Singular Graphs on which the Dihedral Group Acts Vertex Transitively

Abstract: 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 $s \in \mathbb{N}.$