Enumerating Cayley digraphs on dihedral groups
Zai Ping Lu, Jia Yin Xie, Jin-Hua Xie

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 with 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.
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