Circular Planar Electrical Networks I: The Electrical Poset EP_n

2013/09/11 by Alman, Joshua, Lian, Carl, Tran, Brandon
#05A15 #05A16 #06A07 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.2697

Abstract

Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We introduce a poset EP_n of electrical networks with n boundary vertices, giving two equivalent characterizations, one combinatorial and the other topological. We then investigate various properties of the EP_n, proving that it is graded by number of edges of critical representatives. Finally, we answer various enumerative questions related to EP_n, adapting methods of Callan and Stein-Everett.

Related

• The space of circular planar electrical networks
• Electroid varieties and a compactification of the space of electrical\n networks
• Circular Planar Electrical Networks II: Positivity Phenomena
• Circular Planar Resistor Networks with Nonlinear and Signed Conductors
• Simplicial effective resistance and enumeration of spanning trees