Multigraph

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Multigraph

In mathematics, a multigraph or pseudograph is a graph

Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
which is permitted to have multiple edges

Multiple edges

In graph theory, multiple edges , are two or more edge that are incident to the same two vertex . A simple graph has no multiple edges.Depending on the context, a graph may be defined so as to either allow or disallow the presence of 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. Formally, a multigraph G is an ordered pair

Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
G:=(V, E) with

Multigraphs might be used to model the possible flight connections offered by an airline.

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Encyclopedia

In mathematics, a multigraph or pseudograph is a graph

Graph (mathematics)

Multiple edges

Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
G:=(V, E) with

V a set of vertices or nodes,
E a multiset
Multiset

In mathematics, a multiset is a generalization of a Set . A Element of a multiset can have more than one Element , while each member of a set has only one membership....
of unordered pairs of vertices, called edges or lines.

Multigraphs might be used to model the possible flight connections offered by an airline. In this case the pseudograph would be a directed graph

Directed graph

A directed graph or digraph is a pair G= of:* a Set V, whose element are called vertices or nodes,* a set A of ordered pairs of vertices, called arcs, directed edges, or arrows....
with pairs of directed parallel edges connecting cities to show that it is possible to fly both to and from these locations.

Some authors also allow multigraphs to have loops

Loop (graph theory)

In graph theory, a loop is an edge that connects a vertex to itself. A Graph #Simple_Graph contains no loops.Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of 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

Graph (mathematics)

V a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
of vertices or nodes,
A a multiset
Multiset

In mathematics, a multiset is a generalization of a Set . A Element of a multiset can have more than one Element , while each member of a set has only one membership....
of ordered pairs of vertices called directed edges, arcs or arrows.

In category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
a small category

Category (mathematics)

In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "morphism" of any kind that have a few basic properties ....
can be defined as a multidigraph 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.

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

Labeling

Multigraphs and multidigraphs also support the notion of graph labeling

Graph labeling

In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edge or vertex , or both, of a Graph ....
, 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 nodes and arcs. Formally it is an 8-tuple where

V is a set of nodes and A is a multiset of arcs.
and are finite alphabets of the available node and arc labels,
and are two maps indicating the source and target node of an arc,
and are two maps describing the labeling of the nodes and edges.

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

Graph labeling

In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edge or vertex , or both, of a Graph ....
).

Multigraph

Labeling

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