Definition:Degree (Vertex)

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

Also see

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