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Topic: Undirected graph

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	Graph (mathematics) - Wikipedia, the free encyclopedia
		In a graph proper, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A.
		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 routing problems such as the traveling salesman problem.
	en.wikipedia.org /wiki/Undirected_graph (1649 words)

	Graph theory - Wikipedia, the free encyclopedia
		In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection.
		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.
		Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values.
	en.wikipedia.org /wiki/Graph_theory (1279 words)

	Learn more about Graph theory in the online encyclopedia. (Site not responding. Last check: 2007-11-01)
		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.
		Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges).
	www.onlineencyclopedia.org /g/gr/graph_theory.html (486 words)

	Boost Graph Concepts
		By factoring the graph interface into many smaller concepts we provide the graph algorithm writer with a good selection from which to choose the concept that is the closest match for their algorithm.
		The reason the BGL does not provide a separate interface for undirected graphs is that many algorithms on directed graphs also work on undirected graphs, and it would be inconvenient to have to duplicate the algorithms just because of an interface difference.
		In the directed graph, the edges (u,v) and (v,u) can have distinct weight values, whereas in the undirected graph the weight of (u,v) is the same as the weight of (v,u) since they are the same edge.
	www.boost.org /libs/graph/doc/graph_concepts.html (823 words)

	Graph Terminology (Site not responding. Last check: 2007-11-01)
		Graphs and their edges may be directed or undirected depending on whether or not it is important to distinguish direction in the relationship.
		Undirected graphs can be regarded as directed graphs by thinking of each undirected edge as a pair of directed edges in opposite directions.
		In an undirected graph and sometimes in a directed graph, two vertices at the opposite ends of an edge are referred to as neighbors or adjacent vertices.
	www.d.umn.edu /~gshute/cs4521/graphs/terminology.html (1090 words)

	Boost Graph Library: Graph Theory Review
		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.
		undirected graph edges are unordered pairs and connect the two vertices in both directions, hence in an undirected graph (u,v) and (v,u) are two ways of writing the same edge.
		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)

	Graphs (graph)
		An undirected graph may not contain selfloops, i.e., it may not contain an edge whose source is equal to its target.
		In a directed graph an edge is adjacent to its source and in an undirected graph it is adjacent to its source and target.
		In a directed graph a node w is adjacent to a node v if there is an edge (v,w) in E; in an undirected graph w is adjacent to v if there is an edge (v,w) or (w,v) in the graph.
	graphics.stanford.edu /courses/cs368/LEDA/node90.html (3409 words)

	graph
		Formally, a graph is a set of vertices and a binary relation between vertices, adjacency.
		Note: Graphs are so general that many other data structures, such as trees, are just special kinds of graphs.
		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.
	www.nist.gov /dads/HTML/graph.html (545 words)

	undirected graph
		Definition: A graph whose edges are unordered pairs of vertices.
		Formal Definition: A graph G is a pair (V,E), where V is a set of vertices, and E is a set of edges between the vertices E = {{u,v}
		Note: An undirected graph may be represented as a directed graph with two directed edges, one "to" and one "from," for each undirected edge.
	www.nist.gov /dads/HTML/undirectedGraph.html (146 words)

	Cprogramming.com Article: Graphs in Computer Science
		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.
		Undirected Graph: An undirected graph is one in which edges connect nodes bidirectionally (in both directions).
	www.cprogramming.com /tutorial/computersciencetheory/graphtheory.html (541 words)

	Undirected
		Undirected Graphs (ugraph) Definition An instance U of the data type ugraph is an undirected graph as defined in section Graphs.
		An undirected graph is a graph in which the nodes are connected by undirected arcs .
		Decisions about who is to receive an undirected organ are rightly made on the basis of medical criteria, especially on the likelihood of a successful transplant, not the moral worth of the recipient...
	www.sagooptimization.com /undirected (717 words)

