id	sid	eid	entity	type
ejpam-5499	1	1	european journal	ORG
ejpam-5499	1	2	2025	DATE
ejpam-5499	1	3	18	CARDINAL
ejpam-5499	1	4	2	CARDINAL
ejpam-5499	1	5	5499	DATE
ejpam-5499	1	6	1307-5543	DATE
ejpam-5499	1	7	new york	GPE
ejpam-5499	1	8	1	CARDINAL
ejpam-5499	1	9	cyclic2	PERSON
ejpam-5499	1	10	groups3	ORG
ejpam-5499	1	11	jiel mark d. jagmis1	PERSON
ejpam-5499	1	12	daryl m. magpantay2,∗4	PERSON
ejpam-5499	1	13	1	CARDINAL
ejpam-5499	1	14	camarines sur polytechnic colleges	ORG
ejpam-5499	1	15	camarines5	PERSON
ejpam-5499	1	16	2	CARDINAL
ejpam-5499	1	17	batangas state university	ORG
ejpam-5499	1	18	the national engineering university	ORG
ejpam-5499	1	19	pablo borbon	PERSON
ejpam-5499	1	20	batangas city	GPE
ejpam-5499	1	21	batangas	GPE
ejpam-5499	1	22	philippines8 9	ORG
ejpam-5499	2	1	two	CARDINAL
ejpam-5499	3	1	m(g	PERSON
ejpam-5499	3	2	two	CARDINAL
ejpam-5499	6	1	2020	DATE
ejpam-5499	6	2	0510	CARDINAL
ejpam-5499	6	3	groups,11	GPE
ejpam-5499	6	4	properties12	GPE
ejpam-5499	6	5	13 1	DATE
ejpam-5499	6	6	introduction14	ORG
ejpam-5499	6	7	2009	DATE
ejpam-5499	8	1	2	CARDINAL
ejpam-5499	8	2	2015	DATE
ejpam-5499	9	1	3	CARDINAL
ejpam-5499	9	2	4	CARDINAL
ejpam-5499	9	3	further22 ∗corresponding	ORG
ejpam-5499	10	1	j. m. jamis	PERSON
ejpam-5499	10	2	daryl.magpantay@g.batstate-u.edu.ph	GPE
ejpam-5499	10	3	d. m. magpantay https://www.ejpam.com	PERSON
ejpam-5499	10	4	1	CARDINAL
ejpam-5499	11	1	2025	CARDINAL
ejpam-5499	12	1	j. m. jamis	PERSON
ejpam-5499	12	2	d. m. magpantay	PERSON
ejpam-5499	13	1	j. pure	PERSON
ejpam-5499	13	2	18	CARDINAL
ejpam-5499	13	3	2	CARDINAL
ejpam-5499	13	4	2025	CARDINAL
ejpam-5499	13	5	5499 2	DATE
ejpam-5499	13	6	18	CARDINAL
ejpam-5499	13	7	24	CARDINAL
ejpam-5499	13	8	another25	GPE
ejpam-5499	15	1	akiyama, hamada	ORG
ejpam-5499	16	1	5	CARDINAL
ejpam-5499	17	1	32	CARDINAL
ejpam-5499	17	2	two	CARDINAL
ejpam-5499	19	1	6	CARDINAL
ejpam-5499	19	2	1	CARDINAL
ejpam-5499	21	1	2017	DATE
ejpam-5499	21	2	of37	PRODUCT
ejpam-5499	22	1	alib et al	PERSON
ejpam-5499	23	1	7	CARDINAL
ejpam-5499	23	2	of41	PRODUCT
ejpam-5499	24	1	2	CARDINAL
ejpam-5499	24	2	preliminaries44	NORP
ejpam-5499	25	1	are45	DATE
ejpam-5499	25	2	2.1	CARDINAL
ejpam-5499	26	1	theory47	DATE
ejpam-5499	26	2	will48	PERSON
ejpam-5499	28	1	in49	ORG
ejpam-5499	28	2	two	CARDINAL
ejpam-5499	30	1	us51	ORG
ejpam-5499	30	2	1	CARDINAL
ejpam-5499	32	1	2z	CARDINAL
ejpam-5499	33	1	we55	PERSON
ejpam-5499	33	2	2k	CARDINAL
ejpam-5499	33	3	j ∈ s	PERSON
ejpam-5499	33	4	2k+2j	CARDINAL
ejpam-5499	33	5	2(k+	CARDINAL
ejpam-5499	34	1	s.57	GPE
ejpam-5499	34	2	2	CARDINAL
ejpam-5499	35	1	∗	CARDINAL
ejpam-5499	35	2	∗	CARDINAL
ejpam-5499	35	3	j. m. jamis	PERSON
ejpam-5499	35	4	d. m. magpantay	PERSON
ejpam-5499	36	1	j. pure	PERSON
ejpam-5499	36	2	18	CARDINAL
ejpam-5499	36	3	2	CARDINAL
ejpam-5499	36	4	2025	CARDINAL
ejpam-5499	36	5	5499 3	DATE
ejpam-5499	37	1	∗	CARDINAL
ejpam-5499	37	2	∈	ORG
ejpam-5499	37	3	of60	ORG
ejpam-5499	38	1	∈	ORG
ejpam-5499	38	2	∗a	NORP
ejpam-5499	38	3	1	CARDINAL
ejpam-5499	40	1	note64	PERSON
ejpam-5499	40	2	iden-65	ORG
ejpam-5499	40	3	0	DATE
ejpam-5499	40	4	67	DATE
ejpam-5499	41	1	third	ORDINAL
ejpam-5499	41	2	2b	CARDINAL
ejpam-5499	41	3	1	CARDINAL
ejpam-5499	41	4	1	CARDINAL
ejpam-5499	41	5	70	CARDINAL
