Recursive calculation of effective resistances in distance-regular networks based on Bose-Mesner algebra and Christoffel-Darboux identity

TL;DR

This paper presents a recursive algorithm for calculating effective resistances in distance-regular networks using algebraic and spectral techniques, revealing that resistance increases with shortest path distance and linking it to random walk metrics.

Contribution

It introduces a novel recursive method based on Bose-Mesner algebra and Christoffel-Darboux identity for resistance calculation in distance-regular networks.

Findings

Effective resistance increases with shortest path distance.

Resistance distances are equal within the same stratum.

Explicit resistance calculations for key network examples.

Abstract

Recently in \cite{jss1}, the authors have given a method for calculation of the effective resistance (resistance distance) on distance-regular networks, where the calculation was based on stratification introduced in \cite{js} and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. Also, in Ref. \cite{jss1} it has been shown that the resistance distances 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. In this work, an algorithm for recursive calculation of the resistance distances in an arbitrary distance-regular resistor network is provided, where the derivation of the algorithm is based on the Bose-Mesner algebra, stratification of the network, spectral techniques and Christoffel-Darboux identity. It is shown that the effective resistance on a distance-regular network is an strictly increasing function of the shortest path distance defined on the network. In the other words, the two-point resistance $R_{\alpha\beta^{(m+1)}}$ is strictly larger than $R_{\alpha\beta^{(m)}}$. The link between the resistance distance and random walks on distance-regular networks is discussed, where the average commute time (CT) and its square root (called Euclidean commute time (ECT)) as a distance are related to the effective resistance. Finally, for some important examples of finite distance- regular networks, the resistance distances are calculated. {\bf Keywords: resistance distance, association scheme, stratification, distance-regular networks, Christoffel-Darboux identity} {\bf PACs Index: 01.55.+b, 02.10.Yn}