Extremal Copositive Matrices with Zero Supports of Cardinality n-2

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$.