DOCSLIB.ORG

Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Faculty of Mathematics Professorship for Algorithmic and Discrete Mathematics Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Dissertation submitted to the Faculty of Mathematics at Technische Universität Chemnitz in accordance with the requirements for the degree doctor rerum naturalium (Dr. rer. nat.) by MSc. Martin Winter Chemnitz, April 26, 2021 Referees: Prof. Dr. Christoph Helmberg (supervisor) Prof. Dr. Michael Joswig Prof. Dr. Egon Schulte Winter, Martin Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Dissertation, Faculty of Mathematics Technische Universität Chemnitz, 2021 Bibliographical description Martin Winter Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity Dissertation, 184 pages, Technische Universität Chemnitz, Faculty of Mathematics, 2021 Abstract A spectral graph realization is an embedding of a finite simple graph into Euclidean space that is constructed from the eigenvalues and eigenvectors of the graph’s adjacency matrix. It has previously been observed that some polytopes can be reconstructed from their edge-graphs by taking the convex hull of a spectral realization of this edge-graph. These polytopes, which we shall call spectral polytopes, have remarkable rigidity and symmetry properties and are a source for many open questions. In this thesis we aim to further the understanding of this phenomenon by exploring the geometric and combinatorial properties of spectral polytopes on several levels. One of our central questions is whether already “weak” forms of symmetry can be a sufficient reason for a polytope to be spectral. To answer this, we derive a geometric criterion for the identifica- tion of spectral polytopes and apply it to prove that indeed all polytopes of combined vertex- and edge-transitivity are spectral, admit a unique reconstruction from the edge-graph and realize all the symmetries of this edge-graph. We explore the same questions for graph real- izations and find that realizations of combined vertex- and edge-transitivity are not necessar- ily spectral. Instead we show that we require a stronger form of symmetry, called distance- transitivity. Motivated by these findings we take a closer look at the class of edge-transitive polytopes, for which no classification is known. We state a conjecture for a potential classification and pro- vide complete classifications for several sub-classes, such as distance-transitive polytopes and edge-transitive polytopes that are not vertex-transitive. In particular, we show that the latter class contains only polytopes of dimension d 3. As a side result we obtain the complete classification≤ of the vertex-transitive zonotopes and a new characterization for root systems. Keywords Spectral graph realizations, eigenpolytopes, spectral polytopes, edge-transitive polytopes, vertex-transitive zonotopes 3 Acknowledgments Foremost, I am extremely grateful to my doctoral advisor Prof. Dr. Christoph Helmberg, who gave me the freedom to follow my interests, supported me in every instance, read all my papers, improved them by his detailed comments and was always available for questions. I also thank Dr. Thomas Jahn and Dr. Frank Göring for many hours of fruitful discussions. As a result of one such discussion Frank came up with an ingenious argument to significantly shorten the final step in one of the classification proofs presented in this thesis. Special thanks go to Prof. Dr. Achill Schürmann and Prof. Dr. Raman Sanyal who provided valuable feedback and put me on the right track with their literature references. Thanks go also to my numerous proofreaders (foremost Christin, Daniela, Marvin, Thomas) who made my work more readable with every hint after I became blind to the shortcomings myself. Finally, I would like to thank Prof. Dr. Michael Joswig and Prof. Dr. Egon Schulte for the time and effort they invested in reviewing this thesis. The author also gratefully acknowledges the support by the funding of the European Union and the Free State of Saxony who enabled this project in the first place. 5 Contents List of Symbols 11 Introduction 13 I Spectrum and Symmetry 25 1 Point Arrangements 27 1.1 The arrangement space . 28 1.2 Symmetric arrangements . 32 1.3 Separating symmetries . 37 1.4 Deformation and rigidity . 40 1.5 Transitive arrangements . 45 2 Graph Realizations 51 2.1 Spectral and balanced realizations . 53 2.2 Edge- and arc-transitive realizations . 58 2.3 Full local dimension . 62 2.4 Distance-transitive realizations . 65 2.5 Cosine vectors and cosine sequences . 67 3 Eigenpolytopes and Spectral Polytopes 73 3.1 Eigenpolytopes . 74 3.2 Spectral polytopes . 76 3.3 The Theorem of Izmestiev . 80 II Transitivities in Convex Polytopes 85 4 Edge-Transitive Polytopes 87 4.1 An overview of transitivity in polytopes . 88 4.2 Known edge-transitive polytopes . 89 4.3 Simultaneously vertex- and edge-transitive polytopes . 