Combinatorics: a First Encounter
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
1. Directed Graphs Or Quivers What Is Category Theory? • Graph Theory on Steroids • Comic Book Mathematics • Abstract Nonsense • the Secret Dictionary
1. Directed graphs or quivers What is category theory? • Graph theory on steroids • Comic book mathematics • Abstract nonsense • The secret dictionary Sets and classes: For S = fX j X2 = Xg, have S 2 S , S2 = S Directed graph or quiver: C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) Class C0 of objects, vertices, points, . Class C1 of morphisms, (directed) edges, arrows, . For x; y 2 C0, write C(x; y) := ff 2 C1 j @0f = x; @1f = yg f 2C1 tail, domain / @0f / @1f o head, codomain op Opposite or dual graph of C = (C0;C1;@0;@1) is C = (C0;C1;@1;@0) Graph homomorphism F : D ! C has object part F0 : D0 ! C0 and morphism part F1 : D1 ! C1 with @i ◦ F1(f) = F0 ◦ @i(f) for i = 0; 1. Graph isomorphism has bijective object and morphism parts. Poset (X; ≤): set X with reflexive, antisymmetric, transitive order ≤ Hasse diagram of poset (X; ≤): x ! y if y covers x, i.e., x 6= y and [x; y] = fx; yg, so x ≤ z ≤ y ) z = x or z = y. Hasse diagram of (N; ≤) is 0 / 1 / 2 / 3 / ::: Hasse diagram of (f1; 2; 3; 6g; j ) is 3 / 6 O O 1 / 2 1 2 2. Categories Category: Quiver C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) with: • composition: 8 x; y; z 2 C0 ; C(x; y) × C(y; z) ! C(x; z); (f; g) 7! g ◦ f • satisfying associativity: 8 x; y; z; t 2 C0 ; 8 (f; g; h) 2 C(x; y) × C(y; z) × C(z; t) ; h ◦ (g ◦ f) = (h ◦ g) ◦ f y iS qq <SSSS g qq << SSS f qqq h◦g < SSSS qq << SSS qq g◦f < SSS xqq << SS z Vo VV < x VVVV << VVVV < VVVV << h VVVV < h◦(g◦f)=(h◦g)◦f VVVV < VVV+ t • identities: 8 x; y; z 2 C0 ; 9 1y 2 C(y; y) : 8 f 2 C(x; y) ; 1y ◦ f = f and 8 g 2 C(y; z) ; g ◦ 1y = g f y o x MM MM 1y g MM MMM f MMM M& zo g y Example: N0 = fxg ; N1 = N ; 1x = 0 ; 8 m; n 2 N ; n◦m = m+n ; | one object, lots of arrows [monoid of natural numbers under addition] 4 x / x Equation: 3 + 5 = 4 + 4 Commuting diagram: 3 4 x / x 5 ( 1 if m ≤ n; Example: N1 = N ; 8 m; n 2 N ; jN(m; n)j = 0 otherwise | lots of objects, lots of arrows [poset (N; ≤) as a category] These two examples are small categories: have a set of morphisms. -
The Number of Dissimilar Supergraphs of a Linear Graph
Pacific Journal of Mathematics THE NUMBER OF DISSIMILAR SUPERGRAPHS OF A LINEAR GRAPH FRANK HARARY Vol. 7, No. 1 January 1957 THE NUMBER OF DISSIMILAR SUPERGRAPHS OF A LINEAR GRAPH FRANK HARARY 1. Introduction* A (p, q) graph is one with p vertices and q lines. A formula is obtained for the number of dissimilar occurrences of a given (α, β) graph H as a subgraph of all (p, q) graphs G, oc<ip, β <: q, that is, for the number of dissimilar (p, q) supergraphs of H. The enumeration methods are those of Pόlya [7]. This result is then appli- ed to obtain formulas for the number of dissimilar complete subgraphs (cliques) and cycles among all (p, q) graphs. The formula for the num- ber of rooted graphs in [2] is a special case of the number of dissimilar cliques. This note complements [3] in which the number of dissimilar {p, k) subgraphs of a given (p, q) graph is found. We conclude with a discussion of two unsolved problems. A {linear) graph G (see [5] as a general reference) consists of a finite set V of vertices together with a prescribed subset W of the col- lection of all unordered pairs of distinct vertices. The members of W are called lines and two vertices vu v% are adjacent if {vlt v2} e W, that is, if there is a line joining them. By the complement Gr of a graph G, we mean the graph whose vertex-set coincides with that of G, in which two vertices are adjacent if and only if they are not adjacent in G. -
Graph Theory
1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. -
An Application of Graph Theory in Markov Chains Reliability Analysis
