121. MMS Online Graph Theory Course Introduction graph theory and Enumeration Course designed by Dale Winter. The goalsof this project are, firstly, to acquaint you with some of http://www.math.lsa.umich.edu/mmss/coursesONLINE/graph/

Graph Theory and Enumeration Course designed by Dale Winter The goals of this project are, firstly, to acquaint you with some of the ideas and principles involved in the mathematical study of counting and combinatorial graphs, and secondly, to provide a starting point for mathematical explorations of your own. The theory of graphs started in a paper published in 1736 by the Swiss mathematician Leonhard Euler. The idea in Euler's paper, which has blossomed into graph theory, grew out of a now popular problem known as the "Seven Bridges of Königsberg." The problem goes something like this: It was said that people spent their Sundays walking around, trying to find a starting point so that they could walk about the city, cross each bridge exactly once, and then return to their starting point. Can you find a starting point, and a path around the city that allow you to do this? Another famous problem that we will develop tools to help us with is the "Four Color Problem."

122. Search Graph Bibliography At the University of Alberta. http://www.cs.ualberta.ca/~stewart/GRAPH/search/bibsearch.html

Search Graph Biblilography This form will search the graph bibiliography and send output back as either a cite key with a title, or as a set of bibTeX records. See NOTES below. Output format: Cite key bibTeX First Search Term: Second Search Term: Third Search Term: NOTES ON SEARCHING Search Engine The search engine is based on an ancient Perl script written by Dr. Joe Culberson in University of Alberta. A record is reported if the patterns match anywhere in the bibTeX record. All matches are done on lower case after non-alphabetic characters in the bibTeX file are eliminated. Thus, setting the First Search term to "erdos" and the Second to "bollobas" will retrieve a record containing the field as well as two references in a volume honoring Erdos and edited by Bollobas. String Definitions The bibTeX file contains a number of string definitions used throughout. To determine the definition of a string in a bibTeX record, set the output format to bibTeX , then use the string in the First Search Term and the word "string" in the Second. For example, setting the first to "ejor" and the second to "string" results in the output Back to the graph Page stewart@cs.ualberta.ca, July 29, 1995

123. Graph Theory graph theory. To understand the language of graph theory. . In this section,we look at graph theory, perhaps the most fun area of mathematics. http://www.furman.edu/~markus/graph.htm

Graph Theory Learning objectives: To be able to construct graphs depicting various relationships. To use graph theory to solve puzzles. To understand the language of graph theory . In this section, we look at graph theory, perhaps the most fun area of mathematics. A tutorial on Euler paths, Euler circuits, Hamilton paths and Hamilton circuits. Note that you do not need to comment on completing this tutorial - the comments do not go to me! You will need to register to use this. Take a look at the game instant insanity This link is to a mathematical games and recreations page. It includes a write up on Euler's Konigsberg Bridge Problem and Hamilton's around the world game. Click here to return to the Lisa Markus home page. Click here to return to the Finite Mathematics home page.

124. Graph Theory Glossary Alphabetic list of terms by Chris Caldwell. http://www.utm.edu/departments/math/graph/glossary

Graph Theory Glossary Chris Caldwell This glossary is written to supplement the Interactive Tutorials in Graph Theory . Here we define the terms that we introduce in our tutorialsyou may need to go to the library to find the definitions of more advanced terms. Please let me know of any corrections or suggestion! A B C D ... Z adjacent Two vertices are adjacent if they are connected by an edge. arc A synonym for edge. See graph articulation point See cut vertices bipartite A graph is bipartite if its vertices can be partitioned into two disjoint subsets U and V such that each edge connects a vertex from U to one from V. A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has n elements and V has m , then we denote the resulting complete bipartite graph by K n,m . The illustration shows K . See also complete graph and cut vertices chromatic number The chromatic number of a graph is the least number of colors it takes to color its vertices so that adjacent vertices have different colors. For example, this graph has chromatic number three. When applied to a map this is the least number of colors so necessary that countries that share nontrivial borders (borders consisting of more than single points) have different colors. See the

