Q-Polynomial Association Schemes with Irrational Eigenvalues a Major Qualifying Project Report Submitted to the Faculty of The
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Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. -
The Veldkamp Space of GQ(2,4) Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana
The Veldkamp Space of GQ(2,4) Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana To cite this version: Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana. The Veldkamp Space of GQ(2,4). International Journal of Geometric Methods in Modern Physics, World Scientific Publishing, 2010, pp.1133-1145. 10.1142/S0219887810004762. hal-00365656v2 HAL Id: hal-00365656 https://hal.archives-ouvertes.fr/hal-00365656v2 Submitted on 6 Jul 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Veldkamp Space of GQ(2,4) M. Saniga,1 R. M. Green,2 P. L´evay,3 P. Pracna4 and P. Vrana3 1Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´aLomnica, Slovak Republic ([email protected]) 2Department of Mathematics, University of Colorado Campus Box 395, Boulder CO 80309-0395, U. S. A. ([email protected]) 3Department of Theoretical Physics, Institute of Physics Budapest University of Technology and Economics, H-1521 Budapest, Hungary ([email protected] and [email protected]) and 4J. Heyrovsk´yInstitute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejˇskova 3, CZ-182 23 Prague 8, Czech Republic ([email protected]) (6 July 2009) Abstract It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomor- phic to PG(5,2). -
Some Topics Concerning Graphs, Signed Graphs and Matroids
SOME TOPICS CONCERNING GRAPHS, SIGNED GRAPHS AND MATROIDS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Vaidyanathan Sivaraman, M.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Prof. Neil Robertson, Advisor Prof. Akos´ Seress Prof. Matthew Kahle ABSTRACT We discuss well-quasi-ordering in graphs and signed graphs, giving two short proofs of the bounded case of S. B. Rao's conjecture. We give a characterization of graphs whose bicircular matroids are signed-graphic, thus generalizing a theorem of Matthews from the 1970s. We prove a recent conjecture of Zaslavsky on the equality of frus- tration number and frustration index in a certain class of signed graphs. We prove that there are exactly seven signed Heawood graphs, up to switching isomorphism. We present a computational approach to an interesting conjecture of D. J. A. Welsh on the number of bases of matroids. We then move on to study the frame matroids of signed graphs, giving explicit signed-graphic representations of certain families of matroids. We also discuss the cycle, bicircular and even-cycle matroid of a graph and characterize matroids arising as two different such structures. We study graphs in which any two vertices have the same number of common neighbors, giving a quick proof of Shrikhande's theorem. We provide a solution to a problem of E. W. Dijkstra. Also, we discuss the flexibility of graphs on the projective plane. We conclude by men- tioning partial progress towards characterizing signed graphs whose frame matroids are transversal, and some miscellaneous results. -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
Lattices from Group Frames and Vertex Transitive Graphs
Frames Lattices Graphs Examples Lattices from group frames and vertex transitive graphs Lenny Fukshansky Claremont McKenna College (joint work with Deanna Needell, Josiah Park and Jessie Xin) Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: k Let f P R be a nonzero vector, then Gf “ tUf : U P Gu k is called a group frame (or G-frame). If G acts irreducibly on R , Gf is called an irreducible group frame. Irreducible group frames are always tight. Frames Lattices Graphs Examples Group frames Let G be a finite subgroup of Ok pRq, the k-dimensional real orthog- k onal group, then G acts on R by left matrix multiplication. -
Some Applications of Generalized Action Graphs and Monodromy Graphs
