Enumerating Cayley (di-)graphs on dihedral groups

TL;DR

This paper counts the number of Cayley and Cayley di-graphs on dihedral groups of order 2p, using Pólya enumeration, including connected graphs and digraphs with specified out-degree.

Contribution

It provides explicit enumeration formulas for Cayley (di-)graphs on dihedral groups, including connected cases and out-degree specific digraphs, up to isomorphism.

Findings

Number of Cayley and Cayley di-graphs on D_{2p} up to isomorphism.

Enumeration of connected Cayley (di-)graphs.

Counting Cayley digraphs with fixed out-degree k.

Abstract

Let $p$ be an odd prime, and $D_{2p}=\langle \tau,\sigma\mid \tau^p=\sigma^2=e,\sigma\tau\sigma=\tau^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we provide the number of (connected) Cayley (di-)graphs on $D_{2p}$ up to isomorphism by using the P\'{o}lya enumeration theorem. In the process, we also enumerate (connected) Cayley digraphs on $D_{2p}$ of out-degree $k$ up to isomorphism for each $k$.