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 On cliques and Lagrangians of hypergraphs Bull. Korean Math. Soc.Published online 2019 May 16 Qingsong Tang, Xiangde Zhang, and Cheng Zhao Northeastern University, Indiana State University Abstract : 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 Grpah-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 Full-Text :

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