The triangle graph $T_6$ is not SPN

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Stephen Drury


A real symmetric matrix $A$ is copositive if $x'Ax \geq 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative matrix. 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. We show that the triangle graph $T_6$ is not SPN.

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