Some properties of the total graph and regular graph of a commutative ring

Key Findings & Data:The total graph of $R$, $T(\Gamma(R))$, is the simple graph with vertex set $R$ and two distinct vertices are adjacent if their sum is a zero-divisor in $R$.

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Conclusion & Impact:When each of Reg$(\Gamma(R))$ and$Z(\Gamma(R))$ is regular and when it is Eulerian.

🔬Design:Research Article

📍Scope:Comprehensive Sample

⚙️Variables:Total graph of a commutative ring · Regular graph of a commutative ring · Locally connected

Abstract

Let $R$ be a commutative ring with unity. The total graph of $R$, $T(\Gamma(R))$, is the simple graph with vertex set $R$ and two distinct vertices are adjacent if their sum is a zero-divisor in $R$. Let Reg$(\Gamma(R))$ and $Z(\Gamma(R))$ be the subgraphs of $T(\Gamma(R))$ induced by the set of all regular elements and the set of zero-divisors in $R$, respectively. We determine when each of the graphs $T(\Gamma(R))$, Reg$(\Gamma(R))$, and $Z(\Gamma(R))$ is locally connected, and when it is locally homogeneous. When each of Reg$(\Gamma(R))$ and$Z(\Gamma(R))$ is regular and when it is Eulerian.

Keywords

Total graph of a commutative ring Regular graph of a commutative ring Locally connected Locally homogeneous Regular graph Eulerian graph

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Cite This Article

NAZZAL, K., GHANEM, M. (2022). Some properties of the total graph and regular graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics, Cilt: 47 - Sayı: 4, 835–843. https://ajindex.com/articles/530183

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