Some Research On Spectral Radius Of Hypergraphs

Spectral hypergraph theory is an important branch of graph theory.Spectral hypergraph theory is the study of the relationship between eigenvalues of tensors and the hypergraph structure corresponding to hypergraph.Tensor has important applications in physics,mathematics and mechanics.At the same time,it is also a tool for the study of spectral hypergraph theory.The concept of eigenvalues of tensors was proposed by Professor Liqun Qi of Hong Kong Polytechnic University in 2005,Then,Cooper and Dutle defined the adjacency tensor of uniform hypergraph.Liqun Qi defined the Laplacian tensor and signless Laplacian tensor of uniform hypergraph.From this,scholars began to study the relationship between the eigenvalues of tensor corresponding to hypergraph and the structure of hypergraph.The main results of this paper is two parts.The paper gives the characterization of hypergraph structure under tensor spectrum conditions.Characterization of hypergraph structure is given when the spectral radius of signless Laplacian tensor equals to the sum of the first largest degree and the second largest degree of uniform linear connected hypergrah:the hypergraph is a hyperstrar.Let _ix is the maximum component of the eigenvector which corresponding to spectral radius of the signless Laplacian tensor of uniform connected hypergraph.Then the lower bound of the degree of vertex i is given.Characterization of hypergraph structure is given when the largest H eigenvalue of Laplacian tensor equals to the sum of the first largest degree and the second largest degree of uniform linear hypergraph:the hypergraph is a hyperstrar.Let |x_i| is the maximum component of the absolute value of eigenvector which corresponding to the largest H eigenvalue of Laplacian tensor of uniform hypergraph.Then the lower bound of the degree of vertex i is given.In this paper,we study the bounds of spectral radius of uniform hypergraph.For undirected hypergraph,the paper gives the upper buond of spectral radius on adjacency tensor by using the first largest degree and the second largest degree of uniform linear hypergraph.The paper characterizes the lower bound of the difference between spectral radius and the maximum degree by using parameters such as the number of edges,the number of vertexes and the degree of uniform hypergraph.Furthermore,the new upper bound of the largest H eigenvalue is given by using the average 2 outdegree of vertex i of uniform directed hypergraph.