The spectra of lifted 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.