Characterization of digraphs with three complementarity eigenvalues

Abstract

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