Research On Tensors And The Spectrum Of Hypergraphs

The emergence and development of graph theory has gone through hundreds of years,and it has spawned numerous research branches.The spectral graph theory is one of the important directions in graph theory,which based on algebraic theory and matrix theory.The spectral graph theory has a wide range of applications in physics,chemistry and computer science.The spectral graph theory establishes the connection between the topological structure and invariants of the adjacency matrix,?signless?Laplacian matrix,normalized Laplacian matrix,distance matrix.As a natural generalization of graphs,hypergraphs model more general types of relations than graphs do.Therefore,it is a meaningful topic to extend the spectral graph theory to hypergraphs.Chung,Feng,Li and Rodríguez studied the spectral properties of adjacency matrix?or Laplacian matrix?of uniform hypergraph.However,we know that there is no one-to-one correspondence between a hypergraph and its adjacency matrix,and therefore the adjacency matrix sometimes do not reflect the properties of hypergraphs very well.As a generalization of matrix,tensor?hypermatrix?is widely used in mathematics,physics and other fields.In 2005,Qi and Lim independently introduced the definition of eigenvalues of a tensor.Based on which,Cooper and Dutle defined the adjacency tensors for a uniform hypergraph in 2012.Two years later,Qi defined the?signless?Laplacian tensor.These previous works laid the foundation for studying the properties of hypergraphs by tensors.Since then,the study of hypergraph based on tensor has become an active subject.This paper consists of eight chapters.It mainly studies the bounds of the spectral radius of uniform hypergraphs?and spectral radius of signless Laplacian tensor?,the Perron vector and extremal problem in spectral hypergraph theory.This paper is organized as follows:In Chapter 1,we mainly introduces the background of hypergraphs spectral theory and the concepts of hypergraphs and tensors.In Chapter 2,we study the relations between the spectral radius and the degree sequence,maximum?minimum?degree,co-degree.We solve a problem of Nikiforov in[Analytic methods for uniform hypergraphs,Linear Algebra and its Applications,457:455–535,2014].Meanwhile,we disprove a conjecture on the 2-section of a uniform hypergraph posed in the same reference.In addition,we generalize some known results on the spectral radius of graphs to uniform hypergraphs.In Chapter 3,we extend the?-normal labeling method to the p-spectral radius of uniform hypergraphs,and give some applications.In Chapter 4,we study the Perron vectors of hypergraphs using the?-normal labeling method.Based on this method,we estimate x_iand x_i/x_j for the Perron vector x=?x₁,x₂,...,x_n?^Tof a uniform hypergraph.We also estimate the difference of the spectral radius between a connected uniform hypergraph and its proper subhypergraph.Finally,the p-eigenvector of a general hypergraph is considered.In Chapter 5,the irregularity of uniform hypergraphs are considered.For an r-uniform hypergraph H,|V?H?|=n,|E?H?|=m,we define three parameters ??H?:=??H?-rm/n and???,where d_iis the degree of vertex i?[n].Obviously,??H?,s?H?,v?H??0,and equality hold if and only if H is regular.In this chapter we measure the irregularity of uniform hypergraphs by considering the relations among ??H?,s?H? and v?H?.In Chapter 6,we compare the spectral radius of three linear bicyclic hypergraphs in the light of the?-normal labeling method for uniform hypergraphs and the Rayleigh quotient for nonnegative tensors,which solves a conjecture of Fan et al.posed in[Maximizing spectral radii of uniform hypergraphs with few edges,Discussiones Mathematicae Graph Theory,36:845–856,2016]In Chapter 7,we study the p-spectral radius of Berge-G hypergraphs.Let G be a simple graph.We say that a hypergraph H is a Berge-G if there is a bijection ???:E?G??E?H?such that ???.We determine the 3-uniform hypergraphs with maximum p-spectral radius for p?1 among Berge-G hypergraphs when G is a path,a cycle or a star.In Chapter 8,We summarize the paper and pose some questions.