id	sid	tid	token	lemma	pos
cana-726	1	1	communications	communication	NOUN
cana-726	1	2	on	on	ADP
cana-726	1	3	applied	apply	VERB
cana-726	1	4	nonlinear	nonlinear	ADJ
cana-726	1	5	analysis	analysis	NOUN
cana-726	1	6	issn	issn	NOUN
cana-726	1	7	:	:	PUNCT
cana-726	1	8	1074	1074	NUM
cana-726	1	9	-	-	PUNCT
cana-726	1	10	133x	133x	NUM
cana-726	1	11	vol	vol	NOUN
cana-726	1	12	31	31	NUM
cana-726	1	13	no	no	NOUN
cana-726	1	14	.	.	PUNCT
cana-726	2	1	3s	3s	NUM
cana-726	2	2	(	(	PUNCT
cana-726	2	3	2024	2024	NUM
cana-726	2	4	)	)	PUNCT
cana-726	2	5	1	1	NUM
cana-726	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	2	7	bipartite	bipartite	NOUN
cana-726	2	8	through	through	ADP
cana-726	2	9	prescribed	prescribed	ADJ
cana-726	2	10	median	median	NOUN
cana-726	2	11	and	and	CCONJ
cana-726	2	12	antimedian	antimedian	NOUN
cana-726	2	13	of	of	ADP
cana-726	2	14	a	a	DET
cana-726	2	15	commutative	commutative	ADJ
cana-726	2	16	ring	ring	NOUN
cana-726	2	17	with	with	ADP
cana-726	2	18	respect	respect	NOUN
cana-726	2	19	to	to	ADP
cana-726	2	20	an	an	DET
cana-726	2	21	ideal	ideal	ADJ
cana-726	2	22	raghupatruni	raghupatruni	PROPN
cana-726	2	23	sunil	sunil	PROPN
cana-726	2	24	kumar1	kumar1	PROPN
cana-726	2	25	,	,	PUNCT
cana-726	2	26	v.	v.	PROPN
cana-726	2	27	b.	b.	PROPN
cana-726	3	1	v.	v.	ADP
cana-726	3	2	n.	n.	PROPN
cana-726	3	3	prasad2	prasad2	PROPN
cana-726	3	4	,	,	PUNCT
cana-726	3	5	chudamani	chudamani	PROPN
cana-726	3	6	ramineni3	ramineni3	PROPN
cana-726	3	7	,	,	PUNCT
cana-726	3	8	t.	t.	PROPN
cana-726	3	9	rama	rama	PROPN
cana-726	3	10	rao4	rao4	VERB
cana-726	3	11	1research	1research	NUM
cana-726	3	12	scholar	scholar	NOUN
cana-726	3	13	,	,	PUNCT
cana-726	3	14	1,2department	1,2department	NUM
cana-726	3	15	of	of	ADP
cana-726	3	16	engineering	engineering	NOUN
cana-726	3	17	mathematics	mathematic	NOUN
cana-726	3	18	,	,	PUNCT
cana-726	3	19	koneru	koneru	PROPN
cana-726	3	20	lakshmaiah	lakshmaiah	PROPN
cana-726	3	21	education	education	PROPN
cana-726	3	22	foundation	foundation	PROPN
cana-726	3	23	,	,	PUNCT
cana-726	3	24	green	green	ADJ
cana-726	3	25	fields	field	NOUN
cana-726	3	26	,	,	PUNCT
cana-726	3	27	vaddeswaram	vaddeswaram	PROPN
cana-726	3	28	,	,	PUNCT
cana-726	3	29	andhra	andhra	PROPN
cana-726	3	30	pradesh	pradesh	PROPN
cana-726	3	31	,	,	PUNCT
cana-726	3	32	india-522302	india-522302	ADJ
cana-726	3	33	.	.	PUNCT
cana-726	4	1	3department	3department	NUM
cana-726	4	2	of	of	ADP
cana-726	4	3	mathematics	mathematic	NOUN
cana-726	4	4	,	,	PUNCT
cana-726	4	5	p.v.p	p.v.p	PROPN
cana-726	4	6	siddhartha	siddhartha	PROPN
cana-726	4	7	institute	institute	PROPN
cana-726	4	8	of	of	ADP
cana-726	4	9	technology	technology	PROPN
cana-726	4	10	,	,	PUNCT
cana-726	4	11	kanuru	kanuru	PROPN
cana-726	4	12	,	,	PUNCT
cana-726	4	13	vijayawada	vijayawada	PROPN
cana-726	4	14	,	,	PUNCT
cana-726	4	15	andhra	andhra	PROPN
cana-726	4	16	pradesh	pradesh	PROPN
cana-726	4	17	,	,	PUNCT
cana-726	4	18	india520007	india520007	PROPN
cana-726	4	19	.	.	PUNCT
cana-726	5	1	4professor	4professor	NUM
cana-726	5	2	of	of	ADP
cana-726	5	3	mathematics	mathematic	NOUN
cana-726	5	4	,	,	PUNCT
cana-726	5	5	vikas	vikas	PROPN
cana-726	5	6	college	college	PROPN
cana-726	5	7	of	of	ADP
cana-726	5	8	engineering	engineering	NOUN
cana-726	5	9	and	and	CCONJ
cana-726	5	10	technology	technology	NOUN
cana-726	5	11	,	,	PUNCT
cana-726	5	12	vijayawada	vijayawada	PROPN
cana-726	5	13	,	,	PUNCT
cana-726	5	14	andhra	andhra	PROPN
cana-726	5	15	pradesh	pradesh	PROPN
cana-726	5	16	,	,	PUNCT
cana-726	5	17	india-521212	india-521212	ADV
cana-726	5	18	.	.	PUNCT
cana-726	6	1	article	article	NOUN
cana-726	6	2	history	history	NOUN
cana-726	6	3	:	:	PUNCT
cana-726	6	4	received	receive	VERB
cana-726	6	5	:	:	PUNCT
cana-726	6	6	10	10	NUM
cana-726	6	7	-	-	PUNCT
cana-726	6	8	04	04	NUM
cana-726	6	9	-	-	PUNCT
cana-726	6	10	2024	2024	NUM
cana-726	6	11	revised	revise	VERB
cana-726	6	12	:	:	PUNCT
cana-726	6	13	20	20	NUM
cana-726	6	14	-	-	SYM
cana-726	6	15	05	05	NUM
cana-726	6	16	-	-	PUNCT
cana-726	6	17	2024	2024	NUM
cana-726	6	18	accepted	accept	VERB
cana-726	6	19	:	:	PUNCT
cana-726	6	20	02	02	NUM
cana-726	6	21	-	-	PUNCT
cana-726	6	22	06	06	NUM
cana-726	6	23	-	-	PUNCT
cana-726	6	24	2024	2024	NUM
cana-726	6	25	abstract	abstract	NOUN
cana-726	6	26	:	:	PUNCT
cana-726	6	27	introduction	introduction	NOUN
cana-726	6	28	:	:	PUNCT
cana-726	6	29	there	there	PRON
cana-726	6	30	are	be	VERB
cana-726	6	31	plenty	plenty	NOUN
cana-726	6	32	of	of	ADP
cana-726	6	33	ways	way	NOUN
cana-726	6	34	of	of	ADP
cana-726	6	35	partners	partner	NOUN
cana-726	6	36	with	with	ADP
cana-726	6	37	arithmetical	arithmetical	ADJ
cana-726	6	38	constructions	construction	NOUN
cana-726	6	39	.	.	PUNCT
cana-726	7	1	some	some	PRON
cana-726	7	2	of	of	ADP
cana-726	7	3	them	they	PRON
cana-726	7	4	to	to	PART
cana-726	7	5	make	make	VERB
cana-726	7	6	reference	reference	NOUN
cana-726	7	7	to	to	PART
cana-726	7	8	are	are	VERB
cana-726	7	9	bipartite	bipartite	ADJ
cana-726	7	10	from	from	ADP
cana-726	7	11	gatherings	gathering	NOUN
cana-726	7	12	,	,	PUNCT
cana-726	7	13	median	median	NOUN
cana-726	7	14	and	and	CCONJ
cana-726	7	15	anti	anti	ADJ
cana-726	7	16	–	–	PUNCT
cana-726	7	17	median	median	ADJ
cana-726	7	18	from	from	ADP
cana-726	7	19	commutative	commutative	ADJ
cana-726	7	20	rings	ring	NOUN
cana-726	7	21	with	with	ADP
cana-726	7	22	reference	reference	NOUN
cana-726	7	23	to	to	ADP
cana-726	7	24	an	an	DET
cana-726	7	25	ideal	ideal	NOUN
cana-726	7	26	.	.	PUNCT
cana-726	8	1	partnering	partner	VERB
cana-726	8	2	a	a	DET
cana-726	8	3	median	median	ADJ
cana-726	8	4	and	and	CCONJ
cana-726	8	5	anti	anti	ADJ
cana-726	8	6	–	–	PUNCT
cana-726	8	7	median	median	NOUN
cana-726	8	8	of	of	ADP
cana-726	8	9	a	a	DET
cana-726	8	10	commutative	commutative	ADJ
cana-726	8	11	ring	ring	NOUN
cana-726	8	12	was	be	AUX
cana-726	8	13	presented	present	VERB
cana-726	8	14	by	by	ADP
cana-726	8	15	beck	beck	NOUN
cana-726	8	16	in	in	ADP
cana-726	8	17	1988	1988	NUM
cana-726	8	18	.	.	PUNCT
cana-726	9	1	similarly	similarly	ADV
cana-726	9	2	,	,	PUNCT
cana-726	9	3	beck	beck	PROPN
cana-726	9	4	has	have	AUX
cana-726	9	5	researched	research	VERB
cana-726	9	6	the	the	DET
cana-726	9	7	exchange	exchange	NOUN
cana-726	9	8	between	between	ADP
cana-726	9	9	the	the	DET
cana-726	9	10	ring	ring	NOUN
cana-726	9	11	theoretic	theoretic	NOUN
cana-726	9	12	properties	property	NOUN
cana-726	9	13	of	of	ADP
cana-726	9	14	a	a	DET
cana-726	9	15	commutative	commutative	ADJ
cana-726	9	16	ring	ring	NOUN
cana-726	9	17	and	and	CCONJ
cana-726	9	18	related	related	ADJ
cana-726	9	19	median	median	NOUN
cana-726	9	20	and	and	CCONJ
cana-726	9	21	anti	anti	ADJ
cana-726	9	22	–	–	PUNCT
cana-726	9	23	median	median	ADJ
cana-726	9	24	.	.	PUNCT
cana-726	10	1	further	further	PROPN
cana-726	10	2	anderson	anderson	PROPN
cana-726	10	3	and	and	CCONJ
cana-726	10	4	badawi	badawi	PROPN
cana-726	10	5	presented	present	VERB
cana-726	10	6	the	the	DET
cana-726	10	7	idea	idea	NOUN
cana-726	10	8	absolute	absolute	ADJ
cana-726	10	9	of	of	ADP
cana-726	10	10	commutative	commutative	ADJ
cana-726	10	11	rings	ring	NOUN
cana-726	10	12	with	with	ADP
cana-726	10	13	median	median	ADJ
cana-726	10	14	and	and	CCONJ
cana-726	10	15	anti	anti	ADJ
cana-726	10	16	–	–	PUNCT
cana-726	10	17	median	median	NOUN
cana-726	10	18	in	in	ADP
cana-726	10	19	the	the	DET
cana-726	10	20	year	year	NOUN
cana-726	10	21	2008	2008	NUM
cana-726	10	22	.	.	PUNCT
cana-726	11	1	the	the	DET
cana-726	11	2	absolute	absolute	NOUN
cana-726	11	3	of	of	ADP
cana-726	11	4	a	a	DET
cana-726	11	5	commutative	commutative	ADJ
cana-726	11	6	ring	ring	NOUN
cana-726	11	7	r	r	NOUN
cana-726	11	8	is	be	AUX
cana-726	11	9	the	the	DET
cana-726	11	10	undirected	undirected	ADJ
cana-726	11	11	with	with	ADP
cana-726	11	12	r	r	NOUN
cana-726	11	13	as	as	SCONJ
cana-726	11	14	the	the	DET
cana-726	11	15	vertex	vertex	NOUN
cana-726	11	16	set	set	NOUN
cana-726	11	17	and	and	CCONJ
cana-726	11	18	two	two	NUM
cana-726	11	19	particular	particular	ADJ
cana-726	11	20	vertices	vertex	NOUN
cana-726	11	21	in	in	ADP
cana-726	11	22	r	r	NOUN
cana-726	11	23	are	be	AUX
cana-726	11	24	nearby	nearby	ADV
cana-726	11	25	if	if	SCONJ
cana-726	11	26	and	and	CCONJ
cana-726	11	27	provided	provide	VERB
cana-726	11	28	that	that	SCONJ
cana-726	11	29	their	their	PRON
cana-726	11	30	total	total	NOUN
cana-726	11	31	is	be	AUX
cana-726	11	32	a	a	DET
cana-726	11	33	median	median	ADJ
cana-726	11	34	and	and	CCONJ
cana-726	11	35	anti	anti	ADJ
cana-726	11	36	–	–	PUNCT
cana-726	11	37	median	median	NOUN
cana-726	11	38	of	of	ADP
cana-726	11	39	r.	r.	PROPN
cana-726	11	40	as	as	ADP
cana-726	11	41	of	of	ADP
cana-726	11	42	late	late	ADJ
cana-726	11	43	anderson	anderson	PROPN
cana-726	11	44	and	and	CCONJ
cana-726	11	45	badawi	badawi	PROPN
cana-726	11	46	presented	present	VERB
cana-726	11	47	and	and	CCONJ
cana-726	11	48	concentrated	concentrate	VERB
cana-726	11	49	on	on	ADP
cana-726	11	50	the	the	DET
cana-726	11	51	summed	sum	VERB
cana-726	11	52	up	up	ADP
cana-726	11	53	all	all	PRON
cana-726	11	54	out	out	ADP
cana-726	11	55	the	the	PRON
cana-726	11	56	of	of	ADP
cana-726	11	57	commutative	commutative	ADJ
cana-726	11	58	rings	ring	NOUN
cana-726	11	59	concerning	concern	VERB
cana-726	11	60	the	the	DET
cana-726	11	61	multiplicatively	multiplicatively	ADV
cana-726	11	62	prime	prime	ADJ
cana-726	11	63	subset	subset	NOUN
cana-726	11	64	h	h	NOUN
cana-726	11	65	of	of	ADP
cana-726	11	66	r.	r.	PROPN
cana-726	11	67	the	the	DET
cana-726	11	68	summed	sum	VERB
cana-726	11	69	up	up	ADP
cana-726	11	70	complete	complete	ADJ
cana-726	11	71	of	of	ADP
cana-726	11	72	a	a	DET
cana-726	11	73	commutative	commutative	ADJ
cana-726	11	74	ring	ring	NOUN
cana-726	11	75	is	be	AUX
cana-726	11	76	the	the	DET
cana-726	11	77	undirected	undirected	ADJ
cana-726	11	78	with	with	ADP
cana-726	11	79	all	all	DET
cana-726	11	80	components	component	NOUN
cana-726	11	81	of	of	ADP
cana-726	11	82	r	r	NOUN
cana-726	11	83	as	as	ADP
cana-726	11	84	vertices	vertex	NOUN
cana-726	11	85	,	,	PUNCT
cana-726	11	86	and	and	CCONJ
cana-726	11	87	for	for	ADP
cana-726	11	88	two	two	NUM
cana-726	11	89	unmistakable	unmistakable	ADJ
cana-726	11	90	vertices	vertex	NOUN
cana-726	11	91	in	in	ADP
cana-726	11	92	r	r	NOUN
cana-726	11	93	are	be	AUX
cana-726	11	94	nearby	nearby	ADV
cana-726	11	95	if	if	SCONJ
cana-726	11	96	and	and	CCONJ
cana-726	11	97	provided	provide	VERB
cana-726	11	98	that	that	SCONJ
cana-726	11	99	their	their	PRON
cana-726	11	100	total	total	NOUN
cana-726	11	101	is	be	AUX
cana-726	11	102	in	in	ADP
cana-726	11	103	h.	h.	PROPN
cana-726	11	104	objectives	objective	NOUN
cana-726	11	105	:	:	PUNCT
cana-726	11	106	in	in	ADP
cana-726	11	107	this	this	DET
cana-726	11	108	article	article	NOUN
cana-726	11	109	,	,	PUNCT
cana-726	11	110	an	an	DET
cana-726	11	111	endeavour	endeavour	NOUN
cana-726	11	112	has	have	AUX
cana-726	11	113	been	be	AUX
cana-726	11	114	made	make	VERB
cana-726	11	115	to	to	PART
cana-726	11	116	learn	learn	VERB
cana-726	11	117	about	about	ADP
cana-726	11	118	in	in	ADP
cana-726	11	119	hypothetical	hypothetical	ADJ
cana-726	11	120	properties	property	NOUN
cana-726	11	121	and	and	CCONJ
cana-726	11	122	different	different	ADJ
cana-726	11	123	control	control	NOUN
cana-726	11	124	boundaries	boundary	NOUN
cana-726	11	125	of	of	ADP
cana-726	11	126	summed	sum	VERB
cana-726	11	127	up	up	ADP
cana-726	11	128	absolute	absolute	ADJ
cana-726	11	129	of	of	ADP
cana-726	11	130	commutative	commutative	ADJ
cana-726	11	131	rings	ring	NOUN
cana-726	11	132	of	of	ADP
cana-726	11	133	median	median	PROPN
cana-726	11	134	and	and	CCONJ
cana-726	11	135	anti	anti	ADJ
cana-726	11	136	–	–	PUNCT
cana-726	11	137	median	median	ADJ
cana-726	11	138	and	and	CCONJ
cana-726	11	139	its	its	PRON
cana-726	11	140	supplement	supplement	NOUN
cana-726	11	141	.	.	PUNCT
cana-726	12	1	methods	method	NOUN
cana-726	12	2	:	:	PUNCT
cana-726	12	3	theorem	theorem	VERB
cana-726	12	4	2.1	2.1	NUM
cana-726	12	5	.	.	PUNCT
cana-726	13	1	for	for	ADP
cana-726	13	2	a	a	DET
cana-726	13	3	determinate	determinate	ADJ
cana-726	13	4	commutative	commutative	ADJ
cana-726	13	5	semigroup	semigroup	NOUN
cana-726	13	6	s	s	PROPN
cana-726	13	7	,	,	PUNCT
cana-726	13	8	the	the	DET
cana-726	13	9	set	set	NOUN
cana-726	13	10	v(φ(γ(s)))∪{0	v(φ(γ(s)))∪{0	NOUN
cana-726	13	11	}	}	PUNCT
cana-726	13	12	is	be	AUX
cana-726	13	13	an	an	DET
cana-726	13	14	median	median	NOUN
cana-726	13	15	of	of	ADP
cana-726	13	16	s.	s.	PROPN
cana-726	13	17	remark	remark	PROPN
cana-726	13	18	2.1	2.1	NUM
cana-726	13	19	.	.	PUNCT
cana-726	14	1	a	a	DET
cana-726	14	2	subgraph	subgraph	NOUN
cana-726	14	3	h	h	NOUN
cana-726	14	4	of	of	ADP
cana-726	14	5	a	a	DET
cana-726	14	6	graph	graph	NOUN
cana-726	14	7	g	g	NOUN
cana-726	14	8	is	be	AUX
cana-726	14	9	a	a	DET
cana-726	14	10	crossing	crossing	NOUN
cana-726	14	11	subgraph	subgraph	NOUN
cana-726	14	12	of	of	ADP
cana-726	14	13	g	g	PROPN
cana-726	14	14	if	if	SCONJ
cana-726	14	15	v(h	v(h	NOUN
cana-726	14	16	)	)	PUNCT
cana-726	14	17	=	=	SYM
cana-726	14	18	v(g	v(g	ADJ
cana-726	14	19	)	)	PUNCT
cana-726	14	20	.	.	PUNCT
cana-726	15	1	on	on	ADP
cana-726	15	2	the	the	DET
cana-726	15	3	off	off	ADJ
cana-726	15	4	chance	chance	NOUN
cana-726	15	5	that	that	SCONJ
cana-726	15	6	u	u	NOUN
cana-726	15	7	is	be	AUX
cana-726	15	8	a	a	DET
cana-726	15	9	bunch	bunch	NOUN
cana-726	15	10	of	of	ADP
cana-726	15	11	edges	edge	NOUN
cana-726	15	12	of	of	ADP
cana-726	15	13	a	a	DET
cana-726	15	14	graph	graph	NOUN
cana-726	15	15	g	g	NOUN
cana-726	15	16	,	,	PUNCT
cana-726	15	17	g	g	PROPN
cana-726	15	18	\	\	NOUN
cana-726	15	19	u	u	NOUN
cana-726	15	20	is	be	AUX
cana-726	15	21	the	the	DET
cana-726	15	22	crossing	crossing	NOUN
cana-726	15	23	subgraph	subgraph	NOUN
cana-726	15	24	of	of	ADP
cana-726	15	25	g	g	PROPN
cana-726	15	26	acquired	acquire	VERB
cana-726	15	27	by	by	ADP
cana-726	15	28	erasing	erase	VERB
cana-726	15	29	the	the	DET
cana-726	15	30	edges	edge	NOUN
cana-726	15	31	in	in	ADP
cana-726	15	32	u	u	NOUN
cana-726	15	33	from	from	ADP
cana-726	15	34	e(g	e(g	PROPN
cana-726	15	35	)	)	PUNCT
cana-726	15	36	.	.	PUNCT
cana-726	16	1	a	a	DET
cana-726	16	2	subset	subset	ADJ
cana-726	16	3	u	u	NOUN
cana-726	16	4	of	of	ADP
cana-726	16	5	the	the	DET
cana-726	16	6	edge	edge	NOUN
cana-726	16	7	set	set	NOUN
cana-726	16	8	of	of	ADP
cana-726	16	9	an	an	DET
cana-726	16	10	associated	associated	ADJ
cana-726	16	11	chart	chart	NOUN
cana-726	16	12	g	g	PROPN
cana-726	16	13	is	be	AUX
cana-726	16	14	an	an	DET
cana-726	16	15	edge	edge	NOUN
cana-726	16	16	set	set	NOUN
cana-726	16	17	of	of	ADP
cana-726	16	18	g	g	PROPN
cana-726	16	19	if	if	SCONJ
cana-726	16	20	g	g	PROPN
cana-726	16	21	\	\	NOUN
cana-726	16	22	u	u	NOUN
cana-726	16	23	is	be	AUX
cana-726	16	24	separated	separate	VERB
cana-726	16	25	.	.	PUNCT
cana-726	17	1	an	an	DET
cana-726	17	2	edge	edge	NOUN
cana-726	17	3	set	set	NOUN
cana-726	17	4	of	of	ADP
cana-726	17	5	g	g	PROPN
cana-726	17	6	is	be	AUX
cana-726	17	7	negligible	negligible	ADJ
cana-726	17	8	assuming	assume	VERB
cana-726	17	9	no	no	DET
cana-726	17	10	appropriate	appropriate	ADJ
cana-726	17	11	subset	subset	NOUN
cana-726	17	12	of	of	ADP
cana-726	17	13	u	u	NOUN
cana-726	17	14	is	be	AUX
cana-726	17	15	edge	edge	NOUN
cana-726	17	16	set	set	VERB
cana-726	17	17	.	.	PUNCT
cana-726	18	1	assuming	assume	VERB
cana-726	18	2	e	e	NOUN
cana-726	18	3	is	be	AUX
cana-726	18	4	an	an	DET
cana-726	18	5	edge	edge	NOUN
cana-726	18	6	of	of	ADP
cana-726	18	7	g	g	NOUN
cana-726	18	8	,	,	PUNCT
cana-726	18	9	with	with	ADP
cana-726	18	10	the	the	DET
cana-726	18	11	end	end	NOUN
cana-726	18	12	goal	goal	NOUN
cana-726	18	13	that	that	PRON
cana-726	18	14	g	g	PROPN
cana-726	18	15	\	\	PROPN
cana-726	18	16	{	{	PUNCT
cana-726	18	17	e	e	NOUN
cana-726	18	18	}	}	PUNCT
cana-726	18	19	is	be	AUX
cana-726	18	20	detached	detach	VERB
cana-726	18	21	,	,	PUNCT
cana-726	18	22	then	then	ADV
cana-726	18	23	e	e	PROPN
cana-726	18	24	is	be	AUX
cana-726	18	25	known	know	VERB
cana-726	18	26	as	as	ADP
cana-726	18	27	an	an	DET
cana-726	18	28	extension	extension	NOUN
cana-726	18	29	.	.	PUNCT
cana-726	19	1	note	note	VERB
cana-726	19	2	that	that	SCONJ
cana-726	19	3	on	on	ADP
cana-726	19	4	the	the	DET
cana-726	19	5	off	off	ADJ
cana-726	19	6	chance	chance	NOUN
cana-726	19	7	that	that	SCONJ
cana-726	19	8	u	u	NOUN
cana-726	19	9	is	be	AUX
cana-726	19	10	an	an	DET
cana-726	19	11	insignificant	insignificant	ADJ
cana-726	19	12	edge	edge	NOUN
cana-726	19	13	set	set	NOUN
cana-726	19	14	,	,	PUNCT
cana-726	19	15	g	g	PROPN
cana-726	19	16	\	\	NOUN
cana-726	19	17	u	u	NOUN
cana-726	19	18	has	have	VERB
cana-726	19	19	precisely	precisely	ADV
cana-726	19	20	two	two	NUM
cana-726	19	21	associated	associated	ADJ
cana-726	19	22	median	median	ADJ
cana-726	19	23	parts	part	NOUN
cana-726	19	24	.	.	PUNCT
cana-726	20	1	corollary	corollary	ADJ
cana-726	20	2	2.1	2.1	NUM
cana-726	20	3	.	.	PUNCT
cana-726	21	1	let	let	VERB
cana-726	21	2	t	t	NOUN
cana-726	21	3	be	be	AUX
cana-726	21	4	the	the	DET
cana-726	21	5	minimal	minimal	ADJ
cana-726	21	6	edge	edge	NOUN
cana-726	21	7	set	set	NOUN
cana-726	21	8	of	of	ADP
cana-726	21	9	γ(s	γ(	NOUN
cana-726	21	10	)	)	PUNCT
cana-726	21	11	,	,	PUNCT
cana-726	21	12	and	and	CCONJ
cana-726	21	13	g1	g1	NOUN
cana-726	21	14	,	,	PUNCT
cana-726	21	15	g2	g2	PROPN
cana-726	21	16	are	be	AUX
cana-726	21	17	two	two	NUM
cana-726	21	18	median	median	ADJ
cana-726	21	19	parts	part	NOUN
cana-726	21	20	of	of	ADP
cana-726	21	21	g	g	NOUN
cana-726	21	22	\	\	PROPN
cana-726	22	1	t.	t.	NOUN
cana-726	22	2	then	then	ADV
cana-726	22	3	the	the	DET
cana-726	22	4	following	follow	VERB
cana-726	22	5	hold	hold	NOUN
cana-726	22	6	.	.	PUNCT
cana-726	23	1	(	(	PUNCT
cana-726	23	2	i	i	NOUN
cana-726	23	3	)	)	PUNCT
cana-726	23	4	for	for	ADP
cana-726	23	5	any	any	DET
cana-726	23	6	i	i	NOUN
cana-726	23	7	=	=	NOUN
cana-726	23	8	1	1	NUM
cana-726	23	9	,	,	PUNCT
cana-726	23	10	2	2	NUM
cana-726	23	11	,	,	PUNCT
cana-726	23	12	(	(	PUNCT
cana-726	23	13	v(gi	v(gi	NOUN
cana-726	23	14	)	)	PUNCT
cana-726	23	15	∩	∩	NOUN
cana-726	23	16	v(t	v(t	NOUN
cana-726	23	17	)	)	PUNCT
cana-726	23	18	)	)	PUNCT
cana-726	23	19	∪	∪	ADP
cana-726	23	20	{	{	PUNCT
cana-726	23	21	0	0	NUM
cana-726	23	22	}	}	PUNCT
cana-726	23	23	is	be	AUX
cana-726	23	24	ideal	ideal	ADJ
cana-726	23	25	of	of	ADP
cana-726	23	26	s	s	PRON
cana-726	23	27	provided	provide	VERB
cana-726	23	28	gi	gi	NOUN
cana-726	23	29	has	have	VERB
cana-726	23	30	at	at	ADV
cana-726	23	31	least	least	ADV
cana-726	23	32	two	two	NUM
cana-726	23	33	vertices	vertex	NOUN
cana-726	23	34	.	.	PUNCT
cana-726	24	1	(	(	PUNCT
cana-726	24	2	ii	ii	NOUN
cana-726	24	3	)	)	PUNCT
cana-726	24	4	v(t	v(t	NOUN
cana-726	24	5	)	)	PUNCT
cana-726	24	6	∪	∪	X
cana-726	24	7	{	{	PUNCT
cana-726	24	8	0	0	NUM
cana-726	24	9	}	}	PUNCT
cana-726	24	10	is	be	AUX
cana-726	24	11	an	an	DET
cana-726	24	12	ideal	ideal	NOUN
cana-726	24	13	if	if	SCONJ
cana-726	24	14	g_1	g_1	PROPN
cana-726	24	15	or	or	CCONJ
cana-726	24	16	g_2	g_2	PROPN
cana-726	24	17	has	have	VERB
cana-726	24	18	only	only	ADV
cana-726	24	19	one	one	NUM
cana-726	24	20	vertex	vertex	NOUN
cana-726	24	21	.	.	PUNCT
cana-726	25	1	a	a	DET
cana-726	25	2	commutative	commutative	ADJ
cana-726	25	3	semigroup	semigroup	NOUN
cana-726	25	4	is	be	AUX
cana-726	25	5	called	call	VERB
cana-726	25	6	reduced	reduce	VERB
cana-726	25	7	if	if	SCONJ
cana-726	25	8	for	for	ADP
cana-726	25	9	any	any	DET
cana-726	25	10	x	x	SYM
cana-726	25	11	∈	∈	PROPN
cana-726	25	12	s	s	NOUN
cana-726	25	13	,	,	PUNCT
cana-726	25	14	xn	xn	PUNCT
cana-726	25	15	=	=	SYM
cana-726	25	16	0	0	NUM
cana-726	25	17	implies	imply	VERB
cana-726	25	18	x	x	PUNCT
cana-726	25	19	=	=	SYM
cana-726	25	20	0	0	NUM
cana-726	25	21	.	.	PUNCT
cana-726	26	1	the	the	DET
cana-726	26	2	annihilator	annihilator	NOUN
cana-726	26	3	of	of	ADP
cana-726	26	4	x	x	PUNCT
cana-726	26	5	∈	∈	PROPN
cana-726	26	6	s	s	PART
cana-726	26	7	is	be	AUX
cana-726	26	8	denoted	denote	VERB
cana-726	26	9	by	by	ADP
cana-726	26	10	ann	ann	PROPN
cana-726	26	11	(	(	PUNCT
cana-726	26	12	x	x	NOUN
cana-726	26	13	)	)	PUNCT
cana-726	26	14	and	and	CCONJ
cana-726	26	15	it	it	PRON
cana-726	26	16	is	be	AUX
cana-726	26	17	defined	define	VERB
cana-726	26	18	as	as	ADP
cana-726	26	19	ann	ann	PROPN
cana-726	26	20	(	(	PUNCT
cana-726	26	21	x	x	NOUN
cana-726	26	22	)	)	PUNCT
cana-726	27	1	=	=	PRON
cana-726	27	2	{	{	PUNCT
cana-726	27	3	a	a	DET
cana-726	27	4	∈	∈	PROPN
cana-726	27	5	s|ax	s|ax	NOUN
cana-726	27	6	=	=	NOUN
cana-726	27	7	0	0	NUM
cana-726	27	8	}	}	PUNCT
cana-726	27	9	.	.	PUNCT
cana-726	28	1	results	result	NOUN
cana-726	28	2	:	:	PUNCT
cana-726	28	3	theorem	theorem	VERB
cana-726	28	4	2.3	2.3	NUM
cana-726	28	5	.	.	PUNCT
cana-726	29	1	let	let	VERB
cana-726	29	2	s	s	PRON
cana-726	29	3	be	be	AUX
cana-726	29	4	a	a	DET
cana-726	29	5	commutative	commutative	ADJ
cana-726	29	6	ring	ring	NOUN
cana-726	29	7	of	of	ADP
cana-726	29	8	median	median	PROPN
cana-726	29	9	and	and	CCONJ
cana-726	29	10	anti	anti	ADJ
cana-726	29	11	–	–	PUNCT
cana-726	29	12	median	median	ADJ
cana-726	29	13	.	.	PUNCT
cana-726	30	1	then	then	ADV
cana-726	30	2	the	the	DET
cana-726	30	3	subsequent	subsequent	ADJ
cana-726	30	4	results	result	NOUN
cana-726	30	5	hold	hold	VERB
cana-726	30	6	:	:	PUNCT
cana-726	30	7	communications	communication	NOUN
cana-726	30	8	on	on	ADP
cana-726	30	9	applied	apply	VERB
cana-726	30	10	nonlinear	nonlinear	ADJ
cana-726	30	11	analysis	analysis	NOUN
cana-726	30	12	issn	issn	NOUN
cana-726	30	13	:	:	PUNCT
cana-726	30	14	1074	1074	NUM
cana-726	30	15	-	-	PUNCT
cana-726	30	16	133x	133x	NUM
cana-726	30	17	vol	vol	NOUN
cana-726	30	18	31	31	NUM
cana-726	30	19	no	no	NOUN
cana-726	30	20	.	.	PUNCT
cana-726	31	1	3s	3s	NUM
cana-726	31	2	(	(	PUNCT
cana-726	31	3	2024	2024	NUM
cana-726	31	4	)	)	PUNCT
cana-726	31	5	2	2	NUM
cana-726	31	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	31	7	(	(	PUNCT
cana-726	31	8	i	i	NOUN
cana-726	31	9	)	)	PUNCT
cana-726	31	10	if	if	SCONJ
cana-726	31	11	|ass	|ass	PROPN
cana-726	31	12	(	(	PUNCT
cana-726	31	13	s)|	s)|	PROPN
cana-726	31	14	≥	≥	NUM
cana-726	31	15	3	3	NUM
cana-726	31	16	and	and	CCONJ
cana-726	31	17	∅	∅	NOUN
cana-726	31	18	=	=	SYM
cana-726	31	19	ann	ann	X
cana-726	31	20	(	(	PUNCT
cana-726	31	21	x	x	NOUN
cana-726	31	22	)	)	PUNCT
cana-726	31	23	,	,	PUNCT
cana-726	31	24	χ	χ	X
cana-726	31	25	=	=	PUNCT
