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

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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

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References

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• Balakrishnan, V. K.; Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0070054894.
• Bollobas, Bela; Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0387984887.
• Diestel, Reinhard; Graph Theory, Springer; 2nd edition (February 18, 2000). ISBN 0387989765.
• Gross, Jonathon L, and Yellen, Jay; Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0849339820.
• Gross, Jonathon L, and Yellen, Jay; (eds); Handbook of Graph Theory. CRC (December 29, 2003). ISBN 1584880902.
• Zwillinger, Daniel; CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1584882913.

External links

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Graph theory

Circuit (graphe)