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

Temel Bulgular & Veri: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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Sonuç & Katkı:When each of Reg $(\Gamma(R))$ and $Z(\Gamma(R))$ is regular and when it is Eulerian.

🔬Tür:Research Article

📍Kapsam:Genişletilmiş Akademik Örneklem

⚙️Değişkenler: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.

Anahtar kelimeler

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

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Bu Makaleye Atıf Yap

Ghanem, M., Nazzal, K. (2018). Some properties of the total graph and regular graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics, Volume: 47 Issue: 4. https://ajindex.com/articles/1402908

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