cana-726	31	26	ann	ann	PROPN
cana-726	31	27	(	(	PUNCT
cana-726	31	28	y	y	NOUN
cana-726	31	29	)	)	PUNCT
cana-726	31	30	are	be	AUX
cana-726	31	31	two	two	NUM
cana-726	31	32	distinct	distinct	ADJ
cana-726	31	33	elements	element	NOUN
cana-726	31	34	of	of	ADP
cana-726	31	35	ass	ass	NOUN
cana-726	31	36	(	(	PUNCT
cana-726	31	37	s	s	NOUN
cana-726	31	38	)	)	PUNCT
cana-726	31	39	,	,	PUNCT
cana-726	31	40	then	then	ADV
cana-726	31	41	xy	xy	PROPN
cana-726	31	42	=	=	NOUN
cana-726	32	1	0	0	PROPN
cana-726	32	2	.	.	PUNCT
cana-726	32	3	(	(	PUNCT
cana-726	32	4	ii	ii	NOUN
cana-726	32	5	)	)	PUNCT
cana-726	32	6	if	if	SCONJ
cana-726	32	7	|ass	|ass	PROPN
cana-726	32	8	(	(	PUNCT
cana-726	32	9	s)|	s)|	PROPN
cana-726	32	10	≥	≥	NUM
cana-726	32	11	3	3	NUM
cana-726	32	12	,	,	PUNCT
cana-726	32	13	then	then	ADV
cana-726	32	14	girth	girth	ADV
cana-726	32	15	(	(	PUNCT
cana-726	32	16	γ(s	γ(s	PROPN
cana-726	32	17	)	)	PUNCT
cana-726	32	18	)	)	PUNCT
cana-726	33	1	=	=	PUNCT
cana-726	33	2	4	4	X
cana-726	33	3	.	.	PUNCT
cana-726	33	4	(	(	PUNCT
cana-726	33	5	iii	iii	X
cana-726	33	6	)	)	PUNCT
cana-726	33	7	if	if	SCONJ
cana-726	33	8	|ass	|ass	PROPN
cana-726	33	9	(	(	PUNCT
cana-726	33	10	s)|	s)|	PROPN
cana-726	33	11	≥	≥	NUM
cana-726	33	12	6	6	NUM
cana-726	33	13	,	,	PUNCT
cana-726	33	14	then	then	ADV
cana-726	33	15	γ(s	γ(	NOUN
cana-726	33	16	)	)	PUNCT
cana-726	33	17	is	be	AUX
cana-726	33	18	not	not	PART
cana-726	33	19	planar	planar	ADJ
cana-726	33	20	(	(	PUNCT
cana-726	33	21	a	a	DET
cana-726	33	22	graph	graph	NOUN
cana-726	33	23	g	g	NOUN
cana-726	33	24	is	be	AUX
cana-726	33	25	planar	planar	ADJ
cana-726	33	26	if	if	SCONJ
cana-726	33	27	it	it	PRON
cana-726	33	28	can	can	AUX
cana-726	33	29	be	be	AUX
cana-726	33	30	drawn	draw	VERB
cana-726	33	31	in	in	ADP
cana-726	33	32	the	the	DET
cana-726	33	33	plane	plane	NOUN
cana-726	33	34	in	in	ADP
cana-726	33	35	such	such	DET
cana-726	33	36	a	a	DET
cana-726	33	37	way	way	NOUN
cana-726	33	38	that	that	PRON
cana-726	33	39	no	no	DET
cana-726	33	40	two	two	NUM
cana-726	33	41	edges	edge	NOUN
cana-726	33	42	meet	meet	VERB
cana-726	33	43	except	except	SCONJ
cana-726	33	44	at	at	ADP
cana-726	33	45	vertex	vertex	NOUN
cana-726	33	46	with	with	ADP
cana-726	33	47	which	which	PRON
cana-726	33	48	they	they	PRON
cana-726	33	49	are	be	AUX
cana-726	33	50	both	both	PRON
cana-726	33	51	incident	incident	NOUN
cana-726	33	52	)	)	PUNCT
cana-726	33	53	.	.	PUNCT
cana-726	34	1	conclusions	conclusion	NOUN
cana-726	34	2	:	:	PUNCT
cana-726	34	3	theorem	theorem	VERB
cana-726	34	4	2.4	2.4	NUM
cana-726	34	5	.	.	PUNCT
cana-726	35	1	let	let	VERB
cana-726	35	2	s	s	PRON
cana-726	35	3	be	be	AUX
cana-726	35	4	a	a	DET
cana-726	35	5	commutative	commutative	ADJ
cana-726	35	6	semigroup	semigroup	NOUN
cana-726	35	7	,	,	PUNCT
cana-726	35	8	then	then	ADV
cana-726	35	9	the	the	DET
cana-726	35	10	median	median	ADJ
cana-726	35	11	graph	graph	NOUN
cana-726	35	12	of	of	ADP
cana-726	35	13	a	a	DET
cana-726	35	14	bipartite	bipartite	NOUN
cana-726	35	15	graph	graph	NOUN
cana-726	35	16	is	be	AUX
cana-726	35	17	induced	induce	VERB
cana-726	35	18	by	by	ADP
cana-726	35	19	the	the	DET
cana-726	35	20	vertices	vertex	NOUN
cana-726	35	21	of	of	ADP
cana-726	35	22	maximum	maximum	ADJ
cana-726	35	23	degree	degree	NOUN
cana-726	35	24	in	in	ADP
cana-726	35	25	g.	g.	PROPN
cana-726	35	26	proof	proof	NOUN
cana-726	35	27	.	.	PUNCT
cana-726	36	1	s	s	VERB
cana-726	36	2	be	be	AUX
cana-726	36	3	a	a	DET
cana-726	36	4	commutative	commutative	ADJ
cana-726	36	5	semigroup	semigroup	NOUN
cana-726	36	6	and	and	CCONJ
cana-726	36	7	g	g	PROPN
cana-726	36	8	is	be	AUX
cana-726	36	9	a	a	DET
cana-726	36	10	bipartite	bipartite	ADJ
cana-726	36	11	median	median	ADJ
cana-726	36	12	graph	graph	NOUN
cana-726	36	13	,	,	PUNCT
cana-726	36	14	thus	thus	ADV
cana-726	36	15	d(v	d(v	ADJ
cana-726	36	16	,	,	PUNCT
cana-726	36	17	u	u	NOUN
cana-726	36	18	)	)	PUNCT
cana-726	36	19	<	<	X
cana-726	36	20	2	2	NUM
cana-726	36	21	for	for	ADP
cana-726	36	22	any	any	DET
cana-726	36	23	pair	pair	NOUN
cana-726	36	24	of	of	ADP
cana-726	36	25	vertices	vertex	NOUN
cana-726	36	26	u	u	NOUN
cana-726	36	27	,	,	PUNCT
cana-726	36	28	v	v	NOUN
cana-726	36	29	of	of	ADP
cana-726	36	30	g.	g.	PROPN
cana-726	36	31	let	let	VERB
cana-726	36	32	the	the	DET
cana-726	36	33	degree	degree	NOUN
cana-726	36	34	of	of	ADP
cana-726	36	35	v	v	NOUN
cana-726	36	36	in	in	ADP
cana-726	36	37	g	g	PROPN
cana-726	36	38	be	be	PROPN
cana-726	36	39	d.	d.	PROPN
cana-726	36	40	then	then	ADV
cana-726	36	41	,	,	PUNCT
cana-726	36	42	these	these	DET
cana-726	36	43	d	d	NOUN
cana-726	36	44	vertices	vertex	NOUN
cana-726	36	45	are	be	AUX
cana-726	36	46	at	at	ADP
cana-726	36	47	a	a	DET
cana-726	36	48	distance	distance	NOUN
cana-726	36	49	1	1	NUM
cana-726	36	50	from	from	ADP
cana-726	36	51	v.	v.	ADP
cana-726	36	52	so	so	ADV
cana-726	36	53	,	,	PUNCT
cana-726	36	54	there	there	PRON
cana-726	36	55	are	be	VERB
cana-726	36	56	p	p	NOUN
cana-726	36	57	1	1	NUM
cana-726	36	58	d	d	NOUN
cana-726	36	59	vertices	vertice	VERB
cana-726	36	60	u	u	NOUN
cana-726	36	61	in	in	ADP
cana-726	36	62	g	g	PROPN
cana-726	36	63	such	such	ADJ
cana-726	36	64	that	that	SCONJ
cana-726	36	65	d(v	d(v	PROPN
cana-726	36	66	,	,	PUNCT
cana-726	36	67	u	u	NOUN
cana-726	36	68	)	)	PUNCT
cana-726	36	69	=	=	SYM
cana-726	36	70	2	2	NUM
cana-726	36	71	and	and	CCONJ
cana-726	36	72	d(v	d(v	ADJ
cana-726	36	73	)	)	PUNCT
cana-726	36	74	=	=	PUNCT
cana-726	37	1	d	d	PROPN
cana-726	37	2	+	+	CCONJ
cana-726	37	3	2(p	2(p	NUM
cana-726	37	4	1	1	NUM
cana-726	37	5	d	d	NOUN
cana-726	37	6	)	)	PUNCT
cana-726	37	7	=	=	SYM
cana-726	37	8	2(p	2(p	NUM
cana-726	37	9	1	1	X
cana-726	37	10	)	)	PUNCT
cana-726	37	11	d.	d.	NOUN
cana-726	37	12	hence	hence	ADV
cana-726	37	13	the	the	DET
cana-726	37	14	vertices	vertex	NOUN
cana-726	37	15	in	in	ADP
cana-726	37	16	g	g	PROPN
cana-726	37	17	such	such	DET
cana-726	37	18	that	that	DET
cana-726	37	19	d(v)is	d(v)is	ADJ
cana-726	37	20	minimum	minimum	NOUN
cana-726	37	21	are	be	AUX
cana-726	37	22	those	those	PRON
cana-726	37	23	for	for	ADP
cana-726	37	24	which	which	PRON
cana-726	37	25	the	the	DET
cana-726	37	26	degree	degree	NOUN
cana-726	37	27	is	be	AUX
cana-726	37	28	maximum	maximum	ADJ
cana-726	37	29	.	.	PUNCT
cana-726	38	1	hence	hence	ADV
cana-726	38	2	the	the	DET
cana-726	38	3	proof	proof	NOUN
cana-726	38	4	.	.	PUNCT
cana-726	39	1	remark	remark	VERB
cana-726	39	2	2.3	2.3	NUM
cana-726	39	3	the	the	DET
cana-726	39	4	median	median	ADJ
cana-726	39	5	graph	graph	NOUN
cana-726	39	6	of	of	ADP
cana-726	39	7	a	a	DET
cana-726	39	8	bipartite	bipartite	NOUN
cana-726	39	9	graph	graph	NOUN
cana-726	39	10	is	be	AUX
cana-726	39	11	also	also	ADV
cana-726	39	12	a	a	DET
cana-726	39	13	bipartite	bipartite	ADJ
cana-726	39	14	graph	graph	NOUN
cana-726	39	15	.	.	PUNCT
cana-726	40	1	keywords	keyword	NOUN
cana-726	40	2	:	:	PUNCT
cana-726	40	3	median	median	ADJ
cana-726	40	4	,	,	PUNCT
cana-726	40	5	anti	anti	ADJ
cana-726	40	6	median	median	ADJ
cana-726	40	7	,	,	PUNCT
cana-726	40	8	commutative	commutative	ADJ
cana-726	40	9	ring	ring	NOUN
cana-726	40	10	,	,	PUNCT
cana-726	40	11	ideal	ideal	ADJ
cana-726	40	12	.	.	PUNCT
cana-726	41	1	1	1	X
cana-726	41	2	.	.	X
cana-726	41	3	introduction	introduction	NOUN
cana-726	41	4	let	let	VERB
cana-726	41	5	g	g	PROPN
cana-726	41	6	=	=	SYM
cana-726	41	7	(	(	PUNCT
cana-726	41	8	v	v	NOUN
cana-726	41	9	,	,	PUNCT
cana-726	41	10	e	e	NOUN
cana-726	41	11	)	)	PUNCT
cana-726	41	12	be	be	AUX
cana-726	41	13	a	a	PRON
cana-726	41	14	on	on	ADP
cana-726	41	15	n	n	NOUN
cana-726	41	16	vertices	vertex	NOUN
cana-726	41	17	with	with	ADP
cana-726	41	18	vertex	vertex	NOUN
cana-726	41	19	set	set	VERB
cana-726	41	20	v	v	NOUN
cana-726	41	21	and	and	CCONJ
cana-726	41	22	edge	edge	NOUN
cana-726	41	23	set	set	VERB
cana-726	41	24	e.	e.	PROPN
cana-726	41	25	a	a	PROPN
cana-726	41	26	is	be	AUX
cana-726	41	27	bipartite	bipartite	ADJ
cana-726	41	28	if	if	SCONJ
cana-726	41	29	its	its	PRON
cana-726	41	30	vertex	vertex	NOUN
cana-726	41	31	set	set	NOUN
cana-726	41	32	can	can	AUX
cana-726	41	33	be	be	AUX
cana-726	41	34	partitioned	partition	VERB
cana-726	41	35	into	into	ADP
cana-726	41	36	two	two	NUM
cana-726	41	37	nonempty	nonempty	ADJ
cana-726	41	38	subsets	subset	NOUN
cana-726	41	39	x	x	PUNCT
cana-726	41	40	and	and	CCONJ
cana-726	41	41	y	y	PROPN
cana-726	41	42	such	such	ADJ
cana-726	41	43	that	that	SCONJ
cana-726	41	44	each	each	DET
cana-726	41	45	edge	edge	NOUN
cana-726	41	46	of	of	ADP
cana-726	41	47	g	g	PROPN
cana-726	41	48	has	have	VERB
cana-726	41	49	one	one	NUM
cana-726	41	50	end	end	NOUN
cana-726	41	51	in	in	ADP
cana-726	41	52	x	x	X
cana-726	41	53	and	and	CCONJ
cana-726	41	54	the	the	DET
cana-726	41	55	other	other	ADJ
cana-726	41	56	in	in	ADP
cana-726	41	57	y	y	PROPN
cana-726	41	58	,	,	PUNCT
cana-726	41	59	and	and	CCONJ
cana-726	41	60	a	a	PRON
cana-726	41	61	is	be	AUX
cana-726	41	62	k	k	NOUN
cana-726	41	63	-	-	ADJ
cana-726	41	64	partite	partite	ADJ
cana-726	41	65	if	if	SCONJ
cana-726	41	66	its	its	PRON
cana-726	41	67	vertex	vertex	NOUN
cana-726	41	68	set	set	NOUN
cana-726	41	69	can	can	AUX
cana-726	41	70	be	be	AUX
cana-726	41	71	partitioned	partition	VERB
cana-726	41	72	into	into	ADP
cana-726	41	73	k	k	PROPN
cana-726	41	74	nonempty	nonempty	ADJ
cana-726	41	75	subsets	subset	NOUN
cana-726	41	76	such	such	ADJ
cana-726	41	77	that	that	SCONJ
cana-726	41	78	no	no	DET
cana-726	41	79	edge	edge	NOUN
cana-726	41	80	in	in	ADP
cana-726	41	81	g	g	PROPN
cana-726	41	82	has	have	VERB
cana-726	41	83	its	its	PRON
cana-726	41	84	both	both	PRON
cana-726	41	85	ends	end	NOUN
cana-726	41	86	in	in	ADP
cana-726	41	87	the	the	DET
cana-726	41	88	same	same	ADJ
cana-726	41	89	subset	subset	NOUN
cana-726	41	90	.	.	PUNCT
cana-726	42	1	degree	degree	NOUN
cana-726	42	2	of	of	ADP
cana-726	42	3	a	a	DET
cana-726	42	4	vertex	vertex	NOUN
cana-726	42	5	v	v	NOUN
cana-726	42	6	,	,	PUNCT
cana-726	42	7	d(v	d(v	PROPN
cana-726	42	8	)	)	PUNCT
cana-726	42	9	,	,	PUNCT
cana-726	42	10	is	be	AUX
cana-726	42	11	the	the	DET
cana-726	42	12	number	number	NOUN
cana-726	42	13	vertices	vertice	VERB
cana-726	42	14	adjacent	adjacent	ADJ
cana-726	42	15	to	to	ADP
cana-726	42	16	v	v	NOUN
cana-726	42	17	and	and	CCONJ
cana-726	42	18	by	by	ADP
cana-726	42	19	n(v	n(v	PROPN
cana-726	42	20	)	)	PUNCT
cana-726	42	21	we	we	PRON
cana-726	42	22	denote	denote	VERB
cana-726	42	23	the	the	DET
cana-726	42	24	neighbor	neighbor	NOUN
cana-726	42	25	set	set	VERB
cana-726	42	26	of	of	ADP
cana-726	42	27	v.	v.	ADP
cana-726	42	28	the	the	DET
cana-726	42	29	smallest	small	ADJ
cana-726	42	30	and	and	CCONJ
cana-726	42	31	largest	large	ADJ
cana-726	42	32	degrees	degree	NOUN
cana-726	42	33	of	of	ADP
cana-726	42	34	vertices	vertex	NOUN
cana-726	42	35	in	in	ADP
cana-726	42	36	g	g	PROPN
cana-726	42	37	are	be	AUX
cana-726	42	38	respectively	respectively	ADV
cana-726	42	39	denoted	denote	VERB
cana-726	42	40	by	by	ADP
cana-726	42	41	δ(g	δ(g	ADJ
cana-726	42	42	)	)	PUNCT
cana-726	42	43	and	and	CCONJ
cana-726	42	44	∆(g	∆(g	NOUN
cana-726	42	45	)	)	PUNCT
cana-726	42	46	.	.	PUNCT
cana-726	43	1	given	give	VERB
cana-726	43	2	a	a	DET
cana-726	43	3	g	g	NOUN
cana-726	43	4	the	the	DET
cana-726	43	5	issue	issue	NOUN
cana-726	43	6	of	of	ADP
cana-726	43	7	tracking	track	VERB
cana-726	43	8	down	down	ADP
cana-726	43	9	a	a	DET
cana-726	43	10	h	h	NOUN
cana-726	43	11	to	to	ADP
cana-726	43	12	such	such	DET
cana-726	43	13	an	an	DET
cana-726	43	14	extent	extent	NOUN
cana-726	43	15	that	that	SCONJ
cana-726	43	16	m(h	m(h	NOUN
cana-726	43	17	)	)	PUNCT
cana-726	43	18	≅g	≅g	NOUN
cana-726	43	19	is	be	AUX
cana-726	43	20	alluded	allude	VERB
cana-726	43	21	to	to	ADP
cana-726	43	22	as	as	ADP
cana-726	43	23	the	the	DET
cana-726	43	24	median	median	ADJ
cana-726	43	25	issue	issue	NOUN
cana-726	43	26	.	.	PUNCT
cana-726	44	1	in	in	ADP
cana-726	44	2	[	[	X
cana-726	44	3	6	6	NUM
cana-726	44	4	]	]	PUNCT
cana-726	44	5	,	,	PUNCT
cana-726	44	6	it	it	PRON
cana-726	44	7	is	be	AUX
cana-726	44	8	shown	show	VERB
cana-726	44	9	that	that	SCONJ
cana-726	44	10	any	any	DET
cana-726	44	11	g	g	NOUN
cana-726	44	12	=	=	SYM
cana-726	44	13	(	(	PUNCT
cana-726	44	14	v	v	NOUN
cana-726	44	15	,	,	PUNCT
cana-726	44	16	e	e	NOUN
cana-726	44	17	)	)	PUNCT
cana-726	44	18	is	be	AUX
cana-726	44	19	the	the	DET
cana-726	44	20	median	median	NOUN
cana-726	44	21	of	of	ADP
cana-726	44	22	some	some	PRON
cana-726	44	23	associated	associate	VERB
cana-726	44	24	.	.	PUNCT
cana-726	45	1	in	in	ADP
cana-726	45	2	[	[	X
cana-726	45	3	3	3	X
cana-726	45	4	]	]	PUNCT
cana-726	45	5	the	the	DET
cana-726	45	6	thought	thought	NOUN
cana-726	45	7	of	of	ADP
cana-726	45	8	against	against	ADP
cana-726	45	9	median	median	NOUN
cana-726	45	10	of	of	ADP
cana-726	45	11	a	a	PRON
cana-726	45	12	was	be	AUX
cana-726	45	13	presented	present	VERB
cana-726	45	14	and	and	CCONJ
cana-726	45	15	demonstrated	demonstrate	VERB
cana-726	45	16	that	that	SCONJ
cana-726	45	17	each	each	PRON
cana-726	45	18	is	be	AUX
cana-726	45	19	the	the	DET
cana-726	45	20	counter	counter	ADJ
cana-726	45	21	median	median	NOUN
cana-726	45	22	of	of	ADP
cana-726	45	23	some	some	PRON
cana-726	45	24	.	.	PUNCT
cana-726	46	1	the	the	DET
cana-726	46	2	issue	issue	NOUN
cana-726	46	3	of	of	ADP
cana-726	46	4	concurrent	concurrent	ADJ
cana-726	46	5	inserting	inserting	NOUN
cana-726	46	6	of	of	ADP
cana-726	46	7	median	median	ADJ
cana-726	46	8	and	and	CCONJ
cana-726	46	9	hostile	hostile	ADJ
cana-726	46	10	to	to	ADP
cana-726	46	11	median	median	PROPN
cana-726	46	12	is	be	AUX
cana-726	46	13	examined	examine	VERB
cana-726	46	14	in	in	ADP
cana-726	46	15	[	[	X
cana-726	46	16	1	1	NUM
cana-726	46	17	]	]	PUNCT
cana-726	46	18	.	.	PUNCT
cana-726	47	1	another	another	DET
cana-726	47	2	development	development	NOUN
cana-726	47	3	,	,	PUNCT
cana-726	47	4	which	which	PRON
cana-726	47	5	sums	sum	VERB
cana-726	47	6	up	up	ADP
cana-726	47	7	every	every	DET
cana-726	47	8	one	one	NUM
cana-726	47	9	of	of	ADP
cana-726	47	10	the	the	DET
cana-726	47	11	recently	recently	ADV
cana-726	47	12	referenced	reference	VERB
cana-726	47	13	developments	development	NOUN
cana-726	47	14	,	,	PUNCT
cana-726	47	15	can	can	AUX
cana-726	47	16	be	be	AUX
cana-726	47	17	found	find	VERB
cana-726	47	18	in	in	ADP
cana-726	47	19	[	[	X
cana-726	47	20	5	5	NUM
cana-726	47	21	]	]	PUNCT
cana-726	47	22	.	.	PUNCT
cana-726	48	1	the	the	DET
cana-726	48	2	median	median	ADJ
cana-726	48	3	vertices	vertex	NOUN
cana-726	48	4	have	have	VERB
cana-726	48	5	the	the	DET
cana-726	48	6	base	base	NOUN
cana-726	48	7	normal	normal	ADJ
cana-726	48	8	distance	distance	NOUN
cana-726	48	9	in	in	ADP
cana-726	48	10	a	a	PRON
cana-726	48	11	and	and	CCONJ
cana-726	48	12	subsequently	subsequently	ADV
cana-726	48	13	the	the	DET
cana-726	48	14	median	median	ADJ
cana-726	48	15	issue	issue	NOUN
cana-726	48	16	is	be	AUX
cana-726	48	17	huge	huge	ADJ
cana-726	48	18	among	among	ADP
cana-726	48	19	the	the	DET
cana-726	48	20	improvement	improvement	NOUN
cana-726	48	21	issues	issue	NOUN
cana-726	48	22	including	include	VERB
cana-726	48	23	the	the	DET
cana-726	48	24	position	position	NOUN
cana-726	48	25	of	of	ADP
cana-726	48	26	organization	organization	NOUN
cana-726	48	27	servers	server	NOUN
cana-726	48	28	.	.	PUNCT
cana-726	49	1	nonetheless	nonetheless	ADV
cana-726	49	2	,	,	PUNCT
cana-726	49	3	the	the	DET
cana-726	49	4	median	median	ADJ
cana-726	49	5	developments	development	NOUN
cana-726	49	6	for	for	ADP
cana-726	49	7	general	general	ADJ
cana-726	49	8	ca	can	AUX
cana-726	49	9	n't	not	PART
cana-726	49	10	be	be	AUX
cana-726	49	11	straightforwardly	straightforwardly	ADV
cana-726	49	12	applied	apply	VERB
cana-726	49	13	to	to	ADP
cana-726	49	14	a	a	DET
cana-726	49	15	huge	huge	ADJ
cana-726	49	16	number	number	NOUN
cana-726	49	17	as	as	SCONJ
cana-726	49	18	their	their	PRON
cana-726	49	19	fundamental	fundamental	NOUN
cana-726	49	20	has	have	VERB
cana-726	49	21	a	a	DET
cana-726	49	22	place	place	NOUN
cana-726	49	23	with	with	ADP
cana-726	49	24	various	various	ADJ
cana-726	49	25	classes	class	NOUN
cana-726	49	26	of	of	ADP
cana-726	49	27	.	.	PUNCT
cana-726	50	1	it	it	PRON
cana-726	50	2	tends	tend	VERB
cana-726	50	3	to	to	PART
cana-726	50	4	be	be	AUX
cana-726	50	5	seen	see	VERB
cana-726	50	6	that	that	PRON
cana-726	50	7	are	be	AUX
cana-726	50	8	bipartite	bipartite	ADJ
cana-726	50	9	to	to	ADP
cana-726	50	10	basic	basic	ADJ
cana-726	50	11	s	s	NOUN
cana-726	50	12	of	of	ADP
cana-726	50	13	a	a	DET
cana-726	50	14	huge	huge	ADJ
cana-726	50	15	number	number	NOUN
cana-726	50	16	.	.	PUNCT
cana-726	51	1	for	for	ADP
cana-726	51	2	instance	instance	NOUN
cana-726	51	3	,	,	PUNCT
cana-726	51	4	the	the	DET
cana-726	51	5	vast	vast	ADJ
cana-726	51	6	majority	majority	NOUN
cana-726	51	7	of	of	ADP
cana-726	51	8	the	the	DET
cana-726	51	9	examinations	examination	NOUN
cana-726	51	10	in	in	ADP
cana-726	51	11	network	network	NOUN
cana-726	51	12	networks	network	NOUN
cana-726	51	13	are	be	AUX
cana-726	51	14	finished	finish	VERB
cana-726	51	15	utilizing	utilize	VERB
cana-726	51	16	inclination	inclination	NOUN
cana-726	51	17	networks	network	NOUN
cana-726	51	18	[	[	X
cana-726	51	19	4	4	X
cana-726	51	20	]	]	PUNCT
cana-726	51	21	and	and	CCONJ
cana-726	51	22	they	they	PRON
cana-726	51	23	are	be	AUX
cana-726	51	24	displayed	display	VERB
cana-726	51	25	utilizing	utilize	VERB
cana-726	51	26	bipartite	bipartite	NOUN
cana-726	51	27	.	.	PUNCT
cana-726	52	1	it	it	PRON
cana-726	52	2	is	be	AUX
cana-726	52	3	notable	notable	ADJ
cana-726	52	4	that	that	SCONJ
cana-726	52	5	the	the	DET
cana-726	52	6	median	median	NOUN
cana-726	52	7	of	of	ADP
cana-726	52	8	a	a	DET
cana-726	52	9	tree	tree	NOUN
cana-726	52	10	is	be	AUX
cana-726	52	11	a	a	DET
cana-726	52	12	vertex	vertex	NOUN
cana-726	52	13	or	or	CCONJ
cana-726	52	14	an	an	DET
cana-726	52	15	edge	edge	NOUN
cana-726	52	16	.	.	PUNCT
cana-726	53	1	this	this	DET
cana-726	53	2	administrator	administrator	NOUN
cana-726	53	3	was	be	AUX
cana-726	53	4	additionally	additionally	ADV
cana-726	53	5	read	read	VERB
cana-726	53	6	up	up	ADP
cana-726	53	7	for	for	ADP
cana-726	53	8	certain	certain	ADJ
cana-726	53	9	classes	class	NOUN
cana-726	53	10	of	of	ADP
cana-726	53	11	in	in	ADP
cana-726	53	12	[	[	X
cana-726	53	13	7	7	NUM
cana-726	53	14	]	]	PUNCT
cana-726	53	15	and	and	CCONJ
cana-726	53	16	[	[	X
cana-726	53	17	8	8	NUM
cana-726	53	18	]	]	PUNCT
cana-726	53	19	.	.	PUNCT
cana-726	54	1	in	in	ADP
cana-726	54	2	this	this	DET
cana-726	54	3	paper	paper	NOUN
cana-726	54	4	we	we	PRON
cana-726	54	5	show	show	VERB
cana-726	54	6	that	that	SCONJ
cana-726	54	7	any	any	DET
cana-726	54	8	bipartite	bipartite	NOUN
cana-726	54	9	is	be	AUX
cana-726	54	10	the	the	DET
cana-726	54	11	median	median	NOUN
cana-726	54	12	of	of	ADP
cana-726	54	13	another	another	DET
cana-726	54	14	bipartite	bipartite	NOUN
cana-726	54	15	.	.	PUNCT
cana-726	55	1	with	with	ADP
cana-726	55	2	an	an	DET
cana-726	55	3	alternate	alternate	ADJ
cana-726	55	4	development	development	NOUN
cana-726	55	5	,	,	PUNCT
cana-726	55	6	we	we	PRON
cana-726	55	7	show	show	VERB
cana-726	55	8	that	that	SCONJ
cana-726	55	9	the	the	DET
cana-726	55	10	comparative	comparative	ADJ
cana-726	55	11	outcome	outcome	NOUN
cana-726	55	12	additionally	additionally	ADV
cana-726	55	13	hold	hold	VERB
cana-726	55	14	for	for	ADP
cana-726	55	15	k	k	NOUN
cana-726	55	16	-	-	NOUN
cana-726	55	17	partite	partite	ADJ
cana-726	55	18	.	.	PUNCT
cana-726	56	1	the	the	DET
cana-726	56	2	undifferentiated	undifferentiated	ADJ
cana-726	56	3	from	from	ADP
cana-726	56	4	results	result	NOUN
cana-726	56	5	for	for	ADP
cana-726	56	6	against	against	ADP
cana-726	56	7	median	median	ADJ
cana-726	56	8	issue	issue	NOUN
cana-726	56	9	on	on	ADP
cana-726	56	10	these	these	DET
cana-726	56	11	classes	class	NOUN
cana-726	56	12	are	be	AUX
cana-726	56	13	additionally	additionally	ADV
cana-726	56	14	acquired	acquire	VERB
cana-726	56	15	.	.	PUNCT
cana-726	57	1	since	since	SCONJ
cana-726	57	2	any	any	PRON
cana-726	57	3	is	be	AUX
cana-726	57	4	a	a	DET
cana-726	57	5	k	k	NOUN
cana-726	57	6	-	-	NOUN
cana-726	57	7	partite	partite	ADJ
cana-726	57	8	,	,	PUNCT
cana-726	57	9	for	for	ADP
cana-726	57	10	some	some	DET
cana-726	57	11	k	k	NOUN
cana-726	57	12	,	,	PUNCT
cana-726	57	13	these	these	DET
cana-726	57	14	developments	development	NOUN
cana-726	57	15	can	can	AUX
cana-726	57	16	be	be	AUX
cana-726	57	17	applied	apply	VERB
cana-726	57	18	overall	overall	ADV
cana-726	57	19	.	.	PUNCT
cana-726	58	1	for	for	SCONJ
cana-726	58	2	any	any	DET
cana-726	58	3	communications	communication	NOUN
cana-726	58	4	on	on	ADP
cana-726	58	5	applied	apply	VERB
cana-726	58	6	nonlinear	nonlinear	ADJ
cana-726	58	7	analysis	analysis	NOUN
cana-726	58	8	issn	issn	NOUN
cana-726	58	9	:	:	PUNCT
cana-726	58	10	1074	1074	NUM
cana-726	58	11	-	-	PUNCT
cana-726	58	12	133x	133x	NUM
cana-726	58	13	vol	vol	NOUN
cana-726	58	14	31	31	NUM
cana-726	58	15	no	no	NOUN
cana-726	58	16	.	.	PUNCT
cana-726	59	1	3s	3s	NUM
cana-726	59	2	(	(	PUNCT
cana-726	59	3	2024	2024	NUM
cana-726	59	4	)	)	PUNCT
cana-726	59	5	3	3	NUM
cana-726	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	59	7	remaining	remain	VERB
cana-726	59	8	fundamental	fundamental	ADJ
cana-726	59	9	ideas	idea	NOUN
cana-726	59	10	and	and	CCONJ
cana-726	59	11	documentations	documentation	NOUN
cana-726	59	12	not	not	PART
cana-726	59	13	referenced	reference	VERB
cana-726	59	14	in	in	ADP
cana-726	59	15	this	this	DET
cana-726	59	16	paper	paper	NOUN
cana-726	59	17	we	we	PRON
cana-726	59	18	allude	allude	VERB
cana-726	59	19	to	to	ADP
cana-726	59	20	[	[	X
cana-726	59	21	2	2	NUM
cana-726	59	22	]	]	PUNCT
cana-726	59	23	.	.	PUNCT
cana-726	60	1	in	in	ADP
cana-726	60	2	variation	variation	NOUN
cana-726	60	3	to	to	ADP
cana-726	60	4	the	the	DET
cana-726	60	5	concept	concept	NOUN
cana-726	60	6	of	of	ADP
cana-726	60	7	zero	zero	NUM
cana-726	60	8	divisor	divisor	NOUN
cana-726	60	9	,	,	PUNCT
cana-726	60	10	few	few	ADJ
cana-726	60	11	authors	author	NOUN
cana-726	60	12	[	[	X
cana-726	60	13	8	8	NUM
cana-726	60	14	]	]	PUNCT
cana-726	60	15	introduced	introduce	VERB
cana-726	60	16	the	the	DET
cana-726	60	17	total	total	NOUN
cana-726	60	18	of	of	ADP
cana-726	60	19	a	a	DET
cana-726	60	20	commutative	commutative	ADJ
cana-726	60	21	ring	ring	NOUN
cana-726	60	22	.	.	PUNCT
cana-726	61	1	let	let	VERB
cana-726	61	2	r	r	PRON
cana-726	61	3	be	be	AUX
