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

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

The degree sequence of a graph is a list of the degrees of all the vertices of the graph in descending order.

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.