Enumerating Cayley digraphs on dihedral groups

TL;DR

This paper provides an explicit enumeration formula for Cayley digraphs on dihedral groups, especially D_{6p} with prime p>3, using automorphism analysis and the Cauchy-Frobenius Lemma.

Contribution

It introduces a new explicit formula for counting non-isomorphic Cayley digraphs on dihedral groups with the DCI-property, expanding understanding of their automorphism structures.

Findings

Derived an explicit enumeration formula for Cayley digraphs on dihedral groups.

Analyzed automorphism cycle structures to distinguish non-isomorphic digraphs.

Provided counts for Cayley digraphs on D_{6p} with prime p>3.

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.