071	253	4	[	[	SYM
m-1071	253	5	11	11	NUM
m-1071	253	6	]	]	PUNCT
m-1071	253	7	j.-b	j.-b	NOUN
m-1071	253	8	.	.	PUNCT
m-1071	254	1	liu	liu	PROPN
m-1071	254	2	,	,	PUNCT
m-1071	254	3	x.-f	x.-f	PROPN
m-1071	254	4	.	.	PUNCT
m-1071	255	1	pan	pan	PROPN
m-1071	255	2	,	,	PUNCT
m-1071	255	3	j.	j.	PROPN
m-1071	255	4	cao	cao	PROPN
m-1071	255	5	,	,	PUNCT
m-1071	255	6	and	and	CCONJ
m-1071	255	7	x.	x.	PROPN
m-1071	255	8	huang	huang	PROPN
m-1071	255	9	,	,	PUNCT
m-1071	255	10	,	,	PUNCT
m-1071	255	11	e	e	PROPN
m-1071	255	12	kirchhoff	kirchhoff	NOUN
m-1071	255	13	index	index	NOUN
m-1071	255	14	of	of	ADP
m-1071	255	15	toroidal	toroidal	ADJ
m-1071	255	16	meshes	mesh	NOUN
m-1071	255	17	and	and	CCONJ
m-1071	255	18	variant	variant	ADJ
m-1071	255	19	networks	network	NOUN
m-1071	255	20	,	,	PUNCT
m-1071	255	21	mathematical	mathematical	ADJ
m-1071	255	22	problems	problem	NOUN
m-1071	255	23	in	in	ADP
m-1071	255	24	engineering	engineering	NOUN
m-1071	255	25	,	,	PUNCT
m-1071	255	26	vol	vol	NOUN
m-1071	255	27	.	.	PROPN
m-1071	255	28	2014	2014	NUM
m-1071	255	29	,	,	PUNCT
m-1071	255	30	article	article	NOUN
m-1071	255	31	i	i	PROPN
m-1071	255	32	d	d	PROPN
m-1071	255	33	286876	286876	NUM
m-1071	255	34	,	,	PUNCT
m-1071	255	35	8	8	NUM
m-1071	255	36	pages	page	NOUN
m-1071	255	37	,	,	PUNCT
m-1071	255	38	2014	2014	NUM
m-1071	255	39	.	.	PUNCT
m-1071	256	1	[	[	X
m-1071	256	2	12	12	NUM
m-1071	256	3	]	]	PUNCT
m-1071	256	4	j.-b	j.-b	NOUN
m-1071	256	5	.	.	PUNCT
m-1071	257	1	liu	liu	PROPN
m-1071	257	2	,	,	PUNCT
m-1071	257	3	x.-f	x.-f	PROPN
m-1071	257	4	.	.	PUNCT
m-1071	258	1	pan	pan	PROPN
m-1071	258	2	,	,	PUNCT
m-1071	258	3	and	and	CCONJ
m-1071	258	4	j.	j.	PROPN
m-1071	258	5	cao	cao	PROPN
m-1071	258	6	,	,	PUNCT
m-1071	258	7	some	some	DET
m-1071	258	8	properties	property	NOUN
m-1071	258	9	on	on	ADP
m-1071	258	10	estrada	estrada	PROPN
m-1071	258	11	index	index	NOUN
m-1071	258	12	of	of	ADP
m-1071	258	13	folded	fold	VERB
m-1071	258	14	hypercubes	hypercube	NOUN
m-1071	258	15	networks	network	NOUN
m-1071	258	16	,	,	PUNCT
m-1071	258	17	abstract	abstract	ADJ
m-1071	258	18	and	and	CCONJ
m-1071	258	19	applied	apply	VERB
m-1071	258	20	analysis	analysis	NOUN
m-1071	258	21	,	,	PUNCT
m-1071	258	22	vol	vol	NOUN
m-1071	258	23	.	.	PROPN
m-1071	258	24	2014	2014	NUM
m-1071	258	25	,	,	PUNCT
m-1071	258	26	article	article	NOUN
m-1071	258	27	i	i	PROPN
m-1071	258	28	d	d	PROPN
m-1071	258	29	167623	167623	NUM
m-1071	258	30	,	,	PUNCT
m-1071	258	31	6	6	NUM
m-1071	258	32	pages	page	NOUN
m-1071	258	33	,	,	PUNCT
m-1071	258	34	2014	2014	NUM
m-1071	258	35	.	.	PUNCT
m-1071	259	1	[	[	X
m-1071	259	2	13	13	NUM
m-1071	259	3	]	]	X
m-1071	259	4	ghani	ghani	PROPN
m-1071	259	5	,	,	PUNCT
m-1071	259	6	muhammad	muhammad	PROPN
m-1071	259	7	usman	usman	PROPN
m-1071	259	8	,	,	PUNCT
m-1071	259	9	et	et	PROPN
m-1071	259	10	al	al	PROPN
m-1071	259	11	.	.	PUNCT
m-1071	259	12	”	"	PUNCT
m-1071	259	13	a	a	DET
m-1071	259	14	paradigmatic	paradigmatic	ADJ
m-1071	259	15	approach	approach	NOUN
m-1071	259	16	to	to	PART
m-1071	259	17	find	find	VERB
m-1071	259	18	the	the	DET
m-1071	259	19	valencybased	valencybase	VERB
m-1071	259	20	kbanhatti	kbanhatti	NOUN
m-1071	259	21	and	and	CCONJ
m-1071	259	22	redefined	redefine	VERB
m-1071	259	23	zagreb	zagreb	PROPN
m-1071	259	24	entropy	entropy	PROPN
m-1071	259	25	for	for	ADP
m-1071	259	26	niobium	niobium	NOUN
m-1071	259	27	oxide	oxide	NOUN
m-1071	259	28	and	and	CCONJ
m-1071	259	29	a	a	DET
m-1071	259	30	metalorganic	metalorganic	ADJ
m-1071	259	31	framework	framework	NOUN
m-1071	259	32	.	.	PUNCT
m-1071	259	33	”	"	PUNCT
m-1071	260	1	molecules	molecule	NOUN
m-1071	260	2	27.20	27.20	NUM
m-1071	260	3	(	(	PUNCT
m-1071	260	4	2022	2022	NUM
