Distance-Regular Cayley Graphs with Small Valency 3
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Groups and Generalized Polyg.Ons
数理解析研究所講究録 1063 巻 1998 年 1-6 1 Groups and Generalized Polyg.ons Hendrik Van Maldeghem* 1 Introduction Geometric interpretation is a technique that has proved very useful to study certain groups. Especially well disigned for this are the Tits buildings, which are geometric interpretations of groups of Lie type, Chevalley groups, semi simple algebraic groups, groups with a $\mathrm{B}\mathrm{N}$ -pair, Kac-Moody groups, etc. Conversely, given a certain geometry, for instance a special kind of Tits building, one could raise the question whether there is always a group behind it. This is searching for a geometric characterization of the groups in question. Examples are the spherical Tits buildings of rank $\geq 3$ , the affine Tits buildings of rank $\geq 4$ , and certain twin Tits buildings, giving rise to, respectively, semi simple algebraic groups of relative rank $\geq 3$ and groups of mixed type, semi simple algebraic groups and mixed type groups of relative rank $\geq 3$ with a valuation on the root groups in the sense of BRUHAT &TITS [1972], certain Kac-Moody groups. Moreover, a lot of sporadic finite simple groups have been geometrically characterized by geometries which extend Tits buildings. The building bricks in all these cases are the buildings of rank 2, the so-called generalized polygons. The main examples of these are constructed from the parabolic subgroups of a rank 2 Tits system, or $\mathrm{B}\mathrm{N}$ -pair. For instance, in the finite case, one has so-called classical examples related to the linear groups $\mathrm{P}\mathrm{S}\mathrm{L}(3, -
On the Metric Dimension of Imprimitive Distance-Regular Graphs
On the metric dimension of imprimitive distance-regular graphs Robert F. Bailey Abstract. A resolving set for a graph Γ is a collection of vertices S, cho- sen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric di- mension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance- regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs, but also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible. Mathematics Subject Classification (2010). Primary 05E30; Secondary 05C12. Keywords. Metric dimension; resolving set; distance-regular graph; imprimitive; halved graph; folded graph; bipartite double; Taylor graph; incidence graph. 1. Introduction Let Γ = (V; E) be a finite, undirected graph without loops or multiple edges. For u; v 2 V , the distance from u to v is the least number of edges in a path from u to v, and is denoted dΓ(u; v) (or simply d(u; v) if Γ is clear from the context). A resolving set for a graph Γ = (V; E) is a set of vertices R = fv1; : : : ; vkg such that for each vertex w 2 V , the list of distances (d(w; v1);:::; d(w; vk)) uniquely determines w. -
Veldkamp Quadrangles and Polar Spaces
VELDKAMP QUADRANGLES AND POLAR SPACES BERNHARD MÜHLHERR AND RICHARD M. WEISS Abstract. Veldkamp polygons are certain graphs Γ = (V,E) such that for each v ∈ V , Γv is endowed with a symmetric anti-reflexive relation ≡v. These relations are all trivial if and only if Γ is a thick generalized polygon. A Veld- kamp polygon is called flat if no two vertices have the same set of vertices that are opposite in a natural sense. We explore the connection between Veldkamp quadrangles and polar spaces. Using this connection, we give the complete classification of flat Veldkamp quadrangles in which some but not all of the relations ≡v are trivial. 1. Introduction In 2.8, we introduce the notion of a Veldkamp polygon. A Veldkamp polygon is a graph Γ = (V, E) such that for each vertex v ∈ V , the set Γv of vertices adjacent to v is endowed with a symmetric, anti-reflexive relation ≡v satisfying several axioms. The relations ≡v are called the local opposition relations of Γ. If (u,v,w) is a path of length 2 and u ≡v w, we say that the path (u,v,w) is straight. An arbitrary path is straight if every subpath of length 2 is straight. Thus every path is straight if and only if the local opposition relations ≡v are all trivial (in the sense that u ≡v w for all distinct u, w ∈ Γv). This is the case if and only if Γ is a thick generalized polygon. The thick generalized polygons that arise in the theory of algebraic groups all satisfy the Moufang condition. -
