# Definition:Euler Characteristic of Finite Graph

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## Definition

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

Let $X$ be embedded in a surface.

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

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

where:

- $v = \size V$ is the number of vertices
- $e = \size E$ is the number of edges
- $f$ is the number of faces.

### Generalized Formula

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## Also see

## 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 ($\text {1707}$ – $\text {1783}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Euler characteristic** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Euler characteristic** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Euler characteristic**