Research article

A general result on the spectral radii of nonnegative k-uniform tensors

  • Received: 18 November 2019 Accepted: 09 February 2020 Published: 17 February 2020
  • MSC : 05C50, 05C65, 15A69

  • In this paper, we define $k$-uniform tensors for $k\geq 2$, which are more closely related to the $k$-uniform hypergraphs than the general tensors, and introduce the parameter $r^{(q)}_{i}(\mathbb{A})$ for a tensor $\mathbb{A}$, which is the generalization of the $i$-th slice sum $r_ {i}(\mathbb{A})$ (also the $i$-th average 2-slice sum $m_{i}(\mathbb{A})$). By using $r^{(q)}_{i}(\mathbb{A})$ for $q\geq1$, we obtain a general result on the sharp upper bound for the spectral radius of a nonnegative $k$-uniform tensor. When $k = 2, q = 1, 2, 3$, this result deduces the main results for nonnegative matrices in [1,8,27]; when $k\geq 3, q = 1$, this result deduces the main results in [5,20]. We also find that the upper bounds obtained from different $q$ can not be compared. Furthermore, we can obtain some known or new upper bounds by applying the general result to $k$-uniform hypergraphs and $k$-uniform directed hypergraphs, respectively.

    Citation: Chuang Lv, Lihua You, Yufei Huang. A general result on the spectral radii of nonnegative k-uniform tensors[J]. AIMS Mathematics, 2020, 5(3): 1799-1819. doi: 10.3934/math.2020121

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  • In this paper, we define $k$-uniform tensors for $k\geq 2$, which are more closely related to the $k$-uniform hypergraphs than the general tensors, and introduce the parameter $r^{(q)}_{i}(\mathbb{A})$ for a tensor $\mathbb{A}$, which is the generalization of the $i$-th slice sum $r_ {i}(\mathbb{A})$ (also the $i$-th average 2-slice sum $m_{i}(\mathbb{A})$). By using $r^{(q)}_{i}(\mathbb{A})$ for $q\geq1$, we obtain a general result on the sharp upper bound for the spectral radius of a nonnegative $k$-uniform tensor. When $k = 2, q = 1, 2, 3$, this result deduces the main results for nonnegative matrices in [1,8,27]; when $k\geq 3, q = 1$, this result deduces the main results in [5,20]. We also find that the upper bounds obtained from different $q$ can not be compared. Furthermore, we can obtain some known or new upper bounds by applying the general result to $k$-uniform hypergraphs and $k$-uniform directed hypergraphs, respectively.
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    [1] M. Adam, D. Aggeli, A. Aretaki, Some new bounds on the spectral radius of nonnegative matrices, AIMS Mathematics, 5 (2019), 701-716.
    [2] C. Berge, Hypergraph, Combinatorics of Finite Sets, 3 Eds., North-Holland, Amsterdam, 1973.
    [3] R. A. Brualdi, Introductory Combinatorics, 3 Eds., China Machine press, Beijing, 2002.
    [4] K. C. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.
    [5] D. M. Chen, Z. B. Chen, X. D. Zhang, Spectral radius of uniform hypergraphs and degree sequences, Front. Math. China., 6 (2017), 1279-1288.
    [6] Z. M. Chen, L. Q. Qi, Circulant tensors with applications to spectral hypergraph theory and stochastic process, J. Ind. Manag. Optim., 12 (2016), 1227-1247.
    [7] J. Cooper, A. Dutle, Spectral of uniform hypergraph, Linear Algebra Appl., 436 (2012), 3268-3292.
    [8] X. Duan, B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl., 439 (2013), 2961-2970.
    [9] A. Ducournau, A. Bretto, Random walks in directed hypergraphs and application to semisupervised image segmentation, Comput. Vis. Image Und., 120 (2014), 91-102.
    [10] S. Friedland, A. Gaubert, L. Han, Perron-Frobenius theorems for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.
    [11] G. Gallo, G. Longo, S. Pallottino, et al. Directed hypergraphs and applications, Discrete Appl. Math., 42 (1993), 177-201.
    [12] M. Khan, Y. Fan, On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs, Linear Algebra Appl., 480 (2015), 93-106.
    [13] C. Q. Li, Y. T. Li, X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.
    [14] K. Li, L. S. Wang, A polynomial time approximation scheme for embedding a directed hypergraph on a ring, Inform. Process. Lett., 97 (2006), 203-207.
    [15] W. Li, K. N. Michael, Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335.
    [16] 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 (CAMSAP 05), 1 (2005), 129-132.
    [17] L. H. Lim, Foundations of Numerical Multilinear Algebra: Decomposition and Approximation of Tensors, Dissertation, 2007.
    [18] L. H. Lim, Eigenvalues of Tensors and Some Very Basic Spectral Hypergraph Theory, Matrix Computations and Scientific Computing Seminar, 2008.
    [19] H. Y. Lin, B. Mo, B. Zhou, et al. Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs, Appl. Math. Comput., 285 (2016), 217-227.
    [20] C. Lv, L. H. You, X. D. Zhang, A Sharp upper bound on the spectral radius of a nonnegative k-uniform tensor and its applications to (directed) hypergraphs, J. Inequal. Appl., 32 (2020), 1-16.
    [21] H. Minc, Nonnegative Matrices, John and Sons Inc., New York, 1988.
    [22] K. Pearson, T. Zhang, On spectral hypergraph theory of the adjacency tensor, Graphs Combin., 30 (2014), 1233-1248.
    [23] L. Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput, 40 (2005), 1302-1324.
    [24] L. Q. Qi, H+-eigenvalues of Laplacian and signless Lapaclian tensors, Commun. Math. Sci., 12 (2014), 1045-1064.
    [25] J. Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439 (2013), 2350-2366.
    [26] J. Y. Shao, H. Y. Shan, L. Zhang, On some properties of the determinants of tensors, Linear Algebra Appl., 439 (2013), 3057-3069.
    [27] R. Xing, B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl., 449 (2014), 194-209.
    [28] J. S. Xie, L. Q. Qi, Spectral directed hypergraph theory via tensors, Linear and Multilinear Algebra, 64 (2016), 780-794.
    [29] Y. N. Yang, Q. Z. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
    [30] Y. N. Yang, Q. Z. Yang, On some properties of nonnegative weakly irreducible tensors, arXiv: 1111.0713v2, 2011.
    [31] L. H. You, X. H. Huang, X. Y. Yuan, Sharp bounds for spectral radius of nonnegative weakly irreducible tensors, Front. Math. China., 14 (2019), 989-1015.
    [32] X. Y. Yuan, M. Zhang, M. Lu, Some upper bounds on the eigenvalues of uniform hypergraphs, Linear Algebra Appl., 484 (2015), 540-549.

    © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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