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{{about|sets of vertices connected by edges|graphs of mathematical functions|Graph of a function|other uses|Graph (disambiguation)}}

{{short description|Vertices connected in pairs by edges}}

[[File:6n-graf.svg|thumb|250px|A [[graph drawing|drawing]] of a [[labeled graph]] on 6 vertices and 7 edges.]]

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

In [[discrete mathematics]], and more specificallyparticularly in [[graph theory]], a '''graph''' is a structure amountingconsisting toof a [[Set (mathematics)|set]] of objects in whichwhere some pairs of the objects are in some sense "related". The objects correspondare torepresented mathematicalby 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 an ''arclink'' or ''line'').<ref>{{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.|accessdateaccess-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. Graphs are one of the objects of study in [[discrete mathematics]].

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 anyan edge from a person ''A'' to a person ''B'' correspondsmeans tothat ''A'''s admiringowes money to ''B'', then this graph is directed, because admirationowing money is not necessarily reciprocated. The former type of graph is called an ''undirected graph'' and the edges are called ''undirected edges'' while the latter type of graph is called a ''directed graph'' and the edges are called ''directed edges''.

Graphs are the basic subject studied by [[graph theory]]. The word "graph" was first used in this sense by [[James Joseph Sylvester|J. J. Sylvester]] in 1878 due to a direct relation between mathematics and [[chemical structure]] (what he called a chemico-graphical image).<ref>See:

* J. J. Sylvester (February 7, 1878) [https://books.google.com/books?id=KcoKAAAAYAAJ&vqq=Sylvester&pg=PA284#v=onepage&q&f=false "Chemistry and algebra,"], {{Webarchive|url=https://web.archive.org/web/20230204142956/https://books.google.com/books?id=KcoKAAAAYAAJ&vq=Sylvester&pg=PA284 |date=2023-02-04 }} ''Nature'', ''17'' : 284. {{doi|10.1038/017284a0}}. From page 284: "Every invariant and covariant thus becomes expressible by a ''graph'' precisely identical with a Kekuléan diagram or chemicograph."

* J. J. Sylvester (1878) [https://books.google.com/books?id=1q0EAAAAYAAJ&pg=PA64#v=onepage&q&f=false "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, —– with three appendices,"], {{Webarchive|url=https://web.archive.org/web/20230204142957/https://books.google.com/books?id=1q0EAAAAYAAJ&pg=PA64 |date=2023-02-04 }} ''American Journal of Mathematics, Pure and Applied'', ''1'' (1) : 64–90. {{doi|10.2307/2369436}}. {{JSTOR|2369436}}. The term "graph" first appears in this paper on page 65.</ref><ref>{{Cite book

| title = Handbook of graph theory

| first1 = Jonathan L.

Line 16 ⟶ 17:

| publisher = [[CRC Press]]

| year = 2004

| page = [https://books.google.com/books?id=mKkIGIea_BkC&pg=PA35&lpg=PA35 35]

| isbn = 978-1-58488-090-5

| url = https://books.google.com/books?id=mKkIGIea_BkC

| postscriptaccess-date = <!--None2016-02->}}</ref>16

| archive-date = 2023-02-04

| archive-url = https://web.archive.org/web/20230204142959/https://books.google.com/books?id=mKkIGIea_BkC

| url-status = live

}}</ref>

== Definitions ==

Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related [[mathematical structuresstructure]]s.

=== {{anchor|Undirected graph}} Graph ===

[[File:Undirected.svg|thumb|upright|A graph with three vertices and three edges]]

In one very common sense of the term,<ref>See, for instance, Iyanaga and Kawada, ''69 J'', p. 234 or Biggs, p. 4.</ref> a ''graph'' is an [[ordered pair]] {{nobreak|1=''G'' = (''V'', ''E'')}} comprising a [[set (mathematics)|set]] ''V'' of ''vertices'', ''nodes'' or ''points'' together with a set ''E'' of ''edges'', ''arcs'' or ''lines'', which are 2-element subsets of ''V'' (i.e., an edge is associated with two vertices, and the association takes the form of the [[unordered pair]] of the vertices ). To avoid ambiguity, this type of graph may be described precisely as ''[[Graph (discrete mathematics)#Undirected graph|undirected]]'' and ''[[Graph (discrete mathematics)#Simple graph|simple]]''.

