Loop (graph theory)

In

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

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

See also

In graph theory

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

In topology

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

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
  .