m-1071	260	5	):	):	PUNCT
m-1071	260	6	6975	6975	NUM
m-1071	260	7	.	.	PUNCT
m-1071	261	1	[	[	X
m-1071	261	2	14	14	NUM
m-1071	261	3	]	]	X
m-1071	261	4	zhang	zhang	PROPN
m-1071	261	5	,	,	PUNCT
m-1071	261	6	ying	ying	PROPN
m-1071	261	7	-	-	PUNCT
m-1071	261	8	fang	fang	PROPN
m-1071	261	9	,	,	PUNCT
m-1071	261	10	et	et	PROPN
m-1071	261	11	al	al	PROPN
m-1071	261	12	.	.	PUNCT
m-1071	261	13	”	"	PUNCT
m-1071	261	14	connecting	connect	VERB
m-1071	261	15	sio	sio	NOUN
m-1071	261	16	4	4	NUM
m-1071	261	17	in	in	ADP
m-1071	261	18	silicate	silicate	NOUN
m-1071	261	19	and	and	CCONJ
m-1071	261	20	silicate	silicate	ADJ
m-1071	261	21	chain	chain	NOUN
m-1071	261	22	networks	network	NOUN
m-1071	261	23	to	to	PART
m-1071	261	24	compute	compute	VERB
m-1071	261	25	kulli	kulli	PROPN
m-1071	261	26	temperature	temperature	NOUN
m-1071	261	27	indices	index	NOUN
m-1071	261	28	.	.	PUNCT
m-1071	261	29	”	"	PUNCT
m-1071	262	1	molecules	molecule	NOUN
m-1071	262	2	27.21	27.21	NUM
m-1071	262	3	(	(	PUNCT
m-1071	262	4	2022	2022	NUM
m-1071	262	5	):	):	PUNCT
m-1071	262	6	7533	7533	NUM
m-1071	262	7	.	.	PUNCT
m-1071	263	1	[	[	X
m-1071	263	2	15	15	NUM
m-1071	263	3	]	]	X
m-1071	263	4	sourav	sourav	PROPN
m-1071	263	5	mondal	mondal	PROPN
m-1071	263	6	,	,	PUNCT
m-1071	263	7	arindam	arindam	PROPN
m-1071	263	8	dey	dey	PROPN
m-1071	263	9	,	,	PUNCT
m-1071	263	10	nilanjan	nilanjan	NOUN
m-1071	263	11	de	de	PROPN
m-1071	263	12	,	,	PUNCT
m-1071	263	13	and	and	CCONJ
m-1071	263	14	anita	anita	PROPN
m-1071	263	15	pal	pal	NOUN
m-1071	263	16	,	,	PUNCT
m-1071	263	17	qspr	qspr	ADJ
m-1071	263	18	analysis	analysis	NOUN
m-1071	263	19	of	of	ADP
m-1071	263	20	some	some	DET
m-1071	263	21	novel	novel	ADJ
m-1071	263	22	neighbourhood	neighbourhood	NOUN
m-1071	263	23	degree	degree	NOUN
m-1071	263	24	-	-	PUNCT
m-1071	263	25	based	base	VERB
m-1071	263	26	topological	topological	ADJ
m-1071	263	27	descriptors	descriptor	NOUN
m-1071	263	28	,	,	PUNCT
m-1071	263	29	complex	complex	ADJ
m-1071	263	30	&	&	CCONJ
m-1071	263	31	intelligent	intelligent	ADJ
m-1071	263	32	systems	system	NOUN
m-1071	263	33	7	7	NUM
m-1071	263	34	(	(	PUNCT
m-1071	263	35	2021	2021	NUM
m-1071	263	36	)	)	PUNCT
m-1071	263	37	,	,	PUNCT
m-1071	263	38	no	no	INTJ
m-1071	263	39	.	.	NOUN
m-1071	263	40	2	2	NUM
m-1071	263	41	,	,	PUNCT
m-1071	263	42	977996	977996	NUM
m-1071	263	43	.	.	PUNCT
m-1071	264	1	[	[	X
m-1071	264	2	16	16	NUM
m-1071	264	3	]	]	X
m-1071	264	4	alam	alam	PROPN
m-1071	264	5	,	,	PUNCT
m-1071	264	6	ashraful	ashraful	PROPN
m-1071	264	7	,	,	PUNCT
m-1071	264	8	et	et	PROPN
m-1071	264	9	al	al	PROPN
m-1071	264	10	.	.	PUNCT
m-1071	264	11	”	"	PUNCT
m-1071	264	12	degree	degree	NOUN
m-1071	264	13	-	-	PUNCT
m-1071	264	14	based	base	VERB
m-1071	264	15	entropy	entropy	NOUN
m-1071	264	16	for	for	ADP
m-1071	264	17	a	a	DET
m-1071	264	18	non	non	ADJ
m-1071	264	19	-	-	ADJ
m-1071	264	20	kekulean	kekulean	ADJ
m-1071	264	21	benzenoid	benzenoid	NOUN
m-1071	264	22	graph	graph	NOUN
m-1071	264	23	.	.	PUNCT
m-1071	264	24	”	"	PUNCT
m-1071	264	25	journal	journal	NOUN
m-1071	264	26	of	of	ADP
m-1071	264	27	mathematics	mathematics	PROPN
m-1071	264	28	2022	2022	NUM
m-1071	264	29	(	(	PUNCT
m-1071	264	30	2022	2022	NUM
m-1071	264	31	)	)	PUNCT
m-1071	264	32	.	.	PUNCT
m-1071	265	1	[	[	X
m-1071	265	2	17	17	NUM
m-1071	265	3	]	]	X
m-1071	265	4	al	al	PROPN
m-1071	265	5	-	-	PUNCT
m-1071	265	6	ahmadi	ahmadi	PROPN
m-1071	265	7	,	,	PUNCT
m-1071	265	8	bashair	bashair	NOUN
m-1071	265	9	,	,	PUNCT
m-1071	265	10	anwar	anwar	PROPN
m-1071	265	11	saleh	saleh	NOUN
m-1071	265	12	,	,	PUNCT
m-1071	265	13	and	and	CCONJ
m-1071	265	14	wafa	wafa	PROPN
m-1071	265	15	al	al	PROPN
m-1071	265	16	-	-	PUNCT
m-1071	265	17	shammakh	shammakh	PROPN
