# 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.

## 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

**Loops in Graph Theory**

**Loops in Topology**

## References

- Balakrishnan, V. K.;
*Graph Theory*, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4. - Bollobás, Béla;
*Modern Graph Theory*, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7. - Diestel, Reinhard;
*Graph Theory*, Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5. - Gross, Jonathon L, and Yellen, Jay;
*Graph Theory and Its Applications*, CRC Press (December 30, 1998). ISBN 0-8493-3982-0. - Gross, Jonathon L, and Yellen, Jay; (eds);
*Handbook of Graph Theory*. CRC (December 29, 2003). ISBN 1-58488-090-2. - Zwillinger, Daniel;
*CRC Standard Mathematical Tables and Formulae*, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.

## External links

This article incorporates public domain material from the NIST document: Black, Paul E. "Self loop". *Dictionary of Algorithms and Data Structures*.