Betti Numbers of Hypergraphs

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  • This article was downloaded by: [University of Stellenbosch]On: 05 October 2014, At: 05:25Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

    Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

    Betti Numbers of HypergraphsEric Emtander aa Department of Mathematics , Stockholm University , Stockholm, SwedenPublished online: 05 May 2009.

    To cite this article: Eric Emtander (2009) Betti Numbers of Hypergraphs, Communications in Algebra, 37:5, 1545-1571, DOI:10.1080/00927870802098158

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  • Communications in Algebra, 37: 15451571, 2009Copyright Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802098158

    BETTI NUMBERS OF HYPERGRAPHS

    Eric EmtanderDepartment of Mathematics, Stockholm University, Stockholm, Sweden

    In this article, we study some algebraic properties of hypergraphs, in particular theirBetti numbers. We define some different types of complete hypergraphs, which to thebest of our knowledge are not previously considered in the literature. Also, in a naturalway, we define a product on hypergraphs, which in a sense is dual to the join operationon simplicial complexes. For such product, we give a general formula for the Bettinumbers, which specializes neatly in case of linear resolutions.

    Key Words: Betti numbers; Complete hypergraphs; Hypergraphs.

    2000 Mathematics Subject Classification: 05C65; 13D02; 13A99.

    1. INTRODUCTION

    Let be a finite set and = E1 Es a finite collection of nonemptysubsets of . The pair = is called a hypergraph. The elements of arecalled the vertices, and the elements of are called the edges of the hypergraph. Ifwe want to specify what hypergraph we consider, we may write and forthe vertices and edges, respectively.

    The hypergraphs that we will consider can all be seen as naturalgeneralizations of the ordinary complete graph Kn, on n vertices. Our main tools arefamiliar concepts in combinatorial algebra, such as Hochsters formula, the MayerVietoris sequence, and Knneths tensor formula.

    A hypergraph is called simple if: (1) Ei 2 for all i = 1 s and (2) Ej Eiimplies i = j. If the cardinality of is n, we often just use the set n = 1 2 ninstead of .

    We frequently identify a vertex vi of with a variable xi of a polynomialring kx1 xn over some field k, or with its corresponding characteristic vectorvvi = 0 0 1 0 0 in n, consisting of only zeros except in the ithposition were there is a 1. Hence we choose to consider 0 to be a natural number.This also allows us to identify a subset V of n with its characteristic vectorvV = iV vvi. We use bold letters to denote vectors and if w = w1 wn isa squarefree vector in n (i.e., a vector in which 0 wi 1 for i = 1 n), then

    Received January 11, 2008; Revised March 27, 2008. Communicated by R. Villarreal.Address correspondence to Eric Emtander, Department of Mathematics, Stockholm University,

    Stockholm S-10691, Sweden; E-mail: erice@math.su.se

    1545

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  • 1546 EMTANDER

    we define its norm w by w = ni=1 wi. In this way, the cardinality V of V equalsthe norm of the characteristic vector vV.

    Throughout the article, we denote by R the polynomial ring kx1 xn oversome field k, where n is the number of vertices of a hypergraph considered at themoment. We recall that the ring R is in a natural way both - and n-graded.Employing the ideas above, we may think of an edge Ei of a hypergraph as amonomial xEi = jEi xj in R. We use this notion to associate an ideal I Rto a hypergraph . The edge ideal, I, of a hypergraph is the ideal xEi Ei R, generated by the edges of .

    The edge ideal was first introduced by Villarreal (1990), in the case ofsimple graphs. Since then, edge ideals have been studied widely, see for instanceFaridi (2002, 2005), H and Van Tuyl (2006, 2007), Morey et al. (2007), andZheng (2004). In H and Van Tuyl (2006), the authors give some nice recursiveformulas for computing Betti numbers. Furthermore, their techniques illustrate bothsome obstacles that occur when you try to generalize graph theoretical results tohypergraph theoretical, as well as ways of getting around such obstacles.

