The Spectral Radius Of Graphs With Independence Number

As an important research direction of algebraic graph theory,spectral graph theory has attracted more and more attention in recent years.In 1985,Brualdi and J.Hoffman put forward the extreme graphs for the Adjacency eigenvalues.Then the extreme graphs problem of maximal eigenvalues and minimum eigenvalues with given parameters has become a hot issue in graph theory research.At the same time,we further studied the extreme graphs problem of the distance eigenvalues.For spectral radii,we generally study the maximum extreme graphs,and for the distance radii studies,we study more about the minimum extreme graphs.AS they can also reflect the structural information of the graph,so they have good research value.In this paper,we trying to use perturbation and edge transplants to find out the extreme graphs of spectral radius of bipartite unicyclic graphs with given independence number n-3 and the extreme graphs of the minimum distance spectral radius of trees with n-4 pendent vertices.The main arrangements of this paper are as follows.In Chapter 1,we introduce the background,some terms and concepts involved this paper.Next we introduce the research progress and the main conclusions of this paper.In Chapter 2,we discusse the spectral radius of bipartite unicyclic graphs with given independence number n-3.In Chapter 3,we discusse the Laplacian spectral radius of trees with given independence number.In Chapter 4,we discusse the minimum distance spectral radius of trees with n-4 pendent vertices.