Enumerating Cayley digraphs on dihedral groups
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908470968582144 |
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| author | Lu, Zai Ping Xie, Jia Yin Xie, Jin-Hua |
| author_facet | Lu, Zai Ping Xie, Jia Yin Xie, Jin-Hua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumerating Cayley digraphs on dihedral groups Lu, Zai Ping Xie, Jia Yin Xie, Jin-Hua Combinatorics 05C25, 05C30 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. |
| title | Enumerating Cayley digraphs on dihedral groups |
| topic | Combinatorics 05C25, 05C30 |
| url | https://arxiv.org/abs/2507.21658 |