Representation of Zeros of a Copositive Matrix via Maximal Cliques of a Graph

TL;DR

This paper explores the relationship between copositive matrices and graph theory, revealing how zeros of these matrices relate to maximal cliques in a graph, and introduces algorithms for analyzing these zeros.

Contribution

It introduces a novel connection between copositive matrix zeros and maximal cliques, along with algorithms for their construction and analysis.

Findings

Zeros of copositive matrices can be expressed as unions of convex hulls of minimal zeros.

Maximal cliques of a constructed graph correspond to subsets of minimal zeros.

Algorithms for constructing minimal and all normalized zeros are developed.

Abstract

There is a profound connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and solving various challenging graph-related problems. Conversely, graph theory provides a rich set of concepts and techniques that can be applied to analyze key properties of copositive matrices, including their eigenvalues and spectra. In this paper, we present new aspects of the relationship between copositive matrices and graph theory. Focusing on the set of normalized zeros of a copositive matrix, we investigate its properties and demonstrate that this set can be expressed as a union of convex hulls of subsets of minimal zeros. We show that these subsets are connected with the set of maximal cliques of a special graph constructed on the basis of the set of minimal zeros of this matrix. We develop an algorithm for constructing both the set of normalized minimal zeros and the set of all normalized zeros of a copositive matrix.