	Graphs and Networks
		In an undirected graph, the DEGREE of a vertex is the number of vertices it is adjacent with.
		A subgraph of a graph G is a graph G' such that V' is a subset of V and E' is a subset of E. (Of course, only vertices in V' may appear in the pairs in E' if G' is to be a graph).
		C. Recall that we defined a graph in terms of a SET of edges, E. This implies that there cannot be more that one edge connecting any pair of vertices in a graph, or more than one edge connecting any pair of vertices in the same direction in a digraph.
	www.cs.gordon.edu /courses/cs321/lectures/graphs.html (4314 words)

	Graph Attributes
		Note: Some attributes, such as dir or arrowtail, are ambiguous when used in DOT with an undirected graph since the head and tail of an edge are meaningless.
		If the input graph defines the vertices attribute, and output is dot or xdot, this gives the number of points used for a node whose shape is a circle or ellipse.
		To center on the whole graph, use (bbx/2,bby/2), where "bbx,bby" is the value of the bounding box attribute bb.
	www.graphviz.org /doc/info/attrs.html (6841 words)

	Undirected Graphs and Minimum Spanning Trees
		Given a connected, undirected graph G=, the minimum spanning tree problem is to find a tree T= such that E' subset_of E and the cost of T is minimal.
		An unrooted tree can be made into a rooted tree: If the unrooted tree is "floppy" and it is "picked up" by a leaf to make a root, the new root has one child, every internal vertex has at least one child, and every (other) leaf has no children.
		The forest is represented by a partition of the vertices of the graph.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Graph/Undirected (1303 words)

	CHAPTER 11 -- Introduction to Graph Theory (t529)
		The pair G=(V,E) is a graph on V, where V is the set of nodes, and E the set of edges.
		That is, it consists of a subset of vertices and all edges in the original graph between vertices of that subset.
		Isomorphic graphs must have the same number of vertices (otherwise the function f could not be 1-1 and onto), and the same number of edges.
	www.utdallas.edu /~cshields/teach/discrete_fall98/notes/notes_discrete2_fall98_graph.html (3511 words)

	Introduction to Graphs
		Note that in a directed graph, the edges are arrows (are directed from one node to another) while in the undirected graph the edges are plain lines (they have no direction).
		For example, we can use a graph to represent what must be done to finish a CS major, with the nodes representing the courses to be taken, and the edges representing prerequisites.
		Some graphs have a similar property; i.e., there is a special "root" node from which all other nodes are reachable (control-flow graphs often have this property).
	www.cs.wisc.edu /~hasti/cs367-common/notes/GRAPH.html (1791 words)

	Biopathways Graph Data Manager (BGDM)
		I have broad interests in graphs, their role in databases and knowledge representation, graph algorithms (esp. parallel graph algorithms) for biology (phylogenetic tree construction, NMR peak assignment, shotgun sequence assembly, etc.), and graph grammars (for modeling graph query languages, evolution of biopathways, schema evolution and integration).
		Adjacency matrices are binary matrices (for undirected graphs) which contain a 1 for element A(i,j) if there is an edge connecting node i to node j.
		Graph matchings attempt to match entire graphs and are the graph analogs of global sequence alignments.
	hpcrd.lbl.gov /staff/olken/graphdm/graphdm.htm (7829 words)

	Graph Magics
		An undirected graph has an eulerian circuit if and only if it is connected and each vertex has an even degree (degree is the number of edges that are adjacent to that vertex).
		An undirected graph has an eulerian path if and only if it is connected and all vertices except 2 have even degree.
		A directed graph has an eulerian circuit if and only if it is connected and each vertex has the same in-degree as out-degree.
	www.graph-magics.com /articles/euler.php (1280 words)

	Graph Theory Glossary (Site not responding. Last check: 2007-11-01)
		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.
		Edges in graphs are undirected (as opposed to those in digraphs).
	www.utm.edu /departments/math/graph/glossary.html (816 words)

	95.202 Notes 8 - Graphs
		The sum of the degrees of all vertices in a graph is exactly twice the number of edges.
		A path in a graph is a sequence of alternating vertices and edges that starts at a vertex and ends at a vertex such that an edge is incident to its predecessor and successor vertex.
		A traversal is a systematic procedure for exploring a graph by examining all of its vertices and edges.
	www.scs.carleton.ca /~lanthier/teaching/202/Graph/Graph.html (3358 words)