m-1071	265	18	.	.	PUNCT
m-1071	265	19	”	"	PUNCT
m-1071	266	1	downhill	downhill	ADV
m-1071	266	2	zagreb	zagreb	PROPN
m-1071	266	3	topological	topological	ADJ
m-1071	266	4	indices	index	NOUN
m-1071	266	5	and	and	CCONJ
m-1071	266	6	m	m	VERB
m-1071	266	7	dn	dn	ADJ
m-1071	266	8	-	-	ADJ
m-1071	266	9	polynomial	polynomial	ADJ
m-1071	266	10	of	of	ADP
m-1071	266	11	some	some	DET
m-1071	266	12	chemical	chemical	NOUN
m-1071	266	13	structures	structure	NOUN
m-1071	266	14	applied	apply	VERB
m-1071	266	15	for	for	ADP
m-1071	266	16	the	the	DET
m-1071	266	17	treatment	treatment	NOUN
m-1071	266	18	of	of	ADP
m-1071	266	19	covid-19	covid-19	PROPN
m-1071	266	20	patients	patient	NOUN
m-1071	266	21	.	.	PUNCT
m-1071	266	22	”	"	PUNCT
m-1071	267	1	open	open	ADJ
m-1071	267	2	journal	journal	NOUN
m-1071	267	3	of	of	ADP
m-1071	267	4	applied	apply	VERB
m-1071	267	5	sciences	science	NOUN
m-1071	267	6	10.04	10.04	NUM
m-1071	267	7	(	(	PUNCT
m-1071	267	8	2021	2021	NUM
m-1071	267	9	):	):	PUNCT
m-1071	267	10	395	395	NUM
m-1071	267	11	.	.	PUNCT
m-1071	268	1	[	[	X
m-1071	268	2	18	18	NUM
m-1071	268	3	]	]	X
m-1071	268	4	anton	anton	NOUN
m-1071	268	5	b	b	PROPN
m-1071	268	6	zakharov	zakharov	PROPN
m-1071	268	7	,	,	PUNCT
m-1071	268	8	dmytro	dmytro	PROPN
m-1071	268	9	k	k	PROPN
m-1071	268	10	tsarenko	tsarenko	PROPN
m-1071	268	11	,	,	PUNCT
m-1071	268	12	and	and	CCONJ
m-1071	268	13	vladimir	vladimir	PROPN
m-1071	268	14	v	v	PROPN
m-1071	268	15	ivanov	ivanov	PROPN
m-1071	268	16	,	,	PUNCT
m-1071	268	17	topological	topological	ADJ
m-1071	268	18	characteristics	characteristic	NOUN
m-1071	268	19	of	of	ADP
m-1071	268	20	iterated	iterated	ADJ
m-1071	268	21	line	line	NOUN
m-1071	268	22	graphs	graph	NOUN
m-1071	268	23	in	in	ADP
m-1071	268	24	the	the	DET
m-1071	268	25	qsar	qsar	NOUN
m-1071	268	26	problem	problem	NOUN
m-1071	268	27	:	:	PUNCT
m-1071	268	28	a	a	DET
m-1071	268	29	multigraph	multigraph	NOUN
m-1071	268	30	in	in	ADP
m-1071	268	31	the	the	DET
m-1071	268	32	description	description	NOUN
m-1071	268	33	of	of	ADP
m-1071	268	34	properties	property	NOUN
m-1071	268	35	of	of	ADP
m-1071	268	36	unsaturated	unsaturated	ADJ
m-1071	268	37	hydrocarbons	hydrocarbon	NOUN
m-1071	268	38	,	,	PUNCT
m-1071	268	39	structural	structural	ADJ
m-1071	268	40	chemistry	chemistry	NOUN
m-1071	268	41	(	(	PUNCT
m-1071	268	42	2021	2021	NUM
m-1071	268	43	)	)	PUNCT
m-1071	268	44	,	,	PUNCT
m-1071	268	45	111	111	NUM
m-1071	268	46	.	.	PUNCT
m-1071	269	1	[	[	X
m-1071	269	2	19	19	NUM
m-1071	269	3	]	]	X
m-1071	269	4	natarajan	natarajan	PROPN
m-1071	269	5	,	,	PUNCT
m-1071	269	6	vanasundaram	vanasundaram	PROPN
m-1071	269	7	,	,	PUNCT
m-1071	269	8	et	et	PROPN
m-1071	269	9	al	al	PROPN
m-1071	269	10	.	.	PUNCT
m-1071	269	11	”	"	PUNCT
m-1071	269	12	effect	effect	NOUN
m-1071	269	13	of	of	ADP
m-1071	269	14	electron	electron	NOUN
m-1071	269	15	-	-	PUNCT
m-1071	269	16	phonon	phonon	NOUN
m-1071	269	17	interaction	interaction	NOUN
m-1071	269	18	and	and	CCONJ
m-1071	269	19	valence	valence	NOUN
m-1071	269	20	band	band	NOUN
m-1071	269	21	edge	edge	NOUN
m-1071	269	22	shift	shift	NOUN
m-1071	269	23	for	for	ADP
m-1071	269	24	carrier	carrier	NOUN
m-1071	269	25	-	-	PUNCT
m-1071	269	26	type	type	NOUN
m-1071	269	27	reversal	reversal	NOUN
m-1071	269	28	in	in	ADP
m-1071	269	29	layered	layered	ADJ
m-1071	269	30	zns	zns	NOUN
m-1071	269	31	/	/	SYM
m-1071	269	32	rgo	rgo	PROPN
m-1071	269	33	nanocomposites	nanocomposite	NOUN
m-1071	269	34	.	.	PUNCT
m-1071	269	35	”	"	PUNCT
m-1071	269	36	journal	journal	NOUN
m-1071	269	37	of	of	ADP
m-1071	269	38	colloid	colloid	NOUN
m-1071	269	39	and	and	CCONJ
m-1071	269	40	interface	interface	NOUN
m-1071	269	41	science	science	NOUN
m-1071	269	42	586	586	NUM
