On Lagrangians of Hypergraphs Containing Dense Subgraphs

TL;DR

This paper extends the understanding of Lagrangians in hypergraphs, providing upper bounds for hypergraphs with dense subgraphs, supporting conjectures, and generalizing previous results in hypergraph optimization.

Contribution

It offers new upper bounds on hypergraph Lagrangians with dense subgraphs, supporting conjectures and extending prior hypergraph Lagrangian results.

Findings

Established upper bounds for hypergraph Lagrangians with dense subgraphs.

Supported conjectures by Peng and Zhao (2012).

Extended a 2002 result of Talbot on hypergraph Lagrangians.

Abstract

Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). \keywords{Cliques of hypergraphs \and Colex ordering \and Lagrangians of hypergraphs \and Polynomial optimization}