DOCSLIB.ORG

Platonic Hypermaps \Noindentopen Square2 Bracket\Quad 27A Closing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Platonic Hypermaps \Noindentopen Square2 Bracket\Quad 27A Closing Beitr a-dieresis ge zur Algebra und Geometrie nnoindentContributionsB e i tot r Algebra $ nddot andf Geometryag $ ge zur Algebra und Geometrie Volume 42 open parenthesis 2001 closing parenthesis comma No period .. 1 comma 1 hyphen 3 7 period nnoindentBeitr a¨ Contributionsge zur Algebra und to Algebra Geometrie and Geometry PlatonicContributions .. Hypermaps to Algebra and Geometry Volume 42 ( 2001 ) , No . 1 , 1 - 3 7 . VolumeAntonio 42 J period ( 2001 .. Breda ) , Nod quoteright . nquad Azevedo1 , 1 to− the3 7power . of 1 .. Gareth A period Jones Departamento de Matem acute-aPlatonic tica comma .. Universidade Hypermaps de Aveiro n centerline f P l a t o n i c nquad Hypermaps g 3800 Aveiro comma PortugalAntonio J . Breda d ' Azevedo1 Gareth A . Jones Department of MathematicsDepartamento comma .. de University Matem of Southamptona´ tica , Universidade de Aveiro n centerlineSouthamptonf Antonio SO 1 7 1 BJ J .commanquad .. UnitedBreda3800 Kingdom d ' Aveiro $ Azevedo , Portugal ^f 1 g$ nquad Gareth A . Jones g Abstract period .. WeDepartment classify the regular of Mathematics hypermaps open , parenthesis University orientable of Southampton or non hyphen orientable closing parenthesis whose n centerlinefull automorphismfDepartamento group is isomorphicSouthampton de Matem to the $SO symmetrynacute 1 7 1f BJ groupag ,$ of t aUnited ic Platonic a , n Kingdomquad solid periodUniversidade de Aveiro g ThereAbstract are 185 of . them commaWe classify of which the 93 regular are maps hypermaps period .. We ( also orientable classify the or nonregular - orientable hyper hyphen ) whose n centerlinemaps withfull automorphismf3800 automorphism Aveiro group , group Portugal A sub is 5 isomorphic: ..g there are to19of the these symmetry comma all group non hyphen of a Platonic orientable solid comma . and 9 of themThere are maps are period 185 of .. them These , hypermaps of which are 93 constructed are maps . as combinatorial We also classify and topologi the regular hyphen hyper - n centerline fDepartment of Mathematics , nquad University of Southampton g cal objmaps ects comma with automorphism many of them arising group asA coverings5 : there of Platonic are 19 so of lids these and , Kepler all non hyphen - orientable Poinsot , and 9 polyhedraof them open are parenthesis maps . viewed These as hypermaps hypermaps closing are parenthesis constructed comma as combinatorial or their associates and period topologi .. We - conclude by showing that n centerlineany rotarycal obj PfSouthampton latonic ects , hypermap many ofSO i them s 1 regular 7 arising1 BJ comma , asnquad so coverings thereUnited are of no Platonic chiral Kingdom Platonic sog lids hypermaps and Kepler period - Poinsot 1 periodpolyhedra .. Introduction ( viewed as hypermaps ) , or their associates . We conclude by showing that n centerlineThe convexany rotaryf polyhedraAbstract P latonic in R. tonquad hypermapthe powerWe classify of i 3 s with regular the the most , so regular interesting there are hypermaps symmetry no chiral properties (Platonic orientable are hypermaps the orPlatonic non . − orientable ) whose g so1 lids . : ..Introduction these are the t etrahedron T comma the cube C comma the octahedron O comma the dodecahedron D comma and n centerline f full automorphism group is isomorphic to the symmetry group of a Platonic solid . g theThe icosahedron convex polyhedra I comma described in R3 inwith Plato the quoteright most interesting s dialogue .. symmetry Timaeos open properties