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Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs

Received: 25 February 2023     Accepted: 16 March 2023     Published: 28 March 2023
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Abstract

A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let H be a uniform hypergraph. Let A(H) be the adjacency tensor of H. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum E-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.

Published in Mathematics and Computer Science (Volume 8, Issue 2)
DOI 10.11648/j.mcs.20230802.13
Page(s) 51-56
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Uniform Hypergraphs, Adjacency Tensor, Maximal E-Eigenvalue, Degree

References
[1] J. Cooper, A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 3268-3299.
[2] L. H. Lim, Singular values and eigenvalues of tensors, a variational approach, in: Proceedings of the IEEE International Workshop on Computational Advances in Multi- Sensor Adaptive Processing, vol. 1. CAMSAP’05, 2005, 129-132.
[3] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005) 1302-1324.
[4] L. Qi, H+-eigenvalues of Laplacian and signless Laplacian tensors, Commun, Math. Sci. 12 (6) (2014) 1045-1064.
[5] L. Liu, L. Kang, S. Bai, Bounds on the spectral radius of uniform hypergraphs, Discrete Applied Mathematics 259. 10.1016/j.dam.2018.12.007.
[6] S. Friedland, S. Gaubert, L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013) 738-749.
[7] Y. N. Yang, Q. Z. Yang, Further Results for Perron–Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications 31 (2010) 2517-2530.
[8] J. Shao, A general product of tensors with applications, Linear Algebra Appl. 439 (2013) 2350-2366.
[9] G. Yi, Tensor spectral properties of uniform hypergraphs, Fuzhou University 2014.
[10] Y. X. Yuan, M. Zhang, M. Lu, Some upper bounds on the eigenvalues of uniform hypergraphs, Linear Algebra Appl. 2015.
[11] L. Qi, Z. Luo, TENSOR ANALYSIS, Society for Industrial and Applied Mathematics Philadelphia, 2017.
[12] S. R. Bulò, M. Pelillo, New bounds on the clique number of graphs based on spectral hypergraph theory, Springer, Berlin, Heidelberg, 2009.
[13] Y. Yang, Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl. 31 (5) (2010) 2517-2530.
[14] Y. Yang, Q. Yang, On some properties of nonnegative weakly irreducible tensors, arXiv: 1111.0713v3, 2011.
[15] V. T. Sós, E. G. Straus, Extremals of functions on graphs with applications to graphs and hypergraphs, J. Combin. Theory Ser. B, 32 (1982) 246-257.
Cite This Article
  • APA Style

    Hongyu Zhang, Feng Fu, Caoji Yin. (2023). Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs. Mathematics and Computer Science, 8(2), 51-56. https://doi.org/10.11648/j.mcs.20230802.13

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    ACS Style

    Hongyu Zhang; Feng Fu; Caoji Yin. Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs. Math. Comput. Sci. 2023, 8(2), 51-56. doi: 10.11648/j.mcs.20230802.13

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    AMA Style

    Hongyu Zhang, Feng Fu, Caoji Yin. Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs. Math Comput Sci. 2023;8(2):51-56. doi: 10.11648/j.mcs.20230802.13

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  • @article{10.11648/j.mcs.20230802.13,
      author = {Hongyu Zhang and Feng Fu and Caoji Yin},
      title = {Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs},
      journal = {Mathematics and Computer Science},
      volume = {8},
      number = {2},
      pages = {51-56},
      doi = {10.11648/j.mcs.20230802.13},
      url = {https://doi.org/10.11648/j.mcs.20230802.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.13},
      abstract = {A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let H be a uniform hypergraph. Let A(H) be the adjacency tensor of H. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum E-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Some Upper Bounds of Maximum E-Eigenvalues of Uniform Hypergraphs
    AU  - Hongyu Zhang
    AU  - Feng Fu
    AU  - Caoji Yin
    Y1  - 2023/03/28
    PY  - 2023
    N1  - https://doi.org/10.11648/j.mcs.20230802.13
    DO  - 10.11648/j.mcs.20230802.13
    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
    JO  - Mathematics and Computer Science
    SP  - 51
    EP  - 56
    PB  - Science Publishing Group
    SN  - 2575-6028
    UR  - https://doi.org/10.11648/j.mcs.20230802.13
    AB  - A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let H be a uniform hypergraph. Let A(H) be the adjacency tensor of H. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum E-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.
    VL  - 8
    IS  - 2
    ER  - 

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Author Information
  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

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