| Brauer Upper Bound for the Z-Spectral Radius of Nonnegative Tensors |
| Received:June 25, 2018 Revised:April 11, 2019 |
| Key Words:
bound nonnegative tensor Z-eigenvalue hypergraph
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| Fund Project:Supported by the High-Level Innovative Talents of Guizhou Province; Science and Technology Fund Project of GZ; Innovative Talent Team in Guizhou Province (Grant Nos.Zun Ke He Ren Cai[2017]8, Qian Ke He J Zi LKZS [2012]08, Qian Ke HE Pingtai Rencai[2016]5619.) |
| Author Name | Affiliation | | Jun HE | School of Mathematics, Zunyi Normal College, Guizhou 563006, P. R. China | | Hua KE | School of Mathematics, Zunyi Normal College, Guizhou 563006, P. R. China | | Yanmin LIU | School of Mathematics, Zunyi Normal College, Guizhou 563006, P. R. China | | Junkang TIAN | School of Mathematics, Zunyi Normal College, Guizhou 563006, P. R. China |
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| Abstract: |
| In this paper, we have proposed an upper bound for the largest Z-eigenvalue of an irreducible weakly symmetric and nonnegative tensor, which is called the Brauer upper bound: $$\rho_Z(\mathcal{A})\leq \frac{1}{2}\mathop {\max }\limits_{\scriptstyle i,j \in N \hfill \atop \scriptstyle j \ne i \hfill} \Big( {a_{i\cdots i} + a_{j \cdots j} + \sqrt {\left( {a_{i\cdots i} - a_{j\cdots j} } \right)^2 + 4r_i (\mathcal{A})r_j (\mathcal{A})} }\,\Big),$$ where $r_i(\mathcal{A})=\sum\limits_{ii_2\cdots i_m \neq ii\cdots i} a_{ii_2\cdots i_m}$, $i,i_2, \ldots, i_m \in N=\{1,2, \ldots,n\}$. As applications, a bound on the Z-spectral radius of uniform hypergraphs is presented. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2019.04.003 |
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