arXiv subjects
Publications and source records attributed to E. Jabbouri.
In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+y\in Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.
Using the new extension of the zero-divisor graph $\widetildeΓ(R)$ introduced in \cite{Groupe}, we give an approach of the diameter of $Γ(R)$ and $Γ(R[X])$ other than given in \cite{Lucas} thus we give a complete characterization for the possible diameters $1$, $2$ or $3$ of $Γ(R)$ and $Γ(R[x])$.
We continue our study of the new extension of zero-divisor graph. We give a complete characterization for the possible diameters of $\widetildeΓ(R)$ and $\widetildeΓ(R[x_1,\dots,x_n])$, we investigate the relation between the zero-divisor graph, the subgraph of total graph on $Z(R)^{\star}$ and $\widetildeΓ(R)$ and we present some other properties of $\widetildeΓ(R)$.