In mathematics, **graph theory** is the study of * graphs *, 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; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

- Definitions
- Graph
- Directed graph
- Applications
- Computer science
- Linguistics
- Physics and chemistry
- Social sciences
- Biology
- Mathematics
- Other topics
- History
- Graph drawing
- Graph-theoretic data structures
- Problems
- Enumeration
- Subgraphs, induced subgraphs, and minors
- Graph coloring
- Subsumption and unification
- Route problems
- Network flow
- Visibility problems
- Covering problems
- Decomposition problems
- Graph classes
- See also
- Related topics
- Algorithms
- Subareas
- Related areas of mathematics
- Generalizations
- Prominent graph theorists
- Notes
- References
- External links
- Online textbooks

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

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

In one restricted but very common sense of the term,^{ [1] }^{ [2] } a **graph** is an ordered pair comprising:

- , a set of
*vertices*(also called*nodes*or*points*); - , a set of
*edges*(also called*links*or*lines*), which are unordered pairs 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 and are called the *endpoints* of the edge. The edge is said to *join* and and to be *incident* on and on . A vertex may exist in a graph and not belong to an edge. * Multiple edges *, 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,^{ [3] }^{ [4] } a **graph** is an ordered triple comprising:

- , a set of
*vertices*(also called*nodes*or*points*); - , a set of
*edges*(also called*links*or*lines*); - , 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 * 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 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 should be modified to . For undirected multigraphs, the definition of should be modified to . To avoid ambiguity, these types of objects may be called **undirected simple graph permitting loops** and **undirected multigraph permitting loops**, respectively.

and 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, is often assumed to be non-empty, but is allowed to be the empty set. The *order* of a graph is , its number of vertices. The *size* of a graph is , 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.

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.

The edges of an undirected simple graph permitting loops induce a symmetric homogeneous relation ~ on the vertices of that is called the *adjacency relation* of . Specifically, for each edge , its endpoints and are said to be *adjacent* to one another, which is denoted ~ .

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

In one restricted but very common sense of the term,^{ [5] } a **directed graph** is an ordered pair comprising:

- , a set of
*vertices*(also called*nodes*or*points*); - , a set of
*edges*(also called*directed edges*,*directed links*,*directed lines*,*arrows*or*arcs*) which are ordered pairs 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 directed from to , the vertices and are called the *endpoints* of the edge, the *tail* of the edge and the *head* of the edge. The edge is said to *join* and and to be *incident* on and on . A vertex may exist in a graph and not belong to an edge. The edge is called the *inverted edge* of . * 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,^{ [5] } a **directed graph** is an ordered triple comprising:

- , a set of
*vertices*(also called*nodes*or*points*); - , a set of
*edges*(also called*directed edges*,*directed links*,*directed lines*,*arrows*or*arcs*); - , 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 * 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 to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) which is not in . So to allow loops the definitions must be expanded. For directed simple graphs, the definition of should be modified to . For directed multigraphs, the definition of should be modified to . 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 *) respectively.

The edges of a directed simple graph permitting loops is a homogeneous relation ~ on the vertices of that is called the *adjacency relation* of . Specifically, for each edge , its endpoints and are said to be *adjacent* to one another, which is denoted ~ .

Graphs can be used to model many types of relations and processes in physical, biological,^{ [7] }^{ [8] } 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.

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. A similar approach can be taken to problems in social media,^{ [9] } travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases,^{ [10] }^{ [11] } and many other fields. The development of algorithms 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 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-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 typed feature structures, which are directed acyclic graphs. 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. 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. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as TextGraphs, as well as various 'Net' projects, such as WordNet, VerbNet, and others.

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 graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand."^{ [12] } In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. This approach is especially used in computer processing of molecular structures, ranging from chemical editors 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.^{ [13] } 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. 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 lead 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.^{ [14] }^{ [15] }

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. Under the umbrella of social networks are many different types of graphs.^{ [17] } 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.

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 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 ^{ [18] }. Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures. 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 is also used in connectomics;^{ [19] } nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.

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.

A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, 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.

