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

Recently in [1], 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 [2] and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. Also, in Ref. [1] it has been shown that the resistance distances between a node α and all nodes β belonging to the same stratum with respect to the α (Rαβ(i) , β belonging to the i-th stratum with respect to the α) 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 Rαβ(m+1) is strictly larger than Rαβ(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 distanceregular networks, the resistance distances are calculated.