Last modified on 17 April 2011, at 19:16

Transportation Geography and Network Science/Graph theory

A drawing of a graph

Adapted from Wikipedia article [1]

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 here for more detailed definitions and for other variations in the types of graphs that are commonly considered.

Refer here for basic definitions in graph theory.



In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. One practical example: 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. Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.

Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural language often lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the Principle of Compositionality, modeled in a hierarchical graph. Within lexical semantics, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics. Still other methods in phonology (e.g. Optimality Theory, which uses lattice graphs) and morphology (e.g. finite-state morphology, using finite-state transducers) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as TextGraphs, as well as various 'Net' projects, such as WordNet, VerbNet, and others.

Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. 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. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.

Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore w:diffusion mechanisms, notably through the use of w:social network analysis software.

Likewise, graph theory is useful in w: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.

In mathematics, graphs are useful in geometry and certain parts of topology, e.g. Knot Theory. Algebraic graph theory has close links with group theory.

A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or w: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. Network analysis have many practical applications, for example, to model and analyze traffic networks. Applications of network analysis 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.


The Königsberg Bridge problem

The paper written by w:Leonhard Euler on the w: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 w:topology.

More than one century after Euler's paper on the bridges of w:Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from w:differential calculus to study a particular class of graphs, the trees. This study had many implications in theoretical w:chemistry. The involved techniques mainly concerned the w: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.


"[...] Every invariant and co-variant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph. [...] I give a rule for the geometrical multiplication of graphs, i.e. for constructing a graph to the product of in- or co-variants whose separate graphs are given. [...]" (italics as in the original).


The four color problem remained unsolved for more than a century. In 1969 w:Heinrich Heesch published a method for solving the problem using computers.[6] A computer-aided proof produced in 1976 by w:Kenneth Appel and w:Wolfgang Haken makes fundamental use of the notion of "discharging" developed by Heesch.[7][8] The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.[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 w:Gustav Kirchhoff, who published in 1845 his w:Kirchhoff's circuit laws for calculating the w:voltage and current in w:electric circuits.


Drawing graphsEdit

Main page: Graph drawing



Graph-theoretic data structuresEdit

Main page: Graph (data structure)

There are different ways to store graphs in a computer system. The w:data structure used depends on both the graph structure and the w: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 w: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 structuresEdit

w:Incidence list 
The edges are represented by an array containing pairs (w: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.
w: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 structuresEdit

w: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 - incident, 0 - not incident).
w:Adjacency matrix 
This is an n by n matrix A, where n is the number of vertices in the graph. If there is an edge from a vertex x to a vertex y, then the element a_{x, 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.
w:Laplacian matrix or w:Kirchhoff matrix or Admittance matrix 
This is defined as DA, where D is the diagonal w:degree matrix. It explicitly contains both adjacency information and degree information. (However, there are other, similar matrices that are also called "Laplacian matrices" of a graph.)
w:Distance matrix 
A symmetric n by n matrix D whose element d_{x, y} is the length of a w:shortest path between x and y; if there is no such path d_{x, 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 theoryEdit


There is a large literature on w: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 minorsEdit

A common problem, called the w:subgraph isomorphism problem, is finding a fixed graph as a w:subgraph in a given graph. One reason to be interested in such a question is that many w: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 w:NP-complete problem.



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 coloringEdit


Route problemsEdit

Network flowEdit


Visibility graph problemsEdit

Covering problemsEdit

Covering problems are specific instances of subgraph-finding problems, and they tend to be closely related to the w:clique problem or the w:independent set problem.

Graph classesEdit

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:

See alsoEdit

Related topicsEdit



Related areas of mathematicsEdit


Prominent graph theoristsEdit


  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. Retrieved 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. 



External linksEdit

Online textbooksEdit

Other resourcesEdit