# Graph theory

For graphs of mathematical functions, see Graph of a function. Graph theory_sentence_1

For other uses, see Graph (disambiguation). Graph theory_sentence_2

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Graph theory_sentence_3

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). Graph theory_sentence_4

A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graph theory_sentence_5

Graphs are one of the prime objects of study in discrete mathematics. Graph theory_sentence_6

Refer to the glossary of graph theory for basic definitions in graph theory. Graph theory_sentence_7

## Definitions Graph theory_section_0

Definitions in graph theory vary. Graph theory_sentence_8

The following are some of the more basic ways of defining graphs and related mathematical structures. Graph theory_sentence_9

### Graph Graph theory_section_1

To avoid ambiguity, this type of object may be called precisely an undirected simple graph. Graph theory_sentence_10

To avoid ambiguity, this type of object may be called precisely an undirected multigraph. Graph theory_sentence_11

In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1)/2. Graph theory_sentence_12

### Directed graph Graph theory_section_2

Main article: Directed graph Graph theory_sentence_13

A directed graph or digraph is a graph in which edges have orientations. Graph theory_sentence_14

To avoid ambiguity, this type of object may be called precisely a directed simple graph. Graph theory_sentence_15

To avoid ambiguity, this type of object may be called precisely a directed multigraph. Graph theory_sentence_16

## Applications Graph theory_section_3

Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. Graph theory_sentence_17

Many practical problems can be represented by graphs. Graph theory_sentence_18

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. Graph theory_sentence_19

### Computer science Graph theory_section_4

In 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 can be represented by a directed graph, in which the vertices represent web pages and directed edges represent links from one page to another. Graph theory_sentence_20

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. Graph theory_sentence_21

The development of algorithms to handle graphs is therefore of major interest in computer science. Graph theory_sentence_22

The transformation of graphs is often formalized and represented by graph rewrite systems. Graph theory_sentence_23

Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data. Graph theory_sentence_24

### Linguistics Graph theory_section_5

Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural language often lends itself well to discrete structure. Graph theory_sentence_25

Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the principle of compositionality, modeled in a hierarchical graph. Graph theory_sentence_26

More contemporary approaches such as head-driven phrase structure grammar model the syntax of natural language using typed feature structures, which are directed acyclic graphs. Graph theory_sentence_27

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 networks are therefore important in computational linguistics. Graph theory_sentence_28

Still, other methods in phonology (e.g. optimality theory, which uses lattice graphs) and morphology (e.g. finite-state morphology, using finite-state transducers) are common in the analysis of language as a graph. Graph theory_sentence_29

Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as , as well as various 'Net' projects, such as WordNet, VerbNet, and others. Graph theory_sentence_30

### Physics and chemistry Graph theory_section_6

Graph theory is also used to study molecules in chemistry and physics. Graph theory_sentence_31

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. Graph theory_sentence_32

Also, "the Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand." Graph theory_sentence_33

In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. Graph theory_sentence_34

This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. Graph theory_sentence_35

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. Graph theory_sentence_36

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_sentence_37

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. Graph theory_sentence_38

Graphs are also used to represent the micro-scale channels of porous media, in which the vertices represent the pores and the edges represent the smaller channels connecting the pores. Graph theory_sentence_39

Chemical graph theory uses the molecular graph as a means to model molecules. Graph theory_sentence_40

Graphs and networks are excellent models to study and understand phase transitions and critical phenomena. Graph theory_sentence_41

Removal of nodes or edges lead to a critical transition where the network breaks into small clusters which is studied as a phase transition. Graph theory_sentence_42

This breakdown is studied via percolation theory. Graph theory_sentence_43

### Social sciences Graph theory_section_7

Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreading, notably through the use of social network analysis software. Graph theory_sentence_44

Under the umbrella of social networks are many different types of graphs. Graph theory_sentence_45

Acquaintanceship and friendship graphs describe whether people know each other. Graph theory_sentence_46

Influence graphs model whether certain people can influence the behavior of others. Graph theory_sentence_47

Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together. Graph theory_sentence_48

### Biology Graph theory_section_8

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. Graph theory_sentence_49

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. Graph theory_sentence_50

Graphs are also commonly used in molecular biology and genomics to model and analyse datasets with complex relationships. Graph theory_sentence_51

For example, graph-based methods are often used to 'cluster' cells together into cell-types in single-cell transcriptome analysis. Graph theory_sentence_52

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. Graph theory_sentence_53

Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures. Graph theory_sentence_54

Graph-based methods are pervasive that researchers in some fields of biology and these will only become far more widespread as technology develops to leverage this kind of high-throughout multidimensional data. Graph theory_sentence_55

Graph theory is also used in connectomics; nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them. Graph theory_sentence_56

### Mathematics Graph theory_section_9

In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Graph theory_sentence_57

Algebraic graph theory has close links with group theory. Graph theory_sentence_58