m-1071	269	43	(	(	PUNCT
m-1071	269	44	2021	2021	NUM
m-1071	269	45	):	):	PUNCT
m-1071	269	46	39	39	NUM
m-1071	269	47	-	-	SYM
m-1071	269	48	46	46	NUM
m-1071	269	49	.	.	PUNCT
m-1071	270	1	[	[	X
m-1071	270	2	20	20	NUM
m-1071	270	3	]	]	X
m-1071	270	4	zhong	zhong	PROPN
m-1071	270	5	,	,	PUNCT
m-1071	270	6	lingping	lingpe	VERB
m-1071	270	7	.	.	PUNCT
m-1071	270	8	”	"	PUNCT
m-1071	271	1	the	the	DET
m-1071	271	2	harmonic	harmonic	ADJ
m-1071	271	3	index	index	NOUN
m-1071	271	4	for	for	ADP
m-1071	271	5	graphs	graph	NOUN
m-1071	271	6	.	.	PUNCT
m-1071	271	7	”	"	PUNCT
m-1071	271	8	applied	apply	VERB
m-1071	271	9	mathematics	mathematics	NOUN
m-1071	271	10	letters	letter	NOUN
m-1071	271	11	25.3	25.3	NUM
m-1071	271	12	(	(	PUNCT
m-1071	271	13	2012	2012	NUM
m-1071	271	14	):	):	PUNCT
m-1071	271	15	561	561	NUM
m-1071	271	16	-	-	SYM
m-1071	271	17	566	566	NUM
m-1071	271	18	.	.	PUNCT
m-1071	272	1	[	[	X
m-1071	272	2	21	21	NUM
m-1071	272	3	]	]	X
m-1071	272	4	estrada	estrada	PROPN
m-1071	272	5	,	,	PUNCT
m-1071	272	6	ernesto	ernesto	PROPN
m-1071	272	7	,	,	PUNCT
m-1071	272	8	et	et	PROPN
m-1071	272	9	al	al	PROPN
m-1071	272	10	.	.	PUNCT
m-1071	272	11	”	"	PUNCT
m-1071	272	12	an	an	DET
m-1071	272	13	atom	atom	NOUN
m-1071	272	14	-	-	PUNCT
m-1071	272	15	bond	bond	NOUN
m-1071	272	16	connectivity	connectivity	NOUN
m-1071	272	17	index	index	NOUN
m-1071	272	18	:	:	PUNCT
m-1071	272	19	modelling	model	VERB
m-1071	272	20	the	the	DET
m-1071	272	21	enthalpy	enthalpy	NOUN
m-1071	272	22	of	of	ADP
m-1071	272	23	formation	formation	NOUN
m-1071	272	24	of	of	ADP
m-1071	272	25	alkanes	alkane	NOUN
m-1071	272	26	.	.	PUNCT
m-1071	272	27	”	"	PUNCT
m-1071	273	1	(	(	PUNCT
m-1071	273	2	1998	1998	NUM
m-1071	273	3	)	)	PUNCT
m-1071	273	4	.	.	PUNCT
m-1071	274	1	ijo	ijo	PROPN
m-1071	274	2	international	international	PROPN
m-1071	274	3	journal	journal	PROPN
m-1071	274	4	of	of	ADP
m-1071	274	5	mathematics	mathematics	PROPN
m-1071	274	6	(	(	PUNCT
m-1071	274	7	issn	issn	PROPN
m-1071	274	8	:	:	PUNCT
m-1071	274	9	2992	2992	NUM
m-1071	274	10	-	-	SYM
m-1071	274	11	4421	4421	NUM
m-1071	274	12	)	)	PUNCT
m-1071	274	13	ijo	ijo	PROPN
m-1071	274	14	journals	journal	NOUN
m-1071	274	15	volume	volume	NOUN
m-1071	274	16	08	08	NUM
m-1071	275	1	|	|	ADV
m-1071	275	2	issue	issue	VERB
m-1071	275	3	04	04	NUM
m-1071	276	1	|	|	CCONJ
m-1071	276	2	april	april	PROPN
m-1071	276	3	2025	2025	NUM
m-1071	276	4	|	|	ADV
m-1071	276	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	276	6	19	19	NUM
m-1071	276	7	[	[	SYM
m-1071	276	8	22	22	NUM
m-1071	276	9	]	]	PUNCT
m-1071	276	10	furtula	furtula	PROPN
m-1071	276	11	,	,	PUNCT
m-1071	276	12	boris	boris	PROPN
m-1071	276	13	,	,	PUNCT
m-1071	276	14	and	and	CCONJ
m-1071	276	15	ivan	ivan	PROPN
m-1071	276	16	gutman	gutman	PROPN
m-1071	276	17	.	.	PUNCT
m-1071	276	18	”	"	PUNCT
m-1071	276	19	a	a	DET
m-1071	276	20	forgotten	forget	VERB
m-1071	276	21	topological	topological	ADJ
m-1071	276	22	index	index	NOUN
m-1071	276	23	.	.	PUNCT
m-1071	276	24	”	"	PUNCT
m-1071	276	25	journal	journal	NOUN
m-1071	276	26	of	of	ADP
m-1071	276	27	mathematical	mathematical	ADJ
m-1071	276	28	chemistry	chemistry	NOUN
m-1071	276	29	53.4	53.4	NUM
m-1071	276	30	(	(	PUNCT
m-1071	276	31	2015	2015	NUM
m-1071	276	32	):	):	PUNCT
m-1071	276	33	1184	1184	NUM
m-1071	276	34	-	-	SYM
m-1071	276	35	1190	1190	NUM
m-1071	276	36	.	.	PUNCT
m-1071	277	1	[	[	X
m-1071	277	2	23	23	NUM
m-1071	277	3	]	]	PUNCT
m-1071	277	4	vukicevic	vukicevic	NOUN
m-1071	277	5	,	,	PUNCT
m-1071	277	6	damir	damir	PROPN
m-1071	277	7	,	,	PUNCT
m-1071	277	8	and	and	CCONJ
m-1071	277	9	boris	boris	PROPN
m-1071	277	10	furtula	furtula	PROPN
m-1071	277	11	.	.	PUNCT
m-1071	278	1	”	"	PUNCT
