Graph theory

A graph with 6 vertices and 7 edges.

Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be represented by graphs. The link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia and there's a directed edge from article A to article B if and only if A contains a link to B. 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, or by making the edges to the graph directional (A links to B, but B does not necessarily link to A, as in webpages), technically called a digraph. A digraph with weighted edges 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 or to discover the shape of the internet -- see Applications below). However, it should be noted that within network analysis, the definition of the term "network" may differ, and may often refer to a simple graph.

History

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.

Definition

Drawing graphs

See main article graph drawing

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 practise 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.

Graphs as data structures

See main article graph (data structure)

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 node has a list of which nodes it is adjacent to. This can sometimes result in "overkill" in an undirected graph as vertex 3 may be in the list for node 2, then node 2 must be in the list for node 3. Either the programmer may choose to use the unneeded space anyway, or he/she may choose to list the adjacency once. This representation is easier to find all the nodes which are connected to a single node, since these are explicitly listed.
  

  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 is 1, otherwise it is 0. This makes it easier to find subgraphs, and to reverse graphs if needed.

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

  Graph problems

  Finding subgraphs

• 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)
  

  Graph coloring

• 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

• Seven bridges of Königsberg
• Minimum spanning tree
• Steiner tree
• Shortest path problem
• Route inspection problem (also called the "Chinese Postman Problem")
• Traveling salesman problem

  Network flow

• Max flow min cut theorem
• Reconstruction conjecture

  Visibility graph problems

• Museum guard problem

  Important algorithms

• Dijkstra's algorithm
• Kruskal's algorithm
• Nearest neighbour algorithm
• Prim's algorithm

  Related areas of mathematics

• Ramsey theory
• Combinatorics

  Applications

An important application of graph theory can be found in mathematical chemistry where molecules are modelled by graphs.

Glossary

See: Glossary of graph theory

References

[2] class="external">[1

Subareas

Graph theory is composed of several subareas:

• Algebraic graph theory
• Topological graph theory
• Geometric graph theory
• Extremal graph theory
• Metric graph theory
  

  Prominent graph theorists

• Frank Harary
• Paul Erdős

  See also

• Glossary of graph theory
• List of graph theory topics
• Ordered tree data structure - DAGs, binary trees and other special forms of graph.
• Graph (data structure)
• Graph drawing
• Important publications in graph theory
  

  External links

• Graph Theory online textbook
• Graph theory tutorial
• Graph theory algorithm presentation
• Some graph theory algorithm animations
• *Step through the algorithm to understand it.
• The compendium of algorithm visualisation sites
• A search site for finding algorithm implementations, explanations and animations
• Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring
• 1: Image gallery: Some real-life networks
• 2: Example layouts of a graph
• Graph links collection
• Grafos spanish copyleft software