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Topic: Graph theory


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  Graph Theory
Graph Theory was born to study problems of this type.
These pages are not intended to replace the standard texts in Graph Theory, rather to give a place on the web where some of the basic definitions can be found.
In an undirected graph, this is obviously a metric.
www.math.fau.edu /locke/GRAPHTHE.HTM   (1165 words)

  
  Kids.net.au - Encyclopedia Graph theory -   (Site not responding. Last check: 2007-10-27)
Graph theory is the branch of mathematics that examines the properties of graphs.
In computers, a finite directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix: an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.
A subgraph of the graph G is a graph whose vertex set is a subset of the vertex set of G, whose edge set is a subset of the edge set of G, and such that the map w is the restriction of the map from G.
www.kidsseek.com /encyclopedia-wiki/gr/Graph_theory   (1682 words)

  
 Graph theory - Wikipedia, the free encyclopedia
In a graph proper, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A.
Definitions of graphs vary in style and substance, according to the level of abstraction that is approriate to a particular approach or application.
Graph theory is also used to study molecules in chemistry and physics.
en.wikipedia.org /wiki/Graph_theory   (1736 words)

  
 Glossary of graph theory: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-10-27)
In graph theory, the girth of a graph is the length of the shortest cycle contained in the graph....
(a planar graph is a graph of genus 0.
In mathematics a graph invariant or graph property is one of the basic properties of graphs studied in graph theory....
www.absoluteastronomy.com /encyclopedia/g/gl/glossary_of_graph_theory.htm   (6710 words)

  
 Category:Graph theory - Wikipedia, the free encyclopedia
See glossary of graph theory for common terms and their definition.
Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs), which can also have associated directions.
Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges), with an arrowhead on a line representing a directed arc.
en.wikipedia.org /wiki/Category:Graph_theory   (177 words)

  
 Boost Graph Library: Graph Theory Review
This chapter is meant as a refresher on elementary graph theory.
Fundamentally, a graph consists of a set of vertices, and a set of edges, where an edge is something that connects two vertices in the graph.
The primary property of a graph to consider when deciding which data structure to use is sparsity, the number of edges relative to the number of vertices in the graph.
www.boost.org /libs/graph/doc/graph_theory_review.html   (2374 words)

  
 Games on Graphs   (Site not responding. Last check: 2007-10-27)
Graphs are mathematical objects that are made of dots connected by lines.
Graph Theory is the branch of mathematics that involves the study of graphs.
Graph theory has been instrumental for analyzing and solving problems in areas as diverse as computer network design, urban planning, and molecular biology.
www.c3.lanl.gov /mega-math/workbk/graph/graph.html   (88 words)

  
 05C: Graph theory
A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic.
A graph may be viewed as a one-dimensional CW-complex and hence studied with tools from Algebraic Topology, in particular, questions of planarity (and genus).
Determining the genus of a graph is NP-complete.
www.math.niu.edu /~rusin/known-math/index/05CXX.html   (1204 words)

  
 RDF Semantics   (Site not responding. Last check: 2007-10-27)
Readers unfamiliar with model theory may find the glossary in appendix B helpful; throughout the text, uses of terms in a technical sense are linked to their glossary definitions.
If the graphs in the set have no blank nodes in common, then the union of the graphs is a merge; if they do share blank nodes, then it is the union of a set of graphs that is obtained by replacing the graphs in the set by equivalent graphs that share no blank nodes.
Any of the graphs may have some triples added to it; the set of graphs may be extended by extra graphs; or the vocabulary of the graph may be interpreted relative to a stronger notion of vocabulary entailment, i.e.
www.w3.org /TR/rdf-mt   (11716 words)

  
 Ideas, Concepts, and Definitions   (Site not responding. Last check: 2007-10-27)
Graph paper is not particularly useful for drawing the graphs of Graph Theory.
In Graph Theory, a graph is a collection of dots that may or may not be connected to each other by lines.
If you look at a graph and your eyes want to zip all around it like a car on a race course, or if you notice shapes and patterns inside other shapes and patterns, then you are looking at the graph the way a graph theorist does.
www.c3.lanl.gov /mega-math/gloss/graph/gr.html   (215 words)

  
 graph
Formally, a graph is a set of vertices and a binary relation between vertices, adjacency.
Moreover, a mathematical graph is not a comparison chart, nor a diagram with an x- and y-axis, nor a squiggly line on a stock report.
GraphEd -- Graph Editor and Layout Program (C), graph manipulation (C++, C, Mathematica, and Pascal), build, traverse, top sort, etc. weighted, directed graphs (Java), JGraphT (Java) build, traverse, and display directed and undirected graphs, GEF - Graph Editing Framework (Java) a library to edit and display graphs.
www.nist.gov /dads/HTML/graph.html   (545 words)

