Planar electric networks. I.
(Réseaux électriques planaires. I.)

*(French)*Zbl 0816.05052This 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.

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.

Reviewer: H.J.Tiersma (Diemen)

##### 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.) |