Enumerating Cayley (di-)graphs on dihedral groups

Abstract

Let p be an odd prime, and D 2p = τ, σ | τ p = σ 2 = e, στ σ = τ −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. (Q. Huang).2 byÁdám [1]: all circulant graphs are CI-graphs of the corresponding cyclic groups. This conjecture was disproved by Elspas and Turner [9], and however, the conjecture caused a lot of activity on the characterization of CI-graphs (DCI-graphs) or CIgroups (DCI-groups) [3][4][5][6][7][8][11][12][13][14][15]17,[20][21][22]24]. Another motivation for investigating CI-graphs (DCI-graphs) or CI-groups (DCI-groups) is to determine and enumerate the isomorphic classes of Cayley graphs for a given group. By the definition, if Cay(G, S) is a CI-graph (DCI-graph), then to decide whether or not Cay(G, S) is isomorphic to Cay(G, T ), we only need to decide whether or not there exists an automorphism α ∈ Aut(G) such that α(S) = T . For this reason, Mishna [19] (see also [2]) applied the Pólya enumeration theorem to count the isomorphic classes of Cayley graphs (Cayley digraphs) on some CI-groups (DCI-groups). Also, the isomorphic classes of some families of Cayley graphs which are edge-transitive but not arc-transitive were determined in [18,25,26]. For more results about CI-problem (DCI-problem) and determination for isomorphic classes of Cayley graphs, we refer the reader to the review paper [16] and references therein.Let p be an odd prime, and D 2p = τ, σ | τ p = σ 2 = e, στ σ = τ −1 the dihedral group of order 2p. In [3], Babai showed that D 2p is a DCI-group. This remind us the Pólya enumeration theorem could be used to enumerate the isomorphic classes of Cayley (di-)graphs of D 2p . In this paper, inspired by Mishna's work [19], we obtain the number of (connected) Cayley (di-)graphs on D 2p up to isomorphism. In the process, we also enumerate (connected) Cayley digraphs on D 2p of out-degree k up to isomorphism for each k.