Enumerating Cayley (di-)graphs on dihedral groups

Abstract

Let $p$ be an odd prime, and $D_{2p}=⟨\tau ,\sigma \mid {\tau }^p={\sigma }^2=e,\sigma \tau \sigma ={\tau }^{-1}⟩$ 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 Pólya enumeration theorem. In the process, we also enumerate (connected) Cayley digraphs on $D_{2p}$ of out-degree $k$ up to isomorphism for each $k$.

14 pages, 0 figure