Citizendia
Your Ad Here

A drawing of a graph.
A drawing of a graph. Graph drawing, as a branch of Graph theory, applies Topology and Geometry to derive two-dimensional representations of graphs Graph drawing is

In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Mathematics and Computer science, a graph is the basic object of study in Graph theory. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices. For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see graph (mathematics) for more detailed definitions and for other variations in the types of graphs that are commonly considered. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. The graphs studied in graph theory should not be confused with "graphs of functions" and other kinds of graphs. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

Please refer to Glossary of graph theory for some basic definitions in graph theory. Graph theory is a growing area in mathematical research and has a large specialized vocabulary

Contents

History

The Königsberg Bridge problem.
The Königsberg Bridge problem.

The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory. The Seven Bridges of Königsberg is a famous historical problem in mathematics Year 1736 ( MDCCXXXVI) was a Leap year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Leap year [1] This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Alexandre-Théophile Vandermonde ( 28 February 1735 – 1 January 1796) was a French Musician and Chemist who The Knight's Tour is a mathematical problem involving a knight on a Chessboard. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy[2] and L'Huillier,[3] and is at the origin of topology. Simon Antoine Jean L'Huilier (or L'Huillier) ( Geneva, 24 April 1750 - Geneva, 28 March 1840) was a Swiss Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees. Königsberg (Karaliaučius Low German: Königsbarg; Królewiec see also other names) was until 1946 the name of Kaliningrad. Johann Benedict Listing ( July 25, 1808 &ndash December 24 1882) was a German Mathematician. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path. This study had many implications in theoretical chemistry. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties The involved techniques mainly concerned the enumeration of graphs having particular properties. Graph enumeration is a subject of Graph theory that deals with the problems of the following type find how many Non-isomorphic graphs have a given property Enumerative graph theory then rose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in 1959. George Pólya (b December 13, 1887 &ndash d September 7, 1985, in Hungarian Pólya György) was a Hungarian Year 1935 ( MCMXXXV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Year 1937 ( MCMXXXVII) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar. Nicolaas Govert de Bruijn (born 9 July, 1918) is a Dutch Mathematician, affiliated as Professor emeritus The year 1959 ( MCMLIX) was a Common year starting on Thursday (link will display full calendar of the Gregorian calendar. Cayley linked his results on trees with the contemporary studies of chemical composition. [4] The fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a 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. James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician Year 1878 ( MDCCCLXXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common Nature is a prominent Scientific journal, first published on 4 November 1869 [5]

One of the most famous and productive problems of 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?". The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country This problem remained unsolved for more than a century and the proof given by Kenneth Appel and Wolfgang Haken in 1976[6][7] (determination of 1936 types of configurations of which study is sufficient and checking of the properties of these configurations by computer) did not convince all the community. Kenneth Appel (born 1932 is a mathematician who in 1976 with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous Year 1976 ( MCMLXXVI) was a Leap year starting on Thursday (link will display full calendar of the Gregorian calendar. A simpler proof considering far fewer configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas. G Neil Robertson is a mathematician working mainly in Topological graph theory, currently a distinguished professor at The Ohio State University. Paul D Seymour (born July 26, 1950) is a mathematician working in Discrete mathematics, including Combinatorics, Graph theory [8]