125. CAAM 470 - Introduction To Graph Theory CAAM 470 Introduction to graph theory Rice University, Fall 2002. Textbook DouglasB. West, Introduction to graph theory, 2nd edition, Prentice Hall (2000). http://www.caam.rice.edu/~nated/caam470/

CAAM 470 - Introduction to Graph Theory Rice University, Fall 2002 Is it true that in any party of six people there are three mutual acquaintances or three mutual strangers? How many pawns can be placed on the squares of a chessboard so that no three are on a line in any direction? True or false? In a league where each team plays every other team exactly once, there must be a team such that the number of teams that it beats is at most half the number of additional teams that they beat. Instructor Dr. Nathaniel Dean DH 3023, 713-348-6113, nated@caam.rice.edu Office Hours: WF 10:30AM - 11:30AM or by appointment Time and Place: MWF 1:00PM-1:50PM in DH 1046 Accomodations: Any student with a disability requiring accomodations in this class is encouraged to contact me after class or during office hours. All discussions will remain confidential. Additionally, students should contact the Disabled Student Services office in the Ley Student Center. Assistant Cong Teng DH 2107, x2866, cteng@caam.rice.edu

126. Graph Theory WS 03-04 graph theory (37485) WS 03/04. Introduction to graph theory (Textbook) In lecturewe will follow the textbook Introduction to graph theory by Doug West. http://www.ti.inf.ethz.ch/ew/courses/GT03/

Graph Theory (37-485) WS 03/04 Apero You are invited to join the Apero after the whole course. It will take place from 16:00 on February 4, Wednesday at the round area in front of IFW B48.2. Examination Date : March 2 (Tuesday), 2004 Time : 9:00 - 11:00 AM Place : HG F19 (Room F19 of the Main Building of ETH) Detail PDF PS Administration : Mathematics students and graduate students, please send an email to us to tell whether you are going to take the exam or not. Lectures : Wed 10-12 @ IFW B42. Exercises : Wed 15-16 @ IFW B42. Instructors Lecturer: IFW B48.1/ Tel: (01) 632-0858. e-mail: Assistant: Yoshio Okamoto IFW B48.2/ Tel: (01) 632-7148. e-mail: Abstract of the course This course is an introduction to the theory of graphs intended for students in mathematics and computer science/engineering students with an interest in theory. We start from basic definitions and examples, but hope to move on quickly and cover a broad range of topics. Some applications and relations to Computer Science will also be discussed. Emphasis will be given to reading, understanding and developing proofs. There is no prerequisite, other than basic mathematics introduced in the Grundstudium. Possible topics include: degrees, paths, trees, cycles, Eulerian circuits, bipartite graphs, extremality, matchings, connectivity, network flows, vertex and edge colorings, Hamiltonian cycles and planarity. Procedures, exam, exercises

127. CIRM/DONET Graph Theory Workshop The 2000 CIRMDONET WORKSHOP on graph theory May 7-12, 2000 Levico,Trento, Italy List of participants THE PROGRAM -, Robin Thomas http://homepages.cwi.nl/~bgerards/GRAPHSHOP/

The 2000 CIRM-DONET WORKSHOP on GRAPH THEORY May 7-12, 2000 Levico, Trento, Italy List of participants THE PROGRAM Robin Thomas presented the 10-hour course: Structural Graph Theory and Applications to Coloring Robin has set up a web-page with an outline of the course, transparancies of his talks during the workshop, and pointers to relevant literature. Lex Schrijver presented three lectures on: Paths, Matchings, and Algorithms Dominic Welsh presented three lectures on: Counting, Polynomials, and Complexity THE LOCATION The workshop was held at Grand Hotel Bellavista in Levico, a village in the Alps close to Trento. There are many places to sit and work together. The pleasant working atmosphere offered by the hotel is further supported by the scenic surroundings with beautiful hiking possibilities. PARTICIPATION The workshop was open to everyone interested. So not restricted to DONET members. 68 senior and junior researchers in Graph Theory participated at the workshop. They came from all over the world: Brasil, Canada, Czech Republic, France, Germany, Hungary, Italy, Japan, Mexico, The Netherlands, New Zealand, Saudi Arabia, Slovenia, United Kingdom, and the USA (see list of participants ORGANIZATION The meeting has been organized by Michele Conforti (University of Padova

