id	sid	eid	entity	type
ejpam-6076	1	1	european journal	ORG
ejpam-6076	1	2	2025	DATE
ejpam-6076	1	3	18	CARDINAL
ejpam-6076	1	4	3	CARDINAL
ejpam-6076	1	5	6076	DATE
ejpam-6076	1	6	new york	GPE
ejpam-6076	1	7	jamil j. hamja1,∗	PERSON
ejpam-6076	1	8	amy a. laja1	PERSON
ejpam-6076	1	9	b. copel1	PERSON
ejpam-6076	1	10	j. amiruddin-rajik1	PERSON
ejpam-6076	1	11	nurijam hanna m. mohammad1	PERSON
ejpam-6076	1	12	1	CARDINAL
ejpam-6076	1	13	msu	ORG
ejpam-6076	1	14	7500	CARDINAL
ejpam-6076	1	15	philippines	GPE
ejpam-6076	5	1	∈	ORG
ejpam-6076	5	2	at least two	CARDINAL
ejpam-6076	5	3	\j	PRODUCT
ejpam-6076	7	1	first	ORDINAL
ejpam-6076	8	1	secondly	ORDINAL
ejpam-6076	9	1	cartesian	NORP
ejpam-6076	9	2	two	CARDINAL
ejpam-6076	10	1	2020	DATE
ejpam-6076	10	2	05c69	CARDINAL
ejpam-6076	10	3	1	CARDINAL
ejpam-6076	10	4	2020	DATE
ejpam-6076	11	1	1	CARDINAL
ejpam-6076	11	2	2	CARDINAL
ejpam-6076	16	1	j. j. hamja	PERSON
ejpam-6076	16	2	amylaja@msutawi-tawi.edu.ph (a. a. laja	ORG
ejpam-6076	16	3	hounamcopel@msutawi-tawi.edu.ph	ORG
ejpam-6076	16	4	h. b.	PERSON
ejpam-6076	16	5	j. amiruddin	PERSON
ejpam-6076	16	6	n. h. m. mohammad	PERSON
ejpam-6076	16	7	1	CARDINAL
ejpam-6076	17	1	2025	CARDINAL
ejpam-6076	18	1	j. j. hamja	PERSON
ejpam-6076	20	1	j. pure	PERSON
ejpam-6076	20	2	18	CARDINAL
ejpam-6076	20	3	3	CARDINAL
ejpam-6076	20	4	2025	CARDINAL
ejpam-6076	20	5	6076 2	DATE
ejpam-6076	20	6	13	CARDINAL
ejpam-6076	21	1	2023	CARDINAL
ejpam-6076	21	2	3	CARDINAL
ejpam-6076	23	1	⊆	CARDINAL
ejpam-6076	25	1	∈	ORG
ejpam-6076	25	2	one	CARDINAL
ejpam-6076	25	3	∈	ORG
ejpam-6076	30	1	3	CARDINAL
ejpam-6076	30	2	3	CARDINAL
ejpam-6076	34	1	cartesian	NORP
ejpam-6076	37	1	2	CARDINAL
ejpam-6076	37	2	4	CARDINAL
ejpam-6076	38	1	∈	ORG
ejpam-6076	39	1	∈	ORG
ejpam-6076	39	2	∈	ORG
ejpam-6076	41	1	⊆	CARDINAL
ejpam-6076	42	1	j. hamja et al	PERSON
ejpam-6076	43	1	j. pure	PERSON
ejpam-6076	43	2	18	CARDINAL
ejpam-6076	43	3	3	CARDINAL
ejpam-6076	43	4	2025	CARDINAL
ejpam-6076	43	5	6076 3 of 13	DATE
ejpam-6076	46	1	at least two	CARDINAL
ejpam-6076	48	1	∈ l(g	ORG
ejpam-6076	50	1	1	CARDINAL
ejpam-6076	51	1	at least one	CARDINAL
ejpam-6076	51	2	j	GPE
ejpam-6076	51	3	ng[j	ORG
ejpam-6076	56	1	5	CARDINAL
ejpam-6076	57	1	∈	ORG
ejpam-6076	57	2	at least two	CARDINAL
ejpam-6076	59	1	1	CARDINAL
ejpam-6076	60	1	∈	ORG
ejpam-6076	60	2	ab	GPE
ejpam-6076	61	1	α(g	PERSON
ejpam-6076	61	2	paraic et al	PERSON
ejpam-6076	64	1	one	CARDINAL
ejpam-6076	64	2	j	GPE
ejpam-6076	65	1	livingston et al	PERSON
