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

TL;DR

This paper analyzes the zero-divisor graph of a specific finite commutative ring, determining its structural properties, topological indices, and spectral characteristics to understand its algebraic and combinatorial features.

Contribution

It provides a comprehensive analysis of the zero-divisor graph of the ring _p + u_p + u^2_p, including graph invariants, topological indices, and spectral properties, which were not previously studied.

Findings

Determined the clique number, chromatic number, and connectivity of the graph.

Calculated eigenvalues, energy, and spectral radius of adjacency and Laplacian matrices.

Derived parameters of codes from the incidence matrix of the graph.

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 $\Gamma(R).$