m-1071	278	2	topological	topological	ADJ
m-1071	278	3	index	index	NOUN
m-1071	278	4	based	base	VERB
m-1071	278	5	on	on	ADP
m-1071	278	6	the	the	DET
m-1071	278	7	ratios	ratio	NOUN
m-1071	278	8	of	of	ADP
m-1071	278	9	geometrical	geometrical	ADJ
m-1071	278	10	and	and	CCONJ
m-1071	278	11	arithmetical	arithmetical	ADJ
m-1071	278	12	means	mean	NOUN
m-1071	278	13	of	of	ADP
m-1071	278	14	end	end	NOUN
m-1071	278	15	-	-	PUNCT
m-1071	278	16	vertex	vertex	NOUN
m-1071	278	17	degrees	degree	NOUN
m-1071	278	18	of	of	ADP
m-1071	278	19	edges	edge	NOUN
m-1071	278	20	.	.	PUNCT
m-1071	278	21	”	"	PUNCT
m-1071	279	1	journal	journal	NOUN
m-1071	279	2	of	of	ADP
m-1071	279	3	mathematical	mathematical	ADJ
m-1071	279	4	chemistry	chemistry	NOUN
m-1071	279	5	46.4	46.4	NUM
m-1071	279	6	(	(	PUNCT
m-1071	279	7	2009	2009	NUM
m-1071	279	8	):	):	PUNCT
m-1071	279	9	1369	1369	NUM
m-1071	279	10	-	-	SYM
m-1071	279	11	1376	1376	NUM
m-1071	279	12	.	.	PUNCT
m-1071	280	1	[	[	X
m-1071	280	2	24	24	NUM
m-1071	280	3	]	]	SYM
m-1071	280	4	li	li	PROPN
m-1071	280	5	,	,	PUNCT
m-1071	280	6	xu	xu	PROPN
m-1071	280	7	,	,	PUNCT
m-1071	280	8	et	et	PROPN
m-1071	280	9	al	al	PROPN
m-1071	280	10	.	.	PUNCT
m-1071	280	11	”	"	PUNCT
m-1071	280	12	bounds	bound	NOUN
m-1071	280	13	on	on	ADP
m-1071	280	14	general	general	ADJ
m-1071	280	15	randic	randic	ADJ
m-1071	280	16	index	index	NOUN
m-1071	280	17	for	for	ADP
m-1071	280	18	f	f	NOUN
m-1071	280	19	-	-	PUNCT
m-1071	280	20	sum	sum	NOUN
m-1071	280	21	graphs	graph	NOUN
m-1071	280	22	.	.	PUNCT
m-1071	280	23	”	"	PUNCT
m-1071	281	1	journal	journal	NOUN
m-1071	281	2	of	of	ADP
m-1071	281	3	mathematics	mathematics	PROPN
m-1071	281	4	2020	2020	NUM
m-1071	281	5	(	(	PUNCT
m-1071	281	6	2020	2020	NUM
m-1071	281	7	)	)	PUNCT
m-1071	281	8	.	.	PUNCT
m-1071	282	1	kulli	kulli	PROPN
m-1071	282	2	,	,	PUNCT
m-1071	282	3	v.	v.	PROPN
m-1071	282	4	r.	r.	PROPN
m-1071	282	5	,	,	PUNCT
m-1071	282	6	b.	b.	PROPN
m-1071	282	7	chaluvaraju	chaluvaraju	PROPN
m-1071	282	8	,	,	PUNCT
m-1071	282	9	and	and	CCONJ
m-1071	282	10	h.	h.	PROPN
m-1071	282	11	s.	s.	PROPN
m-1071	282	12	boregowda	boregowda	PROPN
m-1071	282	13	.	.	PUNCT
m-1071	283	1	”	"	PUNCT
m-1071	283	2	connectivity	connectivity	NOUN
m-1071	283	3	banhatti	banhatti	ADJ
m-1071	283	4	indices	index	NOUN
m-1071	283	5	for	for	ADP
m-1071	283	6	certain	certain	ADJ
m-1071	283	7	families	family	NOUN
m-1071	283	8	of	of	ADP
m-1071	283	9	benzenoid	benzenoid	NOUN
m-1071	283	10	systems	system	NOUN
m-1071	283	11	.	.	PUNCT
m-1071	283	12	”	"	PUNCT
m-1071	284	1	journal	journal	NOUN
m-1071	284	2	of	of	ADP
m-1071	284	3	ultra	ultra	ADJ
m-1071	284	4	chemistry	chemistry	NOUN
m-1071	284	5	13.4	13.4	NUM
m-1071	284	6	(	(	PUNCT
m-1071	284	7	2017	2017	NUM
m-1071	284	8	):	):	PUNCT
m-1071	284	9	81	81	NUM
m-1071	284	10	[	[	X
m-1071	284	11	25	25	NUM
m-1071	284	12	]	]	X
m-1071	284	13	farrukh	farrukh	PROPN
m-1071	284	14	,	,	PUNCT
m-1071	284	15	fatima	fatima	PROPN
m-1071	284	16	,	,	PUNCT
m-1071	284	17	rashid	rashid	PROPN
m-1071	284	18	farooq	farooq	PROPN
m-1071	284	19	,	,	PUNCT
m-1071	284	20	and	and	CCONJ
m-1071	284	21	mohammad	mohammad	PROPN
m-1071	284	22	r.	r.	PROPN
m-1071	284	23	farahani	farahani	PROPN
m-1071	284	24	.	.	PUNCT
m-1071	285	1	”	"	PUNCT
m-1071	285	2	calculating	calculate	VERB
m-1071	285	3	some	some	DET
m-1071	285	4	topological	topological	ADJ
m-1071	285	5	indices	index	NOUN
m-1071	285	6	of	of	ADP
m-1071	285	7	sio2	sio2	PROPN
m-1071	285	8	layer	layer	NOUN
m-1071	285	9	structure	structure	NOUN
m-1071	285	10	.	.	PUNCT
m-1071	285	11	”	"	PUNCT
m-1071	285	12	journal	journal	NOUN
m-1071	285	13	of	of	ADP
m-1071	285	14	informatics	informatic	NOUN
m-1071	285	15	and	and	CCONJ
