The zero-divisor graph of $2\times 2$ matrix ring and its energies

Abstract

Let $R=M_2(\mathbb F)$ be a $2\times 2$ matrix ring over a finite field $\mathbb F$. The zero-divisor graph of $R$, denoted by $\Gamma^t(R)$, is a simple undirected graph with the vertex set consisting of all nonzero left zero-divisors in $R$, and two vertices $A$ and $B$ being adjacent if and only if $AB^t=0$, where $B^t$ is a transpose of the matrix $B$.
In this paper, we consider a subgraph of $\Gamma^t(R)$ denoted by $IdN(R)$ whose vertex set consists of all non-trivial idempotent and nonzero nilpotent elements in $R$. It has been established that the components of $IdN(R)$ are either complete graphs or complete bipartite graphs. Additionally, a necessary and sufficient condition for the regularity of $IdN(R)$ is obtained. We also analyze the adjacency and Laplacian spectra, as well as the energy and Laplacian energy of $IdN(R)$. Furthermore, it is proved that Beck's conjecture holds for $IdN(R)$.