Circular planar graphs and resistor networks

We consider circular planar graphs and circular planar resistor networks. Associated with each circular planar graph F there is a set n(F) = { (P; Q) } of pairs of sequences of boundary nodes which are connected through F. A graph F is called critical if removing any edge breaks at least one of the connections (P: Q) in n(F). We prove that two critical circular planar graphs are Y-A equivalent if and only if they have the same connections. If a conductivity ;, is assigned to each edge in F, there is a linear from boundary voltages to boundary currents, called the network response. This linear map is represented by a matrix A:. We show that if (F,7) is any circular planar resistor network whose underlying graph F is critical, then the values of all the conductors in F may be calculated from A. Finally, we give an algebraic description of the set ot" network response matrices that can occur for circular planar resistor networks. @ 1998 Published by Elsevier Science inc. All rights reserved. A MS chtss(licathm." 05C40: 05C50; 90B I 0; 94C ! 5 Kcvwm'ds: Graph; Connections; Conductivity: Resistor network; Network response