m-1071	285	16	mathematical	mathematical	ADJ
m-1071	285	17	sciences	science	NOUN
m-1071	285	18	8.3	8.3	NUM
m-1071	285	19	(	(	PUNCT
m-1071	285	20	2016	2016	NUM
m-1071	285	21	):	):	PUNCT
m-1071	285	22	181	181	NUM
m-1071	285	23	-	-	SYM
m-1071	285	24	187	187	NUM
m-1071	285	25	..	..	PUNCT
m-1071	286	1	[	[	X
m-1071	286	2	26	26	NUM
m-1071	286	3	]	]	PUNCT
m-1071	286	4	ghorbani	ghorbani	NOUN
m-1071	286	5	,	,	PUNCT
m-1071	286	6	modjtaba	modjtaba	PROPN
m-1071	286	7	,	,	PUNCT
m-1071	286	8	samaneh	samaneh	PROPN
m-1071	286	9	zangi	zangi	PROPN
m-1071	286	10	,	,	PUNCT
m-1071	286	11	and	and	CCONJ
m-1071	286	12	najaf	najaf	PROPN
m-1071	286	13	amraei	amraei	NOUN
m-1071	286	14	.	.	PUNCT
m-1071	287	1	”	"	PUNCT
m-1071	287	2	new	new	ADJ
m-1071	287	3	results	result	NOUN
m-1071	287	4	on	on	ADP
m-1071	287	5	symmetric	symmetric	ADJ
m-1071	287	6	division	division	NOUN
m-1071	287	7	deg	deg	PROPN
m-1071	287	8	index	index	PROPN
m-1071	287	9	.	.	PUNCT
m-1071	287	10	”	"	PUNCT
m-1071	288	1	journal	journal	NOUN
m-1071	288	2	of	of	ADP
m-1071	288	3	applied	apply	VERB
m-1071	288	4	mathematics	mathematic	NOUN
m-1071	288	5	and	and	CCONJ
m-1071	288	6	computing	compute	VERB
m-1071	288	7	65.1	65.1	NUM
m-1071	288	8	(	(	PUNCT
m-1071	288	9	2021	2021	NUM
m-1071	288	10	):	):	PUNCT
m-1071	288	11	161176	161176	NUM
m-1071	288	12	.	.	PUNCT
m-1071	289	1	[	[	X
m-1071	289	2	27	27	NUM
m-1071	289	3	]	]	X
m-1071	289	4	aguilar	aguilar	PROPN
m-1071	289	5	-	-	PUNCT
m-1071	289	6	snchez	snchez	ADV
m-1071	289	7	,	,	PUNCT
m-1071	289	8	r.	r.	PROPN
m-1071	289	9	,	,	PUNCT
m-1071	289	10	et	et	PROPN
m-1071	289	11	al	al	PROPN
m-1071	289	12	.	.	PUNCT
m-1071	289	13	”	"	PUNCT
m-1071	289	14	analytical	analytical	ADJ
m-1071	289	15	and	and	CCONJ
m-1071	289	16	computational	computational	ADJ
m-1071	289	17	properties	property	NOUN
m-1071	289	18	of	of	ADP
m-1071	289	19	the	the	DET
m-1071	289	20	variable	variable	ADJ
m-1071	289	21	symmetric	symmetric	ADJ
m-1071	289	22	division	division	NOUN
m-1071	289	23	deg	deg	PROPN
m-1071	289	24	index	index	PROPN
m-1071	289	25	.	.	PUNCT
m-1071	289	26	”	"	PUNCT
m-1071	290	1	arxiv	arxiv	PROPN
m-1071	290	2	preprint	preprint	NOUN
m-1071	290	3	arxiv:2106.00913	arxiv:2106.00913	PROPN
m-1071	290	4	(	(	PUNCT
m-1071	290	5	2021	2021	NUM
m-1071	290	6	)	)	PUNCT
m-1071	290	7	.	.	PUNCT
m-1071	291	1	[	[	X
m-1071	291	2	28	28	NUM
m-1071	291	3	]	]	X
m-1071	291	4	shpiz	shpiz	PROPN
m-1071	291	5	,	,	PUNCT
m-1071	291	6	grigory	grigory	PROPN
m-1071	291	7	b.	b.	PROPN
m-1071	291	8	,	,	PUNCT
m-1071	291	9	and	and	CCONJ
m-1071	291	10	alexander	alexander	PROPN
m-1071	291	11	p.	p.	PROPN
m-1071	291	12	kryukov	kryukov	PROPN
m-1071	291	13	.	.	PUNCT
m-1071	292	1	”	"	PUNCT
m-1071	292	2	the	the	DET
m-1071	292	3	method	method	NOUN
m-1071	292	4	of	of	ADP
m-1071	292	5	colored	colored	ADJ
m-1071	292	6	graphs	graph	NOUN
m-1071	292	7	for	for	ADP
m-1071	292	8	simplifying	simplify	VERB
m-1071	292	9	expressions	expression	NOUN
m-1071	292	10	with	with	ADP
m-1071	292	11	indices	index	NOUN
m-1071	292	12	.	.	PUNCT
m-1071	292	13	”	"	PUNCT
m-1071	292	14	programming	programming	NOUN
m-1071	292	15	and	and	CCONJ
m-1071	292	16	computer	computer	NOUN
m-1071	292	17	software	software	NOUN
m-1071	292	18	47.1	47.1	NUM
m-1071	292	19	(	(	PUNCT
m-1071	292	20	2021	2021	NUM
m-1071	292	21	):	):	PUNCT
m-1071	292	22	25	25	NUM
m-1071	292	23	-	-	SYM
m-1071	292	24	28	28	NUM
m-1071	292	25	.	.	PUNCT
m-1071	293	1	[	[	X
m-1071	293	2	29	29	NUM
m-1071	293	3	]	]	X
m-1071	293	4	gutman	gutman	NOUN
m-1071	293	5	,	,	PUNCT
m-1071	293	6	ivan	ivan	PROPN
m-1071	293	7	.	.	PUNCT
m-1071	294	1	”	"	PUNCT
m-1071	294	2	geometric	geometric	ADJ
m-1071	294	3	approach	approach	NOUN
m-1071	294	4	to	to	ADP
m-1071	294	5	degree	degree	NOUN