  
 Graph Theory Open Problems
A graph which can be embedded in the plane so that vertices correspond to points in the plane and edges correspond to unit-length line segments is called a ``unit-distance graph.'' The question above is equivalent to asking what the chromatic number of unit-distance graphs can be.
Paul O'Donnell has found a unit distance graph of girth 12 which cannot be 3-colored, but this graph has an incredibly large number of points.
To get the square of an oriented graph (or any directed graph) you leave the vertex set the same, keep all the arcs, and for each pair of arcs of the form (u,v), (v,w), you add the arc (u,w) if that arc was not already present.
dimacs.rutgers.edu /~hochberg/undopen/graphtheory/graphtheory.html   (705 words)

  
 Amazon.com: Modern Graph Theory: Books: Bela Bollobas   (Site not responding. Last check: 2007-10-27)
The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole.
This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics.
Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest.
www.amazon.com /exec/obidos/tg/detail/-/0387984887?v=glance   (1361 words)

  
 Graph Theory Lesson 8
What this does is to color the vertices of the graph using as few colors as possible and making sure that adjacent vertices always have different colors.
The number of colors used is called the chromatic number of the graph.
Graph coloring can be used to solve problems involving scheduling and assignments.
www.utc.edu /~cpmawata/petersen/lesson8.htm   (354 words)

  
 ``Introduction to Graph Theory'' (2nd edition)
Most research and applications in graph theory concern graphs without multiple edges or loops, and often multiple edges can be modeled by edge weights.
It is convenient in research to use "graph" for whichever model is the current context, but this practice does not work well in a beginning course.
Letting "graph" forbid loops and multiple edges simplifies the first notion for students, making it possible to correctly view the edge set as a set of vertex pairs and avoid the technicalities of an incidence relation in the first definition.
www.math.uiuc.edu /~west/igt   (1070 words)

  
 Graphs
The Graph Theory originates with a 1736 Euler's paper "The Seven Bridges of Königsberg".
The second notion, that of the edges being connections between nodes, is by far too important to the Graph Theory to leave it to one's intuitive perception.
For a graph, the sum of degrees of all its nodes equals twice the number of edges.
www.cut-the-knot.org /do_you_know/graphs.shtml   (1309 words)

  
 graph theory -- graph theory textbooks and resources
The superior explanations, broad coverage, and abundance of illustrations and exercises that positioned this as the premier graph theory text remain, but are now augmented by a broad range of improvements.
Their unparalleled treatment provides a text ideal for a two-semester course and a variety of one-semester classes, from an introductory one-semester course to courses slanted toward classical graph theory, operations research, data structures and algorithms, or algebra and topology.
Authors explore the role of voltage graphs in the derivation of genus formulas, explain the Ringel-Youngs theorem—a proof that revolutionized the field of graph theory—and examine the genus of a group, including imbeddings of Cayley graphs.
www.graphtheory.com   (991 words)

  
 Algorithmic Graph Theory
-- comprehensive graph theory resource for graph theoreticians and students.
The complete project is available for further research.
-- graph drawing tool that supports a variety of layout methods.
www.personal.kent.edu /~rmuhamma/GraphTheory/graphTheory.htm   (171 words)

  
 The Math Forum - Math Library - Graph Theory   (Site not responding. Last check: 2007-10-27)
A graph is a set V of vertices and a set E of edges - pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets.
A series of short interactive tutorials introducing the basic concepts of graph theory, designed with the needs of future high school teachers in mind and currently being used in math courses at the University of Tennessee at Martin.
An Introduction to Graph Theory tutorial uses three motivating problems to introduce the definition of graph along with terms like vertex, arc, degree, and planar.
mathforum.org /library/topics/graph_theory   (2440 words)

  
 CMPE 177 - Classes - Baskin School of Engineering, UCSC
Graph theory and algorithms are developed around applications in computer engineering.
Students are given a graph problem which has no known efficient solution and are asked to develop and implement a method to solve it using the graph algorithms they have learned.
Its coverage of graph algorithms is weak and so the text is supplemented with handouts on graph algorithms.
www.soe.ucsc.edu /classes/cmpe177   (533 words)

