Bulletin of the
Korean Mathematical Society BKMS

Given a graph $G$, the Motzkin and Straus formulation of the maximum clique problem is the quadratic program (QP) formed from the adjacent matrix of the graph $G$ over the standard simplex. It is well-known that the global optimum value of this QP (called Lagrangian) corresponds to the clique number of a graph. It is useful in practice if similar results hold for hypergraphs. In this paper, we attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques when the number of edges is in a certain range. Specifically, we obtain upper bounds for the Lagrangian of a hypergraph when the number of edges is in a certain range. These results further support a conjecture introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). We also establish an upper bound of the clique number in terms of Lagrangians for hypergraphs.

Keywords: cliques of hypergraphs, colex ordering, Lagrangian of hypergraphs, polynomial optimization

MSC numbers: 05C35, 05C65, 05D99, 90C27

Supported by: This research is partially supported by Chinese Universities Scientific Fund(No.N140504004) and the Doctoral Starting up Foundation of Liaoning Province(No.201601011).