ejpam-5499	41	6	3	CARDINAL
ejpam-5499	42	1	∈	ORG
ejpam-5499	42	2	is72	CARDINAL
ejpam-5499	42	3	set73	PERSON
ejpam-5499	43	1	2	CARDINAL
ejpam-5499	46	1	g = r,+	PERSON
ejpam-5499	46	2	4	CARDINAL
ejpam-5499	47	1	polygon	ORG
ejpam-5499	47	2	1	CARDINAL
ejpam-5499	48	1	2n.84	DATE
ejpam-5499	48	2	n ≥ 3	DATE
ejpam-5499	50	1	2π	CARDINAL
ejpam-5499	50	2	polygon	ORG
ejpam-5499	53	1	1	CARDINAL
ejpam-5499	54	1	so88	PRODUCT
ejpam-5499	55	1	90	CARDINAL
ejpam-5499	55	2	j. m. jamis	PERSON
ejpam-5499	55	3	d. m. magpantay	PERSON
ejpam-5499	56	1	j. pure	PERSON
ejpam-5499	56	2	18	CARDINAL
ejpam-5499	56	3	2	CARDINAL
ejpam-5499	56	4	2025	CARDINAL
ejpam-5499	56	5	5499 4	DATE
ejpam-5499	56	6	18 2.2	CARDINAL
ejpam-5499	57	1	of92	ORG
ejpam-5499	57	2	of94	PRODUCT
ejpam-5499	57	3	two	CARDINAL
ejpam-5499	60	1	other.99	LOC
ejpam-5499	60	2	⊆	CARDINAL
ejpam-5499	61	1	1	CARDINAL
ejpam-5499	62	1	d(u	PERSON
ejpam-5499	62	2	∈	ORG
ejpam-5499	62	3	∈	ORG
ejpam-5499	62	4	e(u	PERSON
ejpam-5499	65	1	e(u	PERSON
ejpam-5499	66	1	5	CARDINAL
ejpam-5499	67	1	two	CARDINAL
ejpam-5499	67	2	113	CARDINAL
ejpam-5499	67	3	two	CARDINAL
ejpam-5499	67	4	follows:115	GPE
ejpam-5499	67	5	6	DATE
ejpam-5499	69	1	line(g).118	ORG
ejpam-5499	69	2	7	CARDINAL
ejpam-5499	70	1	three	CARDINAL
ejpam-5499	71	1	123	CARDINAL
ejpam-5499	71	2	5	CARDINAL
ejpam-5499	71	3	8	CARDINAL
ejpam-5499	71	4	1	CARDINAL
ejpam-5499	71	5	9	CARDINAL
ejpam-5499	71	6	3].125	CARDINAL
ejpam-5499	71	7	j. m. jamis	PERSON
ejpam-5499	71	8	d. m. magpantay	PERSON
ejpam-5499	72	1	j. pure	PERSON
ejpam-5499	72	2	18	CARDINAL
ejpam-5499	72	3	2	CARDINAL
ejpam-5499	72	4	2025	CARDINAL
ejpam-5499	72	5	5499 5	DATE
ejpam-5499	72	6	18	CARDINAL
ejpam-5499	72	7	1	CARDINAL
ejpam-5499	73	1	line(cn	ORG
ejpam-5499	74	1	0 and126	MONEY
ejpam-5499	75	1	2	CARDINAL
ejpam-5499	76	1	line(cn	ORG
ejpam-5499	76	2	1	CARDINAL
ejpam-5499	77	1	2	CARDINAL
ejpam-5499	77	2	2	CARDINAL
ejpam-5499	78	1	3(n−1	CARDINAL
ejpam-5499	78	2	2	CARDINAL
ejpam-5499	78	3	2	CARDINAL
ejpam-5499	80	1	129	CARDINAL
ejpam-5499	80	2	3	CARDINAL
ejpam-5499	80	3	∑ v∈v	PERSON
ejpam-5499	81	1	4	CARDINAL
ejpam-5499	86	1	m(g	PERSON
ejpam-5499	86	2	two	CARDINAL
ejpam-5499	86	3	1	CARDINAL
ejpam-5499	86	4	2	CARDINAL
ejpam-5499	86	5	3	CARDINAL
ejpam-5499	88	1	1	CARDINAL
ejpam-5499	88	2	2	DATE
ejpam-5499	88	3	3	DATE
ejpam-5499	88	4	4	CARDINAL
ejpam-5499	88	5	two	CARDINAL
ejpam-5499	88	6	one	CARDINAL
ejpam-5499	90	1	3	CARDINAL
ejpam-5499	90	2	3	CARDINAL
ejpam-5499	93	1	two	CARDINAL
ejpam-5499	93	2	1	CARDINAL
ejpam-5499	93	3	2	CARDINAL
ejpam-5499	93	4	one	CARDINAL
ejpam-5499	93	5	j. m. jamis	PERSON
ejpam-5499	93	6	d. m. magpantay	PERSON
ejpam-5499	94	1	j. pure	PERSON
ejpam-5499	94	2	18	CARDINAL
ejpam-5499	94	3	2	CARDINAL
ejpam-5499	94	4	2025	CARDINAL
ejpam-5499	94	5	5499 6	DATE
ejpam-5499	94	6	18	CARDINAL
ejpam-5499	95	1	1	CARDINAL
ejpam-5499	96	1	n ≥ 2	DATE
ejpam-5499	101	1	⋃	DATE
ejpam-5499	102	1	xi+1]andp	ORG
ejpam-5499	102	2	2	CARDINAL
ejpam-5499	102	3	2	CARDINAL
ejpam-5499	102	4	3	CARDINAL
ejpam-5499	103	1	162	CARDINAL
ejpam-5499	103	2	2	CARDINAL
ejpam-5499	104	1	second	ORDINAL
ejpam-5499	104	2	3	CARDINAL
ejpam-5499	104	3	4	CARDINAL
ejpam-5499	105	1	c3	ORG
ejpam-5499	106	1	3.1	CARDINAL
ejpam-5499	107	1	first	ORDINAL
ejpam-5499	109	1	1	CARDINAL
