Loop (graph theory)

In graph theory, a loop (also called a self-loop) 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 is often called a multigraph. [ For example, see Balakrishnan, p. 1, and Gross (2003), p. 4, Zwillinger, p. 220.]
*Where graphs are defined so as to "disallow" loops and multiple edges, a multigraph or a pseudograph is often defined to mean a "graph" which "can" have loops and multiple edges. [For example, see. Bollobas, p. 7, Diestel, p. 25, and Harary, p. 10.]

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

Notes

References

* Balakrishnan, V. K.; "Graph Theory", McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4.
* Bollobas, Bela; "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

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ee also

* Cycle (graph theory)
* List of cycles

Loops in Topology
* Möbius ladder
* Möbius strip
* Strange loop
* Klein bottle