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On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

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Abstract

Motzkin and Straus established a remarkable connection between the maximum clique and the Graph-Lagrangian of a graph 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 is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Graph-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 Peng and Zhao (Graphs Comb. 29:681–694, 2013) and extend a result of Talbot (Comb. Probab. Comput. 11:199–216, 2002).

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Acknowledgements

We thank two anonymous referees and the editor for helpful and insightful comments.

We also thank Professor Franco Giannessi for suggesting the terminology ‘Graph-Lagrangian’.

This research is partially supported by National Natural Science Foundation of China (No. 11271116).

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Authors and Affiliations

  1. College of Sciences, Northeastern University, Shenyang, 110819, P.R. China

    Qingsong Tang & Xiangde Zhang

  2. School of Mathematics, Jilin University, Changchun, 130012, P.R. China

    Qingsong Tang & Cheng Zhao

  3. College of Mathematics, Hunan University, Changchun, 410082, P.R. China

    Yuejian Peng

  4. Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809, USA

    Cheng Zhao

Authors
  1. Qingsong Tang
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  2. Yuejian Peng
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  3. Xiangde Zhang
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  4. Cheng Zhao
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Corresponding author

Correspondence to Cheng Zhao.

Additional information

Communicated by Horst Martini.

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Tang, Q., Peng, Y., Zhang, X. et al. On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs. J Optim Theory Appl 163, 31–56 (2014). https://doi.org/10.1007/s10957-013-0485-3

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  • DOI: https://doi.org/10.1007/s10957-013-0485-3

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