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Topic: Cycle graph theory

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	Cycle graph -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A directed cycle graph or a dicycle graph is a (Click link for more info and facts about diconnected) diconnected cycle graph, that is all (Click link for more info and facts about directed edge) directed edges in the cycle point in the same direction.
		Any connected graph with a subgraph that is a cycle is not a (A tall perennial woody plant having a main trunk and branches forming a distinct elevated crown; includes both gymnosperms and angiosperms) tree.
		Cycles with an even number of vertices are (Click link for more info and facts about bipartite) bipartite, cycles with an odd number are not.
	www.absoluteastronomy.com /encyclopedia/c/cy/cycle_graph.htm (320 words)

	Cycle graph - Wikipedia, the free encyclopedia
		In graph theory, a cycle graph or cycle is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain.
		Cycles with an even number of vertices are bipartite, cycles with an odd number are not.
		Cycles with an even number of vertices can be decomposed into a minimum of 2 independent sets (that is, α(n) = 2), whereas cycles with an odd number of vertices can be decomposed into a minimum of 3 independent sets (that is, α(n) = 3).
	en.wikipedia.org /wiki/Cycle_graph (323 words)

	Encyclopedia: Graph theory
		Algebraic graph theory is a branch of mathematics.
		In mathematics topological graph theory is a branch of graph theory.
		Extremal graph theory is a branch of mathematics.
	www.nationmaster.com /encyclopedia/Graph-theory (3319 words)

	PlanetMath: graph theory
		Graph theory is the branch of mathematics that concerns itself with graphs.
		It is usually agreed upon that graph theory proper was born in 1736, when Euler formalized the now-famous ``bridges of Königsberg'' problem.
		Now, a (finite) graph is usually thought of as a subset of pairs of elements of a finite set (called vertices), or more generally as a family of arbitrary sets in the case of hypergraphs.
	planetmath.org /encyclopedia/GraphTheory.html (506 words)

	Tree (graph theory) - Wikipedia, the free encyclopedia
		In graph theory, a tree is a graph in which any two vertices are connected by exactly one path.
		G has no simple cycles and, if any edge is added to G, then a simple cycle is formed.
		Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.
	en.wikipedia.org /wiki/Tree_graph (446 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.03/glossary_graph.html (1288 words)

	ipedia.com: Glossary of graph theory Article (Site not responding. Last check: 2007-10-08)
		A graph G is consisted of two types of elements, namely vertex and edge, and can be modeled as a set system of 2-element sets (edges) over a ground set (vertices).
		The complement of a graph G is a graph with the same vertex set as G but with an edge set such that xy is an edge in if and only if xy is not an edge in G.
		A rich theory of the relationship between the properties of this matrix and that of its graph is studied in spectral graph theory.
	www.ipedia.com /glossary_of_graph_theory.html (4395 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)

	Control flow graph - Wikipedia, the free encyclopedia
		A control flow graph (CFG) is an abstract data structure used in compilers.
		A cycle in a CFG may imply that there is a loop in the code (specifically, a cycle caused by a back edge to a dominator).
		A graph for this fragment has edges from A to B, A to C, B to D and C to D. edit]
	en.wikipedia.org /wiki/Control_flow_graph (824 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)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-10-08)
		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)

	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 (1002 words)

	Basic Graph Theory (Site not responding. Last check: 2007-10-08)
		A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex.
		A graph that is not connected is a disconnected graph.
		The sum of the degrees of all the vertices of a graph is twice the number of edges in the graph.
	www.people.vcu.edu /~gasmerom/MAT131/graphs.html (354 words)

	My Graph Theory Page
		Graph Theory is simply a branch of mathematics focusing on the properties of graphs.
		Traversability of graphs was first studied in detail by the mathematician Leonard Euler in 1736, as he tried to solve a problem known as the 'Seven Bridges of Königsberg'.
		A graph is semi-traversable (semi-Eulerian) if it is possible to start at one node and pass over each arc exactly once, finishing at a different node.
	www.angelfire.com /nb/paula/adam.html (1395 words)

	``Introduction to Graph Theory'' (2nd edition)
		"Even graph" is my compromise expression for the condition that all vertex degrees are even, and I will continue to use "cycle" for a 2-regular connected graph, "circuit" for a cyclically-edge-ordered connected even graph, and "circuit" for a minimal dependent set in a matroid.
		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 (1028 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		This is a more difficult question from elementary graph theory: Prove that the maximum number of edges in a graph of order n without an even cycle is [3/2(n-1)].
		If in any of these subsets the associated cycle has an extra edge which passes from one vertex of the subset to another, we are done, because two cycles will be created, one of which must be even.
		Note that the cycle decomposition uses N+1 edges, and so there are at least [3N/2]-N edges not involved in the cycle decomposition.
	mathforum.org /library/drmath/view/52261.html (702 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)

	PlanetMath: graph
		A graph is then simple if there is at most one edge joining each pair of nodes.
		See Also: loop, neighborhood (of a vertex), Euler's polyhedron theorem, digraph, tree, spanning tree, connected graph, cycle, graph theory, graph topology, subgraph, simple path, Euler path, diameter, distance (in a graph), Bethe lattice
		This is version 27 of graph, born on 2001-11-12, modified 2004-04-02.
	planetmath.org /encyclopedia/Graph.html (191 words)

