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Topic: Tree (graph theory)

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	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.
		A tree is called a rooted tree if one vertex has been designated the root, in which case the edges have a natural orientation, towards or away from the root.
		Rooted trees, often with additional structure such as ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure.
	en.wikipedia.org /wiki/Tree_(graph_theory)#Types_of_trees (546 words)

	Graph theory - 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.
		Definitions of graphs vary in style and substance, according to the level of abstraction that is approriate to a particular approach or application.
		Graph theory is also used to study molecules in chemistry and physics.
	en.wikipedia.org /wiki/Graph_theory (1736 words)

	PlanetMath: graph theory (Site not responding. Last check: 2007-11-07)
		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.
	www.planetmath.org /encyclopedia/GraphTheory.html (505 words)

	Category:Graph theory - Wikipedia, the free encyclopedia
		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), which can also have associated directions.
		Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges), with an arrowhead on a line representing a directed arc.
	en.wikipedia.org /wiki/Category:Graph_theory (177 words)

	PlanetMath: tree (Site not responding. Last check: 2007-11-07)
		Therefore a forest or a tree is often used as a data structure.
		Such trees are typically drawn with the root at the top of the diagram, with all other nodes depending down from it (however this is not always the case).
		Given this parent-child relationship, a descendant of a node in a directed tree is defined as any other node reachable from that node (that is, a node's children and all their descendants).
	planetmath.org /encyclopedia/Tree.html (451 words)

	Graph theory - Wikipedia, the free encyclopedia
		Graphs with weights can be used to represent many different concepts; for example if the graph represents a road network, the weights could represent the length of each road
		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.
		Many graph properties are hereditary, which means that a graph has a property if and only if all subgraphs have it too.
	www.wikipedia.org /wiki/Graph_theory (1736 words)

	Boost Graph Library: Graph Theory Review (Site not responding. Last check: 2007-11-07)
		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 (2374 words)

	Whats Graph Theory
		Similarly the tree graph is treated in the literature as an undirected graph -even though the direction of the root is very important.
		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.
	www3.telus.net /public/nstuart/pan/grtheory.htm (3426 words)

	Graph Theory
		Graph Theory was born to study problems of this type.
		In an undirected graph, this is obviously a metric.
		Bound δ (of a graph embedded in on a surface)
	www.math.fau.edu /locke/GRAPHTHE.HTM (1165 words)

	05C: Graph theory
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic.
		A graph may be viewed as a one-dimensional CW-complex and hence studied with tools from Algebraic Topology, in particular, questions of planarity (and genus).
		Determining the genus of a graph is NP-complete.
	www.math.niu.edu /~rusin/known-math/index/05CXX.html (1204 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.
		Most research and applications in graph theory concern graphs without multiple edges or loops, and often multiple edges can be modeled by edge weights.
		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 (1070 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 (545 words)

	tree - Wiktionary
		A large plant, not exactly defined, but typically over four meters in height, a single trunk which grows in girth with age and branches (which also grow in circumference with age).
		An object made from a tree trunk and having multiple hooks or storage platforms.
		(graph theory) A connected graph with no cycles, or a connected graph with N-1 edges and N nodes.
	en.wiktionary.org /wiki/tree (291 words)

	Amazon.com: Modern Graph Theory: Books: Bela Bollobas (Site not responding. Last check: 2007-11-07)
		The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole.
		This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics.
		Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest.
	www.amazon.com /exec/obidos/tg/detail/-/0387984887?v=glance (1400 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-11-07)
		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)

	Graph Theory
		An acyclic graph (also known as a forest) is a graph with no cycles.
		A spanning tree in G is a subgraph of G that includes all the vertices of G and is also a tree.
		A tree with a center is called a central tree, and a tree with a bicenter is called a bicentral tree.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/trees.htm (812 words)

	Graph Theory
		Graph theory is a branch of topology which, although going back to L. Euler, has received particular interest only in recent years, as its applications in electrical engineering and operations research lend themselves readily to algorithmic formulation and solutions on digital computers.
		Graph theory formalizes the relations of entities called graphs, which consist of two sets of objects called nodes (or vertices) and edges, each edge connecting two nodes.
		It is comparatively easy to write an algorithm for the minimum spanning tree [Zahn71], and the concept has been applied in the recognition of tracks from digitizings ([Zahn73], [Cassel80]) and for cluster recognition in multi-dimensional space.
	br.endernet.org /~akrowne/handbook/AN16pp/node112.html (383 words)

