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Topic: Loop (graph theory)

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	Quantum Field Theory Resources
		Quantum field theory is the basis of the Standard Model of particle physics and is the best tested of all physical theories, more general in application and better tested within its range of application than the existing formulation of general relativity (which needs modification to include quantum field effects), describing all electromagnetic and nuclear phenomena.
		This approach, 'loop quantum gravity', is entirely different from that in string theory, which is based on building extra-dimensional speculation upon other speculations, e.g., the speculation that gravity is due to spin-2 gravitons (this is speculative is no experimental evidence for it).
		In loop quantum gravity, by contrast to string theory, the aim is merely to use quantum field theory to derive the framework of general relativity as simply as possible.
	quantumfieldtheory.org (5737 words)

	Glossary of graph theory - Wikipedia, the free encyclopedia
		Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their corresponding vertices are adjacent in H.
		A vertex of degree 0 is an isolated vertex.
		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.
	en.wikipedia.org /wiki/Glossary_of_graph_theory (5364 words)

	PlanetMath: loop
		In graph theory, a loop is an edge which joins a vertex to itself, rather than to some other vertex.
		By definition, a graph cannot contain a loop; a pseudograph, however, may contain both multiple edges and multiple loops.
		This is version 6 of loop, born on 2002-01-25, modified 2003-08-21.
	planetmath.org /encyclopedia/LoopOfAGraph.html (125 words)

	Degree (graph theory) - Wikipedia, the free encyclopedia
		In the mathematical field of graph theory the degree or valency of a vertex v is the number of edges incident to v (with loops being counted twice).
		In a directed graph the indegree of a vertex v is the number of edges terminating at v and the outdegree is the number of edges originating at v.
		If each vertex of the graph has the same degree k the graph is called a k -regular graph and the graph itself is said to have degree k.
	en.wikipedia.org /wiki/Degree_(graph_theory) (213 words)

	Ecology: A graph theory approach to demographic loop analysis
		Loop analysis permits a morph-specific decomposition of the elasticity matrix, and thereby elucidates the total contributions of alternative life-history pathways to the population growth rate.
		A demographic life-cycle graph is "directed" in the sense that the transitions are the probability of an individual moving unidirectionally from one stage to another.
		And lastly, the "loop elasticity" is the sum of the characteristic elasticities on each step in the loop (which is equivalent to the characteristic elasticity multiplied by the number of steps in the loop).
	www.findarticles.com /p/articles/mi_m2120/is_n7_v79/ai_21231396 (1359 words)

	Graph theory
		An undirected graph G consist of a set of vertices,or nodes,V and a set of edges, or arcs, E such that each edge e is associated with an unordered pair of vertices.Thus
		A complete graph is a simple graph with n vertices in which there is an edge between every pair of distinct vertices.
		A graph G =(V, E) is bipartite if there exist subsets V1 and V2 (either possibly empty)of V such that V1 intersect V2 = empty set V1 union V2 = V,and each edge in E is incident on one vertex in V1 and one vertex in V2.
	www.nova.edu /~desir/graph.html (297 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.
		graph is a pair (V,E), where V is a finite set and E is a binary relation on V.
	www.boost.org /libs/graph/doc/graph_theory_review.html (2376 words)

	Notes on Graph Theory (Site not responding. Last check: 2007-10-17)
		Euler when on to describe a general theory which solves the problem of the Bridges of Konisberg and in the process he created a new branch of mathematics called graph theory.
		Graph theory is used today in many fields including city planning, communications, computer science and biology.
		Consider all paths on the graph starting at the beginning of the project and ending at vertex T. Add up the times in the vertices on each path (this is call the length of the path).
	www.ma.utexas.edu /~rodin/302/302s00/graphtheory.html (1609 words)

	graph (Site not responding. Last check: 2007-10-17)
		Glossary of graph theory A loop in a graph or a digraph is an edge e in E...
		Graph theory Graph theory is the branch of mathematics that examines the properties of graphs.
		Graph coloring A 3-coloring of a graph A 3-coloring of a graph Many terms used in this article are defined in the Glossary of graph theory.
	www.wikisearch.net /graph (571 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.nb (Site not responding. Last check: 2007-10-17)
		A loop in a graph or digraph is an edge e in E whose endpoints are the same.
		In a weighted graph or digraph, an additional function E → R associates a value with each edge; such graphs arise in optimal route problems such as the traveling salesman problem.
		In the example graph vertices 1 and 3 have a valency of 2, vertices 2,4 and 5 have a valency of 3 and vertex 6 has a valency of 1.
	www.chapman.edu /~hoshi103/math/Graph_Theory.html (1069 words)

