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

## TL;DR

This paper introduces a new approach to determine the diameter of zero-divisor graphs of a ring and its polynomial extension, providing a complete characterization of possible diameters 1, 2, or 3.

## Contribution

It presents a novel method using an extended zero-divisor graph to characterize the diameters of zero-divisor graphs of rings and their polynomial rings.

## Key findings

- Complete characterization of diameters 1, 2, or 3 for $ ext{G}(R)$ and $ ext{G}(R[X])$
- New approach based on the extended zero-divisor graph $ ilde{ ext{G}}(R)$
- Alternative to previous methods in the literature

## Abstract

Using the new extension of the zero-divisor graph $\widetilde{\Gamma}(R)$ 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 $\Gamma(R[x])$.