Definition:Multigraph

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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 $V$.

Multigraph.png

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

Let $G = \struct {V, E}$ be a multigraph.


A multiple edge is an edge of $G$ which has another edge with the same endvertices.

That is, where there is more than one edge that joins any pair of vertices, each of those edges is called a multiple edge.


Simple Edge

Let $G = \struct {V, E}$ be a multigraph.


A simple edge is an edge $u v$ of $G$ which is the only edge of $G$ which is incident to both $u$ and $v$.


Multiplicity

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


The multiplicity of the above example is $3$.


Also defined as

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.


Also see

  • Results about multigraphs can be found here.


Sources