OtherA senses'''graph''' of(sometimes called an ''undirected graph'' stemto distinguish it from differenta conceptions[[#Directed ofgraph|directed thegraph]], edgeor set.a In''simple onegraph'' moreto generaldistinguish conception,it from a [[multigraph]]){{sfn|Bender|Williamson|2010|p=148}}<ref>See, for instance, GrahamIyanaga etand alKawada, ''69 J'', p. 234 or Biggs, p. 54.</ref> ''E'' is a set[[ordered togetherpair|pair]] with{{math|1=''G'' a= relation of(''V'', ''incidenceE'')}}, thatwhere associates{{mvar|V}} withis eacha edgeset twowhose vertices.elements Inare another generalized notion,called ''Evertices'' (singular: vertex), and {{mvar|E}} is a [[multiset]]set of unordered pairs of<math>\{v_1, (notv_2\}</math> necessarily distinct)of vertices., Manywhose authorselements callare thesecalled types''edges'' of(sometimes object [[multigraph]]s''links'' or pseudographs''lines'').

The vertices {{mvar|u}} and {{mvar|v}} of an edge {{math|{''u'', ''v''} }} are called the edge's ''endpoints''. The edge is said to ''join'' {{mvar|u}} and {{mvar|v}} and to be ''incident'' on them. A vertex may belong to no edge, in which case it is not joined to any other vertex and is called ''isolated''. When an edge <math>\{u,v\}</math> exists, the vertices {{mvar|u}} and {{mvar|v}} are called ''adjacent''.

All of these variants and others are described more fully below.

A [[multigraph]] is a generalization that allows multiple edges to have the same pair of endpoints. In some texts, multigraphs are simply called graphs.{{sfn|Bender|Williamson|2010|p=149}}<ref>Graham et al., p. 5.</ref>

The vertices belonging to an edge are called the ''ends'' or ''end vertices'' of the edge. A vertex may exist in a graph and not belong to an edge.

Sometimes, graphs are allowed to contain ''[[Loop (graph theory)|loop]]s'', which are edges that join a vertex to itself. To allow loops, the pairs of vertices in {{mvar|E}} must be allowed to have the same node twice. Such generalized graphs are called ''graphs with loops'' or simply ''graphs'' when it is clear from the context that loops are allowed.

''V'' and ''E'' are usually taken to be finite, and many of the well-known results are not true (or are rather different) for ''infinite graphs'' because many of the arguments fail in the [[infinite graph|infinite case]]. Moreover, ''V'' is often assumed to be non-empty, but ''E'' is allowed to be the empty set. The ''order'' of a graph is |''V''|, its number of vertices. The ''size'' of a graph is |''E''|, its number of edges. The ''degree'' or ''valency'' of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends (a [[Loop (graph theory)|loop]]) is counted twice.

Generally, the vertex set {{mvar|V}} is taken to be finite (which implies that the edge set {{mvar|E}} is also finite). Sometimes [[infinite graph]]s are considered, but they are usually viewed as a special kind of [[binary relation]], because most results on finite graphs either do not extend to the infinite case or need a rather different proof.

For an edge {{nobreak|{''x'', ''y''}}}, graph theorists usually use the somewhat shorter notation ''xy''.

An [[empty graph]] is a graph that has an [[empty set]] of vertices (and thus an empty set of edges). The ''order'' of a graph is its number {{math|{{abs|''V''}}}} of vertices, usually denoted by {{mvar|n}}. The ''size'' of a graph is its number {{math|{{abs|''E''}}}} of edges, typically denoted by {{mvar|m}}. However, in some contexts, such as for expressing the [[computational complexity]] of algorithms, the term ''size'' is used for the quantity {{math|{{abs|''V''}} + {{abs|''E''}}}} (otherwise, a non-empty graph could have size 0). The ''degree'' or ''valency'' of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice.