91 4.4 Arc-transitive polytopes . 93 4.5 Half-transitive polytopes . 97 4.6 Distance-transitive polytopes . 98 5 Vertex-Transitive Zonotopes 101 5.1 Generators, faces, symmetry and permutahedra . 102 7 Contents 5.2 The case d = 2....................................... 106 5.3 The general case . 108 5.4 The classification . 110 5.5 Related topics . 112 6 Purely Edge-Transitive Polytopes 115 6.1 Bipartite polytopes . 116 6.2 Some easy cases . 119 6.2.1 Bipartite polygons . 119 6.2.2 Inscribed bipartite polytopes . 120 6.2.3 Strictly bipartite polytopes for d 4..................... 121 6.3 Strictly bipartite polyhedra . .≥ . 121 6.4 Adjacent pairs . 128 6.4.1 The case τ1 = (4, 6, 10) ............................. 130 6.4.2 The case τ1 = (4, 6, 8) .............................. 131 6.4.3 The case τ1 = (4, 6, 6) .............................. 131 6.4.4 The case τ1 = (4, 4, 4) .............................. 132 6.4.5 The case τ1 = (4, 4, 2k), 2k 6........................ 133 ≥ 6.4.6 The case τ1 = (4, 4, 6) .............................. 135 6.5 An almost strictly bipartite polyhedron . 135 Conclusions and Outlook 141 Theses 145 Appendix 147 A Matrix Groups and Representations 149 A.1 Matrix groups . 149 A.2 Representations . 151 B Spectral Graph Theory 153 C Polytope Theory 155 C.1 Zonotopes . 156 D Reflection Groups and Root Systems 161 D.1 Reflection groups . 161 D.2 Root systems . 162 D.3 Coxeter-Dynkin diagrams . 162 E Wythoffian, Uniform and Regular Polytopes 165 E.1 Orbit polytopes and Wythoffian polytopes . 165 E.2 Coxeter-Dynkin diagrams for Wythoffian polytopes . 165 E.3 Uniform and regular polytopes . 166 8 Contents F Mathematica Scripts 169 G Additional proofs and computations 171 G.1 Two geometric statements . 171 G.2 A proof for spherical interior angles . 172 9 List of Symbols d R d-dimensional Euclidean space dim dimension of linear or affine space span linear span/hull of a set of vectors aff affine hull of a set of points conv convex hull of a set of points rank rank of a matrix or set of vectors, equivalent to dim span Id identity matrix (of appropriate dimension) d d d GL(R ) group of invertible linear transformations R R d d d! O(R ) group of orthogonal transformations R R d d ! d SO(R ) group of orthogonal transformations R R with determinant one ! πW orthogonal projection onto the subspace W deg(i) vertex degree of the vertex i deg(G) (common) vertex degree of the vertices in a regular graph G dist(i, j) graph theoretic distance between two vertices i, j V diam(G) diameter of the graph G 2 Aut( ) automorphism group of a point arrangement, graph, polytope, etc. · GP edge-graph of the polytope P d skP skeleton of P, a graph realization skP : V (GP ) R of the edge-graph F(P) face lattice of the polytope P ! Fδ(P) set of δ-dimensional faces of the polytope P Zon(R) zonotope generated by R Gen(Z) (standard) generators of the zonotope Z Sym(V ) symmetric group on V , i.e., the group of permutations of the set V n n Perm(R ) group of permutation matrices on R Πσ permutation matrix associated to the permutation σ id identity permutation d Ad (Σ) space of full-dimensional Σ-arrangements in R d Ad (G, Σ) space of full-dimensional Σ-realizations of G in R Spec(G) (adjacency) spectrum of of the graph G EigG(θ) θ-eigenspace of the (adjacency matrix of the) graph G PG(θ) θ-eigenpolytope of the graph G θ eigG θ-realization used to construct the θ-eigenpolytope x Euclidean norm of a vector x k k d x, y standard inner product in R h i Ý(x, y) angle between the vectors x and y d d 1 S d-dimensional sphere (embedded in R + ) 11 List of Symbols Sr (c) sphere with center c and radius r (of appropriate dimension) d vol (relative) volume of a subset of R 12 Introduction The product of mathematics is clarity and understanding. Not theorems, by themselves. William Thurston The content of this thesis was developed in an effort to understand a series of curious ob- servations made while working on so-called spectral graph realizations. Those are a technique to embed a graph (a purely combinatorial object) into Euclidean space by using the eigenval- ues and eigenvectors of its adjacency matrix. Applying this to the edge-graphs of polytopes results in some cases in an embedding that looks just like the skeleton of the original poly- tope. This phenomenon points to connections between polytope theory and spectral graph theory, between convexity and rigidity, between symmetry in geometry and combinatorics, all to be explored in this thesis. A curious observation The best way to motivate the investigation of the described phenomenon is to see it happen. Consider the 3-dimensional cube and let G = (V, E) be its edge-graph with vertex set V = 1, ..., 8 , numbers assigned to the vertices according to the following figure: f g 8 8 To G we can assign a matrix A 0, 1 × , the so-called adjacency matrix, with Ai j = 1 if and only if i j E is an edge.
