
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 is copositive if for every nonnegative vector . 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 . A graph is an SPN graph if every copositive matrix whose graph is 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.
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