	Sharkovsky's Theorem - Graph Theory argument, outline (Site not responding. Last check: 2007-10-08)
		First we will show that if the graph was created by a function and a point of period k, then the graph will have a cycle of period k.
		We can decompose this cycle into two non-repeating cycles, where the sum of their lengths is k.
		Next will will show that if a graph has a periodic point, then it has a fixed point Further, if the periodic point is greater than 1 in order, then this function has a 2 periodic point.
	www.cecm.sfu.ca /~kghare/pm399c/Sharkovsky3c.html (234 words)

	Amazon.com: Books: Graph Theory (Site not responding. Last check: 2007-10-08)
		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
		If you are looking for examples of computer algorithms, look elsewhere; the closest this will get you is to "existence proofs", which is showing that something (such as a hamiltonian cycle) exists in a graph that has thus-and-such number of points or edges, but not tell you which sequence of points/edges make up that something.
	www.amazon.com /exec/obidos/tg/detail/-/0201410338?v=glance (778 words)

	Wiley::Graph Theory
		Intended neither to be a comprehensive overview nor an encyclopedic reference, this focused treatment goes deeply enough into a sufficiently wide variety of topics to illustrate the flavor, elegance, and power of graph theory.
		These strands center, respectively, around matching theory; planar graphs and hamiltonian cycles; topics involving chordal graphs and oriented graphs that naturally emerge from recent developments in the theory of graphic sequences; and an edge coloring strand that embraces both Ramsey theory and a self-contained introduction to Pólya's enumeration of nonisomorphic graphs.
		In the edge coloring strand, the reader is presumed to be familiar with the disjoint cycle factorization of a permutation.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471389250.html (358 words)

	Unsolved Problems
		An (m,n)-cage is an m-regular graph with girth n and, subject to this, with the least possible number of vertices.
		The bandwidth of a graph G is the minimum bandwidth among adjacency matrices of graphs isomorphic to G.
		A graph G is t-tough if, for every vertex cut S, the number of components of G-S is at most S/t.
	www.math.fau.edu /locke/Unsolved.htm (2915 words)

	Graph Theory Lesson 11 (Site not responding. Last check: 2007-10-08)
		A simple path on n vertices is a connected graph with n vertices x
		Is it possible for a bipartite graph to contain a circuit C
		A word of warning: when there is no n-cycle the program could take a long time before it exhausts all possibilities so let us agree that if the program does not find the cycle in 2 min we shall assume there is none.
	www.utc.edu /Faculty/Christopher-Mawata/petersen/lesson11.htm (283 words)

	Articles - Cycle (Site not responding. Last check: 2007-10-08)
		A cycle (Latin "cyclus," from Greek "kuklos" meaning circle) is anything round, in the physical sense (e.g.
		The indiction cycle was a cycle of 15 years used to date medieval documents.
		Biogeochemical cycles such as the Ozone-oxygen cycle (see list there).
	www.wholez.com /articles/Cycle (296 words)

	Graph Theory Course Outline
		We also mentioned a few topics which might be of interest to a few students: automorphism group of a graph, characteristic polynomial of [the adjacency matrix of] a graph, eigenvalues of a graph, similarity tranformations of a matrix, polytopes.
		We dealt with a graph which was k-regular, each pair of adjacent vertices had λ=0 common neighbours, and each pair of non-adjacent vertices had μ=1 common neighbour, and the number of vertices was n=k
		The audience is assumed to be people who have no background in graph theory, and presumably not too much in mathematics (although, I am always happy when people have a better mathematical background than I expect).
	www.math.fau.edu /Locke/courses/GraphTheory/F03Up.htm (975 words)

	Citations: Extremal Graph Theory - Bollobas (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		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)

	Graph Theory Science, Directory (Site not responding. Last check: 2007-10-08)
		Getgrats: General Theory of Graph Transformation Systems A research network funded by the European Commission.
		A Constructive Approach to Graph Theory Notes on a semiotic approach to constructing isomorphism invariants of graphs by John-Tagore Tevet.
		Graph Colorings with Local Constraints A survey by Zsolt Tuza.
	www.indiapolicyinstitute.org /aW5kXzkzMDIy.aspx (366 words)

	Graph-Theoretic Formalization of Balance Theory
		This document summarizes some key ideas in balance theory and details how those ideas are represented within the Cartwright-Harary graph-theoretic formalization of balance theory.
		Structure Theorem: Every balanced stucture can be broken down into two mutually exculsive subsets (one of which may be empty, in which case the theorem is trivially true), such that each positive tie joins two points of the same subset, and each negative tie joins two points from different subsets.
		For instance, according to the Fixed-Point Theorem, we could use all cycles through p, and if all of those cycles are positive (which they are), then we can be confident that the structure is balanced, without assessing the cycles through q and r.
	www.uiowa.edu /~c034220/balance.html (590 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 (8696 words)

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