Lifting Voltages in Graph Covers

TL;DR

This paper investigates conditions under which derived graphs from voltage digraphs with abelian group labels are isomorphic, providing a decidable criterion involving common covers and lifted voltages.

Contribution

It establishes a necessary and sufficient condition for isomorphism of derived graphs from voltage graphs with abelian groups, including a constructive method.

Findings

Derived graphs are isomorphic iff a common cover exists and voltages correspond under lifted groups.

Conditions for isomorphism are decidable and constructive methods are provided.

The paper offers a practical approach to analyze graph covers and voltage assignments.

Abstract

We consider voltage digraphs, here referred to as graphs, whose edges are labeled with elements from a given group, and explore their derived graphs. Given two voltage graphs, with voltages in abelian groups, we establish a necessary and sufficient condition for their two derived graphs to be isomorphic. This condition requires: (1) the existence of a voltage graph that covers both given graphs, and (2) when the two sets of voltages are lifted to the common cover, the correspondence between these sets of voltages determines an isomorphism between the groups generated by these voltages. We show that conditions (1) and (2) are decidable, and provide a method for constructing the common cover and for lifting the voltage assignments.