View source for Graph theory

{{about|sets of vertices connected by edges|graphs of mathematical functions|Graph of a function|other uses|Graph (disambiguation)}}
{{short description|Area of discrete mathematics}}
[[File:Example of simple undirected graph 3.svg|thumb|150px|A [[Graph drawing|drawing]] of a graph with 6 vertices and 7 edges.]]
In [[mathematics]] and [[computer science]], '''graph theory''' is the study of ''[[graph (discrete mathematics)|graph]]s'', which are [[mathematical structures]] used to model pairwise relations between objects. A graph in this context is made up of ''[[Vertex (graph theory)|vertices]]'' (also called ''nodes'' or ''points'') which are connected by ''[[Glossary of graph theory terms#edge|edges]]'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between '''undirected graphs''', where edges link two vertices symmetrically, and '''directed graphs''', where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in [[discrete mathematics]].
== Definitions ==
{{further|Glossary of graph theory}}
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related [[mathematical structure]]s.
=== Graph ===
[[File:Example of simple undirected graph.svg|150 px|thumb|A graph with three vertices and three edges.]]
In one restricted but very common sense of the term,{{sfn|Bender|Williamson|2010|p=148}}<ref>See, for instance, Iyanaga and Kawada, ''69 J'', p. 234 or Biggs, p. 4.</ref> a '''graph''' is an [[ordered pair]] <math>G=(V,E)</math> comprising:
* <math>V</math>, a [[Set (mathematics)|set]] of '''vertices''' (also called '''nodes''' or '''points''');
* <math>E \subseteq \{ \{x, y\} \mid x, y \in V \;\textrm{ and }\; x \neq y \}</math>, a [[Set (mathematics)|set]] of '''edges''' (also called '''links''' or '''lines'''), which are [[unordered pair]]s of vertices (that is, an edge is associated with two distinct vertices).
To avoid ambiguity, this type of object may be called precisely an '''undirected simple graph'''.
In the edge <math>\{x, y\}</math>, the vertices <math>x</math> and <math>y</math> are called the '''endpoints''' of the edge. The edge is said to '''join''' <math>x</math> and <math>y</math> and to be '''incident''' on <math>x</math> and on <math>y</math>. A vertex may exist in a graph and not belong to an edge. Under this definition, '''[[multiple edges]]''', in which two or more edges connect the same vertices, are not allowed.
[[File:Example of simple undirected graph with loops.svg|150px|thumb|Example of an undirected multigraph with 3 vertices, 3 edges and 4 loops.]]
{{Multiple image
| image1 = Example of simple undirected graph 2.svg
| width1 = 75
| caption1 =For vertices A,B,C and D, the degrees are respectively 4,4,5,1
| image2 = Example of simple undirected graph 1.svg
| width2 = 75
| caption2 = For vertices U,V,W and X, the degrees are 2,2,3 and 1 respectively.
| footer = Examples of finding the degree of vertices.
}}
In one more general sense of the term allowing multiple edges,{{sfn|Bender|Williamson|2010|p=149}}<ref>See, for instance, Graham et al., p. 5.</ref> a '''graph''' is an ordered triple <math>G=(V,E,\phi)</math> comprising:
* <math>V</math>, a [[Set (mathematics)|set]] of '''vertices''' (also called '''nodes''' or '''points''');
* <math>E</math>, a [[Set (mathematics)|set]] of '''edges''' (also called '''links''' or '''lines''');
* <math>\phi : E \to \{ \{x, y\} \mid x, y \in V \;\textrm{ and }\; x \neq y \}</math>, an '''incidence function''' mapping every edge to an [[unordered pair]] of vertices (that is, an edge is associated with two distinct vertices).
To avoid ambiguity, this type of object may be called precisely an '''undirected [[multigraph]]'''.
A '''[[Loop (graph theory)|loop]]''' is an edge that joins a vertex to itself. 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 an undirected simple graph) or is incident on (for an undirected multigraph) <math>\{x, x\} = \{x\}</math> which is not in <math>\{ \{x, y\} \mid x, y \in V \;\textrm{ and }\; x \neq y \}</math>. To allow loops, the definitions must be expanded. For undirected simple graphs, the definition of <math>E</math> should be modified to <math>E \subseteq \{ \{x, y\} \mid x, y \in V \}</math>. For undirected multigraphs, the definition of <math>\phi</math> should be modified to <math>\phi : E \to \{ \{x, y\} \mid x, y \in V \}</math>. To avoid ambiguity, these types of objects may be called '''undirected simple graph permitting loops''' and '''undirected multigraph permitting loops''' (sometimes also '''undirected [[pseudograph]]'''), respectively.
<math>V</math> and <math>E</math> 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, <math>V</math> is often assumed to be non-empty, but <math>E</math> is allowed to be the empty set. The '''order''' of a graph is <math>|V|</math>, its number of vertices. The '''size''' of a graph is <math>|E|</math>, its number of edges. The '''degree''' or '''valency''' of a vertex is the number of edges that are incident to it, where a loop is counted twice. The '''degree''' of a graph is the maximum of the degrees of its vertices.
In an undirected simple graph of order ''n'', the maximum degree of each vertex is {{nowrap|''n'' − 1}} and the maximum size of the graph is {{sfrac|''n''(''n'' − 1)|2}}.
