Multiple edges

Multiple edges joining two vertices.

In graph theory, multiple edges (also called parallel edges or a multi-edge), are two or more edges that are incident to the same two vertices. A simple graph has no multiple edges.

Depending on the context, a graph may be defined so as to either allow or disallow the presence of multiple edges (often in concert with allowing or disallowing loops):

• Where graphs are defined so as to allow multiple edges and loops, a graph without loops is often called a multigraph.^[1]
• Where graphs are defined so as to disallow multiple edges and loops, a multigraph or a pseudograph is often defined to mean a "graph" which can have loops and multiple edges.^[2]

Multiple edges are, for example, useful in the consideration of electrical networks, from a graph theoretical point of view.^[3]

A planar graph remains planar if an edge is added between two vertices already joined by an edge, thus adding multiple edges preserves planarity.^[4]

Notes

1. ^ For example, see Balakrishnan, p. 1, and Gross (2003), p. 4, Zwillinger, p. 220.
2. ^ For example, see. Bollobas, p. 7, Diestel, p. 25, and Harary, p. 10.
3. ^ Bollobas, pp. 39, 40.
4. ^ Gross (1998), p. 308.

References

• Balakrishnan, V. K.; Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4.
• Bollobas, Bela; 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.