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Let AβCn be an exceptional extremal copositive nΓn 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 \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,β¦,n}β{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 nβ2, then G must be the cycle graph Cnβ
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