Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs

Abstract

Let G be a simple connected graph with vertex set V(G) and edge set E(G). The resistance distance r(u,v) between two vertices u,v∈V(G) is the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. This function is known to be a metric on the vertex-set of any graph. The sum of resistance distances between pairs of vertices in a G is called Kirchhoff index and is denoted by Kf(G). In this study, we will compute the resistance distance between pairs of vertices of string of diamonds and ring of diamonds by using some methods from electrical network theory such as series and parallel principles, the principle of elimination, the star-triangle transformation, and the delta-wye transformation. Then we determine the exact formulas for the Kirchhoff index of the string of diamonds and ring of diamonds.