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

Abstract

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.

Keywords

• <i>k</i>-uniform tensors
• <i>k</i>-uniform (directed) hypergraphs
• spectral radius
• adjacency tensor
• signless laplacian tensor