Final Copy 2020 03 24 Steve

Total Page:16

File Type:pdf, Size:1020Kb

Final Copy 2020 03 24 Steve This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Stevens, Sophie C C Title: Incidence Geometry in the Plane and Applications to Arithmetic Combinatorics General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. Incidence Geometry in the Plane and Applications to Arithmetic Combinatorics Sophie Clara Charlotte Stevens A dissertation submitted to the University of Bristol in accordance with the requirements for award of the degree of Doctor of Philosophy in the Faculty of Science. School of Mathematics, 12th December 2019 Word count: 34,000 Abstract In this thesis we study the applications of incidence geometry to additive combinatorics, with a particular focus of the setting of the prime residue field. We begin by introducing some context to incidence geometry and its role in combinatorics. In Chapters 4 and 5, we prove a new incidence bound between points and lines in the plane and discuss its applications. The sum-product phenomenon is a motivating problem of this thesis and we discuss this in both the case of an arbitrary field and the specific case of F = R. In this latter setting, in Chapter 6 we prove a bound on the energy formulation of the sum-product problem. A consequence of this is to the sum-product problem itself, as well as to that of expander functions. Finally, in Chapter 8, we prove a new bound on the number of pinned distances in the plane. iii Acknowledgements This document could not and did not happen in isolation and this page presents a chance to begin to express my thanks towards those who have helped me produce it. First and foremost I am indebted to my supervisor Misha Rudnev, for his wisdom, his patience and his kindness. Thank you to Lynne Walling for her accuracy in recommending him as a Good Person, and also for encouraging me to do a PhD. Thanks to Julia Wolf, who provided me with countless opportunities, and to Olly Roche-Newton and Sean Prendiville who are doubtless unaware of the encouragement they gave me. I am grateful of the opportunities that the HARICOT and SPACE seminars provided, and to the combinatorics department at Bristol who provided the material and companionship. I have learnt lots from working with Brendan Murphy, and also with Frank de Zeeuw and Ilya Shkredov. I am grateful to Jozsef Solymosi, especially for the hospitality he provided at the University of British Columbia. My time at Bristol has been greatly enhanced by the people I have met who have kept me sane and happy. Ana Costache, Nick Jones and Luka Rimani´c are the three who have played the greatest role in this task, but Justin D., Charley, Emma, Jenny, Simon, Lydia and Olly K. have all played their parts in my contentment. For those I have have known the longest and who have given me support in all parts of my life, it gives me great pleasure to thank Kats, Cate, Robert and the rest of my (extended) family, but most of all, Tim. v Author's declaration I declare that the work in this dissertation was carried out in accordance with the requirements of the University's Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate's own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author. S.C.C. Stevens December 12th, 2019 vii Publications Some parts of this thesis appear in the following publications: • [108] An Improved Point-Line Incidence Bound over Arbitrary Fields joint work with Frank de Zeeuw Bulletin of the London Mathematical Society 49.5 (2017), pp. 842-858 • [92] On the Energy Variant of the Sum-Product Conjecture joint work with Misha Rudnev and Ilya Shkredov To appear: Revista Matem´aticaIberoamericana, Electronically published on September 10, 2019 • [73] Bisector Energy and Pinned Distances in Positive Characteristic joint work with Brendan Murphy and Misha Rudnev arXiv preprint: 1908.04618 Sections containing reproduction are clearly indicated in the text. ix Contents 1 Introduction 1 2 A beginner's guide to incidence theorems 5 2.1 Incidences in two dimensions . 5 First incidence bounds . 