The edges of an undirected simple graph permitting loops <math>G</math> induce a symmetric [[Binary relation#Homogeneous relation|homogeneous relation]] <math>\sim</math> on the vertices of <math>G</math> that is called the '''adjacency relation''' of <math>G</math>. Specifically, for each edge <math>(x,y)</math>, its endpoints <math>x</math> and <math>y</math> are said to be '''adjacent''' to one another, which is denoted <math>x \sim y</math>.
=== Directed graph ===
{{main|Directed graph}}
[[File:Example of simple directed graph.svg|thumb|150px|A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction).]]
A '''directed graph''' or '''digraph''' is a graph in which edges have orientations.
In one restricted but very common sense of the term,{{sfn|Bender|Williamson|2010|p=161}} a '''directed graph''' is an ordered pair <math>G=(V,E)</math> comprising:
* <math>V</math>, a [[Set (mathematics)|set]] of ''vertices'' (also called ''nodes'' or ''points'');
* <math>E \subseteq \left\{(x,y) \mid (x, y) \in V^2 \;\textrm{ and }\; x \neq y \right\}</math>, a [[Set (mathematics)|set]] of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs'') which are [[ordered pair]]s of vertices (that is, an edge is associated with two distinct vertices).
To avoid ambiguity, this type of object may be called precisely a '''directed simple graph'''. In set theory and graph theory, <math>V^n</math> denotes the set of {{mvar|n}}-[[tuple]]s of elements of <math>V,</math> that is, ordered sequences of <math>n</math> elements that are not necessarily distinct.
In the edge <math>(x, y)</math> directed from <math>x</math> to <math>y</math>, the vertices <math>x</math> and <math>y</math> are called the ''endpoints'' of the edge, <math>x</math> the ''tail'' of the edge and <math>y</math> the ''head'' of the edge. The edge is said to ''join'' <math>x</math> and <math>y</math> and to be ''incident'' on <math>x</math> and on <math>y</math>. A vertex may exist in a graph and not belong to an edge. The edge <math>(y,x)</math> is called the ''inverted edge'' of <math>(x, y)</math>. ''[[Multiple edges]]'', not allowed under the definition above, are two or more edges with both the same tail and the same head.
In one more general sense of the term allowing multiple edges,{{sfn|Bender|Williamson|2010|p=161}} a '''directed graph''' is an ordered triple <math>G=(V,E,\phi)</math> comprising:
* <math>V</math>, a [[Set (mathematics)|set]] of ''vertices'' (also called ''nodes'' or ''points'');
* <math>E</math>, a [[Set (mathematics)|set]] of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs'');
* <math>\phi : E \to \left\{(x,y) \mid (x, y) \in V^2 \;\textrm{ and }\; x \neq y \right\}</math>, an ''incidence function'' mapping every edge to an [[ordered pair]] of vertices (that is, an edge is associated with two distinct vertices).
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>\left\{(x, y) \mid (x, y) \in V^2 \;\textrm{ and }\; x \neq y \right\}</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 \left\{(x, y) \mid (x, y) \in V^2\right\}</math>. For directed multigraphs, the definition of <math>\phi</math> should be modified to <math>\phi : E \to \left\{(x, y) \mid (x, y) \in V^2\right\}</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 <math>G</math> is a [[Binary relation#Homogeneous relation|homogeneous relation]] ~ on the vertices of <math>G</math> that is called the ''adjacency relation'' of <math>G</math>. Specifically, for each edge <math>(x,y)</math>, its endpoints <math>x</math> and <math>y</math> are said to be ''adjacent'' to one another, which is denoted <math>x</math> ~ <math>y</math>.
== Applications ==
[[File:Wikipedia multilingual network graph July 2013.svg|thumb|The network graph formed by Wikipedia editors (edges) contributing to different Wikipedia language versions (vertices) during one month in summer 2013.<ref>{{Cite book|arxiv=1312.0976|last1=Hale|first1=Scott A.|title=Proceedings of the 2014 ACM conference on Web science |chapter=Multilinguals and Wikipedia editing |date=2014 |doi=10.1145/2615569.2615684|pages=99–108|isbn=9781450326223|bibcode=2013arXiv1312.0976H|s2cid=14027025}}</ref>]]
Graphs can be used to model many types of relations and processes in physical, biological,<ref>{{cite journal |last=Mashaghi |first=A. |title=Investigation of a protein complex network |journal=European Physical Journal B |volume=41 |issue=1 |pages=113–121 |year=2004 |doi=10.1140/epjb/e2004-00301-0 |display-authors=etal|arxiv=cond-mat/0304207 |bibcode=2004EPJB...41..113M |s2cid=9233932 }}</ref><ref>{{Cite journal|last1=Shah|first1=Preya|last2=Ashourvan|first2=Arian|last3=Mikhail|first3=Fadi|last4=Pines|first4=Adam|last5=Kini|first5=Lohith|last6=Oechsel|first6=Kelly|last7=Das|first7=Sandhitsu R|last8=Stein|first8=Joel M|last9=Shinohara|first9=Russell T|date=2019-07-01|title=Characterizing the role of the structural connectome in seizure dynamics|journal=Brain|language=en|volume=142|issue=7|pages=1955–1972|doi=10.1093/brain/awz125|pmid=31099821|issn=0006-8950|pmc=6598625}}</ref> social and information systems.<ref>{{Cite journal|last1=Adali|first1=Tulay|last2=Ortega|first2=Antonio|date=May 2018|title=Applications of Graph Theory [Scanning the Issue]|url=https://ieeexplore.ieee.org/document/8349656|journal=Proceedings of the IEEE|volume=106|issue=5|pages=784–786|doi=10.1109/JPROC.2018.2820300|issn=0018-9219|url-access=subscription}}</ref> Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term ''network'' is sometimes defined to mean a graph in which attributes (e.g. names) are associated with the vertices and edges, and the subject that expresses and understands real-world systems as a network is called [[network science]].
