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Topic: Forest graph theory

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Spanning tree (mathematics)

Minimal spanning tree

Connected graph

Biadjacency matrix

Adjacency matrix

Graph theory

Adjacency list

Incidence matrix

Computer science

Minimum spanning tree

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	Graphs Glossary
		A chain in a graph is a sequence of vertices from one vertex to another using the edges.
		The chromatic number of a graph is the smallest k for which the graph is k-colorable.
		An induced (generated) subgraph is a subset of the vertices of the graph together with all the edges of the graph between the vertices of this subset.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.htm (1926 words)

	Kids.Net.Au - Encyclopedia > Graph theory (Site not responding. Last check: 2007-10-20)
		Graph theory is the branch of mathematics that examines the properties of graphs.
		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.
		A subgraph of the graph G is a graph whose vertex set is a subset of the vertex set of G, whose edge set is a subset of the edge set of G, and such that the map w is the restriction of the map from G.
	www.kids.net.au /encyclopedia-wiki/gr/Graph_theory (1657 words)

	PlanetMath: tree
		Formally, a forest is an undirected, acyclic graph.
		As in a graph, a forest is made up of vertices (which are often called nodes interchangeably) and edges.
		Therefore a forest or a tree is often used as a data structure.
	planetmath.org /encyclopedia/Forest.html (451 words)

	Graph Theory Glossary
		For example, Figure 1.3.8 shows a simple graph which is also a bipartite graph because it may be divided into two parts, given by the subsets {1, 2} and {3, 4, 5}, where every edge in the graph goes from a vertex in one part to a vertex in the other part.
		A graph G1 is called a subgraph of graph G2, if every vertex of the first graph is a vertex in the second graph, every edge of the first graph is an edge in the second graph, and every edge of the first graph joins the same vertices as it does in the second graph.
		The vertices of the graph shown in Figure 1.3.29 may be properly colored in four colors: the first color for vertex 1, the second color for vertices 2, and 7, the third color for vertices 4, and 5, and the fourth color for vertices 3, and 6.
	exchange.manifold.net /manifold/manuals/5_userman/mfd50Graph_Theory_Glossary.htm (3582 words)

	[No title]
		Otherwise the graph is said to be disconnected The connected component of a graph G containing a given vertex v an element of V(G) is the largest sub-graph of G that contains v and is a connected graph.
		The girth of a graph is the length of the shortest circuit.
		A subtree of a graph G is a subgraph of G that is a tree.
	web.mit.edu /chungc/urop02/GraphTheory (2652 words)

	forest - Hutchinson encyclopedia article about forest (Site not responding. Last check: 2007-10-20)
		Forests in mountainous areas are often coniferous, because this type of evergreen tree is well suited to the soil, weather, and other environmental factors found at high altitudes.
		A natural, or old-growth, forest has a multistorey canopy and includes young and very old trees (this gives the canopy its range of heights).
		The Pacific forest of the west coast of North America is one of the few remaining old-growth forests in the temperate zone.
	encyclopedia.farlex.com /forest (584 words)

	Common Graphs (Site not responding. Last check: 2007-10-20)
		Euler built the representative graph, observed that it had vertices of odd degree, and proved that this made such a walk impossible (1736).
		Petersen graph: It's the sum of a 1-factor and a 2-factor.
		A factor of a graph G is a spanning subgraph of G (spanning = it contains all the vertices of G), which is not totally disconnected.
	www.cs.utk.edu /~doucet/graph/common.html (270 words)

	Graph Theory Article for Social Measurement
		Graph Theory is the study of properties of graphs.
		Graph Theory notation is not completely standardized and the reader should be aware that vertices may sometimes be called nodes or points, and edges may be called lines or links.
		A subgraph of a graph consists of a subset of the edge set and a subset of the vertex set, with the proviso that both ends of every edge must be included.
	www.math.fau.edu /locke/SocialMeasurement/Article.htm (3814 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.
		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.
		For the algorithm to keep track of where it is in the graph, and which vertex to visit next, BFS needs to color the vertices (see the section on Property Maps for more details about attaching properties to graphs).
	www.boost.org /libs/graph/doc/graph_theory_review.html (2374 words)

	Biography of William Shakespeare
		The authenticity of the order in which the sonnets were printed in 1609 has been doubted; and their subject-matter has been variously explained as being of the nature of a philosophical allegory, of an effort of the dramatic imagination, or of a heartless exercise in the forms of the Petrarchan convention.
		This last theory has been recently and strenuouslymaintained, and may be regarded as the only one which now holds the field in opposition to the autobiographical interpretation.
		There is little plausibility in a theory broached by Mr Sidney Lee, that W. was not the friend of the sonnets at all, but a certain William Hall, who was himself a printer, and might, it is conjectured, have obtained the “copy” of the sonnets for Thorpe.
	www.shakespeare-literature.com /l_biography.html (15100 words)