===Adjacency relation===

The edges ''E'' of an undirected graph ''G'' induce a symmetric binary relation ~ on ''V'' that is called the ''adjacency relation'' of ''G''. Specifically, for each edge {{nobreak|{''x'', ''y''}}}, the vertices ''x'' and ''y'' are said to be ''adjacent'' to one another, which is denoted {{nobreak|''x'' ~ ''y''}}.

In a graph of order {{math|''n''}}, the maximum degree of each vertex is {{math|''n'' − 1}} (or {{math|''n'' + 1}} if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is {{math|''n''(''n'' − 1)/2}} (or {{math|''n''(''n'' + 1)/2}} if loops are allowed).

==Types of graphs==

The edges of a graph define a [[symmetric relation]] on the vertices, called the ''adjacency relation''. Specifically, two vertices {{mvar|x}} and {{mvar|y}} are ''adjacent'' if {{math|{''x'', ''y''} }} is an edge. A graph is fully determined by its [[adjacency matrix]] {{mvar|A}}, which is an {{math|''n'' × ''n''}} square matrix, with {{mvar|A{{sub|ij}}}} specifying the number of connections from vertex {{mvar|i}} to vertex {{mvar|j}}. For a simple graph, {{math|''A{{sub|ij}}''}} is either 0, indicating disconnection, or 1, indicating connection; moreover {{math|1=''A{{sub|ii}}'' = 0}} because an edge in a simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all {{mvar|A{{sub|ii}}}} being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all {{mvar|A{{sub|ij}}}} being equal to a positive integer. Undirected graphs will have a [[symmetric matrix|symmetric]] adjacency matrix (meaning {{math|1=''A{{sub|ij}}'' = ''A{{sub|ji}}''}}).

===Distinction in terms of the main definition===

As stated above, in different contexts it may be useful to refine the term ''graph'' with different degrees of generality. Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise, ''graph'' means "undirected simple finite graph" (see the definitions below).

=== Directed graph ===

{{multiple image

| right

| footer =

| width1 = 125

| image1 = Directed.svg

| caption1 = A directed graph.

| width2 = 125

| image2 = Undirected.svg

| caption2 = A [[Graph (discrete mathematics)#Simple graph|simple]] undirected graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph.

}}

====Undirected graph====

An ''undirected graph'' is a graph in which edges have no orientation. The edge {{nobreak|(''x'', ''y'')}} is identical to the edge {{nobreak|(''y'', ''x'')}}, i.e., they are not ordered pairs, but sets {{nobreak|{''x'', ''y''}}} (or 2-multisets) of vertices. The maximum number of edges in an undirected graph without a loop is {{nobreak|''n''(''n'' − 1)/2}}.

====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 directions]]

A ''directed graph'' or ''digraph'' is a graph in which edges have orientations. It is written as an ordered pair {{nobreak|1=''G'' = (''V'', ''A'')}} (sometimes {{nobreak|1=''G'' = (''V'', ''E'')}}) with

* ''V'' a [[Set (mathematics)|set]] whose [[Element (mathematics)|elements]] are called ''vertices'', ''nodes'', or ''points'';

* ''A'' a set of [[ordered pair]]s of vertices, called ''arrows'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''directed arcs'', or ''directed lines''.

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

An arrow {{nobreak|(''x'', ''y'')}} is considered to be directed ''from'' ''x'' ''to'' ''y''; ''y'' is called the ''head'' and ''x'' is called the ''tail'' of the arrow; ''y'' is said to be a ''direct successor'' of ''x'' and ''x'' is said to be a ''direct predecessor'' of ''y''. If a [[Path (graph theory)|path]] leads from ''x'' to ''y'', then ''y'' is said to be a ''successor'' of ''x'' and ''reachable'' from ''x'', and ''x'' is said to be a ''predecessor'' of ''y''. The arrow {{nobreak|(''y'', ''x'')}} is called the ''inverted arrow'' of {{nobreak|(''x'', ''y'')}}.

