RESEARCH ARTICLE

Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

  • Lihua YOU 1 ,
  • Xiaohua HUANG 1 ,
  • Xiying YUAN , 2
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  • 1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
  • 2. Department of Mathematics, Shanghai University, Shanghai 200444, China

Received date: 18 Jun 2019

Accepted date: 08 Oct 2019

Published date: 15 Oct 2019

Copyright

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature

Abstract

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix, the equality cases of the bounds are completely characterized by graph theory methods. Applying these bounds to a nonnegative irreducible matrix or a connected graph (digraph), we can improve the results of L. H. You, Y. J. Shu, and P. Z. Yuan [Linear Multilinear Algebra, 2017, 65(1): 113–128], and obtain some new or known results. Applying these bounds to a uniform hypergraph, we obtain some new results and improve some known results of X. Y. Yuan, M. Zhang, and M. Lu [Linear Algebra Appl., 2015, 484: 540–549]. Finally, we give a characterization of a strongly connected k-uniform directed hypergraph, and obtain some new results by applying these bounds to a uniform directed hypergraph.

Cite this article

Lihua YOU , Xiaohua HUANG , Xiying YUAN . Sharp bounds for spectral radius of nonnegative weakly irreducible tensors[J]. Frontiers of Mathematics in China, 2019 , 14(5) : 989 -1015 . DOI: 10.1007/s11464-019-0797-1

1
Berge C. Hypergraph: Combinatorics of Finite Sets. Amsterdam: North-Holland, 1973

2
Bermann A, Plemmons R. Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press, 1979

DOI

3
Bozkurt S B, Bozkurt D. On the signless Laplacian spectral radius of digraphs. Ars Combin, 2013, 108: 193–200

4
Chang K C, Pearson K, Zhang T. Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520

DOI

5
Chen Z M, Qi L Q. Circulant tensors with applications to spectral hypergraph theory and stochastic process. J Ind Manag Optim, 2016, 12(4): 1227–1247

DOI

6
Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3299

DOI

7
Cvetković D, Rowlinson P, Simić S K. Signless Laplacians of finite graphs. Linear Algebra Appl, 2007, 423: 155–171

DOI

8
Das K C, Kumar P. Some new bounds on the spectral radius of graphs. Discrete Math, 2004, 281: 149–161

DOI

9
Ducournau A, Bretto A. Random walks in directed hypergraphs and application to semi-supervised image segmentation. Computer Vision and Image Understanding, 2014, 120: 91–102

DOI

10
Friedland S, Gaubert A, Han L. Perron-Frobenius theorems for nonnegative multilinear forms and extensions. Linear Algebra Appl, 2013, 438: 738–749

DOI

11
Gallo G, Longo G, Pallottino S, Nguyen S. Directed hypergraphs and applications. Discrete Appl Math, 1993, 42: 177–201

DOI

12
He C X, Liu Y, Zhao Z H. Some new sharp bounds on the distance spectral radius of graph. MATCH Commum Math Comput Chem, 2010, 63: 783–788

13
Hong W X, You L H. Further results on the spectral radius of matrices and graphs. Appl Math Comput, 2014, 239: 326–332

DOI

14
Hu S L, Hunag Z H, Qi L Q. Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math, 2014, 57(1): 181–195

DOI

15
Li K, Wang L S. A polynomial time approximation scheme for embedding a directed hypergraph on a ring. Inform Process Lett, 2006, 97: 203–207

DOI

16
Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proc of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP 05), Vol 1. 2005, 129–132

17
Lim L H. Foundations of Numerical Multilinear Algebra: Decomposition and Approximation of Tensors. Doctoral Dissertation, Stanford University, 2007

18
Lin H Y, Mo B, Zhou B, Weng W. Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs. Appl Math Comput, 2016, 285: 217–227

DOI

19
Lv C, You L H, Zhang X D. A sharp upper bound on the spectral radius of a non- negative k-uniform tensor and its applications to (directed) hypergraphs. Preprint, 2019

20
Maden A D, Das K C, Cevik A S. Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph. Appl Math Comput, 2013, 219: 5025–5032

DOI

21
Pearson K, Zhang T. On spectral hypergraph theory of the adjacency tensor. Graphs Combin, 2014, 30(5): 1233–1248

DOI

22
Qi L Q. Eigenvalues of a real super symmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

DOI

23
Qi L Q. H+-eigenvalue of Laplacian and signless Laplacian tensors. Commun Math Sci, 2014, 12: 1045–1064

DOI

24
Shao J Y. A general product of tensors with applications. Linear Algebra Appl, 2013, 439: 2350–2366

DOI

25
Stanić Z. Inequalities for Graph Eigenvalues. Cambridge: Cambridge Univ Press, 2015

DOI

26
Xie J S, Qi L Q. Spectral directed hypergraph theory via tensors. Linear Multilinear Algebra, 2016, 64(4): 780–794

DOI

27
Xu G H, Xu C Q. Sharp bounds for the spectral radius of digraphs. Linear Algebra Appl, 2009, 430: 1607–1612

DOI

28
Yang Y N, Yang Q Z. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31(5): 2517–2530

DOI

29
Yang Y N, Yang Q Z. On some properties of nonnegative weakly irreducible tensors. 2011, arXiv: 1111.0713v3

30
You L H, Shu Y J, Yuan P Z. Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and its applications. Linear Multilinear Algebra, 2017, 65(1): 113–128

DOI

31
Yuan X Y, Zhang M, Lu M. Some upper bounds on the eigenvalues of uniform hyper-graphs. Linear Algebra Appl, 2015, 484: 540–549

DOI

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