### 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.

#### Keywords

commutative ring, zero-divisor graph, nilradical graph, non-nilradical graph, chromatic number, planar graph, energy of a graph

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