Some properties of the nilradical and non-nilradical graphs over finite commutative ring $\mathbb{Z}_n$

Abstract

Let $\mathbb{Z}_n$ be the finite commutative ring of residue classes modulo $n$ with identity and $\Gamma(\mathbb{Z}_n)$ be its zero-divisor graph. In this paper, we investigated some properties of nilradical graph, denoted by $N(\mathbb{Z}_n)$ and non-nilradical graph, denoted by $\Omega(\mathbb{Z}_n)$ of $\Gamma(\mathbb{Z}_n)$. In particular, we determined the Chromatic number and Energy of $N(\mathbb{Z}_n)$ and $\Omega(\mathbb{Z}_n)$ for a positive integer $n$. In addition, we have found the conditions in which $N(\mathbb{Z}_n)$ and $\Omega(\mathbb{Z}_n)$ graphs are planar. We have also given MATLAB coding of our calculations.