ABSTRACT The authors continue their study of extremal problems of Turan type for directed graphs with multiple edges, now permitting any finite non-negative integer multiplicity. Having proved earlier (for the case of multiplicity at most 1) that there exists , for any family of “sample” digraphs, a matrix which represents the structure of an “asymptotically extremal sequence” of digraphs (containing none of the sample digraphs, and having a total number of arcs asymptotic to the maximum), they address themselves to the inverse problem: is every matrix so realized for some finite family of sample digraphs? They prove that this is indeed true for “dense” matrices - having certain integer entries, and such that an associated quadratic form attains its maximum for the standard simplex uniquely at an interior point.
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