Idempotent graph of 2x2 matrix ring with involution

Abstract

Let R=M_2(F), where F is a finite field. In this paper, we investigate the idempotent graph of a ring R denoted by I*(R). We demonstrate that I*(R) is disconnected, having the components either complete bipartite graphs or complete graphs. A characterization is obtained for the regularity of I*(R). We determine the adjacency and Laplacian spectrum, the energy of I*(R) and prove Beck's conjecture.