On the Riemann-Hurwitz formula for graph coverings

TL;DR

This paper extends the Riemann-Hurwitz formula to finite connected multigraphs under group actions, relating the genus of the original and quotient graphs with stabilizer orders.

Contribution

It introduces multiple versions of the Riemann-Hurwitz formula for graph coverings, accounting for fixed and invertible edges in the context of group actions.

Findings

Derived formulas relating graph genus and stabilizer groups

Applicable to graphs with fixed and invertible edges

Provides a framework for analyzing graph coverings

Abstract

The aim of this paper is to present a few versions of the Riemann-Hurwitz formula for a regular branched covering of graphs. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first homology group. We consider a finite group acting on a graph, possibly with fixed and invertible edges, and the respective factor graph. Then, the obtained Riemann-Hurwitz formula relates genus of the graph with genus of the factor graph and orders of the vertex and edge stabilisers.