Loop (graph theory)

In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops.

A graph with a loop on vertex 1

Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices):

• Where graphs are defined so as to allow loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph.
• Where graphs are defined so as to disallow loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a multigraph or pseudograph.

In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet.

Degree

For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices.

A special case is a loop, which adds two to the degree. This can be understood by letting each connection of the loop edge count as its own adjacent vertex. In other words, a vertex with a loop "sees" itself as an adjacent vertex from both ends of the edge thus adding two, not one, to the degree.

For a directed graph, a loop adds one to the in degree and one to the out degree.

See also

In graph theory

• Cycle (graph theory)
• Graph theory
• Glossary of graph theory

In topology

• Möbius ladder
• Möbius strip
• Strange loop
• Klein bottle