Enumerating Cayley (di-)graphs on dihedral groups

Abstract: Let $p$ be an odd prime, and $D_{2p}=\langle \tau,\sigma\mid \tau^{p=\sigma2=e,\sigma\tau\sigma=\tau{-1}\rangle$} 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 $k$.