128. 21st LL-Seminar On Graph Theory Translate this page 21st LL-Seminar on graph theory. More detailed information can beobtained by using a browser such as Netscape 2.0 or higher. http://www.uni-klu.ac.at/math-or/LLseminar/

21st LL-Seminar on Graph Theory

129. OUP USA: Graph Theory 1736-1936: Norman L. Biggs Two centuries of graph theory, by Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson. http://www.oup-usa.org/toc/tc_0198539169.html

Check Out Help Search More Search Options Browse Subjects Subjects Agriculture Anthropology ... Sociology Browse Related Subjects Mathematics Applied Mathematics Never Miss an OUP Sale! Free Email Alerts! More Oxford email lists Enter Sales Promo Code Related Links: Title Information Subjects Table of Contents Graph Theory 1736-1936 Graph Theory 1736-1936 Norman L. Biggs, E. Keith Lloyd and Robin J. Wilson paper 256 pages Dec 1986 In Stock Price: $65.00 $5.00 (US) $10.00 (INTL) Paths Circuits Trees Chemical Graphs Euler's Polyhedral Formula The Four-Colour ProblemEarly History Colouring Maps on Surfaces Ideas from Algebra and Topology The Four-Colour Problemto 1936 The Factorization of Graphs Appendix 1: Graph Theory since 1936 Appendix 2: Bibliographical Notes Appendix 3: Bibliography: 1736-1936 Index of Names Contact Us Search Site Map Department Listing ... Employment Opportunities Publication dates and prices are subject to change without notice.

130. Graph Theory Applied To Topology Analysis - CAIDA : ANALYSIS : Topology graph theory Applied to Topology Analysis, Specific methods and definitionsfor analyzing network topology using graph theory are presented below. http://www.caida.org/analysis/topology/graphtheory.xml

ANALYSIS topology : graphtheory.xml Graph Theory Applied to Topology Analysis Network topologies may be treated as a directed graph. Specific methods and definitions for analyzing network topology using graph theory are presented below. Combinatorial Core The combinatorial core of a directed graph is its subset obtained by iterative stripping of nodes that have outdegree and of 2-loops involving nodes that have no other outgoing edges except those connecting them to each other. Note that the indegree of either type of stripped node may be arbitrarily large. In typical skitter snapshots we have studied, the combinatorial core contains approximately 10% of the nodes of the original non-stripped graph. Extended Core The iterative stripping procedure may be selectively applied only to the nodes of outdegree that have indegree 1. In that case, the stubs of the graph, i.e., trees connected to the rest of the graph by a single link, are stripped. We call this superset of the core the extended core , and in our skitter topology snapshots the extended core typically contains 2/3 of the nodes of the original graph, a much greater fraction than in the positive outdegree part (our combinatorial core Reachability We call a node B reachable from a node A if there is a directed path from A to B, i.e. a path in which each directed edge is taken toward B in the proper direction.

131. Extremal Graph Theory Extremal graph theory. The study of how the intrinsic structure ofgraphs ensures certain types of properties (eg, Cliqueformation http://forgodot.new21.org/lecture/graph_theory/contents/Extremal Graph Theory.ht

Extremal Graph Theory The study of how the intrinsic structure of graphs ensures certain types of properties (e.g., Clique-formation and Graph Colorings) under appropriate conditions.

