### AIMS Mathematics

2020, Issue 3: 1799-1819. doi: 10.3934/math.2020121
Research article

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

• Received: 18 November 2019 Accepted: 09 February 2020 Published: 17 February 2020
• MSC : 05C50, 05C65, 15A69

• 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.

Citation: Chuang Lv, Lihua You, Yufei Huang. A general result on the spectral radii of nonnegative k-uniform tensors[J]. AIMS Mathematics, 2020, 5(3): 1799-1819. doi: 10.3934/math.2020121

### Related Papers:

• 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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沈阳化工大学材料科学与工程学院 沈阳 110142

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