Multigraph

 ( Extensions And Generalizations Of Graphs ) 

In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edgesFor example, see Balakrishnan 1997, p. 1 or Chartrand and Zhang 2012, p. 26.), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

There are 2 distinct notions of multiple edges:

• Edges without own identity: The identity of an edge is defined solely by the two nodes it connects. In this case, the term "multiple edges" means that the same edge can occur several times between these two nodes.
• Edges with own identity: Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different 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.

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Undirected multigraph (edges without own identity) A multigraph G is an ordered pair G := ( V, E) with

• V a set of vertices or nodes,
• E a multiset of unordered pairs of vertices, called edges or lines.

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

• V a set of vertices or nodes,
• E a set of edges or lines,
• r : E → {{ x, y} : x, y ∈ V}, assigning to each edge an unordered pair of endpoint nodes.

Some authors allow multigraphs to have loops, that is, an edge that connects a vertex to itself,For example, see Bollobás 2002, p. 7 or Diestel 2010, p. 28. while others call these pseudographs, reserving the term multigraph for the case with no loops.For example, see Wilson 2002, p. 6 or Chartrand and Zhang 2012, pp. 26-27.

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Directed multigraph (edges without own identity) 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

• V a set of vertices or nodes,
• A a multiset of ordered pairs of vertices called directed edges, arcs or arrows.

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

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Directed multigraph (edges with own identity) A multidigraph or quiver G is an ordered tuple G := ( V, A, s, t) with

• V a set of vertices or nodes,
• A a set of edges or lines,
• $s\; :\; A\; \backslash rightarrow\; V$, assigning to each edge its source node,
• $t\; :\; A\; \backslash rightarrow\; V$, assigning to each edge its target node.

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.

In category theory a small category can be defined as a multidigraph (with edges having their own identity) equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition. For this reason, in category theory the term graph is standardly taken to mean "multidigraph", and the underlying multidigraph of a category is called its underlying digraph.

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Labeling Multigraphs and multidigraphs also support the notion of graph labeling, in a similar way. However there is no unity in terminology in this case.

The definitions of labeled multigraphs and labeled multidigraphs are similar, and we define only the latter ones here.

Definition 1: A labeled multidigraph is a labeled graph with labeled arcs.

Formally: A labeled multidigraph G is a multigraph with labeled vertices and arcs. Formally it is an 8-tuple $G=(\backslash Sigma\_V,\; \backslash Sigma\_A,\; V,\; A,\; s,\; t,\; \backslash ell\_V,\; \backslash ell\_A)$ where

• $V$ is a set of vertices and $A$ is a set of arcs.
• $\backslash Sigma\_V$ and $\backslash Sigma\_A$ are finite alphabets of the available vertex and arc labels,
• $s\backslash colon\; A\backslash rightarrow\backslash \; V$ and $t\backslash colon\; A\backslash rightarrow\backslash \; V$ are two maps indicating the source and target vertex of an arc,
• $\backslash ell\_V\backslash colon\; V\backslash rightarrow\backslash Sigma\_V$ and $\backslash ell\_A\backslash colon\; A\backslash rightarrow\backslash Sigma\_A$ are two maps describing the labeling of the vertices and arcs.

Definition 2: A labeled multidigraph is a labeled graph with multiple labeled arcs, i.e. arcs with the same end vertices and the same arc label (note that this notion of a labeled graph is different from the notion given by the article graph labeling).

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See also

• Multidimensional network
• Glossary of graph theory terms
• Graph theory

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Notes

• (1997). 9780070054899, McGraw-Hill. ISBN 9780070054899
• (2025). 9780387984889, Springer. ISBN 9780387984889
• (2025). 9780486483689, Dover. ISBN 9780486483689
• (2025). 9783642142789, Springer. ISBN 9783642142789
• (1998). 9780849339820, CRC Press. ISBN 9780849339820
• (2025). 9781584880905, CRC. ISBN 9781584880905
• (1995). 9780201410334, Addison Wesley. ISBN 9780201410334
• (2025). 9780198510628, Oxford Science Publ.. . ISBN 9780198510628
• (2025). 9781584882916, Chapman & Hall/CRC. ISBN 9781584882916

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