Yuejian Peng

On Lagrangians of r-uniform hypergraphs

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in Can J Math 17:533–540 (1965). 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 Turán 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 Comb Probab Comput 11:199–216 (2002) and a result in Peng and Zhao (Graphs Comb, 29:681–694, 2013).

An extension of the Motzkin-Straus theorem to non-uniform hypergraphs and its applications

In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Turans theorem using the connection. The connection of Lagrangians and Turan densities can be also used to prove the fundamental theorem of Erd?s-Stone-Simonovits on Turan densities of graphs. Very recently, the study of Turan densities of non-uniform hypergraphs has been motivated by extremal poset problems and suggested by Johnston and Lu. In this paper, we attempt to explore the applications of Lagrangian method in determining Turan densities of non-uniform hypergraphs. We first give a definition of the Lagrangian of a non-uniform hypergraph, then give an extension of the Motzkin-Straus theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices. Applying it, we give an extension of the Erd?s-Stone-Simonovits theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices. Our approach follows from the approach in Keevashs paper Keevash (2011).

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533–540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false. Frankl and Füredi conjectured that the r-uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of

Some results on Lagrangians of hypergraphs

{\mathbb N}^{(r)}

Connection between a class of polynomial optimization problems and maximum cliques of non-uniform hypergraphs

N(r) has the largest Lagrangian of all r-uniform hypergraphs with m edges. For

The connection between polynomial optimization, maximum cliques and Turán densities

r=2