# Multiple edges

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] Additionally, they constitute the core differentiating feature of multidimensional networks.

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]

A dipole graph is a graph with two vertices, in which all edges are parallel to each other.

## Notes

## References

- 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.