AIMS Mathematics, 2020, 5(3): 1799-1819. doi: 10.3934/math.2020121

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A general result on the spectral radii of nonnegative k-uniform tensors

1 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China
2 Department of Mathematics, Jilin Medical University, Jilin, 132013, P. R. China
3 Department of Mathematics Teaching, Guangzhou Civil Aviation College, Guangzhou, 510403, P. R. China

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In this paper, we define $k$-uniform tensors for $k\geq 2$, which are more closely related to the $k$-uniform hypergraphs than the general tensors, and introduce the parameter $r^{(q)}_{i}(\mathbb{A})$ for a tensor $\mathbb{A}$, which is the generalization of the $i$-th slice sum $r_ {i}(\mathbb{A})$ (also the $i$-th average 2-slice sum $m_{i}(\mathbb{A})$). By using $r^{(q)}_{i}(\mathbb{A})$ for $q\geq1$, we obtain a general result on the sharp upper bound for the spectral radius of a nonnegative $k$-uniform tensor. When $k=2, q=1, 2, 3$, this result deduces the main results for nonnegative matrices in [1,8,27]; when $k\geq 3, q=1$, this result deduces the main results in [5,20]. We also find that the upper bounds obtained from different $q$ can not be compared. Furthermore, we can obtain some known or new upper bounds by applying the general result to $k$-uniform hypergraphs and $k$-uniform directed hypergraphs, respectively.
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