%0 Journal Article %1 curtis98 %A Curtis, E.B. %A Ingerman, D. %A Morrow, J.A. %D 1998 %J Linear Algebra and its Applications %K determinant dodgson planar %N 1–3 %P 115--150 %R 10.1016/S0024-3795(98)10087-3 %T Circular Planar Graphs and Resistor Networks %V 283 %X We 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.

@article{curtis98, abstract = {We 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. }, added-at = {2016-04-18T15:10:45.000+0200}, author = {Curtis, E.B. and Ingerman, D. and Morrow, J.A.}, biburl = {https://www.bibsonomy.org/bibtex/270e825f4d6b3fcb697eeb72d61b19e19/ytyoun}, doi = {10.1016/S0024-3795(98)10087-3}, interhash = {e3a5cd3fe56410f20bdfeff4f1f0bb56}, intrahash = {70e825f4d6b3fcb697eeb72d61b19e19}, issn = {0024-3795}, journal = {Linear Algebra and its Applications }, keywords = {determinant dodgson planar}, number = {1–3}, pages = {115--150}, timestamp = {2016-05-22T09:14:30.000+0200}, title = {Circular Planar Graphs and Resistor Networks }, volume = 283, year = 1998 }