ejpam-5499	111	1	1	CARDINAL
ejpam-5499	112	1	2	CARDINAL
ejpam-5499	112	2	j. m. jamis	PERSON
ejpam-5499	112	3	d. m. magpantay	PERSON
ejpam-5499	113	1	j. pure	PERSON
ejpam-5499	113	2	18	CARDINAL
ejpam-5499	113	3	2	CARDINAL
ejpam-5499	113	4	2025	CARDINAL
ejpam-5499	113	5	5499 7	DATE
ejpam-5499	113	6	18 .177	CARDINAL
ejpam-5499	113	7	have;178	NORP
ejpam-5499	115	1	1} ⋃	TIME
ejpam-5499	117	1	2	CARDINAL
ejpam-5499	117	2	2	CARDINAL
ejpam-5499	120	1	zj	ORG
ejpam-5499	121	1	1	CARDINAL
ejpam-5499	121	2	1 ≤	QUANTITY
ejpam-5499	121	3	1	CARDINAL
ejpam-5499	121	4	1	CARDINAL
ejpam-5499	121	5	1 ≤	QUANTITY
ejpam-5499	121	6	zi]|p = i+1	PERSON
ejpam-5499	121	7	2	CARDINAL
ejpam-5499	121	8	2 }⋃	TIME
ejpam-5499	121	9	zi+1]|p = i+1	PERSON
ejpam-5499	121	10	2	CARDINAL
ejpam-5499	122	1	2 }⋃	TIME
ejpam-5499	123	1	2	CARDINAL
ejpam-5499	124	1	2 }⋃	TIME
ejpam-5499	124	2	xi+1]|p = i+1	PERSON
ejpam-5499	124	3	2	CARDINAL
ejpam-5499	125	1	2	CARDINAL
ejpam-5499	126	1	j. m. jamis	PERSON
ejpam-5499	126	2	d. m. magpantay	PERSON
ejpam-5499	127	1	j. pure	PERSON
ejpam-5499	127	2	18	CARDINAL
ejpam-5499	127	3	2	CARDINAL
ejpam-5499	127	4	2025	CARDINAL
ejpam-5499	127	5	5499 8	DATE
ejpam-5499	128	1	1183	CARDINAL
ejpam-5499	129	1	2	CARDINAL
ejpam-5499	129	2	1184	CARDINAL
ejpam-5499	129	3	4	CARDINAL
ejpam-5499	129	4	1 ≤	MONEY
ejpam-5499	129	5	2185	CARDINAL
ejpam-5499	129	6	d.	NORP
ejpam-5499	129	7	1	CARDINAL
ejpam-5499	129	8	1.186	CARDINAL
ejpam-5499	129	9	we187	ORG
ejpam-5499	129	10	∑ v∈v	PERSON
ejpam-5499	130	1	1	CARDINAL
ejpam-5499	132	1	1 + 2n−	DATE
ejpam-5499	132	2	2 +	DATE
ejpam-5499	132	3	2 + n2	QUANTITY
ejpam-5499	132	4	3.2	CARDINAL
ejpam-5499	133	1	first	ORDINAL
ejpam-5499	133	2	1	CARDINAL
ejpam-5499	134	1	1	CARDINAL
ejpam-5499	135	1	2	CARDINAL
ejpam-5499	135	2	3	CARDINAL
ejpam-5499	135	3	.194	CARDINAL
ejpam-5499	135	4	have;195	NORP
ejpam-5499	137	1	1} ⋃	TIME
ejpam-5499	139	1	2	CARDINAL
ejpam-5499	139	2	3	CARDINAL
ejpam-5499	142	1	1	CARDINAL
ejpam-5499	142	2	j. m. jamis	PERSON
ejpam-5499	142	3	d. m. magpantay	PERSON
ejpam-5499	143	1	j. pure	PERSON
ejpam-5499	143	2	18	CARDINAL
ejpam-5499	143	3	2	CARDINAL
ejpam-5499	143	4	2025	CARDINAL
ejpam-5499	143	5	5499 9	DATE
ejpam-5499	143	6	18⋃	CARDINAL
ejpam-5499	143	7	zj	ORG
ejpam-5499	144	1	1	CARDINAL
ejpam-5499	144	2	1 ≤	QUANTITY
ejpam-5499	144	3	1	CARDINAL
ejpam-5499	144	4	1	CARDINAL
ejpam-5499	144	5	1 ≤	QUANTITY
ejpam-5499	144	6	zi]|p = i+1	PERSON
ejpam-5499	144	7	2	CARDINAL
ejpam-5499	144	8	3 }⋃	TIME
ejpam-5499	144	9	zi+1]|p = i+1	PERSON
ejpam-5499	144	10	2	CARDINAL
ejpam-5499	145	1	3 }⋃	TIME
ejpam-5499	146	1	2	CARDINAL
ejpam-5499	147	1	3 }⋃	TIME
ejpam-5499	147	2	xi+1]|p = i+1	PERSON
ejpam-5499	147	3	2	CARDINAL
ejpam-5499	148	1	3	CARDINAL
ejpam-5499	148	2	below:199 a. deg(e	ORG
ejpam-5499	149	1	1200	CARDINAL
ejpam-5499	149	2	2	CARDINAL
ejpam-5499	149	3	2201	CARDINAL
ejpam-5499	149	4	1202	DATE
ejpam-5499	149	5	d. deg(yp	PERSON
ejpam-5499	149	6	4	CARDINAL
ejpam-5499	149	7	1 ≤	MONEY
ejpam-5499	149	8	1	CARDINAL
ejpam-5499	149	9	2204	CARDINAL
ejpam-5499	150	1	1	CARDINAL
ejpam-5499	152	1	2(n− 2	ORG
ejpam-5499	152	2	2	CARDINAL
ejpam-5499	152	3	2n+	CARDINAL
ejpam-5499	153	1	4 + 2n−	DATE
ejpam-5499	153	2	4 + n2	QUANTITY
ejpam-5499	153	3	2	CARDINAL