Load more

Recommended publications
  • Systematic Narcissism
    Systematic Narcissism Wendy W. Fok University of Houston Systematic Narcissism is a hybridization through the notion of the plane- tary grid system and the symmetry of narcissism. The installation is based on the obsession of the physical reflection of the object/subject relation- ship, created by the idea of man-made versus planetary projected planes (grids) which humans have made onto the earth. ie: Pierre Charles L’Enfant plan 1791 of the golden section projected onto Washington DC. The fostered form is found based on a triacontahedron primitive and devel- oped according to the primitive angles. Every angle has an increment of 0.25m and is based on a system of derivatives. As indicated, the geometry of the world (or the globe) is based off the planetary grid system, which, according to Bethe Hagens and William S. Becker is a hexakis icosahedron grid system. The formal mathematical aspects of the installation are cre- ated by the offset of that geometric form. The base of the planetary grid and the narcissus reflection of the modular pieces intersect the geometry of the interfacing modular that structures the installation BACKGROUND: Poetics: Narcissism is a fixation with oneself. The Greek myth of Narcissus depicted a prince who sat by a reflective pond for days, self-absorbed by his own beauty. Mathematics: In 1978, Professors William Becker and Bethe Hagens extended the Russian model, inspired by Buckminster Fuller’s geodesic dome, into a grid based on the rhombic triacontahedron, the dual of the icosidodeca- heron Archimedean solid. The triacontahedron has 30 diamond-shaped faces, and possesses the combined vertices of the icosahedron and the dodecahedron.
    [Show full text]
  • On the Archimedean Or Semiregular Polyhedra
    ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces.
    [Show full text]
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
    [Show full text]
  • E8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron Vixra 1301.0150 Frank Dodd (Tony) Smith Jr
    E8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron vixra 1301.0150 Frank Dodd (Tony) Smith Jr. - 2013 The E8 Physics Model (viXra 1108.0027) is based on the Lie Algebra E8. 240 E8 vertices = 112 D8 vertices + 128 D8 half-spinors where D8 is the bivector Lie Algebra of the Real Clifford Algebra Cl(16) = Cl(8)xCl(8). 112 D8 vertices = (24 D4 + 24 D4) = 48 vertices from the D4xD4 subalgebra of D8 plus 64 = 8x8 vertices from the coset space D8 / D4xD4. 128 D8 half-spinor vertices = 64 ++half-half-spinors + 64 --half-half-spinors. An 8-dim Octonionic Spacetime comes from the Cl(8) factors of Cl(16) and a 4+4 = 8-dim Kaluza-Klein M4 x CP2 Spacetime emerges due to the freezing out of a preferred Quaternionic Subspace. Interpreting World-Lines as Strings leads to 26-dim Bosonic String Theory in which 10 dimensions reduce to 4-dim CP2 and a 6-dim Conformal Spacetime from which 4-dim M4 Physical Spacetime emerges. Although the high-dimensional E8 structures are fundamental to the E8 Physics Model it may be useful to see the structures from the point of view of the familiar 3-dim Space where we live. To do that, start by looking the the E8 Root Vector lattice. Algebraically, an E8 lattice corresponds to an Octonion Integral Domain. There are 7 Independent E8 Lattice Octonion Integral Domains corresponding to the 7 Octonion Imaginaries, as described by H. S. M. Coxeter in "Integral Cayley Numbers" (Duke Math. J. 13 (1946) 561-578 and in "Regular and Semi-Regular Polytopes III" (Math.
