| The research of graph theory has been developed for hundreds of years.As an important research direction of graph theory,graph spectra theory has played an important role in the graph theory and other discipline like computer theory.Graph spectra theory is mainly focused on the adjacency matrix,Laplacian matrix,signless Laplacian matrix,and distance matrix related with the graph by algebra and matrix methods and study invariants associated with the graph.After Cooper defined the adjacency tensor of hypergraph,tensor,as a generalization of matrix,has become a main tool for studying the hypergraph.Due to the fact that the characteristic polynomial of a tensor is a resultant expression,research on hypergraph spectra theory based on tensor tends to focus on extreme value problem related with spectral radius and the relationship between the structure and invariants.In 2016,Lu Linyuan et al.provided weighted incidence matrix and α labelling method as a way to compare the adjacency spectral radius of hypergraphs in [Connected hypergraphs with small spectral radius.Linear Algebra and Its Applications,2016,509: 206-227]..Since then,the α label method has become a popular way to study the spectral radius.This paper is also mainly based on the α labelling method on weighted incidence matrix and the generalization of this method to study the graph operation,spectrum and α spectrum of hypergraph.The main contents of each chapter is as follows:1.In the first chapter,we reviewed the development of graph theory,graph spectra theory,hypergraph spectra theory and the research results obtained in recent years.2.In the second chapter,we reviewed the definition of hypergraph,tensor and αtensor.Then we introduce the weighted incidence matrix,α labelling method and related notions.3.In the third chapter,we study the concave-convexity and logarithmic concaveconvexity of the sequence of the component of the weighted incidence matrix corresponding to the Perron vector on symmetrical inner path.We introduce graph operation like vertex-splitting and vertex-releasing and study their effects on the spectral radius.We compare the spectral radius of bicyclic hypergraphs by operation,concave-convexity and logarithmic concave-convexity.Then we characterize the two bicyclic hypergraph with the smallest spectral radius and calculate their spectral radius.In addition,we extend Fan’s definition on the cyclomatic number of hypergraph in [Maximizing spectral radii of uniform hypergraphs with few edges.Discuss.Math.Graph Theory,2016,36(4): 845-856] and give a definition of the cyclomatic number of hypergraph that is also applicable to non-uniform hypergraphs through the K(?)nig representation of the hypergraph.4.In Chapter 4,we use the sequence of the component of the weighted incidence matrix corresponding to the Perron vector on path.to studythe influence of edge grafting on inner path and pendent path.For inner path,we give conditions that edge grafting on inner path will lead to a hypergraph with small spectral radius.We also improve the result about the influence of edge grafting on pendent path on the spectral radius to two vertices whose distance is and get a better conclusion.5.In Chapter 5,we extend α labelling method and its properties to α spectral theory as the(α,)-labelling method and study properties of(α,)-normal hypergraphs.By using(α,)-labelling method we reach a way to calculate the α spectral radius of d-edge regular hypergraphs and compare the α spectral radii of certain kinds of hypergraphs.Then we generalized some graph operations like vertex releasing and their influence to α spectral radius by(α,)labelling method and give a condition under which slitting inner path declines the α spectrtal radius.We characterize the supertree with the first two smallest α spectral radii,the hypergraph with the smallest α spectral radius and the hypergraph with the smallest α spectral radius besides supertrees by graph operations.6.In Chapter 6 we introduce the(α,β)-labelling method and extend Su Li’s conclusion in [The matching polynomials and spectral radii of uniform supertrees.The Electronic Journal of Combinatorics,2018,25(4): 4-13] to α spectral radius.We define the α matching polynomial of the hypertree corresponding to the α tensor and prove that the largest root of the α matching polynomial is the α spectral radius of the hypertree,which gives a way to calculate the α spectral radius of the hypertree.7.In Chapter 7,we summarize this paper and pose some questions. |
