Some Analyses of Erdos Collaboration Graph
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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. -
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. -
Applied Combinatorics 2017 Edition
Keller Trotter Applied Combinatorics 2017 Edition 2017 Edition Mitchel T. Keller William T. Trotter Applied Combinatorics Applied Combinatorics Mitchel T. Keller Washington and Lee University Lexington, Virginia William T. Trotter Georgia Institute of Technology Atlanta, Georgia 2017 Edition Edition: 2017 Edition Website: http://rellek.net/appcomb/ © 2006–2017 Mitchel T. Keller, William T. Trotter This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Interna- tional License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Summary of Contents About the Authors ix Acknowledgements xi Preface xiii Preface to 2017 Edition xv Preface to 2016 Edition xvii Prologue 1 1 An Introduction to Combinatorics 3 2 Strings, Sets, and Binomial Coefficients 17 3 Induction 39 4 Combinatorial Basics 59 5 Graph Theory 69 6 Partially Ordered Sets 113 7 Inclusion-Exclusion 141 8 Generating Functions 157 9 Recurrence Equations 183 10 Probability 213 11 Applying Probability to Combinatorics 229 12 Graph Algorithms 239 vii SUMMARY OF CONTENTS 13 Network Flows 259 14 Combinatorial Applications of Network Flows 279 15 Pólya’s Enumeration Theorem 291 16 The Many Faces of Combinatorics 315 A Epilogue 331 B Background Material for Combinatorics 333 C List of Notation 361 Index 363 viii About the Authors About William T. Trotter William T. Trotter is a Professor in the School of Mathematics at Georgia Tech. He was first exposed to combinatorial mathematics through the 1971 Bowdoin Combi- natorics Conference which featured an array of superstars of that era, including Gian Carlo Rota, Paul Erdős, Marshall Hall, Herb Ryzer, Herb Wilf, William Tutte, Ron Gra- ham, Daniel Kleitman and Ray Fulkerson. -
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. -
Short Cross-Disciplinary Cycles in Co-Authorship Graphs
How small is the center of science? Short cross-disciplinary cycles in co-authorship graphs Chris Fields 528 Zinnia Court Sonoma, CA 95476 USA fi[email protected] October 17, 2014 Abstract Cycles that cross two or more boundaries between disciplines in the co-authorship graph for all of science are used to set upper limits on the number of co-authored pa- pers required to cross 15 disciplines or subdisciplines ranging from macroeconomics to neurology. The upper limits obtained range from one (discrete mathematics, macroe- conomics and nuclear physics) to six (neuroscience). The 15 disciplines or subdis- ciplines examined form a “small world” with an average separation of only 2.0 co- authorship links. It is conjectured that the high-productivity, high average degree centers of all scientific disciplines form a small world, and therefore that the diameter of the co-authorship graph of all of science is only slightly larger than the average diameter of the co-authorship graphs of its subdisciplines. Keywords: Cross-displinary brokers; Field mobility; Graph centrality; Graph diameter; Nobel laureates; Preferential attachment; Small-world networks 1 Introduction Numerous studies have now confirmed the finding of Newman (2001, 2004) that the co- authorship graphs of many scientific disciplines can be characterized as “small worlds” comprising many dense clusters of researchers separated by relatively short (mean length 1 l ≤ 10) minimum paths (reviewed by Mali, Kronegger, Doreian and Ferligoj, 2012). Despite over two decades of efforts to make the world of science as a whole more interconnected by encouraging inter-, multi- or trans-disciplinary collaborations, however, the research enterprise remains organized into disciplines (Jacobs and Frickel, 2009). -
Combinatorics: a First Encounter
Combinatorics: a first encounter Darij Grinberg Thursday 10th January, 2019at 9:29pm (unfinished draft!) Contents 1. Preface1 1.1. Acknowledgments . .4 2. What is combinatorics?4 2.1. Notations and conventions . .4 1. Preface These notes (which are work in progress and will remain so for the foreseeable fu- ture) are meant as an introduction to combinatorics – the mathematical discipline that studies finite sets (roughly speaking). When finished, they will cover topics such as binomial coefficients, the principles of enumeration, permutations, parti- tions and graphs. The emphasis falls on enumerative combinatorics, meaning the art of computing sizes of finite sets (“counting”), and graph theory. I have tried to keep the presentation as self-contained and elementary as possible. The reader is assumed to be familiar with some basics such as induction proofs, equivalence relations and summation signs, as well as have some experience with mathematical proofs. One of the best places to catch up on these basics and to gain said experience is the MIT lecture notes [LeLeMe16] (particularly their first five chapters). Two other resources to familiarize oneself with proofs are [Hammac15] and [Day16]. Generally, most good books about “reading and writing mathemat- ics” or “introductions to abstract mathematics” should convey these skills, although the extent to which they actually do so may differ. These notes are accompanying two classes on combinatorics (Math 4707 and 4990) I am giving at the University of Minneapolis in Fall 2017. Here is a (subjective and somewhat random) list of recommended texts on the kinds of combinatorics that will be considered in these notes: • Enumerative combinatorics (aka counting): 1 Notes on graph theory (Thursday 10th January, 2019, 9:29pm) page 2 – The very basics of the subject can be found in [LeLeMe16, Chapters 14– 15]. -
Structural Balance: a Generalization of Heider's Theory'
VOL. 63, No. 5 SEPTEMBER, 1956 THE PSYCHOLOGICAL REVIEW STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY' DORWIN CARTWRIGHT AND FRANK HARARY Research Center for Group Dynamics, University of Michigan A persistent problem of psychology will arise to change it toward a state has been how to deal conceptually of balance. These pressures may with patterns of interdependent prop- work to change the relation of affec- erties. This problem has been cen- tion between P and 0, the relation of tral, of course, in the theoretical responsibility between 0 and X, or the treatment by Gestalt psychologists of relation of evaluation between P phenomenal or neural configurations andX. or fields (12, 13, 15). It has also The purpose of this paper is to been of concern to social psychologists present and develop the consequences and sociologists who attempt to em- of a formal definition of balance which ploy concepts referring to social sys- is consistent with Heider's conception tems (18). and which may be employed in a more Heider (19), reflecting the general general treatment of empirical con- field-theoretical approach, has con- figurations. The definition is stated sidered certain aspects of cognitive in terms of the mathematical theory fields which contain perceived people of linear graphs (8, 14) and makes use and impersonal objects or events. of a distinction between a given rela- His analysis focuses upon what he calls tion and its opposite relation. Some the P-O-X unit of a cognitive field, of the ramifications of this definition consisting of P (one person), 0 (an- are then examined by means of other person), and X (an impersonal theorems derivable from the definition entity). -
Regular Graphs Containing a Given Graph
Regular graphs containing a given graph Autor(en): Akiyama, Jin / Era, Hiroshi / Harary, Frank Objekttyp: Article Zeitschrift: Elemente der Mathematik Band (Jahr): 38 (1983) Heft 1 PDF erstellt am: 03.10.2021 Persistenter Link: http://doi.org/10.5169/seals-37177 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, www.library.ethz.ch http://www.e-periodica.ch El Math Vol 38, 1983 15 Der Grundgedanke dieses Beweises ist also, zu zeigen, dass diejenigen Paare (a,b), deren Schrittzahl T(a,b) die Abschätzung von Dixon (Lemma 2) erfüllen, den Hauptbeitrag zu dem gesuchten Mittelwert liefern. Zum Abschluss seien noch einige numerische Ergebnisse angegeben, wie man sie mit einem programmierbaren Taschenrechner errechnen kann. -
BEAUTIFUL CONJECTURES in GRAPH THEORY Adrian Bondy