    [Show full text]
  • Some New General Lower Bounds for Mixed Metric Dimension of Graphs
    Some new general lower bounds for mixed metric dimension of graphs Milica Milivojevi´cDanasa, Jozef Kraticab, Aleksandar Savi´cc, Zoran Lj. Maksimovi´cd, aFaculty of Science and Mathematics, University of Kragujevac, Radoja Domanovi´ca 12, Kragujevac, Serbia bMathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11 000 Belgrade, Serbia cFaculty of Mathematics, University of Belgrade, Studentski trg 16/IV, 11000 Belgrade, Serbia dUniversity of Defence, Military Academy, Generala Pavla Juriˇsi´ca Sturmaˇ 33, 11000 Belgrade, Serbia Abstract Let G = (V, E) be a connected simple graph. The distance d(u, v) between vertices u and v from V is the number of edges in the shortest u − v path. If e = uv ∈ E is an edge in G than distance d(w, e) where w is some vertex in G is defined as d(w, e) = min(d(w,u),d(w, v)). Now we can say that vertex w ∈ V resolves two elements x, y ∈ V ∪ E if d(w, x) =6 d(w,y). The mixed resolving set is a set of vertices S, S ⊆ V if and only if any two elements of E ∪ V are resolved by some element of S. A minimal resolving set related to inclusion is called mixed resolving basis, and its cardinality is called the mixed metric dimension of a graph G. This graph invariant is recently introduced and it is of interest to find its general properties and determine its values for various classes of graphs. Since arXiv:2007.05808v1 [math.CO] 11 Jul 2020 the problem of finding mixed metric dimension is a minimization problem, of interest is also to find lower bounds of good quality.
    [Show full text]
  • Distances, Diameter, Girth, and Odd Girth in Generalized Johnson Graphs
    Distances, Diameter, Girth, and Odd Girth in Generalized Johnson Graphs Ari J. Herman Dept. of Mathematics & Statistics Portland State University, Portland, OR, USA Mathematical Literature and Problems project in partial fulfillment of requirements for the Masters of Science in Mathematics Under the direction of Dr. John Caughman with second reader Dr. Derek Garton Abstract Let v > k > i be non-negative integers. The generalized Johnson graph, J(v; k; i), is the graph whose vertices are the k-subsets of a v-set, where vertices A and B are adjacent whenever jA \ Bj = i. In this project, we present the results of the paper \On the girth and diameter of generalized Johnson graphs," by Agong, Amarra, Caughman, Herman, and Terada [1], along with a number of related additional results. In particular, we derive general formulas for the girth, diameter, and odd girth of J(v; k; i). Furthermore, we provide a formula for the distance between any two vertices A and B in terms of the cardinality of their intersection. We close with a number of possible future directions. 2 1. Introduction In this project, we present the results of the paper \On the girth and diameter of generalized Johnson graphs," by Agong, Amarra, Caughman, Herman, and Terada [1], along with a number of related additional results. Let v > k > i be non-negative integers. The generalized Johnson graph, X = J(v; k; i), is the graph whose vertices are the k-subsets of a v-set, with adjacency defined by A ∼ B , jA \ Bj = i: (1) Generalized Johnson graphs were introduced by Chen and Lih in [4].
