On Lagrangians of r-uniform Hypergraphs

TL;DR

This paper investigates the relationship between the Lagrangian of r-uniform hypergraphs and their maximum clique sizes, providing new Motzkin-Straus type results that extend previous findings and refine existing bounds.

Contribution

It introduces new Motzkin-Straus type theorems for r-uniform hypergraphs, generalizing and refining prior results in the field.

Findings

Established Motzkin-Straus type bounds for hypergraphs

Extended Talbot's results to broader hypergraph classes

Refined understanding of Lagrangian and clique size relationship

Abstract

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in [7]. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin-Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus' result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin-Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in [19] and a result in [11].