Research On Spectral Radii And Eigenvectors Of Hypergraphs

Graph is a widely used mathematical model that can reflect the binary relationship of discrete objects.As a generalization of graphs,hypergraphs can reflect the complex and diverse relationships of discrete objects better.Scholars initially used matrices to study hypergraphs,since hypergraphs and matrices do not correspond one by one,matrices do not fully reflect the information of the hypergraphs.In 2005,Qi Liqun and Lek-Heng Lim independenly proposed the concept of eigenvalues of tensors from different angles,Qi Liqun and Zhang Gongqing et al.studied some properties of the spectral on tensors,these results laid the foundation for the study of spectral of hypergraphs.The paper uses tensors which corresponding to hypergraph to study properties of hypergraphs.Combining with some classical results in graph spectrum and spectral properties of tensors to study eigenvalues and eigenvectors of hypergraphs,it mainly includes the bounds of spectral radii on tensors corresponding to hypergraphs and the properties of components corresponding on the eigenvectors of laplacian tensors and signless laplacian tensors.Specifically studied the following,for uniform linear connected hypergraphs,we give the upper bounds of the spectral radii for signless Laplacian tensors,for uniform connected hypergraphs,the spectral radii of adjacency tensors and signless Laplacian tensors are given by the degree sequence,and the structure corresponding to hypergraph is described when the upper and lower bounds of the spectral radii are equal.For general hypergraphs,some properties of spectral on signless Laplacian tensors are studied.According to the properties of components of the eigenvectors on laplacian tensors,a hypergraph with the same Laplacian tensors eigenvalues as the original hypergraph is construct by adding or deleting a hyperedge that satisfies a specific condition.The relationship between eigenvalues and the components of eigenvectors on Laplacian tensors and signless Laplacian tensors are given.For principal eigenvector of signless Laplacian tensor,the bound of the largest component and the smallest component are studied.