132. Quantum Graph Theory Quantum graph theory. Of great concern to physical models using quantum graphtheory is what basis independent quantities characterize a quantum graph. http://members.aol.com/jmtsgibbs/qgraph.htm

Quantum Graph Theory Classical Graphs A classical graph is a set of vertices V and a set E of two element sets of vertices called edges. You can think of this as a set of points with lines connecting pairs of points. The quantum graphs I will define are actually counterparts to directed graphs. A directed graph is a set of vertices V and a set E of ordered pairs of vertices called edges. Imagine putting arrows on all the lines in the figure described above indicating their direction. Also, edges which connect a point to itself are now allowed. Quantum Points A quantum point A co-point Quantum Arrows A quantum arrow There is a natural operation multiplying arrows by points and arrows by arrows represented by ordinary matrix multiplication. The reverse of a quantum arrow is found by taking the Hermitian conjugate of its matrix. A Hermitian arrow is one that is its own reverse. Reversal is a basis independent property of an arrow. Another important type of arrow is a unitary arrow which is represented by a unitary matrix. A unitary arrow may be used to carry out a change of basis on the point space. The property of being unitary is also basis independent. Change of basis effected by the unitary arrow U: Quantum Graphs A quantum graph is an element of the Grassmann algebra over the space of arrows. If the space of points is N dimensional, then the space of arrows is N^2 dimensional, and the space of quantum graphs is 2^(N^2) dimensional. The basis for the space of quantum graphs is the Grassmann product of a number of basis elements of the space of arrows. The Grassmann product is associative, distributive, and antisymmetric. Because of the very limited number of symbols available in HTML, I will use * for the Grassmann product.

133. Problems Ex Cameron's Homepage Maintained by Peter Cameron. http://www.maths.qmw.ac.uk/~pjc/oldprob.html

Problems These are problems which have been on my homepage and are now put out to grass. See also permutation group problems 1. In 1956, Rudin defined a permutation of the integers which maps 3 x to 2 x x +1 to 4 x +1, and 3 x -1 to 4 x -1 for all x Problem: Determine the cycle structure of this permutation. I have just learned (December 1998) that this problem is older: it is the "original Collatz problem" from the 1930s (before the famous 3 x +1 problem). A paper by Jeff Lagarias gives details. 2. Let f k,n ) be the number of rooted trees with n leaves, all at level k (that is, distance k from the root), up to isomorphism of rooted trees. Prove that f k n f k,n ) tends to infinity with n , for fixed k . Is it even true that f k n f k,n ) is at least 1 + ( n k Solution by Peter Johnson. Let r be the maximum number of edges from a vertex on one level to the next level, in a tree with n vertices at level k . Then r k is at least n , so r is at least n k From any tree of height k +1, we obtain at most k different trees of height k by suppressing one level (replacing the paths of length 2 crossing this level by single edges). But there is some tree of height k from which at least p n k trees of height k +1 can be recovered by introducing a new level. (Choose a level where some vertex has at least

134. INTERNATIONAL COLLOQUIUM ON COMBINATORICS AND GRAPH THEORY INTERNATIONAL COLLOQUIUM ON COMBINATORICS AND graph theory. A satellite conferenceof ECM2 ECM2. Date July 1520, 1996, the week before ECM2 Location http://www.math-inst.hu/conferences/comb96/home.html

INTERNATIONAL COLLOQUIUM ON COMBINATORICS AND GRAPH THEORY A satellite conference of Date: July 15-20, 1996, the week before Location: Balatonlelle, Hungary Here is the map Further information First Announcement (HTML version) First Announcement (dvi) Second Announcement (HTML version) Registration form ... The program (100 Kbyte PostScript) Preliminary list of participants (dvi format). e-mail: comb96@math-inst.hu