=== Computer science ===
Within [[computer science]], '[[cybernetics|causal]]' and 'non-causal' linked structures are graphs that are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a [[website]] can be represented by a directed graph, in which the vertices represent web pages and directed edges represent [[Hyperlink|links]] from one page to another. A similar approach can be taken to problems in social media,<ref>{{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| s2cid = 114999767| url = https://hal.archives-ouvertes.fr/hal-01517493/file/A%20social%20network%20analysis%20of%20Twitter%20Mapping%20the%20digital%20humanities%20community.pdf| doi-access = free}}</ref> travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases,<ref>{{Cite journal | volume = 11| issue = 2| last = Vecchio | first = F| title = "Small World" architecture in brain connectivity and hippocampal volume in Alzheimer's disease: a study via graph theory from EEG data| journal =Brain Imaging and Behavior| date = 2017| pages = 473–485| pmid =26960946 | doi = 10.1007/s11682-016-9528-3| s2cid = 3987492}}</ref><ref>{{Cite journal | volume = 81| issue = 2| last = Vecchio | first = F| title = Brain network connectivity assessed using graph theory in frontotemporal dementia| journal = Neurology| date = 2013| pages = 134–143| doi = 10.1212/WNL.0b013e31829a33f8| pmid = 23719145| s2cid = 28334693}}</ref> and many other fields. The development of [[algorithm]]s to [[List of algorithms#Graph algorithms|handle graphs]] is therefore of major interest in computer science. The [[Graph transformation|transformation of graph]]s is often formalized and represented by [[graph rewriting|graph rewrite system]]s. Complementary to [[graph transformation]] systems focusing on rule-based in-memory manipulation of graphs are [[graph database]]s geared towards [[Database transaction|transaction]]-safe, [[Persistence (computer science)|persistent]] storing and querying of [[Graph (data structure)|graph-structured data]].
=== Linguistics ===
Graph-theoretic methods, in various forms, have proven particularly useful in [[linguistics]], since natural language often lends itself well to discrete structure. Traditionally, [[syntax]] and compositional semantics follow tree-based structures, whose expressive power lies in the [[principle of compositionality]], modeled in a hierarchical graph. More contemporary approaches such as [[head-driven phrase structure grammar]] model the syntax of natural language using [[feature structure|typed feature structure]]s, which are [[directed acyclic graph]]s.
Within [[lexical semantics]], especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; [[semantic network]]s are therefore important in [[computational linguistics]]. Still, other methods in phonology (e.g. [[optimality theory]], which uses [[lattice graph]]s) and morphology (e.g. finite-state morphology, using [[finite-state transducer]]s) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as [http://www.textgraphs.org/ TextGraphs], as well as various 'Net' projects, such as [[WordNet]], [[VerbNet]], and others.
=== Physics and chemistry ===
Graph theory is also used to study molecules in [[chemistry]] and [[physics]]. In [[condensed matter physics]], the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the [[Feynman diagram|Feynman graphs and rules of calculation]] summarize [[quantum field theory]] in a form in close contact with the experimental numbers one wants to understand."<ref>{{cite book|first1=J. D.|last1=Bjorken |first2=S. D. |last2=Drell |title=Relativistic Quantum Fields |url=https://archive.org/details/relativisticquan0000bjor_c5q0|url-access=registration|publisher=McGraw-Hill |location=New York |year=1965 |page=viii }}</ref> In chemistry a graph makes a natural model for a molecule, where vertices represent [[atom]]s and edges [[Chemical bond|bond]]s. This approach is especially used in computer processing of molecular structures, ranging from [[Molecule editor|chemical editor]]s to database searching. In [[statistical physics]], graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such
systems. Similarly, in [[computational neuroscience]] graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures.<ref>{{Cite journal|last1=Kumar|first1=Ankush|last2=Kulkarni|first2=G. U.|date=2016-01-04|title=Evaluating conducting network based transparent electrodes from geometrical considerations|journal=Journal of Applied Physics|volume=119|issue=1|pages=015102|doi=10.1063/1.4939280|issn=0021-8979|bibcode=2016JAP...119a5102K}}</ref> Graphs are also used to represent the micro-scale channels of [[Porous medium|porous media]], in which the vertices represent the pores and the edges represent the smaller channels connecting the pores. [[Chemical graph theory]] uses the [[molecular graph]] as a means to model molecules.
Graphs and networks are excellent models to study and understand phase transitions and critical phenomena.