cana-726	61	4	a	a	DET
cana-726	61	5	commutative	commutative	ADJ
cana-726	61	6	ring	ring	NOUN
cana-726	61	7	with	with	ADP
cana-726	61	8	nil(r	nil(r	NOUN
cana-726	61	9	)	)	PUNCT
cana-726	61	10	its	its	PRON
cana-726	61	11	ideal	ideal	NOUN
cana-726	61	12	of	of	ADP
cana-726	61	13	nilpotent	nilpotent	ADJ
cana-726	61	14	elements	element	NOUN
cana-726	61	15	,	,	PUNCT
cana-726	61	16	z(r	z(r	NOUN
cana-726	61	17	)	)	PUNCT
cana-726	61	18	its	its	PRON
cana-726	61	19	set	set	NOUN
cana-726	61	20	of	of	ADP
cana-726	61	21	zero	zero	NUM
cana-726	61	22	-	-	PUNCT
cana-726	61	23	divisors	divisor	NOUN
cana-726	61	24	,	,	PUNCT
cana-726	61	25	and	and	CCONJ
cana-726	61	26	reg(r	reg(r	PROPN
cana-726	61	27	)	)	PUNCT
cana-726	61	28	its	its	PRON
cana-726	61	29	set	set	NOUN
cana-726	61	30	of	of	ADP
cana-726	61	31	regular	regular	ADJ
cana-726	61	32	elements	element	NOUN
cana-726	61	33	.	.	PUNCT
cana-726	62	1	the	the	DET
cana-726	62	2	total	total	NOUN
cana-726	62	3	of	of	ADP
cana-726	62	4	r	r	NOUN
cana-726	62	5	,	,	PUNCT
cana-726	62	6	denoted	denote	VERB
cana-726	62	7	by	by	ADP
cana-726	62	8	t	t	PROPN
cana-726	62	9	(	(	PUNCT
cana-726	62	10	r	r	NOUN
cana-726	62	11	)	)	PUNCT
cana-726	62	12	,	,	PUNCT
cana-726	62	13	is	be	AUX
cana-726	62	14	the	the	DET
cana-726	62	15	undirected	undirected	ADJ
cana-726	62	16	with	with	ADP
cana-726	62	17	all	all	DET
cana-726	62	18	elements	element	NOUN
cana-726	62	19	of	of	ADP
cana-726	62	20	r	r	NOUN
cana-726	62	21	as	as	ADP
cana-726	62	22	vertices	vertex	NOUN
cana-726	62	23	,	,	PUNCT
cana-726	62	24	and	and	CCONJ
cana-726	62	25	for	for	ADP
cana-726	62	26	distinct	distinct	ADJ
cana-726	62	27	x	x	NOUN
cana-726	62	28	,	,	PUNCT
cana-726	62	29	y	y	PROPN
cana-726	62	30	∈	∈	PROPN
cana-726	62	31	r	r	NOUN
cana-726	62	32	,	,	PUNCT
cana-726	62	33	the	the	DET
cana-726	62	34	vertices	vertex	NOUN
cana-726	62	35	x	x	PUNCT
cana-726	62	36	and	and	CCONJ
cana-726	62	37	y	y	PROPN
cana-726	62	38	are	be	AUX
cana-726	62	39	adjacent	adjacent	ADJ
cana-726	62	40	if	if	SCONJ
cana-726	62	41	and	and	CCONJ
cana-726	62	42	only	only	ADV
cana-726	62	43	if	if	SCONJ
cana-726	62	44	x	x	PROPN
cana-726	62	45	+	+	NUM
cana-726	62	46	y	y	PROPN
cana-726	62	47	∈	∈	PROPN
cana-726	62	48	z(r	z(r	PROPN
cana-726	62	49	)	)	PUNCT
cana-726	62	50	.	.	PUNCT
cana-726	63	1	also	also	ADV
cana-726	63	2	they	they	PRON
cana-726	63	3	introduced	introduce	VERB
cana-726	63	4	the	the	DET
cana-726	63	5	three	three	NUM
cana-726	63	6	induce	induce	ADJ
cana-726	63	7	subs	sub	NOUN
cana-726	63	8	nil	nil	NOUN
cana-726	63	9	(	(	PUNCT
cana-726	63	10	r	r	NOUN
cana-726	63	11	)	)	PUNCT
cana-726	63	12	z	z	NOUN
cana-726	63	13	(	(	PUNCT
cana-726	63	14	r	r	NOUN
cana-726	63	15	)	)	PUNCT
cana-726	63	16	and	and	CCONJ
cana-726	63	17	reg	reg	NOUN
cana-726	63	18	(	(	PUNCT
cana-726	63	19	r	r	NOUN
cana-726	63	20	)	)	PUNCT
cana-726	63	21	of	of	ADP
cana-726	63	22	t	t	PROPN
cana-726	63	23	(	(	PUNCT
cana-726	63	24	r	r	NOUN
cana-726	63	25	)	)	PUNCT
cana-726	63	26	with	with	ADP
cana-726	63	27	vertices	vertex	NOUN
cana-726	63	28	nil(r	nil(r	PROPN
cana-726	63	29	)	)	PUNCT
cana-726	63	30	,	,	PUNCT
cana-726	63	31	z(r	z(r	PROPN
cana-726	63	32	)	)	PUNCT
cana-726	63	33	,	,	PUNCT
cana-726	63	34	and	and	CCONJ
cana-726	63	35	reg(r	reg(r	PROPN
cana-726	63	36	)	)	PUNCT
cana-726	63	37	.	.	PUNCT
cana-726	64	1	a	a	DET
cana-726	64	2	graph	graph	NOUN
cana-726	64	3	wherein	wherein	SCONJ
cana-726	64	4	each	each	DET
cana-726	64	5	set	set	NOUN
cana-726	64	6	of	of	ADP
cana-726	64	7	particular	particular	ADJ
cana-726	64	8	vertices	vertex	NOUN
cana-726	64	9	is	be	AUX
cana-726	64	10	joined	join	VERB
cana-726	64	11	by	by	ADP
cana-726	64	12	an	an	DET
cana-726	64	13	edge	edge	NOUN
cana-726	64	14	is	be	AUX
cana-726	64	15	known	know	VERB
cana-726	64	16	as	as	ADP
cana-726	64	17	a	a	DET
cana-726	64	18	total	total	ADJ
cana-726	64	19	graph	graph	NOUN
cana-726	64	20	.	.	PUNCT
cana-726	65	1	we	we	PRON
cana-726	65	2	use	use	VERB
cana-726	65	3	𝐾𝑛	𝐾𝑛	PROPN
cana-726	65	4	for	for	ADP
cana-726	65	5	the	the	DET
cana-726	65	6	total	total	ADJ
cana-726	65	7	graph	graph	NOUN
cana-726	65	8	with	with	ADP
cana-726	65	9	n	n	SYM
cana-726	65	10	vertices	vertex	NOUN
cana-726	65	11	.	.	PUNCT
cana-726	66	1	a	a	DET
cana-726	66	2	r	r	NOUN
cana-726	66	3	-	-	ADJ
cana-726	66	4	partite	partite	ADJ
cana-726	66	5	graph	graph	NOUN
cana-726	66	6	is	be	AUX
cana-726	66	7	a	a	DET
cana-726	66	8	graph	graph	NOUN
cana-726	66	9	whose	whose	DET
cana-726	66	10	vertex	vertex	NOUN
cana-726	66	11	set	set	NOUN
cana-726	66	12	can	can	AUX
cana-726	66	13	be	be	AUX
cana-726	66	14	divided	divide	VERB
cana-726	66	15	into	into	ADP
cana-726	66	16	r	r	NOUN
cana-726	66	17	subsets	subset	NOUN
cana-726	66	18	so	so	SCONJ
cana-726	66	19	that	that	SCONJ
cana-726	66	20	no	no	DET
cana-726	66	21	edge	edge	NOUN
cana-726	66	22	has	have	VERB
cana-726	66	23	the	the	DET
cana-726	66	24	two	two	NUM
cana-726	66	25	vertices	vertex	NOUN
cana-726	66	26	in	in	ADP
cana-726	66	27	any	any	DET
cana-726	66	28	one	one	NUM
cana-726	66	29	subset	subset	NOUN
cana-726	66	30	.	.	PUNCT
cana-726	67	1	a	a	DET
cana-726	67	2	total	total	ADJ
cana-726	67	3	i	i	NOUN
cana-726	67	4	-	-	PUNCT
cana-726	67	5	partite	partite	ADJ
cana-726	67	6	graph	graph	NOUN
cana-726	67	7	is	be	AUX
cana-726	67	8	one	one	NUM
cana-726	67	9	in	in	ADP
cana-726	67	10	which	which	PRON
cana-726	67	11	every	every	DET
cana-726	67	12	vertex	vertex	NOUN
cana-726	67	13	is	be	AUX
cana-726	67	14	joined	join	VERB
cana-726	67	15	to	to	ADP
cana-726	67	16	each	each	DET
cana-726	67	17	vertex	vertex	NOUN
cana-726	67	18	that	that	PRON
cana-726	67	19	is	be	AUX
cana-726	67	20	n't	not	PART
cana-726	67	21	in	in	ADP
cana-726	67	22	a	a	DET
cana-726	67	23	similar	similar	ADJ
cana-726	67	24	subset	subset	NOUN
cana-726	67	25	as	as	ADP
cana-726	67	26	the	the	DET
cana-726	67	27	given	give	VERB
cana-726	67	28	vertex	vertex	NOUN
cana-726	67	29	.	.	PUNCT
cana-726	68	1	the	the	DET
cana-726	68	2	total	total	ADJ
cana-726	68	3	bipartite	bipartite	NOUN
cana-726	68	4	(	(	PUNCT
cana-726	68	5	i.e.	i.e.	X
cana-726	68	6	,	,	PUNCT
cana-726	68	7	complete	complete	ADJ
cana-726	68	8	2	2	NUM
cana-726	68	9	-	-	PUNCT
cana-726	68	10	partite	partite	ADJ
cana-726	68	11	)	)	PUNCT
cana-726	68	12	graph	graph	NOUN
cana-726	68	13	is	be	AUX
cana-726	68	14	signified	signify	VERB
cana-726	68	15	by	by	ADP
cana-726	68	16	𝐾𝑚,𝑛	𝐾𝑚,𝑛	PROPN
cana-726	68	17	where	where	SCONJ
cana-726	68	18	the	the	DET
cana-726	68	19	arrangement	arrangement	NOUN
cana-726	68	20	of	of	ADP
cana-726	68	21	segment	segment	NOUN
cana-726	68	22	has	have	VERB
cana-726	68	23	sizes	size	NOUN
cana-726	68	24	m	m	PROPN
cana-726	68	25	and	and	CCONJ
cana-726	68	26	n.	n.	VERB
cana-726	68	27	the	the	DET
cana-726	68	28	circumference	circumference	NOUN
cana-726	68	29	of	of	ADP
cana-726	68	30	a	a	DET
cana-726	68	31	graph	graph	NOUN
cana-726	68	32	g	g	NOUN
cana-726	68	33	is	be	AUX
cana-726	68	34	the	the	DET
cana-726	68	35	length	length	NOUN
cana-726	68	36	of	of	ADP
cana-726	68	37	a	a	DET
cana-726	68	38	most	most	ADV
cana-726	68	39	limited	limited	ADJ
cana-726	68	40	cycle	cycle	NOUN
cana-726	68	41	in	in	ADP
cana-726	68	42	g	g	PROPN
cana-726	68	43	and	and	CCONJ
cana-726	68	44	is	be	AUX
cana-726	68	45	meant	mean	VERB
cana-726	68	46	by	by	ADP
cana-726	68	47	bigness	bigness	ADV
cana-726	68	48	(	(	PUNCT
cana-726	68	49	g	g	NOUN
cana-726	68	50	)	)	PUNCT
cana-726	68	51	.	.	PUNCT
cana-726	69	1	we	we	PRON
cana-726	69	2	characterize	characterize	VERB
cana-726	69	3	a	a	DET
cana-726	69	4	shading	shading	NOUN
cana-726	69	5	of	of	ADP
cana-726	69	6	a	a	DET
cana-726	69	7	graph	graph	NOUN
cana-726	69	8	g	g	NOUN
cana-726	69	9	to	to	PART
cana-726	69	10	be	be	AUX
cana-726	69	11	a	a	DET
cana-726	69	12	task	task	NOUN
cana-726	69	13	of	of	ADP
cana-726	69	14	tones	tone	NOUN
cana-726	69	15	(	(	PUNCT
cana-726	69	16	components	component	NOUN
cana-726	69	17	of	of	ADP
cana-726	69	18	some	some	DET
cana-726	69	19	set	set	NOUN
cana-726	69	20	)	)	PUNCT
cana-726	69	21	to	to	ADP
cana-726	69	22	the	the	DET
cana-726	69	23	vertices	vertex	NOUN
cana-726	69	24	of	of	ADP
cana-726	69	25	g	g	NOUN
cana-726	69	26	,	,	PUNCT
cana-726	69	27	one	one	NUM
cana-726	69	28	tone	tone	NOUN
cana-726	69	29	to	to	ADP
cana-726	69	30	every	every	DET
cana-726	69	31	vertex	vertex	NOUN
cana-726	69	32	,	,	PUNCT
cana-726	69	33	so	so	SCONJ
cana-726	69	34	nearby	nearby	ADJ
cana-726	69	35	vertices	vertex	NOUN
cana-726	69	36	are	be	AUX
cana-726	69	37	doled	dole	VERB
cana-726	69	38	out	out	ADP
cana-726	69	39	unmistakable	unmistakable	ADJ
cana-726	69	40	tones	tone	NOUN
cana-726	69	41	.	.	PUNCT
cana-726	70	1	in	in	ADP
cana-726	70	2	the	the	DET
cana-726	70	3	event	event	NOUN
cana-726	70	4	that	that	PRON
cana-726	70	5	n	n	NOUN
cana-726	70	6	tones	tone	NOUN
cana-726	70	7	are	be	AUX
cana-726	70	8	utilized	utilize	VERB
cana-726	70	9	,	,	PUNCT
cana-726	70	10	the	the	DET
cana-726	70	11	shading	shading	NOUN
cana-726	70	12	is	be	AUX
cana-726	70	13	alluded	allude	VERB
cana-726	70	14	to	to	ADP
cana-726	70	15	as	as	ADP
cana-726	70	16	a	a	DET
cana-726	70	17	n	n	CCONJ
cana-726	70	18	-	-	PUNCT
cana-726	70	19	shading	shading	NOUN
cana-726	70	20	.	.	PUNCT
cana-726	71	1	on	on	ADP
cana-726	71	2	the	the	DET
cana-726	71	3	off	off	ADJ
cana-726	71	4	chance	chance	NOUN
cana-726	71	5	that	that	SCONJ
cana-726	71	6	there	there	PRON
cana-726	71	7	exists	exist	VERB
cana-726	71	8	a	a	DET
cana-726	71	9	n	n	ADV
cana-726	71	10	-	-	PUNCT
cana-726	71	11	shading	shading	NOUN
cana-726	71	12	of	of	ADP
cana-726	71	13	a	a	DET
cana-726	71	14	graph	graph	NOUN
cana-726	71	15	g	g	NOUN
cana-726	71	16	,	,	PUNCT
cana-726	71	17	g	g	PROPN
cana-726	71	18	is	be	AUX
cana-726	71	19	called	call	VERB
cana-726	71	20	n	n	CCONJ
cana-726	71	21	-	-	ADV
cana-726	71	22	colorable	colorable	ADJ
cana-726	71	23	.	.	PUNCT
cana-726	72	1	the	the	DET
cana-726	72	2	base	base	NOUN
cana-726	72	3	n	n	CCONJ
cana-726	72	4	for	for	ADP
cana-726	72	5	which	which	PRON
cana-726	72	6	a	a	DET
cana-726	72	7	graph	graph	NOUN
cana-726	72	8	g	g	PROPN
cana-726	72	9	is	be	AUX
cana-726	72	10	n	n	CCONJ
cana-726	72	11	-	-	PUNCT
cana-726	72	12	colorable	colorable	ADJ
cana-726	72	13	is	be	AUX
cana-726	72	14	known	know	VERB
cana-726	72	15	as	as	ADP
cana-726	72	16	the	the	DET
cana-726	72	17	chromatic	chromatic	ADJ
cana-726	72	18	number	number	NOUN
cana-726	72	19	of	of	ADP
cana-726	72	20	g	g	NOUN
cana-726	72	21	,	,	PUNCT
cana-726	72	22	and	and	CCONJ
cana-726	72	23	is	be	AUX
cana-726	72	24	indicated	indicate	VERB
cana-726	72	25	by	by	ADP
cana-726	72	26	χ(g	χ(g	PROPN
cana-726	72	27	)	)	PUNCT
cana-726	72	28	.	.	PUNCT
cana-726	73	1	a	a	DET
cana-726	73	2	club	club	NOUN
cana-726	73	3	of	of	ADP
cana-726	73	4	a	a	DET
cana-726	73	5	graph	graph	NOUN
cana-726	73	6	is	be	AUX
cana-726	73	7	a	a	DET
cana-726	73	8	maximal	maximal	ADJ
cana-726	73	9	complete	complete	ADJ
cana-726	73	10	sub	sub	NOUN
cana-726	73	11	and	and	CCONJ
cana-726	73	12	the	the	DET
cana-726	73	13	quantity	quantity	NOUN
cana-726	73	14	of	of	ADP
cana-726	73	15	vertices	vertex	NOUN
cana-726	73	16	in	in	ADP
cana-726	73	17	the	the	DET
cana-726	73	18	biggest	big	ADJ
cana-726	73	19	inner	inner	ADJ
cana-726	73	20	circle	circle	NOUN
cana-726	73	21	of	of	ADP
cana-726	73	22	graph	graph	NOUN
cana-726	73	23	g	g	NOUN
cana-726	73	24	,	,	PUNCT
cana-726	73	25	signified	signify	VERB
cana-726	73	26	by	by	ADP
cana-726	73	27	ω(g	ω(g	NOUN
cana-726	73	28	)	)	PUNCT
cana-726	73	29	,	,	PUNCT
cana-726	73	30	is	be	AUX
cana-726	73	31	known	know	VERB
cana-726	73	32	as	as	ADP
cana-726	73	33	the	the	DET
cana-726	73	34	faction	faction	NOUN
cana-726	73	35	number	number	NOUN
cana-726	73	36	of	of	ADP
cana-726	73	37	g.	g.	PROPN
cana-726	73	38	clearly	clearly	ADV
cana-726	73	39	χ(g	χ(g	PROPN
cana-726	73	40	)	)	PUNCT
cana-726	73	41	≥	≥	NOUN
cana-726	73	42	ω(g	ω(g	NOUN
cana-726	73	43	)	)	PUNCT
cana-726	73	44	for	for	ADP
cana-726	73	45	general	general	ADJ
cana-726	73	46	graph	graph	NOUN
cana-726	73	47	g.	g.	PROPN
cana-726	73	48	assume	assume	VERB
cana-726	73	49	that	that	SCONJ
cana-726	73	50	s	s	VERB
cana-726	73	51	is	be	AUX
cana-726	73	52	a	a	DET
cana-726	73	53	commutative	commutative	ADJ
cana-726	73	54	semigroup	semigroup	NOUN
cana-726	73	55	with	with	ADP
cana-726	73	56	nothing	nothing	PRON
cana-726	73	57	.	.	PUNCT
cana-726	74	1	for	for	ADP
cana-726	74	2	ideal	ideal	ADJ
cana-726	74	3	hypothesis	hypothesis	NOUN
cana-726	74	4	in	in	ADP
cana-726	74	5	commutative	commutative	ADJ
cana-726	74	6	semigroup	semigroup	NOUN
cana-726	74	7	,	,	PUNCT
cana-726	74	8	we	we	PRON
cana-726	74	9	allude	allude	VERB
cana-726	74	10	to	to	ADP
cana-726	74	11	the	the	DET
cana-726	74	12	overview	overview	NOUN
cana-726	74	13	of	of	ADP
cana-726	74	14	median	median	ADJ
cana-726	74	15	and	and	CCONJ
cana-726	74	16	anti	anti	ADJ
cana-726	74	17	–	–	PUNCT
cana-726	74	18	median	median	ADJ
cana-726	75	1	[	[	X
cana-726	75	2	3	3	NUM
cana-726	75	3	]	]	PUNCT
cana-726	75	4	(	(	PUNCT
cana-726	75	5	additionally	additionally	ADV
cana-726	75	6	see	see	VERB
cana-726	75	7	[	[	X
cana-726	75	8	2	2	NUM
cana-726	75	9	]	]	PUNCT
cana-726	75	10	)	)	PUNCT
cana-726	75	11	.	.	PUNCT
cana-726	76	1	here	here	ADV
cana-726	76	2	we	we	PRON
cana-726	76	3	simply	simply	ADV
cana-726	76	4	review	review	VERB
cana-726	76	5	a	a	DET
cana-726	76	6	portion	portion	NOUN
cana-726	76	7	of	of	ADP
cana-726	76	8	the	the	DET
cana-726	76	9	ideas	idea	NOUN
cana-726	76	10	.	.	PUNCT
cana-726	77	1	a	a	DET
cana-726	77	2	non	non	ADJ
cana-726	77	3	-	-	ADJ
cana-726	77	4	void	void	ADJ
cana-726	77	5	subset	subset	VERB
cana-726	77	6	i	i	PRON
cana-726	77	7	of	of	ADP
cana-726	77	8	s	s	PROPN
cana-726	77	9	is	be	AUX
cana-726	77	10	called	call	VERB
cana-726	77	11	ideal	ideal	ADJ
cana-726	77	12	if	if	SCONJ
cana-726	77	13	xs	xs	PROPN
cana-726	77	14	⊆	⊆	NUM
cana-726	77	15	i	i	PRON
cana-726	77	16	for	for	ADP
cana-726	77	17	any	any	DET
cana-726	77	18	x	x	SYM
cana-726	77	19	∈	∈	PROPN
cana-726	77	20	i.	i.	NOUN
cana-726	77	21	an	an	DET
cana-726	77	22	optimal	optimal	ADJ
cana-726	77	23	p	p	NOUN
cana-726	77	24	of	of	ADP
cana-726	77	25	a	a	DET
cana-726	77	26	commutative	commutative	ADJ
cana-726	77	27	semigroup	semigroup	NOUN
cana-726	77	28	is	be	AUX
cana-726	77	29	known	know	VERB
cana-726	77	30	as	as	ADP
cana-726	77	31	an	an	DET
cana-726	77	32	excellent	excellent	ADJ
cana-726	77	33	ideal	ideal	NOUN
cana-726	77	34	of	of	ADP
cana-726	77	35	s	s	PRON
cana-726	77	36	if	if	SCONJ
cana-726	77	37	for	for	ADP
cana-726	77	38	any	any	DET
cana-726	77	39	two	two	NUM
cana-726	77	40	component	component	NOUN
cana-726	77	41	x	x	NOUN
cana-726	77	42	,	,	PUNCT
cana-726	77	43	y	y	PROPN
cana-726	77	44	∈	∈	PROPN
cana-726	77	45	s	s	PROPN
cana-726	77	46	,	,	PUNCT
cana-726	77	47	xy	xy	PROPN
cana-726	77	48	∈	∈	PROPN
cana-726	78	1	p	p	NOUN
cana-726	78	2	infers	infer	NOUN
cana-726	78	3	x	x	PUNCT
cana-726	78	4	∈	∈	PROPN
cana-726	78	5	p	p	NOUN
cana-726	78	6	or	or	CCONJ
cana-726	78	7	y	y	PROPN
cana-726	78	8	∈	∈	PROPN
cana-726	78	9	p.	p.	NOUN
cana-726	78	10	let	let	VERB
cana-726	78	11	z(s	z(s	PROPN
cana-726	78	12	)	)	PUNCT
cana-726	78	13	be	be	AUX
cana-726	78	14	its	its	PRON
cana-726	78	15	arrangement	arrangement	NOUN
cana-726	78	16	of	of	ADP
cana-726	78	17	zero	zero	NUM
cana-726	78	18	-	-	PUNCT
cana-726	78	19	divisors	divisor	NOUN
cana-726	78	20	of	of	ADP
cana-726	78	21	s.	s.	PROPN
cana-726	78	22	all	all	PRON
cana-726	78	23	together	together	ADV
cana-726	78	24	that	that	SCONJ
cana-726	78	25	γ(s	γ(	NOUN
cana-726	78	26	)	)	PUNCT
cana-726	78	27	be	be	VERB
cana-726	78	28	non	non	X
cana-726	78	29	void	void	ADJ
cana-726	78	30	,	,	PUNCT
cana-726	78	31	we	we	PRON
cana-726	78	32	generally	generally	ADV
cana-726	78	33	expect	expect	VERB
cana-726	78	34	s	s	PRON
cana-726	78	35	generally	generally	ADV
cana-726	78	36	contains	contain	VERB
cana-726	78	37	somewhere	somewhere	ADV
cana-726	78	38	around	around	ADP
cana-726	78	39	one	one	NUM
cana-726	78	40	nonzero	nonzero	NOUN
cana-726	78	41	zero	zero	NUM
cana-726	78	42	divisor	divisor	NOUN
cana-726	78	43	.	.	PUNCT
cana-726	79	1	in	in	ADP
cana-726	79	2	[	[	X
cana-726	79	3	14	14	NUM
cana-726	79	4	]	]	PUNCT
cana-726	79	5	we	we	PRON
cana-726	79	6	can	can	AUX
cana-726	79	7	view	view	VERB
cana-726	79	8	that	that	SCONJ
cana-726	79	9	γ(s	γ(	NOUN
cana-726	79	10	)	)	PUNCT
cana-726	79	11	(	(	PUNCT
cana-726	79	12	as	as	ADP
cana-726	79	13	in	in	ADP
cana-726	79	14	the	the	DET
cana-726	79	15	ring	ring	NOUN
cana-726	79	16	case	case	NOUN
cana-726	79	17	)	)	PUNCT
cana-726	79	18	is	be	AUX
cana-726	79	19	generally	generally	ADV
cana-726	79	20	associated	associate	VERB
cana-726	79	21	,	,	PUNCT
cana-726	79	22	and	and	CCONJ
cana-726	79	23	the	the	DET
cana-726	79	24	breadth	breadth	NOUN
cana-726	79	25	of	of	ADP
cana-726	79	26	γ(s	γ(	NOUN
cana-726	79	27	)	)	PUNCT
cana-726	79	28	≤	≤	NOUN
cana-726	79	29	3	3	NUM
cana-726	79	30	.	.	PUNCT
cana-726	80	1	in	in	ADP
cana-726	80	2	the	the	DET
cana-726	80	3	event	event	NOUN
cana-726	80	4	that	that	PRON
cana-726	80	5	γ(s	γ(s	PROPN
cana-726	80	6	)	)	PUNCT
cana-726	80	7	has	have	VERB
cana-726	80	8	a	a	DET
cana-726	80	9	cycle	cycle	NOUN
cana-726	80	10	bigness	bigness	ADV
cana-726	80	11	(	(	PUNCT
cana-726	80	12	γ(s	γ(s	PROPN
cana-726	80	13	)	)	PUNCT
cana-726	80	14	)	)	PUNCT
cana-726	81	1	≤	≤	NUM
cana-726	81	2	4	4	NUM
cana-726	81	3	.	.	PUNCT
cana-726	81	4	they	they	PRON
cana-726	81	5	additionally	additionally	ADV
cana-726	81	6	show	show	VERB
cana-726	81	7	that	that	SCONJ
cana-726	81	8	the	the	DET
cana-726	81	9	quantity	quantity	NOUN
cana-726	81	10	of	of	ADP
cana-726	81	11	insignificant	insignificant	ADJ
cana-726	81	12	beliefs	belief	NOUN
cana-726	81	13	of	of	ADP
cana-726	81	14	s	s	PRON
cana-726	81	15	gives	give	VERB
cana-726	81	16	a	a	DET
cana-726	81	17	lower	lower	ADV
cana-726	81	18	bound	bind	VERB
cana-726	81	19	to	to	ADP
cana-726	81	20	the	the	DET
cana-726	81	21	coterie	coterie	NOUN
cana-726	81	22	number	number	NOUN
cana-726	81	23	of	of	ADP
cana-726	81	24	s.	s.	PROPN
cana-726	81	25	in	in	ADP
cana-726	81	26	[	[	X
cana-726	81	27	26	26	NUM
cana-726	81	28	]	]	PUNCT
cana-726	81	29	authors	author	NOUN
cana-726	81	30	concentrated	concentrate	VERB
cana-726	81	31	on	on	ADP
cana-726	81	32	a	a	DET
cana-726	81	33	graph	graph	NOUN
cana-726	81	34	γ(s	γ(	NOUN
cana-726	81	35	)	)	PUNCT
cana-726	81	36	where	where	SCONJ
cana-726	81	37	the	the	DET
cana-726	81	38	vertex	vertex	NOUN
cana-726	81	39	set	set	NOUN
cana-726	81	40	of	of	ADP
cana-726	81	41	this	this	DET
cana-726	81	42	chart	chart	NOUN
cana-726	81	43	is	be	AUX
cana-726	81	44	z(s	z(s	NUM
cana-726	81	45	)	)	PUNCT
cana-726	81	46	∗	∗	NOUN
cana-726	81	47	and	and	CCONJ
cana-726	81	48	for	for	ADP
cana-726	81	49	particular	particular	ADJ
cana-726	81	50	components	component	NOUN
cana-726	81	51	x	x	NOUN
cana-726	81	52	,	,	PUNCT
cana-726	81	53	y	y	PROPN
cana-726	81	54	∈	∈	PROPN
cana-726	81	55	z(s	z(s	PROPN
cana-726	81	56	)	)	PUNCT
cana-726	81	57	∗	∗	NOUN
cana-726	81	58	,	,	PUNCT
cana-726	81	59	in	in	ADP
cana-726	81	60	the	the	DET
cana-726	81	61	event	event	NOUN
cana-726	81	62	that	that	PRON
cana-726	81	63	xsy	xsy	VERB
cana-726	81	64	=	=	SYM
cana-726	81	65	0	0	NUM
cana-726	81	66	,	,	PUNCT
cana-726	81	67	there	there	PRON
cana-726	81	68	is	be	VERB
cana-726	81	69	an	an	DET
cana-726	81	70	edge	edge	NOUN
cana-726	81	71	interfacing	interface	VERB
cana-726	81	72	x	x	X
cana-726	81	73	and	and	CCONJ
cana-726	81	74	y.	y.	PROPN
cana-726	81	75	note	note	VERB
cana-726	81	76	that	that	SCONJ
cana-726	81	77	γ(s	γ(	NOUN
cana-726	81	78	)	)	PUNCT
cana-726	81	79	is	be	AUX
cana-726	81	80	a	a	DET
cana-726	81	81	subgraph	subgraph	NOUN
cana-726	81	82	of	of	ADP
cana-726	81	83	γ(s	γ(	NOUN
cana-726	81	84	)	)	PUNCT
cana-726	81	85	.	.	PUNCT
cana-726	82	1	as	as	ADP
cana-726	82	2	of	of	ADP
cana-726	82	3	late	late	ADJ
cana-726	82	4	,	,	PUNCT
cana-726	82	5	several	several	ADJ
cana-726	82	6	authors	author	NOUN
cana-726	82	7	concentrated	concentrate	VERB
cana-726	82	8	on	on	ADP
cana-726	82	9	additional	additional	ADJ
cana-726	82	10	the	the	DET
cana-726	82	11	graph	graph	NOUN
cana-726	82	12	γ(s	γ(	NOUN
cana-726	82	13	)	)	PUNCT
cana-726	82	14	and	and	CCONJ
cana-726	82	15	its	its	PRON
cana-726	82	16	augmentation	augmentation	NOUN
cana-726	82	17	to	to	ADP
cana-726	82	18	a	a	DET
cana-726	82	19	simplicial	simplicial	ADJ
cana-726	82	20	complex	complex	NOUN
cana-726	82	21	,	,	PUNCT
cana-726	82	22	cf	cf	NOUN
cana-726	82	23	.	.	PUNCT
cana-726	83	1	[	[	X
cana-726	83	2	13	13	NUM
cana-726	83	3	]	]	PUNCT
cana-726	83	4	.	.	PUNCT
cana-726	84	1	obviously	obviously	ADV
cana-726	84	2	for	for	ADP
cana-726	84	3	any	any	DET
cana-726	84	4	superb	superb	ADJ
cana-726	84	5	ideal	ideal	NOUN
cana-726	84	6	p	p	NOUN
cana-726	84	7	in	in	ADP
cana-726	84	8	the	the	DET
cana-726	84	9	event	event	NOUN
cana-726	84	10	that	that	PRON
cana-726	84	11	x	x	PRON
cana-726	84	12	and	and	CCONJ
cana-726	84	13	y	y	PROPN
cana-726	84	14	are	be	AUX
cana-726	84	15	nearby	nearby	ADV
cana-726	84	16	in	in	ADP
cana-726	84	17	γ(s	γ(	NOUN
cana-726	84	18	)	)	PUNCT
cana-726	84	19	,	,	PUNCT
cana-726	84	20	x	x	PUNCT
cana-726	84	21	∈	∈	PROPN
cana-726	84	22	p	p	NOUN
cana-726	84	23	or	or	CCONJ
cana-726	84	24	y	y	PROPN
cana-726	84	25	∈	∈	PROPN
cana-726	85	1	p.	p.	NOUN
cana-726	86	1	so	so	ADV
cana-726	86	2	,	,	PUNCT
cana-726	86	3	for	for	ADP
cana-726	86	4	each	each	DET
cana-726	86	5	superb	superb	ADJ
cana-726	86	6	ideal	ideal	NOUN
cana-726	86	7	p	p	NOUN
cana-726	86	8	and	and	CCONJ
cana-726	86	9	each	each	DET
cana-726	86	10	edge	edge	NOUN
cana-726	86	11	e	e	NOUN
cana-726	86	12	,	,	PUNCT
cana-726	86	13	one	one	NUM
cana-726	86	14	of	of	ADP
cana-726	86	15	the	the	DET
cana-726	86	16	end	end	NOUN
cana-726	86	17	points	point	NOUN
cana-726	86	18	of	of	ADP
cana-726	86	19	e	e	NOUN
cana-726	86	20	has	have	VERB
cana-726	86	21	a	a	DET
cana-726	86	22	place	place	NOUN
cana-726	86	23	with	with	ADP
cana-726	86	24	p	p	NOUN
cana-726	86	25	,	,	PUNCT
cana-726	86	26	building	build	VERB
cana-726	86	27	graphs	graph	NOUN
cana-726	86	28	from	from	ADP
cana-726	86	29	commutative	commutative	ADJ
cana-726	86	30	rings	ring	NOUN
cana-726	86	31	was	be	AUX
cana-726	86	32	started	start	VERB
cana-726	86	33	by	by	ADP
cana-726	86	34	ivan	ivan	PROPN
cana-726	86	35	beck	beck	PROPN
cana-726	86	36	through	through	ADP
cana-726	86	37	his	his	PRON
cana-726	86	38	work	work	NOUN
cana-726	86	39	on	on	ADP
