Characterization of digraphs with three complementarity eigenvalues

Given a digraph D, the complementarity spectrum of is defined as set eigenvalues its adjacency matrix. This has been shown to be useful in several fields, particularly spectral graph theory. The differences between properties for (undirected) graphs and digraphs, makes study latter particular interest, characterizing strongly connected digraphs with small number non trivial problem. Recently, one two have completely characterized. In this paper we exactly three elements spectrum, ending complete characterization. leads structural characterization general having eigenvalues.