Resistance Distance And Kirchhoff Index Of Some Graphs

The algebraic graph theory is an important branch of discrete mathematics.It mainly uses algebraic methods and techniques to solve graph theory problems.There are three main algebraic methods:the linear algebraic method,the group theory method and the graphical invariant method.The resistance distance r(i,j)(G)between two random vertices i and j of the con-nected graph G refers to the effective resistance between nodes i and j in the electric network obtained when all the edges of G are replaced by unit resistance.The Kirchhoff index Kf(G)of G is the sum of resistance distances between all vertex pairs.The resis-tance distance and the Kirchhoff index of three types of double join graphs and five types of corona graph variants are provided by mainly combining the linear algebraic method and the techniques of the graph theory in this paper.Meanwhile,the Laplacian spectra SpecL(G)of seven types of corona graph variants are depicted.The paper consists of the following four parts:In Chapter 1,some fundamental concepts and lemmas used in this paper are dis-cussed,then some research backgrounds and developments of the resistance distance and the Kirchhoff index of graphs are discussed,and then the main results of this paper are briefly described.In Chapter 2,the resistance distance and the Kirchhoff index of following three kinds of double join graphs,subdivision join graphs,R-subdivision graphs and Q-subdivision graphs are discussed,based on the results of research on join graphs,subdivision graphs,R-graphs and Q-graphs.Inspired by the research methods of adjacency spectrum of the literature[Discus-sions Mathematicae Graph Theory 36(2016)127-140],in Chapter 3,by using the re-lationship between the adjacency matrix and incidence matrix of regular graphs and through appropriate transformation to the Laplacian matrix,the Laplacian spectra of seven types of corona graph variant are described,and a simple application example is provided for the Laplacian spectrum of each type.Based on Chapter 3,the resistance distance and Kirchhoff index of five types of corona graph variants are studied in Chapter 4,and an application example of the Kirch-hoff index formula for each type is provided.