On Motzkin-Straus Type of Results and Frankl-Füredi Conjecture for Hypergraphs

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-Füredi on Graph-Lagrangians of hypergraphs. We show the connection between the Graph-Lagrangian of $\{1,r_1,r_2,\cdots ,r_l\}$-hypergraphs and $\{r_1,r_2,\cdots ,r_l\}$-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.

arXiv comment24 pages

References in corpus (1)

• On Lagrangians of $3$-uniform hypergraphs