Some applications of generalized action graphs and monodromy graphs Tomaˇz Pisanski (University of Primorska and University of Ljubljana) 2015 International Conference on Graph Theory UP FAMNIT, Koper, May 26–28, 2015 Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Thanks Thanks to my former students Alen Orbani´c, Mar´ıa del R´ıo-Francos and their and my co-authors, in particular to Jan Karab´aˇs and many others whose work I am freely using in this talk. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Snub Cube - Chirality in the sense of Conway Snube cube is vertex-transitive (uniform) chiral polyhedron. It comes in two oriented forms. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Motivation We would like to have a single combinatorial mechanism that would enable us to deal with such phenomena. Our goal is to unify several existing structures in such a way that the power of each individual structure is preserved. Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Example: The Cube Example The cube may be considered as an oriented map: R, r, as a map: r0, r1, r2 as a polyhedron .... Question What is the dual of a cube and how can we describe it? Question How can we tell that the cube is a regular map and amphichiral oriented map? Tomaˇz Pisanski (University of Primorska and University of Ljubljana)Some applications of generalized action graphs and monodromy graphs Plan We define generalized action graphs as semi-directed graphs in which the edge set is partitioned into directed 2-factors (forming an action digraph) and undirected 1-factors (forming a monodromy graph) and use them to describe several combinatorial structures, such as maps and oriented maps. -
Symmetric Cubic Graphs of Small Girth 1 Introduction
Symmetric cubic graphs of small girth Marston Conder1 Roman Nedela2 Department of Mathematics Institute of Mathematics University of Auckland Slovak Academy of Science Private Bag 92019 Auckland 975 49 Bansk´aBystrica New Zealand Slovakia Email: [email protected] Email: [email protected] Abstract A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2′-regular (following the notation of Djokovic), or belongs to a small family of exceptional graphs. On the other hand, we show that there are infinitely many 3-regular cubic graphs of girth 10, so that the statement for girth at most 9 cannot be improved to cubic graphs of larger girth. Also we give a characterisation of the 1- or 2′-regular cubic graphs of girth g ≤ 9, proving that with five exceptions these are closely related with quotients of the triangle group ∆(2, 3, g) in each case, or of the group h x,y | x2 = y3 = [x,y]4 = 1 i in the case g = 8. All the 3-transitive cubic graphs and exceptional 1- and 2-regular cubic graphs of girth at most 9 appear in the list of cubic symmetric graphs up to 768 vertices produced by Conder and Dobcs´anyi (2002); the largest is the 3-regular graph F570 of order 570 (and girth 9). -
The Extendability of Matchings in Strongly Regular Graphs
The extendability of matchings in strongly regular graphs Sebastian M. Cioab˘a∗ Weiqiang Li Department of Mathematical Sciences Department of Mathematical Sciences University of Delaware University of Delaware Newark, DE 19707-2553, U.S.A. Newark, DE 19707-2553, U.S.A. [email protected] [email protected] Submitted: Feb 25, 2014; Accepted: Apr 29, 2014; Published: May 13, 2014 Mathematics Subject Classifications: 05E30, 05C50, 05C70 Abstract A graph G of even order v is called t-extendable if it contains a perfect matching, t < v=2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not 3-extendable. We also show that strongly regular graphs of valency k > 3 with λ > 1 are bk=3c-extendable k+1 (when µ 6 k=2) and d 4 e-extendable (when µ > k=2), where λ is the number of common neighbors of any two adjacent vertices and µ is the number of common neighbors of any two non-adjacent vertices. Our results are close to being best pos- sible as there are strongly regular graphs of valency k that are not dk=2e-extendable. We show that the extendability of many strongly regular graphs of valency k is at least dk=2e − 1 and we conjecture that this is true for all primitive strongly regular graphs. We obtain similar results for strongly regular graphs of odd order. -
Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs
mathematics Article Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs Jianqiang Hao 1,*, Yunzhan Gong 2, Jianzhi Sun 1 and Li Tan 1 1 Beijing Key Laboratory of Big Data Technology for Food Safety, Beijing Technology and Business University, No. 11, Fu Cheng Road, Beijing 100048, China 2 State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, No 10, Xitucheng Road, Haidian District, Beijing 100876, China * Correspondence: [email protected]; Tel.: +86-10-6898-5704 Received: 5 July 2019; Accepted: 27 July 2019; Published: 31 July 2019 Abstract: This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to Nauty’s. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its open k-neighborhood subgraph . It defines dif fusion degree sequences and entire dif fusion degree sequence . For each node of a graph G, it designs a characteristic m_NearestNode to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of MaxQ(G). Another theorem presents how to determine the second nodes of MaxQ(G). When computing Cmax(G), if MaxQ(G) already holds the first i nodes u1, u2, ··· , ui, Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of MaxQ(G). Further, it also establishes two theorems to determine the Cmax(G) of disconnected graphs. -