135. Graph Theory -- Graph Theory Textbooks And Resources The purpose of www.graphtheory.com is to provide information about the textbook graph theory and Its Applications and to serve as a comprehensive graph theory resource for graph theoreticians and students. http://www.graphtheory.com/

o Home Page o About the Authors o Jonathan L. Gross o Jay Yellen o ORDER THE BOOKS o Graph Theory Resources o People o Research o Writings o Conferences o Journals o The Four-Color o Theorem o White Pages o White Pages ....o Registration o Feedback o Site Correction Change Request o Request an Evaluation Copy o Graphsong Last Edited 12 Jan 2004 As seen on Yahoo Google AltaVista Hotbot ... MSN , and Lycos More from AltaVista here Aaron D. Gross Webmaster's Homepage Email the Webmaster Graph Theory Textbooks and Resources The purpose of www.graphtheory.com is to provide information about the textbooks The Handbook of Graph Theory Graph Theory and Its Applications and Topological Graph Theory and to serve as a comprehensive graph theory resource for graph theoreticians and students. This site features the textbooks of Jonathan Gross Jay Yellen New in December 2003 zoom cover Order from Amazon Index Handbook of Graph Theory Jonathan L Gross Columbia University, New York, New York, USA

136. Douglas West's Home Page Home page of D. West with description of his books Introduction to graph theory, Mathematical Thinking ProblemSolving and Proofs, Combinatorics A Core Course, The Art of Combinatorics. http://www.math.uiuc.edu/~west/

Douglas B. West's Home Page Professor, Mathematics Dept. Univ. of Illinois .... Urbana, IL 61801-2975 Email: west@math.uiuc.edu ............................. Fax: (217) 333-9576 Office: (217) 333-1863 (226 Illini Hall) .......... Hours: MWF 3:12-4:18, etc. Resources maintained by DBW Links to combinatorial conferences Open problems pages (under construction; more than 30 problems pages now posted) Glossary of terms in combinatorics Open problems archive (past Open Problems columns in postscript) Links about DBW and UIUC Preprints of DBW papers since 1995 Curriculum vita ( ps pdf ) - education and employment, professional activities, etc. Publication list ( ps pdf Abstracts of papers since 1992 ( ps pdf Academic descendants Maria Muyot (spouse) Combinatorics at UIUC and operations research at UIUC Cham-bana hotels and MAP scratch Books and Courses SPRING 2004 COURSES: MATH 312 MATH 417 Book titles link to contents in postscript or to further resources. Course numbers link to corresponding course material. For further information, email west@math.uiuc.edu

137. Basic Research -- Algorithmic Graph Theory Algorithmic graph theory. In the field of Algorithmic graph theorywe are mainly interested in graphs with simple P_4structure. http://www.zpr.uni-koeln.de/GroupBachem/basic_research/index_1.html

Go up to Top Go forward to Scheduling Algorithmic Graph Theory In the field of Algorithmic Graph Theory we are mainly interested in graphs with simple P_4-structure. An induced P_4 of a graph G=(V,E) is an induced minor which is a path of length four, i.e. there are four vertices a, b, c, d in V such that (a,b), (b,c), (c,d) are edges of G but (a,c), (a,d) and (b,d) are non-edges. Figure 1 : Introducing the P_4 In several graphs from applications in scheduling the configuration of a P_4 does not occur very often. For example if we want to schedule courses at the university in such a way that there are no time conflicts for students taking several classes we can model a conflict graph on the courses by making to courses adjacent if there is a student which wants to participate in both. Experience then tells us that conflicting courses cluster. Therefore, graphs with few P_4s are interesting also for applications. P_4-connected components. We did some work on two classes of graphs with a very simple P_4 structure, namely the P_4-sparse and the P_4-extendible graphs. Each of these graphs can be uniquely encoded into a labeled rooted tree where the leaves correspond to small graphs. Exploiting this tree structure several combinatorial optimization problems, which are NP-hard for general graphs are solvable in linear time. We studied the Hamiltonian Cycle Problem, the Hamiltonian Path Problem, the Scattering Number, and the Path Covering Number on these classes.

