Enumerating Cayley digraphs on dihedral groups

Abstract

This paper investigates the enumeration of Cayley digraphs, focusing on counting Cayley digraphs on dihedral groups up to CI-isomorphism. By leveraging the Cauchy-Frobenius Lemma and properties of automorphisms, we derive an explicit formula for the number of non-isomorphic Cayley digraphs on dihedral groups with DCI-property, particularly for the group $\mathrm{D}_{6p}$ with $p>3$ a prime. The enumeration involves detailed analysis of cycle numbers of automorphisms and their actions on the group elements, culminating in a precise count of non-isomorphic digraphs.