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace at VSB Technical University of Ostrava STATISTICS VOLUME: 12 j NUMBER: 2 j 2014 j JUNE An Application of Graph Theory in Markov Chains Reliability Analysis Pavel SKALNY Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VSB–Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava-Poruba, Czech Republic [email protected] Abstract. The paper presents reliability analysis which the production will be equal or greater than the indus- was realized for an industrial company. The aim of the trial partner demand. To deal with the task a Monte paper is to present the usage of discrete time Markov Carlo simulation of Discrete time Markov chains was chains and the flow in network approach. Discrete used. Markov chains a well-known method of stochastic mod- The goal of this paper is to present the Discrete time elling describes the issue. The method is suitable for Markov chain analysis realized on the system, which is many systems occurring in practice where we can easily not distributed in parallel. Since the analysed process distinguish various amount of states. Markov chains is more complicated the graph theory approach seems are used to describe transitions between the states of to be an appropriate tool. the process. The industrial process is described as a graph network. The maximal flow in the network cor- Graph theory has previously been applied to relia- responds to the production. The Ford-Fulkerson algo- bility, but for different purposes than we intend. -
Remembering Frank Harary
Discrete Mathematics Letters Discrete Math. Lett. 6 (2021) 1–7 www.dmlett.com DOI: 10.47443/dml.2021.s101 Editorial Remembering Frank Harary On March 11, 2005, at a special session of the 36th Southeastern International Conference on Combinatorics, Graph The- ory, and Computing held at Florida Atlantic University in Boca Raton, the famous mathematician Ralph Stanton, known for his work in combinatorics and founder of the Institute of Combinatorics and Its Applications, stated that the three mathematicians who had the greatest impact on modern graph theory have now all passed away. Stanton was referring to Frenchman Claude Berge of France, Canadian William Tutte, originally from England, and a third mathematician, an American, who had died only 66 days earlier and to whom this special session was being dedicated. Indeed, that day, March 11, 2005, would have been his 84th birthday. This third mathematician was Frank Harary. Let’s see what led Harary to be so recognized by Stanton. Frank Harary was born in New York City on March 11, 1921. He was the oldest child of Jewish immigrants from Syria and Russia. He earned a B.A. degree from Brooklyn College in 1941, spent a graduate year at Princeton University from 1943 to 1944 in theoretical physics, earned an M.A. degree from Brooklyn College in 1945, spent a year at New York University from 1945 to 1946 in applied mathematics, and then moved to the University of California at Berkeley, where he wrote his Ph.D. Thesis on The Structure of Boolean-like Rings, in 1949, under Alfred Foster. -
Leverages and Constraints for Turkish Foreign Policy in Syrian War: a Structural Balance Approach
Instructions for authors, permissions and subscription information: E-mail: [email protected] Web: www.uidergisi.com.tr Leverages and Constraints for Turkish Foreign Policy in Syrian War: A Structural Balance Approach Serdar Ş. GÜNER* and Dilan E. KOÇ** * Assoc. Prof. Dr., Department of International Relations, Bilkent University ** Student, Department of International Relations, Bilkent University To cite this article: Güner, Serdar Ş. and Koç, Dilan E., “Leverages and Constraints for Turkish Foreign Policy in Syrian War: A Structural Balance Approach”, Uluslararası İlişkiler, Volume 15, No. 59, 2018, pp. 89-103. Copyright @ International Relations Council of Turkey (UİK-IRCT). All rights reserved. No part of this publication may be reproduced, stored, transmitted, or disseminated, in any form, or by any means, without prior written permission from UİK, to whom all requests to reproduce copyright material should be directed, in writing. References for academic and media coverages are beyond this rule. Statements and opinions expressed in Uluslararası İlişkiler are the responsibility of the authors alone unless otherwise stated and do not imply the endorsement by the other authors, the Editors and the Editorial Board as well as the International Relations Council of Turkey. Uluslararası İlişkiler Konseyi Derneği | Uluslararası İlişkiler Dergisi Web: www.uidergisi.com.tr | E- Mail: [email protected] Leverages and Constraints for Turkish Foreign Policy in Syrian War: A Structural Balance Approach Serdar Ş. GÜNER Assoc. Prof. Dr., Department of International Relations, Bilkent University, Ankara, Turkey. E-mail: [email protected] Dilan E. KOÇ Student, Department of International Relations, Bilkent University, Ankara, Turkey. E-mail: [email protected] The authors thank participants of the Seventh Eurasian Peace Science Network Meeting of 2018 and anonymous referees for their comments and suggestions. -