138. Graphical Models Models and Bayesian Networks. By Kevin Murphy, 1998. Graphical modelsare a marriage between probability theory and graph theory. http://www.ai.mit.edu/~murphyk/Bayes/bnintro.html

A Brief Introduction to Graphical Models and Bayesian Networks By Kevin Murphy, 1998. "Graphical models are a marriage between probability theory and graph theory. They provide a natural tool for dealing with two problems that occur throughout applied mathematics and engineering uncertainty and complexity and in particular they are playing an increasingly important role in the design and analysis of machine learning algorithms. Fundamental to the idea of a graphical model is the notion of modularity a complex system is built by combining simpler parts. Probability theory provides the glue whereby the parts are combined, ensuring that the system as a whole is consistent, and providing ways to interface models to data. The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly-interacting sets of variables as well as a data structure that lends itself naturally to the design of efficient general-purpose algorithms. Many of the classical multivariate probabalistic systems studied in fields such as statistics, systems engineering, information theory, pattern recognition and statistical mechanics are special cases of the general graphical model formalism examples include mixture models, factor analysis, hidden Markov models, Kalman filters and Ising models. The graphical model framework provides a way to view all of these systems as instances of a common underlying formalism. This view has many advantages in particular, specialized techniques that have been developed in one field can be transferred between research communities and exploited more widely. Moreover, the graphical model formalism provides a natural framework for the design of new systems."

139. Open Problems In Graph Theory Involving Steiner Distance Open Problems involving Steiner distance. http://www.uwinnipeg.ca/~ooellerm/open_problems/index.html

Some Open Problems in Graph Theory It has been shown by Chartrand, Oellermann, Tian, and Zou that, for a tree T: diam n n This inequality does not hold for graphs in general as was shown by Henning, Oellermann, and Swart . It was shown in the same paper that for a graph G and n=3 and 4: diam n n G. It was shown by Oellermann and Tian that for a tree T: C n-1 (T) is contained in C n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-center of a graph is contained in its Steiner n-center. It was shown by Beineke, Oellermann and Pippert that if T is a tree, then M n-1 (T) is contained in M n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-median of a graph is contained in its Steiner n-median. Oellermann and Tian ). It is known that every graph is the 2-median of some graph (see Holbert ,and Hendry ). Steiner n-medians of trees have been completely characterized by

140. Graph Theory: Links Around The WWW MCS 125 Class Project Maia Borderaux/Dyuti Sengupta. PRACTICAL graph theory. LinkCollection for graph theory. What s wrong? Forget your graph theory? http://www.mills.edu/ACAD_INFO/MCS/CS/MCS125/proj/S00/Practical.Graph.Theory/mat

MCS 125 Class Project Maia Borderaux/Dyuti Sengupta PRACTICAL GRAPH THEORY Graph theory started in the 18th century around the time Leonard Euler solved the Konigsberg Bridge puzzle by proving that if a connected graph has no more than two vertices of odd degree, then some path traverses each edge exactly once. What does this mean? Let's take a look at the picture... GRAPHS ARE FOR REAL It has only been realtively recently that graph theory has been put to wide practical use. For example in chemistry, where atoms are vertices and the bonds between them are edges or in call graphs which have telephone numbers for vertices and calls made between the numbers constitute edges. OH NO, NOT THE WEB AGAIN! Arguably the most widely known and broadly accessed graph being studied today is the World Wide Web. Although not popularly thought of as a graph, one can consider the different Web pages to be vertices and the links between them to be edges, making the Web a graph. Granted, the Web is an enormous graph; it has some 800 million vertices. It has only been recently that graph theory has been thought to be useful in studying such a tremendous structure. Due to the size of the graph of the Web, studying it requires that one move sections of it in and out of memory and disk storage, a procedure that destroys the efficiency models of most algorithms run on these sections. In order to make the study of such large graphs more accessible, they are reduced from directed multigraphs to undirected simple graphs.

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