Circular Planar Electrical Networks I: The Electrical Poset EPn
Abstract
Following de Verdi\`ere-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We introduce a poset EPn of electrical networks with n boundary vertices, giving two equivalent characterizations, one combinatorial and the other topological. We then investigate various properties of the EPn, proving that it is graded by number of edges of critical representatives. Finally, we answer various enumerative questions related to EPn, adapting methods of Callan and Stein-Everett.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.