DOCSLIB.ORG

Covering Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Covering Graphs Chapter 5 Covering Graphs Another central concept in topological crystallography is the “covering map”. Roughly speaking, a covering map in general is a surjective continuous map between topological spaces which preserves the local topological structure (see Notes in this chapter for the exact definition). Figure 5.1 exhibits a covering map from the line R onto the circle S1, which somehow suggests an idea why we use the term “covering”.1 The canonical projection π : Rd −→ Rd/L, introduced in Example 2.5, is another example of a covering map. The theory of covering maps is regarded as a geometric analogue of Galois theory in algebra that describes a beautiful relationship between field extensions2 and Galois groups. Indeed, in a similar manner as Galois theory, covering maps are completely governed by what we call covering transformation groups.Inthis chapter, sacrificing full generality, we shall deal with covering maps for graphs from a combinatorial viewpoint. See [107] for the general theory. Our concern in crystallography is only special covering graphs (more specifically abelian covering graphs introduced in Chap. 6). However the notion of general covering graphs turns out to be of particular importance when we consider a “non- commutative” version of crystals, a mathematical object related to discrete groups [see Notes (IV) in Chap. 9]. Some problems in statistical mechanics are also well set up in terms of covering graph. 1More explicitly, the covering map is given by √ x ∈ R → e2π −1x ∈ S1 = {z ∈ C||z| = 1}. 2A field is an algebraic system with addition, subtraction, multiplication and division such as rational numbers, real numbers and complex numbers. T. Sunada, Topological Crystallography, Surveys and Tutorials in the Applied 53 Mathematical Sciences 6, DOI 10.1007/978-4-431-54177-6 5, © Springer Japan 2013 54 5 Covering Graphs Fig. 5.1 Idea of covering maps S1 5.1 Definition Let X =(V,E) and X0 =(V0,E0) be connected graphs. A morphism ω : X −→ X0 is said to be a covering map if it preserves local adjacency relations between vertices and edges; more precisely, (i) ω : V −→ V0 is surjective. (ii) For every x ∈ V, the restriction ω|Ex : Ex −→ E0,ω(x) is a bijection. Given a covering map ω : X −→ X0, we call X a covering graph over the base graph X0.Thefiber over x ∈ V0 is the inverse image: ω−1(x)={y ∈ V| ω(y)=x}. −1 If |ω (x)| is finite for a vertex x ∈ V0,thenω (or X)issaidtobefinite-fold, otherwise infinite-fold. For a finite-fold covering map ω, the number |ω−1(x)| does not depend on −1 −1 x ∈ V0.Infact,ifweput n(x)=|ω (x)|, then, for e ∈ E0,wefindthat |ω (e)| = |ω−1 ( ) | = ( ) |ω−1( )| = |ω−1 ( ) | = ( ) o e no e ,andalso e t e n t e , and hence 3 n o(e) = n t(e) .SinceX0 is connected, n(x) is constant (Lemma 3.4.1). A covering map ω with n(x)=k for some constant k is said to be k-fold. Example 5.1. In Fig. 5.2, define the map ω by setting ω(1)=ω(7)=a, ω(2)= ω(8)=b, ω(3)=ω(5)=c, ω(4)=ω(6)=d.Thenω is a twofold covering map. A useful aspect of a covering map ω : X −→ X0 is that it has the unique path- lifting property, which claims that for a path c0 =(e0,1,...,e0,n) in X0 and a vertex x in X with ω(x)=o(c0), there exists a unique path c =(e1,...,en) in X (called a lifting of c0) such that ω(c)=c0 and o(c)=x. To see this, we first apply condition 3For an infinite-fold covering map, if we mean by |ω−1(x)| the cardinality of the set ω−1(x),we have the same conclusion. Here two sets A and B in general are said to have the same cardinality, symbolically |A| = |B|, if there exists a one-to-one correspondence between them. 5.1 Definition 55 Fig. 5.2 Example of covering map (ii) to get a unique e1 ∈ Ex with ω(e1)=e0,1, and then successively apply (ii) to ∈ ω( )= , = ,..., obtain ei Et(ei−1) with ei e0 i (i 2 n). The uniqueness of c is obvious from the discussion. In general, a lifting of a morphism f0 : Y =(VY ,EY ) −→ X0 is a morphism f : Y −→ X satisfying f0 = ω ◦ f . Theorem 5.1 (Unique Lifting Theorem). If two liftings f1 and f2 of f satisfy f1(y0)= f2(y0) for some y0 ∈ Y,then f1 = f2. 9X ss ss f ss ss ω ss ss ss / Y X0 f0 Proof. Let e ∈ EY with o(e)=y0.Wehaveω f1(e) = ω f2(e) = f0(e) and o ω f1(e) = o ω f2(e) = o f0(e) = f0 o(e) = f0(y0). Thus applying condition (ii) in the definition of covering map, we find that f1(e)= f2(e) and f1 t(e) = f2 t(e) . Repeating this argument, we observe that f1(e)= f2(e) for every e ∈ EY and f1(y)=f2(y) for every y ∈ VY ; thereby proving f1 = f2 (we use the assumption that Y is connected). In general, a lifting over all of Y does not have to exist (see Sect. 5.3 for a criterion so that a lifting exists). The Euler number behaves well under covering maps; that is, for a k-fold covering graph X =(V,E) over a finite base graph X0,weget χ(X)=kχ(X0) since |V| = k|V0| and |E| = k|E0|. Equivalently, b1(X)=k b1(X0) − 1 + 1. 56 5 Covering Graphs 5.2 Covering Transformation Groups The theory of covering maps is combined with group theory via the notion of covering transformations. This fact is vital to develop topological crystallography. Let ω : X −→ X0 be a covering map. An automorphism σ of X is said to be a covering transformation (or deck transformation) provided that ω ◦ σ = ω.By −1 −1 definition, σ preserves each fiber of ω, i.e., σ ω (x) = ω (x) for every x ∈ V0, and σ is a lifting of the covering map ω as the following commutative diagram indicates: <X yy yy σ yy yy yy ω yy yy yy / X X0 ω The totality of covering transformations, symbolically G(ω), forms a subgroup of Aut(X), which we call the covering transformation group of ω. We show that G(ω) acts freely on the graph X.Letσ =(σV ,σE ) ∈ G(ω). Suppose that there exists a vertex x ∈ V with σV x = x.SinceσV x = IV (x) (IV being the identity map of V), from the unique lifting theorem, it results in σV = IV .To check that G(ω) acts on X without inversion, suppose σE (e)=e.Then ω(e)=ω(σE e)=ω(e)=ω(e), thereby giving rise to a contradiction. A covering map ω : X −→ X0 is said to be regular provided that the action of G(ω) on every fiber ω−1(x) is transitive (hence simply transitive). Thus, if ω is a finite-fold regular covering map, then the order of the group G(ω) (that is, |G(ω)|) is equal to |ω−1(x)|. Example 5.2. A twofold covering map ω : X −→ X0 is always regular. To show this, define the automorphism σ =(σV ,σE ) of X by σV x = y, σV y = x (ω(x)=ω(y), x = y), σE ex = ey, σE ey = ex (ω(ex)=ω(ey), ex ∈ Ex, ey ∈ Ey, x = y). We may check that σ is a covering transformation with σ 2 = I,andthatG(ω)= {I,σ}. Obviously G(ω) acts transitively on each fiber. We shall prove that for a regular covering map ω : X −→ X0, the quotient graph X/G(ω) is isomorphic to X0. From the definition of quotient graphs and regular covering maps, it follows that ω induces a bijection of V/G(ω) onto V0. To prove that the induced map of E/G(ω) into E0 is a bijection, suppose 5.2 Covering Transformation Groups 57 ω(e)=ω(e ) (e,e ∈ E). Since ω o(e) = ω o(e ) , the regularity of ω allows us to find σ ∈ G(ω) such that o(e )=σo(e)=o(σe).Usingω(σe)=ω(e) and condition (ii) for covering maps, we have e = σe. The bijections obtained in this way preserve the adjacency relation of graphs. Thus we have an isomorphism between X/G(ω) and X0. We shall collect several fundamental facts on covering graphs. Theorem 5.2. Suppose that a group G acts freely on a graph X. Then the canonical morphism ω : X −→ X0 = X/G is a regular covering map whose covering transformation group is G (namely, every regular covering map is obtained as a canonical morphism associated with a free action on a graph). Proof. We first show that ω is a covering map. Both ω : V −→ V0 and ω : E −→ E0 are surjective. Thus, we only have to verify that for every x ∈ V,themapω|Ex : −→ ∈ ∈ ω( )= Ex E0ω(x) is a bijection. For e0 E0ω(x),thereise E such that e e0. Observe ω o(e) = o ω(e) = o(e0)=ω(x), so o(e) and x are in the same orbit. This being so, there exists g ∈ G with go(e)=x. We thus have ge ∈ Ex and ω(ge)=ω(e)=e0. This implies that ω|Ex is surjective. To show that ω|Ex is an injection, suppose ω(e)=ω(e ) for e,e ∈ Ex.Letg ∈ G be an element with e = ge.Thenx = o(e )=o(ge)=gx.
Load more

Recommended publications
  • Coverings of Cubic Graphs and 3-Edge Colorability
    Discussiones Mathematicae Graph Theory 41 (2021) 311–334 doi:10.7151/dmgt.2194 COVERINGS OF CUBIC GRAPHS AND 3-EDGE COLORABILITY Leonid Plachta AGH University of Science and Technology Krak´ow, Poland e-mail: [email protected] Abstract Let h: G˜ → G be a finite covering of 2-connected cubic (multi)graphs where G is 3-edge uncolorable. In this paper, we describe conditions under which G˜ is 3-edge uncolorable. As particular cases, we have constructed regular and irregular 5-fold coverings f : G˜ → G of uncolorable cyclically 4-edge connected cubic graphs and an irregular 5-fold covering g : H˜ → H of uncolorable cyclically 6-edge connected cubic graphs. In [13], Steffen introduced the resistance of a subcubic graph, a char- acteristic that measures how far is this graph from being 3-edge colorable. In this paper, we also study the relation between the resistance of the base cubic graph and the covering cubic graph. Keywords: uncolorable cubic graph, covering of graphs, voltage permuta- tion graph, resistance, nowhere-zero 4-flow. 2010 Mathematics Subject Classification: 05C15, 05C10. 1. Introduction 1.1. Motivation and statement of results We will consider proper edge colorings of G. Let ∆(G) be the maximum degree of vertices in a graph G. Denote by χ′(G) the minimum number of colors needed for (proper) edge coloring of G and call it the chromatic index of G. Recall that according to Vizing’s theorem (see [14]), either χ′(G) = ∆(G), or χ′(G) = ∆(G)+1. In this paper, we shall consider only subcubic graphs i.e., graphs G such that ∆(G) ≤ 3.
