A note on strong zero-divisor graphs of near-rings

Abstract

For a near-ring $N$, the strong zero-divisor graph $\Gamma_s(N)$ is a graph with vertices $V^*(N)$, the set of all non-zero left $N$-subset having non-zero annihilators and two vertices $I$ and $J$ are adjacent if and only if $I J=0$. In this paper, we study diameter and girth of the graph $\Gamma_s(N)$ wherein the nilpotent and invariant vertices are playing a significant role. We show that if $\operatorname{diam}\left(\Gamma_s(N)\right)>3$, then $N$ is necessarily a strongly semi-prime near-ring. Also we find the $\chi\left(\Gamma_s(N)\right)$ and investigate some characterizations of cliques and maximal cliques in $\Gamma_s(N)$.