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Topic: Node graph theory

	Graph (mathematics) - Wikipedia, the free encyclopedia
		In mathematics and computer science a graph is the basic object of study in graph theory.
		A quiver is sometimes said to be simply a directed graph, but in practice it is a directed graph with vector spaces attached to the vertices and linear transformations attached to the arcs.
		In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal route problems such as the traveling salesman problem.
	en.wikipedia.org /wiki/Node_(graph_theory) (1370 words)

	Graph (mathematics) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-16)
		In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal route problems such as the (additional info and facts about traveling salesman problem) traveling salesman problem.
		In (additional info and facts about category theory) category theory a (A general concept that marks divisions or coordinations in a conceptual scheme) category can be considered a directed multigraph with the objects as vertices and the (additional info and facts about morphism) morphisms as directed edges.
		Every graph gives rise to a (additional info and facts about matroid) matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.
	www.absoluteastronomy.com /encyclopedia/g/gr/graph_(mathematics).htm (1121 words)

	Ecology: A graph theory approach to demographic loop analysis (Site not responding. Last check: 2007-10-16)
		Graph theory provides a systematic procedure for constructing the loops at step (3) that avoids missing or incorrectly identifying loops, errors that can cause the sum of the loop elasticities to deviate from 1.
		In the case of a demographic model, the stages of the life cycle are the nodes, and the transitions among the stages are the edges of the graph.
		Choose any node to be the first active node, also known as the "reference node" or the "root of the tree." Next, add an edge that connects the active node to a second node.
	www.findarticles.com /p/articles/mi_m2120/is_n7_v79/ai_21231396/pg_2 (1607 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.
		Nodes that are not far, on average, from all other nodes, tend to receive what's flowing through the network sooner than other nodes.
	www.analytictech.com /networks/graphtheory.htm (1221 words)

	Graphs Glossary
		A graph is bipartite if the vertices can be partitioned into two sets, X and Y, so that the only edges of the graph are between the vertices in X and the vertices in Y. Trees are examples of bipartite graphs.
		A chain in a graph is a sequence of vertices from one vertex to another using the edges.
		The closure of a graph G with n vertices, denoted by c(G), is the graph obtained from G by repeatedly adding edges between non-adjacent vertices whose degrees sum to at least n, until this can no longer be done.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.htm (1926 words)

	ipedia.com: Graph theory Article (Site not responding. Last check: 2007-10-16)
		Graph theory is the branch of mathematics that examines the properties of graphs.
		Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs).
		In general, there are four ways to represent a graph in a computer system: The incidence list representation, the incidence matrix representation, adjacency list representation, and the adjacency matrix representation.
	www.ipedia.com /graph_theory.html (901 words)

	A Graph Theory Niche (Site not responding. Last check: 2007-10-16)
		His graph proved that if there are more than two points (his graph had four points) with an odd number of lines to or from, it was not possible to cross all seven bridges just once.
		Graph theory was entirely new to Brigham and Dutton when they started studying it in the early 1980s.
		He is also using graph theory to find the shape of a large molecule consisting of hundreds of thousands of atoms, which has many important applications in the design of pharmaceuticals.
	longwood.cs.ucf.edu /newsletter/vol1/issue_two/graph-theory.html (1629 words)

	Graph Theory
		A graph G is a triple consisting of a vertex set of V(G), an edge set E(G), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints.
		A subgraph of G is a graph all of whose vertices belong to V(G) and all of whose edges belong to E(G).
		A bipartite graph is a graph whose vertex-set can be split into two sets in such a way that each edge of the graph joins a vertex in first set to a vertex in second set.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/defEx.htm (1249 words)

	Control flow graph - Wikipedia, the free encyclopedia
		A control flow graph (CFG) is an abstract data structure used in compilers.
		Each node in the graph represents a basic block, i.e.
		Reachability is another graph property useful in optimization.
	en.wikipedia.org /wiki/Control_flow_graph (824 words)

	Graph theory Article, Graphtheory Information (Site not responding. Last check: 2007-10-16)
		Graph theory is the branch of mathematics that examines theproperties of graphs.
		Informally, a graph is a set of objects called vertices (or nodes) connected bylinks called edges (or arcs).
		Every graph gives rise to a matroid, but in general the graph cannot be recoveredfrom its matroid, so matroids are not truly generalizations of graphs.
	www.anoca.org /graphs/edges/graph_theory.html (837 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 (538 words)