Removal of nodes or edges leads to a critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is studied via [[percolation theory]].<ref>{{Cite book| last = Newman| first = Mark| title = Networks: An Introduction| publisher = Oxford University Press| date = 2010| url = http://math.sjtu.edu.cn/faculty/xiaodong/course/Networks%20An%20introduction.pdf| access-date = 2019-10-30| archive-date = 2020-07-28| archive-url = https://web.archive.org/web/20200728132820/http://math.sjtu.edu.cn/faculty/xiaodong/course/Networks%20An%20introduction.pdf| url-status = dead}}</ref>
=== Social sciences ===
[[File:Moreno Sociogram 2nd Grade.png|thumb|Graph theory in sociology: [[Jacob L. Moreno|Moreno]] [[Sociogram]] (1953).<ref>Grandjean, Martin (2015). [http://www.martingrandjean.ch/social-network-analysis-visualization-morenos-sociograms-revisited/ "Social network analysis and visualization: Moreno’s Sociograms revisited"]. Redesigned network strictly based on Moreno (1934), ''Who Shall Survive''.</ref>]]
Graph theory is also widely used in [[sociology]] as a way, for example, to [[Six Degrees of Kevin Bacon|measure actors' prestige]] or to explore [[Rumor spread in social network|rumor spreading]], notably through the use of [[social network analysis]] software. Under the umbrella of social networks are many different types of graphs.<ref>{{cite book|last=Rosen|first=Kenneth H.|title=Discrete mathematics and its applications|publisher=McGraw-Hill|location=New York|isbn=978-0-07-338309-5|edition=7th|date=2011-06-14}}</ref> Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together.
=== Biology ===
Likewise, graph theory is useful in [[biology]] and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.
Graphs are also commonly used in [[molecular biology]] and [[genomics]] to model and analyse datasets with complex relationships. For example, graph-based methods are often used to 'cluster' cells together into cell-types in [[Single-cell_analysis#Transcriptomics|single-cell transcriptome analysis]]. Another use is to model genes or proteins in a [[Biological pathway|pathway]] and study the relationships between them, such as metabolic pathways and gene regulatory networks.<ref>{{cite journal | last1=Kelly | first1=S. | last2=Black | first2=Michael | title=graphsim: An R package for simulating gene expression data from graph structures of biological pathways | journal=Journal of Open Source Software | publisher=The Open Journal | volume=5 | issue=51 | date=2020-07-09 | issn=2475-9066 | doi=10.21105/joss.02161 | page=2161| bibcode=2020JOSS....5.2161K |biorxiv=10.1101/2020.03.02.972471|s2cid=214722561| doi-access=free | url=https://www.biorxiv.org/content/biorxiv/early/2020/06/30/2020.03.02.972471.full.pdf }}</ref> Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures.
Graph theory is also used in [[connectomics]];<ref>{{Cite journal|last1=Shah|first1=Preya|last2=Ashourvan|first2=Arian|last3=Mikhail|first3=Fadi|last4=Pines|first4=Adam|last5=Kini|first5=Lohith|last6=Oechsel|first6=Kelly|last7=Das|first7=Sandhitsu R|last8=Stein|first8=Joel M|last9=Shinohara|first9=Russell T|date=2019-07-01|title=Characterizing the role of the structural connectome in seizure dynamics|journal=Brain|language=en|volume=142|issue=7|pages=1955–1972|doi=10.1093/brain/awz125|pmid=31099821|issn=0006-8950|pmc=6598625}}</ref> nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.
=== Mathematics ===
In mathematics, graphs are useful in geometry and certain parts of topology such as [[knot theory]]. [[Algebraic graph theory]] has close links with [[group theory]]. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.
=== Other topics ===
A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or [[weighted graph]]s, are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs.
== History ==
[[File:Konigsberg bridges.png|thumb|The Königsberg Bridge problem]]
The paper written by [[Leonhard Euler]] on the [[Seven Bridges of Königsberg]] and published in 1736 is regarded as the first paper in the history of graph theory.<ref name="Biggs">{{Citation|author1=Biggs, N. |author2=Lloyd, E. |author3=Wilson, R. |title=Graph Theory, 1736-1936|title-link=Graph Theory, 1736–1936|publisher=Oxford University Press|year=1986}}</ref> This paper, as well as the one written by [[Alexandre-Théophile Vandermonde|Vandermonde]] on the ''[[Knight's tour|knight problem]],'' carried on with the ''analysis situs'' initiated by [[Gottfried Wilhelm Leibniz|Leibniz]]. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by [[Augustin-Louis Cauchy|Cauchy]]<ref name="Cauchy">{{Citation|author=Cauchy, A. L.|year=1813|title=Recherche sur les polyèdres - premier mémoire|journal=[[Journal de l'École Polytechnique]]|volume= 9 (Cahier 16)|pages=66–86|postscript=.}}</ref> and [[Simon Antoine Jean L'Huilier|L'Huilier]],<ref name="Lhuillier">{{Citation|author=L'Huillier, S.-A.-J.|title=Mémoire sur la polyèdrométrie|journal=Annales de Mathématiques|volume=3|year=1812–1813|pages=169–189|postscript=.}}</ref> and represents the beginning of the branch of mathematics known as [[topology]].
