Some Two-Vertex Resistances of Nested Triangle Network

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 the resistance distances between all pairs of vertices of G. A planar graph is a nested triangle graph with 3n vertices constructed from a sequence of n triangles by joining the pairs of corresponding vertices on consecutive triangles in the sequence. In this paper, some two-vertex resistances of nested triangle network was procured by utilizing techniques from the theory of electrical networks, i.e., the series and parallel principles, the principle of substitution, the star-triangle transformation and the delta-wye transformation. And the Kirchhoff index of nested triangle network is given by the Laplacian eigenvalues.