  
 Graph Theory
  Graph theory is a branch of topology which, although going back to L. Euler, has received particular interest only in recent years, as its applications in electrical engineering and operations research lend themselves readily to algorithmic formulation and solutions on digital computers.
Graph theory formalizes the relations of entities called graphs, which consist of two sets of objects called nodes (or vertices) and edges, each edge connecting two nodes.
It is comparatively easy to write an algorithm for the minimum spanning tree [Zahn71], and the concept has been applied in the recognition of tracks from digitizings ([Zahn73], [Cassel80]) and for cluster recognition in multi-dimensional space.
br.endernet.org /~akrowne/handbook/AN16pp/node112.html   (383 words)

  
 Graph Theory Glossary   (Site not responding. Last check: 2007-10-27)
In a digraph (directed graph) the degree is usually divided into the in-degree and the out-degree (whose sum is the degree of the vertex in the underlying undirected graph).
A digraph (or a directed graph) is a graph in which the edges are directed.
A path is a sequence of consecutive edges in a graph and the length of the path is the number of edges traversed.
www.utm.edu /departments/math/graph/glossary.html   (816 words)

  
 Graphs: Theory - Algorithms - Complexity   (Site not responding. Last check: 2007-10-27)
Groups and Graphs: a software package for graphs, digraphs, combinatorial designs, and their automorphism groups, by B.
Scheinerman, E.R., Ullman, D.H.: Fractional graph theory: a rational approach to the theory of graphs, John Wiley and Sons, New York, 1997.
Graph connections -- relationships between graph theory and other areas of mathematics, Eds.
people.freenet.de /Emden-Weinert/graphs.html   (1244 words)

  
 Amazon.com: Graph Theory: Books: Frank Harary   (Site not responding. Last check: 2007-10-27)
An effort has been made to present the various topics in the theory of graphs in a logical order, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results.
It is no coincidence that graph theory has been independently discovered many times, since it may quite properly be regarded as an area of applied mathematics Read the first page
Graph Theory (Graduate Texts in Mathematics) by Reinhard Diestel
www.amazon.com /exec/obidos/tg/detail/-/0201410338?v=glance   (990 words)

  
  An overview on Graph Theory   (Site not responding. Last check: 2007-10-27)
A graph is a very simple structure consisting of a set of vertices and a family of lines (possibly oriented), called edges (undirected) or arcs (directed), each of them linking some pair of vertices.
The number of concepts that can be defined on graphs is very large, and many generate deep problems or famous conjectures (for instance the four colour problem).
We present in an annex a small bibliographical reference on Graph Theory, and a more precise description of our research topics.
www-leibniz.imag.fr /GRAPH/english/overview.html   (332 words)

  
 Open Directory - Science: Math: Combinatorics: Graph Theory   (Site not responding. Last check: 2007-10-27)
A Constructive Approach to Graph Theory - Notes on a semiotic approach to constructing isomorphism invariants of graphs by John-Tagore Tevet.
Getgrats: General Theory of Graph Transformation Systems - A research network funded by the European Commission.
Sandpiles in Graphs - An application of cellular automata by Angela R. Kerns.
dmoz.org /Science/Math/Combinatorics/Graph_Theory   (440 words)

  
 Open Directory - Science: Math: Combinatorics: Graph Theory: Open Problems   (Site not responding. Last check: 2007-10-27)
Graph Coloring Problems - Archive for the book "Graph Coloring Problems" by Tommy R. Jensen and Bjarne Toft (Wiley Interscience 1995), dedicated to Paul Erdös.
Graph Theory Open Problems - Six problems suitable for undergraduate research projects.
Problems in Topological Graph Theory - Web text by Dan Archdeacon with a list of open questions in topological graph theory.
dmoz.org /Science/Math/Combinatorics/Graph_Theory/Open_Problems   (250 words)

  
 Graph Theory Tutorials   (Site not responding. Last check: 2007-10-27)
This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory.
Starting with three motivating problems, this tutorial introduces the definition of graph along with the related terms: vertex (or node), edge (or arc), loop, degree, adjacent, path, circuit, planar, connected and component.
This question can be changed to "how many colors does it take to color a planar graph?" In this tutorial we explain how to change the map to a graph and then how to answer the question for a graph.
www.utm.edu /departments/math/graph   (282 words)

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