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

For a graph $\Gamma$, the multiplicity of the eigenvalue $0$, denoted by $\eta(\Gamma)$, is called the nullity of $\Gamma$. Also the energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. 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 $\Omega_p(G)$ 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 $\Gamma_p(G):=\mathrm{Cay}(G,\Omega_p(G))$ is an integral graph with $\eta(\Gamma_p(G))=|G|-[G:O_{p^\prime}(G)]$, 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 $\mathcal{E}(\Gamma_p(G))$. Finally, we prove that the diameter of $\Gamma_p(G)$ is at most $|G|_p$.