Algebraic graph theory has been applied to many areas including dynamic systems and complexity. Graph theory_sentence_59

### Other topics Graph theory_section_10

A graph structure can be extended by assigning a weight to each edge of the graph. Graph theory_sentence_60

Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. Graph theory_sentence_61

For example, if a graph represents a road network, the weights could represent the length of each road. Graph theory_sentence_62

There may be several weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. Graph theory_sentence_63

Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs. Graph theory_sentence_64

## History Graph theory_section_11

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. Graph theory_sentence_65

This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Graph theory_sentence_66

Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L'Huilier, and represents the beginning of the branch of mathematics known as topology. Graph theory_sentence_67

More than one century after Euler's paper on the bridges of Königsberg and while Listing was introducing the concept of topology, Cayley was led by an interest in particular analytical forms arising from differential calculus to study a particular class of graphs, the trees. Graph theory_sentence_68

This study had many implications for theoretical chemistry. Graph theory_sentence_69

The techniques he used mainly concern the enumeration of graphs with particular properties. Graph theory_sentence_70

Enumerative graph theory then arose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937. Graph theory_sentence_71

These were generalized by De Bruijn in 1959. Graph theory_sentence_72

Cayley linked his results on trees with contemporary studies of chemical composition. Graph theory_sentence_73

The fusion of ideas from mathematics with those from chemistry began what has become part of the standard terminology of graph theory. Graph theory_sentence_74

In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature, where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams: Graph theory_sentence_75

Graph theory_description_list_0

• "[…] 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).Graph theory_item_0_0

The first textbook on graph theory was written by Dénes Kőnig, and published in 1936. Graph theory_sentence_76

Another book by Frank Harary, 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. Graph theory_sentence_77

Harary donated all of the royalties to fund the Pólya Prize. Graph theory_sentence_78

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?" Graph theory_sentence_79

This problem was first posed by Francis Guthrie in 1852 and its first written record is in a letter of De Morgan addressed to Hamilton the same year. Graph theory_sentence_80

Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. Graph theory_sentence_81

The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. Graph theory_sentence_82

Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Kőnig. Graph theory_sentence_83

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. Graph theory_sentence_84

The four color problem remained unsolved for more than a century. Graph theory_sentence_85

In 1969 Heinrich Heesch published a method for solving the problem using computers. Graph theory_sentence_86

A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of "discharging" developed by Heesch. Graph theory_sentence_87

The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. Graph theory_sentence_88

A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas. Graph theory_sentence_89

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Graph theory_sentence_90

Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. Graph theory_sentence_91

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 current in electric circuits. Graph theory_sentence_92

The introduction of probabilistic methods in graph theory, especially in the study of Erdős and Rényi of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as random graph theory, which has been a fruitful source of graph-theoretic results. Graph theory_sentence_93

## Graph drawing Graph theory_section_12

Main article: Graph drawing Graph theory_sentence_94

Graphs are 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. Graph theory_sentence_95

If the graph is directed, the direction is indicated by drawing an arrow. Graph theory_sentence_96

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. Graph theory_sentence_97

All that matters is which vertices are connected to which others by how many edges and not the exact layout. Graph theory_sentence_98

In practice, it is often difficult to decide if two drawings represent the same graph. Graph theory_sentence_99

Depending on the problem domain some layouts may be better suited and easier to understand than others. Graph theory_sentence_100

The pioneering work of W. Graph theory_sentence_101 T. Tutte was very influential on the subject of graph drawing. Graph theory_sentence_102

Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings. Graph theory_sentence_103

Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. Graph theory_sentence_104

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. Graph theory_sentence_105

For a planar graph, the crossing number is zero by definition. Graph theory_sentence_106

Drawings on surfaces other than the plane are also studied. Graph theory_sentence_107

## Graph-theoretic data structures Graph theory_section_13

Main article: Graph (abstract data type) Graph theory_sentence_108

There are different ways to store graphs in a computer system. Graph theory_sentence_109

The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. Graph theory_sentence_110

Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. Graph theory_sentence_111

List structures are often preferred for sparse graphs as they have smaller memory requirements. Graph theory_sentence_112

Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Graph theory_sentence_113

Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation. Graph theory_sentence_114

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. Graph theory_sentence_115

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. Graph theory_sentence_116

In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. Graph theory_sentence_117

The degree matrix indicates the degree of vertices. Graph theory_sentence_118

The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph. Graph theory_sentence_119

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. Graph theory_sentence_120

## Problems Graph theory_section_14

### Enumeration Graph theory_section_15

There is a large literature on graphical enumeration: the problem of counting graphs meeting specified conditions. Graph theory_sentence_121

Some of this work is found in Harary and Palmer (1973). Graph theory_sentence_122

### Subgraphs, induced subgraphs, and minors Graph theory_section_16

A common problem, called the subgraph isomorphism problem, is finding a fixed graph as a subgraph in a given graph. Graph theory_sentence_123

