# Definition:Degree (Vertex)

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## Definition

Let $G = \struct {V, E}$ be an undirected graph.

Let $v \in V$ be a vertex of $G$.

The **degree of $v$ in $G$** is the number of edges to which it is incident.

It is denoted $\map {\deg_G} v$, or just $\map \deg v$ if it is clear from the context which graph is being referred to.

That is:

- $\map {\deg_G} v = \card {\set {u \in V : \set {u, v} \in E} }$

### Even Vertex

If the degree of $v$ is even, then $v$ is called an **even vertex**.

### Odd Vertex

If the degree of $v$ is odd, then $v$ is an **odd vertex**.

### Isolated Vertex

If the degree of $v$ is zero, then $v$ is an **isolated vertex**.

## Also see

- Out-Degree and In-Degree in the context of directed graphs.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): $\S 2.1$: The Degree of a Vertex - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**degree**(of a vertex of a graph)