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.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11571123, 11871040, 11971180), the Guangdong Provincial Natural Science Foundation (No. 2015A030313377) and Guangdong Engineering Research Center for Data Science.
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You, L., Huang, X. & Yuan, X. Sharp bounds for spectral radius of nonnegative weakly irreducible tensors. Front. Math. China 14, 989–1015 (2019). https://doi.org/10.1007/s11464-019-0797-1
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DOI: https://doi.org/10.1007/s11464-019-0797-1