    [Show full text]
  • COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
    COVERING SPACES, GRAPHS, AND GROUPS CARSON COLLINS Abstract. We introduce the theory of covering spaces, with emphasis on explaining the Galois correspondence of covering spaces and the deck trans- formation group. We focus especially on the topological properties of Cayley graphs and the information these can give us about their corresponding groups. At the end of the paper, we apply our results in topology to prove a difficult theorem on free groups. Contents 1. Covering Spaces 1 2. The Fundamental Group 5 3. Lifts 6 4. The Universal Covering Space 9 5. The Deck Transformation Group 12 6. The Fundamental Group of a Cayley Graph 16 7. The Nielsen-Schreier Theorem 19 Acknowledgments 20 References 20 1. Covering Spaces The aim of this paper is to introduce the theory of covering spaces in algebraic topology and demonstrate a few of its applications to group theory using graphs. The exposition assumes that the reader is already familiar with basic topological terms, and roughly follows Chapter 1 of Allen Hatcher's Algebraic Topology. We begin by introducing the covering space, which will be the main focus of this paper. Definition 1.1. A covering space of X is a space Xe (also called the covering space) equipped with a continuous, surjective map p : Xe ! X (called the covering map) which is a local homeomorphism. Specifically, for every point x 2 X there is some open neighborhood U of x such that p−1(U) is the union of disjoint open subsets Vλ of Xe, such that the restriction pjVλ for each Vλ is a homeomorphism onto U.
    [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]
  • Free Groups and Graphs: the Hanna Neumann Theorem
    FREE GROUPS AND GRAPHS: THE HANNA NEUMANN THEOREM LAUREN COTE Abstract. The relationship between free groups and graphs is one of the many examples of the interaction between seemingly foreign branches of math- ematics. After defining and proving elementary relations between graphs and groups, this paper will present a result of Hanna Neumann on the intersection of free subgroups. Although Neumann's orginal upper bound on the rank of the intersection has been improved upon, her conjecture, that the upper bound is half of the one she originally proved, remains an open question. Contents 1. Groups 1 2. Graphs 3 3. Hanna Neumann Theorem 9 References 11 1. Groups Definition 1.1. Recall that a group G is a set S and a binary operation · with the properties that G is closed under ·, · is associative, there exists an identity e such that for all g 2 (G); g · e = e · g = g, and there exist inverses such that for each g 2 G; there exists g−1 2 G such that g · g−1 = g−1 · g = e. When working with groups, it is helpful to review the definitions of homomor- phism, an operation perserving map. For groups G1 = (S1; ·), G2 = (S2;?), the homomorphism φ : G1 ! G2 has the property that φ(x · y) = φ(x) ? φ(y). An isomorphism is a bijective homomorphism. Definition 1.2. This paper will focus on a particular kind of group, those which have the least possible amount of restriction: free groups. Let i : S ! F . A group F is free on a set S if for all groups G and for all f : S ! G, there exists a unique homomorphism : F ! G so that i ◦ = f.
    [Show full text]
  • Enumerating Branched Orientable Surface Coverings Over a Non-Orientable Surface Ian P
    Discrete Mathematics 303 (2005) 42–55 www.elsevier.com/locate/disc Enumerating branched orientable surface coverings over a non-orientable surface Ian P. Gouldena, Jin Ho Kwakb,1, Jaeun Leec,2 aDepartment of Combinatorics and Optimization, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang 790–784, Korea cMathematics, Yeungnam University, Kyongsan 712–749, Korea Received 21 November 2002; received in revised form 13 October 2003; accepted 29 October 2003 Available online 10 October 2005 Abstract The isomorphism classes ofseveral types ofgraph coverings ofa graph have been enumerated by many authors [M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87–97; S. Hong, J.H. Kwak, J. Lee, Regular graph cover- ings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85–105; J.H. Kwak, J.H. Chun, J.Lee, Enumeration ofregular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273–285; J.H. Kwak, J. Lee, Isomorphism classes ofgraph bundles, Canad. J. Math. XLII (1990) 747–761; J.H. Kwak, J. Lee, Enumeration ofconnected graph coverings, J. Graph Theory 23 (1996) 105–109]. Recently, Kwak et al [Balanced regular coverings ofa signed graph and regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004) 177–193] enumerated the isomorphism classes ofbalanced regular coverings ofa signed graph, as a continuation ofan enumeration work done by Archdeacon et al [Bipartite covering graphs, Discrete Math. 214 (2000) 51–63] the isomorphism classes ofbranched orientable regular surfacecoverings ofa non-orientable surface having a finite abelian covering transformation group.
