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On Lagrangians of r-uniform hypergraphs

Author

Listed:
  • Yuejian Peng

    (Hunan University)

  • Qingsong Tang

    (Northeastern University
    Jilin University)

  • Cheng Zhao

    (Jilin University
    Indiana State University)

Abstract

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).

Suggested Citation

  • Yuejian Peng & Qingsong Tang & Cheng Zhao, 2015. "On Lagrangians of r-uniform hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 812-825, October.
    • Handle: RePEc:spr:jcomop:v:30:y:2015:i:3:d:10.1007_s10878-013-9671-3
    DOI: 10.1007/s10878-013-9671-3
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    References listed on IDEAS

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    1. Luana E. Gibbons & Donald W. Hearn & Panos M. Pardalos & Motakuri V. Ramana, 1997. "Continuous Characterizations of the Maximum Clique Problem," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 754-768, August.
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    1. Qingsong Tang & Yuejian Peng & Xiangde Zhang & Cheng Zhao, 2017. "On Motzkin–Straus type results for non-uniform hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 504-521, August.
    2. Liying Kang & Lele Liu & Erfang Shan, 2019. "The eigenvectors to the p-spectral radius of general hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 556-569, August.

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