# Evaluation of effective resistances in pseudo-distance-regular resistor networks - Condensed Matter > Statistical Mechanics

Abstract: In Refs.1 and 2, calculation of effective resistances on distance-regularnetworks was investigated, where in the first paper, the calculation was basedon the stratification of the network and Stieltjes function associated with thenetwork, whereas in the latter one a recursive formula for effectiveresistances was given based on the Christoffel-Darboux identity. In this paper,evaluation of effective resistances on more general networks calledpseudo-distance-regular networks 21 or QD type networks \cite{obata} isinvestigated, where we use the stratification of these networks and show thatthe effective resistances between a given node such as $\alpha$ and all of thenodes $\beta$ belonging to the same stratum with respect to $\alpha$$R {\alpha\beta^{m}}$, $\beta$ belonging to the $m$-th stratum with respectto the $\alpha$ are the same. Then, based on the spectral techniques, ananalytical formula for effective resistances $R {\alpha\beta^{m}}$ such that$L^{-1} {\alpha\alpha}=L^{-1} {\beta\beta}$ those nodes $\alpha$, $\beta$ ofthe network such that the network is symmetric with respect to them is givenin terms of the first and second orthogonal polynomials associated with thenetwork, where $L^{-1}$ is the pseudo-inverse of the Laplacian of the network.From the fact that in distance-regular networks,$L^{-1} {\alpha\alpha}=L^{-1} {\beta\beta}$ is satisfied for all nodes$\alpha,\beta$ of the network, the effective resistances$R {\alpha\beta^{m}}$ for $m=1,2, .,d$ $d$ is diameter of the network whichis the same as the number of strata are calculated directly, by using thegiven formula.

Source: https://arxiv.org/