Resistance distance in $k$-coalescence of certain graphs

TL;DR

This paper investigates the resistance distance in the $k$-coalescence of complete graphs and related structures, providing formulas and connections to various graph invariants, with applications to specific graph combinations.

Contribution

It introduces new formulas for resistance distances in $k$-coalesced graphs and explores their relation to graph invariants like Kemeny's constant and Kirchhoff indices.

Findings

Derived resistance distance formulas for $k$-coalescence of complete graphs.

Connected resistance distance to graph invariants such as Kemeny's constant.

Calculated resistance distances for specific graph combinations like bipartite and star 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.