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Topic: Spanning tree (mathematics)

Related Topics

Minimal spanning tree

Spanning tree (networks)

Connected graph

Biadjacency matrix

Spanning tree

Adjacency matrix

Graph theory

Adjacency list

Incidence matrix

Traveling salesman problem

Local area network

Ethernet

Berlin Wall

Computer science

Minimum spanning tree

	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.
		If G is a multigraph 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 multigraph obtained by deleting e and G/e is the contraction of G by e, where multiple edges arising from this contraction are not deleted.
		A spanning tree chosen randomly from among all the spanning trees with equal probability is called a uniform spanning tree (UST).
	en.wikipedia.org /wiki/Spanning_tree_(mathematics) (438 words)

	[No title]
		A rooted tree is a directed graph constructed from a tree with one vertex designated as a root, and each edge directed away from the root.
		Exercises in 8.3 8.5 & 8.6 Spanning Trees and Minimum Spanning Trees Given a simple graph G, a spanning tree of G is a subgraph of G that is a tree containing every vertex of G. A simple graph is connected iff it has a spanning tree.
		A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible sum of weights of its edges.
	www.cecs.csulb.edu /~lam/cecs228/ch9.doc (1375 words)

	Mazes
		Trees are the graphs with a single path from every vertex to any other vertex.
		Any connected graph has a spanning tree, i.e., a tree which is a subgraph with exactly same vertices but only some of the edges.
		the graph is bound to be a tree that contains all the vertices of the given graph and, therefore, serves as its spanning tree.
	www.cut-the-knot.org /ctk/Mazes.shtml (1586 words)

	Untitled
		The work involved comes that much closer to "doing real mathematics" when the problem itself arises out of curiosity, is open-ended and, therefore, has not been solved before by scores of previous students, and the path to solving the problem is determined entirely by the solver.
		One student noticed that the sequence of the number of trees appeared to be the same as the well-known Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21,...).
		The students in the Advanced Topics in Mathematics classes are generally those who either have not had much success in mathematics or have not found it interesting or engaging enough to put time, thought, or energy into it.
	www.geocities.com /teego_ab/images/trees/treearticle1.html (1170 words)

	Spanning tree (mathematics) (Site not responding. Last check: 2007-11-06)
		In mathematics and graph theory, a spanning tree of a graph, is a tree which includes every vertex of that graph.
		In certain fields of graph theory, involving weighted graphs, it is often useful to find a minimal spanning tree.
		Cayley's theorem can be used to find the number of labelled spanning trees in a complete graph.
	publicliterature.org /en/wikipedia/s/sp/spanning_tree__mathematics_.html (118 words)

	fwtrees
		A Hamiltonian path is a spanning tree of maximum degree 2.
		A relaxation is to search for spanning trees of small maximum degree.
		Then G contains a spanning tree with maximum degree 3 such that the number of vertices of degree 3 is O(g).
	www.emba.uvm.edu /~archdeac/problems/fwtrees.htm (301 words)

	Seminar in Applied Mathematics - Apr 24, 2000 (Site not responding. Last check: 2007-11-06)
		Abstract: In the design of reliable and invulnerable networks, it is often a goal to maximize the number of spanning trees of a graph with a given number of vertices and edges.
		It is logical to investigate the importance of individual edges to the number of spanning trees of a graph.
		Given a graph G the spanning tree edge density (hereafter density) of an edge e is the fraction of the spanning trees of G that contain e.
	www.stevens-tech.edu /math/Seminars/00S/2000-04-24.html (237 words)

	Mazes
		On every iteration it adds to the existent tree a branch — a vertex and an edge connecting the vertex to the tree, such that the resulting graph remains a tree.
		The resulting tree is known as the minimum spanning tree.
		He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem.
	www.maa.org /editorial/knot/mazes.html (1592 words)