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.^{ [20] } This paper, as well as the one written by Vandermonde on the * knight problem,* carried on with the *analysis situs* initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy ^{ [21] } and L'Huilier,^{ [22] } 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 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 *.^{ [23] } 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 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.^{ [24] } 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 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:^{ [25] }

- "[…] 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, and published in 1936.^{ [26] } Another book by Frank Harary, published in 1969, was "considered the world over to be the definitive textbook on the subject",^{ [27] } 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.^{ [28] }

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 De Morgan addressed to Hamilton 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 and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. 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 published a method for solving the problem using computers.^{ [29] } A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of "discharging" developed by Heesch.^{ [30] }^{ [31] } 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, Sanders and Thomas.^{ [32] }

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and 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 current in electric circuits.

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.

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. If the graph is directed, the direction is indicated by drawing an 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 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 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 graphs as they have smaller memory requirements. 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.^{ [33] }

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 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.

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).

A common problem, called the subgraph isomorphism problem, is finding a fixed graph as a 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 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 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 states:

- A graph is 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 of a given graph. A subdivision or homeomorphism of a graph is any graph obtained by subdividing some (or no) edges. Subdivision containment is related to graph properties such as planarity. For example, Kuratowski's Theorem states:

- A graph is 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 5-vertex-connected graph that is not planar contains a 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:

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 conjecture, also called Behzad's conjecture (unsolved)
- List coloring conjecture (unsolved)
- Hadwiger conjecture (graph theory) (unsolved)

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 compositional, graph unification is the sufficient satisfiability and combination function. Well-known applications include automatic theorem proving and modeling the elaboration of linguistic structure.

- 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)

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

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

- Dominating set problem is the special case of set cover problem where sets are the closed 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, 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. Often, it is required 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 *n* − 1 specified trees having, respectively, 1, 2, 3, ..., *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 matchings as possible
- Graph factorization, a decomposition of a regular graph into regular subgraphs of given degrees

Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:

- Enumerating the members of a class
- Characterizing a class in terms of forbidden substructures
- Ascertaining relationships among classes (e.g. does one property of graphs imply another)
- Finding efficient algorithms to decide membership in a class
- Finding 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
- Publications in graph theory

- Algebraic graph theory
- 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 drawing
- Graph equation
- Graph rewriting
- Graph sandwich problem
- Graph property
- Intersection graph
- Knight's Tour
- Logical graph
- 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

- 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
- Prim's algorithm
- Push–relabel maximum flow algorithm
- Tarjan's strongly connected components algorithm
- Topological sorting