This problem was first posed by Francis Guthrie in 1852 and the first written record of this problem is a letter of De Morgan addressed to Hamilton the same year. Francis Guthrie (b January 22 1831 in London d October 19 1899 in Claremont Cape Town was a South African mathematician and botanist who first posed the Four Year 1852 ( MDCCCLII) was a Leap year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Leap year Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. Sir Alfred Bray Kempe DCL FRS (6 July 1849 Kensington, London – 21 April 1922 London) was a Mathematician The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger has in particular led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. Peter Guthrie Tait ( April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist, best known for the seminal energy Percy John Heawood ( 8 September, 1861 Newport Shropshire, England - 24 January, 1955 Durham, England Frank Plumpton Ramsey ( February 22, 1903 – January 19, 1930) was a British Mathematician who in addition to Hugo Hadwiger (1908 &ndash 1981 was a Swiss Mathematician. He is known for Hadwiger's theorem in Integral geometry, and a number of conjectures In Mathematics, genus has a few different but closely related meanings Topology Orientable surface Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Kőnig. Julius Peter Christian Petersen ( June 16, 1839, Sorø on Zealand &ndash August 5, 1910) was a Danish Mathematician Dénes Kőnig ( September 21, 1884 – October 19, 1944) was a Hungarian Mathematician who worked in and wrote the first The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 is at the origin of another branch of graph theory, the extremal graph theory. Paul (Pál Turán ( ( August 18 1910 &ndash September 26 1976)was a Hungarian Mathematician who worked primarily in Year 1941 ( MCMXLI) was a Common year starting on Wednesday (the link will display 1941 calendar of the Gregorian calendar. Extremal graph theory is a branch of Mathematics. In the narrow sense extremal graph theory studies the graphs which are extremal among graphs with

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Year 1860 ( MDCCLX) was a Leap year starting on Sunday (link will display the full calendar of the Gregorian Calendar (or a Leap year starting Year 1930 ( MCMXXX) was a Common year starting on Wednesday (link will display 1930 calendar of the Gregorian calendar. Marie Ennemond Camille Jordan ( January 5 1838 &ndash January 22 1922) was a French Mathematician, known both for his foundational Kazimierz Kuratowski ( Warsaw, February 2, 1896 &ndash June 18, 1980) was a Polish Mathematician and Logician Hassler Whitney ( 23 March 1907 &ndash 10 May 1989) was an American Mathematician. 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. Gustav Robert Kirchhoff ( March 12, 1824 &ndash October 17, 1887) was a German Physicist who contributed to the fundamental Year 1845 ( MDCCCXLV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common For other laws named after Gustav Kirchhoff, see Kirchhoff's laws. Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage

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. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash Alfréd Rényi (20 March 1921 &ndash 1 February 1970 was a Hungarian Mathematician who made contributions in Combinatorics and Graph theory In Mathematics, a random graph is a graph that is generated by some Random process.

Drawing graphs

Main article: Graph drawing

Graphs are represented graphically by drawing a dot for every vertex, and drawing an arc between two vertices if they are connected by an edge. Graph drawing, as a branch of Graph theory, applies Topology and Geometry to derive two-dimensional representations of graphs Graph drawing is 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-graphical 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.

Graph-theoretic data structures

Main article: Graph (data structure)

There are different ways to store graphs in a computer system. In Computer science, a graph is a kind of Data structure, specifically an Abstract data type (ADT that consists of a set of nodes (also called The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. A data structure in Computer science is a way of storing Data in a computer so that it can be used efficiently In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation 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. In Mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory .

List structures

Incidence list 
The edges are represented by an array containing pairs (ordered if directed) of vertices (that the edge connects) and possibly weight and other data. In Graph theory, an adjacency list is the representation of all edges or arcs in a graph as a list In Computer science an array is a Data structure consisting of a group of elements that are accessed by indexing. Vertices connected by an edge are said to be adjacent.
Adjacency list 
Much like the incidence list, each vertex has a list of which vertices it is adjacent to. In Graph theory, an adjacency list is the representation of all edges or arcs in a graph as a list This causes redundancy in an undirected graph: for example, if vertices A and B are adjacent, A's adjacency list contains B, while B's list contains A. Adjacency queries are faster, at the cost of extra storage space.