    Another way of using hypergraphs to reveal connections between commutativealgebra and combinatorics was introduced by Faridi (2002). There, Faridi considers,the set of facets of a simplicial complex as a hypergraph. In this way a simplicialcomplex may be thought of as a higher dimensional graph. See Faridi (2002, 2005)and Zheng (2004) for details and examples.

    Recall that an (abstract) simplicial complex on vertex set n is a collection

    of subsets of n with the property that F G F G . The elements of

    are called the faces of the complex and the maximal (under inclusion) faces arecalled facets. The dimension dim F of a face F in is defined to be F 1, and thedimension of is defined as dim = maxdim F F . The r-skeleton of is thecollection of faces of dimension at most r. Note that the empty set is the unique1 dimensional face of every complex that is not the void complex which has nofaces. The dimension of the void complex may be defined as .

    The dimension dimR M of a R-module M , is by definition the Krll dimensionof R/AnnM .

    Given a simplicial complex , we denote by its reduced chain complex,and by Hn k = Zn/Bn its nth reduced homology group with coefficientsin the field k. In general, we could use an arbitrary abelian group instead of k, butwe will only consider the case when the coefficients lie in a field. For convenience,we define the homology of the void complex to be zero.

    If X and Y are two sets, we denote their disjoint union by X Y . Thus, supposewe have the two sets n and m. They both contain the number 1, but in n m,these two 1s are considered as distinct objects.

    Let and be simplicial complexes on the disjoint vertex sets x1 xn andy1 ym, respectively. We define the join of and to be the simplicialcomplex on vertex set x1 xn y1 ym having faces xi1 xir yj1 yjs ,where xi1 xir and yj1 yjs are faces of and , respectively.

    If n we denote by n the full simplex on n vertices. That is, the simplicialcomplex on n vertices in which every subset of n is a face. According to this, wemay think of the empty complex as a simplex on zero vertices.

    Given a simplicial complex on n and a subset V n, we denote by Vthe simplicial complex on vertex set V , with faces F F V. We call this the

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  • BETTI NUMBERS OF HYPERGRAPHS 1547

    restriction of to V . If j = j1 jn is a squarefree vector in n, by j we meanthe restriction to the set V n whose characteristic vector is j.

    Now, let be a simplicial complex on n. The StanleyReisner ring R/I of

    is the quotient of the ring R = kx1 xn by the StanleyReisner ideal

    I = xF F

    generated by the nonfaces of .Let n

    k denote the set of all k-subsets (that is, subsets of cardinality k) of

    n. If n < k, we interpret this as being empty. Furthermore, we let nk denote thecardinality of n

    k, so nk = 0 if n < k.

    In Section 2, we recall some basics that we will use throughout the article,while Section 3 is where the main results are found. In Theorems 3.1 and 3.5,respectively, we compute the Betti numbers of the d-complete and the d-completemultipartite hypergraphs, respectively. These results are very natural generalizationsof their graph theoretical counterparts. By considering the independence complexes,the ideas behind the proofs becomes transparent. In Section 3.4, we give a naturaldefinition of a product on hypergraphs. This in turn lets us compute the Bettinumbers of the da1 at-complete hypergraph. All these hypergraphs are in oneway or the other a natural generalization of the ordinary complete graph Kn. Inthe final section, Section 3.6, we define a class of hypergraphs that actually containall the previously considered ones. We show that the hypergraph algebra, R/I,corresponding to such hypergraph, has linear resolution.

    2. PRELIMINARIES

    Here we recall some results and definitions which will be used throughout thearticle.

    2.1. Hypergraphs and Independence Complexes

    Our general reference concerning hypergraphs is Berge (1989). In this article,we will only consider simple hypergraphs, as defined in the introduction. Thus,hypergraph will always mean simple hypergraph.

    Let be a hypergraph. A subhypergraph of is a hypergraph such that , and . If , the induced hypergraph on , , isthe subhypergraph with = and with consisting of the edges of that lies entirely in . A hypergraph is said to be d-uniform if Ei = d for everyedge Ei . Note that a 2-uniform hypergraph is just an ordinary simple graph.

    Let = n be a hypergraph, and consider the edge ideal I R.Note that R/I is precisely the StanleyReisner ring of the simplicial complex

    = F n E FE

    This is called the independence complex of . Note that the edges in are preciselythe minimal nonfaces in .