cana-726	86	40	zero	zero	NUM
cana-726	86	41	-	-	PUNCT
cana-726	86	42	divisor	divisor	NOUN
cana-726	86	43	charts	chart	NOUN
cana-726	86	44	and	and	CCONJ
cana-726	86	45	from	from	ADP
cana-726	86	46	that	that	DET
cana-726	86	47	point	point	NOUN
cana-726	86	48	a	a	DET
cana-726	86	49	few	few	ADJ
cana-726	86	50	graphs	graph	NOUN
cana-726	86	51	developments	development	NOUN
cana-726	86	52	were	be	AUX
cana-726	86	53	made	make	VERB
cana-726	86	54	by	by	ADP
cana-726	86	55	a	a	DET
cana-726	86	56	few	few	ADJ
cana-726	86	57	creators	creator	NOUN
cana-726	86	58	.	.	PUNCT
cana-726	87	1	through	through	ADP
cana-726	87	2	the	the	DET
cana-726	87	3	development	development	NOUN
cana-726	87	4	of	of	ADP
cana-726	87	5	charts	chart	NOUN
cana-726	87	6	from	from	ADP
cana-726	87	7	commutative	commutative	ADJ
cana-726	87	8	rings	ring	NOUN
cana-726	87	9	,	,	PUNCT
cana-726	87	10	exchange	exchange	NOUN
cana-726	87	11	between	between	ADP
cana-726	87	12	mathematical	mathematical	ADJ
cana-726	87	13	properties	property	NOUN
cana-726	87	14	of	of	ADP
cana-726	87	15	commutative	commutative	ADJ
cana-726	87	16	rings	ring	NOUN
cana-726	87	17	and	and	CCONJ
cana-726	87	18	graphs	graph	VERB
cana-726	87	19	hypothetical	hypothetical	ADJ
cana-726	87	20	properties	property	NOUN
cana-726	87	21	of	of	ADP
cana-726	87	22	determined	determined	ADJ
cana-726	87	23	charts	chart	NOUN
cana-726	87	24	are	be	AUX
cana-726	87	25	contemplated	contemplate	VERB
cana-726	87	26	.	.	PUNCT
cana-726	88	1	a	a	DET
cana-726	88	2	portion	portion	NOUN
cana-726	88	3	of	of	ADP
cana-726	88	4	the	the	DET
cana-726	88	5	charts	chart	NOUN
cana-726	88	6	characterized	characterize	VERB
cana-726	88	7	out	out	ADP
cana-726	88	8	of	of	ADP
cana-726	88	9	gatherings	gathering	NOUN
cana-726	88	10	are	be	AUX
cana-726	88	11	cayley	cayley	ADJ
cana-726	88	12	graphs	graph	NOUN
cana-726	88	13	from	from	ADP
cana-726	88	14	bunches	bunche	NOUN
cana-726	88	15	[	[	X
cana-726	88	16	25	25	NUM
cana-726	88	17	]	]	PUNCT
cana-726	88	18	,	,	PUNCT
cana-726	88	19	noncommutating	noncommutate	VERB
cana-726	88	20	chart	chart	NOUN
cana-726	88	21	of	of	ADP
cana-726	88	22	a	a	DET
cana-726	88	23	gathering	gathering	NOUN
cana-726	88	24	[	[	X
cana-726	88	25	2	2	NUM
cana-726	88	26	]	]	PUNCT
cana-726	88	27	,	,	PUNCT
cana-726	88	28	power	power	NOUN
cana-726	88	29	chart	chart	NOUN
cana-726	88	30	of	of	ADP
cana-726	88	31	a	a	DET
cana-726	88	32	limited	limited	ADJ
cana-726	88	33	gathering	gathering	NOUN
cana-726	88	34	[	[	X
cana-726	88	35	29	29	NUM
cana-726	88	36	]	]	PUNCT
cana-726	88	37	.	.	PUNCT
cana-726	89	1	a	a	DET
cana-726	89	2	chart	chart	NOUN
cana-726	89	3	is	be	AUX
cana-726	89	4	characterized	characterize	VERB
cana-726	89	5	out	out	ADP
cana-726	89	6	of	of	ADP
cana-726	89	7	non	non	ADJ
cana-726	89	8	no	no	DET
cana-726	89	9	divisors	divisor	NOUN
cana-726	89	10	of	of	ADP
cana-726	89	11	a	a	DET
cana-726	89	12	ring	ring	NOUN
cana-726	89	13	and	and	CCONJ
cana-726	89	14	is	be	AUX
cana-726	89	15	called	call	VERB
cana-726	89	16	zero	zero	NUM
cana-726	89	17	-	-	PUNCT
cana-726	89	18	divisor	divisor	NOUN
cana-726	89	19	graphs	graph	NOUN
cana-726	89	20	of	of	ADP
cana-726	89	21	a	a	DET
cana-726	89	22	ring	ring	NOUN
cana-726	89	23	[	[	X
cana-726	89	24	12	12	NUM
cana-726	89	25	]	]	PUNCT
cana-726	89	26	.	.	PUNCT
cana-726	90	1	intriguing	intriguing	ADJ
cana-726	90	2	varieties	variety	NOUN
cana-726	90	3	are	be	AUX
cana-726	90	4	likewise	likewise	ADV
cana-726	90	5	characterized	characterize	VERB
cana-726	90	6	like	like	ADP
cana-726	90	7	all	all	PRON
cana-726	90	8	out	out	ADP
cana-726	90	9	graphs	graph	NOUN
cana-726	90	10	[	[	X
cana-726	90	11	8	8	NUM
cana-726	90	12	]	]	PUNCT
cana-726	90	13	,	,	PUNCT
cana-726	90	14	unit	unit	NOUN
cana-726	90	15	charts	chart	NOUN
cana-726	90	16	[	[	X
cana-726	90	17	15	15	NUM
cana-726	90	18	]	]	PUNCT
cana-726	90	19	and	and	CCONJ
cana-726	90	20	co	co	ADJ
cana-726	90	21	maximal	maximal	ADJ
cana-726	90	22	charts	chart	NOUN
cana-726	91	1	[	[	X
cana-726	91	2	33	33	NUM
cana-726	91	3	]	]	PUNCT
cana-726	91	4	related	relate	VERB
cana-726	91	5	with	with	ADP
cana-726	91	6	communications	communication	NOUN
cana-726	91	7	on	on	ADP
cana-726	91	8	applied	apply	VERB
cana-726	91	9	nonlinear	nonlinear	ADJ
cana-726	91	10	analysis	analysis	NOUN
cana-726	91	11	issn	issn	NOUN
cana-726	91	12	:	:	PUNCT
cana-726	91	13	1074	1074	NUM
cana-726	91	14	-	-	PUNCT
cana-726	91	15	133x	133x	NUM
cana-726	91	16	vol	vol	NOUN
cana-726	91	17	31	31	NUM
cana-726	91	18	no	no	NOUN
cana-726	91	19	.	.	PUNCT
cana-726	92	1	3s	3s	NUM
cana-726	92	2	(	(	PUNCT
cana-726	92	3	2024	2024	NUM
cana-726	92	4	)	)	PUNCT
cana-726	92	5	4	4	NUM
cana-726	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	92	7	rings	ring	NOUN
cana-726	92	8	.	.	PUNCT
cana-726	93	1	additionally	additionally	ADV
cana-726	93	2	,	,	PUNCT
cana-726	93	3	charts	chart	NOUN
cana-726	93	4	are	be	AUX
cana-726	93	5	characterized	characterize	VERB
cana-726	93	6	out	out	ADP
cana-726	93	7	of	of	ADP
cana-726	93	8	standards	standard	NOUN
cana-726	93	9	of	of	ADP
cana-726	93	10	a	a	DET
cana-726	93	11	ring	ring	NOUN
cana-726	93	12	,	,	PUNCT
cana-726	93	13	to	to	PART
cana-726	93	14	be	be	AUX
cana-726	93	15	specific	specific	ADJ
cana-726	93	16	obliterating	obliterate	VERB
cana-726	93	17	ideal	ideal	ADJ
cana-726	93	18	graphs	graph	NOUN
cana-726	93	19	of	of	ADP
cana-726	93	20	a	a	DET
cana-726	93	21	ring	ring	NOUN
cana-726	93	22	[	[	X
cana-726	93	23	23	23	NUM
cana-726	93	24	]	]	PUNCT
cana-726	93	25	,	,	PUNCT
cana-726	93	26	convergence	convergence	NOUN
cana-726	93	27	chart	chart	NOUN
cana-726	93	28	of	of	ADP
cana-726	93	29	beliefs	belief	NOUN
cana-726	93	30	of	of	ADP
cana-726	93	31	rings	ring	NOUN
cana-726	93	32	[	[	X
cana-726	93	33	30	30	NUM
cana-726	93	34	,	,	PUNCT
cana-726	93	35	31	31	NUM
cana-726	93	36	]	]	PUNCT
cana-726	93	37	and	and	CCONJ
cana-726	93	38	so	so	ADV
cana-726	93	39	forth	forth	ADV
cana-726	93	40	.	.	PUNCT
cana-726	94	1	connecting	connect	VERB
cana-726	94	2	a	a	DET
cana-726	94	3	graphs	graph	NOUN
cana-726	94	4	with	with	ADP
cana-726	94	5	zero	zero	NUM
cana-726	94	6	-	-	PUNCT
cana-726	94	7	divisors	divisor	NOUN
cana-726	94	8	of	of	ADP
cana-726	94	9	a	a	DET
cana-726	94	10	commutative	commutative	ADJ
cana-726	94	11	ring	ring	NOUN
cana-726	94	12	was	be	AUX
cana-726	94	13	presented	present	VERB
cana-726	94	14	[	[	X
cana-726	94	15	24	24	NUM
cana-726	94	16	]	]	PUNCT
cana-726	94	17	in	in	ADP
cana-726	94	18	1988	1988	NUM
cana-726	94	19	,	,	PUNCT
cana-726	94	20	where	where	SCONJ
cana-726	94	21	the	the	DET
cana-726	94	22	creator	creator	NOUN
cana-726	94	23	discussed	discuss	VERB
cana-726	94	24	shading	shading	NOUN
cana-726	94	25	of	of	ADP
cana-726	94	26	such	such	ADJ
cana-726	94	27	graphs	graph	NOUN
cana-726	94	28	.	.	PUNCT
cana-726	95	1	subsequent	subsequent	ADJ
cana-726	95	2	to	to	ADP
cana-726	95	3	presenting	present	VERB
cana-726	95	4	zero	zero	NUM
cana-726	95	5	-	-	PUNCT
cana-726	95	6	divisor	divisor	NOUN
cana-726	95	7	graphs	graph	NOUN
cana-726	95	8	,	,	PUNCT
cana-726	95	9	i.	i.	PROPN
cana-726	95	10	beck	beck	PROPN
cana-726	95	11	made	make	VERB
cana-726	95	12	a	a	DET
cana-726	95	13	guess	guess	NOUN
cana-726	95	14	that	that	SCONJ
cana-726	95	15	the	the	DET
cana-726	95	16	faction	faction	NOUN
cana-726	95	17	number	number	NOUN
cana-726	95	18	and	and	CCONJ
cana-726	95	19	chromatic	chromatic	ADJ
cana-726	95	20	number	number	NOUN
cana-726	95	21	of	of	ADP
cana-726	95	22	the	the	DET
cana-726	95	23	zero	zero	NUM
cana-726	95	24	-	-	PUNCT
cana-726	95	25	divisor	divisor	NOUN
cana-726	95	26	graphs	graph	NOUN
cana-726	95	27	are	be	AUX
cana-726	95	28	equivalent	equivalent	ADJ
cana-726	95	29	.	.	PUNCT
cana-726	96	1	in	in	ADP
cana-726	96	2	1993	1993	NUM
cana-726	96	3	,	,	PUNCT
cana-726	96	4	few	few	ADJ
cana-726	96	5	authors	author	NOUN
cana-726	96	6	settled	settle	VERB
cana-726	96	7	beck	beck	NOUN
cana-726	96	8	's	's	PART
cana-726	96	9	guess	guess	NOUN
cana-726	96	10	in	in	ADP
cana-726	96	11	negative	negative	ADJ
cana-726	96	12	by	by	ADP
cana-726	96	13	giving	give	VERB
cana-726	96	14	a	a	DET
cana-726	96	15	counter	counter	NOUN
cana-726	96	16	model	model	NOUN
cana-726	96	17	[	[	X
cana-726	96	18	11	11	NUM
cana-726	96	19	]	]	PUNCT
cana-726	96	20	.	.	PUNCT
cana-726	97	1	additionally	additionally	ADV
cana-726	97	2	,	,	PUNCT
cana-726	97	3	they	they	PRON
cana-726	97	4	have	have	AUX
cana-726	97	5	explored	explore	VERB
cana-726	97	6	the	the	DET
cana-726	97	7	interchange	interchange	NOUN
cana-726	97	8	between	between	ADP
cana-726	97	9	the	the	DET
cana-726	97	10	ring	ring	NOUN
cana-726	97	11	hypothetical	hypothetical	ADJ
cana-726	97	12	properties	property	NOUN
cana-726	97	13	of	of	ADP
cana-726	97	14	a	a	DET
cana-726	97	15	commutative	commutative	ADJ
cana-726	97	16	ring	ring	NOUN
cana-726	97	17	and	and	CCONJ
cana-726	97	18	graphs	graph	VERB
cana-726	97	19	hypothetical	hypothetical	ADJ
cana-726	97	20	properties	property	NOUN
cana-726	97	21	of	of	ADP
cana-726	97	22	the	the	DET
cana-726	97	23	zero	zero	NUM
cana-726	97	24	-	-	PUNCT
cana-726	97	25	divisor	divisor	NOUN
cana-726	97	26	chart	chart	NOUN
cana-726	97	27	.	.	PUNCT
cana-726	98	1	the	the	DET
cana-726	98	2	definition	definition	NOUN
cana-726	98	3	alongside	alongside	ADP
cana-726	98	4	name	name	NOUN
cana-726	98	5	for	for	ADP
cana-726	98	6	zero	zero	NUM
cana-726	98	7	-	-	PUNCT
cana-726	98	8	divisor	divisor	NOUN
cana-726	98	9	chart	chart	NOUN
cana-726	98	10	(	(	PUNCT
cana-726	98	11	r	r	NOUN
cana-726	98	12	)	)	PUNCT
cana-726	98	13	was	be	AUX
cana-726	98	14	first	first	ADV
cana-726	98	15	presented	present	VERB
cana-726	98	16	in	in	ADP
cana-726	98	17	1999	1999	NUM
cana-726	98	18	,	,	PUNCT
cana-726	98	19	subsequent	subsequent	ADJ
cana-726	98	20	to	to	ADP
cana-726	98	21	altering	alter	VERB
cana-726	98	22	the	the	DET
cana-726	98	23	meaning	meaning	NOUN
cana-726	98	24	of	of	ADP
cana-726	98	25	d.	d.	PROPN
cana-726	98	26	d.	d.	PROPN
cana-726	98	27	anderson	anderson	PROPN
cana-726	99	1	[	[	X
cana-726	99	2	11	11	NUM
cana-726	99	3	,	,	PUNCT
cana-726	99	4	12	12	NUM
cana-726	99	5	]	]	PUNCT
cana-726	99	6	.	.	PUNCT
cana-726	100	1	example	example	NOUN
cana-726	100	2	of	of	ADP
cana-726	100	3	a	a	DET
cana-726	100	4	zero	zero	NUM
cana-726	100	5	-	-	PUNCT
cana-726	100	6	divisor	divisor	NOUN
cana-726	100	7	graph	graph	NOUN
cana-726	100	8	for	for	ADP
cana-726	100	9	𝑅	𝑅	PROPN
cana-726	100	10	=	=	PROPN
cana-726	100	11	𝑍2𝑋	𝑍2𝑋	PROPN
cana-726	100	12	𝑍2(𝑥	𝑍2(𝑥	X
cana-726	100	13	)	)	PUNCT
cana-726	101	1	<	<	AUX
cana-726	101	2	𝑥2	𝑥2	X
cana-726	101	3	>	>	X
cana-726	101	4	is	be	AUX
cana-726	101	5	shown	show	VERB
cana-726	101	6	in	in	ADP
cana-726	101	7	fig	fig	NOUN
cana-726	101	8	1.1	1.1	NUM
cana-726	101	9	figure	figure	NOUN
cana-726	101	10	1.1	1.1	NUM
cana-726	101	11	:	:	PUNCT
cana-726	101	12	𝑅	𝑅	PROPN
cana-726	101	13	=	=	PUNCT
cana-726	101	14	𝑍2𝑋	𝑍2𝑋	PROPN
cana-726	101	15	𝑍2(𝑥	𝑍2(𝑥	X
cana-726	101	16	)	)	PUNCT
cana-726	102	1	<	<	X
cana-726	102	2	𝑥2	𝑥2	X
cana-726	102	3	>	>	X
cana-726	102	4	2	2	NUM
cana-726	102	5	.	.	PUNCT
cana-726	102	6	objectives	objective	NOUN
cana-726	102	7	definition	definition	NOUN
cana-726	102	8	1.1	1.1	NUM
cana-726	102	9	.	.	PUNCT
cana-726	103	1	[	[	X
cana-726	103	2	2	2	X
cana-726	103	3	]	]	PUNCT
cana-726	103	4	a	a	DET
cana-726	103	5	ring	ring	NOUN
cana-726	103	6	(	(	PUNCT
cana-726	103	7	r	r	NOUN
cana-726	103	8	,	,	PUNCT
cana-726	103	9	+	+	ADJ
cana-726	103	10	,	,	PUNCT
cana-726	103	11	·	·	PUNCT
cana-726	103	12	)	)	PUNCT
cana-726	103	13	is	be	AUX
cana-726	103	14	a	a	DET
cana-726	103	15	nonempty	nonempty	ADJ
cana-726	103	16	set	set	VERB
cana-726	103	17	r	r	NOUN
cana-726	103	18	together	together	ADV
cana-726	103	19	with	with	ADP
cana-726	103	20	binary	binary	ADJ
cana-726	103	21	operations	operation	NOUN
cana-726	103	22	‘	'	PUNCT
cana-726	103	23	+	+	ADJ
cana-726	103	24	’	'	PUNCT
cana-726	103	25	and	and	CCONJ
cana-726	103	26	‘	'	PUNCT
cana-726	103	27	·	·	PUNCT
cana-726	103	28	’	'	PUNCT
cana-726	103	29	defined	define	VERB
cana-726	103	30	on	on	ADP
cana-726	103	31	r	r	NOUN
cana-726	103	32	,	,	PUNCT
cana-726	103	33	which	which	PRON
cana-726	103	34	satisfy	satisfy	VERB
cana-726	103	35	the	the	DET
cana-726	103	36	following	follow	VERB
cana-726	103	37	conditions	condition	NOUN
cana-726	103	38	:	:	PUNCT
cana-726	103	39	(	(	PUNCT
cana-726	103	40	i	i	NOUN
cana-726	103	41	)	)	PUNCT
cana-726	103	42	(	(	PUNCT
cana-726	103	43	r	r	NOUN
cana-726	103	44	,	,	PUNCT
cana-726	103	45	+	+	NOUN
cana-726	103	46	)	)	PUNCT
cana-726	103	47	is	be	AUX
cana-726	103	48	an	an	DET
cana-726	103	49	abelian	abelian	ADJ
cana-726	103	50	group	group	NOUN
cana-726	103	51	(	(	PUNCT
cana-726	103	52	ii	ii	PROPN
cana-726	103	53	)	)	PUNCT
cana-726	103	54	a	a	DET
cana-726	103	55	·	·	PUNCT
cana-726	103	56	(	(	PUNCT
cana-726	103	57	b	b	X
cana-726	103	58	·	·	PUNCT
cana-726	103	59	c	c	X
cana-726	103	60	)	)	PUNCT
cana-726	103	61	=	=	SYM
cana-726	103	62	(	(	PUNCT
cana-726	103	63	a	a	DET
cana-726	103	64	·	·	SYM
cana-726	103	65	b	b	NOUN
cana-726	103	66	)	)	PUNCT
cana-726	103	67	·	·	PUNCT
cana-726	104	1	c	c	X
cana-726	104	2	,	,	PUNCT
cana-726	104	3	∀	∀	X
cana-726	104	4	a	a	PRON
cana-726	104	5	,	,	PUNCT
cana-726	104	6	b	b	NOUN
cana-726	104	7	,	,	PUNCT
cana-726	104	8	c	c	PROPN
cana-726	104	9	∈	∈	PROPN
cana-726	104	10	r	r	NOUN
cana-726	104	11	(	(	PUNCT
cana-726	104	12	iii	iii	NOUN
cana-726	104	13	)	)	PUNCT
cana-726	104	14	a	a	PRON
cana-726	104	15	·	·	PUNCT
cana-726	104	16	(	(	PUNCT
cana-726	104	17	b	b	X
cana-726	104	18	+	+	CCONJ
cana-726	104	19	c	c	X
cana-726	104	20	)	)	PUNCT
cana-726	104	21	=	=	SYM
cana-726	105	1	a	a	DET
cana-726	105	2	·	·	PUNCT
cana-726	105	3	b	b	NOUN
cana-726	105	4	+	+	CCONJ
cana-726	105	5	a	a	DET
cana-726	105	6	·	·	PUNCT
cana-726	105	7	c	c	X
cana-726	105	8	,	,	PUNCT
cana-726	105	9	∀	∀	X
cana-726	105	10	a	a	PRON
cana-726	105	11	,	,	PUNCT
cana-726	105	12	b	b	NOUN
cana-726	105	13	,	,	PUNCT
cana-726	105	14	c	c	PROPN
cana-726	105	15	∈	∈	PROPN
cana-726	105	16	r	r	NOUN
cana-726	105	17	(	(	PUNCT
cana-726	105	18	iv	iv	X
cana-726	105	19	)	)	PUNCT
cana-726	105	20	(	(	PUNCT
cana-726	105	21	a	a	DET
cana-726	105	22	+	+	NOUN
cana-726	105	23	b	b	NOUN
cana-726	105	24	)	)	PUNCT
cana-726	105	25	·	·	PUNCT
cana-726	106	1	c	c	X
cana-726	106	2	=	=	PUNCT
cana-726	107	1	a	a	DET
cana-726	107	2	·	·	PUNCT
cana-726	107	3	c	c	X
cana-726	107	4	+	+	CCONJ
cana-726	107	5	b	b	PROPN
cana-726	107	6	·	·	SYM
cana-726	107	7	c	c	X
cana-726	107	8	,	,	PUNCT
cana-726	107	9	∀	∀	X
cana-726	107	10	a	a	DET
cana-726	107	11	,	,	PUNCT
cana-726	107	12	b	b	NOUN
cana-726	107	13	,	,	PUNCT
cana-726	107	14	c	c	PROPN
cana-726	107	15	∈	∈	PROPN
cana-726	107	16	r.	r.	PROPN
cana-726	107	17	definition	definition	NOUN
cana-726	107	18	1.2	1.2	NUM
cana-726	107	19	.	.	PUNCT
cana-726	108	1	[	[	X
cana-726	108	2	9	9	NUM
cana-726	108	3	]	]	X
cana-726	108	4	a	a	DET
cana-726	108	5	ring	ring	NOUN
cana-726	108	6	r	r	NOUN
cana-726	108	7	is	be	AUX
cana-726	108	8	called	call	VERB
cana-726	108	9	commutative	commutative	ADJ
cana-726	108	10	if	if	SCONJ
cana-726	108	11	for	for	ADP
cana-726	108	12	every	every	DET
cana-726	108	13	a	a	PROPN
cana-726	108	14	,	,	PUNCT
cana-726	108	15	b	b	X
cana-726	108	16	∈	∈	PROPN
cana-726	108	17	r	r	NOUN
cana-726	108	18	,	,	PUNCT
cana-726	108	19	∋	∋	NOUN
cana-726	108	20	a	a	PRON
cana-726	108	21	·	·	PUNCT
cana-726	108	22	b	b	X
cana-726	108	23	=	=	SYM
cana-726	108	24	b	b	PROPN
cana-726	108	25	·	·	PUNCT
cana-726	108	26	a.	a.	NOUN
cana-726	108	27	definition	definition	NOUN
cana-726	108	28	1.3	1.3	NUM
cana-726	108	29	.	.	PUNCT
cana-726	109	1	[	[	X
cana-726	109	2	10	10	NUM
cana-726	109	3	]	]	PUNCT
cana-726	109	4	let	let	VERB
cana-726	109	5	r	r	PRON
cana-726	109	6	be	be	AUX
cana-726	109	7	a	a	DET
cana-726	109	8	ring	ring	NOUN
cana-726	109	9	.	.	PUNCT
cana-726	110	1	an	an	DET
cana-726	110	2	element	element	NOUN
cana-726	110	3	e	e	X
cana-726	110	4	∈	∈	NOUN
cana-726	110	5	r	r	NOUN
cana-726	110	6	is	be	AUX
cana-726	110	7	called	call	VERB
cana-726	110	8	an	an	DET
cana-726	110	9	identity	identity	NOUN
cana-726	110	10	element	element	NOUN
cana-726	110	11	if	if	SCONJ
cana-726	110	12	ea	ea	X
cana-726	110	13	=	=	SYM
cana-726	110	14	ae	ae	PROPN
cana-726	110	15	=	=	PUNCT
cana-726	110	16	a	a	DET
cana-726	110	17	∀	∀	NOUN
cana-726	110	18	a	a	DET
cana-726	110	19	∈	∈	PROPN
cana-726	110	20	r.	r.	NOUN
cana-726	110	21	the	the	DET
cana-726	110	22	identity	identity	NOUN
cana-726	110	23	element	element	NOUN
cana-726	110	24	of	of	ADP
cana-726	110	25	a	a	DET
cana-726	110	26	ring	ring	NOUN
cana-726	110	27	r	r	NOUN
cana-726	110	28	is	be	AUX
cana-726	110	29	denoted	denote	VERB
cana-726	110	30	by	by	ADP
cana-726	110	31	‘	'	PUNCT
cana-726	110	32	1	1	NUM
cana-726	110	33	’	'	PUNCT
cana-726	110	34	.	.	PUNCT
cana-726	111	1	definition	definition	NOUN
cana-726	111	2	1.4	1.4	NUM
cana-726	111	3	.	.	PUNCT
cana-726	112	1	[	[	X
cana-726	112	2	16	16	NUM
cana-726	112	3	]	]	PUNCT
cana-726	112	4	let	let	VERB
cana-726	112	5	r	r	PRON
cana-726	112	6	be	be	AUX
cana-726	112	7	a	a	DET
cana-726	112	8	ring	ring	NOUN
cana-726	112	9	with	with	ADP
cana-726	112	10	identity	identity	NOUN
cana-726	112	11	.	.	PUNCT
cana-726	113	1	an	an	DET
cana-726	113	2	element	element	NOUN
cana-726	113	3	u	u	NOUN
cana-726	113	4	∈	∈	NOUN
cana-726	113	5	r	r	NOUN
cana-726	113	6	is	be	AUX
cana-726	113	7	called	call	VERB
cana-726	113	8	a	a	DET
cana-726	113	9	unit	unit	NOUN
cana-726	113	10	element	element	NOUN
cana-726	113	11	if	if	SCONJ
cana-726	113	12	there	there	PRON
cana-726	113	13	exists	exist	VERB
cana-726	113	14	v	v	ADP
cana-726	113	15	∈	∈	NOUN
cana-726	113	16	r	r	NOUN
cana-726	113	17	such	such	ADJ
cana-726	113	18	that	that	DET
cana-726	113	19	uv	uv	NOUN
cana-726	113	20	=	=	SYM
cana-726	113	21	1	1	NUM
cana-726	113	22	=	=	SYM
cana-726	113	23	vu	vu	NOUN
cana-726	113	24	and	and	CCONJ
cana-726	113	25	the	the	DET
cana-726	113	26	inverse	inverse	NOUN
cana-726	113	27	of	of	ADP
cana-726	113	28	u	u	NOUN
cana-726	113	29	is	be	AUX
cana-726	113	30	often	often	ADV
cana-726	113	31	denoted	denote	VERB
cana-726	113	32	by	by	ADP
cana-726	113	33	u−1	u−1	PROPN
cana-726	113	34	.	.	PUNCT
cana-726	114	1	the	the	DET
cana-726	114	2	collection	collection	NOUN
cana-726	114	3	of	of	ADP
cana-726	114	4	all	all	DET
cana-726	114	5	units	unit	NOUN
cana-726	114	6	in	in	ADP
cana-726	114	7	r	r	NOUN
cana-726	114	8	is	be	AUX
cana-726	114	9	denoted	denote	VERB
cana-726	114	10	by	by	ADP
cana-726	114	11	u(r	u(r	NOUN
cana-726	114	12	)	)	PUNCT
cana-726	114	13	or	or	CCONJ
cana-726	114	14	r×.	r×.	SCONJ
cana-726	114	15	it	it	PRON
cana-726	114	16	is	be	AUX
cana-726	114	17	easy	easy	ADJ
cana-726	114	18	to	to	PART
cana-726	114	19	check	check	VERB
cana-726	114	20	that	that	DET
cana-726	114	21	u(r	u(r	NOUN
cana-726	114	22	)	)	PUNCT
cana-726	114	23	is	be	AUX
cana-726	114	24	a	a	DET
cana-726	114	25	group	group	NOUN
cana-726	114	26	under	under	ADP
cana-726	114	27	multiplication	multiplication	NOUN
cana-726	114	28	and	and	CCONJ
cana-726	114	29	is	be	AUX
cana-726	114	30	called	call	VERB
cana-726	114	31	multiplicative	multiplicative	ADJ
cana-726	114	32	group	group	NOUN
cana-726	114	33	of	of	ADP
cana-726	114	34	r.	r.	PROPN
cana-726	114	35	definition	definition	NOUN
cana-726	114	36	1.5	1.5	NUM
cana-726	114	37	.	.	PUNCT
cana-726	115	1	[	[	X
cana-726	115	2	17	17	NUM
cana-726	115	3	]	]	PUNCT
cana-726	115	4	a	a	DET
cana-726	115	5	ring	ring	NOUN
cana-726	115	6	r	r	NOUN
cana-726	115	7	with	with	ADP
cana-726	115	8	identity	identity	NOUN
cana-726	115	9	is	be	AUX
cana-726	115	10	called	call	VERB
cana-726	115	11	a	a	DET
cana-726	115	12	division	division	NOUN
cana-726	115	13	ring	ring	NOUN
cana-726	115	14	if	if	SCONJ
cana-726	115	15	every	every	DET
cana-726	115	16	nonzero	nonzero	NOUN
cana-726	115	17	element	element	NOUN
cana-726	115	18	of	of	ADP
cana-726	115	19	r	r	NOUN
cana-726	115	20	is	be	AUX
cana-726	115	21	a	a	DET
cana-726	115	22	unit	unit	NOUN
cana-726	115	23	.	.	PUNCT
cana-726	116	1	a	a	DET
cana-726	116	2	commutative	commutative	ADJ
cana-726	116	3	division	division	NOUN
cana-726	116	4	ring	ring	NOUN
cana-726	116	5	r	r	NOUN
cana-726	116	6	is	be	AUX
cana-726	116	7	called	call	VERB
cana-726	116	8	a	a	DET
cana-726	116	9	field	field	NOUN
cana-726	116	10	.	.	PUNCT
cana-726	117	1	definition	definition	NOUN
cana-726	117	2	1.6	1.6	NUM
cana-726	117	3	.	.	PUNCT
cana-726	118	1	[	[	X
cana-726	118	2	18	18	NUM
cana-726	118	3	]	]	PUNCT
cana-726	118	4	an	an	DET
cana-726	118	5	element	element	NOUN
cana-726	118	6	x	x	SYM
cana-726	118	7	∈	∈	NOUN
cana-726	118	8	r	r	NOUN
cana-726	118	9	is	be	AUX
cana-726	118	10	said	say	VERB
cana-726	118	11	to	to	PART
cana-726	118	12	be	be	AUX
cana-726	118	13	a	a	DET
cana-726	118	14	zero	zero	NUM
cana-726	118	15	-	-	PUNCT
cana-726	118	16	divisor	divisor	NOUN
cana-726	118	17	if	if	SCONJ
cana-726	118	18	there	there	PRON
cana-726	118	19	exists	exist	VERB
cana-726	118	20	0	0	NUM
cana-726	118	21	≠	≠	PROPN
cana-726	118	22	y	y	PROPN
cana-726	118	23	∈	∈	PROPN
cana-726	118	24	r	r	NOUN
cana-726	118	25	∋	∋	NOUN
cana-726	118	26	𝑥𝑦	𝑥𝑦	PROPN
cana-726	118	27	=	=	SYM
cana-726	118	28	0	0	NUM
cana-726	118	29	where	where	SCONJ
cana-726	118	30	0	0	NUM
cana-726	118	31	is	be	AUX
cana-726	118	32	the	the	DET
cana-726	118	33	additive	additive	ADJ
cana-726	118	34	identity	identity	NOUN
cana-726	118	35	.	.	PUNCT
cana-726	119	1	the	the	DET
cana-726	119	2	set	set	NOUN
cana-726	119	3	of	of	ADP
cana-726	119	4	all	all	DET
cana-726	119	5	zero	zero	NUM
cana-726	119	6	-	-	PUNCT
cana-726	119	7	divisors	divisor	NOUN
cana-726	119	8	in	in	ADP
cana-726	119	9	r	r	NOUN
cana-726	119	10	is	be	AUX
cana-726	119	11	denoted	denote	VERB
cana-726	119	12	by	by	ADP
cana-726	119	13	z(r	z(r	NOUN
cana-726	119	14	)	)	PUNCT
cana-726	119	15	.	.	PUNCT
cana-726	120	1	communications	communication	NOUN
cana-726	120	2	on	on	ADP
cana-726	120	3	applied	apply	VERB
cana-726	120	4	nonlinear	nonlinear	ADJ
cana-726	120	5	analysis	analysis	NOUN
cana-726	120	6	issn	issn	NOUN
cana-726	120	7	:	:	PUNCT
cana-726	120	8	1074	1074	NUM
cana-726	120	9	-	-	PUNCT
cana-726	120	10	133x	133x	NUM
cana-726	120	11	vol	vol	NOUN
cana-726	120	12	31	31	NUM
cana-726	120	13	no	no	NOUN
cana-726	120	14	.	.	PUNCT
cana-726	121	1	3s	3s	NUM
cana-726	121	2	(	(	PUNCT
cana-726	121	3	2024	2024	NUM
cana-726	121	4	)	)	PUNCT
cana-726	121	5	5	5	NUM