More than one century after Euler's paper on the bridges of [[Königsberg]] and while [[Johann Benedict Listing|Listing]] was introducing the concept of topology, [[Arthur Cayley|Cayley]] was led by an interest in particular analytical forms arising from [[differential calculus]] to study a particular class of graphs, the ''[[tree (graph theory)|tree]]s''.<ref>{{citation|first=A.|last=Cayley|author-link=Arthur Cayley|year=1857|title=On the theory of the analytical forms called trees|journal=[[Philosophical Magazine]]|series=Series IV|volume=13|issue=85|pages=172–176|doi=10.1017/CBO9780511703690.046|isbn=9780511703690|url=https://rcin.org.pl/dlibra/docmetadata?showContent=true&id=173708|url-access=subscription}}</ref> This study had many implications for theoretical [[chemistry]]. The techniques he used mainly concern the [[enumeration of graphs]] with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results published by [[George Pólya|Pólya]] between 1935 and 1937. These were generalized by [[Nicolaas Govert de Bruijn|De Bruijn]] in 1959. Cayley linked his results on trees with contemporary studies of chemical composition.<ref name="Cayley1">{{Citation|author=Cayley, A.|year=1875|journal=Berichte der Deutschen Chemischen Gesellschaft|title=Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen|volume=8|pages=1056–1059|doi=10.1002/cber.18750080252|postscript=.|issue=2|url=https://zenodo.org/record/1425086}}</ref> The fusion of ideas from mathematics with those from chemistry began what has become part of the standard terminology of graph theory.
In particular, the term "graph" was introduced by [[James Joseph Sylvester|Sylvester]] in a paper published in 1878 in ''[[Nature (journal)|Nature]]'', where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:<ref name="Sylvester">{{cite journal | last1 = Sylvester | first1 = James Joseph | year = 1878 | title = Chemistry and Algebra | url = https://archive.org/stream/nature15unkngoog#page/n312/mode/1up | journal = Nature | volume = 17 | issue = 432| page = 284 | doi = 10.1038/017284a0 | bibcode = 1878Natur..17..284S | doi-access = free }}</ref>
:"[…] Every invariant and co-variant thus becomes expressible by a ''graph'' precisely identical with a [[August Kekulé|Kekuléan]] diagram or chemicograph. […] I give a rule for the geometrical multiplication of graphs, ''i.e.'' for constructing a ''graph'' to the product of in- or co-variants whose separate graphs are given. […]" (italics as in the original).
The first textbook on graph theory was written by [[Dénes Kőnig]], and published in 1936.<ref>{{citation|last1=Tutte|first1=W.T.|author-link=W. T. Tutte|title=Graph Theory|publisher=Cambridge University Press|year=2001|isbn=978-0-521-79489-3|page=30|url=https://books.google.com/books?id=uTGhooU37h4C&pg=PA30|access-date=2016-03-14}}</ref> Another book by [[Frank Harary]], published in 1969, was "considered the world over to be the definitive textbook on the subject",<ref>{{citation|page=203|title=Fractal Music, Hypercards, and more…Mathematical Recreations from Scientific American|first=Martin|last=Gardner|author-link=Martin Gardner|publisher=W. H. Freeman and Company|year=1992}}</ref> and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other. Harary donated all of the royalties to fund the [[George Pólya Prize|Pólya Prize]].<ref>{{citation|contribution=The George Polya Prize|url=http://www.siam.org/about/more/siam50.pdf|page=26|title=Looking Back, Looking Ahead: A SIAM History|author=Society for Industrial and Applied Mathematics|year=2002|access-date=2016-03-14|author-link=Society for Industrial and Applied Mathematics|archive-date=2016-03-05|archive-url=https://web.archive.org/web/20160305010030/http://www.siam.org/about/more/siam50.pdf|url-status=dead}}</ref>
One of the most famous and stimulating problems in graph theory is the [[four color problem]]: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by [[Francis Guthrie]] in 1852 and its first written record is in a letter of [[Augustus De Morgan|De Morgan]] addressed to [[William Rowan Hamilton|Hamilton]] the same year. Many incorrect proofs have been proposed, including those by Cayley, [[Alfred Kempe|Kempe]], and others. The study and the generalization of this problem by [[Peter Tait (physicist)|Tait]], [[Percy John Heawood|Heawood]], [[Frank P. Ramsey|Ramsey]] and [[Hugo Hadwiger|Hadwiger]] led to the study of the colorings of the graphs embedded on surfaces with arbitrary [[Genus (mathematics)|genus]]. Tait's reformulation generated a new class of problems, the ''factorization problems'', particularly studied by [[Julius Petersen|Petersen]] and [[Dénes Kőnig|Kőnig]]. The works of Ramsey on colorations and more specially the results obtained by [[Pál Turán|Turán]] in 1941 was at the origin of another branch of graph theory, ''[[extremal graph theory]]''.