In one restricted but very common sense of the term,{{sfn|Bender|Williamson|2010|p=161}} a '''directed graph''' is a pair {{math|1=''G'' = (''V'', ''E'')}} comprising:

A directed graph ''G'' is called ''symmetric'' if, for every arrow in ''G'', the corresponding inverted arrow also belongs to ''G''. A symmetric loopless directed graph {{nobreak|1=''G'' = (''V'', ''A'')}} is equivalent to a simple undirected graph {{nobreak|1=''G′'' = (''V'', ''E'')}}, where the pairs of inverse arrows in ''A'' correspond one-to-one with the edges in ''E''; thus the number of edges in ''G′'' is {{nobreak|1={{abs|''E'' }} = {{abs|''A'' }}/2}}, that is half the number of arrows in ''G''.

* {{mvar|V}}, a [[Set (mathematics)|set]] of ''vertices'' (also called ''nodes'' or ''points'');

* {{mvar|E}}, a [[Set (mathematics)|set]] of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'', or ''arcs''), which are [[ordered pair]]s of distinct vertices: <math>E \subseteq \{(x,y) \mid (x,y) \in V^2 \;\textrm{ and }\; x \neq y \}</math>.

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

In the edge {{math|(''x'', ''y'')}} directed from {{mvar|x}} to {{mvar|y}}, the vertices {{mvar|x}} and {{mvar|y}} are called the ''endpoints'' of the edge, {{mvar|x}} the ''tail'' of the edge and {{mvar|y}} the ''head'' of the edge. The edge is said to ''join'' {{mvar|x}} and {{mvar|y}} and to be ''incident'' on {{mvar|x}} and on {{mvar|y}}. A vertex may exist in a graph and not belong to an edge. The edge {{math|(''y'', ''x'')}} is called the ''inverted edge'' of {{math|(''x'', ''y'')}}. ''[[Multiple edges]]'', not allowed under the definition above, are two or more edges with both the same tail and the same head.

====Oriented graph====

An ''oriented graph'' is a directed graph in which at most one of {{nobreak|(''x'', ''y'')}} and {{nobreak|(''y'', ''x'')}} may be arrows of the graph. That is, it is a directed graph that can be formed as an [[orientation (graph theory)|orientation]] of an undirected graph. However, some authors use "oriented graph" to mean the same as "directed graph".

In one more general sense of the term allowing multiple edges,{{sfn|Bender|Williamson|2010|p=161}} a directed graph is sometimes defined to be an ordered triple {{math|1=''G'' = (''V'', ''E'', ''ϕ'')}} comprising:

====Mixed graph====

* {{mvar|V}}, a [[Set (mathematics)|set]] of ''vertices'' (also called ''nodes'' or ''points'');

{{main|Mixed graph}}

* {{mvar|E}}, a [[Set (mathematics)|set]] of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs'');

A ''mixed graph'' is a graph in which some edges may be directed and some may be undirected. It is written as an ordered triple {{nobreak|1=''G'' = (''V'', ''E'', ''A'')}} with ''V'', ''E'', and ''A'' defined as above. Directed and undirected graphs are special cases.

* {{mvar|ϕ}}, an ''incidence function'' mapping every edge to an [[ordered pair]] of vertices (that is, an edge is associated with two distinct vertices): <math>\phi : E \to \{(x,y) \mid (x,y) \in V^2 \;\textrm{ and }\; x \neq y \}</math>.

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

====Multigraph====

{{main|Multigraph}}

''[[Multiple edge]]s'' are two or more edges that connect the same two vertices. A ''[[Loop (graph theory)|loop]]'' is an edge (directed or undirected) that connects a vertex to itself; it may be permitted or not, according to the application. In this context, an edge with two different ends is called a ''link''.

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 V^2</math>. For directed multigraphs, the definition of <math>\phi</math> should be modified to <math>\phi : E \to 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.

A ''multigraph'', as opposed to a [[#Simple graphs|simple graph]], is an undirected graph in which multiple edges (and sometimes loops) are allowed.

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''}}.

