Let R = M₂ (F) be a 2 2 matrix ring over a finite field F. The zero-divisor graph of R, denoted by 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 ABᵗ = 0, where Bᵗ is a transpose of the matrix B. In this paper, we consider a subgraph of 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).
Lande et al. (Wed,) studied this question.