主 题: The eigenvectors of nonnegative symmetric tensors and hypergraphs associated with spectral radius
内容简介: For a nonnegative symmetric weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, there may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this talk we will show that such projective eigenvariety admits a module structure, which is determined by the support of the tensor and can be characterized explicitly by solving the Smith normal form of the incidence matrix of the tensor. We introduce two parameters: the stabilizing index and the stabilizing dimension of the tensor, where the former is exactly the cardinality of the projective eigenvariety and the latter is the composition length of the projective eigenvariety as a module. We give some upper bounds for the two parameters, and characterize the case that there is only one eigenvector of the tensor corresponding to the spectral radius, i.e. the Perron vector. By applying the above results to the adjacency tensor of a connected uniform hypergraph, we give some upper bounds for the two parameters in terms of the structural parameters of the hypergraph such as path cover number, matching number and the maximum length of paths.
报告人: 范益政 教授 博导
时 间: 2018-05-22 15:30
地 点: 竞慧东楼302
举办单位: 统计与数学学院 统计科学与大数据研究院