CIRCULAR PLANAR GRAPHS AND RESISTOR NETWORKS

We consider circular planar graphs and circular planar resistor networ ks. Associated with each circular planar graph Gamma there is a set pi (Gamma) = {(P; Q)} of pairs of sequences of boundary nodes which are c onnected through Gamma. A graph Gamma is called critical if removing a ny edge breaks at least one of the connections (P; Q) in pi(Gamma). We prove that two critical circular planar graphs are Y-Delta equivalent if and only if they have the same connections. If a conductivity gamm a is assigned to each edge in Gamma, there is a linear from boundary v oltages to boundary currents, called the network response. This linear map is represented by a matrix Lambda(gamma). We show that if(Gamma, gamma) is any circular planar resistor network whose underlying graph Gamma is critical, then the values of all the conductors in Gamma may be calculated from Lambda(gamma). Finally, we give an algebraic descri ption of the set of network response matrices that can occur for circu lar planar resistor networks. (C) 1998 Published by Elsevier Science I nc. All rights reserved.