Relevant Research On A_?-spectral Radii Of Trees With Specified Maximum Degree And Complete Multipartite Graphs

Spectral radius of graph is a hot topic in the research of Graph Theory.It plays an important role in many fields such as medicine,chemistry and computer and other fields.The adjacency matrix,Laplacian matrix,and signless Laplacian matrix of G,have been extensively studies.With the research development,Nikiforov presented the Aa matrix of graph,the adjacency matrix of G is A(G),the degree matrix of G is D(G),??[0,1],then the matrix A?(G)=?D(G)+(1-1?)A(G)is called the A?-matrix of G,and the largest eigenvalue of A?(G)called the A?-spectral radius of G.In this paper,we mainly study the related problems of A?-spectral radius by using the techniques of structural graph theory and matrix theory.This thesis consists of four chapters,the main content as follows:In Chapter 1,we introduce the background and significance of the study,including the research of mathematicians in this field and some related results in recent years.And we mainly give some symbols and basic definitions that we needed in this paper.In Chapter 2,we focus on the sharp bound of A?-spectral radius of trees subject to maximum degree,discuss the characteristics of the extremal graphs and then the extremal graphs are given.In Chapter 3,we study the bounds of the A?-spectral radius of complete multipartite graphs.Furthermore,the extremal graphs are given.In Chapter 4,we summarize the whole paper and make some prospects for the future research.