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How to find G-admissible abelian coverings of a graph?

Haimiao Chen, Hao Shen

Abstract

Given a finite connected simple graph $Γ$, and a subgroup $G$ of its automorphism group, a general method for finding all finite abelian regular coverings of $Γ$ that admit a lift of each element of $G$ is developed. As an application, all connected arc-transitive abelian regular coverings of the Petersen graph are classified up to isomorphism.

How to find G-admissible abelian coverings of a graph?

Abstract

Given a finite connected simple graph , and a subgroup of its automorphism group, a general method for finding all finite abelian regular coverings of that admit a lift of each element of is developed. As an application, all connected arc-transitive abelian regular coverings of the Petersen graph are classified up to isomorphism.

Paper Structure

This paper contains 14 sections, 10 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. General framework
  4. Main goal
  5. An algebraic preparation
  6. The method of finding admissible abelian coverings
  7. Arc-transitive finite abelian coverings of the Petersen graph
  8. The Petersen graph
  9. Arc-transitive abelian coverings
  10. Proof of Theorem \ref{['thm:abelian coverings of Petersen']}
  11. $i_{0}=5$
  12. $i_{0}=4$
  13. $i_{0}=3$
  14. $i_{0}=0$

Key Result

Proposition 2.1

Given two $A$-coverings $\Gamma\times_{\phi}A$ and $\Gamma\times_{\psi}A$. An automorphism $\beta\in\rm Aut(\Gamma)$ can be lifted to $\Gamma\times_{\phi}A\rightarrow\Gamma\times_{\psi}A$ if and only if there exists a group automorphism $\sigma:A\rightarrow A$ such that $E(\psi)\circ\beta_{\ast}=\si

Figures (1)

  • Figure 1: (a) The spanning tree T; (b) the induced subgraph

Theorems & Definitions (29)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Theorem 3.7
  • proof
  • Remark 3.8
  • ...and 19 more