	Some Graph Theory
		A graph is a nonempty set of vertices V and a multiset of edges E that are unordered pairs of elements of V.
		A subgraph is a graph all of whose vertices and edges belong to another graph.
		The complete bipartite graph K m, n is the graph obtained by starting with m vertices in one set and n vertices in another set, and connecting vertices between the two sets in every possible way.
	www.sosu.edu /st/math/faculty/matthews/discrete_folder/graph.htm (879 words)

	Graph Theory
		Two vertices of a graph that are connected by more than one edge are said to contain parallel edges or multiple edges. A graph with multiple edges is called a multigraph.
		of a path in a weighted graph is the sum of the weights of the edges in the path.
		A finite graph G is always a finite union of maximal connected graphs which are called connected components of G.
	www.lv.psu.edu /ojj/courses/ist-230/topics/graphs.html (1660 words)

	Graph Theory Introductions (Site not responding. Last check: 2007-10-17)
		Graph theory was first introduced by Leonard Euler in the eighteenth century.
		Graphs can basically be used to do just about any sort of calculation of funcrtion that involves a grouping of various objects or ideas.
		a, b, and c are the vertices of the graph, and the number in a given cell epresents the number of edges in the graph with the correspnding vertices as endpoints.
	www.mathcs.emory.edu /~whalen/Studentsites/Graphs/graphtheory.html (1077 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 (533 words)

	Graph Theory - The Great Web Directory
		Rutgers, 2000) Extremal and probabilistic graph theory and combinatorics.
		Graph Theory and Combinatorics - Slide 10 of 38
		Combinatorics : Graph Theory (134) Graph Theory, as a branch of Combinatorics, MSC classification 05Cxx.
	thegreatwebdirectory.com /Science/.../Combinatorics/Graph_Theory/Events (883 words)

	Basic Graph Theory (Site not responding. Last check: 2007-10-17)
		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)

	A Survey of Embedding Problems in Topological Graph Theory by Joerg Sawollek
		The main purpose of topological graph theory is to consider geometric realizations of a graph and its embeddings in 3-space or its embeddings or immersions in the plane or surfaces of higher genus.
		A topological graph is a 1-dimensional cell complex where the 0-cells correspond to the vertices and the 1-cells correspond to the edges of an underlying abstract graph.
		as a natural generalization of knot theory, where embeddings of the loop graph consisting of one vertex and one edge are considered, the different embeddings of a given graph and their properties may be investigated.
	at.yorku.ca /t/a/i/c/38.htm (2085 words)

	Application to Graph theory
		Graph Theory is now a major tool in mathematical research, electrical engineering, computer programming and networking, business administration, sociology, economics, marketing, and communications; the list can go on and on.
		In the above graph, the vertices A, C and E have the following property: from each one there is either a 1-step or a 2-step connection to any other vertex in the graph.
		In any dominance-directed graph there is at least one vertex from which there is a 1-step or a 2-step connection to any other vertex in the graph.
	aix1.uottawa.ca /~jkhoury/graph.htm (1112 words)

	Graphs
		A graph is a collection of nodes (also called vertices) and edges each connecting a pair of nodes.
		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.com /do_you_know/graphs.shtml (1005 words)

	Background Theory
		The transformation is spreading in the graph by multiplying the current node transformation with the edge transformation.
		This input for this algorithm is a graph and the output is another graph usually smaller, with less nodes and less edges but with the same geometric data.
		Whenever the user insert a transformation that causes the scene graph to be impossible to solve, the software, automatically, tries to solve the graph with other transformations such as the same transformation combined with a rotation around the X, Y or Z axis.
	www.cs.technion.ac.il /~gotsman/Escher/Html/theory.html (1056 words)

