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Graph theory

From Academic Kids

A graph with 6 vertices and 7 edges.
A graph with 6 vertices and 7 edges.

In mathematics and computer science, graph theory studies the properties of graphs. Informally, a graph is a set of objects called vertices (or Nodes) connected by links called edges (or Arcs) which can be directed. Typically, a graph is designed as a set of dots (the vertices) connected by lines (the 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.

Contents

History

One of the first results in graph theory appeared in Leonhard Euler's paper on Seven Bridges of Knigsberg, 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.

Definition

Main article: Graph (mathematics)

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 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

Main article: Graph (data structure)

There are different ways to store graphs in a computer system. The data structure used depends on 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 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 <math>M_{x, y}<math> is 1, otherwise it is 0. This makes it easier to find subgraphs, and to reverse graphs if needed.

Graph problems

Finding subgraphs

A common problem, called subgraph isomorphism problem, is finding subgraphs in a given graph. Many graph properties are hereditary, which means if a certain subgraph has a property so does the whole graph. For example a graph is non planar if it contains the complete bipartite graph <math>K_{3,3}<math> (See Three cottage problem). Unfortunately, finding maximal subgraphs of a certain kind is often a NP-complete problem.

Graph coloring

Route problems

Network flow

Visibility graph problems

Important algorithms

Related areas of mathematics

Applications

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 whether or not it is a scale-free network, or a small-world network. 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.

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

Subareas

Graph theory is composed of several subareas:

Prominent graph theorists

See also

External links

Topics in mathematics related to structure

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Abstract algebra | Universal algebra | Graph theory | Category theory | Order theory | Model theory | Structural proof theory
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Analysis | Measure theory | Functional analysis | Harmonic analysis
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