Complementarity spectrum of digraphs

TL;DR

This paper investigates the complementarity spectrum of digraphs, focusing on their characterization and the existence of non-isomorphic digraphs sharing the same spectrum, extending the study from undirected graphs to directed graphs.

Contribution

It characterizes digraphs with one or two complementarity eigenvalues and provides examples of non-isomorphic digraphs with identical spectra.

Findings

Characterization of digraphs with one complementarity eigenvalue

Characterization of digraphs with two complementarity eigenvalues

Examples of non-isomorphic digraphs sharing the same spectrum

Abstract

In this paper we study the complementarity spectrum of digraphs, with special attention to the problem of digraph characterization through this complementarity spectrum. That is, whether two non-isomorphic digraphs with the same number of vertices can have the same complementarity eigenvalues. The complementarity eigenvalues of matrices, also called Pareto eigenvalues, has led to the study of the complementarity spectrum of (undirected) graphs and, in particular, the characterization of undirected graphs through these eigenvalues is an open problem. We characterize the digraphs with one and two complementarity eigenvalues, and we give examples of non-isomorphic digraphs with the same complementarity spectrum.