Graph theory
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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 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 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. 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 road^{1}. 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). 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.
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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
- 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 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 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 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 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 K_{3,3} (See Three cottage problem) or if it contains 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
- 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
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:
- Algebraic graph theory
- Topological graph theory
- Geometric graph theory
- Extremal graph theory
- Metric graph theory
- Probabilistic graph theory
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 only present as informational bits of data: 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.
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
- Disjoint-set data structure
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
- 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
- Grafos spanish copyleft software
- Source code for computing neighbor shells in particle systems under periodic boundary conditions
- Graph Theory Journals
- Graph Theory Booksar:نظرية المخططات
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