On quotient digraphs and voltage digraphs

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 ⌈, called a voltage digraph or base, with the spectrum of its lifted digraph ⌈^α . We prove that all the eigenvalues of ⌈ (including multiplicities) are, in addition, eigenvalues of ⌈^α. This study is carried out by introducing several reduced matrix representations of ⌈^α. As an example of our techniques, we study some basic properties of the Alegre digraph and its base.