    [Show full text]
  • Results on Domination Number and Bondage Number for Some Families of Graphs
    International Journal of Pure and Applied Mathematics Volume 107 No. 3 2016, 565-577 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v107i3.5 ijpam.eu RESULTS ON DOMINATION NUMBER AND BONDAGE NUMBER FOR SOME FAMILIES OF GRAPHS G. Hemalatha1, P. Jeyanthi2 § 1Department of Mathematics Shri Andal Alagar College of Engineering Mamandur, Kancheepuram, Tamil Nadu, INDIA 2Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur 628 215, Tamil Nadu, INDIA Abstract: Let G = (V, E) be a simple graph on the vertex set V . In a graph G, A set S ⊆ V is a dominating set of G if every vertex in VS is adjacent to some vertex in S. The domination number of a graph Gγ(G)] is the minimum size of a dominating set of vertices in G. The bondage number of a graph G[Bdγ(G)] is the cardinality of a smallest set of edges whose removal results in a graph with domination number larger than that of G. In this paper we establish domination number and the bondage number for some families of graphs. AMS Subject Classification: 05C69 Key Words: dominating set, bondage number, domination number, cocktail party graph, coxeter graph, crown graph, cubic symmetric graph, doyle graph, folkman graph, levi graph, icosahedral graph 1. Introduction The domination in graphs is one of the major areas in graph theory which attracts many researchers because it has the potential to solve many real life problems involving design and analysis of communication network, social net- Received: October 17, 2015 c 2016 Academic Publications, Ltd.
    [Show full text]
  • Properties of Codes in the Johnson Scheme
    Properties of Codes in the Johnson Scheme Natalia Silberstein Properties of Codes in the Johnson Scheme Research Thesis In Partial Fulfillment of the Requirements for the Degree of Master of Science in Applied Mathematics Natalia Silberstein Submitted to the Senate of the Technion - Israel Institute of Technology Shvat 5767 Haifa February 2007 The Research Thesis Was Done Under the Supervision of Professor Tuvi Etzion in the Faculty of Mathematics I wish to express my sincere gratitude to my supervisor, Prof. Tuvi Etzion for his guidance and kind support. I would also like to thank my family and my friends for thear support. The Generous Financial Help Of The Technion and Israeli Science Foundation Is Gratefully Acknowledged. Contents Abstract 1 List of symbols and abbreviations 3 1 Introduction 4 1.1 Definitions.................................. 5 1.1.1 Blockdesigns............................ 5 1.2 PerfectcodesintheHammingmetric. .. 7 1.3 Perfect codes in the Johnson metric (survey of known results)....... 8 1.4 Organizationofthiswork. 13 2 Perfect codes in J(n, w) 15 2.1 t-designs and codes in J(n, w) ....................... 15 2.1.1 Divisibility conditions for 1-perfect codes in J(n, w) ....... 17 2.1.2 Improvement of Roos’ bound for 1-perfect codes . 20 2.1.3 Number theory’s constraints for size of Φ1(n, w) ......... 24 2.2 Moments .................................. 25 2.2.1 Introduction............................. 25 2.2.1.1 Configurationdistribution . 25 2.2.1.2 Moments. ........................ 26 2.2.2 Binomial moments for 1-perfect codes in J(n, w) ........ 27 2.2.2.1 Applicationsof Binomial momentsfor 1-perfect codes in J(n, w) .......................
    [Show full text]
  • Binary Icosahedral Group and 600-Cell
    Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons.
    [Show full text]
  • Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions
    SQU Journal for Science, 16 (2011) 82-101 © 2011 Sultan Qaboos University Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions Mehmet Koca, Nazife Ozdes Koca* and Muna Al-Shueili Department of Physics, College of Science, Sultan Qaboos University, P.O.Box 36, Al- Khoud, 123 Muscat, Sultanate of Oman, *Email: [email protected]. متعددات السطوح اليدوية المستمدة من أشكال كوكزتر والرباعيات مهمت كوجا، نزيفة كوجا و منى الشعيلي ملخص: هناك متعددات أسطح ٌدوٌان أرخمٌدٌان: المكعب المسطوح والثنعشري اﻷسطح المسطوح مع أشكالها الكتﻻنٌة المزدوجة: رباعٌات السطوح العشرونٌة الخماسٌة مع رباعٌات السطوح السداسٌة الخماسٌة. فً هذا البحث نقوم برسم متعددات السطوح الٌدوٌة مع مزدوجاتها بطرٌقة منظمة. نقوم باستخدام المجموعة الجزئٌة الدورانٌة الحقٌقٌة لمجموعات كوكستر ( WAAAWAWB(1 1 1 ), ( 3 ), ( 3 و WH()3 ﻻشتقاق المدارات الممثلة لﻷشكال. هذه الطرٌقة تقودنا إلى رسم المتعددات اﻷسطح التالٌة: رباعً الوجوه وعشرونً الوجوه و المكعب المسطوح و الثنعشري السطوح المسطوح على التوالً. لقد أثبتنا أنه بإمكان رباعً الوجوه وعشرونً الوجوه التحول إلى صورتها المرآتٌة بالمجموعة الثمانٌة الدورانٌة الحقٌقٌة WBC()/32. لذلك ﻻ ٌمكن تصنٌفها كمتعددات أسطح ٌدوٌة. من المﻻحظ أٌضا أن رؤوس المكعب المسطوح ورؤوس الثنعشري السطوح المسطوح ٌمكن اشتقاقها من المتجهات المجموعة خطٌا من الجذور البسٌطة وذلك باستخدام مجموعة الدوران الحقٌقٌة WBC()/32 و WHC()/32 على التوالً. من الممكن رسم مزدوجاتها بجمع ثﻻثة مدارات من المجموعة قٌد اﻻهتمام. أٌضا نقوم بإنشاء متعددات اﻷسطح الشبه منتظمة بشكل عام بربط متعددات اﻷسطح الٌدوٌة مع صورها المرآتٌة. كنتٌجة لذلك نحصل على مجموعة البٌروهٌدرال كمجموعة جزئٌة من مجموعة الكوكزتر WH()3 ونناقشها فً البحث. الطرٌقة التً نستخدمها تعتمد على الرباعٌات. ABSTRACT: There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron.
