Directed multigraph

In mathematics, and more specifically in graph theory, a multigraph (in contrast to a simple graph) is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

There are two distinct notions of multiple edges:

A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two.

For some authors, the terms pseudograph and multigraph are synonymous. For others, a pseudograph is a multigraph that is permitted to have loops.

A multigraph G is an ordered pair G:=(V, E) with

A multigraph G is an ordered triple G:=(V, E, r) with

Some authors allow multigraphs to have loops, that is, an edge that connects a vertex to itself, while others call these pseudographs, reserving the term multigraph for the case with no loops.

A multidigraph is a directed graph which is permitted to have multiple arcs, i.e., arcs with the same source and target nodes. A multidigraph G is an ordered pair G:=(V,A) with

A mixed multigraph G:=(V,E, A) may be defined in the same way as a mixed graph.

A multidigraph or quiver G is an ordered 4-tuple G:=(V, A, s, t) with

This notion might be used to model the possible flight connections offered by an airline. In this case the multigraph would be a directed graph with pairs of directed parallel edges connecting cities to show that it is possible to fly both to and from these locations.

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