BEAUTIFUL CONJECTURES IN GRAPH THEORY Adrian Bondy What is a beautiful conjecture? The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. G.H. Hardy Some criteria: . Simplicity: short, easily understandable statement relating basic concepts. Element of Surprise: links together seemingly disparate concepts. Generality: valid for a wide variety of objects. Centrality: close ties with a number of existing theorems and/or conjectures. Longevity: at least twenty years old. Fecundity: attempts to prove the conjecture have led to new concepts or new proof techniques. Reconstruction Conjecture P.J. Kelly and S.M. Ulam 1942 Every simple graph on at least three vertices is reconstructible from its vertex-deleted subgraphs STANISLAW ULAM Simple Surprising General Central Old F ertile ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ v1 v2 v6 v3 v5 v4 G G − v1 G − v2 G − v3 PSfrag replacements G − v4 G − v5 G − v6 Edge Reconstruction Conjecture F. Harary 1964 Every simple graph on at least four edges is reconstructible from its edge-deleted subgraphs FRANK HARARY Simple Surprising General Central Old F ertile ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ G H PSfrag replacements G H MAIN FACTS Reconstruction Conjecture False for digraphs. There exist infinite families of nonreconstructible tournaments. P.J. Stockmeyer 1977 Edge Reconstruction Conjecture True for graphs on n vertices and more than n log2 n edges. L. Lovasz´ 1972, V. Muller¨ 1977 Path Decompositions T. Gallai 1968 Every connected simple graph on n vertices can 1 be decomposed into at most 2(n + 1) paths TIBOR GALLAI Simple Surprising General Central Old F ertile ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗ Circuit Decompositions G. -
View This Volume's Front and Back Matter
http://dx.doi.org/10.1090/psapm/026 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME 1 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA Edited by E. Reissner (Brown University, August 1947) VOLUME 2 ELECTROMAGNETIC THEORY Edited by A. H. Taub (Massachusetts Institute of Technology, July 1948) VOLUME 3 ELASTICITY Edited by R. V. Churchill (University of Michigan, June 1949) VOLUME 4 FLUID DYNAMICS Edited by M. H. Martin (University of Maryland, June 1951) VOLUME 5 WAVE MOTION AND VIBRATION THEORY Edited by A. E. Heins (Carnegie Institute of Technology, June 1952) VOLUME 6 NUMERICAL ANALYSIS Edited by /. H. Curtiss (Santa Monica City College, August 1953) VOLUME 7 APPLIED PROBABILITY Edited by L. A. MacColl (Polytechnic Institute of Brooklyn, April 1955) VOLUME 8 CALCULUS OF VARIATIONS AND ITS APPLICATIONS Edited by L. M. Graves (University of Chicago, April 1956) VOLUME 9 ORBIT THEORY Edited by G. Birkhoff and R. E. Longer (New York University, April 1957) VOLUME 10 COMBINATORIAL ANALYSIS Edited by R. Bellman and M. Hall, Jr. (Columbia University, April 1958) VOLUME 11 NUCLEAR REACTOR THEORY Edited by G. Birkhoff and E. P. Wigner (New York City, April 1959) VOLUME 12 STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS Edited by R. Jakobson (New York City, April 1960) VOLUME 13 HYDRODYNAMIC INSTABILITY Edited by R. Bellman, G. Birkhoff, C. C. Lin (New York City, April 1960) VOLUME 14 MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES Edited by R. Bellman (New York City, April 1961) VOLUME 15 EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, AND MATHEMATICS Edited by N. C. Metropolis, A. H. Taub, /. Todd, C. -
John R. Bowles Thesis A
- Math Goes to Hollywood or: How I Learned to Stop Worrying and Love the Math An Honors Thesis (HONRS 499) by John R. Bowles Thesis Advisor -- Dr. Kerry Jones ~~ Ball State University Muncie, IN May 5,2001 Graduation Date: May 5, 2001 speD!1 The:.;I::::; - Abstract Mathematics doesn't put people in the seats at the movie theater, but recently mathematics can be found more and more in popular films. Though Hollywood has shunned the value of mathematics in the past, a trio of excellent films have recently been made with mathematics as an integral character. This paper examines the math topics discussed in three of those movies: Good Will Hunting, Rushmore, and Pi. The mathematics used in each film is analyzed for validity. Also discussed is how each film's depiction of mathematics might effect society's view of mathematics. -- Acknowedgements Thank you Dr. Jones for your patience and guidance with this paper. Thanks also to Dr. Bremigan for your unexpected help. Also, thank you Nate for being there the first time I saw most of these movies and Erin for helping me stay sane while writing this paper. - Bowles I -- John R. Bowles Honors Thesis 5 May 2001 Math Goes to Hollywood or: How I Learned to Stop Worrying and Love the Math "Tough guys don't do math. Tough guys fry chicken for a living!" These are the words that high school math teacher Jaime Escalante (played by Edward James Olmos) sarcastically exclaims to his students in the film Stand and Deliver (IMDB). He is trying to overcome a common stigma shared by many teenagers that math is not cool.