TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, graph theory is the study of ''
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
s'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''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 Discrete mathematics is the study of mathematical structures In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
.

# Definitions

Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related
mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s.

## Graph

In one restricted but very common sense of the term, a graph is an
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$G=\left(V,E\right)$ comprising: * $V$, a set of vertices (also called nodes or points); * $E \subseteq \$, a set of edges (also called links or lines), which are
unordered pairIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 $\$, the vertices $x$ and $y$ are called the endpoints of the edge. The edge is said to join $x$ and $y$ and to be incident on $x$ and on $y$. A vertex may exist in a graph and not belong to an edge.
Multiple edges Multiple edges joining two vertices. In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this conte ...

, not allowed under the definition above, are two or more edges that join the same two vertices. In one more general sense of the term allowing multiple edges, a graph is an ordered triple $G=\left(V,E,\phi\right)$ comprising: * $V$, a set of vertices (also called nodes or points); * $E$, a set of edges (also called links or lines); * $\phi : E \to \$, an incidence function mapping every edge to an
unordered pairIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. A 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 $x$ to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) $\ = \$ which is not in $\$. So to allow loops the definitions must be expanded. For undirected simple graphs, the definition of $E$ should be modified to $E \subseteq \$. For undirected multigraphs, the definition of $\phi$ should be modified to $\phi : E \to \$. To avoid ambiguity, these types of objects may be called undirected simple graph permitting loops and undirected multigraph permitting loops (sometimes also undirected ), respectively. $V$ 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 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 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 and the maximum size of the graph is . The edges of an undirected simple graph permitting loops $G$ induce a symmetric
homogeneous relation Homogeneity and heterogeneity are concepts often used in the sciences and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sc ...
~ on the vertices of $G$ that is called the adjacency relation of $G$. Specifically, for each edge $\left(x,y\right)$, its endpoints $x$ and $y$ are said to be adjacent to one another, which is denoted $x$ ~ $y$.

## Directed graph

A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, a directed graph is an ordered pair $G=\left(V,E\right)$ comprising: * $V$, a set of ''vertices'' (also called ''nodes'' or ''points''); * $E \subseteq \left\$, a set of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs'') which are
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

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 the edge $\left(x, y\right)$ directed from $x$ to $y$, the vertices $x$ and $y$ are called the ''endpoints'' of the edge, $x$ the ''tail'' of the edge and $y$ the ''head'' of the edge. The edge is said to ''join'' $x$ and $y$ and to be ''incident'' on $x$ and on $y$. A vertex may exist in a graph and not belong to an edge. The edge $\left(y,x\right)$ is called the ''inverted edge'' of $\left(x, y\right)$. ''
Multiple edges Multiple edges joining two vertices. In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this conte ...

'', 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, a directed graph is an ordered triple $G=\left(V,E,\phi\right)$ comprising: * $V$, a set of ''vertices'' (also called ''nodes'' or ''points''); * $E$, a set of ''edges'' (also called ''directed edges'', ''directed links'', ''directed lines'', ''arrows'' or ''arcs''); * $\phi : E \to \left\$, an ''incidence function'' mapping every edge to an
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

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'' 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 $x$ to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) $\left(x,x\right)$ which is not in $\left\$. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of $E$ should be modified to $E \subseteq \left\$. For directed multigraphs, the definition of $\phi$ should be modified to $\phi : E \to \left\$. 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 A quiver is a container for holding arrow s and nock. An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight ''shaft'' with a weighty (and usually sharp and pointed) ''arrowh ...
'') respectively. The edges of a directed simple graph permitting loops $G$ is a
homogeneous relation Homogeneity and heterogeneity are concepts often used in the sciences and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sc ...
~ on the vertices of $G$ that is called the ''adjacency relation'' of $G$. Specifically, for each edge $\left(x,y\right)$, its endpoints $x$ and $y$ are said to be ''adjacent'' to one another, which is denoted $x$ ~ $y$.

# Applications

Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. 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 the real-world systems as a network is called
network science Network science is an academic field which studies complex network In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as ...
.