ejpam-5499	153	4	10	CARDINAL
ejpam-5499	153	5	j. m. jamis	PERSON
ejpam-5499	153	6	d. m. magpantay	PERSON
ejpam-5499	154	1	j. pure	PERSON
ejpam-5499	154	2	18	CARDINAL
ejpam-5499	154	3	2	CARDINAL
ejpam-5499	154	4	2025	CARDINAL
ejpam-5499	154	5	5499 10	DATE
ejpam-5499	154	6	18	CARDINAL
ejpam-5499	154	7	1	CARDINAL
ejpam-5499	155	1	2	CARDINAL
ejpam-5499	156	1	5n−3 2	CARDINAL
ejpam-5499	157	1	209	CARDINAL
ejpam-5499	157	2	i. for n	PERSON
ejpam-5499	157	3	3(n− 1	ORG
ejpam-5499	157	4	2	CARDINAL
ejpam-5499	157	5	2n+	CARDINAL
ejpam-5499	158	1	3n−	CARDINAL
ejpam-5499	158	2	5n−	CARDINAL
ejpam-5499	158	3	3 2 ii	QUANTITY
ejpam-5499	159	1	3(n− 2	ORG
ejpam-5499	159	2	2	CARDINAL
ejpam-5499	160	1	2n+ 3n−	CARDINAL
ejpam-5499	160	2	5n−	CARDINAL
ejpam-5499	160	3	the213 theorem below.214 theorem 2	ORG
ejpam-5499	160	4	+5n−6 2	MONEY
ejpam-5499	161	1	216	CARDINAL
ejpam-5499	162	1	two	CARDINAL
ejpam-5499	162	2	i.	PERSON
ejpam-5499	162	3	first	ORDINAL
ejpam-5499	163	1	∑ v∈v	PERSON
ejpam-5499	164	1	+ 5n	CARDINAL
ejpam-5499	164	2	6	CARDINAL
ejpam-5499	165	1	3	CARDINAL
ejpam-5499	165	2	∑ v∈v	PERSON
ejpam-5499	166	1	n2 + 5n− 6	PRODUCT
ejpam-5499	166	2	2	CARDINAL
ejpam-5499	167	1	∑ v∈v	PERSON
ejpam-5499	168	1	3	CARDINAL
ejpam-5499	168	2	∑ v∈v	PERSON
ejpam-5499	170	1	2	CARDINAL
ejpam-5499	171	1	we224	ORG
ejpam-5499	172	1	2	CARDINAL
ejpam-5499	172	2	2 .225	QUANTITY
ejpam-5499	172	3	j. m. jamis	PERSON
ejpam-5499	172	4	d. m. magpantay	PERSON
ejpam-5499	173	1	j. pure	PERSON
ejpam-5499	173	2	18	CARDINAL
ejpam-5499	173	3	2	CARDINAL
ejpam-5499	173	4	2025	CARDINAL
ejpam-5499	173	5	5499 11	DATE
ejpam-5499	173	6	18 4	DATE
ejpam-5499	173	7	understand227	GPE
ejpam-5499	173	8	4.1	CARDINAL
ejpam-5499	174	1	two	CARDINAL
ejpam-5499	174	2	3	CARDINAL
ejpam-5499	176	1	the distance230 i. d(xi	ORG
ejpam-5499	176	2	a) ≤ 3	QUANTITY
ejpam-5499	176	3	∈	ORG
ejpam-5499	177	1	1 ≤	QUANTITY
ejpam-5499	177	2	1,231 ii	TIME
ejpam-5499	178	1	d(yp	ORG
ejpam-5499	178	2	∈	ORG
ejpam-5499	179	1	2	CARDINAL
ejpam-5499	179	2	∈	ORG
ejpam-5499	180	1	1 ≤	QUANTITY
ejpam-5499	180	2	1,234	CARDINAL
ejpam-5499	181	1	d(e	NORP
ejpam-5499	182	1	∈	ORG
ejpam-5499	183	1	four	CARDINAL
ejpam-5499	183	2	cases:236	ORG
ejpam-5499	183	3	∈	ORG
ejpam-5499	185	1	d(xi	GPE
ejpam-5499	185	2	xi+1	PERSON
ejpam-5499	185	3	2	CARDINAL
ejpam-5499	189	1	xj	ORG
ejpam-5499	189	2	zj	ORG
ejpam-5499	190	1	[zi	PERSON
ejpam-5499	190	2	zj	ORG
ejpam-5499	191	1	zj	ORG
ejpam-5499	191	2	xj	ORG
ejpam-5499	191	3	xj	GPE
ejpam-5499	191	4	zj	ORG
ejpam-5499	191	5	xj	ORG
ejpam-5499	191	6	3	CARDINAL
ejpam-5499	192	1	the241	ORG
ejpam-5499	192	2	d(xi	GPE
ejpam-5499	192	3	xj	GPE
ejpam-5499	193	1	3	CARDINAL
ejpam-5499	195	1	d(xi	GPE
ejpam-5499	195	2	yp	ORG
ejpam-5499	195	3	3.242	CARDINAL
ejpam-5499	195	4	d(xi	GPE
ejpam-5499	195	5	1	CARDINAL
ejpam-5499	196	1	zj	ORG
ejpam-5499	198	1	zj	ORG
ejpam-5499	199	1	zj	ORG
ejpam-5499	199	2	zj	ORG
ejpam-5499	200	1	the245	ORG
ejpam-5499	200	2	d(xi	GPE
ejpam-5499	200	3	zj	ORG
ejpam-5499	200	4	2	CARDINAL
ejpam-5499	200	5	zj	ORG
ejpam-5499	201	1	the246	ORG
ejpam-5499	201	2	d(xi	GPE
ejpam-5499	201	3	a) ≤ 3	QUANTITY
ejpam-5499	201	4	∈	ORG
ejpam-5499	203	1	∈	ORG
ejpam-5499	205	1	zj	ORG
ejpam-5499	205	2	zj	ORG
ejpam-5499	206	1	1 since249	QUANTITY
ejpam-5499	206	2	zj	ORG
