Graph Theoretic and Spectral Properties of the Zero-Divisor Graph of $\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p$

Abstract

In this article, we study the zero-divisor graph of the commutative ring with identity $R= \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where $u^2 = 0, v^2 = 0, uv = vu$ and $p$ is an odd prime. We determine several graph-theoretic properties associated with the zero-divisor graph $Γ(R),$ including the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth. In addition, we compute certain topological indices of the graph $Γ(R).$ Furthermore, we find the eigenvalues, energy and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of the zero-divisor graph $(Γ(R).$