Graph (discrete mathematics): Difference between revisions

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[[File:6n-graf.svg|thumb|A graph with six vertices and seven edges]]

In [[discrete mathematics]], particularly in [[graph theory]], a '''graph''' is a structure consisting of a [[Set (mathematics)|set]] of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''[[Vertex (graph theory)|vertices]]'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line'').<ref name=":0">{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|___location=New York|isbn=978-0-486-67870-2|pages=19|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|access-date=8 August 2012|quote=A graph is an object consisting of two sets called its ''vertex set'' and its ''edge set''.|archive-date=5 May 2019|archive-url=https://web.archive.org/web/20190505192352/http://store.doverpublications.com/0486678709.html|url-status=live}}</ref> 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.

The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated.

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=== Directed graph ===

{{main|Directed graph}}

[[File:Directed.svg|thumb|upright|A directed graph with three vertices and four directed edges, where the double arrow represents two directed edges in opposite direction.)directions]]

A '''directed graph''' or '''digraph''' is a graph in which edges have orientations.

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To avoid ambiguity, this type of object may be called precisely a '''directed multigraph'''.

A ''[[Loop (graph theory)|loop]]'' is an edge that joins a vertex to itself. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex <math>x</math> to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) <math>(x,x)</math> which is not in <math>\{(x,y) \mid (x,y) \in V^2 \;\textrm{ and }\; x \neq y \}</math>. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of <math>E</math> should be modified to <math>E \subseteq \{(x,y) \mid (x,y) \in V^2 \}</math>. For directed multigraphs, the definition of <math>\phi</math> should be modified to <math>\phi : E \to \{(x,y) \mid (x,y) \in V^2 \}</math>. To avoid ambiguity, these types of objects may be called precisely a '''directed simple graph permitting loops''' and a '''directed multigraph permitting loops''' (or a ''[[Quiver (mathematics)|quiver]]'') respectively.

The edges of a directed simple graph permitting loops {{mvar|G}} is a [[Binary relation#Homogeneous relation|homogeneous relation]] ~ on the vertices of {{mvar|G}} that is called the ''adjacency relation'' of {{mvar|G}}. Specifically, for each edge {{math|(''x'', ''y'')}}, its endpoints {{mvar|x}} and {{mvar|y}} are said to be ''adjacent'' to one another, which is denoted {{math|''x'' ~ ''y''}}.

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A ''weighted graph'' or a ''network''<ref>{{Citation | last=Strang | first=Gilbert | title=Linear Algebra and Its Applications | publisher=Brooks Cole | edition=4th | year=2005 | isbn=978-0-03-010567-8 }}</ref><ref>{{Citation | last=Lewis | first=John | title=Java Software Structures | publisher=Pearson | edition=4th | year=2013 | isbn=978-0133250121 | page=405 }}</ref> is a graph in which a number (the weight) is assigned to each edge.<ref>{{cite book|last1=Fletcher|first1=Peter|last2=Hoyle|first2=Hughes|last3=Patty|first3=C. Wayne|title=Foundations of Discrete Mathematics|year=1991|publisher=PWS-KENT Pub. Co.| ___location=Boston| isbn=978-0-53492-373-0| pages=463 | edition=International student|quote=A ''weighted graph'' is a graph in which a number ''w''(''e''), called its ''weight'', is assigned to each edge ''e''.}}</ref> Such weights might represent for example costs, lengths or capacities, depending on the problem at hand. Such graphs arise in many contexts, for example in [[shortest path problem]]s such as the [[traveling salesman problem]].

== GraphTypes typesof graphs ==

=== Oriented graph ===

One definition of an ''oriented graph'' is that it is a directed graph in which at most one of {{nowrap|(''x'', ''y'')}} and {{nowrap|(''y'', ''x'')}} may be edges of the graph. That is, it is a directed graph that can be formed as an [[orientation (graph theory)|orientation]] of an undirected (simple) graph.

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* [[strongly regular graph]]s and their generalizations [[distance-regular graph]]s.

== GraphProperties propertiesof graphs ==

{{see also|Glossary of graph theory|Graph property}}

Two edgesvertices of a graph are called ''adjacent'' if they share a common vertexedge. Two edgesvertices of a directed graph are called ''consecutive'' if the head of the first one is the tail of the second one. Similarly, two vertices are called ''adjacent'' if they share a common edge (''consecutive'' if the first one is the tail and the second one is the head of an edge), in which case the common edge is said to ''join'' the two vertices. An edge and a vertex on that edge are called ''incident''.

The graph with only one vertex and no edges is called the ''trivial graph''. A graph with only vertices and no edges is known as an ''edgeless graph''. The graph with no vertices and no edges is sometimes called the ''[[null graph]]'' or ''empty graph'', but the terminology is not consistent and not all mathematicians allow this object.

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Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called ''vertex-labeled''. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called ''edge-labeled''. Graphs with labels attached to edges or vertices are more generally designated as ''labeled''. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called ''unlabeled''. (In the literature, the term ''labeled'' may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)

The [[category theory|category]] of alldirected graphsmultigraphs permitting loops is the [[comma category]] Set ↓ ''D'' where ''D'': Set → Set is the [[functor]] taking a set ''s'' to ''s'' × ''s''.

== Examples ==

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* The diagram is a schematic representation of the graph with vertices <math>V = \{1, 2, 3, 4, 5, 6\}</math> and edges <math>E = \{\{1, 2\}, \{1, 5\}, \{2, 3\}, \{2, 5\}, \{3, 4\}, \{4, 5\}, \{4, 6\}\}.</math>

* In [[computer science]], directed graphs are used to represent knowledge (e.g., [[conceptual graph]]), [[finite -state machine]]s, and many other discrete structures.

* A [[binary relation]] ''R'' on a set ''X'' defines a directed graph. An element ''x'' of ''X'' is a direct predecessor of an element ''y'' of ''X'' if and only if ''xRy''.

* A directed graph can model information networks such as [[Twitter]], with one user following another.<ref name="snatwitter">{{Cite journal| volume = 3| issue = 1| last = Grandjean| first = Martin| title = A social network analysis of Twitter: Mapping the digital humanities community| journal = Cogent Arts & Humanities| date = 2016| pages = 1171458| doi = 10.1080/23311983.2016.1171458| url = https://serval.unil.ch/resource/serval:BIB_81C2C68B1DF5.P001/REF| doi-access = free| access-date = 2019-09-16| archive-date = 2021-03-02| archive-url = https://web.archive.org/web/20210302190117/https://serval.unil.ch/resource/serval:BIB_81C2C68B1DF5.P001/REF| url-status = live}}</ref><ref name="twitterwtf">Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh Zadeh [http://dl.acm.org/citation.cfm?id=2488433 WTF: The who-to-follow system at Twitter] {{Webarchive|url=https://web.archive.org/web/20190712002903/http://dl.acm.org/citation.cfm?id=2488433 |date=2019-07-12 }}, ''Proceedings of the 22nd international conference on World Wide Web''. {{doi|10.1145/2488388.2488433}}.</ref>