ejpam-5499	207	1	d(xi	GPE
ejpam-5499	208	1	250 1	DATE
ejpam-5499	209	1	xj	ORG
ejpam-5499	209	2	zj	ORG
ejpam-5499	210	1	zj	ORG
ejpam-5499	210	2	zj	ORG
ejpam-5499	210	3	xj	ORG
ejpam-5499	210	4	xj	ORG
ejpam-5499	212	1	xj	ORG
ejpam-5499	213	1	2	CARDINAL
ejpam-5499	214	1	where253	PERSON
ejpam-5499	215	1	2	CARDINAL
ejpam-5499	215	2	2	CARDINAL
ejpam-5499	216	1	1	CARDINAL
ejpam-5499	216	2	2	CARDINAL
ejpam-5499	217	1	2	CARDINAL
ejpam-5499	217	2	zj	ORG
ejpam-5499	218	1	zj	ORG
ejpam-5499	218	2	zj	ORG
ejpam-5499	218	3	y i+1 2	ORG
ejpam-5499	220	1	∈	ORG
ejpam-5499	222	1	∈	ORG
ejpam-5499	224	1	d(e	NORP
ejpam-5499	224	2	1	CARDINAL
ejpam-5499	225	1	1 ≤ i ≤ n − 1	QUANTITY
ejpam-5499	225	2	then259 j. m. jamis	ORG
ejpam-5499	225	3	d. m. magpantay	PERSON
ejpam-5499	226	1	j. pure	PERSON
ejpam-5499	226	2	18	CARDINAL
ejpam-5499	226	3	2	CARDINAL
ejpam-5499	226	4	2025	CARDINAL
ejpam-5499	226	5	5499 12	DATE
ejpam-5499	227	1	2	CARDINAL
ejpam-5499	229	1	d(e	NORP
ejpam-5499	230	1	2	CARDINAL
ejpam-5499	230	2	2	CARDINAL
ejpam-5499	231	1	d(e	NORP
ejpam-5499	232	1	2	CARDINAL
ejpam-5499	232	2	d(e	NORP
ejpam-5499	232	3	∈	ORG
ejpam-5499	232	4	4.2	CARDINAL
ejpam-5499	233	1	4	CARDINAL
ejpam-5499	235	1	2	CARDINAL
ejpam-5499	235	2	3 3	CARDINAL
ejpam-5499	235	3	4 267	CARDINAL
ejpam-5499	235	4	1,268 ii	TIME
ejpam-5499	237	1	2	CARDINAL
ejpam-5499	237	2	4 269	CARDINAL
ejpam-5499	237	3	270 iii	QUANTITY
ejpam-5499	237	4	e(zi	PERSON
ejpam-5499	237	5	1	CARDINAL
ejpam-5499	237	6	3 2	CARDINAL
ejpam-5499	237	7	4 271	CARDINAL
ejpam-5499	237	8	1,272	CARDINAL
ejpam-5499	237	9	2273	CARDINAL
ejpam-5499	238	1	γcn .274 i.	PERSON
ejpam-5499	238	2	2	CARDINAL
ejpam-5499	240	1	3	CARDINAL
ejpam-5499	241	1	278	CARDINAL
ejpam-5499	247	1	2	CARDINAL
ejpam-5499	248	1	2	CARDINAL
ejpam-5499	249	1	1	CARDINAL
ejpam-5499	250	1	eccentricity285 e(x1	GPE
ejpam-5499	250	2	2.286 iii	QUANTITY
ejpam-5499	251	1	n ≥ 4	DATE
ejpam-5499	251	2	3.287	CARDINAL
ejpam-5499	251	3	j. m. jamis	PERSON
ejpam-5499	251	4	d. m. magpantay	PERSON
ejpam-5499	252	1	j. pure	PERSON
ejpam-5499	252	2	18	CARDINAL
ejpam-5499	252	3	2	CARDINAL
ejpam-5499	252	4	2025	CARDINAL
ejpam-5499	252	5	5499 13	DATE
ejpam-5499	252	6	18 4.3	CARDINAL
ejpam-5499	253	1	radius288	ORG
ejpam-5499	253	2	5	CARDINAL
ejpam-5499	255	1	1	CARDINAL
ejpam-5499	255	2	2	CARDINAL
ejpam-5499	255	3	n ≥ 3	DATE
ejpam-5499	256	1	2	CARDINAL
ejpam-5499	256	2	3 3	CARDINAL
ejpam-5499	256	3	n ≥ 4	DATE
ejpam-5499	256	4	290	CARDINAL
ejpam-5499	258	1	three	CARDINAL
ejpam-5499	258	2	2	CARDINAL
ejpam-5499	258	3	sincem(γc2	ORG
ejpam-5499	258	4	3	CARDINAL
ejpam-5499	258	5	rad(m(γc2	PERSON
ejpam-5499	259	1	1292	CARDINAL
ejpam-5499	260	1	2.293 ii	MONEY
ejpam-5499	261	1	3	CARDINAL
ejpam-5499	261	2	4	CARDINAL
ejpam-5499	261	3	2	CARDINAL
ejpam-5499	261	4	the294	GPE
ejpam-5499	262	1	2	CARDINAL
ejpam-5499	262	2	diam(m(γc3	GPE
ejpam-5499	263	1	2	CARDINAL
ejpam-5499	263	2	respectively.295 iii.	PERSON
ejpam-5499	264	1	n ≥ 4	DATE
ejpam-5499	264	2	4	CARDINAL
ejpam-5499	264	3	2	CARDINAL
ejpam-5499	264	4	1 and	TIME
ejpam-5499	264	5	2	CARDINAL
ejpam-5499	264	6	is296	ORG
ejpam-5499	265	1	2	CARDINAL
ejpam-5499	265	2	3	CARDINAL
ejpam-5499	266	1	1	CARDINAL
ejpam-5499	266	2	3	CARDINAL
ejpam-5499	266	3	2	CARDINAL
ejpam-5499	266	4	1 ≤	MONEY
