Resistance Distances and Kirchhoff Index in Generalised Join Graphs

Abstract

The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.

Resistance Distances and Kirchhoff Indices Under Graph Operations

“…In [10], Bu et al investigated resistance distance in subdivision-vertex join and subdivision-edge join of graphs. Then, Chen [11] obtained resistance distances and Kirchhoff indices of generalized join of graphs. Liu et al [12] gave resistance distances and Kirchhoff indices of R-vertex join and R-edge join of two graphs.…”