square bracket are the 25 closing Platonic square bracket period .. The rotation groupsso of lids T comma : these of are the t etrahedron T , the cube C , the octahedron O , the dodecahedron D , n centerlineCand .. and the O commaicosahedronfThere .. are and ofI 185 , D described and of I them are isomorphicin , Plato of which ' to s dialogue the 93 alternating are mapsTimaeos and . symmetricnquad[ 25 ]We .groups also The A subclassify rotation 4 comma groups the S sub regular 4 hyper − g andof AT sub , of 5 C semicolon and these O , are and subgroups of D and of index I are 2 isomorphic in the i sometry to the groups alternating of these solids and comma symmetric i somorphic groups to n centerline fmaps with automorphism group $ A f 5 g : $ nquad there are 19 of these , all non − orientable , and 9 g SA sub4;S 4 comma4 S sub 4 times C sub 2 and A sub 5 times C sub 2 period Eachand PlatonicA5; these solid are P can subgroups be regarded of index as a map 2 in comma the i that sometry i s comma groups as a of graph these imbedded solids , in i asomorphic surface to n centerlineopen parenthesisf of them in this are case maps the sphere . nquad closingThese parenthesis hypermaps semicolon are .. constructed more generally as comma combinatorial P can be regarded and topologias a hypermap− g .. open parenthesis a hypergraph n centerline f cal obj ects , many of them arising as coverings of Platonic so lids and Kepler − Poinsot g imbedded in a surface closing parenthesis commaS4;S ..4 and× C within2andA5 either× C2: category P i s a regular obj ect comma .. in Vince quoteright s sense hline n centerline1 subEach The authorf Platonicpolyhedra is grateful solid ( toP viewed canthe Research be as regarded hypermaps and Development as a map ) , or, Joint that their Research i s , associates as aCentre graph of imbedded the . n Commissionquad We in a conclude of surface the by showing that g European( in this Communities case the sphere for financially ) ; more supporting generally this , research P can be period regarded as a hypermap ( a hypergraph n centerline0imbedded 1 38 hyphenfany in 482 a rotary 1 surface slash 93 P ) dollar , latonic and2 period within hypermap 50 circlecopyrt-c either i s category regular 200 1Heldermann P , i s so a regularthere Verlag are obj no ect chiral , in Vince Platonic ' s hypermaps . g sense nnoindent 1 . nquad Introduction nnoindent The convex polyhedra in $ R ^f 3 g$ with the most interesting symmetry properties are the Platonic nnoindent 1soThe author l i d s : isn gratefulquad these to the Research are the and t Development etrahedron Joint T , Research the cube Centre Cof , the Commission octahedron of the O , the dodecahedron D , and theEuropean icosahedron Communities I , describedfor financially in supporting Plato this' s research dialogue . nquad Timaeos [ 25 ] . nquad The rotation groups of T , of C n0138quad− 4821and= O93$2 , :50nquadcirclecopyrtand of− c D2001 andHeldermann I are isomorphic Verlag to the alternating and symmetric groups $ A f 4 g , S f 4 g$ nnoindent and $ A f 5 g ; $ these are subgroups of index 2 in the i sometry groups of these solids , i somorphic to n begin f a l i g n ∗g S f 4 g ,S f 4 g ntimes C f 2 g and A f 5 g ntimes C f 2 g . nendf a l i g n ∗g n hspace ∗fn f i l l gEach Platonic solid P can be regarded as a map , that i s , as a graph imbedded in a surface nnoindent ( in this case the sphere ) ; nquad more generally , P can be regarded as a hypermap nquad ( a hypergraph imbedded in a surface ) , nquad and within either category P i s a regular obj ect , nquad in Vince ' s sense n begin f a l i g n ∗g n r u l e f3emgf0.4 pt g nendf a l i g n ∗g n hspace ∗fn f i l l g $ 1 f The g$ author is grateful to the Research and Development Joint Research Centre of the Commission of the nnoindent European Communities for financially supporting this research . nnoindent $ 0 1 38 − 482 1 / 93 n$ 2 . 