On Zero-Divisor Graph of the ring ${\mathbb{F}}_p+u{\mathbb{F}}_p+u^2{\mathbb{F}}_p$

Abstract

In this article, we discussed the zero-divisor graph of a commutative ring with identity ${\mathbb{F}}_p+u{\mathbb{F}}_p+u^2{\mathbb{F}}_p$ where $u^3=0$ and $p$ is an odd prime. We find the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth of a zero-divisor graph associated with the ring. We find some of topological indices and the main parameters of the code derived from the incidence matrix of the zero-divisor graph $\Gamma (R).$ Also, we find the eigenvalues, energy and spectral radius of both adjacency and Laplacian matrices of $\Gamma (R).$