cana-726	121	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	121	7	definition	definition	NOUN
cana-726	121	8	1.7	1.7	NUM
cana-726	121	9	.	.	PUNCT
cana-726	122	1	[	[	X
cana-726	122	2	19	19	NUM
cana-726	122	3	]	]	X
cana-726	122	4	an	an	DET
cana-726	122	5	ideal	ideal	NOUN
cana-726	122	6	p	p	NOUN
cana-726	122	7	of	of	ADP
cana-726	122	8	a	a	DET
cana-726	122	9	ring	ring	NOUN
cana-726	122	10	r	r	NOUN
cana-726	122	11	is	be	AUX
cana-726	122	12	called	call	VERB
cana-726	122	13	a	a	DET
cana-726	122	14	prime	prime	ADJ
cana-726	122	15	ideal	ideal	NOUN
cana-726	122	16	if	if	SCONJ
cana-726	122	17	𝑃	𝑃	PROPN
cana-726	122	18	≠	≠	PROPN
cana-726	122	19	𝑅	𝑅	PROPN
cana-726	122	20	and	and	CCONJ
cana-726	122	21	∀	∀	NOUN
cana-726	122	22	a	a	PRON
cana-726	122	23	,	,	PUNCT
cana-726	122	24	b	b	X
cana-726	122	25	∈	∈	PROPN
cana-726	122	26	r	r	NOUN
cana-726	122	27	,	,	PUNCT
cana-726	122	28	ab	ab	PROPN
cana-726	122	29	∈	∈	PROPN
cana-726	122	30	p	p	PROPN
cana-726	122	31	implies	imply	VERB
cana-726	122	32	a	a	DET
cana-726	122	33	∈	∈	PROPN
cana-726	122	34	p	p	NOUN
cana-726	122	35	or	or	CCONJ
cana-726	122	36	b	b	PROPN
cana-726	122	37	∈	∈	PROPN
cana-726	122	38	p.	p.	NOUN
cana-726	122	39	definition	definition	NOUN
cana-726	122	40	1.8	1.8	NUM
cana-726	122	41	.	.	PUNCT
cana-726	123	1	[	[	X
cana-726	123	2	20	20	NUM
cana-726	123	3	]	]	PUNCT
cana-726	123	4	a	a	DET
cana-726	123	5	commutative	commutative	ADJ
cana-726	123	6	ring	ring	NOUN
cana-726	123	7	r	r	NOUN
cana-726	123	8	is	be	AUX
cana-726	123	9	called	call	VERB
cana-726	123	10	an	an	DET
cana-726	123	11	integral	integral	ADJ
cana-726	123	12	domain	domain	NOUN
cana-726	123	13	if	if	SCONJ
cana-726	123	14	r	r	NOUN
cana-726	123	15	has	have	VERB
cana-726	123	16	no	no	DET
cana-726	123	17	non	non	ADJ
cana-726	123	18	-	-	ADJ
cana-726	123	19	zero	zero	NUM
cana-726	123	20	zerodivisors	zerodivisor	NOUN
cana-726	123	21	.	.	PUNCT
cana-726	124	1	definition	definition	NOUN
cana-726	124	2	1.9	1.9	NUM
cana-726	124	3	.	.	PUNCT
cana-726	125	1	[	[	X
cana-726	125	2	21	21	NUM
cana-726	125	3	]	]	PUNCT
cana-726	125	4	let	let	VERB
cana-726	125	5	r	r	PRON
cana-726	125	6	be	be	AUX
cana-726	125	7	a	a	DET
cana-726	125	8	ring	ring	NOUN
cana-726	125	9	.	.	PUNCT
cana-726	126	1	the	the	DET
cana-726	126	2	characteristic	characteristic	NOUN
cana-726	126	3	of	of	ADP
cana-726	126	4	r	r	NOUN
cana-726	126	5	is	be	AUX
cana-726	126	6	the	the	DET
cana-726	126	7	least	least	ADV
cana-726	126	8	positive	positive	ADJ
cana-726	126	9	integer	integer	NOUN
cana-726	126	10	n	n	CCONJ
cana-726	126	11	such	such	ADJ
cana-726	126	12	that	that	SCONJ
cana-726	126	13	na	na	ADP
cana-726	126	14	=	=	SYM
cana-726	126	15	0	0	NUM
cana-726	126	16	∀	∀	NOUN
cana-726	126	17	a	a	DET
cana-726	126	18	∈	∈	PROPN
cana-726	126	19	r.	r.	NOUN
cana-726	126	20	if	if	SCONJ
cana-726	126	21	no	no	DET
cana-726	126	22	such	such	ADJ
cana-726	126	23	positive	positive	ADJ
cana-726	126	24	integer	integer	NOUN
cana-726	126	25	exists	exist	VERB
cana-726	126	26	,	,	PUNCT
cana-726	126	27	then	then	ADV
cana-726	126	28	r	r	NOUN
cana-726	126	29	is	be	AUX
cana-726	126	30	said	say	VERB
cana-726	126	31	to	to	PART
cana-726	126	32	be	be	AUX
cana-726	126	33	of	of	ADP
cana-726	126	34	characteristic	characteristic	ADJ
cana-726	126	35	zero	zero	NUM
cana-726	126	36	.	.	PUNCT
cana-726	127	1	let	let	VERB
cana-726	127	2	us	we	PRON
cana-726	127	3	collect	collect	VERB
cana-726	127	4	some	some	DET
cana-726	127	5	basic	basic	ADJ
cana-726	127	6	definitions	definition	NOUN
cana-726	127	7	and	and	CCONJ
cana-726	127	8	results	result	NOUN
cana-726	127	9	on	on	ADP
cana-726	127	10	s	s	NOUN
cana-726	127	11	:	:	PUNCT
cana-726	127	12	definition	definition	NOUN
cana-726	127	13	1.9	1.9	NUM
cana-726	127	14	.	.	PUNCT
cana-726	128	1	[	[	X
cana-726	128	2	5	5	NUM
cana-726	128	3	]	]	PUNCT
cana-726	128	4	given	give	VERB
cana-726	128	5	a	a	DET
cana-726	128	6	bipartite	bipartite	NOUN
cana-726	128	7	g	g	NOUN
cana-726	128	8	of	of	ADP
cana-726	128	9	n	n	PRON
cana-726	128	10	vertices	vertex	NOUN
cana-726	128	11	,	,	PUNCT
cana-726	128	12	there	there	PRON
cana-726	128	13	exists	exist	VERB
cana-726	128	14	a	a	DET
cana-726	128	15	connected	connected	ADJ
cana-726	128	16	bipartite	bipartite	PROPN
cana-726	128	17	h′	h′	PROPN
cana-726	128	18	such	such	ADJ
cana-726	128	19	that	that	SCONJ
cana-726	128	20	g	g	PROPN
cana-726	128	21	is	be	AUX
cana-726	128	22	an	an	DET
cana-726	128	23	induced	induced	ADJ
cana-726	128	24	sub	sub	NOUN
cana-726	128	25	of	of	ADP
cana-726	128	26	h′	h′	PROPN
cana-726	128	27	and	and	CCONJ
cana-726	128	28	all	all	DET
cana-726	128	29	the	the	DET
cana-726	128	30	vertices	vertex	NOUN
cana-726	128	31	of	of	ADP
cana-726	128	32	g	g	PROPN
cana-726	128	33	in	in	ADP
cana-726	128	34	h′	h′	PROPN
cana-726	128	35	have	have	VERB
cana-726	128	36	equal	equal	ADJ
cana-726	128	37	status	status	NOUN
cana-726	128	38	in	in	ADP
cana-726	128	39	h′.	h′.	PROPN
cana-726	128	40	definition	definition	NOUN
cana-726	128	41	1.10	1.10	NUM
cana-726	128	42	.	.	PUNCT
cana-726	129	1	[	[	X
cana-726	129	2	22	22	NUM
cana-726	129	3	]	]	PUNCT
cana-726	129	4	given	give	VERB
cana-726	129	5	a	a	DET
cana-726	129	6	bipartite	bipartite	NOUN
cana-726	129	7	g	g	NOUN
cana-726	129	8	there	there	PRON
cana-726	129	9	exists	exist	VERB
cana-726	129	10	a	a	DET
cana-726	129	11	bipartite	bipartite	NOUN
cana-726	129	12	h	h	NOUN
cana-726	129	13	such	such	ADJ
cana-726	129	14	that	that	DET
cana-726	129	15	m(h	m(h	NOUN
cana-726	129	16	)	)	PUNCT
cana-726	129	17	≅	≅	PROPN
cana-726	129	18	g.	g.	PROPN
cana-726	129	19	definition	definition	NOUN
cana-726	129	20	1.11	1.11	NUM
cana-726	129	21	.	.	PUNCT
cana-726	130	1	[	[	X
cana-726	130	2	27	27	NUM
cana-726	130	3	]	]	PUNCT
cana-726	130	4	given	give	VERB
cana-726	130	5	a	a	DET
cana-726	130	6	bipartite	bipartite	NOUN
cana-726	130	7	g	g	NOUN
cana-726	130	8	there	there	PRON
cana-726	130	9	exists	exist	VERB
cana-726	130	10	a	a	DET
cana-726	130	11	bipartite	bipartite	PROPN
cana-726	130	12	h	h	NOUN
cana-726	130	13	such	such	ADJ
cana-726	130	14	that	that	PRON
cana-726	130	15	am(h	am(h	NUM
cana-726	130	16	)	)	PUNCT
cana-726	130	17	≅	≅	PROPN
cana-726	130	18	g	g	PROPN
cana-726	130	19	definition	definition	NOUN
cana-726	130	20	1.12	1.12	NUM
cana-726	130	21	.	.	PUNCT
cana-726	131	1	[	[	X
cana-726	131	2	28	28	NUM
cana-726	131	3	]	]	PUNCT
cana-726	131	4	given	give	VERB
cana-726	131	5	a	a	DET
cana-726	131	6	k	k	NOUN
cana-726	131	7	-	-	ADJ
cana-726	131	8	partite	partite	ADJ
cana-726	131	9	g	g	NOUN
cana-726	131	10	there	there	PRON
cana-726	131	11	exists	exist	VERB
cana-726	131	12	a	a	DET
cana-726	131	13	k	k	ADJ
cana-726	131	14	-	-	ADJ
cana-726	131	15	partite	partite	ADJ
cana-726	131	16	h	h	NOUN
cana-726	131	17	such	such	ADJ
cana-726	131	18	that	that	DET
cana-726	131	19	m(h	m(h	NOUN
cana-726	131	20	)	)	PUNCT
cana-726	131	21	≅	≅	PROPN
cana-726	131	22	g.	g.	PROPN
cana-726	131	23	definition	definition	NOUN
cana-726	131	24	1.13	1.13	NUM
cana-726	131	25	.	.	PUNCT
cana-726	132	1	[	[	X
cana-726	132	2	32	32	NUM
cana-726	132	3	]	]	PUNCT
cana-726	132	4	given	give	VERB
cana-726	132	5	a	a	DET
cana-726	132	6	k	k	NOUN
cana-726	132	7	-	-	ADJ
cana-726	132	8	partite	partite	ADJ
cana-726	132	9	g	g	NOUN
cana-726	132	10	there	there	PRON
cana-726	132	11	exists	exist	VERB
cana-726	132	12	a	a	DET
cana-726	132	13	k	k	ADJ
cana-726	132	14	-	-	ADJ
cana-726	132	15	partite	partite	ADJ
cana-726	132	16	h′	h′	NOUN
cana-726	132	17	such	such	ADJ
cana-726	132	18	that	that	PRON
cana-726	132	19	am(h′	am(h′	NOUN
cana-726	132	20	)	)	PUNCT
cana-726	133	1	≅	≅	PROPN
cana-726	133	2	g.	g.	PROPN
cana-726	133	3	3	3	NUM
cana-726	133	4	.	.	PUNCT
cana-726	134	1	methods	method	NOUN
cana-726	134	2	theorem	theorem	VERB
cana-726	134	3	2.1	2.1	NUM
cana-726	134	4	.	.	PUNCT
cana-726	135	1	for	for	ADP
cana-726	135	2	a	a	DET
cana-726	135	3	determinate	determinate	ADJ
cana-726	135	4	commutative	commutative	ADJ
cana-726	135	5	semigroup	semigroup	NOUN
cana-726	135	6	s	s	PROPN
cana-726	135	7	,	,	PUNCT
cana-726	135	8	the	the	DET
cana-726	135	9	set	set	NOUN
cana-726	135	10	v(φ(γ(s)))∪{0	v(φ(γ(s)))∪{0	NOUN
cana-726	135	11	}	}	PUNCT
cana-726	135	12	is	be	AUX
cana-726	135	13	a	a	DET
cana-726	135	14	median	median	NOUN
cana-726	135	15	of	of	ADP
cana-726	135	16	s.	s.	PROPN
cana-726	135	17	proof	proof	PROPN
cana-726	135	18	.	.	PUNCT
cana-726	136	1	let	let	VERB
cana-726	136	2	x	x	SYM
cana-726	136	3	∈	∈	PROPN
cana-726	136	4	v(φ	v(φ	PROPN
cana-726	136	5	(	(	PUNCT
cana-726	136	6	γ(s	γ(s	PROPN
cana-726	136	7	)	)	PUNCT
cana-726	136	8	)	)	PUNCT
cana-726	136	9	)	)	PUNCT
cana-726	136	10	,	,	PUNCT
cana-726	136	11	and	and	CCONJ
cana-726	136	12	r	r	NOUN
cana-726	136	13	∈	∈	PROPN
cana-726	136	14	s.	s.	PROPN
cana-726	136	15	suppose	suppose	VERB
cana-726	136	16	that	that	SCONJ
cana-726	136	17	rx	rx	VERB
cana-726	136	18	≠	≠	PROPN
cana-726	136	19	0	0	NUM
cana-726	136	20	.	.	PUNCT
cana-726	137	1	then	then	ADV
cana-726	137	2	e(rx	e(rx	NUM
cana-726	137	3	)	)	PUNCT
cana-726	137	4	=	=	SYM
cana-726	137	5	max{d	max{d	NOUN
cana-726	137	6	(	(	PUNCT
cana-726	137	7	u	u	NOUN
cana-726	137	8	,	,	PUNCT
cana-726	137	9	rx)|u	rx)|u	NOUN
cana-726	137	10	∈	∈	NOUN
cana-726	137	11	v(g	v(g	ADJ
cana-726	137	12	)	)	PUNCT
cana-726	137	13	}	}	PUNCT
cana-726	137	14	≤	≤	NUM
cana-726	137	15	max{d	max{d	NOUN
cana-726	137	16	(	(	PUNCT
cana-726	137	17	u	u	NOUN
cana-726	137	18	,	,	PUNCT
cana-726	137	19	x)|u	x)|u	PROPN
cana-726	137	20	∈	∈	PROPN
cana-726	137	21	v(g	v(g	ADJ
cana-726	137	22	)	)	PUNCT
cana-726	137	23	}	}	PUNCT
cana-726	137	24	=	=	SYM
cana-726	137	25	e(x	e(x	NUM
cana-726	137	26	)	)	PUNCT
cana-726	137	27	.	.	PUNCT
cana-726	138	1	thus	thus	ADV
cana-726	138	2	e(rx	e(rx	NUM
cana-726	138	3	)	)	PUNCT
cana-726	138	4	=	=	SYM
cana-726	138	5	e(x	e(x	NUM
cana-726	138	6	)	)	PUNCT
cana-726	138	7	.	.	PUNCT
cana-726	139	1	hence	hence	ADV
cana-726	139	2	,	,	PUNCT
cana-726	139	3	rx	rx	ADP
cana-726	139	4	∈	∈	PROPN
cana-726	139	5	v(φ	v(φ	PROPN
cana-726	139	6	(	(	PUNCT
cana-726	139	7	γ(s	γ(s	PROPN
cana-726	139	8	)	)	PUNCT
cana-726	139	9	)	)	PUNCT
cana-726	139	10	)	)	PUNCT
cana-726	139	11	∪	∪	ADP
cana-726	139	12	{	{	PUNCT
cana-726	139	13	0	0	NUM
cana-726	139	14	}	}	PUNCT
cana-726	139	15	.	.	PUNCT
cana-726	140	1	remark	remark	VERB
cana-726	140	2	2.1	2.1	NUM
cana-726	140	3	.	.	PUNCT
cana-726	141	1	a	a	DET
cana-726	141	2	subgraph	subgraph	NOUN
cana-726	141	3	h	h	NOUN
cana-726	141	4	of	of	ADP
cana-726	141	5	a	a	DET
cana-726	141	6	graph	graph	NOUN
cana-726	141	7	g	g	NOUN
cana-726	141	8	is	be	AUX
cana-726	141	9	a	a	DET
cana-726	141	10	crossing	crossing	NOUN
cana-726	141	11	subgraph	subgraph	NOUN
cana-726	141	12	of	of	ADP
cana-726	141	13	g	g	PROPN
cana-726	141	14	if	if	SCONJ
cana-726	141	15	v(h	v(h	NOUN
cana-726	141	16	)	)	PUNCT
cana-726	141	17	=	=	SYM
cana-726	141	18	v(g	v(g	ADJ
cana-726	141	19	)	)	PUNCT
cana-726	141	20	.	.	PUNCT
cana-726	142	1	on	on	ADP
cana-726	142	2	the	the	DET
cana-726	142	3	off	off	ADJ
cana-726	142	4	chance	chance	NOUN
cana-726	142	5	that	that	SCONJ
cana-726	142	6	u	u	NOUN
cana-726	142	7	is	be	AUX
cana-726	142	8	a	a	DET
cana-726	142	9	bunch	bunch	NOUN
cana-726	142	10	of	of	ADP
cana-726	142	11	edges	edge	NOUN
cana-726	142	12	of	of	ADP
cana-726	142	13	a	a	DET
cana-726	142	14	graph	graph	NOUN
cana-726	142	15	g	g	NOUN
cana-726	142	16	,	,	PUNCT
cana-726	142	17	g	g	PROPN
cana-726	142	18	\	\	NOUN
cana-726	142	19	u	u	NOUN
cana-726	142	20	is	be	AUX
cana-726	142	21	the	the	DET
cana-726	142	22	crossing	crossing	NOUN
cana-726	142	23	subgraph	subgraph	NOUN
cana-726	142	24	of	of	ADP
cana-726	142	25	g	g	PROPN
cana-726	142	26	acquired	acquire	VERB
cana-726	142	27	by	by	ADP
cana-726	142	28	erasing	erase	VERB
cana-726	142	29	the	the	DET
cana-726	142	30	edges	edge	NOUN
cana-726	142	31	in	in	ADP
cana-726	142	32	u	u	NOUN
cana-726	142	33	from	from	ADP
cana-726	142	34	e(g	e(g	PROPN
cana-726	142	35	)	)	PUNCT
cana-726	142	36	.	.	PUNCT
cana-726	143	1	a	a	DET
cana-726	143	2	subset	subset	ADJ
cana-726	143	3	u	u	NOUN
cana-726	143	4	of	of	ADP
cana-726	143	5	the	the	DET
cana-726	143	6	edge	edge	NOUN
cana-726	143	7	set	set	NOUN
cana-726	143	8	of	of	ADP
cana-726	143	9	an	an	DET
cana-726	143	10	associated	associated	ADJ
cana-726	143	11	chart	chart	NOUN
cana-726	143	12	g	g	PROPN
cana-726	143	13	is	be	AUX
cana-726	143	14	an	an	DET
cana-726	143	15	edge	edge	NOUN
cana-726	143	16	set	set	NOUN
cana-726	143	17	of	of	ADP
cana-726	143	18	g	g	PROPN
cana-726	143	19	if	if	SCONJ
cana-726	143	20	g	g	PROPN
cana-726	143	21	\	\	NOUN
cana-726	143	22	u	u	NOUN
cana-726	143	23	is	be	AUX
cana-726	143	24	separated	separate	VERB
cana-726	143	25	.	.	PUNCT
cana-726	144	1	an	an	DET
cana-726	144	2	edge	edge	NOUN
cana-726	144	3	set	set	NOUN
cana-726	144	4	of	of	ADP
cana-726	144	5	g	g	PROPN
cana-726	144	6	is	be	AUX
cana-726	144	7	negligible	negligible	ADJ
cana-726	144	8	assuming	assume	VERB
cana-726	144	9	no	no	DET
cana-726	144	10	appropriate	appropriate	ADJ
cana-726	144	11	subset	subset	NOUN
cana-726	144	12	of	of	ADP
cana-726	144	13	u	u	NOUN
cana-726	144	14	is	be	AUX
cana-726	144	15	edge	edge	NOUN
cana-726	144	16	set	set	VERB
cana-726	144	17	.	.	PUNCT
cana-726	145	1	assuming	assume	VERB
cana-726	145	2	e	e	NOUN
cana-726	145	3	is	be	AUX
cana-726	145	4	an	an	DET
cana-726	145	5	edge	edge	NOUN
cana-726	145	6	of	of	ADP
cana-726	145	7	g	g	NOUN
cana-726	145	8	,	,	PUNCT
cana-726	145	9	with	with	ADP
cana-726	145	10	the	the	DET
cana-726	145	11	end	end	NOUN
cana-726	145	12	goal	goal	NOUN
cana-726	145	13	that	that	PRON
cana-726	145	14	g	g	PROPN
cana-726	145	15	\	\	PROPN
cana-726	145	16	{	{	PUNCT
cana-726	145	17	e	e	NOUN
cana-726	145	18	}	}	PUNCT
cana-726	145	19	is	be	AUX
cana-726	145	20	detached	detach	VERB
cana-726	145	21	,	,	PUNCT
cana-726	145	22	then	then	ADV
cana-726	145	23	e	e	PROPN
cana-726	145	24	is	be	AUX
cana-726	145	25	known	know	VERB
cana-726	145	26	as	as	ADP
cana-726	145	27	an	an	DET
cana-726	145	28	extension	extension	NOUN
cana-726	145	29	.	.	PUNCT
cana-726	146	1	note	note	VERB
cana-726	146	2	that	that	SCONJ
cana-726	146	3	on	on	ADP
cana-726	146	4	the	the	DET
cana-726	146	5	off	off	ADJ
cana-726	146	6	chance	chance	NOUN
cana-726	146	7	that	that	SCONJ
cana-726	146	8	u	u	NOUN
cana-726	146	9	is	be	AUX
cana-726	146	10	an	an	DET
cana-726	146	11	insignificant	insignificant	ADJ
cana-726	146	12	edge	edge	NOUN
cana-726	146	13	set	set	NOUN
cana-726	146	14	,	,	PUNCT
cana-726	146	15	g	g	PROPN
cana-726	146	16	\	\	NOUN
cana-726	146	17	u	u	NOUN
cana-726	146	18	has	have	VERB
cana-726	146	19	precisely	precisely	ADV
cana-726	146	20	two	two	NUM
cana-726	146	21	associated	associated	ADJ
cana-726	146	22	median	median	ADJ
cana-726	146	23	parts	part	NOUN
cana-726	146	24	.	.	PUNCT
cana-726	147	1	corollary	corollary	ADJ
cana-726	147	2	2.1	2.1	NUM
cana-726	147	3	.	.	PUNCT
cana-726	148	1	let	let	VERB
cana-726	148	2	t	t	NOUN
cana-726	148	3	be	be	AUX
cana-726	148	4	the	the	DET
cana-726	148	5	minimal	minimal	ADJ
cana-726	148	6	edge	edge	NOUN
cana-726	148	7	set	set	NOUN
cana-726	148	8	of	of	ADP
cana-726	148	9	γ(s	γ(	NOUN
cana-726	148	10	)	)	PUNCT
cana-726	148	11	,	,	PUNCT
cana-726	148	12	and	and	CCONJ
cana-726	148	13	g1	g1	NOUN
cana-726	148	14	,	,	PUNCT
cana-726	148	15	g2	g2	PROPN
cana-726	148	16	are	be	AUX
cana-726	148	17	two	two	NUM
cana-726	148	18	median	median	ADJ
cana-726	148	19	parts	part	NOUN
cana-726	148	20	of	of	ADP
cana-726	148	21	g	g	NOUN
cana-726	148	22	\	\	PROPN
cana-726	149	1	t.	t.	NOUN
cana-726	149	2	then	then	ADV
cana-726	149	3	the	the	DET
cana-726	149	4	following	follow	VERB
cana-726	149	5	hold	hold	NOUN
cana-726	149	6	.	.	PUNCT
cana-726	150	1	(	(	PUNCT
cana-726	150	2	i	i	NOUN
cana-726	150	3	)	)	PUNCT
cana-726	150	4	for	for	ADP
cana-726	150	5	any	any	DET
cana-726	150	6	i	i	NOUN
cana-726	150	7	=	=	NOUN
cana-726	150	8	1	1	NUM
cana-726	150	9	,	,	PUNCT
cana-726	150	10	2	2	NUM
cana-726	150	11	,	,	PUNCT
cana-726	150	12	(	(	PUNCT
cana-726	150	13	v(gi	v(gi	NOUN
cana-726	150	14	)	)	PUNCT
cana-726	150	15	∩	∩	NOUN
cana-726	150	16	v(t	v(t	NOUN
cana-726	150	17	)	)	PUNCT
cana-726	150	18	)	)	PUNCT
cana-726	150	19	∪	∪	ADP
cana-726	150	20	{	{	PUNCT
cana-726	150	21	0	0	NUM
cana-726	150	22	}	}	PUNCT
cana-726	150	23	is	be	AUX
cana-726	150	24	ideal	ideal	ADJ
cana-726	150	25	of	of	ADP
cana-726	150	26	s	s	AUX
cana-726	150	27	provided	provide	VERB
cana-726	151	1	g_i	g_i	PROPN
cana-726	151	2	has	have	AUX
cana-726	151	3	at	at	ADV
cana-726	151	4	least	least	ADV
cana-726	151	5	two	two	NUM
cana-726	151	6	vertices	vertex	NOUN
cana-726	151	7	.	.	PUNCT
cana-726	152	1	(	(	PUNCT
cana-726	152	2	ii	ii	NOUN
cana-726	152	3	)	)	PUNCT
cana-726	152	4	v(t	v(t	NOUN
cana-726	152	5	)	)	PUNCT
cana-726	152	6	∪	∪	X
cana-726	152	7	{	{	PUNCT
cana-726	152	8	0	0	NUM
cana-726	152	9	}	}	PUNCT
cana-726	152	10	is	be	AUX
cana-726	152	11	an	an	DET
cana-726	152	12	ideal	ideal	NOUN
cana-726	152	13	if	if	SCONJ
cana-726	152	14	g_1	g_1	PROPN
cana-726	152	15	or	or	CCONJ
cana-726	152	16	g_2	g_2	PROPN
cana-726	152	17	has	have	VERB
cana-726	152	18	only	only	ADV
cana-726	152	19	one	one	NUM
cana-726	152	20	vertex	vertex	NOUN
cana-726	152	21	.	.	PUNCT
cana-726	153	1	a	a	DET
cana-726	153	2	commutative	commutative	ADJ
cana-726	153	3	semigroup	semigroup	NOUN
cana-726	153	4	is	be	AUX
cana-726	153	5	called	call	VERB
cana-726	153	6	reduced	reduce	VERB
cana-726	153	7	if	if	SCONJ
cana-726	153	8	for	for	ADP
cana-726	153	9	any	any	DET
cana-726	153	10	x	x	SYM
cana-726	153	11	∈	∈	PROPN
cana-726	153	12	s	s	NOUN
cana-726	153	13	,	,	PUNCT
cana-726	153	14	x_n	x_n	PUNCT
cana-726	153	15	=	=	SYM
cana-726	153	16	0	0	NUM
cana-726	153	17	implies	imply	VERB
cana-726	153	18	x	x	PUNCT
cana-726	153	19	=	=	SYM
cana-726	153	20	0	0	NUM
cana-726	153	21	.	.	PUNCT
cana-726	154	1	the	the	DET
cana-726	154	2	annihilator	annihilator	NOUN
cana-726	154	3	of	of	ADP
cana-726	154	4	x	x	PUNCT
cana-726	154	5	∈	∈	PROPN
cana-726	154	6	s	s	PART
cana-726	154	7	is	be	AUX
cana-726	154	8	denoted	denote	VERB
cana-726	154	9	by	by	ADP
cana-726	154	10	ann	ann	PROPN
cana-726	154	11	(	(	PUNCT
cana-726	154	12	x	x	NOUN
cana-726	154	13	)	)	PUNCT
cana-726	154	14	and	and	CCONJ
cana-726	154	15	it	it	PRON
cana-726	154	16	is	be	AUX
cana-726	154	17	defined	define	VERB
cana-726	154	18	as	as	ADP
cana-726	154	19	ann	ann	PROPN
cana-726	154	20	(	(	PUNCT
cana-726	154	21	x	x	NOUN
cana-726	154	22	)	)	PUNCT
cana-726	155	1	=	=	PRON
cana-726	155	2	{	{	PUNCT
cana-726	155	3	a	a	DET
cana-726	155	4	∈	∈	PROPN
cana-726	155	5	s|ax	s|ax	NOUN
cana-726	155	6	=	=	NOUN
cana-726	155	7	0	0	NUM
cana-726	155	8	}	}	PUNCT
cana-726	155	9	.	.	PUNCT
cana-726	156	1	4	4	X
cana-726	156	2	.	.	X
cana-726	156	3	results	result	NOUN
cana-726	156	4	theorem	theorem	VERB
cana-726	156	5	2.1	2.1	NUM
cana-726	156	6	.	.	PUNCT
cana-726	157	1	for	for	ADP
cana-726	157	2	a	a	DET
cana-726	157	3	determinate	determinate	ADJ
cana-726	157	4	commutative	commutative	ADJ
cana-726	157	5	semigroup	semigroup	NOUN
cana-726	157	6	s	s	PROPN
cana-726	157	7	,	,	PUNCT
cana-726	157	8	the	the	DET
cana-726	157	9	set	set	NOUN
cana-726	157	10	v(𝜑(γ(s)))∪{0	v(𝜑(γ(s)))∪{0	NOUN
cana-726	157	11	}	}	PUNCT
cana-726	157	12	is	be	AUX
cana-726	157	13	a	a	DET
cana-726	157	14	median	median	NOUN
cana-726	157	15	of	of	ADP
cana-726	157	16	s.	s.	PROPN
cana-726	157	17	remark	remark	PROPN
cana-726	157	18	2.1	2.1	NUM
cana-726	157	19	.	.	PUNCT
cana-726	158	1	a	a	DET
cana-726	158	2	subgraph	subgraph	NOUN
cana-726	158	3	h	h	NOUN
cana-726	158	4	of	of	ADP
cana-726	158	5	a	a	DET
cana-726	158	6	graph	graph	NOUN
cana-726	158	7	g	g	NOUN
cana-726	158	8	is	be	AUX
cana-726	158	9	a	a	DET
cana-726	158	10	crossing	crossing	NOUN
cana-726	158	11	subgraph	subgraph	NOUN
cana-726	158	12	of	of	ADP
cana-726	158	13	g	g	PROPN
cana-726	158	14	if	if	SCONJ
cana-726	158	15	v(h	v(h	NOUN
cana-726	158	16	)	)	PUNCT
cana-726	158	17	=	=	SYM
cana-726	158	18	v(g	v(g	ADJ
cana-726	158	19	)	)	PUNCT
cana-726	158	20	.	.	PUNCT
cana-726	159	1	on	on	ADP
cana-726	159	2	the	the	DET
cana-726	159	3	off	off	ADJ
cana-726	159	4	chance	chance	NOUN
cana-726	159	5	that	that	SCONJ
cana-726	159	6	u	u	NOUN
cana-726	159	7	is	be	AUX
cana-726	159	8	a	a	DET
cana-726	159	9	bunch	bunch	NOUN
cana-726	159	10	of	of	ADP
cana-726	159	11	edges	edge	NOUN
cana-726	159	12	of	of	ADP
cana-726	159	13	a	a	DET
cana-726	159	14	graph	graph	NOUN
cana-726	159	15	g	g	NOUN
cana-726	159	16	,	,	PUNCT
cana-726	159	17	g	g	PROPN
cana-726	159	18	\	\	NOUN
cana-726	159	19	u	u	NOUN
cana-726	159	20	is	be	AUX
cana-726	159	21	the	the	DET
cana-726	159	22	crossing	crossing	NOUN
cana-726	159	23	subgraph	subgraph	NOUN
cana-726	159	24	of	of	ADP
cana-726	159	25	g	g	PROPN
cana-726	159	26	acquired	acquire	VERB
cana-726	159	27	by	by	ADP
cana-726	159	28	erasing	erase	VERB
cana-726	159	29	the	the	DET
cana-726	159	30	edges	edge	NOUN
cana-726	159	31	in	in	ADP
cana-726	159	32	u	u	NOUN
cana-726	159	33	from	from	ADP
cana-726	159	34	e(g	e(g	PROPN
cana-726	159	35	)	)	PUNCT
cana-726	159	36	.	.	PUNCT
cana-726	160	1	a	a	DET
cana-726	160	2	subset	subset	ADJ
cana-726	160	3	u	u	NOUN
cana-726	160	4	of	of	ADP
cana-726	160	5	the	the	DET
cana-726	160	6	edge	edge	NOUN
cana-726	160	7	set	set	NOUN
cana-726	160	8	of	of	ADP
cana-726	160	9	an	an	DET
cana-726	160	10	associated	associated	ADJ
cana-726	160	11	chart	chart	NOUN
cana-726	160	12	g	g	PROPN
cana-726	160	13	is	be	AUX
cana-726	160	14	an	an	DET
cana-726	160	15	edge	edge	NOUN
cana-726	160	16	set	set	NOUN
cana-726	160	17	of	of	ADP
cana-726	160	18	g	g	PROPN
cana-726	160	19	if	if	SCONJ
cana-726	160	20	g	g	PROPN
cana-726	160	21	\	\	NOUN
cana-726	160	22	u	u	NOUN
cana-726	160	23	is	be	AUX
cana-726	160	24	separated	separate	VERB
cana-726	160	25	.	.	PUNCT
cana-726	161	1	an	an	DET
cana-726	161	2	edge	edge	NOUN
cana-726	161	3	set	set	NOUN
cana-726	161	4	of	of	ADP
cana-726	161	5	g	g	PROPN
cana-726	161	6	is	be	AUX
cana-726	161	7	negligible	negligible	ADJ
cana-726	161	8	assuming	assume	VERB
cana-726	161	9	no	no	DET
cana-726	161	10	appropriate	appropriate	ADJ
cana-726	161	11	subset	subset	NOUN
cana-726	161	12	of	of	ADP
cana-726	161	13	u	u	NOUN
cana-726	161	14	is	be	AUX
cana-726	161	15	edge	edge	NOUN
cana-726	161	16	set	set	VERB
cana-726	161	17	.	.	PUNCT
cana-726	162	1	assuming	assume	VERB
cana-726	162	2	e	e	NOUN
cana-726	162	3	is	be	AUX
cana-726	162	4	an	an	DET
cana-726	162	5	edge	edge	NOUN
cana-726	162	6	of	of	ADP
cana-726	162	7	g	g	NOUN
cana-726	162	8	,	,	PUNCT
