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arXiv · 2507.21658

Enumerating Cayley digraphs on dihedral groups

Zai Ping Lu·Jia Yin Xie·Jin-Hua Xie
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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.

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Zai Ping Lu, Jia Yin Xie, Jin-Hua Xie. 2025-07-29. Enumerating Cayley digraphs on dihedral groups. https://arxiv.org/abs/2507.21658

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