Extremals of functions on graphs with applications to graphs and hypergraphs

The method used in an article by T. S. Matzkin and E. G. Straus [Canad. J. Math. 17 (1965), 533–540] is generalized by attaching nonnegative weights to t-tuples of vertices in a hypergraph subject to a suitable normalization condition. The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges. The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P. Turan's Theorem [Colloq. Math. 3 (1954), 19–30: (Hungarian) Mat. Fiz. Lapok 48 (1941), 436–452].