### Abstract

Recently, Jafarizadeh *et al.* [ J. Phys. A: Math. Theor.40, 4949 (2007)] have given a method for calculation of effective resistance (two-point resistance) on distance-regular networks, where the calculation was based on stratification introduced by Jafarizadeh and Salimi [J. Phys. A39, 1 (2006)] and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. Also,Jafarizadeh *et al.* [ J. Phys. A: Math. Theor.40, 4949 (2007)] have shown that effective resistances between a node and all nodes belonging to the same stratum with respect to (, belonging to the stratum with respect to ) are the same. In this work, an algorithm for recursive calculation of the effective resistances in an arbitrary distance-regular resistornetwork 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 a strictly increasing function of the shortest path distance defined on the network. In other words, the effective resistance is strictly larger than . The link between effective resistance and random walks on distance-regular networks is discussed, where average commute time and its square root (called Euclidean commute time) as distance are related to effective resistance. Finally, for some important examples of finite distance-regular networks, effective resistances are calculated.

Received 28 July 2008
Accepted 08 January 2009
Published online 12 February 2009

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
A. Association schemes
B. Stratifications
C. Distance-regular networks and orthogonal polynomials
1. The Christoffel–Darboux identity
III. EFFECTIVE RESISTANCES ON REGULAR RESISTORNETWORKS
A. Random walks and electrical networks
IV. RECURSIVE CALCULATION OF EFFECTIVE RESISTANCE ON DISTANCE-REGULAR NETWORKS BASED ON SPECTRAL ANALYSIS METHODS AND CHRISTOFFEL–DARBOUX IDENTITY
A. Example: Distance-regular networks derived from symmetric group
V. CONCLUSION

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