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