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Topic: Extremal graph theory

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	Graph theory - Wikipedia, the free encyclopedia
		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).
		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.
		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.
	en.wikipedia.org /wiki/Graph_theory (1209 words)

	Encyclopedia: Graph theory (Site not responding. Last check: 2007-10-22)
		In the mathematical discipline of graph theory a covering for a graph is a set of vertices (or edges) so that the elements of the set are close (adjacent) to all edges (or vertices) of the graph.
		Algebraic graph theory is a branch of mathematics.
		In mathematics topological graph theory is a branch of graph theory.
	www.nationmaster.com /encyclopedia/Graph-theory (3291 words)

	Graph theory - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-22)
		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.
		The data structure used depends on the graph structure and the algorithm used for manipulating the graph.
		Incidence matrix - The graph is represented by a matrix of E (edges) by V (vertices), where contains the edge's data (simplest case: 1 - connected, 0 - not connected).
	encyclopedia.worldsearch.com /graph_theory.htm (998 words)

	Extremal graph theory - Wikipedia, the free encyclopedia
		Extremal graph theory is a branch of mathematics.
		In the narrow sense, extremal graph theory studies the graphs which are extremal among graphs with a certain property.
		There are various meanings for the word extremal: with the largest number of edges, the largest minimum degree, the smallest diameter, etc. In a broader sense, various other related questions can be included into extremal graph theory.
	en.wikipedia.org /wiki/Extremal_graph_theory (210 words)

	Graph theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		Informally, a graph is a set of objects called (Click link for more info and facts about vertices) vertices (or Nodes) connected by links called (A sharp side formed by the intersection of two surfaces of an object) edges (or Arcs) which can be directed.
		This illustrates the deep connection between graph theory and (The configuration of a communication network) topology.
		An important application of graph theory can be found in (Click link for more info and facts about mathematical chemistry) mathematical chemistry where ((physics and chemistry) the simplest structural unit of an element or compound) molecules are modelled by graphs.
	www.absoluteastronomy.com /encyclopedia/g/gr/graph_theory.htm (1811 words)

	NSF/CBMS Invited Talks Page
		A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree at most s.
		Extremal Graph Theory is one of the most developed branches of graph theory.
		We develop a theory of linkless embeddings which is analogous to (and generalizes) the theory of planar embeddings.
	www.cs.gc.cuny.edu /~pach/cbms/talks.htm (835 words)

	Publications of Tibor Szabó
		Extremal problems for transversals in graphs with bounded degree, (with G. Tardos), Combinatorica, to appear.
		For a graph property P what is the smallest part size p(d,P) which still guarantees that a transversal inducing a subgraph with property P can be selected from any partitioned graph of maximum degree d.
		The Turán number ex(n,H) of a graph H is the largest integer e such that there exists a graph on n vertices with e edges which does not contain H as a subgraph.
	www.inf.ethz.ch /personal/szabo/extremal.html (1955 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)

	Amazon.com: Books: Modern Graph Theory (Site not responding. Last check: 2007-10-22)
		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 (1196 words)

	Fields Institute - Programs-Thematic -Graph Theory & Optimization
		Extremal graph theory and extremal hypergraph theory are core areas of combinatorics.
		Extremal graph theory was basically started by Turán and was influenced greatly by the results of Paul Erdös: in fact, it was among his favorite fields.
		Both areas, especially the theory of hypergraphs, have developed considerably since the publication of Bollobás' excellent monograph "Extremal graph theory" (1978).
	www.fields.utoronto.ca /programs/scientific/99-00/graph_theory/mini-symposia/extremal_graph_theory (130 words)

	Graph Theory (Site not responding. Last check: 2007-10-22)
		Graph Theory has emerged as an important tool and an exciting branch of applied mathematics in its own right.
		A strong background emphasizing theory and techniques of proof is indispensable for the individual who wishes to apply graphs to other disciplines of mathematical sciences.
		With the growing importance of Graph Theory in mathematics and in applications and its recognition as a legitimate mathematical discipline, I feel that a student, with these exciting skills, will emerge as a practical mathematician in a technical world.
	www.cam.wits.ac.za /cam/gt.html (257 words)

