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

TL;DR

This paper develops a general method to find all finite abelian regular coverings of a graph that admit a specific automorphism subgroup lift, and applies it to classify certain coverings of the Petersen graph.

Contribution

It introduces a new systematic approach for identifying G-admissible abelian coverings of graphs and classifies all such coverings for the Petersen graph.

Findings

Classified all connected arc-transitive abelian regular coverings of the Petersen graph.

Provided a general method for finding G-admissible abelian coverings of any finite connected simple graph.

Abstract

Given a finite connected simple graph $\Gamma$, and a subgroup $G$ of its automorphism group, a general method for finding all finite abelian regular coverings of $\Gamma$ 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.