The four color problem remained unsolved for more than a century. In 1969 [[Heinrich Heesch]] published a method for solving the problem using computers.<ref>Heinrich Heesch: Untersuchungen zum Vierfarbenproblem. Mannheim: Bibliographisches Institut 1969.</ref> A computer-aided proof produced in 1976 by [[Kenneth Appel]] and [[Wolfgang Haken]] makes fundamental use of the notion of "discharging" developed by Heesch.<ref name="AA1">{{Citation|author1=Appel, K. |author2=Haken, W.|title=Every planar map is four colorable. Part I. Discharging|journal=Illinois J. Math.|volume=21|issue=3|year=1977|pages=429–490|postscript=.|doi=10.1215/ijm/1256049011|doi-access=free|url=https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-21/issue-3/Every-planar-map-is-four-colorable-Part-I-Discharging/10.1215/ijm/1256049011.pdf}}</ref><ref name="AA2">{{Citation|author1=Appel, K. |author2=Haken, W.|title=Every planar map is four colorable. Part II. Reducibility|journal=Illinois J. Math.|volume=21|issue=3|year=1977|pages=491–567|postscript=.|doi=10.1215/ijm/1256049012|doi-access=free}}</ref> The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by [[Neil Robertson (mathematician)|Robertson]], [[Paul Seymour (mathematician)|Seymour]], [[Daniel P. Sanders|Sanders]] and [[Robin Thomas (mathematician)|Thomas]].<ref name="RSST">{{Citation|author1=Robertson, N. |author2=Sanders, D. |author3=Seymour, P. |author4=Thomas, R. |title=The four color theorem|journal=Journal of Combinatorial Theory, Series B|volume=70|year=1997|pages=2–44|doi=10.1006/jctb.1997.1750|postscript=.|doi-access=free}}</ref>
The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of [[Camille Jordan|Jordan]], [[Kazimierz Kuratowski|Kuratowski]] and [[Hassler Whitney|Whitney]]. Another important factor of common development of graph theory and [[topology]] came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist [[Gustav Kirchhoff]], who published in 1845 his [[Kirchhoff's circuit laws]] for calculating the [[voltage]] and [[Electric current|current]] in [[electric circuit]]s.
The introduction of probabilistic methods in graph theory, especially in the study of [[Paul Erdős|Erdős]] and [[Alfréd Rényi|Rényi]] of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as ''[[Random graph|random graph theory]]'', which has been a fruitful source of graph-theoretic results.
== Representation ==
A graph is an abstraction of relationships that emerge in nature; hence, it cannot be coupled to a certain representation. The way it is represented depends on the degree of convenience such representation provides for a certain application. The most common representations are the visual, in which, usually, vertices are drawn and connected by edges, and the tabular, in which rows of a table provide information about the relationships between the vertices within the graph.
=== Visual: Graph drawing ===
{{main|Graph drawing}}
Graphs are usually represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow. If the graph is weighted, the weight is added on the arrow.
A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice, it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.
The pioneering work of [[W. T. Tutte]] was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.
Graph drawing also can be said to encompass problems that deal with the [[Crossing number (graph theory)|crossing number]] and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a [[planar graph]], the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied.
There are other techniques to visualize a graph away from vertices and edges, including [[circle packing theorem|circle packings]], [[intersection graph]], and other visualizations of the [[adjacency matrix]].
=== Tabular: Graph data structures ===
{{main|Graph (abstract data type)}}
The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The [[data structure]] used depends on both the graph structure and the [[algorithm]] used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for [[sparse graph]]s as they have smaller memory requirements. [[Matrix (mathematics)|Matrix]] structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation.<ref>{{Cite book | last1 = Kepner | first1 = Jeremy| last2 = Gilbert | first2 = John| title = Graph Algorithms in the Language of Linear Algebra| date = 2011| pages = 1171458|isbn=978-0-898719-90-1| publisher = SIAM | url = https://my.siam.org/Store/Product/viewproduct/?ProductId=106663}}</ref>
List structures include the [[edge list]], an array of pairs of vertices, and the [[adjacency list]], which separately lists the neighbors of each vertex: Much like the edge list, each vertex has a list of which vertices it is adjacent to.
Matrix structures include the [[incidence matrix]], a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the [[adjacency matrix]], in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The [[degree matrix]] indicates the degree of vertices. The [[Laplacian matrix]] is a modified form of the adjacency matrix that incorporates information about the [[degree (graph theory)|degrees]] of the vertices, and is useful in some calculations such as [[Kirchhoff's theorem]] on the number of [[spanning tree]]s of a graph.
The [[distance matrix]], like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a [[shortest path]] between two vertices.
== Problems ==
=== Enumeration ===
There is a large literature on [[graphical enumeration]]: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973).
=== Subgraphs, induced subgraphs, and minors ===
A common problem, called the [[subgraph isomorphism problem]], is finding a fixed graph as a [[Glossary of graph theory#Subgraphs|subgraph]] in a given graph. One reason to be interested in such a question is that many [[graph properties]] are ''hereditary'' for subgraphs, which means that a graph has the property if and only if all subgraphs have it too.
Finding maximal subgraphs of a certain kind is often an [[NP-complete problem]]. For example:
* Finding the largest complete subgraph is called the [[clique problem]] (NP-complete).
One special case of subgraph isomorphism is the [[graph isomorphism problem]]. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time.
A similar problem is finding [[induced subgraph]]s in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example:
* Finding the largest edgeless induced subgraph or [[Independent set (graph theory)|independent set]] is called the [[independent set problem]] (NP-complete).
Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A [[Minor (graph theory)|minor]] or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example, [[Wagner's theorem|Wagner's Theorem]] states:
* A graph is [[planar graph|planar]] if it contains as a minor neither the [[complete bipartite graph]] ''K''<sub>3,3</sub> (see the [[Three-cottage problem]]) nor the complete graph ''K''<sub>5</sub>.
A similar problem, the subdivision containment problem, is to find a fixed graph as a [[Subdivision (graph theory)|subdivision]] of a given graph. A [[Subdivision (graph theory)|subdivision]] or [[Homeomorphism (graph theory)|homeomorphism]] of a graph is any graph obtained by subdividing some (or no) edges. Subdivision containment is related to graph properties such as [[Planarity (graph theory)|planarity]]. For example, [[Kuratowski's theorem|Kuratowski's Theorem]] states:
* A graph is [[Planar graph|planar]] if it contains as a subdivision neither the [[complete bipartite graph]] ''K''<sub>3,3</sub> nor the [[complete graph]] ''K''<sub>5</sub>.
Another problem in subdivision containment is the [[Kelmans–Seymour conjecture]]:
* Every [[K-vertex-connected graph|5-vertex-connected]] graph that is not [[Planar graph|planar]] contains a [[Homeomorphism (graph theory)|subdivision]] of the 5-vertex [[complete graph]] ''K''<sub>5</sub>.
Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their ''point-deleted subgraphs''. For example:
* The [[reconstruction conjecture]]
=== Graph coloring ===
{{main|Graph coloring}}
Many problems and theorems in graph theory have to do with various ways of coloring graphs. Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges (possibly so that no two coincident edges are the same color), or other variations. Among the famous results and conjectures concerning graph coloring are the following:
* [[Four-color theorem]]
* [[Strong perfect graph theorem]]
* [[Erdős–Faber–Lovász conjecture]]
* [[Total coloring|Total coloring conjecture]], also called [[Mehdi Behzad|Behzad]]'s conjecture (unsolved)
* [[List edge-coloring|List coloring conjecture]] (unsolved)
* [[Hadwiger conjecture (graph theory)]] (unsolved)
=== Subsumption and unification ===
Constraint modeling theories concern families of directed graphs related by a [[partial order]]. In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are subsumed by those that are more general. Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph (or the computation thereof) that is consistent with (i.e. contains all of the information in) the inputs, if such a graph exists; efficient unification algorithms are known.
For constraint frameworks which are strictly [[Principle of Compositionality|compositional]], graph unification is the sufficient satisfiability and combination function. Well-known applications include [[Automatic theorem prover|automatic theorem proving]] and modeling the [[Parsing|elaboration of linguistic structure]].
=== Route problems ===
* [[Hamiltonian path problem]]
* [[Minimum spanning tree]]
* [[Route inspection problem]] (also called the "Chinese postman problem")
* [[Seven bridges of Königsberg]]
* [[Shortest path problem]]
* [[Steiner tree]]
* [[Three-cottage problem]]
* [[Traveling salesman problem]] (NP-hard)
=== Network flow ===
There are numerous problems arising especially from applications that have to do with various notions of [[Flow network|flows in networks]], for example:
* [[Max flow min cut theorem]]
=== Visibility problems ===
* [[Museum guard problem]]
=== Covering problems ===
[[Covering problem]]s in graphs may refer to various [[Set cover problem| set cover problems]] on subsets of vertices/subgraphs.
* [[Dominating set]] problem is the special case of set cover problem where sets are the closed [[Neighbourhood (graph theory)|neighborhoods]].
* [[Vertex cover problem]] is the special case of set cover problem where sets to cover are every edges.
* The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph.
=== Decomposition problems ===
Decomposition, defined as partitioning the edge set of a graph (with as many vertices as necessary accompanying the edges of each part of the partition), has a wide variety of questions. Often, the problem is to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles. Other problems specify a family of graphs into which a given graph should be decomposed, for instance, a family of cycles, or decomposing a complete graph ''K''<sub>''n''</sub> into {{nobreak|''n'' − 1}} specified trees having, respectively, 1, 2, 3, ..., {{nobreak|''n'' − 1}} edges.
Some specific decomposition problems that have been studied include:
* [[Arboricity]], a decomposition into as few forests as possible
* [[Cycle double cover]], a decomposition into a collection of cycles covering each edge exactly twice
* [[Edge coloring]], a decomposition into as few [[matching (graph theory)|matching]]s as possible
* [[Graph factorization]], a decomposition of a [[regular graph]] into regular subgraphs of given degrees
=== Graph classes ===
Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:
* [[Graph enumeration|Enumerating]] the members of a class
* Characterizing a class in terms of [[Forbidden graph characterization|forbidden substructure]]s
* Ascertaining relationships among classes (e.g. does one property of graphs imply another)
* Finding efficient [[algorithm]]s to [[Decision problem|decide]] membership in a class
* Finding [[Representation (mathematics)|representation]]s for members of a class
== See also ==
* [[Gallery of named graphs]]
* [[Glossary of graph theory]]
* [[List of graph theory topics]]
* [[List of unsolved problems in graph theory]]
* [[List of publications in mathematics#Graph theory|Publications in graph theory]]
* [[Graph algorithm]]
* [[:Category:Graph theorists|Graph theorists]]
=== Subareas ===
* [[Algebraic graph theory]]
* [[Geometric graph theory]]
* [[Extremal graph theory]]
* [[Random graph|Probabilistic graph theory]]
* [[Topological graph theory]]
* [[Graph drawing]]
== Notes ==
{{Reflist|30em}}
== References ==
* {{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=Berge |first=Claude |date=1958 |title=Théorie des graphes et ses applications |location=Paris |publisher=Dunod }} English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition, Dover, New York 2001.