A Godsil E Thomason I Solomon M Abiad B Van Dam F Balbuena J
8.30–9.15 am A Godsil E Thomason I Solomon M Abiad 9.20–9.40 am 1 Zhan 10 Bukh 19 Srinivasan 24 Reichard 9.45–10.05 am 2 Ye 11 Martin 20 Sumalroj 25 Xu 10.05–10.35 am break break break break 10.35–10.55 am 3 Dalfó 12 Kamat 21 Bencs 26 Peng 11.00–11.20 am 4 McGinnis 13 Timmons 22 Guo 27 Kravitz 11.25–12.10 pm B Van Dam F Balbuena J Muzychuk N Xiang 12.10–2.00 pm lunch lunch lunch lunch 2.00–2.45 pm C Kantor G Füredi K Williford 2.50–3.10 pm 5 Gu 14 Tait 23 Ducey 3.15–3.35 pm 6 Coutinho 15 Kodess L Woldar break 3.40–4.00 pm 7 Greaves 16 Y. Wang 4.00–4.30 pm break break 4.30–4.50 pm 8 Fiol 17 Kronenthal 4:55–5.15 pm 9 W. Wang 18 Moorhouse 5.20–6.10 pm D Haemers H Lazebnik 2 8.30–9.15 am A Chris Godsil Spectral invariants from embeddings 9.20–9.40 am 1 Harmony Zhan Quantum walks and mixing 9.45–10.05 am 2 Dong Ye Median eigenvalues and graph inverse 10.35–10.55 am 3 Cristina Dalfó Characterizing identifying codes from the spectrum of a graph or digraph 11.00–11.20 am 4 Matt McGinnis The smallest eigenvalues of the Hamming graphs 11.25–12:10 pm B Edwin van Dam Partially metric association schemes with a small multiplicity 2.00–2.45 pm C Bill Kantor MUBs 2.50–3.10 pm 5 Xiaofeng Gu Toughness, connectivity and the spectrum of regular graphs 3.15–3.35 pm 6 Gabriel Coutinho Average mixing matrix 3.40–4.00 pm 7 Gary Greaves Edge-regular graphs and regular cliques 4.30–4.50 pm 8 Miguel Angel Fiol An algebraic approach to lifts of digraphs 4.55–5.15 pm 9 Wei Wang A positive proportion of multigraphs are determined by their generalized spectra 5.20–6.10 -
Dynamic Cage Survey
Dynamic Cage Survey Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Robert Jajcay Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Department of Algebra Comenius University Bratislava, Slovakia [email protected] Submitted: May 22, 2008 Accepted: Sep 15, 2008 Version 1 published: Sep 29, 2008 (48 pages) Version 2 published: May 8, 2011 (54 pages) Version 3 published: July 26, 2013 (55 pages) Mathematics Subject Classifications: 05C35, 05C25 Abstract A(k; g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013), #DS16 1 Contents 1 Origins of the Problem 3 2 Known Cages 6 2.1 Small Examples . 6 2.1.1 (3,5)-Cage: Petersen Graph . 7 2.1.2 (3,6)-Cage: Heawood Graph . 7 2.1.3 (3,7)-Cage: McGee Graph . 7 2.1.4 (3,8)-Cage: Tutte-Coxeter Graph . 8 2.1.5 (3,9)-Cages . 8 2.1.6 (3,10)-Cages . 9 2.1.7 (3,11)-Cage: Balaban Graph . 9 2.1.8 (3,12)-Cage: Benson Graph . 9 2.1.9 (4,5)-Cage: Robertson Graph . 9 2.1.10 (5,5)-Cages . -
Distance-Transitive Graphs
Distance-Transitive Graphs Submitted for the module MATH4081 Robert F. Bailey (4MH) Supervisor: Prof. H.D. Macpherson May 10, 2002 2 Robert Bailey Department of Pure Mathematics University of Leeds Leeds, LS2 9JT May 10, 2002 The cover illustration is a diagram of the Biggs-Smith graph, a distance-transitive graph described in section 11.2. Foreword A graph is distance-transitive if, for any two arbitrarily-chosen pairs of vertices at the same distance, there is some automorphism of the graph taking the first pair onto the second. This project studies some of the properties of these graphs, beginning with some relatively simple combinatorial properties (chapter 2), and moving on to dis- cuss more advanced ones, such as the adjacency algebra (chapter 7), and Smith’s Theorem on primitive and imprimitive graphs (chapter 8). We describe four infinite families of distance-transitive graphs, these being the Johnson graphs, odd graphs (chapter 3), Hamming graphs (chapter 5) and Grass- mann graphs (chapter 6). Some group theory used in describing the last two of these families is developed in chapter 4. There is a chapter (chapter 9) on methods for constructing a new graph from an existing one; this concentrates mainly on line graphs and their properties. Finally (chapter 10), we demonstrate some of the ideas used in proving that for a given integer k > 2, there are only finitely many distance-transitive graphs of valency k, concentrating in particular on the cases k = 3 and k = 4. We also (chapter 11) present complete classifications of all distance-transitive graphs with these specific valencies. -
Quantum Automorphism Groups of Finite Graphs
