| In graph theory,many matrices have been introduced into the research of graph theory.At present,some matrices of graphs have been deeply studied,such as incidence matrix,adjacency matrix,Laplacian matrix,and distance matrix etc.In 2013,Aouchiche and Hansen proposed the concepts of distance Laplacian matrix and distance signless Laplacian matrix of a connected graph,and studied their spectral radius of graphs.Let D(G)be the distance matrix of G and Tr(G)be the diagonal matrix of vertex transmissions of G.Then L(G)=Tr(G)-D(G)and Q(G)=Tr(G)+D(G)are the distance Laplacian matrix and distance signless Laplacian matrix of G,respectively.In 2019,inspired by distance Laplacian matrix and distance signless Laplacian matrix,Cui et al.proposed the concept of generalized distance matrix of a graph.The generalized distance matrix of a connected graph G is defined as Dα(G)=αTr(G)+(1-α)D(G),where 0 ≤ α<1,it is clearly that D0(G)=D(G),D1/2(G)=1/2Q(G),Da(G)-Db(G)=(a-b)L(G),0≤a,b<1.The largest eigenvalue of Dα(G)is called the generalized distance spectral radius,which is defined as μα(G).The generalized distance matrix of graph further expands the research scope of matrix spectral radius of graphs.This thesis mainly studies the generalized distance spectral radius of cacti and bicyclic graphs.This thesis is organized as follows.Chapter 1 is an introduction.It introduces the background of graph theory,the basic definitions related to this thesis,the known results of spectral radius of graphs,and the main work of this thesis.Chapter 2 introduced some useful lemmas related to the generalized distance matrix and some graph transformations that decrease the generalized distance spectral radius of graphs.In Chapter 3,we mainly determine the unique graph with minimum generalized distance matrix spectral radius among the cacti of order n with k(k≥ 1)cycles and at least one pendent vertex.We also show that the generalized distance matrix spectral radius of Bn*(the graph obtained from a star of order n by adding two nonadjacent edges)is less than the generalized distance spectral radius of the most of bicyclic graphs of order n.In Chapter 4,we summary the thesis. |
