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 $1\Omega .$ The resistance distance $r_{ij}$ between a pair of vertices $i$ and $j$ in a graph is defined as the effective resistance between $i$ and $j$. This article deals with the resistance distance in the $k$-coalescence of complete graphs. We also present its results in connection with the Kemeny's constant, Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index and mixed degree-Kirchhoff index. Moreover, we obtain the resistance distance in the $k$-coalescence of a complete graph with particular graphs. As an application, 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, pineapple graph, etc.