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

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Authors :

Prohelika Das 1 *

Author Address :

1 Department of Mathematics, Cotton University, Guwahati-781001, India.

*Corresponding author.

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 $IJ=0$. 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 $diam(Gamma_{s}(N))>3$, then $N$ is necessarily a strongly semi-prime near-ring. Also we find the $chi (Gamma_{s}(N))$ and investigate some characterizations of cliques and maximal cliques in $Gamma_{s}(N)$.

Keywords :

Near-ring; essential ideal; diameter; girth; chromatic number.



Article Info :

Received : October 21, 2018; Accepted : January 19, 2019.



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