Cayley graphs on finite groups generated by p-singular elements

A singular graph is a finite graph whose adjacency matrix is singular. Let G be a finite group and p a prime divisor of the order of G. Also let (G, p) be the set of all p-singular elements of G. In this paper, we show that the Cayley graph ₚ (G): =Cay (G, (G, p) ) is a singular graph, if and only if the group G has a non-principal p-block. We apply this result to obtain new classes of singular Cayley graphs (See Corollary 1. 3). We also prove that if G is p-solvable, then the diameter of ₚ (G) is at most the p-part of the order of G.