Let G_1 and G_2 be vertex-disjoint graphs with n_1 and n_2 vertices,m_1 and m_2 edges,respectively.The corona G_1° G_2 of G_1and G_2 was defined as the graph obtained by taking one copy of G_1and n_1 copies of G_...Let G_1 and G_2 be vertex-disjoint graphs with n_1 and n_2 vertices,m_1 and m_2 edges,respectively.The corona G_1° G_2 of G_1and G_2 was defined as the graph obtained by taking one copy of G_1and n_1 copies of G_2,and then joining the ith vertex of G_1 to every vertex in the ith copy of G_2.The edge corona G_1° G_2 of G_1 and G_2was defined as the graph obtained by taking one copy of G_1 and m_1copies of G_2,and then joining two end-vertices of the ith edge of G_1to every vertex in the ith copy of G_2.Kirchhoff index of G_1 ° G_2 was obtained in a much simple proof and the resistance distance of G_1 °G_2 is obtained.Furthermore,the Kirchhoff index and the resistance distance of G_1 ° G_2 were obtained.展开更多
For some complicated graphs obtained by graph operations,it is very difficult to compute resistance distance and Kirchhoff index.Define a new graph operation,and obtain a class of new join graphs:the subdivision-verte...For some complicated graphs obtained by graph operations,it is very difficult to compute resistance distance and Kirchhoff index.Define a new graph operation,and obtain a class of new join graphs:the subdivision-vertex-vertex join G_1* G_2.Then,describe the Laplacian matrix of the graph G_1 * G_2 and use generalized inverse of the Laplacian matrix to get formulas for resistance distance and Kirchhoff index.Through the obtained formulas,the resistance distance of any pairs of vertices and Kirchhoff index of the join graph can be computed.展开更多
The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the comple...The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.展开更多
基金National Natural Science Foundation of China(No.11461020)the Youth Foundation of Hexi University in Gansu Province,China(No.QN2013-07)the Teaching Reform Project of Hexi University,China(No.HXXXJY-2014-011)
文摘Let G_1 and G_2 be vertex-disjoint graphs with n_1 and n_2 vertices,m_1 and m_2 edges,respectively.The corona G_1° G_2 of G_1and G_2 was defined as the graph obtained by taking one copy of G_1and n_1 copies of G_2,and then joining the ith vertex of G_1 to every vertex in the ith copy of G_2.The edge corona G_1° G_2 of G_1 and G_2was defined as the graph obtained by taking one copy of G_1 and m_1copies of G_2,and then joining two end-vertices of the ith edge of G_1to every vertex in the ith copy of G_2.Kirchhoff index of G_1 ° G_2 was obtained in a much simple proof and the resistance distance of G_1 °G_2 is obtained.Furthermore,the Kirchhoff index and the resistance distance of G_1 ° G_2 were obtained.
基金National Natural Science Foundation of China(No.11361033)
文摘For some complicated graphs obtained by graph operations,it is very difficult to compute resistance distance and Kirchhoff index.Define a new graph operation,and obtain a class of new join graphs:the subdivision-vertex-vertex join G_1* G_2.Then,describe the Laplacian matrix of the graph G_1 * G_2 and use generalized inverse of the Laplacian matrix to get formulas for resistance distance and Kirchhoff index.Through the obtained formulas,the resistance distance of any pairs of vertices and Kirchhoff index of the join graph can be computed.
基金Supported by the National Natural Science Foundation of China(11861011,11501133,11661010)。
文摘The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.
