Combined voltage assignments, factored lifts, and their spectra

TL;DR

This paper develops a spectral analysis method for factored lifts of graphs using combined voltage assignments, extending classical concepts to generalized covers with automorphism groups acting freely.

Contribution

It introduces a novel approach to determine spectra of factored lifts via group representations and complex group ring matrices, generalizing existing voltage assignment techniques.

Findings

Derived a complete spectral characterization of factored lift graphs.

Established a sufficient condition for lifting eigenvectors.

Extended voltage assignment concepts to generalized graph covers.

Abstract

We consider lifting eigenvalues and eigenvectors of graphs to their {\em factored lifts}, derived by means of a {\em combined voltage assignment} in a group. The latter extends the concept of (ordinary) voltage assignments known from regular coverings and corresponds to the cases of generalized covers of Poto\v{c}nik and Toledo (2021) in which a group of automorphisms of a lift acts freely on its arc set. With the help of group representations and certain matrices over complex group rings associated with the graphs to be lifted, we develop a method for the determination of the complete spectra of the factored lift graphs and derive a sufficient condition for lifting eigenvectors.