Definition:Euler Characteristic of Finite Graph

Definition
Let $X = \left({V, E}\right)$ be a graph.

Let $X$ be embedded in a surface.

The Euler Characteristic of a $X$ is written $\chi \left({X}\right)$ and is defined as:
 * $\chi \left({X}\right) = v - e + f$

where:
 * $v = \left|{V}\right|$ is the number of vertices
 * $e = \left|{E}\right|$ is the number of edges
 * $f$ is the number of faces.

Euler Characteristic for Planar Graph
The Euler Polyhedron Formula states that for any planar graph (i.e. which can be drawn on a sphere or plane without any two of its edges meeting except at vertices), $\chi = 2$.

Also see

 * Euler Polyhedron Formula