Matrix structures

Incidence matrix 
The graph is represented by a matrix of size |V| (number of vertices) by |E| (number of edges) where the entry [vertex, edge] contains the edge's endpoint data (simplest case: 1 - connected, 0 - not connected). In Mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally
Adjacency matrix 
This is the n by n matrix A, where n is the number of vertices in the graph. In Mathematics and Computer science, the adjacency matrix of a finite directed or undirected graph G on n vertices is the If there is an edge from some vertex x to some vertex y, then the element ax,y is 1 (or in general the number of xy edges), otherwise it is 0. In computing, this matrix makes it easy to find subgraphs, and to reverse a directed graph.
Laplacian matrix or Kirchhoff matrix or Admittance matrix 
This is defined as D − A, where D is the diagonal degree matrix. In the mathematical field of Graph theory the Laplacian matrix, sometimes called admittance matrix or Kirchhoff matrix, is a matrix In the mathematical field of Graph theory the Laplacian matrix, sometimes called admittance matrix or Kirchhoff matrix, is a matrix In the mathematical field of Graph theory the degree matrix is a Diagonal matrix which contains information about the degree of each vertex It explicitly contains both adjacency information and degree information.
Distance matrix 
A symmetric n by n matrix D whose element dx,y is the length of a shortest path between x and y; if there is no such path dx,y = infinity. In Mathematics, a distance matrix is a matrix (two-dimensional Array) containing the Distances taken pairwise of a set of points In Graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes such that the sum of the weights It can be derived from powers of A: d_{x,y}=\min\{n\mid A^n[x,y]\ne 0\}.

Problems in graph theory

Enumeration

There is a large literature on graphical enumeration: the problem of counting graphs meeting specified conditions. Graph enumeration is a subject of Graph theory that deals with the problems of the following type find how many Non-isomorphic graphs have a given property Some of this work is found in Harary and Palmer (1973).

Subgraphs, induced subgraphs, and minors

A common problem, called the subgraph isomorphism problem, is finding a fixed graph as a subgraph in a given graph. In complexity theory, Subgraph-Isomorphism is a Decision problem that is known to be NP-complete. Graph theory is a growing area in mathematical research and has a large specialized vocabulary 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, or all induced subgraphs, have it too. In Graph theory a graph property is any "inherently graph-theoretical" property of graphs (formal definitions follow distinguished from properties of Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem. In Computational complexity theory, the Complexity class NP-complete (abbreviated NP-C or NPC) is a class of problems having two properties

A similar problem is finding induced subgraphs in a given graph. Graph theory is a growing area in mathematical research and has a large specialized vocabulary 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,

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. In Graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. In Graph theory a graph property is any "inherently graph-theoretical" property of graphs (formal definitions follow distinguished from properties of A famous example:

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:

Graph coloring

Many problems have to do with various ways of coloring graphs, for example:

Route problems

Network flow

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

Visibility graph problems

Covering problems

Covering problems are specific instances of subgraph-finding problems, and they tend to be closely related to the clique problem or the independent set problem. Informally the reconstruction conjecture in Graph theory says that graphs are determined uniquely by their subgraphs In Graph theory, graph coloring is a special case of Graph labeling; it is an assignment of labels traditionally called "colors" to elements of a The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country In Graph theory, a perfect graph is a graph in which the chromatic number of every Induced subgraph equals the clique number of that In Graph theory, the Erdős–Faber–Lovász conjecture (1972 is a very deep problem about the coloring of graphs named after Paul Erdős, Vance In Graph theory, total coloring is a type of coloring on the vertices and edges of a graph In Mathematics, list edge-coloring is a type of Graph coloring. In the mathematical field of Graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian Given a connected, Undirected graph, a spanning tree of that graph is a Subgraph which is a tree and connects all the vertices together In Graph theory, a branch of Mathematics, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed The Seven Bridges of Königsberg is a famous historical problem in mathematics In Graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes such that the sum of the weights The Steiner tree problem, named after Jakob Steiner, is a problem in Combinatorial optimization. The classical Mathematical puzzle known as water gas and electricity, the (three utilities problem, or sometimes the three cottage problem, can be stated The Travelling salesman problem ( TSP) in Operations research is a problem in discrete or Combinatorial optimization. In Graph theory, a flow network is a directed graph where each edge has a capacity and each edge receives a Flow. The max-flow min-cut theorem is a statement in optimization theory about maximum flows in Flow networks It derives from Menger's theorem. The art gallery problem or museum problem is a well-studied Visibility problem in Computational geometry. In the mathematical discipline of Graph theory, a covering (or cover) of a graph is a set of vertices (or edges such that each edge (vertex In Computational complexity theory, the clique problem is a graph-theoretic NP-complete problem In Mathematics, the independent set problem ( IS) is a well-known problem in Graph theory and Combinatorics.

