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 $\mathfrak {D}$ (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 $\{e^{\mathbf {i}\frac {2 \pi j}{\ell }}\mathcal {A}:j =0,1, \ldots ,\ell -1\}$, 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 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.