	David Eppstein - Publications
		The complement of a minimum spanning tree is a maximum spanning tree in the dual graph.
		Uses a divide and conquer on the edge set of a graph, together with the idea of replacing subgraphs by sparser certificates, to make various dynamic algorithms as fast on dense graphs as they are on sparse graphs.
		We describe a decomposition of graphs embedded on 2-dimensional manifolds into three subgraphs: a spanning tree, a dual spanning tree, and a set of leftover edges with cardinality determined by the genus of the manifold.
	www.ics.uci.edu /~eppstein/pubs/graph-dyn.html (783 words)

	"Theory of the Urban Web", by Nikos A. Salingaros
		The theory of multiple connectivity is motivated and supported by a major result in physics.
		A result from random graph theory applied to a model in evolutionary biology illustrates what actually happens in creating an organized urban web.
		In graph theory, which we propose as a means of understanding the urban web, paths and edges are the same thing.
	www.acturban.org /biennial/doc_planners/theory_urban_web_salingaros.htm (8276 words)

	[No title]
		In graph theory, a tree is a graph in which any two vertices are connected by exactly one path.
		A forest is a graph in which any two vertices are connected by at most one path.
		Through its coverage of the elements of Graph Theory and algorithms for the solution of Networks, the unit provides a basis for applications of combinatorics in a wide variety of other subjects of immediate practical importance in commerce, industry, and research.
	lycos.cs.cmu.edu /info/graph-theory.html (390 words)

	Spanning tree (mathematics) - Wikipedia, the free encyclopedia
		In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some (or perhaps all) of the edges of G.
		In certain fields of graph theory it is often useful to find a minimum spanning tree of a weighted graph.
		If G is a graph and e is an edge of G, then the number t(G) of spanning trees of G satisfies the deletion-contraction recurrence t(G)=t(G-e)+t(G/e), where G-e is the graph obtained by deleting e and G/e is the contraction of G by e.
	en.wikipedia.org /wiki/Spanning_tree_(mathematics) (428 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)

	[No title] (Site not responding. Last check: 2007-10-20)
		It is known that this subset is rational if and only if the subalphabet of $w$ is a connected graph of the dependency relation (the relation $D$ in the text).
		Our theory is a proper generalization of the theory of finite and infinite words (with explicit termination) and of the theory of finite and infinite (real and complex) traces.
		The set of infinite dependence graphs is bounded complete and (prime) algebraic.
	www.csse.monash.edu.au /mirrors/bibliography/Theory/traces.unique (2060 words)

	Graph Theory
		An acyclic graph (also known as a forest) is a graph with no cycles.
		Thus each component of a forest is tree, and any tree is a connected forest.
		A graph is connected if and only if it has a spanning tree.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/trees.htm (812 words)

	Robertson–Seymour theorem - Wikipedia, the free encyclopedia
		In graph theory, the Robertson–Seymour theorem states that the minor ordering on the finite undirected graphs is a well-quasi-ordering.
		In particular, every (possibly infinite) set of graphs that is downwardly closed according to the minor ordering is represented by a finite set of graphs called its obstruction set, where a graph is in the set if and only if none of its minors is in the obstruction set.
		A graph H is a minor of a graph G if H is isomorphic to a graph obtained from G by contracting edges, deleting edges, and deleting isolated nodes;
	en.wikipedia.org /wiki/Robertson%E2%80%93Seymour_theorem (1354 words)

	Lake Forest College > Academics > Mathematics and Computer Science > Course Descriptio
		The course uses set theory, logic, and language as a foundation for studying a variety of topics central to the development of modern mathematics.
		Theory and applications of the calculus of functions of one variable.
		Graph theory with emphasis on trees, circuits, cut sets, planar graphs, chromatic numbers, and transportation networks.
	www.lakeforest.edu /academics/programs/macs/course.asp (1927 words)

	Forest -- from Wolfram MathWorld
		An acyclic graph (i.e., a graph without any circuits).
		Forests therefore consist only of (possibly disconnected) trees, hence the name "forest." A forest with
		Skiena, S. "Acyclic Graphs." §5.3.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.
	mathworld.wolfram.com /Forest.html (181 words)

	Hamilton College - Academics - Mathematics
		Their research and teaching interests include: logic; mathematical modelling; field independence/dependence; knot theory; mathematics education; lattice theory; topology; graph theory; and group theory.
		Her research interests include graph theory, geometric graph theory and group theory.
		Her doctoral thesis was, "The y-Filtration on Representation Rings of p-Groups." Her research focuses on combinatorial optimization and graph theory.
	www.hamilton.edu /academics/faculty.html?dept=Mathematics (521 words)