cana-726	162	9	with	with	ADP
cana-726	162	10	the	the	DET
cana-726	162	11	end	end	NOUN
cana-726	162	12	goal	goal	NOUN
cana-726	162	13	that	that	PRON
cana-726	162	14	g	g	PROPN
cana-726	162	15	\	\	PROPN
cana-726	162	16	{	{	PUNCT
cana-726	162	17	e	e	NOUN
cana-726	162	18	}	}	PUNCT
cana-726	162	19	is	be	AUX
cana-726	162	20	detached	detach	VERB
cana-726	162	21	,	,	PUNCT
cana-726	162	22	then	then	ADV
cana-726	162	23	e	e	PROPN
cana-726	162	24	is	be	AUX
cana-726	162	25	known	know	VERB
cana-726	162	26	as	as	ADP
cana-726	162	27	an	an	DET
cana-726	162	28	extension	extension	NOUN
cana-726	162	29	.	.	PUNCT
cana-726	163	1	note	note	VERB
cana-726	163	2	that	that	SCONJ
cana-726	163	3	on	on	ADP
cana-726	163	4	the	the	DET
cana-726	163	5	off	off	ADJ
cana-726	163	6	chance	chance	NOUN
cana-726	163	7	that	that	SCONJ
cana-726	163	8	u	u	NOUN
cana-726	163	9	is	be	AUX
cana-726	163	10	an	an	DET
cana-726	163	11	insignificant	insignificant	ADJ
cana-726	163	12	edge	edge	NOUN
cana-726	163	13	set	set	NOUN
cana-726	163	14	,	,	PUNCT
cana-726	163	15	g	g	PROPN
cana-726	163	16	\	\	NOUN
cana-726	163	17	u	u	NOUN
cana-726	163	18	has	have	VERB
cana-726	163	19	precisely	precisely	ADV
cana-726	163	20	two	two	NUM
cana-726	163	21	associated	associated	ADJ
cana-726	163	22	median	median	ADJ
cana-726	163	23	parts	part	NOUN
cana-726	163	24	.	.	PUNCT
cana-726	164	1	communications	communication	NOUN
cana-726	164	2	on	on	ADP
cana-726	164	3	applied	apply	VERB
cana-726	164	4	nonlinear	nonlinear	ADJ
cana-726	164	5	analysis	analysis	NOUN
cana-726	164	6	issn	issn	NOUN
cana-726	164	7	:	:	PUNCT
cana-726	164	8	1074	1074	NUM
cana-726	164	9	-	-	PUNCT
cana-726	164	10	133x	133x	NUM
cana-726	164	11	vol	vol	NOUN
cana-726	164	12	31	31	NUM
cana-726	164	13	no	no	NOUN
cana-726	164	14	.	.	PUNCT
cana-726	165	1	3s	3s	NUM
cana-726	165	2	(	(	PUNCT
cana-726	165	3	2024	2024	NUM
cana-726	165	4	)	)	PUNCT
cana-726	165	5	6	6	NUM
cana-726	165	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	165	7	corollary	corollary	ADJ
cana-726	165	8	2.1	2.1	NUM
cana-726	165	9	.	.	PUNCT
cana-726	166	1	let	let	VERB
cana-726	166	2	t	t	NOUN
cana-726	166	3	be	be	AUX
cana-726	166	4	the	the	DET
cana-726	166	5	minimal	minimal	ADJ
cana-726	166	6	edge	edge	NOUN
cana-726	166	7	set	set	NOUN
cana-726	166	8	of	of	ADP
cana-726	166	9	γ(s	γ(	NOUN
cana-726	166	10	)	)	PUNCT
cana-726	166	11	,	,	PUNCT
cana-726	166	12	and	and	CCONJ
cana-726	166	13	g1	g1	NOUN
cana-726	166	14	,	,	PUNCT
cana-726	166	15	g2	g2	PROPN
cana-726	166	16	are	be	AUX
cana-726	166	17	two	two	NUM
cana-726	166	18	median	median	ADJ
cana-726	166	19	parts	part	NOUN
cana-726	166	20	of	of	ADP
cana-726	166	21	g	g	NOUN
cana-726	166	22	\	\	PROPN
cana-726	167	1	t.	t.	NOUN
cana-726	167	2	then	then	ADV
cana-726	167	3	the	the	DET
cana-726	167	4	following	follow	VERB
cana-726	167	5	hold	hold	NOUN
cana-726	167	6	.	.	PUNCT
cana-726	168	1	(	(	PUNCT
cana-726	168	2	i	i	NOUN
cana-726	168	3	)	)	PUNCT
cana-726	168	4	for	for	ADP
cana-726	168	5	any	any	DET
cana-726	168	6	i	i	NOUN
cana-726	168	7	=	=	NOUN
cana-726	168	8	1	1	NUM
cana-726	168	9	,	,	PUNCT
cana-726	168	10	2	2	NUM
cana-726	168	11	,	,	PUNCT
cana-726	168	12	(	(	PUNCT
cana-726	168	13	v(gi	v(gi	NOUN
cana-726	168	14	)	)	PUNCT
cana-726	168	15	∩	∩	NOUN
cana-726	168	16	v(t	v(t	NOUN
cana-726	168	17	)	)	PUNCT
cana-726	168	18	)	)	PUNCT
cana-726	168	19	∪	∪	ADP
cana-726	168	20	{	{	PUNCT
cana-726	168	21	0	0	NUM
cana-726	168	22	}	}	PUNCT
cana-726	168	23	is	be	AUX
cana-726	168	24	ideal	ideal	ADJ
cana-726	168	25	of	of	ADP
cana-726	168	26	s	s	AUX
cana-726	168	27	provided	provide	VERB
cana-726	168	28	𝐺𝑖	𝐺𝑖	PROPN
cana-726	168	29	has	have	VERB
cana-726	168	30	at	at	ADV
cana-726	168	31	least	least	ADV
cana-726	168	32	two	two	NUM
cana-726	168	33	vertices	vertex	NOUN
cana-726	168	34	.	.	PUNCT
cana-726	169	1	(	(	PUNCT
cana-726	169	2	ii	ii	NOUN
cana-726	169	3	)	)	PUNCT
cana-726	169	4	v(t	v(t	NOUN
cana-726	169	5	)	)	PUNCT
cana-726	169	6	∪	∪	X
cana-726	169	7	{	{	PUNCT
cana-726	169	8	0	0	NUM
cana-726	169	9	}	}	PUNCT
cana-726	169	10	is	be	AUX
cana-726	169	11	an	an	DET
cana-726	169	12	ideal	ideal	NOUN
cana-726	169	13	if	if	SCONJ
cana-726	169	14	𝐺1	𝐺1	PROPN
cana-726	169	15	or	or	CCONJ
cana-726	169	16	𝐺2	𝐺2	NOUN
cana-726	169	17	has	have	VERB
cana-726	169	18	only	only	ADV
cana-726	169	19	one	one	NUM
cana-726	169	20	vertex	vertex	NOUN
cana-726	169	21	.	.	PUNCT
cana-726	170	1	a	a	DET
cana-726	170	2	commutative	commutative	ADJ
cana-726	170	3	semigroup	semigroup	NOUN
cana-726	170	4	is	be	AUX
cana-726	170	5	called	call	VERB
cana-726	170	6	reduced	reduce	VERB
cana-726	170	7	if	if	SCONJ
cana-726	170	8	for	for	ADP
cana-726	170	9	any	any	DET
cana-726	170	10	x	x	SYM
cana-726	170	11	∈	∈	PROPN
cana-726	170	12	s	s	NOUN
cana-726	170	13	,	,	PUNCT
cana-726	170	14	𝑥𝑛	𝑥𝑛	PROPN
cana-726	170	15	=	=	SYM
cana-726	170	16	0	0	NUM
cana-726	170	17	implies	imply	VERB
cana-726	170	18	x	x	PUNCT
cana-726	170	19	=	=	SYM
cana-726	170	20	0	0	NUM
cana-726	170	21	.	.	PUNCT
cana-726	171	1	the	the	DET
cana-726	171	2	annihilator	annihilator	NOUN
cana-726	171	3	of	of	ADP
cana-726	171	4	x	x	PUNCT
cana-726	171	5	∈	∈	PROPN
cana-726	171	6	s	s	PART
cana-726	171	7	is	be	AUX
cana-726	171	8	denoted	denote	VERB
cana-726	171	9	by	by	ADP
cana-726	171	10	ann	ann	PROPN
cana-726	171	11	(	(	PUNCT
cana-726	171	12	x	x	NOUN
cana-726	171	13	)	)	PUNCT
cana-726	171	14	and	and	CCONJ
cana-726	171	15	it	it	PRON
cana-726	171	16	is	be	AUX
cana-726	171	17	defined	define	VERB
cana-726	171	18	as	as	ADP
cana-726	171	19	ann	ann	PROPN
cana-726	171	20	(	(	PUNCT
cana-726	171	21	x	x	NOUN
cana-726	171	22	)	)	PUNCT
cana-726	172	1	=	=	PRON
cana-726	172	2	{	{	PUNCT
cana-726	172	3	a	a	DET
cana-726	172	4	∈	∈	PROPN
cana-726	172	5	s|ax	s|ax	NOUN
cana-726	172	6	=	=	NOUN
cana-726	172	7	0	0	NUM
cana-726	172	8	}	}	PUNCT
cana-726	172	9	.	.	PUNCT
cana-726	173	1	theorem	theorem	VERB
cana-726	173	2	2.2	2.2	NUM
cana-726	173	3	.	.	PUNCT
cana-726	174	1	let	let	VERB
cana-726	174	2	s	s	PRON
cana-726	174	3	be	be	AUX
cana-726	174	4	a	a	DET
cana-726	174	5	commutative	commutative	ADJ
cana-726	174	6	and	and	CCONJ
cana-726	174	7	reduced	reduce	VERB
cana-726	174	8	semigroup	semigroup	NOUN
cana-726	174	9	in	in	ADP
cana-726	174	10	which	which	PRON
cana-726	174	11	γ(s	γ(s	PROPN
cana-726	174	12	)	)	PUNCT
cana-726	174	13	does	do	AUX
cana-726	174	14	not	not	PART
cana-726	174	15	contain	contain	VERB
cana-726	174	16	a	a	DET
cana-726	174	17	median	median	ADJ
cana-726	174	18	and	and	CCONJ
cana-726	174	19	anti	anti	ADJ
cana-726	174	20	–	–	PUNCT
cana-726	174	21	median	median	ADJ
cana-726	174	22	clique	clique	NOUN
cana-726	174	23	.	.	PUNCT
cana-726	175	1	then	then	ADV
cana-726	175	2	s	s	VERB
cana-726	175	3	satisfies	satisfie	NOUN
cana-726	175	4	the	the	DET
cana-726	175	5	ascending	ascend	VERB
cana-726	175	6	chain	chain	NOUN
cana-726	175	7	conditions	condition	NOUN
cana-726	175	8	(	(	PUNCT
cana-726	175	9	a.c.c	a.c.c	NOUN
cana-726	175	10	)	)	PUNCT
cana-726	175	11	on	on	ADP
cana-726	175	12	annihilators	annihilators	PROPN
cana-726	175	13	.	.	PUNCT
cana-726	176	1	theorem	theorem	VERB
cana-726	176	2	2.3	2.3	NUM
cana-726	176	3	.	.	PUNCT
cana-726	177	1	let	let	VERB
cana-726	177	2	s	s	PRON
cana-726	177	3	be	be	AUX
cana-726	177	4	a	a	DET
cana-726	177	5	commutative	commutative	ADJ
cana-726	177	6	rings	ring	NOUN
cana-726	177	7	of	of	ADP
cana-726	177	8	median	median	PROPN
cana-726	177	9	and	and	CCONJ
cana-726	177	10	anti	anti	ADJ
cana-726	177	11	–	–	PUNCT
cana-726	177	12	median	median	ADJ
cana-726	177	13	.	.	PUNCT
cana-726	178	1	then	then	ADV
cana-726	178	2	the	the	DET
cana-726	178	3	subsequent	subsequent	ADJ
cana-726	178	4	results	result	NOUN
cana-726	178	5	hold	hold	VERB
cana-726	178	6	:	:	PUNCT
cana-726	178	7	(	(	PUNCT
cana-726	178	8	i	i	NOUN
cana-726	178	9	)	)	PUNCT
cana-726	178	10	if	if	SCONJ
cana-726	178	11	|ass	|ass	PROPN
cana-726	178	12	(	(	PUNCT
cana-726	178	13	s)|	s)|	PROPN
cana-726	178	14	≥	≥	NUM
cana-726	178	15	3	3	NUM
cana-726	178	16	and	and	CCONJ
cana-726	178	17	∅	∅	NOUN
cana-726	178	18	=	=	SYM
cana-726	178	19	ann	ann	X
cana-726	178	20	(	(	PUNCT
cana-726	178	21	x	x	NOUN
cana-726	178	22	)	)	PUNCT
cana-726	178	23	,	,	PUNCT
cana-726	178	24	𝜒	𝜒	X
cana-726	178	25	=	=	SYM
cana-726	178	26	ann	ann	X
cana-726	178	27	(	(	PUNCT
cana-726	178	28	y	y	NOUN
cana-726	178	29	)	)	PUNCT
cana-726	178	30	are	be	AUX
cana-726	178	31	two	two	NUM
cana-726	178	32	distinct	distinct	ADJ
cana-726	178	33	elements	element	NOUN
cana-726	178	34	of	of	ADP
cana-726	178	35	ass	ass	NOUN
cana-726	178	36	(	(	PUNCT
cana-726	178	37	s	s	NOUN
cana-726	178	38	)	)	PUNCT
cana-726	178	39	,	,	PUNCT
cana-726	178	40	then	then	ADV
cana-726	178	41	xy	xy	PROPN
cana-726	178	42	=	=	NOUN
cana-726	178	43	0	0	PROPN
cana-726	178	44	.	.	PUNCT
cana-726	179	1	(	(	PUNCT
cana-726	179	2	ii	ii	NOUN
cana-726	179	3	)	)	PUNCT
cana-726	179	4	if	if	SCONJ
cana-726	179	5	|ass	|ass	PROPN
cana-726	179	6	(	(	PUNCT
cana-726	179	7	s)|	s)|	PROPN
cana-726	179	8	≥	≥	NUM
cana-726	179	9	3	3	NUM
cana-726	179	10	,	,	PUNCT
cana-726	179	11	then	then	ADV
cana-726	179	12	girth	girth	ADV
cana-726	179	13	(	(	PUNCT
cana-726	179	14	γ(s	γ(s	PROPN
cana-726	179	15	)	)	PUNCT
cana-726	179	16	)	)	PUNCT
cana-726	180	1	=	=	PUNCT
cana-726	180	2	4	4	X
cana-726	180	3	.	.	PUNCT
cana-726	180	4	(	(	PUNCT
cana-726	180	5	iii	iii	X
cana-726	180	6	)	)	PUNCT
cana-726	180	7	if	if	SCONJ
cana-726	180	8	|ass	|ass	PROPN
cana-726	180	9	(	(	PUNCT
cana-726	180	10	s)|	s)|	PROPN
cana-726	180	11	≥	≥	NUM
cana-726	180	12	6	6	NUM
cana-726	180	13	,	,	PUNCT
cana-726	180	14	then	then	ADV
cana-726	180	15	γ(s	γ(	NOUN
cana-726	180	16	)	)	PUNCT
cana-726	180	17	is	be	AUX
cana-726	180	18	not	not	PART
cana-726	180	19	planar	planar	ADJ
cana-726	180	20	(	(	PUNCT
cana-726	180	21	a	a	DET
cana-726	180	22	graph	graph	NOUN
cana-726	180	23	g	g	NOUN
cana-726	180	24	is	be	AUX
cana-726	180	25	planar	planar	ADJ
cana-726	180	26	if	if	SCONJ
cana-726	180	27	it	it	PRON
cana-726	180	28	can	can	AUX
cana-726	180	29	be	be	AUX
cana-726	180	30	drawn	draw	VERB
cana-726	180	31	in	in	ADP
cana-726	180	32	the	the	DET
cana-726	180	33	plane	plane	NOUN
cana-726	180	34	in	in	ADP
cana-726	180	35	such	such	DET
cana-726	180	36	a	a	DET
cana-726	180	37	way	way	NOUN
cana-726	180	38	that	that	PRON
cana-726	180	39	no	no	DET
cana-726	180	40	two	two	NUM
cana-726	180	41	edges	edge	NOUN
cana-726	180	42	meet	meet	VERB
cana-726	180	43	except	except	SCONJ
cana-726	180	44	at	at	ADP
cana-726	180	45	vertex	vertex	NOUN
cana-726	180	46	with	with	ADP
cana-726	180	47	which	which	PRON
cana-726	180	48	they	they	PRON
cana-726	180	49	are	be	AUX
cana-726	180	50	both	both	PRON
cana-726	180	51	incident	incident	NOUN
cana-726	180	52	)	)	PUNCT
cana-726	180	53	.	.	PUNCT
cana-726	181	1	corollary	corollary	ADJ
cana-726	181	2	2.2	2.2	NUM
cana-726	181	3	.	.	PUNCT
cana-726	182	1	let	let	VERB
cana-726	182	2	∅1	∅1	PROPN
cana-726	182	3	=	=	SYM
cana-726	182	4	ann	ann	PROPN
cana-726	182	5	(	(	PUNCT
cana-726	182	6	𝑥1	𝑥1	NOUN
cana-726	182	7	)	)	PUNCT
cana-726	182	8	,	,	PUNCT
cana-726	182	9	∅2	∅2	NOUN
cana-726	182	10	=	=	SYM
cana-726	182	11	ann	ann	PROPN
cana-726	182	12	(	(	PUNCT
cana-726	182	13	𝑥2	𝑥2	PROPN
cana-726	182	14	)	)	PUNCT
cana-726	182	15	,	,	PUNCT
cana-726	182	16	∅3=	∅3=	NUM
cana-726	182	17	ann	ann	PROPN
cana-726	182	18	(	(	PUNCT
cana-726	182	19	𝑥3	𝑥3	NOUN
cana-726	182	20	)	)	PUNCT
cana-726	182	21	and	and	CCONJ
cana-726	182	22	∅4=	∅4=	PROPN
cana-726	182	23	ann	ann	PROPN
cana-726	182	24	(	(	PUNCT
cana-726	182	25	𝑥4	𝑥4	PROPN
cana-726	182	26	)	)	PUNCT
cana-726	182	27	,	,	PUNCT
cana-726	182	28	⋯	⋯	PROPN
cana-726	182	29	∅𝑛=	∅𝑛=	PROPN
cana-726	182	30	ann	ann	PROPN
cana-726	182	31	(	(	PUNCT
cana-726	182	32	𝑥𝑛	𝑥𝑛	NOUN
cana-726	182	33	)	)	PUNCT
cana-726	182	34	belong	belong	VERB
cana-726	182	35	to	to	ADP
cana-726	182	36	ass	ass	NOUN
cana-726	182	37	(	(	PUNCT
cana-726	182	38	s	s	NOUN
cana-726	182	39	)	)	PUNCT
cana-726	182	40	with	with	ADP
cana-726	182	41	reference	reference	NOUN
cana-726	182	42	to	to	ADP
cana-726	182	43	commutative	commutative	ADJ
cana-726	182	44	rings	ring	NOUN
cana-726	182	45	of	of	ADP
cana-726	182	46	median	median	PROPN
cana-726	182	47	and	and	CCONJ
cana-726	182	48	anti	anti	ADJ
cana-726	182	49	–	–	PUNCT
cana-726	182	50	median	median	ADJ
cana-726	182	51	.	.	PUNCT
cana-726	183	1	then	then	ADV
cana-726	183	2	𝑥1	𝑥1	NOUN
cana-726	183	3	−	−	PROPN
cana-726	183	4	𝑥2	𝑥2	PROPN
cana-726	183	5	−	−	PROPN
cana-726	183	6	𝑥3	𝑥3	NOUN
cana-726	183	7	−	−	PROPN
cana-726	183	8	𝑥4	𝑥4	NOUN
cana-726	184	1	−	−	PROPN
cana-726	184	2	⋯	⋯	PROPN
cana-726	184	3	𝑥𝑛	𝑥𝑛	PROPN
cana-726	184	4	is	be	AUX
cana-726	184	5	a	a	DET
cana-726	184	6	cycle	cycle	NOUN
cana-726	184	7	of	of	ADP
cana-726	184	8	length	length	NOUN
cana-726	184	9	n.	n.	PROPN
cana-726	184	10	remark	remark	PROPN
cana-726	184	11	2.2	2.2	NUM
cana-726	184	12	.	.	PUNCT
cana-726	185	1	let	let	VERB
cana-726	185	2	s	s	PRON
cana-726	185	3	be	be	AUX
cana-726	185	4	a	a	DET
cana-726	185	5	commutative	commutative	ADJ
cana-726	185	6	semigroup	semigroup	NOUN
cana-726	185	7	and	and	CCONJ
cana-726	185	8	let	let	VERB
cana-726	185	9	ann	ann	PROPN
cana-726	185	10	a	a	DET
cana-726	185	11	be	be	AUX
cana-726	185	12	a	a	DET
cana-726	185	13	maximal	maximal	ADJ
cana-726	185	14	element	element	NOUN
cana-726	185	15	of	of	ADP
cana-726	185	16	{	{	PUNCT
cana-726	185	17	𝐴𝑛𝑛𝑥	𝐴𝑛𝑛𝑥	PROPN
cana-726	185	18	:	:	PUNCT
cana-726	185	19	0	0	NUM
cana-726	185	20	≠	≠	PROPN
cana-726	185	21	𝑥	𝑥	PRON
cana-726	185	22	∈	∈	PROPN
cana-726	185	23	𝑆	𝑆	PROPN
cana-726	185	24	}	}	PUNCT
cana-726	185	25	.	.	PUNCT
cana-726	186	1	then	then	ADV
cana-726	186	2	ann	ann	PROPN
cana-726	186	3	a	a	PRON
cana-726	186	4	is	be	AUX
cana-726	186	5	a	a	DET
cana-726	186	6	prime	prime	ADJ
cana-726	186	7	ideal	ideal	NOUN
cana-726	186	8	.	.	PUNCT
cana-726	187	1	theorem	theorem	VERB
cana-726	187	2	2.4	2.4	NUM
cana-726	187	3	.	.	PUNCT
cana-726	188	1	let	let	VERB
cana-726	188	2	s	s	PRON
cana-726	188	3	be	be	AUX
cana-726	188	4	a	a	DET
cana-726	188	5	commutative	commutative	ADJ
cana-726	188	6	semigroup	semigroup	NOUN
cana-726	188	7	,	,	PUNCT
cana-726	188	8	then	then	ADV
cana-726	188	9	the	the	DET
cana-726	188	10	median	median	ADJ
cana-726	188	11	graph	graph	NOUN
cana-726	188	12	of	of	ADP
cana-726	188	13	a	a	DET
cana-726	188	14	bipartite	bipartite	NOUN
cana-726	188	15	graph	graph	NOUN
cana-726	188	16	is	be	AUX
cana-726	188	17	induced	induce	VERB
cana-726	188	18	by	by	ADP
cana-726	188	19	the	the	DET
cana-726	188	20	vertices	vertex	NOUN
cana-726	188	21	of	of	ADP
cana-726	188	22	maximum	maximum	ADJ
cana-726	188	23	degree	degree	NOUN
cana-726	188	24	in	in	ADP
cana-726	188	25	g.	g.	PROPN
cana-726	188	26	remark	remark	PROPN
cana-726	188	27	2.3	2.3	NUM
cana-726	188	28	the	the	DET
cana-726	188	29	median	median	ADJ
cana-726	188	30	graph	graph	NOUN
cana-726	188	31	of	of	ADP
cana-726	188	32	a	a	DET
cana-726	188	33	bipartite	bipartite	NOUN
cana-726	188	34	graph	graph	NOUN
cana-726	188	35	is	be	AUX
cana-726	188	36	also	also	ADV
cana-726	188	37	a	a	DET
cana-726	188	38	bipartite	bipartite	ADJ
cana-726	188	39	graph	graph	NOUN
cana-726	188	40	.	.	PUNCT
cana-726	189	1	5	5	X
cana-726	189	2	.	.	X
cana-726	189	3	discussion	discussion	NOUN
cana-726	189	4	theorem	theorem	VERB
cana-726	189	5	2.2	2.2	NUM
cana-726	189	6	.	.	PUNCT
cana-726	190	1	let	let	VERB
cana-726	190	2	s	s	PRON
cana-726	190	3	be	be	AUX
cana-726	190	4	a	a	DET
cana-726	190	5	commutative	commutative	ADJ
cana-726	190	6	and	and	CCONJ
cana-726	190	7	reduced	reduce	VERB
cana-726	190	8	semigroup	semigroup	NOUN
cana-726	190	9	in	in	ADP
cana-726	190	10	which	which	PRON
cana-726	190	11	γ(s	γ(s	PROPN
cana-726	190	12	)	)	PUNCT
cana-726	190	13	does	do	AUX
cana-726	190	14	not	not	PART
cana-726	190	15	contain	contain	VERB
cana-726	190	16	a	a	DET
cana-726	190	17	median	median	ADJ
cana-726	190	18	and	and	CCONJ
cana-726	190	19	anti	anti	ADJ
cana-726	190	20	–	–	PUNCT
cana-726	190	21	median	median	ADJ
cana-726	190	22	clique	clique	NOUN
cana-726	190	23	.	.	PUNCT
cana-726	191	1	then	then	ADV
cana-726	191	2	s	s	VERB
cana-726	191	3	satisfies	satisfie	NOUN
cana-726	191	4	the	the	DET
cana-726	191	5	ascending	ascend	VERB
cana-726	191	6	chain	chain	NOUN
cana-726	191	7	conditions	condition	NOUN
cana-726	191	8	(	(	PUNCT
cana-726	191	9	a.c.c	a.c.c	NOUN
cana-726	191	10	)	)	PUNCT
cana-726	191	11	on	on	ADP
cana-726	191	12	annihilators	annihilators	PROPN
cana-726	191	13	.	.	PUNCT
cana-726	192	1	proof	proof	NOUN
cana-726	192	2	.	.	PUNCT
cana-726	193	1	suppose	suppose	VERB
cana-726	193	2	that	that	SCONJ
cana-726	193	3	ann	ann	PROPN
cana-726	193	4	𝑥1	𝑥1	PROPN
cana-726	193	5	<	<	X
cana-726	193	6	ann	ann	PROPN
cana-726	193	7	𝑥2	𝑥2	PROPN
cana-726	193	8	<	<	X
cana-726	193	9	⋯	⋯	X
cana-726	193	10	<	<	X
cana-726	193	11	𝐴𝑛𝑛	𝐴𝑛𝑛	PROPN
cana-726	193	12	𝑥𝑛	𝑥𝑛	AUX
cana-726	193	13	be	be	AUX
cana-726	193	14	a	a	DET
cana-726	193	15	cumulative	cumulative	ADJ
cana-726	193	16	chain	chain	NOUN
cana-726	193	17	of	of	ADP
cana-726	193	18	ideals	ideal	NOUN
cana-726	193	19	.	.	PUNCT
cana-726	194	1	for	for	ADP
cana-726	194	2	each	each	DET
cana-726	194	3	𝑖	𝑖	PRON
cana-726	194	4	≥	≥	NOUN
cana-726	194	5	2	2	NUM
cana-726	194	6	,	,	PUNCT
cana-726	194	7	select	select	VERB
cana-726	194	8	𝑎	𝑎	PRON
cana-726	194	9	1	1	NUM
cana-726	194	10	∈	∈	PROPN
cana-726	194	11	𝐴𝑛𝑛𝑥1\𝐴𝑛𝑛𝑥𝑖−1	𝐴𝑛𝑛𝑥1\𝐴𝑛𝑛𝑥𝑖−1	PROPN
cana-726	194	12	.	.	PROPN
cana-726	195	1	then	then	ADV
cana-726	195	2	every	every	DET
cana-726	195	3	𝑎𝑛	𝑎𝑛	NOUN
cana-726	195	4	is	be	AUX
cana-726	195	5	nonzero	nonzero	ADJ
cana-726	195	6	,	,	PUNCT
cana-726	195	7	for	for	ADP
cana-726	195	8	𝑛	𝑛	PRON
cana-726	195	9	=	=	SYM
cana-726	195	10	2,3	2,3	NUM
cana-726	195	11	,	,	PUNCT
cana-726	195	12	⋯	⋯	PROPN
cana-726	195	13	.	.	PUNCT
cana-726	196	1	also	also	ADV
cana-726	196	2	𝑦𝑖𝑦𝑗	𝑦𝑖𝑦𝑗	NOUN
cana-726	196	3	for	for	ADP
cana-726	196	4	any	any	DET
cana-726	196	5	𝑖	𝑖	ADP
cana-726	196	6	≠	≠	PROPN
cana-726	196	7	𝑗.	𝑗.	NOUN
cana-726	196	8	since	since	SCONJ
cana-726	196	9	s	s	PROPN
cana-726	196	10	is	be	AUX
cana-726	196	11	a	a	DET
cana-726	196	12	commutative	commutative	ADJ
cana-726	196	13	and	and	CCONJ
cana-726	196	14	condensed	condense	VERB
cana-726	196	15	semigroup	semigroup	NOUN
cana-726	196	16	,	,	PUNCT
cana-726	196	17	we	we	PRON
cana-726	196	18	have	have	VERB
cana-726	196	19	𝑦𝑖	𝑦𝑖	PROPN
cana-726	196	20	≠	≠	PROPN
cana-726	196	21	𝑦𝑗	𝑦𝑗	NOUN
cana-726	196	22	when	when	SCONJ
cana-726	196	23	𝑖	𝑖	X
cana-726	196	24	≠	≠	PROPN
cana-726	196	25	𝑗.	𝑗.	NOUN
cana-726	196	26	thus	thus	ADV
cana-726	196	27	,	,	PUNCT
cana-726	196	28	one	one	PRON
cana-726	196	29	can	can	AUX
cana-726	196	30	obtain	obtain	VERB
cana-726	196	31	a	a	DET
cana-726	196	32	median	median	ADJ
cana-726	196	33	and	and	CCONJ
cana-726	196	34	anti	anti	ADJ
cana-726	196	35	–	–	PUNCT
cana-726	196	36	median	median	NOUN
cana-726	196	37	in	in	ADP
cana-726	196	38	s.	s.	PROPN
cana-726	196	39	this	this	PRON
cana-726	196	40	is	be	AUX
cana-726	196	41	a	a	DET
cana-726	196	42	contradiction	contradiction	NOUN
cana-726	196	43	and	and	CCONJ
cana-726	196	44	so	so	ADV
cana-726	196	45	the	the	DET
cana-726	196	46	assertion	assertion	NOUN
cana-726	196	47	holds	hold	VERB
cana-726	196	48	.	.	PUNCT
cana-726	197	1	theorem	theorem	VERB
cana-726	197	2	2.3	2.3	NUM
cana-726	197	3	.	.	PUNCT
cana-726	198	1	let	let	VERB
cana-726	198	2	s	s	PRON
cana-726	198	3	be	be	AUX
cana-726	198	4	a	a	DET
cana-726	198	5	commutative	commutative	ADJ
cana-726	198	6	ring	ring	NOUN
cana-726	198	7	of	of	ADP
cana-726	198	8	median	median	PROPN
cana-726	198	9	and	and	CCONJ
cana-726	198	10	anti	anti	ADJ
cana-726	198	11	–	–	PUNCT
cana-726	198	12	median	median	ADJ
cana-726	198	13	.	.	PUNCT
cana-726	199	1	then	then	ADV
cana-726	199	2	the	the	DET
cana-726	199	3	subsequent	subsequent	ADJ
cana-726	199	4	results	result	NOUN
cana-726	199	5	hold	hold	VERB
cana-726	199	6	:	:	PUNCT
cana-726	199	7	(	(	PUNCT
cana-726	199	8	i	i	NOUN
cana-726	199	9	)	)	PUNCT
cana-726	199	10	if	if	SCONJ
cana-726	199	11	|ass	|ass	PROPN
cana-726	199	12	(	(	PUNCT
cana-726	199	13	s)|	s)|	PROPN
cana-726	199	14	≥	≥	NUM
cana-726	199	15	3	3	NUM
cana-726	199	16	and	and	CCONJ
cana-726	199	17	∅	∅	NOUN
cana-726	199	18	=	=	SYM
cana-726	199	19	ann	ann	X
cana-726	199	20	(	(	PUNCT
cana-726	199	21	x	x	NOUN
cana-726	199	22	)	)	PUNCT
cana-726	199	23	,	,	PUNCT
cana-726	199	24	χ	χ	X
cana-726	199	25	=	=	PUNCT
cana-726	199	26	ann	ann	PROPN
cana-726	199	27	(	(	PUNCT
cana-726	199	28	y	y	NOUN
cana-726	199	29	)	)	PUNCT
cana-726	199	30	are	be	AUX
cana-726	199	31	two	two	NUM
cana-726	199	32	distinct	distinct	ADJ
cana-726	199	33	elements	element	NOUN
cana-726	199	34	of	of	ADP
cana-726	199	35	ass	ass	NOUN
cana-726	199	36	(	(	PUNCT
cana-726	199	37	s	s	NOUN
cana-726	199	38	)	)	PUNCT
cana-726	199	39	,	,	PUNCT
cana-726	199	40	then	then	ADV
cana-726	199	41	xy	xy	PROPN
cana-726	199	42	=	=	NOUN
cana-726	200	1	0	0	PROPN
cana-726	200	2	.	.	PUNCT
cana-726	200	3	(	(	PUNCT
cana-726	200	4	ii	ii	NOUN
cana-726	200	5	)	)	PUNCT
cana-726	200	6	if	if	SCONJ
cana-726	200	7	|ass	|ass	PROPN
cana-726	200	8	(	(	PUNCT
cana-726	200	9	s)|	s)|	PROPN
cana-726	200	10	≥	≥	NUM
cana-726	200	11	3	3	NUM
cana-726	200	12	,	,	PUNCT
cana-726	200	13	then	then	ADV
cana-726	200	14	girth	girth	ADV
cana-726	200	15	(	(	PUNCT
cana-726	200	16	γ(s	γ(s	PROPN
cana-726	200	17	)	)	PUNCT
cana-726	200	18	)	)	PUNCT
cana-726	201	1	=	=	PUNCT
cana-726	201	2	4	4	X
cana-726	201	3	.	.	PUNCT
cana-726	201	4	(	(	PUNCT
cana-726	201	5	iii	iii	X
cana-726	201	6	)	)	PUNCT
cana-726	201	7	if	if	SCONJ
cana-726	201	8	|ass	|ass	PROPN
cana-726	201	9	(	(	PUNCT
cana-726	201	10	s)|	s)|	PROPN
cana-726	201	11	≥	≥	NUM
cana-726	201	12	6	6	NUM
cana-726	201	13	,	,	PUNCT
cana-726	201	14	then	then	ADV
cana-726	201	15	γ(s	γ(	NOUN
cana-726	201	16	)	)	PUNCT
cana-726	201	17	is	be	AUX
cana-726	201	18	not	not	PART
cana-726	201	19	planar	planar	ADJ
cana-726	201	20	(	(	PUNCT
cana-726	201	21	a	a	DET
cana-726	201	22	graph	graph	NOUN
cana-726	201	23	g	g	NOUN
cana-726	201	24	is	be	AUX
cana-726	201	25	planar	planar	ADJ
cana-726	201	26	if	if	SCONJ
cana-726	201	27	it	it	PRON
cana-726	201	28	can	can	AUX
cana-726	201	29	be	be	AUX
cana-726	201	30	drawn	draw	VERB
cana-726	201	31	in	in	ADP
cana-726	201	32	the	the	DET
cana-726	201	33	plane	plane	NOUN
cana-726	201	34	in	in	ADP
cana-726	201	35	such	such	DET
cana-726	201	36	a	a	DET
cana-726	201	37	way	way	NOUN