A Logical Approach to Asymptotic Combinatorics I. First Order Properties
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATI-EMATlCS 65, 65-96 (1987) A Logical Approach to Asymptotic Combinatorics I. First Order Properties KEVIN J. COMPTON ’ Wesleyan University, Middletown, Connecticut 06457 INTRODUCTION We shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics; this framework provides methods for determining the probability that a large, finite structure, ran- domly chosen from a given class, will have a given property. Our main concern is the asymptotic probability: the limiting value as the size of the structure increases. For example, a common problem in elementary probability texts is to show that the asymptotic probabity that a per- mutation will have no fixed point is l/e. We shall say nothing about the closely related problem of determining rates of convergence, although the methods presented here may extend to such problems. To develop a general approach we must fix a language for specifying properties of structures. Thus, our approach is logical; logic is the branch of mathematics that deals with problems of language. In this paper we con- sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties. In the sequel to this paper we consider properties expressible in the more general language of monadic second order logic. Clearly, we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant. Therefore, we choose to consider only classes closed under disjoint unions and components (see Sect. -
Introduction to Graph Theory: a Discovery Course for Undergraduates
Introduction to Graph Theory: A Discovery Course for Undergraduates James M. Benedict Augusta State University Contents 1 Introductory Concepts 1 1.1 BasicIdeas ...................................... 1 1.2 Graph Theoretic Equality . 2 1.3 DegreesofVertices .................................. 4 1.4 Subgraphs....................................... 6 1.5 The Complement of a Graph . 7 2 Special Subgraphs 9 2.1 Walks ......................................... 9 2.2 Components...................................... 10 2.3 BlocksofaGraph................................... 11 3 Three Famous Results and One Famous Graph 14 3.1 The Four-Color Theorem . 14 3.2 PlanarGraphs..................................... 16 3.3 ThePetersenGraph ................................. 18 3.4 TraceableGraphs ................................... 18 K(3;3;3) = H1 H2 Removing the edge w1v2 destroys the only C4 v 3 u1 H1: H2: w3 v v2 3 u w1 u 2 u u 2 v 1 3 w 1 w 2 2 w3 u3 w1 v2 v1 Place any edge of H1 into H2 or (vice versa) and a C4 is created. i Remarks By reading through this text one can acquire a familiarity with the elementary topics of Graph Theory and the associated (hopefully standard) notation. The notation used here follows that used by Gary Chartrand at Western Michigan University in the last third of the 20th century. His usage of notation was in‡uenced by that of Frank Harary at the University of Michigan beginning in the early 1950’s. The text’s author was Chartrand’s student at WMU from 1973 to 1976. In order to actually learn any graph theory from this text, one must work through and solve the problems found within it. Some of the problems are very easy. -
Interactions of Computational Complexity Theory and Mathematics
Interactions of Computational Complexity Theory and Mathematics Avi Wigderson October 22, 2017 Abstract [This paper is a (self contained) chapter in a new book on computational complexity theory, called Mathematics and Computation, whose draft is available at https://www.math.ias.edu/avi/book]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: • Number Theory: Primality testing • Combinatorial Geometry: Point-line incidences • Operator Theory: The Kadison-Singer problem • Metric Geometry: Distortion of embeddings • Group Theory: Generation and random generation • Statistical Physics: Monte-Carlo Markov chains • Analysis and Probability: Noise stability • Lattice Theory: Short vectors • Invariant Theory: Actions on matrix tuples 1 1 introduction The Theory of Computation (ToC) lays out the mathematical foundations of computer science. I am often asked if ToC is a branch of Mathematics, or of Computer Science. The answer is easy: it is clearly both (and in fact, much more). Ever since Turing's 1936 definition of the Turing machine, we have had a formal mathematical model of computation that enables the rigorous mathematical study of computational tasks, algorithms to solve them, and the resources these require. -
