Definition:Euler Characteristic of Finite Graph

Definition

Let $G = \struct {V, E}$ be a finite graph.

Let $G$ be embedded in a surface.

The Euler characteristic of $G$ is written $\map \chi G$ and is defined as:

$\map \chi G = v - e + f$

where:

$v = \size V$ is the number of vertices

$e = \size E$ is the number of edges

$f$ is the number of faces.

Also see

• Euler's Theorem for Planar Graphs

• Euler Polyhedron Formula

• Definition:Euler Characteristic of Surface

• Results about the Euler characteristic of a finite graph can be found here.

Source of Name

This entry was named for Leonhard Paul Euler.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Euler characteristic
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Euler characteristic