Graph theory
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- 6n-graf.svg|thumb|250px|A drawing of a graph]]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. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices. 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. The graphs studied in graph theory should not be confused with "graphs of functions" and other kinds of graphs.Please refer to Glossary of graph theory for some basic definitions in graph theory.History
Konigsberg bridges.png -
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.
(1) 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(2) and
L'Huillier,
(3) and is at the origin of
topology.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. This study had many implications in theoretical
chemistry. The involved techniques mainly concerned the
enumeration of graphs having particular properties. 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. 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.
(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?" 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. A proof produced in 1976 by
Kenneth Appel and
Wolfgang Haken,
(6)(7) which involved checking the properties of 1,936 configurations by computer, 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.
(8)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.
Drawing graphs
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. 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
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 .
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. 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. 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).
- Adjacency matrix : This is the n by n matrix A, where n is the number of vertices in the graph. 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. 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. It can be derived from powers of A:
Problems in graph theory
Enumeration
There is a large literature on
graphical enumeration: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973).
Subgraphs, induced subgraphs, and minors
A common problem, called the
subgraph isomorphism problem, is finding a fixed graph as a
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.
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,
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. 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.
Applications
Applications of graph theory are primarily, but not exclusively, concerned with labeled graphs and various specializations of these.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 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. There, the
transformation of graphs is often formalized and represented by
graph rewrite systems. They are either directly used or properties of the rewrite systems(e.g. confluence) are studied.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. A digraph with weighted edges in the context of graph theory is called a
network.Networks have many uses in the practical side of graph theory,
network analysis (for example, to model and analyze traffic networks). 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. 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. 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. Thirdly, analysis of
dynamical properties of networks.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. 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. This approach is especially used in computer processing of molecular structures, ranging from
chemical editors to database searching.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.
See also
Related topics
Algorithms
Subareas
Related areas of mathematics
Generalizations
Prominent graph theorists
- Berge, Claude
- Bollobás, Béla
- Chung, Fan
- Dirac, Gabriel Andrew
- Erdős, Paul
- Euler, Leonhard
- Faudree, Ralph
- Graham, Ronald
- Harary, Frank
- Heawood, Percy John
- Kőnig, Dénes
- Lovász, László
- Nešetřil, Jaroslav
- Rényi, Alfréd
- Ringel, Gerhard
- Robertson, Neil
- Seymour, Paul
- Szemerédi, Endre
- Thomassen, Carsten
- Turán, Pál
- Tutte, W. T.
- Tyshkevich, Regina
Notes
-
[BOOK, Biggs, N.; Lloyd, E. and Wilson, R., Graph Theory, 1736-1936, Oxford University Press, 1986, ]
-
[JOURNAL, Cauchy, A.L., 1813, Recherche sur les polyèdres - premier mémoire, Journal de l'Ecole Polytechnique, 9 (Cahier 16), 66–86, ]
-
[JOURNAL, L'Huillier, S.-A.-J., Mémoire sur la polyèdrométrie, Annales de Mathématiques, 3, 1861, 169–189, ]
-
[JOURNAL, Cayley, A., 1875, Berichte der deutschen Chemischen Gesellschaft, Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, 8, 1056–1059, 10.1002/cber.18750080252, ]
-
[JOURNAL, Sylvester, J.J., 1878, Nature, Chemistry and Algebra, 17, 284, 10.1038/017284a0, ]
-
[JOURNAL, Appel, K. and Haken, W., Every planar map is four colorable. Part I. Discharging, Illinois J. Math., 21, 1977, 429–490, ]
-
[JOURNAL, Appel, K. and Haken, W., Every planar map is four colorable. Part II. Reducibility, Illinois J. Math., 21, 1977, 491–567, ]
-
[JOURNAL, Robertson, N.; Sanders, D.; Seymour, P. and Thomas, R., The four color theorem, Journal of Combinatorial Theory Series B, 70, 1997, 2–44, 10.1006/jctb.1997.1750, ]
References
- {{citation|first=Alan|last=Gibbons|authorlink=|title=Algorithmic Graph Theory|year=1985|publisher=Cambridge University Press}}.
- {{citation|authorlink=Claude Berge|last=Berge|first=Claude|title=Théorie des graphes et ses applications|series=Collection Universitaire de Mathématiques|volume=II|publisher=Dunod|location=Paris|year=1958}}. English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition, Dover, New York 2001.
- {{citation|last=Chartrand|first=Gary|authorlink=Gary Theodore Chartrand|title=Introductory Graph Theory|publisher=Dover|isbn=0-486-24775-9|year=1985}}.
- {{citation|last1=Biggs|first1=N.|last2=Lloyd|first2=E.|last3=Wilson|first3=R.|title=Graph Theory, 1736-1936|publisher=Oxford University Press|year=1986}}.
- {{citation|authorlink=Frank Harary|last=Harary|first=Frank|title=Graph Theory|publisher=Addison-Wesley|location=Reading, MA|year=1969}}.
- {{citation|author1-link=Frank Harary|last1=Harary|first1=Frank|last2=Palmer|first2=Edgar M.|title=Graphical Enumeration|year=1973|publisher=Academic Press|location=New York, NY}}.
External links
Online textbooks
Other resources
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