ejpam-5499	268	1	3.300	CARDINAL
ejpam-5499	268	2	4.4	MONEY
ejpam-5499	269	1	central vertices301	ORG
ejpam-5499	269	2	6	CARDINAL
ejpam-5499	269	3	n ≥ 4	DATE
ejpam-5499	269	4	zi ∈	PERSON
ejpam-5499	270	1	4	CARDINAL
ejpam-5499	270	2	2	CARDINAL
ejpam-5499	270	3	1 ≤	QUANTITY
ejpam-5499	271	1	2.304	CARDINAL
ejpam-5499	271	2	5	CARDINAL
ejpam-5499	272	1	2	CARDINAL
ejpam-5499	273	1	1	CARDINAL
ejpam-5499	274	1	the305	NORP
ejpam-5499	274	2	3	CARDINAL
ejpam-5499	275	1	e.307 4.5	CARDINAL
ejpam-5499	276	1	7	CARDINAL
ejpam-5499	278	1	1} ⋃	TIME
ejpam-5499	278	2	4.310	CARDINAL
ejpam-5499	279	1	6.311 4.6	CARDINAL
ejpam-5499	280	1	8	CARDINAL
ejpam-5499	281	1	n ≥ 4	DATE
ejpam-5499	281	2	n .314 j. m. jamis	PERSON
ejpam-5499	281	3	d. m. magpantay	PERSON
ejpam-5499	282	1	j. pure	PERSON
ejpam-5499	282	2	18	CARDINAL
ejpam-5499	282	3	2	CARDINAL
ejpam-5499	282	4	2025	CARDINAL
ejpam-5499	282	5	5499 14	DATE
ejpam-5499	282	6	18	CARDINAL
ejpam-5499	282	7	proof.315	ORG
ejpam-5499	283	1	supposeh	PERSON
ejpam-5499	283	2	by316	NORP
ejpam-5499	283	3	zj	ORG
ejpam-5499	283	4	[zi	PERSON
ejpam-5499	283	5	zj	ORG
ejpam-5499	283	6	degh(zi	GPE
ejpam-5499	284	1	2	CARDINAL
ejpam-5499	285	1	4.7	CARDINAL
ejpam-5499	286	1	circumference321 theorem 9	PERSON
ejpam-5499	288	1	5n−3 2	CARDINAL
ejpam-5499	288	2	5n−6 2	CARDINAL
ejpam-5499	289	1	323	CARDINAL
ejpam-5499	291	1	two	CARDINAL
ejpam-5499	291	2	⊆	CARDINAL
ejpam-5499	292	1	1 ≤	QUANTITY
ejpam-5499	293	1	1,326	CARDINAL
ejpam-5499	295	1	2	CARDINAL
ejpam-5499	295	2	2	CARDINAL
ejpam-5499	296	1	we329	ORDINAL
ejpam-5499	297	1	z3	GPE
ejpam-5499	297	2	x3	GPE
ejpam-5499	297	3	x4,331 z4	PERSON
ejpam-5499	297	4	2	DATE
ejpam-5499	297	5	zi+1	GPE
ejpam-5499	297	6	zn−2	PRODUCT
ejpam-5499	297	7	xn−2	ORG
ejpam-5499	297	8	2	CARDINAL
ejpam-5499	298	1	1 ≤	QUANTITY
ejpam-5499	298	2	1.333	CARDINAL
ejpam-5499	298	3	z3	GPE
ejpam-5499	298	4	x3	GPE
ejpam-5499	298	5	x4	PERSON
ejpam-5499	298	6	z4	GPE
ejpam-5499	298	7	2	DATE
ejpam-5499	298	8	zi+1	GPE
ejpam-5499	299	1	xn−2	ORG
ejpam-5499	299	2	2	CARDINAL
ejpam-5499	299	3	xn−1	GPE
ejpam-5499	299	4	zn−1	GPE
ejpam-5499	299	5	first	ORDINAL
ejpam-5499	301	1	1	CARDINAL
ejpam-5499	301	2	2 .337	CARDINAL
ejpam-5499	301	3	1 +	CARDINAL
ejpam-5499	301	4	2	CARDINAL
ejpam-5499	302	1	5n−3 2	CARDINAL
ejpam-5499	302	2	⊆	CARDINAL
ejpam-5499	303	1	1	CARDINAL
ejpam-5499	305	1	5n−3	CARDINAL
ejpam-5499	305	2	2	CARDINAL
ejpam-5499	306	1	⊆	CARDINAL
ejpam-5499	307	1	1 ≤	QUANTITY
ejpam-5499	308	1	zn−1	ORG
ejpam-5499	308	2	zj	ORG
ejpam-5499	309	1	zn−1	PERSON
ejpam-5499	310	1	zn−1	PERSON
ejpam-5499	311	1	1	CARDINAL
ejpam-5499	311	2	2	CARDINAL
ejpam-5499	311	3	2	CARDINAL
ejpam-5499	312	1	zn−1	PERSON
ejpam-5499	313	1	1	CARDINAL
ejpam-5499	314	1	zn−1	PERSON
ejpam-5499	315	1	we348	PERSON
ejpam-5499	315	2	zn−1	PERSON
ejpam-5499	315	3	1	CARDINAL
ejpam-5499	315	4	z3	GPE
ejpam-5499	315	5	x3	GPE
ejpam-5499	315	6	x4	PERSON
ejpam-5499	315	7	z4	GPE
ejpam-5499	315	8	2	DATE
ejpam-5499	315	9	zi+1	GPE
ejpam-5499	315	10	zn−3	ORG
ejpam-5499	315	11	xn−3,350 yn−2 2	DATE
ejpam-5499	315	12	xn−2	ORG
ejpam-5499	315	13	zn−2	PRODUCT
ejpam-5499	315	14	j. m. jamis	PERSON
ejpam-5499	315	15	d. m. magpantay	PERSON
ejpam-5499	316	1	j. pure	PERSON
ejpam-5499	316	2	18	CARDINAL
ejpam-5499	316	3	2	CARDINAL
