We study the relationship between two key concepts in the theory of digraphs, those of quotient digraphs and voltage digraphs. These techniques contract or expand a given digraph in order to study its characteristics, or to obtain more involved structures. As an application, we relate the spectrum of a digraph Γ, called a voltage digraph or base, with the spectrum of its lifted digraph Γα . We prove that all the eigenvalues of Γ (including multiplicities) are, in addition, eigenvalues of Γα . This study is carried out by introducing several reduced matrix representations of Γ α . As an example of our techniques, we study some basic properties of the Alegre digraph and its base.

Research of the first two authors is supported by MINECO under project MTM2014-60127-P, and by AGAUR under project 2014SGR1147. The first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 734922.