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

- ↑ Bender & Williamson 2010, p. 148.
- ↑ See, for instance, Iyanaga and Kawada,
*69 J*, p. 234 or Biggs, p. 4. - ↑ Bender & Williamson 2010, p. 149.
- ↑ See, for instance, Graham et al., p. 5.
- 1 2 Bender & Williamson 2010, p. 161.
- ↑ Hale, Scott A. (2013). "Multilinguals and Wikipedia Editing".
*Proceedings of the 2014 ACM Conference on Web Science - WebSci '14*: 99–108. arXiv: 1312.0976 . Bibcode:2013arXiv1312.0976H. doi:10.1145/2615569.2615684. ISBN 9781450326223. - ↑ Mashaghi, A.; et al. (2004). "Investigation of a protein complex network".
*European Physical Journal B*.**41**(1): 113–121. arXiv: cond-mat/0304207 . Bibcode:2004EPJB...41..113M. doi:10.1140/epjb/e2004-00301-0. - ↑ Shah, Preya; Ashourvan, Arian; Mikhail, Fadi; Pines, Adam; Kini, Lohith; Oechsel, Kelly; Das, Sandhitsu R; Stein, Joel M; Shinohara, Russell T (2019-07-01). "Characterizing the role of the structural connectome in seizure dynamics".
*Brain*.**142**(7): 1955–1972. doi:10.1093/brain/awz125. ISSN 0006-8950. PMC 6598625 . - ↑ Grandjean, Martin (2016). "A social network analysis of Twitter: Mapping the digital humanities community" (PDF).
*Cogent Arts & Humanities*.**3**(1): 1171458. doi:10.1080/23311983.2016.1171458. - ↑ Vecchio, F (2017). ""Small World" architecture in brain connectivity and hippocampal volume in Alzheimer's disease: a study via graph theory from EEG data".
*Brain Imaging and Behavior*.**11**(2): 473–485. doi:10.1007/s11682-016-9528-3. PMID 26960946. - ↑ Vecchio, F (2013). "Brain network connectivity assessed using graph theory in frontotemporal dementia".
*Neurology*.**81**(2): 134–143. doi:10.1212/WNL.0b013e31829a33f8. PMID 23719145. - ↑ Bjorken, J. D.; Drell, S. D. (1965).
*Relativistic Quantum Fields*. New York: McGraw-Hill. p. viii. - ↑ Kumar, Ankush; Kulkarni, G. U. (2016-01-04). "Evaluating conducting network based transparent electrodes from geometrical considerations".
*Journal of Applied Physics*.**119**(1): 015102. Bibcode:2016JAP...119a5102K. doi:10.1063/1.4939280. ISSN 0021-8979. - ↑ Newman, Mark (2010).
*Networks: An Introduction*(PDF). Oxford University Press. - ↑ Reuven Cohen, Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function Cambridge University Press.
- ↑ Grandjean, Martin (2015). "Social network analysis and visualization: Moreno’s Sociograms revisited". Redesigned network strictly based on Moreno (1934),
*Who Shall Survive*. - ↑ Rosen, Kenneth H. (2011-06-14).
*Discrete mathematics and its applications*(7th ed.). New York: McGraw-Hill. ISBN 978-0-07-338309-5. - ↑ Kelly, S Thomas; Black, Michael A (2 March 2020). "graphsim: An R package for simulating gene expression data from graph structures of biological pathways".
*bioRxiv*. 2020.03.02.972471. doi:10.1101/2020.03.02.972471 . Retrieved 27 May 2020. - ↑ Shah, Preya; Ashourvan, Arian; Mikhail, Fadi; Pines, Adam; Kini, Lohith; Oechsel, Kelly; Das, Sandhitsu R; Stein, Joel M; Shinohara, Russell T (2019-07-01). "Characterizing the role of the structural connectome in seizure dynamics".
*Brain*.**142**(7): 1955–1972. doi:10.1093/brain/awz125. ISSN 0006-8950. PMC 6598625 . - ↑ Biggs, N.; Lloyd, E.; Wilson, R. (1986),
*Graph Theory, 1736-1936*, Oxford University Press - ↑ Cauchy, A. L. (1813), "Recherche sur les polyèdres - premier mémoire",
*Journal de l'École Polytechnique*, 9 (Cahier 16): 66–86. - ↑ L'Huillier, S.-A.-J. (1812–1813), "Mémoire sur la polyèdrométrie",
*Annales de Mathématiques*,**3**: 169–189. - ↑ Cayley, A. (1857), "On the theory of the analytical forms called trees",
*Philosophical Magazine*, Series IV,**13**(85): 172–176, doi:10.1017/CBO9780511703690.046, ISBN 9780511703690 - ↑ Cayley, A. (1875), "Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen",
*Berichte der Deutschen Chemischen Gesellschaft*,**8**(2): 1056–1059, doi:10.1002/cber.18750080252. - ↑ Sylvester, James Joseph (1878). "Chemistry and Algebra".
*Nature*.**17**(432): 284. Bibcode:1878Natur..17..284S. doi: 10.1038/017284a0 . - ↑ Tutte, W.T. (2001),
*Graph Theory*, Cambridge University Press, p. 30, ISBN 978-0-521-79489-3 , retrieved 2016-03-14 - ↑ Gardner, Martin (1992),
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- ↑ Appel, K.; Haken, W. (1977), "Every planar map is four colorable. Part I. Discharging",
*Illinois J. Math.*,**21**(3): 429–490, doi: 10.1215/ijm/1256049011 . - ↑ Appel, K.; Haken, W. (1977), "Every planar map is four colorable. Part II. Reducibility",
*Illinois J. Math.*,**21**(3): 491–567, doi: 10.1215/ijm/1256049012 . - ↑ Robertson, N.; Sanders, D.; Seymour, P.; Thomas, R. (1997), "The four color theorem",
*Journal of Combinatorial Theory, Series B*,**70**: 2–44, doi:10.1006/jctb.1997.1750. - ↑ Kepner, Jeremy; Gilbert, John (2011).
*Graph Algorithms in the Language of Linear Algebra*. SIAM. p. 1171458. ISBN 978-0-898719-90-1.

