selfloops

Selfloops, or loops, are edges that connect a vertex to itself in a graph. In undirected graphs the loop has both endpoints at the same vertex, while in directed graphs the loop is an edge from the vertex to itself. Graphs that allow loops are often called pseudographs; when multiple edges between vertices are also allowed, the term multigraph is used, though conventions vary.

In an undirected graph a loop contributes two to the degree of its vertex; in a directed graph a loop increases both in-degree and out-degree by one. In adjacency matrices, a loop is represented by a nonzero diagonal entry (usually 1) to indicate the presence of the edge.

Loops affect algorithms and properties. Some definitions of a simple graph disallow loops; many algorithms can be adapted by ignoring loops or removing them before processing.

In modeling, loops capture self-influence or persistence. In automata, a state with a self-loop means the machine can stay in that state after reading a particular symbol; in Markov chains, a loop represents staying in the same state with a given probability.

Examples help illustrate their use. A social network might include a self-loop if a node represents a person who is considered connected to themselves in the model. In practice, whether loops are included often depends on the modeling goals and the specific computational framework being used.