# Multigraph

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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 [1] ), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

## Contents

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.

In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges. Therefore, is a subset of , where is the power set of .

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

In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops.

## 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.

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

In mathematics, a multiset is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The positive integer number of instances, given for each element is called the multiplicity of this element in the multiset. As a consequence, an infinite number of multisets exist, which contain only elements a and b, but vary by the multiplicity of their elements:

## 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, yV}, 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, [2] while others call these pseudographs, reserving the term multigraph for the case with no loops. [3]

## 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 multigraphG := (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 G := (V, A, s, t) with

• V a set of vertices or nodes,
• A a set of edges or lines,
• ${\displaystyle s:A\rightarrow V}$, assigning to each edge its source node,
• ${\displaystyle t:A\rightarrow V}$, 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 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 ${\displaystyle G=(\Sigma _{V},\Sigma _{A},V,A,s,t,\ell _{V},\ell _{A})}$ where

• V is a set of vertices and A is a set of arcs.
• ${\displaystyle \Sigma _{V}}$ and ${\displaystyle \Sigma _{A}}$ are finite alphabets of the available vertex and arc labels,
• ${\displaystyle s\colon A\rightarrow \ V}$ and ${\displaystyle t\colon A\rightarrow \ V}$ are two maps indicating the source and target vertex of an arc,
• ${\displaystyle \ell _{V}\colon V\rightarrow \Sigma _{V}}$ and ${\displaystyle \ell _{A}\colon A\rightarrow \Sigma _{A}}$ 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).

## Notes

1. For example, see Balakrishnan 1997, p. 1 or Chartrand and Zhang 2012, p. 26.
2. For example, see Bollobás 2002, p. 7 or Diestel 2010, p. 28.
3. For example, see Wilson 2002, p. 6 or Chartrand and Zhang 2012, pp. 26-27.

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