Resistance distance in k-coalescence of certain graphs

Abstract

Any graph can be considered as a network of resistors, each of which has a resistance of 1Ω. The resistance distance rij between a pair of vertices i and j in a graph is defined as the effective resistance between i and j. Graph operations have been widely used in the analysis of complex networks, with properties abstracted from the real world. This article deals with the resistance distance in the k-coalescence of any two graphs having a clique of order k and also gives results for the particular case of complete graphs. Furthermore, we find Kemeny’s constant, Kirchhoff index, additive degreeKirchhoff index, multiplicative degree-Kirchhoff index and mixed degree-Kirchhoff index of k-coalescence of two complete graphs. Moreover, we obtain the resistance distance in the k-coalescence of a complete graph with particular graphs. Additionally, we provide the resistance distance of certain graphs such as the vertex coalescence of a complete bipartite graph with a complete graph, a complete bipartite graph with a star graph, the windmill graph, dandelion graph, etc.