The dot total graph of a commutative ring without the zero element

Résumé

Let $\mathcal{R}$ be a commutative ring with $1\neq 0$, $Z(\mathcal{R})$ be the set of zero-divisors of $\mathcal{R}$, and $Reg(\mathcal{R})=\mathcal{R}\setminus Z(\mathcal{R})$ be the set of regular elements of $\mathcal{R}$. The dot total graph of $\mathcal{R}$ is the simple (undirected) graph $T_{Z(\mathcal{R})}(\Gamma(\mathcal{R}))$ with vertices all elements of $\mathcal{R}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in Z(\mathcal{R})$. In this paper, we study the (induced) subgraph $T_{Z(\mathcal{R})}(\Gamma(\mathcal{R^{*}}))$ of $T_{Z(\mathcal{R})}(\Gamma(\mathcal{R}))$, with vertices $\mathcal{R^{*}}=\mathcal{R}\setminus \{0\}$. After that, connectivity, clique number, and girth of $T_{Z(\mathcal{R})}(\Gamma(\mathcal{R^{*}}))$ have also been studied. Finally, we determine the cases when $T_{Z(\mathcal{R})}(\Gamma(\mathcal{R^{*}}))$ is Eulerian, Hamiltonian, and $T_{Z(\mathcal{R})}(\Gamma(\mathcal{R^{*}}))$ contains an Eulerian trail.