On Motzkin-Straus Type of Results and Frankl-F\"uredi Conjecture for Hypergraphs

Publication date: 10 December 2013

Abstract: A remarkable connection between the order of a maximum clique and the Graph-Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extension were useful in both combinatorics and optimization. Since then, Graph-Lagrangian has been a useful tool in extremal combinatorics. In this paper, we give a parametrized Graph-Lagrangian for non-uniform hypergraphs and provide several Motzkin-Straus type results for nonuniform hypergraphs which generalize results from [1] and [2]. Another part of the paper concerns a long-standing conjecture of Frankl-F"uredi on Graph-Lagrangians of hypergraphs. We show the connection between the Graph-Lagrangian of

1, r_{1}, r_{2}, c d o t s, r_{l}

-hypergraphs and

r_{1}, r_{2}, c d o t s, r_{l}

-hypergraphs. Some of our results provide solutions to the maximum value of a class of polynomial functions over the standard simplex of the Euclidean space.

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