	Checkpoint - Course 2, Unit 5 (Site not responding. Last check: 2007-11-06)
		Minimal spanning trees and shortest paths are both the "shortest" in some sense, which could mean cheapest, or least in some way.
		But a minimal spanning tree is a shortest tree that includes all vertices as opposed to a shortest path that generally does not include all vertices.
		On the Discrete Mathematics page, you can read an explanation of the main discrete mathematics concepts as they are developed in all four courses.
	www.wmich.edu /cpmp/parentsupport/checkpoints/c2u5.html (703 words)

	PlanetMath: Euler's polyhedron theorem, proof of
		It presents a different approach to the formula, that may be more familiar to modern students who have been exposed to a “Discrete Mathematics” course.
		Recall that in any tree, the number of edges is one less than the number of vertices.
		Cross-references: tree, cycle, path, spanning, connected component, partition, loop, spanning tree, represent, iff, vertices, adjacent, edge, contains, graph, embedding, Jordan curve theorem, topology, algebraic, planar graphs, properties, category, formula, Euler's formula
	planetmath.org /encyclopedia/ProofOfEulersPolyhedronTheorem.html (265 words)

	DMP: Definitions from Discrete Mathematics
		A binary tree is a tree in which each vertex has at most two children.
		A root of a tree is the vertex in a tree such that there is a uniques route from the root to any other vertex in the tree.
		A spanning tree is a tree which is a subgraph of a graph which contains all of the vertices of the graph.
	www.colorado.edu /education/DMP/def.html (3248 words)

	MATHnetBASE: Mathematics Online
		The design of approximation algorithms for spanning tree problems has become an exciting and important area of theoretical computer science and also plays a significant role in emerging fields such as biological sequence alignments and evolutionary tree construction.
		Spanning Trees and Optimization Problems offers the first complete treatment of spanning tree algorithms, from their role in classical computer science to their most modern applications.
		The authors first explain the general properties of spanning trees, then focus on three main categories: minimum spanning trees, shortest-paths trees, and minimum routing cost spanning trees.
	www.mathnetbase.com /ejournals/books/book_summary/summary.asp?id=2677 (251 words)

	Minimum Spanning Tree Animation (Site not responding. Last check: 2007-11-06)
		The minimum spanning tree (MST) of a weighted graph is a spanning tree whose sum of edge weights is minimal.
		The minimum spanning tree describes the cheapest network to connect all of a given set of vertices.
		Kruskal's algorithm for minimum spanning tree works by inserting edges in order of increasing cost, adding as edges to the tree those which connect two previously disjoint components.
	www.cs.sunysb.edu /~skiena/combinatorica/animations/mst.html (146 words)

	EDCI 650 Reacts: Discrete Mathematics
		However, in Principles and Standards for School Mathematics: Discussion Draft (1998), the message of the importance of the integration of discrete mathematics topics and themes throughout the mathematics curriculum is not as strong.
		Despite the careful attention to include discrete mathematical language at each of the grade bands and to thoroughly develop discrete mathematical topics in high school, treatment of such topics in the earlier grade bands is quite superficial.
		Students in Dr. Cooperís Trends in Mathematics Education course at University of Maryland each chose a standard and prepared a summary of the discussion of that standard based on the presentation of that standard in the Pre-K-12 overview, the Pre-K-2 chapter, and the grades 3-5 chapter.
	www.math.umd.edu /users/dac/650old/marcuspaper.html (9257 words)

	Otakar Boruvka on Minimum Spanning Tree Problem (translation of the both 1926 papers, comments, history) - Milkova ... (Site not responding. Last check: 2007-11-06)
		Abstract: O. Boruvka presented in 1926 the first solution of the Minimum Spanning Tree Problem (MST) which is generally regarded as a cornerstone of Combinatorial Optimization.
		A Practical Minimum Spanning Tree Algorithm Using the..
		41 the shortest spanning subtree of a graph and the travelling..
	citeseer.ist.psu.edu /nesetril00otakar.html (1024 words)