Fragments in Symmetric Configurations with Block Size 3
Fragments in symmetric configurations with block size 3 Grahame Erskine∗∗ Terry Griggs∗ Jozef Sir´aˇnˇ †† [email protected] [email protected] [email protected] Abstract We begin the study of collections of three blocks which can occur in a symmetric configuration with block size 3, v3. Formulae are derived for the number of occurrences of these and it is shown that the triangle, i.e. abf, ace, bcd is a basis. It is also shown that symmetric configurations without triangles exist if and only if v = 15 or v ≥ 17. Such configurations containing “many” triangles are also discussed and a complete analysis of the triangle content of those with a cyclic automorphism is given. 1 Introduction In this paper we will be concerned with symmetric configurations with block size 3. First we recall the definitions. A configuration (vr, bk) is a finite incidence structure with v points and b blocks, with the property that there exist positive integers k and r such that (i) each block contains exactly k points; (ii) each point is contained in exactly r blocks; and (iii) any pair of distinct points is contained in at most one block. If v = b (and hence necessarily r = k), the configuration is called symmetric and is usually denoted by vk. We are interested in the case where k = 3. The blocks will also be called triples. A configuration is said to be decomposable or disconnected if it is the union of two configurations on distinct point sets. We are primarily interested in indecomposable (connected) configurations, and so unless otherwise noted, this is assumed throughout the paper. -
Minor Minimal Intrinsically Knotted Graphs with 21 Edges Jamison
Minor Minimal Intrinsically Knotted Graphs with 21 Edges Jamison Barsotti Abstract. In this paper, we prove that the only minor minimal intrinsically knotted graphs with 21 edges are the fourteen graphs that are obtained through a series of Triangle-Y moves from the complete graph on 7 vertices. CHAPTER 1 Introduction A graph G is intrinsically knotted (IK) if every tame embedding of G in R3 contains a non-trivial knotted cycle. In 1983, Conway and Gordon showed that the complete graph on seven vertices, K7, is IK [CG], as well as showing that deleting any edge from K7 will provide a graph that is not IK. This implies that K7 is not only IK but, since no proper minor of K7 will be IK, K7 is said to be minor minimal IK or MMIK. That there is a finite number of MMIK graphs follows from work of Robertson and Seymour [RS]. That an IK graph must have at least 21 edges was shown, independently, by Mattman [M] and Johnson, Kidwell, and Michael [JKM]. Since, K7 has 21 edges, this paper analyzes the graphs with 21 edges and provides a complete list of MMIK graphs on 21 edges. In his paper, Mattman also analyzes graphs with 21 edges and 9 vertices or less, so this paper considers the graphs on 21 edges and more than 9 vertices. When considering IK graphs, a method called the Triangle-Y (TY, or ∆Y ) move is used frequently to obtain new graphs that are also IK [S]. The TY move takes three mutually adjacent vertices and replaces their adjoining edges with a vertex whose neighborhood consists of the three formerly adjacent vertices. -
ON 3-REGULAR 4-ORDERED GRAPHS 1. Introduction The
ON 3-REGULAR 4-ORDERED GRAPHS KAROLA MESZ´ AROS´ Abstract. A simple graph G is k-ordered (respectively, k-ordered hamilton- ian), if for any sequence of k distinct vertices v1,...,vk of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K4 and K3,3. Ng and Schultz observed that a 3-regular 4- ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free, and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4- ordered. Furthermore, we prove that the smallest graph after K4 and K3,3 that is 3-regular 4-ordered hamiltonian is the Heawood graph, and we exhibit for- bidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. Finally, we construct an infinite family of 3-regular 4-ordered graphs. 1. Introduction The concept of k-ordered graphs was introduced in 1997 by Ng and Schultz [8]. A simple graph G is a graph without loops or multiple edges, and it is called hamiltonian if there exists a cycle that contains all vertices of G. In this paper we consider only connected finite simple graphs. A simple graph G is called k- ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v1,...,vk of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order.