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Topic: Clique (graph theory)


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In the News (Mon 28 Dec 09)

  
  Graph Theory
Graph Theory was born to study problems of this type.
In an undirected graph, this is obviously a metric.
Bound δ (of a graph embedded in on a surface)
www.math.fau.edu /locke/GRAPHTHE.HTM   (1165 words)

  
  Graph Theory Glossary
For example, Figure 1.3.8 shows a simple graph which is also a bipartite graph because it may be divided into two parts, given by the subsets {1, 2} and {3, 4, 5}, where every edge in the graph goes from a vertex in one part to a vertex in the other part.
In the graph shown in Figure 1.3.14 a, cycles are represented, for example, by sequences of vertices 1, 5, 4, 2, 3, 4, 1 and 1, 2, 3, 4, 5, 1.
The vertices of the graph shown in Figure 1.3.29 may be properly colored in four colors: the first color for vertex 1, the second color for vertices 2, and 7, the third color for vertices 4, and 5, and the fourth color for vertices 3, and 6.
exchange.manifold.net /manifold/manuals/manifold/networks/graph_theory/graph_theory_glossary.htm   (2620 words)

  
 Graph theory glossary
A coclique in a graph is a clique in its complementary graph (q.v.).
girth (n.): The girth of a graph is the length of the shortest cycle(s) in the graph.
When A,B are graphs, an isomorphism is a bijection from the vertices of A to the vertices of B such that any two vertices of A are adjacent if and only if their images in B are adjacent.
www.math.harvard.edu /~elkies/FS23j.04/glossary_graph.html   (1317 words)

  
 14. Some Graph Theory
A graph G is a collection, E, of distinct unordered pairs of distinct elements of a set V.
Graphs are things that underlie many mathematical structures, and in fact anything that involves pairs of elements, and this includes any kind of relationship between pairs of individual entities.
Another way to state the perfect graph theorem is: If you cannot partition the vertices of G into a number of cliques given by the size of its largest independent set, then G has an induced subgraph H that cannot be partitioned into a number of independent sets given by H’s clique number.
www-math.mit.edu /18.310/some_graph_theory.html   (3368 words)

  
 Intro to Graph Theory
A graph is defined as a set of nodes and a set of lines that connect the nodes.
A subgraph of a graph is a subset of its points together with all the lines connecting members of the subset.
A bridge is an edge whose removal from a graph increases the number of components (disconnects the graph).
www.analytictech.com /mb021/graphtheory.htm   (1984 words)

  
 Analyzing Clique Overlap
The clique graph of a graph G is denoted by K(G).
The last two graphs in Figure 3 are clique graphs; strictly speaking, the first is not since we do not include all the cliques (we ignore those of size 1 and 2).
The cliques 1 to 5 correspond to the actors 1 to 8 and clique 6 to actors 9 to 16.
www.analytictech.com /borgatti/papers/analyzing_clique_overlap.htm   (3329 words)

  
 PlanetMath: clique
A maximal complete subgraph of a graph is a clique, and the clique number
Adapted with permission of the author from Modern Graph Theory by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.
This is version 10 of clique, born on 2002-03-04, modified 2006-09-29.
planetmath.org /encyclopedia/Clique2.html   (100 words)

  
 Graph theory Summary
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.
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.
www.bookrags.com /Graph_theory   (3094 words)

  
 Clique   (Site not responding. Last check: 2007-10-15)
To find a nice maximal clique, sort the vertices from highest degree to lowest degree, put the first vertex in the clique, and then test each of the other vertices in order to see whether it is adjacent to all the clique vertices thus far.
Heuristics for finding large cliques based on randomized techniques   such as simulated annealing are likely to work reasonably well.
This reduction established that clique, vertex cover, and independent set are very closely related problems, so heuristics and programs that solve one of them may also produce reasonable solutions for the other two.
www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE172.HTM   (1020 words)

