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Planar electric networks. I. (Réseaux électriques planaires. I.) (French) Zbl 0816.05052
This paper is the first one of two papers on planar electric networks. An electric network is given as (1) a graph \(\Gamma= (V, V_ 0, E)\) (the network), where \(V\) is the set of nodes and \(V_ 0\) is the set of terminal nodes and \(E\) is the set of edges; loops and multiple edges are allowed, (2) a set of conductors (every edge in the graph can be seen as a conductor having as a measure of conductivity \(\rho_{ij}\) (the inverses of the resistances)), furthermore N is the cardinality of the set of terminal nodes \((V_ 0)\). One can look at the electrical energy associated to the network, it can be given as \(q_ \rho(x)= 1/2 \sum_{(i,j)\in E} \rho_{ij}(x_ i- x_ j)^ 2\), where \(x_ i- x_ j\) is the difference in the electrical potential at the endpoints of an edge. To the network one can associate a linear map L that maps the vector of electrical potentials \((x_ 1,\dots, x_ N)\) on the terminal nodes to the vector of electrical flows \((I_ 1,\dots, I_ N)\) coming out of the terminal nodes.
The first section of the paper shows that L is symmetrical and depends on \(q_ \rho\) only. Hence one can look at the map \(\Phi_ \Gamma\) which maps \(q_ \rho\) to \(L_{\Gamma,\rho}\).
The goal of the two papers is to study the map \(\Phi_ \Gamma\) in cases the graph \(\Gamma\) is a planar graph. The obtained results are generalizations of results by Curtis, Morrow and Mooers for the cases where the graph is rectangular or circular. The main results are: (1) The notions \(\Gamma\)-connected for disjoint subsets of \(V_ 0\) and well- connected for a graph \(\Gamma\) are introduced. These notions are strongly connected to the linear map L associated to \(\Gamma\) and hence to \(\Phi_ \Gamma\). (2) The notion N-critical is introduced (meaning that no edge can be reduced without loosing the well-connectedness of the graph). The graph being N-critical has also consequences for \(\Phi_ \Gamma\). (3) A characterization is given for non-planar networks for which \(V_ 0\) can be cyclically ordered such that \(\Phi_ \Gamma\) is continuous: they appear to be equivalent to planar networks in some sense. (4) Some results of the forthcoming (second) paper are already announced.

MSC:
05C90 Applications of graph theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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