The degree of a graph vertex 
of a graph 
is the number of graph edges
which touch 
.
The vertex degrees are illustrated above for a random graph. The vertex degree is
also called the local degree or valency. The ordered list of vertex degrees in a
given graph is called its degree sequence. A list
of vertex degrees of a graph can be computed in the Wolfram
Language using VertexDegree[g],
and precomputed vertex degrees are available for particular embeddings of many named
graphs via GraphData[graph,
"VertexDegrees"].
The minimum vertex degree in a graph 
is denoted 
, and the maximum
vertex degree is denoted 
(Skiena 1990, p. 157).
The graph vertex degree of a point 
in a graph, denoted 
, satisfies
 |
where 
is the total number of graph edges.
In addition, a connected graph nodes satisfies
 |
where the inequality can be made strict except in the case of the singleton graph 
.
However, while this condition is necessary for a graph
to be connected, it is not sufficient;
an arbitrary graph satisfying the above inequality may be connected
or disconnected. In fact, the criterion is
not useful for connectedness testing since almost all disconnected graphs (with the
exception of some disjoint unions of 
and 
) also satisfy the criterion.
Directed graphs have two types of degrees, known as the indegree and the outdegree.
