Some Results About The Eigenvalues Of Adjacent Matrix Of Graph

Spectral Graph theory is a branch of graph theory that studies the spectral properties of graphs and the relationship between the structure of graphs and their spectral properties.Spectral Graph theory has significant applications in many research fields.The spectral properties of graphs include the spectral properties of adjacency matrix,Laplacian matrix,unsigned Laplacian matrix and some other graph-related matrices.In this paper,the relevant conclusions about the spectral properties of the adjacency matrix of a graph are discussed,mainly involving the zero eigenvalue,the minimum eigenvalue,the spectral radius and the sum of the squares of the positive and negative eigenvalues of the adjacency matrix of the graph.The details are as follows.In Chapter 2,the nullity of the graph is discussed in the upper and lower bounds of nullity given by Wang et al.|V(G)|-2m(G)-c(G)≤η(G)≤|V(G)|-2m(G)+2c(G).It is found that there is graph with nullity |V(G)|-2m(G)+2c(G)-1,that is,the maximum nullity minus one.Moreover,an infinite number of graphs can be constructed so that nullity of the graph can take all the values between the upper and lower bounds of nullity except|V(G)|-2m(G)+2c(G)-1.In Chapter 3,we mainly discuss the graph with the smallest eigenvalue in the connected graph of n vertices and n+k edges,5≤k≤8.This problem stems from a conclusion of Petrovic et al,in 2011,they characterized the graph with minimum smallest eigenvalue among connected graphs with n vertices and n+k edges,0≤k≤4.Through the research and comparison,it can be found that the conclusion in Chapter 3 is the same as the conclusion of Petrovic et al.Therefore,the conclusion of this paper is a generalization of the conclusion of Petrovic et al.In Chapter 4,we give an ordering of starlike graphs in FH(υ),n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths,where FH(υ),n={H(υ)·T|T∈STn｝,and H(υ)·T denotes the graph obtained by identifying vertex υ of H with the root vertex of T.STn be a family of graphs consists of all starlike trees and paths with n+1 vertices.This problem stems from a conclusion given by Oliveira et al in 2018,They gave an ordering of starlike trees by adjacent spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths.In this chapter,our result is a Oliveira a generalization to that of Oliveira et al.Whether it is the conclusion of Oliveira et al.or the research content in Chapter 4,the core idea is the algorithm designed by Jacobs and Trevisan et al.to locate the eigenvalues of the adjacency matrix of the tree.In Chapter 5,some conclusions about the sum of squares of positive and negative eigenvalues of adjacency matrices of graphs are discussed.The original intention of this chapter is to try to solve a famous conjecture put forward by Elphick et al in 2016:For a connected graph G with n vertices,there are min｛S-(G),S+(G)}≥n-1.In this chapter,it is proved that the conjecture is true for all graphs with rank 4,and it is further proved that it is true for most graphs with rank 5.At last,we give some results about the sum of squares of the positive eigenvalues.In Chapter 6,this paper is summarized and prospected,and several questions that can be further researched are proposed.