Circular planar graphs and resistor networks

AbstractWe consider circular planar graphs and circular planar resistor networks. Associated with each circular planar graph Γ there is a set π(Γ) = {(P; Q)} of pairs of sequences of boundary nodes which are connected through Γ. A graph Γ is called critical if removing any edge breaks at least one of the connections (P; Q) in π(Γ). We prove that two critical circular planar graphs are Y−Δ equivalent if and only if they have the same connections. If a conductivity γ is assigned to each edge in Γ, there is a linear from boundary voltages to boundary currents, called the network response. This linear map is represented by a matrix Λγ. We show that if (Γ,γ) is any circular planar resistor network whose underlying graph Γ is critical, then the values of all the conductors in Γ may be calculated from Λγ. Finally, we give an algebraic description of the set of network response matrices that can occur for circular planar resistor networks