  
 Graph Theory (math 224)
A plane graph is a graph which is actually embedded in the plane so that each vertex corresponds to a point and each edge to a simple closed curve (or straight-line segment if you prefer) joining the points corresponding to its endpoints.
The complement of a plane graph is a disjoint union of connected components which are called the _regions_ of the plane graph.
Similarly, for graphs in the _torus_ (think "doughnut" or "inner tube") n-m+r = 0 and the corresponding upper bound on edges is m leq 3n; hence, average degree is at most 6 and so there must be a vertex of degree not exceeding 6 in any toroidal graph.
www.georgetown.edu /faculty/kainen/graph-theory.html   (3496 words)

  
 The Math Forum - Math Library - 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.
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   (2406 words)

  
 Graph Theory
The text is "Introduction to Graph Theory" by Richard J. Trudeau, which is in paperback from Dover Publications, NY, 1994; still in print and available in the bookstore or from amazon.com - here is a picture.
So the emphasis for the final will be on using graph theory as a tool to formulate problems, asking only for you to be familiar with a reasonable proportion of the material we've covered in class, including at least one of the class presentations in addition to that of your own group.
The radius of a graph is the minimum eccentricity of the vertices, while the diameter of a graph is the maximum eccentricity of the vertices.
www.georgetown.edu /faculty/kainen/graphtheory.html   (3531 words)

  
 Open Directory - Science: Math: Combinatorics: Software: Graph Drawing   (Site not responding. Last check: 2007-10-15)
GDToolkit - a Graph Drawing Toolkit - GDToolkit (also known as GDT) is a Graph Drawing Toolkit designed to efficiently manipulate several types of graph, and to automatically draw them according to many different aesthetic criteria and constraints.
Graph Magics - A tool for graph theory, having a generator and offering various algorithms: shortest paths, network flows, maximal clique, optimal coloring etc.
PIGALE - Public Implementation of a Graph Algorithm Library and Editor - PIGALE is a graph editor and an algorithm library essentially concerned with planar graphs.
dmoz.org /Science/Math/Combinatorics/Software/Graph_Drawing   (777 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).
Applications of graph theory to circuitry and networks are included in 94C15: Information theory.
www.math.niu.edu /~rusin/known-math/index/05CXX.html   (1204 words)

  
 Graph Theory: Industrial Drilling
It is clear that these points are "close" to one another, and therefore form a clique.
The size of this clique is a lower bound for the chromatic number.
This is the largest clique of the graph.
links.math.rpi.edu:16080 /devmodules/graph_theory/xhtml/page33.xml   (123 words)

  
 Keith Briggs: : very_nauty - graph theory software
This is a C library of graph algorithms, especially targeted at very fast generation of random graphs, and exact clique number and chromatic number computation.
The name comes from the fact that it is designed to be compatible with Brendan McKay's nauty software, which is mainly concerned with graph generation and isomorphism testing.
In practice, it's possible to use the exact algorithms on graphs with up to a few hundred nodes.
keithbriggs.info /very_nauty.html   (1073 words)

  
 1.5.1 Clique   (Site not responding. Last check: 2007-10-15)
Excerpt from The Algorithm Design Manual: When I went to high school, everybody complained about the ``clique'', a group of friends who all hung around together and seemed to dominate everything social.
Consider a graph whose vertices represent a set of people, with edges between any pair of people who are friends.
A large clique in this graph points to fraud.
www.cs.sunysb.edu /~algorith/files/clique.shtml   (246 words)

  
 Graph Theory: Industrial Drilling
A set of vertices is called a clique if every pair of vertices in the set is adjacent.
a clique) is called maximum if its cardinality is maximal among all independent sets (resp.
A set of edges in a graph is called a matching if no two of them are incident to the same vertex.
www.ibiblio.org /links/devmodules/graph_theory/compat/page24.html   (118 words)

  
 The Graph Theorists' Home Page Guide
First of all, if you're a graph theorist or some person with strong interest in graph theory (you need not to be a mathematician!), and if you have a homepage but don't find a link to it on this page, please contact me as described above.
PIGALE is a graph editor with an interface to the LEDA library and with many algorithms implemented essentially concerning planar graphs.
"Graph Theory and Its Applications" (together with Jay Yellen), "a comprehensive applications-driven textbook that provides material for several different courses in graph theory." This site also provides links to other graph theoretical and mathematical resources.
www.joergzuther.de /math/graph/homes.html   (8736 words)

