Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON CLIQUES AND LAGRANGIANS OF HYPERGRAPHS

  • Tang, Qingsong (College of Sciences Northeastern University) ;
  • Zhang, Xiangde (College of Sciences Northeastern University) ;
  • Zhao, Cheng (Department of Mathematics and Computer Science Indiana State University)
  • Received : 2018.01.30
  • Accepted : 2018.09.19
  • Published : 2019.05.31
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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 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

E1BMAX_2019_v56n3_569_f0001.png 이미지

Figure 1. Hessian Diagram on [t](r)

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