# Definition:Multigraph

## Definition

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

That is, $G = \struct {V, E}$ is a **multigraph** if $V$ is a set and $E$ is a multiset of doubleton subsets of $V$.

The graph above is a **multigraph** because of the double edge between $v_2$ and $v_3$ and the triple edge between $v_5$ and $v_6$.

### 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

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

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Chapter $1$: Mathematical Models: $\S 1.6$: Networks as Mathematical Models - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**graph**:**2.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**graph**:**2.**