Multiple edges

In

simple graph has no multiple edges and no loops

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 or multiple edges is often distinguished from other graphs by calling it a simple graph.^[1]
Where graphs are defined so as to disallow multiple edges and loops, a
pseudograph is often defined to mean a "graph" which can have 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

^ For example, see Balakrishnan, p. 1, and Gross (2003), p. 4, Zwillinger, p. 220.
^ For example, see Bollobás, p. 7; Diestel, p. 28; Harary, p. 10.
^ Bollobás, pp. 39–40.
^ Gross (1998), p. 308.