---
tag: 01M5T0KDG7
kind: definition
name: Multigraph
msc: [05C99]
defines: 0C62PZXX02
uses: [0JD5C82QMB]
sources:
  - url: https://en.wikipedia.org/wiki/Multigraph
    locator: Undirected multigraph (edges with own identity)
---

## Definition

A multigraph is a triple $G = (V, E, r)$ of a set $V$ of vertices, a set $E$ of edges, and a map $r$.
The map $r$ assigns to each edge $e \in E$ a set $r(e) \subseteq V$ of one or two vertices, the ends of $e$.
Two distinct edges may have the same ends. Such edges are parallel, or multiple, edges.
An edge with only one end is a loop.

## Remarks

Some authors do not allow loops in a multigraph, and they call a multigraph with loops a pseudograph.
A [[graph|0JD5C82QMB]] is a multigraph without loops and without parallel edges.
The [[Seven Bridges of Königsberg|0YCW781Q9D]] form a multigraph with $4$ vertices and $7$ edges.

## Explanations

### L1 human

A multigraph is like a graph, but two points can be joined by several lines.
A road map with two different roads between the same two towns is a multigraph.