m-1071	294	6	-	-	PUNCT
m-1071	294	7	based	base	VERB
m-1071	294	8	topological	topological	ADJ
m-1071	294	9	indices	index	NOUN
m-1071	294	10	:	:	PUNCT
m-1071	294	11	sombor	sombor	NOUN
m-1071	294	12	indices	index	NOUN
m-1071	294	13	.	.	PUNCT
m-1071	294	14	”	"	PUNCT
m-1071	294	15	match	match	PROPN
m-1071	294	16	commun	commun	PROPN
m-1071	294	17	.	.	PUNCT
m-1071	295	1	math	math	PROPN
m-1071	295	2	.	.	PUNCT
m-1071	296	1	comput	comput	NOUN
m-1071	296	2	.	.	PUNCT
m-1071	297	1	chem	chem	NOUN
m-1071	297	2	86.1	86.1	NUM
m-1071	297	3	(	(	PUNCT
m-1071	297	4	2021	2021	NUM
m-1071	297	5	):	):	PUNCT
m-1071	297	6	11	11	NUM
m-1071	297	7	-	-	SYM
m-1071	297	8	16	16	NUM
m-1071	297	9	.	.	PUNCT
m-1071	298	1	[	[	X
m-1071	298	2	30	30	NUM
m-1071	298	3	]	]	X
m-1071	298	4	kulli	kulli	PROPN
m-1071	298	5	,	,	PUNCT
m-1071	298	6	v.	v.	PROPN
m-1071	298	7	r.	r.	PROPN
m-1071	298	8	”	"	PUNCT
m-1071	298	9	on	on	ADP
m-1071	298	10	the	the	DET
m-1071	298	11	product	product	NOUN
m-1071	298	12	connectivity	connectivity	NOUN
m-1071	298	13	reverse	reverse	NOUN
m-1071	298	14	index	index	NOUN
m-1071	298	15	of	of	ADP
m-1071	298	16	silicate	silicate	NOUN
m-1071	298	17	and	and	CCONJ
m-1071	298	18	hexagonal	hexagonal	ADJ
m-1071	298	19	networks	network	NOUN
m-1071	298	20	.	.	PUNCT
m-1071	298	21	”	"	PUNCT
m-1071	299	1	rn	rn	PROPN
m-1071	299	2	55	55	NUM
m-1071	299	3	(	(	PUNCT
m-1071	299	4	2017	2017	NUM
m-1071	299	5	):	):	PUNCT
m-1071	299	6	7	7	NUM
m-1071	299	7	.	.	PUNCT
m-1071	300	1	[	[	X
m-1071	300	2	31	31	NUM
m-1071	300	3	]	]	SYM
m-1071	300	4	hu	hu	PROPN
m-1071	300	5	,	,	PUNCT
m-1071	300	6	min	min	PROPN
m-1071	300	7	,	,	PUNCT
m-1071	300	8	et	et	PROPN
m-1071	300	9	al	al	PROPN
m-1071	300	10	.	.	PUNCT
m-1071	300	11	”	"	PUNCT
m-1071	300	12	on	on	ADP
m-1071	300	13	distance	distance	NOUN
m-1071	300	14	-	-	PUNCT
m-1071	300	15	based	base	VERB
m-1071	300	16	topological	topological	ADJ
m-1071	300	17	descriptors	descriptor	NOUN
m-1071	300	18	of	of	ADP
m-1071	300	19	chemical	chemical	ADJ
m-1071	300	20	interconnection	interconnection	NOUN
m-1071	300	21	networks	network	NOUN
m-1071	300	22	.	.	PUNCT
m-1071	300	23	”	"	PUNCT
m-1071	301	1	journal	journal	NOUN
m-1071	301	2	of	of	ADP
m-1071	301	3	mathematics	mathematics	PROPN
m-1071	301	4	2021	2021	NUM
m-1071	301	5	(	(	PUNCT
m-1071	301	6	2021	2021	NUM
m-1071	301	7	)	)	PUNCT
m-1071	301	8	.	.	PUNCT
m-1071	302	1	[	[	X
m-1071	302	2	32	32	NUM
m-1071	302	3	]	]	X
m-1071	302	4	kulli	kulli	PROPN
m-1071	302	5	,	,	PUNCT
m-1071	302	6	v.	v.	PROPN
m-1071	302	7	r.	r.	PROPN
m-1071	302	8	”	"	PUNCT
m-1071	302	9	the	the	DET
m-1071	302	10	sum	sum	NOUN
m-1071	302	11	connectivity	connectivity	NOUN
m-1071	302	12	revan	revan	NOUN
m-1071	302	13	index	index	NOUN
m-1071	302	14	of	of	ADP
m-1071	302	15	silicate	silicate	NOUN
m-1071	302	16	and	and	CCONJ
m-1071	302	17	hexagonal	hexagonal	ADJ
m-1071	302	18	networks	network	NOUN
m-1071	302	19	.	.	PUNCT
m-1071	302	20	”	"	PUNCT
m-1071	302	21	annals	annal	NOUN
m-1071	302	22	of	of	ADP
m-1071	302	23	pure	pure	ADJ
m-1071	302	24	and	and	CCONJ
m-1071	302	25	applied	applied	ADJ
m-1071	302	26	mathematics	mathematic	NOUN
m-1071	302	27	14.3	14.3	NUM
m-1071	302	28	(	(	PUNCT
m-1071	302	29	2017	2017	NUM
m-1071	302	30	):	):	PUNCT
m-1071	302	31	401	401	NUM
m-1071	302	32	-	-	SYM
m-1071	302	33	406	406	NUM
m-1071	302	34	.	.	PUNCT
m-1071	303	1	[	[	X
m-1071	303	2	33	33	NUM
m-1071	303	3	]	]	X
m-1071	303	4	kulli	kulli	PROPN
m-1071	303	5	,	,	PUNCT
m-1071	303	6	v.	v.	PROPN
m-1071	303	7	r.	r.	PROPN
m-1071	303	8	,	,	PUNCT
m-1071	303	9	and	and	CCONJ
m-1071	303	10	ivan	ivan	PROPN
m-1071	303	11	gutman	gutman	PROPN
m-1071	303	12	.	.	PUNCT
m-1071	304	1	”	"	PUNCT
m-1071	304	2	computation	computation	NOUN
