Circular planar electrical networks: posets and positivity. (English) Zbl 1307.05055
Summary: Following Y. Colin de Verdière et al. [Comment. Math. Helv. 71, No. 1, 144–167 (1996; Zbl 0853.05074)] and E. B. Curtis et al. [Linear Algebra Appl. 283, No. 1–3, 115–150 (1998; Zbl 0931.05051)], we prove a host of new results on circular planar electrical networks. We first construct a poset \(EP_n\) of electrical networks with \(n\) boundary vertices, and prove that it is graded by the number of edges of critical representatives. We then answer various enumerative questions related to \(EP_n\), adapting methods of D. Callan [J. Integer Seq. 7, No. 1, Art. 04.1.8, 7 p. (2004; Zbl 1065.05006)] and P. R. Stein and C. J. Everett [Discrete Math. 22, 309–318 (1978; Zbl 0395.05003)]. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of A. Postnikov [“Total positivity, Grassmannians, and networks”, Preprint, arXiv:math/0609764], and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by T. Lam and P. Pylyavskyy [“Laurent phenomenon algebras”, Preprint, arXiv:1206.2611].
MSC:
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 05C82 | Small world graphs, complex networks (graph-theoretic aspects) |
| 05C90 | Applications of graph theory |
Keywords:
planar graph; resistor network; network response; poset; Laurent phenomenon; electrical positroidReferences:
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