    [Show full text]
  • Configuration Spaces of Graphs
    Configuration Spaces of Graphs Daniel Lütgehetmann Master’s Thesis Berlin, October 2014 Supervisor: Prof. Dr. Holger Reich Selbstständigkeitserklärung Hiermit versichere ich, dass ich die vorliegende Masterarbeit selbstständig und nur unter Zuhilfenahme der angegebenen Quellen erstellt habe. Daniel Lütgehetmann Erstkorrektor: Prof. Dr. Holger Reich Zweitkorrektor: Prof. Dr. Elmar Vogt Contents Introduction iii 1 Preliminaries1 1.1 Configuration Spaces . .1 1.2 Reduction to Connected Spaces . .3 1.3 The Rational Cohomology as Σn-Representation . .3 1.4 Some Representation Theory of the Symmetric Group . .4 2 A Deformation Retraction7 2.1 The Poset PnΓ ................................8 2.2 Cube Complexes and Cube Posets . 11 2.3 The Geometric Realization of a Cube Poset . 14 2.4 The Cube Complex KnΓ .......................... 16 2.5 The Embedding into the Configuration Space . 18 2.6 The Deformation Retraction . 21 2.7 The Path Metric on KnΓ .......................... 25 2.8 The Curvature of KnΓ ........................... 26 3 The Homology of Confn(Γ) 29 3.1 Representation Stability . 32 3.2 The Generalized Euler Characteristic of Confn(Γ) ........... 34 3.3 Graphs With At Most One Branched Vertex . 35 3.4 The General Case of Finite Graphs . 38 3.5 Applications . 39 3.6 Next Steps . 41 4 A Sheaf-Theoretic Approach 43 4.1 Sheaf Theory . 43 4.2 The Spectral Sequence . 46 4.3 The Stalks . 50 i Contents 4.4 Decomposition of Rq(inc )(Q) into Easier Sheaves . 52 ∗ 4.5 Sketch of the Proof for Manifolds . 53 BibliographyII Appendix III ii Introduction Let Γ be a graph and n a natural number. We want to understand the ordered configuration space Confn(Γ) consisting of all n-tuples x = (x1; : : : ; xn) of elements in Γ such that x = x for i = j, endowed with the subspace topology induced by the i 6 j 6 inclusion Conf (Γ) Γ n.
    [Show full text]
  • Arxiv:2003.07426V2 [Math.CT] 22 Jul 2021 Modifying Some Constructions of Adams [1] for Connected CW Complexes with Base Point
    BROWN REPRESENTABILITY FOR DIRECTED GRAPHS ZACHARY MCGUIRK † AND BYUNGDO PARK ‡ ABSTRACT. We prove that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is repre- sentable. 1. INTRODUCTION The homotopy theory of directed graphs is a discrete analogue of homotopy theory in algebraic topology. In topology, a homotopy between two continuous maps is defined by an interpolating family of continuous maps parametrized by a closed interval [0; 1]. Its discrete analogue we study uses a directed line graph keeping track of a discrete change of directed graph maps. See for example Grigor’yan, Lin, Muranov, and Yau [8]. Efforts towards a homotopy theory for graphs extend back to the 1970’s and 1980’s with some results by Gianella [6] and Malle [16]. However, the most recent variant of graph homotopy theory appears to have taken off with a paper by Chen, Yau, and Yeh from 2001 [2], before culminating in the 2014 work of Grigor’yan, Lin, Muranov, and Yau [8]. Recently, there is increased interest in this new notion of homotopy for directed graphs because it was shown by Grigor’yan, Jimenez, Muranov, and Yau in [7] that the path-space homology theory of directed graphs is invariant under this version of graph homotopy. The path-space homology and cohomology theories for directed graphs were studied by Grigor’yan, Lin, Muranov, and Yau [9] and by Grigor’yan, Muranov, and Yau in [10] and [11]. The homology theory developed in the references above is quite natural, computable, and can be non-trivial for degrees greater than one, depending on the lengths of admissible, @-invariant paths in a directed graph.