ejpam-6076	67	1	two	CARDINAL
ejpam-6076	69	1	armada et al	ORG
ejpam-6076	69	2	8	CARDINAL
ejpam-6076	70	1	∈	ORG
ejpam-6076	70	2	zero	CARDINAL
ejpam-6076	70	3	at least two	CARDINAL
ejpam-6076	72	1	g j. j. hamja et al	PERSON
ejpam-6076	73	1	j. pure	PERSON
ejpam-6076	73	2	18	CARDINAL
ejpam-6076	73	3	3	CARDINAL
ejpam-6076	73	4	2025	CARDINAL
ejpam-6076	73	5	6076 4 of 13	DATE
ejpam-6076	73	6	∈	ORG
ejpam-6076	73	7	at least two	CARDINAL
ejpam-6076	75	1	hamja	PERSON
ejpam-6076	75	2	3	CARDINAL
ejpam-6076	77	1	3	CARDINAL
ejpam-6076	79	1	1	CARDINAL
ejpam-6076	80	1	3	CARDINAL
ejpam-6076	80	2	1	CARDINAL
ejpam-6076	80	3	n ≡ 0	ORG
ejpam-6076	82	1	2	CARDINAL
ejpam-6076	83	1	3	CARDINAL
ejpam-6076	83	2	γcerp(cn	ORG
ejpam-6076	83	3	n ≡ 0	ORG
ejpam-6076	84	1	3	CARDINAL
ejpam-6076	85	1	3	CARDINAL
ejpam-6076	85	2	n,	ORG
ejpam-6076	85	3	γcerp(kn	GPE
ejpam-6076	85	4	1	CARDINAL
ejpam-6076	85	5	1 or n ≥ 3	DATE
ejpam-6076	85	6	2	DATE
ejpam-6076	85	7	2	CARDINAL
ejpam-6076	85	8	4	CARDINAL
ejpam-6076	86	1	3	CARDINAL
ejpam-6076	86	2	bipartite	NORP
ejpam-6076	86	3	graph km	PERSON
ejpam-6076	86	4	γcerp(km	GPE
ejpam-6076	87	1	1	CARDINAL
ejpam-6076	87	2	1 or n	CARDINAL
ejpam-6076	87	3	4	DATE
ejpam-6076	87	4	2	CARDINAL
ejpam-6076	87	5	2	DATE
ejpam-6076	90	1	∈	ORG
ejpam-6076	91	1	at least one	CARDINAL
ejpam-6076	92	1	∈	ORG
ejpam-6076	93	1	one	CARDINAL
ejpam-6076	95	1	g. j. j. hamja et al	PERSON
ejpam-6076	96	1	j. pure	PERSON
ejpam-6076	96	2	18	CARDINAL
ejpam-6076	96	3	3	CARDINAL
ejpam-6076	96	4	2025	CARDINAL
ejpam-6076	96	5	6076 5	DATE
ejpam-6076	96	6	13	CARDINAL
ejpam-6076	96	7	6	CARDINAL
ejpam-6076	97	1	k. moreover	PERSON
ejpam-6076	97	2	2|ss	CARDINAL
ejpam-6076	99	1	= k.	PERSON
ejpam-6076	101	1	one	CARDINAL
ejpam-6076	103	1	k. moreover	PERSON
ejpam-6076	103	2	at least two	CARDINAL
ejpam-6076	103	3	2|ss	CARDINAL
ejpam-6076	104	1	2|ss	CARDINAL
ejpam-6076	105	1	1	CARDINAL
ejpam-6076	109	1	∈	ORG
ejpam-6076	110	1	dg(a	GPE
ejpam-6076	110	2	∈	ORG
ejpam-6076	112	1	j	PERSON
ejpam-6076	112	2	at least one	CARDINAL
ejpam-6076	112	3	j	GPE
ejpam-6076	113	1	∈	ORG
ejpam-6076	115	1	one	CARDINAL
ejpam-6076	116	1	j	PERSON
ejpam-6076	117	1	j	PERSON
ejpam-6076	120	1	1	CARDINAL
ejpam-6076	121	1	g. hence	GPE
ejpam-6076	122	1	2	CARDINAL
ejpam-6076	122	2	3 ≤	QUANTITY
ejpam-6076	123	1	b. j. j.	PERSON
ejpam-6076	125	1	j. pure	PERSON
ejpam-6076	125	2	18	CARDINAL
ejpam-6076	125	3	3	CARDINAL
ejpam-6076	125	4	2025	CARDINAL
ejpam-6076	125	5	6076 6	DATE
ejpam-6076	125	6	13	CARDINAL