In graph theory, a **tree** is an undirected graph in which any two vertices are connected by *exactly one* path, or equivalently a connected acyclic undirected graph. A **forest** is an undirected graph in which any two vertices are connected by *at most one* path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.

In mathematics, particularly graph theory, and computer science, a **directed acyclic graph** is a directed graph with no directed cycles. That is, it consists of vertices and edges, with each edge directed from one vertex to another, such that there is no way to start at any vertex v and follow a consistently-directed sequence of edges that eventually loops back to v again. Equivalently, a DAG is a directed graph that has a topological ordering, a sequence of the vertices such that every edge is directed from earlier to later in the sequence.

In mathematics, a **hypergraph** is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a hypergraph is a pair where is a set of elements called *nodes* or *vertices*, and is a set of non-empty subsets of called *hyperedges* or *edges*. Therefore, is a subset of , where is the power set of . The size of the vertex set is called the *order of the hypergraph*, and the size of edges set is the *size of the hypergraph*.

In the mathematical field of graph theory, a **bipartite graph** is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the *parts* of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

In graph theory and computer science, an **adjacency matrix** is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.

In graph theory, a **component**, sometimes called a **connected component**, of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph. For example, the graph shown in the illustration has three components. A vertex with no incident edges is itself a component. A graph that is itself connected has exactly one component, consisting of the whole graph.

This is a **glossary of graph theory terms**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

In mathematics, and more specifically in graph theory, a **graph** is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called *vertices* and each of the related pairs of vertices is called an *edge*. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

In graph theory, an **Eulerian trail** is a trail in a finite graph that visits every edge exactly once. Similarly, an **Eulerian circuit** or **Eulerian cycle** is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this:

In the mathematical field of graph theory, a **spanning tree***T* of an undirected graph *G* is a subgraph that is a tree which includes all of the vertices of *G*, with a minimum possible number of edges. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of *G* are also edges of a spanning tree *T* of *G*, then *G* is a tree and is identical to *T*.

In the mathematical area of graph theory, a **clique** is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.

In computer science, a **graph** is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics.

In the mathematical discipline of graph theory, the **line graph** of an undirected graph *G* is another graph L(*G*) that represents the adjacencies between edges of *G*. L(*G*) is constructed in the following way: for each edge in *G*, make a vertex in L(*G*); for every two edges in *G* that have a vertex in common, make an edge between their corresponding vertices in L(*G*).

In the mathematical field of graph theory, a **graph homomorphism** is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices.

In graph theory, the **degree** of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex is denoted or . The **maximum degree** of a graph , denoted by , and the **minimum degree** of a graph, denoted by , are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.

In graph theory, a branch of mathematics, the **circuit rank**, **cyclomatic number**, **cycle rank**, or **nullity** of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It is equal to the number of independent cycles in the graph. Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank r is easily computed using the formula

In mathematics and computer science, **connectivity** is one of the basic concepts of graph theory: it asks for the minimum number of elements that need to be removed to separate the remaining nodes into isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network.

In mathematics, and more specifically in graph theory, a **multigraph** is a graph which is permitted to have multiple edges, that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

In graph theory, a **pseudoforest** is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A **pseudotree** is a connected pseudoforest.

In mathematics, and more specifically in graph theory, a **directed graph** is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.

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Wikimedia Commons has media related to . Graph theory |

- "Graph theory",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Graph theory tutorial
- A searchable database of small connected graphs
- Image gallery: graphs at the Wayback Machine (archived February 6, 2006)
- Concise, annotated list of graph theory resources for researchers
- 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
- Online books , and library resources in your library and in other libraries about graph theory
- A list of graph algorithms with references and links to graph library implementations

- 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
- Graph Theory, by Reinhard Diestel

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