On Zero-Divisor Graph of the Ring $\frac{\mathbb{F}_p[u, v]}{\langle u^2,\, v^2, \, uv-vu\rangle}$

TL;DR

This paper investigates the zero-divisor graph of a specific commutative ring, analyzing its graph-theoretic properties, topological indices, and spectral characteristics.

Contribution

It provides a comprehensive analysis of the zero-divisor graph of a particular non-chain ring, including graph invariants and spectral data, which was not previously studied.

Findings

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

Computed eigenvalues, energy, and spectral radius of various matrices associated with the graph.

Calculated topological indices of the zero-divisor graph.

Abstract

In this article, we study the zero-divisor graph of the commutative non-chain ring with identity $ \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 of the associated zero-divisor graph \(\Gamma(R)\), including clique number, chromatic number, vertex connectivity, edge connectivity, diameter, and girth. In addition, we compute certain topological indices of \(\Gamma(R)\). Furthermore, we obtain the eigenvalues, energy, and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of \(\Gamma(R)\).