Definition:Degree (Vertex)

Undirected Graph

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 a vertex $v$ is even, then $v$ is called an even vertex.

Odd Vertex

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

Isolated Vertex

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

Digraph

Let $G = \left({V, E}\right)$ be a digraph.

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

Out-Degree

The out-degree of $v$ in $G$ is the number of arcs which are incident from $v$.

It is denoted $\operatorname{outdeg}_G \left({v}\right)$, or just $\operatorname{outdeg} \left({v}\right)$ if it is clear from the context which graph is being referred to.

In-Degree

The in-degree of $v$ in $G$ is the number of arcs which are incident to $v$.

It is denoted $\operatorname{indeg}_G \left({v}\right)$, or just $\operatorname{indeg} \left({v}\right)$ if it is clear from the context which graph is being referred to.

Sources

• Gary Chartrand: Introductory Graph Theory (1977): $\S 2.1$