Resistance between two nodes in general position on an $m\times n$ fan network

Abstract

The resistance between two nodes in general position on a fan network with n radial lines and m transverse lines is determined. Also a similar result of Izmailian, Kenna and Wu [7] for an $m\times n$ cobweb network is reproduced but the method used here is significantly different. It avoids the use of the Kirchhoff matrix, requires the solution of just one instead of two eigenvalue problems and results directly in only a single summation. Further the current distribution is given explicitly as a biproduct of the method. The method is the same as that used by Tan, Zhou and Yang [10] to find the cobweb resistance between center and perimeter for $1\le m\le3$ and general $n$. Proof of their conjecture for general $m$ is discussed.