50 circlecopyrt −c 200 1 $ Heldermann Verlag 2 .. A period J period Breda d quoteright Azevedo semicolon G period A period Jones : .. Platonic Hypermaps nnoindentopen square2 bracketnquad 27A closing . J . square Breda bracket d ' Azevedothat the automorphism ; G . A . group Jones Aut : Pnquad acts transitivelyPlatonic on Hypermaps the .. quotedblleft blades quotedblright .. out of2 which A .P J is . Breda d ' Azevedo ; G . A . Jones : Platonic Hypermaps nnoindentconstructed[ period 27 ] that the automorphism group Aut P acts transitively on the nquad ` ` blades ' ' nquad out of which P is constructed[ 27 ] that the . automorphism group Aut P acts transitively on the \ blades " out of which P Ourisaim constructed in this paper . is to classify the regular Platonic hypermaps comma the regular hypermaps H .. open parenthesis orientable .. or .. non hyphen orientable closing parenthesis .. whose .. automorphism .. group .. Aut H .. is .. i somorphic Our aimOur in aim this in paper this paper is to is to classify classify theregular regular Platonic Platonic hypermaps hypermaps, the , the regular regular hypermaps hypermaps .. toH .. the ( orientable or non - orientable ) whose automorphism group Aut H is i H automorphismnquad ( orientable group .. G ..nquad = .. Autor Pn ..quad thicksimnon =− Sorientable sub 4 comma S ) subnquad 4 timeswhose C subn 2quad .. or Aautomorphism sub 5 times C subnquad 2 .. ofgroup some Platonicnquad solidAut H nquad i s nquad i somorphic nquad to nquad the somorphic to the automorphism group G = Aut P ∼= S ;S × C or A × C Pautomorphism semicolon .. for group nquad G nquad $ = $ nquad Aut P nquad $ nsim4 =4 S2 f 4 g5 ,S2 f 4 g ntimes C f 2 g$ of some Platonic solid P ; for t echnical reasons in dealing with A × C it is also useful for us nquadt echnicalor reasons$ A f in5 dealingg ntimes with A subC 5 timesf 2 Cg$ subn 2quad it is alsoof usefulsome for Platonic us to include5 solid2 the P case ; Gnquad = subf thicksim o r A sub 5 period to include the case G = A : The number of G - hypermaps ( regular hypermaps H tThe echnical .. number reasons .. of G hyphen in dealing∼ hypermaps5 with .. open $ Aparenthesisf 5 g regular ntimes .. hypermapsC f 2 .. Hg$ .. with it is.
Load more

Recommended publications
  • Automorphisms of the Double Cover of a Circulant Graph of Valency at Most 7
    AUTOMORPHISMS OF THE DOUBLE COVER OF A CIRCULANT GRAPH OF VALENCY AT MOST 7 ADEMIR HUJDUROVIC,´ ¯DOR¯DE MITROVIC,´ AND DAVE WITTE MORRIS Abstract. A graph X is said to be unstable if the direct product X × K2 (also called the canonical double cover of X) has automorphisms that do not come from automorphisms of its factors X and K2. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most 7. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, because there is a nontrivially unstable circulant graph of valency 8 that does not satisfy the hypotheses of any of Wilson's four instability theorems for circulant graphs. 1. Introduction Let X be a circulant graph. (All graphs in this paper are finite, simple, and undirected.) Definition 1.1 ([16]). The canonical bipartite double cover of X is the bipartite graph BX with V (BX) = V (X) 0; 1 , where × f g (v; 0) is adjacent to (w; 1) in BX v is adjacent to w in X: () Letting S2 be the symmetric group on the 2-element set 0; 1 , it is clear f g that the direct product Aut X S2 is a subgroup of Aut BX. We are interested in cases where this subgroup× is proper: Definition 1.2 ([12, p.
