Effective resistances and Kirchhoff Index in subdivision networks

TL;DR

This paper introduces a method to analyze subdivision networks by expressing their Green kernel, effective resistance, and Kirchhoff index in terms of the original network's parameters, with applications to wheel networks.

Contribution

It provides explicit formulas for the Green kernel, effective resistance, and Kirchhoff index of subdivision networks based on the base network's properties.

Findings

Derived the Green kernel of subdivision networks from the base network.

Expressed effective resistance and Kirchhoff index of subdivision networks in terms of base network.

Applied the results to compute parameters for a wheel network.

Abstract

We define a subdivision network $\Gamma^S$ of a given network $\Gamma,$ by inserting a new vertex in every edge, so that each edge is replaced by two new edges with conductances that fulfill electrical conditions on the new network. In this work, we firstly obtain an expression for the Green kernel of the subdivision network in terms of the Green kernel of the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a wheel.