m-1071	304	3	of	of	ADP
m-1071	304	4	sombor	sombor	NOUN
m-1071	304	5	indices	index	NOUN
m-1071	304	6	of	of	ADP
m-1071	304	7	certain	certain	ADJ
m-1071	304	8	networks	network	NOUN
m-1071	304	9	.	.	PUNCT
m-1071	304	10	”	"	PUNCT
m-1071	305	1	ssrg	ssrg	PROPN
m-1071	305	2	int	int	PROPN
m-1071	305	3	.	.	PUNCT
m-1071	306	1	j.	j.	PROPN
m-1071	306	2	appl	appl	PROPN
m-1071	306	3	.	.	PROPN
m-1071	306	4	chem	chem	PROPN
m-1071	306	5	8.1	8.1	NUM
m-1071	306	6	(	(	PUNCT
m-1071	306	7	2021	2021	NUM
m-1071	306	8	):	):	PUNCT
m-1071	306	9	1	1	NUM
m-1071	306	10	-	-	SYM
m-1071	306	11	5	5	NUM
m-1071	306	12	.	.	PUNCT
m-1071	306	13	ijo	ijo	PROPN
m-1071	306	14	international	international	PROPN
m-1071	306	15	journal	journal	PROPN
m-1071	306	16	of	of	ADP
m-1071	306	17	mathematics	mathematics	PROPN
m-1071	306	18	(	(	PUNCT
m-1071	306	19	issn	issn	PROPN
m-1071	306	20	:	:	PUNCT
m-1071	306	21	2992	2992	NUM
m-1071	306	22	-	-	SYM
m-1071	306	23	4421	4421	NUM
m-1071	306	24	)	)	PUNCT
m-1071	306	25	ijo	ijo	PROPN
m-1071	306	26	journals	journal	NOUN
m-1071	306	27	volume	volume	NOUN
m-1071	306	28	08	08	NUM
m-1071	307	1	|	|	ADV
m-1071	307	2	issue	issue	VERB
m-1071	307	3	04	04	NUM
m-1071	308	1	|	|	CCONJ
m-1071	308	2	april	april	PROPN
m-1071	308	3	2025	2025	NUM
m-1071	309	1	|	|	ADV
m-1071	309	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	309	3	20	20	NUM
m-1071	309	4	[	[	SYM
m-1071	309	5	34	34	NUM
m-1071	309	6	]	]	X
m-1071	309	7	zhou	zhou	PROPN
m-1071	309	8	,	,	PUNCT
m-1071	309	9	bo	bo	PROPN
m-1071	309	10	,	,	PUNCT
m-1071	309	11	and	and	CCONJ
m-1071	309	12	ivan	ivan	PROPN
m-1071	309	13	gutman	gutman	PROPN
m-1071	309	14	.	.	PUNCT
m-1071	309	15	”	"	PUNCT
m-1071	309	16	further	further	ADJ
m-1071	309	17	properties	property	NOUN
m-1071	309	18	of	of	ADP
m-1071	309	19	zagreb	zagreb	PROPN
m-1071	309	20	indices	index	NOUN
m-1071	309	21	.	.	PUNCT
m-1071	309	22	”	"	PUNCT
m-1071	309	23	match	match	PROPN
m-1071	309	24	commun	commun	PROPN
m-1071	309	25	.	.	PUNCT
m-1071	309	26	math	math	PROPN
m-1071	309	27	.	.	PUNCT
m-1071	310	1	comput	comput	NOUN
m-1071	310	2	.	.	PUNCT
m-1071	311	1	chem	chem	NOUN
m-1071	311	2	54.1	54.1	NUM
m-1071	311	3	(	(	PUNCT
m-1071	311	4	2005	2005	NUM
m-1071	311	5	):	):	PUNCT
m-1071	311	6	233	233	NUM
m-1071	311	7	-	-	SYM
m-1071	311	8	239	239	NUM
m-1071	311	9	.	.	PUNCT
m-1071	312	1	ijo	ijo	PROPN
m-1071	312	2	international	international	PROPN
m-1071	312	3	journal	journal	PROPN
m-1071	312	4	of	of	ADP
m-1071	312	5	mathematics	mathematics	PROPN
m-1071	312	6	(	(	PUNCT
m-1071	312	7	issn	issn	PROPN
m-1071	312	8	:	:	PUNCT
m-1071	312	9	2992	2992	NUM
m-1071	312	10	-	-	SYM
m-1071	312	11	4421	4421	NUM
m-1071	312	12	)	)	PUNCT
m-1071	312	13	ijo	ijo	PROPN
m-1071	312	14	journals	journal	NOUN
m-1071	312	15	volume	volume	NOUN
m-1071	312	16	08	08	NUM
m-1071	313	1	|	|	ADV
m-1071	313	2	issue	issue	VERB
m-1071	313	3	04	04	NUM
m-1071	314	1	|	|	CCONJ
m-1071	314	2	april	april	PROPN
m-1071	314	3	2025	2025	NUM
m-1071	315	1	|	|	ADV
m-1071	315	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-1071	315	3	21	21	NUM
m-1071	315	4	introduction	introduction	NOUN
m-1071	315	5	1	1	NUM
m-1071	315	6	chain	chain	NOUN
m-1071	315	7	of	of	ADP
m-1071	315	8	sio4	sio4	PROPN
m-1071	315	9	1.1	1.1	NUM
m-1071	315	10	result	result	NOUN
m-1071	315	11	and	and	CCONJ
m-1071	315	12	discussion	discussion	NOUN
m-1071	315	13	1.2	1.2	NUM
m-1071	315	14	results	result	NOUN
m-1071	315	15	for	for	ADP
m-1071	315	16	p	p	X
m-1071	315	17	<	<	X
m-1071	315	18	q	q	X
m-1071	315	19	and	and	CCONJ
m-1071	315	20	p	p	NOUN
m-1071	315	21	is	be	AUX
m-1071	315	22	odd	odd	ADJ
m-1071	315	23	conclusion	conclusion	NOUN
m-1071	315	24	references	reference	NOUN