    [Show full text]
  • TETRAVALENT SYMMETRIC GRAPHS of ORDER 9P
    J. Korean Math. Soc. 49 (2012), No. 6, pp. 1111–1121 http://dx.doi.org/10.4134/JKMS.2012.49.6.1111 TETRAVALENT SYMMETRIC GRAPHS OF ORDER 9p Song-Tao Guo and Yan-Quan Feng Abstract. A graph is symmetric if its automorphism group acts transi- tively on the set of arcs of the graph. In this paper, we classify tetravalent symmetric graphs of order 9p for each prime p. 1. Introduction Let G be a permutation group on a set Ω and α ∈ Ω. Denote by Gα the stabilizer of α in G, that is, the subgroup of G fixing the point α. We say that G is semiregular on Ω if Gα = 1 for every α ∈ Ω and regular if G is transitive and semiregular. Throughout this paper, we consider undirected finite connected graphs without loops or multiple edges. For a graph X we use V (X), E(X) and Aut(X) to denote its vertex set, edge set, and automorphism group, respectively. For u , v ∈ V (X), denote by {u , v} the edge incident to u and v in X. A graph X is said to be vertex-transitive if Aut(X) acts transitively on V (X). An s-arc in a graph is an ordered (s + 1)-tuple (v0, v1,...,vs−1, vs) of vertices of the graph X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s, and vi−1 = vi+1 for 1 ≤ i ≤ s − 1. In particular, a 1-arc is called an arc for short and a 0-arc is a vertex.
    [Show full text]
  • Constructing Arbitrarily Large Graphs with a Specified Number Of
    Electronic Journal of Graph Theory and Applications 4 (1) (2019), 1–8 Constructing Arbitrarily Large Graphs with a Specified Number of Hamiltonian Cycles Michael Haythorpea aSchool of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA 5042, Australia michael.haythorpe@flinders.edu.au Abstract A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing 5n − 9 vertices and k Hamiltonian cycles for any choice of integers n ≥ k ≥ 4. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when k is chosen to be much smaller than n. Keywords: Hamiltonian cycles, Graph Construction, Broken Crown Mathematics Subject Classification : 05C45 1. Introduction The Hamiltonian cycle problem (HCP) is a famous NP-complete problem in which one must determine whether a given graph contains a simple cycle traversing all vertices of the graph, or not. Such a simple cycle is called a Hamiltonian cycle (HC), and a graph containing at least one arXiv:1902.10351v1 [math.CO] 27 Feb 2019 Hamiltonian cycle is said to be a Hamiltonian graph. Typically, randomly generated graphs (such as Erdos-R˝ enyi´ graphs), if connected, are Hamil- tonian and contain many Hamiltonian cycles. Although HCP is an NP-complete problem, for these graphs it is often fairly easy for a sophisticated heuristic (e.g. see Concorde [1], Keld Helsgaun’s LKH [4] or Snakes-and-ladders Heuristic [2]) to discover one of the multitude of Hamiltonian cy- cles through a clever search.
    [Show full text]