7 The Szemer´edi-Trotter Theorem . 8 The number of k-rich points and lines . 10 Incidences over R3 ......................... 11 2.2 Incidences in arbitrary fields . 11 How is F different to R?...................... 11 Incidences over F2 ......................... 12 Large sets of points . 13 Small sets of points . 13 Incidences over F3 ......................... 13 Points-Lines . 13 Points-Planes . 14 2.3 Collinear triples . 16 2.4 Addendum: Projective transformations . 18 3 A beginner's guide to arithmetic structure 21 3.1 Arithmetic Structure in a Set and the Sum-Product Conjecture 21 Quantifying Structure: Cardinality . 22 Progress on the Sum-Product Phenomenon . 24 Addendum: The Pl¨unnecke-Ruzsa Inequality . 25 x CONTENTS 3.2 Arithmetic Energy . 26 Definitions and Examples . 26 Higher energies . 27 Geometric interpretation of energy . 28 3.3 How Cardinality and Energy Interact . 30 Cauchy-Schwarz inequality . 30 Addendum: The Balog-Szemer´edi-Gowers Theorem . 31 Solymosi's Approach to the Sum-Product Problem . 32 3.4 Other approaches to the sum-product phenomenon . 34 Few sums or few products . 34 On the size of jAA + Aj or jA(A + A)j . 35 4 Incidences between points and lines in arbitrary fields 37 4.1 Introduction . 37 4.2 Incidences over R .......................... 38 4.3 Incidences in Finite Fields . 39 4.4 Main Results . 40 4.5 Discussion . 41 Optimality . 41 Bounding rich points and lines . 42 When to use Theorems 4.4 and 4.5? . 44 4.6 Proof of Theorem 4.5: Cartesian incidence bound . 45 4.7 Proof of Theorem 4.4 . 47 Proof of Theorem 4.4 . 49 4.8 Open questions . 51 5 Applications of point-line incidences 53 5.1 Introduction . 53 5.2 Set expansion and sum-product estimates . 53 The Sum-Product Phenomenon . 53 Sum-Product over F .................... 54 Variations of the sum-product problem . 56 5.3 Geometric applications . 58 Distinct distances and the pinned distance problem . 58 Beck's theorem . 62 5.4 Collinear Quadruples . 65 5.5 Open Questions . 66 xi CONTENTS 6 Energy decomposition of a set 67 6.1 Introduction . 67 Structure of this chapter . 68 6.2 On the energy formulation and Balog{Wooley decomposition . 68 Balog{Wooley decomposition . 70 Main Decomposition Results . 71 6.3 Proof of Decomposition Results . 72 Initial Decomposition . 72 Bootstrapping the initial decomposition and proofs of Theo- rems 6.5 and 6.6 . 79 Proof of Theorems 6.5 and 6.6 . 80 Proof of Corollary 6.7 . 82 7 Applications of energy decomposition 83 7.1 Road-map of this chapter . 83 7.2 Application: Expansion . 83 Expander functions . 83 A new bound on a four-variable expander . 85 Proof of Theorem 7.2 . 85 7.3 The Sum Product Phenomenon . 91 Subsequent Improvements . 92 Proof of Theorem 7.8 . 93 Proof of Theorem 7.8 . 93 Proving Theorem 7.10 . 97 Proof of Theorem 7.10 . 100 Conclusion of the proof of Theorem 7.10 . 102 8 Pinned distances in positive characteristic 103 8.1 Introduction . 103 Structure of this chapter . 104 8.2 Distinct distances over F . 105 What changes? . 105 The case of large sets . 107 Trivial distance bound . 108 8.3 The pinned distance problem . 109 8.4 Literature review . 110 The distinct distances problem over R . 110 xii CONTENTS The Elekes{Sharir Framework . 111 Related work over F . 111 8.5 Main Results . 112 Sub-optimal Result . 113 8.6 Discussion of techniques . 113 High-level overview of pinned distances strategy . 113 Pinned distance strategy: perpendicular bisectors . 114 Counting perpendicular bisectors with incidences . 116 8.7 A toolkit for distinct distances . 117 A framework for distance preserving transformations . 117 Blaschke-Gr¨unwald Kinematic Mapping . 119 Isotropic lines and Perpendicular Bisectors . 120 Axial Symmetries . 121 Incidence Geometry . 123 8.8 Proof of Theorems 8.10 and 8.11 . 124 From pinned distances to isosceles triangles . 124 Bounding isosceles triangles . 126 Count of isosceles triangles with incidence geometry . 127 Proof of Theorem 8.11 . 128 Proof of Theorem 8.10 . 129 8.9 Proof of Proposition 8.12 . 130 Case 1: low multiplicity . 131 Case 2: high multiplicity . 132 8.10 Future work .