  
 Iadis
The maximum clique problem is well known to be NP-hard and is a core problem for a lot of applications in artificial intelligence systems, data mining and many others.
The algorithm is tested on DIMACS graphs to compare it with other well-known algorithms.
Moreover certain modifications of the heuristic colouring strategies described in the article produce even better algorithms for some graph types introducing a need for an artificial intelligence approach in the maximum clique finding algorithms’ implementations.
www.iadis.net /dl/Search_list_open.asp?code=2746   (234 words)

  
 Amazon.com: "extremal graph theory": Key Phrase page   (Site not responding. Last check: 2007-10-15)
See all pages with references to extremal graph theory.
Handbook of Graph Theory (Discrete Mathematics and Its Applications) by Jonathan L. Gross (Editor), Jay Yellen (Editor)
The simplest question of extremal graph theory is this: Consider a graph with n vertices (points).
www.amazon.com /phrase/extremal-graph-theory   (512 words)

  
 clique - OneLook Dictionary Search
Clique, clique, clique (small exclusive group) : Dict.cc Englisch/Deutsch Wörterbuch [home, info]
Phrases that include clique: clique graph, clique problem, chateau clique, château clique, clique du château, more...
Words similar to clique: camp, cliqued, cliquey, cliquing, cliquish, cliquishly, cliquishness, cliquy, coterie, ingroup, pack, gang, inner circle, set, more...
www.onelook.com /?ls=b&w=clique   (275 words)

  
 [No title]
subset of nodes and edges of a graph
presence of a clique in the docking graph proves the existence of a match between ligand and protein provided that non-subset ligand atoms do not clash with the protein
graph is a sparse matrix (typically 1 % filled) determined by distance tolerance and distance minimum
www.bmsc.washington.edu /people/verlinde/BSTR520/clique.html   (241 words)

  
 Open Directory - Science: Math: Combinatorics: Graph Theory
The Clique Algorithm - A polynomial-time algorithm for finding maximal cliques in a graph with new bounds on Ramsey numbers by Ashay Dharwadker.
A Constructive Approach to Graph Theory - Notes on a semiotic approach to constructing isomorphism invariants of graphs by John-Tagore Tevet.
The Vertex Coloring Algorithm - A polynomial-time algorithm for coloring the vertices of a graph with a new constructive proof of Brooks' theorem by Ashay Dharwadker.
dmoz.org /Science/Math/Combinatorics/Graph_Theory   (529 words)

  
 How graph theory relates to your links looking unnatural to Google @ Scatterings
How graph theory relates to your links looking unnatural to Google
When I blogged earlier this month about some things Matt Cutts from Google had to say about linking, I mentioned something called “cliques” from graph theory.
4 comments for How graph theory relates to your links looking unnatural to Google »
www.stephanspencer.com /archives/2005/08/26/how-graph-theory-relates   (629 words)

  
 Graph Theory: Industrial Drilling
To be sure you are ready to tackle this graph theory problem, take a moment to review the basic definitions.
Make sure, in particular, that you fully understand the definitions of an independent set, a clique, and of vertex coloring.
is the size of the maximum clique in a graph
www.ibiblio.org /links/devmodules/graph_theory/compat/page29.html   (105 words)

  
 Problems in Topological Graph Theory
Graphs that quadrangulate both the torus and Klein bottle
Orientable genus of graphs of bounded nonorientable genus
Geometric graphs with each edge crossing at most three others
www.emba.uvm.edu /~archdeac/problems/problems.html   (283 words)

  
 Graph Theory
The earliest paper on graph theory seems to be by Leonhard Euler, Solutio problematis ad geometriam situs pertinentis, Commetarii Academiae Scientiarum Imperialis Petropolitanae 8(1736), 128-140.
Perfect Graphs; Perfect graph pages by V. Chvátal
This page can now be reached from Yahoo.
www.math.fau.edu /locke/graphthe.htm   (1165 words)

  
 Graph Theory: Industrial Drilling
The red vertices in this graph form an independent set of size 4.
Find a clique of size 3 in the graph.
Find a largest matching in the graph above.
links.math.rpi.edu:16080 /devmodules/graph_theory/xhtml/page25.xml   (46 words)

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