On the largest graph-Lagrangians of hypergraphs

TL;DR

This paper investigates the conjecture that certain hypergraphs formed by colex ordering maximize the graph-Lagrangian among all hypergraphs with the same number of edges, providing bounds for specific cases.

Contribution

The paper establishes bounds for the graph-Lagrangians of special hypergraphs, supporting the conjecture by Frankl and F"uredi about maximum Lagrangians.

Findings

Bounds for graph-Lagrangians of specific hypergraphs are established.

Support for the conjecture that colex-ordered hypergraphs maximize the Lagrangian.

Partial results indicating the conjecture's validity in certain cases.

Abstract

Frankl and F\"uredi (1989) conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest graph-Lagrangian of all $r$-graphs with $m$ edges. In this paper, we establish some bounds for graph-Lagrangians of some special $r$-graphs that support this conjecture.