An other approach of the diameter of $\Gamma (R)$ and $\Gamma (R[X])$

Abstract

Using the new extension of the zero-divisor graph $\tilde{\Gamma }(R)$ introduced in \cite{Groupe}, we give an approach of the diameter of $\Gamma (R)$ and $\Gamma (R[X])$ other than given in \cite{Lucas} thus we give a complete characterization for the possible diameters $1$, $2$ or $3$ of $\Gamma (R)$ and $\Gamma (R[x])$.