	Graphs
		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.
		A degree of a node is the number of edges incident to this node.
	www.cut-the-knot.com /do_you_know/graphs.shtml (1002 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)

	Graphs Glossary
		A complete graph is a simple graph in which all pairs of vertices are adjacent.
		A connected graph is one in which every pair of vertices are joined by a walk.
		The diameter of a graph is the length of the longest walk you are forced to use to get from one vertex to another in that graph.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.html (2135 words)

	Graph Theory Glossary
		In a digraph (directed graph) the degree is usually divided into the in-degree and the out-degree (whose sum is the degree of the vertex in the underlying undirected graph).
		A digraph (or a directed graph) is a graph in which the edges are directed.
		A path is a sequence of consecutive edges in a graph and the length of the path is the number of edges traversed.
	www.utm.edu /departments/math/graph/glossary.html (816 words)

	Cprogramming.com Article: Graphs in Computer Science
		Nodes are locations that are connected together by the edges of the graph.
		In addition to the undirected graph, in which the edge is a two-way connection, there are directed graphs, in which edges connect only one way.
		Because graphs are so often used and because they allow the representation of many problems in computer science, such as the Traveling Salesman Problem or something as simple as the relationships between people in a room, they are a convenient means of expressing problems with which many people are comfortable.
	www.cprogramming.com /tutorial/computersciencetheory/graphtheory.html (541 words)

	HR - Automatic Theory Formation In Pure Mathematics (Site not responding. Last check: 2007-10-16)
		HR uses three tables to represent connected graphs, one for the nodes, one for the edges and one for the relation: node a is on edge e.
		To help us understand the concepts HR introduces in graph theory, HR can invoke the DOT program from AT&T. The graphs produced are not particularly eye-catching (it would be better to use the neato program, but I can't get this to work at the moment), but they serve the purpose.
		HR can identify graphs of a given type: eg, in a recent session, HR's 31st concept was a type of graph known as a star (for obvious reasons).
	www.dai.ed.ac.uk /homes/simonco/research/hr/graph_theory.html (345 words)

	PlanetMath: graph
		A graph is then simple if there is at most one edge joining each pair of nodes.
		Adapted with permission of the author from Modern Graph Theory by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.
		This is version 27 of graph, born on 2001-11-12, modified 2004-04-02.
	planetmath.org /encyclopedia/Graph.html (191 words)

	New Page 1
		The number of nodes as defined above is often denoted by the letter p, sometimes called the size of the graph.
		Two graphs are considered isomorphic if there is a labelling of the nodes which preserves adjacency.
		of length 5 from node 2 to node 5.
	srufaculty.sru.edu /david.dailey/graphs/graphs.htm (667 words)

	Programming Concepts - Prof. Holowczak
		Graph theory is based on the notion of Nodes (vertices) and arcs (edges).
		Directed graphs can be used to show a precedence relationship (e.g., which comes first?) or causal relationships (which node "causes" which other node).
		Several interior nodes (nodes that are neither root nor leaves) labeled 3 and 8 appear in the middle of the tree.
	cisnet.baruch.cuny.edu /holowczak/classes/programming (5389 words)

	Hayes00: Graph Theory in Practice (Site not responding. Last check: 2007-10-16)
		A graph is made up of a set of vertices V and a set of edges E. Each edge is a set of two vertices.
		In a directed graph, vertices have an in-degree and an out-degree.
		This is a directed graph because each call originates at one number and is received at a different number.
	cs.colgate.edu /~parks/core/147/reading/Hayes00.html (403 words)

	Graph (mathematics) (Site not responding. Last check: 2007-10-16)
		The distinction between a directed graph and an oriented graph is that if x and y are vertices, a directed graph allows both (x, y) and (y, x) as edges, while only one is permitted in an oriented graph.
		Sometimes E or A are allowed to be multisets, so that there can be more than one edge between the same two vertices.
		Normally, the vertices of a graph by their nature are indistinguishable.
	www.worldhistory.com /wiki/G/Graph-(mathematics).htm (989 words)

