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62-418: 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, a graph is an ordered pair G = ( V , E ) {\displaystyle G=(V,E)} comprising: To avoid ambiguity, this type of object may be called precisely an undirected simple graph . In

124-401: A directed graph is an ordered triple G = ( V , E , ϕ ) {\displaystyle G=(V,E,\phi )} comprising: 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

186-534: A directed simple graph . In set theory and graph theory, V n {\displaystyle V^{n}} denotes the set of n - tuples of elements of V , {\displaystyle V,} that is, ordered sequences of n {\displaystyle n} elements that are not necessarily distinct. In the edge ( x , y ) {\displaystyle (x,y)} directed from x {\displaystyle x} to y {\displaystyle y} ,

248-663: 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 G {\displaystyle G} is a homogeneous relation ~ on the vertices of G {\displaystyle G} that is called the adjacency relation of G {\displaystyle G} . Specifically, for each edge ( x , y ) {\displaystyle (x,y)} , its endpoints x {\displaystyle x} and y {\displaystyle y} are said to be adjacent to one another, which

310-412: A multigraph is a graph which is permitted to have multiple edges (also called parallel edges ), that is, edges that have the same end nodes . Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: A multigraph is different from a hypergraph , which is a graph in which an edge can connect any number of nodes, not just two. For some authors,

372-425: 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 packings , intersection graph , and other visualizations of the adjacency matrix . The tabular representation lends itself well to computational applications. There are different ways to store graphs in

434-464: 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

496-444: A century. In 1969 Heinrich Heesch published a method for solving the problem using computers. A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken 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

558-403: 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

620-571: A critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is studied via percolation theory . 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. Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence

682-432: 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, travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases, and many other fields. The development of algorithms to handle graphs is therefore of major interest in computer science. The transformation of graphs

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744-529: A distinguished self-loop at each vertex serving as the left and right identity for composition. For this reason, in category theory the term graph is standardly taken to mean "multidigraph", and the underlying multidigraph of a category is called its underlying digraph . Multigraphs and multidigraphs also support the notion of graph labeling , in a similar way. However there is no unity in terminology in this case. The definitions of labeled multigraphs and labeled multidigraphs are similar, and we define only

806-403: A fruitful source of graph-theoretic results. 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

868-407: A graph and not belong to an edge. The edge ( y , x ) {\displaystyle (y,x)} is called the inverted edge of ( x , y ) {\displaystyle (x,y)} . 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,

930-469: A graph and not belong to an edge. Under this definition, multiple edges , in which two or more edges connect the same vertices, are not allowed. In one more general sense of the term allowing multiple edges, a graph is an ordered triple G = ( V , E , ϕ ) {\displaystyle G=(V,E,\phi )} comprising: To avoid ambiguity, this type of object may be called precisely an undirected multigraph . A loop

992-403: A graph is | E | {\displaystyle |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 n − 1 and the maximum size of

1054-409: A question is that many graph properties are hereditary for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Finding maximal subgraphs of a certain kind is often an NP-complete problem . For example: 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

1116-438: A set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a topological group . Map between two sets with

1178-587: A structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation , relation , metric , or topology ). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance. A partial list of possible structures are measures , algebraic structures ( groups , fields , etc.), topologies , metric structures ( geometries ), orders , graphs , events , equivalence relations , differential structures , and categories . Sometimes,

1240-500: 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

1302-563: A vertex x {\displaystyle x} to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) ( x , x ) {\displaystyle (x,x)} which is not in { ( x , y ) ∣ ( x , y ) ∈ V 2 and x ≠ y } {\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}} . So to allow loops

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1364-421: 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

1426-581: 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

1488-456: 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: Mathematical structure In Mathematics,

1550-677: 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 {\displaystyle x} to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) { x , x } = { x } {\displaystyle \{x,x\}=\{x\}} which is not in { { x , y } ∣ x , y ∈ V and x ≠ y } {\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}} . To allow loops,

1612-446: Is denoted x {\displaystyle x} ~ y {\displaystyle y} . 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

1674-413: Is denoted x ∼ y {\displaystyle x\sim y} . 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 = ( V , E ) {\displaystyle G=(V,E)} comprising: To avoid ambiguity, this type of object may be called precisely

1736-564: 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

1798-478: The Seven Bridges of Königsberg 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 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 and L'Huilier , and represents

1860-416: 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

1922-669: 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

Graph theory - Misplaced Pages Continue

1984-530: The French group with the pseudonym Nicolas Bourbaki saw structures as the root of mathematics. They first mentioned them in their "Fascicule" of Theory of Sets and expanded it into Chapter IV of the 1957 edition. They identified three mother structures : algebraic, topological, and order . The set of real numbers has several standard structures: There are interfaces among these: Pseudograph In mathematics , and more specifically in graph theory ,

2046-503: 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

2108-506: 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

2170-453: 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 . This study had many implications for theoretical chemistry . The techniques he used mainly concern

