Enumerating Cayley (di-)graphs on dihedral groups

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Let p be an odd prime, and D₂ₚ= τ,σ τᵖ=σ²=e,στσ=τ⁻¹ the dihedral group of order 2p. In this paper, we provide the number of (connected) Cayley (di-)graphs on D₂ₚ up to isomorphism by using the Pólya enumeration theorem. In the process, we also enumerate (connected) Cayley digraphs on D₂ₚ of out-degree k up to isomorphism for each k.