The zero-divisor graph of 2 × 2 matrix ring and its energies

Let R = M_2(F) be a 2 × 2 matrix ring over a finite field F . The zero-divisor graph of R , denoted by Γ^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 Γ^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) .