Some Results For Upper Bound Of The Sum Of Squares Of Positive Eigenvalues Of A Graph

Let G be an undirected simple graph,with vertex set V(G)and edge set E(G).The number of vertices of G is n,and the number of edges of G is m.The adjacency matrix A(G)=(aij)of the graph G is a square matrix of order n,aij=1 if the vertices vi and vj are adjacent and aij=0 otherwise.The eigenvalues of a graph are the eigenvalues of its adjacency matrix.The sum of squares of positive eigenvalues is denoted by S+.The upper bound of spectral radius is an important research subject in graph theory.But the upper bound of the sum of squares of positive eigenvalues has not many researches about it.There is a conjecture:min(S-,S+)?n-1 which has not been fully proved,while it is proved for some special classes of graphs,including bipartite,regular,complete q-partite,hyper-energetic and barbell graphs.This thesis is divided into four chapters.The first chapter introduces the background of graph theory and the overview of the eigenvalues of a graph.The second chapter introduces the sum of squares of positive eigenvalues of a graph and some results of it and its upper bound.The third chapter gives the proof of the main result.At last,this thesis is summered and prospected.