  A singular graph is a finite graph $\Gamma$ whose adjacency matrix is
singular. Let $G$ be a finite group and $p$ a prime divisor of the order of
$G$. Also let $\Omega(G,p)$ be the set of all $p$-singular elements of $G$. In
this paper, we show that the Cayley graph
$\Gamma_p(G):=\mathrm{Cay}(G,\Omega(G,p))$ 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 $\Gamma_p(G)$ is at most the $p$-part
of the order of $G$.