Quantum automorphism groups of finite graphs Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaften der Fakult¨atf¨urMathematik und Informatik der Universit¨atdes Saarlandes Simon Schmidt Saarbr¨ucken, 2020 Datum der Verteidigung: 2. Juli 2020 Dekan: Prof. Dr. Thomas Schuster Pr¨ufungsausschuss: Vorsitzende: Prof. Dr. Gabriela Weitze-Schmith¨usen Berichterstatter: Prof. Dr. Moritz Weber Prof. Dr. Roland Speicher Prof. Dr. Julien Bichon Akademischer Mitarbeiter: Dr. Tobias Mai ii Abstract The present work contributes to the theory of quantum permutation groups. More specifically, we develop techniques for computing quantum automorphism groups of finite graphs and apply those to several examples. Amongst the results, we give a criterion on when a graph has quantum symmetry. By definition, a graph has quantum symmetry if its quantum automorphism group does not coincide with its classical automorphism group. We show that this is the case if the classical automorphism group contains a pair of disjoint automorphisms. Furthermore, we prove that several families of distance-transitive graphs do not have quantum symmetry. This includes the odd graphs, the Hamming graphs H(n; 3), the Johnson graphs J(n; 2), the Kneser graphs K(n; 2) and all cubic distance-transitive graphs of order ≥ 10. In particular, this implies that the Petersen graph does not have quantum symmetry, answering a question asked by Banica and Bichon in 2007. Moreover, we show that the Clebsch graph does have quantum symmetry and −1 prove that its quantum automorphism group is equal to SO5 answering a question asked by Banica, Bichon and Collins. More generally, for odd n, the quantum −1 automorphism group of the folded n-cube graph is SOn . -
On Distance, Geodesic and Arc Transitivity of Graphs
On distance, geodesic and arc transitivity of graphs Alice Devillers, Wei Jin,∗ Cai Heng Li and Cheryl E. Praeger†‡ Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia Abstract We compare three transitivity properties of finite graphs, namely, for a positive integer s, s-distance transitivity, s-geodesic transitivity and s-arc transitivity. It is known that if a finite graph is s-arc transi- tive but not (s + 1)-arc transitive then s ≤ 7 and s 6= 6. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of s, and that these graphs can have arbitrar- ily large diameter if and only if 1 ≤ s ≤ 3. Moreover, for a prime p we prove that there exists a graph of valency p that is 2-geodesic transitive but not 2-arc transitive if and only if p ≡ 1 (mod 4), and for each such prime there is a unique graph with this property: it is an antipodal double cover of the complete graph Kp+1 and is geodesic transitive with automorphism group PSL(2,p) × Z2. Keywords: Graphs; distance-transitivity; geodesic-transitivity; arc-transitivity 1 Introduction arXiv:1110.2235v1 [math.CO] 11 Oct 2011 The study of finite distance-transitive graphs goes back to Higman’s paper [9] in which “groups of maximal diameter” were introduced. These are per- mutation groups which act distance transitively on some graph. The family ∗The second author is supported by the Scholarships for International Research Fees (SIRF) at UWA. -
Distance-Regular Cayley Graphs with Small Valency∗
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 203–222 https://doi.org/10.26493/1855-3974.1964.297 (Also available at http://amc-journal.eu) Distance-regular Cayley graphs with small valency∗ Edwin R. van Dam Department of Econometrics and O.R., Tilburg University, The Netherlands Mojtaba Jazaeri y Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran Received 28 March 2019, accepted 9 May 2019, published online 20 September 2019 Abstract We consider the problem of which distance-regular graphs with small valency are Cay- ley graphs. We determine the distance-regular Cayley graphs with valency at most 4, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 5, and the Cayley graphs among all distance-regular graphs with girth 3 and valency 6 or 7. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some “excep- tional” distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs. Keywords: Cayley graph, distance-regular graph. Math. Subj. Class.: 05E30 ∗We thank Sasha Gavrilyuk, who, in the final stages of writing this paper, at a conference in Plzen,ˇ pointed us to [27] for what he called Higman’s method applied to generalized polygons. -
Arxiv:1708.01095V2 [Math.CO] 30 Mar 2018 Rp Hoy Generalized Theory