Recommended publications
  • On the Theory of Point Weight Designs Royal Holloway
    On the theory of Point Weight Designs Alexander W. Dent Technical Report RHUL–MA–2003–2 26 March 2001 Royal Holloway University of London Department of Mathematics Royal Holloway, University of London Egham, Surrey TW20 0EX, England http://www.rhul.ac.uk/mathematics/techreports Abstract A point-weight incidence structure is a structure of blocks and points where each point is associated with a positive integer weight. A point-weight design is a point-weight incidence structure where the sum of the weights of the points on a block is constant and there exist some condition that specifies the number of blocks that certain sets of points lie on. These structures share many similarities to classical designs. Chapter one provides an introduction to design theory and to some of the existing theory of point-weight designs. Chapter two develops a new type of point-weight design, termed a row-sum point-weight design, that has some of the matrix properties of classical designs. We examine the combinatorial aspects of these designs and show that a Fisher inequality holds and that this is dependent on certain combinatorial properties of the points of minimal weight. We define these points, and the designs containing them, to be either ‘awkward’ or ‘difficult’ depending on these properties. Chapter three extends the combinatorial analysis of row-sum point-weight designs. We examine structures that are simultaneously row-sum and point-sum point-weight designs, paying particular attention to the question of regularity. We also present several general construction techniques and specific examples of row-sum point-weight designs that are generated using these techniques.
    [Show full text]
  • Second Edition Volume I
    Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations .........
    [Show full text]
  • Invariants of Incidence Matrices
    Invariants of Incidence Matrices Chris Godsil (University of Waterloo), Peter Sin (University of Florida), Qing Xiang (University of Delaware). March 29 – April 3, 2009 1 Overview of the Field Incidence matrices arise whenever one attempts to find invariants of a relation between two (usually finite) sets. Researchers in design theory, coding theory, algebraic graph theory, representation theory, and finite geometry all encounter problems about modular ranks and Smith normal forms (SNF) of incidence matrices. For example, the work by Hamada [9] on the dimension of the code generated by r-flats in a projective geometry was motivated by problems in coding theory (Reed-Muller codes) and finite geometry; the work of Wilson [18] on the diagonal forms of subset-inclusion matrices was motivated by questions on existence of designs; and the papers [2] and [5] on p-ranks and Smith normal forms of subspace-inclusion matrices have their roots in representation theory and finite geometry. An impression of the current directions in research can be gained by considering our level of understanding of some fundamental examples. Incidence of subsets of a finite set Let Xr denote the set of subsets of size r in a finite set X. We can consider various incidence relations between Xr and Xs, such as inclusion, empty intersection or, more generally, intersection of fixed size t. These incidence systems are of central importance in the theory of designs, where they play a key role in Wilson’s fundamental work on existence theorems. They also appear in the theory of association schemes. 1 2 Incidence of subspaces of a finite vector space This class of incidence systems is the exact q-analogue of the class of subset incidences.
    [Show full text]
  • 7 Combinatorial Geometry
    7 Combinatorial Geometry Planes Incidence Structure: An incidence structure is a triple (V; B; ∼) so that V; B are disjoint sets and ∼ is a relation on V × B. We call elements of V points, elements of B blocks or lines and we associate each line with the set of points incident with it. So, if p 2 V and b 2 B satisfy p ∼ b we say that p is contained in b and write p 2 b and if b; b0 2 B we let b \ b0 = fp 2 P : p ∼ b; p ∼ b0g. Levi Graph: If (V; B; ∼) is an incidence structure, the associated Levi Graph is the bipartite graph with bipartition (V; B) and incidence given by ∼. Parallel: We say that the lines b; b0 are parallel, and write bjjb0 if b \ b0 = ;. Affine Plane: An affine plane is an incidence structure P = (V; B; ∼) which satisfies the following properties: (i) Any two distinct points are contained in exactly one line. (ii) If p 2 V and ` 2 B satisfy p 62 `, there is a unique line containing p and parallel to `. (iii) There exist three points not all contained in a common line. n Line: Let ~u;~v 2 F with ~v 6= 0 and let L~u;~v = f~u + t~v : t 2 Fg. Any set of the form L~u;~v is called a line in Fn. AG(2; F): We define AG(2; F) to be the incidence structure (V; B; ∼) where V = F2, B is the set of lines in F2 and if v 2 V and ` 2 B we define v ∼ ` if v 2 `.