## Computer science

In
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a
website A website (also written as web site) is a collection of web page A web page (or webpage) is a hypertext File:Douglas Engelbart in 2008.jpg, Douglas Engelbart in 2009, at the 40th anniversary celebrations of "The Mother of All Demos" i ...

can be represented by a directed graph, in which the vertices represent web pages and directed edges represent
links Link or Links may refer to: Places * Link, West Virginia, an unincorporated community in the US * Link River, Klamath Falls, Oregon, US People with the name * Link (singer) (Lincoln Browder, born 1964), American R&B singer * Link (surname) * ...

from one page to another. A similar approach can be taken to problems in social media, travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases, and many other fields. The development of
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s to handle graphs is therefore of major interest in computer science. The transformation of graphs is often formalized and represented by graph rewrite systems. Complementary to
graph transformation In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorit ...
systems focusing on rule-based in-memory manipulation of graphs are
graph database In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwa ...
s geared towards transaction-safe, persistent storing and querying of graph-structured data.

## Linguistics

Graph-theoretic methods, in various forms, have proven particularly useful in
linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the methods for studying ...

, since natural language often lends itself well to discrete structure. Traditionally,
syntax In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the ...

and compositional semantics follow tree-based structures, whose expressive power lies in the
principle of compositionality In semantics Semantics (from grc, wikt:σημαντικός, σημαντικός ''sēmantikós'', "significant") is the study of meaning, reference, or truth. The term can be used to refer to subfields of several distinct disciplines, includi ...
, modeled in a hierarchical graph. More contemporary approaches such as
head-driven phrase structure grammarHead-driven phrase structure grammar (HPSG) is a highly lexicalized, constraint-based grammar developed by Carl Pollard and Ivan Sag. It is a type of phrase structure grammar The term phrase structure grammar was originally introduced by Noam Cho ...
model the syntax of natural language using typed feature structures, which are
directed acyclic graph In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s. Within
lexical semantics Lexical semantics (also known as lexicosemantics), as a subfield of linguistic Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modelin ...
, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words;
semantic network A semantic network, or frame network is a knowledge base A knowledge base (KB) is a technology used to information storage, store complex structured data, structured and unstructured information used by a computer system. The initial use of ...
s are therefore important in
computational linguistics Computational linguistics is an interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, computational linguistics ...
. Still, other methods in phonology (e.g.
optimality theory In linguistics, Optimality Theory (frequently abbreviated OT) is a linguistic model proposing that the observed forms of language arise from the optimal satisfaction of conflicting constraints. OT differs from other approaches to phonological a ...
, which uses
lattice graph A lattice graph, mesh graph, or grid graph, is a graph whose drawing Drawing is a form of visual art in which an artist uses instruments to mark paper Paper is a thin sheet material produced by mechanically and/or chemically processing ...
s) and morphology (e.g. finite-state morphology, using
finite-state transducer A finite-state transducer (FST) is a finite-state machine with two memory ''tapes'', following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. ...
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 a
TextGraphs
as well as various 'Net' projects, such as
WordNet WordNet is a lexical database of semantic relations between words in more than 200 languages. WordNet links words into semantic relations including synonyms, hyponyms, and meronyms. The synonyms are grouped into ''synsets'' with short definitions ...
,
VerbNetThe VerbNet project maps PropBank verb types to their corresponding Beth Levin (linguist), Levin classes. It is a lexical resource that incorporates both semantic and syntactic information about its contents. VerbNet is part of thSemLinkproject in d ...
, and others.

## Physics and chemistry

Graph theory is also used to study molecules in
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

. In
condensed matter physics Condensed matter physics is the field of that deals with the macroscopic and microscopic physical properties of , especially the and which arise from forces between s. More generally, the subject deals with "condensed" phases of matter: syst ...
, 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 graphs and rules of calculation summarize
quantum field theory In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in co ...
in a form in close contact with the experimental numbers one wants to understand." In chemistry a graph makes a natural model for a molecule, where vertices represent
atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

s and edges
bond Bond or bonds may refer to: Common meanings * Bond (finance) In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of ...
s. This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. In
statistical physics Statistical physics is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (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 Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, theoretical analysis and abstractions of the brain to understand the principle ...
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. Graphs are also used to represent the micro-scale channels of
porous media A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid In physics, a fluid is a substance that cont ...

, in which the vertices represent the pores and the edges represent the smaller channels connecting the pores.
Chemical graph theoryChemical graph theory is the topology (chemistry), topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena. The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván ...
uses the
molecular graph In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. A chemical graph is a labeled graph whose vertices correspond ...
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 In statistical physics Statistical physics is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matte ...
.