	UCSB General Catalog - Mathematics (Site not responding. Last check: 2007-11-06)
		Students with a bachelor's degree in mathematics who are interested in pursuing a California Teaching Credential should contact the credential advisor in the Gevirtz Graduate School of Education as soon as possible.
		The remaining 18 units may be in upper-division or graduate-level courses in mathematics or, with the approval of the graduate committee, outside of mathematics, with a limit of 9 units outside the department.
		Prerequisites: Mathematics 5A; and, an upper-division mathematics course (for Mathematics 181A): consent of instructor (for Mathematics 181B).
	www.catalog.ucsb.edu /ls/math.htm (6878 words)

	Topics in Intersection Graph Theory by TA McKee and FR McMorris
		Notice that, by the proof of Lemma 2.2, the left side of (2.1) is less than or equal to the right side for all spanning trees of K(G) (with equality holding if and only if the spanning tree is a clique tree).
		Since the first two terms in (2.1) are fixed and the last term is being maximized, and since equality holds in (2.1) for the clique tree $T$, equality must hold for every maximum spanning tree of $K^w(G)$.
		Then $Q^$ will also be a leaf of the tree* $T^-$ obtained from $T$ by deleting all the vertices of $T(Q^{},Q_3)$; note that $T^-$ is a clique tree for the induced subgraph $G^-$ of $G$ obtained by deleting all the vertices that only occur in vertices of $T(Q^{},Q_3)$.
	www.math.wright.edu /People/Terry_McKee/Button_441_AddCorrect.html (1126 words)

	Amazon.ca: Discrete Mathematics: Books: Richard Johnsonbaugh (Site not responding. Last check: 2007-11-06)
		At the same time, there was a need for a course that extended students' mathematical maturity and ability to deal with abstraction and also included useful topics such as combinatorics, algorithms, and graphs.
		A panel of the Mathematical Association of America (MAA) endorsed a year-long course in discrete mathematics.
		Trees, including binary trees, tree traversals, minimal spanning trees, decision trees, the minimum time for sorting, and tree isomorphisms (Chapter 7).
	www.amazon.ca /Discrete-Mathematics-Richard-Johnsonbaugh/dp/0131176862 (3292 words)

	[No title]
		A low-stretch spanning tree T of a graph G is a spanning tree subgraph in which most edges of G can be routed with small dilation.
		In particular, the stretch of an edge of G in T is the length of the path in T connecting the endpoints of that edge.
		We were motivated to improve their construction because this average-stretch is the dominant term in the complexity of a new solver for diagonally-dominant systems of equations.
	www.cs.yale.edu /homes/kalai/ysem.html (836 words)

	Christmas Tree Ornament - Information (Site not responding. Last check: 2007-11-06)
		The entire tree that was a tradition for a near Bethel particular was probably palm of Deborah an hanging on evergreen tree twigs, sometimes common decoration.
		Made es:Imagen:Binary in array.png Spanning tree mathematics Spanning structure in Spanning tree networks Spanning network protocol Little or Nollaig which Christmas Day was the season and the holidays for s Nollaig na : Image:Juletre da 100px Christmas tree 800 Pope a carol by christmas tree ornament.
		One Hill : One Hill, South One Hill, One Hill, New One Hill, One Hill television One Hill, One Hill, One Hill song One Hill, B is a tree data structure B that is B generically to B compu stub olive The photographer the size.
	www.freewebs.com /information24/christmas-tree-ornament.html (300 words)

	[No title]
		Kruskal's algorithm is another algorithm that produces a minimum spanning tree for any connected weighted graph.
		Kruskal's Algorithm for a Minimum Spanning Tree: Choose an edge with the smallest weight and place it in graph T1.
		While k is less than the number of vertices in the graph G, do the following steps: From among all the edges in G, find an edge G with the smallest weight that does not form a simple circuit when added to Tk.
	www.cecs.csulb.edu /~pompei/cecs228/ch8.doc (421 words)