    [Show full text]
  • New York Journal of Mathematics on Configuration Spaces and Simplicial
    New York Journal of Mathematics New York J. Math. 25 (2019) 723{744. On configuration spaces and simplicial complexes Andrew A. Cooper, Vin de Silva and Radmila Sazdanovic Abstract. The n-point configuration space of a space M is a well- known object in topology, geometry, and combinatorics. We introduce a generalization, the simplicial configuration space MS , which takes as its data a simplicial complex S on n vertices, and explore the properties of its homology, considered as an invariant of S. As in Eastwood-Huggett's geometric categorification of the chromatic polynomial, our construction gives rise to a polynomial invariant of the simplicial complex S, which generalizes and shares several formal prop- erties with the chromatic polynomial. Contents 1. Introduction 723 2. The simplicial configuration space 725 3. Homology of MS 728 4. Properties of the simplicial configuration space 730 5. M matters 733 6. The simplicial chromatic polynomial 737 7. Comparison to other invariants 740 8. Questions and future directions 742 References 743 1. Introduction The configuration space of n points in the space M, n [ i j Confn(M) = M n x = x i6=j was introduced by Fadell and Neuwirth [14]. Its topology has since been the subject of extensive study (e.g. [16, 18, 23]), with particular focus on Received August 15, 2018. 2010 Mathematics Subject Classification. 55U10, 05C15, 05C31. Key words and phrases. simplicial complex, homology, chromatic polynomial, categorification. ISSN 1076-9803/2019 723 724 ANDREW A. COOPER, VIN DE SILVA AND RADMILA SAZDANOVIC how to relate the homology and cohomology of this (noncompact) space to the homology or cohomology of the space M, which may be taken to be a closed manifold or algebraic variety.
    [Show full text]
  • Configuration Spaces, Braids, and Robotics
    CONFIGURATION SPACES, BRAIDS, AND ROBOTICS R. GHRIST ABSTRACT. Braids are intimately related to configuration spaces of points. These configuration spaces give a useful model of autonomous agents (or robots) in an environment. Problems of relevance to autonomous engineering systems (e.g., motion planning, coordination, cooperation, assembly) are di- rectly related to topological and geometric properties of configuration spaces, including their braid groups. These notes detail this correspondence, and ex- plore several novel examples of configuration spaces relevant to applications in robotics. No familiarity with robotics is assumed. These notes shadow a two-hour set of tutorial talks given at the National University of Singapore, June 2007, for the IMS Program on Braids. CONTENTS 1. Configuration spaces and braids 2 2. Planning problems in robotics 3 2.1. Motion planning 3 2.2. Motion planning on tracks 4 2.3. Shape planning for metamorphic robots 5 2.4. Digital microfluidics 5 3. Configuration spaces of graphs 6 3.1. Visualization 6 3.2. Simplification: deformation 8 3.3. Simplification: simplicial 9 3.4. Simplification: discretization 11 4. Discretization 12 4.1. Cubes and collisions 12 4.2. Manifold examples 13 4.3. Everything that rises must converge 15 4.4. Curvature 15 5. Reconfigurable systems 17 5.1. Generators and relations 17 5.2. State complexes 18 5.3. Examples 19 6. Curvature again 24 6.1. State complexes are NPC 24 6.2. Efficient state planning 25 6.3. Back to braids 26 7. Last strands 27 7.1. Configuration spaces 27 1 2 R. GHRIST 7.2. Graph braid groups 29 7.3.