	Graphical Models
		Notice that, if this graph was undirected, the child would always separate the parents; hence when converting a directed graph to an undirected graph, we must add links between "unmarried" parents who share a common child (i.e., "moralize" the graph) to prevent us reading off incorrect independence statements.
		In theory, this runs the risk of double counting, but Yair Weiss and others have proved that in certain cases (e.g., a single loop), events are double counted "equally", and hence "cancel" to give the right answer.
		Classical control theory is mostly concerned with the special case where the graphical model is a Linear Dynamical System and the utility function is negative quadratic loss, e.g., consider a missile tracking an airplane: its goal is to minimize the squared distance between itself and the target.
	www.cs.ubc.ca /~murphyk/Bayes/bnintro.html (6529 words)

	Graph Theory Terminology
		A loop contribute 2 to the degree of the vertex with which it is incident.
		A graph is complete if it has no loops and every pair of distinct vertices is joined by a unique edge.
		A graph is connected if, for each pair of distinct vertices, there is a path from one to the other; a graph which is not connected is disconnected.
	snowwhite.it.brighton.ac.uk /staff/jt40/MM322/MM322Graphtheoryterminologyhandout.html (644 words)

	Graph Theory - The Great Web Directory (Site not responding. Last check: 2007-10-17)
		Graph Theory -- from MathWorld - Graph Theory -- from MathWorld Graph Theory -- from MathWorld The mathematical study of the properties of the formal mathematical structures called graphs.
		Graph theory is the branch of mathematics that concerns itself with graphs.
		LINK: A Combinatorics and Graph Theory Workbench for Applications and Research - LINK: A Combinatorics and Graph Theory Workbench for Applications and Research LINK: A Combinatorics and Graph Theory Workbench for Applications and Research LINK is a set of C++ class libraries that supports applications in discrete...
	www.thegreatwebdirectory.com /Science/Math/Combinatorics/Graph_Theory/Events (1008 words)

	SUO: Graph Theory Redux
		Thanks, Ian ============ Graph Theory ============ (subclass Graph Abstract) (documentation Graph "The and%Class of graphs, where a graph is understood to be a set of and%GraphNodes connected by and%GraphArcs.
		A directed graph is a and%Graph in which all and%GraphArcs have direction, i.e.
		A pseudograph is a and%Graph containing at least one and%GraphLoop.") (<=> (instance ?GRAPH PseudoGraph) (exists (?LOOP) (and (instance ?LOOP GraphLoop) (graphPart ?LOOP ?GRAPH)))) (subclass GraphElement Abstract) (disjoint GraphElement Graph) (partition GraphElement GraphNode GraphArc) (documentation GraphElement "Noncompositional parts of and%Graphs.
	grouper.ieee.org /groups/suo/email/msg07606.html (596 words)

	Graph Theory Lecture Notes 3a (Site not responding. Last check: 2007-10-17)
		Although showing that two graphs are isomorphic is in general very difficult, it is sometimes easy to show that two graphs are not isomorphic.
		A numerical graph invariant is a numerical property of graphs for which any two isomorphic graphs must have the same value.
		The degree sequence of the left graph is <2,2,3,3,3,3> while that for the right graph is <2,2,2,3,3,4>, so they are not isomorphic.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtaln3.html (419 words)

	CSC 270: Graph Theory
		A graph is an ordered pair G=(V,E) where V is a set of vertices and E is a set of edges.
		An isomorphism is a bijective mapping between the two graphs (both the vertices and edges) such that vertices p and q in graph G are adjacent if and only if vertices f(p) and f(q) in graph H are adjacent.
		It might not be feasible to use it for a graph with, say, ten thousand nodes and only about twenty thousand edges, because the adjacency matrix would have a hundred million entries in it (ten thousand squared), which would be about 400 MB of memory.
	www.dgp.toronto.edu /people/ajr/270/eve/notes/graph.html (1874 words)

	3D ANALYSIS OF HUMAN LOCOMOTION USING DUAL-VECTORS AND GRAPH THEORY (Site not responding. Last check: 2007-10-17)
		The elements of [S] are 0 for edges that are not part of the loop, 1 for edges that have loop direction, and -1 otherwise.
		In fact, the translational graph may be used for both domains with the penalty of using additional variables of zero value.
		The directed graph is constructed using mass element m, joints a, k, h, and ground force f.
	www.asb-biomech.org /onlineabs/NACOB98/277 (1257 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 (977 words)

	Graph Theory Tutorials
		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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