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. "Graphs" in this context are not to be confused with "graphs of functions" and other kinds of graphs.

History

One of the first results in graph theory appeared in Leonhard Euler's paper on Seven Bridges of Königsberg, published in 1736. It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. This illustrates the deep connection between graph theory and topology.

In 1845 Gustav Kirchhoff published his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits.

In 1852 Francis Guthrie posed the four color problem which asks if it is possible to color, using only four colors, any map of countries in such a way as to prevent two bordering countries from having the same color. This problem, which was only solved a century later in 1976 by Kenneth Appel and Wolfgang Haken, can be considered the birth of graph theory. While trying to solve it mathematicians invented many fundamental graph theoretic terms and concepts.

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. 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. 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 but can consume huge amounts of memory if the graph is very large.

List structures

• Incidence list - The edges are represented by an array containing pairs (ordered if directed) of vertices (that the edge connects) and eventually weight and other data.

• 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 E (edges) by V (vertices), where [edge, vertex] contains the edge's data (simplest case: 1 - connected, 0 - not connected).

• Adjacency matrix - there is an N by N matrix, 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 M_x,y is 1, otherwise it is 0. This makes it easier to find subgraphs, and to reverse graphs if needed.

• Laplacian matrix or Kirchhoff matrix or Admittance matrix - is defined as degree matrix minus adjacency matrix and thus contains adjacency information and degree information about the vertices

• Distance matrix - A symmetric N by N matrix an element M_x,y of which is the length of shortest path between x and y; if there is no such path M_x,y = infinity. It can be derived from powers of the Adjacency matrix.

Problems in graph theory

Problems about subgraphs

A common problem, called subgraph isomorphism problem, is finding subgraphs in a given graph. Many graph properties are hereditary, which means that a graph has a property if and only if all subgraphs have it too. For example a graph is planar if it contains neither the complete bipartite graph K_3,3 (See Three cottage problem) nor the complete graph K₅. Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem.

• Finding the largest complete graph is called the clique problem (NP-complete)

• Finding the largest independent set is called the independent set problem (NP-complete)

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:

• Reconstruction conjecture

Graph coloring

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

• The four-color theorem
• The strong perfect graph theorem
• The Erdős-Faber-Lovász conjecture (unsolved)
• The total coloring conjecture (unsolved)
• The list coloring conjecture (unsolved)

Route problems

• Hamiltonian path and cycle problems
• Seven Bridges of Königsberg
• Minimum spanning tree
• Steiner tree
• Shortest path problem
• Route inspection problem (also called the "Chinese Postman Problem")
• Traveling salesman problem (NP-Complete)

Network flow

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

• Max flow min cut theorem

Visibility graph problems

• Museum guard problem

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.

• Set cover problem
• Vertex cover 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.

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 two 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.

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.

References

• Harary, Frank, Graph Theory, Addison-Wesley, Reading, MA, 1969.

See also

• Gallery of named graphs
• Glossary of graph theory
• List of graph theory topics
• Publications in graph theory

Related topics

Algorithms

Subareas

Related areas of mathematics

Prominent graph theorists

External links

• More people and publications at: Graph Theory White Pages

Online textbooks

• Graph Theory (1997/2005) by Reinhard Diestel
• Graph Theory with Applications (1976) by Bondy and Murty
• Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs (2006) by Hartmann and Weigt

Other resources

• Graph theory tutorial
• The compendium of algorithm visualisation sites
• Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring
• Image gallery no.1: Some real-life networks
• Image gallery no.2: More real-life graphs
• Graph links collection
• Useful tools and Explanation
• Grafos Spanish copyleft software
• Source code for computing neighbor shells in particle systems under periodic boundary conditions
• Graph Theory Resources
• Weisstein, Eric W., Graph Theory at MathWorld., hosted by the makers of Mathematica[1]