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PreprintarXiv licencearXiv

Generalizing Praeger--Xu graphs

  • Zhaochen Ding
  • Ci Xuan Wu
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Abstract

We uniformly study directed Praeger--Xu graphs and Praeger's generalized family overrightarrowPX(Γ,r,s), without requiring that s| r and Γ is an undirected graph. Assuming that Γ×overrightarrowK2 is connected, we determine the full automorphism group of overrightarrowPX(Γ,r,s) for every r and s, prove connectivity, characterize vertex-transitivity, and give an equivalent criterion for t-arc-transitivity. We also study when overrightarrowPX(Γ,r,s) is a Cayley graph. We also give two applications. First, we give a new family of twin-free connected arc-transitive digraphs of valency $4$ whose vertex stabilizers have exponential growth. We also give a connected regular digraph that is not vertex-transitive but whose direct product with a directed cycle is Cayley.

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Cite

BibTeXRIS
BibTeX
@misc{ding2026generalizing,
  title = {{Generalizing Praeger--Xu graphs}},
  author = {Zhaochen Ding and Ci Xuan Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35623},
  eprint = {2610.35623},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35623},
}
RIS
TY  - UNPB
TI  - Generalizing Praeger--Xu graphs
AU  - Zhaochen Ding
AU  - Ci Xuan Wu
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35623
UR  - https://arxiv.org/abs/2610.35623
AB  - We uniformly study directed Praeger--Xu graphs and Praeger's generalized family overrightarrowPX(Γ,r,s), without requiring that s| r and Γ is an undirected graph. Assuming that Γ×overrightarrowK_2 is connected, we determine the full automorphism group of overrightarrowPX(Γ,r,s) for every r and s, prove connectivity, characterize vertex-transitivity, and give an equivalent criterion for t-arc-transitivity. We also study when overrightarrowPX(Γ,r,s) is a Cayley graph. We also give two applications. First, we give a new family of twin-free connected arc-transitive digraphs of valency $4$ whose vertex stabilizers have exponential growth. We also give a connected regular digraph that is not vertex-transitive but whose direct product with a directed cycle is Cayley.
ER  -