Some Graphs Which Attain Maximum (Signless) Laplacian Spectral Radius

In recent years,the(signless)Laplacian spectral radius of a connected graph has been studied extensively.Based on the previous study,we studied some problems about the maximum signless Laplacian spectral radii of tricyclic graph,among the graphs with given vertex connectivity,among the graphs with given number of blocks and among the graphs with given pendant vertices,respectively.and also determine the Sharp bounds for the Laplacian spectral radius in terms of clique number.Firstly,this paper introduced the achievement and background of spectral theory,the spectrum of adiacent matrix,signless Laplacian matrix and Laplacian matrix of graphs.All graphs considered here are simple,undirected and finite.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 D(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),and the Laplacian matrix of G is L(G)= D(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 μ(G)the largest eigenvalue of L(G)and call it the Laplacian spectral radiusof G.And then we introduced our main results in the next three chapters.Main outcome is as follows:Firstly,in Chapter II,Let gk,nbe the class of all graphs of order n with vertex connectivity k,Gp nbe the class of all graphs of order n with p blocks and Gn(k)be the class of all graphs with order n and k pendant vertices.We determined the graphs with maximum signless Laplacian spectral radius in gk,n,Gp nand Gn(k),respectively.Secondly,in Chapter III,we determine the graphs with maximum signless Laplacian spectral radius in tricyclic graphs.Thirdly,in Chapter IV,we determine the Sharp bounds for the Laplacian spectral radius in terms of clique number.