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
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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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.