    [Show full text]
  • A Generalization of Singer's Theorem: Any Finite Pappian A-Plane Is
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 71, Number 2, September 1978 A GENERALIZATIONOF SINGER'S THEOREM MARK P. HALE, JR. AND DIETER JUNGN1CKEL1 Abstract. Finite pappian Klingenberg planes and finite desarguesian uniform Hjelmslev planes admit abelian collineation groups acting regularly on the point and line sets; this generalizes Singer's theorem on finite desarguesian projective planes. Introduction. Projective Klingenberg and Hjelmslev planes (briefly K- planes, resp., //-planes) are generalizations of ordinary projective planes. The more general A-plane is an incidence structure n whose point and line sets are partitioned into "neighbor classes" such that nonneighbor points (lines) have exactly one line (point) in common and such that the "gross structure" formed by the neighbor classes is an ordinary projective plane IT'. An //-plane is a A-plane whose points are neighbors if and only if they are joined by at least two lines, and dually. A finite A-plane possesses two integer invariants (/, r), where r is the order of IT and t2 the cardinality of any neighbor class of n (cf. [3], [6]). Examples of A-planes are the "desarguesian" A-planes which are constructed by using homogeneous coordinates over a local ring (similar to the construction of the desarguesian projective planes from skewfields); for the precise definition, see Lemma 2. If R is a finite local ring with maximal ideal M, one obtains a (/, r)-K-p\ane with t = \M\ and r = \R/M\. The A-plane will be an //-plane if and only if A is a Hjelmslev ring (see [7] and [»])• We recall that a projective plane is called cyclic if it admits a cyclic collineation group which acts regularly on the point and line sets.
    [Show full text]
  • Certain Properties of Orbits Under Collineation Groups*
    JOURNAL OF COMBINATORIAL THEORY 2, 546-570 (1967) Certain Properties of Orbits under Collineation Groups* DAVID A. FOULSER AND REUBEN SANDLER University of lllinois at Chicago Circle, Chicago, lllinois Communicated by Gian-Carlo Rota l. INTRODUCTION" In this paper, the authors examine some of the combinatorial proper- ties of orbits of projective planes under collineation groups. Dembowski, in [5], has studied in some detail the relations which can hold between a point orbit ~ and a line orbit ~. Our approach here is to associate with the set s of all lines incident with two or more points of ~, and to look at the numerical relationships which occur. In Section 2, the basic concepts are defined, and some necessary condi- tions are derived which must hold if a collineation group with certain properties can exist. Section 3 discusses the insufficiency of these condi- tions and gives some results yielding sufficiency in some special cases; in addition, further necessary conditions are developed. In Section 4, a more detailed analysis is made of the nature of the interaction between and ~(~) under special conditions on the collineation group G. These special conditions are analogous to requiring that G be Abelian -- class (A)--or of prime order -- class (I-I)--and very detailed information is available under these hypotheses. Finally, Section 5 contains a num- ber of examples illustrating the principles discussed in the earlier sections. Standard notation for permutation groups is used throughout this paper. See for example [11]. * This work was partially supported by NSF contract No. GP 5276. 546 CERTAIN PROPERTIES OF ORBITS UNDER COLLINEATION GROUPS 547 2.
    [Show full text]
  • Masterarbeit
    Masterarbeit Solution Methods for the Social Golfer Problem Ausgef¨uhrt am Institut f¨ur Informationssysteme 184/2 Abteilung f¨ur Datenbanken und Artificial Intelligence der Technischen Universit¨at Wien unter der Anleitung von Priv.-Doz. Dr. Nysret Musliu durch Markus Triska Kochgasse 19/7, 1080 Wien Wien, am 20. M¨arz 2008 Abstract The social golfer problem (SGP) is a combinatorial optimisation problem. The task is to schedule g ×p golfers in g groups of p players for w weeks such that no two golfers play in the same group more than once. An instance of the SGP is denoted by the triple g−p−w. The original problem asks for the maximal w such that the instance 8−4−w can be solved. In addition to being an interesting puzzle and hard benchmark problem, the SGP and closely related problems arise in many practical applications such as encoding, encryption and covering tasks. In this thesis, we present and improve upon existing approaches towards solving SGP instances. We prove that the completion problem correspond- ing to the SGP is NP-complete. We correct several mistakes of an existing SAT encoding for the SGP, and propose a modification that yields consid- erably improved running times when solving SGP instances with common SAT solvers. We develop a new and freely available finite domain constraint solver, which lets us experiment with an existing constraint-based formula- tion of the SGP. We use our solver to solve the original problem for 9 weeks, thus matching the best current results of commercial constraint solvers for this instance.
