Clean Graphs over Rings of Order P^2

Heba Adel; Edris; Eman

doi:https://doi.org/10.54216/IJNS.260218

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Volume 26 • Issue 2 • PP: 241-250 • 2025

Clean Graphs over Rings of Order P^2

Heba Adel Abdelkarim ^1*

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,

Edris Rawashdeh ²

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,

Eman Rawshdeh ³

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¹Department of Mathematics, Irbid National University, Irbid, Jordan

²Department of Mathematics, Yarmouk University, Irbid, Jordan

³Department of Basic Scientific Sciences, Al-Huson University College, Al-Balqa Applied University, Irbid, Jordan

* Corresponding Author.

DOI https://doi.org/10.54216/IJNS.260218

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Received: January 10, 2025 Revised: March 02, 2025 Accepted: March 13, 2025

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Abstract

Assume R is a commutative ring with unity. The clean graph CL(R) is defined in which every vertex has the form (a, v), where a is an idempotent in R and v is a unit. In CL(R), two distinct vertices (a1, v1) and (a2, v2) are adjacent if a1a2 = a2a1 = 0 or v1v2 = v2v1 = 1. In this paper, we show that the clean graph CL(R) over a ring of order p2 can be defined only if R is one of the rings: Zp2 ,Zp ⊕Zp,Zp(+)Zp and GF(p2). Then, we study the spectrum, the biclique partition number, and the eigensharp property for the these clean graphs.

Keywords

Commutative Ring Clean Graph Spectrum of graph Biclique partition number Eigensharp graph

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Abdelkarim, Heba Adel, Rawashdeh, Edris, Rawshdeh, Eman. "Clean Graphs over Rings of Order P^2." International Journal of Neutrosophic Science, vol. Volume 26, no. Issue 2, 2025, pp. 241-250. DOI: https://doi.org/10.54216/IJNS.260218

Abdelkarim, H., Rawashdeh, E., Rawshdeh, E. (2025). Clean Graphs over Rings of Order P^2. International Journal of Neutrosophic Science, Volume 26(Issue 2), 241-250. DOI: https://doi.org/10.54216/IJNS.260218

Abdelkarim, Heba Adel, Rawashdeh, Edris, Rawshdeh, Eman. "Clean Graphs over Rings of Order P^2." International Journal of Neutrosophic Science Volume 26, no. Issue 2 (2025): 241-250. DOI: https://doi.org/10.54216/IJNS.260218

Abdelkarim, H., Rawashdeh, E., Rawshdeh, E. (2025) 'Clean Graphs over Rings of Order P^2', International Journal of Neutrosophic Science, Volume 26(Issue 2), pp. 241-250. DOI: https://doi.org/10.54216/IJNS.260218

Abdelkarim H, Rawashdeh E, Rawshdeh E. Clean Graphs over Rings of Order P^2. International Journal of Neutrosophic Science. 2025;Volume 26(Issue 2):241-250. DOI: https://doi.org/10.54216/IJNS.260218

H. Abdelkarim, E. Rawashdeh, E. Rawshdeh, "Clean Graphs over Rings of Order P^2," International Journal of Neutrosophic Science, vol. Volume 26, no. Issue 2, pp. 241-250, 2025. DOI: https://doi.org/10.54216/IJNS.260218

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