Critical groups of covering, voltage, and signed graphs

TL;DR

This paper explores the relationship between graph coverings and their critical groups, providing a detailed description of kernels in terms of voltage and signed graphs, and developing a theory for double covers.

Contribution

It introduces a new framework for understanding kernels of critical group morphisms via voltage and signed graph data, especially for regular and double covers.

Findings

Kernels of critical group surjections are characterized by voltage graph data.

Double covers of signed graphs are analyzed with a new theoretical approach.

The paper establishes a connection between graph coverings and their critical groups.

Abstract

Graph coverings are known to induce surjections of their critical groups. Here we describe the kernels of these morphisms in terms of data parametrizing the covering. Regular coverings are parametrized by voltage graphs, and the above kernel can be identified with a naturally defined voltage graph critical group. For double covers, the voltage graph is a signed graph, and the theory takes a particularly pleasant form, leading also to a theory of double covers of signed graphs.