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. Graph theory_sentence_124

Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem. Graph theory_sentence_125

For example: Graph theory_sentence_126

Graph theory_unordered_list_1

• Finding the largest complete subgraph is called the clique problem (NP-complete).Graph theory_item_1_1

One special case of subgraph isomorphism is the graph isomorphism problem. Graph theory_sentence_127

It asks whether two graphs are isomorphic. Graph theory_sentence_128

It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time. Graph theory_sentence_129

A similar problem is finding induced subgraphs in a given graph. Graph theory_sentence_130

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. Graph theory_sentence_131

Finding maximal induced subgraphs of a certain kind is also often NP-complete. Graph theory_sentence_132

For example: Graph theory_sentence_133

Graph theory_unordered_list_2

Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. Graph theory_sentence_134

A minor or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges. Graph theory_sentence_135

Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. Graph theory_sentence_136

For example, Wagner's Theorem states: Graph theory_sentence_137

Graph theory_unordered_list_3

A similar problem, the subdivision containment problem, is to find a fixed graph as a subdivision of a given graph. Graph theory_sentence_138

A subdivision or homeomorphism of a graph is any graph obtained by subdividing some (or no) edges. Graph theory_sentence_139

Subdivision containment is related to graph properties such as planarity. Graph theory_sentence_140

For example, Kuratowski's Theorem states: Graph theory_sentence_141

Graph theory_unordered_list_4

Another problem in subdivision containment is the Kelmans–Seymour conjecture: Graph theory_sentence_142

Graph theory_unordered_list_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. Graph theory_sentence_143

For example: Graph theory_sentence_144

Graph theory_unordered_list_6

### Graph coloring Graph theory_section_17

Main article: Graph coloring Graph theory_sentence_145

Many problems and theorems in graph theory have to do with various ways of coloring graphs. Graph theory_sentence_146

Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. Graph theory_sentence_147

One may also consider coloring edges (possibly so that no two coincident edges are the same color), or other variations. Graph theory_sentence_148

Among the famous results and conjectures concerning graph coloring are the following: Graph theory_sentence_149

Graph theory_unordered_list_7

### Subsumption and unification Graph theory_section_18

Constraint modeling theories concern families of directed graphs related by a partial order. Graph theory_sentence_150

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. Graph theory_sentence_151

Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. Graph theory_sentence_152

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. Graph theory_sentence_153

For constraint frameworks which are strictly compositional, graph unification is the sufficient satisfiability and combination function. Graph theory_sentence_154

Well-known applications include automatic theorem proving and modeling the elaboration of linguistic structure. Graph theory_sentence_155

### Route problems Graph theory_section_19

Graph theory_unordered_list_8

### Network flow Graph theory_section_20

There are numerous problems arising especially from applications that have to do with various notions of flows in networks, for example: Graph theory_sentence_156

Graph theory_unordered_list_9

### Visibility problems Graph theory_section_21

Graph theory_unordered_list_10

### Covering problems Graph theory_section_22

Covering problems in graphs may refer to various set cover problems on subsets of vertices/subgraphs. Graph theory_sentence_157

Graph theory_unordered_list_11

• Dominating set problem is the special case of set cover problem where sets are the closed neighborhoods.Graph theory_item_11_23
• Vertex cover problem is the special case of set cover problem where sets to cover are every edges.Graph theory_item_11_24
• The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph.Graph theory_item_11_25

### Decomposition problems Graph theory_section_23

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 question. Graph theory_sentence_158

Often, it is required to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles. Graph theory_sentence_159

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 Kn into n − 1 specified trees having, respectively, 1, 2, 3, ..., n − 1 edges. Graph theory_sentence_160

Some specific decomposition problems that have been studied include: Graph theory_sentence_161

Graph theory_unordered_list_12

• Arboricity, a decomposition into as few forests as possibleGraph theory_item_12_26
• Cycle double cover, a decomposition into a collection of cycles covering each edge exactly twiceGraph theory_item_12_27
• Edge coloring, a decomposition into as few matchings as possibleGraph theory_item_12_28
• Graph factorization, a decomposition of a regular graph into regular subgraphs of given degreesGraph theory_item_12_29

### Graph classes Graph theory_section_24

Many problems involve characterizing the members of various classes of graphs. Graph theory_sentence_162

Some examples of such questions are below: Graph theory_sentence_163

Graph theory_unordered_list_13

• Enumerating the members of a classGraph theory_item_13_30
• Characterizing a class in terms of forbidden substructuresGraph theory_item_13_31
• Ascertaining relationships among classes (e.g. does one property of graphs imply another)Graph theory_item_13_32
• Finding efficient algorithms to decide membership in a classGraph theory_item_13_33
• Finding representations for members of a classGraph theory_item_13_34