On The Distance(Signless) Laplacian Spectrum Of Graphs

In recent years,the distance(signless)Laplacian spectrum of a connected graph has been studied extensively.Based on the previous study,in this paper we respectively de?termine the bounds of the sum of the k largest Laplacian、distance、distance(signless)laplacian eigenvalues of graphs.Let G be the graph having the vertex set V(G)={v1,…,un},let A(G)be the ad-jacency matrix of G,and Diag(G)=diag(d1,d2,…,dn)be the diagonal matrix of vertex degrees.Then the signless Laplacian matrix of G is Q(G)=D(G)+A(G).The dis-tance matrix of G is D(G)=(dij)n×n,where dij denote the distance between vi and vj·For 1 ≤i ≤n,Tr(v)is the sum of distance between v and every vertices of G,then Tr(G)=diag(Tr(v1),Tr(v2),…,Tr(vn))is the diagonal matrix of vertex transmissions of G.The distance laplacian matrix of G is n x n matrix defined as LD(G)=Tr(G)-D(G),The distance laplacian matrix of G is n x n matrix defined as QD(G)=Tr(G)+D(G).And then we introduced our main results in the next two chapters.Main outcome are as follows:Firstly,in Chapter Ⅱ,Let n=2r+2t+s+1(r,s≥1,t≥0),and Sn-t be a star with n-t vertices.The firefly graph is such a graph that is obtained by connecting r pairs distinct vertices of Sn-t with r edges,and joining t edges to the other t distinct pendant vertices of Sn-t respectively.Suppose that G is a firefly graph on n vertices.In this chapter we respectively determine the lower bounds of the sum of the largest eigenvalue and the second largest eigenvalue of D(G),LD(G)and QD(G).Secondly,in Chapter Ⅲ,For a simple graph G with n vertices,m edges and having signless laplacian eigenvalues θ1,θ2,…,θn-1,θn,let Sk(G)=∑i=1k θi be the sum of k largest signless laplacian eigenvalues of G.In this chapter,we obtain upper bounds for Sk(G)in terms of the clique number w,the vertex covering number τ and the diameter d of a graph G.The signless laplacian energy LE(G)of a graph G is defined as LE(G)=(?),where d=2m/n is the average degree of G.We obtain an upper bound for the signless laplacian energy LE(G)of a graph G in terms of the number of vertices n,the number of edges m,the clique number w and the vertex covering number τ.