- The current law, yielding
*N-1*equations (with*N*the number of nodes) - The voltage law, for an additional
*L*equations (where*L*is the number of elementary loops)

*B*of them). We also assume that the branches do not cross. Note that writing the current law for the

*N*-th node or the voltage law for a composite loop (consisting of several adjacent elementary loops) does not provide any further information, the resulting equations being linear combinations of the previous ones.

For the problem to be well-posed the number of unknowns and equations is equal, which we can write as:

\[ N + L - B = 1\] This relation is easily proven in plane geometry, but here I would like to show its intimate connection with Euler's formula, which states that, for a convex polyhedron, \[ V + F - E = 2\] where

*V*,

*F*and

*E*are the numbers of vertices, faces and edges, respectively.

Let us start by establishing a correspondence between circuits and polyhedra, as shown in the figure below. Place a sphere on top of the (planar) circuit diagram and connect each node to the North pole by a line segment (this is known as a stereographic projection.) We define the vertices as the intersections of these segments with the sphere; the result is a convex polyhedron.

It is easily seen that, with the notations above, we have the straightforward equivalences

*V = N*and

*E = B*. The number of faces, however,

*F = L + 1*, since the "topmost" face corresponds to the open area surrounding the circuit. Substitution in either of the equations above yields the other one.

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