Calculating two-point resistances in distance-regular resistor networks

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