Definition:Euler Characteristic
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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
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Source of Name
This entry was named for Leonhard Paul Euler.
