nLab Euler formula for planar graphs

Statement

For a connected finite planar graph $\Gamma$ the numbers of edges, vertices and number of regions enclosed by edges (“faces”) always satisfies the relation

$$1 = \vert Vertices(\Gamma)\vert - \vert Edges(\Gamma)\vert + \vert Faces(\Gamma)\vert \,.$$

By passing to the one-point compactification of the plane, which is the 2-sphere, we may think of the planar graph as a polyhedron embedded in the 2-sphere. Under this identification the above is a special case of the general formula for Euler characteristic of CW-complexes. See at Euler characteristic – Of topological spaces.

Applications

In perturbative quantum field theory

That the loop order of a (planar) Feynman diagram is its contribution in powers of Planck's constant to the scattering amplitude is a consequence of Euler’s formula. See at loop order – Relation to powers in Planck’s constant

References

• Wikiedia, Euler’s formula