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

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. A 39, 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 α and all nodes β belonging to the same stratum with respect to α (Rαβ(i), β belonging to the ith stratum with respect to α) are the same. In this work, an algorithm for recursive calculation of the effective resistances 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 a strictly increasing function of the shortest path distance defined on the network. In other words, the effective resistance Rαβ(m+1) is strictly larger than Rαβ(m). 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.