Computation of resistance distance and Kirchhoff index of the two classes of silicate networks

Abstract

The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to G, with each edge being replaced by a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices in G. In this paper, the resistance distance between any two arbitrary vertices of a chain silicate network and a cyclic silicate network was procured by utilizing techniques from the theory of electrical networks, i.e., the series and parallel principles, the principle of elimination, the star-triangle transformation and the delta-wye transformation. Two closed formulae for the Kirchhoff index of the chain silicate network and the cyclic silicate network were obtained respectively.