The spectra of lifted digraphs

TL;DR

This paper introduces a method to determine the complete eigen-spectrum of lifted digraphs using group algebra and character theory, generalizing previous results on Cayley digraph spectra.

Contribution

It provides a novel approach to derive the spectra of lifted digraphs through quotient-like matrices and irreducible characters, extending earlier work on Cayley digraphs.

Findings

Derived the eigenvalues and eigenvectors of lifted digraphs

Connected spectra of lifts to base digraphs and group characters

Generalized previous spectral results for Cayley digraphs

Abstract

We present a method to derive the complete spectrum of the lift $\Gamma^\alpha$ of a base digraph $\Gamma$, with voltage assignments on a (finite) group $G$. The method is based on assigning to $\Gamma$ a quotient-like matrix whose entries are elements of the group algebra $\mathbb{C}[G]$, which fully represents $\Gamma^{\alpha}$. This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of $G$. Thus, our main theorem generalize some previous results of Lov\'az and Babai concerning the spectra of Cayley digraphs.