	Water puzzle, graph theory
		Oystein Ore gave a worldly twist to the Three Glass puzzle and solved it in the framework of the Graph Theory.
		In particular, there is a walk from the starting node (0,0) to the target node (4,0), viz.
		It's perhaps relevant to note that puz(WaterPuzzle) with nodes on the perimeter of the 6x4 rectangle resembles the diagram obtained in describing a slanted cut of the torus with a rational slope.
	www.cut-the-knot.org /wgraph.shtml (304 words)

	[No title]
		move onto the node that is occupied by the robber or the robber (is forced to) move onto the node that is occupied by the cop.
		Suppose that i and j are adjacent nodes of a graph H such that any other node ajacent to i is adjacent to j.
		a graph with one node (looped or not).
	www.math.ucsd.edu /~fan/152/arch/coprob (625 words)

	Math Trek: Splitting Terrorist Cells, Science News Online, Jan. 10, 2004 (Site not responding. Last check: 2007-10-16)
		For graphs of various sorts, it's possible to estimate the probability that the removal of a certain number of nodes would split the graph into two or more separate units.
		A graph model, however, may not be the best one available for representing a typical terrorist organization, mathematician Jonathan D. Farley of the Massachusetts Institute of Technology contends.
		Leaders are represented by the topmost nodes in a diagram of the ordered set representing a cell and foot soldiers are nodes at the bottom.
	www.sciencenews.org /20040110/mathtrek.asp (940 words)

	ONLamp.com: RouteWord: An Interesting Diversion
		To those of you not steeped in graph theory, "graph" in this context refers not to the familiar X-axis and Y-axis plots from high school algebra but instead to a set of "nodes" that may be connected by "edges" to indicate a relationship.
		The field of graph visualization is devoted to finding ways to create meaningful layouts from among the millions of possible node positions.
		Initially, the nodes are plopped down randomly in a plane, and the program computes the force vectors that result as the particles push each other apart while the springs pull them together.
	www.onlamp.com /pub/a/onlamp/2003/11/26/routewords.html (1348 words)

	Graph Theory Background
		A computer network will be modeled as an undirected graph consisting of a set of vertices corresponding to the host computers that are the nodes of the network, and a set of edges corresponding to (bi-directional) communication links that connect the nodes.
		The advantage is that when we show that the probability of network partitioning during a time segment is extremely unlikely, we are simultaneously showing that the mean time between partitionings is a large multiple of a time segment, and therefore that the network is reliable.
		A network consisting of a simple loop, with edges passing from one node to the next until all nodes are visited and the path returns to the original node, is easily seen to be 2-connected; this is known as a loop network.
	www.ccs.neu.edu /home/kenb/etc/partition/node2.html (643 words)

	Maths - Graph Theory - Martin Baker
		Graph Theory has been included here because many 3D standards (such as VRML/X3D) and programming libraries (Java3D) use the concept of a scenegraph.
		If we remove from a graph G (N,U) a subset of its arcs, we are left with a graph of the form:
		If we remove from a graph G = (X,U) a subsei nodes, together with all the arcs incident to or from nodes, we are left with a graph of the form:
	martinb.com /maths/graphTheory (552 words)

	CS267: Notes for Lecture 23, April 9, 1999
		Unfortunately, M is not quite the Laplacian L(G) of the simple graph G of n nodes connected in a chain; the (1,1) and (n,n) entries are 2 instead of 1.
		For a star graph (n-1 nodes all connected to a single, n-th node) or a complete graph (all nodes connected to all other nodes), the lower bound is nearly exact.
		The graph partitioning software described so far, and listed in Lecture 20, consists of libraries to which one passes a graph, and is returned a partitioning.
	www.cs.berkeley.edu /~demmel/cs267/lecture20/lecture20.html (7067 words)

	Social Network Analysis
		The nodes in the network are the people and groups while the links show relationships or flows between the nodes.
		It consists of a number of nodes (each node corresponding to a member of the group) and a number of edges (or ties)÷each one being associated to a communication connection between two actors.
		It was not until the 1970s, therefore--when modern discrete combinatorics (particularly graph theory) experienced rapid development and relatively powerful computers became readily available--that the study of social networks really began to take off as an interdisciplinary specialty.
	lrs.ed.uiuc.edu /tse-portal/analysis/social-network-analysis (1642 words)

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