cana-726	201	38	that	that	PRON
cana-726	201	39	no	no	DET
cana-726	201	40	two	two	NUM
cana-726	201	41	edges	edge	NOUN
cana-726	201	42	meet	meet	VERB
cana-726	201	43	except	except	SCONJ
cana-726	201	44	at	at	ADP
cana-726	201	45	vertex	vertex	NOUN
cana-726	201	46	with	with	ADP
cana-726	201	47	which	which	PRON
cana-726	201	48	they	they	PRON
cana-726	201	49	are	be	AUX
cana-726	201	50	both	both	PRON
cana-726	201	51	incident	incident	NOUN
cana-726	201	52	)	)	PUNCT
cana-726	201	53	.	.	PUNCT
cana-726	202	1	proof	proof	NOUN
cana-726	202	2	.	.	PUNCT
cana-726	203	1	(	(	PUNCT
cana-726	203	2	i	i	NOUN
cana-726	203	3	)	)	PUNCT
cana-726	203	4	.	.	PUNCT
cana-726	204	1	we	we	PRON
cana-726	204	2	can	can	AUX
cana-726	204	3	assume	assume	VERB
cana-726	204	4	that	that	SCONJ
cana-726	204	5	there	there	PRON
cana-726	204	6	exists	exist	VERB
cana-726	204	7	r	r	NOUN
cana-726	204	8	∈	∈	NOUN
cana-726	204	9	∅	∅	NOUN
cana-726	204	10	\	\	NOUN
cana-726	204	11	χ	χ	NOUN
cana-726	204	12	.	.	PUNCT
cana-726	205	1	then	then	ADV
cana-726	205	2	rx	rx	VERB
cana-726	205	3	=	=	NOUN
cana-726	205	4	0	0	NUM
cana-726	205	5	and	and	CCONJ
cana-726	205	6	so	so	ADV
cana-726	205	7	rsx	rsx	ADJ
cana-726	205	8	=	=	SYM
cana-726	205	9	0	0	NUM
cana-726	205	10	∈	∈	PROPN
cana-726	205	11	∅.	∅.	NOUN
cana-726	205	12	since	since	SCONJ
cana-726	205	13	χ	χ	NOUN
cana-726	205	14	is	be	AUX
cana-726	205	15	a	a	DET
cana-726	205	16	prime	prime	ADJ
cana-726	205	17	ideal	ideal	NOUN
cana-726	205	18	,	,	PUNCT
cana-726	205	19	x	x	PUNCT
cana-726	205	20	∈	∈	NOUN
cana-726	205	21	χ	χ	NOUN
cana-726	205	22	and	and	CCONJ
cana-726	205	23	hence	hence	ADV
cana-726	205	24	xy	xy	PROPN
cana-726	206	1	=	=	SYM
cana-726	206	2	0	0	X
cana-726	206	3	.	.	PUNCT
cana-726	207	1	communications	communication	NOUN
cana-726	207	2	on	on	ADP
cana-726	207	3	applied	apply	VERB
cana-726	207	4	nonlinear	nonlinear	ADJ
cana-726	207	5	analysis	analysis	NOUN
cana-726	207	6	issn	issn	NOUN
cana-726	207	7	:	:	PUNCT
cana-726	207	8	1074	1074	NUM
cana-726	207	9	-	-	PUNCT
cana-726	207	10	133x	133x	NUM
cana-726	207	11	vol	vol	NOUN
cana-726	207	12	31	31	NUM
cana-726	207	13	no	no	NOUN
cana-726	207	14	.	.	PUNCT
cana-726	208	1	3s	3s	NUM
cana-726	208	2	(	(	PUNCT
cana-726	208	3	2024	2024	NUM
cana-726	208	4	)	)	PUNCT
cana-726	208	5	7	7	NUM
cana-726	208	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	208	7	(	(	PUNCT
cana-726	208	8	ii	ii	NOUN
cana-726	208	9	)	)	PUNCT
cana-726	208	10	.	.	PUNCT
cana-726	209	1	let	let	VERB
cana-726	209	2	∅_1	∅_1	VERB
cana-726	209	3	=	=	SYM
cana-726	209	4	ann	ann	PROPN
cana-726	209	5	(	(	PUNCT
cana-726	209	6	x_1	x_1	PROPN
cana-726	209	7	)	)	PUNCT
cana-726	209	8	,	,	PUNCT
cana-726	209	9	∅_2	∅_2	X
cana-726	209	10	=	=	SYM
cana-726	209	11	ann	ann	PROPN
cana-726	209	12	(	(	PUNCT
cana-726	209	13	x_2	x_2	PROPN
cana-726	209	14	)	)	PUNCT
cana-726	209	15	,	,	PUNCT
cana-726	209	16	∅_3=	∅_3=	VERB
cana-726	209	17	ann	ann	PROPN
cana-726	209	18	(	(	PUNCT
cana-726	209	19	x_3	x_3	PROPN
cana-726	209	20	)	)	PUNCT
cana-726	209	21	and	and	CCONJ
cana-726	209	22	∅_4=	∅_4=	VERB
cana-726	209	23	ann	ann	PROPN
cana-726	209	24	(	(	PUNCT
cana-726	209	25	x_4	x_4	PROPN
cana-726	209	26	)	)	PUNCT
cana-726	209	27	belong	belong	VERB
cana-726	209	28	to	to	ADP
cana-726	209	29	ass	ass	NOUN
cana-726	209	30	(	(	PUNCT
cana-726	209	31	s	s	NOUN
cana-726	209	32	)	)	PUNCT
cana-726	209	33	.	.	PUNCT
cana-726	210	1	then	then	ADV
cana-726	210	2	x_1	x_1	PROPN
cana-726	210	3	-	-	PUNCT
cana-726	210	4	x_2	x_2	PROPN
cana-726	210	5	-	-	PUNCT
cana-726	210	6	x_3	x_3	PROPN
cana-726	210	7	-	-	PUNCT
cana-726	210	8	x_4	x_4	PROPN
cana-726	210	9	-	-	PROPN
cana-726	210	10	x_1	x_1	PROPN
cana-726	210	11	is	be	AUX
cana-726	210	12	a	a	DET
cana-726	210	13	cycle	cycle	NOUN
cana-726	210	14	of	of	ADP
cana-726	210	15	length	length	NOUN
cana-726	210	16	4	4	NUM
cana-726	210	17	.	.	PUNCT
cana-726	211	1	(	(	PUNCT
cana-726	211	2	iii	iii	NOUN
cana-726	211	3	)	)	PUNCT
cana-726	211	4	.	.	PUNCT
cana-726	212	1	since	since	SCONJ
cana-726	212	2	|ass	|ass	PROPN
cana-726	212	3	(	(	PUNCT
cana-726	212	4	s)|	s)|	PROPN
cana-726	212	5	≥	≥	PROPN
cana-726	212	6	6	6	NUM
cana-726	212	7	,	,	PUNCT
cana-726	212	8	k_5	k_5	PROPN
cana-726	212	9	is	be	AUX
cana-726	212	10	a	a	DET
cana-726	212	11	subgraph	subgraph	NOUN
cana-726	212	12	of	of	ADP
cana-726	212	13	γ(s	γ(	NOUN
cana-726	212	14	)	)	PUNCT
cana-726	212	15	,	,	PUNCT
cana-726	212	16	and	and	CCONJ
cana-726	212	17	hence	hence	ADV
cana-726	212	18	by	by	ADP
cana-726	212	19	kuratowski	kuratowski	PROPN
cana-726	212	20	’s	’s	PART
cana-726	212	21	theorem	theorem	PROPN
cana-726	212	22	γ(s	γ(	NOUN
cana-726	212	23	)	)	PUNCT
cana-726	212	24	is	be	AUX
cana-726	212	25	not	not	PART
cana-726	212	26	planar	planar	ADJ
cana-726	212	27	.	.	PUNCT
cana-726	213	1	corollary	corollary	ADJ
cana-726	213	2	2.2	2.2	NUM
cana-726	213	3	.	.	PUNCT
cana-726	214	1	let	let	VERB
cana-726	214	2	∅_1	∅_1	NOUN
cana-726	214	3	=	=	SYM
cana-726	214	4	ann	ann	PROPN
cana-726	214	5	(	(	PUNCT
cana-726	214	6	x_1	x_1	PROPN
cana-726	214	7	)	)	PUNCT
cana-726	214	8	,	,	PUNCT
cana-726	214	9	∅_2	∅_2	X
cana-726	214	10	=	=	SYM
cana-726	214	11	ann	ann	PROPN
cana-726	214	12	(	(	PUNCT
cana-726	214	13	x_2	x_2	PROPN
cana-726	214	14	)	)	PUNCT
cana-726	214	15	,	,	PUNCT
cana-726	214	16	∅_3=	∅_3=	VERB
cana-726	214	17	ann	ann	PROPN
cana-726	214	18	(	(	PUNCT
cana-726	214	19	x_3	x_3	PROPN
cana-726	214	20	)	)	PUNCT
cana-726	214	21	and	and	CCONJ
cana-726	214	22	∅_4=	∅_4=	VERB
cana-726	214	23	ann	ann	PROPN
cana-726	214	24	(	(	PUNCT
cana-726	214	25	x_4	x_4	PROPN
cana-726	214	26	)	)	PUNCT
cana-726	214	27	,	,	PUNCT
cana-726	214	28	⋯	⋯	PROPN
cana-726	214	29	∅_n=	∅_n=	PROPN
cana-726	214	30	ann	ann	PROPN
cana-726	214	31	(	(	PUNCT
cana-726	214	32	x_n	x_n	NUM
cana-726	214	33	)	)	PUNCT
cana-726	214	34	belong	belong	VERB
cana-726	214	35	to	to	ADP
cana-726	214	36	ass	ass	NOUN
cana-726	214	37	(	(	PUNCT
cana-726	214	38	s	s	NOUN
cana-726	214	39	)	)	PUNCT
cana-726	214	40	with	with	ADP
cana-726	214	41	reference	reference	NOUN
cana-726	214	42	to	to	ADP
cana-726	214	43	commutative	commutative	ADJ
cana-726	214	44	rings	ring	NOUN
cana-726	214	45	of	of	ADP
cana-726	214	46	median	median	PROPN
cana-726	214	47	and	and	CCONJ
cana-726	214	48	anti	anti	ADJ
cana-726	214	49	–	–	PUNCT
cana-726	214	50	median	median	ADJ
cana-726	214	51	.	.	PUNCT
cana-726	215	1	then	then	ADV
cana-726	215	2	x_1	x_1	PROPN
cana-726	215	3	-	-	PUNCT
cana-726	215	4	x_2	x_2	PROPN
cana-726	215	5	-	-	PUNCT
cana-726	215	6	x_3	x_3	PROPN
cana-726	215	7	-	-	PUNCT
cana-726	215	8	x_4-	x_4-	PROPN
cana-726	215	9	…	…	NUM
cana-726	215	10	x_n	x_n	X
cana-726	215	11	is	be	AUX
cana-726	215	12	a	a	DET
cana-726	215	13	cycle	cycle	NOUN
cana-726	215	14	of	of	ADP
cana-726	215	15	length	length	NOUN
cana-726	215	16	n.	n.	PROPN
cana-726	215	17	remark	remark	PROPN
cana-726	215	18	2.2	2.2	NUM
cana-726	215	19	.	.	PUNCT
cana-726	216	1	let	let	VERB
cana-726	216	2	s	s	PRON
cana-726	216	3	be	be	AUX
cana-726	216	4	a	a	DET
cana-726	216	5	commutative	commutative	ADJ
cana-726	216	6	semigroup	semigroup	NOUN
cana-726	216	7	and	and	CCONJ
cana-726	216	8	let	let	VERB
cana-726	216	9	ann	ann	PROPN
cana-726	216	10	a	a	DET
cana-726	216	11	be	be	AUX
cana-726	216	12	a	a	DET
cana-726	216	13	maximal	maximal	ADJ
cana-726	216	14	element	element	NOUN
cana-726	216	15	of	of	ADP
cana-726	216	16	{	{	PUNCT
cana-726	216	17	annx:0≠x∈s	annx:0≠x∈s	NOUN
cana-726	216	18	}	}	PUNCT
cana-726	216	19	.	.	PUNCT
cana-726	217	1	then	then	ADV
cana-726	217	2	ann	ann	PROPN
cana-726	217	3	a	a	PRON
cana-726	217	4	is	be	AUX
cana-726	217	5	a	a	DET
cana-726	217	6	prime	prime	ADJ
cana-726	217	7	ideal	ideal	NOUN
cana-726	217	8	.	.	PUNCT
cana-726	218	1	theorem	theorem	VERB
cana-726	218	2	2.4	2.4	NUM
cana-726	218	3	.	.	PUNCT
cana-726	219	1	let	let	VERB
cana-726	219	2	s	s	PRON
cana-726	219	3	be	be	AUX
cana-726	219	4	a	a	DET
cana-726	219	5	commutative	commutative	ADJ
cana-726	219	6	semigroup	semigroup	NOUN
cana-726	219	7	,	,	PUNCT
cana-726	219	8	then	then	ADV
cana-726	219	9	the	the	DET
cana-726	219	10	median	median	ADJ
cana-726	219	11	graph	graph	NOUN
cana-726	219	12	of	of	ADP
cana-726	219	13	a	a	DET
cana-726	219	14	bipartite	bipartite	NOUN
cana-726	219	15	graph	graph	NOUN
cana-726	219	16	is	be	AUX
cana-726	219	17	induced	induce	VERB
cana-726	219	18	by	by	ADP
cana-726	219	19	the	the	DET
cana-726	219	20	vertices	vertex	NOUN
cana-726	219	21	of	of	ADP
cana-726	219	22	maximum	maximum	ADJ
cana-726	219	23	degree	degree	NOUN
cana-726	219	24	in	in	ADP
cana-726	219	25	g.	g.	PROPN
cana-726	219	26	proof	proof	NOUN
cana-726	219	27	.	.	PUNCT
cana-726	220	1	s	s	VERB
cana-726	220	2	be	be	AUX
cana-726	220	3	a	a	DET
cana-726	220	4	commutative	commutative	ADJ
cana-726	220	5	semigroup	semigroup	NOUN
cana-726	220	6	and	and	CCONJ
cana-726	220	7	g	g	PROPN
cana-726	220	8	is	be	AUX
cana-726	220	9	a	a	DET
cana-726	220	10	bipartite	bipartite	ADJ
cana-726	220	11	median	median	ADJ
cana-726	220	12	graph	graph	NOUN
cana-726	220	13	,	,	PUNCT
cana-726	220	14	thus	thus	ADV
cana-726	220	15	d(v	d(v	ADJ
cana-726	220	16	,	,	PUNCT
cana-726	220	17	u	u	NOUN
cana-726	220	18	)	)	PUNCT
cana-726	220	19	<	<	X
cana-726	220	20	2	2	NUM
cana-726	220	21	for	for	ADP
cana-726	220	22	any	any	DET
cana-726	220	23	pair	pair	NOUN
cana-726	220	24	of	of	ADP
cana-726	220	25	vertices	vertex	NOUN
cana-726	220	26	u	u	NOUN
cana-726	220	27	,	,	PUNCT
cana-726	220	28	v	v	NOUN
cana-726	220	29	of	of	ADP
cana-726	220	30	g.	g.	PROPN
cana-726	220	31	let	let	VERB
cana-726	220	32	the	the	DET
cana-726	220	33	degree	degree	NOUN
cana-726	220	34	of	of	ADP
cana-726	220	35	v	v	NOUN
cana-726	220	36	in	in	ADP
cana-726	220	37	g	g	PROPN
cana-726	220	38	be	be	PROPN
cana-726	220	39	d.	d.	PROPN
cana-726	220	40	then	then	ADV
cana-726	220	41	,	,	PUNCT
cana-726	220	42	these	these	DET
cana-726	220	43	d	d	NOUN
cana-726	220	44	vertices	vertex	NOUN
cana-726	220	45	are	be	AUX
cana-726	220	46	at	at	ADP
cana-726	220	47	a	a	DET
cana-726	220	48	distance	distance	NOUN
cana-726	220	49	1	1	NUM
cana-726	220	50	from	from	ADP
cana-726	220	51	v.	v.	ADP
cana-726	220	52	so	so	ADV
cana-726	220	53	,	,	PUNCT
cana-726	220	54	there	there	PRON
cana-726	220	55	are	be	VERB
cana-726	220	56	p	p	NOUN
cana-726	220	57	1	1	NUM
cana-726	220	58	d	d	NOUN
cana-726	220	59	vertices	vertice	VERB
cana-726	220	60	u	u	NOUN
cana-726	220	61	in	in	ADP
cana-726	220	62	g	g	PROPN
cana-726	220	63	such	such	ADJ
cana-726	220	64	that	that	SCONJ
cana-726	220	65	d(v	d(v	PROPN
cana-726	220	66	,	,	PUNCT
cana-726	220	67	u	u	NOUN
cana-726	220	68	)	)	PUNCT
cana-726	220	69	=	=	SYM
cana-726	220	70	2	2	NUM
cana-726	220	71	and	and	CCONJ
cana-726	220	72	d(v	d(v	ADJ
cana-726	220	73	)	)	PUNCT
cana-726	220	74	=	=	PUNCT
cana-726	221	1	d	d	PROPN
cana-726	221	2	+	+	CCONJ
cana-726	221	3	2(p	2(p	NUM
cana-726	221	4	1	1	NUM
cana-726	221	5	d	d	NOUN
cana-726	221	6	)	)	PUNCT
cana-726	221	7	=	=	SYM
cana-726	221	8	2(p	2(p	NUM
cana-726	221	9	1	1	X
cana-726	221	10	)	)	PUNCT
cana-726	221	11	d.	d.	NOUN
cana-726	221	12	hence	hence	ADV
cana-726	221	13	the	the	DET
cana-726	221	14	vertices	vertex	NOUN
cana-726	221	15	in	in	ADP
cana-726	221	16	g	g	PROPN
cana-726	221	17	such	such	DET
cana-726	221	18	that	that	DET
cana-726	221	19	d(v)is	d(v)is	ADJ
cana-726	221	20	minimum	minimum	NOUN
cana-726	221	21	are	be	AUX
cana-726	221	22	those	those	PRON
cana-726	221	23	for	for	ADP
cana-726	221	24	which	which	PRON
cana-726	221	25	the	the	DET
cana-726	221	26	degree	degree	NOUN
cana-726	221	27	is	be	AUX
cana-726	221	28	maximum	maximum	ADJ
cana-726	221	29	.	.	PUNCT
cana-726	222	1	hence	hence	ADV
cana-726	222	2	proved	prove	VERB
cana-726	222	3	.	.	PUNCT
cana-726	223	1	remark	remark	VERB
cana-726	223	2	2.3	2.3	NUM
cana-726	223	3	.	.	PUNCT
cana-726	224	1	the	the	DET
cana-726	224	2	median	median	ADJ
cana-726	224	3	graph	graph	NOUN
cana-726	224	4	of	of	ADP
cana-726	224	5	a	a	DET
cana-726	224	6	bipartite	bipartite	NOUN
cana-726	224	7	graph	graph	NOUN
cana-726	224	8	is	be	AUX
cana-726	224	9	also	also	ADV
cana-726	224	10	a	a	DET
cana-726	224	11	bipartite	bipartite	ADJ
cana-726	224	12	graph	graph	NOUN
cana-726	224	13	.	.	PUNCT
cana-726	225	1	references	reference	NOUN
cana-726	225	2	[	[	X
cana-726	225	3	1	1	NUM
cana-726	225	4	]	]	PUNCT
cana-726	225	5	k.	k.	PROPN
cana-726	225	6	balakrishnan	balakrishnan	PROPN
cana-726	225	7	,	,	PUNCT
cana-726	225	8	b.	b.	PROPN
cana-726	225	9	breˇsar	breˇsar	PROPN
cana-726	225	10	,	,	PUNCT
cana-726	225	11	m.	m.	NOUN
cana-726	225	12	kovˇse	kovˇse	PROPN
cana-726	225	13	,	,	PUNCT
cana-726	225	14	m.	m.	NOUN
cana-726	225	15	changat	changat	PROPN
cana-726	225	16	,	,	PUNCT
cana-726	225	17	a.r	a.r	PROPN
cana-726	225	18	.	.	PROPN
cana-726	225	19	subhamathi	subhamathi	PROPN
cana-726	225	20	and	and	CCONJ
cana-726	225	21	s.	s.	PROPN
cana-726	225	22	klavˇzar	klavˇzar	PROPN
cana-726	225	23	,	,	PUNCT
cana-726	225	24	simultaneous	simultaneous	ADJ
cana-726	225	25	embeddings	embedding	NOUN
cana-726	225	26	of	of	ADP
cana-726	225	27	s	s	PRON
cana-726	225	28	as	as	ADP
cana-726	225	29	median	median	PROPN
cana-726	225	30	and	and	CCONJ
cana-726	225	31	antimedian	antimedian	PROPN
cana-726	225	32	446	446	NUM
cana-726	225	33	k.	k.	PROPN
cana-726	225	34	pravas	pravas	PROPN
cana-726	225	35	and	and	CCONJ
cana-726	225	36	a.	a.	PROPN
cana-726	225	37	vijayakumar	vijayakumar	PROPN
cana-726	225	38	subs	subs	PROPN
cana-726	225	39	,	,	PUNCT
cana-726	225	40	networks	network	VERB
cana-726	225	41	56	56	NUM
cana-726	225	42	(	(	PUNCT
cana-726	225	43	2010	2010	NUM
cana-726	225	44	)	)	PUNCT
cana-726	225	45	90–94	90–94	NUM
cana-726	225	46	.	.	PUNCT
cana-726	226	1	doi:10.002	doi:10.002	PROPN
cana-726	226	2	/	/	SYM
cana-726	226	3	net.20350	net.20350	PRON
cana-726	227	1	[	[	X
cana-726	227	2	2	2	NUM
cana-726	227	3	]	]	PUNCT
cana-726	227	4	r.	r.	PROPN
cana-726	227	5	balakrishnan	balakrishnan	PROPN
cana-726	227	6	and	and	CCONJ
cana-726	227	7	k.	k.	PROPN
cana-726	227	8	ranganathan	ranganathan	PROPN
cana-726	227	9	,	,	PUNCT
cana-726	227	10	a	a	DET
cana-726	227	11	textbook	textbook	NOUN
cana-726	227	12	of	of	ADP
cana-726	227	13	graph	graph	NOUN
cana-726	227	14	theory	theory	NOUN
cana-726	227	15	,	,	PUNCT
cana-726	227	16	second	second	ADJ
cana-726	227	17	edition	edition	NOUN
cana-726	227	18	(	(	PUNCT
cana-726	227	19	heidelberg	heidelberg	PROPN
cana-726	227	20	,	,	PUNCT
cana-726	227	21	springer	springer	NOUN
cana-726	227	22	,	,	PUNCT
cana-726	227	23	2012	2012	NUM
cana-726	227	24	)	)	PUNCT
cana-726	227	25	.	.	PUNCT
cana-726	228	1	doi:10.1007/978	doi:10.1007/978	ADJ
cana-726	228	2	-	-	PUNCT
cana-726	228	3	1	1	NUM
cana-726	228	4	-	-	PUNCT
cana-726	228	5	4614	4614	NUM
cana-726	228	6	-	-	PUNCT
cana-726	228	7	4529	4529	NUM
cana-726	228	8	-	-	SYM
cana-726	228	9	6	6	NUM
cana-726	229	1	[	[	SYM
cana-726	229	2	3	3	NUM
cana-726	229	3	]	]	X
cana-726	229	4	h.	h.	NOUN
cana-726	229	5	bielak	bielak	PROPN
cana-726	229	6	and	and	CCONJ
cana-726	229	7	m.m	m.m	PROPN
cana-726	229	8	.	.	PROPN
cana-726	229	9	sys	sys	PROPN
cana-726	229	10	lo	lo	PROPN
cana-726	229	11	,	,	PUNCT
cana-726	229	12	peripheral	peripheral	ADJ
cana-726	229	13	vertices	vertex	NOUN
cana-726	229	14	in	in	ADP
cana-726	229	15	s	s	PROPN
cana-726	229	16	,	,	PUNCT
cana-726	229	17	studia	studia	PROPN
cana-726	229	18	sci	sci	PROPN
cana-726	229	19	.	.	PUNCT
cana-726	229	20	math	math	PROPN
cana-726	229	21	.	.	PUNCT
cana-726	230	1	hungar	hungar	NOUN
cana-726	230	2	.	.	PUNCT
cana-726	231	1	18	18	NUM
cana-726	231	2	(	(	PUNCT
cana-726	231	3	1983	1983	NUM
cana-726	231	4	)	)	PUNCT
cana-726	232	1	269–275	269–275	NUM
cana-726	232	2	.	.	PUNCT
cana-726	233	1	[	[	X
cana-726	233	2	4	4	X
cana-726	233	3	]	]	PUNCT
cana-726	233	4	h.	h.	PROPN
cana-726	233	5	kautz	kautz	PROPN
cana-726	233	6	,	,	PUNCT
cana-726	233	7	b.	b.	PROPN
cana-726	233	8	selman	selman	PROPN
cana-726	233	9	and	and	CCONJ
cana-726	233	10	m.	m.	NOUN
cana-726	233	11	shah	shah	PROPN
cana-726	233	12	,	,	PUNCT
cana-726	233	13	referral	referral	ADJ
cana-726	233	14	web	web	NOUN
cana-726	233	15	:	:	PUNCT
cana-726	233	16	combining	combine	VERB
cana-726	233	17	social	social	ADJ
cana-726	233	18	networks	network	NOUN
cana-726	233	19	and	and	CCONJ
cana-726	233	20	collaborative	collaborative	ADJ
cana-726	233	21	filtering	filtering	NOUN
cana-726	233	22	,	,	PUNCT
cana-726	233	23	communications	communication	NOUN
cana-726	233	24	of	of	ADP
cana-726	233	25	the	the	DET
cana-726	233	26	acm	acm	PROPN
cana-726	233	27	40(3	40(3	NUM
cana-726	233	28	)	)	PUNCT
cana-726	233	29	(	(	PUNCT
cana-726	233	30	1997	1997	NUM
cana-726	233	31	)	)	PUNCT
cana-726	233	32	63	63	NUM
cana-726	233	33	–	–	PUNCT
cana-726	233	34	65	65	NUM
cana-726	233	35	.	.	PUNCT
cana-726	234	1	doi:10.1145/245108.245123	doi:10.1145/245108.245123	NOUN
cana-726	235	1	[	[	X
cana-726	235	2	5	5	X
cana-726	235	3	]	]	PUNCT
cana-726	235	4	karuvachery	karuvachery	ADJ
cana-726	235	5	pravas	pravas	PROPN
cana-726	235	6	and	and	CCONJ
cana-726	235	7	ambat	ambat	PROPN
cana-726	235	8	vijayakumar	vijayakumar	PROPN
cana-726	235	9	,	,	PUNCT
cana-726	235	10	the	the	DET
cana-726	235	11	median	median	ADJ
cana-726	235	12	problem	problem	NOUN
cana-726	235	13	on	on	ADP
cana-726	235	14	k	k	X
cana-726	235	15	-	-	ADJ
cana-726	235	16	partite	partite	ADJ
cana-726	235	17	s	s	X
cana-726	235	18	,	,	PUNCT
cana-726	235	19	discussions	discussion	NOUN
cana-726	235	20	mathematical	mathematical	ADJ
cana-726	235	21	theory	theory	NOUN
cana-726	235	22	,	,	PUNCT
cana-726	235	23	35	35	NUM
cana-726	235	24	(	(	PUNCT
cana-726	235	25	2015	2015	NUM
cana-726	235	26	)	)	PUNCT
cana-726	235	27	439–446	439–446	NUM
cana-726	235	28	doi:10.7151	doi:10.7151	NOUN
cana-726	235	29	/	/	SYM
cana-726	235	30	dmgt.1802	dmgt.1802	PROPN
cana-726	235	31	.	.	PUNCT
cana-726	236	1	[	[	X
cana-726	236	2	6	6	NUM
cana-726	236	3	]	]	PUNCT
cana-726	236	4	k.	k.	PROPN
cana-726	236	5	pravas	pravas	PROPN
cana-726	236	6	and	and	CCONJ
cana-726	236	7	a.	a.	NOUN
cana-726	236	8	vijayakumar	vijayakumar	PROPN
cana-726	236	9	,	,	PUNCT
cana-726	236	10	convex	convex	NOUN
cana-726	236	11	median	median	NOUN
cana-726	236	12	and	and	CCONJ
cana-726	236	13	anti	anti	ADJ
cana-726	236	14	-	-	ADJ
cana-726	236	15	median	median	ADJ
cana-726	236	16	at	at	ADP
cana-726	236	17	prescribed	prescribed	ADJ
cana-726	236	18	distance	distance	NOUN
cana-726	236	19	,	,	PUNCT
cana-726	236	20	communicated	communicate	VERB
cana-726	236	21	.	.	PUNCT
cana-726	237	1	[	[	X
cana-726	237	2	7	7	NUM
cana-726	237	3	]	]	X
cana-726	237	4	p.j	p.j	PROPN
cana-726	237	5	.	.	PROPN
cana-726	237	6	slater	slater	PROPN
cana-726	237	7	,	,	PUNCT
cana-726	237	8	medians	median	NOUN
cana-726	237	9	of	of	ADP
cana-726	237	10	arbitrary	arbitrary	ADJ
cana-726	237	11	s	s	NOUN
cana-726	237	12	,	,	PUNCT
cana-726	237	13	j.	j.	PROPN
cana-726	237	14	theory	theory	PROPN
cana-726	237	15	4	4	NUM
cana-726	237	16	(	(	PUNCT
cana-726	237	17	1980	1980	NUM
cana-726	237	18	)	)	PUNCT
cana-726	237	19	389–392	389–392	NUM
cana-726	237	20	.	.	PUNCT
cana-726	237	21	doi:10.1002	doi:10.1002	NOUN
cana-726	237	22	/	/	SYM
cana-726	237	23	jgt.3190040408	jgt.3190040408	NOUN
cana-726	237	24	[	[	X
cana-726	237	25	8	8	NUM
cana-726	237	26	]	]	X
cana-726	237	27	s.b	s.b	PROPN
cana-726	237	28	.	.	PROPN
cana-726	237	29	rao	rao	PROPN
cana-726	237	30	and	and	CCONJ
cana-726	237	31	a.vijayakumar	a.vijayakumar	NOUN
cana-726	237	32	,	,	PUNCT
cana-726	237	33	on	on	ADP
cana-726	237	34	the	the	DET
cana-726	237	35	median	median	NOUN
cana-726	237	36	and	and	CCONJ
cana-726	237	37	the	the	DET
cana-726	237	38	anti	anti	ADJ
cana-726	237	39	-	-	ADJ
cana-726	237	40	median	median	NOUN
cana-726	237	41	of	of	ADP
cana-726	237	42	a	a	DET
cana-726	237	43	co	co	NOUN
cana-726	237	44	,	,	PUNCT
cana-726	237	45	int	int	PROPN
cana-726	237	46	.	.	PUNCT
cana-726	238	1	j.	j.	PROPN
cana-726	238	2	pure	pure	PROPN
cana-726	238	3	appl	appl	PROPN
cana-726	238	4	.	.	PUNCT
cana-726	238	5	math	math	NOUN
cana-726	238	6	.	.	PUNCT
cana-726	239	1	46	46	NUM
cana-726	239	2	(	(	PUNCT
cana-726	239	3	2008	2008	NUM
cana-726	239	4	)	)	PUNCT
cana-726	239	5	703	703	NUM
cana-726	239	6	–	–	PUNCT
cana-726	239	7	710	710	NUM
cana-726	239	8	.	.	PUNCT
cana-726	240	1	[	[	X
cana-726	240	2	9	9	NUM
cana-726	240	3	]	]	X
cana-726	240	4	h.g	h.g	PROPN
cana-726	240	5	.	.	PROPN
cana-726	240	6	yeh	yeh	PROPN
cana-726	240	7	and	and	CCONJ
cana-726	240	8	g.j	g.j	PROPN
cana-726	240	9	.	.	PROPN
cana-726	240	10	chang	chang	PROPN
cana-726	240	11	,	,	PUNCT
cana-726	240	12	centers	center	NOUN
cana-726	240	13	and	and	CCONJ
cana-726	240	14	medians	median	NOUN
cana-726	240	15	of	of	ADP
cana-726	240	16	distance	distance	NOUN
cana-726	240	17	-	-	PUNCT
cana-726	240	18	hereditary	hereditary	NOUN
cana-726	240	19	s	s	NOUN
cana-726	240	20	,	,	PUNCT
cana-726	240	21	discrete	discrete	ADJ
cana-726	240	22	math	math	NOUN
cana-726	240	23	.	.	PUNCT
cana-726	241	1	265	265	NUM
cana-726	241	2	(	(	PUNCT
cana-726	241	3	2003	2003	NUM
cana-726	241	4	)	)	PUNCT
cana-726	241	5	297–310	297–310	NUM
cana-726	241	6	.	.	PUNCT
cana-726	242	1	doi:10.1016	doi:10.1016	PROPN
cana-726	242	2	/	/	SYM
cana-726	242	3	s0012	s0012	PROPN
cana-726	242	4	-	-	PUNCT
cana-726	242	5	365x(02)00630	365x(02)00630	NOUN
cana-726	242	6	-	-	PUNCT
cana-726	242	7	1	1	NUM
cana-726	242	8	.	.	PUNCT
cana-726	243	1	[	[	X
cana-726	243	2	10	10	NUM
cana-726	243	3	]	]	X
cana-726	243	4	s.	s.	PROPN
cana-726	243	5	akbari	akbari	PROPN
cana-726	243	6	,	,	PUNCT
cana-726	243	7	h.	h.	PROPN
cana-726	243	8	r.	r.	PROPN
cana-726	243	9	maimani	maimani	PROPN
cana-726	243	10	,	,	PUNCT
cana-726	243	11	s.	s.	PROPN
cana-726	243	12	yassemi	yassemi	PROPN
cana-726	243	13	,	,	PUNCT
cana-726	243	14	when	when	SCONJ
cana-726	243	15	a	a	DET
cana-726	243	16	zero	zero	NUM
cana-726	243	17	-	-	PUNCT
cana-726	243	18	divisor	divisor	NOUN
cana-726	243	19	graph	graph	NOUN
cana-726	243	20	is	be	AUX
cana-726	243	21	planar	planar	ADJ
cana-726	243	22	or	or	CCONJ
cana-726	243	23	a	a	DET
cana-726	243	24	complete	complete	ADJ
cana-726	243	25	r	r	NOUN
cana-726	243	26	-	-	ADJ
cana-726	243	27	partite	partite	ADJ
cana-726	243	28	graph	graph	NOUN
cana-726	243	29	,	,	PUNCT
cana-726	243	30	j.	j.	PROPN
cana-726	243	31	algebra	algebra	PROPN
cana-726	243	32	270	270	NUM
cana-726	243	33	(	(	PUNCT
cana-726	243	34	2003	2003	NUM
cana-726	243	35	)	)	PUNCT
cana-726	243	36	,	,	PUNCT
cana-726	243	37	169	169	NUM
cana-726	243	38	-180	-180	PROPN