ejpam-6076	129	1	1	CARDINAL
ejpam-6076	132	1	1	CARDINAL
ejpam-6076	134	1	xa	PERSON
ejpam-6076	134	2	j2	ORG
ejpam-6076	136	1	xa	PERSON
ejpam-6076	138	1	j1	ORG
ejpam-6076	138	2	j2	ORG
ejpam-6076	141	1	|j2|	CARDINAL
ejpam-6076	144	1	x3 x4 x5	PERSON
ejpam-6076	144	2	xa−4	NORP
ejpam-6076	144	3	xa−2	GPE
ejpam-6076	144	4	xa−1	GPE
ejpam-6076	146	1	ym−5	GPE
ejpam-6076	146	2	ym−1 ym	PERSON
ejpam-6076	147	1	1	CARDINAL
ejpam-6076	148	1	2	CARDINAL
ejpam-6076	150	1	2	CARDINAL
ejpam-6076	152	1	xa	PERSON
ejpam-6076	152	2	j ′ 2	LAW
ejpam-6076	154	1	xa	PERSON
ejpam-6076	156	1	j ′ 1	LAW
ejpam-6076	156	2	j ′ 2	LAW
ejpam-6076	162	1	2|	CARDINAL
ejpam-6076	166	1	x3 x4 x5	PERSON
ejpam-6076	166	2	xa−4	NORP
ejpam-6076	166	3	xa−5	GPE
ejpam-6076	166	4	xa−2	GPE
ejpam-6076	166	5	xa−1 xa	PERSON
ejpam-6076	167	1	ym−5	GPE
ejpam-6076	167	2	ym−1 ym g′	PERSON
ejpam-6076	170	1	2	CARDINAL
ejpam-6076	172	1	j. j. hamja et al	PERSON
ejpam-6076	173	1	j. pure	PERSON
ejpam-6076	173	2	18	CARDINAL
ejpam-6076	173	3	3	CARDINAL
ejpam-6076	173	4	2025	CARDINAL
ejpam-6076	173	5	6076 7	DATE
ejpam-6076	173	6	2	CARDINAL
ejpam-6076	173	7	1	CARDINAL
ejpam-6076	175	1	graph g. theorem 3	PERSON
ejpam-6076	176	1	n ≥ 1	DATE
ejpam-6076	177	1	1	CARDINAL
ejpam-6076	178	1	1	CARDINAL
ejpam-6076	178	2	2	CARDINAL
ejpam-6076	179	1	2	CARDINAL
ejpam-6076	184	1	1	CARDINAL
ejpam-6076	184	2	j ⊆	PERSON
ejpam-6076	185	1	1 ≤	QUANTITY
ejpam-6076	186	1	1	CARDINAL
ejpam-6076	189	1	1	CARDINAL
ejpam-6076	191	1	1	CARDINAL
ejpam-6076	192	1	j ′	LAW
ejpam-6076	192	2	x′	DATE
ejpam-6076	194	1	y′	DATE
ejpam-6076	194	2	x′	CARDINAL
ejpam-6076	195	1	1	CARDINAL
ejpam-6076	196	1	2	CARDINAL
ejpam-6076	198	1	2	CARDINAL
ejpam-6076	199	1	g = c5.	PERSON
ejpam-6076	201	1	2	CARDINAL
ejpam-6076	202	1	5	CARDINAL
ejpam-6076	204	1	5	CARDINAL
ejpam-6076	204	2	γcer(p5	ORG
ejpam-6076	204	3	2	CARDINAL
ejpam-6076	206	1	4	CARDINAL
ejpam-6076	207	1	3	CARDINAL
ejpam-6076	209	1	gk	PERSON
ejpam-6076	209	2	k ≥ 2.	ORG
ejpam-6076	210	1	i=1 γcerp(gi	PERSON
ejpam-6076	212	1	γcerp(kn	GPE
ejpam-6076	213	1	n. j. j. hamja et al	PERSON
ejpam-6076	214	1	j. pure	PERSON
ejpam-6076	214	2	18	CARDINAL
ejpam-6076	214	3	3	CARDINAL
ejpam-6076	214	4	2025	CARDINAL
ejpam-6076	214	5	6076 8	DATE
ejpam-6076	217	1	3	CARDINAL
ejpam-6076	217	2	1	CARDINAL
ejpam-6076	218	1	4	CARDINAL
ejpam-6076	220	1	1	CARDINAL
ejpam-6076	221	1	3	CARDINAL
ejpam-6076	224	1	f. harary	PERSON
ejpam-6076	224	2	9	CARDINAL
ejpam-6076	224	3	two	CARDINAL
ejpam-6076	224	4	g ◦ h	ORG
