Calculating two-point resistances in distance-regular resistor networks

TL;DR

This paper presents an algorithm for calculating two-point resistances in distance-regular resistor networks using stratification and spectral distribution, providing explicit formulas and bounds for various network types.

Contribution

It introduces a novel algorithm based on stratification and spectral methods to compute resistances in distance-regular networks, including explicit formulas and bounds.

Findings

Resistances between a node and all nodes in the same stratum are equal.

Analytical formulas for two-point resistances are derived in terms of network parameters.

Bounds for resistances in strongly regular networks are established.

Abstract

An algorithm for the calculation of the resistance between two arbitrary nodes in an arbitrary distance-regular resistor network is provided, where the calculation is based on stratification introduced in \cite{js} and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. It is shown that the resistances between a node $\alpha$ and all nodes $\beta$ belonging to the same stratum with respect to the $\alpha$ ($R_{\alpha\beta^{(i)}}$, $\beta$ belonging to the $i$-th stratum with respect to the $\alpha$) are the same. Also, the analytical formulas for two-point resistances $R_{{\alpha\beta^{(i)}}}, i=1,2,3$ are given in terms of the the size of the network and corresponding intersection numbers. In particular, the two-point resistances in a strongly regular network are given in terms of the its parameters ($v,\kappa,\lambda,\mu$). Moreover, the lower and upper bounds for two-point resistances in strongly regular networks are discussed. Keywords:two-point resistance, association scheme, distance-regular networks, Stieltjes function PACs Index: 01.55.+b, 02.10.Yn