Planar electric networks. II
type: Item
instance of: scholarly article
title: Planar electric networks. II
zbMATH Open document ID: 0853.05074
DOI: 10.1007/BF02566413
author: Yves Colin de Verdière
author: Isidoro Gitler
author: Dirk Vertigan
published in: Commentarii Mathematici Helvetici
publication date: 1996-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/140394
review text: A planar electrical network is a planar graph \((V,E)\) having so-called boundary points \(V_0\) and an interior part \(V \backslash V_0\), moreover associated to the edges are the inverses of the resistence \(\rho (e)\) (also called the conductivity). To every boundary point in \(V_0\) there is associated an electrical potential due to which there are incoming and outgoing electrical flows at the boundary points. The authors introduce so called elementary transformations on the interior of electrical planar networks, that when applied can be used to simplify a network by decreasing the number of edges. The planar electrical networks now can be viewed as the nodes of a graph where two nodes (networks) are joined if one can be transformed into the other by an elementary electrical transformation. The connected components in this graph can be used to define an equivalence relation: two networks are equivalent if and only if they are in the same connected component. The first theorem states that for every planar electrical network there exists a sequence of elementary transformations such that after each transformation the number of edges decreases until it reaches a minimum. Moreover two minimal networks in the same equivalence class can be transformed into each by applying a number of elementary transformations during which the number of edges remains constant. In the second theorem well connected planar networks are characterized, i.e. the minimal graphs in the equivalence class of a planar network having \(N\) boundary points have \(N(N - 1)/2\) edges if the network is well connected. As a consequence if the minimal graph in an equivalence class has less than \(N(N - 1)/2\) edges the graphs in this class are not well connected. The most important theorem of the paper is Theorem 4 which states that two planar electrical networks are in the same equivalence class if and only if they have the same number of boundary points and the same in and outgoing flows at these p…
Mathematics Subject Classification ID: 05C90
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 31A25
zbMATH DE Number: 872642
zbMATH Keywords: planar electrical network
zbMATH Keywords: resistence
zbMATH Keywords: electrical flows
zbMATH Keywords: elementary transformations
reviewed by: Herman J. Tiersma
MaRDI profile type: Publication
OpenAlex ID: W2324792688
description: scientific article; zbMATH DE number 872642