Corrigendum to "SPN graphs: when copositive $=$ SPN"

TL;DR

This paper corrects a previous proof regarding the SPN property of certain graphs, providing an alternative proof for a specific case and highlighting open questions for larger graphs.

Contribution

It offers a corrected proof for the case n=5 of a class of graphs being SPN and clarifies unresolved cases for larger graphs.

Findings

Confirmed T_5 is SPN with the new proof.

Open question remains for T_n with n>5.

Unresolved status of K_{2,n} being SPN for n>4.

Abstract

In this corrigendum, an error in the proof of a theorem in [Linear Algebra and its Applications 509 (2016) 82--113] is pointed out. This theorem states that every graph $T_n$ consisting of $n-2$ triangles sharing a common base is SPN. An alternative proof is given here for the case $n=5$, but for all $n>5$ it remains open whether $T_n$ is SPN. As a result, the question whether $K_{2,n}$, $n>4$, is SPN also remains open.