    [Show full text]
  • Incidence Structures Near Configurations of Type (N
    ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 19 (2020) 17–23 https://doi.org/10.26493/1855-3974.1685.395 (Also available at http://amc-journal.eu) Incidence structures near configurations ∗ of type (n3) Peter J. Dukes y , Kaoruko Iwasaki Mathematics and Statistics, University of Victoria, Victoria, Canada Received 24 April 2018, accepted 28 June 2020, published online 26 October 2020 Abstract An (n3) configuration is an incidence structure equivalent to a linear hypergraph on n vertices which is both 3-regular and 3-uniform. We investigate a variant in which one constraint, say 3-regularity, is present, and we allow exactly one line to have size four, exactly one line to have size two, and all other lines to have size three. In particular, we study planar (Euclidean or projective) representations, settling the existence question and adapting Steinitz’ theorem for this setting. Keywords: Geometric configurations, incidence structures. Math. Subj. Class. (2020): 52C30, 05C65 1 Introduction A geometric (nr; mk) configuration is a set of n points and m lines in the Euclidean (or projective) plane such that every line contains exactly k points (uniformity) and every point is incident with exactly r lines (regularity). Grunbaum’s¨ book [3] is an excellent reference for the major results on this topic, including many variations. Our study here is a variation in which uniformity (or, dually, regularity) is mildly re- laxed; see also [1] and [4, x6.8] for prior investigations in this direction. An incidence structure is a triple (P; L; ι), where P is a set of points, L is a set of lines, and ι ⊆ P × L is a relation called incidence.
    [Show full text]
  • Codes from Incidence Matrices and Line Graphs of Paley Graphs
    Advances in Mathematics of Communications doi:10.3934/amc.2011.5.93 Volume 5, No. 1, 2011, 93{108 CODES FROM INCIDENCE MATRICES AND LINE GRAPHS OF PALEY GRAPHS Dina Ghinelli Dipartimento di Matematica Universit`adi Roma `La Sapienza' I-00185 Rome, Italy Jennifer D. Key School of Mathematical Sciences University of KwaZulu-Natal Pietermaritzburg 3209, South Africa (Communicated by Ivan Landjev) Abstract. We examine the p-ary codes from incidence matrices of Paley graphs P (q) where q ≡ 1 (mod 4) is a prime power, and show that the codes q(q−1) q−1 q(q−1) q−1 are [ 4 ; q − 1; 2 ]2 or [ 4 ; q; 2 ]p for p odd. By finding PD-sets we show that for q > 9 the p-ary codes, for any p, can be used for permutation decoding for full error-correction. The binary code from the line graph of P (q) is shown to be the same as the binary code from an incidence matrix for P (q). 1. Introduction The codes from the row span of adjacency matrices of Paley graphs are the well- known quadratic residue codes and studied in many texts. We look here at the codes from the row span of incidence matrices of Paley graphs, following the ideas of some earlier papers that studied such codes for various classes of graphs and their line graphs, including Hamming graphs, complete graphs, and complete bipartite graphs: see [14, 16,6,7,8]. An incidence matrix G for a graph Γ = (V; E) is a jV j × jEj matrix with rows labelled by the vertices and columns by the edges and an entry 1 if a vertex is on edge, 0 otherwise.
