Extremal Copositive Matrices with Zero Supports of Cardinality n-2

Let A ∈ Cn be an exceptional extremal copositive n × n matrix with positive diagonal. A zero u of A is a non-zero nonnegative vector such that uTAu = 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 ⊂ suppu 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, . . . , n} \ {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 Cn.