ejpam-6076	224	5	one	CARDINAL
ejpam-6076	225	1	∈	ORG
ejpam-6076	226	1	9	CARDINAL
ejpam-6076	227	1	5	CARDINAL
ejpam-6076	228	1	3	CARDINAL
ejpam-6076	228	2	2	CARDINAL
ejpam-6076	228	3	m. corollary 2.	PERSON
ejpam-6076	231	1	5	CARDINAL
ejpam-6076	232	1	2	CARDINAL
ejpam-6076	232	2	2	CARDINAL
ejpam-6076	233	1	two	CARDINAL
ejpam-6076	233	2	3	CARDINAL
ejpam-6076	235	1	4	CARDINAL
ejpam-6076	236	1	j. j. hamja et al	PERSON
ejpam-6076	237	1	j. pure	PERSON
ejpam-6076	237	2	18	CARDINAL
ejpam-6076	237	3	3	CARDINAL
ejpam-6076	237	4	2025	CARDINAL
ejpam-6076	237	5	6076 9	DATE
ejpam-6076	237	6	13	CARDINAL
ejpam-6076	237	7	6	CARDINAL
ejpam-6076	238	1	(g)| ≥ 3	DATE
ejpam-6076	240	1	∈	ORG
ejpam-6076	241	1	one	CARDINAL
ejpam-6076	241	2	∈	ORG
ejpam-6076	244	1	γ(g	NORP
ejpam-6076	245	1	∈	ORG
ejpam-6076	245	2	at least two	CARDINAL
ejpam-6076	245	3	v2 ∈ j	GPE
ejpam-6076	247	1	∈	ORG
ejpam-6076	247	2	one	CARDINAL
ejpam-6076	248	1	∈	ORG
ejpam-6076	249	1	one	CARDINAL
ejpam-6076	249	2	j	GPE
ejpam-6076	250	1	∈	ORG
ejpam-6076	250	2	∩	PERSON
ejpam-6076	250	3	1	CARDINAL
ejpam-6076	251	1	one	CARDINAL
ejpam-6076	251	2	j	GPE
ejpam-6076	252	1	∩	PERSON
ejpam-6076	252	2	2	CARDINAL
ejpam-6076	252	3	∈	ORG
ejpam-6076	255	1	1	CARDINAL
ejpam-6076	257	1	3	CARDINAL
ejpam-6076	258	1	(g)| ≥ 3	DATE
ejpam-6076	262	1	∈	ORG
ejpam-6076	263	1	⊆	CARDINAL
ejpam-6076	264	1	2	CARDINAL
ejpam-6076	265	1	2	CARDINAL
ejpam-6076	266	1	j	PERSON
ejpam-6076	267	1	6	CARDINAL
ejpam-6076	268	1	6	CARDINAL
ejpam-6076	268	2	3	CARDINAL
ejpam-6076	270	1	2	CARDINAL
ejpam-6076	271	1	at least two	CARDINAL
ejpam-6076	273	1	5	CARDINAL
ejpam-6076	273	2	two	CARDINAL
ejpam-6076	273	3	two	CARDINAL
ejpam-6076	274	1	f. harary	PERSON
ejpam-6076	274	2	9	CARDINAL
ejpam-6076	274	3	two	CARDINAL
ejpam-6076	275	1	u)(y	ORG
ejpam-6076	275	2	xy ∈	PERSON
ejpam-6076	276	1	⊆	CARDINAL
ejpam-6076	276	2	tx	ORG
ejpam-6076	276	3	⊆	PRODUCT
ejpam-6076	276	4	s. j. j. hamja et al	PERSON
ejpam-6076	277	1	j. pure	PERSON
ejpam-6076	277	2	18	CARDINAL
ejpam-6076	277	3	3	CARDINAL
ejpam-6076	277	4	2025	CARDINAL
ejpam-6076	277	5	6076 10	DATE
ejpam-6076	277	6	13	CARDINAL
ejpam-6076	277	7	7	CARDINAL
ejpam-6076	280	1	g[h	ORG
ejpam-6076	282	1	1	CARDINAL
ejpam-6076	282	2	∈	ORG
ejpam-6076	284	1	g[h	ORG
ejpam-6076	285	1	∈	ORG
ejpam-6076	286	1	∈	ORG
ejpam-6076	287	1	∈	ORG
ejpam-6076	287	2	one	CARDINAL
ejpam-6076	288	1	j	PERSON
ejpam-6076	288	2	∈	ORG
ejpam-6076	290	1	kl ∈ e(h	ORG
ejpam-6076	290	2	q.	PERSON
ejpam-6076	291	1	j	PERSON
