Conference Matrices
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Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M
Codes and modules associated with designs and t-uniform hypergraphs Richard M. Wilson California Institute of Technology (1) Smith and diagonal form (2) Solutions of linear equations in integers (3) Square incidence matrices (4) A chain of codes (5) Self-dual codes; Witt’s theorem (6) Symmetric and quasi-symmetric designs (7) The matrices of t-subsets versus k-subsets, or t-uniform hy- pergaphs (8) Null designs (trades) (9) A diagonal form for Nt (10) A zero-sum Ramsey-type problem (11) Diagonal forms for matrices arising from simple graphs 1. Smith and diagonal form Given an r by m integer matrix A, there exist unimodular matrices E and F ,ofordersr and m,sothatEAF = D where D is an r by m diagonal matrix. Here ‘diagonal’ means that the (i, j)-entry of D is 0 unless i = j; but D is not necessarily square. We call any matrix D that arises in this way a diagonal form for A. As a simple example, ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = . 21 4 −27 050 01−2 The matrix on the right is a diagonal form for the middle matrix on the left. Let the diagonal entries of D be d1,d2,d3,.... ⎛ ⎞ 20 0 ··· ⎜ ⎟ ⎜0240 ···⎟ ⎜ ⎟ ⎝0 0 120 ···⎠ . ... If all diagonal entries di are nonnegative and di divides di+1 for i =1, 2,...,thenD is called the integer Smith normal form of A, or simply the Smith form of A, and the integers di are called the invariant factors,ortheelementary divisors of A.TheSmith form is unique; the unimodular matrices E and F are not. -
Two-Graphs and Skew Two-Graphs in Finite Geometries
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Two-Graphs and Skew Two-Graphs in Finite Geometries G. Eric Moorhouse Department of Mathematics University of Wyoming Laramie Wyoming Dedicated to Professor J. J. Seidel Submitted by Aart Blokhuis ABSTRACT We describe the use of two-graphs and skew (oriented) two-graphs as isomor- phism invariants for translation planes and m-systems (including avoids and spreads) of polar spaces in odd characteristic. 1. INTRODUCTION Two-graphs were first introduced by G. Higman, as natural objects for the action of certain sporadic simple groups. They have since been studied extensively by Seidel, Taylor, and others, in relation to equiangular lines, strongly regular graphs, and other notions; see [26]-[28]. The analogous oriented two-graphs (which we call skew two-graphs) were introduced by Cameron [7], and there is some literature on the equivalent notion of switching classes of tournaments. Our exposition focuses on the use of two-graphs and skew two-graphs as isomorphism invariants of translation planes, and caps, avoids, and spreads of polar spaces in odd characteristic. The earliest precedent for using two-graphs to study ovoids and caps is apparently owing to Shult [29]. The degree sequences of these two-graphs and skew two-graphs yield the invariants known as fingerprints, introduced by J. H. Conway (see Chames [9, lo]). Two LZNEAR ALGEBRA AND ITS APPLZCATZONS 226-228:529-551 (1995) 0 Elsevier Science Inc., 1995 0024-3795/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0024-3795(95)00242-J 530 G. -
Hadamard and Conference Matrices
Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
One Method for Construction of Inverse Orthogonal Matrices
ONE METHOD FOR CONSTRUCTION OF INVERSE ORTHOGONAL MATRICES, P. DIÞÃ National Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania E-mail: [email protected] Received September 16, 2008 An analytical procedure to obtain parametrisation of complex inverse orthogonal matrices starting from complex inverse orthogonal conference matrices is provided. These matrices, which depend on nonzero complex parameters, generalize the complex Hadamard matrices and have applications in spin models and digital signal processing. When the free complex parameters take values on the unit circle they transform into complex Hadamard matrices. 1. INTRODUCTION In a seminal paper [1] Sylvester defined a general class of orthogonal matrices named inverse orthogonal matrices and provided the first examples of what are nowadays called (complex) Hadamard matrices. A particular class of inverse orthogonal matrices Aa()ij are those matrices whose inverse is given 1 t by Aa(1ij ) (1 a ji ) , where t means transpose, and their entries aij satisfy the relation 1 AA nIn (1) where In is the n-dimensional unit matrix. When the entries aij take values on the unit circle A–1 coincides with the Hermitian conjugate A* of A, and in this case (1) is the definition of complex Hadamard matrices. Complex Hadamard matrices have applications in quantum information theory, several branches of combinatorics, digital signal processing, etc. Complex orthogonal matrices unexpectedly appeared in the description of topological invariants of knots and links, see e.g. Ref. [2]. They were called two- weight spin models being related to symmetric statistical Potts models. These matrices have been generalized to two-weight spin models, also called , Paper presented at the National Conference of Physics, September 10–13, 2008, Bucharest-Mãgurele, Romania. -
On Sign-Symmetric Signed Graphs∗
