# Graph theory: Wikis

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# Encyclopedia

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.

Refer to Glossary of graph theory for basic definitions in graph theory.

## History

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.[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, where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:[5]

"[...] 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).

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. In 1969 Heinrich Heesch published a method for solving the problem using computers[6]. A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use the notion of "discharging" developed by Heesch[7][8]. The proof 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.[9]

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-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.

## 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 (tuples if directed) of vertices (that the edge connects) and possibly weight and other data. Vertices connected by an edge are said to be adjacent.
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).
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 DA, 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
$d_{x,y}=\min\{n\mid A^n[x,y]\ne 0\}. \,$

## Problems in graph theory

### Enumeration

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

### Network flow

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

### 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.

### Graph classes

Many problems involve characterizing the members of various classes of graphs. Overlapping significantly with other types in this list, this type of problem includes, for instance:

## 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:

1. First, 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.
2. Second, 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.
3. Third, 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.

Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or habitats) 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.

## 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. doi:10.1002/cber.18750080252.
5. ^ John Joseph Sylvester (1878), Chemistry and Algebra. Nature, volume 17, page 284. doi:10.1038/017284a0. Online version accessed on 2009-12-30.
6. ^ Heinrich Heesch: Untersuchungen zum Vierfarbenproblem. Mannheim: Bibliographisches Institut 1969.
7. ^ Appel, K. and Haken, W. (1977). "Every planar map is four colorable. Part I. Discharging". Illinois J. Math. 21: 429–490.
8. ^ Appel, K. and Haken, W. (1977). "Every planar map is four colorable. Part II. Reducibility". Illinois J. Math. 21: 491–567.
9. ^ Robertson, N.; Sanders, D.; Seymour, P. and Thomas, R. (1997). "The four color theorem". Journal of Combinatorial Theory Series B 70: 2–44. doi:10.1006/jctb.1997.1750.

## References

• Berge, Claude (1958), Théorie des graphes et ses applications, Collection Universitaire de Mathématiques, II, Paris: Dunod . 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.
• Biggs, N.; Lloyd, E.; Wilson, R. (1986), Graph Theory, 1736–1936, Oxford University Press .
• Bondy, J.A.; Murty, U.S.R. (2008), Graph Theory, Springer, ISBN 978-1-84628-969-9 .
• Chartrand, Gary (1985), Introductory Graph Theory, Dover, ISBN 0-486-24775-9 .
• Gibbons, Alan (1985), Algorithmic Graph Theory, Cambridge University Press .
• Golumbic, Martin (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press .
• Harary, Frank; Palmer, Edgar M. (1973), Graphical Enumeration, New York, NY: Academic Press .
• Mahadev, N.V.R.; Peled, Uri N. (1995), Threshold Graphs and Related Topics, North-Holland .

# Simple English

[[File:|thumb|300px|right|An undirected graph.]]

Graph theory is a field of mathematical ideas about graphs. A graph is an abstract representation: A number of points are connected by lines. Each point is usually called vertex (many are called vertices), and the lines are called edges. Graphs are a tool for modelling. They are used to find answers to a number of problems.

Some of these questions are:

• What is the best way for a mailman to get to all of the houses in the area in the least amount of time? The points could represent street corners and lines could represent the houses along the street. (see Chinese postman problem)
• A salesman has to visit different customers, but wants to keep the distance traveled as small as possible. The problem is to find a way so they can do it. This problem is known as Travelling Salesman Problem (and often abbreviated TSP). It is among the hardest problems to solve. If a commonly believed conjecture is true (described as PNP), then an exact solution requires one to try all possible routes to find which is shortest.
• How many colors would be needed to color a map, if countries sharing a border are colored differently? The points could represent the different areas and the lines could represent that two areas are neighboring. (look at the Four color theorem)
• Can a sketch be drawn in one closed line? The lines of the drawing are the lines of the graph and when two or more lines collide, there is a point in the graph. The task is now to find a way through the graph using each line one time. (look at Seven Bridges of Königsberg)

## Different kinds of Graphs

• Graph theory has many aspects. Graphs can be directed or undirected. An example of a directed graph would be the system of roads in a city. Some streets in the city are one way streets. This means, that on those parts there is only one direction to follow.
• Graphs can be weighted. An example would be a road network, with distances, or with tolls (for roads).
• The nodes (the circles in the schematic) of a graph are called vertices. The lines connecting the nodes are called edges. There can be no line between two nodes, there can be one line, or there can be multiple lines.

## History

File:Konigsberg
The Königsberg Bridge problem, on a city map
File:Konigsburg
The same problem, but using a graph

Leonhard Euler used to live in a city that was called Königberg for some time. (The city has been called Kaliningrad since 1946). The city is situated on the river Pregel. There is an island in the river, and there are a number of bridges across the river. Euler loved to go for walks, so he asked himself, whether it was possible to do a walk, use each of the bridges once, and come back to the point where he started. In 1736, he published a scientific article where he showed that this was not possible. Today, this problem is known as the Seven Bridges of Königsberg. The article is seen as the first paper in the history of graph theory.[1]

This article, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula was about 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.

Over a century after Euler, Cayley studied a special class of graphs, called the trees. A tree is a graph where there is only one path between two vertices. Cayley tried to solve a problem from differential calculus. His solution influenced the development of theoretical chemistry. The technique he used mainly focused on listing all graphs which had certain properties. Today, this is called Enumerative graph theory. It developed from Cayleys results, a scientific publication done 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 an article 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.

## Graph theory in perspective

Graph theory is an important part of mathematics and computer science. To many such problems, exact solutions do exist. Many times however, they are very hard to calculate. Therefore, very often, approximations are used. There are two kinds of such approximations, Monte-Carlo algorithms and Las-Vegas algorithms.

Graphs are normally represented by two different sets, typically set a graph G would be represented as the collection of the sets V and E. The set V is a discrete set containing all vertices of the graph. The set E is a binary set, whose pairwise elements are elements of set V. Each pair in set E represents an edge connecting two vertices.

If any two nodes have an edge between them, then the graph is called the complete graph.

I would move forth to define a Path, a Walk, A weighted graph, a directional graph. Then make a comment about how all trees are just subsets of graphs. It could also be proven that all connected graphs may be represented as a Tree. There is more to graph theory than explained here.

## References

1. Biggs, N.; Lloyd, E. and Wilson, R. (1986). Graph Theory, 1736-1936. Oxford University Press.
2. Cauchy, A.L. (1813). [Expression error: Unexpected < operator "Recherche sur les polyèdres - premier mémoire"]. Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.
3. L'Huillier, S.-A.-J. (1861). [Expression error: Unexpected < operator "Mémoire sur la polyèdrométrie"]. Annales de Mathématiques 3: 169–189.
4. Cayley, A. (1875). [Expression error: Unexpected < operator "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. doi:10.1002/cber.18750080252.
5. Sylvester, J.J. (1878). [Expression error: Unexpected < operator "Chemistry and Algebra"]. Nature 17: 284. doi:10.1038/017284a0.
6. Appel, K. and Haken, W. (1977). [Expression error: Unexpected < operator "Every planar map is four colorable. Part I. Discharging"]. Illinois J. Math. 21: 429–490.
7. Appel, K. and Haken, W. (1977). [Expression error: Unexpected < operator "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). [Expression error: Unexpected < operator "The four color theorem"]. Journal of Combinatorial Theory Series B 70: 2–44. doi:10.1006/jctb.1997.1750.