ejpam-6076	292	1	2	CARDINAL
ejpam-6076	292	2	2	CARDINAL
ejpam-6076	292	3	q.	PERSON
ejpam-6076	297	1	1	CARDINAL
ejpam-6076	297	2	∈	ORG
ejpam-6076	298	1	g[h	ORG
ejpam-6076	298	2	1	CARDINAL
ejpam-6076	298	3	∈	ORG
ejpam-6076	299	1	1	CARDINAL
ejpam-6076	299	2	∈	ORG
ejpam-6076	300	1	w	ORG
ejpam-6076	300	2	1	CARDINAL
ejpam-6076	301	1	j	PERSON
ejpam-6076	301	2	∩	PERSON
ejpam-6076	303	1	ng[h](r	ORG
ejpam-6076	303	2	u	NORP
ejpam-6076	304	1	one	CARDINAL
ejpam-6076	305	1	2	CARDINAL
ejpam-6076	305	2	∈	ORG
ejpam-6076	306	1	ng[h](r	ORG
ejpam-6076	306	2	one	CARDINAL
ejpam-6076	306	3	q.	PERSON
ejpam-6076	307	1	1	CARDINAL
ejpam-6076	307	2	g[h	ORG
ejpam-6076	309	1	7	CARDINAL
ejpam-6076	309	2	5	CARDINAL
ejpam-6076	311	1	γcerp(g[h	PERSON
ejpam-6076	313	1	γ(h	ORG
ejpam-6076	314	1	1	CARDINAL
ejpam-6076	314	2	j. j. hamja et al	PERSON
ejpam-6076	315	1	j. pure	PERSON
ejpam-6076	315	2	18	CARDINAL
ejpam-6076	315	3	3	CARDINAL
ejpam-6076	315	4	2025	CARDINAL
ejpam-6076	315	5	6076 11	DATE
ejpam-6076	315	6	13	CARDINAL
ejpam-6076	315	7	6	CARDINAL
ejpam-6076	317	1	g[h	ORG
ejpam-6076	318	1	2	CARDINAL
ejpam-6076	319	1	1	CARDINAL
ejpam-6076	320	1	6 (ii	CARDINAL
ejpam-6076	323	1	g[h	ORG
ejpam-6076	324	1	6	CARDINAL
ejpam-6076	324	2	cartesian	NORP
ejpam-6076	324	3	two	CARDINAL
ejpam-6076	324	4	f. harary	PERSON
ejpam-6076	324	5	9	CARDINAL
ejpam-6076	324	6	cartesian	NORP
ejpam-6076	324	7	two	CARDINAL
ejpam-6076	325	1	two	CARDINAL
ejpam-6076	325	2	uk	GPE
ejpam-6076	325	3	vl	ORG
ejpam-6076	327	1	uk	GPE
ejpam-6076	327	2	∈ e(h	ORG
ejpam-6076	328	1	vl	ORG
ejpam-6076	328	2	∈	ORG
ejpam-6076	331	1	1	CARDINAL
ejpam-6076	331	2	1	CARDINAL
ejpam-6076	334	1	1	CARDINAL
ejpam-6076	335	1	1	CARDINAL
ejpam-6076	335	2	8	CARDINAL
ejpam-6076	336	1	2	CARDINAL
ejpam-6076	337	1	kp	ORG
ejpam-6076	337	2	kp	ORG
ejpam-6076	339	1	kp	ORG
ejpam-6076	340	1	kp	ORG
ejpam-6076	341	1	kp	ORG
ejpam-6076	341	2	kp	ORG
ejpam-6076	342	1	kp	NORP
ejpam-6076	342	2	first	ORDINAL
ejpam-6076	342	3	kp	ORG
ejpam-6076	342	4	second	ORDINAL
ejpam-6076	342	5	kp	ORG
ejpam-6076	343	1	1	CARDINAL
ejpam-6076	343	2	vi	PERSON
ejpam-6076	343	3	vi	PERSON
ejpam-6076	343	4	1	CARDINAL
ejpam-6076	343	5	second	ORDINAL
ejpam-6076	343	6	kp	ORG
ejpam-6076	344	1	2(p− 1	CARDINAL
ejpam-6076	344	2	kp	ORG
ejpam-6076	345	1	2(p	CARDINAL
ejpam-6076	345	2	1	CARDINAL
ejpam-6076	347	1	at least one	CARDINAL
ejpam-6076	347	2	j	GPE
ejpam-6076	348	1	kp	ORG
ejpam-6076	348	2	kp	ORG
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