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 19 (2020) 83–93 https://doi.org/10.26493/1855-3974.2161.f55 (Also available at http://amc-journal.eu) On sign-symmetric signed graphs∗ Ebrahim Ghorbani Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16765-3381, Tehran, Iran Willem H. Haemers Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands Hamid Reza Maimani , Leila Parsaei Majd y Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran Received 24 October 2019, accepted 27 March 2020, published online 10 November 2020 Abstract A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non- bipartite sign-symmetric signed graphs. Sign-symmetric signed graphs have a symmetric spectrum but not the other way around. We present constructions of signed graphs with symmetric spectra which are not sign-symmetric. This, in particular answers a problem posed by Belardo, Cioaba,˘ Koolen, and Wang in 2018. Keywords: Signed graph, spectrum. Math. Subj. Class. (2020): 05C22, 05C50 1 Introduction Let G be a graph with vertex set V and edge set E. All graphs considered in this paper are undirected, finite, and simple (without loops or multiple edges). A signed graph is a graph in which every edge has been declared positive or negative. In fact, a signed graph Γ is a pair (G; σ), where G = (V; E) is a graph, called the underlying ∗The authors would like to thank the anonymous referees for their helpful comments and suggestions. -
Pretty Good State Transfer in Discrete-Time Quantum Walks
Pretty good state transfer in discrete-time quantum walks Ada Chan and Hanmeng Zhan Department of Mathematics and Statistics, York University, Toronto, ON, Canada fssachan, [email protected] Abstract We establish the theory for pretty good state transfer in discrete- time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be m-strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic an- gles, we construct several infinite families of walks that exhibits this phenomenon. 1 Introduction arXiv:2105.03762v1 [math.CO] 8 May 2021 A quantum walk with nice transport properties is desirable in quantum com- putation. For example, Grover's search algorithm [12] is a quantum walk on the looped complete graph, which sends the all-ones vector to a vector that almost \concentrate on" a vertex. In this paper, we study a slightly stronger notion of state transfer, where the target state can be approximated with arbitrary precision. This is called pretty good state transfer. Pretty good state transfer has been extensively studied in continuous-time quantum walks, especially on the paths [9, 22, 1, 5, 21]. However, not much 1 is known for the discrete-time analogues. The biggest difference between these two models is that in a continuous-time quantum walk, the evolution is completely determined by the adjacency or Laplacian matrix of the graph, while in a discrete-time quantum walk, the transition matrix depends on more than just the graph - usually, it is a product of two unitary matrices: U = SC; where S permutes the arcs of the graph, and C, called the coin matrix, send each arc to a linear combination of the outgoing arcs of the same vertex. -
Hadamard Matrices Include
Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
On D. G. Higman's Note on Regular 3-Graphs
Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 99–115 On D. G. Higman’s note on regular 3-graphs Daniel Kalmanovich Department of Mathematics Ben-Gurion University of the Negev 84105 Beer Sheva, Israel Received 17 October 2011, accepted 25 April 2012, published online 4 June 2012 Abstract We introduce the notion of a t-graph and prove that regular 3-graphs are equivalent to cyclic antipodal 3-fold covers of a complete graph. This generalizes the equivalence of regular two-graphs and Taylor graphs. As a consequence, an equivalence between cyclic antipodal distance regular graphs of diameter 3 and certain rank 6 commutative association schemes is proved. New examples of regular 3-graphs are presented. Keywords: Antipodal graph, association scheme, distance regular graph of diameter 3, Godsil- Hensel matrix, group ring, Taylor graph, two-graph. Math. Subj. Class.: 05E30, 05B20, 05E18 1 Introduction This paper is mainly a clarification of [6] — a short draft written by Donald Higman in 1994, entitled “A note on regular 3-graphs”. The considered generalization of two-graphs was introduced by D. G. Higman in [5]. As in the famous correspondence between two-graphs and switching classes of simple graphs, t-graphs are interpreted as equivalence classes of an appropriate switching relation defined on weights, which play the role of simple graphs. In his note Higman uses certain association schemes to characterize regular 3-graphs and to obtain feasibility conditions for their parameters. Specifically, he provides a graph theoretic interpretation of a weight and from the resulted graph he constructs a rank 4 symmetric association scheme and a rank 6 fission of it. -
Lattices from Tight Equiangular Frames Albrecht Böttcher