ejpam-5499	316	4	2025	CARDINAL
ejpam-5499	316	5	5499 15	DATE
ejpam-5499	316	6	18	CARDINAL
ejpam-5499	316	7	⋃	DATE
ejpam-5499	318	1	implies352 that353	GPE
ejpam-5499	318	2	1 +	CARDINAL
ejpam-5499	318	3	1	CARDINAL
ejpam-5499	318	4	2	CARDINAL
ejpam-5499	318	5	2 2	DATE
ejpam-5499	319	1	2 + n−	QUANTITY
ejpam-5499	319	2	2 2	CARDINAL
ejpam-5499	319	3	4 + n−	QUANTITY
ejpam-5499	319	4	2 2	TIME
ejpam-5499	319	5	5n−	CARDINAL
ejpam-5499	320	1	13	CARDINAL
ejpam-5499	320	2	one	CARDINAL
ejpam-5499	320	3	∈	ORG
ejpam-5499	320	4	1	CARDINAL
ejpam-5499	324	1	5n−6 2	DATE
ejpam-5499	325	1	5n−6 2	CARDINAL
ejpam-5499	327	1	4.8	CARDINAL
ejpam-5499	328	1	girth359	PERSON
ejpam-5499	328	2	10	CARDINAL
ejpam-5499	330	1	3	CARDINAL
ejpam-5499	330	2	3.361	CARDINAL
ejpam-5499	331	1	γcn .	GPE
ejpam-5499	332	1	1	CARDINAL
ejpam-5499	333	1	8	CARDINAL
ejpam-5499	334	1	1	CARDINAL
ejpam-5499	334	2	2	CARDINAL
ejpam-5499	336	1	3	CARDINAL
ejpam-5499	338	1	3	CARDINAL
ejpam-5499	338	2	4.9	MONEY
ejpam-5499	339	1	11	CARDINAL
ejpam-5499	341	1	368	CARDINAL
ejpam-5499	341	2	3.369	CARDINAL
ejpam-5499	344	1	8	CARDINAL
ejpam-5499	345	1	2	CARDINAL
ejpam-5499	345	2	1 ≤	QUANTITY
ejpam-5499	345	3	4	CARDINAL
ejpam-5499	345	4	j. m. jamis	PERSON
ejpam-5499	345	5	d. m. magpantay	PERSON
ejpam-5499	346	1	j. pure	PERSON
ejpam-5499	346	2	18	CARDINAL
ejpam-5499	346	3	2	CARDINAL
ejpam-5499	346	4	2025	CARDINAL
ejpam-5499	346	5	5499 16	DATE
ejpam-5499	346	6	18 1 ≤	MONEY
ejpam-5499	346	7	2	CARDINAL
ejpam-5499	348	1	1 ≤ i ≤ n − 1	QUANTITY
ejpam-5499	350	1	2	CARDINAL
ejpam-5499	350	2	1 ≤	MONEY
ejpam-5499	352	1	4.10	CARDINAL
ejpam-5499	353	1	12	CARDINAL
ejpam-5499	354	1	α(m(γcn	CARDINAL
ejpam-5499	357	1	11	DATE
ejpam-5499	357	2	k383	PRODUCT
ejpam-5499	361	1	one	CARDINAL
ejpam-5499	361	2	one	CARDINAL
ejpam-5499	362	1	two	CARDINAL
ejpam-5499	362	2	1	CARDINAL
ejpam-5499	363	1	zj	ORG
ejpam-5499	363	2	xj	ORG
ejpam-5499	363	3	xj	ORG
ejpam-5499	364	1	xj	GPE
ejpam-5499	364	2	∈	ORG
ejpam-5499	364	3	zj	ORG
ejpam-5499	366	1	1	CARDINAL
ejpam-5499	366	2	xj	GPE
ejpam-5499	366	3	xj+1}390	GPE
ejpam-5499	366	4	y = yp|1 ≤	PERSON
ejpam-5499	366	5	2	CARDINAL
ejpam-5499	367	1	every391 yp	PERSON
ejpam-5499	367	2	2 xi′s	DATE
ejpam-5499	368	1	2392	CARDINAL
ejpam-5499	370	1	zj	ORG
ejpam-5499	370	2	xi+1}∪y	ORG
ejpam-5499	372	1	2+	DATE
ejpam-5499	372	2	2	CARDINAL
ejpam-5499	372	3	2	CARDINAL
ejpam-5499	372	4	∈	ORG
ejpam-5499	374	1	zj	ORG
ejpam-5499	375	1	xj+1}|	GPE
ejpam-5499	375	2	|x| =	PERSON
ejpam-5499	375	3	2 + n−	CARDINAL
ejpam-5499	375	4	3	CARDINAL
ejpam-5499	375	5	1.396 2	DATE
ejpam-5499	375	6	∈	ORG
ejpam-5499	376	1	2	CARDINAL
ejpam-5499	376	2	1	CARDINAL
ejpam-5499	376	3	2	CARDINAL
ejpam-5499	377	1	2	CARDINAL
ejpam-5499	379	1	.400	CARDINAL
ejpam-5499	379	2	1 +	CARDINAL
ejpam-5499	379	3	2	CARDINAL
ejpam-5499	379	4	2	CARDINAL
ejpam-5499	380	1	∈	ORG
ejpam-5499	380	2	that401 yp	PERSON
ejpam-5499	383	1	|x| =	PERSON
ejpam-5499	383	2	1+ n−	CARDINAL
ejpam-5499	383	3	1	CARDINAL
ejpam-5499	383	4	n for n	ORG
ejpam-5499	383	5	4.11	CARDINAL
ejpam-5499	384	1	13	CARDINAL
ejpam-5499	385	1	one	CARDINAL
ejpam-5499	385	2	1.408	CARDINAL
ejpam-5499	385	3	j. m. jamis	PERSON
ejpam-5499	385	4	d. m. magpantay	PERSON
ejpam-5499	386	1	j. pure	PERSON
ejpam-5499	386	2	18	CARDINAL
