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Lifting automorphisms along abelian regular coverings of graphs

Haimiao Chen

TL;DR

An effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.

Abstract

This article proposes an effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.

Lifting automorphisms along abelian regular coverings of graphs

TL;DR

An effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.

Abstract

This article proposes an effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.

Paper Structure

This paper contains 4 sections, 5 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$
  3. The criterion for lifting automorphisms
  4. Conclusion and example

Key Result

Proposition 1.1

Let $\tilde{\Gamma}=\Gamma\times_{\phi}A$ be a regular covering. Then an automorphism $\alpha$ of $\Gamma$ can be lifted to an automorphism of $\tilde{\Gamma}$ if and only if, for each closed walk $W$ in $\Gamma$, one has $\phi(\alpha(W))=0$ if $\phi(W)=0$.

Figures (1)

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

Theorems & Definitions (11)

  • Proposition 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 1 more