An algebraic approach to lifts of digraphs

Abstract

We study the relationship between two key concepts in the theory of (di)graphs: the quotient digraph, and the lift $Γ^α$ of a base (voltage) digraph. These techniques contract or expand a given digraph in order to study its characteristics, or obtain more involved structures. This study is carried out by introducing a quotient-like matrix, with complex polynomial entries, which fully represents $Γ^α$. In particular, such a matrix gives the quotient matrix of a regular partition of $Γ^α$, and when the involved group is Abelian, it completely determines the spectrum of $Γ^α$. As some examples of our techniques, we study some basic properties of the Alegre digraph. In addition we completely characterize the spectrum of a new family of digraphs, which contains the generalized Petersen graphs, and that of the Hoffman-Singleton graph.