  Let $\Gamma$ be a simple connect graph on a finite vertex set $V$ and let $A$
be its adjacency matrix. Then $\Gamma$ is said to be \textit{singular} if and
only if $0$ is an eigenvalue of $A.$ The \textit{nullity (singularity)} of
$\Gamma,$ denoted by ${\rm null}(\Gamma),$ is the \textit{algebraic
multiplicity} of the eigenvalue $0$ in the spectrum of $\Gamma.$ The general
problem of characterising singular graphs is easy to state but it seems too
difficult in this time. In this work, we investigate this problem for finite
graphs on which the dihedral group $D_n$ acts vertex transitively as group of
automorphisms. We determine the nullity of such graphs. We show that Cayley
graphs over dihedral groups $D_{p^s} $ is non-singular if $|H \cap C_{p^s}|\neq
|H \cap C_{p^s}b|$ and $|H|<p$ where $p$ is a prime number and $s \in
\mathbb{N}.$