Applications

Applications of graph theory are primarily, but not exclusively, concerned with labeled graphs and various specializations of these. The set covering problem is a classical question in Computer science and Complexity theory. In Computer science, the vertex cover problem or node cover problem is an NP-complete problem and was one of Karp's 21 NP-complete problems.

Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be represented by graphs. The link structure of a website could be represented by a directed graph: the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. A website (alternatively web site or Web site, a back-construction from the Proper noun World Wide Web) is a collection of Web pages A similar approach can be taken to problems in travel, biology, computer chip design, and many other fields. The development of algorithms to handle graphs is therefore of major interest in computer science. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

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. Graph theory is a growing area in mathematical research and has a large specialized vocabulary For example if a graph represents a road network, the weights could represent the length of each road. A digraph with weighted edges in the context of graph theory is called a network. In Graph theory, a network is a digraph with weighted edges These networks have become an especially useful concept in analysing the interaction between Biology

Networks have many uses in the practical side of graph theory, network analysis (for example, to model and analyze traffic networks). Network analysis can refer to Analysis of general networks see Network theory. Within network analysis, the definition of the term "network" varies, and may often refer to a simple graph.

Many applications of graph theory exist in the form of network analysis. Network analysis can refer to Analysis of general networks see Network theory. These split broadly into three categories. Firstly, analysis to determine structural properties of a network, such as the distribution of vertex degrees and the diameter of the graph. In the mathematical field of Graph theory, the distance between two vertices in a graph is the number of edges in a shortest path A vast number of graph measures exist, and the production of useful ones for various domains remains an active area of research. Secondly, analysis to find a measurable quantity within the network, for example, for a transportation network, the level of vehicular flow within any portion of it. A transportation network is a type of directed Weighted graph or network. Thirdly, analysis of dynamical properties of networks. Sequential dynamical systems (SDSs are a class of Discrete Dynamical systems which generalize many aspects of systems such as Cellular automata, and provide

Graph theory is also used to study molecules in chemistry and physics. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. 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. Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. For example, Franzblau's shortest-path (SP) rings. In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny BOND (Building Object Network Databases started development in late 2000 as a Rapid application development tool for the GNOME Desktop by Treshna This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. A molecule editor is a Computer program for creating and modifying representations of Chemical structures There are a number types of molecule editor

Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore diffusion mechanisms, notably through the use of social network analysis software. Sociology (from Latin: socius "companion" and the suffix -ology "the study of" from Greek λόγος lógos "knowledge" Diffusion is the net movement of particles (typically molecules from an area of high concentration to an area of low concentration by uncoordinated random movement A social network is a Social structure made of nodes (which are generally individuals or organizations that are tied by one or more specific types of interdependency such as

References

See also

Related topics

Algorithms

Subareas

Related areas of mathematics

Prominent graph theorists

Notes

  1. ^ Biggs, N. ; Lloyd, E. and Wilson, R. (1986). Graph Theory, 1736-1936. Oxford University Press.  
  2. ^ Cauchy, A. L. (1813). "Recherche sur les polyèdres - premier mémoire". Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.  
  3. ^ L'Huillier, S. -A. -J. (1861). "Mémoire sur la polyèdrométrie". Annales de Mathématiques 3: 169–189.  
  4. ^ 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: 1056–1059.  
  5. ^ Sylvester, J. J. (1878). "Chemistry and Algebra". Nature 17: 284.  
  6. ^ Appel, K. and Haken, W. (1977). "Every planar map is four colorable. Part I. Discharging". Illinois J. Math. 21: 429-490.  
  7. ^ Appel, K. and Haken, W. (1977). "Every planar map is four colorable. Part II. Reducibility". Illinois J. Math. 21: 491-567.  
  8. ^ Robertson, N. ; Sanders, D. ; Seymour, P. and Thomas, R. (1997). "The four color theorem". Journal of Combinatorial Theory Series B 70: 2-44.  

External links

Online textbooks

Other resources

Dictionary

graph theory

-noun

  1. (mathematics) The study of the properties of graphs (in the sense of sets of vertices and sets of ordered or unordered pairs of vertices).
© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic