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Topic: Connectedness (graph theory)

	Connectedness - Wikipedia, the free encyclopedia
		This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the context of graph theory.
		For example, in graph theory, a connected graph is one from which we must remove at least one vertex to create a disconnected graph.
		The connectivity of a graph is the minimum number of vertices that must be removed, to disconnect it.
	en.wikipedia.org /wiki/Connectedness (784 words)

	Connectivity (graph theory) - Wikipedia, the free encyclopedia
		Thus an edge cut of G is a set of edges whose removal renders the graph disconnected, the edge-connectivity κ′(G) is the size of a smallest edge cut, and the local edge-connectivity κ′(u,v) of two vertices u,v is the size of a smallest edge cut disconnecting u from v.
		One of the most important facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices.
		In computational complexity theory, SL is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to L by Omer Reingold in 2004.
	en.wikipedia.org /wiki/Connectivity_(graph_theory) (801 words)

	Whats Graph Theory
		The transitive closure of this graph represents the concept of "is in the overburden of".
		The graph whose directed edges indicate "block Y is in the overburden of a block X" is therefore the transitive closure of the slope graph.
		Graph theory is an approachable area of mathematics.
	www3.telus.net /public/nstuart/pan/grtheory.htm (3426 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)

	Spectral Graph Theory (Site not responding. Last check: 2007-10-12)
		Hoffman, A. J., On the polynomial of a graph, Amer.
		Mowshowitz, A., The characteristic polynomial of a graph.
		Razborov, A. A., The gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear.
	www.sgt.pep.ufrj.br /home_arquivos/articles.html (8896 words)

	Topology - Advanced Topics in Mathematics at CCHS
		A network (or "graph" in the language of graph theory) is simply an idealized version of connectedness between points in a plane.
		Graph Theory has since found applications in Astronomy, Biology, Computer Science, Economics, etc. and has spawned entirely new fields such as "Data Mining".
		These beginnings of Graph Theory led to the development of the field of Topology in general and now includes 3-dimensional and n-dimensional objects and their properties.
	mail.colonial.net /~abeckwith/topo.html (798 words)

	Graph Theory Lecture Notes 8a
		A graph G is k-edge-connected if G is connected and every edge-cut has at least k edges.
		Then the graph that results from a path addition to H is 2-connected.
		Theorem 5.2.3: (Whitney, 1932) A graph G is 2-connected if and only if G is a cycle or a Whitney synthesis from a cycle.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtaln8.html (448 words)

	Module and Programme Catalogue
		Graph theory is an important mathematical tool in such different areas as linguistics, chemistry and, especially, operational research.
		But its origins are in mathematical puzzles such as that of the Bridges of Kvnigsberg, and graph theory continues to have its own intellectual appeal apart from its practical applications.
		The module will provide an introduction to the basic ideas such as connectedness, trees, planar graphs, Eulerian and Hamiltonian graphs, directed graphs and the connection between graph theory and the four colour Problem.
	www.leeds.ac.uk /modules/200203/ug/math3032.htm (158 words)

	Publications of Erdos in Graph Theory
		Zbl 289.05128 • Erdös, Paul; Renyi, Alfréd, On the existence of a factor of degree one of a connected random graph.
		Graph theory in memory of G. Dirac, Pap.
		Zbl 691.05053 • Erdös, Paul; Evans, Anthony B. Representations of graphs and orthogonal Latin square graphs.
	www.zblmath.fiz-karlsruhe.de /MATH/general/erdos/graph.htm (3070 words)

	Evaluating a Dependable Distributed System with Multiple Critical Tasks
		We use graphs to represent the topologies of heterogeneous autonomous decentralized systems and use the residual connectedness reliability (RCR) to characterize the communication capacity among its subsystems connected by its gateways.
		We applied these expressions to several typical graphs and showed that the differences between the upper and lower bounds tend to zero as the sizes of graphs tend to infinite.
		The contributions of this research are twofold, we find an efficient way to model and evaluate the communication capacity of heterogeneous autonomous decentralized systems; we contribute an efficient algorithm to estimate RCR in general graph theory.
	www.public.asu.edu /~ychen10/abstracts/adsjournal.html (300 words)

	19th LL-Seminar on Graph Theory / Abstracts
		A minimum cycle basis in an undirected graph G is a set of simple cycles whose incidence vectors span the cycle space of G and whose overall edge sum is minimal.
		As to the structure PN are bipartite graphs where one kind of vertices - places $p_1,\,p_2,\,...,\,p_n$ represent some subsystems, elementary operations, etc. and the second kind of vertices - transitions $t_1,\,t_2,\,...,\,t_m$ represent discrete events - e.g.
		An automorphism of a map $\cal M$ is an automorphism of the embedded (combinatorial) graph which extends to a self-homeomorphism of the surface.
	www.tbi.univie.ac.at /LL/abstracts.html (3851 words)

	[No title]
		A new combinatorial construction of expander graphs was used recently to resolve a group theoretic question on expansion in Cayley graphs.
		In the 1980's, results from the theory of automorphic forms were used to construct explicit families of Ramanujan graphs, that is, graphs for which Laplace eigenvalues satisfy strong inequalities.
		The theory of arithmetic groups and representation theory of semisimple groups have led to results on the diameters and expansion properties of finite simple groups.
	www.ipam.ucla.edu /programs/agg2004 (814 words)

	Graph Theory, Small Worlds, and the WEB: Selection of some topics for Spring School 2001 (ResearchIndex) (Site not responding. Last check: 2007-10-12)
		Graph Theory, Small Worlds, and the WEB: Selection of some topics for Spring School 2001 (ResearchIndex)
		Graph Theory, Small Worlds, and the WEB: Selection of some topics for Spring School 2001 (2001)
		65 the evolution of ran= dom graphs (context) - os, enyi - 1960
	citeseer.ist.psu.edu /644871.html (474 words)

	620-352 Graph Theory
		This subject introduces the basic concepts of graph theory including isomorphic graphs, subgraphs, connectedness, bipartite graphs, paths and cycles, trees, weighted graphs and distance in graphs, Steiner trees, matchings, flows and eulerian circuits.
		Students should develop the ability to implement algorithms on graphs for finding objects such as minimum spanning trees, maximum matchings and flows; and to implement approximation algorithms.
		This subject demonstrates the variety of applications of graph theory within and outside mathematics.
	www.unimelb.edu.au /HB/subjects/620-352.html (236 words)

	Problems ex Cameron's homepage
		Let s(n) be the number of sequences of elements from the set {1,...,n} for which each term is at least twice the preceding one, and u(n) the number of such sequences in which each term is greater than the sum of its predecessors.
		This was motivated by the question of whether a vertex-transitive cubic graph necessarily has a semiregular group of automorphisms of order greater than 3 (see Problem 50), which is now solved in the affirmative.
		It is known that the only finite or countably infinite graphs G with the property that, for any partition of the vertex set into two parts, at least one of the parts induces a subgraph isomorphic to G, are a single vertex, the countable complete and null graphs, and the countable random graphs.
	www.maths.qmw.ac.uk /~pjc/oldprob.html (7271 words)

	Graph Theory (Site not responding. Last check: 2007-10-12)
		Math 335 provides an introduction to graph theory and its applications.
		Further topics are selected from the theory of finite state machines, Ramsey theory, extremal theory, and graphical enumeration.
		After completing this course, students should have an understanding of the fundamental concepts of graph theory.
	www.calpoly.edu /~math/ugcourses/graphtheory.html (66 words)

	Reinhard Diestel's papers
		Graph Minors I: a short proof of the pathwidth theorem, Combinatorics, Probability and Computing 4 (1995), 27-30; DVI (A better exposition with figures is available here in PDF - an excerpt from the chapter on graph minors in Graph Theory, 1st ed'n.)
		A universal planar graph under the minor relation, J. Graph Theory 32 (1999), 191-206 (with D. Kühn); abstract; PDF
		A separation property of planar triangulations, J. Graph Theory 11 (1987), 43-52.55 (1985), 21-33.
	www.math.uni-hamburg.de /home/diestel/papers/allpapers.html (1505 words)

	John Wiley & Sons, Inc.: Arlinghaus - Graph Theory - ToC
		Graph theory is an ideal launching pad leading to this realm: its basic objects are easy to understand from an intuitive viewpoint, yet it employs the logic and rigor that is characteristic of contemporary pure mathematics.
		We have provided geographic examples from Los Angeles to Berlin and from freeways to pneumatic tube networks, not only to show the synthetic nature of geography as well as of graph theory but also to build the reader's interest so that new applications will ensue.
		We try to reference the theory as we use it, so that the reader can jump right into these problems immediately and return for more detail, using the interactive look-up feature, when it would provide extra insight.
	www.wiley.com /legacy/graph_theory/arlinghaus_toc.html (1415 words)

	Updates to "Introduction to Graph Theory"
		A student learning graph theory should consult this list only when some item in the book seems incorrect, to see whether there is a correction listed here.
		Generalizing to all graphs by requiring preservation of edge multiplicity conflicts with the incidence definition of graph.
		If a graph is defined by its adjacency relation, then a 2-cycle has two automorphisms; if by its incidence relation, then a 2-cycle has four automorphisms (the latter seems to be the proper choice).
	www.math.uiuc.edu /~west/igt/igterr.html (4892 words)

	MATH279 Page (Site not responding. Last check: 2007-10-12)
		Advanced course in graph theory covering graphs, digraphs, trees, networks, connectedness, eulerian circuits, hamiltonian cycles, graph embeddings, matchings, factorizations, graph colorings and Ramsey theory.
		To help students understand the structure of graphs, and the techniques used in analyzing problems in graph theory.
		Through understanding the statement of exercises, students learn definitions and concepts in graph theory.
	www.math.sjsu.edu /~so/math279.html (125 words)

	Graph Theory Lesson 14
		There are several ways to combine two graphs to get a third one.
		Now click operations and union and when asked which graph to form a union with choose graph 1.
		If you draw graphs on the left and right panels of the applet below, you will get the sum of the two graphs in the middle panel.
	www.utc.edu /Faculty/Christopher-Mawata/petersen2/lesson14.htm (136 words)

	G13GRA Introductory Graph Theory
		Wilson, Robin J. Introduction to graph theory 4th ed.
		Further references: Diestel, Reinhard Graph theory 2nd ed.
		Further references: Wilson, Robin J. Introduction to graph theory 4th ed.
	www.maths.nott.ac.uk /personal/pmznd/g13gra/sylgra.html (562 words)

	St. Lawrence University Math Department Patti Frazer Lock (Site not responding. Last check: 2007-10-12)
		My research interests fall into two main areas: graph theory and mathematics education.
		My research in graph theory involves hamiltonian- connectedness properties of graphs, and I
		enjoy working with undergraduates on a variety of research problems involving graph theory.
	it.stlawu.edu /~math/faculty/PFLock.html (103 words)

	connectedness - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "connectedness" is defined.
		Phrases that include connectedness: characterizations of connectedness, visual connectedness
		Words similar to connectedness: connected, connection, connexion, link, more...
	www.onelook.com /?w=connectedness (129 words)

	MAS210, Graph Theory and Applications (Site not responding. Last check: 2007-10-12)
		Graph theory is about counting and searching in finite structures.
		The course examines some of the best known graph-theoretic algorithms and their mathematical basis.
		Wilson & Watkins, Graphs, An Introductory Approach (Wiley).
	www.maths.qmw.ac.uk /~wilfrid/courses/MAS210.html (54 words)

	Graph Theory Lecture Notes (Site not responding. Last check: 2007-10-12)
		Lecture notes for this semester are available for
		Basic Definitions and Graph Families (§§ 1.1 - 1.2)
		Subgraphs and Graph Operations (§§ 2.1 - 2.2)
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln.html (84 words)

	Graph Theory Lessons (Site not responding. Last check: 2007-10-12)
		Lesson 12: Euler Circuits, Hamilton Circuits and Directed Graphs
		Lesson 23: Weighted Graphs, Shortest Paths, and Minimal Spanning Trees
		These pages are optimized for use with the Netscape2.0 and 3.0 family of Web Browsers.
	oneweb.utc.edu /~Christopher-Mawata/petersen2 (62 words)

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