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.

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.

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.

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

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

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• Bollobás, Béla; Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7.
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•  This article incorporates public domain material from Paul E. Black. "Self loop". Dictionary of Algorithms and Data Structures. NIST.