Graph theory

A graph diagram of 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 (assigned a direction). Typically, a graph is designed as a set of nodes (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 a website could be represented by a directed graph: the vertices are the web pages available at the website and there's a directed edge from page A to page 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 of the graph. Graphs with weights can be used to represent many different concepts; for example if the graph represents a road network, the weights could represent the length of each road¹. Another way to extend basic graphs is 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 directed graph or 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). Within network analysis, the definition of the term "network" varies, and may often refer to a simple graph.

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.

Definition

File:Undirected.png

An undirected graph or graph G is an ordered pair G:=(V, E) with

V, a set of vertices or nodes,
E, a set of unordered pairs of distinct vertices, called edges or lines. The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge.

V (and hence E) are usually taken to be finite sets, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case.

A directed graph or digraph G is an ordered pair G:=(V, A) with

V, a set of vertices or nodes,
A, a set of ordered pairs of vertices, called directed edges, arcs, or arrows. An edge e = (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the edge.

There are also some mixed type of graphs with undirected and directed edges.

== 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-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 ==

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 node has a list of which nodes it is adjacent to. This can sometimes result in "overkill" in an undirected graph as node 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 $M_{x,y}$ is 1, otherwise it is 0. This makes it easier to find subgraphs, and to reverse graphs if needed.

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

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 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 $K_{5}$ . 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)

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

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

Visibility graph problems

Museum guard problem

Covering Problems

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

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.

Graph theory is also used to study molecules in science. 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.

Subareas

Graph theory is diverse and contains many identifiable subareas. Some of them are:

Prominent graph theorists

Paul Erdős
Frank Harary
Denes König
W.T. Tutte
Graph theory white pages for more graph theorists and their publications.

Notes

The only information a weighted graph provides as such is (a) the vertices, (b) the edges and (c) the weights. Therefore the example in which the weights represent the roads' lengths doesn't imply that the weights are merely redundant annotations: there is no actual topographical information associated with the graph, so unlike reading a map, measuring the distances between the vertices is completely meaningless -- without the weights, there would be no way of telling what the distance between the vertices is in real life.

External links

Online textbooks

Graph Theory (1997/2005) by Reinhard Diestel
Graph Theory with Applications (1976) by Bondy and Murty

Other resources