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PreprintarXiv licencearXiv

Cubic edge-transitive graphs of order ℓ p2

  • Xue Wang
  • Jun-Jie Huang
  • Da-Wei Yang
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Abstract

A graph is edge-transitive if its automorphism group acts transitively on its edge set. This paper is devoted to a systematic study of cubic edge-transitive graphs of order ℓ p2, where p is a prime and ℓ is an even integer. We first show that either Γ is the unique cubic edge-transitive graph of order $18$, or its full automorphism group Aut(Γ) contains a normal Sylow p-subgroup P which is isomorphic to either Zp2 or Zp2. For the case P≅ Zp2, we prove that Γ is either arc-regular or edge-regular, and provide a complete classification of such graphs. For the case P≅ Zp2, we prove that Γ is a normal cover of certain known graphs. This work generalizes previous classifications by Zhou and Feng (2024), Lu et al.~(2004), and others. As an application, we obtain a classification of cubic edge-transitive graphs of order ℓ p2 in the case where ℓ is square-free.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026cubic,
  title = {{Cubic edge-transitive graphs of order $\ell p^2$}},
  author = {Xue Wang and Jun-Jie Huang and Da-Wei Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32351},
  eprint = {2610.32351},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32351},
}
RIS
TY  - UNPB
TI  - Cubic edge-transitive graphs of order ℓ p^2
AU  - Xue Wang
AU  - Jun-Jie Huang
AU  - Da-Wei Yang
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32351
UR  - https://arxiv.org/abs/2610.32351
AB  - A graph is edge-transitive if its automorphism group acts transitively on its edge set. This paper is devoted to a systematic study of cubic edge-transitive graphs of order ℓ p^2, where p is a prime and ℓ is an even integer. We first show that either Γ is the unique cubic edge-transitive graph of order $18$, or its full automorphism group Aut(Γ) contains a normal Sylow p-subgroup P which is isomorphic to either Z_p^2 or Z_p^2. For the case P≅ Z_p^2, we prove that Γ is either arc-regular or edge-regular, and provide a complete classification of such graphs. For the case P≅ Z_p^2, we prove that Γ is a normal cover of certain known graphs. This work generalizes previous classifications by Zhou and Feng (2024), Lu et al.~(2004), and others. As an application, we obtain a classification of cubic edge-transitive graphs of order ℓ p^2 in the case where ℓ is square-free.
ER  -