2232-449: 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

2294-852: The definitions must be expanded. For directed simple graphs, the definition of E {\displaystyle E} should be modified to E ⊆ { ( x , y ) ∣ ( x , y ) ∈ V 2 } {\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}} . For directed multigraphs, the definition of ϕ {\displaystyle \phi } should be modified to ϕ : E → { ( x , y ) ∣ ( x , y ) ∈ V 2 } {\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}} . To avoid ambiguity, these types of objects may be called precisely

2356-1033: The definitions must be expanded. For undirected simple graphs, the definition of E {\displaystyle E} should be modified to E ⊆ { { x , y } ∣ x , y ∈ V } {\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}} . For undirected multigraphs, the definition of ϕ {\displaystyle \phi } should be modified to ϕ : E → { { x , y } ∣ x , y ∈ V } {\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}} . To avoid ambiguity, these types of objects may be called undirected simple graph permitting loops and undirected multigraph permitting loops (sometimes also undirected pseudograph ), respectively. V {\displaystyle V} and E {\displaystyle E} are usually taken to be finite, and many of

2418-402: 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

2480-499: The edge { x , y } {\displaystyle \{x,y\}} , the vertices x {\displaystyle x} and y {\displaystyle y} are called the endpoints of the edge. The edge is said to join x {\displaystyle x} and y {\displaystyle y} and to be incident on x {\displaystyle x} and on y {\displaystyle y} . A vertex may exist in

2542-426: 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

Graph theory - Misplaced Pages Continue

2604-669: The graph is ⁠ n ( n − 1) / 2 ⁠ . The edges of an undirected simple graph permitting loops G {\displaystyle G} induce a symmetric homogeneous relation ∼ {\displaystyle \sim } on the vertices of G {\displaystyle G} that is called the adjacency relation of G {\displaystyle G} . Specifically, for each edge ( x , y ) {\displaystyle (x,y)} , its endpoints x {\displaystyle x} and y {\displaystyle y} are said to be adjacent to one another, which

2666-538: The latter ones here. Definition 1 : A labeled multidigraph is a labeled graph with labeled arcs. Formally: A labeled multidigraph G is a multigraph with labeled vertices and arcs. Formally it is an 8-tuple G = ( Σ V , Σ A , V , A , s , t , ℓ V , ℓ A ) {\displaystyle G=(\Sigma _{V},\Sigma _{A},V,A,s,t,\ell _{V},\ell _{A})} where Definition 2 : A labeled multidigraph

2728-406: 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

2790-402: The possible flight connections offered by an airline. In this case the multigraph would be a directed graph with pairs of directed parallel edges connecting cities to show that it is possible to fly both to and from these locations. In category theory a small category can be defined as a multidigraph (with edges having their own identity) equipped with an associative composition law and

2852-699: 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 ; 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

2914-430: The same type of structure, which preserve this structure [ morphism : structure in the domain is mapped properly to the (same type) structure in the codomain ] is of special interest in many fields of mathematics. Examples are homomorphisms , which preserve algebraic structures; continuous functions , which preserve topological structures; and differentiable functions , which preserve differential structures. In 1939,

2976-417: 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

3038-461: 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: The first textbook on graph theory was written by Dénes Kőnig , and published in 1936. Another book by Frank Harary , published in 1969, was "considered

3100-450: 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

3162-406: The tabular, in which rows of a table provide information about the relationships between the vertices within the graph. 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

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3224-487: The term multigraph for the case with no loops. A multidigraph is a directed graph which is permitted to have multiple arcs, i.e., arcs with the same source and target nodes. A multidigraph G is an ordered pair G  := ( V , A ) with A mixed multigraph G  := ( V , E , A ) may be defined in the same way as a mixed graph . A multidigraph or quiver G is an ordered 4-tuple G  := ( V , A , s , t ) with This notion might be used to model

3286-430: The terms pseudograph and multigraph are synonymous. For others, a pseudograph is a multigraph that is permitted to have loops . A multigraph G is an ordered pair G  := ( V , E ) with A multigraph G is an ordered triple G  := ( V , E , r ) with Some authors allow multigraphs to have loops , that is, an edge that connects a vertex to itself, while others call these pseudographs , reserving

3348-585: 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." 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

3410-560: The vertices x {\displaystyle x} and y {\displaystyle y} are called the endpoints of the edge, x {\displaystyle x} the tail of the edge and y {\displaystyle y} the head of the edge. The edge is said to join x {\displaystyle x} and y {\displaystyle y} and to be incident on x {\displaystyle x} and on y {\displaystyle y} . A vertex may exist in

3472-413: The vertices and edges, and the subject that expresses and understands real-world systems as a network is called network science . Within computer science , ' causal ' and 'non-causal' linked structures are graphs that are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a website can be represented by

3534-443: 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 {\displaystyle V} is often assumed to be non-empty, but E {\displaystyle E} is allowed to be the empty set. The order of a graph is | V | {\displaystyle |V|} , its number of vertices. The size of

3596-483: The wire segments to obtain electrical properties of network structures. 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 leads to

3658-409: 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

3720-475: 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 : "Is it true that any map drawn in the plane may have its regions colored with four colors, in such

3782-403: Was given twenty years later by Robertson , Seymour , Sanders and Thomas . 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

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3844-423: 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

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