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 $\Gamma (R),$ including the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth. In addition, we compute certain topological indices of the graph $\Gamma (R).$ 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 $(\Gamma (R).$