On regular induced subgraphs of generalized polygons John Bamberg, Anurag Bishnoi, Gordon F. Royle Abstract. The cage problem asks for the smallest number c(k,g) of vertices in a k-regular graph of girth g and graphs meeting this bound are known as cages. While cages are known to exist for all integers k > 2 and g > 3, the exact value of c(k,g) is known only for some small values of k,g and three infinite families where g 6, 8, 12 and k 1 is a prime power. These infinite families come ∈ { } − from the incidence graphs of generalized polygons. Some of the best known upper bounds on c(k,g) for g 6, 8, 12 have been obtained by constructing small regular induced subgraphs of these cages. ∈ { } In this paper, we first use the Expander Mixing Lemma to give a general lower bound on the size of an induced k-regular subgraph of a regular bipartite graph in terms of the second largest eigenvalue of the host graph. We use this bound to show that the known construction of (k, 6)-graphs using Baer subplanes of the Desarguesian projective plane is the best possible. For generalized quadrangles and hexagons, our bounds are new. In particular, we improve the known lower bound on the size of an induced q-regular subgraphs of the classical generalized quadrangle Q(4, q) and show that the known constructions are asymptotically sharp, which answers a question of Metsch [21, Section 6]. For prime powers q, we also improve the known upper bounds on c(q, 8) and c(q, 12) by giving new geometric constructions of q-regular induced subgraphs in the symplectic generalized quadrangle W(3, q) and the split Cayley hexagon H(q), respectively. -
Distance-Regular Graphs Arxiv:1410.6294V2 [Math.CO]
Distance-regular graphs∗ Edwin R. van Dam Jack H. Koolen Department of Econometrics and O.R. School of Mathematical Sciences Tilburg University University of Science and Technology of China The Netherlands and [email protected] Wu Wen-Tsun Key Laboratory of Mathematics of CAS Hefei, Anhui, 230026, China [email protected] Hajime Tanaka Research Center for Pure and Applied Mathematics Graduate School of Information Sciences Tohoku University Sendai 980-8579, Japan [email protected] Mathematics Subject Classifications: 05E30, 05Cxx, 05Exx Abstract This is a survey of distance-regular graphs. We present an introduction to distance- regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph `BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written. Keywords: Distance-regular graph; survey; association scheme; P -polynomial; Q- polynomial; geometric arXiv:1410.6294v2 [math.CO] 15 Apr 2016 ∗This version is published in the Electronic Journal of Combinatorics (2016), #DS22. 1 Contents 1 Introduction6 2 An introduction to distance-regular graphs8 2.1 Definition . .8 2.2 A few examples . .9 2.2.1 The complete graph . .9 2.2.2 The polygons . .9 2.2.3 The Petersen graph and other Odd graphs . .9 2.3 Which graphs are determined by their intersection array? . .9 2.4 Some combinatorial conditions for the intersection array . 11 2.5 The spectrum of eigenvalues and multiplicities . 12 2.6 Association schemes . 15 2.7 The Q-polynomial property . -
Large Incidence-Free Sets in Geometries
Large incidence-free sets in geometries Stefaan De Winter Department of Mathematical Sciences Michigan Technological University Michigan, U.S.A. [email protected] Jeroen Schillewaert Jacques Verstraete ∗ Department of Mathematics University of California at San Diego California, U.S.A. fjschillewaert,[email protected] Submitted: Jan 6, 2012; Accepted: Oct 22, 2012; Published: Nov 8, 2012 Mathematics Subject Classifications: 15A18,05D40,51E12 Abstract Let Π = (P; L; I) denote a rank two geometry. In this paper, we are interested in the largest value of jXjjY j where X ⊂ P and Y ⊂ L are sets such that (X ×Y )\I = ;. Let α(Π) denote this value. We concentrate on the case where P is the point set of PG(n; q) and L is the set of k-spaces in PG(n; q). In the case that Π is the projective plane PG(2; q), Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine α(PG(2; q)) when q is an even power of 2. Therefore, in those cases, p α(Π) = q(q − q + 1)2: We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in generalized polygons. More generally, if P is the point set of PG(n; q) and L is the set of k-spaces in PG(n; q), where 1 6 k 6 n−1, and Πq = (P; L; I), then we show as q ! 1 that 1 q(k+2)(n−k) α(Π) q(k+2)(n−k): 4 .