ejpam-5499	386	3	2	CARDINAL
ejpam-5499	386	4	2025	CARDINAL
ejpam-5499	386	5	5499 17	DATE
ejpam-5499	386	6	18	CARDINAL
ejpam-5499	387	1	1	CARDINAL
ejpam-5499	387	2	1	CARDINAL
ejpam-5499	388	1	1	CARDINAL
ejpam-5499	389	1	2	CARDINAL
ejpam-5499	389	2	8	DATE
ejpam-5499	392	1	2	CARDINAL
ejpam-5499	392	2	w	ORG
ejpam-5499	392	3	x414	CARDINAL
ejpam-5499	392	4	1	CARDINAL
ejpam-5499	393	1	1	CARDINAL
ejpam-5499	393	2	1	CARDINAL
ejpam-5499	393	3	two	CARDINAL
ejpam-5499	393	4	2)417	CARDINAL
ejpam-5499	393	5	8	CARDINAL
ejpam-5499	393	6	r.	NORP
ejpam-5499	393	7	∈	ORG
ejpam-5499	393	8	γcn).418	PERSON
ejpam-5499	393	9	two	CARDINAL
ejpam-5499	394	1	2	CARDINAL
ejpam-5499	394	2	8	DATE
ejpam-5499	394	3	1	CARDINAL
ejpam-5499	395	1	one	CARDINAL
ejpam-5499	395	2	1.422	CARDINAL
ejpam-5499	395	3	14	CARDINAL
ejpam-5499	397	1	15	CARDINAL
ejpam-5499	399	1	14	CARDINAL
ejpam-5499	400	1	4	CARDINAL
ejpam-5499	400	2	16	CARDINAL
ejpam-5499	402	1	9	CARDINAL
ejpam-5499	402	2	5n−3 2	CARDINAL
ejpam-5499	404	1	5	CARDINAL
ejpam-5499	404	2	m(γdn	PRODUCT
ejpam-5499	406	1	m(γdn	PRODUCT
ejpam-5499	407	1	such440	ORG
ejpam-5499	408	1	eulerian	NORP
ejpam-5499	408	2	and442 hamiltonian	PERSON
ejpam-5499	409	1	m(γdn	ORG
ejpam-5499	409	2	j. m. jamis	PERSON
ejpam-5499	409	3	d. m. magpantay	PERSON
ejpam-5499	410	1	j. pure	PERSON
ejpam-5499	410	2	18	CARDINAL
ejpam-5499	410	3	2	CARDINAL
ejpam-5499	410	4	2025	CARDINAL
ejpam-5499	410	5	5499 18	DATE
ejpam-5499	410	6	18	CARDINAL
ejpam-5499	410	7	3	CARDINAL
ejpam-5499	412	1	n γ(m(γcn	ORG
ejpam-5499	413	1	2	CARDINAL
ejpam-5499	414	1	2	CARDINAL
ejpam-5499	414	2	1	CARDINAL
ejpam-5499	414	3	3 ω[m(γdn	PERCENT
ejpam-5499	415	1	2n α(m(γdn	CARDINAL
ejpam-5499	415	2	2n	CARDINAL
ejpam-5499	415	3	3n−1 2	CARDINAL
ejpam-5499	416	1	2n	CARDINAL
ejpam-5499	417	1	2n+	CARDINAL
ejpam-5499	417	2	1 446	CARDINAL
ejpam-5499	419	1	1	CARDINAL
ejpam-5499	419	2	w. b.	PERSON
ejpam-5499	420	1	f. smarandache	PERSON
ejpam-5499	422	1	editura	PERSON
ejpam-5499	422	2	2009.451	CARDINAL
ejpam-5499	423	1	2	CARDINAL
ejpam-5499	423	2	a. d. godase	PERSON
ejpam-5499	425	1	1(1):12–18	DATE
ejpam-5499	425	2	2015.453	CARDINAL
ejpam-5499	425	3	3	CARDINAL
ejpam-5499	425	4	n. f. yalcin	PERSON
ejpam-5499	427	1	university institute of science and technology	ORG
ejpam-5499	427	2	21(2):614–624	CARDINAL
ejpam-5499	427	3	2019.455	CARDINAL
ejpam-5499	427	4	4	CARDINAL
ejpam-5499	427	5	j. u. jeeshma	PERSON
ejpam-5499	429	1	international research456 journal	ORG
ejpam-5499	429	2	7(7):3459–3462	DATE
ejpam-5499	429	3	2020.457	CARDINAL
ejpam-5499	430	1	5	CARDINAL
ejpam-5499	430	2	j. akiyama	PERSON
ejpam-5499	430	3	t. hamada	ORG
ejpam-5499	430	4	i. yoshimora	PERSON
ejpam-5499	431	1	1975.459	CARDINAL
ejpam-5499	432	1	6	CARDINAL
ejpam-5499	432	2	d. s. nair	PERSON
ejpam-5499	434	1	2017.462	CARDINAL
ejpam-5499	435	1	7	CARDINAL
ejpam-5499	435	2	d. magpantay	PERSON
ejpam-5499	437	1	european journal	ORG
ejpam-5499	437	2	15(4):1888–1904	CARDINAL
ejpam-5499	437	3	8	CARDINAL
ejpam-5499	437	4	p. zhang	PERSON
ejpam-5499	438	1	first	ORDINAL
ejpam-5499	439	1	dover publications,466 mineola	ORG
ejpam-5499	439	2	ny	GPE
ejpam-5499	439	3	2012.467	DATE
ejpam-5499	439	4	9	CARDINAL
ejpam-5499	439	5	w. somnuek	PERSON
ejpam-5499	441	1	18(2):148–155	DATE