The spectra of lifted digraphs

Dalfó, C.; Fiol, M. A. and Širáň, J. (2019). The spectra of lifted digraphs. Journal of Algebraic Combinatorics, 50(4) pp. 419–426.

DOI: https://doi.org/10.1007/s10801-018-0862-y

Abstract

We present a method to derive the complete spectrum of the lift \mathrm{\Gamma\alpha} of a base digraph \mathrm{\Gamma}, with voltage assignment α on a (finite) group $\textit{G}$. The method is based on assigning to \mathrm{\Gamma} a quotient-like matrix whose entries are elements of the group algebra \mathds{C}[$\textit{G}$], which fully represents \mathrm{\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 generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs.

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