    [Show full text]
  • Intersection Graphs
    Annals of Pure and Applied Mathematics Annals of Vol. 4, No.1, 2013, 43-91 ISSN: 2279-087X (P), 2279-0888(online) Published on 1October 2013 www.researchmathsci.org Intersection Graphs: An Introduction Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming Vidyasagar University, Midnapore – 721102, India e-mail: [email protected] Received 1 September 2013; accepted 27 September 2013 Abstract. Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation, trapezoid, chordal, disk, circle graphs are more important. In this article, a brief introduction of each of these intersection graphs is given. Some basic properties and algorithmic status of few problems on these graphs are cited. This article will help to the beginners to start work in this direction. Since the article contains a lot of information in a compact form it is also useful for the expert researchers too. Keywords: Design and analysis of algorithms, intersection graphs, interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, tolarence graphs, chordal graphs, circle graphs, string graphs, disk graphs AMS Mathematics Subject Classification (2010): 05C62, 68Q22, 68Q25, 68R10 1. Introduction It is well known that graph is a very useful tool to model problems originated in all most all areas of our life. The geometrical structure of any communication system including Internet is based on graph. The logical setup of a computer is designed with the help of graph. So graph theory is an old as well as young topic of research.
    [Show full text]
  • Zeta Functions of Finite Graphs and Coverings, Iii
    ZETA FUNCTIONS OF FINITE GRAPHS AND COVERINGS, III HAROLD M. STARK AND AUDREY A. TERRAS Abstract. A graph theoretical analogue of Brauer-Siegel theory for zeta func- tions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph. 1. Introduction August 15; 2005 In our previous two papers [12], [13] we developed the theory of zeta and L- functions of graphs and covering graphs. Here zeta and L-functions are reciprocals of polynomials which means these functions have poles not zeros. Just as number theorists are interested in the locations of the zeros of number theoretic zeta and L-functions, thanks to applications to the distribution of primes, we are interested in knowing the locations of the poles of graph-theoretic zeta and L-functions. We study an analog of Brauer-Siegel theory for the zeta functions of number …elds (see Stark [11] or Lang [6]). As explained below, this is a necessary step in the discussion of the distribution of primes. We will always assume that our graphs X are …nite, connected, rank 1 with no danglers (i.e., degree 1 vertices). Let us recall some of the de…nitions basic to Stark and Terras [12], [13]. If X is any connected …nite undirected graph with vertex set V and (undi- rected) edge set E, we orient its edges arbitrarily and obtain 2 E oriented edges j j 1 1 e1; e2; ; en; en+1 = e ; :::; e2n = e .“Primes”[C] in X are equivalence classes 1 n of closed backtrackless tailless primitive paths C.
    [Show full text]
  • Covering Spaces and Subgroups of the Free Group
    COVERING SPACES AND SUBGROUPS OF THE FREE GROUP SAMANTHA NIEVEEN AND ALLISON SMITH Adviser: Dennis Garity Oregon State University Abstract. In this paper we will use the known link between covering spaces of a topological space and subgroups of its fundamental group to investigate the subgroup structures of the free groups on two and more generators. Some of our conclusions, such as Marshall Hall’s formula for the number of subgroups of a free group of finite index, have been shown before. (Hopefully others, such as some calculations for the number of normal subgroups of finite index, have not been. That would be cool.) 1. Introduction The correspondence between the covering spaces of a space X and subgroups of π1(X, x0), the fundamental group of X at a base point x0 is well known from topology. In particular, for any subgroup H of π1(X, x0), there exists a unique (up to covering equivalence) covering −1 p :E→X and a point e0 in p (x0) so that π1(E, e0) is homomorphic to H under a homo- morphism induced by p and E is path connected. Also, each covering space E corresponds −1 in this way to a conjugacy class of subgroups of π1(X, x0) as e0 ranges through p (x0). For more information see Munkres[4] Chapter 8. Throughout this paper we say that a subgroup corresponds to a particular covering and choice of base point or that a covering and choice of base point corresponds to a particular subgroup, referring to the correspondence described here.
    [Show full text]