Characterization of digraphs with three complementarity eigenvalues

TL;DR

This paper characterizes strongly connected digraphs with exactly three complementarity eigenvalues, providing a complete structural description and advancing understanding of their spectral properties in directed graphs.

Contribution

It offers the first complete characterization of strongly connected digraphs with three complementarity eigenvalues, extending previous work on fewer eigenvalues.

Findings

Complete structural characterization of such digraphs.

Identification of properties distinguishing these digraphs.

Extension of spectral graph theory to directed graphs with three 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, makes the study of the latter of particular interest, and characterizing strongly connected digraphs with a small number of complementarity eigenvalues is a non trivial 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.