The Laplacian And Distance Signless Laplacian Spectral Radius Of Graphs

The adjacency matrix,distance matrix,Laplacian matrix,signless Laplacian matrix are all closely related to the structure of the graph.Whether the properties of the graph can be reflected by the eigenvalues of the matrix.In recent years,the Laplacian spectral radius(algebraic connectivity)and distance signless Laplacian spectral radius of a connected graph has been studied extensively by a large number of scholars.Based on the previous study,we characterize the upper bounds of Laplacian algebraic connectivity of n-vertex graphs in terms of matching number,the lower bounds of distance signless Laplacian spectral radius of n-vertex graphs in terms of matching number,and also characterize the graphs with the maximal and minimum distance signless Laplacian spectral radius of has been transformed clique path in clique trees.The graphs studied in this paper are simple,non-directional and finite graphs.Let G be a graph with vertex set V(G)={v1,…,vn},and edge set E(G)={e1,···,em}.Let A(G)be the adjacency matrix of G,and Deg(G)=diag(d1,d2,…,dn)be the diagonal matrix of vertex degrees.Then the signless Laplacian matrix of G is Q(G)=Deg(G)+A(G),and the Laplacian matrix of G is L(G)=Deg(G)-A(G).For a graph G,we denote by q(G)the largest eigenvalue of Q(G)and call it the signless Laplacian spectral radius of G,accordingly,denoted by ?1(L(G))the largest eigenvalue of L(G)and call it the Laplacian spectral radius of G.The distance between u and v in G is denoted by duv and defined as the length of shortest path between u and v in G.The distance matrix of G is denoted by D(G)and defined by D(G)?(duv)u,v?V(G).Since D(G)is a real symmetric matrix,all its eigenvalues are real.The distance spectral radius PD(G)of G is the largest eigenvalue of its distance matrix D(G).The unique normalized eigenvector corresponding to ?D(G)is referred as the Perron vector of D(G).Then we denote by TrG(vi)the sum of distances between vi and other vertices of G,and so TrG(u)=?v?V(G)duv.The largest eigenvalues of Q(G)and LD(G)are called distance singless Laplacian spectral and distance Laplacian spectral radius of G,respectively.The main work and framework of the article are as follows:Firstly,in Chapter ?,Let Gnm be the class of all graphs of order n with matching number m,We characterize the upper bounds of Laplacian algebraic connectivity of n-vertex graphs in terms of matching number.Secondly,in Chapter ?,we determine the lower bounds of distance signless Laplacian spectral radius of n-vertex graphs in terms of matching number.Thirdly,characterize the graphs with the maximal and minimum distance signless Lapla-cian spectral radius of has been transformed clique path in clique trees.