Matrices Of Graphs And Calculation Of Indices

Calculation of the resistance distance of a graph has been an important topic in the past decades.If each edge of a connected graph G is considered as a unit resistance,then the resistance distance between any two vertices in G is defined as the equivalent resistance between the two corresponding nodes in the electrical network.The sum of the resistance distances between all vertex pairs in G is referred to as the Kirchhoff index of G.The first part of this thesis calculates the resistance distance and Kirchhoff index of graphs under a kind of graph operation.In recent years,more and more attention has been paid on a class of new matrices,the so called eccentricity matrices.An eccentricity matrix is obtained from the distance matrix of a graph which keeps the largest value in each column and each row,and all others are zero.The second part of this thesis calculated the spectrum of distance Laplace matrix,distance unsigned Laplace matrix and eccentricity matrix of three generalized windmill graphs.Moreover,their Hosoya polynomials,Harary indices etc.were also calculated.The third part mainly gave a kind of tree of even diameter whose eccentricity matrix had the largest spectrum radius.Chapter one gave the concepts and symbols used in this thesis.The research progress of resistance distance,Kirchhoff index and eccentricity matrix spectrums has been overviewed.Let G be a connected graph.Replace each edge of G by a path of length two,and the resulting graph is denoted by S(G).Furthermore,in S(G),two dangling points are added to each original vertex of G.The newly graph is denoted by RS(G).In Chapter 2,we study the resistance distance and Kirchhoff index of RS(G).In the third chapter,indices of distance Laplace matrix,distance unsigned Laplace matrix and eccentricity matrix of three generalized windmill graphs were calculated.In chapter 4,we analyzed the largest eigenvalue of the eccentricity matrix of trees with even diameter.The trees whose eccentricity matrix had the maximum spectral radius were given.At the end of this thesis,based on what we got in this thesis we put forward some follow-up research questions.