	Graph theory
		Graph theory is used in dealing with problems which have a fairly natural graph/network structure, for example:
		The area of algorithmic graph theory (algorithms to solve graph theory problems) came to prominence in the mid 1970's with the work of Nicos Christofides (Imperial College Management School) and has become more fashionable/well-known since then.
		Applying the shortest path algorithm to such a graph enables us to plan routes for delivery vehicles on the basis of the shortest distance route between stops, or the shortest time route between stops (essentially divide a road into segments, assign a category to each road segment and a vehicle speed for each category).
	people.brunel.ac.uk /~mastjjb/jeb/or/graph.html (3616 words)

	Graph Theory
		he development of graph theory is very similar the development of probability theory, where much of the original work was motivated by efforts to understand games of chance.
		Firstly, since a graph is a very convenient and natural way of representing the relationships between objects we represent objects by vertices and the relationship between them by lines.
		The origin of graph theory can be traced back to Euler's work on the Konigsberg bridges problem (1735), which subsequently led to the concept of an Eulerian graph.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/graphIntro.htm (779 words)

	Graph Theory Lecture Notes 4a
		Suppose that the vertices of a graph represent towns and the edges of the graph are roads between these towns.
		To answer this type of question we need to be able to find a spanning tree in the graph so that the sum of the numbers on the edges is as small as possible (amongst all spanning trees).
		A directed tree is a digraph whose underlying graph is a tree.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtaln4.html (630 words)

	graph theory -- graph theory textbooks and resources
		The superior explanations, broad coverage, and abundance of illustrations and exercises that positioned this as the premier graph theory text remain, but are now augmented by a broad range of improvements.
		Their unparalleled treatment provides a text ideal for a two-semester course and a variety of one-semester classes, from an introductory one-semester course to courses slanted toward classical graph theory, operations research, data structures and algorithms, or algebra and topology.
		Authors explore the role of voltage graphs in the derivation of genus formulas, explain the Ringel-Youngs theorem—a proof that revolutionized the field of graph theory—and examine the genus of a group, including imbeddings of Cayley graphs.
	www.graphtheory.com (991 words)

	Encyclopedia entries starting with INF
		Inferential role semantics is an approach to the theory of meaning associated closely with accounts of proof-theoretic semantics in the semantics of logic into work as the foundational formalism of logic.
		In mathematics, an infinite tree is a partially ordered set (poset) in which there is a single unique minimal element and in which the set of elements less than a given element is well ordered.
		Infinitism is a theory in epistemology the branch of philosophy that treats of the possibility, nature, and means of knowledge.
	encycl.opentopia.com /I/IN/INF (10299 words)

	Graph Theory Lecture Notes 7 (Site not responding. Last check: 2007-11-07)
		Remove an edge from a tree G with k vertices.
		Assume the statement is true for all trees with k or fewer vertices (strong form of induction).
		This gives a graph having all the vertices of G and no circuits, i.e., it is a spanning tree for G. Theorem 3.11: Suppose that G is a graph with n vertices and e edges.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln7.htm (702 words)

	Graph Theory Lesson 13
		A tree is a connected graph that has no circuits.
		In a full n-ary tree we have a special vertex in the tree called the root.
		An n-ary tree with n=3 is called a ternary tree.
	www.utc.edu /Faculty/Christopher-Mawata/petersen/lesson13.htm (578 words)

	The Math Forum - Math Library - Graph Theory (Site not responding. Last check: 2007-11-07)
		A graph is a set V of vertices and a set E of edges - pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets.
		A series of short interactive tutorials introducing the basic concepts of graph theory, designed with the needs of future high school teachers in mind and currently being used in math courses at the University of Tennessee at Martin.
		An Introduction to Graph Theory tutorial uses three motivating problems to introduce the definition of graph along with terms like vertex, arc, degree, and planar.
	mathforum.org /library/topics/graph_theory (2440 words)

	Graphs: Theory - Algorithms - Complexity (Site not responding. Last check: 2007-11-07)
		Groups and Graphs: a software package for graphs, digraphs, combinatorial designs, and their automorphism groups, by B.
		Scheinerman, E.R., Ullman, D.H.: Fractional graph theory: a rational approach to the theory of graphs, John Wiley and Sons, New York, 1997.
		Graph connections -- relationships between graph theory and other areas of mathematics, Eds.
	people.freenet.de /Emden-Weinert/graphs.html (1244 words)

	Muhammad von Aurum's Encyclopedia of Graph Theory - Tree
		Consider a simple undirected graph T with finitely many n vertices then T is a tree if it satisfies any one of the following equivalent statements
		Every tree has either a single vertex as its center or it is bicentral.
		Every tree with at least one edge has at least two vertices of degree 1.
	www.cs.rit.edu /~maa2454/Graphs?T:Tree (118 words)

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