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    Let be an arbitrary simplicial complex on n. We then define the Alexanderdual simplicial complex to by

    = F n n\F

    Note that = .

    2.2. Resolutions and Betti Numbers

    To every finitely generated graded module M over the polynomial ring R =kx1 xn, we may associate a minimal (-)graded free resolution

    0 jRjlj M

    jRjl1j M

    jRj0j M M 0

    where l n and Rj is the R-module obtained by shifting the degrees of R byj. Thus, Rj is the graded R-module in which the grade i component Rji isRij .

    The natural number ijM is called the ijth -graded Betti number of M .If M is multigraded, we may equally well consider the n-graded minimal freeresolution and Betti numbers of M . The difference lies just in the fact that we nowuse multigraded shifts Rj instead of -graded ones. The total ith Betti number isiM =

    j ij . For further details on resolutions, graded rings and Betti numbers,

    we refer the reader to Bruns and Herzog (1998, Sections 1.3 and 1.5).The projective dimension pdM of M is pdM = maxi ijM = 0.The Betti numbers of M occur as the dimensions of certain vector spaces over

    k = R/m, where m is the unique maximal graded ideal in R. Accordingly, the Bettinumbers (and then of course the projective dimension) in general depend on thecharacteristic of k.

    A minimal free resolution of M is said to be linear if for i > 0, ijM = 0,whenever j = i+ d 1 for some fixed natural number d 1. In this article, weonly consider resolutions of quotient rings R/I . Hence, the interesting parts of theresolutions are the degrees greater than zero. In the variuos formulas for Bettinumbers that we give, we thus assume that i > 0.

    In connection to this we mention the EagonReiner theorem.

    Theorem 2.1. Let be a simplicial complex and its Alexander dual complex. ThenR/I is CohenMacaulay if and only if R/I has linear minimal free resolution.

    Proof. See Eagon and Reiner (1998, Theorem 3).

    Since there is a 11 correspondence between StanleyReisner rings (orequivalently squarefree monomial ideals) and simplicial complexes, we get a 11correspondence between simple hypergraphs and StanleyReisner rings as well. Thisenables us to talk about resolutions, Betti numbers, and projective dimensions ofhypergraphs.

    By a resolution, a Betti number, or the projective dimension of a hypergraph , we mean ditto of R/I. Thus ij = ijR/I and pd = pdR/I.

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    One further result which we will use later on is the AuslanderBuchsbaumformula. If R is a finitely generated graded k-algebra for some field k and M = 0 afinitely generated graded R-module with pdM < , then the formula asserts that

    pdM+ depthM = depthRFor a proof, see Bruns and Herzog (1998, Theorem 1.3.3).

    2.3. Hochsters Formula

    In topology one defines Betti numbers in a somewhat different manner.Hochsters formula provides a link between these and the Betti numbers definedabove. Hochsters formula will turn out to be a very useful tool of ours.

    Theorem 2.2 (Hochsters Formula). Let R/I be the StanleyReisner ring of asimplicial complex . The nonzero Betti numbers of R/I are only in squarefree degreesj and may be expressed as

    ijR/I = dimk Hji1j kHence the total ith Betti number may be expressed as

    iR/I =Vn

    dim HV i1V k

    Proof. See Bruns and Herzog (1998, Theorem 5.5.1).

    If one has n-graded Betti numbers, it is easy to obtain the -graded onesvia

    ijR/I =

    jnj =j

    ijR/I

    Thus,

    ijR/I =VnV =j

    dim HV i1V k

    2.4. The MayerVietoris Sequence

    Recall that if we have an exact sequence of complexes,1

    0 L M N 0there is a long exact (reduced) homology sequence associated to it

    HrN Hr1L Hr1M Hr1N

    1That is, complexes of modules over some ring R.

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    Later in this article, we will have great use of this homology sequence in the specialcase where it is associated to a simplicial complex as follows.

    Suppose we have a simplicial complex N and two subcomplexes L and M , suchthat N = L M ....