报告人：安徽大学 范益政 教授

报告时间：2018年11月10日9:30

报告地点：枚乘路校区逸夫楼306

报告简介：

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可约张量的谱对称性和稳定性。谱对称性可由循环指数来刻画，定义为最大的正整数k, 使得张量的谱在复平面旋转$\frac{2\pi}{k}$下保持不变。谱稳定性可由稳定指数来刻画，定义为张量的稳定子的阶。这两个参数与模为谱半径的特征值的数目，以及与谱半径关联的特征向量的数目密切相关。我们应用广义迹给出循环指数的显式公式，证明了对任意整除m的正整数k, 总存在一个m-一致超图G, 使得G的循环指数为k. 如果张量是对称的，我们应用Smith标准型给出稳定指数的求解。我们还证明了非负弱不可约张量的对应于谱半径的特征向量是有限个。）

报告人简介：

范益政，男，教授，博士，博士生导师，教育部新世纪优秀人才，安徽省学术和技术带头人，中国工业与应用数学学会图论组合及应用专业委员会副主任委员，中国运筹学会图论组合学分会常务理事，安徽省数学会常务理事，安徽大学数学科学学院院长。主要研究方向：代数组合与谱图理论。主持国家自然科学基金项目4项，发表论文100多篇，其中SCI收录66篇。