Where graphs are defined so as to ''disallow'' both multiple edges and loops, a multigraph is often defined to mean a graph which can have both multiple edges and loops,<ref>For example, see. Bollobás, p. 7 and Diestel, p. 25.</ref> although many use the term ''[[pseudograph]]'' for this meaning.<ref>Gross (1998), p. 3, Gross (2003), p. 205, Harary, p.10, and Zwillinger, p. 220.</ref> Where graphs are defined so as to ''allow'' both multiple edges and loops, a multigraph is often defined to mean a graph without loops.<ref>For example, see Balakrishnan, p. 1, Gross (2003), p. 4, and Zwillinger, p. 220.</ref>

====Simple Mixed graph= ===

{{main|Mixed graph}}

A simple graph is an undirected graph in which both multiple edges and loops are disallowed. In a simple graph the edges form a ''set'' (rather than a [[multiset]]) and each edge is an unordered pair of ''distinct'' vertices. Thus, we can define a '''simple graph''' to be a set ''V'' of vertices together with a subset ''E'' of the set of 2-element subsets of ''V''.

[[File:Example of simple mixed graph.jpg|thumb|upright|A mixed graph with three vertices, two directed edges, and an undirected edge.]]

A ''mixed graph'' is a graph in which some edges may be directed and some may be undirected. It is an ordered triple {{math|1=''G'' = (''V'', ''E'', ''A'')}} for a ''mixed simple graph'' and {{math|1=''G'' = (''V'', ''E'', ''A'', ''ϕ{{sub|E}}'', ''ϕ{{sub|A}}'')}} for a ''mixed multigraph'' with {{mvar|V}}, {{mvar|E}} (the undirected edges), {{mvar|A}} (the directed edges), {{mvar|ϕ{{sub|E}}}} and {{mvar|ϕ{{sub|A}}}} defined as above. Directed and undirected graphs are special cases.

In a simple graph with ''n'' vertices, the degree of every vertex is at most {{nobreak|''n'' − 1}}.

====Quiver= Weighted graph ===

[[File:Weighted_network.svg|thumb|upright=1.2|A weighted graph with ten vertices and twelve edges]]

{{main|Quiver (mathematics)}}

A ''quiver'' or ''multidigraph'' is a directed multigraph. A quiver may have directed loops in it. Thus, a quiver is a set ''V'' of vertices, a set ''E'' of edges, and two functions <math>s: E \to V</math>, <math>t: E \to V</math>. The map ''s'' assigns to each edge its ''source'' (or ''tail''), while the map ''t'' assigns to each edge its ''target'' (or ''head'').

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]].

====Weighted graph====

A ''weighted graph'' 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=0-53492-373-9| 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. Some authors call such a graph a ''network''.<ref>{{Citation | last=Strang | first=Gilbert | title=Linear Algebra and Its Applications | publisher=Brooks Cole | edition=4th | year=2005 | isbn=0-03-010567-6 | url=https://books.google.com/books?vid=ISBN0030105676 }}</ref> [[Weighted correlation network analysis|Weighted correlation networks]] can be defined by soft-thresholding the pairwise correlations among variables (e.g. gene measurements). Such graphs arise in many contexts, for example in [[shortest path problem]]s such as the [[traveling salesman problem]].

== Types of graphs ==

====Half-edges, loose edges====

=== Oriented graph ===

In certain situations it can be helpful to allow edges with only one end, called ''half-edges'', or no ends, called ''loose edges''; see the articles [[Signed graph]]s and [[Biased graph]]s.

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.

Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a given undirected graph or multigraph.

===Important classes of graph===

==== Regular graph= ===

{{main|Regular graph}}

A ''regular graph'' is a graph in which each vertex has the same number of neighbours, i.e., every vertex has the same degree. A regular graph with vertices of degree ''k'' is called a ''k''‑regular graph or regular graph of degree ''k''.

==== Complete graph= ===

{{main|Complete graph}}

[[File:Complete graph K5.svg|thumb|125pxupright|A complete graph with 5five vertices and ten edges. Each vertex has an edge to every other vertex.]]

A ''complete graph'' is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.

==== Finite graph= ===

A ''finite graph'' is a graph in which the vertex set and the edge set are [[finite set]]s. Otherwise, it is called an ''infinite graph''.

Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.

==== Connected graph= ===

{{main|Connectivity (graph theory)}}

In an undirected graph, an unordered pair of vertices {{nobreaknowrap|{{mset|''x'', ''y''}}}} is called ''connected'' if a path leads from ''x'' to ''y''. Otherwise, the unordered pair is called ''disconnected''.

A ''connected graph'' is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a ''disconnected graph''.

In a directed graph, an ordered pair of vertices {{nobreaknowrap|(''x'', ''y'')}} is called ''strongly connected'' if a directed path leads from ''x'' to ''y''. Otherwise, the ordered pair is called ''weakly connected'' if an undirected path leads from ''x'' to ''y'' after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called ''disconnected''.

A ''strongly connected graph'' is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a ''weakly connected graph'' if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a ''disconnected graph''.

A ''[[k-vertex-connected graph]]'' or ''[[k-edge-connected graph]]'' is a graph in which no set of {{nobreaknowrap|''k'' − 1}} vertices (respectively, edges) exists that, when removed, disconnects the graph. A ''k''-vertex-connected graph is often called simply a ''k-connected graph''.

==== Bipartite graph= ===

{{main|Bipartite graph}}

A ''[[bipartite graph]]'' is a simple graph in which the vertex set can be [[Partition of a set|partitioned]] into two sets, ''W'' and ''X'', so that no two vertices in ''W'' share a common edge and no two vertices in ''X'' share a common edge. Alternatively, it is a graph with a [[chromatic number]] of 2.

In a [[complete bipartite graph]], the vertex set is the union of two disjoint sets, ''W'' and ''X'', so that every vertex in ''W'' is adjacent to every vertex in ''X'' but there are no edges within ''W'' or ''X''.

==== Path graph= ===

{{main|Path graph}}

A ''path graph'' or ''linear graph'' of order {{nobreaknowrap|''n'' ≥ 2}} is a graph in which the vertices can be listed in an order ''v''₁, ''v''₂, …, ''v''_''n'' such that the edges are the {{nobreaknowrap|{{mset|''v''_''i'', ''v''_''i''+1}}}} where ''i'' = 1, 2, …, ''n'' − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a [[Glossary of graph theory#Subgraphs|subgraph]] of another graph, it is a [[Path (graph theory)|path]] in that graph.

==== Planar graph= ===

{{main|Planar graph}}

A ''planar graph'' is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.

==== Cycle graph= ===

{{main|Cycle graph}}

A ''cycle graph'' or ''circular graph'' of order {{nobreaknowrap|''n'' ≥ 3}} is a graph in which the vertices can be listed in an order ''v''₁, ''v''₂, …, ''v''_''n'' such that the edges are the {{nobreaknowrap|{{mset|''v''_''i'', ''v''_''i''+1}}}} where ''i'' = 1, 2, …, ''n'' − 1, plus the edge {{nobreaknowrap|{{mset|''v''_''n'', ''v''₁}}}}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.

==== Tree= ===

{{main|Tree (graph theory)}}

A ''tree'' is an undirected graph in which any two [[Vertex (graph theory)|vertices]] are connected by ''exactly one'' [[Path (graph theory)|path]], or equivalently a [[Connected graph|connected]] [[Cycle (graph theory)|acyclic]] undirected graph.

A ''tree'' is a connected graph with no cycles.

A ''forest'' is aan undirected graph within nowhich cyclesany two vertices are connected by ''at most one'' path, i.e.or theequivalently disjointan acyclic undirected graph, or equivalently a [[Disjoint union of onegraphs|disjoint orunion]] moreof trees.

====Advanced classes=Polytree ===

{{main|Polytree}}

A ''polytree'' (or ''directed tree'' or ''oriented tree'' or ''singly connected network'') is a [[directed acyclic graph]] (DAG) whose underlying undirected graph is a tree.

A ''polyforest'' (or ''directed forest'' or ''oriented forest'') is a directed acyclic graph whose underlying undirected graph is a forest.

