# Definition:Degree (Vertex)

From ProofWiki

## Definition

Let $G = \left({V, E}\right)$ 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 $\deg_G \left({v}\right)$, or just $\deg \left({v}\right)$ if it is clear from the context which graph is being referred to.

That is:

- $\deg_G \left({v}\right) = \left|{\left\{{u \in V : \left\{{u, v}\right\} \in E}\right\}}\right|$.

### 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