    [Show full text]
  • Computational Methods in Finite Geometry
    Computational Methods in Finite Geometry Anton Betten Colorado State University Summer School, Brighton, 2017 Q: What makes a problem interesting? Maybe it is being difficult. If this is true, then Finite Geometry is full of interesting problems. In the words of John F. Kennedy: We choose to go to the moon and do the other things not because they are easy but because they are hard. In the words of Peter Cameron: That a problem is hard does not mean we should not solve it. Prologue The goal is this lecture sequence is to say something meaningful about the problem of constructing and classifying combinatorial objects. (with an emphasis on objects from finite geometry). A large number of problems can be reduced to classifying orbits of groups acting on sets. The sets are often very large and at times not readily available. We need to perform search and isomorph rejection at the same time. Terminology Let G be a group. Let G act on a finite set X. For x; y 2 X; say that x ∼G y if x and y belong to the same G-orbit. Terminology The isomorphism problem is the following: Given x; y 2 X; determine whether x ∼G y or not. Terminology The classification problem is the problem of determining the orbits of G on X, for instance by listing one element from each orbit. Such a list of orbit representatives is called a transversal for the orbits of G on X. From now on, assume that a transversal for the orbits of G on X has been fixed.
    [Show full text]
  • Oriented Hypergraphs I: Introduction and Balance Arxiv:1210.0943V1
    Oriented Hypergraphs I: Introduction and Balance Lucas J. Rusnak∗ Department of Mathematics Texas State University San Marcos, Texas, U.S.A. [email protected] Submitted: Sept 30, 2012; Accepted: MMM DD, YYYY; Published: XX Mathematics Subject Classifications: 05C22, 05C65, 05C75 Abstract An oriented hypergraph is an oriented incidence structure that extends the con- cept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families { balanced, bal- anceable, and unbalanceable { and we obtain a complete classification of the bal- anced circuits of oriented hypergraphs. Keywords: Oriented hypergraph; balanced hypergraph; balanced matrix; signed hypergraph 1 Introduction Oriented hypergraphs have recently appeared in [6] as an extension of the signed graphic incidence, adjacency, and Laplacian matrices to examine walk counting. This paper fur- ther expands the theory of oriented hypergraphs by examining the extension of the cycle arXiv:1210.0943v1 [math.CO] 2 Oct 2012 space of a graph to oriented hypergraphs, and we obtain a classification of the balanced minimally dependent columns of the incidence matrix of an oriented hypergraph. It is known that the cycle space of a graph characterizes the dependencies of the graphic matroid and the minimal dependencies, or circuits, are the edge sets of the simple cycles of the graph. Oriented hypergraphs have a natural division into three categories: balanced, balanceable, and unbalanceable. The family of balanced oriented hypergraphs contain graphs, so a characterization of the balanced circuits of oriented hypergraphs can be regarded as an extension of the following theorem: ∗A special thanks to Thomas Zaslavsky and Gerard Cornu´ejolsfor their feedback.
    [Show full text]
  • Arxiv:2007.01842V3 [Math.CO] 11 Aug 2020 2.3 Box Product for Set System Multigraphs
    Incidence Hypergraphs: Box Products & the Laplacian < Will Grilliettea, Lucas J. Rusnaka, aDepartment of Mathematics, Texas State University, San Marcos, TX 78666, USA Abstract The box product and its associated box exponential are characterized for the categories of quivers (directed graphs), multigraphs, set system hypergraphs, and incidence hypergraphs. It is shown that only the quiver case of the box expo- nential can be characterized via homs entirely within their own category. An asymmetry in the incidence hypergraphic box product is rectified via an incidence dual-closed generalization that effectively treats vertices and edges as real and imaginary parts of a complex number, respectively. This new hypergraphic box product is shown to have a natural interpretation as the canonical box product for graphs via the bipartite representation functor, and its associated box exponential is represented as homs entirely in the category of incidence hypergraphs; with incidences determined by incidence-prism mapping. The evaluation of the box exponential at paths is shown to correspond to the entries in half-powers of the oriented hypergraphic signless Laplacian matrix. Keywords: Box product, incidence hypergraph, set system hypergraph, signless Laplacian, monoidal product. 2010 MSC: 05C76, 05C65, 05E99, 18D10, 18A40 Contents 1 Introduction 2 1.1 Incidence Hypergraphs . .4 1.2 Oriented Hypergraphs . .5 2 Box Products for Graph-like Categories 6 2.1 Box Products for Quivers . .6 2.2 Box Product for Set System Hypergraphs . .9 arXiv:2007.01842v3 [math.CO] 11 Aug 2020 2.3 Box Product for Set System Multigraphs . 11 2.4 Box Product for Incidence Hypergraphs . 12 3 Laplacian Product 15 3.1 Incidence Duality and the Laplacian Product .
    [Show full text]