Singular Graphs on which the Dihedral Group Acts Vertex Transitively

TL;DR

This paper investigates the singularity of finite graphs with vertex transitivity under dihedral group actions, providing conditions for Cayley graphs over dihedral groups to be non-singular based on subgroup intersections.

Contribution

It characterizes the nullity of graphs with dihedral group automorphisms and establishes specific criteria for Cayley graphs over dihedral groups to be non-singular.

Findings

Cayley graphs over dihedral groups $D_{p^s}$ are non-singular under certain subgroup intersection conditions.

The nullity of such graphs can be explicitly determined based on the structure of the acting dihedral group.

Provides new insights into the spectral properties of vertex-transitive graphs with dihedral symmetry.

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 $s \in \mathbb{N}.$