	CAI, Leizhen, CUHK CS (Site not responding. Last check: 2007-11-06)
		Cai, The Complexity of the Locally-Connected Spanning Tree Problem'', Discrete Applied Mathematics, 131(1) 63-75, 2003.
		Cai, "On Spanning 2-Trees in a Graph", Discrete Applied Mathematics, 74(3) 203-216, 1997.
		Cai and D.G. Corneil, "Tree Spanners", SIAM Journal on Discrete Mathematics, 8(3), 359-387, 1995.
	www.cse.cuhk.edu.hk /~lcai (403 words)

	Discrete Mathematics 1 for AQA - Cambridge University Press
		Each chapter starts with a short list of learning objectives and mathematical ideas are explained carefully and clearly, with many stimulating worked examples.
		Discrete Mathematics 1 for AQA introduces algorithms, networks and graphs, spanning tree problems, matchings and a variety of optimisation problems.
		Written specifically for the Discrete module (D1) of the AQA mathematics specification A Mathematical ideas are explained carefully and clearly Plenty of exercises throughout, along with revision exercises and mock exam papers
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521799414 (191 words)

	KSU Department of Computer Science: Latest News (Site not responding. Last check: 2007-11-06)
		Graph is a mathematical object that perfectly models arbitrary binary relationship among data objects.
		A communication network is often modeled as a graph and various algorithms are used to optimize its certain characteristics, solution to which bears a spanning tree.
		With different aspects, there are different measurements of the goodness of a spanning tree.
	www.cs.kent.edu /department/news.html?News_ID=182 (498 words)

	Mathematics Other Homework Help
		Linear Algebra -- Spanning Sets - Proving X as an element of the spanning set
		Spanning Trees and Graphs - Does every graph have a spanning tree?
		Spanning Tree Graph - If you were required by a professor to find a spanning tree of the movie collaboration graph (where each node corresponds to an actor with finite Kevin Bacon number, and two nodes are connected by an...
	www.brainmass.com /homeworkhelp/math/other/2074 (189 words)

	ResearchChannel - Asymptotic Enumeration of Spanning Trees via Traces and Random Walks
		Methods of enumeration of spanning trees in a finite graph and relations to various areas of mathematics and physics have been investigated for more than 150 years.
		Then we will give new formulas for the asymptotics of the number of spanning trees of a graph.
		The general answer involves a quantity for infinite graphs that we call ``tree entropy', which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs.
	www.researchchannel.org /prog/displayseries.asp?collid=926 (167 words)

	Summer REU info- Vic Reiner
		Many of the projects involved spanning trees of graphs, Kirchhoff's Matrix-Tree theorem and its variants, graph Laplacians, chip-firing games and critical groups of graphs.
		Although there are many classes of graphs for which the spanning tree number is known, often through calculation of Laplacian eigenvalues, the structure of the critical group had been computed explicitly for very few examples prior to some of these REU's.
		It was known that there is a simple relation between the number of spanning trees for a regular graph and for its line graph.
	www.math.umn.edu /~reiner/REU/REU.html (948 words)

	Minimum Spanning Trees (MST)
		We describe the greedy method for finding a minimum spanning tree as an edge-coloring process.
		Color Invariant: There is a minimum spanning tree containing all of the blue edges and none of the red ones.
		Coloring step (Prim): Let T be the blue tree containing r.
	www.cs.gsu.edu /~cscagb/csc4520/MSTalgo.htm (444 words)

	DMTCS vol 5 no 1 (2002), pp. 121-126 (Site not responding. Last check: 2007-11-06)
		A spanning tree is colorful if all n-1 colors occur among its edges.
		If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files.
		Gregory Constantine (2002), Multicolored isomorphic spanning trees in complete graphs, Discrete Mathematics and Theoretical Computer Science 5, pp.
	dmtcs.loria.fr /volumes/abstracts/dm050108.abs.html (267 words)

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