校庆60周年系列学术讲座——【范益政 安徽大学 传授】The spectral symmetry and stabilizing property of nonnegative tensors and hypergraphs（非负张量与超图的谱对称性与波动性）-bwin体育平台

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The Perron-Frobenius theorem for nonnegative tensors gives us the symmetric property and stabilizing property of the spectrum of a weakly irreducible nonnegative tensor. The spectral symmetry is reflected by the cyclic index, which is defined to be the maximum positive integer $k$ such that the spectrum of the tensor keeps invariant under a rotation of angle $\frac{2\pi}{k}$ of the complex plane. The spectral stabilizing property is reflected by the stabilizing index, which is the order of a stablizlier group of the tensor. These two parameters are closely related to the the number of eigenvalues with modulus equal to spectral radius, and the number of eigenvectors of the tensor assocaited with the spectral radius, respectively. We give an explicit formula for the cyclic index by using the generalized traces, and proved that for any positive integer $k$ that divides $m$, there always exists an $m$-uniform hypergraph $G$ such that its adjacency tensor has the cyclic index $k$. If the tensor is further symmetric, then we can get the stabilizing index via the Smith normal form. We also show that for a weakly irreducible nonnegative tensor, there are finite many eigenvectors associated with the spectral radius up to a scalar.（Perron-Frobenius定理给出了非负弱不行约张量的谱对称性和波动性。谱对称性可由循环指数来描写，界说为最大的正整数k, 使得张量的谱在复立体旋转$\frac{2\pi}{k}$下坚持稳定。谱波动性可由波动指数来描写，界说为张量的波动子的阶。这两个参数与模为谱半径的特性值的数量，以及与谱半径联系关系的特性向量的数量亲密相干。我们使用狭义迹给出循环指数的显式公式，证明白对恣意整除m的正整数k, 总存在一个m-分歧超图G, 使得G的循环指数为k. 假如张量是对称的，我们使用Smith规范型给出波动指数的求解。我们还证明白非负弱不行约张量的对应于谱半径的特性向量是有限个。）