cana-726	243	39	.	.	PUNCT
cana-726	244	1	[	[	X
cana-726	244	2	11	11	NUM
cana-726	244	3	]	]	X
cana-726	244	4	d.	d.	PROPN
cana-726	244	5	d.	d.	PROPN
cana-726	244	6	anderson	anderson	PROPN
cana-726	244	7	,	,	PUNCT
cana-726	244	8	finitely	finitely	ADV
cana-726	244	9	generated	generate	VERB
cana-726	244	10	multiplicative	multiplicative	ADJ
cana-726	244	11	subsemigroups	subsemigroup	NOUN
cana-726	244	12	of	of	ADP
cana-726	244	13	rings	ring	NOUN
cana-726	244	14	,	,	PUNCT
cana-726	244	15	semigroup	semigroup	PROPN
cana-726	244	16	forum	forum	PROPN
cana-726	244	17	55	55	NUM
cana-726	244	18	(	(	PUNCT
cana-726	244	19	1997	1997	NUM
cana-726	244	20	)	)	PUNCT
cana-726	244	21	,	,	PUNCT
cana-726	244	22	294	294	NUM
cana-726	244	23	-	-	SYM
cana-726	244	24	298	298	NUM
cana-726	244	25	.	.	PUNCT
cana-726	245	1	[	[	X
cana-726	245	2	12	12	NUM
cana-726	245	3	]	]	X
cana-726	245	4	d.	d.	PROPN
cana-726	245	5	d.	d.	PROPN
cana-726	245	6	anderson	anderson	PROPN
cana-726	245	7	,	,	PUNCT
cana-726	245	8	and	and	CCONJ
cana-726	245	9	e.w	e.w	PROPN
cana-726	245	10	.	.	PROPN
cana-726	245	11	johnson	johnson	PROPN
cana-726	245	12	,	,	PUNCT
cana-726	245	13	ideal	ideal	ADJ
cana-726	245	14	theory	theory	NOUN
cana-726	245	15	in	in	ADP
cana-726	245	16	commutative	commutative	ADJ
cana-726	245	17	semigroups	semigroup	NOUN
cana-726	245	18	,	,	PUNCT
cana-726	245	19	semigroup	semigroup	PROPN
cana-726	245	20	forum	forum	PROPN
cana-726	245	21	30	30	NUM
cana-726	245	22	(	(	PUNCT
cana-726	245	23	1984	1984	NUM
cana-726	245	24	)	)	PUNCT
cana-726	245	25	,	,	PUNCT
cana-726	245	26	127158	127158	NUM
cana-726	245	27	.	.	PUNCT
cana-726	246	1	[	[	X
cana-726	246	2	13	13	NUM
cana-726	246	3	]	]	X
cana-726	246	4	d.	d.	PROPN
cana-726	246	5	d.	d.	PROPN
cana-726	246	6	anderson	anderson	PROPN
cana-726	246	7	,	,	PUNCT
cana-726	246	8	m.	m.	PROPN
cana-726	246	9	naseer	naseer	PROPN
cana-726	246	10	,	,	PUNCT
cana-726	246	11	beck	beck	PROPN
cana-726	246	12	's	's	PART
cana-726	246	13	coloring	coloring	NOUN
cana-726	246	14	of	of	ADP
cana-726	246	15	a	a	DET
cana-726	246	16	commutative	commutative	ADJ
cana-726	246	17	ring	ring	NOUN
cana-726	246	18	,	,	PUNCT
cana-726	246	19	j.	j.	PROPN
cana-726	246	20	algebra	algebra	PROPN
cana-726	246	21	159	159	NUM
cana-726	246	22	(	(	PUNCT
cana-726	246	23	1991	1991	NUM
cana-726	246	24	)	)	PUNCT
cana-726	246	25	,	,	PUNCT
cana-726	246	26	500	500	NUM
cana-726	246	27	-	-	SYM
cana-726	246	28	514	514	NUM
cana-726	246	29	.	.	PUNCT
cana-726	247	1	[	[	X
cana-726	247	2	14	14	NUM
cana-726	247	3	]	]	X
cana-726	247	4	d.	d.	PROPN
cana-726	247	5	f.	f.	PROPN
cana-726	247	6	anderson	anderson	PROPN
cana-726	247	7	,	,	PUNCT
cana-726	247	8	a.	a.	PROPN
cana-726	247	9	frazier	frazier	PROPN
cana-726	247	10	,	,	PUNCT
cana-726	247	11	a.	a.	NOUN
cana-726	247	12	lauve	lauve	PROPN
cana-726	247	13	,	,	PUNCT
cana-726	247	14	p.	p.	PROPN
cana-726	247	15	s.	s.	PROPN
cana-726	247	16	livingston	livingston	PROPN
cana-726	247	17	,	,	PUNCT
cana-726	247	18	the	the	DET
cana-726	247	19	zero	zero	NUM
cana-726	247	20	-	-	PUNCT
cana-726	247	21	divisor	divisor	NOUN
cana-726	247	22	graph	graph	NOUN
cana-726	247	23	of	of	ADP
cana-726	247	24	a	a	DET
cana-726	247	25	commutative	commutative	ADJ
cana-726	247	26	ring	ring	PROPN
cana-726	247	27	ii	ii	PROPN
cana-726	247	28	,	,	PUNCT
cana-726	247	29	lecture	lecture	NOUN
cana-726	247	30	notes	note	NOUN
cana-726	247	31	in	in	ADP
cana-726	247	32	pure	pure	ADJ
cana-726	247	33	and	and	CCONJ
cana-726	247	34	appl	appl	NOUN
cana-726	247	35	.	.	PROPN
cana-726	247	36	math	math	NOUN
cana-726	247	37	.	.	PUNCT
cana-726	248	1	,220	,220	PROPN
cana-726	248	2	,	,	PUNCT
cana-726	248	3	dekker	dekker	NOUN
cana-726	248	4	,	,	PUNCT
cana-726	248	5	new	new	PROPN
cana-726	248	6	york	york	PROPN
cana-726	248	7	,	,	PUNCT
cana-726	248	8	2001	2001	NUM
cana-726	248	9	.	.	PUNCT
cana-726	249	1	[	[	X
cana-726	249	2	15	15	NUM
cana-726	249	3	]	]	X
cana-726	249	4	d.	d.	PROPN
cana-726	249	5	f.	f.	PROPN
cana-726	249	6	anderson	anderson	PROPN
cana-726	249	7	,	,	PUNCT
cana-726	249	8	p.	p.	PROPN
cana-726	249	9	s.	s.	PROPN
cana-726	249	10	livingston	livingston	PROPN
cana-726	249	11	,	,	PUNCT
cana-726	249	12	the	the	DET
cana-726	249	13	zero	zero	NUM
cana-726	249	14	-	-	PUNCT
cana-726	249	15	divisor	divisor	NOUN
cana-726	249	16	graph	graph	NOUN
cana-726	249	17	of	of	ADP
cana-726	249	18	a	a	DET
cana-726	249	19	commutative	commutative	ADJ
cana-726	249	20	ring	ring	NOUN
cana-726	249	21	,	,	PUNCT
cana-726	249	22	j.	j.	PROPN
cana-726	249	23	algebra	algebra	PROPN
cana-726	249	24	217	217	NUM
cana-726	249	25	(	(	PUNCT
cana-726	249	26	1999	1999	NUM
cana-726	249	27	)	)	PUNCT
cana-726	249	28	,	,	PUNCT
cana-726	249	29	434	434	NUM
cana-726	249	30	-	-	SYM
cana-726	249	31	447	447	NUM
cana-726	249	32	.	.	PUNCT
cana-726	250	1	communications	communication	NOUN
cana-726	250	2	on	on	ADP
cana-726	250	3	applied	apply	VERB
cana-726	250	4	nonlinear	nonlinear	ADJ
cana-726	250	5	analysis	analysis	NOUN
cana-726	250	6	issn	issn	NOUN
cana-726	250	7	:	:	PUNCT
cana-726	250	8	1074	1074	NUM
cana-726	250	9	-	-	PUNCT
cana-726	250	10	133x	133x	NUM
cana-726	250	11	vol	vol	NOUN
cana-726	250	12	31	31	NUM
cana-726	250	13	no	no	NOUN
cana-726	250	14	.	.	PUNCT
cana-726	251	1	3s	3s	NUM
cana-726	251	2	(	(	PUNCT
cana-726	251	3	2024	2024	NUM
cana-726	251	4	)	)	PUNCT
cana-726	251	5	8	8	NUM
cana-726	251	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-726	252	1	[	[	X
cana-726	252	2	16	16	NUM
cana-726	252	3	]	]	X
cana-726	252	4	d.	d.	PROPN
cana-726	252	5	f.	f.	PROPN
cana-726	252	6	anderson	anderson	PROPN
cana-726	252	7	,	,	PUNCT
cana-726	252	8	r.	r.	PROPN
cana-726	252	9	levy	levy	PROPN
cana-726	252	10	,	,	PUNCT
cana-726	252	11	j.	j.	PROPN
cana-726	252	12	shapiro	shapiro	PROPN
cana-726	252	13	,	,	PUNCT
cana-726	252	14	zero	zero	NUM
cana-726	252	15	-	-	PUNCT
cana-726	252	16	divisor	divisor	NOUN
cana-726	252	17	graphs	graph	NOUN
cana-726	252	18	,	,	PUNCT
cana-726	252	19	von	von	PROPN
cana-726	252	20	neumann	neumann	PROPN
cana-726	252	21	regular	regular	PROPN
cana-726	252	22	rings	ring	NOUN
cana-726	252	23	,	,	PUNCT
cana-726	252	24	and	and	CCONJ
cana-726	252	25	boolean	boolean	ADJ
cana-726	252	26	algebras	algebra	NOUN
cana-726	252	27	,	,	PUNCT
cana-726	252	28	j.	j.	PROPN
cana-726	252	29	pure	pure	PROPN
cana-726	252	30	appl	appl	PROPN
cana-726	252	31	.	.	PUNCT
cana-726	253	1	algebra	algebra	NOUN
cana-726	253	2	180	180	NUM
cana-726	253	3	(	(	PUNCT
cana-726	253	4	2003	2003	NUM
cana-726	253	5	)	)	PUNCT
cana-726	253	6	,	,	PUNCT
cana-726	253	7	221	221	NUM
cana-726	253	8	-	-	SYM
cana-726	253	9	241	241	NUM
cana-726	253	10	.	.	PUNCT
cana-726	254	1	[	[	X
cana-726	254	2	17	17	NUM
cana-726	254	3	]	]	X
cana-726	254	4	i.	i.	PROPN
cana-726	254	5	beck	beck	PROPN
cana-726	254	6	,	,	PUNCT
cana-726	254	7	coloring	coloring	NOUN
cana-726	254	8	of	of	ADP
cana-726	254	9	commutative	commutative	ADJ
cana-726	254	10	rings	ring	NOUN
cana-726	254	11	,	,	PUNCT
cana-726	254	12	j.	j.	PROPN
cana-726	254	13	algebra	algebra	PROPN
cana-726	254	14	116	116	NUM
cana-726	254	15	(	(	PUNCT
cana-726	254	16	1988	1988	NUM
cana-726	254	17	)	)	PUNCT
cana-726	254	18	,	,	PUNCT
cana-726	254	19	208	208	NUM
cana-726	254	20	-	-	SYM
cana-726	254	21	226	226	NUM
cana-726	254	22	.	.	PUNCT
cana-726	255	1	[	[	X
cana-726	255	2	18	18	NUM
cana-726	255	3	]	]	X
cana-726	255	4	r.	r.	PROPN
cana-726	255	5	belsho	belsho	PROPN
cana-726	255	6	and	and	CCONJ
cana-726	255	7	j.	j.	PROPN
cana-726	255	8	chapman	chapman	PROPN
cana-726	255	9	,	,	PUNCT
cana-726	255	10	planar	planar	ADJ
cana-726	255	11	zero	zero	NUM
cana-726	255	12	-	-	PUNCT
cana-726	255	13	divisor	divisor	NOUN
cana-726	255	14	graphs	graph	NOUN
cana-726	255	15	,	,	PUNCT
cana-726	255	16	j.	j.	PROPN
cana-726	255	17	algebra	algebra	PROPN
cana-726	255	18	316	316	NUM
cana-726	255	19	(	(	PUNCT
cana-726	255	20	2007	2007	NUM
cana-726	255	21	)	)	PUNCT
cana-726	255	22	,	,	PUNCT
cana-726	255	23	471480	471480	NUM
cana-726	255	24	.	.	PUNCT
cana-726	256	1	[	[	X
cana-726	256	2	19	19	NUM
cana-726	256	3	]	]	X
cana-726	256	4	g.	g.	PROPN
cana-726	256	5	chartrand	chartrand	PROPN
cana-726	256	6	,	,	PUNCT
cana-726	256	7	o.	o.	PROPN
cana-726	256	8	r.	r.	PROPN
cana-726	256	9	oellermann	oellermann	PROPN
cana-726	256	10	,	,	PUNCT
cana-726	256	11	applied	apply	VERB
cana-726	256	12	and	and	CCONJ
cana-726	256	13	algorithmic	algorithmic	ADJ
cana-726	256	14	graph	graph	NOUN
cana-726	256	15	theory	theory	NOUN
cana-726	256	16	,	,	PUNCT
cana-726	256	17	mcgraw	mcgraw	PROPN
cana-726	256	18	-	-	PUNCT
cana-726	256	19	hill	hill	PROPN
cana-726	256	20	,	,	PUNCT
cana-726	256	21	inc	inc	PROPN
cana-726	256	22	.	.	PROPN
cana-726	256	23	,	,	PUNCT
cana-726	256	24	new	new	PROPN
cana-726	256	25	york	york	PROPN
cana-726	256	26	,	,	PUNCT
cana-726	256	27	1993	1993	NUM
cana-726	256	28	.	.	PUNCT
cana-726	257	1	[	[	X
cana-726	257	2	20	20	NUM
cana-726	257	3	]	]	PUNCT
cana-726	257	4	h.-j	h.-j	PROPN
cana-726	257	5	.	.	PUNCT
cana-726	257	6	chiang	chiang	PROPN
cana-726	257	7	-	-	PUNCT
cana-726	257	8	hsieh	hsieh	PROPN
cana-726	257	9	,	,	PUNCT
cana-726	257	10	classification	classification	NOUN
cana-726	257	11	of	of	ADP
cana-726	257	12	rings	ring	NOUN
cana-726	257	13	with	with	ADP
cana-726	257	14	projective	projective	ADJ
cana-726	257	15	zero	zero	NUM
cana-726	257	16	-	-	PUNCT
cana-726	257	17	divisor	divisor	NOUN
cana-726	257	18	graphs	graph	NOUN
cana-726	257	19	,	,	PUNCT
cana-726	257	20	j.	j.	PROPN
cana-726	257	21	algebra	algebra	PROPN
cana-726	257	22	319	319	NUM
cana-726	257	23	(	(	PUNCT
cana-726	257	24	2008	2008	NUM
cana-726	257	25	)	)	PUNCT
cana-726	257	26	,	,	PUNCT
cana-726	257	27	2789	2789	NUM
cana-726	257	28	-	-	SYM
cana-726	257	29	2802	2802	NUM
cana-726	257	30	.	.	PUNCT
cana-726	258	1	[	[	X
cana-726	258	2	21	21	NUM
cana-726	258	3	]	]	PUNCT
cana-726	258	4	a.	a.	NOUN
cana-726	258	5	h.	h.	PROPN
cana-726	258	6	cliford	cliford	PROPN
cana-726	258	7	and	and	CCONJ
cana-726	258	8	g.	g.	PROPN
cana-726	258	9	b.	b.	PROPN
cana-726	258	10	preston	preston	PROPN
cana-726	258	11	,	,	PUNCT
cana-726	258	12	the	the	DET
cana-726	258	13	algebraic	algebraic	ADJ
cana-726	258	14	theory	theory	NOUN
cana-726	258	15	of	of	ADP
cana-726	258	16	semigroups	semigroup	NOUN
cana-726	258	17	,	,	PUNCT
cana-726	258	18	mathematical	mathematical	ADJ
cana-726	258	19	surveys	survey	NOUN
cana-726	258	20	,	,	PUNCT
cana-726	258	21	no	no	INTJ
cana-726	258	22	.	.	NOUN
cana-726	258	23	7	7	NUM
cana-726	258	24	,	,	PUNCT
cana-726	258	25	vol	vol	NOUN
cana-726	258	26	1	1	NUM
cana-726	258	27	and	and	CCONJ
cana-726	258	28	2	2	NUM
cana-726	258	29	,	,	PUNCT
cana-726	258	30	american	american	PROPN
cana-726	258	31	mathematical	mathematical	ADJ
cana-726	258	32	society	society	NOUN
cana-726	258	33	,	,	PUNCT
cana-726	258	34	providence	providence	NOUN
cana-726	258	35	,	,	PUNCT
cana-726	258	36	ri	ri	PROPN
cana-726	258	37	,	,	PUNCT
cana-726	258	38	1961	1961	NUM
cana-726	258	39	and	and	CCONJ
cana-726	258	40	1967	1967	NUM
cana-726	258	41	.	.	PUNCT
cana-726	259	1	[	[	X
cana-726	259	2	22	22	NUM
cana-726	259	3	]	]	X
cana-726	259	4	f.	f.	PROPN
cana-726	259	5	r.	r.	PROPN
cana-726	259	6	demeyer	demeyer	PROPN
cana-726	259	7	,	,	PUNCT
cana-726	259	8	l.	l.	PROPN
cana-726	259	9	demeyer	demeyer	PROPN
cana-726	259	10	,	,	PUNCT
cana-726	259	11	zero	zero	NUM
cana-726	259	12	-	-	PUNCT
cana-726	259	13	divisor	divisor	NOUN
cana-726	259	14	graphs	graph	NOUN
cana-726	259	15	of	of	ADP
cana-726	259	16	semigroups	semigroup	NOUN
cana-726	259	17	,	,	PUNCT
cana-726	259	18	j.	j.	PROPN
cana-726	259	19	algebra	algebra	PROPN
cana-726	259	20	283	283	NUM
cana-726	259	21	(	(	PUNCT
cana-726	259	22	2005	2005	NUM
cana-726	259	23	)	)	PUNCT
cana-726	259	24	,	,	PUNCT
cana-726	259	25	190	190	NUM
cana-726	259	26	-198	-198	PROPN
cana-726	259	27	.	.	PUNCT
cana-726	260	1	[	[	X
cana-726	260	2	23	23	NUM
cana-726	260	3	]	]	X
cana-726	260	4	f.	f.	PROPN
cana-726	260	5	r.	r.	PROPN
cana-726	260	6	demeyer	demeyer	PROPN
cana-726	260	7	,	,	PUNCT
cana-726	260	8	t.	t.	PROPN
cana-726	260	9	mckenzie	mckenzie	PROPN
cana-726	260	10	,	,	PUNCT
cana-726	260	11	k.	k.	PROPN
cana-726	260	12	schneider	schneider	PROPN
cana-726	260	13	,	,	PUNCT
cana-726	260	14	the	the	DET
cana-726	260	15	zero	zero	NUM
cana-726	260	16	-	-	PUNCT
cana-726	260	17	divisor	divisor	NOUN
cana-726	260	18	graph	graph	NOUN
cana-726	260	19	of	of	ADP
cana-726	260	20	a	a	DET
cana-726	260	21	commutative	commutative	ADJ
cana-726	260	22	semigroup	semigroup	NOUN
cana-726	260	23	,	,	PUNCT
cana-726	260	24	semigroup	semigroup	PROPN
cana-726	260	25	forum	forum	PROPN
cana-726	260	26	65	65	NUM
cana-726	260	27	(	(	PUNCT
cana-726	260	28	2002	2002	NUM
cana-726	260	29	)	)	PUNCT
cana-726	260	30	,	,	PUNCT
cana-726	260	31	206	206	NUM
cana-726	260	32	-	-	SYM
cana-726	260	33	214	214	NUM
cana-726	260	34	.	.	PUNCT
cana-726	261	1	[	[	X
cana-726	261	2	24	24	NUM
cana-726	261	3	]	]	PUNCT
cana-726	261	4	j.	j.	PROPN
cana-726	261	5	m.	m.	PROPN
cana-726	261	6	howie	howie	PROPN
cana-726	261	7	,	,	PUNCT
cana-726	261	8	an	an	DET
cana-726	261	9	introduction	introduction	NOUN
cana-726	261	10	to	to	ADP
cana-726	261	11	semigroup	semigroup	PROPN
cana-726	261	12	theory	theory	PROPN
cana-726	261	13	,	,	PUNCT
cana-726	261	14	l.m.s	l.m.	NOUN
cana-726	261	15	.	.	PUNCT
cana-726	262	1	monographs	monograph	NOUN
cana-726	262	2	,	,	PUNCT
cana-726	262	3	no	no	INTJ
cana-726	262	4	.	.	NOUN
cana-726	263	1	7	7	X
cana-726	263	2	.	.	X
cana-726	263	3	academic	academic	ADJ
cana-726	263	4	press	press	NOUN
cana-726	264	1	[	[	X
cana-726	264	2	harcourt	harcourt	PROPN
cana-726	264	3	brace	brace	PROPN
cana-726	264	4	jovanovich	jovanovich	NOUN
cana-726	264	5	,	,	PUNCT
cana-726	264	6	publishers	publisher	NOUN
cana-726	264	7	]	]	PUNCT
cana-726	264	8	,	,	PUNCT
cana-726	264	9	london	london	PROPN
cana-726	264	10	-	-	PUNCT
cana-726	264	11	new	new	PROPN
cana-726	264	12	york	york	PROPN
cana-726	264	13	,	,	PUNCT
cana-726	264	14	1976	1976	NUM
cana-726	264	15	.	.	PUNCT
cana-726	265	1	[	[	X
cana-726	265	2	25	25	NUM
cana-726	265	3	]	]	PUNCT
cana-726	265	4	j.	j.	PROPN
cana-726	265	5	d.	d.	PROPN
cana-726	265	6	lagrange	lagrange	PROPN
cana-726	265	7	,	,	PUNCT
cana-726	265	8	complemented	complement	VERB
cana-726	265	9	zero	zero	NUM
cana-726	265	10	-	-	PUNCT
cana-726	265	11	divisor	divisor	NOUN
cana-726	265	12	graphs	graph	NOUN
cana-726	265	13	and	and	CCONJ
cana-726	265	14	boolean	boolean	ADJ
cana-726	265	15	rings	ring	NOUN
cana-726	265	16	,	,	PUNCT
cana-726	265	17	j.	j.	PROPN
cana-726	265	18	algebra	algebra	PROPN
cana-726	265	19	315	315	NUM
cana-726	265	20	(	(	PUNCT
cana-726	265	21	2007	2007	NUM
cana-726	265	22	)	)	PUNCT
cana-726	265	23	,	,	PUNCT
cana-726	265	24	600	600	NUM
cana-726	265	25	-	-	SYM
cana-726	265	26	611	611	NUM
cana-726	265	27	.	.	PUNCT
cana-726	266	1	[	[	X
cana-726	266	2	26	26	NUM
cana-726	266	3	]	]	X
cana-726	266	4	r.	r.	PROPN
cana-726	266	5	levy	levy	PROPN
cana-726	266	6	,	,	PUNCT
cana-726	266	7	j.	j.	PROPN
cana-726	266	8	shapiro	shapiro	PROPN
cana-726	266	9	,	,	PUNCT
cana-726	266	10	the	the	DET
cana-726	266	11	zero	zero	NUM
cana-726	266	12	-	-	PUNCT
cana-726	266	13	divisor	divisor	NOUN
cana-726	266	14	graph	graph	NOUN
cana-726	266	15	of	of	ADP
cana-726	266	16	von	von	PROPN
cana-726	266	17	neumann	neumann	PROPN
cana-726	266	18	regular	regular	PROPN
cana-726	266	19	rings	ring	NOUN
cana-726	266	20	,	,	PUNCT
cana-726	266	21	comm	comm	NOUN
cana-726	266	22	.	.	PUNCT
cana-726	267	1	algebra	algebra	NOUN
cana-726	267	2	30	30	NUM
cana-726	267	3	(	(	PUNCT
cana-726	267	4	2002	2002	NUM
cana-726	267	5	)	)	PUNCT
cana-726	267	6	,	,	PUNCT
cana-726	267	7	745	745	NUM
cana-726	267	8	-	-	SYM
cana-726	267	9	750	750	NUM
cana-726	267	10	.	.	PUNCT
cana-726	268	1	[	[	X
cana-726	268	2	27	27	NUM
cana-726	268	3	]	]	PUNCT
cana-726	268	4	t.	t.	PROPN
cana-726	268	5	g.	g.	PROPN
cana-726	268	6	lucas	lucas	PROPN
cana-726	268	7	,	,	PUNCT
cana-726	268	8	the	the	DET
cana-726	268	9	diameter	diameter	NOUN
cana-726	268	10	of	of	ADP
cana-726	268	11	a	a	DET
cana-726	268	12	zero	zero	NUM
cana-726	268	13	-	-	PUNCT
cana-726	268	14	divisor	divisor	NOUN
cana-726	268	15	graph	graph	NOUN
cana-726	268	16	,	,	PUNCT
cana-726	268	17	j.	j.	PROPN
cana-726	268	18	algebra	algebra	PROPN
cana-726	268	19	301	301	NUM
cana-726	268	20	(	(	PUNCT
cana-726	268	21	2006	2006	NUM
cana-726	268	22	)	)	PUNCT
cana-726	268	23	,	,	PUNCT
cana-726	268	24	174	174	NUM
cana-726	268	25	-	-	SYM
cana-726	268	26	193	193	NUM
cana-726	268	27	.	.	PUNCT
cana-726	269	1	[	[	X
cana-726	269	2	28	28	NUM
cana-726	269	3	]	]	X
cana-726	269	4	h.	h.	PROPN
cana-726	269	5	r.	r.	PROPN
cana-726	269	6	maimani	maimani	PROPN
cana-726	269	7	,	,	PUNCT
cana-726	269	8	m.	m.	PROPN
cana-726	269	9	r.	r.	PROPN
cana-726	269	10	pournaki	pournaki	PROPN
cana-726	269	11	,	,	PUNCT
cana-726	269	12	and	and	CCONJ
cana-726	269	13	s.	s.	PROPN
cana-726	269	14	yassemi	yassemi	PROPN
cana-726	269	15	,	,	PUNCT
cana-726	269	16	zero	zero	NUM
cana-726	269	17	-	-	PUNCT
cana-726	269	18	divisor	divisor	NOUN
cana-726	269	19	graph	graph	NOUN
cana-726	269	20	with	with	ADP
cana-726	269	21	respect	respect	NOUN
cana-726	269	22	to	to	ADP
cana-726	269	23	an	an	DET
cana-726	269	24	ideal	ideal	ADJ
cana-726	269	25	,	,	PUNCT
cana-726	269	26	comm	comm	NOUN
cana-726	269	27	.	.	PUNCT
cana-726	270	1	algebra	algebra	NOUN
cana-726	270	2	34	34	NUM
cana-726	270	3	(	(	PUNCT
cana-726	270	4	2006	2006	NUM
cana-726	270	5	)	)	PUNCT
cana-726	270	6	,	,	PUNCT
cana-726	270	7	923	923	NUM
cana-726	270	8	-	-	SYM
cana-726	270	9	929	929	NUM
cana-726	270	10	.	.	PUNCT
cana-726	271	1	[	[	X
cana-726	271	2	29	29	NUM
cana-726	271	3	]	]	X
cana-726	271	4	h.	h.	PROPN
cana-726	271	5	r.	r.	PROPN
cana-726	271	6	maimani	maimani	PROPN
cana-726	271	7	and	and	CCONJ
cana-726	271	8	s.	s.	PROPN
cana-726	271	9	yassemi	yassemi	PROPN
cana-726	271	10	,	,	PUNCT
cana-726	271	11	zero	zero	NUM
cana-726	271	12	-	-	PUNCT
cana-726	271	13	divisor	divisor	NOUN
cana-726	271	14	graphs	graph	NOUN
cana-726	271	15	of	of	ADP
cana-726	271	16	amalgamated	amalgamated	ADJ
cana-726	271	17	duplication	duplication	NOUN
cana-726	271	18	of	of	ADP
cana-726	271	19	a	a	DET
cana-726	271	20	ring	ring	NOUN
cana-726	271	21	along	along	ADP
cana-726	271	22	an	an	DET
cana-726	271	23	ideal	ideal	NOUN
cana-726	271	24	,	,	PUNCT
cana-726	271	25	j.	j.	PROPN
cana-726	271	26	pure	pure	PROPN
cana-726	271	27	appl	appl	PROPN
cana-726	271	28	.	.	PUNCT
cana-726	272	1	algebra	algebra	NOUN
cana-726	272	2	212	212	NUM
cana-726	272	3	(	(	PUNCT
cana-726	272	4	2008	2008	NUM
cana-726	272	5	)	)	PUNCT
cana-726	272	6	,	,	PUNCT
cana-726	272	7	168	168	NUM
cana-726	272	8	-	-	SYM
cana-726	272	9	174	174	NUM
cana-726	272	10	.	.	PUNCT
cana-726	273	1	[	[	X
cana-726	273	2	30	30	NUM
cana-726	273	3	]	]	X
cana-726	273	4	y.	y.	PROPN
cana-726	273	5	s.	s.	PROPN
cana-726	273	6	park	park	PROPN
cana-726	273	7	,	,	PUNCT
cana-726	273	8	j.	j.	PROPN
cana-726	273	9	p.	p.	PROPN
cana-726	273	10	kim	kim	PROPN
cana-726	273	11	,	,	PUNCT
cana-726	273	12	and	and	CCONJ
cana-726	273	13	m.-g	m.-g	PROPN
cana-726	273	14	.	.	PUNCT
cana-726	274	1	sohn	sohn	PROPN
cana-726	274	2	,	,	PUNCT
cana-726	274	3	semiprime	semiprime	NOUN
cana-726	274	4	ideals	ideal	NOUN
cana-726	274	5	in	in	ADP
cana-726	274	6	semigroups	semigroup	NOUN
cana-726	274	7	,	,	PUNCT
cana-726	274	8	math	math	NOUN
cana-726	274	9	.	.	PUNCT
cana-726	275	1	japon	japon	PROPN
cana-726	275	2	.	.	PUNCT
cana-726	276	1	33	33	NUM
cana-726	276	2	(	(	PUNCT
cana-726	276	3	1988	1988	NUM
cana-726	276	4	)	)	PUNCT
cana-726	276	5	,	,	PUNCT
cana-726	276	6	269	269	NUM
cana-726	276	7	-	-	SYM
cana-726	276	8	273	273	NUM
cana-726	276	9	.	.	PUNCT
cana-726	277	1	[	[	X
cana-726	277	2	31	31	NUM
cana-726	277	3	]	]	PUNCT
cana-726	277	4	m.	m.	PROPN
cana-726	277	5	satyanarayana	satyanarayana	PROPN
cana-726	277	6	,	,	PUNCT
cana-726	277	7	semigroups	semigroup	VERB
cana-726	277	8	with	with	ADP
cana-726	277	9	ascending	ascend	VERB
cana-726	277	10	chain	chain	NOUN
cana-726	277	11	condition	condition	NOUN
cana-726	277	12	,	,	PUNCT
cana-726	277	13	j.	j.	PROPN
cana-726	277	14	london	london	PROPN
cana-726	277	15	math	math	PROPN
cana-726	277	16	.	.	PUNCT
cana-726	278	1	soc	soc	PROPN
cana-726	278	2	.	.	PUNCT
cana-726	279	1	5	5	NUM
cana-726	279	2	(	(	PUNCT
cana-726	279	3	1972	1972	NUM
cana-726	279	4	)	)	PUNCT
cana-726	279	5	,	,	PUNCT
cana-726	279	6	11	11	NUM
cana-726	279	7	-	-	SYM
cana-726	279	8	14	14	NUM
cana-726	279	9	.	.	PUNCT
cana-726	280	1	[	[	X
cana-726	280	2	32	32	NUM
cana-726	280	3	]	]	PUNCT
cana-726	280	4	c.	c.	PROPN
cana-726	280	5	wickham	wickham	PROPN
cana-726	280	6	,	,	PUNCT
cana-726	280	7	classification	classification	NOUN
cana-726	280	8	of	of	ADP
cana-726	280	9	rings	ring	NOUN
cana-726	280	10	with	with	ADP
cana-726	280	11	genus	genus	NOUN
cana-726	280	12	one	one	NUM
cana-726	280	13	zero	zero	NUM
cana-726	280	14	-	-	PUNCT
cana-726	280	15	divisor	divisor	NOUN
cana-726	280	16	graphs	graph	NOUN
cana-726	280	17	,	,	PUNCT
cana-726	280	18	comm	comm	NOUN
cana-726	280	19	.	.	PUNCT
cana-726	281	1	algebra	algebra	NOUN
cana-726	281	2	36	36	NUM
cana-726	281	3	(	(	PUNCT
cana-726	281	4	2008	2008	NUM
cana-726	281	5	)	)	PUNCT
cana-726	281	6	,	,	PUNCT
cana-726	281	7	325	325	NUM
cana-726	281	8	-	-	SYM
cana-726	281	9	345	345	NUM
cana-726	281	10	.	.	PUNCT
cana-726	282	1	[	[	X
cana-726	282	2	33	33	NUM
cana-726	282	3	]	]	PUNCT
cana-726	282	4	s.	s.	PROPN
cana-726	282	5	e.	e.	PROPN
cana-726	282	6	wright	wright	PROPN
cana-726	282	7	,	,	PUNCT
cana-726	282	8	lengths	length	NOUN
cana-726	282	9	of	of	ADP
cana-726	282	10	paths	path	NOUN
cana-726	282	11	and	and	CCONJ
cana-726	282	12	cycles	cycle	NOUN
cana-726	282	13	in	in	ADP
cana-726	282	14	zero	zero	NUM
cana-726	282	15	-	-	PUNCT
cana-726	282	16	divisor	divisor	NOUN
cana-726	282	17	graphs	graph	NOUN
cana-726	282	18	and	and	CCONJ
cana-726	282	19	digraphs	digraph	NOUN
cana-726	282	20	of	of	ADP
cana-726	282	21	semi	semi	NOUN
cana-726	282	22	-	-	NOUN
cana-726	282	23	groups	group	NOUN
cana-726	282	24	,	,	PUNCT
cana-726	282	25	comm	comm	NOUN
cana-726	282	26	.	.	PUNCT
cana-726	283	1	algebra	algebra	NOUN
cana-726	283	2	35	35	NUM
cana-726	283	3	(	(	PUNCT
cana-726	283	4	2007	2007	NUM
cana-726	283	5	)	)	PUNCT
cana-726	283	6	,	,	PUNCT
cana-726	283	7	1987	1987	NUM
cana-726	283	8	-	-	SYM
cana-726	283	9	1991	1991	NUM
cana-726	283	10	.	.	PUNCT