Singular Cayley graphs and p-blocks of finite groups

Abstract

For a simple finite graph Γ, the multiplicity of the eigenvalue 0 of the adjacency matrix of Γ is called the nullity of Γ. The energy of Γ is defined as the sum of the absolute values of its eigenvalues. In this paper, we apply the block theory of finite groups to study the Cayley graph ${\Gamma }_p\left(G\right)$ defined on a finite group G in which its connecting set consists of p-singular elements of G. We use this Cayley graph to investigate several methods for constructing singular graphs. Then we assume that G is p-solvable and obtain some nice restrictions on the structure of ${\Gamma }_p\left(G\right)$. We first achieve an explicit formula for the nullity of ${\Gamma }_p\left(G\right)$. In addition, we find a lower bound for the energy of ${\Gamma }_p\left(G\right)$. Finally, we prove that the diameter of ${\Gamma }_p\left(G\right)$ is at most $|G{|}_p$.