	Amazon.com: Books: Algebraic Graph Theory (Site not responding. Last check: 2007-10-22)
		This book is primarily aimed at graduate students and researchers in graph theory, combinatorics, or discrete mathematics in general.
		However, all the necessary graph theory is developed from scratch, so the only pre-requisite for reading it is a first course in linear algebra and a small amount of elementary group theory.
		--It is sketchy on chromatic polynomial, planar graph.
	www.amazon.com /exec/obidos/tg/detail/-/0387952209?v=glance (683 words)

	Extremal Graph Theory:Bollobas, Bela:0486435962:eCampus.com
		The ever-expanding field of extremal graph theory encompasses an array of problem-solving methods, including applications to economics, computer science, and optimization theory.
		Although geared toward mathematicians and research students, much of Extremal Graph Theory is accessible even to undergraduate students of mathematics.
		Pure mathematicians will find it a valuable resource in terms of its unusually large collection of results and concise proofs, and professionals in other fields with an interest in the applications of graph theory will also appreciate its precision and scope.
	www.ecampus.com /bk_detail.asp?isbn=0486435962 (103 words)

	Extremal Graph Theory
		Comprehensive yet concise, this treatment of extremal graph theory is appropriate for advanced undergraduate and graduate students.
		A non-technical introduction to the field of graph theory and its applications.
		Concise, well-written text illustrates development of graph theory and application of its principles in methods both formal and abstract.
	www.doverdirect.com /0486435962.html (128 words)

	BackgroundMaterial (Site not responding. Last check: 2007-10-22)
		These sequences are ordered lexicographically, so among all graphs in F, choose one for which this degree sequence is lexicographically as large as possible.
		Then consider the graph H obtained from G by removing all edges between pairs of vertices in Y and making each vertex in Y adjacent to all vertices adjacent to x_1.
		This graph must have exactly the same degree sequence as G and thus is just G. Now the result follows easily by induction.
	www.math.gatech.edu /~trotter/Section7-Extremal.htm (551 words)

	CMPT 891 Seminar (Site not responding. Last check: 2007-10-22)
		Many models in statistical mechanics and many questions in extremal graph theory can be phrased in these terms.
		We introduce a matrix, which we call the connection matrix, and show that this is positive semidefinite (in statistical mechanics, a related fact is called "reflection positivity").
		Using properties of this matrix, one can define and characterize "convergence" for a sequence of graphs whose size tends to infinity, and construct a limit object from which the limiting values of many graph parameters can be read off.
	www.cs.sfu.ca /CC/891/showseminar.cgi?semid=211 (173 words)

	Extremal graph theory
		A property of a graph is monotone if the whole graph has the property when a subgraph does.
		A property of a graph is non-trivial if the empty graph does not have the property.
		The study of the minimum size of a graph with a monotone, non-trivial property, or the maximum size of a graph without it.
	john.fremlin.org /schoolwork/graph/graph-theory/node7.html (104 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Extremal graph theory deals with graphs satisfying specified constraints and optimizing some criteria.
		Here, a family L of sample graphs is fixed and we consider various conditions on a graph on n vertices that does not contain any graph in L as a subgraph (not necessarily induced).
		In approaching some problems in extremal graph theory, two standard tools are the Regularity Lemma and the Blow-up Lemma.
	dimacs.rutgers.edu /Workshops/Extremal2/announcement.html (533 words)

	Extremal problems
		Galluccio, Anna; Simonovits, Miklós; Simonyi, Gábor On the structure of co-critical graphs.
		Graph theory and its applications: East and West (Jinan, 1986), 155--162, Ann.
		Simonovits, M. A method for solving extremal problems in graph theory, stability problems.
	www.math-inst.hu /~miki/extremal.html (764 words)

	J-H. Kang: Extremal Graph Theory (Site not responding. Last check: 2007-10-22)
		Extremal graph theory is, broadly speaking, the study of relations between various graph invariants, such as order, size, connectivity, minimum/maximum degree, chromatic number, etc., and the values of these invariants that ensure that the graph has certain properties.
		Among all its topics, graph coloring is the most applicable and widely studied.
		Typically, a graph models conflicts, and a good coloring ensures partitions into parts with no conflicts.
	www.math.uiuc.edu /~j-kang5/EGT.html (97 words)

	DIMACS/DIMATIA/Renyi Working Group on Extremal Combinatorics
		A corresponding extremal graph theory notion was developed for certain fixed graphs and very recently extended for all possible fixed graphs by Komlós [28].
		Many problems of extremal graph theory are interesting in the more general setting of extremal set theory or extremal hypergraph theory.
		The theory of group testing arose via testing millions of World War II military draftees for syphilis [13] and it is very relevant to schemes for large-scale blood testing for viruses such as HIV.
	dimacs.rutgers.edu /Workshops/Extremal/main.html (2397 words)

	CS 572 \| Graduate Education \| Computer Science \| UIUC
		Distance and connectivity, matching and factors, vertex and edge colorings, perfect and imperfect graphs, intersection classes and intersection parameters, Turan's theorem, graph Ramsey theory, graph decomposition and other extremal problems.
		Extremal subtrees, shortest paths, diameter, average distance, efficient embedding, bandwidth.
		Turan's theorem, graph Ramsey theory, graph decomposition, representation parameters.
	www.cs.uiuc.edu /graduate/courses.php?course=cs572 (104 words)

	Citations: Extremal Graph Theory - Bollobas (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Thus the situation for graphs H with (H) 3 is quite well understood.
		....14, 2002 Abstract A graph property is called elusive (or evasive) if every algorithm for testing this property has to read in the worst case entries of the adjacency matrix of the given graph.
		Subdivisions of K r+2 in graphs of average degree at least r +..
	citeseer.ist.psu.edu /context/61062/0 (1169 words)

	Ryan Martin's Research Statement
		The focus of my research is extremal graph theory and random combinatorial structures.
		The abstract model is valuable in that the random graph is guaranteed to maintain intrinsic properties of the system; for example, minimum degree.
		The size of a maximum-sized family for bounded-degree graphs G is also known for r≥cn for some constant c depending on the particulars of graph G.
	www.math.cmu.edu /~rymartin/cv/resstate.html (726 words)

	Graphs: Theory - Algorithms - Complexity (Site not responding. Last check: 2007-10-22)
		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)

	CSA Research Seminars -- L. Sunil Chandran (Site not responding. Last check: 2007-10-22)
		Time : 3.30pm Venue : CSA Lecture Hall Coffee : 4.30pm A High Girth Graph Construction and Expanders --------------------------------------------- One of the problems in Extremal Graph Theory is the existence and construction of high girth graphs (girth = length of the shortest cycle in the graph) with large minimal degree.
		Expanders are graphs with the property that every small sized subset of vertices has a large number of neighbours.
		Except for a very basic familiarity with graph theory nothing is assumed.
	ces.iisc.ernet.in /mails/941539404.html (166 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.
		GraphEd, an "interactive, extensible editor for graphs and graph grammars with lots of layout and other algorithms"
	www.joergzuther.de /math/graph/homes.html (8696 words)

	Extremal Graph Theory (L.M.S. monographs) by Bela Bollobas, New, Used Books, Cheap Prices, ISBN 0121117502
		Fundamental Principles in the Theory of Extremal P...
		Graph Theory: Proceedings of the Conference on Gra...
		Graph Theory Singapore 1983: Proceedings of the Fi...
	www.bookfinder4u.com /detail/0121117502.html (230 words)

	Encyclopedia: Extremal graph theory (Site not responding. Last check: 2007-10-22)
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		This graph is known as a Turán graph, it contains
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Extremal-graph-theory (236 words)

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