=== Advanced classes ===

More advanced kinds of graphs are:

* [[Petersen graph]] and its generalizations;

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

== Properties of graphs ==

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

Two edgesvertices of a graph are called ''adjacent'' if they share a common vertexedge. Two arrowsvertices 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 arrowedge), 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.

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''. (Note that inIn 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 [[slicecomma category]] Set ↓ ''D'' where ''D'': Set → Set is the [[functor]] taking a set ''s'' to ''s'' × ''s''.

== Examples ==

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

* The diagram at right is a graphicschematic representation of the following 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.

:: {{nobreak|1=''V'' = {1, 2, 3, 4, 5, 6}}};

:: {{nobreak|1=''E'' = {{1, 2}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {4, 5}, {4, 6}}}}.

* In [[category theory]], a [[category (mathematics)|small category]] can be represented by a directed multigraph in which the objects of the category are represented as vertices and the [[morphism]]s as directed edges. Then, the [[functor]]s between categories induce some, but not necessarily all, of the [[digraph morphism]]s of the graph.

* 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 = httphttps://cogentoaserval.tandfonlineunil.comch/doiresource/full/10serval:BIB_81C2C68B1DF5.1080P001/23311983.2016.1171458REF| doi-access =10 free| access-date = 2019-09-16| archive-date = 2021-03-02| archive-url = https://web.1080archive.org/23311983web/20210302190117/https://serval.2016unil.1171458ch/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>

*Particularly regular examples of directed graphs are given by the [[Cayley graph]]s of finitely-generated groups, as well as [[Schreier coset graph]]s

*In [[category theory]], every [[small category]] has an underlying directed multigraph whose vertices are the objects of the category, and whose edges are the arrows of the category. In the language of category theory, one says that there is a [[forgetful functor]] from the [[category of small categories]] to the [[Quiver (mathematics)|category of quivers]].

== Graph operations ==

{{main|Graph operations}}

There are several operations that produce new graphs from initial ones, which might be classified into the following categories:

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** [[strong product of graphs]],

** [[lexicographic product of graphs]],

** [[series-parallelseries–parallel graph]]s.

== Generalizations ==

In a [[hypergraph]], an edge can join moreany thanpositive twonumber of vertices.

An undirected graph can be seen as a [[simplicial complex]] consisting of 1-[[simplex|simplices]] (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.

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In [[geographic information systems]], [[geometric networks]] are closely modeled after graphs, and borrow many concepts from [[graph theory]] to perform spatial analysis on road networks or utility grids.

== See also ==

* [[Conceptual graph]]

* [[Dual graph]]

* [[Glossary of graph theory]]

* [[Graph (abstract data type)]]

* [[Graph database]]

* [[Graph drawing]]

* [[Graph theory]]

* [[Hypergraph]]

* [[List of graph theory topics]]

* [[List of publications in mathematics#Graph theory|List of publications in graph theory]]

* [[Network theory]]

== Notes ==

{{reflist|30em}}

== References ==

* {{cite book |last=Balakrishnan |first=V. K. |date=1997 |title=Graph Theory|publisher=McGraw-Hill|date=1997-02-01 |edition=1st|isbn=978-0-07-005489-49}}

* {{cite book |last1=Bang-Jensen |first1=J. |last2=Gutin |first2=G. |date=2000 |title=Digraphs: Theory, Algorithms and Applications |url=http://www.cs.rhul.ac.uk/books/dbook/ |publisher=Springer }}

* {{cite book|last=Berge|first=Claude|title=Théorie des graphes et ses applications|publisher=Collection Universitaire de Mathématiques, II|___location=Dunod, Paris|year=1958|pages=viii+277|language=French|authorlink=Claude Berge}} Translation: {{cite book|publisher=Wiley|___location=Dover, New York|year=2001|origyear=1962|title=-}}

* {{cite book |last1=Bender |first1=Edward A. |last2=Williamson |first2=S. Gill |date=2010 |title=Lists, Decisions and Graphs. With an Introduction to Probability |url=https://books.google.com/books?id=vaXv_yhefG8C }}

