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arXiv · 1604.02172

SPN graphs: when copositive=SPN

Naomi Shaked-Monderer
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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.

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Naomi Shaked-Monderer. 2016-04-07. SPN graphs: when copositive=SPN. https://doi.org/10.1016/j.laa.2016.07.018

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