Definition:Multigraph

Definition
A multigraph is a graph that can have more than one edge between a pair of vertices.

That is, $$G = \left({V, E}\right)$$ is a multigraph if $$V$$ is a set and $$E$$ is a multiset of 2-element subsets of $$V$$.



The graph above is a multigraph because of the double edge between $$B$$ and $$C$$ and the triple edge between $$E$$ and $$F$$.

Multiple Edge
Where there is more than one edge between any pair of vertices, each of those edges is called a multiple edge.

Multipliticy
The multiplicity of a multigraph is the maximum multiplicity of its (multiple) edges.

The multiplicity of the above example is $$3$$.

Note
Some sources differ on whether a multigraph must or only may contain multiple edges.

Similarly, sources differ on whether a multigraph may contain loops, and whether a loop counts as a double edge.

If there is any ambiguity, and especially if it matters to the proof, these conditions should be specified.