The K-sum Of The Signless Laplacian Eigenvalues Of Graphs

The theory of graph spectra studies the relationship between the eigenvalues of graphs,the structure properties of graphs and the invariants(such as chromatic number,independent number,diameter and so on)of graph by using algebraic theory and its techniques and combining with the theory of combinatorial mathematics and graph theory.One of the main research problem is whether the properties of graphs can be reflected by algebraic properties of adjacency matrix,the Laplacian matrix and the signless Laplacian matrix of graphs.The signless Laplacian matrix Q(G)is the sum of D(G)and A(G),where D(G)is the degree matrix and A(G)is the adjacency matrix of G.The eigenvalues of Q(G)is called the signless Laplacian eigenvalues of G,and the largest eigenvalue of Q(G)is called the signless Laplacian spectral radius.This paper mainly studies the k-sum of the signless Laplacian eigenvalues of graphs.Details are as follows:In Chapter 1,we state the significance and the background of the topics.In Chapter 2,we consider a problem proposed by Brualdi and Solheid[1]:For a given set of graphs(?),find an upper bound on the spectral radius of this set and char-acterize the graphs in which the maximal spectral radius is attained.In this chapter,we mainly discuss the largest signless Laplacian eigenvalue of the unicyclic graphs with given independence number.And then investigate the influence of graph transforma-tions on the largest signless Laplacian eigenvalue by using the technique of some mod-ifications of graphs.At last,we also characterize the graphs which have the maximum largest signless Laplacian eigenvalue among all unicyclic graphs with given indepen-dence number.In Chapter 3,we start with the conjecture of the k-sum of the signless Laplacian eigenvalues proposed by Ashraf,Omidi and Tayfeh-Rezaie[2],bounding the k-sum of signless Laplacian eigenvalues of graphs of given diameter and maximum degree.Then we investigate the relationship between the k-sum of the signless Laplacian eigenval-ues and the girth and the matching number of graphs and also obtain some sufficient conditions for the conjecture.At last,we give a summary of this paper.