In
graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, a loop (also called a self-loop or a ''buckle'') is an
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
that connects a
vertex to itself. A
simple graph
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Ver ...
contains no loops.
Depending on the context, a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
or a
multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by mor ...
may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing
multiple edges
In graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail verte ...
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
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call ...
, 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
In graph theory
*
Cycle (graph theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal.
A graph wit ...
*
Graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
*
Glossary of graph theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
Symbols
A
B
...
In topology
*
Möbius ladder
In graph theory, the Möbius ladder , for even numbers , is formed from an by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of (the util ...
*
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...
*
Strange loop
*
Klein bottle
In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
References
* Balakrishnan, V. K.; ''Graph Theory'', McGraw-Hill; 1 edition (February 1, 1997). .
* Bollobás, Béla; ''Modern Graph Theory'', Springer; 1st edition (August 12, 2002). .
* Diestel, Reinhard; ''Graph Theory'', Springer; 2nd edition (February 18, 2000). .
* Gross, Jonathon L, and Yellen, Jay; ''Graph Theory and Its Applications'', CRC Press (December 30, 1998). .
* Gross, Jonathon L, and Yellen, Jay; (eds); ''Handbook of Graph Theory''. CRC (December 29, 2003). .
* Zwillinger, Daniel; ''CRC Standard Mathematical Tables and Formulae'', Chapman & Hall/CRC; 31st edition (November 27, 2002). .
External links
* {{DADS, Self loop, selfloop
Graph theory objects