Labeled Multigraph
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is,
edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
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: * ''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.


Undirected multigraph (edges without own identity)

A multigraph ''G'' is an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
''G'' := (''V'', ''E'') with *''V'' a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''vertices'' or ''nodes'', *''E'' a multiset of unordered pairs of vertices, called ''edges'' or ''lines''.


Undirected multigraph (edges with own identity)

A multigraph ''G'' is an ordered
triple Triple is used in several contexts to mean "threefold" or a "treble": Sports * Triple (baseball), a three-base hit * A basketball three-point field goal * A figure skating jump with three rotations * In bowling terms, three strikes in a row * In ...
''G'' := (''V'', ''E'', ''r'') with *''V'' a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''vertices'' or ''nodes'', *''E'' a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''edges'' or ''lines'', *''r'' : ''E'' → , 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, 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.


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.


Directed multigraph (edges with own identity)

A multidigraph or quiver ''G'' is an ordered
4-tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
''G'' := (''V'', ''A'', ''s'', ''t'') with *''V'' a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''vertices'' or ''nodes'', *''A'' a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of ''edges'' or ''lines'', *s : A \rightarrow V, assigning to each edge its source node, *t : A \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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
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.


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=(\Sigma_V, \Sigma_A, V, A, s, t, \ell_V, \ell_A) where *V is a set of vertices and A is a set of arcs. *\Sigma_V and \Sigma_A are finite alphabets of the available vertex and arc labels, *s\colon A\rightarrow\ V and t\colon A\rightarrow\ V are two maps indicating the ''source'' and ''target'' vertex of an arc, *\ell_V\colon V\rightarrow\Sigma_V and \ell_A\colon A\rightarrow\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).


See also

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


Notes


References

* * * * * * * * * *


External links

* {{DADS, Multigraph, multigraph Extensions and generalizations of graphs de:Graph (Graphentheorie)#Multigraph