Extremal Problems For The P-spectral Radius Of Berge Hypergraphs

Spectral graph theory mainly studies the relationship between graphs' structural properties and their eigenvalues of corresponding matrices or other algebraic invariants.In recent years,spectral graph theory has attracted the attention of more and more scholars.It also offers new research methods to researchers in other fields.In 2019,Huang proved the Boolean function sensitivity conjecture,which has remained an open problem for 30 years,very concisely by spectral methods.The extremal problem is one of the core issues in graph theory.It is also widely considered a challenging problem.The classical extremal problems study the maximum number of edges in a given graph class and graphs with this edge number.As a spectral analog,the spectral extremal problems require the characterization for graphs with maximum spectral radius among a family of graphs.As a result of the development of the tensor theory,the spectral theory of hypergraphs,which is a generalization of spectral graph theory,develops rapidly in recent years.In 2005,Qi and Lim introduced the eigenvalues and eigenvectors of tensors independently.In 2012,Cooper and Dutle defined the adjacency tensor of an r-uniform hypergraph as an rth order n-dimensional tensor.They also extended some basic conclusions of the spectral graph theory to hypergraphs.In 2014,Keevash et al.introduced p-spectral radius as a generalization of the spectral radius of hypergraphs.The p-spectral radius of an r-uniform hypergraph H is defined as follows:In the same year,Nikiforov systematically studied the p-spectral radius of hypergraphs by analytical methods,and generalized the spectral theory of hypergraphs to the p-spectral theory of hypergraphs.In 2017,Gerbner and Palmer introduced the concept of Berge hypergraphs as a generalization of Berge-path and Berge-cycle.Let G be a simple graph and H be an r-uniform hypergraph.We say that H is Berge-G if there exists a bijection ?:E(G)?E(H)such that e?f(e)for any e?E(G).In 2018,Kang et al.studied the extremal graphs with the largest p-spectral radius in 3-uniform Berge-path,circle,and star.Let Cn+ be the family of graphs obtained from Cn,the circle with order n,by adding an edge between two non-adjacent vertices.In this paper,we study the p-spectral radius of Berge-G hypergraphs when p>1,G ?Cn+ and g(G)?5.Besides,the hypergraphs with the largest p-spectral radius among this kind of hypergraphs are characterized.The arrangement of this paper is as follows:1.In Chapter 1,we will introduce the research progress,basic knowledge of graph theory and spectral graph theory,as well as used symbols.At the end of this chapter,we give the main result of this paper.2.In Chapter 2,we introduce theorems in spectral graph theory and also methods for studying extremal spectral problems.The extremal spectral problems for F1(u,n,k)type graph,F2(u,n,k)-type graphs,and ?+(l1,l2,l3)are also studied in this chapter.Extreme results on these types of graphs are crucial for us to prove our main theorem.3.In Chapter 3,we first prove that the hypergraphs,which attains the maximum pspectral radius among 3-uniform Berge-G(G ?Cn+ and g(G)>5),satisfies the following two structural properties:(1)They have precisely n vertices;(2)All the edges intersect at a common vertex.Then,using these structural properties,we analyze the core of the spectral extremal graph of this kind of hypergraphs and determine the hypergraph attains the maximum p-spectral radius among them.4.In Chapter 4,we make a summary and put forward a problem worthy of further study.