Loop (graph theory)

A graph with a loop on vertex 1

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.

See also

Loops in Graph Theory

Loops in Topology


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