Graph Algorithms in Bioinformatics an Introduction to Bioinformatics Algorithms Outline
An Introduction to Bioinformatics Algorithms www.bioalgorithms.info Graph Algorithms in Bioinformatics An Introduction to Bioinformatics Algorithms www.bioalgorithms.info Outline 1. Introduction to Graph Theory 2. The Hamiltonian & Eulerian Cycle Problems 3. Basic Biological Applications of Graph Theory 4. DNA Sequencing 5. Shortest Superstring & Traveling Salesman Problems 6. Sequencing by Hybridization 7. Fragment Assembly & Repeats in DNA 8. Fragment Assembly Algorithms An Introduction to Bioinformatics Algorithms www.bioalgorithms.info Section 1: Introduction to Graph Theory An Introduction to Bioinformatics Algorithms www.bioalgorithms.info Knight Tours • Knight Tour Problem: Given an 8 x 8 chessboard, is it possible to find a path for a knight that visits every square exactly once and returns to its starting square? • Note: In chess, a knight may move only by jumping two spaces in one direction, followed by a jump one space in a perpendicular direction. http://www.chess-poster.com/english/laws_of_chess.htm An Introduction to Bioinformatics Algorithms www.bioalgorithms.info 9th Century: Knight Tours Discovered An Introduction to Bioinformatics Algorithms www.bioalgorithms.info 18th Century: N x N Knight Tour Problem • 1759: Berlin Academy of Sciences proposes a 4000 francs prize for the solution of the more general problem of finding a knight tour on an N x N chessboard. • 1766: The problem is solved by Leonhard Euler (pronounced “Oiler”). • The prize was never awarded since Euler was Director of Mathematics at Berlin Academy and was deemed ineligible. Leonhard Euler http://commons.wikimedia.org/wiki/File:Leonhard_Euler_by_Handmann.png An Introduction to Bioinformatics Algorithms www.bioalgorithms.info Introduction to Graph Theory • A graph is a collection (V, E) of two sets: • V is simply a set of objects, which we call the vertices of G. -
Middleware and Application Management Architecture
Middleware and Application Management Architecture vorgelegt vom Diplom-Informatiker Sven van der Meer aus Berlin von der Fakultät IV – Elektrotechnik und Informatik der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften - Dr.-Ing. - genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Gorlatch Berichter: Prof. Dr. Dr. h.c. Radu Popescu-Zeletin Berichter: Prof. Dr. Kurt Geihs Tag der wissenschaftlichen Aussprache: 25.09.2002 Berlin 2002 D 83 Middleware and Application Management Architecture Sven van der Meer Berlin 2002 Preface Abstract This thesis describes a new approach for the integrated management of distributed networks, services, and applications. The main objective of this approach is the realization of software, systems, and services that address composability, scalability, reliability, and robustness as well as autonomous self-adaptation. It focuses on middleware for management, control, and use of fully distributed resources. The term integra- tion refers to the task of unifying existing instrumentations for middleware and management, not their replacement. The rationale of this work stems from the fact, that the current situation of middleware and management systems can be described with the term interworking. The actually needed integration of management and middleware concepts is still an open issue. However, identified trends in communications and computing demand for integrated concepts rather than the definition of new interworking scenarios. Distributed applications need to be prepared to be used and operated in a stable, secure, and efficient way. Middleware and service platforms are employed to solve this task. To guarantee this objective for a long-time operation, the systems needs to be controlled, administered, and maintained in its entirety, supporting the general aim of the system, and for each individual component, to ensure that each part of the system functions perfectly. -
An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical