Editing Loop (graph theory)

{{Short description|Edge that connects a node to itself}}
[[Image:6n-graph2.svg|thumb|A graph with a loop on vertex 1]]

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

Depending on the context, a [[Graph (discrete mathematics)|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 graph|bouquet]].

==Degree==
For an [[undirected graph]], the [[degree (graph theory)|degree]] of a vertex is equal to the number of [[adjacent vertex|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 (graph theory)|in degree]] and one to the [[out degree (graph theory)|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]]

==References==
{{reflist}}

* 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==
* {{DADS|Self loop|selfloop}}

[[Category:Graph theory objects]]