	Brian's Digest: Graph Theory
		However, the graph must contain no redundant arcs (e.g., if an arc exists from node 1 to node 2, and an arc exists from node 2 to node 3, then an arc should not be allowed from node 1 to node 3 because it is redundant).
		A graph without redundant arcs is known as a spanning tree, and there exist efficient (O(n^3)) algorithms to construct a minimal spanning tree from random points in a plane.
		Often some local changes in the graph are allowed so it is of great value to find a small set of vertices apart from which the remainder of the graph can be legally k-colored.
	www.worms.ms.unimelb.edu.au /digest/graph_t97.html (4327 words)

	Forest - OneLook Dictionary Search
		Forest, forest : UltraLingua English Dictionary [home, info]
		Phrases that include Forest: sherwood forest, forest fire, tropical rain forest, petrified forest national park, virgin forest, more...
		Words similar to Forest: afforest, forestal, forestation, forested, forestial, foresting, timber, timberland, wood, woodland, woods, weald, more...
	www.onelook.com /?loc=pub&w=Forest (385 words)

	Algebra and Discrete Math Subfaculty
		The area of algebra and discrete mathematics encompasses both theoretical and applied aspects of mathematics that are foundational for matrix analysis, modern algebra, number theory, combinatorics, and graph theory.
		Students interested in the underlying theory of algebraic and discrete structures will also gain insights into how these concepts are fundamental to a wide array of practical problems.
		Graph theory is the study of paths and networks, connectivity, trees, coverings, and coloring problems.
	virtual.clemson.edu /groups/mathsci/graduate/algebra.html (698 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.
		Research Interests: combinatorics and graph theory, esp. extremal set theory, extremal graph theory, graph coloring, and applications of discrete math to biology, number theory, analysis of algorithms, communications.
	www.joergzuther.de /math/graph/homes.html (8736 words)

	Durango Bill's Home Page
		The minimum number of moves to move a set of ten marbles from one side to the other is 27.
		An unlabeled graph may have an arbitrary number of vertices with none, one, or more edges connecting some or all of the vertices.
		National Forests are a national responsibility - illustrated with a Beetle Bailey cartoon.
	www.durangobill.com (1405 words)

	Problems in Topological Graph Theory
		Graphs that quadrangulate both the torus and Klein bottle
		Orientable genus of graphs of bounded nonorientable genus
		Interpolation conjectures on separating cycles in embedded graphs
	www.emba.uvm.edu /~archdeac/problems/problems.html (283 words)

	Reply to Popular Mechanics re 9/11
		Upon examination it turns out to be a shoddy piece of disinfo produced in a desperate attempt to defend against the fact that Americans are finally waking up and realizing that 9/11 was an inside job, that about 3000 people died at the hands of elements within their own government.
		On that graph, the 8- and 10-second collapses appear--misleadingly--as a pair of sudden spikes.
		Even if one accepts the "pancake" theory, the lower stories would have offered plenty of resistance (they were intact and not damaged by fire), and the collapse would have taken a lot longer than 15 seconds.
	www.serendipity.li /wot/pop_mech/reply_to_popular_mechanics.htm (16873 words)

	Tree-based Parallel Graph Algorithms (Site not responding. Last check: 2007-10-20)
		This paper gives several optimal mesh computer, VLSI, and pyramid computer algorithms for determining properties of an arbitrary undirected graph, where the graph is given as an unordered collection of edges.
		The algorithms first find spanning trees and then use them to determine properties of the graph.
		All of the times are optimal, and the algorithms extend to VLSI and pyramid models.
	www.eecs.umich.edu /~qstout/abs/ICPP85tree.html (155 words)

	Tree (graph theory) - Wikipedia, the free encyclopedia
		A labeled tree with 6 vertices and 5 edges
		Trees are widely used in Computer Science data structures such as binary search trees, heaps, tries, etc.
		G has no simple cycles and has n − 1 edges.
	en.wikipedia.org /wiki/Tree_(graph_theory) (561 words)

	Dictionary of Algorithms and Data Structures
		We need help in automata theory, combinatorics, parallel or randomized algorithms, heuristics, and quantum computing.
		We do not include algorithms particular to business data processing, communications, operating systems or distributed algorithms, programming languages, AI, graphics, or numerical analysis: it is tough enough covering "general" algorithms and data structures.
		Data Structures and Algorithms is a wonderful site with illustrations, explanations, analysis, and code taking the student from arrays and lists through trees, graphs, and intractable problems.
	www.nist.gov /dads (686 words)

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