    [Show full text]
  • Graph Operations and Upper Bounds on Graph Homomorphism Counts
    Graph Operations and Upper Bounds on Graph Homomorphism Counts Luke Sernau March 9, 2017 Abstract We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n-vertex, d-regular graph G and any graph H (possibly with loops), n n d d hom(G, H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 , n o where hom(G, H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G, H) holds for all n-vertex, d-regular G. In particular we show that if HWR is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then n d hom(G, HWR) ≤ hom(Kd+1,HWR) +1 for all n-vertex, d-regular G. This verifies a conjecture of Galvin. arXiv:1510.01833v3 [math.CO] 8 Mar 2017 1 Introduction Graph homomorphisms are an important concept in many areas of graph theory. A graph homomorphism is simply an adjacency-preserving map be- tween a graph G and a graph H. That is, for given graphs G and H, a function φ : V (G) → V (H) is said to be a homomorphism from G to H if for every edge uv ∈ E(G), we have φ(u)φ(v) ∈ E(H) (Here, as throughout, all graphs are simple, meaning without multi-edges, but they are permitted 1 to have loops).
    [Show full text]
  • Lombardi Drawings of Graphs 1 Introduction
    Lombardi Drawings of Graphs Christian A. Duncan1, David Eppstein2, Michael T. Goodrich2, Stephen G. Kobourov3, and Martin Nollenburg¨ 2 1Department of Computer Science, Louisiana Tech. Univ., Ruston, Louisiana, USA 2Department of Computer Science, University of California, Irvine, California, USA 3Department of Computer Science, University of Arizona, Tucson, Arizona, USA Abstract. We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. 1 Introduction The American artist Mark Lombardi [24] was famous for his drawings of social net- works representing conspiracy theories. Lombardi used curved arcs to represent edges, leading to a strong aesthetic quality and high readability. Inspired by this work, we intro- duce the notion of a Lombardi drawing of a graph, in which edges are drawn as circular arcs with perfect angular resolution: consecutive edges are evenly spaced around each vertex. While not all vertices have perfect angular resolution in Lombardi’s work, the even spacing of edges around vertices is clearly one of his aesthetic criteria; see Fig. 1. Traditional graph drawing methods rarely guarantee perfect angular resolution, but poor edge distribution can nevertheless lead to unreadable drawings. Additionally, while some tools provide options to draw edges as curves, most rely on straight-line edges, and it is known that maintaining good angular resolution can result in exponential draw- ing area for straight-line drawings of planar graphs [17,25].
    [Show full text]
  • The Operations Invariant Properties on Graphs Yanzhong Hu, Gang
    International Conference on Education Technology and Information System (ICETIS 2013) The Operations Invariant Properties on Graphs Yanzhong Hu, Gang Cheng School of Computer Science, Hubei University of Technology, Wuhan, 430068, China [email protected], [email protected] Keywords：Invariant property; Hamilton cycle; Cartesian product; Tensor product Abstract. To determine whether or not a given graph has a Hamilton cycle (or is a planar graph), defined the operations invariant properties on graphs, and discussed the various forms of the invariant properties under the circumstance of Cartesian product graph operation and Tensor product graph operation. The main conclusions include: The Hamiltonicity of graph is invariant concerning the Cartesian product, and the non-planarity of the graph is invariant concerning the tensor product. Therefore, when we applied these principles into practice, we testified that Hamilton cycle does exist in hypercube and the Desargues graph is a non-planarity graph. INTRODUCTION The Planarity, Eulerian feature, Hamiltonicity, bipartite property, spectrum, and so on, is often overlooked for a given graph. The traditional research method adopts the idea of direct proof. In recent years, indirect proof method is put forward, (Fang Xie & Yanzhong Hu, 2010) proves non- planarity of the Petersen graph, by use the non-planarity of K3,3, i.e., it induces the properties of the graph G with the properties of the graph H, here H is a sub-graph of G. However, when it comes to induce the other properties, the result is contrary to what we expect. For example, Fig. 1(a) is a Hamiltonian graph which is a sub-graph of Fig.
