- 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)
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.