Extremal copositive matrices with minimal zero supports of cardinality two

TL;DR

This paper characterizes extremal copositive matrices with minimal zero supports of size two, showing they are precisely those with entries in {-1,0,1} obtained via diagonal scaling from specific extremal matrices.

Contribution

It provides a complete characterization of extremal copositive matrices with minimal zero supports of size two, linking them to matrices with entries in {-1,0,1} and known extremal matrices.

Findings

Matrices with minimal zero supports of size two have entries in {-1,0,1}.

Such matrices are obtainable by diagonal scaling from extremal {-1,0,1} matrices.

The characterization extends previous work by Hoffman and Pereira (1973).

Abstract

Let $A \in {\cal C}^n$ be an extremal copositive matrix with unit diagonal. Then the minimal zeros of $A$ all have supports of cardinality two if and only if the elements of $A$ are all from the set $\{-1,0,1\}$. Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal $\{-1,0,1\}$ unit diagonal matrices characterized by Hoffman and Pereira in 1973.