    [Show full text]
  • Not Every Bipartite Double Cover Is Canonical
    Volume 82 BULLETIN of the February 2018 INSTITUTE of COMBINATORICS and its APPLICATIONS Editors-in-Chief: Marco Buratti, Donald Kreher, Tran van Trung Boca Raton, FL, U.S.A. ISSN1182-1278 BULLETIN OF THE ICA Volume 82 (2018), Pages 51{55 Not every bipartite double cover is canonical Tomaˇz Pisanski University of Primorska, FAMNIT, Koper, Slovenia and IMFM, Ljubljana, Slovenia. [email protected]. Abstract: It is explained why the term bipartite double cover should not be used to designate canonical double cover alias Kronecker cover of graphs. Keywords: Kronecker cover, canonical double cover, bipartite double cover, covering graphs. Math. Subj. Class. (2010): 05C76, 05C10. It is not uncommon to use different terminology or notation for the same mathematical concept. There are several reasons for such a phenomenon. Authors independently discover the same object and have to name it. It is• quite natural that they choose different names. Sometimes the same concept appears in different mathematical schools or in• different disciplines. By using particular terminology in a given context makes understanding of such a concept much easier. Sometimes original terminology is not well-chosen, not intuitive and it is difficult• to relate the name of the object to its meaning. A name that is more appropriate for a given concept is preferred. One would expect terminology to be as simple as possible, and as easy to connect it to the concept as possible. However, it should not be too simple. Received: 8 October 2018 51 Accepted: 21 January 2018 In other words it should not introduce ambiguity. Unfortunately, the term bipartite double cover that has been used lately in several places to replace an older term canonical double cover, also known as the Kronecker cover, [1,4], is ambiguous.
    [Show full text]
  • Nearly Tight Approximability Results for Minimum Biclique Cover and Partition
    Nearly Tight Approximability Results for Minimum Biclique Cover and Partition Parinya Chalermsook1, Sandy Heydrich1;3, Eugenia Holm2, and Andreas Karrenbauer1∗ 1 Max Planck Institute for Informatics, Saarbr¨ucken, Germany 2 Dept. for Computer and Information Science, University of Konstanz, Germany 3 Saarbr¨ucken Graduate School of Computer Science {andreas.karrenbauer,parinya.chalermsook,sandy.heydrich}@mpi-inf.mpg.de [email protected] Abstract. In this paper, we consider the minimum biclique cover and minimum biclique partition problems on bipartite graphs. In the min- imum biclique cover problem, we are given an input bipartite graph G = (V; E), and our goal is to compute the minimum number of complete bipartite subgraphs that cover all edges of G. This problem, besides its correspondence to a well-studied notion of bipartite dimension in graph theory, has applications in many other research areas such as artificial intelligence, computer security, automata theory, and biology. Since it is NP-hard, past research has focused on approximation algorithms, fixed parameter tractability, and special graph classes that admit polynomial time exact algorithms. For the minimum biclique partition problem, we are interested in a biclique cover that covers each edge exactly once. We revisit the problems from approximation algorithms' perspectives and give nearly tight lower and upper bound results. We first show that both problems are NP-hard to approximate to within a factor of n1−" (where n is the number of vertices in the input graph). Using a stronger complexity assumption, the hardness becomes Ω~(n), where Ω~(·) hides lower order terms. Then we show that approximation factors of the form n=(log n)γ for some γ > 0 can be obtained.
