Some Studies On Eigenvector And Z Eigenvalue Of Uniform Hypergraphs

The spectral theory of graph is an important topic of graph theory.Compared with the study of the graph spectral theory,the study of the hypergraph spectral theory has attracted the attention of many scholars.On the basis of the development of tensor spectral theory,the research of hypergraph spectral theory has also developed rapidly.Currently,there are many classical results of adjacency tensor、Laplacian tensor and signless Laplacian tensor.Fangrong Jin、Keqin Feng、Linyuan Lu、Wenqing Li、S.Friedman and others have introduced hypergraph adjacency matrix and Laplacian matrix to study the hypergraph properties.However,each hypergraph edge is determined by more than two points,so the adjacency matrix and Laplacian matrix to study can not directly reflect the structural properties of hypergraph.In 2012,J.Cooper and A.Dutle give the definition of the adjacency tensor of uniform hypergraph,tensor representation of hypergraph is given.Since then,the tensor spectral theory of hypergraph has aroused the interest of many scholars.In this paper,through hypergraph tensor representation and combined with some classical results of the graph spectral theory to study the hypergraph of eigenvalues and eigenvectors.It is divided into the following two parts.We proves that the necessary and sufficient conditions for a connected uniform hypergraph to be a bipartite hypergraph;The H-eigenvector corresponding to the largest H-eigenvalue of laplacian tensor in the odd-biparite uniform hypergraph add absolute value is the H-eigenvector corresponding to the largest H-eigenvalue of signless laplacian tensor;if k uniform hypergraph is an unconnected odd bipartite hypergraph,there is a H-eigenvector corresponding to the largest H-eigenvalue of laplacian tensor has a component of zero.This article also gives lower and upper bounds for the largest Z-eigenvalues of adjacent tensor and signless laplacian tensor for uniform directed hypergraphs;all of the Z-eigenvalues of adjacent tensor for uniform directed hyperstar are 0;one of the Z-eigenvalues of laplacian tensor and signless laplacian tensor for uniform directed hyperstar is 1/（k-1）.