## Social sciences

Graph theory is also widely used in
sociology Sociology is a social science Social science is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the scie ...
as a way, for example, to measure actors' prestige or to explore rumor spreading, notably through the use of social network analysis software. Under the umbrella of social networks are many different types of graphs. 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 Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

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 Molecular biology is the branch of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, P ...
and
genomics Genomics is an interdisciplinary field of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interact ...
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 transcriptome analysis. Another use is to model genes or proteins in a pathway and study the relationships between them, such as metabolic pathways and gene regulatory networks. Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures. Graph theory is also used in
connectomics Connectomics is the production and study of connectomes: comprehensive maps of connections within an organism In biology, an organism (from Ancient Greek, Greek: ὀργανισμός, ''organismos'') is any individual contiguous system th ...

; 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 In the mathematical field of topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...
.
Algebraic graph theory Algebraic may refer to any subject related to algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geomet ...
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
. 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 This is a glossary of graph theory. Graph theory is the study of graph (discrete mathematics), graphs, systems of nodes or vertex (graph theory), vertices connected in pairs by lines or #edge, edges. Symbols A ...
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

The paper written by
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

on the
Seven Bridges of Königsberg The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographe ...
and published in 1736 is regarded as the first paper in the history of graph theory. This paper, as well as the one written by Vandermonde on the '','' carried on with the ''analysis situs'' initiated by
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

and L'Huilier, and represents the beginning of the branch of mathematics known as
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

. More than one century after Euler's paper on the bridges of
Königsberg Königsberg (, , ) was the name for the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusade ...

and while Listing was introducing the concept of topology, was led by an interest in particular analytical forms arising from
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
to study a particular class of graphs, the ''
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with se ...
s''. This study had many implications for theoretical
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

. 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 Pólya between 1935 and 1937. These were generalized by De Bruijn in 1959. Cayley linked his results on trees with contemporary studies of chemical composition. 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 in a paper published in 1878 in ''
Nature Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter an ...
'', where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams: :" Every invariant and co-variant thus becomes expressible by a ''graph'' precisely identical with a 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 Dénes Kőnig (September 21, 1884 – October 19, 1944) was a Hungarian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
, and published in 1936. Another book by
Frank Harary Frank Harary (March 11, 1921 – January 4, 2005) was an American mathematician, who specialized in graph theory. He was widely recognized as one of the "fathers" of modern graph theory. Harary was a master of clear exposition and, together with ...
, published in 1969, was "considered the world over to be the definitive textbook on the subject", and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other. Harary donated all of the royalties to fund the Pólya Prize. One of the most famous and stimulating problems in graph theory is the
four color problem In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into wikt:contiguity, contiguous regions, producing a figure called a ''map'', no more than four colors are required to color the r ...
: "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 Francis Guthrie (born 22 January 1831 in London London is the capital city, capital and List of urban areas in the United Kingdom, largest city of England and the United Kingdom. The city stands on the River Thames in the south-east of Eng ...
in 1852 and its first written record is in a letter of
De Morgan De Morgan or de Morgan is a surname, and may refer to: *Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematic ...

HamiltonHamilton may refer to: * Alexander Hamilton (1755–1804), first American Secretary of the Treasury and one of the Founding Fathers of the United States **Hamilton (musical), ''Hamilton'' (musical), a 2015 Broadway musical written by Lin-Manuel Mira ...
the same year. Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait, Heawood,
Ramsey Ramsey may refer to: Geography British Isles * Ramsey, Cambridgeshire Ramsey is a market town and civil parish In England, a civil parish is a type of administrative parish used for local government. It is a territorial designation whic ...
and
Hadwiger Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Ger ...

led to the study of the colorings of the graphs embedded on surfaces with arbitrary
genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology, taxonomy () is the scientific study of naming, defining (Circumscription (taxonomy), circumscribing) and classifying gr ...
. Tait's reformulation generated a new class of problems, the ''factorization problems'', particularly studied by Petersen and Kőnig. The works of Ramsey on colorations and more specially the results obtained by 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 Heinrich Heesch (June 25, 1906 – July 26, 1995) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
published a method for solving the problem using computers. A computer-aided proof produced in 1976 by
Kenneth Appel Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America ...
and
Wolfgang Haken Wolfgang Haken (born June 21, 1928) is a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathem ...

makes fundamental use of the notion of "discharging" developed by Heesch. 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 Robertson,
Seymour Seymour may refer to: Places Australia *Seymour, Victoria, a township *Electoral district of Seymour, a former electoral district in Victoria *Rural City of Seymour, a former local government area in Victoria *Seymour, Tasmania, a locality C ...
, Daniel P. Sanders, Sanders and Robin Thomas (mathematician), Thomas. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 circuits. 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

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, intersection graph, and other visualizations of the adjacency matrix.

