| The spectral theory of graphs is one of the important topics of algebraic graph theory.It is mainly studied the topological structure properties of graphs by using algebraic properties of some matrix of graphs such as adjacency matrix,Laplacian matrix,etc.The multiset of eigenvalues of Laplacian matrix L(G)of graph G is called Laplacian spectrum of G.The resistance distance between vertices i and j of graph G is defined to be the effective resistance when all the edges of G are considered to be unit resistors.The Kirchhoff index Kf(G)is the sum of resistance distances between all pairs of vertices.In this thesis,combing the theory of linear algebra and the techniques of graph theory,the Laplacian spectra of two kinds of new graphs based on Cartesian product and join operation are obtained.As its application,we discuss the(degree-)Kirchhoff index of the regular graph,and the expected values,average values and variance of(degree-)Kirchhoff index of random polygon chains.Finally,we consider the hyper-Kirchhoff index of bicyclic graphs.The main content consists of the five parts as follows:In Chapter 1,some basic concepts,background and progress of Laplacian spectrum and Kirchhoff index of graphs are introduced,and the main results of this thesis are lieted.In Chapter 2,based on the cartesian product and the join operation of graphs,we discuss the Laplacian spectrum of two kinds of new graphs.In Chapter 3,we give the Kirchhoff index,additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index of Q-graph.In Chapter 4,we discuss the expected values,average values and variance of(degree-)Kirchhoff index of random polygon chains.Furthermore,we compare the expected values,average values,the variance between the Wiener index and the Kirchhoff index,and charaterize their correlation coefficient.In Chapter 5,we study the hyper-Kirchhoff index of bicyclic graphs and determine the extreme graph with the largest hyper-Kirchhoff index when the order n≥ 10. |