* {{cite book |last1=Biggs |first1=N. |last2=Lloyd |first2=E. |last3=Wilson |first3=R. |date=1986 |title=Graph Theory, 1736–1936|publisher=Oxford University Press }}
* {{cite book |last1=Bondy |first1=J. A. |last2=Murty |first2=U. S. R. |date=2008 |title=Graph Theory |publisher=Springer |isbn=978-1-84628-969-9 }}
* {{cite book |last1=Bollobás |first1=Béla |first2=O. M. |last2=Riordan |date=2003 |title=Mathematical results on scale-free random graphs in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)) |edition=1st |location=Weinheim |publisher=Wiley VCH }}
* {{cite book |last=Chartrand |first=Gary |date=1985 |title=Introductory Graph Theory |url=https://archive.org/details/introductorygrap0000char|url-access=registration |publisher=Dover |isbn=0-486-24775-9 }}
* {{cite book |last=Deo |first=Narsingh |date=1974 |title=Graph Theory with Applications to Engineering and Computer Science |url=https://www.edutechlearners.com/download/Graphtheory.pdf |archive-url=https://web.archive.org/web/20190517165158/http://www.edutechlearners.com/download/Graphtheory.pdf |archive-date=2019-05-17 |url-status=live |location=Englewood, New Jersey |publisher=Prentice-Hall |isbn=0-13-363473-6 }}
* {{cite book |last=Gibbons |first=Alan |date=1985 |title=Algorithmic Graph Theory |publisher=[[Cambridge University Press]] }}
* {{cite book |last=Golumbic |first=Martin |date=1980 |title=Algorithmic Graph Theory and Perfect Graphs |publisher=[[Academic Press]]}}
* {{cite book |last=Harary |first=Frank |date=1969 |title=Graph Theory |location=Reading, Massachusetts |publisher=Addison-Wesley }}
* {{cite book |last1=Harary |first1=Frank |last2=Palmer |first2=Edgar M. |date=1973 |title=Graphical Enumeration |location=New York, New York |publisher=Academic Press }}
* {{cite book |last1=Mahadev |first1=N. V. R. |last2=Peled |first2=Uri N. |date=1995 |title=Threshold Graphs and Related Topics |publisher=[[North-Holland Publishing Company|North-Holland]] }}
* {{cite book |last=Newman |first=Mark |date=2010 |title=Networks: An Introduction |publisher=Oxford University Press }}
* {{cite book |last1=Kepner |first1=Jeremy |last2=Gilbert |first2=John |date=2011 |title=Graph Algorithms in The Language of Linear Algebra |url=https://my.siam.org/Store/Product/viewproduct/?ProductId=106663 |location=Philadelphia, Pennsylvania |publisher=SIAM |isbn=978-0-898719-90-1 }}
== External links ==
{{Commons}}
* {{springer|title=Graph theory|id=p/g045010}}
* [http://www.utm.edu/departments/math/graph/ Graph theory tutorial] {{Webarchive|url=https://web.archive.org/web/20120116185332/http://www.utm.edu/departments/math/graph/ |date=2012-01-16 }}
* [http://www.gfredericks.com/main/sandbox/graphs A searchable database of small connected graphs]
* {{webarchive |url=https://web.archive.org/web/20060206155001/http://www.nd.edu/~networks/gallery.htm |date=February 6, 2006 |title=Image gallery: graphs }}
* [https://web.archive.org/web/20161114100939/http://www.babelgraph.org/links.html Concise, annotated list of graph theory resources for researchers]
* [http://www.kde.org/applications/education/rocs/ rocs] — a graph theory IDE
* [http://www.orgnet.com/SocialLifeOfRouters.pdf The Social Life of Routers] — non-technical paper discussing graphs of people and computers
* [http://graphtheorysoftware.com/ Graph Theory Software] — tools to teach and learn graph theory
* {{Library resources about |onlinebooks=yes |lcheading=Graph theory |label=graph theory}}
* [http://www.martinbroadhurst.com/Graph-algorithms.html A list of graph algorithms] {{Webarchive|url=https://web.archive.org/web/20190713044421/http://www.martinbroadhurst.com/Graph-algorithms.html |date=2019-07-13 }} with references and links to graph library implementations
=== Online textbooks ===
* [https://arxiv.org/abs/cond-mat/0602129 Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs] (2006) by Hartmann and Weigt
* [http://www.cs.rhul.ac.uk/books/dbook/ Digraphs: Theory Algorithms and Applications] 2007 by Jorgen Bang-Jensen and Gregory Gutin
* [http://diestel-graph-theory.com/index.html Graph Theory, by Reinhard Diestel]
{{Areas of mathematics}}
{{Computer science}}
{{Graph Analysis Software}}
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{{DEFAULTSORT:Graph Theory}}
[[Category:Graph theory| ]]