Some Properties of the Total Graph and Regular Graph of a Commutative Ring

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.