Claremont Colleges Scholarship @ Claremont CMC Faculty Publications and Research CMC Faculty Scholarship 1-1-2016 Lattices from tight equiangular frames Albrecht Böttcher Lenny Fukshansky Claremont McKenna College Stephan Ramon Garcia Pomona College Hiren Maharaj Pomona College Deanna Needell Claremont McKenna College Recommended Citation Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj and Deanna Needell. “Lattices from equiangular tight frames.” Linear Algebra and its Applications, vol. 510, 395–420, 2016. This Article is brought to you for free and open access by the CMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in CMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Lattices from tight equiangular frames Albrecht B¨ottcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell Abstract. We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher’s recent observation that this lattice is perfect. -
2020 Ural Workshop on Group Theory and Combinatorics
Institute of Natural Sciences and Mathematics of the Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center 2020 Ural Workshop on Group Theory and Combinatorics Yekaterinburg – Online, Russia, August 24-30, 2020 Abstracts Yekaterinburg 2020 2020 Ural Workshop on Group Theory and Combinatorics: Abstracts of 2020 Ural Workshop on Group Theory and Combinatorics. Yekaterinburg: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 2020. DOI Editor in Chief Natalia Maslova Editors Vladislav Kabanov Anatoly Kondrat’ev Managing Editors Nikolai Minigulov Kristina Ilenko Booklet Cover Desiner Natalia Maslova c Institute of Natural Sciences and Mathematics of Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center, 2020 2020 Ural Workshop on Group Theory and Combinatorics Conents Contents Conference program 5 Plenary Talks 8 Bailey R. A., Latin cubes . 9 Cameron P. J., From de Bruijn graphs to automorphisms of the shift . 10 Gorshkov I. B., On Thompson’s conjecture for finite simple groups . 11 Ito T., The Weisfeiler–Leman stabilization revisited from the viewpoint of Terwilliger algebras . 12 Ivanov A. A., Densely embedded subgraphs in locally projective graphs . 13 Kabanov V. V., On strongly Deza graphs . 14 Khachay M. Yu., Efficient approximation of vehicle routing problems in metrics of a fixed doubling dimension . -
SYMMETRIC CONFERENCE MATRICES of ORDER Pq2 + 1
Can. J. Math., Vol. XXX, No. 2, 1978, pp. 321-331 SYMMETRIC CONFERENCE MATRICES OF ORDER pq2 + 1 RUDOLF MATHON Introduction and definitions. A conference matrix of order n is a square matrix C with zeros on the diagonal and zbl elsewhere, which satisfies the orthogonality condition CCT = (n — 1)1. If in addition C is symmetric, C = CT, then its order n is congruent to 2 modulo 4 (see [5]). Symmetric conference matrices (C) are related to several important combinatorial configurations such as regular two-graphs, equiangular lines, Hadamard matrices and balanced incomplete block designs [1; 5; and 7, pp. 293-400]. We shall require several definitions. A strongly regular graph with parameters (v, k, X, /z) is an undirected regular graph of order v and degree k with adjacency matrix A = AT satisfying (1.1) A2 + 0* - \)A + (/x - k)I = \xJ, AJ = kJ where / is the all one matrix. Note that in a strongly regular graph any two adjacent (non-adjacent) vertices are adjacent to exactly X(/x) other vertices. An easy counting argument implies the following relation between the param eters v, k, X and /x; (1.2) k(k - X - 1) = (v - k - l)/x. A strongly regular graph is said to be pseudo-cyclic (PC) if v — 1 = 2k = 4/x. From (1.1) and (1.2) it is readily deduced that a PC-graph has parameters of the form (4/ + 1, 2/, t — 1, /), where t > 0 is an integer. We note that the complement of a PC-graph is again pseudo-cyclic with the same parameters as the original graph (though not necessarily isomorphic to it), i.e. -
MATRIX THEORY and APPLICATIONS Edited by Charles R
http://dx.doi.org/10.1090/psapm/040 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures A Subseries of Proceedings of Symposia in Applied Mathematics Volume 40 MATRIX THEORY AND APPLICATIONS Edited by Charles R. Johnson (Phoenix, Arizona, January 1989) Volume 39 CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS Edited by Robert L.. Devaney and Linda Keen (Providence, Rhode Island, August 1988) Volume 38 COMPUTATIONAL COMPLEXITY THEORY Edited by Juris Hartmanis (Atlanta, Georgia, January 1988) Volume 37 MOMENTS IN MATHEMATICS Edited by Henry J. Landau (San Antonio, Texas, January 1987) Volume 36 APPROXIMATION THEORY Edited by Carl de Boor (New Orleans, Louisiana, January 1986) Volume 35 ACTUARIAL MATHEMATICS Edited by Harry H. Panjer (Laramie, Wyoming, August 1985) Volume 34 MATHEMATICS OF INFORMATION PROCESSING Edited by Michael Anshel and William Gewirtz (Louisville, Kentucky, January 1984) Volume 33 FAIR ALLOCATION Edited by H. Peyton Young (Anaheim, California, January 1985) Volume 32 ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS Edited by R. W. McKelvey (Eugene, Oregon, August 1984) Volume 31 COMPUTER COMMUNICATIONS Edited by B. Gopinath (Denver, Colorado, January 1983) Volume 30 POPULATION BIOLOGY Edited by Simon A. Levin (Albany, New York, August 1983) Volume 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton (San Francisco, California, January 1981) Volume 28 STATISTICAL DATA ANALYSIS Edited by R. Gnanadesikan (Toronto, Ontario, August 1982) Volume 27 COMPUTED TOMOGRAPHY Edited by L. A. Shepp (Cincinnati, Ohio, January 1982) Volume 26 THE MATHEMATICS OF NETWORKS Edited by S. A. Burr (Pittsburgh, Pennsylvania, August 1981) Volume 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S.