    [Show full text]
  • Enlarging Properties of Graphs
    Enlarging Properties of Graphs Alan Geoffrey Boshier PhD Thesis, Royal Holloway & Bedford New College, University of London. gv^A y COLLEGE UBRARY O ProQuest Number: 10096264 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10096264 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Abstract The subjects of this Thesis are the enlarging and magnifying properties of graphs. Upper bounds for the isoperimetric number i{G) of a graph G are determined with respect to such elementary graph properties as order, valency, and the number of three and four- cycles. The relationship between z(G) and the genus of G is studied in detail, and a class of graphs called finite element graphs is shown never to supply enlarging families. The magnifying properties of Hamiltonian cubic graphs are investigated, and a class of graphs known as shift graphs is defined. These are shown never to form enlarging families, using a technical lemma derived from Klawe’s Theorem on non-expanding families of graphs. The same lemma is used, in conjunction with some elementary character theory, to prove that several important classes of Cayley graphs do not form enlarging families, and to derive a lower bound on the subdominant eigenvalue of a vertex-transitive graph.
    [Show full text]
  • Bidirected Graph from Wikipedia, the Free Encyclopedia Contents
    Bidirected graph From Wikipedia, the free encyclopedia Contents 1 Bidirected graph 1 1.1 Other meanings ............................................ 1 1.2 See also ................................................ 2 1.3 References ............................................... 2 2 Bipartite double cover 3 2.1 Construction .............................................. 3 2.2 Examples ............................................... 3 2.3 Matrix interpretation ......................................... 4 2.4 Properties ............................................... 4 2.5 Other double covers .......................................... 4 2.6 See also ................................................ 5 2.7 Notes ................................................. 5 2.8 References ............................................... 5 2.9 External links ............................................. 6 3 Complex question 7 3.1 Implication by question ........................................ 7 3.2 Complex question fallacy ....................................... 7 3.2.1 Similar questions and fallacies ................................ 8 3.3 Notes ................................................. 8 4 Directed graph 10 4.1 Basic terminology ........................................... 11 4.2 Indegree and outdegree ........................................ 11 4.3 Degree sequence ............................................ 12 4.4 Digraph connectivity .......................................... 12 4.5 Classes of digraphs .........................................
    [Show full text]
  • Totally Silver Graphs
    Totally Silver Graphs M. Ghebleha, E. S. Mahmoodianb aDepartment of Mathematics, Faculty of Science, Kuwait University, State of Kuwait bDepartment of Mathematical Sciences, Sharif University of Technology, Tehran, Iran Abstract A totally silver coloring of a graph G is a k{coloring of G such that for every vertex v 2 V (G), each color appears exactly once on N[v], the closed neighborhood of v.A totally silver graph is a graph which admits a totally silver coloring. Totally silver coloring are directly related to other areas of graph theory such as distance coloring and domination. In this work, we present several constructive characterizations of totally silver graphs and bipartite totally silver graphs. We give several infinite families of totally silver graphs. We also give cubic totally silver graphs of girth up to 10. Keywords: Graph coloring, distance coloring, silver coloring, domination. 1. Introduction All graphs in this paper are finite and simple. A totally silver coloring of a graph G is a k{coloring of G such that for every vertex v 2 V (G), each color appears exactly once on N[v], the closed neighborhood of v.A totally silver graph is a graph which admits a totally silver coloring. It is immediate from this definition that every graph admitting a totally silver coloring with k colors is (k − 1){regular. Totally silver colorings are introduced in [4]. The term totally silver is inspired by an International Mathematical Olympiad problem on silver matrices [6]. Totally silver colorings appear, under the name perfect colorings, in the study of file seg- mentations in a network in [1].
