Research On A_? Eigenvalues Of Graphs

Algebraic graph theory is a research field to investigate the structural properties of graphs by studying their algebraic properties,and it is one of the most important branches in graph theory.Spectral graph theory,one of an important topic in algebraic graph theory,is devoted to studying the spectral properties(eigenvalues,eigenspaces,characteristic polynomials and spectral radius)of matrices associated with graphs(adjacency matrix,signless Laplace matrix and distance matrix).Aa matrix is a recently defined matrix for graph,results on such a matrix are comparatively few.This thesis mainly studies the spectral properties for Aa matrices for graphs.There are six chapters in this thesis,which are organized as below.In Chapter 1,we first introduce the background of Spectral graph theory and the research progress of related problems involved in this thesis;then we give some basic notions and symbols.In Chapter 2,we determine the first three largest extremal graphs in Halin graphs by their Aa spectral radii.In Chapter 3,we study the Aa spectral radius of the k-th power of a graph.The k-th power of a graph G,denoted by Gk,is a graph with the same set of vertices as G such that two vertices are adjacent in Gk if and only if their distance in G is at most k.Up till now,a few properties on k-th power of a graph have been investigated.For the motivation of the present works,we give upper and lower bounds on the Aa spectral radius of G2.Furthermore,the first three largest Aa spectral radius of the second power of trees are determined in this paper.In Chapter 4,at the first,we study the behavior of the Aa spectral radius under some graph transformations for ??[0,1).As applications,we show that the greedy tree has the maximum Aa spectral radius in BD when D is a tree degree sequence firstly.Furthermore,we determine that the greedy unicyclic graph has the largest Aa spectral radius in BD when D is a unicyclic graphic sequence,In Chapter 5,we focus on the second largest A? eigenvalues of graph.We give a lower bound of ?2(A?(G))at first and then use this bound we show that the star is the unique graph which minimizes the second largest Aa eigenvalues among connected graphs for 1/2 ??<1.In particular,the graphs with ?k(A?(G))=(n-2)? for k?2 are determined.Furthermore,we determine the connected graphs with ?2(A?(G)??+1 are firefly graphs.Moreover,the firefly graphs are shown to be determined by their Aa spectrum for 1/2 ??<1.In Chapter 6,we study Nordhaus-Gaddum type bound for the second largest Aa eigenvalue of a graph.We show that ??(A?(G))+?2(A?(G))?n?-1,and the extremal graphs for which the equality holds are Kn and Sn.If G(?) {Kn,Sn,Kne},then A2(A?(G))+?2(A?(G))?(n-1)?.Moreover,we give two upper Nordhaus-Gaddum type bounds for the second largest Aa eigenvalue of a graph.In Chapter 7,the whole paper is summarized and some problems for further research are discussion.