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The Laplacian of a uniform hypergraph

Author

Listed:
  • Shenglong Hu

    (The Hong Kong Polytechnic University)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

Abstract

In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$ -uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$ , and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H $$^+$$ -eigenvalues of the Laplacian and all the smallest H $$^+$$ -eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H $$^+$$ -eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H $$^+$$ -eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H $$^+$$ -eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H $$^+$$ -eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H $$^+$$ -geometric multiplicity of the zero H $$^+$$ -eigenvalue of the Lapalacian.

Suggested Citation

  • Shenglong Hu & Liqun Qi, 2015. "The Laplacian of a uniform hypergraph," Journal of Combinatorial Optimization, Springer, vol. 29(2), pages 331-366, February.
    • Handle: RePEc:spr:jcomop:v:29:y:2015:i:2:d:10.1007_s10878-013-9596-x
    DOI: 10.1007/s10878-013-9596-x
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    Citations

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    Cited by:

    1. G. H. Shirdel & A. Mortezaee & E. Golpar-raboky, 2021. "$${\mathcal {C}}^k_{m,s}$$ C m , s k as a k-uniform hypergraph and some its properties," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(1), pages 297-303, March.
    2. Alla Kammerdiner & Alexander Veremyev & Eduardo Pasiliao, 2017. "On Laplacian spectra of parametric families of closely connected networks with application to cooperative control," Journal of Global Optimization, Springer, vol. 67(1), pages 187-205, January.
    3. Shi, Tian & Qin, Yi & Yang, Qi & Ma, Zhongjun & Li, Kezan, 2023. "Synchronization of directed uniform hypergraphs via adaptive pinning control," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 615(C).
    4. Liying Kang & Wei Zhang & Erfang Shan, 0. "The spectral radius and domination number in linear uniform hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-12.
    5. Liying Kang & Wei Zhang & Erfang Shan, 2021. "The spectral radius and domination number in linear uniform hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 42(3), pages 581-592, October.
    6. Li, Li & Yan, Xihong & Zhang, Xinzhen, 2022. "An SDP relaxation method for perron pairs of a nonnegative tensor," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    7. Honghai Li & Jia-Yu Shao & Liqun Qi, 2016. "The extremal spectral radii of $$k$$ k -uniform supertrees," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 741-764, October.

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