SPN graphs: when copositive=SPN

TL;DR

This paper investigates the conditions under which copositive matrices associated with a graph are also SPN, providing sufficient and necessary conditions, and proposing a conjecture for a complete characterization of SPN graphs.

Contribution

It introduces new sufficient and necessary conditions for a graph to be SPN and discusses the gap between these conditions, proposing a conjecture for full characterization.

Findings

Identified sufficient conditions for graphs to be SPN.

Established necessary conditions involving forbidden subgraphs.

Proposed a conjecture for the complete characterization of SPN graphs.

Abstract

A real symmetric matrix $A$ is copositive if $x^TAx\ge 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative one. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than $4$. A graph $G$ is an SPN graph if every copositive matrix whose graph is $G$ is SPN. In this paper we present sufficient conditions for a graph to be SPN (in terms of its possible blocks) and necessary conditions for a graph to be SPN (in terms of forbidden subgraphs). We also discuss the remaining gap between these two sets of conditions, and make a conjecture regarding the complete characterization of SPN graphs.