Enumeration of cubic Cayley graphs on dihedral groups

TL;DR

This paper classifies all cubic Cayley graphs on dihedral groups of order 2p using spectral methods, establishing isomorphism criteria and counting classes via quadratic reciprocity.

Contribution

It provides a complete classification of cubic Cayley graphs on dihedral groups and links isomorphism to spectral properties, using number theory techniques.

Findings

Two cubic Cayley graphs are isomorphic iff they are cospectral.

The number of isomorphic classes is determined by quadratic reciprocity.

Spectral methods effectively classify Cayley graphs on dihedral groups.

Abstract

Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on $D_{2p}$ are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on $D_{2p}$ by using Gauss' celebrated law of quadratic reciprocity.