The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

Abstract

Let $\mathcal{A}$ be a weakly irreducible nonnegative tensor with spectral radius $\rho (\mathcal{A})$. Let $\mathfrak{D}$ (respectively, ${\mathfrak{D}}^{(0)}$) be the set of normalized diagonal matrices arising from the eigenvectors of $\mathcal{A}$ corresponding to the eigenvalues with modulus $\rho (\mathcal{A})$ (respectively, 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 $\{e^{\mathbf{i}\frac{2\pi j}{\ell }}\mathcal{A}:j=0,1,\dots ,\ell -1\}$, where $|\mathfrak{D}/{\mathfrak{D}}^{(0)}|=\ell$ 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 the structural information of $\mathcal{A}$ by analyzing the property of $\mathfrak{D}$, 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 show that for an arbitrarily given 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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• A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs
• Spectral bipartite Turan problems on linear hypergraphs
• The trace of uniform hypergraphs with application to Estrada index
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• Leading Coefficients and the Multiplicity of Known Roots
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• Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs
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• Spectral theory of weighted hypergraphs via tensors