Circular Planar Electrical Networks I: The Electrical Poset EP_{n}
Combinatorics
2013-09-12 v1
Abstract
Following de Verdi\`{e}re-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.
Cite
@article{arxiv.1309.2697,
title = {Circular Planar Electrical Networks I: The Electrical Poset EP_{n}},
author = {Joshua Alman and Carl Lian and Brandon Tran},
journal= {arXiv preprint arXiv:1309.2697},
year = {2013}
}
Comments
29 pages, 12 figures