Brauer Upper Bound for the Z-Spectral Radius of Nonnegative Tensors

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.