Definition:Euler Characteristic of Finite Graph

Graphs
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 Polyhedron Formula
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), $$\chi = 2 \ $$.

Generalized Formula
The unproved statements on this page are just resting here until they can be proved on pages of their own.