outdegrees

In graph theory, the outdegree of a vertex in a directed graph is the number of edges that originate at that vertex. Formally, a directed graph G = (V, E) assigns to each vertex v ∈ V a nonnegative integer outdeg(v) = |{(v, u) ∈ E : u ∈ V}|. Each outgoing edge contributes one to the outdegree, and multiple edges or parallel edges are counted separately. A loop from a vertex to itself contributes one to its outdegree as well.

Notation and representations commonly use deg+(v) or outdeg(v) to denote the outdegree. In an adjacency list, the outdegree of a vertex is simply the length of its list of successors. In an adjacency matrix A, where A[i][j] indicates the presence (and possibly weight) of an edge from i to j, the outdegree of vertex i is the sum of the i-th row: ∑_j A[i][j]. For weighted graphs, the term out-strength is sometimes used to refer to the sum of the weights of outgoing edges rather than the count of outgoing edges.

Key properties include that the sum of outdegrees over all vertices equals the total number of edges in the graph (counted with multiplicity, including loops). A directed graph is called out-regular if every vertex has the same outdegree k; similarly, in-regular describes uniform in-degree. The outdegree distribution is often studied in network analysis, where it relates to growth processes, hub structure, and dynamics such as random walks, where transition choices are typically uniform among outbound edges.