* {{cite book|last=Biggs|first=Norman|title=Algebraic Graph Theory|publisher=Cambridge University Press|year=1993|edition=2nd|isbn=0-521-45897-8}}

* {{cite book |last=BollobásBerge |first=BélaClaude |date=1958 |title=ModernThéorie Graphdes graphes et ses applications Theory|publisherlanguage=Springerfr |date___location=2002-08-12Paris |editionpublisher=1st|isbn=0-387-98488-7Dunod }}

* {{cite book |last=Bang-JensenBiggs |first=J.Norman |author2date=Gutin,1993 G.|title=Digraphs:Algebraic Graph Theory, Algorithms and|edition=2nd Applications|publisher=SpringerCambridge University Press |yearisbn=2000|url=http://www.cs.rhul.ac.uk/books/dbook/978-0-521-45897-9 }}

* {{Citecite book |last=Bollobás last1|first=DiestelBéla | first1date=Reinhard2002 | title=Modern Graph Theory | urledition=http://diestel-graph-theory.com/GrTh.html1st | publisher=Springer-Verlag | ___location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 | postscript=<!-0-None387-98488->9 }}.

* {{cite book |last=Diestel |first=Reinhard |date=2005 |title=Graph Theory |url=http://diestel-graph-theory.com/GrTh.html |edition=3rd |___location=Berlin, New York |publisher=Springer-Verlag |isbn=978-3-540-26183-4 }}

* {{cite book|title=Handbook of Combinatorics|editor=Graham, R.L. |editor2=Grötschel, M.|editor2-link= Martin Grötschel |editor3=Lovász, L|publisher=MIT Press|year=1995|isbn=0-262-07169-X}}

* {{cite book |lastlast1=GrossGraham |firstfirst1=JonathanR.L. |last2=Grötschel |first2=M. |last3=Lovász |first3=L. |author2date=Yellen,1995 Jay|title=GraphHandbook Theoryof andCombinatorics Its Applications|publisher=CRCMIT Press|date=1998-12-30 |isbn=978-0-8493262-398207169-07 }}

* {{cite book|title=Handbook of Graph Theory|editorlast1=Gross, |first1=Jonathan L. |editor2last2=Yellen, Jay|publisherfirst2=CRCJay |date=2003-12-291998 |title=Graph Theory and Its Applications |publisher=CRC Press |isbn=1978-584880-0908493-23982-0 }}

* {{cite book |lastlast1=HararyGross |firstfirst1=FrankJonathan L. |titlelast2=GraphYellen Theory|publisherfirst2=AddisonJay Wesley|date=2003 Publishing|title=Handbook of Graph Theory Company|datepublisher=JanuaryCRC 1995|isbn=0978-2011-4103358488-8090-5 }}

* {{cite book |last=IyanagaHarary |first=ShôkichiFrank |author2date=Kawada,1995 Yukiyosi|title=EncyclopedicGraph DictionaryTheory of Mathematics|publisher=MITAddison Wesley Publishing Company Press|year=1977|isbn=978-0-262201-0901641033-34 }}

* {{cite book |last1=Iyanaga |first1=Shôkichi |last2=Kawada |first2=Yukiyosi |date=1977 |title=Encyclopedic Dictionary of Mathematics |url=https://archive.org/details/encyclopedicdict0000niho |publisher=MIT Press |isbn=978-0-262-09016-2 |url-access=registration }}

* {{cite book|last=Zwillinger|first=Daniel|title=CRC Standard Mathematical Tables and Formulae|publisher=Chapman & Hall/CRC|date=2002-11-27|edition=31st|isbn=1-58488-291-3}}

* {{cite book |last=Zwillinger |first=Daniel |date=2002 |title=CRC Standard Mathematical Tables and Formulae |edition=31st |publisher=Chapman & Hall/CRC |isbn=978-1-58488-291-6 }}

== Further reading ==

* {{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=[[Dover Publications]]|___location=New York|isbn=978-0-486-67870-2|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|accessdateaccess-date=8 August 2012}}

== External links ==

{{Library resources box

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* {{Commonscatinline}}

* {{MathWorld | urlname=Graph | title = Graph}}