    [Show full text]
  • An Analysis of the Properties of Various Interconnection Networks K
    Volume 5, No. 4, April 2014 (Special Issue) ISSN No. 0976-5697 International Journal of Advanced Research in Computer Science REVIEW ARTICAL Available Online at www.ijarcs.info An Analysis of the Properties of Various Interconnection Networks K. Karthik Sudarson Jena Dept .of CSE Dept. of IT BVRIT, Narsapur GITAM University [email protected] [email protected] T. Venu Gopal Dept. of CSE JNTUHCEJ, Kondagattu [email protected] Abstract: In this paper, different types of interconnection networks are investigated and some of their properties are analyzed to summarize the differences in their network cost. The various properties of the interconnection networks such as connectivity, routing algorithm, diameter and broadcasting technology are investigated. This analysis gives a framework for the construction of more efficient interconnection networks in future. Key words: Interconnection Networks, Network cost, Routing Algorithms, Diameter, Broad casting Technology Network scales being typically used for comparative I. INTRODUCTION evaluation of an interconnection network due to the said characteristic include network cost [4-10] defined as degree x In parallel computing, to achieve parallelism various diameter of an interconnection network. By virtue of its merit techniques can be employed. One technique of achieving of easily providing a communication network system required parallelism is using MIMD (Multiple Instruction Multiple in applications of all kinds. Hypercube is node-symmetric and Data stream). Machines using MIMD have a number of edge-symmetric, has a simple routing algorithm with maximal processor that functions asynchronously and independently. fault tolerance and a simple reflexive system, and also has a At any time, different processors may be executing different merit that it may be readily embedded with the proposed instructions on different pieces of data.
    [Show full text]
  • Entropy-Based Proofs of Combinatorial Results on Bipartite Graphs
    Entropy-Based Proofs of Combinatorial Results on Bipartite Graphs Igal Sason Andrew and Erna Viterbi Faculty of Electrical and Computer Engineering Technion-Israel Institute of Technology, Haifa 3200003, Israel E-mail: [email protected] Abstract—This work considers new entropy-based proofs of e.g., [7] and [22, Problems 3.25–3.37]). Moore’s bound some known, or otherwise refined, combinatorial bounds for provides an upper bound on the girth of irregular graphs bipartite graphs. These include upper bounds on the number as a function of their number of vertices and their average of the independent sets, lower bounds on the minimal number of colors in constrained edge coloring, and lower bounds on degree [1]. An entropy-based proof of Moore’s bound for the number of walks of a given length in bipartite graphs. The graphs (and bipartite graphs) was introduced in [2]. proofs of these combinatorial results rely on basic properties of 4) Additional combinatorial properties of bipartite graphs: In the Shannon entropy. [12], a new conditional entropy inequality was derived, followed by a study of two of its combinatorial applications I. INTRODUCTION to bipartite graphs. These include a derivation of a lower bound on the minimal number of colors in (rich) graph The Shannon entropy serves as a powerful tool in various coloring, and a derivation of a lower bound on the biclique combinatorial and graph-theoretic applications (see, e.g., the cover number of bipartite graphs. tutorials in [8] and [20], as well as [5], [6], [17], [18], [19]). 5) Although not directly related to bipartite graphs, a variant Combinatorial properties of bipartite graphs are of great of Shearer’s lemma for the relative entropy was introduced interest in graph theory, combinatorics, modern coding theory, in [16] (see Corollary 7 and Remark 9 there), and also and information theory.
    [Show full text]
  • Imprimitive Cometric Association Schemes: Constructions and Analysis
    Imprimitive cometric association schemes: constructions and analysis William J. Martin Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute Worcester, Massachusetts [email protected] Mikhail Muzychuk Department of Computer Science and Mathematics Netanya Academic College Netanya 42365 Israel [email protected] Jason Williford Department of Mathematical Sciences Worcester Polytechnic Institute Worcester, Massachusetts [email protected] June 8, 2006 Dedicated to the memory of Dom de Caen, 1956–2002. Abstract Dualizing the “extended bipartite double” construction for distance-regular graphs, we construct a new family of cometric (or Q-polynomial) association schemes with four associate classes based on linked systems of symmetric designs. The analysis of these new schemes naturally leads to structural questions concerning imprimitive cometric association schemes, some of which we answer with others being left as open prob- lems. In particular, we prove that any Q-antipodal association scheme is dismantlable: the configuration induced on any subset of the equivalence classes in the Q-antipodal imprimitivity system is again a cometric association scheme. Further examples are explored. 1 1 A census of cometric association schemes Very few examples are known of association schemes that are cometric but not metric. Much effort has been devoted to the study and classification of Q-polynomial distance- regular graphs, i.e., association schemes that are both metric and cometric, since the most important families
    [Show full text]