Enumerating Cayley digraphs on dihedral groups

2025/07/29 by Lu, Zai Ping, Xie, Jia Yin, Xie, Jin-Hua
#05C25 #05C30 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.21658

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 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.

Citations

• Dihedral groups with the m-DCI property
• Enumerating Cayley (di-)graphs on dihedral groups

Related

• Enumerating Cayley (di-)graphs on dihedral groups
• Asymptotic enumeration of Cayley digraphs
• Dihedral groups with the m-DCI property
• Dihedral groups of order 2pq or 2pqr are DCI
• On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups