Some Properties Of Spectra Of Uniform Hypergraphs

Hypergraphs are systems of subsets of finite sets,which generalize the concept of graphs.And tensors are a generalization of the matrices.They have structural correspondence.In 2005,Qi and Lim independently proposed the concept of eigenvalues of tensors,in order to study the eigenvalues of tensors,Qi also defined the determinants and characteristic polynomials of tensors.In 2012,Cooper and Dutle defined the adjacency tensors of uniform hypergraphs and gave a number of natural analogs of basic results in spectral graph theory,which provide important tools and methods for the study of the spectral theory of the uniform hypergraphs.In this thesis,we consider the characteristic polynomials and the spectra of the adjacency tensors and the zero eigenvalue of the Laplacian tensor of k-uniform hypergraphs.We present a simplified method to calculate the eigenvalues of completely k-uniform hypergraphs and compute the characteristic polynomials of completely 3uniform hypergraphs.We also determine the algebraic multiplicity of some eigenvalues,which give a partial answer to the questions posed by Cooper and Dutle in 2012.In addition,we study the α-polynomial of the 3-uniform loose cycle and give the characteristics polynomial and α-polynomial of the Fano plane.The thesis is divided into the following seven chapters:In Chapter 1,we first introduce the research background of spectral hypergraph theory and the source of problems studied in this thesis.Then introduce some common notations and basic concepts.Next we introduce some known results and progress in the theory of tensors and hypergraphs.Finally,we introduce the main results of this paper.In Chapter 2,we give a method for computing the eigenvalues of complete kuniform hypergraphs by using symmetric functions.Combining the method and the properties of resultants,we give the eigenvalues of complete 4-uniform hypergraphs,which are the roots of some polynomials of degree five and eleven.We also give some examples to demonstrate the calculation.In Chapter 3,we study the characteristic polynomial of the complete 3-uniform hypergraphs.By transforming the eigenvalue equations of a complete 3-uniform hypergraph and combining with the Poisson formula,we compute the characteristic polynomial of the complete 3-uniform hypergraph,which is the product of some cubic polynomials.We also give the algebraic multiplicities of eigenvalues 0,1,and((n-1)/2).In Chapter 4,by using the eigenvalue theorem(Stickelberger theorem)of multiplication map of finite-dimensional algebra,we study the algebraic multiplicities of the zero Laplacian eigenvalues of uniform hyperstars and loose hyperpaths.Furthermore,we also calculate the Laplacian characteristic polynomials of uniform hyperstars.In Chapter 5,we study the eigenvalues and α-polynomials of 3-uniform loose cycles.We give the method and some examples of calculating α-polynomials of 3-uniform loose cycles.We apply the α-polynomial to compute the matching polynomial for a uniform hypertree.We also give the characteristic polynomial and α-polynomial of the Fano plane.In Chapter 6,we summarize the main content of this paper and give some questions that can be further studied.