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

DOI：10.3770/j.issn:2095-2651.2019.04.003

 作者 单位 何军 遵义师范学院数学学院, 贵州 遵义 56006 柯铧 遵义师范学院数学学院, 贵州 遵义 56006 刘衍民 遵义师范学院数学学院, 贵州 遵义 56006 田俊康 遵义师范学院数学学院, 贵州 遵义 56006

在本文中,我们给出了非负不可约弱对称张量最大Z-特征值的一个Brauer型上界: $$\rho_Z(\mathcal{A})\leq \frac{1}{2}\mathop {\max }\limits_{\scriptstyle i,j \in N \hfill \atop \scriptstyle j \ne i \hfill} ( {a_{i\cdots i} + a_{j \cdots j} + \sqrt {({a_{i\cdots i} - a_{j\cdots j} })^2 + 4r_i (\mathcal{A})r_j (\mathcal{A})} }),$$ 其中$r_i(\mathcal{A})=\sum\limits_{ii_2\cdots i_m \neq ii\cdots i} a_{ii_2\cdots i_m}$, $i\in N=\{1,2, \ldots,n\}$,并将此结果应用到一致超图的Z谱上界中.

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.