Cayley graphs on $p$-solvable groups generated by $p$-singular elements

Abstract

For a graph $\hat{I}“$, the multiplicity of the eigenvalue $0$, denoted by $\hat{I}\cdot (\hat{I}“)$, is called the nullity of $\hat{I}“$. Also the energy of $\hat{I}“$, denoted by $\mathcal{E}(\hat{I}“)$, is defined as the sum of the absolute values of the eigenvalues of $\hat{I}“$. The index of a subgroup $H$ in a group $G$ is denoted by $[G:H]$. For a prime $p$, let $G$ be a finite $p$-solvable group whose order is divisible by $p$. Also let $\hat{I}{\text{c◯}}_p(G)$ be the set of all $p$-singular elements of $G$. In this paper, we apply block theory of finite groups to show that the Cayley graph $\hat{I}{“}_p(G):=\mathrm{Cay}(G,\hat{I}{\text{c◯}}_p(G))$ is an integral graph with $\hat{I}\cdot (\hat{I}{“}_p(G))=|G|-[G:O_{p^{\prime }}(G)]$, where $O_{p^{\prime }}(G)$ is the largest normal subgroup of $G$ whose order is co-prime to $p$. We also find a lower bound for $\mathcal{E}(\hat{I}{“}_p(G))$. Finally, we prove that the diameter of $\hat{I}{“}_p(G)$ is at most $|G{|}_p$.