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Transactions of the American Mathematical Society

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The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs


Authors: Yi-Zheng Fan, Tao Huang, Yan-Hong Bao, Chen-Lu Zhuan-Sun and Ya-Ping Li
Journal: Trans. Amer. Math. Soc. 372 (2019), 2213-2233
MSC (2010): Primary 15A18, 05C65; Secondary 13P15, 05C15
DOI: https://doi.org/10.1090/tran/7741
Published electronically: December 7, 2018
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Abstract: Let $ \mathcal {A}$ be a weakly irreducible nonnegative tensor with spectral radius $ \rho (\mathcal {A})$. Let $ \mathfrak{D}$ (resp., $ \mathfrak{D}^{(0)}$) be the set of normalized diagonal matrices arising from the eigenvectors of $ \mathcal {A}$ corresponding to the eigenvalues with modulus $ \rho (\mathcal {A})$ (resp., the eigenvalue $ \rho (\mathcal {A})$). It is shown that $ \mathfrak{D}$ is an abelian group containing $ \mathfrak{D}^{(0)}$ as a subgroup, which acts transitively on the set $ \{e^{\i \frac {2 \pi j}{\ell }}\mathcal {A}:j =0,1, \ldots ,\ell -1\}$, where $ \vert\mathfrak{D}/\mathfrak{D}^{(0)}\vert=\ell $ and $ \mathfrak{D}^{(0)}$ is the stabilizer of $ \mathcal {A}$. The spectral symmetry of $ \mathcal {A}$ is characterized by the group $ \mathfrak{D}/\mathfrak{D}^{(0)}$, and $ \mathcal {A}$ is called spectral $ \ell $-symmetric. We obtain structural information about $ \mathcal {A}$ by analyzing the property of $ \mathfrak{D}$, and especially for connected hypergraphs we get some results on the edge distribution and coloring. If moreover $ \mathcal {A}$ is symmetric, we prove that $ \mathcal {A}$ is spectral $ \ell $-symmetric if and only if it is $ (m,\ell )$-colorable. We characterize the spectral $ \ell $-symmetry of a tensor by using its generalized traces, and we show that for an arbitrary integer $ m \ge 3$ and each positive integer $ \ell $ with $ \ell \mid m$, there always exists an $ m$-uniform hypergraph $ G$ such that $ G$ is spectral $ \ell $-symmetric.


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Additional Information

Yi-Zheng Fan
Affiliation: School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China
Email: fanyz@ahu.edu.cn

Tao Huang
Affiliation: School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China
Email: huangtao@ahu.edu.cn

Yan-Hong Bao
Affiliation: School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China
Email: baoyh@ahu.edu.cn

Chen-Lu Zhuan-Sun
Affiliation: School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China
Email: zhuansuncl@163.com

Ya-Ping Li
Affiliation: School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China
Email: 18856961415@163.com

DOI: https://doi.org/10.1090/tran/7741
Keywords: Tensor, hypergraphs, adjacency tensor, spectral symmetry, coloring
Received by editor(s): May 28, 2017
Received by editor(s) in revised form: October 25, 2018
Published electronically: December 7, 2018
Additional Notes: The first author was supported by National Natural Science Foundation of China grant #11871073.
The third author was supported by National Natural Science Foundation of China grant #11871071.
Article copyright: © Copyright 2018 American Mathematical Society