## Tabular: Graph data structures

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 In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

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 graphs 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. 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 trees 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. Unfortunately, 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 subgraphs 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''3,3 (see the Three-cottage problem) nor the complete graph ''K''5. 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''3,3 nor the complete graph ''K''5. 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''5. 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

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 (unsolved) * 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 problems 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''''n'' into specified trees having, respectively, 1, 2, 3, ..., 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), matchings 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 substructures * Ascertaining relationships among classes (e.g. does one property of graphs imply another) * Finding efficient
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s to Decision problem, decide membership in a class * Finding Representation (mathematics), representations for members of a class

* 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

## Related topics

*
Algebraic graph theory Algebraic may refer to any subject related to algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geomet ...
* Citation graph * Conceptual graph * Data structure * Disjoint-set data structure * Dual-phase evolution * Entitative graph * Existential graph * Graph algebra * Graph automorphism * Graph coloring * Graph database * Graph (data structure), Graph data structure * Graph drawing * Graph equation * Graph rewriting * Graph sandwich problem * Graph property * Intersection graph * Knight's Tour * Logical graph * Loop (graph theory), Loop * Network theory * Null graph * Pebble motion problems * Percolation * Perfect graph * Quantum graph * Random regular graphs * Semantic networks * Spectral graph theory * Strongly regular graphs * Symmetric graphs * Transitive reduction * Tree (data structure), Tree data structure

## Algorithms

* Bellman–Ford algorithm * Borůvka's algorithm * Breadth-first search * Depth-first search * Dijkstra's algorithm * Edmonds–Karp algorithm * Floyd–Warshall algorithm * Ford–Fulkerson algorithm * Hopcroft–Karp algorithm * Hungarian algorithm * Kosaraju's algorithm * Kruskal's algorithm * Nearest neighbour algorithm * Network simplex algorithm * Planarity testing#Algorithms, Planarity testing algorithms * Prim's algorithm * Push–relabel maximum flow algorithm * Tarjan's strongly connected components algorithm * Topological sorting

## Subareas

*
Algebraic graph theory Algebraic may refer to any subject related to algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geomet ...
* Geometric graph theory * Extremal graph theory * Random graph, Probabilistic graph theory * Topological graph theory

## Related areas of mathematics

* Combinatorics * Group theory * Knot theory * Ramsey theory

## Generalizations

* Hypergraph * Abstract simplicial complex

## Prominent graph theorists

* Noga Alon, Alon, Noga * Claude Berge, Berge, Claude * Béla Bollobás, Bollobás, Béla * John Adrian Bondy, Bondy, Adrian John * Graham Brightwell, Brightwell, Graham * Maria Chudnovsky, Chudnovsky, Maria * Fan Chung, Chung, Fan * Gabriel Andrew Dirac, Dirac, Gabriel Andrew * Paul Erdős, Erdős, Paul * Leonhard Euler, Euler, Leonhard * Ralph Faudree, Faudree, Ralph * Herbert Fleischner, Fleischner, Herbert * Martin Charles Golumbic, Golumbic, Martin * Ronald Graham, Graham, Ronald * Frank Harary, Harary, Frank * Percy John Heawood, Heawood, Percy John * Anton Kotzig, Kotzig, Anton * Dénes Kőnig, Kőnig, Dénes * László Lovász, Lovász, László * U. S. R. Murty, Murty, U. S. R. * Jaroslav Nešetřil, Nešetřil, Jaroslav * Alfréd Rényi, Rényi, Alfréd * Gerhard Ringel, Ringel, Gerhard * Neil Robertson (mathematician), Robertson, Neil * Paul Seymour (mathematician), Seymour, Paul * Benny Sudakov, Sudakov, Benny * Endre Szemerédi, Szemerédi, Endre * Robin Thomas (mathematician), Thomas, Robin * Carsten Thomassen, Thomassen, Carsten * Pál Turán, Turán, Pál * W. T. Tutte, Tutte, W. T. * Hassler Whitney, Whitney, Hassler

# References

* * 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. * * * * * * * * * * * * *

*
Graph theory tutorial

A searchable database of small connected graphs
*

* [http://www.kde.org/applications/education/rocs/ rocs] — a graph theory IDE
The Social Life of Routers
— non-technical paper discussing graphs of people and computers
Graph Theory Software
— tools to teach and learn graph theory *

with references and links to graph library implementations

## Online textbooks

Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs
(2006) by Hartmann and Weigt
Digraphs: Theory Algorithms and Applications
2007 by Jorgen Bang-Jensen and Gregory Gutin

{{DEFAULTSORT:Graph Theory Graph theory,