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nLab

Euler formula for planar graphs

### Context

#### Graph theory

#### Homotopy theory

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

# Contents

## 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

Last revised on August 1, 2018 at 13:36:57.
See the history of this page for a list of all contributions to it.