	Graph Definitions
		graph - a graph G is a collection of two sets:V, a set of nodes called vertices and G, a set of branches (edges) that connect the vertices
		undirected graph or graph - graph in which each edge is associated with an unordered pair of vertices
		Euler circuit - a path in an undirected graph that begins at a vertex v, passes through every edge in the graph exactly once, and terminates at v
	www.mtsu.edu /~csjudy/3110/htmlFiles/graphs.htm (620 words)

	CHAPTER 11 -- Introduction to Graph Theory (t529)
		The associated undirected graph is an undir graph that consists of the vertices of G and the same edges with the directions removed.
		(Since f is one-to-one and onto, and the condition in part (b) is an "iff", there must be the same number of vertices in both graphs and the degrees of the mapped vertices must be the same.) Note that this is the essential property of an undir graph.
		Def: Two loop free undir graphs G1 = (V1, E1), G2 = (V2,E2) are called homeomorphic if either (a) they are isomorphic or (b) they can both be obtained from some loop free undir gr H by a sequence of elementary subdivisions.
	www.utdallas.edu /~cshields/teach/discrete_summ99/notes/discrete2_student_graph.html (4059 words)

	graph (Site not responding. Last check: 2007-11-01)
		The Graph class is intended to model a finite undirected graph.
		The GraphUtil class is filled with a bunch of static methods designed to operate on finite undirected graphs.
		The Vertex class models the vertices of a finite undirected graph.
	www.wku.edu /~leyla.zhuhadar/GraphBase/GraphDocs/graph/package-summary.html (312 words)

	Connectedness of an Undirected Graph
		It is tempting to interpret this figure as a picture of two graphs.
		A traversal of an undirected graph (either depth-first or breadth-first) starting from any vertex will only visit all the other vertices of the graph if that graph is connected.
		Therefore, there is a very simply way to test whether an undirected graph is connected: Count the number of vertices visited during a traversal of the graph.
	www.brpreiss.com /books/opus4/html/page561.html (213 words)

	R: Defining an undirected graph (UG) (Site not responding. Last check: 2007-11-01)
		A simple way to define an undirected graph by means of a single model formula.
		The undirected graph G = (V, E) is defined by a set of nodes V and a set of pairs E.
		The best way is to specify interactions that match the cliques of the undirected graph.
	seldon.it.northwestern.edu /sscc/R/library/ggm/html/UG.html (178 words)

	Bambooweb: Graph (mathematics)
		For example, the link structure of BambooWeb could be represented by a directed graph: the vertices are the articles in BambooWeb, and there's a directed edge from article A to article B if and only if A contains a link to B.
		An undirected graph (or graph for short) is given by
		Very different layouts can correspond to the same graph (see external link #2).
	www.bambooweb.com /articles/g/r/Graph_(mathematics).html (767 words)

	Python: class Graph
		An undirected graph (also called a network) is a data structure that
		graph is undirected, the order of the vertices in the edge does not
		Loads a graph written in GraphViz DOT format from the file filename.
	www.osl.iu.edu /~dgregor/bgl-python/reference/boost.graph.Graph.html (636 words)

	Undirected Graphs ( ugraph ) (Site not responding. Last check: 2007-11-01)
		An instance U of the data type ugraph is an undirected graph as defined in section Graphs.
		creates an instance U of type ugraph and initializes it to the empty undirected graph.
		U(graph G) creates an instance U of type ugraph and initializes it with an undirected copy of G.
	www.pcs.cnu.edu /~riedl/software/leda/ugraph.html (50 words)

	Graph (JUNG 1.7.4 API)
		may contain either directed or undirected edges, but not both.
		to this graph, and returns a reference to the added vertex.
		if each edge of this graph is undirected.
	jung.sourceforge.net /doc/api/edu/uci/ics/jung/graph/Graph.html (113 words)

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